
theorem Th58:
  for G being _finite _Graph, L be MCS:Labeling of G, x being set
  st not x in dom L`1 & dom L`1 <> the_Vertices_of G holds
  (L`2).x <= (L`2).(MCS:PickUnnumbered(L))
proof
  let G be _finite _Graph, L be MCS: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 = MCS:PickUnnumbered(L);
  consider S being finite non empty natural-membered set, F being Function
  such that
A3: S = rng F and
A4: F = V2G | (VG \ dom VLG) and
A5: w = the Element of F " {max S} by A2,Def19;
A6: dom F = dom V2G /\ (VG \ dom VLG) by A4,RELAT_1:61;
  set y = max S;
  y in rng F by A3,XXREAL_2:def 8;
  then
A7: F " {max S} is non empty by FUNCT_1:72;
  then w in dom F by A5,FUNCT_1:def 7;
  then
A8: V2G.w = F.w by A4,FUNCT_1:47;
  F.w in {max S} by A5,A7,FUNCT_1:def 7;
  then
A9: V2G.w = y by A8,TARSKI:def 1;
A10: dom L`2 = 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
A11: x in dom F by A10,A6,XBOOLE_0:def 4;
    then
A12: F.x in S by A3,FUNCT_1:def 3;
    F.x = V2G.x by A4,A11,FUNCT_1:47;
    hence thesis by A9,A12,XXREAL_2:def 8;
  end;
  suppose
    not x in the_Vertices_of G;
    hence thesis by A10,FUNCT_1:def 2;
  end;
end;
