Energy in.
> will keep vibrating ... lose energy slowly
Dissipative process: energy out. Eventually stop vibrating.
Time crystals cycle through their configurations until you put energy in. Otherwise, they are in lowest energy state, so no energy out. Ever.
Practically all we can do with a time crystal is to measure it with the lightest possible touch and hope it doesn't break the periodicity. (So far, afaik, the periodicity has always been broken by the measurement process's energy input).
I don't know what we could do with large numbers of time crystals, though. One can't pick up a snowflake in one's bare hands and use it like a buzz saw to cut a sheet of paper (the snowflake melts on contact), but an avalanche of snowflakes can snap trees. Maybe for time crystals that rotate light a predictable amount at a given time t_x, we could create some sort of interesting lens from a large cloud of such time crystals arranged at different distances from a bright light source -- a sort of "anti-fog".
Theory guides us, but experimentally you can for example make a whole bunch of time crystals (especially straightfoward for driven time crystals, which have a period that's an integer multiple of the driver, the driving force being laser light or microwaves) with an expected set of states it cycles through, and you can test those states once per time crystal. If you reliably get the states theory predicts, that's good evidence.
> could you outline the proof of the existence
The excellent <https://physics.aps.org/articles/v10/5> and decent <https://en.wikipedia.org/wiki/Time_crystal#Experiments> (the former is [5] in that wikipedia article) are good starting points, with ample references.
This reminds me of the anti-zeno effect, in which a particle evolution is increasingly delayed with more frequent observations.