
theorem
  for G being _Graph, V being set, H being addVertices of G, V
  for E being Subset of the_Edges_of G
  holds E is RepDEdgeSelection of G iff E is RepDEdgeSelection of H
proof
  let G be _Graph, V be set, H be addVertices of G,V;
  let E be Subset of the_Edges_of G;
  A1: the_Edges_of G = the_Edges_of H by GLIB_006:def 10;
  hereby
    assume A2: E is RepDEdgeSelection of G;
    now
      let v,w,e0 be object;
      A3: v is set & w is set by TARSKI:1;
      assume e0 DJoins v,w,H;
      then consider e being object such that
        A4: e DJoins v,w,G & e in E and
        A5: for e9 being object st e9 DJoins v,w,G & e9 in E holds e9 = e
        by A2, A3, GLIB_009:41, GLIB_009:def 6;
      take e;
      thus e DJoins v,w,H & e in E by A3, A4, GLIB_009:41;
      let e9 be object;
      assume e9 DJoins v,w,H & e9 in E;
      hence e9 = e by A3, A5, GLIB_009:41;
    end;
    hence E is RepDEdgeSelection of H by A1, GLIB_009:def 6;
  end;
  assume A6: E is RepDEdgeSelection of H;
  now
    let v,w,e0 be object;
    A7: v is set & w is set by TARSKI:1;
    assume e0 DJoins v,w,G;
    then consider e being object such that
      A8: e DJoins v,w,H & e in E and
      A9: for e9 being object st e9 DJoins v,w,H & e9 in E holds e9 = e
      by A6, A7, GLIB_009:41, GLIB_009:def 6;
    take e;
    thus e DJoins v,w,G & e in E by A7, A8, GLIB_009:41;
    let e9 be object;
    assume e9 DJoins v,w,G & e9 in E;
    hence e9 = e by A7, A9, GLIB_009:41;
  end;
  hence E is RepDEdgeSelection of G by GLIB_009:def 6;
end;
