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Consider a finite set of positive variables $P = \{x_1,x_2,\ldots, x_n \}$, and a non-strict order on the expressions $\Sigma_{x_i \subseteq P}x_i$. For example:

$$P = \{x_1,x_2,x_3\}$$

$$x_1 + x_2 + x_3 > x_1 + x_2 > x_2 + x_3 > x_1 = x_2 > x_3$$

Can we claim that if there is a solution in which $\forall i,x_i \in \mathbb{R}^+$, there must be a solution in which $\forall i, x_i \in \mathbb{N}^+$?

Thanks!