# Product of Reciprocals of Real Numbers

## Theorem

$\forall x, y \in \R_{\ne 0}: \dfrac 1 x \times \dfrac 1 y = \dfrac 1 {x \times y}$

## Proof

 $\ds \frac 1 {x \times y} \times \paren {x \times y}$ $=$ $\ds 1$ Real Number Axioms: $\R \text M 4$: Inverses $\ds \leadsto \ \$ $\ds \frac 1 {x \times y} \times \paren {x \times y} \times \frac 1 y$ $=$ $\ds 1 \times \frac 1 y$ as $y \ne 0$ $\ds \leadsto \ \$ $\ds \paren {\frac 1 {x \times y} \times x} \times \paren {y \times \frac 1 y}$ $=$ $\ds 1 \times \frac 1 y$ Real Number Axioms: $\R \text M 1$: Associativity $\ds \leadsto \ \$ $\ds \paren {\frac 1 {x \times y} \times x} \times 1$ $=$ $\ds 1 \times \frac 1 y$ Real Number Axioms: $\R \text M 4$: Inverse $\ds \leadsto \ \$ $\ds \frac 1 {x \times y} \times x$ $=$ $\ds \frac 1 y$ Real Number Axioms: $\R \text M 3$: Identity $\ds \leadsto \ \$ $\ds \paren {\frac 1 {x \times y} \times x} \times \frac 1 x$ $=$ $\ds \frac 1 y \times \frac 1 x$ as $x \ne 0$ $\ds \leadsto \ \$ $\ds \frac 1 {x \times y} \times \paren {x \times \frac 1 x}$ $=$ $\ds \frac 1 y \times \frac 1 x$ Real Number Axioms: $\R \text M 1$: Associativity $\ds \leadsto \ \$ $\ds \frac 1 {x \times y} \times 1$ $=$ $\ds \frac 1 y \times \frac 1 x$ Real Number Axioms: $\R \text M 4$: Inverse $\ds \leadsto \ \$ $\ds \frac 1 {x \times y}$ $=$ $\ds \frac 1 y \times \frac 1 x$ Real Number Axioms: $\R \text M 3$: Identity $\ds \leadsto \ \$ $\ds \frac 1 {x \times y}$ $=$ $\ds \frac 1 x \times \frac 1 y$ Real Number Axioms: $\R \text M 2$: Commutativity

$\blacksquare$