let n be non empty Nat; for J being non empty non void Signature
for T being non-empty MSAlgebra over J
for X being V3() GeneratorSet of T
for S1 being non empty non void b1 -extension n PC-correct QC-correct QCLangSignature over Union X
for x, y being Element of Union X
for x0, y0 being Element of Union (X extended_by ({}, the carrier of S1))
for L being non-empty Language of X extended_by ({}, the carrier of S1),S1
for G1 being QC-theory_with_equality of L
for A being Formula of L
for s being SortSymbol of S1 st 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 & x = x0 & x0 in X . s & y = y0 & y0 in X . s & x <> y & y nin (vf A) . s & X . s is infinite holds
\for (x,(A \iff (\ex (y,((x '=' (y,L)) \and (A / (x0,y0))))))) in G1
let J be non empty non void Signature; for T being non-empty MSAlgebra over J
for X being V3() GeneratorSet of T
for S1 being non empty non void J -extension n PC-correct QC-correct QCLangSignature over Union X
for x, y being Element of Union X
for x0, y0 being Element of Union (X extended_by ({}, the carrier of S1))
for L being non-empty Language of X extended_by ({}, the carrier of S1),S1
for G1 being QC-theory_with_equality of L
for A being Formula of L
for s being SortSymbol of S1 st 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 & x = x0 & x0 in X . s & y = y0 & y0 in X . s & x <> y & y nin (vf A) . s & X . s is infinite holds
\for (x,(A \iff (\ex (y,((x '=' (y,L)) \and (A / (x0,y0))))))) in G1
let T be non-empty MSAlgebra over J; for X being V3() GeneratorSet of T
for S1 being non empty non void J -extension n PC-correct QC-correct QCLangSignature over Union X
for x, y being Element of Union X
for x0, y0 being Element of Union (X extended_by ({}, the carrier of S1))
for L being non-empty Language of X extended_by ({}, the carrier of S1),S1
for G1 being QC-theory_with_equality of L
for A being Formula of L
for s being SortSymbol of S1 st 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 & x = x0 & x0 in X . s & y = y0 & y0 in X . s & x <> y & y nin (vf A) . s & X . s is infinite holds
\for (x,(A \iff (\ex (y,((x '=' (y,L)) \and (A / (x0,y0))))))) in G1
let X be V3() GeneratorSet of T; for S1 being non empty non void J -extension n PC-correct QC-correct QCLangSignature over Union X
for x, y being Element of Union X
for x0, y0 being Element of Union (X extended_by ({}, the carrier of S1))
for L being non-empty Language of X extended_by ({}, the carrier of S1),S1
for G1 being QC-theory_with_equality of L
for A being Formula of L
for s being SortSymbol of S1 st 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 & x = x0 & x0 in X . s & y = y0 & y0 in X . s & x <> y & y nin (vf A) . s & X . s is infinite holds
\for (x,(A \iff (\ex (y,((x '=' (y,L)) \and (A / (x0,y0))))))) in G1
let S1 be non empty non void J -extension n PC-correct QC-correct QCLangSignature over Union X; for x, y being Element of Union X
for x0, y0 being Element of Union (X extended_by ({}, the carrier of S1))
for L being non-empty Language of X extended_by ({}, the carrier of S1),S1
for G1 being QC-theory_with_equality of L
for A being Formula of L
for s being SortSymbol of S1 st 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 & x = x0 & x0 in X . s & y = y0 & y0 in X . s & x <> y & y nin (vf A) . s & X . s is infinite holds
\for (x,(A \iff (\ex (y,((x '=' (y,L)) \and (A / (x0,y0))))))) in G1
let x, y be Element of Union X; for x0, y0 being Element of Union (X extended_by ({}, the carrier of S1))
for L being non-empty Language of X extended_by ({}, the carrier of S1),S1
for G1 being QC-theory_with_equality of L
for A being Formula of L
for s being SortSymbol of S1 st 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 & x = x0 & x0 in X . s & y = y0 & y0 in X . s & x <> y & y nin (vf A) . s & X . s is infinite holds
