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 being Element of Union X st L is vf-qc-correct holds

(\for (x,A)) \imp (\for (x,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 being Element of Union X st L is vf-qc-correct holds

(\for (x,A)) \imp (\for (x,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 being Element of Union X st L is vf-qc-correct holds

(\for (x,A)) \imp (\for (x,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 being Element of Union X st L is vf-qc-correct holds

(\for (x,A)) \imp (\for (x,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 being Element of Union X st L is vf-qc-correct holds

(\for (x,A)) \imp (\for (x,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 being Element of Union X st L is vf-qc-correct holds

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

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

for x being Element of Union X st L is vf-qc-correct holds

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

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

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

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

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

consider a being object such that

A2: ( a in dom X & x in X . a ) by CARD_5:2;

reconsider a = a as Element of J by A2;

set Y = X extended_by ({}, the carrier of S1);

A3: a is SortSymbol of S1 by Th8;

A4: x nin (vf (\for (x,A))) . a by A1, A2, A3, Th113;

(\for (x,A)) \imp (\for (x,A)) in G by Th34;

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

hence (\for (x,A)) \imp (\for (x,x,A)) in G by A2, 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 being Element of Union X st L is vf-qc-correct holds

(\for (x,A)) \imp (\for (x,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 being Element of Union X st L is vf-qc-correct holds

(\for (x,A)) \imp (\for (x,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 being Element of Union X st L is vf-qc-correct holds

(\for (x,A)) \imp (\for (x,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 being Element of Union X st L is vf-qc-correct holds

(\for (x,A)) \imp (\for (x,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 being Element of Union X st L is vf-qc-correct holds

(\for (x,A)) \imp (\for (x,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 being Element of Union X st L is vf-qc-correct holds

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

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

for x being Element of Union X st L is vf-qc-correct holds

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

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

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

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

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

consider a being object such that

A2: ( a in dom X & x in X . a ) by CARD_5:2;

reconsider a = a as Element of J by A2;

set Y = X extended_by ({}, the carrier of S1);

A3: a is SortSymbol of S1 by Th8;

A4: x nin (vf (\for (x,A))) . a by A1, A2, A3, Th113;

(\for (x,A)) \imp (\for (x,A)) in G by Th34;

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

hence (\for (x,A)) \imp (\for (x,x,A)) in G by A2, A4, Th108; :: thesis: verum