set G2 = G .set (OrderingSelector,X);

A1: (G .set (OrderingSelector,X)) . ELabelSelector = G . ELabelSelector by GLIB_003:def 2, GLIB_000:9;

ELabelSelector in dom G by GLIB_003:def 5;

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

then A2: ELabelSelector in dom (G .set (OrderingSelector,X)) by GLIB_000:7;

the_Edges_of G = the_Edges_of (G .set (OrderingSelector,X)) by Lm1, GLIB_000:10;

then ex f being Function st

( (G .set (OrderingSelector,X)) . ELabelSelector = f & dom f c= the_Edges_of (G .set (OrderingSelector,X)) ) by A1, GLIB_003:def 5;

hence G .set (OrderingSelector,X) is [ELabeled] by A2, GLIB_003:def 5; :: thesis: verum

A1: (G .set (OrderingSelector,X)) . ELabelSelector = G . ELabelSelector by GLIB_003:def 2, GLIB_000:9;

ELabelSelector in dom G by GLIB_003:def 5;

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

then A2: ELabelSelector in dom (G .set (OrderingSelector,X)) by GLIB_000:7;

the_Edges_of G = the_Edges_of (G .set (OrderingSelector,X)) by Lm1, GLIB_000:10;

then ex f being Function st

( (G .set (OrderingSelector,X)) . ELabelSelector = f & dom f c= the_Edges_of (G .set (OrderingSelector,X)) ) by A1, GLIB_003:def 5;

hence G .set (OrderingSelector,X) is [ELabeled] by A2, GLIB_003:def 5; :: thesis: verum