Very kind for your to good advice.
I am using a Power-line Networking connection of 'Devolo' brand product (the precise type is: "dLAN 200 AVeasy starter kit")
I just have checked 5 minutes earlier & there is No Problem with the 2nd Computer "Eee PC 901 Notebook" This is what I was fearing mostly as I did a reinitialisation of the Atheros drive on it earlier(after my stupid deletion yesterday...). Also, I need to further tell you that, I did connect the computer "Eee PC 901 Notebook" directly on the Netgear router and it look like it is all fine / I can connect to the internet. But I am keen to use my "Eee PC 901 Notebook" (2nd computer) from my other room, or from a different part of the house, and now it seems that the problem rest with the power line connection which was linking my "Eee PC 901 Notebook" (2nd computer) with a main power socket on the kitchen wall, Or, if it is not set to automatically pick up IP address. I remember that I did do this earlier and I think I set it on ' automatic mode', but I will check again to see if it really is.
Can you give any good advice IF there is a problem with your DHCP server? What should I do if that is the case?
I will see if I can have the same problem from a different room, I was reading on a comment earlier that some time the power line can be a problem (devolo power-line networking kit)
I got some work to do from home over the next days (although it is season's celebrations on the way...) and my kids, who are on school holiday use constantly my main PC all the time :) you know.
Thank you very much for your kind assistance on this.