%   ORIGINAL: h4/nets/SEQ__TENDS
% 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/pair/UNCURRY__DEF: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = f x y
% Assm: h4/bool/TRUTH: T
% Assm: h4/topology/MTOP__OPEN: !m S_27. h4/topology/open (h4/topology/mtop m) S_27 <=> (!x. S_27 x ==> (?e. h4/realax/real__lt (h4/real/real__of__num h4/num/0) e /\ (!y. h4/realax/real__lt (h4/topology/dist m (h4/pair/_2C x y)) e ==> S_27 y)))
% Assm: h4/arithmetic/GREATER__EQ: !n m. h4/arithmetic/_3E_3D n m <=> h4/arithmetic/_3C_3D m n
% Assm: h4/topology/mtop0: !m. h4/topology/mtop m = h4/topology/topology0 (\S_27. !x. S_27 x ==> (?e. h4/realax/real__lt (h4/real/real__of__num h4/num/0) e /\ (!y. h4/realax/real__lt (h4/topology/dist m (h4/pair/_2C x y)) e ==> S_27 y)))
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/arithmetic/GREATER__OR__EQ: !n m. h4/arithmetic/_3E_3D m n <=> h4/arithmetic/_3E m n \/ m = n
% Assm: h4/topology/BALL__NEIGH: !x m e. h4/realax/real__lt (h4/real/real__of__num h4/num/0) e ==> h4/topology/neigh (h4/topology/mtop m) (h4/pair/_2C (h4/topology/B m (h4/pair/_2C x e)) x)
% Assm: h4/topology/BALL__OPEN: !x m e. h4/realax/real__lt (h4/real/real__of__num h4/num/0) e ==> h4/topology/open (h4/topology/mtop m) (h4/topology/B m (h4/pair/_2C x e))
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/nets/MTOP__TENDS: !x0 x g d. h4/nets/tends x x0 (h4/pair/_2C (h4/topology/mtop d) g) <=> (!e. h4/realax/real__lt (h4/real/real__of__num h4/num/0) e ==> (?n. g n n /\ (!m. g m n ==> h4/realax/real__lt (h4/topology/dist d (h4/pair/_2C (x m) x0)) e)))
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/topology/MTOP__LIMPT: !x m S_27. h4/topology/limpt (h4/topology/mtop m) x S_27 <=> (!e. h4/realax/real__lt (h4/real/real__of__num h4/num/0) e ==> (?y. ~(x = y) /\ S_27 y /\ h4/realax/real__lt (h4/topology/dist m (h4/pair/_2C x y)) e))
% Assm: h4/nets/MTOP__TENDS__UNIQ: !x1 x0 x g d. h4/nets/dorder g ==> h4/nets/tends x x0 (h4/pair/_2C (h4/topology/mtop d) g) /\ h4/nets/tends x x1 (h4/pair/_2C (h4/topology/mtop d) g) ==> x0 = x1
% Assm: h4/topology/ball: !x m e. h4/topology/B m (h4/pair/_2C x e) = (\y. h4/realax/real__lt (h4/topology/dist m (h4/pair/_2C x y)) e)
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/nets/tends0: !top s l g. h4/nets/tends s l (h4/pair/_2C top g) <=> (!N. h4/topology/neigh top (h4/pair/_2C N l) ==> (?n. g n n /\ (!m. g m n ==> N (s m))))
% Assm: h4/real/SUM__ZERO: !f N. (!n. h4/arithmetic/_3E_3D n N ==> f n = h4/real/real__of__num h4/num/0) ==> (!m n. h4/arithmetic/_3E_3D m N ==> h4/real/sum (h4/pair/_2C m n) f = h4/real/real__of__num h4/num/0)
% Assm: h4/arithmetic/NOT__GREATER__EQ: !n m. ~h4/arithmetic/_3E_3D m n <=> h4/arithmetic/_3C_3D (h4/num/SUC m) n
% Assm: h4/topology/neigh0: !x top N. h4/topology/neigh top (h4/pair/_2C N x) <=> (?P. h4/topology/open top P /\ h4/pred__set/SUBSET P N /\ P x)
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> t
% Assm: h4/nets/DORDER__NGE: h4/nets/dorder h4/arithmetic/_3E_3D
% Assm: h4/arithmetic/LESS__OR__EQ: !n m. h4/arithmetic/_3C_3D m n <=> h4/prim__rec/_3C m n \/ m = n
% Assm: h4/arithmetic/GREATER__DEF: !n m. h4/arithmetic/_3E m n <=> h4/prim__rec/_3C n m
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/topology/topology__tybij_c1: !r. h4/topology/istopology r <=> h4/topology/open (h4/topology/topology0 r) = r
% Assm: h4/topology/mtop__istopology: !m. h4/topology/istopology (\S_27. !x. S_27 x ==> (?e. h4/realax/real__lt (h4/real/real__of__num h4/num/0) e /\ (!y. h4/realax/real__lt (h4/topology/dist m (h4/pair/_2C x y)) e ==> S_27 y)))
% Assm: h4/arithmetic/NOT__LEQ: !n m. ~h4/arithmetic/_3C_3D m n <=> h4/arithmetic/_3C_3D (h4/num/SUC n) m
% 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/topology/MR1__LIMPT: !x. h4/topology/limpt (h4/topology/mtop h4/topology/mr1) x h4/pred__set/UNIV
% Assm: h4/pred__set/SUBSET__REFL: !s. h4/pred__set/SUBSET s s
% Assm: h4/topology/METRIC__SAME: !x m. h4/topology/dist m (h4/pair/_2C x x) = h4/real/real__of__num h4/num/0
% Assm: h4/topology/METRIC__TRIANGLE: !z y x m. h4/real/real__lte (h4/topology/dist m (h4/pair/_2C x z)) (h4/realax/real__add (h4/topology/dist m (h4/pair/_2C x y)) (h4/topology/dist m (h4/pair/_2C y z)))
% Assm: h4/numeral/numeral__distrib_c30: !n. h4/arithmetic/_3E_3D h4/num/0 n <=> n = h4/num/0
% Assm: h4/topology/METRIC__SYM: !y x m. h4/topology/dist m (h4/pair/_2C x y) = h4/topology/dist m (h4/pair/_2C y x)
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/real/REAL__SUB__LT: !y x. h4/realax/real__lt (h4/real/real__of__num h4/num/0) (h4/real/real__sub x y) <=> h4/realax/real__lt y x
% Assm: h4/real/REAL__LET__TRANS: !z y x. h4/real/real__lte x y /\ h4/realax/real__lt y z ==> h4/realax/real__lt x z
% Assm: h4/real/REAL__ADD__SYM: !y x. h4/realax/real__add x y = h4/realax/real__add y x
% Assm: h4/real/REAL__LT__SUB__LADD: !z y x. h4/realax/real__lt x (h4/real/real__sub y z) <=> h4/realax/real__lt (h4/realax/real__add x z) y
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/pred__set/SPECIFICATION: !x P. h4/bool/IN x P <=> P x
% 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/bool/FALSITY: !t. F ==> t
% Assm: h4/real/REAL__LT__HALF1: !d. h4/realax/real__lt (h4/real/real__of__num h4/num/0) (h4/real/_2F d (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)))) <=> h4/realax/real__lt (h4/real/real__of__num h4/num/0) d
% Assm: h4/arithmetic/LESS__EQ__TRANS: !p n m. h4/arithmetic/_3C_3D m n /\ h4/arithmetic/_3C_3D n p ==> h4/arithmetic/_3C_3D m p
% Assm: h4/arithmetic/LESS__EQ__CASES: !n m. h4/arithmetic/_3C_3D m n \/ h4/arithmetic/_3C_3D n m
% Assm: h4/arithmetic/LESS__EQ__REFL: !m. h4/arithmetic/_3C_3D m m
% Assm: h4/nets/dorder0: !g. h4/nets/dorder g <=> (!x y. g x x /\ g y y ==> (?z. g z z /\ (!w. g w z ==> g w x /\ g w y)))
% Assm: h4/bool/IN__DEF: h4/bool/IN = (\x f. f x)
% Assm: h4/topology/limpt0: !x top S_27. h4/topology/limpt top x S_27 <=> (!N. h4/topology/neigh top (h4/pair/_2C N x) ==> (?y. ~(x = y) /\ S_27 y /\ N y))
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% 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/real/REAL__HALF__DOUBLE: !x. h4/realax/real__add (h4/real/_2F x (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)))) (h4/real/_2F x (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)))) = x
% Assm: h4/real/REAL__NOT__LT: !y x. ~h4/realax/real__lt x y <=> h4/real/real__lte y x
% Assm: h4/real/REAL__LT__ADD2: !z y x w. h4/realax/real__lt w x /\ h4/realax/real__lt y z ==> h4/realax/real__lt (h4/realax/real__add w y) (h4/realax/real__add x z)
% Assm: h4/nets/DORDER__LEMMA: !g. h4/nets/dorder g ==> (!P Q. (?n. g n n /\ (!m. g m n ==> P m)) /\ (?n. g n n /\ (!m. g m n ==> Q m)) ==> (?n. g n n /\ (!m. g m n ==> P m /\ Q m)))
