Equivalence of Definitions of Matroid Circuit Axioms/Lemma 1

Theorem
Let $S$ be a finite set.

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

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 \subseteq \set{x_1, \ldots, x_i}\\ 1 & : \text {otherwise} \end{cases}$

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

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