reserve i,n,m for Nat,
  x,y,X,Y for set,
  r,s for Real;

theorem Th34:
  for M be non empty MetrSpace holds TopSpaceMetr M is compact iff
  TopSpaceMetr M is countably_compact
proof
  let M be non empty MetrSpace;
  set T=TopSpaceMetr M;
  thus T is compact implies T is countably_compact;
  assume
A1: T is countably_compact;
  let F be Subset-Family of T such that
A2: F is Cover of T and
A3: F is open;
  M is totally_bounded by A1,Th31;
  then T is second-countable by Th32;
  then consider G be Subset-Family of T such that
A4: G c= F and
A5: G is Cover of T and
A6: G is countable by A2,A3,Th33;
  G is open by A3,A4;
  then ex H be Subset-Family of T st H c= G & H is Cover of T & H is
  finite by A1,A5,A6;
  hence thesis by A4,XBOOLE_1:1;
end;
