Let we have $V$ a vectorial space, $W_1$ and $W_2$ two subspaces of $V$ and $\mathbb{S}_1=\mathbb{P}(W_1)$ and $\mathbb{S}_2=\mathbb{P}(W_2)$ two projective subspaces in the projective space $\mathbb{P}(V)$.
We need to prove that if $\space \mathbb{S_1+S_2} \space $ is the smallest projective subspace that includes $\mathbb{S_1}$ and $\mathbb{S_2}$, is true that $\space \mathbb{S_1+S_2}=\mathbb{P}(W_1+W_2)$.
I don't even know how to start with this demonstration... can anyone help me?