DET_3

|- (!A. ((det A) = ((real_add ((real_mul ((A $ (NUMERAL (BIT1 _0))) $ (NUMERAL (BIT1 _0)))) ((real_mul ((A $ (NUMERAL (BIT0 (BIT1 _0)))) $ (NUMERAL (BIT0 (BIT1 _0))))) ((A $ (NUMERAL (BIT1 (BIT1 _0)))) $ (NUMERAL (BIT1 (BIT1 _0))))))) ((real_add ((real_mul ((A $ (NUMERAL (BIT1 _0))) $ (NUMERAL (BIT0 (BIT1 _0))))) ((real_mul ((A $ (NUMERAL (BIT0 (BIT1 _0)))) $ (NUMERAL (BIT1 (BIT1 _0))))) ((A $ (NUMERAL (BIT1 (BIT1 _0)))) $ (NUMERAL (BIT1 _0)))))) ((real_sub ((real_sub ((real_sub ((real_mul ((A $ (NUMERAL (BIT1 _0))) $ (NUMERAL (BIT1 (BIT1 _0))))) ((real_mul ((A $ (NUMERAL (BIT0 (BIT1 _0)))) $ (NUMERAL (BIT1 _0)))) ((A $ (NUMERAL (BIT1 (BIT1 _0)))) $ (NUMERAL (BIT0 (BIT1 _0))))))) ((real_mul ((A $ (NUMERAL (BIT1 _0))) $ (NUMERAL (BIT1 _0)))) ((real_mul ((A $ (NUMERAL (BIT0 (BIT1 _0)))) $ (NUMERAL (BIT1 (BIT1 _0))))) ((A $ (NUMERAL (BIT1 (BIT1 _0)))) $ (NUMERAL (BIT0 (BIT1 _0)))))))) ((real_mul ((A $ (NUMERAL (BIT1 _0))) $ (NUMERAL (BIT0 (BIT1 _0))))) ((real_mul ((A $ (NUMERAL (BIT0 (BIT1 _0)))) $ (NUMERAL (BIT1 _0)))) ((A $ (NUMERAL (BIT1 (BIT1 _0)))) $ (NUMERAL (BIT1 (BIT1 _0)))))))) ((real_mul ((A $ (NUMERAL (BIT1 _0))) $ (NUMERAL (BIT1 (BIT1 _0))))) ((real_mul ((A $ (NUMERAL (BIT0 (BIT1 _0)))) $ (NUMERAL (BIT0 (BIT1 _0))))) ((A $ (NUMERAL (BIT1 (BIT1 _0)))) $ (NUMERAL (BIT1 _0))))))))))


EXTREME_POINT_OF_TRANSLATION_EQ

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


SPHERE_EMPTY

|- (!a. (!r. (((real_lt r) (real_of_num (NUMERAL _0))) ==> ((sphere (a , r)) = EMPTY))))


HULL_P

|- (!P. (!s. ((P s) ==> (((hull P) s) = s))))


COUNTABLE_NON_LIMIT_POINTS

|- (!s. (COUNTABLE (s DIFF (GSPEC (\GEN%PVAR%1262. (?x. (((SETSPEC GEN%PVAR%1262) ((limit_point_of x) s)) x)))))))


DOT_RMUL

|- (!c. (!x. (!y. (((dot x) ((% c) y)) = ((real_mul c) ((dot x) y))))))


Lp_ineqs_proofs.y6_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))))) (y6_fan d))))))))))


SPHERE_SING

|- (!x. (!e. ((e = (real_of_num (NUMERAL _0))) ==> ((sphere (x , e)) = (x INSERT EMPTY)))))


PATH_COMPONENT_TRANSLATION

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


COUNTABLE_IMAGE_INJ_EQ

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


FINITE_UNIONS

|- (!s. ((FINITE (UNIONS s)) = ((FINITE s) /\ (!t. ((t IN s) ==> (FINITE t))))))


Kepler_spec.ZERO_SPEC

|- ((vec (NUMERAL _0)) = (real3_of_triple ((real_of_num (NUMERAL _0)) , ((real_of_num (NUMERAL _0)) , (real_of_num (NUMERAL _0))))))


real_measurable_on

|- (!f. (!s. (((real_measurable_on f) s) = ((measurable_on (lift o (f o drop))) ((IMAGE lift) s)))))


REAL_SUMMABLE_CAUCHY

|- (!f. (!s. (((real_summable s) f) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?N. (!m. (!n. ((m >= N) ==> ((real_lt (real_abs ((sum (s INTER (m .. n))) f))) e))))))))))


Ssrnat.addnS

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


COMPLEX_EXPAND

|- (!z. (z = ((vector_add (Cx (Re z))) ((complex_mul ii) (Cx (Im z))))))


INTEGER_POS

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


CARD_EXP_SING

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


Hypermap_and_fan.FINITE_ORBIT_MAP_INVERSE

|- (!f. (!s. (!x. (!n. (!k. (((FINITE s) /\ (((permutes f) s) /\ (((CARD ((orbit_map f) x)) = n) /\ (k <= n)))) ==> ((((POWER (inverse f)) k) x) = (((POWER f) (n - k)) x))))))))


Tame_defs2.marked_list

|- (!L. (!N. (!r. (!x. (((((marked_list L) N) r) x) = ((is_restricted (hypermap_of_list L)) /\ (((normal_list L) N) /\ (((MEM r) N) /\ (((MEM x) r) /\ (((LET (\qs. (LET_END (((all uniq) qs) /\ ((all (\v. ((sizel ((filter (MEM v)) qs)) > (NUMERAL (BIT1 _0))))) (flatten qs)))))) ((quotient_list L) N)) /\ ((((next_el r) x) = ((f_list L) x)) /\ (((LET (\r'. (LET_END ((final_list L) r')))) ((find_list N) (e_list x))) /\ ((((final_list L) r) ==> ((flag_list L) N)) /\ ((~ ((final_list L) r)) ==> ((((s_flag_list L) N) r) x)))))))))))))))


CONVERGENT_BOUNDED_MONOTONE_1

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


Local_lemmas.LOFA_IMP_CARD_FF_V_EQ

|- ((local_fan (V , (E , FF))) ==> ((CARD FF) = (CARD V)))


REAL_SUMMABLE_COMPLEX

|- (((real_summable s) f) = ((summable s) (Cx o f)))


NORM_SUB

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


PRODUCT_CLAUSES_NUMSEG

|- ((!m. (((product (m .. (NUMERAL _0))) f) = (((COND (m = (NUMERAL _0))) (f (NUMERAL _0))) (real_of_num (NUMERAL (BIT1 _0)))))) /\ (!m. (!n. (((product (m .. (SUC n))) f) = (((COND (m <= (SUC n))) ((real_mul ((product (m .. n)) f)) (f (SUC n)))) ((product (m .. n)) f))))))


EIGENVALUES_CHARACTERISTIC_ALT

|- (!A. (!c. ((?v. ((~ (v = (vec (NUMERAL _0)))) /\ (((matrix_vector_mul A) v) = ((% c) v)))) = ((det ((matrix_sub A) ((%% c) (mat (NUMERAL (BIT1 _0)))))) = (real_of_num (NUMERAL _0))))))


ccos

|- (!z. ((ccos z) = ((complex_div ((vector_add (cexp ((complex_mul ii) z))) (cexp ((complex_mul (vector_neg ii)) z)))) (Cx (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))))


Seq.behead

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


MATRIX_INV_COVARIANCE_RMUL

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


HAS_VECTOR_DERIVATIVE_UNIQUE_AT

|- (!f. (!f'. (!x. ((((has_vector_derivative f) f') (at x)) ==> (((vector_derivative f) (at x)) = f')))))


CNJ_CEXP

|- (!z. ((cnj (cexp z)) = (cexp (cnj z))))


LE_EXP

|- (!x. (!m. (!n. (((x EXP m) <= (x EXP n)) = (((COND (x = (NUMERAL _0))) ((m = (NUMERAL _0)) ==> (n = (NUMERAL _0)))) ((x = (NUMERAL (BIT1 _0))) \/ (m <= n)))))))


homeomorphism

|- (!f. (!s. (!g. (!t. (((homeomorphism (s , t)) (f , g)) = ((!x. ((x IN s) ==> ((g (f x)) = x))) /\ ((((IMAGE f) s) = t) /\ (((continuous_on f) s) /\ ((!y. ((y IN t) ==> ((f (g y)) = y))) /\ ((((IMAGE g) t) = s) /\ ((continuous_on g) t)))))))))))


tendsto_real

|- (!f. (!l. (!net. ((((---> f) l) net) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> ((eventually (\x. ((real_lt (real_abs ((real_sub (f x)) l))) e))) net)))))))


ADD_ASSOC

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


IS_INTERVAL_CONVEX_1

|- (!s. ((is_interval s) = (convex s)))


REAL_INV_MUL

|- (!x. (!y. ((real_inv ((real_mul x) y)) = ((real_mul (real_inv x)) (real_inv y)))))


COMPONENTS_LINEAR_IMAGE

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


Hypermap_iso.bij_contracted_dart1

|- (!V. (!E. (((BIJ contracted_dart) (d1_fan ((vec (NUMERAL _0)) , (V , E)))) (dart1_of_fan (V , E)))))


INTERVAL_LOWERBOUND_NONEMPTY

|- (!a. (!b. ((~ ((closed_interval ((CONS (a , b)) NIL)) = EMPTY)) ==> ((interval_lowerbound (closed_interval ((CONS (a , b)) NIL))) = a))))


Fan.set_of_edge

|- (!v. (!E. (!V. ((((set_of_edge v) V) E) = (GSPEC (\GEN%PVAR%5448. (?w. (((SETSPEC GEN%PVAR%5448) (((v INSERT (w INSERT EMPTY)) IN E) /\ (w IN V))) w))))))))


CARD_LE_DIM_SPANNING

|- (!v. (!b. (((v SUBSET (span b)) /\ ((FINITE b) /\ ((CARD b) <= (dim v)))) ==> (independent b))))


Ssrnat.addnC

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


COMPLEX_EQ

|- (!w. (!z. ((w = z) = (((Re w) = (Re z)) /\ ((Im w) = (Im z))))))


PATH_TRANSLATION_EQ

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


path

|- (!g. ((path g) = ((continuous_on g) (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL)))))


CONNECTED_OUTSIDE

|- (!s. ((((NUMERAL (BIT0 (BIT1 _0))) <= (dimindex UNIV)) /\ (bounded s)) ==> (connected (outside s))))


CARD_LE_CONG

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


vec

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


HAS_REAL_VECTOR_DERIVATIVE_WITHIN

|- ((((has_real_derivative f) f') ((within (atreal x)) s)) = (((has_vector_derivative (lift o (f o drop))) (lift f')) ((within (at (lift x))) ((IMAGE lift) s))))


HOLOMORPHIC_ON_DIFFERENTIABLE

|- (!f. (!s. (((holomorphic_on f) s) = (!x. ((x IN s) ==> ((complex_differentiable f) ((within (at x)) s)))))))


CROSS_ORTHOGONAL_MATRIX

|- (!A. (!x. (!y. ((orthogonal_matrix A) ==> (((cross ((matrix_vector_mul A) x)) ((matrix_vector_mul A) y)) = ((% (det A)) ((matrix_vector_mul A) ((cross x) y))))))))


CNJ_CNJ

|- (!z. ((cnj (cnj z)) = z))


Seq.size_subseq_leqif

|- (!s1. (!s2. (((subseq s1) s2) ==> (((leqif (sizel s1)) (sizel s2)) (s1 = s2)))))


LIM_CONTINUOUS

|- (!net. (!f. (!l. ((((continuous f) net) /\ ((f (netlimit net)) = l)) ==> (((--> f) l) net)))))


EQ_ADD_LCANCEL

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


Hypermap.set_of_orbits

|- (!D. (!f. (((set_of_orbits D) f) = (GSPEC (\GEN%PVAR%5395. (?x. (((SETSPEC GEN%PVAR%5395) (x IN D)) ((orbit_map f) x))))))))


HAS_REAL_MEASURE_HAS_MEASURE

|- (!s. (!m. (((has_real_measure s) m) = ((has_measure ((IMAGE lift) s)) m))))


ANR_IMP_ABSOLUTE_NEIGHBOURHOOD_EXTENSOR

|- (!f. (!u. (!t. (!s. (((ANR s) /\ (((continuous_on f) t) /\ ((((IMAGE f) t) SUBSET s) /\ ((closed_in ((subtopology euclidean) u)) t)))) ==> (?v. (?g. ((t SUBSET v) /\ (((open_in ((subtopology euclidean) u)) v) /\ (((continuous_on g) v) /\ ((((IMAGE g) v) SUBSET s) /\ (!x. ((x IN t) ==> ((g x) = (f x)))))))))))))))


Seq2.EL_nth

|- (!x0. (!s. (!i. ((i < (sizel s)) ==> (((EL i) s) = (((nth x0) s) i))))))


SUBSET_CROSS

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


REAL_ADD_LID

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


Fan_misc.EXTENSION_SIGMA_FAN_EQ_RES

|- (!x. (!V. (!E. (!v. (((((extension_sigma_fan x) V) E) v) = ((res ((((sigma_fan x) V) E) v)) (((set_of_edge v) V) E)))))))


AFFINE_INDEPENDENT_IFF_CARD

|- (!s. ((~ (affine_dependent s)) = ((FINITE s) /\ ((aff_dim s) = ((int_sub (int_of_num (CARD s))) (int_of_num (NUMERAL (BIT1 _0))))))))


OPEN_IMP_FSIGMA

|- (!s. ((open s) ==> (fsigma s)))


FRONTIER_TRANSLATION

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


CX_SIN

|- (!x. ((Cx (sin x)) = (csin (Cx x))))


CENTRE_IN_BALL

|- (!x. (!e. ((x IN (ball (x , e))) = ((real_lt (real_of_num (NUMERAL _0))) e))))


path_image

|- (!g. ((path_image g) = ((IMAGE g) (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL)))))


SUMS_INFSUM

|- (!f. (!s. ((((sums f) ((infsum s) f)) s) = ((summable s) f))))


VECTOR_EQ

|- (!x. (!y. ((x = y) = ((((dot x) x) = ((dot x) y)) /\ (((dot y) y) = ((dot x) x))))))


IMAGE_AFFINITY_INTERVAL

|- (!a. (!b. (!m. (!c. (((IMAGE (\x. ((vector_add ((% m) x)) c))) (closed_interval ((CONS (a , b)) NIL))) = (((COND ((closed_interval ((CONS (a , b)) NIL)) = EMPTY)) EMPTY) (((COND ((real_le (real_of_num (NUMERAL _0))) m)) (closed_interval ((CONS (((vector_add ((% m) a)) c) , ((vector_add ((% m) b)) c))) NIL))) (closed_interval ((CONS (((vector_add ((% m) b)) c) , ((vector_add ((% m) a)) c))) NIL)))))))))


CONVEX_BALL

|- (!x. (!e. (convex (ball (x , e)))))


SIN_COS_EQ

|- (!x. (!y. ((((sin y) = (sin x)) /\ ((cos y) = (cos x))) = (?n. ((integer n) /\ (y = ((real_add x) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((real_mul n) pi)))))))))


BINOM_LT

|- (!n. (!k. ((n < k) ==> ((binom (n , k)) = (NUMERAL _0)))))


