reserve M,N for non empty multMagma,
  f for Function of M, N;
reserve M for multMagma;
reserve N,K for multSubmagma of M;
reserve M,N for non empty multMagma,
  A for Subset of M,
  f,g for Function of M,N,
  X for stable Subset of M,
  Y for stable Subset of N;
reserve X for set;
reserve x,y,Y for set;
reserve n,m,p for Nat;
reserve v,v1,v2,w,w1,w2 for Element of free_magma X;
reserve X,Y,Z for non empty set;
reserve M for non empty multMagma;
reserve M,N for non empty multMagma,
      f for Function of M, N,
      H for non empty multSubmagma of N,
      R for compatible Equivalence_Relation of M;
reserve f for Function of X,Y;
reserve g for Function of Y,Z;

theorem
  for M being unital non empty multMagma
  for R being compatible Equivalence_Relation of M
  holds 1_(M ./. R) = Class(R,1_M)
proof
  let M be unital non empty multMagma;
  let R be compatible Equivalence_Relation of M;
  reconsider E = Class(R,1_M) as Element of M ./. R by EQREL_1:def 3;
  for X being Element of M ./. R holds X * E = X & E * X = X by Lm1;
  hence thesis by GROUP_1:def 4;
end;