% Assm: h4/topology/METRIC__NZ: !y x m. ~(x = y) ==> h4/realax/real__lt (h4/real/real__of__num h4/num/0) (h4/topology/dist m (h4/pair/_2C x y))
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/arithmetic/ADD__CLAUSES_c0: !m. h4/arithmetic/_2B h4/num/0 m = m
% Assm: h4/real/abs0: !x. h4/real/abs x = h4/bool/COND (h4/real/real__lte (h4/real/real__of__num h4/num/0) x) x (h4/realax/real__neg x)
% Assm: h4/real/REAL__LT__HALF2: !d. h4/realax/real__lt (h4/real/_2F d (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)))) d <=> h4/realax/real__lt (h4/real/real__of__num h4/num/0) d
% Assm: h4/real/REAL__LT__IMP__NE: !y x. h4/realax/real__lt x y ==> ~(x = y)
% Assm: h4/real/REAL__LT__IMP__LE: !y x. h4/realax/real__lt x y ==> h4/real/real__lte x y
% Assm: h4/bool/ABS__SIMP: !t2 t1. (\x. t1) t2 = t1
% Assm: h4/real/REAL__ADD__RID__UNIQ: !y x. h4/realax/real__add x y = x <=> y = h4/real/real__of__num h4/num/0
% Assm: h4/pred__set/UNIV__DEF: h4/pred__set/UNIV = (\x. T)
% Assm: h4/bool/COND__CLAUSES_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/topology/MR1__ADD: !x d. h4/topology/dist h4/topology/mr1 (h4/pair/_2C x (h4/realax/real__add x d)) = h4/real/abs d
% Assm: h4/real/sum__tupled__primitive: h4/real/sum__tupled = h4/relation/WFREC (h4/min/_40 (\R. h4/relation/WF R /\ (!f m n. R (h4/pair/_2C (h4/pair/_2C n m) f) (h4/pair/_2C (h4/pair/_2C n (h4/num/SUC m)) f)))) (\sum__tupled a. h4/pair/pair__CASE a (\v f. h4/pair/pair__CASE v (\n v3. h4/arithmetic/num__CASE v3 (h4/combin/I (h4/real/real__of__num h4/num/0)) (\m. h4/combin/I (h4/realax/real__add (sum__tupled (h4/pair/_2C (h4/pair/_2C n m) f)) (f (h4/arithmetic/_2B n m)))))))
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/real/sum__curried: !x1 x. h4/real/sum x x1 = h4/real/sum__tupled (h4/pair/_2C x x1)
% Assm: h4/real/REAL__ADD__LID: !x. h4/realax/real__add (h4/real/real__of__num h4/num/0) x = x
% Assm: h4/numeral/numeral__distrib_c27: !n. h4/arithmetic/_3C_3D (h4/arithmetic/NUMERAL n) h4/num/0 <=> h4/arithmetic/_3C_3D n h4/arithmetic/ZERO
% Assm: h4/numeral/numeral__lte_c1: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% Assm: h4/arithmetic/ADD__MONO__LESS__EQ: !p n m. h4/arithmetic/_3C_3D (h4/arithmetic/_2B m n) (h4/arithmetic/_2B m p) <=> h4/arithmetic/_3C_3D n p
% Assm: h4/arithmetic/SUC__ONE__ADD: !n. h4/num/SUC n = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) n
% Assm: h4/arithmetic/LESS__EQUAL__ADD: !n m. h4/arithmetic/_3C_3D m n ==> (?p. n = h4/arithmetic/_2B m p)
% Assm: h4/arithmetic/MULT__CLAUSES_c2: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm: h4/arithmetic/LESS__MONO__ADD__EQ: !p n m. h4/prim__rec/_3C (h4/arithmetic/_2B m p) (h4/arithmetic/_2B n p) <=> h4/prim__rec/_3C m n
% Assm: h4/arithmetic/LESS__EQ__ADD: !n m. h4/arithmetic/_3C_3D m (h4/arithmetic/_2B m n)
% Assm: h4/arithmetic/NOT__LESS: !n m. ~h4/prim__rec/_3C m n <=> h4/arithmetic/_3C_3D n m
% Assm: h4/arithmetic/ADD__SYM: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm: h4/arithmetic/ADD__ASSOC: !p n m. h4/arithmetic/_2B m (h4/arithmetic/_2B n p) = h4/arithmetic/_2B (h4/arithmetic/_2B m n) p
% Assm: h4/arithmetic/num__case__def_c0: !v f. h4/arithmetic/num__CASE h4/num/0 v f = v
% Assm: h4/arithmetic/num__case__def_c1: !v n f. h4/arithmetic/num__CASE (h4/num/SUC n) v f = f n
% Assm: h4/pair/SND0: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm: h4/pair/pair__case__thm: !y x f. h4/pair/pair__CASE (h4/pair/_2C x y) f = f x y
% Assm: h4/pair/FST0: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm: h4/pair/PAIR: !x. h4/pair/_2C (h4/pair/FST x) (h4/pair/SND x) = x
% Assm: h4/prim__rec/WF__LESS: h4/relation/WF h4/prim__rec/_3C
% Assm: h4/num/INDUCTION: !P. P h4/num/0 /\ (!n. P n ==> P (h4/num/SUC n)) ==> (!n. P n)
% Assm: h4/relation/WFREC__COROLLARY: !f R M. f = h4/relation/WFREC R M ==> h4/relation/WF R ==> (!x. f x = M (h4/relation/RESTRICT f R x) x)
% Assm: h4/combin/I__THM: !x. h4/combin/I x = x
% Assm: h4/relation/inv__image__def: !f R. h4/relation/inv__image R f = (\x y. R (f x) (f y))
% Assm: h4/relation/WF__inv__image: !f R. h4/relation/WF R ==> h4/relation/WF (h4/relation/inv__image R f)
% Assm: h4/relation/RESTRICT__LEMMA: !z y f R. R y z ==> h4/relation/RESTRICT f R z y = f y
% Assm: h4/bool/IMP__F__EQ__F: !t. t ==> F <=> t <=> F
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/bool/EQ__CLAUSES_c2: !t. (F <=> t) <=> ~t
% Assm: h4/numeral/iZ0: !x. h4/numeral/iZ x = x
% Assm: h4/bool/EQ__CLAUSES_c3: !t. (t <=> F) <=> ~t
% Assm: h4/arithmetic/MULT__CLAUSES_c1: !m. h4/arithmetic/_2A m h4/num/0 = h4/num/0
% Assm: h4/arithmetic/SUB__0_c1: !m. h4/arithmetic/_2D m h4/num/0 = m
% Assm: h4/arithmetic/MULT__CLAUSES_c0: !m. h4/arithmetic/_2A h4/num/0 m = h4/num/0
% Assm: h4/bool/NOT__FORALL__THM: !P. ~(!x. P x) <=> (?x. ~P x)
% Assm: h4/arithmetic/LESS__0__CASES: !m. h4/num/0 = m \/ h4/prim__rec/_3C h4/num/0 m
% Assm: h4/arithmetic/SUB__0_c0: !m. h4/arithmetic/_2D h4/num/0 m = h4/num/0
% Assm: h4/arithmetic/ADD__CLAUSES_c3: !n m. h4/arithmetic/_2B m (h4/num/SUC n) = h4/num/SUC (h4/arithmetic/_2B m n)
% Assm: h4/bool/EXISTS__OR__THM: !Q P. (?x. P x \/ Q x) <=> (?x. P x) \/ (?x. Q x)
% Assm: h4/arithmetic/ODD0_c0: h4/arithmetic/ODD h4/num/0 <=> F
% Assm: h4/bool/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/arithmetic/EVEN0_c0: h4/arithmetic/EVEN h4/num/0 <=> T
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% Assm: h4/arithmetic/EXP0_c1: !n m. h4/arithmetic/EXP m (h4/num/SUC n) = h4/arithmetic/_2A m (h4/arithmetic/EXP m n)
% Goal: !x0 x d. h4/nets/tends x x0 (h4/pair/_2C (h4/topology/mtop d) h4/arithmetic/_3E_3D) <=> (!e. h4/realax/real__lt (h4/real/real__of__num h4/num/0) e ==> (?N. !n. h4/arithmetic/_3E_3D n N ==> h4/realax/real__lt (h4/topology/dist d (h4/pair/_2C (x n) x0)) e))
%   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_pairs_UNCURRYu_u_DEF]: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = happ (happ f x) y
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_topologys_MTOPu_u_OPEN]: !m S_27. happ (h4/topology/open (h4/topology/mtop m)) S_27 <=> (!x. happ S_27 x ==> (?e. h4/realax/real__lt (h4/real/real__of__num h4/num/0) e /\ (!y. h4/realax/real__lt (h4/topology/dist m (h4/pair/_2C x y)) e ==> happ S_27 y)))
% Assm [h4s_arithmetics_GREATERu_u_EQ]: !n m. happ (happ h4/arithmetic/_3E_3D n) m <=> h4/arithmetic/_3C_3D m n
% Assm [h4s_topologys_mtop0]: !_0. (!m S_27. happ (happ _0 m) S_27 <=> (!x. happ S_27 x ==> (?e. h4/realax/real__lt (h4/real/real__of__num h4/num/0) e /\ (!y. h4/realax/real__lt (h4/topology/dist m (h4/pair/_2C x y)) e ==> happ S_27 y)))) ==> (!m. h4/topology/mtop m = h4/topology/topology0 (happ _0 m))
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_arithmetics_GREATERu_u_ORu_u_EQ]: !n m. happ (happ h4/arithmetic/_3E_3D m) n <=> h4/arithmetic/_3E m n \/ m = n
% Assm [h4s_topologys_BALLu_u_NEIGH]: !x m e. h4/realax/real__lt (h4/real/real__of__num h4/num/0) e ==> h4/topology/neigh (h4/topology/mtop m) (h4/pair/_2C (h4/topology/B m (h4/pair/_2C x e)) x)
% Assm [h4s_topologys_BALLu_u_OPEN]: !x m e. h4/realax/real__lt (h4/real/real__of__num h4/num/0) e ==> happ (h4/topology/open (h4/topology/mtop m)) (h4/topology/B m (h4/pair/_2C x e))
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_netss_MTOPu_u_TENDS]: !x0 x g d. h4/nets/tends x x0 (h4/pair/_2C (h4/topology/mtop d) g) <=> (!e. h4/realax/real__lt (h4/real/real__of__num h4/num/0) e ==> (?n. happ (happ g n) n /\ (!m. happ (happ g m) n ==> h4/realax/real__lt (h4/topology/dist d (h4/pair/_2C (happ x m) x0)) e)))
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_topologys_MTOPu_u_LIMPT]: !x m S_27. h4/topology/limpt (h4/topology/mtop m) x S_27 <=> (!e. h4/realax/real__lt (h4/real/real__of__num h4/num/0) e ==> (?y. ~(x = y) /\ happ S_27 y /\ h4/realax/real__lt (h4/topology/dist m (h4/pair/_2C x y)) e))
% Assm [h4s_netss_MTOPu_u_TENDSu_u_UNIQ]: !x1 x0 x g d. h4/nets/dorder g ==> h4/nets/tends x x0 (h4/pair/_2C (h4/topology/mtop d) g) /\ h4/nets/tends x x1 (h4/pair/_2C (h4/topology/mtop d) g) ==> x0 = x1
% Assm [h4s_topologys_ball]: !x m e x'. happ (h4/topology/B m (h4/pair/_2C x e)) x' <=> h4/realax/real__lt (h4/topology/dist m (h4/pair/_2C x x')) e
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_netss_tends0]: !top s l g. h4/nets/tends s l (h4/pair/_2C top g) <=> (!N. h4/topology/neigh top (h4/pair/_2C N l) ==> (?n. happ (happ g n) n /\ (!m. happ (happ g m) n ==> happ N (happ s m))))
% Assm [h4s_reals_SUMu_u_ZERO]: !f N. (!n. happ (happ h4/arithmetic/_3E_3D n) N ==> happ f n = h4/real/real__of__num h4/num/0) ==> (!m n. happ (happ h4/arithmetic/_3E_3D m) N ==> h4/real/sum (h4/pair/_2C m n) f = h4/real/real__of__num h4/num/0)
% Assm [h4s_arithmetics_NOTu_u_GREATERu_u_EQ]: !n m. ~happ (happ h4/arithmetic/_3E_3D m) n <=> h4/arithmetic/_3C_3D (h4/num/SUC m) n
% Assm [h4s_topologys_neigh0]: !x top N. h4/topology/neigh top (h4/pair/_2C N x) <=> (?P. happ (h4/topology/open top) P /\ h4/pred__set/SUBSET P N /\ happ P x)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% Assm [h4s_netss_DORDERu_u_NGE]: h4/nets/dorder h4/arithmetic/_3E_3D
% Assm [h4s_arithmetics_LESSu_u_ORu_u_EQ]: !n m. h4/arithmetic/_3C_3D m n <=> happ (happ h4/prim__rec/_3C m) n \/ m = n
% Assm [h4s_arithmetics_GREATERu_u_DEF]: !n m. h4/arithmetic/_3E m n <=> happ (happ h4/prim__rec/_3C n) m
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_topologys_topologyu_u_tybiju_c1]: !r. h4/topology/istopology r <=> h4/topology/open (h4/topology/topology0 r) = r
% Assm [h4s_topologys_mtopu_u_istopology]: !_0. (!m S_27. happ (happ _0 m) S_27 <=> (!x. happ S_27 x ==> (?e. h4/realax/real__lt (h4/real/real__of__num h4/num/0) e /\ (!y. h4/realax/real__lt (h4/topology/dist m (h4/pair/_2C x y)) e ==> happ S_27 y)))) ==> (!m. h4/topology/istopology (happ _0 m))
% Assm [h4s_arithmetics_NOTu_u_LEQ]: !n m. ~h4/arithmetic/_3C_3D m n <=> h4/arithmetic/_3C_3D (h4/num/SUC n) m
% 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_topologys_MR1u_u_LIMPT]: !x. h4/topology/limpt (h4/topology/mtop h4/topology/mr1) x h4/pred__set/UNIV
% Assm [h4s_predu_u_sets_SUBSETu_u_REFL]: !s. h4/pred__set/SUBSET s s
% Assm [h4s_topologys_METRICu_u_SAME]: !x m. h4/topology/dist m (h4/pair/_2C x x) = h4/real/real__of__num h4/num/0
% Assm [h4s_topologys_METRICu_u_TRIANGLE]: !z y x m. h4/real/real__lte (h4/topology/dist m (h4/pair/_2C x z)) (h4/realax/real__add (h4/topology/dist m (h4/pair/_2C x y)) (h4/topology/dist m (h4/pair/_2C y z)))
% Assm [h4s_numerals_numeralu_u_distribu_c30]: !n. happ (happ h4/arithmetic/_3E_3D h4/num/0) n <=> n = h4/num/0
% Assm [h4s_topologys_METRICu_u_SYM]: !y x m. h4/topology/dist m (h4/pair/_2C x y) = h4/topology/dist m (h4/pair/_2C y x)
% Assm [h4s_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_reals_REALu_u_SUBu_u_LT]: !y x. h4/realax/real__lt (h4/real/real__of__num h4/num/0) (h4/real/real__sub x y) <=> h4/realax/real__lt y x
% Assm [h4s_reals_REALu_u_LETu_u_TRANS]: !z y x. h4/real/real__lte x y /\ h4/realax/real__lt y z ==> h4/realax/real__lt x z
% Assm [h4s_reals_REALu_u_ADDu_u_SYM]: !y x. h4/realax/real__add x y = h4/realax/real__add y x
% Assm [h4s_reals_REALu_u_LTu_u_SUBu_u_LADD]: !z y x. h4/realax/real__lt x (h4/real/real__sub y z) <=> h4/realax/real__lt (h4/realax/real__add x z) y
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_predu_u_sets_SPECIFICATION]: !x P. h4/bool/IN x P <=> happ P x
% 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_bools_FALSITY]: !t. F ==> t
% Assm [h4s_reals_REALu_u_LTu_u_HALF1]: !d. h4/realax/real__lt (h4/real/real__of__num h4/num/0) (h4/real/_2F d (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)))) <=> h4/realax/real__lt (h4/real/real__of__num h4/num/0) d
% Assm [h4s_arithmetics_LESSu_u_EQu_u_TRANS]: !p n m. h4/arithmetic/_3C_3D m n /\ h4/arithmetic/_3C_3D n p ==> h4/arithmetic/_3C_3D m p
% Assm [h4s_arithmetics_LESSu_u_EQu_u_CASES]: !n m. h4/arithmetic/_3C_3D m n \/ h4/arithmetic/_3C_3D n m
% Assm [h4s_arithmetics_LESSu_u_EQu_u_REFL]: !m. h4/arithmetic/_3C_3D m m
% Assm [h4s_netss_dorder0]: !g. h4/nets/dorder g <=> (!x y. happ (happ g x) x /\ happ (happ g y) y ==> (?z. happ (happ g z) z /\ (!w. happ (happ g w) z ==> happ (happ g w) x /\ happ (happ g w) y)))
% Assm [h4s_bools_INu_u_DEF]: !x x. h4/bool/IN x x <=> happ x x
% Assm [h4s_topologys_limpt0]: !x top S_27. h4/topology/limpt top x S_27 <=> (!N. h4/topology/neigh top (h4/pair/_2C N x) ==> (?y. ~(x = y) /\ happ S_27 y /\ happ N y))
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% 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_reals_REALu_u_HALFu_u_DOUBLE]: !x. h4/realax/real__add (h4/real/_2F x (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)))) (h4/real/_2F x (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)))) = x
% Assm [h4s_reals_REALu_u_NOTu_u_LT]: !y x. ~h4/realax/real__lt x y <=> h4/real/real__lte y x
% Assm [h4s_reals_REALu_u_LTu_u_ADD2]: !z y x w. h4/realax/real__lt w x /\ h4/realax/real__lt y z ==> h4/realax/real__lt (h4/realax/real__add w y) (h4/realax/real__add x z)
% Assm [h4s_netss_DORDERu_u_LEMMA]: !g. h4/nets/dorder g ==> (!P Q. (?n. happ (happ g n) n /\ (!m. happ (happ g m) n ==> happ P m)) /\ (?n. happ (happ g n) n /\ (!m. happ (happ g m) n ==> happ Q m)) ==> (?n. happ (happ g n) n /\ (!m. happ (happ g m) n ==> happ P m /\ happ Q m)))
% Assm [h4s_topologys_METRICu_u_NZ]: !y x m. ~(x = y) ==> h4/realax/real__lt (h4/real/real__of__num h4/num/0) (h4/topology/dist m (h4/pair/_2C x y))
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_arithmetics_ADDu_u_CLAUSESu_c0]: !m. h4/arithmetic/_2B h4/num/0 m = m
% Assm [h4s_reals_abs0]: !x. h4/real/abs x = h4/bool/COND (h4/real/real__lte (h4/real/real__of__num h4/num/0) x) x (h4/realax/real__neg x)
