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 Th25:
  A,B,C is_a_triangle implies angle(B,A,C) - angle(C,B,A) <> PI
  proof
    assume
A1: A,B,C is_a_triangle;
    then B,A,C is_a_triangle by MENELAUS:15;
    then A1A: B,A,C are_mutually_distinct by EUCLID_6:20;
    assume
A2: angle(B,A,C) - angle(C,B,A) = PI;
    per cases by EUCLID11:3;
    suppose
A3:   0 <= angle(B,A,C) < PI;
      angle(B,A,C) - PI = angle(C,B,A) by A2;
      hence contradiction by A3,XREAL_1:49,EUCLID11:2;
    end;
    suppose angle(B,A,C) = PI;
      then angle(C,B,A) = 0 by A2;
      hence contradiction by A1,EUCLID10:30;
    end;
    suppose
A4:   PI < angle(B,A,C) < 2 * PI;
      then angle(B,A,C) - PI < 2 * PI - PI by XREAL_1:14;
      hence contradiction by A1A,A2,A4,EUCLID11:8;
    end;
  end;
