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 Th22:
  A,B,C is_a_triangle implies sin angle(A,B,C) <> 0
  proof
    assume
A1: A,B,C is_a_triangle;
    assume
A2: sin angle(A,B,C) = 0;
    the_area_of_polygon3(C,B,A)
        = |.C-B.| * |.A-B.| * sin angle (A,B,C) /2 by EUCLID_6:5
       .= 0 by A2;
    then not C,B,A is_a_triangle by MENELAUS:9;
    hence thesis by A1,MENELAUS:15;
  end;
