
theorem
  for L being LINE of IncProjSp_of real_projective_plane
  for P,Q being Element of ProjectiveSpace TOP-REAL 3
  for u,v being non zero Element of TOP-REAL 3 st
  P in L & Q in L &
  P = Dir u & Q = Dir v & u`3 <> 0 & v`3 = 0 holds
  P <> Q &
  Dir |[u`1 + v`1, u`2 + v`2, u`3]| in L
  proof
    let L be LINE of IncProjSp_of real_projective_plane;
    let P,Q be Element of ProjectiveSpace TOP-REAL 3;
    let u,v be non zero Element of TOP-REAL 3;
    assume that
A1: P in L and
A2: Q in L and
A3: P = Dir u and
A4: Q = Dir v and
A5: u`3 <> 0 and
A6: v`3 = 0;
    thus
A7: P <> Q
    proof
      assume P = Q;
      then
A8:   are_Prop u,v by A3,A4,ANPROJ_1:22;
      u = |[u`1,u`2,u`3]| & v = |[v`1,v`2,0]| by A6,EUCLID_5:3;
      hence contradiction by A5,A8,Th06;
    end;
    reconsider w = |[u`1 + v`1,u`2 + v`2, u`3]|
      as non zero Element of TOP-REAL 3 by A5,Th05;
    reconsider R = Dir w as Element of ProjectiveSpace TOP-REAL 3
      by ANPROJ_1:26;
    u = |[u`1,u`2,u`3]| & v = |[v`1,v`2,0]| by A6,EUCLID_5:3;
    then |{u,v,w}| = 0 by Th04;
    then P,Q,R are_collinear by A3,A4,BKMODEL1:1;
    then R in Line(P,Q) by COLLSP:11;
    hence thesis by A1,A2,A7,Th07;
  end;
