%   ORIGINAL: h4/transc/DIVISION__EXISTS
% 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/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/AND__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_c2: !t. F ==> t <=> T
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/COND__CLAUSES_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/bool/COND__CLAUSES_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/bool/DE__MORGAN__THM_c0: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm: h4/num/NOT__SUC: !n. ~(h4/num/SUC n = h4/num/0)
% Assm: h4/prim__rec/NOT__LESS__0: !n. ~h4/prim__rec/_3C n h4/num/0
% Assm: h4/prim__rec/LESS__THM: !n m. h4/prim__rec/_3C m (h4/num/SUC n) <=> m = n \/ h4/prim__rec/_3C m n
% Assm: h4/pair/UNCURRY__DEF: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = f x y
% Assm: h4/arithmetic/ONE: h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO) = h4/num/SUC h4/num/0
% Assm: h4/real/REAL__NOT__LE: !y x. ~h4/real/real__lte x y <=> h4/realax/real__lt y x
% Assm: h4/real/REAL__LE__REFL: !x. h4/real/real__lte x x
% Assm: h4/real/REAL__LE__LT: !y x. h4/real/real__lte x y <=> h4/realax/real__lt x y \/ x = y
% Assm: h4/real/REAL__LTE__TRANS: !z y x. h4/realax/real__lt x y /\ h4/real/real__lte y z ==> h4/realax/real__lt x z
% 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__LE__TRANS: !z y x. h4/real/real__lte x y /\ h4/real/real__lte y z ==> h4/real/real__lte x z
% Assm: h4/real/REAL__LT__01: h4/realax/real__lt (h4/real/real__of__num h4/num/0) (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))
% Assm: h4/seq/BOLZANO__LEMMA: !P. (!a b c. h4/real/real__lte a b /\ h4/real/real__lte b c /\ P (h4/pair/_2C a b) /\ P (h4/pair/_2C b c) ==> P (h4/pair/_2C a c)) /\ (!x. ?d. h4/realax/real__lt (h4/real/real__of__num h4/num/0) d /\ (!a b. h4/real/real__lte a x /\ h4/real/real__lte x b /\ h4/realax/real__lt (h4/real/real__sub b a) d ==> P (h4/pair/_2C a b))) ==> (!a b. h4/real/real__lte a b ==> P (h4/pair/_2C a b))
% Assm: h4/transc/tdiv0: !p b a D. h4/transc/tdiv (h4/pair/_2C a b) (h4/pair/_2C D p) <=> h4/transc/division (h4/pair/_2C a b) D /\ (!n. h4/real/real__lte (D n) (p n) /\ h4/real/real__lte (p n) (D (h4/num/SUC n)))
% Assm: h4/transc/gauge0: !g E. h4/transc/gauge E g <=> (!x. E x ==> h4/realax/real__lt (h4/real/real__of__num h4/num/0) (g x))
% Assm: h4/transc/fine0: !p g D. h4/transc/fine g (h4/pair/_2C D p) <=> (!n. h4/prim__rec/_3C n (h4/transc/dsize D) ==> h4/realax/real__lt (h4/real/real__sub (D (h4/num/SUC n)) (D n)) (g (p n)))
% Assm: h4/transc/DIVISION__0: !b a. a = b ==> h4/transc/dsize (\n. h4/bool/COND (n = h4/num/0) a b) = h4/num/0
% Assm: h4/transc/DIVISION__1: !b a. h4/realax/real__lt a b ==> h4/transc/dsize (\n. h4/bool/COND (n = h4/num/0) a b) = h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)
% Assm: h4/transc/DIVISION__SINGLE: !b a. h4/real/real__lte a b ==> h4/transc/division (h4/pair/_2C a b) (\n. h4/bool/COND (n = h4/num/0) a b)
% Assm: h4/transc/DIVISION__APPEND: !g c b a. (?D1 p1. h4/transc/tdiv (h4/pair/_2C a b) (h4/pair/_2C D1 p1) /\ h4/transc/fine g (h4/pair/_2C D1 p1)) /\ (?D2 p2. h4/transc/tdiv (h4/pair/_2C b c) (h4/pair/_2C D2 p2) /\ h4/transc/fine g (h4/pair/_2C D2 p2)) ==> (?D p. h4/transc/tdiv (h4/pair/_2C a c) (h4/pair/_2C D p) /\ h4/transc/fine g (h4/pair/_2C D p))
% Goal: !g b a. h4/real/real__lte a b /\ h4/transc/gauge (\x. h4/real/real__lte a x /\ h4/real/real__lte x b) g ==> (?D p. h4/transc/tdiv (h4/pair/_2C a b) (h4/pair/_2C D p) /\ h4/transc/fine g (h4/pair/_2C D p))
%   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_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_ANDu_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_c2]: !t. F ==> t <=> T
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_CONDu_u_CLAUSESu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_bools_CONDu_u_CLAUSESu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c0]: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm [h4s_nums_NOTu_u_SUC]: !n. ~(h4/num/SUC n = h4/num/0)
% Assm [h4s_primu_u_recs_NOTu_u_LESSu_u_0]: !n. ~h4/prim__rec/_3C n h4/num/0
% Assm [h4s_primu_u_recs_LESSu_u_THM]: !n m. h4/prim__rec/_3C m (h4/num/SUC n) <=> m = n \/ h4/prim__rec/_3C m n
% 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_arithmetics_ONE]: h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO) = h4/num/SUC h4/num/0
% Assm [h4s_reals_REALu_u_NOTu_u_LE]: !y x. ~h4/real/real__lte x y <=> h4/realax/real__lt y x
% Assm [h4s_reals_REALu_u_LEu_u_REFL]: !x. h4/real/real__lte x x
% Assm [h4s_reals_REALu_u_LEu_u_LT]: !y x. h4/real/real__lte x y <=> h4/realax/real__lt x y \/ x = y
% Assm [h4s_reals_REALu_u_LTEu_u_TRANS]: !z y x. h4/realax/real__lt x y /\ h4/real/real__lte y z ==> h4/realax/real__lt x z
% 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_LEu_u_TRANS]: !z y x. h4/real/real__lte x y /\ h4/real/real__lte y z ==> h4/real/real__lte x z
% Assm [h4s_reals_REALu_u_LTu_u_01]: h4/realax/real__lt (h4/real/real__of__num h4/num/0) (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))
% Assm [h4s_seqs_BOLZANOu_u_LEMMA]: !P. (!a b c. h4/real/real__lte a b /\ h4/real/real__lte b c /\ happ P (h4/pair/_2C a b) /\ happ P (h4/pair/_2C b c) ==> happ P (h4/pair/_2C a c)) /\ (!x. ?d. h4/realax/real__lt (h4/real/real__of__num h4/num/0) d /\ (!a b. h4/real/real__lte a x /\ h4/real/real__lte x b /\ h4/realax/real__lt (h4/real/real__sub b a) d ==> happ P (h4/pair/_2C a b))) ==> (!a b. h4/real/real__lte a b ==> happ P (h4/pair/_2C a b))
% Assm [h4s_transcs_tdiv0]: !p b a D. h4/transc/tdiv (h4/pair/_2C a b) (h4/pair/_2C D p) <=> h4/transc/division (h4/pair/_2C a b) D /\ (!n. h4/real/real__lte (happ D n) (happ p n) /\ h4/real/real__lte (happ p n) (happ D (h4/num/SUC n)))
% Assm [h4s_transcs_gauge0]: !g E. h4/transc/gauge E g <=> (!x. happ E x ==> h4/realax/real__lt (h4/real/real__of__num h4/num/0) (happ g x))
% Assm [h4s_transcs_fine0]: !p g D. h4/transc/fine g (h4/pair/_2C D p) <=> (!n. h4/prim__rec/_3C n (h4/transc/dsize D) ==> h4/realax/real__lt (h4/real/real__sub (happ D (h4/num/SUC n)) (happ D n)) (happ g (happ p n)))
% Assm [h4s_transcs_DIVISIONu_u_0]: !_0. (!a b n. ?v. (v <=> n = h4/num/0) /\ happ (happ (happ _0 a) b) n = h4/bool/COND v a b) ==> (!b a. a = b ==> h4/transc/dsize (happ (happ _0 a) b) = h4/num/0)
% Assm [h4s_transcs_DIVISIONu_u_1]: !_0. (!a b n. ?v. (v <=> n = h4/num/0) /\ happ (happ (happ _0 a) b) n = h4/bool/COND v a b) ==> (!b a. h4/realax/real__lt a b ==> h4/transc/dsize (happ (happ _0 a) b) = h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm [h4s_transcs_DIVISIONu_u_SINGLE]: !_0. (!a b n. ?v. (v <=> n = h4/num/0) /\ happ (happ (happ _0 a) b) n = h4/bool/COND v a b) ==> (!b a. h4/real/real__lte a b ==> h4/transc/division (h4/pair/_2C a b) (happ (happ _0 a) b))
% Assm [h4s_transcs_DIVISIONu_u_APPEND]: !g c b a. (?D1 p1. h4/transc/tdiv (h4/pair/_2C a b) (h4/pair/_2C D1 p1) /\ h4/transc/fine g (h4/pair/_2C D1 p1)) /\ (?D2 p2. h4/transc/tdiv (h4/pair/_2C b c) (h4/pair/_2C D2 p2) /\ h4/transc/fine g (h4/pair/_2C D2 p2)) ==> (?D p. h4/transc/tdiv (h4/pair/_2C a c) (h4/pair/_2C D p) /\ h4/transc/fine g (h4/pair/_2C D p))
% Goal: !_0. (!a b x. happ (happ (happ _0 a) b) x <=> h4/real/real__lte a x /\ h4/real/real__lte x b) ==> (!g b a. h4/real/real__lte a b /\ h4/transc/gauge (happ (happ _0 a) b) g ==> (?D p. h4/transc/tdiv (h4/pair/_2C a b) (h4/pair/_2C D p) /\ h4/transc/fine g (h4/pair/_2C D p)))
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_Q196364,TV_Q196360]: ![V_f, V_g]: (![V_x]: s(TV_Q196360,happ(s(t_fun(TV_Q196364,TV_Q196360),V_f),s(TV_Q196364,V_x))) = s(TV_Q196360,happ(s(t_fun(TV_Q196364,TV_Q196360),V_g),s(TV_Q196364,V_x))) => s(t_fun(TV_Q196364,TV_Q196360),V_f) = s(t_fun(TV_Q196364,TV_Q196360),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_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_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_c2, axiom, ![V_t]: ((p(s(t_bool,f)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_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_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) <=> p(s(t_bool,V_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_bools_CONDu_u_CLAUSESu_c1, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,f),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
fof(ah4s_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_nums_NOTu_u_SUC, axiom, ![V_n]: ~ (s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_nums_0))).
fof(ah4s_primu_u_recs_NOTu_u_LESSu_u_0, axiom, ![V_n]: ~ (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_nums_0)))))).
fof(ah4s_primu_u_recs_LESSu_u_THM, axiom, ![V_n, V_m]: (p(s(t_bool,h4s_primu_u_recs_u_3c(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,V_m) = s(t_h4s_nums_num,V_n) | p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))))))).
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_arithmetics_ONE, axiom, 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,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0)))).
fof(ah4s_reals_REALu_u_NOTu_u_LE, axiom, ![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_x)))))).
fof(ah4s_reals_REALu_u_LEu_u_REFL, axiom, ![V_x]: p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_x))))).
fof(ah4s_reals_REALu_u_LEu_u_LT, axiom, ![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_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_LTEu_u_TRANS, axiom, ![V_z, 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_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_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_LEu_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_reals_realu_u_lte(s(t_h4s_realaxs_real,V_y),s(t_h4s_realaxs_real,V_z))))) => p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_z)))))).
fof(ah4s_reals_REALu_u_LTu_u_01, axiom, 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_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))))).
