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
  rng p c= {0,1} & 2*k+1 = min*{N:2*Sum(p|N) > N} implies p|(2*k) is
  dominated_by_0
proof
  set M={N : 2*Sum(p|N) > N};
  set q=p|(2*k);
  assume that
A1: rng p c= {0,1} and
A2: 2 * k + 1 = min*M;
  thus rng q c= {0,1} by A1;
  reconsider M as non empty Subset of NAT by A2,NAT_1:def 1;
  let m;
  assume
A3:  m <= dom q;
  then
A4: Segm m c= Segm len q by NAT_1:39;
  len q <= 2*k by AFINSQ_1:55;
  then Segm len q c= Segm(2*k) by NAT_1:39;
  then Segm m c= Segm(2*k) by A4;
  then m<=2*k by NAT_1:39;
  then
A5: m < 2*k+1 by NAT_1:13;
  assume
A6: 2*Sum(q|m) > m;
  reconsider m as Element of NAT by ORDINAL1:def 12;
  q|m=p|m & m in NAT by A4,RELAT_1:74,XBOOLE_1:1,A3;
  then m in M by A6;
  hence thesis by A2,A5,NAT_1:def 1;
end;
