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

theorem
  for P,G,C being set st C c= G holds P \ C = (P \ G) \/ (P /\ (G \ C))
proof
  let P,G,C be set;
  assume C c= G;
  then
A1: P \ G c= P \ C by Th34;
  thus P \ C c= (P \ G) \/ (P /\ (G \ C))
  proof
    let x be object;
    assume x in P \ C;
    then x in P & not x in G or x in P & x in G & not x in C by XBOOLE_0:def 5;
    then x in P \ G or x in P & x in G \ C by XBOOLE_0:def 5;
    then x in P \ G or x in P /\ (G \ C) by XBOOLE_0:def 4;
    hence thesis by XBOOLE_0:def 3;
  end;
  P /\ (G \ C) = (P /\ G) \ C & (P /\ G) \ C c= P \ C by Th17,Th33,Th49;
  hence thesis by A1,Th8;
end;
