intros.
assert
(a i mX v i ='
a i mX field_inverse (a i) mX (min sum (omit (mult_by_scalars a v) i)) in _).
apply Trans with (mult_by_scalars a v i); auto with algebra.
apply
Trans
with
((a i rX field_inverse (a i)) mX (min sum (omit (mult_by_scalars a v) i)));
auto with algebra.
apply Trans with (one mX (min sum (omit (mult_by_scalars a v) i)));
auto with algebra.
apply Trans with (min sum (omit (mult_by_scalars a v) i)); auto with algebra.
assert
(mult_by_scalars a v i +' sum (omit (mult_by_scalars a v) i) ='
(zero V) in _).
apply Trans with (sum (mult_by_scalars a v)); auto with algebra.
apply Sym.
apply
(seqsum_is_elt_plus_omitseq (M:=V:abelian_monoid) (mult_by_scalars a v)).
2: apply MODULE_comp; auto with algebra.
2: apply Sym.
2: apply FIELD_inverse_r; auto with algebra.
apply Sym.
apply GROUP_law_inverse.
apply Trans with (sum (mult_by_scalars a v)); auto with algebra.
apply
Trans with (mult_by_scalars a v i +' sum (omit (mult_by_scalars a v) i));
auto with algebra.
apply Sym.
apply
(seqsum_is_elt_plus_omitseq (M:=V:abelian_monoid) (mult_by_scalars a v));
auto with algebra.
set (expr := field_inverse (a i) mX (min sum (omit (mult_by_scalars a v) i)))
in *.
cut (one mX v i =' one mX expr in _).
intro.
apply Trans with (one mX expr); auto with algebra.
apply Trans with (one mX v i); auto with algebra.
apply Trans with ((field_inverse (a i) rX a i) mX v i).
apply MODULE_comp; auto with algebra.
apply Sym.
auto with algebra.
apply Trans with ((field_inverse (a i) rX a i) mX expr); auto with algebra.
apply Trans with (field_inverse (a i) mX a i mX expr); auto with algebra.
apply Trans with (field_inverse (a i) mX a i mX v i); auto with algebra.
Qed.

(** - If $y\in W\cup\{x\}$ but $y\neq x$ then $y\in W$ *)
Lemma another_small_lemma :
forall (V : Setoid) (W : part_set V) (x : V),
let Wx := union W (single x) in
forall y : Wx, ~ subtype_elt y =' x in _ -> in_part (subtype_elt y) W.