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

theorem Th31:
  for M be Reflexive symmetric triangle non empty MetrStruct st
  TopSpaceMetr M is countably_compact holds M is totally_bounded
proof
  deffunc P(set)=meet $1;
  let M be Reflexive symmetric triangle non empty MetrStruct such that
A1: TopSpaceMetr M is countably_compact;
  set T=TopSpaceMetr M;
  assume not M is totally_bounded;
  then consider r be Real, A be Subset of M such that
A2: r>0 and
A3: for p,q be Point of M st p <> q & p in A & q in A holds dist(p,q) >= r and
A4: A is infinite by Th30;
  reconsider A as Subset of T;
  set F={{x} where x is Element of T: x in A};
  reconsider F as Subset-Family of T by RELSET_2:16;
A5: now
    let a be Subset of T;
    assume a in F;
    then ex y be Point of T st a={y} & y in A;
    hence card a = 1 by CARD_1:30;
  end;
  set PP={P(y) where y is Subset of T:y in F};
A6: A c= PP
  proof
    let y be object such that
A7: y in A;
    reconsider y9=y as Point of T by A7;
    {y9} in F by A7;
    then P({y9}) in PP;
    hence thesis by SETFAM_1:10;
  end;
  F is locally_finite by A2,A3,Lm7;
  then
A8: F is finite by A1,A5,Th26;
  PP is finite from FRAENKEL:sch 21(A8);
  hence thesis by A4,A6;
end;
