
theorem Th3:
  for X being set, fs being FinSequence of X, fss being Subset of fs,
      m being Element of NAT st m in dom Seq fss holds
      ex n being Element of NAT
  st n in dom fs & m <= n & (Seq fss).m = fs.n
proof
  let X be set, fs be FinSequence of X, fss be Subset of fs, m be Element of
  NAT;
  set f = Sgm(dom fss), n = f.m;
  assume
A2: m in dom Seq fss;
  then
A3: m in dom (fss * f) by FINSEQ_1:def 15;
  then
A4: n in dom fss by FUNCT_1:11;
  Seq fss = fss * f by FINSEQ_1:def 15;
  then (Seq fss).m = fss.n by A2,FUNCT_1:12;
  then
A5: [n, Seq(fss).m] in fss by A4,FUNCT_1:def 2;
  then
A6: n in dom fs by FUNCT_1:1;
A7: m in dom f by A3,FUNCT_1:11;
  (Seq fss).m = fs.n by A5,FUNCT_1:1;
  hence thesis by A7,A6,FINSEQ_3:152;
end;
