%   ORIGINAL: h4/lim/IVT
% 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/lim/CONTL__LIM: !x f. h4/lim/contl f x <=> h4/lim/tends__real__real f (f x) x
% Assm: h4/lim/tends__real__real0: !x0 l f. h4/lim/tends__real__real f l x0 <=> h4/nets/tends f l (h4/pair/_2C (h4/topology/mtop h4/topology/mr1) (h4/nets/tendsto (h4/pair/_2C h4/topology/mr1 x0)))
% Assm: h4/nets/DORDER__TENDSTO: !x m. h4/nets/dorder (h4/nets/tendsto (h4/pair/_2C m x))
% Assm: h4/lim/LIM__MUL: !x m l g f. h4/lim/tends__real__real f l x /\ h4/lim/tends__real__real g m x ==> h4/lim/tends__real__real (\x0. h4/realax/real__mul (f x0) (g x0)) (h4/realax/real__mul l m) x
% Assm: h4/lim/LIM__INV: !x l f. h4/lim/tends__real__real f l x /\ ~(l = h4/real/real__of__num h4/num/0) ==> h4/lim/tends__real__real (\x0. h4/realax/inv (f x0)) (h4/realax/inv l) x
% Assm: h4/lim/LIM__DIV: !x m l g f. h4/lim/tends__real__real f l x /\ h4/lim/tends__real__real g m x /\ ~(m = h4/real/real__of__num h4/num/0) ==> h4/lim/tends__real__real (\x0. h4/real/_2F (f x0) (g x0)) (h4/real/_2F l m) x
% Assm: h4/lim/LIM__SUB: !x m l g f. h4/lim/tends__real__real f l x /\ h4/lim/tends__real__real g m x ==> h4/lim/tends__real__real (\x0. h4/real/real__sub (f x0) (g x0)) (h4/real/real__sub l m) x
% Assm: h4/lim/LIM__ADD: !x m l g f. h4/lim/tends__real__real f l x /\ h4/lim/tends__real__real g m x ==> h4/lim/tends__real__real (\x0. h4/realax/real__add (f x0) (g x0)) (h4/realax/real__add l m) x
% Assm: h4/lim/contl0: !x f. h4/lim/contl f x <=> h4/lim/tends__real__real (\h. f (h4/realax/real__add x h)) (f x) (h4/real/real__of__num h4/num/0)
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/lim/CONT__INV: !x f. h4/lim/contl f x /\ ~(f x = h4/real/real__of__num h4/num/0) ==> h4/lim/contl (\x0. h4/realax/inv (f x0)) x
% Assm: h4/lim/DIFF__CARAT: !x l f. h4/lim/diffl f l x <=> (?g. (!z. h4/real/real__sub (f z) (f x) = h4/realax/real__mul (g z) (h4/real/real__sub z x)) /\ h4/lim/contl g x /\ g x = l)
% Assm: h4/lim/CONT__DIV: !x g f. h4/lim/contl f x /\ h4/lim/contl g x /\ ~(g x = h4/real/real__of__num h4/num/0) ==> h4/lim/contl (\x0. h4/real/_2F (f x0) (g x0)) x
% Assm: h4/lim/CONT__COMPOSE: !x g f. h4/lim/contl f x /\ h4/lim/contl g (f x) ==> h4/lim/contl (\x0. g (f x0)) x
% Assm: h4/lim/CONT__SUB: !x g f. h4/lim/contl f x /\ h4/lim/contl g x ==> h4/lim/contl (\x0. h4/real/real__sub (f x0) (g x0)) x
% Assm: h4/lim/CONT__ADD: !x g f. h4/lim/contl f x /\ h4/lim/contl g x ==> h4/lim/contl (\x0. h4/realax/real__add (f x0) (g x0)) x
% Assm: h4/lim/CONT__MUL: !x g f. h4/lim/contl f x /\ h4/lim/contl g x ==> h4/lim/contl (\x0. h4/realax/real__mul (f x0) (g x0)) x
% Assm: h4/lim/CONT__NEG: !x f. h4/lim/contl f x ==> h4/lim/contl (\x0. h4/realax/real__neg (f x0)) x
% Assm: h4/lim/LIM: !y0 x0 f. h4/lim/tends__real__real f y0 x0 <=> (!e. h4/realax/real__lt (h4/real/real__of__num h4/num/0) e ==> (?d. h4/realax/real__lt (h4/real/real__of__num h4/num/0) d /\ (!x. h4/realax/real__lt (h4/real/real__of__num h4/num/0) (h4/real/abs (h4/real/real__sub x x0)) /\ h4/realax/real__lt (h4/real/abs (h4/real/real__sub x x0)) d ==> h4/realax/real__lt (h4/real/abs (h4/real/real__sub (f x) y0)) e)))
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/lim/LIM__NEG: !x l f. h4/lim/tends__real__real f l x <=> h4/lim/tends__real__real (\x0. h4/realax/real__neg (f x0)) (h4/realax/real__neg l) x
% Assm: h4/lim/DIFF__CONT: !x l f. h4/lim/diffl f l x ==> h4/lim/contl f x
% Assm: h4/lim/diffl0: !x l f. h4/lim/diffl f l x <=> h4/lim/tends__real__real (\h. h4/real/_2F (h4/real/real__sub (f (h4/realax/real__add x h)) (f x)) h) l (h4/real/real__of__num h4/num/0)
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/nets/NET__NULL: !x0 x g. h4/nets/tends x x0 (h4/pair/_2C (h4/topology/mtop h4/topology/mr1) g) <=> h4/nets/tends (\n. h4/real/real__sub (x n) x0) (h4/real/real__of__num h4/num/0) (h4/pair/_2C (h4/topology/mtop h4/topology/mr1) g)
% Assm: h4/lim/LIM__EQUAL: !x0 l g f. (!x. ~(x = x0) ==> f x = g x) ==> (h4/lim/tends__real__real f l x0 <=> h4/lim/tends__real__real g l x0)
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/real/REAL__SUB__RZERO: !x. h4/real/real__sub x (h4/real/real__of__num h4/num/0) = x
% Assm: h4/lim/CONT__CONST: !x k. h4/lim/contl (\x0. k) x
% Assm: h4/real/REAL__SUB__ADD2: !y x. h4/realax/real__add y (h4/real/real__sub x y) = x
% Assm: h4/real/REAL__SUB__0: !y x. h4/real/real__sub x y = h4/real/real__of__num h4/num/0 <=> x = y
% Assm: h4/lim/LIM__UNIQ: !x m l f. h4/lim/tends__real__real f l x /\ h4/lim/tends__real__real f m x ==> l = m
% Assm: h4/lim/LIM__NULL: !x l f. h4/lim/tends__real__real f l x <=> h4/lim/tends__real__real (\x0. h4/real/real__sub (f x0) l) (h4/real/real__of__num h4/num/0) x
% Assm: h4/lim/LIM__TRANSFORM: !x0 l g f. h4/lim/tends__real__real (\x. h4/real/real__sub (f x) (g x)) (h4/real/real__of__num h4/num/0) x0 /\ h4/lim/tends__real__real g l x0 ==> h4/lim/tends__real__real f l x0
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> 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/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/bool/ABS__SIMP: !t2 t1. (\x. t1) t2 = t1
% Assm: h4/lim/LIM__CONST: !x k. h4/lim/tends__real__real (\x0. k) k x
% Assm: h4/nets/NET__INV: !g. h4/nets/dorder g ==> (!x x0. h4/nets/tends x x0 (h4/pair/_2C (h4/topology/mtop h4/topology/mr1) g) /\ ~(x0 = h4/real/real__of__num h4/num/0) ==> h4/nets/tends (\n. h4/realax/inv (x n)) (h4/realax/inv x0) (h4/pair/_2C (h4/topology/mtop h4/topology/mr1) g))
% Assm: h4/nets/NET__NEG: !g. h4/nets/dorder g ==> (!x x0. h4/nets/tends x x0 (h4/pair/_2C (h4/topology/mtop h4/topology/mr1) g) <=> h4/nets/tends (\n. h4/realax/real__neg (x n)) (h4/realax/real__neg x0) (h4/pair/_2C (h4/topology/mtop h4/topology/mr1) g))
% Assm: h4/nets/NET__DIV: !g. h4/nets/dorder g ==> (!x x0 y y0. h4/nets/tends x x0 (h4/pair/_2C (h4/topology/mtop h4/topology/mr1) g) /\ h4/nets/tends y y0 (h4/pair/_2C (h4/topology/mtop h4/topology/mr1) g) /\ ~(y0 = h4/real/real__of__num h4/num/0) ==> h4/nets/tends (\n. h4/real/_2F (x n) (y n)) (h4/real/_2F x0 y0) (h4/pair/_2C (h4/topology/mtop h4/topology/mr1) g))
% Assm: h4/nets/NET__MUL: !g. h4/nets/dorder g ==> (!x y x0 y0. h4/nets/tends x x0 (h4/pair/_2C (h4/topology/mtop h4/topology/mr1) g) /\ h4/nets/tends y y0 (h4/pair/_2C (h4/topology/mtop h4/topology/mr1) g) ==> h4/nets/tends (\n. h4/realax/real__mul (x n) (y n)) (h4/realax/real__mul x0 y0) (h4/pair/_2C (h4/topology/mtop h4/topology/mr1) g))
% Assm: h4/nets/NET__SUB: !g. h4/nets/dorder g ==> (!x x0 y y0. h4/nets/tends x x0 (h4/pair/_2C (h4/topology/mtop h4/topology/mr1) g) /\ h4/nets/tends y y0 (h4/pair/_2C (h4/topology/mtop h4/topology/mr1) g) ==> h4/nets/tends (\n. h4/real/real__sub (x n) (y n)) (h4/real/real__sub x0 y0) (h4/pair/_2C (h4/topology/mtop h4/topology/mr1) g))
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/nets/NET__ADD: !g. h4/nets/dorder g ==> (!x x0 y y0. h4/nets/tends x x0 (h4/pair/_2C (h4/topology/mtop h4/topology/mr1) g) /\ h4/nets/tends y y0 (h4/pair/_2C (h4/topology/mtop h4/topology/mr1) g) ==> h4/nets/tends (\n. h4/realax/real__add (x n) (y n)) (h4/realax/real__add x0 y0) (h4/pair/_2C (h4/topology/mtop h4/topology/mr1) g))
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/real/REAL__SUB__REFL: !x. h4/real/real__sub x x = h4/real/real__of__num h4/num/0
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/real/REAL__ADD__SUB: !y x. h4/real/real__sub (h4/realax/real__add x y) x = y
% Assm: h4/real/REAL__DIV__RMUL: !y x. ~(y = h4/real/real__of__num h4/num/0) ==> h4/realax/real__mul (h4/real/_2F x y) y = x
% Assm: h4/real/REAL__MUL__RZERO: !x. h4/realax/real__mul x (h4/real/real__of__num h4/num/0) = h4/real/real__of__num h4/num/0
% Assm: h4/real/ABS__NZ: !x. ~(x = h4/real/real__of__num h4/num/0) <=> h4/realax/real__lt (h4/real/real__of__num h4/num/0) (h4/real/abs x)
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> t
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/real/ABS__0: h4/real/abs (h4/real/real__of__num h4/num/0) = h4/real/real__of__num h4/num/0
% Assm: h4/nets/LIM__TENDS2: !y0 x0 m2 m1 f. h4/topology/limpt (h4/topology/mtop m1) x0 h4/pred__set/UNIV ==> (h4/nets/tends f y0 (h4/pair/_2C (h4/topology/mtop m2) (h4/nets/tendsto (h4/pair/_2C m1 x0))) <=> (!e. h4/realax/real__lt (h4/real/real__of__num h4/num/0) e ==> (?d. h4/realax/real__lt (h4/real/real__of__num h4/num/0) d /\ (!x. h4/realax/real__lt (h4/real/real__of__num h4/num/0) (h4/topology/dist m1 (h4/pair/_2C x x0)) /\ h4/realax/real__lt (h4/topology/dist m1 (h4/pair/_2C x x0)) d ==> h4/realax/real__lt (h4/topology/dist m2 (h4/pair/_2C (f x) y0)) e))))
