%   ORIGINAL: h4/pred__set/REL__RESTRICT__SUBSET
% 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/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/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/TRUTH: T
% Assm: h4/pred__set/REL__RESTRICT__EMPTY: !R. h4/pred__set/REL__RESTRICT R h4/pred__set/EMPTY = h4/relation/EMPTY__REL
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/relation/RSUBSET__WeakOrder: h4/relation/WeakOrder h4/relation/RSUBSET
% Assm: h4/relation/RSUBSET0: !R2 R1. h4/relation/RSUBSET R1 R2 <=> (!x y. R1 x y ==> R2 x y)
% Assm: h4/bool/FUN__EQ__THM: !g f. f = g <=> (!x. f x = g x)
% Assm: h4/pred__set/EXTENSION: !t s. s = t <=> (!x. h4/bool/IN x s <=> h4/bool/IN x t)
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/relation/EMPTY__REL__DEF: !y x. h4/relation/EMPTY__REL x y <=> F
% Assm: h4/bool/EQ__CLAUSES_c2: !t. (F <=> t) <=> ~t
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> 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/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/pred__set/NOT__IN__EMPTY: !x. ~h4/bool/IN x h4/pred__set/EMPTY
% Assm: h4/bool/AND__CLAUSES_c2: !t. F /\ t <=> F
% Assm: h4/relation/RSUBSET__ANTISYM: !R2 R1. h4/relation/RSUBSET R1 R2 /\ h4/relation/RSUBSET R2 R1 ==> R1 = R2
% Assm: h4/relation/RSUBSET__antisymmetric: h4/relation/antisymmetric h4/relation/RSUBSET
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/pred__set/IN__INTER: !x t s. h4/bool/IN x (h4/pred__set/INTER s t) <=> h4/bool/IN x s /\ h4/bool/IN x t
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% 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/pred__set/PSUBSET__DEF: !t s. h4/pred__set/PSUBSET s t <=> h4/pred__set/SUBSET s t /\ ~(s = t)
% Assm: h4/pred__set/IN__INSERT: !y x s. h4/bool/IN x (h4/pred__set/INSERT y s) <=> x = y \/ h4/bool/IN x s
% 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/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% 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/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/relation/antisymmetric__def: !R. h4/relation/antisymmetric R <=> (!x y. R x y /\ R y x ==> x = y)
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% 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/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__UNION__ABSORPTION: !t s. h4/pred__set/SUBSET s t <=> h4/pred__set/UNION s t = t
% Assm: h4/pred__set/UNION__SUBSET: !u t s. h4/pred__set/SUBSET (h4/pred__set/UNION s t) u <=> h4/pred__set/SUBSET s u /\ h4/pred__set/SUBSET t u
% Assm: h4/pred__set/SUBSET__INTER__ABSORPTION: !t s. h4/pred__set/SUBSET s t <=> h4/pred__set/INTER s t = s
% Assm: h4/pred__set/SUBSET__INTER: !u t s. h4/pred__set/SUBSET s (h4/pred__set/INTER t u) <=> h4/pred__set/SUBSET s t /\ h4/pred__set/SUBSET s u
% Assm: h4/bool/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/bool/DISJ__SYM: !B A. A \/ B <=> B \/ A
% Assm: h4/pred__set/SUBSET__ANTISYM: !t s. h4/pred__set/SUBSET s t /\ h4/pred__set/SUBSET t s ==> s = 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/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/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/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/pred__set/SUBSET__FINITE__I: !t s. h4/pred__set/FINITE s /\ h4/pred__set/SUBSET t s ==> h4/pred__set/FINITE t
% Assm: h4/pred__set/DISJOINT__SUBSET: !u t s. h4/pred__set/DISJOINT s t /\ h4/pred__set/SUBSET u t ==> h4/pred__set/DISJOINT s u
% Assm: h4/pred__set/SUBSET__REFL: !s. h4/pred__set/SUBSET s s
% Assm: h4/pred__set/UNION__DIFF: !t s. h4/pred__set/SUBSET s t ==> h4/pred__set/UNION s (h4/pred__set/DIFF t s) = t /\ h4/pred__set/UNION (h4/pred__set/DIFF t s) s = t
% Assm: h4/relation/WeakOrder__EQ: !R. h4/relation/WeakOrder R ==> (!y z. y = z <=> R y z /\ R z y)
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/pred__set/IN__UNION: !x t s. h4/bool/IN x (h4/pred__set/UNION s t) <=> h4/bool/IN x s \/ h4/bool/IN x t
% Assm: h4/relation/REMPTY__SUBSET_c1: !R. h4/relation/RSUBSET R h4/relation/EMPTY__REL <=> R = h4/relation/EMPTY__REL
% Assm: h4/relation/irreflexive__RSUBSET: !R2 R1. h4/relation/irreflexive R2 /\ h4/relation/RSUBSET R1 R2 ==> h4/relation/irreflexive R1
% Assm: h4/relation/REMPTY__SUBSET_c0: !R. h4/relation/RSUBSET h4/relation/EMPTY__REL R
% Assm: h4/relation/symmetric__inv__RSUBSET: !R. h4/relation/symmetric R <=> h4/relation/RSUBSET (h4/relation/inv R) R
% Assm: h4/relation/transitive__O__RSUBSET: !R. h4/relation/transitive R <=> h4/relation/RSUBSET (h4/relation/O R R) R
% Assm: h4/relation/reflexive__Id__RSUBSET: !R. h4/relation/reflexive R <=> h4/relation/RSUBSET $equals R
% Assm: h4/relation/RUNIV__SUBSET_c0: !R. h4/relation/RSUBSET h4/relation/RUNIV R <=> R = h4/relation/RUNIV
% Assm: h4/relation/RUNIV__SUBSET_c1: !R. h4/relation/RSUBSET R h4/relation/RUNIV
% Assm: h4/relation/O__MONO: !S2 S1 R2 R1. h4/relation/RSUBSET R1 R2 /\ h4/relation/RSUBSET S1 S2 ==> h4/relation/RSUBSET (h4/relation/O R1 S1) (h4/relation/O R2 S2)
% Assm: h4/relation/EqIsBothRSUBSET: !z y. y = z <=> h4/relation/RSUBSET y z /\ h4/relation/RSUBSET z y
% Assm: h4/bool/OR__CLAUSES_c1: !t. t \/ T <=> T
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/relation/reflexive__def: !R. h4/relation/reflexive R <=> (!x. R x x)
% Assm: h4/bool/NOT__FORALL__THM: !P. ~(!x. P x) <=> (?x. ~P x)
% Assm: h4/bool/EQ__CLAUSES_c0: !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/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/bool/DISJ__ASSOC: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm: h4/bool/DE__MORGAN__THM_c1: !B A. ~(A \/ B) <=> ~A /\ ~B
% 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/FINITE__EMPTY: h4/pred__set/FINITE h4/pred__set/EMPTY
% 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__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/SUBSET__EMPTY: !s. h4/pred__set/SUBSET s h4/pred__set/EMPTY <=> s = h4/pred__set/EMPTY
% 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/relation/transitive__def: !R. h4/relation/transitive R <=> (!x y z. R x y /\ R y z ==> R x z)
% Assm: h4/bool/EQ__CLAUSES_c3: !t. (t <=> F) <=> ~t
% 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/RIGHT__OR__EXISTS__THM: !Q P. P \/ (?x. Q x) <=> (?x. P \/ Q x)
% Assm: h4/bool/RIGHT__FORALL__OR__THM: !Q P. (!x. P \/ Q x) <=> P \/ (!x. Q x)
% Assm: h4/bool/SKOLEM__THM: !P. (!x. ?y. P x y) <=> (?f. !x. P x (f x))
% Assm: h4/relation/WeakOrder0: !Z. h4/relation/WeakOrder Z <=> h4/relation/reflexive Z /\ h4/relation/antisymmetric Z /\ h4/relation/transitive Z
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/bool/RIGHT__EXISTS__AND__THM: !Q P. (?x. P /\ Q x) <=> P /\ (?x. Q x)
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> t
% Assm: h4/bool/LEFT__FORALL__OR__THM: !Q P. (!x. P x \/ Q) <=> (!x. P x) \/ Q
% Assm: h4/bool/IMP__DISJ__THM: !B A. A ==> B <=> ~A \/ B
% Assm: h4/bool/UNWIND__FORALL__THM2: !v f. (!x. x = v ==> f x) <=> f v
% Assm: h4/bool/AND__CLAUSES_c4: !t. t /\ t <=> t
% Assm: h4/bool/BOOL__CASES__AX: !t. (t <=> T) \/ (t <=> F)
% Assm: h4/bool/IMP__CLAUSES_c2: !t. F ==> t <=> T
% Assm: h4/pred__set/DELETE__SUBSET: !x s. h4/pred__set/SUBSET (h4/pred__set/DELETE s x) s
% Assm: h4/bool/DE__MORGAN__THM_c0: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm: h4/pred__set/SUBSET__UNIV: !s. h4/pred__set/SUBSET s h4/pred__set/UNIV
% Assm: h4/pred__set/SUBSET__UNION_c0: !t s. h4/pred__set/SUBSET s (h4/pred__set/UNION s t)
% Assm: h4/pred__set/UNIV__SUBSET: !s. h4/pred__set/SUBSET h4/pred__set/UNIV s <=> s = h4/pred__set/UNIV
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/pred__set/INTER__SUBSET_c0: !t s. h4/pred__set/SUBSET (h4/pred__set/INTER s 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/relation/RUNIV0: !y x. h4/relation/RUNIV x y <=> T
% Assm: h4/pred__set/IN__UNIV: !x. h4/bool/IN x h4/pred__set/UNIV
% Assm: h4/bool/NOT__EXISTS__THM: !P. ~(?x. P x) <=> (!x. ~P x)
% Assm: h4/relation/O__DEF: !z x R2 R1. h4/relation/O R1 R2 x z <=> (?y. R2 x y /\ R1 y z)
% Assm: h4/bool/EXISTS__OR__THM: !Q P. (?x. P x \/ Q x) <=> (?x. P x) \/ (?x. Q x)
% Assm: h4/pred__set/DISJOINT__DEF: !t s. h4/pred__set/DISJOINT s t <=> h4/pred__set/INTER s t = h4/pred__set/EMPTY
% Assm: h4/pred__set/IN__DIFF: !x t s. h4/bool/IN x (h4/pred__set/DIFF s t) <=> h4/bool/IN x s /\ ~h4/bool/IN x t
% Assm: h4/relation/irreflexive__def: !R. h4/relation/irreflexive R <=> (!x. ~R x x)
% Assm: h4/relation/inv__DEF: !y x R. h4/relation/inv R x y <=> R y x
% Assm: h4/relation/symmetric__def: !R. h4/relation/symmetric R <=> (!x y. R x y <=> R y x)
% Assm: h4/bool/OR__CLAUSES_c0: !t. T \/ t <=> T
% Assm: h4/pred__set/NOT__EQUAL__SETS: !t s. ~(s = t) <=> (?x. h4/bool/IN x t <=> ~h4/bool/IN x s)
% Goal: !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)
%   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_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_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_TRUTH]: T
% Assm [h4s_predu_u_sets_RELu_u_RESTRICTu_u_EMPTY]: !R. h4/pred__set/REL__RESTRICT R h4/pred__set/EMPTY = h4/relation/EMPTY__REL
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_relations_RSUBSETu_u_WeakOrder]: h4/relation/WeakOrder h4/relation/RSUBSET
% Assm [h4s_relations_RSUBSET0]: !R2 R1. happ (happ h4/relation/RSUBSET R1) R2 <=> (!x y. happ (happ R1 x) y ==> happ (happ R2 x) y)
% Assm [h4s_bools_FUNu_u_EQu_u_THM]: !g f. f = g <=> (!x. happ f x = happ g x)
% Assm [h4s_predu_u_sets_EXTENSION]: !t s. s = t <=> (!x. h4/bool/IN x s <=> h4/bool/IN x t)
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_relations_EMPTYu_u_RELu_u_DEF]: !y x. happ (happ h4/relation/EMPTY__REL x) y <=> F
% Assm [h4s_bools_EQu_u_CLAUSESu_c2]: !t. (F <=> t) <=> ~t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> 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_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_predu_u_sets_NOTu_u_INu_u_EMPTY]: !x. ~h4/bool/IN x h4/pred__set/EMPTY
% Assm [h4s_bools_ANDu_u_CLAUSESu_c2]: !t. F /\ t <=> F
% Assm [h4s_relations_RSUBSETu_u_ANTISYM]: !R2 R1. happ (happ h4/relation/RSUBSET R1) R2 /\ happ (happ h4/relation/RSUBSET R2) R1 ==> R1 = R2
% Assm [h4s_relations_RSUBSETu_u_antisymmetric]: h4/relation/antisymmetric h4/relation/RSUBSET
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_predu_u_sets_INu_u_INTER]: !x t s. h4/bool/IN x (h4/pred__set/INTER s t) <=> h4/bool/IN x s /\ h4/bool/IN x t
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% 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_predu_u_sets_PSUBSETu_u_DEF]: !t s. h4/pred__set/PSUBSET s t <=> h4/pred__set/SUBSET s t /\ ~(s = t)
% Assm [h4s_predu_u_sets_INu_u_INSERT]: !y x s. h4/bool/IN x (h4/pred__set/INSERT y s) <=> x = y \/ h4/bool/IN x s
% 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_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% 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_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% 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_relations_antisymmetricu_u_def]: !R. h4/relation/antisymmetric R <=> (!x y. happ (happ R x) y /\ happ (happ R y) x ==> x = y)
