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I have to prove that if $\int_{a}^{b}fdg$ exist then for every subinterval $[c,d] \subset [a,b]$ also $\int_{c}^{d}fdg$ exist. I was thinking about using theorem that says, if we add some points to our partition of interval then we get fine inequalities for lower and upper sums. But I don't know if it's right and also I do not know how to write it properly.

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