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 Th28:
  A,B,C is_a_triangle & angle(B,A,C) < PI implies
  -PI < angle(B,A,C) - angle(C,B,A) < PI
  proof
    assume that
A1: A,B,C is_a_triangle and
A2: angle(B,A,C) < PI;
    B,A,C is_a_triangle by A1,MENELAUS:15;
    then
A3: B,A,C are_mutually_distinct by EUCLID_6:20;
A4: 0 <= angle(B,A,C) & angle(B,A,C) is non zero by A1,EUCLID10:30,EUCLID11:2;
A5: 0 <= angle(B,A,C) < PI & 0 <= angle(C,B,A) < PI by A2,A4,A3,EUCLID11:5;
A6: 0 < angle(B,A,C) < PI & 0 < angle(C,B,A) < PI by A2,A4,A3,EUCLID11:5;
    0 - PI < angle(B,A,C)-angle(C,B,A) by A6,XREAL_1:14;
    hence -PI < angle(B,A,C)-angle(C,B,A);
    angle(B,A,C) - angle(C,B,A) < PI - 0 by A5,XREAL_1:14;
    hence thesis;
  end;
