g}\prec'_1 ({\mathbf d}, a_1)$.

Given two pencils $H_1(s)\in {\mathbb F}[s]^{(n+p) \times (n+m)}$ and $H(s)\in {\mathbb F}[s]^{ (n+p+x+y)\times (n+m)}$, of $\rank(H_1(s))=n$ and $\rank(H(s))=n+x$, in [@DoSt19 Theorem 4.3] (see also [@Do13 Theorem 2]), necessary and sufficient conditions are given for the existence of a pencil $Y(s)\in {\mathbb F}[s]^{ (x+y)\times (n+m)}$ such that $H(s){\ensuremath{\stackrel{s.e.}{\sim}}}\begin{bmatrix}
  H_1(s)\\Y(s)\end{bmatrix}$. The two following lemmas are particular cases of [@DoSt19 Theorem 4.3] for $x+y=1$. First, we state the result when $x=0$, $y=1$.

\[lemmaDox0\] Let $H_1(s)\in {\mathbb F}[s]^{(n+p) \times (n+m)}$, $H(s)\in {\mathbb F}[s]^{ (n+p+1)\times (n+m)}$ be matrix pencils of $\rank(H_1(s))=\rank(H(s))=n$.

Let $\pi^1_1(s, t)\mid \dots \mid \pi^1_{n}(s, t)$, $g_1 \geq \dots \geq g_{m}\geq 0$, and $w_1 \geq \dots \geq w_{\theta}> 0=w_{\theta+1} \geq \dots \geq w_{p}$ be the homogeneous invariant factors, the column and the row minimal indices of $H_1(s)$, respectively, and let $\pi_1(s, t)\mid \dots \mid \pi_{n}(s, t)$, $c_1 \geq \dots \geq c_{m}\geq 0$, and $u_1 \geq \dots \geq u_{\bar \theta}> 0=u_{\bar \theta+1} \geq \dots \geq u_{p+1}$ be the homogeneous invariant factors, the column and the row minimal indices indices of $H(s)$, respectively.

Let ${\mathbf g}=(g_1, \dots, g_m)$, ${\mathbf w}=(w_1, \dots, w_p)$, ${\mathbf c}=(c_1, \dots, c_m)$, and ${\mathbf u}=(u_1, \dots, u_{p+1})$.

There exists a pencil $h(s)\in{\mathbb F}[s]^{1 \times (n+m)}$ such that 