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

theorem ThEQCOMP4:
  for p be 5_or_greater Prime, z be Element of EC_WParam p,
  P be Element of EC_SetProjCo(z`1,z`2,p) holds
  (P`3_3 = 0 implies P _EQ_ compell_ProjCo(z,p).P)
  proof
    let p be 5_or_greater Prime, z be Element of EC_WParam p,
        P be Element of EC_SetProjCo(z`1,z`2,p);
    set a = z`1;
    set b = z`2;
    set O = [0, 1, 0];
    reconsider O as Element of EC_SetProjCo(a,b,p) by EC_PF_1:42;
    assume A2: P`3_3 = 0;
    A3: compell_ProjCo(z,p).O _EQ_ O by EC_PF_2:40;
    A4: P _EQ_ O by A2,ThEQO;
    then compell_ProjCo(z,p).P _EQ_ compell_ProjCo(z,p).O by EC_PF_2:46;
    then compell_ProjCo(z,p).P _EQ_ O by A3,EC_PF_1:44;
    hence thesis by A4,EC_PF_1:44;
  end;
