%   ORIGINAL: h4/wot/StrongWellOrderExists
% 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/bool/TRUTH: 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/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/bool/AND__CLAUSES_c2: !t. F /\ t <=> F
% Assm: h4/bool/OR__CLAUSES_c0: !t. T \/ t <=> T
% Assm: h4/bool/OR__CLAUSES_c1: !t. t \/ T <=> T
% Assm: h4/bool/OR__CLAUSES_c3: !t. t \/ F <=> t
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> t
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/EQ__CLAUSES_c2: !t. (F <=> t) <=> ~t
% Assm: h4/bool/NOT__IMP: !B A. ~(A ==> B) <=> A /\ ~B
% Assm: h4/bool/DISJ__ASSOC: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm: h4/bool/DE__MORGAN__THM_c0: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm: h4/bool/DE__MORGAN__THM_c1: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm: h4/bool/IMP__DISJ__THM: !B A. A ==> B <=> ~A \/ B
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/EQ__IMP__THM: !t2 t1. (t1 <=> t2) <=> (t1 ==> t2) /\ (t2 ==> t1)
% Assm: h4/bool/EQ__EXPAND: !t2 t1. (t1 <=> t2) <=> t1 /\ t2 \/ ~t1 /\ ~t2
% 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/relation/transitive__def: !R. h4/relation/transitive R <=> (!x y z. R x y /\ R y z ==> R x z)
% Assm: h4/relation/reflexive__def: !R. h4/relation/reflexive R <=> (!x. R x x)
% Assm: h4/relation/antisymmetric__def: !R. h4/relation/antisymmetric R <=> (!x y. R x y /\ R y x ==> x = y)
% Assm: h4/relation/trichotomous0: !R. h4/relation/trichotomous R <=> (!a b. R a b \/ R b a \/ a = b)
% Assm: h4/relation/WF__DEF: !R. h4/relation/WF R <=> (!B. (?w. B w) ==> (?min. B min /\ (!b. R b min ==> ~B b)))
% Assm: h4/relation/Order0: !Z. h4/relation/Order Z <=> h4/relation/antisymmetric Z /\ h4/relation/transitive Z
% Assm: h4/relation/WeakOrder0: !Z. h4/relation/WeakOrder Z <=> h4/relation/reflexive Z /\ h4/relation/antisymmetric Z /\ h4/relation/transitive Z
% Assm: h4/relation/WeakOrd__Ord: !R. h4/relation/WeakOrder R ==> h4/relation/Order R
% Assm: h4/relation/STRORD0: !b a R. h4/relation/STRORD R a b <=> R a b /\ ~(a = b)
% Assm: h4/relation/STRORD__Strong: !R. h4/relation/Order R <=> h4/relation/StrongOrder (h4/relation/STRORD R)
% Assm: h4/relation/trichotomous__STRORD: !R. h4/relation/trichotomous (h4/relation/STRORD R) <=> h4/relation/trichotomous R
% Assm: h4/relation/StrongLinearOrder0: !R. h4/relation/StrongLinearOrder R <=> h4/relation/StrongOrder R /\ h4/relation/trichotomous R
% Assm: h4/relation/WeakLinearOrder0: !R. h4/relation/WeakLinearOrder R <=> h4/relation/WeakOrder R /\ h4/relation/trichotomous R
% Assm: h4/relation/WeakLinearOrder__dichotomy: !R. h4/relation/WeakLinearOrder R <=> h4/relation/WeakOrder R /\ (!a b. R a b \/ R b a)
% Assm: h4/pair/PAIR__EQ: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% Assm: h4/pred__set/SPECIFICATION: !x P. h4/bool/IN x P <=> P x
% Assm: h4/pred__set/EXTENSION: !t s. s = t <=> (!x. h4/bool/IN x s <=> h4/bool/IN x t)
% Assm: h4/pred__set/GSPECIFICATION: !v f. h4/bool/IN v (h4/pred__set/GSPEC f) <=> (?x. h4/pair/_2C v T = f x)
% Assm: h4/pred__set/NOT__IN__EMPTY: !x. ~h4/bool/IN x h4/pred__set/EMPTY
% Assm: h4/pred__set/MEMBER__NOT__EMPTY: !s. (?x. h4/bool/IN x s) <=> ~(s = h4/pred__set/EMPTY)
% Assm: h4/pred__set/IN__UNIV: !x. h4/bool/IN x h4/pred__set/UNIV
% 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/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__REFL: !s. h4/pred__set/SUBSET s s
% 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/PSUBSET__DEF: !t s. h4/pred__set/PSUBSET s t <=> h4/pred__set/SUBSET s t /\ ~(s = 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/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/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/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/NOT__INSERT__EMPTY: !x s. ~(h4/pred__set/INSERT x s = h4/pred__set/EMPTY)
% Assm: h4/pred__set/CHOICE__DEF: !s. ~(s = h4/pred__set/EMPTY) ==> h4/bool/IN (h4/pred__set/CHOICE s) s
% Assm: h4/pred__set/BIGUNION0: !P. h4/pred__set/BIGUNION P = h4/pred__set/GSPEC (\x. h4/pair/_2C x (?s. h4/bool/IN s P /\ h4/bool/IN x s))
% 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/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/BIGINTER0: !P. h4/pred__set/BIGINTER P = h4/pred__set/GSPEC (\x. h4/pair/_2C x (!s. h4/bool/IN s P ==> h4/bool/IN x s))
% Assm: h4/pred__set/IN__BIGINTER: !x B. h4/bool/IN x (h4/pred__set/BIGINTER B) <=> (!P. h4/bool/IN P B ==> h4/bool/IN x P)
% Assm: h4/pred__set/SUBSET__BIGINTER: !X P. h4/pred__set/SUBSET X (h4/pred__set/BIGINTER P) <=> (!Y. h4/bool/IN Y P ==> h4/pred__set/SUBSET X Y)
% Assm: h4/pred__set/COMPL__DEF: !P. h4/pred__set/COMPL P = h4/pred__set/DIFF h4/pred__set/UNIV P
% Assm: h4/pred__set/IN__COMPL: !x s. h4/bool/IN x (h4/pred__set/COMPL s) <=> ~h4/bool/IN x s
% Assm: h4/wot/cpl__def: !B A. h4/wot/cpl A B <=> h4/pred__set/SUBSET A B \/ h4/pred__set/SUBSET B A
% Assm: h4/wot/chain__def: !C. h4/wot/chain C <=> (!a b. h4/bool/IN a C /\ h4/bool/IN b C ==> h4/wot/cpl a b)
% Assm: h4/wot/mex__def: !s. h4/wot/mex s = h4/pred__set/CHOICE (h4/pred__set/COMPL s)
% Assm: h4/wot/setsuc__def: !s. h4/wot/setsuc s = h4/pred__set/INSERT (h4/wot/mex s) s
% Assm: h4/wot/succl__def: !c. h4/wot/succl c <=> (!s. h4/bool/IN s c ==> h4/bool/IN (h4/wot/setsuc s) c)
% Assm: h4/wot/uncl__def: !c. h4/wot/uncl c <=> (!w. h4/pred__set/SUBSET w c /\ h4/wot/chain w ==> h4/bool/IN (h4/pred__set/BIGUNION w) c)
% Assm: h4/wot/tower__def: !A. h4/wot/tower A <=> h4/wot/succl A /\ h4/wot/uncl A
% Assm: h4/wot/t0__def: h4/wot/t0 = h4/pred__set/BIGINTER h4/wot/tower
% Assm: h4/wot/comparable__def: !p. h4/wot/comparable p <=> (!q. h4/bool/IN q h4/wot/t0 ==> h4/wot/cpl p q)
% Assm: h4/wot/U__def: !C. h4/wot/U C = h4/pred__set/GSPEC (\A. h4/pair/_2C A (h4/bool/IN A h4/wot/t0 /\ (h4/pred__set/SUBSET A C \/ h4/pred__set/SUBSET (h4/wot/setsuc C) A)))
% Assm: h4/wot/lub__sub__def: !B. h4/wot/lub__sub B = h4/pred__set/BIGUNION (h4/pred__set/GSPEC (\y. h4/pair/_2C y (h4/bool/IN y h4/wot/t0 /\ (!x. h4/bool/IN x B ==> h4/pred__set/SUBSET y x))))
% Assm: h4/wot/preds__def: !a. h4/wot/preds a = h4/pred__set/BIGUNION (h4/pred__set/GSPEC (\s. h4/pair/_2C s (h4/bool/IN s h4/wot/t0 /\ ~h4/bool/IN a s)))
% Assm: h4/wot/mex__less__eq__def: !b a. h4/wot/mex__less__eq a b <=> h4/pred__set/SUBSET (h4/wot/preds a) (h4/wot/preds b)
% Assm: h4/wot/mex__less__def: h4/wot/mex__less = h4/relation/STRORD h4/wot/mex__less__eq
% Assm: h4/wot/WeakWellOrder__def: !R. h4/wot/WeakWellOrder R <=> h4/relation/WeakOrder R /\ (!B. ~(B = h4/pred__set/EMPTY) ==> (?m. h4/bool/IN m B /\ (!b. h4/bool/IN b B ==> R m b)))
% Assm: h4/wot/preds__image__def: !X. h4/wot/preds__image X = h4/pred__set/GSPEC (\x. h4/pair/_2C (h4/wot/preds x) (h4/bool/IN x X))
% Assm: h4/wot/StrongWellOrder__def: !R. h4/wot/StrongWellOrder R <=> h4/relation/StrongLinearOrder R /\ h4/relation/WF R
% Goal: ?R. h4/relation/StrongLinearOrder R /\ h4/relation/WF R
%   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_bools_TRUTH]: 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_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_Fu_u_IMP]: !t. ~t ==> t ==> F
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c2]: !t. F /\ t <=> F
% Assm [h4s_bools_ORu_u_CLAUSESu_c0]: !t. T \/ t <=> T
% Assm [h4s_bools_ORu_u_CLAUSESu_c1]: !t. t \/ T <=> T
% Assm [h4s_bools_ORu_u_CLAUSESu_c3]: !t. t \/ F <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c2]: !t. (F <=> t) <=> ~t
% Assm [h4s_bools_NOTu_u_IMP]: !B A. ~(A ==> B) <=> A /\ ~B
% Assm [h4s_bools_DISJu_u_ASSOC]: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c0]: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c1]: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm [h4s_bools_IMPu_u_DISJu_u_THM]: !B A. A ==> B <=> ~A \/ B
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_EQu_u_IMPu_u_THM]: !t2 t1. (t1 <=> t2) <=> (t1 ==> t2) /\ (t2 ==> t1)
% Assm [h4s_bools_EQu_u_EXPAND]: !t2 t1. (t1 <=> t2) <=> t1 /\ t2 \/ ~t1 /\ ~t2
% 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_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_relations_reflexiveu_u_def]: !R. h4/relation/reflexive R <=> (!x. happ (happ R x) x)
% 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_relations_trichotomous0]: !R. h4/relation/trichotomous R <=> (!a b. happ (happ R a) b \/ happ (happ R b) a \/ a = b)
% Assm [h4s_relations_WFu_u_DEF]: !R. h4/relation/WF R <=> (!B. (?w. happ B w) ==> (?min. happ B min /\ (!b. happ (happ R b) min ==> ~happ B b)))
% Assm [h4s_relations_Order0]: !Z. h4/relation/Order Z <=> h4/relation/antisymmetric Z /\ h4/relation/transitive Z
% Assm [h4s_relations_WeakOrder0]: !Z. h4/relation/WeakOrder Z <=> h4/relation/reflexive Z /\ h4/relation/antisymmetric Z /\ h4/relation/transitive Z
% Assm [h4s_relations_WeakOrdu_u_Ord]: !R. h4/relation/WeakOrder R ==> h4/relation/Order R
% Assm [h4s_relations_STRORD0]: !b a R. happ (happ (h4/relation/STRORD R) a) b <=> happ (happ R a) b /\ ~(a = b)
% Assm [h4s_relations_STRORDu_u_Strong]: !R. h4/relation/Order R <=> h4/relation/StrongOrder (h4/relation/STRORD R)
% Assm [h4s_relations_trichotomousu_u_STRORD]: !R. h4/relation/trichotomous (h4/relation/STRORD R) <=> h4/relation/trichotomous R
% Assm [h4s_relations_StrongLinearOrder0]: !R. h4/relation/StrongLinearOrder R <=> h4/relation/StrongOrder R /\ h4/relation/trichotomous R
% Assm [h4s_relations_WeakLinearOrder0]: !R. h4/relation/WeakLinearOrder R <=> h4/relation/WeakOrder R /\ h4/relation/trichotomous R
% Assm [h4s_relations_WeakLinearOrderu_u_dichotomy]: !R. h4/relation/WeakLinearOrder R <=> h4/relation/WeakOrder R /\ (!a b. happ (happ R a) b \/ happ (happ R b) a)
% Assm [h4s_pairs_PAIRu_u_EQ]: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% Assm [h4s_predu_u_sets_SPECIFICATION]: !x P. h4/bool/IN x P <=> happ P x
% Assm [h4s_predu_u_sets_EXTENSION]: !t s. s = t <=> (!x. h4/bool/IN x s <=> h4/bool/IN x t)
% Assm [h4s_predu_u_sets_GSPECIFICATION]: !v f. h4/bool/IN v (h4/pred__set/GSPEC f) <=> (?x. h4/pair/_2C v T = happ f x)
% Assm [h4s_predu_u_sets_NOTu_u_INu_u_EMPTY]: !x. ~h4/bool/IN x h4/pred__set/EMPTY
% Assm [h4s_predu_u_sets_MEMBERu_u_NOTu_u_EMPTY]: !s. (?x. h4/bool/IN x s) <=> ~(s = h4/pred__set/EMPTY)
% Assm [h4s_predu_u_sets_INu_u_UNIV]: !x. h4/bool/IN x h4/pred__set/UNIV
% 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_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_REFL]: !s. h4/pred__set/SUBSET s s
% 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_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_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_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_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_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_NOTu_u_INSERTu_u_EMPTY]: !x s. ~(h4/pred__set/INSERT x s = h4/pred__set/EMPTY)
% 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_predu_u_sets_BIGUNION0]: !_0. (!P x. ?v. (v <=> (?s. h4/bool/IN s P /\ h4/bool/IN x s)) /\ happ (happ _0 P) x = h4/pair/_2C x v) ==> (!P. h4/pred__set/BIGUNION P = h4/pred__set/GSPEC (happ _0 P))
% 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_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_BIGINTER0]: !_0. (!P x. ?v. (v <=> (!s. h4/bool/IN s P ==> h4/bool/IN x s)) /\ happ (happ _0 P) x = h4/pair/_2C x v) ==> (!P. h4/pred__set/BIGINTER P = h4/pred__set/GSPEC (happ _0 P))
% Assm [h4s_predu_u_sets_INu_u_BIGINTER]: !x B. h4/bool/IN x (h4/pred__set/BIGINTER B) <=> (!P. h4/bool/IN P B ==> h4/bool/IN x P)
