axiomatic sets 21 (arithmetics in integers)

The addition of two integers is defined as follow.
Given two integers $[(a,b)],[(c,d)]\quad (a,b,c,d\in\mathbb{N})$ ,
\[  [(a,b)]+[(c,d)]=[(a+c,b+d)] .  \]
You will not have any questions.

For example,
\[ [(2,1)]+[(3,1)]=[(5,2)] . \]
This means $1+2=3$ .

\[ [(4,1)]+[(1,4)]=[(5,5)] .  \]
This means $3+(-3)=0$

\[ [(1,3)]+[(1,4)]=[(2,7)] .  \]
This means $(-2)+(-3)=-5$ and equal to
\[  [(21,23)]+[(41,44)]=[(62,67)] . \]

The multiplication of integers is defined same as addition.
\[ [(a,b)]\times [(c,d)]=[(ac+bd,ad+bc)] . \]
Intuitively, as $(a-b)\times(c-d)=(ac+bd)-(ad+bc)$ is true, it will be also true.

You will be able to see the integers are closed under addition (subtraction) and multiplication.

The integers have been constructed on natural numbers $\mathbb{N}$  and some axioms.

(On this definition, unfortunately, $\mathbb{N}\subset\mathbb{Z}$  is not true as a matter of form.  )