Definition expv U n := iterop n.+1.-1 prodv U 1%VS.
Local Notation "A ^+ n" := (expv A n) : vspace_scope.

(* Specifications: *)
Lemma expv0 U : (U ^+ 0 = 1)%VS. Proof. by []. Qed.
Lemma expv1 U : (U ^+ 1 = U)%VS. Proof. by []. Qed.
Lemma expv2 U : (U ^+ 2 = U * U)%VS. Proof. by []. Qed.
Lemma expvSl U n : (U ^+ n.+1 = U * U ^+ n)%VS.
Lemma expv_line u n : (<[u]> ^+ n = <[u ^+ n]>)%VS.
Definition centraliser1_vspace u := lker (amulr u - amull u).