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\floatplacement{figure}{h!}
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\hyphenation{her-ud-over ek-si-stens-va-ri-ab-le an-dre dob-belt-im-pli-ka-ti-on ob-jekt-kvan-tor de-fi-ni-tions-lem-ma ens-be-tyd-en-de 
und-er-af-snit slut-ning-en inde-hol-der for-klar-ing-en si-de-be-ting-el-sen
des-uden be-vis-check-er-en}


\title {\Huge{\AE{}kvivalensrelationer i Logiweb}}
\author {Frederik Eriksen \texttt{(eriksen@diku.dk)}}
\date{22.\ juni 2006}
\maketitle


\newpage

\tableofcontents
\listoffigures



\newpage

\section{Indledning}

Denne rapport indeholder et aksiomsystem for en reduceret version af ZF 
m\ae{}ngdel\ae{}re, som jeg kalder for ``$ZFsub$''. Ud fra dette aksiomsystem vil jeg vise en r\ae{}kke basale s\ae{}tninger inden for udsagnslogik og m\ae{}ngdel\ae{}re. Hovedresultatet er, at en \ae{}kvivalensrelation p\aa{} en m\ae{}ngde ${\cal BS}$ implicit definerer en partition af ${\cal BS}$\footnote{Jeg vil referere til denne s\ae{}tning som ``hovedresultatet''. Oprindeligt var det ogs\aa{} et m\aa{}l at bevise det modsatte resultat --- at enhver partition implicit definerer en \ae{}kvivalensrelation --- men det har der ikke v\ae{}ret tid til.}. Alt det formelle arbejde i rapporten er gennemf\o{}rt med henblik p\aa{} at bevise dette resultat.

Rapporten er udarbejdet ved hj\ae{}lp af Logiweb, som er et system til verifikation og publicering af tekster, der indeholder formel matematik.
Logiweb har verificeret rapportens beviser og publiceret rapporten p\aa{} WWW-adressen

\noindent \resizebox{\textwidth}{!}{\url{http://www.diku.dk/~grue/logiweb/20060417/home/eriksen/equivalence-relations}.}

Det matematiske indhold af denne rapport er ikke synderligt avanceret; f.eks.\ fylder materialet om \ae{}kvivalensrelationer knap to sider i l\ae{}rebogen \cite{kn:hrba} (s.\ 29--31). Ideen med rapporten er snarere at afpr\o{}ve Logiweb end at redeg\o{}re for en matematisk teori. Derfor vil jeg ikke g\o{}re alt for meget ud af at forklare de matematiske begreber; fokus vil i h\o{}jere grad ligge p\aa{} tekniske forhold omkring formaliseringen.

Rapporten er struktureret som f\o{}lger: Afsnit \ref{sec:logiweb} er en kort beskrivelse af nogle detaljer ved Logiweb, som er nyttige at kende for en l\ae{}ser af et Logiweb-dokument som dette\footnote{Dette afsnit er en revideret udgave af afsnit 2 i \cite{kn:peano}.}. Afsnit \ref{sec:syntaks} og \ref{sec:aksiomsystem} omhandler syntaksen og aksiomsystemet for $ZFsub$, og afsnit \ref{sec:makro} indf\o{}rer de makrodefinitioner, jeg vil g\o{}re brug af. S\aa{} kommer vi til lemmaerne og beviserne; vi skal igennem fem afsnit med hj\ae{}lpes\ae{}tninger (afsnit \ref{sec:udsagn}--\ref{sec:lighedslemmaer}), f\o{}r vi endelig kan bevise hovedresultatet i afsnit \ref{sec:hoved}. Afsnit \ref{sec:kon} slutter af med en konklusion p\aa{} det hele.

\section{Lidt om Logiweb} \label{sec:logiweb}

Som n\ae{}vnt i indledningen er denne rapport skrevet ved hj\ae{}lp af Logiweb. Dette betyder dels, 
at rapportens formelle indhold er defineret ud fra nogle andre Logiweb-dokumenter, dels at et par afsnit indeholder nogle s\ae{}rlige definitioner og tabeller, og endelig at et par andre afsnit indeholder en del programkode. I dette afsnit vil jeg kort beskrive disse f\ae{}nomener. For en detaljeret beskrivelse af hele Logiweb systemet vil jeg henvise til \cite{kn:base}; her kan man bla.\ l\ae{}se om den bevischecker, der har verificeret rapportens beviser.

\subsection{Formelle konstruktioner} \label{sec:formelkon}

Det formelle indhold af et Logiweb-dokument er sammensat af en r\ae{}kke formelle konstruktioner. En formel konstruktion kan repr\ae{}sentere alt, hvad der har med formel matematik at g\o{}re: Variable, funktioner, lemmaer, beviser, osv. Der er to kilder til konstruktionerne i et Logiweb-dokument: Dels kan man indf\o{}re sine egne konstruktioner, og dels kan man importere konstruktioner fra andre Logiweb-dokumenter. De importerede konstruktioner i denne rapport har to kilder: Dels 
\cite{kn:base}, og dels de 3 .pdf filer \cite{kn:check}, som tilsammen udg\o{}r \'{e}t Logiweb-dokument.

\subsection{S\ae{}rlige definitioner og tabeller} \label{sec:saerlig}

Den formelle del af et Logiweb-dokument skrives i et formateringssprog ved navn ``pyk''. Hver formel konstruktion, man arbejder med i dokumentet, har tilknyttet en s\aa{}kaldt ``pyk definition''. Dette er en angivelse af, hvad man skal skrive, hvis man i et andet Logiweb-dokument \o{}nsker at benytte den p\aa{}g\ae{}ldende konstruktion. Hvis man indf\o{}rer en ny konstruktion i sit Logiweb-dokument, er det et krav, at man g\o{}r den tilsvarende pyk definition tilg\ae{}ngelig i dokumentet. Denne rapports pyk definitioner er vedlagt i bilag \ref{sec:pyk}.

Logiweb genererer det f\ae{}rdige dokument ved hj\ae{}lp af det kendte formateringssprog  \LaTeX. Derfor har hver formel konstruktion ogs\aa{} tilknyttet en s\aa{}kaldt ``\TeX{} definition'', som angiver, hvordan konstruktionen skal skrives i \LaTeX. Ligesom pyk definitioner skal ogs\aa{} \TeX{} definitioner v\ae{}re tilg\ae{}ngelige i dokumentet. Denne rapports \TeX{} definitioner er vedlagt i bilag \ref{sec:tex}.

Endelig indeholder bilag \ref{sec:prio} en tabel over alle de formelle konstruktioner i denne rapport --- b\aa{}de de importerede og dem, jeg selv har defineret. Denne tabel er f\o{}rst og fremmest med, for at andre Logiweb-dokumenter skal kunne referere til rapporten; uden tabellen kan s\aa{}danne referencer ikke finde sted.

\subsection{L-kode} \label{sec:l}

Logiweb-dokumentet \cite{kn:base} indeholder et funktionelt programmeringssprog, som jeg vil kalde for ``L''. Der er en del forekomster af L-kode i rapporten. For en forklaring af de konstruktioner fra L, som jeg bruger, vil jeg henvise til funktionsbeskrivelserne i appendikset til \cite{kn:check}, afsnit 3.2, s.\ 6. Herudover vil jeg supplere med et par kommentarer undervejs. 

\newpage
\section{Syntaks for $ZFsub$ } \label{sec:syntaks}

Som n\ae{}vnt i indledningen vil jeg arbejde med en reduceret version af ZF m\ae{}ngdel\ae{}re, som jeg kalder for ``$ZFsub$''. Dette afsnit beskriver syntaksen for $ZFsub$. Der er to syntaktiske hovedkategorier i $ZFsub$: Termer og formler. Underafsnit \ref{sec:termer} beskriver syntaksen for termer, og underafsnit \ref{sec:formler} beskriver syntaksen for formler.

\subsection{Termer} \label{sec:termer}

Syntaksen for en term ${\cal T}$ kan beskrives ved den f\o{}lgende BNF-grammatik:
%
\begin{eqnarray*}
{\cal T}  & ::= &  \texttt{V\ae{}rdi} \; | \; \texttt{Variabel} \\
\texttt{V\ae{}rdi}
 & ::= & \mathrm{\O}  \; | \; 
 \{{\cal T}
,{\cal T}
\}  \; | \; 
\cup {\cal T}      \; | \;
P({\cal T}
)      \; | \;
 \{ ph \mathrel{\in} {\cal T}
\mid {\cal F}
\} \\
\texttt{Variabel} & ::=  & \texttt{Objekt-var} \; | \; \texttt{Ex-var} \; | \; 
\texttt{Ph-var}
%\label{eq:term}
\end{eqnarray*}
%
En \texttt{V\ae{}rdi} svarer til en konkret m\ae{}ngde --- v\ae{}rditermer indeholder ingen variable. Den grundl\ae{}ggende v\ae{}rdi er den tomme m\ae{}ngde; herudfra kan vi konstruere par, f\ae{}llesm\ae{}ngder, potensm\ae{}ngder samt delm\ae{}ngder hvis elementer opfylder en bestemt egenskab. Der er s\aa{}ledes ingen individuelle konstanter i $ZFsub$; alt er m\ae{}ngder.

En \texttt{Variabel} kan for det f\o{}rste v\ae{}re en objekt-variabel, som varierer over v\ae{}rdier\footnote{Vi repr\ae{}senterer objektvariable med symbolerne ``$
\mathit{a}$'', ``$
\mathit{b}$'', $\cdots$ ``$
\mathit{z}$'' samt ``$ \mathsf {OBS}$''.}. De to andre typer af variable --- eksistens-variable og pladsholder-variable --- vil jeg vente med at forklare til hhv.\ afsnit \ref{sec:eksistens} og \ref{sec:separation}.

\subsection{Formler} \label{sec:formler}

Syntaksen for en formel ${\cal F}$ kan beskrives ved den f\o{}lgende BNF-grammatik:
%
\begin{eqnarray*}
{\cal F} & ::= & {\cal T}
 \mathrel{\in} {\cal T}   \; | \;  
{\cal T}
\!\mathrel{=}\! {\cal T} \; | \; 
\dot{\neg}\, {\cal F}  \; | \;
{\cal F}
\Rightarrow {\cal F} \; | \;
\forall \texttt{Objekt-var}
\colon {\cal F}
\end{eqnarray*}
%

\noindent Med en formel kan vi alts\aa{} p\aa{}st\aa{}, at en m\ae{}ngde tilh\o{}rer en anden m\ae{}ngde, eller at to m\ae{}ngder er lig hinanden. Desuden kan vi negere formler\footnote{Jeg skriver negationstegnet med en prik over for at undg\aa{} forveksling med konstruktionen $[
{\neg }
\mathsf{x}]$ fra \cite{kn:base}.}, lade formler implicere hinanden, samt kvantificere formler med objektvariable.

\subsection{Objektvariable vs.\ metavariable} \label{sec:objekt}

En metavariabel\footnote{Vi repr\ae{}senterer metavariable med symbolerne ``${\cal A}$'', ``${\cal B}$'', $\cdots$ ``${\cal Z}$'' samt ``${\cal BS}$''.}
 er en variabel, der varierer over vilk\aa{}rlige termer --- alts\aa{} ogs\aa{} over objektvariable. Denne rapport er ikke helt fri for objektvariable; men n\aa{}r vi taler om $ZFsub$ i aksiomer\footnote{Strengt taget burde jeg skrive ``aksiomskemaer'', da vi bruger metavariable. Da der kun er aksiomskemaer i denne rapport, vil jeg imidlertid bruge ordet ``aksiom'' for at g\o{}re teksten lidt lettere.}, definitioner og beviser vil jeg s\aa{} vidt muligt bruge metavariable, da de er mere fleksible end objektvariable. F.eks.\ kan ``$
\mathit{s}$''
i formlen $[
\mathit{s}
\!\mathrel{=}\! {\cal X}]$\footnote{En fodnote om stil: Jeg vil ofte indramme matematiske udtryk i firkantede parenteser ``$[\: ]$'' for at adskille udtrykkene fra den omgivende tekst. De firkantede parenteser har ingen selvst\ae{}ndig betydning.} kun instantieres til en term; vi kan ikke skifte objektvariabel og konkludere $[
\mathit{t}
\!\mathrel{=}\! {\cal X}]$. Derimod kan vi sagtens instantiere formlen $[{\cal S}
\!\mathrel{=}\! {\cal X}]$ til $[
\mathit{t}
\!\mathrel{=}\! {\cal X}]$. En liste over de metavariable, jeg vil bruge, kan ses i bilag \ref{sec:variable}.

%\newpage
\section{Aksiomatisk system} \label{sec:aksiomsystem}

$ZFsub$ er en teori i 1.\ ordens pr\ae{}dikatkalkyle. Vi kan s\aa{}ledes opdele aksiomsystemet for $ZFsub$ i to: En pr\ae{}dikatlogisk del og en m\ae{}ngdeteoretisk del. Underafsnit \ref{sec:1order} gennemg\aa{}r den pr\ae{}dikatlogiske del, og underafsnit \ref{sec:axiomzf} gennemg\aa{}r den m\ae{}ngdeteoretiske del. Bilag \ref{sec:helezf} indeholder en kopi af det samlede aksiomsystem.

\subsection{1.\ ordens pr\ae{}dikatkalkyle} \label{sec:1order}

Her er de f\o{}rste slutningsregler i $ZFsub$:

\display{
$
[ \mathbf{Theory\ } ZFsub
]$

$
[ ZFsub
\mathbf{\ rule\ } MP
\colon 
\Pi {\cal A}
,\linebreak [0] {\cal B}
\colon {\cal A}
\Rightarrow {\cal B}
\vdash {\cal A}
\vdash {\cal B}
]$

$
[ ZFsub
\mathbf{\ rule\ } Gen
\colon 
\Pi {\cal X}
,\linebreak [0] {\cal A}
\colon {\cal A}
\vdash 
\forall {\cal X}
\colon {\cal A}
]$

$
[ ZFsub
\mathbf{\ rule\ } Repetition
\colon 
\Pi {\cal A}
\colon {\cal A}
\vdash {\cal A}
]$

$
[ ZFsub
\mathbf{\ rule\ } Neg
\colon 
\Pi {\cal A}
,\linebreak [0] {\cal B}
\colon \dot{\neg}\, {\cal B}
\Rightarrow {\cal A}
\vdash \dot{\neg}\, {\cal B}
\Rightarrow \dot{\neg}\, {\cal A}
\vdash {\cal B}
]$

$
[ ZFsub
\mathbf{\ rule\ } Ded
\colon 
\Pi {\cal A}
,\linebreak [0] {\cal B}
\colon 
Dedu({\cal A}
,{\cal B}
)
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} {\cal A}
\vdash {\cal B}
]$
}

F\o{}rst et par ord om syntaks: Konstruktionen $[
\mathsf{x}
\vdash 
\mathsf{y}]$ st\aa{}r for inferens, alts\aa{} at vi kan bevise $
\mathsf{y}$, hvis vi har et bevis for $
\mathsf{x}$. Konstruktionen $[
\mathsf{x}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathsf{y}]$ betyder, at      
$
\mathsf{y}$ g\ae{}lder, hvis sidebetingelsen $
\mathsf{x}$ er sand. Endelig er konstruktionen $[
\Pi 
\mathsf{x}
\colon 
\mathsf{y}]$ en meta-alkvantor; betydningen er, at metavariablen $
\mathsf{x}$ kan instantieres til hvad som helst --- selv andre metavariable.

De viste slutningsregler er n\ae{}sten identiske med 
reglerne $
MP$, $
Gen$, $
Neg$ og $
Ded$ i systemet $
S$ fra \cite{kn:check}. Der er dog to \ae{}ndringer. For det f\o{}rste har jeg tilf\o{}jet $[
\Pi {\cal A}
\colon {\cal A}
\vdash {\cal A}]$ som en slutningsregel. (I \cite{kn:check} vises $[
\Pi {\cal A}
\colon {\cal A}
\vdash {\cal A}]$ med et lav-niveau bevis). For det andet har jeg lavet en lille \ae{}ndring i deduktionsreglen $Ded$, som g\o{}r den i stand til at h\aa{}ndtere sidebetingelser bedre. Jeg beskriver \ae{}ndringen i bilag \ref{sec:deduktion}.

Som n\ae{}vnt i \cite{kn:check} kan deduktionsreglen erstatte enhver anvendelse af aksiomskemaerne A4 og A5 fra \cite{kn:mendel} (se evt.\ Mendelsons system p\aa{} s.\ 69). Derfor har jeg ikke medtaget disse aksiomskemaer. P\aa{} denne m\aa{}de f\aa{}r vi ogs\aa{} afpr\o{}vet deduktionsreglens brugervenlighed; vi skal straks se et eksempel, hvor A4 kunne have gjort nytte.

\verb| | 

\subsubsection{H\aa{}ndtering af eksistenskvantorer}
\label{sec:eksistens}

Et sp\o{}rgsm\aa{}l er nemlig, hvordan man skal h\aa{}ndtere introduktion og elimination af eksistenskvantorer --- alts\aa{} slutninger som f.eks.\ 
$[\mathrm{\O}
\!\mathrel{=}\! \mathrm{\O}
\vdash 
\exists 
\mathit{x}
\colon 
\mathit{x}
\!\mathrel{=}\! 
\mathit{x}]$ og
$[
\exists 
\mathit{x}
\colon 
\mathit{x}
\!\mathrel{=}\! 
\mathit{x}
\vdash 
\mathit{c}
\!\mathrel{=}\! 
\mathit{c}]$ (hvor ``$
\mathit{c}$'' er et ikke tidligere anvendt navn p\aa{} en konstant).

Det er muligt at h\aa{}ndtere denne slags slutninger alene med reglerne fra afsnit \ref{sec:1order}, men det er omst\ae{}ndigt og tidskr\ae{}vende (sammenlign  f.eks.\ de to beviser p\aa{} s.\ 81 i \cite{kn:mendel}). Som et minimum kr\ae{}ver det, at man har A4 fra \cite{kn:mendel} til r\aa{}dighed --- og som n\ae{}vnt ovenfor har jeg valgt ikke at medtage dette aksiomskema.

Jeg har i stedet implementeret en l\o{}sning, der er baseret p\aa{} begrebet ``eksistensvariabel''. Vi indf\o{}rer en un\ae{}r operator $[
\mathsf{x}
 ^{Ex}]$ og definerer, at en term er en ekistens-variabel, hviss den har $[
\mathsf{x}
 ^{Ex}]$ som principal operator. Funktionen $[
\mathsf{x}
 ^{Ex}]$ tester, om $
\mathsf{x}$ er en eksistens-variabel:

\display{$
[
\mathsf{x}
 ^{Ex}
\mathrel {\dot {=}}
\mathsf{x}
\stackrel{r}{=} 
\lceil 
\mathsf{x}
 _{Ex}
\rceil 
]$\footnote{Konstruktionen $
[
\mathsf{x}
\mathrel {\dot {=}}
\mathsf{y}
]$ er en s\aa{}kaldt ``v\ae{}rdidefinition'' i L (jvf.\ afsnit \ref{sec:l}). Den svarer til en almindelig funktionsdefinition; vi knytter funktionssignaturen $
\mathsf{x}$ til kroppen $
\mathsf{y}$.}
}

\noindent Vi kan da definere de fire eksistens-variable, som denne rapport vil g\o{}re brug af (jvf.\ bilag \ref{sec:variable}): 

\verb| | 

\display{
$
[Ex_{1}
\mathrel {\ddot {=}}
\mathsf {a}
 _{Ex}
]$\footnote{Konstruktionen $
[
\mathsf{x}
\mathrel {\ddot {=}}
\mathsf{y}
]$ st\aa{}r for ``makrodefinition'' i L. Vi definerer $
\mathsf{x}$ som v\ae{}rende en forkortelse for $
\mathsf{y}$. Ud fra bevischeckerens synspunkt er der ingen forskel p\aa{} $
\mathsf{x}$ og $
\mathsf{y}$.}

$
[Ex_{2}
\mathrel {\ddot {=}}
\mathsf {b}
 _{Ex}
]$

$
[Ex_{10}
\mathrel {\ddot {=}}
\mathsf {j}
 _{Ex}
]$

$
[Ex_{20}
\mathrel {\ddot {=}}
\mathsf {t}
 _{Ex}
]$\footnote{``j'' og ``t'' er hhv.\ bogstav nr.\ 10 og 20 i alfabetet.}
}

\noindent Ideen med disse definitioner er at repr\ae{}sentere en formel som f.eks.\ \mbox{$[
\exists 
\mathit{x}
\colon 
\mathit{x}
\!\mathrel{=}\! 
\mathit{x}]$} ved formlen $[{\cal X}
\!\mathrel{=}\! {\cal X}]$, hvor ``${\cal X}$'' er en eksistensvariabel. P\aa{} denne m\aa{}de kommer eksistensvariablene til at fungere som erstatning for eksistenskvantoren. Dette betyder ogs\aa{}, at der ikke er brug for at nogen regel, der eksplicit fjerner eksistenskvantorer --- de er allerede v\ae{}k.

Til geng\ae{}ld f\aa{}r vi brug for en regel, der kan introducere eksistensvariable. Til den ende definerer vi pr\ae{}dikatet $\langle 
\mathsf {a}
{\equiv} 
\mathsf {b}
| 
\mathsf{x}
{:=} 
\mathsf {t}
\rangle_{Ex} $ i L:

\verb| | 

\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}

\item  $
[\langle 
\mathsf {a}
{\equiv} 
\mathsf {b}
| 
\mathsf{x}
{:=} 
\mathsf {t}
\rangle_{Ex} 
\mathrel {\ddot {=}}\langle 
\lceil 
\mathsf {a}
\rceil 
{\equiv}^0 
\lceil 
\mathsf {b}
\rceil 
| 
\lceil 
\mathsf{x}
\rceil 
{:=} 
\lceil 
\mathsf {t}
\rceil 
\rangle_{Ex} 
]$

\item  $
[\langle 
\mathsf {a}
{\equiv}^0 
\mathsf {b}
| 
\mathsf{x}
{:=} 
\mathsf {t}
\rangle_{Ex} 
\mathrel {\dot {=}}
\lambda 
\mathsf {c}
.
\mathsf{x}
 ^{Ex}
\wedge \langle 
\mathsf {a}
{\equiv}^1 
\mathsf {b}
| 
\mathsf{x}
{:=} 
\mathsf {t}
\rangle_{Ex} 
]$

\item  $
[\langle 
\mathsf {a}
{\equiv}^1 
\mathsf {b}
| 
\mathsf{x}
{:=} 
\mathsf {t}
\rangle_{Ex} 
\mathrel {\dot {=}}
\mathsf {a}
!
\mathsf{x}
!
\mathsf {t}
!
\newline 
{\bf if} \ 
\mathsf {b}
\stackrel{r}{=} 
\lceil 
\forall 
\mathsf {u}
\colon 
\mathsf {v}
\rceil 
\ {\bf then} \ 
\mathsf {F}
\ {\bf else} \ 
\newline 
{\bf if} \ 
\mathsf {b}
 ^{Ex}
\wedge 
\mathsf {b}
\stackrel {t}{=}
\mathsf{x}
\ {\bf then} \ 
\mathsf {a}
\stackrel {t}{=}
\mathsf {t}
\ {\bf else} \ 
\newline 
\mathsf {a}
\stackrel{r}{=} 
\mathsf {b}
\mathrel{\ddot{\wedge}} \langle 
\mathsf {a}
{}^t
{\equiv}^* 
\mathsf {b}
{}^t
| 
\mathsf{x}
{:=} 
\mathsf {t}
\rangle_{Ex} 
]$

\item  $
[\langle 
\mathsf {a}
{\equiv}^* 
\mathsf {b}
| 
\mathsf{x}
{:=} 
\mathsf {t}
\rangle_{Ex} 
\mathrel {\dot {=}}
\mathsf {b}
!
\mathsf{x}
!
\mathsf {t}
!
\mathrm {If}(
\mathsf {a}
,\linebreak [0]
\mathsf {T}
,\linebreak [0]\langle 
\mathsf {a}
{}^h
{\equiv}^1 
\mathsf {b}
{}^h
| 
\mathsf{x}
{:=} 
\mathsf {t}
\rangle_{Ex} 
\mathrel{\ddot{\wedge}} \langle 
\mathsf {a}
{}^t
{\equiv}^* 
\mathsf {b}
{}^t
| 
\mathsf{x}
{:=} 
\mathsf {t}
\rangle_{Ex} 
)
]$

\end{list}

\noindent Pr\ae{}dikatet $\langle 
\mathsf {a}
{\equiv} 
\mathsf {b}
| 
\mathsf{x}
{:=} 
\mathsf {t}
\rangle_{Ex} $ er sandt, hvis $
\mathsf{x}$ er en eksistens-variabel, og hvis formlen $
\mathsf {a}$ er identisk med resultatet af at erstatte alle forekomster af $
\mathsf{x}$ i formlen $
\mathsf {b}$ med termen $
\mathsf {t}$. F.eks.\ er $\langle 
(\mathrm{\O}
\!\mathrel{=}\! \mathrm{\O}
)
{\equiv} 
(Ex_{1}
\!\mathrel{=}\! Ex_{1}
)
| Ex_{1}
{:=} \mathrm{\O}
\rangle_{Ex} $ sand. Herudover er det et krav, at hverken $
\mathsf {a}$ eller $
\mathsf {b}$ m\aa{} indeholde objektkvantorer; s\aa{}ledes er $\langle 
(
\forall 
\mathit{s}
\colon 
\mathit{s}
\!\mathrel{=}\! \mathrm{\O}
)
{\equiv} 
(
\forall 
\mathit{s}
\colon 
\mathit{s}
\!\mathrel{=}\! Ex_{1}
)
| Ex_{1}
{:=} \mathrm{\O}
\rangle_{Ex} $ falsk. Dette krav er udelukkende indf\o{}rt for at g\o{}re koden for $\langle 
\mathsf {a}
{\equiv} 
\mathsf {b}
| 
\mathsf{x}
{:=} 
\mathsf {t}
\rangle_{Ex} $ s\aa{} simpel som mulig; det har ikke v\ae{}ret n\o{}dvendigt at s\ae{}tte $\langle 
\mathsf {a}
{\equiv} 
\mathsf {b}
| 
\mathsf{x}
{:=} 
\mathsf {t}
\rangle_{Ex} $ i stand til at h\aa{}ndtere kvantificering.

Vi kan nu definere den slutningsregel, der st\aa{}r for introduktion af eksistensvariable:

\display{
$
[ ZFsub
\mathbf{\ rule\ } ExistIntro
\colon 
\Pi {\cal X}
,\linebreak [0] {\cal T}
,\linebreak [0] {\cal A}
,\linebreak [0] {\cal B}
\colon \langle {\cal A}
{\equiv} {\cal B}
| {\cal X}
{:=} {\cal T}
\rangle_{Ex} 
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} {\cal A}
\vdash {\cal B}
]$
%
\footnote{I denne definition varierer ``${\cal X}$'' over eksistens-variable.}
%
}

\noindent Med $ExistIntro$ kan vi f.eks.\ slutte $[Ex_{1}
\!\mathrel{=}\! Ex_{1}]$ ud fra $[\mathrm{\O}
\!\mathrel{=}\! \mathrm{\O}]$. Vi har nu defineret seks aksiomer, der tilsammen d\ae{}kker 1. ordens pr\ae{}dikatkalkyle.
 
\verb| | 

\subsection{Selve m\ae{}ngdel\ae{}ren} \label{sec:axiomzf}

De aksiomer i $ZFsub$, der vedr\o{}rer selve m\ae{}ngdel\ae{}ren, har jeg hentet fra kapitel 4.3 og 4.4 i \cite{kn:gold}. Her er de f\o{}rste fem:

\display{ 
$
[ ZFsub
\mathbf{\ rule\ } Extensionality
\colon 
\Pi {\cal X}
,\linebreak [0] {\cal Y}
\colon {\cal X}
\!\mathrel{=}\! {\cal Y}
\mathrel{\dot{\Leftrightarrow}} 
\forall 
\mathit{s}
\colon 
(
\mathit{s}
 \mathrel{\in} {\cal X}
\mathrel{\dot{\Leftrightarrow}} 
\mathit{s}
 \mathrel{\in} {\cal Y}
)
]$

$
[ ZFsub
\mathbf{\ rule\ } \O{}def
\colon 
\Pi {\cal S}
\colon \dot{\neg}\, {\cal S}
 \mathrel{\in} \mathrm{\O}
]$

$
[ ZFsub
\mathbf{\ rule\ } PairDef
\colon 
\Pi {\cal S}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Y}
\colon {\cal S}
 \mathrel{\in} \{{\cal X}
,{\cal Y}
\}
\mathrel{\dot{\Leftrightarrow}} {\cal S}
\!\mathrel{=}\! {\cal X}
 \mathrel{\dot{\vee}} {\cal S}
\!\mathrel{=}\! {\cal Y}
]$

$
[ ZFsub
\mathbf{\ rule\ } UnionDef
\colon 
\Pi {\cal S}
,\linebreak [0] {\cal X}
\colon {\cal S}
 \mathrel{\in} \cup {\cal X}
\mathrel{\dot{\Leftrightarrow}} 
({\cal S}
 \mathrel{\in} Ex_{10}
\mathrel{\dot{\wedge}} Ex_{10}
 \mathrel{\in} {\cal X}
)
]$

$
[ ZFsub
\mathbf{\ rule\ } PowerDef
\colon 
\Pi {\cal S}
,\linebreak [0] {\cal X}
\colon {\cal S}
 \mathrel{\in} P({\cal X}
)
\mathrel{\dot{\Leftrightarrow}} 
\forall 
\mathit{s}
\colon 
(
\mathit{s}
 \mathrel{\in} {\cal S}
\Rightarrow 
\mathit{s}
 \mathrel{\in} {\cal X}
)
]$\footnote{Konstruktionerne $[
\mathsf{x}
\mathrel{\dot{\wedge}} 
\mathsf{y}]$, $[
\mathsf{x}
 \mathrel{\dot{\vee}} 
\mathsf{y}]$ og $[
\mathsf{x}
\mathrel{\dot{\Leftrightarrow}} 
\mathsf{y}]$ svarer til de kendte konnektiver; de bliver defineret formelt i afsnit \ref{sec:konnektiver}.}
}

Reglen $Extensionality$ siger, at to m\ae{}ngder er ens, hviss de har de samme elementer. De \o{}vrige fire regler definerer begreberne ``tom m\ae{}ngde'', ``par'', ``foreningssm\ae{}ngde'' og ``potensm\ae{}ngde''. L\ae{}g m\ae{}rke til, at 
$[\cup {\cal T}]$ er en un\ae{}r operator; ideen er, at $\cup {\cal T}$ er lig med foreningsm\ae{}ngden af alle de m\ae{}ngder, som ${\cal T}$ indeholder. 

Bem\ae{}rk ogs\aa{} at to af aksiomerne indeholder objektvariablen $
\mathit{s}$. Forklaringen herp\aa{} er, at aksiomerne indeholder objektkvantoren $[
\forall 
\mathsf{x}
\colon 
\mathsf{y}]$; og deduktionsreglen fra \cite{kn:check} er ikke egnet til at h\aa{}ndtere kombinationen ``objektkvantor og metavariabel''. Derfor vil $
\mathsf{x}$ i $[
\forall 
\mathsf{x}
\colon 
\mathsf{y}]$ altid v\ae{}re en objektvariabel i denne rapport.

\verb| | 

\subsubsection{Separation og pladsholdervariable}
\label{sec:separation}

Vi mangler stadigv\ae{}k et ``separationsaksiom'' --- dvs.\ et aksiom, der  giver mening til konstruktionen $[ \{ ph \mathrel{\in} {\cal T}
\mid {\cal F}
\}]$ fra afsnit \ref{sec:termer}.  For at implementere separationsaksiomet indf\o{}rer vi begrebet ``pladsholder-variabel''. Fremgangsm\aa{}den er den samme som ved eksistens-variable i afsnit \ref{sec:eksistens}. F\o{}rst indf\o{}rer vi en un\ae{}r operator $[
\mathsf{x}
 _{Ph} ]$ og definerer, at en term er en pladsholder-variabel, hviss den har $[
\mathsf{x}
 _{Ph} ]$ som prim\ae{}r operator. Funktionen $[
\mathsf{x}
 ^{Ph}]$ checker, om $
\mathsf{x}$ er en pladsholder-variabel:

%\verb| |

\display{$
[
\mathsf{x}
 ^{Ph}
\mathrel {\dot {=}}
\mathsf{x}
\stackrel{r}{=} 
\lceil 
\mathsf{x}
 _{Ph} 
\rceil 
]$
}

\noindent Vi kan nu definere de pladsholder-variable, vi f\aa{}r brug for, som f\o{}lger (jvf.\ bilag \ref{sec:variable}):

\verb| |

\display{
$
[ph_{1}
\mathrel {\ddot {=}}
\mathsf {a}
 _{Ph} 
]$

$
[ph_{2}
\mathrel {\ddot {=}}
\mathsf {b}
 _{Ph} 
]$

$
[ph_{3}
\mathrel {\ddot {=}}
\mathsf {c}
 _{Ph} 
]$
}

\noindent S\aa{} definerer vi pr\ae{}dikatet $\langle 
\mathsf {a}
{\equiv} 
\mathsf {b}
| 
\mathsf{x}
{:=} 
\mathsf {t}
\rangle_{Ph} $ i L:

\verb| |

\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}

\item
 $
[\langle 
\mathsf {a}
{\equiv} 
\mathsf {b}
| 
\mathsf{x}
{:=} 
\mathsf {t}
\rangle_{Ph} 
\mathrel {\ddot {=}}\langle 
\lceil 
\mathsf {a}
\rceil 
{\equiv}^0 
\lceil 
\mathsf {b}
\rceil 
| 
\lceil 
\mathsf{x}
\rceil 
{:=} 
\lceil 
\mathsf {t}
\rceil 
\rangle_{Ph} 
]$

\item  $
[\langle 
\mathsf {a}
{\equiv}^0 
\mathsf {b}
| 
\mathsf{x}
{:=} 
\mathsf {t}
\rangle_{Ph} 
\mathrel {\dot {=}}
\lambda 
\mathsf {c}
.
\mathsf{x}
 ^{Ph}
\wedge \langle 
\mathsf {a}
{\equiv}^1 
\mathsf {b}
| 
\mathsf{x}
{:=} 
\mathsf {t}
\rangle_{Ph} 
]$

\item  $
[\langle 
\mathsf {a}
{\equiv}^1 
\mathsf {b}
| 
\mathsf{x}
{:=} 
\mathsf {t}
\rangle_{Ph} 
\mathrel {\dot {=}}
\mathsf {a}
!
\mathsf{x}
!
\mathsf {t}
!
\newline 
{\bf if} \ 
\mathsf {b}
\stackrel{r}{=} 
\lceil 
\forall 
\mathsf {u}
\colon 
\mathsf {v}
\rceil 
\ {\bf then} \ 
\mathsf {F}
\ {\bf else} \ 
\newline 
{\bf if} \ 
\mathsf {b}
 ^{Ph}
\wedge 
\mathsf {b}
\stackrel {t}{=}
\mathsf{x}
\ {\bf then} \ 
\mathsf {a}
\stackrel {t}{=}
\mathsf {t}
\ {\bf else} \ 
\newline 
{\bf if} \ 
\mathsf {b}
 ^{Ex}
\ {\bf then} \ 
\mathsf {a}
\stackrel{r}{=} 
\mathsf {b}
\ {\bf else} \ 
\newline 
\mathsf {a}
\stackrel{r}{=} 
\mathsf {b}
\mathrel{\ddot{\wedge}} \langle 
\mathsf {a}
{}^t
{\equiv}^* 
\mathsf {b}
{}^t
| 
\mathsf{x}
{:=} 
\mathsf {t}
\rangle_{Ph} 
]$

 \item $
[\langle 
\mathsf {a}
{\equiv}^* 
\mathsf {b}
| 
\mathsf{x}
{:=} 
\mathsf {t}
\rangle_{Ph} 
\mathrel {\dot {=}}
\mathsf {b}
!
\mathsf{x}
!
\mathsf {t}
!
\mathrm {If}(
\mathsf {a}
,\linebreak [0]
\mathsf {T}
,\linebreak [0]\langle 
\mathsf {a}
{}^h
{\equiv}^1 
\mathsf {b}
{}^h
| 
\mathsf{x}
{:=} 
\mathsf {t}
\rangle_{Ph} 
\mathrel{\ddot{\wedge}} \langle 
\mathsf {a}
{}^t
{\equiv}^* 
\mathsf {b}
{}^t
| 
\mathsf{x}
{:=} 
\mathsf {t}
\rangle_{Ph} 
)
]$

\end{list}

\noindent Pr\ae{}dikatet $\langle 
\mathsf {a}
{\equiv} 
\mathsf {b}
| 
\mathsf{x}
{:=} 
\mathsf {t}
\rangle_{Ph} $ er sandt, hvis $
\mathsf{x}$ er en pladsholder-variabel, og hvis formlen $
\mathsf {a}$ er identisk med resultatet af at erstatte alle forekomster af $
\mathsf{x}$ i formlen $
\mathsf {b}$ med termen $
\mathsf {t}$. 
Ligesom ved $\langle 
\mathsf {a}
{\equiv} 
\mathsf {b}
| 
\mathsf{x}
{:=} 
\mathsf {t}
\rangle_{Ex} $ kr\ae{}ver vi, at hverken $
\mathsf {a}$ eller $
\mathsf {b}$ indeholder objektkvantorer.
F.eks.\ er $\langle 
(\mathrm{\O}
 \mathrel{\in} {\cal S}
)
{\equiv} 
(ph_{1}
 \mathrel{\in} {\cal S}
)
| ph_{1}
{:=} \mathrm{\O}
\rangle_{Ph} $ sand. Det er i \o{}vrigt tilladt, at $
\mathsf {a}$ og $
\mathsf {b}$ indeholder forskellige ekistensvariable; s\aa{}ledes er $\langle Ex_{1}
{\equiv} Ex_{2}
| ph_{1}
{:=} \mathrm{\O}
\rangle_{Ph} $ sand. P\aa{} denne m\aa{}de kan vi im\o{}dekomme kravet om, at en eksistens-variabel skal v\ae{}re ubrugt, n\aa{}r den introduceres i et bevis.

Vi definerer nu separationsaksiomet $SeparationDef$  ud fra $\langle 
\mathsf {a}
{\equiv} 
\mathsf {b}
| 
\mathsf{x}
{:=} 
\mathsf {t}
\rangle_{Ph} $:

\display{
$
[ ZFsub
\mathbf{\ rule\ } SeparationDef
\colon 
\Pi {\cal A}
,\linebreak [0] {\cal B}
,\linebreak [0] {\cal P}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Z}
\colon {\cal P}
 ^{Ph}
\wedge \langle {\cal B}
{\equiv} {\cal A}
| {\cal P}
{:=} {\cal Z}
\rangle_{Ph} 
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\newline {\cal Z}
 \mathrel{\in}  \{ ph \mathrel{\in} {\cal X}
\mid {\cal A}
\}
\mathrel{\dot{\Leftrightarrow}} {\cal Z}
 \mathrel{\in} {\cal X}
\mathrel{\dot{\wedge}} {\cal B}
]$
}

\noindent Ved f\o{}rste \o{}jekast virker konjunkten $[{\cal P}
 ^{Ph}]$ i denne definition overfl\o{}dig, fordi $\langle {\cal B}
{\equiv} {\cal A}
| {\cal P}
{:=} {\cal Z}
\rangle_{Ph} $ i forvejen kr\ae{}ver, at ${\cal P}$ er en pladsholder-variabel. Forklaringen er, at ``${\cal P}$'' kun optr\ae{}der i definitionens sidebetingelse. Dette betyder, at bevischeckeren har sv\ae{}rt ved at finde ud af, hvordan ``${\cal P}$'' skal instantieres, n\aa{}r vi anvender definitionen. Det ekstra krav $[{\cal P}
 ^{Ph}]$ hj\ae{}lper bevischeckeren til at instantiere ``${\cal P}$'' korrekt.

Med $SeparationDef$ til r\aa{}dighed kan vi nu f.eks.\ definere $ContainsEmpty(
\mathsf{x}
)$ som den delm\ae{}ngde af $
\mathsf{x}$, hvis elementer indeholder $\mathrm{\O}$:

\verb| | 

\display{$
[ContainsEmpty(
\mathsf{x}
)
\mathrel {\ddot {=}} \{ ph \mathrel{\in} 
\mathsf{x}
\mid \mathrm{\O}
 \mathrel{\in} ph_{1}
\}
]$,}

\noindent og vi kan slutte $[{\cal S}
 \mathrel{\in} {\cal X}
\mathrel{\dot{\wedge}} \mathrm{\O}
 \mathrel{\in} {\cal S}]$ ud fra $[{\cal S}
 \mathrel{\in} ContainsEmpty({\cal X}
)]$.

Vi bruger alts\aa{} en pladsholder-variabel som fri variabel i d\'{e}n formel ${\cal F}$, der definerer delm\ae{}ngden $ \{ ph \mathrel{\in} {\cal X}
\mid {\cal F}
\}$. Ideen med at bruge pladsholder-variable (frem for objekt- eller metavariable) til dette form\aa{}l er, at vi ikke \o{}nsker at kvantificere eller instantiere den frie variabel i ${\cal F}$. Det er f.eks.\ noget sludder at skrive ``$
\forall ph_{1}
\colon  \{ ph \mathrel{\in} {\cal X}
\mid \mathrm{\O}
 \mathrel{\in} ph_{1}
\}$'' eller at konkludere ``$ \{ ph \mathrel{\in} {\cal X}
\mid \mathrm{\O}
 \mathrel{\in} \mathrm{\O}
\}$'' ud fra definitionen af $ContainsEmpty({\cal X}
)$. Den frie variabel i ${\cal F}$ skal blot v\ae{}re en pladsholder; derfor pladsholder-variable.

\verb| | 

\section{Makrodefinitioner} \label{sec:makro}

Dette afsnit indeholder d\'{e} makrodefinitioner, som vi vil g\o{}re brug af i resten af rapporten. Definitionerne  drejer sig for det meste om m\ae{}ngdeteoretiske begreber, f.eks.\ ``\ae{}kvivalensklasse'' og ``partition''. Til sidst i afsnittet formulerer vi hovedresultatet --- at der til enhver \ae{}kvivalensrelation svarer en partition --- som et formelt teorem.

\subsection{Konnektiver} \label{sec:konnektiver}

Ud fra de to basale konnektiver $[\dot{\neg}\, 
\mathsf{x}]$ og $[
\mathsf{x}
\Rightarrow 
\mathsf{y}]$ definerer vi konjunktion, disjunktion og dobbeltimplikation:

\display{$
[
\mathsf{x}
\mathrel{\dot{\wedge}} 
\mathsf{y}
\mathrel {\ddot {=}}\dot{\neg}\, 
(
\mathsf{x}
\Rightarrow \dot{\neg}\, 
\mathsf{y}
)
]$

$
[
\mathsf{x}
 \mathrel{\dot{\vee}} 
\mathsf{y}
\mathrel {\ddot {=}}\dot{\neg}\, 
\mathsf{x}
\Rightarrow 
\mathsf{y}
]$

$
[
\mathsf{x}
\mathrel{\dot{\Leftrightarrow}} 
\mathsf{y}
\mathrel {\ddot {=}}
(
\mathsf{x}
\Rightarrow 
\mathsf{y}
)
\mathrel{\dot{\wedge}} 
(
\mathsf{y}
\Rightarrow 
\mathsf{x}
)
]$
}

\subsection{Negerede formler}

Det er ganske enkelt at definere negeret lighed ($
\mathsf{x}
 \mathrel{\neq} 
\mathsf{y}$) og negeret medlemskab ($
\mathsf{x}
 \mathrel{\notin} 
\mathsf{y}$):

\display{
$
[
\mathsf{x}
 \mathrel{\neq} 
\mathsf{y}
\mathrel {\ddot {=}}\dot{\neg}\, 
\mathsf{x}
\!\mathrel{=}\! 
\mathsf{y}
]$

$
[
\mathsf{x}
 \mathrel{\notin} 
\mathsf{y}
\mathrel {\ddot {=}}\dot{\neg}\, 
\mathsf{x}
 \mathrel{\in} 
\mathsf{y}
]$\footnote{H\o{}jresiderne i disse definitioner skal l\ae{}ses som hhv.\ $[\dot{\neg}\, 
(
\mathsf{x}
\!\mathrel{=}\! 
\mathsf{y}
)]$ og $[\dot{\neg}\, 
(
\mathsf{x}
 \mathrel{\in} 
\mathsf{y}
)]$.}
}

\subsection{Delm\ae{}ngde} \label{sec:subset}

M\ae{}ngden $
\mathsf{x}$ er en delm\ae{}ngde af $
\mathsf{y}$ hviss ethvert medlem af $
\mathsf{x}$ ogs\aa{} tilh\o{}rer $
\mathsf{y}$:

\display{
$
[
\mathsf{x}
 \mathrel{\subseteq} 
\mathsf{y}
\mathrel {\ddot {=}}
(
\mathit{s}
 \mathrel{\in} 
\mathsf{x}
\Rightarrow 
\mathit{s}
 \mathrel{\in} 
\mathsf{y}
)
]$ 
}

Vi kommer ikke til at bruge denne definition ret ofte. Man f\aa{}r tit en bedre f\o{}ling med, hvad der foreg\aa{}r i beviserne, hvis man skriver definitionen ud. Desuden bruger denne definition af $[
\mathsf{x}
 \mathrel{\subseteq} 
\mathsf{y}]$  objektvariable og implikation; vi vil ofte foretr\ae{}kke at bruge metavariable og inferens i stedet (som f.eks.\ i \mbox{$[{\cal S}
 \mathrel{\in} 
\mathsf{x}
\vdash {\cal S}
 \mathrel{\in} 
\mathsf{y}]$}). 

\verb| | 

\subsection{Singleton-m\ae{}ngde} \label{sec:single}

$[\{
\mathsf{x}
\}]$ er m\ae{}ngden, der indeholder $
\mathsf{x}$ som sit eneste 
element. Vi definerer $[\{
\mathsf{x}
\}]$ ved at parre $
\mathsf{x}$ med sig selv: 

\display{$
[\{
\mathsf{x}
\}
\mathrel {\ddot {=}}\{
\mathsf{x}
,
\mathsf{x}
\}
]$}

\subsection{Bin\ae{}r foreningsm\ae{}ngde og f\ae{}llesm\ae{}ngde}
\label{sec:makrobin}

Vi definerer foreningsm\ae{}ngden mellem to m\ae{}ngder $
\mathsf{x}$ og $
\mathsf{y}$ som f\o{}lger:

\display{
$
[
\mathsf{x}
 \mathrel{\cup} 
\mathsf{y}
\mathrel {\ddot {=}}\cup \{\{
\mathsf{x}
\}
,\{
\mathsf{y}
\}
\}
]$
}

F\ae{}llesm\ae{}ngden mellem to m\ae{}ngder $
\mathsf{x}$ og $
\mathsf{y}$ er en delm\ae{}ngde af deres foreningsm\ae{}ngde:

\verb| |

\display{
$
[
\mathsf{x}
 \cap 
\mathsf{y}
\mathrel {\ddot {=}} \{ ph \mathrel{\in} 
\mathsf{x}
 \mathrel{\cup} 
\mathsf{y}
\mid ph_{3}
 \mathrel{\in} 
\mathsf{x}
\mathrel{\dot{\wedge}} ph_{3}
 \mathrel{\in} 
\mathsf{y}
\}
]$
}

\subsection{Relation} \label{sec:relation}

Det ordnede par $\langle 
\mathsf{x}
,
\mathsf{y}
\rangle$ indeholder $
\mathsf{x}$ som ``f\o{}rstekomponent'' og $
\mathsf{y}$ som ``andenkomponent''. Den f\o{}lgende definition af $\langle 
\mathsf{x}
,
\mathsf{y}
\rangle$ er den mest udbredte i litteraturen (se f.eks.\ afsnit 4.3 i \cite{kn:gold} og afsnit 2.1 i \cite{kn:hrba}):

\verb| |

\display{$
[\langle 
\mathsf{x}
,
\mathsf{y}
\rangle
\mathrel {\ddot {=}}\{\{
\mathsf{x}
\}
,\{
\mathsf{x}
,
\mathsf{y}
\}
\}
]$}

Vi kan nu definere en ``relation'' som en m\ae{}ngde af ordnede par. Vi udtrykker denne definition ved at formalisere, hvad det vil sige, at $
\mathsf{x}$ er relateret til $
\mathsf{y}$ i kraft af relationen $
\mathsf {r}$:

\verb| |

\display{$
[
\mathsf {r}
(
\mathsf{x}
,
\mathsf{y}
)
\mathrel {\ddot {=}}\langle 
\mathsf{x}
,
\mathsf{y}
\rangle
 \mathrel{\in} 
\mathsf {r}
]$}

Vi kommer faktisk ikke til at bruge disse to definitioner i rapporten; beviserne vil behandle $[
\mathsf {r}
(
\mathsf{x}
,
\mathsf{y}
)]$ som en primitiv konstruktion. Men det er alligevel betryggende at have det formelle grundlag for relationsbegrebet p\aa{} plads.

\newpage 

\subsection{\AE{}kvivalensrelation} \label{sec:eqrel}

At en relation er refleksiv p\aa{} en m\ae{}ngde $
\mathsf{x}$ vil sige, at alle elementer i $
\mathsf{x}$ er relateret til sig selv:

\display{$
[ReflRel(
\mathsf {r}
,
\mathsf{x}
)
\mathrel {\ddot {=}}
\forall 
\mathit{s}
\colon 
(
\mathit{s}
 \mathrel{\in} 
\mathsf{x}
\Rightarrow 
\mathsf {r}
(
\mathit{s}
,
\mathit{s}
)
)
]$}

At en relation er symmetrisk p\aa{} en m\ae{}ngde $
\mathsf{x}$ vil sige, at alle elementer i $
\mathsf{x}$ opfylder den f\o{}lgende implikation:

\verb| | 

\display{$
[SymRel(
\mathsf {r}
,
\mathsf{x}
)
\mathrel {\ddot {=}}
\forall 
\mathit{s}
,\linebreak [0] 
\mathit{t}
\colon 
(
\mathit{s}
 \mathrel{\in} 
\mathsf{x}
\Rightarrow 
\mathit{t}
 \mathrel{\in} 
\mathsf{x}
\Rightarrow 
\mathsf {r}
(
\mathit{s}
,
\mathit{t}
)
\Rightarrow 
\mathsf {r}
(
\mathit{t}
,
\mathit{s}
)
)
]$}

At en relation er transitiv p\aa{} en m\ae{}ngde $
\mathsf{x}$ vil sige, at alle elementer i $
\mathsf{x}$ opfylder den f\o{}lgende implikation:

\verb| | 

\display{$
[TransRel(
\mathsf {r}
,
\mathsf{x}
)
\mathrel {\ddot {=}}
\newline 
\forall 
\mathit{s}
,\linebreak [0] 
\mathit{t}
,\linebreak [0] 
\mathit{u}
\colon 
(
\mathit{s}
 \mathrel{\in} 
\mathsf{x}
\Rightarrow 
\mathit{t}
 \mathrel{\in} 
\mathsf{x}
\Rightarrow 
\mathit{u}
 \mathrel{\in} 
\mathsf{x}
\Rightarrow 
\mathsf {r}
(
\mathit{s}
,
\mathit{t}
)
\Rightarrow 
\mathsf {r}
(
\mathit{t}
,
\mathit{u}
)
\Rightarrow 
\mathsf {r}
(
\mathit{s}
,
\mathit{u}
)
)
]$}

Endelig er en \ae{}kvivalensrelation det samme som en relation, der er refleksiv, symmetrisk og transitiv:

\verb| | 

\display{$
[EqRel(
\mathsf {r}
,
\mathsf{x}
)
\mathrel {\ddot {=}}ReflRel(
\mathsf {r}
,
\mathsf{x}
)
\mathrel{\dot{\wedge}} SymRel(
\mathsf {r}
,
\mathsf{x}
)
\mathrel{\dot{\wedge}} TransRel(
\mathsf {r}
,
\mathsf{x}
)
]$}

\subsection{M\ae{}ngde-variable}

Mange af rapportens beviser sker i forhold til en uspecificeret m\ae{}ngde. Vi vil referere til denne m\ae{}ngde med metavariablen ${\cal BS}$ og objektvariablen $ \mathsf {OBS}$:

\display{
$
[{\cal BS}
\mathrel {\ddot {=}}\underline{\mathsf {bs}}
]$

$
[ \mathsf {OBS}
\mathrel {\ddot {=}}\overline{\mathsf {bs}}
]$\footnote{Navnene ``${\cal BS}$'' og  ``$ \mathsf {OBS}$'' st\aa{}r for hhv.\ for ``big set'' og ``object big set''. Konstruktionerne $[\underline{
\mathsf{x}}]$ og $[\overline{
\mathsf{x}}]$ omdanner $
\mathsf{x}$ til hhv.\ en meta- og en objektvariabel. Variablen $[\mathsf {bs}]$ vil ogs\aa{} blive brugt i nogle af de kommende definitioner, men ikke i selve beviserne.} 
}

\noindent Vi vil s\aa{} vidt muligt bruge metavariablen, men i afsnit \ref{sec:sameinter} og senere bliver det n\o{}dvendigt at g\aa{} over til objektvariablen.

\verb| | 

\subsection{\AE{}kvivalensklasse} \label{sec:eqclass}

Lad $
\mathsf {r}$ v\ae{}re en \ae{}kvivalensrelation defineret p\aa{} $\mathsf {bs}$, og lad $
\mathsf{x}$ v\ae{}re et medlem af $\mathsf {bs}$. Vi definerer \ae{}kvivalensklassen $[
\mathsf{x}
 \mathrel{\in} \mathsf {bs}
]_{
\mathsf {r}
}$ som den delm\ae{}ngde af $\mathsf {bs}$, hvis medlemmer st\aa{}r i forhold til $
\mathsf{x}$:

\display{
$
[[
\mathsf{x}
 \mathrel{\in} \mathsf {bs}
]_{
\mathsf {r}
}
\mathrel {\ddot {=}} \{ ph \mathrel{\in} \mathsf {bs}
\mid 
\mathsf {r}
(ph_{1}
,
\mathsf{x}
)
\}
]$
}

\AE{}kvivalenssystemet $\mathsf {bs}
 / 
\mathsf {r}$  er m\ae{}ngden af alle de \ae{}kvivalensklasser, som $\mathsf {bs}$ definerer p\aa{} $
\mathsf {r}$. Vi definerer $\mathsf {bs}
 / 
\mathsf {r}$ som en delm\ae{}ngde af potensm\ae{}ngden $P(\mathsf {bs}
)$:

\verb| | 

\display{
$
[\mathsf {bs}
 / 
\mathsf {r}
\mathrel {\ddot {=}} \{ ph \mathrel{\in} P(\mathsf {bs}
)
\mid Ex_{20}
 \mathrel{\in} \mathsf {bs}
\mathrel{\dot{\wedge}} [Ex_{20}
 \mathrel{\in} \mathsf {bs}
]_{
\mathsf {r}
}
\!\mathrel{=}\! ph_{2}
\}
]$
}

\newpage
\subsection{Partition} \label{sec:partition}

En partition af en m\ae{}ngde $\mathsf {bs}$ er en m\ae{}ngde $
\mathsf {p}$, som opfylder tre krav:
%
\begin{enumerate}
\item Ingen af m\ae{}ngderne i $
\mathsf {p}$ er tomme.
\item Alle m\ae{}ngderne i $
\mathsf {p}$ er indbyrdes disjunkte.
\item Foreningsm\ae{}ngden af alle m\ae{}ngderne i $
\mathsf {p}$ er lig med $\mathsf {bs}$.
\end{enumerate}
%
Den formelle version af denne definition ser s\aa{}ledes ud:

\display{
$
[Partition(
\mathsf {p}
,\mathsf {bs}
)
\mathrel {\ddot {=}}
(
\forall 
\mathit{s}
\colon 
(
\mathit{s}
 \mathrel{\in} 
\mathsf {p}
\Rightarrow 
\mathit{s}
 \mathrel{\neq} \mathrm{\O}
)
)
\mathrel{\dot{\wedge}} 
\newline 
(
\forall 
\mathit{s}
,\linebreak [0] 
\mathit{t}
\colon 
(
\mathit{s}
 \mathrel{\in} 
\mathsf {p}
\Rightarrow 
\mathit{t}
 \mathrel{\in} 
\mathsf {p}
\Rightarrow 
\mathit{s}
 \mathrel{\neq} 
\mathit{t}
\Rightarrow 
\mathit{s}
 \cap 
\mathit{t}
\!\mathrel{=}\! \mathrm{\O}
)
)
\mathrel{\dot{\wedge}} 
\newline \cup 
\mathsf {p}
\!\mathrel{=}\! \mathsf {bs}
]$
}

Vi kan nu formulere hovedresultatet som et formelt lemma (hvor vi bruger objektvariable):

\verb| |

\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}

\item $
[ ZFsub
\mathbf{\ lemma\ } EqSysIsPartition
\colon EqRel(
\mathit{r}
, \mathsf {OBS}
)
\vdash Partition( \mathsf {OBS}
 / 
\mathit{r}
, \mathsf {OBS}
)
]$

\end{list}

\noindent Form\aa{}let med resten af rapporten er at bevise $EqSysIsPartition$.

