set G1 = G .set (WeightSelector,X);
( dom G c= dom (G .set (WeightSelector,X)) & VLabelSelector in dom G ) by Def6, FUNCT_4:10;
hence VLabelSelector in dom (G .set (WeightSelector,X)) ; :: according to GLIB_003:def 6 :: thesis: ex f being Function st
( (G .set (WeightSelector,X)) . VLabelSelector = f & dom f c= the_Vertices_of (G .set (WeightSelector,X)) )
G == G .set (WeightSelector,X) by Lm3;
then A1: the_Vertices_of G = the_Vertices_of (G .set (WeightSelector,X)) ;
( (G .set (WeightSelector,X)) . VLabelSelector = the_VLabel_of G & dom (the_VLabel_of G) c= the_Vertices_of G ) by Lm2, GLIB_000:9;
hence ex f being Function st
( (G .set (WeightSelector,X)) . VLabelSelector = f & dom f c= the_Vertices_of (G .set (WeightSelector,X)) ) by A1; :: thesis: verum