Equivalence of Definitions of Matroid Circuit Axioms/Condition 2 Implies Condition 4

Theorem
Let $S$ be a finite set.

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

Then:
 * $\mathscr C$ is the set of circuits of a matroid $M = \struct{S, \mathscr I}$ on $S$

Proof
We will define a mapping $\rho$ associated with $\mathscr C$.

It will be shown that $\rho$ is the rank function of a matroid $M$ which has $\mathscr C$ as the set of circuits.

Let $\tuple{x_1, \ldots, x_q}$ be any ordered tuple of elements of $S$.

Define the ordered tuple $\map \theta {\tuple{x_1, \ldots, x_q}}$ by:
 * $\forall i \in \set{1, \ldots, q} : \map \theta {\tuple{x_1, \ldots, x_q}}_i = \begin{cases}

0 & : \exists C \in \mathscr C : x_i \in C \\ 1 & : \text {otherwise} \end{cases}$

Define a mapping $\rho'$ from the set of ordered tuple of $S$ by:
 * $\map {\rho'} {\tuple{x_1, \ldots, x_q}} = \ds \sum_{i = 1}^q \map \theta {\tuple{x_1, \ldots, x_q}}_i$

Lemma 1
Let $\tuple{x_1, \ldots, x_q}$ be any ordered tuple of elements of $S$.

Let $\pi$ be any permutation of $\tuple{x_1, \ldots, x_q}$.

Then:
 * $\map {\rho'} {\tuple{x_1, \ldots, x_q}} = \map {\rho'} {\tuple{x_{\map \pi 1}, \ldots, x_{\map \pi q}}}$

Define a mapping $\rho : \powerset S \to \Z$ by:
 * $\forall A \subseteq S$:
 * $\map \rho A = \map {\rho'} {\tuple{x_1, \ldots, x_q}}$
 * where $A = \set{x_1, \ldots, x_q}$.

From Lemma 1 then $\rho$ is

Define a mapping $\theta : \powerset S \to \powerset S$ by:
 * $\forall X \subseteq S : \map \theta X = \set{x \in X : \nexists C \in \mathscr C : x \in C \subseteq X}$

Define a mapping $\theta : \powerset S \to \powerset S$ by:
 * $\forall X \subseteq S : \map \theta X = X \setminus \bigcup \set{C \in \mathscr C : C \subseteq X}$

Define a mapping $\rho : \powerset S \to \Z$ by:
 * $\forall X \subseteq S : \map \rho X = \size{\map \theta X}$