%   ORIGINAL: h4/lim/DIFF__CONT
% 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/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/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/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/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/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: !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/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__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/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__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__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/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/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/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__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/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__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__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__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/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__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/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/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/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__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/topology/MR1__LIMPT: !x. h4/topology/limpt (h4/topology/mtop h4/topology/mr1) x h4/pred__set/UNIV
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/sat/dc__eq: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% 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/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/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~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/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/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/real/REAL__LE__REFL: !x. h4/real/real__lte x x
% Assm: h4/real/REAL__LT__ADD2: !z y x w. h4/realax/real__lt w x /\ h4/realax/real__lt y z ==> h4/realax/real__lt (h4/realax/real__add w y) (h4/realax/real__add x z)
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> t
% 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__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__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__SUB__RZERO: !x. h4/real/real__sub x (h4/real/real__of__num h4/num/0) = x
% 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__LTE__TRANS: !z y x. h4/realax/real__lt x y /\ h4/real/real__lte y z ==> h4/realax/real__lt x z
% Assm: h4/real/REAL__LET__TRANS: !z y x. h4/real/real__lte x y /\ h4/realax/real__lt y z ==> h4/realax/real__lt x z
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% 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__POS__ID: !x. h4/real/real__lte (h4/real/real__of__num h4/num/0) x ==> h4/real/pos x = x
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% 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/pos__def: !x. h4/real/pos x = h4/bool/COND (h4/real/real__lte (h4/real/real__of__num h4/num/0) x) x (h4/real/real__of__num h4/num/0)
% 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__CONG: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm: h4/real/REAL__ADD2__SUB2: !d c b a. h4/real/real__sub (h4/realax/real__add a b) (h4/realax/real__add c d) = h4/realax/real__add (h4/real/real__sub a c) (h4/real/real__sub b d)
% Assm: h4/real/REAL__ADD__SYM: !y x. h4/realax/real__add x y = h4/realax/real__add y x
% Assm: h4/real/real__sub0: !y x. h4/real/real__sub x y = h4/realax/real__add x (h4/realax/real__neg y)
% Assm: h4/real/REAL__NEG__ADD: !y x. h4/realax/real__neg (h4/realax/real__add x y) = h4/realax/real__add (h4/realax/real__neg x) (h4/realax/real__neg y)
% Assm: h4/real/REAL__ADD__ASSOC: !z y x. h4/realax/real__add x (h4/realax/real__add y z) = h4/realax/real__add (h4/realax/real__add x y) z
% Assm: h4/real/REAL__LTE__ADD: !y x. h4/realax/real__lt (h4/real/real__of__num h4/num/0) x /\ h4/real/real__lte (h4/real/real__of__num h4/num/0) y ==> h4/realax/real__lt (h4/real/real__of__num h4/num/0) (h4/realax/real__add x y)
% Assm: h4/real/REAL__ADD__LID: !x. h4/realax/real__add (h4/real/real__of__num h4/num/0) x = x
% Assm: h4/real/REAL__LTE__ADD2: !z y x w. h4/realax/real__lt w x /\ h4/real/real__lte y z ==> h4/realax/real__lt (h4/realax/real__add w y) (h4/realax/real__add x z)
% Assm: h4/realax/SUP__ALLPOS__LEMMA4: !y. ~h4/realax/real__lt h4/realax/real__0 y ==> (!x. h4/realax/real__lt y (h4/realax/real__of__hreal x))
% Assm: h4/bool/IMP__CLAUSES_c2: !t. F ==> t <=> T
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/quotient/APPLY__RSP: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f g x y. h4/quotient/_3D_3D_3D_3E R1 R2 f g /\ R1 x y ==> R2 (f x) (g y)))
% Assm: h4/quotient/FUN__REL: !g f R2 R1. h4/quotient/_3D_3D_3D_3E R1 R2 f g <=> (!x y. R1 x y ==> R2 (f x) (g y))
% Assm: h4/quotient/EQUALS__PRS: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!x y. x = y <=> R (rep x) (rep y))
