I am pretty sure this lemma is true, but I would like to have a reference. It is a generalization to rectangular matrices of the following fact: the determinant of a block triangular matrix is given by the product of determinants of blocks on the diagonal.
The statement I am looking the proof of is the following: given a block triangular matrix $M$ with diagonal blocks $A_1, \ldots, A_r$, every maximal minor $d$ of $M$ is a product of maximal minors $d_1, \ldots, d_r$ respectively coming from $A_1, \ldots, A_r$.
There could be restrictions on sizes in order for this to be applied, but a few mental experiments seem to suggest that the sizes of $A_1, \ldots, A_r$ does not matter.