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;

theorem
  for g1,g2 being Function of M./.R, N st g1 * nat_hom R = g2 * nat_hom R
  holds g1 = g2
proof
  let g1,g2 be Function of M./.R, N;
  assume A1: g1 * nat_hom R = g2 * nat_hom R;
  set Y = rng nat_hom R;
  rng nat_hom R = the carrier of M ./. R by FUNCT_2:def 3; then
  dom g1 = Y & dom g2 = Y by FUNCT_2:def 1;
  hence g1 = g2 by A1,FUNCT_1:86;
end;
