
theorem SFX:
  for f be complex-valued XFinSequence, x be Nat holds (Sequel f).x = f.x
  proof
    let f be complex-valued XFinSequence, x be Nat;
    x in NAT by ORDINAL1:def 12; then
    A1: x in (dom (NAT --> 0)) \/ (dom f) by XBOOLE_0:def 3;
    per cases;
    suppose
      x in dom f;
      hence thesis by A1,FUNCT_4:def 1;
    end;
    suppose
      B1: not x in dom f; then
      (Sequel f).x = (NAT --> 0).x by A1,FUNCT_4:def 1
      .= 0;
      hence thesis by B1,FUNCT_1:def 2;
    end;
  end;
