
theorem Th82:
  for G1 being _Graph, 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
proof
  let G1 be _Graph, E1 be RepEdgeSelection of G1;
  let G2 be inducedSubgraph of G1, the_Vertices_of G1, E1;
  let E2 be RepEdgeSelection of G2;
  the_Edges_of G1 = G1.edgesBetween(the_Vertices_of G1) &
    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 holds e in E1 implies e in E2
  proof
    let e be object;
    assume A2: e in E1;
    set v = (the_Source_of G2).e, w = (the_Target_of G2).e;
    A3: e Joins v,w,G2 by A1, A2, GLIB_000:def 13;
    then consider e2 being object such that
      A4: e2 Joins v,w,G2 & e2 in E2 and
      for e9 being object st e9 Joins v,w,G2 & e9 in E2 holds e9 = e2
      by Def5;
    A5: e Joins v,w,G1 by A3, GLIB_000:72;
    then consider e1 being object such that
      e1 Joins v,w,G1 & e1 in E1 and
      A6: for e9 being object st e9 Joins v,w,G1 & e9 in E1 holds e9 = e1
      by Def5;
    e = e1 by A2, A5, A6;
    hence e in E2 by A1, A4, A6, GLIB_000:72;
  end;
  then E1 c= E2 by TARSKI:def 3;
  hence thesis by A1, XBOOLE_0:def 10;
end;
