reserve k, m, n, p, K, N for Nat;
reserve i for Integer;
reserve x, y, eps for Real;
reserve seq, seq1, seq2 for Real_Sequence;
reserve sq for FinSequence of REAL;

theorem Th12:
  bseq(k) is convergent & lim(bseq(k))=1/(k!) & lim(bseq(k))=eseq. k
proof
  defpred P[Nat] means bseq($1) is convergent & lim(bseq($1))=1/($1!);
A1: now
    let k;
    assume
A2: P[k];
    thus P[k+1]
    proof
      deffunc F(Nat)=(1/(k+1))*(bseq(k).$1);
      consider seq such that
A3:   for n holds seq.n = F(n) from SEQ_1:sch 1;
      deffunc G(Nat)=seq.$1*(aseq(k).$1);
      consider seq1 such that
A4:   for n holds seq1.n=G(n) from SEQ_1:sch 1;
A5:   for n st n>=1 holds bseq(k+1).n=seq1.n
      proof
        let n;
        assume n>=1;
        hence bseq(k+1).n = (1/(k+1))*(bseq(k).n)*(aseq(k).n) by Th6
          .= (seq.n)*(aseq(k).n) by A3
          .= seq1.n by A4;
      end;
A6:   seq = (1/(k+1))(#)(bseq(k)) by A3,SEQ_1:9;
      then
A7:   seq is convergent by A2;
A8:   lim(seq) = (1/(k+1))*(1/(k!)) by A2,A6,SEQ_2:8
        .= 1/((k+1)!) by Th11;
A9:   aseq(k) is convergent by Th8;
A10:  seq1=seq(#)(aseq(k)) by A4,SEQ_1:8;
      then
A11:  seq1 is convergent by A7,A9;
      hence bseq(k+1) is convergent by A5,SEQ_4:18;
      lim(seq1) = lim(seq)*lim(aseq(k)) by A7,A10,A9,SEQ_2:15
        .= 1/((k+1)!) by A8,Th8;
      hence thesis by A11,A5,SEQ_4:19;
    end;
  end;
A12: P[0]
  proof
    reconsider jj = 1 as Element of REAL by XREAL_0:def 1;
    set bseq0 = seq_const 1;
A13: for n being Nat holds bseq0.n=1 by SEQ_1:57;
A14: for n st n>=1 holds bseq(0).n=bseq0.n
    proof
      let n;
      assume n>=1;
      thus bseq(0).n = 1 by Th10
        .= bseq0.n by SEQ_1:57;
    end;
    hence bseq(0) is convergent by SEQ_4:18;
    lim(bseq0)=1 by A13,Th9;
    hence thesis by A14,NEWTON:12,SEQ_4:19;
  end;
  for k holds P[k] from NAT_1:sch 2(A12,A1);
  hence bseq(k) is convergent & lim(bseq(k))=1/(k!);
  hence thesis by Def5;
end;
