
theorem Th110:
for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2,
  M2 be sigma_Measure of S2,
  E,V be Element of sigma measurable_rectangles(S1,S2),
  P be Set_Sequence of sigma measurable_rectangles(S1,S2),
  x be Element of X1
 st P is non-ascending & lim P = E holds
   ex K be SetSequence of S2 st
    K is non-ascending
  & (for n be Nat holds K.n
      = Measurable-X-section(P.n,x) /\ Measurable-X-section(V,x))
  & lim K = Measurable-X-section(E,x) /\ Measurable-X-section(V,x)
proof
   let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2,
   M2 be sigma_Measure of S2,
   E,V be Element of sigma measurable_rectangles(S1,S2),
   P be Set_Sequence of sigma measurable_rectangles(S1,S2),
   x be Element of X1;
   assume that
A1: P is non-ascending 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 X2 such that
A5: G is non-ascending
  & (for n be Nat holds G.n = X-section(P1.n,x)) by A1,Th39;
   for n be Nat holds G.n in S2
   proof
    let n be Nat;
    P1.n in sigma measurable_rectangles(S1,S2) by A4; then
    X-section(P1.n,x) in S2 by Th44;
    hence G.n in S2 by A5;
   end; then
   reconsider G as Set_Sequence of S2 by MEASURE8:def 2;
   set K = Measurable-X-section(V,x) (/\) G;
A6:G is convergent & lim G = Intersection G by A5,SETLIM_1:64;
   meet rng G = X-section(meet rng P,x) by A5,Th25; then
A7:Intersection G = X-section(meet rng P,x) by SETLIM_1:8
    .= X-section(Intersection P,x) by SETLIM_1:8
    .= Measurable-X-section(E,x) by A1,A2,SETLIM_1:64;
A8:dom K = NAT by FUNCT_2:def 1;
   for n be object st n in NAT holds K.n in S2
   proof
    let n be object;
    assume n in NAT; then
    reconsider n1=n as Element of NAT;
    K.n1 = G.n1 /\ Measurable-X-section(V,x) by SETLIM_2:def 5; then
    K.n1 = Measurable-X-section(P.n1,x) /\ Measurable-X-section(V,x) by A5;
    hence K.n in S2;
   end; then
   reconsider K as SetSequence of S2 by A8,FUNCT_2:3;
A9:for n be Nat holds
     K.n = Measurable-X-section(P.n,x) /\ Measurable-X-section(V,x)
   proof
    let n be Nat;
    K.n = G.n /\ Measurable-X-section(V,x) by SETLIM_2:def 5;
    hence
       K.n = Measurable-X-section(P.n,x) /\ Measurable-X-section(V,x) by A5;
   end;
   take K;
   thus thesis by A9,A7,A6,A5,SETLIM_2:21,92;
end;
