= Solution
For a steady <flow network>, let each origin-destination class $k$ have fixed demand $d_k$. Route flows $h_r\geq0$ satisfy $\sum_{r\in R_k}h_r=d_k$. The link flow is $y_j=\sum_rA_{jr}h_r$, and a link has continuous travel delay $\ell_j(y_j)$. Its route delay is $c_r(h)=\sum_jA_{jr}\ell_j(y_j)$.
A <Wardrop equilibrium> describes nonatomic traffic: each traveler is too small to alter aggregate delays by changing route. For each class there is a minimum delay $\lambda_k$ such that
$$
\boxed{h_r>0\Longrightarrow c_r(h)=\lambda_k,\qquad h_r=0\Longrightarrow c_r(h)\geq\lambda_k}.
$$
Thus all used routes of one class have equal minimum cost, and an unused route cannot offer a shorter trip. The condition concerns private travel time, rather than total network delay.
Equivalently, for every feasible route vector $\widetilde h$,
$$
\sum_r c_r(h)(\widetilde h_r-h_r)\geq0.
$$
Indeed, each used route has class minimum cost, so reallocating demand cannot reduce the cost evaluated at the original flows; conversely a positive flow on a non-minimum route gives an improving transfer. This <variational inequality> form remains meaningful even when route costs are not separable link functions.
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