I'm looking for a *reference* to the following elementary result (or to a generalization of it). 

> **Lemma.** Let $(H, +, \preceq)$ be an (additive) [partially ordered commutative monoid][6] such that 
$$x+z \prec y+z\quad \text{for all }\ x, y, z \in H \ \text{ with }\ x \prec y,
$$
and let $x_1, \ldots, x_n \in H^+$ and $\kappa_1, \ldots, \kappa_n \in \mathbf N$ such that $\kappa_1 x_1 + \cdots + \kappa_i x_i \prec x_{i+1}$ for every $i \in [\![1, n-1]\!]$, under the assumption that $H^+ := \{x \in H: 0_H \prec x\} \ne \emptyset$. If $y$ is an element in the [sumset][1] $\kappa_1 \{0_H, x_1\} + \cdots + \kappa_n\{0_H, x_n\}$,
then there is *uniquely* determined an $n$-tuple $(a_1, \ldots, a_n) \in [\![0, \kappa_1]\!] \times \cdots \times [\![0, \kappa_n]\!]$ for which $y = \sum_{i = 1}^n a_i x_i$.

Any pointer? The result popped up in the study of some [arithmetic invariants][2] (namely, set of distances and set of catenary degrees) of a certain class of non-cancellative BF-monoids. 

For the special case when the monoid under consideration is $(\mathbf N, +)$, I thought I would have found something along the same lines in the literature on the [knapsack problem][3] or the [subset sum problem][4], but I couldn't get to anything and resolved to ask here (after having [first tried at MSE][5]).

Incidentally, the lemma provides, by a simple counting argument, another proof of the existence and uniqueness of the base-$b$ representation of a non-negative integer (for any given base $b \ge 2$).


  [1]: https://en.wikipedia.org/wiki/Sumset
  [2]: http://www.jstor.org/stable/10.4169/amer.math.monthly.123.10.960
  [3]: https://en.wikipedia.org/wiki/Knapsack_problem
  [4]: https://en.wikipedia.org/wiki/Subset_sum_problem
  [5]: https://math.stackexchange.com/questions/2098720/
  [6]: https://en.wikipedia.org/wiki/Ordered_semigroup