let G1 be _Graph; :: thesis: for E1 being RepEdgeSelection of G1
for G2 being inducedSubgraph of G1, the_Vertices_of G1,E1
for E2 being RepEdgeSelection of G2 holds E1 = E2

let E1 be RepEdgeSelection of G1; :: thesis: for G2 being inducedSubgraph of G1, the_Vertices_of G1,E1
for E2 being RepEdgeSelection of G2 holds E1 = E2

let G2 be inducedSubgraph of G1, the_Vertices_of G1,E1; :: thesis: for E2 being RepEdgeSelection of G2 holds E1 = E2
let E2 be RepEdgeSelection of G2; :: thesis: E1 = E2
( the_Edges_of G1 = G1 .edgesBetween () & the_Vertices_of G1 c= the_Vertices_of G1 ) by GLIB_000:34;
then A1: the_Edges_of G2 = E1 by GLIB_000:def 37;
for e being object st e in E1 holds
e in E2
proof
let e be object ; :: thesis: ( e in E1 implies e in E2 )
assume A2: e in E1 ; :: thesis: e in E2
set v = () . e;
set w = () . e;
A3: e Joins () . e,() . e,G2 by ;
then consider e2 being object such that
A4: ( e2 Joins () . e,() . e,G2 & e2 in E2 ) and
for e9 being object st e9 Joins () . e,() . e,G2 & e9 in E2 holds
e9 = e2 by Def5;
A5: e Joins () . e,() . e,G1 by ;
then consider e1 being object such that
( e1 Joins () . e,() . e,G1 & e1 in E1 ) and
A6: for e9 being object st e9 Joins () . e,() . e,G1 & e9 in E1 holds
e9 = e1 by Def5;
e = e1 by A2, A5, A6;
hence e in E2 by ; :: thesis: verum
end;
then E1 c= E2 by TARSKI:def 3;
hence E1 = E2 by ; :: thesis: verum