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 Th48:
  for p be 5_or_greater Prime, z be Element of EC_WParam p,
  P, Q be Element of EC_SetProjCo(z`1,z`2,p)
  st P`3_3 <> 0 & Q`3_3 <> 0 holds
  rep_pt(P) = compell_ProjCo(z,p).(rep_pt(Q)) iff
  P _EQ_ compell_ProjCo(z,p).Q
  proof
    let p be 5_or_greater Prime, z be Element of EC_WParam p,
    P, Q be Element of EC_SetProjCo(z`1,z`2,p) such that
    A1: P`3_3 <> 0 & Q`3_3 <> 0;
    set a = z`1;
    set b = z`2;
    set CQ = compell_ProjCo(z,p).Q;
    reconsider CQ as Element of EC_SetProjCo(a,b,p);
    hereby
      assume A2: rep_pt(P) = compell_ProjCo(z,p).(rep_pt(Q));
      rep_pt(P) = rep_pt(CQ) by A1,A2,Th42;
      hence P _EQ_ compell_ProjCo(z,p).Q by Th39;
    end;
    assume P _EQ_ compell_ProjCo(z,p).Q;
    hence rep_pt(P) = rep_pt(CQ) by Th39
    .= compell_ProjCo(z,p).(rep_pt(Q)) by A1,Th42;
  end;
