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

theorem Th76:
  for U being Subset of Niemytzki-plane for x,r st U = Ball(|[x,r
]|,r) \/ {|[x,0]|} ex f being continuous Function of Niemytzki-plane, I[01] st
f.(|[x,0]|) = 0 & for a,b being Real holds (|[a,b]| in U` implies f.(|[a
,b]|) = 1) & (|[a,b]| in U\{|[x,0]|} implies f.(|[a,b]|) = |.|[x,0]|-|[a,b]|.|
  ^2/(2*r*b))
proof
  let U be Subset of Niemytzki-plane;
  let x,r;
  assume
A1: U = Ball(|[x,r]|,r) \/ {|[x,0]|};
  take f = +(x,r);
  thus f.(|[x,0]|) = 0 by Def5;
  let a,b be Real;
  thus |[a,b]| in U` implies f.(|[a,b]|) = 1
  proof
    assume
A2: |[a,b]| in U`;
    then
A3: not |[a,b]| in U by XBOOLE_0:def 5;
    then
A4: not |[a,b]| in Ball(|[x,r]|,r) by A1,ZFMISC_1:136;
A5: a <> x or b <> 0 by A3,A1,ZFMISC_1:136;
    b >= 0 by A2,Lm1,Th18;
    hence thesis by A4,A5,Def5;
  end;
  assume
A6: |[a,b]| in U\{|[x,0]|};
  then
A7: not |[a,b]| in {|[x,0]|} by XBOOLE_0:def 5;
  |[a,b]| in U by A6,XBOOLE_0:def 5;
  then
A8: |[a,b]| in Ball(|[x,r]|,r) by A7,A1,XBOOLE_0:def 3;
  b >= 0 by A6,Lm1,Th18;
  hence thesis by A8,Def5;
end;
