or tell me the command to test the address?
The address of what?
When you compile code, all names are changed to addresses. client is the name of a variable that contains an instance of the EthernetClient class. while(client) become while(some_address). Testing the address that way makes no sense. It does not make sure that the instance at that address does anything, like keep a connection open.
So, please, am I right in guess: this client.stop(); was in the initial code just for example, not a must, true?
The client.stop() statement is how the client instance tells the hardware that it no longer needs the connection to the server (which may have already closed it's end of the connection).
Considerably different code is needed on the client and on the server for persistent connections. I've never used that model. All the stuff I want to do is true client/server communication, where a new connection is opened for every exchange of data.