theorem Th22:
  for f being Function of A,REAL st f|A is bounded & f is
  integrable holds max-(f) is integrable
proof
  let f be Function of A,REAL;
  assume that
A1: f|A is bounded and
A2: f is integrable;
A3: (-f)|A is bounded by A1,RFUNCT_1:82;
  (-1)(#)f is integrable by A1,A2,INTEGRA2:31;
  then max+(-f) is integrable by A3,Th20;
  hence thesis by Th21;
end;
