is it calling those other apps to open the selected files? if so, it would be nice to have a hook that's called whenever a program is run from the homescreen, seeing as there isn't really anything out there that chooses what to use in opening a file dynamically like that.

Yes, the program checks for the associated app or program and has it loaded (it also installs the necessary hook and restores the parser hook after running). However, since the app does this based on data stored in the file header and not the program, it would be difficult to install a hook to intercept the prgm token to do this. From the homescreen, people can do dim("OPEN( <<file>>) and it will work, so long as the file has the proper file association set.

Looking nice, but can you also use it in basic programs?

Sorunome, that screenshot is of it being used in a BASIC program.

And if yes, is it compatible with BatLib as it uses dim( too?

The answer is yes and no. The two cannot work at the same time, but in this update I will introduce a new command OPEN() that is able to open BatLib programs (among other things). Also, a bunch of BatLib commands are provided at the moment with FileSyst, though some may be removed later.

So as to the update, I have OPEN() working for a handful of file types! These include BASIC, Assembly, xLIB, Omnicalc, BatLib, Celtic 3, Grammer, DoorsCS7 BASIC, DoorsCS7 Assembly, MirageOS, ION, DoorsCS6 Assembly. I will probably be making more adjustments, but it is pretty well functioning at the moment. On top of that it allows you to run programs that are archived and I spent a good portion of today making it so that it would return to the calling program properly. I updated prgmCMD to include the OPEN command and you can see as it opens DK5ION that this should be very useful!
EDIT: Also, I started work on a tutorial.

I know, but FileSyst doesn't use DoorsCS7 except for when the user wants to run a DoorsCS7 program (or some program that DoorsCS7 can run). For example, if the user doesn't have MirageOS on their calculator, but they have DoorsCS7, and they are trying to run a MirageOS program, FileSyst will open up DoorsCS7 and use its RunProg call.

Using the File Association feature, you can open files with certain apps or programs. I even plan to make it so that even BASIC programs can be used to open files o.o (For example, if you try to open up game data, you can make it auto-launch the actual BASIC game, simply by telling FileSyst that the BASIC program is what opens that file type)
EDIT:
To expand a little more, FileSyst stores files with the following format:

1 byte 1 byte n bytes 1 byte 33 bytes 1 byte 2 bytes n 2 bytes n

Type (1Fh indicates it is a folder, all other types are open) Name Size Name Icon (If this is -1, it is a custom icon, 0 is default, anything else is a preset) IconData (If the Icon is not a custom Icon, this is omitted Info (this is for info about Hidden,Locked, Shortcut, or OS var, among other things) Fields Size Fields Data Data Size File Data

Fields have a header byte so there are up to 256 different fields. Field 0 contains information about file associations, Field 1 contains a file description. If the field doesn't have any data, it is removed. If there is no field data, then Field Data is removed and Field Size is set to zero.

Field 0 is where FileSyst figures out which program  or app to open to handle the file.

Also, as an update, here is a screenie where I run the BASIC program MORPION40  (Tic-Tac-Toe) from DCS7 and DK5ION (DonkeyKong) as an ION program from DCS7.

The program in the screenshot uses FileSyst (which is an app) to access the folders and files, otherwise, the program is coded in BASIC.

I won't be releasing an update since I got caught up with other issues (and I didn't have computer access most of the day). I modified the AFA() command (Add File Association) so that it would check if the file association has been made (basically, it won't make the same association twice, that is redundant and takes up memory). I also added RFA() (Remove File Association) and I have started work on the OPEN() function, which opens files based on file associations. I am currently coding in all of the default stuff, so ION programs will open with either DoorsCS7, MirageOS, ION, or Noshell, for example. You can also do something like AFA( Game,'BATB') and then when you do OPEN( Game), FileSyst will install the BatLib hook before running the game. The filetypes I plan to add as defaults:

ION
MOS (MirageOS)
DCS7 (anything DoorsCS7 can run)
DCS6
C3 (Celtic3)
OMNI (Omicalc)
XLIB (xLIB)
BATB (BatLib Basic)
BAT (BatLib assembly)
GRAM (Grammer programs)

I will probably add more later, but those are my main concerns I am testing DCS7 first.

I especially like the second screenshot, and the 3D tracking is awesome o.o

Wow, it isn't faster, but it is still pretty fast and only a few byte bigger than the code I had (excluding the first and last line). That is an awesome use of lists!

I made this code to compute A^K mod P and I was wondering if it could be optmised and still preserve the speediness (or make it faster).
:1→M
:While int(K
:.5K→K
:If int(2fPart(Ans
:AM-Pint(AM/P→M
:A²-Pint(A²/P→A
:End
I feel like it could be optimised in BASIC. And maybe use only three variables (base, exponent, mod).
EDIT: Optimised with suggestions from calc84maniac and Jacobly
EDIT2: Figured out that .5K is faster than K/2