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
  for S, V being Subset of R holds S is bounded & V c= S implies
  diameter V <= diameter S
proof
  let S, V be Subset of R;
  assume that
A1: S is bounded and
A2: V c= S;
A3: V is bounded by A1,A2,Th14;
  per cases;
  suppose
    V = {};
    then diameter V = 0 by Def8;
    hence thesis by A1,Th21;
  end;
  suppose
A4: V <> {};
    for x,y being Point of R st x in V & y in V holds dist(x,y)<=(diameter
    S ) by A1,A2,Def8;
    hence thesis by A3,A4,Def8;
  end;
end;
