
theorem
  for a,b,c be positive Real holds (a+b+c)/a > (a+b)/(a+c) > a/(a+b+c)
  proof
    let a,b,c be positive Real;
    reconsider k = (a+b)/a as heavy positive Real;
    A1: a+b = k*a by XCMPLX_1:87;
    reconsider l = (a+c)/a as heavy positive Real;
    A2: a+c = l*a by XCMPLX_1:87;
    A3: k/l = (a+b)/(a+c) by XCMPLX_1:55;
    A4: k+l-1 > k/l > 1/(k+l-1) by ADB;
    a + b + c = (k+l-1)*a by A1,A2; then
    (a+b+c)/a = k+l-1 by XCMPLX_1:89;
    hence thesis by A3,A4,XCMPLX_1:213;
  end;