% Assm [h4s_reals_REALu_u_LTu_u_HALF2]: !d. h4/realax/real__lt (h4/real/_2F d (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)))) d <=> h4/realax/real__lt (h4/real/real__of__num h4/num/0) d
% Assm [h4s_reals_REALu_u_LTu_u_IMPu_u_NE]: !y x. h4/realax/real__lt x y ==> ~(x = y)
% Assm [h4s_reals_REALu_u_LTu_u_IMPu_u_LE]: !y x. h4/realax/real__lt x y ==> h4/real/real__lte x y
% Assm [h4s_bools_ABSu_u_SIMP]: !t2 t1. t1 = t1
% Assm [h4s_reals_REALu_u_ADDu_u_RIDu_u_UNIQ]: !y x. h4/realax/real__add x y = x <=> y = h4/real/real__of__num h4/num/0
% Assm [h4s_predu_u_sets_UNIVu_u_DEF]: !x. happ h4/pred__set/UNIV x <=> T
% Assm [h4s_bools_CONDu_u_CLAUSESu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_topologys_MR1u_u_ADD]: !x d. h4/topology/dist h4/topology/mr1 (h4/pair/_2C x (h4/realax/real__add x d)) = h4/real/abs d
% Assm [h4s_reals_sumu_u_tupledu_u_primitive]: !_7. (!sum__tupled f n m. happ (happ (happ (happ _7 sum__tupled) f) n) m = h4/combin/I (h4/realax/real__add (happ sum__tupled (h4/pair/_2C (h4/pair/_2C n m) f)) (happ f (h4/arithmetic/_2B n m)))) ==> (!_6. (!sum__tupled f n v3. happ (happ (happ (happ _6 sum__tupled) f) n) v3 = h4/arithmetic/num__CASE v3 (h4/combin/I (h4/real/real__of__num h4/num/0)) (happ (happ (happ _7 sum__tupled) f) n)) ==> (!_5. (!sum__tupled f n. happ (happ (happ _5 sum__tupled) f) n = happ (happ (happ _6 sum__tupled) f) n) ==> (!_4. (!v sum__tupled f. happ (happ (happ _4 v) sum__tupled) f = h4/pair/pair__CASE v (happ (happ _5 sum__tupled) f)) ==> (!_3. (!sum__tupled v. happ (happ _3 sum__tupled) v = happ (happ _4 v) sum__tupled) ==> (!_2. (!sum__tupled a. happ (happ _2 sum__tupled) a = h4/pair/pair__CASE a (happ _3 sum__tupled)) ==> (!_1. (!sum__tupled. happ _1 sum__tupled = happ _2 sum__tupled) ==> (!_0. (!R. happ _0 R <=> h4/relation/WF R /\ (!f m n. happ (happ R (h4/pair/_2C (h4/pair/_2C n m) f)) (h4/pair/_2C (h4/pair/_2C n (h4/num/SUC m)) f))) ==> h4/real/sum__tupled = h4/relation/WFREC (h4/min/_40 _0) _1)))))))
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_reals_sumu_u_curried]: !x1 x. h4/real/sum x x1 = happ h4/real/sum__tupled (h4/pair/_2C x x1)
% Assm [h4s_reals_REALu_u_ADDu_u_LID]: !x. h4/realax/real__add (h4/real/real__of__num h4/num/0) x = x
% Assm [h4s_numerals_numeralu_u_distribu_c27]: !n. h4/arithmetic/_3C_3D (h4/arithmetic/NUMERAL n) h4/num/0 <=> h4/arithmetic/_3C_3D n h4/arithmetic/ZERO
% Assm [h4s_numerals_numeralu_u_lteu_c1]: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% Assm [h4s_arithmetics_ADDu_u_MONOu_u_LESSu_u_EQ]: !p n m. h4/arithmetic/_3C_3D (h4/arithmetic/_2B m n) (h4/arithmetic/_2B m p) <=> h4/arithmetic/_3C_3D n p
% Assm [h4s_arithmetics_SUCu_u_ONEu_u_ADD]: !n. h4/num/SUC n = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) n
% Assm [h4s_arithmetics_LESSu_u_EQUALu_u_ADD]: !n m. h4/arithmetic/_3C_3D m n ==> (?p. n = h4/arithmetic/_2B m p)
% Assm [h4s_arithmetics_MULTu_u_CLAUSESu_c2]: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm [h4s_arithmetics_LESSu_u_MONOu_u_ADDu_u_EQ]: !p n m. happ (happ h4/prim__rec/_3C (h4/arithmetic/_2B m p)) (h4/arithmetic/_2B n p) <=> happ (happ h4/prim__rec/_3C m) n
% Assm [h4s_arithmetics_LESSu_u_EQu_u_ADD]: !n m. h4/arithmetic/_3C_3D m (h4/arithmetic/_2B m n)
% Assm [h4s_arithmetics_NOTu_u_LESS]: !n m. ~happ (happ h4/prim__rec/_3C m) n <=> h4/arithmetic/_3C_3D n m
% Assm [h4s_arithmetics_ADDu_u_SYM]: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm [h4s_arithmetics_ADDu_u_ASSOC]: !p n m. h4/arithmetic/_2B m (h4/arithmetic/_2B n p) = h4/arithmetic/_2B (h4/arithmetic/_2B m n) p
% Assm [h4s_arithmetics_numu_u_caseu_u_defu_c0]: !v f. h4/arithmetic/num__CASE h4/num/0 v f = v
% Assm [h4s_arithmetics_numu_u_caseu_u_defu_c1]: !v n f. h4/arithmetic/num__CASE (h4/num/SUC n) v f = happ f n
% Assm [h4s_pairs_SND0]: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm [h4s_pairs_pairu_u_caseu_u_thm]: !y x f. h4/pair/pair__CASE (h4/pair/_2C x y) f = happ (happ f x) y
% Assm [h4s_pairs_FST0]: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm [h4s_pairs_PAIR]: !x. h4/pair/_2C (h4/pair/FST x) (h4/pair/SND x) = x
% Assm [h4s_primu_u_recs_WFu_u_LESS]: h4/relation/WF h4/prim__rec/_3C
% Assm [h4s_nums_INDUCTION]: !P. happ P h4/num/0 /\ (!n. happ P n ==> happ P (h4/num/SUC n)) ==> (!n. happ P n)
% Assm [h4s_relations_WFRECu_u_COROLLARY]: !f R M. f = h4/relation/WFREC R M ==> h4/relation/WF R ==> (!x. happ f x = happ (happ M (h4/relation/RESTRICT f R x)) x)
% Assm [h4s_combins_Iu_u_THM]: !x. h4/combin/I x = x
% Assm [h4s_relations_invu_u_imageu_u_def]: !f R x x'. happ (happ (h4/relation/inv__image R f) x) x' <=> happ (happ R (happ f x)) (happ f x')
% Assm [h4s_relations_WFu_u_invu_u_image]: !f R. h4/relation/WF R ==> h4/relation/WF (h4/relation/inv__image R f)
% Assm [h4s_relations_RESTRICTu_u_LEMMA]: !z y f R. happ (happ R y) z ==> happ (h4/relation/RESTRICT f R z) y = happ f y
% Assm [h4s_bools_IMPu_u_Fu_u_EQu_u_F]: !t. t ==> F <=> t <=> F
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c2]: !t. (F <=> t) <=> ~t
% Assm [h4s_numerals_iZ0]: !x. h4/numeral/iZ x = x
% Assm [h4s_bools_EQu_u_CLAUSESu_c3]: !t. (t <=> F) <=> ~t
% Assm [h4s_arithmetics_MULTu_u_CLAUSESu_c1]: !m. h4/arithmetic/_2A m h4/num/0 = h4/num/0
% Assm [h4s_arithmetics_SUBu_u_0u_c1]: !m. h4/arithmetic/_2D m h4/num/0 = m
% Assm [h4s_arithmetics_MULTu_u_CLAUSESu_c0]: !m. h4/arithmetic/_2A h4/num/0 m = h4/num/0
% Assm [h4s_bools_NOTu_u_FORALLu_u_THM]: !P. ~(!x. happ P x) <=> (?x. ~happ P x)
% Assm [h4s_arithmetics_LESSu_u_0u_u_CASES]: !m. h4/num/0 = m \/ happ (happ h4/prim__rec/_3C h4/num/0) m
% Assm [h4s_arithmetics_SUBu_u_0u_c0]: !m. h4/arithmetic/_2D h4/num/0 m = h4/num/0
% Assm [h4s_arithmetics_ADDu_u_CLAUSESu_c3]: !n m. h4/arithmetic/_2B m (h4/num/SUC n) = h4/num/SUC (h4/arithmetic/_2B m n)
% 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_arithmetics_ODD0u_c0]: h4/arithmetic/ODD h4/num/0 <=> F
% Assm [h4s_bools_IMPu_u_F]: !t. (t ==> F) ==> ~t
% Assm [h4s_arithmetics_EVEN0u_c0]: h4/arithmetic/EVEN h4/num/0 <=> T
% Assm [h4s_bools_Fu_u_IMP]: !t. ~t ==> t ==> F
% Assm [h4s_arithmetics_EXP0u_c1]: !n m. h4/arithmetic/EXP m (h4/num/SUC n) = h4/arithmetic/_2A m (h4/arithmetic/EXP m n)
% Goal: !x0 x d. h4/nets/tends x x0 (h4/pair/_2C (h4/topology/mtop d) h4/arithmetic/_3E_3D) <=> (!e. h4/realax/real__lt (h4/real/real__of__num h4/num/0) e ==> (?N. !n. happ (happ h4/arithmetic/_3E_3D n) N ==> h4/realax/real__lt (h4/topology/dist d (h4/pair/_2C (happ x n) x0)) e))
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_Q1288908,TV_Q1288904]: ![V_f, V_g]: (![V_x]: s(TV_Q1288904,happ(s(t_fun(TV_Q1288908,TV_Q1288904),V_f),s(TV_Q1288908,V_x))) = s(TV_Q1288904,happ(s(t_fun(TV_Q1288908,TV_Q1288904),V_g),s(TV_Q1288908,V_x))) => s(t_fun(TV_Q1288908,TV_Q1288904),V_f) = s(t_fun(TV_Q1288908,TV_Q1288904),V_g))).
