
theorem Th76:
  for X be non empty set, S be SigmaField of X, M be sigma_Measure
  of S, A be Element of S, F,G be Functional_Sequence of X,ExtREAL, K,L be
ExtREAL_sequence st (for n be Nat holds F.n is_simple_func_in S & dom(F.n)=A )
& (for n be Nat holds F.n is nonnegative) & (for n,m be Nat st n <=m holds for
x be Element of X st x in A holds (F.n).x <= (F.m).x ) & (for n be Nat holds G.
n is_simple_func_in S & dom(G.n)=A) & (for n be Nat holds G.n is nonnegative) &
(for n,m be Nat st n <=m holds for x be Element of X st x in A holds (G.n).x <=
  (G.m).x ) & (for x be Element of X st x in A holds F#x is convergent & G#x is
convergent & lim(F#x) = lim(G#x)) & (for n be Nat holds K.n=integral'(M,F.n) &
  L.n=integral'(M,G.n)) holds K is convergent & L is convergent & lim K = lim L
proof
  let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, A be
Element of S, F,G be Functional_Sequence of X,ExtREAL, K,L be ExtREAL_sequence
  such that
A1: for n be Nat holds F.n is_simple_func_in S & dom(F.n)=A and
A2: for n be Nat holds F.n is nonnegative and
A3: for n,m be Nat st n <=m holds for x be Element of X st x in A holds
  (F.n).x <= (F.m).x and
A4: for n be Nat holds G.n is_simple_func_in S & dom(G.n)=A and
A5: for n be Nat holds G.n is nonnegative and
A6: for n,m be Nat st n <=m holds for x be Element of X st x in A holds
  (G.n).x <= (G.m).x and
A7: for x be Element of X st x in A holds F#x is convergent & G#x is
  convergent & lim(F#x) = lim(G#x) and
A8: for n be Nat holds K.n=integral'(M,F.n) & L.n=integral'(M,G.n);
A9: for n0 be Nat holds L is convergent & sup rng L=lim L & K.n0 <= lim L
  proof
    let n0 be Nat;
    reconsider f=F.n0 as PartFunc of X,ExtREAL;
A10: f is_simple_func_in S by A1;
A11: f is nonnegative by A2;
A12: for x be Element of X st x in dom f holds G#x is convergent & f.x <=
    lim(G#x)
    proof
      let x be Element of X;
A13:  (F#x).n0 <= sup rng (F#x) by Th56;
      assume x in dom f;
      then
A14:  x in A by A1;
      now
        let n,m be Nat;
        assume
A15:    n<=m;
A16:    (F#x).m=(F.m).x by Def13;
        (F#x).n=(F.n).x by Def13;
        hence (F#x).n <= (F#x).m by A3,A14,A15,A16;
      end;
      then
A17:  lim(F#x)=sup rng(F#x) by Th54;
      f.x=(F#x).n0 by Def13;
      hence thesis by A7,A14,A17,A13;
    end;
    dom f = A by A1;
    then consider FF be ExtREAL_sequence such that
A18: for n be Nat holds FF.n = integral'(M,G.n) and
A19: FF is convergent and
A20: sup rng FF = lim FF and
A21: integral'(M,f) <= lim FF by A4,A5,A6,A12,A10,A11,Th75;
    now
      let n be Element of NAT;
      FF.n = integral'(M,G.n) by A18;
      hence FF.n = L.n by A8;
    end;
    then FF=L by FUNCT_2:63;
    hence thesis by A8,A19,A20,A21;
  end;
A22: for n0 be Nat holds K is convergent & sup rng K = lim K & L.n0 <= lim K
  proof
    let n0 be Nat;
    reconsider g=G.n0 as PartFunc of X,ExtREAL;
A23: g is_simple_func_in S by A4;
A24: g is nonnegative by A5;
A25: for x be Element of X st x in dom g holds F#x is convergent & g.x <=
    lim(F#x)
    proof
      let x be Element of X;
A26:  (G#x).n0 <= sup rng(G#x) by Th56;
      assume x in dom g;
      then
A27:  x in A by A4;
      now
        let n,m be Nat;
        assume
A28:    n<=m;
A29:    (G#x).m=(G.m).x by Def13;
        (G#x).n=(G.n).x by Def13;
        hence (G#x).n <= (G#x).m by A6,A27,A28,A29;
      end;
      then
A30:  lim(G#x)=sup rng(G#x) by Th54;
      g.x=(G#x).n0 by Def13;
      hence thesis by A7,A27,A30,A26;
    end;
    dom g = A by A4;
    then consider GG be ExtREAL_sequence such that
A31: for n be Nat holds GG.n = integral'(M,F.n) and
A32: GG is convergent and
A33: sup rng GG = lim GG and
A34: integral'(M,g) <= lim GG by A1,A2,A3,A25,A23,A24,Th75;
    now
      let n be Element of NAT;
      GG.n = integral'(M,F.n) by A31;
      hence GG.n = K.n by A8;
    end;
    then GG=K by FUNCT_2:63;
    hence thesis by A8,A32,A33,A34;
  end;
  hence K is convergent & L is convergent by A9;
A35: lim K <= lim L by A22,A9,Th57;
  lim L <= lim K by A22,A9,Th57;
  hence thesis by A35,XXREAL_0:1;
end;
