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

for a being SortSymbol of J st L is subst-correct & L is vf-qc-correct & x in X . a & x nin (vf A) . a holds

(\ex (x,(A \and B))) \iff (A \and (\ex (x,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

for a being SortSymbol of J st L is subst-correct & L is vf-qc-correct & x in X . a & x nin (vf A) . a holds

(\ex (x,(A \and B))) \iff (A \and (\ex (x,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

for a being SortSymbol of J st L is subst-correct & L is vf-qc-correct & x in X . a & x nin (vf A) . a holds

(\ex (x,(A \and B))) \iff (A \and (\ex (x,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

for a being SortSymbol of J st L is subst-correct & L is vf-qc-correct & x in X . a & x nin (vf A) . a holds

(\ex (x,(A \and B))) \iff (A \and (\ex (x,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

for a being SortSymbol of J st L is subst-correct & L is vf-qc-correct & x in X . a & x nin (vf A) . a holds

(\ex (x,(A \and B))) \iff (A \and (\ex (x,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

for a being SortSymbol of J st L is subst-correct & L is vf-qc-correct & x in X . a & x nin (vf A) . a holds

(\ex (x,(A \and B))) \iff (A \and (\ex (x,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

for a being SortSymbol of J st L is subst-correct & L is vf-qc-correct & x in X . a & x nin (vf A) . a holds

(\ex (x,(A \and B))) \iff (A \and (\ex (x,B))) in G

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

for a being SortSymbol of J st L is subst-correct & L is vf-qc-correct & x in X . a & x nin (vf A) . a holds

(\ex (x,(A \and B))) \iff (A \and (\ex (x,B))) in G

let x be Element of Union X; :: thesis: for a being SortSymbol of J st L is subst-correct & L is vf-qc-correct & x in X . a & x nin (vf A) . a holds

(\ex (x,(A \and B))) \iff (A \and (\ex (x,B))) in G

let a be SortSymbol of J; :: thesis: ( L is subst-correct & L is vf-qc-correct & x in X . a & x nin (vf A) . a implies (\ex (x,(A \and B))) \iff (A \and (\ex (x,B))) in G )

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

assume A1: ( L is subst-correct & L is vf-qc-correct ) ; :: thesis: ( not x in X . a or not x nin (vf A) . a or (\ex (x,(A \and B))) \iff (A \and (\ex (x,B))) in G )

assume A2: x in X . a ; :: thesis: ( not x nin (vf A) . a or (\ex (x,(A \and B))) \iff (A \and (\ex (x,B))) in G )

assume A3: x nin (vf A) . a ; :: thesis: (\ex (x,(A \and B))) \iff (A \and (\ex (x,B))) in G

vf (\not A) = vf A by A1;

then (\for (x,((\not A) \or (\not B)))) \imp ((\not A) \or (\for (x,(\not B)))) in G by A1, A2, A3, Th133;

then A4: (\not ((\not A) \or (\for (x,(\not B))))) \imp (\not (\for (x,((\not A) \or (\not B))))) in G by Th58;

(\for (x,(\not B))) \iff (\not (\ex (x,B))) in G by A1, Th122;

then ( (\not A) \imp (\not A) in G & (\for (x,(\not B))) \imp (\not (\ex (x,B))) in G ) by Th34, Th43;

then ( ((\not A) \or (\not (\ex (x,B)))) \imp (\not (A \and (\ex (x,B)))) in G & ((\not A) \or (\for (x,(\not B)))) \imp ((\not A) \or (\not (\ex (x,B)))) in G & ((\not A) \or (\not (\ex (x,B)))) \imp (\not (A \and (\ex (x,B)))) in G ) by Th59, Th73;

then ( (A \and (\ex (x,B))) \imp (\not ((\not A) \or (\not (\ex (x,B))))) in G & (\not ((\not A) \or (\not (\ex (x,B))))) \imp (\not ((\not A) \or (\for (x,(\not B))))) in G ) by Th58, Th67;

then (A \and (\ex (x,B))) \imp (\not ((\not A) \or (\for (x,(\not B))))) in G by Th45;

then A5: (A \and (\ex (x,B))) \imp (\not (\for (x,((\not A) \or (\not B))))) in G by A4, Th45;

( (\ex (x,(A \and B))) \iff (\not (\for (x,(\not (A \and B))))) in G & (\not (A \and B)) \imp ((\not A) \or (\not B)) in G ) by Th70, Th105;

then ( (\not (\for (x,(\not (A \and B))))) \imp (\ex (x,(A \and B))) in G & \for (x,((\not (A \and B)) \imp ((\not A) \or (\not B)))) in G & (\for (x,((\not (A \and B)) \imp ((\not A) \or (\not B))))) \imp ((\for (x,(\not (A \and B)))) \imp (\for (x,((\not A) \or (\not B))))) in G ) by A1, Def39, Th43, Th109;

then ( (\not (\ex (x,(A \and B)))) \imp (\for (x,(\not (A \and B)))) in G & (\for (x,(\not (A \and B)))) \imp (\for (x,((\not A) \or (\not B)))) in G ) by Def38, Th68;

then (\not (\ex (x,(A \and B)))) \imp (\for (x,((\not A) \or (\not B)))) in G by Th45;

then (\not (\for (x,((\not A) \or (\not B))))) \imp (\ex (x,(A \and B))) in G by Th68;

then A6: (A \and (\ex (x,B))) \imp (\ex (x,(A \and B))) in G by A5, Th45;

(\ex (x,(A \and B))) \imp (A \and (\ex (x,B))) in G by Th134, A1, A2, A3;

hence (\ex (x,(A \and B))) \iff (A \and (\ex (x,B))) in G by A6, Th43; :: 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

for a being SortSymbol of J st L is subst-correct & L is vf-qc-correct & x in X . a & x nin (vf A) . a holds

