
theorem Th58:
for X be set, S be Field_Subset of X, M be Measure of S,
    F be Sep_Sequence of S, n be Nat
 holds union rng(F|Segm(n+1)) in S
     & Partial_Sums(M*F).n = M.(union rng(F|Segm(n+1)))
proof
  let X be set, S be Field_Subset of X, M be Measure of S,
      F be Sep_Sequence of S, n be Nat;
A2: rng(F|Segm(0+1)) = rng(F|Segm 0) \/ {F.0} by Th57
     .= {F.0}; then
A2a:union rng(F|Segm(0+1)) = F.0 by ZFMISC_1:25;
    defpred P2[Nat] means union rng(F|Segm($1+1)) in S;
A14:P2[0] by A2a;
A15:for k be Nat st P2[k] holds P2[k+1]
    proof
     let k be Nat;
     assume A16: P2[k];
     union rng(F|Segm(k+1+1))
      = union(rng(F|Segm(k+1)) \/ {F.(k+1)}) by Th57
     .= union rng(F|Segm(k+1)) \/ union {F.(k+1)} by ZFMISC_1:78
     .= union rng(F|Segm(k+1)) \/ F.(k+1) by ZFMISC_1:25;
     hence union rng(F|Segm(k+1+1)) in S by A16,PROB_1:3;
    end;
P1: for k be Nat holds P2[k] from NAT_1:sch 2(A14,A15);
    hence union rng(F|Segm(n+1)) in S;
    defpred P[Nat] means Partial_Sums(M*F).$1 = M.(union rng (F|Segm ($1+1)));
A1: Partial_Sums(M*F).0 = (M*F).0 by MESFUNC9:def 1
     .= M.(F.0) by FUNCT_2:15;
A3: P[0] by A1,A2,ZFMISC_1:25;
A4: for n be Nat st P[n] holds P[n+1]
    proof
     let n be Nat;
     assume A5: P[n];
A6:  Partial_Sums(M*F).(n+1) = Partial_Sums(M*F).n + (M*F).(n+1)
       by MESFUNC9:def 1
      .= M.(union rng(F|Segm(n+1))) + M.(F.(n+1)) by A5,FUNCT_2:15;
A13: now assume ex x be object st x in union rng(F|Segm(n+1)) /\ F.(n+1); then
      consider x be object such that
A7:    x in union rng(F|Segm(n+1)) /\ F.(n+1);
A8:   x in union rng(F|Segm(n+1)) & x in F.(n+1) by A7,XBOOLE_0:def 4; then
      consider A be set such that
A9:    x in A & A in rng(F|Segm(n+1)) by TARSKI:def 4;
      consider k be object such that
A10:   k in dom(F|Segm(n+1)) & A = (F|Segm(n+1)).k by A9,FUNCT_1:def 3;
      reconsider k as Nat by A10;
A11:  k < n+1 by A10,RELAT_1:57,NAT_1:44;
      A = F.k by A10,FUNCT_1:47; then
      x in F.k /\ F.(n+1) by A8,A9,XBOOLE_0:def 4;
      hence contradiction by A11,PROB_2:def 2,XBOOLE_0:4;
     end;
     union rng(F|Segm(n+1)) in S by P1; then
     M.(union rng(F|Segm(n+1))) + M.(F.(n+1))
      = M.(union rng(F|Segm(n+1)) \/ F.(n+1)) by A13,XBOOLE_0:4,MEASURE1:def 8
     .= M.( union rng(F|Segm(n+1)) \/ union {F.(n+1)} ) by ZFMISC_1:25
     .= M.( union (rng(F|Segm(n+1)) \/ {F.(n+1)}) ) by ZFMISC_1:78
     .= M.( union rng(F|Segm(n+1+1)) ) by Th57;
     hence thesis by A6;
    end;
    for n be Nat holds P[n] from NAT_1:sch 2(A3,A4);
    hence Partial_Sums(M*F).n = M.(union rng (F|Segm (n+1)));
end;
