Light aircraft instructor pilot reported an NMAC with another light aircraft in the vicinity of 7S3 airport.

Narrative

I was instructor pilot in [a light aircraft] doing a left downwind departure from home airport with my learner who seeks to complete a flight review. At our airport; left downwind departures are common off the north-facing runway. We leveled off at pattern altitude which is 1200 MSL (1000 above airport elevation); leaving room for overflights that should maintain at least 1700 MSL until well clear of our traffic pattern.Learner and I observed conflicting traffic coming from our front-left on our ADS-B In display just 200 feet above our altitude; then 100 feet above; close to center-punching us if we did not make a change. Before we could pick the other aircraft up visually; I instructed my learner to make an immediate descent to try to provide some vertical separation. We then saw the other aircraft just a couple of hundred feet ahead of us crossing at right angle to our downwind flight path left-to-right; roughly 200 feet above us.I took a screenshot of the Foreflight display on my iPhone in order to capture offending aircraft's callsign. I then looked it up on Flight Aware and was able to see their flight path and altitude exactly as I just described. I shared all the information with our chief flight instructor; who then said he would call the flight school that offending aircraft belonged to; and discuss the event.