\section{Udsagnslogisk bibliotek} \label{sec:udsagn}

I dette afsnit vil jeg bevise en samling af udsagnslogiske sandheder ( eller ``tautologier''), som vil blive brugt i de f\o{}lgende afsnit. De fleste af disse tautologier har mange andre anvendelser end lige netop m\ae{}ngdel\ae{}re. Beviserne er fordelt p\aa{} syv underafsnit; figur 1 giver et overblik over, hvordan beviserne forholder sig til hinanden. Jeg vil kommentere de fleste af beviserne; dog er nogle af dem s\aa{} tekniske,  at jeg har ladet dem st\aa{} alene.

% figur 1
\begin{figure} 
\resizebox{\textwidth}{!}{
\includegraphics[0cm,0cm][20cm,30cm]{/net/urd/home/disk15/eriksen/.logiweb/grafik/prop.eps}}
\caption[Bevisstrukturen for tautologierne]{Bevisstrukturen for tautologierne. En pil fra lemma $
\mathsf{x}$ til lemma $
\mathsf{y}$ betyder, at $
\mathsf{x}$ bruges i beviset for $
\mathsf{y}$. MP-lemmaerne fra afsnit \ref{sec:mplemmaer} er ikke vist. ``ImpTrans'' st\aa{}r for ``ImplyTransitivity''.}

\end{figure}
\clearpage

\subsection{MP-lemmaer} \label{sec:mplemmaer}

Man f\aa{}r ofte brug for at anvende slutningsreglen $MP$ flere gange i tr\ae{}k. Derfor vil jeg begynde med at vise fire lemmaer, der kan klare mellem 2 og 5 anvendelser af $MP$\footnote{I afsnit \ref{sec:transdef} f\aa{}r vi faktisk brug for at anvende $MP$ 6 gange i tr\ae{}k; men et eller andet sted skal man jo stoppe.}:

\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}

\item $
[ ZFsub
\mathbf{\ lemma\ } MP2
\colon 
\Pi {\cal A}
,\linebreak [0] {\cal B}
,\linebreak [0] {\cal C}
\colon {\cal A}
\Rightarrow {\cal B}
\Rightarrow {\cal C}
\vdash {\cal A}
\vdash {\cal B}
\vdash {\cal C}
]$

\item $
[ ZFsub
\mathbf{\ lemma\ } MP3
\colon 
\Pi {\cal A}
,\linebreak [0] {\cal B}
,\linebreak [0] {\cal C}
,\linebreak [0] {\cal D}
\colon {\cal A}
\Rightarrow {\cal B}
\Rightarrow {\cal C}
\Rightarrow {\cal D}
\vdash {\cal A}
\vdash {\cal B}
\vdash {\cal C}
\vdash {\cal D}
]$

\item $
[ ZFsub
\mathbf{\ lemma\ } MP4
\colon 
\Pi {\cal A}
,\linebreak [0] {\cal B}
,\linebreak [0] {\cal C}
,\linebreak [0] {\cal D}
,\linebreak [0] {\cal E}
\colon 
\newline {\cal A}
\Rightarrow {\cal B}
\Rightarrow {\cal C}
\Rightarrow {\cal D}
\Rightarrow {\cal E}
\vdash {\cal A}
\vdash {\cal B}
\vdash {\cal C}
\vdash {\cal D}
\vdash {\cal E}
]$

\item $
[ ZFsub
\mathbf{\ lemma\ } MP5
\colon 
\Pi {\cal A}
,\linebreak [0] {\cal B}
,\linebreak [0] {\cal C}
,\linebreak [0] {\cal D}
,\linebreak [0] {\cal E}
,\linebreak [0] {\cal F}
\colon 
\newline {\cal A}
\Rightarrow {\cal B}
\Rightarrow {\cal C}
\Rightarrow {\cal D}
\Rightarrow {\cal E}
\Rightarrow {\cal F}
\vdash {\cal A}
\vdash {\cal B}
\vdash {\cal C}
\vdash {\cal D}
\vdash {\cal E}
\vdash {\cal F}
]$

\end{list}

\subsubsection{Det f\o{}rste bevis}

Vi begynder med at bevise $MP2$:

\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}

\item $
[ ZFsub
\mathbf{\ lemma\ } MP2
\colon 
\Pi {\cal A}
,\linebreak [0] {\cal B}
,\linebreak [0] {\cal C}
\colon {\cal A}
\Rightarrow {\cal B}
\Rightarrow {\cal C}
\vdash {\cal A}
\vdash {\cal B}
\vdash {\cal C}
]$

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } MP2
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
,\linebreak [0] {\cal B}
,\linebreak [0] {\cal C}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
\Rightarrow {\cal B}
\Rightarrow {\cal C}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$MP
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal B}
\Rightarrow {\cal C}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$MP
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal C}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\end{list}

\noindent Da dette er rapportens f\o{}rste bevis, vil jeg bringe nogle ekstra kommentarer\footnote{Denne beskrivelse er en revideret udgave af afsnit 5.1 i \cite{kn:peano}.}. Oven over beviset har jeg gentaget definitionen af det, der skal bevises; dette er kun for overblikkets skyld --- det er ikke en formel n\o{}dvendighed. Selve beviset for $MP2$ best\aa{}r af seks linier, nummereret fra 1 til 6. En bevislinie kan have to former. Den f\o{}rste form er:
%
\begin{eqnarray*}
\texttt{Argumentation} & \gg & \texttt{Konklusion} %\label{eq:linie}
\end{eqnarray*}
%
hvor \texttt{Konklusion} er det som linien beviser, mens teksten i \texttt{Argumentation} udg\o{}r en begrundelse for, at \texttt{Konklusion} g\ae{}lder. F.eks.\ siger linie 5, at meta-formlen \mbox{$[{\cal B}
\Rightarrow {\cal C}]$} g\ae{}lder, fordi den kan udledes fra slutningsreglen $MP$ ved substitution. Argumentationen skal l\ae{}ses p\aa{} den m\aa{}de, at konklusionerne fra linie 2 og 3 bliver brugt som pr\ae{}misser til $MP$. Den generelle betydning af konstruktionen $[
\mathsf{x}
\rhd 
\mathsf{y}]$ er, at konklusionen fra linie $
\mathsf{y}$  bliver brugt som pr\ae{}mis i forhold til $
\mathsf{x}$.

Den anden form, en bevislinie kan have, er:
%
\begin{eqnarray*}
\texttt{N\o{}gleord} & \gg & \texttt{Konklusion} %\label{eq:linie2}
\end{eqnarray*}
%
hvor \texttt{N\o{}gleord} er et af de tre ord ``Arbitrary'', ``Premise'' eller ``Side-condition''. Betydningen af ordene ``Premise'' og ``Side-condition'' er \aa{}benlys: De angiver, at liniens konklusion indg\aa{}r som en pr\ae{}mis (hhv.\ sidebetingelse) i den s\ae{}tning, der skal bevises. F.eks.\ siger bevisets linie 2, at $MP2$ bruger meta-formlen \\ \mbox{$[{\cal A}
\Rightarrow {\cal B}
\Rightarrow {\cal C}]$} 
som pr\ae{}mis. N\aa{}r ordet ``Arbitrary'' bruges, best\aa{}r konklusionen af en liste af meta-variable (f.eks. $[{\cal A}
,\linebreak [0] {\cal B}
,\linebreak [0] {\cal C}]$ i linie 1). Ideen hermed er at udtrykke, at vi ikke antager noget om de p\aa{}g\ae{}ldende meta-variable, og at vi derfor har ret til at binde dem med en meta-alkvantor i den s\ae{}tning, der skal bevises. I det forh\aa{}ndenv\ae{}rende bevis berettiger linien med ``Arbitrary'' alts\aa{}, at $MP2$ er kvantificeret med $[
\Pi {\cal A}
,\linebreak [0] {\cal B}
,\linebreak [0] {\cal C}
\colon (\cdots{})]$.

Alt dette har drejet sig om den formelle syntaks for et Logiweb bevis. Der er ikke s\aa{} meget at sige om selve beviset for $MP2$; vi indkapsler simpelthen to p\aa{} hinanden f\o{}lgende anvendelser af $MP$. 

\subsubsection{Beviser for de andre MP-lemmaer}

Beviserne for de \o{}vrige MP-lemmaer er lige ud ad landevejen:

\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}

\item $
[ ZFsub
\mathbf{\ lemma\ } MP3
\colon 
\Pi {\cal A}
,\linebreak [0] {\cal B}
,\linebreak [0] {\cal C}
,\linebreak [0] {\cal D}
\colon {\cal A}
\Rightarrow {\cal B}
\Rightarrow {\cal C}
\Rightarrow {\cal D}
\vdash {\cal A}
\vdash {\cal B}
\vdash {\cal C}
\vdash {\cal D}
]$

\item
$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } MP3
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
,\linebreak [0] {\cal B}
,\linebreak [0] {\cal C}
,\linebreak [0] {\cal D}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_q \else
\if \relax \csname lgwellq\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellq {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellq \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
\Rightarrow {\cal B}
\Rightarrow {\cal C}
\Rightarrow {\cal D}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal C}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$MP2
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_q \else
\if \relax \csname lgwellq\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellq {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellq \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal C}
\Rightarrow {\cal D}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$MP
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal D}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

$
[ ZFsub
\mathbf{\ lemma\ } MP4
\colon 
\Pi {\cal A}
,\linebreak [0] {\cal B}
,\linebreak [0] {\cal C}
,\linebreak [0] {\cal D}
,\linebreak [0] {\cal E}
\colon 
\newline {\cal A}
\Rightarrow {\cal B}
\Rightarrow {\cal C}
\Rightarrow {\cal D}
\Rightarrow {\cal E}
\vdash {\cal A}
\vdash {\cal B}
\vdash {\cal C}
\vdash {\cal D}
\vdash {\cal E}
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } MP4
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
,\linebreak [0] {\cal B}
,\linebreak [0] {\cal C}
,\linebreak [0] {\cal D}
,\linebreak [0] {\cal E}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_q \else
\if \relax \csname lgwellq\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellq {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellq \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
\Rightarrow {\cal B}
\Rightarrow {\cal C}
\Rightarrow {\cal D}
\Rightarrow {\cal E}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal C}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal D}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_r \else
\if \relax \csname lgwellr\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellr {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellr \fi 
$:}$MP2
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_q \else
\if \relax \csname lgwellq\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellq {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellq \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal C}
\Rightarrow {\cal D}
\Rightarrow {\cal E}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$MP2
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_r \else
\if \relax \csname lgwellr\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellr {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellr \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal E}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

$
[ ZFsub
\mathbf{\ lemma\ } MP5
\colon 
\Pi {\cal A}
,\linebreak [0] {\cal B}
,\linebreak [0] {\cal C}
,\linebreak [0] {\cal D}
,\linebreak [0] {\cal E}
,\linebreak [0] {\cal F}
\colon 
\newline {\cal A}
\Rightarrow {\cal B}
\Rightarrow {\cal C}
\Rightarrow {\cal D}
\Rightarrow {\cal E}
\Rightarrow {\cal F}
\vdash {\cal A}
\vdash {\cal B}
\vdash {\cal C}
\vdash {\cal D}
\vdash {\cal E}
\vdash {\cal F}
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } MP5
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
,\linebreak [0] {\cal B}
,\linebreak [0] {\cal C}
,\linebreak [0] {\cal D}
,\linebreak [0] {\cal E}
,\linebreak [0] {\cal F}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_q \else
\if \relax \csname lgwellq\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellq {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellq \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
\Rightarrow {\cal B}
\Rightarrow {\cal C}
\Rightarrow {\cal D}
\Rightarrow {\cal E}
\Rightarrow {\cal F}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal C}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal D}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal E}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_r \else
\if \relax \csname lgwellr\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellr {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellr \fi 
$:}$MP3
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_q \else
\if \relax \csname lgwellq\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellq {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellq \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal D}
\Rightarrow {\cal E}
\Rightarrow {\cal F}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$MP2
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_r \else
\if \relax \csname lgwellr\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellr {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellr \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal F}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\end{list}

\subsection{ Implikation }

Dette afsnit indeholder en r\ae{}kke lemmaer vedr.\ implikation, grupperet i fire under-underafsnit.

\subsubsection{Refleksivitet; blok-konstruktionen}

Lemmaet $AutoImply$ udsiger, at implikations-relationen er refleksiv:

\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}

\item $
[ ZFsub
\mathbf{\ lemma\ } AutoImply
\colon 
\Pi {\cal A}
\colon {\cal A}
\Rightarrow {\cal A}
]$

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } AutoImply
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal A}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal A}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Repetition
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal A}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
\Rightarrow {\cal A}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $


\item Beviset for $AutoImply$ indeholder to nye ting i forhold til de hidtidige beviser: En bevisblok, og en anvendelse af deduktions-reglen. En bevisblok er selvst\ae{}ndig enhed i et bevis; den afh\ae{}nger ikke af den \o{}vrige del af beviset. Den ovenst\aa{}ende bevisblok indeholder et bevis for lemmaet $[
\Pi {\cal A}
\colon {\cal A}
\vdash {\cal A}]$. Pointen er nu, at blokkens sidste linie (linie 5) fungerer som en forkortelse for dette lemma. Vi kan da anvende deduktionsreglen p\aa{} denne linie til at omdanne inferensen $[
\Pi {\cal A}
\colon {\cal A}
\vdash {\cal A}]$ til implikationen $[{\cal A}
\Rightarrow {\cal A}]$. Det vigtigste form\aa{}l med deduktionsreglen er netop, at vi let kan skifte fra inferens til implikation.

\end{list}

\subsubsection{Transitivitet}

Lemmaet $ImplyTransitivity$ udsiger, at implikations-relationen er transitiv:

\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}

\item $
[ ZFsub
\mathbf{\ lemma\ } ImplyTransitivity
\colon 
\Pi {\cal A}
,\linebreak [0] {\cal B}
,\linebreak [0] {\cal C}
\colon {\cal A}
\Rightarrow {\cal B}
\vdash {\cal B}
\Rightarrow {\cal C}
\vdash {\cal A}
\Rightarrow {\cal C}
]$

\item Vi viser $ImplyTransitivity$ ved hj\ae{}lp af $MP$ og deduktionsreglen:

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } ImplyTransitivity
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal A}
,\linebreak [0] {\cal B}
,\linebreak [0] {\cal C}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal A}
\Rightarrow {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal B}
\Rightarrow {\cal C}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal A}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$MP
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$MP
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal C}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
,\linebreak [0] {\cal B}
,\linebreak [0] {\cal C}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
\Rightarrow {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal B}
\Rightarrow {\cal C}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
({\cal A}
\Rightarrow {\cal B}
)
\Rightarrow 
({\cal B}
\Rightarrow {\cal C}
)
\Rightarrow {\cal A}
\Rightarrow {\cal C}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$MP2
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
\Rightarrow {\cal C}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\end{list}

\subsubsection{Sv\ae{}kkelse}

Vi f\aa{}r ofte brug for det f\o{}lgende r\ae{}sonnement: Hvis formlen ${\cal A}$ g\ae{}lder ubetinget, s\aa{} g\ae{}lder den ogs\aa{} under antagelse af en vilk\aa{}rlig anden formel ${\cal B}$. Lemmaet $Weakening$ udtrykker dette r\ae{}sonnement som f\o{}lger:

\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}

\item $
[ ZFsub
\mathbf{\ lemma\ } Weakening
\colon 
\Pi {\cal A}
,\linebreak [0] {\cal B}
\colon {\cal B}
\vdash {\cal A}
\Rightarrow {\cal B}
]$

\item Vi beviser $Weakening$ ved hj\ae{}lp af deduktionsreglen:

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } Weakening
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal A}
,\linebreak [0] {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal A}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Repetition
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
,\linebreak [0] {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal B}
\Rightarrow {\cal A}
\Rightarrow {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$MP
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
\Rightarrow {\cal B}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\end{list}

\subsubsection{Modsigelse}

Det sidste lemma i dette afsnit vedr\o{}rer strengt taget ikke implikation, men derimod inferens  ($
\mathsf{x}
\vdash 
\mathsf{y}$). Lemmaet $FromContradiction$ udsiger, at vi kan bevise hvad som helst, hvis vi har bevist to formler, der modsiger hinanden:

\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}

\item $
[ ZFsub
\mathbf{\ lemma\ } FromContradiction
\colon 
\Pi {\cal A}
,\linebreak [0] {\cal B}
\colon {\cal A}
\vdash \dot{\neg}\, {\cal A}
\vdash {\cal B}
]$

\item Beviset bruger $Weakening$ og slutningsreglen $Neg$:

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } FromContradiction
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
,\linebreak [0] {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\, {\cal A}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$Weakening
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\, {\cal B}
\Rightarrow {\cal A}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$Weakening
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\, {\cal B}
\Rightarrow \dot{\neg}\, {\cal A}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Neg
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal B}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\end{list}

\subsection{H\aa{}ndtering af dobbeltnegationer}

De to lemmaer $RemoveDoubleNeg$ og $AddDoubleNeg$ tillader os hhv.\ at fjerne og tilf\o{}je dobbeltnegationer. Jeg vil ikke kommentere beviserne:

\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}

\item $
[ ZFsub
\mathbf{\ lemma\ } RemoveDoubleNeg
\colon 
\Pi {\cal A}
\colon \dot{\neg}\, \dot{\neg}\, {\cal A}
\vdash {\cal A}
]$

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } RemoveDoubleNeg
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\, \dot{\neg}\, {\cal A}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$Weakening
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\, {\cal A}
\Rightarrow \dot{\neg}\, \dot{\neg}\, {\cal A}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$AutoImply
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\, {\cal A}
\Rightarrow \dot{\neg}\, {\cal A}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Neg
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\item $
[ ZFsub
\mathbf{\ lemma\ } AddDoubleNeg
\colon 
\Pi {\cal A}
\colon {\cal A}
\vdash \dot{\neg}\, \dot{\neg}\, {\cal A}
]$

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } AddDoubleNeg
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal A}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$\dot{\neg}\, \dot{\neg}\, \dot{\neg}\, {\cal A}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$RemoveDoubleNeg
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$\dot{\neg}\, {\cal A}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\, \dot{\neg}\, \dot{\neg}\, {\cal A}
\Rightarrow \dot{\neg}\, {\cal A}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_q \else
\if \relax \csname lgwellq\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellq {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellq \fi 
$:}$Weakening
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\, \dot{\neg}\, \dot{\neg}\, {\cal A}
\Rightarrow {\cal A}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Neg
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_q \else
\if \relax \csname lgwellq\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellq {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellq \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\, \dot{\neg}\, {\cal A}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\end{list}

\subsection{Modus tollens og besl\ae{}gtede lemmaer}

Hovedresultatet fra dette afsnit er slutningsreglen modus tollens, bevist som et lemma:

\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}

\item $
[ ZFsub
\mathbf{\ lemma\ } MT
\colon 
\Pi {\cal A}
,\linebreak [0] {\cal B}
\colon {\cal A}
\Rightarrow {\cal B}
\vdash \dot{\neg}\, {\cal B}
\vdash \dot{\neg}\, {\cal A}
]$

\item For at vise $MT$ begynder vi med et teknisk lemma, der ikke har den store v\ae{}rdi i sig selv: 

\item $
[ ZFsub
\mathbf{\ lemma\ } Technicality
\colon 
\Pi {\cal A}
,\linebreak [0] {\cal B}
\colon {\cal A}
\Rightarrow {\cal B}
\vdash \dot{\neg}\, \dot{\neg}\, {\cal A}
\Rightarrow {\cal B}
]$

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } Technicality
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal A}
,\linebreak [0] {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal A}
\Rightarrow {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$\dot{\neg}\, \dot{\neg}\, {\cal A}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$RemoveDoubleNeg
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal A}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$MP
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
,\linebreak [0] {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
({\cal A}
\Rightarrow {\cal B}
)
\Rightarrow \dot{\neg}\, \dot{\neg}\, {\cal A}
\Rightarrow {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_p \else
\if \relax \csname lgwellp\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellp {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellp \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
\Rightarrow {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$MP
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_p \else
\if \relax \csname lgwellp\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellp {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellp \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\, \dot{\neg}\, {\cal A}
\Rightarrow {\cal B}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\item Uafh\ae{}ngigt af $Technicality$ kan vi vise en version af $MT$, hvor ${\cal A}$ optr\ae{}der i negeret form:

\item $
[ ZFsub
\mathbf{\ lemma\ } NegativeMT
\colon 
\Pi {\cal A}
,\linebreak [0] {\cal B}
\colon \dot{\neg}\, {\cal A}
\Rightarrow {\cal B}
\vdash \dot{\neg}\, {\cal B}
\vdash {\cal A}
]$

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } NegativeMT
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
,\linebreak [0] {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\, {\cal A}
\Rightarrow {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\, {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$Weakening
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\, {\cal A}
\Rightarrow \dot{\neg}\, {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Neg
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\item Ud fra $Technicality$ og $NegativeMT$ kan vi nu vise $MT$:

\item $
[ ZFsub
\mathbf{\ lemma\ } MT
\colon 
\Pi {\cal A}
,\linebreak [0] {\cal B}
\colon {\cal A}
\Rightarrow {\cal B}
\vdash \dot{\neg}\, {\cal B}
\vdash \dot{\neg}\, {\cal A}
]$

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } MT
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
,\linebreak [0] {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
\Rightarrow {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\, {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$Technicality
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\, \dot{\neg}\, {\cal A}
\Rightarrow {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$NegativeMT
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\, {\cal A}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\item Vi slutter dette underafsnit med en variant af $MT$, som erstatter en inferens med en implikation:

\item $
[ ZFsub
\mathbf{\ lemma\ } Contrapositive
\colon 
\Pi {\cal A}
,\linebreak [0] {\cal B}
\colon {\cal A}
\Rightarrow {\cal B}
\vdash \dot{\neg}\, {\cal B}
\Rightarrow \dot{\neg}\, {\cal A}
]$

\item N\aa{}r en inferens skal erstattes med en implikation, er det altid deduktionsreglen, der skal i spil:

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } Contrapositive
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal A}
,\linebreak [0] {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal A}
\Rightarrow {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$\dot{\neg}\, {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$MT
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$\dot{\neg}\, {\cal A}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
,\linebreak [0] {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
\Rightarrow {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
({\cal A}
\Rightarrow {\cal B}
)
\Rightarrow \dot{\neg}\, {\cal B}
\Rightarrow \dot{\neg}\, {\cal A}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$MP
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\, {\cal B}
\Rightarrow \dot{\neg}\, {\cal A}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\end{list}

\subsection{Konjunktion}

Hovedm\aa{}let med dette underafsnit er at konvertere mellem formlerne ${\cal A}$ og ${\cal B}$ og deres konjunktion $[{\cal A}
\mathrel{\dot{\wedge}} {\cal B}]$.

\subsubsection{Forening af konjunkter}

Vi begynder med at sl\aa{} ${\cal A}$ og ${\cal B}$ sammen til $[{\cal A}
\mathrel{\dot{\wedge}} {\cal B}]$:

\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}

\item $
[ ZFsub
\mathbf{\ lemma\ } JoinConjuncts
\colon 
\Pi {\cal A}
,\linebreak [0] {\cal B}
\colon {\cal A}
\vdash {\cal B}
\vdash {\cal A}
\mathrel{\dot{\wedge}} {\cal B}
]$

\item Beviset for $JoinConjuncts$ er af teknisk karakter. Vi viser den makroekspanderede form $[\dot{\neg}\, 
({\cal A}
\Rightarrow \dot{\neg}\, {\cal B}
)]$, som vi i bevisets sidste linie konverterer til $[{\cal A}
\mathrel{\dot{\wedge}} {\cal B}]$. Denne sidste linie er ikke n\o{}dvendig for bevischeckeren, men den g\o{}r beviset lidt nemmere at l\ae{}se:

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } JoinConjuncts
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal A}
,\linebreak [0] {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_z \else
\if \relax \csname lgwellz\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellz {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellz \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal A}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal A}
\Rightarrow \dot{\neg}\, {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$MP
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_z \else
\if \relax \csname lgwellz\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellz {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellz \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$\dot{\neg}\, {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
,\linebreak [0] {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
\Rightarrow 
({\cal A}
\Rightarrow \dot{\neg}\, {\cal B}
)
\Rightarrow \dot{\neg}\, {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}$MP
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
({\cal A}
\Rightarrow \dot{\neg}\, {\cal B}
)
\Rightarrow \dot{\neg}\, {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_q \else
\if \relax \csname lgwellq\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellq {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellq \fi 
$:}$AddDoubleNeg
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\, \dot{\neg}\, {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_Z \else
\if \relax \csname lgwellbigz\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigz {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigz \fi 
$:}$MT
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_q \else
\if \relax \csname lgwellq\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellq {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellq \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\, 
({\cal A}
\Rightarrow \dot{\neg}\, {\cal B}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Repetition
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_Z \else
\if \relax \csname lgwellbigz\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigz {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigz \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
\mathrel{\dot{\wedge}} {\cal B}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\end{list}

\subsubsection{Udskilning af anden konjunkt}

Tautologien $SecondConjunct$ lader os udskille den anden konjunkt fra $[{\cal A}
\mathrel{\dot{\wedge}} {\cal B}]$. Jeg vil ikke kommentere beviset:

\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}

\item $
[ ZFsub
\mathbf{\ lemma\ } SecondConjunct
\colon 
\Pi {\cal A}
,\linebreak [0] {\cal B}
\colon {\cal A}
\mathrel{\dot{\wedge}} {\cal B}
\vdash {\cal B}
]$

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } SecondConjunct
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
$%
\leftskip=1em%
${\cal A}
,\linebreak [0] {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$\dot{\neg}\, {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Weakening
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal A}
\Rightarrow \dot{\neg}\, {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
,\linebreak [0] {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\, {\cal B}
\Rightarrow {\cal A}
\Rightarrow \dot{\neg}\, {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_Z \else
\if \relax \csname lgwellbigz\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigz {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigz \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
\mathrel{\dot{\wedge}} {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}$Repetition
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_Z \else
\if \relax \csname lgwellbigz\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigz {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigz \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\, 
({\cal A}
\Rightarrow \dot{\neg}\, {\cal B}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$NegativeMT
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal B}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\end{list}

\subsubsection{Udskilning af f\o{}rste konjunkt}

For at udskille ${\cal A}$ fra $[{\cal A}
\mathrel{\dot{\wedge}} {\cal B}]$ viser vi f\o{}rst, at $[{\cal A}
\mathrel{\dot{\wedge}} {\cal B}]$ er kommutativ. Jeg vil ikke kommentere beviset:

\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}

\item $
[ ZFsub
\mathbf{\ lemma\ } AndCommutativity
\colon 
\Pi {\cal A}
,\linebreak [0] {\cal B}
\colon {\cal A}
\mathrel{\dot{\wedge}} {\cal B}
\vdash {\cal B}
\mathrel{\dot{\wedge}} {\cal A}
]$

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } AndCommutativity
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
$%
\leftskip=1em%
${\cal A}
,\linebreak [0] {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal B}
\Rightarrow \dot{\neg}\, {\cal A}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal A}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$AddDoubleNeg
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$\dot{\neg}\, \dot{\neg}\, {\cal A}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$MT
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$\dot{\neg}\, {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
,\linebreak [0] {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
({\cal B}
\Rightarrow \dot{\neg}\, {\cal A}
)
\Rightarrow {\cal A}
\Rightarrow \dot{\neg}\, {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_Z \else
\if \relax \csname lgwellbigz\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigz {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigz \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
\mathrel{\dot{\wedge}} {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}$Repetition
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\, 
({\cal A}
\Rightarrow \dot{\neg}\, {\cal B}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_Y \else
\if \relax \csname lgwellbigy\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigy {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigy \fi 
$:}$MT
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\, 
({\cal B}
\Rightarrow \dot{\neg}\, {\cal A}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Repetition
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_Y \else
\if \relax \csname lgwellbigy\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigy {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigy \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal B}
\mathrel{\dot{\wedge}} {\cal A}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\item Nu er det let at udskille den f\o{}rste konjunkt fra $[{\cal A}
\mathrel{\dot{\wedge}} {\cal B}]$: F\o{}rst vender vi konjunktionen om til $[{\cal B}
\mathrel{\dot{\wedge}} {\cal A}]$ ved hj\ae{}lp af $AndCommutativity$, og s\aa{} udskiller vi ${\cal A}$ ved hj\ae{}lp af $SecondConjunct$:

\item $
[ ZFsub
\mathbf{\ lemma\ } FirstConjunct
\colon 
\Pi {\cal A}
,\linebreak [0] {\cal B}
\colon {\cal A}
\mathrel{\dot{\wedge}} {\cal B}
\vdash {\cal A}
]$

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } FirstConjunct
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
,\linebreak [0] {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
\mathrel{\dot{\wedge}} {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$AndCommutativity
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal B}
\mathrel{\dot{\wedge}} {\cal A}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$SecondConjunct
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\end{list}

\subsection{Dobbeltimplikation}

I dette underafsnit viser vi tre enkle resultater vedr.\ dobbeltimplikation.

\subsubsection{Brug sammen med modus ponens}

De f\o{}lgende to tautologier g\o{}r det let at bruge anvende slutningsreglen $MP$ p\aa{} dobbeltimplikationer. Beviserne er enkle og kr\ae{}ver ingen kommentarer:

\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}

\item $
[ ZFsub
\mathbf{\ lemma\ } IffFirst
\colon 
\Pi {\cal A}
,\linebreak [0] {\cal B}
\colon {\cal A}
\mathrel{\dot{\Leftrightarrow}} {\cal B}
\vdash {\cal B}
\vdash {\cal A}
]$

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } IffFirst
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
,\linebreak [0] {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
\mathrel{\dot{\Leftrightarrow}} {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$SecondConjunct
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal B}
\Rightarrow {\cal A}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$MP
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\item$
[ ZFsub
\mathbf{\ lemma\ } IffSecond
\colon 
\Pi {\cal A}
,\linebreak [0] {\cal B}
\colon {\cal A}
\mathrel{\dot{\Leftrightarrow}} {\cal B}
\vdash {\cal A}
\vdash {\cal B}
]$

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } IffSecond
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
,\linebreak [0] {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
\mathrel{\dot{\Leftrightarrow}} {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$FirstConjunct
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
\Rightarrow {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$MP
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal B}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\end{list}

\subsubsection{Kommutativitet}

Lemmaet $IffCommutativity$ f\o{}lger direkte af, at operatoren $[
\mathsf{x}
\mathrel{\dot{\wedge}} 
\mathsf{y}]$
er kommutativ:

\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}

\item $
[ ZFsub
\mathbf{\ lemma\ } IffCommutativity
\colon 
\Pi {\cal A}
,\linebreak [0] {\cal B}
\colon {\cal A}
\mathrel{\dot{\Leftrightarrow}} {\cal B}
\vdash {\cal B}
\mathrel{\dot{\Leftrightarrow}} {\cal A}
]$

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } IffCommutativity
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
,\linebreak [0] {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
\mathrel{\dot{\Leftrightarrow}} {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$Repetition
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
({\cal A}
\Rightarrow {\cal B}
)
\mathrel{\dot{\wedge}} 
({\cal B}
\Rightarrow {\cal A}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$AndCommutativity
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
({\cal B}
\Rightarrow {\cal A}
)
\mathrel{\dot{\wedge}} 
({\cal A}
\Rightarrow {\cal B}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Repetition
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal B}
\mathrel{\dot{\Leftrightarrow}} {\cal A}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\end{list}

\subsection{Disjunktion}

Dette underafsnit indeholder tre lemmaer vedr.\ disjunktion, som vi fordeler p\aa{} to under-underafsnit.

\subsubsection{Sv\ae{}kkelse}

Givet en p\aa{}stand ${\cal B}$ vil vi gerne udlede de svagere p\aa{}stande $[{\cal A}
 \mathrel{\dot{\vee}} {\cal B}]$ og $[{\cal B}
 \mathrel{\dot{\vee}} {\cal A}]$. Den f\o{}rste slutning varetages af lemmaet $WeakenOr1$:

\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}

\item $
[ ZFsub
\mathbf{\ lemma\ } WeakenOr1
\colon 
\Pi {\cal A}
,\linebreak [0] {\cal B}
\colon {\cal B}
\vdash {\cal A}
 \mathrel{\dot{\vee}} {\cal B}
]$

\item Beviset best\aa{}r af en simpel anvendelse af $Weakening$:

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } WeakenOr1
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
,\linebreak [0] {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$Weakening
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\, {\cal A}
\Rightarrow {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Repetition
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
 \mathrel{\dot{\vee}} {\cal B}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\item Slutningen fra ${\cal A}$ til $[{\cal A}
 \mathrel{\dot{\vee}} {\cal B}]$ varetages af lemmaet $WeakenOr2$:

\item $
[ ZFsub
\mathbf{\ lemma\ } WeakenOr2
\colon 
\Pi {\cal A}
,\linebreak [0] {\cal B}
\colon {\cal A}
\vdash {\cal A}
 \mathrel{\dot{\vee}} {\cal B}
]$

\item Kernen i beviset for $WeakenOr2$ er en anvendelse af $FromContradiction$:

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } WeakenOr2
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal A}
,\linebreak [0] {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal A}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$\dot{\neg}\, {\cal A}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$FromContradiction
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
,\linebreak [0] {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
\Rightarrow \dot{\neg}\, {\cal A}
\Rightarrow {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}$MP
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\, {\cal A}
\Rightarrow {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Repetition
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
 \mathrel{\dot{\vee}} {\cal B}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\end{list}

\subsubsection{Slutning ud fra disjunktion}

Lemmaet $FromDisjuncts$ lader os drage slutninger ud fra en disjunktion:

\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}

\item $
[ ZFsub
\mathbf{\ lemma\ } FromDisjuncts
\colon 
\Pi {\cal A}
,\linebreak [0] {\cal B}
,\linebreak [0] {\cal C}
\colon {\cal A}
 \mathrel{\dot{\vee}} {\cal B}
\vdash {\cal A}
\Rightarrow {\cal C}
\vdash {\cal B}
\Rightarrow {\cal C}
\vdash {\cal C}
]$

\item Om beviset vil jeg kun sige, at det er en ret elegant \o{}velse i bevisteknik:

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } FromDisjuncts
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
,\linebreak [0] {\cal B}
,\linebreak [0] {\cal C}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
 \mathrel{\dot{\vee}} {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
\Rightarrow {\cal C}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal B}
\Rightarrow {\cal C}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$Repetition
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\, {\cal A}
\Rightarrow {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$Contrapositive
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\, {\cal B}
\Rightarrow \dot{\neg}\, \dot{\neg}\, {\cal A}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}$Technicality
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\, \dot{\neg}\, {\cal A}
\Rightarrow {\cal C}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
$:}$ImplyTransitivity
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\, {\cal B}
\Rightarrow {\cal C}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
$:}$Contrapositive
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\, {\cal C}
\Rightarrow \dot{\neg}\, \dot{\neg}\, {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_i \else
\if \relax \csname lgwelli\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelli {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelli \fi 
$:}$Contrapositive
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\, {\cal C}
\Rightarrow \dot{\neg}\, {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Neg
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_i \else
\if \relax \csname lgwelli\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelli {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelli \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal C}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\end{list}

\newpage
\section{Regellemaer} \label{sec:regel}

Alle aksiomerne i afsnit \ref{sec:axiomzf} er formuleret som dobbeltimplikationer. Dette er gjort for at holde antallet af aksiomer nede p\aa{} et minimum; men n\aa{}r aksiomerne skal bruges i beviser, er det mere bekvemt at have regler af formen $[{\cal A}
\vdash {\cal B}]$ til r\aa{}dighed. Heldigvis kan vi altid bevise 
$[{\cal A}
\vdash {\cal B}]$,
hvis vi i forvejen har et bevis for $[{\cal A}
\Rightarrow {\cal B}]$ --- der kr\ae{}ves blot en rutinem\ae{}ssig anvendelse af $MP$.

I dette afsnit viser vi regellemmaer svarende til aksiomerne $SeparationDef$, $PairDef$,  $UnionDef$, $PowerDef$ og $Extensionality$.

\subsection{Par} \label{sec:parlemma}

I tilf\ae{}ldet ``$PairDef$'' forl\o{}ber b\aa{}de definition og bevis af regellemmaerne s\aa{} enkelt, som det er muligt. F\o{}rst formulerer vi den ene halvdel af aksiomets dobbelt-implikation som et lemma:

\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}

\item $
[ ZFsub
\mathbf{\ lemma\ } Pair2Formula
\colon 
\Pi {\cal S}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Y}
\colon {\cal S}
 \mathrel{\in} \{{\cal X}
,{\cal Y}
\}
\vdash {\cal S}
\!\mathrel{=}\! {\cal X}
 \mathrel{\dot{\vee}} {\cal S}
\!\mathrel{=}\! {\cal Y}
]$

\item Derefter beviser vi dette lemma ved at anvende $IffSecond$ p\aa{} aksiomet og pr\ae{}missen:

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } Pair2Formula
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
 \mathrel{\in} \{{\cal X}
,{\cal Y}
\}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$PairDef
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
 \mathrel{\in} \{{\cal X}
,{\cal Y}
\}
\mathrel{\dot{\Leftrightarrow}} {\cal S}
\!\mathrel{=}\! {\cal X}
 \mathrel{\dot{\vee}} {\cal S}
\!\mathrel{=}\! {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$IffSecond
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
\!\mathrel{=}\! {\cal X}
 \mathrel{\dot{\vee}} {\cal S}
\!\mathrel{=}\! {\cal Y}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\item Til sidst gentager vi hele processen mht.\ den anden halvdel af dobbeltimplikationen. Den eneste reelle forskel er, at vi bruger $IffFirst$ i beviset (i stedet for $IffSecond$):

\item $
[ ZFsub
\mathbf{\ lemma\ } Formula2Pair
\colon 
\Pi {\cal S}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Y}
\colon {\cal S}
\!\mathrel{=}\! {\cal X}
 \mathrel{\dot{\vee}} {\cal S}
\!\mathrel{=}\! {\cal Y}
\vdash {\cal S}
 \mathrel{\in} \{{\cal X}
,{\cal Y}
\}
]$

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } Formula2Pair
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
\!\mathrel{=}\! {\cal X}
 \mathrel{\dot{\vee}} {\cal S}
\!\mathrel{=}\! {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$PairDef
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
 \mathrel{\in} \{{\cal X}
,{\cal Y}
\}
\mathrel{\dot{\Leftrightarrow}} {\cal S}
\!\mathrel{=}\! {\cal X}
 \mathrel{\dot{\vee}} {\cal S}
\!\mathrel{=}\! {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$IffFirst
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
 \mathrel{\in} \{{\cal X}
,{\cal Y}
\}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\end{list}

\subsection{Foreningsm\ae{}ngde}

I tilf\ae{}ldet ``$UnionDef$'' bruger vi stort set den samme fremgangsm\aa{}de som i afsnit \ref{sec:parlemma}. F\o{}rst h\aa{}ndterer vi den ene halvdel af dobbeltimplikationen:

\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}

\item $
[ ZFsub
\mathbf{\ lemma\ } Union2Formula
\colon 
\Pi {\cal S}
,\linebreak [0] {\cal X}
\colon {\cal S}
 \mathrel{\in} \cup {\cal X}
\vdash {\cal S}
 \mathrel{\in} Ex_{10}
\mathrel{\dot{\wedge}} Ex_{10}
 \mathrel{\in} {\cal X}
]$

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } Union2Formula
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
,\linebreak [0] {\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
 \mathrel{\in} \cup {\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$UnionDef
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
 \mathrel{\in} \cup {\cal X}
\mathrel{\dot{\Leftrightarrow}} 
\newline 
({\cal S}
 \mathrel{\in} Ex_{10}
\mathrel{\dot{\wedge}} Ex_{10}
 \mathrel{\in} {\cal X}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$IffSecond
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
 \mathrel{\in} Ex_{10}
\mathrel{\dot{\wedge}} Ex_{10}
 \mathrel{\in} {\cal X}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\item S\aa{} er turen kommet til den anden halvdel af dobbeltimplikationen:

\item $
[ ZFsub
\mathbf{\ lemma\ } Formula2Union
\colon 
\Pi {\cal S}
,\linebreak [0] {\cal X}
\colon {\cal S}
 \mathrel{\in} Ex_{10}
\vdash Ex_{10}
 \mathrel{\in} {\cal X}
\vdash {\cal S}
 \mathrel{\in} \cup {\cal X}
]$

\item Der er det lille raffinement, at vi i formuleringen af $Formula2Union$ har delt formlen $[{\cal S}
 \mathrel{\in} Ex_{10}
\mathrel{\dot{\wedge}} Ex_{10}
 \mathrel{\in} {\cal X}]$ op i sine konjunkter. Beviset for $Formula2Union$ kr\ae{}ver derfor en ekstra anvendelse af $JoinConjuncts$:

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } Formula2Union
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
,\linebreak [0] {\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
 \mathrel{\in} Ex_{10}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$Ex_{10}
 \mathrel{\in} {\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$JoinConjuncts
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
 \mathrel{\in} Ex_{10}
\mathrel{\dot{\wedge}} Ex_{10}
 \mathrel{\in} {\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$UnionDef
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
 \mathrel{\in} \cup {\cal X}
\mathrel{\dot{\Leftrightarrow}} 
\newline 
({\cal S}
 \mathrel{\in} Ex_{10}
\mathrel{\dot{\wedge}} Ex_{10}
 \mathrel{\in} {\cal X}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$IffFirst
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
 \mathrel{\in} \cup {\cal X}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\end{list}

\subsection{Potensm\ae{}ngde; sidebetingelser} \label{sec:potside}

Tilf\ae{}ldet ``$PowerDef$'' er mere kompliceret end de hidtidige tilf\ae{}lde. Det ene af de to regellemmaer er nemt nok:

\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}

\item $
[ ZFsub
\mathbf{\ lemma\ } SubsetInPower
\colon 
\Pi {\cal S}
,\linebreak [0] {\cal X}
\colon {\cal S}
 \mathrel{\subseteq} {\cal X}
\vdash {\cal S}
 \mathrel{\in} P({\cal X}
)
]$

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } SubsetInPower
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
,\linebreak [0] {\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
 \mathrel{\in} {\cal S}
\Rightarrow 
\mathit{s}
 \mathrel{\in} {\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$Gen
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\forall 
\mathit{s}
\colon 
(
\mathit{s}
 \mathrel{\in} {\cal S}
\Rightarrow 
\mathit{s}
 \mathrel{\in} {\cal X}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$PowerDef
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
 \mathrel{\in} P({\cal X}
)
\mathrel{\dot{\Leftrightarrow}} 
\newline 
\forall 
\mathit{s}
\colon 
(
\mathit{s}
 \mathrel{\in} {\cal S}
\Rightarrow 
\mathit{s}
 \mathrel{\in} {\cal X}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$IffFirst
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
 \mathrel{\in} P({\cal X}
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\end{list}

\noindent Det andet regellemma er ogs\aa{} nemt, hvis vi n\o{}jes med at vise: 

$[
\Pi {\cal S}
,\linebreak [0] {\cal X}
\colon {\cal S}
 \mathrel{\in} P({\cal X}
)
\vdash 
\forall 
\mathit{s}
\colon 
(
\mathit{s}
 \mathrel{\in} {\cal S}
\Rightarrow 
\mathit{s}
 \mathrel{\in} {\cal X}
)]$.

\noindent For at komme videre med konklusionen $[
\forall 
\mathit{s}
\colon 
\mathit{s}
 \mathrel{\in} {\cal S}
\Rightarrow 
\mathit{s}
 \mathrel{\in} {\cal X}]$
m\aa{} vi imidlertid slippe af med objektkvantoren; og dette kan vi lige s\aa{} godt g\o{}re med det samme. Derfor bliver m\aa{}let i stedet at vise:

$[
\Pi {\cal S}
,\linebreak [0] {\cal X}
\colon {\cal S}
 \mathrel{\in} P({\cal X}
)
\vdash {\cal S}
 \mathrel{\subseteq} {\cal X}]$, hvor ${\cal S}
 \mathrel{\subseteq} {\cal X}$ som n\ae{}vnt i afsnit \ref{sec:subset} er en forkortelse for  $[
\mathit{s}
 \mathrel{\in} {\cal S}
\Rightarrow 
\mathit{s}
 \mathrel{\in} {\cal X}]$.

For at n\aa{} dette m\aa{}l m\aa{} vi begynde med et hj\ae{}lpelemma, der ser s\aa{}ledes ud:

\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}

\item $
[ ZFsub
\mathbf{\ lemma\ } HelperPowerIsSub
\colon 
\Pi {\cal S}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Y}
\colon 
\mathit{s}
\#{\cal X}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal Y}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\newline 
\forall 
\mathit{s}
\colon 
(
\mathit{s}
 \mathrel{\in} {\cal X}
\Rightarrow 
\mathit{s}
 \mathrel{\in} {\cal Y}
)
\Rightarrow {\cal S}
 \mathrel{\in} {\cal X}
\Rightarrow {\cal S}
 \mathrel{\in} {\cal Y}
]$

\end{list}

\noindent De to sidebetingelser $[
\mathit{s}
\#{\cal X}]$ og $[
\mathit{s}
\#{\cal Y}]$ er her n\o{}dvendige pga. antecedenten\footnote{Jeg kalder $
\mathsf{x}$ i $[
\mathsf{x}
\Rightarrow 
\mathsf{y}]$ for ``antecedenten''; og jeg kalder $
\mathsf{y}$ for ``konsekvensen''.} $[
\forall 
\mathit{s}
\colon 
(
\mathit{s}
 \mathrel{\in} {\cal X}
\Rightarrow 
\mathit{s}
 \mathrel{\in} {\cal Y}
)]$. Hvis vi i denne formel instantierer ${\cal X}$ eller ${\cal Y}$ til en term, der indeholder frie forekomster af $
\mathit{s}$, s\aa{} \ae{}ndres formlens mening. F.eks.\ er formlen $[
\forall 
\mathit{s}
\colon 
(
\mathit{s}
 \mathrel{\in} 
\mathit{s}
\Rightarrow 
\mathit{s}
 \mathrel{\in} 
\mathit{y}
)]$ n\o{}dvendigvis sand, fordi ingen m\ae{}ngder i $ZFsub$ er medlem af sig selv; men den oprindelige formel $[
\forall 
\mathit{s}
\colon 
(
\mathit{s}
 \mathrel{\in} {\cal X}
\Rightarrow 
\mathit{s}
 \mathrel{\in} {\cal Y}
)]$ er ikke n\o{}dvendigvis sand. Konstruktionen $[
\mathsf{x}
\#
\mathsf{y}]$ (der er defineret i appendikset til \cite{kn:check}) udsiger netop, at objektvariablen $
\mathsf{x}$ ikke forekommer frit i termen $
\mathsf{y}$\footnote{Vi siger at ``$
\mathsf{x}$ undg\aa{}r $
\mathsf{y}$'' (eller at ``$
\mathsf{y}$ undg\aa{}r $
\mathsf{x}$'').}. Vi kommer til at se mange flere eksempler p\aa{} sidebetingelser af formen $[
\mathsf{x}
\#
\mathsf{y}]$: De er n\o{}dvendige, n\aa{}r metavariablen $
\mathsf{y}$ optr\ae{}der i en kontekst, der er kvantificeret med objektvariablen $
\mathsf{x}$.

Indholdet af selve lemmaet er, at vi kan fjerne objektkvantoren fra  
\\ \mbox{$[
\forall 
\mathit{s}
\colon 
(
\mathit{s}
 \mathrel{\in} {\cal X}
\Rightarrow 
\mathit{s}
 \mathrel{\in} {\cal Y}
)]$}, og endda erstatte objektvariablen $
\mathit{s}$ med metavariablen ${\cal S}$. Her er beviset:

\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } HelperPowerIsSub
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal X}
,\linebreak [0] {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{s}
 \mathrel{\in} {\cal X}
\Rightarrow 
\mathit{s}
 \mathrel{\in} {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Repetition
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{s}
 \mathrel{\in} {\cal X}
\Rightarrow 
\mathit{s}
 \mathrel{\in} {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal X}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal Y}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\forall 
\mathit{s}
\colon 
(
\mathit{s}
 \mathrel{\in} {\cal X}
\Rightarrow 
\mathit{s}
 \mathrel{\in} {\cal Y}
)
\Rightarrow {\cal S}
 \mathrel{\in} {\cal X}
\Rightarrow {\cal S}
 \mathrel{\in} {\cal Y}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\end{list}

\noindent Beviset illustrerer styrken af slutningsreglen $Ded$: I dette tilf\ae{}lde omdanner den en inferens (nemlig $[
\Pi {\cal X}
,\linebreak [0] {\cal Y}
\colon 
\mathit{s}
 \mathrel{\in} {\cal X}
\Rightarrow 
\mathit{s}
 \mathrel{\in} {\cal Y}
\vdash 
\mathit{s}
 \mathrel{\in} {\cal X}
\Rightarrow 
\mathit{s}
 \mathrel{\in} {\cal Y}]$) til en implikation (nemlig $[
(
\mathit{s}
 \mathrel{\in} {\cal X}
\Rightarrow 
\mathit{s}
 \mathrel{\in} {\cal Y}
)
\Rightarrow 
\mathit{s}
 \mathrel{\in} {\cal X}
\Rightarrow 
\mathit{s}
 \mathrel{\in} {\cal Y}]$), s\ae{}tter en alkvantor p\aa{} antecedenten (s\aa{} vi f\aa{}r $[
\forall 
\mathit{s}
\colon 
(
\mathit{s}
 \mathrel{\in} {\cal X}
\Rightarrow 
\mathit{s}
 \mathrel{\in} {\cal Y}
)
\Rightarrow 
\mathit{s}
 \mathrel{\in} {\cal X}
\Rightarrow 
\mathit{s}
 \mathrel{\in} {\cal Y}]$), og erstatter den frie forekomst af ``$
\mathit{s}$'' med ``${\cal S}$'' --- alt sammen p\aa{} \'{e}n gang. Dog skal de to sidebetingelser med, for at reglen kan anvendes.

Formuleringen af det andet regellemma bliver nu som f\o{}lger:

\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}

\item $
[ ZFsub
\mathbf{\ lemma\ } PowerIsSub
\colon 
\Pi {\cal S}
,\linebreak [0] {\cal X}
\colon 
\mathit{s}
\#{\cal S}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal X}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} {\cal S}
 \mathrel{\in} P({\cal X}
)
\vdash {\cal S}
 \mathrel{\subseteq} {\cal X}
]$

\end{list}

\noindent Da $PowerIsSub$ benytter sig af $HelperPowerIsSub$, bliver vi n\o{}dt til at lade $PowerIsSub$ ``arve'' de to sidebetingelser fra 
%\\ \mbox{$HelperPowerIsSub$}
dette lemma. Vi kan ikke undg\aa{} sidebetingelserne ved at instantiere de to metavariable ${\cal X}$ og ${\cal Y}$ fra $HelperPowerIsSub$ til nogle andre metavariable; for der er jo ingen garanti for, at disse metavariable vil undg\aa{} $
\mathit{s}$, n\aa{}r de engang bliver instantierede. S\aa{} l\ae{}nge man arbejder med metavariable, er det alts\aa{} meget sv\ae{}rt at slippe af med denne slags sidebetingelser, n\aa{}r de f\o{}rst er blevet indf\o{}rt i en bevisk\ae{}de. 

Her er beviset for $PowerIsSub$:

\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } PowerIsSub
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
,\linebreak [0] {\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal S}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
 \mathrel{\in} P({\cal X}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$PowerDef
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
 \mathrel{\in} P({\cal X}
)
\mathrel{\dot{\Leftrightarrow}} 
\newline 
\forall 
\mathit{s}
\colon 
(
\mathit{s}
 \mathrel{\in} {\cal S}
\Rightarrow 
\mathit{s}
 \mathrel{\in} {\cal X}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$IffSecond
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\forall 
\mathit{s}
\colon 
(
\mathit{s}
 \mathrel{\in} {\cal S}
\Rightarrow 
\mathit{s}
 \mathrel{\in} {\cal X}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$HelperPowerIsSub
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\forall 
\mathit{s}
\colon 
(
\mathit{s}
 \mathrel{\in} {\cal S}
\Rightarrow 
\mathit{s}
 \mathrel{\in} {\cal X}
)
\Rightarrow 
\newline 
\mathit{s}
 \mathrel{\in} {\cal S}
\Rightarrow 
\mathit{s}
 \mathrel{\in} {\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$MP
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
 \mathrel{\in} {\cal S}
\Rightarrow 
\mathit{s}
 \mathrel{\in} {\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Repetition
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
 \mathrel{\subseteq} {\cal X}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\item I linie 7 benytter vi konstruktionen $[
\mathsf{x}
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\mathsf{y}]$ til at fort\ae{}lle bevischeckeren, at de to sidebetingelser er opfyldt, og at vi derfor kan anvende $HelperPowerIsSub$.

\end{list}

\subsection{Potensm\ae{}ngde, variant}

Som n\ae{}vnt er det sv\ae{}rt at slippe af med sidebetingelser, men af og til kan det lade sig g\o{}re.
N\aa{}r vi i afsnit \ref{sec:union2} g\o{}r brug af lemmaerne $HelperPowerIsSub$ og $PowerIsSub$, vil metavariablen ${\cal X}$ f.eks.\ blive instantieret til termer, der undg\aa{}r $
\mathit{s}$. Hermed bliver sidebetingelsen $[
\mathit{s}
\#{\cal X}]$ automatisk opfyldt, mens vi stadigv\ae{}k m\aa{} medregne $[
\mathit{s}
\#{\cal Y}]$ eksplicit. Imidlertid er bevischeckeren s\aa{}ledes indrettet, at de automatisk opfyldte sidebetingelser skal st\aa{} til sidst, n\aa{}r man bruger et lemma, der indeholder sidebetingelser. Derfor viser vi nu nogle varianter af $PowerIsSub$ og  $HelperPowerIsSub$, hvor der er byttet om p\aa{} $[
\mathit{s}
\#{\cal X}]$ og $[
\mathit{s}
\#{\cal Y}]$:

\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}

\item $
[ ZFsub
\mathbf{\ lemma\ } (Switch)HelperPowerIsSub
\colon 
\Pi {\cal S}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Y}
\colon 
\mathit{s}
\#{\cal Y}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal X}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\newline 
\forall 
\mathit{s}
\colon 
(
\mathit{s}
 \mathrel{\in} {\cal X}
\Rightarrow 
\mathit{s}
 \mathrel{\in} {\cal Y}
)
\Rightarrow {\cal S}
 \mathrel{\in} {\cal X}
\Rightarrow {\cal S}
 \mathrel{\in} {\cal Y}
]$

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } (Switch)HelperPowerIsSub
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$HelperPowerIsSub
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\forall 
\mathit{s}
\colon 
(
\mathit{s}
 \mathrel{\in} {\cal X}
\Rightarrow 
\mathit{s}
 \mathrel{\in} {\cal Y}
)
\Rightarrow 
\newline {\cal S}
 \mathrel{\in} {\cal X}
\Rightarrow {\cal S}
 \mathrel{\in} {\cal Y}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\item $
[ ZFsub
\mathbf{\ lemma\ } (Switch)PowerIsSub
\colon 
\Pi {\cal S}
,\linebreak [0] {\cal X}
\colon 
\mathit{s}
\#{\cal X}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal S}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\newline {\cal S}
 \mathrel{\in} P({\cal X}
)
\vdash {\cal S}
 \mathrel{\subseteq} {\cal X}
]$

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } (Switch)PowerIsSub
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
,\linebreak [0] {\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal S}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
 \mathrel{\in} P({\cal X}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$PowerIsSub
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
 \mathrel{\subseteq} {\cal X}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\end{list}

\noindent Vi kunne naturligvis have sparet noget plads ved at vise disse varianter til at begynde med. Jeg har ikke gjort dette, fordi vi nu har f\aa{}et illustreret en af ulemperne ved at arbejde med sidebetingelser.

