 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;
 reserve Q for Extension of V;
 reserve M for Extension of Polish-WFF-set(K,E);
 reserve e for Element of K;
 reserve F,G,H for Formula of M;

theorem
  for K,E,e,M,G,H st E.e = 2 holds
    Polish-ext-head(Polish-binOp(K,M,e).(G,H)) = e
proof
  let K,E,e,M,G,H;
  set F = Polish-binOp(K,M,e).(G,H);
  assume A1: E.e = 2;
  F = e^(G^H) by A1, Def16r;
  then F is K-headed & K-head F = e;
  hence thesis by Th32;
end;
