Hi: I have a running USB memory stick arch linux iso image. It is
archlinux-2018.02.01-x86_64.iso. I want to run it in Live mode. I have
proved, in virtual console 1, several command and seen it executes
them. I can't install it on the hard disk, because there, is Windows 10,
and I can't afford to loose it, as I would loose comunication with
LQ then. In short, I want a Live Arch system. How do you do this?

Oh yes, I've a running Live system, I've just said it! Well, and what now. I want to nave a GUI, but still a live system.

Why not try Manjaro Linux, it is very closely related to Arch Linux but it has gui live, gui installer, gui package manager, etc. So basically it's Arch made easy.

If you're still looking - I have installed Arch to a USB stick many times and used it as a fully functional OS, completely independent of what's on the internal drive. My install instructions are here:

http://www.komputermatrix.com/2018/05/21/114/

You would just install to the external USB rather than an internal drive. And using the systemd bootloader means that you won't mess with the existing installs (as with Grub for example). However, you will need to select your boot drive using the system UEFI. In most cases, I find this very easy. All my laptops have a shortcut key that brings up a boot menu without having to enter the whole UEFI menu.