
theorem Th10:
  for P being Element of BK_model
  for L being LINE of IncProjSp_of real_projective_plane st P in L
  holds ex Q being Element of ProjectiveSpace TOP-REAL 3 st
  P <> Q & Q in L &
  for u being non zero Element of TOP-REAL 3 st Q = Dir u holds u.3 <> 0
  proof
    let P be Element of BK_model;
    let L be LINE of IncProjSp_of real_projective_plane;
    assume
A1: P in L;
    consider u be non zero Element of TOP-REAL 3 such that
A2: P = Dir u & u.3 = 1 and
    BK_to_REAL2 P = |[u.1,u.2]| by Def01;
    consider Q be Element of ProjectiveSpace TOP-REAL 3 such that
A3: (ex v be non zero Element of TOP-REAL 3 st
      Q = Dir v & Q in L & P <> Q & v.3 <> 0) by A1,A2,Th09;
    consider v be non zero Element of TOP-REAL 3 such that
A4: Q = Dir v & Q in L & P <> Q & v.3 <> 0 by A3;
    take Q;
    now
      thus P <> Q & Q in L by A3;
      thus for u being non zero Element of TOP-REAL 3 st Q = Dir u holds
        u.3 <> 0
      proof
        let w be non zero Element of TOP-REAL 3;
        assume Q = Dir w;
        then are_Prop v,w by A4,ANPROJ_1:22;
        then consider a be Real such that
A5:     a <> 0 and
A6:     v = a * w by ANPROJ_1:1;
        a * w = |[ a * w`1, a * w`2, a * w`3 ]| by EUCLID_5:7;
        then v`3 = a * w`3 by A6,EUCLID_5:2;
        then w`3 = v`3 / a by A5,XCMPLX_1:89
                .= v.3 / a by EUCLID_5:def 3;
        hence w.3 <> 0 by A5,A4,EUCLID_5:def 3;
      end;
    end;
    hence thesis;
  end;
