*Let x, y, and z be *partial words* such that z is a *proper prefix* of y. Then x*^{2}* is *compatible* with y*^{m}z, (*x*^{2} ↑* y*^{m}z)*, for some positive integer m if and only if there exist partial words u, v, u*_{0}* , v*_{0}* , . . ., u*_{m - 1}* , v*_{m - 1}* , z*_{x} such that u ≠ ε*, v *≠ ε*, y = uv,*

* x = *(*u*_{0}*v*_{0}) . . .(*u*_{n - 1}*v*_{n - 1})*u*_{n}

= v_{n}(*u*_{n + 1}*v*_{n + 1})*. . .*(*u*_{m - 1}*v*_{m - 1})*z*_{x}

*where *0 ≤ *n * < * m, u *↑* u*_{i} and v ↑* v*_{i} for all 0 ≤ *i* < *m, z *↑* z*_{x} , and where one of the following holds:

*m = *2*n, **|u|* < *|v| , and there exist partial words u', u'*_{n} such that z_{x} = u' u_{n} , z = uu'_{n} , u ↑* u', and u*_{n} ↑* u'*_{n} .*m = *2*n + *1*, **|u|* > *|v| , and there exist partial words v'*_{2n}* and z'*_{x} such that u_{n} = v_{2n}* z*_{x} , u = v'_{2n}* z'*_{x} , v_{2n} ↑* v'*_{2n}* and z*_{x} ↑* z'*_{x} .

*A triple of partial words *(*x, y, z*)* which satisfy these properties we will refer to as a *“*good triple*”*.*