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 Th3:
  for U0 being MSAlgebra over S for B being MSSubset of U0 st B=the
  Sorts of U0 holds B is opers_closed & for o holds o/.B = Den(o,U0)
proof
  let U0 be MSAlgebra over S;
  let B be MSSubset of U0;
  assume
A1: B=the Sorts of U0;
  thus
A2: B is opers_closed
  proof
    let o;
      Result (o,U0) = (B* the ResultSort of S).o by A1,MSUALG_1:def 5;
      hence rng ((Den(o,U0))|((B# * the Arity of S).o)) c= (B * the ResultSort
      of S).o by RELAT_1:def 19;
  end;
  let o;
  B is_closed_on o by A2;
  then
A3: o/.B = (Den(o,U0))|((B# * the Arity of S).o) by Def7;
  per cases;
  suppose
    (B * the ResultSort of S).o = {};
    then Result(o,U0) = {} by A1,MSUALG_1:def 5;
    then Den(o,U0) = {};
    hence thesis by A3;
  end;
  suppose
    (B * the ResultSort of S).o <> {};
    Args(o,U0) = (B# * the Arity of S).o by A1,MSUALG_1:def 4;
    hence thesis by A3;
  end;
end;
