reserve a,b for R_eal;
reserve A,B for Interval;

theorem
  for a,b,c being R_eal st a < b & a < c ex x being R_eal st a < x & x <
  b & x < c & x in REAL
proof
  let a,b,c be R_eal;
A1: min(b,c) = min{b,c} by XXREAL_2:14;
  ex o being R_eal st o in {b,c} & o <= c
  proof
    take c;
    thus thesis by TARSKI:def 2;
  end;
  then
A2: min(b,c) <= c by A1,XXREAL_2:62;
  reconsider m = min(b,c) as R_eal by XXREAL_0:def 1;
  assume a < b & a < c;
  then a < min(b,c) by XXREAL_0:def 9;
  then consider x being R_eal such that
A3: a < x & x < m & x in REAL by Th2;
  take x;
  ex o being R_eal st o in {b,c} & o <= b
  proof
    take b;
    thus thesis by TARSKI:def 2;
  end;
  then min(b,c) <= b by A1,XXREAL_2:62;
  hence thesis by A3,A2,XXREAL_0:2;
end;
