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 & L is vf-qc-correct holds

(\for (x,(A \imp B))) \imp ((\ex (x,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 st L is subst-correct & L is vf-qc-correct holds

(\for (x,(A \imp B))) \imp ((\ex (x,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 st L is subst-correct & L is vf-qc-correct holds

(\for (x,(A \imp B))) \imp ((\ex (x,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 st L is subst-correct & L is vf-qc-correct holds

(\for (x,(A \imp B))) \imp ((\ex (x,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 st L is subst-correct & L is vf-qc-correct holds

(\for (x,(A \imp B))) \imp ((\ex (x,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 st L is subst-correct & L is vf-qc-correct holds

(\for (x,(A \imp B))) \imp ((\ex (x,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 st L is subst-correct & L is vf-qc-correct holds

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

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

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

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

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

A2: (\for (x,(A \imp B))) \imp ((\for (x,(\not B))) \imp (\for (x,(\not A)))) in G by A1, Th117;

((\for (x,(\not B))) \imp (\for (x,(\not A)))) \imp ((\not (\for (x,(\not A)))) \imp (\not (\for (x,(\not B))))) in G by Th57;

then A3: (\for (x,(A \imp B))) \imp ((\not (\for (x,(\not A)))) \imp (\not (\for (x,(\not B))))) in G by A2, Th45;

(\not (\ex (x,A))) \iff (\for (x,(\not A))) in G by Def39;

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

then ( (\not (\not (\ex (x,A)))) \imp (\not (\for (x,(\not A)))) in G & (\ex (x,A)) \imp (\not (\not (\ex (x,A)))) in G ) by Th64, Th58;

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

(\not (\ex (x,B))) \iff (\for (x,(\not B))) in G by Def39;

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

then ( (\not (\for (x,(\not B)))) \imp (\not (\not (\ex (x,B)))) in G & (\not (\not (\ex (x,B)))) \imp (\ex (x,B)) in G ) by Th65, Th58;

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

then ((\not (\for (x,(\not A)))) \imp (\not (\for (x,(\not B))))) \imp ((\ex (x,A)) \imp (\ex (x,B))) in G by A4, Th103;

hence (\for (x,(A \imp B))) \imp ((\ex (x,A)) \imp (\ex (x,B))) in G by A3, 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 st L is subst-correct & L is vf-qc-correct holds

(\for (x,(A \imp B))) \imp ((\ex (x,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 st L is subst-correct & L is vf-qc-correct holds

(\for (x,(A \imp B))) \imp ((\ex (x,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 st L is subst-correct & L is vf-qc-correct holds

(\for (x,(A \imp B))) \imp ((\ex (x,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 st L is subst-correct & L is vf-qc-correct holds

(\for (x,(A \imp B))) \imp ((\ex (x,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 st L is subst-correct & L is vf-qc-correct holds

(\for (x,(A \imp B))) \imp ((\ex (x,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 st L is subst-correct & L is vf-qc-correct holds

(\for (x,(A \imp B))) \imp ((\ex (x,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 st L is subst-correct & L is vf-qc-correct holds

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

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

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

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

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

A2: (\for (x,(A \imp B))) \imp ((\for (x,(\not B))) \imp (\for (x,(\not A)))) in G by A1, Th117;

((\for (x,(\not B))) \imp (\for (x,(\not A)))) \imp ((\not (\for (x,(\not A)))) \imp (\not (\for (x,(\not B))))) in G by Th57;

then A3: (\for (x,(A \imp B))) \imp ((\not (\for (x,(\not A)))) \imp (\not (\for (x,(\not B))))) in G by A2, Th45;

(\not (\ex (x,A))) \iff (\for (x,(\not A))) in G by Def39;

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

then ( (\not (\not (\ex (x,A)))) \imp (\not (\for (x,(\not A)))) in G & (\ex (x,A)) \imp (\not (\not (\ex (x,A)))) in G ) by Th64, Th58;

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

(\not (\ex (x,B))) \iff (\for (x,(\not B))) in G by Def39;

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

then ( (\not (\for (x,(\not B)))) \imp (\not (\not (\ex (x,B)))) in G & (\not (\not (\ex (x,B)))) \imp (\ex (x,B)) in G ) by Th65, Th58;

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

then ((\not (\for (x,(\not A)))) \imp (\not (\for (x,(\not B))))) \imp ((\ex (x,A)) \imp (\ex (x,B))) in G by A4, Th103;

hence (\for (x,(A \imp B))) \imp ((\ex (x,A)) \imp (\ex (x,B))) in G by A3, Th45; :: thesis: verum