%   ORIGINAL: h4/pred__set/SUBSET__PSUBSET__TRANS
% 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
% Assm: h4/pred__set/PSUBSET__DEF: !t s. h4/pred__set/PSUBSET s t <=> h4/pred__set/SUBSET s t /\ ~(s = t)
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/pred__set/SUBSET__FINITE: !s. h4/pred__set/FINITE s ==> (!t. h4/pred__set/SUBSET t s ==> h4/pred__set/FINITE t)
% Assm: h4/pred__set/SUBSET__DEF: !t s. h4/pred__set/SUBSET s t <=> (!x. h4/bool/IN x s ==> h4/bool/IN x t)
% Assm: h4/pred__set/FINITE__PSUBSET__INFINITE: !s. ~h4/pred__set/FINITE s <=> (!t. h4/pred__set/FINITE t ==> h4/pred__set/SUBSET t s ==> h4/pred__set/PSUBSET t s)
% Assm: h4/pred__set/PSUBSET__INSERT__SUBSET: !t s. h4/pred__set/PSUBSET s t <=> (?x. ~h4/bool/IN x s /\ h4/pred__set/SUBSET (h4/pred__set/INSERT x s) t)
% Assm: h4/pred__set/SUBSET__REFL: !s. h4/pred__set/SUBSET s s
% Assm: h4/pred__set/PSUBSET__EQN: !s2 s1. h4/pred__set/PSUBSET s1 s2 <=> h4/pred__set/SUBSET s1 s2 /\ ~h4/pred__set/SUBSET s2 s1
% Assm: h4/pred__set/PSUBSET__SUBSET__TRANS: !u t s. h4/pred__set/PSUBSET s t /\ h4/pred__set/SUBSET t u ==> h4/pred__set/PSUBSET s u
% Assm: h4/pred__set/PSUBSET__MEMBER: !t s. h4/pred__set/PSUBSET s t <=> h4/pred__set/SUBSET s t /\ (?y. h4/bool/IN y t /\ ~h4/bool/IN y s)
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% Assm: h4/pred__set/CARD__PSUBSET: !s. h4/pred__set/FINITE s ==> (!t. h4/pred__set/PSUBSET t s ==> h4/prim__rec/_3C (h4/pred__set/CARD t) (h4/pred__set/CARD s))
% Assm: h4/pred__set/PSUBSET__FINITE: !s. h4/pred__set/FINITE s ==> (!t. h4/pred__set/PSUBSET t s ==> h4/pred__set/FINITE t)
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/pred__set/FINITE__COMPLETE__INDUCTION: !P. (!x. (!y. h4/pred__set/PSUBSET y x ==> P y) ==> h4/pred__set/FINITE x ==> P x) ==> (!x. h4/pred__set/FINITE x ==> P x)
% Assm: h4/pred__set/PSUBSET__TRANS: !u t s. h4/pred__set/PSUBSET s t /\ h4/pred__set/PSUBSET t u ==> h4/pred__set/PSUBSET s u
% Assm: h4/pred__set/PSUBSET__IRREFL: !s. ~h4/pred__set/PSUBSET s s
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/pred__set/SUBSET__TRANS: !u t s. h4/pred__set/SUBSET s t /\ h4/pred__set/SUBSET t u ==> h4/pred__set/SUBSET s u
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/pred__set/SUBSET__ANTISYM: !t s. h4/pred__set/SUBSET s t /\ h4/pred__set/SUBSET t s ==> s = t
% Assm: h4/bool/DISJ__SYM: !B A. A \/ B <=> B \/ A
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/pred__set/INSERT__SUBSET: !x t s. h4/pred__set/SUBSET (h4/pred__set/INSERT x s) t <=> h4/bool/IN x t /\ h4/pred__set/SUBSET s t
% Assm: h4/pred__set/EXTENSION: !t s. s = t <=> (!x. h4/bool/IN x s <=> h4/bool/IN x t)
% Assm: h4/bool/DE__MORGAN__THM_c0: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm: h4/pred__set/PSUBSET__UNIV: !s. h4/pred__set/PSUBSET s h4/pred__set/UNIV <=> (?x. ~h4/bool/IN x s)
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/DISJ__ASSOC: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/bool/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/sat/dc__eq: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm: h4/bool/DE__MORGAN__THM_c1: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/pred__set/NOT__PSUBSET__EMPTY: !s. ~h4/pred__set/PSUBSET s h4/pred__set/EMPTY
% Assm: h4/pred__set/FINITE__PSUBSET__UNIV: ~h4/pred__set/FINITE h4/pred__set/UNIV <=> (!s. h4/pred__set/FINITE s ==> h4/pred__set/PSUBSET s h4/pred__set/UNIV)
% Assm: h4/pred__set/REST__PSUBSET: !s. ~(s = h4/pred__set/EMPTY) ==> h4/pred__set/PSUBSET (h4/pred__set/REST s) s
% Assm: h4/pred__set/NOT__UNIV__PSUBSET: !s. ~h4/pred__set/PSUBSET h4/pred__set/UNIV s
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/NOT__FORALL__THM: !P. ~(!x. P x) <=> (?x. ~P x)
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> t
% Assm: h4/bool/EQ__CLAUSES_c3: !t. (t <=> F) <=> ~t
% Assm: h4/bool/BOOL__CASES__AX: !t. (t <=> T) \/ (t <=> F)
% Assm: h4/pred__set/SUBSET__UNIV: !s. h4/pred__set/SUBSET s h4/pred__set/UNIV
% Assm: h4/bool/COND__CLAUSES_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/arithmetic/LESS__OR__EQ: !n m. h4/arithmetic/_3C_3D m n <=> h4/prim__rec/_3C m n \/ m = n
% Assm: h4/pred__set/CARD__SUBSET: !s. h4/pred__set/FINITE s ==> (!t. h4/pred__set/SUBSET t s ==> h4/arithmetic/_3C_3D (h4/pred__set/CARD t) (h4/pred__set/CARD s))
% Assm: h4/pred__set/CARD__INSERT: !s. h4/pred__set/FINITE s ==> (!x. h4/pred__set/CARD (h4/pred__set/INSERT x s) = h4/bool/COND (h4/bool/IN x s) (h4/pred__set/CARD s) (h4/num/SUC (h4/pred__set/CARD s)))
% Assm: h4/arithmetic/LESS__EQ: !n m. h4/prim__rec/_3C m n <=> h4/arithmetic/_3C_3D (h4/num/SUC m) n
% Assm: h4/bool/RIGHT__FORALL__OR__THM: !Q P. (!x. P \/ Q x) <=> P \/ (!x. Q x)
% Assm: h4/pred__set/NOT__EQUAL__SETS: !t s. ~(s = t) <=> (?x. h4/bool/IN x t <=> ~h4/bool/IN x s)
% Assm: h4/bool/EQ__CLAUSES_c2: !t. (F <=> t) <=> ~t
% Assm: h4/bool/AND__CLAUSES_c4: !t. t /\ t <=> t
% Assm: h4/pred__set/IN__DELETE: !y x s. h4/bool/IN x (h4/pred__set/DELETE s y) <=> h4/bool/IN x s /\ ~(x = y)
% Assm: h4/pred__set/SUBSET__DELETE: !x t s. h4/pred__set/SUBSET s (h4/pred__set/DELETE t x) <=> ~h4/bool/IN x s /\ h4/pred__set/SUBSET s t
% Assm: h4/bool/IMP__CONG: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm: h4/pred__set/SUBSET__DELETE__BOTH: !x s2 s1. h4/pred__set/SUBSET s1 s2 ==> h4/pred__set/SUBSET (h4/pred__set/DELETE s1 x) (h4/pred__set/DELETE s2 x)
% Assm: h4/pred__set/INJ__SUBSET: !t0 t s0 s f. h4/pred__set/INJ f s t /\ h4/pred__set/SUBSET s0 s /\ h4/pred__set/SUBSET t t0 ==> h4/pred__set/INJ f s0 t0
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/bool/LEFT__EXISTS__AND__THM: !Q P. (?x. P x /\ Q) <=> (?x. P x) /\ Q
% Assm: h4/pred__set/pairwise__SUBSET: !t s R. h4/pred__set/pairwise R t /\ h4/pred__set/SUBSET s t ==> h4/pred__set/pairwise R s
% Assm: h4/pred__set/BIGUNION__SUBSET: !X P. h4/pred__set/SUBSET (h4/pred__set/BIGUNION P) X <=> (!Y. h4/bool/IN Y P ==> h4/pred__set/SUBSET Y X)
% Assm: h4/pred__set/SET__EQ__SUBSET: !s2 s1. s1 = s2 <=> h4/pred__set/SUBSET s1 s2 /\ h4/pred__set/SUBSET s2 s1
% Assm: h4/pred__set/CROSS__SUBSET: !Q0 Q P0 P. h4/pred__set/SUBSET (h4/pred__set/CROSS P0 Q0) (h4/pred__set/CROSS P Q) <=> P0 = h4/pred__set/EMPTY \/ Q0 = h4/pred__set/EMPTY \/ h4/pred__set/SUBSET P0 P /\ h4/pred__set/SUBSET Q0 Q
% Assm: h4/pred__set/IN__UNIV: !x. h4/bool/IN x h4/pred__set/UNIV
% Assm: h4/pred__set/REL__RESTRICT__SUBSET: !s2 s1 R. h4/pred__set/SUBSET s1 s2 ==> h4/relation/RSUBSET (h4/pred__set/REL__RESTRICT R s1) (h4/pred__set/REL__RESTRICT R s2)
% Assm: h4/bool/RIGHT__OR__EXISTS__THM: !Q P. P \/ (?x. Q x) <=> (?x. P \/ Q x)
% Assm: h4/pred__set/SUM__IMAGE__SUBSET__LE: !t s f. h4/pred__set/FINITE s /\ h4/pred__set/SUBSET t s ==> h4/arithmetic/_3C_3D (h4/pred__set/SUM__IMAGE f t) (h4/pred__set/SUM__IMAGE f s)
% Assm: h4/pred__set/UNIV__SUBSET: !s. h4/pred__set/SUBSET h4/pred__set/UNIV s <=> s = h4/pred__set/UNIV
% Assm: h4/pred__set/SUBSET__EMPTY: !s. h4/pred__set/SUBSET s h4/pred__set/EMPTY <=> s = h4/pred__set/EMPTY
% Assm: h4/bool/CONJ__ASSOC: !t3 t2 t1. t1 /\ t2 /\ t3 <=> (t1 /\ t2) /\ t3
% Assm: h4/bool/RIGHT__AND__FORALL__THM: !Q P. P /\ (!x. Q x) <=> (!x. P /\ Q x)
% Assm: h4/bool/SKOLEM__THM: !P. (!x. ?y. P x y) <=> (?f. !x. P x (f x))
% Assm: h4/bool/NOT__AND: !t. ~(t /\ ~t)
% Assm: h4/prim__rec/measure__def: h4/prim__rec/measure = h4/relation/inv__image h4/prim__rec/_3C
% Assm: h4/relation/inv__image__def: !f R. h4/relation/inv__image R f = (\x y. R (f x) (f y))
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/prim__rec/WF__measure: !m. h4/relation/WF (h4/prim__rec/measure m)
% Assm: h4/relation/WF__INDUCTION__THM: !R. h4/relation/WF R ==> (!P. (!x. (!y. R y x ==> P y) ==> P x) ==> (!x. P x))
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/bool/IMP__CLAUSES_c2: !t. F ==> t <=> T
% Assm: h4/pred__set/REST__SUBSET: !s. h4/pred__set/SUBSET (h4/pred__set/REST s) s
% Assm: h4/pred__set/REST__DEF: !s. h4/pred__set/REST s = h4/pred__set/DELETE s (h4/pred__set/CHOICE s)
% Assm: h4/pred__set/CHOICE__DEF: !s. ~(s = h4/pred__set/EMPTY) ==> h4/bool/IN (h4/pred__set/CHOICE s) s
% Assm: h4/bool/AND__CLAUSES_c2: !t. F /\ t <=> F
% Assm: h4/bool/NOT__IMP: !B A. ~(A ==> B) <=> A /\ ~B
% Assm: h4/pred__set/INJ__DEF: !t s f. h4/pred__set/INJ f s t <=> (!x. h4/bool/IN x s ==> h4/bool/IN (f x) t) /\ (!x y. h4/bool/IN x s /\ h4/bool/IN y s ==> f x = f y ==> x = y)
% Assm: h4/pred__set/pairwise__def: !s P. h4/pred__set/pairwise P s <=> (!e1 e2. h4/bool/IN e1 s /\ h4/bool/IN e2 s ==> P e1 e2)
% Assm: h4/bool/AND__CLAUSES_c3: !t. t /\ F <=> F
% Assm: h4/bool/RIGHT__EXISTS__AND__THM: !Q P. (?x. P /\ Q x) <=> P /\ (?x. Q x)
% Assm: h4/relation/RSUBSET0: !R2 R1. h4/relation/RSUBSET R1 R2 <=> (!x y. R1 x y ==> R2 x y)
% Assm: h4/pred__set/REL__RESTRICT__DEF: !y x s R. h4/pred__set/REL__RESTRICT R s x y <=> h4/bool/IN x s /\ h4/bool/IN y s /\ R x y
% Assm: h4/bool/NOT__EXISTS__THM: !P. ~(?x. P x) <=> (!x. ~P x)
% Assm: h4/bool/LEFT__FORALL__OR__THM: !Q P. (!x. P x \/ Q) <=> (!x. P x) \/ Q
% Assm: h4/pred__set/IN__BIGUNION: !x sos. h4/bool/IN x (h4/pred__set/BIGUNION sos) <=> (?s. h4/bool/IN x s /\ h4/bool/IN s sos)
% Assm: h4/pred__set/NOT__IN__EMPTY: !x. ~h4/bool/IN x h4/pred__set/EMPTY
% Assm: h4/bool/EXISTS__OR__THM: !Q P. (?x. P x \/ Q x) <=> (?x. P x) \/ (?x. Q x)
% Assm: h4/bool/LEFT__OR__EXISTS__THM: !Q P. (?x. P x) \/ Q <=> (?x. P x \/ Q)
% Assm: h4/pair/SND0: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm: h4/pred__set/IN__CROSS: !x Q P. h4/bool/IN x (h4/pred__set/CROSS P Q) <=> h4/bool/IN (h4/pair/FST x) P /\ h4/bool/IN (h4/pair/SND x) Q
% Assm: h4/pair/FORALL__PROD: !P. (!p. P p) <=> (!p__1 p__2. P (h4/pair/_2C p__1 p__2))
% Assm: h4/pair/FST0: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm: h4/pred__set/FINITE__DELETE: !x s. h4/pred__set/FINITE (h4/pred__set/DELETE s x) <=> h4/pred__set/FINITE s
% Assm: h4/pred__set/FINITE__INSERT: !x s. h4/pred__set/FINITE (h4/pred__set/INSERT x s) <=> h4/pred__set/FINITE s
% Assm: h4/pred__set/FINITE__INDUCT: !P. P h4/pred__set/EMPTY /\ (!s. h4/pred__set/FINITE s /\ P s ==> (!e. ~h4/bool/IN e s ==> P (h4/pred__set/INSERT e s))) ==> (!s. h4/pred__set/FINITE s ==> P s)
% Assm: h4/pred__set/SUBSET__INSERT__DELETE: !x t s. h4/pred__set/SUBSET s (h4/pred__set/INSERT x t) <=> h4/pred__set/SUBSET (h4/pred__set/DELETE s x) t
% Assm: h4/pred__set/DELETE__NON__ELEMENT: !x s. ~h4/bool/IN x s <=> h4/pred__set/DELETE s x = s
% Assm: h4/pred__set/INSERT__DELETE: !x s. h4/bool/IN x s ==> h4/pred__set/INSERT x (h4/pred__set/DELETE s x) = s
% Assm: h4/pred__set/SUBSET__INSERT: !x s. ~h4/bool/IN x s ==> (!t. h4/pred__set/SUBSET s (h4/pred__set/INSERT x t) <=> h4/pred__set/SUBSET s t)
% Assm: h4/numeral/numeral__lte_c1: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% Assm: h4/numeral/numeral__distrib_c27: !n. h4/arithmetic/_3C_3D (h4/arithmetic/NUMERAL n) h4/num/0 <=> h4/arithmetic/_3C_3D n h4/arithmetic/ZERO
% Assm: h4/numeral/numeral__distrib_c1: !n. h4/arithmetic/_2B n h4/num/0 = n
% Assm: h4/arithmetic/SUC__ONE__ADD: !n. h4/num/SUC n = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) n
% Goal: !u t s. h4/pred__set/SUBSET s t /\ h4/pred__set/PSUBSET t u ==> h4/pred__set/PSUBSET s u
%   PROCESSED
% Assm [HLu_TRUTH]: T
% Assm [HLu_FALSITY]: ~F
% Assm [HLu_BOOLu_CASES]: !t. (t <=> T) \/ (t <=> F)
% Assm [HLu_EXT]: !f g. (!x. happ f x = happ g x) ==> f = g
% Assm [h4s_predu_u_sets_PSUBSETu_u_DEF]: !t s. h4/pred__set/PSUBSET s t <=> h4/pred__set/SUBSET s t /\ ~(s = t)
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_predu_u_sets_SUBSETu_u_FINITE]: !s. h4/pred__set/FINITE s ==> (!t. h4/pred__set/SUBSET t s ==> h4/pred__set/FINITE t)
% Assm [h4s_predu_u_sets_SUBSETu_u_DEF]: !t s. h4/pred__set/SUBSET s t <=> (!x. h4/bool/IN x s ==> h4/bool/IN x t)
