|- !f. ASSOC f = (!x y z. f x (f y z) = f (f x y) z)
|- !f. COMM f = (!x y. f x y = f y x)
|- !f g. FCOMM f g = (!x y z. g x (f y z) = f (g x y) z)
|- !f e. RIGHT_ID f e = (!x. f x e = x)
|- !f e. LEFT_ID f e = (!x. f e x = x)
|- !f e. MONOID f e = ASSOC f /\ RIGHT_ID f e /\ LEFT_ID f e
|- ASSOC $/\
|- ASSOC $\/
|- !f. FCOMM f f = ASSOC f
|- MONOID $/\ T
|- MONOID $\/ F