Hypermap.is_node_going

|- (!H. (!L. (!x. (!y. (((((is_node_going H) L) x) y) = (?k. ((y = (((POWER (loop_map L)) k) x)) /\ (!i. ((i <= k) ==> ((((POWER (loop_map L)) i) x) = (((POWER (inverse (node_map H))) i) x)))))))))))


CSQRT_0

|- ((csqrt (Cx (real_of_num (NUMERAL _0)))) = (Cx (real_of_num (NUMERAL _0))))


DIAMETER_BOUNDED_BOUND

|- (!s. (!x. (!y. (((bounded s) /\ ((x IN s) /\ (y IN s))) ==> ((real_le (vector_norm ((vector_sub x) y))) (diameter s))))))


INTEGRABLE_ON_LITTLE_SUBINTERVALS

|- (!f. (!a. (!b. ((!x. ((x IN (closed_interval ((CONS (a , b)) NIL))) ==> (?d. (((real_lt (real_of_num (NUMERAL _0))) d) /\ (!u. (!v. (((x IN (closed_interval ((CONS (u , v)) NIL))) /\ (((closed_interval ((CONS (u , v)) NIL)) SUBSET (ball (x , d))) /\ ((closed_interval ((CONS (u , v)) NIL)) SUBSET (closed_interval ((CONS (a , b)) NIL))))) ==> ((integrable_on f) (closed_interval ((CONS (u , v)) NIL)))))))))) ==> ((integrable_on f) (closed_interval ((CONS (a , b)) NIL)))))))


STC_SYM

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


TENDSTO_REAL

|- (((---> s) l) = ((--> (lift o s)) (lift l)))


CONTINUOUS_WITHIN_ID

|- (!a. (!s. ((continuous (\x. x)) ((within (at a)) s))))


EXTENSION_INTO_AR_LOCAL

|- (!f. (!c. (!s. (!t. ((((continuous_on f) c) /\ ((((IMAGE f) c) SUBSET t) /\ (((retract_of t) UNIV) /\ ((closed_in ((subtopology euclidean) s)) c)))) ==> (?g. (((continuous_on g) s) /\ ((((IMAGE g) UNIV) SUBSET t) /\ (!x. ((x IN c) ==> ((g x) = (f x))))))))))))


EXISTS_PASTECART

|- ((?p. (P p)) = (?x. (?y. (P ((pastecart x) y)))))


NETLIMIT_AT

|- (!a. ((netlimit (at a)) = a))


MEASURABLE_ON_PREIMAGE_OPEN

|- (!f. (((measurable_on f) UNIV) = (!t. ((open t) ==> (lebesgue_measurable (GSPEC (\GEN%PVAR%3534. (?x. (((SETSPEC GEN%PVAR%3534) ((f x) IN t)) x)))))))))


OUTSIDE_COMPACT_IN_OPEN

|- (!s. (!t. (((compact s) /\ ((open t) /\ ((s SUBSET t) /\ (~ (t = EMPTY))))) ==> (~ (((outside s) INTER t) = EMPTY)))))


MEM

|- ((((MEM x) NIL) = F) /\ (((MEM x) ((CONS h) t)) = ((x = h) \/ ((MEM x) t))))


VECTOR_ADD_RID

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


DIST_REFL

|- (!x. ((distance (x , x)) = (real_of_num (NUMERAL _0))))


CONNECTED_PATH_IMAGE

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


IM_COMPLEX_INV_GE_0

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


EXP_EQ_0

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


infsum

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


INTEGRABLE_0

|- (!s. ((integrable_on (\x. (vec (NUMERAL _0)))) s))


LAMBDA_SKOLEM

|- ((!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV))) ==> (?x. ((P i) x)))) = (?x. (!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV))) ==> ((P i) (x $ i))))))


AFF_DIM_INJECTIVE_LINEAR_IMAGE

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


SURJECTIVE_IMAGE

|- (!f. ((!t. (?s. (((IMAGE f) s) = t))) = (!y. (?x. ((f x) = y)))))


Real_ext.REAL_SV_SQRT_0

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


BILINEAR_COMPLEX_MUL

|- (bilinear complex_mul)


COMPACT_UNION

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


Seq.pred1

|- (!a. ((pred1 a) = (\x. (x = a))))


REAL_SUP_UNIQUE

|- (!s. (!b. (((!x. ((x IN s) ==> ((real_le x) b))) /\ (!b'. (((real_lt b') b) ==> (?x. ((x IN s) /\ ((real_lt b') x)))))) ==> ((sup s) = b))))


Misc_defs_and_lemmas.REAL_ADD

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


Functional_equation.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)))))))))))


SYMMETRIC_MATRIX_INV_RMUL

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


AFFINE_HULL_CONVEX_INTER_NONEMPTY_INTERIOR

|- (!s. (!t. (((convex s) /\ (~ ((s INTER (interior t)) = EMPTY))) ==> (((hull affine) (s INTER t)) = ((hull affine) s)))))


BETWEEN_MIDPOINT

|- (!a. (!b. (((between (midpoint (a , b))) (a , b)) /\ ((between (midpoint (a , b))) (b , a)))))


Seq.uniq_size_uniq

|- (!s1. (!s2. ((uniq s1) ==> ((!x. (((MEM x) s1) = ((MEM x) s2))) ==> ((uniq s2) = ((sizel s2) = (sizel s1)))))))


CLOSED_IN_TRANSLATION_EQ

|- (!a. (!s. (!t. (((closed_in ((subtopology euclidean) ((IMAGE (\x. ((vector_add a) x))) t))) ((IMAGE (\x. ((vector_add a) x))) s)) = ((closed_in ((subtopology euclidean) t)) s)))))


FINITE_IMAGE_INJ_EQ

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


INSERT

|- ((x INSERT s) = (GSPEC (\GEN%PVAR%5. (?y. (((SETSPEC GEN%PVAR%5) ((y IN s) \/ (y = x))) y)))))


Nonlinear_lemma.gcy_low

|- (!y. (((real_le y) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) ==> ((gcy y) = (gchi y))))


ANTIDERIVATIVE_CONTINUOUS

|- (!f. (!a. (!b. (((continuous_on f) (closed_interval ((CONS (a , b)) NIL))) ==> (?g. (!x. ((x IN (closed_interval ((CONS (a , b)) NIL))) ==> (((has_vector_derivative g) (f x)) ((within (at x)) (closed_interval ((CONS (a , b)) NIL)))))))))))


PATH_COMPONENT_LINEAR_IMAGE

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


VECTOR_MUL_LCANCEL

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


BETWEEN_DIST_LT

|- (!r. (!a. (!b. (!c. ((((real_lt (distance (c , a))) r) /\ (((real_lt (distance (c , b))) r) /\ ((between x) (a , b)))) ==> ((real_lt (distance (c , x))) r))))))


Sphere.sol0

|- (sol0 = ((real_sub ((real_mul (real_of_num (NUMERAL (BIT1 (BIT1 _0))))) (acs ((real_div (real_of_num (NUMERAL (BIT1 _0)))) (real_of_num (NUMERAL (BIT1 (BIT1 _0)))))))) pi))


ANALYTIC_AT

|- (!f. (!z. (((analytic_on f) (z INSERT EMPTY)) = (?s. ((open s) /\ ((z IN s) /\ ((holomorphic_on f) s)))))))


ABSOLUTELY_INTEGRABLE_CMUL_EQ

|- (!f. (!s. (!c. (((absolutely_integrable_on (\x. ((% c) (f x)))) s) = ((c = (real_of_num (NUMERAL _0))) \/ ((absolutely_integrable_on f) s))))))


IMAGE_LIFT_UNIV

|- (((IMAGE lift) UNIV) = UNIV)


CLOSEST_POINT_UNIQUE

|- (!s. (!a. (!x. (((convex s) /\ ((closed s) /\ ((x IN s) /\ (!z. ((z IN s) ==> ((real_le (distance (a , x))) (distance (a , z)))))))) ==> (x = ((closest_point s) a))))))


FORALL_IN_CLAUSES

|- ((!P. ((!x. ((x IN EMPTY) ==> (P x))) = T)) /\ (!P. (!a. (!s. ((!x. ((x IN (a INSERT s)) ==> (P x))) = ((P a) /\ (!x. ((x IN s) ==> (P x)))))))))


VECTOR_VARIATION_GE_NORM_FUNCTION

|- (!f. (!s. (!a. (!b. ((((has_bounded_variation_on f) s) /\ ((closed_segment ((CONS (a , b)) NIL)) SUBSET s)) ==> ((real_le (vector_norm ((vector_sub (f b)) (f a)))) ((vector_variation s) f)))))))


LIM_WITHIN_OPEN

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


Float.POW_1

|- (!x. (((real_pow x) (NUMERAL (BIT1 _0))) = x))


NORM_NEG

|- (!x. ((vector_norm (vector_neg x)) = (vector_norm x)))


REAL_CONTINUOUS_AT_WITHIN

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


DOT_2

|- (((dot x) y) = ((real_add ((real_mul (x $ (NUMERAL (BIT1 _0)))) (y $ (NUMERAL (BIT1 _0))))) ((real_mul (x $ (NUMERAL (BIT0 (BIT1 _0))))) (y $ (NUMERAL (BIT0 (BIT1 _0)))))))


Seq.cat1s

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


LEBESGUE_MEASURABLE_ALMOST_FSIGMA

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


Trigonometry1.NORM_CAUCHY_SCHWARZ_FRAC

|- (!u. (!v. (((~ (u = (vec (NUMERAL _0)))) /\ (~ (v = (vec (NUMERAL _0))))) ==> (((real_le (real_neg (real_of_num (NUMERAL (BIT1 _0))))) ((real_div ((dot u) v)) ((real_mul (vector_norm u)) (vector_norm v)))) /\ ((real_le ((real_div ((dot u) v)) ((real_mul (vector_norm u)) (vector_norm v)))) (real_of_num (NUMERAL (BIT1 _0))))))))


VECTOR_DIFFERENTIABLE

|- (!f. (!net. (((differentiable f) net) = (?f'. (((has_vector_derivative f) f') net)))))


Sphere.h0

|- (h0 = ((DECIMAL (NUMERAL (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))))


CARD_EXP_CONG

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


FORALL_IN_UNIONS

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


Bump.MCELL_CELL_PARAMETERS_EXIST

|- (!V. (!ul. (!k. (!X. (((k <= (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) /\ ((packing V) /\ ((saturated V) /\ ((X = (((mcell k) V) ul)) /\ ((ul IN ((barV V) (NUMERAL (BIT1 (BIT1 _0))))) /\ (~ (negligible X))))))) ==> ((FST ((cell_params V) X)) = k))))))


Cuxvzoz.vv_azim_le_alt

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


Ssrnat.minnCA

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


IN_CBALL_0

|- (!x. (!e. ((x IN (cball ((vec (NUMERAL _0)) , e))) = ((real_le (vector_norm x)) e))))


Float.ABS_NUM_P

|- (!m. (!n. ((m <= n) ==> ((real_abs ((real_sub (real_of_num n)) (real_of_num m))) = (real_of_num ((m - n) + (n - m)))))))


NUMERAL

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


Real_ext.REAL_PROP_NN_DOUBLE

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


EXP_LOG

|- (!x. (((real_lt (real_of_num (NUMERAL _0))) x) ==> ((exp (log x)) = x)))


PATH_CONNECTED_TRANSLATION_EQ

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


REVERSEPATH_TRANSLATION

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


MEASURE_LOCALLY_LIPSCHITZ_IMAGE

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


Conforming.conforming_solid_angle_fan

|- (!x. (!V. (!E. ((conforming_solid_angle_fan (x , (V , E))) = (!f. ((f IN (face_set (hypermap1_of_fanx (x , (V , E))))) ==> ((LET (\U. (LET_END ((!r. (measurable ((ball (x , r)) INTER U))) /\ (((eventually_radial x) U) /\ (((sol x) U) = ((real_add ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) pi)) ((sum f) (\y. ((real_sub (((((azim_fan x) V) E) (pr2 y)) (pr3 y))) pi)))))))))) ((((dartset_leads_into_fan x) V) E) f))))))))


HYPERPLANE_EQ_EMPTY

|- (!a. (!b. (((GSPEC (\GEN%PVAR%171. (?x. (((SETSPEC GEN%PVAR%171) (((dot a) x) = b)) x)))) = EMPTY) = ((a = (vec (NUMERAL _0))) /\ (~ (b = (real_of_num (NUMERAL _0))))))))


IS_REALINTERVAL_CONVEX

|- (!s. ((is_realinterval s) = (convex ((IMAGE lift) s))))


SUMMABLE_ZETA

|- (!n. (!z. (((real_lt (real_of_num (NUMERAL (BIT1 _0)))) (Re z)) ==> ((summable (from n)) (\k. (complex_inv ((cpow (Cx (real_of_num k))) z)))))))


REAL_SUB_TRIANGLE

|- (!a. (!b. (!c. (((real_add ((real_sub a) b)) ((real_sub b) c)) = ((real_sub a) c)))))


IN_CARD_MUL

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


REAL_ABS_NUM

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


Tame_defs.restricted_hypermap

|- (!H. ((restricted_hypermap H) = ((is_no_double_joins H) /\ ((~ ((dart H) = EMPTY)) /\ ((planar_hypermap H) /\ ((connected_hypermap H) /\ ((plain_hypermap H) /\ ((simple_hypermap H) /\ ((is_edge_nondegenerate H) /\ ((is_node_nondegenerate H) /\ (!f. ((f IN (face_set H)) ==> ((CARD f) >= (NUMERAL (BIT1 (BIT1 _0))))))))))))))))


APPEND_NIL

|- (!l. (((APPEND l) NIL) = l))


BILINEAR_SWAP

|- (!op. ((bilinear (\x. (\y. ((op y) x)))) = (bilinear op)))


triangulation

|- (!tr. ((triangulation tr) = ((FINITE tr) /\ ((!t. ((t IN tr) ==> (?n. ((simplex n) t)))) /\ (!t. (!t'. (((t IN tr) /\ (t' IN tr)) ==> (((face_of (t INTER t')) t) /\ ((face_of (t INTER t')) t')))))))))


Hypermap_and_fan.DART_EXISTS

|- (!V. (!E. (!v. ((v IN V) ==> (?w. ((v , w) IN (dart_of_fan (V , E))))))))


HAS_REAL_INTEGRAL_LBOUND

|- (!f. (!a. (!b. (!i. ((((real_le a) b) /\ ((((has_real_integral f) i) (closed_real_interval ((CONS (a , b)) NIL))) /\ (!x. ((x IN (closed_real_interval ((CONS (a , b)) NIL))) ==> ((real_le B) (f x)))))) ==> ((real_le ((real_mul B) ((real_sub b) a))) i))))))


FSIGMA_IMP_BOREL

|- (!s. ((fsigma s) ==> (borel s)))


REAL_LE_01

|- ((real_le (real_of_num (NUMERAL _0))) (real_of_num (NUMERAL (BIT1 _0))))


Appendix.scs_bm_v39_explicit

|- (!k. (!d. (!a. (!b. (!alpha. (!beta. (!J. (!lo. (!str. ((scs_bm_v39 (scs_v39 (k , (d , (a , (alpha , (beta , (b , (J , (lo , (hi , str))))))))))) = beta))))))))))


NSUM_CLAUSES_NUMSEG

|- ((!m. (((nsum (m .. (NUMERAL _0))) f) = (((COND (m = (NUMERAL _0))) (f (NUMERAL _0))) (NUMERAL _0)))) /\ (!m. (!n. (((nsum (m .. (SUC n))) f) = (((COND (m <= (SUC n))) (((nsum (m .. n)) f) + (f (SUC n)))) ((nsum (m .. n)) f))))))


DEPENDENT_2

|- (!a. (!b. ((dependent (a INSERT (b INSERT EMPTY))) = (((COND (a = b)) (a = (vec (NUMERAL _0)))) (?x. (?y. ((((vector_add ((% x) a)) ((% y) b)) = (vec (NUMERAL _0))) /\ (~ ((x = (real_of_num (NUMERAL _0))) /\ (y = (real_of_num (NUMERAL _0))))))))))))


Seq2.uniq_nthP

|- (!x0. (!s. ((uniq s) = (!i. (!j. (((i < j) /\ (j < (sizel s))) ==> (~ ((((nth x0) s) i) = (((nth x0) s) j)))))))))


Oxl_2012.CARD_INSERT

|- (!x. (!s. ((FINITE s) ==> ((CARD (x INSERT s)) = (((COND (x IN s)) (CARD s)) (SUC (CARD s)))))))


NOT_FORALL_THM

|- (!P. ((~ (!x. (P x))) = (?x. (~ (P x)))))


COMPLEX_MUL_RNEG

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


CONNECTED_INTERMEDIATE_CLOSURE

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


Ssrnat.leq_trans

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


COUNTABLE_OPEN_COMPONENTS

|- (!s. ((open s) ==> (COUNTABLE (components s))))


submetric

|- (!s. (!m. (((submetric m) s) = (metric ((s INTER (mspace m)) , (mdist m))))))


BIJECTIONS_CARD_EQ

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


CONTINUOUS_ABS

|- (!f. (!net. (((continuous f) net) ==> ((continuous (\x. (lambdai. (real_abs ((f x) $ i))))) net))))


LT_REFL

|- (!n. (~ (n < n)))


CONTINUOUS_WITHINREAL

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


CRAMER_MATRIX_LEFT

|- (!A. (!X. (!B. ((~ ((det A) = (real_of_num (NUMERAL _0)))) ==> ((((matrix_mul X) A) = B) = (X = (lambdak. (lambdal. ((real_div (det (lambdai. (lambdaj. (((COND (j = l)) ((B $ k) $ i)) ((A $ j) $ i)))))) (det A))))))))))


lin_combo

|- (!V. (!f. (((lin_combo V) f) = ((vsum V) (\v. ((% (f v)) v))))))


SUB_ELIM_THM

|- ((P (a - b)) = (!d. (((a = (b + d)) \/ ((a < b) /\ (d = (NUMERAL _0)))) ==> (P d))))


REAL_MEASURABLE_COUNTABLE_UNIONS_STRONG

|- (!s. (!B. (((!n. (real_measurable (s n))) /\ (!n. ((real_le (real_measure (UNIONS (GSPEC (\GEN%PVAR%4945. (?k. (((SETSPEC GEN%PVAR%4945) (k <= n)) (s k)))))))) B))) ==> (real_measurable (UNIONS (GSPEC (\GEN%PVAR%4946. (?n. (((SETSPEC GEN%PVAR%4946) (n IN UNIV)) (s n))))))))))


Ssrnat.addn0

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


IM_MUL_II

|- (!z. (((Im ((complex_mul z) ii)) = (Re z)) /\ ((Im ((complex_mul ii) z)) = (Re z))))


BROUWER_INESSENTIAL_ANR

|- (!f. (!s. (((compact s) /\ ((~ (s = EMPTY)) /\ ((ANR s) /\ (((continuous_on f) s) /\ ((((IMAGE f) s) SUBSET s) /\ (?a. ((((homotopic_with (\x. T)) (s , s)) f) (\x. a)))))))) ==> (?x. ((x IN s) /\ ((f x) = x))))))


HOMEOMORPHIC_SIMPLE_PATH_ARC

|- (!g. (!h. (((arc g) /\ ((simple_path h) /\ ((homeomorphic (path_image g)) (path_image h)))) ==> (arc h))))


List_hypermap.good_list_nodes_condition

|- (!L. ((good_list L) ==> ((good_list_nodes L) = ((CARD (node_set (hypermap_of_list L))) = (sizel (list_of_elements L))))))


CLOSURE_INTER_SUBSET

|- (!s. (!t. ((closure (s INTER t)) SUBSET ((closure s) INTER (closure t)))))


aff_gt_def

|- (aff_gt = (affsign sgn_gt))


AFFINE_EQ_SUBSPACE

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


LINSEG_INSEG

|- (!l. (!a. ((woset l) ==> ((inseg ((linseg l) a)) l))))


List_hypermap.mem_list_pairs

|- (!x. (!y. (!l. (((MEM (x , y)) (list_pairs l)) ==> (((MEM x) l) /\ ((MEM y) l))))))


INT_MUL_LID

|- (!x. (((int_mul (int_of_num (NUMERAL (BIT1 _0)))) x) = x))


Lp_ineqs_proofs.lp_fan

|- (!E. (!V. ((lp_fan (V , E)) = ((FAN ((vec (NUMERAL _0)) , (V , E))) /\ ((fully_surrounded (V , E)) /\ ((!v. (!w. (((v IN V) /\ ((w IN V) /\ ((~ (v = w)) /\ (~ ((v INSERT (w INSERT EMPTY)) IN E))))) ==> ((real_le ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) (distance (v , w)))))) /\ ((V SUBSET ball_annulus) /\ (packing V))))))))


REAL_ABS_REFL

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


aff_lt_def

|- (aff_lt = (affsign sgn_lt))


COMPLEX_SUB_POW_R1

|- (!x. (!n. (((NUMERAL (BIT1 _0)) <= n) ==> (((vector_sub ((complex_pow x) n)) (Cx (real_of_num (NUMERAL (BIT1 _0))))) = ((complex_mul ((vector_sub x) (Cx (real_of_num (NUMERAL (BIT1 _0)))))) ((vsum ((NUMERAL _0) .. (n - (NUMERAL (BIT1 _0))))) (\i. ((complex_pow x) i))))))))


Ssrnat.leqif_imp_eq

|- (!m. (!n. (!c. ((((leqif m) n) c) ==> ((m = n) = c)))))


LOG_INV

|- (!x. (((real_lt (real_of_num (NUMERAL _0))) x) ==> ((log (real_inv x)) = (real_neg (log x)))))


REAL_POW_ADD

|- (!x. (!m. (!n. (((real_pow x) (m + n)) = ((real_mul ((real_pow x) m)) ((real_pow x) n))))))


num_gcd

|- (!a. (!b. ((num_gcd (a , b)) = (num_of_int (int_gcd ((int_of_num a) , (int_of_num b)))))))


Appendix.mk_unadorned_v39

|- (!k. (!d. (!a. (!b. (((((mk_unadorned_v39 k) d) a) b) = (scs_v39 (k , (d , (a , (a , (b , (b , ((\i. (\j. F)) , (EMPTY , (EMPTY , EMPTY)))))))))))))))


PASTECART_FST_SND

|- (!z. (((pastecart (fstcart z)) (sndcart z)) = z))


CARD_EQ_OPEN_IN

|- (!u. (!s. ((((locally connected) u) /\ (((open_in ((subtopology euclidean) u)) s) /\ (?x. ((x IN s) /\ ((limit_point_of x) u))))) ==> (s =_c UNIV))))


CLOSURE_SUBSET

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


HOMEOMORPHISM_INJECTIVE_OPEN_MAP_EQ

|- (!f. (!s. (!t. ((((continuous_on f) s) /\ ((((IMAGE f) s) = t) /\ (!x. (!y. (((x IN s) /\ ((y IN s) /\ ((f x) = (f y)))) ==> (x = y)))))) ==> ((?g. ((homeomorphism (s , t)) (f , g))) = (!u. (((open_in ((subtopology euclidean) s)) u) ==> ((open_in ((subtopology euclidean) t)) ((IMAGE f) u)))))))))


HAS_ABSOLUTE_INTEGRAL_CHANGE_OF_VARIABLES

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


vector_angle

|- (!x. (!y. (((vector_angle x) y) = (((COND ((x = (vec (NUMERAL _0))) \/ (y = (vec (NUMERAL _0))))) ((real_div pi) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) (acs ((real_div ((dot x) y)) ((real_mul (vector_norm x)) (vector_norm y))))))))


STARLIKE_IMP_CONTRACTIBLE

|- (!s. ((starlike s) ==> (contractible s)))


WF_ANTISYM

|- (!<<. (!x. (!y. ((WF <<) ==> (~ (((<< x) y) /\ ((<< y) x)))))))


SC_INDUCT

|- (!R. (!SC'. (((!x. (!y. (((R x) y) ==> ((SC' x) y)))) /\ (!x. (!y. (((SC' x) y) ==> ((SC' y) x))))) ==> (!a0. (!a1. ((((SC R) a0) a1) ==> ((SC' a0) a1)))))))


SUM_CONST

|- (!c. (!s. ((FINITE s) ==> (((sum s) (\n. c)) = ((real_mul (real_of_num (CARD s))) c)))))


IMAGE_AFFINITY_REAL_INTERVAL

|- (!a. (!b. (!m. (!c. (((IMAGE (\x. ((real_add ((real_mul m) x)) c))) (closed_real_interval ((CONS (a , b)) NIL))) = (((COND ((closed_real_interval ((CONS (a , b)) NIL)) = EMPTY)) EMPTY) (((COND ((real_le (real_of_num (NUMERAL _0))) m)) (closed_real_interval ((CONS (((real_add ((real_mul m) a)) c) , ((real_add ((real_mul m) b)) c))) NIL))) (closed_real_interval ((CONS (((real_add ((real_mul m) b)) c) , ((real_add ((real_mul m) a)) c))) NIL)))))))))


INTERVAL_SUBSET_IS_INTERVAL

|- (!s. (!a. (!b. ((is_interval s) ==> (((closed_interval ((CONS (a , b)) NIL)) SUBSET s) = (((closed_interval ((CONS (a , b)) NIL)) = EMPTY) \/ ((a IN s) /\ (b IN s))))))))


SIMPLE_PATH_LINEAR_IMAGE_EQ

|- (!f. (!g. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((simple_path (f o g)) = (simple_path g)))))


EXPOSED_POINT_OF_INTER_SUPPORTING_HYPERPLANE_LE

|- (!s. (!a. (!b. (!c. (((!x. ((x IN s) ==> ((real_le ((dot a) x)) b))) /\ ((s INTER (GSPEC (\GEN%PVAR%2202. (?x. (((SETSPEC GEN%PVAR%2202) (((dot a) x) = b)) x))))) = (c INSERT EMPTY))) ==> ((exposed_face_of (c INSERT EMPTY)) s))))))


NOT_COPLANAR_NOT_COLLINEAR

|- (!v1. (!v2. (!v3. (!w. ((~ (coplanar (v1 INSERT (v2 INSERT (v3 INSERT (w INSERT EMPTY)))))) ==> (~ (collinear (v1 INSERT (v2 INSERT (v3 INSERT EMPTY))))))))))


CONNECTED_CHAIN_GEN

|- (!f. (((!s. ((s IN f) ==> ((closed s) /\ (connected s)))) /\ ((?s. ((s IN f) /\ (compact s))) /\ (!s. (!t. (((s IN f) /\ (t IN f)) ==> ((s SUBSET t) \/ (t SUBSET s))))))) ==> (connected (INTERS f))))


CONTENT_POS_LT

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


LINEAR_INJECTIVE_ISOMORPHISM

|- (!f. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> (?f'. ((linear f') /\ ((!x. ((f' (f x)) = x)) /\ (!x. ((f (f' x)) = x)))))))


MEASURABLE_ON_COMPOSE_CONTINUOUS_CLOSED_SET_0

|- (!f. (!g. (!s. (!t. (((closed s) /\ (((measurable_on f) t) /\ ((!x. ((f x) IN s)) /\ (((continuous_on g) s) /\ (((vec (NUMERAL _0)) IN s) /\ ((g (vec (NUMERAL _0))) = (vec (NUMERAL _0)))))))) ==> ((measurable_on (g o f)) t))))))


REAL_INTERVAL_TRANSLATION

|- ((!c. (!a. (!b. ((closed_real_interval ((CONS (((real_add c) a) , ((real_add c) b))) NIL)) = ((IMAGE (\x. ((real_add c) x))) (closed_real_interval ((CONS (a , b)) NIL))))))) /\ (!c. (!a. (!b. ((open_real_interval (((real_add c) a) , ((real_add c) b))) = ((IMAGE (\x. ((real_add c) x))) (open_real_interval (a , b))))))))


BANACH_FIX

|- (!f. (!s. (!c. (((complete s) /\ ((~ (s = EMPTY)) /\ (((real_le (real_of_num (NUMERAL _0))) c) /\ (((real_lt c) (real_of_num (NUMERAL (BIT1 _0)))) /\ ((((IMAGE f) s) SUBSET s) /\ (!x. (!y. (((x IN s) /\ (y IN s)) ==> ((real_le (distance ((f x) , (f y)))) ((real_mul c) (distance (x , y)))))))))))) ==> (?!x. ((x IN s) /\ ((f x) = x)))))))


DIFFERENTIABLE_IMP_CONTINUOUS_ON

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


CX_NEG

|- (!x. ((Cx (real_neg x)) = (vector_neg (Cx x))))


NPRODUCT_EQ_1

|- (!f. (!s. ((!x. ((x IN s) ==> ((f x) = (NUMERAL (BIT1 _0))))) ==> (((nproduct s) f) = (NUMERAL (BIT1 _0))))))


COMPLEX_MUL_CSIN_CCOS

|- (!w. (!z. (((complex_mul (csin w)) (ccos z)) = ((complex_div ((vector_add (csin ((vector_add w) z))) (csin ((vector_sub w) z)))) (Cx (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))))))


SERIES_DIRICHLET

|- (!f. (!g. (!N. (!k. (!m. (((bounded (GSPEC (\GEN%PVAR%1126. (?n. (((SETSPEC GEN%PVAR%1126) (n IN UNIV)) ((vsum (m .. n)) f)))))) /\ ((!n. ((N <= n) ==> ((real_le (g (n + (NUMERAL (BIT1 _0))))) (g n)))) /\ (((--> (lift o g)) (vec (NUMERAL _0))) sequentially))) ==> ((summable (from k)) (\n. ((% (g n)) (f n))))))))))


Nonlinear_lemma.vol3f_palt

|- (!y1. (!y2. (!y3. (!y4. (!y5. (!y6. (!r. (!f. (((y3 = r) /\ ((y4 = r) /\ (y5 = r))) ==> ((((((vol3f y1) y2) y6) r) f) = ((real_sub ((real_mul ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((real_div mm1) pi))) ((real_add ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((((((dih_y y1) y2) y3) y4) y5) y6))) ((real_add ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((((((dih2_y y1) y2) y3) y4) y5) y6))) ((real_add ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((((((dih6_y y1) y2) y3) y4) y5) y6))) ((real_add ((((((dih3_y y1) y2) y3) y4) y5) y6)) ((real_add ((((((dih4_y y1) y2) y3) y4) y5) y6)) ((real_sub ((((((dih5_y y1) y2) y3) y4) y5) y6)) ((real_mul (real_of_num (NUMERAL (BIT1 (BIT1 _0))))) pi))))))))) ((real_mul ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 _0))))))) ((real_div mm2) pi))) ((real_add ((real_mul (f ((real_div y1) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) ((((((dih_y y1) y2) y3) y4) y5) y6))) ((real_add ((real_mul (f ((real_div y2) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) ((((((dih2_y y1) y2) y3) y4) y5) y6))) ((real_mul (f ((real_div y6) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) ((((((dih6_y y1) y2) y3) y4) y5) y6))))))))))))))))


IS_INTERVAL_INTERVAL

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


CLOSURE_BALL

|- (!x. (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> ((closure (ball (x , e))) = (cball (x , e))))))


HULL_HULL

|- (!P. (!s. (((hull P) ((hull P) s)) = ((hull P) s))))


NADD_MUL_SYM

|- (!x. (!y. ((nadd_eq ((nadd_mul x) y)) ((nadd_mul y) x))))


EXP_ADD

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


IM_MUL_CX

|- (!x. (!z. (((Im ((complex_mul (Cx x)) z)) = ((real_mul x) (Im z))) /\ ((Im ((complex_mul z) (Cx x))) = ((real_mul (Im z)) x)))))


MONOIDAL_ADD

|- (monoidal +)


LIM_COMPLEX_MUL

|- (!net. (!f. (!g. (!l. (!m. (((((--> f) l) net) /\ (((--> g) m) net)) ==> (((--> (\x. ((complex_mul (f x)) (g x)))) ((complex_mul l) m)) net)))))))


rotation_matrix

|- (!Q. ((rotation_matrix Q) = ((orthogonal_matrix Q) /\ ((det Q) = (real_of_num (NUMERAL (BIT1 _0)))))))


Ssrnat.leq_addr

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


Ssrnat.halfS

|- (!n. ((half (SUC n)) = (uphalf n)))


SIMPLEX_FURTHEST_LE

|- (!a. (!s. (((FINITE s) /\ (~ (s = EMPTY))) ==> (?y. ((y IN s) /\ (!x. ((x IN ((hull convex) s)) ==> ((real_le (vector_norm ((vector_sub x) a))) (vector_norm ((vector_sub y) a))))))))))


Packing3.VORONOI_SET_2

|- (!V. (!u. (!v. (((voronoi_set V) (u INSERT (v INSERT EMPTY))) = (((voronoi_closed V) v) INTER ((voronoi_closed V) u))))))


RESTRICTION_DEFINED

|- (!s. (!f. (!x. ((x IN s) ==> ((((RESTRICTION s) f) x) = (f x))))))


Trigonometry.JLWZFBH4

|- (!s. ((coplanar s) = (?u. (?v. (?w. (s SUBSET ((hull affine) (u INSERT (v INSERT (w INSERT EMPTY))))))))))


DIV_LE_EXCLUSION

|- (!a. (!b. (!c. (!d. (((~ (b = (NUMERAL _0))) /\ ((b * c) < ((a + (NUMERAL (BIT1 _0))) * d))) ==> ((c DIV d) <= (a DIV b)))))))


ARITH_SUC

|- ((!n. ((SUC (NUMERAL n)) = (NUMERAL (SUC n)))) /\ (((SUC _0) = (BIT1 _0)) /\ ((!n. ((SUC (BIT0 n)) = (BIT1 n))) /\ (!n. ((SUC (BIT1 n)) = (BIT0 (SUC n)))))))


REAL_NEG_RMUL

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


Conforming.SIGMA_FAN_OF_FANADD_AT_POINT2

|- (!x. (!V. (!E. (!E1. (!v. (!u. (!w. (((FAN (x , (V , E))) /\ ((FAN (x , (V , E1))) /\ (((v INSERT (u INSERT EMPTY)) IN E) /\ (((u INSERT (w INSERT EMPTY)) IN E) /\ ((~ ((w INSERT (v INSERT EMPTY)) IN E)) /\ (((((((sigma_fan x) V) E) u) w) = v) /\ ((fan80 (x , (V , E))) /\ ((!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) v) u) = w)))))))))


Add_triangle.pair_expand

|- (!p. (p = ((FST p) , (SND p))))


REAL_LTE_ADD2

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


CARD_MUL_FINITE

|- (!s. (!t. (((FINITE s) /\ (FINITE t)) ==> (FINITE ((*_c s) t)))))


PATHFINISH_IN_PATH_IMAGE

|- (!g. ((pathfinish g) IN (path_image g)))


Pack2.MEASURE_VORONOI_CLOSED_OPEN

|- (!s. (!v. ((measure ((voronoi_closed s) v)) = (measure ((voronoi_open s) v)))))


csin

|- (!z. ((csin z) = ((complex_div ((vector_sub (cexp ((complex_mul ii) z))) (cexp ((complex_mul (vector_neg ii)) z)))) ((complex_mul (Cx (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) ii))))


SUBPATH_SCALING_LEMMA

|- (!u. (!v. (((IMAGE (\x. ((vector_add u) ((% (drop ((vector_sub v) u))) x)))) (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL))) = (closed_segment ((CONS (u , v)) NIL)))))


AFF_DIM_OPEN_IN

|- (!s. (!t. (((~ (s = EMPTY)) /\ (((open_in ((subtopology euclidean) t)) s) /\ (affine t))) ==> ((aff_dim s) = (aff_dim t)))))


NSUM_ADD_NUMSEG

|- (!f. (!g. (!m. (!n. (((nsum (m .. n)) (\i. ((f i) + (g i)))) = (((nsum (m .. n)) f) + ((nsum (m .. n)) g)))))))


EQ_MULT_LCANCEL

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


CONTRACTIBLE_PCROSS_EQ

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


MDIST_SYM

|- (!m. (!x. (!y. (((x IN (mspace m)) /\ (y IN (mspace m))) ==> (((mdist m) (x , y)) = ((mdist m) (y , x)))))))


SUM_POS_LE_NUMSEG

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


FRONTIER_BIJECTIVE_LINEAR_IMAGE

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


DE_MORGAN_THM

|- (!t1. (!t2. (((~ (t1 /\ t2)) = ((~ t1) \/ (~ t2))) /\ ((~ (t1 \/ t2)) = ((~ t1) /\ (~ t2))))))


TRANSLATION_UNIV

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


Real_ext.REAL_PROP_ZERO_SQRT

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


Collect_geom.conv

|- (!S. ((conv S) = (((affsign sgn_ge) EMPTY) S)))


LEFT_FORALL_OR_THM

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


GDELTA_UNION

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


INSIDE_TRANSLATION

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


INVERTIBLE_MATRIX_MUL

|- (!A. (!B. ((invertible ((matrix_mul A) B)) = ((invertible A) /\ (invertible B)))))


PAIR_EQ

|- (!x. (!y. (!a. (!b. (((x , y) = (a , b)) = ((x = a) /\ (y = b)))))))


FSIGMA_INTER

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


COMPLEX_MUL_LID

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


HAS_REAL_INTEGRAL_REFLECT_GEN

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


ORDINAL_CHAINED

|- (!l. (!m. (((ordinal l) /\ (ordinal m)) ==> (((inseg m) l) \/ ((inseg l) m)))))


Ssrnat.minn

|- (!m. (!n. (((minn m) n) = (((COND (m < n)) m) n))))


FORALL_IN_INTERMEDIATE_CLOSURE

|- (!f. (!s. (!s'. (!t. (((closed t) /\ (((continuous_on f) s') /\ ((s SUBSET s') /\ ((s' SUBSET (closure s)) /\ (!x. ((x IN s) ==> ((f x) IN t))))))) ==> (!x. ((x IN s') ==> ((f x) IN t))))))))


HAS_BOUNDED_VARIATION_ABSOLUTELY_INTEGRABLE_DERIVATIVE

|- (!f. (!s. (!a. (!b. (((COUNTABLE s) /\ (((continuous_on f) (closed_interval ((CONS (a , b)) NIL))) /\ (!x. ((x IN ((closed_interval ((CONS (a , b)) NIL)) DIFF s)) ==> ((differentiable f) (at x)))))) ==> (((has_bounded_variation_on f) (closed_interval ((CONS (a , b)) NIL))) = ((absolutely_integrable_on (\x. ((vector_derivative f) (at x)))) (closed_interval ((CONS (a , b)) NIL)))))))))


Appendix.scs_prop_equ_v39

|- (!s. (!i. (((scs_prop_equ_v39 s) i) = ((LET (\shift1. (LET_END ((LET (\shift2. (LET_END ((LET (\shift2b. (LET_END (scs_v39 ((scs_k_v39 s) , ((scs_d_v39 s) , ((shift2 (scs_a_v39 s)) , ((shift2 (scs_am_v39 s)) , ((shift2 (scs_bm_v39 s)) , ((shift2 (scs_b_v39 s)) , ((shift2b (scs_J_v39 s)) , ((shift1 (scs_lo_v39 s)) , ((shift1 (scs_hi_v39 s)) , (shift1 (scs_str_v39 s))))))))))))))) (\f. (\j. (\j'. ((f (i + j)) (i + j'))))))))) (\f. (\j. (\j'. ((f (i + j)) (i + j'))))))))) (\f. (\j. (f (i + j))))))))


COS_PERIODIC

|- (!x. ((cos ((real_add x) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) pi))) = (cos x)))


Hypermap_and_fan.TRIANGULAR_FACE

|- (!V. (!E. (!v. (!w. (((FAN ((vec (NUMERAL _0)) , (V , E))) /\ (((v , w) IN (dart1_of_fan (V , E))) /\ ((CARD ((face (hypermap_of_fan (V , E))) (v , w))) = (NUMERAL (BIT1 (BIT1 _0)))))) ==> ((LET (\w'. (LET_END ((((face (hypermap_of_fan (V , E))) (v , w)) = ((v , w) INSERT ((w , w') INSERT ((w' , v) INSERT EMPTY)))) /\ (((((((sigma_fan (vec (NUMERAL _0))) V) E) w) w') = v) /\ ((((((sigma_fan (vec (NUMERAL _0))) V) E) w') v) = w)))))) (((((sigma_fan (vec (NUMERAL _0))) V) E) v) w)))))))


REAL_SERIES_LMUL

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


CONVEX_TRANSLATION

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


COMPLEX_MUL_LZERO

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


FORALL_IN_PCROSS

|- ((!z. ((z IN (s PCROSS t)) ==> (P z))) = (!x. (!y. (((x IN s) /\ (y IN t)) ==> (P ((pastecart x) y))))))


NORM_LIFT

|- (!x. ((vector_norm (lift x)) = (real_abs x)))


MATRIX_MUL_RCANCEL

|- (!A. (!B. (!C. ((invertible C) ==> ((((matrix_mul A) C) = ((matrix_mul B) C)) = (A = B))))))


SERIES_RATIO

|- (!c. (!a. (!s. (!N. ((((real_lt c) (real_of_num (NUMERAL (BIT1 _0)))) /\ (!n. ((n >= N) ==> ((real_le (vector_norm (a (SUC n)))) ((real_mul c) (vector_norm (a n))))))) ==> (?l. (((sums a) l) s)))))))


LINEAR_CONTINUOUS_WITHIN

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


IMAGE_STRETCH_INTERVAL

|- (!a. (!b. (!m. (((IMAGE (\x. (lambdak. ((real_mul (m k)) (x $ k))))) (closed_interval ((CONS (a , b)) NIL))) = (((COND ((closed_interval ((CONS (a , b)) NIL)) = EMPTY)) EMPTY) (closed_interval ((CONS ((lambdak. ((real_min ((real_mul (m k)) (a $ k))) ((real_mul (m k)) (b $ k)))) , (lambdak. ((real_max ((real_mul (m k)) (a $ k))) ((real_mul (m k)) (b $ k)))))) NIL)))))))


Sphere.rotate3

|- (!f. (!x3. (!x1. (!x2. (!x6. (!x4. (!x5. ((((((((rotate3 f) x1) x2) x3) x4) x5) x6) = ((((((f x3) x1) x2) x6) x4) x5)))))))))


RECTIFIABLE_PATH_TRANSLATION_EQ

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


COVERING_SPACE_FINITE_SHEETS_EQ_CLOSED_MAP_STRONG

|- (!p. (!c. (!s. ((((covering_space (c , p)) s) /\ (!b. ((b IN s) ==> ((limit_point_of b) s)))) ==> ((!b. ((b IN s) ==> (FINITE (GSPEC (\GEN%PVAR%2080. (?x. (((SETSPEC GEN%PVAR%2080) ((x IN c) /\ ((p x) = b))) x))))))) = (!t. (((closed_in ((subtopology euclidean) c)) t) ==> ((closed_in ((subtopology euclidean) s)) ((IMAGE p) t)))))))))


SEGMENT_REFL

|- ((!a. ((closed_segment ((CONS (a , a)) NIL)) = (a INSERT EMPTY))) /\ (!a. ((open_segment (a , a)) = EMPTY)))


REAL_ABS_1

|- ((real_abs (real_of_num (NUMERAL (BIT1 _0)))) = (real_of_num (NUMERAL (BIT1 _0))))


HAS_VECTOR_DERIVATIVE_AT_WITHIN

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


REAL_TENDSTO

|- (((--> s) l) = ((---> (drop o s)) (drop l)))


CLOSED_INTERVAL

|- (!a. (!b. (closed (closed_interval ((CONS (a , b)) NIL)))))


CARD_LE_REFL

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


Lp_ineqs_def.tau_fan

|- (!V. (!E. (!f. (((tau_fan (V , E)) f) = (real_abs (tauVEF (V , (E , f))))))))


Appendix.scs_lo_v39_explicit

|- (!k. (!d. (!a. (!b. (!alpha. (!beta. (!J. (!lo. (!str. ((scs_lo_v39 (scs_v39 (k , (d , (a , (alpha , (beta , (b , (J , (lo , (hi , str))))))))))) = lo))))))))))


Ssrbool.predC

|- (!p. ((predC p) = (\x. (~ (p x)))))


REAL_POW_LE

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


REAL_LE_BETWEEN

|- (!a. (!b. (((real_le a) b) = (?x. (((real_le a) x) /\ ((real_le x) b))))))


LOG_POW

|- (!n. (!x. (((real_lt (real_of_num (NUMERAL _0))) x) ==> ((log ((real_pow x) n)) = ((real_mul (real_of_num n)) (log x))))))


Lvducxu.FACE_NODE_EDGE_ORBIT_INVERSE

|- ((((face H) x) = ((orbit_map (inverse (face_map H))) x)) /\ ((((node H) x) = ((orbit_map (inverse (node_map H))) x)) /\ (((edge H) x) = ((orbit_map (inverse (edge_map H))) x))))


STEINHAUS_SUMS

|- (!s. (!t. (((lebesgue_measurable s) /\ ((~ (negligible s)) /\ ((lebesgue_measurable t) /\ (~ (negligible t))))) ==> (~ ((interior (GSPEC (\GEN%PVAR%4605. (?x. (?y. (((SETSPEC GEN%PVAR%4605) ((x IN s) /\ (y IN t))) ((vector_add x) y))))))) = EMPTY)))))


Ssrnat.subnn

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


CONNECTED_FROM_OPEN_UNION_AND_INTER_LOCAL

|- (!u. (!s. (!t. ((((open_in ((subtopology euclidean) u)) s) /\ (((open_in ((subtopology euclidean) u)) t) /\ ((connected (s UNION t)) /\ (connected (s INTER t))))) ==> ((connected s) /\ (connected t))))))


RETRACT_OF_UNIV

|- (!s. (((retract_of s) UNIV) = ((AR s) /\ (closed s))))


