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

theorem ThEQX:
  for P, Q being Element of EC_SetProjCo(z`1,z`2,p), a being Element of GF(p)
  st a <> 0.GF(p) &
  P`1_3 = a * (Q`1_3) & P`2_3 = a * (Q`2_3) & P`3_3 = a * (Q`3_3)
  holds P _EQ_ Q
  proof
    let P, Q be Element of EC_SetProjCo(z`1,z`2,p), a be Element of GF(p)
    such that
    A1: a <> 0.GF(p) and
    A2: P`1_3 = a * (Q`1_3) & P`2_3 = a * (Q`2_3) & P`3_3 = a * (Q`3_3);
    reconsider PP = P, QQ = Q as Element of ProjCo(GF(p));
    A3: PP`1_3 = a * (Q`1_3) by A2,EC_PF_2:32
    .= a * (QQ`1_3) by EC_PF_2:32;
    A4: PP`2_3 = a * (Q`2_3) by A2,EC_PF_2:32
    .= a * (QQ`2_3) by EC_PF_2:32;
    PP`3_3 = a * (Q`3_3) by A2,EC_PF_2:32
    .= a * (QQ`3_3) by EC_PF_2:32;
    hence thesis by A1,A3,A4,EC_PF_1:def 10;
  end;
