Try this :

variable declaration :

$ result="school/proj_1/file1.txt
school/proj_1/file2.txt
school/proj_1/file3.txt"

Command :

$ grep -oP "school/proj_1/\K.*" "$i" <<< "$result"

or

$ awk -F'/' '{print $NF}' <<< "$result 

Output :

file1.txt
file2.txt
file3.txt

Try this :

variable declaration :

$ result="school/proj_1/file1.txt
school/proj_1/file2.txt
school/proj_1/file3.txt"

Commands :

(all as one-liners)

$ grep -oP "school/proj_1/\K.*" "$i" <<< "$result"

or

$ awk -F'/' '{print $NF}' <<< "$result 

or

$ sed 's|.*/||' <<< "$result"

or if number of sub dirs are fixed :

$ cut -d'/' -f3 <<< "$result"

Output :

file1.txt
file2.txt
file3.txt