Scalar Multiple of Riemann Definite Integral

We adopt the following rules: r, x, y are real numbers, i, j are natural numbers, and p is a finite sequence of elements of R. The following proposition is true (1) For every closed-interval subset A of R and for every x holds x ∈ A iff inf A ¬ x and x ¬ supA. Let I1 be a finite sequence of elements of R. We say that I1 is non-decreasing if and only if the condition (Def. 1) is satisfied. (Def. 1) Let n be a natural number. Suppose n ∈ dom I1 and n+1 ∈ dom I1. Let r, s be real numbers. If r = I1(n) and s = I1(n + 1), then r ¬ s. One can verify that there exists a finite sequence of elements of R which is non-decreasing. The following three propositions are true: