Thank you all for your help, support and adivce.

After a 3 attempt at a fresh installation Debian Jessie is successfully installed and the WiFi connection is working.
As soon as I rebooted after installing the firmware-realtek_0.43_all.deb the WiFi worked right away.

-:-Installing in "Graphical Expert Mode" was the only way that this worked for me.-:-
The other methods of installation either booted to a black screen or repeated the log in over and over again.

evo2, colorpurple21859, EDDY and Widget: thank you all very much! -

You're welcome.

Hey guys, i faced the same problem during debian10 installation. I was using graphical installer for this purpose so after failing 2 to 3 times i just changed the method of install and used simple debian installer and this solved the problem.