Showing the operation of taking the Composition of Paths is Continuous Let $I=[0,1]$.  Let $X$ be a topological space, and denote by $C(I,X)$ the space of all paths in $X$, equipped with the compact-open topology.  We denote by $P_{x,y}(X)$ the fiber of $(x,y)$ along the map $C(I,X)\to X\times X$ defined by $\gamma \mapsto (\gamma(0),\gamma(1))$ equipped with the subspace topology.  
Now for $x,y,z\in X$ we define $$c_{x,y,z}: P_{y,z}(X)\times P_{x,y}(X) \rightarrow P_{x,z}(X)$$ by $$c_{x,y,z}(\beta,\alpha)(t)= \begin{cases} \alpha(2t) & t\in [0,1/2] \\ \beta(2t-1) & t\in [1/2,1] \end{cases}$$
Show that $c_{x,y,z}$ is continuous.
I have found proof of this done by a colleague of mine, but it is a rather involved proof, and I was wondering if there were a simpler approach to this.

Just take two paths $\alpha:x\to y$ and $\beta:y\to z$ and consider a subbasic neighborhood 
$$(K,W)=\{\gamma:I\to X\mid γ[K]\subseteq W\}$$ of $\beta\alpha$. Then $K\subseteq I$ corresponds to
$$K_1=\{2k\mid k\in K\cap[0,1/2]\},\qquad K_2=\{2k-1\mid k\in K\cap[1/2,1]\}$$
Both $K_i$ are compact subsets of $I$. Now $α\in(K_1,W)$ and $β\in(K_2,W)$ if and only if $βα$ is in $(K,W)$. So $(K_1,W)$ is neighborhood of $α$ and $(K_2,W)$ is a neighborhood of $β$, and $c_{x,y,z}[(K_1,W)\times(K_2,W)]\subseteq(K,W)$.
