Section Iso.
Variables (R : ringType) (vT rT : vectType R).
Local Coercion dim : vectType >-> nat.
Fact v2r_subproof : axiom vT vT. Proof. by case: vT => T [bT []]. Qed.
Definition v2r := s2val v2r_subproof.
Let v2r_bij : bijective v2r := s2valP' v2r_subproof.
Fact r2v_subproof : {r2v | cancel r2v v2r}.
Definition r2v := sval r2v_subproof.

Lemma r2vK : cancel r2v v2r.   Proof. exact: svalP r2v_subproof. Qed.
Lemma r2v_inj : injective r2v. Proof. exact: can_inj r2vK. Qed.
Lemma v2rK : cancel v2r r2v.   Proof. by have/bij_can_sym:= r2vK; apply. Qed.
Lemma v2r_inj : injective v2r. Proof. exact: can_inj v2rK. Qed.