
theorem Th10:
  for A being non empty Subset of R^1 ex X being non empty Subset
  of REAL st A = X & lower_bound A = lower_bound X
proof
  let A be non empty Subset of R^1;
  reconsider X = A as non empty Subset of REAL by TOPMETR:17;
  take X;
  lower_bound A = lower_bound [#] A by WEIERSTR:def 3
    .= lower_bound X by WEIERSTR:def 1;
  hence thesis;
end;
