reserve k,n for Nat,
  x,y,z,y1,y2 for object,X,Y for set,
  f,g for Function;
reserve p,q,r,s,t for XFinSequence;
reserve D for set;
reserve i for Nat;
reserve m for Nat,
        D for non empty set;
reserve l for Nat;
reserve M for Nat;
reserve m,n for Nat;
reserve x1,x2,x3,x4 for object;
reserve e,u for object;

theorem Th12:
  for D being set, p being XFinSequence of D holds FS2XFS (XFS2FS p) = p
proof
  let D be set, p be XFinSequence of D;
  A1: len p = len XFS2FS p by Def9A;
  A2: len XFS2FS p = len FS2XFS (XFS2FS p) by Def8;
  for k being Nat st k < len p holds p.k = (FS2XFS (XFS2FS p)).k
  proof
    let k be Nat;
    assume A3: k < len p;
    then 0+1 <= k+1 & k+1 < len p +1 by XREAL_1:6;
    then A4: 1 <= k+1 & k+1 <= len p by NAT_1:13;
    thus p.k = p.(k+1-1)
      .= (XFS2FS p).(k+1) by A4, Def9A
      .= (FS2XFS (XFS2FS p)).k by A1, A3, Def8;
  end;
  hence thesis by A1, A2, Th8;
end;
