Lebesgue Integral of Simple Valued Function

The following propositions are true: (1) Let n, m be natural numbers, a be a function from [: Segn, Segm :] into R, and p, q be finite sequences of elements of R. Suppose that (i) dom p = Seg n, (ii) for every natural number i such that i ∈ dom p there exists a finite sequence r of elements of R such that dom r = Segm and p(i) = ∑ r and for every natural number j such that j ∈ dom r holds r(j) = a(〈i, j〉), (iii) dom q = Segm, and (iv) for every natural number j such that j ∈ dom q there exists a finite sequence s of elements of R such that dom s = Segn and q(j) = ∑ s and for every natural number i such that i ∈ dom s holds s(i) = a(〈i, j〉). Then ∑ p = ∑ q.