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 Th53:
  for p be Prime, a, b be Element of GF(p),
  F1,F2 be set st p > 3 & Disc(a,b,p) <> 0.GF(p)
  & F1 = {Class(R_EllCur(a,b,p),[0,1,0])} & F2 = {Class(R_EllCur(a,b,p),P)
  where P is Element of ProjCo(GF(p)):
  P in EC_SetProjCo(a,b,p) &
  ex X,Y be Element of GF(p) st P=[X,Y,1]} holds F1 misses F2
  proof
    let p be Prime, a, b be Element of GF(p), F1,F2 be set;
    assume A1: p > 3 & Disc(a,b,p) <> 0.GF(p)
    & F1 = {Class(R_EllCur(a,b,p),[0,1,0])}
    & F2 = {Class(R_EllCur(a,b,p),P) where P is Element of ProjCo(GF(p)):
    P in EC_SetProjCo(a,b,p) & ex X,Y be Element of GF(p) st P=[X,Y,1]};
    assume F1 meets F2; then
    F1 /\ F2 <> {} by XBOOLE_0:def 7; then
    consider z be object such that A2: z in F1 /\ F2 by XBOOLE_0:def 1;
    A3: z in F1 & z in F2 by A2,XBOOLE_0:def 4;
    consider P be Element of ProjCo(GF(p)) such that
    A4:z=Class(R_EllCur(a,b,p),P) & P in EC_SetProjCo(a,b,p) &
    ex X,Y be Element of GF(p) st P=[X,Y,1] by A1,A3;
    consider X1,Y1 be Element of GF(p) such that
    A5: P=[X1,Y1,1] by A4;
    A6: z= Class(R_EllCur(a,b,p),[0,1,0]) by A1,A3,TARSKI:def 1;
    reconsider Q=[0,1,0] as Element of ProjCo(GF(p)) by Lm5;
    A7: Q is Element of EC_SetProjCo(a,b,p) by Th42;
    Q in Class(R_EllCur(a,b,p),P) by A4,A6,EQREL_1:23;
    then [P,Q] in R_EllCur(a,b,p) by EQREL_1:18;
    then P _EQ_ Q by Th47,A1,A7,A4;
    then consider a be Element of GF(p) such that
    A8: a <> 0.GF(p) & Q`1_3 = a*(P`1_3) & Q`2_3 = a*(P`2_3)
    & Q`3_3 = a*(P`3_3) by Def10;
    A9: p > 1 by INT_2:def 4;
    A10: 1.GF(p)= 1 by A9,INT_3:14
    .=P`3_3 by A5;
    0.GF(p)= 0 by Th11
    .=a*(P`3_3) by A8
    .=a by A10;
    hence contradiction by A8;
  end;