% Assm: h4/topology/MR1__LIMPT: !x. h4/topology/limpt (h4/topology/mtop h4/topology/mr1) x h4/pred__set/UNIV
% Assm: h4/topology/MR1__DEF: !y x. h4/topology/dist h4/topology/mr1 (h4/pair/_2C x y) = h4/real/abs (h4/real/real__sub y x)
% Assm: h4/real/ABS__SUB: !y x. h4/real/abs (h4/real/real__sub x y) = h4/real/abs (h4/real/real__sub y x)
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/lim/differentiable0: !x f. h4/lim/differentiable f x <=> (?l. h4/lim/diffl f l x)
% Assm: h4/lim/DIFF__UNIQ: !x m l f. h4/lim/diffl f l x /\ h4/lim/diffl f m x ==> l = m
% 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/bool/FALSITY: !t. F ==> 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/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/real/REAL__MUL__RID: !x. h4/realax/real__mul x (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) = x
% 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/real/REAL__MUL__ASSOC: !z y x. h4/realax/real__mul x (h4/realax/real__mul y z) = h4/realax/real__mul (h4/realax/real__mul x y) z
% Assm: h4/real/real__div: !y x. h4/real/_2F x y = h4/realax/real__mul x (h4/realax/inv y)
% Assm: h4/real/REAL__MUL__RINV: !x. ~(x = h4/real/real__of__num h4/num/0) ==> h4/realax/real__mul x (h4/realax/inv x) = h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% 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/ABS__TRIANGLE: !y x. h4/real/real__lte (h4/real/abs (h4/realax/real__add x y)) (h4/realax/real__add (h4/real/abs x) (h4/real/abs y))
% Assm: h4/real/REAL__SUB__TRIANGLE: !c b a. h4/realax/real__add (h4/real/real__sub a b) (h4/real/real__sub b c) = h4/real/real__sub a c
% 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__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/real/REAL__DOWN2: !y x. h4/realax/real__lt (h4/real/real__of__num h4/num/0) x /\ h4/realax/real__lt (h4/real/real__of__num h4/num/0) y ==> (?z. h4/realax/real__lt (h4/real/real__of__num h4/num/0) z /\ h4/realax/real__lt z x /\ h4/realax/real__lt z y)
% 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/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__REFL: !x. h4/real/real__lte x x
% Assm: h4/real/REAL__LT__TRANS: !z y x. h4/realax/real__lt x y /\ h4/realax/real__lt y z ==> h4/realax/real__lt x z
% Assm: h4/real/REAL__MIN__ALT_c1: !y x. h4/real/real__lte y x ==> h4/real/min x y = y
% Assm: h4/real/REAL__LE__NEG2: !y x. h4/real/real__lte (h4/realax/real__neg x) (h4/realax/real__neg y) <=> h4/real/real__lte y x
% Assm: h4/real/REAL__MIN__ALT_c0: !y x. h4/real/real__lte x y ==> h4/real/min x y = x
% Assm: h4/real/REAL__LE__SUP: !x p. (?y. p y) /\ (?y. !z. p z ==> h4/real/real__lte z y) ==> (h4/real/real__lte x (h4/real/sup p) <=> (!y. (!z. p z ==> h4/real/real__lte z y) ==> h4/real/real__lte x y))
% Assm: h4/real/REAL__LT__NEG: !y x. h4/realax/real__lt (h4/realax/real__neg x) (h4/realax/real__neg y) <=> h4/realax/real__lt y x
% Assm: h4/real/REAL__NOT__LT: !y x. ~h4/realax/real__lt x y <=> h4/real/real__lte y 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/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/real/min__def: !y x. h4/real/min x y = h4/bool/COND (h4/real/real__lte x y) x y
% Assm: h4/bool/bool__case__thm_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/bool/COND__CONG: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% Assm: h4/bool/DISJ__ASSOC: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm: h4/bool/DISJ__SYM: !B A. A \/ B <=> B \/ A
% Assm: h4/bool/DE__MORGAN__THM_c1: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm: h4/bool/BOOL__CASES__AX: !t. (t <=> T) \/ (t <=> F)
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/bool/CONJ__ASSOC: !t3 t2 t1. t1 /\ t2 /\ t3 <=> (t1 /\ t2) /\ t3
% Assm: h4/bool/OR__DEF: $or = (\t1 t2. !t. (t1 ==> t) ==> (t2 ==> t) ==> t)
% Assm: h4/real/REAL__LE__ANTISYM: !y x. h4/real/real__lte x y /\ h4/real/real__lte y x <=> x = y
% Assm: h4/bool/F__DEF: F <=> (!t. t)
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/real/REAL__LE__EPSILON: !y x. (!e. h4/realax/real__lt (h4/real/real__of__num h4/num/0) e ==> h4/real/real__lte x (h4/realax/real__add y e)) ==> h4/real/real__lte x y
% Assm: h4/bool/EQ__CLAUSES_c3: !t. (t <=> F) <=> ~t
% Assm: h4/real/real__lt: !y x. h4/realax/real__lt x y <=> ~h4/real/real__lte y x
% Assm: h4/real/REAL__SUP__LE: !P. (?x. P x) /\ (?z. !x. P x ==> h4/real/real__lte x z) ==> (!y. (?x. P x /\ h4/realax/real__lt y x) <=> h4/realax/real__lt y (h4/real/sup P))
% Assm: h4/real/REAL__LE__SUB__RADD: !z y x. h4/real/real__lte (h4/real/real__sub x y) z <=> h4/real/real__lte x (h4/realax/real__add z y)
% Assm: h4/real/REAL__LT__SUB__RADD: !z y x. h4/realax/real__lt (h4/real/real__sub x y) z <=> h4/realax/real__lt x (h4/realax/real__add z y)
% Assm: h4/real/REAL__LE__SUB__LADD: !z y x. h4/real/real__lte x (h4/real/real__sub y z) <=> h4/real/real__lte (h4/realax/real__add x z) y
% Assm: h4/bool/NOT__FORALL__THM: !P. ~(!x. P x) <=> (?x. ~P x)
% Assm: h4/bool/NOT__EXISTS__THM: !P. ~(?x. P x) <=> (!x. ~P x)
% Assm: h4/real/REAL__LT__ADDR: !y x. h4/realax/real__lt x (h4/realax/real__add x y) <=> h4/realax/real__lt (h4/real/real__of__num h4/num/0) y
% Goal: !y f b a. h4/real/real__lte a b /\ (h4/real/real__lte (f a) y /\ h4/real/real__lte y (f b)) /\ (!x. h4/real/real__lte a x /\ h4/real/real__lte x b ==> h4/lim/contl f x) ==> (?x. h4/real/real__lte a x /\ h4/real/real__lte x b /\ f x = y)
%   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_lims_CONTLu_u_LIM]: !x f. h4/lim/contl f x <=> h4/lim/tends__real__real f (happ f x) x
% Assm [h4s_lims_tendsu_u_realu_u_real0]: !x0 l f. h4/lim/tends__real__real f l x0 <=> h4/nets/tends f l (h4/pair/_2C (h4/topology/mtop h4/topology/mr1) (h4/nets/tendsto (h4/pair/_2C h4/topology/mr1 x0)))
% Assm [h4s_netss_DORDERu_u_TENDSTO]: !x m. h4/nets/dorder (h4/nets/tendsto (h4/pair/_2C m x))
% Assm [h4s_lims_LIMu_u_MUL]: !_0. (!f g x0. happ (happ (happ _0 f) g) x0 = h4/realax/real__mul (happ f x0) (happ g x0)) ==> (!x m l g f. h4/lim/tends__real__real f l x /\ h4/lim/tends__real__real g m x ==> h4/lim/tends__real__real (happ (happ _0 f) g) (h4/realax/real__mul l m) x)
% Assm [h4s_lims_LIMu_u_INV]: !_0. (!f x0. happ (happ _0 f) x0 = h4/realax/inv (happ f x0)) ==> (!x l f. h4/lim/tends__real__real f l x /\ ~(l = h4/real/real__of__num h4/num/0) ==> h4/lim/tends__real__real (happ _0 f) (h4/realax/inv l) x)
% Assm [h4s_lims_LIMu_u_DIV]: !_0. (!f g x0. happ (happ (happ _0 f) g) x0 = h4/real/_2F (happ f x0) (happ g x0)) ==> (!x m l g f. h4/lim/tends__real__real f l x /\ h4/lim/tends__real__real g m x /\ ~(m = h4/real/real__of__num h4/num/0) ==> h4/lim/tends__real__real (happ (happ _0 f) g) (h4/real/_2F l m) x)
% Assm [h4s_lims_LIMu_u_SUB]: !_0. (!f g x0. happ (happ (happ _0 f) g) x0 = h4/real/real__sub (happ f x0) (happ g x0)) ==> (!x m l g f. h4/lim/tends__real__real f l x /\ h4/lim/tends__real__real g m x ==> h4/lim/tends__real__real (happ (happ _0 f) g) (h4/real/real__sub l m) x)
% Assm [h4s_lims_LIMu_u_ADD]: !_0. (!f g x0. happ (happ (happ _0 f) g) x0 = h4/realax/real__add (happ f x0) (happ g x0)) ==> (!x m l g f. h4/lim/tends__real__real f l x /\ h4/lim/tends__real__real g m x ==> h4/lim/tends__real__real (happ (happ _0 f) g) (h4/realax/real__add l m) x)
% Assm [h4s_lims_contl0]: !_0. (!f x h. happ (happ (happ _0 f) x) h = happ f (h4/realax/real__add x h)) ==> (!x f. h4/lim/contl f x <=> h4/lim/tends__real__real (happ (happ _0 f) x) (happ f x) (h4/real/real__of__num h4/num/0))
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_lims_CONTu_u_INV]: !_0. (!f x0. happ (happ _0 f) x0 = h4/realax/inv (happ f x0)) ==> (!x f. h4/lim/contl f x /\ ~(happ f x = h4/real/real__of__num h4/num/0) ==> h4/lim/contl (happ _0 f) x)
% Assm [h4s_lims_DIFFu_u_CARAT]: !x l f. h4/lim/diffl f l x <=> (?g. (!z. h4/real/real__sub (happ f z) (happ f x) = h4/realax/real__mul (happ g z) (h4/real/real__sub z x)) /\ h4/lim/contl g x /\ happ g x = l)
% Assm [h4s_lims_CONTu_u_DIV]: !_0. (!f g x0. happ (happ (happ _0 f) g) x0 = h4/real/_2F (happ f x0) (happ g x0)) ==> (!x g f. h4/lim/contl f x /\ h4/lim/contl g x /\ ~(happ g x = h4/real/real__of__num h4/num/0) ==> h4/lim/contl (happ (happ _0 f) g) x)
