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
  |||((r(#)f),(p(#)g),A)||| = r*p*|||(f,g,A)|||
proof
  let f,g be PartFunc of REAL,REAL;
  let A be 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);
  |||((r(#)f),(p(#)g),A)||| = integral((r(#)(f(#)(p(#)g))),A) by RFUNCT_1:12
    .= integral((r(#)(p(#)(f(#)g))),A) by RFUNCT_1:13
    .= integral(((r*p)(#)(f(#)g)),A) by RFUNCT_1:17
    .= (r*p)*integral((f(#)g),A) by A1,INTEGRA6:9;
  hence thesis;
end;
