
theorem Th25:
  for a,b being Real st a < b
   ex c being Real st c in DOM & a < c & c < b
proof
  let a,b be Real;
  assume
A1: a < b;
  per cases;
  suppose
A2: a < 0 & b <= 1;
    now
      per cases;
      case
A3:     b <= 0;
        consider c being Real such that
A4:     a < c and
A5:     c < b by A1,XREAL_1:5;
        reconsider c as Real;
        halfline 0 = {g where g is Real : g<0} by XXREAL_1:229;
        then c in halfline 0 by A3,A5;
        then c in (halfline 0) \/ DYADIC by XBOOLE_0:def 3;
        then c in DOM by URYSOHN1:def 3,XBOOLE_0:def 3;
        hence thesis by A4,A5;
      end;
      case
A6:     0 < b;
        set a1 = 0;
        consider c being Real such that
A7:     c in DYADIC and
A8:     a1 < c & c < b by A2,A6,Th24;
        c in (halfline 0) \/ DYADIC by A7,XBOOLE_0:def 3;
        then c in DOM by URYSOHN1:def 3,XBOOLE_0:def 3;
        hence thesis by A2,A8;
      end;
    end;
    hence thesis;
  end;
  suppose
A9: a < 0 & 1 < b;
    consider a1,b1 being Real such that
A10: a1 = 0 & b1 = 1;
    consider c being Real such that
A11: c in DYADIC and
A12: a1 < c & c < b1 by A10,Th24;
    take c;
    c in (halfline 0) \/ DYADIC by A11,XBOOLE_0:def 3;
    hence thesis by A9,A10,A12,URYSOHN1:def 3,XBOOLE_0:def 3,XXREAL_0:2;
  end;
  suppose
    0 <= a & b <= 1;
    then consider c being Real such that
A13: c in DYADIC and
A14: a < c & c < b by A1,Th24;
    take c;
    c in (halfline 0) \/ DYADIC by A13,XBOOLE_0:def 3;
    hence thesis by A14,URYSOHN1:def 3,XBOOLE_0:def 3;
  end;
  suppose
A15: 0 <= a & 1 < b;
    now
      per cases;
      case
A16:    1 <= a;
        consider c being Real such that
A17:    a < c and
A18:    c < b by A1,XREAL_1:5;
        reconsider c as Real;
        right_open_halfline 1 =
         {g where g is Real : 1<g} & 1 < c by A16,A17,
XXREAL_0:2,XXREAL_1:230;
        then c in right_open_halfline 1;
        then c in DYADIC \/ (right_open_halfline 1) by XBOOLE_0:def 3;
        then c in (halfline 0) \/ (DYADIC \/ right_open_halfline 1) by
XBOOLE_0:def 3;
        then c in DOM by URYSOHN1:def 3,XBOOLE_1:4;
        hence thesis by A17,A18;
      end;
      case
A19:    a < 1;
        set b1 = 1;
        consider c being Real such that
A20:    c in DYADIC and
A21:    a < c and
A22:    c < b1 by A15,A19,Th24;
        c in (halfline 0) \/ DYADIC by A20,XBOOLE_0:def 3;
        then
A23:    c in DOM by URYSOHN1:def 3,XBOOLE_0:def 3;
        c < b by A15,A22,XXREAL_0:2;
        hence thesis by A21,A23;
      end;
    end;
    hence thesis;
  end;
end;
