and lemma 3.2. $\Box$

With lemmas (3.2) and (3.3) at hand, we are now ready for starting the proof of (2.10).

[**Proof of (2.10).**]{} If one replaces the $t$ derivative by the $\tau$ derivative in accordance with (3.20), the trilinear equation (2.10) can be rewritten in the form $$f^*P_3=f_\tau^*P_3^\prime, \eqno(3.25a)$$ with the bilinear forms $P_3$ and $P_3^\prime$ defined respectively by $$P_3=g_{x\tau}f-(f_x-{\rm i}\kappa f)g_\tau-{\rm i\over \kappa}(g_xf-gf_x), \eqno(3.25b)$$ $$P_3^\prime=g_xf-gf_x+{\rm i}\kappa fg. \eqno(3.25c)$$ The trilinear equation (3.25) is proved as follows. Substituting (3.8), (3.10), (3.13), (3.23) and (3.24) into (3.25b) and applying Jacobi’s identity to terms multiplied by $|D|$, $P_3$ is simplified considerably. After some elementary calculations, one finds that $$P_3=\kappa |D({\bf z}^*;{\bf z}_\tau)|\Bigl\{|D|+{1\over\rho^2}|D({\bf z}_t^*;{\bf z})|-{{\rm i}\over \kappa\rho^2}|D({\bf z}_t^*;{\bf z}_x)|\Bigr\}. \eqno(3.26a)$$ Performing the similar calculation for $P_3^\prime$, one obtains $$P_3^\prime={\rm i}\kappa\Bigl\{|D|-|D({\bf z}^*;{\bf z})|\Bigr\}\Bigl\{|D|+{1\over\rho^2}|D({\bf z}_t^*;{\bf z})|-{{\rm i}\over \kappa\rho^2}|D({\bf z}_t^*;{\bf z}_x)|\Bigr\}. \eqno(3.26b)$$ Taking into account the formulas (3.14) and (3.22), the expressions (3.26a) and (3.26b) yield (3.25). The trilinear equation (3.25) coupled with lemma 3.3 now completes the proof of the trilinear equation (2.10). $\Box$

[*3.5. Dark $N$-soliton solution of the derivative NLS equation*]{}

In accordance with the fact that the FL equation is the first negative flow of the Lax hierarchy of the derivative 