PA-28 flight Instructor reported an NMAC event during landing pattern training when another landing aircraft violated 'Right of Way FAR'. The Instructor took evasive action to avoid a collision.

Narrative

My student and I were practicing take-offs and landings at ZZZ1 airport in our training aircraft. There was another instructor also practicing landings with his student in Aircraft X. We were second in line to land behind Aircraft X in downwind. We extended our downwind to give separation to Aircraft X after they landed so they could back taxi and get off the runway. We did not extend to interfere with the traffic pattern. Once we turned final and were established on glide slope; my student yelled out and said a plane just went beneath us. I immediately took the controls; and started a climb. I banked the airplane to the left so I could see beneath me; and there was a plane just about 200 ft. below us. I immediately made a radio call and executed a go-around; there was no response from this aircraft. I made a radio call to Aircraft X to get his tail number which was Aircraft Y. Aircraft Y landed the plane while Aircraft X was still on the active runway.My student and I were actively making our upwind; crosswind; downwind; base and final calls on XXX.X; the local CTAF. I did not hear a single call from Aircraft Y. After we landed I looked up Aircraft Y to see his flight path; this airplane departed ZZZ2 Runway XX; made a left departure straight in for Runway YY where he cut beneath us while we were established on final. I then later listened to the ATC live archives; and Aircraft Y made a radio call final YY after my student made his final call. It sounded like he stepped on my students radio call. The radio calls can be found on day 0; XA00z-XB30z time frame. This person almost created a midair collision below 600 ft. in the pattern; they not only violated 91.113 but also showed careless flying 91.13.