(*** MAKROER BEGYNDER ***)

1 Makrodefinitioner

Dette afsnit indeholder dé makrodefinitioner, som vi vil gøre brug af i resten af rapporten. Definitionerne drejer sig for det meste om mængdeteoretiske begreber, f.eks. “ækvivalensklasse” og “partition”. Til sidst i afsnittet formulerer vi hovedresultatet — at der til enhver ækvivalensrelation svarer en partition — som et formelt teorem.

1.1 Konnektiver

Ud fra de to basale konnektiver [ ˙¬ x] og [x ⇒ y] definerer vi konjunktion, disjunktion og dobbeltimplikation:

[x ˙∧ y →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[x ˙∧ y ¨= ˙¬ (x ⇒ ˙¬ y)]⌉)]
[x ˙∨ y →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[x ˙∨ y ¨= ˙¬ x ⇒ y]⌉)]
[x ˙⇔ y →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[x ˙⇔ y ¨= (x ⇒ y) ˙∧ (y ⇒ x)]⌉)]

1.2 Negerede formler

Det er ganske enkelt at definere negeret lighed ( ˙¬ x==y) og negeret medlemskab ( ˙¬ x ∈ y):

[x ≠ y →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[x ≠ y ¨= ˙¬ x==y]⌉)]
[x ∉ y →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[x ∉ y ¨= ˙¬ x ∈ y]⌉)]^{1}

1.3 Delmængde

Mængden x er en delmængde af y hviss ethvert medlem af x også tilhører y:

[x ⊆ y →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[x ⊆ y ¨= (s ∈ x ⇒ s ∈ y)]⌉)]

1.4 Singleton-mængde

[{x, x}] er mængden, der indeholder x som sit eneste element. Vi definerer [{x, x}] ved at parre x med sig selv:

[{x} →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[{x} ¨= {x, x}]⌉)]

1.5 Binær foreningsmængde og fællesmængde

Vi definerer foreningsmængden mellem to mængder x og y som følger:

[x ∪ y →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[x ∪ y ¨= ∪{{x}, {y}}]⌉)]

1Højresiderne i disse definitioner skal læses som hhv. [ ˙¬ x==y] og [ ˙¬ x ∈ y].

Fællesmængden mellem to mængder x og y er en delmængde af deres foreningsmængde:

[x∩y →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[x∩y ¨= {ph ∈ x ∪ y | ph_{3} ∈ x ˙∧ ph_{3} ∈ y}]⌉)]

1.6 Relation

Det ordnede par {{x, x}, {x, y}} indeholder x som “førstekomponent” og y som “andenkomponent”. Den følgende definition af {{x, x}, {x, y}} er den mest udbredte i litteraturen (se f.eks. afsnit 4.3 i [?] og afsnit 2.1 i [?]):

[⟨x, y⟩ →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[⟨x, y⟩ ¨= {{x}, {x, y}}]⌉)]

Vi kan nu definere en “relation” som en mængde af ordnede par. Vi udtrykker denne definition ved at formalisere, hvad det vil sige, at x er relateret til y i kraft af relationen r:

[r(x, y) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[r(x, y) ¨= ⟨x, y⟩ ∈ r]⌉)]

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

1.7 Ækvivalensrelation

At en relation er refleksiv på en mængde x vil sige, at alle elementer i x er relateret til sig selv:

[ReflRel(r, x) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[ReflRel(r, x) ¨= ∀s: (s ∈ x ⇒
r(s, s))]⌉)]

At en relation er symmetrisk på en mængde x vil sige, at alle elementer i x opfylder den følgende implikation:

[SymRel(r, x) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[SymRel(r, x) ¨= ∀s,
t: (s ∈ x ⇒ t ∈ x ⇒ r(s, t) ⇒ r(t, s))]⌉)]

At en relation er transitiv på en mængde x vil sige, at alle elementer i x opfylder den følgende implikation:

[TransRel(r, x) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[TransRel(r, x) ¨=
∀s, t, u: (s ∈ x ⇒ t ∈ x ⇒ u ∈ x ⇒ r(s, t) ⇒ r(t, u) ⇒ r(s, u))]⌉)]

Endelig er en ækvivalensrelation det samme som en relation, der er refleksiv, symmetrisk og transitiv:

[EqRel(r, x) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[EqRel(r, x) ¨= ReflRel(r, x) ˙∧ SymRel(r, x) ˙∧ TransRel(r, x)]⌉)]

1.8 Mængde-variable

Mange af rapportens beviser sker i forhold til en uspecificeret mængde. Vi vil referere til denne mængde med metavariablen \underline{bs} og objektvariablen \overline{bs}:

[BS →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[BS ¨= \underline{bs}]⌉)]
[OBS →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[OBS ¨= \overline{bs}]⌉)]^{2}

Vi vil så vidt muligt bruge metavariablen, men i afsnit ?? og senere bliver det nødvendigt at gå over til objektvariablen.

1.9 Ækvivalensklasse

Lad r være en ækvivalensrelation defineret på bs, og lad x være et medlem af bs. Vi definerer ækvivalensklassen {ph ∈ bs | {{ph_{1}, ph_{1}}, {ph_{1}, x}} ∈ r} som den delmængde af bs, hvis medlemmer står i forhold til x:

[[x ∈ bs]_{r} →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[[x ∈ bs]_{r} ¨= {ph ∈ bs | r(ph_{1}, x)}]⌉)] Ækvivalenssystemet {ph ∈ P(bs) | ˙¬ t_{Ex} ∈ bs ⇒ ˙¬ {ph ∈ bs | {{ph_{1}, ph_{1}}, {ph_{1}, t_{Ex}}} ∈ r}==ph_{2}} er mængden af alle de ækvivalensklasser, som bs definerer på r. Vi definerer {ph ∈ P(bs) | ˙¬ t_{Ex} ∈ bs ⇒ ˙¬ {ph ∈ bs | {{ph_{1}, ph_{1}}, {ph_{1}, t_{Ex}}} ∈ r}==ph_{2}} som en delmængde af potensmængden P(bs):

2Navnene “\underline{bs}” og “\overline{bs}” står for hhv. for “big set” og “object big set”. Konstruktionerne [\underline{x}] og [\overline{x}] omdanner x til hhv. en meta- og en objektvariabel. Variablen [bs] vil også blive brugt i nogle af de kommende definitioner, men ikke i selve beviserne.

[bs/r →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[bs/r ¨= {ph ∈ P(bs) | Ex_{20} ∈ bs ˙∧ [Ex_{20} ∈

bs]_{r} ==ph_{2}}]⌉)]

1.10 Partition

En partition af en mængde bs er en mængde p, som opfylder tre krav:

1. Ingen af mængderne i p er tomme.

2. Alle mængderne i p er indbyrdes disjunkte.

3. Foreningsmængden af alle mængderne i p er lig med bs.

Den formelle version af denne definition ser således ud:

[Partition(p, bs) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[Partition(p, bs) ¨= (∀s: (s ∈
p ⇒ s ≠ Ø)) ˙∧
(∀s, t: (s ∈ p ⇒ t ∈ p ⇒ s ≠ t ⇒ s ∩ t ==Ø)) ˙∧
∪ p==bs]⌉)]

(*** MAKROER SLUTTER ***)

2 Deduktionsreglen

Dette bilag præsenterer dén version af deduktionsreglen fra [?], som jeg har gjort brug af. Underafsnit ?? forklarer, hvorfor jeg har ændret på den oprindelige regel, og underafsnit 2.1 indeholder selve den ændrede kode (som er skrevet i L).

2.1 Kode

Funktionen [λx.Dedu_{0}(⌈p⌉, ⌈c⌉)] er en kopi af [λx.Ded_{0}(⌈p⌉, ⌈c⌉)] fra [?]:

[Dedu(p, c) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[Dedu(p, c) ¨= λx.Dedu_{0}(⌈p⌉, ⌈c⌉)]⌉)]

Jeg har ændret funktionen [Ded_{0}(p, c)], så den kalder [Dedu_{s}(Dedu_{7}(p), c, T)] i stedet for [Ded_{1}(Ded_{7}(p), c, T)]:

[Dedu_{0}(p, c) →^{val} c!If(Dedu_{8}(p, T), Dedu_{s}(Dedu_{7}(p), c, T), F)]

Funktionen [Dedu_{s}(p, c, s)] giver straks kontrollen videre til [Ded_{1}(p, c, s)] — medmindre p og c begynder med et antal identiske sidebetingelser. I så fald flyttes disse sidebetingelser fra p og c over til listen s, før kontrollen går videre til [Ded_{1}(p, c, s)]:

[Dedu_{s}(p, c, s) →^{val} If(p =^{r} ⌈x ⊢⊢ y⌉, c =^{r} ⌈x ⊢⊢ y⌉ ∧ p^{1} =^{t} c^{1} ∧ Dedu_{s}(p^{2}, c^{2}, c^{1} : : s), Dedu_{1}(p, c, s))]

Fra og med [Ded_{1}(p, c, s)] er koden kopieret fra appendikset til [?]:

[Dedu_{1}(p, c, s) →^{val} If(c =^{r} ⌈x ⊢⊢ y⌉, Dedu_{1}(p, c^{2}, c^{1} : : s), Dedu_{2}(p, c, s))] [Dedu_{2}(p, c, s) →^{val} s!p =^{r} ⌈x ⊢ y⌉ ∧ c =^{r} ⌈x ⇒ y⌉ \{ \begin{matrix}Dedu_{3}(p^{1}, c^{1}, s, T) ∧ Dedu_{2}(p^{2}, c^{2}, s) \\ Dedu_{4}(p, c, s, Dedu_{6}(p, c, T, T)) \end{matrix} ]

[Dedu_{3}(p, c, s, b) →^{val} If(¬c =^{r} ⌈∀_{obj}x: y⌉, Dedu_{4}(p, c, s, b), If(p =^{r} ⌈∀_{obj}x: y⌉ ∧ p^{1} =^{t} c^{1}, Dedu_{4}(p, c, s, b), Dedu_{3}(p, c^{2}, s, c^{1} : : c^{1} : : b)))] [Dedu_{4}(p, c, s, b) →^{val} s!b!If(p =^{r} ⌈\overline{x}⌉, lookup(p, b, T) =^{t} c, If(¬p =^{r} c, F, If(p =^{r} ⌈∀_{obj}x: y⌉, p^{1} =^{t} c^{1} ∧ Dedu_{4}(p^{2}, c^{2}, s, p^{1} : : p^{1} : : b), If(¬p =^{r} ⌈\underline{x}⌉, Dedu_{4}^{∗}(p^{t}, c^{t}, s, b), p^{1} =^{t} c^{1} ∧ Dedu_{5}(p, s, b)))))] [Dedu_{4}^{∗}(p, c, s, b) →^{val} c!s!b!If(p, T, Dedu_{4}(p^{h}, c^{h}, s, b) ∧ Dedu_{4}^{∗}(p^{t}, c^{t}, s, b))] [Dedu_{5}(p, s, b) →^{val} p!s!If(b, T, ⌈⌈x⌉#^{0}⌈y⌉⌉^{h} : : ⌈⌈∗⌉⌉^{h} : : b^{h}^{h} : : T : : ⌈⌈x⌉⌉^{h} : : p : : T : : T ∈_{t} s ∧ Dedu_{5}(p, s, b^{t}))] [Dedu_{6}(p, c, e, b) →^{val} p!c!b!e!If(p =^{r} ⌈\overline{x}⌉, p ∈_{t} e \{ \begin{matrix}b \\ p : : c : : b \end{matrix} , If(¬p =^{r} c, T, If(p =^{r} ⌈\underline{a}⌉, b, If(p =^{r} ⌈∀_{obj}x: y⌉, Dedu_{6}(p^{2}, c^{2}, c^{1} : : e, b), Dedu_{6}^{∗}(p^{t}, c^{t}, e, b)))))] [Dedu_{6}^{∗}(p, c, e, b) →^{val} p!c!b!e!If(p, b, Dedu_{6}^{∗}(p^{t}, c^{t}, e, Dedu_{6}(p^{h}, c^{h}, e, b)))] [Dedu_{7}(p) →^{val} p =^{r} ⌈∀x: y⌉ \{ \begin{matrix}Dedu_{7}(p^{2}) \\ p \end{matrix} ]

[Dedu_{8}(p, b) →^{val} If(p =^{r} ⌈∀x: y⌉, Dedu_{8}(p^{2}, p^{1} : : b), If(p =^{r} ⌈\underline{a}⌉, p ∈_{t} b, Dedu_{8}^{∗}(p^{t}, b)))] [Dedu_{8}^{∗}(p, b) →^{val} b!If(p, T, If(Dedu_{8}(p^{h}, b), Dedu_{8}^{∗}(p^{t}, b), F))] (*** EKSISTENS-VARIABLE ***)

[x^{Ex} →^{val} x =^{r} ⌈x_{Ex}⌉] Vi kan da definere de fire eksistens-variable, som denne rapport vil gøre brug af (jvf. bilag ??):

[Ex_{1} →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[Ex_{1} ¨= a_{Ex}]⌉)] [Ex_{2} →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[Ex_{2} ¨= b_{Ex}]⌉)] [Ex_{10} →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[Ex_{10} ¨= j_{Ex}]⌉)] [Ex_{20} →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[Ex_{20} ¨= t_{Ex}]⌉)] [⟨a≡b|x:==t⟩_{Ex} →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[⟨a≡b|x:==t⟩_{Ex} ¨= ⟨⌈a⌉≡^{0}⌈b⌉|⌈x⌉:==⌈t⌉⟩_{Ex}]⌉)] [⟨a≡^{0}b|x:==t⟩_{Ex} →^{val} λc.x^{Ex} ∧ ⟨a≡^{1}b|x:==t⟩_{Ex}] [⟨a≡^{1}b|x:==t⟩_{Ex} →^{val} a!x!t! If(b =^{r} ⌈∀_{obj}u: v⌉, F, If(b^{Ex} ∧ b =^{t} x, a =^{t} t, If( a =^{r} b, ⟨a^{t}≡^{∗}b^{t}|x:==t⟩_{Ex}, F)))]

[⟨a≡^{∗}b|x:==t⟩_{Ex} →^{val} b!x!t!If(a, T, If(⟨a^{h}≡^{1}b^{h}|x:==t⟩_{Ex}, ⟨a^{t}≡^{∗}b^{t}|x:==t⟩_{Ex}, F))]

(*** AKSIOMATISK SYSTEM ***) [ZFsub →^{stmt} ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} ∗ \underline{y} ∗ \underline{z} = \underline{x} ∗ \underline{y} ∗ \underline{z} ⊕ ∀\underline{(fx)}: ∀\underline{(fy)}: R(\underline{(fx)}) << R(\underline{(fy)}) ⊢ \underline{(fx)} <_{f} \underline{(fy)} ⊕ ∀\underline{m}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: \underline{(fx)} <_{f} \underline{(fy)} ⊢ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⊢ c_{Ex} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= \underline{(fy)}[\underline{m}] + −\underline{(ϵ)} ⊕ ∀\underline{(rx)}: ∀\underline{(ry)}: ∀\underline{(rz)}: \underline{(rx)} == \underline{(ry)} ⊢ \underline{(ry)} == \underline{(rz)} ⊢ \underline{(rx)} == \underline{(rz)} ⊕ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} <= \underline{y} ⇒ \underline{y} <= \underline{z} ⇒ \underline{x} <= \underline{z} ⊕ ∀\underline{x}: ∀\underline{t}: ∀\underline{a}: ∀\underline{b}: ⟨⌈\underline{a}⌉≡^{0}⌈\underline{b}⌉|⌈\underline{x}⌉:==⌈\underline{t}⌉⟩_{Ex} ⊢⊢ \underline{a} ⊢ \underline{b} ⊕ ∀\underline{m}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ c_{Ex} <= \underline{m} ⇒ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) = \underline{(ϵ)} ⊢ SF(\underline{(fx)}, \underline{(fy)}) ⊕ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⇒ \underline{x} + \underline{z} = \underline{y} + \underline{z} ⊕ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} + \underline{y} + \underline{z} = \underline{x}+\underline{y} +\underline{z} ⊕ ∀\underline{s}: ∀\underline{x}: ˙¬ \underline{s} ∈ ∪\underline{x} ⇒ ˙¬ \underline{s} ∈ j_{Ex} ⇒ ˙¬ j_{Ex} ∈ \underline{x} ⇒ ˙¬ ˙¬ \underline{s} ∈ j_{Ex} ⇒ ˙¬ j_{Ex} ∈ \underline{x} ⇒ \underline{s} ∈ ∪\underline{x} ⊕ ∀\underline{m}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(rx)}: ∀\underline{(ry)}: \underline{(rx)} << \underline{(ry)} ⊢ \underline{(fx)} ∈ \underline{(rx)} ⊢ \underline{(fy)} ∈ \underline{(ry)} ⊢ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⊢ a_{Ex} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= \underline{(fy)}[\underline{m}] + −\underline{(ϵ)} ⊕ ∀\underline{a}: ∀\underline{x}: ∀\underline{y}: \underline{a} ⊢ if(\underline{a}, \underline{x}, \underline{y}) = \underline{x} ⊕ ∀\underline{x}: ∀\underline{a}: \underline{a} ⊢ ∀_{obj}\underline{x}: \underline{a} ⊕ ∀\underline{m}: ∀\underline{(fx)}: −_{f} \underline{(fx)}[\underline{m}] = −\underline{(fx)}[\underline{m}] ⊕ ∀\underline{(fx)}: ∀\underline{(fy)}: R(\underline{(fx)}) ∗ ∗R(\underline{(fy)}) == R(\underline{(fx)} ∗_{f} \underline{(fy)}) ⊕ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} ∗ \underline{y} + \underline{z} = \underline{x} ∗ \underline{y} + \underline{x} ∗ \underline{z} ⊕ ∀\underline{x}: \underline{x} + −\underline{x} = 0 ⊕ ∀\underline{a}: ∀\underline{b}: ∀\underline{p}: ∀\underline{x}: ∀\underline{z}: \underline{p}^{Ph} ∧ ⟨\underline{b}≡\underline{a}|\underline{p}:==\underline{z}⟩_{Ph} ⊢⊢ ˙¬ \underline{z} ∈ \{ph ∈ \underline{x} | \underline{a}\} ⇒ ˙¬ \underline{z} ∈ \underline{x} ⇒ ˙¬ \underline{b} ⇒ ˙¬ ˙¬ \underline{z} ∈ \underline{x} ⇒ ˙¬ \underline{b} ⇒ \underline{z} ∈ \{ph ∈ \underline{x} | \underline{a}\} ⊕ ∀\underline{(rx)}: ∀\underline{(ry)}: \underline{(rx)} << \underline{(ry)} ⊢ t_{Ex} ∈ \underline{(ry)} ⊕ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: \underline{(fx)} =_{f} \underline{(fy)} ⊢ \underline{(fx)}[\underline{m}] = \underline{(fy)}[\underline{m}] ⊕ ∀\underline{(rx)}: ∀\underline{(ry)}: \underline{(rx)} == \underline{(ry)} ⊢ \underline{(ry)} == \underline{(rx)} ⊕ ∀\underline{x}: \underline{x} <= \underline{x} ⊕ ∀\underline{a}: ∀\underline{b}: ˙¬ \underline{b} ⇒ \underline{a} ⊢ ˙¬ \underline{b} ⇒ ˙¬ \underline{a} ⊢ \underline{b} ⊕ ∀\underline{m}: 1f[\underline{m}] = 1 ⊕ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⇒ \underline{x} = \underline{z} ⇒ \underline{y} = \underline{z} ⊕ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} <= \underline{y} ⇒ \underline{x} + \underline{z} <= \underline{y} + \underline{z} ⊕ ∀\underline{s}: ˙¬ \underline{s} ∈ Ø ⊕ ∀\underline{(fx)}: ∀\underline{(fy)}: R(\underline{(fx)}) == R(\underline{(fy)}) ⊢ SF(\underline{(fx)}, \underline{(fy)}) ⊕ ∀\underline{(fx)}: ∀\underline{(rx)}: ∀\underline{(ry)}: \underline{(rx)} == \underline{(ry)} ⊢ \underline{(fx)} ∈ \underline{(rx)} ⊢ \underline{(fx)} ∈ \underline{(ry)} ⊕ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: \underline{(fx)} +_{f} \underline{(fy)}[\underline{m}] = \underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}] ⊕ ∀\underline{(fx)}: ∀\underline{(fy)}: \underline{(fx)} ∈ R(\underline{(fy)}) ⊢ SF(\underline{(fx)}, \underline{(fy)}) ⊕ ∀\underline{x}: ˙¬ \underline{x} = 0 ⇒ \underline{x} ∗ rec\underline{x} = 1 ⊕ ∀\underline{x}: \underline{x} + 0 = \underline{x} ⊕ ∀\underline{m}: ∀\underline{n}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(rx)}: ∀\underline{(ry)}: \underline{(fx)} ∈ \underline{(rx)} ⇒ \underline{(fy)} ∈ \underline{(ry)} ⇒ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ a_{Ex} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= \underline{(fy)}[\underline{m}] + −\underline{(ϵ)} ⊢ \underline{(rx)} << \underline{(ry)} ⊕ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: \underline{(fx)}[\underline{m}] = \underline{(fy)}[\underline{m}] ⊢ \underline{(fx)} =_{f} \underline{(fy)} ⊕ ∀\underline{x}: ∀\underline{y}: \underline{x} <= \underline{y} ⇒ \underline{y} <= \underline{x} ⇒ \underline{x} = \underline{y} ⊕ ∀\underline{a}: ∀\underline{b}: λx.Dedu_{0}(⌈\underline{a}⌉, ⌈\underline{b}⌉) ⊢⊢ \underline{a} ⊢ \underline{b} ⊕ ∀\underline{m}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: SF(\underline{(fx)}, \underline{(fy)}) ⊢ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⊢ c_{Ex} <= \underline{m} ⇒ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) = \underline{(ϵ)} ⊕ ∀\underline{x}: ∀\underline{y}: \underline{x} = \underline{y} ⇒ \underline{x} <= \underline{y} ⊕ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: 0 <= \underline{z} ⇒ \underline{x} <= \underline{y} ⇒ \underline{x} ∗ \underline{z} <= \underline{y} ∗ \underline{z} ⊕ ∀\underline{s}: ∀\underline{x}: ∀\underline{y}: ˙¬ \underline{s} ∈ \{\underline{x}, \underline{y}\} ⇒ ˙¬ \underline{s}==\underline{x} ⇒ \underline{s}==\underline{y} ⇒ ˙¬ ˙¬ \underline{s}==\underline{x} ⇒ \underline{s}==\underline{y} ⇒ \underline{s} ∈ \{\underline{x}, \underline{y}\} ⊕ ∀\underline{(fx)}: ∀\underline{(fy)}: SF(\underline{(fx)}, \underline{(fy)}) ⊢ R(\underline{(fx)}) == R(\underline{(fy)}) ⊕ ∀\underline{(fx)}: ∀\underline{(rx)}: ∀\underline{(ry)}: \underline{(rx)} == \underline{(ry)} ⊢ \underline{(fx)} ∈ \underline{(ry)} ⊢ \underline{(fx)} ∈ \underline{(rx)} ⊕ ∀\underline{a}: ∀\underline{b}: \underline{a} ⇒ \underline{b} ⊢ \underline{a} ⊢ \underline{b} ⊕ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: \underline{(fx)} ∗_{f} \underline{(fy)}[\underline{m}] = \underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}] ⊕ ∀\underline{(fx)}: ∀\underline{(fy)}: R((fx) +_{f} (fy)) == R(\underline{(fx)} +_{f} \underline{(fy)}) ⊕ ∀\underline{x}: \underline{x} ∗ 1 = \underline{x} ⊕ ∀\underline{x}: ∀\underline{y}: \underline{x} + \underline{y} = \underline{y} + \underline{x} ⊕ ∀\underline{s}: ∀\underline{x}: ˙¬ \underline{s} ∈ P(\underline{x}) ⇒ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{s} ⇒ \overline{s} ∈ \underline{x} ⇒ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{s} ⇒ \overline{s} ∈ \underline{x} ⇒ \underline{s} ∈ P(\underline{x}) ⊕ ∀\underline{(rx)}: ∀\underline{(ry)}: \underline{(rx)} << \underline{(ry)} ⊢ j_{Ex} ∈ \underline{(rx)} ⊕ ∀\underline{a}: ∀\underline{x}: ∀\underline{y}: ˙¬ \underline{a} ⊢ if(\underline{a}, \underline{x}, \underline{y}) = \underline{y} ⊕ ∀\underline{(rx)}: \underline{(rx)} == \underline{(rx)} ⊕ ∀\underline{a}: \underline{a} ⊢ \underline{a} ⊕ ∀\underline{m}: 0f[\underline{m}] = 0 ⊕ ˙¬ 0 = 1 ⊕ ∀\underline{x}: ∀\underline{y}: ˙¬ \underline{x} <= \underline{y} ⇒ \underline{y} <= \underline{x} ⊕ ∀\underline{x}: ∀\underline{y}: ˙¬ \underline{x}==\underline{y} ⇒ ∀_{obj}\overline{s}: ˙¬ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ⇒ ˙¬ \overline{s} ∈ \underline{y} ⇒ \overline{s} ∈ \underline{x} ⇒ ˙¬ ∀_{obj}\overline{s}: ˙¬ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ⇒ ˙¬ \overline{s} ∈ \underline{y} ⇒ \overline{s} ∈ \underline{x} ⇒ \underline{x}==\underline{y} ⊕ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(rx)}: ∀\underline{(ry)}: \underline{(fx)} ∈ \underline{(rx)} ⇒ \underline{(fy)} ∈ \underline{(ry)} ⇒ SF(\underline{(fx)}, \underline{(fy)}) ⊢ \underline{(rx)} == \underline{(ry)} ⊕ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⇒ \underline{x} ∗ \underline{z} = \underline{y} ∗ \underline{z} ⊕ ∀\underline{m}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ c_{Ex} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= \underline{(fy)}[\underline{m}] + −\underline{(ϵ)} ⊢ \underline{(fx)} <_{f} \underline{(fy)} ⊕ ∀\underline{(fx)}: ∀\underline{(fy)}: \underline{(fx)} <_{f} \underline{(fy)} ⊢ R(\underline{(fx)}) << R(\underline{(fy)}) ⊕ ∀\underline{x}: ∀\underline{y}: \underline{x} ∗ \underline{y} = \underline{y} ∗ \underline{x}]

[MP →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} ⇒ \underline{b} ⊢ \underline{a} ⊢ \underline{b}][MP →^{proof} Rule tactic]

[Gen →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{a}: \underline{a} ⊢ ∀_{obj}\underline{x}: \underline{a}][Gen →^{proof} Rule tactic]

[Repetition →^{stmt} ZFsub ⊢ ∀\underline{a}: \underline{a} ⊢ \underline{a}][Repetition →^{proof} Rule tactic]

[Neg →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ \underline{b} ⇒ \underline{a} ⊢ ˙¬ \underline{b} ⇒ ˙¬ \underline{a} ⊢ \underline{b}][Neg →^{proof} Rule tactic]

[Ded →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: λx.Dedu_{0}(⌈\underline{a}⌉, ⌈\underline{b}⌉) ⊢⊢ \underline{a} ⊢ \underline{b}][Ded →^{proof} Rule tactic]

[ExistIntro →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{t}: ∀\underline{a}: ∀\underline{b}: ⟨⌈\underline{a}⌉≡^{0}⌈\underline{b}⌉|⌈\underline{x}⌉:==⌈\underline{t}⌉⟩_{Ex} ⊢⊢ \underline{a} ⊢ \underline{b}][ExistIntro →^{proof} Rule tactic]

[Extensionality →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ \underline{x}==\underline{y} ⇒ ∀_{obj}\overline{s}: ˙¬ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ⇒ ˙¬ \overline{s} ∈ \underline{y} ⇒ \overline{s} ∈ \underline{x} ⇒ ˙¬ ∀_{obj}\overline{s}: ˙¬ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ⇒ ˙¬ \overline{s} ∈ \underline{y} ⇒ \overline{s} ∈ \underline{x} ⇒ \underline{x}==

\underline{y}][Extensionality →^{proof} Rule tactic]

[Ødef →^{stmt} ZFsub ⊢ ∀\underline{s}: ˙¬ \underline{s} ∈ Ø][Ødef →^{proof} Rule tactic]

[PairDef →^{stmt} ZFsub ⊢ ∀\underline{s}: ∀\underline{x}: ∀\underline{y}: ˙¬ \underline{s} ∈ \{\underline{x}, \underline{y}\} ⇒ ˙¬ \underline{s}==\underline{x} ⇒ \underline{s}==\underline{y} ⇒ ˙¬ ˙¬ \underline{s}==

\underline{x} ⇒ \underline{s}==\underline{y} ⇒ \underline{s} ∈ \{\underline{x}, \underline{y}\}][PairDef →^{proof} Rule tactic]

[UnionDef →^{stmt} ZFsub ⊢ ∀\underline{s}: ∀\underline{x}: ˙¬ \underline{s} ∈ ∪\underline{x} ⇒ ˙¬ \underline{s} ∈ j_{Ex} ⇒ ˙¬ j_{Ex} ∈ \underline{x} ⇒ ˙¬ ˙¬ \underline{s} ∈ j_{Ex} ⇒ ˙¬ j_{Ex} ∈ \underline{x} ⇒ \underline{s} ∈ ∪\underline{x}][UnionDef →^{proof} Rule tactic]

[PowerDef →^{stmt} ZFsub ⊢ ∀\underline{s}: ∀\underline{x}: ˙¬ \underline{s} ∈ P(\underline{x}) ⇒ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{s} ⇒ \overline{s} ∈ \underline{x} ⇒ ˙¬ ∀_{obj}\overline{s}: \overline{s} ∈

\underline{s} ⇒ \overline{s} ∈ \underline{x} ⇒ \underline{s} ∈ P(\underline{x})][PowerDef →^{proof} Rule tactic]

[SeparationDef →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{p}: ∀\underline{x}: ∀\underline{z}: \underline{p}^{Ph} ∧ ⟨\underline{b}≡\underline{a}|\underline{p}:==\underline{z}⟩_{Ph} ⊢⊢ ˙¬ \underline{z} ∈ \{ph ∈ \underline{x} | \underline{a}\} ⇒ ˙¬ \underline{z} ∈ \underline{x} ⇒ ˙¬ \underline{b} ⇒ ˙¬ ˙¬ \underline{z} ∈ \underline{x} ⇒ ˙¬ \underline{b} ⇒ \underline{z} ∈ \{ph ∈ \underline{x} | \underline{a}\}][SeparationDef →^{proof} Rule tactic]

3 Udsagnslogisk bibliotek

I dette afsnit vil jeg bevise en samling af udsagnslogiske sandheder ( eller “tautologier”), som vil blive brugt i de følgende afsnit. De fleste af disse tautologier har mange andre anvendelser end lige netop mængdelære. Beviserne er fordelt på 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 så tekniske, at jeg har ladet dem stå alene.

3.1 MP-lemmaer

Man får ofte brug for at anvende slutningsreglen MP flere gange i træk. Derfor vil jeg begynde med at vise fire lemmaer, der kan klare mellem 2 og 5 anvendelser af MP^{3}:

[MP2 →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⊢ \underline{a} ⊢ \underline{b} ⊢ \underline{c}]

[MP3 →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⇒ \underline{d} ⊢ \underline{a} ⊢ \underline{b} ⊢ \underline{c} ⊢ \underline{d}]

[MP4 →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: ∀\underline{e}: \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⇒ \underline{d} ⇒ \underline{e} ⊢ \underline{a} ⊢ \underline{b} ⊢ \underline{c} ⊢ \underline{d} ⊢ \underline{e}]

[MP5 →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: ∀\underline{e}: ∀\underline{f}: \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⇒ \underline{d} ⇒ \underline{e} ⇒ \underline{f} ⊢ \underline{a} ⊢ \underline{b} ⊢ \underline{c} ⊢ \underline{d} ⊢ \underline{e} ⊢ \underline{f}]

3.1.1 Det første bevis

Vi begynder med at bevise MP2:

[MP2 →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⊢ \underline{a} ⊢ \underline{b} ⊢ \underline{c}]

[MP2 →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⊢ \underline{a} ⊢ \underline{b} ⊢ MP ⊳ \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⊳ \underline{a} ≫ \underline{b} ⇒ \underline{c}; MP ⊳ \underline{b} ⇒ \underline{c} ⊳ \underline{b} ≫ \underline{c}⌉, p_{0}, c)]

Da dette er rapportens første bevis, vil jeg bringe nogle ekstra kommentarer^{4}. Oven over beviset har jeg gentaget definitionen af det, der skal bevises; dette er kun for overblikkets skyld — det er ikke en formel nødvendighed. Selve beviset for MP2 består af seks linier, nummereret fra 1 til 6. En bevislinie kan have to former. Den første form er:

     Argumentation ≫ Konklusion

hvor Konklusion er det som linien beviser, mens teksten i Argumentation udgør en begrundelse for, at Konklusion gælder. F.eks. siger linie 5, at meta-formlen [\underline{b} ⇒ \underline{c}] gælder, fordi den kan udledes fra slutningsreglen MP ved substitution. Argumentationen skal læses på den måde, at konklusionerne fra linie 2 og 3 bliver brugt som præmisser til MP. Den generelle betydning af konstruktionen [x ⊳ y] er, at konklusionen fra linie y bliver brugt som præmis i forhold til x. Den anden form, en bevislinie kan have, er:

3I afsnit ?? får vi faktisk brug for at anvende MP 6 gange i træk; men et eller andet sted skal man jo stoppe.

4Denne beskrivelse er en revideret udgave af afsnit 5.1 i [?].

     Nøgleord ≫ Konklusion

hvor Nøgleord er et af de tre ord “Arbitrary”, “Premise” eller “Side-condition”. Betydningen af ordene “Premise” og “Side-condition” er åbenlys: De angiver, at liniens konklusion indgår som en præmis (hhv. sidebetingelse) i den sætning, der skal bevises. F.eks. siger bevisets linie 2, at MP2 bruger meta-formlen [\underline{a} ⇒ \underline{b} ⇒ \underline{c}] som præmis. Når ordet “Arbitrary” bruges, består konklusionen af en liste af meta-variable (f.eks. [\underline{a}, \underline{b}, \underline{c}] i linie 1). Ideen hermed er at udtrykke, at vi ikke antager noget om de pågældende meta-variable, og at vi derfor har ret til at binde dem med en meta-alkvantor i den sætning, der skal bevises. I det forhåndenværende bevis berettiger linien med “Arbitrary” altså, at MP2 er kvantificeret med [∀\underline{a}: ∀\underline{b}: ∀\underline{c}: (⋯)]. Alt dette har drejet sig om den formelle syntaks for et Logiweb bevis. Der er ikke så meget at sige om selve beviset for MP2; vi indkapsler simpelthen to på hinanden følgende anvendelser af MP.

3.1.2 Beviser for de andre MP-lemmaer

Beviserne for de øvrige MP-lemmaer er lige ud ad landevejen:

[MP3 →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⇒ \underline{d} ⊢ \underline{a} ⊢ \underline{b} ⊢ \underline{c} ⊢ \underline{d}]

[MP3 →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⇒ \underline{d} ⊢ \underline{a} ⊢ \underline{b} ⊢ \underline{c} ⊢ MP2 ⊳ \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⇒ \underline{d} ⊳ \underline{a} ⊳ \underline{b} ≫ \underline{c} ⇒ \underline{d}; MP ⊳ \underline{c} ⇒ \underline{d} ⊳ \underline{c} ≫ \underline{d}⌉, p_{0}, c)]

[MP4 →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: ∀\underline{e}: \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⇒ \underline{d} ⇒ \underline{e} ⊢ \underline{a} ⊢ \underline{b} ⊢ \underline{c} ⊢ \underline{d} ⊢ \underline{e}]

[MP4 →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: ∀\underline{e}: \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⇒ \underline{d} ⇒ \underline{e} ⊢ \underline{a} ⊢ \underline{b} ⊢ \underline{c} ⊢ \underline{d} ⊢ MP2 ⊳ \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⇒ \underline{d} ⇒ \underline{e} ⊳ \underline{a} ⊳ \underline{b} ≫ \underline{c} ⇒ \underline{d} ⇒ \underline{e}; MP2 ⊳ \underline{c} ⇒ \underline{d} ⇒ \underline{e} ⊳ \underline{c} ⊳ \underline{d} ≫ \underline{e}⌉, p_{0}, c)]

[MP5 →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: ∀\underline{e}: ∀\underline{f}: \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⇒ \underline{d} ⇒ \underline{e} ⇒ \underline{f} ⊢ \underline{a} ⊢ \underline{b} ⊢ \underline{c} ⊢ \underline{d} ⊢ \underline{e} ⊢ \underline{f}]

[MP5 →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: ∀\underline{e}: ∀\underline{f}: \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⇒ \underline{d} ⇒ \underline{e} ⇒ \underline{f} ⊢ \underline{a} ⊢ \underline{b} ⊢ \underline{c} ⊢ \underline{d} ⊢ \underline{e} ⊢ MP3 ⊳ \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⇒ \underline{d} ⇒ \underline{e} ⇒ \underline{f} ⊳ \underline{a} ⊳ \underline{b} ⊳ \underline{c} ≫ \underline{d} ⇒ \underline{e} ⇒ \underline{f}; MP2 ⊳ \underline{d} ⇒ \underline{e} ⇒ \underline{f} ⊳ \underline{d} ⊳ \underline{e} ≫ \underline{f}⌉, p_{0}, c)]

3.2 Implikation

Dette afsnit indeholder en række lemmaer vedr. implikation, grupperet i fire under-underafsnit.

3.2.1 Refleksivitet; blok-konstruktionen

Lemmaet AutoImply udsiger, at implikations-relationen er refleksiv:

[AutoImply →^{stmt} ZFsub ⊢ ∀\underline{a}: \underline{a} ⇒ \underline{a}]

[AutoImply →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: \underline{a} ⊢ Repetition ⊳ \underline{a} ≫ \underline{a}; ∀\underline{a}: Ded ⊳ ∀\underline{a}: \underline{a} ⊢ \underline{a} ≫ \underline{a} ⇒ \underline{a}⌉, p_{0}, c)]

Beviset for AutoImply indeholder to nye ting i forhold til de hidtidige beviser: En bevisblok, og en anvendelse af deduktions-reglen. En bevisblok er selvstændig enhed i et bevis; den afhænger ikke af den øvrige del af beviset. Den ovenstående bevisblok indeholder et bevis for lemmaet [∀\underline{a}: \underline{a} ⊢ \underline{a}]. Pointen er nu, at blokkens sidste linie (linie 5) fungerer som en forkortelse for dette lemma. Vi kan da anvende deduktionsreglen på denne linie til at omdanne inferensen [∀\underline{a}: \underline{a} ⊢ \underline{a}] til implikationen [\underline{a} ⇒ \underline{a}]. Det vigtigste formål med deduktionsreglen er netop, at vi let kan skifte fra inferens til implikation.

3.2.2 Transitivitet

Lemmaet ImplyTransitivity udsiger, at implikations-relationen er transitiv:

[ImplyTransitivity →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: \underline{a} ⇒ \underline{b} ⊢ \underline{b} ⇒ \underline{c} ⊢ \underline{a} ⇒ \underline{c}]

Vi viser ImplyTransitivity ved hjælp af MP og deduktionsreglen:

[ImplyTransitivity →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: \underline{a} ⇒ \underline{b} ⊢ \underline{b} ⇒ \underline{c} ⊢ \underline{a} ⊢ MP ⊳ \underline{a} ⇒ \underline{b} ⊳ \underline{a} ≫ \underline{b}; MP ⊳ \underline{b} ⇒ \underline{c} ⊳ \underline{b} ≫ \underline{c}; ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: \underline{a} ⇒ \underline{b} ⊢ \underline{b} ⇒ \underline{c} ⊢ Ded ⊳ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: \underline{a} ⇒ \underline{b} ⊢ \underline{b} ⇒ \underline{c} ⊢ \underline{a} ⊢ \underline{c} ≫ \underline{a} ⇒ \underline{b} ⇒ \underline{b} ⇒ \underline{c} ⇒ \underline{a} ⇒ \underline{c}; MP2 ⊳ \underline{a} ⇒ \underline{b} ⇒ \underline{b} ⇒ \underline{c} ⇒ \underline{a} ⇒ \underline{c} ⊳ \underline{a} ⇒ \underline{b} ⊳ \underline{b} ⇒ \underline{c} ≫ \underline{a} ⇒ \underline{c}⌉, p_{0}, c)]

3.2.3 Svækkelse

Vi får ofte brug for det følgende ræsonnement: Hvis formlen \underline{a} gælder ubetinget, så gælder den også under antagelse af en vilkårlig anden formel \underline{b}. Lemmaet Weakening udtrykker dette ræsonnement som følger:

[Weakening →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: \underline{b} ⊢ \underline{a} ⇒ \underline{b}]

Vi beviser Weakening ved hjælp af deduktionsreglen:

[Weakening →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: \underline{b} ⊢ \underline{a} ⊢ Repetition ⊳ \underline{b} ≫ \underline{b}; ∀\underline{a}: ∀\underline{b}: Ded ⊳ ∀\underline{a}: ∀\underline{b}: \underline{b} ⊢ \underline{a} ⊢ \underline{b} ≫ \underline{b} ⇒ \underline{a} ⇒ \underline{b}; \underline{b} ⊢ MP ⊳ \underline{b} ⇒ \underline{a} ⇒ \underline{b} ⊳ \underline{b} ≫ \underline{a} ⇒ \underline{b}⌉, p_{0}, c)] 3.2.4 Modsigelse

Det sidste lemma i dette afsnit vedrører strengt taget ikke implikation, men derimod inferens (x ⊢ y). Lemmaet FromContradiction udsiger, at vi kan bevise hvad som helst, hvis vi har bevist to formler, der modsiger hinanden:

[FromContradiction →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} ⊢ ˙¬ \underline{a} ⊢ \underline{b}]

Beviset bruger Weakening og slutningsreglen Neg:

[FromContradiction →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} ⊢ ˙¬ \underline{a} ⊢ Weakening ⊳ \underline{a} ≫ ˙¬ \underline{b} ⇒ \underline{a}; Weakening ⊳ ˙¬ \underline{a} ≫ ˙¬ \underline{b} ⇒ ˙¬ \underline{a}; Neg ⊳ ˙¬ \underline{b} ⇒ \underline{a} ⊳ ˙¬ \underline{b} ⇒ ˙¬ \underline{a} ≫ \underline{b}⌉, p_{0}, c)]

3.3 Håndtering af dobbeltnegationer

De to lemmaer RemoveDoubleNeg og AddDoubleNeg tillader os hhv. at fjerne og tilføje dobbeltnegationer. Jeg vil ikke kommentere beviserne:

[RemoveDoubleNeg →^{stmt} ZFsub ⊢ ∀\underline{a}: ˙¬ ˙¬ \underline{a} ⊢ \underline{a}]

[RemoveDoubleNeg →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: ˙¬ ˙¬ \underline{a} ⊢ Weakening ⊳ ˙¬ ˙¬ \underline{a} ≫ ˙¬ \underline{a} ⇒ ˙¬ ˙¬ \underline{a}; AutoImply ≫ ˙¬ \underline{a} ⇒ ˙¬ \underline{a}; Neg⊳ ˙¬ \underline{a} ⇒ ˙¬ \underline{a}⊳ ˙¬ \underline{a} ⇒ ˙¬ ˙¬ \underline{a} ≫ \underline{a}⌉, p_{0}, c)]

[AddDoubleNeg →^{stmt} ZFsub ⊢ ∀\underline{a}: \underline{a} ⊢ ˙¬ ˙¬ \underline{a}]

[AddDoubleNeg →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: ˙¬ ˙¬ ˙¬ \underline{a} ⊢ RemoveDoubleNeg ⊳ ˙¬ ˙¬ ˙¬ \underline{a} ≫ ˙¬ \underline{a}; ∀\underline{a}: Ded⊳∀\underline{a}: ˙¬ ˙¬ ˙¬ \underline{a} ⊢ ˙¬ \underline{a} ≫ ˙¬ ˙¬ ˙¬ \underline{a} ⇒ ˙¬ \underline{a}; \underline{a} ⊢ Weakening ⊳\underline{a} ≫ ˙¬ ˙¬ ˙¬ \underline{a} ⇒ \underline{a}; Neg ⊳ ˙¬ ˙¬ ˙¬ \underline{a} ⇒ \underline{a} ⊳ ˙¬ ˙¬ ˙¬ \underline{a} ⇒ ˙¬ \underline{a} ≫ ˙¬ ˙¬ \underline{a}⌉, p_{0}, c)]

3.4 Modus tollens og beslægtede lemmaer

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

[MT →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} ⇒ \underline{b} ⊢ ˙¬ \underline{b} ⊢ ˙¬ \underline{a}]

For at vise MT begynder vi med et teknisk lemma, der ikke har den store værdi i sig selv:

[Technicality →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} ⇒ \underline{b} ⊢ ˙¬ ˙¬ \underline{a} ⇒ \underline{b}]

[Technicality →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} ⇒ \underline{b} ⊢ ˙¬ ˙¬ \underline{a} ⊢ RemoveDoubleNeg ⊳ ˙¬ ˙¬ \underline{a} ≫ \underline{a}; MP ⊳ \underline{a} ⇒ \underline{b} ⊳ \underline{a} ≫ \underline{b}; ∀\underline{a}: ∀\underline{b}: Ded ⊳ ∀\underline{a}: ∀\underline{b}: \underline{a} ⇒ \underline{b} ⊢ ˙¬ ˙¬ \underline{a} ⊢ \underline{b} ≫ \underline{a} ⇒ \underline{b} ⇒ ˙¬ ˙¬ \underline{a} ⇒ \underline{b}; \underline{a} ⇒ \underline{b} ⊢ MP ⊳ \underline{a} ⇒ \underline{b} ⇒ ˙¬ ˙¬ \underline{a} ⇒ \underline{b} ⊳ \underline{a} ⇒ \underline{b} ≫ ˙¬ ˙¬ \underline{a} ⇒ \underline{b}⌉, p_{0}, c)]

Uafhængigt af Technicality kan vi vise en version af MT, hvor \underline{a} optræder i negeret form:

[NegativeMT →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ \underline{a} ⇒ \underline{b} ⊢ ˙¬ \underline{b} ⊢ \underline{a}]

[NegativeMT →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ \underline{a} ⇒ \underline{b} ⊢ ˙¬ \underline{b} ⊢ Weakening ⊳ ˙¬ \underline{b} ≫ ˙¬ \underline{a} ⇒ ˙¬ \underline{b}; Neg ⊳ ˙¬ \underline{a} ⇒ \underline{b} ⊳ ˙¬ \underline{a} ⇒ ˙¬ \underline{b} ≫ \underline{a}⌉, p_{0}, c)]

Ud fra Technicality og NegativeMT kan vi nu vise MT:

[MT →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} ⇒ \underline{b} ⊢ ˙¬ \underline{b} ⊢ ˙¬ \underline{a}]

[MT →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} ⇒ \underline{b} ⊢ ˙¬ \underline{b} ⊢ Technicality ≫ ˙¬ ˙¬ \underline{a} ⇒ \underline{b}; NegativeMT ⊳ ˙¬ ˙¬ \underline{a} ⇒ \underline{b} ⊳ ˙¬ \underline{b} ≫ ˙¬ \underline{a}⌉, p_{0}, c)]

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

[Contrapositive →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} ⇒ \underline{b} ⊢ ˙¬ \underline{b} ⇒ ˙¬ \underline{a}]

Når en inferens skal erstattes med en implikation, er det altid deduktionsreglen, der skal i spil:

[Contrapositive →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} ⇒ \underline{b} ⊢ ˙¬ \underline{b} ⊢ MT ⊳ \underline{a} ⇒ \underline{b} ⊳ ˙¬ \underline{b} ≫ ˙¬ \underline{a}; ∀\underline{a}: ∀\underline{b}: \underline{a} ⇒ \underline{b} ⊢ Ded ⊳ ∀\underline{a}: ∀\underline{b}: \underline{a} ⇒ \underline{b} ⊢ ˙¬ \underline{b} ⊢ ˙¬ \underline{a} ≫ \underline{a} ⇒ \underline{b} ⇒ ˙¬ \underline{b} ⇒ ˙¬ \underline{a}; MP ⊳ \underline{a} ⇒ \underline{b} ⇒ ˙¬ \underline{b} ⇒ ˙¬ \underline{a} ⊳ \underline{a} ⇒ \underline{b} ≫ ˙¬ \underline{b} ⇒ ˙¬ \underline{a}⌉, p_{0}, c)]

3.5 Konjunktion

Hovedmålet med dette underafsnit er at konvertere mellem formlerne \underline{a} og \underline{b} og deres konjunktion [ ˙¬ \underline{a} ⇒ ˙¬ \underline{b}].

3.5.1 Forening af konjunkter

Vi begynder med at slå \underline{a} og \underline{b} sammen til [ ˙¬ \underline{a} ⇒ ˙¬ \underline{b}]:

[JoinConjuncts →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} ⊢ \underline{b} ⊢ ˙¬ \underline{a} ⇒ ˙¬ \underline{b}]

Beviset for JoinConjuncts er af teknisk karakter. Vi viser den makroekspanderede form [ ˙¬ \underline{a} ⇒ ˙¬ \underline{b}], som vi i bevisets sidste linie konverterer til [ ˙¬ \underline{a} ⇒ ˙¬ \underline{b}]. Denne sidste linie er ikke nødvendig for bevischeckeren, men den gør beviset lidt nemmere at læse:

[JoinConjuncts →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} ⊢ \underline{a} ⇒ ˙¬ \underline{b} ⊢ MP ⊳ \underline{a} ⇒ ˙¬ \underline{b} ⊳ \underline{a} ≫ ˙¬ \underline{b}; ∀\underline{a}: ∀\underline{b}: Ded ⊳ ∀\underline{a}: ∀\underline{b}: \underline{a} ⊢ \underline{a} ⇒ ˙¬ \underline{b} ⊢ ˙¬ \underline{b} ≫ \underline{a} ⇒ \underline{a} ⇒ ˙¬ \underline{b} ⇒ ˙¬ \underline{b}; \underline{a} ⊢ \underline{b} ⊢ MP ⊳ \underline{a} ⇒ \underline{a} ⇒ ˙¬ \underline{b} ⇒ ˙¬ \underline{b} ⊳ \underline{a} ≫ \underline{a} ⇒ ˙¬ \underline{b} ⇒ ˙¬ \underline{b}; AddDoubleNeg ⊳ \underline{b} ≫ ˙¬ ˙¬ \underline{b}; MT ⊳ \underline{a} ⇒ ˙¬ \underline{b} ⇒ ˙¬ \underline{b} ⊳ ˙¬ ˙¬ \underline{b} ≫ ˙¬ \underline{a} ⇒ ˙¬ \underline{b}; Repetition ⊳ ˙¬ \underline{a} ⇒ ˙¬ \underline{b} ≫ ˙¬ \underline{a} ⇒ ˙¬ \underline{b}⌉, p_{0}, c)] 3.5.2 Udskilning af anden konjunkt

Tautologien SecondConjunct lader os udskille den anden konjunkt fra [ ˙¬ \underline{a} ⇒ ˙¬ \underline{b}]. Jeg vil ikke kommentere beviset:

[SecondConjunct →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ \underline{a} ⇒ ˙¬ \underline{b} ⊢ \underline{b}]

[SecondConjunct →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ \underline{b} ⊢ Weakening ⊳ ˙¬ \underline{b} ≫ \underline{a} ⇒ ˙¬ \underline{b}; ∀\underline{a}: ∀\underline{b}: Ded ⊳ ∀\underline{a}: ∀\underline{b}: ˙¬ \underline{b} ⊢ \underline{a} ⇒ ˙¬ \underline{b} ≫ ˙¬ \underline{b} ⇒ \underline{a} ⇒ ˙¬ \underline{b}; ˙¬ \underline{a} ⇒ ˙¬ \underline{b} ⊢ Repetition ⊳ ˙¬ \underline{a} ⇒ ˙¬ \underline{b} ≫ ˙¬ \underline{a} ⇒ ˙¬ \underline{b}; NegativeMT ⊳ ˙¬ \underline{b} ⇒ \underline{a} ⇒ ˙¬ \underline{b} ⊳ ˙¬ \underline{a} ⇒ ˙¬ \underline{b} ≫ \underline{b}⌉, p_{0}, c)]

3.5.3 Udskilning af første konjunkt

For at udskille \underline{a} fra [ ˙¬ \underline{a} ⇒ ˙¬ \underline{b}] viser vi først, at [ ˙¬ \underline{a} ⇒ ˙¬ \underline{b}] er kommutativ. Jeg vil ikke kommentere beviset:

[AndCommutativity →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ \underline{a} ⇒ ˙¬ \underline{b} ⊢ ˙¬ \underline{b} ⇒ ˙¬ \underline{a}]

[AndCommutativity →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: \underline{b} ⇒ ˙¬ \underline{a} ⊢ \underline{a} ⊢ AddDoubleNeg ⊳ \underline{a} ≫ ˙¬ ˙¬ \underline{a}; MT ⊳ \underline{b} ⇒ ˙¬ \underline{a} ⊳ ˙¬ ˙¬ \underline{a} ≫ ˙¬ \underline{b}; ∀\underline{a}: ∀\underline{b}: Ded ⊳ ∀\underline{a}: ∀\underline{b}: \underline{b} ⇒ ˙¬ \underline{a} ⊢ \underline{a} ⊢ ˙¬ \underline{b} ≫ \underline{b} ⇒ ˙¬ \underline{a} ⇒ \underline{a} ⇒ ˙¬ \underline{b}; ˙¬ \underline{a} ⇒ ˙¬ \underline{b} ⊢ Repetition ≫ ˙¬ \underline{a} ⇒ ˙¬ \underline{b}; MT ⊳ \underline{b} ⇒ ˙¬ \underline{a} ⇒ \underline{a} ⇒ ˙¬ \underline{b} ⊳ ˙¬ \underline{a} ⇒ ˙¬ \underline{b} ≫ ˙¬ \underline{b} ⇒ ˙¬ \underline{a}; Repetition ⊳ ˙¬ \underline{b} ⇒ ˙¬ \underline{a} ≫ ˙¬ \underline{b} ⇒ ˙¬ \underline{a}⌉, p_{0}, c)]

Nu er det let at udskille den første konjunkt fra [ ˙¬ \underline{a} ⇒ ˙¬ \underline{b}]: Først vender vi konjunktionen om til [ ˙¬ \underline{b} ⇒ ˙¬ \underline{a}] ved hjælp af AndCommutativity, og så udskiller vi \underline{a} ved hjælp af SecondConjunct:

[FirstConjunct →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ \underline{a} ⇒ ˙¬ \underline{b} ⊢ \underline{a}]

[FirstConjunct →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ \underline{a} ⇒ ˙¬ \underline{b} ⊢ AndCommutativity ⊳ ˙¬ \underline{a} ⇒ ˙¬ \underline{b} ≫ ˙¬ \underline{b} ⇒ ˙¬ \underline{a}; SecondConjunct ⊳ ˙¬ \underline{b} ⇒ ˙¬ \underline{a} ≫ \underline{a}⌉, p_{0}, c)]

3.6 Dobbeltimplikation

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

3.6.1 Brug sammen med modus ponens

De følgende to tautologier gør det let at bruge anvende slutningsreglen MP på dobbeltimplikationer. Beviserne er enkle og kræver ingen kommentarer:

[IffFirst →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ \underline{a} ⇒ \underline{b} ⇒ ˙¬ \underline{b} ⇒ \underline{a} ⊢ \underline{b} ⊢ \underline{a}]

[IffFirst →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ \underline{a} ⇒ \underline{b} ⇒ ˙¬ \underline{b} ⇒ \underline{a} ⊢ \underline{b} ⊢ SecondConjunct ⊳ ˙¬ \underline{a} ⇒ \underline{b} ⇒ ˙¬ \underline{b} ⇒ \underline{a} ≫ \underline{b} ⇒ \underline{a}; MP ⊳ \underline{b} ⇒ \underline{a} ⊳ \underline{b} ≫ \underline{a}⌉, p_{0}, c)] [IffSecond →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ \underline{a} ⇒ \underline{b} ⇒ ˙¬ \underline{b} ⇒ \underline{a} ⊢ \underline{a} ⊢ \underline{b}]

[IffSecond →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ \underline{a} ⇒ \underline{b} ⇒ ˙¬ \underline{b} ⇒ \underline{a} ⊢ \underline{a} ⊢ FirstConjunct ⊳ ˙¬ \underline{a} ⇒ \underline{b} ⇒ ˙¬ \underline{b} ⇒ \underline{a} ≫ \underline{a} ⇒ \underline{b}; MP ⊳ \underline{a} ⇒ \underline{b} ⊳ \underline{a} ≫ \underline{b}⌉, p_{0}, c)]

3.6.2 Kommutativitet

Lemmaet IffCommutativity følger direkte af, at operatoren [ ˙¬ x ⇒ ˙¬ y] er kommutativ:

[IffCommutativity →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ \underline{a} ⇒ \underline{b} ⇒ ˙¬ \underline{b} ⇒ \underline{a} ⊢ ˙¬ \underline{b} ⇒ \underline{a} ⇒ ˙¬ \underline{a} ⇒ \underline{b}]

[IffCommutativity →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ \underline{a} ⇒ \underline{b} ⇒ ˙¬ \underline{b} ⇒ \underline{a} ⊢ Repetition ⊳ ˙¬ \underline{a} ⇒ \underline{b} ⇒ ˙¬ \underline{b} ⇒ \underline{a} ≫ ˙¬ \underline{a} ⇒ \underline{b} ⇒ ˙¬ \underline{b} ⇒ \underline{a}; AndCommutativity ⊳ ˙¬ \underline{a} ⇒ \underline{b} ⇒ ˙¬ \underline{b} ⇒ \underline{a} ≫ ˙¬ \underline{b} ⇒ \underline{a} ⇒ ˙¬ \underline{a} ⇒ \underline{b}; Repetition ⊳ ˙¬ \underline{b} ⇒ \underline{a} ⇒ ˙¬ \underline{a} ⇒ \underline{b} ≫ ˙¬ \underline{b} ⇒ \underline{a} ⇒ ˙¬ \underline{a} ⇒ \underline{b}⌉, p_{0}, c)]

3.7 Disjunktion

Dette underafsnit indeholder tre lemmaer vedr. disjunktion, som vi fordeler på to under-underafsnit.

