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
  L WHEREeq(O,0) = L WHERE NOT O
  proof
    thus L WHEREeq(O,0) c= L WHERE NOT O
    proof
      let z be object; assume z in L WHEREeq(O,0); then
      consider x such that
A1:   z = x & card(x.O) = 0 & x in L;
      x.O = {} by A1; then
      x nin dom O by RELAT_1:170; then
      [x,x] in NOT O by A1,Th36; then
      x,x in NOT O;
      hence thesis by A1;
    end;
    let z be object; assume z in L WHERE NOT O; then
    consider x such that
A2: z = x & ex y st x,y in NOT O & x in L;
    consider y such that
A3: x,y in NOT O & x in L by A2;
    [x,y] in NOT O by A3; then
    x = y & x in X & x nin dom O by Th36; then
    x.O = {} by RELAT_1:170; then
    card(x.O) = 0;
    hence thesis by A2;
  end;
