Prove that, for a > S 0 a > S 0 , and t T t

Prove that, for $a>S_0$, and $t\le T$

$$P\left(T_a\right(S)>T\mid F_t)=1_{\{{max}_{s\le t}{S}_s<a\}}(N\left(d_1\right)-{\left(\frac{a}{S_t}\right)}^{2(r-\delta -{\sigma }^2/2){\sigma }^{-2}}N\left(d_2\right))$$

with

$$\begin{array}{cc}{d}_1& =\frac{1}{\sigma \sqrt{T-t}}(\ln \left(\frac{a}{S_t}\right)-(r-\delta -\frac{{\sigma }^2}{2}))\\ d_2& =\frac{1}{\sigma \sqrt{T-t}}(\ln \left(\frac{S_t}{a}\right)-(r-\delta -\frac{{\sigma }^2}{2})).\end{array}$$