% Assm [h4s_lims_CONTu_u_COMPOSE]: !_0. (!g f x0. happ (happ (happ _0 g) f) x0 = happ g (happ f x0)) ==> (!x g f. h4/lim/contl f x /\ h4/lim/contl g (happ f x) ==> h4/lim/contl (happ (happ _0 g) f) x)
% Assm [h4s_lims_CONTu_u_SUB]: !_0. (!f g x0. happ (happ (happ _0 f) g) x0 = h4/real/real__sub (happ f x0) (happ g x0)) ==> (!x g f. h4/lim/contl f x /\ h4/lim/contl g x ==> h4/lim/contl (happ (happ _0 f) g) x)
% Assm [h4s_lims_CONTu_u_ADD]: !_0. (!f g x0. happ (happ (happ _0 f) g) x0 = h4/realax/real__add (happ f x0) (happ g x0)) ==> (!x g f. h4/lim/contl f x /\ h4/lim/contl g x ==> h4/lim/contl (happ (happ _0 f) g) x)
% Assm [h4s_lims_CONTu_u_MUL]: !_0. (!f g x0. happ (happ (happ _0 f) g) x0 = h4/realax/real__mul (happ f x0) (happ g x0)) ==> (!x g f. h4/lim/contl f x /\ h4/lim/contl g x ==> h4/lim/contl (happ (happ _0 f) g) x)
% Assm [h4s_lims_CONTu_u_NEG]: !_0. (!f x0. happ (happ _0 f) x0 = h4/realax/real__neg (happ f x0)) ==> (!x f. h4/lim/contl f x ==> h4/lim/contl (happ _0 f) x)
% Assm [h4s_lims_LIM]: !y0 x0 f. h4/lim/tends__real__real f y0 x0 <=> (!e. h4/realax/real__lt (h4/real/real__of__num h4/num/0) e ==> (?d. h4/realax/real__lt (h4/real/real__of__num h4/num/0) d /\ (!x. h4/realax/real__lt (h4/real/real__of__num h4/num/0) (h4/real/abs (h4/real/real__sub x x0)) /\ h4/realax/real__lt (h4/real/abs (h4/real/real__sub x x0)) d ==> h4/realax/real__lt (h4/real/abs (h4/real/real__sub (happ f x) y0)) e)))
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_lims_LIMu_u_NEG]: !_0. (!f x0. happ (happ _0 f) x0 = h4/realax/real__neg (happ f x0)) ==> (!x l f. h4/lim/tends__real__real f l x <=> h4/lim/tends__real__real (happ _0 f) (h4/realax/real__neg l) x)
% Assm [h4s_lims_DIFFu_u_CONT]: !x l f. h4/lim/diffl f l x ==> h4/lim/contl f x
% Assm [h4s_lims_diffl0]: !_0. (!f x h. happ (happ (happ _0 f) x) h = h4/real/_2F (h4/real/real__sub (happ f (h4/realax/real__add x h)) (happ f x)) h) ==> (!x l f. h4/lim/diffl f l x <=> h4/lim/tends__real__real (happ (happ _0 f) x) l (h4/real/real__of__num h4/num/0))
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_netss_NETu_u_NULL]: !_0. (!x x0 n. happ (happ (happ _0 x) x0) n = h4/real/real__sub (happ x n) x0) ==> (!x0 x g. h4/nets/tends x x0 (h4/pair/_2C (h4/topology/mtop h4/topology/mr1) g) <=> h4/nets/tends (happ (happ _0 x) x0) (h4/real/real__of__num h4/num/0) (h4/pair/_2C (h4/topology/mtop h4/topology/mr1) g))
% Assm [h4s_lims_LIMu_u_EQUAL]: !x0 l g f. (!x. ~(x = x0) ==> happ f x = happ g x) ==> (h4/lim/tends__real__real f l x0 <=> h4/lim/tends__real__real g l x0)
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_reals_REALu_u_SUBu_u_RZERO]: !x. h4/real/real__sub x (h4/real/real__of__num h4/num/0) = x
% Assm [h4s_lims_CONTu_u_CONST]: !_0. (!k x0. happ (happ _0 k) x0 = k) ==> (!x k. h4/lim/contl (happ _0 k) x)
% Assm [h4s_reals_REALu_u_SUBu_u_ADD2]: !y x. h4/realax/real__add y (h4/real/real__sub x y) = x
% Assm [h4s_reals_REALu_u_SUBu_u_0]: !y x. h4/real/real__sub x y = h4/real/real__of__num h4/num/0 <=> x = y
% Assm [h4s_lims_LIMu_u_UNIQ]: !x m l f. h4/lim/tends__real__real f l x /\ h4/lim/tends__real__real f m x ==> l = m
% Assm [h4s_lims_LIMu_u_NULL]: !_0. (!f l x0. happ (happ (happ _0 f) l) x0 = h4/real/real__sub (happ f x0) l) ==> (!x l f. h4/lim/tends__real__real f l x <=> h4/lim/tends__real__real (happ (happ _0 f) l) (h4/real/real__of__num h4/num/0) x)
% Assm [h4s_lims_LIMu_u_TRANSFORM]: !_0. (!f g x. happ (happ (happ _0 f) g) x = h4/real/real__sub (happ f x) (happ g x)) ==> (!x0 l g f. h4/lim/tends__real__real (happ (happ _0 f) g) (h4/real/real__of__num h4/num/0) x0 /\ h4/lim/tends__real__real g l x0 ==> h4/lim/tends__real__real f l x0)
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> 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_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_bools_ABSu_u_SIMP]: !t2 t1. t1 = t1
% Assm [h4s_lims_LIMu_u_CONST]: !_0. (!k x0. happ (happ _0 k) x0 = k) ==> (!x k. h4/lim/tends__real__real (happ _0 k) k x)
% Assm [h4s_netss_NETu_u_INV]: !_0. (!x n. happ (happ _0 x) n = h4/realax/inv (happ x n)) ==> (!g. h4/nets/dorder g ==> (!x x0. h4/nets/tends x x0 (h4/pair/_2C (h4/topology/mtop h4/topology/mr1) g) /\ ~(x0 = h4/real/real__of__num h4/num/0) ==> h4/nets/tends (happ _0 x) (h4/realax/inv x0) (h4/pair/_2C (h4/topology/mtop h4/topology/mr1) g)))
% Assm [h4s_netss_NETu_u_NEG]: !_0. (!x n. happ (happ _0 x) n = h4/realax/real__neg (happ x n)) ==> (!g. h4/nets/dorder g ==> (!x x0. h4/nets/tends x x0 (h4/pair/_2C (h4/topology/mtop h4/topology/mr1) g) <=> h4/nets/tends (happ _0 x) (h4/realax/real__neg x0) (h4/pair/_2C (h4/topology/mtop h4/topology/mr1) g)))
% Assm [h4s_netss_NETu_u_DIV]: !_0. (!x y n. happ (happ (happ _0 x) y) n = h4/real/_2F (happ x n) (happ y n)) ==> (!g. h4/nets/dorder g ==> (!x x0 y y0. h4/nets/tends x x0 (h4/pair/_2C (h4/topology/mtop h4/topology/mr1) g) /\ h4/nets/tends y y0 (h4/pair/_2C (h4/topology/mtop h4/topology/mr1) g) /\ ~(y0 = h4/real/real__of__num h4/num/0) ==> h4/nets/tends (happ (happ _0 x) y) (h4/real/_2F x0 y0) (h4/pair/_2C (h4/topology/mtop h4/topology/mr1) g)))
% Assm [h4s_netss_NETu_u_MUL]: !_0. (!x y n. happ (happ (happ _0 x) y) n = h4/realax/real__mul (happ x n) (happ y n)) ==> (!g. h4/nets/dorder g ==> (!x y x0 y0. h4/nets/tends x x0 (h4/pair/_2C (h4/topology/mtop h4/topology/mr1) g) /\ h4/nets/tends y y0 (h4/pair/_2C (h4/topology/mtop h4/topology/mr1) g) ==> h4/nets/tends (happ (happ _0 x) y) (h4/realax/real__mul x0 y0) (h4/pair/_2C (h4/topology/mtop h4/topology/mr1) g)))
% Assm [h4s_netss_NETu_u_SUB]: !_0. (!x y n. happ (happ (happ _0 x) y) n = h4/real/real__sub (happ x n) (happ y n)) ==> (!g. h4/nets/dorder g ==> (!x x0 y y0. h4/nets/tends x x0 (h4/pair/_2C (h4/topology/mtop h4/topology/mr1) g) /\ h4/nets/tends y y0 (h4/pair/_2C (h4/topology/mtop h4/topology/mr1) g) ==> h4/nets/tends (happ (happ _0 x) y) (h4/real/real__sub x0 y0) (h4/pair/_2C (h4/topology/mtop h4/topology/mr1) g)))
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_netss_NETu_u_ADD]: !_0. (!x y n. happ (happ (happ _0 x) y) n = h4/realax/real__add (happ x n) (happ y n)) ==> (!g. h4/nets/dorder g ==> (!x x0 y y0. h4/nets/tends x x0 (h4/pair/_2C (h4/topology/mtop h4/topology/mr1) g) /\ h4/nets/tends y y0 (h4/pair/_2C (h4/topology/mtop h4/topology/mr1) g) ==> h4/nets/tends (happ (happ _0 x) y) (h4/realax/real__add x0 y0) (h4/pair/_2C (h4/topology/mtop h4/topology/mr1) g)))
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_reals_REALu_u_SUBu_u_REFL]: !x. h4/real/real__sub x x = h4/real/real__of__num h4/num/0
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_reals_REALu_u_ADDu_u_SUB]: !y x. h4/real/real__sub (h4/realax/real__add x y) x = y
% Assm [h4s_reals_REALu_u_DIVu_u_RMUL]: !y x. ~(y = h4/real/real__of__num h4/num/0) ==> h4/realax/real__mul (h4/real/_2F x y) y = x
% Assm [h4s_reals_REALu_u_MULu_u_RZERO]: !x. h4/realax/real__mul x (h4/real/real__of__num h4/num/0) = h4/real/real__of__num h4/num/0
% Assm [h4s_reals_ABSu_u_NZ]: !x. ~(x = h4/real/real__of__num h4/num/0) <=> h4/realax/real__lt (h4/real/real__of__num h4/num/0) (h4/real/abs x)
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_reals_ABSu_u_0]: h4/real/abs (h4/real/real__of__num h4/num/0) = h4/real/real__of__num h4/num/0
% Assm [h4s_netss_LIMu_u_TENDS2]: !y0 x0 m2 m1 f. h4/topology/limpt (h4/topology/mtop m1) x0 h4/pred__set/UNIV ==> (h4/nets/tends f y0 (h4/pair/_2C (h4/topology/mtop m2) (h4/nets/tendsto (h4/pair/_2C m1 x0))) <=> (!e. h4/realax/real__lt (h4/real/real__of__num h4/num/0) e ==> (?d. h4/realax/real__lt (h4/real/real__of__num h4/num/0) d /\ (!x. h4/realax/real__lt (h4/real/real__of__num h4/num/0) (h4/topology/dist m1 (h4/pair/_2C x x0)) /\ h4/realax/real__lt (h4/topology/dist m1 (h4/pair/_2C x x0)) d ==> h4/realax/real__lt (h4/topology/dist m2 (h4/pair/_2C (happ f x) y0)) e))))
% Assm [h4s_topologys_MR1u_u_LIMPT]: !x. h4/topology/limpt (h4/topology/mtop h4/topology/mr1) x h4/pred__set/UNIV
% Assm [h4s_topologys_MR1u_u_DEF]: !y x. h4/topology/dist h4/topology/mr1 (h4/pair/_2C x y) = h4/real/abs (h4/real/real__sub y x)
% Assm [h4s_reals_ABSu_u_SUB]: !y x. h4/real/abs (h4/real/real__sub x y) = h4/real/abs (h4/real/real__sub y x)
