
theorem SPP:
  for n be Nat, f1,f2 be n-element complex-valued XFinSequence holds
  Sum (f1 + f2) = Sum f1 + Sum f2
  proof
    let n be Nat, f1, f2 be n-element complex-valued XFinSequence;
    defpred P[Nat] means
      for f1,f2 be $1-element complex-valued XFinSequence holds
    Sum (f1 + f2) = Sum f1 + Sum f2;
    A1: for k be Nat st P[k] holds P[k+1]
    proof
      let k be Nat such that
      B1: P[k];
      for f1,f2 be (k+1)-element complex-valued XFinSequence holds
        Sum (f1 + f2) = Sum f1 + Sum f2
      proof
        let f1,f2 be (k+1)-element complex-valued XFinSequence;
        reconsider F = f1 + f2 as (k+1)-element complex-valued XFinSequence;
        reconsider G = (f1 + f2)|k as k-element complex-valued XFinSequence;
        reconsider g1 = f1|k as k-element complex-valued XFinSequence;
        reconsider g2 = f2|k as k-element complex-valued XFinSequence;
        C1: dom f1 = k+1 & dom f2 = k+1 & dom F = k+1 by CARD_1:def 7;
        k + 0 < k + 1 by XREAL_1:6; then
        C2: k in Segm (len f1) & k in Segm (len f2) & k in Segm (len F)
          by C1,NAT_1:44; then
        C3: Sum (f1|(k+1)) = Sum g1 + f1.k & Sum (f2|(k+1)) = Sum g2 + f2.k
          by AFINSQ_2:65;
        C4: Sum (g1 + g2) = Sum g1 + Sum g2 by B1;
        Sum F = Sum (F|(k+1))
        .= Sum G + F.k by C2,AFINSQ_2:65
        .= (Sum g1 + Sum g2) + (f1 + f2).k by C4,SFG
        .= Sum g1 + Sum g2 + (f1.k + f2.k) by C2,VALUED_1:def 1
        .= Sum f1 + Sum f2 by C3;
        hence thesis;
      end;
      hence thesis;
    end;
    A2: P[0];
    for k be Nat holds P[k] from NAT_1:sch 2(A2,A1);
    hence thesis;
  end;
