reserve n,m,k for Element of NAT,
  x,X for set,
  A1 for SetSequence of X,
  Si for SigmaField of X,
  XSeq for SetSequence of Si;
reserve Omega for non empty set,
  Sigma for SigmaField of Omega,
  ASeq for SetSequence of Sigma,
  P for Probability of Sigma;

theorem Th12:
  for seq being sequence of REAL, Eseq being sequence of
  ExtREAL st seq = Eseq & seq is nonnegative summable holds Sum seq = SUM Eseq
proof
  let seq be sequence of REAL, Eseq be sequence of ExtREAL such that
A1: seq = Eseq and
A2: seq is nonnegative summable;
A3: for n being Nat holds seq.n >= 0 by A2,RINFSUP1:def 3;
  Partial_Sums seq is convergent by A2;
  then
A4: Partial_Sums seq is bounded;
  then upper_bound Partial_Sums seq = sup rng Ser Eseq by A1,Th9,Th10;
  then upper_bound Partial_Sums seq = SUM Eseq;
  hence thesis by A4,A3,RINFSUP1:24,SERIES_1:16;
end;
