Ajustice of the supreme court must be by the people to continue her/his term.

Justices of the supreme court are chosen for life, which means that there is no formal requirement for them to continue their term - their work continues unless something is done to dismiss them.

But what they should have is approval or support of the people (one of those two words fits the gap, or "approval and support"), because without it, they can't do their job efficiently. A lack of approval could lead them to leave the office because they might feel that their work is not respected (but the formal process to remove them, impeachment requires them to commit a crime, and not only to loose support)