  This paper presents the derivation of a kinetic-balance condition for
explicitly correlated basis functions employed in semi-classical relativistic
calculations. Such a condition is important to ensure variational stability in
algorithms based on the first-quantized Dirac theory of 1/2-fermions. We
demonstrate that the kinetic-balance condition can be obtained from the row
reduction process commonly applied to solve systems of linear equations. The
resulting form of kinetic balance establishes a relation for the $4^N$
components of the spinor of an $N$-fermion system to the non-relativistic
limit, which is in accordance with recent developments in the field of exact
decoupling in relativistic orbital-based many-electron theory.
