Definition algid U := s2val (algid_subproof U).

(* Specifications: *)
Lemma memv_algid U : algid U \in U.
Lemma algidr A : {in A, right_id (algid A) *%R}.
Lemma unitr_algid1 A u : u \in A -> u \is a GRing.unit -> algid A = 1.
Lemma dim_algid A : \dim <[algid A]> = 1%N.
Lemma asubv A : (A * A <= A)%VS.
Lemma prodv_sub U V A : (U <= A -> V <= A -> U * V <= A)%VS.
Lemma memv_cosetP {U v w} :
reflect (exists2 u, u\in U & w = u * v) (w \in U * <[v]>)%VS.
Lemma dim_cosetv_unit V u : u \is a GRing.unit -> \dim (V * <[u]>) = \dim V.
Lemma memvV A u : (u^-1 \in A) = (u \in A).
Fact aspace_cap_subproof A B : algid A \in B -> is_aspace (A :&: B).