INTEGRAL_INTERVALS_DIFF_INCLUSION_EXCLUSION |- (!f. (!a. (!b. (!c. (!d. ((((integrable_on f) (closed_interval ((CONS (a , b)) NIL))) /\ ((c IN (closed_interval ((CONS (a , b)) NIL))) /\ (d IN (closed_interval ((CONS (a , b)) NIL))))) ==> (((vector_sub ((integral (closed_interval ((CONS (a , d)) NIL))) f)) ((integral (closed_interval ((CONS (a , c)) NIL))) f)) = ((vsum (GSPEC (\GEN%PVAR%2930. (?s. (((SETSPEC GEN%PVAR%2930) ((~ (s = EMPTY)) /\ (s SUBSET ((NUMERAL (BIT1 _0)) .. (dimindex UNIV))))) s))))) (\s. ((% ((real_pow (real_neg (real_of_num (NUMERAL (BIT1 _0))))) (CARD (GSPEC (\GEN%PVAR%2931. (?i. (((SETSPEC GEN%PVAR%2931) ((i IN s) /\ ((real_lt (d $ i)) (c $ i)))) i))))))) ((integral (closed_interval ((CONS ((lambdai. (((COND (i IN s)) ((real_min (c $ i)) (d $ i))) (a $ i))) , (lambdai. (((COND (i IN s)) ((real_max (c $ i)) (d $ i))) (c $ i))))) NIL))) f)))))))))))


Cuxvzoz.vv_split_azim_alt |- (!vv. (!k. (!i. (!j. ((LET (\V. (LET_END ((LET (\E. (LET_END ((LET (\f. (LET_END ((((periodic vv) k) /\ (((NUMERAL (BIT1 (BIT1 _0))) <= k) /\ ((~ (collinear ((vec (NUMERAL _0)) INSERT ((vv i) INSERT ((vv j) INSERT EMPTY))))) /\ ((~ (collinear ((vec (NUMERAL _0)) INSERT ((vv i) INSERT ((vv (i + (k - (NUMERAL (BIT1 _0))))) INSERT EMPTY))))) /\ ((~ (collinear ((vec (NUMERAL _0)) INSERT ((vv i) INSERT ((vv (SUC i)) INSERT EMPTY))))) /\ ((convex_local_fan (V , (E , f))) /\ (!i. (!j. (((i < k) /\ ((j < k) /\ ((vv i) = (vv j)))) ==> (i = j)))))))))) ==> (((((azim (vec (NUMERAL _0))) (vv i)) (vv (SUC i))) (vv (i + (k - (NUMERAL (BIT1 _0)))))) = ((real_add ((((azim (vec (NUMERAL _0))) (vv i)) (vv (SUC i))) (vv j))) ((((azim (vec (NUMERAL _0))) (vv i)) (vv j)) (vv (i + (k - (NUMERAL (BIT1 _0)))))))))))) ((IMAGE (\i. ((vv i) , (vv (SUC i))))) UNIV))))) ((IMAGE (\i. ((vv i) INSERT ((vv (SUC i)) INSERT EMPTY)))) UNIV))))) ((IMAGE vv) UNIV))))))


COMPLEX_IN_SPHERE_0 |- (!v. (!r. ((v IN (sphere ((Cx (real_of_num (NUMERAL _0))) , r))) = ((vector_norm v) = r))))


Tame_list.minimal_min_num |- (minimal = min_num)


Ssrnat.minnAC |- (!m1. (!m2. (!m3. (((minn ((minn m1) m2)) m3) = ((minn ((minn m1) m3)) m2)))))


Lp_ineqs_proofs.dart3_azim3_fan_eq_dih3_y |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!d. ((d IN ((darts_k (NUMERAL (BIT1 (BIT1 _0)))) (hypermap_of_fan (V , E)))) ==> (((azim3_fan (V , E)) d) = ((((((dih3_y (y1_fan d)) (y2_fan d)) ((y3_fan (V , E)) d)) ((y4_fan (V , E)) d)) ((y5_fan (V , E)) d)) (y6_fan d)))))))))))


LOCALLY_CONNECTED_SPHERE |- (!a. (!r. ((locally connected) (sphere (a , r)))))


Merge_ineq.nonf_mud_234_x_v1 |- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (!a. (!b. (!c. (((real_le (real_of_num (NUMERAL _0))) a) ==> ((((((((((mud_234_x_v1 a) b) c) x1) x2) x3) x4) x5) x6) = ((real_mul ((((((mu6_x ((real_mul a) a)) x2) x3) (real_of_num (NUMERAL _0))) (real_of_num (NUMERAL _0))) (real_of_num (NUMERAL _0)))) (sqrt (((((((((delta_234_x ((real_mul a) a)) ((real_mul b) b)) ((real_mul c) c)) x1) x2) x3) x4) x5) x6))))))))))))))


ANTIDERIVATIVE_INTEGRAL_CONTINUOUS |- (!f. (!a. (!b. (((continuous_on f) (closed_interval ((CONS (a , b)) NIL))) ==> (?g. (!u. (!v. (((u IN (closed_interval ((CONS (a , b)) NIL))) /\ ((v IN (closed_interval ((CONS (a , b)) NIL))) /\ ((real_le (drop u)) (drop v)))) ==> (((has_integral f) ((vector_sub (g v)) (g u))) (closed_interval ((CONS (u , v)) NIL)))))))))))


INT_LT_NEGTOTAL |- (!x. ((x = (int_of_num (NUMERAL _0))) \/ (((int_lt (int_of_num (NUMERAL _0))) x) \/ ((int_lt (int_of_num (NUMERAL _0))) (int_neg x)))))


TAN_PERIODIC_NPI |- (!x. (!n. ((tan ((real_add x) ((real_mul (real_of_num n)) pi))) = (tan x))))


CONVEX_ON_IS_INDEFINITE_INTEGRAL |- (!f. (!s. ((((convex_on (drop o f)) s) /\ ((is_interval s) /\ (open s))) ==> (?f'. ((!x. (!y. (((x IN s) /\ ((y IN s) /\ ((real_le (drop x)) (drop y)))) ==> ((real_le (drop (f' x))) (drop (f' y)))))) /\ (!x. (!y. (((x IN s) /\ ((y IN s) /\ ((real_le (drop x)) (drop y)))) ==> (((has_integral f') ((vector_sub (f y)) (f x))) (closed_interval ((CONS (x , y)) NIL)))))))))))


SERIES_DIRICHLET_COMPLEX_EXPLICIT |- (!f. (!g. (!p. (!q. ((((NUMERAL (BIT1 _0)) <= p) /\ ((bounded (GSPEC (\GEN%PVAR%4783. (?n. (((SETSPEC GEN%PVAR%4783) (n IN UNIV)) ((vsum (q .. n)) f)))))) /\ ((!n. ((p <= n) ==> ((real (g n)) /\ ((real_le (real_of_num (NUMERAL _0))) (Re (g n)))))) /\ (!n. ((p <= n) ==> ((real_le (Re (g (n + (NUMERAL (BIT1 _0)))))) (Re (g n)))))))) ==> (?B. (((real_lt (real_of_num (NUMERAL _0))) B) /\ (!m. (!n. ((p <= m) ==> ((real_le (vector_norm ((vsum (m .. n)) (\k. ((complex_mul (f k)) (g k)))))) ((real_mul B) (vector_norm (g m))))))))))))))


REAL_UNIFORMLY_CONTINUOUS_ON_ID |- (!s. ((real_uniformly_continuous_on (\x. x)) s))


AFF_LE_INJECTIVE_LINEAR_IMAGE |- (!f. (!s. (!t. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> (((aff_le ((IMAGE f) s)) ((IMAGE f) t)) = ((IMAGE f) ((aff_le s) t)))))))


Planarity.remark012_fan |- (!x. (!V. (!E. (!u. (!w. (!a. (((FAN (x , (V , E))) /\ (((u INSERT (w INSERT EMPTY)) IN E) /\ (((real_lt (real_of_num (NUMERAL _0))) a) /\ ((real_lt a) (real_of_num (NUMERAL (BIT1 _0))))))) ==> (~ (((vector_add ((% ((real_sub (real_of_num (NUMERAL (BIT1 _0)))) a)) u)) ((% a) w)) IN V)))))))))


Ssrfun.inj_id |- (injective I)


Tame_list.indexf_mem1 |- (!x. (!y. (!s. ((~ ((MEM x) s)) ==> ((((indexf x) y) s) = (sizel s))))))


FIXED_POINT_INESSENTIAL_SPHERE_MAP |- (!f. (!a. (!r. (!c. ((((real_lt (real_of_num (NUMERAL _0))) r) /\ ((((homotopic_with (\x. T)) ((sphere (a , r)) , (sphere (a , r)))) f) (\x. c))) ==> (?x. ((x IN (sphere (a , r))) /\ ((f x) = x))))))))


Lp_ineqs_proofs.y1_def |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!d. ((d IN (dart_of_fan (V , E))) ==> ((y1_fan d) = (yn_fan (FST d)))))))))))


POLYTOPE_LINEAR_IMAGE_EQ |- (!f. (!s. (((linear f) /\ ((!x. (!y. (((f x) = (f y)) ==> (x = y)))) /\ (!y. (?x. ((f x) = y))))) ==> ((polytope ((IMAGE f) s)) = (polytope s)))))


REAL_OPEN_DIFF |- (!s. (!t. (((real_open s) /\ (real_closed t)) ==> (real_open (s DIFF t)))))


Ssrnat.geqE |- (!m. (!n. ((m >= n) = (n <= m))))


TC_RTC_CASES_L |- ((((TC R) x) z) = (?y. ((((RTC R) x) y) /\ ((R y) z))))


REAL_MEASURABLE_EMPTY |- (real_measurable EMPTY)


HAS_BOUNDED_REAL_VARIATION_ON_COMBINE |- (!f. (!a. (!b. (!c. ((((real_le a) c) /\ ((real_le c) b)) ==> (((has_bounded_real_variation_on f) (closed_real_interval ((CONS (a , b)) NIL))) = (((has_bounded_real_variation_on f) (closed_real_interval ((CONS (a , c)) NIL))) /\ ((has_bounded_real_variation_on f) (closed_real_interval ((CONS (c , b)) NIL))))))))))


POSITIVE_SEMIDEFINITE_ZERO_COLUMN |- (!A. (!i. (((positive_semidefinite A) /\ (((NUMERAL (BIT1 _0)) <= i) /\ ((i <= (dimindex UNIV)) /\ (((A $ i) $ i) = (real_of_num (NUMERAL _0)))))) ==> (((column i) A) = (vec (NUMERAL _0))))))


UNCOUNTABLE_HAS_CONDENSATION_POINT |- (!s. ((~ (COUNTABLE s)) ==> (?x. ((condensation_point_of x) s))))


RANK_NEG |- (!A. ((rank (matrix_neg A)) = (rank A)))


LOCALLY_COMPACT_CLOSED_INTER_OPEN |- (!s. (((locally compact) s) = (?t. (?u. ((closed t) /\ ((open u) /\ (s = (t INTER u))))))))


CLOSED_MAP_FROM_COMPOSITION_SURJECTIVE |- (!f. (!g. (!s. (!t. (!u. ((((continuous_on f) s) /\ ((((IMAGE f) s) = t) /\ ((((IMAGE g) t) SUBSET u) /\ (!k. (((closed_in ((subtopology euclidean) s)) k) ==> ((closed_in ((subtopology euclidean) u)) ((IMAGE (g o f)) k))))))) ==> (!k. (((closed_in ((subtopology euclidean) t)) k) ==> ((closed_in ((subtopology euclidean) u)) ((IMAGE g) k))))))))))


HAS_REAL_DERIVATIVE_SERIES |- (!s. (!f. (!f'. (!g'. (!k. (((is_realinterval s) /\ ((!n. (!x. ((x IN s) ==> (((has_real_derivative (f n)) ((f' n) x)) ((within (atreal x)) s))))) /\ ((!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?N. (!n. (!x. (((n >= N) /\ (x IN s)) ==> ((real_le (real_abs ((real_sub ((sum (k INTER ((NUMERAL _0) .. n))) (\i. ((f' i) x)))) (g' x)))) e))))))) /\ (?x. (?l. ((x IN s) /\ (((real_sums (\n. ((f n) x))) l) k))))))) ==> (?g. (!x. ((x IN s) ==> ((((real_sums (\n. ((f n) x))) (g x)) k) /\ (((has_real_derivative g) (g' x)) ((within (atreal x)) s))))))))))))


CONIC_HULL_POINTLESS_AS_IMAGE |- (!s. ((((hull conic) s) DELETE (vec (NUMERAL _0))) = ((IMAGE (\z. ((% (drop (fstcart z))) (sndcart z)))) ((GSPEC (\GEN%PVAR%1459. (?t. (((SETSPEC GEN%PVAR%1459) ((real_lt (real_of_num (NUMERAL _0))) (drop t))) t)))) PCROSS (s DELETE (vec (NUMERAL _0)))))))


HAS_INTEGRAL_COMPLEX_RMUL |- (!f. (!y. (!i. (!c. ((((has_integral f) y) i) ==> (((has_integral (\x. ((complex_mul (f x)) c))) ((complex_mul y) c)) i))))))


SUMMABLE_NEG |- (!x. (!s. (((summable s) x) ==> ((summable s) (\n. (vector_neg (x n)))))))


CENTRE_IN_MCBALL |- (!m. (!x. (!r. ((((real_le (real_of_num (NUMERAL _0))) r) /\ (x IN (mspace m))) ==> (x IN ((mcball m) (x , r)))))))


Nonlinear_lemma.ineq_expand10 |- (!a1. (!a2. (!a3. (!a4. (!a5. (!a6. (!a7. (!a8. (!a9. (!a10. (!b1. (!b2. (!b3. (!b4. (!b5. (!b6. (!b7. (!b8. (!b9. (!b10. (!y1. (!y2. (!y3. (!y4. (!y5. (!y6. (!y7. (!y8. (!y9. (!y10. (!P. (((ineq ((CONS (a1 , (y1 , b1))) ((CONS (a2 , (y2 , b2))) ((CONS (a3 , (y3 , b3))) ((CONS (a4 , (y4 , b4))) ((CONS (a5 , (y5 , b5))) ((CONS (a6 , (y6 , b6))) ((CONS (a7 , (y7 , b7))) ((CONS (a8 , (y8 , b8))) ((CONS (a9 , (y9 , b9))) ((CONS (a10 , (y10 , b10))) NIL))))))))))) P) = ((((real_le a1) y1) /\ ((real_le y1) b1)) ==> ((((real_le a2) y2) /\ ((real_le y2) b2)) ==> ((((real_le a3) y3) /\ ((real_le y3) b3)) ==> ((((real_le a4) y4) /\ ((real_le y4) b4)) ==> ((((real_le a5) y5) /\ ((real_le y5) b5)) ==> ((((real_le a6) y6) /\ ((real_le y6) b6)) ==> ((((real_le a7) y7) /\ ((real_le y7) b7)) ==> ((((real_le a8) y8) /\ ((real_le y8) b8)) ==> ((((real_le a9) y9) /\ ((real_le y9) b9)) ==> ((((real_le a10) y10) /\ ((real_le y10) b10)) ==> P))))))))))))))))))))))))))))))))))))))))))


REALLIM_AT_INFINITY_COMPLEX_0 |- (!f. (!l. ((((---> f) l) at_infinity) = (((---> (f o complex_inv)) l) (at (Cx (real_of_num (NUMERAL _0))))))))


LIM_INFINITY_POSINFINITY_LIFT |- (!f. (!l. ((((--> f) l) at_infinity) ==> (((--> (f o lift)) l) at_posinfinity))))


STONE_WEIERSTRASS_VECTOR_POLYNOMIAL_FUNCTION_AFFINE |- (!f. (!s. (!e. (!t. (((compact s) /\ (((continuous_on f) s) /\ (((real_lt (real_of_num (NUMERAL _0))) e) /\ ((affine t) /\ (((IMAGE f) s) SUBSET t))))) ==> (?g. ((vector_polynomial_function g) /\ ((((IMAGE g) s) SUBSET t) /\ (!x. ((x IN s) ==> ((real_lt (vector_norm ((vector_sub (f x)) (g x)))) e)))))))))))


LEFT_INVERSE_LINEAR |- (!f. (!g. (((linear f) /\ ((g o f) = I)) ==> (linear g))))


Add_triangle.contracted_in_dart |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!d. ((d IN (d_fan ((vec (NUMERAL _0)) , (V , E)))) ==> ((contracted_dart d) IN (dart_of_fan (V , E))))))))


Lp_ineqs_proofs.tau_hi |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> ((lp_tau (V , E)) ==> ((contravening V) ==> (!f. ((f IN (face_set (hypermap_of_fan (V , E)))) ==> ((real_le ((tau_fan (V , E)) f)) tgt)))))))))))


Lp_ineqs_proofs.face_last_dart |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!v. (!w. (((v , w) IN (dart1_of_fan (V , E))) ==> ((((POWER (f_fan_pair_ext (V , E))) ((CARD ((face (hypermap_of_fan (V , E))) (v , w))) - (NUMERAL (BIT1 _0)))) (v , w)) = ((((((sigma_fan (vec (NUMERAL _0))) V) E) v) w) , v))))))))


BINOMIAL_THEOREM |- (!n. (!x. (!y. (((x + y) EXP n) = ((nsum ((NUMERAL _0) .. n)) (\k. ((binom (n , k)) * ((x EXP k) * (y EXP (n - k))))))))))


TRANSLATION_EQ_IMP |- (!P. ((!a. (!s. ((P ((IMAGE (\x. ((vector_add a) x))) s)) = (P s)))) = (!a. (!s. ((P s) ==> (P ((IMAGE (\x. ((vector_add a) x))) s)))))))


Fan.azim_is_zero_fan |- (!x. (!v. (!u. (((~ (v = x)) /\ ((~ (u = x)) /\ (~ (collinear (x INSERT (v INSERT (u INSERT EMPTY))))))) ==> (((((azim x) v) u) u) = (real_of_num (NUMERAL _0)))))))


