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 Th31:
  for U2 being strict non-empty MSSubAlgebra of U0 st U0 |= e holds U2 |= e
proof
  let U2 be strict non-empty MSSubAlgebra of U0 such that
A1: U0 |= e;
  let h be ManySortedFunction of TermAlg S, U2 such that
A2: h is_homomorphism TermAlg S, U2;
A3: the Sorts of TermAlg S is_transformable_to the Sorts of U2;
  the Sorts of U2 is MSSubset of U0 by MSUALG_2:def 9;
  then reconsider f1 = h as ManySortedFunction of TermAlg S, U0 by A3,
EXTENS_1:5;
  f1 is_homomorphism TermAlg S, U0 by A2,MSUALG_9:15;
  hence thesis by A1;
end;
