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
  NOT (F1 AND F2) = (NOT F1) OR (NOT F2)
  proof
A1: F1 = id dom F1 & F2 = id dom F2 by Th45;
    NOT (F1 AND F2) = id(X\dom(F1 AND F2)) by Th37
    .= id (X\dom id ((dom F1) AND dom F2)) by A1,SYSREL:14
    .= id (X\((dom F1) AND dom F2)) by RELAT_1:45
    .= id ((X\dom F1)\/(X\dom F2)) by XBOOLE_1:54
    .= id (X\dom F1) \/ id (X\dom F2) by SYSREL:14
    .= (NOT F1) \/ id (X\dom F2) by Th37;
    hence thesis by Th37;
  end;
