reserve X for non empty set,
        x for Element of X,
        S for SigmaField of X,
        M for sigma_Measure of S,
        f,g,f1,g1 for PartFunc of X,REAL,
        l,m,n,n1,n2 for Nat,
        a,b,c for Real;
reserve k for positive Real;

theorem Th11:
for seq,seq2 be Real_Sequence, k be positive Real st seq is convergent &
  for n be Nat holds
    seq2.n = |. lim seq - seq.n qua Complex .| to_power k
holds seq2 is convergent & lim seq2 = 0
proof
   let seq, seq2 be Real_Sequence, k be positive Real;
   set r = lim seq;
   assume
A1: seq is convergent &
    for n be Nat holds seq2.n = |. r -seq.n qua Complex .| to_power k;
   deffunc U(Nat) = |.r -seq.$1 qua Complex.|;
   consider seq1 be Real_Sequence such that
A2: for n holds seq1.n = U(n) from SEQ_1:sch 1;
   deffunc U(Nat) = r;
   consider seq0 be Real_Sequence such that
A3: for n holds seq0.n = U(n) from SEQ_1:sch 1;
    reconsider r as Element of REAL by XREAL_0:def 1;
   for n be Nat holds seq0.n = r
   by A3; then
A4:seq0 is constant by VALUED_0:def 18; then
A5:seq0 - seq is convergent by A1;
A6:dom seq0 = NAT & dom seq = NAT & dom (seq0 - seq)= NAT
   & dom seq1 = NAT by FUNCT_2:def 1;
A7:dom abs(seq0 - seq) = dom (seq0 - seq) by VALUED_1:def 11;
   for n be Element of NAT holds abs(seq0 - seq).n = seq1.n
   proof
    let n be Element of NAT;
    seq1.n = |. r - seq.n .| by A2; then
    seq1.n = |. seq0.n - seq.n .| by A3; then
    seq1.n = |.(seq0-seq).n.| by A6,VALUED_1:13;
    hence thesis by A6,A7,VALUED_1:def 11;
   end; then
A8:abs(seq0 - seq) = seq1 by FUNCT_2:63; then
A9:seq1 is convergent by A5;
   lim (seq0-seq) = seq0.0 - lim seq by A4,A1,SEQ_4:42; then
   lim (seq0-seq) = r - lim seq by A3; then
A10:lim seq1 = 0 by A5,A8,COMPLEX1:44,SEQ_4:14;
   for n holds seq2.n = (seq1.n) to_power k & seq1.n >= 0
   proof
    let n;
    |. r -seq.n .| = seq1.n by A2;
    hence thesis by A1,COMPLEX1:46;
   end; then
   seq2 is convergent & lim seq2 = (lim seq1) to_power k by A9,HOLDER_1:10;
   hence thesis by A10,POWER:def 2;
end;
