reserve X for non empty set,
  S for SigmaField of X,
  M for sigma_Measure of S,
  f,g for PartFunc of X,ExtREAL,
  E for Element of S;

theorem Th1:
  (for x be Element of X st x in dom f holds f.x <= g.x) implies g-
  f is nonnegative
proof
  assume
A1: for x be Element of X st x in dom f holds f.x <= g.x;
  now
    let y be ExtReal;
    assume y in rng (g-f);
    then consider x being object such that
A2: x in dom (g-f) and
A3: y = (g-f).x by FUNCT_1:def 3;
    reconsider x as set by TARSKI:1;
    dom (g-f) = (dom g /\ dom f)\((g"{+infty} /\ f"{+infty}) \/ (g"{-infty
    } /\ f"{-infty})) by MESFUNC1:def 4;
    then x in dom g /\ dom f by A2,XBOOLE_0:def 5;
    then x in dom f by XBOOLE_0:def 4;
    then 0. <= g.x-f.x by A1,XXREAL_3:40;
    hence 0 <= y by A2,A3,MESFUNC1:def 4;
  end;
  then rng (g-f) is nonnegative by SUPINF_2:def 9;
  hence thesis by SUPINF_2:def 12;
end;
