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

theorem Th81:
  for U being Subset of Niemytzki-plane for x,y,r st y > 0 & U =
Ball(|[x,y]|,r) /\ y>=0-plane ex f being continuous Function of Niemytzki-plane
  , I[01] st f.(|[x,y]|) = 0 & for a,b being Real holds (|[a,b]| in U`
implies f.(|[a,b]|) = 1) & (|[a,b]| in U implies f.(|[a,b]|) = |.|[x,y]|-|[a,b
  ]|.|/r)
proof
  let U be Subset of Niemytzki-plane;
  let x,y,r;
  assume that
A1: y > 0 and
A2: U = Ball(|[x,y]|,r) /\ y>=0-plane;
  reconsider y9 = y as positive Real by A1;
  take f = +(x,y9,r);
  |[x,y9]|`2 = y by EUCLID:52;
  hence f.(|[x,y]|) = 0 by Th77;
  let a,b be Real;
  thus |[a,b]| in U` implies f.(|[a,b]|) = 1
  proof
    assume
A3: |[a,b]| in U`;
    then not |[a,b]| in U by XBOOLE_0:def 5;
    then
A4: not |[a,b]| in Ball(|[x,y]|,r) by A2,A3,Lm1,XBOOLE_0:def 4;
    b >= 0 by A3,Lm1,Th18;
    hence thesis by A4,Def6;
  end;
  assume
A5: |[a,b]| in U;
  then
A6: |[a,b]| in Ball(|[x,y]|,r) by A2,XBOOLE_0:def 4;
  b >= 0 by A5,Lm1,Th18;
  hence thesis by A6,Def6;
end;