% Assm: h4/quotient/IDENTITY__QUOTIENT: h4/quotient/QUOTIENT $equals h4/combin/I h4/combin/I
% Assm: h4/realax/real__QUOTIENT: h4/quotient/QUOTIENT h4/realax/treal__eq h4/realax/real__ABS h4/realax/real__REP
% Assm: h4/quotient/REP__ABS__RSP: !rep abs REL. h4/quotient/QUOTIENT REL abs rep ==> (!x1 x2. REL x1 x2 ==> REL x1 (rep (abs x2)))
% Assm: h4/realax/REAL__POS: !X. h4/realax/real__lt h4/realax/real__0 (h4/realax/real__of__hreal X)
% Assm: h4/realax/TREAL__LT__TRANS: !z y x. h4/realax/treal__lt x y /\ h4/realax/treal__lt y z ==> h4/realax/treal__lt x z
% Assm: h4/realax/TREAL__LT__WELLDEF: !y2 y1 x2 x1. h4/realax/treal__eq x1 x2 /\ h4/realax/treal__eq y1 y2 ==> (h4/realax/treal__lt x1 y1 <=> h4/realax/treal__lt x2 y2)
% Assm: h4/realax/TREAL__EQ__EQUIV: !q p. h4/realax/treal__eq p q <=> h4/realax/treal__eq p = h4/realax/treal__eq q
% Assm: h4/realax/real__lt0: !T2 T1. h4/realax/real__lt T1 T2 <=> h4/realax/treal__lt (h4/realax/real__REP T1) (h4/realax/real__REP T2)
% Assm: h4/realax/TREAL__LT__TOTAL: !y x. h4/realax/treal__eq x y \/ h4/realax/treal__lt x y \/ h4/realax/treal__lt y x
% Assm: h4/quotient/LAMBDA__PRS: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f. (\x. f x) = h4/quotient/_2D_2D_3E rep1 abs2 (\x. rep2 (f (abs1 x)))))
% Assm: h4/quotient/RES__FORALL__RSP: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!f g. h4/quotient/_3D_3D_3D_3E R $equals f g ==> (h4/bool/RES__FORALL (h4/quotient/respects R) f <=> h4/bool/RES__FORALL (h4/quotient/respects R) g))
% Assm: h4/quotient/EQUALS__RSP: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!x1 x2 y1 y2. R x1 x2 /\ R y1 y2 ==> (R x1 y1 <=> R x2 y2))
% Assm: h4/quotient/EQUIV__def: !E. h4/quotient/EQUIV E <=> (!x y. E x y <=> E x = E y)
% Assm: h4/quotient/FORALL__PRS: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!f. $forall f <=> h4/bool/RES__FORALL (h4/quotient/respects R) (h4/quotient/_2D_2D_3E abs h4/combin/I f))
% Assm: h4/quotient/FUN__QUOTIENT: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> h4/quotient/QUOTIENT (h4/quotient/_3D_3D_3D_3E R1 R2) (h4/quotient/_2D_2D_3E rep1 abs2) (h4/quotient/_2D_2D_3E abs1 rep2))
% Assm: h4/combin/I__THM: !x. h4/combin/I x = x
% Assm: h4/quotient/EQUIV__RES__FORALL: !P E. h4/quotient/EQUIV E ==> (h4/bool/RES__FORALL (h4/quotient/respects E) P <=> $forall P)
% Assm: h4/real/REAL__LE__RADD: !z y x. h4/real/real__lte (h4/realax/real__add x z) (h4/realax/real__add y z) <=> h4/real/real__lte x y
% Assm: h4/real/ABS__BOUND: !y x d. h4/realax/real__lt (h4/real/abs (h4/real/real__sub x y)) d ==> h4/realax/real__lt y (h4/realax/real__add x d)
% Assm: h4/bool/IMP__CLAUSES_c4: !t. 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/real/abs0: !x. h4/real/abs x = h4/bool/COND (h4/real/real__lte (h4/real/real__of__num h4/num/0) x) x (h4/realax/real__neg x)
% Assm: h4/real/REAL__LE__NEGR: !x. h4/real/real__lte x (h4/realax/real__neg x) <=> h4/real/real__lte x (h4/real/real__of__num h4/num/0)
% Assm: h4/real/REAL__LE__NEGL: !x. h4/real/real__lte (h4/realax/real__neg x) x <=> h4/real/real__lte (h4/real/real__of__num h4/num/0) x
% Assm: h4/real/REAL__LE__ADD2: !z y x w. h4/real/real__lte w x /\ h4/real/real__lte y z ==> h4/real/real__lte (h4/realax/real__add w y) (h4/realax/real__add x z)
% Assm: h4/real/REAL__LT__IMP__LE: !y x. h4/realax/real__lt x y ==> h4/real/real__lte x y
% Assm: h4/real/REAL__NOT__LE: !y x. ~h4/real/real__lte x 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/real/REAL__LE__LADD: !z y x. h4/real/real__lte (h4/realax/real__add x y) (h4/realax/real__add x z) <=> h4/real/real__lte y z
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/real/ABS__BETWEEN: !y x d. h4/realax/real__lt (h4/real/real__of__num h4/num/0) d /\ h4/realax/real__lt (h4/real/real__sub x d) y /\ h4/realax/real__lt y (h4/realax/real__add x d) <=> h4/realax/real__lt (h4/real/abs (h4/real/real__sub y x)) d
% Assm: h4/real/REAL__SUB__ABS: !y x. h4/real/real__lte (h4/real/real__sub (h4/real/abs x) (h4/real/abs y)) (h4/real/abs (h4/real/real__sub x y))
% Assm: h4/real/REAL__MUL__SYM: !y x. h4/realax/real__mul x y = h4/realax/real__mul y x
% Assm: h4/real/REAL__ADD__SUB: !y x. h4/real/real__sub (h4/realax/real__add x y) x = y
% Assm: h4/real/REAL__SUB__ADD: !y x. h4/realax/real__add (h4/real/real__sub x y) y = x
% Assm: h4/real/REAL__LE__TRANS: !z y x. h4/real/real__lte x y /\ h4/real/real__lte y z ==> h4/real/real__lte x z
% Assm: h4/seq/SER__CDIV: !x0 x c. h4/seq/sums x x0 ==> h4/seq/sums (\n. h4/real/_2F (x n) c) (h4/real/_2F x0 c)
% Assm: h4/seq/SER__CMUL: !x0 x c. h4/seq/sums x x0 ==> h4/seq/sums (\n. h4/realax/real__mul c (x n)) (h4/realax/real__mul c x0)
% 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__EQ__NEG: !y x. h4/realax/real__neg x = h4/realax/real__neg y <=> x = y
% Assm: h4/bool/CONJ__SYM: !t2 t1. t1 /\ t2 <=> t2 /\ t1
% Assm: h4/real/REAL__LE__NEG: !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__LE__ANTISYM: !y x. h4/real/real__lte x y /\ h4/real/real__lte y x <=> x = y
% Assm: h4/real/POW__MUL: !y x n. h4/real/pow (h4/realax/real__mul x y) n = h4/realax/real__mul (h4/real/pow x n) (h4/real/pow y n)
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/real/pow0_c1: !x n. h4/real/pow x (h4/num/SUC n) = h4/realax/real__mul x (h4/real/pow x n)
% Goal: !x l f. h4/lim/diffl f l x ==> h4/lim/contl f x
%   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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_topologys_MR1u_u_LIMPT]: !x. h4/topology/limpt (h4/topology/mtop h4/topology/mr1) x h4/pred__set/UNIV
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_sats_dcu_u_eq]: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% 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_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_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~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_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_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_reals_REALu_u_LEu_u_REFL]: !x. h4/real/real__lte x x
% Assm [h4s_reals_REALu_u_LTu_u_ADD2]: !z y x w. h4/realax/real__lt w x /\ h4/realax/real__lt y z ==> h4/realax/real__lt (h4/realax/real__add w y) (h4/realax/real__add x z)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% 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_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_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_SUBu_u_RZERO]: !x. h4/real/real__sub x (h4/real/real__of__num h4/num/0) = x
% 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_LTEu_u_TRANS]: !z y x. h4/realax/real__lt x y /\ h4/real/real__lte y z ==> h4/realax/real__lt x z
% Assm [h4s_reals_REALu_u_LETu_u_TRANS]: !z y x. h4/real/real__lte x y /\ h4/realax/real__lt y z ==> h4/realax/real__lt x z
% Assm [h4s_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% 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_POSu_u_ID]: !x. h4/real/real__lte (h4/real/real__of__num h4/num/0) x ==> h4/real/pos x = x
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% 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_posu_u_def]: !x. h4/real/pos x = h4/bool/COND (h4/real/real__lte (h4/real/real__of__num h4/num/0) x) x (h4/real/real__of__num h4/num/0)
% 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_CONG]: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm [h4s_reals_REALu_u_ADD2u_u_SUB2]: !d c b a. h4/real/real__sub (h4/realax/real__add a b) (h4/realax/real__add c d) = h4/realax/real__add (h4/real/real__sub a c) (h4/real/real__sub b d)
% Assm [h4s_reals_REALu_u_ADDu_u_SYM]: !y x. h4/realax/real__add x y = h4/realax/real__add y x
% Assm [h4s_reals_realu_u_sub0]: !y x. h4/real/real__sub x y = h4/realax/real__add x (h4/realax/real__neg y)
% Assm [h4s_reals_REALu_u_NEGu_u_ADD]: !y x. h4/realax/real__neg (h4/realax/real__add x y) = h4/realax/real__add (h4/realax/real__neg x) (h4/realax/real__neg y)
% Assm [h4s_reals_REALu_u_ADDu_u_ASSOC]: !z y x. h4/realax/real__add x (h4/realax/real__add y z) = h4/realax/real__add (h4/realax/real__add x y) z
% Assm [h4s_reals_REALu_u_LTEu_u_ADD]: !y x. h4/realax/real__lt (h4/real/real__of__num h4/num/0) x /\ h4/real/real__lte (h4/real/real__of__num h4/num/0) y ==> h4/realax/real__lt (h4/real/real__of__num h4/num/0) (h4/realax/real__add x y)
% Assm [h4s_reals_REALu_u_ADDu_u_LID]: !x. h4/realax/real__add (h4/real/real__of__num h4/num/0) x = x
% Assm [h4s_reals_REALu_u_LTEu_u_ADD2]: !z y x w. h4/realax/real__lt w x /\ h4/real/real__lte y z ==> h4/realax/real__lt (h4/realax/real__add w y) (h4/realax/real__add x z)
% Assm [h4s_realaxs_SUPu_u_ALLPOSu_u_LEMMA4]: !y. ~h4/realax/real__lt h4/realax/real__0 y ==> (!x. h4/realax/real__lt y (h4/realax/real__of__hreal x))
% Assm [h4s_bools_IMPu_u_CLAUSESu_c2]: !t. F ==> t <=> T
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_quotients_APPLYu_u_RSP]: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f g x y. happ (happ (h4/quotient/_3D_3D_3D_3E R1 R2) f) g /\ happ (happ R1 x) y ==> happ (happ R2 (happ f x)) (happ g y)))
% Assm [h4s_quotients_FUNu_u_REL]: !g f R2 R1. happ (happ (h4/quotient/_3D_3D_3D_3E R1 R2) f) g <=> (!x y. happ (happ R1 x) y ==> happ (happ R2 (happ f x)) (happ g y))
% Assm [h4s_quotients_EQUALSu_u_PRS]: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!x y. x = y <=> happ (happ R (happ rep x)) (happ rep y))
% Assm [h4s_quotients_IDENTITYu_u_QUOTIENT]: h4/quotient/QUOTIENT $equals h4/combin/I h4/combin/I
% Assm [h4s_realaxs_realu_u_QUOTIENT]: h4/quotient/QUOTIENT h4/realax/treal__eq h4/realax/real__ABS h4/realax/real__REP
% Assm [h4s_quotients_REPu_u_ABSu_u_RSP]: !rep abs REL. h4/quotient/QUOTIENT REL abs rep ==> (!x1 x2. happ (happ REL x1) x2 ==> happ (happ REL x1) (happ rep (happ abs x2)))
% Assm [h4s_realaxs_REALu_u_POS]: !X. h4/realax/real__lt h4/realax/real__0 (h4/realax/real__of__hreal X)
% Assm [h4s_realaxs_TREALu_u_LTu_u_TRANS]: !z y x. h4/realax/treal__lt x y /\ h4/realax/treal__lt y z ==> h4/realax/treal__lt x z
% Assm [h4s_realaxs_TREALu_u_LTu_u_WELLDEF]: !y2 y1 x2 x1. happ (happ h4/realax/treal__eq x1) x2 /\ happ (happ h4/realax/treal__eq y1) y2 ==> (h4/realax/treal__lt x1 y1 <=> h4/realax/treal__lt x2 y2)
% Assm [h4s_realaxs_TREALu_u_EQu_u_EQUIV]: !q p. happ (happ h4/realax/treal__eq p) q <=> happ h4/realax/treal__eq p = happ h4/realax/treal__eq q
% Assm [h4s_realaxs_realu_u_lt0]: !T2 T1. h4/realax/real__lt T1 T2 <=> h4/realax/treal__lt (happ h4/realax/real__REP T1) (happ h4/realax/real__REP T2)
% Assm [h4s_realaxs_TREALu_u_LTu_u_TOTAL]: !y x. happ (happ h4/realax/treal__eq x) y \/ h4/realax/treal__lt x y \/ h4/realax/treal__lt y x
% Assm [h4s_quotients_LAMBDAu_u_PRS]: !_0. (!rep2 f abs1 x. happ (happ (happ (happ _0 rep2) f) abs1) x = happ rep2 (happ f (happ abs1 x))) ==> (!rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f x. happ f x = happ (happ (h4/quotient/_2D_2D_3E rep1 abs2) (happ (happ (happ _0 rep2) f) abs1)) x)))
% Assm [h4s_quotients_RESu_u_FORALLu_u_RSP]: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!f g. happ (happ (h4/quotient/_3D_3D_3D_3E R $equals) f) g ==> (h4/bool/RES__FORALL (h4/quotient/respects R) f <=> h4/bool/RES__FORALL (h4/quotient/respects R) g))
% Assm [h4s_quotients_EQUALSu_u_RSP]: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!x1 x2 y1 y2. happ (happ R x1) x2 /\ happ (happ R y1) y2 ==> (happ (happ R x1) y1 <=> happ (happ R x2) y2))
% Assm [h4s_quotients_EQUIVu_u_def]: !E. h4/quotient/EQUIV E <=> (!x y. happ (happ E x) y <=> happ E x = happ E y)
% Assm [h4s_quotients_FORALLu_u_PRS]: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!f. $forall f <=> h4/bool/RES__FORALL (h4/quotient/respects R) (happ (h4/quotient/_2D_2D_3E abs h4/combin/I) f))
% Assm [h4s_quotients_FUNu_u_QUOTIENT]: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> h4/quotient/QUOTIENT (h4/quotient/_3D_3D_3D_3E R1 R2) (h4/quotient/_2D_2D_3E rep1 abs2) (h4/quotient/_2D_2D_3E abs1 rep2))
% Assm [h4s_combins_Iu_u_THM]: !x. happ h4/combin/I x = x
% Assm [h4s_quotients_EQUIVu_u_RESu_u_FORALL]: !P E. h4/quotient/EQUIV E ==> (h4/bool/RES__FORALL (h4/quotient/respects E) P <=> $forall P)
% Assm [h4s_reals_REALu_u_LEu_u_RADD]: !z y x. h4/real/real__lte (h4/realax/real__add x z) (h4/realax/real__add y z) <=> h4/real/real__lte x y
% Assm [h4s_reals_ABSu_u_BOUND]: !y x d. h4/realax/real__lt (h4/real/abs (h4/real/real__sub x y)) d ==> h4/realax/real__lt y (h4/realax/real__add x d)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. 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_reals_abs0]: !x. h4/real/abs x = h4/bool/COND (h4/real/real__lte (h4/real/real__of__num h4/num/0) x) x (h4/realax/real__neg x)
% Assm [h4s_reals_REALu_u_LEu_u_NEGR]: !x. h4/real/real__lte x (h4/realax/real__neg x) <=> h4/real/real__lte x (h4/real/real__of__num h4/num/0)
% Assm [h4s_reals_REALu_u_LEu_u_NEGL]: !x. h4/real/real__lte (h4/realax/real__neg x) x <=> h4/real/real__lte (h4/real/real__of__num h4/num/0) x
% Assm [h4s_reals_REALu_u_LEu_u_ADD2]: !z y x w. h4/real/real__lte w x /\ h4/real/real__lte y z ==> h4/real/real__lte (h4/realax/real__add w y) (h4/realax/real__add x z)
% Assm [h4s_reals_REALu_u_LTu_u_IMPu_u_LE]: !y x. h4/realax/real__lt x y ==> h4/real/real__lte x y
% Assm [h4s_reals_REALu_u_NOTu_u_LE]: !y x. ~h4/real/real__lte x y <=> h4/realax/real__lt y x
% Assm [h4s_reals_REALu_u_NOTu_u_LT]: !y x. ~h4/realax/real__lt x y <=> h4/real/real__lte y x
% Assm [h4s_reals_REALu_u_LEu_u_LADD]: !z y x. h4/real/real__lte (h4/realax/real__add x y) (h4/realax/real__add x z) <=> h4/real/real__lte y z
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_reals_ABSu_u_BETWEEN]: !y x d. h4/realax/real__lt (h4/real/real__of__num h4/num/0) d /\ h4/realax/real__lt (h4/real/real__sub x d) y /\ h4/realax/real__lt y (h4/realax/real__add x d) <=> h4/realax/real__lt (h4/real/abs (h4/real/real__sub y x)) d
% Assm [h4s_reals_REALu_u_SUBu_u_ABS]: !y x. h4/real/real__lte (h4/real/real__sub (h4/real/abs x) (h4/real/abs y)) (h4/real/abs (h4/real/real__sub x y))
% Assm [h4s_reals_REALu_u_MULu_u_SYM]: !y x. h4/realax/real__mul x y = h4/realax/real__mul y x
% 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_SUBu_u_ADD]: !y x. h4/realax/real__add (h4/real/real__sub x y) y = x
% Assm [h4s_reals_REALu_u_LEu_u_TRANS]: !z y x. h4/real/real__lte x y /\ h4/real/real__lte y z ==> h4/real/real__lte x z
% Assm [h4s_seqs_SERu_u_CDIV]: !_0. (!x c n. happ (happ (happ _0 x) c) n = h4/real/_2F (happ x n) c) ==> (!x0 x c. h4/seq/sums x x0 ==> h4/seq/sums (happ (happ _0 x) c) (h4/real/_2F x0 c))
% Assm [h4s_seqs_SERu_u_CMUL]: !_0. (!c x n. happ (happ (happ _0 c) x) n = h4/realax/real__mul c (happ x n)) ==> (!x0 x c. h4/seq/sums x x0 ==> h4/seq/sums (happ (happ _0 c) x) (h4/realax/real__mul c x0))
% 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_EQu_u_NEG]: !y x. h4/realax/real__neg x = h4/realax/real__neg y <=> x = y
% Assm [h4s_bools_CONJu_u_SYM]: !t2 t1. t1 /\ t2 <=> t2 /\ t1
% Assm [h4s_reals_REALu_u_LEu_u_NEG]: !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_LEu_u_ANTISYM]: !y x. h4/real/real__lte x y /\ h4/real/real__lte y x <=> x = y
% Assm [h4s_reals_POWu_u_MUL]: !y x n. h4/real/pow (h4/realax/real__mul x y) n = h4/realax/real__mul (h4/real/pow x n) (h4/real/pow y n)
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_reals_pow0u_c1]: !x n. h4/real/pow x (h4/num/SUC n) = h4/realax/real__mul x (h4/real/pow x n)
