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 empty-yielding GeneratorSet of T
for S1 being non empty non void b₁ -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))) \imp (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 empty-yielding 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))) \imp (A \and (\ex (x,B))) in G
let T be non-empty MSAlgebra over J; :: thesis: for X being empty-yielding 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))) \imp (A \and (\ex (x,B))) in G
let X be empty-yielding 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))) \imp (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))) \imp (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))) \imp (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))) \imp (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))) \imp (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))) \imp (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))) \imp (A \and (\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 \and B))) \imp (A \and (\ex (x,B))) in G )
assume A2: ( x in X . a & x nin (vf A) . a ) ; :: thesis: (\ex (x,(A \and B))) \imp (A \and (\ex (x,B))) in G
((\for (x,(\not A))) \or (\for (x,(\not B)))) \imp (\for (x,((\not A) \or (\not B)))) in G by A1, Th127;
then A3: (\not (\for (x,((\not A) \or (\not B))))) \imp (\not ((\for (x,(\not A))) \or (\for (x,(\not B))))) in G by Th58;
((\not A) \or (\not B)) \imp (\not (A \and B)) in G by Th73;
then ( (\ex (x,(A \and B))) \iff (\not (\for (x,(\not (A \and B))))) in G & (\for (x,((\not A) \or (\not B)))) \imp (\for (x,(\not (A \and B)))) in G ) by A1, Th115, Th105;
then ( (\not (\for (x,(\not (A \and B))))) \imp (\not (\for (x,((\not A) \or (\not B))))) in G & (\ex (x,(A \and B))) \imp (\not (\for (x,(\not (A \and B))))) in G ) by Th43, Th58;
then (\ex (x,(A \and B))) \imp (\not (\for (x,((\not A) \or (\not B))))) in G by Th45;
then A4: (\ex (x,(A \and B))) \imp (\not ((\for (x,(\not A))) \or (\for (x,(\not B))))) in G by A3, Th45;
( (\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 ( (\not (\for (x,(\not A)))) \imp (\ex (x,A)) in G & (\not (\for (x,(\not B)))) \imp (\ex (x,B)) in G ) by Th43;
then A5: ((\not (\for (x,(\not A)))) \and (\not (\for (x,(\not B))))) \imp ((\ex (x,A)) \and (\ex (x,B))) in G by Th72;
(\not ((\for (x,(\not A))) \or (\for (x,(\not B))))) \imp ((\not (\for (x,(\not A)))) \and (\not (\for (x,(\not B))))) in G by Th71;
then (\ex (x,(A \and B))) \imp ((\not (\for (x,(\not A)))) \and (\not (\for (x,(\not B))))) in G by A4, Th45;
then A6: (\ex (x,(A \and B))) \imp ((\ex (x,A)) \and (\ex (x,B))) in G by A5, Th45;
A \imp A in G by Th34;
then ( \for (x,(A \imp A)) in G & (\for (x,(A \imp A))) \imp ((\ex (x,A)) \imp A) in G ) by A1, A2, Th120, Def39;
then ( (\ex (x,A)) \imp A in G & (\ex (x,B)) \imp (\ex (x,B)) in G ) by Def38, Th34;
then ((\ex (x,A)) \and (\ex (x,B))) \imp (A \and (\ex (x,B))) in G by Th72;
hence (\ex (x,(A \and B))) \imp (A \and (\ex (x,B))) in G by A6, Th45; :: thesis: verum