theorem Th4:
  x = {} implies x *' y = {}
proof
  set A = DEDEKIND_CUT x, B = DEDEKIND_CUT y;
  assume
A1: x = {};
  not ex e being object st e in { r *' s : r in A & s in B}
  proof
    given e being object such that
A2: e in { r *' s : r in A & s in B};
    ex r,s st e = r *' s & r in A & s in B by A2;
    hence contradiction by A1,Lm10;
  end;
  then { r *' s : r in A & s in B} = {} by XBOOLE_0:def 1;
  hence thesis by Lm11;
end;
