Dimension of the irreducible components of intersection of a projective variety with zero set of homogeneous polynomial.

Question

Let $W\subset \mathbb{P}^n$ be a projective variety and $F_1,F_2,\ldots F_r$ be homogeneous polynomials of degree $>0$. Let $G$ be any irreducible component of $Z(F_1,F_2,\ldots F_r)\cap W$.

Q) Prove that $dim~G\geq dim~W-r$.

Q) Prove that if $W$ is quasi projective and $Z(F_1,F_2,\ldots F_r)\cap W\neq \emptyset$ then also the above result holds true.