
theorem Th24:
  for a,b being Real st a < b & 0 <= a & b <= 1
   ex c being Real st c in DYADIC & a < c & c < b
proof
  let a,b be Real;
  assume that
A1: a < b and
A2: 0 <= a and
A3: b <= 1;
  set eps = b - a;
  consider n being Nat such that
A4: 1 < 2|^n * eps by A1,Th21,XREAL_1:50;
  set aa = 2|^n * a, bb = 2|^n * b;
  1 < bb - aa by A4;
  then consider m being Nat such that
A5: aa < m and
A6: m < bb by A2,Th22;
  set x = m / 2|^n;
  take x;
A7: 0 < 2|^n by NEWTON:83;
  m / 2|^n < b by A6,NEWTON:83,XREAL_1:83;
  then m / 2|^n < 1 by A3,XXREAL_0:2;
  then m < 2|^n by A7,XREAL_1:181;
  then x in dyadic(n) by URYSOHN1:def 1;
  hence thesis by A7,A5,A6,URYSOHN1:def 2,XREAL_1:81,83;
end;
