
theorem Th59:
  for G being _finite _Graph, L be MCS:Labeling of G st dom L`1 <>
  the_Vertices_of G holds not MCS:PickUnnumbered(L) in dom L`1
proof
  let G be _finite _Graph, L be MCS:Labeling of G such that
A1: dom L`1 <> the_Vertices_of G;
  set VG = the_Vertices_of G;
  set V2G = L`2;
  set VLG = L`1;
  set w = MCS:PickUnnumbered(L);
  consider S being finite non empty natural-membered set, F being Function
  such that
A2: S = rng F and
A3: F = V2G | (VG \ dom VLG) and
A4: w = the Element of F " {max S} by A1,Def19;
  set y = max S;
  y in rng F by A2,XXREAL_2:def 8;
  then F " {max S} is non empty by FUNCT_1:72;
  then
A5: w in dom F by A4,FUNCT_1:def 7;
  assume w in dom VLG;
  then
A6: not w in VG \ dom VLG by XBOOLE_0:def 5;
  dom F = dom V2G /\ (VG \ dom VLG) by A3,RELAT_1:61;
  hence contradiction by A5,A6,XBOOLE_0:def 4;
end;
