The Canonical Embedding of the Rational Numbers into an Ordered Field

In the setting of The Integers and the Rational Numbers, with the Natural Numbers and the Integers Identified with Subsets of the Rationals, let $r$, with $+$, $\cdot$, $0_{r}$, $1_{r}$ and $\le$, be an ordered field.

¶ The canonical embedding of $\mathbb{Q}$ into $r$ is the map $\kappa_{r}:\mathbb{Q}\to r$ of The Rational Numbers Embed in Exactly One Way into Every Ordered Field §unique, which is an injective homomorphism of ordered fields by The Rational Numbers Embed in Exactly One Way into Every Ordered Field §homomorphism and The Rational Numbers Embed in Exactly One Way into Every Ordered Field §injective.