RPOW_MINUS1_QUOTIENT_LE |- (!a. (!x. (!y. ((((real_lt (real_of_num (NUMERAL _0))) a) /\ (((real_lt (real_of_num (NUMERAL _0))) x) /\ ((real_le x) y))) ==> ((real_le ((real_div ((real_sub ((rpow a) x)) (real_of_num (NUMERAL (BIT1 _0))))) x)) ((real_div ((real_sub ((rpow a) y)) (real_of_num (NUMERAL (BIT1 _0))))) y))))))


RC_IDEMP |- (!R. ((RC (RC R)) = (RC R)))


Miqmcsn.J_SCS_3T4_prime2 |- ((((scs_J_v39 ((scs_prop_equ_v39 scs_3T4_prime2) i)) i1) j) = F)


LIM_WITHINREAL_LE |- (!f. (!l. (!a. (!s. ((((--> f) l) ((within (atreal a)) s)) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?d. (((real_lt (real_of_num (NUMERAL _0))) d) /\ (!x. (((x IN s) /\ (((real_lt (real_of_num (NUMERAL _0))) (real_abs ((real_sub x) a))) /\ ((real_le (real_abs ((real_sub x) a))) d))) ==> ((real_lt (distance ((f x) , l))) e))))))))))))


HAS_VECTOR_DERIVATIVE_NEG_EQ |- (!f. (!f'. (!net. ((((has_vector_derivative (\x. (vector_neg (f x)))) (vector_neg f')) net) = (((has_vector_derivative f) f') net)))))


RPOW_1_LE |- (!a. (!x. ((((real_le (real_of_num (NUMERAL _0))) x) /\ (((real_le x) (real_of_num (NUMERAL (BIT1 _0)))) /\ ((real_le (real_of_num (NUMERAL _0))) a))) ==> ((real_le ((rpow x) a)) (real_of_num (NUMERAL (BIT1 _0)))))))


CONNECTED_SUMS |- (!s. (!t. (((connected s) /\ (connected t)) ==> (connected (GSPEC (\GEN%PVAR%917. (?x. (?y. (((SETSPEC GEN%PVAR%917) ((x IN s) /\ (y IN t))) ((vector_add x) y))))))))))


REAL_MUL_COS_SIN |- (!x. (!y. (((real_mul (cos x)) (sin y)) = ((real_div ((real_sub (sin ((real_add x) y))) (sin ((real_sub x) y)))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))))


Ssrbool.implyNb |- (!a. (!b. (((~ a) ==> b) = (a \/ b))))


Merge_ineq.nonf_mud_135_x_v1 |- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (!a. (!b. (!c. (((real_le (real_of_num (NUMERAL _0))) a) ==> ((((((((((mud_135_x_v1 a) b) c) x1) x2) x3) x4) x5) x6) = ((real_mul ((((((mu6_x ((real_mul a) a)) x1) x3) (real_of_num (NUMERAL _0))) (real_of_num (NUMERAL _0))) (real_of_num (NUMERAL _0)))) (sqrt (((((((((delta_135_x ((real_mul a) a)) ((real_mul b) b)) ((real_mul c) c)) x1) x2) x3) x4) x5) x6))))))))))))))


LEBESGUE_MEASURABLE_PREIMAGE_OPEN |- (!f. (!t. ((((measurable_on f) UNIV) /\ (open t)) ==> (lebesgue_measurable (GSPEC (\GEN%PVAR%3550. (?x. (((SETSPEC GEN%PVAR%3550) ((f x) IN t)) x))))))))


Nonlinear_lemma.edge_quadratic |- (!x1. (!x2. (!x3. (!x5. (!x6. ((quadratic_root_plus (x1 , (((real_add ((real_mul x1) x1)) ((real_sub ((real_mul ((real_sub x3) x5)) ((real_sub x2) x6))) ((real_mul x1) ((real_add x2) ((real_add x3) ((real_add x5) x6)))))) , ((real_add ((real_mul x1) ((real_mul x3) x5))) ((real_sub ((real_sub ((real_mul x1) ((real_mul x2) x6))) ((real_mul x3) ((real_mul ((real_add x1) ((real_add ((real_sub x2) x3)) ((real_sub x5) x6)))) x6)))) ((real_mul x2) ((real_mul x5) ((real_add ((real_sub x1) x2)) ((real_add ((real_sub x3) x5)) x6))))))))) = ((real_div ((real_add (real_neg ((real_mul x1) x1))) ((real_add ((real_mul x1) x2)) ((real_add ((real_sub ((real_mul x1) x3)) ((real_mul x2) x3))) ((real_add ((real_mul x1) x5)) ((real_add ((real_mul x2) x5)) ((real_add ((real_mul x1) x6)) ((real_add ((real_sub ((real_mul x3) x6)) ((real_mul x5) x6))) (sqrt ((real_mul (((ups_x x1) x3) x5)) (((ups_x x1) x2) x6))))))))))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) x1))))))))


Seq.leq_size_perm |- (!s1. (!s2. ((uniq s1) ==> ((!x. (((MEM x) s1) ==> ((MEM x) s2))) ==> (((sizel s2) <= (sizel s1)) ==> ((!x. (((MEM x) s1) = ((MEM x) s2))) /\ ((sizel s1) = (sizel s2))))))))


LIM_CONG_ATREAL |- ((!x. ((~ (x = a)) ==> ((f x) = (g x)))) ==> ((((--> (\x. (f x))) l) (atreal a)) = (((--> g) l) (atreal a))))


HOMEOMORPHIC_SCALING_RIGHT |- (!c. (((real_lt (real_of_num (NUMERAL _0))) c) ==> (!s. (!t. (((homeomorphic s) ((IMAGE (\x. ((% c) x))) t)) = ((homeomorphic s) t))))))


REAL_INTEGRAL_REFLECT |- (!f. (!a. (!b. (((real_integral (closed_real_interval ((CONS ((real_neg b) , (real_neg a))) NIL))) (\x. (f (real_neg x)))) = ((real_integral (closed_real_interval ((CONS (a , b)) NIL))) f)))))


Ssrnat.ltn_pexp2l |- (!m. (!n1. (!n2. (((NUMERAL _0) < m) ==> (((m EXP n1) < (m EXP n2)) ==> (n1 < n2))))))


STC_INC_TC |- (!R. (!x. (!y. ((((TC R) x) y) ==> (((STC R) x) y)))))


OPEN_MAP_IFF_CLOSED_MAP_BIJECTIVE |- (!f. (!s. (!t. (((((IMAGE f) s) = t) /\ (!x. (!y. (((x IN s) /\ ((y IN s) /\ ((f x) = (f y)))) ==> (x = y))))) ==> ((!u. (((open_in ((subtopology euclidean) s)) u) ==> ((open_in ((subtopology euclidean) t)) ((IMAGE f) u)))) = (!c. (((closed_in ((subtopology euclidean) s)) c) ==> ((closed_in ((subtopology euclidean) t)) ((IMAGE f) c)))))))))


Nonlinear_lemma.vol2r_y |- (!y. (!r. (((vol2r y) r) = ((real_sub ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((real_mul pi) ((real_mul r) ((real_mul r) (real_inv (real_of_num (NUMERAL (BIT1 (BIT1 _0)))))))))) ((real_mul ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) ((real_mul pi) ((real_mul (real_inv (real_of_num (NUMERAL (BIT1 (BIT1 _0)))))) ((real_pow y) (NUMERAL (BIT0 (BIT1 _0)))))))))))


LOCALLY_CONNECTED_SPHERE_GEN |- (!s. (((bounded s) /\ (convex s)) ==> ((locally connected) (relative_frontier s))))


Wrgcvdr_cizmrrh.hypermap |- (!H. (((tuple_hypermap H) = (D , (e , (n , f)))) ==> (((permutes e) D) /\ (((permutes n) D) /\ (((permutes f) D) /\ ((e o (n o f)) = I))))))


SUPERADMISSIBLE_MATCH_SEQPATTERN |- (!<<. (!p. (!s. (!c1. (!c2. (!e. ((((((admissible <<) p) s) (\f. (\x. (?y. ((((c1 f) x) ((e f) x)) y))))) /\ (((((superadmissible <<) (\f. (\x. (((p f) x) /\ (?y. ((((c1 f) x) ((e f) x)) y)))))) s) (\f. (\x. ((_MATCH ((e f) x)) ((c1 f) x))))) /\ ((((superadmissible <<) (\f. (\x. (((p f) x) /\ (~ (?y. ((((c1 f) x) ((e f) x)) y))))))) s) (\f. (\x. ((_MATCH ((e f) x)) ((c2 f) x))))))) ==> ((((superadmissible <<) p) s) (\f. (\x. ((_MATCH ((e f) x)) ((_SEQPATTERN ((c1 f) x)) ((c2 f) x)))))))))))))


SEPARATION_NORMAL_COMPACT |- (!s. (!t. (((compact s) /\ ((closed t) /\ ((s INTER t) = EMPTY))) ==> (?u. (?v. ((open u) /\ ((compact (closure u)) /\ ((open v) /\ ((s SUBSET u) /\ ((t SUBSET v) /\ ((u INTER v) = EMPTY)))))))))))


Upfzbzm_support_lemmas.john_harrison_lemma1 |- ((GABS (\f. (!x. (!y. ((GEQ (f (x , y))) ((P x) y)))))) = (\p. ((P (FST p)) (SND p))))


INT_LT_ADDL |- (!x. (!y. (((int_lt y) ((int_add x) y)) = ((int_lt (int_of_num (NUMERAL _0))) x))))


ORTHOGONAL_BASIS_EXISTS |- (!v. (?b. ((independent b) /\ ((b SUBSET (span v)) /\ ((v SUBSET (span b)) /\ ((b HAS_SIZE (dim v)) /\ ((pairwise orthogonal) b)))))))


FSIGMA_IMP_LEBESGUE_MEASURABLE |- (!s. ((fsigma s) ==> (lebesgue_measurable s)))


NPRODUCT_DELTA |- (!s. (!a. (((nproduct s) (\x. (((COND (x = a)) b) (NUMERAL (BIT1 _0))))) = (((COND (a IN s)) b) (NUMERAL (BIT1 _0))))))


CLOSED_MAP_CLOSURES |- (!f. ((!s. ((closed s) ==> (closed ((IMAGE f) s)))) = (!s. ((closure ((IMAGE f) s)) SUBSET ((IMAGE f) (closure s))))))


Ssrnat.sqrn_add_sub |- (!m. (!n. ((n <= m) ==> ((((m + n) EXP (NUMERAL (BIT0 (BIT1 _0)))) - ((NUMERAL (BIT0 (BIT0 (BIT1 _0)))) * (m * n))) = ((m - n) EXP (NUMERAL (BIT0 (BIT1 _0))))))))


COMPLEX_IN_BALL_0 |- (!v. (!r. ((v IN (ball ((Cx (real_of_num (NUMERAL _0))) , r))) = ((real_lt (vector_norm v)) r))))


NORM_CCOS_POW_2 |- (!z. (((real_pow (vector_norm (ccos z))) (NUMERAL (BIT0 (BIT1 _0)))) = ((real_add ((real_pow (cos (Re z))) (NUMERAL (BIT0 (BIT1 _0))))) ((real_div ((real_pow ((real_sub (exp (Im z))) (real_inv (exp (Im z))))) (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))))))


Seq.unzip2_zip |- (!s. (!t. (((sizel t) <= (sizel s)) ==> ((unzip2 ((zip s) t)) = t))))


EXPOSED_POINT_OF_FURTHEST_POINT |- (!s. (!a. (!b. (((b IN s) /\ (!x. ((x IN s) ==> ((real_le (distance (a , x))) (distance (a , b)))))) ==> ((exposed_face_of (b INSERT EMPTY)) s)))))


CONVEX_NORM |- (!s. ((convex_on vector_norm) s))


Tame_defs2.good_list_nodes |- (!L. ((good_list_nodes L) = ((node_set (hypermap_of_list L)) = (set_of_list (nodes_of_list L)))))


DEPENDENT_LINEAR_IMAGE |- (!f. (!s. (((linear f) /\ ((!x. (!y. (((x IN s) /\ ((y IN s) /\ ((f x) = (f y)))) ==> (x = y)))) /\ (dependent s))) ==> (dependent ((IMAGE f) s)))))


SUBSPACE_REAL |- (subspace real)


ANR_INSERT |- (!s. (!a. (((closed s) /\ (ANR s)) ==> (ANR (a INSERT s)))))


HAS_VECTOR_DERIVATIVE_ID |- (!net. (((has_vector_derivative (\x. x)) (vec (NUMERAL (BIT1 _0)))) net))


REAL_INTEGRABLE_REAL_BOUNDED_VARIATION_PRODUCT |- (!f. (!g. (!a. (!b. ((((real_integrable_on f) (closed_real_interval ((CONS (a , b)) NIL))) /\ ((has_bounded_real_variation_on g) (closed_real_interval ((CONS (a , b)) NIL)))) ==> ((real_integrable_on (\x. ((real_mul (g x)) (f x)))) (closed_real_interval ((CONS (a , b)) NIL))))))))


HAS_REAL_INTEGRAL_NEGLIGIBLE_EQ |- (!f. (!s. ((!x. ((x IN s) ==> ((real_le (real_of_num (NUMERAL _0))) (f x)))) ==> ((((has_real_integral f) (real_of_num (NUMERAL _0))) s) = (real_negligible (GSPEC (\GEN%PVAR%4964. (?x. (((SETSPEC GEN%PVAR%4964) ((x IN s) /\ (~ ((f x) = (real_of_num (NUMERAL _0)))))) x)))))))))


CARD_EXP_C |- (!s. (!t. (((FINITE s) /\ (FINITE t)) ==> ((CARD ((^_c s) t)) = ((CARD s) EXP (CARD t))))))


Nonlinear_lemma.gcy_high_hplus |- (!y. (((real_le ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) hplus)) y) ==> ((gcy y) = ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) ((real_div mm1) pi)))))


NPRODUCT_FACT |- (!n. (((nproduct ((NUMERAL (BIT1 _0)) .. n)) (\m. m)) = (FACT n)))


INT_NEG_LT0 |- (!x. (((int_lt (int_neg x)) (int_of_num (NUMERAL _0))) = ((int_lt (int_of_num (NUMERAL _0))) x)))


CONVEX_CONE_PCROSS_EQ |- (!s. (!t. ((convex_cone (s PCROSS t)) = ((convex_cone s) /\ (convex_cone t)))))


REAL_UNIFORMLY_CONTINUOUS_ON_COMPOSE |- (!f. (!g. (!s. ((((real_uniformly_continuous_on f) s) /\ ((real_uniformly_continuous_on g) ((IMAGE f) s))) ==> ((real_uniformly_continuous_on (g o f)) s)))))


TAGGED_DIVISION_OF_NONTRIVIAL |- (!s. (!a. (!b. ((((tagged_division_of s) (closed_interval ((CONS (a , b)) NIL))) /\ (~ ((content (closed_interval ((CONS (a , b)) NIL))) = (real_of_num (NUMERAL _0))))) ==> ((tagged_division_of (GSPEC (\GEN%PVAR%2765. (?x. (?k. (((SETSPEC GEN%PVAR%2765) (((x , k) IN s) /\ (~ ((content k) = (real_of_num (NUMERAL _0)))))) (x , k))))))) (closed_interval ((CONS (a , b)) NIL)))))))


Lp_gen_theory.iso_face_trans |- (!H. (!G. (!g. (((hyp_iso g) (G , H)) ==> (!P. ((!f. ((f IN (face_set H)) ==> (P f))) ==> (!f. ((f IN (face_set G)) ==> (P ((IMAGE g) f))))))))))


Seq.mem_seq4 |- (!x. (!y1. (!y2. (!y3. (!y4. (((MEM x) ((CONS y1) ((CONS y2) ((CONS y3) ((CONS y4) NIL))))) = ((x = y1) \/ ((x = y2) \/ ((x = y3) \/ (x = y4))))))))))


REAL_OPEN_REAL_CLOSED |- (!s. ((real_open s) = (real_closed (UNIV DIFF s))))


Ssrbool.implybE |- (!a. (!b. ((a ==> b) = ((~ a) \/ b))))


IMAGE_MATRIX_INV |- (!A. (((IMAGE (\x. ((matrix_vector_mul (matrix_inv A)) x))) UNIV) = ((IMAGE (\x. ((matrix_vector_mul (transp A)) x))) UNIV)))


ANR_DELETE |- (!s. (!a. ((ANR s) ==> (ANR (s DELETE a)))))


RECTIFIABLE_PATH_DIFFERENTIABLE |- (!g. (!s. (((COUNTABLE s) /\ ((path g) /\ (!t. ((t IN ((closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL)) DIFF s)) ==> ((differentiable g) (at t)))))) ==> ((rectifiable_path g) = ((absolutely_integrable_on (\t. ((vector_derivative g) (at t)))) (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL)))))))


COVERING_SPACE_COUNTABLE_SHEETS |- (!p. (!c. (!s. (!b. (((covering_space (c , p)) s) ==> (COUNTABLE (GSPEC (\GEN%PVAR%2073. (?x. (((SETSPEC GEN%PVAR%2073) ((x IN c) /\ ((p x) = b))) x))))))))))


COUNTABLE_DIFF_FINITE |- (!s. (!t. ((FINITE s) ==> ((COUNTABLE (t DIFF s)) = (COUNTABLE t)))))


Hypermap_and_fan.F_HAS_NO_FIXED_POINTS_IN_D1 |- (!V. (!E. (!x. (((FAN ((vec (NUMERAL _0)) , (V , E))) /\ (x IN (dart1_of_fan (V , E)))) ==> (~ (((f_fan_pair (V , E)) x) = x))))))


NOT_EMPTY_INSERT |- (!x. (!s. (~ (EMPTY = (x INSERT s)))))


Lp_ineqs_proofs.azim3_hi |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!d. ((d IN (dart_of_fan (V , E))) ==> ((real_le ((azim3_fan (V , E)) d)) pi)))))))))


