= Solution
Fix a deterministic $a\geq0$. By the <Strong Markov property> at the finite <Brownian first-passage time> $H_a$, the process $\widetilde B_s=B_{H_a+s}-a$ is a standard <Brownian motion> started at zero. Such a process takes positive values arbitrarily soon <almost surely>: for every $\varepsilon>0$, the <Brownian reflection principle> makes its maximum on $[0,\varepsilon]$ have the distribution of $|B_\varepsilon|$, so the probability that the maximum is zero is zero. Intersecting these <events> for $\varepsilon=1/n$ gives an infimum of positive-crossing times equal to zero. Therefore
$$
\boxed{T_a=H_a\quad\text{almost surely for each fixed }a\geq0.}
$$
The infimum defining the strict crossing time need not itself be a time when $B_t>a$; continuity gives $B_{T_a}=a$. The quantifier here is fixed-level <almost sure equality>, not <indistinguishability of stochastic processes> as the level varies.
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