
theorem Th21:
  for X be non empty set, f,g be PartFunc of X,ExtREAL st (for x
  be set st x in dom f /\ dom g holds g.x <= f.x & -infty < g.x & f.x < +infty)
  holds f-g is nonnegative
proof
  let X be non empty set, f,g be PartFunc of X,ExtREAL;
  assume
A1: for x be set st x in dom f /\ dom g holds g.x <= f.x & -infty < g.x
  & f.x < +infty;
  now
    let x be object;
    assume
A2: x in dom(f-g);
    dom(f-g) = (dom f /\ dom g)\(f"{+infty}/\g"{+infty} \/ f"{-infty}/\g"{
    -infty}) by MESFUNC1:def 4;
    then dom(f-g) c= dom f /\ dom g by XBOOLE_1:36;
    then 0 <= f.x - g.x by A1,A2,XXREAL_3:40;
    hence 0 <= (f-g).x by A2,MESFUNC1:def 4;
  end;
  hence thesis by SUPINF_2:52;
end;
