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
  for D being set, p being FinSequence of D, n being Nat
  holds (FS2XFS p)|n = FS2XFS(p|n) & (FS2XFS p)/^n = FS2XFS(p/^n)
proof
  let D be set, p be FinSequence of D, n be Nat;
  thus (FS2XFS p)|n = FS2XFS XFS2FS((FS2XFS p)|n)
    .= FS2XFS((XFS2FS FS2XFS p)|n) by Th17
    .= FS2XFS(p|n);
  thus (FS2XFS p)/^n = FS2XFS XFS2FS((FS2XFS p)/^n)
    .= FS2XFS((XFS2FS FS2XFS p)/^n) by Th17
    .= FS2XFS(p/^n);
end;
