PAOE06
Premises:
(forall ?X (<=> (even ?X) (exists ?Y (= ?X (+ ?Y ?Y))))) (forall ?X (<=> (odd ?X) (not (even ?X)))) (even 0) (even (+ 0 0)) (not (even (succ 0))) (even (succ (succ 0))) (odd (succ 0)) (not (odd 0)) (forall ?X (=> (even ?X) (even (+ 0 ?X)))) (forall ?X (even (+ ?X ?X))) (forall ?X (<=> (even ?X) (not (odd ?X)))) (forall ?X (=> (even ?X) (even (+ ?X ?X)))) (forall ?X (or (even ?X) (odd ?X))) (forall ?X (=> (even ?X) (even (succ (succ ?X))))) (forall ?X (=> (even ?X) (even (+ (succ (succ 0)) ?X)))) (forall ?X (=> (even ?X) (odd (succ ?X)))) (forall ?X (<=> (even ?X) (not (odd ?X)))) (forall ?X (=> (odd ?X) (even (succ ?X)))) (forall ?X (=> (odd ?X) (odd (succ (succ ?X))))) (forall ?X (<=> (odd ?X) (not (even ?X)))) (forall (?X ?Y) (=> (and (even ?X) (even ?Y)) (even (+ ?X ?Y)))) (forall (?X ?Y) (=> (even (+ ?X ?Y)) (even (+ (succ ?X) (succ ?Y))))) (forall (?X ?Y) (=> (and (odd ?X) (odd ?Y)) (even (+ ?X ?Y)))) (forall (?X ?Y) (=> (and (odd ?X) (even ?Y)) (odd (+ ?X ?Y)))) (forall (?X ?Y) (=> (odd (+ ?X ?Y)) (or (odd ?X) (odd ?Y)))) (forall (?X ?Y) (= (succ (succ (+ ?X ?Y))) (+ (succ ?X) (succ ?Y)))) (forall ?X (=> (even (succ (succ ?X))) (even ?X))) (forall ?X(or (not (odd ?X)) (not (even ?X)))) (forall (?X ?Y) (<=> (= ?Y (+ (+ ?X ?X) (succ 0))) (odd ?Y))) (forall (?X ?Y) (=> (even (+ ?X ?Y)) (even (+ ?Y ?X)))) (forall (?X ?Y) (=> (and (odd ?X) (odd ?Y)) (odd (* ?X ?Y)))) (forall (?X ?Y) (=> (even ?X) (even (* ?X ?Y)))) (forall (?X ?Y) (=> (odd (* ?X ?Y)) (even (* (succ ?X) ?Y))))
Conclusion:
(forall ?X (forall ?Y (=> (and (odd ?X) (odd ?Y)) (not (even (* ?X ?Y))))))