Proving Natural Predecessor existence using Well Order

Question

I know that the Principle of Induction is equivalent to Well Order at least in Natural Numbers, but I have seen that the demonstration of Induction using Well Order uses the existence of Predecessor:

1. [Assume Induction]: Suppose $P(0)$ is true, and $P(n+1)$ is true whenever $P(n)$ is true.
2. [$S$ definition]: Let $S$ be the non-empty set of $k$ for which $P(k)$ is not true.
3. [Well-Order]: By well-ordering $S$ has a least element, which cannot be $k = 0$.
4. [Predecessor & Contradiction]: But then $P(k-1)$ is true, and so $P(k)$ is true.

When using $P(k-1)$ at [4.] it is assumed that $k-1$ exists as a Predecessor of $k$, but how would one demonstrate Predecesor existence using Peano Axioms without Induction Principle? That is to say, all Peano Axioms but replacing the Principle of Induction with the Principle of Well Order.