3.7.1 Svækkelse

Givet en påstand \underline{b} vil vi gerne udlede de svagere påstande [ ˙¬ \underline{a} ⇒ \underline{b}] og [ ˙¬ \underline{b} ⇒ \underline{a}]. Den første slutning varetages af lemmaet WeakenOr1:

[WeakenOr1 →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: \underline{b} ⊢ ˙¬ \underline{a} ⇒ \underline{b}]

Beviset består af en simpel anvendelse af Weakening:

[WeakenOr1 →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: \underline{b} ⊢ Weakening ⊳ \underline{b} ≫ ˙¬ \underline{a} ⇒ \underline{b}; Repetition ⊳ ˙¬ \underline{a} ⇒ \underline{b} ≫ ˙¬ \underline{a} ⇒ \underline{b}⌉, p_{0}, c)]

Slutningen fra \underline{a} til [ ˙¬ \underline{a} ⇒ \underline{b}] varetages af lemmaet WeakenOr2:

[WeakenOr2 →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} ⊢ ˙¬ \underline{a} ⇒ \underline{b}]

Kernen i beviset for WeakenOr2 er en anvendelse af FromContradiction:

[WeakenOr2 →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} ⊢ ˙¬ \underline{a} ⊢ FromContradiction ⊳ \underline{a} ⊳ ˙¬ \underline{a} ≫ \underline{b}; ∀\underline{a}: ∀\underline{b}: Ded ⊳ ∀\underline{a}: ∀\underline{b}: \underline{a} ⊢ ˙¬ \underline{a} ⊢ \underline{b} ≫ \underline{a} ⇒ ˙¬ \underline{a} ⇒ \underline{b}; \underline{a} ⊢ MP⊳\underline{a} ⇒ ˙¬ \underline{a} ⇒ \underline{b}⊳\underline{a} ≫ ˙¬ \underline{a} ⇒ \underline{b}; Repetition⊳ ˙¬ \underline{a} ⇒ \underline{b} ≫ ˙¬ \underline{a} ⇒ \underline{b}⌉, p_{0}, c)] 3.7.2 Slutning ud fra disjunktion

Lemmaet FromDisjuncts lader os drage slutninger ud fra en disjunktion:

[FromDisjuncts →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ˙¬ \underline{a} ⇒ \underline{b} ⊢ \underline{a} ⇒ \underline{c} ⊢ \underline{b} ⇒ \underline{c} ⊢ \underline{c}]

Om beviset vil jeg kun sige, at det er en ret elegant øvelse i bevisteknik:

[FromDisjuncts →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ˙¬ \underline{a} ⇒ \underline{b} ⊢ \underline{a} ⇒ \underline{c} ⊢ \underline{b} ⇒ \underline{c} ⊢ Repetition ⊳ ˙¬ \underline{a} ⇒ \underline{b} ≫ ˙¬ \underline{a} ⇒ \underline{b}; Contrapositive ⊳ ˙¬ \underline{a} ⇒ \underline{b} ≫ ˙¬ \underline{b} ⇒ ˙¬ ˙¬ \underline{a}; Technicality ⊳ \underline{a} ⇒ \underline{c} ≫ ˙¬ ˙¬ \underline{a} ⇒ \underline{c}; ImplyTransitivity ⊳ ˙¬ \underline{b} ⇒ ˙¬ ˙¬ \underline{a} ⊳ ˙¬ ˙¬ \underline{a} ⇒ \underline{c} ≫ ˙¬ \underline{b} ⇒ \underline{c}; Contrapositive ⊳ ˙¬ \underline{b} ⇒ \underline{c} ≫ ˙¬ \underline{c} ⇒ ˙¬ ˙¬ \underline{b}; Contrapositive ⊳ \underline{b} ⇒ \underline{c} ≫ ˙¬ \underline{c} ⇒ ˙¬ \underline{b}; Neg ⊳ ˙¬ \underline{c} ⇒ ˙¬ \underline{b} ⊳ ˙¬ \underline{c} ⇒ ˙¬ ˙¬ \underline{b} ≫ \underline{c}⌉, p_{0}, c)]

(********************)

[am →^{prio}

Preassociative [am], [base], [bracket ∗ end bracket], [big bracket ∗ end bracket], [ $ ∗ $ ], [flush left [∗]], [x], [y], [z], [[∗ ▷◁ ∗]], [[∗ →^{∗} ∗]], [pyk], [tex], [name], [prio], [∗], [T], [if(∗, ∗, ∗)], [[∗ ⇒^{∗} ∗]], [val], [claim], [⊥], [f(∗)], [(∗)^{I}], [F], [\underline{0}], [\underline{1}], [\underline{2}], [\underline{3}], [\underline{4}], [\underline{5}], [\underline{6}], [\underline{7}], [\underline{8}], [\underline{9}], [0], [1], [2], [3], [4], [5], [6], [7], [8], [9], [a], [b], [c], [d], [e], [f], [g], [h], [i], [j], [k], [l], [m], [n], [o], [p], [q], [r], [s], [t], [u], [v], [w], [(∗)^{M}], [If(∗, ∗, ∗)], [array{∗} ∗ end array], [l], [c], [r], [empty], [⟨∗ | ∗:= ∗⟩], [M(∗)], [Ũ(∗)], [U(∗)], [U^{M}(∗)], [apply(∗, ∗)], [apply_{1}(∗, ∗)], [identifier(∗)], [identifier_{1}(∗, ∗)], [arrayplus(∗, ∗)], [array-remove(∗, ∗, ∗)], [array-put(∗, ∗, ∗, ∗)], [array-add(∗, ∗, ∗, ∗, ∗)], [bit(∗, ∗)], [bit_{1}(∗, ∗)], [rack], ["vector"], ["bibliography"], ["dictionary"], ["body"], ["codex"], ["expansion"], ["code"], ["cache"], ["diagnose"], ["pyk"], ["tex"], ["texname"], ["value"], ["message"], ["macro"], ["definition"], ["unpack"], ["claim"], ["priority"], ["lambda"], ["apply"], ["true"], ["if"], ["quote"], ["proclaim"], ["define"], ["introduce"], ["hide"], ["pre"], ["post"], [E(∗, ∗, ∗)], [E_{2}(∗, ∗, ∗, ∗, ∗)], [E_{3}(∗, ∗, ∗, ∗)], [E_{4}(∗, ∗, ∗, ∗)], [lookup(∗, ∗, ∗)], [abstract(∗, ∗, ∗, ∗)], [⌈∗⌉], [M(∗, ∗, ∗)], [M_{2}(∗, ∗, ∗, ∗)], [M^{∗}(∗, ∗, ∗)], [macro], [s_{0}], [zip(∗, ∗)], [assoc_{1}(∗, ∗, ∗)], [(∗)^{p}], [self], [[∗ ¨= ∗]], [[∗ ˙= ∗]], [[∗ ´= ∗]], [[∗ =^{pyk} ∗]], [[∗ =^{tex} ∗]], [[∗ =^{name} ∗]], [Priority table[∗]], [M̃_{1}], [M̃_{2}(∗)], [M̃_{3}(∗)], [M̃_{4}(∗, ∗, ∗, ∗)], [M̃(∗, ∗, ∗)], [Q̃(∗, ∗, ∗)], [Q̃_{2}(∗, ∗, ∗)], [Q̃_{3}(∗, ∗, ∗, ∗)], [Q̃^{∗}(∗, ∗, ∗)], [(∗)], [(∗)], [display(∗)], [statement(∗)], [[∗]^{·}], [[∗]^{−}], [aspect(∗, ∗)], [aspect(∗, ∗, ∗)], [⟨∗⟩], [tuple_{1}(∗)], [tuple_{2}(∗)], [let_{2}(∗, ∗)], [let_{1}(∗, ∗)], [[∗ =^{claim} ∗]], [checker], [check(∗, ∗)], [check_{2}(∗, ∗, ∗)], [check_{3}(∗, ∗, ∗)], [check^{∗}(∗, ∗)], [check_{2}^{∗}(∗, ∗, ∗)], [[∗]^{·}], [[∗]^{−}], [[∗]^{∘}], [msg], [[∗ =^{msg} ∗]], [<stmt>], [stmt], [[∗ =^{stmt} ∗]], [HeadNil′], [HeadPair′], [Transitivity′], [⊥ ⊥], [Contra′], [T′_{E}], [L_{1}], [\underline{∗}], [A], [B], [C], [D], [E], [F], [G], [H], [I], [J], [K], [L], [M], [N], [O], [P], [Q], [R], [S], [T ], [U], [V], [W], [X], [Y], [Z], [⟨∗ | ∗:= ∗⟩], [⟨^{∗}∗ | ∗:= ∗⟩], [∅], [Remainder], [(∗)^{v}], [intro(∗, ∗, ∗, ∗)], [intro(∗, ∗, ∗)], [error(∗, ∗)], [error_{2}(∗, ∗)], [proof(∗, ∗, ∗)], [proof_{2}(∗, ∗)], [S(∗, ∗)], [S^{I}(∗, ∗)], [S^{⊳} (∗, ∗)], [S_{1}^{⊳} (∗, ∗, ∗)], [S^{E}(∗, ∗)], [S_{1}^{E}(∗, ∗, ∗)], [S^{+}(∗, ∗)], [S_{1}^{+}(∗, ∗, ∗)], [S^{−}(∗, ∗)], [S_{1}^{−}(∗, ∗, ∗)], [S^{∗}(∗, ∗)], [S_{1}^{∗}(∗, ∗, ∗)], [S_{2}^{∗}(∗, ∗, ∗, ∗)], [S^{@}(∗, ∗)], [S_{1}^{@}(∗, ∗, ∗)], [S^{⊢}(∗, ∗)], [S_{1}^{⊢}(∗, ∗, ∗, ∗)], [S⊢ ⊢(∗, ∗)],

[S⊢ ⊢ ; 1(∗, ∗, ∗, ∗)], [S^{i.e.}(∗, ∗)], [S_{1}^{i.e.}(∗, ∗, ∗, ∗)], [S_{2}^{i.e.}(∗, ∗, ∗, ∗, ∗)], [S^{∀}(∗, ∗)],

[S_{1}^{∀}(∗, ∗, ∗, ∗)], [S^{;}(∗, ∗)], [S_{1}^{;}(∗, ∗, ∗)], [S_{2}^{;}(∗, ∗, ∗, ∗)], [T (∗)], [claims(∗, ∗, ∗)], [claims_{2}(∗, ∗, ∗)], [<proof>], [proof], [[Lemma ∗: ∗]], [[Proof of ∗: ∗]], [[∗ lemma ∗: ∗]], [[∗ antilemma ∗: ∗]], [[∗ rule ∗: ∗]], [[∗ antirule ∗: ∗]], [verifier], [V_{1}(∗)], [V_{2}(∗, ∗)], [V_{3}(∗, ∗, ∗, ∗)], [V_{4}(∗, ∗)], [V_{5}(∗, ∗, ∗, ∗)], [V_{6}(∗, ∗, ∗, ∗)], [V_{7}(∗, ∗, ∗, ∗)], [Cut(∗, ∗)], [Head_{⊕}(∗)], [Tail_{⊕}(∗)], [rule_{1}(∗, ∗)], [rule(∗, ∗)], [Rule tactic], [Plus(∗, ∗)], [[Theory ∗]], [theory_{2}(∗, ∗)], [theory_{3}(∗, ∗)], [theory_{4}(∗, ∗, ∗)], [HeadNil′′], [HeadPair′′], [Transitivity′′], [Contra′′], [HeadNil], [HeadPair], [Transitivity], [Contra], [T_{E}], [ragged right], [ragged right expansion ], [parm(∗, ∗, ∗)], [parm^{∗}(∗, ∗, ∗)], [inst(∗, ∗)], [inst^{∗}(∗, ∗)], [occur(∗, ∗, ∗)], [occur^{∗}(∗, ∗, ∗)], [unify(∗ = ∗, ∗)], [unify^{∗}(∗ = ∗, ∗)], [unify_{2}(∗ = ∗, ∗)], [L_{a}], [L_{b}], [L_{c}], [L_{d}], [L_{e}], [L_{f}], [L_{g}], [L_{h}], [L_{i}], [L_{j}], [L_{k}], [L_{l}], [L_{m}], [L_{n}], [L_{o}], [L_{p}], [L_{q}], [L_{r}], [L_{s}], [L_{t}], [L_{u}], [L_{v}], [L_{w}], [L_{x}], [L_{y}], [L_{z}], [L_{A}], [L_{B}], [L_{C}], [L_{D}], [L_{E}], [L_{F}], [L_{G}], [L_{H}], [L_{I}], [L_{J}], [L_{K}], [L_{L}], [L_{M}], [L_{N}], [L_{O}], [L_{P}], [L_{Q}], [L_{R}], [L_{S}], [L_{T}], [L_{U}], [L_{V}], [L_{W}], [L_{X}], [L_{Y}], [L_{Z}], [L_{?}], [Reflexivity], [Reflexivity_{1}], [Commutativity], [Commutativity_{1}], [<tactic>], [tactic], [[∗ =^{tactic} ∗]], [P(∗, ∗, ∗)], [P^{∗}(∗, ∗, ∗)], [p_{0}], [conclude_{1}(∗, ∗)], [conclude_{2}(∗, ∗, ∗)], [conclude_{3}(∗, ∗, ∗, ∗)], [conclude_{4}(∗, ∗)], [check], [[∗ =^{∘} ∗]], [RootVisible(∗)], [A], [R], [C], [T], [L], [\{∗\}], [\overline{∗}], [a], [b], [c], [d], [e], [f ], [g], [h], [i], [j ], [k], [l], [m], [n], [o], [p], [q], [r], [s], [t], [u], [v], [w], [x], [y], [z], [⟨∗≡ ∗ | ∗:=∗⟩], [⟨∗≡^{0} ∗ | ∗:=∗⟩], [⟨∗≡^{1} ∗ | ∗:=∗⟩], [⟨∗≡^{∗} ∗ | ∗:=∗⟩], [Ded(∗, ∗)], [Ded_{0}(∗, ∗)], [Ded_{1}(∗, ∗, ∗)], [Ded_{2}(∗, ∗, ∗)], [Ded_{3}(∗, ∗, ∗, ∗)], [Ded_{4}(∗, ∗, ∗, ∗)], [Ded_{4}^{∗}(∗, ∗, ∗, ∗)], [Ded_{5}(∗, ∗, ∗)], [Ded_{6}(∗, ∗, ∗, ∗)], [Ded_{6}^{∗}(∗, ∗, ∗, ∗)], [Ded_{7}(∗)], [Ded_{8}(∗, ∗)], [Ded_{8}^{∗}(∗, ∗)], [S], [Neg], [MP], [Gen], [Ded], [S1], [S2], [S3], [S4], [S5], [S6], [S7], [S8], [S9], [Repetition], [A1′], [A2′], [A4′], [A5′], [Prop 3.2a], [Prop 3.2b], [Prop 3.2c], [Prop 3.2d], [Prop 3.2e_{1}], [Prop 3.2e_{2}], [Prop 3.2e], [Prop 3.2f_{1}], [Prop 3.2f_{2}], [Prop 3.2f], [Prop 3.2g_{1}], [Prop 3.2g_{2}], [Prop 3.2g], [Prop 3.2h_{1}], [Prop 3.2h_{2}], [Prop 3.2h], [Block_{1}(∗, ∗, ∗)], [Block_{2}(∗)], [(⋯)], [Objekt-var], [Ex-var], [Ph-var], [Værdi], [Variabel], [Op(∗)], [Op(∗, ∗)], [∗ ¨ == ∗], [ContainsEmpty(∗)], [Dedu(∗, ∗)], [Dedu_{0}(∗, ∗)], [Dedu_{s}(∗, ∗, ∗)], [Dedu_{1}(∗, ∗, ∗)], [Dedu_{2}(∗, ∗, ∗)], [Dedu_{3}(∗, ∗, ∗, ∗)], [Dedu_{4}(∗, ∗, ∗, ∗)], [Dedu_{4}^{∗}(∗, ∗, ∗, ∗)], [Dedu_{5}(∗, ∗, ∗)], [Dedu_{6}(∗, ∗, ∗, ∗)], [Dedu_{6}^{∗}(∗, ∗, ∗, ∗)], [Dedu_{7}(∗)], [Dedu_{8}(∗, ∗)], [Dedu_{8}^{∗}(∗, ∗)], [Ex_{1}], [Ex_{2}], [Ex3], [Ex_{10}], [Ex_{20}], [∗_{Ex}], [∗^{Ex}], [⟨∗≡ ∗ | ∗:==∗⟩_{Ex}], [⟨∗≡^{0} ∗ | ∗:==∗⟩_{Ex}], [⟨∗≡^{1} ∗ | ∗:==∗⟩_{Ex}], [⟨∗≡^{∗} ∗ | ∗:==∗⟩_{Ex}], [ph_{1}], [ph_{2}], [ph_{3}], [∗_{Ph}], [∗^{Ph}], [⟨∗≡ ∗ | ∗:==∗⟩_{Ph}], [⟨∗≡^{0} ∗ | ∗:==∗⟩_{Ph}], [⟨∗≡^{1} ∗ | ∗:==∗⟩_{Ph}], [⟨∗≡^{∗} ∗ | ∗:==∗⟩_{Ph}], [bs], [OBS], [BS], [Ø], [ZFsub], [MP], [Gen], [Repetition], [Neg], [Ded], [ExistIntro], [Extensionality], [Ødef], [PairDef], [UnionDef], [PowerDef], [SeparationDef], [AddDoubleNeg], [RemoveDoubleNeg], [AndCommutativity], [AutoImply], [Contrapositive], [FirstConjunct], [SecondConjunct], [FromContradiction], [FromDisjuncts], [IffCommutativity], [IffFirst], [IffSecond], [ImplyTransitivity], [JoinConjuncts], [MP2], [MP3], [MP4], [MP5], [MT], [NegativeMT], [Technicality], [Weakening], [WeakenOr1], [WeakenOr2], [Formula2Pair], [Pair2Formula], [Formula2Union], [Union2Formula], [Formula2Sep], [Sep2Formula], [SubsetInPower], [HelperPowerIsSub], [PowerIsSub], [(Switch)HelperPowerIsSub], [(Switch)PowerIsSub], [ToSetEquality], [HelperToSetEquality(t)], [ToSetEquality(t)], [HelperFromSetEquality], [FromSetEquality], [HelperReflexivity], [Reflexivity], [HelperSymmetry], [Symmetry], [HelperTransitivity], [Transitivity], [ERisReflexive], [ERisSymmetric], [ERisTransitive], [ØisSubset], [HelperMemberNotØ], [MemberNotØ], [HelperUniqueØ], [UniqueØ], [==Reflexivity], [==Symmetry], [Helper==Transitivity], [==Transitivity], [HelperTransferNotEq], [TransferNotEq], [HelperPairSubset], [Helper(2)PairSubset], [PairSubset], [SamePair], [SameSingleton], [UnionSubset], [SameUnion], [SeparationSubset], [SameSeparation], [SameBinaryUnion], [IntersectionSubset], [SameIntersection], [AutoMember], [HelperEqSysNotØ], [EqSysNotØ], [HelperEqSubset], [EqSubset], [HelperEqNecessary], [EqNecessary], [HelperNoneEqNecessary], [Helper(2)NoneEqNecessary], [NoneEqNecessary], [EqClassIsSubset], [EqClassesAreDisjoint], [AllDisjoint], [AllDisjointImply], [BSsubset], [Union(BS/R)subset], [UnionIdentity], [EqSysIsPartition], [(ϵ)], [(fx)], [(fy)], [(fz)], [(fv)], [var fv], [(rx)], [(ry)], [(rz)], [(ru)], [ϵ], [FX], [FY], [FZ], [FU], [FV], [RX], [RY], [RZ], [RU], [0], [1], [(−1)], [2], [1/2], [0f], [1f], [00], [01], [leqReflexivity], [leqAntisymmetryAxiom], [leqTransitivityAxiom], [leqTotality], [leqAdditionAxiom], [leqMultiplicationAxiom], [plusAssociativity], [plusCommutativity], [Negative], [plus0], [timesAssociativity], [timesCommutativity], [ReciprocalAxiom], [times1], [Distribution], [0not1], [equalityAxiom], [eqLeqAxiom], [eqAdditionAxiom], [eqMultiplicationAxiom], [SENC1], [SENC2], [IfThenElse(T)], [IfThenElse(F)], [From = f], [To = f], [From < f], [To < f], [PlusF], [TimesF], [MinusF], [0f], [1f], [FromSF], [ToSF], [To == XX], [From ==], [To ==], [From << XX], [From << (1)], [From << (2)], [to << XX], [From <<], [To <<], [FromInR], [PlusR], [TimesR], [leqAntisymmetry], [leqTransitivity], [leqAddition], [leqMultiplication], [Reciprocal], [Equality], [eqLeq], [eqAddition], [eqMultiplication], [ToNegatedImply], [TND], [ImplyNegation], [FromNegations], [From3Disjuncts], [From2 ∗ 2Disjuncts], [NegateDisjunct1], [NegateDisjunct2], [ExpandDisjuncts], [eqReflexivity], [eqSymmetry], [eqTransitivity], [eqTransitivity4], [eqTransitivity5], [eqTransitivity6], [plus0Left], [times1Left], [lemma eqAdditionLeft], [EqMultiplicationLeft], [DistributionOut], [Three2twoTerms], [Three2threeTerms], [Three2threeFactors], [AddEquations], [SubtractEquations], [SubtractEquationsLeft], [EqNegated], [PositiveToRight(Eq)], [PositiveToLeft(Eq)(1term)], [NegativeToLeft(Eq)], [LessNeq], [NeqSymmetry], [NeqNegated], [SubNeqRight], [SubNeqLeft], [NeqAddition], [NeqMultiplication], [UniqueNegative], [DoubleMinus], [LeqLessEq], [LessLeq], [FromLeqGeq], [subLeqRight], [subLeqLeft], [Leq + 1], [PositiveToRight(Leq)], [PositiveToRight(Leq)(1term)], [negativeToLeft(Leq)], [LeqAdditionLeft], [leqSubtraction], [leqSubtractionLeft], [thirdGeq], [LeqNegated], [AddEquations(Leq)], [ThirdGeqSeries], [LeqNeqLess], [FromLess], [ToLess], [fromNotLess], [toNotLess], [NegativeLessPositive], [leqLessTransitivity], [LessLeqTransitivity], [LessTransitivity], [LessTotality], [SubLessRight], [SubLessLeft], [LessAddition], [LessAdditionLeft], [LessMultiplication], [LessMultiplicationLeft], [LessDivision], [AddEquations(Less)], [LessNegated], [PositiveNegated], [NonpositiveNegated], [NegativeNegated], [NonnegativeNegated], [PositiveHalved], [NonnegativeNumerical], [NegativeNumerical], [PositiveNumerical], [lemma nonpositiveNumerical], [|0| = 0], [0 <= |x|], [SameNumerical], [SignNumerical(+)], [SignNumerical], [NumericalDifference], [SplitNumericalSumHelper], [splitNumericalSum(++)], [splitNumericalSum(−−)], [splitNumericalSum(+ − small)], [splitNumericalSum(+ − big)], [splitNumericalSum(+−)], [splitNumericalSum(−+)], [splitNumericalSum], [insertMiddleTerm(Numerical)], [x + y = zBackwards], [x ∗ y = zBackwards], [x = x + (y − y)], [x = x + y − y], [], [insertMiddleTerm(Sum)], [insertMiddleTerm(Difference)], [x ∗ 0 + x = x], [x ∗ 0 = 0], [(−1) ∗ (−1) + (−1) ∗ 1 = 0], [(−1) ∗ (−1) = 1], [0 < 1Helper], [0 < 1], [0 < 2], [0 < 1/2], [TwoWholes], [TwoHalves], [−x − y = −(x + y)], [MinusNegated], [Times(−1)], [Times(−1)Left], [−0 = 0], [SFsymmetry], [SFtransitivity], [= fToSameF], [PlusF(Sym)], [TimesF(Sym)], [f2R(Plus)], [f2R(Times)], [PlusR(Sym)], [TimesR(Sym)], [LessLeq(R)], [eqLeq(R)], [SubLessRight(R)], [SubLessLeft(R)], [<< TransitivityHelper(Q)], [<< Transitivity], [<<== Reflexivity], [<<== AntisymmetryHelper(Q)], [<<== Antisymmetry], [<<== Transitivity], [Plus0f], [Plus00], [== Addition], [== AdditionLeft], [<< Addition], [<<== Addition], [PlusAssociativity(F)], [PlusAssociativity(R)], [Negative(R)], [PlusCommutativity(F)], [PlusCommutativity(R)], [TimesAssociativity(F)], [TimesAssociativity(R)], [Times1f], [Times01], [TimesCommutativity(F)], [TimesCommutativity(R)], [Distribution(F)], [Distribution(R)]; Preassociative [∗ {∗}], [∗/indexintro(∗, ∗, ∗, ∗)], [∗/intro(∗, ∗, ∗)], [∗/bothintro(∗, ∗, ∗, ∗, ∗)], [∗/nameintro(∗, ∗, ∗, ∗)], [∗′], [∗[ ∗]], [∗[∗→∗]], [∗[∗⇒∗]], [∗0], [∗1], [0b], [∗-color(∗)], [∗-color^{∗}(∗)], [∗^{H}], [∗^{T}], [∗^{U}], [∗^{h}], [∗^{t}], [∗^{s}], [∗^{c}], [∗^{d}], [∗^{a}], [∗^{C}], [∗^{M}], [∗^{B}], [∗^{r}], [∗^{i}], [∗^{d}], [∗^{R}], [∗^{0}], [∗^{1}], [∗^{2}], [∗^{3}], [∗^{4}], [∗^{5}], [∗^{6}], [∗^{7}], [∗^{8}], [∗^{9}], [∗^{E}], [∗^{V}], [∗^{C}], [∗^{C^{∗}}], [∗^{hide}]; Preassociative [“ ∗ ”], [], [(∗)^{t}], [string(∗) + ∗], [string(∗) ++ ∗], [ ∗], [ ∗], [!∗], ["∗], [#∗], [$∗], [%∗], [&∗], [’∗], [(∗], [)∗], [∗∗], [+∗], [, ∗], [-∗], [.∗], [/∗], [0∗], [1∗], [2∗], [3∗], [4∗], [5∗], [6∗], [7∗], [8∗], [9∗], [:∗], [; ∗], [<∗], [=∗], [>∗], [?∗], [@∗], [A∗], [B∗], [C∗], [D∗], [E∗], [F∗], [G∗], [H∗], [I∗], [J∗], [K∗], [L∗], [M∗], [N∗], [O∗], [P∗], [Q∗], [R∗], [S∗], [T∗], [U∗], [V∗], [W∗], [X∗], [Y∗], [Z∗], [[∗], [\∗], []∗], [ˆ ∗], [ ∗], [‘∗], [a∗], [b∗], [c∗], [d∗], [e∗], [f∗], [g∗], [h∗], [i∗], [j∗], [k∗], [l∗], [m∗], [n∗], [o∗], [p∗], [q∗], [r∗], [s∗], [t∗], [u∗], [v∗], [w∗], [x∗], [y∗], [z∗], [{∗], [|∗], [}∗], [˜ ∗], [Preassociative ∗; ∗], [Postassociative ∗; ∗], [[∗], ∗], [priority ∗ end], [newline ∗], [macro newline ∗], [MacroIndent(∗)]; Preassociative [∗ ’ ∗], [∗ ‘ ∗]; Preassociative [∗′], [R(∗)], [− − R(∗)], [rec∗]; Preassociative [∗/∗], [∗ ∩ ∗], [∗[∗]]; Preassociative [∪∗], [∗ ∪ ∗], [P(∗)]; Preassociative [{∗}]; Preassociative [{∗, ∗}], [⟨∗, ∗⟩], [−∗], [−_{f}∗]; Preassociative [∗ ∈ ∗], [∗(∗, ∗)], [ReflRel(∗, ∗)], [SymRel(∗, ∗)], [TransRel(∗, ∗)], [EqRel(∗, ∗)], [[∗ ∈ ∗]_{∗}], [Partition(∗, ∗)]; Preassociative [∗ · ∗], [∗ ·_{0} ∗], [∗ ∗ ∗], [∗ ∗_{f} ∗], [∗ ∗ ∗∗]; Preassociative [∗ + ∗], [∗ +_{0} ∗], [∗ +_{1} ∗], [∗ − ∗], [∗ −_{0} ∗], [∗ −_{1} ∗], [∗ + ∗], [∗ − ∗], [∗ +_{f} ∗], [∗ −_{f} ∗], [∗ + +∗], [R(∗) − −R(∗)]; Preassociative [| ∗ |], [if(∗, ∗, ∗)]; Preassociative [∗ = ∗], [∗ ≠ ∗], [∗ <= ∗], [∗ < ∗], [∗ =_{f} ∗], [∗ <_{f} ∗], [SF(∗, ∗)], [∗ == ∗], [∗ << ∗], [∗ <<== ∗]; Preassociative [∗ ∪ {∗}], [∗ ∪ ∗], [∗\{∗}]; Postassociative [∗ ˙. . ∗], [∗ \underline{˙. .} ∗], [∗ \underline{: :} ∗], [∗ \underline{+2∗} ∗], [∗ : : ∗], [∗ +2∗ ∗]; Postassociative [∗, ∗]; Preassociative

[∗ ≈^{B} ∗], [∗ ≈^{D} ∗], [∗ ≈^{C} ∗], [∗ ≈^{P} ∗], [∗ ≈ ∗], [∗ = ∗], [∗ →^{+} ∗], [∗ =^{t} ∗], [∗ =^{t^{∗}} ∗], [∗ =^{r} ∗],

[∗ ∈_{t} ∗], [∗ ⊆_{T} ∗], [∗ =^{T} ∗], [∗ =^{s} ∗], [∗ free in ∗], [∗ free in^{∗} ∗], [∗ free for ∗ in ∗], [∗ free for^{∗} ∗ in ∗], [∗ ∈_{c} ∗], [∗ < ∗], [∗ <′ ∗], [∗ ≤′ ∗], [∗ = ∗], [∗ ≠ ∗], [∗^{var}], [∗#^{0}∗], [∗#^{1}∗], [∗#^{∗}∗], [∗==∗], [∗ ⊆ ∗]; Preassociative [¬∗], [ ˙¬ ∗], [∗ ∉ ∗], [∗ ≠ ∗]; Preassociative [∗ ∧ ∗], [∗ ¨∧ ∗], [∗ ˜∧ ∗], [∗ ∧_{c} ∗], [∗ ˙∧ ∗]; Preassociative [∗ ∨ ∗], [∗ ∥ ∗], [∗ ¨∨ ∗]; Postassociative [∗ ˙∨ ∗]; Preassociative [∃∗: ∗], [∀∗: ∗], [∀_{obj}∗: ∗]; Postassociative [∗ ¨⇒ ∗], [∗ ⇒ ∗], [∗ ⇔ ∗], [∗ ˙⇔ ∗]; Preassociative [{ph ∈ ∗ | ∗}]; Postassociative [∗ : ∗], [∗ spy ∗], [∗!∗]; Preassociative

[∗ \{ \begin{matrix}∗ \\ ∗ \end{matrix} ];

Preassociative [λ ∗ .∗], [Λ ∗ .∗], [Λ∗], [if ∗ then ∗ else ∗], [let ∗ = ∗ in ∗], [let ∗ ¨= ∗ in ∗]; Preassociative [∗#∗]; Preassociative [∗^{I}], [∗^{⊳} ], [∗^{V}], [∗^{+}], [∗^{−}], [∗^{∗}]; Preassociative [∗ @ ∗], [∗ ⊳ ∗], [∗ ⊳ ⊳ ∗], [∗ ≫ ∗], [∗ ⊵ ∗]; Postassociative [∗ ⊢ ∗], [∗ ⊢⊢ ∗], [∗ i.e. ∗]; Preassociative [∀∗: ∗], [Π∗: ∗]; Postassociative [∗ ⊕ ∗]; Postassociative [∗; ∗]; Preassociative [∗ proves ∗]; Preassociative [∗ proof of ∗ : ∗], [Line ∗ : ∗ ≫ ∗; ∗], [Last line ∗ ≫ ∗ □], [Line ∗ : Premise ≫ ∗; ∗], [Line ∗ : Side-condition ≫ ∗; ∗], [Arbitrary ≫ ∗; ∗], [Local ≫ ∗ = ∗; ∗], [Begin ∗; ∗ : End; ∗], [Last block line ∗ ≫ ∗; ], [Arbitrary ≫ ∗; ∗]; Postassociative [∗ | ∗]; Postassociative [∗ , ∗], [∗[ ∗]∗]; Preassociative [∗&∗]; Preassociative [∗\\∗], [∗ linebreak[4] ∗], [∗\\∗]; ]

A Pyk definitioner

[(⋯) →^{pyk} “cdots”] [Objekt-var →^{pyk} “object-var”] [Ex-var →^{pyk} “ex-var”] [Ph-var →^{pyk} “ph-var”] [Værdi →^{pyk} “vaerdi”] [Variabel →^{pyk} “variabel”] [Op(∗) →^{pyk} “op " end op”] [Op(∗, ∗) →^{pyk} “op2 " comma " end op2”] [∗ ¨ == ∗ →^{pyk} “define-equal " comma " end equal”] [ContainsEmpty(∗) →^{pyk} “contains-empty " end empty”] [Dedu(∗, ∗) →^{pyk} “1deduction " conclude " end 1deduction”] [Dedu_{0}(∗, ∗) →^{pyk} “1deduction zero " conclude " end 1deduction”] [Dedu_{s}(∗, ∗, ∗) →^{pyk} “1deduction side " conclude " condition " end 1deduction”] [Dedu_{1}(∗, ∗, ∗) →^{pyk} “1deduction one " conclude " condition " end 1deduction”] [Dedu_{2}(∗, ∗, ∗) →^{pyk} “1deduction two " conclude " condition " end 1deduction”] [Dedu_{3}(∗, ∗, ∗, ∗) →^{pyk} “1deduction three " conclude " condition " bound " end 1deduction”] [Dedu_{4}(∗, ∗, ∗, ∗) →^{pyk} “1deduction four " conclude " condition " bound " end 1deduction”] [Dedu_{4}^{∗}(∗, ∗, ∗, ∗) →^{pyk} “1deduction four star " conclude " condition " bound " end 1deduction”] [Dedu_{5}(∗, ∗, ∗) →^{pyk} “1deduction five " condition " bound " end 1deduction”] [Dedu_{6}(∗, ∗, ∗, ∗) →^{pyk} “1deduction six " conclude " exception " bound " end 1deduction”] [Dedu_{6}^{∗}(∗, ∗, ∗, ∗) →^{pyk} “1deduction six star " conclude " exception " bound " end 1deduction”] [Dedu_{7}(∗) →^{pyk} “1deduction seven " end 1deduction”] [Dedu_{8}(∗, ∗) →^{pyk} “1deduction eight " bound " end 1deduction”] [Dedu_{8}^{∗}(∗, ∗) →^{pyk} “1deduction eight star " bound " end 1deduction”]

[Ex_{1} →^{pyk} “ex1”]

[Ex_{2} →^{pyk} “ex2”]

[Ex3 →^{pyk} “ex3”]

[Ex_{10} →^{pyk} “ex10”]

[Ex_{20} →^{pyk} “ex20”]

[∗_{Ex} →^{pyk} “existential var " end var”] [∗^{Ex} →^{pyk} “" is existential var”] [⟨∗≡ ∗ | ∗:==∗⟩_{Ex} →^{pyk} “exist-sub " is " where " is " end sub”] [⟨∗≡^{0} ∗ | ∗:==∗⟩_{Ex} →^{pyk} “exist-sub0 " is " where " is " end sub”] [⟨∗≡^{1} ∗ | ∗:==∗⟩_{Ex} →^{pyk} “exist-sub1 " is " where " is " end sub”] [⟨∗≡^{∗} ∗ | ∗:==∗⟩_{Ex} →^{pyk} “exist-sub∗ " is " where " is " end sub”] [ph_{1} →^{pyk} “placeholder-var1”] [ph_{2} →^{pyk} “placeholder-var2”] [ph_{3} →^{pyk} “placeholder-var3”] [∗_{Ph} →^{pyk} “placeholder-var " end var”] [∗^{Ph} →^{pyk} “" is placeholder-var”] [⟨∗≡ ∗ | ∗:==∗⟩_{Ph} →^{pyk} “ph-sub " is " where " is " end sub”] [⟨∗≡^{0} ∗ | ∗:==∗⟩_{Ph} →^{pyk} “ph-sub0 " is " where " is " end sub”] [⟨∗≡^{1} ∗ | ∗:==∗⟩_{Ph} →^{pyk} “ph-sub1 " is " where " is " end sub”] [⟨∗≡^{∗} ∗ | ∗:==∗⟩_{Ph} →^{pyk} “ph-sub∗ " is " where " is " end sub”] [bs →^{pyk} “var big set”] [OBS →^{pyk} “object big set”] [BS →^{pyk} “meta big set”] [Ø →^{pyk} “zermelo empty set”] [ZFsub →^{pyk} “system Q”] [MP →^{pyk} “1rule mp”] [Gen →^{pyk} “1rule gen”] [Repetition →^{pyk} “1rule repetition”] [Neg →^{pyk} “1rule ad absurdum”] [Ded →^{pyk} “1rule deduction”] [ExistIntro →^{pyk} “1rule exist intro”] [Extensionality →^{pyk} “axiom extensionality”] [Ødef →^{pyk} “axiom empty set”] [PairDef →^{pyk} “axiom pair definition”] [UnionDef →^{pyk} “axiom union definition”] [PowerDef →^{pyk} “axiom power definition”] [SeparationDef →^{pyk} “axiom separation definition”] [AddDoubleNeg →^{pyk} “prop lemma add double neg”] [RemoveDoubleNeg →^{pyk} “prop lemma remove double neg”] [AndCommutativity →^{pyk} “prop lemma and commutativity”] [AutoImply →^{pyk} “prop lemma auto imply”] [Contrapositive →^{pyk} “prop lemma contrapositive”] [FirstConjunct →^{pyk} “prop lemma first conjunct”] [SecondConjunct →^{pyk} “prop lemma second conjunct”] [FromContradiction →^{pyk} “prop lemma from contradiction”] [FromDisjuncts →^{pyk} “prop lemma from disjuncts”] [IffCommutativity →^{pyk} “prop lemma iff commutativity”] [IffFirst →^{pyk} “prop lemma iff first”] [IffSecond →^{pyk} “prop lemma iff second”] [ImplyTransitivity →^{pyk} “prop lemma imply transitivity”] [JoinConjuncts →^{pyk} “prop lemma join conjuncts”] [MP2 →^{pyk} “prop lemma mp2”] [MP3 →^{pyk} “prop lemma mp3”] [MP4 →^{pyk} “prop lemma mp4”] [MP5 →^{pyk} “prop lemma mp5”] [MT →^{pyk} “prop lemma mt”] [NegativeMT →^{pyk} “prop lemma negative mt”] [Technicality →^{pyk} “prop lemma technicality”] [Weakening →^{pyk} “prop lemma weakening”] [WeakenOr1 →^{pyk} “prop lemma weaken or first”] [WeakenOr2 →^{pyk} “prop lemma weaken or second”] [Formula2Pair →^{pyk} “lemma formula2pair”] [Pair2Formula →^{pyk} “lemma pair2formula”] [Formula2Union →^{pyk} “lemma formula2union”] [Union2Formula →^{pyk} “lemma union2formula”] [Formula2Sep →^{pyk} “lemma formula2separation”] [Sep2Formula →^{pyk} “lemma separation2formula”] [SubsetInPower →^{pyk} “lemma subset in power set”] [HelperPowerIsSub →^{pyk} “lemma power set is subset0”] [PowerIsSub →^{pyk} “lemma power set is subset”] [(Switch)HelperPowerIsSub →^{pyk} “lemma power set is subset0-switch”] [(Switch)PowerIsSub →^{pyk} “lemma power set is subset-switch”] [ToSetEquality →^{pyk} “lemma set equality suff condition”] [HelperToSetEquality(t) →^{pyk} “lemma set equality suff condition(t)0”] [ToSetEquality(t) →^{pyk} “lemma set equality suff condition(t)”] [HelperFromSetEquality →^{pyk} “lemma set equality skip quantifier”] [FromSetEquality →^{pyk} “lemma set equality nec condition”] [HelperReflexivity →^{pyk} “lemma reflexivity0”] [Reflexivity →^{pyk} “lemma reflexivity”] [HelperSymmetry →^{pyk} “lemma symmetry0”] [Symmetry →^{pyk} “lemma symmetry”] [HelperTransitivity →^{pyk} “lemma transitivity0”] [Transitivity →^{pyk} “lemma transitivity”] [ERisReflexive →^{pyk} “lemma er is reflexive”] [ERisSymmetric →^{pyk} “lemma er is symmetric”] [ERisTransitive →^{pyk} “lemma er is transitive”] [ØisSubset →^{pyk} “lemma empty set is subset”] [HelperMemberNotØ →^{pyk} “lemma member not empty0”] [MemberNotØ →^{pyk} “lemma member not empty”] [HelperUniqueØ →^{pyk} “lemma unique empty set0”] [UniqueØ →^{pyk} “lemma unique empty set”] [==Reflexivity →^{pyk} “lemma ==Reflexivity”] [==Symmetry →^{pyk} “lemma ==Symmetry”] [Helper==Transitivity →^{pyk} “lemma ==Transitivity0”] [==Transitivity →^{pyk} “lemma ==Transitivity”] [HelperTransferNotEq →^{pyk} “lemma transfer ˜ is0”] [TransferNotEq →^{pyk} “lemma transfer ˜ is”] [HelperPairSubset →^{pyk} “lemma pair subset0”] [Helper(2)PairSubset →^{pyk} “lemma pair subset1”] [PairSubset →^{pyk} “lemma pair subset”] [SamePair →^{pyk} “lemma same pair”] [SameSingleton →^{pyk} “lemma same singleton”] [UnionSubset →^{pyk} “lemma union subset”] [SameUnion →^{pyk} “lemma same union”] [SeparationSubset →^{pyk} “lemma separation subset”] [SameSeparation →^{pyk} “lemma same separation”] [SameBinaryUnion →^{pyk} “lemma same binary union”] [IntersectionSubset →^{pyk} “lemma intersection subset”] [SameIntersection →^{pyk} “lemma same intersection”] [AutoMember →^{pyk} “lemma auto member”] [HelperEqSysNotØ →^{pyk} “lemma eq-system not empty0”] [EqSysNotØ →^{pyk} “lemma eq-system not empty”] [HelperEqSubset →^{pyk} “lemma eq subset0”] [EqSubset →^{pyk} “lemma eq subset”] [HelperEqNecessary →^{pyk} “lemma equivalence nec condition0”] [EqNecessary →^{pyk} “lemma equivalence nec condition”] [HelperNoneEqNecessary →^{pyk} “lemma none-equivalence nec condition0”] [Helper(2)NoneEqNecessary →^{pyk} “lemma none-equivalence nec condition1”] [NoneEqNecessary →^{pyk} “lemma none-equivalence nec condition”] [EqClassIsSubset →^{pyk} “lemma equivalence class is subset”] [EqClassesAreDisjoint →^{pyk} “lemma equivalence classes are disjoint”] [AllDisjoint →^{pyk} “lemma all disjoint”] [AllDisjointImply →^{pyk} “lemma all disjoint-imply”] [BSsubset →^{pyk} “lemma bs subset union(bs/r)”] [Union(BS/R)subset →^{pyk} “lemma union(bs/r) subset bs”] [UnionIdentity →^{pyk} “lemma union(bs/r) is bs”] [EqSysIsPartition →^{pyk} “theorem eq-system is partition”] [(ϵ) →^{pyk} “var ep”] [(fx) →^{pyk} “var fx”] [(fy) →^{pyk} “var fy”] [(fz) →^{pyk} “var fz”] [(fv) →^{pyk} “var fu”] [var fv →^{pyk} “var fv”] [(rx) →^{pyk} “var rx”] [(ry) →^{pyk} “var ry”] [(rz) →^{pyk} “var rz”] [(ru) →^{pyk} “var ru”] [ϵ →^{pyk} “meta ep”] [FX →^{pyk} “meta fx”] [FY →^{pyk} “meta fy”] [FZ →^{pyk} “meta fz”] [FU →^{pyk} “meta fu”] [FV →^{pyk} “meta fv”] [RX →^{pyk} “meta rx”] [RY →^{pyk} “meta ry”] [RZ →^{pyk} “meta rz”] [RU →^{pyk} “meta ru”]

[0 →^{pyk} “0”]

[1 →^{pyk} “1”]

[(−1) →^{pyk} “(-1)”]

[2 →^{pyk} “2”]

[1/2 →^{pyk} “1/2”]

[0f →^{pyk} “0f”]

[1f →^{pyk} “1f”]

[00 →^{pyk} “00”]

[01 →^{pyk} “01”]

[leqReflexivity →^{pyk} “axiom leqReflexivity”] [leqAntisymmetryAxiom →^{pyk} “axiom leqAntisymmetry”] [leqTransitivityAxiom →^{pyk} “axiom leqTransitivity”] [leqTotality →^{pyk} “axiom leqTotality”] [leqAdditionAxiom →^{pyk} “axiom leqAddition”] [leqMultiplicationAxiom →^{pyk} “axiom leqMultiplication”] [plusAssociativity →^{pyk} “axiom plusAssociativity”] [plusCommutativity →^{pyk} “axiom plusCommutativity”] [Negative →^{pyk} “axiom negative”] [plus0 →^{pyk} “axiom plus0”] [timesAssociativity →^{pyk} “axiom timesAssociativity”] [timesCommutativity →^{pyk} “axiom timesCommutativity”] [ReciprocalAxiom →^{pyk} “axiom reciprocal”] [times1 →^{pyk} “axiom times1”] [Distribution →^{pyk} “axiom distribution”] [0not1 →^{pyk} “axiom 0not1”] [equalityAxiom →^{pyk} “axiom equality”] [eqLeqAxiom →^{pyk} “axiom eqLeq”] [eqAdditionAxiom →^{pyk} “axiom eqAddition”] [eqMultiplicationAxiom →^{pyk} “axiom eqMultiplication”] [SENC1 →^{pyk} “lemma set equality nec condition(1)”] [SENC2 →^{pyk} “lemma set equality nec condition(2)”] [IfThenElse(T) →^{pyk} “1rule ifThenElse true”] [IfThenElse(F) →^{pyk} “1rule ifThenElse false”] [From = f →^{pyk} “1rule from=f”] [To = f →^{pyk} “1rule to=f”] [From < f →^{pyk} “1rule from<f”] [To < f →^{pyk} “1rule to<f”] [PlusF →^{pyk} “axiom plusF”] [TimesF →^{pyk} “axiom timesF”] [MinusF →^{pyk} “axiom minusF”] [0f →^{pyk} “axiom 0f”] [1f →^{pyk} “axiom 1f”] [FromSF →^{pyk} “1rule fromSameF”] [ToSF →^{pyk} “1rule toSameF”] [To == XX →^{pyk} “1rule to==XX”] [From == →^{pyk} “1rule from==”] [To == →^{pyk} “1rule to==”] [From << XX →^{pyk} “1rule from<<XX”] [From << (1) →^{pyk} “1rule from<<XX(1)”] [From << (2) →^{pyk} “1rule from<<XX(2)”] [to << XX →^{pyk} “1rule to<<XX”] [From << →^{pyk} “1rule from<<”] [To << →^{pyk} “1rule to<<”] [FromInR →^{pyk} “1rule fromInR”] [PlusR →^{pyk} “axiom plusR”] [TimesR →^{pyk} “axiom timesR”] [leqAntisymmetry →^{pyk} “lemma leqAntisymmetry”] [leqTransitivity →^{pyk} “lemma leqTransitivity”] [leqAddition →^{pyk} “lemma leqAddition”] [leqMultiplication →^{pyk} “lemma leqMultiplication”] [Reciprocal →^{pyk} “lemma reciprocal”] [Equality →^{pyk} “lemma equality”] [eqLeq →^{pyk} “lemma eqLeq”] [eqAddition →^{pyk} “lemma eqAddition”] [eqMultiplication →^{pyk} “lemma eqMultiplication”] [ToNegatedImply →^{pyk} “prop lemma to negated imply”] [TND →^{pyk} “prop lemma tertium non datur”] [ImplyNegation →^{pyk} “prop lemma imply negation”] [FromNegations →^{pyk} “prop lemma from negations”] [From3Disjuncts →^{pyk} “prop lemma from three disjuncts”] [From2 ∗ 2Disjuncts →^{pyk} “prop lemma from two times two disjuncts”] [NegateDisjunct1 →^{pyk} “prop lemma negate first disjunct”] [NegateDisjunct2 →^{pyk} “prop lemma negate second disjunct”] [ExpandDisjuncts →^{pyk} “prop lemma expand disjuncts”] [eqReflexivity →^{pyk} “lemma eqReflexivity”] [eqSymmetry →^{pyk} “lemma eqSymmetry”] [eqTransitivity →^{pyk} “lemma eqTransitivity”] [eqTransitivity4 →^{pyk} “lemma eqTransitivity4”] [eqTransitivity5 →^{pyk} “lemma eqTransitivity5”] [eqTransitivity6 →^{pyk} “lemma eqTransitivity6”] [plus0Left →^{pyk} “lemma plus0Left”] [times1Left →^{pyk} “lemma times1Left”] [lemma eqAdditionLeft →^{pyk} “lemma eqAdditionLeft”] [EqMultiplicationLeft →^{pyk} “lemma eqMultiplicationLeft”] [DistributionOut →^{pyk} “lemma distributionOut”] [Three2twoTerms →^{pyk} “lemma three2twoTerms”] [Three2threeTerms →^{pyk} “lemma three2threeTerms”] [Three2threeFactors →^{pyk} “lemma three2twoFactors”] [AddEquations →^{pyk} “lemma addEquations”] [SubtractEquations →^{pyk} “lemma subtractEquations”] [SubtractEquationsLeft →^{pyk} “lemma subtractEquationsLeft”] [EqNegated →^{pyk} “lemma eqNegated”] [PositiveToRight(Eq) →^{pyk} “lemma positiveToRight(Eq)”] [PositiveToLeft(Eq)(1term) →^{pyk} “lemma positiveToLeft(Eq)(1 term)”] [NegativeToLeft(Eq) →^{pyk} “lemma negativeToLeft(Eq)”] [LessNeq →^{pyk} “lemma lessNeq”] [NeqSymmetry →^{pyk} “lemma neqSymmetry”] [NeqNegated →^{pyk} “lemma neqNegated”] [SubNeqRight →^{pyk} “lemma subNeqRight”] [SubNeqLeft →^{pyk} “lemma subNeqLeft”] [NeqAddition →^{pyk} “lemma neqAddition”] [NeqMultiplication →^{pyk} “lemma neqMultiplication”] [UniqueNegative →^{pyk} “lemma uniqueNegative”] [DoubleMinus →^{pyk} “lemma doubleMinus”] [LeqLessEq →^{pyk} “lemma leqLessEq”] [LessLeq →^{pyk} “lemma lessLeq”] [FromLeqGeq →^{pyk} “lemma from leqGeq”] [subLeqRight →^{pyk} “lemma subLeqRight”] [subLeqLeft →^{pyk} “lemma subLeqLeft”] [Leq + 1 →^{pyk} “lemma leqPlus1”] [PositiveToRight(Leq) →^{pyk} “lemma positiveToRight(Leq)”] [PositiveToRight(Leq)(1term) →^{pyk} “lemma positiveToRight(Leq)(1 term)”] [negativeToLeft(Leq) →^{pyk} “lemma negativeToLeft(Leq)”] [LeqAdditionLeft →^{pyk} “lemma leqAdditionLeft”] [leqSubtraction →^{pyk} “lemma leqSubtraction”] [leqSubtractionLeft →^{pyk} “lemma leqSubtractionLeft”] [thirdGeq →^{pyk} “lemma thirdGeq”] [LeqNegated →^{pyk} “lemma leqNegated”] [AddEquations(Leq) →^{pyk} “lemma addEquations(Leq)”] [ThirdGeqSeries →^{pyk} “lemma thirdGeqSeries”] [LeqNeqLess →^{pyk} “lemma leqNeqLess”] [FromLess →^{pyk} “lemma fromLess”] [ToLess →^{pyk} “lemma toLess”] [fromNotLess →^{pyk} “lemma fromNotLess”] [toNotLess →^{pyk} “lemma toNotLess”] [NegativeLessPositive →^{pyk} “lemma negativeLessPositive”] [leqLessTransitivity →^{pyk} “lemma leqLessTransitivity”] [LessLeqTransitivity →^{pyk} “lemma lessLeqTransitivity”] [LessTransitivity →^{pyk} “lemma lessTransitivity”] [LessTotality →^{pyk} “lemma lessTotality”] [SubLessRight →^{pyk} “lemma subLessRight”] [SubLessLeft →^{pyk} “lemma subLessLeft”] [LessAddition →^{pyk} “lemma lessAddition”] [LessAdditionLeft →^{pyk} “lemma lessAdditionLeft”] [LessMultiplication →^{pyk} “lemma lessMultiplication”] [LessMultiplicationLeft →^{pyk} “lemma lessMultiplicationLeft”] [LessDivision →^{pyk} “lemma lessDivision”] [AddEquations(Less) →^{pyk} “lemma addEquations(Less)”] [LessNegated →^{pyk} “lemma lessNegated”] [PositiveNegated →^{pyk} “lemma positiveNegated”] [NonpositiveNegated →^{pyk} “lemma nonpositiveNegated”] [NegativeNegated →^{pyk} “lemma negativeNegated”] [NonnegativeNegated →^{pyk} “lemma nonnegativeNegated”] [PositiveHalved →^{pyk} “lemma positiveHalved”] [NonnegativeNumerical →^{pyk} “lemma nonnegativeNumerical”] [NegativeNumerical →^{pyk} “lemma negativeNumerical”] [PositiveNumerical →^{pyk} “lemma positiveNumerical”] [lemma nonpositiveNumerical →^{pyk} “lemma nonpositiveNumerical”] [|0| = 0 →^{pyk} “lemma |0|=0”] [0 <= |x| →^{pyk} “lemma 0<=|x|”] [SameNumerical →^{pyk} “lemma sameNumerical”] [SignNumerical(+) →^{pyk} “lemma signNumerical(+)”] [SignNumerical →^{pyk} “lemma signNumerical”] [NumericalDifference →^{pyk} “lemma numericalDifference”] [SplitNumericalSumHelper →^{pyk} “lemma splitNumericalSumHelper”] [splitNumericalSum(++) →^{pyk} “lemma splitNumericalSum(++)”] [splitNumericalSum(−−) →^{pyk} “lemma splitNumericalSum(--)”] [splitNumericalSum(+ − small) →^{pyk} “lemma splitNumericalSum(+-, smallNegative)”] [splitNumericalSum(+ − big) →^{pyk} “lemma splitNumericalSum(+-, bigNegative)”] [splitNumericalSum(+−) →^{pyk} “lemma splitNumericalSum(+-)”] [splitNumericalSum(−+) →^{pyk} “lemma splitNumericalSum(-+)”] [splitNumericalSum →^{pyk} “lemma splitNumericalSum”] [insertMiddleTerm(Numerical) →^{pyk} “lemma insertMiddleTerm(Numerical)”] [x + y = zBackwards →^{pyk} “lemma x+y=zBackwards”] [x ∗ y = zBackwards →^{pyk} “lemma x∗y=zBackwards”] [x = x + (y − y) →^{pyk} “lemma x=x+(y-y)”] [x = x + y − y →^{pyk} “lemma x=x+y-y”] [ →^{pyk} “lemma x=x∗y∗(1/y)”] [insertMiddleTerm(Sum) →^{pyk} “lemma insertMiddleTerm(Sum)”] [insertMiddleTerm(Difference) →^{pyk} “lemma insertMiddleTerm(Difference)”] [x ∗ 0 + x = x →^{pyk} “lemma x∗0+x=x”] [x ∗ 0 = 0 →^{pyk} “lemma x∗0=0”] [(−1) ∗ (−1) + (−1) ∗ 1 = 0 →^{pyk} “lemma (-1)∗(-1)+(-1)∗1=0”] [(−1) ∗ (−1) = 1 →^{pyk} “lemma (-1)∗(-1)=1”] [0 < 1Helper →^{pyk} “lemma 0<1Helper”] [0 < 1 →^{pyk} “lemma 0<1”] [0 < 2 →^{pyk} “lemma 0<2”] [0 < 1/2 →^{pyk} “lemma 0<1/2”] [TwoWholes →^{pyk} “lemma x+x=2∗x”] [TwoHalves →^{pyk} “lemma (1/2)x+(1/2)x=x”] [−x − y = −(x + y) →^{pyk} “lemma -x-y=-(x+y)”] [MinusNegated →^{pyk} “lemma minusNegated”] [Times(−1) →^{pyk} “lemma times(-1)”] [Times(−1)Left →^{pyk} “lemma times(-1)Left”] [−0 = 0 →^{pyk} “lemma -0=0”] [SFsymmetry →^{pyk} “lemma sameFsymmetry”] [SFtransitivity →^{pyk} “lemma sameFtransitivity”] [= fToSameF →^{pyk} “lemma =f to sameF”] [PlusF(Sym) →^{pyk} “lemma plusF(Sym)”] [TimesF(Sym) →^{pyk} “lemma timesF(Sym)”] [f2R(Plus) →^{pyk} “lemma f2R(Plus)”] [f2R(Times) →^{pyk} “lemma f2R(Times)”] [PlusR(Sym) →^{pyk} “lemma plusR(Sym)”] [TimesR(Sym) →^{pyk} “lemma timesR(Sym)”] [LessLeq(R) →^{pyk} “lemma lessLeq(R)”] [eqLeq(R) →^{pyk} “lemma eqLeq(R)”] [SubLessRight(R) →^{pyk} “lemma subLessRight(R)”] [SubLessLeft(R) →^{pyk} “lemma subLessLeft(R)”] [<< TransitivityHelper(Q) →^{pyk} “lemma <<TransitivityHelper(Q)”] [<< Transitivity →^{pyk} “lemma <<Transitivity”] [<<== Reflexivity →^{pyk} “lemma <<==Reflexivity”] [<<== AntisymmetryHelper(Q) →^{pyk} “lemma <<==AntisymmetryHelper(Q)”] [<<== Antisymmetry →^{pyk} “lemma <<==Antisymmetry”] [<<== Transitivity →^{pyk} “lemma <<==Transitivity”] [Plus0f →^{pyk} “lemma plus0f”] [Plus00 →^{pyk} “lemma plus00”] [== Addition →^{pyk} “lemma ==Addition”] [== AdditionLeft →^{pyk} “lemma ==AdditionLeft”] [<< Addition →^{pyk} “lemma <<Addition”] [<<== Addition →^{pyk} “lemma <<==Addition”] [PlusAssociativity(F) →^{pyk} “lemma plusAssociativity(F)”] [PlusAssociativity(R) →^{pyk} “lemma plusAssociativity(R)”] [Negative(R) →^{pyk} “lemma negative(R)”] [PlusCommutativity(F) →^{pyk} “lemma plusCommutativity(F)”] [PlusCommutativity(R) →^{pyk} “lemma plusCommutativity(R)”] [TimesAssociativity(F) →^{pyk} “lemma timesAssociativity(F)”] [TimesAssociativity(R) →^{pyk} “lemma timesAssociativity(R)”] [Times1f →^{pyk} “lemma times1f”] [Times01 →^{pyk} “lemma times01”] [TimesCommutativity(F) →^{pyk} “lemma timesCommutativity(F)”] [TimesCommutativity(R) →^{pyk} “lemma timesCommutativity(R)”] [Distribution(F) →^{pyk} “lemma distribution(F)”] [Distribution(R) →^{pyk} “lemma distribution(R)”]

[R(∗) →^{pyk} “R( " )”]

[− − R(∗) →^{pyk} “--R( " )”]

[rec∗ →^{pyk} “1/ "”]

[∗/∗ →^{pyk} “eq-system of " modulo "”] [∗ ∩ ∗ →^{pyk} “intersection " comma " end intersection”]

[∗[∗] →^{pyk} “[ " ; " ]”]

[∪∗ →^{pyk} “union " end union”] [∗ ∪ ∗ →^{pyk} “binary-union " comma " end union”] [P(∗) →^{pyk} “power " end power”] [{∗} →^{pyk} “zermelo singleton " end singleton”] [{∗, ∗} →^{pyk} “zermelo pair " comma " end pair”] [⟨∗, ∗⟩ →^{pyk} “zermelo ordered pair " comma " end pair”]

[−∗ →^{pyk} “- "”]

[−_{f}∗ →^{pyk} “-f "”]

[∗ ∈ ∗ →^{pyk} “" in0 "”]

[∗(∗, ∗) →^{pyk} “" is related to " under "”] [ReflRel(∗, ∗) →^{pyk} “" is reflexive relation in "”] [SymRel(∗, ∗) →^{pyk} “" is symmetric relation in "”] [TransRel(∗, ∗) →^{pyk} “" is transitive relation in "”] [EqRel(∗, ∗) →^{pyk} “" is equivalence relation in "”] [[∗ ∈ ∗]_{∗} →^{pyk} “equivalence class of " in " modulo "”] [Partition(∗, ∗) →^{pyk} “" is partition of "”]

[∗ ∗ ∗ →^{pyk} “" ∗ "”]

[∗ ∗_{f} ∗ →^{pyk} “" ∗f "”]

[∗ ∗ ∗∗ →^{pyk} “" ∗∗ "”]

[∗ + ∗ →^{pyk} “" + "”]

[∗ − ∗ →^{pyk} “" - "”]

[∗ +_{f} ∗ →^{pyk} “" +f "”]

[∗ −_{f} ∗ →^{pyk} “" -f "”]

[∗ + +∗ →^{pyk} “" ++ "”]

[R(∗) − −R(∗) →^{pyk} “R( " ) -- R( " )”]

[| ∗ | →^{pyk} “| " |”]

[if(∗, ∗, ∗) →^{pyk} “if( " , " , " )”]

[∗ = ∗ →^{pyk} “" = "”]

[∗ ≠ ∗ →^{pyk} “" != "”]

[∗ <= ∗ →^{pyk} “" <= "”]

[∗ < ∗ →^{pyk} “" < "”]

[∗ =_{f} ∗ →^{pyk} “" =f "”]

[∗ <_{f} ∗ →^{pyk} “" <f "”]

[SF(∗, ∗) →^{pyk} “" sameF "”]

[∗ == ∗ →^{pyk} “" == "”]

[∗ << ∗ →^{pyk} “" << "”]

[∗ <<== ∗ →^{pyk} “" <<== "”]

[∗==∗ →^{pyk} “" zermelo is "”] [∗ ⊆ ∗ →^{pyk} “" is subset of "”]

[ ˙¬ ∗ →^{pyk} “not0 "”]

[∗ ∉ ∗ →^{pyk} “" zermelo ˜ in "”] [∗ ≠ ∗ →^{pyk} “" zermelo ˜ is "”]

[∗ ˙∧ ∗ →^{pyk} “" and0 "”]

[∗ ˙∨ ∗ →^{pyk} “" or0 "”]

[∗ ˙⇔ ∗ →^{pyk} “" iff "”]

[{ph ∈ ∗ | ∗} →^{pyk} “the set of ph in " such that " end set”]

[am →^{pyk} “am”]

B TEX definitioner

[am →^{tex} “am”]

[(⋯) →^{tex} “(\cdots{})”]

[Objekt-var →^{tex} “\texttt{Objekt-var}”]

[Ex-var →^{tex} “\texttt{Ex-var}”]

[Ph-var →^{tex} “\texttt{Ph-var}”]

[Værdi →^{tex} “\texttt{V\ae{}rdi}”]

[Variabel →^{tex} “\texttt{Variabel}”]

[Op(x) →^{tex} “Op(#1. )”]

[Op(x, y) →^{tex} “Op(#1. , #2. )”]

[x ¨ == y →^{tex} “#1.

