# Intro to Metamathematics Kleene $\S54$ Lemma IId

In this section, Kleene builds a formal system for primitive recursive functions. The beginning of the proof for lemma IId is skipped because it comes for general properties, but I must be missing something.

We easily see, by general properties of $$\vdash$$, that $$E^{\psi_{1}...\psi_{l}}_{f_{1}...f_{l}},E_{l+1}...E_k\vdash f_i($$x$$_1,...,$$x$$_{n_{i}}) =$$ x, if

$$f_i($$x$$_1,...,$$x$$_{n_{i}}) =$$ x $$\in E^{\phi_i}_{f_{i}}$$

What is the justification in this?

(Sorry, I couldn't put more background information this is lemma four of five lemmas. It would be too much to describe in one question, so just refer to the text. Thanks)