The problem of R-n-observability is considered for the simplest linear time-delay differential-algebraic system consisting of differential and difference equations. A determining equation system is introduced and a number of algebraic properties of the determining equation solutions is established, in particular, the well-known Hamilton-Cayley matrix theorem is generalized to the solutions of determining equation. As a result, an effective parametric rank criterion for the R-n-observability is given. A dual controllability result is also formulated.