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 Th18:
  for A be MSSubset of U0 for B be MSSubset of U0 st B in SubSort(
  A) holds rng (Den(o,U0)|(((MSSubSort A)# * (the Arity of S)).o)) c= (B * (the
  ResultSort of S)).o
proof
  let A be MSSubset of U0, B be MSSubset of U0;
  set m = ((MSSubSort A)# * (the Arity of S)).o, b = (B# * (the Arity of S)).o
  , d = Den(o,U0);
  assume
A1: B in SubSort(A);
  then b /\ m = m by Th17,XBOOLE_1:28;
  then d|m = (d|b)|m by RELAT_1:71;
  then
A2: rng (d|m) c= rng(d|b) by RELAT_1:70;
  B is opers_closed by A1,Th13;
  then B is_closed_on o;
  then rng (d|b) c= (B * (the ResultSort of S)).o;
  hence thesis by A2;
end;
