reserve n,n1,m,m1,k for Nat;
reserve x,X,X1 for set;
reserve g,g1,g2,t,x0,x1,x2 for Complex;
reserve s1,s2,q1,seq,seq1,seq2,seq3 for Complex_Sequence;
reserve Y for Subset of COMPLEX;
reserve f,f1,f2,h,h1,h2 for PartFunc of COMPLEX,COMPLEX;
reserve p,r,s for Real;
reserve Ns,Nseq for increasing sequence of NAT;

theorem Th24:
  seq is convergent & lim seq<>0 implies ex k st (seq ^\k) is non-zero
proof
  assume that
A1: seq is convergent and
A2: lim seq<>0;
  consider n1 such that
A3: for m st n1<=m holds |.(lim seq).|/2<|.(seq.m).| by A1,A2,COMSEQ_2:33;
  take k=n1;
  now
    let n be Element of NAT;
    0+k<=n+k by XREAL_1:7;
    then |.(lim seq).|/2<|.(seq.(n+k)).| by A3;
    then
A4: |.(lim seq).|/2<|.((seq ^\k).n).| by NAT_1:def 3;
    0<|.(lim seq).| by A2,COMPLEX1:47;
    then 0/2<|.(lim seq).|/2 by XREAL_1:74;
    then 0 <|.((seq ^\k).n).| by A4;
    hence (seq ^\k).n<>0c by COMPLEX1:47;
  end;
  hence thesis by COMSEQ_1:4;
end;
