reserve i,j,k,n,m for Nat,
        X for set,
        b,s for bag of X,
        x for object;
reserve O for Ordinal,
        R for right_zeroed add-associative right_complementable
          right_unital distributive non trivial doubleLoopStr,
        p for Polynomial of O, R;
reserve O for Ordinal,
        R for right_zeroed add-associative right_complementable
             right_unital distributive non trivial doubleLoopStr,
        p for Polynomial of O, R;

theorem Th54:
   count_reps({},n) = EmptyBag n
proof
  set s = count_reps({},n), E= EmptyBag n;
A1: dom s = n = dom E by PARTFUN1:def 2;
  for x be object st x in dom s holds s.x=E.x
  proof
    let x be object such that
A2: x in dom s;
    x in Segm n by A2;
    then reconsider i=x as Nat;
    thus s.x = card ({}"{i+1}) by Def8,A2
    .= E.x;
  end;
  hence thesis by A1;
end;
