
theorem Th7:
  for x being Real, n being Nat st x in dyadic(n) holds
  inf_number_dyadic(x) <= n
proof
  let x be Real;
  let n be Nat;
A1: dyadic n c= DYADIC by URYSOHN2:23;
  defpred P[Nat] means x in dyadic($1);
  assume
A2: x in dyadic(n);
  then
A3: ex s being Nat st P[s];
  ex q being Nat st P[q] & for i being Nat st P[i] holds q <= i from NAT_1
  :sch 5(A3);
  then consider q being Nat such that
A4: x in dyadic(q) and
A5: for i being Nat st x in dyadic(i) holds q <= i;
A6: q <= n by A2,A5;
  now
    per cases;
    case
      q = 0;
      hence thesis by A2,A1,A4,Def3;
    end;
    case
      0 < q;
      then consider m being Nat such that
A7:   q = m + 1 by NAT_1:6;
      reconsider m as Nat;
      not x in dyadic(m)
      proof
        assume x in dyadic(m);
        then m + 1 <= m + 0 by A5,A7;
        hence thesis by XREAL_1:6;
      end;
      hence thesis by A2,A1,A4,A6,A7,Def3;
    end;
  end;
  hence thesis;
end;
