HAS_REAL_INTEGRAL_NEGLIGIBLE

|- (!f. (!s. (!t. (((real_negligible s) /\ (!x. ((x IN (t DIFF s)) ==> ((f x) = (real_of_num (NUMERAL _0)))))) ==> (((has_real_integral f) (real_of_num (NUMERAL _0))) t)))))


sum

|- (sum = (iterate real_add))


MEASURE_COUNTABLE_UNIONS_LE

|- (!d. (!B. (((!n. (measurable (d n))) /\ (!n. ((real_le ((sum ((NUMERAL _0) .. n)) (\k. (measure (d k))))) B))) ==> ((measurable (UNIONS (GSPEC (\GEN%PVAR%3383. (?n. (((SETSPEC GEN%PVAR%3383) (n IN UNIV)) (d n))))))) /\ ((real_le (measure (UNIONS (GSPEC (\GEN%PVAR%3384. (?n. (((SETSPEC GEN%PVAR%3384) (n IN UNIV)) (d n)))))))) B)))))


IN_CBALL

|- (!x. (!y. (!e. ((y IN (cball (x , e))) = ((real_le (distance (x , y))) e)))))


Seq.cat0s

|- (!s. (((cat NIL) s) = s))


Tame_defs.tame_8

|- (!H. ((tame_8 H) = ((number_of_faces H) >= (NUMERAL (BIT1 (BIT1 _0))))))


MEASURABLE_LEBESGUE_MEASURABLE_SUBSET

|- (!s. (!t. (((lebesgue_measurable s) /\ ((measurable t) /\ (s SUBSET t))) ==> (measurable s))))


Hypermap_iso.finite_surj_eq_bij

|- (!s. (!t. ((FINITE s) ==> ((FINITE t) ==> (((CARD s) = (CARD t)) ==> (!f. ((((SURJ f) s) t) = (((BIJ f) s) t))))))))


INTEGRABLE_COMPLEX_LMUL_EQ

|- (!f. (!s. (!c. (((integrable_on (\x. ((complex_mul c) (f x)))) s) = ((c = (Cx (real_of_num (NUMERAL _0)))) \/ ((integrable_on f) s))))))


Fan.remark1_fan

|- (!x. (!V. (!E. (!u. (!v. ((FAN (x , (V , E))) ==> ((FINITE (((set_of_edge v) V) E)) /\ ((((v INSERT (u INSERT EMPTY)) IN E) = (u IN (((set_of_edge v) V) E))) /\ (((v INSERT (u INSERT EMPTY)) IN E) ==> ((~ (collinear (x INSERT (v INSERT (u INSERT EMPTY))))) /\ ((~ (x = v)) /\ ((~ (x = u)) /\ ((~ (v = u)) /\ (((DISJOINT (x INSERT (v INSERT EMPTY))) (u INSERT EMPTY)) /\ (((DISJOINT (x INSERT (u INSERT EMPTY))) (v INSERT EMPTY)) /\ (((DISJOINT (x INSERT EMPTY)) (v INSERT (u INSERT EMPTY))) /\ ((~ (u IN (aff (x INSERT (v INSERT EMPTY))))) /\ ((v IN V) /\ ((~ (collinear (x INSERT (v INSERT (u INSERT EMPTY))))) = (~ (u IN (aff (x INSERT (v INSERT EMPTY))))))))))))))))))))))))


LOCALLY_PATH_CONNECTED_SPHERE_GEN

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


INFSUM_UNIQUE

|- (!f. (!l. (!s. ((((sums f) l) s) ==> (((infsum s) f) = l)))))


closure

|- (!s. ((closure s) = (s UNION (GSPEC (\GEN%PVAR%568. (?x. (((SETSPEC GEN%PVAR%568) ((limit_point_of x) s)) x)))))))


CONNECTED_COMPONENT_MAXIMAL

|- (!s. (!t. (!x. (((x IN t) /\ ((connected t) /\ (t SUBSET s))) ==> (t SUBSET ((connected_component s) x))))))


REAL_DIFFERENTIABLE_COMPOSE_WITHIN

|- (!f. (!g. (!x. (!s. ((((real_differentiable f) ((within (atreal x)) s)) /\ ((real_differentiable g) ((within (atreal (f x))) ((IMAGE f) s)))) ==> ((real_differentiable (g o f)) ((within (atreal x)) s)))))))


VECTOR_VARIATION_CONTINUOUS

|- (!f. (!a. (!b. (!c. ((((has_bounded_variation_on f) (closed_interval ((CONS (a , b)) NIL))) /\ (c IN (closed_interval ((CONS (a , b)) NIL)))) ==> (((continuous (\x. (lift ((vector_variation (closed_interval ((CONS (a , x)) NIL))) f)))) ((within (at c)) (closed_interval ((CONS (a , b)) NIL)))) = ((continuous f) ((within (at c)) (closed_interval ((CONS (a , b)) NIL))))))))))


Oxl_2012.CARD_EMPTY

|- ((CARD EMPTY) = (NUMERAL _0))


Tame_lemmas.reflection

|- (!x. (!n. (((reflection n) x) = ((vector_sub x) ((% ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((real_div ((dot x) n)) ((dot n) n)))) n)))))


Tame_defs.rotate_to

|- (!v. (!vs. (((rotate_to vs) v) = ((CONS v) ((APPEND (SND ((split_at v) vs))) (FST ((split_at v) vs)))))))


ORTHOGONAL_MATRIX_TRANSP

|- (!A. ((orthogonal_matrix (transp A)) = (orthogonal_matrix A)))


POLYHEDRON_SUMS

|- (!s. (!t. (((polyhedron s) /\ (polyhedron t)) ==> (polyhedron (GSPEC (\GEN%PVAR%2421. (?x. (?y. (((SETSPEC GEN%PVAR%2421) ((x IN s) /\ (y IN t))) ((vector_add x) y))))))))))


int_mul_th

|- (!x. (!y. ((real_of_int ((int_mul x) y)) = ((real_mul (real_of_int x)) (real_of_int y)))))


CONNECTED_COMPONENT_EQ_SELF

|- (!s. (!x. (((connected s) /\ (x IN s)) ==> (((connected_component s) x) = s))))


support

|- (!s. (!f. (!op. ((((support op) f) s) = (GSPEC (\GEN%PVAR%158. (?x. (((SETSPEC GEN%PVAR%158) ((x IN s) /\ (~ ((f x) = (neutral op))))) x))))))))


Functional_equation.proj_x6

|- (((((((proj_x6 x1) x2) x3) x4) x5) x6) = x6)


DIAMETER_LINEAR_IMAGE

|- (!f. (!s. (((linear f) /\ (!x. ((vector_norm (f x)) = (vector_norm x)))) ==> ((diameter ((IMAGE f) s)) = (diameter s)))))


Ssrnat.eqn_addr

|- (!p. (!m. (!n. (((m + p) = (n + p)) = (m = n)))))


LENGTH_EQ_NIL

|- (!l. (((LENGTH l) = (NUMERAL _0)) = (l = NIL)))


INTERVAL_NE_EMPTY

|- (((~ ((closed_interval ((CONS (a , b)) NIL)) = EMPTY)) = (!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV))) ==> ((real_le (a $ i)) (b $ i))))) /\ ((~ ((open_interval (a , b)) = EMPTY)) = (!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV))) ==> ((real_lt (a $ i)) (b $ i))))))


division_of

|- (!s. (!i. (((division_of s) i) = ((FINITE s) /\ ((!k. ((k IN s) ==> ((k SUBSET i) /\ ((~ (k = EMPTY)) /\ (?a. (?b. (k = (closed_interval ((CONS (a , b)) NIL))))))))) /\ ((!k1. (!k2. (((k1 IN s) /\ ((k2 IN s) /\ (~ (k1 = k2)))) ==> (((interior k1) INTER (interior k2)) = EMPTY)))) /\ ((UNIONS s) = i)))))))


IN_CROSS

|- (!x. (!y. (!s. (!t. (((x , y) IN (s CROSS t)) = ((x IN s) /\ (y IN t)))))))


Real_ext.REAL_PROP_ZERO_POW

|- (!x. (!n. ((((real_pow x) n) = (real_of_num (NUMERAL _0))) = ((x = (real_of_num (NUMERAL _0))) /\ (~ (n = (NUMERAL _0)))))))


VECTOR_VARIATION_MONOTONE

|- (!f. (!s. (!t. ((((has_bounded_variation_on f) s) /\ (t SUBSET s)) ==> ((real_le ((vector_variation t) f)) ((vector_variation s) f))))))


Hypermap.edge_set

|- (!H. ((edge_set H) = ((set_of_orbits (dart H)) (edge_map H))))


ADD1

|- (!m. ((SUC m) = (m + (NUMERAL (BIT1 _0)))))


VECTOR_MUL_EQ_0

|- (!a. (!x. ((((% a) x) = (vec (NUMERAL _0))) = ((a = (real_of_num (NUMERAL _0))) \/ (x = (vec (NUMERAL _0)))))))


nproduct

|- (nproduct = (iterate *))


Lp_ineqs_proofs.y2_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 (BIT0 (BIT1 _0))))) (y2_fan d))))))))))


CARD_EXP_0

|- (!s. (!c. (((^_c s) EMPTY) =_c (c INSERT EMPTY))))


SUM_EQ_0_NUMSEG

|- (!f. (!m. (!n. ((!i. (((m <= i) /\ (i <= n)) ==> ((f i) = (real_of_num (NUMERAL _0))))) ==> (((sum (m .. n)) f) = (real_of_num (NUMERAL _0)))))))


REAL_ABS_POS

|- (!x. ((real_le (real_of_num (NUMERAL _0))) (real_abs x)))


INSIDE_LINEAR_IMAGE

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


VSUM_CASES

|- (!s. (!P. (!f. (!g. ((FINITE s) ==> (((vsum s) (\x. (((COND (P x)) (f x)) (g x)))) = ((vector_add ((vsum (GSPEC (\GEN%PVAR%112. (?x. (((SETSPEC GEN%PVAR%112) ((x IN s) /\ (P x))) x))))) f)) ((vsum (GSPEC (\GEN%PVAR%113. (?x. (((SETSPEC GEN%PVAR%113) ((x IN s) /\ (~ (P x)))) x))))) g))))))))


REAL_CONTINUOUS_ON_EQ

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


CLOSED_PATH_IMAGE

|- (!g. ((path g) ==> (closed (path_image g))))


STARLIKE_LINEAR_IMAGE_EQ

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


HOMOTOPIC_ON_COMPONENTS

|- (!s. (!t. (!f. (!g. ((((locally connected) s) /\ (!c. ((c IN (components s)) ==> ((((homotopic_with (\x. T)) (c , t)) f) g)))) ==> ((((homotopic_with (\x. T)) (s , t)) f) g))))))


VECTOR_NEG_NEG

|- ((vector_neg (vector_neg x)) = x)


REAL_ABS_DIV

|- (!x. (!y. ((real_abs ((real_div x) y)) = ((real_div (real_abs x)) (real_abs y)))))


DISJOINT

|- (!s. (!t. (((DISJOINT s) t) = ((s INTER t) = EMPTY))))


ABSOLUTELY_SETCONTINUOUS_INDEFINITE_INTEGRAL

|- (!f. (!s. ((((absolutely_integrable_on f) s) /\ (lebesgue_measurable s)) ==> ((absolutely_setcontinuous_on (\k. ((integral k) f))) s))))


Hypermap.lemmaZHQCZLX

|- (!H. (((simple_hypermap H) /\ ((plain_hypermap H) /\ (!x. ((x IN (dart H)) ==> ((NUMERAL (BIT1 (BIT1 _0))) <= (CARD ((face H) x))))))) ==> (!x. ((x IN (dart H)) ==> (~ (((node_map H) x) = x))))))


APPEND

|- ((!l. (((APPEND NIL) l) = l)) /\ (!h. (!t. (!l. (((APPEND ((CONS h) t)) l) = ((CONS h) ((APPEND t) l)))))))


BALL_TRANSLATION

|- (!a. (!x. (!r. ((ball (((vector_add a) x) , r)) = ((IMAGE (\y. ((vector_add a) y))) (ball (x , r)))))))


MAT_0_COMPONENT

|- (!i. (!j. ((((mat (NUMERAL _0)) $ i) $ j) = (real_of_num (NUMERAL _0)))))


drop

|- (!x. ((drop x) = (x $ (NUMERAL (BIT1 _0)))))


SUBSPACE_0

|- ((subspace s) ==> ((vec (NUMERAL _0)) IN s))


real_continuous_atreal

|- (((real_continuous f) (atreal x)) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?d. (((real_lt (real_of_num (NUMERAL _0))) d) /\ (!x'. (((real_lt (real_abs ((real_sub x') x))) d) ==> ((real_lt (real_abs ((real_sub (f x')) (f x)))) e))))))))


real_summable

|- (!f. (!s. (((real_summable s) f) = (?l. (((real_sums f) l) s)))))


ROOT_MONO_LE_EQ

|- (!n. (!x. (!y. ((~ (n = (NUMERAL _0))) ==> (((real_le ((root n) x)) ((root n) y)) = ((real_le x) y))))))


VSUM_0

|- (((vsum s) (\x. (vec (NUMERAL _0)))) = (vec (NUMERAL _0)))


Add_triangle.fully_surrounded_imp_fan80

|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> (fan80 ((vec (NUMERAL _0)) , (V , E)))))))


RELPOW_R

|- ((((((RELPOW (NUMERAL _0)) R) x) y) = (x = y)) /\ (((((RELPOW (SUC n)) R) x) y) = (?z. (((R x) z) /\ ((((RELPOW n) R) z) y)))))


NORM_REAL

|- (!x. ((vector_norm x) = (real_abs (x $ (NUMERAL (BIT1 _0))))))


Seq.sumn0

|- ((sumn NIL) = (NUMERAL _0))


CONVEX_CLOSURE_RELATIVE_INTERIOR

|- (!s. ((convex s) ==> ((closure (relative_interior s)) = (closure s))))


LE_ANTISYM

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


COMPLEX_POLY_NEG_CLAUSES

|- ((!x. ((vector_neg x) = ((complex_mul (Cx (real_neg (real_of_num (NUMERAL (BIT1 _0)))))) x))) /\ (!x. (!y. (((vector_sub x) y) = ((vector_add x) ((complex_mul (Cx (real_neg (real_of_num (NUMERAL (BIT1 _0)))))) y))))))


MEM_LINEAR_IMAGE

|- (!f. (!x. (!l. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> (((MEM (f x)) ((MAP f) l)) = ((MEM x) l))))))


PATH_LENGTH_TRANSLATION

|- (!a. (!g. ((path_length ((\x. ((vector_add a) x)) o g)) = (path_length g))))


INT_LE_TRANS

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


HAS_REAL_INTEGRAL_REFL

|- (!f. (!a. (((has_real_integral f) (real_of_num (NUMERAL _0))) (closed_real_interval ((CONS (a , a)) NIL)))))


EXISTS_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)))))))))


HAS_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)))))) ==> ((has_real_measure (UNIONS f)) ((sum f) m)))))


DIMINDEX_2

|- ((dimindex UNIV) = (NUMERAL (BIT0 (BIT1 _0))))


Miqmcsn.scs_3M1_prime

|- (scs_3M1_prime = ((((mk_unadorned_v39 (NUMERAL (BIT1 (BIT1 _0)))) ((DECIMAL (NUMERAL (BIT1 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))) (((funlist_v39 ((CONS (((NUMERAL _0) , (NUMERAL (BIT1 _0))) , cstab)) NIL)) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (NUMERAL (BIT1 (BIT1 _0))))) (((funlist_v39 ((CONS (((NUMERAL _0) , (NUMERAL (BIT1 _0))) , cstab)) NIL)) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) (NUMERAL (BIT1 (BIT1 _0))))))


HOLOMORPHIC_ON_ADD

|- (!f. (!g. (!s. ((((holomorphic_on f) s) /\ ((holomorphic_on g) s)) ==> ((holomorphic_on (\z. ((vector_add (f z)) (g z)))) s)))))


INT_OF_NUM_LT

|- (!m. (!n. (((int_lt (int_of_num m)) (int_of_num n)) = (m < n))))


Ssrbool.andTb

|- (!b. ((T /\ b) = b))


REAL_ARCH_SIMPLE

|- (!x. (?n. ((real_le x) (real_of_num n))))


real_max

|- (!n. (!m. (((real_max m) n) = (((COND ((real_le m) n)) n) m))))


REAL_UNIFORMLY_CONTINUOUS_ON

|- (!f. (!s. (((real_uniformly_continuous_on f) s) = ((uniformly_continuous_on (lift o (f o drop))) ((IMAGE lift) s)))))


Asfutbf.IN_NEG

|- (!v. ((v IN ((IMAGE vector_neg) V)) = ((vector_neg v) IN V)))


ITER_ALT

|- ((!f. (!x. ((((ITER (NUMERAL _0)) f) x) = x))) /\ (!f. (!n. (!x. ((((ITER (SUC n)) f) x) = (((ITER n) f) (f x)))))))


SPAN_SUPERSET

|- (!x. ((x IN s) ==> (x IN (span s))))


CARD_LT_CONG

|- (!s. (!s'. (!t. (!t'. (((s =_c s') /\ (t =_c t')) ==> ((s <_c t) = (s' <_c t')))))))


SUBSET_EMPTY

|- (!s. ((s SUBSET EMPTY) = (s = EMPTY)))


ANR_INTERVAL

|- ((!a. (!b. (ANR (closed_interval ((CONS (a , b)) NIL))))) /\ (!a. (!b. (ANR (open_interval (a , b))))))


TRUTH

|- T


Ssrbool.contraNN

|- (!c. (!b. ((c ==> b) ==> ((~ b) ==> (~ c)))))


CLOSED_UNIV

|- (closed UNIV)


Ssrnat.mulnn

|- (!m. ((m * m) = (m EXP (NUMERAL (BIT0 (BIT1 _0))))))


TAGGED_PARTIAL_DIVISION_OF_TRIVIAL

|- (!p. (((tagged_partial_division_of p) EMPTY) = (p = EMPTY)))


LIM_SEQUENTIALLY

|- (!s. (!l. ((((--> s) l) sequentially) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?N. (!n. ((N <= n) ==> ((real_lt (distance ((s n) , l))) e)))))))))


REAL_SERIES_0

|- (!s. (((real_sums (\n. (real_of_num (NUMERAL _0)))) (real_of_num (NUMERAL _0))) s))


COMPONENTS_EQ_SING_EXISTS

|- (!s. ((?a. ((components s) = (a INSERT EMPTY))) = ((connected s) /\ (~ (s = EMPTY)))))


SEGMENT_1

|- ((!a. (!b. ((closed_segment ((CONS (a , b)) NIL)) = (((COND ((real_le (drop a)) (drop b))) (closed_interval ((CONS (a , b)) NIL))) (closed_interval ((CONS (b , a)) NIL)))))) /\ (!a. (!b. ((open_segment (a , b)) = (((COND ((real_le (drop a)) (drop b))) (open_interval (a , b))) (open_interval (b , a)))))))


HOLOMORPHIC_ON_IMP_CONTINUOUS_ON

|- (!f. (!s. (((holomorphic_on f) s) ==> ((continuous_on f) s))))


summable

|- (!f. (!s. (((summable s) f) = (?l. (((sums f) l) s)))))


BOUNDED_EMPTY

|- (bounded EMPTY)


Sphere.dihatn_x

|- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((((((dihatn_x x1) x2) x3) x4) x5) x6) = ((LET (\d_x4. (LET_END ((LET (\d. (LET_END ((real_add ((real_div pi) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) (atn ((real_div (real_neg d_x4)) (sqrt ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) ((real_mul x1) d))))))))) ((((((delta_x x1) x2) x3) x4) x5) x6))))) ((((((delta_x4 x1) x2) x3) x4) x5) x6)))))))))


less

|- (!l. (!x. (!y. (((less l) (x , y)) = ((l (x , y)) /\ (~ (x = y)))))))


Real_ext.REAL_PROP_POS_RADD

|- (!x. (!y. ((((real_lt (real_of_num (NUMERAL _0))) x) /\ ((real_le (real_of_num (NUMERAL _0))) y)) ==> ((real_lt (real_of_num (NUMERAL _0))) ((real_add x) y)))))


IS_REALINTERVAL_UNIV

|- (is_realinterval UNIV)


INTEGRAL_INTERVALS_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))))) ==> (((integral (closed_interval ((CONS (a , d)) NIL))) f) = ((vsum (GSPEC (\GEN%PVAR%2926. (?s. (((SETSPEC GEN%PVAR%2926) (s SUBSET ((NUMERAL (BIT1 _0)) .. (dimindex UNIV)))) s))))) (\s. ((% ((real_pow (real_neg (real_of_num (NUMERAL (BIT1 _0))))) (CARD (GSPEC (\GEN%PVAR%2927. (?i. (((SETSPEC GEN%PVAR%2927) ((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)))))))))))


Seq.undup

|- (((undup NIL) = NIL) /\ ((undup ((CONS x) s')) = (((COND ((MEM x) s')) (undup s')) ((CONS x) (undup s')))))


Hypermap.number_of_edges

|- (!H. ((number_of_edges H) = (CARD (edge_set H))))


Functional_equation.gamma2_x1_div_a

|- (!m. ((gamma2_x1_div_a m) = (promote1_to_6 (gamma2_x_div_azim m))))


REAL_SERIES_NEG

|- (!x. (!x0. (!s. ((((real_sums x) x0) s) ==> (((real_sums (\n. (real_neg (x n)))) (real_neg x0)) s)))))


FINITE_IMP_BOUNDED

|- (!s. ((FINITE s) ==> (bounded s)))


DISJOINT_INSERT

|- (!x. (!s. (!t. (((DISJOINT (x INSERT s)) t) = (((DISJOINT s) t) /\ (~ (x IN t)))))))


ABSOLUTELY_REAL_INTEGRABLE_DECREASING_PRODUCT

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


HAS_COMPLEX_DERIVATIVE_CCOS

|- (!z. (((has_complex_derivative ccos) (vector_neg (csin z))) (at z)))


ABSOLUTELY_INTEGRABLE_CONVOLUTION_SYM

|- (!bop. (!f. (!g. (!x. (((absolutely_integrable_on (\y. ((bop (f ((vector_sub x) y))) (g y)))) UNIV) = ((absolutely_integrable_on (\y. ((bop (f y)) (g ((vector_sub x) y))))) UNIV))))))


INDEPENDENT_LINEAR_IMAGE_EQ

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


CONNECTED_INSERT

|- (!s. (!a. ((connected s) ==> ((connected (a INSERT s)) = ((s = EMPTY) \/ (a IN (closure s)))))))


LT_IMP_LE

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


Sphere.sqrt8

|- (sqrt8 = (sqrt (real_of_num (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 _0))))))))


REAL_SUM_INTEGRAL_UBOUND_INCREASING

|- (!f. (!g. (!m. (!n. (((m <= n) /\ ((!x. ((x IN (closed_real_interval ((CONS ((real_of_num m) , ((real_add (real_of_num n)) (real_of_num (NUMERAL (BIT1 _0)))))) NIL))) ==> (((has_real_derivative g) (f x)) ((within (atreal x)) (closed_real_interval ((CONS ((real_of_num m) , ((real_add (real_of_num n)) (real_of_num (NUMERAL (BIT1 _0)))))) NIL)))))) /\ (!x. (!y. ((((real_le (real_of_num m)) x) /\ (((real_le x) y) /\ ((real_le y) ((real_add (real_of_num n)) (real_of_num (NUMERAL (BIT1 _0))))))) ==> ((real_le (f x)) (f y))))))) ==> ((real_le ((sum (m .. n)) (\k. (f (real_of_num k))))) ((real_sub (g ((real_add (real_of_num n)) (real_of_num (NUMERAL (BIT1 _0)))))) (g (real_of_num m)))))))))


COMPACT_SIMPLEX

|- (!n. (!s. (((simplex n) s) ==> (compact s))))


Hypermap_iso.bij_contracted_dart2

|- (!V. (!E. (((BIJ contracted_dart) (d20_fan ((vec (NUMERAL _0)) , (V , E)))) (GSPEC (\GEN%PVAR%7631. (?v. (((SETSPEC GEN%PVAR%7631) ((v IN V) /\ ((((set_of_edge v) V) E) = EMPTY))) (v , v))))))))


Lp_ineqs_proofs.azim_fan_eq_dih_y

|- (!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))) ==> (((azim_dart (V , E)) d) = ((((((dih_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)))))))))))


Sphere.packing_lt

|- ((packing V) = (!u. (!v. (((u IN V) /\ ((v IN V) /\ ((real_lt (distance (u , v))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) ==> (u = v)))))


DIFF

|- (!s. (!t. ((s DIFF t) = (GSPEC (\GEN%PVAR%4. (?x. (((SETSPEC GEN%PVAR%4) ((x IN s) /\ (~ (x IN t)))) x)))))))


LINEAR_ROTATE2D

|- (!t. (linear (rotate2d t)))


CX_POW

|- (!x. (!n. ((Cx ((real_pow x) n)) = ((complex_pow (Cx x)) n))))


CONTINUOUS_ON_CLOSED_GEN

|- (!f. (!s. (!t. ((((IMAGE f) s) SUBSET t) ==> (((continuous_on f) s) = (!u. (((closed_in ((subtopology euclidean) t)) u) ==> ((closed_in ((subtopology euclidean) s)) (GSPEC (\GEN%PVAR%616. (?x. (((SETSPEC GEN%PVAR%616) ((x IN s) /\ ((f x) IN u))) x))))))))))))


REAL_CONTINUOUS_ATREAL_WITHINREAL

|- (!f. (!s. (!x. (((real_continuous f) (atreal x)) ==> ((real_continuous f) ((within (atreal x)) s))))))


LOCALLY_CONNECTED_PCROSS

|- (!s. (!t. ((((locally connected) s) /\ ((locally connected) t)) ==> ((locally connected) (s PCROSS t)))))


REAL_SUMS

|- (((real_sums f) l) = ((sums (lift o f)) (lift l)))


REAL_LT_MIN

|- (!x. (!y. (!z. (((real_lt z) ((real_min x) y)) = (((real_lt z) x) /\ ((real_lt z) y))))))


FACE_OF_LINEAR_IMAGE

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


NEGLIGIBLE_SUBSET

|- (!s. (!t. (((negligible s) /\ (t SUBSET s)) ==> (negligible t))))


IN_IMAGE

|- (!y. (!s. (!f. ((y IN ((IMAGE f) s)) = (?x. ((y = (f x)) /\ (x IN s)))))))


OPEN_EXISTS_IN

|- (!P. (!Q. ((!a. ((P a) ==> (open (GSPEC (\GEN%PVAR%542. (?x. (((SETSPEC GEN%PVAR%542) ((Q a) x)) x))))))) ==> (open (GSPEC (\GEN%PVAR%543. (?x. (((SETSPEC GEN%PVAR%543) (?a. ((P a) /\ ((Q a) x)))) x))))))))


Hypermap_and_fan.IN_DART_OF_FAN

|- (!V. (!E. (!x. (((FAN ((vec (NUMERAL _0)) , (V , E))) /\ (x IN (dart_of_fan (V , E)))) ==> (?v. (?w. ((x = (v , w)) /\ (((v , w) IN (dart_of_fan (V , E))) /\ ((v IN V) /\ (w IN V))))))))))


compact_in

|- (!top. (!s. (((compact_in s) top) = ((s SUBSET (topspace top)) /\ (!U. (((!u. ((u IN U) ==> ((open_in top) u))) /\ (s SUBSET (UNIONS U))) ==> (?V. ((FINITE V) /\ ((V SUBSET U) /\ (s SUBSET (UNIONS V)))))))))))


Collect_geom2.ETA_Y_POW2_EQ

|- (((real_div ((real_mul ((real_pow (distance (v1 , v2))) (NUMERAL (BIT0 (BIT1 _0))))) ((real_mul ((real_pow (distance (v1 , v3))) (NUMERAL (BIT0 (BIT1 _0))))) ((real_pow (distance (v2 , v3))) (NUMERAL (BIT0 (BIT1 _0))))))) (((ups_x ((real_pow (distance (v1 , v2))) (NUMERAL (BIT0 (BIT1 _0))))) ((real_pow (distance (v1 , v3))) (NUMERAL (BIT0 (BIT1 _0))))) ((real_pow (distance (v2 , v3))) (NUMERAL (BIT0 (BIT1 _0)))))) = ((real_pow (((eta_y ((dist3 v2) v3)) ((dist3 v1) v3)) ((dist3 v1) v2))) (NUMERAL (BIT0 (BIT1 _0)))))


REAL_INTEGRABLE_BY_PARTS

|- (!f. (!g. (!f'. (!g'. (!a. (!b. (!c. (((COUNTABLE c) /\ (((real_continuous_on (\x. ((real_mul (f x)) (g x)))) (closed_real_interval ((CONS (a , b)) NIL))) /\ ((!x. ((x IN ((open_real_interval (a , b)) DIFF c)) ==> ((((has_real_derivative f) (f' x)) (atreal x)) /\ (((has_real_derivative g) (g' x)) (atreal x))))) /\ ((real_integrable_on (\x. ((real_mul (f x)) (g' x)))) (closed_real_interval ((CONS (a , b)) NIL)))))) ==> ((real_integrable_on (\x. ((real_mul (f' x)) (g x)))) (closed_real_interval ((CONS (a , b)) NIL)))))))))))


RELATIVE_BOUNDARY_OF_TRIANGLE

|- (!a. (!b. (!c. ((~ (collinear (a INSERT (b INSERT (c INSERT EMPTY))))) ==> ((((hull convex) (a INSERT (b INSERT (c INSERT EMPTY)))) DIFF (relative_interior ((hull convex) (a INSERT (b INSERT (c INSERT EMPTY)))))) = ((closed_segment ((CONS (a , b)) NIL)) UNION ((closed_segment ((CONS (b , c)) NIL)) UNION (closed_segment ((CONS (c , a)) NIL)))))))))


ANR_SING

|- (!a. (ANR (a INSERT EMPTY)))


SC_RULES

|- (!R. ((!x. (!y. (((R x) y) ==> (((SC R) x) y)))) /\ (!x. (!y. ((((SC R) x) y) ==> (((SC R) y) x))))))


INTERVAL_REAL_INTERVAL

|- (((closed_interval ((CONS (a , b)) NIL)) = ((IMAGE lift) (closed_real_interval ((CONS ((drop a) , (drop b))) NIL)))) /\ ((open_interval (a , b)) = ((IMAGE lift) (open_real_interval ((drop a) , (drop b))))))


VECTOR_VARIATION_TRANSLATION2

|- (!a. (!f. (!s. (((vector_variation ((IMAGE (\x. ((vector_add (vector_neg a)) x))) s)) (\x. (f ((vector_add a) x)))) = ((vector_variation s) f)))))


Seq.in_cons

|- (!y. (!s. (!x. (((MEM x) ((CONS y) s)) = ((x = y) \/ ((MEM x) s))))))


TRACE_COVARIANCE_POS_LE

|- (!A. ((real_le (real_of_num (NUMERAL _0))) (trace ((matrix_mul (transp A)) A))))


MCBALL_SUBSET_MSPACE

|- (!m. (!x. (!r. (((mcball m) (x , r)) SUBSET (mspace m)))))


CONIC_RELATIVE_INTERIOR_INSERT

|- (!s. ((conic s) ==> (conic ((vec (NUMERAL _0)) INSERT (relative_interior s)))))


EXTREME_POINTS_OF_CONVEX_HULL

|- (!s. ((GSPEC (\GEN%PVAR%2195. (?x. (((SETSPEC GEN%PVAR%2195) ((extreme_point_of x) ((hull convex) s))) x)))) SUBSET s))


ADD_CLAUSES

|- ((!n. (((NUMERAL _0) + n) = n)) /\ ((!m. ((m + (NUMERAL _0)) = m)) /\ ((!m. (!n. (((SUC m) + n) = (SUC (m + n))))) /\ (!m. (!n. ((m + (SUC n)) = (SUC (m + n))))))))


ONORM_TRIANGLE

|- (!f. (!g. (((linear f) /\ (linear g)) ==> ((real_le (onorm (\x. ((vector_add (f x)) (g x))))) ((real_add (onorm f)) (onorm g))))))


SETDIST_LE_DIST

|- (!s. (!t. (!x. (!y. (((x IN s) /\ (y IN t)) ==> ((real_le (setdist (s , t))) (distance (x , y))))))))


ABSOLUTELY_INTEGRABLE_TRANSLATION

|- (!f. (!s. (!a. (((absolutely_integrable_on (\x. (f ((vector_add a) x)))) s) = ((absolutely_integrable_on f) ((IMAGE (\x. ((vector_add a) x))) s))))))


AFFINE_TRANSLATION_EQ

|- (!a. (!s. ((affine ((IMAGE (\x. ((vector_add a) x))) s)) = (affine s))))


LOCALLY_CONNECTED_LINEAR_IMAGE_EQ

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


Nonlin_def.delta_x1

|- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((((((delta_x1 x1) x2) x3) x4) x5) x6) = ((real_add ((real_mul (real_neg x1)) x4)) ((real_add ((real_sub ((real_sub ((real_mul x2) x5)) ((real_mul x3) x5))) ((real_mul x2) x6))) ((real_add ((real_mul x3) x6)) ((real_mul x4) ((real_add (real_neg x1)) ((real_add x2) ((real_add ((real_sub x3) x4)) ((real_add x5) x6)))))))))))))))


Packing3.BARV_CONS

|- (!V. (!k. (!ul. ((((barV V) k) ul) ==> (?h. (?t. ((ul = ((CONS h) t)) /\ (h = (HD ul)))))))))


AFFINE_HULL_FINITE_STEP_GEN

|- (!P. (((?u. ((!x. ((x IN EMPTY) ==> ((P x) (u x)))) /\ ((((sum EMPTY) u) = w) /\ (((vsum EMPTY) (\x. ((% (u x)) x))) = y)))) = ((w = (real_of_num (NUMERAL _0))) /\ (y = (vec (NUMERAL _0))))) /\ (((FINITE s) /\ ((!y. (((a IN s) /\ ((P a) y)) ==> ((P a) ((real_div y) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))))) /\ (!x. (!y. (((a IN s) /\ (((P a) x) /\ ((P a) y))) ==> ((P a) ((real_add x) y))))))) ==> ((?u. ((!x. ((x IN (a INSERT s)) ==> ((P x) (u x)))) /\ ((((sum (a INSERT s)) u) = w) /\ (((vsum (a INSERT s)) (\x. ((% (u x)) x))) = y)))) = (?v. (?u. (((P a) v) /\ ((!x. ((x IN s) ==> ((P x) (u x)))) /\ ((((sum s) u) = ((real_sub w) v)) /\ (((vsum s) (\x. ((% (u x)) x))) = ((vector_sub y) ((% v) a))))))))))))


MAP_REVERSE

|- (!f. (!l. ((REVERSE ((MAP f) l)) = ((MAP f) (REVERSE l)))))


ORDER_EXISTENCE_CARD

|- (!R. (!f. (!z. (!k. (((FINITE (GSPEC (\GEN%PVAR%74. (?n. (((SETSPEC GEN%PVAR%74) (n IN UNIV)) (R (((ITER n) f) z))))))) /\ (((CARD (GSPEC (\GEN%PVAR%75. (?n. (((SETSPEC GEN%PVAR%75) (n IN UNIV)) (R (((ITER n) f) z))))))) <= k) /\ (((R z) z) /\ ((!x. (!y. (((R x) y) ==> ((R y) x)))) /\ ((!x. (!y. (!z. ((((R x) y) /\ ((R y) z)) ==> ((R x) z))))) /\ (!x. (!y. (((R (f x)) (f y)) = ((R x) y))))))))) ==> (?d. (((NUMERAL _0) < d) /\ ((d <= k) /\ (!n. (((R (((ITER n) f) z)) z) = ((num_divides d) n)))))))))))


SERIES_GOESTOZERO

|- (!s. (!x. (((summable s) x) ==> (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> ((eventually (\n. ((n IN s) ==> ((real_lt (vector_norm (x n))) e)))) sequentially))))))


MAP_APPEND

|- (!f. (!l1. (!l2. (((MAP f) ((APPEND l1) l2)) = ((APPEND ((MAP f) l1)) ((MAP f) l2))))))


TC_CASES_L

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


Sphere.dih5atn_x

|- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((((((dih5atn_x x1) x2) x3) x4) x5) x6) = ((LET (\e. (LET_END ((LET (\d. (LET_END ((real_add ((real_div pi) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) (atn ((real_div (real_neg e)) (sqrt ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) ((real_mul x5) d))))))))) ((((((delta_x x1) x2) x3) x4) x5) x6))))) (((((((rotate5 delta_x4) x1) x2) x3) x4) x5) x6)))))))))


Ssrnat.leq_pmul2r

|- (!m. (!n1. (!n2. (((NUMERAL _0) < m) ==> (((n1 * m) <= (n2 * m)) = (n1 <= n2))))))


IN_SET_OF_LIST

|- (!x. (!l. ((x IN (set_of_list l)) = ((MEM x) l))))


IN_FROM

|- (!m. (!n. ((m IN (from n)) = (n <= m))))


OPEN_IN_OPEN_INTER

|- (!u. (!s. ((open s) ==> ((open_in ((subtopology euclidean) u)) (u INTER s)))))


LT_MULT_LCANCEL

|- (!m. (!n. (!p. (((m * n) < (m * p)) = ((~ (m = (NUMERAL _0))) /\ (n < p))))))


CONTRACTIBLE_SING

|- (!a. (contractible (a INSERT EMPTY)))


CLOG_NEG_1

|- ((clog (vector_neg (Cx (real_of_num (NUMERAL (BIT1 _0)))))) = ((complex_mul ii) (Cx pi)))


ABSOLUTELY_CONTINUOUS_ON_IMP_CONTINUOUS

|- (!f. (!s. ((((absolutely_continuous_on f) s) /\ (is_interval s)) ==> ((continuous_on f) s))))


FRONTIER_INTERIORS

|- (!s. ((frontier s) = ((UNIV DIFF (interior s)) DIFF (interior (UNIV DIFF s)))))


MBALL_SUBSET_MCBALL

|- (!m. (!x. (!r. (((mball m) (x , r)) SUBSET ((mcball m) (x , r))))))


Fnjlbxs.vector_of_list_left_inverse

|- (!d. (!list. (((dimindex UNIV) = d) ==> (((d * (LENGTH list)) <= (dimindex UNIV)) ==> ((((list_of_vector d) (LENGTH list)) ((vector_of_list d) list)) = list)))))


real_log_convex_on

|- (!s. (!f. (((real_log_convex_on f) s) = (!x. (!y. (!u. (!v. (((x IN s) /\ ((y IN s) /\ (((real_le (real_of_num (NUMERAL _0))) u) /\ (((real_le (real_of_num (NUMERAL _0))) v) /\ (((real_add u) v) = (real_of_num (NUMERAL (BIT1 _0)))))))) ==> (((real_le (real_of_num (NUMERAL _0))) (f ((real_add ((real_mul u) x)) ((real_mul v) y)))) /\ ((real_le (f ((real_add ((real_mul u) x)) ((real_mul v) y)))) ((real_mul ((rpow (f x)) u)) ((rpow (f y)) v))))))))))))


COMPACT_PATH_IMAGE

|- (!g. ((path g) ==> (compact (path_image g))))


GDELTA_TRANSLATION

|- (!a. (!s. ((gdelta ((IMAGE (\x. ((vector_add a) x))) s)) = (gdelta s))))


Topology.complement_set

|- (!x. (!v. ((complement_set (x INSERT (v INSERT EMPTY))) = (GSPEC (\GEN%PVAR%5482. (?y. (((SETSPEC GEN%PVAR%5482) (~ (y IN (aff (x INSERT (v INSERT EMPTY)))))) y)))))))


Seq.nth_cat

|- (!x0. (!s1. (!s2. (!n. ((((nth x0) ((cat s1) s2)) n) = (((COND (n < (sizel s1))) (((nth x0) s1) n)) (((nth x0) s2) (n - (sizel s1)))))))))


CONVEX_INTERVAL

|- (!a. (!b. ((convex (closed_interval ((CONS (a , b)) NIL))) /\ (convex (open_interval (a , b))))))


TC_CASES_R

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


AFFINE_AFFINE_HULL

|- (!s. (affine ((hull affine) s)))


AFF_DIM_DIM

|- (!s. ((aff_dim s) = (((COND ((vec (NUMERAL _0)) IN ((hull affine) s))) (int_of_num (dim s))) ((int_sub (int_of_num (dim s))) (int_of_num (NUMERAL (BIT1 _0)))))))


CONTINUOUS_CLOSED_IN_PREIMAGE_GEN

|- (!f. (!s. (!t. (!u. ((((continuous_on f) s) /\ ((((IMAGE f) s) SUBSET t) /\ ((closed_in ((subtopology euclidean) t)) u))) ==> ((closed_in ((subtopology euclidean) s)) (GSPEC (\GEN%PVAR%618. (?x. (((SETSPEC GEN%PVAR%618) ((x IN s) /\ ((f x) IN u))) x))))))))))


aff_ge_def

|- (aff_ge = (affsign sgn_ge))


METRIC_TOPCONTINUOUS

|- (!m. (!m'. (!f. ((((topcontinuous (mtopology m)) (mtopology m')) f) = ((!x. ((x IN (mspace m)) ==> ((f x) IN (mspace m')))) /\ (!a. (!e. ((((real_lt (real_of_num (NUMERAL _0))) e) /\ (a IN (mspace m))) ==> (?d. (((real_lt (real_of_num (NUMERAL _0))) d) /\ (!x. (((x IN (mspace m)) /\ ((real_lt ((mdist m) (a , x))) d)) ==> ((real_lt ((mdist m') ((f a) , (f x)))) e)))))))))))))


FACE_OF_SIMPLEX_SUBSET

|- (!n. (!s. (!f. ((((simplex n) s) /\ ((face_of f) s)) ==> (?c. ((c SUBSET (GSPEC (\GEN%PVAR%2470. (?x. (((SETSPEC GEN%PVAR%2470) ((extreme_point_of x) s)) x))))) /\ (f = ((hull convex) c))))))))


EMPTY_SUBSET

|- (!s. (EMPTY SUBSET s))


Nonlin_def.sqrt_x2

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


measurable_on

|- (!s. (!f. (((measurable_on f) s) = (?k. (?g. ((negligible k) /\ ((!n. ((continuous_on (g n)) UNIV)) /\ (!x. ((~ (x IN k)) ==> (((--> (\n. ((g n) x))) (((COND (x IN s)) (f x)) (vec (NUMERAL _0)))) sequentially))))))))))


VECTOR_VARIATION_CONTINUOUS_RIGHT

|- (!f. (!a. (!b. (!c. ((((has_bounded_variation_on f) (closed_interval ((CONS (a , b)) NIL))) /\ (c IN (closed_interval ((CONS (a , b)) NIL)))) ==> (((continuous (\x. (lift ((vector_variation (closed_interval ((CONS (a , x)) NIL))) f)))) ((within (at c)) (closed_interval ((CONS (c , b)) NIL)))) = ((continuous f) ((within (at c)) (closed_interval ((CONS (c , b)) NIL))))))))))


SUBSPACE_IMP_AFFINE

|- (!s. ((subspace s) ==> (affine s)))


Nonlin_def.sqrt_x1

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


CONTINUOUS_LIFT_ABS_COMPONENT

|- (!net. (!f. (!k. (((continuous f) net) ==> ((continuous (\x. (lift (real_abs ((f x) $ k))))) net)))))


Nonlin_def.uni

|- (!f. (!x. (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. ((((((((uni (f , x)) x1) x2) x3) x4) x5) x6) = (f ((((((x x1) x2) x3) x4) x5) x6)))))))))))


NORM_BASIS

|- (!k. ((((NUMERAL (BIT1 _0)) <= k) /\ (k <= (dimindex UNIV))) ==> ((vector_norm (basis k)) = (real_of_num (NUMERAL (BIT1 _0))))))


TRACE_COVARIANCE_POSITIVE_SEMIDEFINITE_LE

|- (!A. ((positive_semidefinite A) ==> ((real_le (trace ((matrix_mul (transp A)) A))) ((real_pow (trace A)) (NUMERAL (BIT0 (BIT1 _0)))))))


RC_REFL

|- (!R. (!x. (((RC R) x) x)))


DEPENDENT_LINEAR_IMAGE_EQ

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


Seq.has_count

|- (!a. (!s. (((has a) s) = ((NUMERAL _0) < ((count a) s)))))


REAL_LE_ADDR

|- (!x. (!y. (((real_le x) ((real_add x) y)) = ((real_le (real_of_num (NUMERAL _0))) y))))


arc

|- (!g. ((arc g) = ((path g) /\ (!x. (!y. (((x IN (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL))) /\ ((y IN (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL))) /\ ((g x) = (g y)))) ==> (x = y)))))))


NEGLIGIBLE_IMP_MEASURABLE

|- (!s. ((negligible s) ==> (measurable s)))


POLYTOPE_EQ_BOUNDED_POLYHEDRON

|- (!s. ((polytope s) = ((polyhedron s) /\ (bounded s))))


INT_GT

|- (!x. (!y. (((int_gt x) y) = ((int_lt y) x))))


INTEGER_ADD

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


UNBOUNDED_OUTSIDE

|- (!s. ((bounded s) ==> (~ (bounded (outside s)))))


TAYLOR_CCOS

|- (!n. (!z. ((real_le (vector_norm ((vector_sub (ccos z)) ((vsum ((NUMERAL _0) .. n)) (\k. ((complex_mul ((complex_pow (vector_neg (Cx (real_of_num (NUMERAL (BIT1 _0)))))) k)) ((complex_div ((complex_pow z) ((NUMERAL (BIT0 (BIT1 _0))) * k))) (Cx (real_of_num (FACT ((NUMERAL (BIT0 (BIT1 _0))) * k))))))))))) ((real_mul (exp (real_abs (Im z)))) ((real_div ((real_pow (vector_norm z)) (((NUMERAL (BIT0 (BIT1 _0))) * n) + (NUMERAL (BIT0 (BIT1 _0)))))) (real_of_num (FACT (((NUMERAL (BIT0 (BIT1 _0))) * n) + (NUMERAL (BIT1 _0))))))))))


Ssrnat.distn

|- (!m. (!n. (((distn m) n) = ((m - n) + (n - m)))))


VECTOR_ADD_SUB

|- (((vector_sub ((vector_add x) y)) x) = y)


CEXP_CONVERGES

|- (!z. (((sums (\n. ((complex_div ((complex_pow z) n)) (Cx (real_of_num (FACT n)))))) (cexp z)) (from (NUMERAL _0))))


REAL

|- (!z. ((real z) = ((Cx (Re z)) = z)))


UNION_ACI

|- (((p UNION q) = (q UNION p)) /\ ((((p UNION q) UNION r) = (p UNION (q UNION r))) /\ (((p UNION (q UNION r)) = (q UNION (p UNION r))) /\ (((p UNION p) = p) /\ ((p UNION (p UNION q)) = (p UNION q))))))


LINEAR_CX_IM

|- (linear (Cx o Im))


CONNECTED_DISJOINT_UNIONS_OPEN_UNIQUE

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


EVENTUALLY_AT_INFINITY

|- (!p. (((eventually p) at_infinity) = (?b. (!x. (((real_ge (vector_norm x)) b) ==> (p x))))))


CONNECTED_INTERVAL

|- ((!a. (!b. (connected (closed_interval ((CONS (a , b)) NIL))))) /\ (!a. (!b. (connected (open_interval (a , b))))))


Miqmcsn.SCS_4M8_02_SLICE_02

|- ((scs_arrow_v39 (scs_4M8_02 INSERT EMPTY)) (((scs_prop_equ_v39 (scs_opp_v39 scs_3T7)) (NUMERAL (BIT0 (BIT1 _0)))) INSERT EMPTY))


CBALL_TRIVIAL

|- (!x. ((cball (x , (real_of_num (NUMERAL _0)))) = (x INSERT EMPTY)))


Lp_ineqs_proofs.f_fan_pair_ext_in_darts_k

|- (!V. (!E. (!k. (!d. (((FAN ((vec (NUMERAL _0)) , (V , E))) /\ (d IN ((darts_k k) (hypermap_of_fan (V , E))))) ==> ((((f_fan_pair_ext (V , E)) d) IN ((darts_k k) (hypermap_of_fan (V , E)))) /\ (((inverse (f_fan_pair_ext (V , E))) d) IN ((darts_k k) (hypermap_of_fan (V , E))))))))))


MATRIX_MUL_LID

|- (!A. (((matrix_mul (mat (NUMERAL (BIT1 _0)))) A) = A))


HOMEOMORPHIC_INJECTIVE_LINEAR_IMAGE_RIGHT_EQ

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


BARYCENTRE_TRANSLATION

|- (!a. (!s. ((barycentre ((IMAGE (\x. ((vector_add a) x))) s)) = ((vector_add (((COND ((FINITE s) /\ (~ (s = EMPTY)))) a) (vec (NUMERAL _0)))) (barycentre s)))))


NORM_TRIANGLE_LE

|- (!x. (!y. (((real_le ((real_add (vector_norm x)) (vector_norm y))) e) ==> ((real_le (vector_norm ((vector_add x) y))) e))))


PATHSTART_TRANSLATION

|- (!a. (!g. ((pathstart ((\x. ((vector_add a) x)) o g)) = ((vector_add a) (pathstart g)))))


CLOG_II

|- ((clog ii) = ((complex_mul ii) (Cx ((real_div pi) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))))


NSUM_CONST

|- (!c. (!s. ((FINITE s) ==> (((nsum s) (\n. c)) = ((CARD s) * c)))))


CONTINUOUS_ON_PASTECART

|- (!f. (!g. (!s. ((((continuous_on f) s) /\ ((continuous_on g) s)) ==> ((continuous_on (\x. ((pastecart (f x)) (g x)))) s)))))


Hypermap_iso.hyp_iso_comm

|- (!G. (!f. (!H. (((hyp_iso f) (H , G)) ==> (!x. ((x IN (dart H)) ==> ((((edge_map G) (f x)) = (f ((edge_map H) x))) /\ ((((node_map G) (f x)) = (f ((node_map H) x))) /\ (((face_map G) (f x)) = (f ((face_map H) x)))))))))))


HAS_COMPLEX_DERIVATIVE_CLOG

|- (!z. ((((Im z) = (real_of_num (NUMERAL _0))) ==> ((real_lt (real_of_num (NUMERAL _0))) (Re z))) ==> (((has_complex_derivative clog) (complex_inv z)) (at z))))


JACOBIAN_WORKS

|- (!f. (!net. (((differentiable f) net) = (((has_derivative f) (\h. ((matrix_vector_mul ((jacobian f) net)) h))) net))))


HOMOTOPY_EQUIVALENT_TRANSLATION_RIGHT_EQ

|- (!a. (!s. (!t. (((homotopy_equivalent s) ((IMAGE (\x. ((vector_add a) x))) t)) = ((homotopy_equivalent s) t)))))


Add_triangle.f_fan_pair_eq_inv_n_o_e

|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!d. ((d IN (dart1_of_fan (V , E))) ==> (((f_fan_pair (V , E)) d) = ((inverse (n_fan_pair_ext (V , E))) ((e_fan_pair (V , E)) d))))))))


CONTINUOUS_ON_EQ_CONTINUOUS_AT

|- (!f. (!s. ((open s) ==> (((continuous_on f) s) = (!x. ((x IN s) ==> ((continuous f) (at x))))))))


Ssrnat.odd_add

|- (!m. (!n. ((odd (m + n)) = ((XOR (odd m)) (odd n)))))


REAL_MEASURABLE_ON_COMPOSE_CONTINUOUS_0

|- (!f. (!g. (!s. ((((real_measurable_on f) s) /\ (((real_continuous_on g) UNIV) /\ ((g (real_of_num (NUMERAL _0))) = (real_of_num (NUMERAL _0))))) ==> ((real_measurable_on (g o f)) s)))))


COMPACT_CONTINUOUS_IMAGE

|- (!f. (!s. ((((continuous_on f) s) /\ (compact s)) ==> (compact ((IMAGE f) s)))))


Seq2.list_sum_perm_eq

|- (!s1. (!s2. (!f. (((perm_eq s1) s2) ==> (((list_sum s1) f) = ((list_sum s2) f))))))


FLOOR_POS

|- (!x. (((real_le (real_of_num (NUMERAL _0))) x) ==> (?n. ((floor x) = (real_of_num n)))))


CONTINUOUS_CLOSED_IN_PREIMAGE_EQ

|- (!f. (!s. (((continuous_on f) s) = (!t. ((closed t) ==> ((closed_in ((subtopology euclidean) s)) (GSPEC (\GEN%PVAR%630. (?x. (((SETSPEC GEN%PVAR%630) ((x IN s) /\ ((f x) IN t))) x))))))))))


Seq.sumn

|- (sumn = ((foldr +) (NUMERAL _0)))


TRACE_POSITIVE_SEMIDEFINITE

|- (!A. ((positive_semidefinite A) ==> ((real_le (real_of_num (NUMERAL _0))) (trace A))))


Nonlin_def.unit6

|- (((((((unit6 x1) x2) x3) x4) x5) x6) = (real_of_num (NUMERAL (BIT1 _0))))


COLLINEAR_LINEAR_IMAGE_EQ

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


Hypermap.lemma_power_loop_map_in_loop

|- (!L. (!x. (!k. (((in_dart_of_loop x) L) ==> ((in_dart_of_loop (((POWER (loop_map L)) k) x)) L)))))


Ssrnat.sub0n

|- (!n. (((NUMERAL _0) - n) = (NUMERAL _0)))


Nonlinear_lemma.halfbump_x4

|- (!x1. (!x2. (!x3. (!x5. (!x6. (!x4. (((((((halfbump_x4 x1) x2) x3) x4) x5) x6) = (halfbump_x x4))))))))


Seq.constant_nseq

|- (!n. (!x. (constant ((nseq n) x))))


REAL_OF_NUM_SUC

|- (!n. (((real_add (real_of_num n)) (real_of_num (NUMERAL (BIT1 _0)))) = (real_of_num (SUC n))))


Lp_ineqs_def.list_defs_th

|- ((!fan. (!g. ((azim_list (g , fan)) = ((azim_dart fan) o g)))) /\ ((!fan. (!g. ((azim2_list (g , fan)) = ((azim2_fan fan) o g)))) /\ ((!fan. (!g. ((azim3_list (g , fan)) = ((azim3_fan fan) o g)))) /\ ((!h. ((yn_list h) = (yn_fan o h))) /\ ((!h. ((ln_list h) = (ln_fan o h))) /\ ((!h. ((rho_list h) = (rho_fan o h))) /\ ((!fan. (!g. ((rhazim_list (g , fan)) = ((rhazim_fan fan) o g)))) /\ ((!fan. (!g. ((rhazim2_list (g , fan)) = ((rhazim2_fan fan) o g)))) /\ ((!fan. (!g. ((rhazim3_list (g , fan)) = ((rhazim3_fan fan) o g)))) /\ ((!g. ((ye_list g) = (ye_fan o g))) /\ ((!fan. (!g. ((sol_list (g , fan)) = ((sol_fan fan) o (IMAGE g))))) /\ ((!fan. (!g. ((tau_list (g , fan)) = ((tau_fan fan) o (IMAGE g))))) /\ ((!g. ((y1_list g) = (y1_fan o g))) /\ ((!g. ((y2_list g) = (y2_fan o g))) /\ ((!fan. (!g. ((y3_list (g , fan)) = ((y3_fan fan) o g)))) /\ ((!fan. (!g. ((y4_list (g , fan)) = ((y4_fan fan) o g)))) /\ ((!fan. (!g. ((y5_list (g , fan)) = ((y5_fan fan) o g)))) /\ ((!g. ((y6_list g) = (y6_fan o g))) /\ ((!fan. (!g. ((y8_list (g , fan)) = ((y8_fan fan) o g)))) /\ ((!fan. (!g. ((y9_list (g , fan)) = ((y9_fan fan) o g)))) /\ (!fan. (!g. ((y4'_list (g , fan)) = ((y4'_fan fan) o g))))))))))))))))))))))))


Fan.n_fan

|- (!x. (!V. (!E. ((((n_fan x) V) E) = (GABS (\f. (!x. (!v. (!w. (!w1. ((GEQ (f (x , (v , (w , w1))))) (x , (v , ((((((sigma_fan x) V) E) v) w) , (((((sigma_fan x) V) E) v) (((((sigma_fan x) V) E) v) w))))))))))))))))


Sphere.rotate5

|- (!f. (!x5. (!x3. (!x4. (!x2. (!x6. (!x1. ((((((((rotate5 f) x1) x2) x3) x4) x5) x6) = ((((((f x5) x3) x4) x2) x6) x1)))))))))


INVERTIBLE_FIXPOINT_PROPERTY

|- (!s. (!t. (!i. (!r. ((((continuous_on i) t) /\ ((((IMAGE i) t) SUBSET s) /\ (((continuous_on r) s) /\ ((((IMAGE r) s) SUBSET t) /\ (!y. ((y IN t) ==> ((r (i y)) = y))))))) ==> ((!f. ((((continuous_on f) s) /\ (((IMAGE f) s) SUBSET s)) ==> (?x. ((x IN s) /\ ((f x) = x))))) ==> (!g. ((((continuous_on g) t) /\ (((IMAGE g) t) SUBSET t)) ==> (?y. ((y IN t) /\ ((g y) = y)))))))))))


RC_TC

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


DIST_0

|- (!x. (((distance (x , (vec (NUMERAL _0)))) = (vector_norm x)) /\ ((distance ((vec (NUMERAL _0)) , x)) = (vector_norm x))))


VECTOR_ANGLE_RANGE

|- (!x. (!y. (((real_le (real_of_num (NUMERAL _0))) ((vector_angle x) y)) /\ ((real_le ((vector_angle x) y)) pi))))


REAL_MUL_LID

|- (!x. (((real_mul (real_of_num (NUMERAL (BIT1 _0)))) x) = x))


Ssrnat.dist0n

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


Hypermap.lemma_dart_inveriant_under_inverse_maps

|- (!H. (!x. ((x IN (dart H)) ==> ((((inverse (edge_map H)) x) IN (dart H)) /\ ((((inverse (node_map H)) x) IN (dart H)) /\ (((inverse (face_map H)) x) IN (dart H)))))))


ordinal

|- (!l. ((ordinal l) = ((woset l) /\ (!x. (((fl l) x) ==> (x = (@y. (~ ((less l) (y , x))))))))))


LOG_MUL

|- (!x. (!y. ((((real_lt (real_of_num (NUMERAL _0))) x) /\ ((real_lt (real_of_num (NUMERAL _0))) y)) ==> ((log ((real_mul x) y)) = ((real_add (log x)) (log y))))))


SUBSET_DIFF

|- (!s. (!t. ((s DIFF t) SUBSET s)))


CONNECTED_IMP_PERFECT

|- (!s. (!x. (((connected s) /\ ((~ (?a. (s = (a INSERT EMPTY)))) /\ (x IN s))) ==> ((limit_point_of x) s))))


Imjxphr.HYPER_MM_COLLINEAR

|- ((((scs_k_v39 s) = k) /\ ((is_scs_v39 s) /\ (((MMs_v39 s) w) /\ (((NUMERAL (BIT1 (BIT1 _0))) < k) /\ (((w l) = v2) /\ (((w (SUC l)) = v1) /\ ((((real_pow (vector_norm v1)) (NUMERAL (BIT0 (BIT1 _0)))) = x1) /\ ((((real_pow (vector_norm v2)) (NUMERAL (BIT0 (BIT1 _0)))) = x2) /\ (((real_pow (vector_norm ((vector_sub v1) v2))) (NUMERAL (BIT0 (BIT1 _0)))) = x6))))))))) ==> (((real_lt (real_of_num (NUMERAL _0))) x1) /\ (((real_lt (real_of_num (NUMERAL _0))) x2) /\ (((real_lt (real_of_num (NUMERAL _0))) x6) /\ (~ (collinear ((vec (NUMERAL _0)) INSERT (v1 INSERT (v2 INSERT EMPTY)))))))))


Fnjlbxs.lim_not_equal

|- (!a. (!b. (!f. (!g. (((~ (a = b)) /\ ((((--> f) a) sequentially) /\ (((--> g) b) sequentially))) ==> (?N. (!n. ((N <= n) ==> (~ ((f n) = (g n)))))))))))


Ssrnat.distnEl

|- (!m. (!n. ((n <= m) ==> (((distn m) n) = (m - n)))))


HOMEOMORPHISM_GRAPH_EXPLICIT

|- (!f. (!s. (((homeomorphism (s , (GSPEC (\GEN%PVAR%1033. (?x. (((SETSPEC GEN%PVAR%1033) (x IN s)) ((pastecart x) (f x)))))))) ((\x. ((pastecart x) (f x))) , fstcart)) = ((continuous_on f) s))))


COUNTABLE_EMPTY

|- (COUNTABLE EMPTY)


HAS_COMPLEX_DERIVATIVE_CTAN

|- (!z. ((~ ((ccos z) = (Cx (real_of_num (NUMERAL _0))))) ==> (((has_complex_derivative ctan) (complex_inv ((complex_pow (ccos z)) (NUMERAL (BIT0 (BIT1 _0)))))) (at z))))


outside

|- (!s. ((outside s) = (GSPEC (\GEN%PVAR%2007. (?x. (((SETSPEC GEN%PVAR%2007) ((~ (x IN s)) /\ (~ (bounded ((connected_component (UNIV DIFF s)) x))))) x))))))


INT_FORALL_ABS

|- (!P. ((!n. (P (int_of_num n))) = (!x. (P (int_abs x)))))


DIST_LIFT

|- (!x. (!y. ((distance ((lift x) , (lift y))) = (real_abs ((real_sub x) y)))))


Collect_geom.DELTA_X34

|- (!x12. (!x13. (!x14. (!x23. (!x24. (!x34. (((((((delta x12) x13) x14) x23) x24) x34) = ((real_add ((real_mul (real_neg x12)) ((real_pow x34) (NUMERAL (BIT0 (BIT1 _0)))))) ((real_add ((real_mul ((real_add ((real_mul (real_neg x13)) x14)) ((real_add ((real_mul (real_neg x23)) x24)) ((real_add ((real_mul x13) x24)) ((real_add ((real_mul x14) x23)) ((real_add ((real_mul (real_neg x12)) x12)) ((real_add ((real_mul x12) x14)) ((real_add ((real_mul x12) x24)) ((real_add ((real_mul x12) x13)) ((real_mul x12) x23)))))))))) x34)) ((real_add ((real_mul ((real_mul x14) x23)) x12)) ((real_add ((real_mul ((real_mul x14) x23)) x13)) ((real_add ((real_mul ((real_mul (real_neg x14)) x23)) x14)) ((real_add ((real_mul ((real_mul (real_neg x14)) x23)) x23)) ((real_add ((real_mul ((real_mul x14) x23)) x24)) ((real_add ((real_mul ((real_mul (real_neg x12)) x13)) x23)) ((real_add ((real_mul ((real_mul (real_neg x12)) x14)) x24)) ((real_add ((real_mul ((real_mul x13) x24)) x12)) ((real_add ((real_mul ((real_mul (real_neg x13)) x24)) x13)) ((real_add ((real_mul ((real_mul x13) x24)) x14)) ((real_add ((real_mul ((real_mul x13) x24)) x23)) ((real_mul ((real_mul (real_neg x13)) x24)) x24)))))))))))))))))))))


PATH_COMPONENT_IN

|- (!s. (!x. (!y. ((((path_component s) x) y) ==> ((x IN s) /\ (y IN s))))))


Seq.rotr

|- (!n. (!s. (((rotr n) s) = ((rot ((sizel s) - n)) s))))


CARD_LE_CONNECTED_COMPONENTS

|- (!f. (!s. (((continuous_on f) s) ==> ((GSPEC (\GEN%PVAR%1819. (?y. (((SETSPEC GEN%PVAR%1819) (y IN ((IMAGE f) s))) ((connected_component ((IMAGE f) s)) y))))) <=_c (GSPEC (\GEN%PVAR%1820. (?x. (((SETSPEC GEN%PVAR%1820) (x IN s)) ((connected_component s) x)))))))))


REAL_LE_MUL2

|- (!w. (!x. (!y. (!z. ((((real_le (real_of_num (NUMERAL _0))) w) /\ (((real_le w) x) /\ (((real_le (real_of_num (NUMERAL _0))) y) /\ ((real_le y) z)))) ==> ((real_le ((real_mul w) y)) ((real_mul x) z)))))))


HOMEOMORPHIC_TRANSLATION_LEFT_EQ

|- (!a. (!s. (!t. (((homeomorphic ((IMAGE (\x. ((vector_add a) x))) s)) t) = ((homeomorphic s) t)))))


rcone_gt

|- (rcone_gt = (rconesgn real_gt))


SPAN_LINEAR_IMAGE

|- (!f. (!s. ((linear f) ==> ((span ((IMAGE f) s)) = ((IMAGE f) (span s))))))


LIFT_NUM

|- (!n. ((lift (real_of_num n)) = (vec n)))


HAS_INTEGRAL_INTEGRAL

|- (!f. (!s. (((integrable_on f) s) = (((has_integral f) ((integral s) f)) s))))


MULT_EQ_0

|- (!m. (!n. (((m * n) = (NUMERAL _0)) = ((m = (NUMERAL _0)) \/ (n = (NUMERAL _0))))))


HAS_REAL_DERIVATIVE_LMUL_ATREAL

|- (!f. (!f'. (!c. (!x. ((((has_real_derivative f) f') (atreal x)) ==> (((has_real_derivative (\x. ((real_mul c) (f x)))) ((real_mul c) f')) (atreal x)))))))


REAL_COMPACT_CONTINUOUS_IMAGE

|- (!f. (!s. ((((real_continuous_on f) s) /\ (real_compact s)) ==> (real_compact ((IMAGE f) s)))))


INTER_COMM

|- (!s. (!t. ((s INTER t) = (t INTER s))))


Ssrnat.addn_eq0

|- (!m. (!n. (((m + n) = (NUMERAL _0)) = ((m = (NUMERAL _0)) /\ (n = (NUMERAL _0))))))


LOCALLY_PATH_CONNECTED_SPHERE

|- (!a. (!r. ((locally path_connected) (sphere (a , r)))))


COMPACT_EMPTY

|- (compact EMPTY)


Fnjlbxs.list_of_vector

|- (!v. (!d. (!n. ((((list_of_vector d) n) v) = ((TABLE (\k. (lambdai. (v $ ((k * d) + i))))) n)))))


COMPLEX_SUB_REFL

|- (!x. (((vector_sub x) x) = (Cx (real_of_num (NUMERAL _0)))))


LT_SUC_LE

|- (!m. (!n. ((m < (SUC n)) = (m <= n))))


FRONTIER_SUBSET_EQ

|- (!s. (((frontier s) SUBSET s) = (closed s)))


Hypermap_iso.iso_face_set

|- (!G. (!f. (!H. (((hyp_iso f) (H , G)) ==> ((face_set G) = ((IMAGE (IMAGE f)) (face_set H)))))))


MEASURABLE_SCALING

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


SPAN_MONO

|- (!s. (!t. ((s SUBSET t) ==> ((span s) SUBSET (span t)))))


More_list_hypermap.node_hypermap_of_list_explicit

|- (!L. (!n. (!d. (((good_list_nodes L) /\ (((MEM n) (list_of_nodes L)) /\ ((MEM d) n))) ==> (((node (hypermap_of_list L)) d) = (set_of_list n))))))


CLOSED_UNION

|- (!s. (!t. (((closed s) /\ (closed t)) ==> (closed (s UNION t)))))


IMAGE_CLAUSES

|- ((((IMAGE f) EMPTY) = EMPTY) /\ (((IMAGE f) (x INSERT s)) = ((f x) INSERT ((IMAGE f) s))))


HAS_REAL_DERIVATIVE_ID

|- (!net. (((has_real_derivative (\x. x)) (real_of_num (NUMERAL (BIT1 _0)))) net))


BOUNDED_UNION

|- (!s. (!t. ((bounded (s UNION t)) = ((bounded s) /\ (bounded t)))))


AFFINE_SCALING

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


CARD_LT_IMP_LE

|- (!s. (!t. ((s <_c t) ==> (s <=_c t))))


Ssrbool.contraLR

|- (!c. (!b. (((~ c) ==> (~ b)) ==> (b ==> c))))


JOINPATHS_LINEAR_IMAGE

|- (!f. (!g1. (!g2. ((linear f) ==> (((++ (f o g1)) (f o g2)) = (f o ((++ g1) g2)))))))


AFFINE_DEPENDENT_LINEAR_IMAGE_EQ

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


MULT_AC

|- (((m * n) = (n * m)) /\ ((((m * n) * p) = (m * (n * p))) /\ ((m * (n * p)) = (n * (m * p)))))


ORTHOGONAL_TRANSFORMATION_NEG

|- (!f. ((orthogonal_transformation (\x. (vector_neg (f x)))) = (orthogonal_transformation f)))


Miqmcsn.PSORT_5_EXPLICIT

|- ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL _0) , (NUMERAL _0))) = ((NUMERAL _0) , (NUMERAL _0))) /\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT1 _0)) , (NUMERAL (BIT1 _0)))) = ((NUMERAL (BIT1 _0)) , (NUMERAL (BIT1 _0)))) /\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT0 (BIT1 _0))) , (NUMERAL (BIT0 (BIT1 _0))))) = ((NUMERAL (BIT0 (BIT1 _0))) , (NUMERAL (BIT0 (BIT1 _0))))) /\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT1 (BIT1 _0))) , (NUMERAL (BIT1 (BIT1 _0))))) = ((NUMERAL (BIT1 (BIT1 _0))) , (NUMERAL (BIT1 (BIT1 _0))))) /\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT0 (BIT0 (BIT1 _0)))) , (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) = ((NUMERAL (BIT0 (BIT0 (BIT1 _0)))) , (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) /\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL _0) , (NUMERAL (BIT1 _0)))) = ((NUMERAL _0) , (NUMERAL (BIT1 _0)))) /\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL _0) , (NUMERAL (BIT0 (BIT1 _0))))) = ((NUMERAL _0) , (NUMERAL (BIT0 (BIT1 _0))))) /\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL _0) , (NUMERAL (BIT1 (BIT1 _0))))) = ((NUMERAL _0) , (NUMERAL (BIT1 (BIT1 _0))))) /\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL _0) , (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) = ((NUMERAL _0) , (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) /\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT1 _0)) , (NUMERAL _0))) = ((NUMERAL _0) , (NUMERAL (BIT1 _0)))) /\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT1 _0)) , (NUMERAL (BIT0 (BIT1 _0))))) = ((NUMERAL (BIT1 _0)) , (NUMERAL (BIT0 (BIT1 _0))))) /\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT1 _0)) , (NUMERAL (BIT1 (BIT1 _0))))) = ((NUMERAL (BIT1 _0)) , (NUMERAL (BIT1 (BIT1 _0))))) /\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT1 _0)) , (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) = ((NUMERAL (BIT1 _0)) , (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) /\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT0 (BIT1 _0))) , (NUMERAL _0))) = ((NUMERAL _0) , (NUMERAL (BIT0 (BIT1 _0))))) /\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT0 (BIT1 _0))) , (NUMERAL (BIT1 _0)))) = ((NUMERAL (BIT1 _0)) , (NUMERAL (BIT0 (BIT1 _0))))) /\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT0 (BIT1 _0))) , (NUMERAL (BIT1 (BIT1 _0))))) = ((NUMERAL (BIT0 (BIT1 _0))) , (NUMERAL (BIT1 (BIT1 _0))))) /\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT0 (BIT1 _0))) , (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) = ((NUMERAL (BIT0 (BIT1 _0))) , (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) /\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT1 (BIT1 _0))) , (NUMERAL _0))) = ((NUMERAL _0) , (NUMERAL (BIT1 (BIT1 _0))))) /\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT1 (BIT1 _0))) , (NUMERAL (BIT1 _0)))) = ((NUMERAL (BIT1 _0)) , (NUMERAL (BIT1 (BIT1 _0))))) /\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT1 (BIT1 _0))) , (NUMERAL (BIT0 (BIT1 _0))))) = ((NUMERAL (BIT0 (BIT1 _0))) , (NUMERAL (BIT1 (BIT1 _0))))) /\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT1 (BIT1 _0))) , (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) = ((NUMERAL (BIT1 (BIT1 _0))) , (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) /\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT0 (BIT0 (BIT1 _0)))) , (NUMERAL _0))) = ((NUMERAL _0) , (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) /\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT0 (BIT0 (BIT1 _0)))) , (NUMERAL (BIT1 _0)))) = ((NUMERAL (BIT1 _0)) , (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) /\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT0 (BIT0 (BIT1 _0)))) , (NUMERAL (BIT0 (BIT1 _0))))) = ((NUMERAL (BIT0 (BIT1 _0))) , (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) /\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT0 (BIT0 (BIT1 _0)))) , (NUMERAL (BIT1 (BIT1 _0))))) = ((NUMERAL (BIT1 (BIT1 _0))) , (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) /\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT0 (BIT0 (BIT1 _0)))) , (NUMERAL (BIT1 (BIT0 (BIT1 _0)))))) = ((NUMERAL _0) , (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) /\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT1 (BIT1 _0))) , (NUMERAL (BIT1 (BIT0 (BIT1 _0)))))) = ((NUMERAL _0) , (NUMERAL (BIT1 (BIT1 _0))))) /\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT0 (BIT1 _0))) , (NUMERAL (BIT1 (BIT0 (BIT1 _0)))))) = ((NUMERAL _0) , (NUMERAL (BIT0 (BIT1 _0))))) /\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT1 _0)) , (NUMERAL (BIT1 (BIT0 (BIT1 _0)))))) = ((NUMERAL _0) , (NUMERAL (BIT1 _0)))) /\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT1 (BIT0 (BIT1 _0)))) , (NUMERAL (BIT1 _0)))) = ((NUMERAL _0) , (NUMERAL (BIT1 _0)))) /\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT1 (BIT0 (BIT1 _0)))) , (NUMERAL (BIT0 (BIT1 _0))))) = ((NUMERAL _0) , (NUMERAL (BIT0 (BIT1 _0))))) /\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT1 (BIT0 (BIT1 _0)))) , (NUMERAL (BIT1 (BIT1 _0))))) = ((NUMERAL _0) , (NUMERAL (BIT1 (BIT1 _0))))) /\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT1 (BIT0 (BIT1 _0)))) , (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) = ((NUMERAL _0) , (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) /\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT1 (BIT0 (BIT1 _0)))) , (NUMERAL (BIT1 (BIT0 (BIT1 _0)))))) = ((NUMERAL _0) , (NUMERAL _0))) /\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT1 (BIT0 (BIT1 _0)))) , (NUMERAL (BIT0 (BIT1 (BIT1 _0)))))) = ((NUMERAL _0) , (NUMERAL (BIT1 _0)))) /\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT1 (BIT0 (BIT1 _0)))) , (NUMERAL (BIT1 (BIT1 (BIT1 _0)))))) = ((NUMERAL _0) , (NUMERAL (BIT0 (BIT1 _0))))) /\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT0 (BIT0 (BIT1 _0)))) , (NUMERAL (BIT0 (BIT1 (BIT1 _0)))))) = ((NUMERAL (BIT1 _0)) , (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) /\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT0 (BIT1 (BIT1 _0)))) , (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) = ((NUMERAL (BIT1 _0)) , (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) /\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT0 (BIT1 (BIT1 _0)))) , (NUMERAL (BIT1 (BIT0 (BIT1 _0)))))) = ((NUMERAL _0) , (NUMERAL (BIT1 _0)))) /\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT0 (BIT1 (BIT1 _0)))) , (NUMERAL (BIT1 (BIT1 (BIT1 _0)))))) = ((NUMERAL (BIT1 _0)) , (NUMERAL (BIT0 (BIT1 _0))))) /\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT1 (BIT1 (BIT1 _0)))) , (NUMERAL (BIT1 (BIT0 (BIT1 _0)))))) = ((NUMERAL _0) , (NUMERAL (BIT0 (BIT1 _0))))) /\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT1 (BIT1 (BIT1 _0)))) , (NUMERAL (BIT0 (BIT1 (BIT1 _0)))))) = ((NUMERAL (BIT1 _0)) , (NUMERAL (BIT0 (BIT1 _0))))) /\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT1 (BIT1 (BIT1 _0)))) , (NUMERAL (BIT1 (BIT1 (BIT1 _0)))))) = ((NUMERAL (BIT0 (BIT1 _0))) , (NUMERAL (BIT0 (BIT1 _0))))) /\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT0 (BIT1 (BIT1 _0)))) , (NUMERAL (BIT0 (BIT1 (BIT1 _0)))))) = ((NUMERAL (BIT1 _0)) , (NUMERAL (BIT1 _0)))) /\ ((((psort (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT1 (BIT1 _0))) , (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) = ((NUMERAL _0) , (NUMERAL (BIT1 (BIT1 _0))))) /\ ((((psort (NUMERAL (BIT1 (BIT1 _0)))) ((NUMERAL (BIT0 (BIT1 _0))) , (NUMERAL _0))) = ((NUMERAL _0) , (NUMERAL (BIT0 (BIT1 _0))))) /\ ((((psort (NUMERAL (BIT1 (BIT1 _0)))) ((NUMERAL (BIT0 (BIT1 _0))) , (NUMERAL (BIT1 _0)))) = ((NUMERAL (BIT1 _0)) , (NUMERAL (BIT0 (BIT1 _0))))) /\ ((((psort (NUMERAL (BIT1 (BIT1 _0)))) ((NUMERAL (BIT1 _0)) , (NUMERAL _0))) = ((NUMERAL _0) , (NUMERAL (BIT1 _0)))) /\ ((((psort (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) ((NUMERAL _0) , (NUMERAL _0))) = ((NUMERAL _0) , (NUMERAL _0))) /\ ((((psort (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT1 _0)) , (NUMERAL (BIT1 _0)))) = ((NUMERAL (BIT1 _0)) , (NUMERAL (BIT1 _0)))) /\ ((((psort (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT0 (BIT1 _0))) , (NUMERAL (BIT0 (BIT1 _0))))) = ((NUMERAL (BIT0 (BIT1 _0))) , (NUMERAL (BIT0 (BIT1 _0))))) /\ ((((psort (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT1 (BIT1 _0))) , (NUMERAL (BIT1 (BIT1 _0))))) = ((NUMERAL (BIT1 (BIT1 _0))) , (NUMERAL (BIT1 (BIT1 _0))))) /\ ((((psort (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT0 (BIT0 (BIT1 _0)))) , (NUMERAL (BIT1 (BIT1 _0))))) = ((NUMERAL _0) , (NUMERAL (BIT1 (BIT1 _0))))) /\ ((((psort (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT0 (BIT0 (BIT1 _0)))) , (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) = ((NUMERAL _0) , (NUMERAL _0))) /\ ((((psort (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) ((NUMERAL _0) , (NUMERAL (BIT0 (BIT1 _0))))) = ((NUMERAL _0) , (NUMERAL (BIT0 (BIT1 _0))))) /\ ((((psort (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT0 (BIT0 (BIT1 _0)))) , (NUMERAL (BIT1 (BIT0 (BIT1 _0)))))) = ((NUMERAL _0) , (NUMERAL (BIT1 _0)))) /\ ((((psort (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT1 (BIT0 (BIT1 _0)))) , (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) = ((NUMERAL _0) , (NUMERAL (BIT1 _0)))) /\ ((((psort (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT1 (BIT0 (BIT1 _0)))) , (NUMERAL (BIT1 (BIT0 (BIT1 _0)))))) = ((NUMERAL (BIT1 _0)) , (NUMERAL (BIT1 _0)))) /\ ((((psort (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT1 _0)) , (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) = ((NUMERAL _0) , (NUMERAL (BIT1 _0)))) /\ ((((psort (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT0 (BIT1 _0))) , (NUMERAL (BIT1 (BIT0 (BIT1 _0)))))) = ((NUMERAL (BIT1 _0)) , (NUMERAL (BIT0 (BIT1 _0))))) /\ (((psort (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT1 (BIT1 _0))) , (NUMERAL (BIT0 (BIT1 (BIT1 _0)))))) = ((NUMERAL (BIT0 (BIT1 _0))) , (NUMERAL (BIT1 (BIT1 _0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))


RESTRICTION_IN_EXTENSIONAL

|- (!s. (!f. (((RESTRICTION s) f) IN (EXTENSIONAL s))))


NOT_EVEN

|- (!n. ((~ (EVEN n)) = (ODD n)))


between

|- (!a. (!x. (!b. (((between x) (a , b)) = ((distance (a , b)) = ((real_add (distance (a , x))) (distance (x , b))))))))


PATHFINISH_TRANSLATION

|- (!a. (!g. ((pathfinish ((\x. ((vector_add a) x)) o g)) = ((vector_add a) (pathfinish g)))))


CARD_EXP_POWERSET

|- (!s. (((^_c UNIV) s) =_c (GSPEC (\GEN%PVAR%40. (?t. (((SETSPEC GEN%PVAR%40) (t SUBSET s)) t))))))


RPOW_POW

|- (!x. (!n. (((rpow x) (real_of_num n)) = ((real_pow x) n))))


Tame_general.CONTRAVENING_FAN

|- (!V. ((contravening V) ==> (FAN ((vec (NUMERAL _0)) , (V , (ESTD V))))))


VECTOR_MUL_COMPONENT

|- (!c. (!x. (!i. ((((% c) x) $ i) = ((real_mul c) (x $ i))))))


LOCALLY_EMPTY

|- (!P. ((locally P) EMPTY))


ABSOLUTELY_REAL_INTEGRABLE_INCREASING_PRODUCT

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


MEMBER_NOT_EMPTY

|- (!s. ((?x. (x IN s)) = (~ (s = EMPTY))))


LAMBDA_ETA

|- (!g. ((lambdai. (g $ i)) = g))


CARD_LE_INJ

|- (!s. (!t. (((FINITE s) /\ ((FINITE t) /\ ((CARD s) <= (CARD t)))) ==> (?f. ((((IMAGE f) s) SUBSET t) /\ (!x. (!y. (((x IN s) /\ ((y IN s) /\ ((f x) = (f y)))) ==> (x = y)))))))))


CARD_EQ_CONG

|- (!s. (!s'. (!t. (!t'. (((s =_c s') /\ (t =_c t')) ==> ((s =_c t) = (s' =_c t')))))))


midpoint

|- (!a. (!b. ((midpoint (a , b)) = ((% (real_inv (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) ((vector_add a) b)))))


CONTINUOUS_ON_COMPOSE

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


SYMMETRIC_MATRIX_INV

|- (!A. (((transp (matrix_inv A)) = (matrix_inv A)) = ((transp A) = A)))


AFFINE_PCROSS

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


Nonlin_def.delta_x4

|- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((((((delta_x4 x1) x2) x3) x4) x5) x6) = ((real_add ((real_sub ((real_mul (real_neg x2)) x3)) ((real_mul x1) x4))) ((real_add ((real_mul x2) x5)) ((real_add ((real_sub ((real_mul x3) x6)) ((real_mul x5) x6))) ((real_mul x1) ((real_add (real_neg x1)) ((real_add x2) ((real_add ((real_sub x3) x4)) ((real_add x5) x6)))))))))))))))


HAS_BOUNDED_VARIATION_ON_COMPONENTWISE

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


INSIDE_EMPTY

|- ((inside EMPTY) = EMPTY)


CONTINUOUS_ON_CX_LIFT

|- (!f. (!s. (((continuous_on (\x. (Cx (f x)))) s) = ((continuous_on (\x. (lift (f x)))) s))))


HOMOTOPIC_PATHS_IMP_PATH

|- (!s. (!p. (!q. ((((homotopic_paths s) p) q) ==> ((path p) /\ (path q))))))


CONVEX_HALFSPACE_LE

|- (!a. (!b. (convex (GSPEC (\GEN%PVAR%1428. (?x. (((SETSPEC GEN%PVAR%1428) ((real_le ((dot a) x)) b)) x)))))))


ORTHOGONAL_MUL

|- ((!a. (!x. (!y. (((orthogonal ((% a) x)) y) = ((a = (real_of_num (NUMERAL _0))) \/ ((orthogonal x) y)))))) /\ (!a. (!x. (!y. (((orthogonal x) ((% a) y)) = ((a = (real_of_num (NUMERAL _0))) \/ ((orthogonal x) y)))))))


Seq.rot_add_mod

|- (!m. (!n. (!s. ((n <= (sizel s)) ==> ((m <= (sizel s)) ==> (((rot m) ((rot n) s)) = ((rot (((COND ((m + n) <= (sizel s))) (m + n)) ((m + n) - (sizel s)))) s)))))))


REAL_MUL_RINV

|- (!x. ((~ (x = (real_of_num (NUMERAL _0)))) ==> (((real_mul x) (real_inv x)) = (real_of_num (NUMERAL (BIT1 _0))))))


