Lattices with top element deleted.

Question

Suppose $L$ is a lattice with top element $1$. Suppose further that if $1$ was removed from the lattice, we would have a partial order with a new top element, call it $1'$. Then, must the partial order with $1'$ itself be a sublattice of $L$?

Answer 1

Yes, it would be a sublattice because $\cap$ does not change and $\cup$ of two elements in the new POS is the same as their $\cup$ in the initial lattice.