theorem
  for L, E for F being Polish-WFF of L, E holds
    F in Polish-atoms(L, E) iff Polish-arity F = 0
proof
  let L, E;
  let F be Polish-WFF of L, E;
  thus F in Polish-atoms(L, E) implies Polish-arity F = 0
  proof
    assume A1: F in Polish-atoms(L, E);
    then F in L by Def7;
    then Polish-WFF-head F = F by Th53;
    hence Polish-arity F = 0 by A1, Def7;
  end;
  assume A10: Polish-arity F = 0;
  then L-tail F in Polish-WFF-set(L, E)^^0 by Th67;
  then L-tail F in {{}} by Th6;
  then L-tail F = {} by TARSKI:def 1;
  then F = (L-head F)^{};
  then F = Polish-WFF-head F by FINSEQ_1:34;
  hence F in Polish-atoms(L, E) by A10, Def7;
end;
