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
  for seq1 st seq1 is subsequence of seq & seq1 is non-zero
  holds lim (seq1")=(lim seq)"
proof
  assume that
A1: seq is convergent and
A2: lim seq<>0;
  let seq1 such that
A3: seq1 is subsequence of seq and
A4: seq1 is non-zero;
  lim seq1=lim seq by A1,A3,Th17;
  hence thesis by A1,A2,A3,A4,Th16,SEQ_2:22;
end;
