I do not understand how doing something for somebody is liberty for the benefitted person.
You really have an obsession with aid to the poor, don't you? Why do you think that's important or in any way defining?
Equality of wealth is the basis of Liberty?
It is indeed. And I'm not going to go into Thoreau, who had a wonderful heart and some great insights but who should not be taken as offering practical suggestions for anyone (nor as having followed them himself; look into how he was supported while he was at Walden sometime).
Not, as I said already,
complete equality, but limited inequality. You cannot be free without the economic means to be free. You cannot be free, simply put, unless you can survive without holding a job working for someone else. As long as you are working for someone else, you are under that person's authority, you have a boss, and so you are not free. Now, there's a qualifier: If you don't NEED the job, if you can get by fine without it, and are therefore genuinely holding the job by your own voluntary choice, then yes, you are free. But for the overwhelming majority of people, that's not the case.
Excessive inequality of wealth reduces most people to servitude. When wages are very high, a person can go a long time between jobs and his freedom is thereby increased. When they are not, he is at the mercy of his employer and is thus not free. Moreover, huge inequality of wealth means a huge imbalance of political power, too, and the very wealthy suborn the government to act in service to their own ends -- which is the very definition of tyranny.
There is no liberty without property. (I forget who said that, but it's true.) This means that, if most people are denied property, then they are also denied liberty. And please, please, PLEASE don't go off on a tangent again about people being "given" property who haven't worked for it; the problem we have is people being denied the property they HAVE worked for, having it stolen by rich and powerful men who take their labor and keep the property it makes, too. While there are certainly people who cannot earn their way, the real problem we face is that too many people are not being allowed to earn their way, in order to maximize the wealth accruing to the richest.