  In this paper we provide a proof of the multiplicative kinematic description
of crystal elastoplasticity in the setting of large deformations, i.e.
$\mathbf{F}=\mathbf{F}^e\mathbf{F}^p$, for a two dimensional single crystal.
The proof starts by considering a general configuration at the mesoscopic
scale, where the dislocations are discrete line defects (points in the
two-dimensional description used here) and the displacement field can be
considered continuous everywhere in the domain except at the slip surfaces,
over which there is a displacement jump. At such scale, as previously shown by
two of the authors, there exists unique physically-based definitions of the
total deformation tensor $\mathbf{F}$ and the elastic and plastic tensors
$\mathbf{F}^e$ and $\mathbf{F}^p$ that do not require the consideration of any
non-realizable intermediate configuration and do not assume any a priori
relation between them of the form $\mathbf{F}=\mathbf{F}^e\mathbf{F}^p$. This
mesoscopic description is then passed to the continuum limit via homogenization
i.e., by increasing the number of slip surfaces to infinity and reducing the
lattice parameter to zero. We show for two-dimensional deformations of
initially perfect single crystals that the classical continuum formulation is
recovered in the limit with $\mathbf{F}=\mathbf{F}^e\mathbf{F}^p$, $\det
\mathbf{F}^p= 1$ and $\mathbf{G}=\text{Curl}\ \mathbf{F}^p$ the dislocation
density tensor.
