reserve a, b, r, s for Real;

theorem Th9:
  upper_bound left_closed_halfline(a) = a
proof
  set X = left_closed_halfline(a);
A1: for s st 0 < s ex r st r in X & a-s < r
  proof
    let s;
    assume 0 < s;
    then
A2: a-s < a-0 by XREAL_1:15;
    take a;
    thus a in X by XXREAL_1:234;
    thus thesis by A2;
  end;
  for r st r in X holds r <= a by XXREAL_1:234;
  hence thesis by A1,SEQ_4:def 1;
end;
