
theorem
for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2,
 F be FinSequence of sigma measurable_rectangles(S1,S2),
 Fx be FinSequence of S1, p be set st
  dom F = dom Fx
& ( for n be Nat st n in dom Fx holds Fx.n = Measurable-Y-section(F.n,p) )
  holds Measurable-Y-section(Union F,p) = Union Fx
proof
   let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2,
   F be FinSequence of sigma measurable_rectangles(S1,S2),
   Fx be FinSequence of S1, p be set;
   assume that
A1: dom F = dom Fx and
A2: for n be Nat st n in dom Fx holds Fx.n = Measurable-Y-section(F.n,p);
A3:union rng F = Union F by CARD_3:def 4;
   reconsider F1 = F as FinSequence of bool [:X1,X2:] by FINSEQ_2:24;
   reconsider F1x = Fx as FinSequence of bool X1 by FINSEQ_2:24;
   for n be Nat st n in dom F1x holds F1x.n = Y-section(F1.n,p)
   proof
    let n be Nat;
    assume n in dom F1x; then
    Fx.n = Measurable-Y-section(F.n,p) by A2;
    hence thesis;
   end; then
   Y-section(union rng F1,p) = union rng F1x by A1,Th23;
   hence thesis by A3,CARD_3:def 4;
end;