% Assm [h4s_predu_u_sets_FINITEu_u_PSUBSETu_u_INFINITE]: !s. ~h4/pred__set/FINITE s <=> (!t. h4/pred__set/FINITE t ==> h4/pred__set/SUBSET t s ==> h4/pred__set/PSUBSET t s)
% Assm [h4s_predu_u_sets_PSUBSETu_u_INSERTu_u_SUBSET]: !t s. h4/pred__set/PSUBSET s t <=> (?x. ~h4/bool/IN x s /\ h4/pred__set/SUBSET (h4/pred__set/INSERT x s) t)
% Assm [h4s_predu_u_sets_SUBSETu_u_REFL]: !s. h4/pred__set/SUBSET s s
% Assm [h4s_predu_u_sets_PSUBSETu_u_EQN]: !s2 s1. h4/pred__set/PSUBSET s1 s2 <=> h4/pred__set/SUBSET s1 s2 /\ ~h4/pred__set/SUBSET s2 s1
% Assm [h4s_predu_u_sets_PSUBSETu_u_SUBSETu_u_TRANS]: !u t s. h4/pred__set/PSUBSET s t /\ h4/pred__set/SUBSET t u ==> h4/pred__set/PSUBSET s u
% Assm [h4s_predu_u_sets_PSUBSETu_u_MEMBER]: !t s. h4/pred__set/PSUBSET s t <=> h4/pred__set/SUBSET s t /\ (?y. h4/bool/IN y t /\ ~h4/bool/IN y s)
% Assm [h4s_bools_Fu_u_IMP]: !t. ~t ==> t ==> F
% Assm [h4s_predu_u_sets_CARDu_u_PSUBSET]: !s. h4/pred__set/FINITE s ==> (!t. h4/pred__set/PSUBSET t s ==> happ (happ h4/prim__rec/_3C (h4/pred__set/CARD t)) (h4/pred__set/CARD s))
% Assm [h4s_predu_u_sets_PSUBSETu_u_FINITE]: !s. h4/pred__set/FINITE s ==> (!t. h4/pred__set/PSUBSET t s ==> h4/pred__set/FINITE t)
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_predu_u_sets_FINITEu_u_COMPLETEu_u_INDUCTION]: !P. (!x. (!y. h4/pred__set/PSUBSET y x ==> happ P y) ==> h4/pred__set/FINITE x ==> happ P x) ==> (!x. h4/pred__set/FINITE x ==> happ P x)
% Assm [h4s_predu_u_sets_PSUBSETu_u_TRANS]: !u t s. h4/pred__set/PSUBSET s t /\ h4/pred__set/PSUBSET t u ==> h4/pred__set/PSUBSET s u
% Assm [h4s_predu_u_sets_PSUBSETu_u_IRREFL]: !s. ~h4/pred__set/PSUBSET s s
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_predu_u_sets_SUBSETu_u_TRANS]: !u t s. h4/pred__set/SUBSET s t /\ h4/pred__set/SUBSET t u ==> h4/pred__set/SUBSET s u
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_predu_u_sets_SUBSETu_u_ANTISYM]: !t s. h4/pred__set/SUBSET s t /\ h4/pred__set/SUBSET t s ==> s = t
% Assm [h4s_bools_DISJu_u_SYM]: !B A. A \/ B <=> B \/ A
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_predu_u_sets_INSERTu_u_SUBSET]: !x t s. h4/pred__set/SUBSET (h4/pred__set/INSERT x s) t <=> h4/bool/IN x t /\ h4/pred__set/SUBSET s t
% Assm [h4s_predu_u_sets_EXTENSION]: !t s. s = t <=> (!x. h4/bool/IN x s <=> h4/bool/IN x t)
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c0]: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm [h4s_predu_u_sets_PSUBSETu_u_UNIV]: !s. h4/pred__set/PSUBSET s h4/pred__set/UNIV <=> (?x. ~h4/bool/IN x s)
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_DISJu_u_ASSOC]: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_bools_IMPu_u_F]: !t. (t ==> F) ==> ~t
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_sats_dcu_u_eq]: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c1]: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_predu_u_sets_NOTu_u_PSUBSETu_u_EMPTY]: !s. ~h4/pred__set/PSUBSET s h4/pred__set/EMPTY
% Assm [h4s_predu_u_sets_FINITEu_u_PSUBSETu_u_UNIV]: ~h4/pred__set/FINITE h4/pred__set/UNIV <=> (!s. h4/pred__set/FINITE s ==> h4/pred__set/PSUBSET s h4/pred__set/UNIV)
% Assm [h4s_predu_u_sets_RESTu_u_PSUBSET]: !s. ~(s = h4/pred__set/EMPTY) ==> h4/pred__set/PSUBSET (h4/pred__set/REST s) s
% Assm [h4s_predu_u_sets_NOTu_u_UNIVu_u_PSUBSET]: !s. ~h4/pred__set/PSUBSET h4/pred__set/UNIV s
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_NOTu_u_FORALLu_u_THM]: !P. ~(!x. happ P x) <=> (?x. ~happ P x)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c3]: !t. (t <=> F) <=> ~t
% Assm [h4s_bools_BOOLu_u_CASESu_u_AX]: !t. (t <=> T) \/ (t <=> F)
% Assm [h4s_predu_u_sets_SUBSETu_u_UNIV]: !s. h4/pred__set/SUBSET s h4/pred__set/UNIV
% Assm [h4s_bools_CONDu_u_CLAUSESu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_arithmetics_LESSu_u_ORu_u_EQ]: !n m. h4/arithmetic/_3C_3D m n <=> happ (happ h4/prim__rec/_3C m) n \/ m = n
% Assm [h4s_predu_u_sets_CARDu_u_SUBSET]: !s. h4/pred__set/FINITE s ==> (!t. h4/pred__set/SUBSET t s ==> h4/arithmetic/_3C_3D (h4/pred__set/CARD t) (h4/pred__set/CARD s))
% Assm [h4s_predu_u_sets_CARDu_u_INSERT]: !s. h4/pred__set/FINITE s ==> (!x. h4/pred__set/CARD (h4/pred__set/INSERT x s) = h4/bool/COND (h4/bool/IN x s) (h4/pred__set/CARD s) (h4/num/SUC (h4/pred__set/CARD s)))
% Assm [h4s_arithmetics_LESSu_u_EQ]: !n m. happ (happ h4/prim__rec/_3C m) n <=> h4/arithmetic/_3C_3D (h4/num/SUC m) n
% Assm [h4s_bools_RIGHTu_u_FORALLu_u_ORu_u_THM]: !Q P. (!x. P \/ happ Q x) <=> P \/ (!x. happ Q x)
% Assm [h4s_predu_u_sets_NOTu_u_EQUALu_u_SETS]: !t s. ~(s = t) <=> (?x. h4/bool/IN x t <=> ~h4/bool/IN x s)
% Assm [h4s_bools_EQu_u_CLAUSESu_c2]: !t. (F <=> t) <=> ~t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c4]: !t. t /\ t <=> t
% Assm [h4s_predu_u_sets_INu_u_DELETE]: !y x s. h4/bool/IN x (h4/pred__set/DELETE s y) <=> h4/bool/IN x s /\ ~(x = y)
% Assm [h4s_predu_u_sets_SUBSETu_u_DELETE]: !x t s. h4/pred__set/SUBSET s (h4/pred__set/DELETE t x) <=> ~h4/bool/IN x s /\ h4/pred__set/SUBSET s t
% Assm [h4s_bools_IMPu_u_CONG]: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm [h4s_predu_u_sets_SUBSETu_u_DELETEu_u_BOTH]: !x s2 s1. h4/pred__set/SUBSET s1 s2 ==> h4/pred__set/SUBSET (h4/pred__set/DELETE s1 x) (h4/pred__set/DELETE s2 x)
% Assm [h4s_predu_u_sets_INJu_u_SUBSET]: !t0 t s0 s f. h4/pred__set/INJ f s t /\ h4/pred__set/SUBSET s0 s /\ h4/pred__set/SUBSET t t0 ==> h4/pred__set/INJ f s0 t0
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_bools_LEFTu_u_EXISTSu_u_ANDu_u_THM]: !Q P. (?x. happ P x /\ Q) <=> (?x. happ P x) /\ Q
% Assm [h4s_predu_u_sets_pairwiseu_u_SUBSET]: !t s R. h4/pred__set/pairwise R t /\ h4/pred__set/SUBSET s t ==> h4/pred__set/pairwise R s
% Assm [h4s_predu_u_sets_BIGUNIONu_u_SUBSET]: !X P. h4/pred__set/SUBSET (h4/pred__set/BIGUNION P) X <=> (!Y. h4/bool/IN Y P ==> h4/pred__set/SUBSET Y X)
% Assm [h4s_predu_u_sets_SETu_u_EQu_u_SUBSET]: !s2 s1. s1 = s2 <=> h4/pred__set/SUBSET s1 s2 /\ h4/pred__set/SUBSET s2 s1
% Assm [h4s_predu_u_sets_CROSSu_u_SUBSET]: !Q0 Q P0 P. h4/pred__set/SUBSET (h4/pred__set/CROSS P0 Q0) (h4/pred__set/CROSS P Q) <=> P0 = h4/pred__set/EMPTY \/ Q0 = h4/pred__set/EMPTY \/ h4/pred__set/SUBSET P0 P /\ h4/pred__set/SUBSET Q0 Q
% Assm [h4s_predu_u_sets_INu_u_UNIV]: !x. h4/bool/IN x h4/pred__set/UNIV
% Assm [h4s_predu_u_sets_RELu_u_RESTRICTu_u_SUBSET]: !s2 s1 R. h4/pred__set/SUBSET s1 s2 ==> h4/relation/RSUBSET (h4/pred__set/REL__RESTRICT R s1) (h4/pred__set/REL__RESTRICT R s2)
% Assm [h4s_bools_RIGHTu_u_ORu_u_EXISTSu_u_THM]: !Q P. P \/ (?x. happ Q x) <=> (?x. P \/ happ Q x)
% Assm [h4s_predu_u_sets_SUMu_u_IMAGEu_u_SUBSETu_u_LE]: !t s f. h4/pred__set/FINITE s /\ h4/pred__set/SUBSET t s ==> h4/arithmetic/_3C_3D (h4/pred__set/SUM__IMAGE f t) (h4/pred__set/SUM__IMAGE f s)
% Assm [h4s_predu_u_sets_UNIVu_u_SUBSET]: !s. h4/pred__set/SUBSET h4/pred__set/UNIV s <=> s = h4/pred__set/UNIV
% Assm [h4s_predu_u_sets_SUBSETu_u_EMPTY]: !s. h4/pred__set/SUBSET s h4/pred__set/EMPTY <=> s = h4/pred__set/EMPTY
% Assm [h4s_bools_CONJu_u_ASSOC]: !t3 t2 t1. t1 /\ t2 /\ t3 <=> (t1 /\ t2) /\ t3
% Assm [h4s_bools_RIGHTu_u_ANDu_u_FORALLu_u_THM]: !Q P. P /\ (!x. happ Q x) <=> (!x. P /\ happ Q x)
% Assm [h4s_bools_SKOLEMu_u_THM]: !P. (!x. ?y. happ (happ P x) y) <=> (?f. !x. happ (happ P x) (happ f x))
% Assm [h4s_bools_NOTu_u_AND]: !t. ~(t /\ ~t)
% Assm [h4s_primu_u_recs_measureu_u_def]: h4/prim__rec/measure = h4/relation/inv__image h4/prim__rec/_3C
% Assm [h4s_relations_invu_u_imageu_u_def]: !f R x x'. happ (happ (happ (h4/relation/inv__image R) f) x) x' <=> happ (happ R (happ f x)) (happ f x')
% Assm [h4s_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% Assm [h4s_primu_u_recs_WFu_u_measure]: !m. h4/relation/WF (happ h4/prim__rec/measure m)
% Assm [h4s_relations_WFu_u_INDUCTIONu_u_THM]: !R. h4/relation/WF R ==> (!P. (!x. (!y. happ (happ R y) x ==> happ P y) ==> happ P x) ==> (!x. happ P x))
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_bools_IMPu_u_CLAUSESu_c2]: !t. F ==> t <=> T
% Assm [h4s_predu_u_sets_RESTu_u_SUBSET]: !s. h4/pred__set/SUBSET (h4/pred__set/REST s) s
% Assm [h4s_predu_u_sets_RESTu_u_DEF]: !s. h4/pred__set/REST s = h4/pred__set/DELETE s (h4/pred__set/CHOICE s)
% Assm [h4s_predu_u_sets_CHOICEu_u_DEF]: !s. ~(s = h4/pred__set/EMPTY) ==> h4/bool/IN (h4/pred__set/CHOICE s) s
% Assm [h4s_bools_ANDu_u_CLAUSESu_c2]: !t. F /\ t <=> F
% Assm [h4s_bools_NOTu_u_IMP]: !B A. ~(A ==> B) <=> A /\ ~B
% Assm [h4s_predu_u_sets_INJu_u_DEF]: !t s f. h4/pred__set/INJ f s t <=> (!x. h4/bool/IN x s ==> h4/bool/IN (happ f x) t) /\ (!x y. h4/bool/IN x s /\ h4/bool/IN y s ==> happ f x = happ f y ==> x = y)
% Assm [h4s_predu_u_sets_pairwiseu_u_def]: !s P. h4/pred__set/pairwise P s <=> (!e1 e2. h4/bool/IN e1 s /\ h4/bool/IN e2 s ==> happ (happ P e1) e2)
% Assm [h4s_bools_ANDu_u_CLAUSESu_c3]: !t. t /\ F <=> F
% Assm [h4s_bools_RIGHTu_u_EXISTSu_u_ANDu_u_THM]: !Q P. (?x. P /\ happ Q x) <=> P /\ (?x. happ Q x)
% Assm [h4s_relations_RSUBSET0]: !R2 R1. h4/relation/RSUBSET R1 R2 <=> (!x y. happ (happ R1 x) y ==> happ (happ R2 x) y)
% Assm [h4s_predu_u_sets_RELu_u_RESTRICTu_u_DEF]: !y x s R. happ (happ (h4/pred__set/REL__RESTRICT R s) x) y <=> h4/bool/IN x s /\ h4/bool/IN y s /\ happ (happ R x) y
% Assm [h4s_bools_NOTu_u_EXISTSu_u_THM]: !P. ~(?x. happ P x) <=> (!x. ~happ P x)
% Assm [h4s_bools_LEFTu_u_FORALLu_u_ORu_u_THM]: !Q P. (!x. happ P x \/ Q) <=> (!x. happ P x) \/ Q
% Assm [h4s_predu_u_sets_INu_u_BIGUNION]: !x sos. h4/bool/IN x (h4/pred__set/BIGUNION sos) <=> (?s. h4/bool/IN x s /\ h4/bool/IN s sos)
% Assm [h4s_predu_u_sets_NOTu_u_INu_u_EMPTY]: !x. ~h4/bool/IN x h4/pred__set/EMPTY
% Assm [h4s_bools_EXISTSu_u_ORu_u_THM]: !Q P. (?x. happ P x \/ happ Q x) <=> (?x. happ P x) \/ (?x. happ Q x)
% Assm [h4s_bools_LEFTu_u_ORu_u_EXISTSu_u_THM]: !Q P. (?x. happ P x) \/ Q <=> (?x. happ P x \/ Q)
% Assm [h4s_pairs_SND0]: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm [h4s_predu_u_sets_INu_u_CROSS]: !x Q P. h4/bool/IN x (h4/pred__set/CROSS P Q) <=> h4/bool/IN (h4/pair/FST x) P /\ h4/bool/IN (h4/pair/SND x) Q
% Assm [h4s_pairs_FORALLu_u_PROD]: !P. (!p. happ P p) <=> (!p__1 p__2. happ P (h4/pair/_2C p__1 p__2))
% Assm [h4s_pairs_FST0]: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm [h4s_predu_u_sets_FINITEu_u_DELETE]: !x s. h4/pred__set/FINITE (h4/pred__set/DELETE s x) <=> h4/pred__set/FINITE s
% Assm [h4s_predu_u_sets_FINITEu_u_INSERT]: !x s. h4/pred__set/FINITE (h4/pred__set/INSERT x s) <=> h4/pred__set/FINITE s
% Assm [h4s_predu_u_sets_FINITEu_u_INDUCT]: !P. happ P h4/pred__set/EMPTY /\ (!s. h4/pred__set/FINITE s /\ happ P s ==> (!e. ~h4/bool/IN e s ==> happ P (h4/pred__set/INSERT e s))) ==> (!s. h4/pred__set/FINITE s ==> happ P s)
% Assm [h4s_predu_u_sets_SUBSETu_u_INSERTu_u_DELETE]: !x t s. h4/pred__set/SUBSET s (h4/pred__set/INSERT x t) <=> h4/pred__set/SUBSET (h4/pred__set/DELETE s x) t
% Assm [h4s_predu_u_sets_DELETEu_u_NONu_u_ELEMENT]: !x s. ~h4/bool/IN x s <=> h4/pred__set/DELETE s x = s
% Assm [h4s_predu_u_sets_INSERTu_u_DELETE]: !x s. h4/bool/IN x s ==> h4/pred__set/INSERT x (h4/pred__set/DELETE s x) = s
% Assm [h4s_predu_u_sets_SUBSETu_u_INSERT]: !x s. ~h4/bool/IN x s ==> (!t. h4/pred__set/SUBSET s (h4/pred__set/INSERT x t) <=> h4/pred__set/SUBSET s t)
% Assm [h4s_numerals_numeralu_u_lteu_c1]: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% Assm [h4s_numerals_numeralu_u_distribu_c27]: !n. h4/arithmetic/_3C_3D (h4/arithmetic/NUMERAL n) h4/num/0 <=> h4/arithmetic/_3C_3D n h4/arithmetic/ZERO
% Assm [h4s_numerals_numeralu_u_distribu_c1]: !n. h4/arithmetic/_2B n h4/num/0 = n
% Assm [h4s_arithmetics_SUCu_u_ONEu_u_ADD]: !n. h4/num/SUC n = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) n
% Goal: !u t s. h4/pred__set/SUBSET s t /\ h4/pred__set/PSUBSET t u ==> h4/pred__set/PSUBSET s u
fof(aHLu_TRUTH, axiom, p(s(t_bool,t0))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t0) | s(t_bool,V_t) = s(t_bool,f))).
