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 Th25:
  for P being Element of absolute holds
  ex u being non zero Element of TOP-REAL 3 st u.3 = 1 & P = Dir u
  proof
    let P be Element of absolute;
    consider u be Element of TOP-REAL 3 such that
A1: u is not zero and
A2: P = Dir u by ANPROJ_1:26;
    u.3 <> 0 by A1,A2,BKMODEL1:83;
    then
A3: u`3 <> 0 by EUCLID_5:def 3;
    reconsider v = |[u`1/u`3,u`2/u`3,1]| as non zero Element of TOP-REAL 3;
    take v;
    thus v.3 = v`3 by EUCLID_5:def 3
            .= 1 by EUCLID_5:2;
    u`3 * v = |[u`3 * (u`1/u`3), u`3 * (u`2/u`3), u`3 * 1]| by EUCLID_5:8
           .= |[ u`1,u`3 * (u`2/u`3), u`3 * 1]| by A3,XCMPLX_1:87
           .= |[ u`1,u`2, u`3 * 1]| by A3,XCMPLX_1:87
           .= u by EUCLID_5:3;
    then are_Prop u,v by A3,ANPROJ_1:1;
    hence P = Dir v by A1,A2,ANPROJ_1:22;
  end;