% Assm [h4s_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% Assm [h4s_lims_differentiable0]: !x f. h4/lim/differentiable f x <=> (?l. h4/lim/diffl f l x)
% Assm [h4s_lims_DIFFu_u_UNIQ]: !x m l f. h4/lim/diffl f l x /\ h4/lim/diffl f m x ==> l = m
% 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_bools_FALSITY]: !t. F ==> 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_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_reals_REALu_u_MULu_u_RID]: !x. h4/realax/real__mul x (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) = x
% 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_reals_REALu_u_MULu_u_ASSOC]: !z y x. h4/realax/real__mul x (h4/realax/real__mul y z) = h4/realax/real__mul (h4/realax/real__mul x y) z
% Assm [h4s_reals_realu_u_div]: !y x. h4/real/_2F x y = h4/realax/real__mul x (h4/realax/inv y)
% Assm [h4s_reals_REALu_u_MULu_u_RINV]: !x. ~(x = h4/real/real__of__num h4/num/0) ==> h4/realax/real__mul x (h4/realax/inv x) = h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% 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_ABSu_u_TRIANGLE]: !y x. h4/real/real__lte (h4/real/abs (h4/realax/real__add x y)) (h4/realax/real__add (h4/real/abs x) (h4/real/abs y))
% Assm [h4s_reals_REALu_u_SUBu_u_TRIANGLE]: !c b a. h4/realax/real__add (h4/real/real__sub a b) (h4/real/real__sub b c) = h4/real/real__sub a c
% 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_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_reals_REALu_u_DOWN2]: !y x. h4/realax/real__lt (h4/real/real__of__num h4/num/0) x /\ h4/realax/real__lt (h4/real/real__of__num h4/num/0) y ==> (?z. h4/realax/real__lt (h4/real/real__of__num h4/num/0) z /\ h4/realax/real__lt z x /\ h4/realax/real__lt z y)
% 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_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_REFL]: !x. h4/real/real__lte x x
% Assm [h4s_reals_REALu_u_LTu_u_TRANS]: !z y x. h4/realax/real__lt x y /\ h4/realax/real__lt y z ==> h4/realax/real__lt x z
% Assm [h4s_reals_REALu_u_MINu_u_ALTu_c1]: !y x. h4/real/real__lte y x ==> h4/real/min x y = y
% Assm [h4s_reals_REALu_u_LEu_u_NEG2]: !y x. h4/real/real__lte (h4/realax/real__neg x) (h4/realax/real__neg y) <=> h4/real/real__lte y x
% Assm [h4s_reals_REALu_u_MINu_u_ALTu_c0]: !y x. h4/real/real__lte x y ==> h4/real/min x y = x
% Assm [h4s_reals_REALu_u_LEu_u_SUP]: !x p. (?y. happ p y) /\ (?y. !z. happ p z ==> h4/real/real__lte z y) ==> (h4/real/real__lte x (h4/real/sup p) <=> (!y. (!z. happ p z ==> h4/real/real__lte z y) ==> h4/real/real__lte x y))
% Assm [h4s_reals_REALu_u_LTu_u_NEG]: !y x. h4/realax/real__lt (h4/realax/real__neg x) (h4/realax/real__neg y) <=> h4/realax/real__lt y 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_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_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_reals_minu_u_def]: !y x. h4/real/min x y = h4/bool/COND (h4/real/real__lte x y) x y
% Assm [h4s_bools_boolu_u_caseu_u_thmu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_bools_CONDu_u_CONG]: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_IMPu_u_F]: !t. (t ==> F) ==> ~t
% Assm [h4s_bools_Fu_u_IMP]: !t. ~t ==> t ==> F
% Assm [h4s_bools_DISJu_u_ASSOC]: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm [h4s_bools_DISJu_u_SYM]: !B A. A \/ B <=> B \/ A
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c1]: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm [h4s_bools_BOOLu_u_CASESu_u_AX]: !t. (t <=> T) \/ (t <=> F)
% Assm [h4s_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_bools_CONJu_u_ASSOC]: !t3 t2 t1. t1 /\ t2 /\ t3 <=> (t1 /\ t2) /\ t3
% Assm [h4s_bools_ORu_u_DEF]: !x x'. $or x x' <=> (!t. (x ==> t) ==> (x' ==> t) ==> t)
% Assm [h4s_reals_REALu_u_LEu_u_ANTISYM]: !y x. h4/real/real__lte x y /\ h4/real/real__lte y x <=> x = y
% Assm [h4s_bools_Fu_u_DEF]: F <=> (!t. t)
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_reals_REALu_u_LEu_u_EPSILON]: !y x. (!e. h4/realax/real__lt (h4/real/real__of__num h4/num/0) e ==> h4/real/real__lte x (h4/realax/real__add y e)) ==> h4/real/real__lte x y
% Assm [h4s_bools_EQu_u_CLAUSESu_c3]: !t. (t <=> F) <=> ~t
% Assm [h4s_reals_realu_u_lt]: !y x. h4/realax/real__lt x y <=> ~h4/real/real__lte y x
% Assm [h4s_reals_REALu_u_SUPu_u_LE]: !P. (?x. happ P x) /\ (?z. !x. happ P x ==> h4/real/real__lte x z) ==> (!y. (?x. happ P x /\ h4/realax/real__lt y x) <=> h4/realax/real__lt y (h4/real/sup P))
% Assm [h4s_reals_REALu_u_LEu_u_SUBu_u_RADD]: !z y x. h4/real/real__lte (h4/real/real__sub x y) z <=> h4/real/real__lte x (h4/realax/real__add z y)
% Assm [h4s_reals_REALu_u_LTu_u_SUBu_u_RADD]: !z y x. h4/realax/real__lt (h4/real/real__sub x y) z <=> h4/realax/real__lt x (h4/realax/real__add z y)
% Assm [h4s_reals_REALu_u_LEu_u_SUBu_u_LADD]: !z y x. h4/real/real__lte x (h4/real/real__sub y z) <=> h4/real/real__lte (h4/realax/real__add x z) y
% Assm [h4s_bools_NOTu_u_FORALLu_u_THM]: !P. ~(!x. happ P x) <=> (?x. ~happ P x)
% Assm [h4s_bools_NOTu_u_EXISTSu_u_THM]: !P. ~(?x. happ P x) <=> (!x. ~happ P x)
% Assm [h4s_reals_REALu_u_LTu_u_ADDR]: !y x. h4/realax/real__lt x (h4/realax/real__add x y) <=> h4/realax/real__lt (h4/real/real__of__num h4/num/0) y
% Goal: !y f b a. h4/real/real__lte a b /\ (h4/real/real__lte (happ f a) y /\ h4/real/real__lte y (happ f b)) /\ (!x. h4/real/real__lte a x /\ h4/real/real__lte x b ==> h4/lim/contl f x) ==> (?x. h4/real/real__lte a x /\ h4/real/real__lte x b /\ happ f x = y)
fof(aHLu_TRUTH, axiom, p(s(t_bool,t))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f0)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f0))).
fof(aHLu_EXT, axiom, ![TV_Q1376615,TV_Q1376611]: ![V_f, V_g]: (![V_x]: s(TV_Q1376611,happ(s(t_fun(TV_Q1376615,TV_Q1376611),V_f),s(TV_Q1376615,V_x))) = s(TV_Q1376611,happ(s(t_fun(TV_Q1376615,TV_Q1376611),V_g),s(TV_Q1376615,V_x))) => s(t_fun(TV_Q1376615,TV_Q1376611),V_f) = s(t_fun(TV_Q1376615,TV_Q1376611),V_g))).
fof(ah4s_lims_CONTLu_u_LIM, axiom, ![V_x, V_f]: s(t_bool,h4s_lims_contl(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_x))) = s(t_bool,h4s_lims_tendsu_u_realu_u_real(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_x))),s(t_h4s_realaxs_real,V_x)))).
fof(ah4s_lims_tendsu_u_realu_u_real0, axiom, ![V_x0, V_l, V_f]: s(t_bool,h4s_lims_tendsu_u_realu_u_real(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_l),s(t_h4s_realaxs_real,V_x0))) = s(t_bool,h4s_netss_tends(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_l),s(t_h4s_pairs_prod(t_h4s_topologys_topology(t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_bool))),h4s_pairs_u_2c(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_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_bool)),h4s_netss_tendsto(s(t_h4s_pairs_prod(t_h4s_topologys_metric(t_h4s_realaxs_real),t_h4s_realaxs_real),h4s_pairs_u_2c(s(t_h4s_topologys_metric(t_h4s_realaxs_real),h4s_topologys_mr1),s(t_h4s_realaxs_real,V_x0)))))))))).
fof(ah4s_netss_DORDERu_u_TENDSTO, axiom, ![TV_u_27a]: ![V_x, V_m]: p(s(t_bool,h4s_netss_dorder(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_netss_tendsto(s(t_h4s_pairs_prod(t_h4s_topologys_metric(TV_u_27a),TV_u_27a),h4s_pairs_u_2c(s(t_h4s_topologys_metric(TV_u_27a),V_m),s(TV_u_27a,V_x))))))))).
fof(ah4s_lims_LIMu_u_MUL, axiom, ![V_uu_0]: (![V_f, V_g, V_x0]: s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real)),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real))),V_uu_0),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f))),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_g))),s(t_h4s_realaxs_real,V_x0))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_x0))),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_x0))))) => ![V_x, V_m, V_l, V_g, V_f]: ((p(s(t_bool,h4s_lims_tendsu_u_realu_u_real(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_l),s(t_h4s_realaxs_real,V_x)))) & p(s(t_bool,h4s_lims_tendsu_u_realu_u_real(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_g),s(t_h4s_realaxs_real,V_m),s(t_h4s_realaxs_real,V_x))))) => p(s(t_bool,h4s_lims_tendsu_u_realu_u_real(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real)),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real))),V_uu_0),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f))),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_g))),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,V_l),s(t_h4s_realaxs_real,V_m))),s(t_h4s_realaxs_real,V_x))))))).