\mathrel {\ddot {==}} #2.”]

[ContainsEmpty(x) →^{tex} “ContainsEmpty(#1. )”]

[Dedu(x, y) →^{tex} “ Dedu(#1. , #2. )”]

[Dedu_{0}(x, y) →^{tex} “ Dedu 0(#1. , #2. )”]

[Dedu_{s}(x, y, z) →^{tex} “Dedu {s}(#1. , #2. , #3. )”]

[Dedu_{1}(x, y, z) →^{tex} “ Dedu 1(#1. , #2. , #3. )”] [Dedu_{2}(x, y, z) →^{tex} “ Dedu 2(#1. , #2. , #3. )”]

[Dedu_{3}(x, y, z, u) →^{tex} “ Dedu 3(#1. , #2. , #3. , #4. )”]

[Dedu_{4}(x, y, z, u) →^{tex} “ Dedu 4(#1. , #2. , #3. , #4. )”]

[Dedu_{4}^{∗}(x, y, z, u) →^{tex} “ Dedu 4ˆ ∗(#1. , #2. , #3. , #4. )”]

[Dedu_{5}(x, y, z) →^{tex} “ Dedu 5(#1. , #2. , #3. )”]

[Dedu_{6}(p, c, e, b) →^{tex} “ Dedu 6(#1. , #2. , #3. , #4. )”]

[Dedu_{6}^{∗}(p, c, e, b) →^{tex} “ Dedu 6ˆ ∗(#1. , #2. , #3. , #4. )”]

[Dedu_{7}(p) →^{tex} “ Dedu 7(#1.

)”]

[Dedu_{8}(p, b) →^{tex} “ Dedu 8(#1. , #2. )”]

[Dedu_{8}^{∗}(p, b) →^{tex} “ Dedu 8ˆ ∗(#1. , #2. )”]

[Ex_{1} →^{tex} “Ex {1}”]

[Ex_{2} →^{tex} “Ex {2}”]

[Ex_{10} →^{tex} “Ex {10}”]

[Ex_{20} →^{tex} “Ex {20}”]

[x_{Ex} →^{tex} “#1.

{Ex}”]

[x^{Ex} →^{tex} “#1.

ˆ {Ex}”]

[⟨x≡y|z:==u⟩_{Ex} →^{tex} “\langle #1. {\equiv} #2. | #3. {:==} #4. \rangle {Ex} ”]

[⟨x≡^{0}y|z:==u⟩_{Ex} →^{tex} “\langle #1. {\equiv}ˆ 0 #2. | #3. {:==} #4. \rangle {Ex} ”]

[⟨x≡^{1}y|z:==u⟩_{Ex} →^{tex} “\langle #1. {\equiv}ˆ 1 #2. | #3. {:==} #4. \rangle {Ex} ”]

[⟨x≡^{∗}y|z:==u⟩_{Ex} →^{tex} “\langle #1. {\equiv}ˆ ∗ #2. | #3. {:==} #4. \rangle {Ex} ”]

[ph_{1} →^{tex} “ph {1}”]

[ph_{2} →^{tex} “ph {2}”]

[ph_{3} →^{tex} “ph {3}”]

[x_{Ph} →^{tex} “#1.

{Ph} ”]

[x^{Ph} →^{tex} “#1.

ˆ {Ph}”]

[⟨x≡y|z:==u⟩_{Ph} →^{tex} “\langle #1. {\equiv} #2. | #3. {:==} #4. \rangle {Ph} ”]

[⟨x≡^{0}y|z:==u⟩_{Ph} →^{tex} “\langle #1. {\equiv}ˆ 0 #2. | #3. {:==} #4. \rangle {Ph} ”]

[⟨x≡^{1}y|z:==u⟩_{Ph} →^{tex} “\langle #1. {\equiv}ˆ 1 #2. | #3. {:==} #4. \rangle {Ph} ”]

[⟨x≡^{∗}y|z:==u⟩_{Ph} →^{tex} “\langle #1. {\equiv}ˆ ∗ #2. | #3. {:==} #4. \rangle {Ph} ”]

[bs →^{tex} “\mathsf {bs}”]

[OBS →^{tex} “ \mathsf {OBS}”]

[BS →^{tex} “{\cal BS}”]

[Ø →^{tex} “\mathrm{\O}”]

[ZFsub →^{tex} “ZFsub”]

[MP →^{tex} “MP”]

[Gen →^{tex} “Gen”] [Repetition →^{tex} “Repetition”] [Neg →^{tex} “Neg”] [Ded →^{tex} “Ded”] [ExistIntro →^{tex} “ExistIntro”] [Extensionality →^{tex} “Extensionality”] [Ødef →^{tex} “\O{}def”] [PairDef →^{tex} “PairDef”] [UnionDef →^{tex} “UnionDef”] [PowerDef →^{tex} “PowerDef”] [SeparationDef →^{tex} “SeparationDef”] [AddDoubleNeg →^{tex} “AddDoubleNeg”] [RemoveDoubleNeg →^{tex} “RemoveDoubleNeg”] [AndCommutativity →^{tex} “AndCommutativity”] [AutoImply →^{tex} “AutoImply”] [Contrapositive →^{tex} “Contrapositive”] [FirstConjunct →^{tex} “FirstConjunct”] [SecondConjunct →^{tex} “SecondConjunct”] [FromContradiction →^{tex} “FromContradiction”] [FromDisjuncts →^{tex} “FromDisjuncts”] [IffCommutativity →^{tex} “IffCommutativity”] [IffFirst →^{tex} “IffFirst”] [IffSecond →^{tex} “IffSecond”] [ImplyTransitivity →^{tex} “ImplyTransitivity”] [JoinConjuncts →^{tex} “JoinConjuncts”]

[MP2 →^{tex} “MP2”]

[MP3 →^{tex} “MP3”]

[MP4 →^{tex} “MP4”]

[MP5 →^{tex} “MP5”]

[MT →^{tex} “MT”]

[NegativeMT →^{tex} “NegativeMT”] [Technicality →^{tex} “Technicality”] [Weakening →^{tex} “Weakening”] [WeakenOr1 →^{tex} “WeakenOr1”] [WeakenOr2 →^{tex} “WeakenOr2”] [Pair2Formula →^{tex} “Pair2Formula”] [Formula2Pair →^{tex} “Formula2Pair”] [Union2Formula →^{tex} “Union2Formula”] [Formula2Union →^{tex} “Formula2Union”] [Sep2Formula →^{tex} “Sep2Formula”] [Formula2Sep →^{tex} “Formula2Sep”] [SubsetInPower →^{tex} “SubsetInPower”] [HelperPowerIsSub →^{tex} “HelperPowerIsSub”] [PowerIsSub →^{tex} “PowerIsSub”] [(Switch)HelperPowerIsSub →^{tex} “(Switch)HelperPowerIsSub”] [(Switch)PowerIsSub →^{tex} “(Switch)PowerIsSub”] [ToSetEquality →^{tex} “ToSetEquality”] [HelperToSetEquality(t) →^{tex} “HelperToSetEquality(t)”] [ToSetEquality(t) →^{tex} “ToSetEquality(t)”] [HelperFromSetEquality →^{tex} “HelperFromSetEquality”] [FromSetEquality →^{tex} “FromSetEquality”] [HelperReflexivity →^{tex} “HelperReflexivity”] [Reflexivity →^{tex} “Reflexivity”] [HelperSymmetry →^{tex} “HelperSymmetry”] [Symmetry →^{tex} “Symmetry”] [HelperTransitivity →^{tex} “HelperTransitivity”] [Transitivity →^{tex} “Transitivity”], [ERisReflexive →^{tex} “ERisReflexive”] [ERisSymmetric →^{tex} “ERisSymmetric”] [ERisTransitive →^{tex} “ERisTransitive”] [ØisSubset →^{tex} “\O{}isSubset”] [HelperMemberNotØ →^{tex} “HelperMemberNot\O{}”] [MemberNotØ →^{tex} “MemberNot\O{}”] [HelperUniqueØ →^{tex} “HelperUnique\O{}”] [UniqueØ →^{tex} “Unique\O{}”] [==Reflexivity →^{tex} “==\!{}Reflexivity”] [==Symmetry →^{tex} “==\!{}Symmetry”] [Helper==Transitivity →^{tex} “Helper\!{}==\!{}Transitivity”] [==Transitivity →^{tex} “\!{}==\!{}Transitivity”] [HelperTransferNotEq →^{tex} “HelperTransferNotEq”] [TransferNotEq →^{tex} “TransferNotEq”] [HelperPairSubset →^{tex} “HelperPairSubset”] [Helper(2)PairSubset →^{tex} “Helper(2)PairSubset”] [PairSubset →^{tex} “PairSubset”] [SamePair →^{tex} “SamePair”] [SameSingleton →^{tex} “SameSingleton”] [UnionSubset →^{tex} “UnionSubset”] [SameUnion →^{tex} “SameUnion”] [SeparationSubset →^{tex} “SeparationSubset”] [SameSeparation →^{tex} “SameSeparation”] [SameBinaryUnion →^{tex} “SameBinaryUnion”] [IntersectionSubset →^{tex} “IntersectionSubset”] [SameIntersection →^{tex} “SameIntersection”] [AutoMember →^{tex} “AutoMember”] [HelperEqSysNotØ →^{tex} “HelperEqSysNot\O{}”] [EqSysNotØ →^{tex} “EqSysNot\O{}”] [HelperEqSubset →^{tex} “HelperEqSubset”] [EqSubset →^{tex} “EqSubset”] [EqNecessary →^{tex} “EqNecessary”] [HelperEqNecessary →^{tex} “HelperEqNecessary”] [HelperNoneEqNecessary →^{tex} “HelperNoneEqNecessary”] [Helper(2)NoneEqNecessary →^{tex} “Helper(2)NoneEqNecessary”] [NoneEqNecessary →^{tex} “NoneEqNecessary”] [EqClassIsSubset →^{tex} “EqClassIsSubset”] [EqClassesAreDisjoint →^{tex} “EqClassesAreDisjoint”] [AllDisjoint →^{tex} “AllDisjoint”] [AllDisjointImply →^{tex} “AllDisjointImply”] [BSsubset →^{tex} “BSsubset”] [Union(BS/R)subset →^{tex} “Union(BS/R)subset”] [UnionIdentity →^{tex} “UnionIdentity”] [EqSysIsPartition →^{tex} “EqSysIsPartition”]

[x/y →^{tex} “#1.

/ #2.”]

[x ∩ y →^{tex} “#1.

\cap #2.”]

[∪x →^{tex} “\cup #1.”]

[x ∪ y →^{tex} “#1.

\mathrel{\cup} #2.”]

[P(x) →^{tex} “P(#1.

)”]

[{x} →^{tex} “\{#1.

\}”]

[{x, y} →^{tex} “\{#1.

, #2. \}”]

[⟨x, y⟩ →^{tex} “\langle #1. , #2. \rangle”],

[x ∈ y →^{tex} “#1.

\mathrel{\in} #2.”]

[z(x, y) →^{tex} “#3.

(#1. , #2. )”]

[ReflRel(r, x) →^{tex} “ReflRel(#1. , #2. )”]

[SymRel(r, x) →^{tex} “SymRel(#1. , #2. )”]

[TransRel(r, x) →^{tex} “TransRel(#1. , #2. )”]

[EqRel(r, x) →^{tex} “EqRel(#1. , #2. )”]

[[x ∈ bs]_{r} →^{tex} “[#1.

\mathrel{\in} #2. ] {#3. }”]

[Partition(x, y) →^{tex} “Partition(#1. , #2. )”]

[x==y →^{tex} “#1.

\!\mathrel{==}\! #2.”]

[x ⊆ y →^{tex} “#1.

\mathrel{\subseteq} #2.”]

[ ˙¬ x →^{tex} “\dot{\neg}\, #1.”]

[x ∉ y →^{tex} “#1.

\mathrel{\notin} #2.”]

[x ≠ y →^{tex} “#1.

\mathrel{\neq} #2.”]

[x ˙∧ y →^{tex} “#1.

\mathrel{\dot{\wedge}} #2.”]

[x ˙∨ y →^{tex} “#1.

\mathrel{\dot{\vee}} #2.”]

[x ˙⇔ y →^{tex} “#1.

\mathrel{\dot{\Leftrightarrow}} #2.”]

[{ph ∈ x | a} →^{tex} “ \{ ph \mathrel{\in} #1. \mid #2. \}”]

————- RRRRRRRRRRRRRRR ————- (*** aksiomer ***) [leqReflexivity →^{stmt} ZFsub ⊢ ∀\underline{x}: \underline{x} <= \underline{x}][leqReflexivity →^{proof} Rule tactic] [leqAntisymmetryAxiom →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} <= \underline{y} ⇒ \underline{y} <= \underline{x} ⇒ \underline{x} =

\underline{y}][leqAntisymmetryAxiom →^{proof} Rule tactic]

[leqTransitivityAxiom →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} <= \underline{y} ⇒ \underline{y} <= \underline{z} ⇒ \underline{x} <=

\underline{z}][leqTransitivityAxiom →^{proof} Rule tactic] [leqTotality →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ \underline{x} <= \underline{y} ⇒ \underline{y} <= \underline{x}][leqTotality →^{proof} Rule tactic] [leqAdditionAxiom →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} <= \underline{y} ⇒ \underline{x} + \underline{z} <= \underline{y} + \underline{z}][leqAdditionAxiom →^{proof} Rule tactic] [leqMultiplicationAxiom →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: 0 <= \underline{z} ⇒ \underline{x} <= \underline{y} ⇒ \underline{x} ∗ \underline{z} <= \underline{y} ∗ \underline{z}][leqMultiplicationAxiom →^{proof} Rule tactic] [plusAssociativity →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} + \underline{y} + \underline{z} = \underline{x} + \underline{y} + \underline{z}][plusAssociativity →^{proof} Rule tactic] [plusCommutativity →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} + \underline{y} = \underline{y} + \underline{x}][plusCommutativity →^{proof} Rule tactic] [Negative →^{stmt} ZFsub ⊢ ∀\underline{x}: \underline{x} + −\underline{x} = 0][Negative →^{proof} Rule tactic] [plus0 →^{stmt} ZFsub ⊢ ∀\underline{x}: \underline{x} + 0 = \underline{x}][plus0 →^{proof} Rule tactic] [timesAssociativity →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} ∗ \underline{y} ∗ \underline{z} = \underline{x} ∗ \underline{y} ∗ \underline{z}][timesAssociativity →^{proof} Rule tactic] [timesCommutativity →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} ∗ \underline{y} = \underline{y} ∗ \underline{x}][timesCommutativity →^{proof} Rule tactic] [ReciprocalAxiom →^{stmt} ZFsub ⊢ ∀\underline{x}: ˙¬ \underline{x} = 0 ⇒ \underline{x} ∗ rec\underline{x} = 1][ReciprocalAxiom →^{proof} Rule tactic] [times1 →^{stmt} ZFsub ⊢ ∀\underline{x}: \underline{x} ∗ 1 = \underline{x}][times1 →^{proof} Rule tactic] [Distribution →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} ∗ \underline{y} + \underline{z} = \underline{x} ∗ \underline{y} + \underline{x} ∗ \underline{z}][Distribution →^{proof} Rule tactic] [0not1 →^{stmt} ZFsub ⊢ ˙¬ 0 = 1][0not1 →^{proof} Rule tactic] [equalityAxiom →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⇒ \underline{x} = \underline{z} ⇒ \underline{y} = \underline{z}][equalityAxiom →^{proof} Rule tactic] [eqLeqAxiom →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} = \underline{y} ⇒ \underline{x} <= \underline{y}][eqLeqAxiom →^{proof} Rule tactic] [eqAdditionAxiom →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⇒ \underline{x} + \underline{z} = \underline{y} + \underline{z}][eqAdditionAxiom →^{proof} Rule tactic] [eqMultiplicationAxiom →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⇒ \underline{x} ∗ \underline{z} = \underline{y} ∗ \underline{z}][eqMultiplicationAxiom →^{proof} Rule tactic] (*** XX snydeaksiomer ***) [==Reflexivity →^{stmt} ZFsub ⊢ ∀\underline{(rx)}: \underline{(rx)} == \underline{(rx)}][==Reflexivity →^{proof} Rule tactic] [==Symmetry →^{stmt} ZFsub ⊢ ∀\underline{(rx)}: ∀\underline{(ry)}: \underline{(rx)} == \underline{(ry)} ⊢ \underline{(ry)} == \underline{(rx)}][== Symmetry →^{proof} Rule tactic] [==Transitivity →^{stmt} ZFsub ⊢ ∀\underline{(rx)}: ∀\underline{(ry)}: ∀\underline{(rz)}: \underline{(rx)} == \underline{(ry)} ⊢ \underline{(ry)} == \underline{(rz)} ⊢ \underline{(rx)} == \underline{(rz)}][==Transitivity →^{proof} Rule tactic] XX ikke 100procent identisk med originalen fra equivalence-relations [SENC1 →^{stmt} ZFsub ⊢ ∀\underline{(fx)}: ∀\underline{(rx)}: ∀\underline{(ry)}: \underline{(rx)} == \underline{(ry)} ⊢ \underline{(fx)} ∈ \underline{(rx)} ⊢ \underline{(fx)} ∈ \underline{(ry)}][SENC1 →^{proof} Rule tactic] XX boer bevises ud fra nummer 1 [SENC2 →^{stmt} ZFsub ⊢ ∀\underline{(fx)}: ∀\underline{(rx)}: ∀\underline{(ry)}: \underline{(rx)} == \underline{(ry)} ⊢ \underline{(fx)} ∈ \underline{(ry)} ⊢ \underline{(fx)} ∈ \underline{(rx)}][SENC2 →^{proof} Rule tactic] [IfThenElse(T) →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{x}: ∀\underline{y}: \underline{a} ⊢ if(\underline{a}, \underline{x}, \underline{y}) = \underline{x}][IfThenElse(T) →^{proof} Rule tactic] [IfThenElse(F) →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{x}: ∀\underline{y}: ˙¬ \underline{a} ⊢ if(\underline{a}, \underline{x}, \underline{y}) = \underline{y}][IfThenElse(F) →^{proof} Rule tactic] [FromSF →^{stmt} ZFsub ⊢ ∀\underline{m}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: SF(\underline{(fx)}, \underline{(fy)}) ⊢ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⊢ c_{Ex} <= \underline{m} ⇒ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) = \underline{(ϵ)}][FromSF →^{proof} Rule tactic] [ToSF →^{stmt} ZFsub ⊢ ∀\underline{m}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ c_{Ex} <= \underline{m} ⇒ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) = \underline{(ϵ)} ⊢ SF(\underline{(fx)}, \underline{(fy)})][ToSF →^{proof} Rule tactic] [From = f →^{stmt} ZFsub ⊢ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: \underline{(fx)} =_{f} \underline{(fy)} ⊢ \underline{(fx)}[\underline{m}] = \underline{(fy)}[\underline{m}]][From = f →^{proof} Rule tactic] XX hm... det er nok med bare 1 meta m XX loesning: objektkvantor [To = f →^{stmt} ZFsub ⊢ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: \underline{(fx)}[\underline{m}] = \underline{(fy)}[\underline{m}] ⊢ \underline{(fx)} =_{f} \underline{(fy)}][To = f →^{proof} Rule tactic] [From < f →^{stmt} ZFsub ⊢ ∀\underline{m}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: \underline{(fx)} <_{f} \underline{(fy)} ⊢ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⊢ c_{Ex} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= \underline{(fy)}[\underline{m}]+−\underline{(ϵ)}][From < f →^{proof} Rule tactic] [To < f →^{stmt} ZFsub ⊢ ∀\underline{m}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ c_{Ex} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= \underline{(fy)}[\underline{m}] + −\underline{(ϵ)} ⊢ \underline{(fx)} <_{f} \underline{(fy)}][To < f →^{proof} Rule tactic] [PlusF →^{stmt} ZFsub ⊢ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: \underline{(fx)} +_{f} \underline{(fy)}[\underline{m}] = \underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}]][PlusF →^{proof} Rule tactic] [MinusF →^{stmt} ZFsub ⊢ ∀\underline{m}: ∀\underline{(fx)}: −_{f} \underline{(fx)}[\underline{m}] = −\underline{(fx)}[\underline{m}]][MinusF →^{proof} Rule tactic] [TimesF →^{stmt} ZFsub ⊢ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: \underline{(fx)} ∗_{f} \underline{(fy)}[\underline{m}] = \underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}]][TimesF →^{proof} Rule tactic] [0f →^{stmt} ZFsub ⊢ ∀\underline{m}: 0f[\underline{m}] = 0][0f →^{proof} Rule tactic] [1f →^{stmt} ZFsub ⊢ ∀\underline{m}: 1f[\underline{m}] = 1][1f →^{proof} Rule tactic] [To == XX →^{stmt} ZFsub ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(rx)}: ∀\underline{(ry)}: \underline{(fx)} ∈ \underline{(rx)} ⇒ \underline{(fy)} ∈ \underline{(ry)} ⇒ SF(\underline{(fx)}, \underline{(fy)}) ⊢ \underline{(rx)} == \underline{(ry)}][To == XX →^{proof} Rule tactic] [From == →^{stmt} ZFsub ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: R(\underline{(fx)}) == R(\underline{(fy)}) ⊢ SF(\underline{(fx)}, \underline{(fy)})][From == →^{proof} Rule tactic] [To == →^{stmt} ZFsub ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: SF(\underline{(fx)}, \underline{(fy)}) ⊢ R(\underline{(fx)}) == R(\underline{(fy)})][To == →^{proof} Rule tactic] [From << XX →^{stmt} ZFsub ⊢ ∀\underline{m}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(rx)}: ∀\underline{(ry)}: \underline{(rx)} << \underline{(ry)} ⊢ \underline{(fx)} ∈ \underline{(rx)} ⊢ \underline{(fy)} ∈ \underline{(ry)} ⊢ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⊢ a_{Ex} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= \underline{(fy)}[\underline{m}] + −\underline{(ϵ)}][From << XX →^{proof} Rule tactic] [From << (1) →^{stmt} ZFsub ⊢ ∀\underline{(rx)}: ∀\underline{(ry)}: \underline{(rx)} << \underline{(ry)} ⊢ j_{Ex} ∈ \underline{(rx)}][From << (1) →^{proof} Rule tactic] [From << (2) →^{stmt} ZFsub ⊢ ∀\underline{(rx)}: ∀\underline{(ry)}: \underline{(rx)} << \underline{(ry)} ⊢ t_{Ex} ∈ \underline{(ry)}][From << (2) →^{proof} Rule tactic] [to << XX →^{stmt} ZFsub ⊢ ∀\underline{m}: ∀\underline{n}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(rx)}: ∀\underline{(ry)}: \underline{(fx)} ∈ \underline{(rx)} ⇒ \underline{(fy)} ∈ \underline{(ry)} ⇒ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ a_{Ex} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= \underline{(fy)}[\underline{m}] + −\underline{(ϵ)} ⊢ \underline{(rx)} << \underline{(ry)}][to << XX →^{proof} Rule tactic] [From << →^{stmt} ZFsub ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: R(\underline{(fx)}) << R(\underline{(fy)}) ⊢ \underline{(fx)} <_{f} \underline{(fy)}][From << →^{proof} Rule tactic] [To << →^{stmt} ZFsub ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: \underline{(fx)} <_{f} \underline{(fy)} ⊢ R(\underline{(fx)}) << R(\underline{(fy)})][To << →^{proof} Rule tactic] [FromInR →^{stmt} ZFsub ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: \underline{(fx)} ∈ R(\underline{(fy)}) ⊢ SF(\underline{(fx)}, \underline{(fy)})][FromInR →^{proof} Rule tactic] [PlusR →^{stmt} ZFsub ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: R((fx) +_{f} (fy)) == R(\underline{(fx)} +_{f} \underline{(fy)})][PlusR →^{proof} Rule tactic] [TimesR →^{stmt} ZFsub ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: R(\underline{(fx)}) ∗ ∗R(\underline{(fy)}) == R(\underline{(fx)} ∗_{f} \underline{(fy)})][TimesR →^{proof} Rule tactic] (*** makroer ***)

[ϵ →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[ϵ ¨= \underline{(ϵ)}]⌉)]

[FX →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[FX ¨= \underline{(fx)}]⌉)] [FY →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[FY ¨= \underline{(fy)}]⌉)] [FZ →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[FZ ¨= \underline{(fz)}]⌉)] [FU →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[FU ¨= \underline{(fv)}]⌉)] [FV →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[FV ¨= \underline{var fv}]⌉)] [RX →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[RX ¨= \underline{(rx)}]⌉)] [RY →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[RY ¨= \underline{(ry)}]⌉)] [RZ →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[RZ ¨= \underline{(rz)}]⌉)] [RU →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[RU ¨= \underline{(ru)}]⌉)] [Ex3 →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[Ex3 ¨= c_{Ex}]⌉)] [x <<== y →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[x <<== y ¨= x << y ˙∨ x == y]⌉)] [(−1) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[(−1) ¨= −1]⌉)] [2 →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[2 ¨= (1 + 1)]⌉)] [1/2 →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[1/2 ¨= rec2]⌉)] [x < y →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[x < y ¨= x <= y ˙∧ x ≠ y]⌉)] [x ≠ y →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[x ≠ y ¨= ˙¬ x = y]⌉)] [x − y →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[x − y ¨= x + (−y)]⌉)] [|x| →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[|x| ¨= if(0 <= x, x, −x)]⌉)] [00 →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[00 ¨= R(0f)]⌉)] [01 →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[01 ¨= R(1f)]⌉)] [R((fx)) + +R((fy)) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[R((fx)) + +R((fy)) ¨= R((fx) +_{f} (fy))]⌉)] [− − R((fx)) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[− − R((fx)) ¨= R(−_{f}(fx))]⌉)] XX noedvendig? [R((fx)) − −R((fy)) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[R((fx)) − −R((fy)) ¨= R((fx)) + +R(−_{f}(fy))]⌉)] XX noedvendigt med [R( ) – R( )] konstruktionen? (*** REGELLEMMAER ***) [leqTransitivity →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} <= \underline{y} ⊢ \underline{y} <= \underline{z} ⊢ \underline{x} <= \underline{z}]

[leqTransitivity →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} <= \underline{y} ⊢ \underline{y} <= \underline{z} ⊢ leqTransitivityAxiom ≫ \underline{x} <= \underline{y} ⇒ \underline{y} <= \underline{z} ⇒ \underline{x} <= \underline{z}; MP2 ⊳ \underline{x} <= \underline{y} ⇒ \underline{y} <= \underline{z} ⇒ \underline{x} <= \underline{z} ⊳ \underline{x} <= \underline{y} ⊳ \underline{y} <= \underline{z} ≫ \underline{x} <= \underline{z}⌉, p_{0}, c)]

[leqAntisymmetry →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} <= \underline{y} ⊢ \underline{y} <= \underline{x} ⊢ \underline{x} = \underline{y}]

[leqAntisymmetry →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} <= \underline{y} ⊢ \underline{y} <= \underline{x} ⊢ leqAntisymmetryAxiom ≫ \underline{x} <= \underline{y} ⇒ \underline{y} <= \underline{x} ⇒ \underline{x} = \underline{y}; MP2 ⊳ \underline{x} <= \underline{y} ⇒ \underline{y} <= \underline{x} ⇒ \underline{x} = \underline{y} ⊳ \underline{x} <= \underline{y} ⊳ \underline{y} <= \underline{x} ≫ \underline{x} = \underline{y}⌉, p_{0}, c)]

[leqAddition →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} <= \underline{y} ⊢ \underline{x} + \underline{z} <= \underline{y} + \underline{z}]

[leqAddition →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} <= \underline{y} ⊢ leqAdditionAxiom ≫ \underline{x} <= \underline{y} ⇒ \underline{x} + \underline{z} <= \underline{y} + \underline{z}; MP ⊳ \underline{x} <= \underline{y} ⇒ \underline{x} + \underline{z} <= \underline{y} + \underline{z} ⊳ \underline{x} <= \underline{y} ≫ \underline{x} + \underline{z} <= \underline{y} + \underline{z}⌉, p_{0}, c)]

[leqMultiplication →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: 0 <= \underline{z} ⊢ \underline{x} <= \underline{y} ⊢ \underline{x} ∗ \underline{z} <= \underline{y} ∗ \underline{z}]

[leqMultiplication →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: 0 <= \underline{z} ⊢ \underline{x} <= \underline{y} ⊢ leqMultiplicationAxiom ≫ 0 <= \underline{z} ⇒ \underline{x} <= \underline{y} ⇒ \underline{x} ∗ \underline{z} <= \underline{y} ∗ \underline{z}; MP2 ⊳ 0 <= \underline{z} ⇒ \underline{x} <= \underline{y} ⇒ \underline{x} ∗ \underline{z} <= \underline{y} ∗ \underline{z} ⊳ 0 <= \underline{z} ⊳ \underline{x} <= \underline{y} ≫ \underline{x} ∗ \underline{z} <= \underline{y} ∗ \underline{z}⌉, p_{0}, c)]

[Reciprocal →^{stmt} ZFsub ⊢ ∀\underline{x}: ˙¬ \underline{x} = 0 ⊢ \underline{x} ∗ rec\underline{x} = 1] [Reciprocal →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ˙¬ \underline{x} = 0 ⊢ ReciprocalAxiom ≫ ˙¬ \underline{x} = 0 ⇒ \underline{x} ∗ rec\underline{x} = 1; MP ⊳ ˙¬ \underline{x} = 0 ⇒ \underline{x} ∗ rec\underline{x} = 1 ⊳ ˙¬ \underline{x} = 0 ≫ \underline{x} ∗ rec\underline{x} = 1⌉, p_{0}, c)] [Equality →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ \underline{x} = \underline{z} ⊢ \underline{y} = \underline{z}] [Equality →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ \underline{x} = \underline{z} ⊢ equalityAxiom ≫ \underline{x} = \underline{y} ⇒ \underline{x} = \underline{z} ⇒ \underline{y} = \underline{z}; MP2 ⊳ \underline{x} = \underline{y} ⇒ \underline{x} = \underline{z} ⇒ \underline{y} = \underline{z} ⊳ \underline{x} = \underline{y} ⊳ \underline{x} = \underline{z} ≫ \underline{y} = \underline{z}⌉, p_{0}, c)] [eqLeq →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} = \underline{y} ⊢ \underline{x} <= \underline{y}]

[eqLeq →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} = \underline{y} ⊢ eqLeqAxiom ≫ \underline{x} = \underline{y} ⇒ \underline{x} <= \underline{y}; MP ⊳ \underline{x} = \underline{y} ⇒ \underline{x} <= \underline{y} ⊳ \underline{x} = \underline{y} ≫ \underline{x} <= \underline{y}⌉, p_{0}, c)]

[eqAddition →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ \underline{x} + \underline{z} = \underline{y} + \underline{z}]

[eqAddition →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ eqAdditionAxiom ≫ \underline{x} = \underline{y} ⇒ \underline{x} + \underline{z} = \underline{y} + \underline{z}; MP ⊳ \underline{x} = \underline{y} ⇒ \underline{x} + \underline{z} = \underline{y} + \underline{z} ⊳ \underline{x} = \underline{y} ≫ \underline{x} + \underline{z} = \underline{y} + \underline{z}⌉, p_{0}, c)]

[eqMultiplication →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ \underline{x} ∗ \underline{z} = \underline{y} ∗ \underline{z}]

[eqMultiplication →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ eqMultiplicationAxiom ≫ \underline{x} = \underline{y} ⇒ \underline{x} ∗ \underline{z} = \underline{y} ∗ \underline{z}; MP ⊳ \underline{x} = \underline{y} ⇒ \underline{x} ∗ \underline{z} = \underline{y} ∗ \underline{z} ⊳ \underline{x} = \underline{y} ≫ \underline{x} ∗ \underline{z} = \underline{y} ∗ \underline{z}⌉, p_{0}, c)] (*** UDSAGNSLOGIK ***) [ToNegatedImply →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} ⊢ ˙¬ \underline{b} ⊢ ˙¬ \underline{a} ⇒ \underline{b}] [ToNegatedImply →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} ⊢ ˙¬ \underline{b} ⊢ ˙¬ ˙¬ \underline{a} ⇒ \underline{b} ⊢ RemoveDoubleNeg ⊳ ˙¬ ˙¬ \underline{a} ⇒ \underline{b} ≫ \underline{a} ⇒ \underline{b}; MP ⊳ \underline{a} ⇒ \underline{b} ⊳ \underline{a} ≫ \underline{b}; FromContradiction ⊳ \underline{b} ⊳ ˙¬ \underline{b} ≫ ˙¬ \underline{a} ⇒ \underline{b}; ∀\underline{a}: ∀\underline{b}: Ded ⊳ ∀\underline{a}: ∀\underline{b}: \underline{a} ⊢ ˙¬ \underline{b} ⊢ ˙¬ ˙¬ \underline{a} ⇒ \underline{b} ⊢ ˙¬ \underline{a} ⇒ \underline{b} ≫ \underline{a} ⇒ ˙¬ \underline{b} ⇒ ˙¬ ˙¬ \underline{a} ⇒ \underline{b} ⇒ ˙¬ \underline{a} ⇒ \underline{b}; \underline{a} ⊢ ˙¬ \underline{b} ⊢ MP2 ⊳ \underline{a} ⇒ ˙¬ \underline{b} ⇒ ˙¬ ˙¬ \underline{a} ⇒ \underline{b} ⇒ ˙¬ \underline{a} ⇒ \underline{b} ⊳ \underline{a} ⊳ ˙¬ \underline{b} ≫ ˙¬ ˙¬ \underline{a} ⇒ \underline{b} ⇒ ˙¬ \underline{a} ⇒ \underline{b}; AutoImply ≫ ˙¬ ˙¬ \underline{a} ⇒ \underline{b} ⇒ ˙¬ ˙¬ \underline{a} ⇒ \underline{b}; Neg ⊳ ˙¬ ˙¬ \underline{a} ⇒ \underline{b} ⇒ ˙¬ \underline{a} ⇒ \underline{b} ⊳ ˙¬ ˙¬ \underline{a} ⇒ \underline{b} ⇒ ˙¬ ˙¬ \underline{a} ⇒ \underline{b} ≫ ˙¬ \underline{a} ⇒ \underline{b}⌉, p_{0}, c)] [TND →^{stmt} ZFsub ⊢ ∀\underline{a}: ˙¬ \underline{a} ⇒ ˙¬ \underline{a}] [TND →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: AutoImply ≫ ˙¬ \underline{a} ⇒ ˙¬ \underline{a}; Repetition ⊳ ˙¬ \underline{a} ⇒ ˙¬ \underline{a} ≫ ˙¬ \underline{a} ⇒ ˙¬ \underline{a}⌉, p_{0}, c)] [FromNegations →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} ⇒ \underline{b} ⊢ ˙¬ \underline{a} ⇒ \underline{b} ⊢ \underline{b}] [FromNegations →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} ⇒ \underline{b} ⊢ ˙¬ \underline{a} ⇒ \underline{b} ⊢ TND ≫ ˙¬ \underline{a} ⇒ ˙¬ \underline{a}; FromDisjuncts ⊳ ˙¬ \underline{a} ⇒ ˙¬ \underline{a} ⊳ \underline{a} ⇒ \underline{b} ⊳ ˙¬ \underline{a} ⇒ \underline{b} ≫ \underline{b}⌉, p_{0}, c)] [ImplyNegation →^{stmt} ZFsub ⊢ ∀\underline{a}: \underline{a} ⇒ ˙¬ \underline{a} ⊢ ˙¬ \underline{a}] [ImplyNegation →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: \underline{a} ⇒ ˙¬ \underline{a} ⊢ AutoImply ≫ ˙¬ \underline{a} ⇒ ˙¬ \underline{a}; TND ≫ ˙¬ \underline{a} ⇒ ˙¬ \underline{a}; FromDisjuncts ⊳ ˙¬ \underline{a} ⇒ ˙¬ \underline{a} ⊳ \underline{a} ⇒ ˙¬ \underline{a} ⊳ ˙¬ \underline{a} ⇒ ˙¬ \underline{a} ≫ ˙¬ \underline{a}⌉, p_{0}, c)] [From3Disjuncts →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: ˙¬ \underline{a} ⇒ ˙¬ \underline{b} ⇒ \underline{c} ⊢ \underline{a} ⇒ \underline{d} ⊢ \underline{b} ⇒ \underline{d} ⊢ \underline{c} ⇒ \underline{d} ⊢ \underline{d}] [From3Disjuncts →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: ˙¬ \underline{a} ⇒ ˙¬ \underline{b} ⇒ \underline{c} ⊢ \underline{b} ⇒ \underline{d} ⊢ \underline{c} ⇒ \underline{d} ⊢ ˙¬ \underline{a} ⊢ Repetition ⊳ ˙¬ \underline{a} ⇒ ˙¬ \underline{b} ⇒ \underline{c} ≫ ˙¬ \underline{a} ⇒ ˙¬ \underline{b} ⇒ \underline{c}; MP ⊳ ˙¬ \underline{a} ⇒ ˙¬ \underline{b} ⇒ \underline{c} ⊳ ˙¬ \underline{a} ≫ ˙¬ \underline{b} ⇒ \underline{c}; FromDisjuncts ⊳ ˙¬ \underline{b} ⇒ \underline{c} ⊳ \underline{b} ⇒ \underline{d} ⊳ \underline{c} ⇒ \underline{d} ≫ \underline{d}; ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: Ded ⊳ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: ˙¬ \underline{a} ⇒ ˙¬ \underline{b} ⇒ \underline{c} ⊢ \underline{b} ⇒ \underline{d} ⊢ \underline{c} ⇒ \underline{d} ⊢ ˙¬ \underline{a} ⊢ \underline{d} ≫ ˙¬ \underline{a} ⇒ ˙¬ \underline{b} ⇒ \underline{c} ⇒ \underline{b} ⇒ \underline{d} ⇒ \underline{c} ⇒ \underline{d} ⇒ ˙¬ \underline{a} ⇒ \underline{d}; AutoImply ≫ \underline{a} ⇒ \underline{d} ⇒ \underline{a} ⇒ \underline{d}; ˙¬ \underline{a} ⇒ ˙¬ \underline{b} ⇒ \underline{c} ⊢ \underline{a} ⇒ \underline{d} ⊢ \underline{b} ⇒ \underline{d} ⊢ \underline{c} ⇒ \underline{d} ⊢ MP3 ⊳ ˙¬ \underline{a} ⇒ ˙¬ \underline{b} ⇒ \underline{c} ⇒ \underline{b} ⇒ \underline{d} ⇒ \underline{c} ⇒ \underline{d} ⇒ ˙¬ \underline{a} ⇒ \underline{d} ⊳ ˙¬ \underline{a} ⇒ ˙¬ \underline{b} ⇒ \underline{c} ⊳ \underline{b} ⇒ \underline{d} ⊳ \underline{c} ⇒ \underline{d} ≫ ˙¬ \underline{a} ⇒ \underline{d}; MP ⊳ \underline{a} ⇒ \underline{d} ⇒ \underline{a} ⇒ \underline{d} ⊳ \underline{a} ⇒ \underline{d} ≫ \underline{a} ⇒ \underline{d}; FromNegations ⊳ \underline{a} ⇒ \underline{d} ⊳ ˙¬ \underline{a} ⇒ \underline{d} ≫ \underline{d}⌉, p_{0}, c)] [NegateDisjunct1 →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ \underline{a} ⇒ \underline{b} ⊢ ˙¬ \underline{a} ⊢ \underline{b}] [NegateDisjunct1 →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ \underline{a} ⇒ \underline{b} ⊢ ˙¬ \underline{a} ⊢ Repetition ⊳ ˙¬ \underline{a} ⇒ \underline{b} ≫ ˙¬ \underline{a} ⇒ \underline{b}; MP ⊳ ˙¬ \underline{a} ⇒ \underline{b} ⊳ ˙¬ \underline{a} ≫ \underline{b}⌉, p_{0}, c)] [NegateDisjunct2 →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ \underline{a} ⇒ \underline{b} ⊢ ˙¬ \underline{b} ⊢ \underline{a}] [NegateDisjunct2 →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ \underline{a} ⇒ \underline{b} ⊢ ˙¬ \underline{b} ⊢ Repetition ⊳ ˙¬ \underline{a} ⇒ \underline{b} ≫ ˙¬ \underline{a} ⇒ \underline{b}; NegativeMT ⊳ ˙¬ \underline{a} ⇒ \underline{b} ⊳ ˙¬ \underline{b} ≫ \underline{a}⌉, p_{0}, c)] (***) [ExpandDisjuncts →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: ˙¬ \underline{a} ⇒ \underline{b} ⊢ ˙¬ \underline{c} ⇒ \underline{d} ⊢ ˙¬ \underline{b} ⇒ ˙¬ \underline{d} ⇒ ˙¬ \underline{a} ⇒ ˙¬ \underline{c}] [ExpandDisjuncts →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: ˙¬ \underline{a} ⇒ \underline{b} ⊢ ˙¬ \underline{c} ⇒ \underline{d} ⊢ ˙¬ \underline{b} ⊢ ˙¬ \underline{d} ⊢ NegateDisjunct2 ⊳ ˙¬ \underline{a} ⇒ \underline{b} ⊳ ˙¬ \underline{b} ≫ \underline{a}; NegateDisjunct2 ⊳ ˙¬ \underline{c} ⇒ \underline{d} ⊳ ˙¬ \underline{d} ≫ \underline{c}; JoinConjuncts ⊳ \underline{a} ⊳ \underline{c} ≫ ˙¬ \underline{a} ⇒ ˙¬ \underline{c}; ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: Ded ⊳ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: ˙¬ \underline{a} ⇒ \underline{b} ⊢ ˙¬ \underline{c} ⇒ \underline{d} ⊢ ˙¬ \underline{b} ⊢ ˙¬ \underline{d} ⊢ ˙¬ \underline{a} ⇒ ˙¬ \underline{c} ≫ ˙¬ \underline{a} ⇒ \underline{b} ⇒ ˙¬ \underline{c} ⇒ \underline{d} ⇒ ˙¬ \underline{b} ⇒ ˙¬ \underline{d} ⇒ ˙¬ \underline{a} ⇒ ˙¬ \underline{c}; ˙¬ \underline{a} ⇒ \underline{b} ⊢ ˙¬ \underline{c} ⇒ \underline{d} ⊢ MP2 ⊳ ˙¬ \underline{a} ⇒ \underline{b} ⇒ ˙¬ \underline{c} ⇒ \underline{d} ⇒ ˙¬ \underline{b} ⇒ ˙¬ \underline{d} ⇒ ˙¬ \underline{a} ⇒ ˙¬ \underline{c} ⊳ ˙¬ \underline{a} ⇒ \underline{b} ⊳ ˙¬ \underline{c} ⇒ \underline{d} ≫ ˙¬ \underline{b} ⇒ ˙¬ \underline{d} ⇒ ˙¬ \underline{a} ⇒ ˙¬ \underline{c}; Repetition ⊳ ˙¬ \underline{b} ⇒ ˙¬ \underline{d} ⇒ ˙¬ \underline{a} ⇒ ˙¬ \underline{c} ≫ ˙¬ \underline{b} ⇒ ˙¬ \underline{d} ⇒ ˙¬ \underline{a} ⇒ ˙¬ \underline{c}⌉, p_{0}, c)] [From2 ∗ 2Disjuncts →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: ∀\underline{e}: ˙¬ \underline{a} ⇒ \underline{b} ⊢ ˙¬ \underline{c} ⇒ \underline{d} ⊢ \underline{a} ⇒ \underline{c} ⇒ \underline{e} ⊢ \underline{a} ⇒ \underline{d} ⇒ \underline{e} ⊢ \underline{b} ⇒ \underline{c} ⇒ \underline{e} ⊢ \underline{b} ⇒ \underline{d} ⇒ \underline{e} ⊢ \underline{e}] [From2 ∗ 2Disjuncts →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: ∀\underline{e}: ˙¬ \underline{c} ⇒ \underline{d} ⊢ \underline{a} ⇒ \underline{c} ⇒ \underline{e} ⊢ \underline{a} ⇒ \underline{d} ⇒ \underline{e} ⊢ \underline{a} ⊢ MP ⊳ \underline{a} ⇒ \underline{c} ⇒ \underline{e} ⊳ \underline{a} ≫ \underline{c} ⇒ \underline{e}; MP ⊳ \underline{a} ⇒ \underline{d} ⇒ \underline{e} ⊳ \underline{a} ≫ \underline{d} ⇒ \underline{e}; FromDisjuncts ⊳ ˙¬ \underline{c} ⇒ \underline{d} ⊳ \underline{c} ⇒ \underline{e} ⊳ \underline{d} ⇒ \underline{e} ≫ \underline{e}; ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: ∀\underline{e}: ˙¬ \underline{a} ⇒ \underline{b} ⊢ ˙¬ \underline{c} ⇒ \underline{d} ⊢ \underline{b} ⇒ \underline{c} ⇒ \underline{e} ⊢ \underline{b} ⇒ \underline{d} ⇒ \underline{e} ⊢ ˙¬ \underline{a} ⊢ NegateDisjunct1 ⊳ ˙¬ \underline{a} ⇒ \underline{b} ⊳ ˙¬ \underline{a} ≫ \underline{b}; MP ⊳ \underline{b} ⇒ \underline{c} ⇒ \underline{e} ⊳ \underline{b} ≫ \underline{c} ⇒ \underline{e}; MP ⊳ \underline{b} ⇒ \underline{d} ⇒ \underline{e} ⊳ \underline{b} ≫ \underline{d} ⇒ \underline{e}; FromDisjuncts ⊳ ˙¬ \underline{c} ⇒ \underline{d} ⊳ \underline{c} ⇒ \underline{e} ⊳ \underline{d} ⇒ \underline{e} ≫ \underline{e}; ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: ∀\underline{e}: Ded ⊳ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: ∀\underline{e}: ˙¬ \underline{c} ⇒ \underline{d} ⊢ \underline{a} ⇒ \underline{c} ⇒ \underline{e} ⊢ \underline{a} ⇒ \underline{d} ⇒ \underline{e} ⊢ \underline{a} ⊢ \underline{e} ≫ ˙¬ \underline{c} ⇒ \underline{d} ⇒ \underline{a} ⇒ \underline{c} ⇒ \underline{e} ⇒ \underline{a} ⇒ \underline{d} ⇒ \underline{e} ⇒ \underline{a} ⇒ \underline{e}; Ded ⊳ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: ∀\underline{e}: ˙¬ \underline{a} ⇒ \underline{b} ⊢ ˙¬ \underline{c} ⇒ \underline{d} ⊢ \underline{b} ⇒ \underline{c} ⇒ \underline{e} ⊢ \underline{b} ⇒ \underline{d} ⇒ \underline{e} ⊢ ˙¬ \underline{a} ⊢ \underline{e} ≫ ˙¬ \underline{a} ⇒ \underline{b} ⇒ ˙¬ \underline{c} ⇒ \underline{d} ⇒ \underline{b} ⇒ \underline{c} ⇒ \underline{e} ⇒ \underline{b} ⇒ \underline{d} ⇒ \underline{e} ⇒ ˙¬ \underline{a} ⇒ \underline{e}; ˙¬ \underline{a} ⇒ \underline{b} ⊢ ˙¬ \underline{c} ⇒ \underline{d} ⊢ \underline{a} ⇒ \underline{c} ⇒ \underline{e} ⊢ \underline{a} ⇒ \underline{d} ⇒ \underline{e} ⊢ \underline{b} ⇒ \underline{c} ⇒ \underline{e} ⊢ \underline{b} ⇒ \underline{d} ⇒ \underline{e} ⊢ MP3 ⊳ ˙¬ \underline{c} ⇒ \underline{d} ⇒ \underline{a} ⇒ \underline{c} ⇒ \underline{e} ⇒ \underline{a} ⇒ \underline{d} ⇒ \underline{e} ⇒ \underline{a} ⇒ \underline{e} ⊳ ˙¬ \underline{c} ⇒ \underline{d} ⊳ \underline{a} ⇒ \underline{c} ⇒ \underline{e} ⊳ \underline{a} ⇒ \underline{d} ⇒ \underline{e} ≫ \underline{a} ⇒ \underline{e}; MP4 ⊳ ˙¬ \underline{a} ⇒ \underline{b} ⇒ ˙¬ \underline{c} ⇒ \underline{d} ⇒ \underline{b} ⇒ \underline{c} ⇒ \underline{e} ⇒ \underline{b} ⇒ \underline{d} ⇒ \underline{e} ⇒ ˙¬ \underline{a} ⇒ \underline{e} ⊳ ˙¬ \underline{a} ⇒ \underline{b} ⊳ ˙¬ \underline{c} ⇒ \underline{d} ⊳ \underline{b} ⇒ \underline{c} ⇒ \underline{e} ⊳ \underline{b} ⇒ \underline{d} ⇒ \underline{e} ≫ ˙¬ \underline{a} ⇒ \underline{e}; FromNegations ⊳ \underline{a} ⇒ \underline{e} ⊳ ˙¬ \underline{a} ⇒ \underline{e} ≫ \underline{e}⌉, p_{0}, c)] (*** EQUALITY ***) [eqReflexivity →^{stmt} ZFsub ⊢ ∀\underline{x}: \underline{x} = \underline{x}] [eqReflexivity →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: leqReflexivity ≫ \underline{x} <= \underline{x}; leqAntisymmetry ⊳ \underline{x} <= \underline{x} ⊳ \underline{x} <= \underline{x} ≫ \underline{x} = \underline{x}⌉, p_{0}, c)] [eqSymmetry →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} = \underline{y} ⊢ \underline{y} = \underline{x}]

[eqSymmetry →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} = \underline{y} ⊢ eqReflexivity ≫ \underline{x} = \underline{x}; Equality ⊳ \underline{x} = \underline{y} ⊳ \underline{x} = \underline{x} ≫ \underline{y} = \underline{x}⌉, p_{0}, c)]

[eqTransitivity →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ \underline{y} = \underline{z} ⊢ \underline{x} = \underline{z}] [eqTransitivity →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ \underline{y} = \underline{z} ⊢ eqSymmetry ⊳ \underline{x} = \underline{y} ≫ \underline{y} = \underline{x}; Equality ⊳ \underline{y} = \underline{x} ⊳ \underline{y} = \underline{z} ≫ \underline{x} = \underline{z}⌉, p_{0}, c)]

[eqTransitivity4 →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: \underline{x} = \underline{y} ⊢ \underline{y} = \underline{z} ⊢ \underline{z} = \underline{u} ⊢ \underline{x} = \underline{u}]

