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;
