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
  seq is convergent & lim seq<>0 implies ex seq1 st seq1 is subsequence
  of seq & seq1 is non-zero
proof
  assume seq is convergent & lim seq <>0;
  then consider k such that
A1: seq ^\k is non-zero by Th24;
  take seq ^\k;
  thus thesis by A1;
end;
