STUDENT ABOARD DA40 ON DOWNWIND EXPERIENCES CLOSE ENCOUNTER WITH AN INBOUND CESSNA CLEARED TO ENTER ON THE PATTERN ON BASE LEG.

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I WAS FLYING ACFT X WITH MY CFI IN THE R SEAT. I AM A STUDENT PLT WITH CLASS 3 MEDICAL AND 1.6H SOLO TIME. WE WERE PRACTICING XWIND LNDGS AFTER A QUICK RUN-THROUGH OF DIVERSIONS. AHEAD OF US IN THE PATTERN WAS A REMOS LIGHT SPORT; FLOWN BY AN INSTRUCTOR. THE REMOS IS SLOWER AND WAS MAKING TIGHT PATTERNS. WE HAD TO WIDEN A LITTLE TO KEEP FROM OVERTAKING IT. IMMEDIATELY PRIOR TO OUR DEP; A C152 OR C172 DEPARTED FOR PRACTICE ABOUT 15 MI W. THINGS HAPPENED QUICKLY; SO I MAY HAVE THE ACFT SWAPPED; BUT I BELIEVE THE REMOS WAS ALREADY ON SHORT FINAL. WE; IN THE DA40; WERE ON DOWNWIND; FIRST NOTCH OF FLAPS; JUST DSNDING THROUGH 1500 FT; AND ABOUT TO GO WING UP FOR BASE. WE HEAR TWR SPEAKING TO THE CESSNA. THEY WERE RETURNING FROM THE W. THE TWR CLRED THEM TO CONTINUE IN ON BASE; THEN TURN FINAL. I SLOWED THE DA40 A BIT; TO ABOUT 78 KIAS. THE TWR ASKED US TO WIDEN OUR BASE. WE WERE ALREADY NEAR THE END OF OUR DOWNWIND. AT THIS POINT; WE SPOT THE CESSNA ON A 90 DEG COLLISION COURSE. MY CFI; TOOK THE CTLS AND STEEPLY TURNED L SINCE TURNING R WOULD HAVE PUT US INSIDE THE CESSNA'S TURNS. THE CESSNA MADE NO EVASIVE MANEUVERS AND CONTINUED ON-COURSE. AFTERWARDS; WE TURNED BACK AND WERE A LITTLE WIDE AND A LITTLE DOWNWIND. THE REST OF THE APCH AND LNDG WAS UNEVENTFUL. THE CESSNA AND OUR DA40 CAME WITHIN 50 FT VERT AND ABOUT 500 FT HORIZ.