
theorem Th68:
  for F be disjoint_valued FinSequence of Family_of_Intervals
    st Union F in Family_of_Intervals holds
      pre-Meas.(Union F) = Sum(pre-Meas*F)
proof
    let F be disjoint_valued FinSequence of Family_of_Intervals;
    assume Union F in Family_of_Intervals; then
    consider G be disjoint_valued FinSequence of Family_of_Intervals
     such that
A1:  F,G are_fiberwise_equipotent and
A2:  for n be Nat st n in dom G holds Union(G|n) in Family_of_Intervals
       & pre-Meas.(Union(G|n)) = Sum(pre-Meas*(G|n)) by Th63;
    per cases;
    suppose A3: F = {}; then
      union rng F = {} by ZFMISC_1:2; then
      Union F = {} & {} c= REAL by CARD_3:def 4; then
      pre-Meas.(Union F) = diameter {} by Th59
        .= 0 by MEASURE5:def 6;
      hence pre-Meas.(Union F) = Sum(pre-Meas*F) by A3,EXTREAL1:7;
    end;
    suppose F <> {}; then
A4:  1 <= len F by FINSEQ_1:20;
A5:  len F = len G & dom F = dom G by A1,RFINSEQ:3;
  rng F = rng G by A1,CLASSES1:75; then
     Union F = union rng G by CARD_3:def 4; then
A6: Union F = Union G by CARD_3:def 4;
A7:  G|len F = G by A5,FINSEQ_1:58;
     len F in dom G by A4,A5,FINSEQ_3:25; then
A8: pre-Meas.(Union G) = Sum(pre-Meas*G) by A7,A2;
A9: pre-Meas*G is nonnegative & pre-Meas*F is nonnegative by MEASURE9:47;
A10: dom(pre-Meas) = Family_of_Intervals by FUNCT_2:def 1;
     rng G c= Family_of_Intervals & rng F c= Family_of_Intervals;
     hence pre-Meas.(Union F) = Sum(pre-Meas*F)
       by A6,A8,A9,Th67,A1,A5,A10,CLASSES1:83;
    end;
end;
