
theorem Th30:
  for G being _finite _Graph, L be LexBFS:Labeling of G st dom L`1
  <> the_Vertices_of G holds not LexBFS:PickUnnumbered(L) in dom L`1
proof
  let G be _finite _Graph, L be LexBFS: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 = LexBFS:PickUnnumbered(L);
  consider S being non empty finite Subset of bool NAT, B being non empty
  finite Subset of Bags NAT, F being Function such that
A2: S = rng F and
A3: F = V2G | (VG \ dom VLG) and
  for x being finite Subset of NAT holds x in S implies (x,1)-bag in B and
A4: for x being set holds x in B implies ex y being finite Subset of NAT
  st y in S & x = (y,1)-bag and
A5: w = the Element of F " {support max(B,InvLexOrder NAT)} by A1,Def11;
  set mw = max(B,InvLexOrder NAT);
  mw in B by Def4;
  then consider y being finite Subset of NAT such that
A6: y in S and
A7: mw = (y,1)-bag by A4;
  y = support mw by A7,UPROOTS:8;
  then F " {support mw} is non empty by A2,A6,FUNCT_1:72;
  then
A8: w in dom F by A5,FUNCT_1:def 7;
  assume w in dom VLG;
  then
A9: 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 A8,A9,XBOOLE_0:def 4;
end;
