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