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Jun 27 at 12:41 comment added Elías Guisado Villalgordo $\def\Im{\operatorname{Im}}\def\Ker{\operatorname{Ker}}$In McCleary's User's Guide to Spectral Sequences the exactness of \eqref{seq} is proven with the sequence of equalities:$$\Ker f' =\Ker f/\Im d=\Im g/\Im d=\Im g'.$$Among these, the last one is the only one that I think is non-trivial because of the way $g'$ is defined.
Jun 27 at 8:25 history asked Elías Guisado Villalgordo CC BY-SA 4.0