Some Properties of the Intervals

(1) There exists a function F from N into [:N, N :] such thatF is one-to-one and dom F = N and rngF = [:N, N :]. (2) For every functionF from N into R such thatF is non-negative holds 0 R ≤ ∑F. (3) Let F be a function fromN into R andx be an extended real number. Suppose there exists a natural number n such thatx≤ F(n) andF is non-negative. Then x≤ ∑F. (8)1 For all extended real numbers x, y such thatx is a real number holds (y− x)+ x = y and (y+x)−x = y. (10)2 For all extended real numbers x, y, zsuch thatz∈R andy< x holds(z+x)−(z+y) = x−y.