reserve a, b, r, s for Real;

theorem Th12:
  lower_bound right_open_halfline(a) = a
proof
  set X = right_open_halfline(a);
A1: for s st 0 < s ex r st r in X & r < a+s
  proof
    let s;
    assume 0 < s;
    then
A2: a+0 < a+s by XREAL_1:6;
    take (a+a+s)/2;
    a < (a+(a+s))/2 by A2,XREAL_1:226;
    hence thesis by A2,XREAL_1:226,XXREAL_1:235;
  end;
  for r st r in X holds a <= r by XXREAL_1:235;
  hence thesis by A1,SEQ_4:def 2;
end;
