Record mixin_of (K : AbelianGroup) := Mixin {
  mult : K -> K -> K ;
  one : K ;
  ax1 : forall x y z, mult x (mult y z) = mult (mult x y) z ;
  ax2 : forall x, mult x one = x ;
  ax3 : forall x, mult one x = x ;
  ax4 : forall x y z, mult (plus x y) z = plus (mult x z) (mult y z) ;
  ax5 : forall x y z, mult x (plus y z) = plus (mult x y) (mult x z)
}.Record mixin_of (K : AbelianGroup) := Mixin {
  mult : K -> K -> K ;
  one : K ;
  ax1 : forall x y z, mult x (mult y z) = mult (mult x y) z ;
  ax2 : forall x, mult x one = x ;
  ax3 : forall x, mult one x = x ;
  ax4 : forall x y z, mult (plus x y) z = plus (mult x z) (mult y z) ;
  ax5 : forall x y z, mult x (plus y z) = plus (mult x y) (mult x z)
}.Record mixin_of (K : AbelianGroup) := Mixin {
  mult : K -> K -> K ;
  one : K ;
  ax1 : forall x y z, mult x (mult y z) = mult (mult x y) z ;
  ax2 : forall x, mult x one = x ;
  ax3 : forall x, mult one x = x ;
  ax4 : forall x y z, mult (plus x y) z = plus (mult x z) (mult y z) ;
  ax5 : forall x y z, mult x (plus y z) = plus (mult x y) (mult x z)
}.Record mixin_of (K : AbelianGroup) := Mixin {
  mult : K -> K -> K ;
  one : K ;
  ax1 : forall x y z, mult x (mult y z) = mult (mult x y) z ;
  ax2 : forall x, mult x one = x ;
  ax3 : forall x, mult one x = x ;
  ax4 : forall x y z, mult (plus x y) z = plus (mult x z) (mult y z) ;
  ax5 : forall x y z, mult x (plus y z) = plus (mult x y) (mult x z)
}.Record mixin_of (K : AbelianGroup) := Mixin {
  mult : K -> K -> K ;
  one : K ;
  ax1 : forall x y z, mult x (mult y z) = mult (mult x y) z ;
  ax2 : forall x, mult x one = x ;
  ax3 : forall x, mult one x = x ;
  ax4 : forall x y z, mult (plus x y) z = plus (mult x z) (mult y z) ;
  ax5 : forall x y z, mult x (plus y z) = plus (mult x y) (mult x z)
}.Record mixin_of (K : AbelianGroup) := Mixin {
  mult : K -> K -> K ;
  one : K ;
  ax1 : forall x y z, mult x (mult y z) = mult (mult x y) z ;
  ax2 : forall x, mult x one = x ;
  ax3 : forall x, mult one x = x ;
  ax4 : forall x y z, mult (plus x y) z = plus (mult x z) (mult y z) ;
  ax5 : forall x y z, mult x (plus y z) = plus (mult x y) (mult x z)
}.Record mixin_of (K : AbelianGroup) := Mixin {
  mult : K -> K -> K ;
  one : K ;
  ax1 : forall x y z, mult x (mult y z) = mult (mult x y) z ;
  ax2 : forall x, mult x one = x ;
  ax3 : forall x, mult one x = x ;
  ax4 : forall x y z, mult (plus x y) z = plus (mult x z) (mult y z) ;
  ax5 : forall x y z, mult x (plus y z) = plus (mult x y) (mult x z)
}.Record mixin_of (K : AbelianGroup) := Mixin {
  mult : K -> K -> K ;
  one : K ;
  ax1 : forall x y z, mult x (mult y z) = mult (mult x y) z ;
  ax2 : forall x, mult x one = x ;
  ax3 : forall x, mult one x = x ;
  ax4 : forall x y z, mult (plus x y) z = plus (mult x z) (mult y z) ;
  ax5 : forall x y z, mult x (plus y z) = plus (mult x y) (mult x z)
}.Record mixin_of (K : AbelianGroup) := Mixin {
  mult : K -> K -> K ;
  one : K ;
  ax1 : forall x y z, mult x (mult y z) = mult (mult x y) z ;
  ax2 : forall x, mult x one = x ;
  ax3 : forall x, mult one x = x ;
  ax4 : forall x y z, mult (plus x y) z = plus (mult x z) (mult y z) ;
  ax5 : forall x y z, mult x (plus y z) = plus (mult x y) (mult x z)
}.Record class_of (K : Type) := Class {
  base : AbelianGroup.class_of K ;
  mixin : mixin_of (AbelianGroup.Pack _ base K)
}.Record class_of (K : Type) := Class {
  base : AbelianGroup.class_of K ;
  mixin : mixin_of (AbelianGroup.Pack _ base K)
}.Record class_of (K : Type) := Class {
  base : AbelianGroup.class_of K ;
  mixin : mixin_of (AbelianGroup.Pack _ base K)
}.Record class_of (K : Type) := Class {
  base : AbelianGroup.class_of K ;
  mixin : mixin_of (AbelianGroup.Pack _ base K)
}.Structure type := Pack { sort; _ : class_of sort ; _ : Type }.Structure type := Pack { sort; _ : class_of sort ; _ : Type }.Structure type := Pack { sort; _ : class_of sort ; _ : Type }.Notation xclass := (class : class_of xT).Definition AbelianGroup := AbelianGroup.Pack cT xclass xT.Notation Ring := type.Definition mult : K -> K -> K := Ring.mult _ (Ring.class K).Definition one : K := Ring.one _ (Ring.class K).

Lemma sum_n_m_mult_r : forall (a : K) (u : nat -> K) (n m : nat), sum_n_m (fun k => mult (u k) a) n m = mult (sum_n_m u n m) a.