[eqTransitivity4 →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: \underline{x} = \underline{y} ⊢ \underline{y} = \underline{z} ⊢ \underline{z} = \underline{u} ⊢ eqTransitivity ⊳ \underline{x} = \underline{y} ⊳ \underline{y} = \underline{z} ≫ \underline{x} = \underline{z}; eqTransitivity ⊳ \underline{x} = \underline{z} ⊳ \underline{z} = \underline{u} ≫ \underline{x} = \underline{u}⌉, p_{0}, c)] [eqTransitivity5 →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: ∀\underline{v}: \underline{x} = \underline{y} ⊢ \underline{y} = \underline{z} ⊢ \underline{z} = \underline{u} ⊢ \underline{u} = \underline{v} ⊢ \underline{x} = \underline{v}] [eqTransitivity5 →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: ∀\underline{v}: \underline{x} = \underline{y} ⊢ \underline{y} = \underline{z} ⊢ \underline{z} = \underline{u} ⊢ \underline{u} = \underline{v} ⊢ eqTransitivity4 ⊳ \underline{x} = \underline{y} ⊳ \underline{y} = \underline{z} ⊳ \underline{z} = \underline{u} ≫ \underline{x} = \underline{u}; eqTransitivity ⊳ \underline{x} = \underline{u} ⊳ \underline{u} = \underline{v} ≫ \underline{x} = \underline{v}⌉, p_{0}, c)] [eqTransitivity6 →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: ∀\underline{v}: ∀\underline{w}: \underline{x} = \underline{y} ⊢ \underline{y} = \underline{z} ⊢ \underline{z} = \underline{u} ⊢ \underline{u} = \underline{v} ⊢ \underline{v} = \underline{w} ⊢ \underline{x} = \underline{w}] [eqTransitivity6 →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: ∀\underline{v}: ∀\underline{w}: \underline{x} = \underline{y} ⊢ \underline{y} = \underline{z} ⊢ \underline{z} = \underline{u} ⊢ \underline{u} = \underline{v} ⊢ \underline{v} = \underline{w} ⊢ eqTransitivity5 ⊳ \underline{x} = \underline{y} ⊳ \underline{y} = \underline{z} ⊳ \underline{z} = \underline{u} ⊳ \underline{u} = \underline{v} ≫ \underline{x} = \underline{v}; eqTransitivity ⊳ \underline{x} = \underline{v} ⊳ \underline{v} = \underline{w} ≫ \underline{x} = \underline{w}⌉, p_{0}, c)] [plus0Left →^{stmt} ZFsub ⊢ ∀\underline{x}: 0 + \underline{x} = \underline{x}] [plus0Left →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: plus0 ≫ \underline{x} + 0 = \underline{x}; plusCommutativity ≫ 0 + \underline{x} = \underline{x} + 0; eqTransitivity ⊳ 0 + \underline{x} = \underline{x} + 0 ⊳ \underline{x} + 0 = \underline{x} ≫ 0 + \underline{x} = \underline{x}⌉, p_{0}, c)] [times1Left →^{stmt} ZFsub ⊢ ∀\underline{x}: 1 ∗ \underline{x} = \underline{x}] [times1Left →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: times1 ≫ \underline{x} ∗ 1 = \underline{x}; timesCommutativity ≫ 1 ∗ \underline{x} = \underline{x} ∗ 1; eqTransitivity ⊳ 1 ∗ \underline{x} = \underline{x} ∗ 1 ⊳ \underline{x} ∗ 1 = \underline{x} ≫ 1 ∗ \underline{x} = \underline{x}⌉, p_{0}, c)] [lemma eqAdditionLeft →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ \underline{z} + \underline{x} = \underline{z} + \underline{y}]

[lemma eqAdditionLeft →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ eqAddition ⊳ \underline{x} = \underline{y} ≫ \underline{x} + \underline{z} = \underline{y} + \underline{z}; plusCommutativity ≫ \underline{z} + \underline{x} = \underline{x} + \underline{z}; plusCommutativity ≫ \underline{y} + \underline{z} = \underline{z} + \underline{y}; eqTransitivity4 ⊳ \underline{z} + \underline{x} = \underline{x} + \underline{z} ⊳ \underline{x} + \underline{z} = \underline{y} + \underline{z} ⊳ \underline{y} + \underline{z} = \underline{z} + \underline{y} ≫ \underline{z} + \underline{x} = \underline{z} + \underline{y}⌉, p_{0}, c)]

[EqMultiplicationLeft →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ \underline{z} ∗ \underline{x} = \underline{z} ∗ \underline{y}]

[EqMultiplicationLeft →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ eqMultiplication ⊳ \underline{x} = \underline{y} ≫ \underline{x} ∗ \underline{z} = \underline{y} ∗ \underline{z}; timesCommutativity ≫ \underline{z} ∗ \underline{x} = \underline{x} ∗ \underline{z}; timesCommutativity ≫ \underline{y} ∗ \underline{z} = \underline{z} ∗ \underline{y}; eqTransitivity4 ⊳ \underline{z} ∗ \underline{x} = \underline{x} ∗ \underline{z} ⊳ \underline{x} ∗ \underline{z} = \underline{y} ∗ \underline{z} ⊳ \underline{y} ∗ \underline{z} = \underline{z} ∗ \underline{y} ≫ \underline{z} ∗ \underline{x} = \underline{z} ∗ \underline{y}⌉, p_{0}, c)]

[DistributionOut →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} ∗ \underline{y} + \underline{x} ∗ \underline{z} = \underline{x} ∗ \underline{y} + \underline{z}]

[DistributionOut →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: Distribution ≫ \underline{x} ∗ \underline{y} + \underline{z} = \underline{x} ∗ \underline{y} + \underline{x} ∗ \underline{z}; eqSymmetry ⊳ \underline{x} ∗ \underline{y} + \underline{z} = \underline{x} ∗ \underline{y} + \underline{x} ∗ \underline{z} ≫ \underline{x} ∗ \underline{y} + \underline{x} ∗ \underline{z} = \underline{x} ∗ \underline{y} + \underline{z}⌉, p_{0}, c)]

[Three2twoTerms →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: \underline{y} + \underline{z} = \underline{u} ⊢ \underline{x} + \underline{y} + \underline{z} = \underline{x} + \underline{u}]

[Three2twoTerms →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: \underline{y} + \underline{z} = \underline{u} ⊢ lemma eqAdditionLeft ⊳ \underline{y} + \underline{z} = \underline{u} ≫ \underline{x} + \underline{y} + \underline{z} = \underline{x} + \underline{u}; plusAssociativity ≫ \underline{x} + \underline{y} + \underline{z} = \underline{x} + \underline{y} + \underline{z}; eqTransitivity ⊳ \underline{x} + \underline{y} + \underline{z} = \underline{x} + \underline{y} + \underline{z} ⊳ \underline{x} + \underline{y} + \underline{z} = \underline{x} + \underline{u} ≫ \underline{x} + \underline{y} + \underline{z} = \underline{x} + \underline{u}⌉, p_{0}, c)]

[Three2threeTerms →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} + \underline{y} + \underline{z} = \underline{x} + \underline{z} + \underline{y}]

[Three2threeTerms →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: plusCommutativity ≫ \underline{y} + \underline{z} = \underline{z} + \underline{y}; Three2twoTerms ⊳ \underline{y} + \underline{z} = \underline{z} + \underline{y} ≫ \underline{x} + \underline{y} + \underline{z} = \underline{x} + \underline{z} + \underline{y}; plusAssociativity ≫ \underline{x} + \underline{z} + \underline{y} = \underline{x} + \underline{z} + \underline{y}; eqSymmetry ⊳ \underline{x} + \underline{z} + \underline{y} = \underline{x} + \underline{z} + \underline{y} ≫ \underline{x} + \underline{z} + \underline{y} = \underline{x} + \underline{z} + \underline{y}; eqTransitivity ⊳ \underline{x} + \underline{y} + \underline{z} = \underline{x} + \underline{z} + \underline{y} ⊳ \underline{x} + \underline{z} + \underline{y} = \underline{x} + \underline{z} + \underline{y} ≫ \underline{x} + \underline{y} + \underline{z} = \underline{x} + \underline{z} + \underline{y}⌉, p_{0}, c)]

[Three2threeFactors →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: \underline{y} ∗ \underline{z} = \underline{u} ⊢ \underline{x} ∗ \underline{y} ∗ \underline{z} = \underline{x} ∗ \underline{u}]

[Three2threeFactors →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: \underline{y} ∗ \underline{z} = \underline{u} ⊢ EqMultiplicationLeft⊳\underline{y}∗\underline{z} = \underline{u} ≫ \underline{x}∗\underline{y}∗\underline{z} = \underline{x}∗\underline{u}; timesAssociativity ≫ \underline{x}∗\underline{y}∗\underline{z} = \underline{x}∗\underline{y}∗\underline{z}; eqTransitivity⊳\underline{x}∗\underline{y}∗\underline{z} = \underline{x}∗\underline{y}∗\underline{z}⊳\underline{x}∗\underline{y}∗\underline{z} = \underline{x}∗\underline{u} ≫ \underline{x}∗\underline{y}∗\underline{z} = \underline{x}∗\underline{u}⌉, p_{0}, c)]

[AddEquations →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: \underline{x} = \underline{y} ⊢ \underline{z} = \underline{u} ⊢ \underline{x} + \underline{z} = \underline{y} + \underline{u}]

[AddEquations →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: \underline{x} = \underline{y} ⊢ \underline{z} = \underline{u} ⊢ eqAddition ⊳ \underline{x} = \underline{y} ≫ \underline{x} + \underline{z} = \underline{y} + \underline{z}; lemma eqAdditionLeft ⊳ \underline{z} = \underline{u} ≫ \underline{y} + \underline{z} = \underline{y} + \underline{u}; eqTransitivity ⊳ \underline{x} + \underline{z} = \underline{y} + \underline{z} ⊳ \underline{y} + \underline{z} = \underline{y} + \underline{u} ≫ \underline{x} + \underline{z} = \underline{y} + \underline{u}⌉, p_{0}, c)]

[SubtractEquations →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: \underline{x} + \underline{z} = \underline{y} + \underline{u} ⊢ \underline{z} = \underline{u} ⊢ \underline{x} = \underline{y}]

[SubtractEquations →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: \underline{x} + \underline{z} = \underline{y} + \underline{u} ⊢ \underline{z} = \underline{u} ⊢ eqAddition ⊳ \underline{x} + \underline{z} = \underline{y} + \underline{u} ≫ \underline{x} + \underline{z} + −\underline{z} = \underline{y} + \underline{u} + −\underline{z}; plus0Left ≫ 0 + \underline{z} = \underline{z}; eqTransitivity ⊳ 0 + \underline{z} = \underline{z} ⊳ \underline{z} = \underline{u} ≫ 0 + \underline{z} = \underline{u}; PositiveToRight(Eq) ⊳ 0 + \underline{z} = \underline{u} ≫ 0 = \underline{u} + −\underline{z}; eqSymmetry ⊳ 0 = \underline{u} + −\underline{z} ≫ \underline{u} + −\underline{z} = 0; lemma eqAdditionLeft ⊳ \underline{u} + −\underline{z} = 0 ≫ \underline{y} + \underline{u} + −\underline{z} = \underline{y} + 0; plusAssociativity ≫ \underline{y} + \underline{u} + −\underline{z} = \underline{y} + \underline{u} + −\underline{z}; plus0 ≫ \underline{y} + 0 = \underline{y}; eqTransitivity4 ⊳ \underline{y} + \underline{u} + −\underline{z} = \underline{y} + \underline{u} + −\underline{z} ⊳ \underline{y} + \underline{u} + −\underline{z} = \underline{y} + 0 ⊳ \underline{y} + 0 = \underline{y} ≫ \underline{y} + \underline{u} + −\underline{z} = \underline{y}; x = x + y − y ≫ \underline{x} = \underline{x} + \underline{z} + −\underline{z}; eqTransitivity4 ⊳ \underline{x} = \underline{x} + \underline{z} + −\underline{z} ⊳ \underline{x} + \underline{z} + −\underline{z} = \underline{y} + \underline{u} + −\underline{z} ⊳ \underline{y} + \underline{u} + −\underline{z} = \underline{y} ≫ \underline{x} = \underline{y}⌉, p_{0}, c)]

[SubtractEquationsLeft →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: \underline{x}+\underline{z} = \underline{y}+\underline{u} ⊢ \underline{x} = \underline{y} ⊢ \underline{z} = \underline{u}]

[SubtractEquationsLeft →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: \underline{x} + \underline{z} = \underline{y} + \underline{u} ⊢ \underline{x} = \underline{y} ⊢ plusCommutativity ≫ \underline{z} + \underline{x} = \underline{x} + \underline{z}; plusCommutativity ≫ \underline{y} + \underline{u} = \underline{u} + \underline{y}; eqTransitivity4 ⊳ \underline{z} + \underline{x} = \underline{x} + \underline{z} ⊳ \underline{x} + \underline{z} = \underline{y} + \underline{u} ⊳ \underline{y} + \underline{u} = \underline{u} + \underline{y} ≫ \underline{z} + \underline{x} = \underline{u} + \underline{y}; SubtractEquations ⊳ \underline{z} + \underline{x} = \underline{u} + \underline{y} ⊳ \underline{x} = \underline{y} ≫ \underline{z} = \underline{u}⌉, p_{0}, c)]

[EqNegated →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} = \underline{y} ⊢ −\underline{x} = −\underline{y}]

[EqNegated →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} = \underline{y} ⊢ Negative ≫ \underline{x} + −\underline{x} = 0; Negative ≫ \underline{y} + −\underline{y} = 0; eqSymmetry ⊳ \underline{y} + −\underline{y} = 0 ≫ 0 = \underline{y} + −\underline{y}; eqTransitivity ⊳ \underline{x} + −\underline{x} = 0 ⊳ 0 = \underline{y} + −\underline{y} ≫ \underline{x} + −\underline{x} = \underline{y} + −\underline{y}; SubtractEquationsLeft ⊳ \underline{x} + −\underline{x} = \underline{y} + −\underline{y} ⊳ \underline{x} = \underline{y} ≫ −\underline{x} = −\underline{y}⌉, p_{0}, c)]

[PositiveToRight(Eq) →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} + \underline{y} = \underline{z} ⊢ \underline{x} = \underline{z} + −\underline{y}]

[PositiveToRight(Eq) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} + \underline{y} = \underline{z} ⊢ eqAddition ⊳ \underline{x} + \underline{y} = \underline{z} ≫ \underline{x} + \underline{y} + −\underline{y} = \underline{z} + −\underline{y}; x = x + y − y ≫ \underline{x} = \underline{x}+\underline{y}+−\underline{y}; eqTransitivity⊳\underline{x} = \underline{x}+\underline{y}+−\underline{y}⊳\underline{x}+\underline{y}+−\underline{y} = \underline{z}+−\underline{y} ≫ \underline{x} = \underline{z}+−\underline{y}⌉, p_{0}, c)] [PositiveToLeft(Eq)(1term) →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} = \underline{y} ⊢ \underline{x} + −\underline{y} = 0]

[PositiveToLeft(Eq)(1term) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} = \underline{y} ⊢ eqAddition ⊳ \underline{x} = \underline{y} ≫ \underline{x} + −\underline{y} = \underline{y} + −\underline{y}; Negative ≫ \underline{y} + −\underline{y} = 0; eqTransitivity ⊳ \underline{x} + −\underline{y} = \underline{y} + −\underline{y} ⊳ \underline{y} + −\underline{y} = 0 ≫ \underline{x} + −\underline{y} = 0⌉, p_{0}, c)]

[PositiveToRight(Leq)(1term) →^{stmt} ZFsub ⊢ ∀\underline{y}: ∀\underline{z}: \underline{y} <= \underline{z} ⊢ 0 <= \underline{z} + −\underline{y}]

[PositiveToRight(Leq)(1term) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{y}: ∀\underline{z}: \underline{y} <= \underline{z} ⊢ plus0Left ≫ 0 + \underline{y} = \underline{y}; eqSymmetry ⊳ 0 + \underline{y} = \underline{y} ≫ \underline{y} = 0 + \underline{y}; subLeqLeft ⊳ \underline{y} = 0 + \underline{y} ⊳ \underline{y} <= \underline{z} ≫ 0 + \underline{y} <= \underline{z}; PositiveToRight(Leq) ⊳ 0 + \underline{y} <= \underline{z} ≫ 0 <= \underline{z} + −\underline{y}⌉, p_{0}, c)]

[NegativeToLeft(Eq) →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} + −\underline{z} ⊢ \underline{x} + \underline{z} = \underline{y}]

[NegativeToLeft(Eq) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} + −\underline{z} ⊢ eqAddition ⊳ \underline{x} = \underline{y} + −\underline{z} ≫ \underline{x} + \underline{z} = \underline{y} + −\underline{z} + \underline{z}; Three2threeTerms ≫ \underline{y} + −\underline{z} + \underline{z} = \underline{y} + \underline{z} + −\underline{z}; x = x + y − y ≫ \underline{y} = \underline{y} + \underline{z} + −\underline{z}; eqSymmetry ⊳ \underline{y} = \underline{y} + \underline{z} + −\underline{z} ≫ \underline{y} + \underline{z} + −\underline{z} = \underline{y}; eqTransitivity4 ⊳ \underline{x} + \underline{z} = \underline{y} + −\underline{z} + \underline{z} ⊳ \underline{y} + −\underline{z} + \underline{z} = \underline{y} + \underline{z} + −\underline{z} ⊳ \underline{y} + \underline{z} + −\underline{z} = \underline{y} ≫ \underline{x} + \underline{z} = \underline{y}⌉, p_{0}, c)] (*** NO EQUALITY ***) [LessNeq →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊢ ˙¬ \underline{x} = \underline{y}]

[LessNeq →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊢ Repetition ⊳ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ≫ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y}; SecondConjunct ⊳ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ≫ ˙¬ \underline{x} = \underline{y}⌉, p_{0}, c)]

[NeqSymmetry →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ \underline{x} = \underline{y} ⊢ ˙¬ \underline{y} = \underline{x}]

[NeqSymmetry →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \underline{y} = \underline{x} ⊢ eqSymmetry ⊳ \underline{y} = \underline{x} ≫ \underline{x} = \underline{y}; ∀\underline{x}: ∀\underline{y}: Ded ⊳ ∀\underline{x}: ∀\underline{y}: \underline{y} = \underline{x} ⊢ \underline{x} = \underline{y} ≫ \underline{y} = \underline{x} ⇒ \underline{x} = \underline{y}; ˙¬ \underline{x} = \underline{y} ⊢ MT ⊳ \underline{y} = \underline{x} ⇒ \underline{x} = \underline{y} ⊳ ˙¬ \underline{x} = \underline{y} ≫ ˙¬ \underline{y} = \underline{x}⌉, p_{0}, c)]

[NeqNegated →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ \underline{x} = \underline{y} ⊢ ˙¬ − \underline{x} = −\underline{y}]

[NeqNegated →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ \underline{x} = \underline{y} ⊢ −\underline{x} = −\underline{y} ⊢ EqNegated ⊳ −\underline{x} = −\underline{y} ≫ − − \underline{x} = − − \underline{y}; DoubleMinus ≫ − − \underline{x} = \underline{x}; eqSymmetry ⊳ − − \underline{x} = \underline{x} ≫ \underline{x} = − − \underline{x}; DoubleMinus ≫ − − \underline{y} = \underline{y}; eqTransitivity4 ⊳ \underline{x} = − − \underline{x} ⊳ − − \underline{x} = − − \underline{y} ⊳ − − \underline{y} = \underline{y} ≫ \underline{x} = \underline{y}; FromContradiction⊳\underline{x} = \underline{y}⊳ ˙¬ \underline{x} = \underline{y} ≫ ˙¬ −\underline{x} = −\underline{y}; ∀\underline{x}: ∀\underline{y}: Ded⊳∀\underline{x}: ∀\underline{y}: ˙¬ \underline{x} = \underline{y} ⊢ −\underline{x} = −\underline{y} ⊢ ˙¬ − \underline{x} = −\underline{y} ≫ ˙¬ \underline{x} = \underline{y} ⇒ −\underline{x} = −\underline{y} ⇒ ˙¬ − \underline{x} = −\underline{y}; ˙¬ \underline{x} = \underline{y} ⊢ MP ⊳ ˙¬ \underline{x} = \underline{y} ⇒ −\underline{x} = −\underline{y} ⇒ ˙¬ − \underline{x} = −\underline{y} ⊳ ˙¬ \underline{x} = \underline{y} ≫ −\underline{x} = −\underline{y} ⇒ ˙¬ − \underline{x} = −\underline{y}; ImplyNegation ⊳ −\underline{x} = −\underline{y} ⇒ ˙¬ − \underline{x} = −\underline{y} ≫ ˙¬ − \underline{x} = −\underline{y}⌉, p_{0}, c)]

[SubNeqRight →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ ˙¬ \underline{z} = \underline{x} ⊢ ˙¬ \underline{z} = \underline{y}]

[SubNeqRight →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ ˙¬ \underline{z} = \underline{x} ⊢ NeqSymmetry ⊳ ˙¬ \underline{z} = \underline{x} ≫ ˙¬ \underline{x} = \underline{z}; SubNeqLeft ⊳ \underline{x} = \underline{y} ⊳ ˙¬ \underline{x} = \underline{z} ≫ ˙¬ \underline{y} = \underline{z}; NeqSymmetry ⊳ ˙¬ \underline{y} = \underline{z} ≫ ˙¬ \underline{z} = \underline{y}⌉, p_{0}, c)]

[SubNeqLeft →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ ˙¬ \underline{x} = \underline{z} ⊢ ˙¬ \underline{y} = \underline{z}]

[SubNeqLeft →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ ˙¬ \underline{x} = \underline{z} ⊢ equalityAxiom ≫ \underline{y} = \underline{x} ⇒ \underline{y} = \underline{z} ⇒ \underline{x} = \underline{z}; eqSymmetry ⊳ \underline{x} = \underline{y} ≫ \underline{y} = \underline{x}; MP ⊳ \underline{y} = \underline{x} ⇒ \underline{y} = \underline{z} ⇒ \underline{x} = \underline{z} ⊳ \underline{y} = \underline{x} ≫ \underline{y} = \underline{z} ⇒ \underline{x} = \underline{z}; Contrapositive ⊳ \underline{y} = \underline{z} ⇒ \underline{x} = \underline{z} ≫ ˙¬ \underline{x} = \underline{z} ⇒ ˙¬ \underline{y} = \underline{z}; MP ⊳ ˙¬ \underline{x} = \underline{z} ⇒ ˙¬ \underline{y} = \underline{z} ⊳ ˙¬ \underline{x} = \underline{z} ≫ ˙¬ \underline{y} = \underline{z}⌉, p_{0}, c)]

[NeqAddition →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ \underline{x} = \underline{y} ⊢ ˙¬ \underline{x} + \underline{z} = \underline{y} + \underline{z}]

[NeqAddition →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ \underline{x} = \underline{y} ⊢ \underline{x} + \underline{z} = \underline{y} + \underline{z} ⊢ eqReflexivity ≫ \underline{z} = \underline{z}; SubtractEquations ⊳ \underline{x} + \underline{z} = \underline{y} + \underline{z} ⊳ \underline{z} = \underline{z} ≫ \underline{x} = \underline{y}; FromContradiction ⊳ \underline{x} = \underline{y} ⊳ ˙¬ \underline{x} = \underline{y} ≫ ˙¬ \underline{x} + \underline{z} = \underline{y} + \underline{z}; ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: Ded ⊳ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ \underline{x} = \underline{y} ⊢ \underline{x} + \underline{z} = \underline{y} + \underline{z} ⊢ ˙¬ \underline{x} + \underline{z} = \underline{y} + \underline{z} ≫ ˙¬ \underline{x} = \underline{y} ⇒ \underline{x} + \underline{z} = \underline{y} + \underline{z} ⇒ ˙¬ \underline{x} + \underline{z} = \underline{y} + \underline{z}; ˙¬ \underline{x} = \underline{y} ⊢ MP ⊳ ˙¬ \underline{x} = \underline{y} ⇒ \underline{x} + \underline{z} = \underline{y} + \underline{z} ⇒ ˙¬ \underline{x} + \underline{z} = \underline{y} + \underline{z} ⊳ ˙¬ \underline{x} = \underline{y} ≫ \underline{x} + \underline{z} = \underline{y} + \underline{z} ⇒ ˙¬ \underline{x} + \underline{z} = \underline{y} + \underline{z}; ImplyNegation ⊳ \underline{x} + \underline{z} = \underline{y} + \underline{z} ⇒ ˙¬ \underline{x} + \underline{z} = \underline{y} + \underline{z} ≫ ˙¬ \underline{x} + \underline{z} = \underline{y} + \underline{z}⌉, p_{0}, c)]

[NeqMultiplication →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ \underline{z} = 0 ⊢ ˙¬ \underline{x} = \underline{y} ⊢ ˙¬ \underline{x} ∗ \underline{z} = \underline{y} ∗ \underline{z}]

[NeqMultiplication →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ \underline{z} = 0 ⊢ ˙¬ \underline{x} = \underline{y} ⊢ \underline{x} ∗ \underline{z} = \underline{y} ∗ \underline{z} ⊢ ⊳ ˙¬ \underline{z} = 0 ≫ \underline{x} = \underline{x} ∗ \underline{z} ∗ rec\underline{z}; eqMultiplication ⊳ \underline{x} ∗ \underline{z} = \underline{y} ∗ \underline{z} ≫ \underline{x} ∗ \underline{z} ∗ rec\underline{z} = \underline{y}∗\underline{z}∗rec\underline{z}; ⊳ ˙¬ \underline{z} = 0 ≫ \underline{y} = \underline{y}∗\underline{z}∗rec\underline{z}; eqSymmetry⊳\underline{y} = \underline{y}∗\underline{z}∗rec\underline{z} ≫ \underline{y}∗\underline{z}∗rec\underline{z} = \underline{y}; eqTransitivity4 ⊳ \underline{x} = \underline{x} ∗ \underline{z} ∗ rec\underline{z} ⊳ \underline{x} ∗ \underline{z} ∗ rec\underline{z} = \underline{y} ∗ \underline{z} ∗ rec\underline{z} ⊳ \underline{y} ∗ \underline{z} ∗ rec\underline{z} = \underline{y} ≫ \underline{x} = \underline{y}; FromContradiction ⊳ \underline{x} = \underline{y} ⊳ ˙¬ \underline{x} = \underline{y} ≫ ˙¬ \underline{x} ∗ \underline{z} = \underline{y}∗\underline{z}; ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: Ded⊳∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ \underline{z} = 0 ⊢ ˙¬ \underline{x} = \underline{y} ⊢ \underline{x}∗\underline{z} = \underline{y}∗\underline{z} ⊢ ˙¬ \underline{x}∗\underline{z} = \underline{y}∗\underline{z} ≫ ˙¬ \underline{z} = 0 ⇒ ˙¬ \underline{x} = \underline{y} ⇒ \underline{x}∗\underline{z} = \underline{y}∗\underline{z} ⇒ ˙¬ \underline{x}∗\underline{z} = \underline{y}∗\underline{z}; ˙¬ \underline{z} = 0 ⊢ ˙¬ \underline{x} = \underline{y} ⊢ MP2⊳ ˙¬ \underline{z} = 0 ⇒ ˙¬ \underline{x} = \underline{y} ⇒ \underline{x} ∗ \underline{z} = \underline{y} ∗ \underline{z} ⇒ ˙¬ \underline{x} ∗ \underline{z} = \underline{y} ∗ \underline{z} ⊳ ˙¬ \underline{z} = 0 ⊳ ˙¬ \underline{x} = \underline{y} ≫ \underline{x} ∗ \underline{z} = \underline{y} ∗ \underline{z} ⇒ ˙¬ \underline{x}∗\underline{z} = \underline{y}∗\underline{z}; ImplyNegation⊳\underline{x}∗\underline{z} = \underline{y}∗\underline{z} ⇒ ˙¬ \underline{x}∗\underline{z} = \underline{y}∗\underline{z} ≫ ˙¬ \underline{x}∗\underline{z} = \underline{y}∗\underline{z}⌉, p_{0}, c)] (*** NEGATIVE ***) [UniqueNegative →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} + \underline{y} = 0 ⊢ \underline{x} + \underline{z} = 0 ⊢ \underline{y} = \underline{z}]

[UniqueNegative →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} + \underline{y} = 0 ⊢ \underline{x} + \underline{z} = 0 ⊢ plusCommutativity ≫ \underline{y} + \underline{x} = \underline{x} + \underline{y}; eqTransitivity ⊳ \underline{y} + \underline{x} = \underline{x} + \underline{y} ⊳ \underline{x} + \underline{y} = 0 ≫ \underline{y} + \underline{x} = 0; PositiveToRight(Eq) ⊳ \underline{y} + \underline{x} = 0 ≫ \underline{y} = 0 + −\underline{x}; plusCommutativity ≫ \underline{z} + \underline{x} = \underline{x} + \underline{z}; eqTransitivity ⊳ \underline{z} + \underline{x} = \underline{x} + \underline{z} ⊳ \underline{x} + \underline{z} = 0 ≫ \underline{z} + \underline{x} = 0; PositiveToRight(Eq) ⊳ \underline{z} + \underline{x} = 0 ≫ \underline{z} = 0 + −\underline{x}; eqSymmetry ⊳ \underline{z} = 0 + −\underline{x} ≫ 0 + −\underline{x} = \underline{z}; eqTransitivity ⊳ \underline{y} = 0 + −\underline{x} ⊳ 0 + −\underline{x} = \underline{z} ≫ \underline{y} = \underline{z}⌉, p_{0}, c)]

[DoubleMinus →^{stmt} ZFsub ⊢ ∀\underline{x}: − − \underline{x} = \underline{x}] [DoubleMinus →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: Negative ≫ −\underline{x} + − − \underline{x} = 0; x + y = zBackwards ⊳ −\underline{x} + − − \underline{x} = 0 ≫ 0 = − − \underline{x} + −\underline{x}; NegativeToLeft(Eq) ⊳ 0 = − − \underline{x} + −\underline{x} ≫ 0 + \underline{x} = − − \underline{x}; plus0Left ≫ 0 + \underline{x} = \underline{x}; Equality ⊳ 0 + \underline{x} = − − \underline{x} ⊳ 0 + \underline{x} = \underline{x} ≫ − − \underline{x} = \underline{x}⌉, p_{0}, c)] (*** LEQ, nummer 1 af 2 ***) [LeqLessEq →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} <= \underline{y} ⊢ ˙¬ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⇒ \underline{x} = \underline{y}]

[LeqLessEq →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} <= \underline{y} ⊢ ˙¬ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊢ fromNotLess ⊳ ˙¬ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ≫ \underline{y} <= \underline{x}; leqAntisymmetry ⊳ \underline{x} <= \underline{y} ⊳ \underline{y} <= \underline{x} ≫ \underline{x} = \underline{y}; ∀\underline{x}: ∀\underline{y}: Ded ⊳ ∀\underline{x}: ∀\underline{y}: \underline{x} <= \underline{y} ⊢ ˙¬ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊢ \underline{x} = \underline{y} ≫ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⇒ \underline{x} = \underline{y}; \underline{x} <= \underline{y} ⊢ MP ⊳ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⇒ \underline{x} = \underline{y} ⊳ \underline{x} <= \underline{y} ≫ ˙¬ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⇒ \underline{x} = \underline{y}; Repetition ⊳ ˙¬ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⇒ \underline{x} = \underline{y} ≫ ˙¬ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⇒ \underline{x} = \underline{y}⌉, p_{0}, c)] [LessLeq →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊢ \underline{x} <= \underline{y}]

[LessLeq →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊢ Repetition ⊳ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ≫ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y}; FirstConjunct ⊳ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ≫ \underline{x} <= \underline{y}⌉, p_{0}, c)]

[FromLeqGeq →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{x}: ∀\underline{y}: \underline{x} <= \underline{y} ⇒ \underline{a} ⊢ \underline{y} <= \underline{x} ⇒ \underline{a} ⊢ \underline{a}]

[FromLeqGeq →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: ∀\underline{x}: ∀\underline{y}: \underline{x} <= \underline{y} ⇒ \underline{a} ⊢ \underline{y} <= \underline{x} ⇒ \underline{a} ⊢ leqTotality ≫ ˙¬ \underline{x} <= \underline{y} ⇒ \underline{y} <= \underline{x}; FromDisjuncts ⊳ ˙¬ \underline{x} <= \underline{y} ⇒ \underline{y} <= \underline{x} ⊳ \underline{x} <= \underline{y} ⇒ \underline{a} ⊳ \underline{y} <= \underline{x} ⇒ \underline{a} ≫ \underline{a}⌉, p_{0}, c)]

[subLeqRight →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ \underline{z} <= \underline{x} ⊢ \underline{z} <= \underline{y}]

[subLeqRight →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ \underline{z} <= \underline{x} ⊢ eqLeq ⊳ \underline{x} = \underline{y} ≫ \underline{x} <= \underline{y}; leqTransitivity ⊳ \underline{z} <= \underline{x} ⊳ \underline{x} <= \underline{y} ≫ \underline{z} <= \underline{y}⌉, p_{0}, c)]

[subLeqLeft →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ \underline{x} <= \underline{z} ⊢ \underline{y} <= \underline{z}]

[subLeqLeft →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ \underline{x} <= \underline{z} ⊢ eqSymmetry ⊳ \underline{x} = \underline{y} ≫ \underline{y} = \underline{x}; eqLeq ⊳ \underline{y} = \underline{x} ≫ \underline{y} <= \underline{x}; leqTransitivity ⊳ \underline{y} <= \underline{x} ⊳ \underline{x} <= \underline{z} ≫ \underline{y} <= \underline{z}⌉, p_{0}, c)]

[Leq + 1 →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} <= \underline{y} ⊢ ˙¬ \underline{x} <= \underline{y} + 1 ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} + 1]

[Leq + 1 →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} <= \underline{y} ⊢ 0 < 1 ≫ ˙¬ 0 <= 1 ⇒ ˙¬ ˙¬ 0 = 1; LessAdditionLeft ⊳ ˙¬ 0 <= 1 ⇒ ˙¬ ˙¬ 0 = 1 ≫ ˙¬ \underline{y} + 0 <= \underline{y} + 1 ⇒ ˙¬ ˙¬ \underline{y} + 0 = \underline{y} + 1; plus0 ≫ \underline{y} + 0 = \underline{y}; SubLessLeft ⊳ \underline{y} + 0 = \underline{y} ⊳ ˙¬ \underline{y} + 0 <= \underline{y} + 1 ⇒ ˙¬ ˙¬ \underline{y} + 0 = \underline{y} + 1 ≫ ˙¬ \underline{y} <= \underline{y} + 1 ⇒ ˙¬ ˙¬ \underline{y} = \underline{y} + 1; leqLessTransitivity ⊳ \underline{x} <= \underline{y} ⊳ ˙¬ \underline{y} <= \underline{y} + 1 ⇒ ˙¬ ˙¬ \underline{y} = \underline{y} + 1 ≫ ˙¬ \underline{x} <= \underline{y} + 1 ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} + 1⌉, p_{0}, c)]

[PositiveToRight(Leq) →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} + \underline{y} <= \underline{z} ⊢ \underline{x} <= \underline{z} + −\underline{y}]

[PositiveToRight(Leq) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} + \underline{y} <= \underline{z} ⊢ leqAddition ⊳ \underline{x} + \underline{y} <= \underline{z} ≫ \underline{x} + \underline{y} + −\underline{y} <= \underline{z} + −\underline{y}; x = x + y − y ≫ \underline{x} = \underline{x} + \underline{y} + −\underline{y}; eqSymmetry ⊳ \underline{x} = \underline{x} + \underline{y} + −\underline{y} ≫ \underline{x} + \underline{y} + −\underline{y} = \underline{x}; subLeqLeft ⊳ \underline{x} + \underline{y} + −\underline{y} = \underline{x} ⊳ \underline{x} + \underline{y} + −\underline{y} <= \underline{z} + −\underline{y} ≫ \underline{x} <= \underline{z} + −\underline{y}⌉, p_{0}, c)]

[LeqAdditionLeft →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} <= \underline{y} ⊢ \underline{z} + \underline{x} <= \underline{z} + \underline{y}]

[LeqAdditionLeft →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} <= \underline{y} ⊢ leqAddition ⊳ \underline{x} <= \underline{y} ≫ \underline{x} + \underline{z} <= \underline{y} + \underline{z}; plusCommutativity ≫ \underline{x} + \underline{z} = \underline{z} + \underline{x}; plusCommutativity ≫ \underline{y}+\underline{z} = \underline{z}+\underline{y}; subLeqLeft⊳\underline{x}+\underline{z} = \underline{z}+\underline{x}⊳\underline{x}+\underline{z} <= \underline{y}+\underline{z} ≫ \underline{z}+\underline{x} <= \underline{y}+\underline{z}; subLeqRight⊳\underline{y}+\underline{z} = \underline{z}+\underline{y}⊳\underline{z}+\underline{x} <= \underline{y}+\underline{z} ≫ \underline{z}+\underline{x} <= \underline{z}+\underline{y}⌉, p_{0}, c)]

[leqSubtraction →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} + \underline{z} <= \underline{y} + \underline{z} ⊢ \underline{x} <= \underline{y}]

[leqSubtraction →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} + \underline{z} <= \underline{y} + \underline{z} ⊢ leqAddition ⊳ \underline{x} + \underline{z} <= \underline{y} + \underline{z} ≫ \underline{x} + \underline{z} + −\underline{z} <= \underline{y} + \underline{z} + −\underline{z}; x = x + y − y ≫ \underline{x} = \underline{x} + \underline{z} + −\underline{z}; eqSymmetry ⊳ \underline{x} = \underline{x} + \underline{z} + −\underline{z} ≫ \underline{x} + \underline{z} + −\underline{z} = \underline{x}; x = x + y − y ≫ \underline{y} = \underline{y} + \underline{z} + −\underline{z}; eqSymmetry ⊳ \underline{y} = \underline{y} + \underline{z} + −\underline{z} ≫ \underline{y} + \underline{z} + −\underline{z} = \underline{y}; subLeqLeft ⊳ \underline{x} + \underline{z} + −\underline{z} = \underline{x} ⊳ \underline{x} + \underline{z} + −\underline{z} <= \underline{y} + \underline{z} + −\underline{z} ≫ \underline{x} <= \underline{y} + \underline{z} + −\underline{z}; subLeqRight ⊳ \underline{y} + \underline{z} + −\underline{z} = \underline{y} ⊳ \underline{x} <= \underline{y} + \underline{z} + −\underline{z} ≫ \underline{x} <= \underline{y}⌉, p_{0}, c)]

[leqSubtractionLeft →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{z} + \underline{x} <= \underline{z} + \underline{y} ⊢ \underline{x} <= \underline{y}] [leqSubtractionLeft →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{z} + \underline{x} <= \underline{z} + \underline{y} ⊢ plusCommutativity ≫ \underline{z} + \underline{x} = \underline{x} + \underline{z}; plusCommutativity ≫ \underline{z} + \underline{y} = \underline{y} + \underline{z}; subLeqLeft ⊳ \underline{z} + \underline{x} = \underline{x} + \underline{z} ⊳ \underline{z} + \underline{x} <= \underline{z} + \underline{y} ≫ \underline{x} + \underline{z} <= \underline{z} + \underline{y}; subLeqRight ⊳ \underline{z} + \underline{y} = \underline{y} + \underline{z} ⊳ \underline{x} + \underline{z} <= \underline{z} + \underline{y} ≫ \underline{x} + \underline{z} <= \underline{y} + \underline{z}; leqSubtraction ⊳ \underline{x} + \underline{z} <= \underline{y} + \underline{z} ≫ \underline{x} <= \underline{y}⌉, p_{0}, c)]

[thirdGeq →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ \underline{x} <= c_{Ex} ⇒ ˙¬ \underline{y} <= c_{Ex}]

[thirdGeq →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} <= \underline{y} ⊢ leqReflexivity ≫ \underline{y} <= \underline{y}; JoinConjuncts ⊳ \underline{x} <= \underline{y} ⊳ \underline{y} <= \underline{y} ≫ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ \underline{y} <= \underline{y}; ExistIntro @ c_{Ex} @ \underline{y} ⊳ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ \underline{y} <= \underline{y} ≫ ˙¬ \underline{x} <= c_{Ex} ⇒ ˙¬ \underline{y} <= c_{Ex}; ∀\underline{x}: ∀\underline{y}: \underline{y} <= \underline{x} ⊢ leqReflexivity ≫ \underline{x} <= \underline{x}; JoinConjuncts ⊳ \underline{x} <= \underline{x} ⊳ \underline{y} <= \underline{x} ≫ ˙¬ \underline{x} <= \underline{x} ⇒ ˙¬ \underline{y} <= \underline{x}; ExistIntro @ c_{Ex} @ \underline{x} ⊳ ˙¬ \underline{x} <= \underline{x} ⇒ ˙¬ \underline{y} <= \underline{x} ≫ ˙¬ \underline{x} <= c_{Ex} ⇒ ˙¬ \underline{y} <= c_{Ex}; ∀\underline{x}: ∀\underline{y}: Ded ⊳ ∀\underline{x}: ∀\underline{y}: \underline{x} <= \underline{y} ⊢ ˙¬ \underline{x} <= c_{Ex} ⇒ ˙¬ \underline{y} <= c_{Ex} ≫ \underline{x} <= \underline{y} ⇒ ˙¬ \underline{x} <= c_{Ex} ⇒ ˙¬ \underline{y} <= c_{Ex}; Ded ⊳ ∀\underline{x}: ∀\underline{y}: \underline{y} <= \underline{x} ⊢ ˙¬ \underline{x} <= c_{Ex} ⇒ ˙¬ \underline{y} <= c_{Ex} ≫ \underline{y} <= \underline{x} ⇒ ˙¬ \underline{x} <= c_{Ex} ⇒ ˙¬ \underline{y} <= c_{Ex}; leqTotality ≫ ˙¬ \underline{x} <= \underline{y} ⇒ \underline{y} <= \underline{x}; FromDisjuncts ⊳ ˙¬ \underline{x} <= \underline{y} ⇒ \underline{y} <= \underline{x} ⊳ \underline{x} <= \underline{y} ⇒ ˙¬ \underline{x} <= c_{Ex} ⇒ ˙¬ \underline{y} <= c_{Ex} ⊳ \underline{y} <= \underline{x} ⇒ ˙¬ \underline{x} <= c_{Ex} ⇒ ˙¬ \underline{y} <= c_{Ex} ≫ ˙¬ \underline{x} <= c_{Ex} ⇒ ˙¬ \underline{y} <= c_{Ex}⌉, p_{0}, c)]

[LeqNegated →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} <= \underline{y} ⊢ −\underline{y} <= −\underline{x}]

[LeqNegated →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} <= \underline{y} ⊢ leqAddition ⊳ \underline{x} <= \underline{y} ≫ \underline{x} + −\underline{x} <= \underline{y} + −\underline{x}; Negative ≫ \underline{x} + −\underline{x} = 0; subLeqLeft ⊳ \underline{x} + −\underline{x} = 0 ⊳ \underline{x} + −\underline{x} <= \underline{y} + −\underline{x} ≫ 0 <= \underline{y} + −\underline{x}; plusCommutativity ≫ \underline{y} + −\underline{x} = −\underline{x} + \underline{y}; subLeqRight ⊳ \underline{y} + −\underline{x} = −\underline{x} + \underline{y} ⊳ 0 <= \underline{y} + −\underline{x} ≫ 0 <= −\underline{x} + \underline{y}; leqAddition ⊳ 0 <= −\underline{x} + \underline{y} ≫ 0 + −\underline{y} <= −\underline{x} + \underline{y} + −\underline{y}; plus0Left ≫ 0 + −\underline{y} = −\underline{y}; x = x + y − y ≫ −\underline{x} = −\underline{x} + \underline{y} + −\underline{y}; eqSymmetry ⊳ −\underline{x} = −\underline{x} + \underline{y} + −\underline{y} ≫ −\underline{x} + \underline{y} + −\underline{y} = −\underline{x}; subLeqLeft ⊳ 0 + −\underline{y} = −\underline{y} ⊳ 0 + −\underline{y} <= −\underline{x} + \underline{y} + −\underline{y} ≫ −\underline{y} <= −\underline{x} + \underline{y} + −\underline{y}; subLeqRight ⊳ −\underline{x} + \underline{y} + −\underline{y} = −\underline{x} ⊳ −\underline{y} <= −\underline{x} + \underline{y} + −\underline{y} ≫ −\underline{y} <= −\underline{x}⌉, p_{0}, c)]

[AddEquations(Leq) →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: \underline{x} <= \underline{y} ⊢ \underline{z} <= \underline{u} ⊢ \underline{x} + \underline{z} <= \underline{y} + \underline{u}]

[AddEquations(Leq) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: \underline{x} <= \underline{y} ⊢ \underline{z} <= \underline{u} ⊢ leqAddition ⊳ \underline{x} <= \underline{y} ≫ \underline{x} + \underline{z} <= \underline{y} + \underline{z}; LeqAdditionLeft ⊳ \underline{z} <= \underline{u} ≫ \underline{y} + \underline{z} <= \underline{y} + \underline{u}; leqTransitivity ⊳ \underline{x} + \underline{z} <= \underline{y} + \underline{z} ⊳ \underline{y} + \underline{z} <= \underline{y} + \underline{u} ≫ \underline{x} + \underline{z} <= \underline{y} + \underline{u}⌉, p_{0}, c)] (*** LESS ***) [LeqNeqLess →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} <= \underline{y} ⊢ ˙¬ \underline{x} = \underline{y} ⊢ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y}]

[LeqNeqLess →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} <= \underline{y} ⊢ ˙¬ \underline{x} = \underline{y} ⊢ JoinConjuncts ⊳ \underline{x} <= \underline{y} ⊳ ˙¬ \underline{x} = \underline{y} ≫ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y}; Repetition ⊳ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ≫ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y}⌉, p_{0}, c)]

[FromLess →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊢ ˙¬ \underline{y} <= \underline{x}]

[FromLess →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \underline{y} <= \underline{x} ⊢ toNotLess ⊳ \underline{y} <= \underline{x} ≫ ˙¬ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y}; ∀\underline{x}: ∀\underline{y}: Ded ⊳ ∀\underline{x}: ∀\underline{y}: \underline{y} <= \underline{x} ⊢ ˙¬ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ≫ \underline{y} <= \underline{x} ⇒ ˙¬ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y}; ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊢ AddDoubleNeg ⊳ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ≫ ˙¬ ˙¬ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y}; MT ⊳ \underline{y} <= \underline{x} ⇒ ˙¬ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊳ ˙¬ ˙¬ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ≫ ˙¬ \underline{y} <= \underline{x}⌉, p_{0}, c)]

[ToLess →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ \underline{x} <= \underline{y} ⊢ ˙¬ \underline{y} <= \underline{x} ⇒ ˙¬ ˙¬ \underline{y} = \underline{x}]

[ToLess →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ ˙¬ \underline{y} <= \underline{x} ⇒ ˙¬ ˙¬ \underline{y} = \underline{x} ⊢ fromNotLess ⊳ ˙¬ ˙¬ \underline{y} <= \underline{x} ⇒ ˙¬ ˙¬ \underline{y} = \underline{x} ≫ \underline{x} <= \underline{y}; ∀\underline{x}: ∀\underline{y}: Ded ⊳ ∀\underline{x}: ∀\underline{y}: ˙¬ ˙¬ \underline{y} <= \underline{x} ⇒ ˙¬ ˙¬ \underline{y} = \underline{x} ⊢ \underline{x} <= \underline{y} ≫ ˙¬ ˙¬ \underline{y} <= \underline{x} ⇒ ˙¬ ˙¬ \underline{y} = \underline{x} ⇒ \underline{x} <= \underline{y}; ˙¬ \underline{x} <= \underline{y} ⊢ NegativeMT ⊳ ˙¬ ˙¬ \underline{y} <= \underline{x} ⇒ ˙¬ ˙¬ \underline{y} = \underline{x} ⇒ \underline{x} <= \underline{y} ⊳ ˙¬ \underline{x} <= \underline{y} ≫ ˙¬ \underline{y} <= \underline{x} ⇒ ˙¬ ˙¬ \underline{y} = \underline{x}⌉, p_{0}, c)]

[fromNotLess →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊢ \underline{y} <= \underline{x}]

[fromNotLess →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊢ \underline{x} <= \underline{y} ⊢ Repetition ⊳ ˙¬ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ≫ ˙¬ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y}; RemoveDoubleNeg ⊳ ˙¬ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ≫ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y}; MP ⊳ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊳ \underline{x} <= \underline{y} ≫ ˙¬ ˙¬ \underline{x} = \underline{y}; RemoveDoubleNeg ⊳ ˙¬ ˙¬ \underline{x} = \underline{y} ≫ \underline{x} = \underline{y}; eqSymmetry ⊳ \underline{x} = \underline{y} ≫ \underline{y} = \underline{x}; eqLeq ⊳ \underline{y} = \underline{x} ≫ \underline{y} <= \underline{x}; ∀\underline{x}: ∀\underline{y}: Ded ⊳ ∀\underline{x}: ∀\underline{y}: ˙¬ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊢ \underline{x} <= \underline{y} ⊢ \underline{y} <= \underline{x} ≫ ˙¬ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⇒ \underline{x} <= \underline{y} ⇒ \underline{y} <= \underline{x}; ˙¬ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊢ MP ⊳ ˙¬ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⇒ \underline{x} <= \underline{y} ⇒ \underline{y} <= \underline{x} ⊳ ˙¬ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ≫ \underline{x} <= \underline{y} ⇒ \underline{y} <= \underline{x}; AutoImply ≫ \underline{y} <= \underline{x} ⇒ \underline{y} <= \underline{x}; leqTotality ≫ ˙¬ \underline{x} <= \underline{y} ⇒ \underline{y} <= \underline{x}; FromDisjuncts ⊳ ˙¬ \underline{x} <= \underline{y} ⇒ \underline{y} <= \underline{x} ⊳ \underline{x} <= \underline{y} ⇒ \underline{y} <= \underline{x} ⊳ \underline{y} <= \underline{x} ⇒ \underline{y} <= \underline{x} ≫ \underline{y} <= \underline{x}⌉, p_{0}, c)]

[toNotLess →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} <= \underline{y} ⊢ ˙¬ ˙¬ \underline{y} <= \underline{x} ⇒ ˙¬ ˙¬ \underline{y} = \underline{x}]

[toNotLess →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} <= \underline{y} ⊢ \underline{y} <= \underline{x} ⊢ leqAntisymmetry ⊳ \underline{y} <= \underline{x} ⊳ \underline{x} <= \underline{y} ≫ \underline{y} = \underline{x}; AddDoubleNeg ⊳ \underline{y} = \underline{x} ≫ ˙¬ ˙¬ \underline{y} = \underline{x}; ∀\underline{x}: ∀\underline{y}: Ded ⊳ ∀\underline{x}: ∀\underline{y}: \underline{x} <= \underline{y} ⊢ \underline{y} <= \underline{x} ⊢ ˙¬ ˙¬ \underline{y} = \underline{x} ≫ \underline{x} <= \underline{y} ⇒ \underline{y} <= \underline{x} ⇒ ˙¬ ˙¬ \underline{y} = \underline{x}; \underline{x} <= \underline{y} ⊢ MP ⊳ \underline{x} <= \underline{y} ⇒ \underline{y} <= \underline{x} ⇒ ˙¬ ˙¬ \underline{y} = \underline{x} ⊳ \underline{x} <= \underline{y} ≫ \underline{y} <= \underline{x} ⇒ ˙¬ ˙¬ \underline{y} = \underline{x}; AddDoubleNeg ⊳ \underline{y} <= \underline{x} ⇒ ˙¬ ˙¬ \underline{y} = \underline{x} ≫ ˙¬ ˙¬ \underline{y} <= \underline{x} ⇒ ˙¬ ˙¬ \underline{y} = \underline{x}; Repetition ⊳ ˙¬ ˙¬ \underline{y} <= \underline{x} ⇒ ˙¬ ˙¬ \underline{y} = \underline{x} ≫ ˙¬ ˙¬ \underline{y} <= \underline{x} ⇒ ˙¬ ˙¬ \underline{y} = \underline{x}; Repetition ⊳ ˙¬ ˙¬ \underline{y} <= \underline{x} ⇒ ˙¬ ˙¬ \underline{y} = \underline{x} ≫ ˙¬ ˙¬ \underline{y} <= \underline{x} ⇒ ˙¬ ˙¬ \underline{y} = \underline{x}⌉, p_{0}, c)]

[LessAddition →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊢ ˙¬ \underline{x} + \underline{z} <= \underline{y} + \underline{z} ⇒ ˙¬ ˙¬ \underline{x} + \underline{z} = \underline{y} + \underline{z}]

[LessAddition →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊢ LessLeq ⊳ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ≫ \underline{x} <= \underline{y}; leqAddition ⊳ \underline{x} <= \underline{y} ≫ \underline{x} + \underline{z} <= \underline{y} + \underline{z}; LessNeq ⊳ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ≫ ˙¬ \underline{x} = \underline{y}; NeqAddition ⊳ ˙¬ \underline{x} = \underline{y} ≫ ˙¬ \underline{x} + \underline{z} = \underline{y} + \underline{z}; JoinConjuncts ⊳ \underline{x} + \underline{z} <= \underline{y} + \underline{z} ⊳ ˙¬ \underline{x} + \underline{z} = \underline{y} + \underline{z} ≫ ˙¬ \underline{x} + \underline{z} <= \underline{y} + \underline{z} ⇒ ˙¬ ˙¬ \underline{x} + \underline{z} = \underline{y} + \underline{z}⌉, p_{0}, c)]

[LessAdditionLeft →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊢ ˙¬ \underline{z} + \underline{x} <= \underline{z} + \underline{y} ⇒ ˙¬ ˙¬ \underline{z} + \underline{x} = \underline{z} + \underline{y}]

[LessAdditionLeft →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊢ LessAddition ⊳ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ≫ ˙¬ \underline{x} + \underline{z} <= \underline{y} + \underline{z} ⇒ ˙¬ ˙¬ \underline{x} + \underline{z} = \underline{y} + \underline{z}; plusCommutativity ≫ \underline{x} + \underline{z} = \underline{z} + \underline{x}; SubLessLeft ⊳ \underline{x} + \underline{z} = \underline{z} + \underline{x} ⊳ ˙¬ \underline{x} + \underline{z} <= \underline{y} + \underline{z} ⇒ ˙¬ ˙¬ \underline{x} + \underline{z} = \underline{y} + \underline{z} ≫ ˙¬ \underline{z} + \underline{x} <= \underline{y} + \underline{z} ⇒ ˙¬ ˙¬ \underline{z} + \underline{x} = \underline{y}+\underline{z}; plusCommutativity ≫ \underline{y}+\underline{z} = \underline{z}+\underline{y}; SubLessRight⊳\underline{y}+\underline{z} = \underline{z}+\underline{y}⊳ ˙¬ \underline{z}+\underline{x} <= \underline{y} + \underline{z} ⇒ ˙¬ ˙¬ \underline{z} + \underline{x} = \underline{y} + \underline{z} ≫ ˙¬ \underline{z} + \underline{x} <= \underline{z} + \underline{y} ⇒ ˙¬ ˙¬ \underline{z} + \underline{x} = \underline{z} + \underline{y}⌉, p_{0}, c)]

[LessMultiplication →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ 0 <= \underline{z} ⇒ ˙¬ ˙¬ 0 = \underline{z} ⊢ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊢ ˙¬ \underline{x} ∗ \underline{z} <= \underline{y} ∗ \underline{z} ⇒ ˙¬ ˙¬ \underline{x} ∗ \underline{z} = \underline{y} ∗ \underline{z}]

[LessMultiplication →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ 0 <= \underline{z} ⇒ ˙¬ ˙¬ 0 = \underline{z} ⊢ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊢ LessLeq ⊳ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ≫ \underline{x} <= \underline{y}; LessLeq ⊳ ˙¬ 0 <= \underline{z} ⇒ ˙¬ ˙¬ 0 = \underline{z} ≫ 0 <= \underline{z}; leqMultiplication ⊳ 0 <= \underline{z} ⊳ \underline{x} <= \underline{y} ≫ \underline{x} ∗ \underline{z} <= \underline{y} ∗ \underline{z}; LessNeq ⊳ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ≫ ˙¬ \underline{x} = \underline{y}; LessNeq ⊳ ˙¬ 0 <= \underline{z} ⇒ ˙¬ ˙¬ 0 = \underline{z} ≫ ˙¬ 0 = \underline{z}; NeqSymmetry ⊳ ˙¬ 0 = \underline{z} ≫ ˙¬ \underline{z} = 0; NeqMultiplication ⊳ ˙¬ \underline{z} = 0 ⊳ ˙¬ \underline{x} = \underline{y} ≫ ˙¬ \underline{x} ∗ \underline{z} = \underline{y} ∗ \underline{z}; LeqNeqLess ⊳ \underline{x} ∗ \underline{z} <= \underline{y} ∗ \underline{z} ⊳ ˙¬ \underline{x} ∗ \underline{z} = \underline{y} ∗ \underline{z} ≫ ˙¬ \underline{x} ∗ \underline{z} <= \underline{y} ∗ \underline{z} ⇒ ˙¬ ˙¬ \underline{x} ∗ \underline{z} = \underline{y} ∗ \underline{z}⌉, p_{0}, c)]

[LessMultiplicationLeft →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ 0 <= \underline{z} ⇒ ˙¬ ˙¬ 0 = \underline{z} ⊢ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊢ ˙¬ \underline{z} ∗ \underline{x} <= \underline{z} ∗ \underline{y} ⇒ ˙¬ ˙¬ \underline{z} ∗ \underline{x} = \underline{z} ∗ \underline{y}]

[LessMultiplicationLeft →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ 0 <= \underline{z} ⇒ ˙¬ ˙¬ 0 = \underline{z} ⊢ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊢ LessMultiplication⊳ ˙¬ 0 <= \underline{z} ⇒ ˙¬ ˙¬ 0 = \underline{z}⊳ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ≫ ˙¬ \underline{x}∗\underline{z} <= \underline{y}∗\underline{z} ⇒ ˙¬ ˙¬ \underline{x}∗\underline{z} = \underline{y}∗\underline{z}; timesCommutativity ≫ \underline{x}∗\underline{z} = \underline{z} ∗ \underline{x}; timesCommutativity ≫ \underline{y} ∗ \underline{z} = \underline{z} ∗ \underline{y}; SubLessLeft ⊳ \underline{x} ∗ \underline{z} = \underline{z} ∗ \underline{x} ⊳ ˙¬ \underline{x} ∗ \underline{z} <= \underline{y}∗\underline{z} ⇒ ˙¬ ˙¬ \underline{x}∗\underline{z} = \underline{y}∗\underline{z} ≫ ˙¬ \underline{z}∗\underline{x} <= \underline{y}∗\underline{z} ⇒ ˙¬ ˙¬ \underline{z}∗\underline{x} = \underline{y}∗\underline{z}; SubLessRight⊳\underline{y}∗\underline{z} = \underline{z}∗\underline{y}⊳ ˙¬ \underline{z}∗\underline{x} <= \underline{y}∗\underline{z} ⇒ ˙¬ ˙¬ \underline{z}∗\underline{x} = \underline{y}∗\underline{z} ≫ ˙¬ \underline{z}∗\underline{x} <= \underline{z}∗\underline{y} ⇒ ˙¬ ˙¬ \underline{z}∗\underline{x} = \underline{z}∗\underline{y}⌉, p_{0}, c)]

