reserve P for Element of BK_model;
reserve N,N1,N2 for invertible Matrix of 3,F_Real;
reserve l,l1,l2 for Element of the Lines of IncProjSp_of real_projective_plane;
reserve P for Point of ProjectiveSpace TOP-REAL 3,
        l for LINE of IncProjSp_of real_projective_plane;

theorem Th26:
  for a,b,c,d being Real
  for u,v being non zero Element of TOP-REAL 3 st u = |[a,b,1]| &
  v = |[c,d,0]| holds Dir u <> Dir v
  proof
    let a,b,c,d be Real;
    let u,v be non zero Element of TOP-REAL 3;
    assume that
A1: u = |[a,b,1]| and
A2: v = |[c,d,0]|;
    assume Dir u = Dir v;
    then are_Prop u,v by ANPROJ_1:22;
    then consider x be Real such that
    x <> 0 and
A3: u = x * v by ANPROJ_1:1;
    1 = (x * v)`3 by A3,A1,EUCLID_5:2
     .= |[x * v`1,x * v`2,x * v`3]|`3 by EUCLID_5:7
     .= x * v`3 by EUCLID_5:2
     .= x * 0 by A2,EUCLID_5:2
     .= 0;
    hence contradiction;
  end;
