Natural Numbers Read in an Ordered Field Agree with the Canonical Embedding of the Rationals

In the setting of The Integers and the Rational Numbers, with the Natural Numbers and the Integers Identified with Subsets of the Rationals, let $F$ be an ordered field, $\kappa_{F}:\mathbb{Q}\to F$ the canonical embedding, and $n\in\mathbb{N}_{0}$.

¶ The element $n$ of $F$, read as in Commutative Rings, Fields and Ordered Fields: Standard Notation §numerals, is $\kappa_{F}(n)$, with $n$ read as a rational number as in The Integers and the Rational Numbers, with the Natural Numbers and the Integers Identified with Subsets of the Rationals §identification.