Well-Ordering Axiom - Strong Induction

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Strong induction:

Assume there is a proposition that is a function of a single natural number, that is P(x).

If P(1) is true
and
P(1) and P(2) and so on through P(n) imply P(n+1)
then
P(i) is true for all natural numbers.

Proof of Well-ordering axiom by strong induction:

We will do induction based on the size of the subset.

That is, the property P(i) in strong induction will ...