theorem
  f1 is_bounded_on X & f2 is_bounded_on Y implies f1-f2 is_bounded_on X /\ Y
proof
  assume that
A1: f1 is_bounded_on X and
A2: f2 is_bounded_on Y;
  -f2 is_bounded_on Y by A2,Th45;
  then f1+-f2 is_bounded_on (X /\ Y) by A1,Th46;
  hence thesis by Th25;
end;
