PA28 INSTRUCTOR AND STUDENT HAD CLRNC TO CROSS A RWY. THE STUDENT STOPPED SHORT OF THE RWY WHEN HE NOTICED AN ACR ON TKOF ROLL ON THE INTERSECTING RWY.

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I WAS ACTING AS INSTRUCTOR ON AN IFR TRAINING FLT FROM MDW TO VYS. FROM THE W RAMP; WE WERE INSTRUCTED TO TAXI TO RWY 31L VIA TXWYS W; J; F; F4. NO HOLD SHORT WAS MENTIONED. THIS TAXI RTE CROSSES RWYS 4L/22R AND 4R/22L; NEITHER OF WHICH WAS ADVERTISED AS ACTIVE ON THE ATIS. AS WE APCHED RWY 22L; MY STUDENT (WHO WAS IN THE L SEAT); NOTICED AN ACFT ON RWY 22L AND DECIDED TO STOP. ABOUT THIS TIME; I OBSERVED THAT THE ACFT WAS ON ITS TKOF ROLL. ALTHOUGH WE WERE PAST THE HOLD SHORT LINES; ALL PARTS OF OUR ACFT WERE SHORT OF THE EDGE OF RWY 22L; SO WE HELD POS. AFTER THE OTHER ACFT WAS CLR; I QUERIED GND AS TO WHETHER WE COULD NOW CROSS RWY 22L. FROM THE TONE OF THEIR RESPONSE; IT SOUNDED LIKE THEY HAD EITHER FORGOTTEN ABOUT US OR HAD NOT COORDINATED OUR TAXI RTE WITH LCL CTL'S DECISION TO USE RWY 22L FOR A DEP. WHILE WE DID FOLLOW OUR ASSIGNED TAXI RTE; I WOULD HAVE BEEN MORE VIGILANT IN LOOKING FOR TFC ON RWY 22L/R IF ATIS HAD ADVERTISED THEIR USE. I BELIEVE THAT IF EXPLICIT PERMISSION FROM GND CTL HAD BEEN REQUIRED TO CROSS RWYS; ALONG WITH BETTER COORD BTWN GND AND LCL CTLRS; THIS WOULD NOT HAVE HAPPENED. IN ADDITION; GND RADAR MIGHT ALSO HAVE WARNED THE CTLRS IN TIME.