reserve A,B,C,D,E,F,G for Point of TOP-REAL 2;

theorem
  A,B,C are_mutually_distinct & angle(A,B,C) > PI implies
  angle(A,B,C) + angle(B,C,A) + angle(C,A,B) = 5*PI
  proof
    assume that
A1: A,B,C are_mutually_distinct and
A2: angle(A,B,C) > PI;
    angle(A,B,C) <> 0 & angle(B,C,A) <> 0 & angle(C,A,B) <> 0
    by A1,A2,COMPTRIG:5,EUCLID_6:24;
    then
A3: angle(A,B,C) = 2*PI - angle(C,B,A) & angle(B,C,A) = 2*PI - angle(A,C,B) &
    angle(C,A,B) = 2*PI - angle(B,A,C) by EUCLID_3:38;
    2*PI - angle(A,B,C) < 2*PI - PI by A2,XREAL_1:15;
    then angle(C,B,A) + angle(B,A,C) + angle(A,C,B) = PI
    by A1,A3,EUCLID_3:47;
    hence thesis by A3;
  end;
