diff --git a/CHANGELOG_UNRELEASED.md b/CHANGELOG_UNRELEASED.md
index 5b7f660af..b44ac9e97 100644
--- a/CHANGELOG_UNRELEASED.md
+++ b/CHANGELOG_UNRELEASED.md
@@ -207,6 +207,19 @@
 - moved from `theories` to `theories/topology_theory`:
   + file `function_spaces.v`
 
+- moved from `theories` to `theories/normedtype_theory`:
+  + file `tvs.v`
+
+- moved from `tvs.v` to `pseudometric_normed_Zmodule.v`:
+  + definitions `NbhsNmodule`, `NbhsZmodule`, `PreTopologicalNmodule`, `PreTopologicalZmodule`,
+    `PreUniformNmodule`, `PreUniformZmodule`
+
+- in `tvs.v`, turned into `Let`'s:
+  + local lemmas `standard_add_continuous`, `standard_scale_continuous`, `standard_locally_convex`
+
+- in `normed_module.v`, turned into `Let`'s:
+  + local lemmas `add_continuous`, `scale_continuous`, `locally_convex`
+
 ### Renamed
 
 - in `topology_structure.v`
diff --git a/_CoqProject b/_CoqProject
index 44d0d9852..8b90d4861 100644
--- a/_CoqProject
+++ b/_CoqProject
@@ -75,12 +75,12 @@ theories/homotopy_theory/continuous_path.v
 
 theories/ess_sup_inf.v
 theories/cantor.v
-theories/tvs.v
 
 theories/normedtype_theory/num_normedtype.v
 theories/normedtype_theory/matrix_normedtype.v
 theories/normedtype_theory/ereal_normedtype.v
 theories/normedtype_theory/pseudometric_normed_Zmodule.v
+theories/normedtype_theory/tvs.v
 theories/normedtype_theory/normed_module.v
 theories/normedtype_theory/complete_normed_module.v
 theories/normedtype_theory/urysohn.v
diff --git a/theories/Make b/theories/Make
index 9c1128c72..75ab71c74 100644
--- a/theories/Make
+++ b/theories/Make
@@ -41,12 +41,12 @@ homotopy_theory/continuous_path.v
 
 ess_sup_inf.v
 cantor.v
-tvs.v
 
 normedtype_theory/num_normedtype.v
 normedtype_theory/matrix_normedtype.v
 normedtype_theory/ereal_normedtype.v
 normedtype_theory/pseudometric_normed_Zmodule.v
+normedtype_theory/tvs.v
 normedtype_theory/normed_module.v
 normedtype_theory/complete_normed_module.v
 normedtype_theory/urysohn.v
diff --git a/theories/normedtype_theory/normed_module.v b/theories/normedtype_theory/normed_module.v
index 3e5b5d482..d119bce20 100644
--- a/theories/normedtype_theory/normed_module.v
+++ b/theories/normedtype_theory/normed_module.v
@@ -93,15 +93,13 @@ HB.factory Record PseudoMetricNormedZmod_Lmodule_isNormedModule (K : numFieldTyp
 
 HB.builders Context K V & PseudoMetricNormedZmod_Lmodule_isNormedModule K V.
 
-(* NB: These lemmas are done later with more machinery. They should
-   be simplified once normedtype is split, and the present section can
-   depend on near lemmas on pseudometricnormedzmodtype? *)
 (* add_continuous has been moved to pseudometric_normed_Zmodule.v,
   scale_continuous is proved but is not proved again anymore later in this file *)
-Lemma add_continuous : continuous (fun x : V * V => x.1 + x.2).
+Let add_continuous : continuous (fun x : V * V => x.1 + x.2).
 Proof. exact: add_continuous. Qed.
 
-Lemma scale_continuous : continuous (fun z : K^o * V => z.1 *: z.2).
+(** NB: we have almost the same proof in `tvs.v` *)
+Let scale_continuous : continuous (fun z : K^o * V => z.1 *: z.2).
 Proof.
 move=> [/= k x]; apply/cvgrPdist_lt => _/posnumP[e]; near +oo_K => M.
 near=> l z => /=; have M0 : 0 < M by [].
@@ -113,7 +111,8 @@ rewrite (@le_lt_trans _ _ (`|k - l| * M)) ?ler_wpM2l -?ltr_pdivlMr//.
 by near: l; apply: cvgr_dist_lt; rewrite // divr_gt0.
 Unshelve. all: by end_near. Qed.
 
