
theorem
  for f be complex-valued XFinSequence holds Sequel f = f^(seq_const 0)
  proof
    let f be complex-valued XFinSequence;
    A1: dom (Sequel f) = dom (f^(seq_const 0))
    proof
      dom (Sequel f) = dom (NAT --> 0) \/ dom f by FUNCT_4:def 1
      .= NAT \/ dom f
      .= dom (f^(seq_const 0)) by COMSEQ_1:1;
      hence thesis;
    end;
    for x be Nat holds (Sequel f).x = (f^(seq_const 0)).x
    proof
      let x be Nat;
      (Sequel f).x = f.x & (f^(seq_const 0)).x = f.x by SCX,SFX;
      hence thesis;
    end;
    hence thesis by A1;
  end;
