reserve n,n1,m,k for Nat;
reserve x,y for set;
reserve s,g,g1,g2,r,p,p2,q,t for Real;
reserve s1,s2,s3 for Real_Sequence;
reserve Nseq for increasing sequence of NAT;
reserve X for Subset of REAL;

theorem
  for X st X<>{} & X is compact holds upper_bound X in X & lower_bound X in X
proof
  let X such that
A1: X<>{} and
A2: X is compact;
  X is real-bounded closed by A2,Th10;
  hence thesis by A1,Th12,Th13;
end;
