COMPLEX_DIFFERENTIABLE_DIV_AT |- (!f. (!g. (!z. ((((complex_differentiable f) (at z)) /\ (((complex_differentiable g) (at z)) /\ (~ ((g z) = (Cx (real_of_num (NUMERAL _0))))))) ==> ((complex_differentiable (\z. ((complex_div (f z)) (g z)))) (at z))))))


NSUM_PERMUTE_NUMSEG |- (!f. (!p. (!m. (!n. (((permutes p) (m .. n)) ==> (((nsum (m .. n)) f) = ((nsum (m .. n)) (f o p))))))))


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


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


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


Ckqowsa_3_points.pos_vector_angle_bounds |- (!v. (!u. (((real_gt ((dot v) u)) (real_of_num (NUMERAL _0))) ==> (((real_le (real_of_num (NUMERAL _0))) ((vector_angle v) u)) /\ ((real_le ((vector_angle v) u)) ((real_div pi) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))))))


LIPSCHITZ_EXTENSION_EXISTS |- (!f. (!s. ((?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. ((?B. (!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)))))))))


Pack2.sum_measure_voronoi_closed_le_ball |- (!S. (!p. (!r. ((((real_le (real_of_num (NUMERAL _0))) r) /\ ((packing S) /\ (saturated S))) ==> ((real_le ((sum (S INTER (ball (p , ((real_add r) (real_of_num (NUMERAL (BIT1 _0)))))))) (\v. (measure ((voronoi_closed S) v))))) (measure (ball (p , ((real_add r) (real_of_num (NUMERAL (BIT1 (BIT1 _0)))))))))))))


Lp_ineqs_proofs.dart4_azim2_eq_sum |- (!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)))) ==> (((azim2_fan (V , E)) d) = ((real_add ((((((dih_y (y2_fan d)) ((y3_fan (V , E)) d)) (y1_fan d)) ((y5_fan (V , E)) d)) (y6_fan d)) ((y4'_fan (V , E)) d))) ((((((dih_y (y2_fan d)) ((y7_fan (V , E)) d)) ((y3_fan (V , E)) d)) ((y8_fan (V , E)) d)) ((y4'_fan (V , E)) d)) ((y9_fan (V , E)) d))))))))))))


Float.INT_POW_ODD_NEG1 |- (!x. (!n. (((int_pow (int_neg (int_of_num x))) (NUMERAL (BIT1 n))) = (int_neg ((int_pow (int_of_num x)) (NUMERAL (BIT1 n)))))))


PATH_CONNECTED_SUMS |- (!s. (!t. (((path_connected s) /\ (path_connected t)) ==> (path_connected (GSPEC (\GEN%PVAR%1816. (?x. (?y. (((SETSPEC GEN%PVAR%1816) ((x IN s) /\ (y IN t))) ((vector_add x) y))))))))))


FUNDAMENTAL_GROUP_SIMPLY_CONNECTED |- (!s. (!a. (((simply_connected s) /\ (a IN s)) ==> ((fundamental_group (s , a)) = (((homotopic_paths s) (linepath (a , a))) INSERT EMPTY)))))


HAS_REAL_MEASURE_SCALING_EQ |- (!s. (!m. (!c. ((~ (c = (real_of_num (NUMERAL _0)))) ==> (((has_real_measure ((IMAGE (\x. ((real_mul c) x))) s)) ((real_mul (real_abs c)) m)) = ((has_real_measure s) m))))))


PATH_CONNECTED_COMPLEMENT_CARD_LT |- (!s. ((((NUMERAL (BIT0 (BIT1 _0))) <= (dimindex UNIV)) /\ (s <_c UNIV)) ==> (path_connected (UNIV DIFF s))))


LIM_NULL_COMPLEX_LMUL |- (!net. (!f. (!c. ((((--> f) (Cx (real_of_num (NUMERAL _0)))) net) ==> (((--> (\x. ((complex_mul c) (f x)))) (Cx (real_of_num (NUMERAL _0)))) net)))))


REAL_NEGLIGIBLE_DIFF |- (!s. (!t. ((real_negligible s) ==> (real_negligible (s DIFF t)))))


CLOSED_CONNECTED_PREIMAGES_IMP_CONTINUOUS_ON |- (!f. (!s. (!t. (((((IMAGE f) s) SUBSET t) /\ ((compact t) /\ ((!y. (connected (GSPEC (\GEN%PVAR%1942. (?x. (((SETSPEC GEN%PVAR%1942) ((x IN s) /\ ((f x) = y))) x)))))) /\ ((!c. (((closed_in ((subtopology euclidean) s)) c) ==> ((closed_in ((subtopology euclidean) t)) ((IMAGE f) c)))) /\ (!c. (((c SUBSET s) /\ (connected c)) ==> (connected ((IMAGE f) c)))))))) ==> ((continuous_on f) s)))))


NOT_ON_PATH_CBALL |- (!g. (!z. (((path g) /\ (~ (z IN (path_image g)))) ==> (?e. (((real_lt (real_of_num (NUMERAL _0))) e) /\ (((cball (z , e)) INTER (path_image g)) = EMPTY))))))


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


ARG_ROTATE2D_UNIQUE_2PI |- (!s. (!t. (!z. (((~ (z = (Cx (real_of_num (NUMERAL _0))))) /\ (((real_le (real_of_num (NUMERAL _0))) s) /\ (((real_lt s) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) pi)) /\ (((real_le (real_of_num (NUMERAL _0))) t) /\ (((real_lt t) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) pi)) /\ ((Arg ((rotate2d s) z)) = (Arg ((rotate2d t) z)))))))) ==> (s = t)))))


INSIDE_CONNECTED_COMPONENT_LE |- (!s. ((((NUMERAL (BIT0 (BIT1 _0))) <= (dimindex UNIV)) /\ (bounded s)) ==> ((inside s) = (GSPEC (\GEN%PVAR%2016. (?x. (((SETSPEC GEN%PVAR%2016) ((~ (x IN s)) /\ (!B. (?y. (((real_le B) (vector_norm y)) /\ (~ (((connected_component (UNIV DIFF s)) x) y))))))) x)))))))


CLOG_EQ |- (!w. (!z. (((~ (w = (Cx (real_of_num (NUMERAL _0))))) /\ (~ (z = (Cx (real_of_num (NUMERAL _0)))))) ==> (((clog w) = (clog z)) = (w = z)))))


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


REAL_SGN_SIGN |- (!p. ((real_sgn (sign p)) = (sign p)))


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


NORM_EQ_0_DOT |- (!x. (((vector_norm x) = (real_of_num (NUMERAL _0))) = (((dot x) x) = (real_of_num (NUMERAL _0)))))


COMPLEX_SUB_NEG2 |- (!x. (!y. (((vector_sub (vector_neg x)) (vector_neg y)) = ((vector_sub y) x))))


BILINEAR_CROSS |- (bilinear cross)


RATIONAL_ALT |- (!x. ((rational x) = (?p. (?q. ((~ (q = (NUMERAL _0))) /\ ((real_abs x) = ((real_div (real_of_num p)) (real_of_num q))))))))


Merge_ineq.nonf_mudLs_126_x |- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (!a. (!b. (!c. (!d. (!e. ((((((((((((mudLs_126_x a) b) c) d) e) x1) x2) x3) x4) x5) x6) = ((real_mul ((((((mu6_x ((real_mul c) c)) x1) x2) (real_of_num (NUMERAL _0))) (real_of_num (NUMERAL _0))) (real_of_num (NUMERAL _0)))) ((real_add ((real_mul ((real_div (real_of_num (NUMERAL (BIT1 _0)))) ((real_add a) b))) ((real_sub (((((((((delta_126_x ((real_mul c) c)) ((real_mul d) d)) ((real_mul e) e)) x1) x2) x3) x4) x5) x6)) ((real_mul a) a)))) a))))))))))))))


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


REAL_CONVEX_ON_DERIVATIVE_INCREASING_IMP |- (!f. (!f'x. (!f'y. (!s. (!x. (!y. ((((real_convex_on f) s) /\ (((closed_real_interval ((CONS (x , y)) NIL)) SUBSET s) /\ ((((has_real_derivative f) f'x) ((within (atreal x)) s)) /\ ((((has_real_derivative f) f'y) ((within (atreal y)) s)) /\ ((real_lt x) y))))) ==> ((real_le f'x) f'y))))))))


CONTINUOUS_ON_DIST_CLOSEST_POINT |- (!s. (!t. (((closed s) /\ (~ (s = EMPTY))) ==> ((continuous_on (\x. (lift (distance (x , ((closest_point s) x)))))) t))))


Lp_ineqs_proofs.dart3_azim_fan_eq_dih_y |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!d. ((d IN ((darts_k (NUMERAL (BIT1 (BIT1 _0)))) (hypermap_of_fan (V , E)))) ==> (((azim_dart (V , E)) d) = ((((((dih_y (y1_fan d)) (y2_fan d)) ((y3_fan (V , E)) d)) ((y4_fan (V , E)) d)) ((y5_fan (V , E)) d)) (y6_fan d)))))))))))


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


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


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


SETDIST_RELATIVE_INTERIOR |- ((!s. (!t. ((convex s) ==> ((setdist ((relative_interior s) , t)) = (setdist (s , t)))))) /\ (!s. (!t. ((convex t) ==> ((setdist (s , (relative_interior t))) = (setdist (s , t)))))))


ABSOLUTELY_INTEGRABLE_CONVOLUTION_L2 |- (!bop. (!f. (!g. (!x. (((bilinear bop) /\ (((measurable_on f) UNIV) /\ (((measurable_on g) UNIV) /\ (((absolutely_integrable_on (\x. (lift ((real_pow (vector_norm (f x))) (NUMERAL (BIT0 (BIT1 _0))))))) UNIV) /\ ((absolutely_integrable_on (\x. (lift ((real_pow (vector_norm (g x))) (NUMERAL (BIT0 (BIT1 _0))))))) UNIV))))) ==> ((absolutely_integrable_on (\y. ((bop (f ((vector_sub x) y))) (g y)))) UNIV))))))


CCOS_CSIN_CSQRT |- (!z. ((((real_lt (real_of_num (NUMERAL _0))) (cos (Re z))) \/ (((cos (Re z)) = (real_of_num (NUMERAL _0))) /\ ((real_le ((real_mul (Im z)) (sin (Re z)))) (real_of_num (NUMERAL _0))))) ==> ((ccos z) = (csqrt ((vector_sub (Cx (real_of_num (NUMERAL (BIT1 _0))))) ((complex_pow (csin z)) (NUMERAL (BIT0 (BIT1 _0)))))))))


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


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


Hijqaha.J_SCS_4M5 |- ((((scs_J_v39 ((scs_prop_equ_v39 scs_4M5') i)) i1) j) = F)


ANALYTIC_ON_SUB |- (!f. (!g. (!s. ((((analytic_on f) s) /\ ((analytic_on g) s)) ==> ((analytic_on (\z. ((vector_sub (f z)) (g z)))) s)))))


LOG_ROOT |- (!n. (!x. (((~ (n = (NUMERAL _0))) /\ ((real_lt (real_of_num (NUMERAL _0))) x)) ==> ((log ((root n) x)) = ((real_div (log x)) (real_of_num n))))))


CONTINUOUS_ON_ABS |- (!f. (!s. (((continuous_on f) s) ==> ((continuous_on (\x. (lambdai. (real_abs ((f x) $ i))))) s))))


COMPLEX_DIFFERENTIABLE_WITHIN_CSIN |- (!s. (!z. ((complex_differentiable csin) ((within (at z)) s))))


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


DIV_MULT2 |- (!m. (!n. (!p. ((~ ((m * p) = (NUMERAL _0))) ==> (((m * n) DIV (m * p)) = (n DIV p))))))


Seq.mask_cons |- (!b. (!m. (!x. (!s. (((mask ((CONS b) m)) ((CONS x) s)) = ((cat ((nseq (((COND b) (NUMERAL (BIT1 _0))) (NUMERAL _0))) x)) ((mask m) s)))))))


REAL_INTEGRAL_RESTRICT |- (!f. (!s. (!t. ((s SUBSET t) ==> (((real_integral t) (\x. (((COND (x IN s)) (f x)) (real_of_num (NUMERAL _0))))) = ((real_integral s) f))))))


AFF_LE_SPECIAL_SCALE |- (!s. (!t. (!a. (!v. (((FINITE s) /\ ((FINITE t) /\ ((~ ((vec (NUMERAL _0)) IN t)) /\ ((~ (v IN t)) /\ ((~ (((% a) v) IN t)) /\ ((real_lt (real_of_num (NUMERAL _0))) a)))))) ==> (((aff_le ((vec (NUMERAL _0)) INSERT (((% a) v) INSERT s))) t) = ((aff_le ((vec (NUMERAL _0)) INSERT (v INSERT s))) t)))))))


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


AFF_DIM_PSUBSET |- (!s. (!t. ((((hull affine) s) PSUBSET ((hull affine) t)) ==> ((int_lt (aff_dim s)) (aff_dim t)))))


