
theorem SCX:
  for f be complex-valued XFinSequence, x be Nat holds
    (f^(seq_const 0)).x = f.x
  proof
    let f be complex-valued XFinSequence, x be Nat;
    NAT = dom (f^(seq_const 0)) by COMSEQ_1:1
    .= dom (f \/ (Shift (seq_const 0, dom f))); then
    (dom f) \/ (dom (Shift (seq_const 0, dom f))) = NAT by XTUPLE_0:23; then
    A1: x in (dom f) \/ (dom (Shift (seq_const 0, dom f)))
      by ORDINAL1:def 12;
    A1a: Shift(seq_const 0,len f) \/ (f^(seq_const 0)) = (f^(seq_const 0))
      by XBOOLE_1:6;
    (dom (Shift (seq_const 0,len f))) /\ ((dom f) \/
      (dom (Shift (seq_const 0,len f)))) =
        dom (Shift (seq_const 0,len f)) by XBOOLE_1:21; then
    A2: dom (Shift (seq_const 0,len f)) /\ dom (f^(seq_const 0)) =
      dom (Shift (seq_const 0,len f)) by XTUPLE_0:23;
    per cases;
    suppose
      x in dom f; then
      (f^(seq_const 0)).x = ((f^(seq_const 0))|(len f)).x by FUNCT_1:49
      .= f.x;
      hence thesis;
    end;
    suppose
      B1: not x in dom f; then
      B2: x in dom (Shift (seq_const 0, dom f)) by A1,XBOOLE_0:def 3;
      f.x = (seq_const 0).(len f - x) by B1,FUNCT_1:def 2
      .= (Shift (seq_const 0, len f)).x
      .= (f^(seq_const 0)).x by A1a,A2,B2,PARTFUN1:2;
      hence thesis;
    end;
  end;
