
theorem Th7:
  for X be ComplexNormSpace, seq,seq1 be sequence of X holds seq1
  is subsequence of seq & seq is convergent implies seq1 is convergent
proof
  let X be ComplexNormSpace;
  let seq,seq1 be sequence of X;
  assume that
A1: seq1 is subsequence of seq and
A2: seq is convergent;
  consider g1 be Element of X such that
A3: for p be Real st 0<p ex n be Nat st
   for m be Nat st n<=m holds ||.(seq.m)-g1.||< p by A2;
  take g1;
  let p be Real;
  assume 0<p;
  then consider n1 be Nat such that
A4: for m be Nat st n1<=m holds ||.(seq.m)-g1.||<p by A3;
  take n=n1;
  let m be Nat such that
A5: n<=m;
A6: m in NAT by ORDINAL1:def 12;
  consider Nseq be increasing sequence of NAT such that
A7: seq1=seq*Nseq by A1,VALUED_0:def 17;
  m<=Nseq.m by SEQM_3:14;
  then
A8: n<=Nseq.m by A5,XXREAL_0:2;
  seq1.m=seq.(Nseq.m) by A7,FUNCT_2:15,A6;
  hence thesis by A4,A8;
end;
