For arbitrary rational Cauchy sequences \((x_n)_{n\in\mathbb N}\), \((y_n)_{n\in\mathbb N}\), and \((z_n)_{n\in\mathbb N}\) with the binary operations addition "\( + \)" and multiplication "\(\cdot\)", the following distributivity laws hold:
"left-distributivity property": $$(x_n)_{n\in\mathbb N}\cdot\lbrack (y_n)_{n\in\mathbb N}+(z_n)_{n\in\mathbb N}\rbrack =\lbrack (x_n)_{n\in\mathbb N}\cdot (y_n)_{n\in\mathbb N}\rbrack +\lbrack (x_n)_{n\in\mathbb N}\cdot (z_n)_{n\in\mathbb N}\rbrack,$$ "right-distributivity property": $$\lbrack (y_n)_{n\in\mathbb N}+(z_n)_{n\in\mathbb N}\rbrack \cdot (x_n)_{n\in\mathbb N}=\lbrack (y_n)_{n\in\mathbb N}\cdot (x_n)_{n\in\mathbb N}]+\lbrack (z_n)_{n\in\mathbb N}\cdot (x_n)_{n\in\mathbb N}\rbrack.$$
Proofs: 1
