Lemma

Let X, Y, Z be topological spaces and let $g: Y\rightarrow Z$ be a continuous mapping.

a) The mapping $f\mapsto g\circ f$ of $\mathscr{C}_c(X; Y)$ into $\mathscr{C}_c(X; Z)$ is continuous.

b) If the topological space Z is separated, the mapping $h\mapsto h\circ g$ of $\mathscr{C}_c(Z; X)$ into $\mathscr{C}_c(Y; X)$ is continuous.

Given a compact subset K of X, an open subset U of Z and a continuous mapping $f$ of X into Y, in order that one have $(g\circ f)(K)\subset U$, it is necessary and sufficient that one have $f(K)\subset \overset{-1}{g}(U)$. The first assertion therefore follows from the definition of the topology of compact convergence (TG, X, p. 26, def. 1).

Analogously, let K be a compact subset of Y and U an open subset of X. Since Z is supposed separated, the set $g(K)$ is compact (TG, I, p. 63, cor. 1). If $h$ is a mapping of Z into X, the condition $(h\circ g)(K)\subset U$ is nothing other than the condition $h(g(K))\subset U$, whence the second assertion.

Let us now prove proposition 9. By the lemma, the mapping $\widetilde{p}$ is continuous; by remark 1 of TG, X, p. 27, the mappings $e_E$ and $e_B$ are continuous. For every mapping $g\in \mathscr{C}_c(T; E)$, one has

so that the square diagram of the proposition is commutative.

Let $\varphi :\mathscr{C}_c(T; E)\rightarrow \mathscr{C}_c(T; B)\times_BE$ be the continuous mapping defined by $\varphi (g) = (p\circ g, g(t))$ for every $g\in \mathscr{C}_c(T; E)$. By proposition 3 of I, p. 125, it is bijective. In fact, for every pair $(f, x)\in \mathscr{C}_c(T; B)\times_BE,\varphi^{-1}(f, x)$ is the unique continuous lifting $g$ of $f$ to E such that $g(t) =x$. Since the space T is locally compact, the mapping $\psi : (\mathscr{C}_c(T; B)\times_BE)\times T\rightarrow B$ defined by $\psi ((f, x), u) =f(u)$ is continuous (TG, X, p. 28, cor. 1). By corollary 3 above, the mapping $\psi$ has a unique continuous lifting $\theta : (\mathscr{C}_c(T; B)\times_B$ $E)\times T\rightarrow E$ such that $\theta ((f, x), t) =x$ for $(f, x)\in \mathscr{C}_c(T; B)\times_BE$. Thus $\theta ((f, x), u) =\varphi^{-1}(f, x)(u)$ for $(f, x)\in \mathscr{C}_c(T; B)\times_BE$ and $u\in T$. By theorem 3 of TG, X, p. 28, the mapping $(f, x)\mapsto \theta ((f, x),\cdot )$ of $\mathscr{C}_c(T; B)\times_BE$ into $\mathscr{C}_c(T; E)$ is continuous, that is to say $\varphi^{-1}$ is continuous.

Thus, the mapping $\varphi$ is a homeomorphism of $\mathscr{C}_c(T; E)$ onto the fibred product $\mathscr{C}_c(T; B)\times_BE$, whence the proposition (I, p. 8, prop. 2).