reserve n for Nat;
reserve i for Integer;
reserve r,s,t for Real;
reserve An,Bn,Cn,Dn for Point of TOP-REAL n;
reserve L1,L2 for Element of line_of_REAL n;
reserve A,B,C for Point of TOP-REAL 2;

theorem Th21:
  A,B,C is_a_triangle implies sin (angle(A,B,C)/2) > 0
  proof
    assume
A1: A,B,C is_a_triangle;
    0 <= angle(A,B,C) < 2 * PI by EUCLID11:2;
    then 0 < angle(A,B,C) < 2 * PI by A1,EUCLID10:30;
    then 0 / 2 < angle(A,B,C) / 2 < 2 * PI / 2 by XREAL_1:74;
    then 2*PI*0 < angle(A,B,C)/2 < PI + 2*PI*0;
    hence thesis by SIN_COS6:11;
  end;
