theorem
  f1 is_bounded_on X & f2|Y is constant implies f1+f2 is_bounded_on (X /\ Y)
proof
  assume that
A1: f1 is_bounded_on X and
A2: f2|Y is constant;
  f2 is_bounded_on Y by A2,Th54;
  hence thesis by A1,Th46;
end;
