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

(\ex (x,y,A)) \imp (\ex (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

(\ex (x,y,A)) \imp (\ex (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

(\ex (x,y,A)) \imp (\ex (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

(\ex (x,y,A)) \imp (\ex (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

(\ex (x,y,A)) \imp (\ex (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

(\ex (x,y,A)) \imp (\ex (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

(\ex (x,y,A)) \imp (\ex (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

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

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

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

then (\for (y,x,(\not A))) \imp (\for (x,y,(\not A))) in G by Th138;

then A2: (\not (\for (x,y,(\not A)))) \imp (\not (\for (y,x,(\not A)))) in G by Th58;

( (\ex (x,y,A)) \iff (\not (\for (x,y,(\not A)))) in G & (\ex (y,x,A)) \iff (\not (\for (y,x,(\not A)))) in G ) by A1, Th111;

then ( (\ex (x,y,A)) \imp (\not (\for (y,x,(\not A)))) in G & (\not (\for (y,x,(\not A)))) \iff (\ex (y,x,A)) in G ) by A2, Th90, Th92;

hence (\ex (x,y,A)) \imp (\ex (y,x,A)) in G by Th93; :: 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

(\ex (x,y,A)) \imp (\ex (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

(\ex (x,y,A)) \imp (\ex (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

(\ex (x,y,A)) \imp (\ex (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

(\ex (x,y,A)) \imp (\ex (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

(\ex (x,y,A)) \imp (\ex (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

(\ex (x,y,A)) \imp (\ex (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

(\ex (x,y,A)) \imp (\ex (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

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

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

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

then (\for (y,x,(\not A))) \imp (\for (x,y,(\not A))) in G by Th138;

then A2: (\not (\for (x,y,(\not A)))) \imp (\not (\for (y,x,(\not A)))) in G by Th58;

( (\ex (x,y,A)) \iff (\not (\for (x,y,(\not A)))) in G & (\ex (y,x,A)) \iff (\not (\for (y,x,(\not A)))) in G ) by A1, Th111;

then ( (\ex (x,y,A)) \imp (\not (\for (y,x,(\not A)))) in G & (\not (\for (y,x,(\not A)))) \iff (\ex (y,x,A)) in G ) by A2, Th90, Th92;

hence (\ex (x,y,A)) \imp (\ex (y,x,A)) in G by Th93; :: thesis: verum