
theorem Th56:
for A be non empty Subset of REAL, F be Interval_Covering of A,
 G be one-to-one FinSequence of bool REAL,
 H be FinSequence of ExtREAL st
  rng G c= rng F & dom G = dom H &
 (for n be Nat holds H.n = diameter(G.n)) holds Sum H <= vol F
proof
    let A be non empty Subset of REAL, F be Interval_Covering of A,
    G be one-to-one FinSequence of bool REAL,
    H be FinSequence of ExtREAL;
    assume that
A1:  rng G c= rng F and
A2:  dom G = dom H and
A3:  for n be Nat holds H.n = diameter(G.n);

    consider F1 be Interval_Covering of A such that
A4:  (for n be Nat st n in dom G holds G.n = F1.n) &
     vol F1 = vol F by A1,Th55;

    consider S be sequence of ExtREAL such that
A5:  Sum H = S.(len H) & S.0 = 0 &
     for n be Nat st n < len H holds S.(n+1) = S.n + H.(n+1) by EXTREAL1:def 2;

    defpred P[Nat] means $1 <= len H implies S.$1 <= (Ser(F1 vol)).$1;

    F1 vol is nonnegative by MEASURE7:12; then
A6: P[0] by A5,SUPINF_2:40;

A7: for n be Nat st P[n] holds P[n+1]
    proof
     let n be Nat;
     assume
A8:   P[n];
     assume
A9:   n+1 <= len H; then
A10: n+1 in dom G by A2,FINSEQ_3:25,NAT_1:11;

     S.(n+1) = S.n + H.(n+1) by A5,A9,NAT_1:13; then
     S.(n+1) = S.n + diameter(G.(n+1)) by A3; then
     S.(n+1) = S.n + diameter(F1.(n+1)) by A4,A10; then
A11: S.(n+1) = S.n + (F1 vol).(n+1) by MEASURE7:def 4;

     S.n + (F1 vol).(n+1) <= (Ser(F1 vol)).n + (F1 vol).(n+1)
       by A8,A9,NAT_1:13,XXREAL_3:35;
     hence S.(n+1) <= (Ser(F1 vol)).(n+1) by A11,SUPINF_2:def 11;
    end;

    for n be Nat holds P[n] from NAT_1:sch 2(A6,A7); then
A12:Sum H <= (Ser(F1 vol)).(len H) by A5;
    (Ser(F1 vol)).(len H) <= SUM(F1 vol) by MEASURE7:6,12; then
    Sum H <= SUM(F1 vol) by A12,XXREAL_0:2;
    hence Sum H <= vol F by A4,MEASURE7:def 6;
end;
