reserve a,b,c for positive Real,
  m,x,y,z for Real,
  n for Nat,
  s,s1,s2,s3,s4,s5 for Real_Sequence;

theorem
  (for n holds s.n>0 & s.n>s.(n-1)) implies (n+1)*s.(n+1) > ( Partial_Sums s).n
proof
  defpred X[Nat] means ($1+1)*s.($1+1)>(Partial_Sums s).$1;
  assume
A1: for n holds s.n>0 & s.n>s.(n-1);
A2: for n st X[n] holds X[n+1]
  proof
    let n;
    assume (n+1)*s.(n+1)>(Partial_Sums s).n;
    then
A3: (Partial_Sums(s)).n+s.(n+1)<(n+1)*s.(n+1)+s.(n+1) by XREAL_1:6;
    s.(n+2)>s.((n+2)-1) by A1;
    then (n+2)*s.(n+2)>(n+2)*s.(n+1) by XREAL_1:68;
    then (Partial_Sums s).n+s.(n+1)<(n+2)*s.(n+2) by A3,XXREAL_0:2;
    hence thesis by SERIES_1:def 1;
  end;
  s.1>s.(1-1) by A1;
  then
A4: X[0] by SERIES_1:def 1;
  for n holds X[n] from NAT_1:sch 2(A4,A2);
  hence thesis;
end;
