let n be non empty Nat; :: thesis:
let J be non empty non void Signature; :: thesis:
let T be non-empty MSAlgebra over J; :: thesis:
let X be empty-yielding GeneratorSet of T; :: thesis:
let S1 be non empty non void J -extension n PC-correct QC-correct QCLangSignature over Union X; :: thesis:
let L be non-empty Language of X extended_by ({}, the carrier of S1),S1; :: thesis:
let G be QC-theory of L; :: thesis:
let A be Formula of L; :: thesis:
let x be Element of Union X; :: thesis:
set Y = X extended_by ({}, the carrier of S1);
assume A1: L is subst-correct ; :: thesis:
consider a being object such that
A2: ( a in dom X & x in X . a ) by CARD_5:2;
J is Subsignature of S1 by Def2;
then A3: ( the carrier of J c= the carrier of S1 & dom (X extended_by ({}, the carrier of S1)) = the carrier of S1 & dom X = the carrier of J ) by PARTFUN1:def 2, INSTALG1:10;
then reconsider a = a as SortSymbol of S1 by A2;
A4: x in (X extended_by ({}, the carrier of S1)) . a by A2, A3, Th1;
then reconsider x0 = x as Element of Union (X extended_by ({}, the carrier of S1)) by A3, CARD_5:2;
X c= the Sorts of T by PBOOLE:def 18;
then ( X . a c= the Sorts of T . a & the Sorts of T . a = the Sorts of L . a ) by A2, Th16;
then reconsider t = x as Element of the Sorts of L . a by A2;
X extended_by ({}, the carrier of S1) is ManySortedSubset of the Sorts of L by Th23;
then A / (x0,t) = A / (x0,x0) by A4, Th14
.= A by A4, A1 ;
hence (\for (x,A)) \imp A in G by A2, A4, Def39; :: thesis: