 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;
 reserve A for Polish-arity-function of T;
 reserve C for Extension of B;
 reserve Z for B-closed Polish-language;
 reserve J for Polish-ext-set of B;
 reserve V for full Polish-language of T;
 reserve U for T-extending Polish-language;
 reserve W for full Polish-language of U;
 reserve M for Polish-ext-set of C;
 reserve e for Element of dom C;
 reserve F, G, H for Formula of M;

theorem
  for T,U,V,W holds V c= W iff W is (Polish-arity V)-closed
proof
  let T,U,V,W;
  set A = Polish-arity V;
  set B = Polish-arity W;
  thus V c= W implies W is A-closed
  proof
    assume V c= W;
    then A2: A c= B by Th22;
    W = Polish-WFF-set(U,B);
    then A3: W is B-closed;
    let p,n,q;
    assume that
      A5: p in T and A6: n = A.p and A7: q in W^^n;
    A9: T c= U by Def9;
    p in dom A by A5, FUNCT_2:def 1;
    then n = B.p by A2, A6, GRFUNC_1:2;
    hence p^q in W by A3, A5, A7, A9;
  end;
  V = Polish-expression-set(T,A);
  hence thesis by POLNOT_1:37;
end;
