reserve a, b, n for Nat,
  r for Real,
  f for FinSequence of REAL;
reserve p for Prime;

theorem Th9:
  f |^ 0 = (len f) |-> 1
proof
A1: len (f|^0) = len f by Def1;
A2: for k being Nat st 1 <= k & k <= len (f|^0) holds (f|^0).k = (len f |->
  1).k
  proof
    let k be Nat;
    assume
A3: 1 <= k & k <= len (f|^0);
    then
A4: k in dom (f|^0) by FINSEQ_3:25;
A5: k in Seg len f by A1,A3;
    thus (f|^0).k = f.k |^ 0 by A4,Def1
      .= 1 by NEWTON:4
      .= (len f |-> 1).k by A5,FUNCOP_1:7;
  end;
  len (len f |-> 1) = len f by CARD_1:def 7;
  hence thesis by A1,A2;
end;
