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 Th20:
  for A be MSSubset of U0 holds MSSubSort(A) is opers_closed & A
  c= MSSubSort(A)
proof
  let A be MSSubset of U0;
  thus MSSubSort(A) is opers_closed
  proof
    let o be OperSymbol of S;
    rng ((Den(o,U0))|(((MSSubSort A)# * (the Arity of S)).o)) c= ((
    MSSubSort A) * (the ResultSort of S)).o by Th19;
    hence thesis;
  end;
  A c= Constants(U0) (\/) A & Constants(U0) (\/) A c= MSSubSort(A) by Th15,
PBOOLE:14;
  hence thesis by PBOOLE:13;
end;
