
theorem
  for eps being Real st 0 < eps
  for d being Real st 0 < d
  ex r1,r2 being Real
    st r1 in DYADIC \/ (right_open_halfline 1) & r2 in DYADIC \/ (
  right_open_halfline 1) & 0 < r1 & r1 < d & d < r2 & r2 - r1 < eps
proof
  let eps be Real;
  assume 0 < eps;
  then consider eps1 being Real such that
A1: 0 < eps1 and
A2: eps1 < eps by XREAL_1:5;
  reconsider eps1 as Real;
  let d be Real;
  assume
A3: 0 < d;
  per cases;
  suppose
    eps1 < d;
    then
A4: 0 < d - eps1 by XREAL_1:50;
    d - eps1 < d - 0 by A1,XREAL_1:15;
    then ex c being Real st c in DOM & d - eps1 < c & c < d by Th25;
    then consider r1 being Real such that
A5: d - eps1 < r1 and
A6: r1 < d and
A7: r1 in DOM;
    r1 in (halfline 0) \/ DYADIC or r1 in right_open_halfline 1 by A7,
URYSOHN1:def 3,XBOOLE_0:def 3;
    then
A8: r1 in halfline 0 or r1 in DYADIC or r1 in right_open_halfline 1 by
XBOOLE_0:def 3;
    eps - eps < eps - eps1 by A2,XREAL_1:15;
    then d + 0 < d + (eps - eps1) by XREAL_1:8;
    then ex c being Real st c in DOM & d < c & c < d + (eps - eps1)
           by Th25;
    then consider r2 being Real such that
A9: d < r2 and
A10: r2 < d + (eps - eps1) and
A11: r2 in DOM;
    r2 in halfline 0 \/ DYADIC or r2 in right_open_halfline 1 by A11,
URYSOHN1:def 3,XBOOLE_0:def 3;
    then
A12: r2 in halfline 0 or r2 in DYADIC or r2 in right_open_halfline 1 by
XBOOLE_0:def 3;
A13: r1 < r2 by A6,A9,XXREAL_0:2;
    then
A14: d - eps1 < r2 by A5,XXREAL_0:2;
    take r1,r2;
A15: (d + (eps - eps1)) - (d - eps1) = eps;
    r1 < d + (eps - eps1) by A10,A13,XXREAL_0:2;
    then
A16: |.r2 - r1 qua Complex.| < eps by A5,A10,A14,A15,Th30;
    0 <= r2 - r1 by A13,XREAL_1:48;
    hence thesis by A5,A6,A9,A4,A8,A12,A16,ABSVALUE:def 1,XBOOLE_0:def 3
,XXREAL_1:233;
  end;
  suppose
A17: d <= eps1;
    consider r1 being Real such that
A18: r1 in DOM and
A19: 0 < r1 and
A20: r1 < d by A3,Th25;
A21: r1 in halfline 0 \/ DYADIC or r1 in right_open_halfline 1 by A18,
URYSOHN1:def 3,XBOOLE_0:def 3;
    0 + d < r1 + eps1 by A17,A19,XREAL_1:8;
    then ex c being Real st c in DOM & d < c & c < r1 + eps1 by Th25;
    then consider r2 being Real such that
A22: d < r2 and
A23: r2 < r1 + eps1 and
A24: r2 in DOM;
    take r1,r2;
A25: r2 in halfline 0 \/ DYADIC or r2 in right_open_halfline 1 by A24,
URYSOHN1:def 3,XBOOLE_0:def 3;
    not r1 in halfline 0 by A19,XXREAL_1:233;
    then
A26: r1 in DYADIC or r1 in right_open_halfline 1 by A21,XBOOLE_0:def 3;
    not r2 in halfline 0 by A3,A22,XXREAL_1:233;
    then
A27: r2 in DYADIC or r2 in right_open_halfline 1 by A25,XBOOLE_0:def 3;
    r2 - r1 < r1 + eps1 - r1 by A23,XREAL_1:9;
    hence thesis by A2,A19,A20,A22,A26,A27,XBOOLE_0:def 3,XXREAL_0:2;
  end;
end;
