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 Th15:
  for P being Subset of T holds P = {t1} implies P is bounded
proof
  let P be Subset of T;
  assume
A1: P = {t1};
  {t1} is Subset of Ball(t1,1) by Th11,SUBSET_1:41;
  hence thesis by A1,Th14;
end;