Jeg vil ikke kommentere f\ae{}nomenet ``sidebetingelser af formen $[
\mathsf{x}
\#
\mathsf{y}]$'' yderligere --- indtil afsnit \ref{sec:binaerforen}, hvor antallet af sidebetingelser bliver s\aa{} stort, at det ikke kan ignoreres.

\subsection{Separation} \label{sec:seplemma}

Tilf\ae{}ldet ``$SeparationDef$'' er n\ae{}sten lige s\aa{} enkelt som tilf\ae{}ldet ``$PairDef$'' fra afsnit \ref{sec:parlemma}. Vi skal blot huske at overf\o{}re aksiomets sidebetingelse til de to regellemmaer:  

\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}

\item $
[ ZFsub
\mathbf{\ lemma\ } Sep2Formula
\colon 
\Pi {\cal A}
,\linebreak [0] {\cal B}
,\linebreak [0] {\cal P}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Y}
\colon {\cal P}
 ^{Ph}
\wedge \langle {\cal B}
{\equiv} {\cal A}
| {\cal P}
{:=} {\cal Y}
\rangle_{Ph} 
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\newline {\cal Y}
 \mathrel{\in}  \{ ph \mathrel{\in} {\cal X}
\mid {\cal A}
\}
\vdash {\cal Y}
 \mathrel{\in} {\cal X}
\mathrel{\dot{\wedge}} {\cal B}
]$

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } Sep2Formula
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
,\linebreak [0] {\cal B}
,\linebreak [0] {\cal P}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal P}
 ^{Ph}
\wedge \langle {\cal B}
{\equiv} {\cal A}
| {\cal P}
{:=} {\cal Y}
\rangle_{Ph} 
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal Y}
 \mathrel{\in}  \{ ph \mathrel{\in} {\cal X}
\mid {\cal A}
\}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_q \else
\if \relax \csname lgwellq\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellq {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellq \fi 
$:}$SeparationDef
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal Y}
 \mathrel{\in}  \{ ph \mathrel{\in} {\cal X}
\mid {\cal A}
\}
\mathrel{\dot{\Leftrightarrow}} 
\newline {\cal Y}
 \mathrel{\in} {\cal X}
\mathrel{\dot{\wedge}} {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$FirstConjunct
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_q \else
\if \relax \csname lgwellq\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellq {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellq \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal Y}
 \mathrel{\in}  \{ ph \mathrel{\in} {\cal X}
\mid {\cal A}
\}
\Rightarrow 
\newline {\cal Y}
 \mathrel{\in} {\cal X}
\mathrel{\dot{\wedge}} {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$MP
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal Y}
 \mathrel{\in} {\cal X}
\mathrel{\dot{\wedge}} {\cal B}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\item $
[ ZFsub
\mathbf{\ lemma\ } Formula2Sep
\colon 
\Pi {\cal A}
,\linebreak [0] {\cal B}
,\linebreak [0] {\cal P}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Y}
\colon {\cal P}
 ^{Ph}
\wedge \langle {\cal B}
{\equiv} {\cal A}
| {\cal P}
{:=} {\cal Y}
\rangle_{Ph} 
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\newline {\cal Y}
 \mathrel{\in} {\cal X}
\vdash {\cal B}
\vdash {\cal Y}
 \mathrel{\in}  \{ ph \mathrel{\in} {\cal X}
\mid {\cal A}
\}
]$

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } Formula2Sep
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
,\linebreak [0] {\cal B}
,\linebreak [0] {\cal P}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_s \else
\if \relax \csname lgwells\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwells {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwells \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal P}
 ^{Ph}
\wedge \langle {\cal B}
{\equiv} {\cal A}
| {\cal P}
{:=} {\cal Y}
\rangle_{Ph} 
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal Y}
 \mathrel{\in} {\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_r \else
\if \relax \csname lgwellr\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellr {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellr \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_t \else
\if \relax \csname lgwellt\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellt {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellt \fi 
$:}$JoinConjuncts
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_r \else
\if \relax \csname lgwellr\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellr {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellr \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal Y}
 \mathrel{\in} {\cal X}
\mathrel{\dot{\wedge}} {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_q \else
\if \relax \csname lgwellq\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellq {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellq \fi 
$:}$SeparationDef
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_s \else
\if \relax \csname lgwells\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwells {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwells \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
({\cal Y}
 \mathrel{\in}  \{ ph \mathrel{\in} {\cal X}
\mid {\cal A}
\}
\Rightarrow {\cal Y}
 \mathrel{\in} {\cal X}
\mathrel{\dot{\wedge}} {\cal B}
)
\mathrel{\dot{\wedge}} 
({\cal Y}
 \mathrel{\in} {\cal X}
\mathrel{\dot{\wedge}} {\cal B}
\Rightarrow 
\newline {\cal Y}
 \mathrel{\in}  \{ ph \mathrel{\in} {\cal X}
\mid {\cal A}
\}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$SecondConjunct
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_q \else
\if \relax \csname lgwellq\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellq {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellq \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal Y}
 \mathrel{\in} {\cal X}
\mathrel{\dot{\wedge}} {\cal B}
\Rightarrow 
\newline {\cal Y}
 \mathrel{\in}  \{ ph \mathrel{\in} {\cal X}
\mid {\cal A}
\}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$MP
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_t \else
\if \relax \csname lgwellt\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellt {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellt \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal Y}
 \mathrel{\in}  \{ ph \mathrel{\in} {\cal X}
\mid {\cal A}
\}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\end{list}

\subsection{Ekstentionalitet}

Tilf\ae{}ldet ``$Extensionality$'' er lidt mere kompliceret end det almindelige tilf\ae{}lde, fordi vi skal forholde os til objektkvantoren i meta-formlen $[{\cal X}
\!\mathrel{=}\! {\cal Y}
\mathrel{\dot{\Leftrightarrow}} 
\forall 
\mathit{s}
\colon 
(
\mathit{s}
 \mathrel{\in} {\cal X}
\mathrel{\dot{\Leftrightarrow}} 
\mathit{s}
 \mathrel{\in} {\cal Y}
)]$. Derfor er der to underafsnit; lemmaerne i underafsnit \ref{sec:tillighed} er afledt af implikationen $[
\forall 
\mathit{s}
\colon 
(
\mathit{s}
 \mathrel{\in} {\cal X}
\mathrel{\dot{\Leftrightarrow}} 
\mathit{s}
 \mathrel{\in} {\cal Y}
)
\Rightarrow {\cal X}
\!\mathrel{=}\! {\cal Y}]$, mens lemmaerne i underafsnit \ref{sec:fralighed} er afledt af den omvendte implikation.

\subsubsection{Tilstr\ae{}kkelig betingelse for lighed}
\label{sec:tillighed}

To m\ae{}ngder er lig hinanden, hvis de er hinandens delm\ae{}ngder:

\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}

\item $
[ ZFsub
\mathbf{\ lemma\ } ToSetEquality
\colon 
\Pi {\cal X}
,\linebreak [0] {\cal Y}
\colon 
\newline 
\mathit{s}
 \mathrel{\in} {\cal X}
\Rightarrow 
\mathit{s}
 \mathrel{\in} {\cal Y}
\vdash 
\mathit{s}
 \mathrel{\in} {\cal Y}
\Rightarrow 
\mathit{s}
 \mathrel{\in} {\cal X}
\vdash {\cal X}
\!\mathrel{=}\! {\cal Y}
]$

\item Beviset for $ToSetEquality$ er ret enkelt. Vi skal blot huske at s\ae{}tte en objektkvantor p\aa{} pr\ae{}misserne, hvilket g\o{}res med slutningsreglen $Gen$:

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } ToSetEquality
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal X}
,\linebreak [0] {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
 \mathrel{\in} {\cal X}
\Rightarrow 
\mathit{s}
 \mathrel{\in} {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
 \mathrel{\in} {\cal Y}
\Rightarrow 
\mathit{s}
 \mathrel{\in} {\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$JoinConjuncts
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
 \mathrel{\in} {\cal X}
\mathrel{\dot{\Leftrightarrow}} 
\mathit{s}
 \mathrel{\in} {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$Gen
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\forall 
\mathit{s}
\colon 
(
\mathit{s}
 \mathrel{\in} {\cal X}
\mathrel{\dot{\Leftrightarrow}} 
\mathit{s}
 \mathrel{\in} {\cal Y}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$Extensionality
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal X}
\!\mathrel{=}\! {\cal Y}
\mathrel{\dot{\Leftrightarrow}} 
\forall 
\mathit{s}
\colon 
(
\mathit{s}
 \mathrel{\in} {\cal X}
\mathrel{\dot{\Leftrightarrow}} 
\mathit{s}
 \mathrel{\in} {\cal Y}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}$SecondConjunct
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\forall 
\mathit{s}
\colon 
(
\mathit{s}
 \mathrel{\in} {\cal X}
\mathrel{\dot{\Leftrightarrow}} 
\mathit{s}
 \mathrel{\in} {\cal Y}
)
\Rightarrow {\cal X}
\!\mathrel{=}\! {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$MP
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal X}
\!\mathrel{=}\! {\cal Y}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\item I afsnit \ref{sec:parlig} f\aa{}r vi brug for en version af dette lemma, hvor objektvariablen $
\mathit{s}$ er erstattet af en anden objektvariabel (jeg har valgt $
\mathit{t}$). For at vise denne alternative version viser vi f\o{}rst lemmaet $HelperToSetEquality(t)$, som indkapsler en anvendelse af deduktionsreglen:

\item $
[ ZFsub
\mathbf{\ lemma\ } HelperToSetEquality(t)
\colon 
\Pi {\cal X}
,\linebreak [0] {\cal Y}
\colon 
\mathit{t}
\#{\cal X}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#{\cal Y}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\newline 
\forall 
\mathit{t}
\colon 
(
\mathit{t}
 \mathrel{\in} {\cal X}
\Rightarrow 
\mathit{t}
 \mathrel{\in} {\cal Y}
)
\Rightarrow 
(
\mathit{s}
 \mathrel{\in} {\cal X}
\Rightarrow 
\mathit{s}
 \mathrel{\in} {\cal Y}
)
]$

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } HelperToSetEquality(t)
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal X}
,\linebreak [0] {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{t}
 \mathrel{\in} {\cal X}
\Rightarrow 
\mathit{t}
 \mathrel{\in} {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Repetition
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{t}
 \mathrel{\in} {\cal X}
\Rightarrow 
\mathit{t}
 \mathrel{\in} {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal X}
,\linebreak [0] {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{t}
\#{\cal X}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#{\cal Y}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\forall 
\mathit{t}
\colon 
(
\mathit{t}
 \mathrel{\in} {\cal X}
\Rightarrow 
\mathit{t}
 \mathrel{\in} {\cal Y}
)
\Rightarrow 
(
\mathit{s}
 \mathrel{\in} {\cal X}
\Rightarrow 
\mathit{s}
 \mathrel{\in} {\cal Y}
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\item Hovedlemmaet hedder $ToSetEquality(t)$: 

\item $
[ ZFsub
\mathbf{\ lemma\ } ToSetEquality(t)
\colon 
\Pi {\cal X}
,\linebreak [0] {\cal Y}
\colon 
\mathit{t}
\#{\cal X}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#{\cal Y}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\newline 
\mathit{t}
 \mathrel{\in} {\cal X}
\Rightarrow 
\mathit{t}
 \mathrel{\in} {\cal Y}
\vdash 
\mathit{t}
 \mathrel{\in} {\cal Y}
\Rightarrow 
\mathit{t}
 \mathrel{\in} {\cal X}
\vdash {\cal X}
\!\mathrel{=}\! {\cal Y}
]$

\item I beviset herfor kombinerer vi $Gen$ med hj\ae{}lpelemmaet til at omdanne pr\ae{}missen $[
\mathit{t}
 \mathrel{\in} {\cal X}
\Rightarrow 
\mathit{t}
 \mathrel{\in} {\cal Y}]$ til $[
\mathit{s}
 \mathrel{\in} {\cal X}
\Rightarrow 
\mathit{s}
 \mathrel{\in} {\cal Y}]$ (linie 6--8). Vi gentager denne procedure mht.\ pr\ae{}missen $[
\mathit{t}
 \mathrel{\in} {\cal Y}
\Rightarrow 
\mathit{t}
 \mathrel{\in} {\cal X}]$ (linie 9-11). Vi kan da slutte af med en anvendelse af det oprindelige lemma $ToSetEquality$ (linie 12):

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } ToSetEquality(t)
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal X}
,\linebreak [0] {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{t}
\#{\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{t}
\#{\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{t}
 \mathrel{\in} {\cal X}
\Rightarrow 
\mathit{t}
 \mathrel{\in} {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{t}
 \mathrel{\in} {\cal Y}
\Rightarrow 
\mathit{t}
 \mathrel{\in} {\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$Gen
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\forall 
\mathit{t}
\colon 
(
\mathit{t}
 \mathrel{\in} {\cal X}
\Rightarrow 
\mathit{t}
 \mathrel{\in} {\cal Y}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$HelperToSetEquality(t)
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\forall 
\mathit{t}
\colon 
(
\mathit{t}
 \mathrel{\in} {\cal X}
\Rightarrow 
\mathit{t}
 \mathrel{\in} {\cal Y}
)
\Rightarrow 
\newline 
\mathit{s}
 \mathrel{\in} {\cal X}
\Rightarrow 
\mathit{s}
 \mathrel{\in} {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}$MP
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
 \mathrel{\in} {\cal X}
\Rightarrow 
\mathit{s}
 \mathrel{\in} {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$Gen
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\forall 
\mathit{t}
\colon 
(
\mathit{t}
 \mathrel{\in} {\cal Y}
\Rightarrow 
\mathit{t}
 \mathrel{\in} {\cal X}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
$:}$HelperToSetEquality(t)
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\forall 
\mathit{t}
\colon 
(
\mathit{t}
 \mathrel{\in} {\cal Y}
\Rightarrow 
\mathit{t}
 \mathrel{\in} {\cal X}
)
\Rightarrow 
\newline 
\mathit{s}
 \mathrel{\in} {\cal Y}
\Rightarrow 
\mathit{s}
 \mathrel{\in} {\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
$:}$MP
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
 \mathrel{\in} {\cal Y}
\Rightarrow 
\mathit{s}
 \mathrel{\in} {\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$ToSetEquality
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal X}
\!\mathrel{=}\! {\cal Y}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\end{list}

Arbejdsfordelingen mellem $HelperToSetEquality(t)$ og $ToSetEquality(t)$ er temmelig uj\ae{}vn; det meste af arbejdet foreg\aa{}r i beviset for $ToSetEquality(t)$.  At der er to lemmaer i stedet for \'{e}t, har en teknisk forklaring: Logiwebs bevischecker kan ikke arbejde med konklusioner af formen $[
\mathsf{x}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathsf{y}]$. Vi kan derfor kun referere til konklusionen $[
\mathit{t}
\#{\cal X}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#{\cal Y}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} (\cdots{})]$ ved at lade den v\ae{}re konklusionen p\aa{} et bevis; og derfor m\aa{} vi stoppe op, n\aa{}r vi n\aa{}r til linie 7 i $HelperToSetEquality(t)$. Vi vil flere gange senere komme ud for lemmaer, hvis eksistens har denne tekniske forklaring. Jeg vil bruge ordet ``lemmastump'' om disse lemmaer.


\subsubsection{N\o{}dvendig betingelse for lighed} \label{sec:fralighed}

Vi begynder dette underafsnit med et hj\ae{}lpelemma, som vi bruger til at fjerne objektkvantoren fra $Extensionality$: 

\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}

\item $
[ ZFsub
\mathbf{\ lemma\ } HelperFromSetEquality
\colon 
\Pi {\cal S}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Y}
\colon 
\mathit{s}
\#{\cal X}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal Y}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\newline 
\forall 
\mathit{s}
\colon 
(
\mathit{s}
 \mathrel{\in} {\cal X}
\mathrel{\dot{\Leftrightarrow}} 
\mathit{s}
 \mathrel{\in} {\cal Y}
)
\Rightarrow 
({\cal S}
 \mathrel{\in} {\cal X}
\mathrel{\dot{\Leftrightarrow}} {\cal S}
 \mathrel{\in} {\cal Y}
)
]$

\item Beviset for $HelperFromSetEquality$ er en simpel anvendelse af deduktionsreglen:

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } HelperFromSetEquality
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal X}
,\linebreak [0] {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{s}
 \mathrel{\in} {\cal X}
\mathrel{\dot{\Leftrightarrow}} 
\mathit{s}
 \mathrel{\in} {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Repetition
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{s}
 \mathrel{\in} {\cal X}
\mathrel{\dot{\Leftrightarrow}} 
\mathit{s}
 \mathrel{\in} {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal X}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal Y}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\forall 
\mathit{s}
\colon 
(
\mathit{s}
 \mathrel{\in} {\cal X}
\mathrel{\dot{\Leftrightarrow}} 
\mathit{s}
 \mathrel{\in} {\cal Y}
)
\Rightarrow 
({\cal S}
 \mathrel{\in} {\cal X}
\mathrel{\dot{\Leftrightarrow}} {\cal S}
 \mathrel{\in} {\cal Y}
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\item Hovelemmaet i dette underafsnit udsiger, at hvis ${\cal X}$ er lig med ${\cal Y}$, s\aa{} er ${\cal X}$ en delm\ae{}ngde af ${\cal Y}$\footnote{Naturligvis kan man ogs\aa{} vise, at hvis ${\cal X}$ er lig med ${\cal Y}$, s\aa{} er ${\cal Y}$ en delm\ae{}ngde af ${\cal X}$. Det har imidlertid ikke v\ae{}ret n\o{}dvendigt at bruge dette lemma.}:

\item $
[ ZFsub
\mathbf{\ lemma\ } FromSetEquality
\colon 
\Pi {\cal S}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Y}
\colon 
\mathit{s}
\#{\cal X}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal Y}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\newline {\cal X}
\!\mathrel{=}\! {\cal Y}
\vdash {\cal S}
 \mathrel{\in} {\cal X}
\vdash {\cal S}
 \mathrel{\in} {\cal Y}
]$

\item Beviset er en enkel anvendelse af $Extensionality$ og $HelperFromSetEquality$:

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } FromSetEquality
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal X}
\!\mathrel{=}\! {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
 \mathrel{\in} {\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$Extensionality
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal X}
\!\mathrel{=}\! {\cal Y}
\mathrel{\dot{\Leftrightarrow}} 
\forall 
\mathit{s}
\colon 
(
\mathit{s}
 \mathrel{\in} {\cal X}
\mathrel{\dot{\Leftrightarrow}} 
\mathit{s}
 \mathrel{\in} {\cal Y}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$IffSecond
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\forall 
\mathit{s}
\colon 
(
\mathit{s}
 \mathrel{\in} {\cal X}
\mathrel{\dot{\Leftrightarrow}} 
\mathit{s}
 \mathrel{\in} {\cal Y}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}$HelperFromSetEquality
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\forall 
\mathit{s}
\colon 
(
\mathit{s}
 \mathrel{\in} {\cal X}
\mathrel{\dot{\Leftrightarrow}} 
\mathit{s}
 \mathrel{\in} {\cal Y}
)
\Rightarrow 
\newline 
({\cal S}
 \mathrel{\in} {\cal X}
\mathrel{\dot{\Leftrightarrow}} {\cal S}
 \mathrel{\in} {\cal Y}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
$:}$MP
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
 \mathrel{\in} {\cal X}
\mathrel{\dot{\Leftrightarrow}} {\cal S}
 \mathrel{\in} {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$IffSecond
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
 \mathrel{\in} {\cal Y}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\end{list}

\section{Definitionslemmaer} \label{sec:def}

Regellemmaerne fra afsnit \ref{sec:regel} blev afledt fra aksiomer. Lemmaerne i dette afsnit bliver afledt fra makrodefinitionerne i afsnit \ref{sec:eqrel}; heraf navnet ``definitionslemmaer''. Form\aa{}let med definitionslemmaerne er at  muligg\o{}re effektiv anvendelse af makrodefinitionerne i beviser.

\subsection{Refleksiv relation} \label{sec:refldef}

Den f\o{}rste definition, vi tager under behandling, er definitionen 
\\ \mbox{$[ReflRel({\cal R}
,{\cal BS}
)
 \mathrel {\ddot {=}} 
\forall 
\mathit{s}
\colon 
(
\mathit{s}
 \mathrel{\in} {\cal BS}
\Rightarrow {\cal R}
(
\mathit{s}
,
\mathit{s}
)
)]$}. Vi bruger her en lemmastump ved navn $HelperReflexivity$, som lader os fjerne objektkvantoren fra denne definitions h\o{}jreside:  

\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}

\item $
[ ZFsub
\mathbf{\ lemma\ } HelperReflexivity
\colon 
\Pi {\cal R}
,\linebreak [0] {\cal S}
,\linebreak [0] {\cal BS}
\colon 
\mathit{s}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\newline 
\forall 
\mathit{s}
\colon 
(
\mathit{s}
 \mathrel{\in} {\cal BS}
\Rightarrow {\cal R}
(
\mathit{s}
,
\mathit{s}
)
)
\Rightarrow {\cal S}
 \mathrel{\in} {\cal BS}
\Rightarrow {\cal R}
({\cal S}
,{\cal S}
)
]$

\item Vi viser $HelperReflexivity$ ved hj\ae{}lp af deduktionsreglen:

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } HelperReflexivity
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal R}
,\linebreak [0] {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{s}
 \mathrel{\in} {\cal BS}
\Rightarrow {\cal R}
(
\mathit{s}
,
\mathit{s}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Repetition
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{s}
 \mathrel{\in} {\cal BS}
\Rightarrow {\cal R}
(
\mathit{s}
,
\mathit{s}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal R}
,\linebreak [0] {\cal S}
,\linebreak [0] {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\forall 
\mathit{s}
\colon 
(
\mathit{s}
 \mathrel{\in} {\cal BS}
\Rightarrow {\cal R}
(
\mathit{s}
,
\mathit{s}
)
)
\Rightarrow {\cal S}
 \mathrel{\in} {\cal BS}
\Rightarrow {\cal R}
({\cal S}
,{\cal S}
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\item I hovedlemmaet $Reflexivity$ omdanner vi lemmastumpens implikationer til inferenser:

\item $
[ ZFsub
\mathbf{\ lemma\ } Reflexivity
\colon 
\Pi {\cal R}
,\linebreak [0] {\cal S}
,\linebreak [0] {\cal BS}
\colon 
\mathit{s}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} ReflRel({\cal R}
,{\cal BS}
)
\vdash 
\newline {\cal S}
 \mathrel{\in} {\cal BS}
\vdash {\cal R}
({\cal S}
,{\cal S}
)
]$

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } Reflexivity
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal R}
,\linebreak [0] {\cal S}
,\linebreak [0] {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal R}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$ReflRel({\cal R}
,{\cal BS}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
 \mathrel{\in} {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$HelperReflexivity
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$ReflRel({\cal R}
,{\cal BS}
)
\Rightarrow {\cal S}
 \mathrel{\in} {\cal BS}
\Rightarrow {\cal R}
({\cal S}
,{\cal S}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$MP2
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal R}
({\cal S}
,{\cal S}
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\end{list}

\noindent Vi skal straks se flere eksempler p\aa{} netop denne arbejdsdeling mellem lemmastump og hovedlemma. 

\subsection{Symmetrisk relation} \label{sec:symreldef}

De to lemmaer i dette underafsnit er afledt fra definitionen af ``symmetrisk relation'' i afsnit \ref{sec:eqrel}. Fremgangsm\aa{}den er den samme som i afsnit \ref{sec:refldef}. Lemmaet $HelperSymmetry$ fungerer som lemmastump i forhold til $Symmetry$:

\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}

\item $
[ ZFsub
\mathbf{\ lemma\ } HelperSymmetry
\colon 
\Pi {\cal R}
,\linebreak [0] {\cal S}
,\linebreak [0] {\cal T}
,\linebreak [0] {\cal BS}
\colon 
\newline 
\mathit{s}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} SymRel({\cal R}
,{\cal BS}
)
\Rightarrow 
\newline {\cal S}
 \mathrel{\in} {\cal BS}
\Rightarrow {\cal T}
 \mathrel{\in} {\cal BS}
\Rightarrow {\cal R}
({\cal S}
,{\cal T}
)
\Rightarrow {\cal R}
({\cal T}
,{\cal S}
)
]$

\item $
[ ZFsub
\mathbf{\ lemma\ } Symmetry
\colon 
\Pi {\cal R}
,\linebreak [0] {\cal S}
,\linebreak [0] {\cal T}
,\linebreak [0] {\cal BS}
\colon 
\mathit{s}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} SymRel({\cal R}
,{\cal BS}
)
\vdash {\cal S}
 \mathrel{\in} {\cal BS}
\vdash {\cal T}
 \mathrel{\in} {\cal BS}
\vdash {\cal R}
({\cal S}
,{\cal T}
)
\vdash {\cal R}
({\cal T}
,{\cal S}
)
]$


\item Beviset for $HelperSymmetry$ fungerer ligesom beviset for  $HelperReflexivity$ fra afsnit \ref{sec:refldef} (dog skriver vi ikke definitionen af $SymRel({\cal R}
,{\cal BS}
)$ ud):

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } HelperSymmetry
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal R}
,\linebreak [0] {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{s}
 \mathrel{\in} {\cal BS}
\Rightarrow 
\mathit{t}
 \mathrel{\in} {\cal BS}
\Rightarrow {\cal R}
(
\mathit{s}
,
\mathit{t}
)
\Rightarrow {\cal R}
(
\mathit{t}
,
\mathit{s}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Repetition
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{s}
 \mathrel{\in} {\cal BS}
\Rightarrow 
\mathit{t}
 \mathrel{\in} {\cal BS}
\Rightarrow {\cal R}
(
\mathit{s}
,
\mathit{t}
)
\Rightarrow {\cal R}
(
\mathit{t}
,
\mathit{s}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal R}
,\linebreak [0] {\cal S}
,\linebreak [0] {\cal T}
,\linebreak [0] {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} SymRel({\cal R}
,{\cal BS}
)
\Rightarrow {\cal S}
 \mathrel{\in} {\cal BS}
\Rightarrow {\cal T}
 \mathrel{\in} {\cal BS}
\Rightarrow {\cal R}
({\cal S}
,{\cal T}
)
\Rightarrow {\cal R}
({\cal T}
,{\cal S}
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $


\item I beviset for $Symmetry$ skifter vi fra implikation til inferens:

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } Symmetry
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal R}
,\linebreak [0] {\cal S}
,\linebreak [0] {\cal T}
,\linebreak [0] {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal R}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{t}
\#{\cal R}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{t}
\#{\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$SymRel({\cal R}
,{\cal BS}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
 \mathrel{\in} {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal T}
 \mathrel{\in} {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal R}
({\cal S}
,{\cal T}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_i \else
\if \relax \csname lgwelli\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelli {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelli \fi 
$:}$HelperSymmetry
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$SymRel({\cal R}
,{\cal BS}
)
\Rightarrow {\cal S}
 \mathrel{\in} {\cal BS}
\Rightarrow {\cal T}
 \mathrel{\in} {\cal BS}
\Rightarrow {\cal R}
({\cal S}
,{\cal T}
)
\Rightarrow {\cal R}
({\cal T}
,{\cal S}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$MP4
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_i \else
\if \relax \csname lgwelli\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelli {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelli \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal R}
({\cal T}
,{\cal S}
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\end{list}

\subsection{Transitiv relation} \label{sec:transdef}

De to lemmaer i dette underafsnit er afledt fra definitionen af ``transitiv relation'' i afsnit \ref{sec:eqrel}. Vi bruger n\o{}jagtig den samme fremgangsm\aa{}de som i afsnit \ref{sec:refldef} og \ref{sec:symreldef}: Lemmastumpen ($HelperTransitivity$) fjerner definitionens objektkvantor ved hj\ae{}lp af $Ded$, og hovedlemmaet ($Transitivity$) skifter fra implikation til inferens:

\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}

\item $
[ ZFsub
\mathbf{\ lemma\ } HelperTransitivity
\colon 
\Pi {\cal R}
,\linebreak [0] {\cal S}
,\linebreak [0] {\cal T}
,\linebreak [0] {\cal U}
,\linebreak [0] {\cal BS}
\colon 
\newline 
\mathit{s}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{u}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{u}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} TransRel({\cal R}
,{\cal BS}
)
\Rightarrow 
\newline {\cal S}
 \mathrel{\in} {\cal BS}
\Rightarrow {\cal T}
 \mathrel{\in} {\cal BS}
\Rightarrow {\cal U}
 \mathrel{\in} {\cal BS}
\Rightarrow {\cal R}
({\cal S}
,{\cal T}
)
\Rightarrow {\cal R}
({\cal T}
,{\cal U}
)
\Rightarrow {\cal R}
({\cal S}
,{\cal U}
)
]$


\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } HelperTransitivity
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal R}
,\linebreak [0] {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{s}
 \mathrel{\in} {\cal BS}
\Rightarrow 
\mathit{t}
 \mathrel{\in} {\cal BS}
\Rightarrow 
\newline 
\mathit{u}
 \mathrel{\in} {\cal BS}
\Rightarrow {\cal R}
(
\mathit{s}
,
\mathit{t}
)
\Rightarrow {\cal R}
(
\mathit{t}
,
\mathit{u}
)
\Rightarrow {\cal R}
(
\mathit{s}
,
\mathit{u}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Repetition
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{s}
 \mathrel{\in} {\cal BS}
\Rightarrow 
\mathit{t}
 \mathrel{\in} {\cal BS}
\Rightarrow 
\newline 
\mathit{u}
 \mathrel{\in} {\cal BS}
\Rightarrow {\cal R}
(
\mathit{s}
,
\mathit{t}
)
\Rightarrow {\cal R}
(
\mathit{t}
,
\mathit{u}
)
\Rightarrow {\cal R}
(
\mathit{s}
,
\mathit{u}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal R}
,\linebreak [0] {\cal S}
,\linebreak [0] {\cal T}
,\linebreak [0] {\cal U}
,\linebreak [0] {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{u}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{u}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} TransRel({\cal R}
,{\cal BS}
)
\Rightarrow {\cal S}
 \mathrel{\in} {\cal BS}
\Rightarrow {\cal T}
 \mathrel{\in} {\cal BS}
\Rightarrow {\cal U}
 \mathrel{\in} {\cal BS}
\Rightarrow {\cal R}
({\cal S}
,{\cal T}
)
\Rightarrow {\cal R}
({\cal T}
,{\cal U}
)
\Rightarrow {\cal R}
({\cal S}
,{\cal U}
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\item $
[ ZFsub
\mathbf{\ lemma\ } Transitivity
\colon 
\Pi {\cal R}
,\linebreak [0] {\cal S}
,\linebreak [0] {\cal T}
,\linebreak [0] {\cal U}
,\linebreak [0] {\cal BS}
\colon 
\newline 
\mathit{s}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{u}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{u}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} TransRel({\cal R}
,{\cal BS}
)
\vdash 
\newline {\cal S}
 \mathrel{\in} {\cal BS}
\vdash {\cal T}
 \mathrel{\in} {\cal BS}
\vdash {\cal U}
 \mathrel{\in} {\cal BS}
\vdash {\cal R}
({\cal S}
,{\cal T}
)
\vdash {\cal R}
({\cal T}
,{\cal U}
)
\vdash {\cal R}
({\cal S}
,{\cal U}
)
]$

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } Transitivity
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal R}
,\linebreak [0] {\cal S}
,\linebreak [0] {\cal T}
,\linebreak [0] {\cal U}
,\linebreak [0] {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal R}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{t}
\#{\cal R}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{t}
\#{\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{u}
\#{\cal R}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{u}
\#{\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$TransRel({\cal R}
,{\cal BS}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
 \mathrel{\in} {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_i \else
\if \relax \csname lgwelli\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelli {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelli \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal T}
 \mathrel{\in} {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_j \else
\if \relax \csname lgwellj\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellj {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellj \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal U}
 \mathrel{\in} {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_k \else
\if \relax \csname lgwellk\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellk {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellk \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal R}
({\cal S}
,{\cal T}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_l \else
\if \relax \csname lgwelll\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelll {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelll \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal R}
({\cal T}
,{\cal U}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_m \else
\if \relax \csname lgwellm\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellm {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellm \fi 
$:}$HelperTransitivity
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$TransRel({\cal R}
,{\cal BS}
)
\Rightarrow {\cal S}
 \mathrel{\in} {\cal BS}
\Rightarrow {\cal T}
 \mathrel{\in} {\cal BS}
\Rightarrow {\cal U}
 \mathrel{\in} {\cal BS}
\Rightarrow {\cal R}
({\cal S}
,{\cal T}
)
\Rightarrow {\cal R}
({\cal T}
,{\cal U}
)
\Rightarrow {\cal R}
({\cal S}
,{\cal U}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_n \else
\if \relax \csname lgwelln\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelln {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelln \fi 
$:}$MP5
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_m \else
\if \relax \csname lgwellm\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellm {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellm \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_i \else
\if \relax \csname lgwelli\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelli {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelli \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_j \else
\if \relax \csname lgwellj\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellj {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellj \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_k \else
\if \relax \csname lgwellk\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellk {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellk \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal R}
({\cal T}
,{\cal U}
)
\Rightarrow {\cal R}
({\cal S}
,{\cal U}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$MP
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_n \else
\if \relax \csname lgwelln\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelln {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelln \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_l \else
\if \relax \csname lgwelll\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelll {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelll \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal R}
({\cal S}
,{\cal U}
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\end{list}

\subsection{\AE{}kvivalensrelation}

De tre lemmaer i dette underafsnit udsiger, at en \ae{}kvivalensrelation er hhv.\ refleksiv, symmetrisk og transitiv. Beviserne er simple; vi anvender et par tautologier p\aa{} definitionen af $EqRel({\cal R}
,{\cal BS}
)$ fra afsnit \ref{sec:eqrel}:

\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}

\item $
[ ZFsub
\mathbf{\ lemma\ } ERisReflexive
\colon 
\Pi {\cal R}
\colon EqRel({\cal R}
,{\cal BS}
)
\vdash ReflRel({\cal R}
,{\cal BS}
)
]$

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } ERisReflexive
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal R}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$EqRel({\cal R}
,{\cal BS}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$Repetition
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$ReflRel({\cal R}
,{\cal BS}
)
\mathrel{\dot{\wedge}} SymRel({\cal R}
,{\cal BS}
)
\mathrel{\dot{\wedge}} TransRel({\cal R}
,{\cal BS}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$FirstConjunct
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$ReflRel({\cal R}
,{\cal BS}
)
\mathrel{\dot{\wedge}} SymRel({\cal R}
,{\cal BS}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$FirstConjunct
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$ReflRel({\cal R}
,{\cal BS}
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\item $
[ ZFsub
\mathbf{\ lemma\ } ERisSymmetric
\colon 
\Pi {\cal R}
\colon EqRel({\cal R}
,{\cal BS}
)
\vdash SymRel({\cal R}
,{\cal BS}
)
]$

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } ERisSymmetric
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal R}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$EqRel({\cal R}
,{\cal BS}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$Repetition
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$ReflRel({\cal R}
,{\cal BS}
)
\mathrel{\dot{\wedge}} SymRel({\cal R}
,{\cal BS}
)
\mathrel{\dot{\wedge}} TransRel({\cal R}
,{\cal BS}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$FirstConjunct
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$ReflRel({\cal R}
,{\cal BS}
)
\mathrel{\dot{\wedge}} SymRel({\cal R}
,{\cal BS}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$SecondConjunct
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$SymRel({\cal R}
,{\cal BS}
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\item $
[ ZFsub
\mathbf{\ lemma\ } ERisTransitive
\colon 
\Pi {\cal R}
\colon EqRel({\cal R}
,{\cal BS}
)
\vdash TransRel({\cal R}
,{\cal BS}
)
]$

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } ERisTransitive
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal R}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$EqRel({\cal R}
,{\cal BS}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$Repetition
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$ReflRel({\cal R}
,{\cal BS}
)
\mathrel{\dot{\wedge}} SymRel({\cal R}
,{\cal BS}
)
\mathrel{\dot{\wedge}} TransRel({\cal R}
,{\cal BS}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$SecondConjunct
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$TransRel({\cal R}
,{\cal BS}
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\end{list}

\section{Basale s\ae{}tninger i m\ae{}ngdel\ae{}re} \label{sec:basal}

Lemmaerne i dette afsnit er en l\o{}s samling af m\ae{}ngdeteoretiske s\ae{}tninger, som bliver brugt i beviset for hovedresultatet i afsnit \ref{sec:hoved}. Lemmaerne er grupperet efter emne: Underafsnit \ref{sec:empty} omhandler den tomme m\ae{}ngde, underafsnit \ref{sec:lighed} handler om lighedsrelationen $[
\mathsf{x}
\!\mathrel{=}\! 
\mathsf{y}]$, og underafsnit \ref{sec:notequal} handler om negeret lighed ($
\mathsf{x}
 \mathrel{\neq} 
\mathsf{y}$). Figur 2 giver et overblik over lemmaerne.

\clearpage
% figur 2
\begin{figure} 
\resizebox{\textwidth}{!}{
\includegraphics[0cm,0cm][20cm,22cm]{/net/urd/home/disk15/eriksen/.logiweb/grafik/diverse.eps}}

\caption[Bevisstrukturen for afsnit \ref{sec:basal}]{Bevisstrukturen for afsnit \ref{sec:basal}. Kun lemmaer fra dette afsnit er vist. Tallene i parentes angiver, hvor mange sidebetingelser de forskellige lemmaer indeholder. En pil fra lemma $
\mathsf{x}$ til lemma $
\mathsf{y}$ betyder, at $
\mathsf{x}$ bruges i beviset for $
\mathsf{y}$. Tallene ved pilene angiver, hvor mange sidebetingelser $
\mathsf{y}$ modtager fra $
\mathsf{x}$. Hvis der f.eks.\ st\aa{}r ``$2 \cdot \mathbf{4}$'', betyder det, at $
\mathsf{x}$ bliver anvendt to gange i beviset for $
\mathsf{y}$, begge gange med 4 sidebetingelser. 
Der er overlap mellem nogle af sidebetingelserne, og lemmaer kan generere deres egne sidebetingelser. Derfor er der ikke n\o{}dvendigvis overensstemmelse mellem tallene i parentes og tallene ved lemmaerne.}

\end{figure}
\clearpage


\subsection{Den tomme m\ae{}ngde} \label{sec:empty}

Dette underafsnit indeholder 3 lemmaer, der hver har f\aa{}et sit under-underafsnit.

\subsubsection{En delm\ae{}ngde af alle m\ae{}ngder}

Vi viser f\o{}rst, at den tomme m\ae{}ngde er en delm\ae{}ngde af alle m\ae{}ngder:

\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}

\item $
[ ZFsub
\mathbf{\ lemma\ } \O{}isSubset
\colon 
\Pi {\cal S}
,\linebreak [0] {\cal X}
\colon {\cal S}
 \mathrel{\in} \mathrm{\O}
\Rightarrow {\cal S}
 \mathrel{\in} {\cal X}
]$

\item Beviset for $\O{}isSubset$ benytter sig af deduktionsreglen og tautologien \\ \mbox{$FromContradiction$}:

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } \O{}isSubset
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
,\linebreak [0] {\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
 \mathrel{\in} \mathrm{\O}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$\O{}def
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
 \mathrel{\notin} \mathrm{\O}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$FromContradiction
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
 \mathrel{\in} {\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
,\linebreak [0] {\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
 \mathrel{\in} \mathrm{\O}
\Rightarrow {\cal S}
 \mathrel{\in} {\cal X}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\end{list}

\subsubsection{Der er kun \'{e}n tom m\ae{}ngde}

Det n\ae{}ste lemma beviser p\aa{}standen ``der er kun \'{e}n tom m\ae{}ngde''. Vi formulerer denne p\aa{}stand som f\o{}lger: ``Hvis en m\ae{}ngde ${\cal X}$ ikke har nogen medlemmer, s\aa{} 
er ${\cal X}$ lig med $\mathrm{\O}$'':

\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}

\item $
[ ZFsub
\mathbf{\ lemma\ } Unique\O{}
\colon 
\Pi {\cal X}
\colon 
\mathit{s}
 \mathrel{\notin} {\cal X}
\vdash {\cal X}
\!\mathrel{=}\! \mathrm{\O}
]$

\item Vi ved allerede fra lemmaet $\O{}isSubset$, at $\mathrm{\O}$ er en delm\ae{}ngde af ${\cal X}$. Alts\aa{} mangler vi blot at vise, at ${\cal X}$ er en delm\ae{}ngde af $\mathrm{\O}$. Dette klarer vi med hj\ae{}lpelemmaet $HelperUnique\O{}$:

\item $
[ ZFsub
\mathbf{\ lemma\ } HelperUnique\O{}
\colon 
\Pi {\cal S}
,\linebreak [0] {\cal X}
\colon {\cal S}
 \mathrel{\notin} {\cal X}
\vdash {\cal S}
 \mathrel{\in} {\cal X}
\Rightarrow {\cal S}
 \mathrel{\in} \mathrm{\O}
]$

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } HelperUnique\O{}
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
,\linebreak [0] {\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
 \mathrel{\notin} {\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
 \mathrel{\in} {\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$FromContradiction
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
 \mathrel{\in} \mathrm{\O}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
,\linebreak [0] {\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
 \mathrel{\notin} {\cal X}
\Rightarrow {\cal S}
 \mathrel{\in} {\cal X}
\Rightarrow {\cal S}
 \mathrel{\in} \mathrm{\O}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
 \mathrel{\notin} {\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$MP
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
 \mathrel{\in} {\cal X}
\Rightarrow {\cal S}
 \mathrel{\in} \mathrm{\O}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\item Med $HelperUnique\O{}$ til r\aa{}dighed er det let at vise $Unique\O{}$:

\item $
[ ZFsub
\mathbf{\ lemma\ } Unique\O{}
\colon 
\Pi {\cal X}
\colon 
\mathit{s}
 \mathrel{\notin} {\cal X}
\vdash {\cal X}
\!\mathrel{=}\! \mathrm{\O}
]$

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } Unique\O{}
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
 \mathrel{\notin} {\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$HelperUnique\O{}
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
 \mathrel{\in} {\cal X}
\Rightarrow 
\mathit{s}
 \mathrel{\in} \mathrm{\O}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$\O{}isSubset
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
 \mathrel{\in} \mathrm{\O}
\Rightarrow 
\mathit{s}
 \mathrel{\in} {\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$ToSetEquality
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal X}
\!\mathrel{=}\! \mathrm{\O}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\end{list}

\subsubsection{Lemmaet ``$MemberNot\O{}$''}
\label{sec:membernotempty}

Som det sidste resultat i dette underafsnit viser vi, at en m\ae{}ngde, der har et medlem, ikke er lig med den tomme m\ae{}ngde:

\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}

\item $
[ ZFsub
\mathbf{\ lemma\ } MemberNot\O{}
\colon 
\Pi {\cal S}
,\linebreak [0] {\cal X}
\colon 
\mathit{s}
\#{\cal X}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} {\cal S}
 \mathrel{\in} {\cal X}
\vdash {\cal X}
 \mathrel{\neq} \mathrm{\O}
]$

\item Beviset for $MemberNot\O{}$ kan gengives som f\o{}lger: 
%
\begin{enumerate}
\item ``Antag at  ${\cal S}$  tilh\o{}rer ${\cal X}$. Hvis ${\cal X}$ var lig med $\mathrm{\O}$, s\aa{} ville ${\cal S}$ ogs\aa{} tilh\o{}re $\mathrm{\O}$''. 

\item ``Men ${\cal S}$ tilh\o{}rer ikke $\mathrm{\O}$. Derfor er ${\cal X}$ ikke lig med $\mathrm{\O}$''. 
\end{enumerate}

\item Vi viser punkt 1 som lemmastumpen $HelperMemberNot\O{}$; herefter gennemf\o{}rer vi punkt 2 i selve beviset for $MemberNot\O{}$:

\item $
[ ZFsub
\mathbf{\ lemma\ } HelperMemberNot\O{}
\colon 
\Pi {\cal S}
,\linebreak [0] {\cal X}
\colon 
\newline 
\mathit{s}
\#{\cal X}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} {\cal S}
 \mathrel{\in} {\cal X}
\Rightarrow {\cal X}
\!\mathrel{=}\! \mathrm{\O}
\Rightarrow {\cal S}
 \mathrel{\in} \mathrm{\O}
]$

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } HelperMemberNot\O{}
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
,\linebreak [0] {\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{s}
\#{\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
 \mathrel{\in} {\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal X}
\!\mathrel{=}\! \mathrm{\O}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$FromSetEquality
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
 \mathrel{\in} \mathrm{\O}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
,\linebreak [0] {\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal X}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\newline {\cal S}
 \mathrel{\in} {\cal X}
\Rightarrow {\cal X}
\!\mathrel{=}\! \mathrm{\O}
\Rightarrow {\cal S}
 \mathrel{\in} \mathrm{\O}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\item $
[ ZFsub
\mathbf{\ lemma\ } MemberNot\O{}
\colon 
\Pi {\cal S}
,\linebreak [0] {\cal X}
\colon 
\mathit{s}
\#{\cal X}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} {\cal S}
 \mathrel{\in} {\cal X}
\vdash {\cal X}
 \mathrel{\neq} \mathrm{\O}
]$

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } MemberNot\O{}
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
,\linebreak [0] {\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
 \mathrel{\in} {\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$HelperMemberNot\O{}
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
 \mathrel{\in} {\cal X}
\Rightarrow {\cal X}
\!\mathrel{=}\! \mathrm{\O}
\Rightarrow {\cal S}
 \mathrel{\in} \mathrm{\O}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$MP
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal X}
\!\mathrel{=}\! \mathrm{\O}
\Rightarrow {\cal S}
 \mathrel{\in} \mathrm{\O}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$\O{}def
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
 \mathrel{\notin} \mathrm{\O}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$MT
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal X}
 \mathrel{\neq} \mathrm{\O}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\end{list}

\subsection{ $[
\mathsf{x}
\!\mathrel{=}\! 
\mathsf{y}]$ er en \ae{}kvivalensrelation } \label{sec:lighed}

M\aa{}let med dette underafsnit er at vise, at $[
\mathsf{x}
\!\mathrel{=}\! 
\mathsf{y}]$ er en refleksiv, symmetrisk og transitiv relation.

At $[
\mathsf{x}
\!\mathrel{=}\! 
\mathsf{y}]$ er refleksiv, f\o{}lger af, at $[
\mathsf{x}
\Rightarrow 
\mathsf{y}]$ er refleksiv:

\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}

\item $
[ ZFsub
\mathbf{\ lemma\ } =\!{}Reflexivity
\colon 
\Pi {\cal S}
\colon {\cal S}
\!\mathrel{=}\! {\cal S}
]$

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } =\!{}Reflexivity
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}$AutoImply
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
 \mathrel{\in} {\cal S}
\Rightarrow 
\mathit{s}
 \mathrel{\in} {\cal S}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$ToSetEquality
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
\!\mathrel{=}\! {\cal S}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\item \indent \verb|  | Symmetri-lemmaet for $[
\mathsf{x}
\!\mathrel{=}\! 
\mathsf{y}]$ ser s\aa{}ledes ud:

\item $
[ ZFsub
\mathbf{\ lemma\ } =\!{}Symmetry
\colon 
\Pi {\cal X}
,\linebreak [0] {\cal Y}
\colon 
\mathit{s}
\#{\cal X}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal Y}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} {\cal X}
\!\mathrel{=}\! {\cal Y}
\vdash {\cal Y}
\!\mathrel{=}\! {\cal X}
]$

\item Vi beviser $=\!{}Symmetry$ ved at reducere termen $[{\cal X}
\!\mathrel{=}\! {\cal Y}]$  til de to implikationer  $[
\mathit{s}
 \mathrel{\in} {\cal X}
\Rightarrow 
\mathit{s}
 \mathrel{\in} {\cal Y}]$ og $[
\mathit{s}
 \mathrel{\in} {\cal Y}
\Rightarrow 
\mathit{s}
 \mathrel{\in} {\cal X}]$. Herefter s\ae{}tter vi de to implikationer sammen i omvendt r\ae{}kkef\o{}lge, hvilket giver os $[{\cal Y}
\!\mathrel{=}\! {\cal X}]$:

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } =\!{}Symmetry
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal X}
,\linebreak [0] {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal X}
\!\mathrel{=}\! {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$Extensionality
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal X}
\!\mathrel{=}\! {\cal Y}
\mathrel{\dot{\Leftrightarrow}} 
\forall 
\mathit{s}
\colon 
(
\mathit{s}
 \mathrel{\in} {\cal X}
\mathrel{\dot{\Leftrightarrow}} 
\mathit{s}
 \mathrel{\in} {\cal Y}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$IffSecond
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\forall 
\mathit{s}
\colon 
(
\mathit{s}
 \mathrel{\in} {\cal X}
\mathrel{\dot{\Leftrightarrow}} 
\mathit{s}
 \mathrel{\in} {\cal Y}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$HelperFromSetEquality
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\forall 
\mathit{s}
\colon 
(
\mathit{s}
 \mathrel{\in} {\cal X}
\mathrel{\dot{\Leftrightarrow}} 
\mathit{s}
 \mathrel{\in} {\cal Y}
)
\Rightarrow 
\newline 
(
\mathit{s}
 \mathrel{\in} {\cal X}
\mathrel{\dot{\Leftrightarrow}} 
\mathit{s}
 \mathrel{\in} {\cal Y}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$MP
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
 \mathrel{\in} {\cal X}
\mathrel{\dot{\Leftrightarrow}} 
\mathit{s}
 \mathrel{\in} {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}$FirstConjunct
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
 \mathrel{\in} {\cal X}
\Rightarrow 
\mathit{s}
 \mathrel{\in} {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
$:}$SecondConjunct
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
 \mathrel{\in} {\cal Y}
\Rightarrow 
\mathit{s}
 \mathrel{\in} {\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$ToSetEquality
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal Y}
\!\mathrel{=}\! {\cal X}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\item \indent \verb|  | Transitivitets-lemmaet for $[
\mathsf{x}
\!\mathrel{=}\! 
\mathsf{y}]$ ser s\aa{}ledes ud:

\item $
[ ZFsub
\mathbf{\ lemma\ } \!{}=\!{}Transitivity
\colon 
\Pi {\cal X}
,\linebreak [0] {\cal Y}
,\linebreak [0] {\cal Z}
\colon 
\mathit{s}
\#{\cal X}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal Y}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal Z}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\newline {\cal X}
\!\mathrel{=}\! {\cal Y}
\vdash {\cal Y}
\!\mathrel{=}\! {\cal Z}
\vdash {\cal X}
\!\mathrel{=}\! {\cal Z}
]$

\item Vi viser f\o{}rst den svagere p\aa{}stand, at ${\cal X}$ er en delm\ae{}ngde af ${\cal Z}$:

\item $
[ ZFsub
\mathbf{\ lemma\ } Helper\!{}=\!{}Transitivity
\colon 
\Pi {\cal S}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Y}
,\linebreak [0] {\cal Z}
\colon 
\mathit{s}
\#{\cal X}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal Y}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal Z}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\newline {\cal X}
\!\mathrel{=}\! {\cal Y}
\Rightarrow {\cal Y}
\!\mathrel{=}\! {\cal Z}
\Rightarrow {\cal S}
 \mathrel{\in} {\cal X}
\Rightarrow {\cal S}
 \mathrel{\in} {\cal Z}
]$

\item Bevisets kerne best\aa{}r af to anvendelser af $FromSetEquality$ (linie 8--9):

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } Helper\!{}=\!{}Transitivity
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Y}
,\linebreak [0] {\cal Z}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{s}
\#{\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{s}
\#{\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_C \else
\if \relax \csname lgwellbigc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigc \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{s}
\#{\cal Z}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal X}
\!\mathrel{=}\! {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal Y}
\!\mathrel{=}\! {\cal Z}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
 \mathrel{\in} {\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$FromSetEquality
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
 \mathrel{\in} {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$FromSetEquality
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_C \else
\if \relax \csname lgwellbigc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
 \mathrel{\in} {\cal Z}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Y}
,\linebreak [0] {\cal Z}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal X}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal Y}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal Z}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} {\cal X}
\!\mathrel{=}\! {\cal Y}
\Rightarrow {\cal Y}
\!\mathrel{=}\! {\cal Z}
\Rightarrow {\cal S}
 \mathrel{\in} {\cal X}
\Rightarrow {\cal S}
 \mathrel{\in} {\cal Z}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\item Vi bruger nu $Helper\!{}=\!{}Transitivity$ til at vise, at ${\cal X}$ er en delm\ae{}ngde af ${\cal Z}$ (linie 5--8), og vice versa (line 9--12): 

\item $
[ ZFsub
\mathbf{\ lemma\ } \!{}=\!{}Transitivity
\colon 
\Pi {\cal X}
,\linebreak [0] {\cal Y}
,\linebreak [0] {\cal Z}
\colon 
\mathit{s}
\#{\cal X}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal Y}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal Z}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\newline {\cal X}
\!\mathrel{=}\! {\cal Y}
\vdash {\cal Y}
\!\mathrel{=}\! {\cal Z}
\vdash {\cal X}
\!\mathrel{=}\! {\cal Z}
]$

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } \!{}=\!{}Transitivity
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal X}
,\linebreak [0] {\cal Y}
,\linebreak [0] {\cal Z}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_C \else
\if \relax \csname lgwellbigc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigc \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal Z}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal X}
\!\mathrel{=}\! {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal Y}
\!\mathrel{=}\! {\cal Z}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$Helper\!{}=\!{}Transitivity
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_C \else
\if \relax \csname lgwellbigc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal X}
\!\mathrel{=}\! {\cal Y}
\Rightarrow {\cal Y}
\!\mathrel{=}\! {\cal Z}
\Rightarrow 
\newline 
\mathit{s}
 \mathrel{\in} {\cal X}
\Rightarrow 
\mathit{s}
 \mathrel{\in} {\cal Z}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$MP2
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
 \mathrel{\in} {\cal X}
\Rightarrow 
\mathit{s}
 \mathrel{\in} {\cal Z}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$=\!{}Symmetry
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal Y}
\!\mathrel{=}\! {\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}$=\!{}Symmetry
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_C \else
\if \relax \csname lgwellbigc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal Z}
\!\mathrel{=}\! {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
$:}$Helper\!{}=\!{}Transitivity
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_C \else
\if \relax \csname lgwellbigc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigc \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal Z}
\!\mathrel{=}\! {\cal Y}
\Rightarrow {\cal Y}
\!\mathrel{=}\! {\cal X}
\Rightarrow 
\newline 
\mathit{s}
 \mathrel{\in} {\cal Z}
\Rightarrow 
\mathit{s}
 \mathrel{\in} {\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
$:}$MP2
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
 \mathrel{\in} {\cal Z}
\Rightarrow 
\mathit{s}
 \mathrel{\in} {\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$ToSetEquality
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal X}
\!\mathrel{=}\! {\cal Z}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\end{list}

\subsection{Negeret lighed} \label{sec:notequal}

I dette sidste underafsnit viser vi, at lige st\o{}rrelser kan erstatte hinanden i en negeret lighed:

\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}

\item $
[ ZFsub
\mathbf{\ lemma\ } TransferNotEq
\colon 
\Pi {\cal X}
,\linebreak [0] {\cal Y}
,\linebreak [0] {\cal V}
,\linebreak [0] {\cal W}
\colon 
\mathit{s}
\#{\cal X}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal Y}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal V}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal W}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} {\cal X}
 \mathrel{\neq} {\cal Y}
\vdash {\cal X}
\!\mathrel{=}\! {\cal V}
\vdash {\cal Y}
\!\mathrel{=}\! {\cal W}
\vdash {\cal V}
 \mathrel{\neq} {\cal W}
]$

\item Beviset for $TransferNotEq$ kan gengives som f\o{}lger: 
%
\begin{enumerate}
\item ``Antag at  ${\cal V}$  er lig med  ${\cal W}$. Ud fra pr\ae{}misserne $[{\cal X}
\!\mathrel{=}\! {\cal V}]$ og $[{\cal Y}
\!\mathrel{=}\! {\cal W}]$ f\aa{}r vi da $[{\cal X}
\!\mathrel{=}\! {\cal Y}]$.'' 

