reserve X for non empty set,
  Y for set,
  S for SigmaField of X,
  M for sigma_Measure of S,
  f,g for PartFunc of X,COMPLEX,
  r for Real,
  k for Real,
  n for Nat,
  E for Element of S;
reserve x,A for set;

theorem Th41:
  for f,g be PartFunc of X,REAL st (for x be set st x in dom f
  holds f.x <= g.x) holds g-f is nonnegative
proof
  let f,g be PartFunc of X,REAL;
  assume
A1: for x be set st x in dom f holds f.x <= g.x;
  now
    let x be object;
    assume
A2: x in dom(g-f);
    then x in dom g /\ dom f by VALUED_1:12;
    then x in dom f by XBOOLE_0:def 4;
    then 0 <= g.x - f.x by A1,XREAL_1:48;
    hence 0 <=(g-f).x by A2,VALUED_1:13;
  end;
  hence thesis by MESFUNC6:52;
end;
