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

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

assume A2: ( x in X . a & x nin (vf A) . a ) ; :: thesis: (\ex (x,(A \imp B))) \imp (A \imp (\ex (x,B))) in G

A \imp A in G by Th34;

then \for (x,(A \imp A)) in G by Def39;

then ( A \imp (\for (x,A)) in G & (\for (x,(\not B))) \imp (\for (x,(\not B))) in G ) by A2, Th108, Th34;

then A3: (A \and (\for (x,(\not B)))) \imp ((\for (x,A)) \and (\for (x,(\not B)))) in G by Th72;

((\for (x,A)) \and (\for (x,(\not B)))) \imp (\for (x,(A \and (\not B)))) in G by A1, Th126;

then (A \and (\for (x,(\not B)))) \imp (\for (x,(A \and (\not B)))) in G by A3, Th45;

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

( (\not A) \imp (\not A) in G & B \imp (\not (\not B)) in G ) by Th34, Th64;

then A5: ((\not A) \or B) \imp ((\not A) \or (\not (\not B))) in G by Th59;

((\not A) \or (\not (\not B))) \imp (\not (A \and (\not B))) in G by Th73;

then ( (A \and (\not B)) \imp (\not (\not (A \and (\not B)))) in G & (\not (\not (A \and (\not B)))) \imp (\not ((\not A) \or (\not (\not B)))) in G ) by Th58, Th64;

then ( (A \imp B) \imp ((\not A) \or B) in G & (\not ((\not A) \or (\not (\not B)))) \imp (\not ((\not A) \or B)) in G & (A \and (\not B)) \imp (\not ((\not A) \or (\not (\not B)))) in G ) by A5, Th82, Th58, Th45;

then ( (\not ((\not A) \or B)) \imp (\not (A \imp B)) in G & (A \and (\not B)) \imp (\not ((\not A) \or B)) in G ) by Th58, Th45;

then (A \and (\not B)) \imp (\not (A \imp B)) in G by Th45;

then (\for (x,(A \and (\not B)))) \imp (\for (x,(\not (A \imp B)))) in G by A1, Th115;

then ( (\ex (x,(A \imp B))) \iff (\not (\for (x,(\not (A \imp B))))) in G & (\not (\for (x,(\not (A \imp B))))) \imp (\not (\for (x,(A \and (\not B))))) in G ) by Th105, Th58;

then (\ex (x,(A \imp B))) \imp (\not (\for (x,(A \and (\not B))))) in G by Th92;

then A6: (\ex (x,(A \imp B))) \imp (\not (A \and (\for (x,(\not B))))) in G by A4, Th45;

(\ex (x,B)) \iff (\not (\for (x,(\not B)))) in G by Th105;

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

then ( (\not (A \and (\for (x,(\not B))))) \imp ((\not A) \or (\not (\for (x,(\not B))))) in G & ((\not A) \or (\not (\for (x,(\not B))))) \imp ((\not A) \or (\ex (x,B))) in G ) by Th59, Th70;

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

then A7: (\ex (x,(A \imp B))) \imp ((\not A) \or (\ex (x,B))) in G by A6, Th45;

( A \imp (\not (\not A)) in G & (\ex (x,B)) \imp (\ex (x,B)) in G ) by Th34, Th64;

then ( ((\not A) \or (\ex (x,B))) \imp ((\not (\not A)) \imp (\ex (x,B))) in G & ((\not (\not A)) \imp (\ex (x,B))) \imp (A \imp (\ex (x,B))) in G ) by Th62, Th103;

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

hence (\ex (x,(A \imp B))) \imp (A \imp (\ex (x,B))) in G by A7, 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, 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 \imp B))) \imp (A \imp (\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 \imp B))) \imp (A \imp (\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 \imp B))) \imp (A \imp (\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 \imp B))) \imp (A \imp (\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 \imp B))) \imp (A \imp (\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 \imp B))) \imp (A \imp (\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 \imp B))) \imp (A \imp (\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 \imp B))) \imp (A \imp (\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 \imp B))) \imp (A \imp (\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 \imp B))) \imp (A \imp (\ex (x,B))) in G )

