  For the class of solvable groups of homeomorphisms of the line preserving
orientation and containing a freely acting element, we establish the
metabelianity of the quotient group $G/H_G$, where the elements of the normal
subgroup $H_G$ are stabilizers of the minimal set. This fact is an important
element in the classification theorem, used, in particular, in the study of the
Thompson's group $F$.
