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 Th15:
  for E be finite non empty set, ASeq being SetSequence of E st
  ASeq is non-ascending
    ex N be Nat st for m be Nat
  st N<=m holds Intersection ASeq = ASeq.m
proof
  let E be finite non empty set, ASeq being SetSequence of E;
  assume
A1: ASeq is non-ascending;
  then consider N0 be Nat such that
A2: for m be Nat st N0<=m holds ASeq.N0 = ASeq.m by Th14;
  take N0;
  let N be Nat;
  assume N0 <= N;
  then
A3: ASeq.N= ASeq.N0 by A2;
  thus Intersection ASeq = ASeq.N
  proof
    for x be object st x in Intersection ASeq holds x in ASeq.N by PROB_1:13;
    hence Intersection ASeq c= ASeq.N;
    let x be object;
    assume
A4: x in ASeq.N;
    now
      let n be Nat;
      per cases;
      suppose
        n <= N0;
        then ASeq.N0 c= ASeq.n by A1,PROB_1:def 4;
        hence x in ASeq.n by A3,A4;
      end;
      suppose
        N0 < n;
        hence x in ASeq.n by A2,A3,A4;
      end;
    end;
    hence x in Intersection ASeq by PROB_1:13;
  end;
end;
