  We define a linear homogeneous equation to be strongly r-regular if, when a
finite number of inequalities is added to the equation, the system of the
equation and inequalities is still r-regular. In this paper, we show that, if a
linear homogeneous equation is r-regular, then it is strongly r-regular. In
2009, Alexeev and Tsimerman introduced a family of equations, each of which is
(n-1)-regular but not n-regular, verifying a conjecture of Rado from 1933.
These equations are actually strongly (n-1)-regular as an immediate corollary
of our results.
