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

(\for (x,(A \or B))) \imp (A \or (\for (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

(\for (x,(A \or B))) \imp (A \or (\for (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

(\for (x,(A \or B))) \imp (A \or (\for (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

(\for (x,(A \or B))) \imp (A \or (\for (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

(\for (x,(A \or B))) \imp (A \or (\for (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

(\for (x,(A \or B))) \imp (A \or (\for (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

(\for (x,(A \or B))) \imp (A \or (\for (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

(\for (x,(A \or B))) \imp (A \or (\for (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

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

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

set c = a;

set a = \not A;

set b = B;

x nin (vf (\not A)) . a by A1, A2;

then A3: (\for (x,((\not A) \imp B))) \imp ((\not A) \imp (\for (x,B))) in G by A2, Def39;

(A \or B) \imp ((\not A) \imp B) in G by Th62;

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

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

( (\not (\not A)) \imp A in G & (\for (x,B)) \imp (\for (x,B)) in G ) by Th34, Th65;

then ( ((\not (\not A)) \or (\for (x,B))) \imp (A \or (\for (x,B))) in G & ((\not A) \imp (\for (x,B))) \imp ((\not (\not A)) \or (\for (x,B))) in G ) by Th59, Th82;

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

hence (\for (x,(A \or B))) \imp (A \or (\for (x,B))) in G by A4, 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

(\for (x,(A \or B))) \imp (A \or (\for (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

(\for (x,(A \or B))) \imp (A \or (\for (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

(\for (x,(A \or B))) \imp (A \or (\for (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

(\for (x,(A \or B))) \imp (A \or (\for (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

(\for (x,(A \or B))) \imp (A \or (\for (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

(\for (x,(A \or B))) \imp (A \or (\for (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

(\for (x,(A \or B))) \imp (A \or (\for (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

(\for (x,(A \or B))) \imp (A \or (\for (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

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

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

set c = a;

set a = \not A;

set b = B;

x nin (vf (\not A)) . a by A1, A2;

then A3: (\for (x,((\not A) \imp B))) \imp ((\not A) \imp (\for (x,B))) in G by A2, Def39;

(A \or B) \imp ((\not A) \imp B) in G by Th62;

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

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

( (\not (\not A)) \imp A in G & (\for (x,B)) \imp (\for (x,B)) in G ) by Th34, Th65;

then ( ((\not (\not A)) \or (\for (x,B))) \imp (A \or (\for (x,B))) in G & ((\not A) \imp (\for (x,B))) \imp ((\not (\not A)) \or (\for (x,B))) in G ) by Th59, Th82;

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

hence (\for (x,(A \or B))) \imp (A \or (\for (x,B))) in G by A4, Th45; :: thesis: verum