CARD_EQ_CARD_IMP

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


MDIST_REFL

|- (!m. (!x. ((x IN (mspace m)) ==> (((mdist m) (x , x)) = (real_of_num (NUMERAL _0))))))


tendsto

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


ADD_AC

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


Conforming.conforming_fan

|- (!x. (!V. (!E. ((conforming_fan (x , (V , E))) = ((!v. ((v IN V) ==> ((CARD (((set_of_edge v) V) E)) > (NUMERAL (BIT1 _0))))) /\ ((fan80 (x , (V , E))) /\ ((conforming_bijection_fan (x , (V , E))) /\ ((conforming_half_space_fan (x , (V , E))) /\ ((conforming_solid_angle_fan (x , (V , E))) /\ (conforming_diagonal_fan (x , (V , E))))))))))))


IM_DEF

|- (!z. ((Im z) = (z $ (NUMERAL (BIT0 (BIT1 _0))))))


CLOSURE_HALFSPACE_LT

|- (!a. (!b. ((~ (a = (vec (NUMERAL _0)))) ==> ((closure (GSPEC (\GEN%PVAR%806. (?x. (((SETSPEC GEN%PVAR%806) ((real_lt ((dot a) x)) b)) x))))) = (GSPEC (\GEN%PVAR%807. (?x. (((SETSPEC GEN%PVAR%807) ((real_le ((dot a) x)) b)) x))))))))


INT_EQ_IMP_LE

|- (!x. (!y. ((x = y) ==> ((int_le x) y))))


LEBESGUE_MEASURABLE_LEBESGUE_MEASURABLE_PREIMAGE_BOREL

|- (!f. (!s. (!t. ((((measurable_on f) s) /\ ((lebesgue_measurable s) /\ (borel t))) ==> (lebesgue_measurable (GSPEC (\GEN%PVAR%3660. (?x. (((SETSPEC GEN%PVAR%3660) ((x IN s) /\ ((f x) IN t))) x)))))))))


IN_UNION

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


REALLIM_ATREAL

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


INF

|- (!s. (((~ (s = EMPTY)) /\ (?b. (!x. ((x IN s) ==> ((real_le b) x))))) ==> ((!x. ((x IN s) ==> ((real_le (inf s)) x))) /\ (!b. ((!x. ((x IN s) ==> ((real_le b) x))) ==> ((real_le b) (inf s)))))))


Collect_geom2.VBVYGGT

|- (!v1. (!v2. (!v3. (!v4. ((((CARD (v1 INSERT (v2 INSERT (v3 INSERT (v4 INSERT EMPTY))))) = (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) /\ (~ (coplanar_alt (v1 INSERT (v2 INSERT (v3 INSERT (v4 INSERT EMPTY))))))) ==> (((circumcenter (v1 INSERT (v2 INSERT (v3 INSERT (v4 INSERT EMPTY))))) IN ((hull affine) (v1 INSERT (v2 INSERT (v3 INSERT (v4 INSERT EMPTY)))))) /\ ((?r. (!v. ((v IN (v1 INSERT (v2 INSERT (v3 INSERT (v4 INSERT EMPTY))))) ==> (r = (distance ((circumcenter (v1 INSERT (v2 INSERT (v3 INSERT (v4 INSERT EMPTY))))) , v)))))) /\ ((!p. (((p IN ((hull affine) (v1 INSERT (v2 INSERT (v3 INSERT (v4 INSERT EMPTY)))))) /\ (?r. (!v. ((v IN (v1 INSERT (v2 INSERT (v3 INSERT (v4 INSERT EMPTY))))) ==> (r = (distance (p , v))))))) ==> (p = (circumcenter (v1 INSERT (v2 INSERT (v3 INSERT (v4 INSERT EMPTY)))))))) /\ ((LET (\x12. (LET_END ((LET (\x13. (LET_END ((LET (\x14. (LET_END ((LET (\x23. (LET_END ((LET (\x24. (LET_END ((LET (\x34. (LET_END ((LET (\chi11. (LET_END ((LET (\chi22. (LET_END ((LET (\chi33. (LET_END ((LET (\chi44. (LET_END ((circumcenter (v1 INSERT (v2 INSERT (v3 INSERT (v4 INSERT EMPTY))))) = ((% ((real_div (real_of_num (NUMERAL (BIT1 _0)))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((((((delta x12) x13) x14) x23) x24) x34)))) ((vector_add ((% chi11) v1)) ((vector_add ((% chi22) v2)) ((vector_add ((% chi33) v3)) ((% chi44) v4))))))))) ((((((chi x34) x24) x14) x23) x13) x12))))) ((((((chi x34) x13) x23) x14) x24) x12))))) ((((((chi x12) x24) x23) x14) x13) x34))))) ((((((chi x12) x13) x14) x23) x24) x34))))) ((real_pow (distance (v3 , v4))) (NUMERAL (BIT0 (BIT1 _0)))))))) ((real_pow (distance (v2 , v4))) (NUMERAL (BIT0 (BIT1 _0)))))))) ((real_pow (distance (v2 , v3))) (NUMERAL (BIT0 (BIT1 _0)))))))) ((real_pow (distance (v1 , v4))) (NUMERAL (BIT0 (BIT1 _0)))))))) ((real_pow (distance (v1 , v3))) (NUMERAL (BIT0 (BIT1 _0)))))))) ((real_pow (distance (v1 , v2))) (NUMERAL (BIT0 (BIT1 _0)))))))))))))


Hypermap.lemma_in_face

|- (!H. (!x. (!n. ((((POWER (face_map H)) n) x) IN ((face H) x)))))


VSUM_SING_NUMSEG

|- (((vsum (n .. n)) f) = (f n))


MEASURABLE_ON_MEASURABLE_PREIMAGE_CLOSED

|- (!f. (!s. ((measurable s) ==> (((measurable_on f) s) = (!t. ((closed t) ==> (measurable (GSPEC (\GEN%PVAR%3646. (?x. (((SETSPEC GEN%PVAR%3646) ((x IN s) /\ ((f x) IN t))) x)))))))))))


SUBSET_ANTISYM

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


Ssrnat.mul0n

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


CAUCHY_CONTINUOUS_EXTENDS_TO_CLOSURE

|- (!f. (!s. ((!x. (((cauchy x) /\ (!n. ((x n) IN s))) ==> (cauchy (f o x)))) ==> (?g. (((continuous_on g) (closure s)) /\ (!x. ((x IN s) ==> ((g x) = (f x)))))))))


Appendix.BBprime2_v39

|- (!vv. (!s. (((BBprime2_v39 s) vv) = (((BBprime_v39 s) vv) /\ (((BBindex_v39 s) vv) = (BBindex_min_v39 s))))))


SERIES_DIRICHLET_COMPLEX_VERY_EXPLICIT

|- (!f. (!g. (!B. (!p. ((((real_lt (real_of_num (NUMERAL _0))) B) /\ (((NUMERAL (BIT1 _0)) <= p) /\ ((!m. (!n. ((p <= m) ==> ((real_le (vector_norm ((vsum (m .. n)) f))) B)))) /\ ((!n. ((p <= n) ==> ((real (g n)) /\ ((real_le (real_of_num (NUMERAL _0))) (Re (g n)))))) /\ (!n. ((p <= n) ==> ((real_le (Re (g (n + (NUMERAL (BIT1 _0)))))) (Re (g n))))))))) ==> (!m. (!n. ((p <= m) ==> ((real_le (vector_norm ((vsum (m .. n)) (\k. ((complex_mul (f k)) (g k)))))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((real_mul B) (vector_norm (g m)))))))))))))


REAL_MAX_ACI

|- ((((real_max x) y) = ((real_max y) x)) /\ ((((real_max ((real_max x) y)) z) = ((real_max x) ((real_max y) z))) /\ ((((real_max x) ((real_max y) z)) = ((real_max y) ((real_max x) z))) /\ ((((real_max x) x) = x) /\ (((real_max x) ((real_max x) y)) = ((real_max x) y))))))


MEASURABLE_NEGLIGIBLE_SYMDIFF_EQ

|- (!s. (!t. ((negligible ((s DIFF t) UNION (t DIFF s))) ==> ((measurable s) = (measurable t)))))


Nonlin_def.dih_x_div_sqrtdelta_posbranch

|- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((((((dih_x_div_sqrtdelta_posbranch x1) x2) x3) x4) x5) x6) = ((LET (\d_x4. (LET_END ((LET (\d. (LET_END ((real_mul ((real_div (sqrt ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) x1))) d_x4)) (matan ((real_div ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) ((real_mul x1) d))) ((real_pow d_x4) (NUMERAL (BIT0 (BIT1 _0)))))))))) ((((((delta_x x1) x2) x3) x4) x5) x6))))) ((((((delta_x4 x1) x2) x3) x4) x5) x6)))))))))


IM_COMPLEX_INV_LE_0

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


REAL_SUB_ADD2

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


REAL_INTEGRAL

|- (((real_integrable_on f) s) ==> (((real_integral s) f) = (drop ((integral ((IMAGE lift) s)) (lift o (f o drop))))))


REAL_ADD_AC

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


absolutely_real_integrable_on

|- (!f. (!s. (((absolutely_real_integrable_on f) s) = (((real_integrable_on f) s) /\ ((real_integrable_on (\x. (real_abs (f x)))) s)))))


real_convex_on

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


CONTINUOUS_LIFT_PRODUCT

|- (!net. (!f. (!t. (((FINITE t) /\ (!i. ((i IN t) ==> ((continuous (\x. (lift ((f x) i)))) net)))) ==> ((continuous (\x. (lift ((product t) (f x))))) net)))))


MIN

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


CONTINUOUS_AT_CEXP

|- (!z. ((continuous cexp) (at z)))


Collect_geom.DELTA_POS_4POINTS

|- (!x1. (!x2. (!x3. (!x4. ((real_le (real_of_num (NUMERAL _0))) ((((((delta ((real_pow (distance (x1 , x2))) (NUMERAL (BIT0 (BIT1 _0))))) ((real_pow (distance (x1 , x3))) (NUMERAL (BIT0 (BIT1 _0))))) ((real_pow (distance (x1 , x4))) (NUMERAL (BIT0 (BIT1 _0))))) ((real_pow (distance (x2 , x3))) (NUMERAL (BIT0 (BIT1 _0))))) ((real_pow (distance (x2 , x4))) (NUMERAL (BIT0 (BIT1 _0))))) ((real_pow (distance (x3 , x4))) (NUMERAL (BIT0 (BIT1 _0))))))))))


CSQRT_PRINCIPAL

|- (!z. (((real_lt (real_of_num (NUMERAL _0))) (Re (csqrt z))) \/ (((Re (csqrt z)) = (real_of_num (NUMERAL _0))) /\ ((real_le (real_of_num (NUMERAL _0))) (Im (csqrt z))))))


Hypermap_iso.hyp_iso_card_components

|- (!H. (!G. (!f. (((hyp_iso f) (H , G)) ==> (!x. ((x IN (dart H)) ==> (((CARD ((face H) x)) = (CARD ((face G) (f x)))) /\ (((CARD ((node H) x)) = (CARD ((node G) (f x)))) /\ ((CARD ((edge H) x)) = (CARD ((edge G) (f x))))))))))))


DIM_EQ_SPAN

|- (!s. (!t. (((s SUBSET t) /\ ((dim t) <= (dim s))) ==> ((span s) = (span t)))))


CLOSED_TRANSLATION

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


Lp_ineqs_def.y7_fan

|- (!V. (!E. (!x. (((y7_fan (V , E)) x) = (vector_norm (FST (((POWER (inverse (f_fan_pair_ext (V , E)))) (NUMERAL (BIT0 (BIT1 _0)))) x)))))))


IN_ELIM_PAIR_THM

|- (!P. (!a. (!b. (((a , b) IN (GSPEC (\GEN%PVAR%18. (?x. (?y. (((SETSPEC GEN%PVAR%18) ((P x) y)) (x , y))))))) = ((P a) b)))))


Hypermap_and_fan.NODE_HYPERMAP_OF_FAN_ALT

|- (!V. (!E. (!v. (!w. (((FAN ((vec (NUMERAL _0)) , (V , E))) /\ ((v , w) IN (dart1_of_fan (V , E)))) ==> (((node (hypermap_of_fan (V , E))) (v , w)) = (GSPEC (\GEN%PVAR%5635. (?u. (((SETSPEC GEN%PVAR%5635) (u IN (((set_of_edge v) V) E))) (v , u)))))))))))


REAL_INTEGRABLE_EQ

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


PAIR

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


COMPLEX_NORM_NZ

|- (!z. (((real_lt (real_of_num (NUMERAL _0))) (vector_norm z)) = (~ (z = (Cx (real_of_num (NUMERAL _0)))))))


SYMMETRIC_MATRIX

|- (!A. (((transp A) = A) = ((adjoint (\x. ((matrix_vector_mul A) x))) = (\x. ((matrix_vector_mul A) x)))))


AFFINE_BOUNDED_EQ_LOWDIM

|- (!s. ((affine s) ==> ((bounded s) = ((int_le (aff_dim s)) (int_of_num (NUMERAL _0))))))


Float.FLOAT_ADD_PP

|- (!a. (!b. (!m. (!n. ((n <= m) ==> (((real_add ((float a) (int_of_num m))) ((float b) (int_of_num n))) = ((float ((int_add ((int_mul (int_of_num ((NUMERAL (BIT0 (BIT1 _0))) EXP (m - n)))) a)) b)) (int_of_num n))))))))


DIVISION_OF_NONTRIVIAL

|- (!s. (!a. (!b. ((((division_of s) (closed_interval ((CONS (a , b)) NIL))) /\ (~ ((content (closed_interval ((CONS (a , b)) NIL))) = (real_of_num (NUMERAL _0))))) ==> ((division_of (GSPEC (\GEN%PVAR%2756. (?k. (((SETSPEC GEN%PVAR%2756) ((k IN s) /\ (~ ((content k) = (real_of_num (NUMERAL _0)))))) k))))) (closed_interval ((CONS (a , b)) NIL)))))))


TRIVIAL_LIMIT_SEQUENTIALLY

|- (~ (trivial_limit sequentially))


TC_TRANS_R

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


COUNTABLE

|- (!t. ((COUNTABLE t) = (UNIV >=_c t)))


