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;
reserve D for Point of TOP-REAL 2;
reserve a,b,c,d for Real;

theorem Th59:
  A,B,C is_a_triangle & angle(B,A,C) - angle(C,B,A) <> PI &
  angle(B,A,C) - angle(C,B,A) <> -PI
  implies
  cos ((angle(B,A,C) - angle(C,B,A))/2) <> 0
  proof
    assume that
A1: A,B,C is_a_triangle and
A2: angle(B,A,C) - angle(C,B,A) <> PI and
A3: angle(B,A,C) - angle(C,B,A) <> -PI;
    assume
A4: cos((angle(B,A,C) - angle(C,B,A))/2) = 0;
    consider i0 be Integer such that
A5: (angle(B,A,C) - angle(C,B,A))/2 = PI/2 + PI * i0 by A4,BORSUK_7:8;
    set a = angle(B,A,C) - angle(C,B,A);
A6: 0 <= angle(B,A,C) < 2 * PI & 0 <= angle(C,B,A) < 2 * PI
      by EUCLID11:2;
    then
A7: 0 < angle(B,A,C) < 2 * PI & 0 < angle(C,B,A) < 2 * PI
      by A1,EUCLID10:30;
A8: 0 - 2*PI < angle(B,A,C)-angle(C,B,A) by A7,XREAL_1:14;
    angle(B,A,C) - angle(C,B,A) < 2 * PI - 0 by A6,XREAL_1:14;
    then (-2)*PI/PI<(1+2*i0)*PI/PI<2*PI/PI & PI <> 0
           by A8,A5,COMPTRIG:5,XREAL_1:74;
    then (-2)*(PI/PI)<(1+2*i0)*(PI/PI)<2*(PI/PI) & PI/PI = 1 by XCMPLX_1:60;
    then (-2) - 1 < 1 + 2 * i0 - 1< 2 - 1 by XREAL_1:14;
    then (-3) / 2 < 2 * i0 / 2 < 1 / 2 by XREAL_1:74;
    then i0 = 0 or i0 = -1 by Th2;
    hence contradiction by A5,A2,A3;
  end;