INSIDE_EQ_OUTSIDE |- (!s. (((inside s) = (outside s)) = (s = UNIV)))


VSUM_SUPPORT |- (!f. (!s. (((vsum (GSPEC (\GEN%PVAR%110. (?x. (((SETSPEC GEN%PVAR%110) ((x IN s) /\ (~ ((f x) = (vec (NUMERAL _0)))))) x))))) f) = ((vsum s) f))))


DIFFERENTIABLE_ON_NORM |- (!s. ((~ ((vec (NUMERAL _0)) IN s)) ==> ((differentiable_on (\x. (lift (vector_norm x)))) s)))


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


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


Misc_defs_and_lemmas.SURJ_SECTION |- (!a. (!b. (!f. ((((SURJ f) a) b) ==> (?g. (((INJ g) b) a))))))


Nonlinear_lemma.sqrt_x4 |- (((((((sqrt_x4 x1) x2) x3) x4) x5) x6) = (sqrt x4))


INT_MIN_MIN |- (!x. (!y. (((int_le ((int_min x) y)) x) /\ ((int_le ((int_min x) y)) y))))


ANALYTIC_AT_MUL |- (!f. (!g. (!z. ((((analytic_on f) (z INSERT EMPTY)) /\ ((analytic_on g) (z INSERT EMPTY))) ==> ((analytic_on (\w. ((complex_mul (f w)) (g w)))) (z INSERT EMPTY))))))


CASN_UNIQUE |- (!w. (!z. ((((csin z) = w) /\ (((real_lt (real_abs (Re z))) ((real_div pi) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) \/ (((real_abs (Re z)) = ((real_div pi) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) /\ ((Im z) = (real_of_num (NUMERAL _0)))))) ==> ((casn w) = z))))


COUNTABLE_CASES |- (!s. ((COUNTABLE s) = ((FINITE s) \/ (s =_c UNIV))))


AFF_LOWDIM_SUBSET_HYPERPLANE |- (!s. (((int_lt (aff_dim s)) (int_of_num (dimindex UNIV))) ==> (?a. (?b. ((~ (a = (vec (NUMERAL _0)))) /\ (s SUBSET (GSPEC (\GEN%PVAR%1597. (?x. (((SETSPEC GEN%PVAR%1597) (((dot a) x) = b)) x))))))))))


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


AFFINE_SCALING_EQ |- (!s. (!c. ((~ (c = (real_of_num (NUMERAL _0)))) ==> ((affine ((IMAGE (\x. ((% c) x))) s)) = (affine s)))))


ABSOLUTELY_CONTINUOUS_ON_VMUL |- (!f. (!s. (!v. (((absolutely_continuous_on (\x. (lift (f x)))) s) ==> ((absolutely_continuous_on (\x. ((% (f x)) v))) s)))))


MANHATTAN |- (((mspace manhattan) = UNIV) /\ (!x. (!y. (((mdist manhattan) (x , y)) = ((sum ((NUMERAL (BIT1 _0)) .. (dimindex UNIV))) (\i. (real_abs ((real_sub (x $ i)) (y $ i)))))))))


VECTOR_POLYNOMIAL_FUNCTION_o |- (!f. (!g. (((vector_polynomial_function f) /\ (vector_polynomial_function g)) ==> (vector_polynomial_function (g o f)))))


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


CONTINUOUS_ON_LIFT_ABS_COMPONENT |- (!f. (!s. (((continuous_on f) s) ==> ((continuous_on (\x. (lift (real_abs ((f x) $ k))))) s))))


REAL_SERIES_BOUND |- (!f. (!g. (!s. (!a. (!b. (((((real_sums f) a) s) /\ ((((real_sums g) b) s) /\ (!i. ((i IN s) ==> ((real_le (real_abs (f i))) (g i)))))) ==> ((real_le (real_abs a)) b)))))))


MEASURE_CBALL_SCALING |- (!a. (!c. (!r. (((real_le (real_of_num (NUMERAL _0))) c) ==> ((measure (cball (a , ((real_mul c) r)))) = ((real_mul ((real_pow c) (dimindex UNIV))) (measure (cball (a , r)))))))))


REAL_MEASURABLE_SCALING_EQ |- (!s. (!c. ((~ (c = (real_of_num (NUMERAL _0)))) ==> ((real_measurable ((IMAGE (\x. ((real_mul c) x))) s)) = (real_measurable s)))))


LIM_INFINITY_SEQUENTIALLY_COMPLEX |- (!f. (!l. ((((--> f) l) at_infinity) ==> (((--> (\n. (f (Cx (real_of_num n))))) l) sequentially))))


VECTOR_AFFINITY_EQ |- (!m. (!c. (!x. (!y. ((~ (m = (real_of_num (NUMERAL _0)))) ==> ((((vector_add ((% m) x)) c) = y) = (x = ((vector_add ((% (real_inv m)) y)) (vector_neg ((% (real_inv m)) c))))))))))


IS_INTERVAL_IMP_FSIGMA |- (!s. ((is_interval s) ==> (fsigma s)))


FORALL_TRIPLED_THM |- (!P. ((! (GABS (\f. (!x. (!y. (!z. ((GEQ (f (x , (y , z)))) (((P x) y) z)))))))) = (!x. (!y. (!z. (((P x) y) z))))))


INT_SUB_SUB2 |- (!x. (!y. (((int_sub x) ((int_sub x) y)) = y)))


CONTINUOUS_ON_MDIST |- (!m. (!a. ((a IN (mspace m)) ==> (((topcontinuous (mtopology m)) euclideanreal) (\x. ((mdist m) (a , x)))))))


SUBSET_CONVEX_HULL_RELATIVE_FRONTIER |- (!s. (((bounded s) /\ (~ (?a. (s = (a INSERT EMPTY))))) ==> (s SUBSET ((hull convex) (relative_frontier s)))))


Tactics_jordan.expand_prod_le |- (!B1. (!B2. (!C. ((((real_le (real_of_num (NUMERAL _0))) ((real_mul B1) B2)) ==> C) ==> (((real_le (real_of_num (NUMERAL _0))) B1) ==> (((real_le (real_of_num (NUMERAL _0))) B2) ==> C))))))


Ssrnat.half_gt0 |- (!n. (((NUMERAL _0) < (half n)) = ((NUMERAL (BIT1 _0)) < n)))


Hypermap_iso.finite_inj_eq_bij |- (!s. (!t. ((FINITE s) ==> ((FINITE t) ==> (((CARD s) = (CARD t)) ==> (!f. ((((INJ f) s) t) = (((BIJ f) s) t))))))))


FLOOR_DIV_DIV |- (!m. (!n. ((~ (m = (NUMERAL _0))) ==> ((floor ((real_div (real_of_num n)) (real_of_num m))) = (real_of_num (n DIV m))))))


CARD_EQ_FULLSIZE_POWERSET |- (!s. ((INFINITE s) ==> ((GSPEC (\GEN%PVAR%51. (?t. (((SETSPEC GEN%PVAR%51) ((t SUBSET s) /\ (t =_c s))) t)))) =_c (GSPEC (\GEN%PVAR%52. (?t. (((SETSPEC GEN%PVAR%52) (t SUBSET s)) t)))))))


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


Nonlinear_lemma.c2001 |- (!y. (((real_le y) ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))) ==> ((real_le y) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0))))


REAL_ABS_SIN_BOUND_LE |- (!x. ((real_le (real_abs (sin x))) (real_abs x)))


FACES_OF_LINEAR_IMAGE |- (!f. (!s. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((GSPEC (\GEN%PVAR%2138. (?t. (((SETSPEC GEN%PVAR%2138) ((face_of t) ((IMAGE f) s))) t)))) = ((IMAGE (IMAGE f)) (GSPEC (\GEN%PVAR%2139. (?t. (((SETSPEC GEN%PVAR%2139) ((face_of t) s)) t)))))))))


VECTOR_DIFF_CHAIN_AT |- (!f. (!g. (!f'. (!g'. (!x. (((((has_vector_derivative f) f') (at x)) /\ (((has_vector_derivative g) g') (at (f x)))) ==> (((has_vector_derivative (g o f)) ((% (drop f')) g')) (at x))))))))


GRASSMANN_PLUCKER_3 |- (!x1. (!x2. (!x3. (!y1. (!y2. (!y3. (((real_mul (det (vector ((CONS x1) ((CONS x2) ((CONS x3) NIL)))))) (det (vector ((CONS y1) ((CONS y2) ((CONS y3) NIL)))))) = ((real_add ((real_mul (det (vector ((CONS y1) ((CONS x2) ((CONS x3) NIL)))))) (det (vector ((CONS x1) ((CONS y2) ((CONS y3) NIL))))))) ((real_add ((real_mul (det (vector ((CONS y2) ((CONS x2) ((CONS x3) NIL)))))) (det (vector ((CONS y1) ((CONS x1) ((CONS y3) NIL))))))) ((real_mul (det (vector ((CONS y3) ((CONS x2) ((CONS x3) NIL)))))) (det (vector ((CONS y1) ((CONS y2) ((CONS x1) NIL)))))))))))))))


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


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


Float.ABS_NUM |- (!m. (!n. ((real_abs ((real_sub (real_of_num n)) (real_of_num m))) = (real_of_num ((m - n) + (n - m))))))


COMPLEX_DIFFERENTIABLE_WITHIN_SUBSET |- (!f. (!s. (!t. ((((complex_differentiable f) ((within (at x)) s)) /\ (t SUBSET s)) ==> ((complex_differentiable f) ((within (at x)) t))))))


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


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


NPRODUCT_CLAUSES_LEFT |- (!f. (!m. (!n. ((m <= n) ==> (((nproduct (m .. n)) f) = ((f m) * ((nproduct ((m + (NUMERAL (BIT1 _0))) .. n)) f)))))))


Add_triangle.fully_surrounded_card_set_of_edge |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!v. (((fully_surrounded (V , E)) /\ (v IN V)) ==> ((CARD (((set_of_edge v) V) E)) >= (NUMERAL (BIT1 (BIT1 _0)))))))))


DIM_SPECIAL_SUBSPACE |- (!k. ((dim (GSPEC (\GEN%PVAR%216. (?x. (((SETSPEC GEN%PVAR%216) (!i. ((((NUMERAL (BIT1 _0)) <= i) /\ ((i <= (dimindex UNIV)) /\ (i IN k))) ==> ((x $ i) = (real_of_num (NUMERAL _0)))))) x))))) = (CARD (((NUMERAL (BIT1 _0)) .. (dimindex UNIV)) DIFF k))))


HOMEOMORPHIC_COMPACT_ARNESS |- (!s. (!s'. (((homeomorphic s) s') ==> (((compact s) /\ ((retract_of s) UNIV)) = ((compact s') /\ ((retract_of s') UNIV))))))


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


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


INT_POW_LT2 |- (!n. (!x. (!y. (((~ (n = (NUMERAL _0))) /\ (((int_le (int_of_num (NUMERAL _0))) x) /\ ((int_lt x) y))) ==> ((int_lt ((int_pow x) n)) ((int_pow y) n))))))


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


CART_EQ_FULL |- (!x. (!y. ((x = y) = (!i. ((x $ i) = (y $ i))))))


MEASURABLE_BOUNDED_BY_INTEGRABLE_IMP_ABSOLUTELY_INTEGRABLE_AE |- (!f. (!g. (!s. (!t. ((((measurable_on f) s) /\ (((integrable_on g) s) /\ ((negligible t) /\ (!x. ((x IN (s DIFF t)) ==> ((real_le (vector_norm (f x))) (drop (g x)))))))) ==> ((absolutely_integrable_on f) s))))))


DOT_4 |- (((dot x) y) = ((real_add ((real_mul (x $ (NUMERAL (BIT1 _0)))) (y $ (NUMERAL (BIT1 _0))))) ((real_add ((real_mul (x $ (NUMERAL (BIT0 (BIT1 _0))))) (y $ (NUMERAL (BIT0 (BIT1 _0)))))) ((real_add ((real_mul (x $ (NUMERAL (BIT1 (BIT1 _0))))) (y $ (NUMERAL (BIT1 (BIT1 _0)))))) ((real_mul (x $ (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) (y $ (NUMERAL (BIT0 (BIT0 (BIT1 _0))))))))))


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


Nonlinear_lemma.rad2_y_e |- (!y1. (!y2. (!y3. (!y4. (!y5. (!y6. (((((((rad2_y y1) y2) y3) y4) y5) y6) = (((((((y_of_x rad2_x) y1) y2) y3) y4) y5) y6))))))))


MIDPOINTS_IN_CONVEX_HULL |- (!x. (!s. (((x IN ((hull convex) s)) /\ (y IN ((hull convex) s))) ==> ((midpoint (x , y)) IN ((hull convex) s)))))


Asfutbf.INTER_NEG |- (!A. (!B. ((((IMAGE vector_neg) A) INTER ((IMAGE vector_neg) B)) = ((IMAGE vector_neg) (A INTER B)))))


