An extension of Harrington's noncupping theorem

(1) Call a c.e. degree b anti-cupping relative to x, if there is a c.e. a < b such that for any c.e. w, w not greater than or equal to x implies a U w not greater than or equal to b U x. (ii) Call a c.e. degree b everywhere anti-cupping (e.a.c.), if it is anti-cupping relative to x for each c.e. degree x. By a tree method, we prove that every high c.e. degree has e.a.c. property by extending Harrington's anti-cupping theorem.