theorem
  not FirstNotIn L in L
proof
  set FNI = FirstNotIn L;
  consider sn being non empty Subset of NAT such that
A1: FNI = intloc min sn and
A2: sn = {k where k is Element of NAT : not intloc k in L} by Def3;
  min sn in sn by XXREAL_2:def 7;
  then ex k being Element of NAT st k = min sn & not intloc k in L by A2;
  hence thesis by A1;
end;
