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
  seq is summable implies a * seq is summable & Sum(a * seq) = a * Sum( seq)
proof
  assume seq is summable;
  then
A1: Partial_Sums(seq) is convergent;
  then a * Partial_Sums(seq) is convergent by BHSP_2:5;
  then Partial_Sums(a * seq) is convergent by Th3;
  hence a * seq is summable;
  thus Sum(a * seq) = lim (a * Partial_Sums(seq)) by Th3
    .= a * Sum(seq) by A1,BHSP_2:15;
end;
