
theorem Th56:
for X be non empty set, F be without_+infty-valued FinSequence
  of Funcs(X,ExtREAL) holds Partial_Sums F is without_+infty-valued
proof
   let X be non empty set, F be without_+infty-valued FinSequence of
    Funcs(X,ExtREAL);
   defpred P[Nat] means $1 in dom (Partial_Sums F) implies
     (Partial_Sums F).$1 is without+infty;
A1:P[0] by FINSEQ_3:24;
A2:for n be Nat st P[n] holds P[n+1]
   proof
    let n be Nat;
    assume A3: P[n];
B1: len F = len (Partial_Sums F) by DEF13; then
A4: dom F = dom (Partial_Sums F) by FINSEQ_3:29;
    assume A5: n+1 in dom(Partial_Sums F);
    per cases;
    suppose A6: n = 0; then
     F.1 is without+infty by A4,A5,DEF10;
     hence (Partial_Sums F).(n+1) is without+infty by A6,DEF13;
    end;
    suppose A7: n <> 0; then
A8:  n >= 1 by NAT_1:14;
     n+1 <= len F by A5,B1,FINSEQ_3:25; then
A9:  n < len F by NAT_1:13;
     F.(n+1) is without+infty by A4,A5,DEF10; then
     reconsider p = (Partial_Sums F)/.n, q = F/.(n+1)
       as without+infty Function of X,ExtREAL
         by A3,A4,A5,A8,A9,FINSEQ_3:25,PARTFUN1:def 6;
     p+q is without+infty Function of X,ExtREAL;
     hence (Partial_Sums F).(n+1) is without+infty by A7,A9,DEF13,NAT_1:14;
    end;
   end;
   for n be Nat holds P[n] from NAT_1:sch 2(A1,A2);
   hence thesis;
end;
