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 st L is subst-correct & L is vf-qc-correct holds
(\for (x,(A \and B))) \imp ((\for (x,A)) \and (\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 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 st L is subst-correct & L is vf-qc-correct holds
(\for (x,(A \and B))) \imp ((\for (x,A)) \and (\for (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 st L is subst-correct & L is vf-qc-correct holds
(\for (x,(A \and B))) \imp ((\for (x,A)) \and (\for (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 st L is subst-correct & L is vf-qc-correct holds
(\for (x,(A \and B))) \imp ((\for (x,A)) \and (\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 st L is subst-correct & L is vf-qc-correct holds
(\for (x,(A \and B))) \imp ((\for (x,A)) \and (\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 st L is subst-correct & L is vf-qc-correct holds
(\for (x,(A \and B))) \imp ((\for (x,A)) \and (\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 st L is subst-correct & L is vf-qc-correct holds
(\for (x,(A \and B))) \imp ((\for (x,A)) \and (\for (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 \and B))) \imp ((\for (x,A)) \and (\for (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 \and B))) \imp ((\for (x,A)) \and (\for (x,B))) in G )
assume A1: ( L is subst-correct & L is vf-qc-correct ) ; :: thesis: (\for (x,(A \and B))) \imp ((\for (x,A)) \and (\for (x,B))) in G
(A \and B) \imp A in G by Def38;
then A2: (\for (x,(A \and B))) \imp (\for (x,A)) in G by A1, Th115;
(A \and B) \imp B in G by Def38;
then A3: (\for (x,(A \and B))) \imp (\for (x,B)) in G by A1, Th115;
((\for (x,(A \and B))) \imp (\for (x,A))) \imp (((\for (x,(A \and B))) \imp (\for (x,B))) \imp ((\for (x,(A \and B))) \imp ((\for (x,A)) \and (\for (x,B))))) in G by Th49;
then ((\for (x,(A \and B))) \imp (\for (x,B))) \imp ((\for (x,(A \and B))) \imp ((\for (x,A)) \and (\for (x,B)))) in G by A2, Def38;
hence (\for (x,(A \and B))) \imp ((\for (x,A)) \and (\for (x,B))) in G by A3, Def38; :: thesis: verum