reserve P for Element of BK_model;
reserve N,N1,N2 for invertible Matrix of 3,F_Real;
reserve l,l1,l2 for Element of the Lines of IncProjSp_of real_projective_plane;
reserve P for Point of ProjectiveSpace TOP-REAL 3,
        l for LINE of IncProjSp_of real_projective_plane;

theorem Th28:
  for a,b being Real st a^2 + b^2 <= 1 holds
  Dir |[a,b,1]| in BK_model \/ absolute
  proof
    let a,b be Real;
    assume a^2 + b^2 <= 1;
    then per cases by XXREAL_0:1;
    suppose
A1:   a^2 + b^2 = 1;
      reconsider u = |[a,b,1]| as non zero Element of TOP-REAL 3;
      reconsider P = Dir u as Point of ProjectiveSpace TOP-REAL 3
        by ANPROJ_1:26;
      now
B1:     u.1 = u`1 by EUCLID_5:def 1
                  .= a by EUCLID_5:2;
        u.2 = u`2 by EUCLID_5:def 2
                  .= b by EUCLID_5:2;
        hence |[u.1,u.2]| in circle(0,0,1) by A1,B1,BKMODEL1:13;
        thus u.3 = |[a,b,1]|`3 by EUCLID_5:def 3
                .= 1 by EUCLID_5:2;
      end;
      then P is Element of absolute by BKMODEL1:86;
      hence thesis by XBOOLE_0:def 3;
    end;
    suppose
A2:   a^2 + b^2 < 1;
      reconsider w = |[a,b,1]| as non zero Element of TOP-REAL 3;
      w`1 = a & w`2 = b & w`3 = 1 by EUCLID_5:2;
      then w.1 = a & w.2 = b & w.3 = 1 by EUCLID_5:def 1,def 2,def 3;
      then Dir |[a,b,1]| is Element of BK_model by A2,Th27;
      hence thesis by XBOOLE_0:def 3;
    end;
  end;
