let A, B, C be Point of (TOP-REAL 2); :: thesis: ( A,B,C is_a_triangle implies ( not angle (A,B,C) is zero & not angle (B,C,A) is zero & not angle (C,A,B) is zero & not angle (A,C,B) is zero & not angle (C,B,A) is zero & not angle (B,A,C) is zero ) )
assume A1: A,B,C is_a_triangle ; :: thesis: ( not angle (A,B,C) is zero & not angle (B,C,A) is zero & not angle (C,A,B) is zero & not angle (A,C,B) is zero & not angle (C,B,A) is zero & not angle (B,A,C) is zero )
then ( B,C,A is_a_triangle & C,A,B is_a_triangle & B,A,C is_a_triangle & A,C,B is_a_triangle & C,B,A is_a_triangle ) by MENELAUS:15;
hence ( not angle (A,B,C) is zero & not angle (B,C,A) is zero & not angle (C,A,B) is zero & not angle (A,C,B) is zero & not angle (C,B,A) is zero & not angle (B,A,C) is zero ) by A1, Lm7; :: thesis: verum