  We prove the following: Let $A$ be a C$^{\ast}$-algebra. Then for $a, b \in
A^+ \setminus\{ 0 \}$, we have $a b = 0$ if and only is $\Vert \Vert c
\Vert^{-1} c + \Vert d \Vert^{-1} d \Vert = 1$ whenever $0 < c \le a$ and $0 <
d \le b$ in $A^+$.
