Logiweb aspects of lemma lessMultiplication(F) helper in pyk

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The predefined "pyk" aspect

define pyk of lemma lessMultiplication(F) helper as text unicode start of text unicode small l unicode small e unicode small m unicode small m unicode small a unicode space unicode small l unicode small e unicode small s unicode small s unicode capital m unicode small u unicode small l unicode small t unicode small i unicode small p unicode small l unicode small i unicode small c unicode small a unicode small t unicode small i unicode small o unicode small n unicode left parenthesis unicode capital f unicode right parenthesis unicode space unicode small h unicode small e unicode small l unicode small p unicode small e unicode small r unicode end of text end unicode text end text end define

The predefined "tex" aspect

define tex of lemma lessMultiplication(F) helper as text unicode start of text unicode capital l unicode small e unicode small s unicode small s unicode capital m unicode small u unicode small l unicode small t unicode small i unicode small p unicode small l unicode small i unicode small c unicode small a unicode small t unicode small i unicode small o unicode small n unicode left parenthesis unicode capital f unicode right parenthesis unicode left parenthesis unicode capital h unicode small e unicode small l unicode small p unicode small e unicode small r unicode right parenthesis unicode end of text end unicode text end text end define

The user defined "the statement aspect" aspect

define statement of lemma lessMultiplication(F) helper as system Q infer all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var ep1 end metavar indeed all metavar var ep2 end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep1 end metavar imply not0 not0 0 = metavar var ep1 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep1 end metavar imply for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep2 end metavar imply not0 not0 0 = metavar var ep2 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] <= [ metavar var fz end metavar ; metavar var m end metavar ] + - metavar var ep2 end metavar imply not0 not0 0 <= metavar var ep1 end metavar * metavar var ep2 end metavar imply not0 not0 0 = metavar var ep1 end metavar * metavar var ep2 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + - metavar var ep1 end metavar * metavar var ep2 end metavar end define

