reserve x,y for object;
reserve S for non void non empty ManySortedSign,
  o for OperSymbol of S,
  U0,U1, U2 for MSAlgebra over S;

theorem
  the MSAlgebra of U0 = the MSAlgebra of U1 implies U0 is MSSubAlgebra of U1
proof assume
A1: the MSAlgebra of U0 = the MSAlgebra of U1;
  hence the Sorts of U0 is MSSubset of U1 by PBOOLE:def 18;
  let B be MSSubset of U1;
  set f1 = the Charact of U0, f2 = Opers(U1,B);
  assume
A2: B=the Sorts of U0;
    thus
 B is opers_closed
    proof let o be OperSymbol of S;
      (Den(o,U1))|((B#*the Arity of S).o) c= Den(o,U1) by RELAT_1:59; then
      rng((Den(o,U1))|((B#*the Arity of S).o)) c= rng Den(o,U1) & rng Den(o,U1)
      c= Result(o,U1) & Result(o,U1) = (B*the ResultSort of S).o
      by A1,A2,MSUALG_1:def 5,RELAT_1:11,def 19;
      hence rng ((Den(o,U1))|((B#*the Arity of S).o)) c=
      (B*the ResultSort of S).o;
    end;
  for x being object st x in (the carrier' of S) holds f1.x = f2.x
  by A1,A2,Th4;
  hence thesis;
end;
