reserve c, c1, c2, d, d1, d2, e, y for Real,
  k, n, m, N, n1, N0, N1, N2, N3, M for Element of NAT,
  x for set;

theorem
  for n st n >= 1 holds for f being Real_Sequence, k being Element of
NAT st (for n holds f.n = Sum(seq_n^(k), n)) holds f.n >= (n to_power (k+1)) /
  (k+1)
proof
  defpred P[Nat] means for f being Real_Sequence, k being Element of NAT st (
for n holds f.n = Sum(seq_n^(k), n)) holds f.$1 >= ($1 to_power (k+1)) / (k+1);
A1: for n being Nat st n >= 1 & P[n] holds P[n+1]
  proof
    let n be Nat such that
    n >= 1 and
A2: for f being Real_Sequence, k being Element of NAT st (for n holds
    f.n = Sum(seq_n^(k), n)) holds f.n >= (n to_power (k+1)) / (k+1);
    reconsider n as Element of NAT by ORDINAL1:def 12;
    let f be Real_Sequence, k be Element of NAT such that
A3: for n holds f.n = Sum(seq_n^(k), n);
    set R3 = ((n,1) In_Power (k+1));
    len R3 = (k+1)+1 by NEWTON:def 4
      .= k+2;
    then reconsider R3 as Element of (k+2)-tuples_on REAL by FINSEQ_2:92;
    set R2 = ((k+1)")*((n,1) In_Power (k+1));
    len R2 = len ((n,1) In_Power (k+1)) by NEWTON:2
      .= (k+1)+1 by NEWTON:def 4
      .= k+2;
    then reconsider R2 as Element of (k+2)-tuples_on REAL by FINSEQ_2:92;
    reconsider nk = (n to_power (k+1)) / (k+1) as Element of REAL
            by XREAL_0:def 1;
    set R1 = <*nk*>^((n,1) In_Power k);
A4: len <*(n to_power (k+1)) / (k+1)*> = 1 by FINSEQ_1:40;
    set g = seq_n^(k);
    f.n >= (n to_power (k+1)) / (k+1) by A2,A3;
    then Sum(g,n) >= (n to_power (k+1)) / (k+1) by A3;
    then
A5: (Partial_Sums(g)).n >= (n to_power (k+1)) / (k+1) by SERIES_1:def 5;
    reconsider nk =(n to_power (k+1)) / (k+1) as Element of REAL
             by XREAL_0:def 1;
    g.(n+1) = (n+1) to_power k by Def3
      .= Sum((n,1) In_Power k) by NEWTON:30;
    then
A6: (n to_power (k+1)) / (k+1) + g.(n+1)
      = Sum (<*nk*>^((n,1) In_Power k)) by RVSUM_1:76;
    len ((n,1) In_Power k) = k+1 by NEWTON:def 4;
    then
A7: len R1 = (k+1)+1 by A4,FINSEQ_1:22
      .= k+2;
    then reconsider R1 as Element of (k+2)-tuples_on REAL
           by FINSEQ_2:92;
A8: for i being Nat st i in Seg (k+2) holds R2.i<=R1.i
    proof
      set k1 = (k+1)";
      let i be Nat such that
A9:   i in Seg (k+2);
A10:  1 <= i by A9,FINSEQ_1:1;
      set r2 = R2.i, r1 = R1.i;
A11:  i <= (k+2) by A9,FINSEQ_1:1;
      per cases by A10,XXREAL_0:1;
      suppose
A12:    i = 1;
        n|^(k+1) = R3.1 by NEWTON:28;
        then r2 = k1*(n|^(k+1)) by A12,RVSUM_1:45
          .= (n to_power (k+1)) / (k+1);
        hence thesis by A12,FINSEQ_1:41;
      end;
      suppose
A13:    i > 1;
        set i0 = i-1;
        set m = i0-1;
A14:    i-1 > 1-1 by A13,XREAL_1:9;
