
theorem Th42:
  for G being _Graph, v,w being Vertex of G
  holds v,w are_adjacent iff w in v.allNeighbors()
proof
  let G be _Graph, v,w be Vertex of G;
  hereby
    assume v,w are_adjacent;
    then ex e being object st e Joins v,w,G by CHORD:def 3;
    hence w in v.allNeighbors() by GLIB_000:71;
  end;
  assume w in v.allNeighbors();
  then ex e being object st e Joins v,w,G by GLIB_000:71;
  hence thesis by CHORD:def 3;
end;