fof(ah4s_lims_LIMu_u_INV, axiom, ![V_uu_0]: (![V_f, V_x0]: s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real)),V_uu_0),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f))),s(t_h4s_realaxs_real,V_x0))) = s(t_h4s_realaxs_real,h4s_realaxs_inv(s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_x0))))) => ![V_x, V_l, V_f]: ((p(s(t_bool,h4s_lims_tendsu_u_realu_u_real(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_l),s(t_h4s_realaxs_real,V_x)))) & ~ (s(t_h4s_realaxs_real,V_l) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))))) => p(s(t_bool,h4s_lims_tendsu_u_realu_u_real(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real)),V_uu_0),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f))),s(t_h4s_realaxs_real,h4s_realaxs_inv(s(t_h4s_realaxs_real,V_l))),s(t_h4s_realaxs_real,V_x))))))).
fof(ah4s_lims_LIMu_u_DIV, axiom, ![V_uu_0]: (![V_f, V_g, V_x0]: s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real)),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real))),V_uu_0),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f))),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_g))),s(t_h4s_realaxs_real,V_x0))) = s(t_h4s_realaxs_real,h4s_reals_u_2f(s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_x0))),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_x0))))) => ![V_x, V_m, V_l, V_g, V_f]: ((p(s(t_bool,h4s_lims_tendsu_u_realu_u_real(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_l),s(t_h4s_realaxs_real,V_x)))) & (p(s(t_bool,h4s_lims_tendsu_u_realu_u_real(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_g),s(t_h4s_realaxs_real,V_m),s(t_h4s_realaxs_real,V_x)))) & ~ (s(t_h4s_realaxs_real,V_m) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))))) => p(s(t_bool,h4s_lims_tendsu_u_realu_u_real(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real)),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real))),V_uu_0),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f))),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_g))),s(t_h4s_realaxs_real,h4s_reals_u_2f(s(t_h4s_realaxs_real,V_l),s(t_h4s_realaxs_real,V_m))),s(t_h4s_realaxs_real,V_x))))))).
fof(ah4s_lims_LIMu_u_SUB, axiom, ![V_uu_0]: (![V_f, V_g, V_x0]: s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real)),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real))),V_uu_0),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f))),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_g))),s(t_h4s_realaxs_real,V_x0))) = s(t_h4s_realaxs_real,h4s_reals_realu_u_sub(s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_x0))),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_x0))))) => ![V_x, V_m, V_l, V_g, V_f]: ((p(s(t_bool,h4s_lims_tendsu_u_realu_u_real(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_l),s(t_h4s_realaxs_real,V_x)))) & p(s(t_bool,h4s_lims_tendsu_u_realu_u_real(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_g),s(t_h4s_realaxs_real,V_m),s(t_h4s_realaxs_real,V_x))))) => p(s(t_bool,h4s_lims_tendsu_u_realu_u_real(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real)),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real))),V_uu_0),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f))),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_g))),s(t_h4s_realaxs_real,h4s_reals_realu_u_sub(s(t_h4s_realaxs_real,V_l),s(t_h4s_realaxs_real,V_m))),s(t_h4s_realaxs_real,V_x))))))).
fof(ah4s_lims_LIMu_u_ADD, axiom, ![V_uu_0]: (![V_f, V_g, V_x0]: s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real)),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real))),V_uu_0),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f))),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_g))),s(t_h4s_realaxs_real,V_x0))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_x0))),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_x0))))) => ![V_x, V_m, V_l, V_g, V_f]: ((p(s(t_bool,h4s_lims_tendsu_u_realu_u_real(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_l),s(t_h4s_realaxs_real,V_x)))) & p(s(t_bool,h4s_lims_tendsu_u_realu_u_real(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_g),s(t_h4s_realaxs_real,V_m),s(t_h4s_realaxs_real,V_x))))) => p(s(t_bool,h4s_lims_tendsu_u_realu_u_real(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real)),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real))),V_uu_0),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f))),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_g))),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,V_l),s(t_h4s_realaxs_real,V_m))),s(t_h4s_realaxs_real,V_x))))))).
fof(ah4s_lims_contl0, axiom, ![V_uu_0]: (![V_f, V_x, V_h]: s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real)),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real))),V_uu_0),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f))),s(t_h4s_realaxs_real,V_x))),s(t_h4s_realaxs_real,V_h))) = s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_h))))) => ![V_x, V_f]: s(t_bool,h4s_lims_contl(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_x))) = s(t_bool,h4s_lims_tendsu_u_realu_u_real(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real)),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real))),V_uu_0),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f))),s(t_h4s_realaxs_real,V_x))),s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),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_nums_0))))))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_EQu_u_SYMu_u_EQ, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) <=> s(TV_u_27a,V_y) = s(TV_u_27a,V_x))).
fof(ah4s_lims_CONTu_u_INV, axiom, ![V_uu_0]: (![V_f, V_x0]: s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real)),V_uu_0),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f))),s(t_h4s_realaxs_real,V_x0))) = s(t_h4s_realaxs_real,h4s_realaxs_inv(s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_x0))))) => ![V_x, V_f]: ((p(s(t_bool,h4s_lims_contl(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_x)))) & ~ (s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),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_nums_0))))) => p(s(t_bool,h4s_lims_contl(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real)),V_uu_0),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f))),s(t_h4s_realaxs_real,V_x))))))).
fof(ah4s_lims_DIFFu_u_CARAT, axiom, ![V_x, V_l, V_f]: (p(s(t_bool,h4s_lims_diffl(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_l),s(t_h4s_realaxs_real,V_x)))) <=> ?[V_g]: (![V_z]: s(t_h4s_realaxs_real,h4s_reals_realu_u_sub(s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_z))),s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_x))))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(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_z))),s(t_h4s_realaxs_real,h4s_reals_realu_u_sub(s(t_h4s_realaxs_real,V_z),s(t_h4s_realaxs_real,V_x))))) & (p(s(t_bool,h4s_lims_contl(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_g),s(t_h4s_realaxs_real,V_x)))) & 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))) = s(t_h4s_realaxs_real,V_l))))).
fof(ah4s_lims_CONTu_u_DIV, axiom, ![V_uu_0]: (![V_f, V_g, V_x0]: s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real)),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real))),V_uu_0),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f))),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_g))),s(t_h4s_realaxs_real,V_x0))) = s(t_h4s_realaxs_real,h4s_reals_u_2f(s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_x0))),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_x0))))) => ![V_x, V_g, V_f]: ((p(s(t_bool,h4s_lims_contl(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_x)))) & (p(s(t_bool,h4s_lims_contl(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_g),s(t_h4s_realaxs_real,V_x)))) & ~ (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))) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))))) => p(s(t_bool,h4s_lims_contl(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real)),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real))),V_uu_0),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f))),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_g))),s(t_h4s_realaxs_real,V_x))))))).
fof(ah4s_lims_CONTu_u_COMPOSE, axiom, ![V_uu_0]: (![V_g, V_f, V_x0]: s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real)),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real))),V_uu_0),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_g))),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f))),s(t_h4s_realaxs_real,V_x0))) = 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_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_x0))))) => ![V_x, V_g, V_f]: ((p(s(t_bool,h4s_lims_contl(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_x)))) & p(s(t_bool,h4s_lims_contl(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_g),s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_x))))))) => p(s(t_bool,h4s_lims_contl(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real)),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real))),V_uu_0),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_g))),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f))),s(t_h4s_realaxs_real,V_x))))))).
fof(ah4s_lims_CONTu_u_SUB, axiom, ![V_uu_0]: (![V_f, V_g, V_x0]: s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real)),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real))),V_uu_0),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f))),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_g))),s(t_h4s_realaxs_real,V_x0))) = s(t_h4s_realaxs_real,h4s_reals_realu_u_sub(s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_x0))),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_x0))))) => ![V_x, V_g, V_f]: ((p(s(t_bool,h4s_lims_contl(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_x)))) & p(s(t_bool,h4s_lims_contl(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_g),s(t_h4s_realaxs_real,V_x))))) => p(s(t_bool,h4s_lims_contl(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real)),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real))),V_uu_0),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f))),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_g))),s(t_h4s_realaxs_real,V_x))))))).
fof(ah4s_lims_CONTu_u_ADD, axiom, ![V_uu_0]: (![V_f, V_g, V_x0]: s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real)),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real))),V_uu_0),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f))),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_g))),s(t_h4s_realaxs_real,V_x0))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_x0))),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_x0))))) => ![V_x, V_g, V_f]: ((p(s(t_bool,h4s_lims_contl(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_x)))) & p(s(t_bool,h4s_lims_contl(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_g),s(t_h4s_realaxs_real,V_x))))) => p(s(t_bool,h4s_lims_contl(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real)),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real))),V_uu_0),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f))),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_g))),s(t_h4s_realaxs_real,V_x))))))).
fof(ah4s_lims_CONTu_u_MUL, axiom, ![V_uu_0]: (![V_f, V_g, V_x0]: s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real)),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real))),V_uu_0),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f))),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_g))),s(t_h4s_realaxs_real,V_x0))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_x0))),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_x0))))) => ![V_x, V_g, V_f]: ((p(s(t_bool,h4s_lims_contl(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_x)))) & p(s(t_bool,h4s_lims_contl(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_g),s(t_h4s_realaxs_real,V_x))))) => p(s(t_bool,h4s_lims_contl(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real)),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real))),V_uu_0),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f))),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_g))),s(t_h4s_realaxs_real,V_x))))))).
fof(ah4s_lims_CONTu_u_NEG, axiom, ![V_uu_0]: (![V_f, V_x0]: s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real)),V_uu_0),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f))),s(t_h4s_realaxs_real,V_x0))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_neg(s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_x0))))) => ![V_x, V_f]: (p(s(t_bool,h4s_lims_contl(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_x)))) => p(s(t_bool,h4s_lims_contl(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real)),V_uu_0),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f))),s(t_h4s_realaxs_real,V_x))))))).
fof(ah4s_lims_LIM, axiom, ![V_y0, V_x0, V_f]: (p(s(t_bool,h4s_lims_tendsu_u_realu_u_real(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_y0),s(t_h4s_realaxs_real,V_x0)))) <=> ![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_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_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,h4s_reals_abs(s(t_h4s_realaxs_real,h4s_reals_realu_u_sub(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_x0)))))))) & p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,h4s_reals_abs(s(t_h4s_realaxs_real,h4s_reals_realu_u_sub(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_x0))))),s(t_h4s_realaxs_real,V_d))))) => p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,h4s_reals_abs(s(t_h4s_realaxs_real,h4s_reals_realu_u_sub(s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_x))),s(t_h4s_realaxs_real,V_y0))))),s(t_h4s_realaxs_real,V_e))))))))).
