
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),
 Fy be FinSequence of S2, p be set st
  dom F = dom Fy
& ( for n be Nat st n in dom Fy holds Fy.n = Measurable-X-section(F.n,p) )
  holds Measurable-X-section(Union F,p) = Union Fy
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),
   Fy be FinSequence of S2, p be set;
   assume that
A1: dom F = dom Fy and
A2: for n be Nat st n in dom Fy holds Fy.n = Measurable-X-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 F1y = Fy as FinSequence of bool X2 by FINSEQ_2:24;
   for n be Nat st n in dom F1y holds F1y.n = X-section(F1.n,p)
   proof
    let n be Nat;
    assume n in dom F1y; then
    Fy.n = Measurable-X-section(F.n,p) by A2;
    hence thesis;
   end; then
   X-section(union rng F1,p) = union rng F1y by A1,Th22;
   hence thesis by A3,CARD_3:def 4;
end;