COMPLEX_POW_INV

|- (!x. (!n. (((complex_pow (complex_inv x)) n) = (complex_inv ((complex_pow x) n)))))


DOT_CAUCHY_SCHWARZ_EQUAL

|- (!x. (!y. ((((real_pow ((dot x) y)) (NUMERAL (BIT0 (BIT1 _0)))) = ((real_mul ((dot x) x)) ((dot y) y))) = (collinear ((vec (NUMERAL _0)) INSERT (x INSERT (y INSERT EMPTY)))))))


CONTINUOUS_ATREAL

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


HAS_REAL_INTEGRAL_RESTRICT_INTER

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


HAS_REAL_DERIVATIVE_ADD

|- (!f. (!f'. (!g. (!g'. (!net. (((((has_real_derivative f) f') net) /\ (((has_real_derivative g) g') net)) ==> (((has_real_derivative (\x. ((real_add (f x)) (g x)))) ((real_add f') g')) net)))))))


ROW_TRANSP

|- (!A. (!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV))) ==> (((row i) (transp A)) = ((column i) A)))))


Trigonometry2.LEMMA_SUM_ALL_OVER_CYCLIC_SET

|- (!W. (!xicm. (((FINITE W) /\ (((CARD W) = n) /\ ((!x. ((W x) ==> (~ (x = (vec (NUMERAL _0)))))) /\ (((polar_cycle_on xicm) W) /\ (W p))))) ==> ((?p. (?q. ((W p) /\ ((W q) /\ (~ (p = q)))))) ==> (((sum ((NUMERAL _0) .. (n - (NUMERAL (BIT1 _0))))) (\i. ((arg_diff (((ITER i) xicm) p)) (((ITER (i + (NUMERAL (BIT1 _0)))) xicm) p)))) = ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) pi))))))


VECTOR_NEG_COMPONENT

|- (!x. (!i. (((vector_neg x) $ i) = (real_neg (x $ i)))))


Hypermap_iso.hyp_iso_inverse_comm

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


AR_TRANSLATION

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


AFFINE_INTERS

|- ((!s. ((s IN f) ==> (affine s))) ==> (affine (INTERS f)))


Pack2.KIUMVTC

|- (!p. (!r. (!S. ((packing S) ==> (FINITE (S INTER (ball (p , r))))))))


LIM_COMPLEX_RMUL

|- (!f. (!l. (!c. ((((--> f) l) net) ==> (((--> (\x. ((complex_mul (f x)) c))) ((complex_mul l) c)) net)))))


REAL_LE_MUL

|- (!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_mul x) y)))))


CLOSED_INTER

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


Seq.subseq_uniq

|- (!s1. (!s2. (((subseq s1) s2) ==> ((uniq s2) ==> (uniq s1)))))


Seq.filter_subseq

|- (!a. (!s. ((subseq ((filter a) s)) s)))


SIMPLEX_TRANSLATION_EQ

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


DIFFERENTIABLE_NORM_AT

|- (!a. ((~ (a = (vec (NUMERAL _0)))) ==> ((differentiable (\x. (lift (vector_norm x)))) (at a))))


CARD_EXP_CANTOR

|- (!s. (s <_c ((^_c UNIV) s)))


NOT_IMP

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


Ssrnat.muln0

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


Sphere.acs_sqrt_x1_d4

|- (!x2. (!x3. (!x4. (!x5. (!x6. (!x1. (((((((acs_sqrt_x1_d4 x1) x2) x3) x4) x5) x6) = (acs ((real_div (sqrt x1)) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0))))))))))))))


LE_ADDR

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


Audit_formal_proof.chk_ball_annulus_def

|- (!v. ((v IN ball_annulus) = (((real_le (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (vector_norm v)) /\ ((real_le (vector_norm v)) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))))


COMPACT_INTER

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


Add_triangle.f_split_fan_face

|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> (!v. (!w. ((((v , w) IN (dart_of_fan (V , E))) /\ ((NUMERAL (BIT1 (BIT1 _0))) < (CARD ((face (hypermap_of_fan (V , E))) (v , w))))) ==> ((((LET (\w1. (\d2. (\f. (LET_END (((f (v , w)) = (w , w1)) /\ (((f (w , w1)) = (w1 , v)) /\ (((f (w1 , w)) = d2) /\ (((f ((((((sigma_fan (vec (NUMERAL _0))) V) E) w1) v) , w1)) = (w1 , w)) /\ (!d. (((d IN (dart_of_fan ((split_fan_face (V , E)) (v , w)))) /\ (~ ((d = (v , w)) \/ ((d = (w , w1)) \/ ((d = (w1 , w)) \/ (d = ((((((sigma_fan (vec (NUMERAL _0))) V) E) w1) v) , w1))))))) ==> ((f d) = ((f_fan_pair (V , E)) d))))))))))))) (((((sigma_fan (vec (NUMERAL _0))) V) E) v) w)) ((f_fan_pair (V , E)) (v , w))) (f_fan_pair ((split_fan_face (V , E)) (v , w)))))))))))


Lp_ineqs_proofs.f_fan_pair_ext_power_in_dart

|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!k. (!d. ((d IN (dart_of_fan (V , E))) ==> ((((POWER (f_fan_pair_ext (V , E))) k) d) IN (dart_of_fan (V , E)))))))))


ORTHOGONAL_TRANSFORMATION_INJECTIVE

|- (!f. ((orthogonal_transformation f) ==> (!x. (!y. (((f x) = (f y)) ==> (x = y))))))


CLOSURE_EQ_EMPTY

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


Fan.MONO_SIGMA_FAN

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


Hypermap.node

|- (!H. (!x. (((node H) x) = ((orbit_map (node_map H)) x))))


Trigonometry2.ups_x

|- (!x1. (!x2. (!x6. ((((ups_x x1) x2) x6) = ((real_add ((real_sub ((real_sub ((real_mul (real_neg x1)) x1)) ((real_mul x2) x2))) ((real_mul x6) x6))) ((real_add ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((real_mul x1) x6))) ((real_add ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((real_mul x1) x2))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((real_mul x2) x6)))))))))


Sphere.hplus

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


real_measurable

|- (!s. ((real_measurable s) = (?m. ((has_real_measure s) m))))


SPAN_0

|- ((vec (NUMERAL _0)) IN (span s))


DIMINDEX_4

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


FORALL_DROP

|- ((!x. (P x)) = (!x. (P (drop x))))


MIDPOINT_IN_SEGMENT

|- ((!a. (!b. ((midpoint (a , b)) IN (closed_segment ((CONS (a , b)) NIL))))) /\ (!a. (!b. (((midpoint (a , b)) IN (open_segment (a , b))) = (~ (a = b))))))


RELATIVE_ORIENTATION

|- (!A. (!f. (!s. (((rows A) = (GSPEC (\GEN%PVAR%2550. (?v. (((SETSPEC GEN%PVAR%2550) ((extreme_point_of v) s)) v))))) ==> (((relative_orientation f) s) = (real_sgn ((real_div (det ((maprows f) A))) (det A))))))))


REAL_POLY_NEG_CLAUSES

|- ((!x. ((real_neg x) = ((real_mul (real_neg (real_of_num (NUMERAL (BIT1 _0))))) x))) /\ (!x. (!y. (((real_sub x) y) = ((real_add x) ((real_mul (real_neg (real_of_num (NUMERAL (BIT1 _0))))) y))))))


has_bounded_variation_on

|- (!f. (!s. (((has_bounded_variation_on f) s) = ((has_bounded_setvariation_on (\k. ((vector_sub (f (interval_upperbound k))) (f (interval_lowerbound k))))) s))))


REAL_SUB_RDISTRIB

|- (!x. (!y. (!z. (((real_mul ((real_sub x) y)) z) = ((real_sub ((real_mul x) z)) ((real_mul y) z))))))


Seq.nseq

|- (!n. (!x. (((nseq n) x) = (((ncons n) x) NIL))))


Hypermap_and_fan.FULLY_SURROUNDED

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


Ssrnat.mul1n

|- (!n. (((NUMERAL (BIT1 _0)) * n) = n))


ORTHOGONAL_MATRIX_ORTHONORMAL_ROWS_PAIRWISE

|- (!A. ((orthogonal_matrix A) = (((CARD (rows A)) = (dimindex UNIV)) /\ ((!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV))) ==> ((vector_norm ((row i) A)) = (real_of_num (NUMERAL (BIT1 _0)))))) /\ ((pairwise orthogonal) (rows A))))))


Appendix.s_init_list_v39

|- (s_init_list_v39 = ((LET (\upperbd. (LET_END ((LET (\a_pro. (LET_END ((CONS ((((mk_unadorned_v39 (NUMERAL (BIT0 (BIT1 (BIT1 _0))))) (d_tame (NUMERAL (BIT0 (BIT1 (BIT1 _0)))))) (((cs_adj (NUMERAL (BIT0 (BIT1 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0))) (((cs_adj (NUMERAL (BIT0 (BIT1 (BIT1 _0))))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) upperbd))) ((CONS ((((mk_unadorned_v39 (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) (d_tame (NUMERAL (BIT1 (BIT0 (BIT1 _0)))))) (((cs_adj (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0))) (((cs_adj (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) upperbd))) ((CONS ((((mk_unadorned_v39 (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) (d_tame (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) (((cs_adj (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0))) (((cs_adj (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) upperbd))) ((CONS ((((mk_unadorned_v39 (NUMERAL (BIT1 (BIT1 _0)))) (d_tame (NUMERAL (BIT1 (BIT1 _0))))) (((cs_adj (NUMERAL (BIT1 (BIT1 _0)))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0))) (((cs_adj (NUMERAL (BIT1 (BIT1 _0)))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) upperbd))) ((CONS ((((mk_unadorned_v39 (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 _0)))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))) (((cs_adj (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) sqrt8)) (((cs_adj (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) upperbd))) ((CONS ((((mk_unadorned_v39 (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) ((DECIMAL (NUMERAL (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 _0))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))) (((cs_adj (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT1 (BIT1 _0)))))) (((cs_adj (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) upperbd))) ((CONS ((((mk_unadorned_v39 (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 _0)))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))) ((((a_pro (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0))) ((((a_pro (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) sqrt8) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) upperbd))) ((CONS ((((mk_unadorned_v39 (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 _0))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))) ((((a_pro (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) sqrt8)) ((((a_pro (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) sqrt8) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) upperbd))) NIL))))))))))) (\k. (\p. (\a1. (\a2. (\i. (\j. (((COND ((i MOD k) = (j MOD k))) (real_of_num (NUMERAL _0))) (((COND (((i MOD k) INSERT ((j MOD k) INSERT EMPTY)) = ((NUMERAL _0) INSERT ((NUMERAL (BIT1 _0)) INSERT EMPTY)))) p) (((COND (((j MOD k) = ((SUC i) MOD k)) \/ (((SUC j) MOD k) = (i MOD k)))) a1) a2))))))))))))) (real_of_num (NUMERAL (BIT0 (BIT1 (BIT1 _0)))))))


NUMSEG_OFFSET_IMAGE

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


CONVERGENT_IMP_BOUNDED

|- (!s. (!l. ((((--> s) l) sequentially) ==> (bounded ((IMAGE s) UNIV)))))


HAS_DERIVATIVE_CONST

|- (!c. (!net. (((has_derivative (\x. c)) (\h. (vec (NUMERAL _0)))) net)))


Collect_geom.DELTA_COEFS

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


LIM_NULL_COMPLEX_SUB

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


CLOSED_IN_SUBSET_TRANS

|- (!s. (!t. (!u. ((((closed_in ((subtopology euclidean) u)) s) /\ ((s SUBSET t) /\ (t SUBSET u))) ==> ((closed_in ((subtopology euclidean) t)) s)))))


REAL_CONTINUOUS_AT_ID

|- (!x. ((real_continuous (\x. x)) (atreal x)))


AFFINE_HULL_LINEAR_IMAGE

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


int_sgn_th

|- (!x. ((real_of_int (int_sgn x)) = (real_sgn (real_of_int x))))


ABS_DROP

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


ARC_LINEAR_IMAGE_EQ

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


higher_complex_derivative

|- ((((higher_complex_derivative (NUMERAL _0)) f) = f) /\ (!n. (((higher_complex_derivative (SUC n)) f) = (complex_derivative ((higher_complex_derivative n) f)))))


ANR_FROM_UNION_AND_INTER_LOCAL

|- (!s. (!t. ((((closed_in ((subtopology euclidean) (s UNION t))) s) /\ (((closed_in ((subtopology euclidean) (s UNION t))) t) /\ ((ANR (s UNION t)) /\ (ANR (s INTER t))))) ==> ((ANR s) /\ (ANR t)))))


CONTINUOUS_ON_CLOSEST_POINT

|- (!s. (!t. (((convex s) /\ ((closed s) /\ (~ (s = EMPTY)))) ==> ((continuous_on (closest_point s)) t))))


REAL_MIN_LE

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


Functional_equation.functional_arclength_x_123'

|- (arclength_x_123' = (uni (acs , ((div6 ((add6 proj_x1) ((sub6 proj_x2) proj_x3))) (uni (sqrt , ((scalar6 ((mul6 proj_x1) proj_x2)) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0))))))))))))


NEGLIGIBLE_IMP_LEBESGUE_MEASURABLE

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


CONVEX_HULL_LINEAR_IMAGE

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


Xwitccn.PERIODIC_PROPERTY

|- (((~ (k = (NUMERAL _0))) /\ ((periodic vv) k)) ==> (!i. ((vv (i MOD k)) = (vv i))))


DOT_POS_LT

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


LE_0

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


HAS_VECTOR_DERIVATIVE_CONST

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


REAL_SGN_POW

|- (!x. (!n. ((real_sgn ((real_pow x) n)) = ((real_pow (real_sgn x)) n))))


Misc_defs_and_lemmas.min_least

|- (!X. (!c. ((X c) ==> ((X (min_num X)) /\ ((min_num X) <= c)))))


