Meaning of $x_c$ and $\hat x_c$ in proof of Hammersley–Clifford theorem

Question

I'm reading this paper http://image.diku.dk/igel/paper/AItRBM-proof.pdf and I got stuck in page 4 with equation (1) that's based on Hammersley–Clifford theorem. I'm not good in reading set theory notation and also I wasn't able to understand the point from the equation.

I couldn't understand what are $x_c$ and $\hat{x}_c$

Here's the section, I'm stuck just at equation (1):

Answer 1

Note that $\mathbf x$ and $\hat{\mathbf x}$ are vectors indexed by the vertex set $V$. The assumption in (1) is that every entry of $\mathbf x$ and $\hat{\mathbf x}$ corresponding to some vertex in the clique $C$ coïncide.

Example: if $V=\{1,2,3,4,5\}$ and $C=\{1,2\}$, condition (1) asks that some function $g$ on $\Lambda^5$ is such that $g(x,y,z,t,u)=g(x,y,z',t',u')$ for every $x$, $y$, $z$, $t$, $u$, $z'$, $t'$, $u'$ in $\Lambda$.