
theorem Th19:
  for X be non empty set, S be SigmaField of X, M be sigma_Measure
of S, f,g be PartFunc of X,ExtREAL st f is nonnegative & g is nonnegative holds
  f+g is nonnegative
proof
  let X be non empty set;
  let S be SigmaField of X;
  let M be sigma_Measure of S;
  let f,g be PartFunc of X,ExtREAL;
  assume that
A1: f is nonnegative and
A2: g is nonnegative;
  for x be object st x in dom(f+g) holds 0 <= (f+g).x
  proof
    let x be object;
    assume
A3: x in dom(f+g);
    0 <= f.x by A1,SUPINF_2:51;
    then
A4: g.x <= f.x + g.x by XXREAL_3:39;
    0 <= g.x by A2,SUPINF_2:51;
    hence thesis by A3,A4,MESFUNC1:def 3;
  end;
  hence thesis by SUPINF_2:52;
end;
