
theorem Th1:
  for T being non empty normal TopSpace, A,B being closed Subset of
  T st A <> {} & A misses B holds for n being Nat holds ex G being
Function of dyadic(n),bool the carrier of T st A c= G.0 & B = [#]T \ G.1 & for
r1,r2 being Element of dyadic(n) st r1 < r2 holds G.r1 is open & G.r2 is open &
  Cl(G.r1) c= G.r2
proof
  let T be non empty normal TopSpace;
  let A,B be closed Subset of T;
  assume that
A1: A <> {} and
A2: A misses B;
  defpred P[Nat] means
  ex G being Function of dyadic($1),bool the
carrier of T st A c= G.0 & B = [#]T \ G.1 & (for r1,r2 being Element of dyadic(
  $1) st r1 < r2 holds (G.r1 is open & G.r2 is open & Cl(G.r1) c= G.r2));
A3: for n being Nat st P[n] holds P[n+1]
  proof
    let n be Nat;
    given G being Function of dyadic(n),bool the carrier of T such that
A4: A c= G.0 & B = [#]T \ G.1 & for r1,r2 being Element of dyadic(n)
    st r1 < r2 holds G.r1 is open & G.r2 is open & Cl(G.r1) c= G.r2;
    consider F being Function of dyadic(n+1),bool the carrier of T such that
A5: A c= F.0 & B = [#]T \ F.1 & for r1,r2,r being Element of dyadic(n+
    1) st r1 < r2 holds F.r1 is open & F.r2 is open & Cl(F.r1) c= F.r2 & (r in
    dyadic(n) implies F.r = G.r) by A1,A4,URYSOHN1:24;
    take F;
    thus thesis by A5;
  end;
A6: P[0] by A1,A2,Lm1;
  thus for n being Nat holds P[n] from NAT_1:sch 2(A6,A3);
end;
