
theorem Th2:
  for X being non empty bounded_above real-membered set, Y being
  closed Subset of REAL st X c= Y holds upper_bound X in Y
proof
  let X be non empty bounded_above real-membered set;
  let Y be closed Subset of REAL;
  assume
A1: X c= Y;
  reconsider X as non empty bounded_above Subset of REAL by MEMBERED:3;
A2: upper_bound X = upper_bound Cl X & upper_bound Cl X in Cl X by RCOMP_1:12
,TOPREAL6:69;
  Cl X c= Y by A1,MEASURE6:57;
  hence thesis by A2;
end;
