reserve x,y,z,X,Y for set;
reserve X,Y for non empty set,
  f for Function of X,Y;
reserve X, Y for non empty set,
  F for (BinOp of Y),
  B for (Element of Fin X),
  f for Function of X,Y;

theorem Th28:
  F is commutative & F is associative & F is having_a_unity
  implies for f holds F$$({}.X,f) = the_unity_wrt F
proof
  assume
A1: F is commutative & F is associative & F is having_a_unity;
  let f;
  the_unity_wrt F is_a_unity_wrt F & ex G being Function of Fin X, Y st F
$$ ( {}.X,f) = G.{}.X & (for e being Element of Y st e is_a_unity_wrt F holds G
.{} = e) & (for x being Element of X holds G.{x} = f.x) & for B9 being Element
  of Fin X st B9 c= {}.X & B9 <> {} for x being Element of X st x in {}.X \ B9
  holds G.( B9 \/ {x}) = F.(G.B9,f.x) by A1,Def3,Th11;
  hence thesis;
end;
