
theorem Th118:
  for G1 being _Graph, G2 being GraphComplement of G1
  for v1 being Vertex of G1, v2 being Vertex of G2 st v1 = v2
  holds v2.allNeighbors() = the_Vertices_of G2 \ (v1.allNeighbors() \/ {v2})
proof
  let G1 be _Graph, G2 be GraphComplement of G1;
  let v1 be Vertex of G1, v2 be Vertex of G2;
  assume A1: v1 = v2;
  consider G9 being LGraphComplement of G1 such that
    A2: G2 is removeLoops of G9 by Def9;
  reconsider v9 = v1 as Vertex of G9 by Def7;
  thus v2.allNeighbors() = v9.allNeighbors() \ {v2} by A1, A2, GLIBPRE0:63
    .= (the_Vertices_of G9 \ v1.allNeighbors()) \ {v2} by Th77
    .= the_Vertices_of G9 \ (v1.allNeighbors() \/ {v2}) by XBOOLE_1:41
    .= the_Vertices_of G2 \ (v1.allNeighbors() \/ {v2}) by A2, GLIB_000:53;
end;
