The notation and terminology used here are introduced in the following papers: [13], [14], [3], [9], [2], [7], [1], [4], [5], [10], [6], [12], [11], and [15]. The following proposition is true (1) For all real numbers x, y such that 0 ≤ x < y there exists a real number s0 such that 0 < s0 and x < y 1+s0 < y. The scheme RecExD3 deals with a non empty set A, an element B of A, an element C of A, ...

A mapping f : X → X from a set X to itself has a fixed point if there is an x ∈ X such that f(x) = x. The simplest fixed point theorem is that a continuous function f : [a, b] → [a, b] has at least one fixed point. This is a consequence of the intermediate value theorem from calculus, as follows. Since f(a) ≥ a and f(b) ≤ b, we have f(b) − b ≤ 0 ≤ f(a) − a. The difference f(x)−x is continuous, ...