NMAC. RWY TRANSGRESSION BY ACR CAUSED TKOF ACFT TO TAKE EVASIVE ACTION.

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THE INCIDENT I WAS INVOLVED IN OCCURRED AT MIDWAY ARPT ON OCT/WED/95. THE TIME WAS ABOUT XX00 LCL. I WAS THE PLT OF A C172 GIVING A FLT REVIEW WITH A STUDENT. THE OTHER ACFT INVOLVED WAS A B737 OPERATED BY ACR X AIRLINES. MY STUDENT AND I TAXIED TO THE END OF RWY 22L AND COMPLETED OUR RUN-UP/BEFORE TKOF CHKLIST. AFTER CONTACTING MIDWAY TWR; WE WERE INSTRUCTED TO POS AND HOLD RWY 22L. WE ACKNOWLEDGED THE 'POS AND HOLD' INSTRUCTIONS AND TOOK THE ACTIVE. SHORTLY AFTER; WE WERE CLRED FOR DEP. AFTER READING BACK OUR TKOF INSTRUCTIONS AND CLRNC; I BEGAN THE GND ROLL. WHILE ON THE TKOF ROLL; I GLANCED DOWN TO CHK THE ENG GAUGES AND INSTS. UPON LOOKING UP; I NOTICED THE B737 HAD BEGUN TO TAXI ON TO OUR RWY 22L. I QUICKLY TOLD MY STUDENT (WHO WAS DOING THE TKOF) THAT I HAD THE PLANE. AN ABORTED TKOF WOULD HAVE CERTAINLY CAUSED GND CONTACT WITH THE B737; SO I ELECTED TO CONTINUE WITH THE TKOF. I ROTATED THE ACFT AND IMMEDIATELY PITCHED FOR BEST ANGLE OF CLB. SECONDS LATER; I CROSSED OVER THE TAXIING ACFT AND PROCEEDED ON COURSE ASSIGNED BY ATC. UPON LNDG; I WAS ASKED TO TELEPHONE MIDWAY TWR. AFTER CALLING; I FOUND OUT THAT THE B737 WAS INSTRUCTED TO HOLD SHORT OF RWY 22L. HE ACKNOWLEDGED HIS ASSIGNED INSTRUCTIONS BUT UNFORTUNATELY FAILED TO COMPLY WITH THE INSTRUCTIONS. ADDITIONALLY MIDWAY TWR CONFIRMED THAT I WAS INDEED CLRED FOR TKOF AS DESCRIBED EARLIER. POSSIBLE CONTRIBUTING FACTORS: INTERSECTING RWY/TXWYS AT MIDWAY ARE OFTEN DIFFICULT TO DISTINGUISH BTWN. COMPLACENCY ON BEHALF OF THE B737. LACK OF MONITORING BTWN GND CTL AND TWR. LACK OF MONITORING BTWN GND CTL AND TAXIING ACFT.