 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 = 2 & Polish-ext-head F = e ex G,H
    st F = Polish-binOp(C,M,e).(G,H)
proof
  let C,e,M,F;
  set K = dom C;
  assume that A1: C.e = 2 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 A7: n = C.e & g in M^^n by A5;
  reconsider g as Element of M^^(1+1) by A1, A7;
  M^^(1+1) = (M^^1)^M by POLNOT_1:6;
  then consider p,q such that A8: g = p^q and A9: p in M and A10: q in M
    by POLNOT_1:def 2;
  reconsider G = p, H = q as Formula of M by A9, A10;
  take G,H;
  thus F = e^(p^q) by A6, A8 .= Polish-binOp(C,M,e).(G,H) by A1, Def16;
  thus thesis;
end;
