let n be non empty Nat; :: thesis: for J being non empty non void Signature

for T being non-empty MSAlgebra over J

for X being V3() GeneratorSet of T

for S1 being non empty non void b_{1} -extension n PC-correct QC-correct QCLangSignature over Union X

for L being non-empty Language of X extended_by ({}, the carrier of S1),S1

for G being QC-theory of L

for A being Formula of L

for x, y being Element of Union X st L is vf-qc-correct & L is subst-correct holds

(\for (x,y,A)) \imp (\for (y,x,A)) in G

let J be non empty non void Signature; :: thesis: for T being non-empty MSAlgebra over J

for X being V3() GeneratorSet of T

for S1 being non empty non void J -extension n PC-correct QC-correct QCLangSignature over Union X

for L being non-empty Language of X extended_by ({}, the carrier of S1),S1

for G being QC-theory of L

for A being Formula of L

for x, y being Element of Union X st L is vf-qc-correct & L is subst-correct holds

(\for (x,y,A)) \imp (\for (y,x,A)) in G

let T be non-empty MSAlgebra over J; :: thesis: for X being V3() GeneratorSet of T

for S1 being non empty non void J -extension n PC-correct QC-correct QCLangSignature over Union X

for L being non-empty Language of X extended_by ({}, the carrier of S1),S1

for G being QC-theory of L

for A being Formula of L

for x, y being Element of Union X st L is vf-qc-correct & L is subst-correct holds

(\for (x,y,A)) \imp (\for (y,x,A)) in G

let X be V3() GeneratorSet of T; :: thesis: for S1 being non empty non void J -extension n PC-correct QC-correct QCLangSignature over Union X

for L being non-empty Language of X extended_by ({}, the carrier of S1),S1

for G being QC-theory of L

for A being Formula of L

for x, y being Element of Union X st L is vf-qc-correct & L is subst-correct holds

(\for (x,y,A)) \imp (\for (y,x,A)) in G

let S1 be non empty non void J -extension n PC-correct QC-correct QCLangSignature over Union X; :: thesis: for L being non-empty Language of X extended_by ({}, the carrier of S1),S1

for G being QC-theory of L

for A being Formula of L

for x, y being Element of Union X st L is vf-qc-correct & L is subst-correct holds

(\for (x,y,A)) \imp (\for (y,x,A)) in G

let L be non-empty Language of X extended_by ({}, the carrier of S1),S1; :: thesis: for G being QC-theory of L

for A being Formula of L

for x, y being Element of Union X st L is vf-qc-correct & L is subst-correct holds

(\for (x,y,A)) \imp (\for (y,x,A)) in G

let G be QC-theory of L; :: thesis: for A being Formula of L

for x, y being Element of Union X st L is vf-qc-correct & L is subst-correct holds

(\for (x,y,A)) \imp (\for (y,x,A)) in G

let A be Formula of L; :: thesis: for x, y being Element of Union X st L is vf-qc-correct & L is subst-correct holds

(\for (x,y,A)) \imp (\for (y,x,A)) in G

let x, y be Element of Union X; :: thesis: ( L is vf-qc-correct & L is subst-correct implies (\for (x,y,A)) \imp (\for (y,x,A)) in G )

assume A1: ( L is vf-qc-correct & L is subst-correct ) ; :: thesis: (\for (x,y,A)) \imp (\for (y,x,A)) in G

then (\for (y,A)) \imp A in G by Th104;

then A2: \for (x,((\for (y,A)) \imp A)) in G by Def39;

(\for (x,((\for (y,A)) \imp A))) \imp ((\for (x,(\for (y,A)))) \imp (\for (x,A))) in G by A1, Th109;

then (\for (x,(\for (y,A)))) \imp (\for (x,A)) in G by A2, Def38;

then A3: \for (y,((\for (x,(\for (y,A)))) \imp (\for (x,A)))) in G by Def39;

consider a being object such that

A4: ( a in dom X & y in X . a ) by CARD_5:2;

consider b being object such that

A5: ( b in dom X & x in X . b ) by CARD_5:2;

J is Subsignature of S1 by Def2;

then ( dom X = the carrier of J & the carrier of J c= the carrier of S1 ) by PARTFUN1:def 2, INSTALG1:10;

then reconsider a = a, b = b as Element of S1 by A4, A5;

reconsider c = a as Element of J by A4;

vf (\for (x,y,A)) = (vf (\for (y,A))) (\) (b -singleton x) by A1, A5;

then (vf (\for (x,y,A))) . a = ((vf (\for (y,A))) . a) \ ((b -singleton x) . a) by PBOOLE:def 6;

then y nin (vf (\for (x,y,A))) . c by A1, A4, Th113;

hence (\for (x,y,A)) \imp (\for (y,x,A)) in G by A3, A4, Th108; :: thesis: verum

for T being non-empty MSAlgebra over J

for X being V3() GeneratorSet of T

for S1 being non empty non void b

for L being non-empty Language of X extended_by ({}, the carrier of S1),S1

for G being QC-theory of L

for A being Formula of L

for x, y being Element of Union X st L is vf-qc-correct & L is subst-correct holds

