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In this paper we prove uniqueness theorems for mappings 
$F\in W_{\text{loc}}^{1,n}(\mathbb{B}^{n};\mathbb{R}^{n})$ of finite distortion 
$1\leq K(x)=\Vert \mathit{DF}(x)\Vert ^{n}/J_{F}(x)$ satisfying some integrability conditions. These types of theorems fundamentally state that if a mapping defined in 
$\mathbb{B}^{n}$ has the same boundary limit 
$a$ on a ‘relatively large’ set 
$E\subset \unicode[STIX]{x2202}\mathbb{B}^{n}$, then the mapping is constant. Here the size of the set 
$E$ is measured in terms of its 
$p$-capacity or equivalently its Hausdorff dimension.
