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 Th52:
  (for n being Nat holds s.n>0 & s1.n=1/s.n)
  implies for n holds Partial_Sums(s1).n>0
proof
  defpred X[Nat] means Partial_Sums(s1).$1>0;
  assume
A1: for n being Nat holds s.n>0 & s1.n=1/s.n;
  then
A2: s1.0=1/s.0;
A3: for n st X[n] holds X[n+1]
  proof
    let n;
    assume
A4: Partial_Sums(s1).n>0;
A5: Partial_Sums(s1).(n+1)=Partial_Sums(s1).n+s1.(n+1) by SERIES_1:def 1;
A6: s.(n+1)>0 by A1;
    s1.(n+1)=1/s.(n+1) by A1;
    hence thesis by A4,A5,A6;
  end;
  s.0>0 by A1;
  then
A7: X[0] by A2,SERIES_1:def 1;
  for n holds X[n] from NAT_1:sch 2(A7,A3);
  hence thesis;
end;
