# Proof

(related to Proposition: Addition of Integers)

Let $$x$$ and $$y$$ be integers. By definition, it means that they are equivalence classes represented by some natural numbers $$x=\lbrack a,b\rbrack$$, $$y=\lbrack c,d\rbrack$$, $$a,b,c,d\in \mathbb N$$.

Note that $$a+b$$, $$c+d$$ are all natural numbers, since they are the sums of the respective natural numbers $$a,c$$ and $$b,d$$. Therefore, the sum $$\begin{array}{rcl} x+y=\lbrack a+c,b+d\rbrack . \end{array}$$ exists, because it denotes some new integer, as it is represented by the natural numbers $$a+c$$ and $$b+d$$.

It remains to be shown that the addition of integers does not depend on the specific representatives of the numbers $$x$$ and $$y$$. Suppose, we have different representatives $$\begin{array}{rcl} x=\lbrack a_1,b_1\rbrack =\lbrack a_2,b_2\rbrack ,~y=\lbrack c_1,d_1\rbrack =\lbrack c_2,d_2\rbrack .&&(*) \end{array}$$ Without loss of generality, we can assume $$a_1\ge a_2$$ and $$c_1\ge c_2$$. It follows from the definition of integers that there exist some natural numbers $$i,j$$ with $$\begin{array}{rl} a_1=a_2+i,&c_1=c_2+j,\\ b_1=b_2+i,&d_1=d_2+j.\\ \end{array}\quad (*)$$

We have to show that $$x+y=\lbrack a_1+c_1,b_1+d_1\rbrack =\lbrack a_2+c_2,b_2+d_2\rbrack .$$ In the following, we will use the following mathematical definitions and concepts: * definition of integers, * definition of adding integers (hypothesis), * commutativity law of adding natural numbers, * cancellation law of adding natural numbers, and * associativity law of adding natural numbers:

$$\begin{array}{rcll} x+y&=&\lbrack a_1,b_1\rbrack +\lbrack c_1,d_1\rbrack &\text{by definition of integers}\\ &=&\lbrack a_1+c_1,b_1+d_1\rbrack &\text{by hypothesis}\\ &=&\lbrack (a_2+i)+(c_2+j),(b_2+i)+(d_2+j)\rbrack &\text{according to }(*)\\ &=&\lbrack a_2+i+c_2+j,b_2+i+d_2+j\rbrack &\text{by associativity law for adding natural numbers}\\ &=&\lbrack a_2+c_2+i+j,b_2+d_2+i+j\rbrack &\text{by commutativity law for adding natural numbers}\\ &=&\lbrack a_2+c_2+(i+j),b_2+d_2+(i+j)\rbrack &\text{by associativity law for adding natural numbers}\\ &=&\lbrack a_2+c_2+\cancel{(i+j)},b_2+d_2+\cancel{(i+j)}\rbrack &\text{by cancellation law for adding natural numbers}\\ &=&\lbrack a_2+c_2,b_2+d_2\rbrack &\text{by definition of integers}\\ &=&\lbrack a_2,b_2\rbrack +\lbrack c_2,d_2\rbrack &\text{by hypothesis}\\ \end{array}$$

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### References

#### Bibliography

1. Kramer Jürg, von Pippich, Anna-Maria: "Von den natürlichen Zahlen zu den Quaternionen", Springer-Spektrum, 2013