[FOM] Torsion subgroup,SOSOA

[Jan Pax]

 Hi,
could someone please answer me a question on Simpson's book SOSOA?
Page 118, III.6.2
I'm unable to decode what says (and why it is there) the formula in the proof
All m (m<|n_i|-->i <>f(m))
and the line above, why term (2m+1)x_f(m) .
In the same proof X exists already by Sigma_0^0 comprehention right?

And page 108, definition III.2.3 compactness:
how do I see that this definition mimics the usual topological definition (saying
that topological space is compact if a finite subcovering can be chosen from
any open covering)

Thank you, JP