
theorem SumA:
  for f be real-valued FinSequence,
      i be Nat,
      a be Real st i in dom f holds
    Sum (f+*(i,a)) = Sum f - f.i + a
  proof
    let f be real-valued FinSequence,
        i be Nat,
        a be Real;
    reconsider w = f as FinSequence of REAL by RVSUM_1:145;
    reconsider aa = a as Element of REAL by XREAL_0:def 1;
    assume
A1: i in dom f; then
Z1: Sum (w+*(i,a)) = Sum ((w | (i-'1)) ^ <*aa*> ^ (w/^i)) by CopyForValued
    .= Sum ((w | (i-'1)) ^ <*aa*>) + Sum (w/^i) by RVSUM_1:75
    .= Sum (w | (i-'1)) + Sum <*aa*> + Sum (w/^i) by RVSUM_1:75
    .= Sum (w | (i-'1)) + aa + Sum (w/^i) by RVSUM_1:73
    .= aa + (Sum (w | (i-'1)) + Sum (w/^i))
    .= aa + Sum ((w | (i-'1)) ^ (w/^i)) by RVSUM_1:75;
    reconsider fi = f.i as Real;
    1 <= i & i <= len w by A1,FINSEQ_3:25; then
    Sum w = Sum ((w | (i-'1)) ^ <*f.i*> ^ (w/^i)) by FINSEQ_5:84
    .= Sum ((w | (i-'1)) ^ <*f.i*>) + Sum (w/^i) by RVSUM_1:75
    .= Sum (w | (i-'1)) + Sum <*f.i*> + Sum (w/^i) by RVSUM_1:75
    .= Sum (w | (i-'1)) + fi + Sum (w/^i) by RVSUM_1:73
    .= fi + (Sum (w | (i-'1)) + Sum (w/^i))
    .= fi + Sum ((w | (i-'1)) ^ (w/^i)) by RVSUM_1:75;
    hence thesis by Z1;
  end;