(\for (x,y,A)) \imp (\for (y,x,A)) in G

let J be non empty non void Signature; :: thesis: for T being non-empty MSAlgebra over J

for X being V3() GeneratorSet of T

for S1 being non empty non void J -extension n PC-correct QC-correct QCLangSignature over Union X

for L being non-empty Language of X extended_by ({}, the carrier of S1),S1

for G being QC-theory of L

for A being Formula of L

for x, y being Element of Union X st L is vf-qc-correct & L is subst-correct holds

(\for (x,y,A)) \imp (\for (y,x,A)) in G

let T be non-empty MSAlgebra over J; :: thesis: for X being V3() GeneratorSet of T

for S1 being non empty non void J -extension n PC-correct QC-correct QCLangSignature over Union X

for L being non-empty Language of X extended_by ({}, the carrier of S1),S1

for G being QC-theory of L

for A being Formula of L

for x, y being Element of Union X st L is vf-qc-correct & L is subst-correct holds

(\for (x,y,A)) \imp (\for (y,x,A)) in G

let X be V3() GeneratorSet of T; :: thesis: for S1 being non empty non void J -extension n PC-correct QC-correct QCLangSignature over Union X

for L being non-empty Language of X extended_by ({}, the carrier of S1),S1

for G being QC-theory of L

for A being Formula of L

for x, y being Element of Union X st L is vf-qc-correct & L is subst-correct holds

(\for (x,y,A)) \imp (\for (y,x,A)) in G

let S1 be non empty non void J -extension n PC-correct QC-correct QCLangSignature over Union X; :: thesis: for L being non-empty Language of X extended_by ({}, the carrier of S1),S1

for G being QC-theory of L

for A being Formula of L

for x, y being Element of Union X st L is vf-qc-correct & L is subst-correct holds

(\for (x,y,A)) \imp (\for (y,x,A)) in G

let L be non-empty Language of X extended_by ({}, the carrier of S1),S1; :: thesis: for G being QC-theory of L

for A being Formula of L

for x, y being Element of Union X st L is vf-qc-correct & L is subst-correct holds

(\for (x,y,A)) \imp (\for (y,x,A)) in G

let G be QC-theory of L; :: thesis: for A being Formula of L

for x, y being Element of Union X st L is vf-qc-correct & L is subst-correct holds

(\for (x,y,A)) \imp (\for (y,x,A)) in G

let A be Formula of L; :: thesis: for x, y being Element of Union X st L is vf-qc-correct & L is subst-correct holds

(\for (x,y,A)) \imp (\for (y,x,A)) in G

let x, y be Element of Union X; :: thesis: ( L is vf-qc-correct & L is subst-correct implies (\for (x,y,A)) \imp (\for (y,x,A)) in G )

assume A1: ( L is vf-qc-correct & L is subst-correct ) ; :: thesis: (\for (x,y,A)) \imp (\for (y,x,A)) in G

then (\for (y,A)) \imp A in G by Th104;

then A2: \for (x,((\for (y,A)) \imp A)) in G by Def39;

(\for (x,((\for (y,A)) \imp A))) \imp ((\for (x,(\for (y,A)))) \imp (\for (x,A))) in G by A1, Th109;

then (\for (x,(\for (y,A)))) \imp (\for (x,A)) in G by A2, Def38;

then A3: \for (y,((\for (x,(\for (y,A)))) \imp (\for (x,A)))) in G by Def39;

consider a being object such that

A4: ( a in dom X & y in X . a ) by CARD_5:2;

consider b being object such that

A5: ( b in dom X & x in X . b ) by CARD_5:2;

J is Subsignature of S1 by Def2;

then ( dom X = the carrier of J & the carrier of J c= the carrier of S1 ) by PARTFUN1:def 2, INSTALG1:10;

then reconsider a = a, b = b as Element of S1 by A4, A5;

reconsider c = a as Element of J by A4;

vf (\for (x,y,A)) = (vf (\for (y,A))) (\) (b -singleton x) by A1, A5;

then (vf (\for (x,y,A))) . a = ((vf (\for (y,A))) . a) \ ((b -singleton x) . a) by PBOOLE:def 6;

then y nin (vf (\for (x,y,A))) . c by A1, A4, Th113;

hence (\for (x,y,A)) \imp (\for (y,x,A)) in G by A3, A4, Th108; :: thesis: verum