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 L being non-empty Language of X extended_by ({}, the carrier of S1),S1
for G1 being QC-theory_with_equality of L
for s1 being SortSymbol of S1
for t1, t2, t3 being Element of L,s1 st L is subst-eq-correct & L is vf-finite & L is subst-correct2 & L is subst-correct3 & s1 in the carrier of J & X . s1 is infinite holds
((t1 '=' (t2,L)) \and (t2 '=' (t3,L))) \imp (t1 '=' (t3,L)) 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 L being non-empty Language of X extended_by ({}, the carrier of S1),S1
for G1 being QC-theory_with_equality of L
for s1 being SortSymbol of S1
for t1, t2, t3 being Element of L,s1 st L is subst-eq-correct & L is vf-finite & L is subst-correct2 & L is subst-correct3 & s1 in the carrier of J & X . s1 is infinite holds
((t1 '=' (t2,L)) \and (t2 '=' (t3,L))) \imp (t1 '=' (t3,L)) 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 L being non-empty Language of X extended_by ({}, the carrier of S1),S1
for G1 being QC-theory_with_equality of L
for s1 being SortSymbol of S1
for t1, t2, t3 being Element of L,s1 st L is subst-eq-correct & L is vf-finite & L is subst-correct2 & L is subst-correct3 & s1 in the carrier of J & X . s1 is infinite holds
((t1 '=' (t2,L)) \and (t2 '=' (t3,L))) \imp (t1 '=' (t3,L)) 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 L being non-empty Language of X extended_by ({}, the carrier of S1),S1
for G1 being QC-theory_with_equality of L
for s1 being SortSymbol of S1
for t1, t2, t3 being Element of L,s1 st L is subst-eq-correct & L is vf-finite & L is subst-correct2 & L is subst-correct3 & s1 in the carrier of J & X . s1 is infinite holds
((t1 '=' (t2,L)) \and (t2 '=' (t3,L))) \imp (t1 '=' (t3,L)) in G1
let S1 be 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 G1 being QC-theory_with_equality of L
for s1 being SortSymbol of S1
for t1, t2, t3 being Element of L,s1 st L is subst-eq-correct & L is vf-finite & L is subst-correct2 & L is subst-correct3 & s1 in the carrier of J & X . s1 is infinite holds
((t1 '=' (t2,L)) \and (t2 '=' (t3,L))) \imp (t1 '=' (t3,L)) 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 s1 being SortSymbol of S1
for t1, t2, t3 being Element of L,s1 st L is subst-eq-correct & L is vf-finite & L is subst-correct2 & L is subst-correct3 & s1 in the carrier of J & X . s1 is infinite holds
((t1 '=' (t2,L)) \and (t2 '=' (t3,L))) \imp (t1 '=' (t3,L)) in G1
let G1 be QC-theory_with_equality of L; for s1 being SortSymbol of S1
for t1, t2, t3 being Element of L,s1 st L is subst-eq-correct & L is vf-finite & L is subst-correct2 & L is subst-correct3 & s1 in the carrier of J & X . s1 is infinite holds
((t1 '=' (t2,L)) \and (t2 '=' (t3,L))) \imp (t1 '=' (t3,L)) in G1
let s1 be SortSymbol of S1; for t1, t2, t3 being Element of L,s1 st L is subst-eq-correct & L is vf-finite & L is subst-correct2 & L is subst-correct3 & s1 in the carrier of J & X . s1 is infinite holds
((t1 '=' (t2,L)) \and (t2 '=' (t3,L))) \imp (t1 '=' (t3,L)) in G1
let t1, t2, t3 be Element of L,s1; ( L is subst-eq-correct & L is vf-finite & L is subst-correct2 & L is subst-correct3 & s1 in the carrier of J & X . s1 is infinite implies ((t1 '=' (t2,L)) \and (t2 '=' (t3,L))) \imp (t1 '=' (t3,L)) in G1 )
assume that
A0:
( L is subst-eq-correct & L is vf-finite & L is subst-correct2 & L is subst-correct3 )
and
A1:
s1 in the carrier of J
and
A3:
X . s1 is infinite
; ((t1 '=' (t2,L)) \and (t2 '=' (t3,L))) \imp (t1 '=' (t3,L)) in G1
set Y = X extended_by ({}, the carrier of S1);
vf t3 is V31()
by A0;
then
(vf t3) . s1 is finite
by FINSET_1:def 5;
then A4:
( the Element of (X . s1) \ ((vf t3) . s1) in X . s1 & the Element of (X . s1) \ ((vf t3) . s1) nin (vf t3) . s1 )
by A3, XBOOLE_0:def 5;
( dom X = the carrier of J & dom (X extended_by ({}, the carrier of S1)) = the carrier of S1 )
by PARTFUN1:def 2;
then A8:
the Element of (X . s1) \ ((vf t3) . s1) in (X extended_by ({}, the carrier of S1)) . s1
by A1, A4, Th1;
( dom X = the carrier of J & (X . s1) \ ((vf t1) . s1) c= X . s1 )
by PARTFUN1:def 2;
then reconsider x = the Element of (X . s1) \ ((vf t3) . s1) as Element of Union X by A1, A4, CARD_5:2;
reconsider x0 = x as Element of Union (X extended_by ({}, the carrier of S1)) by Th24;
( X . s1 c= the Sorts of T . s1 & the Sorts of T . s1 = the Sorts of L . s1 )
by A1, Th16, PBOOLE:def 2, PBOOLE:def 18;
then reconsider t = x as Element of L,s1 by A4;
A5:
(t2 '=' (t1,L)) \imp (((t '=' (t3,L)) / (x0,t2)) \imp ((t '=' (t3,L)) / (x0,t1))) in G1
by A4, Def42;
(t1 '=' (t2,L)) \imp (t2 '=' (t1,L)) in G1
by A0, A1, A3, ThOne;
then A6:
(t1 '=' (t2,L)) \imp (((t '=' (t3,L)) / (x0,t2)) \imp ((t '=' (t3,L)) / (x0,t1))) in G1
by A5, Th45;
A7: (t '=' (t3,L)) / (x0,t2) =
(t / (x0,t2)) '=' ((t3 / (x0,t2)),L)
by A0, A4
.=
(t / (x0,t2)) '=' (t3,L)
by A0, A8
.=
t2 '=' (t3,L)
by A0, A8
;
A2: (t '=' (t3,L)) / (x0,t1) =
(t / (x0,t1)) '=' ((t3 / (x0,t1)),L)
by A0, A4
.=
(t / (x0,t1)) '=' (t3,L)
by A0, A8
.=
t1 '=' (t3,L)
by A0, A8
;
((t1 '=' (t2,L)) \imp ((t2 '=' (t3,L)) \imp (t1 '=' (t3,L)))) \imp (((t1 '=' (t2,L)) \and (t2 '=' (t3,L))) \imp (t1 '=' (t3,L))) in G1
by Th48;
hence
((t1 '=' (t2,L)) \and (t2 '=' (t3,L))) \imp (t1 '=' (t3,L)) in G1
by A7, A6, A2, Def38; verum