reserve a, b, r, M2 for Real;
reserve Rseq,Rseq1,Rseq2 for Real_Sequence;
reserve k, n, m, m1, m2 for Nat;
reserve X for RealUnitarySpace;
reserve g for Point of X;
reserve seq, seq1, seq2 for sequence of X;

theorem
  seq1 is summable & seq2 is summable implies seq1 + seq2 is summable &
  Sum(seq1 + seq2) = Sum(seq1) + Sum(seq2)
proof
  assume seq1 is summable & seq2 is summable;
  then
A1: Partial_Sums(seq1) is convergent & Partial_Sums(seq2) is convergent;
  then Partial_Sums(seq1) + Partial_Sums(seq2) is convergent by BHSP_2:3;
  then Partial_Sums(seq1 + seq2) is convergent by Th1;
  hence seq1 + seq2 is summable;
  thus Sum(seq1 + seq2) = lim (Partial_Sums(seq1) + Partial_Sums(seq2)) by Th1
    .= Sum(seq1) + Sum(seq2) by A1,BHSP_2:13;
end;
