intros.
generalize (basis_prop (Build_basis H) x).
intro.
generalize (span_ind_eqv_span (seq_set v) x).
intros.
inversion_clear H1.
generalize (H2 H0).
clear H0 H2 H3; intro H'.
inversion_clear H'.
generalize H0; clear H0.
generalize x; clear x.
elim x0.
intros.
exists (const_seq n (zero F)).
simpl in H0.
apply Trans with (zero V); auto with algebra.
apply sum_of_zeros; intro; simpl in |- *; auto with algebra arith.
intro c.
simpl in c.
elim c.
simpl in |- *.
intros.
inversion_clear subtype_prf.
generalize H1; clear H1.
elim x1.
intros x2 y H1.
exists (Basisvec_Fn F y).
apply Trans with subtype_elt; auto with algebra.
apply Trans with (v (Build_finiteT y)); auto with algebra.
apply Sym.
apply projection_via_mult_by_scalars.
intros.
generalize (H0 (span_ind_injection s) (Refl (span_ind_injection s))).
generalize (H1 (span_ind_injection s0) (Refl (span_ind_injection s0))).
intros.
clear H0 H1.
inversion_clear H3.
inversion_clear H4.
exists (pointwise (sgroup_law_map F:Map _ _) x1 x2).
simpl in H2.
apply
Trans
with
(sum
(pointwise (sgroup_law_map V) (mult_by_scalars x1 v)
(mult_by_scalars x2 v))).
apply sum_comp.
simpl in |- *.
red in |- *.
intro.
simpl in |- *.
apply Trans with ((x1 x3 +' x2 x3) mX v x3); auto with algebra.
apply Trans with (x1 x3 mX v x3 +' x2 x3 mX v x3); auto with algebra.
apply Trans with (sum (mult_by_scalars x1 v) +' sum (mult_by_scalars x2 v)).
generalize sum_of_sums.
intro.
generalize (H3 n V (mult_by_scalars x1 v) (mult_by_scalars x2 v)).
intros.
simpl in H4.
apply H4.
apply Trans with (span_ind_injection s0 +' span_ind_injection s);
auto with algebra.
apply Trans with (span_ind_injection s +' span_ind_injection s0);
auto with algebra.
intros.
generalize (H0 (span_ind_injection s) (Refl (span_ind_injection s))).
intro.
clear H0.
inversion_clear H2.
exists (pointwise (uncurry (RING_comp (R:=F))) (const_seq n c) x1).
apply
Trans with (sum (mult_by_scalars (const_seq n c) (mult_by_scalars x1 v))).
apply sum_comp.
simpl in |- *.
red in |- *.
intro.
simpl in |- *.
auto with algebra.
apply Trans with (c mX sum (mult_by_scalars x1 v)); auto with algebra.
simpl in H1.
apply Trans with (c mX span_ind_injection s); auto with algebra.
Qed.
Section Nice_basis_properties.
Variable x : V.
Variable n : Nat.
Variable b : seq n V.
Variable Db : distinct b.
Variable Bb : is_basis (seq_set b).
Let difference_seq : forall (G : group) (a a' : seq n G), seq n G.
intros.
apply (Build_Map (Ap:=fun i : fin n => a i +' (min a' i))); auto with algebra.
red in |- *.
intros.
apply SGROUP_comp; auto with algebra.
Defined.

(* Specifications: *)
(** - %\intoc{\bf Theorem 1.7}% *)
Lemma basis_expansion_uniqueness :
forall a a' : seq n F,
sum (mult_by_scalars a b) =' x in _ ->
sum (mult_by_scalars a' b) =' x in _ -> a =' a' in _.