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

assume A2: ( x in X . a & x nin (vf A) . a ) ; :: thesis: (\ex (x,(A \imp B))) \imp (A \imp (\ex (x,B))) in G

A \imp A in G by Th34;

then \for (x,(A \imp A)) in G by Def39;

then ( A \imp (\for (x,A)) in G & (\for (x,(\not B))) \imp (\for (x,(\not B))) in G ) by A2, Th108, Th34;

then A3: (A \and (\for (x,(\not B)))) \imp ((\for (x,A)) \and (\for (x,(\not B)))) in G by Th72;

((\for (x,A)) \and (\for (x,(\not B)))) \imp (\for (x,(A \and (\not B)))) in G by A1, Th126;

then (A \and (\for (x,(\not B)))) \imp (\for (x,(A \and (\not B)))) in G by A3, Th45;

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

( (\not A) \imp (\not A) in G & B \imp (\not (\not B)) in G ) by Th34, Th64;

then A5: ((\not A) \or B) \imp ((\not A) \or (\not (\not B))) in G by Th59;

((\not A) \or (\not (\not B))) \imp (\not (A \and (\not B))) in G by Th73;

then ( (A \and (\not B)) \imp (\not (\not (A \and (\not B)))) in G & (\not (\not (A \and (\not B)))) \imp (\not ((\not A) \or (\not (\not B)))) in G ) by Th58, Th64;

then ( (A \imp B) \imp ((\not A) \or B) in G & (\not ((\not A) \or (\not (\not B)))) \imp (\not ((\not A) \or B)) in G & (A \and (\not B)) \imp (\not ((\not A) \or (\not (\not B)))) in G ) by A5, Th82, Th58, Th45;

then ( (\not ((\not A) \or B)) \imp (\not (A \imp B)) in G & (A \and (\not B)) \imp (\not ((\not A) \or B)) in G ) by Th58, Th45;

then (A \and (\not B)) \imp (\not (A \imp B)) in G by Th45;

then (\for (x,(A \and (\not B)))) \imp (\for (x,(\not (A \imp B)))) in G by A1, Th115;

then ( (\ex (x,(A \imp B))) \iff (\not (\for (x,(\not (A \imp B))))) in G & (\not (\for (x,(\not (A \imp B))))) \imp (\not (\for (x,(A \and (\not B))))) in G ) by Th105, Th58;

then (\ex (x,(A \imp B))) \imp (\not (\for (x,(A \and (\not B))))) in G by Th92;

then A6: (\ex (x,(A \imp B))) \imp (\not (A \and (\for (x,(\not B))))) in G by A4, Th45;

(\ex (x,B)) \iff (\not (\for (x,(\not B)))) in G by Th105;

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

then ( (\not (A \and (\for (x,(\not B))))) \imp ((\not A) \or (\not (\for (x,(\not B))))) in G & ((\not A) \or (\not (\for (x,(\not B))))) \imp ((\not A) \or (\ex (x,B))) in G ) by Th59, Th70;

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

then A7: (\ex (x,(A \imp B))) \imp ((\not A) \or (\ex (x,B))) in G by A6, Th45;

( A \imp (\not (\not A)) in G & (\ex (x,B)) \imp (\ex (x,B)) in G ) by Th34, Th64;

then ( ((\not A) \or (\ex (x,B))) \imp ((\not (\not A)) \imp (\ex (x,B))) in G & ((\not (\not A)) \imp (\ex (x,B))) \imp (A \imp (\ex (x,B))) in G ) by Th62, Th103;

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

hence (\ex (x,(A \imp B))) \imp (A \imp (\ex (x,B))) in G by A7, Th45; :: thesis: verum