-Lemma locally_convex :
+(** NB: we have almost the same proof in `tvs.v` *)
+Let locally_convex :
   exists2 B : set (set V), (forall b, b \in B -> convex b) & basis B.
 Proof.
 exists [set B | exists x r, B = ball x r].
@@ -419,9 +418,6 @@ exists e%:num => //= y xe_y; rewrite lt_def => /andP [xney _].
 by apply: xe_A => //; rewrite eq_sym.
 Qed.
 
-Lemma scale_continuous : continuous (fun z : K * V => z.1 *: z.2).
-Proof. exact: scale_continuous. Abort.
-
 Arguments scale_continuous _ _ : clear implicits.
 
 Lemma scaler_continuous k : continuous (fun x : V => k *: x).
diff --git a/theories/normedtype_theory/normedtype.v b/theories/normedtype_theory/normedtype.v
index 35815ff15..40a00783b 100644
--- a/theories/normedtype_theory/normedtype.v
+++ b/theories/normedtype_theory/normedtype.v
@@ -1,7 +1,7 @@
 (* mathcomp analysis (c) 2025 Inria and AIST. License: CeCILL-C.              *)
 From mathcomp Require Export num_normedtype.
-From mathcomp Require Export ereal_normedtype.
 From mathcomp Require Export pseudometric_normed_Zmodule.
+From mathcomp Require Export ereal_normedtype.
 From mathcomp Require Export normed_module.
 From mathcomp Require Export matrix_normedtype.
 From mathcomp Require Export complete_normed_module.
diff --git a/theories/normedtype_theory/num_normedtype.v b/theories/normedtype_theory/num_normedtype.v
index 38b10494c..e5f1cf5a5 100644
--- a/theories/normedtype_theory/num_normedtype.v
+++ b/theories/normedtype_theory/num_normedtype.v
@@ -5,7 +5,7 @@ From mathcomp Require Import interval interval_inference archimedean rat.
 From mathcomp Require Import unstable.
 From mathcomp Require Import mathcomp_extra boolp classical_sets functions.
 From mathcomp Require Import cardinality set_interval reals real_interval.
-From mathcomp Require Import topology prodnormedzmodule tvs.
+From mathcomp Require Import topology prodnormedzmodule.
 
 (**md**************************************************************************)
 (* # Preliminaries for norm-related notions                                   *)
diff --git a/theories/normedtype_theory/pseudometric_normed_Zmodule.v b/theories/normedtype_theory/pseudometric_normed_Zmodule.v
index dd4ed670d..ab179df0b 100644
--- a/theories/normedtype_theory/pseudometric_normed_Zmodule.v
+++ b/theories/normedtype_theory/pseudometric_normed_Zmodule.v
@@ -4,7 +4,7 @@ From mathcomp Require Import all_ssreflect_compat finmap ssralg ssrnum ssrint.
 From mathcomp Require Import interval interval_inference archimedean.
 From mathcomp Require Import boolp classical_sets functions cardinality.
 From mathcomp Require Import set_interval ereal reals topology.
-From mathcomp Require Import prodnormedzmodule tvs num_normedtype.
+From mathcomp Require Import prodnormedzmodule num_normedtype.
 
 (**md**************************************************************************)
 (* # Normed topological abelian groups                                        *)
@@ -124,6 +124,15 @@ Unshelve. all: by end_near. Qed.
 
 End at_left_right_topologicalType.
 
