  We show that Nash-Williams' theorem asserting that the countable transfinite
sequences of elements of a better-quasi-ordering ordered by embeddability form
a better-quasi-ordering is provable in the subsystem of second order arithmetic
Pi^1_1-CA_0 but is not equivalent to Pi^1_1-CA_0. We obtain some partial
results towards the proof of this theorem in the weaker subsystem ATR_0 and we
show that the minimality lemmas typical of wqo and bqo theory imply Pi^1_1-CA_0
and hence cannot be used in such a proof.
