Analysing proofs in analysis

Using a suitable representation of real numbers as Cauchy sequences of rational numbers with fixed rate of convergency (e.g. 2−n) the predicate ≤ becomes Π1 while < is Σ1. Hence A :≡ ( |F (x)| ≤ 2−n → |G(x)| < 2−k ) can be prenexed into a formula ∨ l ∈ INA0(x, n, k, l) with decidable A0. Furthermore for a suitable standard representation of X (such that the elements of X are represented by functions f ∈ IN and every function ∈ IN can be conceived as a representative of some element ∈ X)the quantification over X reduces to quantification over IN. Therefore (2)

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