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 Th52:
  for p be Prime,
  a, b,d1,Y1,d2,Y2 be Element of GF(p)
  st p > 3 & Disc(a,b,p) <> 0.GF(p)
  & [d1,Y1,1] in EC_SetProjCo(a,b,p)
  & [d2,Y2,1] in EC_SetProjCo(a,b,p) holds
  Class(R_EllCur(a,b,p),[d1,Y1,1]) = Class(R_EllCur(a,b,p),[d2,Y2,1])
  iff d1=d2 & Y1=Y2
  proof
    let p be Prime, a, b,d1,Y1,d2,Y2 be Element of GF(p);
    assume
    A1: p > 3 & Disc(a,b,p) <> 0.GF(p)
    & [d1,Y1,1] in EC_SetProjCo(a,b,p) & [d2,Y2,1] in EC_SetProjCo(a,b,p);
    hereby assume
      Class(R_EllCur(a,b,p),[d1,Y1,1])
      = Class(R_EllCur(a,b,p),[d2,Y2,1]); then
      [d2,Y2,1] in Class(R_EllCur(a,b,p),[d1,Y1,1]) by A1,EQREL_1:23; then
      A2: [[d1,Y1,1],[d2,Y2,1]] in R_EllCur(a,b,p) by EQREL_1:18;
      reconsider P=[d1,Y1,1], Q=[d2,Y2,1] as Element of ProjCo(GF(p)) by A1;
      P _EQ_ Q by Th47,A1,A2; then
      consider a be Element of GF(p) such that
      A3: 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;
      A4: p > 1 by INT_2:def 4;
      A5: 1.GF(p)= 1 by A4,INT_3:14
      .=P`3_3;
      A6: 1.GF(p)= 1 by A4,INT_3:14
      .=a*(P`3_3) by A3
      .=a by A5;
      thus d2 = a*(P`1_3) by A3
      .= P`1_3 by A6
      .= d1;
      thus Y2 = a*(P`2_3) by A3
      .= P`2_3 by A6
      .=Y1;
    end;
    assume d1=d2 & Y1=Y2;
    hence thesis;
  end;
