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 Th44:
  for p be Prime, P,Q,R be Element of ProjCo(GF(p)) holds
  ( P _EQ_ Q & Q _EQ_ R implies P _EQ_ R)
  proof
    let p be Prime, P,Q,R be Element of ProjCo(GF(p));
    assume A1:P _EQ_ Q & Q _EQ_ R; then
    consider a be Element of GF(p) such that
    A2: a <> 0.GF(p) & P`1_3 = a*(Q`1_3) & P`2_3 = a*(Q`2_3)
    & P`3_3 = a*(Q`3_3);
    consider b be Element of GF(p) such that
    A3: b <> 0.GF(p) & Q`1_3 = b*(R`1_3) & Q`2_3 = b*(R`2_3)
    & Q`3_3 = b*(R`3_3) by A1;
    take d = a*b;
    thus thesis by A2,A3,GROUP_1:def 3,VECTSP_1:12;
  end;
