The setting is that we are given a set of inputs (the matrix S), and we want to (adaptively) choose which inputs (rows of S) to obtain the output (corresponding row of T ) for.
proposition 4 states that under the linear system formulation, the set of safe inputs F LS is exactly the same as the row space of S. if we ask for an input that is already in the row space of S, this will not affect F LS at all.