%   ORIGINAL: 'h4/thm/bool/LCOMM_THM_'
% Assm: HL_TRUTH: T
% Assm: HL_FALSITY: ~F
% Assm: HL_BOOL_CASES: !t. (t <=> T) \/ (t <=> F)
% Assm: HL_EXT: !f g. (!x. f x = g x) ==> f = g
% Goal: !f. (!x y z. f x (f y z) = f (f x y) z) ==> (!x y. f x y = f y x) ==> (!x y z. f x (f y z) = f y (f x z))
%   PROCESSED
% Assm ['HL_TRUTH']: T
% Assm ['HL_FALSITY']: ~F
% Assm ['HL_BOOL_CASES']: !t. (t <=> T) \/ (t <=> F)
% Assm ['HL_EXT']: !f g. (!x. happ f x = happ g x) ==> f = g
% Goal: !f. (!x y z. happ (happ f x) (happ (happ f y) z) = happ (happ f (happ (happ f x) y)) z) ==> (!x y. happ (happ f x) y = happ (happ f y) x) ==> (!x y z. happ (happ f x) (happ (happ f y) z) = happ (happ f y) (happ (happ f x) z))
fof('HL_TRUTH', axiom, p(s(bool,'T'))).
fof('HL_FALSITY', axiom, ~ (p(s(bool,'F')))).
fof('HL_BOOL_CASES', axiom, ![T]: (s(bool,T) = s(bool,'T') | s(bool,T) = s(bool,'F'))).
fof('HL_EXT', axiom, ![V_3f35529,V_3f35525]: ![F, G]: (![X]: s(V_3f35525,happ(s(fun(V_3f35529,V_3f35525),F),s(V_3f35529,X))) = s(V_3f35525,happ(s(fun(V_3f35529,V_3f35525),G),s(V_3f35529,X))) => s(fun(V_3f35529,V_3f35525),F) = s(fun(V_3f35529,V_3f35525),G))).
fof('h4/thm/bool/LCOMM_THM_', conjecture, ![A]: ![F]: (![X, Y, Z]: s(A,happ(s(fun(A,A),happ(s(fun(A,fun(A,A)),F),s(A,X))),s(A,happ(s(fun(A,A),happ(s(fun(A,fun(A,A)),F),s(A,Y))),s(A,Z))))) = s(A,happ(s(fun(A,A),happ(s(fun(A,fun(A,A)),F),s(A,happ(s(fun(A,A),happ(s(fun(A,fun(A,A)),F),s(A,X))),s(A,Y))))),s(A,Z))) => (![X, Y]: s(A,happ(s(fun(A,A),happ(s(fun(A,fun(A,A)),F),s(A,X))),s(A,Y))) = s(A,happ(s(fun(A,A),happ(s(fun(A,fun(A,A)),F),s(A,Y))),s(A,X))) => ![X, Y, Z]: s(A,happ(s(fun(A,A),happ(s(fun(A,fun(A,A)),F),s(A,X))),s(A,happ(s(fun(A,A),happ(s(fun(A,fun(A,A)),F),s(A,Y))),s(A,Z))))) = s(A,happ(s(fun(A,A),happ(s(fun(A,fun(A,A)),F),s(A,Y))),s(A,happ(s(fun(A,A),happ(s(fun(A,fun(A,A)),F),s(A,X))),s(A,Z)))))))).
