Strong Induction: Every Integer Above 1 Factors into Primes

State the idea of the new principle, assuming all of $P(M), \ldots, P(k)$ instead of $P(k)$ alone, in the domino picture where every domino up to the $k$th has already fallen.

State the principle for $T \subseteq \mathbb{Z}$ with $M \in T$ and $\{M, M+1, \ldots, k\} \subseteq T$ implying $k + 1 \in T$, and write the proof outline with its stronger inductive hypothesis.

Write the complete proof: basis step $P(2)$, the assumption $P(2), \ldots, P(k)$, Case 1 for prime $k + 1$, and Case 2 with $k + 1 = a \cdot b$, $1 < a, b \leq k$, invoking $P(a)$ and $P(b)$.