reserve a,b,c,d,e,f for Real,
        g           for positive Real,
        x,y         for Complex,
        S,T         for Element of REAL 2,
        u,v,w       for Element of TOP-REAL 3;
reserve a,b,c for Element of F_Real,
          M,N for Matrix of 3,F_Real;
reserve D        for non empty set;
reserve d1,d2,d3 for Element of D;
reserve A        for Matrix of 1,3,D;
reserve B        for Matrix of 3,1,D;
reserve u,v for non zero Element of TOP-REAL 3;
reserve P,Q,R for POINT of IncProjSp_of real_projective_plane,
            L for LINE of IncProjSp_of real_projective_plane,
        p,q,r for Point of real_projective_plane;
reserve u,v,w for non zero Element of TOP-REAL 3;

theorem Th67:
  for P being Element of absolute st P = Dir u holds u.3 <> 0
  proof
    let P be Element of absolute;
    assume
A1: P = Dir u;
    P in conic(1,1,-1,0,0,0);
    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) = 0} by PASCAL:def 2;
    then consider Q be Point of ProjectiveSpace TOP-REAL 3 such that
A2: P = Q and
A3: 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) = 0;
A4: qfconic(1,1,-1,0,0,0,u) = 0 by A1,A2,A3;
    thus u.3 <> 0
    proof
      assume
A5:   u.3 = 0;
A6:   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 = 0 by A4,PASCAL:def 1;
      reconsider u1 = u.1, u2 = u.2 as Real;
      u1 ^2 + u2 ^2 = 0 by A5,A6;
      then u1 = 0 & u2 = 0 by COMPLEX1:1;
      then u`1 = 0 & u`2 = 0 & u`3 = 0 by A5,EUCLID_5:def 1,def 2,def 3;
      hence contradiction by EUCLID_5:3,4;
    end;
  end;
