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

theorem
  for a,b being R_eal holds (a <= b implies diameter [.a,b.] = b - a) &
  (b < a implies diameter [.a,b.] = 0.)
proof
  let a,b being R_eal;
  hereby
    assume
A1: a <= b;
    then
A2: sup [.a,b.] = b by XXREAL_2:29;
    [.a,b.] <> {} & inf [.a,b.] = a by A1,XXREAL_1:30,XXREAL_2:25;
    hence diameter [.a,b.] = b - a by A2,Def6;
  end;
  assume b < a;
  then [.a,b.] = {} by XXREAL_1:29;
  hence thesis by Def6;
end;
