reserve x for set;
reserve i,j for Integer;
reserve n,n1,n2,n3 for Nat;
reserve K,K1,K2,K3 for Field;
reserve SK1,SK2 for Subfield of K;
reserve ek,ek1,ek2 for Element of K;
reserve p for Prime;
reserve a,b,c for Element of GF(p);
reserve F for FinSequence of GF(p);
reserve Px,Py,Pz for Element of GF(p);

theorem Th47:
  for p be Prime, a, b be Element of GF(p),
  P,Q be Element of ProjCo(GF(p))
  st Disc(a,b,p) <> 0.GF(p) &
  P in EC_SetProjCo(a,b,p) & Q in EC_SetProjCo(a,b,p) holds
  ( P _EQ_ Q iff [P,Q] in R_EllCur (a,b,p))
  proof
    let p be Prime, a, b be Element of GF(p),
    P,Q be Element of ProjCo(GF(p));
    assume A1: Disc(a,b,p) <> 0.GF(p)
    & P in EC_SetProjCo(a,b,p) & Q in EC_SetProjCo(a,b,p);
    hereby assume P _EQ_ Q; then
  A2: [P,Q] in R_ProjCo p;
      [P,Q] in [:EC_SetProjCo(a,b,p),EC_SetProjCo(a,b,p):]
      by A1,ZFMISC_1:87;
      hence [P,Q] in R_EllCur (a,b,p) by A2,XBOOLE_0:def 4;
    end;
    assume [P,Q] in R_EllCur (a,b,p); then
    [P,Q] in (R_ProjCo p) by XBOOLE_0:def 4;
    hence P _EQ_ Q by Th46;
  end;
