Definition of Integrability for Partial Functions from R to R and Integrability for Continuous Functions

(1) Let F , F1, F2 be finite sequences of elements of R and given r1, r2. If F1 = 〈r1〉 F or F1 = F a 〈r1〉 and if F2 = 〈r2〉 F or F2 = F a 〈r2〉, then ∑(F1−F2) = r1− r2. (2) Let F1, F2 be finite sequences of elements of R. If lenF1 = lenF2, then len(F1 +F2) = lenF1 and len(F1−F2) = lenF1 and ∑(F1 +F2) = ∑F1 +∑F2 and ∑(F1−F2) = ∑F1−∑F2. (3) Let F1, F2 be finite sequences of elements of R. If lenF1 = lenF2 and for every i such that i ∈ domF1 holds F1(i)≤ F2(i), then ∑F1 ≤ ∑F2.