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_lims_LIMu_u_NEG, axiom, ![V_uu_0]: (![V_f, V_x0]: s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real)),V_uu_0),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f))),s(t_h4s_realaxs_real,V_x0))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_neg(s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_x0))))) => ![V_x, V_l, V_f]: s(t_bool,h4s_lims_tendsu_u_realu_u_real(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_l),s(t_h4s_realaxs_real,V_x))) = s(t_bool,h4s_lims_tendsu_u_realu_u_real(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real)),V_uu_0),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f))),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_neg(s(t_h4s_realaxs_real,V_l))),s(t_h4s_realaxs_real,V_x))))).
fof(ah4s_lims_DIFFu_u_CONT, axiom, ![V_x, V_l, V_f]: (p(s(t_bool,h4s_lims_diffl(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_l),s(t_h4s_realaxs_real,V_x)))) => p(s(t_bool,h4s_lims_contl(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_x)))))).
fof(ah4s_lims_diffl0, axiom, ![V_uu_0]: (![V_f, V_x, V_h]: s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real)),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real))),V_uu_0),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f))),s(t_h4s_realaxs_real,V_x))),s(t_h4s_realaxs_real,V_h))) = s(t_h4s_realaxs_real,h4s_reals_u_2f(s(t_h4s_realaxs_real,h4s_reals_realu_u_sub(s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_h))))),s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_x))))),s(t_h4s_realaxs_real,V_h))) => ![V_x, V_l, V_f]: s(t_bool,h4s_lims_diffl(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_l),s(t_h4s_realaxs_real,V_x))) = s(t_bool,h4s_lims_tendsu_u_realu_u_real(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real)),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real))),V_uu_0),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f))),s(t_h4s_realaxs_real,V_x))),s(t_h4s_realaxs_real,V_l),s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))))))).
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_netss_NETu_u_NULL, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_x, V_x0, V_n]: s(t_h4s_realaxs_real,happ(s(t_fun(TV_u_27a,t_h4s_realaxs_real),happ(s(t_fun(t_h4s_realaxs_real,t_fun(TV_u_27a,t_h4s_realaxs_real)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_fun(TV_u_27a,t_h4s_realaxs_real))),V_uu_0),s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_x))),s(t_h4s_realaxs_real,V_x0))),s(TV_u_27a,V_n))) = s(t_h4s_realaxs_real,h4s_reals_realu_u_sub(s(t_h4s_realaxs_real,happ(s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_x),s(TV_u_27a,V_n))),s(t_h4s_realaxs_real,V_x0))) => ![V_x0, V_x, V_g]: s(t_bool,h4s_netss_tends(s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_x),s(t_h4s_realaxs_real,V_x0),s(t_h4s_pairs_prod(t_h4s_topologys_topology(t_h4s_realaxs_real),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_pairs_u_2c(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_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_g))))) = s(t_bool,h4s_netss_tends(s(t_fun(TV_u_27a,t_h4s_realaxs_real),happ(s(t_fun(t_h4s_realaxs_real,t_fun(TV_u_27a,t_h4s_realaxs_real)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_fun(TV_u_27a,t_h4s_realaxs_real))),V_uu_0),s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_x))),s(t_h4s_realaxs_real,V_x0))),s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_pairs_prod(t_h4s_topologys_topology(t_h4s_realaxs_real),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_pairs_u_2c(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_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_g))))))).
fof(ah4s_lims_LIMu_u_EQUAL, axiom, ![V_x0, V_l, V_g, V_f]: (![V_x]: (~ (s(t_h4s_realaxs_real,V_x) = s(t_h4s_realaxs_real,V_x0)) => s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_x))) = 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)))) => s(t_bool,h4s_lims_tendsu_u_realu_u_real(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_l),s(t_h4s_realaxs_real,V_x0))) = s(t_bool,h4s_lims_tendsu_u_realu_u_real(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_g),s(t_h4s_realaxs_real,V_l),s(t_h4s_realaxs_real,V_x0))))).
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_reals_REALu_u_SUBu_u_RZERO, axiom, ![V_x]: s(t_h4s_realaxs_real,h4s_reals_realu_u_sub(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_nums_0))))) = s(t_h4s_realaxs_real,V_x)).
fof(ah4s_lims_CONTu_u_CONST, axiom, ![V_uu_0]: (![V_k, V_x0]: s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real)),V_uu_0),s(t_h4s_realaxs_real,V_k))),s(t_h4s_realaxs_real,V_x0))) = s(t_h4s_realaxs_real,V_k) => ![V_x, V_k]: p(s(t_bool,h4s_lims_contl(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real)),V_uu_0),s(t_h4s_realaxs_real,V_k))),s(t_h4s_realaxs_real,V_x)))))).
fof(ah4s_reals_REALu_u_SUBu_u_ADD2, axiom, ![V_y, V_x]: s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,V_y),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_h4s_realaxs_real,V_x)).
fof(ah4s_reals_REALu_u_SUBu_u_0, axiom, ![V_y, V_x]: (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_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_y))).
fof(ah4s_lims_LIMu_u_UNIQ, axiom, ![V_x, V_m, V_l, V_f]: ((p(s(t_bool,h4s_lims_tendsu_u_realu_u_real(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_l),s(t_h4s_realaxs_real,V_x)))) & p(s(t_bool,h4s_lims_tendsu_u_realu_u_real(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_m),s(t_h4s_realaxs_real,V_x))))) => s(t_h4s_realaxs_real,V_l) = s(t_h4s_realaxs_real,V_m))).
fof(ah4s_lims_LIMu_u_NULL, axiom, ![V_uu_0]: (![V_f, V_l, V_x0]: s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real)),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real))),V_uu_0),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f))),s(t_h4s_realaxs_real,V_l))),s(t_h4s_realaxs_real,V_x0))) = s(t_h4s_realaxs_real,h4s_reals_realu_u_sub(s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_x0))),s(t_h4s_realaxs_real,V_l))) => ![V_x, V_l, V_f]: s(t_bool,h4s_lims_tendsu_u_realu_u_real(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_l),s(t_h4s_realaxs_real,V_x))) = s(t_bool,h4s_lims_tendsu_u_realu_u_real(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real)),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real))),V_uu_0),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f))),s(t_h4s_realaxs_real,V_l))),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))))).
fof(ah4s_lims_LIMu_u_TRANSFORM, axiom, ![V_uu_0]: (![V_f, V_g, V_x]: s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real)),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real))),V_uu_0),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f))),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_g))),s(t_h4s_realaxs_real,V_x))) = s(t_h4s_realaxs_real,h4s_reals_realu_u_sub(s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_x))),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))))) => ![V_x0, V_l, V_g, V_f]: ((p(s(t_bool,h4s_lims_tendsu_u_realu_u_real(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real)),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real))),V_uu_0),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f))),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_g))),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_x0)))) & p(s(t_bool,h4s_lims_tendsu_u_realu_u_real(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_g),s(t_h4s_realaxs_real,V_l),s(t_h4s_realaxs_real,V_x0))))) => p(s(t_bool,h4s_lims_tendsu_u_realu_u_real(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_l),s(t_h4s_realaxs_real,V_x0))))))).
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,f0))) <=> (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f0)))))).
fof(ah4s_sats_ANDu_u_INV2, axiom, ![V_A]: ((~ (p(s(t_bool,V_A))) => p(s(t_bool,f0))) => ((p(s(t_bool,V_A)) => p(s(t_bool,f0))) => p(s(t_bool,f0))))).
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_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,f0))) <=> ((p(s(t_bool,V_A)) => p(s(t_bool,f0))) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f0)))))).
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,f0))))).
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_lims_LIMu_u_CONST, axiom, ![V_uu_0]: (![V_k, V_x0]: s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real)),V_uu_0),s(t_h4s_realaxs_real,V_k))),s(t_h4s_realaxs_real,V_x0))) = s(t_h4s_realaxs_real,V_k) => ![V_x, V_k]: p(s(t_bool,h4s_lims_tendsu_u_realu_u_real(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real)),V_uu_0),s(t_h4s_realaxs_real,V_k))),s(t_h4s_realaxs_real,V_k),s(t_h4s_realaxs_real,V_x)))))).
fof(ah4s_netss_NETu_u_INV, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_x, V_n]: s(t_h4s_realaxs_real,happ(s(t_fun(TV_u_27a,t_h4s_realaxs_real),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_realaxs_real),t_fun(TV_u_27a,t_h4s_realaxs_real)),V_uu_0),s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_x))),s(TV_u_27a,V_n))) = s(t_h4s_realaxs_real,h4s_realaxs_inv(s(t_h4s_realaxs_real,happ(s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_x),s(TV_u_27a,V_n))))) => ![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_x0]: ((p(s(t_bool,h4s_netss_tends(s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_x),s(t_h4s_realaxs_real,V_x0),s(t_h4s_pairs_prod(t_h4s_topologys_topology(t_h4s_realaxs_real),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_pairs_u_2c(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_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_g)))))) & ~ (s(t_h4s_realaxs_real,V_x0) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))))) => p(s(t_bool,h4s_netss_tends(s(t_fun(TV_u_27a,t_h4s_realaxs_real),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_realaxs_real),t_fun(TV_u_27a,t_h4s_realaxs_real)),V_uu_0),s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_x))),s(t_h4s_realaxs_real,h4s_realaxs_inv(s(t_h4s_realaxs_real,V_x0))),s(t_h4s_pairs_prod(t_h4s_topologys_topology(t_h4s_realaxs_real),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_pairs_u_2c(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_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_g)))))))))).
fof(ah4s_netss_NETu_u_NEG, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_x, V_n]: s(t_h4s_realaxs_real,happ(s(t_fun(TV_u_27a,t_h4s_realaxs_real),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_realaxs_real),t_fun(TV_u_27a,t_h4s_realaxs_real)),V_uu_0),s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_x))),s(TV_u_27a,V_n))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_neg(s(t_h4s_realaxs_real,happ(s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_x),s(TV_u_27a,V_n))))) => ![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_x0]: s(t_bool,h4s_netss_tends(s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_x),s(t_h4s_realaxs_real,V_x0),s(t_h4s_pairs_prod(t_h4s_topologys_topology(t_h4s_realaxs_real),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_pairs_u_2c(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_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_g))))) = s(t_bool,h4s_netss_tends(s(t_fun(TV_u_27a,t_h4s_realaxs_real),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_realaxs_real),t_fun(TV_u_27a,t_h4s_realaxs_real)),V_uu_0),s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_x))),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_neg(s(t_h4s_realaxs_real,V_x0))),s(t_h4s_pairs_prod(t_h4s_topologys_topology(t_h4s_realaxs_real),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_pairs_u_2c(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_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_g)))))))).
