reserve r,r1,g for Real,
  n,m,k for Nat,
  seq,seq1, seq2 for Real_Sequence,
  f,f1,f2 for PartFunc of REAL,REAL,
  x for set;
reserve r,r1,r2,g,g1,g2 for Real;

theorem
  seq is divergent_to+infty or seq is divergent_to-infty implies ex n st
  for m st n<=m holds seq^\m is non-zero
proof
  assume
A1: seq is divergent_to+infty or seq is divergent_to-infty;
  now
    per cases by A1;
    suppose
      seq is divergent_to+infty;
      then consider n such that
A2:   for m st n<=m holds 0<seq.m;
      take n;
      let m such that
A3:   n<=m;
      now
        let k;
        0<seq.(k+m) by A2,A3,NAT_1:12;
        hence 0<>(seq^\m).k by NAT_1:def 3;
      end;
      hence seq^\m is non-zero by SEQ_1:5;
    end;
    suppose
      seq is divergent_to-infty;
      then consider n such that
A4:   for m st n<=m holds seq.m<0;
      take n;
      let m such that
A5:   n<=m;
      now
        let k;
        seq.(k+m)<0 by A4,A5,NAT_1:12;
        hence (seq^\m).k<>0 by NAT_1:def 3;
      end;
      hence seq^\m is non-zero by SEQ_1:5;
    end;
  end;
  hence thesis;
end;
