reserve x1,x2,y1,a,b,c for Real;

theorem Th11:
  for p be Real
   st p>=1 holds for rseq be Real_Sequence st (for n
  be Nat holds rseq.n=0) holds rseq rto_power p is summable & ( Sum(
  rseq rto_power p) ) to_power (1/p)=0
proof
  let p be Real such that
A1: p>=1;
A2: 1/p > 0 by A1,XREAL_1:139;
  let rseq be Real_Sequence such that
A3: for n be Nat holds rseq.n=0;
A4: for n be Nat holds (rseq rto_power p).n=0
  proof
    let n be Nat;
    rseq.n=0 by A3;
    then |.(rseq).n.| =0 by ABSVALUE:2;
    then |.(rseq).n.| to_power p =0 by A1,POWER:def 2;
    hence thesis by Def1;
  end;
A5: for m be Nat holds Partial_Sums (rseq rto_power p).m = 0
  proof
    defpred P[Nat] means
   (rseq rto_power p).$1 = (Partial_Sums(rseq rto_power p)).$1;
    let m be Nat;
A6: for k be Nat st P[k] holds P[k+1]
    proof
      let k be Nat such that
A7:   (rseq rto_power p).k = (Partial_Sums (rseq rto_power p)).k;
      thus (rseq rto_power p).(k+1) = 0 + (rseq rto_power p).(k+1)
        .= (rseq rto_power p).k + (rseq rto_power p).(k+1) by A4
        .= (Partial_Sums ((rseq rto_power p))).(k+1) by A7,SERIES_1:def 1;
    end;
A8: P[0] by SERIES_1:def 1;
    for n be Nat holds P[n] from NAT_1:sch 2(A8,A6);
    hence (Partial_Sums ((rseq rto_power p))).m = (rseq rto_power p).m
      .= 0 by A4;
  end;
A9: for e be Real st 0<e ex n be Nat st
   for m be Nat st n<=m holds |.(Partial_Sums (rseq rto_power p)).m-0 .|<e
  proof
    let e be Real such that
A10: 0<e;
    take 0;
    let m be Nat such that
    0<=m;
    |.(Partial_Sums (rseq rto_power p)).m-0 .| = |.0-0 .| by A5
      .= 0 by ABSVALUE:def 1;
    hence thesis by A10;
  end;
  then
A11: Partial_Sums (rseq rto_power p) is convergent by SEQ_2:def 6;
  then lim (Partial_Sums (rseq rto_power p)) =0 by A9,SEQ_2:def 7;
  then Sum(rseq rto_power p) =0 by SERIES_1:def 3;
  hence thesis by A2,A11,POWER:def 2,SERIES_1:def 2;
end;