fof(ah4s_pairs_UNCURRYu_u_DEF, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_y, V_x, V_f]: s(TV_u_27c,h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_topologys_MTOPu_u_OPEN, axiom, ![TV_u_27a]: ![V_m, V_Su_27]: (p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_topologys_open(s(t_h4s_topologys_topology(TV_u_27a),h4s_topologys_mtop(s(t_h4s_topologys_metric(TV_u_27a),V_m))))),s(t_fun(TV_u_27a,t_bool),V_Su_27)))) <=> ![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Su_27),s(TV_u_27a,V_x)))) => ?[V_e]: (p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_realaxs_real,V_e)))) & ![V_y]: (p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,h4s_topologys_dist(s(t_h4s_topologys_metric(TV_u_27a),V_m),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_y))))),s(t_h4s_realaxs_real,V_e)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Su_27),s(TV_u_27a,V_y))))))))).
fof(ah4s_arithmetics_GREATERu_u_EQ, axiom, ![V_n, V_m]: s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_arithmetics_u_3eu_3d),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_m))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))).
fof(ah4s_topologys_mtop0, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_m, V_Su_27]: (p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),happ(s(t_fun(t_h4s_topologys_metric(TV_u_27a),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_uu_0),s(t_h4s_topologys_metric(TV_u_27a),V_m))),s(t_fun(TV_u_27a,t_bool),V_Su_27)))) <=> ![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Su_27),s(TV_u_27a,V_x)))) => ?[V_e]: (p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_realaxs_real,V_e)))) & ![V_y]: (p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,h4s_topologys_dist(s(t_h4s_topologys_metric(TV_u_27a),V_m),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_y))))),s(t_h4s_realaxs_real,V_e)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Su_27),s(TV_u_27a,V_y)))))))) => ![V_m]: s(t_h4s_topologys_topology(TV_u_27a),h4s_topologys_mtop(s(t_h4s_topologys_metric(TV_u_27a),V_m))) = s(t_h4s_topologys_topology(TV_u_27a),h4s_topologys_topology0(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),happ(s(t_fun(t_h4s_topologys_metric(TV_u_27a),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_uu_0),s(t_h4s_topologys_metric(TV_u_27a),V_m))))))).
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_arithmetics_GREATERu_u_ORu_u_EQ, axiom, ![V_n, V_m]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_arithmetics_u_3eu_3d),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n)))) <=> (p(s(t_bool,h4s_arithmetics_u_3e(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) | s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,V_n)))).
fof(ah4s_topologys_BALLu_u_NEIGH, axiom, ![TV_u_27a]: ![V_x, V_m, V_e]: (p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_realaxs_real,V_e)))) => p(s(t_bool,h4s_topologys_neigh(s(t_h4s_topologys_topology(TV_u_27a),h4s_topologys_mtop(s(t_h4s_topologys_metric(TV_u_27a),V_m))),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27a),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),h4s_topologys_b(s(t_h4s_topologys_metric(TV_u_27a),V_m),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_realaxs_real),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_h4s_realaxs_real,V_e))))),s(TV_u_27a,V_x)))))))).
fof(ah4s_topologys_BALLu_u_OPEN, axiom, ![TV_u_27a]: ![V_x, V_m, V_e]: (p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_realaxs_real,V_e)))) => p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_topologys_open(s(t_h4s_topologys_topology(TV_u_27a),h4s_topologys_mtop(s(t_h4s_topologys_metric(TV_u_27a),V_m))))),s(t_fun(TV_u_27a,t_bool),h4s_topologys_b(s(t_h4s_topologys_metric(TV_u_27a),V_m),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_realaxs_real),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_h4s_realaxs_real,V_e)))))))))).
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_netss_MTOPu_u_TENDS, axiom, ![TV_u_27b,TV_u_27a]: ![V_x0, V_x, V_g, V_d]: (p(s(t_bool,h4s_netss_tends(s(t_fun(TV_u_27b,TV_u_27a),V_x),s(TV_u_27a,V_x0),s(t_h4s_pairs_prod(t_h4s_topologys_topology(TV_u_27a),t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool))),h4s_pairs_u_2c(s(t_h4s_topologys_topology(TV_u_27a),h4s_topologys_mtop(s(t_h4s_topologys_metric(TV_u_27a),V_d))),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_g)))))) <=> ![V_e]: (p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_realaxs_real,V_e)))) => ?[V_n]: (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_g),s(TV_u_27b,V_n))),s(TV_u_27b,V_n)))) & ![V_m]: (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_g),s(TV_u_27b,V_m))),s(TV_u_27b,V_n)))) => p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,h4s_topologys_dist(s(t_h4s_topologys_metric(TV_u_27a),V_d),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_x),s(TV_u_27b,V_m))),s(TV_u_27a,V_x0))))),s(t_h4s_realaxs_real,V_e))))))))).
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_topologys_MTOPu_u_LIMPT, axiom, ![TV_u_27a]: ![V_x, V_m, V_Su_27]: (p(s(t_bool,h4s_topologys_limpt(s(t_h4s_topologys_topology(TV_u_27a),h4s_topologys_mtop(s(t_h4s_topologys_metric(TV_u_27a),V_m))),s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_Su_27)))) <=> ![V_e]: (p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_realaxs_real,V_e)))) => ?[V_y]: (~ (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),V_Su_27),s(TV_u_27a,V_y)))) & p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,h4s_topologys_dist(s(t_h4s_topologys_metric(TV_u_27a),V_m),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_y))))),s(t_h4s_realaxs_real,V_e))))))))).
fof(ah4s_netss_MTOPu_u_TENDSu_u_UNIQ, axiom, ![TV_u_27b,TV_u_27a]: ![V_x1, V_x0, V_x, V_g, V_d]: (p(s(t_bool,h4s_netss_dorder(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_g)))) => ((p(s(t_bool,h4s_netss_tends(s(t_fun(TV_u_27b,TV_u_27a),V_x),s(TV_u_27a,V_x0),s(t_h4s_pairs_prod(t_h4s_topologys_topology(TV_u_27a),t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool))),h4s_pairs_u_2c(s(t_h4s_topologys_topology(TV_u_27a),h4s_topologys_mtop(s(t_h4s_topologys_metric(TV_u_27a),V_d))),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_g)))))) & p(s(t_bool,h4s_netss_tends(s(t_fun(TV_u_27b,TV_u_27a),V_x),s(TV_u_27a,V_x1),s(t_h4s_pairs_prod(t_h4s_topologys_topology(TV_u_27a),t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool))),h4s_pairs_u_2c(s(t_h4s_topologys_topology(TV_u_27a),h4s_topologys_mtop(s(t_h4s_topologys_metric(TV_u_27a),V_d))),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_g))))))) => s(TV_u_27a,V_x0) = s(TV_u_27a,V_x1)))).
fof(ah4s_topologys_ball, axiom, ![TV_u_27a]: ![V_x, V_m, V_e, V_xi_]: s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),h4s_topologys_b(s(t_h4s_topologys_metric(TV_u_27a),V_m),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_realaxs_real),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_h4s_realaxs_real,V_e))))),s(TV_u_27a,V_xi_))) = s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,h4s_topologys_dist(s(t_h4s_topologys_metric(TV_u_27a),V_m),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_xi_))))),s(t_h4s_realaxs_real,V_e)))).
