(** - The type of formal expressions *)
Inductive span_ind_formal : Type :=
  | Zerovector : span_ind_formal
  | Immediately : S -> span_ind_formal
  | Plusvector : span_ind_formal -> span_ind_formal -> span_ind_formal
  | Multvector : F -> span_ind_formal -> span_ind_formal.(** - The type of formal expressions *)
Inductive span_ind_formal : Type :=
  | Zerovector : span_ind_formal
  | Immediately : S -> span_ind_formal
  | Plusvector : span_ind_formal -> span_ind_formal -> span_ind_formal
  | Multvector : F -> span_ind_formal -> span_ind_formal.(** - The type of formal expressions *)
Inductive span_ind_formal : Type :=
  | Zerovector : span_ind_formal
  | Immediately : S -> span_ind_formal
  | Plusvector : span_ind_formal -> span_ind_formal -> span_ind_formal
  | Multvector : F -> span_ind_formal -> span_ind_formal.(** - The type of formal expressions *)
Inductive span_ind_formal : Type :=
  | Zerovector : span_ind_formal
  | Immediately : S -> span_ind_formal
  | Plusvector : span_ind_formal -> span_ind_formal -> span_ind_formal
  | Multvector : F -> span_ind_formal -> span_ind_formal.(** - The type of formal expressions *)
Inductive span_ind_formal : Type :=
  | Zerovector : span_ind_formal
  | Immediately : S -> span_ind_formal
  | Plusvector : span_ind_formal -> span_ind_formal -> span_ind_formal
  | Multvector : F -> span_ind_formal -> span_ind_formal.Definition span_ind_set : part_set V.(** %\tocdef{span\_ind}% *)
Definition span_ind : subspace V.Definition lin_comb_ind (v : V) : Prop := in_part v span_ind.

Lemma span_ind_eqv_span : forall (S : part_set V) (v : V), is_lin_comb v S <-> lin_comb_ind S v.