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

theorem Th30:
  for p being Point of Niemytzki-plane for r being positive Real
 ex a being Point of TOP-REAL 2, U being open Subset of Niemytzki-plane
st p in U & a in U & for b being Point of TOP-REAL 2 st b in U holds |.b-a.| <
  r
proof
  let p be Point of Niemytzki-plane;
  let r be positive Real;
  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
A2: 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;
A3: the carrier of Niemytzki-plane = y>=0-plane by Def3;
  p in the carrier of Niemytzki-plane;
  then reconsider p9 = p as Point of TOP-REAL 2 by A3;
A4: p = |[p9`1,p9`2]| by EUCLID:53;
  per cases by A3,A4,Th18;
  suppose
A5: p9`2 = 0;
    then
    BB.p = {Ball(|[p9`1,q]|,q) \/ {|[p9`1,0]|} where q is Real:
    q > 0} by A1,A4;
    then
A6: Ball(|[p9`1,r/2]|,r/2) \/ {|[p9`1,0]|} in BB.p;
    BB.p c= the topology of Niemytzki-plane by TOPS_2:64;
    then reconsider U = Ball(|[p9`1,r/2]|,r/2) \/ {p} as open Subset of
    Niemytzki-plane by A6,A4,A5,PRE_TOPC:def 2;
    take a = |[p9`1,r/2]|, U;
    thus p in U by ZFMISC_1:136;
A7: r/2 < r/2+r/2 by XREAL_1:29;
    a in Ball(a,r/2) by Th13;
    hence a in U by XBOOLE_0:def 3;
    let b be Point of TOP-REAL 2;
    assume b in U;
    then
A8: b in Ball(a,r/2) or b = p by ZFMISC_1:136;
    p9-a = |[p9`1-p9`1,0-r/2]| by A4,A5,EUCLID:62;
    then |.p9-a.| = |.0-r/2.| by TOPREAL6:23
      .= |.r/2-0 .| by COMPLEX1:60
      .= r/2 by ABSVALUE:def 1;
    then |.b-a.| <= r/2 by A8,TOPREAL9:7;
    hence thesis by A7,XXREAL_0:2;
  end;
  suppose
A9: p9`2 > 0;
    then BB.p = {Ball(|[p9`1,p9`2]|,q) /\ y>=0-plane
   where q is Real: q > 0} by A2,A4;
    then
A10: Ball(p9,r/2) /\ y>=0-plane in BB.p by A4;
    BB.p c= the topology of Niemytzki-plane by TOPS_2:64;
    then reconsider U = Ball(p9,r/2) /\ y>=0-plane as open Subset of
    Niemytzki-plane by A10,PRE_TOPC:def 2;
    take a = p9, U;
A11: p in Ball(a,r/2) by Th13;
    p in y>=0-plane by A4,A9;
    hence p in U & a in U by A11,XBOOLE_0:def 4;
    let b be Point of TOP-REAL 2;
    assume b in U;
    then b in Ball(a,r/2) by XBOOLE_0:def 4;
    then
A12: |.b-a.| <= r/2 by TOPREAL9:7;
    r/2 < r/2+r/2 by XREAL_1:29;
    hence thesis by A12,XXREAL_0:2;
  end;
end;
