2 SMA TRAINING ACFT EXPERIENCE NMAC IN UNCTLED TFC PATTERN AT PVU.

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I WAS INVOLVED IN A NEAR MISS SIT IN THE TFC PATTERN AT THE PROVO ARPT (PVU). I AM AN INSTRUCTOR AND WAS WITH A NEW PVT PLT STUDENT. WE ENTERED THE TFC PATTERN WITH A 45 DEG ENTRY TO THE DOWNWIND LEG. WHILE ON THE 45 DEGS WE WERE LOOKING FOR TFC IN THE PATTERN. WE SAW ONE ON XWIND AND ONE ON A LATE DOWNWIND GETTING READY TO TURN BASE. APPARENTLY THERE WAS ONE MORE ACFT ON DOWNWIND WE DID NOT SEE. ON DOWNWIND WE STARTED OUR DSCNT AND AS WE CALLED OUR TURN TO BASE ANOTHER ACFT CALLED BACK SAYING THAT WE HAD CUT IN FRONT OF IT AND WERE ABOUT 100 FT BELOW THE OTHER AIRPLANE. WE NEVER SAW HIM. THE PATTERN WAS BUSY AND WE MAY HAVE HEARD RADIO CALLS FROM ONE AIRPLANE AND ASSUMED IT WAS THE ONE WE WERE LOOKING AT. EVASIVE ACTION WAS NOT REQUIRED. WE JUST EXITED THE PATTERN TO THE E AND SET UP FOR A RE-ENTRY. I THINK THAT SOME THINGS WHICH CAUSED THIS OCCURRENCE WAS IMPROPER LOOKOUT AND COM BTWN THE PLTS OF THE 2 AIRPLANES. I TALKED WITH THE OTHER PLT AFTERWARDS AND HE TOLD ME THAT HE SAW US ON THE 45 DEGS BUT HE DIDN'T MAKE ANY CALLS UNTIL WE WERE RIGHT UNDER HIM. OUR ARPT IS UNCTLED AND WITH 3+ FLT SCHOOLS ON THE FIELD THE PATTERN CAN GET VERY BUSY. THIS TYPE OF SIT IS PREVENTABLE BY USING STANDARDIZED PROCS AND BEING EXTRA VIGILANT WHEN IN HIGH TFC AREAS.