
theorem Th29:
  for G being _finite _Graph, L be LexBFS:Labeling of G, x being
set st not x in dom L`1 & dom L`1 <> the_Vertices_of G holds (L`2.x,1)-bag <= (
  L`2.(LexBFS:PickUnnumbered(L)),1)-bag, InvLexOrder NAT
proof
  let G be _finite _Graph, L be LexBFS:Labeling of G, x be set such that
A1: not x in dom L`1 and
A2: 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
A3: S = rng F and
A4: F = V2G | (VG \ dom VLG) and
A5: for x being finite Subset of NAT holds x in S implies (x,1)-bag in B and
A6: 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
A7: w = the Element of F " {support max(B,InvLexOrder NAT)} by A2,Def11;
A8: dom F = dom V2G /\ (VG \ dom VLG) by A4,RELAT_1:61;
  set mw = max(B,InvLexOrder NAT);
  mw in B by Def4;
  then consider y being finite Subset of NAT such that
A9: y in S and
A10: mw = (y,1)-bag by A6;
A11: y = support mw by A10,UPROOTS:8;
  then
A12: F " {support mw} is non empty by A3,A9,FUNCT_1:72;
  then w in dom F by A7,FUNCT_1:def 7;
  then
A13: V2G.w = F.w by A4,FUNCT_1:47;
  F.w in {support mw} by A7,A12,FUNCT_1:def 7;
  then
A14: (V2G.w,1)-bag = mw by A10,A11,A13,TARSKI:def 1;
A15: dom V2G = the_Vertices_of G by FUNCT_2:def 1;
  per cases;
  suppose
    x in the_Vertices_of G;
    then x in VG \ dom VLG by A1,XBOOLE_0:def 5;
    then
A16: x in dom F by A15,A8,XBOOLE_0:def 4;
    then
A17: F.x in S by A3,FUNCT_1:def 3;
    F.x = V2G.x by A4,A16,FUNCT_1:47;
    then (V2G.x,1)-bag in B by A5,A17;
    hence thesis by A14,Def4;
  end;
  suppose
    not x in the_Vertices_of G;
    then V2G.x = {} by A15,FUNCT_1:def 2;
    then (V2G.x,1)-bag = EmptyBag NAT by UPROOTS:9;
    hence thesis by TERMORD:9;
  end;
end;
