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 x, y be Element of Union X; :: thesis:
let x0, y0 be Element of Union (X extended_by ({}, the carrier of S1)); :: thesis:
let L be non-empty Language of X extended_by ({}, the carrier of S1),S1; :: thesis:
let G1 be QC-theory_with_equality of L; :: thesis:
let A be Formula of L; :: thesis:
let s be SortSymbol of S1; :: thesis:
set Y = X extended_by ({}, the carrier of S1);
assume that
A0: ( L is subst-correct3 & L is vf-finite & L is subst-correct2 & L is subst-correct & L is subst-eq-correct & L is vf-qc-correct & L is vf-eq-correct ) and
A1: ( x = x0 & x0 in X . s & y = y0 & y0 in X . s & x <> y & y nin (vf A) . s & X . s is infinite ) ; :: thesis:
A2: ( s in dom X & dom X = the carrier of J ) by A1, FUNCT_1:def 2, PARTFUN1:def 2;
then ( X . s c= the Sorts of T . s & the Sorts of T . s = the Sorts of L . s ) by Th16, PBOOLE:def 2, PBOOLE:def 18;
then reconsider t1 = x0, t2 = y0 as Element of L,s by A1;
reconsider j = s as SortSymbol of J by A2;
reconsider q1 = t1, q2 = t2 as Element of T,j by Th16;
A3: (t1 '=' (t2,L)) \imp (((A / (x0,y0)) / (y0,t1)) \imp ((A / (x0,y0)) / (y0,t2))) in G1 by A1, Def42;
dom (X extended_by ({}, the carrier of S1)) = the carrier of S1 by PARTFUN1:def 2;
then A4: ( X . s = (X extended_by ({}, the carrier of S1)) . s & the Sorts of L . the formula-sort of S1 <> {} & X extended_by ({}, the carrier of S1) is ManySortedSubset of the Sorts of L ) by A2, Th1, Th23;
then A5: (A / (x0,y0)) / (y0,t1) = (A / (x0,y0)) / (y0,x0) by A1, Th14
.= A by A0, A1, A4 ;
A6: (A / (x0,y0)) / (y0,t2) = (A / (x0,y0)) / (y0,y0) by A1, A4, Th14
.= A / (x0,y0) by A0, A1, A4 ;
A \imp ((t1 '=' (t2,L)) \imp (A / (x0,y0))) in G1 by A3, A5, A6, Th38;
then \for (y,(A \imp ((t1 '=' (t2,L)) \imp (A / (x0,y0))))) in G1 by Def39;
then A9: A \imp (\for (y,((t1 '=' (t2,L)) \imp (A / (x0,y0))))) in G1 by A1, A2, Th108;
B1: (((t1 '=' (t2,L)) \and (\not (A / (x0,y0)))) / (y0,x0)) \imp (\ex (y,((t1 '=' (t2,L)) \and (\not (A / (x0,y0)))))) in G1 by A0, A1, A2, Th110;
vf t1 = s -singleton x0 by A0, A1;
then (vf t1) . s = {x0} by AOFA_A00:6;
then B2: y0 nin (vf t1) . s by A1, TARSKI:def 1;
B3: ((t1 '=' (t2,L)) \and (\not (A / (x0,y0)))) / (y0,x0) = ((t1 '=' (t2,L)) / (y0,x0)) \and ((\not (A / (x0,y0))) / (y0,x0)) by A0, A1, A4, Th27
.= ((t1 '=' (t2,L)) / (y0,x0)) \and (\not ((A / (x0,y0)) / (y0,x0))) by A0, A1, A4, Th27
.= ((t1 '=' (t2,L)) / (y0,x0)) \and (\not A) by A0, A1, A4
.= ((t1 '=' (t2,L)) / (y0,t1)) \and (\not A) by A1, A4, Th14
.= ((t1 / (y0,t1)) '=' ((t2 / (y0,t1)),L)) \and (\not A) by A0, A1
.= (t1 '=' ((t2 / (y0,t1)),L)) \and (\not A) by B2, A0, A1, A4
.= (t1 '=' (t1,L)) \and (\not A) by A0, A1, A4 ;
q1 '=' (q1,L) in G1 by Def42;
then ( (\not A) \imp (t1 '=' (t1,L)) in G1 & (\not A) \imp (\not A) in G1 ) by Th34, Th44;
then (\not A) \imp ((t1 '=' (t1,L)) \and (\not A)) in G1 by Th201;
then (\not A) \imp (\ex (y,((t1 '=' (t2,L)) \and (\not (A / (x0,y0)))))) in G1 by B1, B3, Th45;
then B4: (\not (\ex (y,((t1 '=' (t2,L)) \and (\not (A / (x0,y0))))))) \imp A in G1 by Th68;
( (A / (x0,y0)) \imp (\not (\not (A / (x0,y0)))) in G1 & (\not (t1 '=' (t2,L))) \imp (\not (t1 '=' (t2,L))) in G1 ) by Th34, Th64;
then ( ((\not (t1 '=' (t2,L))) \or (\not (\not (A / (x0,y0))))) \imp (\not ((t1 '=' (t2,L)) \and (\not (A / (x0,y0))))) in G1 & ((\not (t1 '=' (t2,L))) \or (A / (x0,y0))) \imp ((\not (t1 '=' (t2,L))) \or (\not (\not (A / (x0,y0))))) in G1 ) by Th73, Th59;
then ( ((\not (t1 '=' (t2,L))) \or (A / (x0,y0))) \imp (\not ((t1 '=' (t2,L)) \and (\not (A / (x0,y0))))) in G1 & ((t1 '=' (t2,L)) \imp (A / (x0,y0))) \imp ((\not (t1 '=' (t2,L))) \or (A / (x0,y0))) in G1 ) by Th45, Th82;
then ((t1 '=' (t2,L)) \imp (A / (x0,y0))) \imp (\not ((t1 '=' (t2,L)) \and (\not (A / (x0,y0))))) in G1 by Th45;
then ((t1 '=' (t2,L)) \and (\not (A / (x0,y0)))) \imp (\not ((t1 '=' (t2,L)) \imp (A / (x0,y0)))) in G1 by Th67;
then (\ex (y,((t1 '=' (t2,L)) \and (\not (A / (x0,y0)))))) \imp (\ex (y,(\not ((t1 '=' (t2,L)) \imp (A / (x0,y0)))))) in G1 by A0, Th128;
then (\not (\ex (y,(\not ((t1 '=' (t2,L)) \imp (A / (x0,y0))))))) \imp (\not (\ex (y,((t1 '=' (t2,L)) \and (\not (A / (x0,y0))))))) in G1 by Th58;
then for G being QC-theory of L holds
( (\not (\ex (y,(\not ((t1 '=' (t2,L)) \imp (A / (x0,y0))))))) \imp A in G1 & (\for (y,((t1 '=' (t2,L)) \imp (A / (x0,y0))))) \iff (\not (\ex (y,(\not ((t1 '=' (t2,L)) \imp (A / (x0,y0))))))) in G ) by B4, Th45, Th106;
then (\for (y,((t1 '=' (t2,L)) \imp (A / (x0,y0))))) \imp A in G1 by Th92;
then A \iff (\for (y,((t1 '=' (t2,L)) \imp (A / (x0,y0))))) in G1 by A9, Th43;
hence \for (x,(A \iff (\for (y,((x '=' (y,L)) \imp (A / (x0,y0))))))) in G1 by A1, Def39; :: thesis: