reserve i, j, k, c, m, n for Nat,
  a, x, y, z, X, Y for set,
  D, E for non empty set,
  R for Relation,
  f, g for Function,
  p, q for FinSequence;

theorem Th19:
  for f being homogeneous PartFunc of X*,Y holds dom f c= (arity f)-tuples_on X
proof
  let f be homogeneous PartFunc of X*, Y;
  let x be object;
  assume
A1: x in dom f;
  per cases;
  suppose
A2: X is empty;
    then x = <*>(X*) by A1,FUNCT_7:17,TARSKI:def 1;
    then
A3: arity f = len <*>(X*) by A1,MARGREL1:def 25;
    0-tuples_on X = {{}} by Th5;
    hence thesis by A1,A2,A3,FUNCT_7:17;
  end;
  suppose
    X is non empty;
    then reconsider X9 = X as non empty set;
    reconsider x9 = x as FinSequence of X9 by A1,FINSEQ_1:def 11;
    len x9 = arity f by A1,MARGREL1:def 25;
    then x9 is Element of (arity f)-tuples_on X9 by FINSEQ_2:92;
    hence thesis;
  end;
end;