fof(aHLu_EXT, axiom, ![TV_Q1243834,TV_Q1243830]: ![V_f, V_g]: (![V_x]: s(TV_Q1243830,happ(s(t_fun(TV_Q1243834,TV_Q1243830),V_f),s(TV_Q1243834,V_x))) = s(TV_Q1243830,happ(s(t_fun(TV_Q1243834,TV_Q1243830),V_g),s(TV_Q1243834,V_x))) => s(t_fun(TV_Q1243834,TV_Q1243830),V_f) = s(t_fun(TV_Q1243834,TV_Q1243830),V_g))).
fof(ah4s_predu_u_sets_PSUBSETu_u_DEF, axiom, ![TV_u_27a]: ![V_t, V_s]: (p(s(t_bool,h4s_predu_u_sets_psubset(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t)))) <=> (p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t)))) & ~ (s(t_fun(TV_u_27a,t_bool),V_s) = s(t_fun(TV_u_27a,t_bool),V_t))))).
fof(ah4s_bools_IMPu_u_ANTISYMu_u_AX, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) => ((p(s(t_bool,V_t2)) => p(s(t_bool,V_t1))) => s(t_bool,V_t1) = s(t_bool,V_t2)))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t0))).
fof(ah4s_bools_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t0) <=> p(s(t_bool,V_t)))).
fof(ah4s_predu_u_sets_SUBSETu_u_FINITE, axiom, ![TV_u_27a]: ![V_s]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) => ![V_t]: (p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_t),s(t_fun(TV_u_27a,t_bool),V_s)))) => p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_t))))))).
fof(ah4s_predu_u_sets_SUBSETu_u_DEF, axiom, ![TV_u_27a]: ![V_t, V_s]: (p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t)))) <=> ![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) => p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_t))))))).
fof(ah4s_predu_u_sets_FINITEu_u_PSUBSETu_u_INFINITE, axiom, ![TV_u_27a]: ![V_s]: (~ (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s))))) <=> ![V_t]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_t)))) => (p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_t),s(t_fun(TV_u_27a,t_bool),V_s)))) => p(s(t_bool,h4s_predu_u_sets_psubset(s(t_fun(TV_u_27a,t_bool),V_t),s(t_fun(TV_u_27a,t_bool),V_s)))))))).
fof(ah4s_predu_u_sets_PSUBSETu_u_INSERTu_u_SUBSET, axiom, ![TV_u_27a]: ![V_t, V_s]: (p(s(t_bool,h4s_predu_u_sets_psubset(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t)))) <=> ?[V_x]: (~ (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))))) & p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_fun(TV_u_27a,t_bool),V_t))))))).
fof(ah4s_predu_u_sets_SUBSETu_u_REFL, axiom, ![TV_u_27a]: ![V_s]: p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_s))))).
fof(ah4s_predu_u_sets_PSUBSETu_u_EQN, axiom, ![TV_u_27a]: ![V_s2, V_s1]: (p(s(t_bool,h4s_predu_u_sets_psubset(s(t_fun(TV_u_27a,t_bool),V_s1),s(t_fun(TV_u_27a,t_bool),V_s2)))) <=> (p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_s1),s(t_fun(TV_u_27a,t_bool),V_s2)))) & ~ (p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_s2),s(t_fun(TV_u_27a,t_bool),V_s1)))))))).
fof(ah4s_predu_u_sets_PSUBSETu_u_SUBSETu_u_TRANS, axiom, ![TV_u_27a]: ![V_u, V_t, V_s]: ((p(s(t_bool,h4s_predu_u_sets_psubset(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t)))) & p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_t),s(t_fun(TV_u_27a,t_bool),V_u))))) => p(s(t_bool,h4s_predu_u_sets_psubset(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_u)))))).
fof(ah4s_predu_u_sets_PSUBSETu_u_MEMBER, axiom, ![TV_u_27a]: ![V_t, V_s]: (p(s(t_bool,h4s_predu_u_sets_psubset(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t)))) <=> (p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t)))) & ?[V_y]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_y),s(t_fun(TV_u_27a,t_bool),V_t)))) & ~ (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_y),s(t_fun(TV_u_27a,t_bool),V_s))))))))).
fof(ah4s_bools_Fu_u_IMP, axiom, ![V_t]: (~ (p(s(t_bool,V_t))) => (p(s(t_bool,V_t)) => p(s(t_bool,f))))).
fof(ah4s_predu_u_sets_CARDu_u_PSUBSET, axiom, ![TV_u_27a]: ![V_s]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) => ![V_t]: (p(s(t_bool,h4s_predu_u_sets_psubset(s(t_fun(TV_u_27a,t_bool),V_t),s(t_fun(TV_u_27a,t_bool),V_s)))) => p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,h4s_predu_u_sets_card(s(t_fun(TV_u_27a,t_bool),V_t))))),s(t_h4s_nums_num,h4s_predu_u_sets_card(s(t_fun(TV_u_27a,t_bool),V_s))))))))).
fof(ah4s_predu_u_sets_PSUBSETu_u_FINITE, axiom, ![TV_u_27a]: ![V_s]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) => ![V_t]: (p(s(t_bool,h4s_predu_u_sets_psubset(s(t_fun(TV_u_27a,t_bool),V_t),s(t_fun(TV_u_27a,t_bool),V_s)))) => p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_t))))))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t0)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_predu_u_sets_FINITEu_u_COMPLETEu_u_INDUCTION, axiom, ![TV_u_27a]: ![V_P]: (![V_x]: (![V_y]: (p(s(t_bool,h4s_predu_u_sets_psubset(s(t_fun(TV_u_27a,t_bool),V_y),s(t_fun(TV_u_27a,t_bool),V_x)))) => p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P),s(t_fun(TV_u_27a,t_bool),V_y))))) => (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_x)))) => p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P),s(t_fun(TV_u_27a,t_bool),V_x)))))) => ![V_x]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_x)))) => p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P),s(t_fun(TV_u_27a,t_bool),V_x))))))).
