
theorem Th4:
  for X being set, fs being FinSequence of X, fss being Subset of fs
  holds len Seq fss = card fss
proof
  let X be set, fs be FinSequence of X, fss be Subset of fs;
A1: Seq fss = fss * Sgm(dom fss) by FINSEQ_1:def 15;
  rng Sgm(dom fss) = dom fss by FINSEQ_1:def 14;
  then dom (Seq fss) = dom (Sgm (dom fss)) by A1,RELAT_1:27;
  then dom (Seq fss) = Seg (card dom fss) by FINSEQ_3:40;
  then len Seq fss = card dom fss by FINSEQ_1:def 3;
  hence thesis by CARD_1:62;
end;