(\ex (x,(A \and B))) \iff (A \and (\ex (x,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

for a being SortSymbol of J st L is subst-correct & L is vf-qc-correct & x in X . a & x nin (vf A) . a holds

(\ex (x,(A \and B))) \iff (A \and (\ex (x,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

for a being SortSymbol of J st L is subst-correct & L is vf-qc-correct & x in X . a & x nin (vf A) . a holds

(\ex (x,(A \and B))) \iff (A \and (\ex (x,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

for a being SortSymbol of J st L is subst-correct & L is vf-qc-correct & x in X . a & x nin (vf A) . a holds

(\ex (x,(A \and B))) \iff (A \and (\ex (x,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

for a being SortSymbol of J st L is subst-correct & L is vf-qc-correct & x in X . a & x nin (vf A) . a holds

(\ex (x,(A \and B))) \iff (A \and (\ex (x,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

for a being SortSymbol of J st L is subst-correct & L is vf-qc-correct & x in X . a & x nin (vf A) . a holds

(\ex (x,(A \and B))) \iff (A \and (\ex (x,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

for a being SortSymbol of J st L is subst-correct & L is vf-qc-correct & x in X . a & x nin (vf A) . a holds

(\ex (x,(A \and B))) \iff (A \and (\ex (x,B))) in G

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

for a being SortSymbol of J st L is subst-correct & L is vf-qc-correct & x in X . a & x nin (vf A) . a holds

(\ex (x,(A \and B))) \iff (A \and (\ex (x,B))) in G

let x be Element of Union X; :: thesis: for a being SortSymbol of J st L is subst-correct & L is vf-qc-correct & x in X . a & x nin (vf A) . a holds

(\ex (x,(A \and B))) \iff (A \and (\ex (x,B))) in G

let a be SortSymbol of J; :: thesis: ( L is subst-correct & L is vf-qc-correct & x in X . a & x nin (vf A) . a implies (\ex (x,(A \and B))) \iff (A \and (\ex (x,B))) in G )

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

assume A1: ( L is subst-correct & L is vf-qc-correct ) ; :: thesis: ( not x in X . a or not x nin (vf A) . a or (\ex (x,(A \and B))) \iff (A \and (\ex (x,B))) in G )

assume A2: x in X . a ; :: thesis: ( not x nin (vf A) . a or (\ex (x,(A \and B))) \iff (A \and (\ex (x,B))) in G )

assume A3: x nin (vf A) . a ; :: thesis: (\ex (x,(A \and B))) \iff (A \and (\ex (x,B))) in G

vf (\not A) = vf A by A1;

then (\for (x,((\not A) \or (\not B)))) \imp ((\not A) \or (\for (x,(\not B)))) in G by A1, A2, A3, Th133;

then A4: (\not ((\not A) \or (\for (x,(\not B))))) \imp (\not (\for (x,((\not A) \or (\not B))))) in G by Th58;

(\for (x,(\not B))) \iff (\not (\ex (x,B))) in G by A1, Th122;

then ( (\not A) \imp (\not A) in G & (\for (x,(\not B))) \imp (\not (\ex (x,B))) in G ) by Th34, Th43;

then ( ((\not A) \or (\not (\ex (x,B)))) \imp (\not (A \and (\ex (x,B)))) in G & ((\not A) \or (\for (x,(\not B)))) \imp ((\not A) \or (\not (\ex (x,B)))) in G & ((\not A) \or (\not (\ex (x,B)))) \imp (\not (A \and (\ex (x,B)))) in G ) by Th59, Th73;

then ( (A \and (\ex (x,B))) \imp (\not ((\not A) \or (\not (\ex (x,B))))) in G & (\not ((\not A) \or (\not (\ex (x,B))))) \imp (\not ((\not A) \or (\for (x,(\not B))))) in G ) by Th58, Th67;

then (A \and (\ex (x,B))) \imp (\not ((\not A) \or (\for (x,(\not B))))) in G by Th45;

then A5: (A \and (\ex (x,B))) \imp (\not (\for (x,((\not A) \or (\not B))))) in G by A4, Th45;

( (\ex (x,(A \and B))) \iff (\not (\for (x,(\not (A \and B))))) in G & (\not (A \and B)) \imp ((\not A) \or (\not B)) in G ) by Th70, Th105;

then ( (\not (\for (x,(\not (A \and B))))) \imp (\ex (x,(A \and B))) in G & \for (x,((\not (A \and B)) \imp ((\not A) \or (\not B)))) in G & (\for (x,((\not (A \and B)) \imp ((\not A) \or (\not B))))) \imp ((\for (x,(\not (A \and B)))) \imp (\for (x,((\not A) \or (\not B))))) in G ) by A1, Def39, Th43, Th109;

then ( (\not (\ex (x,(A \and B)))) \imp (\for (x,(\not (A \and B)))) in G & (\for (x,(\not (A \and B)))) \imp (\for (x,((\not A) \or (\not B)))) in G ) by Def38, Th68;

then (\not (\ex (x,(A \and B)))) \imp (\for (x,((\not A) \or (\not B)))) in G by Th45;

then (\not (\for (x,((\not A) \or (\not B))))) \imp (\ex (x,(A \and B))) in G by Th68;

then A6: (A \and (\ex (x,B))) \imp (\ex (x,(A \and B))) in G by A5, Th45;

(\ex (x,(A \and B))) \imp (A \and (\ex (x,B))) in G by Th134, A1, A2, A3;

hence (\ex (x,(A \and B))) \iff (A \and (\ex (x,B))) in G by A6, Th43; :: thesis: verum