fof(ah4s_predu_u_sets_PSUBSETu_u_TRANS, axiom, ![TV_u_27a]: ![V_u, V_t, V_s]: ((p(s(t_bool,h4s_predu_u_sets_psubset(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t)))) & p(s(t_bool,h4s_predu_u_sets_psubset(s(t_fun(TV_u_27a,t_bool),V_t),s(t_fun(TV_u_27a,t_bool),V_u))))) => p(s(t_bool,h4s_predu_u_sets_psubset(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_u)))))).
fof(ah4s_predu_u_sets_PSUBSETu_u_IRREFL, axiom, ![TV_u_27a]: ![V_s]: ~ (p(s(t_bool,h4s_predu_u_sets_psubset(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_s)))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t0)))).
fof(ah4s_bools_REFLu_u_CLAUSE, axiom, ![TV_u_27a]: ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_x) <=> p(s(t_bool,t0)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_FORALLu_u_SIMP, axiom, ![TV_u_27a]: ![V_t]: (![V_x]: p(s(t_bool,V_t)) <=> p(s(t_bool,V_t)))).
fof(ah4s_predu_u_sets_SUBSETu_u_TRANS, axiom, ![TV_u_27a]: ![V_u, V_t, V_s]: ((p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t)))) & p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_t),s(t_fun(TV_u_27a,t_bool),V_u))))) => p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_u)))))).
fof(ah4s_bools_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_EQu_u_SYMu_u_EQ, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) <=> s(TV_u_27a,V_y) = s(TV_u_27a,V_x))).
fof(ah4s_predu_u_sets_SUBSETu_u_ANTISYM, axiom, ![TV_u_27a]: ![V_t, V_s]: ((p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t)))) & p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_t),s(t_fun(TV_u_27a,t_bool),V_s))))) => s(t_fun(TV_u_27a,t_bool),V_s) = s(t_fun(TV_u_27a,t_bool),V_t))).
fof(ah4s_bools_DISJu_u_SYM, axiom, ![V_B, V_A]: ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) <=> (p(s(t_bool,V_B)) | p(s(t_bool,V_A))))).
fof(ah4s_bools_EQu_u_CLAUSESu_c0, axiom, ![V_t]: (s(t_bool,t0) = s(t_bool,V_t) <=> p(s(t_bool,V_t)))).
fof(ah4s_predu_u_sets_INSERTu_u_SUBSET, axiom, ![TV_u_27a]: ![V_x, V_t, V_s]: (p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_fun(TV_u_27a,t_bool),V_t)))) <=> (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_t)))) & p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t))))))).
fof(ah4s_predu_u_sets_EXTENSION, axiom, ![TV_u_27a]: ![V_t, V_s]: (s(t_fun(TV_u_27a,t_bool),V_s) = s(t_fun(TV_u_27a,t_bool),V_t) <=> ![V_x]: s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_t))))).
fof(ah4s_bools_DEu_u_MORGANu_u_THMu_c0, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) & p(s(t_bool,V_B)))) <=> (~ (p(s(t_bool,V_A))) | ~ (p(s(t_bool,V_B)))))).
fof(ah4s_predu_u_sets_PSUBSETu_u_UNIV, axiom, ![TV_u_27a]: ![V_s]: (p(s(t_bool,h4s_predu_u_sets_psubset(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_univ)))) <=> ?[V_x]: ~ (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))))))).
fof(ah4s_sats_ANDu_u_INV2, axiom, ![V_A]: ((~ (p(s(t_bool,V_A))) => p(s(t_bool,f))) => ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => p(s(t_bool,f))))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_DISJu_u_ASSOC, axiom, ![V_C, V_B, V_A]: ((p(s(t_bool,V_A)) | (p(s(t_bool,V_B)) | p(s(t_bool,V_C)))) <=> ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) | p(s(t_bool,V_C))))).
fof(ah4s_sats_ORu_u_DUAL3, axiom, ![V_B, V_A]: ((~ ((~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_sats_dcu_u_conj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) & p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_r))))) & ((p(s(t_bool,V_q)) | ~ (p(s(t_bool,V_p)))) & (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p)))))))).
fof(ah4s_sats_ANDu_u_INVu_u_IMP, axiom, ![V_A]: (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_A))) => p(s(t_bool,f))))).
fof(ah4s_bools_IMPu_u_F, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) => ~ (p(s(t_bool,V_t))))).
fof(ah4s_sats_dcu_u_imp, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) => p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (~ (p(s(t_bool,V_q))) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_sats_dcu_u_disj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_q)))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (p(s(t_bool,V_q)) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_sats_dcu_u_neg, axiom, ![V_q, V_p]: ((p(s(t_bool,V_p)) <=> ~ (p(s(t_bool,V_q)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p))))))).
fof(ah4s_sats_dcu_u_eq, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> s(t_bool,V_q) = s(t_bool,V_r)) <=> ((p(s(t_bool,V_p)) | (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) & ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_q))))) & ((p(s(t_bool,V_q)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_p))))) & (p(s(t_bool,V_r)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p)))))))))).
fof(ah4s_bools_DEu_u_MORGANu_u_THMu_c1, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) <=> (~ (p(s(t_bool,V_A))) & ~ (p(s(t_bool,V_B)))))).
fof(ah4s_sats_ORu_u_DUAL2, axiom, ![V_B, V_A]: ((~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_sats_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_predu_u_sets_NOTu_u_PSUBSETu_u_EMPTY, axiom, ![TV_u_27a]: ![V_s]: ~ (p(s(t_bool,h4s_predu_u_sets_psubset(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)))))).
fof(ah4s_predu_u_sets_FINITEu_u_PSUBSETu_u_UNIV, axiom, ![TV_u_27a]: (~ (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_univ))))) <=> ![V_s]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) => p(s(t_bool,h4s_predu_u_sets_psubset(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_univ))))))).
fof(ah4s_predu_u_sets_RESTu_u_PSUBSET, axiom, ![TV_u_27a]: ![V_s]: (~ (s(t_fun(TV_u_27a,t_bool),V_s) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)) => p(s(t_bool,h4s_predu_u_sets_psubset(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_rest(s(t_fun(TV_u_27a,t_bool),V_s))),s(t_fun(TV_u_27a,t_bool),V_s)))))).
fof(ah4s_predu_u_sets_NOTu_u_UNIVu_u_PSUBSET, axiom, ![TV_u_27a]: ![V_s]: ~ (p(s(t_bool,h4s_predu_u_sets_psubset(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_univ),s(t_fun(TV_u_27a,t_bool),V_s)))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t0))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_ANDu_u_IMPu_u_INTRO, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) => (p(s(t_bool,V_t2)) => p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) => p(s(t_bool,V_t3))))).
fof(ah4s_bools_NOTu_u_FORALLu_u_THM, axiom, ![TV_u_27a]: ![V_P]: (~ (![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) <=> ?[V_x]: ~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t0)) => p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c3, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,f) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_BOOLu_u_CASESu_u_AX, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t0) | s(t_bool,V_t) = s(t_bool,f))).
fof(ah4s_predu_u_sets_SUBSETu_u_UNIV, axiom, ![TV_u_27a]: ![V_s]: p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_univ))))).
fof(ah4s_bools_CONDu_u_CLAUSESu_c1, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,f),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
fof(ah4s_arithmetics_LESSu_u_ORu_u_EQ, axiom, ![V_n, V_m]: (p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) <=> (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n)))) | s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,V_n)))).
fof(ah4s_predu_u_sets_CARDu_u_SUBSET, axiom, ![TV_u_27a]: ![V_s]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) => ![V_t]: (p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_t),s(t_fun(TV_u_27a,t_bool),V_s)))) => p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_predu_u_sets_card(s(t_fun(TV_u_27a,t_bool),V_t))),s(t_h4s_nums_num,h4s_predu_u_sets_card(s(t_fun(TV_u_27a,t_bool),V_s))))))))).
fof(ah4s_predu_u_sets_CARDu_u_INSERT, axiom, ![TV_u_27a]: ![V_s]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) => ![V_x]: s(t_h4s_nums_num,h4s_predu_u_sets_card(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))))) = s(t_h4s_nums_num,h4s_bools_cond(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_h4s_nums_num,h4s_predu_u_sets_card(s(t_fun(TV_u_27a,t_bool),V_s))),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_predu_u_sets_card(s(t_fun(TV_u_27a,t_bool),V_s))))))))).
fof(ah4s_arithmetics_LESSu_u_EQ, axiom, ![V_n, V_m]: s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n)))).