ABSOLUTELY_INTEGRABLE_ON_MUL_BERNOULLI_FRAC |- (!f. (!s. (!n. (((absolutely_integrable_on f) s) ==> ((absolutely_integrable_on (\x. ((% ((bernoulli n) (frac (drop x)))) (f x)))) s)))))


REAL_SUMMABLE_LMUL |- (!s. (!x. (!c. (((real_summable s) x) ==> ((real_summable s) (\n. ((real_mul c) (x n))))))))


HAS_BOUNDED_REAL_VARIATION_REFLECT_EQ_INTERVAL |- (!f. (!u. (!v. (((has_bounded_real_variation_on (\x. (f (real_neg x)))) (closed_real_interval ((CONS (u , v)) NIL))) = ((has_bounded_real_variation_on f) (closed_real_interval ((CONS ((real_neg v) , (real_neg u))) NIL)))))))


REAL_INTEGRABLE_SPIKE_INTERIOR |- (!f. (!g. (!a. (!b. ((!x. ((x IN (open_real_interval (a , b))) ==> ((g x) = (f x)))) ==> (((real_integrable_on f) (closed_real_interval ((CONS (a , b)) NIL))) ==> ((real_integrable_on g) (closed_real_interval ((CONS (a , b)) NIL)))))))))


Float.FLOAT_ADD_PPv2 |- (!a. (!b. (!m. (!n. ((m < n) ==> (((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 (n - m)))) b)) a)) (int_of_num m))))))))


More_list_hypermap.nodes_hypermap_of_list_all |- (!L. ((good_list_nodes L) ==> ((ALL (\n. ((ALL (\d. (((node (hypermap_of_list L)) d) = (set_of_list n)))) n))) (list_of_nodes L))))


SUP_INSERT_INSERT |- (!a. (!b. (!s. ((sup (b INSERT (a INSERT s))) = (sup (((real_max a) b) INSERT s))))))


AFF_LT_EQ_AFFINE_HULL |- (!s. (!t. (((aff_lt s) EMPTY) = ((hull affine) s))))


Nonlinear_lemma.lmdih5_ldih5' |- (!y1. (!y2. (!y3. (!y4. (!y5. (!y6. ((((real_le (real_of_num (NUMERAL _0))) y5) /\ ((real_le y5) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0))) ==> (((((((lmdih5_x_div_sqrtdelta_posbranch ((real_mul y1) y1)) ((real_mul y2) y2)) ((real_mul y3) y3)) ((real_mul y4) y4)) ((real_mul y5) y5)) ((real_mul y6) y6)) = ((((((ldih5_x_div_sqrtdelta_posbranch ((real_mul y1) y1)) ((real_mul y2) y2)) ((real_mul y3) y3)) ((real_mul y4) y4)) ((real_mul y5) y5)) ((real_mul y6) y6))))))))))


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


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


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


BOUNDED_RELATIVE_INTERIOR |- (!s. ((bounded s) ==> (bounded (relative_interior s))))


Ardbzye.J_SCS_3T3 |- ((((scs_J_v39 ((scs_prop_equ_v39 scs_3T3) i)) i1) j) = F)


HOMOTOPY_EQUIVALENT_SING |- (!s. (!a. (((homotopy_equivalent s) (a INSERT EMPTY)) = ((~ (s = EMPTY)) /\ (contractible s)))))


Packing3.MINIMAL_INTERS_EXISTS |- (!s. (!f. (((FINITE f) /\ (s = (INTERS f))) ==> (?g. ((g SUBSET f) /\ ((s = (INTERS g)) /\ (!g'. ((g' PSUBSET g) ==> (s PSUBSET (INTERS g'))))))))))


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


DIVISION_OF_UNIONS |- (!f. (((FINITE f) /\ ((!p. ((p IN f) ==> ((division_of p) (UNIONS p)))) /\ (!k1. (!k2. (((k1 IN (UNIONS f)) /\ ((k2 IN (UNIONS f)) /\ (~ (k1 = k2)))) ==> (((interior k1) INTER (interior k2)) = EMPTY)))))) ==> ((division_of (UNIONS f)) (UNIONS (UNIONS f)))))


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


REAL_CONTINUOUS_AT_RPOW_RIGHT |- (!a. (!x. (((real_lt (real_of_num (NUMERAL _0))) a) ==> ((real_continuous (\x. ((rpow a) x))) (atreal x)))))


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


CONTINUOUS_IMP_REAL_MEASURABLE_ON |- (!f. (((real_continuous_on f) UNIV) ==> ((real_measurable_on f) UNIV)))


DIM_NONEMPTY_INTERIOR |- (!s. ((~ ((interior s) = EMPTY)) ==> ((dim s) = (dimindex UNIV))))


SIMPLE_PATH_CONTINUOUS_IMAGE |- (!f. (!g. (((simple_path g) /\ (((continuous_on f) (path_image g)) /\ (!x. (!y. (((x IN (path_image g)) /\ ((y IN (path_image g)) /\ ((f x) = (f y)))) ==> (x = y)))))) ==> (simple_path (f o g)))))


CONVEX_ON_EPIGRAPH_SLICE_LT |- (!f. (!s. (!a. ((((convex_on f) s) /\ (convex s)) ==> (convex (GSPEC (\GEN%PVAR%1705. (?x. (((SETSPEC GEN%PVAR%1705) ((x IN s) /\ ((real_lt (f x)) a))) x)))))))))


TRANSP_ROWVECTOR |- (!v. ((transp (rowvector v)) = (columnvector v)))


ALL_IMP |- (!P. (!Q. (!l. (((!x. ((((MEM x) l) /\ (P x)) ==> (Q x))) /\ ((ALL P) l)) ==> ((ALL Q) l)))))


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


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


SUBSPACE_EQ_AFFINE |- (!s. ((subspace s) = ((affine s) /\ ((vec (NUMERAL _0)) IN s))))


EXTREME_POINT_OF_CONVEX_HULL_INSERT_EQ |- (!s. (!a. (!x. (((FINITE s) /\ (~ (a IN ((hull affine) s)))) ==> (((extreme_point_of x) ((hull convex) (a INSERT s))) = ((x = a) \/ ((extreme_point_of x) ((hull convex) s))))))))


REAL_INTEGRAL_EQ_0 |- (!f. (!s. ((!x. ((x IN s) ==> ((f x) = (real_of_num (NUMERAL _0))))) ==> (((real_integral s) f) = (real_of_num (NUMERAL _0))))))


Miqmcsn.STAB_4M8_13_SCS |- ((((scs_diag (scs_k_v39 scs_4M8_13)) i) j) ==> ((is_scs_v39 (((scs_stab_diag_v39 scs_4M8_13) i) j)) /\ (scs_basic_v39 (((scs_stab_diag_v39 scs_4M8_13) i) j))))


LINEAR_OPEN_MAPPING |- (!f. (!g. (((linear f) /\ ((linear g) /\ ((f o g) = I))) ==> (!s. ((open s) ==> (open ((IMAGE f) s)))))))


Tactics_jordan.strip_lt_lemma |- (!B1. (!B2. (!C. ((((real_lt (real_of_num (NUMERAL _0))) ((real_add B1) B2)) ==> C) ==> (((real_lt (real_of_num (NUMERAL _0))) B2) ==> (((real_le (real_of_num (NUMERAL _0))) B1) ==> C))))))


Lp_ineqs_proofs.sol_y_sym_reflect |- (!y1. (!y3. (!y2. (!y4. (!y6. (!y5. (((((((sol_y y1) y2) y3) y4) y5) y6) = ((((((sol_y y1) y3) y2) y4) y6) y5))))))))


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


CARD_FINITE_IMAGE |- (!s. ((CARD UNIV) = (dimindex s)))


TAGGED_DIVISION_OF_TRIVIAL |- (!p. (((tagged_division_of p) EMPTY) = (p = EMPTY)))


ROTATE2D_ADD_VECTORS |- (!t. (!w. (!z. (((rotate2d t) ((vector_add w) z)) = ((vector_add ((rotate2d t) w)) ((rotate2d t) z))))))


INTERVAL_IMAGE_STRETCH_INTERVAL |- (!a. (!b. (!m. (?u. (?v. (((IMAGE (\x. (lambdak. ((real_mul (m k)) (x $ k))))) (closed_interval ((CONS (a , b)) NIL))) = (closed_interval ((CONS (u , v)) NIL))))))))


SIMPLEX_IMP_CLOSED |- (!s. (!n. (((simplex n) s) ==> (closed s))))


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


LOG_SQRT |- (!x. (((real_lt (real_of_num (NUMERAL _0))) x) ==> ((log (sqrt x)) = ((real_div (log x)) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))))


Vol1.RADIAL_VS_RADIAL_NORM |- (!x. (!r. (!C. ((((radial r) x) C) = (((radial_norm r) x) C)))))


Seq.mask1 |- (!b. (!x. (((mask ((CONS b) NIL)) ((CONS x) NIL)) = ((nseq (((COND b) (NUMERAL (BIT1 _0))) (NUMERAL _0))) x))))


ALL2_MAP |- (!P. (!f. (!l. ((((ALL2 P) ((MAP f) l)) l) = ((ALL (\a. ((P (f a)) a))) l)))))


Lp_main_estimate.ineq_tau6_tauVEF |- (!V. (!E. ((lp_fan (V , E)) ==> ((!V. (!E. (!FF. (((convex_local_fan (V , (E , FF))) /\ ((packing V) /\ ((V SUBSET ball_annulus) /\ (((NUMERAL (BIT0 (BIT0 (BIT1 _0)))) <= (CARD V)) /\ (((CARD V) <= (NUMERAL (BIT0 (BIT1 (BIT1 _0))))) /\ ((!v. (!w. (((~ (v = w)) /\ ((v IN V) /\ ((w IN V) /\ (~ ((v INSERT (w INSERT EMPTY)) IN E))))) ==> ((real_le ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) (distance (v , w)))))) /\ (!v. (!w. (((v INSERT (w INSERT EMPTY)) IN E) ==> (((real_le (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (distance (v , w))) /\ ((real_le (distance (v , w))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))))))))))) ==> ((real_le (d_tame (CARD V))) (((tau_fun V) E) FF)))))) ==> (!d. ((d IN ((darts_k (NUMERAL (BIT0 (BIT1 (BIT1 _0))))) (hypermap_of_fan (V , E)))) ==> ((((real_le ((y5_fan (V , E)) d)) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) /\ (((real_le (y6_fan d)) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) /\ (((real_le ((y8_fan (V , E)) d)) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) /\ (((real_le ((y9_fan (V , E)) d)) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) /\ (((real_le (ye_fan (((POWER (f_fan_pair_ext (V , E))) (NUMERAL (BIT0 (BIT1 _0)))) d))) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) /\ ((real_le (ye_fan (((POWER (f_fan_pair_ext (V , E))) (NUMERAL (BIT1 (BIT1 _0)))) d))) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))))))))) ==> ((real_le ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 _0)))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))) (tauVEF (V , (E , ((face (hypermap_of_fan (V , E))) d))))))))))))


Trigonometry1.VECTOR_NEG_COMPONENT_3 |- (!x. ((((vector_neg x) $ (NUMERAL (BIT1 _0))) = (real_neg (x $ (NUMERAL (BIT1 _0))))) /\ ((((vector_neg x) $ (NUMERAL (BIT0 (BIT1 _0)))) = (real_neg (x $ (NUMERAL (BIT0 (BIT1 _0)))))) /\ (((vector_neg x) $ (NUMERAL (BIT1 (BIT1 _0)))) = (real_neg (x $ (NUMERAL (BIT1 (BIT1 _0)))))))))


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


POLYTOPE_VERTEX_LOWER_BOUND |- (!p. ((polytope p) ==> ((int_le ((int_add (aff_dim p)) (int_of_num (NUMERAL (BIT1 _0))))) (int_of_num (CARD (GSPEC (\GEN%PVAR%2456. (?v. (((SETSPEC GEN%PVAR%2456) ((extreme_point_of v) p)) v)))))))))


INT_LT_ADDNEG2 |- (!x. (!y. (!z. (((int_lt ((int_add x) (int_neg y))) z) = ((int_lt x) ((int_add z) y))))))


Hypermap_iso.image_res |- (!f. (!s. (!u. ((s SUBSET u) ==> (((IMAGE f) s) = ((IMAGE ((res f) u)) s))))))


Lp_ineqs_proofs.rhazim3c |- (!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))) ==> (((rhazim3_fan (V , E)) d) = ((rhazim_fan (V , E)) ((inverse (f_fan_pair_ext (V , E))) d)))))))))))


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


FACE_OF_SLICE |- (!f. (!s. (!t. ((((face_of f) s) /\ (convex t)) ==> ((face_of (f INTER t)) (s INTER t))))))


INT_MIN_LT |- (!x. (!y. (!z. (((int_lt ((int_min x) y)) z) = (((int_lt x) z) \/ ((int_lt y) z))))))


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


SUMMABLE_ADD |- (!x. (!y. (!s. ((((summable s) x) /\ ((summable s) y)) ==> ((summable s) (\n. ((vector_add (x n)) (y n))))))))