% Goal: !x l f. h4/lim/diffl f l x ==> h4/lim/contl f x
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_Q1376334,TV_Q1376330]: ![V_f, V_g]: (![V_x]: s(TV_Q1376330,happ(s(t_fun(TV_Q1376334,TV_Q1376330),V_f),s(TV_Q1376334,V_x))) = s(TV_Q1376330,happ(s(t_fun(TV_Q1376334,TV_Q1376330),V_g),s(TV_Q1376334,V_x))) => s(t_fun(TV_Q1376334,TV_Q1376330),V_f) = s(t_fun(TV_Q1376334,TV_Q1376330),V_g))).
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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_bools_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_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_sats_dcu_u_eq, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> s(t_bool,V_q) = s(t_bool,V_r)) <=> ((p(s(t_bool,V_p)) | (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) & ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_q))))) & ((p(s(t_bool,V_q)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_p))))) & (p(s(t_bool,V_r)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p)))))))))).
fof(ah4s_sats_ANDu_u_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_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_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_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_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_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_dcu_u_neg, axiom, ![V_q, V_p]: ((p(s(t_bool,V_p)) <=> ~ (p(s(t_bool,V_q)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p))))))).
fof(ah4s_sats_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_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_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_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_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_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_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_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_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_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_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_LTEu_u_TRANS, axiom, ![V_z, V_y, V_x]: ((p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y)))) & p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,V_y),s(t_h4s_realaxs_real,V_z))))) => p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_z)))))).
fof(ah4s_reals_REALu_u_LETu_u_TRANS, axiom, ![V_z, V_y, V_x]: ((p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y)))) & p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,V_y),s(t_h4s_realaxs_real,V_z))))) => p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_z)))))).
fof(ah4s_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_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_POSu_u_ID, axiom, ![V_x]: (p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_realaxs_real,V_x)))) => s(t_h4s_realaxs_real,h4s_reals_pos(s(t_h4s_realaxs_real,V_x))) = s(t_h4s_realaxs_real,V_x))).
fof(ah4s_bools_REFLu_u_CLAUSE, axiom, ![TV_u_27a]: ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_x) <=> p(s(t_bool,t)))).
fof(ah4s_bools_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_posu_u_def, axiom, ![V_x]: s(t_h4s_realaxs_real,h4s_reals_pos(s(t_h4s_realaxs_real,V_x))) = s(t_h4s_realaxs_real,h4s_bools_cond(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_realaxs_real,V_x))),s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))))).
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_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_reals_REALu_u_ADD2u_u_SUB2, axiom, ![V_d, V_c, V_b, V_a]: 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_a),s(t_h4s_realaxs_real,V_b))),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,V_c),s(t_h4s_realaxs_real,V_d))))) = 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_c))),s(t_h4s_realaxs_real,h4s_reals_realu_u_sub(s(t_h4s_realaxs_real,V_b),s(t_h4s_realaxs_real,V_d)))))).
fof(ah4s_reals_REALu_u_ADDu_u_SYM, axiom, ![V_y, V_x]: s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,V_y),s(t_h4s_realaxs_real,V_x)))).
fof(ah4s_reals_realu_u_sub0, 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_realaxs_realu_u_add(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_neg(s(t_h4s_realaxs_real,V_y)))))).
fof(ah4s_reals_REALu_u_NEGu_u_ADD, axiom, ![V_y, V_x]: s(t_h4s_realaxs_real,h4s_realaxs_realu_u_neg(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_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)))))).
fof(ah4s_reals_REALu_u_ADDu_u_ASSOC, axiom, ![V_z, 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,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,V_y),s(t_h4s_realaxs_real,V_z))))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(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_z)))).
fof(ah4s_reals_REALu_u_LTEu_u_ADD, 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_reals_realu_u_lte(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_realaxs_real,V_y))))) => p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y)))))))).
