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 Th68:
  for P being Element of absolute st P = Dir u & u.3 = 1 holds
  |[u.1,u.2]| in circle(0,0,1)
  proof
    let P be Element of absolute;
    assume that
A1: P = Dir u and
A2: u.3 = 1;
    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
A3: P = Q and
A4: for u being Element of TOP-REAL 3 st u is non zero & Q = Dir u holds
    qfconic(1,1,-1,0,0,0,u) = 0;
    qfconic(1,1,-1,0,0,0,u) = 0 by A1,A3,A4;
    then
A5: 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 PASCAL:def 1;
    reconsider u1 = u.1, u2 = u.2 as Real;
    u1^2 + u2^2 = 1 by A2,A5;
    hence thesis by Th11;
  end;
