Definition C_CompleteSpace_mixin :=
  CompleteSpace.Mixin _ C_complete_lim C_complete_ax1 C_complete_ax2.(* on C *)

Canonical C_CompleteNormedModule :=
  CompleteNormedModule.Pack _ C (CompleteNormedModule.Class C_AbsRing _ (NormedModule.class _ C_NormedModule) C_CompleteSpace_mixin) C.(* on R *)

Canonical C_R_CompleteNormedModule :=
  CompleteNormedModule.Pack _ C (CompleteNormedModule.Class R_AbsRing _ (NormedModule.class _ C_R_NormedModule) C_CompleteSpace_mixin) C.

Lemma locally_C x P : locally (T := C_UniformSpace) x P <-> locally (T := AbsRing_UniformSpace C_AbsRing) x P.