Pairwise Disjoint Subsets in Semiring Part of Partition

Lemma
Let $$\mathbb S$$ be a semiring of sets.

Let $$A, A_1, A_2, \ldots, A_n$$ all belong to $$\mathbb S$$.

Let $$A_1, A_2, \ldots, A_n$$ all be pairwise disjoint subsets of $$A$$.

Then there exists a finite expansion of $$A$$:
 * $$\exists s \ge n: A = \bigcup_{k=1}^s A_k$$

with $$A_1, \ldots, A_n$$ as its first $$n$$ elements, such that:
 * $$\forall k, 1 \le k \le s: A_k \in \mathbb S$$;
 * $$\forall k, l, 1 \le k \le s, 1 \le l \le s: k \ne l \implies A_k \cap A_l = \varnothing$$.

That is, the nature of a semiring is such that every collection of pairwise disjoint subsets of a given set $$A$$ of that semiring is part of a larger collection of pairwise disjoint subsets of $$A$$ which forms a complete partition of $$A$$.

Proof
By the definition of a semiring of sets, the lemma holds for $$n = 1$$.

Now we suppose that the lemma holds for $$n = m$$, and we attempt to show it consequently holds for $$n = m+1$$.

So, let $$A_1, A_2, \ldots, A_m, A_{m+1}$$ all be pairwise disjoint subsets of $$A$$.

By hypothesis:
 * $$A = A_1 \cup A_2 \cup \cdots \cup A_m \cup B_1 \cup \cdots \cup B_p$$

where $$A_1, A_2, \ldots, A_m, B_1, \ldots, B_p$$ are pairwise disjoint subsets of $$A$$, all belonging to $$\mathbb S$$.

Let $$B_{q1} = A_{m+1} \cap B_q$$.

By the definition of a semiring of sets:
 * $$B_q = B_{q1} \cup \cdots \cup B_{q r_q}$$

where all the $$B_{qj}$$ are pairwise disjoint subsets of $$B_q$$, all belonging to $$\mathbb S$$.

But then we see that:
 * $$A = A_1 \cup A_2 \cup \cdots \cup A_m \cup A_{m+1} \cup \bigcup_{q=1}^p \left({\bigcup_{j=2}^{r_q} B_{qj}}\right)$$

and so the lemma is true for $$m+1$$.

The result follows by induction.