set G3 = G2 .set (VLabelSelector,(the_VLabel_of G1));
VLabelSelector in {VLabelSelector} by TARSKI:def 1;
then VLabelSelector in (dom G2) \/ {VLabelSelector} by XBOOLE_0:def 3;
hence VLabelSelector in dom (G2 .set (VLabelSelector,(the_VLabel_of G1))) by GLIB_000:7; :: according to GLIB_010:def 2 :: thesis: ex f being ManySortedSet of the_Vertices_of (G2 .set (VLabelSelector,(the_VLabel_of G1))) st (G2 .set (VLabelSelector,(the_VLabel_of G1))) . VLabelSelector = f
consider f being ManySortedSet of the_Vertices_of G1 such that
A1: G1 . VLabelSelector = f by Def2;
G2 == G2 .set (VLabelSelector,(the_VLabel_of G1)) by GLIB_003:7;
then the_Vertices_of G2 = the_Vertices_of (G2 .set (VLabelSelector,(the_VLabel_of G1))) by GLIB_000:def 34;
then the_Vertices_of G1 = the_Vertices_of (G2 .set (VLabelSelector,(the_VLabel_of G1))) by GLIB_007:4;
then reconsider f = f as ManySortedSet of the_Vertices_of (G2 .set (VLabelSelector,(the_VLabel_of G1))) ;
take f ; :: thesis: (G2 .set (VLabelSelector,(the_VLabel_of G1))) . VLabelSelector = f
thus (G2 .set (VLabelSelector,(the_VLabel_of G1))) . VLabelSelector = the_VLabel_of G1 by GLIB_000:8
.= f by A1, GLIB_003:def 9 ; :: thesis: verum