fof(ah4s_seqs_BOLZANOu_u_LEMMA, axiom, ![V_P]: ((![V_a, V_b, V_c]: ((p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,V_a),s(t_h4s_realaxs_real,V_b)))) & (p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,V_b),s(t_h4s_realaxs_real,V_c)))) & (p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),t_bool),V_P),s(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_pairs_u_2c(s(t_h4s_realaxs_real,V_a),s(t_h4s_realaxs_real,V_b)))))) & p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),t_bool),V_P),s(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_pairs_u_2c(s(t_h4s_realaxs_real,V_b),s(t_h4s_realaxs_real,V_c))))))))) => p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),t_bool),V_P),s(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_pairs_u_2c(s(t_h4s_realaxs_real,V_a),s(t_h4s_realaxs_real,V_c))))))) & ![V_x]: ?[V_d]: (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_d)))) & ![V_a, V_b]: ((p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,V_a),s(t_h4s_realaxs_real,V_x)))) & (p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_b)))) & p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,h4s_reals_realu_u_sub(s(t_h4s_realaxs_real,V_b),s(t_h4s_realaxs_real,V_a))),s(t_h4s_realaxs_real,V_d)))))) => p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),t_bool),V_P),s(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_pairs_u_2c(s(t_h4s_realaxs_real,V_a),s(t_h4s_realaxs_real,V_b))))))))) => ![V_a, V_b]: (p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,V_a),s(t_h4s_realaxs_real,V_b)))) => p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),t_bool),V_P),s(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_pairs_u_2c(s(t_h4s_realaxs_real,V_a),s(t_h4s_realaxs_real,V_b))))))))).
fof(ah4s_transcs_tdiv0, axiom, ![V_p, V_b, V_a, V_D]: (p(s(t_bool,h4s_transcs_tdiv(s(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_pairs_u_2c(s(t_h4s_realaxs_real,V_a),s(t_h4s_realaxs_real,V_b))),s(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),h4s_pairs_u_2c(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_D),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_p)))))) <=> (p(s(t_bool,h4s_transcs_division(s(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_pairs_u_2c(s(t_h4s_realaxs_real,V_a),s(t_h4s_realaxs_real,V_b))),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_D)))) & ![V_n]: (p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_D),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_p),s(t_h4s_nums_num,V_n)))))) & p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_p),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_D),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n)))))))))))).
fof(ah4s_transcs_gauge0, axiom, ![V_g, V_E]: (p(s(t_bool,h4s_transcs_gauge(s(t_fun(t_h4s_realaxs_real,t_bool),V_E),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_g)))) <=> ![V_x]: (p(s(t_bool,happ(s(t_fun(t_h4s_realaxs_real,t_bool),V_E),s(t_h4s_realaxs_real,V_x)))) => 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,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_g),s(t_h4s_realaxs_real,V_x))))))))).
fof(ah4s_transcs_fine0, axiom, ![V_p, V_g, V_D]: (p(s(t_bool,h4s_transcs_fine(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_g),s(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),h4s_pairs_u_2c(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_D),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_p)))))) <=> ![V_n]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_transcs_dsize(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_D)))))) => p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,h4s_reals_realu_u_sub(s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_D),s(t_h4s_nums_num,h4s_nums_suc(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_D),s(t_h4s_nums_num,V_n))))),s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_g),s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_p),s(t_h4s_nums_num,V_n))))))))))).
fof(ah4s_transcs_DIVISIONu_u_0, axiom, ![V_uu_0]: (![V_a, V_b, V_n]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_nums_0)) & s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_fun(t_h4s_nums_num,t_h4s_realaxs_real))),V_uu_0),s(t_h4s_realaxs_real,V_a))),s(t_h4s_realaxs_real,V_b))),s(t_h4s_nums_num,V_n))) = s(t_h4s_realaxs_real,h4s_bools_cond(s(t_bool,V_v),s(t_h4s_realaxs_real,V_a),s(t_h4s_realaxs_real,V_b)))) => ![V_b, V_a]: (s(t_h4s_realaxs_real,V_a) = s(t_h4s_realaxs_real,V_b) => s(t_h4s_nums_num,h4s_transcs_dsize(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_fun(t_h4s_nums_num,t_h4s_realaxs_real))),V_uu_0),s(t_h4s_realaxs_real,V_a))),s(t_h4s_realaxs_real,V_b))))) = s(t_h4s_nums_num,h4s_nums_0)))).
fof(ah4s_transcs_DIVISIONu_u_1, axiom, ![V_uu_0]: (![V_a, V_b, V_n]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_nums_0)) & s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_fun(t_h4s_nums_num,t_h4s_realaxs_real))),V_uu_0),s(t_h4s_realaxs_real,V_a))),s(t_h4s_realaxs_real,V_b))),s(t_h4s_nums_num,V_n))) = s(t_h4s_realaxs_real,h4s_bools_cond(s(t_bool,V_v),s(t_h4s_realaxs_real,V_a),s(t_h4s_realaxs_real,V_b)))) => ![V_b, V_a]: (p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,V_a),s(t_h4s_realaxs_real,V_b)))) => s(t_h4s_nums_num,h4s_transcs_dsize(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_fun(t_h4s_nums_num,t_h4s_realaxs_real))),V_uu_0),s(t_h4s_realaxs_real,V_a))),s(t_h4s_realaxs_real,V_b))))) = s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero)))))))).
