reserve p for 5_or_greater Prime;
reserve z for Element of EC_WParam p;

theorem ThEQO:
  for p be 5_or_greater Prime, z be Element of EC_WParam p,
      P, O be Element of EC_SetProjCo(z`1,z`2,p)
  st O = [0, 1, 0] holds
  (P`3_3 = 0 iff P _EQ_ O)
  proof
    let p be 5_or_greater Prime, z be Element of EC_WParam p,
        P, O be Element of EC_SetProjCo(z`1,z`2,p) such that
    A2: O = [0, 1, 0];
    set a = z`1;
    set b = z`2;
    consider PP be Element of ProjCo(GF(p)) such that
    A3: PP = P & PP in EC_SetProjCo(a,b,p);
    hereby
      assume P`3_3 = 0; then
      PP`3_3 = 0 by A3,EC_PF_2:32;
      then A5: rep_pt(P) = [0, 1, 0] by A3,EC_PF_2:def 7;
      rep_pt(O) = [0, 1, 0] by A2,ThRepPoint5;
      hence P _EQ_ O by A5,EC_PF_2:39;
    end;
    assume P _EQ_ O; then
    rep_pt(P) = rep_pt(O) by EC_PF_2:39
    .= O by A2,ThRepPoint5;
    then (rep_pt(PP))`3_3 = 0 by A2,A3,MCART_1:def 7;
    then PP`3_3 = 0 by EC_PF_2:37;
    hence thesis by A3,EC_PF_2:32;
  end;
