
theorem Th4: :: SPpart:
for X, Y being set, F being a_partition of Y
 st Y c< X holds F \/ { X \ Y } is a_partition of X
proof
  let X, Y be set, F be a_partition of Y;
  assume A1: Y c< X;
  then A2: X \ Y <> {} by XBOOLE_1:105;
    Y c= X by A1;
  then A3: Y \/ (X \ Y) = X by XBOOLE_1:45;
   { X \ Y } is a_partition of X \ Y by A2,EQREL_1:39;
  hence F \/ { X \ Y } is a_partition of X by A3,Th3,XBOOLE_1:79;
end;
