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;
reserve a,b,c for Element of F_Real,
          M,N for Matrix of 3,F_Real;
reserve D        for non empty set;
reserve d1,d2,d3 for Element of D;
reserve A        for Matrix of 1,3,D;
reserve B        for Matrix of 3,1,D;
reserve u,v for non zero Element of TOP-REAL 3;
reserve P,Q,R for POINT of IncProjSp_of real_projective_plane,
            L for LINE of IncProjSp_of real_projective_plane,
        p,q,r for Point of real_projective_plane;
reserve u,v,w for non zero Element of TOP-REAL 3;

theorem
  p = Dir u & q = Dir v & R = Dir w & L = Line(p,q) implies
    (R on L iff |{u,v,w}| = 0)
  proof
    assume that
A1: p = Dir u and
A2: q = Dir v and
A3: R = Dir w and
A4: L = Line(p,q);
    reconsider r = Dir w as Point of real_projective_plane by ANPROJ_1:26;
    hereby
      assume R on L;
      then p,q,r are_collinear by A3,A4,Th61;
      then consider u1,v1,w1 be Element of TOP-REAL 3 such that
A5:   p = Dir u1 and
A6:   q = Dir v1 and
A7:   r = Dir w1 and
A8:   u1 is non zero & v1 is non zero & w1 is non zero and
A9:   u1 = v1 or u1 = w1 or v1 = w1 or
        {u1,v1,w1} is linearly-dependent by ANPROJ_8:10;
      u1,v1,w1 are_LinDep by A9,ANPROJ_8:9;
      then |{u1,v1,w1}| = 0 by ANPROJ_8:43;
      then |{u,v1,w1}| = 0 by A1,A5,A8,ANPROJ_8:58;
      then |{u,v,w1}| = 0 by A2,A6,A8,ANPROJ_8:59;
      hence |{u,v,w}| = 0 by A8,A7,ANPROJ_8:60;
    end;
    assume |{u,v,w}| = 0;
    then u = v or u = w or v = w or {u,v,w} is linearly-dependent
      by ANPROJ_8:9,ANPROJ_8:43;
    then p,q,r are_collinear by A1,A2,ANPROJ_8:10;
    hence R on L by A3,A4,Th61;
  end;
