(related to Proposition: Distributivity Law For Real Numbers)
Because the multiplication of real numbers is commutative, it is without loss of generality sufficient to show the left-distributivity law \[x\cdot(y+z)=(x\cdot y)+(x\cdot z).\]
By definition of real numbers, the numbers \(x,y,z\) are some equivalence classes \[\begin{array}{rcl}x&:=&(x_n)_{n\in\mathbb N}+I,\\y&:=&(y_n)_{n\in\mathbb N}+I,\\z&:=&(z_n)_{n\in\mathbb N}+I,\\\end{array}\]
in which \((x_n)_{n\in\mathbb N}\), \((y_n)_{n\in\mathbb N}\), and \((z_n)_{n\in\mathbb N}\) denote rational Cauchy sequences, which represent the real numbers \(x\) and \(y\) and \(x\), while \(I\) denotes the set of all rational sequences, which converge to \(0\).
The left-distributivity law can be proven using the following mathematical definitions and concepts: * definition of adding real numbers, * definition of multiplying real numbers, * distributivity law for rational numbers. The proof follows:
\[\begin{array}{ccll} x\cdot(y+z)&=&((x_n)_{n\in\mathbb N} + I)\cdot[((y_n)_{n\in\mathbb N} + I) + ((z_n)_{n\in\mathbb N} + I)]&\text{by definition of real numbers}\\ &=&((x_n)_{n\in\mathbb N} + I)\cdot((y_n + z_n)_{n\in\mathbb N} + I)&\text{by definition of adding real numbers}\\ &=&(x_n\cdot(y_n + z_n))_{n\in\mathbb N} + I&\text{by definition of multiplying real numbers}\\ &=&((x_n \cdot y_n) + (x_n \cdot z_n))_{n\in\mathbb N} + I&\text{by distributivity law for multiplying rational numbers}\\ &=&[(x_n \cdot y_n)_{n\in\mathbb N} + I] + [(x_n \cdot z_n))_{n\in\mathbb N} + I]&\text{by definition of adding real numbers}\\ &=&[((x_n)_{n\in\mathbb N} + I) \cdot ((y_n)_{n\in\mathbb N} + I)] + [((x_n)_{n\in\mathbb N} + I) \cdot ((z_n)_{n\in\mathbb N} + I)]&\text{by definition of multiplying real numbers}\\ &=&(x\cdot y)+(x\cdot z)&\text{by definition of real numbers}\\ \end{array}\]