        then reconsider i0 as Element of NAT by INT_1:3;
        set l = k-m;
A15:    i0 >= 0+1 by A14,INT_1:7;
        then reconsider m as Element of NAT by INT_1:3;
        set i3 = (k+1)-i0;
        len (((n,1) In_Power k)) = k+1 by NEWTON:def 4;
        then
A16:    dom(((n,1) In_Power k)) = Seg(k+1) by FINSEQ_1:def 3;
        i-1 <= (k+2)-1 by A11,XREAL_1:9;
        then
A17:    i0 in dom(((n,1) In_Power k)) by A15,A16,FINSEQ_1:1;
        m = i-2;
        then
A18:    k >= m+0 by A11,XREAL_1:20;
        then l >= 0 by XREAL_1:19;
        then reconsider l as Element of NAT by INT_1:3;
A19:    i3 = l;
        then
A20:    i0 + 0 <= (k+1) by XREAL_1:19;
        reconsider i3 as Element of NAT by A19;
        len(((n,1) In_Power (k+1))) = (k+1)+1 by NEWTON:def 4;
        then dom(((n,1) In_Power (k+1))) = Seg(k+2) by FINSEQ_1:def 3;
        then R3.i = ((k+1) choose i0)*(n|^i3)*(1|^i0) by A9,NEWTON:def 4;
        then
A21:    r2 = k1*(((k+1) choose i0)*(n|^i3)*(1|^i0)) by RVSUM_1:45
          .= k1*(((k+1) choose l)*(n|^l)*(1|^i0)) by A20,NEWTON:20
          .= k1*(((k+1) choose l)*(n|^l)*1) by NEWTON:10
          .= k1*((k+1) choose l)*(n to_power l);
        k-m <= k-0 by XREAL_1:13;
        then
A22:    ((k+1) choose l) / (k+1) <= (k choose l) by Lm43;
        r1 = ((n,1) In_Power k).i0 by A4,A7,A11,A13,FINSEQ_1:24
          .= (k choose m)*(n|^l)*(1|^m) by A17,NEWTON:def 4
          .= (k choose l)*(n|^l)*(1|^m) by A18,NEWTON:20
          .= (k choose l)*(n|^l)*1 by NEWTON:10
          .= (k choose l)*(n to_power l);
        hence thesis by A21,A22,XREAL_1:64;
      end;
    end;
    ((n+1) to_power (k+1)) / (k+1) = ((n+1)|^(k+1))*(k+1)"
      .= Sum((n,1) In_Power (k+1))*(k+1)" by NEWTON:30
      .= Sum(((k+1)")*((n,1) In_Power (k+1))) by RVSUM_1:87;
    then
A23: ((n+1) to_power (k+1)) / (k+1) <= Sum R1 by A8,RVSUM_1:82;
    f.(n+1) = Sum(g,n+1) by A3
      .= (Partial_Sums(g)).(n+1) by SERIES_1:def 5
      .= (Partial_Sums(g)).n + g.(n+1) by SERIES_1:def 1;
    then f.(n+1) >= (n to_power (k+1)) / (k+1) + g.(n+1) by A5,XREAL_1:6;
    hence thesis by A6,A23,XXREAL_0:2;
  end;
A24: P[1]
  proof
    let f be Real_Sequence, k be Element of NAT such that
A25: for n holds f.n = Sum(seq_n^(k), n);
    set g = seq_n^(k);
A26: (1 to_power (k+1)) / (k+1) = 1 / (k+1) by POWER:26;
A27: 0+1 <= k+1 by XREAL_1:6;
    f.1 = Sum(g,1) by A25
      .= (Partial_Sums(g)).(0+1) by SERIES_1:def 5
      .= (Partial_Sums(g)).0 + g.1 by SERIES_1:def 1
      .= g.1 + g.0 by SERIES_1:def 1
      .= (1 to_power k) + g.0 by Def3
      .= 1 + g.0 by POWER:26
      .= 1 + 0 by Def3
      .= 1/1;
    hence thesis by A26,A27,XREAL_1:85;
  end;
  for n be Nat st n >= 1 holds P[n] from NAT_1:sch 8(A24, A1);
  hence thesis;
end;
