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 Th70:
  for u being Element of TOP-REAL 3 st qfconic(1,1,-1,0,0,0,u) = 0 & u.3 = 1
  holds|[u.1,u.2]| in Sphere(0.TOP-REAL 2,1)
  proof
    let u be Element of TOP-REAL 3;
    assume that
A1: qfconic(1,1,-1,0,0,0,u) = 0 and
A2: u.3 = 1;
    0 = 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 A1,PASCAL:def 1
     .= (u.1)^2 + (u.2)^2 - 1 by A2;
    then |[u.1,u.2]| in circle(0,0,1) by Th11;
    hence thesis by EUCLID:54,TOPREAL9:52;
  end;
