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;
