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

ELabelSelector in dom G by Def1;

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

hence ELabelSelector in dom (G .set (OrderingSelector,X)) by GLIB_000:7; :: according to GLIB_010:def 1 :: thesis: ex f being ManySortedSet of the_Edges_of (G .set (OrderingSelector,X)) st (G .set (OrderingSelector,X)) . ELabelSelector = f

consider f being ManySortedSet of the_Edges_of G such that

A1: G . ELabelSelector = f by Def1;

G == G .set (OrderingSelector,X) by Th3;

then the_Edges_of G = the_Edges_of (G .set (OrderingSelector,X)) by GLIB_000:def 34;

then reconsider f = f as ManySortedSet of the_Edges_of (G .set (OrderingSelector,X)) ;

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

OrderingSelector <> ELabelSelector by GLIB_003:def 2;

hence (G .set (OrderingSelector,X)) . ELabelSelector = f by A1, GLIB_000:9; :: thesis: verum

ELabelSelector in dom G by Def1;

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

hence ELabelSelector in dom (G .set (OrderingSelector,X)) by GLIB_000:7; :: according to GLIB_010:def 1 :: thesis: ex f being ManySortedSet of the_Edges_of (G .set (OrderingSelector,X)) st (G .set (OrderingSelector,X)) . ELabelSelector = f

consider f being ManySortedSet of the_Edges_of G such that

A1: G . ELabelSelector = f by Def1;

G == G .set (OrderingSelector,X) by Th3;

then the_Edges_of G = the_Edges_of (G .set (OrderingSelector,X)) by GLIB_000:def 34;

then reconsider f = f as ManySortedSet of the_Edges_of (G .set (OrderingSelector,X)) ;

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

OrderingSelector <> ELabelSelector by GLIB_003:def 2;

hence (G .set (OrderingSelector,X)) . ELabelSelector = f by A1, GLIB_000:9; :: thesis: verum