Prove that in presence of many goods, the law of DMRS implies that at least one good must be a net substitute for any other good I've been reading a book where the author claims that "if there are two goods they are always net substitutes. Along an indifference curve an increase in price of good 2 leads to less of good 2 and more of good 1 being consumed when utility is maximized. When there are more than two goods, the law of diminishing marginal rate of substitution implies that at least one good must be a net substitute for any other good." 

The first part can be proved easily using graphs, it's the second that I couldn't prove mathematically.

Source of the quotation has been attached here, Pg-104.
I don't know what is the definition of the &quot;law of DMRS&quot;, but I think what the author meant is summarized in these lines of MWG:

Proposition 3.G.2: suppose that $u(\cdot)$ is a continuous utility function representing a locally nonsatiated and strictly convex preference relation $\succsim$ defined on $\mathbb{R}^L_+$. Suppose also that $h(\cdot, u)$ [the hicksian demand function] is continuously differentiable at $(p, u)$, and denote its $L\times L$ derivative matrix [with respect to each $p$] by $D_p h(p, u)$. Then

$D_p h(p, u)$ is a negative semidefinite matrix.

$D_p h(p, u) p = 0$



Well, 1 implies that the elements of the main diagonal of the matrix are all non positive, i.e. $\frac{\partial h_l (p, u)}{\partial p_l} \leq 0$ for each $l$.
We say that two goods are substitutes if $\frac{\partial h_l (p, u)}{\partial p_k} \geq 0$ (a positive variation in the price of $k$ implies an increase in the compensated demand of $l$).
Part 2 of the proposition says that $D_p h(p, u) p = 0$. But as $\frac{\partial h_l (p, u)}{\partial p_l} \leq 0$, for every $l$ we must have a good $m$ such that $\frac{\partial h_l (p, u)}{\partial p_m} \geq 0$, which means exactly that every good has a substitute.