$\pi_{R-S}(r) - \pi_{R-S} \left (\pi_{R-S} (r) \times s - \pi_{R-S,S}(r)\right )$
$\quad= \pi_{a,b}(r) - \pi_{a,b} \left (\pi_{a,b} (r) \times s - \pi_{\color{\red}{a,b,c}}(r) \right)$ [See here I have written a,b,c because $R-S$ tuple returning a,b and now adding column of S, here comma operation doing union of columns]
Now, $\pi_{a,b} (r) \times s$ what it returns?
It is doing nothing but concatenation of $a,b$ column of $r$ and $c$ column of $s$
So, it is returning
a |
b |
c |
Arj |
TY |
12 |
Arj |
TY |
14
|
Cell |
TR |
12 |
Cell |
TR |
14 |
Tom |
TW |
12 |
Tom |
TW |
14 |
BEN |
TE |
12 |
BEN |
TE |
14 |
Now, we can do this part of question, which returning nothing but the rows of $r$, which is not in original table
i.e. $ \pi_{R-S} \left (\pi_{R-S} (r) \times s - \pi_{R-S,S}(r)\right )$
And finally,
$\pi_{R-S}(r) - \pi_{R-S} \left (\pi_{R-S} (r) \times s - \pi_{R-S,S}(r)\right )$
$\quad=(r/s) $