LIM_NULL_COMPLEX_RMUL |- (!net. (!f. (!c. ((((--> f) (Cx (real_of_num (NUMERAL _0)))) net) ==> (((--> (\x. ((complex_mul (f x)) c))) (Cx (real_of_num (NUMERAL _0)))) net)))))


Ssrnat.succn_inj |- (!n. (!m. (((SUC n) = (SUC m)) ==> (n = m))))


Seq.zip_unzip |- (!s. (((zip (unzip1 s)) (unzip2 s)) = s))


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


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


OPEN_INTERVAL_EQ |- ((!a. (!b. ((open (closed_interval ((CONS (a , b)) NIL))) = ((closed_interval ((CONS (a , b)) NIL)) = EMPTY)))) /\ (!a. (!b. (open (open_interval (a , b))))))


FINITE_FUNSPACE_UNIV |- (((FINITE UNIV) /\ (FINITE UNIV)) ==> (FINITE UNIV))


INT_MAX |- (!x. (!y. (((int_max x) y) = (((COND ((int_le x) y)) y) x))))


Pack2.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_closed S) v)))))))))))


INSIDE_INSIDE_EQ_EMPTY |- (!s. (((closed s) /\ (connected s)) ==> ((inside (inside s)) = EMPTY)))


INT_MIN_MAX |- (!x. (!y. (((int_min x) y) = (int_neg ((int_max (int_neg x)) (int_neg y))))))


WOSET_FINITE_TOSET |- (!l. (((toset l) /\ (FINITE (fl l))) ==> (woset l)))


EXISTS_DOUBLE_ARC |- (!g. (!a. (!b. (((simple_path g) /\ (((pathfinish g) = (pathstart g)) /\ ((a IN (path_image g)) /\ ((b IN (path_image g)) /\ (~ (a = b)))))) ==> (?u. (?d. ((arc u) /\ ((arc d) /\ (((pathstart u) = a) /\ (((pathfinish u) = b) /\ (((pathstart d) = b) /\ (((pathfinish d) = a) /\ ((((path_image u) INTER (path_image d)) = (a INSERT (b INSERT EMPTY))) /\ (((path_image u) UNION (path_image d)) = (path_image g)))))))))))))))


Lp_ineqs_proofs.y4'_list_def |- (!h. (!V. (!E. (!g. (!L. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((good_list L) ==> (((hyp_iso g) ((hypermap_of_list L) , (hypermap_of_fan (V , E)))) ==> ((!x. ((g x) = ((h (FST x)) , (h (SND x))))) ==> ((ALL (\d. (((y4'_list (g , (V , E))) d) = ((ye_list g) ((SND d) , (FST ((inverse (f_list_ext L)) d))))))) (list_of_darts L)))))))))))


COUNTABLE_CLOPEN_IN |- (!u. ((compact u) ==> (COUNTABLE (GSPEC (\GEN%PVAR%1247. (?c. (((SETSPEC GEN%PVAR%1247) (((closed_in ((subtopology euclidean) u)) c) /\ ((open_in ((subtopology euclidean) u)) c))) c)))))))


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


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


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


HOMEOMORPHIC_OPEN_INTERVALS |- (!a. (!b. (!c. (!d. ((((open_interval (a , b)) = EMPTY) = ((open_interval (c , d)) = EMPTY)) ==> ((homeomorphic (open_interval (a , b))) (open_interval (c , d))))))))


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


INT_LT_MAX |- (!x. (!y. (!z. (((int_lt z) ((int_max x) y)) = (((int_lt z) x) \/ ((int_lt z) y))))))


Seq.all_predT |- (!s. ((all predT) s))


REAL_SUMMABLE_IMP_BOUNDED |- (!f. (!k. (((real_summable k) f) ==> (real_bounded ((IMAGE f) k)))))


Optimize.split_2h0 |- (!a. (!b. (!y. ((((real_le a) y) /\ ((real_le y) b)) ==> ((((real_le a) y) /\ ((real_le y) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0))) \/ (((real_le ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) y) /\ ((real_le y) b)))))))


MEASURABLE_COUNTABLE_UNIONS |- (!s. (!B. (((!n. (measurable (s n))) /\ (!n. ((real_le ((sum ((NUMERAL _0) .. n)) (\k. (measure (s k))))) B))) ==> (measurable (UNIONS (GSPEC (\GEN%PVAR%3387. (?n. (((SETSPEC GEN%PVAR%3387) (n IN UNIV)) (s n))))))))))


Tame_lemmas.dart_leads_into_same |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> (!f. (!v. (!w. (!x. (!y. (((f IN (face_set (hypermap_of_fan (V , E)))) /\ (((v , w) IN f) /\ ((x , y) IN f))) ==> ((((((dart_leads_into (vec (NUMERAL _0))) V) E) v) w) = (((((dart_leads_into (vec (NUMERAL _0))) V) E) x) y))))))))))))


MEASURE_INTERVAL_1_ALT |- ((!a. (!b. ((measure (closed_interval ((CONS (a , b)) NIL))) = (((COND ((real_lt (drop a)) (drop b))) ((real_sub (drop b)) (drop a))) (real_of_num (NUMERAL _0)))))) /\ (!a. (!b. ((measure (open_interval (a , b))) = (((COND ((real_lt (drop a)) (drop b))) ((real_sub (drop b)) (drop a))) (real_of_num (NUMERAL _0)))))))


NSUM_PAIR |- (!f. (!m. (!n. (((nsum (((NUMERAL (BIT0 (BIT1 _0))) * m) .. (((NUMERAL (BIT0 (BIT1 _0))) * n) + (NUMERAL (BIT1 _0))))) f) = ((nsum (m .. n)) (\i. ((f ((NUMERAL (BIT0 (BIT1 _0))) * i)) + (f (((NUMERAL (BIT0 (BIT1 _0))) * i) + (NUMERAL (BIT1 _0)))))))))))


ABSOLUTELY_CONTINUOUS_ON_LIFT_ABS |- (!f. (((absolutely_continuous_on (\x. (lift (f x)))) s) ==> ((absolutely_continuous_on (\x. (lift (real_abs (f x))))) s)))


CONTINUOUS_CARD_LT_RANGE_CONSTANT |- (!f. (!s. (((connected s) /\ (((continuous_on f) s) /\ (((IMAGE f) s) <_c UNIV))) ==> (?a. (!x. ((x IN s) ==> ((f x) = a)))))))


IN_CBALL_IM |- (!x. (!z. (!e. ((x IN (cball (z , e))) ==> ((real_le (real_abs ((real_sub (Im x)) (Im z)))) e)))))


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


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


Lp_ineqs_proofs.azim2_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 ((azim2_fan (V , E)) d)) pi)))))))))


REFLECT_ALONG_EQ |- (!v. (!x. (!y. ((((reflect_along v) x) = ((reflect_along v) y)) = (x = y)))))


Rogers.KHEJKCI |- (!V. (!k. (!ul. (((saturated V) /\ ((packing V) /\ (((barV V) k) ul))) ==> ((face_of ((voronoi_list V) ul)) ((voronoi_closed V) (HD ul)))))))


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


SEPARATING_HYPERPLANE_RELATIVE_INTERIORS |- (!s. (!t. (((convex s) /\ ((convex t) /\ ((~ (((s = EMPTY) /\ (t = UNIV)) \/ ((s = UNIV) /\ (t = EMPTY)))) /\ ((DISJOINT (relative_interior s)) (relative_interior t))))) ==> (?a. (?b. ((~ (a = (vec (NUMERAL _0)))) /\ ((!x. ((x IN s) ==> ((real_le ((dot a) x)) b))) /\ (!x. ((x IN t) ==> ((real_ge ((dot a) x)) b))))))))))


Localization.localization |- (!V. (!E. (!FF. (((localization (V , E)) FF) = (((v_prime V) FF) , ((e_prime E) FF))))))


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


REAL_DIFFERENTIABLE_WITHIN_TAN |- (!s. (!x. ((~ ((cos x) = (real_of_num (NUMERAL _0)))) ==> ((real_differentiable tan) ((within (atreal x)) s)))))


CCOS_CONVERGES |- (!z. (((sums (\n. ((complex_mul ((complex_pow (vector_neg (Cx (real_of_num (NUMERAL (BIT1 _0)))))) n)) ((complex_div ((complex_pow z) ((NUMERAL (BIT0 (BIT1 _0))) * n))) (Cx (real_of_num (FACT ((NUMERAL (BIT0 (BIT1 _0))) * n)))))))) (ccos z)) (from (NUMERAL _0))))


MEASURABLE_TRIANGLE |- (!a. (!b. (!c. (measurable ((hull convex) (a INSERT (b INSERT (c INSERT EMPTY))))))))


DIFFERENTIABLE_ON_NEG |- (!f. (!s. (((differentiable_on f) s) ==> ((differentiable_on (\z. (vector_neg (f z)))) s))))


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


Ssrnat.minn0 |- (!n. (((minn n) (NUMERAL _0)) = (NUMERAL _0)))


REAL_MEASURABLE_COUNTABLE_UNIONS |- (!s. (!B. (((!n. (real_measurable (s n))) /\ (!n. ((real_le ((sum ((NUMERAL _0) .. n)) (\k. (real_measure (s k))))) B))) ==> (real_measurable (UNIONS (GSPEC (\GEN%PVAR%4952. (?n. (((SETSPEC GEN%PVAR%4952) (n IN UNIV)) (s n))))))))))


NEGLIGIBLE_CONVEX_HULL_1 |- (!a. (negligible ((hull convex) (a INSERT EMPTY))))


COMPLEX_INV_II |- ((complex_inv ii) = (vector_neg ii))


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


Seq2.prev_el_rot |- (!x. (!s. (((MEM x) s) ==> (((prev_el s) x) = ((headl x) ((rotr (NUMERAL (BIT1 _0))) ((rot ((indexl x) s)) s)))))))


HAS_REAL_MEASURE_IMP_REAL_MEASURABLE |- (!s. (!m. (((has_real_measure s) m) ==> (real_measurable s))))


RIGHT_POLAR_DECOMPOSITION_INVERTIBLE_UNIQUE |- (!A. (!U. (!V. (!P. (!S. (((invertible A) /\ ((orthogonal_matrix U) /\ ((positive_semidefinite P) /\ ((orthogonal_matrix V) /\ ((positive_semidefinite S) /\ ((((matrix_mul U) P) = A) /\ (((matrix_mul V) S) = A))))))) ==> ((P = S) /\ (U = V))))))))


HAS_BOUNDED_VARIATION_ON_INDEFINITE_INTEGRAL_LEFT |- (!f. (!a. (!b. (((absolutely_integrable_on f) (closed_interval ((CONS (a , b)) NIL))) ==> ((has_bounded_variation_on (\c. ((integral (closed_interval ((CONS (c , b)) NIL))) f))) (closed_interval ((CONS (a , b)) NIL)))))))


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


COLLINEAR_DESCALE |- (!a. (!b. (!c. (!x. (!y. (!z. (((~ (a = (real_of_num (NUMERAL _0)))) /\ ((~ (c = (real_of_num (NUMERAL _0)))) /\ ((collinear (((% a) x) INSERT (((% b) y) INSERT (((% c) z) INSERT EMPTY)))) /\ (~ ((vec (NUMERAL _0)) IN ((hull affine) (x INSERT (y INSERT (z INSERT EMPTY))))))))) ==> (collinear (x INSERT (y INSERT (z INSERT EMPTY)))))))))))


COVERING_SPACE_FINITE_SHEETS_EQ_CLOSED_MAP |- (!p. (!c. (!s. ((((covering_space (c , p)) s) /\ ((connected s) /\ (~ (?a. (s = (a INSERT EMPTY)))))) ==> ((!b. ((b IN s) ==> (FINITE (GSPEC (\GEN%PVAR%2081. (?x. (((SETSPEC GEN%PVAR%2081) ((x IN c) /\ ((p x) = b))) x))))))) = (!t. (((closed_in ((subtopology euclidean) c)) t) ==> ((closed_in ((subtopology euclidean) s)) ((IMAGE p) t)))))))))


REAL_DIFFERENTIABLE_CARATHEODORY_WITHINREAL |- (!f. (!z. (!s. (((real_differentiable f) ((within (atreal z)) s)) = (?g. ((!w. (((real_sub (f w)) (f z)) = ((real_mul (g w)) ((real_sub w) z)))) /\ ((real_continuous g) ((within (atreal z)) s))))))))


FACET_OF_HALFSPACE_GE |- (!f. (!a. (!b. (((facet_of f) (GSPEC (\GEN%PVAR%2218. (?x. (((SETSPEC GEN%PVAR%2218) ((real_ge ((dot a) x)) b)) x))))) = ((~ (a = (vec (NUMERAL _0)))) /\ (f = (GSPEC (\GEN%PVAR%2219. (?x. (((SETSPEC GEN%PVAR%2219) (((dot a) x) = b)) x))))))))))


