Order_Embedding_between_Quotient_Fields_is_Unique

Let $\struct {R_1, +_1, \circ_1, \le_1}$ and $\struct {S, +_2, \circ_2, \le_2}$ be totally ordered integral domains. Let $K, L$ be totally ordered fields of quotients of $\struct {R_1, +_1, \circ_1, \le_1}$ and $\struct {S, +_2, \circ_2, \le_2}$ respectively. Let $\phi: R \to S$ be a order embedding. Then there is one and only one order embedding $\psi: K \to L$ extending $\phi$. Also $\forall x \in R, y \in R_{\ne 0}: \map \psi {\dfrac x y} = \dfrac {\map \phi x} {\map \phi y}$ If $\phi: R \to S$ is an order isomorphism, then so is $\psi$.