reserve n for Nat;
reserve i for Integer;
reserve r,s,t for Real;
reserve An,Bn,Cn,Dn for Point of TOP-REAL n;
reserve L1,L2 for Element of line_of_REAL n;
reserve A,B,C for Point of TOP-REAL 2;
reserve D for Point of TOP-REAL 2;
reserve a,b,c,d for Real;

theorem
  A,C,B is_a_triangle & angle(A,C,B) < PI implies
  angle(B,A,C) = arctan (cot (angle(A,C,B) / 2)
    * ((|.C-B.|-|.C-A.|) / (|.C-B.| + |.C-A.|))) + PI/2 - angle(A,C,B)/2
    &
  angle(C,B,A) = PI/2 - angle(A,C,B)/2 - arctan (cot (angle(A,C,B) / 2)
    * ((|.C-B.|-|.C-A.|) / (|.C-B.| + |.C-A.|)))
  proof
    assume A,C,B is_a_triangle & angle(A,C,B) < PI;
    then angle(B,A,C) - angle(C,B,A)
          = (2 * (arctan (cot (angle(A,C,B) / 2)
            * ((|.C-B.|-|.C-A.|) / (|.C-B.| + |.C-A.|))))) &
         angle(B,A,C) + angle(C,B,A) = (PI - angle(A,C,B)) by Th62,Th24;
    hence thesis;
  end;