SUM_CASES

|- (!s. (!P. (!f. (!g. ((FINITE s) ==> (((sum s) (\x. (((COND (P x)) (f x)) (g x)))) = ((real_add ((sum (GSPEC (\GEN%PVAR%213. (?x. (((SETSPEC GEN%PVAR%213) ((x IN s) /\ (P x))) x))))) f)) ((sum (GSPEC (\GEN%PVAR%214. (?x. (((SETSPEC GEN%PVAR%214) ((x IN s) /\ (~ (P x)))) x))))) g))))))))


independent

|- (!s. ((independent s) = (~ (dependent s))))


Seq.foldr

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


REAL_SUMMABLE

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


DOT_RADD

|- (!x. (!y. (!z. (((dot x) ((vector_add y) z)) = ((real_add ((dot x) y)) ((dot x) z))))))


Real_ext.REAL_LE_LMUL_IMP

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


CNJ_MUL

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


Fatugpd.norm_mul_normalize

|- (!v. (((% (vector_norm v)) (normalize v)) = v))


Hypermap.lemma_transitive_permutation

|- (!L. (!x. (((in_dart_of_loop x) L) ==> ((dart_of_loop L) = ((orbit_map (loop_map L)) x)))))


COMPLEX_ADD_LID

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


Hypermap_and_fan.FINITE_ORBIT_MAP

|- (!s. (!f. (!x. (!n. (((FINITE s) /\ (((permutes f) s) /\ ((CARD ((orbit_map f) x)) = n))) ==> (((orbit_map f) x) = (GSPEC (\GEN%PVAR%5631. (?k. (((SETSPEC GEN%PVAR%5631) (k < n)) (((POWER f) k) x)))))))))))


LEBESGUE_MEASURABLE_CLOSED_IN

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


COMPLEX_NORM_POW

|- (!z. (!n. ((vector_norm ((complex_pow z) n)) = ((real_pow (vector_norm z)) n))))


Conforming.SIGMA_FAN_OF_FANADD_AT_POINT3

|- (!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) w) v) = u)))))))))


SUBSET_UNION

|- ((!s. (!t. (s SUBSET (s UNION t)))) /\ (!s. (!t. (s SUBSET (t UNION s)))))


REVERSEPATH_LINEAR_IMAGE

|- (!f. (!g. ((linear f) ==> ((reversepath (f o g)) = (f o (reversepath g))))))


TRANSITIVE_STEPWISE_LE

|- (!R. (((!x. ((R x) x)) /\ ((!x. (!y. (!z. ((((R x) y) /\ ((R y) z)) ==> ((R x) z))))) /\ (!n. ((R n) (SUC n))))) ==> (!m. (!n. ((m <= n) ==> ((R m) n))))))


VECTOR_SUB_LZERO

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


UNIONS_MONO_IMAGE

|- ((!x. ((x IN s) ==> ((f x) SUBSET (g x)))) ==> ((UNIONS ((IMAGE f) s)) SUBSET (UNIONS ((IMAGE g) s))))


OPEN_INTER

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


Functional_equation.functional_gchi4_x'

|- (gchi4_x' = ((mul6 (uni (gchi , proj_y4))) dih4_x'))


SETDIST_SYM

|- (!s. (!t. ((setdist (s , t)) = (setdist (t , s)))))


PI_NZ

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


NSUM_SING_NUMSEG

|- (!f. (!n. (((nsum (n .. n)) f) = (f n))))


AR_IMP_ABSOLUTE_RETRACT

|- (!s. (!u. (!s'. (((AR s) /\ (((homeomorphic s) s') /\ ((closed_in ((subtopology euclidean) u)) s'))) ==> ((retract_of s') u)))))


HAS_BOUNDED_VARIATION_DARBOUX_STRICT

|- (!f. (!a. (!b. (((has_bounded_variation_on f) (closed_interval ((CONS (a , b)) NIL))) = (?g. (?h. ((!x. (!y. (((x IN (closed_interval ((CONS (a , b)) NIL))) /\ ((y IN (closed_interval ((CONS (a , b)) NIL))) /\ ((real_lt (drop x)) (drop y)))) ==> ((real_lt (drop (g x))) (drop (g y)))))) /\ ((!x. (!y. (((x IN (closed_interval ((CONS (a , b)) NIL))) /\ ((y IN (closed_interval ((CONS (a , b)) NIL))) /\ ((real_lt (drop x)) (drop y)))) ==> ((real_lt (drop (h x))) (drop (h y)))))) /\ (!x. ((f x) = ((vector_sub (g x)) (h x))))))))))))


PATH_CONNECTED_PUNCTURED_CONVEX

|- (!s. (!a. (((convex s) /\ (~ ((aff_dim s) = (int_of_num (NUMERAL (BIT1 _0)))))) ==> (path_connected (s DELETE a)))))


CONTINUOUS_AT_CSQRT

|- (!z. ((((Im z) = (real_of_num (NUMERAL _0))) ==> ((real_lt (real_of_num (NUMERAL _0))) (Re z))) ==> ((continuous csqrt) (at z))))


Bump.MCELL2

|- (!V. (!ul. (((mcell2 V) ul) = (((mcell (NUMERAL (BIT0 (BIT1 _0)))) V) ul))))


INT_OF_NUM_MUL

|- (!m. (!n. (((int_mul (int_of_num m)) (int_of_num n)) = (int_of_num (m * n)))))


REAL_LEBESGUE_MEASURABLE_COMPL

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


PATH_CONNECTED_PCROSS

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


DIFFERENTIAL_COMPONENT_ZERO_AT_MAXMIN

|- (!f. (!f'. (!x. (!s. (!k. ((((NUMERAL (BIT1 _0)) <= k) /\ ((k <= (dimindex UNIV)) /\ ((x IN s) /\ ((open s) /\ ((((has_derivative f) f') (at x)) /\ ((!w. ((w IN s) ==> ((real_le ((f w) $ k)) ((f x) $ k)))) \/ (!w. ((w IN s) ==> ((real_le ((f x) $ k)) ((f w) $ k)))))))))) ==> (!h. (((f' h) $ k) = (real_of_num (NUMERAL _0))))))))))


CONTINUOUS_ON_LIFT_COMPONENT

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


AR_STRONG_DEFORMATION_RETRACT_OF_AR

|- (!s. (!t. (((AR s) /\ ((AR t) /\ ((closed_in ((subtopology euclidean) s)) t))) ==> (?r. (((((homotopic_with (\h. (!x. ((x IN t) ==> ((h x) = x))))) (s , s)) (\x. x)) r) /\ ((retraction (s , t)) r))))))


MEASURABLE_CBALL

|- (!a. (!r. (measurable (cball (a , r)))))


Lp_ineqs_proofs.split_list_list

|- (!L. (!ds. (((split_list_list L) ds) = (((foldr (\d. (\L. ((split_list_hyp L) d)))) L) ds))))


CARD_EQ_TRANS

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


Nonlinear_lemma.lmfun_lfun

|- (!y. (((real_le y) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) ==> ((lmfun ((real_div y) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) = (lfun ((real_div y) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))))))


ITERATE_DELTA

|- (!op. ((monoidal op) ==> (!f. (!a. (!s. ((((iterate op) s) (\x. (((COND (x = a)) (f x)) (neutral op)))) = (((COND (a IN s)) (f a)) (neutral op))))))))


HAS_SIZE_FUNSPACE_UNIV

|- (!m. (!n. (((UNIV HAS_SIZE m) /\ (UNIV HAS_SIZE n)) ==> (UNIV HAS_SIZE (n EXP m)))))


Tame_list.POWER_TO_ITER

|- (!n. (((POWER f) n) = ((ITER n) f)))


Ssrnat.addnn

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


INT_LE_SUB_LADD

|- (!x. (!y. (!z. (((int_le x) ((int_sub y) z)) = ((int_le ((int_add x) z)) y)))))


INT_OF_NUM_EQ

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


Tame_defs2.core

|- (!L. (!r. (((core L) r) = ((MAP (FST o HD)) ((parts L) r)))))


MEASURABLE_IMP_LEBESGUE_MEASURABLE

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


Tame_defs.scriptL

|- (!V. ((scriptL V) = ((sum V) (\v. (lmfun ((real_div (vector_norm v)) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))))))


EXISTS_THM

|- (? = (\P. (P (@ P))))


POLYTOPE_SEGMENT

|- (!a. (!b. (polytope (closed_segment ((CONS (a , b)) NIL)))))


COMPLEX_MUL_SYM

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


Tame_defs.tame_1

|- (!H. ((tame_1 H) = ((plain_hypermap H) /\ (planar_hypermap H))))


CONVEX_HALFSPACE_LT

|- (!a. (!b. (convex (GSPEC (\GEN%PVAR%1439. (?x. (((SETSPEC GEN%PVAR%1439) ((real_lt ((dot a) x)) b)) x)))))))


BOUNDED_SUBSET

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


LET_END_DEF

|- (!t. ((LET_END t) = t))


LIM_NULL_COMPARISON

|- (!net. (!f. (!g. ((((eventually (\x. ((real_le (vector_norm (f x))) (g x)))) net) /\ (((--> (\x. (lift (g x)))) (vec (NUMERAL _0))) net)) ==> (((--> f) (vec (NUMERAL _0))) net)))))


ABSOLUTELY_REAL_INTEGRABLE_0

|- (!s. ((absolutely_real_integrable_on (\x. (real_of_num (NUMERAL _0)))) s))


ANR_TRANSLATION

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


GABS_DEF

|- (!P. ((GABS P) = (@ P)))


extreme_point_of

|- (!s. (!x. (((extreme_point_of x) s) = ((x IN s) /\ (!a. (!b. (((a IN s) /\ (b IN s)) ==> (~ (x IN (open_segment (a , b)))))))))))


RC_SC

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


FINITE_EMPTY

|- (FINITE EMPTY)


HAS_INTEGRAL_COMPLEX_LMUL

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


INTEGRABLE_DECREASING_PRODUCT

|- (!f. (!g. (!a. (!b. ((((integrable_on f) (closed_interval ((CONS (a , b)) NIL))) /\ (!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 (g y)) (g x)))))) ==> ((integrable_on (\x. ((% (g x)) (f x)))) (closed_interval ((CONS (a , b)) NIL))))))))


List_hypermap_iso.fan_list_iso

|- (!v. (!f. (!V. (!E. (!w. (((fan_list_iso (f , (V , E))) (v , w)) = (((fan_list_nodes_iso (f , (V , E))) v) , ((fan_list_nodes_iso (f , (V , E))) w))))))))


HAS_BOUNDED_SETVARIATION_WORKS_ON_INTERVAL

|- (!f. (!a. (!b. (((has_bounded_setvariation_on f) (closed_interval ((CONS (a , b)) NIL))) ==> ((!d. (((division_of d) (closed_interval ((CONS (a , b)) NIL))) ==> ((real_le ((sum d) (\k. (vector_norm (f k))))) ((set_variation (closed_interval ((CONS (a , b)) NIL))) f)))) /\ (!B. ((!d. (((division_of d) (closed_interval ((CONS (a , b)) NIL))) ==> ((real_le ((sum d) (\k. (vector_norm (f k))))) B))) ==> ((real_le ((set_variation (closed_interval ((CONS (a , b)) NIL))) f)) B))))))))


LIM_INV_N_OFFSET

|- (!z. (((--> (\n. (complex_inv ((vector_add (Cx (real_of_num n))) z)))) (Cx (real_of_num (NUMERAL _0)))) sequentially))


LEBESGUE_MEASURABLE_NEGLIGIBLE_SYMDIFF

|- (!s. (!t. (((lebesgue_measurable s) /\ (negligible ((s DIFF t) UNION (t DIFF s)))) ==> (lebesgue_measurable t))))


COMPLEX_SUB_RZERO

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


OPEN_IN_OPEN

|- (!s. (!u. (((open_in ((subtopology euclidean) u)) s) = (?t. ((open t) /\ (s = (u INTER t)))))))


Calc_derivative.invert_den_eq

|- (!a. (!b. ((((real_div a) b) = (real_of_num (NUMERAL _0))) = (((real_mul a) b) = (real_of_num (NUMERAL _0))))))


CONVEX_LINEAR_IMAGE_EQ

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


HAS_SIZE_0

|- (!s. ((s HAS_SIZE (NUMERAL _0)) = (s = EMPTY)))


Fan.sigma_fan_in_set_of_edge

|- (!x. (!V. (!E. (!v. (!u. (((FAN (x , (V , E))) /\ (u IN (((set_of_edge v) V) E))) ==> ((((((sigma_fan x) V) E) v) u) IN (((set_of_edge v) V) E))))))))


Localization.has_orders

|- (!k. (!f. (((has_orders f) k) = ((!i. ((((NUMERAL _0) < i) /\ (i < k)) ==> (~ (((ITER i) f) = I)))) /\ (((ITER k) f) = I)))))


SETDIST_SUBSETS_EQ

|- (!s. (!t. (!s'. (!t'. (((s' SUBSET s) /\ ((t' SUBSET t) /\ (!x. (!y. (((x IN s) /\ (y IN t)) ==> (?x'. (?y'. ((x' IN s') /\ ((y' IN t') /\ ((real_le (distance (x' , y'))) (distance (x , y)))))))))))) ==> ((setdist (s' , t')) = (setdist (s , t))))))))


SUBSET_INTERIOR_EQ

|- (!s. ((s SUBSET (interior s)) = (open s)))


REAL_LE_INV2

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


Add_triangle.azim_dart1_eq_azim

|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!v. (!w. ((((v , w) IN (dart1_of_fan (V , E))) /\ ((CARD (((set_of_edge v) V) E)) > (NUMERAL (BIT1 _0)))) ==> (((azim_dart (V , E)) (v , w)) = ((((azim (vec (NUMERAL _0))) v) w) (((((sigma_fan (vec (NUMERAL _0))) V) E) v) w)))))))))


FSTCART_SUB

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


OPEN_SEGMENT_LINEAR_IMAGE

|- (!f. (!a. (!b. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((open_segment ((f a) , (f b))) = ((IMAGE f) (open_segment (a , b))))))))


AFF_DIM_OPEN

|- (!s. (((open s) /\ (~ (s = EMPTY))) ==> ((aff_dim s) = (int_of_num (dimindex UNIV)))))


SUM_DEGENERATE

|- (!f. (!s. ((~ (FINITE (GSPEC (\GEN%PVAR%194. (?x. (((SETSPEC GEN%PVAR%194) ((x IN s) /\ (~ ((f x) = (real_of_num (NUMERAL _0)))))) x)))))) ==> (((sum s) f) = (real_of_num (NUMERAL _0))))))


