%  problem-set/algebra/category.theory/p14.ver2.clauses
%  created : 07/07/89
%  revised : 07/07/89

% description: 
%
% Lemma 14: cod(cod(x)) = cod(x).

% representation:
%
% declare_predicate(2,=).
% declare_function(2,star).
% declare_functions(1,[dom,cod]). 
%
% x = y : usual classical equality
% star(x,y) : x composed with y
% dom(x) : the object which is the domain of x
% cod(x) : the object which is the codomain of x 


% equality axiom
(all x (x = x)).

% category theory axioms
(all x (cod(dom(x)) = dom(x))).
(all x (dom(cod(x)) = cod(x))).
(all x (star(dom(x),x) = x)).
(all x (star(x,cod(x)) = x)).
(all x all y ((cod(x) = dom(y)) -> (dom(star(x,y)) = dom(x)))).
(all x all y ((cod(x) = dom(y)) -> (cod(star(x,y)) = cod(y)))).
(all x all y all z (((cod(x) = dom(y)) & (cod(y) = dom(z))) -> 
	(star(x,star(y,z)) = star(star(x,y),z)))).

% denial of lemma 14

(cod(cod(a)) != cod(a)).