
theorem Th20:
  for X be RealNormSpace for seq be summable sequence of X for z
  be Real holds z *seq is summable & Sum(z *seq)= z * Sum(seq)
proof
  let X be RealNormSpace;
  let seq be summable sequence of X;
  let z be Real;
A1: Partial_Sums(z *seq)=(z *Partial_Sums(seq)) by Th19;
A2: Partial_Sums(seq) is convergent by Def1;
  then
A3: (z *Partial_Sums(seq)) is convergent by NORMSP_1:22;
  Sum(z *seq)=lim((z *Partial_Sums(seq))) by Th19
    .=z * Sum(seq) by A2,NORMSP_1:28;
  hence thesis by A3,A1;
end;
