
theorem Th44:
  for G being complete _Graph, v being Vertex of G
  holds the_Vertices_of G \ {v} c= v.allNeighbors()
proof
  let G be complete _Graph, v be Vertex of G;
  now
    let x be object;
    assume x in the_Vertices_of G \ {v};
    then A1: x in the_Vertices_of G & not x in {v} by XBOOLE_0:def 5;
    then reconsider w = x as Vertex of G;
    v <> w by A1, TARSKI:def 1;
    then ex e being object st e Joins v,w,G by CHORD:def 3, CHORD:def 6;
    hence x in v.allNeighbors() by GLIB_000:71;
  end;
  hence thesis by TARSKI:def 3;
end;
