Answer
Pythagorean Theorem is verified.
Work Step by Step
Given the vectors ${u}=(4,1,-5), \quad {v}=(2,-3,1)$, then their lengths can be calculated as follows
$$\|{u}\|^2=16+1+25=42, \quad \|{v}\|^2=4+9+1=14.$$
Since $u+v=(6,-2,-4)$, then
$$\|{u+v}\|^2=36+4+16=56.$$
One can see that,
$$\|{u}+{v}\|^{2}=\|{u}\|^{2}+\|{v}\|^{2}$$
Hence, Pythagorean Theorem is verified.