Let $\displaystyle E,F,K$ be fields such that $\displaystyle F \subset E \subset K$. Prove that if $\displaystyle A=\{a_1,...,a_n\}$ is a basis for $\displaystyle K$ over $\displaystyle E$ and $\displaystyle B=\{b_1,...b_m\}$ is a basis for $\displaystyle E$ over $\displaystyle F$, then $\displaystyle C=\{a_ib_j, 1 \leq i \leq n, 1 \leq j \leq m\}$ is a basis for $\displaystyle K$ over $\displaystyle F$. Conclude that if $\displaystyle \dim_E(K)$ and $\displaystyle \dim_F(E)$ are finite, then $\displaystyle \dim_F(K)=\dim_E(K)\dim_F(E)$.

I know how to do the proof, but I'm not sure how to conclude that $\displaystyle \dim_F(K)=\dim_E(K)\dim_F(E)$. Some help please.