Functional_equation.functional_ups234_x |- (ups234_x = (((((((compose6 ups_126) proj_x2) proj_x3) dummy6) dummy6) dummy6) proj_x4))


Add_triangle.fan_hypermap_map2 |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!d. ((d IN (d1_fan ((vec (NUMERAL _0)) , (V , E)))) ==> ((((face_map (hypermap1_of_fanx ((vec (NUMERAL _0)) , (V , E)))) d) = ((((f1_fan (vec (NUMERAL _0))) V) E) d)) /\ ((((node_map (hypermap1_of_fanx ((vec (NUMERAL _0)) , (V , E)))) d) = ((((n_fan (vec (NUMERAL _0))) V) E) d)) /\ (((edge_map (hypermap1_of_fanx ((vec (NUMERAL _0)) , (V , E)))) d) = ((((e_fan (vec (NUMERAL _0))) V) E) d)))))))))


Float.INT_POS_NEG |- (!n. (((int_lt (int_of_num (NUMERAL _0))) (int_neg (int_of_num n))) = F))


Ssrnat.distn0 |- (!n. (((distn n) (NUMERAL _0)) = n))


VOLUME_FRUSTT |- (!v0. (!v1. (!h. (!a. (((real_lt (real_of_num (NUMERAL _0))) a) ==> ((measurable ((((frustt v0) v1) h) a)) /\ ((measure ((((frustt v0) v1) h) a)) = (((COND ((v1 = v0) \/ (((real_le (real_of_num (NUMERAL (BIT1 _0)))) a) \/ ((real_lt h) (real_of_num (NUMERAL _0)))))) (real_of_num (NUMERAL _0))) ((real_mul pi) ((real_mul ((real_sub ((real_pow ((real_div h) a)) (NUMERAL (BIT0 (BIT1 _0))))) ((real_pow h) (NUMERAL (BIT0 (BIT1 _0)))))) ((real_div h) (real_of_num (NUMERAL (BIT1 (BIT1 _0)))))))))))))))


LIMPT_OF_OPEN |- (!s. (!x. (((open s) /\ (x IN s)) ==> ((limit_point_of x) s))))


COMPLEX_DIFFERENTIABLE_CARATHEODORY_WITHIN |- (!f. (!z. (!s. (((complex_differentiable f) ((within (at z)) s)) = (?g. ((!w. (((vector_sub (f w)) (f z)) = ((complex_mul (g w)) ((vector_sub w) z)))) /\ ((continuous g) ((within (at z)) s))))))))


Pack2.POLYTOPE_VORONOI_CLOSED |- (!s. (!v. (((packing s) /\ ((saturated s) /\ (v IN s))) ==> (polytope ((voronoi_closed s) v)))))


PROPER_MAP_FROM_COMPOSITION_LEFT |- (!f. (!g. (!s. (!t. (!u. ((((continuous_on f) s) /\ ((((IMAGE f) s) = t) /\ (((continuous_on g) t) /\ ((((IMAGE g) t) SUBSET u) /\ (!k. (((k SUBSET u) /\ (compact k)) ==> (compact (GSPEC (\GEN%PVAR%738. (?x. (((SETSPEC GEN%PVAR%738) ((x IN s) /\ (((g o f) x) IN k))) x))))))))))) ==> (!k. (((k SUBSET u) /\ (compact k)) ==> (compact (GSPEC (\GEN%PVAR%739. (?x. (((SETSPEC GEN%PVAR%739) ((x IN t) /\ ((g x) IN k))) x)))))))))))))


UNIFORMLY_CONTINUOUS_ON_COMPLEX_MUL |- (!f. (!g. (!s. ((((uniformly_continuous_on f) s) /\ (((uniformly_continuous_on g) s) /\ ((bounded ((IMAGE f) s)) /\ (bounded ((IMAGE g) s))))) ==> ((uniformly_continuous_on (\x. ((complex_mul (f x)) (g x)))) s)))))


NSUM_EQ_GENERAL_INVERSES |- (!s. (!t. (!f. (!g. (!h. (!k. (((!y. ((y IN t) ==> (((k y) IN s) /\ ((h (k y)) = y)))) /\ (!x. ((x IN s) ==> (((h x) IN t) /\ (((k (h x)) = x) /\ ((g (h x)) = (f x))))))) ==> (((nsum s) f) = ((nsum t) g)))))))))


HAS_BOUNDED_REAL_VARIATION_ON_SUB |- (!f. (!g. (!s. ((((has_bounded_real_variation_on f) s) /\ ((has_bounded_real_variation_on g) s)) ==> ((has_bounded_real_variation_on (\x. ((real_sub (f x)) (g x)))) s)))))


RECTIFIABLE_PATH_REPARAMETRIZATION |- (!g. (!h. (!h'. (((rectifiable_path g) /\ ((homeomorphism ((closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL)) , (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL)))) (h , h'))) ==> (rectifiable_path (g o h))))))


Tame_list.hideDups_eq_nil |- (!s. (((hideDups s) = NIL) = (s = NIL)))


Nonlinear_lemma.sqrt_x2 |- (((((((sqrt_x2 x1) x2) x3) x4) x5) x6) = (sqrt x2))


HAS_REAL_DERIVATIVE_DERIVATIVE |- (!f. (!f'. (!x. ((((has_real_derivative f) f') (atreal x)) ==> (((real_derivative f) x) = f')))))


CONVEX_CONE_HULL_CONVEX_HULL |- (!s. (((hull convex_cone) s) = ((vec (NUMERAL _0)) INSERT (GSPEC (\GEN%PVAR%1701. (?c. (?x. (((SETSPEC GEN%PVAR%1701) (((real_le (real_of_num (NUMERAL _0))) c) /\ (x IN ((hull convex) s)))) ((% c) x)))))))))


Pqcsxwg.REAL_CONTINUOUS_WITHIN_SQRT_COMPOSE |- (!f. (!s. (!a. ((((real_continuous f) ((within (at a)) s)) /\ (((real_lt (real_of_num (NUMERAL _0))) (f a)) \/ (!x. ((x IN s) ==> ((real_le (real_of_num (NUMERAL _0))) (f x)))))) ==> ((real_continuous (\x. (sqrt (f x)))) ((within (at a)) s))))))


Lp_ineqs_proofs.rhazim_lo |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!d. ((d IN (dart_of_fan (V , E))) ==> ((real_le (real_of_num (NUMERAL _0))) ((rhazim_fan (V , E)) d))))))))))


REAL_INFSUM_RESTRICT |- (!k. (!a. (((real_infsum UNIV) (\n. (((COND (n IN k)) (a n)) (real_of_num (NUMERAL _0))))) = ((real_infsum k) a))))


Tame_lemmas.reflection_eq2 |- (!n. (!x. ((((reflection n) x) = x) = (((dot x) n) = (real_of_num (NUMERAL _0))))))


Tame_list.nextElem_ALT |- (((((nextElem NIL) b) x) = b) /\ ((((nextElem ((CONS a) as)) b) x) = (((COND (x = a)) (((COND (as = NIL)) b) ((LET (\a_. (LET_END ((LET (\as_. (LET_END a_))) (TL as))))) (HD as)))) (((nextElem as) b) x))))


Seq.has_mask_cons |- (!a. (!b. (!m. (!x. (!s. (((has a) ((mask ((CONS b) m)) ((CONS x) s))) = ((b /\ (a x)) \/ ((has a) ((mask m) s)))))))))


ORTHOGONAL_MATRIX_NORM_EQ |- (!A. ((orthogonal_matrix A) = (!x. ((vector_norm ((matrix_vector_mul A) x)) = (vector_norm x)))))


Tame_general.RUNOQPQ |- (!H. ((tame_planar_hypermap H) ==> (restricted_hypermap H)))


Bump.MCELL2_PARAM_UL |- (!V. (!ul. (!vl. (!X. (((packing V) /\ ((saturated V) /\ ((X = ((mcell2 V) ul)) /\ ((ul IN ((barV V) (NUMERAL (BIT1 (BIT1 _0))))) /\ ((~ (negligible X)) /\ (vl = (SND ((cell_params V) X)))))))) ==> ((X = ((mcell2 V) vl)) /\ (vl IN ((barV V) (NUMERAL (BIT1 (BIT1 _0)))))))))))


DECREASING_LEFT_LIMIT |- (!f. (!a. (!b. (!c. (((!x. (!y. (((x IN (closed_real_interval ((CONS (a , b)) NIL))) /\ ((y IN (closed_real_interval ((CONS (a , b)) NIL))) /\ ((real_le x) y))) ==> ((real_le (f y)) (f x))))) /\ (c IN (closed_real_interval ((CONS (a , b)) NIL)))) ==> (?l. (((---> f) l) ((within (atreal c)) (closed_real_interval ((CONS (a , c)) NIL))))))))))


MEASURE_LIPSCHITZ_IMAGE |- (!f. (!s. (!B. ((((dimindex UNIV) <= (dimindex UNIV)) /\ ((measurable s) /\ (!x. (!y. (((x IN s) /\ (y IN s)) ==> ((real_le (vector_norm ((vector_sub (f x)) (f y)))) ((real_mul B) (vector_norm ((vector_sub x) y))))))))) ==> ((real_le (measure ((IMAGE f) s))) ((real_mul ((real_pow B) (dimindex UNIV))) (measure s)))))))


IM_COMPLEX_INV_EQ_0 |- (!z. (((Im (complex_inv z)) = (real_of_num (NUMERAL _0))) = ((Im z) = (real_of_num (NUMERAL _0)))))


CARD_EQ_COUNTABLE_SUBSETS_REAL |- ((GSPEC (\GEN%PVAR%73. (?s. (((SETSPEC GEN%PVAR%73) (COUNTABLE s)) s)))) =_c UNIV)


QUOTIENT_MAP_FROM_SUBSET |- (!f. (!s. (!t. (!u. ((((continuous_on f) t) /\ ((((IMAGE f) t) SUBSET u) /\ ((s SUBSET t) /\ ((((IMAGE f) s) = u) /\ (!v. ((v SUBSET u) ==> (((open_in ((subtopology euclidean) s)) (GSPEC (\GEN%PVAR%669. (?x. (((SETSPEC GEN%PVAR%669) ((x IN s) /\ ((f x) IN v))) x))))) = ((open_in ((subtopology euclidean) u)) v)))))))) ==> (!v. ((v SUBSET u) ==> (((open_in ((subtopology euclidean) t)) (GSPEC (\GEN%PVAR%670. (?x. (((SETSPEC GEN%PVAR%670) ((x IN t) /\ ((f x) IN v))) x))))) = ((open_in ((subtopology euclidean) u)) v)))))))))


REAL_DIFFERENTIABLE_BOUND |- (!f. (!f'. (!s. (!B. (((is_realinterval s) /\ (!x. ((x IN s) ==> ((((has_real_derivative f) (f' x)) ((within (atreal x)) s)) /\ ((real_le (real_abs (f' x))) B))))) ==> (!x. (!y. (((x IN s) /\ (y IN s)) ==> ((real_le (real_abs ((real_sub (f x)) (f y)))) ((real_mul B) (real_abs ((real_sub x) y))))))))))))


RELATIVE_ORIENTATION_COMPOSE |- (!f. (!g. (!c. ((((simplex ((int_sub (int_of_num (dimindex UNIV))) (int_of_num (NUMERAL (BIT1 _0))))) c) /\ (~ (((relative_orientation f) c) = (real_of_num (NUMERAL _0))))) ==> (((relative_orientation (g o f)) c) = ((real_mul ((relative_orientation g) ((vertex_image f) c))) ((relative_orientation f) c)))))))


DIFFERENTIABLE_ON_MUL |- (!f. (!g. (!s. ((((differentiable_on (lift o f)) s) /\ ((differentiable_on g) s)) ==> ((differentiable_on (\x. ((% (f x)) (g x)))) s)))))


CONTINUOUS_ON_CX_IM |- (!s. ((continuous_on (Cx o Im)) s))


ORTHOGONAL_TRANSFORMATION_BALL |- (!f. (!r. ((orthogonal_transformation f) ==> (((IMAGE f) (ball ((vec (NUMERAL _0)) , r))) = (ball ((vec (NUMERAL _0)) , r))))))


REAL_SUMMABLE_IMP_REAL_SUMS_BOUNDED |- (!f. (!k. (((real_summable (from k)) f) ==> (real_bounded (GSPEC (\GEN%PVAR%4863. (?n. (((SETSPEC GEN%PVAR%4863) (n IN UNIV)) ((sum (k .. n)) f)))))))))


PATH_CONNECTED_SCALING |- (!s. (!c. ((path_connected s) ==> (path_connected ((IMAGE (\x. ((% c) x))) s)))))


INT_LE_SQUARE_ABS |- (!x. (!y. (((int_le (int_abs x)) (int_abs y)) = ((int_le ((int_pow x) (NUMERAL (BIT0 (BIT1 _0))))) ((int_pow y) (NUMERAL (BIT0 (BIT1 _0))))))))


INT_LTE_TOTAL |- (!x. (!y. (((int_lt x) y) \/ ((int_le y) x))))


CLOG_NEG |- (!z. ((~ (z = (Cx (real_of_num (NUMERAL _0))))) ==> ((clog (vector_neg z)) = (((COND (((real_le (Im z)) (real_of_num (NUMERAL _0))) /\ (~ (((real_lt (Re z)) (real_of_num (NUMERAL _0))) /\ ((Im z) = (real_of_num (NUMERAL _0))))))) ((vector_add (clog z)) ((complex_mul ii) (Cx pi)))) ((vector_sub (clog z)) ((complex_mul ii) (Cx pi)))))))


Seq2.next_el_hd_belast |- (!h. (!t. (((NUMERAL (BIT1 _0)) < (sizel t)) ==> (((next_el ((belast h) t)) h) = (HD t)))))


HOMOTOPIC_NEARBY_PATHS |- (!g. (!s. (((path g) /\ ((open s) /\ ((path_image g) SUBSET s))) ==> (?e. (((real_lt (real_of_num (NUMERAL _0))) e) /\ (!h. (((path h) /\ (((pathstart h) = (pathstart g)) /\ (((pathfinish h) = (pathfinish g)) /\ (!t. ((t IN (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL))) ==> ((real_lt (vector_norm ((vector_sub (h t)) (g t)))) e)))))) ==> (((homotopic_paths s) g) h))))))))


REAL_INTEGRABLE_RESTRICT_INTER |- (!f. (!s. (!t. (((real_integrable_on (\x. (((COND (x IN s)) (f x)) (real_of_num (NUMERAL _0))))) t) = ((real_integrable_on f) (s INTER t))))))


PSUBSET_INSERT_SUBSET |- (!s. (!t. ((s PSUBSET t) = (?x. ((~ (x IN s)) /\ ((x INSERT s) SUBSET t))))))


LIM_UNION_UNIV |- (!f. (!x. (!l. (!s. (!t. (((((--> f) l) ((within (at x)) s)) /\ ((((--> f) l) ((within (at x)) t)) /\ ((s UNION t) = UNIV))) ==> (((--> f) l) (at x))))))))


DIST_FSTCART |- (!x. (!y. ((real_le (distance ((fstcart x) , (fstcart y)))) (distance (x , y)))))


INT_DIFFSQ |- (!x. (!y. (((int_mul ((int_add x) y)) ((int_sub x) y)) = ((int_sub ((int_mul x) x)) ((int_mul y) y)))))


INT_NEG_SUB |- (!x. (!y. ((int_neg ((int_sub x) y)) = ((int_sub y) x))))


COMPLEX_EQ_ADD_RCANCEL |- (!x. (!y. (!z. ((((vector_add x) z) = ((vector_add y) z)) = (x = y)))))


PRODUCT_NEG_NUMSEG_1 |- (!f. (!n. (((product ((NUMERAL (BIT1 _0)) .. n)) (\i. (real_neg (f i)))) = ((real_mul ((real_pow (real_neg (real_of_num (NUMERAL (BIT1 _0))))) n)) ((product ((NUMERAL (BIT1 _0)) .. n)) f)))))


REAL_MEASURE_DISJOINT_UNIONS_IMAGE_STRONG |- (!f. (!s. (((FINITE (GSPEC (\GEN%PVAR%4934. (?x. (((SETSPEC GEN%PVAR%4934) ((x IN s) /\ (~ ((f x) = EMPTY)))) x))))) /\ ((!x. ((x IN s) ==> (real_measurable (f x)))) /\ (!x. (!y. (((x IN s) /\ ((y IN s) /\ (~ (x = y)))) ==> ((DISJOINT (f x)) (f y))))))) ==> ((real_measure (UNIONS ((IMAGE f) s))) = ((sum s) (\x. (real_measure (f x))))))))


HELLY_COMPACT |- (!f. ((((FINITE f) ==> ((CARD f) >= ((dimindex UNIV) + (NUMERAL (BIT1 _0))))) /\ ((!s. ((s IN f) ==> ((convex s) /\ (compact s)))) /\ (!t. (((t SUBSET f) /\ ((FINITE t) /\ ((CARD t) = ((dimindex UNIV) + (NUMERAL (BIT1 _0)))))) ==> (~ ((INTERS t) = EMPTY)))))) ==> (~ ((INTERS f) = EMPTY))))


Tame_list.BUTLAST_NIL |- (!xs. (((NUMERAL (BIT1 _0)) < (sizel xs)) ==> (~ ((BUTLAST xs) = NIL))))


REAL_SUMMABLE_ZETA_INTEGER |- (!n. (!m. (((NUMERAL (BIT0 (BIT1 _0))) <= m) ==> ((real_summable (from n)) (\k. (real_inv ((real_pow (real_of_num k)) m)))))))


REAL_CLOSED_OPEN_INTERVAL |- (!a. (!b. (((real_le a) b) ==> ((closed_real_interval ((CONS (a , b)) NIL)) = ((open_real_interval (a , b)) UNION (a INSERT (b INSERT EMPTY)))))))


