The Definition of the Riemann Definite Integral and some Related Lemmas

For simplicity, we follow the rules: a, b are real numbers, F , G, H are finite sequences of elements of R, i, j, k are natural numbers, X is a non empty set, and x1 is a set. Let I1 be a subset of R. We say that I1 is closed-interval if and only if: (Def. 1) There exist real numbers a, b such that a ≤ b and I1 = [a, b]. Let us mention that there exists a subset of R which is closed-interval. In the sequel A is a closed-interval subset of R. One can prove the following propositions:

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