
theorem Th9:
  for T being non empty normal TopSpace, A,B being closed Subset
  of T st A <> {} & A misses B holds for G being Rain of A,B, x being Real st x
  in DYADIC holds ex y being Subset of T st for n being Nat st x in
  dyadic(n) holds y = (G.n).x
proof
  let T be non empty normal TopSpace;
  let A,B be closed Subset of T;
  assume
A1: A <> {} & A misses B;
  let G be Rain of A,B;
  let x be Real;
  assume
A2: x in DYADIC;
  reconsider s = inf_number_dyadic(x) as Nat;
  G.s is Drizzle of A,B,s by A1,Def2;
  then reconsider y = (G.s).x as Subset of T by A2,Th5,FUNCT_2:5;
  take y;
  let n be Nat;
  assume x in dyadic(n);
  then consider m being Nat such that
A3: n = s + m by Th7,NAT_1:10;
  thus thesis by A1,A2,A3,Th8;
end;
