SubClass BinOp (A : Setoid) := Map2 A A A.Infix "+_M" := ApMop (at level 40, left associativity).Definition Monoid_ass := forall x y z : A, x +_M (y +_M z) =_S x +_M y +_M z.Definition Monoid_idl := forall x : A, e +_M x =_S x.Definition Monoid_idr := forall x : A, x =_S x +_M e.Structure Monoid : Type := 
  {Mcarrier :> Setoid;
   Mop : BinOp Mcarrier;
   Munit : Mcarrier;
   Prf_monoid_ass : Monoid_ass Mop;
   Prf_monoid_idl : Monoid_idl Mop Munit;
   Prf_monoid_idr : Monoid_idr Mop Munit}.Structure Monoid : Type := 
  {Mcarrier :> Setoid;
   Mop : BinOp Mcarrier;
   Munit : Mcarrier;
   Prf_monoid_ass : Monoid_ass Mop;
   Prf_monoid_idl : Monoid_idl Mop Munit;
   Prf_monoid_idr : Monoid_idr Mop Munit}.Structure Monoid : Type := 
  {Mcarrier :> Setoid;
   Mop : BinOp Mcarrier;
   Munit : Mcarrier;
   Prf_monoid_ass : Monoid_ass Mop;
   Prf_monoid_idl : Monoid_idl Mop Munit;
   Prf_monoid_idr : Monoid_idr Mop Munit}.Structure Monoid : Type := 
  {Mcarrier :> Setoid;
   Mop : BinOp Mcarrier;
   Munit : Mcarrier;
   Prf_monoid_ass : Monoid_ass Mop;
   Prf_monoid_idl : Monoid_idl Mop Munit;
   Prf_monoid_idr : Monoid_idr Mop Munit}.Structure Monoid : Type := 
  {Mcarrier :> Setoid;
   Mop : BinOp Mcarrier;
   Munit : Mcarrier;
   Prf_monoid_ass : Monoid_ass Mop;
   Prf_monoid_idl : Monoid_idl Mop Munit;
   Prf_monoid_idr : Monoid_idr Mop Munit}.Structure Monoid : Type := 
  {Mcarrier :> Setoid;
   Mop : BinOp Mcarrier;
   Munit : Mcarrier;
   Prf_monoid_ass : Monoid_ass Mop;
   Prf_monoid_idl : Monoid_idl Mop Munit;
   Prf_monoid_idr : Monoid_idr Mop Munit}.Structure Monoid : Type := 
  {Mcarrier :> Setoid;
   Mop : BinOp Mcarrier;
   Munit : Mcarrier;
   Prf_monoid_ass : Monoid_ass Mop;
   Prf_monoid_idl : Monoid_idl Mop Munit;
   Prf_monoid_idr : Monoid_idr Mop Munit}.Definition ApMop (m : Monoid) (x y : m) := Mop m x y.

Lemma Midl1 : forall (M : Monoid) (x : M), x =_S Munit M +_M x.