fof(ah4s_bools_RIGHTu_u_FORALLu_u_ORu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,V_P)) | p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> (p(s(t_bool,V_P)) | ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_predu_u_sets_NOTu_u_EQUALu_u_SETS, axiom, ![TV_u_27a]: ![V_t, V_s]: (~ (s(t_fun(TV_u_27a,t_bool),V_s) = s(t_fun(TV_u_27a,t_bool),V_t)) <=> ?[V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_t)))) <=> ~ (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))))))).
fof(ah4s_bools_EQu_u_CLAUSESu_c2, axiom, ![V_t]: (s(t_bool,f) = s(t_bool,V_t) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_predu_u_sets_INu_u_DELETE, axiom, ![TV_u_27a]: ![V_y, V_x, V_s]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_delete(s(t_fun(TV_u_27a,t_bool),V_s),s(TV_u_27a,V_y)))))) <=> (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) & ~ (s(TV_u_27a,V_x) = s(TV_u_27a,V_y))))).
fof(ah4s_predu_u_sets_SUBSETu_u_DELETE, axiom, ![TV_u_27a]: ![V_x, V_t, V_s]: (p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_delete(s(t_fun(TV_u_27a,t_bool),V_t),s(TV_u_27a,V_x)))))) <=> (~ (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))))) & p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t))))))).
fof(ah4s_bools_IMPu_u_CONG, axiom, ![V_yu_27, V_y, V_xu_27, V_x]: ((s(t_bool,V_x) = s(t_bool,V_xu_27) & (p(s(t_bool,V_xu_27)) => s(t_bool,V_y) = s(t_bool,V_yu_27))) => ((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) <=> (p(s(t_bool,V_xu_27)) => p(s(t_bool,V_yu_27)))))).
fof(ah4s_predu_u_sets_SUBSETu_u_DELETEu_u_BOTH, axiom, ![TV_u_27a]: ![V_x, V_s2, V_s1]: (p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_s1),s(t_fun(TV_u_27a,t_bool),V_s2)))) => p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_delete(s(t_fun(TV_u_27a,t_bool),V_s1),s(TV_u_27a,V_x))),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_delete(s(t_fun(TV_u_27a,t_bool),V_s2),s(TV_u_27a,V_x)))))))).
fof(ah4s_predu_u_sets_INJu_u_SUBSET, axiom, ![TV_u_27a,TV_u_27b]: ![V_t0, V_t, V_s0, V_s, V_f]: ((p(s(t_bool,h4s_predu_u_sets_inj(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27b,t_bool),V_t)))) & (p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_s0),s(t_fun(TV_u_27a,t_bool),V_s)))) & p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27b,t_bool),V_t),s(t_fun(TV_u_27b,t_bool),V_t0)))))) => p(s(t_bool,h4s_predu_u_sets_inj(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),V_s0),s(t_fun(TV_u_27b,t_bool),V_t0)))))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,t0))) <=> p(s(t_bool,t0)))).
fof(ah4s_bools_LEFTu_u_EXISTSu_u_ANDu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (?[V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & p(s(t_bool,V_Q))) <=> (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & p(s(t_bool,V_Q))))).
fof(ah4s_predu_u_sets_pairwiseu_u_SUBSET, axiom, ![TV_u_27a]: ![V_t, V_s, V_R]: ((p(s(t_bool,h4s_predu_u_sets_pairwise(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,t_bool),V_t)))) & p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t))))) => p(s(t_bool,h4s_predu_u_sets_pairwise(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,t_bool),V_s)))))).
fof(ah4s_predu_u_sets_BIGUNIONu_u_SUBSET, axiom, ![TV_u_27a]: ![V_X, V_P]: (p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_bigunion(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P))),s(t_fun(TV_u_27a,t_bool),V_X)))) <=> ![V_Y]: (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_Y),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P)))) => p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_Y),s(t_fun(TV_u_27a,t_bool),V_X))))))).
fof(ah4s_predu_u_sets_SETu_u_EQu_u_SUBSET, axiom, ![TV_u_27a]: ![V_s2, V_s1]: (s(t_fun(TV_u_27a,t_bool),V_s1) = s(t_fun(TV_u_27a,t_bool),V_s2) <=> (p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_s1),s(t_fun(TV_u_27a,t_bool),V_s2)))) & p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_s2),s(t_fun(TV_u_27a,t_bool),V_s1))))))).
fof(ah4s_predu_u_sets_CROSSu_u_SUBSET, axiom, ![TV_u_27a,TV_u_27b]: ![V_Q0, V_Q, V_P0, V_P]: (p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_predu_u_sets_cross(s(t_fun(TV_u_27a,t_bool),V_P0),s(t_fun(TV_u_27b,t_bool),V_Q0))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_predu_u_sets_cross(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27b,t_bool),V_Q)))))) <=> (s(t_fun(TV_u_27a,t_bool),V_P0) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty) | (s(t_fun(TV_u_27b,t_bool),V_Q0) = s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_empty) | (p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_P0),s(t_fun(TV_u_27a,t_bool),V_P)))) & p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27b,t_bool),V_Q0),s(t_fun(TV_u_27b,t_bool),V_Q))))))))).
fof(ah4s_predu_u_sets_INu_u_UNIV, axiom, ![TV_u_27a]: ![V_x]: p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_univ))))).
fof(ah4s_predu_u_sets_RELu_u_RESTRICTu_u_SUBSET, axiom, ![TV_u_27a]: ![V_s2, V_s1, V_R]: (p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_s1),s(t_fun(TV_u_27a,t_bool),V_s2)))) => p(s(t_bool,h4s_relations_rsubset(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_predu_u_sets_relu_u_restrict(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,t_bool),V_s1))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_predu_u_sets_relu_u_restrict(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,t_bool),V_s2)))))))).
fof(ah4s_bools_RIGHTu_u_ORu_u_EXISTSu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: ((p(s(t_bool,V_P)) | ?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> ?[V_x]: (p(s(t_bool,V_P)) | p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_predu_u_sets_SUMu_u_IMAGEu_u_SUBSETu_u_LE, axiom, ![TV_u_27a]: ![V_t, V_s, V_f]: ((p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) & p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_t),s(t_fun(TV_u_27a,t_bool),V_s))))) => p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_predu_u_sets_sumu_u_image(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(t_fun(TV_u_27a,t_bool),V_t))),s(t_h4s_nums_num,h4s_predu_u_sets_sumu_u_image(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(t_fun(TV_u_27a,t_bool),V_s)))))))).
fof(ah4s_predu_u_sets_UNIVu_u_SUBSET, axiom, ![TV_u_27a]: ![V_s]: (p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_univ),s(t_fun(TV_u_27a,t_bool),V_s)))) <=> s(t_fun(TV_u_27a,t_bool),V_s) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_univ))).
fof(ah4s_predu_u_sets_SUBSETu_u_EMPTY, axiom, ![TV_u_27a]: ![V_s]: (p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)))) <=> s(t_fun(TV_u_27a,t_bool),V_s) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty))).
fof(ah4s_bools_CONJu_u_ASSOC, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) & (p(s(t_bool,V_t2)) & p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) & p(s(t_bool,V_t3))))).
fof(ah4s_bools_RIGHTu_u_ANDu_u_FORALLu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: ((p(s(t_bool,V_P)) & ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> ![V_x]: (p(s(t_bool,V_P)) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_SKOLEMu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_P]: (![V_x]: ?[V_y]: p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_P),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))) <=> ?[V_f]: ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_P),s(TV_u_27a,V_x))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x)))))))).
fof(ah4s_bools_NOTu_u_AND, axiom, ![V_t]: ~ ((p(s(t_bool,V_t)) & ~ (p(s(t_bool,V_t)))))).
fof(ah4s_primu_u_recs_measureu_u_def, axiom, ![TV_u_27a]: s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_primu_u_recs_measure) = s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_relations_invu_u_image(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c)))).
fof(ah4s_relations_invu_u_imageu_u_def, axiom, ![TV_u_27b,TV_u_27a]: ![V_f, V_R, V_x, V_xi_]: s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_relations_invu_u_image(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R))),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(TV_u_27a,V_x))),s(TV_u_27a,V_xi_))) = s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_xi_)))))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,t0))) <=> p(s(t_bool,V_t)))).
fof(ah4s_primu_u_recs_WFu_u_measure, axiom, ![TV_u_27a]: ![V_m]: p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_primu_u_recs_measure),s(t_fun(TV_u_27a,t_h4s_nums_num),V_m))))))).
fof(ah4s_relations_WFu_u_INDUCTIONu_u_THM, axiom, ![TV_u_27a]: ![V_R]: (p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) => ![V_P]: (![V_x]: (![V_y]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_y))),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_y))))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) => ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t0)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t0)))).
