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
  X is bounded_below iff q3++X is bounded_below
proof
  -q3++(q3++X) = (-q3+q3)++X by MEMBER_1:147
    .= X by MEMBER_1:146;
  hence thesis by Lm7;
end;
