His $\displaystyle \text{LT}(x) = x \mod 100$, so there is no need to verify distributivity.
The LT() proof and the congruence proofs are actually the same.
$\displaystyle x \mod 100$ can be split into two congruences, namely $\displaystyle x \mod 25$ and $\displaystyle x \mod 4$ since they are relatively prime
Abhishek's proof realizes that we can just focus on $\displaystyle x \mod 25$ part