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 Th43:
  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) holds
  P = Q iff compell_ProjCo(z,p).P = 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);
    thus P = Q implies
    compell_ProjCo(z,p).P = compell_ProjCo(z,p).Q;
    assume A1: compell_ProjCo(z,p).P = compell_ProjCo(z,p).Q;
    thus P = compell_ProjCo(z,p).(compell_ProjCo(z,p).Q) by A1,Th41
    .= Q by Th41;
  end;
