Does $A=B$ imply that the rank of $A$ equals the rank of $B$?

Question

I am trying to understand a poof of the clubs of oddtown theorem, and I am stuck at this step:

If $A$ is an $m \times n$ matrix and $AA^T=I_m$, then the rank of $AA^T$ is AT LEAST $m$.

Now if $A=B$ does not this imply that the rank of $A$ equals the rank of $B$? So I thought the rank of $AA^T$ would be exactly $m$, which is the rank of $I_m$.

Answer 1

It should be a typo: the proof probably meant that the rank of $A$ is at least $m$ (since the rank of a product of two matrices is at most the minimum of the rank of each factor).