
theorem
  for f be Complex_Sequence, n be Nat holds
    XProduct (f|(n+1)) = (Partial_Product f).n
  proof
    let f be Complex_Sequence, n be Nat;
    defpred P[Nat] means XProduct (f|($1+1)) = (Partial_Product f).$1;
    A1: P[0]
    proof
      XProduct (f|(0+1)) = XProduct (f|0) * f.0 by C265
      .= 1*f.0 by PFO;
      hence thesis by PP;
    end;
    A2: for k be Nat st P[k] holds P[k+1]
    proof
      let k be Nat such that
      B1: XProduct (f|(k+1)) = (Partial_Product f).k;
      XProduct (f|((k+1)+1)) = (Partial_Product f).k * f.(k+1) by B1,C265;
      hence thesis by PP;
    end;
    for x be Nat holds P[x] from NAT_1:sch 2(A1,A2);
    hence thesis;
  end;
