Logiweb aspects of lemma sameFmultiplication helper in pyk

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The predefined "pyk" aspect

define pyk of lemma sameFmultiplication 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 s unicode small a unicode small m unicode small e unicode capital f unicode small 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 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 sameFmultiplication helper as text unicode start of text unicode capital s unicode small a unicode small m unicode small e unicode capital f unicode small 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 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 sameFmultiplication helper as system Q infer all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var ep 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 0 <= metavar var ep end metavar * 1/ metavar var v1 end metavar imply not0 not0 0 = metavar var ep end metavar * 1/ metavar var v1 end metavar imply metavar var n end metavar <= metavar var m end metavar imply not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= metavar var ep end metavar * 1/ metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = metavar var ep end metavar * 1/ metavar var v1 end metavar imply for all objects metavar var v2 end metavar indeed not0 | [ metavar var fz end metavar ; metavar var v2 end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fz end metavar ; metavar var v2 end metavar ] | = metavar var v1 end metavar imply not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n end metavar <= metavar var m end metavar imply not0 | [ 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 ep end metavar imply not0 not0 | [ 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 ep end metavar end define

The user defined "the proof aspect" aspect

define proof of lemma sameFmultiplication helper as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var ep 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 0 <= metavar var ep end metavar * 1/ metavar var v1 end metavar imply not0 not0 0 = metavar var ep end metavar * 1/ metavar var v1 end metavar imply metavar var n end metavar <= metavar var m end metavar imply not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= metavar var ep end metavar * 1/ metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = metavar var ep end metavar * 1/ metavar var v1 end metavar infer for all objects metavar var v2 end metavar indeed not0 | [ metavar var fz end metavar ; metavar var v2 end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fz end metavar ; metavar var v2 end metavar ] | = metavar var v1 end metavar infer not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar infer metavar var n end metavar <= metavar var m end metavar infer lemma a4 at metavar var v2 end metavar modus ponens for all objects metavar var v2 end metavar indeed not0 | [ metavar var fz end metavar ; metavar var v2 end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fz end metavar ; metavar var v2 end metavar ] | = metavar var v1 end metavar conclude not0 | [ metavar var fz end metavar ; metavar var v2 end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fz end metavar ; metavar var v2 end metavar ] | = metavar var v1 end metavar cut lemma 0<=|x| conclude 0 <= | [ metavar var fz end metavar ; metavar var v2 end metavar ] | cut lemma leqLessTransitivity modus ponens 0 <= | [ metavar var fz end metavar ; metavar var v2 end metavar ] | modus ponens not0 | [ metavar var fz end metavar ; metavar var v2 end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fz end metavar ; metavar var v2 end metavar ] | = metavar var v1 end metavar conclude not0 0 <= metavar var v1 end metavar imply not0 not0 0 = metavar var v1 end metavar cut lemma positiveInverted modus ponens not0 0 <= metavar var v1 end metavar imply not0 not0 0 = metavar var v1 end metavar conclude not0 0 <= 1/ metavar var v1 end metavar imply not0 not0 0 = 1/ metavar var v1 end metavar cut lemma positiveFactors modus ponens not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar modus ponens not0 0 <= 1/ metavar var v1 end metavar imply not0 not0 0 = 1/ metavar var v1 end metavar conclude not0 0 <= metavar var ep end metavar * 1/ metavar var v1 end metavar imply not0 not0 0 = metavar var ep end metavar * 1/ metavar var v1 end metavar cut lemma a4 at metavar var m end metavar modus ponens for all objects metavar var m end metavar indeed not0 0 <= metavar var ep end metavar * 1/ metavar var v1 end metavar imply not0 not0 0 = metavar var ep end metavar * 1/ metavar var v1 end metavar imply metavar var n end metavar <= metavar var m end metavar imply not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= metavar var ep end metavar * 1/ metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = metavar var ep end metavar * 1/ metavar var v1 end metavar conclude not0 0 <= metavar var ep end metavar * 1/ metavar var v1 end metavar imply not0 not0 0 = metavar var ep end metavar * 1/ metavar var v1 end metavar imply metavar var n end metavar <= metavar var m end metavar imply not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= metavar var ep end metavar * 1/ metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = metavar var ep end metavar * 1/ metavar var v1 end metavar cut prop lemma mp2 modus ponens not0 0 <= metavar var ep end metavar * 1/ metavar var v1 end metavar imply not0 not0 0 = metavar var ep end metavar * 1/ metavar var v1 end metavar imply metavar var n end metavar <= metavar var m end metavar imply not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= metavar var ep end metavar * 1/ metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = metavar var ep end metavar * 1/ metavar var v1 end metavar modus ponens not0 0 <= metavar var ep end metavar * 1/ metavar var v1 end metavar imply not0 not0 0 = metavar var ep end metavar * 1/ metavar var v1 end metavar modus ponens metavar var n end metavar <= metavar var m end metavar conclude not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= metavar var ep end metavar * 1/ metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = metavar var ep end metavar * 1/ metavar var v1 end metavar cut lemma reciprocalToLeft(Less) modus ponens not0 0 <= metavar var v1 end metavar imply not0 not0 0 = metavar var v1 end metavar modus ponens not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= metavar var ep end metavar * 1/ metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = metavar var ep end metavar * 1/ metavar var v1 end metavar conclude not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | * metavar var v1 end metavar <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | * metavar var v1 end metavar = metavar var ep end metavar cut lemma timesF 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 fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] cut lemma timesF 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 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 fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] cut lemma eqNegated 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 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 fy end metavar ; metavar var m end metavar ] * [ metavar var fz 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 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 fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] cut lemma -x*y=-(x*y) conclude - [ metavar var fy 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 ] cut lemma eqSymmetry modus ponens - [ metavar var fy 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 ] conclude - [ metavar var fy 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 ] cut lemma eqTransitivity 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 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 fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] modus ponens - [ metavar var fy 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 ] 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 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 fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] cut lemma addEquations 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 fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m 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 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 fy end metavar ; metavar var m end metavar ] * [ metavar var fz 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 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 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 ] cut lemma distributionOutLeft 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 fz end metavar ; metavar var m end metavar ] * [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] cut lemma eqTransitivity 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 ] + - [ 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 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 ] 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 fz end metavar ; metavar var m end metavar ] * [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy 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 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 fz end metavar ; metavar var m end metavar ] * [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] cut lemma sameNumerical 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 ] + - [ 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 fz end metavar ; metavar var m end metavar ] * [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy 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 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 fz end metavar ; metavar var m end metavar ] * [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | cut lemma splitNumericalProduct conclude | [ metavar var fz end metavar ; metavar var m end metavar ] * [ metavar var fx 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 fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | cut axiom timesCommutativity conclude | [ metavar var fz end metavar ; metavar var m end metavar ] | * | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = | [ metavar var fx 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 ] | cut lemma eqTransitivity4 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 ] + - [ 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 fz end metavar ; metavar var m end metavar ] * [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | modus ponens | [ metavar var fz end metavar ; metavar var m end metavar ] * [ metavar var fx 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 fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | modus ponens | [ metavar var fz end metavar ; metavar var m end metavar ] | * | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = | [ metavar var fx 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 ] | 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 fx 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 ] | cut lemma eqSymmetry 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 ] + - [ 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 fx 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 ] | conclude | [ metavar var fx 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 ] | = | [ 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 ] | cut lemma a4 at [ metavar var fz end metavar ; metavar var m end metavar ] modus ponens for all objects metavar var v2 end metavar indeed not0 | [ metavar var fz end metavar ; metavar var v2 end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fz end metavar ; metavar var v2 end metavar ] | = metavar var v1 end metavar conclude | [ metavar var fz end metavar ; metavar var m end metavar ] | <= metavar var v1 end metavar cut lemma 0<=|x| conclude 0 <= | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | cut lemma leqMultiplicationLeft modus ponens 0 <= | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | modus ponens | [ metavar var fz end metavar ; metavar var m end metavar ] | <= metavar var v1 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 fz end metavar ; metavar var m end metavar ] | <= | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | * metavar var v1 end metavar cut lemma leqLessTransitivity modus ponens | [ metavar var fx 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 fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | * metavar var v1 end metavar modus ponens not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | * metavar var v1 end metavar <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | * metavar var v1 end metavar = metavar var ep end metavar conclude not0 | [ metavar var fx 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 ep end metavar imply not0 not0 | [ metavar var fx 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 ep end metavar cut lemma subLessLeft modus ponens | [ metavar var fx 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 ] | = | [ 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 ] | modus ponens not0 | [ metavar var fx 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 ep end metavar imply not0 not0 | [ metavar var fx 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 ep end metavar conclude not0 | [ 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 ep end metavar imply not0 not0 | [ 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 ep end metavar cut all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var ep 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 v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var ep 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 0 <= metavar var ep end metavar * 1/ metavar var v1 end metavar imply not0 not0 0 = metavar var ep end metavar * 1/ metavar var v1 end metavar imply metavar var n end metavar <= metavar var m end metavar imply not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= metavar var ep end metavar * 1/ metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = metavar var ep end metavar * 1/ metavar var v1 end metavar infer for all objects metavar var v2 end metavar indeed not0 | [ metavar var fz end metavar ; metavar var v2 end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fz end metavar ; metavar var v2 end metavar ] | = metavar var v1 end metavar infer not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar infer metavar var n end metavar <= metavar var m end metavar infer not0 | [ 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 ep end metavar imply not0 not0 | [ 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 ep end metavar conclude for all objects metavar var m end metavar indeed not0 0 <= metavar var ep end metavar * 1/ metavar var v1 end metavar imply not0 not0 0 = metavar var ep end metavar * 1/ metavar var v1 end metavar imply metavar var n end metavar <= metavar var m end metavar imply not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= metavar var ep end metavar * 1/ metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = metavar var ep end metavar * 1/ metavar var v1 end metavar imply for all objects metavar var v2 end metavar indeed not0 | [ metavar var fz end metavar ; metavar var v2 end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fz end metavar ; metavar var v2 end metavar ] | = metavar var v1 end metavar imply not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n end metavar <= metavar var m end metavar imply not0 | [ 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 ep end metavar imply not0 not0 | [ 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 ep end metavar end quote state proof state cache var c end expand end define