reserve X,Y,z,s for set, L,L1,L2,A,B for List of X, x for Element of X,
  O,O1,O2,O3 for Operation of X, a,b,y for Element of X, n,m for Nat;
reserve F,F1,F2 for filtering Operation of X;

theorem Th52:
  O AND NOT O = {}
  proof
    dom (O AND NOT O) = dom(O /\ id(X\dom O)) by Th37; then
A1: dom (O AND NOT O) c= (dom O) /\ dom id(X\dom O) by XTUPLE_0:24;
    (dom O) /\ dom id(X\dom O) = (dom O) /\ (X\dom O) by RELAT_1:45
    .= ((dom O) /\ X)\dom O by XBOOLE_1:49
    .= (dom O)\dom O by XBOOLE_1:28
    .= {} by XBOOLE_1:37;
    hence thesis by A1,RELAT_1:41,XBOOLE_1:3;
  end;
