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 Th37:
  NOT O = id(X\dom O)
  proof
    let z,s be object;
    thus [z,s] in NOT O implies [z,s] in id(X\dom O)
    proof
      assume [z,s] in NOT O; then
A1:   z = s & z in X & z nin dom O by Th36; then
      z in X\dom O by XBOOLE_0:def 5;
      hence thesis by A1,RELAT_1:def 10;
    end;
    assume [z,s] in id(X\dom O); then
A2: z = s & z in X\dom O by RELAT_1:def 10; then
    z in X & z nin dom O by XBOOLE_0:def 5;
    hence thesis by A2,Th36;
  end;
