Relative Matrix of Composition of Linear Transformations/Proof 2

Proof
Let:
 * $\AA = \sequence {a_m}$
 * $\BB = \sequence {b_n}$
 * $\CC = \sequence {c_p}$

Let:
 * $\sqbrk \alpha_{m n} = \sqbrk {f; \sequence {b_n}, \sequence {a_m} }$

and:
 * $\sqbrk \beta_{n p} = \sqbrk {g; \sequence {c_p}, \sequence {b_n} }$

Then: