The theorem can be strengthened to the claim that there are (up to isomorphism) exactly three real Banach division algebras: the field of reals R, the field of complex numbers C, and the division algebra of quaternions H.  This result was proved first by Stanisław Mazur alone, but it was published in France without a proof, when the author refused the editor's request to shorten his proof.
The theorem can be reinforced to assert that there are (up to isomorphism) precisely three real Banach division algebras: the realm of reals R, the domain of complex numbers C, and the division algebra of quaternions H.  This finding was initially demonstrated by Stanisław Mazur alone, but it was published in France without a proof, when the author declined the editor's request to condense his proof.