reserve i,j,k,l for natural Number;
reserve A for set, a,b,x,x1,x2,x3 for object;
reserve D,D9,E for non empty set;
reserve d,d1,d2,d3 for Element of D;
reserve d9,d19,d29,d39 for Element of D9;
reserve p,q,r for FinSequence;
reserve s for Element of D*;

theorem
  for z being Tuple of i,D for f being Permutation of Seg i
  holds z*f is Tuple of i,D
proof
  let z be Tuple of i,D;
  let f be Permutation of Seg i;
  rng f = Seg i by FUNCT_2:def 3;
  hence thesis by Th112;
end;
