 reserve a,b for object;
 reserve k,l,m,n for Nat;
 reserve p,q,r,s for FinSequence;
 reserve P for non empty FinSequence-membered set;
 reserve S,T for Polish-language;
 reserve V for Polish-language of T;
 reserve K for non trivial Polish-language;
 reserve E for Polish-arity-function of K;
 reserve B for Polish-arity-function;

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;
