Market for lemons derivation
A type-1 seller will trade her car only if the car's measure of quality $x$ (privately known) is less than or equal to the average car quality in the market, $\mu$. $x \in [0,2]$ since no buyer values a car more than $2$ and a car cannot be sold for less than $0$. To find the probability of a car having quality less than or equal to $\mu$, we consider the CDF of $x$, $F(x)$, assuming $x$ is uniformly distributed over the support $[0,2]$:

$$
F(x)=
\begin{cases}
0 &amp;&amp; x \leq 0 \\ 
\frac{x}{2} &amp;&amp; x \in  [0,2] \\ 
1 &amp;&amp; x \geq 2.
\end{cases}
$$

So the probability that a car has $x \leq \mu$ is $P(x \leq \mu) = F(\mu) = \frac{\mu}{2}$. A seller facing a price $p$ for her car becomes indifferent towards selling at $\mu = p \implies F = \frac{p}{2}$. Scaling up by a total of $N$ cars possessed by type-1 sellers implies that the market supply from type-1 sellers is $S_1(p) = \frac{p}{2}N$.