Packing3.HULL_INTER_EQ_INTER |- (!P. (!s. (!t. (((P s) /\ (P t)) ==> (((hull P) (s INTER t)) = (s INTER t))))))


Constants_approx.tgt_interval |- ((interval_arith tgt) (((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))))) , ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))))


Lp_main_estimate.ineq_tau4_diag3_std |- (!V. (!E. ((lp_fan (V , E)) ==> ((!V. (!E. (!FF. (((convex_local_fan (V , (E , FF))) /\ ((packing V) /\ ((V SUBSET ball_annulus) /\ (((CARD V) = (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) /\ ((!v. (!w. (((~ (v = w)) /\ ((v IN V) /\ ((w IN V) /\ (~ ((v INSERT (w INSERT EMPTY)) IN E))))) ==> ((real_le (real_of_num (NUMERAL (BIT1 (BIT1 _0))))) (distance (v , w)))))) /\ (!v. (!w. (((v INSERT (w INSERT EMPTY)) IN E) ==> (((real_le (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (distance (v , w))) /\ ((real_le (distance (v , w))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0))))))))))) ==> ((real_le ((DECIMAL (NUMERAL (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 _0))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))) (((tau_fun V) E) FF)))))) ==> ((E = (ESTD V)) ==> (!d. ((d IN ((darts_k (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) (hypermap_of_fan (V , E)))) ==> ((((real_le (real_of_num (NUMERAL (BIT1 (BIT1 _0))))) ((y4'_fan (V , E)) d)) /\ ((real_le (real_of_num (NUMERAL (BIT1 (BIT1 _0))))) ((y4'_fan (V , E)) ((f_fan_pair_ext (V , E)) d)))) ==> ((real_le ((DECIMAL (NUMERAL (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 _0))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))) ((tau_fan (V , E)) ((face (hypermap_of_fan (V , E))) d)))))))))))


ADMISSIBLE_MATCH_SEQPATTERN |- (!<<. (!p. (!s. (!c1. (!c2. (!e. ((((((admissible <<) p) s) (\f. (\x. (?y. ((((c1 f) x) ((e f) x)) y))))) /\ (((((admissible <<) (\f. (\x. (((p f) x) /\ (?y. ((((c1 f) x) ((e f) x)) y)))))) s) (\f. (\x. ((_MATCH ((e f) x)) ((c1 f) x))))) /\ ((((admissible <<) (\f. (\x. (((p f) x) /\ (~ (?y. ((((c1 f) x) ((e f) x)) y))))))) s) (\f. (\x. ((_MATCH ((e f) x)) ((c2 f) x))))))) ==> ((((admissible <<) p) s) (\f. (\x. ((_MATCH ((e f) x)) ((_SEQPATTERN ((c1 f) x)) ((c2 f) x)))))))))))))


REAL_DIFFERENTIABLE_WITHIN |- (!f. (!a. (!s. (((real_differentiable f) ((within (atreal x)) s)) = ((differentiable (lift o (f o drop))) ((within (at (lift x))) ((IMAGE lift) s)))))))


REAL_LOG_CONVEX_PRODUCT |- (!f. (!s. (!k. (((FINITE k) /\ (!i. ((i IN k) ==> ((real_log_convex_on (\x. ((f x) i))) s)))) ==> ((real_log_convex_on (\x. ((product k) (f x)))) s)))))


HOLOMORPHIC_ON_CLOG |- (!s. ((!z. (((z IN s) /\ ((Im z) = (real_of_num (NUMERAL _0)))) ==> ((real_lt (real_of_num (NUMERAL _0))) (Re z)))) ==> ((holomorphic_on clog) s)))


REAL_INTEGRABLE_SPIKE_EQ |- (!f. (!g. (!s. (!t. (((real_negligible s) /\ (!x. ((x IN (t DIFF s)) ==> ((g x) = (f x))))) ==> (((real_integrable_on f) t) = ((real_integrable_on g) t)))))))


AFFINE_PCROSS_EQ |- (!s. (!t. ((affine (s PCROSS t)) = ((s = EMPTY) \/ ((t = EMPTY) \/ ((affine s) /\ (affine t)))))))


Rdwkarc.SATURATED_TRANS |- (!V. (!x. ((saturated V) ==> (saturated (GSPEC (\GEN%PVAR%6925. (?u. (((SETSPEC GEN%PVAR%6925) (((vector_add u) x) IN V)) u))))))))


EMPTY_INTERIOR_AFF_DIM |- (!s. (((int_lt (aff_dim s)) (int_of_num (dimindex UNIV))) ==> ((interior s) = EMPTY)))


CONVEX_AFF_LT |- (!s. (!t. (convex ((aff_lt s) t))))


Ssrbool.iffRLn |- (!Q. (!P. ((P = Q) ==> ((~ Q) ==> (~ P)))))


REAL_SUMMABLE_ADD |- (!x. (!y. (!s. ((((real_summable s) x) /\ ((real_summable s) y)) ==> ((real_summable s) (\n. ((real_add (x n)) (y n))))))))


Seq.all_predI |- (!a1. (!a2. (!s. (((all ((predI a1) a2)) s) = (((all a1) s) /\ ((all a2) s))))))


REAL_CONTINUOUS_INV_ATREAL |- (!f. (!a. ((((real_continuous f) (atreal a)) /\ (~ ((f a) = (real_of_num (NUMERAL _0))))) ==> ((real_continuous (\x. (real_inv (f x)))) (atreal a)))))


COMPLEX_ADD_SUB |- (!x. (!y. (((vector_sub ((vector_add x) y)) x) = y)))


SUM_PERMUTE_NUMSEG |- (!f. (!p. (!m. (!n. (((permutes p) (m .. n)) ==> (((sum (m .. n)) f) = ((sum (m .. n)) (f o p))))))))


FARKAS_LEMMA_ALT |- (!A. (!b. ((?x. (!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV))) ==> ((real_le (((matrix_vector_mul A) x) $ i)) (b $ i))))) = (~ (?y. ((!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV))) ==> ((real_le (real_of_num (NUMERAL _0))) (y $ i)))) /\ ((((vector_matrix_mul y) A) = (vec (NUMERAL _0))) /\ ((real_lt ((dot b) y)) (real_of_num (NUMERAL _0))))))))))


LOCALLY_CONNECTED_PATH_IMAGE |- (!p. ((path p) ==> ((locally connected) (path_image p))))


HAS_BOUNDED_VARIATION_ON_LINEAR_IMAGE |- (!f. (!g. (!a. (!b. (((linear f) /\ ((has_bounded_variation_on g) ((IMAGE f) (closed_interval ((CONS (a , b)) NIL))))) ==> ((has_bounded_variation_on (g o f)) (closed_interval ((CONS (a , b)) NIL))))))))


COMPACT_CONTINUOUS_IMAGE_EQ |- (!f. (!s. ((!x. (!y. (((x IN s) /\ ((y IN s) /\ ((f x) = (f y)))) ==> (x = y)))) ==> (((continuous_on f) s) = (!t. (((compact t) /\ (t SUBSET s)) ==> (compact ((IMAGE f) t))))))))


EMPTY_INSIDE_PSUBSET_CONVEX_FRONTIER |- (!s. (!t. (((convex s) /\ (t PSUBSET (frontier s))) ==> ((inside t) = EMPTY))))


Geomdetail.IN_SET3 |- ((x IN (a INSERT (b INSERT (c INSERT EMPTY)))) = ((x = a) \/ ((x = b) \/ (x = c))))


MBOUNDED_INSERT |- (!m. (!a. (!s. (((mbounded m) (a INSERT s)) = ((a IN (mspace m)) /\ ((mbounded m) s))))))


INSIDE_SUBSET_CONVEX_HULL |- (!s. ((inside s) SUBSET ((hull convex) s)))


Hdplygy.INDEX_J1_SY |- (!s. ((x IN (J1_SY s)) ==> (((NUMERAL (BIT1 _0)) <= (FST x)) /\ ((FST x) <= (k_sy s)))))


Fan.remark_fan1 |- (!v. (!w. (!V. (!E. (((v IN V) /\ (w IN V)) ==> ((w IN (((set_of_edge v) V) E)) = (v IN (((set_of_edge w) V) E))))))))


Float.REAL_EQ_RCANCEL_IMP' |- (!x. (!y. (!z. ((((real_mul x) z) = ((real_mul y) z)) ==> ((~ (z = (real_of_num (NUMERAL _0)))) ==> (x = y))))))


COMPLEX_POW_DIV |- (!x. (!y. (!n. (((complex_pow ((complex_div x) y)) n) = ((complex_div ((complex_pow x) n)) ((complex_pow y) n))))))


Fan.FAN5_LINEAR_IMAGE_EQ |- (!f. (!x. (!V. (!E. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((fan5 ((f x) , (((IMAGE f) V) , ((IMAGE (IMAGE f)) E)))) = (fan5 (x , (V , E)))))))))


REAL_CONVEX_DISTANCE |- (!s. (!a. ((real_convex_on (\x. (real_abs ((real_sub a) x)))) s)))


CONIC_HALFSPACE_LE |- (!a. (conic (GSPEC (\GEN%PVAR%1464. (?x. (((SETSPEC GEN%PVAR%1464) ((real_le ((dot a) x)) (real_of_num (NUMERAL _0)))) x))))))


HAS_REAL_DERIVATIVE_DIFFERENTIABLE |- (!f. (!x. ((((has_real_derivative f) ((real_derivative f) x)) (atreal x)) = ((real_differentiable f) (atreal x)))))


Lp_ineqs_quads.ineq119_std |- (!V. (!E. ((lp_fan (V , E)) ==> (quad_nonlinear_v10 ==> ((!y1. (!y2. (!y3. (!y4. (!y5. (!y6. ((ineq ((CONS ((real_of_num (NUMERAL (BIT0 (BIT1 _0)))) , (y1 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) ((CONS ((real_of_num (NUMERAL (BIT0 (BIT1 _0)))) , (y2 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) ((CONS ((real_of_num (NUMERAL (BIT0 (BIT1 _0)))) , (y3 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) ((CONS ((real_of_num (NUMERAL (BIT0 (BIT1 _0)))) , (y4 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) ((CONS (((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0) , (y5 , ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 _0))))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))) ((CONS ((real_of_num (NUMERAL (BIT0 (BIT1 _0)))) , (y6 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) NIL))))))) ((real_gt ((real_add ((real_sub ((((((taum y1) y2) y3) y4) y5) y6)) ((real_mul ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 _0))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))) ((((((dih_y y1) y2) y3) y4) y5) y6)))) ((real_div ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) ((DECIMAL (NUMERAL _0)) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0)))))))))))))) ==> ((!y1. (!y2. (!y3. (!y4. (!y5. (!y6. ((ineq ((CONS (((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 (BIT1 _0))))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))) , (y1 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) h0)))) ((CONS (((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0)))))) , (y2 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) ((CONS (((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0)))))) , (y3 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) ((CONS (((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0) , (y4 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) h0)))) ((CONS (((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0)))))) , (y5 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) ((CONS (((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0)))))) , (y6 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) NIL))))))) ((real_lt ((((((delta_y y1) y2) y3) y4) y5) y6)) (real_of_num (NUMERAL _0)))))))))) ==> ((!V. (!E. (!FF. (((convex_local_fan (V , (E , FF))) /\ ((packing V) /\ ((V SUBSET ball_annulus) /\ (((NUMERAL (BIT0 (BIT0 (BIT1 _0)))) <= (CARD V)) /\ (((CARD V) <= (NUMERAL (BIT0 (BIT1 (BIT1 _0))))) /\ ((!v. (!w. (((~ (v = w)) /\ ((v IN V) /\ ((w IN V) /\ (~ ((v INSERT (w INSERT EMPTY)) IN E))))) ==> ((real_le ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) (distance (v , w)))))) /\ (!v. (!w. (((v INSERT (w INSERT EMPTY)) IN E) ==> (((real_le (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (distance (v , w))) /\ ((real_le (distance (v , w))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))))))))))) ==> ((real_le (d_tame (CARD V))) (((tau_fun V) E) FF)))))) ==> ((lp_tau (V , E)) ==> ((E = (ESTD V)) ==> (!d. ((d IN ((darts_k (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) (hypermap_of_fan (V , E)))) ==> ((real_ge ((real_add ((real_sub ((tau_fan (V , E)) ((face (hypermap_of_fan (V , E))) d))) ((real_mul ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 _0))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))) ((azim_dart (V , E)) d)))) ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))))))) (real_of_num (NUMERAL _0))))))))))))))


NEGATIONS_BALL |- (!r. (((IMAGE vector_neg) (ball ((vec (NUMERAL _0)) , r))) = (ball ((vec (NUMERAL _0)) , r))))


SUMMABLE_COMPONENT |- (!f. (!s. (!k. ((((summable s) f) /\ (((NUMERAL (BIT1 _0)) <= k) /\ (k <= (dimindex UNIV)))) ==> ((summable s) (\i. (lift ((f i) $ k))))))))


REAL_VARIATION_AFFINITY2 |- (!m. (!c. (!f. (!s. (((real_variation ((IMAGE (\x. ((real_add ((real_mul (real_inv m)) x)) (real_neg ((real_mul (real_inv m)) c))))) s)) (\x. (f ((real_add ((real_mul m) x)) c)))) = (((COND (m = (real_of_num (NUMERAL _0)))) (real_of_num (NUMERAL _0))) ((real_variation s) f)))))))


NSUM_RESTRICT |- (!f. (!s. ((FINITE s) ==> (((nsum s) (\x. (((COND (x IN s)) (f x)) (NUMERAL _0)))) = ((nsum s) f)))))


FORALL_SUBSET_IMAGE_INJ |- (!P. (!f. (!s. ((!t. ((t SUBSET ((IMAGE f) s)) ==> (P t))) = (!t. (((t SUBSET s) /\ (!x. (!y. (((x IN t) /\ (y IN t)) ==> (((f x) = (f y)) = (x = y)))))) ==> (P ((IMAGE f) t))))))))


CONTINUOUS_ON_COMPONENTS_OPEN_EQ |- (!f. (!s. ((open s) ==> (((continuous_on f) s) = (!c. ((c IN (components s)) ==> ((continuous_on f) c)))))))


Leaf_cell.AFF_GE_MONO_TRANS |- (!X. (!Y. (!S. ((S SUBSET X) ==> (((aff_ge (X DIFF S)) (Y UNION S)) SUBSET ((aff_ge X) Y))))))


REALLIM_1_OVER_LOG |- (((---> (\n. (real_inv (log (real_of_num n))))) (real_of_num (NUMERAL _0))) sequentially)


CONTINUOUS_CONTINUOUS_ATREAL |- (!f. (!x. (((continuous f) (atreal x)) = ((continuous (f o drop)) (at (lift x))))))


DIAMETER_ATTAINED_FRONTIER |- (!s. (((bounded s) /\ (~ ((diameter s) = (real_of_num (NUMERAL _0))))) ==> (?x. (?y. ((x IN (frontier s)) /\ ((y IN (frontier s)) /\ ((vector_norm ((vector_sub x) y)) = (diameter s))))))))


CARD_EQ_ARC_IMAGE |- (!g. ((arc g) ==> ((path_image g) =_c UNIV)))


COMPLEX_DERIVATIVE_LMUL_AT |- (!f. (!c. (!z. (((analytic_on f) (z INSERT EMPTY)) ==> (((complex_derivative (\w. ((complex_mul c) (f w)))) z) = ((complex_mul c) ((complex_derivative f) z)))))))


INT_EQ_ADD_LCANCEL |- (!x. (!y. (!z. ((((int_add x) y) = ((int_add x) z)) = (y = z)))))


Collect_geom2.INSERT_SUBSET |- ((EMPTY SUBSET s) /\ (((a INSERT s) SUBSET ss) = ((a IN ss) /\ (s SUBSET ss))))


EXISTS_PATH_SUBPATH_TO_FRONTIER_CLOSED |- (!g. (!s. (((closed s) /\ ((path g) /\ (((pathstart g) IN s) /\ (~ ((pathfinish g) IN s))))) ==> (?h. ((path h) /\ (((pathstart h) = (pathstart g)) /\ (((path_image h) SUBSET ((path_image g) INTER s)) /\ ((pathfinish h) IN (frontier s)))))))))


OPEN_CONTAINS_CBALL_EQ |- (!s. ((open s) ==> (!x. ((x IN s) = (?e. (((real_lt (real_of_num (NUMERAL _0))) e) /\ ((cball (x , e)) SUBSET s)))))))


TAN_BOUND_PI2 |- (!x. (((real_lt (real_abs x)) ((real_div pi) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0))))))) ==> ((real_lt (real_abs (tan x))) (real_of_num (NUMERAL (BIT1 _0))))))


REAL_MEASURABLE_REAL_NEGLIGIBLE_SYMDIFF |- (!s. (!t. (((real_measurable s) /\ (real_negligible ((s DIFF t) UNION (t DIFF s)))) ==> (real_measurable t))))


OPEN_MAP_INTERIORS |- (!f. ((!s. ((open s) ==> (open ((IMAGE f) s)))) = (!s. (((IMAGE f) (interior s)) SUBSET (interior ((IMAGE f) s))))))


INTERIOR_BALL |- (!a. (!r. ((interior (ball (a , r))) = (ball (a , r)))))


INT_LE_TRANS_LT |- (!x. (!y. (((int_le x) y) = (!z. (((int_lt y) z) ==> ((int_lt x) z))))))


LOCALLY_NEGLIGIBLE |- (!s. (((locally negligible) s) = (negligible s)))