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_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_netss_tends0, axiom, ![TV_u_27a,TV_u_27b]: ![V_top, V_s, V_l, V_g]: (p(s(t_bool,h4s_netss_tends(s(t_fun(TV_u_27b,TV_u_27a),V_s),s(TV_u_27a,V_l),s(t_h4s_pairs_prod(t_h4s_topologys_topology(TV_u_27a),t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool))),h4s_pairs_u_2c(s(t_h4s_topologys_topology(TV_u_27a),V_top),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_g)))))) <=> ![V_N]: (p(s(t_bool,h4s_topologys_neigh(s(t_h4s_topologys_topology(TV_u_27a),V_top),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27a),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_N),s(TV_u_27a,V_l)))))) => ?[V_n]: (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_g),s(TV_u_27b,V_n))),s(TV_u_27b,V_n)))) & ![V_m]: (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_g),s(TV_u_27b,V_m))),s(TV_u_27b,V_n)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_N),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_s),s(TV_u_27b,V_m))))))))))).
fof(ah4s_reals_SUMu_u_ZERO, axiom, ![V_f, V_N]: (![V_n]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_arithmetics_u_3eu_3d),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_N)))) => s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_f),s(t_h4s_nums_num,V_n))) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))) => ![V_m, V_n]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_arithmetics_u_3eu_3d),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_N)))) => s(t_h4s_realaxs_real,h4s_reals_sum(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_f))) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))))).
fof(ah4s_arithmetics_NOTu_u_GREATERu_u_EQ, axiom, ![V_n, V_m]: (~ (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_arithmetics_u_3eu_3d),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n))))) <=> p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_topologys_neigh0, axiom, ![TV_u_27a]: ![V_x, V_top, V_N]: (p(s(t_bool,h4s_topologys_neigh(s(t_h4s_topologys_topology(TV_u_27a),V_top),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27a),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_N),s(TV_u_27a,V_x)))))) <=> ?[V_P]: (p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_topologys_open(s(t_h4s_topologys_topology(TV_u_27a),V_top))),s(t_fun(TV_u_27a,t_bool),V_P)))) & (p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27a,t_bool),V_N)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))))))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_netss_DORDERu_u_NGE, axiom, p(s(t_bool,h4s_netss_dorder(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_arithmetics_u_3eu_3d))))).
fof(ah4s_arithmetics_LESSu_u_ORu_u_EQ, axiom, ![V_n, V_m]: (p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) <=> (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n)))) | s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,V_n)))).
fof(ah4s_arithmetics_GREATERu_u_DEF, axiom, ![V_n, V_m]: s(t_bool,h4s_arithmetics_u_3e(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_m)))).
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_topologys_topologyu_u_tybiju_c1, axiom, ![TV_u_27a]: ![V_r]: (p(s(t_bool,h4s_topologys_istopology(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_r)))) <=> s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_topologys_open(s(t_h4s_topologys_topology(TV_u_27a),h4s_topologys_topology0(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_r))))) = s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_r))).
fof(ah4s_topologys_mtopu_u_istopology, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_m, V_Su_27]: (p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),happ(s(t_fun(t_h4s_topologys_metric(TV_u_27a),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_uu_0),s(t_h4s_topologys_metric(TV_u_27a),V_m))),s(t_fun(TV_u_27a,t_bool),V_Su_27)))) <=> ![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Su_27),s(TV_u_27a,V_x)))) => ?[V_e]: (p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_realaxs_real,V_e)))) & ![V_y]: (p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,h4s_topologys_dist(s(t_h4s_topologys_metric(TV_u_27a),V_m),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_y))))),s(t_h4s_realaxs_real,V_e)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Su_27),s(TV_u_27a,V_y)))))))) => ![V_m]: p(s(t_bool,h4s_topologys_istopology(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),happ(s(t_fun(t_h4s_topologys_metric(TV_u_27a),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_uu_0),s(t_h4s_topologys_metric(TV_u_27a),V_m)))))))).
fof(ah4s_arithmetics_NOTu_u_LEQ, axiom, ![V_n, V_m]: (~ (p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))))) <=> p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_m)))))).
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_topologys_MR1u_u_LIMPT, axiom, ![V_x]: p(s(t_bool,h4s_topologys_limpt(s(t_h4s_topologys_topology(t_h4s_realaxs_real),h4s_topologys_mtop(s(t_h4s_topologys_metric(t_h4s_realaxs_real),h4s_topologys_mr1))),s(t_h4s_realaxs_real,V_x),s(t_fun(t_h4s_realaxs_real,t_bool),h4s_predu_u_sets_univ))))).
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_topologys_METRICu_u_SAME, axiom, ![TV_u_27a]: ![V_x, V_m]: s(t_h4s_realaxs_real,h4s_topologys_dist(s(t_h4s_topologys_metric(TV_u_27a),V_m),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_x))))) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))).
fof(ah4s_topologys_METRICu_u_TRIANGLE, axiom, ![TV_u_27a]: ![V_z, V_y, V_x, V_m]: p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,h4s_topologys_dist(s(t_h4s_topologys_metric(TV_u_27a),V_m),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_z))))),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,h4s_topologys_dist(s(t_h4s_topologys_metric(TV_u_27a),V_m),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_y))))),s(t_h4s_realaxs_real,h4s_topologys_dist(s(t_h4s_topologys_metric(TV_u_27a),V_m),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_y),s(TV_u_27a,V_z))))))))))).
fof(ah4s_numerals_numeralu_u_distribu_c30, axiom, ![V_n]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_arithmetics_u_3eu_3d),s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_nums_num,V_n)))) <=> s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_nums_0))).
fof(ah4s_topologys_METRICu_u_SYM, axiom, ![TV_u_27a]: ![V_y, V_x, V_m]: s(t_h4s_realaxs_real,h4s_topologys_dist(s(t_h4s_topologys_metric(TV_u_27a),V_m),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_y))))) = s(t_h4s_realaxs_real,h4s_topologys_dist(s(t_h4s_topologys_metric(TV_u_27a),V_m),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_y),s(TV_u_27a,V_x)))))).
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_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
fof(ah4s_reals_REALu_u_SUBu_u_LT, axiom, ![V_y, V_x]: s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_realaxs_real,h4s_reals_realu_u_sub(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y))))) = s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,V_y),s(t_h4s_realaxs_real,V_x)))).
fof(ah4s_reals_REALu_u_LETu_u_TRANS, axiom, ![V_z, V_y, V_x]: ((p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y)))) & p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,V_y),s(t_h4s_realaxs_real,V_z))))) => p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_z)))))).
fof(ah4s_reals_REALu_u_ADDu_u_SYM, axiom, ![V_y, V_x]: s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,V_y),s(t_h4s_realaxs_real,V_x)))).
fof(ah4s_reals_REALu_u_LTu_u_SUBu_u_LADD, axiom, ![V_z, V_y, V_x]: s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,h4s_reals_realu_u_sub(s(t_h4s_realaxs_real,V_y),s(t_h4s_realaxs_real,V_z))))) = s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_z))),s(t_h4s_realaxs_real,V_y)))).
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_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_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_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_reals_REALu_u_LTu_u_HALF1, axiom, ![V_d]: s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_realaxs_real,h4s_reals_u_2f(s(t_h4s_realaxs_real,V_d),s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))))) = s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_realaxs_real,V_d)))).
fof(ah4s_arithmetics_LESSu_u_EQu_u_TRANS, axiom, ![V_p, V_n, V_m]: ((p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) & p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))))) => p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p)))))).
fof(ah4s_arithmetics_LESSu_u_EQu_u_CASES, axiom, ![V_n, V_m]: (p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) | p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))))).
fof(ah4s_arithmetics_LESSu_u_EQu_u_REFL, axiom, ![V_m]: p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_m))))).
fof(ah4s_netss_dorder0, axiom, ![TV_u_27a]: ![V_g]: (p(s(t_bool,h4s_netss_dorder(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_g)))) <=> ![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_g),s(TV_u_27a,V_x))),s(TV_u_27a,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_g),s(TV_u_27a,V_y))),s(TV_u_27a,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_g),s(TV_u_27a,V_z))),s(TV_u_27a,V_z)))) & ![V_w]: (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_g),s(TV_u_27a,V_w))),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_g),s(TV_u_27a,V_w))),s(TV_u_27a,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_g),s(TV_u_27a,V_w))),s(TV_u_27a,V_y)))))))))).
fof(ah4s_bools_INu_u_DEF, axiom, ![TV_u_27a]: ![V_x, V_x0]: s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_x0))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_x0),s(TV_u_27a,V_x)))).
fof(ah4s_topologys_limpt0, axiom, ![TV_u_27a]: ![V_x, V_top, V_Su_27]: (p(s(t_bool,h4s_topologys_limpt(s(t_h4s_topologys_topology(TV_u_27a),V_top),s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_Su_27)))) <=> ![V_N]: (p(s(t_bool,h4s_topologys_neigh(s(t_h4s_topologys_topology(TV_u_27a),V_top),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27a),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_N),s(TV_u_27a,V_x)))))) => ?[V_y]: (~ (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),V_Su_27),s(TV_u_27a,V_y)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_N),s(TV_u_27a,V_y))))))))).
