
theorem
  for G1 being _Graph, G2 being DLGraphComplement of G1
  for v1 being Vertex of G1, v2 being Vertex of G2
  st v1 = v2 & v1 is isolated holds
    v2.inNeighbors() = the_Vertices_of G2 &
    v2.outNeighbors() = the_Vertices_of G2 &
    v2.allNeighbors() = the_Vertices_of G2
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 & v1 is isolated;
  then v1.allNeighbors() = {} by GLIB_000:113;
  then A2: v1.inNeighbors() = {} & v1.outNeighbors() = {};
  hence A3: v2.inNeighbors() = the_Vertices_of G2 \ {} by A1, Th60
    .= the_Vertices_of G2;
  thus v2.outNeighbors() = the_Vertices_of G2 \ {} by A1, A2, Th60
    .= the_Vertices_of G2;
  hence v2.allNeighbors() = the_Vertices_of G2 by A3;
end;
