
theorem Th65:
  for G1 being _Graph, G2 being removeParallelEdges of G1
  for G3 being LGraphComplement of G1
  holds G3 is LGraphComplement of G2
proof
  let G1 be _Graph, G2 be removeParallelEdges of G1;
  let G3 be LGraphComplement of G1;
  consider E being RepEdgeSelection of G1 such that
    A1: G2 is inducedSubgraph of G1, the_Vertices_of G1, E by GLIB_009:def 7;
  the_Vertices_of G1 c= the_Vertices_of G1 &
    the_Edges_of G1 = G1.edgesBetween(the_Vertices_of G1) by GLIB_000:34;
  then A2: the_Vertices_of G2 = the_Vertices_of G1 & the_Edges_of G2 = E
    by A1, GLIB_000:def 37;
  then A3: the_Vertices_of G3 = the_Vertices_of G2 by Def7;
  the_Edges_of G3 misses the_Edges_of G1 by Def7;
  then A4: the_Edges_of G3 misses the_Edges_of G2 by XBOOLE_1:63;
  now
    let v,w be Vertex of G2;
    A5: v is Vertex of G1 & w is Vertex of G1 by GLIB_000:def 33;
    reconsider v1=v, w1=w as Vertex of G1 by GLIB_000:def 33;
    hereby
      given e2 being object such that
        A6: e2 Joins v,w,G2;
      A7: e2 Joins v,w,G1 by A6, GLIB_000:72;
      given e3 being object such that
        A8: e3 Joins v,w,G3;
      thus contradiction by A7, A8, Th64;
    end;
    assume not ex e3 being object st e3 Joins v,w,G3;
    then consider e1 being object such that
      A9: e1 Joins v,w,G1 by A5, Def7;
    consider e2 being object such that
      A10: e2 Joins v,w,G1 & e2 in E and
      for e9 being object st e9 Joins v,w,G1 & e9 in E holds e9 = e2
      by A9, GLIB_009:def 5;
    take e2;
    thus e2 Joins v,w,G2 by A2, A10, GLIB_000:73;
  end;
  hence thesis by A3, A4, Def7;
end;
