Arguments qisom_inj {gT G H} eqGH [x1 x2].
Arguments morphim_qisom_inj {gT G H} eqGH [x1 x2].
Section FirstIsomorphism.
Variables aT rT : finGroupType.

Lemma first_isom (G : {group aT}) (f : {morphism G >-> rT}) :
{g : {morphism G / 'ker f >-> rT} | 'injm g &
forall A : {set aT}, g @* (A / 'ker f) = f @* A}.