The user defined "the proof aspect" aspect

define proof of lemma lessMultiplication(F) helper as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var ep1 end metavar indeed all metavar var ep2 end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep1 end metavar imply not0 not0 0 = metavar var ep1 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep1 end metavar infer for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep2 end metavar imply not0 not0 0 = metavar var ep2 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] <= [ metavar var fz end metavar ; metavar var m end metavar ] + - metavar var ep2 end metavar infer lemma a4 at metavar var m end metavar modus ponens for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep1 end metavar imply not0 not0 0 = metavar var ep1 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep1 end metavar conclude not0 not0 0 <= metavar var ep1 end metavar imply not0 not0 0 = metavar var ep1 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep1 end metavar cut lemma a4 at metavar var m end metavar modus ponens for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep2 end metavar imply not0 not0 0 = metavar var ep2 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] <= [ metavar var fz end metavar ; metavar var m end metavar ] + - metavar var ep2 end metavar conclude not0 not0 0 <= metavar var ep2 end metavar imply not0 not0 0 = metavar var ep2 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] <= [ metavar var fz end metavar ; metavar var m end metavar ] + - metavar var ep2 end metavar cut prop lemma first conjunct modus ponens not0 not0 0 <= metavar var ep1 end metavar imply not0 not0 0 = metavar var ep1 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep1 end metavar conclude not0 0 <= metavar var ep1 end metavar imply not0 not0 0 = metavar var ep1 end metavar cut prop lemma first conjunct modus ponens not0 not0 0 <= metavar var ep2 end metavar imply not0 not0 0 = metavar var ep2 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] <= [ metavar var fz end metavar ; metavar var m end metavar ] + - metavar var ep2 end metavar conclude not0 0 <= metavar var ep2 end metavar imply not0 not0 0 = metavar var ep2 end metavar cut lemma positiveFactors modus ponens not0 0 <= metavar var ep1 end metavar imply not0 not0 0 = metavar var ep1 end metavar modus ponens not0 0 <= metavar var ep2 end metavar imply not0 not0 0 = metavar var ep2 end metavar conclude not0 0 <= metavar var ep1 end metavar * metavar var ep2 end metavar imply not0 not0 0 = metavar var ep1 end metavar * metavar var ep2 end metavar cut all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var ep1 end metavar indeed all metavar var ep2 end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep1 end metavar imply not0 not0 0 = metavar var ep1 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep1 end metavar infer for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep2 end metavar imply not0 not0 0 = metavar var ep2 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] <= [ metavar var fz end metavar ; metavar var m end metavar ] + - metavar var ep2 end metavar infer metavar var n end metavar <= metavar var m end metavar infer lemma a4 at metavar var m end metavar modus ponens for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep1 end metavar imply not0 not0 0 = metavar var ep1 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep1 end metavar conclude not0 not0 0 <= metavar var ep1 end metavar imply not0 not0 0 = metavar var ep1 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep1 end metavar cut lemma a4 at metavar var m end metavar modus ponens for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep2 end metavar imply not0 not0 0 = metavar var ep2 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] <= [ metavar var fz end metavar ; metavar var m end metavar ] + - metavar var ep2 end metavar conclude not0 not0 0 <= metavar var ep2 end metavar imply not0 not0 0 = metavar var ep2 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] <= [ metavar var fz end metavar ; metavar var m end metavar ] + - metavar var ep2 end metavar cut prop lemma first conjunct modus ponens not0 not0 0 <= metavar var ep1 end metavar imply not0 not0 0 = metavar var ep1 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep1 end metavar conclude not0 0 <= metavar var ep1 end metavar imply not0 not0 0 = metavar var ep1 end metavar cut prop lemma first conjunct modus ponens not0 not0 0 <= metavar var ep2 end metavar imply not0 not0 0 = metavar var ep2 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] <= [ metavar var fz end metavar ; metavar var m end metavar ] + - metavar var ep2 end metavar conclude not0 0 <= metavar var ep2 end metavar imply not0 not0 0 = metavar var ep2 end metavar cut prop lemma second conjunct modus ponens not0 not0 0 <= metavar var ep1 end metavar imply not0 not0 0 = metavar var ep1 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep1 end metavar conclude metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep1 end metavar cut 1rule mp modus ponens metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep1 end metavar modus ponens metavar var n end metavar <= metavar var m end metavar conclude [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep1 end metavar cut prop lemma second conjunct modus ponens not0 not0 0 <= metavar var ep2 end metavar imply not0 not0 0 = metavar var ep2 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] <= [ metavar var fz end metavar ; metavar var m end metavar ] + - metavar var ep2 end metavar conclude metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] <= [ metavar var fz end metavar ; metavar var m end metavar ] + - metavar var ep2 end metavar cut 1rule mp modus ponens metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] <= [ metavar var fz end metavar ; metavar var m end metavar ] + - metavar var ep2 end metavar modus ponens metavar var n end metavar <= metavar var m end metavar conclude [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] <= [ metavar var fz end metavar ; metavar var m end metavar ] + - metavar var ep2 end metavar cut axiom natType conclude metavar var m end metavar in0 N cut lemma 0f modus ponens metavar var m end metavar in0 N conclude [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] = 0 cut lemma subLeqLeft modus ponens [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] = 0 modus ponens [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] <= [ metavar var fz end metavar ; metavar var m end metavar ] + - metavar var ep2 end metavar conclude 0 <= [ metavar var fz end metavar ; metavar var m end metavar ] + - metavar var ep2 end metavar cut lemma lessMultiplication(F) helper2 modus ponens not0 0 <= metavar var ep1 end metavar imply not0 not0 0 = metavar var ep1 end metavar modus