On images of complete topologized subsemilattices in sequential semitopological semilattices

A topologized semilattice $X$ is called complete if each non-empty chain $C\subset X$ has $\inf C\in \bar{C}$ and $\sup C\in \bar{C}$. We prove that for any continuous homomorphism $h:X\to Y$ from a complete topologized semilattice $X$ to a sequential Hausdorff semitopological semilattice $Y$ the image $h(X)$ is closed in $Y$.