
theorem Th84:
  for G1 being _Graph, E1 being RepDEdgeSelection of G1
  for G2 being inducedSubgraph of G1, the_Vertices_of G1, E1
  for E2 being RepEdgeSelection of G2
  holds E2 c= E1 & E2 is RepEdgeSelection of G1
proof
  let G1 be _Graph, E1 be RepDEdgeSelection of G1;
  let G2 be inducedSubgraph of G1, the_Vertices_of G1, E1;
  let E2 be RepEdgeSelection of G2;
  the_Vertices_of G1 c= the_Vertices_of G1 &
    the_Edges_of G1 = G1.edgesBetween(the_Vertices_of G1) by GLIB_000:34;
  then A1: the_Vertices_of G2 = the_Vertices_of G1 &
    the_Edges_of G2 = E1 by GLIB_000:def 37;
  hence E2 c= E1;
  then A2: E2 c= the_Edges_of G1 by XBOOLE_1:1;
  now
    let v,w,e0 be object;
    A3: v is set & w is set by TARSKI:1;
    assume A4: e0 Joins v,w,G1;
    ex e1 being object st e1 Joins v,w,G2
    proof
      per cases by A4, GLIB_000:16;
      suppose e0 DJoins v,w,G1;
        then consider e1 being object such that
          A5: e1 DJoins v,w,G1 & e1 in E1 and
          for e9 being object st e9 DJoins v,w,G1 & e9 in E1 holds e9 = e1
          by Def6;
        take e1;
        e1 DJoins v,w,G2 by A1, A3, A5, GLIB_000:73;
        hence e1 Joins v,w,G2 by GLIB_000:16;
      end;
      suppose e0 DJoins w,v,G1;
        then consider e1 being object such that
          A6: e1 DJoins w,v,G1 & e1 in E1 and
          for e9 being object st e9 DJoins w,v,G1 & e9 in E1 holds e9 = e1
          by Def6;
        take e1;
        e1 DJoins w,v,G2 by A3, A1, A6, GLIB_000:73;
        hence e1 Joins v,w,G2 by GLIB_000:16;
      end;
    end;
    then consider e1 being object such that
      A7: e1 Joins v,w,G2;
    consider e2 being object such that
      A8: e2 Joins v,w,G2 & e2 in E2 and
      A9: for e9 being object st e9 Joins v,w,G2 & e9 in E2 holds e9 = e2
      by A7, Def5;
    take e2;
    thus e2 Joins v,w,G1 & e2 in E2 by A3, A8, GLIB_000:72;
    let e9 be object;
    assume A10: e9 Joins v,w,G1 & e9 in E2;
    e9 Joins v,w,G2 by A3, A10, GLIB_000:73;
    hence e9 = e2 by A9, A10;
  end;
  hence thesis by A2, Def5;
end;
