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

theorem
  (for n holds s1.n = n-root (abs(s).n)) & s1 is convergent & lim s1 < 1
  implies s is absolutely_summable
proof
  assume that
A1: for n holds s1.n = n-root (abs(s).n) and
A2: s1 is convergent & lim s1 < 1;
  now
    let n;
    abs(s).n = |.s.n.| by SEQ_1:12;
    hence abs(s).n >= 0 by COMPLEX1:46;
    thus s1.n = n-root (abs(s).n) by A1;
  end;
  then abs(s) is summable by A2,Th28;
  hence thesis;
end;
