$

Replacing in the equality 3.4.1(1) the elements $ x, y, z \in
V(u)~$ respectively with $~ x-u, y-u, z-u \in V(u),~$ we obtain the following equality\
$(f)~~ (x - u)\odot (( y - u) +  (z - u ))= (x - u)\odot  (y - u)
+ (x - u )\odot (z-u) - (x - u), $\
since $ \beta(x-u) = 0.$

Using the relation (f), the equality $(e.1)$ becomes\
$(e.2)~~ x\boxdot ( y \boxplus z \boxplus ( \boxminus \beta(x))) =
(x-u)\odot (y-u) + (x-u)\odot (z-u)+ 2 u - x.$\
On the other hand we have\
$(e.3)~~~ ( x\boxdot y ) \boxplus (  x\boxdot z  ) \boxplus (
\boxminus x))=( ( x\boxdot y ) \boxplus (  x\boxdot z  )) \boxplus
( 2 u - x)=$\
$= (  x\boxdot y  +  x\boxdot z - u  ) \boxplus ( 2 u - x) =
x\boxdot y  +  x\boxdot z -  x =$\
$= (x-u)\odot (y-u) + (x-u)\odot (z-u)+ 2 u - x.$\
Using (e.2) and (e.3) we obtain the equality (e). Hence, the relation 3.4.1(1) from Definition 3.1 holds.

In the same manner we can prove that the relations 3.1.4(2) - 3.1.4(4) from Definition 3.1 are verified.

We call $(V(u), \boxplus, \boxtimes, 