reserve C for Simple_closed_curve,
  p,q,p1 for Point of TOP-REAL 2,
  i,j,k,n for Nat,
  r,s for Real;

theorem Th2:
for A being compact Subset of REAL, B being non empty Subset of REAL st B c= A
  holds lower_bound B in A
proof
  let A be compact Subset of REAL, B be non empty Subset of REAL such that
A1: B c= A;
A2: A is real-bounded by RCOMP_1:10;
  then
A3: B is bounded_below by A1,XXREAL_2:44;
A4: Cl B c= A by A1,MEASURE6:57;
  Cl B is bounded_below by A1,A2,MEASURE6:57,XXREAL_2:44;
  then lower_bound Cl B in Cl B by RCOMP_1:13;
  then lower_bound Cl B in A by A4;
  hence thesis by A3,TOPREAL6:68;
end;
