Proof of Associativity of the Matrix Product

Let $\alpha\in\{1,\dots,m\}$ and $\beta\in\{1,\dots,q\}$ be arbitrary indices. By the matrix product definition, applied twice,

Since the real numbers $\mathbb{R}$ form an ordered field and hence a field, the distributive law (axiom 9 of Field) gives $\bigl(\sum_{i=1}^{n}x_i\bigr)y=\sum_{i=1}^{n}(x_i y)$ for finite sums, so

Associativity of multiplication in $\mathbb{R}$ (axiom 5 of Field) gives $(A_{\alpha i}\,B_{ik})C_{k\beta}=A_{\alpha i}(B_{ik}\,C_{k\beta})$. Exchanging the order of the finite double sum using commutativity and associativity of addition (axioms 4 and 1 of Field), and then factoring $A_{\alpha i}$ from the inner sum by the distributive law again,

where the penultimate step applies the matrix product definition to $BC$. Since $\alpha$ and $\beta$ are arbitrary, $(AB)C=A(BC)$.