reserve a,b,c,d,e,f for Real,
        g           for positive Real,
        x,y         for Complex,
        S,T         for Element of REAL 2,
        u,v,w       for Element of TOP-REAL 3;

theorem Th01:
  for P1,P2,P3 being Element of ProjectiveSpace TOP-REAL 3 st
  u is non zero & v is non zero & w is non zero &
  P1 = Dir u & P2 = Dir v & P3 = Dir w holds
  (P1,P2,P3 are_collinear iff |{u,v,w}| = 0)
  proof
    let P1,P2,P3 be Element of ProjectiveSpace TOP-REAL 3;
    assume that
A1: u is non zero and
A2: v is non zero and
A3: w is non zero and
A4: P1 = Dir u and
A5: P2 = Dir v and
A6: P3 = Dir w;
    hereby
      assume P1,P2,P3 are_collinear;
      then consider u9,v9,w9 be Element of TOP-REAL 3 such that
A7:   P1 = Dir(u9) and
A8:   P2 = Dir(v9) and
A9:   P3 = Dir(w9) & u9 is not zero & v9 is not zero & w9 is not zero &
        u9,v9,w9 are_LinDep by ANPROJ_2:23;
      [ Dir u9,Dir v9,Dir w9 ] in the Collinearity of
        ProjectiveSpace TOP-REAL 3 by A9,ANPROJ_1:25;
      then u,v,w are_LinDep by A7,A8,A9,A1,A2,A3,A4,A5,A6,ANPROJ_1:25;
      hence |{u,v,w}| = 0 by ANPROJ_8:43;
    end;
    thus thesis by A1,A2,A3,A4,A5,A6,ANPROJ_2:23,ANPROJ_8:43;
  end;