[LessDivision →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: 0 <= \underline{z} ⊢ ˙¬ \underline{x} ∗ \underline{z} <= \underline{y} ∗ \underline{z} ⇒ ˙¬ ˙¬ \underline{x} ∗ \underline{z} = \underline{y} ∗ \underline{z} ⊢ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y}]

[LessDivision →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: 0 <= \underline{z} ⊢ ˙¬ \underline{x} ∗ \underline{z} <= \underline{y} ∗ \underline{z} ⇒ ˙¬ ˙¬ \underline{x} ∗ \underline{z} = \underline{y} ∗ \underline{z} ⊢ FromLess ⊳ ˙¬ \underline{x} ∗ \underline{z} <= \underline{y} ∗ \underline{z} ⇒ ˙¬ ˙¬ \underline{x} ∗ \underline{z} = \underline{y} ∗ \underline{z} ≫ ˙¬ \underline{y} ∗ \underline{z} <= \underline{x} ∗ \underline{z}; leqMultiplicationAxiom ≫ 0 <= \underline{z} ⇒ \underline{y} <= \underline{x} ⇒ \underline{y} ∗ \underline{z} <= \underline{x} ∗ \underline{z}; MP ⊳ 0 <= \underline{z} ⇒ \underline{y} <= \underline{x} ⇒ \underline{y} ∗ \underline{z} <= \underline{x} ∗ \underline{z} ⊳ 0 <= \underline{z} ≫ \underline{y} <= \underline{x} ⇒ \underline{y} ∗ \underline{z} <= \underline{x} ∗ \underline{z}; Contrapositive ⊳ \underline{y} <= \underline{x} ⇒ \underline{y} ∗ \underline{z} <= \underline{x} ∗ \underline{z} ≫ ˙¬ \underline{y} ∗ \underline{z} <= \underline{x} ∗ \underline{z} ⇒ ˙¬ \underline{y} <= \underline{x}; MP ⊳ ˙¬ \underline{y} ∗ \underline{z} <= \underline{x} ∗ \underline{z} ⇒ ˙¬ \underline{y} <= \underline{x} ⊳ ˙¬ \underline{y} ∗ \underline{z} <= \underline{x} ∗ \underline{z} ≫ ˙¬ \underline{y} <= \underline{x}; ToLess ⊳ ˙¬ \underline{y} <= \underline{x} ≫ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y}⌉, p_{0}, c)]

[AddEquations(Less) →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊢ ˙¬ \underline{z} <= \underline{u} ⇒ ˙¬ ˙¬ \underline{z} = \underline{u} ⊢ ˙¬ \underline{x} + \underline{z} <= \underline{y} + \underline{u} ⇒ ˙¬ ˙¬ \underline{x} + \underline{z} = \underline{y} + \underline{u}]

[AddEquations(Less) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊢ ˙¬ \underline{z} <= \underline{u} ⇒ ˙¬ ˙¬ \underline{z} = \underline{u} ⊢ LessAddition ⊳ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ≫ ˙¬ \underline{x} + \underline{z} <= \underline{y}+\underline{z} ⇒ ˙¬ ˙¬ \underline{x}+\underline{z} = \underline{y}+\underline{z}; LessAdditionLeft⊳ ˙¬ \underline{z} <= \underline{u} ⇒ ˙¬ ˙¬ \underline{z} = \underline{u} ≫ ˙¬ \underline{y}+\underline{z} <= \underline{y}+\underline{u} ⇒ ˙¬ ˙¬ \underline{y}+\underline{z} = \underline{y}+\underline{u}; LessTransitivity⊳ ˙¬ \underline{x}+\underline{z} <= \underline{y}+\underline{z} ⇒ ˙¬ ˙¬ \underline{x}+\underline{z} = \underline{y}+\underline{z}⊳ ˙¬ \underline{y}+\underline{z} <= \underline{y}+\underline{u} ⇒ ˙¬ ˙¬ \underline{y}+\underline{z} = \underline{y}+\underline{u} ≫ ˙¬ \underline{x}+\underline{z} <= \underline{y}+\underline{u} ⇒ ˙¬ ˙¬ \underline{x}+\underline{z} = \underline{y}+\underline{u}⌉, p_{0}, c)]

[leqLessTransitivity →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} <= \underline{y} ⊢ ˙¬ \underline{y} <= \underline{z} ⇒ ˙¬ ˙¬ \underline{y} = \underline{z} ⊢ ˙¬ \underline{x} <= \underline{z} ⇒ ˙¬ ˙¬ \underline{x} = \underline{z}] [leqLessTransitivity →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} <= \underline{y} ⊢ ˙¬ \underline{y} <= \underline{z} ⇒ ˙¬ ˙¬ \underline{y} = \underline{z} ⊢ \underline{x} = \underline{z} ⊢ FirstConjunct ⊳ ˙¬ \underline{y} <= \underline{z} ⇒ ˙¬ ˙¬ \underline{y} = \underline{z} ≫ \underline{y} <= \underline{z}; SecondConjunct⊳ ˙¬ \underline{y} <= \underline{z} ⇒ ˙¬ ˙¬ \underline{y} = \underline{z} ≫ ˙¬ \underline{y} = \underline{z}; subLeqLeft⊳\underline{x} = \underline{z}⊳\underline{x} <= \underline{y} ≫ \underline{z} <= \underline{y}; leqAntisymmetry⊳\underline{y} <= \underline{z}⊳\underline{z} <= \underline{y} ≫ \underline{y} = \underline{z}; FromContradiction⊳ \underline{y} = \underline{z} ⊳ ˙¬ \underline{y} = \underline{z} ≫ ˙¬ \underline{x} = \underline{z}; ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: Ded ⊳ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} <= \underline{y} ⊢ ˙¬ \underline{y} <= \underline{z} ⇒ ˙¬ ˙¬ \underline{y} = \underline{z} ⊢ \underline{x} = \underline{z} ⊢ ˙¬ \underline{x} = \underline{z} ≫ \underline{x} <= \underline{y} ⇒ ˙¬ \underline{y} <= \underline{z} ⇒ ˙¬ ˙¬ \underline{y} = \underline{z} ⇒ \underline{x} = \underline{z} ⇒ ˙¬ \underline{x} = \underline{z}; \underline{x} <= \underline{y} ⊢ ˙¬ \underline{y} <= \underline{z} ⇒ ˙¬ ˙¬ \underline{y} = \underline{z} ⊢ MP2 ⊳ \underline{x} <= \underline{y} ⇒ ˙¬ \underline{y} <= \underline{z} ⇒ ˙¬ ˙¬ \underline{y} = \underline{z} ⇒ \underline{x} = \underline{z} ⇒ ˙¬ \underline{x} = \underline{z} ⊳ \underline{x} <= \underline{y} ⊳ ˙¬ \underline{y} <= \underline{z} ⇒ ˙¬ ˙¬ \underline{y} = \underline{z} ≫ \underline{x} = \underline{z} ⇒ ˙¬ \underline{x} = \underline{z}; ImplyNegation ⊳ \underline{x} = \underline{z} ⇒ ˙¬ \underline{x} = \underline{z} ≫ ˙¬ \underline{x} = \underline{z}; FirstConjunct ⊳ ˙¬ \underline{y} <= \underline{z} ⇒ ˙¬ ˙¬ \underline{y} = \underline{z} ≫ \underline{y} <= \underline{z}; leqTransitivity ⊳ \underline{x} <= \underline{y} ⊳ \underline{y} <= \underline{z} ≫ \underline{x} <= \underline{z}; JoinConjuncts ⊳ \underline{x} <= \underline{z} ⊳ ˙¬ \underline{x} = \underline{z} ≫ ˙¬ \underline{x} <= \underline{z} ⇒ ˙¬ ˙¬ \underline{x} = \underline{z}⌉, p_{0}, c)] [LessLeqTransitivity →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊢ \underline{y} <= \underline{z} ⊢ ˙¬ \underline{x} <= \underline{z} ⇒ ˙¬ ˙¬ \underline{x} = \underline{z}] [LessLeqTransitivity →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊢ \underline{y} <= \underline{z} ⊢ \underline{z} = \underline{x} ⊢ FirstConjunct ⊳ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ≫ \underline{x} <= \underline{y}; SecondConjunct ⊳ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ≫ ˙¬ \underline{x} = \underline{y}; subLeqRight ⊳ \underline{z} = \underline{x} ⊳ \underline{y} <= \underline{z} ≫ \underline{y} <= \underline{x}; leqAntisymmetry ⊳ \underline{x} <= \underline{y} ⊳ \underline{y} <= \underline{x} ≫ \underline{x} = \underline{y}; FromContradiction ⊳ \underline{x} = \underline{y} ⊳ ˙¬ \underline{x} = \underline{y} ≫ ˙¬ \underline{z} = \underline{x}; ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: Ded ⊳ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊢ \underline{y} <= \underline{z} ⊢ \underline{z} = \underline{x} ⊢ ˙¬ \underline{z} = \underline{x} ≫ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⇒ \underline{y} <= \underline{z} ⇒ \underline{z} = \underline{x} ⇒ ˙¬ \underline{z} = \underline{x}; ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊢ \underline{y} <= \underline{z} ⊢ MP2 ⊳ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⇒ \underline{y} <= \underline{z} ⇒ \underline{z} = \underline{x} ⇒ ˙¬ \underline{z} = \underline{x}⊳ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y}⊳\underline{y} <= \underline{z} ≫ \underline{z} = \underline{x} ⇒ ˙¬ \underline{z} = \underline{x}; ImplyNegation⊳\underline{z} = \underline{x} ⇒ ˙¬ \underline{z} = \underline{x} ≫ ˙¬ \underline{z} = \underline{x}; NeqSymmetry ⊳ ˙¬ \underline{z} = \underline{x} ≫ ˙¬ \underline{x} = \underline{z}; FirstConjunct ⊳ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ≫ \underline{x} <= \underline{y}; leqTransitivity ⊳ \underline{x} <= \underline{y} ⊳ \underline{y} <= \underline{z} ≫ \underline{x} <= \underline{z}; JoinConjuncts ⊳ \underline{x} <= \underline{z} ⊳ ˙¬ \underline{x} = \underline{z} ≫ ˙¬ \underline{x} <= \underline{z} ⇒ ˙¬ ˙¬ \underline{x} = \underline{z}⌉, p_{0}, c)] [LessTransitivity →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊢ ˙¬ \underline{y} <= \underline{z} ⇒ ˙¬ ˙¬ \underline{y} = \underline{z} ⊢ ˙¬ \underline{x} <= \underline{z} ⇒ ˙¬ ˙¬ \underline{x} = \underline{z}]

[LessTransitivity →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊢ ˙¬ \underline{y} <= \underline{z} ⇒ ˙¬ ˙¬ \underline{y} = \underline{z} ⊢ FirstConjunct ⊳ ˙¬ \underline{y} <= \underline{z} ⇒ ˙¬ ˙¬ \underline{y} = \underline{z} ≫ \underline{y} <= \underline{z}; LessLeqTransitivity ⊳ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊳ \underline{y} <= \underline{z} ≫ ˙¬ \underline{x} <= \underline{z} ⇒ ˙¬ ˙¬ \underline{x} = \underline{z}⌉, p_{0}, c)] [LessTotality →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⇒ ˙¬ \underline{x} = \underline{y} ⇒ ˙¬ \underline{y} <= \underline{x} ⇒ ˙¬ ˙¬ \underline{y} = \underline{x}]

[LessTotality →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊢ ˙¬ \underline{x} = \underline{y} ⊢ fromNotLess ⊳ ˙¬ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ≫ \underline{y} <= \underline{x}; NeqSymmetry ⊳ ˙¬ \underline{x} = \underline{y} ≫ ˙¬ \underline{y} = \underline{x}; LeqNeqLess ⊳ \underline{y} <= \underline{x} ⊳ ˙¬ \underline{y} = \underline{x} ≫ ˙¬ \underline{y} <= \underline{x} ⇒ ˙¬ ˙¬ \underline{y} = \underline{x}; ∀\underline{x}: ∀\underline{y}: Ded ⊳ ∀\underline{x}: ∀\underline{y}: ˙¬ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊢ ˙¬ \underline{x} = \underline{y} ⊢ ˙¬ \underline{y} <= \underline{x} ⇒ ˙¬ ˙¬ \underline{y} = \underline{x} ≫ ˙¬ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⇒ ˙¬ \underline{x} = \underline{y} ⇒ ˙¬ \underline{y} <= \underline{x} ⇒ ˙¬ ˙¬ \underline{y} = \underline{x}; Repetition ⊳ ˙¬ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⇒ ˙¬ \underline{x} = \underline{y} ⇒ ˙¬ \underline{y} <= \underline{x} ⇒ ˙¬ ˙¬ \underline{y} = \underline{x} ≫ ˙¬ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⇒ ˙¬ \underline{x} = \underline{y} ⇒ ˙¬ \underline{y} <= \underline{x} ⇒ ˙¬ ˙¬ \underline{y} = \underline{x}⌉, p_{0}, c)]

[SubLessRight →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ ˙¬ \underline{z} <= \underline{x} ⇒ ˙¬ ˙¬ \underline{z} = \underline{x} ⊢ ˙¬ \underline{z} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{z} = \underline{y}]

[SubLessRight →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ ˙¬ \underline{z} <= \underline{x} ⇒ ˙¬ ˙¬ \underline{z} = \underline{x} ⊢ Repetition ⊳ ˙¬ \underline{z} <= \underline{x} ⇒ ˙¬ ˙¬ \underline{z} = \underline{x} ≫ ˙¬ \underline{z} <= \underline{x} ⇒ ˙¬ ˙¬ \underline{z} = \underline{x}; FirstConjunct ⊳ ˙¬ \underline{z} <= \underline{x} ⇒ ˙¬ ˙¬ \underline{z} = \underline{x} ≫ \underline{z} <= \underline{x}; subLeqRight ⊳ \underline{x} = \underline{y} ⊳ \underline{z} <= \underline{x} ≫ \underline{z} <= \underline{y}; SecondConjunct ⊳ ˙¬ \underline{z} <= \underline{x} ⇒ ˙¬ ˙¬ \underline{z} = \underline{x} ≫ ˙¬ \underline{z} = \underline{x}; SubNeqRight ⊳ \underline{x} = \underline{y} ⊳ ˙¬ \underline{z} = \underline{x} ≫ ˙¬ \underline{z} = \underline{y}; JoinConjuncts ⊳ \underline{z} <= \underline{y} ⊳ ˙¬ \underline{z} = \underline{y} ≫ ˙¬ \underline{z} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{z} = \underline{y}⌉, p_{0}, c)] [SubLessLeft →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ ˙¬ \underline{x} <= \underline{z} ⇒ ˙¬ ˙¬ \underline{x} = \underline{z} ⊢ ˙¬ \underline{y} <= \underline{z} ⇒ ˙¬ ˙¬ \underline{y} = \underline{z}]

[SubLessLeft →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ ˙¬ \underline{x} <= \underline{z} ⇒ ˙¬ ˙¬ \underline{x} = \underline{z} ⊢ Repetition ⊳ ˙¬ \underline{x} <= \underline{z} ⇒ ˙¬ ˙¬ \underline{x} = \underline{z} ≫ ˙¬ \underline{x} <= \underline{z} ⇒ ˙¬ ˙¬ \underline{x} = \underline{z}; FirstConjunct ⊳ ˙¬ \underline{x} <= \underline{z} ⇒ ˙¬ ˙¬ \underline{x} = \underline{z} ≫ \underline{x} <= \underline{z}; subLeqLeft ⊳ \underline{x} = \underline{y} ⊳ \underline{x} <= \underline{z} ≫ \underline{y} <= \underline{z}; SecondConjunct⊳ ˙¬ \underline{x} <= \underline{z} ⇒ ˙¬ ˙¬ \underline{x} = \underline{z} ≫ ˙¬ \underline{x} = \underline{z}; SubNeqLeft⊳\underline{x} = \underline{y}⊳ ˙¬ \underline{x} = \underline{z} ≫ ˙¬ \underline{y} = \underline{z}; JoinConjuncts ⊳ \underline{y} <= \underline{z} ⊳ ˙¬ \underline{y} = \underline{z} ≫ ˙¬ \underline{y} <= \underline{z} ⇒ ˙¬ ˙¬ \underline{y} = \underline{z}⌉, p_{0}, c)]

[NegativeLessPositive →^{stmt} ZFsub ⊢ ∀\underline{x}: ˙¬ 0 <= \underline{x} ⇒ ˙¬ ˙¬ 0 = \underline{x} ⊢ ˙¬ − \underline{x} <= \underline{x} ⇒ ˙¬ ˙¬ − \underline{x} = \underline{x}] [NegativeLessPositive →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ˙¬ 0 <= \underline{x} ⇒ ˙¬ ˙¬ 0 = \underline{x} ⊢ FirstConjunct ⊳ ˙¬ 0 <= \underline{x} ⇒ ˙¬ ˙¬ 0 = \underline{x} ≫ 0 <= \underline{x}; leqAddition ⊳ 0 <= \underline{x} ≫ 0 + −\underline{x} <= \underline{x} + −\underline{x}; plus0Left ≫ 0 + −\underline{x} = −\underline{x}; Negative ≫ \underline{x} + −\underline{x} = 0; subLeqLeft⊳0+−\underline{x} = −\underline{x}⊳0+−\underline{x} <= \underline{x}+−\underline{x} ≫ −\underline{x} <= \underline{x}+−\underline{x}; subLeqRight⊳ \underline{x} + −\underline{x} = 0 ⊳ −\underline{x} <= \underline{x} + −\underline{x} ≫ −\underline{x} <= 0; leqLessTransitivity ⊳ −\underline{x} <= 0 ⊳ ˙¬ 0 <= \underline{x} ⇒ ˙¬ ˙¬ 0 = \underline{x} ≫ ˙¬ − \underline{x} <= \underline{x} ⇒ ˙¬ ˙¬ − \underline{x} = \underline{x}⌉, p_{0}, c)] [LessNegated →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊢ ˙¬ − \underline{y} <= −\underline{x} ⇒ ˙¬ ˙¬ − \underline{y} = −\underline{x}]

[LessNegated →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊢ LessLeq ⊳ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ≫ \underline{x} <= \underline{y}; LeqNegated ⊳ \underline{x} <= \underline{y} ≫ −\underline{y} <= −\underline{x}; LessNeq ⊳ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ≫ ˙¬ \underline{x} = \underline{y}; NeqNegated ⊳ ˙¬ \underline{x} = \underline{y} ≫ ˙¬ − \underline{x} = −\underline{y}; NeqSymmetry ⊳ ˙¬ − \underline{x} = −\underline{y} ≫ ˙¬ − \underline{y} = −\underline{x}; LeqNeqLess ⊳ −\underline{y} <= −\underline{x} ⊳ ˙¬ − \underline{y} = −\underline{x} ≫ ˙¬ − \underline{y} <= −\underline{x} ⇒ ˙¬ ˙¬ − \underline{y} = −\underline{x}⌉, p_{0}, c)]

[PositiveNegated →^{stmt} ZFsub ⊢ ∀\underline{x}: ˙¬ 0 <= \underline{x} ⇒ ˙¬ ˙¬ 0 = \underline{x} ⊢ ˙¬ − \underline{x} <= 0 ⇒ ˙¬ ˙¬ − \underline{x} = 0] [PositiveNegated →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ˙¬ 0 <= \underline{x} ⇒ ˙¬ ˙¬ 0 = \underline{x} ⊢ LessNegated ⊳ ˙¬ 0 <= \underline{x} ⇒ ˙¬ ˙¬ 0 = \underline{x} ≫ ˙¬ − \underline{x} <= −0 ⇒ ˙¬ ˙¬ − \underline{x} = −0; −0 = 0 ≫ −0 = 0; SubLessRight ⊳ −0 = 0 ⊳ ˙¬ − \underline{x} <= −0 ⇒ ˙¬ ˙¬ − \underline{x} = −0 ≫ ˙¬ − \underline{x} <= 0 ⇒ ˙¬ ˙¬ − \underline{x} = 0⌉, p_{0}, c)] [NonpositiveNegated →^{stmt} ZFsub ⊢ ∀\underline{x}: \underline{x} <= 0 ⊢ 0 <= −\underline{x}] [NonpositiveNegated →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: \underline{x} <= 0 ⊢ LeqNegated ⊳ \underline{x} <= 0 ≫ −0 <= −\underline{x}; −0 = 0 ≫ −0 = 0; subLeqLeft ⊳ −0 = 0 ⊳ −0 <= −\underline{x} ≫ 0 <= −\underline{x}⌉, p_{0}, c)] [NegativeNegated →^{stmt} ZFsub ⊢ ∀\underline{x}: ˙¬ \underline{x} <= 0 ⇒ ˙¬ ˙¬ \underline{x} = 0 ⊢ ˙¬ 0 <= −\underline{x} ⇒ ˙¬ ˙¬ 0 = −\underline{x}] [NegativeNegated →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ˙¬ \underline{x} <= 0 ⇒ ˙¬ ˙¬ \underline{x} = 0 ⊢ LessNegated ⊳ ˙¬ \underline{x} <= 0 ⇒ ˙¬ ˙¬ \underline{x} = 0 ≫ ˙¬ − 0 <= −\underline{x} ⇒ ˙¬ ˙¬ − 0 = −\underline{x}; −0 = 0 ≫ −0 = 0; SubLessLeft ⊳ −0 = 0 ⊳ ˙¬ − 0 <= −\underline{x} ⇒ ˙¬ ˙¬ − 0 = −\underline{x} ≫ ˙¬ 0 <= −\underline{x} ⇒ ˙¬ ˙¬ 0 = −\underline{x}⌉, p_{0}, c)] [NonnegativeNegated →^{stmt} ZFsub ⊢ ∀\underline{x}: 0 <= \underline{x} ⊢ −\underline{x} <= 0] [NonnegativeNegated →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: 0 <= \underline{x} ⊢ LeqNegated ⊳ 0 <= \underline{x} ≫ −\underline{x} <= −0; −0 = 0 ≫ −0 = 0; subLeqRight ⊳ −0 = 0 ⊳ −\underline{x} <= −0 ≫ −\underline{x} <= 0⌉, p_{0}, c)] [PositiveHalved →^{stmt} ZFsub ⊢ ∀\underline{x}: ˙¬ 0 <= \underline{x} ⇒ ˙¬ ˙¬ 0 = \underline{x} ⊢ ˙¬ 0 <= rec1 + 1 ∗ \underline{x} ⇒ ˙¬ ˙¬ 0 = rec1 + 1 ∗ \underline{x}] [PositiveHalved →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ˙¬ 0 <= \underline{x} ⇒ ˙¬ ˙¬ 0 = \underline{x} ⊢ 0 < 1/2 ≫ ˙¬ 0 <= rec1 + 1 ⇒ ˙¬ ˙¬ 0 = rec1 + 1; LessMultiplicationLeft ⊳ ˙¬ 0 <= rec1 + 1 ⇒ ˙¬ ˙¬ 0 = rec1 + 1 ⊳ ˙¬ 0 <= \underline{x} ⇒ ˙¬ ˙¬ 0 = \underline{x} ≫ ˙¬ rec1 + 1 ∗ 0 <= rec1 + 1 ∗ \underline{x} ⇒ ˙¬ ˙¬ rec1 + 1 ∗ 0 = rec1 + 1 ∗ \underline{x}; x ∗ 0 = 0 ≫ rec1 + 1 ∗ 0 = 0; SubLessLeft ⊳ rec1 + 1 ∗ 0 = 0 ⊳ ˙¬ rec1 + 1 ∗ 0 <= rec1 + 1 ∗ \underline{x} ⇒ ˙¬ ˙¬ rec1+1∗0 = rec1+1∗\underline{x} ≫ ˙¬ 0 <= rec1+1∗\underline{x} ⇒ ˙¬ ˙¬ 0 = rec1+1∗\underline{x}⌉, p_{0}, c)] (*** NUMERISK ***) [NonnegativeNumerical →^{stmt} ZFsub ⊢ ∀\underline{x}: 0 <= \underline{x} ⊢ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = \underline{x}] [NonnegativeNumerical →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: 0 <= \underline{x} ⊢ IfThenElse(T) ⊳ 0 <= \underline{x} ≫ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = \underline{x}; Repetition ⊳ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = \underline{x} ≫ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = \underline{x}⌉, p_{0}, c)] [PositiveNumerical →^{stmt} ZFsub ⊢ ∀\underline{x}: ˙¬ 0 <= \underline{x} ⇒ ˙¬ ˙¬ 0 = \underline{x} ⊢ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = \underline{x}] [PositiveNumerical →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ˙¬ 0 <= \underline{x} ⇒ ˙¬ ˙¬ 0 = \underline{x} ⊢ LessLeq ⊳ ˙¬ 0 <= \underline{x} ⇒ ˙¬ ˙¬ 0 = \underline{x} ≫ 0 <= \underline{x}; NonnegativeNumerical ⊳ 0 <= \underline{x} ≫ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = \underline{x}⌉, p_{0}, c)] [NegativeNumerical →^{stmt} ZFsub ⊢ ∀\underline{x}: ˙¬ \underline{x} <= 0 ⇒ ˙¬ ˙¬ \underline{x} = 0 ⊢ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = −\underline{x}] [NegativeNumerical →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ˙¬ \underline{x} <= 0 ⇒ ˙¬ ˙¬ \underline{x} = 0 ⊢ FromLess ⊳ ˙¬ \underline{x} <= 0 ⇒ ˙¬ ˙¬ \underline{x} = 0 ≫ ˙¬ 0 <= \underline{x}; IfThenElse(F) ⊳ ˙¬ 0 <= \underline{x} ≫ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = −\underline{x}; Repetition ⊳ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = −\underline{x} ≫ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = −\underline{x}⌉, p_{0}, c)] [lemma nonpositiveNumerical →^{stmt} ZFsub ⊢ ∀\underline{x}: \underline{x} <= 0 ⊢ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = −\underline{x}] [lemma nonpositiveNumerical →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ˙¬ \underline{x} <= 0 ⇒ ˙¬ ˙¬ \underline{x} = 0 ⊢ NegativeNumerical ⊳ ˙¬ \underline{x} <= 0 ⇒ ˙¬ ˙¬ \underline{x} = 0 ≫ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = −\underline{x}; ∀\underline{x}: \underline{x} = 0 ⊢ eqSymmetry ⊳ \underline{x} = 0 ≫ 0 = \underline{x}; eqLeq ⊳ 0 = \underline{x} ≫ 0 <= \underline{x}; NonnegativeNumerical ⊳ 0 <= \underline{x} ≫ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = \underline{x}; −0 = 0 ≫ −0 = 0; eqSymmetry ⊳ −0 = 0 ≫ 0 = −0; EqNegated ⊳ 0 = \underline{x} ≫ −0 = −\underline{x}; eqTransitivity5⊳if(0 <= \underline{x}, \underline{x}, −\underline{x}) = \underline{x}⊳\underline{x} = 0⊳0 = −0⊳−0 = −\underline{x} ≫ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = −\underline{x}; ∀\underline{x}: Ded ⊳ ∀\underline{x}: ˙¬ \underline{x} <= 0 ⇒ ˙¬ ˙¬ \underline{x} = 0 ⊢ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = −\underline{x} ≫ ˙¬ \underline{x} <= 0 ⇒ ˙¬ ˙¬ \underline{x} = 0 ⇒ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = −\underline{x}; Ded ⊳ ∀\underline{x}: \underline{x} = 0 ⊢ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = −\underline{x} ≫ \underline{x} = 0 ⇒ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = −\underline{x}; \underline{x} <= 0 ⊢ LeqLessEq ⊳ \underline{x} <= 0 ≫ ˙¬ ˙¬ \underline{x} <= 0 ⇒ ˙¬ ˙¬ \underline{x} = 0 ⇒ \underline{x} = 0; FromDisjuncts ⊳ ˙¬ ˙¬ \underline{x} <= 0 ⇒ ˙¬ ˙¬ \underline{x} = 0 ⇒ \underline{x} = 0 ⊳ ˙¬ \underline{x} <= 0 ⇒ ˙¬ ˙¬ \underline{x} = 0 ⇒ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = −\underline{x} ⊳ \underline{x} = 0 ⇒ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = −\underline{x} ≫ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = −\underline{x}⌉, p_{0}, c)] [|0| = 0 →^{stmt} ZFsub ⊢ if(0 <= 0, 0, −0) = 0] [|0| = 0 →^{proof} λc.λx.P(⌈ZFsub ⊢ leqReflexivity ≫ 0 <= 0; NonnegativeNumerical ⊳ 0 <= 0 ≫ if(0 <= 0, 0, −0) = 0⌉, p_{0}, c)] [0 <= |x| →^{stmt} ZFsub ⊢ ∀\underline{x}: 0 <= if(0 <= \underline{x}, \underline{x}, −\underline{x})] [0 <= |x| →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: 0 <= \underline{x} ⊢ NonnegativeNumerical ⊳ 0 <= \underline{x} ≫ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = \underline{x}; eqSymmetry ⊳ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = \underline{x} ≫ \underline{x} = if(0 <= \underline{x}, \underline{x}, −\underline{x}); subLeqRight ⊳ \underline{x} = if(0 <= \underline{x}, \underline{x}, −\underline{x}) ⊳ 0 <= \underline{x} ≫ 0 <= if(0 <= \underline{x}, \underline{x}, −\underline{x}); ∀\underline{x}: ˙¬ 0 <= \underline{x} ⊢ ToLess ⊳ ˙¬ 0 <= \underline{x} ≫ ˙¬ \underline{x} <= 0 ⇒ ˙¬ ˙¬ \underline{x} = 0; NegativeNumerical ⊳ ˙¬ \underline{x} <= 0 ⇒ ˙¬ ˙¬ \underline{x} = 0 ≫ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = −\underline{x}; eqSymmetry ⊳ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = −\underline{x} ≫ −\underline{x} = if(0 <= \underline{x}, \underline{x}, −\underline{x}); NegativeNegated ⊳ ˙¬ \underline{x} <= 0 ⇒ ˙¬ ˙¬ \underline{x} = 0 ≫ ˙¬ 0 <= −\underline{x} ⇒ ˙¬ ˙¬ 0 = −\underline{x}; LessLeq ⊳ ˙¬ 0 <= −\underline{x} ⇒ ˙¬ ˙¬ 0 = −\underline{x} ≫ 0 <= −\underline{x}; subLeqRight ⊳ −\underline{x} = if(0 <= \underline{x}, \underline{x}, −\underline{x}) ⊳ 0 <= −\underline{x} ≫ 0 <= if(0 <= \underline{x}, \underline{x}, −\underline{x}); ∀\underline{x}: Ded ⊳ ∀\underline{x}: 0 <= \underline{x} ⊢ 0 <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) ≫ 0 <= \underline{x} ⇒ 0 <= if(0 <= \underline{x}, \underline{x}, −\underline{x}); Ded ⊳ ∀\underline{x}: ˙¬ 0 <= \underline{x} ⊢ 0 <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) ≫ ˙¬ 0 <= \underline{x} ⇒ 0 <= if(0 <= \underline{x}, \underline{x}, −\underline{x}); FromNegations ⊳ 0 <= \underline{x} ⇒ 0 <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) ⊳ ˙¬ 0 <= \underline{x} ⇒ 0 <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) ≫ 0 <= if(0 <= \underline{x}, \underline{x}, −\underline{x})⌉, p_{0}, c)] [SameNumerical →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} = \underline{y} ⊢ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= \underline{y}, \underline{y}, −\underline{y})]

[SameNumerical →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: 0 <= \underline{x} ⊢ \underline{x} = \underline{y} ⊢ NonnegativeNumerical ⊳ 0 <= \underline{x} ≫ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = \underline{x}; subLeqRight ⊳ \underline{x} = \underline{y} ⊳ 0 <= \underline{x} ≫ 0 <= \underline{y}; NonnegativeNumerical ⊳ 0 <= \underline{y} ≫ if(0 <= \underline{y}, \underline{y}, −\underline{y}) = \underline{y}; eqSymmetry ⊳ if(0 <= \underline{y}, \underline{y}, −\underline{y}) = \underline{y} ≫ \underline{y} = if(0 <= \underline{y}, \underline{y}, −\underline{y}); eqTransitivity4 ⊳ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = \underline{x} ⊳ \underline{x} = \underline{y} ⊳ \underline{y} = if(0 <= \underline{y}, \underline{y}, −\underline{y}) ≫ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= \underline{y}, \underline{y}, −\underline{y}); ∀\underline{x}: ∀\underline{y}: ˙¬ 0 <= \underline{x} ⊢ \underline{x} = \underline{y} ⊢ ToLess ⊳ ˙¬ 0 <= \underline{x} ≫ ˙¬ \underline{x} <= 0 ⇒ ˙¬ ˙¬ \underline{x} = 0; NegativeNumerical ⊳ ˙¬ \underline{x} <= 0 ⇒ ˙¬ ˙¬ \underline{x} = 0 ≫ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = −\underline{x}; SubLessLeft ⊳ \underline{x} = \underline{y} ⊳ ˙¬ \underline{x} <= 0 ⇒ ˙¬ ˙¬ \underline{x} = 0 ≫ ˙¬ \underline{y} <= 0 ⇒ ˙¬ ˙¬ \underline{y} = 0; NegativeNumerical ⊳ ˙¬ \underline{y} <= 0 ⇒ ˙¬ ˙¬ \underline{y} = 0 ≫ if(0 <= \underline{y}, \underline{y}, −\underline{y}) = −\underline{y}; eqSymmetry ⊳ if(0 <= \underline{y}, \underline{y}, −\underline{y}) = −\underline{y} ≫ −\underline{y} = if(0 <= \underline{y}, \underline{y}, −\underline{y}); EqNegated ⊳ \underline{x} = \underline{y} ≫ −\underline{x} = −\underline{y}; eqTransitivity4 ⊳ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = −\underline{x} ⊳ −\underline{x} = −\underline{y} ⊳ −\underline{y} = if(0 <= \underline{y}, \underline{y}, −\underline{y}) ≫ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= \underline{y}, \underline{y}, −\underline{y}); ∀\underline{x}: ∀\underline{y}: \underline{x} = \underline{y} ⊢ Ded ⊳ ∀\underline{x}: ∀\underline{y}: 0 <= \underline{x} ⊢ \underline{x} = \underline{y} ⊢ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= \underline{y}, \underline{y}, −\underline{y}) ≫ 0 <= \underline{x} ⇒ \underline{x} = \underline{y} ⇒ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= \underline{y}, \underline{y}, −\underline{y}); Ded ⊳ ∀\underline{x}: ∀\underline{y}: ˙¬ 0 <= \underline{x} ⊢ \underline{x} = \underline{y} ⊢ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= \underline{y}, \underline{y}, −\underline{y}) ≫ ˙¬ 0 <= \underline{x} ⇒ \underline{x} = \underline{y} ⇒ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= \underline{y}, \underline{y}, −\underline{y}); FromNegations ⊳ 0 <= \underline{x} ⇒ \underline{x} = \underline{y} ⇒ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= \underline{y}, \underline{y}, −\underline{y}) ⊳ ˙¬ 0 <= \underline{x} ⇒ \underline{x} = \underline{y} ⇒ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= \underline{y}, \underline{y}, −\underline{y}) ≫ \underline{x} = \underline{y} ⇒ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= \underline{y}, \underline{y}, −\underline{y}); MP ⊳ \underline{x} = \underline{y} ⇒ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= \underline{y}, \underline{y}, −\underline{y}) ⊳ \underline{x} = \underline{y} ≫ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= \underline{y}, \underline{y}, −\underline{y})⌉, p_{0}, c)]

[SignNumerical(+) →^{stmt} ZFsub ⊢ ∀\underline{x}: ˙¬ 0 <= \underline{x} ⇒ ˙¬ ˙¬ 0 = \underline{x} ⊢ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= −\underline{x}, −\underline{x}, − − \underline{x})] [SignNumerical(+) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ˙¬ 0 <= \underline{x} ⇒ ˙¬ ˙¬ 0 = \underline{x} ⊢ PositiveNumerical ⊳ ˙¬ 0 <= \underline{x} ⇒ ˙¬ ˙¬ 0 = \underline{x} ≫ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = \underline{x}; PositiveNegated ⊳ ˙¬ 0 <= \underline{x} ⇒ ˙¬ ˙¬ 0 = \underline{x} ≫ ˙¬ − \underline{x} <= 0 ⇒ ˙¬ ˙¬ − \underline{x} = 0; NegativeNumerical ⊳ ˙¬ − \underline{x} <= 0 ⇒ ˙¬ ˙¬ − \underline{x} = 0 ≫ if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}) = −−\underline{x}; DoubleMinus ≫ −−\underline{x} = \underline{x}; eqTransitivity⊳if(0 <= −\underline{x}, −\underline{x}, −−\underline{x}) = −−\underline{x}⊳ −−\underline{x} = \underline{x} ≫ if(0 <= −\underline{x}, −\underline{x}, −−\underline{x}) = \underline{x}; eqSymmetry⊳if(0 <= −\underline{x}, −\underline{x}, −−\underline{x}) = \underline{x} ≫ \underline{x} = if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}); eqTransitivity ⊳ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = \underline{x} ⊳ \underline{x} = if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}) ≫ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= −\underline{x}, −\underline{x}, − − \underline{x})⌉, p_{0}, c)] [SignNumerical →^{stmt} ZFsub ⊢ ∀\underline{x}: if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= −\underline{x}, −\underline{x}, − − \underline{x})] [SignNumerical →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ˙¬ \underline{x} <= 0 ⇒ ˙¬ ˙¬ \underline{x} = 0 ⊢ NegativeNegated ⊳ ˙¬ \underline{x} <= 0 ⇒ ˙¬ ˙¬ \underline{x} = 0 ≫ ˙¬ 0 <= −\underline{x} ⇒ ˙¬ ˙¬ 0 = −\underline{x}; SignNumerical(+) ⊳ ˙¬ 0 <= −\underline{x} ⇒ ˙¬ ˙¬ 0 = −\underline{x} ≫ if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}) = if(0 <= −−\underline{x}, −−\underline{x}, −−−\underline{x}); DoubleMinus ≫ −−\underline{x} = \underline{x}; SameNumerical⊳−−\underline{x} = \underline{x} ≫ if(0 <= − − \underline{x}, − − \underline{x}, − − −\underline{x}) = if(0 <= \underline{x}, \underline{x}, −\underline{x}); eqTransitivity ⊳ if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}) = if(0 <= − − \underline{x}, − − \underline{x}, − − −\underline{x}) ⊳ if(0 <= − − \underline{x}, − − \underline{x}, − − −\underline{x}) = if(0 <= \underline{x}, \underline{x}, −\underline{x}) ≫ if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}) = if(0 <= \underline{x}, \underline{x}, −\underline{x}); eqSymmetry ⊳ if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}) = if(0 <= \underline{x}, \underline{x}, −\underline{x}) ≫ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}); ∀\underline{x}: \underline{x} = 0 ⊢ EqNegated ⊳ \underline{x} = 0 ≫ −\underline{x} = −0; −0 = 0 ≫ −0 = 0; eqSymmetry ⊳ \underline{x} = 0 ≫ 0 = \underline{x}; eqTransitivity4 ⊳ −\underline{x} = −0 ⊳ −0 = 0 ⊳ 0 = \underline{x} ≫ −\underline{x} = \underline{x}; eqSymmetry ⊳ −\underline{x} = \underline{x} ≫ \underline{x} = −\underline{x}; SameNumerical ⊳ \underline{x} = −\underline{x} ≫ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}); ∀\underline{x}: ˙¬ 0 <= \underline{x} ⇒ ˙¬ ˙¬ 0 = \underline{x} ⊢ SignNumerical(+) ⊳ ˙¬ 0 <= \underline{x} ⇒ ˙¬ ˙¬ 0 = \underline{x} ≫ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}); ∀\underline{x}: Ded ⊳ ∀\underline{x}: ˙¬ \underline{x} <= 0 ⇒ ˙¬ ˙¬ \underline{x} = 0 ⊢ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}) ≫ ˙¬ \underline{x} <= 0 ⇒ ˙¬ ˙¬ \underline{x} = 0 ⇒ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}); Ded ⊳ ∀\underline{x}: \underline{x} = 0 ⊢ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}) ≫ \underline{x} = 0 ⇒ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}); Ded ⊳ ∀\underline{x}: ˙¬ 0 <= \underline{x} ⇒ ˙¬ ˙¬ 0 = \underline{x} ⊢ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}) ≫ ˙¬ 0 <= \underline{x} ⇒ ˙¬ ˙¬ 0 = \underline{x} ⇒ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}); LessTotality ≫ ˙¬ ˙¬ \underline{x} <= 0 ⇒ ˙¬ ˙¬ \underline{x} = 0 ⇒ ˙¬ \underline{x} = 0 ⇒ ˙¬ 0 <= \underline{x} ⇒ ˙¬ ˙¬ 0 = \underline{x}; From3Disjuncts ⊳ ˙¬ ˙¬ \underline{x} <= 0 ⇒ ˙¬ ˙¬ \underline{x} = 0 ⇒ ˙¬ \underline{x} = 0 ⇒ ˙¬ 0 <= \underline{x} ⇒ ˙¬ ˙¬ 0 = \underline{x} ⊳ ˙¬ \underline{x} <= 0 ⇒ ˙¬ ˙¬ \underline{x} = 0 ⇒ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}) ⊳ \underline{x} = 0 ⇒ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}) ⊳ ˙¬ 0 <= \underline{x} ⇒ ˙¬ ˙¬ 0 = \underline{x} ⇒ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}) ≫ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= −\underline{x}, −\underline{x}, − − \underline{x})⌉, p_{0}, c)] [NumericalDifference →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: if(0 <= \underline{x} + −\underline{y}, \underline{x} + −\underline{y}, −\underline{x} + −\underline{y}) = if(0 <= \underline{y} + −\underline{x}, \underline{y} + −\underline{x}, −\underline{y} + −\underline{x})]

[NumericalDifference →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: SignNumerical ≫ if(0 <= \underline{x}+−\underline{y}, \underline{x}+−\underline{y}, −\underline{x}+−\underline{y}) = if(0 <= −\underline{x}+−\underline{y}, −\underline{x}+−\underline{y}, −−\underline{x}+−\underline{y}); MinusNegated ≫ −\underline{x}+−\underline{y} = \underline{y}+−\underline{x}; SameNumerical⊳−\underline{x}+−\underline{y} = \underline{y}+−\underline{x} ≫ if(0 <= −\underline{x}+−\underline{y}, −\underline{x}+ −\underline{y}, − − \underline{x} + −\underline{y}) = if(0 <= \underline{y} + −\underline{x}, \underline{y} + −\underline{x}, −\underline{y} + −\underline{x}); eqTransitivity ⊳ if(0 <= \underline{x} + −\underline{y}, \underline{x} + −\underline{y}, −\underline{x} + −\underline{y}) = if(0 <= −\underline{x} + −\underline{y}, −\underline{x} + −\underline{y}, − − \underline{x} + −\underline{y}) ⊳ if(0 <= −\underline{x} + −\underline{y}, −\underline{x} + −\underline{y}, − − \underline{x} + −\underline{y}) = if(0 <= \underline{y} + −\underline{x}, \underline{y} + −\underline{x}, −\underline{y} + −\underline{x}) ≫ if(0 <= \underline{x} + −\underline{y}, \underline{x} + −\underline{y}, −\underline{x} + −\underline{y}) = if(0 <= \underline{y} + −\underline{x}, \underline{y} + −\underline{x}, −\underline{y} + −\underline{x})⌉, p_{0}, c)]

[SplitNumericalSumHelper →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: if(0 <= −\underline{x}+−\underline{y}, −\underline{x}+−\underline{y}, −−\underline{x}+−\underline{y}) <= if(0 <= −\underline{x}, −\underline{x}, −−\underline{x})+if(0 <= −\underline{y}, −\underline{y}, −−\underline{y}) ⊢ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y})]

[SplitNumericalSumHelper →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: if(0 <= −\underline{x}+−\underline{y}, −\underline{x}+−\underline{y}, −−\underline{x}+−\underline{y}) <= if(0 <= −\underline{x}, −\underline{x}, −−\underline{x})+if(0 <= −\underline{y}, −\underline{y}, −−\underline{y}) ⊢ SignNumerical ≫ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}); SignNumerical ≫ if(0 <= \underline{y}, \underline{y}, −\underline{y}) = if(0 <= −\underline{y}, −\underline{y}, − − \underline{y}); AddEquations ⊳ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}) ⊳ if(0 <= \underline{y}, \underline{y}, −\underline{y}) = if(0 <= −\underline{y}, −\underline{y}, − − \underline{y}) ≫ if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}) = if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}) + if(0 <= −\underline{y}, −\underline{y}, − − \underline{y}); eqSymmetry ⊳ if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}) = if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}) + if(0 <= −\underline{y}, −\underline{y}, − − \underline{y}) ≫ if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}) + if(0 <= −\underline{y}, −\underline{y}, − − \underline{y}) = if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}); −x − y = −(x + y) ≫ −\underline{x} + −\underline{y} = −\underline{x} + \underline{y}; SameNumerical ⊳ −\underline{x} + −\underline{y} = −\underline{x} + \underline{y} ≫ if(0 <= −\underline{x} + −\underline{y}, −\underline{x} + −\underline{y}, − − \underline{x} + −\underline{y}) = if(0 <= −\underline{x} + \underline{y}, −\underline{x} + \underline{y}, − − \underline{x} + \underline{y}); SignNumerical ≫ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) = if(0 <= −\underline{x} + \underline{y}, −\underline{x} + \underline{y}, − − \underline{x} + \underline{y}); eqSymmetry ⊳ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) = if(0 <= −\underline{x}+\underline{y}, −\underline{x}+\underline{y}, −−\underline{x}+\underline{y}) ≫ if(0 <= −\underline{x}+\underline{y}, −\underline{x}+\underline{y}, −−\underline{x}+\underline{y}) = if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}); eqTransitivity ⊳ if(0 <= −\underline{x} + −\underline{y}, −\underline{x} + −\underline{y}, − − \underline{x} + −\underline{y}) = if(0 <= −\underline{x} + \underline{y}, −\underline{x} + \underline{y}, − − \underline{x} + \underline{y}) ⊳ if(0 <= −\underline{x} + \underline{y}, −\underline{x} + \underline{y}, − − \underline{x} + \underline{y}) = if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) ≫ if(0 <= −\underline{x} + −\underline{y}, −\underline{x} + −\underline{y}, − − \underline{x} + −\underline{y}) = if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}); subLeqRight ⊳ if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}) + if(0 <= −\underline{y}, −\underline{y}, − − \underline{y}) = if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}) ⊳ if(0 <= −\underline{x} + −\underline{y}, −\underline{x} + −\underline{y}, − − \underline{x} + −\underline{y}) <= if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}) + if(0 <= −\underline{y}, −\underline{y}, − − \underline{y}) ≫ if(0 <= −\underline{x} + −\underline{y}, −\underline{x} + −\underline{y}, − − \underline{x} + −\underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}); subLeqLeft ⊳ if(0 <= −\underline{x} + −\underline{y}, −\underline{x} + −\underline{y}, − − \underline{x} + −\underline{y}) = if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) ⊳ if(0 <= −\underline{x} + −\underline{y}, −\underline{x} + −\underline{y}, − − \underline{x} + −\underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}) ≫ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y})⌉, p_{0}, c)]

[splitNumericalSum(++) →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: 0 <= \underline{x} ⊢ 0 <= \underline{y} ⊢ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y})]

[splitNumericalSum(++) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: 0 <= \underline{x} ⊢ 0 <= \underline{y} ⊢ AddEquations(Leq) ⊳ 0 <= \underline{x} ⊳ 0 <= \underline{y} ≫ 0 + 0 <= \underline{x} + \underline{y}; plus0 ≫ 0 + 0 = 0; subLeqLeft⊳0+0 = 0⊳0+0 <= \underline{x}+\underline{y} ≫ 0 <= \underline{x}+\underline{y}; NonnegativeNumerical⊳ 0 <= \underline{x}+\underline{y} ≫ if(0 <= \underline{x}+\underline{y}, \underline{x}+\underline{y}, −\underline{x}+\underline{y}) = \underline{x}+\underline{y}; NonnegativeNumerical⊳0 <= \underline{x} ≫ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = \underline{x}; NonnegativeNumerical ⊳ 0 <= \underline{y} ≫ if(0 <= \underline{y}, \underline{y}, −\underline{y}) = \underline{y}; AddEquations ⊳ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = \underline{x} ⊳ if(0 <= \underline{y}, \underline{y}, −\underline{y}) = \underline{y} ≫ if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}) = \underline{x} + \underline{y}; eqSymmetry ⊳ if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}) = \underline{x} + \underline{y} ≫ \underline{x} + \underline{y} = if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}); eqTransitivity ⊳ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) = \underline{x} + \underline{y} ⊳ \underline{x} + \underline{y} = if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}) ≫ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) = if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}); eqLeq ⊳ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) = if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}) ≫ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y})⌉, p_{0}, c)]

[splitNumericalSum(−−) →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} <= 0 ⊢ \underline{y} <= 0 ⊢ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y})]

[splitNumericalSum(−−) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} <= 0 ⊢ \underline{y} <= 0 ⊢ NonpositiveNegated ⊳ \underline{x} <= 0 ≫ 0 <= −\underline{x}; NonpositiveNegated ⊳ \underline{y} <= 0 ≫ 0 <= −\underline{y}; splitNumericalSum(++) ⊳ 0 <= −\underline{x} ⊳ 0 <= −\underline{y} ≫ if(0 <= −\underline{x} + −\underline{y}, −\underline{x} + −\underline{y}, − − \underline{x} + −\underline{y}) <= if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}) + if(0 <= −\underline{y}, −\underline{y}, − − \underline{y}); SplitNumericalSumHelper ⊳ if(0 <= −\underline{x} + −\underline{y}, −\underline{x} + −\underline{y}, − − \underline{x} + −\underline{y}) <= if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}) + if(0 <= −\underline{y}, −\underline{y}, − − \underline{y}) ≫ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y})⌉, p_{0}, c)] [splitNumericalSum(+ − small) →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: 0 <= \underline{x} ⊢ \underline{y} <= 0 ⊢ if(0 <= \underline{y}, \underline{y}, −\underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) ⊢ if(0 <= \underline{x}+\underline{y}, \underline{x}+\underline{y}, −\underline{x}+\underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x})]

[splitNumericalSum(+ − small) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: 0 <= \underline{x} ⊢ \underline{y} <= 0 ⊢ if(0 <= \underline{y}, \underline{y}, −\underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) ⊢ LeqAdditionLeft ⊳ \underline{y} <= 0 ≫ \underline{x} + \underline{y} <= \underline{x} + 0; plus0 ≫ \underline{x} + 0 = \underline{x}; subLeqRight ⊳ \underline{x} + 0 = \underline{x} ⊳ \underline{x} + \underline{y} <= \underline{x} + 0 ≫ \underline{x} + \underline{y} <= \underline{x}; PositiveToRight(Leq)(1term) ⊳ if(0 <= \underline{y}, \underline{y}, −\underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) ≫ 0 <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + −if(0 <= \underline{y}, \underline{y}, −\underline{y}); lemma nonpositiveNumerical ⊳ \underline{y} <= 0 ≫ if(0 <= \underline{y}, \underline{y}, −\underline{y}) = −\underline{y}; EqNegated ⊳ if(0 <= \underline{y}, \underline{y}, −\underline{y}) = −\underline{y} ≫ −if(0 <= \underline{y}, \underline{y}, −\underline{y}) = − − \underline{y}; DoubleMinus ≫ − − \underline{y} = \underline{y}; eqTransitivity ⊳ −if(0 <= \underline{y}, \underline{y}, −\underline{y}) = − − \underline{y} ⊳ −−\underline{y} = \underline{y} ≫ −if(0 <= \underline{y}, \underline{y}, −\underline{y}) = \underline{y}; NonnegativeNumerical⊳0 <= \underline{x} ≫ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = \underline{x}; AddEquations ⊳ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = \underline{x} ⊳ −if(0 <= \underline{y}, \underline{y}, −\underline{y}) = \underline{y} ≫ if(0 <= \underline{x}, \underline{x}, −\underline{x}) + −if(0 <= \underline{y}, \underline{y}, −\underline{y}) = \underline{x} + \underline{y}; subLeqRight ⊳ if(0 <= \underline{x}, \underline{x}, −\underline{x}) + −if(0 <= \underline{y}, \underline{y}, −\underline{y}) = \underline{x} + \underline{y} ⊳ 0 <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + −if(0 <= \underline{y}, \underline{y}, −\underline{y}) ≫ 0 <= \underline{x} + \underline{y}; NonnegativeNumerical ⊳ 0 <= \underline{x} + \underline{y} ≫ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) = \underline{x} + \underline{y}; eqSymmetry ⊳ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) = \underline{x} + \underline{y} ≫ \underline{x} + \underline{y} = if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}); eqSymmetry ⊳ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = \underline{x} ≫ \underline{x} = if(0 <= \underline{x}, \underline{x}, −\underline{x}); subLeqLeft ⊳ \underline{x} + \underline{y} = if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) ⊳ \underline{x} + \underline{y} <= \underline{x} ≫ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= \underline{x}; subLeqRight ⊳ \underline{x} = if(0 <= \underline{x}, \underline{x}, −\underline{x}) ⊳ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= \underline{x} ≫ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x})⌉, p_{0}, c)]

[splitNumericalSum(+ − big) →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: 0 <= \underline{x} ⊢ \underline{y} <= 0 ⊢ ˙¬ if(0 <= \underline{x}, \underline{x}, −\underline{x}) <= if(0 <= \underline{y}, \underline{y}, −\underline{y}) ⇒ ˙¬ ˙¬ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= \underline{y}, \underline{y}, −\underline{y}) ⊢ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{y}, \underline{y}, −\underline{y})]

[splitNumericalSum(+ − big) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: 0 <= \underline{x} ⊢ \underline{y} <= 0 ⊢ ˙¬ if(0 <= \underline{x}, \underline{x}, −\underline{x}) <= if(0 <= \underline{y}, \underline{y}, −\underline{y}) ⇒ ˙¬ ˙¬ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= \underline{y}, \underline{y}, −\underline{y}) ⊢ NonnegativeNegated ⊳ 0 <= \underline{x} ≫ −\underline{x} <= 0; NonpositiveNegated ⊳ \underline{y} <= 0 ≫ 0 <= −\underline{y}; SignNumerical ≫ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}); SubLessLeft ⊳ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}) ⊳ ˙¬ if(0 <= \underline{x}, \underline{x}, −\underline{x}) <= if(0 <= \underline{y}, \underline{y}, −\underline{y}) ⇒ ˙¬ ˙¬ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= \underline{y}, \underline{y}, −\underline{y}) ≫ ˙¬ if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}) <= if(0 <= \underline{y}, \underline{y}, −\underline{y}) ⇒ ˙¬ ˙¬ if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}) = if(0 <= \underline{y}, \underline{y}, −\underline{y}); SignNumerical ≫ if(0 <= \underline{y}, \underline{y}, −\underline{y}) = if(0 <= −\underline{y}, −\underline{y}, − − \underline{y}); SubLessRight ⊳ if(0 <= \underline{y}, \underline{y}, −\underline{y}) = if(0 <= −\underline{y}, −\underline{y}, − − \underline{y}) ⊳ ˙¬ if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}) <= if(0 <= \underline{y}, \underline{y}, −\underline{y}) ⇒ ˙¬ ˙¬ if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}) = if(0 <= \underline{y}, \underline{y}, −\underline{y}) ≫ ˙¬ if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}) <= if(0 <= −\underline{y}, −\underline{y}, − − \underline{y}) ⇒ ˙¬ ˙¬ if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}) = if(0 <= −\underline{y}, −\underline{y}, −−\underline{y}); LessLeq⊳ ˙¬ if(0 <= −\underline{x}, −\underline{x}, −−\underline{x}) <= if(0 <= −\underline{y}, −\underline{y}, −−\underline{y}) ⇒ ˙¬ ˙¬ if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}) = if(0 <= −\underline{y}, −\underline{y}, − − \underline{y}) ≫ if(0 <= −\underline{x}, −\underline{x}, −−\underline{x}) <= if(0 <= −\underline{y}, −\underline{y}, −−\underline{y}); splitNumericalSum(+−small)⊳0 <= −\underline{y} ⊳ −\underline{x} <= 0 ⊳ if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}) <= if(0 <= −\underline{y}, −\underline{y}, − − \underline{y}) ≫ if(0 <= −\underline{y} + −\underline{x}, −\underline{y} + −\underline{x}, − − \underline{y} + −\underline{x}) <= if(0 <= −\underline{y}, −\underline{y}, − − \underline{y}); SignNumerical ≫ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) = if(0 <= −\underline{x} + \underline{y}, −\underline{x} + \underline{y}, − − \underline{x} + \underline{y}); −x − y = −(x + y) ≫ −\underline{x} + −\underline{y} = −\underline{x} + \underline{y}; plusCommutativity ≫ −\underline{x} + −\underline{y} = −\underline{y} + −\underline{x}; Equality ⊳ −\underline{x} + −\underline{y} = −\underline{x} + \underline{y} ⊳ −\underline{x} + −\underline{y} = −\underline{y} + −\underline{x} ≫ −\underline{x} + \underline{y} = −\underline{y}+−\underline{x}; SameNumerical⊳−\underline{x}+\underline{y} = −\underline{y}+−\underline{x} ≫ if(0 <= −\underline{x}+\underline{y}, −\underline{x}+\underline{y}, −−\underline{x}+\underline{y}) = if(0 <= −\underline{y} +−\underline{x}, −\underline{y} +−\underline{x}, −−\underline{y} +−\underline{x}); eqTransitivity ⊳if(0 <= \underline{x}+\underline{y}, \underline{x}+\underline{y}, −\underline{x}+ \underline{y}) = if(0 <= −\underline{x} + \underline{y}, −\underline{x} + \underline{y}, − − \underline{x} + \underline{y}) ⊳ if(0 <= −\underline{x} + \underline{y}, −\underline{x} + \underline{y}, − − \underline{x} + \underline{y}) = if(0 <= −\underline{y} + −\underline{x}, −\underline{y} + −\underline{x}, − − \underline{y} + −\underline{x}) ≫ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) = if(0 <= −\underline{y}+−\underline{x}, −\underline{y}+−\underline{x}, −−\underline{y}+−\underline{x}); eqSymmetry⊳if(0 <= \underline{x}+\underline{y}, \underline{x}+\underline{y}, −\underline{x}+\underline{y}) = if(0 <= −\underline{y}+−\underline{x}, −\underline{y}+−\underline{x}, −−\underline{y}+−\underline{x}) ≫ if(0 <= −\underline{y}+−\underline{x}, −\underline{y}+−\underline{x}, −−\underline{y}+−\underline{x}) = if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}); eqSymmetry ⊳ if(0 <= \underline{y}, \underline{y}, −\underline{y}) = if(0 <= −\underline{y}, −\underline{y}, − − \underline{y}) ≫ if(0 <= −\underline{y}, −\underline{y}, − − \underline{y}) = if(0 <= \underline{y}, \underline{y}, −\underline{y}); subLeqLeft ⊳ if(0 <= −\underline{y} + −\underline{x}, −\underline{y} + −\underline{x}, − − \underline{y} + −\underline{x}) = if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) ⊳ if(0 <= −\underline{y} + −\underline{x}, −\underline{y} + −\underline{x}, − − \underline{y} + −\underline{x}) <= if(0 <= −\underline{y}, −\underline{y}, − − \underline{y}) ≫ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= −\underline{y}, −\underline{y}, − − \underline{y}); subLeqRight ⊳ if(0 <= −\underline{y}, −\underline{y}, − − \underline{y}) = if(0 <= \underline{y}, \underline{y}, −\underline{y}) ⊳ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= −\underline{y}, −\underline{y}, − − \underline{y}) ≫ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{y}, \underline{y}, −\underline{y})⌉, p_{0}, c)]

