set G1 = G .set WeightSelector ,X;
( dom G c= dom (G .set WeightSelector ,X) & ELabelSelector in dom G ) by Def5, FUNCT_4:11;
hence ELabelSelector in dom (G .set WeightSelector ,X) ; :: according to GLIB_003:def 5 :: thesis: ex f being Function st
( (G .set WeightSelector ,X) . ELabelSelector = f & dom f c= the_Edges_of (G .set WeightSelector ,X) )
G == G .set WeightSelector ,X by Lm3;
then A1: the_Edges_of G = the_Edges_of (G .set WeightSelector ,X) by GLIB_000:def 36;
( (G .set WeightSelector ,X) . ELabelSelector = the_ELabel_of G & dom (the_ELabel_of G) c= the_Edges_of G ) by Lm1, GLIB_000:12;
hence ex f being Function st
( (G .set WeightSelector ,X) . ELabelSelector = f & dom f c= the_Edges_of (G .set WeightSelector ,X) ) by A1; :: thesis: verum