Answer
The electron's speed relative to the proton is $~~0.994~c~~$ toward the left.
Work Step by Step
Let $u'$ be the electron's velocity relative to the lab reference frame.
Let $v$ be the lab's velocity relative to the proton.
Let $u$ be the electron's velocity relative to the proton.
We can find $u$:
$u = \frac{u'+v}{1+\frac{u'v}{c^2}}$
$u = \frac{-0.9c-0.9c}{1+\frac{(-0.9c)(-0.9c)}{c^2}}$
$u = \frac{-1.8c}{1+0.81}$
$u = -0.994~c$
The electron's speed relative to the proton is $~~0.994~c~~$ toward the left.