
theorem Th13:
  for F being sequence of (bool REAL) holds for G being
Interval_Covering of F holds for n being Element of NAT holds 0. <= (vol(G)).n
proof
  let F be sequence of  bool REAL;
  let G be Interval_Covering of F;
  let n be Element of NAT;
  for k being Element of NAT holds 0. <= ((G.n) vol).k
  proof
    let k be Element of NAT;
    0. <= diameter((G.n).k) by MEASURE5:13;
    hence thesis by Def4;
  end;
  then
A1: ((G.n) vol) is nonnegative by SUPINF_2:39;
  (vol(G)).n = vol(G.n) by Def7;
  hence thesis by A1,MEASURE6:2;
end;
