let A, B, C be Point of (TOP-REAL 2); ( A,B,C is_a_triangle & angle (B,A,C) < PI implies ( - (PI / 2) < ((angle (B,A,C)) - (angle (C,B,A))) / 2 & ((angle (B,A,C)) - (angle (C,B,A))) / 2 < PI / 2 ) )
assume
( A,B,C is_a_triangle & angle (B,A,C) < PI )
; ( - (PI / 2) < ((angle (B,A,C)) - (angle (C,B,A))) / 2 & ((angle (B,A,C)) - (angle (C,B,A))) / 2 < PI / 2 )
then
( - PI < (angle (B,A,C)) - (angle (C,B,A)) & (angle (B,A,C)) - (angle (C,B,A)) < PI )
by Th28;
then
( (- PI) / 2 < ((angle (B,A,C)) - (angle (C,B,A))) / 2 & ((angle (B,A,C)) - (angle (C,B,A))) / 2 < PI / 2 )
by XREAL_1:74;
hence
( - (PI / 2) < ((angle (B,A,C)) - (angle (C,B,A))) / 2 & ((angle (B,A,C)) - (angle (C,B,A))) / 2 < PI / 2 )
; verum