ponens not0 0 <= metavar var ep2 end metavar imply not0 not0 0 = metavar var ep2 end metavar modus ponens [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep1 end metavar modus ponens 0 <= [ metavar var fz end metavar ; metavar var m end metavar ] + - metavar var ep2 end metavar conclude [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] + - metavar var ep1 end metavar * metavar var ep2 end metavar cut lemma timesF(Sym) conclude [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] = [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] cut lemma subLeqLeft modus ponens [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] = [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] modus ponens [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] + - metavar var ep1 end metavar * metavar var ep2 end metavar conclude [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] + - metavar var ep1 end metavar * metavar var ep2 end metavar cut lemma timesF(Sym) conclude [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] = [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] cut lemma eqAddition modus ponens [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] = [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] conclude [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] + - metavar var ep1 end metavar * metavar var ep2 end metavar = [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + - metavar var ep1 end metavar * metavar var ep2 end metavar cut lemma subLeqRight modus ponens [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] + - metavar var ep1 end metavar * metavar var ep2 end metavar = [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + - metavar var ep1 end metavar * metavar var ep2 end metavar modus ponens [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] + - metavar var ep1 end metavar * metavar var ep2 end metavar conclude [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + - metavar var ep1 end metavar * metavar var ep2 end metavar cut all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var ep1 end metavar indeed all metavar var ep2 end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed 1rule deduction modus ponens all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var ep1 end metavar indeed all metavar var ep2 end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep1 end metavar imply not0 not0 0 = metavar var ep1 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep1 end metavar infer for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep2 end metavar imply not0 not0 0 = metavar var ep2 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] <= [ metavar var fz end metavar ; metavar var m end metavar ] + - metavar var ep2 end metavar infer not0 0 <= metavar var ep1 end metavar * metavar var ep2 end metavar imply not0 not0 0 = metavar var ep1 end metavar * metavar var ep2 end metavar conclude for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep1 end metavar imply not0 not0 0 = metavar var ep1 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep1 end metavar imply for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep2 end metavar imply not0 not0 0 = metavar var ep2 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] <= [ metavar var fz end metavar ; metavar var m end metavar ] + - metavar var ep2 end metavar imply not0 0 <= metavar var ep1 end metavar * metavar var ep2 end metavar imply not0 not0 0 = metavar var ep1 end metavar * metavar var ep2 end metavar cut 1rule deduction modus ponens all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var ep1 end metavar indeed all metavar var ep2 end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep1 end metavar imply not0 not0 0 = metavar var ep1 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep1 end metavar infer for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep2 end metavar imply not0 not0 0 = metavar var ep2 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] <= [ metavar var fz end metavar ; metavar var m end metavar ] + - metavar var ep2 end metavar infer metavar var n end metavar <= metavar var m end metavar infer [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + - metavar var ep1 end metavar * metavar var ep2 end metavar conclude for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep1 end metavar imply not0 not0 0 = metavar var ep1 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep1 end metavar imply for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep2 end metavar imply not0 not0 0 = metavar var ep2 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] <= [ metavar var fz end metavar ; metavar var m end metavar ] + - metavar var ep2 end metavar imply metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + - metavar var ep1 end metavar * metavar var ep2 end metavar cut prop lemma doubly conditioned join conjuncts modus ponens for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep1 end metavar imply not0 not0 0 = metavar var ep1 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep1 end metavar imply for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep2 end metavar imply not0 not0 0 = metavar var ep2 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] <= [ metavar var fz end metavar ; metavar var m end metavar ] + - metavar var ep2 end metavar imply not0 0 <= metavar var ep1 end metavar * metavar var ep2 end metavar imply not0 not0 0 = metavar var ep1 end metavar * metavar var ep2 end metavar modus ponens for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep1 end metavar imply not0 not0 0 = metavar var ep1 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep1 end metavar imply for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep2 end metavar imply not0 not0 0 = metavar var ep2 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] <= [ metavar var fz end metavar ; metavar var m end metavar ] + - metavar var ep2 end metavar imply metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + - metavar var ep1 end metavar * metavar var ep2 end metavar conclude for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep1 end metavar imply not0 not0 0 = metavar var ep1 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep1 end metavar imply for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep2 end metavar imply not0 not0 0 = metavar var ep2 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] <= [ metavar var fz end metavar ; metavar var m end metavar ] + - metavar var ep2 end metavar imply not0 not0 0 <= metavar var ep1 end metavar * metavar var ep2 end metavar imply not0 not0 0 = metavar var ep1 end metavar * metavar var ep2 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + - metavar var ep1 end metavar * metavar var ep2 end metavar end quote state proof state cache var c end expand end define