Induction proof

This page explains how to proove with induction method on the natural number with Rodin tools. In other words, how to proove :
$P(0)$
$\forall i.i \in \mathbb{N}\land P(i) \Rightarrow P(i+1)$
$\vdash$
$\forall i. i \in \mathbb{N} \Rightarrow P(i+1)$

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The proof key is the following theorem:
$\forall s.s \subseteq \mathbb{N} \land 0 \in s \land (\forall n.n \in s => n+1 \in s)\Rightarrow N \subseteq s$

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The proof of the previous theorem is given by the following.
Instanciate the key theorem with : $\{x|x\in \mathbb{N} \land P(x)\}$