reserve a,b,c,d for positive Real,
  m,u,w,x,y,z for Real,
  n,k for Nat,
  s,s1 for Real_Sequence;

theorem Th50:
  (for n holds s.n<1) implies for n holds Partial_Sums(s).n<n+1
proof
  defpred X[Nat] means Partial_Sums(s).$1<$1+1;
  assume
A1: for n holds s.n<1;
A2: for n st X[n] holds X[n+1]
  proof
    let n;
A3: Partial_Sums(s).(n+1) = Partial_Sums(s).n+s.(n+1) by SERIES_1:def 1;
    assume Partial_Sums(s).n<n+1;
    hence thesis by A1,A3,XREAL_1:8;
  end;
  Partial_Sums(s).0 = s.0 by SERIES_1:def 1;
  then
A4: X[0] by A1;
  for n holds X[n] from NAT_1:sch 2(A4,A2);
  hence thesis;
end;