fof(ah4s_reals_REALu_u_ADDu_u_LID, axiom, ![V_x]: s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_realaxs_real,V_x))) = s(t_h4s_realaxs_real,V_x)).
fof(ah4s_reals_REALu_u_LTEu_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_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,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_realaxs_SUPu_u_ALLPOSu_u_LEMMA4, axiom, ![V_y]: (~ (p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,h4s_realaxs_realu_u_0),s(t_h4s_realaxs_real,V_y))))) => ![V_x]: p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,V_y),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_ofu_u_hreal(s(t_h4s_hreals_hreal,V_x)))))))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f0)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f0)))).
fof(ah4s_quotients_APPLYu_u_RSP, axiom, ![TV_u_27c,TV_u_27d,TV_u_27b,TV_u_27a]: ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27c,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(t_fun(TV_u_27d,TV_u_27b),V_rep2)))) => ![V_f, V_g, V_x, V_y]: ((p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_fun(TV_u_27a,TV_u_27b),V_g)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))))) => p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_y)))))))))).
fof(ah4s_quotients_FUNu_u_REL, axiom, ![TV_u_27b,TV_u_27a]: ![V_g, V_f, V_R2, V_R1]: (p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_fun(TV_u_27a,TV_u_27b),V_g)))) <=> ![V_x, V_y]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(TV_u_27a,V_x))),s(TV_u_27a,V_y)))) => p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_y))))))))).
fof(ah4s_quotients_EQUALSu_u_PRS, axiom, ![TV_u_27a,TV_u_27b]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ![V_x, V_y]: (s(TV_u_27b,V_x) = s(TV_u_27b,V_y) <=> p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_x))))),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_y))))))))).
fof(ah4s_quotients_IDENTITYu_u_QUOTIENT, axiom, ![TV_u_27a]: p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),d_equals),s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_i),s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_i))))).
fof(ah4s_realaxs_realu_u_QUOTIENT, axiom, p(s(t_bool,h4s_quotients_quotient(s(t_fun(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),t_fun(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),t_bool)),h4s_realaxs_trealu_u_eq),s(t_fun(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),t_h4s_realaxs_real),h4s_realaxs_realu_u_abs),s(t_fun(t_h4s_realaxs_real,t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal)),h4s_realaxs_realu_u_rep))))).
fof(ah4s_quotients_REPu_u_ABSu_u_RSP, axiom, ![TV_u_27b,TV_u_27a]: ![V_rep, V_abs, V_REL]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_REL),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ![V_x1, V_x2]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_REL),s(TV_u_27a,V_x1))),s(TV_u_27a,V_x2)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_REL),s(TV_u_27a,V_x1))),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,V_x2))))))))))).
fof(ah4s_realaxs_REALu_u_POS, axiom, ![V_X]: p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,h4s_realaxs_realu_u_0),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_ofu_u_hreal(s(t_h4s_hreals_hreal,V_X))))))).
fof(ah4s_realaxs_TREALu_u_LTu_u_TRANS, axiom, ![V_z, V_y, V_x]: ((p(s(t_bool,h4s_realaxs_trealu_u_lt(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_x),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_y)))) & p(s(t_bool,h4s_realaxs_trealu_u_lt(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_y),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_z))))) => p(s(t_bool,h4s_realaxs_trealu_u_lt(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_x),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_z)))))).
fof(ah4s_realaxs_TREALu_u_LTu_u_WELLDEF, axiom, ![V_y2, V_y1, V_x2, V_x1]: ((p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),t_fun(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),t_bool)),h4s_realaxs_trealu_u_eq),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_x1))),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_x2)))) & p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),t_fun(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),t_bool)),h4s_realaxs_trealu_u_eq),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_y1))),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_y2))))) => s(t_bool,h4s_realaxs_trealu_u_lt(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_x1),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_y1))) = s(t_bool,h4s_realaxs_trealu_u_lt(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_x2),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_y2))))).
fof(ah4s_realaxs_TREALu_u_EQu_u_EQUIV, axiom, ![V_q, V_p]: (p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),t_fun(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),t_bool)),h4s_realaxs_trealu_u_eq),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_p))),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_q)))) <=> s(t_fun(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),t_fun(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),t_bool)),h4s_realaxs_trealu_u_eq),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_p))) = s(t_fun(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),t_fun(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),t_bool)),h4s_realaxs_trealu_u_eq),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_q))))).
fof(ah4s_realaxs_realu_u_lt0, axiom, ![V_T2, V_T1]: s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,V_T1),s(t_h4s_realaxs_real,V_T2))) = s(t_bool,h4s_realaxs_trealu_u_lt(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),happ(s(t_fun(t_h4s_realaxs_real,t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal)),h4s_realaxs_realu_u_rep),s(t_h4s_realaxs_real,V_T1))),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),happ(s(t_fun(t_h4s_realaxs_real,t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal)),h4s_realaxs_realu_u_rep),s(t_h4s_realaxs_real,V_T2)))))).
fof(ah4s_realaxs_TREALu_u_LTu_u_TOTAL, axiom, ![V_y, V_x]: (p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),t_fun(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),t_bool)),h4s_realaxs_trealu_u_eq),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_x))),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_y)))) | (p(s(t_bool,h4s_realaxs_trealu_u_lt(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_x),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_y)))) | p(s(t_bool,h4s_realaxs_trealu_u_lt(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_y),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_x))))))).
