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 Th58:
  B <> A implies
  sin ((angle(B,A,C) + angle(C,B,A))/2) *
  cos ((angle(B,A,C) - angle(C,B,A))/2) * (|.C-B.| - |.C-A.|) =
  sin ((angle(B,A,C) - angle(C,B,A))/2) *
  cos ((angle(B,A,C) + angle(C,B,A))/2) * (|.C-B.| + |.C-A.|)
  proof
    assume B<>A;
    then
A1: (sin angle(B,A,C) + sin angle(C,B,A)) * (|.C-B.| - |.C-A.|) =
    (sin angle(B,A,C) - sin angle(C,B,A)) * (|.C-B.| + |.C-A.|) by Th57;
    set BAC = angle(B,A,C), CBA = angle(C,B,A);
    sin BAC + sin CBA = 2 * (cos((BAC-CBA)/2)*sin((BAC+CBA)/2)) by SIN_COS4:15;
    then 2 * (cos ((BAC-CBA)/2) * sin((BAC+CBA)/2)) * (|.C-B.| - |.C-A.|)=
    2 * (cos ((BAC+CBA)/2) * sin((BAC-CBA)/2))*(|.C-B.| + |.C-A.|)
       by A1,SIN_COS4:16;
    hence thesis;
  end;