HAS_BOUNDED_REAL_VARIATION_ON_EMPTY |- (!f. ((has_bounded_real_variation_on f) EMPTY))


Functional_equation.rotate5_dihatn_x |- ((rotate5 dihatn_x) = dih5atn_x)


INT_LT_NEG |- (!x. (!y. (((int_lt (int_neg x)) (int_neg y)) = ((int_lt y) x))))


NPRODUCT_POS_LT_NUMSEG |- (!f. (!m. (!n. ((!x. (((m <= x) /\ (x <= n)) ==> ((NUMERAL _0) < (f x)))) ==> ((NUMERAL _0) < ((nproduct (m .. n)) f))))))


Lp_main_estimate.ineq_tau5_diags_std |- (!V. (!E. ((lp_fan (V , E)) ==> ((!V. (!E. (!FF. (((convex_local_fan (V , (E , FF))) /\ ((packing V) /\ ((V SUBSET ball_annulus) /\ (((CARD V) = (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) /\ ((!v. (!w. (((~ (v = w)) /\ ((v IN V) /\ ((w IN V) /\ (~ ((v INSERT (w INSERT EMPTY)) IN E))))) ==> ((real_le sqrt8) (distance (v , w)))))) /\ (!v. (!w. (((v INSERT (w INSERT EMPTY)) IN E) ==> (((real_le (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (distance (v , w))) /\ ((real_le (distance (v , w))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0))))))))))) ==> ((real_le ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 _0)))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))) (((tau_fun V) E) FF)))))) ==> ((E = (ESTD V)) ==> (!d. ((d IN ((darts_k (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) (hypermap_of_fan (V , E)))) ==> ((((real_le sqrt8) ((y4'_fan (V , E)) d)) /\ (((real_le sqrt8) ((y4'_fan (V , E)) ((f_fan_pair_ext (V , E)) d))) /\ (((real_le sqrt8) ((y4'_fan (V , E)) (((POWER (f_fan_pair_ext (V , E))) (NUMERAL (BIT0 (BIT1 _0)))) d))) /\ (((real_le sqrt8) ((y4'_fan (V , E)) (((POWER (f_fan_pair_ext (V , E))) (NUMERAL (BIT1 (BIT1 _0)))) d))) /\ ((real_le sqrt8) ((y4'_fan (V , E)) ((inverse (f_fan_pair_ext (V , E))) d))))))) ==> ((real_le ((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))))))))))))) ((tau_fan (V , E)) ((face (hypermap_of_fan (V , E))) d)))))))))))


CONVEX_HALFSPACE_RE_GT |- (!b. (convex (GSPEC (\GEN%PVAR%4769. (?z. (((SETSPEC GEN%PVAR%4769) ((real_gt (Re z)) b)) z))))))


Functional_equation.functional_x1_delta_x |- (x1_delta_x = ((mul6 proj_x1) delta_x))


INT_SGN_INT_SGN |- (!x. ((int_sgn (int_sgn x)) = (int_sgn x)))


REAL_VARIATION_TRANSLATION2 |- (!a. (!f. (!s. (((real_variation ((IMAGE (\x. ((real_add (real_neg a)) x))) s)) (\x. (f ((real_add a) x)))) = ((real_variation s) f)))))


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


RANK_TRIANGLE |- (!A. (!B. ((rank ((matrix_add A) B)) <= ((rank A) + (rank B)))))


HAS_COMPLEX_DERIVATIVE_SEQUENCE |- (!s. (!f. (!f'. (!g'. (((convex s) /\ ((!n. (!x. ((x IN s) ==> (((has_complex_derivative (f n)) ((f' n) x)) ((within (at x)) s))))) /\ ((!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?N. (!n. (!x. (((n >= N) /\ (x IN s)) ==> ((real_le (vector_norm ((vector_sub ((f' n) x)) (g' x)))) e))))))) /\ (?x. (?l. ((x IN s) /\ (((--> (\n. ((f n) x))) l) sequentially))))))) ==> (?g. (!x. ((x IN s) ==> ((((--> (\n. ((f n) x))) (g x)) sequentially) /\ (((has_complex_derivative g) (g' x)) ((within (at x)) s)))))))))))


Merge_ineq.nonf_arclength_x_123' |- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((((((arclength_x_123' x1) x2) x3) x4) x5) x6) = (acs ((real_div ((real_add x1) ((real_sub x2) x3))) (sqrt ((real_mul ((real_mul x1) x2)) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0))))))))))))))))


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


INT_POW_ZERO |- (!n. (((int_pow (int_of_num (NUMERAL _0))) n) = (((COND (n = (NUMERAL _0))) (int_of_num (NUMERAL (BIT1 _0)))) (int_of_num (NUMERAL _0)))))


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


CONVEX_HULL_REDUNDANT_POINT |- (!s. (!a. ((((hull convex) (s DELETE a)) = ((hull convex) s)) = (~ ((extreme_point_of a) ((hull convex) s))))))


CONTINUOUS_DECREASING_IMAGE_INTERVAL_1 |- (!f. (!a. (!b. (((~ ((closed_interval ((CONS (a , b)) NIL)) = EMPTY)) /\ (((continuous_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 (drop (f y))) (drop (f x)))))))) ==> (((IMAGE f) (closed_interval ((CONS (a , b)) NIL))) = (closed_interval ((CONS ((f b) , (f a))) NIL)))))))


MATRIX_NEG_0 |- ((matrix_neg (mat (NUMERAL _0))) = (mat (NUMERAL _0)))


SIN_EQ_1 |- (!x. (((sin x) = (real_of_num (NUMERAL (BIT1 _0)))) = (?n. ((integer n) /\ (x = ((real_mul ((real_add ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) n)) ((real_div (real_of_num (NUMERAL (BIT1 _0)))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) pi))))))


COMPACT_ARC_IMAGE |- (!g. ((arc g) ==> (compact (path_image g))))


ENR_DELETE |- (!s. (!a. ((ENR s) ==> (ENR (s DELETE a)))))


Reduction3.size_mapAt |- (!ns. (!xs. (!f. ((sizel (((mapAt ns) f) xs)) = (sizel xs)))))


FRONTIER_FRONTIER_FRONTIER |- (!s. ((frontier (frontier (frontier s))) = (frontier (frontier s))))


INT_EQ_ADD_RCANCEL_0 |- (!x. (!y. ((((int_add x) y) = y) = (x = (int_of_num (NUMERAL _0))))))


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


EULER_ROTATION_THEOREM_GEN |- (!A. (((ODD (dimindex UNIV)) /\ (rotation_matrix A)) ==> (?v. (((vector_norm v) = (real_of_num (NUMERAL (BIT1 _0)))) /\ (((matrix_vector_mul A) v) = v)))))


INDEPENDENT_2 |- (!a. (!b. (!x. (!y. (((independent (a INSERT (b INSERT EMPTY))) /\ (~ (a = b))) ==> ((((vector_add ((% x) a)) ((% y) b)) = (vec (NUMERAL _0))) = ((x = (real_of_num (NUMERAL _0))) /\ (y = (real_of_num (NUMERAL _0))))))))))


CONTINUOUS_ON_COMPLEX_DIV |- (!f. (!g. (!s. ((((continuous_on f) s) /\ (((continuous_on g) s) /\ (!x. ((x IN s) ==> (~ ((g x) = (Cx (real_of_num (NUMERAL _0))))))))) ==> ((continuous_on (\x. ((complex_div (f x)) (g x)))) s)))))


CONVEX_HULL_REDUNDANT_SUBSET |- (!s. (!t. (((compact s) /\ ((t SUBSET s) /\ ((s DIFF t) SUBSET (interior ((hull convex) s))))) ==> (((hull convex) s) = ((hull convex) t)))))


NORM_SEGMENT_LOWERBOUND |- (!a. (!b. (!x. (!r. (!d. ((((real_lt (real_of_num (NUMERAL _0))) r) /\ (((vector_norm a) = r) /\ (((vector_norm b) = r) /\ ((x IN (closed_segment ((CONS (a , b)) NIL))) /\ (((dot a) b) = ((real_mul d) ((real_pow r) (NUMERAL (BIT0 (BIT1 _0)))))))))) ==> ((real_le ((real_mul (sqrt ((real_div ((real_sub (real_of_num (NUMERAL (BIT1 _0)))) (real_abs d))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) r)) (vector_norm x))))))))


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


Functional_equation.functional_delta_x1 |- (delta_x1 = (rotate4 delta_x4))


Functional_equation.nonf_gamma2_x1_div_a |- (!m. (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. ((((((((gamma2_x1_div_a m) x1) x2) x3) x4) x5) x6) = ((gamma2_x_div_azim m) x1)))))))))


CONTINUOUS_ON_COMPOSE_ARG |- (!s. (!p. ((((continuous_on (p o drop)) (closed_interval ((CONS ((vec (NUMERAL _0)) , (lift ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) pi)))) NIL))) /\ (((p ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) pi)) = (p (real_of_num (NUMERAL _0)))) /\ (~ ((Cx (real_of_num (NUMERAL _0))) IN s)))) ==> ((continuous_on (\z. (p (Arg z)))) s))))


Add_triangle.f_split_fan_face_eq |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> (!v. (!w. (!d. ((((v , w) IN (dart_of_fan (V , E))) /\ ((d IN (dart_of_fan (V , E))) /\ (~ (d IN ((face (hypermap_of_fan (V , E))) (v , w)))))) ==> (((f_fan_pair ((split_fan_face (V , E)) (v , w))) d) = ((f_fan_pair (V , E)) d))))))))))


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


SECOND_MEAN_VALUE_THEOREM_BONNET |- (!f. (!g. (!a. (!b. (((~ ((closed_interval ((CONS (a , b)) NIL)) = EMPTY)) /\ (((integrable_on f) (closed_interval ((CONS (a , b)) NIL))) /\ ((!x. ((x IN (closed_interval ((CONS (a , b)) NIL))) ==> ((real_le (real_of_num (NUMERAL _0))) (g x)))) /\ (!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))) /\ (((integral (closed_interval ((CONS (a , b)) NIL))) (\x. ((% (g x)) (f x)))) = ((% (g b)) ((integral (closed_interval ((CONS (c , b)) NIL))) f))))))))))


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


BOUNDED_ARC_IMAGE |- (!g. ((arc g) ==> (bounded (path_image g))))


CNJ_CSQRT |- (!z. ((((Im z) = (real_of_num (NUMERAL _0))) ==> ((real_le (real_of_num (NUMERAL _0))) (Re z))) ==> ((cnj (csqrt z)) = (csqrt (cnj z)))))


Ssrnat.minn_maxr |- (!m. (!n1. (!n2. (((minn m) ((maxn n1) n2)) = ((maxn ((minn m) n1)) ((minn m) n2))))))


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


AFFINE_HULL_EQ_SING |- (!s. (!a. ((((hull affine) s) = (a INSERT EMPTY)) = (s = (a INSERT EMPTY)))))


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


DIV_ADD_MOD |- (!a. (!b. (!n. ((~ (n = (NUMERAL _0))) ==> ((((a + b) MOD n) = ((a MOD n) + (b MOD n))) = (((a + b) DIV n) = ((a DIV n) + (b DIV n))))))))


Lp_ineqs_proofs.sol_hi |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!f. ((f IN (face_set (hypermap_of_fan (V , E)))) ==> ((real_le ((sol_fan (V , E)) f)) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) pi))))))))))


ORTHOGONAL_TRANSFORMATION_ONTO_SUBSPACE |- (!s. (!t. (((subspace s) /\ ((subspace t) /\ ((dim s) = (dim t)))) ==> (?f. ((orthogonal_transformation f) /\ (((IMAGE f) s) = t))))))


PROPER_LOCAL_HOMEOMORPHISM_GLOBAL |- (!f. (!s. (!t. (((path_connected s) /\ ((simply_connected t) /\ (((continuous_on f) s) /\ ((((IMAGE f) s) = t) /\ ((!k. (((k SUBSET t) /\ (compact k)) ==> (compact (GSPEC (\GEN%PVAR%2104. (?x. (((SETSPEC GEN%PVAR%2104) ((x IN s) /\ ((f x) IN k))) x))))))) /\ (!x. ((x IN s) ==> (?u. (?v. (?q. ((x IN u) /\ (((open_in ((subtopology euclidean) s)) u) /\ (((open_in ((subtopology euclidean) t)) v) /\ ((homeomorphism (u , v)) (f , q))))))))))))))) ==> (?g. ((homeomorphism (s , t)) (f , g)))))))


FCONS_UNDO |- (!f. (f = ((FCONS (f (NUMERAL _0))) (f o SUC))))


ISOMETRY_IMP_AFFINITY |- (!f. ((!x. (!y. ((distance ((f x) , (f y))) = (distance (x , y))))) ==> (?h. ((linear h) /\ (!x. ((f x) = ((vector_add (f (vec (NUMERAL _0)))) (h x))))))))


IS_AFFINE_HULL |- (!s. ((affine s) = (?t. (s = ((hull affine) t)))))


