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 Th69:
  for P being Point of ProjectiveSpace TOP-REAL 3 st
  P = Dir u & u.3 = 1 & |[u.1,u.2]| in circle(0,0,1) holds
  P is Element of absolute
  proof
    let P be Point of ProjectiveSpace TOP-REAL 3;
    assume that
A1: P = Dir u and
A2: u.3 = 1 and
A3: |[u.1,u.2]| in circle(0,0,1);
    reconsider u1 = u.1, u2 = u.2, u3 = u.3 as Real;
A4: u1^2 + u2^2 = |. |[u1 - 0,u2 - 0]| .|^2 by TOPGEN_5:9
               .= |. |[u1,u2]| - |[0,0]| .|^2 by EUCLID:62
               .= 1^2 by A3,TOPREAL9:43
               .= 1;
    now
      let v be Element of TOP-REAL 3;
      assume that
A5:   v is non zero and
A6:   P = Dir v;
      are_Prop u,v by A1,A5,A6,ANPROJ_1:22;
      then consider a be Real such that
A7:   a <> 0 and
A8:   u = a * v by ANPROJ_1:1;
      reconsider v1 = v.1, v2 = v.2,v3 = v.3 as Real;
      u1 = a * v1 & u2 = a * v2 & u3 = a * v3 by A8,RVSUM_1:44;
      then
A9:   v1 = u1 / a & v2 = u2 / a & v3 = u3 / a by A7,XCMPLX_1:89;
      qfconic(1,1,-1,0,0,0,v)
        = 1 * v1 * v1 + 1 * v2 * v2 + (-1) *  v3 * v3 + 0 * v1 * v2
        + 0 * v1 * v3 + 0 * v2 * v3 by PASCAL:def 1
       .= (u1 / a) * (u1 / a) + (u2 / a) * (u2 / a) - (u3 / a) * (u3 / a)
          by A9
       .= 1/a * u1 * (u1 / a) + (u2 / a) * (u2 / a) - (u3 / a) * (u3 / a)
          by XCMPLX_1:99
        .= 1/a * u1 * (1/a * u1) + (u2 / a) * (u2 / a) - (u3 / a) * (u3 / a)
          by XCMPLX_1:99
        .= ((1/a) * (1/a)) * u1 * u1 + 1 / a * u2 * (u2 / a)
          - (u3 / a) * (u3 / a) by XCMPLX_1:99
        .= ((1/a) * (1/a)) * u1 * u1 + 1 / a * u2 * (1/a * u2)
          - (u3 / a) * (u3 / a) by XCMPLX_1:99
        .= ((1/a) * (1/a)) * u1 * u1 + ((1 / a) * (1 / a)) * u2 * u2
          - 1/a * u3 * (u3 / a) by XCMPLX_1:99
        .= ((1/a) * (1/a)) * u1 * u1 + ((1 / a) * (1 / a)) * u2 * u2
          - 1/a * u3 * (1/a * u3) by XCMPLX_1:99
        .= ((1/a) * (1/a)) * (u1^2 + u2 * u2 - u3 * u3)
        .= 0 by A2,A4;
      hence qfconic(1,1,-1,0,0,0,v)=0;
    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) = 0};
    hence thesis by PASCAL:def 2;
  end;
