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;