[splitNumericalSum(+−) →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: 0 <= \underline{x} ⊢ \underline{y} <= 0 ⊢ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y})]

[splitNumericalSum(+−) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: if(0 <= \underline{y}, \underline{y}, −\underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) ⊢ 0 <= \underline{x} ⊢ \underline{y} <= 0 ⊢ splitNumericalSum(+ − small) ⊳ 0 <= \underline{x} ⊳ \underline{y} <= 0 ⊳ if(0 <= \underline{y}, \underline{y}, −\underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) ≫ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}); 0 <= |x| ≫ 0 <= if(0 <= \underline{y}, \underline{y}, −\underline{y}); LeqAdditionLeft ⊳ 0 <= if(0 <= \underline{y}, \underline{y}, −\underline{y}) ≫ if(0 <= \underline{x}, \underline{x}, −\underline{x}) + 0 <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}); plus0 ≫ if(0 <= \underline{x}, \underline{x}, −\underline{x}) + 0 = if(0 <= \underline{x}, \underline{x}, −\underline{x}); subLeqLeft ⊳ if(0 <= \underline{x}, \underline{x}, −\underline{x}) + 0 = if(0 <= \underline{x}, \underline{x}, −\underline{x}) ⊳ if(0 <= \underline{x}, \underline{x}, −\underline{x}) + 0 <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}) ≫ if(0 <= \underline{x}, \underline{x}, −\underline{x}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}); leqTransitivity ⊳ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) ⊳ if(0 <= \underline{x}, \underline{x}, −\underline{x}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}) ≫ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}); ∀\underline{x}: ∀\underline{y}: ˙¬ if(0 <= \underline{y}, \underline{y}, −\underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) ⊢ 0 <= \underline{x} ⊢ \underline{y} <= 0 ⊢ ToLess ⊳ ˙¬ if(0 <= \underline{y}, \underline{y}, −\underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) ≫ ˙¬ if(0 <= \underline{x}, \underline{x}, −\underline{x}) <= if(0 <= \underline{y}, \underline{y}, −\underline{y}) ⇒ ˙¬ ˙¬ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= \underline{y}, \underline{y}, −\underline{y}); splitNumericalSum(+ − big) ⊳ 0 <= \underline{x} ⊳ \underline{y} <= 0 ⊳ ˙¬ if(0 <= \underline{x}, \underline{x}, −\underline{x}) <= if(0 <= \underline{y}, \underline{y}, −\underline{y}) ⇒ ˙¬ ˙¬ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= \underline{y}, \underline{y}, −\underline{y}) ≫ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{y}, \underline{y}, −\underline{y}); 0 <= |x| ≫ 0 <= if(0 <= \underline{x}, \underline{x}, −\underline{x}); leqAddition ⊳ 0 <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) ≫ 0 + if(0 <= \underline{y}, \underline{y}, −\underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}); plus0Left ≫ 0 + if(0 <= \underline{y}, \underline{y}, −\underline{y}) = if(0 <= \underline{y}, \underline{y}, −\underline{y}); subLeqLeft ⊳ 0 + if(0 <= \underline{y}, \underline{y}, −\underline{y}) = if(0 <= \underline{y}, \underline{y}, −\underline{y}) ⊳ 0 + if(0 <= \underline{y}, \underline{y}, −\underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}) ≫ if(0 <= \underline{y}, \underline{y}, −\underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}); leqTransitivity ⊳ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{y}, \underline{y}, −\underline{y}) ⊳ if(0 <= \underline{y}, \underline{y}, −\underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}) ≫ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}); ∀\underline{x}: ∀\underline{y}: Ded ⊳ ∀\underline{x}: ∀\underline{y}: if(0 <= \underline{y}, \underline{y}, −\underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) ⊢ 0 <= \underline{x} ⊢ \underline{y} <= 0 ⊢ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}) ≫ if(0 <= \underline{y}, \underline{y}, −\underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) ⇒ 0 <= \underline{x} ⇒ \underline{y} <= 0 ⇒ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}); Ded ⊳ ∀\underline{x}: ∀\underline{y}: ˙¬ if(0 <= \underline{y}, \underline{y}, −\underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) ⊢ 0 <= \underline{x} ⊢ \underline{y} <= 0 ⊢ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}) ≫ ˙¬ if(0 <= \underline{y}, \underline{y}, −\underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) ⇒ 0 <= \underline{x} ⇒ \underline{y} <= 0 ⇒ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}); 0 <= \underline{x} ⊢ \underline{y} <= 0 ⊢ FromNegations ⊳ if(0 <= \underline{y}, \underline{y}, −\underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) ⇒ 0 <= \underline{x} ⇒ \underline{y} <= 0 ⇒ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}) ⊳ ˙¬ if(0 <= \underline{y}, \underline{y}, −\underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) ⇒ 0 <= \underline{x} ⇒ \underline{y} <= 0 ⇒ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}) ≫ 0 <= \underline{x} ⇒ \underline{y} <= 0 ⇒ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}); MP2 ⊳ 0 <= \underline{x} ⇒ \underline{y} <= 0 ⇒ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}) ⊳ 0 <= \underline{x} ⊳ \underline{y} <= 0 ≫ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y})⌉, p_{0}, c)]

[splitNumericalSum(−+) →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} <= 0 ⊢ 0 <= \underline{y} ⊢ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y})]

[splitNumericalSum(−+) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} <= 0 ⊢ 0 <= \underline{y} ⊢ NonpositiveNegated ⊳ \underline{x} <= 0 ≫ 0 <= −\underline{x}; NonnegativeNegated ⊳ 0 <= \underline{y} ≫ −\underline{y} <= 0; splitNumericalSum(+−) ⊳ 0 <= −\underline{x} ⊳ −\underline{y} <= 0 ≫ if(0 <= −\underline{x} + −\underline{y}, −\underline{x} + −\underline{y}, − − \underline{x} + −\underline{y}) <= if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}) + if(0 <= −\underline{y}, −\underline{y}, − − \underline{y}); SplitNumericalSumHelper ⊳ if(0 <= −\underline{x} + −\underline{y}, −\underline{x} + −\underline{y}, − − \underline{x} + −\underline{y}) <= if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}) + if(0 <= −\underline{y}, −\underline{y}, − − \underline{y}) ≫ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y})⌉, p_{0}, c)]

[splitNumericalSum →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y})]

[splitNumericalSum →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: 0 <= \underline{x} ⊢ 0 <= \underline{y} ⊢ splitNumericalSum(++) ⊳ 0 <= \underline{x} ⊳ 0 <= \underline{y} ≫ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}); ∀\underline{x}: ∀\underline{y}: 0 <= \underline{x} ⊢ \underline{y} <= 0 ⊢ splitNumericalSum(+−) ⊳ 0 <= \underline{x} ⊳ \underline{y} <= 0 ≫ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}); ∀\underline{x}: ∀\underline{y}: \underline{x} <= 0 ⊢ 0 <= \underline{y} ⊢ splitNumericalSum(−+) ⊳ \underline{x} <= 0 ⊳ 0 <= \underline{y} ≫ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}); ∀\underline{x}: ∀\underline{y}: \underline{x} <= 0 ⊢ \underline{y} <= 0 ⊢ splitNumericalSum(−−) ⊳ \underline{x} <= 0 ⊳ \underline{y} <= 0 ≫ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}); ∀\underline{x}: ∀\underline{y}: Ded ⊳ ∀\underline{x}: ∀\underline{y}: 0 <= \underline{x} ⊢ 0 <= \underline{y} ⊢ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}) ≫ 0 <= \underline{x} ⇒ 0 <= \underline{y} ⇒ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}); Ded ⊳ ∀\underline{x}: ∀\underline{y}: 0 <= \underline{x} ⊢ \underline{y} <= 0 ⊢ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}) ≫ 0 <= \underline{x} ⇒ \underline{y} <= 0 ⇒ if(0 <= \underline{x}+\underline{y}, \underline{x}+\underline{y}, −\underline{x}+\underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x})+if(0 <= \underline{y}, \underline{y}, −\underline{y}); Ded⊳∀\underline{x}: ∀\underline{y}: \underline{x} <= 0 ⊢ 0 <= \underline{y} ⊢ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}) ≫ \underline{x} <= 0 ⇒ 0 <= \underline{y} ⇒ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}); Ded⊳∀\underline{x}: ∀\underline{y}: \underline{x} <= 0 ⊢ \underline{y} <= 0 ⊢ if(0 <= \underline{x}+\underline{y}, \underline{x}+\underline{y}, −\underline{x}+\underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}) ≫ \underline{x} <= 0 ⇒ \underline{y} <= 0 ⇒ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}); FromLeqGeq ⊳ 0 <= \underline{x} ⇒ 0 <= \underline{y} ⇒ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}) ⊳ \underline{x} <= 0 ⇒ 0 <= \underline{y} ⇒ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}) ≫ 0 <= \underline{y} ⇒ if(0 <= \underline{x}+\underline{y}, \underline{x}+\underline{y}, −\underline{x}+\underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x})+if(0 <= \underline{y}, \underline{y}, −\underline{y}); FromLeqGeq ⊳ 0 <= \underline{x} ⇒ \underline{y} <= 0 ⇒ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}) ⊳ \underline{x} <= 0 ⇒ \underline{y} <= 0 ⇒ if(0 <= \underline{x}+\underline{y}, \underline{x}+\underline{y}, −\underline{x}+\underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x})+if(0 <= \underline{y}, \underline{y}, −\underline{y}) ≫ \underline{y} <= 0 ⇒ if(0 <= \underline{x}+\underline{y}, \underline{x}+\underline{y}, −\underline{x}+\underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x})+if(0 <= \underline{y}, \underline{y}, −\underline{y}); FromLeqGeq⊳0 <= \underline{y} ⇒ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}) ⊳ \underline{y} <= 0 ⇒ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}) ≫ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y})⌉, p_{0}, c)]

[insertMiddleTerm(Numerical) →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: if(0 <= \underline{x}+\underline{y}, \underline{x}+\underline{y}, −\underline{x}+\underline{y}) <= if(0 <= \underline{x}+−\underline{z}, \underline{x}+−\underline{z}, −\underline{x}+−\underline{z})+if(0 <= \underline{z}+\underline{y}, \underline{z}+\underline{y}, −\underline{z}+\underline{y})]

[insertMiddleTerm(Numerical) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: splitNumericalSum ≫ if(0 <= \underline{x} + −\underline{z} + \underline{z} + \underline{y}, \underline{x} + −\underline{z} + \underline{z} + \underline{y}, −\underline{x} + −\underline{z} + \underline{z} + \underline{y}) <= if(0 <= \underline{x} + −\underline{z}, \underline{x} + −\underline{z}, −\underline{x} + −\underline{z}) + if(0 <= \underline{z} + \underline{y}, \underline{z} + \underline{y}, −\underline{z} + \underline{y}); insertMiddleTerm(Sum) ≫ \underline{x} + \underline{y} = \underline{x}+−\underline{z}+\underline{z}+\underline{y}; SameNumerical⊳\underline{x}+\underline{y} = \underline{x}+−\underline{z}+\underline{z}+\underline{y} ≫ if(0 <= \underline{x}+\underline{y}, \underline{x}+\underline{y}, −\underline{x}+\underline{y}) = if(0 <= \underline{x} + −\underline{z} + \underline{z} + \underline{y}, \underline{x} + −\underline{z} + \underline{z} + \underline{y}, −\underline{x} + −\underline{z} + \underline{z} + \underline{y}); eqSymmetry ⊳ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) = if(0 <= \underline{x} + −\underline{z} + \underline{z} + \underline{y}, \underline{x} + −\underline{z} + \underline{z} + \underline{y}, −\underline{x} + −\underline{z} + \underline{z} + \underline{y}) ≫ if(0 <= \underline{x}+−\underline{z}+\underline{z}+\underline{y}, \underline{x}+−\underline{z}+\underline{z}+\underline{y}, −\underline{x}+−\underline{z}+\underline{z}+\underline{y}) = if(0 <= \underline{x}+\underline{y}, \underline{x}+\underline{y}, −\underline{x}+ \underline{y}); subLeqLeft⊳if(0 <= \underline{x}+−\underline{z}+\underline{z}+\underline{y}, \underline{x}+−\underline{z}+\underline{z}+\underline{y}, −\underline{x}+−\underline{z}+\underline{z}+\underline{y}) = if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) ⊳ if(0 <= \underline{x} + −\underline{z} + \underline{z} + \underline{y}, \underline{x} + −\underline{z} + \underline{z} + \underline{y}, −\underline{x} + −\underline{z} + \underline{z} + \underline{y}) <= if(0 <= \underline{x}+−\underline{z}, \underline{x}+−\underline{z}, −\underline{x}+−\underline{z})+if(0 <= \underline{z}+\underline{y}, \underline{z}+\underline{y}, −\underline{z}+\underline{y}) ≫ if(0 <= \underline{x}+\underline{y}, \underline{x}+ \underline{y}, −\underline{x}+\underline{y}) <= if(0 <= \underline{x}+−\underline{z}, \underline{x}+−\underline{z}, −\underline{x}+−\underline{z})+if(0 <= \underline{z}+\underline{y}, \underline{z}+\underline{y}, −\underline{z}+\underline{y})⌉, p_{0}, c)] (*** REGNESTYKKER ***) [x + y = zBackwards →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} + \underline{y} = \underline{z} ⊢ \underline{z} = \underline{y} + \underline{x}]

[x + y = zBackwards →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} + \underline{y} = \underline{z} ⊢ plusCommutativity ≫ \underline{x} + \underline{y} = \underline{y} + \underline{x}; Equality ⊳ \underline{x} + \underline{y} = \underline{z} ≫ \underline{z} = \underline{y} + \underline{x}⌉, p_{0}, c)]

[x ∗ y = zBackwards →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} ∗ \underline{y} = \underline{z} ⊢ \underline{z} = \underline{y} ∗ \underline{x}]

[x ∗ y = zBackwards →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} ∗ \underline{y} = \underline{z} ⊢ timesCommutativity ≫ \underline{x} ∗ \underline{y} = \underline{y} ∗ \underline{x}; Equality ⊳ \underline{x} ∗ \underline{y} = \underline{z} ≫ \underline{z} = \underline{y} ∗ \underline{x}⌉, p_{0}, c)]

[x = x + (y − y) →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} = \underline{x} + \underline{y} + −\underline{y}]

[x = x + (y − y) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: plus0 ≫ \underline{x} + 0 = \underline{x}; Negative ≫ \underline{y} + −\underline{y} = 0; eqSymmetry ⊳ \underline{y} + −\underline{y} = 0 ≫ 0 = \underline{y} + −\underline{y}; lemma eqAdditionLeft ⊳ 0 = \underline{y} + −\underline{y} ≫ \underline{x} + 0 = \underline{x} + \underline{y} + −\underline{y}; Equality ⊳ \underline{x} + 0 = \underline{x} ⊳ \underline{x} + 0 = \underline{x} + \underline{y} + −\underline{y} ≫ \underline{x} = \underline{x} + \underline{y} + −\underline{y}⌉, p_{0}, c)]

[x = x + y − y →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} = \underline{x} + \underline{y} + −\underline{y}]

[x = x + y − y →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: x = x + (y − y) ≫ \underline{x} = \underline{x}+\underline{y}+−\underline{y}; plusAssociativity ≫ \underline{x}+\underline{y}+−\underline{y} = \underline{x}+\underline{y}+−\underline{y}; eqSymmetry⊳\underline{x}+\underline{y}+−\underline{y} = \underline{x} + \underline{y} + −\underline{y} ≫ \underline{x} + \underline{y} + −\underline{y} = \underline{x} + \underline{y} + −\underline{y}; eqTransitivity ⊳ \underline{x} = \underline{x} + \underline{y} + −\underline{y} ⊳ \underline{x} + \underline{y} + −\underline{y} = \underline{x} + \underline{y} + −\underline{y} ≫ \underline{x} = \underline{x} + \underline{y} + −\underline{y}⌉, p_{0}, c)]

[ →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ \underline{y} = 0 ⊢ \underline{x} = \underline{x} ∗ \underline{y} ∗ rec\underline{y}]

[ →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ \underline{y} = 0 ⊢ times1 ≫ \underline{x} ∗ 1 = \underline{x}; Reciprocal ⊳ ˙¬ \underline{y} = 0 ≫ \underline{y} ∗ rec\underline{y} = 1; Three2threeFactors ⊳ \underline{y} ∗ rec\underline{y} = 1 ≫ \underline{x} ∗ \underline{y} ∗ rec\underline{y} = \underline{x} ∗ 1; eqTransitivity ⊳ \underline{x} ∗ \underline{y} ∗ rec\underline{y} = \underline{x} ∗ 1 ⊳ \underline{x} ∗ 1 = \underline{x} ≫ \underline{x} ∗ \underline{y} ∗ rec\underline{y} = \underline{x}; eqSymmetry ⊳ \underline{x} ∗ \underline{y} ∗ rec\underline{y} = \underline{x} ≫ \underline{x} = \underline{x} ∗ \underline{y} ∗ rec\underline{y}⌉, p_{0}, c)]

[insertMiddleTerm(Sum) →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} + \underline{y} = \underline{x} + −\underline{z} + \underline{z} + \underline{y}]

[insertMiddleTerm(Sum) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: x = x + y − y ≫ \underline{x} = \underline{x} + \underline{z} + −\underline{z}; Three2threeTerms ≫ \underline{x} + \underline{z} + −\underline{z} = \underline{x} + −\underline{z} + \underline{z}; eqTransitivity ⊳ \underline{x} = \underline{x} + \underline{z} + −\underline{z} ⊳ \underline{x} + \underline{z} + −\underline{z} = \underline{x} + −\underline{z} + \underline{z} ≫ \underline{x} = \underline{x} + −\underline{z} + \underline{z}; eqAddition ⊳ \underline{x} = \underline{x} + −\underline{z} + \underline{z} ≫ \underline{x} + \underline{y} = \underline{x} + −\underline{z} + \underline{z} + \underline{y}; plusAssociativity ≫ \underline{x} + −\underline{z} + \underline{z} + \underline{y} = \underline{x} + −\underline{z} + \underline{z} + \underline{y}; eqTransitivity ⊳ \underline{x} + \underline{y} = \underline{x} + −\underline{z} + \underline{z} + \underline{y} ⊳ \underline{x} + −\underline{z} + \underline{z} + \underline{y} = \underline{x} + −\underline{z} + \underline{z} + \underline{y} ≫ \underline{x} + \underline{y} = \underline{x} + −\underline{z} + \underline{z} + \underline{y}⌉, p_{0}, c)]

[insertMiddleTerm(Difference) →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} + −\underline{y} = \underline{x} + \underline{z} + −\underline{y} + \underline{z}]

[insertMiddleTerm(Difference) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: insertMiddleTerm(Sum) ≫ \underline{x} + −\underline{y} = \underline{x}+−−\underline{z}+−\underline{z}+−\underline{y}; DoubleMinus ≫ −−\underline{z} = \underline{z}; lemma eqAdditionLeft⊳−−\underline{z} = \underline{z} ≫ \underline{x} + − − \underline{z} = \underline{x} + \underline{z}; plusCommutativity ≫ −\underline{z} + −\underline{y} = −\underline{y} + −\underline{z}; −x − y = −(x + y) ≫ −\underline{y} + −\underline{z} = −\underline{y} + \underline{z}; eqTransitivity ⊳ −\underline{z} + −\underline{y} = −\underline{y} + −\underline{z} ⊳ −\underline{y} + −\underline{z} = −\underline{y} + \underline{z} ≫ −\underline{z} + −\underline{y} = −\underline{y} + \underline{z}; AddEquations ⊳ \underline{x} + − − \underline{z} = \underline{x} + \underline{z} ⊳ −\underline{z} + −\underline{y} = −\underline{y}+\underline{z} ≫ \underline{x}+−−\underline{z}+−\underline{z}+−\underline{y} = \underline{x}+\underline{z}+−\underline{y}+\underline{z}; eqTransitivity⊳\underline{x}+−\underline{y} = \underline{x}+−−\underline{z}+ −\underline{z}+−\underline{y} ⊳\underline{x}+−−\underline{z}+−\underline{z}+−\underline{y} = \underline{x}+\underline{z}+−\underline{y} +\underline{z} ≫ \underline{x}+−\underline{y} = \underline{x}+\underline{z}+−\underline{y} +\underline{z}⌉, p_{0}, c)]

[x ∗ 0 + x = x →^{stmt} ZFsub ⊢ ∀\underline{x}: \underline{x} ∗ 0 + \underline{x} = \underline{x}] [x∗0+x = x →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: times1 ≫ \underline{x}∗1 = \underline{x}; eqSymmetry⊳\underline{x}∗1 = \underline{x} ≫ \underline{x} = \underline{x} ∗ 1; lemma eqAdditionLeft ⊳ \underline{x} = \underline{x} ∗ 1 ≫ \underline{x} ∗ 0 + \underline{x} = \underline{x} ∗ 0 + \underline{x} ∗ 1; Distribution ≫ \underline{x} ∗ 0 + 1 = \underline{x} ∗ 0 + \underline{x} ∗ 1; eqSymmetry ⊳ \underline{x} ∗ 0 + 1 = \underline{x} ∗ 0 + \underline{x} ∗ 1 ≫ \underline{x} ∗ 0 + \underline{x} ∗ 1 = \underline{x} ∗ 0 + 1; plus0Left ≫ 0 + 1 = 1; EqMultiplicationLeft⊳0+1 = 1 ≫ \underline{x}∗0+1 = \underline{x}∗1; eqTransitivity5⊳\underline{x}∗0+\underline{x} = \underline{x}∗0+\underline{x}∗1⊳\underline{x}∗0+\underline{x}∗1 = \underline{x}∗0+1⊳\underline{x}∗0+1 = \underline{x}∗1⊳\underline{x}∗1 = \underline{x} ≫ \underline{x}∗0+\underline{x} = \underline{x}⌉, p_{0}, c)] [x ∗ 0 = 0 →^{stmt} ZFsub ⊢ ∀\underline{x}: \underline{x} ∗ 0 = 0] [x ∗ 0 = 0 →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: x = x + (y − y) ≫ \underline{x} ∗ 0 = \underline{x} ∗ 0 + \underline{x} + −\underline{x}; plusAssociativity ≫ \underline{x} ∗ 0 + \underline{x} + −\underline{x} = \underline{x} ∗ 0 + \underline{x} + −\underline{x}; eqSymmetry ⊳ \underline{x} ∗ 0 + \underline{x} + −\underline{x} = \underline{x} ∗ 0 + \underline{x} + −\underline{x} ≫ \underline{x} ∗ 0 + \underline{x} + −\underline{x} = \underline{x}∗0+\underline{x}+−\underline{x}; x∗0+x = x ≫ \underline{x}∗0+\underline{x} = \underline{x}; eqAddition⊳\underline{x}∗0+\underline{x} = \underline{x} ≫ \underline{x}∗0+\underline{x}+−\underline{x} = \underline{x}+−\underline{x}; Negative ≫ \underline{x}+−\underline{x} = 0; eqTransitivity5⊳\underline{x}∗0 = \underline{x}∗0+\underline{x}+−\underline{x}⊳\underline{x}∗0+\underline{x}+−\underline{x} = \underline{x} ∗ 0 + \underline{x} + −\underline{x} ⊳ \underline{x} ∗ 0 + \underline{x} + −\underline{x} = \underline{x} + −\underline{x} ⊳ \underline{x} + −\underline{x} = 0 ≫ \underline{x} ∗ 0 = 0⌉, p_{0}, c)] [(−1) ∗ (−1) + (−1) ∗ 1 = 0 →^{stmt} ZFsub ⊢ −1 ∗ −1 + −1 ∗ 1 = 0] [(−1) ∗ (−1) + (−1) ∗ 1 = 0 →^{proof} λc.λx.P(⌈ZFsub ⊢ DistributionOut ≫ −1 ∗ −1 + −1 ∗ 1 = −1 ∗ −1 + 1; Negative ≫ 1 + −1 = 0; plusCommutativity ≫ −1 + 1 = 1 + −1; eqTransitivity ⊳ −1 + 1 = 1 + −1 ⊳ 1 + −1 = 0 ≫ −1 + 1 = 0; EqMultiplicationLeft ⊳ −1 + 1 = 0 ≫ −1 ∗ −1 + 1 = −1 ∗ 0; x ∗ 0 = 0 ≫ −1 ∗ 0 = 0; eqTransitivity4 ⊳ −1 ∗ −1 + −1 ∗ 1 = −1 ∗ −1 + 1 ⊳ −1 ∗ −1 + 1 = −1 ∗ 0 ⊳ −1 ∗ 0 = 0 ≫ −1 ∗ −1 + −1 ∗ 1 = 0⌉, p_{0}, c)] [(−1) ∗ (−1) = 1 →^{stmt} ZFsub ⊢ −1 ∗ −1 = 1] [(−1) ∗ (−1) = 1 →^{proof} λc.λx.P(⌈ZFsub ⊢ x = x + (y − y) ≫ −1 ∗ −1 = −1 ∗ −1 + 1 + −1; times1 ≫ −1 ∗ 1 = −1; eqSymmetry ⊳ −1 ∗ 1 = −1 ≫ −1 = −1 ∗ 1; lemma eqAdditionLeft ⊳ −1 = −1 ∗ 1 ≫ 1 + −1 = 1 + −1 ∗ 1; lemma eqAdditionLeft ⊳ 1 + −1 = 1 + −1 ∗ 1 ≫ −1 ∗ −1 + 1 + −1 = −1 ∗ −1 + 1 + −1 ∗ 1; plusCommutativity ≫ 1 + −1 ∗ 1 = −1∗1+1; lemma eqAdditionLeft⊳1+−1∗1 = −1∗1+1 ≫ −1∗−1+1+−1∗1 = −1 ∗ −1 + −1 ∗ 1 + 1; plusAssociativity ≫ −1 ∗ −1 + −1 ∗ 1 + 1 = −1∗−1+−1∗1+1; eqSymmetry⊳−1∗−1+−1∗1+1 = −1∗−1+−1∗1+1 ≫ −1 ∗ −1 + −1 ∗ 1 + 1 = −1 ∗ −1 + −1 ∗ 1 + 1; (−1) ∗ (−1) + (−1) ∗ 1 = 0 ≫ −1 ∗ −1 + −1 ∗ 1 = 0; eqAddition ⊳ −1 ∗ −1 + −1 ∗ 1 = 0 ≫ −1 ∗ −1 + −1 ∗ 1 + 1 = 0 + 1; plus0Left ≫ 0 + 1 = 1; eqTransitivity5 ⊳ −1 ∗ −1 = −1∗−1+1+−1⊳−1∗−1+1+−1 = −1∗−1+1+−1∗1⊳−1∗−1+1+−1∗1 = −1 ∗ −1 + −1 ∗ 1 + 1 ⊳ −1 ∗ −1 + −1 ∗ 1 + 1 = −1 ∗ −1 + −1 ∗ 1 + 1 ≫ −1 ∗ −1 = −1 ∗ −1 + −1 ∗ 1 + 1; eqTransitivity4 ⊳ −1 ∗ −1 = −1∗−1+−1∗1+1⊳−1∗−1+−1∗1+1 = 0+1⊳0+1 = 1 ≫ −1∗−1 = 1⌉, p_{0}, c)] [0 < 1Helper →^{stmt} ZFsub ⊢ 1 <= 0 ⇒ 0 <= 1] [0 < 1Helper →^{proof} λc.λx.P(⌈ZFsub ⊢ 1 <= 0 ⊢ leqAddition ⊳ 1 <= 0 ≫ 1+−1 <= 0+−1; Negative ≫ 1+−1 = 0; subLeqLeft⊳1+−1 = 0⊳1+−1 <= 0 + −1 ≫ 0 <= 0 + −1; plus0Left ≫ 0 + −1 = −1; subLeqRight ⊳ 0 + −1 = −1 ⊳ 0 <= 0 + −1 ≫ 0 <= −1; leqMultiplication ⊳ 0 <= −1 ⊳ 0 <= −1 ≫ 0 ∗ −1 <= −1 ∗ −1; x ∗ 0 = 0 ≫ −1 ∗ 0 = 0; timesCommutativity ≫ 0 ∗ −1 = −1 ∗ 0; eqTransitivity ⊳ 0 ∗ −1 = −1 ∗ 0 ⊳ −1 ∗ 0 = 0 ≫ 0 ∗ −1 = 0; subLeqLeft ⊳ 0 ∗ −1 = 0 ⊳ 0 ∗ −1 <= −1 ∗ −1 ≫ 0 <= −1 ∗ −1; (−1) ∗ (−1) = 1 ≫ −1 ∗ −1 = 1; subLeqRight ⊳ −1 ∗ −1 = 1 ⊳ 0 <= −1 ∗ −1 ≫ 0 <= 1; Ded ⊳ 1 <= 0 ⊢ 0 <= 1 ≫ 1 <= 0 ⇒ 0 <= 1⌉, p_{0}, c)] [0 < 1 →^{stmt} ZFsub ⊢ ˙¬ 0 <= 1 ⇒ ˙¬ ˙¬ 0 = 1] [0 < 1 →^{proof} λc.λx.P(⌈ZFsub ⊢ leqTotality ≫ ˙¬ 0 <= 1 ⇒ 1 <= 0; AutoImply ≫ 0 <= 1 ⇒ 0 <= 1; 0 < 1Helper ≫ 1 <= 0 ⇒ 0 <= 1; FromDisjuncts ⊳ ˙¬ 0 <= 1 ⇒ 1 <= 0 ⊳ 0 <= 1 ⇒ 0 <= 1 ⊳ 1 <= 0 ⇒ 0 <= 1 ≫ 0 <= 1; 0not1 ≫ ˙¬ 0 = 1; JoinConjuncts ⊳ 0 <= 1 ⊳ ˙¬ 0 = 1 ≫ ˙¬ 0 <= 1 ⇒ ˙¬ ˙¬ 0 = 1⌉, p_{0}, c)] [0 < 2 →^{stmt} ZFsub ⊢ ˙¬ 0 <= 1 + 1 ⇒ ˙¬ ˙¬ 0 = 1 + 1] [0 < 2 →^{proof} λc.λx.P(⌈ZFsub ⊢ 0 < 1 ≫ ˙¬ 0 <= 1 ⇒ ˙¬ ˙¬ 0 = 1; LessAddition ⊳ ˙¬ 0 <= 1 ⇒ ˙¬ ˙¬ 0 = 1 ≫ ˙¬ 0 + 1 <= 1 + 1 ⇒ ˙¬ ˙¬ 0 + 1 = 1 + 1; plus0Left ≫ 0 + 1 = 1; SubLessLeft ⊳ 0 + 1 = 1 ⊳ ˙¬ 0 + 1 <= 1 + 1 ⇒ ˙¬ ˙¬ 0+1 = 1+1 ≫ ˙¬ 1 <= 1+1 ⇒ ˙¬ ˙¬ 1 = 1+1; LessTransitivity⊳ ˙¬ 0 <= 1 ⇒ ˙¬ ˙¬ 0 = 1⊳ ˙¬ 1 <= 1+1 ⇒ ˙¬ ˙¬ 1 = 1+1 ≫ ˙¬ 0 <= 1+1 ⇒ ˙¬ ˙¬ 0 = 1+1⌉, p_{0}, c)] [0 < 1/2 →^{stmt} ZFsub ⊢ ˙¬ 0 <= rec1 + 1 ⇒ ˙¬ ˙¬ 0 = rec1 + 1] [0 < 1/2 →^{proof} λc.λx.P(⌈ZFsub ⊢ 0 < 2 ≫ ˙¬ 0 <= 1 + 1 ⇒ ˙¬ ˙¬ 0 = 1 + 1; FirstConjunct ⊳ ˙¬ 0 <= 1 + 1 ⇒ ˙¬ ˙¬ 0 = 1 + 1 ≫ 0 <= 1 + 1; SecondConjunct ⊳ ˙¬ 0 <= 1 + 1 ⇒ ˙¬ ˙¬ 0 = 1 + 1 ≫ ˙¬ 0 = 1 + 1; NeqSymmetry ⊳ ˙¬ 0 = 1 + 1 ≫ ˙¬ 1 + 1 = 0; 0 < 1 ≫ ˙¬ 0 <= 1 ⇒ ˙¬ ˙¬ 0 = 1; x ∗ 0 = 0 ≫ 1 + 1 ∗ 0 = 0; x ∗ y = zBackwards ⊳ 1 + 1 ∗ 0 = 0 ≫ 0 = 0 ∗ 1 + 1; SubLessLeft ⊳ 0 = 0 ∗ 1 + 1 ⊳ ˙¬ 0 <= 1 ⇒ ˙¬ ˙¬ 0 = 1 ≫ ˙¬ 0 ∗ 1 + 1 <= 1 ⇒ ˙¬ ˙¬ 0 ∗ 1 + 1 = 1; Reciprocal ⊳ ˙¬ 1 + 1 = 0 ≫ 1 + 1 ∗ rec1 + 1 = 1; x ∗ y = zBackwards ⊳ 1 + 1 ∗ rec1 + 1 = 1 ≫ 1 = rec1 + 1 ∗ 1 + 1; SubLessRight ⊳ 1 = rec1 + 1 ∗ 1 + 1 ⊳ ˙¬ 0 ∗ 1 + 1 <= 1 ⇒ ˙¬ ˙¬ 0 ∗ 1 + 1 = 1 ≫ ˙¬ 0 ∗ 1 + 1 <= rec1 + 1 ∗ 1 + 1 ⇒ ˙¬ ˙¬ 0 ∗ 1 + 1 = rec1 + 1 ∗ 1 + 1; LessDivision ⊳ 0 <= 1 + 1 ⊳ ˙¬ 0 ∗ 1 + 1 <= rec1 + 1 ∗ 1 + 1 ⇒ ˙¬ ˙¬ 0 ∗ 1 + 1 = rec1 + 1 ∗ 1 + 1 ≫ ˙¬ 0 <= rec1 + 1 ⇒ ˙¬ ˙¬ 0 = rec1 + 1⌉, p_{0}, c)] [TwoWholes →^{stmt} ZFsub ⊢ ∀\underline{x}: \underline{x} + \underline{x} = 1 + 1 ∗ \underline{x}] [TwoWholes →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: times1 ≫ \underline{x} ∗ 1 = \underline{x}; eqSymmetry ≫ \underline{x} = \underline{x} ∗ 1; lemma eqAdditionLeft ⊳ \underline{x} = \underline{x} ∗ 1 ≫ \underline{x} + \underline{x} = \underline{x} + \underline{x} ∗ 1; eqAddition ⊳ \underline{x} = \underline{x} ∗ 1 ≫ \underline{x} + \underline{x} ∗ 1 = \underline{x} ∗ 1 + \underline{x} ∗ 1; eqTransitivity ⊳ \underline{x} + \underline{x} = \underline{x} + \underline{x} ∗ 1 ⊳ \underline{x} + \underline{x} ∗ 1 = \underline{x} ∗ 1 + \underline{x} ∗ 1 ≫ \underline{x} + \underline{x} = \underline{x} ∗ 1 + \underline{x} ∗ 1; DistributionOut ≫ \underline{x} ∗ 1 + \underline{x} ∗ 1 = \underline{x} ∗ 1 + 1; Repetition ⊳ \underline{x} ∗ 1 + \underline{x} ∗ 1 = \underline{x} ∗ 1 + 1 ≫ \underline{x} ∗ 1 + \underline{x} ∗ 1 = \underline{x} ∗ 1 + 1; timesCommutativity ≫ \underline{x} ∗ 1 + 1 = 1 + 1 ∗ \underline{x}; eqTransitivity4 ⊳ \underline{x} + \underline{x} = \underline{x}∗1+\underline{x}∗1⊳\underline{x}∗1+\underline{x}∗1 = \underline{x}∗1+1⊳\underline{x}∗1+1 = 1+1∗\underline{x} ≫ \underline{x}+\underline{x} = 1+1∗\underline{x}⌉, p_{0}, c)] [TwoHalves →^{stmt} ZFsub ⊢ ∀\underline{x}: rec1 + 1 ∗ \underline{x} + rec1 + 1 ∗ \underline{x} = \underline{x}] [TwoHalves →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: 0 < 2 ≫ ˙¬ 0 <= 1 + 1 ⇒ ˙¬ ˙¬ 0 = 1 + 1; LessNeq ⊳ ˙¬ 0 <= 1 + 1 ⇒ ˙¬ ˙¬ 0 = 1 + 1 ≫ ˙¬ 0 = 1 + 1; NeqSymmetry ⊳ ˙¬ 0 = 1 + 1 ≫ ˙¬ 1 + 1 = 0; TwoWholes ≫ rec1 + 1 ∗ \underline{x} + rec1 + 1 ∗ \underline{x} = 1 + 1 ∗ rec1 + 1 ∗ \underline{x}; timesAssociativity ≫ 1 + 1 ∗ rec1 + 1 ∗ \underline{x} = 1 + 1 ∗ rec1 + 1 ∗ \underline{x}; eqSymmetry ⊳ 1 + 1 ∗ rec1 + 1 ∗ \underline{x} = 1+1∗rec1+1∗\underline{x} ≫ 1+1∗rec1+1∗\underline{x} = 1+1∗rec1+1∗\underline{x}; Reciprocal⊳ ˙¬ 1+1 = 0 ≫ 1 + 1 ∗ rec1 + 1 = 1; eqMultiplication ⊳ 1 + 1 ∗ rec1 + 1 = 1 ≫ 1 + 1 ∗ rec1 + 1 ∗ \underline{x} = 1 ∗ \underline{x}; times1Left ≫ 1 ∗ \underline{x} = \underline{x}; eqTransitivity5 ⊳ rec1 + 1 ∗ \underline{x} + rec1 + 1 ∗ \underline{x} = 1 + 1 ∗ rec1 + 1 ∗ \underline{x} ⊳ 1 + 1 ∗ rec1 + 1 ∗ \underline{x} = 1 + 1 ∗ rec1 + 1 ∗ \underline{x} ⊳ 1 + 1 ∗ rec1 + 1 ∗ \underline{x} = 1 ∗ \underline{x} ⊳ 1 ∗ \underline{x} = \underline{x} ≫ rec1 + 1 ∗ \underline{x} + rec1 + 1 ∗ \underline{x} = \underline{x}⌉, p_{0}, c)] [Times(−1) →^{stmt} ZFsub ⊢ ∀\underline{x}: \underline{x} ∗ −1 = −\underline{x}][Times(−1) →^{proof} Rule tactic] [Times(−1) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: Negative ≫ 1 + −1 = 0; plusCommutativity ≫ −1 + 1 = 1 + −1; eqTransitivity ⊳ −1 + 1 = 1 + −1 ⊳ 1 + −1 = 0 ≫ −1 + 1 = 0; EqMultiplicationLeft ⊳ −1 + 1 = 0 ≫ \underline{x} ∗ −1 + 1 = \underline{x} ∗ 0; x ∗ 0 = 0 ≫ \underline{x} ∗ 0 = 0; eqTransitivity ⊳ \underline{x} ∗ −1 + 1 = \underline{x} ∗ 0 ⊳ \underline{x} ∗ 0 = 0 ≫ \underline{x} ∗ −1 + 1 = 0; Distribution ≫ \underline{x} ∗ −1 + 1 = \underline{x} ∗ −1 + \underline{x} ∗ 1; eqSymmetry ⊳ \underline{x} ∗ −1 + 1 = \underline{x} ∗ −1 + \underline{x} ∗ 1 ≫ \underline{x} ∗ −1 + \underline{x} ∗ 1 = \underline{x}∗−1+1; eqTransitivity⊳\underline{x}∗−1+\underline{x}∗1 = \underline{x}∗−1+1⊳\underline{x}∗−1+1 = 0 ≫ \underline{x}∗−1+\underline{x}∗1 = 0; PositiveToRight(Eq) ⊳ \underline{x} ∗ −1 + \underline{x} ∗ 1 = 0 ≫ \underline{x} ∗ −1 = 0 + −\underline{x} ∗ 1; plus0Left ≫ 0 + −\underline{x} ∗ 1 = −\underline{x} ∗ 1; eqTransitivity ⊳ \underline{x} ∗ −1 = 0 + −\underline{x} ∗ 1 ⊳ 0 + −\underline{x} ∗ 1 = −\underline{x} ∗ 1 ≫ \underline{x} ∗ −1 = −\underline{x} ∗ 1; times1 ≫ \underline{x} ∗ 1 = \underline{x}; EqNegated ⊳ \underline{x} ∗ 1 = \underline{x} ≫ −\underline{x} ∗ 1 = −\underline{x}; eqTransitivity ⊳ \underline{x} ∗ −1 = −\underline{x} ∗ 1 ⊳ −\underline{x} ∗ 1 = −\underline{x} ≫ \underline{x} ∗ −1 = −\underline{x}⌉, p_{0}, c)] [Times(−1)Left →^{stmt} ZFsub ⊢ ∀\underline{x}: −1 ∗ \underline{x} = −\underline{x}] [Times(−1)Left →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: Times(−1) ≫ \underline{x} ∗ −1 = −\underline{x}; timesCommutativity ≫ −1 ∗ \underline{x} = \underline{x} ∗ −1; eqTransitivity ⊳ −1 ∗ \underline{x} = \underline{x} ∗ −1 ⊳ \underline{x} ∗ −1 = −\underline{x} ≫ −1 ∗ \underline{x} = −\underline{x}⌉, p_{0}, c)] [−x − y = −(x + y) →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: −\underline{x} + −\underline{y} = −\underline{x} + \underline{y}]

[−x − y = −(x + y) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: Times(−1)Left ≫ −1 ∗ \underline{x} = −\underline{x}; Times(−1)Left ≫ −1∗\underline{y} = −\underline{y}; AddEquations⊳−1∗\underline{x} = −\underline{x}⊳−1∗\underline{y} = −\underline{y} ≫ −1∗\underline{x}+−1∗\underline{y} = −\underline{x}+−\underline{y}; eqSymmetry⊳−1∗\underline{x}+−1∗\underline{y} = −\underline{x}+−\underline{y} ≫ −\underline{x}+−\underline{y} = −1∗\underline{x}+−1∗\underline{y}; DistributionOut ≫ −1∗\underline{x}+−1∗\underline{y} = −1∗\underline{x}+\underline{y}; Times(−1)Left ≫ −1∗\underline{x}+\underline{y} = −\underline{x}+\underline{y}; eqTransitivity4⊳−\underline{x}+−\underline{y} = −1∗\underline{x}+−1∗\underline{y}⊳−1∗\underline{x}+−1∗\underline{y} = −1 ∗ \underline{x} + \underline{y} ⊳ −1 ∗ \underline{x} + \underline{y} = −\underline{x} + \underline{y} ≫ −\underline{x} + −\underline{y} = −\underline{x} + \underline{y}⌉, p_{0}, c)]

[MinusNegated →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: −\underline{x} + −\underline{y} = \underline{y} + −\underline{x}]

[MinusNegated →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: DoubleMinus ≫ − − \underline{y} = \underline{y}; eqAddition ⊳ − − \underline{y} = \underline{y} ≫ − − \underline{y} + −\underline{x} = \underline{y} + −\underline{x}; eqSymmetry ⊳ − − \underline{y} + −\underline{x} = \underline{y} + −\underline{x} ≫ \underline{y} + −\underline{x} = − − \underline{y} + −\underline{x}; −x − y = −(x + y) ≫ − − \underline{y} + −\underline{x} = − − \underline{y} + \underline{x}; plusCommutativity ≫ −\underline{y} + \underline{x} = \underline{x} + −\underline{y}; EqNegated ⊳ −\underline{y} + \underline{x} = \underline{x} + −\underline{y} ≫ − − \underline{y} + \underline{x} = −\underline{x} + −\underline{y}; eqTransitivity4 ⊳ \underline{y} + −\underline{x} = − − \underline{y} + −\underline{x} ⊳ − − \underline{y} + −\underline{x} = − − \underline{y} + \underline{x} ⊳ − − \underline{y} + \underline{x} = −\underline{x} + −\underline{y} ≫ \underline{y} + −\underline{x} = −\underline{x} + −\underline{y}; eqSymmetry ⊳ \underline{y} + −\underline{x} = −\underline{x} + −\underline{y} ≫ −\underline{x} + −\underline{y} = \underline{y} + −\underline{x}⌉, p_{0}, c)]

[−0 = 0 →^{stmt} ZFsub ⊢ −0 = 0] [−0 = 0 →^{proof} λc.λx.P(⌈ZFsub ⊢ Negative ≫ 0 + −0 = 0; plus0 ≫ 0 + 0 = 0; UniqueNegative ⊳ 0 + −0 = 0 ⊳ 0 + 0 = 0 ≫ −0 = 0⌉, p_{0}, c)] (*** LEQ, nummer 2 af 2 ***) [negativeToLeft(Leq) →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} <= \underline{y} + −\underline{z} ⊢ \underline{x} + \underline{z} <= \underline{y}]

[negativeToLeft(Leq) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} <= \underline{y} + −\underline{z} ⊢ leqAddition ⊳ \underline{x} <= \underline{y} + −\underline{z} ≫ \underline{x} + \underline{z} <= \underline{y} + −\underline{z} + \underline{z}; x = x + y − y ≫ \underline{y} = \underline{y} + \underline{z} + −\underline{z}; Three2threeTerms ≫ \underline{y} + \underline{z} + −\underline{z} = \underline{y} + −\underline{z} + \underline{z}; eqTransitivity ⊳ \underline{y} = \underline{y} + \underline{z} + −\underline{z} ⊳ \underline{y} + \underline{z} + −\underline{z} = \underline{y} + −\underline{z} + \underline{z} ≫ \underline{y} = \underline{y} + −\underline{z} + \underline{z}; eqSymmetry ⊳ \underline{y} = \underline{y} + −\underline{z} + \underline{z} ≫ \underline{y} + −\underline{z} + \underline{z} = \underline{y}; subLeqRight ⊳ \underline{y} + −\underline{z} + \underline{z} = \underline{y} ⊳ \underline{x} + \underline{z} <= \underline{y} + −\underline{z} + \underline{z} ≫ \underline{x} + \underline{z} <= \underline{y}⌉, p_{0}, c)] (*** SAME-F ***) [SFsymmetry →^{stmt} ZFsub ⊢ ∀\underline{(ϵ)}: ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: SF(\underline{(fx)}, \underline{(fy)}) ⊢ SF(\underline{(fy)}, \underline{(fx)})]

[SFsymmetry →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{(ϵ)}: ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: SF(\underline{(fx)}, \underline{(fy)}) ⊢ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⊢ c_{Ex} <= \underline{m} ⊢ FromSF ⊳ SF(\underline{(fx)}, \underline{(fy)}) ⊳ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ≫ c_{Ex} <= \underline{m} ⇒ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) = \underline{(ϵ)}; MP ⊳ c_{Ex} <= \underline{m} ⇒ ˙¬ if(0 <= \underline{(fx)}[\underline{m}]+−\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}]+−\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}]+−\underline{(fy)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) = \underline{(ϵ)} ⊳ c_{Ex} <= \underline{m} ≫ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) = \underline{(ϵ)}; NumericalDifference ≫ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) = if(0 <= \underline{(fy)}[\underline{m}]+−\underline{(fx)}[\underline{m}], \underline{(fy)}[\underline{m}]+−\underline{(fx)}[\underline{m}], −\underline{(fy)}[\underline{m}]+−\underline{(fx)}[\underline{m}]); SubLessLeft⊳if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) = if(0 <= \underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}], \underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}], −\underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}]) ⊳ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) = \underline{(ϵ)} ≫ ˙¬ if(0 <= \underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}], \underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}], −\underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}], \underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}], −\underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}]) = \underline{(ϵ)}; ∀\underline{(ϵ)}: ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: Ded ⊳ ∀\underline{(ϵ)}: ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: SF(\underline{(fx)}, \underline{(fy)}) ⊢ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⊢ c_{Ex} <= \underline{m} ⊢ ˙¬ if(0 <= \underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}], \underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}], −\underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}], \underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}], −\underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}]) = \underline{(ϵ)} ≫ SF(\underline{(fx)}, \underline{(fy)}) ⇒ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ c_{Ex} <= \underline{m} ⇒ ˙¬ if(0 <= \underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}], \underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}], −\underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}], \underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}], −\underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}]) = \underline{(ϵ)}; SF(\underline{(fx)}, \underline{(fy)}) ⊢ MP ⊳ SF(\underline{(fx)}, \underline{(fy)}) ⇒ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ c_{Ex} <= \underline{m} ⇒ ˙¬ if(0 <= \underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}], \underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}], −\underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}], \underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}], −\underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}]) = \underline{(ϵ)} ⊳ SF(\underline{(fx)}, \underline{(fy)}) ≫ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ c_{Ex} <= \underline{m} ⇒ ˙¬ if(0 <= \underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}], \underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}], −\underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}], \underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}], −\underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}]) = \underline{(ϵ)}; ToSF ⊳ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ c_{Ex} <= \underline{m} ⇒ ˙¬ if(0 <= \underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}], \underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}], −\underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}], \underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}], −\underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}]) = \underline{(ϵ)} ≫ SF(\underline{(fy)}, \underline{(fx)})⌉, p_{0}, c)]

[PlusF(Sym) →^{stmt} ZFsub ⊢ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: \underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}] = \underline{(fx)} +_{f} \underline{(fy)}[\underline{m}]]

[PlusF(Sym) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: PlusF ≫ \underline{(fx)} +_{f} \underline{(fy)}[\underline{m}] = \underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}]; eqSymmetry ⊳ \underline{(fx)} +_{f} \underline{(fy)}[\underline{m}] = \underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}] ≫ \underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}] = \underline{(fx)} +_{f} \underline{(fy)}[\underline{m}]⌉, p_{0}, c)]

[TimesF(Sym) →^{stmt} ZFsub ⊢ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: \underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}] = \underline{(fx)} ∗_{f} \underline{(fy)}[\underline{m}]]

[TimesF(Sym) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: TimesF ≫ \underline{(fx)} ∗_{f} \underline{(fy)}[\underline{m}] = \underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}]; eqSymmetry ⊳ \underline{(fx)} ∗_{f} \underline{(fy)}[\underline{m}] = \underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}] ≫ \underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}] = \underline{(fx)} ∗_{f} \underline{(fy)}[\underline{m}]⌉, p_{0}, c)]

[Plus0f →^{stmt} ZFsub ⊢ ∀\underline{m}: ∀\underline{(fx)}: \underline{(fx)} +_{f} 0f =_{f} \underline{(fx)}]

[Plus0f →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{m}: ∀\underline{(fx)}: PlusF ≫ \underline{(fx)} +_{f} 0f[\underline{m}] = \underline{(fx)}[\underline{m}] + 0f[\underline{m}]; 0f ≫ 0f[\underline{m}] = 0; lemma eqAdditionLeft ⊳ 0f[\underline{m}] = 0 ≫ \underline{(fx)}[\underline{m}] + 0f[\underline{m}] = \underline{(fx)}[\underline{m}] + 0; plus0 ≫ \underline{(fx)}[\underline{m}] + 0 = \underline{(fx)}[\underline{m}]; eqTransitivity4 ⊳ \underline{(fx)} +_{f} 0f[\underline{m}] = \underline{(fx)}[\underline{m}] + 0f[\underline{m}] ⊳ \underline{(fx)}[\underline{m}] + 0f[\underline{m}] = \underline{(fx)}[\underline{m}] + 0 ⊳ \underline{(fx)}[\underline{m}] + 0 = \underline{(fx)}[\underline{m}] ≫ \underline{(fx)}+_{f}0f[\underline{m}] = \underline{(fx)}[\underline{m}]; To = f⊳\underline{(fx)}+_{f}0f[\underline{m}] = \underline{(fx)}[\underline{m}] ≫ \underline{(fx)}+_{f}0f =_{f} \underline{(fx)}⌉, p_{0}, c)]

[SFtransitivity →^{stmt} ZFsub ⊢ ∀\underline{m}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: SF(\underline{(fx)}, \underline{(fy)}) ⊢ SF(\underline{(fy)}, \underline{(fz)}) ⊢ SF(\underline{(fx)}, \underline{(fz)})]

[SFtransitivity →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{m}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: SF(\underline{(fx)}, \underline{(fy)}) ⊢ SF(\underline{(fy)}, \underline{(fz)}) ⊢ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⊢ c_{Ex} <= \underline{m} ⊢ PositiveHalved ⊳ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ≫ ˙¬ 0 <= rec1 + 1 ∗ \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = rec1 + 1 ∗ \underline{(ϵ)}; FromSF ⊳ SF(\underline{(fx)}, \underline{(fy)}) ⊳ ˙¬ 0 <= rec1 + 1 ∗ \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = rec1 + 1 ∗ \underline{(ϵ)} ≫ c_{Ex} <= \underline{m} ⇒ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) <= rec1 + 1 ∗ \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) = rec1 + 1 ∗ \underline{(ϵ)}; MP ⊳ c_{Ex} <= \underline{m} ⇒ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) <= rec1 + 1 ∗ \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) = rec1 + 1 ∗ \underline{(ϵ)} ⊳ c_{Ex} <= \underline{m} ≫ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) <= rec1 + 1 ∗ \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) = rec1 + 1 ∗ \underline{(ϵ)}; FromSF ⊳ SF(\underline{(fy)}, \underline{(fz)}) ⊳ ˙¬ 0 <= rec1 + 1 ∗ \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = rec1 + 1 ∗ \underline{(ϵ)} ≫ c_{Ex} <= \underline{m} ⇒ ˙¬ if(0 <= \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], −\underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}]) <= rec1 + 1 ∗ \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], −\underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}]) = rec1 + 1 ∗ \underline{(ϵ)}; MP ⊳ c_{Ex} <= \underline{m} ⇒ ˙¬ if(0 <= \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], −\underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}]) <= rec1 + 1 ∗ \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], −\underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}]) = rec1 + 1 ∗ \underline{(ϵ)} ⊳ c_{Ex} <= \underline{m} ≫ ˙¬ if(0 <= \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], −\underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}]) <= rec1 + 1 ∗ \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], −\underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}]) = rec1 + 1 ∗ \underline{(ϵ)}; AddEquations(Less) ⊳ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) <= rec1 + 1 ∗ \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) = rec1 + 1 ∗ \underline{(ϵ)} ⊳ ˙¬ if(0 <= \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], −\underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}]) <= rec1 + 1 ∗ \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], −\underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}]) = rec1 + 1 ∗ \underline{(ϵ)} ≫ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) + if(0 <= \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], −\underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}]) <= rec1 + 1 ∗ \underline{(ϵ)} + rec1 + 1 ∗ \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) + if(0 <= \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], −\underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}]) = rec1 + 1 ∗ \underline{(ϵ)} + rec1 + 1 ∗ \underline{(ϵ)}; TwoHalves ≫ rec1 + 1 ∗ \underline{(ϵ)} + rec1 + 1 ∗ \underline{(ϵ)} = \underline{(ϵ)}; SubLessRight ⊳ rec1 + 1 ∗ \underline{(ϵ)} + rec1 + 1 ∗ \underline{(ϵ)} = \underline{(ϵ)} ⊳ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) + if(0 <= \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], −\underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}]) <= rec1 + 1 ∗ \underline{(ϵ)} + rec1 + 1 ∗ \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) + if(0 <= \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], \underline{(fy)}[\underline{m}]+−\underline{(fz)}[\underline{m}], −\underline{(fy)}[\underline{m}]+−\underline{(fz)}[\underline{m}]) = rec1+1∗\underline{(ϵ)}+rec1+1∗\underline{(ϵ)} ≫ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) + if(0 <= \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], −\underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}]+−\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}]+−\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}]+−\underline{(fy)}[\underline{m}])+if(0 <= \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], −\underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}]) = \underline{(ϵ)}; insertMiddleTerm(Numerical) ≫ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}]) <= if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) + if(0 <= \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], −\underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}]); leqLessTransitivity ⊳ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}]) <= if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) + if(0 <= \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], −\underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}]) ⊳ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) + if(0 <= \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], −\underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}]+−\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}]+−\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}]+−\underline{(fy)}[\underline{m}])+if(0 <= \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], −\underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}]) = \underline{(ϵ)} ≫ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}]) = \underline{(ϵ)}; ∀\underline{m}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: Ded ⊳ ∀\underline{m}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: SF(\underline{(fx)}, \underline{(fy)}) ⊢ SF(\underline{(fy)}, \underline{(fz)}) ⊢ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⊢ c_{Ex} <= \underline{m} ⊢ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}]) = \underline{(ϵ)} ≫ SF(\underline{(fx)}, \underline{(fy)}) ⇒ SF(\underline{(fy)}, \underline{(fz)}) ⇒ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ c_{Ex} <= \underline{m} ⇒ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}]) = \underline{(ϵ)}; SF(\underline{(fx)}, \underline{(fy)}) ⊢ SF(\underline{(fy)}, \underline{(fz)}) ⊢ MP2 ⊳ SF(\underline{(fx)}, \underline{(fy)}) ⇒ SF(\underline{(fy)}, \underline{(fz)}) ⇒ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ c_{Ex} <= \underline{m} ⇒ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}]) = \underline{(ϵ)} ⊳ SF(\underline{(fx)}, \underline{(fy)}) ⊳ SF(\underline{(fy)}, \underline{(fz)}) ≫ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ c_{Ex} <= \underline{m} ⇒ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}]) = \underline{(ϵ)}; ToSF ⊳ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ c_{Ex} <= \underline{m} ⇒ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}]) = \underline{(ϵ)} ≫ SF(\underline{(fx)}, \underline{(fz)})⌉, p_{0}, c)]

