reserve X for set;
reserve S for Subset-Family of X;

theorem Lem1: for X,Y be set holds (X\/Y)\(Y\X)=X
proof
  let X,Y be set;
A3: X\/(X/\Y) = X by XBOOLE_1:22;
A4: X\(Y\X)=(X\Y)\/ X/\X by XBOOLE_1:52
  .=X by XBOOLE_1:7,8;
  Y\(Y\X)=(Y\Y)\/ Y/\X by XBOOLE_1:52
  .={} \/ (Y/\X) by XBOOLE_1:37
  .=X/\Y;
  hence thesis by XBOOLE_1:42,A3,A4;
end;
