
theorem Th5:
  for x being Real st x in DYADIC holds x in dyadic( inf_number_dyadic(x))
proof
  let x be Real;
  set s = inf_number_dyadic(x);
  defpred P[Nat] means not x in dyadic(s + 1 + $1);
  assume
A1: x in DYADIC;
  then consider k being Nat such that
A2: x in dyadic(k) by URYSOHN1:def 2;
A3: for i being Nat st P[i] holds P[(i+1)]
  proof
    let i be Nat;
    assume
A4: not x in dyadic(s + 1 + i);
    assume x in dyadic(s + 1 + (i + 1));
    then x in dyadic(s + 1 + i + 1);
    then s + 0 = s + (i + 2) by A1,A4,Def3;
    hence thesis;
  end;
  assume
A5: not x in dyadic(s);
A6: s < k
  proof
    assume not s < k;
    then dyadic(k) c= dyadic(s) by URYSOHN2:29;
    hence thesis by A5,A2;
  end;
  then consider i being Nat such that
A7: k = i + 1 by NAT_1:6;
  s <= i by A6,A7,NAT_1:13;
  then consider m being Nat such that
A8: i = s + m by NAT_1:10;
  reconsider m as Nat;
A9: P[0] by A1,A5,Def3;
  for i being Nat holds P[i] from NAT_1:sch 2(A9,A3);
  then not x in dyadic(s + 1 + m);
  hence thesis by A2,A7,A8;
end;
