
theorem Th1:
  for f being FinSequence st 1 <= len f holds f | Seg 1 = <*f.1*>
proof
  let f be FinSequence;
  assume 1 <= len f;
  then Seg 1 c= Seg len f by FINSEQ_1:5;
  then
A1: Seg 1 c= dom f by FINSEQ_1:def 3;
  reconsider f1 = f | Seg 1 as FinSequence by FINSEQ_1:15;
  0+1 in Seg 1 by FINSEQ_1:4;
  then
A2: (f | Seg 1).1 = f.1 by FUNCT_1:49;
  dom f1 = Seg 1 by A1,RELAT_1:62;
  then len f1 = 1 by FINSEQ_1:def 3;
  hence thesis by A2,FINSEQ_1:40;
end;
