reserve i,j,k,n,m for Nat,
  x,y,z,y1,y2 for object, X,Y,D for set,
  p,q for XFinSequence;
reserve k1,k2 for Nat;
reserve D for non empty set,
  F,G for XFinSequence of D,
  b for BinOp of D,
  d,d1,d2 for Element of D;
reserve F for XFinSequence,
        rF,rF1,rF2 for real-valued XFinSequence,
        r for Real,
        cF,cF1,cF2 for complex-valued XFinSequence,
        c,c1,c2 for Complex;
reserve r,s for XFinSequence;

theorem Th5: :: from BALLOT_1:5
  for D being set
  for d be FinSequence of D holds XFS2FS (FS2XFS d) = d
proof
  let D be set;
  let d be FinSequence of D;
  set Xd=FS2XFS d;
A1: len d = len Xd by AFINSQ_1:def 8;
A2: len Xd = len XFS2FS Xd by AFINSQ_1:def 9;
  now let i such that
A3: 1 <= i and
A4: i <= len d;
    reconsider i1=i-1 as Nat by A3,NAT_1:21;
A5: i1+1 = i;
    thus d.i = Xd.i1 by A4,A5,NAT_1:13,AFINSQ_1:def 8
            .= (XFS2FS Xd).i by A3,A4,A1,AFINSQ_1:def 9;
  end;
  hence thesis by A1,A2;
end;
