If your Rainbow vacuum suddenly shuts down and after you let it sit for 5-15 minutes it would turn on for another few minutes of vacuuming and shut off again then this is a part that you may need. However it just may be serving its purpose and saving your Rainbow vacuum field and armature from overheating because something else is wrong (excessive sparking between commutator and carbon motor brushes, problem in the fan system or insufficient motor cooling).
BEFORE removing the old thermal fuse note how it is installed. You will notice that one end of the fuse is slightly different from another.