fof(ah4s_netss_NETu_u_DIV, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_x, V_y, V_n]: s(t_h4s_realaxs_real,happ(s(t_fun(TV_u_27a,t_h4s_realaxs_real),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_realaxs_real),t_fun(TV_u_27a,t_h4s_realaxs_real)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_realaxs_real),t_fun(t_fun(TV_u_27a,t_h4s_realaxs_real),t_fun(TV_u_27a,t_h4s_realaxs_real))),V_uu_0),s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_x))),s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_y))),s(TV_u_27a,V_n))) = s(t_h4s_realaxs_real,h4s_reals_u_2f(s(t_h4s_realaxs_real,happ(s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_x),s(TV_u_27a,V_n))),s(t_h4s_realaxs_real,happ(s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_y),s(TV_u_27a,V_n))))) => ![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_x0, V_y, V_y0]: ((p(s(t_bool,h4s_netss_tends(s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_x),s(t_h4s_realaxs_real,V_x0),s(t_h4s_pairs_prod(t_h4s_topologys_topology(t_h4s_realaxs_real),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_pairs_u_2c(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_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_g)))))) & (p(s(t_bool,h4s_netss_tends(s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_y),s(t_h4s_realaxs_real,V_y0),s(t_h4s_pairs_prod(t_h4s_topologys_topology(t_h4s_realaxs_real),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_pairs_u_2c(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_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_g)))))) & ~ (s(t_h4s_realaxs_real,V_y0) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))))) => p(s(t_bool,h4s_netss_tends(s(t_fun(TV_u_27a,t_h4s_realaxs_real),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_realaxs_real),t_fun(TV_u_27a,t_h4s_realaxs_real)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_realaxs_real),t_fun(t_fun(TV_u_27a,t_h4s_realaxs_real),t_fun(TV_u_27a,t_h4s_realaxs_real))),V_uu_0),s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_x))),s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_y))),s(t_h4s_realaxs_real,h4s_reals_u_2f(s(t_h4s_realaxs_real,V_x0),s(t_h4s_realaxs_real,V_y0))),s(t_h4s_pairs_prod(t_h4s_topologys_topology(t_h4s_realaxs_real),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_pairs_u_2c(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_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_g)))))))))).
fof(ah4s_netss_NETu_u_MUL, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_x, V_y, V_n]: s(t_h4s_realaxs_real,happ(s(t_fun(TV_u_27a,t_h4s_realaxs_real),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_realaxs_real),t_fun(TV_u_27a,t_h4s_realaxs_real)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_realaxs_real),t_fun(t_fun(TV_u_27a,t_h4s_realaxs_real),t_fun(TV_u_27a,t_h4s_realaxs_real))),V_uu_0),s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_x))),s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_y))),s(TV_u_27a,V_n))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,happ(s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_x),s(TV_u_27a,V_n))),s(t_h4s_realaxs_real,happ(s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_y),s(TV_u_27a,V_n))))) => ![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, V_x0, V_y0]: ((p(s(t_bool,h4s_netss_tends(s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_x),s(t_h4s_realaxs_real,V_x0),s(t_h4s_pairs_prod(t_h4s_topologys_topology(t_h4s_realaxs_real),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_pairs_u_2c(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_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_g)))))) & p(s(t_bool,h4s_netss_tends(s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_y),s(t_h4s_realaxs_real,V_y0),s(t_h4s_pairs_prod(t_h4s_topologys_topology(t_h4s_realaxs_real),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_pairs_u_2c(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_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_g))))))) => p(s(t_bool,h4s_netss_tends(s(t_fun(TV_u_27a,t_h4s_realaxs_real),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_realaxs_real),t_fun(TV_u_27a,t_h4s_realaxs_real)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_realaxs_real),t_fun(t_fun(TV_u_27a,t_h4s_realaxs_real),t_fun(TV_u_27a,t_h4s_realaxs_real))),V_uu_0),s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_x))),s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_y))),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,V_x0),s(t_h4s_realaxs_real,V_y0))),s(t_h4s_pairs_prod(t_h4s_topologys_topology(t_h4s_realaxs_real),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_pairs_u_2c(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_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_g)))))))))).
fof(ah4s_netss_NETu_u_SUB, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_x, V_y, V_n]: s(t_h4s_realaxs_real,happ(s(t_fun(TV_u_27a,t_h4s_realaxs_real),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_realaxs_real),t_fun(TV_u_27a,t_h4s_realaxs_real)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_realaxs_real),t_fun(t_fun(TV_u_27a,t_h4s_realaxs_real),t_fun(TV_u_27a,t_h4s_realaxs_real))),V_uu_0),s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_x))),s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_y))),s(TV_u_27a,V_n))) = s(t_h4s_realaxs_real,h4s_reals_realu_u_sub(s(t_h4s_realaxs_real,happ(s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_x),s(TV_u_27a,V_n))),s(t_h4s_realaxs_real,happ(s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_y),s(TV_u_27a,V_n))))) => ![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_x0, V_y, V_y0]: ((p(s(t_bool,h4s_netss_tends(s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_x),s(t_h4s_realaxs_real,V_x0),s(t_h4s_pairs_prod(t_h4s_topologys_topology(t_h4s_realaxs_real),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_pairs_u_2c(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_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_g)))))) & p(s(t_bool,h4s_netss_tends(s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_y),s(t_h4s_realaxs_real,V_y0),s(t_h4s_pairs_prod(t_h4s_topologys_topology(t_h4s_realaxs_real),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_pairs_u_2c(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_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_g))))))) => p(s(t_bool,h4s_netss_tends(s(t_fun(TV_u_27a,t_h4s_realaxs_real),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_realaxs_real),t_fun(TV_u_27a,t_h4s_realaxs_real)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_realaxs_real),t_fun(t_fun(TV_u_27a,t_h4s_realaxs_real),t_fun(TV_u_27a,t_h4s_realaxs_real))),V_uu_0),s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_x))),s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_y))),s(t_h4s_realaxs_real,h4s_reals_realu_u_sub(s(t_h4s_realaxs_real,V_x0),s(t_h4s_realaxs_real,V_y0))),s(t_h4s_pairs_prod(t_h4s_topologys_topology(t_h4s_realaxs_real),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_pairs_u_2c(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_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_g)))))))))).
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_netss_NETu_u_ADD, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_x, V_y, V_n]: s(t_h4s_realaxs_real,happ(s(t_fun(TV_u_27a,t_h4s_realaxs_real),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_realaxs_real),t_fun(TV_u_27a,t_h4s_realaxs_real)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_realaxs_real),t_fun(t_fun(TV_u_27a,t_h4s_realaxs_real),t_fun(TV_u_27a,t_h4s_realaxs_real))),V_uu_0),s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_x))),s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_y))),s(TV_u_27a,V_n))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,happ(s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_x),s(TV_u_27a,V_n))),s(t_h4s_realaxs_real,happ(s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_y),s(TV_u_27a,V_n))))) => ![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_x0, V_y, V_y0]: ((p(s(t_bool,h4s_netss_tends(s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_x),s(t_h4s_realaxs_real,V_x0),s(t_h4s_pairs_prod(t_h4s_topologys_topology(t_h4s_realaxs_real),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_pairs_u_2c(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_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_g)))))) & p(s(t_bool,h4s_netss_tends(s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_y),s(t_h4s_realaxs_real,V_y0),s(t_h4s_pairs_prod(t_h4s_topologys_topology(t_h4s_realaxs_real),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_pairs_u_2c(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_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_g))))))) => p(s(t_bool,h4s_netss_tends(s(t_fun(TV_u_27a,t_h4s_realaxs_real),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_realaxs_real),t_fun(TV_u_27a,t_h4s_realaxs_real)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_realaxs_real),t_fun(t_fun(TV_u_27a,t_h4s_realaxs_real),t_fun(TV_u_27a,t_h4s_realaxs_real))),V_uu_0),s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_x))),s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_y))),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,V_x0),s(t_h4s_realaxs_real,V_y0))),s(t_h4s_pairs_prod(t_h4s_topologys_topology(t_h4s_realaxs_real),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_pairs_u_2c(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_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_g)))))))))).
fof(ah4s_bools_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
fof(ah4s_reals_REALu_u_SUBu_u_REFL, axiom, ![V_x]: s(t_h4s_realaxs_real,h4s_reals_realu_u_sub(s(t_h4s_realaxs_real,V_x),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_nums_0)))).
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_reals_REALu_u_ADDu_u_SUB, axiom, ![V_y, V_x]: s(t_h4s_realaxs_real,h4s_reals_realu_u_sub(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)).
fof(ah4s_reals_REALu_u_DIVu_u_RMUL, axiom, ![V_y, 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)))) => s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,h4s_reals_u_2f(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y))),s(t_h4s_realaxs_real,V_y))) = s(t_h4s_realaxs_real,V_x))).
fof(ah4s_reals_REALu_u_MULu_u_RZERO, axiom, ![V_x]: s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(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_nums_0))))) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))).
fof(ah4s_reals_ABSu_u_NZ, axiom, ![V_x]: (~ (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_nums_0)))) <=> 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_abs(s(t_h4s_realaxs_real,V_x)))))))).
fof(ah4s_sats_dcu_u_neg, axiom, ![V_q, V_p]: ((p(s(t_bool,V_p)) <=> ~ (p(s(t_bool,V_q)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p))))))).
fof(ah4s_sats_dcu_u_imp, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) => p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (~ (p(s(t_bool,V_q))) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_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_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_reals_ABSu_u_0, axiom, s(t_h4s_realaxs_real,h4s_reals_abs(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_nums_0)))).
fof(ah4s_netss_LIMu_u_TENDS2, axiom, ![TV_u_27a,TV_u_27b]: ![V_y0, V_x0, V_m2, V_m1, V_f]: (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_m1))),s(TV_u_27a,V_x0),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_univ)))) => (p(s(t_bool,h4s_netss_tends(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27b,V_y0),s(t_h4s_pairs_prod(t_h4s_topologys_topology(TV_u_27b),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_pairs_u_2c(s(t_h4s_topologys_topology(TV_u_27b),h4s_topologys_mtop(s(t_h4s_topologys_metric(TV_u_27b),V_m2))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_netss_tendsto(s(t_h4s_pairs_prod(t_h4s_topologys_metric(TV_u_27a),TV_u_27a),h4s_pairs_u_2c(s(t_h4s_topologys_metric(TV_u_27a),V_m1),s(TV_u_27a,V_x0)))))))))) <=> ![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_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_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,h4s_topologys_dist(s(t_h4s_topologys_metric(TV_u_27a),V_m1),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_x0)))))))) & 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_m1),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_x0))))),s(t_h4s_realaxs_real,V_d))))) => p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,h4s_topologys_dist(s(t_h4s_topologys_metric(TV_u_27b),V_m2),s(t_h4s_pairs_prod(TV_u_27b,TV_u_27b),h4s_pairs_u_2c(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,V_y0))))),s(t_h4s_realaxs_real,V_e)))))))))).