UNION_COMM

|- (!s. (!t. ((s UNION t) = (t UNION s))))


REAL_OF_NUM_EQ

|- (!m. (!n. (((real_of_num m) = (real_of_num n)) = (m = n))))


CONTINUOUS_ADD

|- (!f. (!g. (!net. ((((continuous f) net) /\ ((continuous g) net)) ==> ((continuous (\x. ((vector_add (f x)) (g x)))) net)))))


Ssrbool.predU

|- (!p1. (!p2. (((predU p1) p2) = (\x. ((p1 x) \/ (p2 x))))))


HOMOTOPIC_PATHS_JOIN

|- (!s. (!p. (!q. (!p'. (!q'. (((((homotopic_paths s) p) p') /\ ((((homotopic_paths s) q) q') /\ ((pathfinish p) = (pathstart q)))) ==> (((homotopic_paths s) ((++ p) q)) ((++ p') q'))))))))


REAL_SUM_INTEGRAL_LBOUND_INCREASING

|- (!f. (!g. (!m. (!n. (((m <= n) /\ ((!x. ((x IN (closed_real_interval ((CONS (((real_sub (real_of_num m)) (real_of_num (NUMERAL (BIT1 _0)))) , (real_of_num n))) NIL))) ==> (((has_real_derivative g) (f x)) ((within (atreal x)) (closed_real_interval ((CONS (((real_sub (real_of_num m)) (real_of_num (NUMERAL (BIT1 _0)))) , (real_of_num n))) NIL)))))) /\ (!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_of_num n)))) ==> ((real_le (f x)) (f y))))))) ==> ((real_le ((real_sub (g (real_of_num n))) (g ((real_sub (real_of_num m)) (real_of_num (NUMERAL (BIT1 _0))))))) ((sum (m .. n)) (\k. (f (real_of_num k))))))))))


DIMINDEX_NONZERO

|- (!s. (~ ((dimindex s) = (NUMERAL _0))))


Seq2.butlast

|- (!h. (!h2. (!t. (((butlast NIL) = NIL) /\ (((butlast ((CONS h) NIL)) = NIL) /\ ((butlast ((CONS h) ((CONS h2) t))) = ((CONS h) (butlast ((CONS h2) t)))))))))


Ssrnat.leq_eqVlt

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


Seq.size

|- (sizel = LENGTH)


FINITE_NUMSEG

|- (!m. (!n. (FINITE (m .. n))))


Ssrnat.sqrn_sub

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


FORALL_COMPLEX

|- ((!z. (P z)) = (!x. (!y. (P (complex (x , y))))))


Add_triangle.fan_split_fan_face

|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> (!v. (!w. (((v , w) IN (dart_of_fan (V , E))) ==> ((FAN ((vec (NUMERAL _0)) , ((split_fan_face (V , E)) (v , w)))) /\ (((dart_of_fan ((split_fan_face (V , E)) (v , w))) = ((dart1_of_fan (V , E)) UNION (((((((sigma_fan (vec (NUMERAL _0))) V) E) v) w) , w) INSERT ((w , (((((sigma_fan (vec (NUMERAL _0))) V) E) v) w)) INSERT EMPTY)))) /\ (fully_surrounded ((split_fan_face (V , E)) (v , w))))))))))))


AFFINE_HULL_EQ_EMPTY

|- (!s. ((((hull affine) s) = EMPTY) = (s = EMPTY)))


CEXP_CLOG

|- (!z. ((~ (z = (Cx (real_of_num (NUMERAL _0))))) ==> ((cexp (clog z)) = z)))


Appendix.periodic_empty

|- (!n. ((periodic EMPTY) n))


Collect_geom.DELTA_RRR_INTERPRETE

|- (((((((delta r) r) r) a) b) c) = ((real_add ((real_mul (real_neg a)) ((real_mul b) c))) ((real_mul r) (((ups_x a) b) c))))


MEASURE_TRANSLATION

|- (!a. (!s. ((measure ((IMAGE (\x. ((vector_add a) x))) s)) = (measure s))))


REAL_INV_0

|- ((real_inv (real_of_num (NUMERAL _0))) = (real_of_num (NUMERAL _0)))


MBOUNDED

|- (!m. (!s. (((mbounded m) s) = ((s = EMPTY) \/ ((!x. ((x IN s) ==> (x IN (mspace m)))) /\ (?c. (?b. ((c IN (mspace m)) /\ (!x. ((x IN s) ==> ((real_le ((mdist m) (c , x))) b)))))))))))


CLOSED_CLOSURE

|- (!s. (closed (closure s)))


LIM_TRANSFORM_WITHIN_OPEN

|- (!f. (!g. (!s. (!a. (!l. (((open s) /\ ((a IN s) /\ ((!x. (((x IN s) /\ (~ (x = a))) ==> ((f x) = (g x)))) /\ (((--> f) l) (at a))))) ==> (((--> g) l) (at a))))))))


ADD_0

|- (!m. ((m + (NUMERAL _0)) = m))


Hypermap.face_refl

|- (!H. (!x. (x IN ((face H) x))))


SUPPORTING_HYPERPLANE_FRONTIER

|- (!s. (!x. (((convex s) /\ (x IN (frontier s))) ==> (?a. ((~ (a = (vec (NUMERAL _0)))) /\ (!y. ((y IN (closure s)) ==> ((real_le ((dot a) x)) ((dot a) y)))))))))


LIM_WITHIN

|- (!f. (!l. (!a. (!s. ((((--> f) l) ((within (at 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))) (distance (x , a))) /\ ((real_lt (distance (x , a))) d))) ==> ((real_lt (distance ((f x) , l))) e))))))))))))


PASTECART_ADD

|- (!x1. (!y1. (!x2. (!y2. (((vector_add ((pastecart x1) y1)) ((pastecart x2) y2)) = ((pastecart ((vector_add x1) x2)) ((vector_add y1) y2)))))))


RIGHT_FORALL_OR_THM

|- (!P. (!Q. ((!x. (P \/ (Q x))) = (P \/ (!x. (Q x))))))


VECTOR_ADD_AC

|- ((((vector_add m) n) = ((vector_add n) m)) /\ ((((vector_add ((vector_add m) n)) p) = ((vector_add m) ((vector_add n) p))) /\ (((vector_add m) ((vector_add n) p)) = ((vector_add n) ((vector_add m) p)))))


SETDIST_POS_LE

|- (!s. (!t. ((real_le (real_of_num (NUMERAL _0))) (setdist (s , t)))))


SERIES_DROP_LE

|- (!f. (!g. (!s. (!a. (!b. (((((sums f) a) s) /\ ((((sums g) b) s) /\ (!x. ((x IN s) ==> ((real_le (drop (f x))) (drop (g x))))))) ==> ((real_le (drop a)) (drop b))))))))


IN_INTERVAL

|- ((!x. ((x IN (open_interval (a , b))) = (!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV))) ==> (((real_lt (a $ i)) (x $ i)) /\ ((real_lt (x $ i)) (b $ i))))))) /\ (!x. ((x IN (closed_interval ((CONS (a , b)) NIL))) = (!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV))) ==> (((real_le (a $ i)) (x $ i)) /\ ((real_le (x $ i)) (b $ i))))))))


has_derivative

|- (!f. (!f'. (!net. ((((has_derivative f) f') net) = ((linear f') /\ (((--> (\y. ((% (real_inv (vector_norm ((vector_sub y) (netlimit net))))) ((vector_sub (f y)) ((vector_add (f (netlimit net))) (f' ((vector_sub y) (netlimit net)))))))) (vec (NUMERAL _0))) net))))))


INTEGER_SUB

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


rconesgn

|- (!sgn. (!w. (!v. (!h. (((((rconesgn sgn) v) w) h) = (GSPEC (\GEN%PVAR%5166. (?x. (((SETSPEC GEN%PVAR%5166) ((sgn ((dot ((vector_sub x) v)) ((vector_sub w) v))) ((real_mul (distance (x , v))) ((real_mul (distance (w , v))) h)))) x)))))))))


OPERATIVE_FUNCTION_ENDPOINT_DIFF

|- (!f. ((operative vector_add) (\k. ((vector_sub (f (interval_upperbound k))) (f (interval_lowerbound k))))))


Kepler_spec.atn_1_series

|- (!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))))))))


REAL_INTEGRABLE_SPIKE

|- (!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)))))))


Seq2.mem_delete1_uniq

|- (!x. (!s. (((uniq s) /\ ((MEM x) s)) ==> (!y. (((MEM y) ((delete1 x) s)) = ((~ (y = x)) /\ ((MEM y) s)))))))


Ssrbool.andbA

|- (!b. (!c. (!p. ((b /\ (c /\ p)) = ((b /\ c) /\ p)))))


Ssrnat.subn_eq0

|- (!m. (!n. (((m - n) = (NUMERAL _0)) = (m <= n))))


REAL_LE_LSQRT

|- (!x. (!y. ((((real_le (real_of_num (NUMERAL _0))) x) /\ (((real_le (real_of_num (NUMERAL _0))) y) /\ ((real_le x) ((real_pow y) (NUMERAL (BIT0 (BIT1 _0))))))) ==> ((real_le (sqrt x)) y))))


REAL_NEG_ADD

|- (!x. (!y. ((real_neg ((real_add x) y)) = ((real_add (real_neg x)) (real_neg y)))))


INTEGRAL_COMBINE

|- (!f. (!a. (!b. (!c. ((((real_le (drop a)) (drop c)) /\ (((real_le (drop c)) (drop b)) /\ ((integrable_on f) (closed_interval ((CONS (a , b)) NIL))))) ==> (((vector_add ((integral (closed_interval ((CONS (a , c)) NIL))) f)) ((integral (closed_interval ((CONS (c , b)) NIL))) f)) = ((integral (closed_interval ((CONS (a , b)) NIL))) f)))))))


Pack2.PACKING

|- (!s. ((packing s) = (!u. (!v. (((u IN s) /\ ((v IN s) /\ ((real_lt (distance (u , v))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) ==> (u = v))))))


COMPLEX_CMUL

|- (!c. (!x. (((% c) x) = ((complex_mul (Cx c)) x))))


Kepler_spec.abs_spec

|- (!x. ((((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)))))


VECTOR_VARIATION_COMBINE

|- (!f. (!a. (!b. (!c. ((((real_le (drop a)) (drop c)) /\ (((real_le (drop c)) (drop b)) /\ ((has_bounded_variation_on f) (closed_interval ((CONS (a , b)) NIL))))) ==> (((real_add ((vector_variation (closed_interval ((CONS (a , c)) NIL))) f)) ((vector_variation (closed_interval ((CONS (c , b)) NIL))) f)) = ((vector_variation (closed_interval ((CONS (a , b)) NIL))) f)))))))


OUTSIDE_LINEAR_IMAGE

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


PERMUTATION_PERMUTES

|- (!p. ((permutation p) = (?s. ((FINITE s) /\ ((permutes p) s)))))


Appendix.scs_b_v39_explicit

|- (!k. (!d. (!a. (!b. (!alpha. (!beta. (!J. (!lo. (!str. ((scs_b_v39 (scs_v39 (k , (d , (a , (alpha , (beta , (b , (J , (lo , (hi , str))))))))))) = b))))))))))


Functional_equation.sub6

|- (!f. (!g. (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((((((((sub6 f) g) x1) x2) x3) x4) x5) x6) = ((real_sub ((((((f x1) x2) x3) x4) x5) x6)) ((((((g x1) x2) x3) x4) x5) x6)))))))))))


PATH_CONNECTED_CONTINUOUS_IMAGE

|- (!f. (!s. ((((continuous_on f) s) /\ (path_connected s)) ==> (path_connected ((IMAGE f) s)))))


FSIGMA_PCROSS

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


REAL_EXP_MONO_LE

|- (!x. (!y. (((real_le (exp x)) (exp y)) = ((real_le x) y))))


TAN_PERIODIC_PI

|- (!x. ((tan ((real_add x) pi)) = (tan x)))


SUBSPACE_UNIV

|- (subspace UNIV)


HAS_BOUNDED_REAL_VARIATION_REFLECT_EQ

|- (!f. (!s. (((has_bounded_real_variation_on (\x. (f (real_neg x)))) s) = ((has_bounded_real_variation_on f) ((IMAGE real_neg) s)))))


MDIST

|- (!s. (!d. ((is_metric_space (s , d)) ==> ((mdist (metric (s , d))) = d))))


CONTINUOUS_ON_LIFT_DOT

|- (!s. ((continuous_on (lift o (\y. ((dot a) y)))) s))


Fan.POWER_0

|- (!f. (((POWER f) (NUMERAL _0)) = I))


COMPLEX_DIFFERENTIABLE_CPOW_RIGHT

|- (!w. (!z. ((complex_differentiable (\z. ((cpow w) z))) (at z))))


ENDS_IN_REAL_INTERVAL

|- ((!a. (!b. ((a IN (closed_real_interval ((CONS (a , b)) NIL))) = (~ ((closed_real_interval ((CONS (a , b)) NIL)) = EMPTY))))) /\ ((!a. (!b. ((b IN (closed_real_interval ((CONS (a , b)) NIL))) = (~ ((closed_real_interval ((CONS (a , b)) NIL)) = EMPTY))))) /\ ((!a. (!b. (~ (a IN (open_real_interval (a , b)))))) /\ (!a. (!b. (~ (b IN (open_real_interval (a , b)))))))))


Add_triangle.fully_surrounded_dart_of_fan_eq

|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((dart_of_fan (V , E)) = (dart1_of_fan (V , E)))))))


LIMPT_APPROACHABLE

|- (!x. (!s. (((limit_point_of x) s) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?x'. ((x' IN s) /\ ((~ (x' = x)) /\ ((real_lt (distance (x' , x))) e)))))))))


URYSOHN_LOCAL

|- (!s. (!t. (!u. (!a. (!b. ((((closed_in ((subtopology euclidean) u)) s) /\ (((closed_in ((subtopology euclidean) u)) t) /\ ((s INTER t) = EMPTY))) ==> (?f. (((continuous_on f) u) /\ ((!x. ((x IN u) ==> ((f x) IN (closed_segment ((CONS (a , b)) NIL))))) /\ ((!x. ((x IN s) ==> ((f x) = a))) /\ (!x. ((x IN t) ==> ((f x) = b)))))))))))))


REAL_LT_SUB_RADD

|- (!x. (!y. (!z. (((real_lt ((real_sub x) y)) z) = ((real_lt x) ((real_add z) y))))))


inside

|- (!s. ((inside s) = (GSPEC (\GEN%PVAR%2006. (?x. (((SETSPEC GEN%PVAR%2006) ((~ (x IN s)) /\ (bounded ((connected_component (UNIV DIFF s)) x)))) x))))))


COMPONENT_LE_NORM

|- (!x. (!i. ((real_le (real_abs (x $ i))) (vector_norm x))))


OPEN_IN_PCROSS_EQ

|- (!s. (!s'. (!t. (!t'. (((open_in ((subtopology euclidean) (s PCROSS t))) (s' PCROSS t')) = ((s' = EMPTY) \/ ((t' = EMPTY) \/ (((open_in ((subtopology euclidean) s)) s') /\ ((open_in ((subtopology euclidean) t)) t')))))))))


ABSOLUTELY_SETCONTINUOUS_COMPARISON

|- (!f. (!g. (!s. ((((absolutely_setcontinuous_on f) s) /\ (!a. (!b. (((~ ((closed_interval ((CONS (a , b)) NIL)) = EMPTY)) /\ ((closed_interval ((CONS (a , b)) NIL)) SUBSET s)) ==> ((real_le (vector_norm (g (closed_interval ((CONS (a , b)) NIL))))) (vector_norm (f (closed_interval ((CONS (a , b)) NIL))))))))) ==> ((absolutely_setcontinuous_on g) s)))))


Seq.belast

|- (!x. (!h. (!t. ((((belast x) NIL) = NIL) /\ (((belast x) ((CONS h) t)) = ((CONS x) ((belast h) t)))))))


SUM_0

|- (!s. (((sum s) (\n. (real_of_num (NUMERAL _0)))) = (real_of_num (NUMERAL _0))))


AGM_2

|- (!x. (!y. (!u. (!v. ((((real_le (real_of_num (NUMERAL _0))) x) /\ (((real_le (real_of_num (NUMERAL _0))) y) /\ (((real_le (real_of_num (NUMERAL _0))) u) /\ (((real_le (real_of_num (NUMERAL _0))) v) /\ (((real_add u) v) = (real_of_num (NUMERAL (BIT1 _0)))))))) ==> ((real_le ((real_mul ((rpow x) u)) ((rpow y) v))) ((real_add ((real_mul u) x)) ((real_mul v) y))))))))


ALL2

|- (((((ALL2 P) NIL) NIL) = T) /\ (((((ALL2 P) ((CONS h1) t1)) NIL) = F) /\ (((((ALL2 P) NIL) ((CONS h2) t2)) = F) /\ ((((ALL2 P) ((CONS h1) t1)) ((CONS h2) t2)) = (((P h1) h2) /\ (((ALL2 P) t1) t2))))))


RPOW_LE2

|- (!x. (!y. (!z. ((((real_le (real_of_num (NUMERAL _0))) x) /\ (((real_le x) y) /\ ((real_le (real_of_num (NUMERAL _0))) z))) ==> ((real_le ((rpow x) z)) ((rpow y) z))))))


