reserve r,p,x for Real;
reserve n for Element of NAT;
reserve A for non empty closed_interval Subset of REAL;
reserve Z for open Subset of REAL;

theorem
  for f,g being PartFunc of REAL,REAL,
  A being non empty closed_interval Subset of
REAL st (f(#)g)|A is bounded & f(#)g is_integrable_on A & A c= dom(f(#)g) holds
  |||(-f,g,A)||| = - |||(f,g,A)|||
proof
  let f,g being PartFunc of REAL,REAL;
  let A being non empty closed_interval Subset of REAL;
  assume
A1: (f(#)g)|A is bounded & f(#)g is_integrable_on A & A c= dom(f(#)g);
  |||(-f,g,A)||| = integral(((-1)(#)(f(#)g)),A) by RFUNCT_1:12
    .= (-1)*integral((f(#)g),A) by A1,INTEGRA6:9
    .= - |||(f,g,A)|||;
  hence thesis;
end;
