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 Th14:
  F is commutative & F is associative implies for b being Element
  of X holds F $$ ({.b.},f) = f.b
proof
  assume
A1: F is commutative & F is associative;
  let b be Element of X;
  ex G being Function of Fin X, Y st F $$ ({.b.},f) = G.{b} & (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= {b} & B9 <> {} for
x being Element of X st x in {b} \ B9 holds G.(B9 \/ {x}) = F.(G.B9,f.x) by A1
,Def3;
  hence thesis;
end;