fof(ah4s_quotients_LAMBDAu_u_PRS, axiom, ![TV_u_27b,TV_u_27d,TV_u_27a,TV_u_27c]: ![V_uu_0]: (![V_rep2, V_f, V_abs1, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27d,TV_u_27b),t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b)))),V_uu_0),s(t_fun(TV_u_27d,TV_u_27b),V_rep2))),s(t_fun(TV_u_27c,TV_u_27d),V_f))),s(t_fun(TV_u_27a,TV_u_27c),V_abs1))),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27d,TV_u_27b),V_rep2),s(TV_u_27d,happ(s(t_fun(TV_u_27c,TV_u_27d),V_f),s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(TV_u_27a,V_x))))))) => ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27c,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(t_fun(TV_u_27d,TV_u_27b),V_rep2)))) => ![V_f, V_x]: s(TV_u_27d,happ(s(t_fun(TV_u_27c,TV_u_27d),V_f),s(TV_u_27c,V_x))) = s(TV_u_27d,happ(s(t_fun(TV_u_27c,TV_u_27d),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27c,TV_u_27d)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27c,TV_u_27a),V_rep1),s(t_fun(TV_u_27b,TV_u_27d),V_abs2))),s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27d,TV_u_27b),t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b)))),V_uu_0),s(t_fun(TV_u_27d,TV_u_27b),V_rep2))),s(t_fun(TV_u_27c,TV_u_27d),V_f))),s(t_fun(TV_u_27a,TV_u_27c),V_abs1))))),s(TV_u_27c,V_x))))))).
fof(ah4s_quotients_RESu_u_FORALLu_u_RSP, axiom, ![TV_u_27b,TV_u_27a]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ![V_f, V_g]: (p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(t_bool,t_fun(t_bool,t_bool)),d_equals))),s(t_fun(TV_u_27a,t_bool),V_f))),s(t_fun(TV_u_27a,t_bool),V_g)))) => s(t_bool,h4s_bools_resu_u_forall(s(t_fun(TV_u_27a,t_bool),h4s_quotients_respects(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_fun(TV_u_27a,t_bool),V_f))) = s(t_bool,h4s_bools_resu_u_forall(s(t_fun(TV_u_27a,t_bool),h4s_quotients_respects(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_fun(TV_u_27a,t_bool),V_g)))))).
fof(ah4s_quotients_EQUALSu_u_RSP, axiom, ![TV_u_27b,TV_u_27a]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ![V_x1, V_x2, V_y1, V_y2]: ((p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x1))),s(TV_u_27a,V_x2)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_y1))),s(TV_u_27a,V_y2))))) => s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x1))),s(TV_u_27a,V_y1))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x2))),s(TV_u_27a,V_y2)))))).
fof(ah4s_quotients_EQUIVu_u_def, axiom, ![TV_u_27a]: ![V_E]: (p(s(t_bool,h4s_quotients_equiv(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_E)))) <=> ![V_x, V_y]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_E),s(TV_u_27a,V_x))),s(TV_u_27a,V_y)))) <=> s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_E),s(TV_u_27a,V_x))) = s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_E),s(TV_u_27a,V_y)))))).
fof(ah4s_quotients_FORALLu_u_PRS, axiom, ![TV_u_27a,TV_u_27b]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ![V_f]: s(t_bool,d_forall(s(t_fun(TV_u_27b,t_bool),V_f))) = s(t_bool,h4s_bools_resu_u_forall(s(t_fun(TV_u_27a,t_bool),h4s_quotients_respects(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27a,t_bool)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(t_bool,t_bool),h4s_combins_i))),s(t_fun(TV_u_27b,t_bool),V_f))))))).
fof(ah4s_quotients_FUNu_u_QUOTIENT, axiom, ![TV_u_27a,TV_u_27c,TV_u_27d,TV_u_27b]: ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27c,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(t_fun(TV_u_27d,TV_u_27b),V_rep2)))) => p(s(t_bool,h4s_quotients_quotient(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))),s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27c,TV_u_27d)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27c,TV_u_27a),V_rep1),s(t_fun(TV_u_27b,TV_u_27d),V_abs2))),s(t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(TV_u_27a,TV_u_27b)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27d,TV_u_27b),V_rep2))))))))).
fof(ah4s_combins_Iu_u_THM, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_i),s(TV_u_27a,V_x))) = s(TV_u_27a,V_x)).
fof(ah4s_quotients_EQUIVu_u_RESu_u_FORALL, axiom, ![TV_u_27a]: ![V_P, V_E]: (p(s(t_bool,h4s_quotients_equiv(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_E)))) => s(t_bool,h4s_bools_resu_u_forall(s(t_fun(TV_u_27a,t_bool),h4s_quotients_respects(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_E))),s(t_fun(TV_u_27a,t_bool),V_P))) = s(t_bool,d_forall(s(t_fun(TV_u_27a,t_bool),V_P))))).
fof(ah4s_reals_REALu_u_LEu_u_RADD, axiom, ![V_z, V_y, V_x]: 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,h4s_realaxs_realu_u_add(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,V_y)))).
fof(ah4s_reals_ABSu_u_BOUND, axiom, ![V_y, V_x, 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,V_x),s(t_h4s_realaxs_real,V_y))))),s(t_h4s_realaxs_real,V_d)))) => p(s(t_bool,h4s_realaxs_realu_u_lt(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_d)))))))).
