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

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

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

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

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

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

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

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

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

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

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

then (\for (x,((\not A) \and (\not B)))) \imp ((\for (x,(\not A))) \and (\for (x,(\not B)))) in G by Th125;

then A2: (\not ((\for (x,(\not A))) \and (\for (x,(\not B))))) \imp (\not (\for (x,((\not A) \and (\not B))))) in G by Th58;

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

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

(\not (A \or B)) \imp ((\not A) \and (\not B)) in G by Th71;

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

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

then A4: ((\not (\for (x,(\not A)))) \or (\not (\for (x,(\not B))))) \imp (\not (\for (x,(\not (A \or B))))) in G by A3, Th45;

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

then (\not (\for (x,(\not (A \or B))))) \iff (\ex (x,(A \or B))) in G by Th90;

then A5: ((\not (\for (x,(\not A)))) \or (\not (\for (x,(\not B))))) \imp (\ex (x,(A \or B))) in G by A4, Th93;

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

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

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

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

A7: ( (\ex (x,A)) \iff (\not (\for (x,(\not A)))) in G & (\ex (x,B)) \iff (\not (\for (x,(\not B)))) in G ) by Th105;

A8: ( (\not (\for (x,(\not A)))) \imp (\ex (x,A)) in G & (\not (\for (x,(\not B)))) \imp (\ex (x,B)) in G ) by A7, Th43;

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

then A9: (\ex (x,(A \or B))) \imp (\not (\for (x,(\not (A \or B))))) in G by Th43;

((\not A) \and (\not B)) \imp (\not (A \or B)) in G by Th74;

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

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

then A10: (\ex (x,(A \or B))) \imp (\not (\for (x,((\not A) \and (\not B))))) in G by A9, Th45;

A11: ((\not (\for (x,(\not A)))) \or (\not (\for (x,(\not B))))) \imp ((\ex (x,A)) \or (\ex (x,B))) in G by A8, Th59;

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

then A12: (\not ((\for (x,(\not A))) \and (\for (x,(\not B))))) \imp ((\ex (x,A)) \or (\ex (x,B))) in G by A11, Th45;

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

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

then (\ex (x,(A \or B))) \imp (\not ((\for (x,(\not A))) \and (\for (x,(\not B))))) in G by A10, Th45;

then (\ex (x,(A \or B))) \imp ((\ex (x,A)) \or (\ex (x,B))) in G by A12, Th45;

hence ((\ex (x,A)) \or (\ex (x,B))) \iff (\ex (x,(A \or 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 st L is subst-correct & L is vf-qc-correct holds

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

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

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

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

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

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

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

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

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

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

then (\for (x,((\not A) \and (\not B)))) \imp ((\for (x,(\not A))) \and (\for (x,(\not B)))) in G by Th125;

then A2: (\not ((\for (x,(\not A))) \and (\for (x,(\not B))))) \imp (\not (\for (x,((\not A) \and (\not B))))) in G by Th58;

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

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

(\not (A \or B)) \imp ((\not A) \and (\not B)) in G by Th71;

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

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

then A4: ((\not (\for (x,(\not A)))) \or (\not (\for (x,(\not B))))) \imp (\not (\for (x,(\not (A \or B))))) in G by A3, Th45;

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

then (\not (\for (x,(\not (A \or B))))) \iff (\ex (x,(A \or B))) in G by Th90;

then A5: ((\not (\for (x,(\not A)))) \or (\not (\for (x,(\not B))))) \imp (\ex (x,(A \or B))) in G by A4, Th93;

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

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

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

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

A7: ( (\ex (x,A)) \iff (\not (\for (x,(\not A)))) in G & (\ex (x,B)) \iff (\not (\for (x,(\not B)))) in G ) by Th105;

A8: ( (\not (\for (x,(\not A)))) \imp (\ex (x,A)) in G & (\not (\for (x,(\not B)))) \imp (\ex (x,B)) in G ) by A7, Th43;

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

then A9: (\ex (x,(A \or B))) \imp (\not (\for (x,(\not (A \or B))))) in G by Th43;

((\not A) \and (\not B)) \imp (\not (A \or B)) in G by Th74;

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

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

then A10: (\ex (x,(A \or B))) \imp (\not (\for (x,((\not A) \and (\not B))))) in G by A9, Th45;

A11: ((\not (\for (x,(\not A)))) \or (\not (\for (x,(\not B))))) \imp ((\ex (x,A)) \or (\ex (x,B))) in G by A8, Th59;

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

then A12: (\not ((\for (x,(\not A))) \and (\for (x,(\not B))))) \imp ((\ex (x,A)) \or (\ex (x,B))) in G by A11, Th45;

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

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

then (\ex (x,(A \or B))) \imp (\not ((\for (x,(\not A))) \and (\for (x,(\not B))))) in G by A10, Th45;

then (\ex (x,(A \or B))) \imp ((\ex (x,A)) \or (\ex (x,B))) in G by A12, Th45;

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