Normed Division Algebra is Division Algebra

Theorem
Let $A = \struct {A_F, \oplus}$ be a normed divison algebra over a field $F$.

Let the unit of $A$ be $1_A$, and the zero of $A$ be $0_A$.

Then $A$ is a unitary division algebra.

Also:
 * $\norm {1_A} = 1$

where $\norm {1_A}$ denotes the norm of $1_A$.

Proof
Let $A = \struct {A_F, \oplus}$ be a normed divison algebra as defined in the hypothesis.

The fact that $A$ is a unitary algebra is a consequence of the definition of normed divison algebra.

From the definition of a norm, we have that:
 * $\forall a \in A: \norm a = 0 \iff a = 0_A$

So, let $a, b \in A \setminus \set {0_A}$.

We have:

Thus for any arbitrary $a, b \ne 0_A$ we have shown that $a \oplus b \ne 0_A$.

Thus $A$ is a division algebra.

Next:

demonstrating that $\norm {1_A} = 1$.