Fnjlbxs.closedS |- (!k. (!S. ((((NUMERAL (BIT1 (BIT1 _0))) * k) <= (dimindex UNIV)) ==> ((!l. ((l IN S) = (((set_of_list l) SUBSET ball_annulus) /\ ((packing (set_of_list l)) /\ (((LENGTH l) = k) /\ (uniq l)))))) ==> (closed ((IMAGE (vector_of_list (NUMERAL (BIT1 (BIT1 _0))))) S))))))


REAL_MEASURABLE_TRANSLATION |- (!s. ((real_measurable ((IMAGE (\x. ((real_add a) x))) s)) = (real_measurable s)))


ZERO_AE_DERIVATIVE_IMP_CONSTANT |- (!f. (!a. (!b. (!s. ((((absolutely_continuous_on f) (closed_interval ((CONS (a , b)) NIL))) /\ ((negligible s) /\ (!x. ((x IN ((closed_interval ((CONS (a , b)) NIL)) DIFF s)) ==> (((has_vector_derivative f) (vec (NUMERAL _0))) (at x)))))) ==> (!x. ((x IN (closed_interval ((CONS (a , b)) NIL))) ==> ((f x) = (f a)))))))))


CLOSURE_OF_EMPTY |- (!top. (((closure_of top) EMPTY) = EMPTY))


BETWEEN_COLLINEAR_DIST_EQ |- (!a. (!b. (!x. (((between x) (a , b)) = ((collinear (a INSERT (x INSERT (b INSERT EMPTY)))) /\ (((real_le (distance (x , a))) (distance (a , b))) /\ ((real_le (distance (x , b))) (distance (a , b)))))))))


EXPOSED_POINT_OF_INTER_SUPPORTING_HYPERPLANE_GE |- (!s. (!a. (!b. (!c. (((!x. ((x IN s) ==> ((real_ge ((dot a) x)) b))) /\ ((s INTER (GSPEC (\GEN%PVAR%2203. (?x. (((SETSPEC GEN%PVAR%2203) (((dot a) x) = b)) x))))) = (c INSERT EMPTY))) ==> ((exposed_face_of (c INSERT EMPTY)) s))))))


Ssrbool.addbT |- (!b. (((XOR b) T) = (~ b)))


Miqmcsn.SCS_4M8_13_ARROW_3T7_OPP |- ((scs_arrow_v39 (scs_4M8_02 INSERT EMPTY)) ((scs_opp_v39 scs_3T7) INSERT EMPTY))


Seq.constantP |- (!s. ((?x. (s = ((nseq (sizel s)) x))) = (constant s)))


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


LIPSCHITZ_VECTOR_POLYNOMIAL_FUNCTION |- (!f. (!s. (((vector_polynomial_function f) /\ (bounded s)) ==> (?B. (((real_lt (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))))))))))))


CONVEX_MAX |- (!f. (!g. (!s. ((((convex_on f) s) /\ ((convex_on g) s)) ==> ((convex_on (\x. ((real_max (f x)) (g x)))) s)))))


Dih2k_hypermap.EE_OF_HYP_HAS_ORDERS |- (((FAN ((vec (NUMERAL _0)) , ((V_SY (vecmats l)) , (E_SY (vecmats l))))) /\ (((NUMERAL (BIT0 (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)))))) ==> ((has_orders (ee_of_hyp ((vec (NUMERAL _0)) , ((V_SY (vecmats l)) , (E_SY (vecmats l)))))) (NUMERAL (BIT0 (BIT1 _0)))))


EXPLICIT_SUBSET_INTERIOR_CONVEX_HULL_MINIMAL |- (!s. (((FINITE s) /\ (((hull affine) s) = UNIV)) ==> ((GSPEC (\GEN%PVAR%1795. (?y. (((SETSPEC GEN%PVAR%1795) (?u. ((!x. ((x IN s) ==> ((real_lt (real_of_num (NUMERAL _0))) (u x)))) /\ ((((sum s) u) = (real_of_num (NUMERAL (BIT1 _0)))) /\ (((vsum s) (\x. ((% (u x)) x))) = y))))) y)))) SUBSET (interior ((hull convex) s)))))


INT_LE_ADD |- (!x. (!y. ((((int_le (int_of_num (NUMERAL _0))) x) /\ ((int_le (int_of_num (NUMERAL _0))) y)) ==> ((int_le (int_of_num (NUMERAL _0))) ((int_add x) y)))))


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


Hypermap.lemma_map_loop_map |- (!H. (!NF. (!L. (!x. ((((is_normal H) NF) /\ ((L IN NF) /\ (((in_dart_of_loop x) L) /\ (((loop_map L) x) IN (((atom H) L) x))))) ==> (((loop_map L) x) = ((inverse (node_map H)) x)))))))


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


Collect_geom2.c_coef |- (!x12. (!x13. (!x14. (!x23. (!x24. (((((((b_coef x12) x13) x14) x23) x24) = ((real_add ((real_mul (real_neg x13)) x14)) ((real_add ((real_mul (real_neg x23)) x24)) ((real_add ((real_mul x13) x24)) ((real_add ((real_mul x14) x23)) ((real_add ((real_mul (real_neg x12)) x12)) ((real_add ((real_mul x12) x14)) ((real_add ((real_mul x12) x24)) ((real_add ((real_mul x12) x13)) ((real_mul x12) x23)))))))))) /\ ((((((c_coef x12) x13) x14) x23) x24) = ((real_add ((real_mul ((real_mul x14) x23)) x12)) ((real_add ((real_mul ((real_mul x14) x23)) x13)) ((real_add ((real_mul ((real_mul (real_neg x14)) x23)) x14)) ((real_add ((real_mul ((real_mul (real_neg x14)) x23)) x23)) ((real_add ((real_mul ((real_mul x14) x23)) x24)) ((real_add ((real_mul ((real_mul (real_neg x12)) x13)) x23)) ((real_add ((real_mul ((real_mul (real_neg x12)) x14)) x24)) ((real_add ((real_mul ((real_mul x13) x24)) x12)) ((real_add ((real_mul ((real_mul (real_neg x13)) x24)) x13)) ((real_add ((real_mul ((real_mul x13) x24)) x14)) ((real_add ((real_mul ((real_mul x13) x24)) x23)) ((real_mul ((real_mul (real_neg x13)) x24)) x24)))))))))))))))))))


CROSS_JACOBI |- (!x. (!y. (!z. (((vector_add ((cross x) ((cross y) z))) ((vector_add ((cross y) ((cross z) x))) ((cross z) ((cross x) y)))) = (vec (NUMERAL _0))))))


COVERING_SPACE_COMPACT |- (!p. (!c. (!s. (((covering_space (c , p)) s) ==> ((compact c) = ((compact s) /\ (!b. ((b IN s) ==> (FINITE (GSPEC (\GEN%PVAR%2085. (?x. (((SETSPEC GEN%PVAR%2085) ((x IN c) /\ ((p x) = b))) x)))))))))))))


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


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


Calc_derivative.imp_eq |- (!p. (!p'. (!x. (!a. (!a'. (!b. (!b'. (((x = (real_of_num (NUMERAL _0))) /\ (((((ratform p) x) a) b) /\ ((p = p') /\ ((a = a') /\ (b = b'))))) ==> ((p' /\ (~ (b' = (real_of_num (NUMERAL _0))))) ==> (a' = (real_of_num (NUMERAL _0))))))))))))


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


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


UPPER_HEMICONTINUOUS |- (!f. (!t. (!s. ((!x. ((x IN s) ==> ((f x) SUBSET t))) ==> ((!u. (((open_in ((subtopology euclidean) t)) u) ==> ((open_in ((subtopology euclidean) s)) (GSPEC (\GEN%PVAR%880. (?x. (((SETSPEC GEN%PVAR%880) ((x IN s) /\ ((f x) SUBSET u))) x))))))) = (!u. (((closed_in ((subtopology euclidean) t)) u) ==> ((closed_in ((subtopology euclidean) s)) (GSPEC (\GEN%PVAR%881. (?x. (((SETSPEC GEN%PVAR%881) ((x IN s) /\ (~ (((f x) INTER u) = EMPTY)))) x))))))))))))


CONNECTED_OPEN_DIFF_CARD_LT |- (!s. (!t. ((((NUMERAL (BIT0 (BIT1 _0))) <= (dimindex UNIV)) /\ ((open s) /\ ((connected s) /\ (t <_c UNIV)))) ==> (connected (s DIFF t)))))


OPEN_PATH_CONNECTED_COMPONENT |- (!s. (!x. ((open s) ==> (((path_component s) x) = ((connected_component s) x)))))


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


COUNTABLE_ENR_PATH_COMPONENTS |- (!s. ((ENR s) ==> (COUNTABLE (GSPEC (\GEN%PVAR%2725. (?x. (((SETSPEC GEN%PVAR%2725) (x IN s)) ((path_component s) x))))))))


Lp_gen_theory.card_face_dart_iso |- (!H. (!g. (!G. (((hyp_iso g) (G , H)) ==> (!d. ((d IN (dart G)) ==> ((CARD ((IMAGE g) ((face G) d))) = (CARD ((face G) d)))))))))


SETDIST_UNIV |- ((!s. ((setdist (s , UNIV)) = (real_of_num (NUMERAL _0)))) /\ (!t. ((setdist (UNIV , t)) = (real_of_num (NUMERAL _0)))))


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


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


CARD_EQ_PERFECT_SET |- (!s. (((closed s) /\ ((!x. ((x IN s) ==> ((limit_point_of x) s))) /\ (~ (s = EMPTY)))) ==> (s =_c UNIV)))


HAUSDIST_STILL_NONEMPTY_INTERIOR |- (!s. (((bounded s) /\ ((convex s) /\ (~ ((interior s) = EMPTY)))) ==> (?e. (((real_lt (real_of_num (NUMERAL _0))) e) /\ (!s'. (((bounded s') /\ ((convex s') /\ ((~ (s' = EMPTY)) /\ ((real_lt (hausdist (s , s'))) e)))) ==> (~ ((interior s') = EMPTY))))))))


STARLIKE_IMP_CONNECTED |- (!s. ((starlike s) ==> (connected s)))


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


CONVEX_HAS_BOUNDED_VARIATION |- (!f. (!a. (!b. (((convex_on f) (closed_interval ((CONS (a , b)) NIL))) ==> ((has_bounded_variation_on (lift o f)) (closed_interval ((CONS (a , b)) NIL)))))))


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


REAL_MEASURABLE_ON_MAX |- (!f. (!g. (!s. ((((real_measurable_on f) s) /\ ((real_measurable_on g) s)) ==> ((real_measurable_on (\x. ((real_max (f x)) (g x)))) s)))))


UNBOUNDED_HALFSPACE_COMPONENT_GT |- (!a. (!k. (~ (bounded (GSPEC (\GEN%PVAR%839. (?x. (((SETSPEC GEN%PVAR%839) ((real_gt (x $ k)) a)) x))))))))


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


Seq.eqseq_rcons |- (!s1. (!s2. (!x1. (!x2. ((((rcons s1) x1) = ((rcons s2) x2)) = ((s1 = s2) /\ (x1 = x2)))))))


REAL_CONTINUOUS_ON_DIV |- (!f. (!g. (!s. ((((real_continuous_on f) s) /\ (((real_continuous_on g) s) /\ (!x. ((x IN s) ==> (~ ((g x) = (real_of_num (NUMERAL _0)))))))) ==> ((real_continuous_on (\x. ((real_div (f x)) (g x)))) s)))))


Asfutbf.affine_hull_neg |- (!v. (((IMAGE vector_neg) ((hull affine) ((vec (NUMERAL _0)) INSERT (v INSERT EMPTY)))) = ((hull affine) ((vec (NUMERAL _0)) INSERT ((vector_neg v) INSERT EMPTY)))))


INTERIOR_STD_SIMPLEX |- ((interior ((hull convex) ((vec (NUMERAL _0)) INSERT (GSPEC (\GEN%PVAR%1807. (?i. (((SETSPEC GEN%PVAR%1807) (((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV)))) (basis i)))))))) = (GSPEC (\GEN%PVAR%1808. (?x. (((SETSPEC GEN%PVAR%1808) ((!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV))) ==> ((real_lt (real_of_num (NUMERAL _0))) (x $ i)))) /\ ((real_lt ((sum ((NUMERAL (BIT1 _0)) .. (dimindex UNIV))) (\i. (x $ i)))) (real_of_num (NUMERAL (BIT1 _0)))))) x)))))


HIGHER_COMPLEX_DERIVATIVE_1 |- (!f. (!z. ((((higher_complex_derivative (NUMERAL (BIT1 _0))) f) z) = ((complex_derivative f) z))))


CROSS_ROTATION_MATRIX |- (!A. (!x. (!y. ((rotation_matrix A) ==> (((cross ((matrix_vector_mul A) x)) ((matrix_vector_mul A) y)) = ((matrix_vector_mul A) ((cross x) y)))))))