SUM_INTEGRAL_BOUNDS_INCREASING |- (!f. (!g. (!m. (!n. (((m <= n) /\ ((!x. ((x IN (closed_segment ((CONS ((Cx ((real_sub (real_of_num m)) (real_of_num (NUMERAL (BIT1 _0))))) , (Cx ((real_add (real_of_num n)) (real_of_num (NUMERAL (BIT1 _0))))))) NIL))) ==> (((has_complex_derivative g) (f x)) (at x)))) /\ (!x. (!y. ((((real_le ((real_sub (real_of_num m)) (real_of_num (NUMERAL (BIT1 _0))))) x) /\ (((real_le x) y) /\ ((real_le y) ((real_add (real_of_num n)) (real_of_num (NUMERAL (BIT1 _0))))))) ==> ((real_le (Re (f (Cx x)))) (Re (f (Cx y))))))))) ==> (((real_le (Re ((vector_sub (g (Cx (real_of_num n)))) (g (Cx ((real_sub (real_of_num m)) (real_of_num (NUMERAL (BIT1 _0))))))))) ((sum (m .. n)) (\k. (Re (f (Cx (real_of_num k))))))) /\ ((real_le ((sum (m .. n)) (\k. (Re (f (Cx (real_of_num k))))))) (Re ((vector_sub (g (Cx ((real_add (real_of_num n)) (real_of_num (NUMERAL (BIT1 _0))))))) (g (Cx (real_of_num m))))))))))))


COND_ABS |- (!b. (!f. (!g. ((\x. (((COND b) (f x)) (g x))) = (((COND b) f) g)))))


ABSOLUTELY_CONTINUOUS_ON_NEG |- (!f. (!s. (((absolutely_continuous_on f) s) ==> ((absolutely_continuous_on (\x. (vector_neg (f x)))) s))))


Ssrnat.distn_eq0 |- (!m. (!n. ((((distn m) n) = (NUMERAL _0)) = (m = n))))


HOLOMORPHIC_ON_COMPOSE_GEN |- (!f. (!g. (!s. (!t. ((((holomorphic_on f) s) /\ (((holomorphic_on g) t) /\ (!z. ((z IN s) ==> ((f z) IN t))))) ==> ((holomorphic_on (g o f)) s))))))


REAL_DIFFERENTIABLE_WITHIN_SIN |- (!s. (!x. ((real_differentiable sin) ((within (atreal x)) s))))


REAL_MEASURABLE_ON_CASES |- (!P. (!f. (!g. (!s. (((real_lebesgue_measurable (GSPEC (\GEN%PVAR%5045. (?x. (((SETSPEC GEN%PVAR%5045) (P x)) x))))) /\ (((real_measurable_on f) s) /\ ((real_measurable_on g) s))) ==> ((real_measurable_on (\x. (((COND (P x)) (f x)) (g x)))) s))))))


REAL_INTEGRABLE_DECREASING_PRODUCT |- (!f. (!g. (!a. (!b. ((((real_integrable_on f) (closed_real_interval ((CONS (a , b)) NIL))) /\ (!x. (!y. (((x IN (closed_real_interval ((CONS (a , b)) NIL))) /\ ((y IN (closed_real_interval ((CONS (a , b)) NIL))) /\ ((real_le x) y))) ==> ((real_le (g y)) (g x)))))) ==> ((real_integrable_on (\x. ((real_mul (g x)) (f x)))) (closed_real_interval ((CONS (a , b)) NIL))))))))


Ckqowsa_3_points.in_aff_ge_0_2_imp_dot_pos |- (!v1. (!v2. (!w. ((((real_lt (real_of_num (NUMERAL _0))) ((dot v1) v2)) /\ ((w IN ((aff_ge ((vec (NUMERAL _0)) INSERT EMPTY)) (v1 INSERT (v2 INSERT EMPTY)))) /\ (~ (w = (vec (NUMERAL _0)))))) ==> (((real_lt (real_of_num (NUMERAL _0))) ((dot v1) w)) /\ ((real_lt (real_of_num (NUMERAL _0))) ((dot v2) w)))))))


REAL_LE_ROOT |- (!n. (!x. (!y. (((~ (n = (NUMERAL _0))) /\ ((real_le (real_of_num (NUMERAL _0))) x)) ==> (((real_le x) ((root n) y)) = ((real_le ((real_pow x) n)) y))))))


CROSS_MONO |- (!s. (!t. (!s'. (!t'. (((s SUBSET s') /\ (t SUBSET t')) ==> ((s CROSS t) SUBSET (s' CROSS t')))))))


REAL_INTEGRAL_SUBSET_LE |- (!f. (!s. (!t. (((s SUBSET t) /\ (((real_integrable_on f) s) /\ (((real_integrable_on f) t) /\ (!x. ((x IN t) ==> ((real_le (real_of_num (NUMERAL _0))) (f x))))))) ==> ((real_le ((real_integral s) f)) ((real_integral t) f))))))


Audit_formal_proof.chk_basic_definitions |- ((((real_le (real_of_num (NUMERAL _0))) x) ==> ((real_abs x) = x)) /\ ((((real_lt x) (real_of_num (NUMERAL _0))) ==> ((real_abs x) = (real_neg x))) /\ ((((real_pow x) (NUMERAL _0)) = (real_of_num (NUMERAL (BIT1 _0)))) /\ ((((real_pow x) (SUC n)) = ((real_mul x) ((real_pow x) n))) /\ ((((real_le (real_of_num (NUMERAL _0))) x) ==> (((real_le (real_of_num (NUMERAL _0))) (sqrt x)) /\ (((real_pow (sqrt x)) (NUMERAL (BIT0 (BIT1 _0)))) = x))) /\ ((((sum ((NUMERAL _0) .. (NUMERAL _0))) f) = (f (NUMERAL _0))) /\ ((((sum ((NUMERAL _0) .. (SUC n))) f) = ((real_add ((sum ((NUMERAL _0) .. n)) f)) (f (SUC n)))) /\ (((real_le (real_abs ((real_sub ((real_div pi) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0))))))) ((sum ((NUMERAL _0) .. n)) (\i. ((real_div ((real_pow (real_neg (real_of_num (NUMERAL (BIT1 _0))))) i)) (real_of_num (((NUMERAL (BIT0 (BIT1 _0))) * i) + (NUMERAL (BIT1 _0)))))))))) ((real_div (real_of_num (NUMERAL (BIT1 _0)))) (real_of_num (((NUMERAL (BIT0 (BIT1 _0))) * n) + (NUMERAL (BIT1 (BIT1 _0))))))) /\ (((A HAS_SIZE n) = ((FINITE A) /\ ((CARD A) = n))) /\ ((EMPTY HAS_SIZE (NUMERAL _0)) /\ (((a INSERT EMPTY) HAS_SIZE (NUMERAL (BIT1 _0))) /\ ((((A HAS_SIZE m) /\ ((B HAS_SIZE n) /\ ((A INTER B) HAS_SIZE p))) ==> ((A UNION B) HAS_SIZE (m + (n - p)))) /\ (((triple_of_real3 o real3_of_triple) = (\w. w)) /\ (((real3_of_triple o triple_of_real3) = (\v. v)) /\ (((vec (NUMERAL _0)) = (real3_of_triple ((real_of_num (NUMERAL _0)) , ((real_of_num (NUMERAL _0)) , (real_of_num (NUMERAL _0)))))) /\ (((distance ((real3_of_triple (x , (y , z))) , (real3_of_triple (x' , (y' , z'))))) = (sqrt ((real_add ((real_pow ((real_sub x) x')) (NUMERAL (BIT0 (BIT1 _0))))) ((real_add ((real_pow ((real_sub y) y')) (NUMERAL (BIT0 (BIT1 _0))))) ((real_pow ((real_sub z) z')) (NUMERAL (BIT0 (BIT1 _0)))))))) /\ ((packing V) = (!u. (!v. (((u IN V) /\ ((v IN V) /\ ((real_lt (distance (u , v))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) ==> (u = v)))))))))))))))))))))


REAL_POLYNOMIAL_FUNCTION_EXPLICIT_NZ |- (!f. ((real_polynomial_function f) = (?s. (?a. ((FINITE s) /\ ((!k. ((k IN s) ==> (~ ((a k) = (real_of_num (NUMERAL _0)))))) /\ (f = (\x. ((sum s) (\k. ((real_mul (a k)) ((product ((NUMERAL (BIT1 _0)) .. (dimindex UNIV))) (\i. ((real_pow (x $ i)) (k $ i)))))))))))))))


HAS_BOUNDED_REAL_VARIATION_ON_RMUL |- (!f. (!c. (!s. (((has_bounded_real_variation_on f) s) ==> ((has_bounded_real_variation_on (\x. ((real_mul (f x)) c))) s)))))


MATRIX_INV_EQ_0 |- (!A. (((matrix_inv A) = (mat (NUMERAL _0))) = (A = (mat (NUMERAL _0)))))


SIMPLICIAL_COMPLEX_DISJOINT_RELATIVE_INTERIORS |- (!t. (!c. (!d. (((simplicial_complex t) /\ ((c IN t) /\ ((d IN t) /\ (~ (c = d))))) ==> (((relative_interior c) INTER (relative_interior d)) = EMPTY)))))


HOMEOMORPHIC_CBALLS |- (!a. (!b. (!d. (!e. ((((real_lt (real_of_num (NUMERAL _0))) d) /\ ((real_lt (real_of_num (NUMERAL _0))) e)) ==> ((homeomorphic (cball (a , d))) (cball (b , e))))))))


CTAN_DOUBLE |- (!z. (((~ ((ccos z) = (Cx (real_of_num (NUMERAL _0))))) /\ (~ ((ccos ((complex_mul (Cx (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) z)) = (Cx (real_of_num (NUMERAL _0)))))) ==> ((ctan ((complex_mul (Cx (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) z)) = ((complex_div ((complex_mul (Cx (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) (ctan z))) ((vector_sub (Cx (real_of_num (NUMERAL (BIT1 _0))))) ((complex_pow (ctan z)) (NUMERAL (BIT0 (BIT1 _0)))))))))


EVENTUALLY_AT_INFINITY_POS |- (!p. (((eventually p) at_infinity) = (?b. (((real_lt (real_of_num (NUMERAL _0))) b) /\ (!x. (((real_ge (vector_norm x)) b) ==> (p x)))))))


TAGGED_DIVISION_UNIONS_EXISTS |- (!d. (!iset. (!i. (((FINITE iset) /\ ((!i. ((i IN iset) ==> (?p. (((tagged_division_of p) i) /\ ((fine d) p))))) /\ ((!i1. (!i2. (((i1 IN iset) /\ ((i2 IN iset) /\ (~ (i1 = i2)))) ==> (((interior i1) INTER (interior i2)) = EMPTY)))) /\ ((UNIONS iset) = i)))) ==> (?p. (((tagged_division_of p) i) /\ ((fine d) p)))))))


Betwn_corek_z_x.core_rot |- (((good_list L) /\ (((all (\x. ((MEM x) (list_of_darts L)))) r) /\ ((is_edge_nondegenerate (hypermap_of_list L)) /\ ((uniq r) /\ ((final_list L) r))))) ==> (((core L) ((rot n) r)) = ((rot n) ((core L) r))))


COMPONENTS_OPEN_UNIQUE |- (!f. (!s. (((!c. ((c IN f) ==> ((open c) /\ ((connected c) /\ (~ (c = EMPTY)))))) /\ (((pairwise DISJOINT) f) /\ ((UNIONS f) = s))) ==> ((components s) = f))))


Ssrbool.andbK |- (!a. (!b. (((a /\ b) \/ a) = a)))


INT_MIN |- (!x. (!y. (((int_min x) y) = (((COND ((int_le x) y)) x) y))))


NORM_CPOW |- (!w. (!z. (((real w) /\ ((real_lt (real_of_num (NUMERAL _0))) (Re w))) ==> ((vector_norm ((cpow w) z)) = ((rpow (vector_norm w)) (Re z))))))


COMPLEX_DIFFSQ |- (!x. (!y. (((complex_mul ((vector_add x) y)) ((vector_sub x) y)) = ((vector_sub ((complex_mul x) x)) ((complex_mul y) y)))))


LIM_NULL_COMPLEX_MUL |- (!net. (!f. (!g. (((((--> f) (Cx (real_of_num (NUMERAL _0)))) net) /\ (((--> g) (Cx (real_of_num (NUMERAL _0)))) net)) ==> (((--> (\x. ((complex_mul (f x)) (g x)))) (Cx (real_of_num (NUMERAL _0)))) net)))))


REAL_NEGLIGIBLE_TRANSLATION_EQ |- (!s. (!a. ((real_negligible ((IMAGE (\x. ((real_add a) x))) s)) = (real_negligible s))))


NSUM_OFFSET_0 |- (!f. (!m. (!n. ((m <= n) ==> (((nsum (m .. n)) f) = ((nsum ((NUMERAL _0) .. (n - m))) (\i. (f (i + m)))))))))


BOUNDED_HYPERPLANE_EQ_TRIVIAL |- (!a. (!b. ((bounded (GSPEC (\GEN%PVAR%1578. (?x. (((SETSPEC GEN%PVAR%1578) (((dot a) x) = b)) x))))) = (((COND (a = (vec (NUMERAL _0)))) (~ (b = (real_of_num (NUMERAL _0))))) ((dimindex UNIV) = (NUMERAL (BIT1 _0)))))))


FRONTIER_INTER_SUBSET_INTER |- (!s. (!t. ((frontier (s INTER t)) SUBSET (((closure s) INTER (frontier t)) UNION ((frontier s) INTER (closure t))))))


INTEGRABLE_SUBINTERVALS_IMP_REAL_MEASURABLE |- (!f. ((!a. (!b. ((real_integrable_on f) (closed_real_interval ((CONS (a , b)) NIL))))) ==> ((real_measurable_on f) UNIV)))


Functional_equation.functional_proj_y3 |- (proj_y3 = (rotate3 proj_y1))


REAL_MEASURABLE_ON_PREIMAGE_HALFSPACE_GE |- (!f. (((real_measurable_on f) UNIV) = (!a. (real_lebesgue_measurable (GSPEC (\GEN%PVAR%5049. (?x. (((SETSPEC GEN%PVAR%5049) ((real_ge (f x)) a)) x))))))))


HAS_REAL_INTEGRAL_EQ_EQ |- (!f. (!g. (!k. (!s. ((!x. ((x IN s) ==> ((f x) = (g x)))) ==> ((((has_real_integral f) k) s) = (((has_real_integral g) k) s)))))))


PATH_IMAGE_SYM |- (!p. (!q. ((((pathfinish p) = (pathstart q)) /\ ((pathfinish q) = (pathstart p))) ==> ((path_image ((++ p) q)) = (path_image ((++ q) p))))))


INTEGRAL_PASTECART_CONTINUOUS |- (!f. (!a. (!b. (!c. (!d. (((continuous_on f) (closed_interval ((CONS (((pastecart a) c) , ((pastecart b) d))) NIL))) ==> (((integral (closed_interval ((CONS (((pastecart a) c) , ((pastecart b) d))) NIL))) f) = ((integral (closed_interval ((CONS (a , b)) NIL))) (\x. ((integral (closed_interval ((CONS (c , d)) NIL))) (\y. (f ((pastecart x) y)))))))))))))


PROPER_MAP_FROM_COMPOSITION_RIGHT |- (!f. (!g. (!s. (!t. (!u. ((((continuous_on f) s) /\ ((((IMAGE f) s) SUBSET t) /\ (((continuous_on g) t) /\ ((((IMAGE g) t) SUBSET u) /\ (!k. (((k SUBSET u) /\ (compact k)) ==> (compact (GSPEC (\GEN%PVAR%740. (?x. (((SETSPEC GEN%PVAR%740) ((x IN s) /\ (((g o f) x) IN k))) x))))))))))) ==> (!k. (((k SUBSET t) /\ (compact k)) ==> (compact (GSPEC (\GEN%PVAR%741. (?x. (((SETSPEC GEN%PVAR%741) ((x IN s) /\ ((f x) IN k))) x)))))))))))))


Localization.wedge_in_fan_ge |- (!E. (!v. (!w. (((wedge_in_fan_ge (v , w)) E) = (((COND ((CARD ((EE v) E)) > (NUMERAL (BIT1 _0)))) ((((wedge_ge (vec (NUMERAL _0))) v) w) ((((azim_cycle ((EE v) E)) (vec (NUMERAL _0))) v) w))) (GSPEC (\GEN%PVAR%7533. (?x. (((SETSPEC GEN%PVAR%7533) T) x)))))))))


LEBESGUE_MEASURABLE_ALMOST_GDELTA |- (!s. ((lebesgue_measurable s) ==> (?c. (?t. ((gdelta c) /\ ((negligible t) /\ (((s UNION t) = c) /\ ((DISJOINT t) s))))))))


AR_INTERVAL |- ((!a. (!b. ((AR (closed_interval ((CONS (a , b)) NIL))) = (~ ((closed_interval ((CONS (a , b)) NIL)) = EMPTY))))) /\ (!a. (!b. ((AR (open_interval (a , b))) = (~ ((open_interval (a , b)) = EMPTY))))))


RELATIVE_INTERIOR_RELATIVE_INTERIOR |- (!s. ((relative_interior (relative_interior s)) = (relative_interior s)))


MEASURABLE_SMALL_IMP_NEGLIGIBLE |- (!s. (((measurable s) /\ (s <_c UNIV)) ==> (negligible s)))


LEBESGUE_DENSITY_THEOREM_INTEGRAL_BALL |- (!f. (!s. ((!a. (!b. ((absolutely_integrable_on f) (s INTER (closed_interval ((CONS (a , b)) NIL)))))) ==> (?k. ((negligible k) /\ (!x. ((~ (x IN k)) ==> (((--> (\e. ((% (real_inv (measure (ball (x , (drop e)))))) ((integral (s INTER (ball (x , (drop e))))) f)))) (((COND (x IN s)) (f x)) (vec (NUMERAL _0)))) ((within (at (vec (NUMERAL _0)))) (GSPEC (\GEN%PVAR%3725. (?t. (((SETSPEC GEN%PVAR%3725) ((real_lt (real_of_num (NUMERAL _0))) (drop t))) t)))))))))))))


CCOS_DOUBLE_CCOS |- (!z. ((ccos ((complex_mul (Cx (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) z)) = ((vector_sub ((complex_mul (Cx (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) ((complex_pow (ccos z)) (NUMERAL (BIT0 (BIT1 _0)))))) (Cx (real_of_num (NUMERAL (BIT1 _0)))))))


