 Solution

Mathematical induction consists of an induction base case $P(1)$ and an inductive step $P(n)\Rightarrow P(n+1)$ for every positive integer $n$. To derive it from the well-ordering principle, suppose the set of counterexamples were nonempty and take its least element $n$. The base case excludes $n=1$. If $n>1$, minimality gives $P(n-1)$, and the step then gives $P(n)$, a contradiction. Hence there are no counterexamples.

For the stated integer congruence, the base case is $9\equiv 2(\mathrm{mod}7)$. If $9^n\equiv 2^n(\mathrm{mod}7)$, multiplication and $9\equiv 2(\mathrm{mod}7)$ givecompleting the induction.

The purported shifted-sum proof has a correct conditional successor calculation, but the base case is missing and false: at $n=1$ it would assert $1=1+126$. An inductive step without a true induction base case cannot establish any instance.