CONIC_HULL_EXPLICIT

|- (!s. (((hull conic) s) = (GSPEC (\GEN%PVAR%1457. (?c. (?x. (((SETSPEC GEN%PVAR%1457) (((real_le (real_of_num (NUMERAL _0))) c) /\ (x IN s))) ((% c) x))))))))


Ssrnat.add0n

|- (!n. (((NUMERAL _0) + n) = n))


BALL_SUBSET_CBALL

|- (!x. (!e. ((ball (x , e)) SUBSET (cball (x , e)))))


ACS_COS

|- (!x. ((((real_le (real_of_num (NUMERAL _0))) x) /\ ((real_le x) pi)) ==> ((acs (cos x)) = x)))


LENGTH_LIST_OF_SET

|- (!s. ((FINITE s) ==> ((LENGTH (list_of_set s)) = (CARD s))))


CARD_EXP_ABSORB

|- (!s. (!t. (((INFINITE t) /\ ((UNIV <=_c s) /\ (s <=_c ((^_c UNIV) t)))) ==> (((^_c s) t) =_c ((^_c UNIV) t)))))


HAS_REAL_INTEGRAL_COMBINE

|- (!f. (!i. (!j. (!a. (!b. (!c. ((((real_le a) c) /\ (((real_le c) b) /\ ((((has_real_integral f) i) (closed_real_interval ((CONS (a , c)) NIL))) /\ (((has_real_integral f) j) (closed_real_interval ((CONS (c , b)) NIL)))))) ==> (((has_real_integral f) ((real_add i) j)) (closed_real_interval ((CONS (a , b)) NIL))))))))))


BOUNDED_INSERT

|- (!x. (!s. ((bounded (x INSERT s)) = (bounded s))))


CARD_LE_LT

|- (!s. (!t. ((s <=_c t) = ((s <_c t) \/ (s =_c t)))))


ABSOLUTELY_INTEGRABLE_CONVOLUTION_L1_LINF

|- (!bop. (!f. (!g. (!x. (((bilinear bop) /\ (((absolutely_integrable_on f) UNIV) /\ (((measurable_on g) UNIV) /\ (bounded ((IMAGE g) UNIV))))) ==> ((absolutely_integrable_on (\y. ((bop (f ((vector_sub x) y))) (g y)))) UNIV))))))


LINEAR_ID

|- (linear (\x. x))


CONTINUOUS_COUNTABLE_RANGE_CONSTANT_EQ

|- (!s. ((connected s) = (!f. ((((continuous_on f) s) /\ (COUNTABLE ((IMAGE f) s))) ==> (?a. (!x. ((x IN s) ==> ((f x) = a))))))))


REAL_EQ_LDIV_EQ

|- (!x. (!y. (!z. (((real_lt (real_of_num (NUMERAL _0))) z) ==> ((((real_div x) z) = y) = (x = ((real_mul y) z)))))))


Conforming.SIGMA_FAN_OF_FANADD_AT_POINT5

|- (!x. (!V. (!E. (!v. (!u. (!w. (!w1. (((FAN (x , (V , E))) /\ ((FAN (x , (V , E1))) /\ ((fan80 (x , (V , E))) /\ (((v INSERT (u INSERT EMPTY)) IN E) /\ (((u INSERT (w INSERT EMPTY)) IN E) /\ ((~ ((w INSERT (v INSERT EMPTY)) IN E)) /\ ((~ (w1 = (((((inverse1_sigma_fan x) V) E) w) u))) /\ (((w INSERT (w1 INSERT EMPTY)) IN E) /\ (((((((sigma_fan x) V) E) u) w) = v) /\ ((!v. ((v IN V) ==> ((CARD (((set_of_edge v) V) E)) > (NUMERAL (BIT1 _0))))) /\ ((E UNION ((v INSERT (w INSERT EMPTY)) INSERT EMPTY)) = E1))))))))))) ==> ((((((sigma_fan x) V) E1) w) w1) = (((((sigma_fan x) V) E) w) w1))))))))))


DIAMETER_EMPTY

|- ((diameter EMPTY) = (real_of_num (NUMERAL _0)))


vector_add

|- (!x. (!y. (((vector_add x) y) = (lambdai. ((real_add (x $ i)) (y $ i))))))


AFFINE_LINEAR_IMAGE_EQ

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


DOT_RZERO

|- (!x. (((dot x) (vec (NUMERAL _0))) = (real_of_num (NUMERAL _0))))


Hypermap_iso.ext_dart

|- (!V. (!E. (!v. (!w. (((ext_dart (V , E)) (v , w)) = ((vec (NUMERAL _0)) , (v , (w , (((((extension_sigma_fan (vec (NUMERAL _0))) V) E) v) w)))))))))


CLOSURE_UNIONS

|- (!f. ((FINITE f) ==> ((closure (UNIONS f)) = (UNIONS (GSPEC (\GEN%PVAR%573. (?s. (((SETSPEC GEN%PVAR%573) (s IN f)) (closure s)))))))))


REAL_CONTINUOUS_WITHIN

|- (!f. (!x. (!s. (((real_continuous f) ((within (at x)) s)) = (((---> f) (f x)) ((within (at x)) s))))))


NADD_LE_LMUL

|- (!x. (!y. (!z. (((nadd_le y) z) ==> ((nadd_le ((nadd_mul x) y)) ((nadd_mul x) z))))))


Functional_equation.constant6

|- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (!c. ((((((((constant6 c) x1) x2) x3) x4) x5) x6) = c))))))))


OPEN_IMP_LOCALLY_PATH_CONNECTED

|- (!s. ((open s) ==> ((locally path_connected) s)))


Lp_main_estimate.quad_diags_imp

|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!v. (!w. (!a. (!b. ((LET (\f. (LET_END ((LET (\w'. (LET_END ((LET (\u. (LET_END ((((v , w) IN ((darts_k (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) (hypermap_of_fan (V , E)))) /\ ((~ (a = b)) /\ ((a IN ((v_prime V) f)) /\ ((b IN ((v_prime V) f)) /\ (~ ((a INSERT (b INSERT EMPTY)) IN ((e_prime E) f))))))) ==> ((((a , b) = (w , w')) \/ ((a , b) = (w' , w))) \/ (((a , b) = (v , u)) \/ ((a , b) = (u , v)))))))) (((((sigma_fan (vec (NUMERAL _0))) V) E) w') v))))) (((((sigma_fan (vec (NUMERAL _0))) V) E) v) w))))) ((face (hypermap_of_fan (V , E))) (v , w))))))))))


SUM_4

|- (!t. (((sum ((NUMERAL (BIT1 _0)) .. (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) t) = ((real_add (t (NUMERAL (BIT1 _0)))) ((real_add (t (NUMERAL (BIT0 (BIT1 _0))))) ((real_add (t (NUMERAL (BIT1 (BIT1 _0))))) (t (NUMERAL (BIT0 (BIT0 (BIT1 _0))))))))))


ANALYTIC_ON_COMPOSE

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


CLOSED_INJECTIVE_LINEAR_IMAGE_EQ

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


DIM_UNIQUE

|- (!v. (!b. (((b SUBSET v) /\ ((v SUBSET (span b)) /\ ((independent b) /\ (b HAS_SIZE n)))) ==> ((dim v) = n))))


BOOL_CASES_AX

|- (!t. ((t = T) \/ (t = F)))


TRACE_MATRIX_INV_LMUL

|- (!A. ((trace ((matrix_mul (matrix_inv A)) A)) = (real_of_num (rank A))))


INTEGRABLE_SPIKE

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


IMAGE_EQ_EMPTY

|- (!f. (!s. ((((IMAGE f) s) = EMPTY) = (s = EMPTY))))


MATCH_SEQPATTERN

|- (((_MATCH x) ((_SEQPATTERN r) s)) = (((COND (?y. ((r x) y))) ((_MATCH x) r)) ((_MATCH x) s)))


CONVEX_IMP_SIMPLY_CONNECTED

|- (!s. ((convex s) ==> (simply_connected s)))


ORTHOGONAL_TRANSFORMATION

|- (!f. ((orthogonal_transformation f) = ((linear f) /\ (!v. ((vector_norm (f v)) = (vector_norm v))))))


Lp_ineqs_proofs.y4'_eq_y4

|- (!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)))) ==> (((y4'_fan (V , E)) d) = ((y4_fan (V , E)) d))))))))))


Appendix.scs_am_v39_explicit

|- (!k. (!d. (!a. (!b. (!alpha. (!beta. (!J. (!lo. (!str. ((scs_am_v39 (scs_v39 (k , (d , (a , (alpha , (beta , (b , (J , (lo , (hi , str))))))))))) = alpha))))))))))


Add_triangle.split_fan_face

|- (!V. (!v. (!w. (!E. (((split_fan_face (V , E)) (v , w)) = (V , ((w INSERT ((((((sigma_fan (vec (NUMERAL _0))) V) E) v) w) INSERT EMPTY)) INSERT E)))))))


LIM_AT_INFINITY_WITHIN

|- (!f. (!l. (!s. ((((--> f) l) ((within at_infinity) s)) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?b. (!x. (((x IN s) /\ ((real_ge (vector_norm x)) b)) ==> ((real_lt (distance ((f x) , l))) e))))))))))


ARZELA_ASCOLI

|- (!f. (!s. (!M. (((compact s) /\ ((!n. (!x. ((x IN s) ==> ((real_le (vector_norm ((f n) x))) M)))) /\ (!x. (!e. (((x IN s) /\ ((real_lt (real_of_num (NUMERAL _0))) e)) ==> (?d. (((real_lt (real_of_num (NUMERAL _0))) d) /\ (!n. (!y. (((y IN s) /\ ((real_lt (vector_norm ((vector_sub x) y))) d)) ==> ((real_lt (vector_norm ((vector_sub ((f n) x)) ((f n) y)))) e))))))))))) ==> (?g. (((continuous_on g) s) /\ (?r. ((!m. (!n. ((m < n) ==> ((r m) < (r n))))) /\ (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?N. (!n. (!x. (((n >= N) /\ (x IN s)) ==> ((real_lt (vector_norm ((vector_sub ((f (r n)) x)) (g x)))) e)))))))))))))))


real_min

|- (!m. (!n. (((real_min m) n) = (((COND ((real_le m) n)) m) n))))


Nonlin_def.add6

|- (!f. (!g. (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((((((((add6 f) g) x1) x2) x3) x4) x5) x6) = ((real_add ((((((f x1) x2) x3) x4) x5) x6)) ((((((g x1) x2) x3) x4) x5) x6)))))))))))


REAL_LE_ADD2

|- (!w. (!x. (!y. (!z. ((((real_le w) x) /\ ((real_le y) z)) ==> ((real_le ((real_add w) y)) ((real_add x) z)))))))


REAL_SGN_NEG

|- (!x. ((real_sgn (real_neg x)) = (real_neg (real_sgn x))))


COND_ELIM_THM

|- ((P (((COND c) x) y)) = ((c ==> (P x)) /\ ((~ c) ==> (P y))))


Lp_ineqs_proofs.y9_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 (BIT0 (BIT1 _0))))) ((y9_fan (V , E)) d))))))))))


CONTINUOUS_ON_ID

|- (!s. ((continuous_on (\x. x)) s))


transp

|- (!A. ((transp A) = (lambdai. (lambdaj. ((A $ j) $ i)))))


ARC_SIMPLE_PATH_SUBPATH

|- (!g. (!u. (!v. (((simple_path g) /\ ((u IN (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL))) /\ ((v IN (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL))) /\ (~ ((g u) = (g v)))))) ==> (arc (((subpath u) v) g))))))


CONTINUOUS_INJECTIVE_IMAGE_SEGMENT_1

|- (!f. (!a. (!b. ((((continuous_on f) (closed_segment ((CONS (a , b)) NIL))) /\ (!x. (!y. (((x IN (closed_segment ((CONS (a , b)) NIL))) /\ ((y IN (closed_segment ((CONS (a , b)) NIL))) /\ ((f x) = (f y)))) ==> (x = y))))) ==> (((IMAGE f) (closed_segment ((CONS (a , b)) NIL))) = (closed_segment ((CONS ((f a) , (f b))) NIL)))))))


Hypermap.face_map_and_darts

|- (!H. ((FINITE (dart H)) /\ ((permutes (face_map H)) (dart H))))


bounded

|- (!s. ((bounded s) = (?a. (!x. ((x IN s) ==> ((real_le (vector_norm x)) a))))))


linear

|- (!f. ((linear f) = ((!x. (!y. ((f ((vector_add x) y)) = ((vector_add (f x)) (f y))))) /\ (!c. (!x. ((f ((% c) x)) = ((% c) (f x))))))))


RSTC

|- (!R. ((RSTC R) = (RC (TC (SC R)))))


ABSOLUTELY_REAL_INTEGRABLE_ADD

|- (!f. (!g. (!s. ((((absolutely_real_integrable_on f) s) /\ ((absolutely_real_integrable_on g) s)) ==> ((absolutely_real_integrable_on (\x. ((real_add (f x)) (g x)))) s)))))


EDGE_OF_TRANSLATION_EQ

|- (!a. (!f. (!s. (((edge_of ((IMAGE (\x. ((vector_add a) x))) f)) ((IMAGE (\x. ((vector_add a) x))) s)) = ((edge_of f) s)))))


ITERATE_SUPPORT

|- (!op. (!f. (!s. ((((iterate op) (((support op) f) s)) f) = (((iterate op) s) f)))))


MEASURABLE_ON_CONVOLUTION

|- (!bop. (!f. (!g. (!x. (((bilinear bop) /\ (((measurable_on f) UNIV) /\ ((measurable_on g) UNIV))) ==> ((measurable_on (\y. ((bop (f ((vector_sub x) y))) (g y)))) UNIV))))))


VECTOR_ADD_COMPONENT

|- (!x. (!y. (!i. ((((vector_add x) y) $ i) = ((real_add (x $ i)) (y $ i))))))


real_bounded

|- (!s. ((real_bounded s) = (?B. (!x. ((x IN s) ==> ((real_le (real_abs x)) B))))))


NEGLIGIBLE_COUNTABLE

|- (!s. ((COUNTABLE s) ==> (negligible s)))


CSIN_CIRCLE

|- (!z. (((vector_add ((complex_pow (csin z)) (NUMERAL (BIT0 (BIT1 _0))))) ((complex_pow (ccos z)) (NUMERAL (BIT0 (BIT1 _0))))) = (Cx (real_of_num (NUMERAL (BIT1 _0))))))


SURJECTIVE_IFF_INJECTIVE_GEN

|- (!s. (!t. (!f. (((FINITE s) /\ ((FINITE t) /\ (((CARD s) = (CARD t)) /\ (((IMAGE f) s) SUBSET t)))) ==> ((!y. ((y IN t) ==> (?x. ((x IN s) /\ ((f x) = y))))) = (!x. (!y. (((x IN s) /\ ((y IN s) /\ ((f x) = (f y)))) ==> (x = y)))))))))


OPEN_TRANSLATION_EQ

|- (!a. (!s. ((open ((IMAGE (\x. ((vector_add a) x))) s)) = (open s))))


real_gt

|- (!y. (!x. (((real_gt x) y) = ((real_lt y) x))))


CONVEX_CONE_PCROSS

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


HAS_DERIVATIVE_INVERSE_STRONG

|- (!f. (!g. (!f'. (!g'. (!s. (!x. (((open s) /\ ((x IN s) /\ (((continuous_on f) s) /\ ((!x. ((x IN s) ==> ((g (f x)) = x))) /\ ((((has_derivative f) f') (at x)) /\ ((f' o g') = I)))))) ==> (((has_derivative g) g') (at (f x))))))))))


ALTERNATING_SUM_BOUND

|- (!a. (((!n. ((real_le (real_abs (a (SUC n)))) (real_abs (a n)))) /\ ((!n. ((EVEN n) ==> ((real_le (real_of_num (NUMERAL _0))) (a n)))) /\ (!n. ((ODD n) ==> ((real_le (a n)) (real_of_num (NUMERAL _0))))))) ==> (!m. (!n. ((real_le (real_abs ((sum (m .. n)) a))) (real_abs (a m)))))))


Hypermap_iso.bij_res_inv

|- (!f. (!s. (!t. ((((BIJ f) s) t) ==> (((BIJ ((res_inv f) s)) t) s)))))


Seq.map_rot

|- (!n0. (!f. (!s. (((map f) ((rot n0) s)) = ((rot n0) ((map f) s))))))


inverse

|- (!f. ((inverse f) = (\y. (@x. ((f x) = y)))))


AFFINE_HULL_FINITE

|- (!s. (((hull affine) s) = (GSPEC (\GEN%PVAR%1412. (?y. (((SETSPEC GEN%PVAR%1412) (?u. ((((sum s) u) = (real_of_num (NUMERAL (BIT1 _0)))) /\ (((vsum s) (\v. ((% (u v)) v))) = y)))) y))))))


CEXP_EULER

|- (!z. ((cexp ((complex_mul ii) z)) = ((vector_add (ccos z)) ((complex_mul ii) (csin z)))))


NOT_BOUNDED_UNIV

|- (~ (bounded UNIV))


