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 Th46:
  for p be Prime, P,Q be Element of ProjCo(GF(p)) holds
  P _EQ_ Q iff [P,Q] in R_ProjCo p
  proof
    let p be Prime, P,Q be Element of ProjCo(GF(p));
    thus P _EQ_ Q implies [P,Q] in R_ProjCo p;
    assume [P,Q] in R_ProjCo p; then
    consider X0,Y0 be Element of ProjCo(GF(p)) such that
A1: [P,Q] = [X0,Y0] & X0 _EQ_ Y0;
    P=X0 & Q=Y0 by A1,XTUPLE_0:1;
    hence P _EQ_ Q by A1;
  end;
