A man may die. The institution he represents endures.

"The King is dead" = the man who was king is dead.

"Long live the King" = the institution lives, and there is another King even if he hasn't been informed yet, has been informed but not been crowned or is a child too young to understand. Implicit in the statement is the idea that the authority of "being King" transfers from one to the next at the moment of death. Not notification. Not coronation. As soon as one dies, the next is recognised.

For example, Henry III died in November 1272. His son Edward was his heir...but Edward was away on campaign in the Holy Land when Henry died. Edward could not be immediately contacted. When he was contacted, he could not immediately return. Even if he left for home as soon as he knew, it would have taken him months to return.

Edward would not set foot in England again until August 1274 - just over a year and a half later - and once he arrived he could not be crowned as Edward I for another two weeks.

Who do you think was king between Henry's death and Edward's coronation? How would you solve that problem? It's a year and a half's gap, there must be a king somewhere.

"The King is dead, long live the King" is a way of solving it. Edward is the king...whether he's aware of it yet or not.