theorem Th29:
  A,B,C is_a_triangle & angle(B,A,C) < PI implies
  -PI/2 < (angle(B,A,C) - angle(C,B,A))/2 < PI/2
  proof
    assume A,B,C is_a_triangle & angle(B,A,C) < PI;
    then -PI < angle(B,A,C) - angle(C,B,A) < PI by Th28;
    then (-PI) / 2 < (angle(B,A,C) - angle(C,B,A))/2 < PI / 2 by XREAL_1:74;
    hence thesis;
  end;
