reserve x, x1, x2, y, X, D for set,
  i, j, k, l, m, n, N for Nat,
  p, q for XFinSequence of NAT,
  q9 for XFinSequence,
  pd, qd for XFinSequence of D;

theorem Th9:
  p is dominated_by_0 & n <= len p-2*Sum p implies p^(n-->1) is dominated_by_0
proof
  set q=(n-->1);
  assume that
A1: p is dominated_by_0 and
A2: n <= len p- 2 * Sum p;
  rng q c= {1} & {1} c= {0,1} by FUNCOP_1:13,ZFMISC_1:7;
  then
A3: rng q c= {0,1};
  rng p c= {0,1} by A1;
  then rng p \/ rng q c= {0,1} by A3,XBOOLE_1:8;
  hence rng (p^q) c= {0,1} by AFINSQ_1:26;
  let m such that
A4: m <= dom (p^q);
  now
    per cases;
    suppose
      m <= dom p;
      then (p^q)|m=p|m by AFINSQ_1:58;
      hence thesis by A1,Th2;
    end;
    suppose
      m > dom p;
      then reconsider md=m-dom p as Nat by NAT_1:21;
A5:   m=dom p+ md;
      Sum (md-->1)=md*1 by AFINSQ_2:58;
      then
A6:   Sum (p^(md-->1))=Sum p+ md by AFINSQ_2:55;
      dom q = n & len q = dom q;
      then dom (p^q)= len p + n by AFINSQ_1:def 3;
      then md+dom p<=n+dom p by A4;
      then
A7:   md <= n by XREAL_1:6;
      then q|md = md-->1 by Lm1;
      then (p^q)|m=p^(md-->1) by A5,AFINSQ_1:59;
      then 2*Sum ((p^q)|m)= 2 * Sum p + m-dom p +md by A6;
      then
A8:   2*Sum ((p^q)|m) <=2 * Sum p+m-dom p+n by A7,XREAL_1:6;
      n-n<=len p- 2 * Sum p-n by A2,XREAL_1:9;
      then m-(len p-2 * Sum p-n)<=m-0 by XREAL_1:10;
      hence thesis by A8,XXREAL_0:2;
    end;
  end;
  hence thesis;
end;
