Suppose that the set $S$ is linearly independent. Apply the Gram-Schmidt procedure (Theorem GSP) twice, creating first the linearly independent set $T_1$ from $S$, and then creating $T_2$ from $T_1$. As a consequence of Exercise O.T30, prove that $T_1=T_2$. In other words, it is pointless to apply the Gram-Schmidt procedure twice.