% Assm [h4s_predu_u_sets_SUBSETu_u_BIGINTER]: !X P. h4/pred__set/SUBSET X (h4/pred__set/BIGINTER P) <=> (!Y. h4/bool/IN Y P ==> h4/pred__set/SUBSET X Y)
% Assm [h4s_predu_u_sets_COMPLu_u_DEF]: !P. h4/pred__set/COMPL P = h4/pred__set/DIFF h4/pred__set/UNIV P
% Assm [h4s_predu_u_sets_INu_u_COMPL]: !x s. h4/bool/IN x (h4/pred__set/COMPL s) <=> ~h4/bool/IN x s
% Assm [h4s_wots_cplu_u_def]: !B A. h4/wot/cpl A B <=> h4/pred__set/SUBSET A B \/ h4/pred__set/SUBSET B A
% Assm [h4s_wots_chainu_u_def]: !C. h4/wot/chain C <=> (!a b. h4/bool/IN a C /\ h4/bool/IN b C ==> h4/wot/cpl a b)
% Assm [h4s_wots_mexu_u_def]: !s. h4/wot/mex s = h4/pred__set/CHOICE (h4/pred__set/COMPL s)
% Assm [h4s_wots_setsucu_u_def]: !s. h4/wot/setsuc s = h4/pred__set/INSERT (h4/wot/mex s) s
% Assm [h4s_wots_succlu_u_def]: !c. h4/wot/succl c <=> (!s. h4/bool/IN s c ==> h4/bool/IN (h4/wot/setsuc s) c)
% Assm [h4s_wots_unclu_u_def]: !c. h4/wot/uncl c <=> (!w. h4/pred__set/SUBSET w c /\ h4/wot/chain w ==> h4/bool/IN (h4/pred__set/BIGUNION w) c)
% Assm [h4s_wots_toweru_u_def]: !A. happ h4/wot/tower A <=> h4/wot/succl A /\ h4/wot/uncl A
% Assm [h4s_wots_t0u_u_def]: h4/wot/t0 = h4/pred__set/BIGINTER h4/wot/tower
% Assm [h4s_wots_comparableu_u_def]: !p. h4/wot/comparable p <=> (!q. h4/bool/IN q h4/wot/t0 ==> h4/wot/cpl p q)
% Assm [h4s_wots_Uu_u_def]: !_0. (!C A. ?v. (v <=> h4/bool/IN A h4/wot/t0 /\ (h4/pred__set/SUBSET A C \/ h4/pred__set/SUBSET (h4/wot/setsuc C) A)) /\ happ (happ _0 C) A = h4/pair/_2C A v) ==> (!C. h4/wot/U C = h4/pred__set/GSPEC (happ _0 C))
% Assm [h4s_wots_lubu_u_subu_u_def]: !_0. (!B y. ?v. (v <=> h4/bool/IN y h4/wot/t0 /\ (!x. h4/bool/IN x B ==> h4/pred__set/SUBSET y x)) /\ happ (happ _0 B) y = h4/pair/_2C y v) ==> (!B. h4/wot/lub__sub B = h4/pred__set/BIGUNION (h4/pred__set/GSPEC (happ _0 B)))
% Assm [h4s_wots_predsu_u_def]: !_0. (!a s. ?v. (v <=> h4/bool/IN s h4/wot/t0 /\ ~h4/bool/IN a s) /\ happ (happ _0 a) s = h4/pair/_2C s v) ==> (!a. h4/wot/preds a = h4/pred__set/BIGUNION (h4/pred__set/GSPEC (happ _0 a)))
% Assm [h4s_wots_mexu_u_lessu_u_equ_u_def]: !b a. happ (happ h4/wot/mex__less__eq a) b <=> h4/pred__set/SUBSET (h4/wot/preds a) (h4/wot/preds b)
% Assm [h4s_wots_mexu_u_lessu_u_def]: h4/wot/mex__less = h4/relation/STRORD h4/wot/mex__less__eq
% Assm [h4s_wots_WeakWellOrderu_u_def]: !R. h4/wot/WeakWellOrder R <=> h4/relation/WeakOrder R /\ (!B. ~(B = h4/pred__set/EMPTY) ==> (?m. h4/bool/IN m B /\ (!b. h4/bool/IN b B ==> happ (happ R m) b)))
% Assm [h4s_wots_predsu_u_imageu_u_def]: !_0. (!X x. happ (happ _0 X) x = h4/pair/_2C (h4/wot/preds x) (h4/bool/IN x X)) ==> (!X. h4/wot/preds__image X = h4/pred__set/GSPEC (happ _0 X))
% Assm [h4s_wots_StrongWellOrderu_u_def]: !R. h4/wot/StrongWellOrder R <=> h4/relation/StrongLinearOrder R /\ h4/relation/WF R
% Goal: ?R. h4/relation/StrongLinearOrder R /\ h4/relation/WF R
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_Q318309,TV_Q318305]: ![V_f, V_g]: (![V_x]: s(TV_Q318305,happ(s(t_fun(TV_Q318309,TV_Q318305),V_f),s(TV_Q318309,V_x))) = s(TV_Q318305,happ(s(t_fun(TV_Q318309,TV_Q318305),V_g),s(TV_Q318309,V_x))) => s(t_fun(TV_Q318309,TV_Q318305),V_f) = s(t_fun(TV_Q318309,TV_Q318305),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,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_EXCLUDEDu_u_MIDDLE, 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_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_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_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_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_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_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_ORu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) | p(s(t_bool,f))) <=> p(s(t_bool,V_t)))).
