# 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 "law of DMRS", 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

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

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

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.