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

theorem Th10:
  s is summable implies r(#)s is summable & Sum(r(#)s) =r*Sum(s)
proof
  assume s is summable;
  then
A1: Partial_Sums(s) is convergent;
  then r(#)Partial_Sums(s) is convergent;
  then Partial_Sums(r(#)s) is convergent by Th9;
  hence r(#)s is summable;
  thus Sum(r(#)s) =lim (r(#)Partial_Sums(s)) by Th9
    .=r*Sum(s) by A1,SEQ_2:8;
end;
