Section Def.
Variable R : RING.
Variable Mod Mod2 : MODULE R.
Variable f : Hom Mod Mod2.
Definition Ker : submodule Mod.
apply (Build_submodule (R:=R) (M:=Mod) (submodule_subgroup:=Ker f)).
simpl in |- *.
intros a x H'; try assumption.
apply Trans with (module_mult a (f x)); auto with algebra.
apply Trans with (module_mult a (monoid_unit (module_carrier Mod2)));
auto with algebra.
Defined.
Definition coKer : submodule Mod2.
apply (Build_submodule (R:=R) (M:=Mod2) (submodule_subgroup:=coKer f)).
simpl in |- *.
intros a x H'; try assumption.
elim H'; intros x0 E; elim E; intros H'0 H'1; try exact H'1; clear E H'.
exists (module_mult a x0); split; [ try assumption | idtac ].
apply Trans with (module_mult a (f x0)); auto with algebra.
Defined.

Lemma Ker_prop :
forall x : Mod, in_part x Ker -> Equal (f x) (monoid_unit Mod2).