\item ``Men vi har antaget $[{\cal X}
 \mathrel{\neq} {\cal Y}]$. Derfor kan ${\cal V}$ ikke v\ae{}re lig med ${\cal W}$.''
\end{enumerate}

\item Vi viser punkt 1 som lemmastumpen $HelperTransferNotEq$;  punkt 2 klares i selve beviset for $TransferNotEq$:

\item $
[ ZFsub
\mathbf{\ lemma\ } HelperTransferNotEq
\colon 
\Pi {\cal X}
,\linebreak [0] {\cal Y}
,\linebreak [0] {\cal V}
,\linebreak [0] {\cal W}
\colon 
\newline 
\mathit{s}
\#{\cal X}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal Y}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal V}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal W}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} {\cal X}
\!\mathrel{=}\! {\cal V}
\Rightarrow {\cal Y}
\!\mathrel{=}\! {\cal W}
\Rightarrow {\cal V}
\!\mathrel{=}\! {\cal W}
\Rightarrow {\cal X}
\!\mathrel{=}\! {\cal Y}
]$

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } HelperTransferNotEq
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal X}
,\linebreak [0] {\cal Y}
,\linebreak [0] {\cal V}
,\linebreak [0] {\cal W}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_X \else
\if \relax \csname lgwellbigx\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigx {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigx \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{s}
\#{\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_Y \else
\if \relax \csname lgwellbigy\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigy {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigy \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{s}
\#{\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_V \else
\if \relax \csname lgwellbigv\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigv {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigv \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{s}
\#{\cal V}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_W \else
\if \relax \csname lgwellbigw\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigw {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigw \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{s}
\#{\cal W}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal X}
\!\mathrel{=}\! {\cal V}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal Y}
\!\mathrel{=}\! {\cal W}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal V}
\!\mathrel{=}\! {\cal W}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$\!{}=\!{}Transitivity
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_X \else
\if \relax \csname lgwellbigx\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigx {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigx \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_V \else
\if \relax \csname lgwellbigv\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigv {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigv \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_W \else
\if \relax \csname lgwellbigw\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigw {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigw \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal X}
\!\mathrel{=}\! {\cal W}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$=\!{}Symmetry
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_Y \else
\if \relax \csname lgwellbigy\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigy {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
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\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_W \else
\if \relax \csname lgwellbigw\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigw {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigw \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal W}
\!\mathrel{=}\! {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$\!{}=\!{}Transitivity
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_X \else
\if \relax \csname lgwellbigx\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigx {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigx \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_W \else
\if \relax \csname lgwellbigw\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigw {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigw \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_Y \else
\if \relax \csname lgwellbigy\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigy {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigy \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal X}
\!\mathrel{=}\! {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal X}
,\linebreak [0] {\cal Y}
,\linebreak [0] {\cal V}
,\linebreak [0] {\cal W}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal X}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal Y}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal V}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal W}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} {\cal X}
\!\mathrel{=}\! {\cal V}
\Rightarrow {\cal Y}
\!\mathrel{=}\! {\cal W}
\Rightarrow {\cal V}
\!\mathrel{=}\! {\cal W}
\Rightarrow {\cal X}
\!\mathrel{=}\! {\cal Y}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\newpage

\item $
[ ZFsub
\mathbf{\ lemma\ } TransferNotEq
\colon 
\Pi {\cal X}
,\linebreak [0] {\cal Y}
,\linebreak [0] {\cal V}
,\linebreak [0] {\cal W}
\colon 
\mathit{s}
\#{\cal X}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal Y}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal V}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal W}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} {\cal X}
 \mathrel{\neq} {\cal Y}
\vdash {\cal X}
\!\mathrel{=}\! {\cal V}
\vdash {\cal Y}
\!\mathrel{=}\! {\cal W}
\vdash {\cal V}
 \mathrel{\neq} {\cal W}
]$

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } TransferNotEq
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal X}
,\linebreak [0] {\cal Y}
,\linebreak [0] {\cal V}
,\linebreak [0] {\cal W}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_X \else
\if \relax \csname lgwellbigx\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigx {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigx \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_Y \else
\if \relax \csname lgwellbigy\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigy {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigy \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_V \else
\if \relax \csname lgwellbigv\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigv {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigv \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal V}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_W \else
\if \relax \csname lgwellbigw\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigw {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigw \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal W}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal X}
 \mathrel{\neq} {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal X}
\!\mathrel{=}\! {\cal V}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal Y}
\!\mathrel{=}\! {\cal W}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$HelperTransferNotEq
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_X \else
\if \relax \csname lgwellbigx\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigx {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigx \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_Y \else
\if \relax \csname lgwellbigy\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigy {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigy \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_V \else
\if \relax \csname lgwellbigv\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigv {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigv \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_W \else
\if \relax \csname lgwellbigw\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigw {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigw \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal X}
\!\mathrel{=}\! {\cal V}
\Rightarrow {\cal Y}
\!\mathrel{=}\! {\cal W}
\Rightarrow 
\newline {\cal V}
\!\mathrel{=}\! {\cal W}
\Rightarrow {\cal X}
\!\mathrel{=}\! {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$MP2
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal V}
\!\mathrel{=}\! {\cal W}
\Rightarrow {\cal X}
\!\mathrel{=}\! {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$MT
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal V}
 \mathrel{\neq} {\cal W}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\end{list}

\section{Lighedslemmaer} \label{sec:lighedslemmaer}

Ved et ``un\ae{}rt lighedslemma'' vil jeg forst\aa{} et lemma af f\o{}lgende form: ``Hvis der g\ae{}lder $[{\cal X}
\!\mathrel{=}\! {\cal Y}]$, s\aa{} g\ae{}lder $[Op({\cal X}
)
\!\mathrel{=}\! Op({\cal Y}
)]$'' --- hvor `` $[Op({\cal X}
)]$'' st\aa{}r for en un\ae{}r syntaktisk konstruktion som f.eks.\ $[\cup 
\mathsf{x}]$ eller $[\{
\mathsf{x}
\}]$.

Tilsvarende kan vi definere et ``bin\ae{}rt lighedslemma'' som et lemma med indholdet: ``Hvis der g\ae{}lder $[{\cal X}
\!\mathrel{=}\! {\cal Y}]$ og $[{\cal V}
\!\mathrel{=}\! {\cal W}]$, s\aa{} g\ae{}lder 
$[Op({\cal X}
,{\cal V}
)
\!\mathrel{=}\! Op({\cal Y}
,{\cal W}
)]$'' --- her kan  ``$[Op(
\mathsf{x}
,
\mathsf{y}
)]$'' f.eks.\ st\aa{} for $[\{
\mathsf{x}
,
\mathsf{y}
\}]$ eller $[
\mathsf{x}
 \cap 
\mathsf{y}]$.

I dette afsnit beviser vi lighedslemmaerne for konstruktionerne 
$[\{
\mathsf{x}
,
\mathsf{y}
\}]$, $[\{
\mathsf{x}
\}]$, $[\cup 
\mathsf{x}]$, $[
\mathsf{x}
 \mathrel{\cup} 
\mathsf{y}]$, $[ \{ ph \mathrel{\in} 
\mathsf{x}
\mid 
\mathsf {f}
\}]$ og $[
\mathsf{x}
 \cap 
\mathsf{y}]$. Figur 3 giver et overblik over afsnittets lemmaer. Hele arbejdet i dette afsnit munder ud i \'{e}n eneste anvendelse af det sidste lighedslemma (``$SameIntersection$'') i beviset for hovedresultatet (i afsnit \ref{sec:alldisjoint}). Man kan alts\aa{} godt sige, at vi skyder gr\aa{}spurve med kanoner. Lighedslemmaer har dog ogs\aa{} interesse derved, at de kan bruges til at vise, at $ZFsub$ er en teori med lighed (jvf.\ afsnit 2.8 i \cite{kn:mendel}); men det er alts\aa{} ikke et s\aa{}dant bevis, som er form\aa{}let med dette afsnit.

\clearpage
% figur 3 (30cm)
\begin{figure} 
\resizebox{\textwidth}{!}{
\includegraphics[0cm,0cm][20cm,25cm]{/net/urd/home/disk15/eriksen/.logiweb/grafik/same.eps}}
\caption[Bevisstrukturen for lighedslemmaerne]{Bevisstrukturen for lighedslemmaerne. Kun lighedslemmaerne er vist. Figuren skal l\ae{}ses ligesom figur 2.}
\end{figure}
\clearpage


\subsection{Par} \label{sec:parlig}

N\aa{}r vi skal vise et lighedslemma med konklusionen $[Op({\cal X}
,{\cal V}
)
\!\mathrel{=}\! Op({\cal Y}
,{\cal W}
)]$, er det ofte en god ide f\o{}rst at vise et ``delm\ae{}ngdelemma'' --- dvs.\ et lemma, der konkluderer, at $Op({\cal X}
,{\cal V}
)$ er en delm\ae{}ngde af $Op({\cal Y}
,{\cal W}
)$. Vi kan da vise lighedslemmaet ud fra $ToSetEquality$ og to anvendelser af delm\ae{}ngdelemmaet. I dette underafsnit begynder vi med det f\o{}lgende delm\ae{}ngdelemma:

\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}

\item $
[ ZFsub
\mathbf{\ lemma\ } PairSubset
\colon 
\Pi {\cal S}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Y}
,\linebreak [0] {\cal V}
,\linebreak [0] {\cal W}
\colon 
\newline 
\mathit{s}
\#{\cal X}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal Y}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal V}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal W}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal S}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\newline {\cal X}
\!\mathrel{=}\! {\cal Y}
\Rightarrow {\cal V}
\!\mathrel{=}\! {\cal W}
\Rightarrow {\cal S}
 \mathrel{\in} \{{\cal X}
,{\cal V}
\}
\Rightarrow {\cal S}
 \mathrel{\in} \{{\cal Y}
,{\cal W}
\}
]$

\item Beviset for $PairSubset$ kan gengives som f\o{}lger:
\begin{enumerate}

\item ``Antag $[{\cal X}
\!\mathrel{=}\! {\cal Y}]$, $[{\cal V}
\!\mathrel{=}\! {\cal W}]$ og at  ${\cal S}$ tilh\o{}rer $\{{\cal X}
,{\cal Y}
\}$. Der g\ae{}lder da ${\cal S}
\!\mathrel{=}\! {\cal X}$ eller ${\cal S}
\!\mathrel{=}\! {\cal Y}$.''

\item ``${\cal S}
\!\mathrel{=}\! {\cal X}$ medf\o{}rer $[{\cal S}
\!\mathrel{=}\! {\cal Y}
 \mathrel{\dot{\vee}} {\cal S}
\!\mathrel{=}\! {\cal W}]$ (under antagelse af $[{\cal X}
\!\mathrel{=}\! {\cal Y}]$).''

\item ``${\cal S}
\!\mathrel{=}\! {\cal Y}$ medf\o{}rer ogs\aa{} $[{\cal S}
\!\mathrel{=}\! {\cal Y}
 \mathrel{\dot{\vee}} {\cal S}
\!\mathrel{=}\! {\cal W}]$.''

\item ``Derfor m\aa{} $[{\cal S}
\!\mathrel{=}\! {\cal Y}
 \mathrel{\dot{\vee}} {\cal S}
\!\mathrel{=}\! {\cal W}]$ g\ae{}lde ubetinget. Dette betyder igen, at ${\cal S}$ tilh\o{}rer $\{{\cal Y}
,{\cal W}
\}$, QED.''

\end{enumerate}

\item Punkt 2 og 3 varetages af de to lemmastumper $HelperPairSubset$ og \\ \mbox{$Helper(2)PairSubset$}. Punkt 1 og 4 vises i selve beviset for $PairSubset$:

\item $
[ ZFsub
\mathbf{\ lemma\ } HelperPairSubset
\colon 
\Pi {\cal S}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Y}
,\linebreak [0] {\cal W}
\colon 
\newline 
\mathit{s}
\#{\cal S}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal X}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal Y}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal W}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\newline {\cal X}
\!\mathrel{=}\! {\cal Y}
\Rightarrow {\cal S}
\!\mathrel{=}\! {\cal X}
\Rightarrow {\cal S}
\!\mathrel{=}\! {\cal Y}
 \mathrel{\dot{\vee}} {\cal S}
\!\mathrel{=}\! {\cal W}
]$

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } HelperPairSubset
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Y}
,\linebreak [0] {\cal W}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_S \else
\if \relax \csname lgwellbigs\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigs {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigs \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{s}
\#{\cal S}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_X \else
\if \relax \csname lgwellbigx\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigx {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigx \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{s}
\#{\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_Y \else
\if \relax \csname lgwellbigy\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigy {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigy \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{s}
\#{\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_W \else
\if \relax \csname lgwellbigw\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigw {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigw \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{s}
\#{\cal W}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal X}
\!\mathrel{=}\! {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
\!\mathrel{=}\! {\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
$:}$\!{}=\!{}Transitivity
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_S \else
\if \relax \csname lgwellbigs\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigs {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigs \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_X \else
\if \relax \csname lgwellbigx\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigx {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigx \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_Y \else
\if \relax \csname lgwellbigy\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigy {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigy \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
\!\mathrel{=}\! {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$WeakenOr2
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
\!\mathrel{=}\! {\cal Y}
 \mathrel{\dot{\vee}} {\cal S}
\!\mathrel{=}\! {\cal W}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Y}
,\linebreak [0] {\cal W}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal S}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal X}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal Y}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal W}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} {\cal X}
\!\mathrel{=}\! {\cal Y}
\Rightarrow {\cal S}
\!\mathrel{=}\! {\cal X}
\Rightarrow 
\newline {\cal S}
\!\mathrel{=}\! {\cal Y}
 \mathrel{\dot{\vee}} {\cal S}
\!\mathrel{=}\! {\cal W}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\item $
[ ZFsub
\mathbf{\ lemma\ } Helper(2)PairSubset
\colon 
\Pi {\cal S}
,\linebreak [0] {\cal Y}
,\linebreak [0] {\cal V}
,\linebreak [0] {\cal W}
\colon 
\newline 
\mathit{s}
\#{\cal S}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal Y}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal V}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal W}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\newline {\cal V}
\!\mathrel{=}\! {\cal W}
\Rightarrow {\cal S}
\!\mathrel{=}\! {\cal V}
\Rightarrow {\cal S}
\!\mathrel{=}\! {\cal Y}
 \mathrel{\dot{\vee}} {\cal S}
\!\mathrel{=}\! {\cal W}
]$

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } Helper(2)PairSubset
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
,\linebreak [0] {\cal Y}
,\linebreak [0] {\cal V}
,\linebreak [0] {\cal W}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_S \else
\if \relax \csname lgwellbigs\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigs {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigs \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{s}
\#{\cal S}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_Y \else
\if \relax \csname lgwellbigy\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigy {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigy \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{s}
\#{\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_V \else
\if \relax \csname lgwellbigv\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigv {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigv \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{s}
\#{\cal V}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_W \else
\if \relax \csname lgwellbigw\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigw {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigw \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{s}
\#{\cal W}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal V}
\!\mathrel{=}\! {\cal W}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
\!\mathrel{=}\! {\cal V}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
$:}$\!{}=\!{}Transitivity
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_S \else
\if \relax \csname lgwellbigs\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigs {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigs \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_V \else
\if \relax \csname lgwellbigv\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigv {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigv \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_W \else
\if \relax \csname lgwellbigw\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigw {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigw \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
\!\mathrel{=}\! {\cal W}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$WeakenOr1
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
\!\mathrel{=}\! {\cal Y}
 \mathrel{\dot{\vee}} {\cal S}
\!\mathrel{=}\! {\cal W}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
,\linebreak [0] {\cal Y}
,\linebreak [0] {\cal V}
,\linebreak [0] {\cal W}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal S}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal Y}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal V}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal W}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} {\cal V}
\!\mathrel{=}\! {\cal W}
\Rightarrow {\cal S}
\!\mathrel{=}\! {\cal V}
\Rightarrow 
\newline {\cal S}
\!\mathrel{=}\! {\cal Y}
 \mathrel{\dot{\vee}} {\cal S}
\!\mathrel{=}\! {\cal W}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\item $
[ ZFsub
\mathbf{\ lemma\ } PairSubset
\colon 
\Pi {\cal S}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Y}
,\linebreak [0] {\cal V}
,\linebreak [0] {\cal W}
\colon 
\newline 
\mathit{s}
\#{\cal X}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal Y}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal V}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal W}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal S}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\newline {\cal X}
\!\mathrel{=}\! {\cal Y}
\Rightarrow {\cal V}
\!\mathrel{=}\! {\cal W}
\Rightarrow {\cal S}
 \mathrel{\in} \{{\cal X}
,{\cal V}
\}
\Rightarrow {\cal S}
 \mathrel{\in} \{{\cal Y}
,{\cal W}
\}
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } PairSubset
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Y}
,\linebreak [0] {\cal V}
,\linebreak [0] {\cal W}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_X \else
\if \relax \csname lgwellbigx\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigx {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigx \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{s}
\#{\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_Y \else
\if \relax \csname lgwellbigy\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigy {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigy \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{s}
\#{\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_V \else
\if \relax \csname lgwellbigv\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigv {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigv \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{s}
\#{\cal V}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_W \else
\if \relax \csname lgwellbigw\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigw {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigw \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{s}
\#{\cal W}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_S \else
\if \relax \csname lgwellbigs\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigs {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigs \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{s}
\#{\cal S}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal X}
\!\mathrel{=}\! {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal V}
\!\mathrel{=}\! {\cal W}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
 \mathrel{\in} \{{\cal X}
,{\cal V}
\}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_T \else
\if \relax \csname lgwellbigt\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigt {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigt \fi 
$:}$Pair2Formula
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
\!\mathrel{=}\! {\cal X}
 \mathrel{\dot{\vee}} {\cal S}
\!\mathrel{=}\! {\cal V}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$HelperPairSubset
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_S \else
\if \relax \csname lgwellbigs\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigs {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigs \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_X \else
\if \relax \csname lgwellbigx\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigx {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigx \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_Y \else
\if \relax \csname lgwellbigy\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigy {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigy \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_W \else
\if \relax \csname lgwellbigw\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigw {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigw \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal X}
\!\mathrel{=}\! {\cal Y}
\Rightarrow {\cal S}
\!\mathrel{=}\! {\cal X}
\Rightarrow 
\newline {\cal S}
\!\mathrel{=}\! {\cal Y}
 \mathrel{\dot{\vee}} {\cal S}
\!\mathrel{=}\! {\cal W}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_i \else
\if \relax \csname lgwelli\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelli {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelli \fi 
$:}$MP
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
\!\mathrel{=}\! {\cal X}
\Rightarrow {\cal S}
\!\mathrel{=}\! {\cal Y}
 \mathrel{\dot{\vee}} {\cal S}
\!\mathrel{=}\! {\cal W}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
$:}$Helper(2)PairSubset
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_S \else
\if \relax \csname lgwellbigs\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigs {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigs \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_Y \else
\if \relax \csname lgwellbigy\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigy {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigy \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_V \else
\if \relax \csname lgwellbigv\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigv {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigv \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_W \else
\if \relax \csname lgwellbigw\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigw {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigw \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal V}
\!\mathrel{=}\! {\cal W}
\Rightarrow {\cal S}
\!\mathrel{=}\! {\cal V}
\Rightarrow 
\newline {\cal S}
\!\mathrel{=}\! {\cal Y}
 \mathrel{\dot{\vee}} {\cal S}
\!\mathrel{=}\! {\cal W}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_n \else
\if \relax \csname lgwelln\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelln {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelln \fi 
$:}$MP
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
\!\mathrel{=}\! {\cal V}
\Rightarrow {\cal S}
\!\mathrel{=}\! {\cal Y}
 \mathrel{\dot{\vee}} {\cal S}
\!\mathrel{=}\! {\cal W}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
$:}$FromDisjuncts
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_T \else
\if \relax \csname lgwellbigt\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigt {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigt \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_i \else
\if \relax \csname lgwelli\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelli {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelli \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_n \else
\if \relax \csname lgwelln\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelln {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelln \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
\!\mathrel{=}\! {\cal Y}
 \mathrel{\dot{\vee}} {\cal S}
\!\mathrel{=}\! {\cal W}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Formula2Pair
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
 \mathrel{\in} \{{\cal Y}
,{\cal W}
\}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_p \else
\if \relax \csname lgwellp\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellp {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellp \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Y}
,\linebreak [0] {\cal V}
,\linebreak [0] {\cal W}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_p \else
\if \relax \csname lgwellp\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellp {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellp \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal X}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal Y}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal V}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal W}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal S}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} {\cal X}
\!\mathrel{=}\! {\cal Y}
\Rightarrow {\cal V}
\!\mathrel{=}\! {\cal W}
\Rightarrow 
\newline {\cal S}
 \mathrel{\in} \{{\cal X}
,{\cal V}
\}
\Rightarrow {\cal S}
 \mathrel{\in} \{{\cal Y}
,{\cal W}
\}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\item Kommet s\aa{} vidt kan vi bevise lighedslemmaet $SamePair$. Vi viser, at $\{{\cal X}
,{\cal V}
\}$ er en delm\ae{}ngde af $\{{\cal Y}
,{\cal W}
\}$ (linie 8--11), og vice versa (linie 12--14):

\item $
[ ZFsub
\mathbf{\ lemma\ } SamePair
\colon 
\Pi {\cal X}
,\linebreak [0] {\cal Y}
,\linebreak [0] {\cal V}
,\linebreak [0] {\cal W}
\colon 
\newline 
\mathit{s}
\#{\cal X}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal Y}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal V}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal W}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#\{{\cal X}
,{\cal V}
\}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#\{{\cal Y}
,{\cal W}
\}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\newline {\cal X}
\!\mathrel{=}\! {\cal Y}
\vdash {\cal V}
\!\mathrel{=}\! {\cal W}
\vdash \{{\cal X}
,{\cal V}
\}
\!\mathrel{=}\! \{{\cal Y}
,{\cal W}
\}
]$

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } SamePair
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal X}
,\linebreak [0] {\cal Y}
,\linebreak [0] {\cal V}
,\linebreak [0] {\cal W}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_X \else
\if \relax \csname lgwellbigx\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigx {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigx \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_Y \else
\if \relax \csname lgwellbigy\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigy {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigy \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_V \else
\if \relax \csname lgwellbigv\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigv {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigv \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal V}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_W \else
\if \relax \csname lgwellbigw\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigw {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigw \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal W}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_T \else
\if \relax \csname lgwellbigt\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigt {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigt \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{t}
\#\{{\cal X}
,{\cal V}
\}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_U \else
\if \relax \csname lgwellbigu\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigu {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigu \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{t}
\#\{{\cal Y}
,{\cal W}
\}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal X}
\!\mathrel{=}\! {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal V}
\!\mathrel{=}\! {\cal W}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$PairSubset
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_X \else
\if \relax \csname lgwellbigx\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigx {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigx \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_Y \else
\if \relax \csname lgwellbigy\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigy {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigy \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_V \else
\if \relax \csname lgwellbigv\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigv {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigv \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_W \else
\if \relax \csname lgwellbigw\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigw {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigw \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal X}
\!\mathrel{=}\! {\cal Y}
\Rightarrow {\cal V}
\!\mathrel{=}\! {\cal W}
\Rightarrow 
\newline 
\mathit{t}
 \mathrel{\in} \{{\cal X}
,{\cal V}
\}
\Rightarrow 
\mathit{t}
 \mathrel{\in} \{{\cal Y}
,{\cal W}
\}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
$:}$MP2
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{t}
 \mathrel{\in} \{{\cal X}
,{\cal V}
\}
\Rightarrow 
\mathit{t}
 \mathrel{\in} \{{\cal Y}
,{\cal W}
\}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$=\!{}Symmetry
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_X \else
\if \relax \csname lgwellbigx\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigx {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigx \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_Y \else
\if \relax \csname lgwellbigy\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigy {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigy \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal Y}
\!\mathrel{=}\! {\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$=\!{}Symmetry
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_V \else
\if \relax \csname lgwellbigv\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigv {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigv \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_W \else
\if \relax \csname lgwellbigw\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigw {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigw \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal W}
\!\mathrel{=}\! {\cal V}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_F \else
\if \relax \csname lgwellbigf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigf \fi 
$:}$PairSubset
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_Y \else
\if \relax \csname lgwellbigy\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigy {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigy \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_X \else
\if \relax \csname lgwellbigx\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigx {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigx \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_W \else
\if \relax \csname lgwellbigw\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigw {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigw \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_V \else
\if \relax \csname lgwellbigv\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigv {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigv \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal Y}
\!\mathrel{=}\! {\cal X}
\Rightarrow {\cal W}
\!\mathrel{=}\! {\cal V}
\Rightarrow 
\newline 
\mathit{t}
 \mathrel{\in} \{{\cal Y}
,{\cal W}
\}
\Rightarrow 
\mathit{t}
 \mathrel{\in} \{{\cal X}
,{\cal V}
\}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_G \else
\if \relax \csname lgwellbigg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigg \fi 
$:}$MP2
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_F \else
\if \relax \csname lgwellbigf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigf \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{t}
 \mathrel{\in} \{{\cal Y}
,{\cal W}
\}
\Rightarrow 
\mathit{t}
 \mathrel{\in} \{{\cal X}
,{\cal V}
\}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$ToSetEquality(t)
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_T \else
\if \relax \csname lgwellbigt\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigt {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigt \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_U \else
\if \relax \csname lgwellbigu\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigu {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigu \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_G \else
\if \relax \csname lgwellbigg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigg \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\{{\cal X}
,{\cal V}
\}
\!\mathrel{=}\! \{{\cal Y}
,{\cal W}
\}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\end{list}

\subsection{Singleton-m\ae{}ngde}

Lighedslemmaet for singleton-konstruktionen ser s\aa{}ledes ud:

\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}

\item $
[ ZFsub
\mathbf{\ lemma\ } SameSingleton
\colon 
\Pi {\cal X}
,\linebreak [0] {\cal Y}
\colon 
\newline 
\mathit{s}
\#{\cal X}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal Y}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#\{{\cal X}
,{\cal X}
\}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#\{{\cal Y}
,{\cal Y}
\}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} {\cal X}
\!\mathrel{=}\! {\cal Y}
\vdash \{{\cal X}
\}
\!\mathrel{=}\! \{{\cal Y}
\}
]$

\item Da $[\{
\mathsf{x}
\}]$ er makrodefineret som $[\{
\mathsf{x}
,
\mathsf{x}
\}]$, kan vi let vise $SameSingleton$ ud fra $SamePair$:

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } SameSingleton
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal X}
,\linebreak [0] {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_C \else
\if \relax \csname lgwellbigc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigc \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{t}
\#\{{\cal X}
,{\cal X}
\}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_D \else
\if \relax \csname lgwellbigd\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigd {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigd \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{t}
\#\{{\cal Y}
,{\cal Y}
\}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal X}
\!\mathrel{=}\! {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$SamePair
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_C \else
\if \relax \csname lgwellbigc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigc \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_D \else
\if \relax \csname lgwellbigd\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigd {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigd \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\{{\cal X}
,{\cal X}
\}
\!\mathrel{=}\! \{{\cal Y}
,{\cal Y}
\}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Repetition
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\{{\cal X}
\}
\!\mathrel{=}\! \{{\cal Y}
\}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\end{list}

\subsection{Foreningsm\ae{}ngde}

Delm\ae{}ngdelemmaet mht.\ $[\cup 
\mathsf{x}]$ ser s\aa{}ledes ud:

\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}

\item $
[ ZFsub
\mathbf{\ lemma\ } UnionSubset
\colon 
\Pi {\cal S}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Y}
\colon 
\mathit{s}
\#{\cal X}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal Y}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\newline {\cal X}
\!\mathrel{=}\! {\cal Y}
\Rightarrow {\cal S}
 \mathrel{\in} \cup {\cal X}
\Rightarrow {\cal S}
 \mathrel{\in} \cup {\cal Y}
]$

\item Bevisets kerne (linie 7--11) er en simpel anvendelse af definitionen af $[\cup 
\mathsf{x}]$:

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } UnionSubset
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{s}
\#{\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{s}
\#{\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal X}
\!\mathrel{=}\! {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
 \mathrel{\in} \cup {\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$Union2Formula
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
 \mathrel{\in} Ex_{10}
\mathrel{\dot{\wedge}} Ex_{10}
 \mathrel{\in} {\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$FirstConjunct
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
 \mathrel{\in} Ex_{10}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$SecondConjunct
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$Ex_{10}
 \mathrel{\in} {\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}$FromSetEquality
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$Ex_{10}
 \mathrel{\in} {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Formula2Union
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
 \mathrel{\in} \cup {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal X}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal Y}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\newline {\cal X}
\!\mathrel{=}\! {\cal Y}
\Rightarrow {\cal S}
 \mathrel{\in} \cup {\cal X}
\Rightarrow {\cal S}
 \mathrel{\in} \cup {\cal Y}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\item I beviset for lighedslemmaet $SameUnion$ viser vi, at $\cup {\cal X}$ er en delm\ae{}ngde af $\cup {\cal Y}$ (linie 4--6), og vice versa (line 7--9): 

\item $
[ ZFsub
\mathbf{\ lemma\ } SameUnion
\colon 
\Pi {\cal X}
,\linebreak [0] {\cal Y}
\colon 
\mathit{s}
\#{\cal X}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal Y}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} {\cal X}
\!\mathrel{=}\! {\cal Y}
\vdash \cup {\cal X}
\!\mathrel{=}\! \cup {\cal Y}
]$

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } SameUnion
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal X}
,\linebreak [0] {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal X}
\!\mathrel{=}\! {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$UnionSubset
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal X}
\!\mathrel{=}\! {\cal Y}
\Rightarrow 
\mathit{s}
 \mathrel{\in} \cup {\cal X}
\Rightarrow 
\mathit{s}
 \mathrel{\in} \cup {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$MP
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
 \mathrel{\in} \cup {\cal X}
\Rightarrow 
\mathit{s}
 \mathrel{\in} \cup {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$=\!{}Symmetry
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal Y}
\!\mathrel{=}\! {\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$UnionSubset
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal Y}
\!\mathrel{=}\! {\cal X}
\Rightarrow 
\mathit{s}
 \mathrel{\in} \cup {\cal Y}
\Rightarrow 
\mathit{s}
 \mathrel{\in} \cup {\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}$MP
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
 \mathrel{\in} \cup {\cal Y}
\Rightarrow 
\mathit{s}
 \mathrel{\in} \cup {\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$ToSetEquality
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\cup {\cal X}
\!\mathrel{=}\! \cup {\cal Y}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\end{list}

\subsection{Bin\ae{}r foreningsm\ae{}ngde} \label{sec:binaerforen}

I lighedslemmaet for $[
\mathsf{x}
 \mathrel{\cup} 
\mathsf{y}]$ bliver problemet med sidebetingelserne meget tydeligt:

\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}

\item$
[ ZFsub
\mathbf{\ lemma\ } SameBinaryUnion
\colon 
\Pi {\cal X}
,\linebreak [0] {\cal Y}
,\linebreak [0] {\cal V}
,\linebreak [0] {\cal W}
\colon 
\mathit{s}
\#{\cal X}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal Y}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#\{{\cal X}
,{\cal X}
\}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#\{{\cal Y}
,{\cal Y}
\}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal V}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal W}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#\{{\cal V}
,{\cal V}
\}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#\{{\cal W}
,{\cal W}
\}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#\{{\cal X}
\}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#\{{\cal Y}
\}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#\{{\cal V}
\}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#\{{\cal W}
\}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#\{\{{\cal X}
\}
,\{{\cal V}
\}
\}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#\{\{{\cal Y}
\}
,\{{\cal W}
\}
\}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#\{\{{\cal X}
\}
,\{{\cal V}
\}
\}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#\{\{{\cal Y}
\}
,\{{\cal W}
\}
\}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} {\cal X}
\!\mathrel{=}\! {\cal Y}
\vdash {\cal V}
\!\mathrel{=}\! {\cal W}
\vdash {\cal X}
 \mathrel{\cup} {\cal V}
\!\mathrel{=}\! {\cal Y}
 \mathrel{\cup} {\cal W}
]$

\end{list}

Alt i alt er der 16 sidebetingelser. Et s\aa{}dant lemma er jo ikke til at arbejde med. Derfor vil vi bruge objektvariable i formuleringen af de lemmaer, der afh\ae{}nger af $SameBinaryUnion$. P\aa{} d\'{e}n m\aa{}de bliver alle sidebetingelserne automatisk opfyldt, og vi beh\o{}ver ikke at behandle dem eksplicit. 

Problemet med $SameBinaryUnion$ er, at det nedarver sidebetingelser fra tre forskellige lemmaer (jvf.\ figur 3), og at der ikke er noget overlap mellem sidebetingelserne. Mange af sidebetingelserne er ensbetydende; f.eks.\ medf\o{}rer $[
\mathit{s}
\#{\cal X}]$ og $[
\mathit{s}
\#\{{\cal X}
\}]$ hinanden. Bevischeckeren er imidlertid ikke i stand til at udnytte dette; vi kan ikke g\o{}re andet end at opremse alle sidebetingelserne.

Ironisk nok er selve beviset for $SameBinaryUnion$ ret enkelt. Vi s\ae{}tter blot lighedslemmaerne $SameSingleton$, $SamePair$ og $SameUnion$ sammen:

\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } SameBinaryUnion
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal X}
,\linebreak [0] {\cal Y}
,\linebreak [0] {\cal V}
,\linebreak [0] {\cal W}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_C \else
\if \relax \csname lgwellbigc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigc \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{t}
\#\{{\cal X}
,{\cal X}
\}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_D \else
\if \relax \csname lgwellbigd\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigd {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigd \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{t}
\#\{{\cal Y}
,{\cal Y}
\}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_E \else
\if \relax \csname lgwellbige\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbige {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbige \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal V}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_F \else
\if \relax \csname lgwellbigf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigf \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal W}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_G \else
\if \relax \csname lgwellbigg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigg \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{t}
\#\{{\cal V}
,{\cal V}
\}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_H \else
\if \relax \csname lgwellbigh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigh \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{t}
\#\{{\cal W}
,{\cal W}
\}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_I \else
\if \relax \csname lgwellbigi\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigi {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigi \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#\{{\cal X}
\}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_J \else
\if \relax \csname lgwellbigj\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigj {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigj \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#\{{\cal Y}
\}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_K \else
\if \relax \csname lgwellbigk\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigk {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigk \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#\{{\cal V}
\}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_L \else
\if \relax \csname lgwellbigl\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigl {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigl \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#\{{\cal W}
\}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_O \else
\if \relax \csname lgwellbigo\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigo {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigo \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#\{\{{\cal X}
\}
,\{{\cal V}
\}
\}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_P \else
\if \relax \csname lgwellbigp\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigp {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigp \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#\{\{{\cal Y}
\}
,\{{\cal W}
\}
\}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_M \else
\if \relax \csname lgwellbigm\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigm {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigm \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{t}
\#\{\{{\cal X}
\}
,\{{\cal V}
\}
\}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_N \else
\if \relax \csname lgwellbign\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbign {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbign \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{t}
\#\{\{{\cal Y}
\}
,\{{\cal W}
\}
\}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal X}
\!\mathrel{=}\! {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal V}
\!\mathrel{=}\! {\cal W}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$SameSingleton
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_C \else
\if \relax \csname lgwellbigc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigc \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_D \else
\if \relax \csname lgwellbigd\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigd {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigd \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\{{\cal X}
\}
\!\mathrel{=}\! \{{\cal Y}
\}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$SameSingleton
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_E \else
\if \relax \csname lgwellbige\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbige {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbige \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_F \else
\if \relax \csname lgwellbigf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigf \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_G \else
\if \relax \csname lgwellbigg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigg \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_H \else
\if \relax \csname lgwellbigh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigh \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\{{\cal V}
\}
\!\mathrel{=}\! \{{\cal W}
\}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$SamePair
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_I \else
\if \relax \csname lgwellbigi\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigi {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigi \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_J \else
\if \relax \csname lgwellbigj\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigj {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigj \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_K \else
\if \relax \csname lgwellbigk\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigk {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigk \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_L \else
\if \relax \csname lgwellbigl\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigl {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigl \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_M \else
\if \relax \csname lgwellbigm\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigm {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigm \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_N \else
\if \relax \csname lgwellbign\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbign {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbign \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\{\{{\cal X}
\}
,\{{\cal V}
\}
\}
\!\mathrel{=}\! \{\{{\cal Y}
\}
,\{{\cal W}
\}
\}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}$SameUnion
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_O \else
\if \relax \csname lgwellbigo\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigo {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigo \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_P \else
\if \relax \csname lgwellbigp\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigp {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigp \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\cup \{\{{\cal X}
\}
,\{{\cal V}
\}
\}
\!\mathrel{=}\! \cup \{\{{\cal Y}
\}
,\{{\cal W}
\}
\}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Repetition
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal X}
 \mathrel{\cup} {\cal V}
\!\mathrel{=}\! {\cal Y}
 \mathrel{\cup} {\cal W}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\end{list}

\subsection{Separation}

Delm\ae{}ngdelemmaet for konstruktionen $[ \{ ph \mathrel{\in} {\cal T}
\mid {\cal F}
\}]$ ser s\aa{}ledes ud:

\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}

\item $
[ ZFsub
\mathbf{\ lemma\ } SeparationSubset
\colon 
\Pi {\cal A}
,\linebreak [0] {\cal B}
,\linebreak [0] {\cal S}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Y}
\colon 
\mathit{s}
\#{\cal X}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal Y}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\newline {\cal X}
\!\mathrel{=}\! {\cal Y}
\Rightarrow 
({\cal A}
\mathrel{\dot{\Leftrightarrow}} {\cal B}
)
\Rightarrow {\cal S}
 \mathrel{\in}  \{ ph \mathrel{\in} {\cal X}
\mid {\cal A}
\}
\Rightarrow {\cal S}
 \mathrel{\in}  \{ ph \mathrel{\in} {\cal Y}
\mid {\cal B}
\}
]$

\item I beviset g\aa{}r vi fra $[{\cal S}
 \mathrel{\in} {\cal X}]$ til $[{\cal S}
 \mathrel{\in} {\cal Y}]$ (linie 9--10), og fra $[{\cal A}]$ til $[{\cal B}]$ (linie 11--12):

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } SeparationSubset
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal A}
,\linebreak [0] {\cal B}
,\linebreak [0] {\cal S}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{s}
\#{\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{s}
\#{\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal X}
\!\mathrel{=}\! {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal A}
\mathrel{\dot{\Leftrightarrow}} {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
 \mathrel{\in}  \{ ph \mathrel{\in} {\cal X}
\mid {\cal A}
\}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$Sep2Formula
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
 \mathrel{\in} {\cal X}
\mathrel{\dot{\wedge}} {\cal A}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$FirstConjunct
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
 \mathrel{\in} {\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}$FromSetEquality
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
 \mathrel{\in} {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
$:}$SecondConjunct
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal A}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
$:}$IffSecond
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Formula2Sep
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
 \mathrel{\in}  \{ ph \mathrel{\in} {\cal Y}
\mid {\cal B}
\}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_j \else
\if \relax \csname lgwellj\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellj {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellj \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
,\linebreak [0] {\cal B}
,\linebreak [0] {\cal S}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_j \else
\if \relax \csname lgwellj\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellj {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellj \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal X}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal Y}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} {\cal X}
\!\mathrel{=}\! {\cal Y}
\Rightarrow 
\newline 
({\cal A}
\mathrel{\dot{\Leftrightarrow}} {\cal B}
)
\Rightarrow {\cal S}
 \mathrel{\in}  \{ ph \mathrel{\in} {\cal X}
\mid {\cal A}
\}
\Rightarrow {\cal S}
 \mathrel{\in}  \{ ph \mathrel{\in} {\cal Y}
\mid {\cal B}
\}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\item I beviset for lighedslemmaet $SameSeparation$  viser vi, at $ \{ ph \mathrel{\in} {\cal X}
\mid {\cal A}
\}$ er en delm\ae{}ngde af $ \{ ph \mathrel{\in} {\cal Y}
\mid {\cal B}
\}$ (linie 4--7), og vice versa (linie 8--11):

\item $
[ ZFsub
\mathbf{\ lemma\ } SameSeparation
\colon 
\Pi {\cal A}
,\linebreak [0] {\cal B}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Y}
\colon 
\mathit{s}
\#{\cal X}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal Y}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\newline {\cal X}
\!\mathrel{=}\! {\cal Y}
\vdash {\cal A}
\mathrel{\dot{\Leftrightarrow}} {\cal B}
\vdash  \{ ph \mathrel{\in} {\cal X}
\mid {\cal A}
\}
\!\mathrel{=}\!  \{ ph \mathrel{\in} {\cal Y}
\mid {\cal B}
\}
]$

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } SameSeparation
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
,\linebreak [0] {\cal B}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal X}
\!\mathrel{=}\! {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
\mathrel{\dot{\Leftrightarrow}} {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$SeparationSubset
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal X}
\!\mathrel{=}\! {\cal Y}
\Rightarrow 
({\cal A}
\mathrel{\dot{\Leftrightarrow}} {\cal B}
)
\Rightarrow 
\newline 
\mathit{s}
 \mathrel{\in}  \{ ph \mathrel{\in} {\cal X}
\mid {\cal A}
\}
\Rightarrow 
\newline 
\mathit{s}
 \mathrel{\in}  \{ ph \mathrel{\in} {\cal Y}
\mid {\cal B}
\}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$MP2
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
 \mathrel{\in}  \{ ph \mathrel{\in} {\cal X}
\mid {\cal A}
\}
\Rightarrow 
\newline 
\mathit{s}
 \mathrel{\in}  \{ ph \mathrel{\in} {\cal Y}
\mid {\cal B}
\}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$=\!{}Symmetry
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal Y}
\!\mathrel{=}\! {\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}$IffCommutativity
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal B}
\mathrel{\dot{\Leftrightarrow}} {\cal A}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
$:}$SeparationSubset
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal Y}
\!\mathrel{=}\! {\cal X}
\Rightarrow 
({\cal B}
\mathrel{\dot{\Leftrightarrow}} {\cal A}
)
\Rightarrow 
\newline 
\mathit{s}
 \mathrel{\in}  \{ ph \mathrel{\in} {\cal Y}
\mid {\cal B}
\}
\Rightarrow 
\newline 
\mathit{s}
 \mathrel{\in}  \{ ph \mathrel{\in} {\cal X}
\mid {\cal A}
\}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
$:}$MP2
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
 \mathrel{\in}  \{ ph \mathrel{\in} {\cal Y}
\mid {\cal B}
\}
\Rightarrow 
\newline 
\mathit{s}
 \mathrel{\in}  \{ ph \mathrel{\in} {\cal X}
\mid {\cal A}
\}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$ToSetEquality
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$ \{ ph \mathrel{\in} {\cal X}
\mid {\cal A}
\}
\!\mathrel{=}\!  \{ ph \mathrel{\in} {\cal Y}
\mid {\cal B}
\}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\end{list}

\subsection{Bin\ae{}r f\ae{}llesm\ae{}ngde} \label{sec:sameinter}

Delm\ae{}ngdelemmaet for konstruktionen $[
\mathsf{x}
 \cap 
\mathsf{y}]$ adskiller sig fra normen ved ikke at n\ae{}vne $[
\mathsf{x}
 \cap 
\mathsf{y}]$ eksplicit:

\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}

\item $
[ ZFsub
\mathbf{\ lemma\ } IntersectionSubset
\colon 
\Pi {\cal S}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Y}
,\linebreak [0] {\cal V}
,\linebreak [0] {\cal W}
\colon 
\newline 
\mathit{s}
\#{\cal X}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal Y}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal V}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal W}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\newline {\cal X}
\!\mathrel{=}\! {\cal Y}
\Rightarrow {\cal V}
\!\mathrel{=}\! {\cal W}
\Rightarrow {\cal S}
 \mathrel{\in} {\cal X}
\mathrel{\dot{\wedge}} {\cal S}
 \mathrel{\in} {\cal V}
\Rightarrow {\cal S}
 \mathrel{\in} {\cal Y}
\mathrel{\dot{\wedge}} {\cal S}
 \mathrel{\in} {\cal W}
]$

\item Bevisets kerne best\aa{}r af to anvendelser af $FromSetEquality$ (linie 10--11 og \mbox{12--13}):

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } IntersectionSubset
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Y}
,\linebreak [0] {\cal V}
,\linebreak [0] {\cal W}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{s}
\#{\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{s}
\#{\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_C \else
\if \relax \csname lgwellbigc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigc \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{s}
\#{\cal V}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_D \else
\if \relax \csname lgwellbigd\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigd {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigd \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{s}
\#{\cal W}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal X}
\!\mathrel{=}\! {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal V}
\!\mathrel{=}\! {\cal W}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
 \mathrel{\in} {\cal X}
\mathrel{\dot{\wedge}} {\cal S}
 \mathrel{\in} {\cal V}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$FirstConjunct
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
 \mathrel{\in} {\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$FromSetEquality
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
 \mathrel{\in} {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}$SecondConjunct
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
 \mathrel{\in} {\cal V}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
$:}$FromSetEquality
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_C \else
\if \relax \csname lgwellbigc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigc \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_D \else
\if \relax \csname lgwellbigd\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigd {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigd \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
 \mathrel{\in} {\cal W}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$JoinConjuncts
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
 \mathrel{\in} {\cal Y}
\mathrel{\dot{\wedge}} {\cal S}
 \mathrel{\in} {\cal W}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Y}
,\linebreak [0] {\cal V}
,\linebreak [0] {\cal W}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal X}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal Y}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal V}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal W}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} {\cal X}
\!\mathrel{=}\! {\cal Y}
\Rightarrow {\cal V}
\!\mathrel{=}\! {\cal W}
\Rightarrow {\cal S}
 \mathrel{\in} {\cal X}
\mathrel{\dot{\wedge}} {\cal S}
 \mathrel{\in} {\cal V}
\Rightarrow {\cal S}
 \mathrel{\in} {\cal Y}
\mathrel{\dot{\wedge}} {\cal S}
 \mathrel{\in} {\cal W}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\end{list}

S\aa{} er vi kommet til lighedslemmaet $SameIntersection$. Da dette lemma afh\ae{}nger af $SameBinaryUnion$, sl\aa{}r vi her over til objektvariable (jvf.\ afsnit \ref{sec:binaerforen}). Den store ulempe herved er, at objektvariable ikke kan instantieres til andre variable. Derfor vil en formulering som f.eks.
$[
\mathit{x}
\!\mathrel{=}\! 
\mathit{y}
\vdash 
\mathit{v}
\!\mathrel{=}\! 
\mathit{w}
\vdash 
\mathit{x}
 \cap 
\mathit{v}
\!\mathrel{=}\! 
\mathit{y}
 \cap 
\mathit{w}]$
sandsynligvis ikke kunne anvendes senere hen i rapporten, da vi ikke kan instantiere $
\mathit{x}$, $
\mathit{y}$, $
\mathit{v}$ eller $
\mathit{w}$.

Vi m\aa{} alts\aa{} unders\o{}ge, hvad $SameIntersection$ egentlig skal bruges til, f\o{}r vi formulerer lemmaet. Det viser sig, at den f\o{}lgende formulering kan bruges:

\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}

\item $
[ ZFsub
\mathbf{\ lemma\ } SameIntersection
\colon 
\mathit{x}
\!\mathrel{=}\! [Ex_{1}
 \mathrel{\in}  \mathsf {OBS}
]_{
\mathit{r}
}
\vdash 
\mathit{y}
\!\mathrel{=}\! [Ex_{2}
 \mathrel{\in}  \mathsf {OBS}
]_{
\mathit{r}
}
\vdash 
\mathit{x}
 \cap 
\mathit{y}
\!\mathrel{=}\! [Ex_{1}
 \mathrel{\in}  \mathsf {OBS}
]_{
\mathit{r}
}
 \cap [Ex_{2}
 \mathrel{\in}  \mathsf {OBS}
]_{
\mathit{r}
}
]$

\item Man kan sige, at vi har instantieret objektvariablene ``p\aa{} forh\aa{}nd''.

\item Beviset for $SameIntersection$  kan beskrives som f\o{}lger:
%
\item
\begin{enumerate}
\item Fra $SameBinaryUnion$ f\aa{}r vi, at de to bin\ae{}re foreningsm\ae{}ngder er \'{e}ns (linie 3).
\item De to formler $[ph_{3}
 \mathrel{\in} 
\mathit{x}
\mathrel{\dot{\wedge}} ph_{3}
 \mathrel{\in} 
\mathit{y}]$ og \\ \mbox{$[ph_{3}
 \mathrel{\in} [Ex_{1}
 \mathrel{\in}  \mathsf {OBS}
]_{
\mathit{r}
}
\mathrel{\dot{\wedge}} ph_{3}
 \mathrel{\in} [Ex_{2}
 \mathrel{\in}  \mathsf {OBS}
]_{
\mathit{r}
}]$} implicerer hinanden (linie 4--5 og 6--9).
\item Fra punkt 1, punkt 2 og $SameSeparation$ f\aa{}r vi da, at de to bin\ae{}re f\ae{}llesm\ae{}ngder er \'{e}ns (linie 10--12).
\end{enumerate}
%
\item Her er beviset:

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } SameIntersection
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{x}
\!\mathrel{=}\! [Ex_{1}
 \mathrel{\in}  \mathsf {OBS}
]_{
\mathit{r}
}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{y}
\!\mathrel{=}\! [Ex_{2}
 \mathrel{\in}  \mathsf {OBS}
]_{
\mathit{r}
}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$SameBinaryUnion
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{x}
 \mathrel{\cup} 
\mathit{y}
\!\mathrel{=}\! 
\newline [Ex_{1}
 \mathrel{\in}  \mathsf {OBS}
]_{
\mathit{r}
}
 \mathrel{\cup} [Ex_{2}
 \mathrel{\in}  \mathsf {OBS}
]_{
\mathit{r}
}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$IntersectionSubset
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{x}
\!\mathrel{=}\! [Ex_{1}
 \mathrel{\in}  \mathsf {OBS}
]_{
\mathit{r}
}
\Rightarrow 
\newline 
\mathit{y}
\!\mathrel{=}\! [Ex_{2}
 \mathrel{\in}  \mathsf {OBS}
]_{
\mathit{r}
}
\Rightarrow 
\newline ph_{3}
 \mathrel{\in} 
\mathit{x}
\mathrel{\dot{\wedge}} ph_{3}
 \mathrel{\in} 
\mathit{y}
\Rightarrow 
\newline ph_{3}
 \mathrel{\in} [Ex_{1}
 \mathrel{\in}  \mathsf {OBS}
]_{
\mathit{r}
}
\mathrel{\dot{\wedge}} 
\newline ph_{3}
 \mathrel{\in} [Ex_{2}
 \mathrel{\in}  \mathsf {OBS}
]_{
\mathit{r}
}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$MP2
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$ph_{3}
 \mathrel{\in} 
\mathit{x}
\mathrel{\dot{\wedge}} ph_{3}
 \mathrel{\in} 
\mathit{y}
\Rightarrow 
\newline ph_{3}
 \mathrel{\in} [Ex_{1}
 \mathrel{\in}  \mathsf {OBS}
]_{
\mathit{r}
}
\mathrel{\dot{\wedge}} 
\newline ph_{3}
 \mathrel{\in} [Ex_{2}
 \mathrel{\in}  \mathsf {OBS}
]_{
\mathit{r}
}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}$=\!{}Symmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$[Ex_{1}
 \mathrel{\in}  \mathsf {OBS}
]_{
\mathit{r}
}
\!\mathrel{=}\! 
\mathit{x}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
$:}$=\!{}Symmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$[Ex_{2}
 \mathrel{\in}  \mathsf {OBS}
]_{
\mathit{r}
}
\!\mathrel{=}\! 
\mathit{y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
$:}$IntersectionSubset
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$[Ex_{1}
 \mathrel{\in}  \mathsf {OBS}
]_{
\mathit{r}
}
\!\mathrel{=}\! 
\mathit{x}
\Rightarrow 
\newline [Ex_{2}
 \mathrel{\in}  \mathsf {OBS}
]_{
\mathit{r}
}
\!\mathrel{=}\! 
\mathit{y}
\Rightarrow 
\newline ph_{3}
 \mathrel{\in} [Ex_{1}
 \mathrel{\in}  \mathsf {OBS}
]_{
\mathit{r}
}
\mathrel{\dot{\wedge}} 
\newline ph_{3}
 \mathrel{\in} [Ex_{2}
 \mathrel{\in}  \mathsf {OBS}
]_{
\mathit{r}
}
\Rightarrow 
\newline ph_{3}
 \mathrel{\in} 
\mathit{x}
\mathrel{\dot{\wedge}} ph_{3}
 \mathrel{\in} 
\mathit{y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_i \else
\if \relax \csname lgwelli\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelli {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelli \fi 
$:}$MP2
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$ph_{3}
 \mathrel{\in} [Ex_{1}
 \mathrel{\in}  \mathsf {OBS}
]_{
\mathit{r}
}
\mathrel{\dot{\wedge}} 
\newline ph_{3}
 \mathrel{\in} [Ex_{2}
 \mathrel{\in}  \mathsf {OBS}
]_{
\mathit{r}
}
\Rightarrow 
\newline ph_{3}
 \mathrel{\in} 
\mathit{x}
\mathrel{\dot{\wedge}} ph_{3}
 \mathrel{\in} 
\mathit{y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_j \else
\if \relax \csname lgwellj\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellj {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellj \fi 
$:}$JoinConjuncts
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_i \else
\if \relax \csname lgwelli\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelli {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelli \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$ph_{3}
 \mathrel{\in} 
\mathit{x}
\mathrel{\dot{\wedge}} ph_{3}
 \mathrel{\in} 
\mathit{y}
\mathrel{\dot{\Leftrightarrow}} 
\newline ph_{3}
 \mathrel{\in} [Ex_{1}
 \mathrel{\in}  \mathsf {OBS}
]_{
\mathit{r}
}
\mathrel{\dot{\wedge}} 
\newline ph_{3}
 \mathrel{\in} [Ex_{2}
 \mathrel{\in}  \mathsf {OBS}
]_{
\mathit{r}
}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_k \else
\if \relax \csname lgwellk\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellk {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellk \fi 
$:}$SameSeparation
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_j \else
\if \relax \csname lgwellj\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellj {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellj \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$ \{ ph \mathrel{\in} 
\mathit{x}
 \mathrel{\cup} 
\mathit{y}
\mid ph_{3}
 \mathrel{\in} 
\mathit{x}
\mathrel{\dot{\wedge}} ph_{3}
 \mathrel{\in} 
\mathit{y}
\}
\!\mathrel{=}\! 
\newline  \{ ph \mathrel{\in} [Ex_{1}
 \mathrel{\in}  \mathsf {OBS}
]_{
\mathit{r}
}
 \mathrel{\cup} [Ex_{2}
 \mathrel{\in}  \mathsf {OBS}
]_{
\mathit{r}
}
\mid ph_{3}
 \mathrel{\in} [Ex_{1}
 \mathrel{\in}  \mathsf {OBS}
]_{
\mathit{r}
}
\mathrel{\dot{\wedge}} ph_{3}
 \mathrel{\in} [Ex_{2}
 \mathrel{\in}  \mathsf {OBS}
]_{
\mathit{r}
}
\}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Repetition
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_k \else
\if \relax \csname lgwellk\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellk {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellk \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{x}
 \cap 
\mathit{y}
\!\mathrel{=}\! 
\newline [Ex_{1}
 \mathrel{\in}  \mathsf {OBS}
]_{
\mathit{r}
}
 \cap [Ex_{2}
 \mathrel{\in}  \mathsf {OBS}
]_{
\mathit{r}
}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\end{list}

\section{Hovedresultatet} \label{sec:hoved}

Vi kan nu endelig g\aa{} i gang med hovedresultatet, som vi allerede udtrykte formelt i afsnit \ref{sec:partition}:

\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}

\item $
[ ZFsub
\mathbf{\ lemma\ } EqSysIsPartition
\colon EqRel(
\mathit{r}
, \mathsf {OBS}
)
\vdash Partition( \mathsf {OBS}
 / 
\mathit{r}
, \mathsf {OBS}
)
]$

\end{list}

Vi skal vise tre ting:
\begin{itemize}
\item Ingen \ae{}kvivalensklasser er tomme (underafsnit \ref{sec:nummer1}),
\item Alle \ae{}kvivalensklasser er disjunkte (underafsnit \ref{sec:nummer2}), og
\item F\ae{}llesm\ae{}ngden af alle \ae{}kvivalensklasserne er lig den oprindelige m\ae{}ngde $ \mathsf {OBS}$ (underafsnit \ref{sec:nummer3}).
\end{itemize}

\noindent Til sidst binder vi alle tr\aa{}dene sammen i underafsnit \ref{sec:sidste}.

