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;

theorem
  (NOT O1) OR NOT O2 c= NOT (O1 AND O2)
  proof
    let z,s be object;
    assume [z,s] in (NOT O1) OR NOT O2; then
    [z,s] in NOT O1 or [z,s] in NOT O2 by XBOOLE_0:def 3; then
A1: s = z & z in X & (z nin dom O1 or z nin dom O2) by Th36; then
    z nin (dom O1) /\ dom O2 & dom(O1 AND O2) c= (dom O1)/\dom O2
    by XBOOLE_0:def 4,XTUPLE_0:24; then
    z nin dom(O1 AND O2);
    hence thesis by A1,Th36;
  end;
