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

theorem
  for a,b,c being R_eal st a < c & b < c ex x being R_eal st a < x & b <
  x & x < c & x in REAL
proof
  let a,b,c be R_eal;
  reconsider m = max(a,b) as R_eal by XXREAL_0:def 1;
A1: b in {a,b} by TARSKI:def 2;
  assume a < c & b < c;
  then max(a,b) < c by XXREAL_0:def 10;
  then consider x being R_eal such that
A2: m < x & x < c & x in REAL by Th2;
  take x;
  max(a,b) = max{a,b} & a in {a,b} by TARSKI:def 2,XXREAL_2:12;
  hence thesis by A2,A1,XXREAL_2:61;
end;
