
theorem Th56:
 for X be non empty set, S be SigmaField of X, f be PartFunc of X,ExtREAL
   st (ex A be Element of S st A = dom f & f is A-measurable)
      & f qua ext-real-valued Function is nonpositive
  ex F be Functional_Sequence of X,ExtREAL st (for n be Nat
holds F.n is_simple_func_in S & dom(F.n) = dom f) & (for n be Nat holds F.n is
nonpositive) & (for n,m be Nat st n <=m holds for x be Element of X st x in dom
f holds (F.n).x >= (F.m).x ) & for x be Element of X st x in dom f holds (F#x)
  is convergent & lim(F#x) = f.x
proof
    let X be non empty set, S be SigmaField of X, 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 qua ext-real-valued Function is nonpositive;
    set g = -f;
    consider A be Element of S such that
A3:  A = dom f & f is A-measurable by A1;
A4: A = dom g by A3,MESFUNC1:def 7; then
    consider G be Functional_Sequence of X,ExtREAL such that
A6:  (for n be Nat holds G.n is_simple_func_in S & dom(G.n) = dom g)
   & (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 dom g holds
         (G.n).x <= (G.m).x )
   & for x be Element of X st x in dom g holds (G#x) is convergent
   & lim(G#x) = g.x by A2,A3,MEASUR11:63,MESFUNC5:64;
    take F=-G;
    thus
A8: for n be Nat holds F.n is_simple_func_in S & dom(F.n) = dom f
    proof
     let n be Nat;
A9:  dom(G.n) = dom g by A6;
A10: F.n = -(G.n) by Th37;
     hence F.n is_simple_func_in S by A6,Th30;
     thus dom(F.n) = dom f by A3,A4,A9,A10,MESFUNC1:def 7;
    end;
    thus for n be Nat holds F.n is nonpositive
    proof
     let n be Nat;
A12: G.n is nonnegative by A6;
     F.n = -(G.n) by Th37;
     hence F.n is nonpositive by A12;
    end;
    thus for n,m be Nat st n <=m holds
     for x be Element of X st x in dom f holds (F.n).x >= (F.m).x
    proof
     let n,m be Nat;
     assume A14: n <= m;
     let x be Element of X;
     assume A15: x in dom f;
     dom(F.n) = dom f & F.n = -(G.n)
   & dom(F.m) = dom f & F.m = -(G.m) by A8,Th37; then
     (F.n).x = -((G.n).x) & (F.m).x = -((G.m).x) by A15,MESFUNC1:def 7;
     hence (F.n).x >= (F.m).x by A15,A3,A4,A6,A14,XXREAL_3:38;
    end;
    thus for x be Element of X st x in dom f
      holds (F#x) is convergent & lim(F#x) = f.x
    proof
     let x be Element of X;
     assume A17: x in dom f; then
A18: (G#x) is convergent & lim(G#x) = g.x by A3,A4,A6;
     hence (F#x) is convergent by Th39;
     lim(F#x) = - g.x by A18,Th39; then
     lim(F#x) = - (-f.x) by A3,A4,A17,MESFUNC1:def 7;
     hence lim(F#x) = f.x;
    end;
end;
