
theorem
  for G1 being _Graph, G2 being DGraphComplement 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} &
    v2.outNeighbors() = the_Vertices_of G2 \ {v2} &
    v2.allNeighbors() = the_Vertices_of G2 \ {v2}
proof
  let G1 be _Graph, G2 be DGraphComplement 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() = {};
  thus A3: v2.inNeighbors() = the_Vertices_of G2 \ ({}\/{v2}) by A1, A2, Th95
    .= the_Vertices_of G2 \ {v2};
  thus v2.outNeighbors() = the_Vertices_of G2 \ ({} \/ {v2}) by A1, A2, Th95
    .= the_Vertices_of G2 \ {v2};
  hence thesis by A3;
end;