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_topologys_MR1u_u_DEF, axiom, ![V_y, V_x]: 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,V_y))))) = s(t_h4s_realaxs_real,h4s_reals_abs(s(t_h4s_realaxs_real,h4s_reals_realu_u_sub(s(t_h4s_realaxs_real,V_y),s(t_h4s_realaxs_real,V_x)))))).
fof(ah4s_reals_ABSu_u_SUB, axiom, ![V_y, V_x]: s(t_h4s_realaxs_real,h4s_reals_abs(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_h4s_realaxs_real,h4s_reals_abs(s(t_h4s_realaxs_real,h4s_reals_realu_u_sub(s(t_h4s_realaxs_real,V_y),s(t_h4s_realaxs_real,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_lims_differentiable0, axiom, ![V_x, V_f]: (p(s(t_bool,h4s_lims_differentiable(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_x)))) <=> ?[V_l]: p(s(t_bool,h4s_lims_diffl(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_l),s(t_h4s_realaxs_real,V_x)))))).
fof(ah4s_lims_DIFFu_u_UNIQ, axiom, ![V_x, V_m, V_l, V_f]: ((p(s(t_bool,h4s_lims_diffl(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_l),s(t_h4s_realaxs_real,V_x)))) & p(s(t_bool,h4s_lims_diffl(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_m),s(t_h4s_realaxs_real,V_x))))) => s(t_h4s_realaxs_real,V_l) = s(t_h4s_realaxs_real,V_m))).
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_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f0)) => 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,f0),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f0))) <=> p(s(t_bool,t)))).
fof(ah4s_sats_pthu_u_ni1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => p(s(t_bool,V_p)))).
fof(ah4s_sats_pthu_u_ni2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_reals_REALu_u_MULu_u_RID, axiom, ![V_x]: s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(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_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))) = s(t_h4s_realaxs_real,V_x)).
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_reals_REALu_u_MULu_u_ASSOC, axiom, ![V_z, V_y, V_x]: s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,V_y),s(t_h4s_realaxs_real,V_z))))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y))),s(t_h4s_realaxs_real,V_z)))).
fof(ah4s_reals_realu_u_div, axiom, ![V_y, V_x]: s(t_h4s_realaxs_real,h4s_reals_u_2f(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,h4s_realaxs_inv(s(t_h4s_realaxs_real,V_y)))))).
fof(ah4s_reals_REALu_u_MULu_u_RINV, axiom, ![V_x]: (~ (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_nums_0)))) => s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,h4s_realaxs_inv(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_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))).
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_ABSu_u_TRIANGLE, axiom, ![V_y, V_x]: p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,h4s_reals_abs(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,h4s_reals_abs(s(t_h4s_realaxs_real,V_x))),s(t_h4s_realaxs_real,h4s_reals_abs(s(t_h4s_realaxs_real,V_y))))))))).
fof(ah4s_reals_REALu_u_SUBu_u_TRIANGLE, axiom, ![V_c, V_b, V_a]: s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,h4s_reals_realu_u_sub(s(t_h4s_realaxs_real,V_a),s(t_h4s_realaxs_real,V_b))),s(t_h4s_realaxs_real,h4s_reals_realu_u_sub(s(t_h4s_realaxs_real,V_b),s(t_h4s_realaxs_real,V_c))))) = s(t_h4s_realaxs_real,h4s_reals_realu_u_sub(s(t_h4s_realaxs_real,V_a),s(t_h4s_realaxs_real,V_c)))).
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_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_reals_REALu_u_DOWN2, axiom, ![V_y, 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,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,V_y))))) => ?[V_z]: (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_z)))) & (p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,V_z),s(t_h4s_realaxs_real,V_x)))) & p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,V_z),s(t_h4s_realaxs_real,V_y)))))))).
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_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_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_LTu_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_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_MINu_u_ALTu_c1, axiom, ![V_y, V_x]: (p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,V_y),s(t_h4s_realaxs_real,V_x)))) => s(t_h4s_realaxs_real,h4s_reals_min(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y))) = s(t_h4s_realaxs_real,V_y))).
fof(ah4s_reals_REALu_u_LEu_u_NEG2, axiom, ![V_y, V_x]: s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,h4s_realaxs_realu_u_neg(s(t_h4s_realaxs_real,V_x))),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_neg(s(t_h4s_realaxs_real,V_y))))) = 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_MINu_u_ALTu_c0, 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)))) => s(t_h4s_realaxs_real,h4s_reals_min(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y))) = s(t_h4s_realaxs_real,V_x))).
fof(ah4s_reals_REALu_u_LEu_u_SUP, axiom, ![V_x, V_p]: ((?[V_y]: p(s(t_bool,happ(s(t_fun(t_h4s_realaxs_real,t_bool),V_p),s(t_h4s_realaxs_real,V_y)))) & ?[V_y]: ![V_z]: (p(s(t_bool,happ(s(t_fun(t_h4s_realaxs_real,t_bool),V_p),s(t_h4s_realaxs_real,V_z)))) => p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,V_z),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,h4s_reals_sup(s(t_fun(t_h4s_realaxs_real,t_bool),V_p)))))) <=> ![V_y]: (![V_z]: (p(s(t_bool,happ(s(t_fun(t_h4s_realaxs_real,t_bool),V_p),s(t_h4s_realaxs_real,V_z)))) => p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,V_z),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_reals_REALu_u_LTu_u_NEG, axiom, ![V_y, V_x]: s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,h4s_realaxs_realu_u_neg(s(t_h4s_realaxs_real,V_x))),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_neg(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_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_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_IMPu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,t))) <=> p(s(t_bool,t)))).
fof(ah4s_reals_minu_u_def, axiom, ![V_y, V_x]: s(t_h4s_realaxs_real,h4s_reals_min(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y))) = s(t_h4s_realaxs_real,h4s_bools_cond(s(t_bool,h4s_reals_realu_u_lte(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_bools_boolu_u_caseu_u_thmu_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_CONG, axiom, ![TV_u_27a]: ![V_yu_27, V_y, V_xu_27, V_x, V_Q, V_P]: ((s(t_bool,V_P) = s(t_bool,V_Q) & ((p(s(t_bool,V_Q)) => s(TV_u_27a,V_x) = s(TV_u_27a,V_xu_27)) & (~ (p(s(t_bool,V_Q))) => s(TV_u_27a,V_y) = s(TV_u_27a,V_yu_27)))) => s(TV_u_27a,h4s_bools_cond(s(t_bool,V_P),s(TV_u_27a,V_x),s(TV_u_27a,V_y))) = s(TV_u_27a,h4s_bools_cond(s(t_bool,V_Q),s(TV_u_27a,V_xu_27),s(TV_u_27a,V_yu_27))))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f0))) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_IMPu_u_F, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f0))) => ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_Fu_u_IMP, axiom, ![V_t]: (~ (p(s(t_bool,V_t))) => (p(s(t_bool,V_t)) => p(s(t_bool,f0))))).
fof(ah4s_bools_DISJu_u_ASSOC, axiom, ![V_C, V_B, V_A]: ((p(s(t_bool,V_A)) | (p(s(t_bool,V_B)) | p(s(t_bool,V_C)))) <=> ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) | p(s(t_bool,V_C))))).
fof(ah4s_bools_DISJu_u_SYM, axiom, ![V_B, V_A]: ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) <=> (p(s(t_bool,V_B)) | p(s(t_bool,V_A))))).
fof(ah4s_bools_DEu_u_MORGANu_u_THMu_c1, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) <=> (~ (p(s(t_bool,V_A))) & ~ (p(s(t_bool,V_B)))))).
fof(ah4s_bools_BOOLu_u_CASESu_u_AX, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f0))).
fof(ah4s_sats_dcu_u_disj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_q)))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (p(s(t_bool,V_q)) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_sats_dcu_u_conj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) & p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_r))))) & ((p(s(t_bool,V_q)) | ~ (p(s(t_bool,V_p)))) & (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p)))))))).
fof(ah4s_bools_CONJu_u_ASSOC, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) & (p(s(t_bool,V_t2)) & p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) & p(s(t_bool,V_t3))))).
fof(ah4s_bools_ORu_u_DEF, axiom, ![V_x, V_xi_]: (p(s(t_bool,d_or(s(t_bool,V_x),s(t_bool,V_xi_)))) <=> ![V_t]: ((p(s(t_bool,V_x)) => p(s(t_bool,V_t))) => ((p(s(t_bool,V_xi_)) => p(s(t_bool,V_t))) => p(s(t_bool,V_t)))))).
fof(ah4s_reals_REALu_u_LEu_u_ANTISYM, 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_reals_realu_u_lte(s(t_h4s_realaxs_real,V_y),s(t_h4s_realaxs_real,V_x))))) <=> s(t_h4s_realaxs_real,V_x) = s(t_h4s_realaxs_real,V_y))).
fof(ah4s_bools_Fu_u_DEF, axiom, (p(s(t_bool,f0)) <=> ![V_t]: p(s(t_bool,V_t)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f0)))).
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_reals_REALu_u_LEu_u_EPSILON, axiom, ![V_y, 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)))) => p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,V_y),s(t_h4s_realaxs_real,V_e))))))) => 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_EQu_u_CLAUSESu_c3, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,f0) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_reals_realu_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_SUPu_u_LE, axiom, ![V_P]: ((?[V_x]: p(s(t_bool,happ(s(t_fun(t_h4s_realaxs_real,t_bool),V_P),s(t_h4s_realaxs_real,V_x)))) & ?[V_z]: ![V_x]: (p(s(t_bool,happ(s(t_fun(t_h4s_realaxs_real,t_bool),V_P),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_z)))))) => ![V_y]: (?[V_x]: (p(s(t_bool,happ(s(t_fun(t_h4s_realaxs_real,t_bool),V_P),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_x))))) <=> p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,V_y),s(t_h4s_realaxs_real,h4s_reals_sup(s(t_fun(t_h4s_realaxs_real,t_bool),V_P))))))))).
fof(ah4s_reals_REALu_u_LEu_u_SUBu_u_RADD, axiom, ![V_z, V_y, V_x]: s(t_bool,h4s_reals_realu_u_lte(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_h4s_realaxs_real,V_z))) = s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,V_z),s(t_h4s_realaxs_real,V_y)))))).
fof(ah4s_reals_REALu_u_LTu_u_SUBu_u_RADD, axiom, ![V_z, V_y, V_x]: s(t_bool,h4s_realaxs_realu_u_lt(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_h4s_realaxs_real,V_z))) = s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,V_z),s(t_h4s_realaxs_real,V_y)))))).
fof(ah4s_reals_REALu_u_LEu_u_SUBu_u_LADD, axiom, ![V_z, V_y, V_x]: s(t_bool,h4s_reals_realu_u_lte(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_reals_realu_u_lte(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_NOTu_u_FORALLu_u_THM, axiom, ![TV_u_27a]: ![V_P]: (~ (![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) <=> ?[V_x]: ~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_NOTu_u_EXISTSu_u_THM, axiom, ![TV_u_27a]: ![V_P]: (~ (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) <=> ![V_x]: ~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))))).
fof(ah4s_reals_REALu_u_LTu_u_ADDR, axiom, ![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_realaxs_realu_u_add(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,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_realaxs_real,V_y)))).
fof(ch4s_lims_IVT, conjecture, ![V_y, V_f, 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_reals_realu_u_lte(s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_a))),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,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_b))))))) & ![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))))) => p(s(t_bool,h4s_lims_contl(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_x))))))) => ?[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)))) & s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_x))) = s(t_h4s_realaxs_real,V_y))))).
