
theorem Th2:
  for T being non empty normal TopSpace, A,B being closed Subset of
T st A <> {} & A misses B holds for n being Nat, G being Drizzle of
A,B,n holds ex F being Drizzle of A,B,n+1 st for r being Element of dyadic(n+1)
  st r in dyadic(n) holds F.r = G.r
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;
  let n be Nat;
  let G be Drizzle of A,B,n;
A3: 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 by A1,A2,Def1;
  A c= G.0 & B = [#]T \ G.1 by A1,A2,Def1;
  then consider
  F being Function of dyadic(n+1),bool the carrier of T such that
A4: A c= F.0 & B = [#]T \ F.1 and
A5: 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,A3,URYSOHN1:24;
  for r1,r2 being Element of dyadic(n+1) st r1 < r2 holds F.r1 is open & F
  .r2 is open & Cl(F.r1) c= F.r2 by A5;
  then reconsider F as Drizzle of A,B,n+1 by A1,A2,A4,Def1;
  take F;
  let r be Element of dyadic(n+1);
A6: 0 in dyadic(n+1) & 1 in dyadic(n+1) by URYSOHN1:6;
  assume r in dyadic(n);
  hence thesis by A5,A6;
end;
