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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)

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