set G0 = G .set (WeightSelector,X);
VLabelSelector in dom G by Def2;
then VLabelSelector in (dom G) \/ {WeightSelector} by XBOOLE_0:def 3;
hence VLabelSelector in dom (G .set (WeightSelector,X)) by GLIB_000:7; :: according to GLIB_010:def 2 :: thesis: ex f being ManySortedSet of the_Vertices_of (G .set (WeightSelector,X)) st (G .set (WeightSelector,X)) . VLabelSelector = f
consider f being ManySortedSet of the_Vertices_of G such that
A1: G . VLabelSelector = f by Def2;
G == G .set (WeightSelector,X) by GLIB_003:7;
then the_Vertices_of G = the_Vertices_of (G .set (WeightSelector,X)) by GLIB_000:def 34;
then reconsider f = f as ManySortedSet of the_Vertices_of (G .set (WeightSelector,X)) ;
take f ; :: thesis: (G .set (WeightSelector,X)) . VLabelSelector = f
WeightSelector <> VLabelSelector by GLIB_003:def 1, GLIB_003:def 3;
hence (G .set (WeightSelector,X)) . VLabelSelector = f by A1, GLIB_000:9; :: thesis: verum