I have a problem to solve but my intuitive solution seems far too easy
Given three random Vectors X, Y, Z.
If we know that X, Y are independent, and Z and (X´, Y´)´ are independent, I have to show that X, Y, and Z are independent.
My idea:
$$P(X<x, Y<y, Z<z)=P(X<x)*P(Y<y,Z<z)=P(X<x)*P(Y<y)*P(Z<z)$$
which would show independence. But I am not sure if this would prove it. It feels like I am missing a point here.
You have considered random variables, not random vectors. Your first step is not correct. The correct argument (in the case of random variables) is $$P(X<x,Y<y,Z<z)=P(X<x,Y<y)P(Z<z)$$ $$=P(X<x)P(Y<y)P(Z<z)$$ where the first equality holds because $Z$ is in dependent of $(X,Y)$ and the second equality is because $X$ and $Y$ are independent.