 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 C,e,M,F st C.e = 1 & Polish-ext-head F = e
      ex G st F = Polish-unOp(C,M,e).G
proof
  let C,e,M,F;
  set K = dom C;
  assume that A1: C.e = 1 and A2: Polish-ext-head F = e;
  set G = K-tail F;
  A5: M is C-compatible;
  A6: F is K-headed & K-head F = e by A2, Th10;
  then consider n such that A8: n = C.e & G in M^^n by A5;
  reconsider G as Element of M by A1, A8;
  take G;
  thus F = e^G by A6 .= Polish-unOp(C,M,e).G by A1, Def15;
  thus thesis;
end;
