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 Th36:
 for z,s being object holds  [z,s] in NOT O iff z = s & z in X & z nin dom O
  proof let z,s be object;
    thus [z,s] in NOT O implies z = s & z in X & z nin dom O
    proof
      assume
A1:   [z,s] in NOT O; then
      s in Im(NOT O,z) & z in X by RELAT_1:169,ZFMISC_1:87; then
      s in (NOT O).:{z} & {z} c= X by ZFMISC_1:31; then
      s in union {IFEQ(x.O, {}, {x}, {}): x in {z}} by Def15; then
      consider Y such that
A2:   s in Y & Y in {IFEQ(x.O, {}, {x}, {}): x in {z}} by TARSKI:def 4;
      consider x such that
A3:   Y = IFEQ(x.O, {}, {x}, {}) & x in {z} by A2;
A4:   x = z by A3,TARSKI:def 1;
A5:   x.O = {} by A2,A3,FUNCOP_1:def 8; then
      s in {x} by A2,A3,FUNCOP_1:def 8; then
A6:      s = x &
       for s being object holds [x,s] nin O by A5,RELAT_1:169,TARSKI:def 1;
      hence z = s by A4;
      thus z in X by A1,ZFMISC_1:87;
          x in dom O iff
          ex y being object st [x,y] in O by XTUPLE_0:def 12;
      hence z nin dom O by A4,A6;
    end;
    assume
A7: z = s & z in X & z nin dom O;
    then reconsider z as Element of X;
    z.O c= {}
    proof
      let y be object; assume y in z.O; then
      [z,y] in O by RELAT_1:169;
      hence thesis by A7,XTUPLE_0:def 12;
    end; then
    z.O = {}; then
A8: IFEQ(z.O, {}, {z}, {}) = {z} by FUNCOP_1:def 8;
A9: z in {z} by TARSKI:def 1;
    reconsider L = {z} as Subset of X by A7,ZFMISC_1:31;
    {z} in {IFEQ(x.O, {}, {x}, {}): x in {z}} by A8,A9; then
    z in union {IFEQ(x.O, {}, {x}, {}): x in {z}} by A9,TARSKI:def 4; then
    z in L|(NOT O) by Def15; then
    z in z.NOT O;
    hence thesis by A7,RELAT_1:169;
  end;
