
theorem
  for P being Element of ProjectiveSpace TOP-REAL 3
  for u being non zero Element of TOP-REAL 3 st
  P = Dir u & P in BK_model holds u.3 <> 0
  proof
    let P be Element of ProjectiveSpace TOP-REAL 3;
    let u be non zero Element of TOP-REAL 3;
    assume that
A1: P = Dir u and
A2: P in BK_model;
    assume
A3: u.3 = 0;
    consider Q be Point of ProjectiveSpace TOP-REAL 3 such that
A4: P = Q and
A5: for u be Element of TOP-REAL 3 st u is non zero & Q = Dir u holds
      qfconic(1,1,-1,0,0,0,u) is negative by A2;
A6: qfconic(1,1,-1,0,0,0,u) is negative by A1,A4,A5;
    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 by A3;
    then u.1 = 0 & u.2 = 0 by A6,BKMODEL1:19;
    then u`1 = 0 & u`2 = 0 & u`3 = 0 by A3,EUCLID_5:def 1,def 2,def 3;
    hence contradiction by EUCLID_5:3,4;
  end;
