  A partial conservation of the seniority quantum number in $j=9/2$ shells has
been found recently in a numerical application. In this paper a complete
analytic proof for this problem is derived as an extension of the work by
Zamick and P. Van Isacker [Phys. Rev. C 78 (2008) 044327]. We analyze the
properties of the non-diagonal matrix elements with the help of the
one-particle and two-particle coefficients of fractional parentage (cfp's). It
is found that all non-diagonal (and the relevant diagonal) matrix elements can
be re-expressed in simple ways and are proportional to certain one-particle
cfp's. This remarkable occurrence of partial dynamic symmetry is the
consequence of the peculiar property of the $j=9/2$ shell, where all $v=3$ and
5 states are uniquely defined.
