reserve n,m,k for Nat;
reserve a,p,r for Real;
reserve s,s1,s2,s3 for Real_Sequence;

theorem
  s1 is summable & s2 is summable implies s1-s2 is summable & Sum(s1-s2)
  = Sum(s1) - Sum(s2)
proof
  assume s1 is summable & s2 is summable;
  then
A1: Partial_Sums(s1) is convergent & Partial_Sums(s2) is convergent;
  then Partial_Sums(s1) - Partial_Sums(s2) is convergent;
  then Partial_Sums(s1-s2) is convergent by Th6;
  hence s1-s2 is summable;
  thus Sum(s1-s2) =lim (Partial_Sums(s1) - Partial_Sums(s2)) by Th6
    .=Sum(s1) - Sum(s2) by A1,SEQ_2:12;
end;
