
theorem
  for G1 being _Graph, G2 being reverseEdgeDirections of G1
  for G3 being DGraphComplement of G1, G4 being reverseEdgeDirections of G3
  holds G4 is DGraphComplement of G2
proof
  let G1 be _Graph, G2 be reverseEdgeDirections of G1;
  let G3 be DGraphComplement of G1, G4 be reverseEdgeDirections of G3;
  A1: the_Vertices_of G4 = the_Vertices_of G3 by GLIB_007:4
    .= the_Vertices_of G1 by Th80
    .= the_Vertices_of G2 by GLIB_007:4;
  the_Edges_of G3 misses the_Edges_of G1 by Th80;
  then the_Edges_of G4 misses the_Edges_of G1 by GLIB_007:4;
  then A2: the_Edges_of G4 misses the_Edges_of G2 by GLIB_007:4;
  now
    let v,w be Vertex of G2;
    assume A3: v <> w;
    A4: v is Vertex of G1 & w is Vertex of G1 by GLIB_007:4;
    hereby
      given e2 being object such that
        A5: e2 DJoins v,w,G2;
      e2 in the_Edges_of G2 by A5, GLIB_000:def 14;
      then e2 in the_Edges_of G1 by GLIB_007:4;
      then A6: e2 DJoins w,v,G1 by A5, GLIB_007:7;
      given e4 being object such that
        A7: e4 DJoins v,w,G4;
      e4 in the_Edges_of G4 by A7, GLIB_000:def 14;
      then e4 in the_Edges_of G3 by GLIB_007:4;
      then e4 DJoins w,v,G3 by A7, GLIB_007:7;
      hence contradiction by A6, Th81;
    end;
    assume A8: not ex e4 being object st e4 DJoins v,w,G4;
    not ex e3 being object st e3 DJoins w,v,G3
    proof
      given e3 being object such that
        A9: e3 DJoins w,v,G3;
      e3 in the_Edges_of G3 by A9, GLIB_000:def 14;
      then e3 DJoins v,w,G4 by A9, GLIB_007:7;
      hence contradiction by A8;
    end;
    then consider e1 being object such that
      A10: e1 DJoins w,v,G1 by A3, A4, Th80;
    take e1;
    e1 in the_Edges_of G1 by A10, GLIB_000:def 14;
    hence e1 DJoins v,w,G2 by A10, GLIB_007:7;
  end;
  hence thesis by A1, A2, Th80;
end;
