IN A 'SEE AND AVOID' EVENT; PA28 INSTRUCTOR ADVISES STUDENT TO WATCH AS A CONVERGING; SAME ALT CONFLICT IS DEVELOPING. EVENT STARTLES INSTRUCTOR.

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WE WERE APCHING INITIAL APCH FIX; 240 DEG RADIAL; DME ARC FOR ILS RWY 13; HUT; FROM THE SE AT 3500 FT WHEN I (THE INSTRUCTOR) SAW THE APCHING ACFT 12 O'CLOCK POS; ABOUT 1/2 MI. IT SEEMED TO ME WE WOULD PASS SAFELY BEHIND THIS ACFT. I TOLD THE STUDENT TO LOOK AT IT. IN ABOUT 4-5 SECONDS; WE PASSED; BUT MUCH CLOSER THAN I HAD EXPECTED. IT STARTLED ME AND I LIFTED THE R WING. THE OTHER ACFT WAS ON A SE HEADING. WHEN I LIFTED THE WING; THE OTHER ACFT SAW US FOR THE FIRST TIME. THE LESSON I THINK I LEARNED HERE; AND THE PURPOSE FOR SENDING THIS RPT; IS THAT IF I OR ANY PLT SEES ANOTHER ACFT APCHING IN CLOSE PROX -- EVASIVE ACTION SHOULD BE EXECUTED IMMEDIATELY. A FEW DEGS HEADING DIFFERENCE OF EITHER OF OUR ACFT COULD HAVE CAUSED A MIDAIR COLLISION. WE WERE AT THE OUTER EDGE OF WICHITA APCH'S RADAR COVERAGE. WICHITA NEVER GAVE US TFC AND SHORTLY AFTER OUR NEAR MISS; WE WERE INFORMED THAT 'RADAR CONTACT LOST; RPT ESTABLISHED ON THE APCH.' EVASIVE ACTION SERVES 2 PURPOSES: 1) YOU AVOID THE OTHER ACFT; 2) ANY MOVEMENT OF EITHER ACFT MAY GET THE ATTN OF OTHER PLT.