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 Th21:
  for S being Subset of R st S is bounded holds 0 <= diameter S
proof
  let S be Subset of R;
  assume
A1: S is bounded;
  per cases;
  suppose
    S = {};
    hence thesis by Def8;
  end;
  suppose
A2: S <> {};
    set x = the Element of S;
    reconsider x as Element of R by A2,TARSKI:def 3;
    dist(x,x)<=diameter S by A1,A2,Def8;
    hence thesis by METRIC_1:1;
  end;
end;
