Statistical independence of the Boolean function superposition. II

Let $x,y,z$ be sets of different Boolean variables, $f(x,y)$, $f_1(x,y)$, $f_2(x,y)$, $f_1(x,y)\oplus f_2(x,y)$ are Boolean functions being statistically independent on the variables in $x$, and $h(x_1,x_2,z)$, $g(x)$ are any Boolean functions. Then the function $h(f_1(x,y),f_2(x,y),z)$ is statistically independent on the variables in $x$; and the same is true for the function $f(x,y)\oplus g(x)$ iff $f$ is balanced or $g=\mathrm{const}$.