\subsection{Ingen \ae{}kvivalensklasser er tomme} \label{sec:nummer1}

Lemmaet $AutoMember$ udsiger, at \ae{}kvivalensklassen $[{\cal S}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}$ indeholder ${\cal S}$ selv som medlem:

\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}

\item $
[ ZFsub
\mathbf{\ lemma\ } AutoMember
\colon 
\Pi {\cal R}
,\linebreak [0] {\cal S}
,\linebreak [0] {\cal BS}
\colon 
\mathit{s}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} EqRel({\cal R}
,{\cal BS}
)
\vdash 
\newline {\cal S}
 \mathrel{\in} {\cal BS}
\vdash {\cal S}
 \mathrel{\in} [{\cal S}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
]$

\item $AutoMember$ f\o{}lger af, at vi har ${\cal R}
({\cal S}
,{\cal S}
)$, fordi \ae{}kvivalensrelationen ${\cal R}$ er refleksiv:

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } AutoMember
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal R}
,\linebreak [0] {\cal S}
,\linebreak [0] {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal R}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$EqRel({\cal R}
,{\cal BS}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
 \mathrel{\in} {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$ERisReflexive
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$ReflRel({\cal R}
,{\cal BS}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$Reflexivity
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal R}
({\cal S}
,{\cal S}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Formula2Sep
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
 \mathrel{\in}  \{ ph \mathrel{\in} {\cal BS}
\mid {\cal R}
(ph_{1}
,{\cal S}
)
\}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\item Ud fra $AutoMember$ kan vi nu vise, at ingen medlemmer af et \ae{}kvivalenssystem er tomme:

\item $
[ ZFsub
\mathbf{\ lemma\ } EqSysNot\O{}
\colon 
\Pi {\cal R}
,\linebreak [0] {\cal S}
,\linebreak [0] {\cal BS}
\colon 
\mathit{s}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal S}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\newline 
\mathit{s}
\#[Ex_{1}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{u}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{u}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} EqRel({\cal R}
,{\cal BS}
)
\vdash 
\newline {\cal S}
 \mathrel{\in} {\cal BS}
 / {\cal R}
\Rightarrow {\cal S}
 \mathrel{\neq} \mathrm{\O}
]$

\item Pga.\ sidebetingelserne deler vi beviset op i to, med $HelperEqSysNot\O{}$ som lemmastump:

\item $
[ ZFsub
\mathbf{\ lemma\ } HelperEqSysNot\O{}
\colon 
\Pi {\cal R}
,\linebreak [0] {\cal S}
,\linebreak [0] {\cal BS}
\colon 
\mathit{s}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal S}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\newline 
\mathit{s}
\#[Ex_{1}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{u}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{u}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} EqRel({\cal R}
,{\cal BS}
)
\Rightarrow 
\newline {\cal S}
 \mathrel{\in} {\cal BS}
 / {\cal R}
\Rightarrow {\cal S}
 \mathrel{\neq} \mathrm{\O}
]$


\item N\ae{}sten hele beviset varetages af $HelperEqSysNot\O{}$. Beviset forl\o{}ber som f\o{}lger: Lad ${\cal S}$ v\ae{}re et medlem af et \ae{}kvivalenssystem (linie 8). Pr. definition er ${\cal S}$ en \ae{}kvivalensklasse (linie 9--12). Da enhver \ae{}kvivalensklasse har et medlem, har ${\cal S}$ et medlem (linie 13--14). Da enhver m\ae{}ngde med et medlem ikke er tom, er ${\cal S}$ ikke tom, QED (linie 15). 

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } HelperEqSysNot\O{}
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal R}
,\linebreak [0] {\cal S}
,\linebreak [0] {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{s}
\#{\cal R}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_C \else
\if \relax \csname lgwellbigc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigc \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{s}
\#{\cal S}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{s}
\#{\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_D \else
\if \relax \csname lgwellbigd\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigd {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigd \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{s}
\#[Ex_{1}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$EqRel(
$%
\leftskip=1em%
${\cal R}
,{\cal BS}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
 \mathrel{\in} {\cal BS}
 / {\cal R}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$Sep2Formula
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
 \mathrel{\in} P({\cal BS}
)
\mathrel{\dot{\wedge}} 
(Ex_{1}
 \mathrel{\in} {\cal BS}
\mathrel{\dot{\wedge}} [Ex_{1}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
\!\mathrel{=}\! {\cal S}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$SecondConjunct
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$Ex_{1}
 \mathrel{\in} {\cal BS}
\mathrel{\dot{\wedge}} [Ex_{1}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
\!\mathrel{=}\! {\cal S}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$FirstConjunct
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$Ex_{1}
 \mathrel{\in} {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}$SecondConjunct
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$[Ex_{1}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
\!\mathrel{=}\! {\cal S}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
$:}$AutoMember
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$Ex_{1}
 \mathrel{\in} [Ex_{1}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
$:}$FromSetEquality
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_D \else
\if \relax \csname lgwellbigd\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigd {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigd \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_C \else
\if \relax \csname lgwellbigc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$Ex_{1}
 \mathrel{\in} {\cal S}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$MemberNot\O{}
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_C \else
\if \relax \csname lgwellbigc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
 \mathrel{\neq} \mathrm{\O}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_i \else
\if \relax \csname lgwelli\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelli {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelli \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal R}
,\linebreak [0] {\cal S}
,\linebreak [0] {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_i \else
\if \relax \csname lgwelli\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelli {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelli \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal S}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#[Ex_{1}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{u}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{u}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} EqRel({\cal R}
,{\cal BS}
)
\Rightarrow {\cal S}
 \mathrel{\in} {\cal BS}
 / {\cal R}
\Rightarrow {\cal S}
 \mathrel{\neq} \mathrm{\O}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\item I hovedlemmaet skifter vi en implikation ud med en inferens:

\item $
[ ZFsub
\mathbf{\ lemma\ } EqSysNot\O{}
\colon 
\Pi {\cal R}
,\linebreak [0] {\cal S}
,\linebreak [0] {\cal BS}
\colon 
\mathit{s}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal S}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\newline 
\mathit{s}
\#[Ex_{1}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{u}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{u}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} EqRel({\cal R}
,{\cal BS}
)
\vdash 
\newline {\cal S}
 \mathrel{\in} {\cal BS}
 / {\cal R}
\Rightarrow {\cal S}
 \mathrel{\neq} \mathrm{\O}
]$

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } EqSysNot\O{}
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal R}
,\linebreak [0] {\cal S}
,\linebreak [0] {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal R}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_C \else
\if \relax \csname lgwellbigc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigc \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal S}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_D \else
\if \relax \csname lgwellbigd\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigd {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigd \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#[Ex_{1}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_E \else
\if \relax \csname lgwellbige\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbige {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbige \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{t}
\#{\cal R}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_F \else
\if \relax \csname lgwellbigf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigf \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{t}
\#{\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_G \else
\if \relax \csname lgwellbigg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigg \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{u}
\#{\cal R}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_H \else
\if \relax \csname lgwellbigh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigh \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{u}
\#{\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$EqRel({\cal R}
,{\cal BS}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$HelperEqSysNot\O{}
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_C \else
\if \relax \csname lgwellbigc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigc \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_D \else
\if \relax \csname lgwellbigd\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigd {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigd \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_E \else
\if \relax \csname lgwellbige\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbige {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbige \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_F \else
\if \relax \csname lgwellbigf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigf \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_G \else
\if \relax \csname lgwellbigg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigg \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_H \else
\if \relax \csname lgwellbigh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigh \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$EqRel({\cal R}
,{\cal BS}
)
\Rightarrow 
\newline {\cal S}
 \mathrel{\in} {\cal BS}
 / {\cal R}
\Rightarrow {\cal S}
 \mathrel{\neq} \mathrm{\O}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$MP
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
 \mathrel{\in} {\cal BS}
 / {\cal R}
\Rightarrow {\cal S}
 \mathrel{\neq} \mathrm{\O}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\end{list}

\subsection{Alle \ae{}kvivalensklasser er disjunkte} \label{sec:nummer2}

Dette underafsnit indeholder den mest komplicerede del af beviset for hovedresultatet. Som det kan ses af figur 4, har jeg m\aa{}ttet tage et ``snyde-aksiom'' til hj\ae{}lp for at blive f\ae{}rdig. 

\clearpage
% figur 4
\begin{figure} 
\resizebox{\textwidth}{!}{
\includegraphics[0cm,5cm][20cm,30cm]{/net/urd/home/disk15/eriksen/.logiweb/grafik/nummer2.eps}}
\caption[Bevisstrukturen for underafsnit \ref{sec:nummer2}]{Bevisstrukturen for underafsnit \ref{sec:nummer2}. Kun lemmaerne fra dette afsnit er vist. Figuren skal l\ae{}ses ligesom figur 2. Kassen m\ae{}rket ``$CheatAllDisjoint$'' repr\ae{}senterer et ``snyde-aksiom'', s\aa{} der g\aa{}r ingen pile ind i denne kasse.}
\end{figure}
\clearpage

\subsubsection{Lemmaet ``$EqSubset$''}

Indholdet af det f\o{}rste lemma, $EqSubset$, kan udtrykkes som f\o{}lger: ``Lad ${\cal X}$ og ${\cal Y}$ v\ae{}re to medlemmer af ${\cal BS}$. Hvis der g\ae{}lder ${\cal R}
({\cal X}
,{\cal Y}
)$, s\aa{} er \ae{}kvivalensklassen $[{\cal X}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}$ en delm\ae{}ngde af \ae{}kvivalensklassen $[{\cal Y}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}$'':

\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}

\item $
[ ZFsub
\mathbf{\ lemma\ } EqSubset
\colon 
\Pi {\cal R}
,\linebreak [0] {\cal S}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Y}
,\linebreak [0] {\cal BS}
\colon 
\newline 
\mathit{s}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{u}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{u}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\newline {\cal X}
 \mathrel{\in} {\cal BS}
\vdash {\cal Y}
 \mathrel{\in} {\cal BS}
\vdash EqRel({\cal R}
,{\cal BS}
)
\vdash {\cal R}
({\cal X}
,{\cal Y}
)
\vdash 
\newline {\cal S}
 \mathrel{\in} [{\cal X}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
\Rightarrow {\cal S}
 \mathrel{\in} [{\cal Y}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
]$

\item Beviset forl\o{}ber som f\o{}lger: 
%
\begin{enumerate}
\item Lad ${\cal S}$ v\ae{}re et vilk\aa{}rligt medlem af $[{\cal X}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}$. Vi har da ${\cal R}
({\cal S}
,{\cal X}
)$ \\
\mbox{(linie 13--16)}.
\item Ud fra ${\cal R}
({\cal S}
,{\cal X}
)$
og antagelsen ${\cal R}
({\cal X}
,{\cal Y}
)$
f\aa{}r vi ${\cal R}
({\cal S}
,{\cal Y}
)$, da ${\cal R}$ er transitiv (linie 17--19).
\item ${\cal R}
({\cal S}
,{\cal Y}
)$ medf\o{}rer igen ${\cal S}
 \mathrel{\in} [{\cal Y}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}$, QED (linie 20).
\end{enumerate}
%

\item Hele dette r\ae{}sonnement gennemf\o{}res i lemmastumpen $HelperEqSubset$. Selve beviset for $EqSubset$ best\aa{}r af den s\ae{}dvanlige omskrivning fra implikation til inferens. Her er beviserne:

\item $
[ ZFsub
\mathbf{\ lemma\ } HelperEqSubset
\colon 
\Pi {\cal R}
,\linebreak [0] {\cal S}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Y}
,\linebreak [0] {\cal BS}
\colon 
\newline 
\mathit{s}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{u}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{u}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\newline {\cal X}
 \mathrel{\in} {\cal BS}
\Rightarrow {\cal Y}
 \mathrel{\in} {\cal BS}
\Rightarrow EqRel({\cal R}
,{\cal BS}
)
\Rightarrow {\cal R}
({\cal X}
,{\cal Y}
)
\Rightarrow 
\newline {\cal S}
 \mathrel{\in} [{\cal X}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
\Rightarrow {\cal S}
 \mathrel{\in} [{\cal Y}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
]$

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } HelperEqSubset
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal R}
,\linebreak [0] {\cal S}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Y}
,\linebreak [0] {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{s}
\#{\cal R}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_E \else
\if \relax \csname lgwellbige\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbige {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbige \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{s}
\#{\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_F \else
\if \relax \csname lgwellbigf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigf \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{t}
\#{\cal R}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_J \else
\if \relax \csname lgwellbigj\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigj {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigj \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{t}
\#{\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_K \else
\if \relax \csname lgwellbigk\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigk {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigk \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{u}
\#{\cal R}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_O \else
\if \relax \csname lgwellbigo\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigo {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigo \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{u}
\#{\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_r \else
\if \relax \csname lgwellr\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellr {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellr \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal X}
 \mathrel{\in} {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_s \else
\if \relax \csname lgwells\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwells {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwells \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal Y}
 \mathrel{\in} {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_Z \else
\if \relax \csname lgwellbigz\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigz {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigz \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$EqRel(
$%
\leftskip=1em%
${\cal R}
,{\cal BS}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_q \else
\if \relax \csname lgwellq\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellq {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellq \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal R}
(
$%
\leftskip=1em%
${\cal X}
,{\cal Y}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
 \mathrel{\in} [{\cal X}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_R \else
\if \relax \csname lgwellbigr\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigr {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigr \fi 
$:}$Repetition
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
 \mathrel{\in}  \{ ph \mathrel{\in} {\cal BS}
\mid {\cal R}
(ph_{1}
,{\cal X}
)
\}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$Sep2Formula
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_R \else
\if \relax \csname lgwellbigr\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigr {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigr \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
 \mathrel{\in} {\cal BS}
\mathrel{\dot{\wedge}} {\cal R}
({\cal S}
,{\cal X}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$SecondConjunct
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal R}
(
$%
\leftskip=1em%
${\cal S}
,{\cal X}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_t \else
\if \relax \csname lgwellt\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellt {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellt \fi 
$:}$ERisTransitive
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_Z \else
\if \relax \csname lgwellbigz\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigz {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigz \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$TransRel(
$%
\leftskip=1em%
${\cal R}
,{\cal BS}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$FirstConjunct
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
 \mathrel{\in} {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
$:}$Transitivity
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_E \else
\if \relax \csname lgwellbige\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbige {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbige \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_F \else
\if \relax \csname lgwellbigf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigf \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_J \else
\if \relax \csname lgwellbigj\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigj {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigj \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_K \else
\if \relax \csname lgwellbigk\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigk {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigk \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_O \else
\if \relax \csname lgwellbigo\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigo {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigo \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_t \else
\if \relax \csname lgwellt\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellt {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellt \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_r \else
\if \relax \csname lgwellr\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellr {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellr \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_s \else
\if \relax \csname lgwells\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwells {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwells \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_q \else
\if \relax \csname lgwellq\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellq {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellq \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal R}
(
$%
\leftskip=1em%
${\cal S}
,{\cal Y}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Formula2Sep
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
 \mathrel{\in} [{\cal Y}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_i \else
\if \relax \csname lgwelli\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelli {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelli \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal R}
,\linebreak [0] {\cal S}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Y}
,\linebreak [0] {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_i \else
\if \relax \csname lgwelli\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelli {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelli \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{u}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{u}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} {\cal X}
 \mathrel{\in} {\cal BS}
\Rightarrow {\cal Y}
 \mathrel{\in} {\cal BS}
\Rightarrow EqRel({\cal R}
,{\cal BS}
)
\Rightarrow {\cal R}
({\cal X}
,{\cal Y}
)
\Rightarrow {\cal S}
 \mathrel{\in} [{\cal X}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
\Rightarrow 
\newline {\cal S}
 \mathrel{\in} [{\cal Y}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\item $
[ ZFsub
\mathbf{\ lemma\ } EqSubset
\colon 
\Pi {\cal R}
,\linebreak [0] {\cal S}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Y}
,\linebreak [0] {\cal BS}
\colon 
\newline 
\mathit{s}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{u}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{u}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\newline {\cal X}
 \mathrel{\in} {\cal BS}
\vdash {\cal Y}
 \mathrel{\in} {\cal BS}
\vdash EqRel({\cal R}
,{\cal BS}
)
\vdash {\cal R}
({\cal X}
,{\cal Y}
)
\vdash 
\newline {\cal S}
 \mathrel{\in} [{\cal X}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
\Rightarrow {\cal S}
 \mathrel{\in} [{\cal Y}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
]$

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } EqSubset
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal R}
,\linebreak [0] {\cal S}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Y}
,\linebreak [0] {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal R}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_E \else
\if \relax \csname lgwellbige\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbige {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbige \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_F \else
\if \relax \csname lgwellbigf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigf \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{t}
\#{\cal R}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_J \else
\if \relax \csname lgwellbigj\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigj {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigj \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{t}
\#{\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_K \else
\if \relax \csname lgwellbigk\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigk {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigk \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{u}
\#{\cal R}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_O \else
\if \relax \csname lgwellbigo\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigo {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigo \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{u}
\#{\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal X}
 \mathrel{\in} {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal Y}
 \mathrel{\in} {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$EqRel({\cal R}
,{\cal BS}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal R}
({\cal X}
,{\cal Y}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$HelperEqSubset
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_E \else
\if \relax \csname lgwellbige\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbige {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbige \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_F \else
\if \relax \csname lgwellbigf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigf \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_J \else
\if \relax \csname lgwellbigj\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigj {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigj \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_K \else
\if \relax \csname lgwellbigk\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigk {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigk \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_O \else
\if \relax \csname lgwellbigo\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigo {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigo \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal X}
 \mathrel{\in} {\cal BS}
\Rightarrow {\cal Y}
 \mathrel{\in} {\cal BS}
\Rightarrow EqRel({\cal R}
,{\cal BS}
)
\Rightarrow {\cal R}
({\cal X}
,{\cal Y}
)
\Rightarrow {\cal S}
 \mathrel{\in} [{\cal X}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
\Rightarrow 
\newline {\cal S}
 \mathrel{\in} [{\cal Y}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$MP4
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
 \mathrel{\in} [{\cal X}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
\Rightarrow 
\newline {\cal S}
 \mathrel{\in} [{\cal Y}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\end{list}

\subsubsection{Lemmaet ``$EqNecessary$''}

Det er nu en let sag at vise den st\ae{}rkere p\aa{}stand, at $[{\cal X}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}$ rent faktisk er lig med $[{\cal Y}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}$\footnote{Dette er lemma 4.4.a.1 i \cite{kn:hrba}.}. Som s\aa{} ofte f\o{}r ligger den interessante del af beviset i en lemmastump, der her hedder ``$HelperEqNecessary$'':

\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}

\item $
[ ZFsub
\mathbf{\ lemma\ } HelperEqNecessary
\colon 
\Pi {\cal R}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Y}
,\linebreak [0] {\cal BS}
\colon 
\newline 
\mathit{s}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{u}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{u}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\newline {\cal X}
 \mathrel{\in} {\cal BS}
\Rightarrow {\cal Y}
 \mathrel{\in} {\cal BS}
\Rightarrow EqRel({\cal R}
,{\cal BS}
)
\Rightarrow {\cal R}
({\cal X}
,{\cal Y}
)
\Rightarrow [{\cal X}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
\!\mathrel{=}\! [{\cal Y}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
]$

\item Beviset kan beskrives som f\o{}lger:
\begin{enumerate}
\item Fra $EqSubset$ ved vi, at antagelsen ${\cal R}
({\cal X}
,{\cal Y}
)$ medf\o{}rer, at $[{\cal X}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}$ er en delm\ae{}ngde af $[{\cal X}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}$. (Linie 13).
\item Da ${\cal R}$ er symmetrisk, g\ae{}lder der ogs\aa{} ${\cal R}
({\cal Y}
,{\cal X}
)$. (Linie 14--15).
\item Vi kan derfor bruge $EqSubset$ \'{e}n gang til --- vi f\aa{}r, at $[{\cal Y}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}$ er en delm\ae{}ngde af $[{\cal X}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}$. (Linie 16).
\item Ud fra punkt 1 og 3 f\o{}lger $[[{\cal X}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
\!\mathrel{=}\! [{\cal Y}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}]$, QED. (Linie 17).
\end{enumerate}

\item Her er selve beviset:

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } HelperEqNecessary
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal R}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Y}
,\linebreak [0] {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{s}
\#{\cal R}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{s}
\#{\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_C \else
\if \relax \csname lgwellbigc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigc \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{t}
\#{\cal R}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_D \else
\if \relax \csname lgwellbigd\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigd {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigd \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{t}
\#{\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_E \else
\if \relax \csname lgwellbige\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbige {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbige \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{u}
\#{\cal R}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_F \else
\if \relax \csname lgwellbigf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigf \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{u}
\#{\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal X}
 \mathrel{\in} {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal Y}
 \mathrel{\in} {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$EqRel(
$%
\leftskip=1em%
${\cal R}
,{\cal BS}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal R}
(
$%
\leftskip=1em%
${\cal X}
,{\cal Y}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}$EqSubset
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_C \else
\if \relax \csname lgwellbigc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigc \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_D \else
\if \relax \csname lgwellbigd\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigd {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigd \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_E \else
\if \relax \csname lgwellbige\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbige {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbige \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_F \else
\if \relax \csname lgwellbigf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigf \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{s}
 \mathrel{\in} [{\cal X}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
\Rightarrow 
\newline 
\mathit{s}
 \mathrel{\in} [{\cal Y}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
$:}$ERisSymmetric
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$SymRel(
$%
\leftskip=1em%
${\cal R}
,{\cal BS}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
$:}$Symmetry
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_C \else
\if \relax \csname lgwellbigc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigc \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_D \else
\if \relax \csname lgwellbigd\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigd {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigd \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal R}
(
$%
\leftskip=1em%
${\cal Y}
,{\cal X}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_i \else
\if \relax \csname lgwelli\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelli {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelli \fi 
$:}$EqSubset
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_C \else
\if \relax \csname lgwellbigc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigc \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_D \else
\if \relax \csname lgwellbigd\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigd {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigd \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_E \else
\if \relax \csname lgwellbige\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbige {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbige \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_F \else
\if \relax \csname lgwellbigf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigf \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{s}
 \mathrel{\in} [{\cal Y}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
\Rightarrow 
\newline 
\mathit{s}
 \mathrel{\in} [{\cal X}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$ToSetEquality
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_i \else
\if \relax \csname lgwelli\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelli {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelli \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$[{\cal X}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
\!\mathrel{=}\! [{\cal Y}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_j \else
\if \relax \csname lgwellj\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellj {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellj \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal R}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Y}
,\linebreak [0] {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_j \else
\if \relax \csname lgwellj\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellj {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellj \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{u}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{u}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} {\cal X}
 \mathrel{\in} {\cal BS}
\Rightarrow {\cal Y}
 \mathrel{\in} {\cal BS}
\Rightarrow EqRel({\cal R}
,{\cal BS}
)
\Rightarrow {\cal R}
({\cal X}
,{\cal Y}
)
\Rightarrow [{\cal X}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
\!\mathrel{=}\! [{\cal Y}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\item Vi mangler nu blot at skifte fra implikationer til inferenser:

\item $
[ ZFsub
\mathbf{\ lemma\ } EqNecessary
\colon 
\Pi {\cal R}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Y}
,\linebreak [0] {\cal BS}
\colon 
\newline 
\mathit{s}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{u}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{u}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\newline {\cal X}
 \mathrel{\in} {\cal BS}
\vdash {\cal Y}
 \mathrel{\in} {\cal BS}
\vdash EqRel({\cal R}
,{\cal BS}
)
\vdash {\cal R}
({\cal X}
,{\cal Y}
)
\Rightarrow [{\cal X}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
\!\mathrel{=}\! [{\cal Y}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
]$

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } EqNecessary
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal R}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Y}
,\linebreak [0] {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal R}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_C \else
\if \relax \csname lgwellbigc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigc \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{t}
\#{\cal R}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_D \else
\if \relax \csname lgwellbigd\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigd {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigd \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{t}
\#{\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_E \else
\if \relax \csname lgwellbige\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbige {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbige \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{u}
\#{\cal R}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_F \else
\if \relax \csname lgwellbigf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigf \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{u}
\#{\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal X}
 \mathrel{\in} {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal Y}
 \mathrel{\in} {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$EqRel({\cal R}
,{\cal BS}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$HelperEqNecessary
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_C \else
\if \relax \csname lgwellbigc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigc \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_D \else
\if \relax \csname lgwellbigd\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigd {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigd \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_E \else
\if \relax \csname lgwellbige\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbige {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbige \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_F \else
\if \relax \csname lgwellbigf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigf \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal X}
 \mathrel{\in} {\cal BS}
\Rightarrow {\cal Y}
 \mathrel{\in} {\cal BS}
\Rightarrow EqRel({\cal R}
,{\cal BS}
)
\Rightarrow {\cal R}
({\cal X}
,{\cal Y}
)
\Rightarrow [{\cal X}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
\!\mathrel{=}\! [{\cal Y}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$MP3
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal R}
({\cal X}
,{\cal Y}
)
\Rightarrow 
\newline [{\cal X}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
\!\mathrel{=}\! [{\cal Y}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\end{list}

\subsubsection{Lemmaet ``$NoneEqNecessary$''}

Resultatet fra dette under-underafsnit kan formuleres som f\o{}lger: ``Lad ${\cal X}$ og ${\cal Y}$ v\ae{}re to medlemmer af       
${\cal BS}$, som ikke opfylder ${\cal R}
({\cal X}
,{\cal Y}
)$. Da er de to \ae{}kvivalensklasser $[{\cal X}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}$ og $[{\cal Y}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}$ indbyrdes disjunkte.''\footnote{Dette er lemma 4.4.b.1 i \cite{kn:hrba}.}

Vi viser dette ved at vise den kontrapositive p\aa{}stand: Hvis \ae{}kvivalensklasserne har et medlem tilf\ae{}lles, s\aa{} g\ae{}lder ${\cal R}
({\cal X}
,{\cal Y}
)$. Dette varetages af lemmaet $HelperNoneEqNecessary$:

\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}


\item $
[ ZFsub
\mathbf{\ lemma\ } HelperNoneEqNecessary
\colon 
\Pi {\cal R}
,\linebreak [0] {\cal S}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Y}
,\linebreak [0] {\cal BS}
\colon 
\newline 
\mathit{s}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{u}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{u}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\newline {\cal X}
 \mathrel{\in} {\cal BS}
\Rightarrow {\cal Y}
 \mathrel{\in} {\cal BS}
\Rightarrow EqRel({\cal R}
,{\cal BS}
)
\Rightarrow 
\newline {\cal S}
 \mathrel{\in} [{\cal X}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
 \cap [{\cal Y}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
\Rightarrow {\cal R}
({\cal X}
,{\cal Y}
)
]$

\item Beviset for $HelperEqNecessary$ kan beskrives som f\o{}lger:
%
\begin{enumerate}
\item Antag at ${\cal S}$ tilh\o{}rer b\aa{}de $[{\cal X}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}$ og $[{\cal Y}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}$. Da g\ae{}lder ${\cal R}
({\cal S}
,{\cal X}
)$ (linie 17--19) og ${\cal R}
({\cal S}
,{\cal Y}
)$ (linie 20--22).
\item Da ${\cal R}$ er symmetrisk, f\o{}lger ${\cal R}
({\cal X}
,{\cal S}
)$ af ${\cal R}
({\cal S}
,{\cal X}
)$. (Linie 24).
\item Da ${\cal R}$ ogs\aa{} er transitiv, medf\o{}rer ${\cal R}
({\cal X}
,{\cal S}
)$ og ${\cal R}
({\cal S}
,{\cal Y}
)$ tilsammen ${\cal R}
({\cal X}
,{\cal Y}
)$, QED. (Linie 25).
\end{enumerate}
%
Her er selve beviset:

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } HelperNoneEqNecessary
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal R}
,\linebreak [0] {\cal S}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Y}
,\linebreak [0] {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_E \else
\if \relax \csname lgwellbige\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbige {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbige \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal R}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_F \else
\if \relax \csname lgwellbigf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigf \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_G \else
\if \relax \csname lgwellbigg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigg \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{t}
\#{\cal R}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_H \else
\if \relax \csname lgwellbigh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigh \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{t}
\#{\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_I \else
\if \relax \csname lgwellbigi\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigi {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigi \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{u}
\#{\cal R}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_J \else
\if \relax \csname lgwellbigj\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigj {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigj \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{u}
\#{\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_X \else
\if \relax \csname lgwellbigx\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigx {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigx \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal X}
 \mathrel{\in} {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_Y \else
\if \relax \csname lgwellbigy\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigy {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigy \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal Y}
 \mathrel{\in} {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$EqRel({\cal R}
,{\cal BS}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
 \mathrel{\in} [{\cal X}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
 \cap [{\cal Y}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
$:}$ERisSymmetric
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$SymRel({\cal R}
,{\cal BS}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_C \else
\if \relax \csname lgwellbigc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigc \fi 
$:}$ERisTransitive
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$TransRel({\cal R}
,{\cal BS}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$Sep2Formula
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
 \mathrel{\in} [{\cal X}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
 \mathrel{\cup} [{\cal Y}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
\mathrel{\dot{\wedge}} 
({\cal S}
 \mathrel{\in} [{\cal X}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
\mathrel{\dot{\wedge}} 
\newline {\cal S}
 \mathrel{\in} [{\cal Y}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$SecondConjunct
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
 \mathrel{\in} [{\cal X}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
\mathrel{\dot{\wedge}} 
\newline {\cal S}
 \mathrel{\in} [{\cal Y}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$FirstConjunct
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
 \mathrel{\in} [{\cal X}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$Sep2Formula
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
 \mathrel{\in} {\cal BS}
\mathrel{\dot{\wedge}} {\cal R}
({\cal S}
,{\cal X}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}$SecondConjunct
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal R}
({\cal S}
,{\cal X}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
$:}$SecondConjunct
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
 \mathrel{\in} [{\cal Y}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
$:}$Sep2Formula
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
 \mathrel{\in} {\cal BS}
\mathrel{\dot{\wedge}} {\cal R}
({\cal S}
,{\cal Y}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_i \else
\if \relax \csname lgwelli\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelli {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelli \fi 
$:}$SecondConjunct
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal R}
({\cal S}
,{\cal Y}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_S \else
\if \relax \csname lgwellbigs\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigs {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigs \fi 
$:}$FirstConjunct
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
 \mathrel{\in} {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_j \else
\if \relax \csname lgwellj\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellj {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellj \fi 
$:}$Symmetry
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_E \else
\if \relax \csname lgwellbige\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbige {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbige \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_F \else
\if \relax \csname lgwellbigf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigf \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_G \else
\if \relax \csname lgwellbigg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigg \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_H \else
\if \relax \csname lgwellbigh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigh \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_S \else
\if \relax \csname lgwellbigs\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigs {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigs \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_X \else
\if \relax \csname lgwellbigx\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigx {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigx \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal R}
({\cal X}
,{\cal S}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Transitivity
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_E \else
\if \relax \csname lgwellbige\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbige {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbige \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_F \else
\if \relax \csname lgwellbigf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigf \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_G \else
\if \relax \csname lgwellbigg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigg \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_H \else
\if \relax \csname lgwellbigh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigh \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_I \else
\if \relax \csname lgwellbigi\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigi {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigi \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_J \else
\if \relax \csname lgwellbigj\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigj {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigj \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_C \else
\if \relax \csname lgwellbigc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_X \else
\if \relax \csname lgwellbigx\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigx {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigx \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_S \else
\if \relax \csname lgwellbigs\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigs {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigs \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_Y \else
\if \relax \csname lgwellbigy\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigy {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigy \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_j \else
\if \relax \csname lgwellj\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellj {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellj \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_i \else
\if \relax \csname lgwelli\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelli {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelli \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal R}
({\cal X}
,{\cal Y}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_k \else
\if \relax \csname lgwellk\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellk {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellk \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal R}
,\linebreak [0] {\cal S}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Y}
,\linebreak [0] {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_k \else
\if \relax \csname lgwellk\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellk {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellk \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{u}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{u}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} {\cal X}
 \mathrel{\in} {\cal BS}
\Rightarrow {\cal Y}
 \mathrel{\in} {\cal BS}
\Rightarrow EqRel({\cal R}
,{\cal BS}
)
\Rightarrow 
\newline {\cal S}
 \mathrel{\in} [{\cal X}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
 \cap [{\cal Y}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
\Rightarrow {\cal R}
({\cal X}
,{\cal Y}
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\item Vi bruger nu modus tollens til at vise en kontrapositiv version af 
\\ \mbox{$HelperNoneEqNecessary$}. Dette sker i beviset for en lemmastump ved navn $Helper(2)NoneEqNecessary$:

\item $
[ ZFsub
\mathbf{\ lemma\ } Helper(2)NoneEqNecessary
\colon 
\Pi {\cal R}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Y}
,\linebreak [0] {\cal BS}
\colon 
\newline 
\mathit{s}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{u}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{u}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\newline {\cal X}
 \mathrel{\in} {\cal BS}
\Rightarrow {\cal Y}
 \mathrel{\in} {\cal BS}
\Rightarrow EqRel({\cal R}
,{\cal BS}
)
\Rightarrow 
\newline \dot{\neg}\, {\cal R}
({\cal X}
,{\cal Y}
)
\Rightarrow [{\cal X}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
 \cap [{\cal Y}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
\!\mathrel{=}\! \mathrm{\O}
]$

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } Helper(2)NoneEqNecessary
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal R}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Y}
,\linebreak [0] {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_E \else
\if \relax \csname lgwellbige\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbige {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbige \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{s}
\#{\cal R}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_F \else
\if \relax \csname lgwellbigf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigf \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{s}
\#{\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_G \else
\if \relax \csname lgwellbigg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigg \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{t}
\#{\cal R}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_H \else
\if \relax \csname lgwellbigh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigh \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{t}
\#{\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_I \else
\if \relax \csname lgwellbigi\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigi {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigi \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{u}
\#{\cal R}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_J \else
\if \relax \csname lgwellbigj\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigj {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigj \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{u}
\#{\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_X \else
\if \relax \csname lgwellbigx\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigx {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigx \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal X}
 \mathrel{\in} {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_Y \else
\if \relax \csname lgwellbigy\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigy {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigy \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal Y}
 \mathrel{\in} {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$EqRel(
$%
\leftskip=1em%
${\cal R}
,{\cal BS}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$\dot{\neg}\, {\cal R}
({\cal X}
,{\cal Y}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$HelperNoneEqNecessary
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_E \else
\if \relax \csname lgwellbige\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbige {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbige \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_F \else
\if \relax \csname lgwellbigf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigf \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_G \else
\if \relax \csname lgwellbigg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigg \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_H \else
\if \relax \csname lgwellbigh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigh \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_I \else
\if \relax \csname lgwellbigi\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigi {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigi \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_J \else
\if \relax \csname lgwellbigj\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigj {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigj \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal X}
 \mathrel{\in} {\cal BS}
\Rightarrow {\cal Y}
 \mathrel{\in} {\cal BS}
\Rightarrow EqRel({\cal R}
,{\cal BS}
)
\Rightarrow 
\newline 
\mathit{s}
 \mathrel{\in} [{\cal X}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
 \cap [{\cal Y}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
\Rightarrow {\cal R}
({\cal X}
,{\cal Y}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$MP3
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_X \else
\if \relax \csname lgwellbigx\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigx {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigx \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_Y \else
\if \relax \csname lgwellbigy\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigy {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigy \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{s}
 \mathrel{\in} [{\cal X}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
 \cap [{\cal Y}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
\Rightarrow {\cal R}
({\cal X}
,{\cal Y}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$MT
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{s}
 \mathrel{\notin} [{\cal X}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
 \cap [{\cal Y}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Unique\O{}
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$[{\cal X}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
 \cap [{\cal Y}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
\!\mathrel{=}\! \mathrm{\O}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal R}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Y}
,\linebreak [0] {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{u}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{u}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\newline {\cal X}
 \mathrel{\in} {\cal BS}
\Rightarrow {\cal Y}
 \mathrel{\in} {\cal BS}
\Rightarrow EqRel({\cal R}
,{\cal BS}
)
\Rightarrow 
\newline \dot{\neg}\, {\cal R}
({\cal X}
,{\cal Y}
)
\Rightarrow 
\newline [{\cal X}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
 \cap [{\cal Y}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
\!\mathrel{=}\! \mathrm{\O}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\item I beviset for hovedlemmaet  $NoneEqNecessary$ foretager vi den s\ae{}dvanlige konvertering fra $[
\mathsf{x}
\Rightarrow 
\mathsf{y}]$ til $[
\mathsf{x}
\vdash 
\mathsf{y}]$:

\item $
[ ZFsub
\mathbf{\ lemma\ } NoneEqNecessary
\colon 
\Pi {\cal R}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Y}
,\linebreak [0] {\cal BS}
\colon 
\newline 
\mathit{s}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{u}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{u}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\newline {\cal X}
 \mathrel{\in} {\cal BS}
\vdash {\cal Y}
 \mathrel{\in} {\cal BS}
\vdash EqRel({\cal R}
,{\cal BS}
)
\vdash 
\newline \dot{\neg}\, {\cal R}
({\cal X}
,{\cal Y}
)
\Rightarrow [{\cal X}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
 \cap [{\cal Y}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
\!\mathrel{=}\! \mathrm{\O}
]$

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } NoneEqNecessary
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal R}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Y}
,\linebreak [0] {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_E \else
\if \relax \csname lgwellbige\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbige {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbige \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal R}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_F \else
\if \relax \csname lgwellbigf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigf \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_G \else
\if \relax \csname lgwellbigg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigg \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{t}
\#{\cal R}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_H \else
\if \relax \csname lgwellbigh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigh \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{t}
\#{\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_I \else
\if \relax \csname lgwellbigi\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigi {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigi \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{u}
\#{\cal R}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_J \else
\if \relax \csname lgwellbigj\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigj {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigj \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{u}
\#{\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_X \else
\if \relax \csname lgwellbigx\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigx {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigx \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal X}
 \mathrel{\in} {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_Y \else
\if \relax \csname lgwellbigy\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigy {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigy \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal Y}
 \mathrel{\in} {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_R \else
\if \relax \csname lgwellbigr\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigr {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigr \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$EqRel({\cal R}
,{\cal BS}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$Helper(2)NoneEqNecessary
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_E \else
\if \relax \csname lgwellbige\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbige {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbige \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_F \else
\if \relax \csname lgwellbigf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigf \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_G \else
\if \relax \csname lgwellbigg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigg \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_H \else
\if \relax \csname lgwellbigh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigh \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_I \else
\if \relax \csname lgwellbigi\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigi {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigi \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_J \else
\if \relax \csname lgwellbigj\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigj {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigj \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal X}
 \mathrel{\in} {\cal BS}
\Rightarrow {\cal Y}
 \mathrel{\in} {\cal BS}
\Rightarrow EqRel({\cal R}
,{\cal BS}
)
\Rightarrow \dot{\neg}\, {\cal R}
({\cal X}
,{\cal Y}
)
\Rightarrow 
\newline [{\cal X}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
 \cap [{\cal Y}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
\!\mathrel{=}\! \mathrm{\O}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$MP3
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_X \else
\if \relax \csname lgwellbigx\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigx {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigx \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_Y \else
\if \relax \csname lgwellbigy\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigy {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigy \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_R \else
\if \relax \csname lgwellbigr\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigr {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigr \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\, {\cal R}
({\cal X}
,{\cal Y}
)
\Rightarrow 
\newline [{\cal X}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
 \cap [{\cal Y}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
\!\mathrel{=}\! \mathrm{\O}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\end{list}

\subsubsection{To \ae{}kvivalensklasser er disjunkte}

Vi kan nu bevise, at to forskellige \ae{}kvivalensklasser er disjunkte:

\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}

\item $
[ ZFsub
\mathbf{\ lemma\ } EqClassesAreDisjoint
\colon 
\Pi {\cal R}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Y}
,\linebreak [0] {\cal BS}
\colon 
\newline 
\mathit{s}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{u}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{u}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\newline {\cal X}
 \mathrel{\in} {\cal BS}
\vdash {\cal Y}
 \mathrel{\in} {\cal BS}
\vdash EqRel({\cal R}
,{\cal BS}
)
\vdash 
\newline [{\cal X}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
 \mathrel{\neq} [{\cal Y}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
\vdash [{\cal X}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
 \cap [{\cal Y}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
\!\mathrel{=}\! \mathrm{\O}
]$

\item Beviset for $EqClassesAreDisjoint$ forl\o{}ber som f\o{}lger:
%
\begin{enumerate}
\item Vi antager, at $[{\cal X}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}$ og $[{\cal Y}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}$ ikke er lig med hinanden. Fra $EqNecessary$ ved vi, at hvis der gjaldt ${\cal R}
({\cal X}
,{\cal Y}
)$, s\aa{} ville $[{\cal X}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}$ og $[{\cal Y}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}$ v\ae{}re lig med hinanden. Derfor m\aa{} der g\ae{}lde  $\dot{\neg}\, {\cal R}
({\cal X}
,{\cal Y}
)$. \mbox{(Linie 12--13)}.
\item Fra $NoneEqNecessary$ ved vi, at $\dot{\neg}\, {\cal R}
({\cal X}
,{\cal Y}
)$ medf\o{}rer, at de to \ae{}kvivalensklasser er disjunkte; QED. (Linie 14--15).
\end{enumerate}
%
Her er beviset:

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } EqClassesAreDisjoint
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal R}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Y}
,\linebreak [0] {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal R}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_C \else
\if \relax \csname lgwellbigc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigc \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{t}
\#{\cal R}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_D \else
\if \relax \csname lgwellbigd\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigd {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigd \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{t}
\#{\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_E \else
\if \relax \csname lgwellbige\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbige {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbige \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{u}
\#{\cal R}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_F \else
\if \relax \csname lgwellbigf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigf \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{u}
\#{\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal X}
 \mathrel{\in} {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal Y}
 \mathrel{\in} {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$EqRel({\cal R}
,{\cal BS}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$[{\cal X}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
 \mathrel{\neq} [{\cal Y}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$EqNecessary
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_C \else
\if \relax \csname lgwellbigc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigc \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_D \else
\if \relax \csname lgwellbigd\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigd {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigd \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_E \else
\if \relax \csname lgwellbige\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbige {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbige \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_F \else
\if \relax \csname lgwellbigf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigf \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal R}
({\cal X}
,{\cal Y}
)
\Rightarrow 
\newline [{\cal X}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
\!\mathrel{=}\! [{\cal Y}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}$MT
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\, {\cal R}
({\cal X}
,{\cal Y}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
$:}$NoneEqNecessary
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_C \else
\if \relax \csname lgwellbigc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigc \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_D \else
\if \relax \csname lgwellbigd\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigd {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigd \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_E \else
\if \relax \csname lgwellbige\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbige {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbige \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_F \else
\if \relax \csname lgwellbigf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigf \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\, {\cal R}
({\cal X}
,{\cal Y}
)
\Rightarrow 
\newline [{\cal X}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
 \cap [{\cal Y}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
\!\mathrel{=}\! \mathrm{\O}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$MP
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$[{\cal X}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
 \cap [{\cal Y}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
\!\mathrel{=}\! \mathrm{\O}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\end{list}

\subsubsection{Alle \ae{}kvivalensklasser er disjunkte} 
\label{sec:alldisjoint}

Ud fra $EqClassesAreDisjoint$ kan vi nu vise, at alle medlemmer af et \ae{}kvivalenssystem er parvis disjunkte. Da beviset afh\ae{}nger af $SameIntersection$, som igen afh\ae{}nger af $SameBinaryUnion$, bruger vi objektvariable (jvf.\ afsnit \ref{sec:binaerforen}):

\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}

\item $
[ ZFsub
\mathbf{\ lemma\ } AllDisjoint
\colon EqRel(
\mathit{r}
, \mathsf {OBS}
)
\vdash 
\newline 
\mathit{x}
 \mathrel{\in}  \mathsf {OBS}
 / 
\mathit{r}
\vdash 
\mathit{y}
 \mathrel{\in}  \mathsf {OBS}
 / 
\mathit{r}
\vdash 
\mathit{x}
 \mathrel{\neq} 
\mathit{y}
\vdash 
\mathit{x}
 \cap 
\mathit{y}
\!\mathrel{=}\! \mathrm{\O}
]$

\item Beviset forl\o{}ber som f\o{}lger:
%
\begin{enumerate}
\item Lad $
\mathit{x}$ og $
\mathit{y}$ v\ae{}re to forskellige medlemmer af et \ae{}kvivalenssystem $ \mathsf {OBS}
 / 
\mathit{r}$ (linie 2--4).
\item M\ae{}ngden $
\mathit{x}$ m\aa{} v\ae{}re en \ae{}kvivalensklasse; vi kan alts\aa{} skrive \\
$
\mathit{x}
\!\mathrel{=}\! [Ex_{1}
 \mathrel{\in}  \mathsf {OBS}
]_{
\mathit{r}
}$ for et $Ex_{1}$, som tilh\o{}rer $ \mathsf {OBS}$. (Linie 5--9).
\item Det samme g\ae{}lder for $
\mathit{y}$; her skriver vi 
$
\mathit{y}
\!\mathrel{=}\! [Ex_{2}
 \mathrel{\in}  \mathsf {OBS}
]_{
\mathit{r}
}$. (Linie 10--14).
\item Da $
\mathit{x}$ og $
\mathit{y}$ er forskellige, m\aa{} $[Ex_{1}
 \mathrel{\in}  \mathsf {OBS}
]_{
\mathit{r}
}$ og $[Ex_{2}
 \mathrel{\in}  \mathsf {OBS}
]_{
\mathit{r}
}$ ogs\aa{} v\ae{}re forskellige. (Linie 15).
\item Heraf f\o{}lger, at f\ae{}llesm\ae{}ngden mellem $[Ex_{1}
 \mathrel{\in}  \mathsf {OBS}
]_{
\mathit{r}
}$ og $[Ex_{2}
 \mathrel{\in}  \mathsf {OBS}
]_{
\mathit{r}
}$ er tom. (Linie 16).
\item Af lighedslemmaet $SameIntersection$ f\aa{}r vi \\ \mbox{$
\mathit{x}
 \cap 
\mathit{y}
\!\mathrel{=}\! [Ex_{1}
 \mathrel{\in}  \mathsf {OBS}
]_{
\mathit{r}
}
 \cap [Ex_{1}
 \mathrel{\in}  \mathsf {OBS}
]_{
\mathit{r}
}$}. (Linie 17).
\item Da $[Ex_{1}
 \mathrel{\in}  \mathsf {OBS}
]_{
\mathit{r}
}
 \cap [Ex_{1}
 \mathrel{\in}  \mathsf {OBS}
]_{
\mathit{r}
}$ er tom (punkt 5), m\aa{} $
\mathit{x}
 \cap 
\mathit{y}$   derfor ogs\aa{} v\ae{}re tom, QED. (Linie 18).
\end{enumerate}
%
Her er beviset:

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } AllDisjoint
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$EqRel(
\mathit{r}
, \mathsf {OBS}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{x}
 \mathrel{\in}  \mathsf {OBS}
 / 
\mathit{r}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{y}
 \mathrel{\in}  \mathsf {OBS}
 / 
\mathit{r}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{x}
 \mathrel{\neq} 
\mathit{y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$Sep2Formula
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{x}
 \mathrel{\in} P( \mathsf {OBS}
)
\mathrel{\dot{\wedge}} 
(Ex_{1}
 \mathrel{\in}  \mathsf {OBS}
\mathrel{\dot{\wedge}} [Ex_{1}
 \mathrel{\in}  \mathsf {OBS}
]_{
\mathit{r}
}
\!\mathrel{=}\! 
\mathit{x}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}$SecondConjunct
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Ex_{1}
 \mathrel{\in}  \mathsf {OBS}
\mathrel{\dot{\wedge}} [Ex_{1}
 \mathrel{\in}  \mathsf {OBS}
]_{
\mathit{r}
}
\!\mathrel{=}\! 
\mathit{x}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_S \else
\if \relax \csname lgwellbigs\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigs {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigs \fi 
$:}$FirstConjunct
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Ex_{1}
 \mathrel{\in}  \mathsf {OBS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
$:}$SecondConjunct
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$[Ex_{1}
 \mathrel{\in}  \mathsf {OBS}
]_{
\mathit{r}
}
\!\mathrel{=}\! 
\mathit{x}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_R \else
\if \relax \csname lgwellbigr\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigr {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigr \fi 
$:}$=\!{}Symmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{x}
\!\mathrel{=}\! [Ex_{1}
 \mathrel{\in}  \mathsf {OBS}
]_{
\mathit{r}
}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
$:}$Sep2Formula
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{y}
 \mathrel{\in} P( \mathsf {OBS}
)
\mathrel{\dot{\wedge}} 
(Ex_{2}
 \mathrel{\in}  \mathsf {OBS}
\mathrel{\dot{\wedge}} [Ex_{2}
 \mathrel{\in}  \mathsf {OBS}
]_{
\mathit{r}
}
\!\mathrel{=}\! 
\mathit{y}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_i \else
\if \relax \csname lgwelli\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelli {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelli \fi 
$:}$SecondConjunct
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Ex_{2}
 \mathrel{\in}  \mathsf {OBS}
\mathrel{\dot{\wedge}} [Ex_{2}
 \mathrel{\in}  \mathsf {OBS}
]_{
\mathit{r}
}
\!\mathrel{=}\! 
\mathit{y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_T \else
\if \relax \csname lgwellbigt\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigt {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigt \fi 
$:}$FirstConjunct
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_i \else
\if \relax \csname lgwelli\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelli {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelli \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Ex_{2}
 \mathrel{\in}  \mathsf {OBS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_j \else
\if \relax \csname lgwellj\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellj {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellj \fi 
$:}$SecondConjunct
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_i \else
\if \relax \csname lgwelli\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelli {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelli \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$[Ex_{2}
 \mathrel{\in}  \mathsf {OBS}
]_{
\mathit{r}
}
\!\mathrel{=}\! 
\mathit{y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_U \else
\if \relax \csname lgwellbigu\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigu {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigu \fi 
$:}$=\!{}Symmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_j \else
\if \relax \csname lgwellj\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellj {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellj \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{y}
\!\mathrel{=}\! [Ex_{2}
 \mathrel{\in}  \mathsf {OBS}
]_{
\mathit{r}
}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_r \else
\if \relax \csname lgwellr\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellr {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellr \fi 
$:}$TransferNotEq
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_R \else
\if \relax \csname lgwellbigr\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigr {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigr \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_U \else
\if \relax \csname lgwellbigu\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigu {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigu \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$[Ex_{1}
 \mathrel{\in}  \mathsf {OBS}
]_{
\mathit{r}
}
 \mathrel{\neq} [Ex_{2}
 \mathrel{\in}  \mathsf {OBS}
]_{
\mathit{r}
}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_s \else
\if \relax \csname lgwells\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwells {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwells \fi 
$:}$EqClassesAreDisjoint
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_S \else
\if \relax \csname lgwellbigs\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigs {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigs \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_T \else
\if \relax \csname lgwellbigt\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigt {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigt \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_r \else
\if \relax \csname lgwellr\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellr {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellr \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$[Ex_{1}
 \mathrel{\in}  \mathsf {OBS}
]_{
\mathit{r}
}
 \cap [Ex_{2}
 \mathrel{\in}  \mathsf {OBS}
]_{
\mathit{r}
}
\!\mathrel{=}\! \mathrm{\O}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_t \else
\if \relax \csname lgwellt\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellt {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellt \fi 
$:}$SameIntersection
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_R \else
\if \relax \csname lgwellbigr\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigr {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigr \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_U \else
\if \relax \csname lgwellbigu\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigu {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigu \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{x}
 \cap 
\mathit{y}
\!\mathrel{=}\! 
\newline [Ex_{1}
 \mathrel{\in}  \mathsf {OBS}
]_{
\mathit{r}
}
 \cap [Ex_{2}
 \mathrel{\in}  \mathsf {OBS}
]_{
\mathit{r}
}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$\!{}=\!{}Transitivity
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_t \else
\if \relax \csname lgwellt\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellt {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellt \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_s \else
\if \relax \csname lgwells\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwells {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwells \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{x}
 \cap 
\mathit{y}
\!\mathrel{=}\! \mathrm{\O}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\item Bem\ae{}rk at vi i dette bevis introducerer b\aa{}de $Ex_{1}$ og $Ex_{2}$ ud fra definitionen af $[ \mathsf {OBS}
 / 
\mathit{r}]$, selvom denne definition bruger eksistensvariablen $Ex_{20}$. Vi nyder her gavn af, at separationsaksiomet tillader introduktion af ubrugte eksistens-variable (som n\ae{}vnt i afsnit \ref{sec:separation}).