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_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_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_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,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_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_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_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_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_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_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_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_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_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_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_EQu_u_IMPu_u_THM, axiom, ![V_t2, V_t1]: (s(t_bool,V_t1) = s(t_bool,V_t2) <=> ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) & (p(s(t_bool,V_t2)) => p(s(t_bool,V_t1)))))).
fof(ah4s_bools_EQu_u_EXPAND, axiom, ![V_t2, V_t1]: (s(t_bool,V_t1) = s(t_bool,V_t2) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) | (~ (p(s(t_bool,V_t1))) & ~ (p(s(t_bool,V_t2))))))).
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_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_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_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_relations_trichotomous0, axiom, ![TV_u_27a]: ![V_R]: (p(s(t_bool,h4s_relations_trichotomous(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) <=> ![V_a, V_b]: (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_a))),s(TV_u_27a,V_b)))) | (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_b))),s(TV_u_27a,V_a)))) | s(TV_u_27a,V_a) = s(TV_u_27a,V_b))))).
fof(ah4s_relations_WFu_u_DEF, 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_B]: (?[V_w]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_B),s(TV_u_27a,V_w)))) => ?[V_min]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_B),s(TV_u_27a,V_min)))) & ![V_b]: (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_b))),s(TV_u_27a,V_min)))) => ~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_B),s(TV_u_27a,V_b)))))))))).
fof(ah4s_relations_Order0, axiom, ![TV_u_27g]: ![V_Z]: (p(s(t_bool,h4s_relations_order(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_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_relations_WeakOrdu_u_Ord, 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)))) => p(s(t_bool,h4s_relations_order(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))))).
fof(ah4s_relations_STRORD0, axiom, ![TV_u_27a]: ![V_b, V_a, 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_relations_strord(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(TV_u_27a,V_a))),s(TV_u_27a,V_b)))) <=> (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_a))),s(TV_u_27a,V_b)))) & ~ (s(TV_u_27a,V_a) = s(TV_u_27a,V_b))))).
fof(ah4s_relations_STRORDu_u_Strong, axiom, ![TV_u_27a]: ![V_R]: s(t_bool,h4s_relations_order(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))) = s(t_bool,h4s_relations_strongorder(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_strord(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))))).
fof(ah4s_relations_trichotomousu_u_STRORD, axiom, ![TV_u_27a]: ![V_R]: s(t_bool,h4s_relations_trichotomous(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_strord(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))))) = s(t_bool,h4s_relations_trichotomous(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))).
fof(ah4s_relations_StrongLinearOrder0, axiom, ![TV_u_27a]: ![V_R]: (p(s(t_bool,h4s_relations_stronglinearorder(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) <=> (p(s(t_bool,h4s_relations_strongorder(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) & p(s(t_bool,h4s_relations_trichotomous(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))))))).
fof(ah4s_relations_WeakLinearOrder0, axiom, ![TV_u_27a]: ![V_R]: (p(s(t_bool,h4s_relations_weaklinearorder(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) <=> (p(s(t_bool,h4s_relations_weakorder(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) & p(s(t_bool,h4s_relations_trichotomous(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))))))).
fof(ah4s_relations_WeakLinearOrderu_u_dichotomy, axiom, ![TV_u_27a]: ![V_R]: (p(s(t_bool,h4s_relations_weaklinearorder(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),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_a, V_b]: (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_a))),s(TV_u_27a,V_b)))) | 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_b))),s(TV_u_27a,V_a)))))))).
fof(ah4s_pairs_PAIRu_u_EQ, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x, V_b, V_a]: (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(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_a),s(TV_u_27b,V_b))) <=> (s(TV_u_27a,V_x) = s(TV_u_27a,V_a) & s(TV_u_27b,V_y) = s(TV_u_27b,V_b)))).
fof(ah4s_predu_u_sets_SPECIFICATION, axiom, ![TV_u_27a]: ![V_x, V_P]: s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_P))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),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_predu_u_sets_GSPECIFICATION, axiom, ![TV_u_27a,TV_u_27b]: ![V_v, V_f]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_v),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,t_bool)),V_f)))))) <=> ?[V_x]: s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_v),s(t_bool,t))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,t_bool)),V_f),s(TV_u_27b,V_x))))).
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_predu_u_sets_MEMBERu_u_NOTu_u_EMPTY, axiom, ![TV_u_27a]: ![V_s]: (?[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)))) <=> ~ (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_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_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_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_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_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_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_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_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_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_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_NOTu_u_INSERTu_u_EMPTY, axiom, ![TV_u_27a]: ![V_x, 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_s))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty))).
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_predu_u_sets_BIGUNION0, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_P, V_x]: ?[V_v]: ((p(s(t_bool,V_v)) <=> ?[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_P)))) & 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_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),V_uu_0),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P))),s(TV_u_27a,V_x))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_bool,V_v)))) => ![V_P]: 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),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),V_uu_0),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P))))))).
