
theorem Th69:
  for K be disjoint_valued Function of NAT,Family_of_Intervals
    st Union K in Family_of_Intervals holds
      pre-Meas.(Union K) <= SUM(pre-Meas*K)
proof
    let K be disjoint_valued Function of NAT,Family_of_Intervals;
    assume
A1: Union K in Family_of_Intervals;
    reconsider F = K as sequence of bool REAL by FUNCT_2:7;
    pre-Meas.(Union K) = OS_Meas.(Union F) by A1,FUNCT_1:49
     .= OS_Meas.(union rng F) by CARD_3:def 4; then
A2: pre-Meas.(Union K) <= SUM(OS_Meas*F) by MEASURE4:def 1;
    for n be Element of NAT holds (OS_Meas*F).n = (pre-Meas*K).n
    proof
      let n be Element of NAT;
      reconsider A = F.n as Subset of REAL;
A3:   dom F = NAT & dom K = NAT by FUNCT_2:def 1; then
      (pre-Meas*K).n = pre-Meas.(K.n) by FUNCT_1:13
        .= OS_Meas.(K.n) by FUNCT_1:49;
      hence thesis by A3,FUNCT_1:13;
    end;
    hence pre-Meas.(Union K) <= SUM(pre-Meas*K) by A2,FUNCT_2:def 8;
end;
