reserve r, s, t for Real;
reserve seq for Real_Sequence,
  X, Y for Subset of REAL;
reserve r3, r1, q3, p3 for Real;

theorem
  for X being non empty Subset of REAL st X is bounded_below
   holds lower_bound (q3++X) = q3+lower_bound X
proof
  let X be non empty Subset of REAL such that
A1: X is bounded_below;
  set i = q3+lower_bound X;
A2: now
    let t;
    assume
A3: for s st s in q3++X holds s >= t;
    now
      let s;
      assume s in X;
      then t <= q3+s by A3,MEMBER_1:141;
      hence t-q3 <= s by XREAL_1:20;
    end;
    then lower_bound X >= t-q3 by SEQ_4:43;
    hence t <= i by XREAL_1:20;
  end;
  now
    let s;
    assume s in q3++X; then
    s in { q3+r3 : r3 in X} by Lm5;
    then consider r3 such that
A4: q3+r3 = s and
A5: r3 in X;
    r3 >= lower_bound X by A1,A5,SEQ_4:def 2;
    hence s >= i by A4,XREAL_1:6;
  end;
  hence thesis by A2,SEQ_4:44;
end;
