Mathematical induction

Formally, assume $n\ \text{nat}$ means that $n$ is a natural number. To prove that whenever $n\ \text{nat}$, we have $\mathcal{P}(n)$, it's sufficient to show that:

1. $\mathcal{P}(\text{zero})$.
2. For every $n$ such that $n\ \text{nat}$, if $\mathcal{P}(n)$ then $\mathcal{P}(\text{succ}(n))$.