
theorem Th42:
  for I be Element of Family_of_Intervals st I is open_interval holds
    OS_Meas.I <= diameter I
proof
    let I be Element of Family_of_Intervals;
    assume
     I is open_interval; then
    consider F be Open_Interval_Covering of I such that
A1:  F.0 = I &
     (for n being Nat st n <> 0 holds F.n = {}) &
     union rng F = I &
     SUM(F vol) = diameter I by Th36;
    vol F = diameter I by A1,MEASURE7:def 6; then
A2: diameter I in Svc2(I) by Def7;
    inf(Svc2(I)) is LowerBound of Svc2(I) by XXREAL_2:def 4; then
A3: inf(Svc2(I)) <= diameter I by A2,XXREAL_2:def 2;
    inf(Svc I) <= inf(Svc2 I) by Th30; then
    inf(Svc I) <= diameter I by A3,XXREAL_0:2;
    hence thesis by MEASURE7:def 10;
end;
