
theorem Th10:
for X be non empty set, F be summable FinSequence of Funcs(X,ExtREAL) holds
   dom F = dom (Partial_Sums F)
 & (for n be Nat st n in dom F holds (Partial_Sums F)/.n = (Partial_Sums F).n)
 & (for n be Nat, x be Element of X st 1 <= n < len F holds
     ((Partial_Sums F)/.(n+1)).x = ((Partial_Sums F)/.n).x + (F/.(n+1)).x)
proof
   let X be non empty set, F be summable FinSequence of Funcs(X,ExtREAL);
   len F = len (Partial_Sums F) by MEASUR11:def 11;
   hence A1: dom F = dom(Partial_Sums F) by FINSEQ_3:29;
   hence for n be Nat st n in dom F holds
     (Partial_Sums F)/.n = (Partial_Sums F).n by PARTFUN1:def 6;
   thus for n be Nat, x be Element of X st 1 <= n < len F holds
     ((Partial_Sums F)/.(n+1)).x = ((Partial_Sums F)/.n).x + (F/.(n+1)).x
   proof
    let n be Nat, x be Element of X;
    assume A3: 1 <= n < len F; then
    1 <= n+1 <= len F by NAT_1:13; then
A4: (Partial_Sums F)/.(n+1) = (Partial_Sums F).(n+1)
      by A1,PARTFUN1:def 6,FINSEQ_3:25
     .= ((Partial_Sums F)/.n) + F/.(n+1) by A3,MEASUR11:def 11;
    dom ((Partial_Sums F)/.(n+1)) = X by FUNCT_2:def 1;
    hence ((Partial_Sums F)/.(n+1)).x = ((Partial_Sums F)/.n).x + (F/.(n+1)).x
      by A4,MESFUNC1:def 3;
   end;
end;
