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 vf-qc-correct & L is subst-correct holds
((\for (x,A)) \and (\for (x,B))) \imp (\for (x,(A \and 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 vf-qc-correct & L is subst-correct holds
((\for (x,A)) \and (\for (x,B))) \imp (\for (x,(A \and 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 vf-qc-correct & L is subst-correct holds
((\for (x,A)) \and (\for (x,B))) \imp (\for (x,(A \and 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 vf-qc-correct & L is subst-correct holds
((\for (x,A)) \and (\for (x,B))) \imp (\for (x,(A \and 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 vf-qc-correct & L is subst-correct holds
((\for (x,A)) \and (\for (x,B))) \imp (\for (x,(A \and 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 vf-qc-correct & L is subst-correct holds
((\for (x,A)) \and (\for (x,B))) \imp (\for (x,(A \and 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 vf-qc-correct & L is subst-correct holds
((\for (x,A)) \and (\for (x,B))) \imp (\for (x,(A \and B))) in G
let A, B be Formula of L; :: thesis: for x being Element of Union X st L is vf-qc-correct & L is subst-correct holds
((\for (x,A)) \and (\for (x,B))) \imp (\for (x,(A \and B))) in G
let x be Element of Union X; :: thesis: ( L is vf-qc-correct & L is subst-correct implies ((\for (x,A)) \and (\for (x,B))) \imp (\for (x,(A \and B))) in G )
set Y = X extended_by ({}, the carrier of S1);
assume A1: ( L is vf-qc-correct & L is subst-correct ) ; :: thesis: ((\for (x,A)) \and (\for (x,B))) \imp (\for (x,(A \and B))) in G
then ( (\for (x,A)) \imp A in G & (\for (x,B)) \imp B in G ) by Th104;
then ((\for (x,A)) \and (\for (x,B))) \imp (A \and B) in G by Th72;
then A2: \for (x,(((\for (x,A)) \and (\for (x,B))) \imp (A \and B))) in G by Def39;
consider a being object such that
A3: ( a in dom X & x in X . a ) by CARD_5:2;
J is Subsignature of S1 by Def2;
then A4: ( dom X = the carrier of J & the carrier of J c= the carrier of S1 & the carrier of S1 = dom (X extended_by ({}, the carrier of S1)) ) by INSTALG1:10, PARTFUN1:def 2;
reconsider a = a as SortSymbol of J by A3;
( x nin (vf (\for (x,A))) . a & x nin (vf (\for (x,B))) . a ) by A1, A3, A4, Th113;
then x nin ((vf (\for (x,A))) . a) \/ ((vf (\for (x,B))) . a) by XBOOLE_0:def 3;
then x nin ((vf (\for (x,A))) (\/) (vf (\for (x,B)))) . a by A4, PBOOLE:def 4;
then x nin (vf ((\for (x,A)) \and (\for (x,B)))) . a by A1;
hence ((\for (x,A)) \and (\for (x,B))) \imp (\for (x,(A \and B))) in G by A3, A2, Th108; :: thesis: verum