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 Th46:
  for f,g being non empty quasi_total Element of HFuncs D
   st arity f = 0 & arity g = 0 & f.{} = g.{} holds f = g
proof
  set X = D;
  let f,g be non empty quasi_total Element of HFuncs X;
  assume that
A1: arity f = 0 and
A2: arity g = 0 and
A3: f.{} = g.{};
  now
    thus dom f = 0-tuples_on X by A1,Th21
      .= {<*>X} by FINSEQ_2:94;
    thus dom g = 0-tuples_on X by A2,Th21
      .= {<*>X} by FINSEQ_2:94;
    let x be object;
    assume x in {<*>X};
    then x = {} by TARSKI:def 1;
    hence f.x = g.x by A3;
  end;
  hence thesis;
end;
