theorem Th10:
  Partial_Intersection A1 is non-ascending
proof
  now
    let n be Nat;
    (Partial_Intersection A1).(n+1) = (Partial_Intersection A1).n /\ A1.(n
    +1) by Def1;
    hence (Partial_Intersection A1).(n+1) c= (Partial_Intersection A1).n by
XBOOLE_1:17;
  end;
  hence thesis by PROB_2:6;
end;
