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 O2) = (NOT O1) AND NOT O2
  proof
    let z,s be object;
    thus [z,s] in NOT (O1 OR O2) implies [z,s] in (NOT O1) AND NOT O2
    proof
      assume [z,s] in NOT (O1 OR O2); then
A1:   z = s & z in X & z nin dom (O1 OR O2) by Th36; then
      z nin (dom O1)\/dom O2 by XTUPLE_0:23; then
      z nin dom O1 & z nin dom O2 by XBOOLE_0:def 3; then
      [z,s] in NOT O1 & [z,s] in NOT O2 by A1,Th36;
      hence thesis by XBOOLE_0:def 4;
    end;
    assume [z,s] in (NOT O1) AND NOT O2; then
    [z,s] in NOT O1 & [z,s] in NOT O2 by XBOOLE_0:def 4; then
A2: z = s & z in X & z nin dom O1 & z nin dom O2 by Th36; then
    z nin (dom O1)\/dom O2 by XBOOLE_0:def 3; then
    z nin dom (O1 OR O2) by XTUPLE_0:23;
    hence thesis by A2,Th36;
  end;
