reserve i for Nat, x,y for set;
reserve S for non empty non void ManySortedSign;
reserve X for non-empty ManySortedSet of S;

theorem Th44:
  for A being non empty set, n being Nat
  for f being Function of n-tuples_on A,A holds
  f is homogeneous quasi_total non empty PartFunc of A*,A &
  for g being homogeneous Function st f = g holds g is n-ary
  proof
    let A be non empty set;
    let n be Nat;
    let f be Function of n-tuples_on A,A;
    n-tuples_on A c= A*  by FINSEQ_2:134; then
    reconsider f as PartFunc of A*,A by RELSET_1:7;
A2: dom f = n-tuples_on A by FUNCT_2:def 1; then
    reconsider f as homogeneous PartFunc of A*,A by COMPUT_1:16;
    set t = the Element of n-tuples_on A;
    arity f = len t by A2,MARGREL1:def 25 .= n by FINSEQ_2:133;
    hence thesis by A2,COMPUT_1:def 21,22;
  end;
