set G3 = G2 .set (ELabelSelector,(the_ELabel_of G1));

ELabelSelector in {ELabelSelector} by TARSKI:def 1;

then ELabelSelector in (dom G2) \/ {ELabelSelector} by XBOOLE_0:def 3;

hence ELabelSelector in dom (G2 .set (ELabelSelector,(the_ELabel_of G1))) by GLIB_000:7; :: according to GLIB_010:def 1 :: thesis: ex f being ManySortedSet of the_Edges_of (G2 .set (ELabelSelector,(the_ELabel_of G1))) st (G2 .set (ELabelSelector,(the_ELabel_of G1))) . ELabelSelector = f

consider f being ManySortedSet of the_Edges_of G1 such that

A1: G1 . ELabelSelector = f by Def1;

G2 == G2 .set (ELabelSelector,(the_ELabel_of G1)) by GLIB_003:7;

then the_Edges_of G2 = the_Edges_of (G2 .set (ELabelSelector,(the_ELabel_of G1))) by GLIB_000:def 34;

then the_Edges_of G1 = the_Edges_of (G2 .set (ELabelSelector,(the_ELabel_of G1))) by GLIB_007:4;

then reconsider f = f as ManySortedSet of the_Edges_of (G2 .set (ELabelSelector,(the_ELabel_of G1))) ;

take f ; :: thesis: (G2 .set (ELabelSelector,(the_ELabel_of G1))) . ELabelSelector = f

thus (G2 .set (ELabelSelector,(the_ELabel_of G1))) . ELabelSelector = the_ELabel_of G1 by GLIB_000:8

.= f by A1, GLIB_003:def 8 ; :: thesis: verum

ELabelSelector in {ELabelSelector} by TARSKI:def 1;

then ELabelSelector in (dom G2) \/ {ELabelSelector} by XBOOLE_0:def 3;

hence ELabelSelector in dom (G2 .set (ELabelSelector,(the_ELabel_of G1))) by GLIB_000:7; :: according to GLIB_010:def 1 :: thesis: ex f being ManySortedSet of the_Edges_of (G2 .set (ELabelSelector,(the_ELabel_of G1))) st (G2 .set (ELabelSelector,(the_ELabel_of G1))) . ELabelSelector = f

consider f being ManySortedSet of the_Edges_of G1 such that

A1: G1 . ELabelSelector = f by Def1;

G2 == G2 .set (ELabelSelector,(the_ELabel_of G1)) by GLIB_003:7;

then the_Edges_of G2 = the_Edges_of (G2 .set (ELabelSelector,(the_ELabel_of G1))) by GLIB_000:def 34;

then the_Edges_of G1 = the_Edges_of (G2 .set (ELabelSelector,(the_ELabel_of G1))) by GLIB_007:4;

then reconsider f = f as ManySortedSet of the_Edges_of (G2 .set (ELabelSelector,(the_ELabel_of G1))) ;

take f ; :: thesis: (G2 .set (ELabelSelector,(the_ELabel_of G1))) . ELabelSelector = f

thus (G2 .set (ELabelSelector,(the_ELabel_of G1))) . ELabelSelector = the_ELabel_of G1 by GLIB_000:8

.= f by A1, GLIB_003:def 8 ; :: thesis: verum