Improve further

Signed-off-by: Marcello Seri <[email protected]>

6-differentiaforms.tex

@@ -919,7 +919,8 @@ \section{De Rham cohomology and Poincar\'e lemma}
 Given any continuous mapping $G \in C^0(M,N)$, there exists $F \in C^\infty(M,N)$ which is homotopic to $G$. Moreover, if $G$ is smooth\footnote{Note that a function $f : M \to N$ is defined to be smooth on a subset $A \subset M$ if there is some smooth function $g: U \to N$, defined on an open $U\supset A$ such that $g = f$ on $A$.} on a closed subset $U\subset M$, then one can choose $F$ so that $F=G$ on $U$.
 \end{theorem}
 %
-In particular, if two smooth maps are homotopic then they are also smoothly homotopic, in the sense that the map $K$ is smooth.
+In particular, if two smooth maps are homotopic then they are also smoothly homotopic: we can assume the map $K$ to be smooth.
+
 To see this, continously extend $K$ to a mapping $\widetilde K : \R \times M \to N$ by defining
 \begin{align}
   \widetilde K(t, p) = K(0, p) \mbox{ if } t \leq 0,