LINEAR_FRECHET_DERIVATIVE |- (!f. (!net. (((differentiable f) net) ==> (linear ((frechet_derivative f) net)))))


HOLOMORPHIC_ON_CACS |- (!s. ((!z. (((z IN s) /\ ((Im z) = (real_of_num (NUMERAL _0)))) ==> ((real_lt (real_abs (Re z))) (real_of_num (NUMERAL (BIT1 _0)))))) ==> ((holomorphic_on cacs) s)))


DIFFERENTIABLE_REAL_COMPLEX |- (!f. (!a. (((complex_differentiable f) (at (Cx (drop a)))) ==> ((differentiable (\x. (f (Cx (drop x))))) (at a)))))


COMPLEX_L1_LE_NORM |- (!z. ((real_le ((real_mul ((real_div (sqrt (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) ((real_add (real_abs (Re z))) (real_abs (Im z))))) (vector_norm z)))


Merge_ineq.nonf_sol_euler156_x_div_sqrtdelta |- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((((((sol_euler156_x_div_sqrtdelta x1) x2) x3) x4) x5) x6) = ((LET (\a. (LET_END ((real_div (matan ((real_div ((((((delta_x x1) x2) x3) x4) x5) x6)) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) ((real_pow a) (NUMERAL (BIT0 (BIT1 _0)))))))) a)))) ((real_add (sqrt ((real_mul x6) ((real_mul x1) x5)))) ((real_add ((real_mul (sqrt x6)) ((real_div ((real_add x1) ((real_sub x5) x3))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) ((real_add ((real_mul (sqrt x1)) ((real_div ((real_add x6) ((real_sub x5) x4))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) ((real_mul (sqrt x5)) ((real_div ((real_add x6) ((real_sub x1) x2))) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))))))))))))))


INDEFINITE_INTEGRAL_UNIFORMLY_CONTINUOUS_EXPLICIT |- (!f. (!a. (!b. (!e. ((((integrable_on f) (closed_interval ((CONS (a , b)) NIL))) /\ ((real_lt (real_of_num (NUMERAL _0))) e)) ==> (?k. (((real_lt (real_of_num (NUMERAL _0))) k) /\ (!c. (!d. (!c'. (!d'. (((c IN (closed_interval ((CONS (a , b)) NIL))) /\ ((d IN (closed_interval ((CONS (a , b)) NIL))) /\ ((c' IN (closed_interval ((CONS (a , b)) NIL))) /\ ((d' IN (closed_interval ((CONS (a , b)) NIL))) /\ (((real_le (vector_norm ((vector_sub c') c))) k) /\ ((real_le (vector_norm ((vector_sub d') d))) k)))))) ==> ((real_lt (vector_norm ((vector_sub ((integral (closed_interval ((CONS (c' , d')) NIL))) f)) ((integral (closed_interval ((CONS (c , d)) NIL))) f)))) e)))))))))))))


MULT_DIV_LE |- (!m. (!n. (!p. ((~ (p = (NUMERAL _0))) ==> ((m * (n DIV p)) <= ((m * n) DIV p))))))


ABSOLUTELY_REAL_INTEGRABLE_INF |- (!fs. (!s. (!k. (((FINITE k) /\ ((~ (k = EMPTY)) /\ (!i. ((i IN k) ==> ((absolutely_real_integrable_on (\x. ((fs x) i))) s))))) ==> ((absolutely_real_integrable_on (\x. (inf ((IMAGE (fs x)) k)))) s)))))


REAL_CONVEX_ON_EQ |- (!f. (!g. (!s. (((is_realinterval s) /\ ((!x. ((x IN s) ==> ((f x) = (g x)))) /\ ((real_convex_on f) s))) ==> ((real_convex_on g) s)))))


HAS_BOUNDED_REAL_VARIATION_DARBOUX_STRICT |- (!f. (!a. (!b. (((has_bounded_real_variation_on f) (closed_real_interval ((CONS (a , b)) NIL))) = (?g. (?h. ((!x. (!y. (((x IN (closed_real_interval ((CONS (a , b)) NIL))) /\ ((y IN (closed_real_interval ((CONS (a , b)) NIL))) /\ ((real_lt x) y))) ==> ((real_lt (g x)) (g y))))) /\ ((!x. (!y. (((x IN (closed_real_interval ((CONS (a , b)) NIL))) /\ ((y IN (closed_real_interval ((CONS (a , b)) NIL))) /\ ((real_lt x) y))) ==> ((real_lt (h x)) (h y))))) /\ (!x. ((f x) = ((real_sub (g x)) (h x))))))))))))


ORTHOGONAL_TO_ORTHOGONAL_2D |- (!x. (!y. (!z. (((~ (x = (vec (NUMERAL _0)))) /\ (((orthogonal x) y) /\ ((orthogonal x) z))) ==> (collinear ((vec (NUMERAL _0)) INSERT (y INSERT (z INSERT EMPTY))))))))


PERMUTES_INDUCT |- (!P. (!s. (((FINITE s) /\ ((P I) /\ (!a. (!b. (!p. (((a IN s) /\ ((b IN s) /\ ((P p) /\ (permutation p)))) ==> (P ((swap (a , b)) o p)))))))) ==> (!p. (((permutes p) s) ==> (P p))))))


Nonlinear_lemma.halfbump_x_expand |- (!x. (((real_le (real_of_num (NUMERAL _0))) x) ==> ((halfbump_x x) = ((real_add (real_neg ((real_div (real_of_num (NUMERAL (BIT1 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 _0)))))))))))))))))))))))))) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 _0))))))))))))))))))))))))))) ((real_sub ((real_mul ((real_div (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 _0)))))))))))))))))) (real_of_num (NUMERAL (BIT1 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT1 _0)))))))))))))))))) (sqrt x))) ((real_mul ((real_div (real_of_num (NUMERAL (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))))))))))) (real_of_num (NUMERAL (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 (BIT1 (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 _0))))))))))))))))))))) x))))))


Functional_equation.functional_proj_y1 |- (proj_y1 = (promote1_to_6 sqrt))


MEASURABLE_CLOSED_IN |- (!s. (!t. ((((closed_in ((subtopology euclidean) s)) t) /\ (measurable s)) ==> (measurable t))))


REAL_MEASURABLE_ON_MIN |- (!f. (!g. (!s. ((((real_measurable_on f) s) /\ ((real_measurable_on g) s)) ==> ((real_measurable_on (\x. ((real_min (f x)) (g x)))) s)))))


DIFFERENTIABLE_ON_EQ |- (!f. (!g. (!s. (((!x. ((x IN s) ==> ((f x) = (g x)))) /\ ((differentiable_on f) s)) ==> ((differentiable_on g) s)))))


NORMAL_MATRIX_KERNEL_TRANSP |- (!A. ((((matrix_mul (transp A)) A) = ((matrix_mul A) (transp A))) ==> ((GSPEC (\GEN%PVAR%234. (?x. (((SETSPEC GEN%PVAR%234) (((matrix_vector_mul (transp A)) x) = (vec (NUMERAL _0)))) x)))) = (GSPEC (\GEN%PVAR%235. (?x. (((SETSPEC GEN%PVAR%235) (((matrix_vector_mul A) x) = (vec (NUMERAL _0)))) x)))))))


MATRIX_CMUL_ADD_RDISTRIB |- (!A. (!b. (!c. (((%% ((real_add b) c)) A) = ((matrix_add ((%% b) A)) ((%% c) A))))))


Lp_ineqs_proofs.tau_sum5 |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> ((lp_tau (V , E)) ==> (!f. ((f IN (face_set (hypermap_of_fan (V , E)))) ==> (((CARD f) = (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ==> (((sum f) (rhazim_fan (V , E))) = ((real_add ((tau_fan (V , E)) f)) ((real_mul (real_of_num (NUMERAL (BIT1 (BIT1 _0))))) ((real_add pi) sol0))))))))))))))


REALLIM_NULL_LMUL |- (!net. (!f. (!c. ((((---> f) (real_of_num (NUMERAL _0))) net) ==> (((---> (\x. ((real_mul c) (f x)))) (real_of_num (NUMERAL _0))) net)))))


Lp_gen_theory.ALL_list_dart_pairs_split |- (!L. (!P. (((ALL P) (list_dart_pairs L)) = ((ALL (\d1. ((ALL (\d2. (P (d1 , d2)))) (list_of_darts L)))) (list_of_darts L)))))


REAL_INTEGRABLE_RMUL_EQ |- (!f. (!s. (!c. (((real_integrable_on (\x. ((real_mul (f x)) c))) s) = ((c = (real_of_num (NUMERAL _0))) \/ ((real_integrable_on f) s))))))


AR_BALL |- (!a. (!r. ((AR (ball (a , r))) = ((real_lt (real_of_num (NUMERAL _0))) r))))


Functional_equation.functional_proj_y2 |- (proj_y2 = (rotate2 proj_y1))


Lp_ineqs_proofs.y5_hi_std |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> ((E = (ESTD V)) ==> (!d. ((d IN (dart_of_fan (V , E))) ==> ((real_le ((y5_fan (V , E)) d)) (real_of_num (NUMERAL (BIT1 (BIT1 _0))))))))))))))


ABSOLUTELY_INTEGRABLE_CHANGE_OF_VARIABLES |- (!f. (!g. (!g'. (!s. (((lebesgue_measurable s) /\ ((!x. ((x IN s) ==> (((has_derivative g) (g' x)) ((within (at x)) s)))) /\ (!x. (!y. (((x IN s) /\ ((y IN s) /\ ((g x) = (g y)))) ==> (x = y)))))) ==> (((absolutely_integrable_on f) ((IMAGE g) s)) = ((absolutely_integrable_on (\x. ((% (real_abs (det (matrix (g' x))))) (f (g x))))) s)))))))


UNION_INTERIOR_SUBSET |- (!s. (!t. (((interior s) UNION (interior t)) SUBSET (interior (s UNION t)))))


NEGATIONS_SPHERE |- (!r. (((IMAGE vector_neg) (sphere ((vec (NUMERAL _0)) , r))) = (sphere ((vec (NUMERAL _0)) , r))))


CARD_UNION_OVERLAP |- (!s. (!t. (((FINITE s) /\ ((FINITE t) /\ ((CARD (s UNION t)) < ((CARD s) + (CARD t))))) ==> (~ ((s INTER t) = EMPTY)))))


NEGLIGIBLE_IFF_MEASURABLE_SUBSETS |- (!s. ((negligible s) = (!t. ((t SUBSET s) ==> (measurable t)))))


ARC_JOIN_EQ_ALT |- (!g1. (!g2. (((pathfinish g1) = (pathstart g2)) ==> ((arc ((++ g1) g2)) = ((arc g1) /\ ((arc g2) /\ (((path_image g1) INTER (path_image g2)) = ((pathstart g2) INSERT EMPTY))))))))


Appendix.LENGTH_s_init_list |- ((LENGTH s_init_list_v39) = (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 _0))))))


Lp_ineqs_proofs.sol_sum3 |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!f. ((f IN (face_set (hypermap_of_fan (V , E)))) ==> (((CARD f) = (NUMERAL (BIT1 (BIT1 _0)))) ==> (((sum f) (azim_dart (V , E))) = ((real_add ((sol_fan (V , E)) f)) pi)))))))))))


VSUM_INJECTION |- (!f. (!p. (!s. (((FINITE s) /\ ((!x. ((x IN s) ==> ((p x) IN s))) /\ (!x. (!y. (((x IN s) /\ ((y IN s) /\ ((p x) = (p y)))) ==> (x = y)))))) ==> (((vsum s) (f o p)) = ((vsum s) f))))))


NEIGHBOURHOOD_EXTENSION_INTO_ANR |- (!f. (!s. (!t. ((((continuous_on f) s) /\ ((((IMAGE f) s) SUBSET t) /\ ((ANR t) /\ (closed s)))) ==> (?v. (?g. ((s SUBSET v) /\ ((open v) /\ (((continuous_on g) v) /\ ((((IMAGE g) v) SUBSET t) /\ (!x. ((x IN s) ==> ((g x) = (f x))))))))))))))


BERNOULLI_NUMBER_ALT |- (!n. (((sum ((NUMERAL _0) .. n)) (\k. ((real_mul (real_of_num (binom ((n + (NUMERAL (BIT1 _0))) , k)))) ((bernoulli k) (real_of_num (NUMERAL _0)))))) = (((COND (n = (NUMERAL _0))) (real_of_num (NUMERAL (BIT1 _0)))) (real_of_num (NUMERAL _0)))))


LIM_WITHIN_EMPTY |- (!f. (!l. (!x. (((--> f) l) ((within (at x)) EMPTY)))))


Miqmcsn.J_SCS_4M8_13 |- ((((scs_J_v39 ((scs_prop_equ_v39 scs_4M8_13) i)) i1) j) = F)


STC_IDEMP |- (!R. ((STC (STC R)) = (STC R)))


ROTATE2D_SUB_ARG |- (!w. (!z. (((~ (w = (Cx (real_of_num (NUMERAL _0))))) /\ (~ (z = (Cx (real_of_num (NUMERAL _0)))))) ==> ((rotate2d ((real_sub (Arg w)) (Arg z))) = (rotate2d (Arg ((complex_div w) z)))))))


HOMOTOPIC_COMPOSE |- (!f. (!f'. (!g. (!g'. (!s. (!t. (!u. ((((((homotopic_with (\x. T)) (s , t)) f) f') /\ ((((homotopic_with (\x. T)) (t , u)) g) g')) ==> ((((homotopic_with (\x. T)) (s , u)) (g o f)) (g' o f'))))))))))


CLOSEST_POINT_TRANSLATION |- (!s. (!a. (((convex s) /\ ((closed s) /\ (~ (s = EMPTY)))) ==> (((closest_point ((IMAGE (\x. ((vector_add a) x))) s)) ((vector_add a) x)) = ((vector_add a) ((closest_point s) x))))))


LT_MULT2 |- (!m. (!n. (!p. (!q. (((m < n) /\ (p < q)) ==> ((m * p) < (n * q)))))))


INT_MIN_LE |- (!x. (!y. (!z. (((int_le ((int_min x) y)) z) = (((int_le x) z) \/ ((int_le y) z))))))


COLLINEAR_TRANSLATION |- (!s. (!a. ((collinear s) ==> (collinear ((IMAGE (\x. ((vector_add a) x))) s)))))


Functional_equation.functional_proj_x2 |- (proj_x2 = (rotate2 proj_x1))


TIETZE_CLOSED_INTERVAL_1 |- (!f. (!u. (!s. (!a. (!b. ((((real_le (drop a)) (drop b)) /\ (((closed_in ((subtopology euclidean) u)) s) /\ (((continuous_on f) s) /\ (!x. ((x IN s) ==> ((f x) IN (closed_interval ((CONS (a , b)) NIL)))))))) ==> (?g. (((continuous_on g) u) /\ ((!x. ((x IN s) ==> ((g x) = (f x)))) /\ (!x. ((x IN u) ==> ((g x) IN (closed_interval ((CONS (a , b)) NIL))))))))))))))


Trigonometry.QAFHJNM |- (!v. (!w. (!x. ((LET (\e3. (LET_END ((LET (\r. (LET_END ((LET (\phi. (LET_END (((~ (v = x)) /\ (~ (v = w))) ==> (?u'. ((((dot u') e3) = (real_of_num (NUMERAL _0))) /\ (x = ((vector_add v) ((vector_add u') ((% ((real_mul r) (cos phi))) e3)))))))))) (((arcV v) w) x))))) (vector_norm ((vector_sub x) v)))))) ((% ((real_div (real_of_num (NUMERAL (BIT1 _0)))) (vector_norm ((vector_sub w) v)))) ((vector_sub w) v))))))


HAS_BOUNDED_REAL_VARIATION_AFFINITY_EQ |- (!m. (!c. (!f. (!s. (((has_bounded_real_variation_on (\x. (f ((real_add ((real_mul m) x)) c)))) s) = ((m = (real_of_num (NUMERAL _0))) \/ ((has_bounded_real_variation_on f) ((IMAGE (\x. ((real_add ((real_mul m) x)) c))) s))))))))


Lp_ineqs_proofs.SURJ_pair |- (!h1. (!h2. (!g. ((!x. ((g x) = ((h1 (FST x)) , (h2 (SND x))))) ==> (!s1. (!t1. (!s2. (!t2. (((((SURJ h1) s1) t1) /\ (((SURJ h2) s2) t2)) ==> (((SURJ g) (s1 CROSS s2)) (t1 CROSS t2)))))))))))


CARD_LE_COMPONENTS |- (!f. (!s. (((continuous_on f) s) ==> ((components ((IMAGE f) s)) <=_c (components s)))))


Tame_inequalities.DELTA_X4_MONO_LE_4 |- (!x1. (!x2. (!x3. (!a. (!x5. (!x6. (!b. ((((real_lt a) b) /\ ((real_le (real_of_num (NUMERAL _0))) x1)) ==> ((real_le ((((((delta_x4 x1) x2) x3) b) x5) x6)) ((((((delta_x4 x1) x2) x3) a) x5) x6))))))))))


REAL_DIFFERENTIABLE_WITHIN_EXP |- (!s. (!x. ((real_differentiable exp) ((within (atreal x)) s))))


Seq.eq_filter |- (!a1. (!a2. ((!x. ((a1 x) = (a2 x))) ==> (!s. (((filter a1) s) = ((filter a2) s))))))


SUMMABLE_SUB |- (!x. (!y. (!s. ((((summable s) x) /\ ((summable s) y)) ==> ((summable s) (\n. ((vector_sub (x n)) (y n))))))))


HOMOTOPIC_INVERTIBLE_LINEAR_MAPS |- (!f. (!g. (((((homotopic_with (\h. ((linear h) /\ (invertible (matrix h))))) (UNIV , UNIV)) f) g) = ((linear f) /\ ((linear g) /\ ((real_lt (real_of_num (NUMERAL _0))) ((real_mul (det (matrix f))) (det (matrix g)))))))))


