INSTRUCTOR PILOT AND STUDENT ON TOUCH AND GO FROM RWY 12L REPORT NMAC WITH ACFT DEPARTING RWY 12R AT CPS AT 200 FEET AGL.

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IT WAS APPROX XA46 LCL TIME. MY STUDENT AND I WERE CLRED FOR A TOUCH AND GO ON RWY 12L AT THE ST LOUIS DOWNTOWN ARPT. RWY 12R WAS ALSO IN USE FOR TKOFS AND LNDGS. AS WE WERE ROTATING ON OUR TOUCH AND GO I LOOKED OVER TO THE R AND SAW ANOTHER ACFT THAT HAD JUST TAKEN OFF FROM RWY 12R. IT WAS CLBING AT ROUGHLY THE SAME RATE AND ANGLE AS OUR ACFT WAS. NEITHER ACFT WERE GIVEN TA'S BY THE CTL TWR. I THEN TURNED TO MY STUDENT TO GET HIS ATTN TO POINT OUT THE ACFT AND TO ENSURE THAT HE MAINTAINED EXTENDED CTRLINE TO NOT CREATE A COLLISION HAZARD. WHEN WE LOOKED OVER THE SECOND TIME THE OTHER ACFT THAT DEPARTED RWY 12R WAS IN A L BANK COMING IN OUR DIRECTION AT APPROX OUR SAME ALT; WHICH WAS ABOUT 200 FT AGL. I IMMEDIATELY TOOK THE CTLS AND PUSHED THE NOSE OF OUR ACFT DOWN TO AVOID A COLLISION. IN THE ATTEMPT TO AVOID THE COLLISION WE GOT AS LOW AS APPROX 100 FT AGL. THE OTHER ACFT FLEW DIRECTLY OVER THE TOP OF US AT AN ESTIMATED VERT DISTANCE OF 100 FT AND HORIZ DISTANCE OF 0 FT. HAD WE NOT SEEN THE OTHER ACFT THERE IS NO DOUBT THAT THERE WOULD HAVE BEEN A COLLISION.