gr.group theory – Left-side cosets of an open subgroup

Let $G$ be a topological group and $H$ its closed subgroup. $K$ and $L$ are open subgroups of $G$ and $H$ respectively. Let $g_{1}, g_{2}in G$. We assume $Lcap g_{1}Kneq phi$ and $Lcap g_{2}Kneq phi$. Then Is there an element $xin H$ (also $xin L$) such that $Lcap g_{1}K=x(Lcap g_{2}K)$ ?