
theorem Th64:
for X be non empty set, S be SigmaField of X, P be Element of S,
 F be summable FinSequence of Funcs(X,ExtREAL)
 st (for n be Nat st n in dom F holds F/.n is P-measurable) holds
 for n be Nat st n in dom F holds (Partial_Sums F)/.n is P-measurable
proof
   let X be non empty set, S be SigmaField of X, P be Element of S,
   F be summable FinSequence of Funcs(X,ExtREAL);
   assume A1: for n be Nat st n in dom F holds F/.n is P-measurable;
A2:P c= X;
A3:len F = len (Partial_Sums F) by DEF13; then
A4:dom F = dom (Partial_Sums F) by FINSEQ_3:29;
   defpred P[Nat] means
     $1 in dom F implies (Partial_Sums F)/.$1 is P-measurable;
   per cases by DEF12;
   suppose A5: F is without_+infty-valued;
A6: P[0] by FINSEQ_3:24;
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 in dom F;
     per cases;
     suppose A10: n = 0; then
      (Partial_Sums F)/.(n+1) = (Partial_Sums F).1 by A4,A9,PARTFUN1:def 6
       .= F.1 by DEF13
       .= F/.1 by A9,A10,PARTFUN1:def 6;
      hence (Partial_Sums F)/.(n+1) is P-measurable by A1,A9,A10;
     end;
     suppose A11: n <> 0; then
A12:  n >= 1 by NAT_1:14;
      n+1 <= len F by A9,FINSEQ_3:25; then
A13:  n < len F by NAT_1:13; then
A15:  F/.(n+1) = F.(n+1) & (Partial_Sums F)/.n = (Partial_Sums F).n
    & (Partial_Sums F)/.(n+1) = (Partial_Sums F).(n+1)
         by A4,A9,A12,FINSEQ_3:25,PARTFUN1:def 6; then
A16:  (Partial_Sums F)/.(n+1) = (Partial_Sums F)/.n + F/.(n+1)
         by A11,A13,NAT_1:14,DEF13;
      Partial_Sums F is without_+infty-valued by A5,Th56; then
A17:  F/.(n+1) is without+infty & (Partial_Sums F)/.n is without+infty
         by A5,A9,A12,A15,A13,A3,FINSEQ_3:25; then
A19:  dom((Partial_Sums F)/.n + F/.(n+1))
       = dom((Partial_Sums F)/.n) /\ dom(F/.(n+1)) by MESFUNC9:1;
A18:  P c= dom((Partial_Sums F)/.n) & P c= dom(F/.(n+1)) by A2,FUNCT_2:def 1;
      F/.(n+1) is P-measurable by A9,A1;
      hence (Partial_Sums F)/.(n+1) is P-measurable
        by A8,A12,A13,A16,A17,A18,A19,Th61,FINSEQ_3:25,XBOOLE_1:19;
     end;
    end;
    for n be Nat holds P[n] from NAT_1:sch 2(A6,A7);
    hence
     for n be Nat st n in dom F holds (Partial_Sums F)/.n is P-measurable;
   end;
   suppose A19: F is without_-infty-valued;
A20:P[0] by FINSEQ_3:24;
A21:for n be Nat st P[n] holds P[n+1]
    proof
     let n be Nat;
     assume A24: P[n];
     assume A25: n+1 in dom F;
     per cases;
     suppose A26: n = 0; then
      (Partial_Sums F)/.(n+1) = (Partial_Sums F).1 by A25,A4,PARTFUN1:def 6
       .= F.1 by DEF13
       .= F/.1 by A25,A26,PARTFUN1:def 6;
      hence (Partial_Sums F)/.(n+1) is P-measurable by A1,A25,A26;
     end;
     suppose A27: n <> 0; then
A28:  n >= 1 by NAT_1:14;
      n+1 <= len F by A25,FINSEQ_3:25; then
A29:  n < len F by NAT_1:13; then
A30:  F/.(n+1) = F.(n+1) & (Partial_Sums F)/.n = (Partial_Sums F).n
    & (Partial_Sums F)/.(n+1) = (Partial_Sums F).(n+1)
         by A4,A25,A28,FINSEQ_3:25,PARTFUN1:def 6; then
A31:  (Partial_Sums F)/.(n+1) = (Partial_Sums F)/.n + F/.(n+1)
         by A27,A29,DEF13,NAT_1:14;
      Partial_Sums F is without_-infty-valued by A19,Th57; then
A32:  F/.(n+1) is without-infty & (Partial_Sums F)/.n is without-infty
          by A19,A25,A29,A3,A28,A30,FINSEQ_3:25;
      F/.(n+1) is P-measurable by A25,A1;
      hence (Partial_Sums F)/.(n+1) is P-measurable
        by A31,A32,A29,A24,A28,FINSEQ_3:25,MESFUNC5:31;
     end;
    end;
    for n be Nat holds P[n] from NAT_1:sch 2(A20,A21);
    hence
     for n be Nat st n in dom F holds (Partial_Sums F)/.n is P-measurable;
   end;
end;
