 reserve Rseq, Rseq1, Rseq2 for Function of [:NAT,NAT:],REAL;

theorem SH5:
  for s being Real_Sequence, n being Nat holds
    (Partial_Sums s).n = Sum(Shift(s|Segm(n+1),1))
proof
   let s be Real_Sequence, n be Nat;
   defpred P[Nat] means (Partial_Sums s).$1 = Sum(Shift(s|Segm($1+1),1));
A1:(Partial_Sums s).0 = s.0 by SERIES_1:def 1;
   Shift(s|Segm(0+1),1) = <*s.0*> by SH4; then
A2:P[0] by A1,RVSUM_1:73;
A3:for k being Nat st P[k] holds P[k+1]
   proof
    let k be Nat;
    assume A4: P[k];
A5: (Partial_Sums s).(k+1) = (Partial_Sums s).k + s.(k+1) by SERIES_1:def 1;
    Shift(s|Segm(k+1+1),1) = Shift(s|Segm(k+1),1)^<*s.(k+1)*> by SH4;
    hence P[k+1] by A4,A5,RVSUM_1:74;
   end;
   for k be Nat holds P[k] from NAT_1:sch 2(A2,A3);
   hence thesis;
end;
