reserve x,y for Real,
  u,v,w for set,
  r for positive Real;

theorem Th27:
  for x being Real, r being positive Real holds
  Ball(|[x,r]|,r) \/ {|[x,0]|} is open Subset of Niemytzki-plane
proof
  let x be Real;
  let r be positive Real;
  the carrier of Niemytzki-plane = y>=0-plane by Def3;
  then reconsider a = |[x,0]| as Point of Niemytzki-plane by Th18;
  consider BB being Neighborhood_System of Niemytzki-plane such that
A1: for x holds BB.(|[x,0]|) = {Ball(|[x,q]|,q) \/ {|[x,0]|} where q is
  Real: q > 0} and
  for x,y st y > 0 holds BB.(|[x,y]|) = {Ball(|[x,y]|,q) /\ y>=0-plane
  where q is Real: q > 0} by Def3;
  BB.(|[x,0]|) = {Ball(|[x,q]|,q) \/ {|[x,0]|} where q is Real:
  q > 0} by A1;
  then Ball(|[x,r]|,r) \/ {|[x,0]|} in BB.a;
  hence thesis by YELLOW_8:12;
end;
