reserve X,Y for set, x,y,z for object, i,j,n for natural number;
reserve
  n for non empty Nat,
  S for non empty non void n PC-correct PCLangSignature,
  L for language MSAlgebra over S,
  F for PC-theory of L,
  A,B,C,D for Formula of L;
reserve
  J for non empty non void Signature,
  T for non-empty MSAlgebra over J,
  X for non empty-yielding GeneratorSet of T,
  S1 for J-extension non empty non void n PC-correct QC-correct
  QCLangSignature over Union X,
  L for non-empty Language of X extended_by ({},the carrier of S1), S1,
  G for QC-theory of L,
  A,B,C,D for Formula of L;
reserve x,y,z for Element of Union X;
reserve x0,y0,z0 for Element of Union (X extended_by ({},the carrier of S1));
reserve a for SortSymbol of J;
reserve
  L for
    non-empty T-extension Language of X extended_by ({},the carrier of S1), S1,
  G for QC-theory of L,
  G1 for QC-theory_with_equality of L,
  A,B,C,D for Formula of L,
  s,s1 for SortSymbol of S1,
  t,t9 for Element of L,s,
  t1,t2,t3 for Element of L,s1;

theorem
  L is subst-eq-correct vf-finite subst-correct2 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
  proof
    assume that
A0: L is subst-eq-correct vf-finite subst-correct2 subst-correct3 and
A1: s1 in the carrier of J and
A3: X.s1 is infinite;
    set Y = X extended_by({}, the carrier of S1);
    vf t3 is finite-yielding 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 Y = the carrier of S1 by PARTFUN1:def 2;
    then
A8: the Element of X.s1\((vf t3).s1) in Y.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 Y by Th24;
    X.s1 c= (the Sorts of T).s1 = (the Sorts of L).s1
    by A1,Th16,PBOOLE:def 2,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 thesis by A7,A6,A2,Def38;
  end;
