  We show how the existence of three objects, $\Omega_{\rm trap}$, ${\bf W}$,
and $C$, for a continuous piecewise-linear map $f$ on $\mathbb{R}^N$, implies
that $f$ has a topological attractor with a positive Lyapunov exponent. First,
$\Omega_{\rm trap} \subset \mathbb{R}^N$ is trapping region for $f$. Second,
${\bf W}$ is a finite set of words that encodes the forward orbits of all
points in $\Omega_{\rm trap}$. Finally, $C \subset T \mathbb{R}^N$ is an
invariant expanding cone for derivatives of compositions of $f$ formed by the
words in ${\bf W}$. We develop an algorithm that identifies these objects for
two-dimensional homeomorphisms comprised of two affine pieces. The main effort
is in the explicit construction of $\Omega_{\rm trap}$ and $C$. Their existence
is equated to a set of computable conditions in a general way. This results in
a computer-assisted proof of chaos throughout a relatively large regime of
parameter space. We also observe how the failure of $C$ to be expanding can
coincide with a bifurcation of $f$. Lyapunov exponents are evaluated using
one-sided directional derivatives so that forward orbits that intersect a
switching manifold (where $f$ is not differentiable) can be included in the
analysis.
