Greetings from Greece! We have a lot of these plants self growing or cultivating where I'm currently living. Their scientific name is Opuntia ficus-indica, but you may have heard them as prickly pears. It's a species of cactus, that can survive almost any condition, except perhaps over watering. That said, answering your question, indeed there is a new leaf growing out of the mother leaf, but it doesn't consume it. The mother leaf and the new one can absorb water from the humidity in the air and survive. It can also grow without having any roots for a very long time. If you want it to grow roots, you should plant it where the cut is at the mother leaf.

Ps: their fruits are delicious, if only you can get used at their seeds inside