INT_LT_POW2 |- (!n. ((int_lt (int_of_num (NUMERAL _0))) ((int_pow (int_of_num (NUMERAL (BIT0 (BIT1 _0))))) n)))


INT_SGN_POW |- (!x. (!n. ((int_sgn ((int_pow x) n)) = ((int_pow (int_sgn x)) n))))


CONTINUOUS_TRANSFORM_WITHINREAL_SET_IMP |- (!f. (!a. (!s. (!t. ((((eventually (\x. ((x IN t) ==> (x IN s)))) (atreal a)) /\ ((continuous f) ((within (atreal a)) s))) ==> ((continuous f) ((within (atreal a)) t)))))))


CONIC_IMP_SIMPLY_CONNECTED |- (!s. ((conic s) ==> (simply_connected s)))


More_list_hypermap.contravening_dist_dart_le |- (!V. (!g. (!L. ((contravening V) ==> ((good_list L) ==> (((hyp_iso g) ((hypermap_of_list L) , (hypermap_of_fan (V , (ESTD V))))) ==> ((ALL (\d. ((real_le (distance (g d))) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))) (list_of_darts L))))))))


AFFINE_EXPLICIT |- (!s. ((affine s) = (!t. (!u. (((FINITE t) /\ ((t SUBSET s) /\ (((sum t) u) = (real_of_num (NUMERAL (BIT1 _0)))))) ==> (((vsum t) (\x. ((% (u x)) x))) IN s))))))


CONVEX |- (!V. ((convex V) = (!s. (!u. (((FINITE s) /\ ((~ (s = EMPTY)) /\ ((s SUBSET V) /\ ((!x. ((x IN s) ==> ((real_le (real_of_num (NUMERAL _0))) (u x)))) /\ (((sum s) u) = (real_of_num (NUMERAL (BIT1 _0)))))))) ==> (((vsum s) (\x. ((% (u x)) x))) IN V))))))


COMPLEX_NEG_RMUL |- (!x. (!y. ((vector_neg ((complex_mul x) y)) = ((complex_mul x) (vector_neg y)))))


INT_EQ_MUL_LCANCEL |- (!x. (!y. (!z. ((((int_mul x) y) = ((int_mul x) z)) = ((x = (int_of_num (NUMERAL _0))) \/ (y = z))))))


Seq2.butlast_cons |- (!h. (!t. ((butlast ((CONS h) t)) = (((COND (t = NIL)) NIL) ((CONS h) (butlast t))))))


FRONTIER_SPHERE |- (!a. (!r. ((frontier (sphere (a , r))) = (sphere (a , r)))))


SUMMABLE_COMPLEX_RMUL |- (!f. (!c. (!s. (((summable s) f) ==> ((summable s) (\x. ((complex_mul (f x)) c)))))))


HOMOTOPIC_PATHS_RCANCEL_EQ |- (!p. (!q. (!r. (!s. ((((pathfinish p) = (pathstart r)) /\ ((pathfinish q) = (pathstart r))) ==> ((((homotopic_paths s) ((++ p) r)) ((++ q) r)) = ((((homotopic_paths s) p) q) /\ ((path r) /\ ((path_image r) SUBSET s)))))))))


DEFORMATION_RETRACT_OF_CONTRACTIBLE_SING |- (!s. (!a. (((contractible s) /\ (a IN s)) ==> (?r. (((((homotopic_with (\x. T)) (s , s)) (\x. x)) r) /\ ((retraction (s , (a INSERT EMPTY))) r))))))


IS_INTERVAL_SIMPLY_CONNECTED_1 |- (!s. ((is_interval s) = (simply_connected s)))


REGULAR_OPEN_INTER |- (!s. (!t. ((((interior (closure s)) = s) /\ ((interior (closure t)) = t)) ==> ((interior (closure (s INTER t))) = (s INTER t)))))


HAS_INTEGRAL_LIM_SEQUENTIALLY |- (!f. (!l. (((((--> (f o lift)) (vec (NUMERAL _0))) at_posinfinity) /\ ((!n. ((integrable_on f) (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec n))) NIL)))) /\ (((--> (\n. ((integral (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec n))) NIL))) f))) l) sequentially))) ==> (((has_integral f) l) (GSPEC (\GEN%PVAR%2954. (?t. (((SETSPEC GEN%PVAR%2954) ((real_le (real_of_num (NUMERAL _0))) (drop t))) t))))))))


INT_LE_NEGR |- (!x. (((int_le x) (int_neg x)) = ((int_le x) (int_of_num (NUMERAL _0)))))


Lp_ineqs_proofs.rho_lo |- (!E. (!V. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!v. ((v IN V) ==> ((real_le (real_of_num (NUMERAL (BIT1 _0)))) (rho_fan v))))))))))


VITALI_COVERING_LEMMA_CBALLS |- (!s. (!a. (!r. (!k. (!B. (((s SUBSET (UNIONS ((IMAGE (\i. (cball ((a i) , (r i))))) k))) /\ (!i. ((i IN k) ==> (((real_lt (real_of_num (NUMERAL _0))) (r i)) /\ ((real_le (r i)) B))))) ==> (?c. ((COUNTABLE c) /\ ((c SUBSET k) /\ (((pairwise (\i. (\j. ((DISJOINT (cball ((a i) , (r i)))) (cball ((a j) , (r j))))))) c) /\ (s SUBSET (UNIONS ((IMAGE (\i. (cball ((a i) , ((real_mul (real_of_num (NUMERAL (BIT1 (BIT0 (BIT1 _0)))))) (r i)))))) c)))))))))))))


VECTOR_EXPAND_1 |- (!x. (x = (vector ((CONS (x $ (NUMERAL (BIT1 _0)))) NIL))))


STEINHAUS_DIFFS |- (!s. (!t. (((lebesgue_measurable s) /\ ((~ (negligible s)) /\ ((lebesgue_measurable t) /\ (~ (negligible t))))) ==> (~ ((interior (GSPEC (\GEN%PVAR%4608. (?x. (?y. (((SETSPEC GEN%PVAR%4608) ((x IN s) /\ (y IN t))) ((vector_sub x) y))))))) = EMPTY)))))


INTEGRABLE_UNION_EQ |- (!f. (!s. (!t. ((((integrable_on f) s) /\ ((integrable_on f) t)) ==> (((integrable_on f) (s UNION t)) = ((integrable_on f) (s INTER t)))))))


SING_STRONG_DEFORMATION_RETRACT_OF_AR |- (!s. (!a. (((AR s) /\ (a IN s)) ==> (?r. (((((homotopic_with (\h. ((h a) = a))) (s , s)) (\x. x)) r) /\ ((retraction (s , (a INSERT EMPTY))) r))))))


INT_SUB_ADD2 |- (!x. (!y. (((int_add y) ((int_sub x) y)) = x)))


DIFFERENTIABLE_ON_EMPTY |- (!f. ((differentiable_on f) EMPTY))


REAL_MEASURE_DISJOINT_UNIONS |- (!m. (!f. (((FINITE f) /\ ((!s. ((s IN f) ==> ((has_real_measure s) (m s)))) /\ (!s. (!t. (((s IN f) /\ ((t IN f) /\ (~ (s = t)))) ==> ((DISJOINT s) t)))))) ==> ((real_measure (UNIONS f)) = ((sum f) m)))))


IN_BALL_RE |- (!x. (!z. (!e. ((x IN (ball (z , e))) ==> ((real_lt (real_abs ((real_sub (Re x)) (Re z)))) e)))))


Tame_list.rev2 |- (!x. (!y. ((rev ((CONS x) ((CONS y) NIL))) = ((CONS y) ((CONS x) NIL)))))


INT_SGN_INEQS |- ((!x. (((int_le (int_of_num (NUMERAL _0))) (int_sgn x)) = ((int_le (int_of_num (NUMERAL _0))) x))) /\ ((!x. (((int_lt (int_of_num (NUMERAL _0))) (int_sgn x)) = ((int_lt (int_of_num (NUMERAL _0))) x))) /\ ((!x. (((int_ge (int_of_num (NUMERAL _0))) (int_sgn x)) = ((int_ge (int_of_num (NUMERAL _0))) x))) /\ ((!x. (((int_gt (int_of_num (NUMERAL _0))) (int_sgn x)) = ((int_gt (int_of_num (NUMERAL _0))) x))) /\ ((!x. (((int_of_num (NUMERAL _0)) = (int_sgn x)) = ((int_of_num (NUMERAL _0)) = x))) /\ ((!x. (((int_le (int_sgn x)) (int_of_num (NUMERAL _0))) = ((int_le x) (int_of_num (NUMERAL _0))))) /\ ((!x. (((int_lt (int_sgn x)) (int_of_num (NUMERAL _0))) = ((int_lt x) (int_of_num (NUMERAL _0))))) /\ ((!x. (((int_ge (int_sgn x)) (int_of_num (NUMERAL _0))) = ((int_ge x) (int_of_num (NUMERAL _0))))) /\ ((!x. (((int_gt (int_sgn x)) (int_of_num (NUMERAL _0))) = ((int_gt x) (int_of_num (NUMERAL _0))))) /\ (!x. (((int_sgn x) = (int_of_num (NUMERAL _0))) = (x = (int_of_num (NUMERAL _0))))))))))))))


CONVEX_CONE_HULL_NONEMPTY |- (!s. (~ (((hull convex_cone) s) = EMPTY)))


INT_LT_ADD1 |- (!x. (!y. (((int_le x) y) ==> ((int_lt x) ((int_add y) (int_of_num (NUMERAL (BIT1 _0))))))))


Seq.subseq_uniqP |- (!s1. (!s2. ((uniq s2) ==> (((subseq s1) s2) = (s1 = ((filter (\x. ((MEM x) s1))) s2))))))


Lp_ineqs_proofs.azim3c |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!d. ((d IN (dart_of_fan (V , E))) ==> (((azim3_fan (V , E)) d) = ((azim_dart (V , E)) ((inverse (f_fan_pair_ext (V , E))) d)))))))))))


Tactics_jordan.switch_lemma_le |- (!A. (!B. (!C. ((((real_le (real_of_num (NUMERAL _0))) A) ==> (((real_le (real_of_num (NUMERAL _0))) B) ==> C)) = (((real_le (real_of_num (NUMERAL _0))) B) ==> (((real_le (real_of_num (NUMERAL _0))) A) ==> C))))))


HOLOMORPHIC_ON_DIV |- (!f. (!g. (!s. ((((holomorphic_on f) s) /\ (((holomorphic_on g) s) /\ (!z. ((z IN s) ==> (~ ((g z) = (Cx (real_of_num (NUMERAL _0))))))))) ==> ((holomorphic_on (\z. ((complex_div (f z)) (g z)))) s)))))


ABSOLUTE_RETRACT_FROM_UNION_AND_INTER |- (!s. (!t. ((((retract_of (s UNION t)) UNIV) /\ (((retract_of (s INTER t)) UNIV) /\ ((closed s) /\ (closed t)))) ==> ((retract_of s) UNIV))))


INTEGRABLE_SPIKE_EQ |- (!f. (!g. (!s. (!t. (((negligible s) /\ (!x. ((x IN (t DIFF s)) ==> ((g x) = (f x))))) ==> (((integrable_on f) t) = ((integrable_on g) t)))))))


CSQRT_EQ_0 |- (!z. (((csqrt z) = (Cx (real_of_num (NUMERAL _0)))) = (z = (Cx (real_of_num (NUMERAL _0))))))


LIM_COMPLEX_REAL_0 |- (!f. (!g. ((((eventually (\n. ((Re (g n)) = (f n)))) sequentially) /\ (((--> g) (Cx (real_of_num (NUMERAL _0)))) sequentially)) ==> (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?N. (!n. ((N <= n) ==> ((real_lt (real_abs (f n))) e)))))))))


CONTINUOUS_ON_CX_NORM |- (!s. ((continuous_on (\z. (Cx (vector_norm z)))) s))


Add_triangle.ext_in_dart1 |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!d. ((d IN (dart1_of_fan (V , E))) ==> (((ext_dart (V , E)) d) IN (d1_fan ((vec (NUMERAL _0)) , (V , E)))))))))


INFSUM_0 |- (((infsum s) (\i. (vec (NUMERAL _0)))) = (vec (NUMERAL _0)))


Local_lemmas1.CON_ATREAL_REAL_CON2 |- (!f. ((((continuous f) (atreal t)) /\ ((continuous g) (atreal t))) ==> ((real_continuous (\t. (distance ((f t) , (g t))))) (atreal t))))


Seq2.foldr |- (!x. (!s. (!f. (!z0. (((((foldr f) z0) ((CONS x) s)) = ((f x) (((foldr f) z0) s))) /\ ((((foldr f) z0) NIL) = z0))))))


REAL_INTEGRABLE_SPIKE_FINITE |- (!f. (!g. (!s. (((FINITE s) /\ (!x. ((x IN (t DIFF s)) ==> ((g x) = (f x))))) ==> (((real_integrable_on f) t) ==> ((real_integrable_on g) t))))))


NSUM_UNIONS_NONZERO |- (!f. (!s. (((FINITE s) /\ ((!t. ((t IN s) ==> (FINITE t))) /\ (!t1. (!t2. (!x. (((t1 IN s) /\ ((t2 IN s) /\ ((~ (t1 = t2)) /\ ((x IN t1) /\ (x IN t2))))) ==> ((f x) = (NUMERAL _0)))))))) ==> (((nsum (UNIONS s)) f) = ((nsum s) (\t. ((nsum t) f)))))))


Localization.order |- (!f. (!x. (!y. ((((order f) x) y) = (@n. (((((ITER n) f) x) = y) /\ (!i. ((((NUMERAL _0) < i) /\ (i < n)) ==> (~ ((((ITER i) f) x) = y))))))))))


COMPLEX_DIFFERENTIABLE_WITHIN_CLOG |- (!s. (!z. ((((Im z) = (real_of_num (NUMERAL _0))) ==> ((real_lt (real_of_num (NUMERAL _0))) (Re z))) ==> ((complex_differentiable clog) ((within (at z)) s)))))


INTEGRAL_PASTECART_CONST |- (!a. (!b. (!c. (!d. (!k. (((integral (closed_interval ((CONS (((pastecart a) c) , ((pastecart b) d))) NIL))) (\x. k)) = ((integral (closed_interval ((CONS (a , b)) NIL))) (\x. ((integral (closed_interval ((CONS (c , d)) NIL))) (\y. k))))))))))


FSIGMA_PCROSS_EQ |- (!s. (!t. ((fsigma (s PCROSS t)) = ((s = EMPTY) \/ ((t = EMPTY) \/ ((fsigma s) /\ (fsigma t)))))))


FSIGMA_LINEAR_IMAGE |- (!f. (!s. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((fsigma ((IMAGE f) s)) = (fsigma s)))))


INTEGER_SUB_EQ |- ((!x. (!y. ((integer x) ==> ((integer ((real_sub x) y)) = (integer y))))) /\ (!x. (!y. ((integer y) ==> ((integer ((real_sub x) y)) = (integer x))))))


Topology.union_aff |- (!x. (!v. (UNIV = ((aff (x INSERT (v INSERT EMPTY))) UNION (complement_set (x INSERT (v INSERT EMPTY)))))))


ANR_IMP_CLOSED_NEIGHBOURHOOD_RETRACT |- (!s. (!u. (((ANR s) /\ ((closed_in ((subtopology euclidean) u)) s)) ==> (?v. (?w. (((open_in ((subtopology euclidean) u)) v) /\ (((closed_in ((subtopology euclidean) u)) w) /\ ((s SUBSET v) /\ ((v SUBSET w) /\ ((retract_of s) w))))))))))


DIVMOD_ELIM_THM' |- (((P (m DIV n)) (m MOD n)) = (?q. (?r. ((((n = (NUMERAL _0)) /\ ((q = (NUMERAL _0)) /\ (r = m))) \/ ((m = ((q * n) + r)) /\ (r < n))) /\ ((P q) r)))))


SUMS_COMPLEX_0 |- (!f. (!s. ((!n. ((n IN s) ==> ((f n) = (Cx (real_of_num (NUMERAL _0)))))) ==> (((sums f) (Cx (real_of_num (NUMERAL _0)))) s))))


MBOUNDED_SUBSET_MSPACE |- (!m. (!s. (((mbounded m) s) ==> (s SUBSET (mspace m)))))


INT_ADD_LDISTRIB |- (!x. (!y. (!z. (((int_mul x) ((int_add y) z)) = ((int_add ((int_mul x) y)) ((int_mul x) z))))))


TC_TC_RTC_CASES |- (!R. (!x. (!z. ((((TC R) x) z) = (?y. ((((TC R) x) y) /\ (((RTC R) y) z)))))))


LOCALLY_COMPACT_IMP_BOREL |- (!s. (((locally compact) s) ==> (borel s)))


LIM_CONG_WITHINREAL |- ((!x. ((~ (x = a)) ==> ((f x) = (g x)))) ==> ((((--> (\x. (f x))) l) ((within (atreal a)) s)) = (((--> g) l) ((within (atreal a)) s))))


NOT_PSUBSET_EMPTY |- (!s. (~ (s PSUBSET EMPTY)))


LOG_LE |- (!x. (((real_le (real_of_num (NUMERAL _0))) x) ==> ((real_le (log ((real_add (real_of_num (NUMERAL (BIT1 _0)))) x))) x)))


CONVERGENT_REAL_BOUNDED_MONOTONE |- (!s. (((real_bounded ((IMAGE s) UNIV)) /\ ((!n. ((real_le (s n)) (s (SUC n)))) \/ (!n. ((real_le (s (SUC n))) (s n))))) ==> (?l. (((---> s) l) sequentially))))


CENTRE_IN_MCBALL_EQ |- (!m. (!x. (!r. ((x IN ((mcball m) (x , r))) = ((x IN (mspace m)) /\ ((real_le (real_of_num (NUMERAL _0))) r))))))


SETDIST_FRONTIERS |- (!s. (!t. ((setdist (s , t)) = (((COND ((DISJOINT s) t)) (setdist ((frontier s) , (frontier t)))) (real_of_num (NUMERAL _0))))))


