关于论文《Topologically Trivial Legendrian Knot》中Lemma 1.3关键假设的理解疑问

我最近在啃Eliashberg和Fraser的论文《Topologically Trivial Legendrian Knot》，这篇论文在Archive上能找到电子版。目前大部分内容我都捋顺了，唯独卡在了Lemma 1.3上——这个引理要证明的是两个微分同胚的Legendrian树必然是同痕的。

在证明过程里有这么一段表述：

By the assumption there is a (germ of a) diffeomorphism $h \colon \Pi \to \Pi$ which sends $\tilde{L}'$ into $\tilde{L}$. One can assume that $h$ is fixed at the points of the projection of $L'_1$, preserves the area form $dx\wedge dy$, and is isotopic to the identity in the space of diffeomorphisms with these properties.

我知道这么提问可能不太符合规范，但我实在找不到别的头绪搞懂这段假设的合理性了……

备注：内容来源于stack exchange，提问作者Mirko