set f = (the_Vertices_of G) --> {};
A1: rng {} c= NAT ;
{} c= NAT ;
then {} in Fin NAT by FINSUB_1:def 5;
then {{}} c= Fin NAT by ZFMISC_1:31;
then A2: rng ((the_Vertices_of G) --> {}) c= Fin NAT ;
dom ((the_Vertices_of G) --> {}) = the_Vertices_of G ;
then A3: (the_Vertices_of G) --> {} in Funcs ((the_Vertices_of G),(Fin NAT)) by A2, FUNCT_2:def 2;
dom {} c= the_Vertices_of G ;
then {} in PFuncs ((the_Vertices_of G),NAT) by A1, PARTFUN1:def 3;
hence [{},((the_Vertices_of G) --> {})] is LexBFS:Labeling of G by A3, ZFMISC_1:def 2; :: thesis: verum