fof(ah4s_transcs_DIVISIONu_u_SINGLE, axiom, ![V_uu_0]: (![V_a, V_b, V_n]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_nums_0)) & s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_fun(t_h4s_nums_num,t_h4s_realaxs_real))),V_uu_0),s(t_h4s_realaxs_real,V_a))),s(t_h4s_realaxs_real,V_b))),s(t_h4s_nums_num,V_n))) = s(t_h4s_realaxs_real,h4s_bools_cond(s(t_bool,V_v),s(t_h4s_realaxs_real,V_a),s(t_h4s_realaxs_real,V_b)))) => ![V_b, V_a]: (p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,V_a),s(t_h4s_realaxs_real,V_b)))) => p(s(t_bool,h4s_transcs_division(s(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_pairs_u_2c(s(t_h4s_realaxs_real,V_a),s(t_h4s_realaxs_real,V_b))),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_fun(t_h4s_nums_num,t_h4s_realaxs_real))),V_uu_0),s(t_h4s_realaxs_real,V_a))),s(t_h4s_realaxs_real,V_b))))))))).
fof(ah4s_transcs_DIVISIONu_u_APPEND, axiom, ![V_g, V_c, V_b, V_a]: ((?[V_D1, V_p1]: (p(s(t_bool,h4s_transcs_tdiv(s(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_pairs_u_2c(s(t_h4s_realaxs_real,V_a),s(t_h4s_realaxs_real,V_b))),s(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),h4s_pairs_u_2c(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_D1),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_p1)))))) & p(s(t_bool,h4s_transcs_fine(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_g),s(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),h4s_pairs_u_2c(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_D1),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_p1))))))) & ?[V_D2, V_p2]: (p(s(t_bool,h4s_transcs_tdiv(s(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_pairs_u_2c(s(t_h4s_realaxs_real,V_b),s(t_h4s_realaxs_real,V_c))),s(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),h4s_pairs_u_2c(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_D2),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_p2)))))) & p(s(t_bool,h4s_transcs_fine(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_g),s(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),h4s_pairs_u_2c(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_D2),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_p2)))))))) => ?[V_D, V_p]: (p(s(t_bool,h4s_transcs_tdiv(s(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_pairs_u_2c(s(t_h4s_realaxs_real,V_a),s(t_h4s_realaxs_real,V_c))),s(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),h4s_pairs_u_2c(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_D),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_p)))))) & p(s(t_bool,h4s_transcs_fine(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_g),s(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),h4s_pairs_u_2c(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_D),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_p))))))))).
fof(ch4s_transcs_DIVISIONu_u_EXISTS, conjecture, ![V_uu_0]: (![V_a, V_b, V_x]: (p(s(t_bool,happ(s(t_fun(t_h4s_realaxs_real,t_bool),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_bool)),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_bool))),V_uu_0),s(t_h4s_realaxs_real,V_a))),s(t_h4s_realaxs_real,V_b))),s(t_h4s_realaxs_real,V_x)))) <=> (p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,V_a),s(t_h4s_realaxs_real,V_x)))) & p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_b)))))) => ![V_g, V_b, V_a]: ((p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,V_a),s(t_h4s_realaxs_real,V_b)))) & p(s(t_bool,h4s_transcs_gauge(s(t_fun(t_h4s_realaxs_real,t_bool),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_bool)),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_bool))),V_uu_0),s(t_h4s_realaxs_real,V_a))),s(t_h4s_realaxs_real,V_b))),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_g))))) => ?[V_D, V_p]: (p(s(t_bool,h4s_transcs_tdiv(s(t_h4s_pairs_prod(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_pairs_u_2c(s(t_h4s_realaxs_real,V_a),s(t_h4s_realaxs_real,V_b))),s(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),h4s_pairs_u_2c(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_D),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_p)))))) & p(s(t_bool,h4s_transcs_fine(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_g),s(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),h4s_pairs_u_2c(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_D),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_p)))))))))).
