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 Th112:
  for z being Tuple of i,D for f being Function of
  Seg i,Seg i st rng f = Seg i holds z*f is Element of i-tuples_on D
proof
  let z be Tuple of i,D;
  let f be Function of Seg i,Seg i;
A1: len z = i by CARD_1:def 7;
  then
A2: Seg i = dom z by FINSEQ_1:def 3;
  then reconsider t = z*f as FinSequence of D by Th45;
  assume rng f = Seg i;
  then len t = len z by A2,Th40;
  hence thesis by A1,Th90;
end;