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_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_BIGINTER0, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_P, V_x]: ?[V_v]: ((p(s(t_bool,V_v)) <=> ![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_P)))) => 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_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),V_uu_0),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P))),s(TV_u_27a,V_x))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_bool,V_v)))) => ![V_P]: s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_biginter(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_gspec(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),V_uu_0),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P))))))).
fof(ah4s_predu_u_sets_INu_u_BIGINTER, axiom, ![TV_u_27a]: ![V_x, V_B]: (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_biginter(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_B)))))) <=> ![V_P]: (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_B)))) => p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_P))))))).
fof(ah4s_predu_u_sets_SUBSETu_u_BIGINTER, 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),V_X),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_biginter(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P)))))) <=> ![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_X),s(t_fun(TV_u_27a,t_bool),V_Y))))))).
fof(ah4s_predu_u_sets_COMPLu_u_DEF, axiom, ![TV_u_27a]: ![V_P]: s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_compl(s(t_fun(TV_u_27a,t_bool),V_P))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_diff(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_univ),s(t_fun(TV_u_27a,t_bool),V_P)))).
fof(ah4s_predu_u_sets_INu_u_COMPL, 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),h4s_predu_u_sets_compl(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_s))))))).
fof(ah4s_wots_cplu_u_def, axiom, ![TV_u_27x]: ![V_B, V_A]: (p(s(t_bool,h4s_wots_cpl(s(t_fun(TV_u_27x,t_bool),V_A),s(t_fun(TV_u_27x,t_bool),V_B)))) <=> (p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27x,t_bool),V_A),s(t_fun(TV_u_27x,t_bool),V_B)))) | p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27x,t_bool),V_B),s(t_fun(TV_u_27x,t_bool),V_A))))))).
fof(ah4s_wots_chainu_u_def, axiom, ![TV_u_27x]: ![V_C]: (p(s(t_bool,h4s_wots_chain(s(t_fun(t_fun(TV_u_27x,t_bool),t_bool),V_C)))) <=> ![V_a, V_b]: ((p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27x,t_bool),V_a),s(t_fun(t_fun(TV_u_27x,t_bool),t_bool),V_C)))) & p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27x,t_bool),V_b),s(t_fun(t_fun(TV_u_27x,t_bool),t_bool),V_C))))) => p(s(t_bool,h4s_wots_cpl(s(t_fun(TV_u_27x,t_bool),V_a),s(t_fun(TV_u_27x,t_bool),V_b))))))).
fof(ah4s_wots_mexu_u_def, axiom, ![TV_u_27x]: ![V_s]: s(TV_u_27x,h4s_wots_mex(s(t_fun(TV_u_27x,t_bool),V_s))) = s(TV_u_27x,h4s_predu_u_sets_choice(s(t_fun(TV_u_27x,t_bool),h4s_predu_u_sets_compl(s(t_fun(TV_u_27x,t_bool),V_s)))))).
fof(ah4s_wots_setsucu_u_def, axiom, ![TV_u_27x]: ![V_s]: s(t_fun(TV_u_27x,t_bool),h4s_wots_setsuc(s(t_fun(TV_u_27x,t_bool),V_s))) = s(t_fun(TV_u_27x,t_bool),h4s_predu_u_sets_insert(s(TV_u_27x,h4s_wots_mex(s(t_fun(TV_u_27x,t_bool),V_s))),s(t_fun(TV_u_27x,t_bool),V_s)))).
fof(ah4s_wots_succlu_u_def, axiom, ![TV_u_27x]: ![V_c]: (p(s(t_bool,h4s_wots_succl(s(t_fun(t_fun(TV_u_27x,t_bool),t_bool),V_c)))) <=> ![V_s]: (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27x,t_bool),V_s),s(t_fun(t_fun(TV_u_27x,t_bool),t_bool),V_c)))) => p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27x,t_bool),h4s_wots_setsuc(s(t_fun(TV_u_27x,t_bool),V_s))),s(t_fun(t_fun(TV_u_27x,t_bool),t_bool),V_c))))))).
fof(ah4s_wots_unclu_u_def, axiom, ![TV_u_27x]: ![V_c]: (p(s(t_bool,h4s_wots_uncl(s(t_fun(t_fun(TV_u_27x,t_bool),t_bool),V_c)))) <=> ![V_w]: ((p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(t_fun(TV_u_27x,t_bool),t_bool),V_w),s(t_fun(t_fun(TV_u_27x,t_bool),t_bool),V_c)))) & p(s(t_bool,h4s_wots_chain(s(t_fun(t_fun(TV_u_27x,t_bool),t_bool),V_w))))) => p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27x,t_bool),h4s_predu_u_sets_bigunion(s(t_fun(t_fun(TV_u_27x,t_bool),t_bool),V_w))),s(t_fun(t_fun(TV_u_27x,t_bool),t_bool),V_c))))))).
fof(ah4s_wots_toweru_u_def, axiom, ![TV_u_27x]: ![V_A]: (p(s(t_bool,happ(s(t_fun(t_fun(t_fun(TV_u_27x,t_bool),t_bool),t_bool),h4s_wots_tower),s(t_fun(t_fun(TV_u_27x,t_bool),t_bool),V_A)))) <=> (p(s(t_bool,h4s_wots_succl(s(t_fun(t_fun(TV_u_27x,t_bool),t_bool),V_A)))) & p(s(t_bool,h4s_wots_uncl(s(t_fun(t_fun(TV_u_27x,t_bool),t_bool),V_A))))))).
fof(ah4s_wots_t0u_u_def, axiom, ![TV_u_27x]: s(t_fun(t_fun(TV_u_27x,t_bool),t_bool),h4s_wots_t0) = s(t_fun(t_fun(TV_u_27x,t_bool),t_bool),h4s_predu_u_sets_biginter(s(t_fun(t_fun(t_fun(TV_u_27x,t_bool),t_bool),t_bool),h4s_wots_tower)))).
