reserve a for set,
  i for Nat;

theorem Th5:
  for f being FinSequence of {0} holds f = i |-> 0 iff len f = i
proof
  let f be FinSequence of {0};
  thus f = i |-> 0 implies len f = i by CARD_1:def 7;
  assume len f = i;
  then
A1: dom f = Seg i by FINSEQ_1:def 3;
  per cases;
  suppose
A2: Seg i = {};
    hence f = {} by A1
      .= 0 |-> 0
      .= i |-> 0 by A2;
  end;
  suppose
A3: Seg i <> {};
    rng f c= {0} by FINSEQ_1:def 4;
    then rng f = {0} or rng f = {} by ZFMISC_1:33;
    hence thesis by A1,A3,FUNCOP_1:9,RELAT_1:42;
  end;
end;
