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 Th14:
  for C, D being Subset of N holds C is bounded & D c= C implies D is bounded
proof
  let C, D be Subset of N;
  assume that
A1: C is bounded and
A2: D c= C;
  consider r such that
A3: 0<r and
A4: for x,y being Point of N st x in C & y in C holds dist(x,y)<=r by A1;
  ex r st 0<r & for x,y being Point of N st x in D & y in D holds dist(x,y )<=r
  proof
    take r;
    thus 0<r by A3;
    let x,y be Point of N;
    assume x in D & y in D;
    hence thesis by A2,A4;
  end;
  hence thesis;
end;