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


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


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


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


Conforming.lemma_card_node_eq_set_of_orbits |- (!x. (!V. (!E. (!f. (!y. (((FAN (x , (V , E))) /\ ((!v. ((v IN V) ==> ((CARD (((set_of_edge v) V) E)) > (NUMERAL (BIT1 _0))))) /\ ((f IN (node_set (hypermap1_of_fanx (x , (V , E))))) /\ (y IN f)))) ==> ((CARD (GSPEC (\GEN%PVAR%5670. (?z. (((SETSPEC GEN%PVAR%5670) (?i. ((i >= (NUMERAL _0)) /\ (z = (x , ((pr2 y) , ((((((((power_map_points sigma_fan) x) V) E) (pr2 y)) (pr3 y)) i) , (((((((power_map_points sigma_fan) x) V) E) (pr2 y)) (pr3 y)) (SUC i))))))))) z))))) = (CARD (((((set_of_orbits_points_fan x) V) E) (pr2 y)) (pr3 y))))))))))


NSUM_UNION_LZERO |- (!f. (!u. (!v. (((FINITE v) /\ (!x. (((x IN u) /\ (~ (x IN v))) ==> ((f x) = (NUMERAL _0))))) ==> (((nsum (u UNION v)) f) = ((nsum v) f))))))


Lp_gen_theory.iso_dart_pairs_trans |- (!H. (!G. (!g. (((hyp_iso g) (G , H)) ==> (!P. ((!p. ((p IN ((dart H) CROSS (dart H))) ==> (P p))) ==> (!p. ((p IN ((dart G) CROSS (dart G))) ==> (P ((g (FST p)) , (g (SND p))))))))))))


EVEN_AND_ODD |- (!n. (~ ((EVEN n) /\ (ODD n))))


EXISTS_TRIPLED_THM |- (!P. ((? (GABS (\f. (!x. (!y. (!z. ((GEQ (f (x , (y , z)))) (((P x) y) z)))))))) = (?x. (?y. (?z. (((P x) y) z))))))


COMPLEX_DERIVATIVE_TRANSFORM_WITHIN_OPEN |- (!f. (!g. (!s. (!z. (((open s) /\ (((holomorphic_on f) s) /\ (((holomorphic_on g) s) /\ ((z IN s) /\ (!w. ((w IN s) ==> ((f w) = (g w)))))))) ==> (((complex_derivative f) z) = ((complex_derivative g) z)))))))


REAL_DIFFERENTIABLE_WITHIN_LOG |- (!s. (!x. (((real_lt (real_of_num (NUMERAL _0))) x) ==> ((real_differentiable log) ((within (atreal x)) s)))))


FRECHET_DERIVATIVE_WITHIN_CLOSED_INTERVAL |- (!f. (!f'. (!x. (!a. (!b. (((!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV))) ==> ((real_lt (a $ i)) (b $ i)))) /\ ((x IN (closed_interval ((CONS (a , b)) NIL))) /\ (((has_derivative f) f') ((within (at x)) (closed_interval ((CONS (a , b)) NIL)))))) ==> (((frechet_derivative f) ((within (at x)) (closed_interval ((CONS (a , b)) NIL)))) = f')))))))


RESTRICTION_IDEMP |- (!s. (!f. (((RESTRICTION s) ((RESTRICTION s) f)) = ((RESTRICTION s) f))))


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


REAL_STEINHAUS |- (!s. (((real_measurable s) /\ ((real_lt (real_of_num (NUMERAL _0))) (real_measure s))) ==> (?d. (((real_lt (real_of_num (NUMERAL _0))) d) /\ ((open_real_interval ((real_neg d) , d)) SUBSET (GSPEC (\GEN%PVAR%4987. (?x. (?y. (((SETSPEC GEN%PVAR%4987) ((x IN s) /\ (y IN s))) ((real_sub x) y)))))))))))


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


Seq2.all_subseq |- (!s1. (!s2. (!P. ((((subseq s1) s2) /\ ((all P) s2)) ==> ((all P) s1)))))


Trigonometry.UIVNNRR1 |- (!V. (((aff_ge V) EMPTY) = ((aff_gt V) EMPTY)))


SYMMETRIC_LINEAR_IMAGE |- (!f. (!s. (((!x. ((x IN s) ==> ((vector_neg x) IN s))) /\ (linear f)) ==> (!x. ((x IN ((IMAGE f) s)) ==> ((vector_neg x) IN ((IMAGE f) s)))))))


Oxl_def.BIJ_NUMSEG |- (!s. (!a. (!b. (((BIJ (\j. (j + s))) (a .. b)) ((a + s) .. (b + s))))))


NORM_CSIN_POW_2 |- (!z. (((real_pow (vector_norm (csin z))) (NUMERAL (BIT0 (BIT1 _0)))) = ((real_div ((real_add (exp ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (Im z)))) ((real_sub (real_inv (exp ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (Im z))))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (cos ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (Re z))))))) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0))))))))


HAS_REAL_DERIVATIVE_ZERO_UNIQUE_STRONG_INTERVAL |- (!f. (!a. (!b. (!k. (!y. (((COUNTABLE k) /\ (((real_continuous_on f) (closed_real_interval ((CONS (a , b)) NIL))) /\ (((f a) = y) /\ (!x. ((x IN ((closed_real_interval ((CONS (a , b)) NIL)) DIFF k)) ==> (((has_real_derivative f) (real_of_num (NUMERAL _0))) ((within (atreal x)) (closed_real_interval ((CONS (a , b)) NIL))))))))) ==> (!x. ((x IN (closed_real_interval ((CONS (a , b)) NIL))) ==> ((f x) = y)))))))))


REAL_INTEGRABLE_DECREASING_PRODUCT_UNIV |- (!f. (!g. (!B. ((((real_integrable_on f) UNIV) /\ ((!x. (!y. (((real_le x) y) ==> ((real_le (g y)) (g x))))) /\ (!x. ((real_le (real_abs (g x))) B)))) ==> ((real_integrable_on (\x. ((real_mul (g x)) (f x)))) UNIV)))))


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


CONIC_IMP_CONNECTED |- (!s. ((conic s) ==> (connected s)))


OUTSIDE_CONNECTED_COMPONENT_LE |- (!s. ((outside s) = (GSPEC (\GEN%PVAR%2012. (?x. (((SETSPEC GEN%PVAR%2012) (!B. (?y. (((real_le B) (vector_norm y)) /\ (((connected_component (UNIV DIFF s)) x) y))))) x))))))


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


INT_LE_ADDR |- (!x. (!y. (((int_le x) ((int_add x) y)) = ((int_le (int_of_num (NUMERAL _0))) y))))


Seq2.take_belast |- (!s. (!x. (!n. (((n < (sizel s)) /\ ((NUMERAL _0) < n)) ==> (((take n) ((belast x) s)) = ((CONS x) ((take (n - (NUMERAL (BIT1 _0)))) s)))))))


CLOSED_UNION_COMPLEMENT_COMPONENTS |- (!s. (!c. (((closed s) /\ (c SUBSET (components (UNIV DIFF s)))) ==> (closed (s UNION (UNIONS c))))))


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


AR_PCROSS_EQ |- (!s. (!t. ((AR (s PCROSS t)) = ((AR s) /\ ((AR t) /\ ((~ (s = EMPTY)) /\ (~ (t = EMPTY))))))))


Odxlstc.DEFORMATION_DIST_LE_ALL_COM_LE3 |- (((is_scs_v39 s) /\ (((MMs_v39 s) w) /\ (((scs_k_v39 s) = k) /\ (((NUMERAL (BIT1 (BIT1 _0))) <= k) /\ (!i. ((~ ((i MOD k) = (l MOD k))) ==> ((real_lt (((scs_a_v39 s) l) i)) (distance ((w l) , (w i)))))))))) ==> (?e. (((real_lt (real_of_num (NUMERAL _0))) e) /\ (!t. (!i. ((((real_lt (real_of_num (NUMERAL _0))) t) /\ (((real_lt t) e) /\ (~ ((i MOD k) = (l MOD k))))) ==> ((real_lt (((scs_a_v39 s) l) i)) (distance (((% ((real_sub (real_of_num (NUMERAL (BIT1 _0)))) t)) (w l)) , (w i))))))))))


NPRODUCT_SUPPORT |- (!f. (!s. (((nproduct (((support *) f) s)) f) = ((nproduct s) f))))


SEPARATING_HYPERPLANE_AFFINE_HULLS |- (!s. (!t. (((~ (s = EMPTY)) /\ ((~ (t = EMPTY)) /\ ((DISJOINT ((hull affine) s)) ((hull affine) t)))) ==> (?a. (?b. (?c. ((~ (a = (vec (NUMERAL _0)))) /\ (((real_lt b) c) /\ ((!x. ((x IN s) ==> (((dot a) x) = b))) /\ (!x. ((x IN t) ==> (((dot a) x) = c))))))))))))


EVENTUALLY_FALSE |- (!net. (((eventually (\x. F)) net) = (trivial_limit net)))


PATH_COMPONENT_EMPTY |- (!x. (((path_component EMPTY) x) = EMPTY))


INFSUM_EQ |- (!f. (!g. (!k. ((((summable k) f) /\ (((summable k) g) /\ (!x. ((x IN k) ==> ((f x) = (g x)))))) ==> (((infsum k) f) = ((infsum k) g))))))


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


MEASURABLE_ON_LEBESGUE_MEASURABLE_PREIMAGE_BOREL_EQ |- (!f. (!s. ((((measurable_on f) s) /\ (lebesgue_measurable s)) = (!t. ((borel t) ==> (lebesgue_measurable (GSPEC (\GEN%PVAR%3662. (?x. (((SETSPEC GEN%PVAR%3662) ((x IN s) /\ ((f x) IN t))) x))))))))))


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


REALLIM_1_OVER_POW |- (!k. (((NUMERAL (BIT1 _0)) <= k) ==> (((---> (\n. (real_inv ((real_pow (real_of_num n)) k)))) (real_of_num (NUMERAL _0))) sequentially)))


COMPLEX_STONE_WEIERSTRASS |- (!P. (!s. (((compact s) /\ ((!f. ((P f) ==> ((continuous_on f) s))) /\ ((!c. (P (\x. c))) /\ ((!f. ((P f) ==> (P (\x. (cnj (f x)))))) /\ ((!f. (!g. (((P f) /\ (P g)) ==> (P (\x. ((vector_add (f x)) (g x))))))) /\ ((!f. (!g. (((P f) /\ (P g)) ==> (P (\x. ((complex_mul (f x)) (g x))))))) /\ (!x. (!y. (((x IN s) /\ ((y IN s) /\ (~ (x = y)))) ==> (?f. ((P f) /\ (~ ((f x) = (f y)))))))))))))) ==> (!f. (!e. ((((continuous_on f) s) /\ ((real_lt (real_of_num (NUMERAL _0))) e)) ==> (?g. ((P g) /\ (!x. ((x IN s) ==> ((real_lt (vector_norm ((vector_sub (f x)) (g x)))) e)))))))))))


Nonlinear_lemma.lmdih5_0' |- (!y1. (!y2. (!y3. (!y4. (!y5. (!y6. (((real_le ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) y5) ==> (((((((lmdih5_x_div_sqrtdelta_posbranch ((real_mul y1) y1)) ((real_mul y2) y2)) ((real_mul y3) y3)) ((real_mul y4) y4)) ((real_mul y5) y5)) ((real_mul y6) y6)) = (real_of_num (NUMERAL _0))))))))))


WF_EREC |- ((WF <<) ==> (!H. ((!f. (!g. (!x. ((!z. (((<< z) x) ==> ((f z) = (g z)))) ==> (((H f) x) = ((H g) x)))))) ==> (?!f. (!x. ((f x) = ((H f) x)))))))


Functional_equation.functional_asn797k |- (asn797k = ((mul6 proj_x1) (uni (asn , ((mul6 (constant6 cos797)) (uni (sin , ((div6 (constant6 pi)) proj_x1))))))))


ITSET_EQ |- (!s. (!f. (!g. (!b. (((FINITE s) /\ ((!x. ((x IN s) ==> ((f x) = (g x)))) /\ ((!x. (!y. (!s. ((~ (x = y)) ==> (((f x) ((f y) s)) = ((f y) ((f x) s))))))) /\ (!x. (!y. (!s. ((~ (x = y)) ==> (((g x) ((g y) s)) = ((g y) ((g x) s)))))))))) ==> ((((ITSET f) s) b) = (((ITSET g) s) b)))))))


CLOSED_SIMPLE_PATH_IMAGE |- (!g. ((simple_path g) ==> (closed (path_image g))))


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


Lp_ineqs_proofs.rho_fan_eq_rho |- (!E. (!V. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!v. ((v IN V) ==> ((rho_fan v) = (rho (vector_norm v)))))))))))


MEASURE_DISJOINT_UNIONS |- (!m. (!f. (((FINITE f) /\ ((!s. ((s IN f) ==> ((has_measure s) (m s)))) /\ (!s. (!t. (((s IN f) /\ ((t IN f) /\ (~ (s = t)))) ==> ((DISJOINT s) t)))))) ==> ((measure (UNIONS f)) = ((sum f) m)))))


