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

(\for (x,A)) \imp 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

(\for (x,A)) \imp 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

(\for (x,A)) \imp 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

(\for (x,A)) \imp 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

(\for (x,A)) \imp 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

(\for (x,A)) \imp 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

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

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

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

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

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

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

consider a being object such that

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

J is Subsignature of S1 by Def2;

then A3: ( the carrier of J c= the carrier of S1 & dom (X extended_by ({}, the carrier of S1)) = the carrier of S1 & dom X = the carrier of J ) by PARTFUN1:def 2, INSTALG1:10;

then reconsider a = a as SortSymbol of S1 by A2;

A4: x in (X extended_by ({}, the carrier of S1)) . a by A2, A3, Th1;

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

X c= the Sorts of T by PBOOLE:def 18;

then ( X . a c= the Sorts of T . a & the Sorts of T . a = the Sorts of L . a ) by A2, Th16;

then reconsider t = x as Element of the Sorts of L . a by A2;

X extended_by ({}, the carrier of S1) is ManySortedSubset of the Sorts of L by Th23;

then A / (x0,t) = A / (x0,x0) by A4, Th14

.= A by A4, A1 ;

hence (\for (x,A)) \imp A in G by A2, A4, Def39; :: 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

(\for (x,A)) \imp 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

(\for (x,A)) \imp 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

(\for (x,A)) \imp 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

(\for (x,A)) \imp 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

(\for (x,A)) \imp 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

(\for (x,A)) \imp 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

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

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

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

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

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

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

consider a being object such that

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

J is Subsignature of S1 by Def2;

then A3: ( the carrier of J c= the carrier of S1 & dom (X extended_by ({}, the carrier of S1)) = the carrier of S1 & dom X = the carrier of J ) by PARTFUN1:def 2, INSTALG1:10;

then reconsider a = a as SortSymbol of S1 by A2;

A4: x in (X extended_by ({}, the carrier of S1)) . a by A2, A3, Th1;

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

X c= the Sorts of T by PBOOLE:def 18;

then ( X . a c= the Sorts of T . a & the Sorts of T . a = the Sorts of L . a ) by A2, Th16;

then reconsider t = x as Element of the Sorts of L . a by A2;

X extended_by ({}, the carrier of S1) is ManySortedSubset of the Sorts of L by Th23;

then A / (x0,t) = A / (x0,x0) by A4, Th14

.= A by A4, A1 ;

hence (\for (x,A)) \imp A in G by A2, A4, Def39; :: thesis: verum