A splitting theorem for $n-REA$ degrees

We prove that, for any $D$, $A$ and $U$ with $D>_{T}A\oplus U$ and r.e., in $A\oplus U$, there are pairs $X_{0},X_{1}$ and $Y_{0},Y_{1}$ such that $D\equiv_{T}X_{0}\oplus X_{1}$; $D\equiv_{T}Y_{0}\oplus Y_{1}$; and, for any $i$ and $j$ from $\{0,1\}$ and any set $B$, if $X_{i}\oplus A\geq_{T}B$ and $Y_{j}\oplus A\geq_{T}B$, then $A\geq_{T}B$. We then deduce that for any degrees $\mathbf{d}$, $\mathbf{a}$, and $\mathbf{b}$ such that $\mathbf{a}$and $\mathbf{b}$ are recursive in $\mathbf{d}$, $\mathbf{a}\not \geq _{T}\mathbf{b}$, and $\mathbf{d}$ is $n-REA$ into $\mathbf{a}$, $\mathbf{d}$can be split over $\mathbf{a}$ avoiding $\mathbf{b}$. This shows that the Main Theorem of Cooper (Bull. Amer. Math. Soc. 23 (1990), 151-158) is false.