
theorem Th40:
  for P being Point of ProjectiveSpace TOP-REAL 3 st
  ex u being non zero Element of TOP-REAL 3 st
  P = Dir u & u`3 <> 0 holds P is non point_at_infty
  proof
    let P be Point of ProjectiveSpace TOP-REAL 3;
    given u be non zero Element of TOP-REAL 3 such that
A1: P = Dir u and
A2: u`3 <> 0;
    now
      let v be non zero Element of TOP-REAL 3;
      assume P = Dir v;
      then are_Prop u,v by A1,ANPROJ_1:22;
      then consider a be Real such that
A3:   a <> 0 and
A4:   v = a * u by ANPROJ_1:1;
      v`3 = a * u`3 by A4,EUCLID_5:9;
      hence v`3 <> 0 by A2,A3;
    end;
    hence thesis;
  end;