% Assm [h4s_bools_Fu_u_IMP]: !t. ~t ==> t ==> F
% 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_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_UNIONu_u_ABSORPTION]: !t s. h4/pred__set/SUBSET s t <=> h4/pred__set/UNION s t = t
% Assm [h4s_predu_u_sets_UNIONu_u_SUBSET]: !u t s. h4/pred__set/SUBSET (h4/pred__set/UNION s t) u <=> h4/pred__set/SUBSET s u /\ h4/pred__set/SUBSET t u
% Assm [h4s_predu_u_sets_SUBSETu_u_INTERu_u_ABSORPTION]: !t s. h4/pred__set/SUBSET s t <=> h4/pred__set/INTER s t = s
% Assm [h4s_predu_u_sets_SUBSETu_u_INTER]: !u t s. h4/pred__set/SUBSET s (h4/pred__set/INTER t u) <=> h4/pred__set/SUBSET s t /\ h4/pred__set/SUBSET s u
% Assm [h4s_bools_IMPu_u_F]: !t. (t ==> F) ==> ~t
% Assm [h4s_bools_DISJu_u_SYM]: !B A. A \/ B <=> B \/ A
% 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_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_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_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_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_predu_u_sets_SUBSETu_u_FINITEu_u_I]: !t s. h4/pred__set/FINITE s /\ h4/pred__set/SUBSET t s ==> h4/pred__set/FINITE t
% Assm [h4s_predu_u_sets_DISJOINTu_u_SUBSET]: !u t s. h4/pred__set/DISJOINT s t /\ h4/pred__set/SUBSET u t ==> h4/pred__set/DISJOINT s u
% Assm [h4s_predu_u_sets_SUBSETu_u_REFL]: !s. h4/pred__set/SUBSET s s
% Assm [h4s_predu_u_sets_UNIONu_u_DIFF]: !t s. h4/pred__set/SUBSET s t ==> h4/pred__set/UNION s (h4/pred__set/DIFF t s) = t /\ h4/pred__set/UNION (h4/pred__set/DIFF t s) s = t
% Assm [h4s_relations_WeakOrderu_u_EQ]: !R. h4/relation/WeakOrder R ==> (!y z. y = z <=> happ (happ R y) z /\ happ (happ R z) y)
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_predu_u_sets_INu_u_UNION]: !x t s. h4/bool/IN x (h4/pred__set/UNION s t) <=> h4/bool/IN x s \/ h4/bool/IN x t
% Assm [h4s_relations_REMPTYu_u_SUBSETu_c1]: !R. happ (happ h4/relation/RSUBSET R) h4/relation/EMPTY__REL <=> R = h4/relation/EMPTY__REL
% Assm [h4s_relations_irreflexiveu_u_RSUBSET]: !R2 R1. h4/relation/irreflexive R2 /\ happ (happ h4/relation/RSUBSET R1) R2 ==> h4/relation/irreflexive R1
% Assm [h4s_relations_REMPTYu_u_SUBSETu_c0]: !R. happ (happ h4/relation/RSUBSET h4/relation/EMPTY__REL) R
% Assm [h4s_relations_symmetricu_u_invu_u_RSUBSET]: !R. h4/relation/symmetric R <=> happ (happ h4/relation/RSUBSET (h4/relation/inv R)) R
% Assm [h4s_relations_transitiveu_u_Ou_u_RSUBSET]: !R. h4/relation/transitive R <=> happ (happ h4/relation/RSUBSET (h4/relation/O R R)) R
% Assm [h4s_relations_reflexiveu_u_Idu_u_RSUBSET]: !R. h4/relation/reflexive R <=> happ (happ h4/relation/RSUBSET $equals) R
% Assm [h4s_relations_RUNIVu_u_SUBSETu_c0]: !R. happ (happ h4/relation/RSUBSET h4/relation/RUNIV) R <=> R = h4/relation/RUNIV
% Assm [h4s_relations_RUNIVu_u_SUBSETu_c1]: !R. happ (happ h4/relation/RSUBSET R) h4/relation/RUNIV
% Assm [h4s_relations_Ou_u_MONO]: !S2 S1 R2 R1. happ (happ h4/relation/RSUBSET R1) R2 /\ happ (happ h4/relation/RSUBSET S1) S2 ==> happ (happ h4/relation/RSUBSET (h4/relation/O R1 S1)) (h4/relation/O R2 S2)
% Assm [h4s_relations_EqIsBothRSUBSET]: !z y. y = z <=> happ (happ h4/relation/RSUBSET y) z /\ happ (happ h4/relation/RSUBSET z) y
% Assm [h4s_bools_ORu_u_CLAUSESu_c1]: !t. t \/ T <=> T
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_relations_reflexiveu_u_def]: !R. h4/relation/reflexive R <=> (!x. happ (happ R x) x)
% Assm [h4s_bools_NOTu_u_FORALLu_u_THM]: !P. ~(!x. happ P x) <=> (?x. ~happ P x)
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !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_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_bools_DISJu_u_ASSOC]: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c1]: !B A. ~(A \/ B) <=> ~A /\ ~B
