Let $Y_1$ and $Y_2$ be two distinct points in a projective space $\mathbb{P}^n.$ Also let $I(Y_1)$ and $I(Y_2)$ be the corresponding homogeneous ideals in $R=K[X_0,\ldots,X_n].$ Then how can I show that $\dim(R/I(Y_1) \cap I(Y_2))= \dim R/I(Y_1)$ without looking at the dimension of the projective algebraic sets.
Help me. Thanks.