\end{list}

\subsubsection{Implikation i stedet for inferens} \label{sec:konvertering}

Det sidste skridt i dette afsnit er at skifte inferenserne fra $AllDisjoint$ ud med implikationer. (Dette er p\aa{}kr\ae{}vet for, at resultatet fra $AllDisjoint$  kan bruges i beviset for hovedresultatet). Vi plejer jo at bruge deduktionsreglen til d\'{e}n slags konverteringer; men der et problem. Vi kan ikke konkludere 

\display{
$[EqRel(
\mathit{r}
, \mathsf {OBS}
)
\Rightarrow 
\mathit{x}
 \mathrel{\in}  \mathsf {OBS}
 / 
\mathit{r}
\Rightarrow 
\newline 
\mathit{y}
 \mathrel{\in}  \mathsf {OBS}
 / 
\mathit{r}
\Rightarrow 
\mathit{x}
 \mathrel{\neq} 
\mathit{y}
\Rightarrow 
\mathit{x}
 \cap 
\mathit{y}
\!\mathrel{=}\! \mathrm{\O}]$}

\noindent ud fra $Ded$; vi kan kun konkludere

\verb| | 

\display{$[
\forall 
\mathit{r}
,\linebreak [0]  \mathsf {OBS}
\colon EqRel(
\mathit{r}
, \mathsf {OBS}
)
\Rightarrow 
\forall 
\mathit{r}
,\linebreak [0] 
\mathit{x}
,\linebreak [0]  \mathsf {OBS}
\colon 
\mathit{x}
 \mathrel{\in}  \mathsf {OBS}
 / 
\mathit{r}
\Rightarrow 
\newline 
\forall 
\mathit{r}
,\linebreak [0] 
\mathit{y}
,\linebreak [0]  \mathsf {OBS}
\colon 
\mathit{y}
 \mathrel{\in}  \mathsf {OBS}
 / 
\mathit{r}
\Rightarrow 
\forall 
\mathit{x}
,\linebreak [0] 
\mathit{y}
\colon 
\mathit{x}
 \mathrel{\neq} 
\mathit{y}
\Rightarrow 
\mathit{x}
 \cap 
\mathit{y}
\!\mathrel{=}\! \mathrm{\O}]$.}

\noindent Deduktionsreglen tillader alts\aa{} ikke frie objektvariable i antecedenterne. Dette er ogs\aa{} et rimeligt krav; ellers ville vi f.eks.\ kunne slutte den falske implikation $[
\mathit{s}
\!\mathrel{=}\! \mathrm{\O}
\Rightarrow 
\forall 
\mathit{s}
\colon 
\mathit{s}
\!\mathrel{=}\! \mathrm{\O}]$ ud fra lemmaet $[
\mathit{s}
\!\mathrel{=}\! \mathrm{\O}
\vdash 
\forall 
\mathit{s}
\colon 
\mathit{s}
\!\mathrel{=}\! \mathrm{\O}]$. Problemet er, at beviset for hovedresultatet kr\ae{}ver, at implikationen er renset for objektkvantorer. S\aa{} vi er alts\aa{} havnet i en blindgyde.

Vi l\o{}ser problemet ved at indf\o{}re f\o{}lgende ``snyde-aksiom'':

\verb| | 

\display{$
[ ZFsub
\mathbf{\ rule\ } CheatAllDisjoint
\colon 
\Pi {\cal R}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Y}
,\linebreak [0] {\cal BS}
\colon EqRel({\cal R}
,{\cal BS}
)
\vdash 
\newline {\cal X}
 \mathrel{\in} {\cal BS}
 / {\cal R}
\vdash {\cal Y}
 \mathrel{\in} {\cal BS}
 / {\cal R}
\vdash {\cal X}
 \mathrel{\neq} {\cal Y}
\vdash {\cal X}
 \cap {\cal Y}
\!\mathrel{=}\! \mathrm{\O}
]$}

\noindent I $CheatAllDisjoint$ formulerer vi $AllDisjoint$ ved hj\ae{}lp af metavariable. Vi kan opfatte $CheatAllDisjoint$ som d\'{e}t resultat, vi var n\aa{}et frem til, hvis vi havde k\ae{}mpet videre med de mange sidebetingelser (bortset fra at der ikke er nogen sidebetingelser i formuleringen af $CheatAllDisjoint$). Ud fra denne formulering af $AllDisjoint$ er det nemt at aflede en implikation:

\verb| | 

\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}

\item $
[ ZFsub
\mathbf{\ lemma\ } AllDisjointImply
\colon 
\Pi {\cal R}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Y}
,\linebreak [0] {\cal BS}
\colon 
\newline 
\mathit{s}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{u}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{u}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} EqRel({\cal R}
,{\cal BS}
)
\Rightarrow 
\newline {\cal X}
 \mathrel{\in} {\cal BS}
 / {\cal R}
\Rightarrow {\cal Y}
 \mathrel{\in} {\cal BS}
 / {\cal R}
\Rightarrow {\cal X}
 \mathrel{\neq} {\cal Y}
\Rightarrow {\cal X}
 \cap {\cal Y}
\!\mathrel{=}\! \mathrm{\O}
]$

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } AllDisjointImply
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal R}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Y}
,\linebreak [0] {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$EqRel(
$%
\leftskip=1em%
${\cal R}
,{\cal BS}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal X}
 \mathrel{\in} {\cal BS}
 / {\cal R}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal Y}
 \mathrel{\in} {\cal BS}
 / {\cal R}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal X}
 \mathrel{\neq} {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$CheatAllDisjoint
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal X}
 \cap {\cal Y}
\!\mathrel{=}\! \mathrm{\O}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal R}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Y}
,\linebreak [0] {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{u}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{u}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} EqRel({\cal R}
,{\cal BS}
)
\Rightarrow {\cal X}
 \mathrel{\in} {\cal BS}
 / {\cal R}
\Rightarrow {\cal Y}
 \mathrel{\in} {\cal BS}
 / {\cal R}
\Rightarrow {\cal X}
 \mathrel{\neq} {\cal Y}
\Rightarrow 
\newline {\cal X}
 \cap {\cal Y}
\!\mathrel{=}\! \mathrm{\O}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\item Vores brug af metavariable betyder, at sidebetingelserne straks vender tilbage --- dog i et t\aa{}leligt omfang. Det er lemmaet $AllDisjointImply$, som vi kommer til at bruge i beviset for $EqSysIsPartition$.

\end{list}

\subsection{\AE{}kvivalensklassernes foreningsm\ae{}ngde}
\label{sec:nummer3}

I dette underafsnit viser vi f\o{}rst, at ${\cal BS}$ er en delm\ae{}ngde af \ae{}kvivalensklassernes foreningsm\ae{}ngde $\cup 
({\cal BS}
 / {\cal R}
)$ (under-underafsnit \ref{sec:union1}), og vice versa (under-underafsnit \ref{sec:union2}). Herudfra kan vi let vise, at de to m\ae{}ngder er lig hinanden (under-underafsnit \ref{sec:union3}).

\subsubsection{Den ene halvdel} \label{sec:union1}


Lemmaet $EqClassIsSubset$ udsiger, at en \ae{}kvivalensklasse $[{\cal X}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}$ defineret p\aa{} en m\ae{}ngde ${\cal BS}$ er en delm\ae{}ngde af ${\cal BS}$:

\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}

\item $
[ ZFsub
\mathbf{\ lemma\ } EqClassIsSubset
\colon 
\Pi {\cal R}
,\linebreak [0] {\cal S}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal BS}
\colon {\cal S}
 \mathrel{\in} [{\cal X}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
\Rightarrow {\cal S}
 \mathrel{\in} {\cal BS}
]$

\item Lemmaet f\o{}lger direkte af definitionen af $[[{\cal X}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}]$ fra afsnit \ref{sec:eqclass}:

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } EqClassIsSubset
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal R}
,\linebreak [0] {\cal S}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
 \mathrel{\in} [{\cal X}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$Sep2Formula
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
 \mathrel{\in} {\cal BS}
\mathrel{\dot{\wedge}} {\cal R}
({\cal S}
,{\cal X}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$FirstConjunct
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
 \mathrel{\in} {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal R}
,\linebreak [0] {\cal S}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
 \mathrel{\in} [{\cal X}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
\Rightarrow {\cal S}
 \mathrel{\in} {\cal BS}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\item Kommet s\aa{} vidt kan vi bevise, at ${\cal BS}$ er en delm\ae{}ngde af $\cup 
({\cal BS}
 / {\cal R}
)$:

\item $
[ ZFsub
\mathbf{\ lemma\ } BSsubset
\colon 
\Pi {\cal R}
,\linebreak [0] {\cal S}
,\linebreak [0] {\cal BS}
\colon 
\newline 
\mathit{s}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{u}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{u}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} EqRel({\cal R}
,{\cal BS}
)
\Rightarrow 
\newline {\cal S}
 \mathrel{\in} {\cal BS}
\Rightarrow {\cal S}
 \mathrel{\in} \cup 
({\cal BS}
 / {\cal R}
)
]$

\item Beviset for $BSsubset$ falder i tre dele:
%
\begin{enumerate}
\item Lad ${\cal S}$ v\ae{}re et vilk\aa{}rligt medlem af ${\cal BS}$. Ud fra $AutoMember$ har vi, at ${\cal S}$ tilh\o{}rer sin egen \ae{}kvivalensklasse $[{\cal S}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}$ (linie 6--7).
\item Videre har vi, at denne \ae{}kvivalensklasse tilh\o{}rer \ae{}kvivalenssystemet \\
\mbox{${\cal BS}
 / {\cal R}$} 
(linie 8--13).
\item Ud fra punkt 1 og 2 og definitionen af ``foreningsm\ae{}ngde'' f\aa{}r vi, at ${\cal S}$ tilh\o{}rer \ae{}kvivalenssystemets foreningsm\ae{}ngde $\cup 
({\cal BS}
 / {\cal R}
)$, QED \\ \mbox{(linie 14--16)}.
\end{enumerate}
%
Her er selve beviset:

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } BSsubset
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal R}
,\linebreak [0] {\cal S}
,\linebreak [0] {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{s}
\#{\cal R}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{s}
\#{\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$EqRel(
$%
\leftskip=1em%
${\cal R}
,{\cal BS}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
 \mathrel{\in} {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$AutoMember
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
 \mathrel{\in} [{\cal S}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
$:}$EqClassIsSubset
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{s}
 \mathrel{\in} [{\cal S}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
\Rightarrow 
\mathit{s}
 \mathrel{\in} {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
$:}$SubsetInPower
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$[{\cal S}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
 \mathrel{\in} P({\cal BS}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$=\!{}Reflexivity
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$[{\cal S}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
\!\mathrel{=}\! [{\cal S}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$JoinConjuncts
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
 \mathrel{\in} {\cal BS}
\mathrel{\dot{\wedge}} 
\newline [{\cal S}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
\!\mathrel{=}\! [{\cal S}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}$ExistIntro
\mathop {\char64} Ex_{1}
\mathop {\char64} {\cal S}
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$Ex_{1}
 \mathrel{\in} {\cal BS}
\mathrel{\dot{\wedge}} 
\newline [Ex_{1}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
\!\mathrel{=}\! [{\cal S}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_j \else
\if \relax \csname lgwellj\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellj {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellj \fi 
$:}$Formula2Sep
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$[{\cal S}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
 \mathrel{\in} {\cal BS}
 / {\cal R}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_k \else
\if \relax \csname lgwellk\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellk {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellk \fi 
$:}$ExistIntro
\mathop {\char64} Ex_{10}
\mathop {\char64} 
\newline [{\cal S}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
 \mathrel{\in} Ex_{10}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_l \else
\if \relax \csname lgwelll\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelll {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelll \fi 
$:}$ExistIntro
\mathop {\char64} Ex_{10}
\mathop {\char64} 
\newline [{\cal S}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_j \else
\if \relax \csname lgwellj\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellj {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellj \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$Ex_{10}
 \mathrel{\in} {\cal BS}
 / {\cal R}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Formula2Union
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_k \else
\if \relax \csname lgwellk\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellk {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellk \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_l \else
\if \relax \csname lgwelll\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelll {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelll \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
 \mathrel{\in} \cup 
({\cal BS}
 / {\cal R}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_m \else
\if \relax \csname lgwellm\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellm {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellm \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal R}
,\linebreak [0] {\cal S}
,\linebreak [0] {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_m \else
\if \relax \csname lgwellm\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellm {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellm \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{u}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{u}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} EqRel({\cal R}
,{\cal BS}
)
\Rightarrow {\cal S}
 \mathrel{\in} {\cal BS}
\Rightarrow {\cal S}
 \mathrel{\in} \cup 
({\cal BS}
 / {\cal R}
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\end{list}

\noindent I beviset bruger vi for f\o{}rste gang slutningsreglen $ExistIntro$. Vi anvender her konstruktionen $[
\mathsf{x}
\mathop {\char64} 
\mathsf{y}]$ til at fort\ae{}lle bevischeckeren, hvordan metavariablene ${\cal X}$ og ${\cal T}$ i $ExistIntro$  skal instantieres. Dette kan bevischeckeren ikke selv finde frem til, da b\aa{}de ``${\cal X}$'' og ``${\cal T}$'' 
kun optr\ae{}der i reglens sidebetingelse. Det er med andre ord det samme problem som ved separationsaksiomet i afsnit \ref{sec:separation}, som vi her har l\o{}st p\aa{} en anden m\aa{}de.

\subsubsection{Den anden halvdel} \label{sec:union2}

Vi viser nu det modsatte resultat af  $BSsubset$:  ${\cal BS}$ er en delm\ae{}ngde af $\cup 
({\cal BS}
 / {\cal R}
)$:

\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}

\item $
[ ZFsub
\mathbf{\ lemma\ } Union(BS/R)subset
\colon 
\Pi {\cal R}
,\linebreak [0] {\cal S}
,\linebreak [0] {\cal BS}
\colon 
\newline 
\mathit{s}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{u}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{u}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} EqRel({\cal R}
,{\cal BS}
)
\Rightarrow 
\newline {\cal S}
 \mathrel{\in} \cup 
({\cal BS}
 / {\cal R}
)
\Rightarrow {\cal S}
 \mathrel{\in} {\cal BS}
]$

\item Beviset for $Union(BS/R)subset$ kan opdeles i fire dele:
%
\begin{enumerate}
\item Lad ${\cal S}$ v\ae{}re et medlem af $\cup 
({\cal BS}
 / {\cal R}
)$. $[{\cal S}]$ m\aa{} tilh\o{}re en \ae{}kvivalensklasse, som vi kalder for $Ex_{10}$ (linie 5--7).
\item $[Ex_{10}]$ tilh\o{}rer \ae{}kvivalenssystemet ${\cal BS}
 / {\cal R}$, som igen er en delm\ae{}ngde af potensm\ae{}ngden $P({\cal BS}
)$. Derfor m\aa{} $Ex_{10}$ tilh\o{}re $P({\cal BS}
)$ (linie 8--10).
\item Dette medf\o{}rer igen, at $Ex_{10}$ er en delm\ae{}ngde af ${\cal BS}$ (linie 11).
\item Ud fra punkt 1 og 3 f\aa{}r vi, at ${\cal S}$ tilh\o{}rer ${\cal BS}$, QED (linie 12--14).
\end{enumerate}
%
Her er selve beviset:

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } Union(BS/R)subset
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal R}
,\linebreak [0] {\cal S}
,\linebreak [0] {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\mathit{s}
\#{\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$EqRel(
$%
\leftskip=1em%
${\cal R}
,{\cal BS}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
 \mathrel{\in} \cup 
({\cal BS}
 / {\cal R}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$Union2Formula
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
 \mathrel{\in} Ex_{10}
\mathrel{\dot{\wedge}} Ex_{10}
 \mathrel{\in} {\cal BS}
 / {\cal R}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$FirstConjunct
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
 \mathrel{\in} Ex_{10}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$SecondConjunct
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$Ex_{10}
 \mathrel{\in} {\cal BS}
 / {\cal R}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}$Sep2Formula
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$Ex_{10}
 \mathrel{\in} P({\cal BS}
)
\mathrel{\dot{\wedge}} 
\newline 
(Ex_{1}
 \mathrel{\in} {\cal BS}
\mathrel{\dot{\wedge}} 
\newline [Ex_{1}
 \mathrel{\in} {\cal BS}
]_{{\cal R}
}
\!\mathrel{=}\! Ex_{10}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
$:}$FirstConjunct
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$Ex_{10}
 \mathrel{\in} P({\cal BS}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
$:}$(Switch)PowerIsSub
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$Ex_{10}
 \mathrel{\subseteq} {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_i \else
\if \relax \csname lgwelli\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelli {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelli \fi 
$:}$Gen
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\forall 
\mathit{s}
\colon 
(
\mathit{s}
 \mathrel{\in} Ex_{10}
\Rightarrow 
\mathit{s}
 \mathrel{\in} {\cal BS}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_j \else
\if \relax \csname lgwellj\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellj {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellj \fi 
$:}$(Switch)HelperPowerIsSub
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
$
\forall 
\mathit{s}
\colon 
(
\mathit{s}
 \mathrel{\in} Ex_{10}
\Rightarrow 
\mathit{s}
 \mathrel{\in} {\cal BS}
)
\Rightarrow {\cal S}
 \mathrel{\in} Ex_{10}
\Rightarrow {\cal S}
 \mathrel{\in} {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$MP2
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_j \else
\if \relax \csname lgwellj\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellj {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellj \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_i \else
\if \relax \csname lgwelli\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelli {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelli \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
$%
\leftskip=1em%
${\cal S}
 \mathrel{\in} {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_k \else
\if \relax \csname lgwellk\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellk {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellk \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal R}
,\linebreak [0] {\cal S}
,\linebreak [0] {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_k \else
\if \relax \csname lgwellk\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellk {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellk \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{u}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{u}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} EqRel({\cal R}
,{\cal BS}
)
\Rightarrow 
\newline {\cal S}
 \mathrel{\in} \cup 
({\cal BS}
 / {\cal R}
)
\Rightarrow {\cal S}
 \mathrel{\in} {\cal BS}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\end{list}

\subsubsection{De to halvdele s\ae{}ttes sammen} \label{sec:union3}

Nu hvor vi har vist, at ${\cal BS}$ og $\cup 
({\cal BS}
 / {\cal R}
)$ er hinandens delm\ae{}ngder, er det let at vise, at de er lig med hinanden:

\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}

\item $
[ ZFsub
\mathbf{\ lemma\ } UnionIdentity
\colon 
\Pi {\cal R}
,\linebreak [0] {\cal BS}
\colon 
\newline 
\mathit{s}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{s}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{t}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{u}
\#{\cal R}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathit{u}
\#{\cal BS}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} EqRel({\cal R}
,{\cal BS}
)
\vdash 
\newline \cup 
({\cal BS}
 / {\cal R}
)
\!\mathrel{=}\! {\cal BS}
]$

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } UnionIdentity
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal R}
,\linebreak [0] {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal R}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
\#{\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_C \else
\if \relax \csname lgwellbigc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigc \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{t}
\#{\cal R}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_D \else
\if \relax \csname lgwellbigd\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigd {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigd \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{t}
\#{\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_E \else
\if \relax \csname lgwellbige\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbige {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbige \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{u}
\#{\cal R}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_F \else
\if \relax \csname lgwellbigf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigf \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{u}
\#{\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$EqRel({\cal R}
,{\cal BS}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$BSsubset
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_C \else
\if \relax \csname lgwellbigc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigc \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_D \else
\if \relax \csname lgwellbigd\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigd {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigd \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_E \else
\if \relax \csname lgwellbige\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbige {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbige \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_F \else
\if \relax \csname lgwellbigf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigf \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$EqRel({\cal R}
,{\cal BS}
)
\Rightarrow 
\newline 
\mathit{s}
 \mathrel{\in} {\cal BS}
\Rightarrow 
\mathit{s}
 \mathrel{\in} \cup 
({\cal BS}
 / {\cal R}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$MP
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
 \mathrel{\in} {\cal BS}
\Rightarrow 
\mathit{s}
 \mathrel{\in} \cup 
({\cal BS}
 / {\cal R}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$Union(BS/R)subset
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_C \else
\if \relax \csname lgwellbigc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigc \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_D \else
\if \relax \csname lgwellbigd\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigd {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigd \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_E \else
\if \relax \csname lgwellbige\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbige {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbige \fi 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_F \else
\if \relax \csname lgwellbigf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigf \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$EqRel({\cal R}
,{\cal BS}
)
\Rightarrow 
\newline 
\mathit{s}
 \mathrel{\in} \cup 
({\cal BS}
 / {\cal R}
)
\Rightarrow 
\mathit{s}
 \mathrel{\in} {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$MP
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
 \mathrel{\in} \cup 
({\cal BS}
 / {\cal R}
)
\Rightarrow 
\mathit{s}
 \mathrel{\in} {\cal BS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$ToSetEquality
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\cup 
({\cal BS}
 / {\cal R}
)
\!\mathrel{=}\! {\cal BS}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\end{list}

\subsection{Det sidste bevis} \label{sec:sidste}

Rapportens sidste bevis vedr\o{}rer naturligvis hovedresultatet:

\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}

\item $
[ ZFsub
\mathbf{\ lemma\ } EqSysIsPartition
\colon EqRel(
\mathit{r}
, \mathsf {OBS}
)
\vdash Partition( \mathsf {OBS}
 / 
\mathit{r}
, \mathsf {OBS}
)
]$

\item I beviset sammens\ae{}tter vi resultaterne fra underafsnit \ref{sec:nummer1} (linie 2--4), underafsnit \ref{sec:nummer2} (linie 5--8) og underafsnit \ref{sec:nummer3} (linie 9):

\item $
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 ZFsub
\mathbf {\ proof\ of\ } EqSysIsPartition
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$EqRel(
\mathit{r}
, \mathsf {OBS}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$EqSysNot\O{}
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{x}
 \mathrel{\in}  \mathsf {OBS}
 / 
\mathit{r}
\Rightarrow 
\mathit{x}
 \mathrel{\neq} \mathrm{\O}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
 \mathrel{\in}  \mathsf {OBS}
 / 
\mathit{r}
\Rightarrow 
\mathit{s}
 \mathrel{\neq} \mathrm{\O}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$Gen
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\forall 
\mathit{s}
\colon 
(
\mathit{s}
 \mathrel{\in}  \mathsf {OBS}
 / 
\mathit{r}
\Rightarrow 
\mathit{s}
 \mathrel{\neq} \mathrm{\O}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$AllDisjointImply
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$EqRel(
\mathit{r}
, \mathsf {OBS}
)
\Rightarrow 
\newline 
\mathit{s}
 \mathrel{\in}  \mathsf {OBS}
 / 
\mathit{r}
\Rightarrow 
\mathit{t}
 \mathrel{\in}  \mathsf {OBS}
 / 
\mathit{r}
\Rightarrow 
\mathit{s}
 \mathrel{\neq} 
\mathit{t}
\Rightarrow 
\mathit{s}
 \cap 
\mathit{t}
\!\mathrel{=}\! \mathrm{\O}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_Z \else
\if \relax \csname lgwellbigz\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigz {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigz \fi 
$:}$MP
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\mathit{s}
 \mathrel{\in}  \mathsf {OBS}
 / 
\mathit{r}
\Rightarrow 
\mathit{t}
 \mathrel{\in}  \mathsf {OBS}
 / 
\mathit{r}
\Rightarrow 
\mathit{s}
 \mathrel{\neq} 
\mathit{t}
\Rightarrow 
\mathit{s}
 \cap 
\mathit{t}
\!\mathrel{=}\! \mathrm{\O}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$Gen
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_Z \else
\if \relax \csname lgwellbigz\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigz {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigz \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\forall 
\mathit{t}
\colon 
(
\mathit{s}
 \mathrel{\in}  \mathsf {OBS}
 / 
\mathit{r}
\Rightarrow 
\newline 
\mathit{t}
 \mathrel{\in}  \mathsf {OBS}
 / 
\mathit{r}
\Rightarrow 
\mathit{s}
 \mathrel{\neq} 
\mathit{t}
\Rightarrow 
\mathit{s}
 \cap 
\mathit{t}
\!\mathrel{=}\! \mathrm{\O}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}$Gen
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\forall 
\mathit{s}
,\linebreak [0] 
\mathit{t}
\colon 
(
\mathit{s}
 \mathrel{\in}  \mathsf {OBS}
 / 
\mathit{r}
\Rightarrow 
\newline 
\mathit{t}
 \mathrel{\in}  \mathsf {OBS}
 / 
\mathit{r}
\Rightarrow 
\mathit{s}
 \mathrel{\neq} 
\mathit{t}
\Rightarrow 
\mathit{s}
 \cap 
\mathit{t}
\!\mathrel{=}\! \mathrm{\O}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
$:}$UnionIdentity
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\cup 
( \mathsf {OBS}
 / 
\mathit{r}
)
\!\mathrel{=}\!  \mathsf {OBS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
$:}$JoinConjuncts
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
(
\forall 
\mathit{s}
\colon 
(
\mathit{s}
 \mathrel{\in}  \mathsf {OBS}
 / 
\mathit{r}
\Rightarrow 
\mathit{s}
 \mathrel{\neq} \mathrm{\O}
)
)
\mathrel{\dot{\wedge}} 
\forall 
\mathit{s}
,\linebreak [0] 
\mathit{t}
\colon 
(
\mathit{s}
 \mathrel{\in}  \mathsf {OBS}
 / 
\mathit{r}
\Rightarrow 
\newline 
\mathit{t}
 \mathrel{\in}  \mathsf {OBS}
 / 
\mathit{r}
\Rightarrow 
\newline 
\mathit{s}
 \mathrel{\neq} 
\mathit{t}
\Rightarrow 
\mathit{s}
 \cap 
\mathit{t}
\!\mathrel{=}\! \mathrm{\O}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_i \else
\if \relax \csname lgwelli\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelli {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelli \fi 
$:}$JoinConjuncts
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
(
\forall 
\mathit{s}
\colon 
(
\mathit{s}
 \mathrel{\in}  \mathsf {OBS}
 / 
\mathit{r}
\Rightarrow 
\mathit{s}
 \mathrel{\neq} \mathrm{\O}
)
)
\mathrel{\dot{\wedge}} 
(
\forall 
\mathit{s}
,\linebreak [0] 
\mathit{t}
\colon 
(
\mathit{s}
 \mathrel{\in}  \mathsf {OBS}
 / 
\mathit{r}
\Rightarrow 
\newline 
\mathit{t}
 \mathrel{\in}  \mathsf {OBS}
 / 
\mathit{r}
\Rightarrow 
\newline 
\mathit{s}
 \mathrel{\neq} 
\mathit{t}
\Rightarrow 
\mathit{s}
 \cap 
\mathit{t}
\!\mathrel{=}\! \mathrm{\O}
)
)
\mathrel{\dot{\wedge}} \cup 
( \mathsf {OBS}
 / 
\mathit{r}
)
\!\mathrel{=}\!  \mathsf {OBS}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Repetition
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_i \else
\if \relax \csname lgwelli\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelli {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelli \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Partition( \mathsf {OBS}
 / 
\mathit{r}
, \mathsf {OBS}
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

\end{list}

\noindent L\ae{}g m\ae{}rke til, at vi i linie 2 instantierer meta-variablen ${\cal S}$ i $EqSysNot\O{}$ til objektvariablen $
\mathit{x}$, selvom det egentlig er $
\mathit{s}$, vi gerne ville have fat i. Dette skyldes, at $EqSysNot\O{}$ indeholder sidebetingelsen $[
\mathit{s}
\#{\cal S}]$ (jvf.\ afsnit \ref{sec:nummer1}). Vi l\o{}ser problemet i linie 3, hvor deduktionsreglen skifter $
\mathit{x}$ ud med $
\mathit{s}$. Deduktionsreglen tillader kun s\aa{}danne variabelskift, n\aa{}r der ikke er nogen metavariable i n\ae{}rheden; det er derfor, at vi ikke tidligere har brugt reglen p\aa{} denne m\aa{}de.

\newpage
\section{Konklusion} \label{sec:kon}

Vi har ikke haft 100\% succes med at bevise hovedresultatet; der m\aa{}tte et ``snyde-aksiom'' til hj\ae{}lp. Problemet er, at systemet kr\ae{}ver et stort antal sidebetingelser for at sikre, at objekt- og metavariable undg\aa{}r hinanden. Et sp\o{}rgsm\aa{}l er, om vi kunne have undg\aa{}et dette problem ved at h\aa{}ndtere aksiomerne og definitionerne fra afsnit \ref{sec:aksiomsystem} og \ref{sec:makro} anderledes. Her er en kort diskussion af et par alternative strategier:

Det er n\ae{}ppe holdbart kun at arbejde med objektvariable. Deduktionsreglen kan gennemf\o{}re variabelskift som f.eks.\ ``$
\mathit{y}$ i stedet for $
\mathit{x}$'', men ikke instantieringer som f.eks.\ ``$\{
\mathit{y}
,
\mathit{z}
\}$ i stedet for $
\mathit{x}$''. (Jvf.\ formuleringen af $SameIntersection$ i afsnit \ref{sec:sameinter}, hvor vi m\aa{}tte instantiere objektvariablene ``p\aa{} forh\aa{}nd''). Vi s\aa{} ogs\aa{} i afsnit \ref{sec:konvertering}, at det er sv\ae{}rt at skifte fra inferens til implikation med objektvariable; deduktionsreglen kan ikke anvendes.

Det andet ekstrem er kun at arbejde med metavariable og -kvantorer. Man kunne da f.eks.\ formulere den ene halvdel af $Extensionality$ som 

\display{$[
\Pi {\cal X}
,\linebreak [0] {\cal Y}
\colon 
(
\Pi {\cal S}
\colon {\cal S}
 \mathrel{\in} {\cal X}
\mathrel{\dot{\Leftrightarrow}} {\cal S}
 \mathrel{\in} {\cal Y}
)
\vdash {\cal X}
\!\mathrel{=}\! {\cal Y}]$.} 

\noindent Logiwebs bevischecker er imidlertid ikke gearet til denne l\o{}sning, og det virker heller ikke som om, at vi p\aa{} denne m\aa{}de l\o{}ser det egentlige problem: F.eks.\ giver det noget sludder, hvis vi instantierer ``${\cal S}$''  til ``${\cal X}$'' i det ovenst\aa{}ende. Den samme kritik kan rettes mod forslaget ``lad objektkvantorerne binde  metavariable i stedet for objektvariable''.

Den bedste l\o{}sning m\aa{} derfor v\ae{}re at lade bevischeckeren f\aa{} ansvaret for at administrere sidebetingelserne, s\aa{}ledes at brugeren kun l\ae{}gger m\ae{}rke til dem, hvis han overtr\ae{}der dem. Jeg har imidlertid ikke noget overblik over, hvor kr\ae{}vende det er at implementere denne l\o{}sning.

Min anden hovedkonklusion er, at deduktionsreglen skal suppleres med A4 fra \cite{kn:mendel}, hvis eksistenskvantorer skal h\aa{}ndteres tilfredsstillende. Min l\o{}sning med eksistens-variable har fungeret i denne rapport, men det er ikke nogen sikker metode. Der er f.eks.\ intet, som forhindrer brugeren i at introducere allerede brugte eksistens-variable og dermed opn\aa{} systemets godkendelse af et ukorrekt bevis. Det er et \aa{}bent sp\o{}rgsm\aa{}l, hvor brugervenlig kombinationen ``$Ded$ og A4'' er; den interesserede l\ae{}ser skal v\ae{}re velkommen til at g\o{}re sine egne erfaringer.

\newpage

\section*{}
\addcontentsline{toc}{section}{Litteratur}
\bibliography{/net/urd/home/disk15/eriksen/.logiweb/equivalence-relations/page}

\appendix

\section{Oversigt over variabelnavne} \label{sec:variable}

\begin{itemize}
\item De fire vigtigste objektvariable er: $[
\mathit{s}]$, $[
\mathit{t}]$, $[
\mathit{u}]$ og $[ \mathsf {OBS}]$. 
\item Eksistens-variablen $[Ex_{10}]$ bruges i aksiomet 
$[UnionDef]$ fra afsnit \ref{sec:axiomzf}.
\item Ekistens-variablen $[Ex_{20}]$ bruges i definitionen
af $[\mathsf {bs}
 / 
\mathsf {r}]$ fra afsnit 
\ref{sec:eqclass}.
\item Eksistens-variablene $[Ex_{1}]$ og $[Ex_{2}]$ bruges i beviser, n\aa{}r $[Ex_{10}]$ eller $[Ex_{20}]$ ikke er p\aa{}kr\ae{}vede.
\item Pladsholder-variablen $[ph_{1}]$ bruges i definitionen af $[[
\mathsf{x}
 \mathrel{\in} \mathsf {bs}
]_{
\mathsf {r}
}]$ fra afsnit \ref{sec:eqclass}.
\item Pladsholder-variablen $[ph_{2}]$ bruges i definitionen 
af $[\mathsf {bs}
 / 
\mathsf {r}]$ fra afsnit 
\ref{sec:eqclass}.
\item Pladsholder-variablen $[ph_{3}]$ bruges i definitionen af $[
\mathsf{x}
 \cap 
\mathsf{y}]$ fra afsnit \ref{sec:makrobin}.
\item Metavariablene $[{\cal A}]$, $[{\cal B}]$ $\cdots$ $[{\cal F}]$ varierer over formler.
\item Metavariabelene $[{\cal R}]$, $[{\cal S}]$ $\cdots$ $[{\cal Z}]$ samt $[{\cal BS}]$ varierer over termer. ($[{\cal R}]$ st\aa{}r altid for en relation).
\item Metavariablen $[{\cal P}]$ varierer over pladsholder-variable.
\item Endelig kan de ``r\aa{}'' variable $[
\mathsf {a}]$, $[
\mathsf {b}]$ $\cdots$ $[
\mathsf{z}]$ samt $[\mathsf {bs}]$ variere over hvad som helst; disse variable er ikke tilknyttet nogen semantik.
\end{itemize}

\section{Det samlede aksiomsystem} \label{sec:helezf}

\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}

\item $
[ ZFsub
\mathbf{\ rule\ } MP
\colon 
\Pi {\cal A}
,\linebreak [0] {\cal B}
\colon {\cal A}
\Rightarrow {\cal B}
\vdash {\cal A}
\vdash {\cal B}
]$

\item $
[ ZFsub
\mathbf{\ rule\ } Gen
\colon 
\Pi {\cal X}
,\linebreak [0] {\cal A}
\colon {\cal A}
\vdash 
\forall {\cal X}
\colon {\cal A}
]$

\item $
[ ZFsub
\mathbf{\ rule\ } Repetition
\colon 
\Pi {\cal A}
\colon {\cal A}
\vdash {\cal A}
]$

\item $
[ ZFsub
\mathbf{\ rule\ } Neg
\colon 
\Pi {\cal A}
,\linebreak [0] {\cal B}
\colon \dot{\neg}\, {\cal B}
\Rightarrow {\cal A}
\vdash \dot{\neg}\, {\cal B}
\Rightarrow \dot{\neg}\, {\cal A}
\vdash {\cal B}
]$

\item $
[ ZFsub
\mathbf{\ rule\ } Ded
\colon 
\Pi {\cal A}
,\linebreak [0] {\cal B}
\colon 
Dedu({\cal A}
,{\cal B}
)
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} {\cal A}
\vdash {\cal B}
]$

\item $
[ ZFsub
\mathbf{\ rule\ } ExistIntro
\colon 
\Pi {\cal X}
,\linebreak [0] {\cal T}
,\linebreak [0] {\cal A}
,\linebreak [0] {\cal B}
\colon \langle {\cal A}
{\equiv} {\cal B}
| {\cal X}
{:=} {\cal T}
\rangle_{Ex} 
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} {\cal A}
\vdash {\cal B}
]$

\item $
[ ZFsub
\mathbf{\ rule\ } Extensionality
\colon 
\Pi {\cal X}
,\linebreak [0] {\cal Y}
\colon {\cal X}
\!\mathrel{=}\! {\cal Y}
\mathrel{\dot{\Leftrightarrow}} 
\forall 
\mathit{s}
\colon 
(
\mathit{s}
 \mathrel{\in} {\cal X}
\mathrel{\dot{\Leftrightarrow}} 
\mathit{s}
 \mathrel{\in} {\cal Y}
)
]$

\item $
[ ZFsub
\mathbf{\ rule\ } \O{}def
\colon 
\Pi {\cal S}
\colon \dot{\neg}\, {\cal S}
 \mathrel{\in} \mathrm{\O}
]$

\item $
[ ZFsub
\mathbf{\ rule\ } PairDef
\colon 
\Pi {\cal S}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Y}
\colon {\cal S}
 \mathrel{\in} \{{\cal X}
,{\cal Y}
\}
\mathrel{\dot{\Leftrightarrow}} {\cal S}
\!\mathrel{=}\! {\cal X}
 \mathrel{\dot{\vee}} {\cal S}
\!\mathrel{=}\! {\cal Y}
]$

\item $
[ ZFsub
\mathbf{\ rule\ } UnionDef
\colon 
\Pi {\cal S}
,\linebreak [0] {\cal X}
\colon {\cal S}
 \mathrel{\in} \cup {\cal X}
\mathrel{\dot{\Leftrightarrow}} 
({\cal S}
 \mathrel{\in} Ex_{10}
\mathrel{\dot{\wedge}} Ex_{10}
 \mathrel{\in} {\cal X}
)
]$

\item $
[ ZFsub
\mathbf{\ rule\ } PowerDef
\colon 
\Pi {\cal S}
,\linebreak [0] {\cal X}
\colon {\cal S}
 \mathrel{\in} P({\cal X}
)
\mathrel{\dot{\Leftrightarrow}} 
\forall 
\mathit{s}
\colon 
(
\mathit{s}
 \mathrel{\in} {\cal S}
\Rightarrow 
\mathit{s}
 \mathrel{\in} {\cal X}
)
]$

\item $
[ ZFsub
\mathbf{\ rule\ } SeparationDef
\colon 
\Pi {\cal A}
,\linebreak [0] {\cal B}
,\linebreak [0] {\cal P}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Z}
\colon {\cal P}
 ^{Ph}
\wedge \langle {\cal B}
{\equiv} {\cal A}
| {\cal P}
{:=} {\cal Z}
\rangle_{Ph} 
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\newline {\cal Z}
 \mathrel{\in}  \{ ph \mathrel{\in} {\cal X}
\mid {\cal A}
\}
\mathrel{\dot{\Leftrightarrow}} {\cal Z}
 \mathrel{\in} {\cal X}
\mathrel{\dot{\wedge}} {\cal B}
]$

\item $
[ ZFsub
\mathbf{\ rule\ } CheatAllDisjoint
\colon 
\Pi {\cal R}
,\linebreak [0] {\cal X}
,\linebreak [0] {\cal Y}
,\linebreak [0] {\cal BS}
\colon EqRel({\cal R}
,{\cal BS}
)
\vdash 
\newline {\cal X}
 \mathrel{\in} {\cal BS}
 / {\cal R}
\vdash {\cal Y}
 \mathrel{\in} {\cal BS}
 / {\cal R}
\vdash {\cal X}
 \mathrel{\neq} {\cal Y}
\vdash {\cal X}
 \cap {\cal Y}
\!\mathrel{=}\! \mathrm{\O}
]$

\end{list}

\newpage
\section{Deduktionsreglen} \label{sec:deduktion}

Dette bilag pr\ae{}senterer d\'{e}n version af deduktionsreglen fra \cite{kn:check}, som jeg har brugt af. Underafsnit \ref{sec:motivering} forklarer, hvorfor jeg har \ae{}ndret p\aa{} den oprindelige regel, og underafsnit \ref{sec:kode} indeholder selve den \ae{}ndrede kode (som er skrevet i L). 

\subsection{Motivering} \label{sec:motivering}

I beviset for $HelperMemberNot\O{}$ i afsnit \ref{sec:membernotempty} konkluderer vi \\
$[
\mathit{s}
\#{\cal X}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} {\cal S}
 \mathrel{\in} {\cal X}
\Rightarrow {\cal X}
\!\mathrel{=}\! \mathrm{\O}
\Rightarrow {\cal S}
 \mathrel{\in} \mathrm{\O}]$ ud fra pr\ae{}missen \\
$[
\mathit{s}
\#{\cal X}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} {\cal S}
 \mathrel{\in} {\cal X}
\vdash {\cal X}
\!\mathrel{=}\! \mathrm{\O}
\vdash {\cal S}
 \mathrel{\in} \mathrm{\O}]$ ved hj\ae{}lp af deduktionsreglen; det er et klassisk skift fra inferens til implikation. Denne slutning kan imidlertid ikke gennemf\o{}res med deduktionsreglen fra \cite{kn:check}, fordi denne regel fjerner alle sidebetingelser fra konklusionen, f\o{}r pr\ae{}mis og konklusion sammenlignes. Da pr\ae{}missen stadigv\ae{}k begynder med sidebetingelsen $[
\mathit{s}
\#{\cal X}]$, matcher pr\ae{}mis og konklusion ikke hinanden, og slutningen fra pr\ae{}mis til konklusion forkastes. Deduktionsreglen fra dette bilag tillader derimod, at pr\ae{}mis og konklusion begynder med et antal identiske sidebetingelser. Dermed kan vi uden problemer gennemf\o{}re slutninger som den ovenst\aa{}ende, hvor pr\ae{}missen indeholder sidebetingelser.

\subsection{Kode} \label{sec:kode}

Funktionen $[
Dedu(
\mathsf {p}
,
\mathsf {c}
)]$ er en kopi af $[
Ded(
\mathsf {p}
,
\mathsf {c}
)]$ fra \cite{kn:check}:

\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}

\item $
[
Dedu(
\mathsf {p}
,
\mathsf {c}
)
\mathrel {\ddot {=}}
\lambda 
\mathsf{x}
.
Dedu_0(
\lceil 
\mathsf {p}
\rceil 
,
\lceil 
\mathsf {c}
\rceil 
)
]$

%(***   AENDRING BEGYNDER   ***)

\item Jeg har \ae{}ndret funktionen $[
Ded_0(
\mathsf {p}
,
\mathsf {c}
)]$, s\aa{} den kalder $[Dedu_{s}(
Dedu_7(
\mathsf {p}
)
,
\mathsf {c}
,
\mathsf {T}
)]$ i stedet for $[
Ded_1(
Ded_7(
\mathsf {p}
)
,
\mathsf {c}
,
\mathsf {T}
)]$:

\item $
[
Dedu_0(
\mathsf {p}
,
\mathsf {c}
)
\mathrel {\dot {=}}
\mathsf {c}
!
Dedu_8(
\mathsf {p}
,
\mathsf {T}
)
\mathrel{\ddot{\wedge}} Dedu_{s}(
Dedu_7(
\mathsf {p}
)
,
\mathsf {c}
,
\mathsf {T}
)
]$

\item Funktionen $[Dedu_{s}(
\mathsf {p}
,
\mathsf {c}
,
\mathsf {s}
)]$ giver straks kontrollen videre til $[
Ded_1(
\mathsf {p}
,
\mathsf {c}
,
\mathsf {s}
)]$ --- medmindre $
\mathsf {p}$ og $
\mathsf {c}$ begynder med et antal identiske sidebetingelser. I s\aa{} fald flyttes disse sidebetingelser fra $
\mathsf {p}$ og $
\mathsf {c}$ over til listen $
\mathsf {s}$, f\o{}r kontrollen g\aa{}r videre til $[
Ded_1(
\mathsf {p}
,
\mathsf {c}
,
\mathsf {s}
)]$:

\item $
[Dedu_{s}(
\mathsf {p}
,
\mathsf {c}
,
\mathsf {s}
)
\mathrel {\dot {=}}
{\bf if} \ 
\mathsf {p}
\stackrel{r}{=} 
\lceil 
\mathsf{x}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathsf{y}
\rceil 
\ {\bf then} \ 
\newline 
\mathsf {c}
\stackrel{r}{=} 
\lceil 
\mathsf{x}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathsf{y}
\rceil 
\wedge 
\mathsf {p}
{}^{1}
\stackrel {t}{=}
\mathsf {c}
{}^{1}
\wedge Dedu_{s}(
\mathsf {p}
{}^{2}
,
\mathsf {c}
{}^{2}
,
\mathsf {c}
{}^{1}
\mathrel { : \, : }
\mathsf {s}
)
\ {\bf else} \ 
\newline 
Dedu_1(
\mathsf {p}
,
\mathsf {c}
,
\mathsf {s}
)
]$
  
%(***   AENDRING SLUTTER   ***)

\item Fra og med $[
Ded_1(
\mathsf {p}
,
\mathsf {c}
,
\mathsf {s}
)]$ er koden kopieret fra appendikset til \cite{kn:check}:

\item
$
[
Dedu_1(
\mathsf {p}
,
\mathsf {c}
,
\mathsf {s}
)
\mathrel {\dot {=}}
{\bf if} \ 
\mathsf {c}
\stackrel{r}{=} 
\lceil 
\mathsf{x}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\mathsf{y}
\rceil 
\ {\bf then} \ 
Dedu_1(
\mathsf {p}
,
\mathsf {c}
{}^{2}
,
\mathsf {c}
{}^{1}
\mathrel { : \, : }
\mathsf {s}
)
\ {\bf else} \ 
Dedu_2(
\mathsf {p}
,
\mathsf {c}
,
\mathsf {s}
)
]$

\item $
[
Dedu_2(
\mathsf {p}
,
\mathsf {c}
,
\mathsf {s}
)
\mathrel {\dot {=}}
\mathsf {s}
!
\newline 
\mathsf {p}
\stackrel{r}{=} 
\lceil 
\mathsf{x}
\vdash 
\mathsf{y}
\rceil 
\wedge 
\mathsf {c}
\stackrel{r}{=} 
\lceil 
\mathsf{x}
\Rightarrow 
\mathsf{y}
\rceil 
\left\{\protect \begin {array}{l}
Dedu_3(
\mathsf {p}
{}^{1}
,
\mathsf {c}
{}^{1}
,
\mathsf {s}
,
\mathsf {T}
)
\wedge 
Dedu_2(
\mathsf {p}
{}^{2}
,
\mathsf {c}
{}^{2}
,
\mathsf {s}
)
\\
Dedu_4(
\mathsf {p}
,
\mathsf {c}
,
\mathsf {s}
,
Dedu_6(
\mathsf {p}
,
\mathsf {c}
,
\mathsf {T}
,
\mathsf {T}
)
)
\protect \end {array}\right.
]$

\item $
[
Dedu_3(
\mathsf {p}
,
\mathsf {c}
,
\mathsf {s}
,
\mathsf {b}
)
\mathrel {\dot {=}}
{\bf if} \ 
{\neg }
\mathsf {c}
\stackrel{r}{=} 
\lceil 
\forall 
\mathsf{x}
\colon 
\mathsf{y}
\rceil 
\ {\bf then} \ 
Dedu_4(
\mathsf {p}
,
\mathsf {c}
,
\mathsf {s}
,
\mathsf {b}
)
\ {\bf else} \ 
\newline 
{\bf if} \ 
\mathsf {p}
\stackrel{r}{=} 
\lceil 
\forall 
\mathsf{x}
\colon 
\mathsf{y}
\rceil 
\wedge 
\mathsf {p}
{}^{1}
\stackrel {t}{=}
\mathsf {c}
{}^{1}
\ {\bf then} \ 
Dedu_4(
\mathsf {p}
,
\mathsf {c}
,
\mathsf {s}
,
\mathsf {b}
)
\ {\bf else} \ 
\newline 
Dedu_3(
\mathsf {p}
,
\mathsf {c}
{}^{2}
,
\mathsf {s}
,
(
\mathsf {c}
{}^{1}
\mathrel { : \, : }
\mathsf {c}
{}^{1}
)
\mathrel { : \, : }
\mathsf {b}
)
]$ 

\item $
[
Dedu_4(
\mathsf {p}
,
\mathsf {c}
,
\mathsf {s}
,
\mathsf {b}
)
\mathrel {\dot {=}}
\mathsf {s}
!
\mathsf {b}
!
\newline 
{\bf if} \ 
\mathsf {p}
\stackrel{r}{=} 
\lceil 
\mathit{x}
\rceil 
\ {\bf then} \ 
\mathbf {lookup}(
\mathsf {p}
,
\mathsf {b}
,
\mathsf {T}
)
\stackrel {t}{=}
\mathsf {c}
\ {\bf else} \ 
\newline 
{\bf if} \ 
{\neg }
\mathsf {p}
\stackrel{r}{=} 
\mathsf {c}
\ {\bf then} \ 
\mathsf {F}
\ {\bf else} \ 
\newline 
{\bf if} \ 
\mathsf {p}
\stackrel{r}{=} 
\lceil 
\forall 
\mathsf{x}
\colon 
\mathsf{y}
\rceil 
\ {\bf then} \ 
\mathsf {p}
{}^{1}
\stackrel {t}{=}
\mathsf {c}
{}^{1}
\wedge 
Dedu_4(
\mathsf {p}
{}^{2}
,
\mathsf {c}
{}^{2}
,
\mathsf {s}
,
(
\mathsf {p}
{}^{1}
\mathrel { : \, : }
\mathsf {p}
{}^{1}
)
\mathrel { : \, : }
\mathsf {b}
)
\ {\bf else} \ 
\newline 
{\bf if} \ 
{\neg }
\mathsf {p}
\stackrel{r}{=} 
\lceil {\cal X}
\rceil 
\ {\bf then} \ 
Dedu_4^*(
\mathsf {p}
{}^t
,
\mathsf {c}
{}^t
,
\mathsf {s}
,
\mathsf {b}
)
\ {\bf else} \ 
\newline 
\mathsf {p}
{}^{1}
\stackrel {t}{=}
\mathsf {c}
{}^{1}
\wedge 
Dedu_5(
\mathsf {p}
,
\mathsf {s}
,
\mathsf {b}
)
]$

\item $
[
Dedu_4^*(
\mathsf {p}
,
\mathsf {c}
,
\mathsf {s}
,
\mathsf {b}
)
\mathrel {\dot {=}}
\mathsf {c}
!
\mathsf {s}
!
\mathsf {b}
!
{\bf if} \ 
\mathsf {p}
\ {\bf then} \ 
\mathsf {T}
\ {\bf else} \ 
Dedu_4(
\mathsf {p}
{}^h
,
\mathsf {c}
{}^h
,
\mathsf {s}
,
\mathsf {b}
)
\wedge 
Dedu_4^*(
\mathsf {p}
{}^t
,
\mathsf {c}
{}^t
,
\mathsf {s}
,
\mathsf {b}
)
]$

\item $
[
Dedu_5(
\mathsf {p}
,
\mathsf {s}
,
\mathsf {b}
)
\mathrel {\dot {=}}
\mathsf {p}
!
\mathsf {s}
!
{\bf if} \ 
\mathsf {b}
\ {\bf then} \ 
\mathsf {T}
\ {\bf else} \ 
\newline 
\langle 
\lceil 
\mathsf{x}
\#
\mathsf{y}
\rceil 
{}^h
,\linebreak [0] 
\langle 
\lceil 
\lceil 
\ast 
\rceil 
\rceil 
{}^h
,\linebreak [0] 
\mathsf {b}
{}^h
{}^h
\rangle 
,\linebreak [0] 
\langle 
\lceil 
\lceil 
\mathsf{x}
\rceil 
\rceil 
{}^h
,\linebreak [0] 
\mathsf {p}
\rangle 
\rangle 
\in_t 
\mathsf {s}
\wedge 
Dedu_5(
\mathsf {p}
,
\mathsf {s}
,
\mathsf {b}
{}^t
)
]$

\item $
[
Dedu_6(
\mathsf {p}
,
\mathsf {c}
,
\mathsf {e}
,
\mathsf {b}
)
\mathrel {\dot {=}}
\mathsf {p}
!
\mathsf {c}
!
\mathsf {b}
!
\mathsf {e}
!
\newline 
{\bf if} \ 
\mathsf {p}
\stackrel{r}{=} 
\lceil 
\mathit{x}
\rceil 
\ {\bf then} \ 
\mathsf {p}
\in_t 
\mathsf {e}
\left\{\protect \begin {array}{l}
\mathsf {b}
\\
(
\mathsf {p}
\mathrel { : \, : }
\mathsf {c}
)
\mathrel { : \, : }
\mathsf {b}
\protect \end {array}\right.
\ {\bf else} \ 
\newline 
{\bf if} \ 
{\neg }
\mathsf {p}
\stackrel{r}{=} 
\mathsf {c}
\ {\bf then} \ 
\mathsf {T}
\ {\bf else} \ 
\newline 
{\bf if} \ 
\mathsf {p}
\stackrel{r}{=} 
\lceil {\cal A}
\rceil 
\ {\bf then} \ 
\mathsf {b}
\ {\bf else} \ 
\newline 
{\bf if} \ 
\mathsf {p}
\stackrel{r}{=} 
\lceil 
\forall 
\mathsf{x}
\colon 
\mathsf{y}
\rceil 
\ {\bf then} \ 
Dedu_6(
\mathsf {p}
{}^{2}
,
\mathsf {c}
{}^{2}
,
\mathsf {c}
{}^{1}
\mathrel { : \, : }
\mathsf {e}
,
\mathsf {b}
)
\ {\bf else} \ 
\newline 
Dedu_6^*(
\mathsf {p}
{}^t
,
\mathsf {c}
{}^t
,
\mathsf {e}
,
\mathsf {b}
)
]$

\item $
[
Dedu_6^*(
\mathsf {p}
,
\mathsf {c}
,
\mathsf {e}
,
\mathsf {b}
)
\mathrel {\dot {=}}
\mathsf {p}
!
\mathsf {c}
!
\mathsf {b}
!
\mathsf {e}
!
{\bf if} \ 
\mathsf {p}
\ {\bf then} \ 
\mathsf {b}
\ {\bf else} \ 
Dedu_6^*(
\mathsf {p}
{}^t
,
\mathsf {c}
{}^t
,
\mathsf {e}
,
Dedu_6(
\mathsf {p}
{}^h
,
\mathsf {c}
{}^h
,
\mathsf {e}
,
\mathsf {b}
)
)
]$

\item $
[
Dedu_7(
\mathsf {p}
)
\mathrel {\dot {=}}
\mathsf {p}
\stackrel{r}{=} 
\lceil 
\Pi 
\mathsf{x}
\colon 
\mathsf{y}
\rceil 
\left\{\protect \begin {array}{l}
Dedu_7(
\mathsf {p}
{}^{2}
)
\\
\mathsf {p}
\protect \end {array}\right.
]$

\item $
[
Dedu_8(
\mathsf {p}
,
\mathsf {b}
)
\mathrel {\dot {=}}
\newline 
{\bf if} \ 
\mathsf {p}
\stackrel{r}{=} 
\lceil 
\Pi 
\mathsf{x}
\colon 
\mathsf{y}
\rceil 
\ {\bf then} \ 
Dedu_8(
\mathsf {p}
{}^{2}
,
\mathsf {p}
{}^{1}
\mathrel { : \, : }
\mathsf {b}
)
\ {\bf else} \ 
\newline 
{\bf if} \ 
\mathsf {p}
\stackrel{r}{=} 
\lceil {\cal A}
\rceil 
\ {\bf then} \ 
\mathsf {p}
\in_t 
\mathsf {b}
\ {\bf else} \ 
Dedu_8^*(
\mathsf {p}
{}^t
,
\mathsf {b}
)
]$

\item $
[
Dedu_8^*(
\mathsf {p}
,
\mathsf {b}
)
\mathrel {\dot {=}}
\mathsf {b}
!
{\bf if} \ 
\mathsf {p}
\ {\bf then} \ 
\mathsf {T}
\ {\bf else} \ 
Dedu_8(
\mathsf {p}
{}^h
,
\mathsf {b}
)
\mathrel{\ddot{\wedge}} 
Dedu_8^*(
\mathsf {p}
{}^t
,
\mathsf {b}
)
]$

