Flight Instructor reported an NMAC joining the traffic pattern at S43 non-towered airport. Flight Instructor took evasive action to avoid a possible collision.

Narrative

I was conducting a stage check for a stage 1 private pilot student. We just finished our final stage check maneuver and we were on our way to S43. S43 was landing 15L and we were arriving from the north. We descended to 1000'MSL and maneuvered west of the field to enter the 45 right downwind for 15L. When we were roughly 2 miles west of the field we noticed ADSB traffic departing out of PAE off of Runway 16R/L on the left crosswind eastbound at 2300'. PAE is located just 7NM west of S43 on a 600' bluff. We had the aircraft visually in sight and came to the conclusion that they were just transitioning eastbound and were not a factor as they moved from right to left in our windscreen and they were not reporting any calls on S43's CTAF. The student I was working with started to finish their before landing checklist as we approached the 45 right downwind for 15L. I soon noticed the previous airplane we had in sight was now directly in front of us in a descending right banked turn. They appeared to be maneuvering to join the pattern exactly where we were at that time. Their relative movement stopped so I took flight controls immediately and maneuvered our airplane to the right and descended to divert our flight paths. When we were clear of the conflict we re joined the traffic pattern and landed. And so did they; after they rejoined the pattern.I believe a few factors lead to this situation. 1. I fell into expectation bias as we routinely see aircraft departing PAE and flying a eastbound course south of S43 to avoid Seattle's class B airspace. This lead to me making the assumption that the airplane was just doing what I was expecting; and I started to assume that they were not going to be a factor. 2. When I fly from PAE to S43; it is a very short flight and what I notice is that there is not a lot of time between leaving the class D airspace and joining the S43 pattern. Because of this; I've noticed a lot of people staying on Tower frequency until Tower tells them to change frequency. I think a possibility is that the other airplane was still with Tower and potentially forgot to change to S43's 123.0 CTAF to make their voluntary position reports. I think that if the other airplane was communicating their intentions more clearly; we would have both worked out a solution.In the future I will try to not fall into expectation bias and monitor traffic objectively especially when I have not established communications with them. I can not assume another aircraft flight paths objectively unless I am talking to the pilot myself or if they are on an ATC clearance. The space to maneuver between S43 and PAE class D surface area is very tight and leads to a lot of squeezing traffic during busy periods. I just need to pay better attention and be more proactive.