Section FirstIsomorphism.
Variables aT rT : finGroupType.
Variables (G H : {group aT}) (f : {morphism G >-> rT}).
Hypothesis sHG : H \subset G.

Lemma first_isom_loc : {g : {morphism H / 'ker_H f >-> rT} |
'injm g & forall A : {set aT}, A \subset H -> g @* (A / 'ker_H f) = f @* A}.