theorem
  (for i holds s.i = s.(i+1)) implies s is constant
proof
  assume
A1: for i holds s.i = s.(i+1);
  now
    let i,j;
A2: now
      let i,j such that
A3:   i <= j;
      defpred P[Nat] means i <= $1 implies s.i = s.$1;
A4:   for j being Nat st P[j] holds P[j+1]
      proof
        let j being Nat such that
A5:     P[j];
        assume i <= j+1;
        then i < j+1 or i = j+1 by XXREAL_0:1;
        hence thesis by A1,A5,NAT_1:13;
      end;
A6:   P[0] by NAT_1:3;
      for j being Nat holds P[j] from NAT_1:sch 2(A6,A4);
      hence s.i = s.j by A3;
    end;
    i <= j or j <= i;
    hence s.i = s.j by A2;
  end;
  hence thesis;
end;
