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.

(** - %\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 _.