+HB.structure Definition NbhsNmodule := {M of Nbhs M & GRing.Nmodule M}.
+HB.structure Definition NbhsZmodule := {M of Nbhs M & GRing.Zmodule M}.
+HB.structure Definition PreTopologicalNmodule :=
+  {M of Topological M & GRing.Nmodule M}.
+HB.structure Definition PreTopologicalZmodule :=
+  {M of Topological M & GRing.Zmodule M}.
+HB.structure Definition PreUniformNmodule := {M of Uniform M & GRing.Nmodule M}.
+HB.structure Definition PreUniformZmodule := {M of Uniform M & GRing.Zmodule M}.
+
 HB.mixin Record NormedZmod_PseudoMetric_eq (R : numDomainType) T
     & Num.NormedZmodule R T & PseudoPointedMetric R T := {
   pseudo_metric_ball_norm : ball = ball_ (fun x : T => `| x |)
diff --git a/theories/tvs.v b/theories/normedtype_theory/tvs.v
similarity index 86%
rename from theories/tvs.v
rename to theories/normedtype_theory/tvs.v
index bd6f428b1..09c6de58a 100644
--- a/theories/tvs.v
+++ b/theories/normedtype_theory/tvs.v
@@ -3,7 +3,8 @@ From HB Require Import structures.
 From mathcomp Require Import all_ssreflect_compat ssralg ssrnum vector.
 From mathcomp Require Import interval_inference.
 From mathcomp Require Import boolp classical_sets functions cardinality.
-From mathcomp Require Import set_interval reals topology.
+From mathcomp Require Import set_interval reals topology num_normedtype.
+From mathcomp Require Import pseudometric_normed_Zmodule.
 
 (**md**************************************************************************)
 (* # Topological vector spaces                                                *)
@@ -73,14 +74,9 @@ Local Open Scope ring_scope.
 (* HB.structure Definition FilteredZmodule := {M of Filtered M M & GRing.Zmodule M}. *)
 (* HB.structure Definition FilteredLmodule (K : numDomainType) := *)
 (*   {M of Filtered M M & GRing.Lmodule K M}. *)
-HB.structure Definition NbhsNmodule := {M of Nbhs M & GRing.Nmodule M}.
-HB.structure Definition NbhsZmodule := {M of Nbhs M & GRing.Zmodule M}.
 HB.structure Definition NbhsLmodule (K : numDomainType) :=
   {M of Nbhs M & GRing.Lmodule K M}.
 
-HB.structure Definition PreTopologicalNmodule :=
-  {M of Topological M & GRing.Nmodule M}.
-
 HB.mixin Record PreTopologicalNmodule_isTopologicalNmodule M
     & PreTopologicalNmodule M := {
   add_continuous : continuous (fun x : M * M => x.1 + x.2) ;
@@ -89,9 +85,6 @@ HB.mixin Record PreTopologicalNmodule_isTopologicalNmodule M
 HB.structure Definition TopologicalNmodule :=
   {M of PreTopologicalNmodule M & PreTopologicalNmodule_isTopologicalNmodule M}.
 
-HB.structure Definition PreTopologicalZmodule :=
-  {M of Topological M & GRing.Zmodule M}.
-
 HB.mixin Record TopologicalNmodule_isTopologicalZmodule M
     & Topological M & GRing.Zmodule M := {
   opp_continuous : continuous (-%R : M -> M) ;
@@ -191,8 +184,6 @@ HB.instance Definition _ :=
 
 HB.end.
 
-HB.structure Definition PreUniformNmodule := {M of Uniform M & GRing.Nmodule M}.
-
 HB.mixin Record PreUniformNmodule_isUniformNmodule M & PreUniformNmodule M := {
   add_unif_continuous : unif_continuous (fun x : M * M => x.1 + x.2)
 }.
@@ -200,8 +191,6 @@ HB.mixin Record PreUniformNmodule_isUniformNmodule M & PreUniformNmodule M := {
 HB.structure Definition UniformNmodule :=
   {M of PreUniformNmodule M & PreUniformNmodule_isUniformNmodule M}.
 
-HB.structure Definition PreUniformZmodule := {M of Uniform M & GRing.Zmodule M}.
-
 HB.mixin Record UniformNmodule_isUniformZmodule M
     & Uniform M & GRing.Zmodule M := {
   opp_unif_continuous : unif_continuous (-%R : M -> M)
@@ -503,84 +492,44 @@ End Tvs_numField.
 Section standard_topology.
 Variable R : numFieldType.
 
