
theorem
  for N be Subset of REAL holds
    N is compact
  iff
    (for F be Subset-Family of REAL
       st F is Cover of N
        & (for P be Subset of REAL st P in F holds P is open) holds
       ex G be Subset-Family of REAL
         st G c= F & G is Cover of N & G is finite)
  proof
    let N be Subset of REAL;
    reconsider M = N as Subset of R^1;
    M is compact iff
    (for F be Subset-Family of REAL
       st F is Cover of N
        & for P be Subset of REAL st P in F holds P is open holds
       ex G be Subset-Family of REAL
         st G c= F & G is Cover of N & G is finite) by Th27;
    hence thesis by JORDAN5A:25;
  end;
