Decomposing global isotopies into local ones

Question

I'm looking for a reference for the following basic-looking statement:

Let $X$ be a smooth manifold covered by open sets $U_1$ and $U_2$. Let $f:X \rightarrow X$ be a map isotopic to identity via an isotopy with compact support. Then one can represent $f$ as a composition $f_1\circ f_2 \circ \cdots f_n$ so that each $f_i$ is isotopic to identity via an isotopy whose support is compact and contained in $U_1$ or $U_2$.

Answer 1

This is called "The Fragmentation Lemma", Lemma 2.1.8 in Banyaga's book "The Structure of Classical Diffeomorphism Groups".