let A, B, C be Point of (TOP-REAL 2); :: thesis: ( 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).|)))) )
assume ( A,C,B is_a_triangle & angle (A,C,B) < PI ) ; :: thesis: (angle (B,A,C)) - (angle (C,B,A)) = 2 * (arctan ((cot ((angle (A,C,B)) / 2)) * ((|.(C - B).| - |.(C - A).|) / (|.(C - B).| + |.(C - A).|))))
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 (angle (B,A,C)) - (angle (C,B,A)) = 2 * (arctan ((cot ((angle (A,C,B)) / 2)) * ((|.(C - B).| - |.(C - A).|) / (|.(C - B).| + |.(C - A).|)))) ; :: thesis: verum