
theorem Th59:
  Dir |[1,0,1]| is Element of absolute
  proof
    reconsider u = |[1,0,1]| as non zero Element of TOP-REAL 3;
    reconsider P = Dir u as Element of ProjectiveSpace TOP-REAL 3
      by ANPROJ_1:26;
    now
      thus u.3 = u`3 by EUCLID_5:def 3
              .= 1 by EUCLID_5:2;
      now
        thus u.1 = u`1 by EUCLID_5:def 1
                .= 1 by EUCLID_5:2;
        thus u.2 = u`2 by EUCLID_5:def 2
                .= 0 by EUCLID_5:2;
      end;
      then u.1 * u.1 = 1 & u.2 * u.2 = 0;
      then (u.1)^2 = 1 & (u.2)^2 = 0 by SQUARE_1:def 1;
      then (u.1)^2 + (u.2)^2 = 1;
      hence |[u.1,u.2]| in circle(0,0,1) by BKMODEL1:13;
    end;
    then P is Element of absolute by BKMODEL1:86;
    hence thesis;
  end;