\for (x,(A \iff (\ex (y,((x '=' (y,L)) \and (A / (x0,y0))))))) in G1
let x0, y0 be Element of Union (X extended_by ({}, the carrier of S1)); for L being non-empty Language of X extended_by ({}, the carrier of S1),S1
for G1 being QC-theory_with_equality of L
for A being Formula of L
for s being SortSymbol of S1 st 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 & x = x0 & x0 in X . s & y = y0 & y0 in X . s & x <> y & y nin (vf A) . s & X . s is infinite holds
\for (x,(A \iff (\ex (y,((x '=' (y,L)) \and (A / (x0,y0))))))) in G1
let L be non-empty Language of X extended_by ({}, the carrier of S1),S1; for G1 being QC-theory_with_equality of L
for A being Formula of L
for s being SortSymbol of S1 st 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 & x = x0 & x0 in X . s & y = y0 & y0 in X . s & x <> y & y nin (vf A) . s & X . s is infinite holds
\for (x,(A \iff (\ex (y,((x '=' (y,L)) \and (A / (x0,y0))))))) in G1
let G1 be QC-theory_with_equality of L; for A being Formula of L
for s being SortSymbol of S1 st 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 & x = x0 & x0 in X . s & y = y0 & y0 in X . s & x <> y & y nin (vf A) . s & X . s is infinite holds
\for (x,(A \iff (\ex (y,((x '=' (y,L)) \and (A / (x0,y0))))))) in G1
let A be Formula of L; for s being SortSymbol of S1 st 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 & x = x0 & x0 in X . s & y = y0 & y0 in X . s & x <> y & y nin (vf A) . s & X . s is infinite holds
\for (x,(A \iff (\ex (y,((x '=' (y,L)) \and (A / (x0,y0))))))) in G1
let s be SortSymbol of S1; ( 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 & x = x0 & x0 in X . s & y = y0 & y0 in X . s & x <> y & y nin (vf A) . s & X . s is infinite implies \for (x,(A \iff (\ex (y,((x '=' (y,L)) \and (A / (x0,y0))))))) in G1 )
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 )
; \for (x,(A \iff (\ex (y,((x '=' (y,L)) \and (A / (x0,y0))))))) in G1
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:
(t2 '=' (t1,L)) \imp (((A / (x0,y0)) / (y0,t2)) \imp ((A / (x0,y0)) / (y0,t1))) in G1
by A1, Def42;
(t1 '=' (t2,L)) \imp (t2 '=' (t1,L)) in G1
by A0, A1, A2, ThOne;
then A8:
(t1 '=' (t2,L)) \imp (((A / (x0,y0)) / (y0,t2)) \imp ((A / (x0,y0)) / (y0,t1))) in G1
by A3, Th45;
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
;
((t1 '=' (t2,L)) \imp ((A / (x0,y0)) \imp A)) \imp (((t1 '=' (t2,L)) \and (A / (x0,y0))) \imp A) in G1
by Th48;
then
((t1 '=' (t2,L)) \and (A / (x0,y0))) \imp A in G1
by A8, A5, A6, Def38;
then
( \for (y,(((t1 '=' (t2,L)) \and (A / (x0,y0))) \imp A)) in G1 & (\for (y,(((t1 '=' (t2,L)) \and (A / (x0,y0))) \imp A))) \imp ((\ex (y,((t1 '=' (t2,L)) \and (A / (x0,y0))))) \imp A) in G1 )
by A0, A1, A2, Th120, Def39;
then A9:
(\ex (y,((t1 '=' (t2,L)) \and (A / (x0,y0))))) \imp A in G1
by Def38;
B1:
(((t1 '=' (t2,L)) \and (A / (x0,y0))) / (y0,x0)) \imp (\ex (y,((t1 '=' (t2,L)) \and (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 (A / (x0,y0))) / (y0,x0) =
((t1 '=' (t2,L)) / (y0,x0)) \and ((A / (x0,y0)) / (y0,x0))
by A0, A1, A4, Th27
.=
((t1 '=' (t2,L)) / (y0,x0)) \and A
by A0, A1, A4
.=
((t1 '=' (t2,L)) / (y0,t1)) \and A
by A1, A4, Th14
.=
((t1 / (y0,t1)) '=' ((t2 / (y0,t1)),L)) \and A
by A0, A1
.=
(t1 '=' ((t2 / (y0,t1)),L)) \and A
by B2, A0, A1, A4
.=
(t1 '=' (t1,L)) \and A
by A0, A1, A4
;
q1 '=' (q1,L) in G1
by Def42;
then
( A \imp (t1 '=' (t1,L)) in G1 & A \imp A in G1 )
by Th34, Th44;
then
A \imp ((t1 '=' (t1,L)) \and A) in G1
by Th201;
then
A \imp (\ex (y,((t1 '=' (t2,L)) \and (A / (x0,y0))))) in G1
by B1, B3, Th45;
then
A \iff (\ex (y,((t1 '=' (t2,L)) \and (A / (x0,y0))))) in G1
by A9, Th43;
hence
\for (x,(A \iff (\ex (y,((x '=' (y,L)) \and (A / (x0,y0))))))) in G1
by A1, Def39; verum