
theorem
  for G being _Graph, V being non empty Subset of the_Vertices_of G
  for H being (inducedSubgraph of G, V), E being RepEdgeSelection of G
  holds E /\ G.edgesBetween(V) is RepEdgeSelection of H
proof
  let G be _Graph, V be non empty Subset of the_Vertices_of G;
  let H be (inducedSubgraph of G, V), E be RepEdgeSelection of G;
  G.edgesBetween(V) = the_Edges_of H by GLIB_000:def 37;
  then reconsider E9 = E /\ G.edgesBetween(V) as Subset of the_Edges_of H
    by XBOOLE_1:17;
  now
    let v,w,e0 be object;
    A1: v is set & w is set by TARSKI:1;
    assume A2: e0 Joins v,w,H;
    then e0 Joins v,w,G by A1, GLIB_000:72;
    then consider e being object such that
      A3: e Joins v,w,G & e in E and
      A4: for e9 being object st e9 Joins v,w,G & e9 in E holds e9 = e
      by GLIB_009:def 5;
    take e;
    v in the_Vertices_of H & w in the_Vertices_of H by A2, GLIB_000:13;
    then v in V & w in V by GLIB_000:def 37;
    then A5: e in G.edgesBetween(V) by A3, GLIB_000:32;
    then e in the_Edges_of H by GLIB_000:def 37;
    hence e Joins v,w,H by A1, A3, GLIB_000:73;
    thus e in E9 by A3, A5, XBOOLE_0:def 4;
    let e9 be object;
    assume e9 Joins v,w,H & e9 in E9;
    then e9 Joins v,w,G & e9 in E by A1, GLIB_000:72, XBOOLE_0:def 4;
    hence e9 = e by A4;
  end;
  hence thesis by GLIB_009:def 5;
end;
