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 Th33:
  U0, U1 are_isomorphic & U0 |= e implies U1 |= e
proof
  assume that
A1: U0, U1 are_isomorphic and
A2: U0 |= e;
  consider F be ManySortedFunction of U0, U1 such that
A3: F is_isomorphism U0, U1 by A1;
  consider G be ManySortedFunction of U1, U0 such that
A4: G = F"";
  F is "1-1" & F is "onto" by A3,MSUALG_3:13;
  then
A5: (G.s) = (F.s)" by A4,MSUALG_3:def 4;
  F is "onto" by A3,MSUALG_3:13;
  then
A6: rng (F.s) = (the Sorts of U1).s;
  let h1 be ManySortedFunction of TermAlg S, U1 such that
A7: h1 is_homomorphism TermAlg S, U1;
  set F1 = G ** h1;
  G is_isomorphism U1, U0 by A3,A4,MSUALG_3:14;
  then G is_homomorphism U1, U0 by MSUALG_3:13;
  then
A8: F1 is_homomorphism TermAlg S, U0 by A7,MSUALG_3:10;
  F is "1-1" by A3,MSUALG_3:13;
  then
A9: (F.s) is one-to-one by MSUALG_3:1;
  (F1.s) = (G.s) * (h1.s) by MSUALG_3:2;
  then
A10: (F.s) * (F1.s) = (F.s) * (G.s) * (h1.s) by RELAT_1:36
    .= id ((the Sorts of U1).s) * (h1.s) by A5,A6,A9,FUNCT_2:29
    .= h1.s by FUNCT_2:17;
A11: dom (F1.s) = (the Sorts of TermAlg S).s by FUNCT_2:def 1;
  hence h1.s.(e`1) = (F.s).((F1.s).(e`1)) by A10,Th29,FUNCT_1:13
    .= (F.s).((F1.s).(e`2)) by A2,A8
    .= h1.s.(e`2) by A10,A11,Th30,FUNCT_1:13;
end;
