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 Th27:
  A,B,C is_a_triangle implies (-2) * PI < angle(B,A,C) - angle(C,B,A) < 2 * PI
  proof
    assume
A1: A,B,C is_a_triangle;
A2: 0 <= angle(B,A,C) < 2 * PI & 0 <= angle(C,B,A) < 2 * PI
    by EUCLID11:2;
    then
A3: 0 < angle(B,A,C) < 2 * PI & 0 < angle(C,B,A) < 2 * PI
    by A1,EUCLID10:30;
    0 - 2*PI < angle(B,A,C)-angle(C,B,A) by A3,XREAL_1:14;
    hence (-2) * PI < angle(B,A,C)-angle(C,B,A);
    angle(B,A,C) - angle(C,B,A) < 2 * PI - 0 by A2,XREAL_1:14;
    hence thesis;
  end;
