theorem
  X c= Y & h|Y is increasing implies h|X is increasing
proof
  assume that
A1: X c= Y and
A2: h|Y is increasing;
  now
A3: X /\ dom h c= Y /\ dom h by A1,XBOOLE_1:26;
    let r1,r2;
    assume r1 in X /\ dom h & r2 in X /\ dom h & r1 < r2;
    hence h.r1 < h.r2 by A2,A3,Th20;
  end;
  hence thesis by Th20;
end;