% 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_FINITEu_u_EMPTY]: h4/pred__set/FINITE h4/pred__set/EMPTY
% 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_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_SUBSETu_u_EMPTY]: !s. h4/pred__set/SUBSET s h4/pred__set/EMPTY <=> s = h4/pred__set/EMPTY
% 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_relations_transitiveu_u_def]: !R. h4/relation/transitive R <=> (!x y z. happ (happ R x) y /\ happ (happ R y) z ==> happ (happ R x) z)
% Assm [h4s_bools_EQu_u_CLAUSESu_c3]: !t. (t <=> F) <=> ~t
% 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_RIGHTu_u_ORu_u_EXISTSu_u_THM]: !Q P. P \/ (?x. happ Q x) <=> (?x. P \/ happ Q x)
% Assm [h4s_bools_RIGHTu_u_FORALLu_u_ORu_u_THM]: !Q P. (!x. P \/ happ Q x) <=> P \/ (!x. 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_relations_WeakOrder0]: !Z. h4/relation/WeakOrder Z <=> h4/relation/reflexive Z /\ h4/relation/antisymmetric Z /\ h4/relation/transitive Z
% Assm [h4s_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% Assm [h4s_bools_RIGHTu_u_EXISTSu_u_ANDu_u_THM]: !Q P. (?x. P /\ happ Q x) <=> P /\ (?x. happ Q x)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% Assm [h4s_bools_LEFTu_u_FORALLu_u_ORu_u_THM]: !Q P. (!x. happ P x \/ Q) <=> (!x. happ P x) \/ Q
% Assm [h4s_bools_IMPu_u_DISJu_u_THM]: !B A. A ==> B <=> ~A \/ B
% Assm [h4s_bools_UNWINDu_u_FORALLu_u_THM2]: !v f. (!x. x = v ==> happ f x) <=> happ f v
% Assm [h4s_bools_ANDu_u_CLAUSESu_c4]: !t. t /\ t <=> t
% Assm [h4s_bools_BOOLu_u_CASESu_u_AX]: !t. (t <=> T) \/ (t <=> F)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c2]: !t. F ==> t <=> T
% Assm [h4s_predu_u_sets_DELETEu_u_SUBSET]: !x s. h4/pred__set/SUBSET (h4/pred__set/DELETE s x) s
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c0]: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm [h4s_predu_u_sets_SUBSETu_u_UNIV]: !s. h4/pred__set/SUBSET s h4/pred__set/UNIV
% Assm [h4s_predu_u_sets_SUBSETu_u_UNIONu_c0]: !t s. h4/pred__set/SUBSET s (h4/pred__set/UNION s t)
% 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_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_predu_u_sets_INTERu_u_SUBSETu_c0]: !t s. h4/pred__set/SUBSET (h4/pred__set/INTER s 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_relations_RUNIV0]: !y x. happ (happ h4/relation/RUNIV x) y <=> T
% Assm [h4s_predu_u_sets_INu_u_UNIV]: !x. h4/bool/IN x h4/pred__set/UNIV
% Assm [h4s_bools_NOTu_u_EXISTSu_u_THM]: !P. ~(?x. happ P x) <=> (!x. ~happ P x)
% Assm [h4s_relations_Ou_u_DEF]: !z x R2 R1. happ (happ (h4/relation/O R1 R2) x) z <=> (?y. happ (happ R2 x) y /\ happ (happ R1 y) z)
% 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_predu_u_sets_DISJOINTu_u_DEF]: !t s. h4/pred__set/DISJOINT s t <=> h4/pred__set/INTER s t = h4/pred__set/EMPTY
% Assm [h4s_predu_u_sets_INu_u_DIFF]: !x t s. h4/bool/IN x (h4/pred__set/DIFF s t) <=> h4/bool/IN x s /\ ~h4/bool/IN x t
% Assm [h4s_relations_irreflexiveu_u_def]: !R. h4/relation/irreflexive R <=> (!x. ~happ (happ R x) x)
% Assm [h4s_relations_invu_u_DEF]: !y x R. happ (happ (h4/relation/inv R) x) y <=> happ (happ R y) x
% Assm [h4s_relations_symmetricu_u_def]: !R. h4/relation/symmetric R <=> (!x y. happ (happ R x) y <=> happ (happ R y) x)
% Assm [h4s_bools_ORu_u_CLAUSESu_c0]: !t. T \/ t <=> T
% 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)
% Goal: !s2 s1 R. h4/pred__set/SUBSET s1 s2 ==> happ (happ h4/relation/RSUBSET (h4/pred__set/REL__RESTRICT R s1)) (h4/pred__set/REL__RESTRICT R s2)
fof(aHLu_TRUTH, axiom, p(s(t_bool,t))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f))).
