What responsibility should the gym bear if it falls down in the gym?

That depends on how you feel. If the fitness equipment is unqualified, the gym safety measures are not complete, and the human factors of the gym people cause you to be injured, then basically they will take full responsibility. If it is because you use the wrong equipment, don't listen to the guidance of the gym staff and exercise in the wrong way, then you are mainly responsible. Specific circumstances, specifically.