reserve Omega for non empty set;
reserve r for Real;
reserve Sigma for SigmaField of Omega;
reserve P for Probability of Sigma;
reserve E for finite non empty set;

theorem
  for E be finite non empty set, ASeq being SetSequence of E st ASeq is
  non-descending holds ex N be Nat st for m be Nat st N<=m holds Union ASeq =
  ASeq.m
proof
  let E be finite non empty set, ASeq being SetSequence of E;
  assume
A1: ASeq is non-descending;
  then consider N0 being Nat such that
A2: for m be Nat st N0 <= m holds ASeq.N0 = ASeq.m by Th16;
  reconsider N = N0 as Nat;
  take N;
  let m be Nat;
  reconsider M = m as Element of NAT by ORDINAL1:def 12;
  assume
A3: N <= m;
  now
    let x be object;
    assume x in Union ASeq;
    then consider n be Nat such that
A4: x in ASeq.n by PROB_1:12;
    per cases;
    suppose
A5:   n < N;
A6:   ASeq.N c= ASeq.M by A2,A3;
      ASeq.n c= ASeq.N by A1,A5,PROB_1:def 5;
      then ASeq.n c= ASeq.m by A6;
      hence x in ASeq.m by A4;
    end;
    suppose
      N <= n;
      then ASeq.N = ASeq.n by A2;
      then ASeq.n c= ASeq.M by A2,A3;
      hence x in ASeq.m by A4;
    end;
  end;
  then
A7: Union ASeq c= ASeq.m;
  ASeq.m c= Union ASeq by ABCMIZ_1:1;
  hence ASeq.m = Union ASeq by A7;
end;
