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

theorem Th12:
  for p be Real st 1 <=p for rseq be Real_Sequence st rseq
rto_power p is summable & ( Sum(rseq rto_power p) ) to_power (1/p)=0 holds for
  n be Nat holds rseq.n = 0
proof
  let p be Real such that
A1: 1 <=p;
A2: 1/p > 0 by A1,XREAL_1:139;
  let rseq being Real_Sequence such that
A3: rseq rto_power p is summable and
A4: Sum(rseq rto_power p) to_power (1/p)=0;
A5: for i be Nat holds ((rseq) rto_power p).i >= 0
  proof
    let i be Nat;
    ( (rseq) rto_power p).i =|.(rseq).i.| to_power p by Def1;
    hence thesis by A1,Lm1,COMPLEX1:46;
  end;
  then ( (Sum(rseq rto_power p) ) to_power (1/p)) to_power p =(Sum(rseq
  rto_power p) ) to_power((1/p)*p) by A1,A2,A3,HOLDER_1:2,SERIES_1:18
    .=(Sum(rseq rto_power p) ) to_power(1) by A1,XCMPLX_1:106
    .=(Sum(rseq rto_power p) ) by POWER:25;
  then
A6: (Sum(rseq rto_power p) ) = 0 by A1,A4,POWER:def 2;
  now
    let n be Nat;
    reconsider n9=n as Nat;
A7: 0 to_power (1/p) =0 by A2,POWER:def 2;
    (rseq rto_power p).n9 =|.rseq.n9.| to_power p by Def1;
    then
A8: |.rseq.n.| to_power p =0 by A3,A5,A6,RSSPACE:17;
    (|.rseq.n.| to_power p) to_power (1/p) =|.rseq.n.| to_power (p*(1/p
    )) by A1,A2,COMPLEX1:46,HOLDER_1:2
      .=|.rseq.n.| to_power (1) by A1,XCMPLX_1:106
      .=|.rseq.n.| by POWER:25;
    hence rseq.n =0 by A8,A7,ABSVALUE:2;
  end;
  hence thesis;
end;
