
theorem
  for X being Relation, Y, Z being set st Z c= Y & Y \ Z is
  without_pairs holds X \ Y = X \ Z
proof
  let X be Relation;
  let Y,Z be set;
  assume
A1: Z c= Y;
  assume Y \ Z is without_pairs;
  then not(ex x being object st x in X /\ (Y \ Z));
  then X /\ (Y \ Z) = {} by XBOOLE_0:7;
  then X misses Y \ Z by XBOOLE_0:def 7;
  then
A2: X \ (Y \ Z) = X by XBOOLE_1:83;
  X \ Y = X \ ( (Y \ Z)\/Z ) by A1,XBOOLE_1:45
    .= X \ Z by A2,XBOOLE_1:41;
  hence thesis;
end;
