# Proof

(related to Proposition: Multiplication 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 $$ac$$, $$bd$$, $$ad$$ and $$cb$$ are all natural numbers, since they are the products of the respective natural numbers $$a,b,c,d\in \mathbb N$$. Also, $$ac + bd$$ and $$ad + cb$$ are natural numbers, since they are the sums of the respective natural numbers $$ac ,bd$$ and $$ad ,cb$$. Therefore, the product $$\begin{array}{ccl} x\cdot y:=\lbrack ac + bd,~ ad + bc\rbrack \end{array}$$ exists, because it denotes some new integer, as it is represented by the natural numbers $$ac + bd$$ and $$ad + cb$$.

It remains to be shown that the multiplication 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 $$x\cdot y=\lbrack a_1c_1 + b_1d_1,~ a_1d_1 + b_1c_1\rbrack =\lbrack a_2c_2 + b_2d_2,~ a_2d_2 + b_2c_2\rbrack .$$

In the following, we will use the following mathematical definitions and concepts: * definition of integers, * definition of adding integers (hypothesis), * associativity law of adding natural numbers, * distributivity law for adding natural numbers, * commutativity law of adding natural numbers, and * cancellation law of adding natural numbers:

$$\begin{array}{rcll} x\cdot y&=&\lbrack a_1,b_1\rbrack \cdot \lbrack c_1,d_1\rbrack &\text{by definition of integers}\\ &=&\lbrack a_1c_1 + b_1d_1,~ a_1d_1 + b_1c_1\rbrack &\text{by hypothesis}\\ &=&\lbrack (a_2+i)\cdot (c_2+j)+(b_2+i)\cdot (d_2+j),\\ &&~(a_2+i)\cdot (d_2+j) + (b_2+i)\cdot (c_2+j)\rbrack &\text{according to }(*)\\ &=&\lbrack a_2c_2+a_2j+ic_2+ij+b_2d_2+b_2j+id_2+ij,\\ &&~a_2d_2+a_2j+id_2+ij+b_2c_2+b_2j+ic_2+ij\rbrack &\text{by associativity and distributivity law for adding natural numbers}\\ &=&\lbrack a_2c_2+b_2d_2+a_2j+ic_2+ij+b_2j+id_2+ij,\\ &&~a_2d_2+b_2c_2+a_2j+ic_2+ij+b_2j+id_2+ij\rbrack &\text{by commutativity law for adding natural numbers}\\ &=&\lbrack a_2c_2+b_2d_2+(a_2j+ic_2+ij+b_2j+id_2+ij),\\ &&~a_2d_2+b_2c_2+(a_2j+ic_2+ij+b_2j+id_2+ij)\rbrack &\text{by associativity law for adding natural numbers}\\ &=&\lbrack a_2c_2+b_2d_2+\cancel{(a_2j+ic_2+ij+b_2j+id_2+ij)},\\ &&~a_2d_2+b_2c_2+\cancel{(a_2j+ic_2+ij+b_2j+id_2+ij)}\rbrack &\text{by cancellation law for adding natural numbers}\\ &=&\lbrack a_2c_2+b_2d_2,a_2d_2+b_2c_2\rbrack &\text{by definition of integers}\\ &=&\lbrack a_2,b_2\rbrack \cdot \lbrack c_2,d_2\rbrack &\text{by hypothesis}\\ \end{array}$$

Github: ### References

#### Bibliography

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