reserve k,m,n for Nat,
  a, b, c for object,
  x, y, X, Y, Z for set,
  D for non empty set;
reserve p, q, r, s, t, u, v for FinSequence;
reserve P, Q, R, P1, P2, Q1, Q2, R1, R2 for FinSequence-membered set;
reserve S, T for non empty FinSequence-membered set;
reserve A for Function of P, NAT;
reserve U, V, W for Subset of P*;
reserve k,l,m,n,i,j for Nat,
  a, b, c for object,
  x, y, z, X, Y, Z for set,
  D, D1, D2 for non empty set;
reserve p, q, r, s, t, u, v for FinSequence;
reserve P, Q, R for FinSequence-membered set;
reserve B, C for antichain;
reserve S, T for Polish-language;
reserve A for Polish-arity-function of T;
reserve U, V, W for Polish-language of T;
reserve F, G for Polish-WFF of T, A;
reserve f for Polish-recursion-function of A, D;
reserve K, K1, K2 for Function of Polish-WFF-set(T, A), D;
reserve L for non trivial Polish-language;
reserve E for Polish-arity-function of L;
reserve g for Polish-recursion-function of E, D;
reserve J, J1, J2, J3 for Subset of Polish-WFF-set(L, E);
reserve H for Function of J, D;
reserve H1 for Function of J1, D;
reserve H2 for Function of J2, D;
reserve H3 for Function of J3, D;

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;
