reserve A,B,O for Ordinal,
        o for object,
        x,y,z for Surreal,
        n,m for Nat;

theorem Th18:
  for r be Rational holds
    r is dyadic-like iff ex i be Integer, n be Nat st r = i / (2|^n)
proof
  let r be Rational;
  thus r is dyadic-like implies ex i be Integer, n be Nat st r = i / (2|^n)
  proof
    assume r is dyadic-like;
    then consider n be Nat such that
A1: denominator(r) = 2 |^ n;
    take numerator(r), n;
    thus thesis by A1,RAT_1:15;
  end;
  given i be Integer, n be Nat such that
A2: r = i / (2|^n);
  consider w be Nat such that
A3:  i=numerator(r)*w & 2|^n=denominator(r)*w by A2,RAT_1:27;
  consider t be Element of NAT such that
A4:w = 2 |^ t & t <= n by INT_2:28,A3,INT_1:def 3,PEPIN:34;
  reconsider nt=n-t as Nat by A4,NAT_1:21;
  n = t+nt;
  then denominator(r)* (2 |^ t) = 2|^nt * 2|^t by A4,A3,NEWTON:8;
  hence thesis by XCMPLX_1:5;
end;
