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

theorem LmRepPoint9:
  for P being Element of ProjCo(GF(p)) holds
  rep_pt(rep_pt(P)) = rep_pt(P)
  proof
    let P be Element of ProjCo(GF(p));
    set rP = rep_pt(P);
    per cases;
    suppose A1: P`3_3 = 0;
      then rep_pt(P) = [0, 1, 0] by EC_PF_2:def 7;
      then rP`3_3 = 0 by MCART_1:def 7;
      hence rep_pt(rep_pt(P)) = [0, 1, 0] by EC_PF_2:def 7
      .= rep_pt(P) by A1,EC_PF_2:def 7;
    end;
    suppose P`3_3 <> 0;
      then rep_pt(P) = [P`1_3*(P`3_3)",P`2_3*(P`3_3)",1] by EC_PF_2:def 7;
      then rP`3_3 = 1 by MCART_1:def 7;
      hence thesis by ThRepPoint5;
    end;
  end;
