User:Leigh.Samphier/Matroids/Circuits of Matroid iff Matroid Circuit Axioms/Lemma 5

Theorem
Let $S$ be a finite set.

Let $\mathscr C$ be a non-empty set of subsets of $S$ that satisfies the circuit axioms:

For any ordered tuple $\tuple{x_1, \ldots, x_q}$ of elements of $S$, let $\map \theta {x_1, \ldots, x_q}$ be the ordered tuple defined by:
 * $\forall i \in \set{1, \ldots, q} : \map \theta {x_1, \ldots, x_q}_i = \begin{cases}

0 & : \exists C \in \mathscr C : x_i \in C \subseteq \set{x_1, \ldots, x_i}\\ 1 & : \text {otherwise} \end{cases}$

Let $t$ be the mapping from the set of ordered tuple of $S$ defined by:
 * $\map t {x_1, \ldots, x_q} = \ds \sum_{i = 1}^q \map \theta {x_1, \ldots, x_q}_i$

Let $\rho : \powerset S \to \Z$ be the mapping satisfying the matroid rank axioms defined by:
 * $\forall A \subseteq S$:
 * $\map \rho A = \begin{cases}

0 & : \text{if } A = \O \\ \map t {x_1, \ldots, x_q } & : \text{if } A = \set{x_1, \ldots, x_q} \end{cases}$

Let $M = \struct{S, \mathscr I}$ be the matroid corresponding to the rank function $\rho$.

Let $\mathscr C_M$ be the set of circuits of $M$

Then:
 * $\forall C \in \mathscr C : \exists C' \in \mathscr C_M : C' \subseteq C$

Proof
Let $C \in \mathscr C$.

From circuit axiom $(\text C 1)$:
 * $\exists y \in C$

We have:
 * $C \subseteq C = \paren{C \setminus \set y} \cup \set y$

Lemma 2
We have:

From Rank of Independent Subset Equals Cardinality:
 * $C \notin \mathscr I$

That is, $C$ is dependent.

From Dependent Subset Contains a Circuit:
 * $\exists C' \in \mathscr C_M$ such that $C' \subseteq C$