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_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_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_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_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_conj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) & p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_r))))) & ((p(s(t_bool,V_q)) | ~ (p(s(t_bool,V_p)))) & (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p)))))))).
fof(ah4s_sats_ANDu_u_INVu_u_IMP, axiom, ![V_A]: (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_A))) => p(s(t_bool,f))))).
fof(ah4s_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_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
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_reals_REALu_u_HALFu_u_DOUBLE, axiom, ![V_x]: s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,h4s_reals_u_2f(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))),s(t_h4s_realaxs_real,h4s_reals_u_2f(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))))) = s(t_h4s_realaxs_real,V_x)).
fof(ah4s_reals_REALu_u_NOTu_u_LT, axiom, ![V_y, V_x]: (~ (p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y))))) <=> p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,V_y),s(t_h4s_realaxs_real,V_x)))))).
fof(ah4s_reals_REALu_u_LTu_u_ADD2, axiom, ![V_z, V_y, V_x, V_w]: ((p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,V_w),s(t_h4s_realaxs_real,V_x)))) & p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,V_y),s(t_h4s_realaxs_real,V_z))))) => p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,V_w),s(t_h4s_realaxs_real,V_y))),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_z)))))))).
fof(ah4s_netss_DORDERu_u_LEMMA, axiom, ![TV_u_27a]: ![V_g]: (p(s(t_bool,h4s_netss_dorder(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_g)))) => ![V_P, V_Q]: ((?[V_n]: (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_g),s(TV_u_27a,V_n))),s(TV_u_27a,V_n)))) & ![V_m]: (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_g),s(TV_u_27a,V_m))),s(TV_u_27a,V_n)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_m)))))) & ?[V_n]: (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_g),s(TV_u_27a,V_n))),s(TV_u_27a,V_n)))) & ![V_m]: (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_g),s(TV_u_27a,V_m))),s(TV_u_27a,V_n)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_m))))))) => ?[V_n]: (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_g),s(TV_u_27a,V_n))),s(TV_u_27a,V_n)))) & ![V_m]: (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_g),s(TV_u_27a,V_m))),s(TV_u_27a,V_n)))) => (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_m)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_m)))))))))).
fof(ah4s_topologys_METRICu_u_NZ, axiom, ![TV_u_27a]: ![V_y, V_x, V_m]: (~ (s(TV_u_27a,V_x) = s(TV_u_27a,V_y)) => p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_realaxs_real,h4s_topologys_dist(s(t_h4s_topologys_metric(TV_u_27a),V_m),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27a,V_y)))))))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_arithmetics_ADDu_u_CLAUSESu_c0, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_reals_abs0, axiom, ![V_x]: s(t_h4s_realaxs_real,h4s_reals_abs(s(t_h4s_realaxs_real,V_x))) = s(t_h4s_realaxs_real,h4s_bools_cond(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_realaxs_real,V_x))),s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_neg(s(t_h4s_realaxs_real,V_x)))))).
fof(ah4s_reals_REALu_u_LTu_u_HALF2, axiom, ![V_d]: s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,h4s_reals_u_2f(s(t_h4s_realaxs_real,V_d),s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))),s(t_h4s_realaxs_real,V_d))) = s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_realaxs_real,V_d)))).
fof(ah4s_reals_REALu_u_LTu_u_IMPu_u_NE, axiom, ![V_y, V_x]: (p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y)))) => ~ (s(t_h4s_realaxs_real,V_x) = s(t_h4s_realaxs_real,V_y)))).
fof(ah4s_reals_REALu_u_LTu_u_IMPu_u_LE, axiom, ![V_y, V_x]: (p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y)))) => p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y)))))).
fof(ah4s_bools_ABSu_u_SIMP, axiom, ![TV_u_27b,TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,V_t1) = s(TV_u_27a,V_t1)).
fof(ah4s_reals_REALu_u_ADDu_u_RIDu_u_UNIQ, axiom, ![V_y, V_x]: (s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y))) = s(t_h4s_realaxs_real,V_x) <=> s(t_h4s_realaxs_real,V_y) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))))).
fof(ah4s_predu_u_sets_UNIVu_u_DEF, axiom, ![TV_u_27a]: ![V_x]: s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_univ),s(TV_u_27a,V_x))) = s(t_bool,t)).
fof(ah4s_bools_CONDu_u_CLAUSESu_c0, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,t),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t1)).
fof(ah4s_topologys_MR1u_u_ADD, axiom, ![V_x, V_d]: s(t_h4s_realaxs_real,h4s_topologys_dist(s(t_h4s_topologys_metric(t_h4s_realaxs_real),h4s_topologys_mr1),s(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_pairs_u_2c(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_d))))))) = s(t_h4s_realaxs_real,h4s_reals_abs(s(t_h4s_realaxs_real,V_d)))).
fof(ah4s_reals_sumu_u_tupledu_u_primitive, axiom, ![V_uu_7]: (![V_sumu_u_tupled, V_f, V_n, V_m]: s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_realaxs_real))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_h4s_realaxs_real),t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_realaxs_real)))),V_uu_7),s(t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_h4s_realaxs_real),V_sumu_u_tupled))),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_f))),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_m))) = s(t_h4s_realaxs_real,h4s_combins_i(s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_h4s_realaxs_real),V_sumu_u_tupled),s(t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),h4s_pairs_u_2c(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m))),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_f))))),s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_f),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m))))))))) => ![V_uu_6]: (![V_sumu_u_tupled, V_f, V_n, V_v3]: s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_realaxs_real))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_h4s_realaxs_real),t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_realaxs_real)))),V_uu_6),s(t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_h4s_realaxs_real),V_sumu_u_tupled))),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_f))),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_v3))) = s(t_h4s_realaxs_real,h4s_arithmetics_numu_u_case(s(t_h4s_nums_num,V_v3),s(t_h4s_realaxs_real,h4s_combins_i(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))))),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_realaxs_real))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_h4s_realaxs_real),t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_realaxs_real)))),V_uu_7),s(t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_h4s_realaxs_real),V_sumu_u_tupled))),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_f))),s(t_h4s_nums_num,V_n))))) => ![V_uu_5]: (![V_sumu_u_tupled, V_f, V_n]: s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_realaxs_real))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_h4s_realaxs_real),t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_realaxs_real)))),V_uu_5),s(t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_h4s_realaxs_real),V_sumu_u_tupled))),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_f))),s(t_h4s_nums_num,V_n))) = s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_realaxs_real))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_h4s_realaxs_real),t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_realaxs_real)))),V_uu_6),s(t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_h4s_realaxs_real),V_sumu_u_tupled))),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_f))),s(t_h4s_nums_num,V_n))) => ![V_uu_4]: (![V_v, V_sumu_u_tupled, V_f]: s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_h4s_realaxs_real),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_h4s_realaxs_real),t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_h4s_realaxs_real)),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_h4s_realaxs_real),t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_h4s_realaxs_real))),V_uu_4),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_v))),s(t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_h4s_realaxs_real),V_sumu_u_tupled))),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_f))) = s(t_h4s_realaxs_real,h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_v),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_realaxs_real))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_h4s_realaxs_real),t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_realaxs_real)))),V_uu_5),s(t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_h4s_realaxs_real),V_sumu_u_tupled))),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_f))))) => ![V_uu_3]: (![V_sumu_u_tupled, V_v]: s(t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_h4s_realaxs_real),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_h4s_realaxs_real)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_h4s_realaxs_real),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_h4s_realaxs_real))),V_uu_3),s(t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_h4s_realaxs_real),V_sumu_u_tupled))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_v))) = s(t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_h4s_realaxs_real),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_h4s_realaxs_real),t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_h4s_realaxs_real)),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_h4s_realaxs_real),t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_h4s_realaxs_real))),V_uu_4),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_v))),s(t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_h4s_realaxs_real),V_sumu_u_tupled))) => ![V_uu_2]: (![V_sumu_u_tupled, V_a]: s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_h4s_realaxs_real),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_h4s_realaxs_real),t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_h4s_realaxs_real)),V_uu_2),s(t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_h4s_realaxs_real),V_sumu_u_tupled))),s(t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),V_a))) = s(t_h4s_realaxs_real,h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),V_a),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_h4s_realaxs_real)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_h4s_realaxs_real),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_h4s_realaxs_real))),V_uu_3),s(t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_h4s_realaxs_real),V_sumu_u_tupled))))) => ![V_uu_1]: (![V_sumu_u_tupled]: s(t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_h4s_realaxs_real),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_h4s_realaxs_real),t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_h4s_realaxs_real)),V_uu_1),s(t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_h4s_realaxs_real),V_sumu_u_tupled))) = s(t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_h4s_realaxs_real),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_h4s_realaxs_real),t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_h4s_realaxs_real)),V_uu_2),s(t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_h4s_realaxs_real),V_sumu_u_tupled))) => ![V_uu_0]: (![V_R]: (p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_bool)),t_bool),V_uu_0),s(t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_bool)),V_R)))) <=> (p(s(t_bool,h4s_relations_wf(s(t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_bool)),V_R)))) & ![V_f, V_m, V_n]: p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_bool)),V_R),s(t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),h4s_pairs_u_2c(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m))),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_f))))),s(t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),h4s_pairs_u_2c(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))))),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_f)))))))) => s(t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_h4s_realaxs_real),h4s_reals_sumu_u_tupled) = s(t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_h4s_realaxs_real),h4s_relations_wfrec(s(t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_bool)),h4s_mins_u_40(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_bool)),t_bool),V_uu_0))),s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_h4s_realaxs_real),t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_h4s_realaxs_real)),V_uu_1)))))))))))).
