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

theorem ThUnityProjCo:
  for P, O being Element of EC_SetProjCo(z`1,z`2,p) st O = [0, 1, 0]
  holds addell_ProjCo(z,p).(P,O) _EQ_ P & addell_ProjCo(z,p).(O,P) _EQ_ P
  proof
    let P, O be Element of EC_SetProjCo(z`1,z`2,p) such that
    A1: O = [0, 1, 0];
    addell_ProjCo(z,p).(P,O) _EQ_ P
    proof
      per cases;
      suppose B1: P _EQ_ O;
        then addell_ProjCo(z,p).(P,O) = O by A1,EC_PF_2:def 9;
        hence addell_ProjCo(z,p).(P,O) _EQ_ P by B1;
      end;
      suppose not P _EQ_ O;
        hence addell_ProjCo(z,p).(P,O) _EQ_ P by A1,EC_PF_2:def 9;
      end;
    end;
    hence thesis by A1,EC_PF_2:def 9;
  end;