[= fToSameF →^{stmt} ZFsub ⊢ ∀\underline{(ϵ)}: ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: \underline{(fx)} =_{f} \underline{(fy)} ⊢ SF(\underline{(fx)}, \underline{(fy)})]

[= fToSameF →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{(ϵ)}: ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: \underline{(fx)} =_{f} \underline{(fy)} ⊢ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⊢ From = f ⊳ \underline{(fx)} =_{f} \underline{(fy)} ≫ \underline{(fx)}[\underline{m}] = \underline{(fy)}[\underline{m}]; PositiveToLeft(Eq)(1term) ⊳ \underline{(fx)}[\underline{m}] = \underline{(fy)}[\underline{m}] ≫ \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}] = 0; SameNumerical ⊳ \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}] = 0 ≫ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) = if(0 <= 0, 0, −0); |0| = 0 ≫ if(0 <= 0, 0, −0) = 0; eqTransitivity ⊳ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) = if(0 <= 0, 0, −0) ⊳ if(0 <= 0, 0, −0) = 0 ≫ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) = 0; eqSymmetry ⊳ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) = 0 ≫ 0 = if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]); SubLessLeft ⊳ 0 = if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) ⊳ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ≫ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) = \underline{(ϵ)}; Weakening ⊳ ˙¬ if(0 <= \underline{(fx)}[\underline{m}]+−\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}]+−\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}]+−\underline{(fy)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) = \underline{(ϵ)} ≫ 0 <= \underline{m} ⇒ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) = \underline{(ϵ)}; ExistIntro @ c_{Ex} @ 0 ⊳ 0 <= \underline{m} ⇒ ˙¬ if(0 <= \underline{(fx)}[\underline{m}]+−\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}]+−\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}]+−\underline{(fy)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) = \underline{(ϵ)} ≫ c_{Ex} <= \underline{m} ⇒ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) = \underline{(ϵ)}; ∀\underline{(ϵ)}: ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: Ded ⊳ ∀\underline{(ϵ)}: ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: \underline{(fx)} =_{f} \underline{(fy)} ⊢ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⊢ c_{Ex} <= \underline{m} ⇒ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) = \underline{(ϵ)} ≫ \underline{(fx)} =_{f} \underline{(fy)} ⇒ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ c_{Ex} <= \underline{m} ⇒ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) = \underline{(ϵ)}; \underline{(fx)} =_{f} \underline{(fy)} ⊢ MP ⊳ \underline{(fx)} =_{f} \underline{(fy)} ⇒ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ c_{Ex} <= \underline{m} ⇒ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) = \underline{(ϵ)} ⊳ \underline{(fx)} =_{f} \underline{(fy)} ≫ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ c_{Ex} <= \underline{m} ⇒ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) = \underline{(ϵ)}; ToSF ⊳ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ c_{Ex} <= \underline{m} ⇒ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) = \underline{(ϵ)} ≫ SF(\underline{(fx)}, \underline{(fy)})⌉, p_{0}, c)]

[f2R(Plus) →^{stmt} ZFsub ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: ∀\underline{(fv)}: ∀\underline{var fv}: SF(\underline{(fx)} +_{f} \underline{(fy)}, \underline{(fz)}) ⊢ R((fx) +_{f} (fy)) == R(\underline{(fz)})]

[f2R(Plus) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: ∀\underline{(fv)}: ∀\underline{var fv}: SF(\underline{(fx)} +_{f} \underline{(fy)}, \underline{(fz)}) ⊢ \underline{(fv)} ∈ R(\underline{(fx)} +_{f} \underline{(fy)}) ⊢ \underline{var fv} ∈ R(\underline{(fz)}) ⊢ FromInR ⊳ \underline{(fv)} ∈ R(\underline{(fx)} +_{f} \underline{(fy)}) ≫ SF(\underline{(fv)}, \underline{(fx)} +_{f} \underline{(fy)}); FromInR ⊳ \underline{var fv} ∈ R(\underline{(fz)}) ≫ SF(\underline{var fv}, \underline{(fz)}); SFsymmetry ⊳ SF(\underline{var fv}, \underline{(fz)}) ≫ SF(\underline{(fz)}, \underline{var fv}); SFtransitivity ⊳ SF(\underline{(fv)}, \underline{(fx)} +_{f} \underline{(fy)}) ⊳ SF(\underline{(fx)} +_{f} \underline{(fy)}, \underline{(fz)}) ≫ SF(\underline{(fv)}, \underline{(fz)}); SFtransitivity ⊳ SF(\underline{(fv)}, \underline{(fz)}) ⊳ SF(\underline{(fz)}, \underline{var fv}) ≫ SF(\underline{(fv)}, \underline{var fv}); ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: ∀\underline{(fv)}: ∀\underline{var fv}: Ded ⊳ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: ∀\underline{(fv)}: ∀\underline{var fv}: SF(\underline{(fx)} +_{f} \underline{(fy)}, \underline{(fz)}) ⊢ \underline{(fv)} ∈ R(\underline{(fx)} +_{f} \underline{(fy)}) ⊢ \underline{var fv} ∈ R(\underline{(fz)}) ⊢ SF(\underline{(fv)}, \underline{var fv}) ≫ SF(\underline{(fx)} +_{f} \underline{(fy)}, \underline{(fz)}) ⇒ \underline{(fv)} ∈ R(\underline{(fx)} +_{f} \underline{(fy)}) ⇒ \underline{var fv} ∈ R(\underline{(fz)}) ⇒ SF(\underline{(fv)}, \underline{var fv}); SF(\underline{(fx)} +_{f} \underline{(fy)}, \underline{(fz)}) ⊢ MP ⊳ SF(\underline{(fx)} +_{f} \underline{(fy)}, \underline{(fz)}) ⇒ \underline{(fv)} ∈ R(\underline{(fx)} +_{f} \underline{(fy)}) ⇒ \underline{var fv} ∈ R(\underline{(fz)}) ⇒ SF(\underline{(fv)}, \underline{var fv}) ⊳ SF(\underline{(fx)} +_{f} \underline{(fy)}, \underline{(fz)}) ≫ \underline{(fv)} ∈ R(\underline{(fx)} +_{f} \underline{(fy)}) ⇒ \underline{var fv} ∈ R(\underline{(fz)}) ⇒ SF(\underline{(fv)}, \underline{var fv}); To == XX ⊳ \underline{(fv)} ∈ R(\underline{(fx)} +_{f} \underline{(fy)}) ⇒ \underline{var fv} ∈ R(\underline{(fz)}) ⇒ SF(\underline{(fv)}, \underline{var fv}) ≫ R(\underline{(fx)} +_{f} \underline{(fy)}) == R(\underline{(fz)}); PlusR ≫ R((fx) +_{f} (fy)) == R(\underline{(fx)} +_{f} \underline{(fy)}); ==Transitivity ⊳ R((fx) +_{f} (fy)) == R(\underline{(fx)}+_{f} \underline{(fy)})⊳R(\underline{(fx)}+_{f} \underline{(fy)}) == R(\underline{(fz)}) ≫ R((fx)+_{f} (fy)) == R(\underline{(fz)})⌉, p_{0}, c)]

[f2R(Times) →^{stmt} ZFsub ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: SF(\underline{(fx)} ∗_{f} \underline{(fy)}, \underline{(fz)}) ⊢ R(\underline{(fx)}) ∗ ∗R(\underline{(fy)}) == R(\underline{(fz)})]

[f2R(Times) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: SF(\underline{(fx)} ∗_{f} \underline{(fy)}, \underline{(fz)}) ⊢ To == ⊳SF(\underline{(fx)} ∗_{f} \underline{(fy)}, \underline{(fz)}) ≫ R(\underline{(fx)} ∗_{f} \underline{(fy)}) == R(\underline{(fz)}); TimesR ≫ R(\underline{(fx)}) ∗ ∗R(\underline{(fy)}) == R(\underline{(fx)} ∗_{f} \underline{(fy)}); ==Transitivity ⊳ R(\underline{(fx)}) ∗ ∗R(\underline{(fy)}) == R(\underline{(fx)} ∗_{f} \underline{(fy)}) ⊳ R(\underline{(fx)} ∗_{f} \underline{(fy)}) == R(\underline{(fz)}) ≫ R(\underline{(fx)}) ∗ ∗R(\underline{(fy)}) == R(\underline{(fz)})⌉, p_{0}, c)] (*** R-AFDELINGEN ***)

[PlusR(Sym) →^{stmt} ZFsub ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: R(\underline{(fx)} +_{f} \underline{(fy)}) == R((fx) +_{f} (fy))]

[PlusR(Sym) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: PlusR ≫ R((fx) +_{f} (fy)) == R(\underline{(fx)} +_{f} \underline{(fy)}); ==Symmetry ⊳ R((fx) +_{f} (fy)) == R(\underline{(fx)} +_{f} \underline{(fy)}) ≫ R(\underline{(fx)} +_{f} \underline{(fy)}) == R((fx) +_{f} (fy))⌉, p_{0}, c)]

[TimesR(Sym) →^{stmt} ZFsub ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: R(\underline{(fx)} ∗_{f} \underline{(fy)}) == R(\underline{(fx)}) ∗ ∗R(\underline{(fy)})]

[TimesR(Sym) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: TimesR ≫ R(\underline{(fx)}) ∗ ∗R(\underline{(fy)}) == R(\underline{(fx)} ∗_{f} \underline{(fy)}); ==Symmetry ⊳ R(\underline{(fx)}) ∗ ∗R(\underline{(fy)}) == R(\underline{(fx)} ∗_{f} \underline{(fy)}) ≫ R(\underline{(fx)} ∗_{f} \underline{(fy)}) == R(\underline{(fx)}) ∗ ∗R(\underline{(fy)})⌉, p_{0}, c)] (*** LEQ-R ***) [eqLeq(R) →^{stmt} ZFsub ⊢ ∀\underline{(rx)}: ∀\underline{(ry)}: \underline{(rx)} == \underline{(ry)} ⊢ ˙¬ \underline{(rx)} << \underline{(ry)} ⇒ \underline{(rx)} == \underline{(ry)}]

[eqLeq(R) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{(rx)}: ∀\underline{(ry)}: \underline{(rx)} == \underline{(ry)} ⊢ WeakenOr1 ⊳ \underline{(rx)} == \underline{(ry)} ≫ ˙¬ \underline{(rx)} << \underline{(ry)} ⇒ \underline{(rx)} == \underline{(ry)}; Repetition ⊳ ˙¬ \underline{(rx)} << \underline{(ry)} ⇒ \underline{(rx)} == \underline{(ry)} ≫ ˙¬ \underline{(rx)} << \underline{(ry)} ⇒ \underline{(rx)} == \underline{(ry)}⌉, p_{0}, c)]

[ThirdGeqSeries →^{stmt} ZFsub ⊢ ∀\underline{m}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: ∀\underline{(fv)}: ∀\underline{(rx)}: ∀\underline{(ry)}: ∀\underline{(rz)}: ∀\underline{(ru)}: \underline{(rx)} << \underline{(ry)} ⊢ \underline{(rz)} << \underline{(ru)} ⊢ \underline{(fx)} ∈ \underline{(rx)} ⊢ \underline{(fy)} ∈ \underline{(ry)} ⊢ \underline{(fz)} ∈ \underline{(rz)} ⊢ \underline{(fv)} ∈ \underline{(ru)} ⊢ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⊢ c_{Ex} <= \underline{m} ⊢ ˙¬ \underline{(fx)}[\underline{m}] <= \underline{(fy)}[\underline{m}] + −\underline{(ϵ)} ⇒ ˙¬ \underline{(fz)}[\underline{m}] <= \underline{(fv)}[\underline{m}] + −\underline{(ϵ)}]

[ThirdGeqSeries →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{m}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: ∀\underline{(fv)}: ∀\underline{(rx)}: ∀\underline{(ry)}: ∀\underline{(rz)}: ∀\underline{(ru)}: \underline{(rx)} << \underline{(ry)} ⊢ \underline{(rz)} << \underline{(ru)} ⊢ \underline{(fx)} ∈ \underline{(rx)} ⊢ \underline{(fy)} ∈ \underline{(ry)} ⊢ \underline{(fz)} ∈ \underline{(rz)} ⊢ \underline{(fv)} ∈ \underline{(ru)} ⊢ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⊢ c_{Ex} <= \underline{m} ⊢ From << XX ⊳ \underline{(rx)} << \underline{(ry)} ⊳ \underline{(fx)} ∈ \underline{(rx)} ⊳ \underline{(fy)} ∈ \underline{(ry)} ⊳ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ≫ a_{Ex} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= \underline{(fy)}[\underline{m}] + −\underline{(ϵ)}; From << XX ⊳ \underline{(rz)} << \underline{(ru)} ⊳ \underline{(fz)} ∈ \underline{(rz)} ⊳ \underline{(fv)} ∈ \underline{(ru)} ⊳ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ≫ a_{Ex} <= \underline{m} ⇒ \underline{(fz)}[\underline{m}] <= \underline{(fv)}[\underline{m}] + −\underline{(ϵ)}; ExistIntro @ b_{Ex} @ a_{Ex} ⊳ a_{Ex} <= \underline{m} ⇒ \underline{(fz)}[\underline{m}] <= \underline{(fv)}[\underline{m}] + −\underline{(ϵ)} ≫ b_{Ex} <= \underline{m} ⇒ \underline{(fz)}[\underline{m}] <= \underline{(fv)}[\underline{m}] + −\underline{(ϵ)}; thirdGeq ≫ ˙¬ a_{Ex} <= c_{Ex} ⇒ ˙¬ b_{Ex} <= c_{Ex}; FirstConjunct ⊳ ˙¬ a_{Ex} <= c_{Ex} ⇒ ˙¬ b_{Ex} <= c_{Ex} ≫ a_{Ex} <= c_{Ex}; leqTransitivity ⊳ a_{Ex} <= c_{Ex} ⊳ c_{Ex} <= \underline{m} ≫ a_{Ex} <= \underline{m}; MP ⊳ a_{Ex} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= \underline{(fy)}[\underline{m}] + −\underline{(ϵ)} ⊳ a_{Ex} <= \underline{m} ≫ \underline{(fx)}[\underline{m}] <= \underline{(fy)}[\underline{m}] + −\underline{(ϵ)}; SecondConjunct ⊳ ˙¬ a_{Ex} <= c_{Ex} ⇒ ˙¬ b_{Ex} <= c_{Ex} ≫ b_{Ex} <= c_{Ex}; leqTransitivity ⊳ b_{Ex} <= c_{Ex} ⊳ c_{Ex} <= \underline{m} ≫ b_{Ex} <= \underline{m}; MP ⊳ b_{Ex} <= \underline{m} ⇒ \underline{(fz)}[\underline{m}] <= \underline{(fv)}[\underline{m}] + −\underline{(ϵ)} ⊳ b_{Ex} <= \underline{m} ≫ \underline{(fz)}[\underline{m}] <= \underline{(fv)}[\underline{m}] + −\underline{(ϵ)}; JoinConjuncts ⊳ \underline{(fx)}[\underline{m}] <= \underline{(fy)}[\underline{m}] + −\underline{(ϵ)} ⊳ \underline{(fz)}[\underline{m}] <= \underline{(fv)}[\underline{m}] + −\underline{(ϵ)} ≫ ˙¬ \underline{(fx)}[\underline{m}] <= \underline{(fy)}[\underline{m}] + −\underline{(ϵ)} ⇒ ˙¬ \underline{(fz)}[\underline{m}] <= \underline{(fv)}[\underline{m}] + −\underline{(ϵ)}⌉, p_{0}, c)] (*** LESS-R ***) XX lidt grimt med de ekstra variable [SubLessLeft(R) →^{stmt} ZFsub ⊢ ∀\underline{(ϵ)}: ∀\underline{m}: ∀\underline{(fy)}: ∀\underline{(fz)}: ∀\underline{(rx)}: ∀\underline{(ry)}: ∀\underline{(rz)}: \underline{(rx)} == \underline{(ry)} ⊢ \underline{(rx)} << \underline{(rz)} ⊢ \underline{(ry)} << \underline{(rz)}]

[SubLessLeft(R) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{(ϵ)}: ∀\underline{m}: ∀\underline{(fy)}: ∀\underline{(fz)}: ∀\underline{(rx)}: ∀\underline{(ry)}: ∀\underline{(rz)}: \underline{(rx)} == \underline{(ry)} ⊢ \underline{(rx)} << \underline{(rz)} ⊢ \underline{(fy)} ∈ \underline{(ry)} ⊢ \underline{(fz)} ∈ \underline{(rz)} ⊢ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⊢ SENC2 ⊳ \underline{(rx)} == \underline{(ry)} ⊳ \underline{(fy)} ∈ \underline{(ry)} ≫ \underline{(fy)} ∈ \underline{(rx)}; From << XX ⊳ \underline{(rx)} << \underline{(rz)} ⊳ \underline{(fy)} ∈ \underline{(rx)} ⊳ \underline{(fz)} ∈ \underline{(rz)} ⊳ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ≫ a_{Ex} <= \underline{m} ⇒ \underline{(fy)}[\underline{m}] <= \underline{(fz)}[\underline{m}] + −\underline{(ϵ)}; ∀\underline{(ϵ)}: ∀\underline{m}: ∀\underline{(fy)}: ∀\underline{(fz)}: ∀\underline{(rx)}: ∀\underline{(ry)}: ∀\underline{(rz)}: Ded ⊳ ∀\underline{(ϵ)}: ∀\underline{m}: ∀\underline{(fy)}: ∀\underline{(fz)}: ∀\underline{(rx)}: ∀\underline{(ry)}: ∀\underline{(rz)}: \underline{(rx)} == \underline{(ry)} ⊢ \underline{(rx)} << \underline{(rz)} ⊢ \underline{(fy)} ∈ \underline{(ry)} ⊢ \underline{(fz)} ∈ \underline{(rz)} ⊢ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⊢ a_{Ex} <= \underline{m} ⇒ \underline{(fy)}[\underline{m}] <= \underline{(fz)}[\underline{m}] + −\underline{(ϵ)} ≫ \underline{(rx)} == \underline{(ry)} ⇒ \underline{(rx)} << \underline{(rz)} ⇒ \underline{(fy)} ∈ \underline{(ry)} ⇒ \underline{(fz)} ∈ \underline{(rz)} ⇒ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ a_{Ex} <= \underline{m} ⇒ \underline{(fy)}[\underline{m}] <= \underline{(fz)}[\underline{m}] + −\underline{(ϵ)}; \underline{(rx)} == \underline{(ry)} ⊢ \underline{(rx)} << \underline{(rz)} ⊢ MP2 ⊳ \underline{(rx)} == \underline{(ry)} ⇒ \underline{(rx)} << \underline{(rz)} ⇒ \underline{(fy)} ∈ \underline{(ry)} ⇒ \underline{(fz)} ∈ \underline{(rz)} ⇒ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ a_{Ex} <= \underline{m} ⇒ \underline{(fy)}[\underline{m}] <= \underline{(fz)}[\underline{m}] + −\underline{(ϵ)} ⊳ \underline{(rx)} == \underline{(ry)} ⊳ \underline{(rx)} << \underline{(rz)} ≫ \underline{(fy)} ∈ \underline{(ry)} ⇒ \underline{(fz)} ∈ \underline{(rz)} ⇒ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ a_{Ex} <= \underline{m} ⇒ \underline{(fy)}[\underline{m}] <= \underline{(fz)}[\underline{m}] + −\underline{(ϵ)}; to << XX ⊳ \underline{(fy)} ∈ \underline{(ry)} ⇒ \underline{(fz)} ∈ \underline{(rz)} ⇒ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ a_{Ex} <= \underline{m} ⇒ \underline{(fy)}[\underline{m}] <= \underline{(fz)}[\underline{m}] + −\underline{(ϵ)} ≫ \underline{(ry)} << \underline{(rz)}⌉, p_{0}, c)]

[SubLessRight(R) →^{stmt} ZFsub ⊢ ∀\underline{(ϵ)}: ∀\underline{m}: ∀\underline{(fy)}: ∀\underline{(fz)}: ∀\underline{(rx)}: ∀\underline{(ry)}: ∀\underline{(rz)}: \underline{(rx)} == \underline{(ry)} ⊢ \underline{(rz)} << \underline{(rx)} ⊢ \underline{(rz)} << \underline{(ry)}]

[SubLessRight(R) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{(ϵ)}: ∀\underline{m}: ∀\underline{(fy)}: ∀\underline{(fz)}: ∀\underline{(rx)}: ∀\underline{(ry)}: ∀\underline{(rz)}: \underline{(rx)} == \underline{(ry)} ⊢ \underline{(rz)} << \underline{(rx)} ⊢ \underline{(fz)} ∈ \underline{(rz)} ⊢ \underline{(fy)} ∈ \underline{(ry)} ⊢ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⊢ SENC2 ⊳ \underline{(rx)} == \underline{(ry)} ≫ \underline{(fy)} ∈ \underline{(rx)}; From << XX ⊳ \underline{(rz)} << \underline{(rx)} ⊳ \underline{(fz)} ∈ \underline{(rz)} ⊳ \underline{(fy)} ∈ \underline{(rx)} ⊳ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ≫ a_{Ex} <= \underline{m} ⇒ \underline{(fz)}[\underline{m}] <= \underline{(fy)}[\underline{m}] + −\underline{(ϵ)}; ∀\underline{(ϵ)}: ∀\underline{m}: ∀\underline{(fy)}: ∀\underline{(fz)}: ∀\underline{(rx)}: ∀\underline{(ry)}: ∀\underline{(rz)}: Ded ⊳ ∀\underline{(ϵ)}: ∀\underline{m}: ∀\underline{(fy)}: ∀\underline{(fz)}: ∀\underline{(rx)}: ∀\underline{(ry)}: ∀\underline{(rz)}: \underline{(rx)} == \underline{(ry)} ⊢ \underline{(rz)} << \underline{(rx)} ⊢ \underline{(fz)} ∈ \underline{(rz)} ⊢ \underline{(fy)} ∈ \underline{(ry)} ⊢ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⊢ a_{Ex} <= \underline{m} ⇒ \underline{(fz)}[\underline{m}] <= \underline{(fy)}[\underline{m}] + −\underline{(ϵ)} ≫ \underline{(rx)} == \underline{(ry)} ⇒ \underline{(rz)} << \underline{(rx)} ⇒ \underline{(fz)} ∈ \underline{(rz)} ⇒ \underline{(fy)} ∈ \underline{(ry)} ⇒ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ a_{Ex} <= \underline{m} ⇒ \underline{(fz)}[\underline{m}] <= \underline{(fy)}[\underline{m}] + −\underline{(ϵ)}; \underline{(rx)} == \underline{(ry)} ⊢ \underline{(rz)} << \underline{(rx)} ⊢ MP2 ⊳ \underline{(rx)} == \underline{(ry)} ⇒ \underline{(rz)} << \underline{(rx)} ⇒ \underline{(fz)} ∈ \underline{(rz)} ⇒ \underline{(fy)} ∈ \underline{(ry)} ⇒ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ a_{Ex} <= \underline{m} ⇒ \underline{(fz)}[\underline{m}] <= \underline{(fy)}[\underline{m}] + −\underline{(ϵ)} ⊳ \underline{(rx)} == \underline{(ry)} ⊳ \underline{(rz)} << \underline{(rx)} ≫ \underline{(fz)} ∈ \underline{(rz)} ⇒ \underline{(fy)} ∈ \underline{(ry)} ⇒ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ a_{Ex} <= \underline{m} ⇒ \underline{(fz)}[\underline{m}] <= \underline{(fy)}[\underline{m}] + −\underline{(ϵ)}; to << XX ⊳ \underline{(fz)} ∈ \underline{(rz)} ⇒ \underline{(fy)} ∈ \underline{(ry)} ⇒ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ a_{Ex} <= \underline{m} ⇒ \underline{(fz)}[\underline{m}] <= \underline{(fy)}[\underline{m}] + −\underline{(ϵ)} ≫ \underline{(rz)} << \underline{(ry)}⌉, p_{0}, c)]

[LessLeq(R) →^{stmt} ZFsub ⊢ ∀\underline{(rx)}: ∀\underline{(ry)}: \underline{(rx)} << \underline{(ry)} ⊢ ˙¬ \underline{(rx)} << \underline{(ry)} ⇒ \underline{(rx)} == \underline{(ry)}]

[LessLeq(R) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{(rx)}: ∀\underline{(ry)}: \underline{(rx)} << \underline{(ry)} ⊢ WeakenOr2 ⊳ \underline{(rx)} << \underline{(ry)} ≫ ˙¬ \underline{(rx)} << \underline{(ry)} ⇒ \underline{(rx)} == \underline{(ry)}; Repetition ⊳ ˙¬ \underline{(rx)} << \underline{(ry)} ⇒ \underline{(rx)} == \underline{(ry)} ≫ ˙¬ \underline{(rx)} << \underline{(ry)} ⇒ \underline{(rx)} == \underline{(ry)}⌉, p_{0}, c)]

[<< Transitivity →^{stmt} ZFsub ⊢ ∀\underline{m}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fz)}: ∀\underline{(rx)}: ∀\underline{(ry)}: ∀\underline{(rz)}: \underline{(rx)} << \underline{(ry)} ⊢ \underline{(ry)} << \underline{(rz)} ⊢ \underline{(rx)} << \underline{(rz)}]

[<< Transitivity →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{m}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fz)}: ∀\underline{(rx)}: ∀\underline{(ry)}: ∀\underline{(rz)}: \underline{(rx)} << \underline{(ry)} ⊢ \underline{(ry)} << \underline{(rz)} ⊢ \underline{(fx)} ∈ \underline{(rx)} ⊢ \underline{(fz)} ∈ \underline{(rz)} ⊢ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⊢ c_{Ex} <= \underline{m} ⊢ From <<

(2) ⊳ \underline{(rx)} << \underline{(ry)} ≫ t_{Ex} ∈ \underline{(ry)}; ThirdGeqSeries ⊳ \underline{(rx)} << \underline{(ry)} ⊳ \underline{(ry)} << \underline{(rz)} ⊳ \underline{(fx)} ∈ \underline{(rx)} ⊳ t_{Ex} ∈ \underline{(ry)} ⊳ t_{Ex} ∈ \underline{(ry)} ⊳ \underline{(fz)} ∈ \underline{(rz)} ⊳ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⊳ c_{Ex} <= \underline{m} ≫ ˙¬ \underline{(fx)}[\underline{m}] <= t_{Ex}[\underline{m}] + −\underline{(ϵ)} ⇒ ˙¬ t_{Ex}[\underline{m}] <= \underline{(fz)}[\underline{m}] + −\underline{(ϵ)}; FirstConjunct ⊳ ˙¬ \underline{(fx)}[\underline{m}] <= t_{Ex}[\underline{m}] + −\underline{(ϵ)} ⇒ ˙¬ t_{Ex}[\underline{m}] <= \underline{(fz)}[\underline{m}] + −\underline{(ϵ)} ≫ \underline{(fx)}[\underline{m}] <= t_{Ex}[\underline{m}] + −\underline{(ϵ)}; SecondConjunct ⊳ ˙¬ \underline{(fx)}[\underline{m}] <= t_{Ex}[\underline{m}] + −\underline{(ϵ)} ⇒ ˙¬ t_{Ex}[\underline{m}] <= \underline{(fz)}[\underline{m}] + −\underline{(ϵ)} ≫ t_{Ex}[\underline{m}] <= \underline{(fz)}[\underline{m}] + −\underline{(ϵ)}; << TransitivityHelper(Q) ⊳ \underline{(fx)}[\underline{m}] <= t_{Ex}[\underline{m}] + −\underline{(ϵ)} ⊳ t_{Ex}[\underline{m}] <= \underline{(fz)}[\underline{m}] + −\underline{(ϵ)} ⊳ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ≫ \underline{(fx)}[\underline{m}] <= \underline{(fz)}[\underline{m}] + −\underline{(ϵ)}; ∀\underline{m}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fz)}: ∀\underline{(rx)}: ∀\underline{(ry)}: ∀\underline{(rz)}: Ded ⊳ ∀\underline{m}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fz)}: ∀\underline{(rx)}: ∀\underline{(ry)}: ∀\underline{(rz)}: \underline{(rx)} << \underline{(ry)} ⊢ \underline{(ry)} << \underline{(rz)} ⊢ \underline{(fx)} ∈ \underline{(rx)} ⊢ \underline{(fz)} ∈ \underline{(rz)} ⊢ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⊢ c_{Ex} <= \underline{m} ⊢ \underline{(fx)}[\underline{m}] <= \underline{(fz)}[\underline{m}] + −\underline{(ϵ)} ≫ \underline{(rx)} << \underline{(ry)} ⇒ \underline{(ry)} << \underline{(rz)} ⇒ \underline{(fx)} ∈ \underline{(rx)} ⇒ \underline{(fz)} ∈ \underline{(rz)} ⇒ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ c_{Ex} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= \underline{(fz)}[\underline{m}] + −\underline{(ϵ)}; \underline{(rx)} << \underline{(ry)} ⊢ \underline{(ry)} << \underline{(rz)} ⊢ MP2 ⊳ \underline{(rx)} << \underline{(ry)} ⇒ \underline{(ry)} << \underline{(rz)} ⇒ \underline{(fx)} ∈ \underline{(rx)} ⇒ \underline{(fz)} ∈ \underline{(rz)} ⇒ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ c_{Ex} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= \underline{(fz)}[\underline{m}] + −\underline{(ϵ)} ⊳ \underline{(rx)} << \underline{(ry)} ⊳ \underline{(ry)} << \underline{(rz)} ≫ \underline{(fx)} ∈ \underline{(rx)} ⇒ \underline{(fz)} ∈ \underline{(rz)} ⇒ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ c_{Ex} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= \underline{(fz)}[\underline{m}] + −\underline{(ϵ)}; ExistIntro @ a_{Ex} @ c_{Ex} ⊳ \underline{(fx)} ∈ \underline{(rx)} ⇒ \underline{(fz)} ∈ \underline{(rz)} ⇒ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ c_{Ex} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= \underline{(fz)}[\underline{m}] + −\underline{(ϵ)} ≫ \underline{(fx)} ∈ \underline{(rx)} ⇒ \underline{(fz)} ∈ \underline{(rz)} ⇒ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ a_{Ex} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= \underline{(fz)}[\underline{m}] + −\underline{(ϵ)}; to << XX ⊳ \underline{(fx)} ∈ \underline{(rx)} ⇒ \underline{(fz)} ∈ \underline{(rz)} ⇒ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ a_{Ex} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= \underline{(fz)}[\underline{m}] + −\underline{(ϵ)} ≫ \underline{(rx)} << \underline{(rz)}⌉, p_{0}, c)] (*** NUMMER 1 ***) [<<== Reflexivity →^{stmt} ZFsub ⊢ ∀\underline{(rx)}: ˙¬ \underline{(rx)} << \underline{(rx)} ⇒ \underline{(rx)} == \underline{(rx)}]

[<<== Reflexivity →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{(rx)}: ==Reflexivity ≫ \underline{(rx)} == \underline{(rx)}; eqLeq(R) ⊳ \underline{(rx)} == \underline{(rx)} ≫ ˙¬ \underline{(rx)} << \underline{(rx)} ⇒ \underline{(rx)} == \underline{(rx)}⌉, p_{0}, c)] (*** NUMMER 2 ***) [<<== AntisymmetryHelper(Q) →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ 0 <= \underline{z} ⇒ ˙¬ ˙¬ 0 = \underline{z} ⊢ \underline{x} <= \underline{y} + −\underline{z} ⊢ \underline{y} <= \underline{x} + −\underline{z} ⊢ \underline{a}]

[<<== AntisymmetryHelper(Q) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ 0 <= \underline{z} ⇒ ˙¬ ˙¬ 0 = \underline{z} ⊢ \underline{x} <= \underline{y} + −\underline{z} ⊢ \underline{y} <= \underline{x} + −\underline{z} ⊢ leqAddition ⊳ \underline{x} <= \underline{y} + −\underline{z} ≫ \underline{x} + \underline{z} <= \underline{y} + −\underline{z} + \underline{z}; plusAssociativity ≫ \underline{y} + −\underline{z} + \underline{z} = \underline{y} + −\underline{z} + \underline{z}; plusCommutativity ≫ −\underline{z} + \underline{z} = \underline{z} + −\underline{z}; lemma eqAdditionLeft ⊳ −\underline{z} + \underline{z} = \underline{z} + −\underline{z} ≫ \underline{y} + −\underline{z} + \underline{z} = \underline{y} + \underline{z} + −\underline{z}; x = x + (y − y) ≫ \underline{y} = \underline{y} + \underline{z} + −\underline{z}; eqSymmetry ⊳ \underline{y} = \underline{y} + \underline{z} + −\underline{z} ≫ \underline{y} + \underline{z} + −\underline{z} = \underline{y}; eqTransitivity4⊳\underline{y}+−\underline{z}+\underline{z} = \underline{y}+−\underline{z}+\underline{z}⊳\underline{y}+−\underline{z}+\underline{z} = \underline{y}+\underline{z}+−\underline{z}⊳\underline{y}+\underline{z}+−\underline{z} = \underline{y} ≫ \underline{y} + −\underline{z} + \underline{z} = \underline{y}; subLeqRight ⊳ \underline{y} + −\underline{z} + \underline{z} = \underline{y} ⊳ \underline{x} + \underline{z} <= \underline{y} + −\underline{z} + \underline{z} ≫ \underline{x} + \underline{z} <= \underline{y}; leqTransitivity ⊳ \underline{x} + \underline{z} <= \underline{y} ⊳ \underline{y} <= \underline{x} + −\underline{z} ≫ \underline{x} + \underline{z} <= \underline{x} + −\underline{z}; leqSubtractionLeft ⊳ \underline{x} + \underline{z} <= \underline{x} + −\underline{z} ≫ \underline{z} <= −\underline{z}; toNotLess ⊳ \underline{z} <= −\underline{z} ≫ ˙¬ ˙¬ − \underline{z} <= \underline{z} ⇒ ˙¬ ˙¬ − \underline{z} = \underline{z}; NegativeLessPositive ⊳ ˙¬ 0 <= \underline{z} ⇒ ˙¬ ˙¬ 0 = \underline{z} ≫ ˙¬ − \underline{z} <= \underline{z} ⇒ ˙¬ ˙¬ − \underline{z} = \underline{z}; FromContradiction ⊳ ˙¬ − \underline{z} <= \underline{z} ⇒ ˙¬ ˙¬ − \underline{z} = \underline{z} ⊳ ˙¬ ˙¬ − \underline{z} <= \underline{z} ⇒ ˙¬ ˙¬ − \underline{z} = \underline{z} ≫ \underline{a}⌉, p_{0}, c)] [<<== Antisymmetry →^{stmt} ZFsub ⊢ ∀\underline{(rx)}: ∀\underline{(ry)}: ˙¬ \underline{(rx)} << \underline{(ry)} ⇒ \underline{(rx)} == \underline{(ry)} ⊢ ˙¬ \underline{(ry)} << \underline{(rx)} ⇒ \underline{(ry)} == \underline{(rx)} ⊢ \underline{(rx)} == \underline{(ry)}] [<<== Antisymmetry →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{(rx)}: ∀\underline{(ry)}: ˙¬ \underline{(rx)} << \underline{(ry)} ⇒ \underline{(rx)} == \underline{(ry)} ⊢ ˙¬ \underline{(ry)} << \underline{(rx)} ⇒ \underline{(ry)} == \underline{(rx)} ⊢ Repetition ⊳ ˙¬ \underline{(rx)} << \underline{(ry)} ⇒ \underline{(rx)} == \underline{(ry)} ≫ ˙¬ \underline{(rx)} << \underline{(ry)} ⇒ \underline{(rx)} == \underline{(ry)}; Repetition ⊳ ˙¬ \underline{(ry)} << \underline{(rx)} ⇒ \underline{(ry)} == \underline{(rx)} ≫ ˙¬ \underline{(ry)} << \underline{(rx)} ⇒ \underline{(ry)} == \underline{(rx)}; ExpandDisjuncts ⊳ ˙¬ \underline{(rx)} << \underline{(ry)} ⇒ \underline{(rx)} == \underline{(ry)} ⊳ ˙¬ \underline{(ry)} << \underline{(rx)} ⇒ \underline{(ry)} == \underline{(rx)} ≫ ˙¬ \underline{(rx)} == \underline{(ry)} ⇒ ˙¬ \underline{(ry)} == \underline{(rx)} ⇒ ˙¬ \underline{(rx)} << \underline{(ry)} ⇒ ˙¬ \underline{(ry)} << \underline{(rx)}; AutoImply ≫ \underline{(rx)} == \underline{(ry)} ⇒ \underline{(rx)} == \underline{(ry)}; ∀\underline{(rx)}: ∀\underline{(ry)}: \underline{(ry)} == \underline{(rx)} ⊢==Symmetry ⊳ \underline{(ry)} == \underline{(rx)} ≫ \underline{(rx)} == \underline{(ry)}; Ded ⊳ ∀\underline{(rx)}: ∀\underline{(ry)}: \underline{(ry)} == \underline{(rx)} ⊢ \underline{(rx)} == \underline{(ry)} ≫ \underline{(ry)} == \underline{(rx)} ⇒ \underline{(rx)} == \underline{(ry)}; ∀\underline{(rx)}: ∀\underline{(ry)}: ˙¬ \underline{(rx)} << \underline{(ry)} ⇒ ˙¬ \underline{(ry)} << \underline{(rx)} ⊢ FirstConjunct ⊳ ˙¬ \underline{(rx)} << \underline{(ry)} ⇒ ˙¬ \underline{(ry)} << \underline{(rx)} ≫ \underline{(rx)} << \underline{(ry)}; From << (1) ⊳ \underline{(rx)} << \underline{(ry)} ≫ j_{Ex} ∈ \underline{(rx)}; From << (2) ⊳ \underline{(rx)} << \underline{(ry)} ≫ t_{Ex} ∈ \underline{(ry)}; 0 < 1 ≫ ˙¬ 0 <= 1 ⇒ ˙¬ ˙¬ 0 = 1; From << XX ⊳ \underline{(rx)} << \underline{(ry)} ⊳ j_{Ex} ∈ \underline{(rx)} ⊳ t_{Ex} ∈ \underline{(ry)} ⊳ ˙¬ 0 <= 1 ⇒ ˙¬ ˙¬ 0 = 1 ≫ a_{Ex} <= c_{Ex} ⇒ j_{Ex}[c_{Ex}] <= t_{Ex}[c_{Ex}] + −1; SecondConjunct ⊳ ˙¬ \underline{(rx)} << \underline{(ry)} ⇒ ˙¬ \underline{(ry)} << \underline{(rx)} ≫ \underline{(ry)} << \underline{(rx)}; From << XX ⊳ \underline{(ry)} << \underline{(rx)} ⊳ t_{Ex} ∈ \underline{(ry)} ⊳ j_{Ex} ∈ \underline{(rx)} ⊳ ˙¬ 0 <= 1 ⇒ ˙¬ ˙¬ 0 = 1 ≫ a_{Ex} <= c_{Ex} ⇒ t_{Ex}[c_{Ex}] <= j_{Ex}[c_{Ex}] + −1; ExistIntro @ b_{Ex} @ a_{Ex} ⊳ a_{Ex} <= c_{Ex} ⇒ t_{Ex}[c_{Ex}] <= j_{Ex}[c_{Ex}] + −1 ≫ b_{Ex} <= c_{Ex} ⇒ t_{Ex}[c_{Ex}] <= j_{Ex}[c_{Ex}] + −1; thirdGeq ≫ ˙¬ a_{Ex} <= c_{Ex} ⇒ ˙¬ b_{Ex} <= c_{Ex}; FirstConjunct ⊳ ˙¬ a_{Ex} <= c_{Ex} ⇒ ˙¬ b_{Ex} <= c_{Ex} ≫ a_{Ex} <= c_{Ex}; MP ⊳ a_{Ex} <= c_{Ex} ⇒ j_{Ex}[c_{Ex}] <= t_{Ex}[c_{Ex}] + −1 ⊳ a_{Ex} <= c_{Ex} ≫ j_{Ex}[c_{Ex}] <= t_{Ex}[c_{Ex}] + −1; SecondConjunct ⊳ ˙¬ a_{Ex} <= c_{Ex} ⇒ ˙¬ b_{Ex} <= c_{Ex} ≫ b_{Ex} <= c_{Ex}; MP ⊳ b_{Ex} <= c_{Ex} ⇒ t_{Ex}[c_{Ex}] <= j_{Ex}[c_{Ex}] + −1 ⊳ b_{Ex} <= c_{Ex} ≫ t_{Ex}[c_{Ex}] <= j_{Ex}[c_{Ex}] + −1; <<== AntisymmetryHelper(Q) ⊳ ˙¬ 0 <= 1 ⇒ ˙¬ ˙¬ 0 = 1 ⊳ j_{Ex}[c_{Ex}] <= t_{Ex}[c_{Ex}] + −1 ⊳ t_{Ex}[c_{Ex}] <= j_{Ex}[c_{Ex}] + −1 ≫ \underline{(rx)} == \underline{(ry)}; Ded ⊳ ∀\underline{(rx)}: ∀\underline{(ry)}: ˙¬ \underline{(rx)} << \underline{(ry)} ⇒ ˙¬ \underline{(ry)} << \underline{(rx)} ⊢ \underline{(rx)} == \underline{(ry)} ≫ ˙¬ \underline{(rx)} << \underline{(ry)} ⇒ ˙¬ \underline{(ry)} << \underline{(rx)} ⇒ \underline{(rx)} == \underline{(ry)}; From3Disjuncts ⊳ ˙¬ \underline{(rx)} == \underline{(ry)} ⇒ ˙¬ \underline{(ry)} == \underline{(rx)} ⇒ ˙¬ \underline{(rx)} << \underline{(ry)} ⇒ ˙¬ \underline{(ry)} << \underline{(rx)} ⊳ \underline{(rx)} == \underline{(ry)} ⇒ \underline{(rx)} == \underline{(ry)} ⊳ \underline{(ry)} == \underline{(rx)} ⇒ \underline{(rx)} == \underline{(ry)}⊳ ˙¬ \underline{(rx)} << \underline{(ry)} ⇒ ˙¬ \underline{(ry)} << \underline{(rx)} ⇒ \underline{(rx)} == \underline{(ry)} ≫ \underline{(rx)} == \underline{(ry)}⌉, p_{0}, c)] (*** NUMMER 3 ***) [<< TransitivityHelper(Q) →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: \underline{x} <= \underline{y} + −\underline{u} ⊢ \underline{y} <= \underline{z} + −\underline{u} ⊢ ˙¬ 0 <= \underline{u} ⇒ ˙¬ ˙¬ 0 = \underline{u} ⊢ \underline{x} <= \underline{z} + −\underline{u}] [<< TransitivityHelper(Q) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: \underline{x} <= \underline{y} + −\underline{u} ⊢ \underline{y} <= \underline{z} + −\underline{u} ⊢ ˙¬ 0 <= \underline{u} ⇒ ˙¬ ˙¬ 0 = \underline{u} ⊢ PositiveNegated ⊳ ˙¬ 0 <= \underline{u} ⇒ ˙¬ ˙¬ 0 = \underline{u} ≫ ˙¬ − \underline{u} <= 0 ⇒ ˙¬ ˙¬ − \underline{u} = 0; LessAdditionLeft ⊳ ˙¬ − \underline{u} <= 0 ⇒ ˙¬ ˙¬ − \underline{u} = 0 ≫ ˙¬ \underline{y} + −\underline{u} <= \underline{y} + 0 ⇒ ˙¬ ˙¬ \underline{y} + −\underline{u} = \underline{y} + 0; plus0 ≫ \underline{y} + 0 = \underline{y}; SubLessRight ⊳ \underline{y} + 0 = \underline{y} ⊳ ˙¬ \underline{y} + −\underline{u} <= \underline{y} + 0 ⇒ ˙¬ ˙¬ \underline{y} + −\underline{u} = \underline{y} + 0 ≫ ˙¬ \underline{y}+−\underline{u} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{y}+−\underline{u} = \underline{y}; leqLessTransitivity⊳\underline{x} <= \underline{y}+−\underline{u}⊳ ˙¬ \underline{y}+−\underline{u} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{y} + −\underline{u} = \underline{y} ≫ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y}; LessLeqTransitivity ⊳ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊳ \underline{y} <= \underline{z} + −\underline{u} ≫ ˙¬ \underline{x} <= \underline{z} + −\underline{u} ⇒ ˙¬ ˙¬ \underline{x} = \underline{z} + −\underline{u}; LessLeq ⊳ ˙¬ \underline{x} <= \underline{z} + −\underline{u} ⇒ ˙¬ ˙¬ \underline{x} = \underline{z} + −\underline{u} ≫ \underline{x} <= \underline{z} + −\underline{u}⌉, p_{0}, c)] [<<== Transitivity →^{stmt} ZFsub ⊢ ∀\underline{(rx)}: ∀\underline{(ry)}: ∀\underline{(rz)}: ˙¬ \underline{(rx)} << \underline{(ry)} ⇒ \underline{(rx)} == \underline{(ry)} ⊢ ˙¬ \underline{(ry)} << \underline{(rz)} ⇒ \underline{(ry)} == \underline{(rz)} ⊢ ˙¬ \underline{(rx)} << \underline{(rz)} ⇒ \underline{(rx)} == \underline{(rz)}] [<<== Transitivity →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{(rx)}: ∀\underline{(ry)}: ∀\underline{(rz)}: \underline{(rx)} << \underline{(ry)} ⊢ \underline{(ry)} << \underline{(rz)} ⊢<< Transitivity ⊳ \underline{(rx)} << \underline{(ry)} ⊳ \underline{(ry)} << \underline{(rz)} ≫ \underline{(rx)} << \underline{(rz)}; LessLeq(R) ⊳ \underline{(rx)} << \underline{(rz)} ≫ ˙¬ \underline{(rx)} << \underline{(rz)} ⇒ \underline{(rx)} == \underline{(rz)}; ∀\underline{(rx)}: ∀\underline{(ry)}: ∀\underline{(rz)}: \underline{(rx)} << \underline{(ry)} ⊢ \underline{(ry)} == \underline{(rz)} ⊢ SubLessRight(R) ⊳ \underline{(ry)} == \underline{(rz)} ⊳ \underline{(rx)} << \underline{(ry)} ≫ \underline{(rx)} << \underline{(rz)}; LessLeq(R) ⊳ \underline{(rx)} << \underline{(rz)} ≫ ˙¬ \underline{(rx)} << \underline{(rz)} ⇒ \underline{(rx)} == \underline{(rz)}; ∀\underline{(rx)}: ∀\underline{(ry)}: ∀\underline{(rz)}: \underline{(rx)} == \underline{(ry)} ⊢ \underline{(ry)} << \underline{(rz)} ⊢==Symmetry ⊳ \underline{(rx)} == \underline{(ry)} ≫ \underline{(ry)} == \underline{(rx)}; SubLessLeft(R) ⊳ \underline{(ry)} == \underline{(rx)} ⊳ \underline{(ry)} << \underline{(rz)} ≫ \underline{(rx)} << \underline{(rz)}; LessLeq(R) ⊳ \underline{(rx)} << \underline{(rz)} ≫ ˙¬ \underline{(rx)} << \underline{(rz)} ⇒ \underline{(rx)} == \underline{(rz)}; ∀\underline{(rx)}: ∀\underline{(ry)}: ∀\underline{(rz)}: \underline{(rx)} == \underline{(ry)} ⊢ \underline{(ry)} == \underline{(rz)} ⊢ == Transitivity ⊳ \underline{(rx)} == \underline{(ry)} ⊳ \underline{(ry)} == \underline{(rz)} ≫ \underline{(rx)} == \underline{(rz)}; eqLeq(R) ⊳ \underline{(rx)} == \underline{(rz)} ≫ ˙¬ \underline{(rx)} << \underline{(rz)} ⇒ \underline{(rx)} == \underline{(rz)}; ∀\underline{(rx)}: ∀\underline{(ry)}: ∀\underline{(rz)}: ˙¬ \underline{(rx)} << \underline{(ry)} ⇒ \underline{(rx)} == \underline{(ry)} ⊢ ˙¬ \underline{(ry)} << \underline{(rz)} ⇒ \underline{(ry)} == \underline{(rz)} ⊢ Repetition ⊳ ˙¬ \underline{(rx)} << \underline{(ry)} ⇒ \underline{(rx)} == \underline{(ry)} ≫ ˙¬ \underline{(rx)} << \underline{(ry)} ⇒ \underline{(rx)} == \underline{(ry)}; Repetition ⊳ ˙¬ \underline{(ry)} << \underline{(rz)} ⇒ \underline{(ry)} == \underline{(rz)} ≫ ˙¬ \underline{(ry)} << \underline{(rz)} ⇒ \underline{(ry)} == \underline{(rz)}; Ded ⊳ ∀\underline{(rx)}: ∀\underline{(ry)}: ∀\underline{(rz)}: \underline{(rx)} << \underline{(ry)} ⊢ \underline{(ry)} << \underline{(rz)} ⊢ ˙¬ \underline{(rx)} << \underline{(rz)} ⇒ \underline{(rx)} == \underline{(rz)} ≫ \underline{(rx)} << \underline{(ry)} ⇒ \underline{(ry)} << \underline{(rz)} ⇒ ˙¬ \underline{(rx)} << \underline{(rz)} ⇒ \underline{(rx)} == \underline{(rz)}; Ded ⊳ ∀\underline{(rx)}: ∀\underline{(ry)}: ∀\underline{(rz)}: \underline{(rx)} << \underline{(ry)} ⊢ \underline{(ry)} == \underline{(rz)} ⊢ ˙¬ \underline{(rx)} << \underline{(rz)} ⇒ \underline{(rx)} == \underline{(rz)} ≫ \underline{(rx)} << \underline{(ry)} ⇒ \underline{(ry)} == \underline{(rz)} ⇒ ˙¬ \underline{(rx)} << \underline{(rz)} ⇒ \underline{(rx)} == \underline{(rz)}; Ded ⊳ ∀\underline{(rx)}: ∀\underline{(ry)}: ∀\underline{(rz)}: \underline{(rx)} == \underline{(ry)} ⊢ \underline{(ry)} << \underline{(rz)} ⊢ ˙¬ \underline{(rx)} << \underline{(rz)} ⇒ \underline{(rx)} == \underline{(rz)} ≫ \underline{(rx)} == \underline{(ry)} ⇒ \underline{(ry)} << \underline{(rz)} ⇒ ˙¬ \underline{(rx)} << \underline{(rz)} ⇒ \underline{(rx)} == \underline{(rz)}; Ded ⊳ ∀\underline{(rx)}: ∀\underline{(ry)}: ∀\underline{(rz)}: \underline{(rx)} == \underline{(ry)} ⊢ \underline{(ry)} == \underline{(rz)} ⊢ ˙¬ \underline{(rx)} << \underline{(rz)} ⇒ \underline{(rx)} == \underline{(rz)} ≫ \underline{(rx)} == \underline{(ry)} ⇒ \underline{(ry)} == \underline{(rz)} ⇒ ˙¬ \underline{(rx)} << \underline{(rz)} ⇒ \underline{(rx)} == \underline{(rz)}; From2 ∗ 2Disjuncts ⊳ ˙¬ \underline{(rx)} << \underline{(ry)} ⇒ \underline{(rx)} == \underline{(ry)} ⊳ ˙¬ \underline{(ry)} << \underline{(rz)} ⇒ \underline{(ry)} == \underline{(rz)} ⊳ \underline{(rx)} << \underline{(ry)} ⇒ \underline{(ry)} << \underline{(rz)} ⇒ ˙¬ \underline{(rx)} << \underline{(rz)} ⇒ \underline{(rx)} == \underline{(rz)} ⊳ \underline{(rx)} << \underline{(ry)} ⇒ \underline{(ry)} == \underline{(rz)} ⇒ ˙¬ \underline{(rx)} << \underline{(rz)} ⇒ \underline{(rx)} == \underline{(rz)} ⊳ \underline{(rx)} == \underline{(ry)} ⇒ \underline{(ry)} << \underline{(rz)} ⇒ ˙¬ \underline{(rx)} << \underline{(rz)} ⇒ \underline{(rx)} == \underline{(rz)} ⊳ \underline{(rx)} == \underline{(ry)} ⇒ \underline{(ry)} == \underline{(rz)} ⇒ ˙¬ \underline{(rx)} << \underline{(rz)} ⇒ \underline{(rx)} == \underline{(rz)} ≫ ˙¬ \underline{(rx)} << \underline{(rz)} ⇒ \underline{(rx)} == \underline{(rz)}⌉, p_{0}, c)] (*** NUMMER 5 ***) [== Addition →^{stmt} ZFsub ⊢ ∀\underline{m}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: R(\underline{(fx)}) == R(\underline{(fy)}) ⊢ R((fx) +_{f} (fy)) == R((fx) +_{f} (fy))] [== Addition →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{m}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: R(\underline{(fx)}) == R(\underline{(fy)}) ⊢ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⊢ c_{Ex} <= \underline{m} ⊢ From == ⊳R(\underline{(fx)}) == R(\underline{(fy)}) ≫ SF(\underline{(fx)}, \underline{(fy)}); FromSF ⊳ SF(\underline{(fx)}, \underline{(fy)}) ⊳ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ≫ c_{Ex} <= \underline{m} ⇒ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) = \underline{(ϵ)}; MP ⊳ c_{Ex} <= \underline{m} ⇒ ˙¬ if(0 <= \underline{(fx)}[\underline{m}]+−\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}]+−\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}]+−\underline{(fy)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) = \underline{(ϵ)} ⊳ c_{Ex} <= \underline{m} ≫ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) = \underline{(ϵ)}; insertMiddleTerm(Difference) ≫ \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}] = \underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}] + −\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}]; PlusF ≫ \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] = \underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}]; PlusF ≫ \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] = \underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}]; EqNegated ⊳ \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] = \underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}] ≫ −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] = −\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}]; AddEquations ⊳ \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] = \underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}] ⊳ −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] = −\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}] ≫ \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] = \underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}] + −\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}]; eqSymmetry ⊳ \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] = \underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}] + −\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}] ≫ \underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}] + −\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}] = \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}]; eqTransitivity ⊳ \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}] = \underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}] + −\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}] ⊳ \underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}] + −\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}] = \underline{(fx)}+_{f} \underline{(fz)}[\underline{m}]+−\underline{(fy)}+_{f} \underline{(fz)}[\underline{m}] ≫ \underline{(fx)}[\underline{m}]+−\underline{(fy)}[\underline{m}] = \underline{(fx)}+_{f} \underline{(fz)}[\underline{m}]+−\underline{(fy)}+_{f} \underline{(fz)}[\underline{m}]; SameNumerical⊳\underline{(fx)}[\underline{m}]+−\underline{(fy)}[\underline{m}] = \underline{(fx)}+_{f} \underline{(fz)}[\underline{m}]+−\underline{(fy)}+_{f} \underline{(fz)}[\underline{m}] ≫ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) = if(0 <= \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}], \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}], −\underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}]); SubLessLeft ⊳ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}]+−\underline{(fy)}[\underline{m}]) = if(0 <= \underline{(fx)}+_{f} \underline{(fz)}[\underline{m}]+−\underline{(fy)}+_{f} \underline{(fz)}[\underline{m}], \underline{(fx)}+_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}], −\underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}]) ⊳ ˙¬ if(0 <= \underline{(fx)}[\underline{m}]+−\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}]+−\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}]+−\underline{(fy)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) = \underline{(ϵ)} ≫ ˙¬ if(0 <= \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}], \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}], −\underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}], \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}], −\underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}]) = \underline{(ϵ)}; ∀\underline{m}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: Ded ⊳ ∀\underline{m}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: R(\underline{(fx)}) == R(\underline{(fy)}) ⊢ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⊢ c_{Ex} <= \underline{m} ⊢ ˙¬ if(0 <= \underline{(fx)}+_{f} \underline{(fz)}[\underline{m}]+−\underline{(fy)}+_{f} \underline{(fz)}[\underline{m}], \underline{(fx)}+_{f} \underline{(fz)}[\underline{m}]+−\underline{(fy)}+_{f} \underline{(fz)}[\underline{m}], −\underline{(fx)}+_{f} \underline{(fz)}[\underline{m}]+ −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}], \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}], −\underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}]) = \underline{(ϵ)} ≫ R(\underline{(fx)}) == R(\underline{(fy)}) ⇒ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ c_{Ex} <= \underline{m} ⇒ ˙¬ if(0 <= \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}], \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}], −\underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}], \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}], −\underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}]) = \underline{(ϵ)}; R(\underline{(fx)}) == R(\underline{(fy)}) ⊢ MP ⊳ R(\underline{(fx)}) == R(\underline{(fy)}) ⇒ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ c_{Ex} <= \underline{m} ⇒ ˙¬ if(0 <= \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}], \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}], −\underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}], \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}], −\underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}]) = \underline{(ϵ)} ⊳ R(\underline{(fx)}) == R(\underline{(fy)}) ≫ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ c_{Ex} <= \underline{m} ⇒ ˙¬ if(0 <= \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}], \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}], −\underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}], \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}], −\underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}]) = \underline{(ϵ)}; ToSF ⊳ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ c_{Ex} <= \underline{m} ⇒ ˙¬ if(0 <= \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}], \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}], −\underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}], \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}], −\underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}]) = \underline{(ϵ)} ≫ SF(\underline{(fx)} +_{f} \underline{(fz)}, \underline{(fy)} +_{f} \underline{(fz)}); To == ⊳SF(\underline{(fx)} +_{f} \underline{(fz)}, \underline{(fy)} +_{f} \underline{(fz)}) ≫ R(\underline{(fx)} +_{f} \underline{(fz)}) == R(\underline{(fy)} +_{f} \underline{(fz)}); PlusR ≫ R((fx) +_{f} (fy)) == R(\underline{(fx)} +_{f} \underline{(fz)}); PlusR ≫ R((fx) +_{f} (fy)) == R(\underline{(fy)} +_{f} \underline{(fz)}); ==Symmetry ⊳ R((fx) +_{f} (fy)) == R(\underline{(fy)} +_{f} \underline{(fz)}) ≫ R(\underline{(fy)} +_{f} \underline{(fz)}) == R((fx) +_{f} (fy)); ==Transitivity ⊳ R((fx) +_{f} (fy)) == R(\underline{(fx)} +_{f} \underline{(fz)}) ⊳ R(\underline{(fx)} +_{f} \underline{(fz)}) == R(\underline{(fy)} +_{f} \underline{(fz)}) ≫ R((fx) +_{f} (fy)) == R(\underline{(fy)} +_{f} \underline{(fz)}); ==Transitivity ⊳ R((fx) +_{f} (fy)) == R(\underline{(fy)} +_{f} \underline{(fz)}) ⊳ R(\underline{(fy)} +_{f} \underline{(fz)}) == R((fx) +_{f} (fy)) ≫ R((fx) +_{f} (fy)) == R((fx) +_{f} (fy))⌉, p_{0}, c)]

