reserve M for non empty MetrSpace,
  c,g1,g2 for Element of M;
reserve N for non empty MetrStruct,
  w for Element of N,
  G for Subset-Family of N,
  C for Subset of N;
reserve R for Reflexive non empty MetrStruct;
reserve T for Reflexive symmetric triangle non empty MetrStruct,
  t1 for Element of T,
  Y for Subset-Family of T,
  P for Subset of T;
reserve f for Function,
  n,m,p,n1,n2,k for Nat,
  r,s,L for Real,
  x,y for set;
reserve S1 for sequence of M,
  S2 for sequence of N;

theorem
  T is totally_bounded implies T is bounded
proof
  assume T is totally_bounded;
  then consider Y such that
A1: Y is finite & the carrier of T = union Y and
A2: for P st P in Y ex x being Element of T st P = Ball(x,1);
  for P being Subset of T st P in Y holds P is bounded
  proof
    let P be Subset of T;
    assume P in Y;
    then ex x being Element of T st P = Ball(x,1) by A2;
    hence thesis;
  end;
  then [#]T is bounded by A1,Th17;
  hence thesis by Th18;
end;
