  We present a derivation of the low-energy effective meson Lagrangian of the
Nambu -- Jona-Lasinio (NJL) model on the basis of Schwinger's proper-time
regularization of the one-loop fermion determinant. We consider the case in
which the $SU(2)\times SU(2)$ chiral symmetry of the NJL Lagrangian is broken
by the current quark mass matrix with $\hat{m}_u\ne\hat{m}_d$. The
non-degeneracy of $d$ and $u$ masses destroys one of the most crucial features
of the proper-time expansion -- the chiral-invariant structure of Seeley --
DeWitt coefficients. We show however that systematic resummations inside the
proper-time expansion are still possible and derive a result which is in full
agreement with the chiral Ward -- Takahashi identities.
