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

for x0, y0 being Element of Union (X extended_by ({}, the carrier of S1)) st L is subst-correct holds

for a being SortSymbol of J st x in X . a & y in X . a & x0 = x & y0 = y holds

(A / (x0,y0)) \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, y being Element of Union X

for x0, y0 being Element of Union (X extended_by ({}, the carrier of S1)) st L is subst-correct holds

for a being SortSymbol of J st x in X . a & y in X . a & x0 = x & y0 = y holds

(A / (x0,y0)) \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, y being Element of Union X

for x0, y0 being Element of Union (X extended_by ({}, the carrier of S1)) st L is subst-correct holds

for a being SortSymbol of J st x in X . a & y in X . a & x0 = x & y0 = y holds

(A / (x0,y0)) \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, y being Element of Union X

for x0, y0 being Element of Union (X extended_by ({}, the carrier of S1)) st L is subst-correct holds

for a being SortSymbol of J st x in X . a & y in X . a & x0 = x & y0 = y holds

(A / (x0,y0)) \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, y being Element of Union X

for x0, y0 being Element of Union (X extended_by ({}, the carrier of S1)) st L is subst-correct holds

for a being SortSymbol of J st x in X . a & y in X . a & x0 = x & y0 = y holds

(A / (x0,y0)) \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, y being Element of Union X

for x0, y0 being Element of Union (X extended_by ({}, the carrier of S1)) st L is subst-correct holds

for a being SortSymbol of J st x in X . a & y in X . a & x0 = x & y0 = y holds

(A / (x0,y0)) \imp (\ex (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

for x0, y0 being Element of Union (X extended_by ({}, the carrier of S1)) st L is subst-correct holds

for a being SortSymbol of J st x in X . a & y in X . a & x0 = x & y0 = y holds

(A / (x0,y0)) \imp (\ex (x,A)) in G

let A be Formula of L; :: thesis: for x, y being Element of Union X

for x0, y0 being Element of Union (X extended_by ({}, the carrier of S1)) st L is subst-correct holds

for a being SortSymbol of J st x in X . a & y in X . a & x0 = x & y0 = y holds

(A / (x0,y0)) \imp (\ex (x,A)) in G

let x, y be Element of Union X; :: thesis: for x0, y0 being Element of Union (X extended_by ({}, the carrier of S1)) st L is subst-correct holds

for a being SortSymbol of J st x in X . a & y in X . a & x0 = x & y0 = y holds

(A / (x0,y0)) \imp (\ex (x,A)) in G

let x0, y0 be Element of Union (X extended_by ({}, the carrier of S1)); :: thesis: ( L is subst-correct implies for a being SortSymbol of J st x in X . a & y in X . a & x0 = x & y0 = y holds

(A / (x0,y0)) \imp (\ex (x,A)) in G )

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

assume A1: L is subst-correct ; :: thesis: for a being SortSymbol of J st x in X . a & y in X . a & x0 = x & y0 = y holds

(A / (x0,y0)) \imp (\ex (x,A)) in G

let a be SortSymbol of J; :: thesis: ( x in X . a & y in X . a & x0 = x & y0 = y implies (A / (x0,y0)) \imp (\ex (x,A)) in G )

assume A2: ( x in X . a & y in X . a & x0 = x & y0 = y ) ; :: thesis: (A / (x0,y0)) \imp (\ex (x,A)) in G

J is Subsignature of S1 by Def2;

then the carrier of J c= the carrier of S1 by INSTALG1:10;

then A3: ( a in the carrier of S1 & X c= the Sorts of T & dom the Sorts of L = the carrier of S1 ) by PARTFUN1:def 2, PBOOLE:def 18;

then ( the Sorts of L . a in rng the Sorts of L & the Sorts of L . a = the Sorts of T . a ) by Th16, FUNCT_1:def 3;

then A4: ( X . a c= the Sorts of T . a & the Sorts of T . a = the Sorts of L . a & the Sorts of L . a c= Union the Sorts of L ) by A3, ZFMISC_1:74;

then reconsider t = y as Element of Union the Sorts of L by A2;

A5: a is SortSymbol of S1 by Th8;

then A6: ( x in (X extended_by ({}, the carrier of S1)) . a & y in (X extended_by ({}, the carrier of S1)) . a ) by A2, Th2;

A7: X extended_by ({}, the carrier of S1) is ManySortedSubset of the Sorts of L by Th23;

(\for (x,(\not A))) \imp ((\not A) / (x0,t)) in G by A2, A4, A6, Def39;

then (\for (x,(\not A))) \imp ((\not A) / (x0,y0)) in G by A2, A6, A7, A5, Th14;

then (\not ((\not A) / (x0,y0))) \imp (\not (\for (x,(\not A)))) in G by Th58;

then ( (\not (\not (A / (x0,y0)))) \imp (\not (\for (x,(\not A)))) in G & (A / (x0,y0)) \imp (\not (\not (A / (x0,y0)))) in G & (\ex (x,A)) \iff (\not (\for (x,(\not A)))) in G & ((\ex (x,A)) \iff (\not (\for (x,(\not A))))) \imp ((\not (\for (x,(\not A)))) \imp (\ex (x,A))) in G ) by A1, A2, A6, A7, A5, Th27, Def38, Th64, Th105;

then ( (A / (x0,y0)) \imp (\not (\for (x,(\not A)))) in G & (\not (\for (x,(\not A)))) \imp (\ex (x,A)) in G ) by Th45, Def38;

hence (A / (x0,y0)) \imp (\ex (x,A)) in G by Th45; :: 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