UNIONS_IMAGE

|- (!f. (!s. ((UNIONS ((IMAGE f) s)) = (GSPEC (\GEN%PVAR%29. (?y. (((SETSPEC GEN%PVAR%29) (?x. ((x IN s) /\ (y IN (f x))))) y)))))))


PAIRWISE_DISJOINT_COMPONENTS

|- (!u. ((pairwise DISJOINT) (components u)))


Localization.v_prime

|- (!V. (!FF. (((v_prime V) FF) = (GSPEC (\GEN%PVAR%7534. (?v. (((SETSPEC GEN%PVAR%7534) ((v IN V) /\ (?w. ((v , w) IN FF)))) v)))))))


matrix_neg

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


DIMINDEX_1

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


DISJOINT_RELATIVE_INTERIOR_CONVEX_HULL

|- (!s. (((~ (affine_dependent s)) /\ (~ (?a. (s = (a INSERT EMPTY))))) ==> (((relative_interior ((hull convex) s)) INTER s) = EMPTY)))


Collect_geom2.NOT_COL_EQ_UPS_X_POS

|- (!v1. (!v2. (!v3. ((~ (collinear (v1 INSERT (v2 INSERT (v3 INSERT EMPTY))))) = ((real_lt (real_of_num (NUMERAL _0))) (((ups_x ((real_pow (distance (v1 , v2))) (NUMERAL (BIT0 (BIT1 _0))))) ((real_pow (distance (v1 , v3))) (NUMERAL (BIT0 (BIT1 _0))))) ((real_pow (distance (v2 , v3))) (NUMERAL (BIT0 (BIT1 _0))))))))))


COUNTABLE_DELETE

|- (!x. (!s. ((COUNTABLE (s DELETE x)) = (COUNTABLE s))))


PATH_SUBPATH

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


RC_CLOSED

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


CROSS_RADD

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


Tame_defs2.transform_assumption

|- (!L. ((transform_assumption_v2 L) = (!N. (!r. (!x. (((good_list L) /\ ((good_list_nodes L) /\ ((((marked_list L) N) r) x))) ==> ((((marked_list L) (((((ntrans L) N) r) x) (NUMERAL (BIT1 _0)))) (((((rtrans L) N) r) x) (NUMERAL (BIT1 _0)))) x)))))))


EXTREME_POINTS_OF_CONVEX_HULL_AFFINE_INDEPENDENT

|- (!s. ((~ (affine_dependent s)) ==> ((GSPEC (\GEN%PVAR%2247. (?x. (((SETSPEC GEN%PVAR%2247) ((extreme_point_of x) ((hull convex) s))) x)))) = s)))


INSEG_SUBSET_FL

|- (!l. (!m. (((inseg m) l) ==> (!x. (((fl m) x) ==> ((fl l) x))))))


CARD_ADD_C

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


LINEAR_1

|- (!f. ((linear f) = (?c. (f = (\x. ((% c) x))))))


EXCLUDED_MIDDLE

|- (!t. (t \/ (~ t)))


REAL_DIFFSQ

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


CSIN_ADD

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


CARD_PSUBSET

|- (!a. (!b. (((a PSUBSET b) /\ (FINITE b)) ==> ((CARD a) < (CARD b)))))


OPEN_IN_TRANS

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


Add_triangle.dart_switch

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


DET_DEPENDENT_ROWS

|- (!A. ((dependent (rows A)) ==> ((det A) = (real_of_num (NUMERAL _0)))))


ADD_SUB

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


poset

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


HAS_REAL_MEASURE_EMPTY

|- ((has_real_measure EMPTY) (real_of_num (NUMERAL _0)))


MEM_LIST_OF_SET

|- (!s. ((FINITE s) ==> (!x. (((MEM x) (list_of_set s)) = (x IN s)))))


Functional_equation.functional_taum_x'

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


Fan.d_fan

|- (!x. (!V. (!E. ((d_fan (x , (V , E))) = ((d1_fan (x , (V , E))) UNION (d20_fan (x , (V , E))))))))


Nonlin_def.ldih_x_div_sqrtdelta_posbranch

|- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((((((ldih_x_div_sqrtdelta_posbranch x1) x2) x3) x4) x5) x6) = ((real_mul (lfun ((real_div (sqrt x1)) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) ((((((dih_x_div_sqrtdelta_posbranch x1) x2) x3) x4) x5) x6)))))))))


REAL_ADD_SYM

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


Add_triangle.fan_map_ext

|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!d. ((d IN (dart1_of_fan (V , E))) ==> ((((((f1_fan (vec (NUMERAL _0))) V) E) ((ext_dart (V , E)) d)) = ((ext_dart (V , E)) ((f_fan_pair (V , E)) d))) /\ ((((((n_fan (vec (NUMERAL _0))) V) E) ((ext_dart (V , E)) d)) = ((ext_dart (V , E)) ((n_fan_pair (V , E)) d))) /\ (((((e_fan (vec (NUMERAL _0))) V) E) ((ext_dart (V , E)) d)) = ((ext_dart (V , E)) ((e_fan_pair (V , E)) d))))))))))


RPOW_RPOW

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


Seq2.list_sum_set_of_list

|- (!s. (!f. ((uniq s) ==> (((list_sum s) f) = ((sum (set_of_list s)) f)))))


AFFSIGN_SPECIAL_SCALE

|- (!sgn. (!s. (!t. (!a. (!v. (((FINITE s) /\ ((FINITE t) /\ ((~ ((vec (NUMERAL _0)) IN t)) /\ ((~ (v IN t)) /\ ((~ (((% a) v) IN t)) /\ ((!x. ((sgn x) ==> (sgn ((real_div x) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))))) /\ ((!x. (!y. (((sgn x) /\ (sgn y)) ==> (sgn ((real_add x) y))))) /\ ((real_lt (real_of_num (NUMERAL _0))) a)))))))) ==> ((((affsign sgn) ((vec (NUMERAL _0)) INSERT (((% a) v) INSERT s))) t) = (((affsign sgn) ((vec (NUMERAL _0)) INSERT (v INSERT s))) t))))))))


CLOSEST_POINT_IN_RELATIVE_FRONTIER

|- (!s. (!x. (((closed s) /\ ((~ (s = EMPTY)) /\ (x IN (((hull affine) s) DIFF (relative_interior s))))) ==> (((closest_point s) x) IN (relative_frontier s)))))


Betwn_corek_z_x.parts_rot

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


Seq.uniq_leq_size

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


LIM_NULL_CMUL

|- (!net. (!f. (!c. ((((--> f) (vec (NUMERAL _0))) net) ==> (((--> (\x. ((% c) (f x)))) (vec (NUMERAL _0))) net)))))


INT_OF_NUM_GE

|- (!m. (!n. (((int_ge (int_of_num m)) (int_of_num n)) = (m >= n))))


REAL_CONTINUOUS_CONTINUOUS1

|- (((real_continuous f) net) = ((continuous (lift o f)) net))


REAL_CONTINUOUS_CONTINUOUS_ATREAL

|- (((real_continuous f) (atreal x)) = ((continuous (lift o (f o drop))) (at (lift x))))


SECOND_MEAN_VALUE_THEOREM_GEN_FULL

|- (!f. (!g. (!a. (!b. (!u. (!v. (((~ ((closed_interval ((CONS (a , b)) NIL)) = EMPTY)) /\ (((integrable_on f) (closed_interval ((CONS (a , b)) NIL))) /\ ((!x. ((x IN (open_interval (a , b))) ==> (((real_le u) (g x)) /\ ((real_le (g x)) v)))) /\ (!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 (g x)) (g y)))))))) ==> (?c. ((c IN (closed_interval ((CONS (a , b)) NIL))) /\ (((has_integral (\x. ((% (g x)) (f x)))) ((vector_add ((% u) ((integral (closed_interval ((CONS (a , c)) NIL))) f))) ((% v) ((integral (closed_interval ((CONS (c , b)) NIL))) f)))) (closed_interval ((CONS (a , b)) NIL))))))))))))


affine

|- (!s. ((affine s) = (!x. (!y. (!u. (!v. (((x IN s) /\ ((y IN s) /\ (((real_add u) v) = (real_of_num (NUMERAL (BIT1 _0)))))) ==> (((vector_add ((% u) x)) ((% v) y)) IN s))))))))


UNIONS_0

|- ((UNIONS EMPTY) = EMPTY)


COMPLEX_SUB_RDISTRIB

|- (!x. (!y. (!z. (((complex_mul ((vector_sub x) y)) z) = ((vector_sub ((complex_mul x) z)) ((complex_mul y) z))))))


LIMPT_INJECTIVE_LINEAR_IMAGE_EQ

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


MOD_MOD_REFL

|- (!m. (!n. ((~ (n = (NUMERAL _0))) ==> (((m MOD n) MOD n) = (m MOD n)))))


REAL_AFFINITY_EQ

|- (!m. (!c. (!x. (!y. ((~ (m = (real_of_num (NUMERAL _0)))) ==> ((((real_add ((real_mul m) x)) c) = y) = (x = ((real_add ((real_mul (real_inv m)) y)) (real_neg ((real_div c) m))))))))))


NOT_CLAUSES

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


CTAN_SUB

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


ENDS_NOT_IN_SEGMENT

|- (!a. (!b. ((~ (a IN (open_segment (a , b)))) /\ (~ (b IN (open_segment (a , b)))))))


TRIANGULATION_DISJOINT_RELATIVE_INTERIORS

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


eq_c

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


MEASURABLE_ON_PREIMAGE_HALFSPACE_COMPONENT_LE_DENSE

|- (!f. (!r. (((closure ((IMAGE lift) r)) = UNIV) ==> (((measurable_on f) UNIV) = (!a. (!k. ((((NUMERAL (BIT1 _0)) <= k) /\ ((k <= (dimindex UNIV)) /\ (a IN r))) ==> (lebesgue_measurable (GSPEC (\GEN%PVAR%3594. (?x. (((SETSPEC GEN%PVAR%3594) ((real_le ((f x) $ k)) a)) x))))))))))))


Planarity.properties1_inside_fan

|- (!x. (!u. (!w. ((((DISJOINT (x INSERT EMPTY)) (u INSERT (w INSERT EMPTY))) /\ (((real_lt (real_of_num (NUMERAL _0))) a) /\ ((real_lt a) (real_of_num (NUMERAL (BIT1 _0)))))) ==> (((vector_add ((% ((real_sub (real_of_num (NUMERAL (BIT1 _0)))) a)) u)) ((% a) w)) IN ((aff_ge (x INSERT EMPTY)) (u INSERT (w INSERT EMPTY))))))))


CONVEX_HULL_INSERT_SEGMENTS

|- (!s. (!a. (((hull convex) (a INSERT s)) = (((COND (s = EMPTY)) (a INSERT EMPTY)) (UNIONS (GSPEC (\GEN%PVAR%1485. (?x. (((SETSPEC GEN%PVAR%1485) (x IN ((hull convex) s))) (closed_segment ((CONS (a , x)) NIL)))))))))))


REALLIM_SUB

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


POLYHEDRON_POSITIVE_ORTHANT

|- (polyhedron (GSPEC (\GEN%PVAR%2275. (?x. (((SETSPEC GEN%PVAR%2275) (!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV))) ==> ((real_le (real_of_num (NUMERAL _0))) (x $ i))))) x)))))


NSUM_PERMUTE

|- (!f. (!p. (!s. (((permutes p) s) ==> (((nsum s) f) = ((nsum s) (f o p)))))))


DIFFERENTIABLE_TRANSFORM_WITHIN

|- (!f. (!g. (!x. (!s. (!d. ((((real_lt (real_of_num (NUMERAL _0))) d) /\ ((x IN s) /\ ((!x'. (((x' IN s) /\ ((real_lt (distance (x' , x))) d)) ==> ((f x') = (g x')))) /\ ((differentiable f) ((within (at x)) s))))) ==> ((differentiable g) ((within (at x)) s))))))))


real_negligible

|- (!s. ((real_negligible s) = (negligible ((IMAGE lift) s))))


REAL_LE_LDIV_EQ

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


LOCALLY_PATH_CONNECTED_LINEAR_IMAGE_EQ

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


HOLOMORPHIC_ON_IMP_DIFFERENTIABLE_AT

|- (!f. (!s. (!x. ((((holomorphic_on f) s) /\ ((open s) /\ (x IN s))) ==> ((complex_differentiable f) (at x))))))


CLOSED_SUBSET

|- (!s. (!t. (((s SUBSET t) /\ (closed s)) ==> ((closed_in ((subtopology euclidean) t)) s))))


CLOSED_IN_IMP_SUBSET

|- (!top. (!s. (!t. (((closed_in ((subtopology top) s)) t) ==> (t SUBSET s)))))


Functional_equation.proj_y4

|- (!x1. (!x2. (!x3. (!x5. (!x6. (!x4. (((((((proj_y4 x1) x2) x3) x4) x5) x6) = (sqrt x4))))))))


INTER_SUBSET

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


ROOT_POS_LT

|- (!n. (!x. (((real_lt (real_of_num (NUMERAL _0))) x) ==> ((real_lt (real_of_num (NUMERAL _0))) ((root n) x)))))


VSUM_RESTRICT_SET

|- (!P. (!s. (!f. (((vsum (GSPEC (\GEN%PVAR%111. (?x. (((SETSPEC GEN%PVAR%111) ((x IN s) /\ (P x))) x))))) f) = ((vsum s) (\x. (((COND (P x)) (f x)) (vec (NUMERAL _0)))))))))


REAL_POLY_CLAUSES

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


CARD_NUMSEG_1

|- (!n. ((CARD ((NUMERAL (BIT1 _0)) .. n)) = n))


MDIST_REVERSE_TRIANGLE

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


MEASURABLE_ON_COMPOSE_CONTINUOUS_OPEN_INTERVAL

|- (!f. (!g. (!a. (!b. ((((measurable_on f) UNIV) /\ ((!x. ((f x) IN (open_interval (a , b)))) /\ ((continuous_on g) (open_interval (a , b))))) ==> ((measurable_on (g o f)) UNIV))))))