Nonlinear_lemma.sqrt_x1 |- (((((((sqrt_x1 x1) x2) x3) x4) x5) x6) = (sqrt x1))


Ssrnat.odd_double |- (!n. ((odd (double n)) = F))


Seq.nth_incr_nth |- (!v. (!i. (!j. ((((nth (NUMERAL _0)) ((incr_nth v) i)) j) = ((((COND (i = j)) (NUMERAL (BIT1 _0))) (NUMERAL _0)) + (((nth (NUMERAL _0)) v) j))))))


Lp_main_estimate.ineq_tau6 |- (!V. (!E. ((lp_fan (V , E)) ==> ((!V. (!E. (!FF. (((convex_local_fan (V , (E , FF))) /\ ((packing V) /\ ((V SUBSET ball_annulus) /\ (((NUMERAL (BIT0 (BIT0 (BIT1 _0)))) <= (CARD V)) /\ (((CARD V) <= (NUMERAL (BIT0 (BIT1 (BIT1 _0))))) /\ ((!v. (!w. (((~ (v = w)) /\ ((v IN V) /\ ((w IN V) /\ (~ ((v INSERT (w INSERT EMPTY)) IN E))))) ==> ((real_le ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) (distance (v , w)))))) /\ (!v. (!w. (((v INSERT (w INSERT EMPTY)) IN E) ==> (((real_le (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (distance (v , w))) /\ ((real_le (distance (v , w))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))))))))))) ==> ((real_le (d_tame (CARD V))) (((tau_fun V) E) FF)))))) ==> (!d. ((d IN ((darts_k (NUMERAL (BIT0 (BIT1 (BIT1 _0))))) (hypermap_of_fan (V , E)))) ==> ((((real_le ((y5_fan (V , E)) d)) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) /\ (((real_le (y6_fan d)) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) /\ (((real_le ((y8_fan (V , E)) d)) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) /\ (((real_le ((y9_fan (V , E)) d)) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) /\ (((real_le (ye_fan (((POWER (f_fan_pair_ext (V , E))) (NUMERAL (BIT0 (BIT1 _0)))) d))) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) /\ ((real_le (ye_fan (((POWER (f_fan_pair_ext (V , E))) (NUMERAL (BIT1 (BIT1 _0)))) d))) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))))))))) ==> ((real_le ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 _0)))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))) ((tau_fan (V , E)) ((face (hypermap_of_fan (V , E))) d))))))))))


INT_POW_LT |- (!x. (!n. (((int_lt (int_of_num (NUMERAL _0))) x) ==> ((int_lt (int_of_num (NUMERAL _0))) ((int_pow x) n)))))


NPRODUCT_CONST_NUMSEG_1 |- (!c. (!n. (((nproduct ((NUMERAL (BIT1 _0)) .. n)) (\x. c)) = (c EXP n))))


COMPLEX_NEG_LMUL |- (!x. (!y. ((vector_neg ((complex_mul x) y)) = ((complex_mul (vector_neg x)) y))))


COMPLEX_DIFFERENTIABLE_INV_AT |- (!f. (!z. ((((complex_differentiable f) (at z)) /\ (~ ((f z) = (Cx (real_of_num (NUMERAL _0)))))) ==> ((complex_differentiable (\z. (complex_inv (f z)))) (at z)))))


REAL_VARIATION_TRIANGLE |- (!f. (!g. (!s. ((((has_bounded_real_variation_on f) s) /\ ((has_bounded_real_variation_on g) s)) ==> ((real_le ((real_variation s) (\x. ((real_add (f x)) (g x))))) ((real_add ((real_variation s) f)) ((real_variation s) g)))))))


Packing3.VORONOI_SET_2_BIS_LE |- (!V. (!u. (!v. (((u IN V) /\ (v IN V)) ==> (((voronoi_set V) (u INSERT (v INSERT EMPTY))) = (((voronoi_closed V) v) INTER ((bis_le u) v)))))))


INSIDE_INSIDE_COMPACT_CONNECTED |- (!s. (!t. (((closed s) /\ ((compact t) /\ ((s SUBSET (inside t)) /\ (connected t)))) ==> ((inside s) SUBSET (inside t)))))


Lp_ineqs_proofs.rhazim_fan_eq_rhazim |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!d. ((d IN (dart_of_fan (V , E))) ==> (((rhazim_fan (V , E)) d) = ((((((rhazim (y1_fan d)) (y2_fan d)) ((y3_fan (V , E)) d)) ((y4'_fan (V , E)) d)) ((y5_fan (V , E)) d)) (y6_fan d)))))))))))


Ssrnat.neq_ltn |- (!m. (!n. ((~ (m = n)) = ((m < n) \/ (n < m)))))


INT_MUL_ASSOC |- (!x. (!y. (!z. (((int_mul x) ((int_mul y) z)) = ((int_mul ((int_mul x) y)) z)))))


Ssrbool.orKb |- (!a. (!b. ((a /\ (b \/ a)) = a)))


ATN_LE_X |- (!x. (((real_le (real_of_num (NUMERAL _0))) x) ==> ((real_le (atn x)) x)))


FORALL_ALL |- (!P. (!l. ((!x. ((ALL (P x)) l)) = ((ALL (\s. (!x. ((P x) s)))) l))))


REAL_SUMMABLE_ZETA |- (!n. (!x. (((real_lt (real_of_num (NUMERAL (BIT1 _0)))) x) ==> ((real_summable (from n)) (\k. (real_inv ((rpow (real_of_num k)) x)))))))


Seq2.list_of_empty_set |- ((list_of_set EMPTY) = NIL)


CONVEX_STARCENTRES |- (!s. (convex (GSPEC (\GEN%PVAR%1721. (?a. (((SETSPEC GEN%PVAR%1721) ((a IN s) /\ (!x. ((x IN s) ==> ((closed_segment ((CONS (a , x)) NIL)) SUBSET s))))) a))))))


RE_POS_SEGMENT |- (!a. (!b. (!x. (((x IN (closed_segment ((CONS (a , b)) NIL))) /\ (((real_lt (real_of_num (NUMERAL _0))) (Re a)) /\ ((real_lt (real_of_num (NUMERAL _0))) (Re b)))) ==> ((real_lt (real_of_num (NUMERAL _0))) (Re x))))))


Merge_ineq.nonf_flat_term2_234_x |- (!d. (!a. (!b. (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. ((((((((((flat_term2_234_x d) a) b) x1) x2) x3) x4) x5) x6) = (flat_term_x (((((((((edge2_234_x d) a) b) x1) x2) x3) x4) x5) x6))))))))))))


SUM_MULTICOUNT |- (!R. (!s. (!t. (!k. (((FINITE s) /\ ((FINITE t) /\ (!j. ((j IN t) ==> ((CARD (GSPEC (\GEN%PVAR%205. (?i. (((SETSPEC GEN%PVAR%205) ((i IN s) /\ ((R i) j))) i))))) = k))))) ==> (((sum s) (\i. (real_of_num (CARD (GSPEC (\GEN%PVAR%206. (?j. (((SETSPEC GEN%PVAR%206) ((j IN t) /\ ((R i) j))) j)))))))) = (real_of_num (k * (CARD t)))))))))


ANALYTIC_ON_COMPOSE_GEN |- (!f. (!g. (!s. (!t. ((((analytic_on f) s) /\ (((analytic_on g) t) /\ (!z. ((z IN s) ==> ((f z) IN t))))) ==> ((analytic_on (g o f)) s))))))


Merge_ineq.ineq_mp_simple |- (!a. (!b. (!c. (((ineq a) (b ==> c)) = (b ==> ((ineq a) c))))))


MEASURABLE_ON_PREIMAGE_HALFSPACE_COMPONENT_GT_DENSE |- (!f. (!r. (((closure ((IMAGE lift) r)) = UNIV) ==> (((measurable_on f) UNIV) = (!a. (!k. ((((NUMERAL (BIT1 _0)) <= k) /\ ((k <= (dimindex UNIV)) /\ (a IN r))) ==> (lebesgue_measurable (GSPEC (\GEN%PVAR%3601. (?x. (((SETSPEC GEN%PVAR%3601) ((real_gt ((f x) $ k)) a)) x))))))))))))


CROSS_INTER |- ((!s. (!t. (!u. ((s CROSS (t INTER u)) = ((s CROSS t) INTER (s CROSS u)))))) /\ (!s. (!t. (!u. (((s INTER t) CROSS u) = ((s CROSS u) INTER (t CROSS u)))))))


REAL_POW |- (!z. (!n. ((real z) ==> (real ((complex_pow z) n)))))


BOUNDED_SPHERE |- (!a. (!r. (bounded (sphere (a , r)))))


NSUM_IMAGE_NONZERO |- (!d. (!i. (!s. (((FINITE s) /\ (!x. (!y. (((x IN s) /\ ((y IN s) /\ ((~ (x = y)) /\ ((i x) = (i y))))) ==> ((d (i x)) = (NUMERAL _0)))))) ==> (((nsum ((IMAGE i) s)) d) = ((nsum s) (d o i)))))))


STC_REFL |- (!R. ((!x. ((R x) x)) ==> (!x. (((STC R) x) x))))


HAS_DERIVATIVE_VSUM_NUMSEG |- (!f. (!net. (!m. (!n. ((!i. (((m <= i) /\ (i <= n)) ==> (((has_derivative (f i)) (f' i)) net))) ==> (((has_derivative (\x. ((vsum (m .. n)) (\i. ((f i) x))))) (\h. ((vsum (m .. n)) (\i. ((f' i) h))))) net))))))


EXTENSION_INTO_AR |- (!f. (!s. (!t. ((((continuous_on f) s) /\ ((((IMAGE f) s) SUBSET t) /\ (((retract_of t) UNIV) /\ (closed s)))) ==> (?g. (((continuous_on g) UNIV) /\ ((((IMAGE g) UNIV) SUBSET t) /\ (!x. ((x IN s) ==> ((g x) = (f x)))))))))))


HULL_UNIONS_SUBSET |- (!P. (!f. ((UNIONS (GSPEC (\GEN%PVAR%96. (?s. (((SETSPEC GEN%PVAR%96) (s IN f)) ((hull P) s)))))) SUBSET ((hull P) (UNIONS f)))))


CONTINUOUS_WITHIN_CTAN |- (!s. (!z. ((~ ((ccos z) = (Cx (real_of_num (NUMERAL _0))))) ==> ((continuous ctan) ((within (at z)) s)))))


Upfzbzm_support_lemmas.FINITE_PERMUTE_3 |- (FINITE (GSPEC (\GEN%PVAR%6211. (?p. (((SETSPEC GEN%PVAR%6211) ((permutes p) ((NUMERAL _0) INSERT ((NUMERAL (BIT1 _0)) INSERT ((NUMERAL (BIT0 (BIT1 _0))) INSERT EMPTY))))) p)))))


INTEGRAL_COMBINE_TAGGED_DIVISION_BOTTOMUP |- (!f. (!p. (!a. (!b. ((((tagged_division_of p) (closed_interval ((CONS (a , b)) NIL))) /\ (!x. (!k. (((x , k) IN p) ==> ((integrable_on f) k))))) ==> (((integral (closed_interval ((CONS (a , b)) NIL))) f) = ((vsum p) (GABS (\f. (!x. (!k. ((GEQ (f (x , k))) ((integral k) f)))))))))))))


INT_LE_LT |- (!x. (!y. (((int_le x) y) = (((int_lt x) y) \/ (x = y)))))


Nonlinear_lemma.sqrt_x6 |- (((((((sqrt_x6 x1) x2) x3) x4) x5) x6) = (sqrt x6))


AFFINE_DEPENDENT_BIGGERSET_GENERAL |- (!s. (((FINITE s) ==> ((CARD s) >= ((dim s) + (NUMERAL (BIT0 (BIT1 _0)))))) ==> (affine_dependent s)))


Lp_gen_theory.components_iso_image_face_map |- (!H. (!g. (!G. (((hyp_iso g) (G , H)) ==> (!d. ((d IN (dart G)) ==> ((((node H) (g ((face_map G) d))) = ((IMAGE g) ((node G) ((face_map G) d)))) /\ ((((face H) (g ((face_map G) d))) = ((IMAGE g) ((face G) ((face_map G) d)))) /\ ((((node H) (g ((inverse (face_map G)) d))) = ((IMAGE g) ((node G) ((inverse (face_map G)) d)))) /\ (((face H) (g ((inverse (face_map G)) d))) = ((IMAGE g) ((face G) ((inverse (face_map G)) d)))))))))))))


IM_COMPLEX_INV_GT_0 |- (!z. (((real_lt (real_of_num (NUMERAL _0))) (Im (complex_inv z))) = ((real_lt (Im z)) (real_of_num (NUMERAL _0)))))


List_hypermap.flatten_filter_empty |- (!L. ((flatten ((filter (\l. (~ (l = NIL)))) L)) = (flatten L)))


Collect_geom.IN_SET4 |- ((x IN (a INSERT (b INSERT (c INSERT (d INSERT EMPTY))))) = ((x = a) \/ ((x = b) \/ ((x = c) \/ (x = d)))))


CFUNSPACE_MDIST_LT |- (!top. (!m. (!f. (!g. (!a. (!x. ((((compact_in (topspace top)) top) /\ ((f IN (mspace ((cfunspace top) m))) /\ ((g IN (mspace ((cfunspace top) m))) /\ (((real_lt ((mdist ((cfunspace top) m)) (f , g))) a) /\ (x IN (topspace top)))))) ==> ((real_lt ((mdist m) ((f x) , (g x)))) a))))))))


COMPLEX_SUB_TRIANGLE |- (!a. (!b. (!c. (((vector_add ((vector_sub a) b)) ((vector_sub b) c)) = ((vector_sub a) c)))))


REAL_SECOND_MEAN_VALUE_THEOREM_GEN |- (!f. (!g. (!a. (!b. (!u. (!v. (((~ ((closed_real_interval ((CONS (a , b)) NIL)) = EMPTY)) /\ (((real_integrable_on f) (closed_real_interval ((CONS (a , b)) NIL))) /\ ((!x. ((x IN (open_real_interval (a , b))) ==> (((real_le u) (g x)) /\ ((real_le (g x)) v)))) /\ (!x. (!y. (((x IN (closed_real_interval ((CONS (a , b)) NIL))) /\ ((y IN (closed_real_interval ((CONS (a , b)) NIL))) /\ ((real_le x) y))) ==> ((real_le (g x)) (g y)))))))) ==> (?c. ((c IN (closed_real_interval ((CONS (a , b)) NIL))) /\ (((real_integral (closed_real_interval ((CONS (a , b)) NIL))) (\x. ((real_mul (g x)) (f x)))) = ((real_add ((real_mul u) ((real_integral (closed_real_interval ((CONS (a , c)) NIL))) f))) ((real_mul v) ((real_integral (closed_real_interval ((CONS (c , b)) NIL))) f)))))))))))))


REALLIM_AT_NEGINFINITY |- (!f. (!l. ((((---> f) l) at_neginfinity) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?b. (!x. (((real_le x) b) ==> ((real_lt (real_abs ((real_sub (f x)) l))) e)))))))))


REAL_NEGLIGIBLE_ON_INTERVALS |- (!s. ((real_negligible s) = (!a. (!b. (real_negligible (s INTER (closed_real_interval ((CONS (a , b)) NIL))))))))


REALLIM_WITHIN_OPEN |- (!f. (!l. (!a. (!s. (((a IN s) /\ (open s)) ==> ((((---> f) l) ((within (at a)) s)) = (((---> f) l) (at a))))))))


NO_RETRACTION_FRONTIER_BOUNDED |- (!s. (((bounded s) /\ (~ ((interior s) = EMPTY))) ==> (~ ((retract_of (frontier s)) s))))


Upfzbzm_support_lemmas.FINITE_PERMUTE_4 |- (FINITE (GSPEC (\GEN%PVAR%6212. (?p. (((SETSPEC GEN%PVAR%6212) ((permutes p) ((NUMERAL _0) INSERT ((NUMERAL (BIT1 _0)) INSERT ((NUMERAL (BIT0 (BIT1 _0))) INSERT ((NUMERAL (BIT1 (BIT1 _0))) INSERT EMPTY)))))) p)))))


SUMMABLE_SUBSET_ABSCONV |- (!x. (!s. (!t. ((((summable s) (\n. (lift (vector_norm (x n))))) /\ (t SUBSET s)) ==> ((summable t) (\n. (lift (vector_norm (x n)))))))))


BOHL_ALT |- (!f. (!s. (!a. ((((continuous_on f) s) /\ ((convex s) /\ ((compact s) /\ ((a IN (interior s)) /\ (((IMAGE f) s) SUBSET (UNIV DELETE a)))))) ==> (?x. ((x IN (frontier s)) /\ (a IN (open_segment (x , (f x))))))))))


Ssrbool.negbFE |- (!b. (((~ b) = F) ==> b))


Nonlinear_lemma.rho_sqrtx |- (!x. ((rho (sqrt x)) = ((real_add (real_of_num (NUMERAL (BIT1 _0)))) ((real_mul const1) ((real_div ((real_sub (sqrt x)) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 _0)))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))))))))


OPEN_OPEN_LEFT_PROJECTION |- (!s. (!t. (((open s) /\ (open t)) ==> (open (GSPEC (\GEN%PVAR%1073. (?x. (((SETSPEC GEN%PVAR%1073) ((x IN s) /\ (?y. (((pastecart x) y) IN t)))) x))))))))


Lp_ineqs_proofs.y8_def |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!d. ((d IN (dart_of_fan (V , E))) ==> (((y8_fan (V , E)) d) = ((y5_fan (V , E)) ((inverse (f_fan_pair_ext (V , E))) d)))))))))))


HOLOMORPHIC_ON_CTAN |- (!s. ((!z. ((z IN s) ==> (~ ((ccos z) = (Cx (real_of_num (NUMERAL _0))))))) ==> ((holomorphic_on ctan) s)))