Second Principle of Mathematical Induction/One-Based

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

Suppose that:


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


 * $(2): \quad \forall k \in \N_{>0}: \map P 1 \land \map P 2 \land \ldots \land \map P {k - 1} \land \map P k \implies \map P {k + 1}$

Then:


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