reserve p,p1,p2,q,r,F,G,G1,G2,H,H1,H2 for ZF-formula,
  x,x1,x2,y,y1,y2,z,z1,z2,s,t for Variable,
  a,X for set;
reserve M for non empty set,
  m,m9 for Element of M,
  v,v9 for Function of VAR,M;
reserve i,j for Element of NAT;

theorem Th156:
  'not' F = ('not' H)/(x,y) iff F = H/(x,y)
proof
  set N = ('not' H)/(x,y);
A1: len <*2*> = 1 by FINSEQ_1:39;
A2: dom 'not' F = Seg len 'not' F & dom 'not' H = Seg len 'not' H by
FINSEQ_1:def 3;
A3: len 'not' F = len <*2*> + len F & len 'not' H = len <*2*> + len H by
FINSEQ_1:22;
A4: dom F = Seg len F & dom H = Seg len H by FINSEQ_1:def 3;
  thus 'not' F = ('not' H)/(x,y) implies F = H/(x,y)
  proof
    assume
A5: 'not' F = N;
    then
A6: dom 'not' F = dom 'not' H by Def3;
    then
A7: len 'not' F = len 'not' H by FINSEQ_3:29;
A8: now
      let a be object;
      assume
A9:   a in dom F;
      then reconsider i = a as Element of NAT;
A10:  F.a = N.(1+i) & 1+i in dom N by A1,A5,A9,FINSEQ_1:28,def 7;
A11:  ('not' H).(1+i) = H.a by A1,A4,A3,A7,A9,FINSEQ_1:def 7;
      then
A12:  H.a = x implies F.a = y by A5,A6,A10,Def3;
A13:  H.a <> x implies F.a = H.a by A5,A6,A10,A11,Def3;
      H.a = x implies H/(x,y).a = y by A4,A3,A7,A9,Def3;
      hence F.a = H/(x,y).a by A4,A3,A7,A9,A12,A13,Def3;
    end;
A14: dom H = dom (H/(x,y)) by Def3;
    dom F = dom H by A3,A7,FINSEQ_3:29;
    hence thesis by A14,A8,FUNCT_1:2;
  end;
  assume
A15: F = H/(x,y);
  then
A16: dom F = dom H by Def3;
  then
A17: len F = len H by FINSEQ_3:29;
A18: dom <*2*> = {1} by FINSEQ_1:2,def 8;
A19: now
    let a be object;
    assume
A20: a in dom 'not' F;
    then reconsider i = a as Element of NAT;
A21: now
A22:  ('not' H).a <> x implies N.a = ('not' H).a by A2,A3,A17,A20,Def3;
      given j be Nat such that
A23:  j in dom F and
A24:  i = 1+j;
A25:  H.j = ('not' H).i & F.j = ('not' F).i by A1,A16,A23,A24,FINSEQ_1:def 7;
      then
A26:  ('not' H).a = x implies ('not' F).a = y by A15,A16,A23,Def3;
      ('not' H).a <> x implies ('not' F).a = ('not' H).a by A15,A16,A23,A25
,Def3;
      hence ('not' F).a = N.a by A2,A3,A17,A20,A26,A22,Def3;
    end;
    now
A27:  ('not' H).1 = 2 & 2 <> x by Th135,FINSEQ_1:41;
      assume i in {1};
      then
A28:  i = 1 by TARSKI:def 1;
      then ('not' F).i = 2 by FINSEQ_1:41;
      hence ('not' F).a = N.a by A2,A3,A17,A20,A28,A27,Def3;
    end;
    hence ('not' F).a = N.a by A1,A18,A20,A21,FINSEQ_1:25;
  end;
  dom 'not' F = dom N by A2,A3,A17,Def3;
  hence thesis by A19,FUNCT_1:2;
end;
