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 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,Th18;
  hence thesis by A1,A2,A3,A4,Th17,COMSEQ_2:35;
end;
