reserve a, b, r, s for Real;

theorem Th10:
  upper_bound left_open_halfline(a) = a
proof
  set X = left_open_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-s+a)/2;
    (a-s+a)/2 < a by A2,XREAL_1:226;
    hence thesis by A2,XREAL_1:226,XXREAL_1:233;
  end;
  for r st r in X holds r <= a by XXREAL_1:233;
  hence thesis by A1,SEQ_4:def 1;
end;
