reserve n,m,k for Element of NAT,
  x,X for set,
  A1 for SetSequence of X,
  Si for SigmaField of X,
  XSeq for SetSequence of Si;
reserve Omega for non empty set,
  Sigma for SigmaField of Omega,
  ASeq for SetSequence of Sigma,
  P for Probability of Sigma;

theorem Th18:
  A1 is non-descending implies
  for n being Nat holds (Partial_Diff_Union A1).(n+1) misses A1.n
proof
  assume
A1: A1 is non-descending;
  let n be Nat;
  (Partial_Diff_Union A1).(n+1) = A1.(n+1) \ A1.n by A1,Th16;
  hence thesis by XBOOLE_1:79;
end;
