let G be _finite _Graph; :: thesis: for L being LexBFS:Labeling of G st dom (L `1) <> the_Vertices_of G holds
not LexBFS:PickUnnumbered L in dom (L `1)
let L be LexBFS:Labeling of G; :: thesis: ( dom (L `1) <> the_Vertices_of G implies not LexBFS:PickUnnumbered L in dom (L `1) )
assume A1: dom (L `1) <> the_Vertices_of G ; :: thesis: not LexBFS:PickUnnumbered L in dom (L `1)
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 = (L `2) | ((the_Vertices_of G) \ (dom (L `1))) and
for x being finite Subset of NAT st x in S holds
(x,1) -bag in B and
A4: for x being set st x in B holds
ex y being finite Subset of NAT st
( y in S & x = (y,1) -bag ) and
A5: LexBFS:PickUnnumbered L = the Element of F " {(support (max (B,(InvLexOrder NAT))))} by A1, Def11;
set mw = max (B,(InvLexOrder NAT));
max (B,(InvLexOrder NAT)) in B by Def4;
then consider y being finite Subset of NAT such that
A6: y in S and
A7: max (B,(InvLexOrder NAT)) = (y,1) -bag by A4;
y = support (max (B,(InvLexOrder NAT))) by A7, UPROOTS:8;
then not F " {(support (max (B,(InvLexOrder NAT))))} is empty by A2, A6, FUNCT_1:72;
then A8: LexBFS:PickUnnumbered L in dom F by A5, FUNCT_1:def 7;
assume LexBFS:PickUnnumbered L in dom (L `1) ; :: thesis: contradiction
then A9: not LexBFS:PickUnnumbered L in (the_Vertices_of G) \ (dom (L `1)) by XBOOLE_0:def 5;
dom F = (dom (L `2)) /\ ((the_Vertices_of G) \ (dom (L `1))) by A3, RELAT_1:61;
hence contradiction by A8, A9, XBOOLE_0:def 4; :: thesis: verum