Does not start, full battery, all lamps are on when I turn the start key?

Yep, it does sound like the starter motor is not working. You can test to confirm by checking for power at the trigger wire, this guide can help you and I have included the how to change it out as well with the starter relay location so you can check it out.

https://www.2carpros.com/articles/starter-not-working-repair

This guide will help you test the starter relay:

https://www.2carpros.com/articles/how-to-check-an-electrical-relay-and-wiring-control-circuit

Check out the images (below). Let us know what happens and please upload pictures or videos of the problem.

Thank you for your tips. However, is it any fuses involved in the start procedure? In such cases which fuses and location of the fuses.

Yes, there is a fusible link (3) and dedicated fuse #12, here is the location of the fuse and the starter wiring diagrams so you can see how the system works.

This guide should help as well:

https://www.2carpros.com/articles/how-to-check-a-car-fuse

Please check out the images below for the diagrams needed.

Thak you very much for the replies. My car is a Mitsubishi Galant S/W V6 1998 120 kW 03 combi petrol with a manual gear box and left-hand drive. The specifications are M1, LNEA5WWY000206, E4*95/54*0014*01, ST511.
Are the diagrams for such a car? In the electronic schemes the automatic drive is indicated!
What is the Key Remember Switch? Please check this!

Motor code is 6A13.

They do not show the manual version but I would think you would just swap out the clutch switch for the park/neutral switch, do you have power at the trigger wire of the starter when in the crank position of the key?

Thank you for all your replies. Still, I am confused about the pictures of the fuses for my Mitsubishi Galant V6 combi 1998 120 kW EA5W 6A13 motor. I have found the attached pictures of the fuse boxes. I have not found a starting circuit diagram for my car. Please check if your starting circuit diagram is the correct for my car version.

If this is a euro car the wiring could be a little different, here are the starter wiring diagrams from an alternative source.