fof(ah4s_wots_comparableu_u_def, axiom, ![TV_u_27x]: ![V_p]: (p(s(t_bool,h4s_wots_comparable(s(t_fun(TV_u_27x,t_bool),V_p)))) <=> ![V_q]: (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27x,t_bool),V_q),s(t_fun(t_fun(TV_u_27x,t_bool),t_bool),h4s_wots_t0)))) => p(s(t_bool,h4s_wots_cpl(s(t_fun(TV_u_27x,t_bool),V_p),s(t_fun(TV_u_27x,t_bool),V_q))))))).
fof(ah4s_wots_Uu_u_def, axiom, ![TV_u_27x]: ![V_uu_0]: (![V_C, V_A]: ?[V_v]: ((p(s(t_bool,V_v)) <=> (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27x,t_bool),V_A),s(t_fun(t_fun(TV_u_27x,t_bool),t_bool),h4s_wots_t0)))) & (p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27x,t_bool),V_A),s(t_fun(TV_u_27x,t_bool),V_C)))) | p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27x,t_bool),h4s_wots_setsuc(s(t_fun(TV_u_27x,t_bool),V_C))),s(t_fun(TV_u_27x,t_bool),V_A))))))) & s(t_h4s_pairs_prod(t_fun(TV_u_27x,t_bool),t_bool),happ(s(t_fun(t_fun(TV_u_27x,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27x,t_bool),t_bool)),happ(s(t_fun(t_fun(TV_u_27x,t_bool),t_fun(t_fun(TV_u_27x,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27x,t_bool),t_bool))),V_uu_0),s(t_fun(TV_u_27x,t_bool),V_C))),s(t_fun(TV_u_27x,t_bool),V_A))) = s(t_h4s_pairs_prod(t_fun(TV_u_27x,t_bool),t_bool),h4s_pairs_u_2c(s(t_fun(TV_u_27x,t_bool),V_A),s(t_bool,V_v)))) => ![V_C]: s(t_fun(t_fun(TV_u_27x,t_bool),t_bool),h4s_wots_u(s(t_fun(TV_u_27x,t_bool),V_C))) = s(t_fun(t_fun(TV_u_27x,t_bool),t_bool),h4s_predu_u_sets_gspec(s(t_fun(t_fun(TV_u_27x,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27x,t_bool),t_bool)),happ(s(t_fun(t_fun(TV_u_27x,t_bool),t_fun(t_fun(TV_u_27x,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27x,t_bool),t_bool))),V_uu_0),s(t_fun(TV_u_27x,t_bool),V_C))))))).
fof(ah4s_wots_lubu_u_subu_u_def, axiom, ![TV_u_27x]: ![V_uu_0]: (![V_B, V_y]: ?[V_v]: ((p(s(t_bool,V_v)) <=> (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27x,t_bool),V_y),s(t_fun(t_fun(TV_u_27x,t_bool),t_bool),h4s_wots_t0)))) & ![V_x]: (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27x,t_bool),V_x),s(t_fun(t_fun(TV_u_27x,t_bool),t_bool),V_B)))) => p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27x,t_bool),V_y),s(t_fun(TV_u_27x,t_bool),V_x))))))) & s(t_h4s_pairs_prod(t_fun(TV_u_27x,t_bool),t_bool),happ(s(t_fun(t_fun(TV_u_27x,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27x,t_bool),t_bool)),happ(s(t_fun(t_fun(t_fun(TV_u_27x,t_bool),t_bool),t_fun(t_fun(TV_u_27x,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27x,t_bool),t_bool))),V_uu_0),s(t_fun(t_fun(TV_u_27x,t_bool),t_bool),V_B))),s(t_fun(TV_u_27x,t_bool),V_y))) = s(t_h4s_pairs_prod(t_fun(TV_u_27x,t_bool),t_bool),h4s_pairs_u_2c(s(t_fun(TV_u_27x,t_bool),V_y),s(t_bool,V_v)))) => ![V_B]: s(t_fun(TV_u_27x,t_bool),h4s_wots_lubu_u_sub(s(t_fun(t_fun(TV_u_27x,t_bool),t_bool),V_B))) = s(t_fun(TV_u_27x,t_bool),h4s_predu_u_sets_bigunion(s(t_fun(t_fun(TV_u_27x,t_bool),t_bool),h4s_predu_u_sets_gspec(s(t_fun(t_fun(TV_u_27x,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27x,t_bool),t_bool)),happ(s(t_fun(t_fun(t_fun(TV_u_27x,t_bool),t_bool),t_fun(t_fun(TV_u_27x,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27x,t_bool),t_bool))),V_uu_0),s(t_fun(t_fun(TV_u_27x,t_bool),t_bool),V_B))))))))).
