  Using a Toda bracket computation $\langle \theta_4, 2, \sigma^2\rangle$ due
to Daniel C. Isaksen [11], we investigate the $45$-stem more thoroughly. We
prove that $\theta_4^2=0$ using a $4$-fold Toda bracket. By [2], this implies
that $\theta_5$ exists and there exists a $\theta_5$ such that $2\theta_5=0$.
Based on $\theta_4^2=0$, we simplify significantly the $9$-cell complex
construction in [1] to a $4$-cell complex, which leads to another proof that
$\theta_5$ exists.
