of vector space over $K$.

$(ii)~~$ The vector space $~( V(u), \boxplus, \boxtimes )~$ together with the restrictions of structure functions $~\alpha,
\beta, \iota~$ to $~V(u)~$ and the multiplication\
$~\boxdot:V(u)_{(2)}= V(u)\times V(u) \to V(u)~ $ given by: $$x\boxdot y = (x-u)\odot(y-u) + u ,~~~\forall~x,y\in
V(u)\label{3.3}$$ has a structure of vector groupoid with a single unit over $K$.

$(i)~$ Using the linearity of the functions $\alpha$ and $\beta$ we verify that the laws $\boxplus$ and $\boxtimes$ given by (3.1) and (3.2) are well-defined. For instance, for $x,y\in V(u)$ we have $ \alpha(x\boxplus y) = \alpha(x+y-u)=\alpha(x)+ \alpha(y) -
\alpha(u) = u,$ since $ \alpha(x)=\alpha(y)=\alpha(u)=u.$ Similarly, $\beta( x\boxplus y ) =u.$ Hence $ x\boxplus y \in
V(u).$ It is easy to verify that $ (V(u), \boxplus ) $ is a commutative group. Its null vector is the element $ u\in V(u)$. The opposite $~\boxminus x~$ of $~x\in V(u)~$ is $~\boxminus x = 2 u -
 x$.

For any $ x, y\in V(u) $ and $ k, k_{1}, k_{2} \in K $, the law $\boxtimes $ verify the following relations:

$(a)~~~k\boxtimes ( x\boxplus y ) = ( k\boxtimes  x ) \boxplus (k
\boxplus y ),$\
$(b)~~~(k_{1} + k_{2})\boxtimes  x = (k_{1} \boxtimes x ) 