Pull back and intersection of divisors

Question

Let $X$ be a smooth projective variety over complex numbers. Let $Z$ be a smooth closed subscheme of $X$. Let $L$ be a very ample line bundle on $X$. Then $L|_Z=E$ is a very ample line bundle on $Z$. If $D,D'\in |E|$ are divisors on $Z$ such that $D\cap D'$ has codimension 2 in $Z$, can we say that $D$ and $D' $ come from divisors in $|L|$ on $X$ whose intersection has codimension 2?