\end{list}

\section{Pyk definitioner} \label{sec:pyk}

\begin{flushleft}
$(
[(\cdots{})
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}cdots
\mbox{''}
]
\linebreak[4] 
[\texttt{Objekt-var}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}object
\mbox{-}var
\mbox{''}
]
\linebreak[4] 
[\texttt{Ex-var}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}ex
\mbox{-}var
\mbox{''}
]
\linebreak[4] 
[\texttt{Ph-var}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}ph
\mbox{-}var
\mbox{''}
]
\linebreak[4] 
[\texttt{V\ae{}rdi}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}vaerdi
\mbox{''}
]
\linebreak[4] 
[\texttt{Variabel}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}variabel
\mbox{''}
]
\linebreak[4] 
[Op(
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}op
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}op
\mbox{''}
]
\linebreak[4] 
[Op(
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}op2
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}comma
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}op2
\mbox{''}
]
\linebreak[4] 
[
\ast 
 \mathrel {\ddot {=}} 
\ast 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}define
\mbox{-}equal
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}comma
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}equal
\mbox{''}
]
\linebreak[4] 
[ContainsEmpty(
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}contains
\mbox{-}empty
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}empty
\mbox{''}
]
\linebreak[4] 
[
Dedu(
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1deduction
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}conclude
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}1deduction
\mbox{''}
]
\linebreak[4] 
[
Dedu_0(
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1deduction
\linebreak [0]\ \hskip0em plus2.0em{}zero
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}conclude
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}1deduction
\mbox{''}
]
\linebreak[4] 
[Dedu_{s}(
\ast 
,
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1deduction
\linebreak [0]\ \hskip0em plus2.0em{}side
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}conclude
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}condition
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}1deduction
\mbox{''}
]
\linebreak[4] 
[
Dedu_1(
\ast 
,
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1deduction
\linebreak [0]\ \hskip0em plus2.0em{}one
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}conclude
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}condition
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}1deduction
\mbox{''}
]
\linebreak[4] 
[
Dedu_2(
\ast 
,
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1deduction
\linebreak [0]\ \hskip0em plus2.0em{}two
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}conclude
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}condition
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}1deduction
\mbox{''}
]
\linebreak[4] 
[
Dedu_3(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1deduction
\linebreak [0]\ \hskip0em plus2.0em{}three
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}conclude
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}condition
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}bound
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}1deduction
\mbox{''}
]
\linebreak[4] 
[
Dedu_4(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1deduction
\linebreak [0]\ \hskip0em plus2.0em{}four
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}conclude
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}condition
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}bound
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}1deduction
\mbox{''}
]
\linebreak[4] 
[
Dedu_4^*(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1deduction
\linebreak [0]\ \hskip0em plus2.0em{}four
\linebreak [0]\ \hskip0em plus2.0em{}star
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}conclude
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}condition
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}bound
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}1deduction
\mbox{''}
]
\linebreak[4] 
[
Dedu_5(
\ast 
,
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1deduction
\linebreak [0]\ \hskip0em plus2.0em{}five
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}condition
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}bound
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}1deduction
\mbox{''}
]
\linebreak[4] 
[
Dedu_6(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1deduction
\linebreak [0]\ \hskip0em plus2.0em{}six
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}conclude
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}exception
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}bound
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}1deduction
\mbox{''}
]
\linebreak[4] 
[
Dedu_6^*(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1deduction
\linebreak [0]\ \hskip0em plus2.0em{}six
\linebreak [0]\ \hskip0em plus2.0em{}star
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}conclude
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}exception
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}bound
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}1deduction
\mbox{''}
]
\linebreak[4] 
[
Dedu_7(
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1deduction
\linebreak [0]\ \hskip0em plus2.0em{}seven
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}1deduction
\mbox{''}
]
\linebreak[4] 
[
Dedu_8(
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1deduction
\linebreak [0]\ \hskip0em plus2.0em{}eight
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}bound
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}1deduction
\mbox{''}
]
\linebreak[4] 
[
Dedu_8^*(
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1deduction
\linebreak [0]\ \hskip0em plus2.0em{}eight
\linebreak [0]\ \hskip0em plus2.0em{}star
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}bound
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}1deduction
\mbox{''}
]
\linebreak[4] 
[Ex_{1}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}ex1
\mbox{''}
]
\linebreak[4] 
[Ex_{2}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}ex2
\mbox{''}
]
\linebreak[4] 
[Ex_{10}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}ex10
\mbox{''}
]
\linebreak[4] 
[Ex_{20}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}ex20
\mbox{''}
]
\linebreak[4] 
[
\ast 
 _{Ex}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}existential
\linebreak [0]\ \hskip0em plus2.0em{}var
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}var
\mbox{''}
]
\linebreak[4] 
[
\ast 
 ^{Ex}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}existential
\linebreak [0]\ \hskip0em plus2.0em{}var
\mbox{''}
]
\linebreak[4] 
[\langle 
\ast 
{\equiv} 
\ast 
| 
\ast 
{:=} 
\ast 
\rangle_{Ex} 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}exist
\mbox{-}sub
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}where
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}sub
\mbox{''}
]
\linebreak[4] 
[\langle 
\ast 
{\equiv}^0 
\ast 
| 
\ast 
{:=} 
\ast 
\rangle_{Ex} 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}exist
\mbox{-}sub0
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}where
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}sub
\mbox{''}
]
\linebreak[4] 
[\langle 
\ast 
{\equiv}^1 
\ast 
| 
\ast 
{:=} 
\ast 
\rangle_{Ex} 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}exist
\mbox{-}sub1
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}where
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}sub
\mbox{''}
]
\linebreak[4] 
[\langle 
\ast 
{\equiv}^* 
\ast 
| 
\ast 
{:=} 
\ast 
\rangle_{Ex} 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}exist
\mbox{-}sub
{*}
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}where
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}sub
\mbox{''}
]
\linebreak[4] 
[ph_{1}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}placeholder
\mbox{-}var1
\mbox{''}
]
\linebreak[4] 
[ph_{2}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}placeholder
\mbox{-}var2
\mbox{''}
]
\linebreak[4] 
[ph_{3}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}placeholder
\mbox{-}var3
\mbox{''}
]
\linebreak[4] 
[
\ast 
 _{Ph} 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}placeholder
\mbox{-}var
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}var
\mbox{''}
]
\linebreak[4] 
[
\ast 
 ^{Ph}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}placeholder
\mbox{-}var
\mbox{''}
]
\linebreak[4] 
[\langle 
\ast 
{\equiv} 
\ast 
| 
\ast 
{:=} 
\ast 
\rangle_{Ph} 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}ph
\mbox{-}sub
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}where
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}sub
\mbox{''}
]
\linebreak[4] 
[\langle 
\ast 
{\equiv}^0 
\ast 
| 
\ast 
{:=} 
\ast 
\rangle_{Ph} 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}ph
\mbox{-}sub0
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}where
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}sub
\mbox{''}
]
\linebreak[4] 
[\langle 
\ast 
{\equiv}^1 
\ast 
| 
\ast 
{:=} 
\ast 
\rangle_{Ph} 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}ph
\mbox{-}sub1
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}where
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}sub
\mbox{''}
]
\linebreak[4] 
[\langle 
\ast 
{\equiv}^* 
\ast 
| 
\ast 
{:=} 
\ast 
\rangle_{Ph} 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}ph
\mbox{-}sub
{*}
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}where
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}sub
\mbox{''}
]
\linebreak[4] 
[\mathsf {bs}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}var
\linebreak [0]\ \hskip0em plus2.0em{}big
\linebreak [0]\ \hskip0em plus2.0em{}set
\mbox{''}
]
\linebreak[4] 
[ \mathsf {OBS}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}object
\linebreak [0]\ \hskip0em plus2.0em{}big
\linebreak [0]\ \hskip0em plus2.0em{}set
\mbox{''}
]
\linebreak[4] 
[{\cal BS}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}meta
\linebreak [0]\ \hskip0em plus2.0em{}big
\linebreak [0]\ \hskip0em plus2.0em{}set
\mbox{''}
]
\linebreak[4] 
[\mathrm{\O}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}zermelo
\linebreak [0]\ \hskip0em plus2.0em{}empty
\linebreak [0]\ \hskip0em plus2.0em{}set
\mbox{''}
]
\linebreak[4] 
[ZFsub
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}system
\linebreak [0]\ \hskip0em plus2.0em{}zf
\mbox{''}
]
\linebreak[4] 
[MP
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1rule
\linebreak [0]\ \hskip0em plus2.0em{}mp
\mbox{''}
]
\linebreak[4] 
[Gen
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1rule
\linebreak [0]\ \hskip0em plus2.0em{}gen
\mbox{''}
]
\linebreak[4] 
[Repetition
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1rule
\linebreak [0]\ \hskip0em plus2.0em{}repetition
\mbox{''}
]
\linebreak[4] 
[Neg
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1rule
\linebreak [0]\ \hskip0em plus2.0em{}ad
\linebreak [0]\ \hskip0em plus2.0em{}absurdum
\mbox{''}
]
\linebreak[4] 
[Ded
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1rule
\linebreak [0]\ \hskip0em plus2.0em{}deduction
\mbox{''}
]
\linebreak[4] 
[ExistIntro
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1rule
\linebreak [0]\ \hskip0em plus2.0em{}exist
\linebreak [0]\ \hskip0em plus2.0em{}intro
\mbox{''}
]
\linebreak[4] 
[Extensionality
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}axiom
\linebreak [0]\ \hskip0em plus2.0em{}extensionality
\mbox{''}
]
\linebreak[4] 
[\O{}def
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}axiom
\linebreak [0]\ \hskip0em plus2.0em{}empty
\linebreak [0]\ \hskip0em plus2.0em{}set
\mbox{''}
]
\linebreak[4] 
[PairDef
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}axiom
\linebreak [0]\ \hskip0em plus2.0em{}pair
\linebreak [0]\ \hskip0em plus2.0em{}definition
\mbox{''}
]
\linebreak[4] 
[UnionDef
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}axiom
\linebreak [0]\ \hskip0em plus2.0em{}union
\linebreak [0]\ \hskip0em plus2.0em{}definition
\mbox{''}
]
\linebreak[4] 
[PowerDef
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}axiom
\linebreak [0]\ \hskip0em plus2.0em{}power
\linebreak [0]\ \hskip0em plus2.0em{}definition
\mbox{''}
]
\linebreak[4] 
[SeparationDef
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}axiom
\linebreak [0]\ \hskip0em plus2.0em{}separation
\linebreak [0]\ \hskip0em plus2.0em{}definition
\mbox{''}
]
\linebreak[4] 
[CheatAllDisjoint
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}cheating
\linebreak [0]\ \hskip0em plus2.0em{}axiom
\linebreak [0]\ \hskip0em plus2.0em{}all
\linebreak [0]\ \hskip0em plus2.0em{}disjoint
\mbox{''}
]
\linebreak[4] 
[AddDoubleNeg
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}add
\linebreak [0]\ \hskip0em plus2.0em{}double
\linebreak [0]\ \hskip0em plus2.0em{}neg
\mbox{''}
]
\linebreak[4] 
[RemoveDoubleNeg
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}remove
\linebreak [0]\ \hskip0em plus2.0em{}double
\linebreak [0]\ \hskip0em plus2.0em{}neg
\mbox{''}
]
\linebreak[4] 
[AndCommutativity
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}and
\linebreak [0]\ \hskip0em plus2.0em{}commutativity
\mbox{''}
]
\linebreak[4] 
[AutoImply
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}auto
\linebreak [0]\ \hskip0em plus2.0em{}imply
\mbox{''}
]
\linebreak[4] 
[Contrapositive
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}contrapositive
\mbox{''}
]
\linebreak[4] 
[FirstConjunct
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}first
\linebreak [0]\ \hskip0em plus2.0em{}conjunct
\mbox{''}
]
\linebreak[4] 
[SecondConjunct
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}second
\linebreak [0]\ \hskip0em plus2.0em{}conjunct
\mbox{''}
]
\linebreak[4] 
[FromContradiction
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}from
\linebreak [0]\ \hskip0em plus2.0em{}contradiction
\mbox{''}
]
\linebreak[4] 
[FromDisjuncts
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}from
\linebreak [0]\ \hskip0em plus2.0em{}disjuncts
\mbox{''}
]
\linebreak[4] 
[IffCommutativity
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}iff
\linebreak [0]\ \hskip0em plus2.0em{}commutativity
\mbox{''}
]
\linebreak[4] 
[IffFirst
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}iff
\linebreak [0]\ \hskip0em plus2.0em{}first
\mbox{''}
]
\linebreak[4] 
[IffSecond
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}iff
\linebreak [0]\ \hskip0em plus2.0em{}second
\mbox{''}
]
\linebreak[4] 
[ImplyTransitivity
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}imply
\linebreak [0]\ \hskip0em plus2.0em{}transitivity
\mbox{''}
]
\linebreak[4] 
[JoinConjuncts
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}join
\linebreak [0]\ \hskip0em plus2.0em{}conjuncts
\mbox{''}
]
\linebreak[4] 
[MP2
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}mp2
\mbox{''}
]
\linebreak[4] 
[MP3
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}mp3
\mbox{''}
]
\linebreak[4] 
[MP4
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}mp4
\mbox{''}
]
\linebreak[4] 
[MP5
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}mp5
\mbox{''}
]
\linebreak[4] 
[MT
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}mt
\mbox{''}
]
\linebreak[4] 
[NegativeMT
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}negative
\linebreak [0]\ \hskip0em plus2.0em{}mt
\mbox{''}
]
\linebreak[4] 
[Technicality
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}technicality
\mbox{''}
]
\linebreak[4] 
[Weakening
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}weakening
\mbox{''}
]
\linebreak[4] 
[WeakenOr1
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}weaken
\linebreak [0]\ \hskip0em plus2.0em{}or
\linebreak [0]\ \hskip0em plus2.0em{}first
\mbox{''}
]
\linebreak[4] 
[WeakenOr2
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}weaken
\linebreak [0]\ \hskip0em plus2.0em{}or
\linebreak [0]\ \hskip0em plus2.0em{}second
\mbox{''}
]
\linebreak[4] 
[Formula2Pair
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}formula2pair
\mbox{''}
]
\linebreak[4] 
[Pair2Formula
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}pair2formula
\mbox{''}
]
\linebreak[4] 
[Formula2Union
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}formula2union
\mbox{''}
]
\linebreak[4] 
[Union2Formula
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}union2formula
\mbox{''}
]
\linebreak[4] 
[Formula2Sep
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}formula2separation
\mbox{''}
]
\linebreak[4] 
[Sep2Formula
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}separation2formula
\mbox{''}
]
\linebreak[4] 
[SubsetInPower
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}subset
\linebreak [0]\ \hskip0em plus2.0em{}in
\linebreak [0]\ \hskip0em plus2.0em{}power
\linebreak [0]\ \hskip0em plus2.0em{}set
\mbox{''}
]
\linebreak[4] 
[HelperPowerIsSub
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}power
\linebreak [0]\ \hskip0em plus2.0em{}set
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}subset0
\mbox{''}
]
\linebreak[4] 
[PowerIsSub
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}power
\linebreak [0]\ \hskip0em plus2.0em{}set
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}subset
\mbox{''}
]
\linebreak[4] 
[(Switch)HelperPowerIsSub
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}power
\linebreak [0]\ \hskip0em plus2.0em{}set
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}subset0
\mbox{-}switch
\mbox{''}
]
\linebreak[4] 
[(Switch)PowerIsSub
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}power
\linebreak [0]\ \hskip0em plus2.0em{}set
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}subset
\mbox{-}switch
\mbox{''}
]
\linebreak[4] 
[ToSetEquality
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}set
\linebreak [0]\ \hskip0em plus2.0em{}equality
\linebreak [0]\ \hskip0em plus2.0em{}suff
\linebreak [0]\ \hskip0em plus2.0em{}condition
\mbox{''}
]
\linebreak[4] 
[HelperToSetEquality(t)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}set
\linebreak [0]\ \hskip0em plus2.0em{}equality
\linebreak [0]\ \hskip0em plus2.0em{}suff
\linebreak [0]\ \hskip0em plus2.0em{}condition(t)0
\mbox{''}
]
\linebreak[4] 
[ToSetEquality(t)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}set
\linebreak [0]\ \hskip0em plus2.0em{}equality
\linebreak [0]\ \hskip0em plus2.0em{}suff
\linebreak [0]\ \hskip0em plus2.0em{}condition(t)
\mbox{''}
]
\linebreak[4] 
[HelperFromSetEquality
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}set
\linebreak [0]\ \hskip0em plus2.0em{}equality
\linebreak [0]\ \hskip0em plus2.0em{}skip
\linebreak [0]\ \hskip0em plus2.0em{}quantifier
\mbox{''}
]
\linebreak[4] 
[FromSetEquality
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}set
\linebreak [0]\ \hskip0em plus2.0em{}equality
\linebreak [0]\ \hskip0em plus2.0em{}nec
\linebreak [0]\ \hskip0em plus2.0em{}condition
\mbox{''}
]
\linebreak[4] 
[HelperReflexivity
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}reflexivity0
\mbox{''}
]
\linebreak[4] 
[Reflexivity
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}reflexivity
\mbox{''}
]
\linebreak[4] 
[HelperSymmetry
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}symmetry0
\mbox{''}
]
\linebreak[4] 
[Symmetry
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}symmetry
\mbox{''}
]
\linebreak[4] 
[HelperTransitivity
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}transitivity0
\mbox{''}
]
\linebreak[4] 
[Transitivity
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}transitivity
\mbox{''}
]
\linebreak[4] 
[ERisReflexive
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}er
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}reflexive
\mbox{''}
]
\linebreak[4] 
[ERisSymmetric
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}er
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}symmetric
\mbox{''}
]
\linebreak[4] 
[ERisTransitive
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}er
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}transitive
\mbox{''}
]
\linebreak[4] 
[\O{}isSubset
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}empty
\linebreak [0]\ \hskip0em plus2.0em{}set
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}subset
\mbox{''}
]
\linebreak[4] 
[HelperMemberNot\O{}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}member
\linebreak [0]\ \hskip0em plus2.0em{}not
\linebreak [0]\ \hskip0em plus2.0em{}empty0
\mbox{''}
]
\linebreak[4] 
[MemberNot\O{}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}member
\linebreak [0]\ \hskip0em plus2.0em{}not
\linebreak [0]\ \hskip0em plus2.0em{}empty
\mbox{''}
]
\linebreak[4] 
[HelperUnique\O{}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}unique
\linebreak [0]\ \hskip0em plus2.0em{}empty
\linebreak [0]\ \hskip0em plus2.0em{}set0
\mbox{''}
]
\linebreak[4] 
[Unique\O{}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}unique
\linebreak [0]\ \hskip0em plus2.0em{}empty
\linebreak [0]\ \hskip0em plus2.0em{}set
\mbox{''}
]
\linebreak[4] 
[=\!{}Reflexivity
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}
{=}reflexivity
\mbox{''}
]
\linebreak[4] 
[=\!{}Symmetry
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}
{=}symmetry
\mbox{''}
]
\linebreak[4] 
[Helper\!{}=\!{}Transitivity
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}
{=}transitivity0
\mbox{''}
]
\linebreak[4] 
[\!{}=\!{}Transitivity
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}
{=}transitivity
\mbox{''}
]
\linebreak[4] 
[HelperTransferNotEq
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}transfer
\linebreak [0]\ \hskip0em plus2.0em{}
\char126 is0
\mbox{''}
]
\linebreak[4] 
[TransferNotEq
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}transfer
\linebreak [0]\ \hskip0em plus2.0em{}
\char126 is
\mbox{''}
]
\linebreak[4] 
[HelperPairSubset
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}pair
\linebreak [0]\ \hskip0em plus2.0em{}subset0
\mbox{''}
]
\linebreak[4] 
[Helper(2)PairSubset
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}pair
\linebreak [0]\ \hskip0em plus2.0em{}subset1
\mbox{''}
]
\linebreak[4] 
[PairSubset
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}pair
\linebreak [0]\ \hskip0em plus2.0em{}subset
\mbox{''}
]
\linebreak[4] 
[SamePair
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}same
\linebreak [0]\ \hskip0em plus2.0em{}pair
\mbox{''}
]
\linebreak[4] 
[SameSingleton
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}same
\linebreak [0]\ \hskip0em plus2.0em{}singleton
\mbox{''}
]
\linebreak[4] 
[UnionSubset
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}union
\linebreak [0]\ \hskip0em plus2.0em{}subset
\mbox{''}
]
\linebreak[4] 
[SameUnion
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}same
\linebreak [0]\ \hskip0em plus2.0em{}union
\mbox{''}
]
\linebreak[4] 
[SeparationSubset
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}separation
\linebreak [0]\ \hskip0em plus2.0em{}subset
\mbox{''}
]
\linebreak[4] 
[SameSeparation
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}same
\linebreak [0]\ \hskip0em plus2.0em{}separation
\mbox{''}
]
\linebreak[4] 
[SameBinaryUnion
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}same
\linebreak [0]\ \hskip0em plus2.0em{}binary
\linebreak [0]\ \hskip0em plus2.0em{}union
\mbox{''}
]
\linebreak[4] 
[IntersectionSubset
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}intersection
\linebreak [0]\ \hskip0em plus2.0em{}subset
\mbox{''}
]
\linebreak[4] 
[SameIntersection
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}same
\linebreak [0]\ \hskip0em plus2.0em{}intersection
\mbox{''}
]
\linebreak[4] 
[AutoMember
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}auto
\linebreak [0]\ \hskip0em plus2.0em{}member
\mbox{''}
]
\linebreak[4] 
[HelperEqSysNot\O{}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}eq
\mbox{-}system
\linebreak [0]\ \hskip0em plus2.0em{}not
\linebreak [0]\ \hskip0em plus2.0em{}empty0
\mbox{''}
]
\linebreak[4] 
[EqSysNot\O{}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}eq
\mbox{-}system
\linebreak [0]\ \hskip0em plus2.0em{}not
\linebreak [0]\ \hskip0em plus2.0em{}empty
\mbox{''}
]
\linebreak[4] 
[HelperEqSubset
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}eq
\linebreak [0]\ \hskip0em plus2.0em{}subset0
\mbox{''}
]
\linebreak[4] 
[EqSubset
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}eq
\linebreak [0]\ \hskip0em plus2.0em{}subset
\mbox{''}
]
\linebreak[4] 
[HelperEqNecessary
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}equivalence
\linebreak [0]\ \hskip0em plus2.0em{}nec
\linebreak [0]\ \hskip0em plus2.0em{}condition0
\mbox{''}
]
\linebreak[4] 
[EqNecessary
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}equivalence
\linebreak [0]\ \hskip0em plus2.0em{}nec
\linebreak [0]\ \hskip0em plus2.0em{}condition
\mbox{''}
]
\linebreak[4] 
[HelperNoneEqNecessary
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}none
\mbox{-}equivalence
\linebreak [0]\ \hskip0em plus2.0em{}nec
\linebreak [0]\ \hskip0em plus2.0em{}condition0
\mbox{''}
]
\linebreak[4] 
[Helper(2)NoneEqNecessary
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}none
\mbox{-}equivalence
\linebreak [0]\ \hskip0em plus2.0em{}nec
\linebreak [0]\ \hskip0em plus2.0em{}condition1
\mbox{''}
]
\linebreak[4] 
[NoneEqNecessary
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}none
\mbox{-}equivalence
\linebreak [0]\ \hskip0em plus2.0em{}nec
\linebreak [0]\ \hskip0em plus2.0em{}condition
\mbox{''}
]
\linebreak[4] 
[EqClassIsSubset
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}equivalence
\linebreak [0]\ \hskip0em plus2.0em{}class
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}subset
\mbox{''}
]
\linebreak[4] 
[EqClassesAreDisjoint
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}equivalence
\linebreak [0]\ \hskip0em plus2.0em{}classes
\linebreak [0]\ \hskip0em plus2.0em{}are
\linebreak [0]\ \hskip0em plus2.0em{}disjoint
\mbox{''}
]
\linebreak[4] 
[AllDisjoint
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}all
\linebreak [0]\ \hskip0em plus2.0em{}disjoint
\mbox{''}
]
\linebreak[4] 
[AllDisjointImply
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}all
\linebreak [0]\ \hskip0em plus2.0em{}disjoint
\mbox{-}imply
\mbox{''}
]
\linebreak[4] 
[BSsubset
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}bs
\linebreak [0]\ \hskip0em plus2.0em{}subset
\linebreak [0]\ \hskip0em plus2.0em{}union(bs/r)
\mbox{''}
]
\linebreak[4] 
[Union(BS/R)subset
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}union(bs/r)
\linebreak [0]\ \hskip0em plus2.0em{}subset
\linebreak [0]\ \hskip0em plus2.0em{}bs
\mbox{''}
]
\linebreak[4] 
[UnionIdentity
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}union(bs/r)
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}bs
\mbox{''}
]
\linebreak[4] 
[EqSysIsPartition
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}theorem
\linebreak [0]\ \hskip0em plus2.0em{}eq
\mbox{-}system
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}partition
\mbox{''}
]
\linebreak[4] 
[
\ast 
 / 
\ast 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}eq
\mbox{-}system
\linebreak [0]\ \hskip0em plus2.0em{}of
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}modulo
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]
\linebreak[4] 
[
\ast 
 \cap 
\ast 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}intersection
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}comma
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}intersection
\mbox{''}
]
\linebreak[4] 
[\cup 
\ast 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}union
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}union
\mbox{''}
]
\linebreak[4] 
[
\ast 
 \mathrel{\cup} 
\ast 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}binary
\mbox{-}union
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}comma
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}union
\mbox{''}
]
\linebreak[4] 
[P(
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}power
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}power
\mbox{''}
]
\linebreak[4] 
[\{
\ast 
\}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}zermelo
\linebreak [0]\ \hskip0em plus2.0em{}singleton
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}singleton
\mbox{''}
]
\linebreak[4] 
[\{
\ast 
,
\ast 
\}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}zermelo
\linebreak [0]\ \hskip0em plus2.0em{}pair
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}comma
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}pair
\mbox{''}
]
\linebreak[4] 
[\langle 
\ast 
,
\ast 
\rangle
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}zermelo
\linebreak [0]\ \hskip0em plus2.0em{}ordered
\linebreak [0]\ \hskip0em plus2.0em{}pair
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}comma
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}pair
\mbox{''}
]
\linebreak[4] 
[
\ast 
 \mathrel{\in} 
\ast 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}zermelo
\linebreak [0]\ \hskip0em plus2.0em{}in
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]
\linebreak[4] 
[
\ast 
(
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}related
\linebreak [0]\ \hskip0em plus2.0em{}to
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}under
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]
\linebreak[4] 
[ReflRel(
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}reflexive
\linebreak [0]\ \hskip0em plus2.0em{}relation
\linebreak [0]\ \hskip0em plus2.0em{}in
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]
\linebreak[4] 
[SymRel(
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}symmetric
\linebreak [0]\ \hskip0em plus2.0em{}relation
\linebreak [0]\ \hskip0em plus2.0em{}in
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]
\linebreak[4] 
[TransRel(
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}transitive
\linebreak [0]\ \hskip0em plus2.0em{}relation
\linebreak [0]\ \hskip0em plus2.0em{}in
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]
\linebreak[4] 
[EqRel(
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}equivalence
\linebreak [0]\ \hskip0em plus2.0em{}relation
\linebreak [0]\ \hskip0em plus2.0em{}in
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]
\linebreak[4] 
[[
\ast 
 \mathrel{\in} 
\ast 
]_{
\ast 
}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}equivalence
\linebreak [0]\ \hskip0em plus2.0em{}class
\linebreak [0]\ \hskip0em plus2.0em{}of
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}in
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}modulo
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]
\linebreak[4] 
[Partition(
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}partition
\linebreak [0]\ \hskip0em plus2.0em{}of
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]
\linebreak[4] 
[
\ast 
\!\mathrel{=}\! 
\ast 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}zermelo
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]
\linebreak[4] 
[
\ast 
 \mathrel{\subseteq} 
\ast 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}subset
\linebreak [0]\ \hskip0em plus2.0em{}of
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]
\linebreak[4] 
[\dot{\neg}\, 
\ast 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}not0
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]
\linebreak[4] 
[
\ast 
 \mathrel{\notin} 
\ast 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}zermelo
\linebreak [0]\ \hskip0em plus2.0em{}
\char126 in
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]
\linebreak[4] 
[
\ast 
 \mathrel{\neq} 
\ast 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}zermelo
\linebreak [0]\ \hskip0em plus2.0em{}
\char126 is
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]
\linebreak[4] 
[
\ast 
\mathrel{\dot{\wedge}} 
\ast 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}and0
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]
\linebreak[4] 
[
\ast 
 \mathrel{\dot{\vee}} 
\ast 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}or0
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]
\linebreak[4] 
[
\ast 
\mathrel{\dot{\Leftrightarrow}} 
\ast 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}iff
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]
\linebreak[4] 
[ \{ ph \mathrel{\in} 
\ast 
\mid 
\ast 
\}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}the
\linebreak [0]\ \hskip0em plus2.0em{}set
\linebreak [0]\ \hskip0em plus2.0em{}of
\linebreak [0]\ \hskip0em plus2.0em{}ph
\linebreak [0]\ \hskip0em plus2.0em{}in
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}such
\linebreak [0]\ \hskip0em plus2.0em{}that
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}set
\mbox{''}
]
\linebreak[4] 
[EquivalenceRelations
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}equivalence
\mbox{-}relations
\mbox{''}
]
\linebreak[4] 
)^{\mathbf {p}}$
\end{flushleft}

\newpage
\section{Prioritetstabel} \label{sec:prio}

Den nedenst\aa{}ende tabel indeholder alle de formelle konstruktioner, som er til r\aa{}dighed i dette dokument. Konstruktionerne er grupperet efter prioritet; de f\o{}rstn\ae{}vnte grupper har st\o{}rst prioritet. Hver gruppe er markeret med et af de to ord ``$\mathbf{Preassociative}$'' eller ``$\mathbf{Postassociative}$'', der angiver, om konstruktionerne er venstre- eller h\o{}jreassociative. 


\begin {flushleft}$
\mathbf{Priority\ table} 
\newline \mathbf {Preassociative} \newline 
[EquivalenceRelations
] , \linebreak [0] 
[
\mbox{base}
] , \linebreak [0] 
[
\mbox{bracket $
\ast $ end bracket}
] , \linebreak [0] 
[
\mbox{big bracket $
\ast $ end bracket}
] , \linebreak [0] 
[
\ \$
\ast \$\linebreak[0]\ 
] , \linebreak [0] 
[
\mathbf{flush\ left\ }[ 
\ast 
]
] , \linebreak [0] 
[
\mathsf{x}
] , \linebreak [0] 
[
\mathsf{y}
] , \linebreak [0] 
[
\mathsf{z}
] , \linebreak [0] 
[
[
\ast 
\bowtie
\ast 
]
] , \linebreak [0] 
[
[
\ast 
\stackrel{
\ast 
}{\rightarrow}
\ast 
]
] , \linebreak [0] 
[
\mathrm{pyk}
] , \linebreak [0] 
[
\mathrm{tex}
] , \linebreak [0] 
[
\mathrm{name}
] , \linebreak [0] 
[
\mathrm{prio}
] , \linebreak [0] 
[
\ast 
] , \linebreak [0] 
[
\mathsf {T}
] , \linebreak [0] 
[
\mathrm {if}(
\ast 
,\linebreak [0]
\ast 
,\linebreak [0]
\ast 
)
] , \linebreak [0] 
[
[
\ast 
\stackrel {
\ast 
}{\Rightarrow }
\ast 
]
] , \linebreak [0] 
[
\mathrm {val}
] , \linebreak [0] 
[
\mathrm {claim}
] , \linebreak [0] 
[
\bot 
] , \linebreak [0] 
[
f(
\ast 
)
] , \linebreak [0] 
[
(
\ast 
){}^{I} 
] , \linebreak [0] 
[
\mathsf {F}
] , \linebreak [0] 
[
\underline {0}
] , \linebreak [0] 
[
\underline {1}
] , \linebreak [0] 
[
\underline {2}
] , \linebreak [0] 
[
\underline {3}
] , \linebreak [0] 
[
\underline {4}
] , \linebreak [0] 
[
\underline {5}
] , \linebreak [0] 
[
\underline {6}
] , \linebreak [0] 
[
\underline {7}
] , \linebreak [0] 
[
\underline {8}
] , \linebreak [0] 
[
\underline {9}
] , \linebreak [0] 
[
0
] , \linebreak [0] 
[
1
] , \linebreak [0] 
[
2
] , \linebreak [0] 
[
3
] , \linebreak [0] 
[
4
] , \linebreak [0] 
[
5
] , \linebreak [0] 
[
6
] , \linebreak [0] 
[
7
] , \linebreak [0] 
[
8
] , \linebreak [0] 
[
9
] , \linebreak [0] 
[
\mathsf {a}
] , \linebreak [0] 
[
\mathsf {b}
] , \linebreak [0] 
[
\mathsf {c}
] , \linebreak [0] 
[
\mathsf {d}
] , \linebreak [0] 
[
\mathsf {e}
] , \linebreak [0] 
[
\mathsf {f}
] , \linebreak [0] 
[
\mathsf {g}
] , \linebreak [0] 
[
\mathsf {h}
] , \linebreak [0] 
[
\mathsf {i}
] , \linebreak [0] 
[
\mathsf {j}
] , \linebreak [0] 
[
\mathsf {k}
] , \linebreak [0] 
[
\mathsf {l}
] , \linebreak [0] 
[
\mathsf {m}
] , \linebreak [0] 
[
\mathsf {n}
] , \linebreak [0] 
[
\mathsf {o}
] , \linebreak [0] 
[
\mathsf {p}
] , \linebreak [0] 
[
\mathsf {q}
] , \linebreak [0] 
[
\mathsf {r}
] , \linebreak [0] 
[
\mathsf {s}
] , \linebreak [0] 
[
\mathsf {t}
] , \linebreak [0] 
[
\mathsf {u}
] , \linebreak [0] 
[
\mathsf {v}
] , \linebreak [0] 
[
\mathsf {w}
] , \linebreak [0] 
[
(
\ast 
)^M
] , \linebreak [0] 
[
\mathrm {If}(
\ast 
,\linebreak [0]
\ast 
,\linebreak [0]
\ast 
)
] , \linebreak [0] 
[\mathrm {array}\{
\ast 
\}
\ast 
\mathrm {end\ array}
] , \linebreak [0] 
[
l
] , \linebreak [0] 
[
c
] , \linebreak [0] 
[
r
] , \linebreak [0] 
[
\mathrm {empty}
] , \linebreak [0] 
[
\langle 
\ast 
\,{|}
\ast 
{:=}\,
\ast 
\rangle 
] , \linebreak [0] 
[
{\cal M}(
\ast 
)
] , \linebreak [0] 
[
\tilde {{\cal U}}(
\ast 
)
] , \linebreak [0] 
[
{\cal U}(
\ast 
)
] , \linebreak [0] 
[
{\cal U}^M(
\ast 
)
] , \linebreak [0] 
[
\mathbf {apply}(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\mathbf {apply}_1(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
identifier( 
\ast 
)
] , \linebreak [0] 
[
identifier_{1}( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
array\mbox{-}\linebreak[0]plus( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
array\mbox{-}\linebreak[0]remove( 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
array\mbox{-}\linebreak[0]put( 
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
array\mbox{-}\linebreak[0]add( 
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
bit( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
bit_1( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
rack
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {vector}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {bibliography}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {dictionary}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {body}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {codex}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {expansion}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {code}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {cache}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {diagnose}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {pyk}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {tex}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {texname}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {value}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {message}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {macro}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {definition}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {unpack}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {claim}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {priority}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {lambda}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {apply}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {true}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {if}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {quote}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {proclaim}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {define}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {introduce}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {hide}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {pre}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {post}\mbox {\tt \char34}
] , \linebreak [0] 
[
{\cal E}(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
{\cal E}_2(
\ast 
,
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
{\cal E}_3(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
{\cal E}_4(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\mathbf {lookup}(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\mathbf {abstract}(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\lceil 
\ast 
\rceil 
] , \linebreak [0] 
[
{\cal M}(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
{\cal M}_2(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
{\cal M}^*(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\mathrm {macro}
] , \linebreak [0] 
[
s_0
] , \linebreak [0] 
[
\mathbf {zip}(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\mathbf {assoc}_1(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[(
\ast 
)^{\mathbf {p}}
] , \linebreak [0] 
[
\mbox{self}
] , \linebreak [0] 
[
[
\ast 
\mathrel {\ddot {=}}
\ast 
]
] , \linebreak [0] 
[
[
\ast 
\mathrel {\dot {=}}
\ast 
]
] , \linebreak [0] 
[
[
\ast 
\mathrel {\acute {=}}
\ast 
]
] , \linebreak [0] 
[
[
\ast 
\stackrel {\mathrm {pyk}}{=} 
\ast 
]
] , \linebreak [0] 
[
[
\ast 
\stackrel {\mathrm {tex}}{=} 
\ast 
]
] , \linebreak [0] 
[
[
\ast 
\stackrel {\mathrm {name}}{=} 
\ast 
]
] , \linebreak [0] 
[
\mathbf{Priority\ table} [ 
\ast 
]
] , \linebreak [0] 
[
\tilde {{\cal M}}_1
] , \linebreak [0] 
[
\tilde {{\cal M}}_2(
\ast 
)
] , \linebreak [0] 
[
\tilde {{\cal M}}_3(
\ast 
)
] , \linebreak [0] 
[
\tilde {{\cal M}}_4(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\tilde {{\cal M}}(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\tilde {{\cal Q}}(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\tilde {{\cal Q}}_2(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\tilde {{\cal Q}}_3(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\tilde {{\cal Q}}^*(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
(
\ast 
)
] , \linebreak [0] 
[
\left( 
\ast 
\right) 
] , \linebreak [0] 
[
display(
\ast 
)
] , \linebreak [0] 
[
statement(
\ast 
)
] , \linebreak [0] 
[
[
\ast 
]^{\cdot}
] , \linebreak [0] 
[
[
\ast 
]^-
] , \linebreak [0] 
[
\mathbf{aspect}( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
\mathbf{aspect}( 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
\langle 
\ast 
\rangle 
] , \linebreak [0] 
[
\mathbf {tuple}_1( 
\ast 
) 
] , \linebreak [0] 
[
\mathbf {tuple}_2( 
\ast 
) 
] , \linebreak [0] 
[
let_2( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
let_1( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
[
\ast 
\stackrel {claim}{=}
\ast 
]
] , \linebreak [0] 
[
\mbox{checker}
] , \linebreak [0] 
[
\mathbf{check}( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
\mathbf{check}_2( 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
\mathbf{check}_3( 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
\mathbf{check}^*( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
\mathbf{check}^*_2( 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
\relax [ 
\ast 
\relax ]^{\cdot} 
] , \linebreak [0] 
[
\relax [ 
\ast 
\relax ]^{-} 
] , \linebreak [0] 
[
\relax [ 
\ast 
\relax ]^{\circ} 
] , \linebreak [0] 
[
msg
] , \linebreak [0] 
[
[
\ast 
\stackrel {msg}{=} 
\ast 
]
] , \linebreak [0] 
[
{<}stmt{>}
] , \linebreak [0] 
[
stmt
] , \linebreak [0] 
[
[
\ast 
\stackrel {stmt}{=}
\ast 
]
] , \linebreak [0] 
[
HeadNil'
] , \linebreak [0] 
[
HeadPair'
] , \linebreak [0] 
[
Transitivity'
] , \linebreak [0] 
[
{\makebox [0mm][l]{$\bot $}\,{\bot }}
] , \linebreak [0] 
[
Contra'
] , \linebreak [0] 
[
T'_{E}
] , \linebreak [0] 
[
L_{1}
] , \linebreak [0] 
[\underline{
\ast }
] , \linebreak [0] 
[{\cal A}
] , \linebreak [0] 
[{\cal B}
] , \linebreak [0] 
[{\cal C}
] , \linebreak [0] 
[{\cal D}
] , \linebreak [0] 
[{\cal E}
] , \linebreak [0] 
[{\cal F}
] , \linebreak [0] 
[{\cal G}
] , \linebreak [0] 
[{\cal H}
] , \linebreak [0] 
[{\cal I}
] , \linebreak [0] 
[{\cal J}
] , \linebreak [0] 
[{\cal K}
] , \linebreak [0] 
[{\cal L}
] , \linebreak [0] 
[{\cal M}
] , \linebreak [0] 
[{\cal N}
] , \linebreak [0] 
[{\cal O}
] , \linebreak [0] 
[{\cal P}
] , \linebreak [0] 
[{\cal Q}
] , \linebreak [0] 
[{\cal R}
] , \linebreak [0] 
[{\cal S}
] , \linebreak [0] 
[{\cal T}
] , \linebreak [0] 
[{\cal U}
] , \linebreak [0] 
[{\cal V}
] , \linebreak [0] 
[{\cal W}
] , \linebreak [0] 
[{\cal X}
] , \linebreak [0] 
[{\cal Y}
] , \linebreak [0] 
[{\cal Z}
] , \linebreak [0] 
[
\langle 
\ast 
\,{\protect\vert}
\ast 
{:=}\,
\ast 
\rangle 
] , \linebreak [0] 
[
\langle ^ { \ast } 
\ast 
\,{\protect\vert}
\ast 
{:=}\,
\ast 
\rangle 
] , \linebreak [0] 
[
\emptyset 
] , \linebreak [0] 
[
Remainder
] , \linebreak [0] 
[
( 
\ast 
)^{\bf v}
] , \linebreak [0] 
[
intro(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
intro(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
error(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
error_{2}(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
proof( 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
proof_{2}( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}^{I}(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}^{\rhd}(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{1}^{\rhd}(
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}^{E}(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{1}^{E}(
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}^{+}(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{1}^{+}(
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}^{-}(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{1}^{-}(
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}^{\ast}(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{1}^{\ast}(
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{2}^{\ast}(
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}^{\char64}(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{1}^{\char64}(
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}^{\vdash}(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{1}^{\vdash}(
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}^{\makebox [0mm][l]{\scriptsize $\vdash $}\,{\vdash }}(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{1}^{\makebox [0mm][l]{\scriptsize $\vdash $}\,{\vdash }}(
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}^{i.e.}(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{1}^{i.e.}(
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{2}^{i.e.}(
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}^{\forall}(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{1}^{\forall}(
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}^{;}(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{1}^{;}(
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{2}^{;}(
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal T}( 
\ast 
)
] , \linebreak [0] 
[
claims( 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
claims_2( 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{<}proof{>}
] , \linebreak [0] 
[
proof
] , \linebreak [0] 
[
[ \mathbf{Lemma\ } 
\ast 
\colon 
\ast 
]
] , \linebreak [0] 
[
[ \mathbf{Proof\ of\ }
\ast 
\colon 
\ast 
]
] , \linebreak [0] 
[
[ 
\ast 
\mathbf{\ lemma\ } 
\ast 
\colon 
\ast 
]
] , \linebreak [0] 
[
[ 
\ast 
\mathbf{\ antilemma\ } 
\ast 
\colon 
\ast 
]
] , \linebreak [0] 
[
[ 
\ast 
\mathbf{\ rule\ } 
\ast 
\colon 
\ast 
]
] , \linebreak [0] 
[
[ 
\ast 
\mathbf{\ antirule\ } 
\ast 
\colon 
\ast 
]
] , \linebreak [0] 
[
\mbox{verifier}
] , \linebreak [0] 
[
{\cal V}_1( 
\ast 
)
] , \linebreak [0] 
[
{\cal V}_2( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal V}_3( 
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal V}_4( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal V}_5( 
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal V}_6( 
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal V}_7( 
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
Cut( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
Head_{\oplus} ( 
\ast 
)
] , \linebreak [0] 
[
Tail_{\oplus} ( 
\ast 
)
] , \linebreak [0] 
[
rule_1( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
rule( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
Rule\ tactic
] , \linebreak [0] 
[
Plus( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
[ \mathbf{Theory\ } 
\ast 
]
] , \linebreak [0] 
[
theory_2( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
theory_3( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
theory_4( 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
HeadNil''
] , \linebreak [0] 
[
HeadPair''
] , \linebreak [0] 
[
Transitivity''
] , \linebreak [0] 
[
Contra''
] , \linebreak [0] 
[
HeadNil
] , \linebreak [0] 
[
HeadPair
] , \linebreak [0] 
[
Transitivity
] , \linebreak [0] 
[
Contra
] , \linebreak [0] 
[
T_E
] , \linebreak [0] 
[
ragged\ right
] , \linebreak [0] 
[
ragged\ right\ expansion\ 
] , \linebreak [0] 
[
parm(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
parm^*(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
inst(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
inst^*(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
occur(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
occur^*(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
unify(
\ast 
=
\ast 
,
\ast 
)
] , \linebreak [0] 
[
unify^*(
\ast 
=
\ast 
,
\ast 
)
] , \linebreak [0] 
[
unify_2(
\ast 
=
\ast 
,
\ast 
)
] , \linebreak [0] 
[L_a
] , \linebreak [0] 
[L_b
] , \linebreak [0] 
[L_c
] , \linebreak [0] 
[L_d
] , \linebreak [0] 
[L_e
] , \linebreak [0] 
[L_f
] , \linebreak [0] 
[L_g
] , \linebreak [0] 
[L_h
] , \linebreak [0] 
[L_i
] , \linebreak [0] 
[L_j
] , \linebreak [0] 
[L_k
] , \linebreak [0] 
[L_l
] , \linebreak [0] 
[L_m
] , \linebreak [0] 
[L_n
] , \linebreak [0] 
[L_o
] , \linebreak [0] 
[L_p
] , \linebreak [0] 
[L_q
] , \linebreak [0] 
[L_r
] , \linebreak [0] 
[L_s
] , \linebreak [0] 
[L_t
] , \linebreak [0] 
[L_u
] , \linebreak [0] 
[L_v
] , \linebreak [0] 
[L_w
] , \linebreak [0] 
[L_x
] , \linebreak [0] 
[L_y
] , \linebreak [0] 
[L_z
] , \linebreak [0] 
[L_A
] , \linebreak [0] 
[L_B
] , \linebreak [0] 
[L_C
] , \linebreak [0] 
[L_D
] , \linebreak [0] 
[L_E
] , \linebreak [0] 
[L_F
] , \linebreak [0] 
[L_G
] , \linebreak [0] 
[L_H
] , \linebreak [0] 
[L_I
] , \linebreak [0] 
[L_J
] , \linebreak [0] 
[L_K
] , \linebreak [0] 
[L_L
] , \linebreak [0] 
[L_M
] , \linebreak [0] 
[L_N
] , \linebreak [0] 
[L_O
] , \linebreak [0] 
[L_P
] , \linebreak [0] 
[L_Q
] , \linebreak [0] 
[L_R
] , \linebreak [0] 
[L_S
] , \linebreak [0] 
[L_T
] , \linebreak [0] 
[L_U
] , \linebreak [0] 
[L_V
] , \linebreak [0] 
[L_W
] , \linebreak [0] 
[L_X
] , \linebreak [0] 
[L_Y
] , \linebreak [0] 
[L_Z
] , \linebreak [0] 
[L_?
] , \linebreak [0] 
[
Reflexivity
] , \linebreak [0] 
[
Reflexivity_1
] , \linebreak [0] 
[
Commutativity
] , \linebreak [0] 
[
Commutativity_1
] , \linebreak [0] 
[
{<}tactic{>}
] , \linebreak [0] 
[
tactic
] , \linebreak [0] 
[
[
\ast 
\stackrel {tactic}{=}
\ast 
]
] , \linebreak [0] 
[
{\cal P}( 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal P}^*( 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
p_0
] , \linebreak [0] 
[
conclude_1 ( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
conclude_2 ( 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
conclude_3 ( 
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
conclude_4 ( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
\mbox{check}
] , \linebreak [0] 
[
[
\ast 
\stackrel {\circ}{=}
\ast 
]
] , \linebreak [0] 
[
RootVisible(
\ast 
)
] , \linebreak [0] 
[
A
] , \linebreak [0] 
[
R
] , \linebreak [0] 
[
C
] , \linebreak [0] 
[
T
] , \linebreak [0] 
[
L
] , \linebreak [0] 
[
\{
\ast 
\}
] , \linebreak [0] 
[\overline{
\ast }
] , \linebreak [0] 
[
\mathit{a}
] , \linebreak [0] 
[
\mathit{b}
] , \linebreak [0] 
[
\mathit{c}
] , \linebreak [0] 
[
\mathit{d}
] , \linebreak [0] 
[
\mathit{e}
] , \linebreak [0] 
[
\mathit{f}
] , \linebreak [0] 
[
\mathit{g}
] , \linebreak [0] 
[
\mathit{h}
] , \linebreak [0] 
[
\mathit{i}
] , \linebreak [0] 
[
\mathit{j}
] , \linebreak [0] 
[
\mathit{k}
] , \linebreak [0] 
[
\mathit{l}
] , \linebreak [0] 
[
\mathit{m}
] , \linebreak [0] 
[
\mathit{n}
] , \linebreak [0] 
[
\mathit{o}
] , \linebreak [0] 
[
\mathit{p}
] , \linebreak [0] 
[
\mathit{q}
] , \linebreak [0] 
[
\mathit{r}
] , \linebreak [0] 
[
\mathit{s}
] , \linebreak [0] 
[
\mathit{t}
] , \linebreak [0] 
[
\mathit{u}
] , \linebreak [0] 
[
\mathit{v}
] , \linebreak [0] 
[
\mathit{w}
] , \linebreak [0] 
[
\mathit{x}
] , \linebreak [0] 
[
\mathit{y}
] , \linebreak [0] 
[
\mathit{z}
] , \linebreak [0] 
[
\langle 
\ast 
{\equiv} 
\ast 
| 
\ast 
{:=} 
\ast 
\rangle 
] , \linebreak [0] 
[
\langle 
\ast 
{\equiv}^0 
\ast 
| 
\ast 
{:=} 
\ast 
\rangle 
] , \linebreak [0] 
[
\langle 
\ast 
{\equiv}^1 
\ast 
| 
\ast 
{:=} 
\ast 
\rangle 
] , \linebreak [0] 
[
\langle 
\ast 
{\equiv}^* 
\ast 
| 
\ast 
{:=} 
\ast 
\rangle 
] , \linebreak [0] 
[
Ded(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Ded_0(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Ded_1(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Ded_2(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Ded_3(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Ded_4(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Ded_4^*(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Ded_5(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Ded_6(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Ded_6^*(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Ded_7(
\ast 
)
] , \linebreak [0] 
[
Ded_8(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Ded_8^*(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
S
] , \linebreak [0] 
[
Neg
] , \linebreak [0] 
[
MP
] , \linebreak [0] 
[
Gen
] , \linebreak [0] 
[
Ded
] , \linebreak [0] 
[
S1
] , \linebreak [0] 
[
S2
] , \linebreak [0] 
[
S3
] , \linebreak [0] 
[
S4
] , \linebreak [0] 
[
S5
] , \linebreak [0] 
[
S6
] , \linebreak [0] 
[
S7
] , \linebreak [0] 
[
S8
] , \linebreak [0] 
[
S9
] , \linebreak [0] 
[
Repetition
] , \linebreak [0] 
[
A1'
] , \linebreak [0] 
[
A2'
] , \linebreak [0] 
[
A4'
] , \linebreak [0] 
[
A5'
] , \linebreak [0] 
[
Prop\ 3.2a
] , \linebreak [0] 
[
Prop\ 3.2b
] , \linebreak [0] 
[
Prop\ 3.2c
] , \linebreak [0] 
[
Prop\ 3.2d
] , \linebreak [0] 
[
Prop\ 3.2e_1
] , \linebreak [0] 
[
Prop\ 3.2e_2
] , \linebreak [0] 
[
Prop\ 3.2e
] , \linebreak [0] 
[
Prop\ 3.2f_1
] , \linebreak [0] 
[
Prop\ 3.2f_2
] , \linebreak [0] 
[
Prop\ 3.2f
] , \linebreak [0] 
[
Prop\ 3.2g_1
] , \linebreak [0] 
[
Prop\ 3.2g_2
] , \linebreak [0] 
[
Prop\ 3.2g
] , \linebreak [0] 
[
Prop\ 3.2h_1
] , \linebreak [0] 
[
Prop\ 3.2h_2
] , \linebreak [0] 
[
Prop\ 3.2h
] , \linebreak [0] 
[
Block_1(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Block_2(
\ast 
)
] , \linebreak [0] 
[(\cdots{})
] , \linebreak [0] 
[\texttt{Objekt-var}
] , \linebreak [0] 
[\texttt{Ex-var}
] , \linebreak [0] 
[\texttt{Ph-var}
] , \linebreak [0] 
[\texttt{V\ae{}rdi}
] , \linebreak [0] 
[\texttt{Variabel}
] , \linebreak [0] 
[Op(
\ast 
)
] , \linebreak [0] 
[Op(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\ast 
 \mathrel {\ddot {=}} 
\ast 
] , \linebreak [0] 
[ContainsEmpty(
\ast 
)
] , \linebreak [0] 
[
Dedu(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Dedu_0(
\ast 
,
\ast 
)
] , \linebreak [0] 
[Dedu_{s}(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Dedu_1(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Dedu_2(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Dedu_3(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Dedu_4(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Dedu_4^*(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Dedu_5(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Dedu_6(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Dedu_6^*(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Dedu_7(
\ast 
)
] , \linebreak [0] 
[
Dedu_8(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Dedu_8^*(
\ast 
,
\ast 
)
] , \linebreak [0] 
[Ex_{1}
] , \linebreak [0] 
[Ex_{2}
] , \linebreak [0] 
[Ex_{10}
] , \linebreak [0] 
[Ex_{20}
] , \linebreak [0] 
[
\ast 
 _{Ex}
] , \linebreak [0] 
[
\ast 
 ^{Ex}
] , \linebreak [0] 
[\langle 
\ast 
{\equiv} 
\ast 
| 
\ast 
{:=} 
\ast 
\rangle_{Ex} 
] , \linebreak [0] 
[\langle 
\ast 
{\equiv}^0 
\ast 
| 
\ast 
{:=} 
\ast 
\rangle_{Ex} 
] , \linebreak [0] 
[\langle 
\ast 
{\equiv}^1 
\ast 
| 
\ast 
{:=} 
\ast 
\rangle_{Ex} 
] , \linebreak [0] 
[\langle 
\ast 
{\equiv}^* 
\ast 
| 
\ast 
{:=} 
\ast 
\rangle_{Ex} 
] , \linebreak [0] 
[ph_{1}
] , \linebreak [0] 
[ph_{2}
] , \linebreak [0] 
[ph_{3}
] , \linebreak [0] 
[
\ast 
 _{Ph} 
] , \linebreak [0] 
[
\ast 
 ^{Ph}
] , \linebreak [0] 
[\langle 
\ast 
{\equiv} 
\ast 
| 
\ast 
{:=} 
\ast 
\rangle_{Ph} 
] , \linebreak [0] 
[\langle 
\ast 
{\equiv}^0 
\ast 
| 
\ast 
{:=} 
\ast 
\rangle_{Ph} 
] , \linebreak [0] 
[\langle 
\ast 
{\equiv}^1 
\ast 
| 
\ast 
{:=} 
\ast 
\rangle_{Ph} 
] , \linebreak [0] 
[\langle 
\ast 
{\equiv}^* 
\ast 
| 
\ast 
{:=} 
\ast 
\rangle_{Ph} 
] , \linebreak [0] 
[\mathsf {bs}
] , \linebreak [0] 
[ \mathsf {OBS}
] , \linebreak [0] 
[{\cal BS}
] , \linebreak [0] 
[\mathrm{\O}
] , \linebreak [0] 
[ZFsub
] , \linebreak [0] 
[MP
] , \linebreak [0] 
[Gen
] , \linebreak [0] 
[Repetition
] , \linebreak [0] 
[Neg
] , \linebreak [0] 
[Ded
] , \linebreak [0] 
[ExistIntro
] , \linebreak [0] 
[Extensionality
] , \linebreak [0] 
[\O{}def
] , \linebreak [0] 
[PairDef
] , \linebreak [0] 
[UnionDef
] , \linebreak [0] 
[PowerDef
] , \linebreak [0] 
[SeparationDef
] , \linebreak [0] 
[CheatAllDisjoint
] , \linebreak [0] 
[AddDoubleNeg
] , \linebreak [0] 
[RemoveDoubleNeg
] , \linebreak [0] 
[AndCommutativity
] , \linebreak [0] 
[AutoImply
] , \linebreak [0] 
[Contrapositive
] , \linebreak [0] 
[FirstConjunct
] , \linebreak [0] 
[SecondConjunct
] , \linebreak [0] 
[FromContradiction
] , \linebreak [0] 
[FromDisjuncts
] , \linebreak [0] 
[IffCommutativity
] , \linebreak [0] 
[IffFirst
] , \linebreak [0] 
[IffSecond
] , \linebreak [0] 
[ImplyTransitivity
] , \linebreak [0] 
[JoinConjuncts
] , \linebreak [0] 
[MP2
] , \linebreak [0] 
[MP3
] , \linebreak [0] 
[MP4
] , \linebreak [0] 
[MP5
] , \linebreak [0] 
[MT
] , \linebreak [0] 
[NegativeMT
] , \linebreak [0] 
[Technicality
] , \linebreak [0] 
[Weakening
] , \linebreak [0] 
[WeakenOr1
] , \linebreak [0] 
[WeakenOr2
] , \linebreak [0] 
[Formula2Pair
] , \linebreak [0] 
[Pair2Formula
] , \linebreak [0] 
[Formula2Union
] , \linebreak [0] 
[Union2Formula
] , \linebreak [0] 
[Formula2Sep
] , \linebreak [0] 
[Sep2Formula
] , \linebreak [0] 
[SubsetInPower
] , \linebreak [0] 
[HelperPowerIsSub
] , \linebreak [0] 
[PowerIsSub
] , \linebreak [0] 
[(Switch)HelperPowerIsSub
] , \linebreak [0] 
[(Switch)PowerIsSub
] , \linebreak [0] 
[ToSetEquality
] , \linebreak [0] 
[HelperToSetEquality(t)
] , \linebreak [0] 
[ToSetEquality(t)
] , \linebreak [0] 
[HelperFromSetEquality
] , \linebreak [0] 
[FromSetEquality
] , \linebreak [0] 
[HelperReflexivity
] , \linebreak [0] 
[Reflexivity
] , \linebreak [0] 
[HelperSymmetry
] , \linebreak [0] 
[Symmetry
] , \linebreak [0] 
[HelperTransitivity
] , \linebreak [0] 
[Transitivity
] , \linebreak [0] 
[ERisReflexive
] , \linebreak [0] 
[ERisSymmetric
] , \linebreak [0] 
[ERisTransitive
] , \linebreak [0] 
[\O{}isSubset
] , \linebreak [0] 
[HelperMemberNot\O{}
] , \linebreak [0] 
[MemberNot\O{}
] , \linebreak [0] 
[HelperUnique\O{}
] , \linebreak [0] 
[Unique\O{}
] , \linebreak [0] 
[=\!{}Reflexivity
] , \linebreak [0] 
[=\!{}Symmetry
] , \linebreak [0] 
[Helper\!{}=\!{}Transitivity
] , \linebreak [0] 
[\!{}=\!{}Transitivity
] , \linebreak [0] 
[HelperTransferNotEq
] , \linebreak [0] 
[TransferNotEq
] , \linebreak [0] 
[HelperPairSubset
] , \linebreak [0] 
[Helper(2)PairSubset
] , \linebreak [0] 
[PairSubset
] , \linebreak [0] 
[SamePair
] , \linebreak [0] 
[SameSingleton
] , \linebreak [0] 
[UnionSubset
] , \linebreak [0] 
[SameUnion
] , \linebreak [0] 
[SeparationSubset
] , \linebreak [0] 
[SameSeparation
] , \linebreak [0] 
[SameBinaryUnion
] , \linebreak [0] 
[IntersectionSubset
] , \linebreak [0] 
[SameIntersection
] , \linebreak [0] 
[AutoMember
] , \linebreak [0] 
[HelperEqSysNot\O{}
] , \linebreak [0] 
[EqSysNot\O{}
] , \linebreak [0] 
[HelperEqSubset
] , \linebreak [0] 
[EqSubset
] , \linebreak [0] 
[HelperEqNecessary
] , \linebreak [0] 
[EqNecessary
] , \linebreak [0] 
[HelperNoneEqNecessary
] , \linebreak [0] 
[Helper(2)NoneEqNecessary
] , \linebreak [0] 
[NoneEqNecessary
] , \linebreak [0] 
[EqClassIsSubset
] , \linebreak [0] 
[EqClassesAreDisjoint
] , \linebreak [0] 
[AllDisjoint
] , \linebreak [0] 
[AllDisjointImply
] , \linebreak [0] 
[BSsubset
] , \linebreak [0] 
[Union(BS/R)subset
] , \linebreak [0] 
[UnionIdentity
] , \linebreak [0] 
[EqSysIsPartition
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\_\{
\ast 
\}
] , \linebreak [0] 
[
\ast 
/indexintro(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\ast 
/intro(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\ast 
/bothintro(
\ast 
,
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\ast 
/nameintro(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\ast '
] , \linebreak [0] 
[
\ast 
{[}
\ast 
{]}
] , \linebreak [0] 
[
\ast 
[ 
\ast 
{\rightarrow} 
\ast 
]
] , \linebreak [0] 
[
\ast 
[ 
\ast 
{\Rightarrow} 
\ast 
]
] , \linebreak [0] 
[
\ast 
0
] , \linebreak [0] 
[
\ast 
1
] , \linebreak [0] 
[
0 \mathrm {b}
] , \linebreak [0] 
[
\ast 
\mbox {-color}( 
\ast 
)
] , \linebreak [0] 
[
\ast 
\mbox {-color}^{\ast}( 
\ast 
)
] , \linebreak [0] 
[
\ast 
{}^H
] , \linebreak [0] 
[
\ast 
{}^T
] , \linebreak [0] 
[
\ast 
{}^U
] , \linebreak [0] 
[
\ast 
{}^h
] , \linebreak [0] 
[
\ast 
{}^t
] , \linebreak [0] 
[
\ast 
{}^s
] , \linebreak [0] 
[
\ast 
{}^c
] , \linebreak [0] 
[
\ast 
{}^d
] , \linebreak [0] 
[
\ast 
{}^a
] , \linebreak [0] 
[
\ast 
{}^C
] , \linebreak [0] 
[
\ast 
{}^M
] , \linebreak [0] 
[
\ast 
{}^B
] , \linebreak [0] 
[
\ast 
{}^{r}
] , \linebreak [0] 
[
\ast 
{}^{i}
] , \linebreak [0] 
[
\ast 
{}^{d}
] , \linebreak [0] 
[
\ast 
{}^{R}
] , \linebreak [0] 
[
\ast 
{}^{0}
] , \linebreak [0] 
[
\ast 
{}^{1}
] , \linebreak [0] 
[
\ast 
{}^{2}
] , \linebreak [0] 
[
\ast 
{}^{3}
] , \linebreak [0] 
[
\ast 
{}^{4}
] , \linebreak [0] 
[
\ast 
{}^{5}
] , \linebreak [0] 
[
\ast 
{}^{6}
] , \linebreak [0] 
[
\ast 
{}^{7}
] , \linebreak [0] 
[
\ast 
{}^{8}
] , \linebreak [0] 
[
\ast 
{}^{9}
] , \linebreak [0] 
[
\ast 
{} ^ { E }
] , \linebreak [0] 
[
\ast 
{} ^ {\cal V}
] , \linebreak [0] 
[
\ast 
{} ^ {\cal C}
] , \linebreak [0] 
[
\ast 
{} ^ {{\cal C} ^ { \ast }}
] , \linebreak [0] 
[
\ast 
{}^{hide}
]
; 
\newline \mathbf {Preassociative} \newline 
[
\mbox{``}
\ast 
\mbox{''}
] , \linebreak [0] 
[
] , \linebreak [0] 
[
(
\ast 
)^{\bf t}
] , \linebreak [0] 
[
\mbox{string}(
\ast 
)+
\ast 
] , \linebreak [0] 
[\mbox{string}(
\ast 
)\mathrel{++}
\ast 
] , \linebreak [0] 
[
\newline 
\ast 
] , \linebreak [0] 
[
\linebreak [0]\ \hskip0em plus2.0em{}
\ast 
] , \linebreak [0] 
[!
\ast 
] , \linebreak [0] 
[
\mbox {\tt \char34}
\ast 
] , \linebreak [0] 
[
\#
\ast 
] , \linebreak [0] 
[
\$
\ast 
] , \linebreak [0] 
[
\%
\ast 
] , \linebreak [0] 
[
\&
\ast 
] , \linebreak [0] 
[
\mbox {'}
\ast 
] , \linebreak [0] 
[(
\ast 
] , \linebreak [0] 
[)
\ast 
] , \linebreak [0] 
[
{*}
\ast 
] , \linebreak [0] 
[
{+}
\ast 
] , \linebreak [0] 
[,
\ast 
] , \linebreak [0] 
[
\mbox{-}
\ast 
] , \linebreak [0] 
[.
\ast 
] , \linebreak [0] 
[/
\ast 
] , \linebreak [0] 
[0
\ast 
] , \linebreak [0] 
[1
\ast 
] , \linebreak [0] 
[2
\ast 
] , \linebreak [0] 
[3
\ast 
] , \linebreak [0] 
[4
\ast 
] , \linebreak [0] 
[5
\ast 
] , \linebreak [0] 
[6
\ast 
] , \linebreak [0] 
[7
\ast 
] , \linebreak [0] 
[8
\ast 
] , \linebreak [0] 
[9
\ast 
] , \linebreak [0] 
[
{:}
\ast 
] , \linebreak [0] 
[;
\ast 
] , \linebreak [0] 
[
{<}
\ast 
] , \linebreak [0] 
[
{=}
\ast 
] , \linebreak [0] 
[
{>}
\ast 
] , \linebreak [0] 
[?
\ast 
] , \linebreak [0] 
[@
\ast 
] , \linebreak [0] 
[A
\ast 
] , \linebreak [0] 
[B
\ast 
] , \linebreak [0] 
[C
\ast 
] , \linebreak [0] 
[D
\ast 
] , \linebreak [0] 
[E
\ast 
] , \linebreak [0] 
[F
\ast 
] , \linebreak [0] 
[G
\ast 
] , \linebreak [0] 
[H
\ast 
] , \linebreak [0] 
[I
\ast 
] , \linebreak [0] 
[J
\ast 
] , \linebreak [0] 
[K
\ast 
] , \linebreak [0] 
[L
\ast 
] , \linebreak [0] 
[M
\ast 
] , \linebreak [0] 
[N
\ast 
] , \linebreak [0] 
[O
\ast 
] , \linebreak [0] 
[P
\ast 
] , \linebreak [0] 
[Q
\ast 
] , \linebreak [0] 
[R
\ast 
] , \linebreak [0] 
[S
\ast 
] , \linebreak [0] 
[T
\ast 
] , \linebreak [0] 
[U
\ast 
] , \linebreak [0] 
[V
\ast 
] , \linebreak [0] 
[W
\ast 
] , \linebreak [0] 
[X
\ast 
] , \linebreak [0] 
[Y
\ast 
] , \linebreak [0] 
[Z
\ast 
] , \linebreak [0] 
[[
\ast 
] , \linebreak [0] 
[
\mbox {$\backslash $}
\ast 
] , \linebreak [0] 
[]
\ast 
] , \linebreak [0] 
[
{\char94}
\ast 
] , \linebreak [0] 
[
\_
\ast 
] , \linebreak [0] 
[
\mbox {`}
\ast 
] , \linebreak [0] 
[a
\ast 
] , \linebreak [0] 
[b
\ast 
] , \linebreak [0] 
[c
\ast 
] , \linebreak [0] 
[d
\ast 
] , \linebreak [0] 
[e
\ast 
] , \linebreak [0] 
[f
\ast 
] , \linebreak [0] 
[g
\ast 
] , \linebreak [0] 
[h
\ast 
] , \linebreak [0] 
[i
\ast 
] , \linebreak [0] 
[j
\ast 
] , \linebreak [0] 
[k
\ast 
] , \linebreak [0] 
[l
\ast 
] , \linebreak [0] 
[m
\ast 
] , \linebreak [0] 
[n
\ast 
] , \linebreak [0] 
[o
\ast 
] , \linebreak [0] 
[p
\ast 
] , \linebreak [0] 
[q
\ast 
] , \linebreak [0] 
[r
\ast 
] , \linebreak [0] 
[s
\ast 
] , \linebreak [0] 
[t
\ast 
] , \linebreak [0] 
[u
\ast 
] , \linebreak [0] 
[v
\ast 
] , \linebreak [0] 
[w
\ast 
] , \linebreak [0] 
[x
\ast 
] , \linebreak [0] 
[y
\ast 
] , \linebreak [0] 
[z
\ast 
] , \linebreak [0] 
[
\{
\ast 
] , \linebreak [0] 
[|
\ast 
] , \linebreak [0] 
[
\}
\ast 
] , \linebreak [0] 
[
\char126 
\ast 
] , \linebreak [0] 
[
\mathbf{Preassociative}\,
\ast 
; 
\ast 
] , \linebreak [0] 
[
\mathbf{Postassociative}\,
\ast 
; 
\ast 
] , \linebreak [0] 
[
[
\ast 
] , \linebreak [0] 
\ast 
] , \linebreak [0] 
[
\mathrm{priority} \,
\ast 
\, \mathrm{end}
] , \linebreak [0] 
[
newline\ 
\ast 
] , \linebreak [0] 
[
macro\ newline\ 
\ast 
] , \linebreak [0] 
[
MacroIndent(
\ast 
)
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\mathbin {\mbox {'}}
\ast 
] , \linebreak [0] 
[
\ast 
\mathbin {\mbox {`}}
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
{}'
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
 / 
\ast 
] , \linebreak [0] 
[
\ast 
 \cap 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[\cup 
\ast 
] , \linebreak [0] 
[
\ast 
 \mathrel{\cup} 
\ast 
] , \linebreak [0] 
[P(
\ast 
)
]
; 
\newline \mathbf {Preassociative} \newline 
[\{
\ast 
\}
]
; 
\newline \mathbf {Preassociative} \newline 
[\{
\ast 
,
\ast 
\}
] , \linebreak [0] 
[\langle 
\ast 
,
\ast 
\rangle
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
 \mathrel{\in} 
\ast 
] , \linebreak [0] 
[
\ast 
(
\ast 
,
\ast 
)
] , \linebreak [0] 
[ReflRel(
\ast 
,
\ast 
)
] , \linebreak [0] 
[SymRel(
\ast 
,
\ast 
)
] , \linebreak [0] 
[TransRel(
\ast 
,
\ast 
)
] , \linebreak [0] 
[EqRel(
\ast 
,
\ast 
)
] , \linebreak [0] 
[[
\ast 
 \mathrel{\in} 
\ast 
]_{
\ast 
}
] , \linebreak [0] 
[Partition(
\ast 
,
\ast 
)
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\cdot 
\ast 
] , \linebreak [0] 
[
\ast 
\cdot_0 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
+ 
\ast 
] , \linebreak [0] 
[
\ast 
\mathop{+_0} 
\ast 
] , \linebreak [0] 
[
\ast 
\mathop{+_1} 
\ast 
] , \linebreak [0] 
[
\ast 
- 
\ast 
] , \linebreak [0] 
[
\ast 
\mathop{-_0} 
\ast 
] , \linebreak [0] 
[
\ast 
\mathop{-_1} 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\cup \{ 
\ast 
\}
] , \linebreak [0] 
[
\ast 
\cup 
\ast 
] , \linebreak [0] 
[
\ast 
\backslash \{ 
\ast 
\}
]
; 
\newline\mathbf{Postassociative} \newline 
[
\ast 
\mathrel { \dot { . \, . } }
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel { \underline { \dot { . \, . } } }
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel { \underline { : \, : } }
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel { \underline { {+} 2 \ast } }
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel { : \, : }
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel { {+} 2 \ast }
\ast 
]
; 
\newline\mathbf{Postassociative} \newline 
[
\ast 
,\linebreak [0] 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\stackrel {B}{\approx }
\ast 
] , \linebreak [0] 
[
\ast 
\stackrel {D}{\approx }
\ast 
] , \linebreak [0] 
[
\ast 
\stackrel {C}{\approx }
\ast 
] , \linebreak [0] 
[
\ast 
\stackrel {P}{\approx }
\ast 
] , \linebreak [0] 
[
\ast 
\approx 
\ast 
] , \linebreak [0] 
[
\ast 
=
\ast 
] , \linebreak [0] 
[
\ast 
\stackrel {+}{\rightarrow }
\ast 
] , \linebreak [0] 
[
\ast 
\stackrel {t}{=}
\ast 
] , \linebreak [0] 
[
\ast 
\stackrel {t^*}{=}
\ast 
] , \linebreak [0] 
[
\ast 
\stackrel{r}{=} 
\ast 
] , \linebreak [0] 
[
\ast 
\in_t 
\ast 
] , \linebreak [0] 
[
\ast 
\subseteq_T 
\ast 
] , \linebreak [0] 
[
\ast 
\stackrel{T}{=} 
\ast 
] , \linebreak [0] 
[
\ast 
\stackrel{s}{=} 
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel {free\ in} 
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel {free\ in}^{\ast} 
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel {free\ for} 
\ast 
\mathrel {in} 
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel {free\ for}^{\ast} 
\ast 
\mathrel {in} 
\ast 
] , \linebreak [0] 
[
\ast 
\in_c 
\ast 
] , \linebreak [0] 
[
\ast 
< 
\ast 
] , \linebreak [0] 
[
\ast 
<' 
\ast 
] , \linebreak [0] 
[
\ast 
\le' 
\ast 
] , \linebreak [0] 
[
\ast 
= 
\ast 
] , \linebreak [0] 
[
\ast 
\neq 
\ast 
] , \linebreak [0] 
[
\ast 
{}^{var}
] , \linebreak [0] 
[
\ast 
\#^0
\ast 
] , \linebreak [0] 
[
\ast 
\#^1
\ast 
] , \linebreak [0] 
[
\ast 
\#^*
\ast 
] , \linebreak [0] 
[
\ast 
\!\mathrel{=}\! 
\ast 
] , \linebreak [0] 
[
\ast 
 \mathrel{\subseteq} 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
{\neg }
\ast 
] , \linebreak [0] 
[\dot{\neg}\, 
\ast 
] , \linebreak [0] 
[
\ast 
 \mathrel{\notin} 
\ast 
] , \linebreak [0] 
[
\ast 
 \mathrel{\neq} 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\wedge 
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel{\ddot{\wedge}} 
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel{\tilde{\wedge}} 
\ast 
] , \linebreak [0] 
[
\ast 
\wedge_c 
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel{\dot{\wedge}} 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\vee 
\ast 
] , \linebreak [0] 
[
\ast 
\parallel 
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel{\ddot{\vee}} 
\ast 
] , \linebreak [0] 
[
\ast 
 \mathrel{\dot{\vee}} 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\exists 
\ast 
\colon 
\ast 
] , \linebreak [0] 
[
\forall 
\ast 
\colon 
\ast 
] , \linebreak [0] 
[
\forall_{obj} 
\ast 
\colon 
\ast 
]
; 
\newline\mathbf{Postassociative} \newline 
[
\ast 
\mathrel{\ddot{\Rightarrow}} 
\ast 
] , \linebreak [0] 
[
\ast 
\Rightarrow 
\ast 
] , \linebreak [0] 
[
\ast 
\Leftrightarrow 
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel{\dot{\Leftrightarrow}} 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[ \{ ph \mathrel{\in} 
\ast 
\mid 
\ast 
\}
]
; 
\newline\mathbf{Postassociative} \newline 
[
\ast 
:
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel{spy}
\ast 
] , \linebreak [0] 
[
\ast 
!
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\left\{\protect \begin {array}{l}
\ast 
\\
\ast 
\protect \end {array}\right.
]
; 
\newline \mathbf {Preassociative} \newline 
[
\lambda 
\ast 
.
\ast 
] , \linebreak [0] 
[
\Lambda 
\ast 
. 
\ast 
] , \linebreak [0] 
[
\Lambda 
\ast 
] , \linebreak [0] 
[
{\bf if} \ 
\ast 
\ {\bf then} \ 
\ast 
\ {\bf else} \ 
\ast 
] , \linebreak [0] 
[
\mathbf{let\ } 
\ast 
= 
\ast 
\mathbf{\ in\ } 
\ast 
] , \linebreak [0] 
[
\mathbf{let\ } 
\ast 
\mathrel{\ddot{=}} 
\ast 
\mathrel{\ in\ } 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\#
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
{} ^ { I }
] , \linebreak [0] 
[
\ast 
{} ^ { \rhd }
] , \linebreak [0] 
[
\ast 
{} ^ { V } 
] , \linebreak [0] 
[
\ast 
{} ^ { + } 
] , \linebreak [0] 
[
\ast 
{} ^ { - } 
] , \linebreak [0] 
[
\ast 
{} ^ { \ast } 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\mathop {\char64} 
\ast 
] , \linebreak [0] 
[
\ast 
\rhd 
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\ast 
] , \linebreak [0] 
[
\ast 
\gg 
\ast 
] , \linebreak [0] 
[
\ast 
\unrhd 
\ast 
]
; 
\newline\mathbf{Postassociative} \newline 
[
\ast 
\vdash 
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel {i.e.} 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\forall 
\ast 
\colon 
\ast 
] , \linebreak [0] 
[
\Pi 
\ast 
\colon 
\ast 
]
; 
\newline\mathbf{Postassociative} \newline 
[
\ast 
\mathrel {\oplus} 
\ast 
]
; 
\newline\mathbf{Postassociative} \newline 
[
\ast 
; 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\ proves\ 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\mathbf{\ proof\ of\ } 
\ast 
: 
\ast 
] , \linebreak [0] 
[
Line \, 
\ast 
: 
\ast 
\gg 
\ast 
; 
\ast 
] , \linebreak [0] 
[
Last\ line \, 
\ast 
\gg 
\ast 
\,\Box
] , \linebreak [0] 
[
Line \, 
\ast 
: Premise \gg 
\ast 
;
\ast 
] , \linebreak [0] 
[
Line \, 
\ast 
: \mbox{Side-condition} \gg 
\ast 
; 
\ast 
] , \linebreak [0] 
[
Arbitrary \gg 
\ast 
;
\ast 
] , \linebreak [0] 
[
Local \gg 
\ast 
= 
\ast 
; 
\ast 
] , \linebreak [0] 
[
Begin \, 
\ast 
 ; 
\ast 
 : End ; 
\ast 
] , \linebreak [0] 
[
Last\ block\ line \, 
\ast 
\gg 
\ast 
\,;
] , \linebreak [0] 
[
Arbitrary \gg 
\ast 
;
\ast 
]
; 
\newline\mathbf{Postassociative} \newline 
[
\ast 
\mathrel{|} 
\ast 
]
; 
\newline\mathbf{Postassociative} \newline 
[
\ast 
\mathrel{\mathrm{,}}
\ast 
] , \linebreak [0] 
[
\ast 
{[}
\ast 
{]}
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\& 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\backslash \backslash 
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel{linebreak[4]} 
\ast 
] , \linebreak [0] 
[
\ast 
\backslash \backslash 
\ast 
]
; 
\mathbf{End\ table}$
\end {flushleft}

