 Solution

The context-free languages $L_1=\{a^n{b}^n{c}^m:n,m\ge 0\}$ and $L_2=\{a^m{b}^n{c}^n:n,m\ge 0\}$ have context-free grammars $S\to TC$, $T\to aTb\mid \epsilon$, $C\to cC\mid \epsilon$, and $S\to AT$, $A\to aA\mid \epsilon$, $T\to bTc\mid \epsilon$, respectively.

Their intersection is $L=\{a^n{b}^n{c}^n:n\ge 0\}$. Suppose $L$ satisfied the pumping lemma for context-free languages with length $p$. In a decomposition $a^p{b}^p{c}^p=uvwxy$ with $|vwx|\le p$ and $|vx|>0$, the window $vwx$ cannot meet both the $a$ and $c$ blocks. Pumping down changes at least one count and leaves at least one count unchanged. Thus the three counts are unequal, or the word order is invalid, contrary to the pumping lemma for context-free languages. Hence CFLs are not closed under intersection.