
theorem Th77:
  for G1 being _Graph, G2 being LGraphComplement 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()
proof
  let G1 be _Graph, G2 be LGraphComplement of G1;
  let v1 be Vertex of G1, v2 be Vertex of G2;
  assume A1: v1 = v2;
  now
    let x be object;
    hereby
      assume x in v2.allNeighbors();
      then consider e being object such that
        A2: e Joins v2,x,G2 by GLIB_000:71;
      A3: x in the_Vertices_of G2 by A2, GLIB_000:13;
      then reconsider w = x as Vertex of G1 by Def7;
      not ex e0 being object st e0 Joins v1,w,G1 by A1, A2, Def7;
      then not x in v1.allNeighbors() by GLIB_000:71;
      hence x in the_Vertices_of G2 \ v1.allNeighbors() by A3, XBOOLE_0:def 5;
    end;
    assume x in the_Vertices_of G2 \ v1.allNeighbors();
    then A4: x in the_Vertices_of G2 & not x in v1.allNeighbors()
      by XBOOLE_0:def 5;
    then reconsider w = x as Vertex of G1 by Def7;
    not ex e0 being object st e0 Joins v1,w,G1 by A4, GLIB_000:71;
    then consider e being object such that
      A5: e Joins v1,w,G2 by Def7;
    thus x in v2.allNeighbors() by A1, A5, GLIB_000:71;
  end;
  hence v2.allNeighbors() = the_Vertices_of G2 \ v1.allNeighbors() by TARSKI:2;
end;
