
theorem Th8:
  for X,Z being set, Y being Relation st Z c= Y & X \ Z is
  without_pairs holds X \ Y = X \ Z
proof
  let X,Z being set;
  let Y being Relation;
  assume
A1: Z c= Y;
  assume X \ Z is without_pairs;
  then not(ex x being object st x in (Y \ Z) /\ (X \ Z));
  then (Y \ Z) /\ (X \ Z) = {} by XBOOLE_0:7;
  then Y \ Z misses X \ Z by XBOOLE_0:def 7;
  then
A2: (X \ Z) \ (Y \ Z) = (X \ Z) 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;