fof(ah4s_wots_predsu_u_def, axiom, ![TV_u_27x]: ![V_uu_0]: (![V_a, V_s]: ?[V_v]: ((p(s(t_bool,V_v)) <=> (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27x,t_bool),V_s),s(t_fun(t_fun(TV_u_27x,t_bool),t_bool),h4s_wots_t0)))) & ~ (p(s(t_bool,h4s_bools_in(s(TV_u_27x,V_a),s(t_fun(TV_u_27x,t_bool),V_s))))))) & s(t_h4s_pairs_prod(t_fun(TV_u_27x,t_bool),t_bool),happ(s(t_fun(t_fun(TV_u_27x,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27x,t_bool),t_bool)),happ(s(t_fun(TV_u_27x,t_fun(t_fun(TV_u_27x,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27x,t_bool),t_bool))),V_uu_0),s(TV_u_27x,V_a))),s(t_fun(TV_u_27x,t_bool),V_s))) = s(t_h4s_pairs_prod(t_fun(TV_u_27x,t_bool),t_bool),h4s_pairs_u_2c(s(t_fun(TV_u_27x,t_bool),V_s),s(t_bool,V_v)))) => ![V_a]: s(t_fun(TV_u_27x,t_bool),h4s_wots_preds(s(TV_u_27x,V_a))) = s(t_fun(TV_u_27x,t_bool),h4s_predu_u_sets_bigunion(s(t_fun(t_fun(TV_u_27x,t_bool),t_bool),h4s_predu_u_sets_gspec(s(t_fun(t_fun(TV_u_27x,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27x,t_bool),t_bool)),happ(s(t_fun(TV_u_27x,t_fun(t_fun(TV_u_27x,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27x,t_bool),t_bool))),V_uu_0),s(TV_u_27x,V_a))))))))).
fof(ah4s_wots_mexu_u_lessu_u_equ_u_def, axiom, ![TV_u_27x]: ![V_b, V_a]: s(t_bool,happ(s(t_fun(TV_u_27x,t_bool),happ(s(t_fun(TV_u_27x,t_fun(TV_u_27x,t_bool)),h4s_wots_mexu_u_lessu_u_eq),s(TV_u_27x,V_a))),s(TV_u_27x,V_b))) = s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27x,t_bool),h4s_wots_preds(s(TV_u_27x,V_a))),s(t_fun(TV_u_27x,t_bool),h4s_wots_preds(s(TV_u_27x,V_b)))))).
fof(ah4s_wots_mexu_u_lessu_u_def, axiom, ![TV_u_27x]: s(t_fun(TV_u_27x,t_fun(TV_u_27x,t_bool)),h4s_wots_mexu_u_less) = s(t_fun(TV_u_27x,t_fun(TV_u_27x,t_bool)),h4s_relations_strord(s(t_fun(TV_u_27x,t_fun(TV_u_27x,t_bool)),h4s_wots_mexu_u_lessu_u_eq)))).
fof(ah4s_wots_WeakWellOrderu_u_def, axiom, ![TV_u_27a]: ![V_R]: (p(s(t_bool,h4s_wots_weakwellorder(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),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_B]: (~ (s(t_fun(TV_u_27a,t_bool),V_B) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)) => ?[V_m]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_m),s(t_fun(TV_u_27a,t_bool),V_B)))) & ![V_b]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_b),s(t_fun(TV_u_27a,t_bool),V_B)))) => 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_m))),s(TV_u_27a,V_b)))))))))).
fof(ah4s_wots_predsu_u_imageu_u_def, axiom, ![TV_u_27x]: ![V_uu_0]: (![V_X, V_x]: s(t_h4s_pairs_prod(t_fun(TV_u_27x,t_bool),t_bool),happ(s(t_fun(TV_u_27x,t_h4s_pairs_prod(t_fun(TV_u_27x,t_bool),t_bool)),happ(s(t_fun(t_fun(TV_u_27x,t_bool),t_fun(TV_u_27x,t_h4s_pairs_prod(t_fun(TV_u_27x,t_bool),t_bool))),V_uu_0),s(t_fun(TV_u_27x,t_bool),V_X))),s(TV_u_27x,V_x))) = s(t_h4s_pairs_prod(t_fun(TV_u_27x,t_bool),t_bool),h4s_pairs_u_2c(s(t_fun(TV_u_27x,t_bool),h4s_wots_preds(s(TV_u_27x,V_x))),s(t_bool,h4s_bools_in(s(TV_u_27x,V_x),s(t_fun(TV_u_27x,t_bool),V_X))))) => ![V_X]: s(t_fun(t_fun(TV_u_27x,t_bool),t_bool),h4s_wots_predsu_u_image(s(t_fun(TV_u_27x,t_bool),V_X))) = s(t_fun(t_fun(TV_u_27x,t_bool),t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27x,t_h4s_pairs_prod(t_fun(TV_u_27x,t_bool),t_bool)),happ(s(t_fun(t_fun(TV_u_27x,t_bool),t_fun(TV_u_27x,t_h4s_pairs_prod(t_fun(TV_u_27x,t_bool),t_bool))),V_uu_0),s(t_fun(TV_u_27x,t_bool),V_X))))))).
fof(ah4s_wots_StrongWellOrderu_u_def, axiom, ![TV_u_27a]: ![V_R]: (p(s(t_bool,h4s_wots_strongwellorder(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) <=> (p(s(t_bool,h4s_relations_stronglinearorder(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) & p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))))))).
fof(ch4s_wots_StrongWellOrderExists, conjecture, ![TV_u_27a]: ?[V_R]: (p(s(t_bool,h4s_relations_stronglinearorder(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) & p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))))).
