
theorem Th44:
for I be Element of Family_of_Intervals st I is Interval
  holds OS_Meas.I <= diameter I
proof
    let I be Element of Family_of_Intervals;
    assume
A1:  I is Interval;
    per cases;
    suppose A2: I = {};
     OS_Meas is zeroed by MEASURE4:def 1; then
     OS_Meas.I = 0 by A2,VALUED_0:def 19;
     hence OS_Meas.I <= diameter I by A2,MEASURE5:def 6;
    end;
    suppose A3: I <> {};
     I is open_interval or I is closed_interval or
     I is right_open_interval or I is left_open_interval by A1,MEASURE5:1;
     hence OS_Meas.I <= diameter I
       by A3,Th42,Th43,Lm10,Lm11;
    end;
end;
