reserve x for set;
reserve i,j for Integer;
reserve n,n1,n2,n3 for Nat;
reserve p for Prime;
reserve a,b,c,d for Element of GF(p);
reserve K for Ring;
reserve a1,a2,a3,a4,a5,a6 for Element of K;
reserve px,py,pz for object;
reserve Px,Py,Pz for Element of GF(p);
reserve P for Element of ProjCo(GF(p));
reserve O for Element of EC_SetProjCo(a,b,p);

theorem Th38:
  for p be Prime, a, b be Element of GF(p),
  P be Element of ProjCo(GF(p))
  holds (rep_pt(P))`3_3 <> 0 implies
  rep_pt(P) = [(P`1_3)*(P`3_3)", (P`2_3)*(P`3_3)", 1] & P`3_3 <> 0
  proof
    let p be Prime, a, b be Element of GF(p),
    P be Element of ProjCo(GF(p));
    assume A1: (rep_pt(P))`3_3 <> 0;
    hereby
      assume A2: rep_pt(P) <> [(P`1_3)*(P`3_3)", (P`2_3)*(P`3_3)", 1];
      rep_pt(P) = [0, 1, 0] by A2,Def7;
      hence contradiction by A1;
    end;
    assume A3: P`3_3 = 0;
    rep_pt(P) = [0, 1, 0] by A3,Def7;
    hence contradiction by A1;
  end;
