theorem
  f1|X is bounded & f2|Y is constant implies (f1-f2)|(X /\ Y) is bounded
  & (f2-f1)|(X /\ Y) is bounded & (f1(#)f2)|(X /\ Y) is bounded
proof
  assume that
A1: f1|X is bounded and
A2: f2|Y is constant;
  (-f2)|Y is constant by A2,Th79;
  hence (f1-f2)|(X /\ Y) is bounded by A1,Th82;
A3: f2|Y is bounded by A2,Th80;
  hence (f2-f1)|(X /\ Y) is bounded by A1,Th75;
  thus thesis by A1,A3,Th75;
end;