fof(aHLu_EXT, axiom, ![TV_Q1238922,TV_Q1238918]: ![V_f, V_g]: (![V_x]: s(TV_Q1238918,happ(s(t_fun(TV_Q1238922,TV_Q1238918),V_f),s(TV_Q1238922,V_x))) = s(TV_Q1238918,happ(s(t_fun(TV_Q1238922,TV_Q1238918),V_g),s(TV_Q1238922,V_x))) => s(t_fun(TV_Q1238922,TV_Q1238918),V_f) = s(t_fun(TV_Q1238922,TV_Q1238918),V_g))).
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_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_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_predu_u_sets_RELu_u_RESTRICTu_u_EMPTY, axiom, ![TV_u_27a]: ![V_R]: 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),h4s_predu_u_sets_empty))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_emptyu_u_rel)).
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_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_bools_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_relations_RSUBSETu_u_WeakOrder, axiom, ![TV_u_27a,TV_u_27b]: p(s(t_bool,h4s_relations_weakorder(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),t_bool)),h4s_relations_rsubset))))).
fof(ah4s_relations_RSUBSET0, axiom, ![TV_u_27a,TV_u_27b]: ![V_R2, V_R1]: (p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),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_bools_FUNu_u_EQu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_g, V_f]: (s(t_fun(TV_u_27a,TV_u_27b),V_f) = s(t_fun(TV_u_27a,TV_u_27b),V_g) <=> ![V_x]: 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_g),s(TV_u_27a,V_x))))).
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_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_relations_EMPTYu_u_RELu_u_DEF, axiom, ![TV_u_27a]: ![V_y, V_x]: 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_relations_emptyu_u_rel),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))) = s(t_bool,f)).
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_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,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_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_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_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_relations_RSUBSETu_u_ANTISYM, axiom, ![TV_u_27a,TV_u_27b]: ![V_R2, V_R1]: ((p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),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)))) & p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),t_bool)),h4s_relations_rsubset),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R2))),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_R1) = s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R2))).
fof(ah4s_relations_RSUBSETu_u_antisymmetric, axiom, ![TV_u_27a,TV_u_27b]: p(s(t_bool,h4s_relations_antisymmetric(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),t_bool)),h4s_relations_rsubset))))).
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_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_predu_u_sets_INu_u_INTER, axiom, ![TV_u_27a]: ![V_x, V_t, 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_inter(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_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_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,t)))).
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_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_predu_u_sets_INu_u_INSERT, 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_insert(s(TV_u_27a,V_y),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))))))).
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_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_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_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_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_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_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_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_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_relations_antisymmetricu_u_def, axiom, ![TV_u_27a]: ![V_R]: (p(s(t_bool,h4s_relations_antisymmetric(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) <=> ![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_x))),s(TV_u_27a,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))))) => s(TV_u_27a,V_x) = s(TV_u_27a,V_y)))).
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_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_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_UNIONu_u_ABSORPTION, 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)))) <=> s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_union(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_t))).
fof(ah4s_predu_u_sets_UNIONu_u_SUBSET, 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),h4s_predu_u_sets_union(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_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)))) & 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))))))).
fof(ah4s_predu_u_sets_SUBSETu_u_INTERu_u_ABSORPTION, 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)))) <=> s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_inter(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))).
fof(ah4s_predu_u_sets_SUBSETu_u_INTER, 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),h4s_predu_u_sets_inter(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_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_u))))))).
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_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_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_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_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_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_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_predu_u_sets_SUBSETu_u_FINITEu_u_I, axiom, ![TV_u_27a]: ![V_t, 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_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_DISJOINTu_u_SUBSET, axiom, ![TV_u_27a]: ![V_u, V_t, V_s]: ((p(s(t_bool,h4s_predu_u_sets_disjoint(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_u),s(t_fun(TV_u_27a,t_bool),V_t))))) => p(s(t_bool,h4s_predu_u_sets_disjoint(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_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_UNIONu_u_DIFF, 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)))) => (s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_union(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_diff(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) & s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_union(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_diff(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_relations_WeakOrderu_u_EQ, axiom, ![TV_u_27a]: ![V_R]: (p(s(t_bool,h4s_relations_weakorder(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) => ![V_y, V_z]: (s(TV_u_27a,V_y) = s(TV_u_27a,V_z) <=> (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_z)))) & 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_z))),s(TV_u_27a,V_y)))))))).
