Principle of Mathematical Induction/Zero-Based

Theorem
Let $\map P n$ be a propositional function depending on $n \in \N$.

Suppose that:


 * $(1): \quad \map P 0$ is true


 * $(2): \quad \forall k \in \N: k \ge 0 : \map P k \implies \map P {k + 1}$

Then:


 * $\map P n$ is true for all $n \in \N$.

Proof
Consider $\N$ defined as a Peano structure.

The result follows from Principle of Mathematical Induction for Peano Structure.