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 ;

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 ;

then {} in PFuncs ((the_Vertices_of G),NAT) by A1, PARTFUN1:def 3;

hence [{},((the_Vertices_of G) --> 0)] is MCS:Labeling of G by A3, ZFMISC_1:def 2; :: thesis: verum

A1: rng {} c= NAT ;

A2: rng ((the_Vertices_of G) --> 0) c= NAT ;

dom ((the_Vertices_of G) --> 0) = the_Vertices_of G ;

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 ;

then {} in PFuncs ((the_Vertices_of G),NAT) by A1, PARTFUN1:def 3;

hence [{},((the_Vertices_of G) --> 0)] is MCS:Labeling of G by A3, ZFMISC_1:def 2; :: thesis: verum