AN INSTRUCTOR AND STUDENT IN A C172 HAVE NMAC WITH ANOTHER C172; ALSO BASED AT EVB.

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I WAS CONDUCTING A DISCOVERY FLT (INTRODUCTORY FLT LESSON) IN A 2001 C172SP. I TOOK OFF FROM NEW SMYRNA BEACH ARPT (EVB). I DEPARTED ON RWY 11 STRAIGHT OUT; HAVING ANNOUNCED MY INTENTIONS TO DO SO ON THE CTAF; 122.8; JUST BEFORE COMMENCING THE TKOF. WE LEVELED OFF AT 1000 FT; 200 FT BELOW THE CLASS C AIRSPACE FOR DAYTONA BEACH. THE STUDENT; A LOW TIME; PRE-SOLO STUDENT PLT WS GIVEN THE CTLS. WE FLEW STRAIGHT AND LEVEL ON COURSE 110 DEGS UNTIL REACHING THE COASTLINE; ABOUT 2 1/2 MI FROM THE ARPT. WE THEN TURNED N TOWARDS DAYTONA BEACH; PARALLELING THE COASTLINE. WE FLEW N FOR ABOUT 1 MI; CONTINUING TO MONITOR THE CTAF. THERE HAD BEEN NO XMISSIONS FROM ANY ACFT OTHER THAN ONE THAT HAD JUST LANDED AT NEW SMYRNA AFTER I HAD DEPARTED. THE ACFT WAS SET TO CRUISE AT 90 KTS. STILL MONITORING CTAF; HEARING NO ACFT IN THE AREA; I CONTINUED SCANNING VISUALLY AND OBSERVED A C172 OFF MY L WING. IT WAS COMMENCING A L TURN WHILE OVERTAKING ME IN A GRADUAL CLB. IT WAS AT MY ALT AT A DISTANCE OF ABOUT 100 FT. I IMMEDIATELY CALLED 'MY PLANE' TO MY STUDENT; TOOK PHYSICAL CTL OF THE ACFT AND BROKE OFF TO THE R; DSNDING TO ABOUT 800 FT OVER THE WATER. I ANNOUNCED MY POS OVER THE CTAF AND ATTEMPTED TO ESTABLISH CONTACT WITH THE PLT OF THE OTHER CESSNA USING HIS TAIL NUMBER. I RECEIVED NO RESPONSE. I THEN OBSERVED THE ACFT DIRECTLY ENTERING A L DOWNWIND LEG FOR RWY 11 (WINDS HAD BEEN FAVORING RWY 11 ALL DAY). I THEN RESUMED MY FLT; SWITCHING FREQS TO DAYTONA APCH AND RECEIVING A XPONDER CODE AND VECTORS INTO CLASS C AIRSPACE. THE NEXT DAY I INTERVIEWED THE PLT OF THE OTHER ACFT AND HIS FLT INSTRUCTOR. THEY HAD BEEN INBOUND IN THE NDB 29 APCH BUT HAD NEGLECTED TO ANNOUNCE THEIR POS ON THE CTAF. UPON REACHING THE COASTLINE; THEY BROKE OFF THE APCH; TURNED TO THE N; PASSED BEHIND ME AND THEN SPOTTED ME AS THEY ROLLED WINGS LEVEL. THEY THEN TURNED TO THEIR L TO AVOID ME AT THE SAME TIME THAT I SAW THEM AND TURNED R TO AVOID A COLLISION. THEY INDICATED THAT THEY HAD HEARD MY CALL ANNOUNCING MY TKOF AND STRAIGHT OUT DEP; BUT DIDN'T SEE ME UNTIL NEARLY TURNING INTO ME. MY ACFT HAD ITS BEACON AND LNDG LIGHT ILLUMINATED; THE OTHER HAD ITS BEACON ILLUMINATED. THE FIRST POS RPT BY THE OTHER ACFT WAS MADE AS IT ENTERED THE DOWNWIND LEG. CONTRIBUTING WAS THE FACT THAT BOTH OF THE OTHER PLTS SPEAK ITALIAN AS A PRIMARY LANGUAGE AND ARE NOT TOO PROFICIENT IN ENGLISH.