First-Order Equation Operator on a Hilbert Triple

In the setting of Hilbert Triples: Standing Notation and Background, let $U\subseteq H$ be open in $H$, with $W=D(A)\cap U$ as in Hilbert Triples: Standing Notation and Background (v), let $\mathrm{Sym}(V)$ be as in Hilbert Triples: Standing Notation and Background (vi), and let $F$ be a second-order equation operator on $U$ relative to $(H,V,A)$.

(i) The operator $F$ is first order if

$$F(x,r,p,X)=F(x,r,p,X')$$

for every $x\in W$, every $r\in\mathbb{R}$, every $p\in H$ and all $X,X'\in\mathrm{Sym}(V)$.