  We prove a result about extension of a minimal AF-equivalence relation R on
the Cantor set X, the extension being `small' in the sense that we modify R on
a thin closed subset Y of X. We show that the resulting extended equivalence
relation S is orbit equivalent to the original R, and so, in particular, S is
affable. Even in the simplest case--when Y is a finite set--this result is
highly non-trivial. The result itself--called the absorption theorem--is a
powerful and crucial tool for the study of the orbit structure of minimal
Z^n-actions on the Cantor set [GMPS]. The absorption theorem is a significant
generalization of the main theorem proved in [GPS2]. However, we shall need a
few key results from [GPS2] in order to prove the absorption theorem.
