reserve x,A,B,X,X9,Y,Y9,Z,V for set;

theorem
  X \ (Y \/ Z) = (X \ Y) /\ (X \ Z)
proof
  X \(Y \/ Z) c= X \ Y & X \ (Y \/ Z) c= X \ Z by Th7,Th34;
  hence X \ (Y \/ Z) c= (X \ Y) /\ (X \ Z) by Th19;
  let x be object;
  assume
A1: x in (X \ Y) /\ (X \ Z);
  then
A2: x in (X \ Y) by XBOOLE_0:def 4;
  then
A3: x in X by XBOOLE_0:def 5;
  x in (X \ Z) by A1,XBOOLE_0:def 4;
  then
A4: not x in Z by XBOOLE_0:def 5;
  not x in Y by A2,XBOOLE_0:def 5;
  then not x in (Y \/ Z) by A4,XBOOLE_0:def 3;
  hence thesis by A3,XBOOLE_0:def 5;
end;
