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_2:32;
    ].a,b.[ <> {} & inf ].a,b.[ = a by A1,XXREAL_1:33,XXREAL_2:28;
    hence diameter ].a,b.[ = b - a by A2,Def6;
  end;
  assume b <= a;
  then ].a,b.[ = {} by XXREAL_1:28;
  hence thesis by Def6;
end;
