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 Th31:
  for p be Prime, a, b be Element of GF(p),
  P be Element of EC_SetProjCo(a,b,p)
  holds P = [P`1_3,P`2_3,P`3_3]
  proof
    let p be Prime, a, b be Element of GF(p),
    P be Element of EC_SetProjCo(a,b,p);
    P is Element of ProjCo(GF(p));
    then consider Px, Py, Pz be object such that
    Px in the carrier of GF(p) & Py in the carrier of GF(p)
    & Pz in the carrier of GF(p) and
    A1: P = [Px, Py, Pz] by MCART_1:68;
    thus P = [P`1_3, Py, Pz] by A1,Def3
    .= [P`1_3, P`2_3, Pz] by A1,Def4
    .= [P`1_3, P`2_3, P`3_3] by A1,Def5;
  end;
