= Solution
Let $A=\max_{0\leq t\leq1}B_t$, a random level. The <Brownian reflection principle> gives $A\overset d=|B_1|$, so $A>0$ <almost surely>. Also $A-B_1\overset d=|B_1|$: this follows by applying reflection to the reversed increment process $(B_1-B_{1-s})_{0\leq s\leq1}$, which has the law of a standard <Brownian motion>. Thus $B_1<A$ <almost surely> as well.
By continuity and <compactness>, $A$ is attained at some time strictly between zero and one, so $H_A<1$. It is not exceeded anywhere on $[0,1]$; since $B_1<A$, continuity even excludes an exceedance in a positive interval just after time one. Consequently $T_A>1$. Both conclusions hold on a single probability-one <event>, giving the <fixed-level versus simultaneous Brownian passage-time equality> distinction:
$$
\boxed{\mathbb P\bigl(T_a=H_a\text{ for every }a\geq0\bigr)=0.}
$$
Thus \b[the simultaneous assertion is false]. Equality for each deterministic level, and even simultaneously for all rational levels, cannot be extended to all real levels by an uncountable intersection.
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