
theorem Th79:
  for X be non empty set, S be SigmaField of X, M be sigma_Measure
  of S, f be PartFunc of X,ExtREAL st (ex A be Element of S st A = dom f & f
  is A-measurable) & f is nonnegative holds 0 <= integral+(M,f)
proof
  let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be
  PartFunc of X,ExtREAL;
  assume that
A1: ex A be Element of S st A = dom f & f is A-measurable and
A2: f is nonnegative;
  consider F1 be Functional_Sequence of X,ExtREAL, K1 be ExtREAL_sequence such
  that
A3: for n be Nat holds F1.n is_simple_func_in S & dom(F1.n) = dom f and
A4: for n be Nat holds F1.n is nonnegative and
A5: for n,m be Nat st n <=m holds for x be Element of X st x in dom f
  holds (F1.n).x <= (F1.m).x and
  for x be Element of X st x in dom f holds F1#x is convergent & lim( F1#x
  ) = f.x and
A6: for n be Nat holds K1.n=integral'(M,F1.n) and
  K1 is convergent and
A7: integral+(M,f)=lim K1 by A1,A2,Def15;
  for n,m be Nat st n<=m holds K1.n <= K1.m
  proof
    let n,m be Nat;
A8: F1.m is_simple_func_in S by A3;
A9: dom(F1.m) = dom f by A3;
A10: K1.m = integral'(M,F1.m) by A6;
A11: dom(F1.n) = dom f by A3;
    assume
A12: n<=m;
A13: for x be object st x in dom(F1.m - F1.n) holds (F1.n).x <= (F1.m).x
    proof
      let x be object;
      assume x in dom(F1.m - F1.n);
      then x in (dom(F1.m) /\ dom(F1.n))\ (((F1.m)"{+infty}/\(F1.n)"{+infty})
      \/((F1.m)"{-infty}/\(F1.n)"{-infty})) by MESFUNC1:def 4;
      then x in dom(F1.m) /\ dom(F1.n) by XBOOLE_0:def 5;
      hence thesis by A5,A12,A11,A9;
    end;
A14: F1.m is nonnegative by A4;
A15: F1.n is nonnegative by A4;
A16: F1.n is_simple_func_in S by A3;
    then
A17: dom(F1.m - F1.n) = dom(F1.m) /\ dom(F1.n) by A8,A15,A14,A13,Th69;
    then
A18: F1.n|dom(F1.m - F1.n) = F1.n by A11,A9,GRFUNC_1:23;
A19: F1.m|dom(F1.m - F1.n) = F1.m by A11,A9,A17,GRFUNC_1:23;
    integral'(M,F1.n|dom(F1.m - F1.n)) <= integral'(M,F1.m|dom(F1.m - F1.
    n )) by A16,A8,A15,A14,A13,Th70;
    hence thesis by A6,A10,A18,A19;
  end;
  then lim K1 =sup rng K1 by Th54;
  then
A20: K1.0 <= lim K1 by Th56;
  for n be Nat holds 0 <= K1.n
  proof
    let n be Nat;
A21: F1.n is_simple_func_in S by A3;
    K1.n = integral'(M,F1.n) by A6;
    hence thesis by A4,A21,Th68;
  end;
  hence thesis by A7,A20;
end;
