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 subst-correct holds

A \imp (\ex (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 subst-correct holds

A \imp (\ex (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 subst-correct holds

A \imp (\ex (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 subst-correct holds

A \imp (\ex (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 subst-correct holds

A \imp (\ex (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 subst-correct holds

A \imp (\ex (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 subst-correct holds

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

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

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

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

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

assume A1: L is subst-correct ; :: thesis: A \imp (\ex (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 SortSymbol of J by A2;

A3: ( x in X . a & a is SortSymbol of S1 ) by A2, Th8;

then A4: ( x in (X extended_by ({}, the carrier of S1)) . a & dom (X extended_by ({}, the carrier of S1)) = the carrier of S1 ) by Th2, PARTFUN1:def 2;

then reconsider x0 = x as Element of Union (X extended_by ({}, the carrier of S1)) by A3, CARD_5:2;

(A / (x0,x0)) \imp (\ex (x,A)) in G by A1, A2, Th110;

hence A \imp (\ex (x,A)) in G by A1, A3, A4; :: 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 subst-correct holds

A \imp (\ex (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 subst-correct holds

A \imp (\ex (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 subst-correct holds

A \imp (\ex (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 subst-correct holds

A \imp (\ex (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 subst-correct holds

A \imp (\ex (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 subst-correct holds

A \imp (\ex (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 subst-correct holds

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

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

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

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

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

assume A1: L is subst-correct ; :: thesis: A \imp (\ex (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 SortSymbol of J by A2;

A3: ( x in X . a & a is SortSymbol of S1 ) by A2, Th8;

then A4: ( x in (X extended_by ({}, the carrier of S1)) . a & dom (X extended_by ({}, the carrier of S1)) = the carrier of S1 ) by Th2, PARTFUN1:def 2;

then reconsider x0 = x as Element of Union (X extended_by ({}, the carrier of S1)) by A3, CARD_5:2;

(A / (x0,x0)) \imp (\ex (x,A)) in G by A1, A2, Th110;

hence A \imp (\ex (x,A)) in G by A1, A3, A4; :: thesis: verum