EXPLICIT_SUBSET_RELATIVE_INTERIOR_CONVEX_HULL_MINIMAL

|- (!s. ((FINITE s) ==> ((GSPEC (\GEN%PVAR%1790. (?y. (((SETSPEC GEN%PVAR%1790) (?u. ((!x. ((x IN s) ==> ((real_lt (real_of_num (NUMERAL _0))) (u x)))) /\ ((((sum s) u) = (real_of_num (NUMERAL (BIT1 _0)))) /\ (((vsum s) (\x. ((% (u x)) x))) = y))))) y)))) SUBSET (relative_interior ((hull convex) s)))))


LEBESGUE_MEASURABLE_INTERVAL

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


HOMOTOPIC_PATHS_NEARBY_EXPLICIT

|- (!g. (!s. (!h. (((path g) /\ ((path h) /\ (((pathstart h) = (pathstart g)) /\ (((pathfinish h) = (pathfinish g)) /\ (!t. (!x. (((t IN (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL))) /\ (~ (x IN s))) ==> ((real_lt (vector_norm ((vector_sub (h t)) (g t)))) (vector_norm ((vector_sub (g t)) x)))))))))) ==> (((homotopic_paths s) g) h)))))


HOMOTOPIC_CONSTANT_MAPS

|- (!s. (!t. (!a. (!b. (((((homotopic_with (\x. T)) (s , t)) (\x. a)) (\x. b)) = ((s = EMPTY) \/ (((path_component t) a) b)))))))


NORM_LE_INFNORM

|- (!x. ((real_le (vector_norm x)) ((real_mul (sqrt (real_of_num (dimindex UNIV)))) (infnorm x))))


Tame_defs2.C_EL

|- (!n. (!l. (((#!# l) n) = ((EL n) l))))


Hypermap_and_fan.INVERSE_F_FAN_PAIR_EXT_EXPLICIT

|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((inverse (f_fan_pair_ext (V , E))) = ((res (GABS (\f. (!v. (!w. ((GEQ (f (v , w))) ((((((sigma_fan (vec (NUMERAL _0))) V) E) v) w) , v))))))) (dart1_of_fan (V , E)))))))


Ssrbool.andb_id2l

|- (!a. (!b. (!c. ((a ==> (b = c)) ==> ((a /\ b) = (a /\ c))))))


Seq.uniq_perm_eq

|- (!s1. (!s2. ((uniq s1) ==> ((uniq s2) ==> ((!x. (((MEM x) s1) = ((MEM x) s2))) ==> ((perm_eq s1) s2))))))


FLOOR

|- (!x. ((integer (floor x)) /\ (((real_le (floor x)) x) /\ ((real_lt x) ((real_add (floor x)) (real_of_num (NUMERAL (BIT1 _0))))))))


HAS_SIZE_UNION

|- (!s. (!t. (!m. (!n. (((s HAS_SIZE m) /\ ((t HAS_SIZE n) /\ ((DISJOINT s) t))) ==> ((s UNION t) HAS_SIZE (m + n)))))))


CARD_EQ_RESTRICTED_POWERSET

|- (!s. (!t. ((INFINITE s) ==> ((GSPEC (\GEN%PVAR%48. (?k. (((SETSPEC GEN%PVAR%48) ((k SUBSET s) /\ (k =_c t))) k)))) =_c (((COND (t <=_c s)) ((^_c s) t)) EMPTY)))))


toset

|- (!l. ((toset l) = ((poset l) /\ (!x. (!y. (((x IN (fl l)) /\ (y IN (fl l))) ==> ((l (x , y)) \/ (l (y , x)))))))))


ABSOLUTELY_CONTINUOUS_COMPARISON

|- (!f. (!g. (!s. ((((absolutely_continuous_on f) s) /\ (!x. (!y. (((x IN s) /\ ((y IN s) /\ ((real_lt (drop x)) (drop y)))) ==> ((real_le (distance ((g x) , (g y)))) (distance ((f x) , (f y)))))))) ==> ((absolutely_continuous_on g) s)))))


EQ_ADD_RCANCEL

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


POLYHEDRON_LINEAR_IMAGE

|- (!f. (!s. (((linear f) /\ (polyhedron s)) ==> (polyhedron ((IMAGE f) s)))))


one_RECURSION

|- (!e. (?fn. ((fn one) = e)))


ITERATE_CLOSED

|- (!op. ((monoidal op) ==> (!P. (((P (neutral op)) /\ (!x. (!y. (((P x) /\ (P y)) ==> (P ((op x) y)))))) ==> (!f. (!s. ((!x. (((x IN s) /\ (~ ((f x) = (neutral op)))) ==> (P (f x)))) ==> (P (((iterate op) s) f)))))))))


HOMEOMORPHIC_MINIMAL

|- (!s. (!t. (((homeomorphic s) t) = (?f. (?g. ((!x. ((x IN s) ==> (((f x) IN t) /\ ((g (f x)) = x)))) /\ ((!y. ((y IN t) ==> (((g y) IN s) /\ ((f (g y)) = y)))) /\ (((continuous_on f) s) /\ ((continuous_on g) t)))))))))


Imjxphr.EXISTS_SMALL_LE_CONST

|- ((((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))))) /\ ((((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) /\ ((a = (real_neg (real_of_num (NUMERAL (BIT1 _0))))) /\ ((real_lt c) (distance (v2 , w))))))))))) ==> (?e. (((real_lt (real_of_num (NUMERAL _0))) e) /\ (!t. ((((real_lt (real_of_num (NUMERAL _0))) t) /\ ((real_lt t) e)) ==> ((real_lt c) (distance (((((((((v3_defor_v1 a) v1) v2) x1) x2) x6) x6) ((real_sub x2) t)) , w))))))))


CLOSED_SEQUENTIAL_LIMITS

|- (!s. ((closed s) = (!x. (!l. (((!n. ((x n) IN s)) /\ (((--> x) l) sequentially)) ==> (l IN s))))))


Functional_equation.proj_x4

|- (((((((proj_x4 x1) x2) x3) x4) x5) x6) = x4)


BUTLAST

|- (((BUTLAST NIL) = NIL) /\ ((BUTLAST ((CONS h) t)) = (((COND (t = NIL)) NIL) ((CONS h) (BUTLAST t)))))


FINITE_NUMSEG_LT

|- (!n. (FINITE (GSPEC (\GEN%PVAR%119. (?m. (((SETSPEC GEN%PVAR%119) (m < n)) m))))))


CONTINUOUS_AT_DIST_CLOSEST_POINT

|- (!s. (!x. (((closed s) /\ (~ (s = EMPTY))) ==> ((continuous (\x. (lift (distance (x , ((closest_point s) x)))))) (at x)))))


NOT_ODD

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


NEGLIGIBLE_SING

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


Real_ext.REAL_PROP_NN_SQRT

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


REAL_EQ_MUL_LCANCEL

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


Seq.unzip1

|- (unzip1 = (map FST))


LAMBDA_UNPAIR_THM

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


Sphere.asn797k

|- (!x2. (!x3. (!x4. (!x5. (!x6. (!k. (((((((asn797k k) x2) x3) x4) x5) x6) = ((real_mul k) (asn ((real_mul (cos ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))))))) (sin ((real_div pi) k))))))))))))


TIETZE_CLOSED_INTERVAL

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


FUNDAMENTAL_THEOREM_OF_CALCULUS_STRONG

|- (!f. (!f'. (!s. (!a. (!b. (((COUNTABLE s) /\ (((real_le (drop a)) (drop b)) /\ (((continuous_on f) (closed_interval ((CONS (a , b)) NIL))) /\ (!x. ((x IN ((closed_interval ((CONS (a , b)) NIL)) DIFF s)) ==> (((has_vector_derivative f) (f' x)) ((within (at x)) (closed_interval ((CONS (a , b)) NIL))))))))) ==> (((has_integral f') ((vector_sub (f b)) (f a))) (closed_interval ((CONS (a , b)) NIL)))))))))


REAL_CONTINUOUS_WITHINREAL_SUBSET

|- (!f. (!s. (!t. ((((real_continuous f) ((within (atreal x)) s)) /\ (t SUBSET s)) ==> ((real_continuous f) ((within (atreal x)) t))))))


SERIES_CMUL

|- (!x. (!x0. (!c. (!s. ((((sums x) x0) s) ==> (((sums (\n. ((% c) (x n)))) ((% c) x0)) s))))))


VECTOR_1

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


List_hypermap.e_list

|- (!d. ((e_list d) = ((SND d) , (FST d))))


Ssrnat.addnA

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


UNION

|- (!s. (!t. ((s UNION t) = (GSPEC (\GEN%PVAR%0. (?x. (((SETSPEC GEN%PVAR%0) ((x IN s) \/ (x IN t))) x)))))))


DOT_SQUARE_NORM

|- (!x. (((dot x) x) = ((real_pow (vector_norm x)) (NUMERAL (BIT0 (BIT1 _0))))))


Seq.rcons

|- (!x. (!t. (!z. ((((rcons NIL) z) = ((CONS z) NIL)) /\ (((rcons ((CONS x) t)) z) = ((CONS x) ((rcons t) z)))))))


Functional_equation.dih4_x'

|- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. ((((real_le (real_of_num (NUMERAL _0))) x1) /\ (((real_le (real_of_num (NUMERAL _0))) x2) /\ (((real_le (real_of_num (NUMERAL _0))) x3) /\ (((real_le (real_of_num (NUMERAL _0))) x4) /\ (((real_le (real_of_num (NUMERAL _0))) x5) /\ ((real_le (real_of_num (NUMERAL _0))) x6)))))) ==> (((((((dih4_x x1) x2) x3) x4) x5) x6) = ((((((dih4_x' x1) x2) x3) x4) x5) x6)))))))))


LINEAR_0

|- (!f. ((linear f) ==> ((f (vec (NUMERAL _0))) = (vec (NUMERAL _0)))))


INTERIOR_CBALL

|- (!x. (!e. ((interior (cball (x , e))) = (ball (x , e)))))


DOT_BASIS

|- (!x. (!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV))) ==> ((((dot (basis i)) x) = (x $ i)) /\ (((dot x) (basis i)) = (x $ i))))))


NSUM_LE

|- (!f. (!g. (!s. (((FINITE s) /\ (!x. ((x IN s) ==> ((f x) <= (g x))))) ==> (((nsum s) f) <= ((nsum s) g))))))


CONTINUOUS_GDELTA_PREIMAGE

|- (!f. (!s. (!t. ((((continuous_on f) s) /\ ((gdelta s) /\ (gdelta t))) ==> (gdelta (GSPEC (\GEN%PVAR%1321. (?x. (((SETSPEC GEN%PVAR%1321) ((x IN s) /\ ((f x) IN t))) x)))))))))


Seq.mem_undup

|- (!s. (!x. (((MEM x) (undup s)) = ((MEM x) s))))


REFLECT_UNIV

|- (((IMAGE vector_neg) UNIV) = UNIV)


SUBSET_INTERVAL_1

|- (!a. (!b. (!c. (!d. ((((closed_interval ((CONS (a , b)) NIL)) SUBSET (closed_interval ((CONS (c , d)) NIL))) = (((real_lt (drop b)) (drop a)) \/ (((real_le (drop c)) (drop a)) /\ (((real_le (drop a)) (drop b)) /\ ((real_le (drop b)) (drop d)))))) /\ ((((closed_interval ((CONS (a , b)) NIL)) SUBSET (open_interval (c , d))) = (((real_lt (drop b)) (drop a)) \/ (((real_lt (drop c)) (drop a)) /\ (((real_le (drop a)) (drop b)) /\ ((real_lt (drop b)) (drop d)))))) /\ ((((open_interval (a , b)) SUBSET (closed_interval ((CONS (c , d)) NIL))) = (((real_le (drop b)) (drop a)) \/ (((real_le (drop c)) (drop a)) /\ (((real_lt (drop a)) (drop b)) /\ ((real_le (drop b)) (drop d)))))) /\ (((open_interval (a , b)) SUBSET (open_interval (c , d))) = (((real_le (drop b)) (drop a)) \/ (((real_le (drop c)) (drop a)) /\ (((real_lt (drop a)) (drop b)) /\ ((real_le (drop b)) (drop d)))))))))))))


CONTINUOUS_AT_WITHIN

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


STC_CASES_L

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


RAT_LEMMA5

|- ((((real_lt (real_of_num (NUMERAL _0))) y1) /\ ((real_lt (real_of_num (NUMERAL _0))) y2)) ==> ((((real_div x1) y1) = ((real_div x2) y2)) = (((real_mul x1) y2) = ((real_mul x2) y1))))


REAL_MIN_LT

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


NORM_CAUCHY_SCHWARZ_ABS

|- (!x. (!y. ((real_le (real_abs ((dot x) y))) ((real_mul (vector_norm x)) (vector_norm y)))))


BOREL_UNIONS

|- (!u. (((COUNTABLE u) /\ (!s. ((s IN u) ==> (borel s)))) ==> (borel (UNIONS u))))


HAS_COMPLEX_DERIVATIVE_DIFFERENTIABLE

|- (!f. (!x. ((((has_complex_derivative f) ((complex_derivative f) x)) (at x)) = ((complex_differentiable f) (at x)))))


HOMEOMORPHISM_IMP_CLOSED_MAP

|- (!f. (!g. (!s. (!t. (!u. ((((homeomorphism (s , t)) (f , g)) /\ ((closed_in ((subtopology euclidean) s)) u)) ==> ((closed_in ((subtopology euclidean) t)) ((IMAGE f) u))))))))


LIFT_SUM

|- (!k. (!x. ((lift ((sum k) x)) = ((vsum k) (lift o x)))))


CONTINUOUS_AT_IMP_CONTINUOUS_ON

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


IMAGE_o

|- (!f. (!g. (!s. (((IMAGE (f o g)) s) = ((IMAGE f) ((IMAGE g) s))))))


SIN_PI

|- ((sin pi) = (real_of_num (NUMERAL _0)))


CONTRACTIBLE_EMPTY

|- (contractible EMPTY)


VECTOR_MUL_LID

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


RECTIFIABLE_PATH_JOIN

|- (!g1. (!g2. (((pathfinish g1) = (pathstart g2)) ==> ((rectifiable_path ((++ g1) g2)) = ((rectifiable_path g1) /\ (rectifiable_path g2))))))


HAS_MEASURE_MEASURABLE_MEASURE

|- (!s. (!m. (((has_measure s) m) = ((measurable s) /\ ((measure s) = m)))))


INTEGRAL_NORM_BOUND_INTEGRAL

|- (!f. (!g. (!s. ((((integrable_on f) s) /\ (((integrable_on g) s) /\ (!x. ((x IN s) ==> ((real_le (vector_norm (f x))) (drop (g x))))))) ==> ((real_le (vector_norm ((integral s) f))) (drop ((integral s) g)))))))


Tame_list.IMP_IMP_THM

|- (!a. (!b. ((a ==> (b ==> c)) = ((a /\ b) ==> c))))


DIAMETER_CLOSURE

|- (!s. ((diameter (closure s)) = (diameter s)))


REAL_LT_01

|- ((real_lt (real_of_num (NUMERAL _0))) (real_of_num (NUMERAL (BIT1 _0))))


bool_elim

|- (!P. (((P T) /\ (P F)) ==> (!x. (P x))))


