set G0 = G .set (ELabelSelector,X);

VLabelSelector in dom G by Def2;

then VLabelSelector in (dom G) \/ {ELabelSelector} by XBOOLE_0:def 3;

hence VLabelSelector in dom (G .set (ELabelSelector,X)) by GLIB_000:7; :: according to GLIB_010:def 2 :: thesis: ex f being ManySortedSet of the_Vertices_of (G .set (ELabelSelector,X)) st (G .set (ELabelSelector,X)) . VLabelSelector = f

consider f being ManySortedSet of the_Vertices_of G such that

A2: G . VLabelSelector = f by Def2;

G == G .set (ELabelSelector,X) by GLIB_003:7;

then the_Vertices_of G = the_Vertices_of (G .set (ELabelSelector,X)) by GLIB_000:def 34;

then reconsider f = f as ManySortedSet of the_Vertices_of (G .set (ELabelSelector,X)) ;

take f ; :: thesis: (G .set (ELabelSelector,X)) . VLabelSelector = f

ELabelSelector <> VLabelSelector by GLIB_003:def 3, GLIB_003:def 2;

hence (G .set (ELabelSelector,X)) . VLabelSelector = f by A2, GLIB_000:9; :: thesis: verum

VLabelSelector in dom G by Def2;

then VLabelSelector in (dom G) \/ {ELabelSelector} by XBOOLE_0:def 3;

hence VLabelSelector in dom (G .set (ELabelSelector,X)) by GLIB_000:7; :: according to GLIB_010:def 2 :: thesis: ex f being ManySortedSet of the_Vertices_of (G .set (ELabelSelector,X)) st (G .set (ELabelSelector,X)) . VLabelSelector = f

consider f being ManySortedSet of the_Vertices_of G such that

A2: G . VLabelSelector = f by Def2;

G == G .set (ELabelSelector,X) by GLIB_003:7;

then the_Vertices_of G = the_Vertices_of (G .set (ELabelSelector,X)) by GLIB_000:def 34;

then reconsider f = f as ManySortedSet of the_Vertices_of (G .set (ELabelSelector,X)) ;

take f ; :: thesis: (G .set (ELabelSelector,X)) . VLabelSelector = f

ELabelSelector <> VLabelSelector by GLIB_003:def 3, GLIB_003:def 2;

hence (G .set (ELabelSelector,X)) . VLabelSelector = f by A2, GLIB_000:9; :: thesis: verum