
theorem C265:
  for n be Nat, f be Complex_Sequence holds
    XProduct (f|n) * f.n = XProduct (f|(n+1))
  proof
    let n be Nat, f be Complex_Sequence;
    reconsider g = f|(n+1) as complex-valued XFinSequence;
    n + 0 < n + 1 by XREAL_1:6; then
    A1: n in dom g by AFINSQ_1:86; then
    XProduct (g|(n+1)) = (XProduct(g|n))*g.n by A265
    .= (XProduct(f|((n+1)/\n)))*(f|(n+1)).n by RELAT_1:71
    .= XProduct (f|n)*(f|(n+1)).n;
    hence thesis by A1,FUNCT_1:47;
  end;
