Third Principle of Mathematical Induction

Theorem
Let $P \left({n}\right)$ be a propositional function depending on $n \in \N$.

If:


 * $(1): \quad P \left({n}\right)$ is true for all $n \le d$ for some $d \in \N$
 * $(2): \quad \forall m \in \N: \left({\forall k \in \N, m \le k < m + d: P \left({k}\right)}\right) \implies P \left({m + d}\right)$

then $P \left({n}\right)$ is true for all $n \in \N$.