[== AdditionLeft →^{stmt} ZFsub ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: R(\underline{(fx)}) == R(\underline{(fy)}) ⊢ R((fx) +_{f} (fy)) == R((fx) +_{f} (fy))] [== AdditionLeft →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: R(\underline{(fx)}) == R(\underline{(fy)}) ⊢== Addition ⊳ R(\underline{(fx)}) == R(\underline{(fy)}) ≫ R((fx) +_{f} (fy)) == R((fx) +_{f} (fy)); PlusCommutativity(R) ≫ R((fx) +_{f} (fy)) == R((fx)+_{f} (fy)); PlusCommutativity(R) ≫ R((fx)+_{f} (fy)) == R((fx)+_{f} (fy)); == Transitivity ⊳ R((fx) +_{f} (fy)) == R((fx) +_{f} (fy)) ⊳ R((fx) +_{f} (fy)) == R((fx) +_{f} (fy)) ≫ R((fx) +_{f} (fy)) == R((fx) +_{f} (fy)); == Transitivity ⊳ R((fx) +_{f} (fy)) == R((fx) +_{f} (fy)) ⊳ R((fx) +_{f} (fy)) == R((fx) +_{f} (fy)) ≫ R((fx) +_{f} (fy)) == R((fx) +_{f} (fy))⌉, p_{0}, c)] (***) [<< Addition →^{stmt} ZFsub ⊢ ∀\underline{m}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: R(\underline{(fx)}) << R(\underline{(fy)}) ⊢ R((fx) +_{f} (fy)) << R((fx) +_{f} (fy))] [<< Addition →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{m}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: R(\underline{(fx)}) << R(\underline{(fy)}) ⊢ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⊢ c_{Ex} <= \underline{m} ⊢ From << ⊳R(\underline{(fx)}) << R(\underline{(fy)}) ≫ \underline{(fx)} <_{f} \underline{(fy)}; From < f ⊳ \underline{(fx)} <_{f} \underline{(fy)} ⊳ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ≫ c_{Ex} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= \underline{(fy)}[\underline{m}] + −\underline{(ϵ)}; MP ⊳ c_{Ex} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= \underline{(fy)}[\underline{m}]+−\underline{(ϵ)}⊳c_{Ex} <= \underline{m} ≫ \underline{(fx)}[\underline{m}] <= \underline{(fy)}[\underline{m}]+−\underline{(ϵ)}; leqAddition⊳\underline{(fx)}[\underline{m}] <= \underline{(fy)}[\underline{m}] + −\underline{(ϵ)} ≫ \underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}] <= \underline{(fy)}[\underline{m}] + −\underline{(ϵ)} + \underline{(fz)}[\underline{m}]; PlusF(Sym) ≫ \underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}] = \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}]; subLeqLeft ⊳ \underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}] = \underline{(fx)}+_{f} \underline{(fz)}[\underline{m}]⊳\underline{(fx)}[\underline{m}]+\underline{(fz)}[\underline{m}] <= \underline{(fy)}[\underline{m}]+−\underline{(ϵ)}+\underline{(fz)}[\underline{m}] ≫ \underline{(fx)}+_{f} \underline{(fz)}[\underline{m}] <= \underline{(fy)}[\underline{m}] + −\underline{(ϵ)} + \underline{(fz)}[\underline{m}]; Three2threeTerms ≫ \underline{(fy)}[\underline{m}] + −\underline{(ϵ)} + \underline{(fz)}[\underline{m}] = \underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}] + −\underline{(ϵ)}; PlusF(Sym) ≫ \underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}] = \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}]; eqAddition ⊳ \underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}] = \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] ≫ \underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}]+−\underline{(ϵ)} = \underline{(fy)}+_{f} \underline{(fz)}[\underline{m}]+−\underline{(ϵ)}; eqTransitivity ⊳\underline{(fy)}[\underline{m}]+−\underline{(ϵ)}+\underline{(fz)}[\underline{m}] = \underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}] + −\underline{(ϵ)} ⊳ \underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}] + −\underline{(ϵ)} = \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(ϵ)} ≫ \underline{(fy)}[\underline{m}] + −\underline{(ϵ)} + \underline{(fz)}[\underline{m}] = \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(ϵ)}; subLeqRight ⊳ \underline{(fy)}[\underline{m}] + −\underline{(ϵ)} + \underline{(fz)}[\underline{m}] = \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(ϵ)} ⊳ \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] <= \underline{(fy)}[\underline{m}] + −\underline{(ϵ)} + \underline{(fz)}[\underline{m}] ≫ \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] <= \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(ϵ)}; ∀\underline{m}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: Ded ⊳ ∀\underline{m}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: R(\underline{(fx)}) << R(\underline{(fy)}) ⊢ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⊢ c_{Ex} <= \underline{m} ⊢ \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] <= \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(ϵ)} ≫ R(\underline{(fx)}) << R(\underline{(fy)}) ⇒ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ c_{Ex} <= \underline{m} ⇒ \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] <= \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(ϵ)}; R(\underline{(fx)}) << R(\underline{(fy)}) ⊢ MP ⊳ R(\underline{(fx)}) << R(\underline{(fy)}) ⇒ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ c_{Ex} <= \underline{m} ⇒ \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] <= \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(ϵ)} ⊳ R(\underline{(fx)}) << R(\underline{(fy)}) ≫ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ c_{Ex} <= \underline{m} ⇒ \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] <= \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(ϵ)}; To < f ⊳ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ c_{Ex} <= \underline{m} ⇒ \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] <= \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(ϵ)} ≫ \underline{(fx)}+_{f} \underline{(fz)} <_{f} \underline{(fy)}+_{f} \underline{(fz)}; To << ⊳\underline{(fx)}+_{f} \underline{(fz)} <_{f} \underline{(fy)}+_{f} \underline{(fz)} ≫ R(\underline{(fx)}+_{f} \underline{(fz)}) << R(\underline{(fy)}+_{f} \underline{(fz)}); PlusR(Sym) ≫ R(\underline{(fx)}+_{f} \underline{(fz)}) == R((fx)+_{f} (fy)); PlusR(Sym) ≫ R(\underline{(fy)} +_{f} \underline{(fz)}) == R((fx) +_{f} (fy)); SubLessLeft(R) ⊳ R(\underline{(fx)} +_{f} \underline{(fz)}) == R((fx) +_{f} (fy)) ⊳ R(\underline{(fx)} +_{f} \underline{(fz)}) << R(\underline{(fy)} +_{f} \underline{(fz)}) ≫ R((fx) +_{f} (fy)) << R(\underline{(fy)} +_{f} \underline{(fz)}); SubLessRight(R) ⊳ R(\underline{(fy)} +_{f} \underline{(fz)}) == R((fx) +_{f} (fy)) ⊳ R((fx) +_{f} (fy)) << R(\underline{(fy)} +_{f} \underline{(fz)}) ≫ R((fx) +_{f} (fy)) << R((fx) +_{f} (fy))⌉, p_{0}, c)] (***) [<<== Addition →^{stmt} ZFsub ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: ˙¬ R(\underline{(fx)}) << R(\underline{(fy)}) ⇒ R(\underline{(fx)}) == R(\underline{(fy)}) ⊢ ˙¬ R((fx) +_{f} (fy)) << R((fx) +_{f} (fy)) ⇒ R((fx) +_{f} (fy)) == R((fx) +_{f} (fy))] [<<== Addition →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: R(\underline{(fx)}) << R(\underline{(fy)}) ⊢<< Addition ⊳ R(\underline{(fx)}) << R(\underline{(fy)}) ≫ R((fx) +_{f} (fy)) << R((fx) +_{f} (fy)); LessLeq(R) ⊳ R((fx) +_{f} (fy)) << R((fx) +_{f} (fy)) ≫ ˙¬ R((fx) +_{f} (fy)) << R((fx) +_{f} (fy)) ⇒ R((fx) +_{f} (fy)) == R((fx) +_{f} (fy)); ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: R(\underline{(fx)}) == R(\underline{(fy)}) ⊢== Addition ⊳ R(\underline{(fx)}) == R(\underline{(fy)}) ≫ R((fx) +_{f} (fy)) == R((fx) +_{f} (fy)); eqLeq(R) ⊳ R((fx) +_{f} (fy)) == R((fx) +_{f} (fy)) ≫ ˙¬ R((fx) +_{f} (fy)) << R((fx) +_{f} (fy)) ⇒ R((fx) +_{f} (fy)) == R((fx) +_{f} (fy)); ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: Ded ⊳ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: R(\underline{(fx)}) << R(\underline{(fy)}) ⊢ ˙¬ R((fx) +_{f} (fy)) << R((fx) +_{f} (fy)) ⇒ R((fx) +_{f} (fy)) == R((fx) +_{f} (fy)) ≫ R(\underline{(fx)}) << R(\underline{(fy)}) ⇒ ˙¬ R((fx) +_{f} (fy)) << R((fx) +_{f} (fy)) ⇒ R((fx) +_{f} (fy)) == R((fx) +_{f} (fy)); Ded ⊳ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: R(\underline{(fx)}) == R(\underline{(fy)}) ⊢ ˙¬ R((fx) +_{f} (fy)) << R((fx) +_{f} (fy)) ⇒ R((fx) +_{f} (fy)) == R((fx) +_{f} (fy)) ≫ R(\underline{(fx)}) == R(\underline{(fy)}) ⇒ ˙¬ R((fx) +_{f} (fy)) << R((fx) +_{f} (fy)) ⇒ R((fx) +_{f} (fy)) == R((fx) +_{f} (fy)); ˙¬ R(\underline{(fx)}) << R(\underline{(fy)}) ⇒ R(\underline{(fx)}) == R(\underline{(fy)}) ⊢ Repetition ⊳ ˙¬ R(\underline{(fx)}) << R(\underline{(fy)}) ⇒ R(\underline{(fx)}) == R(\underline{(fy)}) ≫ ˙¬ R(\underline{(fx)}) << R(\underline{(fy)}) ⇒ R(\underline{(fx)}) == R(\underline{(fy)}); FromDisjuncts ⊳ ˙¬ R(\underline{(fx)}) << R(\underline{(fy)}) ⇒ R(\underline{(fx)}) == R(\underline{(fy)}) ⊳ R(\underline{(fx)}) << R(\underline{(fy)}) ⇒ ˙¬ R((fx) +_{f} (fy)) << R((fx) +_{f} (fy)) ⇒ R((fx) +_{f} (fy)) == R((fx) +_{f} (fy)) ⊳ R(\underline{(fx)}) == R(\underline{(fy)}) ⇒ ˙¬ R((fx) +_{f} (fy)) << R((fx) +_{f} (fy)) ⇒ R((fx) +_{f} (fy)) == R((fx) +_{f} (fy)) ≫ ˙¬ R((fx)+_{f} (fy)) << R((fx)+_{f} (fy)) ⇒ R((fx)+_{f} (fy)) == R((fx)+_{f} (fy))⌉, p_{0}, c)] (*** NUMMER 7 ***) [PlusAssociativity(F) →^{stmt} ZFsub ⊢ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: \underline{(fx)} +_{f} \underline{(fy)} +_{f} \underline{(fz)} =_{f} \underline{(fx)} +_{f} \underline{(fy)} +_{f} \underline{(fz)}]

[PlusAssociativity(F) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: PlusF ≫ \underline{(fx)} +_{f} \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] = \underline{(fx)} +_{f} \underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}]; PlusF ≫ \underline{(fx)} +_{f} \underline{(fy)}[\underline{m}] = \underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}]; eqAddition ⊳ \underline{(fx)} +_{f} \underline{(fy)}[\underline{m}] = \underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}] ≫ \underline{(fx)} +_{f} \underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}] = \underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}]; plusAssociativity ≫ \underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}] = \underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}]; PlusF(Sym) ≫ \underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}] = \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}]; lemma eqAdditionLeft ⊳ \underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}] = \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] ≫ \underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}] = \underline{(fx)}[\underline{m}] + \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}]; PlusF(Sym) ≫ \underline{(fx)}[\underline{m}] + \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] = \underline{(fx)} +_{f} \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}]; eqTransitivity6 ⊳ \underline{(fx)} +_{f} \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] = \underline{(fx)} +_{f} \underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}] ⊳ \underline{(fx)} +_{f} \underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}] = \underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}] ⊳ \underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}]+\underline{(fz)}[\underline{m}] = \underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}]+\underline{(fz)}[\underline{m}]⊳\underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}]+\underline{(fz)}[\underline{m}] = \underline{(fx)}[\underline{m}] + \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] ⊳ \underline{(fx)}[\underline{m}] + \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] = \underline{(fx)} +_{f} \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] ≫ \underline{(fx)} +_{f} \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] = \underline{(fx)} +_{f} \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}]; To = f ⊳ \underline{(fx)} +_{f} \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] = \underline{(fx)} +_{f} \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] ≫ \underline{(fx)} +_{f} \underline{(fy)} +_{f} \underline{(fz)} =_{f} \underline{(fx)} +_{f} \underline{(fy)} +_{f} \underline{(fz)}⌉, p_{0}, c)] [PlusAssociativity(R) →^{stmt} ZFsub ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: R(\underline{(fx)} +_{f} \underline{(fy)} +_{f} \underline{(fz)}) == R(\underline{(fx)} +_{f} \underline{(fy)} +_{f} \underline{(fz)})]

[PlusAssociativity(R) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: PlusAssociativity(F) ≫ \underline{(fx)} +_{f} \underline{(fy)} +_{f} \underline{(fz)} =_{f} \underline{(fx)} +_{f} \underline{(fy)} +_{f} \underline{(fz)}; = fToSameF ⊳ \underline{(fx)} +_{f} \underline{(fy)} +_{f} \underline{(fz)} =_{f} \underline{(fx)} +_{f} \underline{(fy)} +_{f} \underline{(fz)} ≫ SF(\underline{(fx)}+_{f} \underline{(fy)}+_{f} \underline{(fz)}, \underline{(fx)}+_{f} \underline{(fy)}+_{f} \underline{(fz)}); f2R(Plus)⊳SF(\underline{(fx)}+_{f} \underline{(fy)}+_{f} \underline{(fz)}, \underline{(fx)}+_{f} \underline{(fy)} +_{f} \underline{(fz)}) ≫ R((fx) +_{f} (fy)) == R(\underline{(fx)} +_{f} \underline{(fy)} +_{f} \underline{(fz)}); PlusR(Sym) ≫ R(\underline{(fx)} +_{f} \underline{(fy)} +_{f} \underline{(fz)}) == R((fx) +_{f} (fy)); == Transitivity ⊳ R(\underline{(fx)} +_{f} \underline{(fy)} +_{f} \underline{(fz)}) == R((fx) +_{f} (fy)) ⊳ R((fx) +_{f} (fy)) == R(\underline{(fx)} +_{f} \underline{(fy)} +_{f} \underline{(fz)}) ≫ R(\underline{(fx)} +_{f} \underline{(fy)} +_{f} \underline{(fz)}) == R(\underline{(fx)} +_{f} \underline{(fy)} +_{f} \underline{(fz)})⌉, p_{0}, c)] (*** NUMMER 8 ***) [Plus00 →^{stmt} ZFsub ⊢ ∀\underline{(fx)}: R((fx) +_{f} (fy)) == R(\underline{(fx)})]

[Plus00 →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{(fx)}: Plus0f ≫ \underline{(fx)} +_{f} 0f =_{f} \underline{(fx)}; = fToSameF ⊳ \underline{(fx)} +_{f} 0f =_{f} \underline{(fx)} ≫ SF(\underline{(fx)} +_{f} 0f, \underline{(fx)}); f2R(Plus) ⊳ SF(\underline{(fx)} +_{f} 0f, \underline{(fx)}) ≫ R((fx) +_{f} (fy)) == R(\underline{(fx)}); Repetition ⊳ R((fx) +_{f} (fy)) == R(\underline{(fx)}) ≫ R((fx) +_{f} (fy)) == R(\underline{(fx)})⌉, p_{0}, c)] (*** NUMMER 9 ***) [Negative(R) →^{stmt} ZFsub ⊢ ∀\underline{m}: ∀\underline{(fx)}: R((fx) +_{f} (fy)) == R(0f)]

[Negative(R) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{m}: ∀\underline{(fx)}: PlusF ≫ \underline{(fx)} +_{f} −_{f} \underline{(fx)}[\underline{m}] = \underline{(fx)}[\underline{m}] + −_{f} \underline{(fx)}[\underline{m}]; MinusF ≫ −_{f} \underline{(fx)}[\underline{m}] = −\underline{(fx)}[\underline{m}]; lemma eqAdditionLeft⊳−_{f} \underline{(fx)}[\underline{m}] = −\underline{(fx)}[\underline{m}] ≫ \underline{(fx)}[\underline{m}]+−_{f} \underline{(fx)}[\underline{m}] = \underline{(fx)}[\underline{m}] + −\underline{(fx)}[\underline{m}]; Negative ≫ \underline{(fx)}[\underline{m}] + −\underline{(fx)}[\underline{m}] = 0; 0f ≫ 0f[\underline{m}] = 0; eqSymmetry ⊳ 0f[\underline{m}] = 0 ≫ 0 = 0f[\underline{m}]; eqTransitivity5 ⊳ \underline{(fx)} +_{f} −_{f} \underline{(fx)}[\underline{m}] = \underline{(fx)}[\underline{m}] + −_{f} \underline{(fx)}[\underline{m}] ⊳ \underline{(fx)}[\underline{m}] + −_{f} \underline{(fx)}[\underline{m}] = \underline{(fx)}[\underline{m}] + −\underline{(fx)}[\underline{m}] ⊳ \underline{(fx)}[\underline{m}] + −\underline{(fx)}[\underline{m}] = 0⊳0 = 0f[\underline{m}] ≫ \underline{(fx)}+_{f} −_{f} \underline{(fx)}[\underline{m}] = 0f[\underline{m}]; To = f⊳\underline{(fx)}+_{f} −_{f} \underline{(fx)}[\underline{m}] = 0f[\underline{m}] ≫ \underline{(fx)} +_{f} −_{f} \underline{(fx)} =_{f} 0f; = fToSameF ⊳ \underline{(fx)} +_{f} −_{f} \underline{(fx)} =_{f} 0f ≫ SF(\underline{(fx)} +_{f} −_{f} \underline{(fx)}, 0f); f2R(Plus) ⊳ SF(\underline{(fx)} +_{f} −_{f} \underline{(fx)}, 0f) ≫ R((fx) +_{f} (fy)) == R(0f); Repetition⊳R((fx)+_{f} (fy)) == R(0f) ≫ R((fx)+_{f} (fy)) == R(0f)⌉, p_{0}, c)] (*** NUMMER 10 ***) [PlusCommutativity(F) →^{stmt} ZFsub ⊢ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: \underline{(fx)}+_{f} \underline{(fy)} =_{f} \underline{(fy)}+_{f} \underline{(fx)}]

[PlusCommutativity(F) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: PlusF ≫ \underline{(fx)} +_{f} \underline{(fy)}[\underline{m}] = \underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}]; plusCommutativity ≫ \underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}] = \underline{(fy)}[\underline{m}] + \underline{(fx)}[\underline{m}]; PlusF(Sym) ≫ \underline{(fy)}[\underline{m}] + \underline{(fx)}[\underline{m}] = \underline{(fy)} +_{f} \underline{(fx)}[\underline{m}]; eqTransitivity4 ⊳ \underline{(fx)} +_{f} \underline{(fy)}[\underline{m}] = \underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}] ⊳ \underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}] = \underline{(fy)}[\underline{m}] + \underline{(fx)}[\underline{m}] ⊳ \underline{(fy)}[\underline{m}] + \underline{(fx)}[\underline{m}] = \underline{(fy)} +_{f} \underline{(fx)}[\underline{m}] ≫ \underline{(fx)} +_{f} \underline{(fy)}[\underline{m}] = \underline{(fy)} +_{f} \underline{(fx)}[\underline{m}]; To = f ⊳ \underline{(fx)} +_{f} \underline{(fy)}[\underline{m}] = \underline{(fy)} +_{f} \underline{(fx)}[\underline{m}] ≫ \underline{(fx)} +_{f} \underline{(fy)} =_{f} \underline{(fy)} +_{f} \underline{(fx)}⌉, p_{0}, c)]

[PlusCommutativity(R) →^{stmt} ZFsub ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: R((fx) +_{f} (fy)) == R((fx) +_{f} (fy))] [PlusCommutativity(R) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: PlusCommutativity(F) ≫ \underline{(fx)} +_{f} \underline{(fy)} =_{f} \underline{(fy)} +_{f} \underline{(fx)}; = fToSameF ⊳ \underline{(fx)} +_{f} \underline{(fy)} =_{f} \underline{(fy)} +_{f} \underline{(fx)} ≫ SF(\underline{(fx)} +_{f} \underline{(fy)}, \underline{(fy)} +_{f} \underline{(fx)}); f2R(Plus) ⊳ SF(\underline{(fx)} +_{f} \underline{(fy)}, \underline{(fy)} +_{f} \underline{(fx)}) ≫ R((fx) +_{f} (fy)) == R(\underline{(fy)} +_{f} \underline{(fx)}); PlusR(Sym) ≫ R(\underline{(fy)} +_{f} \underline{(fx)}) == R((fx) +_{f} (fy)); == Transitivity ⊳ R((fx) +_{f} (fy)) == R(\underline{(fy)} +_{f} \underline{(fx)}) ⊳ R(\underline{(fy)} +_{f} \underline{(fx)}) == R((fx) +_{f} (fy)) ≫ R((fx) +_{f} (fy)) == R((fx) +_{f} (fy))⌉, p_{0}, c)] (*** NUMMER 11 ***) [TimesAssociativity(F) →^{stmt} ZFsub ⊢ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: \underline{(fx)} ∗_{f} \underline{(fy)} ∗_{f} \underline{(fz)} =_{f} \underline{(fx)} ∗_{f} \underline{(fy)} ∗_{f} \underline{(fz)}]

[TimesAssociativity(F) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: TimesF ≫ \underline{(fx)} ∗_{f} \underline{(fy)} ∗_{f} \underline{(fz)}[\underline{m}] = \underline{(fx)} ∗_{f} \underline{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}]; TimesF ≫ \underline{(fx)} ∗_{f} \underline{(fy)}[\underline{m}] = \underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}]; eqMultiplication ⊳ \underline{(fx)} ∗_{f} \underline{(fy)}[\underline{m}] = \underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}] ≫ \underline{(fx)} ∗_{f} \underline{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}] = \underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}]; timesAssociativity ≫ \underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}] = \underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}]; TimesF(Sym) ≫ \underline{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}] = \underline{(fy)} ∗_{f} \underline{(fz)}[\underline{m}]; EqMultiplicationLeft ⊳ \underline{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}] = \underline{(fy)} ∗_{f} \underline{(fz)}[\underline{m}] ≫ \underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}] = \underline{(fx)}[\underline{m}] ∗ \underline{(fy)} ∗_{f} \underline{(fz)}[\underline{m}]; TimesF(Sym) ≫ \underline{(fx)}[\underline{m}] ∗ \underline{(fy)} ∗_{f} \underline{(fz)}[\underline{m}] = \underline{(fx)} ∗_{f} \underline{(fy)} ∗_{f} \underline{(fz)}[\underline{m}]; eqTransitivity6 ⊳ \underline{(fx)} ∗_{f} \underline{(fy)} ∗_{f} \underline{(fz)}[\underline{m}] = \underline{(fx)} ∗_{f} \underline{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}] ⊳ \underline{(fx)} ∗_{f} \underline{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}] = \underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}] ⊳ \underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}] = \underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}] ⊳ \underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}] = \underline{(fx)}[\underline{m}] ∗ \underline{(fy)} ∗_{f} \underline{(fz)}[\underline{m}] ⊳ \underline{(fx)}[\underline{m}] ∗ \underline{(fy)} ∗_{f} \underline{(fz)}[\underline{m}] = \underline{(fx)} ∗_{f} \underline{(fy)} ∗_{f} \underline{(fz)}[\underline{m}] ≫ \underline{(fx)} ∗_{f} \underline{(fy)} ∗_{f} \underline{(fz)}[\underline{m}] = \underline{(fx)} ∗_{f} \underline{(fy)} ∗_{f} \underline{(fz)}[\underline{m}]; To = f ⊳ \underline{(fx)} ∗_{f} \underline{(fy)} ∗_{f} \underline{(fz)}[\underline{m}] = \underline{(fx)} ∗_{f} \underline{(fy)} ∗_{f} \underline{(fz)}[\underline{m}] ≫ \underline{(fx)} ∗_{f} \underline{(fy)} ∗_{f} \underline{(fz)} =_{f} \underline{(fx)} ∗_{f} \underline{(fy)} ∗_{f} \underline{(fz)}⌉, p_{0}, c)]

[TimesAssociativity(R) →^{stmt} ZFsub ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: R(\underline{(fx)} ∗_{f} \underline{(fy)} ∗_{f} \underline{(fz)}) == R(\underline{(fx)} ∗_{f} \underline{(fy)} ∗_{f} \underline{(fz)})]

[TimesAssociativity(R) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: TimesAssociativity(F) ≫ \underline{(fx)} ∗_{f} \underline{(fy)} ∗_{f} \underline{(fz)} =_{f} \underline{(fx)} ∗_{f} \underline{(fy)} ∗_{f} \underline{(fz)}; = fToSameF ⊳ \underline{(fx)} ∗_{f} \underline{(fy)} ∗_{f} \underline{(fz)} =_{f} \underline{(fx)} ∗_{f} \underline{(fy)} ∗_{f} \underline{(fz)} ≫ SF(\underline{(fx)} ∗_{f} \underline{(fy)} ∗_{f} \underline{(fz)}, \underline{(fx)} ∗_{f} \underline{(fy)} ∗_{f} \underline{(fz)}); f2R(Times) ⊳ SF(\underline{(fx)} ∗_{f} \underline{(fy)} ∗_{f} \underline{(fz)}, \underline{(fx)} ∗_{f} \underline{(fy)}∗_{f} \underline{(fz)}) ≫ R(\underline{(fx)}∗_{f} \underline{(fy)})∗∗R(\underline{(fz)}) == R(\underline{(fx)}∗_{f} \underline{(fy)}∗_{f} \underline{(fz)}); TimesR(Sym) ≫ R(\underline{(fx)} ∗_{f} \underline{(fy)} ∗_{f} \underline{(fz)}) == R(\underline{(fx)} ∗_{f} \underline{(fy)}) ∗ ∗R(\underline{(fz)}); ==Transitivity ⊳ R(\underline{(fx)} ∗_{f} \underline{(fy)} ∗_{f} \underline{(fz)}) == R(\underline{(fx)} ∗_{f} \underline{(fy)}) ∗ ∗R(\underline{(fz)}) ⊳ R(\underline{(fx)} ∗_{f} \underline{(fy)}) ∗ ∗R(\underline{(fz)}) == R(\underline{(fx)} ∗_{f} \underline{(fy)} ∗_{f} \underline{(fz)}) ≫ R(\underline{(fx)} ∗_{f} \underline{(fy)} ∗_{f} \underline{(fz)}) == R(\underline{(fx)} ∗_{f} \underline{(fy)} ∗_{f} \underline{(fz)})⌉, p_{0}, c)] (*** NUMMER 12 ***) [Times1f →^{stmt} ZFsub ⊢ ∀\underline{m}: ∀\underline{(fx)}: \underline{(fx)} ∗_{f} 1f =_{f} \underline{(fx)}]

[Times1f →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{m}: ∀\underline{(fx)}: TimesF ≫ \underline{(fx)} ∗_{f} 1f[\underline{m}] = \underline{(fx)}[\underline{m}] ∗ 1f[\underline{m}]; 1f ≫ 1f[\underline{m}] = 1; EqMultiplicationLeft ⊳ 1f[\underline{m}] = 1 ≫ \underline{(fx)}[\underline{m}] ∗ 1f[\underline{m}] = \underline{(fx)}[\underline{m}] ∗ 1; times1 ≫ \underline{(fx)}[\underline{m}] ∗ 1 = \underline{(fx)}[\underline{m}]; eqTransitivity4 ⊳ \underline{(fx)} ∗_{f} 1f[\underline{m}] = \underline{(fx)}[\underline{m}] ∗ 1f[\underline{m}] ⊳ \underline{(fx)}[\underline{m}] ∗ 1f[\underline{m}] = \underline{(fx)}[\underline{m}] ∗ 1 ⊳ \underline{(fx)}[\underline{m}] ∗ 1 = \underline{(fx)}[\underline{m}] ≫ \underline{(fx)}∗_{f} 1f[\underline{m}] = \underline{(fx)}[\underline{m}]; To = f ⊳\underline{(fx)}∗_{f} 1f[\underline{m}] = \underline{(fx)}[\underline{m}] ≫ \underline{(fx)}∗_{f} 1f =_{f} \underline{(fx)}⌉, p_{0}, c)]

[Times01 →^{stmt} ZFsub ⊢ ∀\underline{(fx)}: R(\underline{(fx)}) ∗ ∗R(1f) == R(\underline{(fx)})]

[Times01 →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{(fx)}: Times1f ≫ \underline{(fx)} ∗_{f} 1f =_{f} \underline{(fx)}; = fToSameF ⊳ \underline{(fx)} ∗_{f} 1f =_{f} \underline{(fx)} ≫ SF(\underline{(fx)} ∗_{f} 1f, \underline{(fx)}); f2R(Times) ⊳ SF(\underline{(fx)} ∗_{f} 1f, \underline{(fx)}) ≫ R(\underline{(fx)}) ∗ ∗R(1f) == R(\underline{(fx)}); Repetition ⊳ R(\underline{(fx)}) ∗ ∗R(1f) == R(\underline{(fx)}) ≫ R(\underline{(fx)}) ∗ ∗R(1f) == R(\underline{(fx)})⌉, p_{0}, c)] (*** NUMMER 14 ***) [TimesCommutativity(F) →^{stmt} ZFsub ⊢ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: \underline{(fx)}∗_{f} \underline{(fy)} =_{f} \underline{(fy)}∗_{f} \underline{(fx)}]

[TimesCommutativity(F) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: TimesF ≫ \underline{(fx)} ∗_{f} \underline{(fy)}[\underline{m}] = \underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}]; timesCommutativity ≫ \underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}] = \underline{(fy)}[\underline{m}] ∗ \underline{(fx)}[\underline{m}]; TimesF(Sym) ≫ \underline{(fy)}[\underline{m}] ∗ \underline{(fx)}[\underline{m}] = \underline{(fy)} ∗_{f} \underline{(fx)}[\underline{m}]; eqTransitivity4 ⊳ \underline{(fx)} ∗_{f} \underline{(fy)}[\underline{m}] = \underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}] ⊳ \underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}] = \underline{(fy)}[\underline{m}] ∗ \underline{(fx)}[\underline{m}] ⊳ \underline{(fy)}[\underline{m}] ∗ \underline{(fx)}[\underline{m}] = \underline{(fy)} ∗_{f} \underline{(fx)}[\underline{m}] ≫ \underline{(fx)} ∗_{f} \underline{(fy)}[\underline{m}] = \underline{(fy)} ∗_{f} \underline{(fx)}[\underline{m}]; To = f ⊳ \underline{(fx)} ∗_{f} \underline{(fy)}[\underline{m}] = \underline{(fy)} ∗_{f} \underline{(fx)}[\underline{m}] ≫ \underline{(fx)} ∗_{f} \underline{(fy)} =_{f} \underline{(fy)} ∗_{f} \underline{(fx)}⌉, p_{0}, c)]

[TimesCommutativity(R) →^{stmt} ZFsub ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: R(\underline{(fx)}) ∗ ∗R(\underline{(fy)}) == R(\underline{(fy)}) ∗ ∗R(\underline{(fx)})]

[TimesCommutativity(R) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: TimesCommutativity(F) ≫ \underline{(fx)} ∗_{f} \underline{(fy)} =_{f} \underline{(fy)} ∗_{f} \underline{(fx)}; = fToSameF ⊳ \underline{(fx)} ∗_{f} \underline{(fy)} =_{f} \underline{(fy)} ∗_{f} \underline{(fx)} ≫ SF(\underline{(fx)} ∗_{f} \underline{(fy)}, \underline{(fy)} ∗_{f} \underline{(fx)}); f2R(Times) ⊳ SF(\underline{(fx)} ∗_{f} \underline{(fy)}, \underline{(fy)} ∗_{f} \underline{(fx)}) ≫ R(\underline{(fx)}) ∗ ∗R(\underline{(fy)}) == R(\underline{(fy)} ∗_{f} \underline{(fx)}); TimesR(Sym) ≫ R(\underline{(fy)} ∗_{f} \underline{(fx)}) == R(\underline{(fy)}) ∗ ∗R(\underline{(fx)}); == Transitivity ⊳ R(\underline{(fx)}) ∗ ∗R(\underline{(fy)}) == R(\underline{(fy)} ∗_{f} \underline{(fx)}) ⊳ R(\underline{(fy)} ∗_{f} \underline{(fx)}) == R(\underline{(fy)}) ∗ ∗R(\underline{(fx)}) ≫ R(\underline{(fx)}) ∗ ∗R(\underline{(fy)}) == R(\underline{(fy)}) ∗ ∗R(\underline{(fx)})⌉, p_{0}, c)] (*** NUMMER 15 ***) [Distribution(F) →^{stmt} ZFsub ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: \underline{(fx)} ∗_{f} \underline{(fy)} +_{f} \underline{(fz)} =_{f} \underline{(fx)} ∗_{f} \underline{(fy)} +_{f} \underline{(fx)} ∗_{f} \underline{(fz)}]

[Distribution(F) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: TimesF ≫ \underline{(fx)} ∗_{f} \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] = \underline{(fx)}[\underline{m}] ∗ \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}]; PlusF ≫ \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] = \underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}]; EqMultiplicationLeft ⊳ \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] = \underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}] ≫ \underline{(fx)}[\underline{m}] ∗ \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] = \underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}]; Distribution ≫ \underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}] = \underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}] + \underline{(fx)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}]; TimesF(Sym) ≫ \underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}] = \underline{(fx)} ∗_{f} \underline{(fy)}[\underline{m}]; TimesF(Sym) ≫ \underline{(fx)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}] = \underline{(fx)}∗_{f} \underline{(fz)}[\underline{m}]; AddEquations⊳\underline{(fx)}[\underline{m}]∗\underline{(fy)}[\underline{m}] = \underline{(fx)}∗_{f} \underline{(fy)}[\underline{m}]⊳\underline{(fx)}[\underline{m}]∗\underline{(fz)}[\underline{m}] = \underline{(fx)} ∗_{f} \underline{(fz)}[\underline{m}] ≫ \underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}] + \underline{(fx)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}] = \underline{(fx)} ∗_{f} \underline{(fy)}[\underline{m}] + \underline{(fx)} ∗_{f} \underline{(fz)}[\underline{m}]; PlusF(Sym) ≫ \underline{(fx)} ∗_{f} \underline{(fy)}[\underline{m}] + \underline{(fx)} ∗_{f} \underline{(fz)}[\underline{m}] = \underline{(fx)} ∗_{f} \underline{(fy)} +_{f} \underline{(fx)} ∗_{f} \underline{(fz)}[\underline{m}]; eqTransitivity6 ⊳ \underline{(fx)} ∗_{f} \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] = \underline{(fx)}[\underline{m}] ∗ \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] ⊳ \underline{(fx)}[\underline{m}] ∗ \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] = \underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}] ⊳ \underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}] = \underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}] + \underline{(fx)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}] ⊳ \underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}] + \underline{(fx)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}] = \underline{(fx)} ∗_{f} \underline{(fy)}[\underline{m}] + \underline{(fx)} ∗_{f} \underline{(fz)}[\underline{m}] ⊳ \underline{(fx)} ∗_{f} \underline{(fy)}[\underline{m}] + \underline{(fx)} ∗_{f} \underline{(fz)}[\underline{m}] = \underline{(fx)} ∗_{f} \underline{(fy)} +_{f} \underline{(fx)} ∗_{f} \underline{(fz)}[\underline{m}] ≫ \underline{(fx)} ∗_{f} \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] = \underline{(fx)} ∗_{f} \underline{(fy)} +_{f} \underline{(fx)} ∗_{f} \underline{(fz)}[\underline{m}]; To = f ⊳ \underline{(fx)} ∗_{f} \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] = \underline{(fx)}∗_{f} \underline{(fy)}+_{f} \underline{(fx)}∗_{f} \underline{(fz)}[\underline{m}] ≫ \underline{(fx)}∗_{f} \underline{(fy)}+_{f} \underline{(fz)} =_{f} \underline{(fx)}∗_{f} \underline{(fy)}+_{f} \underline{(fx)}∗_{f} \underline{(fz)}⌉, p_{0}, c)]

[Distribution(R) →^{stmt} ZFsub ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: R(\underline{(fx)}) ∗ ∗R(\underline{(fy)} +_{f} \underline{(fz)}) == R((fx) +_{f} (fy))] [Distribution(R) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: Distribution(F) ≫ \underline{(fx)}∗_{f} \underline{(fy)}+_{f} \underline{(fz)} =_{f} \underline{(fx)}∗_{f} \underline{(fy)}+_{f} \underline{(fx)}∗_{f} \underline{(fz)}; = fToSameF⊳\underline{(fx)}∗_{f} \underline{(fy)}+_{f} \underline{(fz)} =_{f} \underline{(fx)} ∗_{f} \underline{(fy)} +_{f} \underline{(fx)} ∗_{f} \underline{(fz)} ≫ SF(\underline{(fx)} ∗_{f} \underline{(fy)} +_{f} \underline{(fz)}, \underline{(fx)} ∗_{f} \underline{(fy)} +_{f} \underline{(fx)} ∗_{f} \underline{(fz)}); f2R(Times) ⊳ SF(\underline{(fx)} ∗_{f} \underline{(fy)} +_{f} \underline{(fz)}, \underline{(fx)} ∗_{f} \underline{(fy)} +_{f} \underline{(fx)} ∗_{f} \underline{(fz)}) ≫ R(\underline{(fx)}) ∗ ∗R(\underline{(fy)} +_{f} \underline{(fz)}) == R(\underline{(fx)} ∗_{f} \underline{(fy)} +_{f} \underline{(fx)} ∗_{f} \underline{(fz)}); PlusR(Sym) ≫ R(\underline{(fx)} ∗_{f} \underline{(fy)} +_{f} \underline{(fx)} ∗_{f} \underline{(fz)}) == R((fx) +_{f} (fy)); ==Transitivity ⊳ R(\underline{(fx)}) ∗ ∗R(\underline{(fy)}+_{f} \underline{(fz)}) == R(\underline{(fx)}∗_{f} \underline{(fy)}+_{f} \underline{(fx)}∗_{f} \underline{(fz)})⊳R(\underline{(fx)}∗_{f} \underline{(fy)}+_{f} \underline{(fx)}∗_{f} \underline{(fz)}) == R((fx) +_{f} (fy)) ≫ R(\underline{(fx)}) ∗ ∗R(\underline{(fy)} +_{f} \underline{(fz)}) == R((fx) +_{f} (fy))⌉, p_{0}, c)] (*** tex-definitioner ***) [(ϵ) →^{tex} “(\epsilon)”]

[(fx) →^{tex} “(fx)”]

[(fy) →^{tex} “(fy)”]

[(fz) →^{tex} “(fz)”]

[(fv) →^{tex} “(fu)”]

[(fv) →^{tex} “(fv)”]

[(rx) →^{tex} “(rx)”]

[(ry) →^{tex} “(ry)”]

[(rz) →^{tex} “(rz)”]

[(ru) →^{tex} “(ru)”]

[ϵ →^{tex} “\epsilon”]

[FX →^{tex} “FX”]

[FY →^{tex} “FY”]

[FZ →^{tex} “FZ”]

[FU →^{tex} “FU”]

[FV →^{tex} “FV”]

[RX →^{tex} “RX”]

[RY →^{tex} “RY”]

[RZ →^{tex} “RZ”]

[RU →^{tex} “RU”]

[Ex3 →^{tex} “Ex3”]

[0 →^{tex} “0”]

[1 →^{tex} “1”]

[(−1) →^{tex} “(-1)”]

[2 →^{tex} “2”]

[1/2 →^{tex} “1/2”]

[0f →^{tex} “0f”]

[00 →^{tex} “00”]

[x = y →^{tex} “#1.

= #2.”]

[x ≠ y →^{tex} “#1.

\neq #2.”]

[x < y →^{tex} “#1.

< #2.”]

[x <= y →^{tex} “#1.

<= #2.”]

[x =_{f} y →^{tex} “#1.

= {f}#2.”]

[x <_{f} y →^{tex} “#1.

< {f}#2.”] [SF(x, y) →^{tex} “SF(#1. , #2. )”]

[x == y →^{tex} “#1.

== #2.”]

[x << y →^{tex} “#1.

<< #2.”]

[x <<== y →^{tex} “#1.

<<== #2.”]

[x[y] →^{tex} “#1.

[#2. ]”]

[−x →^{tex} “-#1.”]

[−_{f}x →^{tex} “- {f}#1.”]

[x + y →^{tex} “#1.

+#2.”]

[x − y →^{tex} “#1.

-#2.”]

[(fx) +_{f} (fy) →^{tex} “#1. + {f}#2.”]

[(fx) −_{f} (fy) →^{tex} “#1.

- {f}#2.”]

[(fx) ∗_{f} (fy) →^{tex} “#1.

∗ {f}#2.”]

[x + +y →^{tex} “#1.

++#2.”]

[R((fx)) − −R((fy)) →^{tex} “R(#1. ) -- R(#2. )”]

[x ∗ y →^{tex} “#1.

∗#2.”]

[x ∗ ∗y →^{tex} “#1.

∗∗#2.”]

[leqReflexivity →^{tex} “leqReflexivity”]

[recx →^{tex} “rec#1.”]

[|x| →^{tex} “|#1.|”]

[if(x, y, z) →^{tex} “if(#1. , #2. , #3. )”]

[R(x) →^{tex} “R(#1.

)”]

[− − R(x) →^{tex} “--R(#1.

)”] [leqAntisymmetryAxiom →^{tex} “leqAntisymmetryAxiom”] [leqTransitivityAxiom →^{tex} “leqTransitivityAxiom”] [leqTotality →^{tex} “leqTotality”] [leqAdditionAxiom →^{tex} “leqAdditionAxiom”] [leqMultiplicationAxiom →^{tex} “leqMultiplicationAxiom”] [plusAssociativity →^{tex} “plusAssociativity”] [plusCommutativity →^{tex} “plusCommutativity”] [Negative →^{tex} “Negative”] [plus0 →^{tex} “plus0”] [timesAssociativity →^{tex} “timesAssociativity”] [timesCommutativity →^{tex} “timesCommutativity”] [ReciprocalAxiom →^{tex} “ReciprocalAxiom”] [times1 →^{tex} “times1”] [plusAssociativity →^{tex} “plusAssociativity”] [plusCommutativity →^{tex} “plusCommutativity”] [Negative →^{tex} “Negative”] [Distribution →^{tex} “Distribution”] [0not1 →^{tex} “0not1”] [equalityAxiom →^{tex} “equalityAxiom”] [eqLeqAxiom →^{tex} “eqLeqAxiom”] [eqAdditionAxiom →^{tex} “eqAdditionAxiom”] [eqMultiplicationAxiom →^{tex} “eqMultiplicationAxiom”] [SENC1 →^{tex} “SENC1”] [SENC2 →^{tex} “SENC2”] [IfThenElse(T) →^{tex} “IfThenElse(T)”] [IfThenElse(F) →^{tex} “IfThenElse(F)”] [From = f →^{tex} “From=f”] [To = f →^{tex} “To=f”] [From < f →^{tex} “From<f”] [To < f →^{tex} “To<f”] [PlusF →^{tex} “PlusF”] [MinusF →^{tex} “MinusF”] [TimesF →^{tex} “TimesF”]

[0f →^{tex} “0f”]

[1f →^{tex} “1f”]

[FromSF →^{tex} “FromSF”] [ToSF →^{tex} “ToSF”] [To == XX →^{tex} “To==XX”] [From ==→^{tex} “From==”]

[To ==→^{tex} “To==”]

[From << XX →^{tex} “From<<XX”] [From << (1) →^{tex} “From<<(1)”] [From << (2) →^{tex} “From<<(2)”] [to << XX →^{tex} “to<<XX”] [From <<→^{tex} “From<<”]

[To <<→^{tex} “To<<”]

[FromInR →^{tex} “FromInR”] [PlusR →^{tex} “PlusR”] [TimesR →^{tex} “TimesR”] [leqAntisymmetry →^{tex} “leqAntisymmetry”] [leqTransitivity →^{tex} “leqTransitivity”] [leqAddition →^{tex} “leqAddition”] [Reciprocal →^{tex} “Reciprocal”] [Equality →^{tex} “Equality”] [eqLeq →^{tex} “eqLeq”] [eqAddition →^{tex} “eqAddition”] [eqMultiplication →^{tex} “eqMultiplication”] [ToNegatedImply →^{tex} “ToNegatedImply”] [TND →^{tex} “TND”] [ImplyNegation →^{tex} “ImplyNegation”] [FromNegations →^{tex} “FromNegations”] [From3Disjuncts →^{tex} “From3Disjuncts”] [NegateDisjunct1 →^{tex} “NegateDisjunct1”] [NegateDisjunct2 →^{tex} “NegateDisjunct2”] [ExpandDisjuncts →^{tex} “ExpandDisjuncts”] [From2 ∗ 2Disjuncts →^{tex} “From2∗2Disjuncts”] [eqReflexivity →^{tex} “eqReflexivity”] [eqSymmetry →^{tex} “eqSymmetry”] [eqTransitivity →^{tex} “eqTransitivity”] [eqTransitivity4 →^{tex} “eqTransitivity4”] [eqTransitivity5 →^{tex} “eqTransitivity5”] [eqTransitivity6 →^{tex} “eqTransitivity6”] [plus0Left →^{tex} “plus0Left”] [times1Left →^{tex} “times1Left”] [EqMultiplicationLeft →^{tex} “EqMultiplicationLeft”] [DistributionOut →^{tex} “DistributionOut”] [Three2twoTerms →^{tex} “Three2twoTerms”] [Three2threeTerms →^{tex} “Three2threeTerms”] [Three2threeFactors →^{tex} “Three2threeFactors”] [AddEquations →^{tex} “AddEquations”] [SubtractEquations →^{tex} “SubtractEquations”] [SubtractEquationsLeft →^{tex} “SubtractEquationsLeft”] [EqNegated →^{tex} “EqNegated”] [PositiveToRight(Eq) →^{tex} “PositiveToRight(Eq)”] [PositiveToLeft(Eq)(1term) →^{tex} “PositiveToLeft(Eq)(1 term)”] [NegativeToLeft(Eq) →^{tex} “NegativeToLeft(Eq)”] [UniqueNegative →^{tex} “UniqueNegative”] [DoubleMinus →^{tex} “DoubleMinus”] [LessNeq →^{tex} “LessNeq”] [NeqSymmetry →^{tex} “NeqSymmetry”] [NeqNegated →^{tex} “NeqNegated”] [SubNeqRight →^{tex} “SubNeqRight”] [SubNeqLeft →^{tex} “SubNeqLeft”] [NeqAddition →^{tex} “NeqAddition”] [NeqMultiplication →^{tex} “NeqMultiplication”] [LeqLessEq →^{tex} “LeqLessEq”] [LessLeq →^{tex} “LessLeq”] [FromLeqGeq →^{tex} “FromLeqGeq”] [subLeqRight →^{tex} “subLeqRight”] [subLeqLeft →^{tex} “subLeqLeft”] [Leq + 1 →^{tex} “Leq+1”] [PositiveToRight(Leq) →^{tex} “PositiveToRight(Leq)”] [PositiveToRight(Leq)(1term) →^{tex} “PositiveToRight(Leq)(1 term)”] [LeqAdditionLeft →^{tex} “LeqAdditionLeft”] [leqSubtraction →^{tex} “leqSubtraction”] [leqSubtractionLeft →^{tex} “leqSubtractionLeft”] [leqMultiplication →^{tex} “leqMultiplication”] [thirdGeq →^{tex} “thirdGeq”] [LeqNegated →^{tex} “LeqNegated”] [AddEquations(Leq) →^{tex} “AddEquations(Leq)”] [LeqNeqLess →^{tex} “LeqNeqLess”] [FromLess →^{tex} “FromLess”] [ToLess →^{tex} “ToLess”] [fromNotLess →^{tex} “fromNotLess”] [toNotLess →^{tex} “toNotLess”] [LessAddition →^{tex} “LessAddition”] [LessAdditionLeft →^{tex} “LessAdditionLeft”] [LessMultiplication →^{tex} “LessMultiplication”] [LessMultiplicationLeft →^{tex} “LessMultiplicationLeft”] [LessDivision →^{tex} “LessDivision”] [AddEquations(Less) →^{tex} “AddEquations(Less)”] [NegativeLessPositive →^{tex} “NegativeLessPositive”] [leqLessTransitivity →^{tex} “leqLessTransitivity”] [LessLeqTransitivity →^{tex} “LessLeqTransitivity”] [LessTransitivity →^{tex} “LessTransitivity”] [LessTotality →^{tex} “LessTotality”] [SubLessRight →^{tex} “SubLessRight”] [SubLessLeft →^{tex} “SubLessLeft”] [LessNegated →^{tex} “LessNegated”] [PositiveNegated →^{tex} “PositiveNegated”] [NonpositiveNegated →^{tex} “NonpositiveNegated”] [NegativeNegated →^{tex} “NegativeNegated”] [NonnegativeNegated →^{tex} “NonnegativeNegated”] [PositiveHalved →^{tex} “PositiveHalved”] [NonnegativeNumerical →^{tex} “NonnegativeNumerical”] [NegativeNumerical →^{tex} “NegativeNumerical”] [PositiveNumerical →^{tex} “PositiveNumerical”]

[|0| = 0 →^{tex} “|0|=0”]

[0 <= |x| →^{tex} “0<=|x|”]

[SameNumerical →^{tex} “SameNumerical”] [SignNumerical(+) →^{tex} “SignNumerical(+)”] [SignNumerical →^{tex} “SignNumerical”] [NumericalDifference →^{tex} “NumericalDifference”] [SplitNumericalSumHelper →^{tex} “SplitNumericalSumHelper”] [splitNumericalSum(++) →^{tex} “splitNumericalSum(++)”] [splitNumericalSum(−−) →^{tex} “splitNumericalSum(--)”] [splitNumericalSum(+ − small) →^{tex} “splitNumericalSum(+-small)”] [splitNumericalSum(+ − big) →^{tex} “splitNumericalSum(+-big)”] [splitNumericalSum(+−) →^{tex} “splitNumericalSum(+-)”] [splitNumericalSum(−+) →^{tex} “splitNumericalSum(-+)”] [splitNumericalSum →^{tex} “splitNumericalSum”] [insertMiddleTerm(Numerical) →^{tex} “insertMiddleTerm(Numerical)”] [x + y = zBackwards →^{tex} “x+y=zBackwards”] [x ∗ y = zBackwards →^{tex} “x∗y=zBackwards”] [x = x + (y − y) →^{tex} “x=x+(y-y)”] [x = x + y − y →^{tex} “x=x+y-y”]

[→^{tex} “ ”]

[insertMiddleTerm(Sum) →^{tex} “insertMiddleTerm(Sum)”] [insertMiddleTerm(Difference) →^{tex} “insertMiddleTerm(Difference)”] [x ∗ 0 + x = x →^{tex} “x∗0+x=x”]

[x ∗ 0 = 0 →^{tex} “x∗0=0”]

[(−1) ∗ (−1) + (−1) ∗ 1 = 0 →^{tex} “(-1)∗(-1)+(-1)∗1=0”]

[(−1) ∗ (−1) = 1 →^{tex} “(-1)∗(-1)=1”]

[0 < 1Helper →^{tex} “0<1Helper”]

[0 < 1 →^{tex} “0<1”]

[0 < 2 →^{tex} “0<2”]

[0 < 1/2 →^{tex} “0<1/2”]

[TwoWholes →^{tex} “TwoWholes”] [TwoHalves →^{tex} “TwoHalves”] [−x − y = −(x + y) →^{tex} “-x-y=-(x+y)”] [MinusNegated →^{tex} “MinusNegated”] [Times(−1) →^{tex} “Times(-1)”] [Times(−1)Left →^{tex} “Times(-1)Left”]

[−0 = 0 →^{tex} “-0=0”]

[negativeToLeft(Leq) →^{tex} “negativeToLeft(Leq)”] [SFsymmetry →^{tex} “SFsymmetry”] [SFtransitivity →^{tex} “SFtransitivity”] [= fToSameF →^{tex} “=fToSameF ”] [PlusF(Sym) →^{tex} “PlusF(Sym)”] [TimesF(Sym) →^{tex} “TimesF(Sym)”] [f2R(Plus) →^{tex} “f2R(Plus)”] [f2R(Times) →^{tex} “f2R(Times)”] [PlusR(Sym) →^{tex} “PlusR(Sym)”] [TimesR(Sym) →^{tex} “TimesR(Sym)”] [LessLeq(R) →^{tex} “LessLeq(R)”] [eqLeq(R) →^{tex} “eqLeq(R)”] [ThirdGeqSeries →^{tex} “ThirdGeqSeries”] [SubLessRight(R) →^{tex} “SubLessRight(R)”] [SubLessLeft(R) →^{tex} “SubLessLeft(R)”] [<< TransitivityHelper(Q) →^{tex} “<<TransitivityHelper(Q)”] [<< Transitivity →^{tex} “<<Transitivity”] [<<== Reflexivity →^{tex} “<<==Reflexivity”] [<<== AntisymmetryHelper(Q) →^{tex} “<<==AntisymmetryHelper(Q)”] [<<== Antisymmetry →^{tex} “<<==Antisymmetry”] [<<== Transitivity →^{tex} “<<==Transitivity”] [Plus0f →^{tex} “Plus0f”] [Plus00 →^{tex} “Plus00”] [== Addition →^{tex} “==Addition”] [== AdditionLeft →^{tex} “==AdditionLeft”] [<< Addition →^{tex} “<<Addition”] [<<== Addition →^{tex} “<<==Addition”] [PlusAssociativity(F) →^{tex} “PlusAssociativity(F)”] [PlusAssociativity(R) →^{tex} “PlusAssociativity(R)”] [Negative(R) →^{tex} “Negative(R)”] [PlusCommutativity(F) →^{tex} “PlusCommutativity(F)”] [PlusCommutativity(R) →^{tex} “PlusCommutativity(R)”] [Times1f →^{tex} “Times1f”] [Times01 →^{tex} “Times01”] [TimesAssociativity(F) →^{tex} “TimesAssociativity(F)”] [TimesAssociativity(R) →^{tex} “TimesAssociativity(R)”] [TimesCommutativity(F) →^{tex} “TimesCommutativity(F)”] [TimesCommutativity(R) →^{tex} “TimesCommutativity(R)”] [Distribution(F) →^{tex} “Distribution(F)”] [Distribution(R) →^{tex} “Distribution(R)”]