
theorem
  for D be set, f be FinSequence, g be XFinSequence of D holds
    (f^g)/^(len f) = XFS2FS g
  proof
    let D be set, f be FinSequence, g be XFinSequence of D;
    len f + 0 <= len f + len g by XREAL_1:6; then
    A0: len f <= len (f^g) by XL1;
    A1: len g = (len f + len g) - len f
    .= len (f^g) - len f by XL1
    .= len ((f^g)/^(len f)) by A0,RFINSEQ:def 1;
    len XFS2FS g = len g by AFINSQ_1:def 9; then
    A3: dom XFS2FS g = Seg (len g) by FINSEQ_1:def 3
    .= dom ((f^g)/^(len f)) by A1,FINSEQ_1:def 3;
    for i be Nat st i in dom ((f^g)/^(len f)) holds
    ((f^g)/^(len f)).i = (XFS2FS g).i
    proof
      let i be Nat; assume
      B1: i in dom ((f^g)/^(len f)); then
      i in Seg len g by A1,FINSEQ_1:def 3; then
      B3: 1 <= i <= len g by FINSEQ_1:1; then
      reconsider j = i - 1 as Nat;
      j+1 in Seg (len g) by B1,A1,FINSEQ_1:def 3; then
      B4:j in Segm len g by NEWTON02:106;
      (XFS2FS g).(j+1) = g.((j+1)-'1) by B3,AFINSQ_1:def 9
      .= (f^g).(len f + j + 1) by B4,Def2
      .= (f^g).(len f + i)
      .= ((f^g)/^(len f)).i by A0,B1,RFINSEQ:def 1;
      hence thesis;
    end;
    hence thesis by A3;
  end;
