reserve X for ARS, a,b,c,u,v,w,x,y,z for Element of X;
reserve i,j,k for Element of ARS_01;
reserve l,m,n for Element of ARS_02;
reserve A for set;

theorem Th0:
  for n being natural number, X being non empty set, x being Element of X
  ex f being non empty homogeneous quasi_total PartFunc of X*, X st
  arity f = n & f = (n-tuples_on X) --> x
  proof
    let n be natural number, X be non empty set;
    let x be Element of X;
    set f = (n-tuples_on X) --> x;
    dom f = n-tuples_on X & rng f = {x} & n in omega
    by FUNCOP_1:8,ORDINAL1:def 12; then
    dom f c= X* & rng f c= X by ZFMISC_1:31,FINSEQ_2:134; then
    reconsider f as non empty PartFunc of X*, X by RELSET_1:4;
A2: f is quasi_total
    proof
      let x,y be FinSequence of X; assume
      len x = len y & x in dom f; then
      len x = n & len y = n by FINSEQ_2:132;
      hence thesis by FINSEQ_2:133;
    end;
    f is homogeneous
    proof
      let x,y be FinSequence; assume
      x in dom f & y in dom f; then
      reconsider x,y as Element of n-tuples_on X;
      len x = n & len y = n by FINSEQ_2:132;
      hence thesis;
    end; then
    reconsider f as non empty homogeneous quasi_total PartFunc of X*, X by A2;
    take f;
    set y = the Element of n-tuples_on X;
A3: for x being FinSequence st x in dom f holds n = len x by FINSEQ_2:132;
    y in dom f;
    hence arity f = n by A3,MARGREL1:def 25;
    thus thesis;
  end;
