Question about Lebesque Integral

The Lebesque integral ,I, of function f with respect to measure u is
defined by

I (f du) = I (f+ du) - I (f- du)

where f+ and f- are the positive and negative parts of f, respectively.
Now my text (Bartle) states that "it is easy to see that if" f = v -
w, with v and w non-negative, then

I (f du)= I (v du) - I ( w du)

I don't see how this follows from the basic definition of I as supremum
of integrals of simple functions. Can someone please help with a proof?
Thanks.

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