II_NZ

|- (~ (ii = (Cx (real_of_num (NUMERAL _0)))))


REAL_CONTINUOUS_AT_AZIM_SHARP

|- (!v. (!w. (!w1. (!w2. (((~ (collinear (v INSERT (w INSERT (w1 INSERT EMPTY))))) /\ (~ (w2 IN ((aff_ge (v INSERT (w INSERT EMPTY))) (w1 INSERT EMPTY))))) ==> ((real_continuous (((azim v) w) w1)) (at w2)))))))


Misc_defs_and_lemmas.min_num

|- (!X. ((min_num X) = (@m. ((m IN X) /\ (!n. ((n IN X) ==> (m <= n)))))))


real

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


CSQRT

|- (!z. (((complex_pow (csqrt z)) (NUMERAL (BIT0 (BIT1 _0)))) = z))


CONVEX_HAS_BOUNDED_VARIATION_EQ

|- (!f. (!s. ((((convex_on (drop o f)) s) /\ (is_interval s)) ==> (((has_bounded_variation_on f) s) = (bounded ((IMAGE f) s))))))


DIFFERENTIABLE_CHAIN_WITHIN

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


LE_TRANS

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


REAL_LE_TOTAL

|- (!x. (!y. (((real_le x) y) \/ ((real_le y) x))))


REAL_CLOSED_IN

|- (!s. ((real_closed s) = ((closed_in euclideanreal) s)))


Dih2k_hypermap.IMAGE_NN_OF_HYP_F_SY

|- (((FAN ((vec (NUMERAL _0)) , ((V_SY (vecmats l)) , (E_SY (vecmats l))))) /\ (((NUMERAL (BIT1 _0)) < (dimindex UNIV)) /\ (!i. (!j. ((((NUMERAL (BIT1 _0)) <= i) /\ ((i <= (dimindex UNIV)) /\ (((NUMERAL (BIT1 _0)) <= j) /\ ((j <= (dimindex UNIV)) /\ (((row i) (vecmats l)) = ((row j) (vecmats l))))))) ==> (i = j)))))) ==> (((IMAGE (nn_of_hyp ((vec (NUMERAL _0)) , ((V_SY (vecmats l)) , (E_SY (vecmats l)))))) (F_SY (vecmats l))) = (GSPEC (\GEN%PVAR%7498. (?i. (((SETSPEC GEN%PVAR%7498) (((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV)))) (((row (SUC (i MOD (dimindex UNIV)))) (vecmats l)) , ((row i) (vecmats l)))))))))


FROM_INTER_NUMSEG

|- (!k. (!n. (((from k) INTER ((NUMERAL _0) .. n)) = (k .. n))))


REAL_INV_INV

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


COMPLEX_EQ_SUB_LADD

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


POLYHEDRON_IMP_CONVEX

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


Hypermap.lemma_loop_outside_node

|- (!H. (!NF. (!L. (!x. ((((is_normal H) NF) /\ (L IN NF)) ==> (~ ((dart_of_loop L) SUBSET ((node H) x))))))))


orthogonal_matrix

|- (!Q. ((orthogonal_matrix Q) = ((((matrix_mul (transp Q)) Q) = (mat (NUMERAL (BIT1 _0)))) /\ (((matrix_mul Q) (transp Q)) = (mat (NUMERAL (BIT1 _0)))))))


ITERATE_OP_GEN

|- (!op. ((monoidal op) ==> (!f. (!g. (!s. (((FINITE (((support op) f) s)) /\ (FINITE (((support op) g) s))) ==> ((((iterate op) s) (\x. ((op (f x)) (g x)))) = ((op (((iterate op) s) f)) (((iterate op) s) g)))))))))


Lp_ineqs_proofs.y7_hi

|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!d. ((d IN (dart_of_fan (V , E))) ==> ((real_le ((y7_fan (V , E)) d)) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))))))))))))


Fan_defs.n_fan_pair_ext

|- (!V. (!E. (!x. (((n_fan_pair_ext (V , E)) x) = (((COND (x IN (dart1_of_fan (V , E)))) ((n_fan_pair (V , E)) x)) x)))))


Seq2.list_sum_add

|- (!s. (!f1. (!f2. (((list_sum s) (\x. ((real_add (f1 x)) (f2 x)))) = ((real_add ((list_sum s) f1)) ((list_sum s) f2))))))


Tame_lemmas.tauVEF_alt3

|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> (!v. (!w. (((v , w) IN (dart_of_fan (V , E))) ==> ((LET (\f. (LET_END ((tauVEF (V , (E , f))) = ((real_add ((sum f) (\x. ((real_sub ((real_sub ((real_mul ((real_add (real_of_num (NUMERAL (BIT1 _0)))) ((real_mul ((real_div sol0) pi)) ((real_sub (real_of_num (NUMERAL (BIT1 _0)))) (lmfun (h_dart x)))))) ((azim_dart (V , E)) x))) pi)) sol0)))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((real_add pi) sol0))))))) ((face (hypermap_of_fan (V , E))) (v , w))))))))))


REAL_SUP_LE

|- (!b. (((~ (s = EMPTY)) /\ (!x. ((x IN s) ==> ((real_le x) b)))) ==> ((real_le (sup s)) b)))


DOT_SYM

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


REAL_ABS_ABS

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


CONNECTED_COMPONENT_EQ_EMPTY

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


REAL_BOUNDED

|- ((real_bounded s) = (bounded ((IMAGE lift) s)))


SPAN_CARD_GE_DIM

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


REAL_DIFFERENTIABLE_AT_EXP

|- (!x. ((real_differentiable exp) (atreal x)))


LIM_COMPLEX_LMUL

|- (!f. (!l. (!c. ((((--> f) l) net) ==> (((--> (\x. ((complex_mul c) (f x)))) ((complex_mul c) l)) net)))))


REAL_LT_INF_FINITE

|- (!s. (!a. (((FINITE s) /\ (~ (s = EMPTY))) ==> (((real_lt a) (inf s)) = (!x. ((x IN s) ==> ((real_lt a) x)))))))


Fan.properties_of_set_of_edge_fan

|- (!x. (!V. (!E. (!v. (!u. ((FAN (x , (V , E))) ==> (((v INSERT (u INSERT EMPTY)) IN E) = (u IN (((set_of_edge v) V) E)))))))))


LIFT_CMUL

|- (!x. (!c. ((lift ((real_mul c) x)) = ((% c) (lift x)))))


Hypermap.NON_ZERO

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


ARITH_LE

|- ((!m. (!n. (((NUMERAL m) <= (NUMERAL n)) = (m <= n)))) /\ (((_0 <= _0) = T) /\ ((!n. (((BIT0 n) <= _0) = (n <= _0))) /\ ((!n. (((BIT1 n) <= _0) = F)) /\ ((!n. ((_0 <= (BIT0 n)) = T)) /\ ((!n. ((_0 <= (BIT1 n)) = T)) /\ ((!m. (!n. (((BIT0 m) <= (BIT0 n)) = (m <= n)))) /\ ((!m. (!n. (((BIT0 m) <= (BIT1 n)) = (m <= n)))) /\ ((!m. (!n. (((BIT1 m) <= (BIT0 n)) = (m < n)))) /\ (!m. (!n. (((BIT1 m) <= (BIT1 n)) = (m <= n)))))))))))))


HOMEOMORPHISM_INJECTIVE_CLOSED_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. (((closed_in ((subtopology euclidean) s)) u) ==> ((closed_in ((subtopology euclidean) t)) ((IMAGE f) u)))))))))


Appendix.scs_k_v39_explicit

|- (!k. (!d. (!a. (!b. (!alpha. (!beta. (!J. (!lo. (!str. ((scs_k_v39 (scs_v39 (k , (d , (a , (alpha , (beta , (b , (J , (lo , (hi , str))))))))))) = k))))))))))


ORTHOGONAL_LINEAR_IMAGE_EQ

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


OUTSIDE_CONNECTED_COMPONENT_LT

|- (!s. ((outside s) = (GSPEC (\GEN%PVAR%2011. (?x. (((SETSPEC GEN%PVAR%2011) (!B. (?y. (((real_lt B) (vector_norm y)) /\ (((connected_component (UNIV DIFF s)) x) y))))) x))))))


TRACE_TRANSP

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


REALLIM_COMPLEX

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


DIFFERENTIABLE_BOUND

|- (!f. (!f'. (!s. (!B. (((convex s) /\ ((!x. ((x IN s) ==> (((has_derivative f) (f' x)) ((within (at x)) s)))) /\ (!x. ((x IN s) ==> ((real_le (onorm (f' x))) B))))) ==> (!x. (!y. (((x IN s) /\ (y IN s)) ==> ((real_le (vector_norm ((vector_sub (f x)) (f y)))) ((real_mul B) (vector_norm ((vector_sub x) y))))))))))))


CLOSED_LIMPT

|- (!s. ((closed s) = (!x. (((limit_point_of x) s) ==> (x IN s)))))


Ssrbool.implybF

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


Functional_equation.domain6

|- (!h. (!f. (!g. ((((domain6 h) f) g) = (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((((((h x1) x2) x3) x4) x5) x6) ==> (((((((f x1) x2) x3) x4) x5) x6) = ((((((g x1) x2) x3) x4) x5) x6)))))))))))))


FINITE_INDEX_INRANGE

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


INTEGRABLE_CONST

|- (!a. (!b. (!c. ((integrable_on (\x. c)) (closed_interval ((CONS (a , b)) NIL))))))


SIN_0

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


REAL_SGN_REAL_SGN

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


CENTRE_IN_CBALL

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


REAL_LEBESGUE_MEASURABLE_INTER

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


DOT_LSUB

|- (!x. (!y. (!z. (((dot ((vector_sub x) y)) z) = ((real_sub ((dot x) z)) ((dot y) z))))))


EXP

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


TAGGED_PARTIAL_DIVISION_SUBSET

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


CARD_EQ_IMP_LE

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


MOD_MOD

|- (!m. (!n. (!p. ((~ ((n * p) = (NUMERAL _0))) ==> (((m MOD (n * p)) MOD n) = (m MOD n))))))


IN_INTERIOR

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


Lp_ineqs_def.sol_fan

|- (!f. (!V. (!E. (((sol_fan (V , E)) f) = (real_abs ((real_add ((sum f) (\x. ((real_sub ((azim_dart (V , E)) x)) pi)))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) pi)))))))


INTERIOR_INTER

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


CONTINUOUS_AT_TRANSLATION

|- (!a. (!z. (!f. (((continuous f) (at ((vector_add a) z))) = ((continuous (\x. (f ((vector_add a) x)))) (at z))))))


Seq.cat

|- (!x. (!t. (!s2. ((((cat NIL) s2) = s2) /\ (((cat ((CONS x) t)) s2) = ((CONS x) ((cat t) s2)))))))


LE_SUC_LT

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


Hypermap_iso.hyp_iso

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


sign

|- (!p. ((sign p) = (((COND (evenperm p)) (real_of_num (NUMERAL (BIT1 _0)))) (real_neg (real_of_num (NUMERAL (BIT1 _0)))))))


CLOSED_IN_UNION_COMPLEMENT_COMPONENTS

|- (!u. (!s. (!c. ((((locally connected) u) /\ (((closed_in ((subtopology euclidean) u)) s) /\ (c SUBSET (components (u DIFF s))))) ==> ((closed_in ((subtopology euclidean) u)) (s UNION (UNIONS c)))))))


Pack1.ineq_lm5_3_step3

|- (!S. (!p. (!r. (!A. ((((real_le (real_of_num (NUMERAL _0))) r) /\ ((packing S) /\ ((saturated S) /\ ((fcc_compatible A) S)))) ==> ((real_le ((real_mul (sqrt (real_of_num (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 _0)))))))))) (real_of_num (CARD (S INTER (ball (p , ((real_add r) (real_of_num (NUMERAL (BIT1 _0))))))))))) ((sum (S INTER (ball (p , ((real_add r) (real_of_num (NUMERAL (BIT1 _0)))))))) (\v. ((real_add (A v)) (measure ((voronoi_open S) v)))))))))))


HAS_REAL_INTEGRAL_LMUL

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


HAS_COMPLEX_DERIVATIVE_POW_AT

|- (!f. (!f'. (!x. (!n. ((((has_complex_derivative f) f') (at x)) ==> (((has_complex_derivative (\x. ((complex_pow (f x)) n))) ((complex_mul (Cx (real_of_num n))) ((complex_mul ((complex_pow (f x)) (n - (NUMERAL (BIT1 _0))))) f'))) (at x)))))))


CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN

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


SUM_GP

|- (!x. (!m. (!n. (((sum (m .. n)) (\i. ((real_pow x) i))) = (((COND (n < m)) (real_of_num (NUMERAL _0))) (((COND (x = (real_of_num (NUMERAL (BIT1 _0))))) (real_of_num ((n + (NUMERAL (BIT1 _0))) - m))) ((real_div ((real_sub ((real_pow x) m)) ((real_pow x) (SUC n)))) ((real_sub (real_of_num (NUMERAL (BIT1 _0)))) x))))))))


REAL_LT_IMP_LE

|- (!x. (!y. (((real_lt x) y) ==> ((real_le x) y))))


CONTENT_1

|- (!a. (!b. (((real_le (drop a)) (drop b)) ==> ((content (closed_interval ((CONS (a , b)) NIL))) = ((real_sub (drop b)) (drop a))))))


TAN_NEG

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


REAL_LTE_TRANS

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


SIMPLE_IMAGE

|- (!f. (!s. ((GSPEC (\GEN%PVAR%11. (?x. (((SETSPEC GEN%PVAR%11) (x IN s)) (f x))))) = ((IMAGE f) s))))


BOUNDED_POS

|- (!s. ((bounded s) = (?b. (((real_lt (real_of_num (NUMERAL _0))) b) /\ (!x. ((x IN s) ==> ((real_le (vector_norm x)) b)))))))


int_abs_th

|- (!x. ((real_of_int (int_abs x)) = (real_abs (real_of_int x))))


Hypermap_iso.orbit_comm

|- (!s. (!g2. (!f. (!g1. ((!x. ((x IN s) ==> ((g1 x) IN s))) ==> ((!x. ((x IN s) ==> ((f (g1 x)) = (g2 (f x))))) ==> (!x. ((x IN s) ==> (((orbit_map g2) (f x)) = ((IMAGE f) ((orbit_map g1) x)))))))))))


