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;
