
theorem Th18:
  for X be RealNormSpace for seq1,seq2 be sequence of X st seq1 is
summable & seq2 is summable holds seq1-seq2 is summable & Sum(seq1-seq2)= Sum(
  seq1)-Sum(seq2)
proof
  let X be RealNormSpace;
  let seq1,seq2 be sequence of X;
  assume seq1 is summable & seq2 is summable;
  then
A1: Partial_Sums(seq1) is convergent & Partial_Sums(seq2) is convergent;
  then
A2: Partial_Sums(seq1)-Partial_Sums(seq2) is convergent by NORMSP_1:20;
A3: Partial_Sums(seq1)-Partial_Sums(seq2) =Partial_Sums(seq1-seq2) by Th16;
  Sum(seq1-seq2)=lim(Partial_Sums(seq1)-Partial_Sums(seq2)) by Th16
    .=lim(Partial_Sums(seq1))-lim(Partial_Sums(seq2)) by A1,NORMSP_1:26;
  hence thesis by A2,A3;
end;
