
theorem Th65:
  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 holds dom(f+g) = dom
f /\ dom g & integral'(M,f+g) = integral'(M,f|dom(f+g)) + integral'(M,g|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;
A5: g|dom(f+g) is nonnegative by A4,Th15;
A:  f|dom(f+g) is nonnegative by A3,Th15;
  not -infty in rng g by A4,Def3;
  then
A7: g"{-infty} = {} by FUNCT_1:72;
  not -infty in rng f by A3,Def3;
  then
A8: f"{-infty} = {} by FUNCT_1:72;
  then
A9: (dom f /\ dom g) \ (f"{-infty} /\ g"{+infty} \/ f"{+infty} /\ g"{ -infty
  }) = dom f /\ dom g by A7;
  hence
A10: dom(f+g) = dom f /\ dom g by MESFUNC1:def 3;
A11: dom(f+g) is Element of S by A1,A2,Th37,Th38;
  then
A12: f|dom(f+g) is_simple_func_in S by A1,Th34;
A13: g|dom(f+g) is_simple_func_in S by A2,A11,Th34;
  dom(f|dom(f+g)) = dom f /\ dom(f+g) by RELAT_1:61;
  then
A14: dom(f|dom(f+g)) = dom f /\ dom f /\ dom g by A10,XBOOLE_1:16;
  dom(g|dom(f+g)) = dom g /\ dom(f+g) by RELAT_1:61;
  then
A15: dom(g|dom(f+g)) = dom g /\ dom g /\ dom f by A10,XBOOLE_1:16;
  per cases;
  suppose
A16: dom(f+g) = {};
    dom(g|dom(f+g)) = dom g /\ dom(f+g) by RELAT_1:61;
    then
A17: integral'(M,g|dom(f+g)) = 0 by A16,Def14;
    dom(f|dom(f+g)) = dom f /\ dom(f+g) by RELAT_1:61;
    then
A18: integral'(M,f|dom(f+g)) = 0 by A16,Def14;
    integral'(M,f+g) = 0 by A16,Def14;
    hence thesis by A18,A17;
  end;
  suppose
A19: dom(f+g) <> {};
A20: (g|dom(f+g))"{-infty} = dom(f+g) /\ g"{-infty} by FUNCT_1:70
      .= {} by A7;
    (f|dom(f+g))"{-infty} = dom(f+g) /\ f"{-infty} by FUNCT_1:70
      .= {} by A8;
    then
    (dom(f|dom(f+g)) /\ dom(g|dom(f+g))) \ ( (f|dom(f+g))"{-infty} /\ (g|
dom(f+g))"{+infty} \/ (f|dom(f+g))"{+infty} /\ (g|dom(f+g))"{-infty} ) = dom(f+
    g) by A9,A14,A15,A20,MESFUNC1:def 3;
    then
A21: dom(f|dom(f+g) + g|dom(f+g)) = dom(f+g) by MESFUNC1:def 3;
A22: for x be Element of X st x in dom(f|dom(f+g) + g|dom(f+g)) holds (f|
    dom(f+g) + g|dom(f+g)).x = (f+g).x
    proof
      let x be Element of X;
      assume
A23:  x in dom(f|dom(f+g) + g|dom(f+g));
      then (f|dom(f+g) + g|dom(f+g)).x = (f|dom(f+g)).x + (g|dom(f+g)).x by
MESFUNC1:def 3
        .= f.x + (g|dom(f+g)).x by A21,A23,FUNCT_1:49
        .= f.x + g.x by A21,A23,FUNCT_1:49;
      hence thesis by A21,A23,MESFUNC1:def 3;
    end;
    integral(M,(f|dom(f+g) + g|dom(f+g))) = integral(M,f|dom(f+g)
    )+integral(M,g|dom(f+g)) by A10,A12,A13,A14,A15,A19,MESFUNC4:5,A,A5;
    then
A24: integral(M,f+g) = integral(M,f|dom(f+g)) + integral(M,g|
    dom(f+g)) by A21,A22,PARTFUN1:5;
A25: integral(M,g|dom(f+g)) = integral'(M,g|dom(f+g)) by A10,A15,A19,Def14;
    integral(M,f|dom(f+g)) = integral'(M,f|dom(f+g)) by A10,A14,A19,Def14;
    hence thesis by A19,A24,A25,Def14;
  end;
end;