fof(ah4s_bools_SELECTu_u_AX, axiom, ![TV_u_27a]: ![V_x, V_P]: (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_P),s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_P)))))))).
fof(ah4s_reals_sumu_u_curried, axiom, ![V_x1, V_x]: s(t_h4s_realaxs_real,h4s_reals_sum(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_x1))) = s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_h4s_realaxs_real),h4s_reals_sumu_u_tupled),s(t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),h4s_pairs_u_2c(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_x1)))))).
fof(ah4s_reals_REALu_u_ADDu_u_LID, axiom, ![V_x]: s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_realaxs_real,V_x))) = s(t_h4s_realaxs_real,V_x)).
fof(ah4s_numerals_numeralu_u_distribu_c27, axiom, ![V_n]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_nums_0))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_zero)))).
fof(ah4s_numerals_numeralu_u_lteu_c1, axiom, ![V_n]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_zero))) = s(t_bool,f)).
fof(ah4s_arithmetics_ADDu_u_MONOu_u_LESSu_u_EQ, axiom, ![V_p, V_n, V_m]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p))))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p)))).
fof(ah4s_arithmetics_SUCu_u_ONEu_u_ADD, axiom, ![V_n]: s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_n)))).
fof(ah4s_arithmetics_LESSu_u_EQUALu_u_ADD, axiom, ![V_n, V_m]: (p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) => ?[V_p]: s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p))))).
fof(ah4s_arithmetics_MULTu_u_CLAUSESu_c2, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_arithmetics_LESSu_u_MONOu_u_ADDu_u_EQ, axiom, ![V_p, V_n, V_m]: s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p))))),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))))) = s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n)))).
fof(ah4s_arithmetics_LESSu_u_EQu_u_ADD, axiom, ![V_n, V_m]: p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))))))).
fof(ah4s_arithmetics_NOTu_u_LESS, axiom, ![V_n, V_m]: (~ (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n))))) <=> p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))))).
fof(ah4s_arithmetics_ADDu_u_SYM, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))).
fof(ah4s_arithmetics_ADDu_u_ASSOC, axiom, ![V_p, V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_p)))).
fof(ah4s_arithmetics_numu_u_caseu_u_defu_c0, axiom, ![TV_u_27a]: ![V_v, V_f]: s(TV_u_27a,h4s_arithmetics_numu_u_case(s(t_h4s_nums_num,h4s_nums_0),s(TV_u_27a,V_v),s(t_fun(t_h4s_nums_num,TV_u_27a),V_f))) = s(TV_u_27a,V_v)).
fof(ah4s_arithmetics_numu_u_caseu_u_defu_c1, axiom, ![TV_u_27a]: ![V_v, V_n, V_f]: s(TV_u_27a,h4s_arithmetics_numu_u_case(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))),s(TV_u_27a,V_v),s(t_fun(t_h4s_nums_num,TV_u_27a),V_f))) = s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_f),s(t_h4s_nums_num,V_n)))).
fof(ah4s_pairs_SND0, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x]: s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27b,V_y)).
fof(ah4s_pairs_pairu_u_caseu_u_thm, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_y, V_x, V_f]: s(TV_u_27a,h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(TV_u_27b,TV_u_27c),h4s_pairs_u_2c(s(TV_u_27b,V_x),s(TV_u_27c,V_y))),s(t_fun(TV_u_27b,t_fun(TV_u_27c,TV_u_27a)),V_f))) = s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27c,TV_u_27a)),V_f),s(TV_u_27b,V_x))),s(TV_u_27c,V_y)))).
fof(ah4s_pairs_FST0, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x]: s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27a,V_x)).
fof(ah4s_pairs_PAIR, axiom, ![TV_u_27a,TV_u_27b]: ![V_x]: s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))),s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))))) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x)).
fof(ah4s_primu_u_recs_WFu_u_LESS, axiom, p(s(t_bool,h4s_relations_wf(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c))))).
fof(ah4s_nums_INDUCTION, axiom, ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,h4s_nums_0)))) & ![V_n]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_n)))) => p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n)))))))) => ![V_n]: p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_relations_WFRECu_u_COROLLARY, axiom, ![TV_u_27b,TV_u_27a]: ![V_f, V_R, V_M]: (s(t_fun(TV_u_27a,TV_u_27b),V_f) = s(t_fun(TV_u_27a,TV_u_27b),h4s_relations_wfrec(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_M))) => (p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) => ![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),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_M),s(t_fun(TV_u_27a,TV_u_27b),h4s_relations_restrict(s(t_fun(TV_u_27a,TV_u_27b),V_f),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_combins_Iu_u_THM, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,h4s_combins_i(s(TV_u_27a,V_x))) = s(TV_u_27a,V_x)).
fof(ah4s_relations_invu_u_imageu_u_def, axiom, ![TV_u_27b,TV_u_27a]: ![V_f, V_R, V_x, V_xi_]: s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_invu_u_image(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(TV_u_27a,V_x))),s(TV_u_27a,V_xi_))) = s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_xi_)))))).
fof(ah4s_relations_WFu_u_invu_u_image, axiom, ![TV_u_27a,TV_u_27b]: ![V_f, V_R]: (p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R)))) => p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_invu_u_image(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_f)))))))).
fof(ah4s_relations_RESTRICTu_u_LEMMA, axiom, ![TV_u_27b,TV_u_27a]: ![V_z, V_y, V_f, 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)),V_R),s(TV_u_27a,V_y))),s(TV_u_27a,V_z)))) => s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),h4s_relations_restrict(s(t_fun(TV_u_27a,TV_u_27b),V_f),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))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_y))))).
fof(ah4s_bools_IMPu_u_Fu_u_EQu_u_F, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) <=> s(t_bool,V_t) = s(t_bool,f))).
fof(ah4s_bools_ETAu_u_AX, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f)))).
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_c2, axiom, ![V_t]: (s(t_bool,f) = s(t_bool,V_t) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_numerals_iZ0, axiom, ![V_x]: s(t_h4s_nums_num,h4s_numerals_iz(s(t_h4s_nums_num,V_x))) = s(t_h4s_nums_num,V_x)).
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_arithmetics_MULTu_u_CLAUSESu_c1, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,h4s_nums_0)).
fof(ah4s_arithmetics_SUBu_u_0u_c1, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_arithmetics_MULTu_u_CLAUSESu_c0, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,h4s_nums_0)).
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_arithmetics_LESSu_u_0u_u_CASES, axiom, ![V_m]: (s(t_h4s_nums_num,h4s_nums_0) = s(t_h4s_nums_num,V_m) | p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_nums_num,V_m)))))).
fof(ah4s_arithmetics_SUBu_u_0u_c0, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,h4s_nums_0)).
fof(ah4s_arithmetics_ADDu_u_CLAUSESu_c3, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))) = s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))))).
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_arithmetics_ODD0u_c0, axiom, s(t_bool,h4s_arithmetics_odd(s(t_h4s_nums_num,h4s_nums_0))) = s(t_bool,f)).
fof(ah4s_bools_IMPu_u_F, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) => ~ (p(s(t_bool,V_t))))).
fof(ah4s_arithmetics_EVEN0u_c0, axiom, s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,h4s_nums_0))) = s(t_bool,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_arithmetics_EXP0u_c1, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_exp(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_arithmetics_exp(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))))).
fof(ch4s_netss_SEQu_u_TENDS, conjecture, ![TV_u_27a]: ![V_x0, V_x, V_d]: (p(s(t_bool,h4s_netss_tends(s(t_fun(t_h4s_nums_num,TV_u_27a),V_x),s(TV_u_27a,V_x0),s(t_h4s_pairs_prod(t_h4s_topologys_topology(TV_u_27a),t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool))),h4s_pairs_u_2c(s(t_h4s_topologys_topology(TV_u_27a),h4s_topologys_mtop(s(t_h4s_topologys_metric(TV_u_27a),V_d))),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_arithmetics_u_3eu_3d)))))) <=> ![V_e]: (p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_realaxs_real,V_e)))) => ?[V_N]: ![V_n]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_arithmetics_u_3eu_3d),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_N)))) => p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,h4s_topologys_dist(s(t_h4s_topologys_metric(TV_u_27a),V_d),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_x),s(t_h4s_nums_num,V_n))),s(TV_u_27a,V_x0))))),s(t_h4s_realaxs_real,V_e)))))))).
