reserve E for non empty set;
reserve a for Element of E;
reserve A, B for Subset of E;
reserve Y for set;
reserve p for FinSequence;
reserve e, e1, e2 for Singleton of E;

theorem
  e = A \/ B & A <> B implies A = {} & B = e or A = e & B = {}
proof
  assume that
A1: e = A \/ B and
A2: A <> B;
  A c= e by A1,XBOOLE_1:7;
  then
A3: A = {} or A = e by Th1;
  B c= e by A1,XBOOLE_1:7;
  hence thesis by A2,A3,Th1;
end;