\section{\TeX{} definitioner} \label{sec:tex}

\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}

\item $
[EquivalenceRelations
\stackrel {\mathrm {tex}}{=} 
\mbox{``}EquivalenceRelations
\mbox{''}
]$ 

\item $
[(\cdots{})
\stackrel {\mathrm {tex}}{=} 
\mbox{``}(
\mbox {$\backslash $}cdots
\{
\})
\mbox{''}
]$ 

\item $
[\texttt{Objekt-var}
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\mbox {$\backslash $}texttt
\{Objekt
\mbox{-}var
\}
\mbox{''}
]$ 

\item $
[\texttt{Ex-var}
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\mbox {$\backslash $}texttt
\{Ex
\mbox{-}var
\}
\mbox{''}
]$ 

\item $
[\texttt{Ph-var}
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\mbox {$\backslash $}texttt
\{Ph
\mbox{-}var
\}
\mbox{''}
]$ 

\item $
[\texttt{V\ae{}rdi}
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\mbox {$\backslash $}texttt
\{V
\mbox {$\backslash $}ae
\{
\}rdi
\}
\mbox{''}
]$ 

\item $
[\texttt{Variabel}
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\mbox {$\backslash $}texttt
\{Variabel
\}
\mbox{''}
]$

\item $
[Op(
\mathsf{x}
)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}Op(
\#1.
\newline )
\mbox{''}
]$ 

\item $
[Op(
\mathsf{x}
,
\mathsf{y}
)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}Op(
\#1.
\newline ,
\#2.
\newline )
\mbox{''}
]$ 

\item $
[
\mathsf{x}
 \mathrel {\ddot {=}} 
\mathsf{y}
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\#1.
\newline 
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}mathrel
\linebreak [0]\ \hskip0em plus2.0em{}
\{
\mbox {$\backslash $}ddot
\linebreak [0]\ \hskip0em plus2.0em{}
\{
{=}
\}
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\mbox{''}
]$

\item $
[ContainsEmpty(
\mathsf{x}
)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}ContainsEmpty(
\#1.
\newline )
\mbox{''}
]$ 

$
[
Dedu(
\mathsf{x}
,
\mathsf{y}
)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\newline Dedu(
\#1.
\newline ,
\#2.
\newline )
\mbox{''}
]$

 $
[
Dedu_0(
\mathsf{x}
,
\mathsf{y}
)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\newline Dedu
\_0(
\#1.
\newline ,
\#2.
\newline )
\mbox{''}
]$

 $
[Dedu_{s}(
\mathsf{x}
,
\mathsf{y}
,
\mathsf{z}
)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}Dedu
\_
\{s
\}(
\#1.
\newline ,
\#2.
\newline ,
\#3.
\newline )
\mbox{''}
]$

 $
[
Dedu_1(
\mathsf{x}
,
\mathsf{y}
,
\mathsf{z}
)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\newline Dedu
\_1(
\#1.
\newline ,
\#2.
\newline ,
\#3.
\newline )
\mbox{''}
]$

 $
[
Dedu_2(
\mathsf{x}
,
\mathsf{y}
,
\mathsf{z}
)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\newline Dedu
\_2(
\#1.
\newline ,
\#2.
\newline ,
\#3.
\newline )
\mbox{''}
]$

 $
[
Dedu_3(
\mathsf{x}
,
\mathsf{y}
,
\mathsf{z}
,
\mathsf {u}
)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\newline Dedu
\_3(
\#1.
\newline ,
\#2.
\newline ,
\#3.
\newline ,
\#4.
\newline )
\mbox{''}
]$

 $
[
Dedu_4(
\mathsf{x}
,
\mathsf{y}
,
\mathsf{z}
,
\mathsf {u}
)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\newline Dedu
\_4(
\#1.
\newline ,
\#2.
\newline ,
\#3.
\newline ,
\#4.
\newline )
\mbox{''}
]$

 $
[
Dedu_4^*(
\mathsf{x}
,
\mathsf{y}
,
\mathsf{z}
,
\mathsf {u}
)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\newline Dedu
\_4
{\char94}
{*}(
\#1.
\newline ,
\#2.
\newline ,
\#3.
\newline ,
\#4.
\newline )
\mbox{''}
]$

 $
[
Dedu_5(
\mathsf{x}
,
\mathsf{y}
,
\mathsf{z}
)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\newline Dedu
\_5(
\#1.
\newline ,
\#2.
\newline ,
\#3.
\newline )
\mbox{''}
]$

 $
[
Dedu_6(
\mathsf {p}
,
\mathsf {c}
,
\mathsf {e}
,
\mathsf {b}
)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\newline Dedu
\_6(
\#1.
\newline ,
\#2.
\newline ,
\#3.
\newline ,
\#4.
\newline )
\mbox{''}
]$

 $
[
Dedu_6^*(
\mathsf {p}
,
\mathsf {c}
,
\mathsf {e}
,
\mathsf {b}
)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\newline Dedu
\_6
{\char94}
{*}(
\#1.
\newline ,
\#2.
\newline ,
\#3.
\newline ,
\#4.
\newline )
\mbox{''}
]$

 $
[
Dedu_7(
\mathsf {p}
)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\newline Dedu
\_7(
\#1.
\newline )
\mbox{''}
]$

 $
[
Dedu_8(
\mathsf {p}
,
\mathsf {b}
)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\newline Dedu
\_8(
\#1.
\newline ,
\#2.
\newline )
\mbox{''}
]$

 $
[
Dedu_8^*(
\mathsf {p}
,
\mathsf {b}
)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\newline Dedu
\_8
{\char94}
{*}(
\#1.
\newline ,
\#2.
\newline )
\mbox{''}
]$

\item $
[Ex_{1}
\stackrel {\mathrm {tex}}{=} 
\mbox{``}Ex
\_
\{1
\}
\mbox{''}
]$ 

\item $
[Ex_{2}
\stackrel {\mathrm {tex}}{=} 
\mbox{``}Ex
\_
\{2
\}
\mbox{''}
]$ 

\item $
[Ex_{10}
\stackrel {\mathrm {tex}}{=} 
\mbox{``}Ex
\_
\{10
\}
\mbox{''}
]$ 

\item $
[Ex_{20}
\stackrel {\mathrm {tex}}{=} 
\mbox{``}Ex
\_
\{20
\}
\mbox{''}
]$ 

\item $
[
\mathsf{x}
 _{Ex}
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\#1.
\newline 
\linebreak [0]\ \hskip0em plus2.0em{}
\_
\{Ex
\}
\mbox{''}
]$ 

\item $
[
\mathsf{x}
 ^{Ex}
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\#1.
\newline 
\linebreak [0]\ \hskip0em plus2.0em{}
{\char94}
\{Ex
\}
\mbox{''}
]$ 

$
[\langle 
\mathsf{x}
{\equiv} 
\mathsf{y}
| 
\mathsf{z}
{:=} 
\mathsf {u}
\rangle_{Ex} 
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\mbox {$\backslash $}langle
\linebreak [0]\ \hskip0em plus2.0em{}
\#1.
\newline 
\{
\mbox {$\backslash $}equiv
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\newline |
\linebreak [0]\ \hskip0em plus2.0em{}
\#3.
\newline 
\{
{:}
{=}
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\#4.
\newline 
\mbox {$\backslash $}rangle
\_
\{Ex
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox{''}
]$

$
[\langle 
\mathsf{x}
{\equiv}^0 
\mathsf{y}
| 
\mathsf{z}
{:=} 
\mathsf {u}
\rangle_{Ex} 
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\mbox {$\backslash $}langle
\linebreak [0]\ \hskip0em plus2.0em{}
\#1.
\newline 
\{
\mbox {$\backslash $}equiv
\}
{\char94}0
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\newline |
\linebreak [0]\ \hskip0em plus2.0em{}
\#3.
\newline 
\{
{:}
{=}
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\#4.
\newline 
\mbox {$\backslash $}rangle
\_
\{Ex
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox{''}
]$

$
[\langle 
\mathsf{x}
{\equiv}^1 
\mathsf{y}
| 
\mathsf{z}
{:=} 
\mathsf {u}
\rangle_{Ex} 
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\mbox {$\backslash $}langle
\linebreak [0]\ \hskip0em plus2.0em{}
\#1.
\newline 
\{
\mbox {$\backslash $}equiv
\}
{\char94}1
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\newline |
\linebreak [0]\ \hskip0em plus2.0em{}
\#3.
\newline 
\{
{:}
{=}
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\#4.
\newline 
\mbox {$\backslash $}rangle
\_
\{Ex
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox{''}
]$

$
[\langle 
\mathsf{x}
{\equiv}^* 
\mathsf{y}
| 
\mathsf{z}
{:=} 
\mathsf {u}
\rangle_{Ex} 
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\mbox {$\backslash $}langle
\linebreak [0]\ \hskip0em plus2.0em{}
\#1.
\newline 
\{
\mbox {$\backslash $}equiv
\}
{\char94}
{*}
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\newline |
\linebreak [0]\ \hskip0em plus2.0em{}
\#3.
\newline 
\{
{:}
{=}
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\#4.
\newline 
\mbox {$\backslash $}rangle
\_
\{Ex
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox{''}
]$

\item $
[ph_{1}
\stackrel {\mathrm {tex}}{=} 
\mbox{``}ph
\_
\{1
\}
\mbox{''}
]$ 

\item $
[ph_{2}
\stackrel {\mathrm {tex}}{=} 
\mbox{``}ph
\_
\{2
\}
\mbox{''}
]$ 

\item $
[ph_{3}
\stackrel {\mathrm {tex}}{=} 
\mbox{``}ph
\_
\{3
\}
\mbox{''}
]$ 

\item $
[
\mathsf{x}
 _{Ph} 
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\#1.
\newline 
\linebreak [0]\ \hskip0em plus2.0em{}
\_
\{Ph
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox{''}
]$ 

\item $
[
\mathsf{x}
 ^{Ph}
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\#1.
\newline 
\linebreak [0]\ \hskip0em plus2.0em{}
{\char94}
\{Ph
\}
\mbox{''}
]$ 

$
[\langle 
\mathsf{x}
{\equiv} 
\mathsf{y}
| 
\mathsf{z}
{:=} 
\mathsf {u}
\rangle_{Ph} 
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\mbox {$\backslash $}langle
\linebreak [0]\ \hskip0em plus2.0em{}
\#1.
\newline 
\{
\mbox {$\backslash $}equiv
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\newline |
\linebreak [0]\ \hskip0em plus2.0em{}
\#3.
\newline 
\{
{:}
{=}
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\#4.
\newline 
\mbox {$\backslash $}rangle
\_
\{Ph
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox{''}
]$

$
[\langle 
\mathsf{x}
{\equiv}^0 
\mathsf{y}
| 
\mathsf{z}
{:=} 
\mathsf {u}
\rangle_{Ph} 
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\mbox {$\backslash $}langle
\linebreak [0]\ \hskip0em plus2.0em{}
\#1.
\newline 
\{
\mbox {$\backslash $}equiv
\}
{\char94}0
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\newline |
\linebreak [0]\ \hskip0em plus2.0em{}
\#3.
\newline 
\{
{:}
{=}
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\#4.
\newline 
\mbox {$\backslash $}rangle
\_
\{Ph
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox{''}
]$

$
[\langle 
\mathsf{x}
{\equiv}^1 
\mathsf{y}
| 
\mathsf{z}
{:=} 
\mathsf {u}
\rangle_{Ph} 
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\mbox {$\backslash $}langle
\linebreak [0]\ \hskip0em plus2.0em{}
\#1.
\newline 
\{
\mbox {$\backslash $}equiv
\}
{\char94}1
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\newline |
\linebreak [0]\ \hskip0em plus2.0em{}
\#3.
\newline 
\{
{:}
{=}
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\#4.
\newline 
\mbox {$\backslash $}rangle
\_
\{Ph
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox{''}
]$

$
[\langle 
\mathsf{x}
{\equiv}^* 
\mathsf{y}
| 
\mathsf{z}
{:=} 
\mathsf {u}
\rangle_{Ph} 
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\mbox {$\backslash $}langle
\linebreak [0]\ \hskip0em plus2.0em{}
\#1.
\newline 
\{
\mbox {$\backslash $}equiv
\}
{\char94}
{*}
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\newline |
\linebreak [0]\ \hskip0em plus2.0em{}
\#3.
\newline 
\{
{:}
{=}
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\#4.
\newline 
\mbox {$\backslash $}rangle
\_
\{Ph
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox{''}
]$

\item $
[\mathsf {bs}
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\mbox {$\backslash $}mathsf
\linebreak [0]\ \hskip0em plus2.0em{}
\{bs
\}
\mbox{''}
]$ 
\item $
[ \mathsf {OBS}
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}mathsf
\linebreak [0]\ \hskip0em plus2.0em{}
\{OBS
\}
\mbox{''}
]$ 


 $
[{\cal BS}
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\{
\mbox {$\backslash $}cal
\linebreak [0]\ \hskip0em plus2.0em{}BS
\}
\mbox{''}
]$

\item $
[\mathrm{\O}
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\mbox {$\backslash $}mathrm
\{
\mbox {$\backslash $}O
\}
\mbox{''}
]$

\item $
[ZFsub
\stackrel {\mathrm {tex}}{=} 
\mbox{``}ZFsub
\mbox{''}
]$

\item $
[MP
\stackrel {\mathrm {tex}}{=} 
\mbox{``}MP
\mbox{''}
]$
\item $
[Gen
\stackrel {\mathrm {tex}}{=} 
\mbox{``}Gen
\mbox{''}
]$
\item $
[Repetition
\stackrel {\mathrm {tex}}{=} 
\mbox{``}Repetition
\mbox{''}
]$ 
\item $
[Neg
\stackrel {\mathrm {tex}}{=} 
\mbox{``}Neg
\mbox{''}
]$
\item $
[Ded
\stackrel {\mathrm {tex}}{=} 
\mbox{``}Ded
\mbox{''}
]$
\item $
[ExistIntro
\stackrel {\mathrm {tex}}{=} 
\mbox{``}ExistIntro
\mbox{''}
]$ 



\item $
[Extensionality
\stackrel {\mathrm {tex}}{=} 
\mbox{``}Extensionality
\mbox{''}
]$
\item $
[\O{}def
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\mbox {$\backslash $}O
\{
\}def
\mbox{''}
]$ 
\item $
[PairDef
\stackrel {\mathrm {tex}}{=} 
\mbox{``}PairDef
\mbox{''}
]$ 
\item $
[UnionDef
\stackrel {\mathrm {tex}}{=} 
\mbox{``}UnionDef
\mbox{''}
]$
\item $
[PowerDef
\stackrel {\mathrm {tex}}{=} 
\mbox{``}PowerDef
\mbox{''}
]$ 
\item $
[SeparationDef
\stackrel {\mathrm {tex}}{=} 
\mbox{``}SeparationDef
\mbox{''}
]$
\item $
[CheatAllDisjoint
\stackrel {\mathrm {tex}}{=} 
\mbox{``}CheatAllDisjoint
\mbox{''}
]$ 

\item $
[AddDoubleNeg
\stackrel {\mathrm {tex}}{=} 
\mbox{``}AddDoubleNeg
\mbox{''}
]$ 
\item $
[RemoveDoubleNeg
\stackrel {\mathrm {tex}}{=} 
\mbox{``}RemoveDoubleNeg
\mbox{''}
]$ 
\item $
[AndCommutativity
\stackrel {\mathrm {tex}}{=} 
\mbox{``}AndCommutativity
\mbox{''}
]$ 
\item $
[AutoImply
\stackrel {\mathrm {tex}}{=} 
\mbox{``}AutoImply
\mbox{''}
]$ 
\item $
[Contrapositive
\stackrel {\mathrm {tex}}{=} 
\mbox{``}Contrapositive
\mbox{''}
]$ 
\item $
[FirstConjunct
\stackrel {\mathrm {tex}}{=} 
\mbox{``}FirstConjunct
\mbox{''}
]$ 
\item $
[SecondConjunct
\stackrel {\mathrm {tex}}{=} 
\mbox{``}SecondConjunct
\mbox{''}
]$ 
\item $
[FromContradiction
\stackrel {\mathrm {tex}}{=} 
\mbox{``}FromContradiction
\mbox{''}
]$ 
\item $
[FromDisjuncts
\stackrel {\mathrm {tex}}{=} 
\mbox{``}FromDisjuncts
\mbox{''}
]$ 
\item $
[IffCommutativity
\stackrel {\mathrm {tex}}{=} 
\mbox{``}IffCommutativity
\mbox{''}
]$ 
\item $
[IffFirst
\stackrel {\mathrm {tex}}{=} 
\mbox{``}IffFirst
\mbox{''}
]$ 

\item $
[IffSecond
\stackrel {\mathrm {tex}}{=} 
\mbox{``}IffSecond
\mbox{''}
]$ 
\item $
[ImplyTransitivity
\stackrel {\mathrm {tex}}{=} 
\mbox{``}ImplyTransitivity
\mbox{''}
]$ 



\item $
[JoinConjuncts
\stackrel {\mathrm {tex}}{=} 
\mbox{``}JoinConjuncts
\mbox{''}
]$ 

\item $
[MP2
\stackrel {\mathrm {tex}}{=} 
\mbox{``}MP2
\mbox{''}
]$ 

\item $
[MP3
\stackrel {\mathrm {tex}}{=} 
\mbox{``}MP3
\mbox{''}
]$ 

\item $
[MP4
\stackrel {\mathrm {tex}}{=} 
\mbox{``}MP4
\mbox{''}
]$ 

\item $
[MP5
\stackrel {\mathrm {tex}}{=} 
\mbox{``}MP5
\mbox{''}
]$ 
\item $
[MT
\stackrel {\mathrm {tex}}{=} 
\mbox{``}MT
\mbox{''}
]$ 

\item $
[NegativeMT
\stackrel {\mathrm {tex}}{=} 
\mbox{``}NegativeMT
\mbox{''}
]$ 
\item $
[Technicality
\stackrel {\mathrm {tex}}{=} 
\mbox{``}Technicality
\mbox{''}
]$ 


\item $
[Weakening
\stackrel {\mathrm {tex}}{=} 
\mbox{``}Weakening
\mbox{''}
]$ 

\item $
[WeakenOr1
\stackrel {\mathrm {tex}}{=} 
\mbox{``}WeakenOr1
\mbox{''}
]$ 

\item $
[WeakenOr2
\stackrel {\mathrm {tex}}{=} 
\mbox{``}WeakenOr2
\mbox{''}
]$ 

\item $
[Pair2Formula
\stackrel {\mathrm {tex}}{=} 
\mbox{``}Pair2Formula
\mbox{''}
]$ 

\item $
[Formula2Pair
\stackrel {\mathrm {tex}}{=} 
\mbox{``}Formula2Pair
\mbox{''}
]$ 

\item $
[Union2Formula
\stackrel {\mathrm {tex}}{=} 
\mbox{``}Union2Formula
\mbox{''}
]$ 

\item $
[Formula2Union
\stackrel {\mathrm {tex}}{=} 
\mbox{``}Formula2Union
\mbox{''}
]$ 

\item $
[Sep2Formula
\stackrel {\mathrm {tex}}{=} 
\mbox{``}Sep2Formula
\mbox{''}
]$ 

\item $
[Formula2Sep
\stackrel {\mathrm {tex}}{=} 
\mbox{``}Formula2Sep
\mbox{''}
]$ 

\item $
[SubsetInPower
\stackrel {\mathrm {tex}}{=} 
\mbox{``}SubsetInPower
\mbox{''}
]$ 

\item $
[HelperPowerIsSub
\stackrel {\mathrm {tex}}{=} 
\mbox{``}HelperPowerIsSub
\mbox{''}
]$ 

\item $
[PowerIsSub
\stackrel {\mathrm {tex}}{=} 
\mbox{``}PowerIsSub
\mbox{''}
]$ 

\item $
[(Switch)HelperPowerIsSub
\stackrel {\mathrm {tex}}{=} 
\mbox{``}(Switch)HelperPowerIsSub
\mbox{''}
]$ 

\item $
[(Switch)PowerIsSub
\stackrel {\mathrm {tex}}{=} 
\mbox{``}(Switch)PowerIsSub
\mbox{''}
]$ 

\item $
[ToSetEquality
\stackrel {\mathrm {tex}}{=} 
\mbox{``}ToSetEquality
\mbox{''}
]$ 

\item $
[HelperToSetEquality(t)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}HelperToSetEquality(t)
\mbox{''}
]$ 

\item $
[ToSetEquality(t)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}ToSetEquality(t)
\mbox{''}
]$ 

\item $
[HelperFromSetEquality
\stackrel {\mathrm {tex}}{=} 
\mbox{``}HelperFromSetEquality
\mbox{''}
]$ 

\item $
[FromSetEquality
\stackrel {\mathrm {tex}}{=} 
\mbox{``}FromSetEquality
\mbox{''}
]$ 

\item $
[HelperReflexivity
\stackrel {\mathrm {tex}}{=} 
\mbox{``}HelperReflexivity
\mbox{''}
]$ 

\item $
[Reflexivity
\stackrel {\mathrm {tex}}{=} 
\mbox{``}Reflexivity
\mbox{''}
]$ 

\item $
[HelperSymmetry
\stackrel {\mathrm {tex}}{=} 
\mbox{``}HelperSymmetry
\mbox{''}
]$ 

\item $
[Symmetry
\stackrel {\mathrm {tex}}{=} 
\mbox{``}Symmetry
\mbox{''}
]$ 

\item $
[HelperTransitivity
\stackrel {\mathrm {tex}}{=} 
\mbox{``}HelperTransitivity
\mbox{''}
]$

\item $
[Transitivity
\stackrel {\mathrm {tex}}{=} 
\mbox{``}Transitivity
\mbox{''}
]$,

\item $
[ERisReflexive
\stackrel {\mathrm {tex}}{=} 
\mbox{``}ERisReflexive
\mbox{''}
]$ 

\item $
[ERisSymmetric
\stackrel {\mathrm {tex}}{=} 
\mbox{``}ERisSymmetric
\mbox{''}
]$ 

\item $
[ERisTransitive
\stackrel {\mathrm {tex}}{=} 
\mbox{``}ERisTransitive
\mbox{''}
]$ 

\item $
[\O{}isSubset
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\mbox {$\backslash $}O
\{
\}isSubset
\mbox{''}
]$ 

\item $
[HelperMemberNot\O{}
\stackrel {\mathrm {tex}}{=} 
\mbox{``}HelperMemberNot
\mbox {$\backslash $}O
\{
\}
\mbox{''}
]$ 

\item $
[MemberNot\O{}
\stackrel {\mathrm {tex}}{=} 
\mbox{``}MemberNot
\mbox {$\backslash $}O
\{
\}
\mbox{''}
]$ 

\item $
[HelperUnique\O{}
\stackrel {\mathrm {tex}}{=} 
\mbox{``}HelperUnique
\mbox {$\backslash $}O
\{
\}
\mbox{''}
]$ 

\item $
[Unique\O{}
\stackrel {\mathrm {tex}}{=} 
\mbox{``}Unique
\mbox {$\backslash $}O
\{
\}
\mbox{''}
]$ 

\item $
[=\!{}Reflexivity
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
{=}
\mbox {$\backslash $}!
\{
\}Reflexivity
\mbox{''}
]$ 

\item $
[=\!{}Symmetry
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
{=}
\mbox {$\backslash $}!
\{
\}Symmetry
\mbox{''}
]$ 

\item $
[Helper\!{}=\!{}Transitivity
\stackrel {\mathrm {tex}}{=} 
\mbox{``}Helper
\mbox {$\backslash $}!
\{
\}
{=}
\mbox {$\backslash $}!
\{
\}Transitivity
\mbox{''}
]$ 

\item $
[\!{}=\!{}Transitivity
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\mbox {$\backslash $}!
\{
\}
{=}
\mbox {$\backslash $}!
\{
\}Transitivity
\mbox{''}
]$ 

\item $
[HelperTransferNotEq
\stackrel {\mathrm {tex}}{=} 
\mbox{``}HelperTransferNotEq
\mbox{''}
]$ 

\item $
[TransferNotEq
\stackrel {\mathrm {tex}}{=} 
\mbox{``}TransferNotEq
\mbox{''}
]$ 

\item $
[HelperPairSubset
\stackrel {\mathrm {tex}}{=} 
\mbox{``}HelperPairSubset
\mbox{''}
]$ 

\item $
[Helper(2)PairSubset
\stackrel {\mathrm {tex}}{=} 
\mbox{``}Helper(2)PairSubset
\mbox{''}
]$ 

\item $
[PairSubset
\stackrel {\mathrm {tex}}{=} 
\mbox{``}PairSubset
\mbox{''}
]$ 

\item $
[SamePair
\stackrel {\mathrm {tex}}{=} 
\mbox{``}SamePair
\mbox{''}
]$ 

\item $
[SameSingleton
\stackrel {\mathrm {tex}}{=} 
\mbox{``}SameSingleton
\mbox{''}
]$ 


\item $
[UnionSubset
\stackrel {\mathrm {tex}}{=} 
\mbox{``}UnionSubset
\mbox{''}
]$ 

\item $
[SameUnion
\stackrel {\mathrm {tex}}{=} 
\mbox{``}SameUnion
\mbox{''}
]$ 



\item $
[SeparationSubset
\stackrel {\mathrm {tex}}{=} 
\mbox{``}SeparationSubset
\mbox{''}
]$

\item $
[SameSeparation
\stackrel {\mathrm {tex}}{=} 
\mbox{``}SameSeparation
\mbox{''}
]$ 

\item $
[SameBinaryUnion
\stackrel {\mathrm {tex}}{=} 
\mbox{``}SameBinaryUnion
\mbox{''}
]$ 

\item $
[IntersectionSubset
\stackrel {\mathrm {tex}}{=} 
\mbox{``}IntersectionSubset
\mbox{''}
]$ 

\item $
[SameIntersection
\stackrel {\mathrm {tex}}{=} 
\mbox{``}SameIntersection
\mbox{''}
]$ 

\item $
[AutoMember
\stackrel {\mathrm {tex}}{=} 
\mbox{``}AutoMember
\mbox{''}
]$ 

\item $
[HelperEqSysNot\O{}
\stackrel {\mathrm {tex}}{=} 
\mbox{``}HelperEqSysNot
\mbox {$\backslash $}O
\{
\}
\mbox{''}
]$ 

\item $
[EqSysNot\O{}
\stackrel {\mathrm {tex}}{=} 
\mbox{``}EqSysNot
\mbox {$\backslash $}O
\{
\}
\mbox{''}
]$ 

\item $
[HelperEqSubset
\stackrel {\mathrm {tex}}{=} 
\mbox{``}HelperEqSubset
\mbox{''}
]$ 

\item $
[EqSubset
\stackrel {\mathrm {tex}}{=} 
\mbox{``}EqSubset
\mbox{''}
]$ 

\item $
[EqNecessary
\stackrel {\mathrm {tex}}{=} 
\mbox{``}EqNecessary
\mbox{''}
]$ 

\item $
[HelperEqNecessary
\stackrel {\mathrm {tex}}{=} 
\mbox{``}HelperEqNecessary
\mbox{''}
]$ 

\item $
[HelperNoneEqNecessary
\stackrel {\mathrm {tex}}{=} 
\mbox{``}HelperNoneEqNecessary
\mbox{''}
]$ 

\item $
[Helper(2)NoneEqNecessary
\stackrel {\mathrm {tex}}{=} 
\mbox{``}Helper(2)NoneEqNecessary
\mbox{''}
]$ 

\item $
[NoneEqNecessary
\stackrel {\mathrm {tex}}{=} 
\mbox{``}NoneEqNecessary
\mbox{''}
]$ 

\item $
[EqClassIsSubset
\stackrel {\mathrm {tex}}{=} 
\mbox{``}EqClassIsSubset
\mbox{''}
]$ 

\item $
[EqClassesAreDisjoint
\stackrel {\mathrm {tex}}{=} 
\mbox{``}EqClassesAreDisjoint
\mbox{''}
]$ 


\item $
[AllDisjoint
\stackrel {\mathrm {tex}}{=} 
\mbox{``}AllDisjoint
\mbox{''}
]$ 

\item $
[AllDisjointImply
\stackrel {\mathrm {tex}}{=} 
\mbox{``}AllDisjointImply
\mbox{''}
]$ 

\item $
[BSsubset
\stackrel {\mathrm {tex}}{=} 
\mbox{``}BSsubset
\mbox{''}
]$ 

\item $
[Union(BS/R)subset
\stackrel {\mathrm {tex}}{=} 
\mbox{``}Union(BS/R)subset
\mbox{''}
]$ 

\item $
[UnionIdentity
\stackrel {\mathrm {tex}}{=} 
\mbox{``}UnionIdentity
\mbox{''}
]$ 

\item $
[EqSysIsPartition
\stackrel {\mathrm {tex}}{=} 
\mbox{``}EqSysIsPartition
\mbox{''}
]$

\item $
[
\mathsf{x}
 / 
\mathsf{y}
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\#1.
\newline 
\linebreak [0]\ \hskip0em plus2.0em{}/
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\mbox{''}
]$ 

\item $
[
\mathsf{x}
 \cap 
\mathsf{y}
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\#1.
\newline 
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}cap
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\mbox{''}
]$

\item $
[\cup 
\mathsf{x}
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\mbox {$\backslash $}cup
\linebreak [0]\ \hskip0em plus2.0em{}
\#1.
\mbox{''}
]$

\item $
[
\mathsf{x}
 \mathrel{\cup} 
\mathsf{y}
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\#1.
\newline 
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}mathrel
\{
\mbox {$\backslash $}cup
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\mbox{''}
]$ 

\item $
[P(
\mathsf{x}
)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}P(
\#1.
\newline )
\mbox{''}
]$ 

\item $
[\{
\mathsf{x}
\}
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\mbox {$\backslash $}
\{
\#1.
\newline 
\mbox {$\backslash $}
\}
\mbox{''}
]$

\item $
[\{
\mathsf{x}
,
\mathsf{y}
\}
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\mbox {$\backslash $}
\{
\#1.
\newline ,
\#2.
\newline 
\mbox {$\backslash $}
\}
\mbox{''}
]$

\item $
[\langle 
\mathsf{x}
,
\mathsf{y}
\rangle
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\mbox {$\backslash $}langle
\linebreak [0]\ \hskip0em plus2.0em{}
\#1.
\newline ,
\#2.
\newline 
\mbox {$\backslash $}rangle
\mbox{''}
]$,

\item $
[
\mathsf{x}
 \mathrel{\in} 
\mathsf{y}
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\#1.
\newline 
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}mathrel
\{
\mbox {$\backslash $}in
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\mbox{''}
]$

\item $
[
\mathsf{z}
(
\mathsf{x}
,
\mathsf{y}
)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\#3.
\newline (
\#1.
\newline ,
\#2.
\newline )
\mbox{''}
]$

\item $
[ReflRel(
\mathsf {r}
,
\mathsf{x}
)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}ReflRel(
\#1.
\newline ,
\#2.
\newline )
\mbox{''}
]$

\item $
[SymRel(
\mathsf {r}
,
\mathsf{x}
)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}SymRel(
\#1.
\newline ,
\#2.
\newline )
\mbox{''}
]$

\item $
[TransRel(
\mathsf {r}
,
\mathsf{x}
)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}TransRel(
\#1.
\newline ,
\#2.
\newline )
\mbox{''}
]$


\item $
[EqRel(
\mathsf {r}
,
\mathsf{x}
)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}EqRel(
\#1.
\newline ,
\#2.
\newline )
\mbox{''}
]$


\item $
[[
\mathsf{x}
 \mathrel{\in} \mathsf {bs}
]_{
\mathsf {r}
}
\stackrel {\mathrm {tex}}{=} 
\mbox{``}[
\#1.
\newline 
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}mathrel
\{
\mbox {$\backslash $}in
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\newline ]
\_
\{
\#3.
\newline 
\}
\mbox{''}
]$ 

\item $
[Partition(
\mathsf{x}
,
\mathsf{y}
)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}Partition(
\#1.
\newline ,
\#2.
\newline )
\mbox{''}
]$ 

\item $
[
\mathsf{x}
\!\mathrel{=}\! 
\mathsf{y}
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\#1.
\newline 
\mbox {$\backslash $}!
\mbox {$\backslash $}mathrel
\{
{=}
\}
\mbox {$\backslash $}!
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\mbox{''}
]$

\item $
[
\mathsf{x}
 \mathrel{\subseteq} 
\mathsf{y}
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\#1.
\newline 
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}mathrel
\{
\mbox {$\backslash $}subseteq
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\mbox{''}
]$ 

\item $
[\dot{\neg}\, 
\mathsf{x}
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\mbox {$\backslash $}dot
\{
\mbox {$\backslash $}neg
\}
\mbox {$\backslash $},
\linebreak [0]\ \hskip0em plus2.0em{}
\#1.
\mbox{''}
]$

\item $
[
\mathsf{x}
 \mathrel{\notin} 
\mathsf{y}
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\#1.
\newline 
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}mathrel
\{
\mbox {$\backslash $}notin
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\mbox{''}
]$ 

\item $
[
\mathsf{x}
 \mathrel{\neq} 
\mathsf{y}
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\#1.
\newline 
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}mathrel
\{
\mbox {$\backslash $}neq
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\mbox{''}
]$ 

\item $
[
\mathsf{x}
\mathrel{\dot{\wedge}} 
\mathsf{y}
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\#1.
\newline 
\mbox {$\backslash $}mathrel
\{
\mbox {$\backslash $}dot
\{
\mbox {$\backslash $}wedge
\}
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\mbox{''}
]$

\item $
[
\mathsf{x}
 \mathrel{\dot{\vee}} 
\mathsf{y}
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\#1.
\newline 
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}mathrel
\{
\mbox {$\backslash $}dot
\{
\mbox {$\backslash $}vee
\}
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\mbox{''}
]$ 

$
[
\mathsf{x}
\mathrel{\dot{\Leftrightarrow}} 
\mathsf{y}
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\#1.
\newline 
\mbox {$\backslash $}mathrel
\{
\mbox {$\backslash $}dot
\{
\mbox {$\backslash $}Leftrightarrow
\}
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\mbox{''}
]$

\item $
[ \{ ph \mathrel{\in} 
\mathsf{x}
\mid 
\mathsf {a}
\}
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}
\{
\linebreak [0]\ \hskip0em plus2.0em{}ph
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}mathrel
\{
\mbox {$\backslash $}in
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\#1.
\newline 
\mbox {$\backslash $}mid
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\newline 
\mbox {$\backslash $}
\}
\mbox{''}
]$

\end{list}

\end{document}