ABSOLUTELY_REAL_INTEGRABLE_MIN

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


HD

|- ((HD ((CONS h) t)) = h)


Lp_ineqs_def.y6_fan

|- (!x. ((y6_fan x) = (distance x)))


REAL_DIV_RMUL

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


int_le

|- (!x. (!y. (((int_le x) y) = ((real_le (real_of_int x)) (real_of_int y)))))


Hypermap.FACE_NOT_EMPTY

|- (!H. (!x. ((NUMERAL (BIT1 _0)) <= (CARD ((face H) x)))))


CONVEX_IMP_CONTRACTIBLE

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


MIDPOINT_EQ_ENDPOINT

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


MEASURABLE_MEASURABLE_PREIMAGE_CLOSED

|- (!f. (!s. (!t. ((((measurable_on f) s) /\ ((measurable s) /\ (closed t))) ==> (measurable (GSPEC (\GEN%PVAR%3655. (?x. (((SETSPEC GEN%PVAR%3655) ((x IN s) /\ ((f x) IN t))) x)))))))))


PATH_IMAGE_JOIN

|- (!g1. (!g2. (((pathfinish g1) = (pathstart g2)) ==> ((path_image ((++ g1) g2)) = ((path_image g1) UNION (path_image g2))))))


POWER_SERIES_CONV_IMP_ABSCONV_WEAK

|- (!a. (!k. (!w. (!z. ((((summable k) (\n. ((complex_mul (a n)) ((complex_pow z) n)))) /\ ((real_lt (vector_norm w)) (vector_norm z))) ==> ((summable k) (\n. ((complex_mul (Cx (vector_norm (a n)))) ((complex_pow w) n)))))))))


MATRIX_VECTOR_MUL_LINEAR

|- (!A. (linear (\x. ((matrix_vector_mul A) x))))


HOMOTOPY_EQUIVALENT_INJECTIVE_LINEAR_IMAGE_RIGHT_EQ

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


CONVEX_HULL_TRANSLATION

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


REAL_LEBESGUE_MEASURABLE

|- (!s. ((real_lebesgue_measurable s) = (lebesgue_measurable ((IMAGE lift) s))))


Yxionxl.YXIONXL3

|- (!s. (!i. ((is_scs_v39 s) ==> ((scs_arrow_v39 (s INSERT EMPTY)) (((scs_prop_equ_v39 s) i) INSERT EMPTY)))))


RIGHT_POLAR_DECOMPOSITION_UNIQUE

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


span

|- (!s. ((span s) = ((hull subspace) s)))


INTEGRABLE_CMUL

|- (!f. (!c. (!s. (((integrable_on f) s) ==> ((integrable_on (\x. ((% c) (f x)))) s)))))


ABSOLUTELY_CONTINUOUS_DIFFERENTIABLE_BV_GEN

|- (!f. (!s. (!t. (((is_interval s) /\ ((bounded s) /\ (((continuous_on f) s) /\ (((has_bounded_variation_on f) s) /\ ((COUNTABLE t) /\ (!x. ((x IN (s DIFF t)) ==> ((differentiable f) ((within (at x)) s))))))))) ==> ((absolutely_continuous_on f) s)))))


MEASURABLE_ON_LEBESGUE_MEASURABLE_PREIMAGE_HALFSPACE_COMPONENT_GT

|- (!f. (!s. ((lebesgue_measurable s) ==> (((measurable_on f) s) = (!a. (!k. ((((NUMERAL (BIT1 _0)) <= k) /\ (k <= (dimindex UNIV))) ==> (lebesgue_measurable (GSPEC (\GEN%PVAR%3635. (?x. (((SETSPEC GEN%PVAR%3635) ((x IN s) /\ ((real_gt ((f x) $ k)) a))) x))))))))))))


List_hypermap_iso.fan_imp_finite_set

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


VECTOR_MUL_ASSOC

|- (!a. (!b. (!x. (((% a) ((% b) x)) = ((% ((real_mul a) b)) x)))))


HAS_BOUNDED_VARIATION_ON_ID

|- (!s. ((bounded s) ==> ((has_bounded_variation_on (\x. x)) s)))


Ssrbool.andb_idr

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


PATH_COMPONENT_OF_SUBSET

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


Lp_ineqs_proofs.dart4_yf_eq

|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!d. ((d IN ((darts_k (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) (hypermap_of_fan (V , E)))) ==> (((y1_fan ((f_fan_pair_ext (V , E)) d)) = (y2_fan d)) /\ (((y2_fan ((f_fan_pair_ext (V , E)) d)) = ((y7_fan (V , E)) d)) /\ ((((y3_fan (V , E)) ((f_fan_pair_ext (V , E)) d)) = (y1_fan d)) /\ ((((y5_fan (V , E)) ((f_fan_pair_ext (V , E)) d)) = (y6_fan d)) /\ (((y6_fan ((f_fan_pair_ext (V , E)) d)) = ((y9_fan (V , E)) d)) /\ ((((y7_fan (V , E)) ((f_fan_pair_ext (V , E)) d)) = ((y3_fan (V , E)) d)) /\ ((((y8_fan (V , E)) ((f_fan_pair_ext (V , E)) d)) = ((y5_fan (V , E)) d)) /\ (((y9_fan (V , E)) ((f_fan_pair_ext (V , E)) d)) = ((y8_fan (V , E)) d)))))))))))))))))


REAL_LE_ANTISYM

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


Lp_ineqs_def.yn_fan

|- (!v. ((yn_fan v) = (vector_norm v)))


POLYTOPE_IMP_CONVEX

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


COHOMOTOPICALLY_TRIVIAL_ON_COMPONENTS

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


BILINEAR_BOUNDED_POS

|- (!h. ((bilinear h) ==> (?B. (((real_lt (real_of_num (NUMERAL _0))) B) /\ (!x. (!y. ((real_le (vector_norm ((h x) y))) ((real_mul B) ((real_mul (vector_norm x)) (vector_norm y))))))))))


Functional_equation.proj_x1

|- (((((((proj_x1 x1) x2) x3) x4) x5) x6) = x1)


DIAGONAL_POSITIVE_SEMIDEFINITE

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


azim_def

|- (!w1. (!w2. (!w. (!v. (((((azim v) w) w1) w2) = (((COND ((collinear (v INSERT (w INSERT (w1 INSERT EMPTY)))) \/ (collinear (v INSERT (w INSERT (w2 INSERT EMPTY)))))) (real_of_num (NUMERAL _0))) (@theta. (((real_le (real_of_num (NUMERAL _0))) theta) /\ (((real_lt theta) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) pi)) /\ (?h1. (?h2. (!e1. (!e2. (!e3. (((((orthonormal e1) e2) e3) /\ ((((% (distance (w , v))) e3) = ((vector_sub w) v)) /\ (~ (w = v)))) ==> (?psi. (?r1. (?r2. ((((vector_sub w1) v) = ((vector_add ((% ((real_mul r1) (cos psi))) e1)) ((vector_add ((% ((real_mul r1) (sin psi))) e2)) ((% h1) ((vector_sub w) v))))) /\ ((((vector_sub w2) v) = ((vector_add ((% ((real_mul r2) (cos ((real_add psi) theta)))) e1)) ((vector_add ((% ((real_mul r2) (sin ((real_add psi) theta)))) e2)) ((% h2) ((vector_sub w) v))))) /\ (((real_lt (real_of_num (NUMERAL _0))) r1) /\ ((real_lt (real_of_num (NUMERAL _0))) r2))))))))))))))))))))))


HOMEOMORPHIC_REFL

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


Functional_equation.functional_dih4_x_div_sqrtdelta_posbranch

|- (dih4_x_div_sqrtdelta_posbranch = (rotate4 dih_x_div_sqrtdelta_posbranch))


Nonlinear_lemma.hminus_lt_h0

|- ((real_lt hminus) h0)


INFINITE

|- (!s. ((INFINITE s) = (~ (FINITE s))))


MBOUNDED_SUBSET

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


ENR_TRANSLATION

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


I_THM

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


Seq.leq_size_uniq

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


BOUNDED_BALL

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


Seq.all_pred1P

|- (!x. (!s. ((s = ((nseq (sizel s)) x)) = ((all (pred1 x)) s))))


FORALL_1

|- ((!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (NUMERAL (BIT1 _0)))) ==> (P i))) = (P (NUMERAL (BIT1 _0))))


ALL_MEM

|- (!P. (!l. ((!x. (((MEM x) l) ==> (P x))) = ((ALL P) l))))


OPEN_CLOSED

|- (!s. ((open s) = (closed (UNIV DIFF s))))


PASTECART_IN_PCROSS

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


CLOSURE_UNION

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


EULER

|- (!z. ((cexp z) = ((complex_mul (Cx (exp (Re z)))) ((vector_add (Cx (cos (Im z)))) ((complex_mul ii) (Cx (sin (Im z))))))))


COMPLEX_DIFFERENTIABLE_AT_CASN

|- (!z. ((((Im z) = (real_of_num (NUMERAL _0))) ==> ((real_lt (real_abs (Re z))) (real_of_num (NUMERAL (BIT1 _0))))) ==> ((complex_differentiable casn) (at z))))


HAS_REAL_INTEGRAL_STRADDLE_NULL

|- (!f. (!g. (!s. (((!x. ((x IN s) ==> (((real_le (real_of_num (NUMERAL _0))) (f x)) /\ ((real_le (f x)) (g x))))) /\ (((has_real_integral g) (real_of_num (NUMERAL _0))) s)) ==> (((has_real_integral f) (real_of_num (NUMERAL _0))) s)))))


Conforming.STEP3_REDUCE_FAN

|- (!x. (!V. (!E. (!E1. (!ds. (!f1. (!f2. (!f3. (!v. (!u. (!w. (((FAN (x , (V , E))) /\ ((!v. ((v IN V) ==> ((CARD (((set_of_edge v) V) E)) > (NUMERAL (BIT1 _0))))) /\ ((fan80 (x , (V , E))) /\ ((ds IN (face_set (hypermap1_of_fanx (x , (V , E))))) /\ (((CARD ds) > (NUMERAL (BIT1 (BIT1 _0)))) /\ (((f1 INSERT (f2 INSERT (f3 INSERT EMPTY))) SUBSET ds) /\ ((((((f1_fan x) V) E) f1) = f2) /\ ((((((f1_fan x) V) E) f2) = f3) /\ ((~ (((((f1_fan x) V) E) f3) = f1)) /\ (((pr2 f1) = v) /\ (((pr2 f2) = u) /\ (((pr2 f3) = w) /\ (((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) /\ ((E UNION ((v INSERT (w INSERT EMPTY)) INSERT EMPTY)) = E1))))))))))))))))) ==> (FAN (x , (V , E1)))))))))))))))


ALL

|- ((((ALL P) NIL) = T) /\ (((ALL P) ((CONS h) t)) = ((P h) /\ ((ALL P) t))))


LET_DEF

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


HAUSDIST_EMPTY

|- ((!t. ((hausdist (EMPTY , t)) = (real_of_num (NUMERAL _0)))) /\ (!s. ((hausdist (s , EMPTY)) = (real_of_num (NUMERAL _0)))))


CLOSED_SING

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


int_ge

|- (!x. (!y. (((int_ge x) y) = ((real_ge (real_of_int x)) (real_of_int y)))))


Lp_ineqs_def.azim3_fan

|- (!V. (!E. (!x. (((azim3_fan (V , E)) x) = ((azim_dart (V , E)) ((inverse (f_fan_pair_ext (V , E))) x))))))


Lp_main_estimate.in_e_prime

|- (!E. (!f. (!v. (!w. (((v INSERT (w INSERT EMPTY)) IN ((e_prime E) f)) = (((v INSERT (w INSERT EMPTY)) IN E) /\ (((v , w) IN f) \/ ((w , v) IN f))))))))


REAL_CLOSED

|- (!s. ((real_closed s) = (closed ((IMAGE lift) s))))


CONVEX_EMPTY

|- (convex EMPTY)


CONNECTED_WITH_OUTSIDE

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


SUPERADMISSIBLE_COND

|- (!<<. (!p. (!P. (!s. (!h. (!k. ((((((admissible <<) p) s) P) /\ (((((superadmissible <<) (\f. (\x. (((p f) x) /\ ((P f) x))))) s) h) /\ ((((superadmissible <<) (\f. (\x. (((p f) x) /\ (~ ((P f) x)))))) s) k))) ==> ((((superadmissible <<) p) s) (\f. (\x. (((COND ((P f) x)) ((h f) x)) ((k f) x))))))))))))


PATH_CONNECTED_OPEN_DELETE

|- (!s. (!a. ((((NUMERAL (BIT0 (BIT1 _0))) <= (dimindex UNIV)) /\ ((open s) /\ (connected s))) ==> (path_connected (s DELETE a)))))


CLOSURE_TRANSLATION

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


COMPACT_SPHERE

|- (!a. (!r. (compact (GSPEC (\GEN%PVAR%555. (?y. (((SETSPEC GEN%PVAR%555) ((vector_norm ((vector_sub y) a)) = r)) y)))))))


NPRODUCT_CLAUSES_NUMSEG

|- ((!m. (((nproduct (m .. (NUMERAL _0))) f) = (((COND (m = (NUMERAL _0))) (f (NUMERAL _0))) (NUMERAL (BIT1 _0))))) /\ (!m. (!n. (((nproduct (m .. (SUC n))) f) = (((COND (m <= (SUC n))) (((nproduct (m .. n)) f) * (f (SUC n)))) ((nproduct (m .. n)) f))))))


REAL_MEASURE_UNIQUE

|- (!s. (!m. (((has_real_measure s) m) ==> ((real_measure s) = m))))


LINEAR_COMPOSE_ADD

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


orthogonal

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


REAL_SEGMENT_SEGMENT

|- ((!a. (!b. ((closed_real_segment ((CONS (a , b)) NIL)) = ((IMAGE drop) (closed_segment ((CONS ((lift a) , (lift b))) NIL)))))) /\ (!a. (!b. ((open_real_segment (a , b)) = ((IMAGE drop) (open_segment ((lift a) , (lift b))))))))


