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

theorem
  for P, Q being Element of EC_SetProjCo(z`1,z`2,p) holds
  compell_ProjCo(z,p).(addell_ProjCo(z,p).(P,Q)) _EQ_
  addell_ProjCo(z,p).(compell_ProjCo(z,p).P,compell_ProjCo(z,p).Q)
  proof
    let P, Q be Element of EC_SetProjCo(z`1,z`2,p);
    reconsider O = [0, 1, 0] as Element of EC_SetProjCo(z`1,z`2,p)
      by EC_PF_1:42;
    per cases;
    suppose A1: P _EQ_ O;
      then A2: compell_ProjCo(z,p).(addell_ProjCo(z,p).(P,Q))
      = compell_ProjCo(z,p).Q by EC_PF_2:def 9;
      compell_ProjCo(z,p).P _EQ_ O by A1,ThCompellO;
      hence thesis by A2,EC_PF_2:def 9;
    end;
    suppose A1: not P _EQ_ O & Q _EQ_ O;
      then A2: compell_ProjCo(z,p).(addell_ProjCo(z,p).(P,Q))
      = compell_ProjCo(z,p).P by EC_PF_2:def 9;
      A3: not compell_ProjCo(z,p).P _EQ_ O by A1,ThCompellO;
      compell_ProjCo(z,p).Q _EQ_ O by A1,ThCompellO;
      hence thesis by A2,A3,EC_PF_2:def 9;
    end;
    suppose A1: not P _EQ_ O & not Q _EQ_ O;
      reconsider rP = rep_pt(P), rQ = rep_pt(Q)
      as Element of EC_SetProjCo(z`1,z`2,p) by EC_PF_2:36;
      (rep_pt(P))`3_3 = 1 by A1,ThRepPoint7;
      then A3: rP`3_3 = 1 by EC_PF_2:32;
      (rep_pt(Q))`3_3 = 1 by A1,ThRepPoint7;
      then rQ`3_3 = 1 by EC_PF_2:32;
      then A5: compell_ProjCo(z,p).(addell_ProjCo(z,p).(rP,rQ)) _EQ_
      addell_ProjCo(z,p).(compell_ProjCo(z,p).rP,compell_ProjCo(z,p).rQ)
        by A3,LmCompAddCom;
      rP _EQ_ P & rQ _EQ_ Q by EC_PF_2:36;
      then addell_ProjCo(z,p).(rP,rQ) _EQ_ addell_ProjCo(z,p).(P,Q)
        by ThAdd3;
      then compell_ProjCo(z,p).(addell_ProjCo(z,p).(rP,rQ)) _EQ_
      compell_ProjCo(z,p).(addell_ProjCo(z,p).(P,Q)) by EC_PF_2:46;
      then A6: compell_ProjCo(z,p).(addell_ProjCo(z,p).(P,Q)) _EQ_
      addell_ProjCo(z,p).(compell_ProjCo(z,p).rP,compell_ProjCo(z,p).rQ)
      by A5,EC_PF_1:44;
      P`3_3 <> 0 by A1,ThEQO;
      then A7: compell_ProjCo(z,p).rP = rep_pt(compell_ProjCo(z,p).P)
      by EC_PF_2:42;
      Q`3_3 <> 0 by A1,ThEQO;
      then A8: compell_ProjCo(z,p).rQ = rep_pt(compell_ProjCo(z,p).Q)
      by EC_PF_2:42;
      A9: rep_pt(compell_ProjCo(z,p).P) _EQ_ compell_ProjCo(z,p).P
      by EC_PF_2:36;
      rep_pt(compell_ProjCo(z,p).Q) _EQ_ compell_ProjCo(z,p).Q
      by EC_PF_2:36;
      then addell_ProjCo(z,p).(compell_ProjCo(z,p).rP,compell_ProjCo(z,p).rQ)
      _EQ_ addell_ProjCo(z,p).(compell_ProjCo(z,p).P,compell_ProjCo(z,p).Q)
      by A7,A8,A9,ThAdd3;
      hence thesis by A6,EC_PF_1:44;
    end;
  end;
