intros.
apply GROUP_reg_right with (s i); auto with algebra.
apply Trans with (sum s +' ((min s i) +' s i)); auto with algebra.
apply Trans with (sum s +' (zero AG)); auto with algebra.
apply Trans with (sum s); auto with algebra.
apply Trans with (s i +' sum (omit s i)); auto with algebra.
generalize seqsum_is_elt_plus_omitseq.
intro p.
apply (p n AG s i).
Qed.
Hint Resolve omit_head_is_seqtl omit_tlelt seqsum_is_elt_plus_omitseq
seqsum_min_elt_is_omitseq: algebra.

(* Specifications: *)
Lemma omit_mult_by_scalars :
forall (n : Nat) (F : ring) (V : module F) (a : seq n F)
(v : seq n V) (i : fin n),
omit (mult_by_scalars a v) i =' mult_by_scalars (omit a i) (omit v i) in _.