Set of subgradients at the intersection of k hyperplanes

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How to find the set of subgradients at the $(n-k)$-dimensional "edge" created by the intersection of $k$ hyperplanes $(k \lt n)$, say:

$a_{11}x_1+...+a_{1n}x_n=b_1,\\ a_{21}x_1+...+a_{2n}x_n=b_2,\\...\\ a_{k1}x_1+...+a_{kn}x_n=b_k;\\ a_{ij} \in \Bbb R$

For two hyperplanes (take the first and second hyperplanes above, e.g.) I think it's everything in between the subgradients of the two hyperplanes, i.e. the set

$S=[a_{11},a_{21}] \times [a_{12},a_{22}] \times...\times[a_{1n},a_{2n}]$

(Assuming $a_{1j} \leq a_{2j} \forall j=1(1)n$)

Is that correct. Thank you for your help in advance.