DivisorChain.second_of_chain_is_irreducible

theorem DivisorChain.second_of_chain_is_irreducible : ∀ {M : Type u_1} [inst : CancelCommMonoidWithZero M] {q : Associates M} {n : ℕ}, n ≠ 0 → ∀ {c : Fin (n + 1) → Associates M}, StrictMono c → (∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i) → q ≠ 0 → Irreducible (c 1)

The theorem states that the second element of a "chain" is irreducible. More precisely, given a type `M` that forms a `CancelCommMonoidWithZero` (an algebraic structure with commutative multiplication and cancellation), a non-zero `Associates M` (which are elements of `M` related by the `Associated` relation) denoted as `q`, a natural number `n` not equal to zero, and a strictly monotone function `c` mapping from zero-based indices to `Associates M` (i.e., a "chain"), such that any `Associates M` `r` is less than or equal to `q` if and only if it equals `c i` for some index `i`, and `q` is not zero, then the second element in the chain (i.e., `c 1`) is irreducible. An irreducible element is one that cannot be factored into other elements, except for units and associates of itself.

More concisely: Given a `CancelCommMonoidWithZero` `M` and a strictly monotone chain `c` of non-zero associates `q` such that every associate is equal to `q` for some index or is zero, the second element in the chain is an irreducible element.