COMPLEX_MUL_RINV

|- (!z. ((~ (z = (Cx (real_of_num (NUMERAL _0))))) ==> (((complex_mul z) (complex_inv z)) = (Cx (real_of_num (NUMERAL (BIT1 _0)))))))


PATH_COMPONENT_TRANS

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


FRONTIER_CLOSED

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


Seq.undup_uniq

|- (!s. (uniq (undup s)))


DEPENDENT_IMP_AFFINE_DEPENDENT

|- (!a. (!s. (((dependent (GSPEC (\GEN%PVAR%1511. (?x. (((SETSPEC GEN%PVAR%1511) (x IN s)) ((vector_sub x) a)))))) /\ (~ (a IN s))) ==> (affine_dependent (a INSERT s)))))


BOUNDED_LINEAR_IMAGE_EQ

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


GDELTA_EMPTY

|- (gdelta EMPTY)


EVENTUALLY_WITHIN

|- (!s. (!a. (!p. (((eventually p) ((within (at a)) s)) = (?d. (((real_lt (real_of_num (NUMERAL _0))) d) /\ (!x. (((x IN s) /\ (((real_lt (real_of_num (NUMERAL _0))) (distance (x , a))) /\ ((real_lt (distance (x , a))) d))) ==> (p x)))))))))


HULL_ANTIMONO

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


Sphere.vol2r

|- (!r. (!y. (((vol2r y) r) = ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((real_mul pi) ((real_div ((real_sub ((real_mul r) r)) ((real_pow ((real_div y) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) (NUMERAL (BIT0 (BIT1 _0)))))) (real_of_num (NUMERAL (BIT1 (BIT1 _0))))))))))


Ssrnat.uphalfS

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


Functional_equation.functional_gchi4_x

|- (gchi4_x = ((mul6 (uni (gchi , proj_y4))) dih4_x))


Lp_ineqs_proofs.f_fan_pair_ext_power_inverse

|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!k. (!d. (((d IN (dart_of_fan (V , E))) /\ (k < (CARD ((face (hypermap_of_fan (V , E))) d)))) ==> ((((POWER (f_fan_pair_ext (V , E))) k) d) = (((POWER (inverse (f_fan_pair_ext (V , E)))) ((CARD ((face (hypermap_of_fan (V , E))) d)) - k)) d))))))))


Seq.cats1

|- (!s. (!z. (((cat s) ((CONS z) NIL)) = ((rcons s) z))))


RELATIVE_INTERIOR_INTERIOR

|- (!s. ((((hull affine) s) = UNIV) ==> ((relative_interior s) = (interior s))))


SUB_ADD

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


HAS_BOUNDED_VARIATION_COMPOSE_HOMEOMORPHISM

|- (!f. (!f'. (!g. (!s. (!t. (((is_interval s) /\ (((homeomorphism (s , t)) (f , f')) /\ ((has_bounded_variation_on g) t))) ==> ((has_bounded_variation_on (g o f)) s)))))))


CONNECTED_IMP_PERFECT_AFF_DIM

|- (!s. (!x. (((connected s) /\ ((~ ((aff_dim s) = (int_of_num (NUMERAL _0)))) /\ (x IN s))) ==> ((limit_point_of x) s))))


GDELTA_UNIV

|- (gdelta UNIV)


LENGTH_MAP

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


Hypermap_and_fan.FACE_LAST_POINT

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


Hypermap.LT1_NZ

|- (!n. (((NUMERAL (BIT1 _0)) <= n) = ((NUMERAL _0) < n)))


SETDIST_ZERO

|- (!s. (!t. ((~ ((DISJOINT s) t)) ==> ((setdist (s , t)) = (real_of_num (NUMERAL _0))))))


CONVEX_DISTANCE

|- (!s. (!a. ((convex_on (\x. (distance (a , x)))) s)))


POLYHEDRON_TRANSLATION_EQ

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


IS_HULL

|- (!P. (!s. ((!f. ((!s. ((s IN f) ==> (P s))) ==> (P (INTERS f)))) ==> ((P s) = (?t. (s = ((hull P) t)))))))


Miqmcsn.scs_3T4_prime2

|- (scs_3T4_prime2 = ((((mk_unadorned_v39 (NUMERAL (BIT1 (BIT1 _0)))) ((DECIMAL (NUMERAL (BIT1 (BIT1 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 (BIT0 (BIT1 _0)))))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 _0))))))))))))))))) (((funlist_v39 ((CONS (((NUMERAL _0) , (NUMERAL (BIT1 _0))) , (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) ((CONS (((NUMERAL _0) , (NUMERAL (BIT0 (BIT1 _0)))) , cstab)) NIL))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) (NUMERAL (BIT1 (BIT1 _0))))) (((funlist_v39 ((CONS (((NUMERAL _0) , (NUMERAL (BIT1 _0))) , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0))) NIL)) cstab) (NUMERAL (BIT1 (BIT1 _0))))))


AFFINE_DEPENDENT_EXPLICIT

|- (!p. ((affine_dependent p) = (?s. (?u. ((FINITE s) /\ ((s SUBSET p) /\ ((((sum s) u) = (real_of_num (NUMERAL _0))) /\ ((?v. ((v IN s) /\ (~ ((u v) = (real_of_num (NUMERAL _0)))))) /\ (((vsum s) (\v. ((% (u v)) v))) = (vec (NUMERAL _0)))))))))))


PATH_VECTOR_POLYNOMIAL_FUNCTION

|- (!g. ((vector_polynomial_function g) ==> (path g)))


NORM_CEXP

|- (!z. ((vector_norm (cexp z)) = (exp (Re z))))


ARC_IMP_PATH

|- (!g. ((arc g) ==> (path g)))


Seq.can_inj

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


REAL_POW_NZ

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


Hypermap_iso.bij_ext

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


SUM_DIFF

|- (!f. (!s. (!t. (((FINITE s) /\ (t SUBSET s)) ==> (((sum (s DIFF t)) f) = ((real_sub ((sum s) f)) ((sum t) f)))))))


REAL_SUB_RZERO

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


TC_MONO

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


Hypermap_iso.iso_number_of_nodes

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


HAS_SIZE

|- (!s. (!n. ((s HAS_SIZE n) = ((FINITE s) /\ ((CARD s) = n)))))


SPHERE_UNION_BALL

|- (!a. (!r. (((sphere (a , r)) UNION (ball (a , r))) = (cball (a , r)))))


BOHL

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


REFLECT_REAL_INTERVAL

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


Seq.count_predT

|- (!s. (((count predT) s) = (sizel s)))


LINEAR_CNJ

|- (linear cnj)


INTERIOR_SPHERE

|- (!a. (!r. ((interior (sphere (a , r))) = EMPTY)))


EXISTS_PAIR_THM

|- (!P. ((?p. (P p)) = (?p1. (?p2. (P (p1 , p2))))))


GEQ_DEF

|- (!a. (!b. (((GEQ a) b) = (a = b))))


CONIC_IMP_CONTRACTIBLE

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


RE_CLOG_POS_LE

|- (!z. ((~ (z = (Cx (real_of_num (NUMERAL _0))))) ==> (((real_le (real_abs (Im (clog z)))) ((real_div pi) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) = ((real_le (real_of_num (NUMERAL _0))) (Re z)))))


ABSOLUTELY_SETCONTINUOUS_ON_NEG

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


EXP_2

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


EXP_MONO_LE

|- (!x. (!y. (!n. (((x EXP n) <= (y EXP n)) = ((x <= y) \/ (n = (NUMERAL _0)))))))


FORALL_DOT_EQ_0

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


PASTECART_SUB

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


REAL_MUL_RID

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


epigraph

|- (!s. (!f. (((epigraph s) f) = (GSPEC (\GEN%PVAR%1703. (?xy. (((SETSPEC GEN%PVAR%1703) (((fstcart xy) IN s) /\ ((real_le (f (fstcart xy))) (drop (sndcart xy))))) xy)))))))


real_differentiable

|- (!f. (!net. (((real_differentiable f) net) = (?f'. (((has_real_derivative f) f') net)))))


INT_NOT_LE

|- (!x. (!y. ((~ ((int_le x) y)) = ((int_lt y) x))))


HAS_FRECHET_DERIVATIVE_UNIQUE_AT

|- (!f. (!f'. (!x. ((((has_derivative f) f') (at x)) ==> (((frechet_derivative f) (at x)) = f')))))


LT_SUC

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


ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE

|- (!f. (!s. (((absolutely_integrable_on f) s) ==> ((integrable_on f) s))))


PATHFINISH_LINEAR_IMAGE

|- (!f. (!g. ((linear f) ==> ((pathfinish (f o g)) = (f (pathfinish g))))))


Hypermap_iso.image_ext_eq

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


HAS_COMPLEX_DERIVATIVE_CATN

|- (!z. ((((Re z) = (real_of_num (NUMERAL _0))) ==> ((real_lt (real_abs (Im z))) (real_of_num (NUMERAL (BIT1 _0))))) ==> (((has_complex_derivative catn) (complex_inv ((vector_add (Cx (real_of_num (NUMERAL (BIT1 _0))))) ((complex_pow z) (NUMERAL (BIT0 (BIT1 _0))))))) (at z))))


NEGLIGIBLE_LINEAR_IMAGE_EQ

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


SIN_CIRCLE

|- (!x. (((real_add ((real_pow (sin x)) (NUMERAL (BIT0 (BIT1 _0))))) ((real_pow (cos x)) (NUMERAL (BIT0 (BIT1 _0))))) = (real_of_num (NUMERAL (BIT1 _0)))))


BIT1

|- (!n. ((BIT1 n) = (SUC (n + n))))


Lp_ineqs_proofs.face4_eq

|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!v. (!w. ((LET (\w'. (LET_END ((LET (\u. (LET_END (((v , w) IN ((darts_k (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) (hypermap_of_fan (V , E)))) ==> (((((f_fan_pair_ext (V , E)) (v , w)) = (w , u)) /\ (((((POWER (f_fan_pair_ext (V , E))) (NUMERAL (BIT0 (BIT1 _0)))) (v , w)) = (u , w')) /\ (((((POWER (f_fan_pair_ext (V , E))) (NUMERAL (BIT1 (BIT1 _0)))) (v , w)) = (w' , v)) /\ ((((f_fan_pair_ext (V , E)) (w , u)) = (u , w')) /\ ((((f_fan_pair_ext (V , E)) (u , w')) = (w' , v)) /\ (((f_fan_pair_ext (V , E)) (w' , v)) = (v , w))))))) /\ ((((v , w) IN (dart1_of_fan (V , E))) /\ (((w , u) IN (dart1_of_fan (V , E))) /\ (((u , w') IN (dart1_of_fan (V , E))) /\ ((w' , v) IN (dart1_of_fan (V , E)))))) /\ (((face (hypermap_of_fan (V , E))) (v , w)) = ((v , w) INSERT ((w , u) INSERT ((u , w') INSERT ((w' , v) INSERT EMPTY))))))))))) (((((sigma_fan (vec (NUMERAL _0))) V) E) w') v))))) (((((sigma_fan (vec (NUMERAL _0))) V) E) v) w)))))))


rational

|- (!x. ((rational x) = (?m. (?n. ((integer m) /\ ((integer n) /\ ((~ (n = (real_of_num (NUMERAL _0)))) /\ (x = ((real_div m) n)))))))))


complex_add

|- (((vector_add w) z) = (complex (((real_add (Re w)) (Re z)) , ((real_add (Im w)) (Im z)))))


CONNECTED_LIMIT_POINTS_EQ_CLOSURE

|- (!s. (((connected s) /\ (~ (?a. (s = (a INSERT EMPTY))))) ==> ((GSPEC (\GEN%PVAR%584. (?x. (((SETSPEC GEN%PVAR%584) ((limit_point_of x) s)) x)))) = (closure s))))


SERIES_ADD

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


Fan.power_map_points

|- (((((((((power_map_points f) x) V) E) v) w) (NUMERAL _0)) = w) /\ ((((((((power_map_points f) x) V) E) v) w) (SUC n)) = (((((f x) V) E) v) (((((((power_map_points f) x) V) E) v) w) n))))


TOPSPACE_EUCLIDEAN_SUBTOPOLOGY

|- (!s. ((topspace ((subtopology euclidean) s)) = s))


Nonlinear_lemma.sqrtxx

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


FINITE_IMP_COMPACT

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


REAL_POW_NEG

|- (!x. (!n. (((real_pow (real_neg x)) n) = (((COND (EVEN n)) ((real_pow x) n)) (real_neg ((real_pow x) n))))))


EQ_MULT_RCANCEL

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


FINITE_DIFF

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


Fnjlbxs.vector_of_list

|- (!list. (!d. (((vector_of_list d) list) = (lambdai. (((COND (((i - (NUMERAL (BIT1 _0))) DIV d) < (LENGTH list))) (((EL ((i - (NUMERAL (BIT1 _0))) DIV d)) list) $ (((i - (NUMERAL (BIT1 _0))) MOD d) + (NUMERAL (BIT1 _0))))) (real_of_num (NUMERAL _0)))))))


Hypermap_and_fan.NOT_IN_DART1_OF_FAN

|- (!V. (!E. (!v. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (~ ((v , v) IN (dart1_of_fan (V , E))))))))


IN_REAL_INTERVAL

|- (!a. (!b. (!x. (((x IN (closed_real_interval ((CONS (a , b)) NIL))) = (((real_le a) x) /\ ((real_le x) b))) /\ ((x IN (open_real_interval (a , b))) = (((real_lt a) x) /\ ((real_lt x) b)))))))


SPAN_OPEN

|- (!s. (((open s) /\ (~ (s = EMPTY))) ==> ((span s) = UNIV)))


DIVISION

|- (!m. (!n. ((~ (n = (NUMERAL _0))) ==> ((m = (((m DIV n) * n) + (m MOD n))) /\ ((m MOD n) < n)))))


Seq.mask

|- ((((mask NIL) s') = NIL) /\ ((((mask m') NIL) = NIL) /\ (((mask ((CONS b) m')) ((CONS x) s')) = (((COND b) ((CONS x) ((mask m') s'))) ((mask m') s')))))


