\boxplus
( k_{2}\boxtimes x ),$\
$(c)~~~k_{1}\boxtimes (  k_{2}\boxtimes  x ) = (k_{1}k_{2})\boxtimes x ,$\
$(d)~~~ 1 \boxtimes  x  = x $ ( here $1$ is the unit of the field $K$ ).\
Indeed, we have $~k\boxtimes ( x\boxplus y ) = k ( x\boxplus y ) +
(1 -k )u = k ( x + y ) + (1 -2 k )u ~$ and $~( k\boxtimes  x )
\boxplus (k \boxplus y )= ( k\boxtimes  x ) + (k \boxplus y ) - u
= k ( x +  y ) + (1 - 2k )u ~$. Hence the equality (a) holds.

In the same manner we prove that the equalities (b) - (d) hold. Therefore $( V, \boxplus, \boxtimes ) $ is a vector space.

$(ii)~$ From the above assertion follows that the condition (3.1.1) from Definition 3.1 is satisfied.

The restrictions of the linear maps $ \alpha $ and $\beta $ to $V(u)$ are linear maps, and so the condition (3.1.2) from Definition 3.1 holds.

Also, the restriction of the linear maps $ \iota $ to $V(u)$ is linear map. Applying the equality 3.1.3(1) from Definition 3.1, for any $ x\in V(u)$ we have\
$ x \boxplus \iota(x) 