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 Th27:
  for u being non zero Element of TOP-REAL 3 st
  (u.1)^2 + (u.2)^2 < 1 & u.3 = 1 holds Dir u is Element of BK_model
  proof
    let u be non zero Element of TOP-REAL 3;
    assume that
A1: (u.1)^2 + (u.2)^2 < 1 and
A2: u.3 = 1;
    reconsider P = Dir u as Point of ProjectiveSpace TOP-REAL 3 by ANPROJ_1:26;
    now
      let v be Element of TOP-REAL 3;
      assume that
A3:   v is non zero and
A4:   P = Dir v;
      qfconic(1,1,-1,0,0,0,u) = 1 * u.1 * u.1 + 1 * u.2 * u.2
            + (-1) * u.3 * u.3
            + 0 * u.1 * u.2 + 0 * u.1 * u.3 + 0 * u.2 * u.3 by PASCAL:def 1
        .= (u.1)^2 + (u.2)^2 + (-1) by A2;
      then qfconic(1,1,-1,0,0,0,u) < 1 + (-1) by A1,XREAL_1:8;
      hence qfconic(1,1,-1,0,0,0,v) is negative by A3,A4,BKMODEL1:81;
    end;
    then P in {P where P is Point of ProjectiveSpace TOP-REAL 3:
      for u being Element of TOP-REAL 3 st u is non zero & P = Dir u holds
      qfconic(1,1,-1,0,0,0,u) is negative};
    hence thesis by BKMODEL2:def 1;
  end;