Sphere.rhazim3

|- (rhazim3 = (node3_y rhazim))


Fan.POWER_1

|- (!f. (((POWER f) (NUMERAL (BIT1 _0))) = f))


COMPLEX_DERIVATIVE_LMUL

|- (!f. (!c. (!z. (((complex_differentiable f) (at z)) ==> (((complex_derivative (\w. ((complex_mul c) (f w)))) z) = ((complex_mul c) ((complex_derivative f) z)))))))


COVERING_SPACE_HOMEOMORPHISM

|- (!p. (!c. (!s. ((((covering_space (c , p)) s) /\ ((path_connected c) /\ (simply_connected s))) ==> (?q. ((homeomorphism (c , s)) (p , q)))))))


SUBSET_INTER

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


CFUNSPACE_IMP_BOUNDED2

|- (!top. (!m. (!f. (!g. (((f IN (mspace ((cfunspace top) m))) /\ (g IN (mspace ((cfunspace top) m)))) ==> (?b. (!x. ((x IN (topspace top)) ==> ((real_le ((mdist m) ((f x) , (g x)))) b)))))))))


LIPSCHITZ_REAL_POLYNOMIAL_FUNCTION

|- (!f. (!s. (((real_polynomial_function f) /\ (bounded s)) ==> (?B. (((real_lt (real_of_num (NUMERAL _0))) B) /\ (!x. (!y. (((x IN s) /\ (y IN s)) ==> ((real_le (real_abs ((real_sub (f x)) (f y)))) ((real_mul B) (vector_norm ((vector_sub x) y))))))))))))


PRESERVES_NORM_PRESERVES_DOT

|- (!f. (!x. (!y. (((linear f) /\ (!x. ((vector_norm (f x)) = (vector_norm x)))) ==> (((dot (f x)) (f y)) = ((dot x) y))))))


Ssrnat.factS

|- (!n. ((FACT (SUC n)) = ((SUC n) * (FACT n))))


REAL_SUB_ADD

|- (!x. (!y. (((real_add ((real_sub x) y)) y) = x)))


EVENTUALLY_AT_NEGINFINITY

|- (!p. (((eventually p) at_neginfinity) = (?b. (!x. (((real_le x) b) ==> (p x))))))


CONNECTED_INTER_FRONTIER

|- (!s. (!t. (((connected s) /\ ((~ ((s INTER t) = EMPTY)) /\ (~ ((s DIFF t) = EMPTY)))) ==> (~ ((s INTER (frontier t)) = EMPTY)))))


PROPER_LOCALLY_INJECTIVE_OPEN_IMP_COVERING_MAP

|- (!p. (!c. (!s. ((((continuous_on p) c) /\ ((((IMAGE p) c) = s) /\ ((!k. (((k SUBSET s) /\ (compact k)) ==> (compact (GSPEC (\GEN%PVAR%2101. (?x. (((SETSPEC GEN%PVAR%2101) ((x IN c) /\ ((p x) IN k))) x))))))) /\ ((!u. (((open_in ((subtopology euclidean) c)) u) ==> ((open_in ((subtopology euclidean) s)) ((IMAGE p) u)))) /\ (!x. ((x IN c) ==> (?t. ((x IN t) /\ (((open_in ((subtopology euclidean) c)) t) /\ (!y. (!z. (((y IN t) /\ ((z IN t) /\ ((p y) = (p z)))) ==> (y = z))))))))))))) ==> ((covering_space (c , p)) s)))))


REAL_CONVEX_ON_SUBSET

|- (!f. (!s. (!t. ((((real_convex_on f) t) /\ (s SUBSET t)) ==> ((real_convex_on f) s)))))


Fan.FAN

|- (!x. (!V. (!E. ((FAN (x , (V , E))) = (((UNIONS E) SUBSET V) /\ ((graph E) /\ ((fan1 (x , (V , E))) /\ ((fan2 (x , (V , E))) /\ ((fan6 (x , (V , E))) /\ (fan7 (x , (V , E))))))))))))


Appendix.is_scs_v39

|- (!s. ((is_scs_v39 s) = (((real_lt (scs_d_v39 s)) ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT0 (BIT1 _0)))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) /\ (((NUMERAL (BIT1 (BIT1 _0))) <= (scs_k_v39 s)) /\ (((scs_k_v39 s) <= (NUMERAL (BIT0 (BIT1 (BIT1 _0))))) /\ (((periodic (scs_lo_v39 s)) (scs_k_v39 s)) /\ (((periodic (scs_hi_v39 s)) (scs_k_v39 s)) /\ (((periodic (scs_str_v39 s)) (scs_k_v39 s)) /\ (((periodic (scs_str_v39 s)) (scs_k_v39 s)) /\ (((periodic2 (scs_a_v39 s)) (scs_k_v39 s)) /\ (((periodic2 (scs_am_v39 s)) (scs_k_v39 s)) /\ (((periodic2 (scs_bm_v39 s)) (scs_k_v39 s)) /\ (((periodic2 (scs_b_v39 s)) (scs_k_v39 s)) /\ (((periodic2 (scs_J_v39 s)) (scs_k_v39 s)) /\ ((!i. (!j. (((((scs_a_v39 s) i) j) = (((scs_a_v39 s) j) i)) /\ (((((scs_am_v39 s) i) j) = (((scs_am_v39 s) j) i)) /\ (((((scs_bm_v39 s) i) j) = (((scs_bm_v39 s) j) i)) /\ (((((scs_b_v39 s) i) j) = (((scs_b_v39 s) j) i)) /\ ((((scs_J_v39 s) i) j) = (((scs_J_v39 s) j) i)))))))) /\ ((!i. (!j. (((real_le (((scs_a_v39 s) i) j)) (((scs_am_v39 s) i) j)) /\ (((real_le (((scs_am_v39 s) i) j)) (((scs_bm_v39 s) i) j)) /\ ((real_le (((scs_bm_v39 s) i) j)) (((scs_b_v39 s) i) j)))))) /\ ((!i. ((((scs_a_v39 s) i) i) = (real_of_num (NUMERAL _0)))) /\ ((!i. (!j. (((i < (scs_k_v39 s)) /\ ((j < (scs_k_v39 s)) /\ (~ (i = j)))) ==> ((real_le (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (((scs_a_v39 s) i) j))))) /\ ((!i. (((scs_k_v39 s) = (NUMERAL (BIT1 (BIT1 _0)))) ==> ((real_lt (((scs_b_v39 s) i) (SUC i))) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))))) /\ ((!i. (((NUMERAL (BIT1 (BIT1 _0))) < (scs_k_v39 s)) ==> ((real_le (((scs_b_v39 s) i) (SUC i))) cstab))) /\ ((!i. (!j. ((((scs_J_v39 s) i) j) ==> (((j MOD (scs_k_v39 s)) = ((SUC i) MOD (scs_k_v39 s))) \/ ((i MOD (scs_k_v39 s)) = ((SUC j) MOD (scs_k_v39 s))))))) /\ ((!i. (!j. ((((scs_J_v39 s) i) j) ==> (((((scs_a_v39 s) i) j) = sqrt8) /\ ((((scs_b_v39 s) i) j) = cstab))))) /\ (((CARD (GSPEC (\GEN%PVAR%7668. (?i. (((SETSPEC GEN%PVAR%7668) ((i < (scs_k_v39 s)) /\ (((real_lt ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) (((scs_b_v39 s) i) (SUC i))) \/ ((real_lt (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (((scs_a_v39 s) i) (SUC i)))))) i))))) + (scs_k_v39 s)) <= (NUMERAL (BIT0 (BIT1 (BIT1 _0)))))))))))))))))))))))))))


REAL_POW_EQ_ABS

|- (!n. (!x. (!y. (((~ (n = (NUMERAL _0))) /\ (((real_pow x) n) = ((real_pow y) n))) ==> ((real_abs x) = (real_abs y))))))


CONNECTED_SUBSET_SEGMENT

|- (!s. (!a. (!b. (((connected s) /\ ((s SUBSET (closed_segment ((CONS (a , b)) NIL))) /\ ((a IN s) /\ (b IN s)))) ==> (s = (closed_segment ((CONS (a , b)) NIL)))))))


Tame_defs.contravening

|- (!V. ((contravening V) = ((packing V) /\ ((V SUBSET ball_annulus) /\ (((real_gt (scriptL V)) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 _0))))))) /\ ((!W. (((packing W) /\ (W SUBSET ball_annulus)) ==> ((real_le (scriptL W)) (scriptL V)))) /\ ((((CARD V) = (NUMERAL (BIT1 (BIT0 (BIT1 (BIT1 _0)))))) \/ (((CARD V) = (NUMERAL (BIT0 (BIT1 (BIT1 (BIT1 _0)))))) \/ ((CARD V) = (NUMERAL (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))) /\ ((!v. ((v IN V) ==> ((surrounded_node (V , (ESTD V))) v))) /\ (!v. ((v IN V) ==> (((surrounded_node (V , (ECTC V))) v) \/ ((vector_norm v) = (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))))))))))))


Calc_derivative.ratform

|- (!p. (!r. (!a. (!b. (((((ratform p) r) a) b) = (p ==> ((~ (b = (real_of_num (NUMERAL _0)))) /\ (r = ((real_div a) b)))))))))


EVENTUALLY_MONO

|- (!net. (!p. (!q. (((!x. ((p x) ==> (q x))) /\ ((eventually p) net)) ==> ((eventually q) net)))))


IN_CLOSURE_DELETE

|- (!s. (!x. ((x IN (closure (s DELETE x))) = ((limit_point_of x) s))))


Trigonometry.UTNKIAC

|- (!x. (!y. ((((real_lt (real_neg ((real_div pi) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) x) /\ (((real_lt x) y) /\ ((real_lt y) ((real_div pi) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))))) ==> ((real_lt (tan x)) (tan y)))))


LEBESGUE_DENSITY_THEOREM_LIFT_CBALL

|- (!s. ((lebesgue_measurable s) ==> (?k. ((negligible k) /\ (!x. ((~ (x IN k)) ==> (((--> (\e. (lift ((real_div (measure (s INTER (cball (x , (drop e)))))) (measure (cball (x , (drop e)))))))) (((COND (x IN s)) (vec (NUMERAL (BIT1 _0)))) (vec (NUMERAL _0)))) ((within (at (vec (NUMERAL _0)))) (GSPEC (\GEN%PVAR%3726. (?t. (((SETSPEC GEN%PVAR%3726) ((real_lt (real_of_num (NUMERAL _0))) (drop t))) t))))))))))))


Conforming.SUM_SOL_IN_FACE_SET_EQ_4PI

|- (!x. (!V. (!E. (((FAN (x , (V , E))) /\ (conforming_fan (x , (V , E)))) ==> (((sum (face_set (hypermap1_of_fanx (x , (V , E))))) (\f. ((sol x) ((((dartset_leads_into_fan x) V) E) f)))) = ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) pi))))))


VECTOR_SUB_REFL

|- (!x. (((vector_sub x) x) = (vec (NUMERAL _0))))


IMAGE_LIFT_DROP

|- ((!s. (((IMAGE (lift o drop)) s) = s)) /\ (!s. (((IMAGE (drop o lift)) s) = s)))


IN_RELATIVE_INTERIOR

|- (!x. (!s. ((x IN (relative_interior s)) = ((x IN s) /\ (?e. (((real_lt (real_of_num (NUMERAL _0))) e) /\ (((ball (x , e)) INTER ((hull affine) s)) SUBSET s)))))))


REAL_CONTINUOUS_CONST

|- (!net. (!c. ((real_continuous (\x. c)) net)))


COMPONENTS_EQ_SING

|- (!s. (((components s) = (s INSERT EMPTY)) = ((connected s) /\ (~ (s = EMPTY)))))


MULT_ASSOC

|- (!m. (!n. (!p. ((m * (n * p)) = ((m * n) * p)))))


RANK_CMUL

|- (!A. (!c. ((rank ((%% c) A)) = (((COND (c = (real_of_num (NUMERAL _0)))) (NUMERAL _0)) (rank A)))))


Planarity.dartset_leads_into_is_topological_component_yfan

|- (!x. (!V. (!E. (!ds. (((FAN (x , (V , E))) /\ ((!v. ((v IN V) ==> ((CARD (((set_of_edge v) V) E)) > (NUMERAL (BIT1 _0))))) /\ ((fan80 (x , (V , E))) /\ (ds IN (face_set (hypermap1_of_fanx (x , (V , E)))))))) ==> (((((dartset_leads_into_fan x) V) E) ds) IN (topological_component_yfan (x , (V , E)))))))))


Dih2k_hypermap.INDEX_VECMAT

|- (((i < (dimindex UNIV)) /\ (((NUMERAL (BIT1 _0)) <= i') /\ (i' <= (dimindex UNIV)))) ==> (((NUMERAL (BIT1 _0)) <= ((i * (dimindex UNIV)) + i')) /\ (((i * (dimindex UNIV)) + i') <= (dimindex UNIV))))


LIMPT_TRANSLATION_EQ

|- (!a. (!s. (!x. (((limit_point_of ((vector_add a) x)) ((IMAGE (\y. ((vector_add a) y))) s)) = ((limit_point_of x) s)))))


minimal

|- (!P. ((minimal P) = (@n. ((P n) /\ (!m. ((m < n) ==> (~ (P m))))))))


HAS_REAL_MEASURE_REAL_NEGLIGIBLE_UNIONS_IMAGE

|- (!f. (!s. (((FINITE s) /\ ((!x. ((x IN s) ==> (real_measurable (f x)))) /\ (!x. (!y. (((x IN s) /\ ((y IN s) /\ (~ (x = y)))) ==> (real_negligible ((f x) INTER (f y)))))))) ==> ((has_real_measure (UNIONS ((IMAGE f) s))) ((sum s) (\x. (real_measure (f x))))))))


LINEAR_INDEPENDENT_EXTEND

|- (!f. (!b. ((independent b) ==> (?g. ((linear g) /\ (!x. ((x IN b) ==> ((g x) = (f x)))))))))


GT

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


IN_UNIONS

|- (!s. (!x. ((x IN (UNIONS s)) = (?t. ((t IN s) /\ (x IN t))))))


OPEN_CONVEX_HULL

|- (!s. ((open s) ==> (open ((hull convex) s))))


MAT_CMUL

|- (!a. ((mat a) = ((%% (real_of_num a)) (mat (NUMERAL (BIT1 _0))))))


REAL_ABS_SIGN2

|- (!x. (!y. (((real_lt (real_abs ((real_sub x) y))) (real_neg y)) ==> ((real_lt x) (real_of_num (NUMERAL _0))))))


CONVEX_CONE_CONVEX_CONE_HULL

|- (!s. (convex_cone ((hull convex_cone) s)))


LIM_WITHINREAL_WITHIN

|- ((((--> f) a) ((within (atreal x)) s)) = (((--> (f o drop)) a) ((within (at (lift x))) ((IMAGE lift) s))))


COMPLEX_NEG_SUB

|- (!x. (!y. ((vector_neg ((vector_sub x) y)) = ((vector_sub y) x))))


IMAGE_I

|- (!s. (((IMAGE I) s) = s))


LINEAR_NEGATION

|- (linear vector_neg)


CONTINUOUS_WITHIN_SQRT_COMPOSE

|- (!f. (!s. (!a. ((((continuous (\x. (lift (f x)))) ((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)))))) ==> ((continuous (\x. (lift (sqrt (f x))))) ((within (at a)) s))))))


COMPACT_EQ_BOUNDED_CLOSED

|- (!s. ((compact s) = ((bounded s) /\ (closed s))))


Collect_geom.COPLANAR_3

|- (!a. (!b. (!c. (((NUMERAL (BIT0 (BIT1 _0))) <= (dimindex UNIV)) ==> (coplanar_alt (a INSERT (b INSERT (c INSERT EMPTY))))))))


SUBSET_IMAGE

|- (!f. (!s. (!t. ((s SUBSET ((IMAGE f) t)) = (?u. ((u SUBSET t) /\ (s = ((IMAGE f) u))))))))


Functional_equation.functional_taum_x

|- (taum_x = ((add6 rhazim_x) ((add6 rhazim2_x) ((sub6 rhazim3_x) (constant6 ((real_mul ((real_add (real_of_num (NUMERAL (BIT1 _0)))) const1)) pi))))))


Collect_geom.RADV_FORMULAR

|- (!va. (!vb. (!vc. ((~ (collinear (va INSERT (vb INSERT (vc INSERT EMPTY))))) ==> ((radV (va INSERT (vb INSERT (vc INSERT EMPTY)))) = (((eta_y ((dist3 vb) vc)) ((dist3 va) vc)) ((dist3 va) vb)))))))


Pack1.fcc_compatible

|- (!S. (!f. (((fcc_compatible f) S) = (!v. ((v IN S) ==> ((real_le (sqrt (real_of_num (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 _0)))))))))) ((real_add (measure ((voronoi_open S) v))) (f v))))))))


REAL_OPEN_SET_RATIONAL

|- (!s. (((real_open s) /\ (~ (s = EMPTY))) ==> (?x. ((rational x) /\ (x IN s)))))


FROM_INTER_NUMSEG_MAX

|- (!m. (!n. (!p. (((from p) INTER (m .. n)) = (((MAX p) m) .. n)))))