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 Th7:
  (seq^\k is divergent_to+infty implies seq is divergent_to+infty)
  & (seq^\k is divergent_to-infty implies seq is divergent_to-infty)
proof
  thus seq^\k is divergent_to+infty implies seq is divergent_to+infty
  proof
    assume
A1: seq^\k is divergent_to+infty;
    let r;
    consider n1 be Nat such that
A2: for m st n1<=m holds r<(seq^\k).m by A1;
    take n=n1+k;
    let m;
    assume n<=m;
    then consider n2 be Nat such that
A3: m=n+n2 by NAT_1:10;
    reconsider n2 as Element of NAT by ORDINAL1:def 12;
A4: r<(seq^\k).(n1+n2) by A2,NAT_1:12;
    n1+n2+k=m by A3;
    hence thesis by A4,NAT_1:def 3;
  end;
  assume
A5: seq^\k is divergent_to-infty;
  let r;
  consider n1 be Nat such that
A6: for m st n1<=m holds (seq^\k).m<r by A5;
  take n=n1+k;
  let m;
  assume n<=m;
  then consider n2 be Nat such that
A7: m=n+n2 by NAT_1:10;
  reconsider n2 as Element of NAT by ORDINAL1:def 12;
A8: (seq^\k).(n1+n2)< r by A6,NAT_1:12;
  n1+n2+k=m by A7;
  hence thesis by A8,NAT_1:def 3;
end;
