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;
