
theorem CopyForValued:
  for w being real-valued FinSequence,
      r being Real,
      i being Nat st
    i in dom w holds w+*(i,r) = (w | (i-'1)) ^ <*r*> ^ (w/^i)
  proof
    let w be real-valued FinSequence, r be Real, i be Nat;
    reconsider ww = w as FinSequence of REAL by RVSUM_1:145;
    reconsider rr = r as Element of REAL by XREAL_0:def 1;
    assume i in dom w; then
    ww +* (i,rr) = (ww | (i-'1)) ^ <*rr*> ^ (ww/^i) by FUNCT_7:98;
    hence thesis;
  end;
