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I have been trying to learn some stability theory lately, and I have been reading "Geometric Stability Theory" by Pillay. There is a lemma he states without proving which I am trying to prove, but I am getting stuck. The lemma states that if $b\in acl(aA)$ then $$R^{\infty}(tp(b/A))\leq R^\infty(tp(ab/A))=R^\infty(tp(a/A)).$$ Here is what I want to say: since $tp(ab/A)\models tp(a/A)$, we have $R^\infty(ab/A)\leq R^\infty(a/A)$. On the other hand, since $b\in acl(aA)$, it follows that $tp(a/A)\models tp(ab/A)$, and so $R^\infty(a/A)\leq R^\infty(ab/A)$, this takes care of the equality. But then since $tp(ab/A)\models tp(b/A)$, we must have $R^\infty(ab/A)\leq R^\infty(b/A)$, so that if the inequality were true, we would necessarily have equality throughout, which I find rather unlikely. So I am pretty sure that there is something wrong with my reasoning, but I am not sure exactly what it is.

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