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 Th62:
  A,C,B is_a_triangle & angle(A,C,B) < PI implies
  angle(B,A,C) - angle(C,B,A) = 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.|))) by Th61;
    hence thesis;
  end;
