
theorem Th109:
for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2,
  M1 be sigma_Measure of S1,
  E,V be Element of sigma measurable_rectangles(S1,S2),
  P be Set_Sequence of sigma measurable_rectangles(S1,S2),
  y be Element of X2
 st P is non-descending & lim P = E
holds
   ex K be SetSequence of S1 st
    K is non-descending
  & (for n be Nat holds K.n
      = Measurable-Y-section(P.n,y) /\ Measurable-Y-section(V,y))
  & lim K = Measurable-Y-section(E,y) /\ Measurable-Y-section(V,y)
proof
   let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2,
   M1 be sigma_Measure of S1,
   E,V be Element of sigma measurable_rectangles(S1,S2),
   P be Set_Sequence of sigma measurable_rectangles(S1,S2),
   x be Element of X2;
   assume that
A1: P is non-descending and
A2: lim P = E;
A4: for n be Nat holds P.n in sigma measurable_rectangles(S1,S2);
    reconsider P1 = P as SetSequence of [:X1,X2:];
    consider G be SetSequence of X1 such that
A5:  G is non-descending
   & (for n be Nat holds G.n = Y-section(P1.n,x)) by A1,Th38;
    for n be Nat holds G.n in S1
    proof
     let n be Nat;
     P1.n in sigma measurable_rectangles(S1,S2) by A4; then
     Y-section(P1.n,x) in S1 by Th44;
     hence G.n in S1 by A5;
    end; then
    reconsider G as Set_Sequence of S1 by MEASURE8:def 2;
    set K = Measurable-Y-section(V,x) (/\) G;
A6: G is convergent & lim G = Union G by A5,SETLIM_1:63;
    union rng G = Y-section(union rng P,x) by A5,Th26; then
A7: Union G = Y-section(union rng P,x) by CARD_3:def 4
     .= Y-section(Union P,x) by CARD_3:def 4
     .= Measurable-Y-section(E,x) by A1,A2,SETLIM_1:63;
A8: dom K = NAT by FUNCT_2:def 1;
    for n be object st n in NAT holds K.n in S1
    proof
     let n be object;
     assume n in NAT; then
     reconsider n1=n as Element of NAT;
     K.n1 = G.n1 /\ Measurable-Y-section(V,x) by SETLIM_2:def 5; then
     K.n1 = Measurable-Y-section(P.n1,x) /\ Measurable-Y-section(V,x) by A5;
     hence K.n in S1;
    end; then
    reconsider K as SetSequence of S1 by A8,FUNCT_2:3;
A9: for n be Nat holds
      K.n = Measurable-Y-section(P.n,x) /\ Measurable-Y-section(V,x)
    proof
     let n be Nat;
     K.n = G.n /\ Measurable-Y-section(V,x) by SETLIM_2:def 5;
     hence
       K.n = Measurable-Y-section(P.n,x) /\ Measurable-Y-section(V,x) by A5;
    end;
    take K;
    thus thesis by A9,A7,A5,A6,SETLIM_2:22,92;
end;
