set G1 = G .set VLabelSelector ,X;
( dom (G .set VLabelSelector ,X) = (dom G) \/ {VLabelSelector } & VLabelSelector in {VLabelSelector } ) by GLIB_000:9, TARSKI:def 1;
hence VLabelSelector in dom (G .set VLabelSelector ,X) by XBOOLE_0:def 3; :: according to GLIB_003:def 6 :: thesis: ex f being Function st
( (G .set VLabelSelector ,X) . VLabelSelector = f & dom f c= the_Vertices_of (G .set VLabelSelector ,X) )
G == G .set VLabelSelector ,X by Lm3;
then A1: the_Vertices_of G = the_Vertices_of (G .set VLabelSelector ,X) by GLIB_000:def 36;
dom X c= the_Vertices_of G ;
hence ex f being Function st
( (G .set VLabelSelector ,X) . VLabelSelector = f & dom f c= the_Vertices_of (G .set VLabelSelector ,X) ) by A1, GLIB_000:11; :: thesis: verum