A splitting theorem for n-REA degrees

We prove that, for any D, A and U with D > (T) A circle plus U and r.e., in A circle plus U, there are pairs X-0, X-1 and Y-0, Y-1 such that D = (T) X -0 circle plus X-1; D = (T) Y-0 circle plus Y-1; and, for any i and j from {0, 1} and any set B, if X-i circle plus A greater than or equal to (T) B a nd Y-j circle plus A greater than or equal to (T) B, then A greater than or equal to (T) B. We then deduce that for any degrees d, a, and b such that a and b are recursive in d, a not greater than or equal to (T) b, and d is n-REA into a, d can be split over a avoiding b. This shows that the Main Th eorem of Cooper (Bull. Amer. Math. Soc. 23 (1990), 151-158) is false.