-Local Lemma standard_add_continuous : continuous (fun x : R^o * R^o => x.1 + x.2).
+(** NB: we have almost the same proof in `pseudometric_normed_Zmodule.v` *)
+Let standard_add_continuous : continuous (fun x : R^o * R^o => x.1 + x.2).
 Proof.
-(* NB(rei): almost the same code as in normedtype.v *)
-move=> [/= x y]; apply/cvg_ballP => e e0 /=.
-exists (ball x (e / 2), ball y (e / 2)) => /=.
-  by split; apply: nbhsx_ballx; rewrite divr_gt0.
-rewrite /ball /ball_/= => xy /= [nx ny].
-by rewrite opprD addrACA (le_lt_trans (ler_normD _ _)) // (splitr e) ltrD.
-Qed.
+move=> [/= x y]; apply/cvgrPdist_lt=> _/posnumP[e]; near=> a b => /=.
+by rewrite opprD addrACA normm_lt_split.
+Unshelve. all: by end_near. Qed.
 
-Local Lemma standard_scale_continuous : continuous (fun z : R^o * R^o => z.1 *: z.2).
+Let standard_scale_continuous : continuous (fun z : R^o * R^o => z.1 *: z.2).
 Proof.
-(* NB: This lemma is proved once again in normedtype, in a shorter way with much more machinery *)
-(*     To be rewritten once normedtype is split and tvs can depend on these lemmas *)
-move=> [k x]; apply/cvg_ballP => e le0 /=.
-pose M : R := maxr (`|e| + 1) (maxr `|k| (`|x| + `|x| + 2^-1 + 1)).
-have M0l : 0 < `|e| + 1 by rewrite ltr_wpDl.
-have M0r : 0 < maxr `|k| (`|x| + `|x| + 2^-1 + 1).
-  rewrite comparable_lt_max; last exact/real_comparable.
-  by rewrite normr_gt0; case: eqP => /=.
-have Me : `|e| < M.
-  rewrite (@lt_le_trans _ _ (`|e| + 1)) ?ltrDl//.
-  have /= -> := num_le_max (`|e| + 1) (PosNum M0l) (PosNum M0r).
-  by rewrite lexx.
-have M0 : 0 < M by apply: le_lt_trans Me.
-pose r := `|e| / 2 / M.
-exists (ball k r, ball x r) => [|[z1 z2] [k1r k2r]].
-  by split; exists r; rewrite //= ?divr_gt0 //= normr_gt0 gt_eqF.
-have := @ball_split _ _ (k * z2)  (k * x)  (z1 * z2) `|e|.
-rewrite /ball/= real_lter_normr ?gtr0_real//.
-rewrite (_ : _ < - e = false) ?orbF; last by rewrite ltr_nnorml// oppr_le0 ltW.
-apply.
-  rewrite -mulrBr normrM (@le_lt_trans _ _ (M * `|x - z2|)) ?ler_wpM2r//.
-    have /= -> := num_le_max `|k| (PosNum M0l) (PosNum M0r).
-    by apply/orP; right; rewrite /maxr; case: ifPn => // /ltW.
-  by rewrite -ltr_pdivlMl ?(lt_le_trans k2r)// mulrC.
-rewrite -mulrBl normrM (@le_lt_trans _ _ (`|k - z1| * M)) ?ler_wpM2l//.
-  rewrite (@le_trans _ _ (`|z2| + `|x|))// ?lerDl ?normr_ge0//.
-  have z2xe : `|z2| <= `|x| + r.
-    by rewrite -lerBlDl -(normrN x) (le_trans (lerB_normD _ _))// distrC ltW.
-  rewrite (@le_trans _ _ (`|x| + r + `|x|)) ?lerD// addrC.
-  rewrite [leRHS](_ : _ = M^-1 * (M *  M)); last first.
-    by rewrite mulKf// gt_eqF.
-  rewrite [leLHS](_ : _ = M^-1 * (M * (`|x| + `|x|) + `|e| / 2)); last first.
-    by rewrite mulrDr mulKf ?gt_eqF// mulrC addrA.
-  rewrite ler_wpM2l// ?invr_ge0// ?ltW// -ltrBrDl -mulrBr ltr_pM// ltrBrDl//.
-  rewrite (@lt_le_trans _ _ (`|x| + `|x| + 2^-1 + 1)) //.
-    by rewrite ltrDl ltr01.
-  rewrite (num_le_max _ (PosNum M0l) (PosNum M0r))//=.
-  apply/orP; right; have [->|k0] := eqVneq k 0.
-    by rewrite normr0 comparable_le_max ?real_comparable// lexx orbT.
-  have nk0 : 0 < `|k| by rewrite normr_gt0.
-  have xx21 : 0 < `|x| + `|x| + 2^-1 + 1%R by rewrite addr_gt0.
-  by rewrite (num_le_max _ (PosNum nk0) (PosNum xx21))// lexx orbT.
-by rewrite -ltr_pdivlMr ?(lt_le_trans k1r) ?normr_gt0.
-Qed.
+move=> [/= k x]; apply/cvgrPdist_lt => _/posnumP[e]; near +oo_R => M.
+near=> l z => /=; have M0 : 0 < M by [].
+rewrite (@distm_lt_split _ _ (k *: z)) // -?(scalerBr, scalerBl) normrM.
+  rewrite (@le_lt_trans _ _ (M * `|x - z|)) ?ler_wpM2r -?ltr_pdivlMl//.
+  by near: z; apply: cvgr_dist_lt; rewrite // mulr_gt0 ?invr_gt0.
+rewrite (@le_lt_trans _ _ (`|k - l| * M)) ?ler_wpM2l -?ltr_pdivlMr//.
+  by near: z; near: M; exact: (@cvg_bounded _ R^o _ _ _ _ _ (@cvg_refl _ _)).
+by near: l; apply: cvgr_dist_lt; rewrite // divr_gt0.
+Unshelve. all: by end_near. Qed.
 
