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, B being Formula of L

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

(\for (x,(A \imp B))) \imp ((\for (x,A)) \imp B) 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, B being Formula of L

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

(\for (x,(A \imp B))) \imp ((\for (x,A)) \imp B) 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, B being Formula of L

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

(\for (x,(A \imp B))) \imp ((\for (x,A)) \imp B) 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, B being Formula of L

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

(\for (x,(A \imp B))) \imp ((\for (x,A)) \imp B) 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, B being Formula of L

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

(\for (x,(A \imp B))) \imp ((\for (x,A)) \imp B) 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, B being Formula of L

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

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

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

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

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

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

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

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

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

consider a being object such that

A1: ( a in dom X & x in X . a ) by CARD_5:2;

J is Subsignature of S1 by Def2;

then A2: ( 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;

reconsider a = a as SortSymbol of J by A1;

A3: x in (X extended_by ({}, the carrier of S1)) . a by A1, A2, Th1;

A4: X . a is Subset of ( the Sorts of T . a) by Th13;

A5: X extended_by ({}, the carrier of S1) is ManySortedSubset of the Sorts of L by Th23;

then ( x in Union X & Union X = Union (X extended_by ({}, the carrier of S1)) & Union (X extended_by ({}, the carrier of S1)) c= Union the Sorts of L ) by Th24, MSAFREE4:1, PBOOLE:def 18;

then reconsider t = x as Element of Union the Sorts of L ;

A6: the Sorts of T . a = the Sorts of L . a by Th16;

reconsider x0 = x as Element of Union (X extended_by ({}, the carrier of S1)) by Th24;

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

then ( A / (x0,x0) = A & (A \imp B) / (x0,x0) = A \imp B ) by A1, A2, A3;

then ( A / (x0,t) = A & (A \imp B) / (x0,t) = A \imp B ) by A5, A1, A2, A3, Th14;

then ( (\for (x,(A \imp B))) \imp (A \imp B) in G & (\for (x,A)) \imp A in G ) by A6, A1, A4, A3, Def39;

hence (\for (x,(A \imp B))) \imp ((\for (x,A)) \imp B) in G by Th96; :: 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, B being Formula of L

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

(\for (x,(A \imp B))) \imp ((\for (x,A)) \imp B) 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, B being Formula of L

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

(\for (x,(A \imp B))) \imp ((\for (x,A)) \imp B) 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, B being Formula of L

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

(\for (x,(A \imp B))) \imp ((\for (x,A)) \imp B) 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, B being Formula of L

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

(\for (x,(A \imp B))) \imp ((\for (x,A)) \imp B) 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, B being Formula of L

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

(\for (x,(A \imp B))) \imp ((\for (x,A)) \imp B) 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, B being Formula of L

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

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

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

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

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

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

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

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

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

consider a being object such that

A1: ( a in dom X & x in X . a ) by CARD_5:2;

J is Subsignature of S1 by Def2;

then A2: ( 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;

reconsider a = a as SortSymbol of J by A1;

A3: x in (X extended_by ({}, the carrier of S1)) . a by A1, A2, Th1;

A4: X . a is Subset of ( the Sorts of T . a) by Th13;

A5: X extended_by ({}, the carrier of S1) is ManySortedSubset of the Sorts of L by Th23;

then ( x in Union X & Union X = Union (X extended_by ({}, the carrier of S1)) & Union (X extended_by ({}, the carrier of S1)) c= Union the Sorts of L ) by Th24, MSAFREE4:1, PBOOLE:def 18;

then reconsider t = x as Element of Union the Sorts of L ;

A6: the Sorts of T . a = the Sorts of L . a by Th16;

reconsider x0 = x as Element of Union (X extended_by ({}, the carrier of S1)) by Th24;

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

then ( A / (x0,x0) = A & (A \imp B) / (x0,x0) = A \imp B ) by A1, A2, A3;

then ( A / (x0,t) = A & (A \imp B) / (x0,t) = A \imp B ) by A5, A1, A2, A3, Th14;

then ( (\for (x,(A \imp B))) \imp (A \imp B) in G & (\for (x,A)) \imp A in G ) by A6, A1, A4, A3, Def39;

hence (\for (x,(A \imp B))) \imp ((\for (x,A)) \imp B) in G by Th96; :: thesis: verum