# Principle of Mathematical Induction/One-Based/Proof 3

## Theorem

Let $\map P n$ be a propositional function depending on $n \in \N_{>0}$.

Suppose that:

$(1): \quad \map P 1$ is true
$(2): \quad \forall k \in \N_{>0}: k \ge 1 : \map P k \implies \map P {k + 1}$

Then:

$\map P n$ is true for all $n \in \N_{>0}$.

## Proof

We have that Natural Numbers are Non-Negative Integers.

Then we have that Integers form Well-Ordered Integral Domain.

The result follows from Induction on Well-Ordered Integral Domain.

$\blacksquare$