reserve n,k,k1,m,m1,n1,n2,l for Nat;
reserve r,r1,r2,p,p1,g,g1,g2,s,s1,s2,t for Real;
reserve seq,seq1,seq2 for Real_Sequence;
reserve Nseq for increasing sequence of NAT;
reserve x for set;
reserve X,Y for Subset of REAL;

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 Th23;
  take seq ^\k;
  thus thesis by A1;
end;
