Are order-preserving field embeddings unique?

Question

Let $K$ be an ordered field with an embedding into $\mathbb R$,

$$f:K\hookrightarrow\mathbb R,$$

where $f$ is order preserving. Is $f$ unique?

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Follow up from this question of mine (same question but without ordering assumption). This question is related, but not quite the same.

Answer 1

Yes, in that case it is unique, because for any $x\in K$, its image $f(x)$ is completely characterized by the sets $A=\{a\in \mathbb{Q}|a\leqslant f(x)\}$ and $B=\{b\in \mathbb{Q}|f(x)\leqslant b\}$, and we see that $A=\{a\in \mathbb{Q}|a\leqslant x\}$ and $B=\{b\in \mathbb{Q}|x\leqslant b\}$ so they only depend on the order on $K$.