fof(ah4s_predu_u_sets_RESTu_u_SUBSET, axiom, ![TV_u_27a]: ![V_s]: p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_rest(s(t_fun(TV_u_27a,t_bool),V_s))),s(t_fun(TV_u_27a,t_bool),V_s))))).
fof(ah4s_predu_u_sets_RESTu_u_DEF, axiom, ![TV_u_27a]: ![V_s]: s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_rest(s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_delete(s(t_fun(TV_u_27a,t_bool),V_s),s(TV_u_27a,h4s_predu_u_sets_choice(s(t_fun(TV_u_27a,t_bool),V_s)))))).
fof(ah4s_predu_u_sets_CHOICEu_u_DEF, axiom, ![TV_u_27a]: ![V_s]: (~ (s(t_fun(TV_u_27a,t_bool),V_s) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)) => p(s(t_bool,h4s_bools_in(s(TV_u_27a,h4s_predu_u_sets_choice(s(t_fun(TV_u_27a,t_bool),V_s))),s(t_fun(TV_u_27a,t_bool),V_s)))))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) & p(s(t_bool,V_t))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_NOTu_u_IMP, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) => p(s(t_bool,V_B)))) <=> (p(s(t_bool,V_A)) & ~ (p(s(t_bool,V_B)))))).
fof(ah4s_predu_u_sets_INJu_u_DEF, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_s, V_f]: (p(s(t_bool,h4s_predu_u_sets_inj(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27b,t_bool),V_t)))) <=> (![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) => p(s(t_bool,h4s_bools_in(s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))),s(t_fun(TV_u_27b,t_bool),V_t))))) & ![V_x, V_y]: ((p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) & p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_y),s(t_fun(TV_u_27a,t_bool),V_s))))) => (s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_y))) => s(TV_u_27a,V_x) = s(TV_u_27a,V_y)))))).
fof(ah4s_predu_u_sets_pairwiseu_u_def, axiom, ![TV_u_27a]: ![V_s, V_P]: (p(s(t_bool,h4s_predu_u_sets_pairwise(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_P),s(t_fun(TV_u_27a,t_bool),V_s)))) <=> ![V_e1, V_e2]: ((p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_e1),s(t_fun(TV_u_27a,t_bool),V_s)))) & p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_e2),s(t_fun(TV_u_27a,t_bool),V_s))))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_P),s(TV_u_27a,V_e1))),s(TV_u_27a,V_e2))))))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,f))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_RIGHTu_u_EXISTSu_u_ANDu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (?[V_x]: (p(s(t_bool,V_P)) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> (p(s(t_bool,V_P)) & ?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_relations_RSUBSET0, axiom, ![TV_u_27a,TV_u_27b]: ![V_R2, V_R1]: (p(s(t_bool,h4s_relations_rsubset(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R1),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R2)))) <=> ![V_x, V_y]: (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R1),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))) => p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R2),s(TV_u_27a,V_x))),s(TV_u_27b,V_y))))))).
fof(ah4s_predu_u_sets_RELu_u_RESTRICTu_u_DEF, axiom, ![TV_u_27a]: ![V_y, V_x, V_s, V_R]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_predu_u_sets_relu_u_restrict(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,t_bool),V_s))),s(TV_u_27a,V_x))),s(TV_u_27a,V_y)))) <=> (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) & (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_y),s(t_fun(TV_u_27a,t_bool),V_s)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x))),s(TV_u_27a,V_y)))))))).
fof(ah4s_bools_NOTu_u_EXISTSu_u_THM, axiom, ![TV_u_27a]: ![V_P]: (~ (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) <=> ![V_x]: ~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_LEFTu_u_FORALLu_u_ORu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) | p(s(t_bool,V_Q))) <=> (![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) | p(s(t_bool,V_Q))))).
fof(ah4s_predu_u_sets_INu_u_BIGUNION, axiom, ![TV_u_27a]: ![V_x, V_sos]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_bigunion(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_sos)))))) <=> ?[V_s]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) & p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_sos))))))).
fof(ah4s_predu_u_sets_NOTu_u_INu_u_EMPTY, axiom, ![TV_u_27a]: ![V_x]: ~ (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)))))).
fof(ah4s_bools_EXISTSu_u_ORu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (?[V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) | p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) | ?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_LEFTu_u_ORu_u_EXISTSu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: ((?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) | p(s(t_bool,V_Q))) <=> ?[V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) | p(s(t_bool,V_Q))))).
fof(ah4s_pairs_SND0, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x]: s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27b,V_y)).
fof(ah4s_predu_u_sets_INu_u_CROSS, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_Q, V_P]: (p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_predu_u_sets_cross(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27b,t_bool),V_Q)))))) <=> (p(s(t_bool,h4s_bools_in(s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))),s(t_fun(TV_u_27a,t_bool),V_P)))) & p(s(t_bool,h4s_bools_in(s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))),s(t_fun(TV_u_27b,t_bool),V_Q))))))).
fof(ah4s_pairs_FORALLu_u_PROD, axiom, ![TV_u_27a,TV_u_27b]: ![V_P]: (![V_p]: p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),V_P),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p)))) <=> ![V_pu_u_1, V_pu_u_2]: p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),V_P),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_pu_u_1),s(TV_u_27b,V_pu_u_2)))))))).
fof(ah4s_pairs_FST0, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x]: s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27a,V_x)).
fof(ah4s_predu_u_sets_FINITEu_u_DELETE, axiom, ![TV_u_27a]: ![V_x, V_s]: s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_delete(s(t_fun(TV_u_27a,t_bool),V_s),s(TV_u_27a,V_x))))) = s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))).
fof(ah4s_predu_u_sets_FINITEu_u_INSERT, axiom, ![TV_u_27a]: ![V_x, V_s]: s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))))) = s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))).
fof(ah4s_predu_u_sets_FINITEu_u_INDUCT, axiom, ![TV_u_27a]: ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)))) & ![V_s]: ((p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) & p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P),s(t_fun(TV_u_27a,t_bool),V_s))))) => ![V_e]: (~ (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_e),s(t_fun(TV_u_27a,t_bool),V_s))))) => p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_e),s(t_fun(TV_u_27a,t_bool),V_s))))))))) => ![V_s]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) => p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P),s(t_fun(TV_u_27a,t_bool),V_s))))))).
fof(ah4s_predu_u_sets_SUBSETu_u_INSERTu_u_DELETE, axiom, ![TV_u_27a]: ![V_x, V_t, V_s]: s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_t))))) = s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_delete(s(t_fun(TV_u_27a,t_bool),V_s),s(TV_u_27a,V_x))),s(t_fun(TV_u_27a,t_bool),V_t)))).
fof(ah4s_predu_u_sets_DELETEu_u_NONu_u_ELEMENT, axiom, ![TV_u_27a]: ![V_x, V_s]: (~ (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))))) <=> s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_delete(s(t_fun(TV_u_27a,t_bool),V_s),s(TV_u_27a,V_x))) = s(t_fun(TV_u_27a,t_bool),V_s))).
fof(ah4s_predu_u_sets_INSERTu_u_DELETE, axiom, ![TV_u_27a]: ![V_x, V_s]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) => s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_delete(s(t_fun(TV_u_27a,t_bool),V_s),s(TV_u_27a,V_x))))) = s(t_fun(TV_u_27a,t_bool),V_s))).
fof(ah4s_predu_u_sets_SUBSETu_u_INSERT, axiom, ![TV_u_27a]: ![V_x, V_s]: (~ (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))))) => ![V_t]: s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_t))))) = s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t))))).
fof(ah4s_numerals_numeralu_u_lteu_c1, axiom, ![V_n]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_zero))) = s(t_bool,f)).
fof(ah4s_numerals_numeralu_u_distribu_c27, axiom, ![V_n]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_nums_0))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_zero)))).
fof(ah4s_numerals_numeralu_u_distribu_c1, axiom, ![V_n]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,V_n)).
fof(ah4s_arithmetics_SUCu_u_ONEu_u_ADD, axiom, ![V_n]: s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_n)))).
fof(ch4s_predu_u_sets_SUBSETu_u_PSUBSETu_u_TRANS, conjecture, ![TV_u_27a]: ![V_u, V_t, V_s]: ((p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t)))) & p(s(t_bool,h4s_predu_u_sets_psubset(s(t_fun(TV_u_27a,t_bool),V_t),s(t_fun(TV_u_27a,t_bool),V_u))))) => p(s(t_bool,h4s_predu_u_sets_psubset(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_u)))))).
