Relative Matrix of Composition of Linear Transformations/Proof 2

Proof
Let $\mathcal A = \left \langle {a_m} \right \rangle$, $\mathcal B = \left \langle {b_n} \right \rangle$, $\mathcal C = \left \langle {c_p} \right \rangle$.

Let $\left[{\alpha}\right]_{m n} = \left[{f; \left \langle {b_n} \right \rangle, \left \langle {a_m} \right \rangle}\right]$ and $\left[{\beta}\right]_{n p} = \left[{g; \left \langle {c_p} \right \rangle, \left \langle {b_n} \right \rangle}\right]$.

Then: