
theorem
  for m being Element of NAT, X being finite Subset of [:NAT,{m}:] ex k
being Element of NAT st for l being Element of NAT st l >= k holds not [l,m] in
  X
proof
  let m be Element of NAT;
  let X be finite Subset of [:NAT,{m}:];
  per cases;
  suppose
    X is non empty;
    then reconsider X9 = X as finite non empty Subset of [:NAT,{m}:];
    consider n being non zero Element of NAT such that
A1: X9 c= [:Seg n \/ {0},{m}:] by Th3;
    take k = n+1;
    let l be Element of NAT;
    assume
A2: l >= k;
    assume [l,m] in X;
    then
A3: l in Seg n \/ {0} by A1,ZFMISC_1:87;
    now
      per cases by A3,XBOOLE_0:def 3;
      suppose
        l in Seg n;
        then l <= n by FINSEQ_1:1;
        hence contradiction by A2,NAT_1:13;
      end;
      suppose
        l in {0};
        hence contradiction by A2,TARSKI:def 1;
      end;
    end;
    hence thesis;
  end;
  suppose
    X is empty;
    then for l being Element of NAT st l >= 0 holds not [l,m] in X;
    hence thesis;
  end;
end;