for x0, y0 being Element of Union (X extended_by ({}, the carrier of S1)) st L is subst-correct holds

for a being SortSymbol of J st x in X . a & y in X . a & x0 = x & y0 = y holds

(A / (x0,y0)) \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, y being Element of Union X

for x0, y0 being Element of Union (X extended_by ({}, the carrier of S1)) st L is subst-correct holds

for a being SortSymbol of J st x in X . a & y in X . a & x0 = x & y0 = y holds

(A / (x0,y0)) \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, y being Element of Union X

for x0, y0 being Element of Union (X extended_by ({}, the carrier of S1)) st L is subst-correct holds

for a being SortSymbol of J st x in X . a & y in X . a & x0 = x & y0 = y holds

(A / (x0,y0)) \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, y being Element of Union X

for x0, y0 being Element of Union (X extended_by ({}, the carrier of S1)) st L is subst-correct holds

for a being SortSymbol of J st x in X . a & y in X . a & x0 = x & y0 = y holds

(A / (x0,y0)) \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, y being Element of Union X

for x0, y0 being Element of Union (X extended_by ({}, the carrier of S1)) st L is subst-correct holds

for a being SortSymbol of J st x in X . a & y in X . a & x0 = x & y0 = y holds

(A / (x0,y0)) \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, y being Element of Union X

for x0, y0 being Element of Union (X extended_by ({}, the carrier of S1)) st L is subst-correct holds

for a being SortSymbol of J st x in X . a & y in X . a & x0 = x & y0 = y holds

(A / (x0,y0)) \imp (\ex (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

for x0, y0 being Element of Union (X extended_by ({}, the carrier of S1)) st L is subst-correct holds

for a being SortSymbol of J st x in X . a & y in X . a & x0 = x & y0 = y holds

(A / (x0,y0)) \imp (\ex (x,A)) in G

let A be Formula of L; :: thesis: for x, y being Element of Union X

for x0, y0 being Element of Union (X extended_by ({}, the carrier of S1)) st L is subst-correct holds

for a being SortSymbol of J st x in X . a & y in X . a & x0 = x & y0 = y holds

(A / (x0,y0)) \imp (\ex (x,A)) in G

let x, y be Element of Union X; :: thesis: for x0, y0 being Element of Union (X extended_by ({}, the carrier of S1)) st L is subst-correct holds

for a being SortSymbol of J st x in X . a & y in X . a & x0 = x & y0 = y holds

(A / (x0,y0)) \imp (\ex (x,A)) in G

let x0, y0 be Element of Union (X extended_by ({}, the carrier of S1)); :: thesis: ( L is subst-correct implies for a being SortSymbol of J st x in X . a & y in X . a & x0 = x & y0 = y holds

(A / (x0,y0)) \imp (\ex (x,A)) in G )

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

assume A1: L is subst-correct ; :: thesis: for a being SortSymbol of J st x in X . a & y in X . a & x0 = x & y0 = y holds

(A / (x0,y0)) \imp (\ex (x,A)) in G

let a be SortSymbol of J; :: thesis: ( x in X . a & y in X . a & x0 = x & y0 = y implies (A / (x0,y0)) \imp (\ex (x,A)) in G )

assume A2: ( x in X . a & y in X . a & x0 = x & y0 = y ) ; :: thesis: (A / (x0,y0)) \imp (\ex (x,A)) in G

J is Subsignature of S1 by Def2;

then the carrier of J c= the carrier of S1 by INSTALG1:10;

then A3: ( a in the carrier of S1 & X c= the Sorts of T & dom the Sorts of L = the carrier of S1 ) by PARTFUN1:def 2, PBOOLE:def 18;

then ( the Sorts of L . a in rng the Sorts of L & the Sorts of L . a = the Sorts of T . a ) by Th16, FUNCT_1:def 3;

then A4: ( X . a c= the Sorts of T . a & the Sorts of T . a = the Sorts of L . a & the Sorts of L . a c= Union the Sorts of L ) by A3, ZFMISC_1:74;

then reconsider t = y as Element of Union the Sorts of L by A2;

A5: a is SortSymbol of S1 by Th8;

then A6: ( x in (X extended_by ({}, the carrier of S1)) . a & y in (X extended_by ({}, the carrier of S1)) . a ) by A2, Th2;

A7: X extended_by ({}, the carrier of S1) is ManySortedSubset of the Sorts of L by Th23;

(\for (x,(\not A))) \imp ((\not A) / (x0,t)) in G by A2, A4, A6, Def39;

then (\for (x,(\not A))) \imp ((\not A) / (x0,y0)) in G by A2, A6, A7, A5, Th14;

then (\not ((\not A) / (x0,y0))) \imp (\not (\for (x,(\not A)))) in G by Th58;

then ( (\not (\not (A / (x0,y0)))) \imp (\not (\for (x,(\not A)))) in G & (A / (x0,y0)) \imp (\not (\not (A / (x0,y0)))) in G & (\ex (x,A)) \iff (\not (\for (x,(\not A)))) in G & ((\ex (x,A)) \iff (\not (\for (x,(\not A))))) \imp ((\not (\for (x,(\not A)))) \imp (\ex (x,A))) in G ) by A1, A2, A6, A7, A5, Th27, Def38, Th64, Th105;

then ( (A / (x0,y0)) \imp (\not (\for (x,(\not A)))) in G & (\not (\for (x,(\not A)))) \imp (\ex (x,A)) in G ) by Th45, Def38;

hence (A / (x0,y0)) \imp (\ex (x,A)) in G by Th45; :: thesis: verum