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 Th32:
  for p be Prime, a, b be Element of GF(p),
  P be Element of EC_SetProjCo(a,b,p),
  Q be Element of ProjCo(GF(p))
  holds P = Q iff P`1_3 = Q`1_3 & P`2_3 = Q`2_3 & P`3_3 = Q`3_3
  proof
    let p be Prime, a, b be Element of GF(p),
    P be Element of EC_SetProjCo(a,b,p),
    Q be Element of ProjCo(GF(p));
    A1: P = [P`1_3, P`2_3, P`3_3] by Th31;
    A2: Q = [Q`1_3, Q`2_3, Q`3_3] by AA;
    thus P = Q implies
    P`1_3 = Q`1_3 & P`2_3 = Q`2_3 & P`3_3 = Q`3_3 by A1;
    assume A3:  P`1_3 = Q`1_3 & P`2_3 = Q`2_3 & P`3_3 = Q`3_3;
    thus P = Q by A2,A3,Th31;
  end;
