Logical OR in my shell script

I think you can avoid a loop entirely here:

cp *TH[xy]*.ats /home/milenko/procmt

There's no need to loop through the results and then do a separate comparison; a single glob will expand to the list of files that you want.

There were a couple of problems with your original approach:

• Firstly, you were trying to test for exact matches, so the condition would never be true.
• Also, take care with spaces: ]] is a keyword in the compound command [[, so it needs to be a separate word (i.e. surrounded by spaces).

What about using extglob for extended globbing? This way you can use the for itself to get the required extensions:

shopt -s extglobfor file in *TH?(x|y)*.ats; do   # do things with "$file" ...done

*TH?(x|y)*.ats expands to those files containing <something> + TH + either x or y + <something> + .ats

Your script fails because you have a typo in it:

if [[ ("${file}" = THx) || ("${file}" = THy)]]#                                          ^#                              missing space

This is fine:

$ d="hi"$ [[ ($d == hi) || ($d == ha) ]] && echo "yes"yes

Although the parentheses are superfluous:

$ [[ $d == hi || $d == ha ]] && echo "yes"yes

Your question specifies "files that contain THx or THy" ... but your code specifies that the file name is THx or THy.