set f = (the_Vertices_of G) --> 0 ;
A1: rng {} c= NAT ;
A2: rng ((the_Vertices_of G) --> 0 ) c= NAT ;
dom ((the_Vertices_of G) --> 0 ) = the_Vertices_of G by FUNCOP_1:19;
then A3: (the_Vertices_of G) --> 0 in Funcs (the_Vertices_of G),NAT by A2, FUNCT_2:def 2;
dom {} c= the_Vertices_of G by XBOOLE_1:2;
then {} in PFuncs (the_Vertices_of G),NAT by A1, PARTFUN1:def 5;
hence [{} ,((the_Vertices_of G) --> 0 )] is MCS:Labeling of G by A3, ZFMISC_1:def 2; :: thesis: verum