-Local Lemma standard_locally_convex :
+Let standard_locally_convex :
   exists2 B : set (set R^o), (forall b, b \in B -> convex b) & basis B.
 Proof.
-(* NB(rei): almost the same code as in normedtype.v *)
 exists [set B | exists x r, B = ball x r].
   move=> b; rewrite inE /= => [[x]] [r] -> z y l.
-  rewrite !inE /ball /= => zx yx l0; rewrite -subr_gt0 => l1.
-  have -> : x = l *: x + (1 - l) *: x by rewrite scalerBl addrC subrK scale1r.
+  rewrite !inE -!ball_normE /= => zx yx l0; rewrite -subr_gt0 => l1.
+  have -> : x = l *: x + (1 - l) *: x by rewrite addrC scalerBl subrK scale1r.
   rewrite [X in `|X|](_ : _ = l *: (x - z) + (1 - l) *: (x - y)); last first.
-    by rewrite opprD addrACA -mulrBr -mulrBr.
+    by rewrite opprD addrACA -scalerBr -scalerBr.
   rewrite (@le_lt_trans _ _ (`|l| * `|x - z| + `|1 - l| * `|x - y|))//.
     by rewrite -!normrM ler_normD.
   rewrite (@lt_le_trans _ _ (`|l| * r + `|1 - l| * r ))//.
-    by rewrite ltr_leD// -lter_pdivlMl ?mulKf// ?normrE ?gt_eqF// ltW.
-  suff-> : `|1 - l| = 1 - `|l| by rewrite -mulrDl addrC subrK mul1r.
-  by move: l0 l1 => /ltW/normr_idP -> /ltW/normr_idP ->.
+    by rewrite ltr_leD// lter_pM2l// ?normrE ?gt_eqF// ltW.
+  by rewrite !gtr0_norm// -mulrDl addrC subrK mul1r.
 split.
   move=> B [x] [r] ->.
-  rewrite openE/= /ball/= /interior => y /= bxy; rewrite -nbhs_ballE.
+  rewrite openE/= -ball_normE/= /interior => y /= bxy; rewrite -nbhs_ballE.
   exists (r - `|x - y|) => /=; first by rewrite subr_gt0.
-  move=> z; rewrite /ball/= ltrBrDr.
+  move=> z; rewrite -ball_normE/= ltrBrDr.
   by apply: le_lt_trans; rewrite [in leRHS]addrC ler_distD.
 move=> x B; rewrite -nbhs_ballE/= => -[r] r0 Bxr /=.
 by exists (ball x r) => //; split; [exists x, r|exact: ballxx].