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 Th48:
  for p be Prime, a, b be Element of GF(p),
  P be Element of ProjCo(GF(p))
  st p > 3 & Disc(a,b,p) <> 0.GF(p) &
  P in EC_SetProjCo(a,b,p) & P`3_3 <> 0 holds
  ex Q be Element of ProjCo(GF(p)) st Q in EC_SetProjCo(a,b,p)
  & Q _EQ_ P & Q`3_3 = 1
  proof
    let p be Prime, a, b be Element of GF(p), P be Element of ProjCo(GF(p));
    assume
    A1: p > 3 & Disc(a,b,p) <> 0.GF(p) &
    P in EC_SetProjCo(a,b,p) & P`3_3 <> 0;
    set d=(P`3_3)";
A2: P`3_3 <> 0.GF(p) by A1,Th11;
A3: d <> 0.GF(p)
    proof
      assume A4: d = 0.GF(p);
  A5: d*(P`3_3) = 1_GF(p) by A2,VECTSP_1:def 10
      .=1 by Th12;
      d*(P`3_3) = 0.GF(p) by A4
      .= 0 by Th11;
      hence contradiction by A5;
    end;
    reconsider Q =[d*(P`1_3),d*(P`2_3),d*(P`3_3)] as Element of
    [:the carrier of GF(p), the carrier of GF(p), the carrier of GF(p):];
A6: Q`1_3 = d*(P`1_3) & Q`2_3 = d*(P`2_3) & Q`3_3 = d*(P`3_3);
    then
    Q in EC_SetProjCo(a,b,p) by A1,A3,Th45; then
    consider PP be Element of ProjCo(GF(p)) such that
A7: Q = PP & EC_WEqProjCo(a,b,p).PP = 0.GF(p);
    reconsider Q as Element of ProjCo(GF(p)) by A7;
    take Q;
    thus Q in EC_SetProjCo(a,b,p) by A6,A1,A3,Th45;
    thus Q _EQ_ P by A3;
    thus Q`3_3 = d*(P`3_3)
    .= 1_GF(p) by A2,VECTSP_1:def 10
    .= 1 by Th12;
  end;
