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

theorem Th25:
  y>=0-plane \ y=0-line is open Subset of Niemytzki-plane
proof
  consider BB being Neighborhood_System of Niemytzki-plane such that
  for x holds BB.(|[x,0]|) = {Ball(|[x,r]|,r) \/ {|[x,0]|} where r is
  Real: r > 0} and
A1: for x,y st y > 0 holds BB.(|[x,y]|) = {Ball(|[x,y]|,r) /\ y>=0-plane
  where r is Real: r > 0} by Def3;
A2: the carrier of Niemytzki-plane = y>=0-plane by Def3;
  then reconsider A = y>=0-plane \ y=0-line as Subset of Niemytzki-plane by
XBOOLE_1:36;
  now
    let a be Point of Niemytzki-plane;
    assume
A3: a in A;
    then a in y>=0-plane by XBOOLE_0:def 5;
    then consider x,y such that
A4: a = |[x,y]| and
A5: y >= 0;
    set B = Ball(|[x,y]|,y)/\y>=0-plane;
    reconsider B as Subset of Niemytzki-plane by A2,XBOOLE_1:17;
    not a in y=0-line by A3,XBOOLE_0:def 5;
    then
A6: y <> 0 by A4;
    then B in {Ball(|[x,y]|,r) /\ y>=0-plane where r is Real: r >
    0} by A5;
    then
A7: B in BB.a by A1,A4,A5,A6;
    take B;
    dom BB = the carrier of Niemytzki-plane by PARTFUN1:def 2;
    hence B in Union BB by A7,CARD_5:2;
    thus a in B by A7,YELLOW_8:12;
    Ball(|[x,y]|,y) c= y>=0-plane by A5,A6,Th20;
    then B = Ball(|[x,y]|,y) by XBOOLE_1:28;
    then B misses y=0-line by A5,A6,Th21;
    hence B c= A by A2,XBOOLE_1:86;
  end;
  hence thesis by YELLOW_9:31;
end;
