NMAC WITH LNDG ARR TFC. UNAUTH UNCOORD RWY ENTRY AT NON TWR ARPT.

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I WAS ON MY FIRST TRAINING FLT WITH A PARTICULAR STUDENT THAT I HADN'T FLOWN WITH BEFORE. I WAS GOING TO EVALUATE HIS READINESS FOR HIS PRIVATE PLT CHK RIDE. WE HAD COMPLETED THE BEFORE TKOF CHKLIST. WHILE GOING THROUGH THE RUNUP; WE HAD HEARD AN SMT PLT MAKE AN 8 MI OUT AND 4 MI OUT RADIO CALL; INBOUND FOR LNDG AND EVENTUALLY AN ABEAM-THE-FIELD CALL. THE LAST CALL WAS A L BASE AND 1 OF OUR FLT SCHOOL ACFT CHKED THE TFC PATTERN AND THEN QUICKLY DEPARTED. MY STUDENT AND I PULLED UP TO THE HOLD SHORT LINES AND ANGLED THE ACFT FOR A POSITIVE VIEW OF THE L BASE AND FINAL LEG FOR THE ACTIVE RWY. I EXPECTED TO READILY FIND THE SMT BUT HE WAS NOWHERE IN SIGHT; SO I TOLD THE STUDENT TO EXPEDITE A TKOF. HOWEVER; HE RADIOED TAKING THE ACTIVE; THEN SLOWLY TAXIED OUT ONTO THE DISPLACED THRESHOLD AND ALIGNED THE ACFT WITH THE CTRLINE. AS A LAST PRECAUTION; I LOOKED OUT OF THE REAR WINDOW OF OUR ACFT AND THE SMT WAS ONLY ABOUT 100 FT AWAY. I KNEW IT WAS TOO LATE TO TAXI CLR SO I MADE SURE MY STUDENT DIDN'T MOVE US. THE SMT FLEW DOWN TO OUR LEVEL; OVER THE GRASS; AND THEN EXECUTED A GAR BACK INTO THE PATTERN. MY THOUGHT IS THAT BECAUSE THE SMT WAS BEIGE WITH BLUE STRIPES; HE MUST HAVE BLENDED IN WITH THE HAZE AND ALREADY BEEN ON FINAL. WE COULD HAVE AVOIDED THIS SIT HAD MY STUDENT AND I WAITED; IF THE SMT PLT HAD CALLED FINAL AND HAD HIS LNDG LIGHT ON.