 reserve n,k for Nat;
 reserve L for comRing;
 reserve R for domRing;
 reserve x0 for positive Real;

theorem Th13:
  for I be Ideal of L
  for F be FinSequence of L
    st for i being Nat st i in dom F holds F.i in I holds Sum F in I
    proof
      let I be Ideal of L;
      defpred P[Nat] means
      for F being FinSequence of L st len F = $1 &
      for i being Nat st i in dom F holds F.i in I holds Sum F in I;
A1:   P[0]
      proof
        let F be FinSequence of L;
        assume len F = 0 &
        for i being Nat st i in dom F holds F.i in I; then
        F = <*> the carrier of L; then
        Sum F = 0.L by RLVECT_1:43;
        hence Sum F in I by IDEAL_1:2;
      end;
A3:   for n being Nat st P[n] holds P[n+1]
      proof
        let n be Nat;
        assume
A4:     P[n];
        let F be FinSequence of L;
        assume
A5:     len F = n+1 &
        for i being Nat st i in dom F holds F.i in I;
        reconsider F0 = F| n as FinSequence of L;
        n+1 in Seg (n+1) by FINSEQ_1:4; then
        n+1 in dom F by A5,FINSEQ_1:def 3; then
        reconsider af = F.(n+1) as Element of I by A5;
A6:     len F0 = n by FINSEQ_1:59,A5,NAT_1:11; then
A7:     dom F0 = Seg n by FINSEQ_1:def 3;
A8:     len F = (len F0) + 1 by A5,FINSEQ_1:59,NAT_1:11;
A9:     F0 = F | dom F0 by A6,FINSEQ_1:def 3;
A10:    for i being Nat st i in dom F0 holds F0.i in I
        proof
          let i be Nat;
          assume
A11:      i in dom F0;
          dom F = Seg (n+1) by A5,FINSEQ_1:def 3; then
          dom F0 c= dom F by A7,FINSEQ_1:5,NAT_1:11; then
          F.i in I by A5,A11;
          hence thesis by A11,FUNCT_1:47;
        end;
        reconsider i1 = Sum F0 as Element of I by A6,A4,A10;
        reconsider i2 = af as Element of I;
        Sum F = i1 + i2 by A5,A8,A9,RLVECT_1:38;
        hence Sum F in I by IDEAL_1:def 1;
      end;
A12:  for n being Nat holds P[n] from NAT_1:sch 2(A1,A3);
      let F be FinSequence of L;
      assume
A13:  for i being Nat st i in dom F holds F.i in I;
      len F is Nat;
      hence Sum F in I by A12,A13;
    end;
