reserve i,n,m for Nat,
  x,y,X,Y for set,
  r,s for Real;

theorem Th16:
  for M be non empty MetrSpace,A be non empty Subset of M for B be
Subset of M, B9 be Subset of M|A st B = B9 & B is bounded holds diameter B9 <=
  diameter B
proof
  let M be non empty MetrSpace,A be non empty Subset of M;
  let B be Subset of M, B9 be Subset of M|A such that
A1: B = B9 and
A2: B is bounded;
A3: B9 is bounded by A1,A2,Th15;
  per cases;
  suppose
A4: B9={}(M|A);
    then diameter B9=0 by TBSP_1:def 8;
    hence thesis by A1,A4,TBSP_1:def 8;
  end;
  suppose
A5: B9<>{}(M|A);
    now
      let x,y be Point of (M|A) such that
A6:   x in B9 and
A7:   y in B9;
      reconsider x9=x,y9=y as Point of M by TOPMETR:8;
      dist(x,y) = dist(x9,y9) by TOPMETR:def 1;
      hence dist(x,y)<=diameter B by A1,A2,A6,A7,TBSP_1:def 8;
    end;
    hence thesis by A3,A5,TBSP_1:def 8;
  end;
end;
