now :: thesis: ( ELabelSelector in dom (G2 .set (ELabelSelector,(the_ELabel_of G1))) & ex f being Function st
( (G2 .set (ELabelSelector,(the_ELabel_of G1))) . ELabelSelector = f & dom f c= the_Edges_of (G2 .set (ELabelSelector,(the_ELabel_of G1))) ) )

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; :: thesis: ex f being Function st
( (G2 .set (ELabelSelector,(the_ELabel_of G1))) . ELabelSelector = f & dom f c= the_Edges_of (G2 .set (ELabelSelector,(the_ELabel_of G1))) )
consider f being Function such that
A1: ( G1 . ELabelSelector = f & dom f c= the_Edges_of G1 ) by GLIB_003:def 5;
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 A2: the_Edges_of G1 = the_Edges_of (G2 .set (ELabelSelector,(the_ELabel_of G1))) by GLIB_007:4;
take f = f; :: thesis: ( (G2 .set (ELabelSelector,(the_ELabel_of G1))) . ELabelSelector = f & dom f c= the_Edges_of (G2 .set (ELabelSelector,(the_ELabel_of G1))) )
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: dom f c= the_Edges_of (G2 .set (ELabelSelector,(the_ELabel_of G1)))
thus dom f c= the_Edges_of (G2 .set (ELabelSelector,(the_ELabel_of G1))) by A1, A2; :: thesis: verum

end;

hence G2 .set (ELabelSelector,(the_ELabel_of G1)) is [ELabeled] by GLIB_003:def 5; :: thesis: verum