reserve I for set;
reserve S for non empty non void ManySortedSign,
  U0, U1 for non-empty MSAlgebra over S;
reserve s for SortSymbol of S;
reserve e for Element of (Equations S).s;
reserve E for EqualSet of S;

theorem Th35:
  for R being MSCongruence of U0 st U0 |= e holds QuotMSAlg (U0,R) |= e
proof
  let R be MSCongruence of U0 such that
A1: U0 |= e;
  set n = (MSNat_Hom(U0,R)).s;
  let h be ManySortedFunction of TermAlg S, QuotMSAlg (U0,R);
  assume h is_homomorphism TermAlg S, QuotMSAlg (U0,R);
  then consider h0 be ManySortedFunction of TermAlg S, U0 such that
A2: h0 is_homomorphism TermAlg S, U0 and
A3: h = (MSNat_Hom(U0,R)) ** h0 by Th24,MSUALG_4:3;
A4: dom (h0.s) = (the Sorts of TermAlg S).s by FUNCT_2:def 1;
  thus h.s.(e`1) = (n*(h0.s)).(e`1) by A3,MSUALG_3:2
    .= n.((h0.s).(e`1)) by A4,Th29,FUNCT_1:13
    .= n.((h0.s).(e`2)) by A1,A2
    .= (n*(h0.s)).(e`2) by A4,Th30,FUNCT_1:13
    .= h.s.(e`2) by A3,MSUALG_3:2;
end;