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_bools_IMPu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,t))) <=> p(s(t_bool,t)))).
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,t)))).
fof(ah4s_predu_u_sets_INu_u_UNION, axiom, ![TV_u_27a]: ![V_x, V_t, 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_union(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_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_relations_REMPTYu_u_SUBSETu_c1, axiom, ![TV_u_27a]: ![V_R]: (p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_bool)),h4s_relations_rsubset),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_emptyu_u_rel)))) <=> s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_emptyu_u_rel))).
fof(ah4s_relations_irreflexiveu_u_RSUBSET, axiom, ![TV_u_27a]: ![V_R2, V_R1]: ((p(s(t_bool,h4s_relations_irreflexive(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R2)))) & p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_bool)),h4s_relations_rsubset),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R2))))) => p(s(t_bool,h4s_relations_irreflexive(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1)))))).
fof(ah4s_relations_REMPTYu_u_SUBSETu_c0, axiom, ![TV_u_27a]: ![V_R]: p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_bool)),h4s_relations_rsubset),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_emptyu_u_rel))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))))).
fof(ah4s_relations_symmetricu_u_invu_u_RSUBSET, axiom, ![TV_u_27a]: ![V_R]: s(t_bool,h4s_relations_symmetric(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))) = s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_bool)),h4s_relations_rsubset),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_inv(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))).
fof(ah4s_relations_transitiveu_u_Ou_u_RSUBSET, axiom, ![TV_u_27a]: ![V_R]: s(t_bool,h4s_relations_transitive(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))) = s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_bool)),h4s_relations_rsubset),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_o(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))).
fof(ah4s_relations_reflexiveu_u_Idu_u_RSUBSET, axiom, ![TV_u_27a]: ![V_R]: s(t_bool,h4s_relations_reflexive(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))) = s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_bool)),h4s_relations_rsubset),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),d_equals))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))).
fof(ah4s_relations_RUNIVu_u_SUBSETu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_R]: (p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),t_bool)),h4s_relations_rsubset),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),h4s_relations_runiv))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R)))) <=> s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R) = s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),h4s_relations_runiv))).
fof(ah4s_relations_RUNIVu_u_SUBSETu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_R]: p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),t_bool)),h4s_relations_rsubset),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),h4s_relations_runiv))))).
fof(ah4s_relations_Ou_u_MONO, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_S2, V_S1, V_R2, V_R1]: ((p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),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)))) & p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_bool),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),t_bool)),h4s_relations_rsubset),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_S1))),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_S2))))) => p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27b,t_bool)),t_bool),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27b,t_bool)),t_fun(t_fun(TV_u_27c,t_fun(TV_u_27b,t_bool)),t_bool)),h4s_relations_rsubset),s(t_fun(TV_u_27c,t_fun(TV_u_27b,t_bool)),h4s_relations_o(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R1),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_S1))))),s(t_fun(TV_u_27c,t_fun(TV_u_27b,t_bool)),h4s_relations_o(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27c,t_fun(TV_u_27a,t_bool)),V_S2)))))))).
fof(ah4s_relations_EqIsBothRSUBSET, axiom, ![TV_u_27a,TV_u_27b]: ![V_z, V_y]: (s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_y) = s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_z) <=> (p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),t_bool)),h4s_relations_rsubset),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_y))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_z)))) & p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),t_bool)),h4s_relations_rsubset),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_z))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_y))))))).
fof(ah4s_bools_ORu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) | p(s(t_bool,t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
fof(ah4s_relations_reflexiveu_u_def, axiom, ![TV_u_27a]: ![V_R]: (p(s(t_bool,h4s_relations_reflexive(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) <=> ![V_x]: 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_x)))))).
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_EQu_u_CLAUSESu_c0, axiom, ![V_t]: (s(t_bool,t) = s(t_bool,V_t) <=> p(s(t_bool,V_t)))).
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_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_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_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_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_FINITEu_u_EMPTY, 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_empty))))).
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_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_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_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_relations_transitiveu_u_def, axiom, ![TV_u_27a]: ![V_R]: (p(s(t_bool,h4s_relations_transitive(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) <=> ![V_x, V_y, V_z]: ((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)))) & 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_z))))) => 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_z))))))).