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_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_reals_abs0, axiom, ![V_x]: s(t_h4s_realaxs_real,h4s_reals_abs(s(t_h4s_realaxs_real,V_x))) = s(t_h4s_realaxs_real,h4s_bools_cond(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_realaxs_real,V_x))),s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_neg(s(t_h4s_realaxs_real,V_x)))))).
fof(ah4s_reals_REALu_u_LEu_u_NEGR, axiom, ![V_x]: s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_neg(s(t_h4s_realaxs_real,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_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))))).
fof(ah4s_reals_REALu_u_LEu_u_NEGL, axiom, ![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,V_x))) = s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_realaxs_real,V_x)))).
fof(ah4s_reals_REALu_u_LEu_u_ADD2, axiom, ![V_z, V_y, V_x, V_w]: ((p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,V_w),s(t_h4s_realaxs_real,V_x)))) & p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,V_y),s(t_h4s_realaxs_real,V_z))))) => p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,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_LTu_u_IMPu_u_LE, axiom, ![V_y, V_x]: (p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y)))) => p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y)))))).
fof(ah4s_reals_REALu_u_NOTu_u_LE, axiom, ![V_y, V_x]: (~ (p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y))))) <=> p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,V_y),s(t_h4s_realaxs_real,V_x)))))).
fof(ah4s_reals_REALu_u_NOTu_u_LT, axiom, ![V_y, V_x]: (~ (p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y))))) <=> p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,V_y),s(t_h4s_realaxs_real,V_x)))))).
fof(ah4s_reals_REALu_u_LEu_u_LADD, axiom, ![V_z, V_y, V_x]: 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_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))))) = s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,V_y),s(t_h4s_realaxs_real,V_z)))).
fof(ah4s_bools_IMPu_u_ANTISYMu_u_AX, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) => ((p(s(t_bool,V_t2)) => p(s(t_bool,V_t1))) => s(t_bool,V_t1) = s(t_bool,V_t2)))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f0)) => p(s(t_bool,V_t)))).
fof(ah4s_bools_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
fof(ah4s_reals_ABSu_u_BETWEEN, axiom, ![V_y, V_x, V_d]: ((p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_realaxs_real,V_d)))) & (p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,h4s_reals_realu_u_sub(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_d))),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,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,V_x),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,V_y),s(t_h4s_realaxs_real,V_x))))),s(t_h4s_realaxs_real,V_d)))))).
fof(ah4s_reals_REALu_u_SUBu_u_ABS, axiom, ![V_y, V_x]: p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,h4s_reals_realu_u_sub(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))))),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))))))))).
fof(ah4s_reals_REALu_u_MULu_u_SYM, axiom, ![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,V_y))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,V_y),s(t_h4s_realaxs_real,V_x)))).
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_SUBu_u_ADD, axiom, ![V_y, V_x]: 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_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_LEu_u_TRANS, axiom, ![V_z, V_y, V_x]: ((p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y)))) & p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,V_y),s(t_h4s_realaxs_real,V_z))))) => p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_z)))))).
fof(ah4s_seqs_SERu_u_CDIV, axiom, ![V_uu_0]: (![V_x, V_c, V_n]: s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_fun(t_h4s_nums_num,t_h4s_realaxs_real))),V_uu_0),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_x))),s(t_h4s_realaxs_real,V_c))),s(t_h4s_nums_num,V_n))) = s(t_h4s_realaxs_real,h4s_reals_u_2f(s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_x),s(t_h4s_nums_num,V_n))),s(t_h4s_realaxs_real,V_c))) => ![V_x0, V_x, V_c]: (p(s(t_bool,h4s_seqs_sums(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_x),s(t_h4s_realaxs_real,V_x0)))) => p(s(t_bool,h4s_seqs_sums(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_fun(t_h4s_nums_num,t_h4s_realaxs_real))),V_uu_0),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_x))),s(t_h4s_realaxs_real,V_c))),s(t_h4s_realaxs_real,h4s_reals_u_2f(s(t_h4s_realaxs_real,V_x0),s(t_h4s_realaxs_real,V_c))))))))).
fof(ah4s_seqs_SERu_u_CMUL, axiom, ![V_uu_0]: (![V_c, V_x, V_n]: s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_fun(t_h4s_nums_num,t_h4s_realaxs_real))),V_uu_0),s(t_h4s_realaxs_real,V_c))),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_x))),s(t_h4s_nums_num,V_n))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,V_c),s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_x),s(t_h4s_nums_num,V_n))))) => ![V_x0, V_x, V_c]: (p(s(t_bool,h4s_seqs_sums(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_x),s(t_h4s_realaxs_real,V_x0)))) => p(s(t_bool,h4s_seqs_sums(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_fun(t_h4s_nums_num,t_h4s_realaxs_real))),V_uu_0),s(t_h4s_realaxs_real,V_c))),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_x))),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,V_c),s(t_h4s_realaxs_real,V_x0))))))))).
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_EQu_u_NEG, axiom, ![V_y, V_x]: (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_h4s_realaxs_real,V_x) = s(t_h4s_realaxs_real,V_y))).
fof(ah4s_bools_CONJu_u_SYM, 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))))).
fof(ah4s_reals_REALu_u_LEu_u_NEG, 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_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_reals_POWu_u_MUL, axiom, ![V_y, V_x, V_n]: s(t_h4s_realaxs_real,h4s_reals_pow(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_nums_num,V_n))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,h4s_reals_pow(s(t_h4s_realaxs_real,V_x),s(t_h4s_nums_num,V_n))),s(t_h4s_realaxs_real,h4s_reals_pow(s(t_h4s_realaxs_real,V_y),s(t_h4s_nums_num,V_n)))))).
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_pow0u_c1, axiom, ![V_x, V_n]: s(t_h4s_realaxs_real,h4s_reals_pow(s(t_h4s_realaxs_real,V_x),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,h4s_reals_pow(s(t_h4s_realaxs_real,V_x),s(t_h4s_nums_num,V_n)))))).
fof(ch4s_lims_DIFFu_u_CONT, conjecture, ![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)))))).
