Proof of eqn.(8) on pg.367 in Royden & Fitzpatrick "Real Analysis" fourth edition.

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The equation is given below:

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And the definition of integrale for nonnegative functions is given below:

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And we already proved the following:

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**My Proof is: **

It follows directly from the definition of the integral and from $(5)$ in proposition $(8).$

My question is:

Is this proof correct or there are some precise details that should be included ?