Another proof the $\operatorname{rank} (AB) \leq \operatorname{rank} (A)$

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Given a matrix $A$, if the rank of $A$ is defined to be the highest order of non-vanishing minor of $A$.

If we apply the above definition, how to check that $\operatorname{rank}(AB) \leq \operatorname{rank}(A)$?