
theorem Th14:
  for P,Q being Element of real_projective_plane
  st P in absolute & Q in BK_model
  holds
  ex R being Element of real_projective_plane st R in BK_model &
  Q <> R & R,Q,P are_collinear
  proof
    let P,Q being Element of real_projective_plane;
    assume that
A1: P in absolute and
A2: Q in BK_model;
    reconsider QBK = Q as Element of BK_model by A2;
    consider u be non zero Element of TOP-REAL 3 such that
    (u.1)^2 + (u.2)^2 = 1 and
A3: u.3 = 1 and
A4: P = Dir u by A1,BKMODEL1:89;
    consider v be non zero Element of TOP-REAL 3 such that
A5: Dir v = QBK & v.3 = 1 & BK_to_REAL2 QBK = |[v.1,v.2]| by Def01;
    |[ (v.1 + u.1)/2,(v.2 + u.2)/2,1 ]| is non zero by EUCLID_5:4,FINSEQ_1:78;
    then reconsider w = |[ (v.1 + u.1)/2,(v.2 + u.2)/2,1 ]| as
      non zero Element of TOP-REAL 3;
    reconsider R = Dir w as Element of real_projective_plane by ANPROJ_1:26;
    take R;
    now
      u = |[u`1,u`2,u`3]| & v = |[v`1,v`2,v`3]| by EUCLID_5:3;
      then u = |[u.1,u`2,u`3]| & v = |[v.1,v`2,v`3]|;
      then u = |[u.1,u.2,u`3]| & v = |[v.1,v.2,v`3]|;
      hence u = |[u.1,u.2,1]| & v = |[v.1,v.2,1]| by A3,A5;
    end;
    then R in BK_model & R <> Q & Q,R,P are_collinear by A1,A4,A5,Th13;
    hence thesis by COLLSP:4;
  end;
