theorem
  (for x be set st x in dom(f-g) holds g.x <= f.x) implies f-g is nonnegative
proof
A1: dom (f-g) = dom f /\ dom g by VALUED_1:12;
  assume for x be set st x in dom(f-g) holds g.x <= f.x;
  hence thesis by A1,Th58;
end;