COLLINEAR_SUBSET

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


PERMUTES_INVERSE_EQ

|- (!p. (!s. (((permutes p) s) ==> (!x. (!y. ((((inverse p) y) = x) = ((p x) = y)))))))


Sphere.packing

|- (!S. ((packing S) = (!u. (!v. (((S u) /\ ((S v) /\ (~ (u = v)))) ==> ((real_le (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (distance (u , v))))))))


FINE_UNIONS

|- (!d. (!ps. ((!p. ((p IN ps) ==> ((fine d) p))) ==> ((fine d) (UNIONS ps)))))


Trigonometry2.WHEN_K_POS_ARCV_STABLE

|- (((real_lt (real_of_num (NUMERAL _0))) k) ==> ((((arcV (vec (NUMERAL _0))) u) v) = (((arcV (vec (NUMERAL _0))) u) ((% k) v))))


HAS_REAL_MEASURE_DIFF_SUBSET

|- (!s1. (!s2. (!m1. (!m2. ((((has_real_measure s1) m1) /\ (((has_real_measure s2) m2) /\ (s2 SUBSET s1))) ==> ((has_real_measure (s1 DIFF s2)) ((real_sub m1) m2)))))))


HAS_REAL_DERIVATIVE_SUM

|- (!f. (!net. (!s. (((FINITE s) /\ (!a. ((a IN s) ==> (((has_real_derivative (f a)) (f' a)) net)))) ==> (((has_real_derivative (\x. ((sum s) (\a. ((f a) x))))) ((sum s) f')) net)))))


NEGLIGIBLE_CONVEX_FRONTIER

|- (!s. ((convex s) ==> (negligible (frontier s))))


VECTOR_NEG_EQ_0

|- (((vector_neg x) = (vec (NUMERAL _0))) = (x = (vec (NUMERAL _0))))


SUM_INTEGRAL_LBOUND_INCREASING

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


CX_ADD

|- (!x. (!y. ((Cx ((real_add x) y)) = ((vector_add (Cx x)) (Cx y)))))


VECTOR_EQ_NEG2

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


STD_SIMPLEX

|- (((hull convex) ((vec (NUMERAL _0)) INSERT (GSPEC (\GEN%PVAR%1805. (?i. (((SETSPEC GEN%PVAR%1805) (((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV)))) (basis i))))))) = (GSPEC (\GEN%PVAR%1806. (?x. (((SETSPEC GEN%PVAR%1806) ((!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV))) ==> ((real_le (real_of_num (NUMERAL _0))) (x $ i)))) /\ ((real_le ((sum ((NUMERAL (BIT1 _0)) .. (dimindex UNIV))) (\i. (x $ i)))) (real_of_num (NUMERAL (BIT1 _0)))))) x)))))


AFFINE_DEPENDENT_TRANSLATION_EQ

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


REAL_LT_SUB_LADD

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


fsigma

|- (!s. ((fsigma s) = (?g. ((COUNTABLE g) /\ ((!c. ((c IN g) ==> (closed c))) /\ ((UNIONS g) = s))))))


DROP_ADD

|- (!x. (!y. ((drop ((vector_add x) y)) = ((real_add (drop x)) (drop y)))))


absolutely_setcontinuous_on

|- (!s. (!f. (((absolutely_setcontinuous_on f) s) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?r. (((real_lt (real_of_num (NUMERAL _0))) r) /\ (!d. (!t. ((((division_of d) t) /\ ((t SUBSET s) /\ ((real_lt ((sum d) content)) r))) ==> ((real_lt ((sum d) (\k. (vector_norm (f k))))) e)))))))))))


IM

|- ((Im (complex (x , y))) = y)


NORM_0

|- ((vector_norm (vec (NUMERAL _0))) = (real_of_num (NUMERAL _0)))


MULT_CLAUSES

|- ((!n. (((NUMERAL _0) * n) = (NUMERAL _0))) /\ ((!m. ((m * (NUMERAL _0)) = (NUMERAL _0))) /\ ((!n. (((NUMERAL (BIT1 _0)) * n) = n)) /\ ((!m. ((m * (NUMERAL (BIT1 _0))) = m)) /\ ((!m. (!n. (((SUC m) * n) = ((m * n) + n)))) /\ (!m. (!n. ((m * (SUC n)) = (m + (m * n))))))))))


IN_INTERVAL_1

|- (!a. (!b. (!x. (((x IN (closed_interval ((CONS (a , b)) NIL))) = (((real_le (drop a)) (drop x)) /\ ((real_le (drop x)) (drop b)))) /\ ((x IN (open_interval (a , b))) = (((real_lt (drop a)) (drop x)) /\ ((real_lt (drop x)) (drop b))))))))


FUNSPACE

|- (!s. (!m. (((mspace ((funspace s) m)) = (GSPEC (\GEN%PVAR%526. (?f. (((SETSPEC GEN%PVAR%526) ((!x. ((x IN s) ==> ((f x) IN (mspace m)))) /\ ((f IN (EXTENSIONAL s)) /\ ((mbounded m) ((IMAGE f) s))))) f))))) /\ (!f. (!g. (((mdist ((funspace s) m)) (f , g)) = (((COND (s = EMPTY)) (real_of_num (NUMERAL _0))) (sup (GSPEC (\GEN%PVAR%527. (?x. (((SETSPEC GEN%PVAR%527) (x IN s)) ((mdist m) ((f x) , (g x)))))))))))))))


NORM_MUL

|- (!a. (!x. ((vector_norm ((% a) x)) = ((real_mul (real_abs a)) (vector_norm x)))))


VOLUME_FRUSTT_STRONG

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


MATRIX_ID

|- ((matrix (\x. x)) = (mat (NUMERAL (BIT1 _0))))


LOCALLY_PATH_CONNECTED_TRANSLATION_EQ

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


LIM_AT_ID

|- (!a. (((--> (\x. x)) a) (at a)))


CONVEX_CLOSURE_INTERIOR

|- (!s. (((convex s) /\ (~ ((interior s) = EMPTY))) ==> ((closure (interior s)) = (closure s))))


MATRIX_VECTOR_MUL_INV_EQ_0

|- (!A. ((((matrix_vector_mul (matrix_inv A)) x) = (vec (NUMERAL _0))) = (((matrix_vector_mul (transp A)) x) = (vec (NUMERAL _0)))))


CLOSURE_MINIMAL

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


Hypermap_and_fan.RES_EMPTY

|- (!f. (((res f) EMPTY) = I))


SUM_ADD_NUMSEG

|- (!f. (!g. (!m. (!n. (((sum (m .. n)) (\i. ((real_add (f i)) (g i)))) = ((real_add ((sum (m .. n)) f)) ((sum (m .. n)) g)))))))


Seq.all_count

|- (!a. (!s. (((all a) s) = (((count a) s) = (sizel s)))))


pathstart

|- (!g. ((pathstart g) = (g (vec (NUMERAL _0)))))


Lp_ineqs_proofs.face_eq_power

|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!k. (!d. ((d IN ((darts_k k) (hypermap_of_fan (V , E)))) ==> (((face (hypermap_of_fan (V , E))) d) = (GSPEC (\GEN%PVAR%8730. (?i. (((SETSPEC GEN%PVAR%8730) (i < k)) (((POWER (f_fan_pair_ext (V , E))) i) d))))))))))))


Tame_defs.split_at_rec

|- (((((split_at_rec c) bs) NIL) = (bs , NIL)) /\ ((((split_at_rec c) bs) ((CONS a) xs)) = (((COND (a = c)) (bs , xs)) (((split_at_rec c) ((APPEND bs) ((CONS a) NIL))) xs))))


INTERIOR_SEGMENT

|- ((!a. (!b. ((interior (closed_segment ((CONS (a , b)) NIL))) = (((COND ((NUMERAL (BIT0 (BIT1 _0))) <= (dimindex UNIV))) EMPTY) (open_segment (a , b)))))) /\ (!a. (!b. ((interior (open_segment (a , b))) = (((COND ((NUMERAL (BIT0 (BIT1 _0))) <= (dimindex UNIV))) EMPTY) (open_segment (a , b)))))))


Add_triangle.ext_contracted

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


HAS_REAL_INTEGRAL_INTEGRAL

|- (!f. (!s. (((real_integrable_on f) s) = (((has_real_integral f) ((real_integral s) f)) s))))


INDEFINITE_INTEGRAL_UNIFORMLY_CONTINUOUS

|- (!f. (!a. (!b. (((integrable_on f) (closed_interval ((CONS (a , b)) NIL))) ==> ((uniformly_continuous_on (\y. ((integral (closed_interval ((CONS ((fstcart y) , (sndcart y))) NIL))) f))) (closed_interval ((CONS (((pastecart a) a) , ((pastecart b) b))) NIL)))))))


ENDS_IN_SEGMENT

|- (!a. (!b. ((a IN (closed_segment ((CONS (a , b)) NIL))) /\ (b IN (closed_segment ((CONS (a , b)) NIL))))))


Sphere.rho

|- (!y. ((rho y) = ((real_add (real_of_num (NUMERAL (BIT1 _0)))) ((real_sub const1) ((real_mul const1) (ly y))))))


integrable_on

|- (!f. (!i. (((integrable_on f) i) = (?y. (((has_integral f) y) i)))))


REAL_SERIES_SUB

|- (!x. (!x0. (!y. (!y0. (!s. (((((real_sums x) x0) s) /\ (((real_sums y) y0) s)) ==> (((real_sums (\n. ((real_sub (x n)) (y n)))) ((real_sub x0) y0)) s)))))))


NPRODUCT_EQ

|- (!f. (!g. (!s. ((!x. ((x IN s) ==> ((f x) = (g x)))) ==> (((nproduct s) f) = ((nproduct s) g))))))


FSTCART_PASTECART

|- (!x. (!y. ((fstcart ((pastecart x) y)) = x)))


Ssrbool.iffLR

|- (!P. (!Q. ((P = Q) ==> (P ==> Q))))


Seq2.list_sum_cons

|- (!h. (!t. (!f. (((list_sum ((CONS h) t)) f) = ((real_add (f h)) ((list_sum t) f))))))


PATH_IMAGE_NONEMPTY

|- (!g. (~ ((path_image g) = EMPTY)))


Fan_defs.fully_surrounded

|- (!V. (!E. ((fully_surrounded (V , E)) = (!x. ((x IN (dart_of_fan (V , E))) ==> ((real_lt ((azim_dart (V , E)) x)) pi))))))


HAS_BOUNDED_VARIATION_COMPOSE_DECREASING

|- (!f. (!g. (!a. (!b. (((!x. (!y. (((x IN (closed_interval ((CONS (a , b)) NIL))) /\ ((y IN (closed_interval ((CONS (a , b)) NIL))) /\ ((real_le (drop x)) (drop y)))) ==> ((real_le (drop (f y))) (drop (f x)))))) /\ ((has_bounded_variation_on g) (closed_interval ((CONS ((f b) , (f a))) NIL)))) ==> ((has_bounded_variation_on (g o f)) (closed_interval ((CONS (a , b)) NIL))))))))


Lp_ineqs_def.y4_fan

|- (!V. (!E. (!x. (((y4_fan (V , E)) x) = (distance ((f_fan_pair_ext (V , E)) x))))))


EXPOSED_FACE_OF_TRANSLATION_EQ

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


HAS_REAL_INTEGRAL_EQ

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


SUBSET_UNIV

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


TOPSPACE_EUCLIDEANREAL

|- ((topspace euclideanreal) = UNIV)


COVERING_SPACE_FINITE_SHEETS_EQ_PROPER_MAP

|- (!p. (!c. (!s. (((covering_space (c , p)) s) ==> ((!b. ((b IN s) ==> (FINITE (GSPEC (\GEN%PVAR%2082. (?x. (((SETSPEC GEN%PVAR%2082) ((x IN c) /\ ((p x) = b))) x))))))) = (!k. (((k SUBSET s) /\ (compact k)) ==> (compact (GSPEC (\GEN%PVAR%2083. (?x. (((SETSPEC GEN%PVAR%2083) ((x IN c) /\ ((p x) IN k))) x))))))))))))


Tame_defs.d_tame

|- (!n. ((d_tame n) = (((COND (n = (NUMERAL (BIT1 (BIT1 _0))))) (real_of_num (NUMERAL _0))) (((COND (n = (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) ((DECIMAL (NUMERAL (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))) (((COND (n = (NUMERAL (BIT1 (BIT0 (BIT1 _0)))))) ((DECIMAL (NUMERAL (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 _0))))))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 _0))))))))))))))))) (((COND (n = (NUMERAL (BIT0 (BIT1 (BIT1 _0)))))) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 _0)))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))) tgt))))))


MEASURABLE_MIDPOINT_CONVEX_IMP_CONVEX_GEN

|- (!f. (!s. ((((measurable_on (lift o f)) s) /\ ((convex s) /\ (!x. (!y. (((x IN s) /\ (y IN s)) ==> ((real_le (f (midpoint (x , y)))) ((real_div ((real_add (f x)) (f y))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))))))) ==> (!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)))) /\ (((closed_segment ((CONS (x , y)) NIL)) SUBSET (frontier s)) ==> (x = y))))))) ==> ((real_le (f ((vector_add ((% u) x)) ((% v) y)))) ((real_add ((real_mul u) (f x))) ((real_mul v) (f y))))))))))))


Ssrbool.predI

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


COMPLEX_NORM_INV

|- (!z. ((vector_norm (complex_inv z)) = (real_inv (vector_norm z))))


SIMPLY_CONNECTED_TRANSLATION

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


REAL_SECOND_MEAN_VALUE_THEOREM_GEN_FULL

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


IMAGE_UNION

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


VECTOR_EQ_ADDR

|- ((((vector_add x) y) = x) = (y = (vec (NUMERAL _0))))


COMPACT_CLOSED_POINTIMAGES_IMP_CONTINUOUS_ON

|- (!f. (!s. (((!c. (((c SUBSET s) /\ (compact c)) ==> (compact ((IMAGE f) c)))) /\ (!y. ((closed_in ((subtopology euclidean) s)) (GSPEC (\GEN%PVAR%1929. (?x. (((SETSPEC GEN%PVAR%1929) ((x IN s) /\ ((f x) = y))) x))))))) ==> ((continuous_on f) s))))


REAL_CNJ

|- (!z. ((real z) = ((cnj z) = z)))


REAL_LT_TRANS

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


real_differentiable_on

|- (!f. (!s. (((real_differentiable_on f) s) = (!x. ((x IN s) ==> (?f'. (((has_real_derivative f) f') ((within (atreal x)) s))))))))


LINEAR_NEG

|- (!f. (!x. ((linear f) ==> ((f (vector_neg x)) = (vector_neg (f x))))))


HOMOTOPY_EQUIVALENT_CONTRACTIBILITY

|- (!s. (!t. (((homotopy_equivalent s) t) ==> ((contractible s) = (contractible t)))))


CONTINUOUS_AE_IMP_MEASURABLE_ON_LEBESGUE_MEASURABLE_SUBSET

|- (!f. (!s. (!m. ((((continuous_on f) (s DIFF m)) /\ ((lebesgue_measurable s) /\ (negligible m))) ==> ((measurable_on f) s)))))


FLOOR_UNIQUE

|- (!x. (!a. (((integer a) /\ (((real_le a) x) /\ ((real_lt x) ((real_add a) (real_of_num (NUMERAL (BIT1 _0))))))) = ((floor x) = a))))


