theorem Th1:
  for T,B holds B is Polish-arity-function of T iff T = dom B
proof
  let T,B;
  thus B is Polish-arity-function of T implies T = dom B by FUNCT_2:def 1;
  consider U being Polish-language such that
  A1: B is Polish-arity-function of U by Def1;
  assume T = dom B;
  hence thesis by A1, FUNCT_2:def 1;
end;