RELATIVE_FRONTIER_TRANSLATION

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


CONVEX_IMP_LOCALLY_CONNECTED

|- (!s. ((convex s) ==> ((locally connected) s)))


vector_sub

|- (!x. (!y. (((vector_sub x) y) = (lambdai. ((real_sub (x $ i)) (y $ i))))))


PAIRWISE_ORTHOGONAL_INDEPENDENT

|- (!s. ((((pairwise orthogonal) s) /\ (~ ((vec (NUMERAL _0)) IN s))) ==> (independent s)))


INTERIOR_CLOSURE_INTER_OPEN

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


Lvducxu.FIRST_AAUHTVE2

|- ((FAN (x , (V , E))) ==> ((FINITE ((darts_of_hyp E) V)) /\ (((permutes (ee_of_hyp (x , (V , E)))) ((darts_of_hyp E) V)) /\ (((permutes (nn_of_hyp (x , (V , E)))) ((darts_of_hyp E) V)) /\ (((permutes (ff_of_hyp (x , (V , E)))) ((darts_of_hyp E) V)) /\ ((((ee_of_hyp (x , (V , E))) o ((nn_of_hyp (x , (V , E))) o (ff_of_hyp (x , (V , E))))) = I) /\ (((ee_of_hyp (x , (V , E))) o (ee_of_hyp (x , (V , E)))) = I)))))))


CONTINUOUS_COMPLEX_DIV

|- (!net. (!f. (!g. ((((continuous f) net) /\ (((continuous g) net) /\ (~ ((g (netlimit net)) = (Cx (real_of_num (NUMERAL _0))))))) ==> ((continuous (\x. ((complex_div (f x)) (g x)))) net)))))


SEPARATION_NORMAL

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


FORALL_IN_GSPEC

|- ((!P. (!f. ((!z. ((z IN (GSPEC (\GEN%PVAR%21. (?x. (((SETSPEC GEN%PVAR%21) (P x)) (f x)))))) ==> (Q z))) = (!x. ((P x) ==> (Q (f x))))))) /\ ((!P. (!f. ((!z. ((z IN (GSPEC (\GEN%PVAR%22. (?x. (?y. (((SETSPEC GEN%PVAR%22) ((P x) y)) ((f x) y))))))) ==> (Q z))) = (!x. (!y. (((P x) y) ==> (Q ((f x) y)))))))) /\ ((!P. (!f. ((!z. ((z IN (GSPEC (\GEN%PVAR%23. (?w. (?x. (?y. (((SETSPEC GEN%PVAR%23) (((P w) x) y)) (((f w) x) y)))))))) ==> (Q z))) = (!w. (!x. (!y. ((((P w) x) y) ==> (Q (((f w) x) y))))))))) /\ (!P. (!f. ((!z. ((z IN (GSPEC (\GEN%PVAR%24. (?v. (?w. (?x. (?y. (((SETSPEC GEN%PVAR%24) ((((P v) w) x) y)) ((((f v) w) x) y))))))))) ==> (Q z))) = (!v. (!w. (!x. (!y. (((((P v) w) x) y) ==> (Q ((((f v) w) x) y)))))))))))))


CX_INV

|- (!x. ((Cx (real_inv x)) = (complex_inv (Cx x))))


SIMPLY_CONNECTED_INJECTIVE_LINEAR_IMAGE

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


DIAMETER_LE

|- (!s. ((((~ (s = EMPTY)) \/ ((real_le (real_of_num (NUMERAL _0))) d)) /\ (!x. (!y. (((x IN s) /\ (y IN s)) ==> ((real_le (vector_norm ((vector_sub x) y))) d))))) ==> ((real_le (diameter s)) d)))


SUM_LT

|- (!f. (!g. (!s. (((FINITE s) /\ ((!x. ((x IN s) ==> ((real_le (f x)) (g x)))) /\ (?x. ((x IN s) /\ ((real_lt (f x)) (g x)))))) ==> ((real_lt ((sum s) f)) ((sum s) g))))))


HELLY_CLOSED

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


HAS_DERIVATIVE_ZERO_UNIQUE_STRONG_INTERVAL

|- (!f. (!a. (!b. (!k. (!y. (((COUNTABLE k) /\ (((continuous_on f) (closed_interval ((CONS (a , b)) NIL))) /\ (((f a) = y) /\ (!x. ((x IN ((closed_interval ((CONS (a , b)) NIL)) DIFF k)) ==> (((has_derivative f) (\h. (vec (NUMERAL _0)))) ((within (at x)) (closed_interval ((CONS (a , b)) NIL))))))))) ==> (!x. ((x IN (closed_interval ((CONS (a , b)) NIL))) ==> ((f x) = y)))))))))


List_hypermap.darts_of_list

|- (!L. ((darts_of_list L) = (set_of_list (list_of_darts L))))


Seq.size_nil

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


INTERS_IMAGE

|- (!f. (!s. ((INTERS ((IMAGE f) s)) = (GSPEC (\GEN%PVAR%30. (?y. (((SETSPEC GEN%PVAR%30) (!x. ((x IN s) ==> (y IN (f x))))) y)))))))


mbounded

|- (!s. (!m. (((mbounded m) s) = (?c. (?b. (s SUBSET ((mcball m) (c , b))))))))


SEGMENT_TRANSLATION

|- ((!c. (!a. (!b. ((closed_segment ((CONS (((vector_add c) a) , ((vector_add c) b))) NIL)) = ((IMAGE (\x. ((vector_add c) x))) (closed_segment ((CONS (a , b)) NIL))))))) /\ (!c. (!a. (!b. ((open_segment (((vector_add c) a) , ((vector_add c) b))) = ((IMAGE (\x. ((vector_add c) x))) (open_segment (a , b))))))))


HAS_REAL_DERIVATIVE_FROM_COMPLEX_AT

|- (!f. (!f'. (!x. (((((has_complex_derivative f) f') (at (Cx x))) /\ (!z. ((real z) ==> (real (f z))))) ==> (((has_real_derivative (Re o (f o Cx))) (Re f')) (atreal x))))))


CLOSURE_EQ

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


I_DEF

|- (I = (\x. x))


TAGGED_DIVISION_UNIONS

|- (!iset. (!pfn. (((FINITE iset) /\ ((!i. ((i IN iset) ==> ((tagged_division_of (pfn i)) i))) /\ (!i1. (!i2. (((i1 IN iset) /\ ((i2 IN iset) /\ (~ (i1 = i2)))) ==> (((interior i1) INTER (interior i2)) = EMPTY)))))) ==> ((tagged_division_of (UNIONS ((IMAGE pfn) iset))) (UNIONS iset)))))


RTC

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


NSUM_UNION_NONZERO

|- (!f. (!s. (!t. (((FINITE s) /\ ((FINITE t) /\ (!x. ((x IN (s INTER t)) ==> ((f x) = (NUMERAL _0)))))) ==> (((nsum (s UNION t)) f) = (((nsum s) f) + ((nsum t) f)))))))


IMAGE_SNDCART_PCROSS

|- (!s. (!t. (((IMAGE sndcart) (s PCROSS t)) = (((COND (s = EMPTY)) EMPTY) t))))


retraction

|- (!s. (!t. (!r. (((retraction (s , t)) r) = ((t SUBSET s) /\ (((continuous_on r) s) /\ ((((IMAGE r) s) SUBSET t) /\ (!x. ((x IN t) ==> ((r x) = x))))))))))


ABSOLUTELY_INTEGRABLE_NORM

|- (!f. (!s. (((absolutely_integrable_on f) s) ==> ((absolutely_integrable_on (\x. (lift (vector_norm (f x))))) s))))


HAS_MEASURE_LINEAR_IMAGE_SAME

|- (!f. (!s. (((linear f) /\ ((measurable s) /\ ((real_abs (det (matrix f))) = (real_of_num (NUMERAL (BIT1 _0)))))) ==> ((has_measure ((IMAGE f) s)) (measure s)))))


ITERATE_DIFF

|- (!op. ((monoidal op) ==> (!f. (!s. (!t. (((FINITE s) /\ (t SUBSET s)) ==> (((op (((iterate op) (s DIFF t)) f)) (((iterate op) t) f)) = (((iterate op) s) f))))))))


REAL_DIV_1

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


EVENTUALLY_AT

|- (!a. (!p. (((eventually p) (at a)) = (?d. (((real_lt (real_of_num (NUMERAL _0))) d) /\ (!x. ((((real_lt (real_of_num (NUMERAL _0))) (distance (x , a))) /\ ((real_lt (distance (x , a))) d)) ==> (p x))))))))


REAL_LET_TRANS

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


CARD_LE_ANTISYM

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


VECTOR_SUB_COMPONENT

|- (!x. (!y. (!i. ((((vector_sub x) y) $ i) = ((real_sub (x $ i)) (y $ i))))))


Hypermap.EQ_SUC

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


Packing3.VORONOI_LIST_SING

|- (!V. (!u. (((voronoi_list V) ((CONS u) NIL)) = ((voronoi_closed V) u))))


LIM_COMPONENTWISE

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


CONVEX_SUMS

|- (!s. (!t. (((convex s) /\ (convex t)) ==> (convex (GSPEC (\GEN%PVAR%1471. (?x. (?y. (((SETSPEC GEN%PVAR%1471) ((x IN s) /\ (y IN t))) ((vector_add x) y))))))))))


CONTENT_PASTECART

|- (!a. (!b. (!c. (!d. ((content (closed_interval ((CONS (((pastecart a) c) , ((pastecart b) d))) NIL))) = ((real_mul (content (closed_interval ((CONS (a , b)) NIL)))) (content (closed_interval ((CONS (c , d)) NIL)))))))))


NSUM_CLAUSES

|- ((!f. (((nsum EMPTY) f) = (NUMERAL _0))) /\ (!x. (!f. (!s. ((FINITE s) ==> (((nsum (x INSERT s)) f) = (((COND (x IN s)) ((nsum s) f)) ((f x) + ((nsum s) f)))))))))


REAL_CONTINUOUS_AT_SQRT

|- (!x. (((real_lt (real_of_num (NUMERAL _0))) x) ==> ((real_continuous sqrt) (atreal x))))


CROSS_LADD

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


CEXP_NEG

|- (!z. ((cexp (vector_neg z)) = (complex_inv (cexp z))))


Lp_ineqs_proofs.dart3_yf_eq

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


LIM_TRANSFORM_WITHINREAL_SET_IMP

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


INTERS_0

|- ((INTERS EMPTY) = UNIV)


INTEGER_CLOSED

|- ((!n. (integer (real_of_num n))) /\ ((!x. (!y. (((integer x) /\ (integer y)) ==> (integer ((real_add x) y))))) /\ ((!x. (!y. (((integer x) /\ (integer y)) ==> (integer ((real_sub x) y))))) /\ ((!x. (!y. (((integer x) /\ (integer y)) ==> (integer ((real_mul x) y))))) /\ ((!x. (!r. ((integer x) ==> (integer ((real_pow x) r))))) /\ ((!x. ((integer x) ==> (integer (real_neg x)))) /\ ((!x. ((integer x) ==> (integer (real_abs x)))) /\ ((!x. (!y. (((integer x) /\ (integer y)) ==> (integer ((real_max x) y))))) /\ (!x. (!y. (((integer x) /\ (integer y)) ==> (integer ((real_min x) y)))))))))))))


Conforming.add_edge_imp_card_set_edge_ge1_fan

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


OPEN_IN_DIFF

|- (!top. (!s. (!t. ((((open_in top) s) /\ ((closed_in top) t)) ==> ((open_in top) (s DIFF t))))))


Sphere.bump

|- (!h. ((bump h) = ((real_mul ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))) ((real_sub (real_of_num (NUMERAL (BIT1 _0)))) ((real_div ((real_pow ((real_sub h) h0)) (NUMERAL (BIT0 (BIT1 _0))))) ((real_pow ((real_sub hplus) h0)) (NUMERAL (BIT0 (BIT1 _0)))))))))


FACE_OF_CONIC

|- (!s. (!f. (((conic s) /\ ((face_of f) s)) ==> (conic f))))


KIRSZBRAUN

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


Hypermap.res

|- (!f. (!s. (!x. ((((res f) s) x) = (((COND (x IN s)) (f x)) x)))))


Lp_ineqs_def.y9_fan

|- (!V. (!E. (!x. (((y9_fan (V , E)) x) = (distance ((f_fan_pair_ext (V , E)) x))))))


REAL_EXP_NEG

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


Sphere.rotate2

|- (!f. (!x2. (!x3. (!x1. (!x5. (!x6. (!x4. ((((((((rotate2 f) x1) x2) x3) x4) x5) x6) = ((((((f x2) x3) x1) x5) x6) x4)))))))))


Trigonometry2.AFF2

|- (!u. (!v. ((aff (u INSERT (v INSERT EMPTY))) = (GSPEC (\GEN%PVAR%5314. (?x. (((SETSPEC GEN%PVAR%5314) (?t. (x = ((vector_add ((% t) u)) ((% ((real_sub (real_of_num (NUMERAL (BIT1 _0)))) t)) v))))) x)))))))


AFFINE

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


PATHSTART_IN_PATH_IMAGE

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


MEM_EL

|- (!l. (!n. ((n < (LENGTH l)) ==> ((MEM ((EL n) l)) l))))


LAMBDA_UNIQUE

|- (!f. (!g. ((!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV))) ==> ((f $ i) = (g i)))) = ((lambda g) = f))))


INJ

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


Appendix.MMs_v39

|- (!vv. (!s. (((MMs_v39 s) vv) = (((BBprime2_v39 s) vv) /\ ((!i. (((scs_str_v39 s) i) ==> (((((azim (vec (NUMERAL _0))) (vv i)) (vv (SUC i))) (vv (i + ((scs_k_v39 s) - (NUMERAL (BIT1 _0)))))) = pi))) /\ ((!i. (((scs_lo_v39 s) i) ==> ((vector_norm (vv i)) = (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) /\ ((!i. (((scs_hi_v39 s) i) ==> ((vector_norm (vv i)) = ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) /\ ((!i. (!j. ((real_le (((scs_am_v39 s) i) j)) (distance ((vv i) , (vv j)))))) /\ (!i. (!j. ((real_le (distance ((vv i) , (vv j)))) (((scs_bm_v39 s) i) j))))))))))))