
theorem Th70:
  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_simple_func_in S & g
is_simple_func_in S & f is nonnegative & g is nonnegative &
(for x be object st x
  in dom(f-g) holds g.x <= f.x) holds integral'(M,g|dom(f-g)) <= integral'(M,f|
  dom(f-g))
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_simple_func_in S and
A2: g is_simple_func_in S and
A3: f is nonnegative and
A4: g is nonnegative and
A5: for x be object st x in dom(f-g) holds g.x <= f.x;
  (-jj)(#)g is_simple_func_in S by A2,Th39;
  then -g is_simple_func_in S by MESFUNC2:9;
  then f+(-g) is_simple_func_in S by A1,Th38;
  then
A6: f-g is_simple_func_in S by MESFUNC2:8;
A7: integral'(M,f|dom(f-g)) = integral'(M,f-g)+integral'(M,g|dom(f-g)) by A1,A2
,A3,A4,A5,Th69;
  now
    assume integral'(M,f|dom(f-g)) <> +infty;
    0 <= integral'(M,f-g) by A1,A2,A5,A6,Th40,Th68;
    hence thesis by A7,XXREAL_3:39;
  end;
  hence thesis by XXREAL_0:4;
end;
