
theorem DSS:
  for D be set, f be XFinSequence of D holds dom Shift (f,1) = Seg len f
  proof
    let D be set, f be XFinSequence of D;
    A1: for x be object st x in Seg len f holds x in dom Shift (f,1)
    proof
      let x be object; assume
      B1: x in Seg len f; then
      reconsider x as Nat;
      x >= 1 by B1,FINSEQ_1:1; then
      reconsider y = x - 1 as Nat;
      y + 1 in Seg len f by B1; then
      y in Segm (len f) by NEWTON02:106; then
      y + 1 in dom (Shift (f,1)) by VALUED_1:24;
      hence thesis;
    end;
    for x be object st x in dom Shift (f,1) holds x in Seg len f
    proof
      let x be object; assume
      B1: x in dom Shift (f,1); then
      reconsider x as Nat;
      consider y be Nat such that
      B2: y in dom f & x = y + 1 by B1,VALUED_1:39;
      y in Segm (len f) by B2;
      hence thesis by B2,NEWTON02:106;
    end;
    hence thesis by A1,TARSKI:2;
  end;
