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 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; :: thesis: 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; :: thesis: 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; :: thesis: 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; :: thesis: 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; :: thesis: 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)); :: thesis: 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; :: thesis: 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; :: thesis: 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; :: thesis: 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; :: thesis: ( 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 ) ; :: thesis: \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 ;
then ( X . s c= the Sorts of T . s & the Sorts of T . s = the Sorts of L . s ) by ;
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 ;
(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 ;
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 ;
then A5: (A / (x0,y0)) / (y0,t1) = (A / (x0,y0)) / (y0,x0) by
.= 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 ;
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 ;
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 ;
hence \for (x,(A \iff (\ex (y,((x '=' (y,L)) \and (A / (x0,y0))))))) in G1 by ; :: thesis: verum