
theorem Th1:
  for X being non empty bounded_below real-membered set, Y being
  closed Subset of REAL st X c= Y holds lower_bound X in Y
proof
  let X be non empty bounded_below real-membered set;
  let Y be closed Subset of REAL;
  assume
A1: X c= Y;
  reconsider X as non empty bounded_below Subset of REAL by MEMBERED:3;
A2: lower_bound X = lower_bound Cl X & lower_bound Cl X in Cl X by RCOMP_1:13
,TOPREAL6:68;
  Cl X c= Y by A1,MEASURE6:57;
  hence thesis by A2;
end;
