@MartinSleziak
I want to show that if $(I_a)_{a \in A}$ is a family of ideals in $K[x_1, \dots , x_2]$, then $V \left ( \sum_{a \in A} I_a \right )=\cap_{a \in A} V(I_a)$
I tried the following:
$x \in I_a$
$\sum_{a \in A} I_a$ consists of all finite sums, where summands belong to the ideals $I_{a} $.
We could see $x$ as a sum.
So, $x \in \sum_{a \in A} I_a$.
Therefore, $I_a \subset \sum_{a \in A} I_a \Rightarrow V \left ( \sum_{a \in A} I_a \right ) \subset V(I_a)$
Is it right? Or have I done something wrong?