
theorem Th8:
  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 for n being Nat holds (G.inf_number_dyadic(x)).x = (G.(
  inf_number_dyadic(x) + 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;
  set s = inf_number_dyadic(x);
  defpred Q[Nat] means (G.s).x = (G.(s + $1)).x;
  assume
A2: x in DYADIC;
A3: for n being Nat st Q[n] holds Q[(n+1)]
  proof
    let n be Nat;
    assume
A4: (G.s).x = (G.(s + n)).x;
    s <= s + (n + 1) by NAT_1:11;
    then
A5: x in dyadic(s + n + 1) by A2,Th6;
    G.(s + n) is Drizzle of A,B,s + n by A1,Def2;
    then
A6: dom(G.(s + n)) = dyadic(s + n) by FUNCT_2:def 1;
    x in dyadic(s + n) by A2,Th6,NAT_1:11;
    hence thesis by A1,A4,A5,A6,Def2;
  end;
A7: Q[0];
  for i be Nat holds Q[i] from NAT_1:sch 2(A7,A3);
  hence thesis;
end;
