C172 INSTRUCTOR AND STUDENT HAD LESS THAN LEGAL SEPARATION WITH A PA28 AT AN UNCTLED ARPT TFC PATTERN.

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MY STUDENT AND I WERE ON A LEFT DOWNWIND FOR THE RWY. WE HAD HEARD A CESSNA 206 CALL L UPWIND FOR THE RWY. OUT OF NOWHERE; A PIPER DAKOTA FLEW RIGHT BELOW US ON A XWIND. THE PLTS RADIOED AND SAID HE WOULD FOLLOW US IN. AS WE TURNED BASE LEG; I LOOKED OVER AND THE DAKOTA WAS TURNING INTO US; PROBABLY 500 FT OR LESS FROM US. THE DAKOTA THEN QUESTIONED US; IMPLYING WE WERE AT FAULT. WE TURNED FINAL AS THE DAKOTA FLEW RIGHT UNDERNEATH US. HE ASKED IF WE WERE ON FINAL. I STATED 'WE WERE TRYING TO FLY A FINAL; IF HE WOULD NOT CUT US OFF.' THE DAKOTA EXECUTED A GAR ON SHORT FINAL. WE PROCEEDED TO DO A TOUCH-AND-GO. AFTER WE TOOK OF AGAIN; THE CESSNA 206 HAD CALLED MIDFIELD DOWNWIND. WE WATCHED THE DAKOTA TURN DOWNWIND BEHIND THE CESSNA 206. THE DAKOTA PROCEEDED TO CUT OFF THE CESSNA 206. THE DAKOTA TOLD THE CESSNA 206 OVER THE RADIO; 'I AM INSIDE YOU ON DOWNWIND; AND LOWER THAN YOU; SO I AM GOING TO LAND IN FRONT OF YOU.' THE CESSNA 206 REPLIED; 'I AM #1 FOR LNDG; I AM LNDG FIRST.' THE CESSNA 206 CLRLY HAD THE RIGHT-OF-WAY FROM OUR VIEWPOINT. THE DAKOTA TURNED BASE IN FRONT OF THE CESSNA 206 AND LANDED AHEAD OF THE CESSNA 206. THE GUY IN THE CESSNA 206 WAS VERY UPSET; TELLING THE DAKOTA THEY WOULD TALK ON THE GND. BOTH AIRPLANES AND US LANDED WITHOUT FURTHER TROUBLE. THE PERSON IN THE PIPER DAKOTA WAS CLRLY AT FAULT IN BOTH SITS.