Translations preserve fast finite-energy planes

Let $S = H^{-1}(1) \subset V_1 = \mathbb{R}^4$ be a strictly star-shaped level with a compatible complex structure $J$ on $\xi$. If $u : \mathbb{C} \to V_1 \setminus \{0\}$ is a fast finite-energy plane (Hryniewicz, Definition 2.14), then so is its translate $c \cdot u = e^{2c} u$ for every $c \in \mathbb{R}$.

This is used to transport statements about fast planes along translations in the $\mathbb{R}$-direction of the symplectization.

Formalization Note: shiftA c u is the map $z \mapsto e^{2c} \bullet u(z)$, which is how the translation $(a + c, w)$ of $\tilde u = (a, w)$ appears in the model $V_1 \setminus \{0\} \cong \mathbb{R} \times S$.