 reserve i,n for Nat;
 reserve r for Real;
 reserve ra for Element of F_Real;
 reserve a,b,c for non zero Element of F_Real;
 reserve u,v for Element of TOP-REAL 3;
 reserve p1 for FinSequence of (1-tuples_on REAL);
 reserve pf,uf for FinSequence of F_Real;
 reserve N for Matrix of 3,F_Real;
 reserve K for Field;
 reserve k for Element of K;
 reserve N,N1,N2 for invertible Matrix of 3,F_Real;
 reserve P,P1,P2,P3 for Point of ProjectiveSpace TOP-REAL 3;

theorem Th22:
  for P being Point of ProjectiveSpace TOP-REAL 3
  ex a,b,c being Element of F_Real st P = Dir |[a,b,c]| &
  (a <> 0 or b <> 0 or c <> 0)
  proof
    let P be Point of ProjectiveSpace TOP-REAL 3;
    consider u be Element of TOP-REAL 3 such that
A1: u is non zero and
A2: P = Dir u by ANPROJ_1:26;
A3: u = |[u`1,u`2,u`3]| by EUCLID_5:3
     .= |[u.1,u`2,u`3]| by EUCLID_5:def 1
     .= |[u.1,u.2,u`3]| by EUCLID_5:def 2
     .= |[u.1,u.2,u.3]| by EUCLID_5:def 3;
    reconsider a = u.1,b = u.2,c = u.3 as Element of F_Real by XREAL_0:def 1;
    take a,b,c;
    thus P = Dir |[a,b,c]| by A2,A3;
    thus a <> 0 or b <> 0 or c <> 0 by A1,A3,EUCLID_5:4;
  end;