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_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_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_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_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_relations_WeakOrder0, axiom, ![TV_u_27g]: ![V_Z]: (p(s(t_bool,h4s_relations_weakorder(s(t_fun(TV_u_27g,t_fun(TV_u_27g,t_bool)),V_Z)))) <=> (p(s(t_bool,h4s_relations_reflexive(s(t_fun(TV_u_27g,t_fun(TV_u_27g,t_bool)),V_Z)))) & (p(s(t_bool,h4s_relations_antisymmetric(s(t_fun(TV_u_27g,t_fun(TV_u_27g,t_bool)),V_Z)))) & p(s(t_bool,h4s_relations_transitive(s(t_fun(TV_u_27g,t_fun(TV_u_27g,t_bool)),V_Z)))))))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,t))) <=> p(s(t_bool,V_t)))).
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_bools_IMPu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
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_bools_IMPu_u_DISJu_u_THM, 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_bools_UNWINDu_u_FORALLu_u_THM2, axiom, ![TV_u_27a]: ![V_v, V_f]: (![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_v) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_f),s(TV_u_27a,V_x))))) <=> p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_f),s(TV_u_27a,V_v)))))).
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_bools_BOOLu_u_CASESu_u_AX, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f))).
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,t)))).
fof(ah4s_predu_u_sets_DELETEu_u_SUBSET, axiom, ![TV_u_27a]: ![V_x, V_s]: 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_s),s(TV_u_27a,V_x))),s(t_fun(TV_u_27a,t_bool),V_s))))).
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_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_predu_u_sets_SUBSETu_u_UNIONu_c0, 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),h4s_predu_u_sets_union(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_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_sats_pthu_u_ni1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => p(s(t_bool,V_p)))).
fof(ah4s_sats_pthu_u_ni2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_predu_u_sets_INTERu_u_SUBSETu_c0, 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),h4s_predu_u_sets_inter(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))))).
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_relations_RUNIV0, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x]: 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)),h4s_relations_runiv),s(TV_u_27a,V_x))),s(TV_u_27b,V_y))) = s(t_bool,t)).
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_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_relations_Ou_u_DEF, axiom, ![TV_u_27g,TV_u_27h,TV_u_27k]: ![V_z, V_x, V_R2, V_R1]: (p(s(t_bool,happ(s(t_fun(TV_u_27k,t_bool),happ(s(t_fun(TV_u_27g,t_fun(TV_u_27k,t_bool)),h4s_relations_o(s(t_fun(TV_u_27h,t_fun(TV_u_27k,t_bool)),V_R1),s(t_fun(TV_u_27g,t_fun(TV_u_27h,t_bool)),V_R2))),s(TV_u_27g,V_x))),s(TV_u_27k,V_z)))) <=> ?[V_y]: (p(s(t_bool,happ(s(t_fun(TV_u_27h,t_bool),happ(s(t_fun(TV_u_27g,t_fun(TV_u_27h,t_bool)),V_R2),s(TV_u_27g,V_x))),s(TV_u_27h,V_y)))) & p(s(t_bool,happ(s(t_fun(TV_u_27k,t_bool),happ(s(t_fun(TV_u_27h,t_fun(TV_u_27k,t_bool)),V_R1),s(TV_u_27h,V_y))),s(TV_u_27k,V_z))))))).
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_predu_u_sets_DISJOINTu_u_DEF, axiom, ![TV_u_27a]: ![V_t, V_s]: (p(s(t_bool,h4s_predu_u_sets_disjoint(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),h4s_predu_u_sets_inter(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),h4s_predu_u_sets_empty))).
fof(ah4s_predu_u_sets_INu_u_DIFF, axiom, ![TV_u_27a]: ![V_x, V_t, 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_diff(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_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_relations_irreflexiveu_u_def, axiom, ![TV_u_27a]: ![V_R]: (p(s(t_bool,h4s_relations_irreflexive(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) <=> ![V_x]: ~ (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_x))))))).
fof(ah4s_relations_invu_u_DEF, axiom, ![TV_u_27a]: ![V_y, V_x, V_R]: 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_relations_inv(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))) = 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)))).
fof(ah4s_relations_symmetricu_u_def, axiom, ![TV_u_27a]: ![V_R]: (p(s(t_bool,h4s_relations_symmetric(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) <=> ![V_x, V_y]: 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))) = 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))))).
fof(ah4s_bools_ORu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) | p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
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(ch4s_predu_u_sets_RELu_u_RESTRICTu_u_SUBSET, conjecture, ![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,happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),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)))))))).