INTEGRAL_CONST

|- (!a. (!b. (!c. (((integral (closed_interval ((CONS (a , b)) NIL))) (\x. c)) = ((% (content (closed_interval ((CONS (a , b)) NIL)))) c)))))


SUBSPACE_IMP_CONVEX

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


AFF_DIM_SUBSET

|- (!s. (!t. ((s SUBSET t) ==> ((int_le (aff_dim s)) (aff_dim t)))))


INTEGRABLE_DECREASING_PRODUCT_UNIV

|- (!f. (!g. (!B. ((((integrable_on f) UNIV) /\ ((!x. (!y. (((real_le (drop x)) (drop y)) ==> ((real_le (g y)) (g x))))) /\ (!x. ((real_le (real_abs (g x))) B)))) ==> ((integrable_on (\x. ((% (g x)) (f x)))) UNIV)))))


ANALYTIC_AT_TWO

|- (!f. (!g. (!z. ((((analytic_on f) (z INSERT EMPTY)) /\ ((analytic_on g) (z INSERT EMPTY))) = (?s. ((open s) /\ ((z IN s) /\ (((holomorphic_on f) s) /\ ((holomorphic_on g) s)))))))))


EMPTY_GSPEC

|- ((GSPEC (\GEN%PVAR%14. (?x. (((SETSPEC GEN%PVAR%14) F) x)))) = EMPTY)


DIMINDEX_3

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


num_cases

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


List_hypermap.darts_k

|- (!H. (!k. (((darts_k k) H) = (GSPEC (\GEN%PVAR%8595. (?d. (((SETSPEC GEN%PVAR%8595) ((d IN (dart H)) /\ ((CARD ((face H) d)) = k))) d)))))))


Seq.cat_take_drop

|- (!n0. (!s. (((cat ((take n0) s)) ((dropl n0) s)) = s)))


MSPACE

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


Ssrnat.leq_exp2l

|- (!m. (!n1. (!n2. (((NUMERAL (BIT1 _0)) < m) ==> (((m EXP n1) <= (m EXP n2)) = (n1 <= n2))))))


Fan.th3

|- (!x. (!v. (!u. ((~ (collinear (x INSERT (v INSERT (u INSERT EMPTY))))) ==> ((~ (x = v)) /\ ((~ (x = 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))))))))))))))


COUNTABLE_SUBSET

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


pair_RECURSION

|- (!PAIR'. (?fn. (!a0. (!a1. ((fn (a0 , a1)) = ((PAIR' a0) a1))))))


RE_NEG

|- (!x. ((Re (vector_neg x)) = (real_neg (Re x))))


RIGHT_POLAR_DECOMPOSITION_INVERTIBLE

|- (!A. ((invertible A) = (?U. (?P. ((orthogonal_matrix U) /\ ((positive_definite P) /\ (((matrix_mul U) P) = A)))))))


CX_MUL

|- (!x. (!y. ((Cx ((real_mul x) y)) = ((complex_mul (Cx x)) (Cx y)))))


POLYHEDRON_UNIV

|- (polyhedron UNIV)


Topology.IMAGE_SEG

|- (!n. (!f. (((IMAGE f) (GSPEC (\GEN%PVAR%5485. (?i. (((SETSPEC GEN%PVAR%5485) (i < n)) i))))) = (GSPEC (\GEN%PVAR%5486. (?i. (((SETSPEC GEN%PVAR%5486) (i < n)) (f i))))))))


HAS_REAL_INTEGRAL

|- ((((has_real_integral f) y) (closed_real_interval ((CONS (a , b)) NIL))) = (((has_integral (lift o (f o drop))) (lift y)) (closed_interval ((CONS ((lift a) , (lift b))) NIL))))


Hypermap_and_fan.NODE_SET_AS_IMAGE

|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (?f. ((!v. (!w. (((f v) = (f w)) ==> (v = w)))) /\ ((!v. (!x. ((x IN (f v)) ==> ((FST x) = v)))) /\ ((!v. ((FST (CHOICE (f v))) = v)) /\ ((node_set (hypermap_of_fan (V , E))) = ((IMAGE f) V)))))))))


REAL_OPEN_IN

|- (!s. ((real_open s) = ((open_in euclideanreal) s)))


REAL_NEGNEG

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


Localization.convex_local_fan

|- (!FF. (!V. (!E. ((convex_local_fan (V , (E , FF))) = ((local_fan (V , (E , FF))) /\ (!x. ((x IN FF) ==> (((real_le ((azim_in_fan x) E)) pi) /\ (V SUBSET ((wedge_in_fan_ge x) E))))))))))


CARD_ADD_FINITE

|- (!s. (!t. (((FINITE s) /\ (FINITE t)) ==> (FINITE ((+_c s) t)))))


VECTOR_3

|- ((((vector ((CONS x) ((CONS y) ((CONS z) NIL)))) $ (NUMERAL (BIT1 _0))) = x) /\ ((((vector ((CONS x) ((CONS y) ((CONS z) NIL)))) $ (NUMERAL (BIT0 (BIT1 _0)))) = y) /\ (((vector ((CONS x) ((CONS y) ((CONS z) NIL)))) $ (NUMERAL (BIT1 (BIT1 _0)))) = z)))


SYMMETRIC_MATRIX_EQ_DIAGONALIZABLE_ALT

|- (!A. (((transp A) = A) = (?P. (?D. ((orthogonal_matrix P) /\ ((diagonal_matrix D) /\ (A = ((matrix_mul (transp P)) ((matrix_mul D) P)))))))))


LEBESGUE_MEASURABLE_OPEN_IN

|- (!s. (!t. ((((open_in ((subtopology euclidean) s)) t) /\ (lebesgue_measurable s)) ==> (lebesgue_measurable t))))


INTER_ASSOC

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


CONVEX_ON_EQ

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


NUMSEG_EMPTY

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


MATRIX_VECTOR_MUL_ASSOC

|- (!A. (!B. (!x. (((matrix_vector_mul A) ((matrix_vector_mul B) x)) = ((matrix_vector_mul ((matrix_mul A) B)) x)))))


Tame_defs.tame_5a

|- (!H. ((tame_5a H) = (is_no_double_joins H)))


UNIV_NOT_EMPTY

|- (~ (UNIV = EMPTY))


Nonlin_def.ldih6_x_div_sqrtdelta_posbranch

|- (ldih6_x_div_sqrtdelta_posbranch = (rotate6 ldih_x_div_sqrtdelta_posbranch))


AFFINE_HYPERPLANE

|- (!a. (!b. (affine (GSPEC (\GEN%PVAR%1401. (?x. (((SETSPEC GEN%PVAR%1401) (((dot a) x) = b)) x)))))))


CLOSED_HALFSPACE_IM_LE

|- (!b. (closed (GSPEC (\GEN%PVAR%4757. (?z. (((SETSPEC GEN%PVAR%4757) ((real_le (Im z)) b)) z))))))


Hypermap_and_fan.FACE_SUBSET_DART1_OF_FAN

|- (!V. (!E. (!x. (((FAN ((vec (NUMERAL _0)) , (V , E))) /\ (x IN (dart1_of_fan (V , E)))) ==> (((face (hypermap_of_fan (V , E))) x) SUBSET (dart1_of_fan (V , E)))))))


POSITIVE_SEMIDEFINITE_MUL_EQ

|- (!A. (!B. (((positive_semidefinite A) /\ (positive_semidefinite B)) ==> ((positive_semidefinite ((matrix_mul A) B)) = (((matrix_mul A) B) = ((matrix_mul B) A))))))


real_compact

|- (!s. ((real_compact s) = (compact ((IMAGE lift) s))))


matrix_cmul

|- (!c. (!A. (((%% c) A) = (lambdai. (lambdaj. ((real_mul c) ((A $ i) $ j)))))))


Wrgcvdr_cizmrrh.X_IN_ITS_ORBIT

|- (x IN ((orbit_map f) x))


DIAGONAL_MATRIX_MUL

|- (!A. (!B. (((diagonal_matrix A) /\ (diagonal_matrix B)) ==> (diagonal_matrix ((matrix_mul A) B)))))


INSEG_WOSET

|- (!l. (!m. ((((inseg m) l) /\ (woset l)) ==> (woset m))))


INTEGRABLE_ON_EMPTY

|- (!f. ((integrable_on f) EMPTY))


Functional_equation.mk_126

|- (!f. ((mk_126 f) = (((((((compose6 f) proj_x1) proj_x2) two6) two6) two6) proj_x6)))


SUM_RESTRICT_SET

|- (!P. (!s. (!f. (((sum (GSPEC (\GEN%PVAR%199. (?x. (((SETSPEC GEN%PVAR%199) ((x IN s) /\ (P x))) x))))) f) = ((sum s) (\x. (((COND (P x)) (f x)) (real_of_num (NUMERAL _0)))))))))


MONOIDAL_MUL

|- (monoidal *)


IMAGE_LIFT_REAL_INTERVAL

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


RIGHT_IMP_EXISTS_THM

|- (!P. (!Q. ((P ==> (?x. (Q x))) = (?x. (P ==> (Q x))))))


COMPLEX_NORM_GE_RE_IM

|- (!z. (((real_le (real_abs (Re z))) (vector_norm z)) /\ ((real_le (real_abs (Im z))) (vector_norm z))))


MEASURABLE_NONNEGLIGIBLE_IMP_LARGE

|- (!s. (((measurable s) /\ ((real_lt (real_of_num (NUMERAL _0))) (measure s))) ==> (s =_c UNIV)))


SUP_EQ

|- (!s. (!t. ((!b. ((!x. ((x IN s) ==> ((real_le x) b))) = (!x. ((x IN t) ==> ((real_le x) b))))) ==> ((sup s) = (sup t)))))


UNION_EMPTY

|- ((!s. ((EMPTY UNION s) = s)) /\ (!s. ((s UNION EMPTY) = s)))


absolutely_continuous_on

|- (!f. (!s. (((absolutely_continuous_on f) s) = ((absolutely_setcontinuous_on (\k. ((vector_sub (f (interval_upperbound k))) (f (interval_lowerbound k))))) s))))


CFUNSPACE

|- ((!top. (!m. ((mspace ((cfunspace top) m)) = (GSPEC (\GEN%PVAR%534. (?f. (((SETSPEC GEN%PVAR%534) ((!x. ((x IN (topspace top)) ==> ((f x) IN (mspace m)))) /\ ((f IN (EXTENSIONAL (topspace top))) /\ (((mbounded m) ((IMAGE f) (topspace top))) /\ (((topcontinuous top) (mtopology m)) f))))) f))))))) /\ (!f. (!g. (((mdist ((cfunspace top) m)) (f , g)) = (((COND ((topspace top) = EMPTY)) (real_of_num (NUMERAL _0))) (sup (GSPEC (\GEN%PVAR%535. (?x. (((SETSPEC GEN%PVAR%535) (x IN (topspace top))) ((mdist m) ((f x) , (g x)))))))))))))


ABSOLUTELY_INTEGRABLE_REFLECT_GEN

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


CONTINUOUS_AT_ID

|- (!a. ((continuous (\x. x)) (at a)))


Real_ext.pow

|- ((((real_pow x) (NUMERAL _0)) = (real_of_num (NUMERAL (BIT1 _0)))) /\ (!n. (((real_pow x) (SUC n)) = ((real_mul x) ((real_pow x) n)))))


Functional_equation.proj_y2

|- (!x1. (!x3. (!x4. (!x5. (!x6. (!x2. (((((((proj_y2 x1) x2) x3) x4) x5) x6) = (sqrt x2))))))))


SELECT_UNIQUE

|- (!P. (!x. ((!y. ((P y) = (y = x))) ==> ((@ P) = x))))


Sphere.voronoi_closed

|- (!S. (!v. (((voronoi_closed S) v) = (GSPEC (\GEN%PVAR%5242. (?x. (((SETSPEC GEN%PVAR%5242) (!w. ((S w) ==> ((real_le (distance (x , v))) (distance (x , w)))))) x)))))))


ANALYTIC_ON_INV

|- (!f. (!s. ((((analytic_on f) s) /\ (!z. ((z IN s) ==> (~ ((f z) = (Cx (real_of_num (NUMERAL _0)))))))) ==> ((analytic_on (\z. (complex_inv (f z)))) s))))


FRONTIER_INTER_SUBSET

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


LIM_1_OVER_LOG

|- (((--> (\n. ((complex_div (Cx (real_of_num (NUMERAL (BIT1 _0))))) (clog (Cx (real_of_num n)))))) (Cx (real_of_num (NUMERAL _0)))) sequentially)


REAL_OF_INT_OF_REAL

|- (!x. ((integer x) ==> ((real_of_int (int_of_real x)) = x)))


SETDIST_EQ_0_COMPACT_CLOSED

|- (!s. (!t. (((compact s) /\ (closed t)) ==> (((setdist (s , t)) = (real_of_num (NUMERAL _0))) = ((s = EMPTY) \/ ((t = EMPTY) \/ (~ ((s INTER t) = EMPTY))))))))


SETDIST_FRONTIER

|- ((!s. (!t. (((DISJOINT s) t) ==> ((setdist ((frontier s) , t)) = (setdist (s , t)))))) /\ (!s. (!t. (((DISJOINT s) t) ==> ((setdist (s , (frontier t))) = (setdist (s , t)))))))


OPEN_IMP_ANR

|- (!s. ((open s) ==> (ANR s)))


SPAN_VSUM

|- (!s. (!f. (!t. (((FINITE t) /\ (!x. ((x IN t) ==> ((f x) IN (span s))))) ==> (((vsum t) f) IN (span s))))))


o_THM

|- (!f. (!g. (!x. (((f o g) x) = (f (g x))))))


ANR_PCROSS_EQ

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


REAL_POW_MONO

|- (!m. (!n. (!x. ((((real_le (real_of_num (NUMERAL (BIT1 _0)))) x) /\ (m <= n)) ==> ((real_le ((real_pow x) m)) ((real_pow x) n))))))


HAS_BOUNDED_REAL_VARIATION_ON_LMUL

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


CLOSED_IN_UNIONS

|- (!top. (!s. (((FINITE s) /\ (!t. ((t IN s) ==> ((closed_in top) t)))) ==> ((closed_in top) (UNIONS s)))))


Lp_main_estimate.tau_fun_local

|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!f. ((f IN (face_set (hypermap_of_fan (V , E)))) ==> ((((tau_fun ((v_prime V) f)) ((e_prime E) f)) f) = (((tau_fun V) E) f)))))))


COMPLEX_DIFF_CHAIN_WITHIN

|- (!f. (!g. (!f'. (!g'. (!x. (!s. (((((has_complex_derivative f) f') ((within (at x)) s)) /\ (((has_complex_derivative g) g') ((within (at (f x))) ((IMAGE f) s)))) ==> (((has_complex_derivative (g o f)) ((complex_mul g') f')) ((within (at x)) s)))))))))


NEGLIGIBLE_AFFINE_HULL

|- (!s. (((FINITE s) /\ ((CARD s) <= (dimindex UNIV))) ==> (negligible ((hull affine) s))))


Functional_equation.proj_x3

|- (((((((proj_x3 x1) x2) x3) x4) x5) x6) = x3)


has_real_integral

|- (!f. (!y. (!s. ((((has_real_integral f) y) s) = (((has_integral (lift o (f o drop))) (lift y)) ((IMAGE lift) s))))))


ONORM

|- (!f. ((linear f) ==> ((!x. ((real_le (vector_norm (f x))) ((real_mul (onorm f)) (vector_norm x)))) /\ (!b. ((!x. ((real_le (vector_norm (f x))) ((real_mul b) (vector_norm x)))) ==> ((real_le (onorm f)) b))))))


