reserve X for set,
  F for Field_Subset of X,
  M for Measure of F,
  A,B for Subset of X,
  Sets for SetSequence of X,
  seq,seq1,seq2 for ExtREAL_sequence,
  n,k for Nat;

theorem Th2:
  seq is nonnegative implies seq is summable & SUM seq = Sum seq
proof
  assume seq is nonnegative;
  then
A1: Partial_Sums seq is non-decreasing by MESFUNC9:16;
  then Partial_Sums seq is convergent by RINFSUP2:37;
  hence seq is summable;
  lim Partial_Sums seq = sup Partial_Sums seq by A1,RINFSUP2:37;
  hence Sum seq = SUM seq by Th1;
end;
