Normed Division Ring Completions are Isometric and Isomorphic

Theorem
Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring.

Let $\struct {S_1, \norm {\, \cdot \,}_1 }$ and $\struct {S_2, \norm {\, \cdot \,}_2 }$ be normed division ring completions of $\struct {R, \norm {\, \cdot \,} }$

Then there exists an isometric isomorphism $\psi: \struct {S_1, \norm {\, \cdot \,}_1 } \to \struct {S_2, \norm {\, \cdot \,}_2 }$

Proof
By the definition of a normed division ring completion then:
 * $\quad$there exists a distance-preserving ring monomorphisms $\phi_1: R \to S_1$
 * $\quad R_1 = \map {\phi_1} R$ is a dense subring of $S_1$
 * $\quad S_1$ is a complete metric space
 * $\quad$there exists a distance-preserving ring monomorphisms $\phi_2: R \to S_2$
 * $\quad R_2 = \map {\phi_2} R$ is a dense subring of $S_2$
 * $\quad S_2$ is a complete metric space