Tame_general.CONTRAVENING_DART_DIST

|- (!V. (!x. (((contravening V) /\ (x IN (dart_of_fan (V , (ESTD V))))) ==> (((real_le ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (distance x)) /\ ((real_le (distance x)) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))))))))


REAL_NORM

|- (!z. ((real z) ==> ((vector_norm z) = (real_abs (Re z)))))


INTERIOR_TRANSLATION

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


INSIDE_OUTSIDE

|- (!s. ((inside s) = (UNIV DIFF (s UNION (outside s)))))


rowvector

|- (!v. ((rowvector v) = (lambdai. (lambdaj. (v $ j)))))


FRONTIER_COMPLEMENT

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


CONTINUOUS_ON_CONST

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


Topology.set_of_orbits_points_fan

|- (!x. (!V. (!E. (!v. (!u. ((((((set_of_orbits_points_fan x) V) E) v) u) = (GSPEC (\GEN%PVAR%5483. (?i. (((SETSPEC GEN%PVAR%5483) ((NUMERAL _0) <= i)) (((((((power_map_points sigma_fan) x) V) E) v) u) i)))))))))))


Tame_defs.split_at

|- (!c. (!xs. (((split_at c) xs) = (((split_at_rec c) NIL) xs))))


BOREL_IMP_LEBESGUE_MEASURABLE

|- (!s. ((borel s) ==> (lebesgue_measurable s)))


Seq.rev_cons

|- (!x. (!s. ((rev ((CONS x) s)) = ((rcons (rev s)) x))))


REAL_MUL

|- (!w. (!z. (((real w) /\ (real z)) ==> (real ((complex_mul w) z)))))


POSITIVE_DEFINITE_MUL_EQ

|- (!A. (!B. (((positive_definite A) /\ (positive_definite B)) ==> ((positive_definite ((matrix_mul A) B)) = (((matrix_mul A) B) = ((matrix_mul B) A))))))


REAL_NEG_MINUS1

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


Add_triangle.f_fan_pair_power

|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!d. (!i. ((d IN (dart1_of_fan (V , E))) ==> ((((POWER (f_fan_pair (V , E))) i) d) = (((POWER (f_fan_pair_ext (V , E))) i) d))))))))


AFFINE_LINEAR_IMAGE

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


Sphere.acs_sqrt_x2_d4

|- (!x1. (!x3. (!x4. (!x5. (!x6. (!x2. (((((((acs_sqrt_x2_d4 x1) x2) x3) x4) x5) x6) = (acs ((real_div (sqrt x2)) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0))))))))))))))


CNJ_EQ_CX

|- (!x. (!z. (((cnj z) = (Cx x)) = (z = (Cx x)))))


polytope

|- (!s. ((polytope s) = (?v. ((FINITE v) /\ (s = ((hull convex) v))))))


PATH_CONTINUOUS_IMAGE

|- (!f. (!g. (((path g) /\ ((continuous_on f) (path_image g))) ==> (path (f o g)))))


HAS_REAL_INTEGRAL_SUBSTITUTION_SIMPLE

|- (!f. (!g. (!g'. (!a. (!b. (!c. (!d. ((((real_continuous_on f) (closed_real_interval ((CONS (c , d)) NIL))) /\ ((!x. ((x IN (closed_real_interval ((CONS (a , b)) NIL))) ==> (((has_real_derivative g) (g' x)) ((within (atreal x)) (closed_real_interval ((CONS (a , b)) NIL)))))) /\ ((((IMAGE g) (closed_real_interval ((CONS (a , b)) NIL))) SUBSET (closed_real_interval ((CONS (c , d)) NIL))) /\ (((real_le a) b) /\ (((real_le c) d) /\ ((real_le (g a)) (g b))))))) ==> (((has_real_integral (\x. ((real_mul (f (g x))) (g' x)))) ((real_integral (closed_real_interval ((CONS ((g a) , (g b))) NIL))) f)) (closed_real_interval ((CONS (a , b)) NIL)))))))))))


REALLIM_CONG_AT

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


relative_frontier

|- (!s. ((relative_frontier s) = ((closure s) DIFF (relative_interior s))))


REAL_INV_EQ_0

|- (!x. (((real_inv x) = (real_of_num (NUMERAL _0))) = (x = (real_of_num (NUMERAL _0)))))


FUN_EQ_THM

|- (!f. (!g. ((f = g) = (!x. ((f x) = (g x))))))


Ssrnat.distnEr

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


Lp_ineqs_def.azim2_fan

|- (!V. (!E. (!x. (((azim2_fan (V , E)) x) = ((azim_dart (V , E)) ((f_fan_pair_ext (V , E)) x))))))


Lp_ineqs_proofs.y7_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))))) ((y7_fan (V , E)) d))))))))))


Nonlinear_lemma.dih_y_sym

|- (!y1. (!y2. (!y3. (!y4. (!y5. (!y6. (((((((dih_y y1) y2) y3) y4) y5) y6) = ((((((dih_y y1) y3) y2) y4) y6) y5))))))))


SUM_OFFSET

|- (!p. (!f. (!m. (!n. (((sum ((m + p) .. (n + p))) f) = ((sum (m .. n)) (\i. (f (i + p)))))))))


Real_ext.REAL_PROP_POS_LADD

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


Float.POW_2

|- (!x. (((real_pow x) (NUMERAL (BIT0 (BIT1 _0)))) = ((real_mul x) x)))


LINEAR_ADD

|- (!f. (!x. (!y. ((linear f) ==> ((f ((vector_add x) y)) = ((vector_add (f x)) (f y)))))))


REAL_NEG_LMUL

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


COLLINEAR_3_EXPLICIT

|- (!x. (!y. (!z. ((collinear (x INSERT (y INSERT (z INSERT EMPTY)))) = (?a. (?b. (?c. ((((vector_add ((% a) x)) ((vector_add ((% b) y)) ((% c) z))) = (vec (NUMERAL _0))) /\ ((((real_add a) ((real_add b) c)) = (real_of_num (NUMERAL _0))) /\ (~ ((a = (real_of_num (NUMERAL _0))) /\ ((b = (real_of_num (NUMERAL _0))) /\ (c = (real_of_num (NUMERAL _0)))))))))))))))


Real_ext.REAL_PROP_NN_ADD2

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


IM_CLOG_POS_LE

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


Hypermap.lemma_in_face_set

|- (!H. (!x. ((x IN (dart H)) = (((face H) x) IN (face_set H)))))


Lp_ineqs_proofs.y9_eq

|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!v. (!w. ((((v , w) IN (dart_of_fan (V , E))) /\ ((CARD ((face (hypermap_of_fan (V , E))) (v , w))) = (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) ==> ((LET (\w'. (LET_END ((LET (\u. (LET_END (((y9_fan (V , E)) (v , w)) = (distance (w , u)))))) (((((sigma_fan (vec (NUMERAL _0))) V) E) w') v))))) (((((sigma_fan (vec (NUMERAL _0))) V) E) v) w)))))))))))


CONTINUOUS_ON_COMPONENTWISE_LIFT

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


Seq.size_rot

|- (!n0. (!s. ((sizel ((rot n0) s)) = (sizel s))))


HAS_COMPLEX_DERIVATIVE_ID

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


HOMEOMORPHIC_SYM

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


conic

|- (!s. ((conic s) = (!x. (!c. (((x IN s) /\ ((real_le (real_of_num (NUMERAL _0))) c)) ==> (((% c) x) IN s))))))


FRAC_FLOOR

|- (!x. ((frac x) = ((real_sub x) (floor x))))


Hypermap_iso.fan_hypermaps_iso_explicit

|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((hyp_iso contracted_dart) ((hypermap1_of_fanx ((vec (NUMERAL _0)) , (V , E))) , (hypermap_of_fan (V , E)))))))


Sphere.INITIAL_SUBLIST

|- (!zl. (!xl. (((initial_sublist xl) zl) = (?yl. (zl = ((APPEND xl) yl))))))


INT_OF_NUM_OF_INT

|- (!x. (((int_le (int_of_num (NUMERAL _0))) x) ==> ((int_of_num (num_of_int x)) = x)))


COMPLEX_POW_1

|- (!x. (((complex_pow x) (NUMERAL (BIT1 _0))) = x))


HULL_SUBSET

|- (!P. (!s. (s SUBSET ((hull P) s))))


CONNECTED_SEGMENT

|- ((!a. (!b. (connected (closed_segment ((CONS (a , b)) NIL))))) /\ (!a. (!b. (connected (open_segment (a , b))))))


SUM_CLAUSES_NUMSEG

|- ((!m. (((sum (m .. (NUMERAL _0))) f) = (((COND (m = (NUMERAL _0))) (f (NUMERAL _0))) (real_of_num (NUMERAL _0))))) /\ (!m. (!n. (((sum (m .. (SUC n))) f) = (((COND (m <= (SUC n))) ((real_add ((sum (m .. n)) f)) (f (SUC n)))) ((sum (m .. n)) f))))))


COMPACT_CONVEX_COLLINEAR_SEGMENT

|- (!s. (((~ (s = EMPTY)) /\ ((compact s) /\ ((convex s) /\ (collinear s)))) ==> (?a. (?b. (s = (closed_segment ((CONS (a , b)) NIL)))))))


DOT_EQ_0

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


LEBESGUE_MEASURABLE_LINEAR_IMAGE_EQ

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


LT

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


IN_CLOSURE_OF

|- (!top. (!s. (!x. ((x IN ((closure_of top) s)) = ((x IN (topspace top)) /\ (!t. (((x IN t) /\ ((open_in top) t)) ==> (?y. ((y IN s) /\ (y IN t))))))))))


HAUSDIST_LINEAR_IMAGE

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


FACE_OF_SING

|- (!x. (!s. (((face_of (x INSERT EMPTY)) s) = ((extreme_point_of x) s))))


Functional_equation.div6

|- (!f. (!g. (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((((((((div6 f) g) x1) x2) x3) x4) x5) x6) = ((real_div ((((((f x1) x2) x3) x4) x5) x6)) ((((((g x1) x2) x3) x4) x5) x6)))))))))))


RE_VSUM

|- (!f. (!s. ((FINITE s) ==> ((Re ((vsum s) f)) = ((sum s) (\x. (Re (f x))))))))


real_integrable_on

|- (!f. (!i. (((real_integrable_on f) i) = (?y. (((has_real_integral f) y) i)))))


FUBINI_INTEGRAL_INTERVAL

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


Seq.head

|- (!x0. (!h. (!t. ((((headl x0) NIL) = x0) /\ (((headl x0) ((CONS h) t)) = h)))))


HAS_REAL_INTEGRAL_AFFINITY

|- (!f. (!i. (!a. (!b. (!m. (!c. (((((has_real_integral f) i) (closed_real_interval ((CONS (a , b)) NIL))) /\ (~ (m = (real_of_num (NUMERAL _0))))) ==> (((has_real_integral (\x. (f ((real_add ((real_mul m) x)) c)))) ((real_mul (real_inv (real_abs m))) i)) ((IMAGE (\x. ((real_mul (real_inv m)) ((real_sub x) c)))) (closed_real_interval ((CONS (a , b)) NIL)))))))))))


REAL_SGN_POW_2

|- (!x. ((real_sgn ((real_pow x) (NUMERAL (BIT0 (BIT1 _0))))) = (real_sgn (real_abs x))))


LINEPATH_LINEAR_IMAGE

|- (!f. ((linear f) ==> (!b. (!c. ((linepath ((f b) , (f c))) = (f o (linepath (b , c))))))))


VECTOR_SUB_RDISTRIB

|- (((% ((real_sub a) b)) x) = ((vector_sub ((% a) x)) ((% b) x)))


Packing3.SING_UNION_EQ_INSERT

|- (!s. (!x. (((x INSERT EMPTY) UNION s) = (x INSERT s))))


REAL_EXP_POS_LE

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


HAS_BOUNDED_VARIATION_AFFINITY_EQ

|- (!m. (!c. (!f. (!s. (((has_bounded_variation_on (\x. (f ((vector_add ((% m) x)) c)))) s) = ((m = (real_of_num (NUMERAL _0))) \/ ((has_bounded_variation_on f) ((IMAGE (\x. ((vector_add ((% m) x)) c))) s))))))))


Fan.fan5

|- (!V. (!E. (!x. ((fan5 (x , (V , E))) = (!e. (!f. (((e IN E) /\ ((f IN E) /\ (~ (e = f)))) ==> ((((aff_gt (x INSERT EMPTY)) e) INTER ((aff_gt (x INSERT EMPTY)) f)) = EMPTY))))))))


CARD_EXP_ADD

|- (!s. (!t. (!u. (((^_c s) ((+_c t) u)) =_c ((*_c ((^_c s) t)) ((^_c s) u))))))


RELATIVE_INTERIOR_SEGMENT

|- ((!a. (!b. ((relative_interior (closed_segment ((CONS (a , b)) NIL))) = (((COND (a = b)) (a INSERT EMPTY)) (open_segment (a , b)))))) /\ (!a. (!b. ((relative_interior (open_segment (a , b))) = (open_segment (a , b))))))


Hypermap_iso.surj_res_inv_right

|- (!f. (!s. (!t. (!y. (((((SURJ f) s) t) /\ (y IN t)) ==> (((f (((res_inv f) s) y)) = y) /\ ((((res_inv f) s) y) IN s)))))))


REAL_DIFFERENTIABLE_ATREAL_WITHIN

|- (!f. (!s. (!z. (((real_differentiable f) (atreal z)) ==> ((real_differentiable f) ((within (atreal z)) s))))))


COMPLEX_INV_EQ_0

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


HOMOTOPIC_WITH

|- ((!h. (!k. ((!x. ((x IN X) ==> ((h x) = (k x)))) ==> ((P h) = (P k))))) ==> (((((homotopic_with P) (X , Y)) p) q) = (?h. (((continuous_on h) ((closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL)) PCROSS X)) /\ ((((IMAGE h) ((closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL)) PCROSS X)) SUBSET Y) /\ ((!x. ((x IN X) ==> ((h ((pastecart (vec (NUMERAL _0))) x)) = (p x)))) /\ ((!x. ((x IN X) ==> ((h ((pastecart (vec (NUMERAL (BIT1 _0)))) x)) = (q x)))) /\ (!t. ((t IN (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL))) ==> (P (\x. (h ((pastecart t) x)))))))))))))


LINEAR_SURJECTIVE_IMP_INJECTIVE

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


Fan_defs.e_fan_pair

|- (!V. (!E. (!w. (!v. (((e_fan_pair (V , E)) (v , w)) = (w , v))))))


CBALL_SING

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


DIST_EQ_0

|- (!x. (!y. (((distance (x , y)) = (real_of_num (NUMERAL _0))) = (x = y))))


Kepler_spec.dist_spec

|- (!x. (!y. (!z. (!x'. (!y'. (!z'. ((distance ((real3_of_triple (x , (y , z))) , (real3_of_triple (x' , (y' , z'))))) = (sqrt ((real_add ((real_pow ((real_sub x) x')) (NUMERAL (BIT0 (BIT1 _0))))) ((real_add ((real_pow ((real_sub y) y')) (NUMERAL (BIT0 (BIT1 _0))))) ((real_pow ((real_sub z) z')) (NUMERAL (BIT0 (BIT1 _0))))))))))))))