reserve e,u for set;
reserve X, Y for non empty TopSpace;

theorem Th8:
  for x being Point of X, y being Point of Y, V being
  a_neighborhood of x, W being a_neighborhood of y holds [:V,W:] is
  a_neighborhood of [x,y]
proof
  let x be Point of X, y be Point of Y, V be a_neighborhood of x, W be
  a_neighborhood of y;
  x in Int V & y in Int W by CONNSP_2:def 1;
  then [x,y] in [:Int V, Int W:] by ZFMISC_1:87;
  hence [x,y] in Int [:V,W:] by Th7;
end;
