It appears that an inequality of this kind has been used here. Note: both sequences ($x_i$ and $y_i$) are non-negative, and $p\geq 1$.
In fact, the author then claims that last expression is less than or equal to $$(\sum_{i=1}^n x_i+ y_i)^p.$$
I think I've been able to verify this claim: if we call $$a:=\sum_{i=1}^n x_i$$ and $$b:=\sum_{i=1}^n y_i$$ then the claim becomes $$a^p+b^p\leq (a+b)^p.$$
I think I can prove this as follows: \begin{align*} (a^p+b^p)^{1/p}=&\left((a^{1/p}a^{1-1/p})^p+(b^{1/p}b^{1-1/p})^p\right)^{1/p}\\ \leq& \max\{a,b\}^{1-1/p}(a+b)^{1/p}\\ \leq &(a+b)^{1-1/p}(a+b)^p\\ = &a+b. \end{align*}
Raising both sides to the power of $p$ establishes the claim.
However, I haven't used Hölder's inequality, or the intermediate step (which I included in the title). Can you see what the author has done?