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

theorem
  0 <= angle(A,B,C) < PI or angle(A,B,C) = PI or PI < angle(A,B,C) < 2*PI
  proof
    set alpha = angle(A,B,C);
    0 <= alpha < 2*PI by Th2;
    then alpha in [. 0 , PI .[ \/ { PI } \/ ]. PI , 2*PI .[
    by Lm1,XXREAL_1:3;
    then alpha in [.0 , PI.[ \/ {PI} or alpha in ].PI,2*PI.[
    by XBOOLE_0:def 3;
    then alpha in [.0 , PI.[ or alpha in {PI} or alpha in ].PI,2*PI.[
    by XBOOLE_0:def 3;
    hence thesis by XXREAL_1:3,TARSKI:def 1,XXREAL_1:4;
  end;
