
theorem
  for G1 being _Graph, G2 being DLGraphComplement of G1
  for v1 being Vertex of G1, v2 being Vertex of G2 st v1 = v2 holds
    (v1 is isolated implies v2 is non isolated) &
    (v1 is endvertex implies v2 is non endvertex)
proof
  let G1 be _Graph, G2 be DLGraphComplement of G1;
  let v1 be Vertex of G1, v2 be Vertex of G2;
  assume A1: v1 = v2;
  hereby
    assume A2: v1 is isolated;
    not ex e1 being object st e1 DJoins v1,v1,G1
    proof
      given e1 being object such that
        A3: e1 DJoins v1,v1,G1;
      e1 Joins v1,v1,G1 by A3, GLIB_000:16;
      hence contradiction by A2, GLIB_000:143;
    end;
    then consider e2 being object such that
      A4: e2 DJoins v1,v1,G2 by Def6;
    e2 Joins v2,v2,G2 by A1, A4, GLIB_000:16;
    hence v2 is non isolated by GLIB_000:143;
  end;
  hereby
    assume A5: v1 is endvertex;
    not ex e1 being object st e1 DJoins v1,v1,G1
    proof
      given e1 being object such that
        A6: e1 DJoins v1,v1,G1;
      e1 Joins v1,v1,G1 by A6, GLIB_000:16;
      hence contradiction by A5, GLIB_000:146;
    end;
    then consider e2 being object such that
      A7: e2 DJoins v1,v1,G2 by Def6;
    e2 Joins v2,v2,G2 by A1, A7, GLIB_000:16;
    hence v2 is non endvertex by GLIB_000:146;
  end;
end;
