
theorem
  for D be set, f be XFinSequence, g be FinSequence of D holds
  (f^g)/^(len f) = FS2XFS g
  proof
    let D be set, f be XFinSequence, g be FinSequence of D;
    len (f^g) = len f + len g by Def1; then
    A1: len ((f^g)/^(len f)) = len f + len g -' len f by AFINSQ_2:def 2
    .= len g;
    for i be Nat st i in dom ((f^g)/^(len f)) holds
    ((f^g)/^(len f)).i = (FS2XFS g).i
    proof
      let i be Nat; assume
      B1: i in dom ((f^g)/^(len f)); then
      B2: i in Segm len g by A1; then
      i+1 in Seg len g by NEWTON02:106; then
      B4: i+1 in dom g by FINSEQ_1:def 3;
      (FS2XFS g).i = g.(i+1) by B2,NAT_1:44,AFINSQ_1:def 8
      .= (f^g).(len f + (i + 1) - 1) by B4,Def1
      .= (f^g).(len f + i)
      .= ((f^g)/^(len f)).i by B1,AFINSQ_2:def 2;
      hence thesis;
    end;
    hence thesis by A1,AFINSQ_1:def 8;
  end;
