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 Th57:
  B <> A implies
  (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.|)
  proof
    assume B<>A;
    then |.C-B.| * sin angle(C,B,A) = |.C-A.| * sin angle(B,A,C) by EUCLID_6:6;
    hence thesis;
  end;
