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

theorem
  for a,b being R_eal holds a = -infty & b = +infty & (A = ].a,b.[ or A
  = [.a,b.] or A = [.a,b.[ or A = ].a,b.]) implies diameter(A) = +infty
proof
  let a,b be R_eal;
  assume that
A1: a = -infty & b = +infty and
A2: A = ].a,b.[ or A = [.a,b.] or A = [.a,b.[ or A = ].a,b.];
A3: sup A = +infty & inf A = -infty by A1,A2,XXREAL_2:25,26,27,28,29,30,31,32;
  then A is non empty by XXREAL_2:40;
  then diameter(A) = b - a by A1,A3,Def6
    .= +infty by A1,XXREAL_3:13;
  hence thesis;
end;
