I wish if someone send me the proof of the famous theorem of rank equalities due to G. Marsaglia which cited in many publications on ranks for block matrices involving generalized inverses. For instence, Lemma 1.1 ([3]) in the following paper of Y. Tian:
https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.431.5605&rep=rep1&type=pdf
I wish that the proof is based on the ranges of the submatrices of the block matrix they form.