CONVEX_HULL_INTERIOR_SUBSET |- (!s. (((hull convex) (interior s)) SUBSET (interior ((hull convex) s))))


Hypermap_iso.fan_hypermaps_iso2 |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((iso (hypermap_of_fan (V , E))) (hypermap1_of_fanx ((vec (NUMERAL _0)) , (V , E)))))))


LOG_DIV |- (!x. (!y. ((((real_lt (real_of_num (NUMERAL _0))) x) /\ ((real_lt (real_of_num (NUMERAL _0))) y)) ==> ((log ((real_div x) y)) = ((real_sub (log x)) (log y))))))


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


CONTINUOUS_FROM_CLOSED_GRAPH |- (!f. (!s. (!t. (((compact t) /\ ((((IMAGE f) s) SUBSET t) /\ (closed (GSPEC (\GEN%PVAR%967. (?x. (((SETSPEC GEN%PVAR%967) (x IN s)) ((pastecart x) (f x))))))))) ==> ((continuous_on f) s)))))


FINITE_CBALL |- (!a. (!r. ((FINITE (cball (a , r))) = ((real_le r) (real_of_num (NUMERAL _0))))))


Polar_fan.AZIM_CYCLE_SING |- (!x. (!u. (!v. (((((azim_cycle (v INSERT EMPTY)) x) u) v) = v))))


CONTINUOUS_WITHIN_CNJ |- (!s. (!z. ((continuous cnj) ((within (at z)) s))))


ODD_MOD |- (!n. ((ODD n) = ((n MOD (NUMERAL (BIT0 (BIT1 _0)))) = (NUMERAL (BIT1 _0)))))


LIPSCHITZ_IMP_RECTIFIABLE_PATH |- (!g. (!b. ((!x. (!y. (((x IN (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL))) /\ (y IN (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL)))) ==> ((real_le (vector_norm ((vector_sub (g x)) (g y)))) ((real_mul b) (vector_norm ((vector_sub x) y))))))) ==> (rectifiable_path g))))


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


MATRIX_ADD_LNEG |- (!A. (((matrix_add (matrix_neg A)) A) = (mat (NUMERAL _0))))


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


UNIFORMLY_CONTINUOUS_ON_MUL |- (!f. (!g. (!s. ((((uniformly_continuous_on (lift o f)) s) /\ (((uniformly_continuous_on g) s) /\ ((bounded ((IMAGE (lift o f)) s)) /\ (bounded ((IMAGE g) s))))) ==> ((uniformly_continuous_on (\x. ((% (f x)) (g x)))) s)))))


Tame_list.mapAt_ALT |- (((((mapAt NIL) f) xs) = xs) /\ ((((mapAt ((CONS n) ns)) f) xs) = (((COND (n < (LENGTH xs))) (((mapAt ns) f) (((list_update xs) n) (f ((#!# xs) n))))) (((mapAt ns) f) xs))))


COUNTABLE_OPEN_CONNECTED_COMPONENTS |- (!s. (!t. ((open s) ==> (COUNTABLE (GSPEC (\GEN%PVAR%1917. (?x. (((SETSPEC GEN%PVAR%1917) (x IN t)) ((connected_component s) x)))))))))


WOSET_INSEG_ORDINAL |- (woset (GABS (\f. (!x. (!y. ((GEQ (f (x , y))) ((ordinal x) /\ ((ordinal y) /\ ((inseg x) y)))))))))


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


Merge_ineq.delta_pos_4_784 |- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. ((((real_le (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) x1) /\ (((real_le x1) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) /\ (((real_le (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) x2) /\ (((real_le x2) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) /\ (((real_le (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) x3) /\ (((real_le x3) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) /\ (((real_le (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) x4) /\ (((real_le x4) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) /\ (((real_le (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) x5) /\ (((real_le x5) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) /\ (((real_le (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) x6) /\ ((real_le x6) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))))))))))))))) ==> (((real_lt (real_of_num (NUMERAL _0))) ((((((delta_x x1) x2) x3) x4) x5) x6)) /\ (((real_lt (real_of_num (NUMERAL _0))) x1) /\ (((real_lt (real_of_num (NUMERAL _0))) x2) /\ (((real_lt (real_of_num (NUMERAL _0))) x3) /\ (((real_lt (real_of_num (NUMERAL _0))) x4) /\ (((real_lt (real_of_num (NUMERAL _0))) x5) /\ ((real_lt (real_of_num (NUMERAL _0))) x6))))))))))))))


INT_POW_EQ_0 |- (!x. (!n. ((((int_pow x) n) = (int_of_num (NUMERAL _0))) = ((x = (int_of_num (NUMERAL _0))) /\ (~ (n = (NUMERAL _0)))))))


FACE_OF_HALFSPACE_GE |- (!f. (!a. (!b. (((face_of f) (GSPEC (\GEN%PVAR%2156. (?x. (((SETSPEC GEN%PVAR%2156) ((real_ge ((dot a) x)) b)) x))))) = ((f = EMPTY) \/ ((f = (GSPEC (\GEN%PVAR%2157. (?x. (((SETSPEC GEN%PVAR%2157) (((dot a) x) = b)) x))))) \/ (f = (GSPEC (\GEN%PVAR%2158. (?x. (((SETSPEC GEN%PVAR%2158) ((real_ge ((dot a) x)) b)) x)))))))))))


MEASURABLE_MIDPOINT_CONVEX_IMP_CONVEX_1D |- (!f. (!s. ((((measurable_on (lift o f)) s) /\ ((convex s) /\ (!x. (!y. (((x IN s) /\ (y IN s)) ==> ((real_le (f (midpoint (x , y)))) ((real_div ((real_add (f x)) (f y))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))))))) ==> ((convex_on f) s))))


CONTINUOUS_ON_LIFT_PRODUCT |- (!f. (!s. (!t. (((FINITE t) /\ (!i. ((i IN t) ==> ((continuous_on (\x. (lift ((f x) i)))) s)))) ==> ((continuous_on (\x. (lift ((product t) (f x))))) s)))))


CONNECTED_NEST_GEN |- (!s. (((!n. ((closed (s n)) /\ (connected (s n)))) /\ ((?n. (compact (s n))) /\ (!m. (!n. ((m <= n) ==> ((s n) SUBSET (s m))))))) ==> (connected (INTERS (GSPEC (\GEN%PVAR%1155. (?n. (((SETSPEC GEN%PVAR%1155) (n IN UNIV)) (s n)))))))))


REALLIM_ATREAL_ID |- (((---> (\x. x)) a) (atreal a))


ABSOLUTELY_CONTINUOUS_ISOMETRIC_COMPOSE |- (!f. (!g. (!s. ((!x. (!y. ((distance ((f x) , (f y))) = (distance (x , y))))) ==> (((absolutely_continuous_on (f o g)) s) = ((absolutely_continuous_on g) s))))))


MEASURABLE_SCALING_EQ |- (!s. (!c. ((~ (c = (real_of_num (NUMERAL _0)))) ==> ((measurable ((IMAGE (\x. ((% c) x))) s)) = (measurable s)))))


NEGATIONS_CBALL |- (!r. (((IMAGE vector_neg) (cball ((vec (NUMERAL _0)) , r))) = (cball ((vec (NUMERAL _0)) , r))))


TAN_SUB |- (!x. (!y. (((~ ((cos x) = (real_of_num (NUMERAL _0)))) /\ ((~ ((cos y) = (real_of_num (NUMERAL _0)))) /\ (~ ((cos ((real_sub x) y)) = (real_of_num (NUMERAL _0)))))) ==> ((tan ((real_sub x) y)) = ((real_div ((real_sub (tan x)) (tan y))) ((real_add (real_of_num (NUMERAL (BIT1 _0)))) ((real_mul (tan x)) (tan y))))))))


MEASURABLE_TETRAHEDRON |- (!a. (!b. (!c. (!d. (measurable ((hull convex) (a INSERT (b INSERT (c INSERT (d INSERT EMPTY))))))))))


SUBSET_CONVEX_HULL_FRONTIER |- (!s. ((bounded s) ==> (s SUBSET ((hull convex) (frontier s)))))


RTC_NE_IMP_TC |- (!R. (!x. (!y. (((((RTC R) x) y) /\ (~ (x = y))) ==> (((TC R) x) y)))))


NPRODUCT_CLOSED |- (!P. (!f. (!s. (((P (NUMERAL (BIT1 _0))) /\ ((!x. (!y. (((P x) /\ (P y)) ==> (P (x * y))))) /\ (!a. ((a IN s) ==> (P (f a)))))) ==> (P ((nproduct s) f))))))


CX_COSH |- ((Cx ((real_div ((real_add (exp x)) (real_inv (exp x)))) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) = (ccos ((complex_mul ii) (Cx x))))


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


POSITIVE_DEFINITE_2 |- (!A. ((positive_definite A) = ((((A $ (NUMERAL (BIT0 (BIT1 _0)))) $ (NUMERAL (BIT1 _0))) = ((A $ (NUMERAL (BIT1 _0))) $ (NUMERAL (BIT0 (BIT1 _0))))) /\ (((real_lt (real_of_num (NUMERAL _0))) ((A $ (NUMERAL (BIT1 _0))) $ (NUMERAL (BIT1 _0)))) /\ (((real_lt (real_of_num (NUMERAL _0))) ((A $ (NUMERAL (BIT0 (BIT1 _0)))) $ (NUMERAL (BIT0 (BIT1 _0))))) /\ ((real_lt ((real_pow ((A $ (NUMERAL (BIT1 _0))) $ (NUMERAL (BIT0 (BIT1 _0))))) (NUMERAL (BIT0 (BIT1 _0))))) ((real_mul ((A $ (NUMERAL (BIT1 _0))) $ (NUMERAL (BIT1 _0)))) ((A $ (NUMERAL (BIT0 (BIT1 _0)))) $ (NUMERAL (BIT0 (BIT1 _0)))))))))))


Misc_defs_and_lemmas.UNIONS_SUBSET |- (!U. (!X. ((!A. ((A IN U) ==> (A SUBSET X))) ==> ((UNIONS U) SUBSET X))))


CARD_LT_REFL |- (!s. (~ (s <_c s)))


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


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


REAL_OPEN_RATIONAL |- (!P. (((real_open (GSPEC (\GEN%PVAR%4978. (?x. (((SETSPEC GEN%PVAR%4978) (P x)) x))))) /\ (?x. (P x))) ==> (?x. ((rational x) /\ (P x)))))


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


Packing3.INTERS_INTER_INTERS_ALT |- (!f. (!g. (!P. (((INTERS (GSPEC (\GEN%PVAR%6019. (?x. (((SETSPEC GEN%PVAR%6019) (x IN f)) (P x)))))) INTER (INTERS (GSPEC (\GEN%PVAR%6020. (?y. (((SETSPEC GEN%PVAR%6020) (y IN g)) (P y))))))) = (INTERS (GSPEC (\GEN%PVAR%6021. (?u. (((SETSPEC GEN%PVAR%6021) (u IN (f UNION g))) (P u))))))))))


Trigonometry.ISRTTNZ |- (((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)) /\ (!p. (!i. (!j. (((W p) /\ (((NUMERAL _0) <= i) /\ ((i <= j) /\ (j < n)))) ==> (((real_add ((arg_diff p) (((ITER i) xicm) p))) ((arg_diff (((ITER i) xicm) p)) (((ITER j) xicm) p))) = ((arg_diff p) (((ITER j) xicm) p)))))))))


ANR_COVERING_SPACE |- (!p. (!s. (!c. ((((covering_space (c , p)) s) /\ (ANR c)) ==> (ANR s)))))


Miqmcsn.mod_prep |- (((i + (NUMERAL (BIT1 (BIT1 _0)))) = (((NUMERAL (BIT1 _0)) * (NUMERAL (BIT1 (BIT1 _0)))) + i)) /\ (((i + (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) = (((NUMERAL (BIT1 _0)) * (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) + i)) /\ (((i + (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) = (((NUMERAL (BIT1 _0)) * (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) + i)) /\ (((i + (NUMERAL (BIT0 (BIT1 (BIT1 _0))))) = (((NUMERAL (BIT1 _0)) * (NUMERAL (BIT0 (BIT1 (BIT1 _0))))) + i)) /\ ((SUC (((NUMERAL (BIT1 _0)) * k) + i)) = (((NUMERAL (BIT1 _0)) * k) + (SUC i)))))))


DIFFERENTIAL_ZERO_MAXMIN_COMPONENT |- (!f. (!x. (!e. (!k. ((((NUMERAL (BIT1 _0)) <= k) /\ ((k <= (dimindex UNIV)) /\ (((real_lt (real_of_num (NUMERAL _0))) e) /\ (((!y. ((y IN (ball (x , e))) ==> ((real_le ((f y) $ k)) ((f x) $ k)))) \/ (!y. ((y IN (ball (x , e))) ==> ((real_le ((f x) $ k)) ((f y) $ k))))) /\ ((differentiable f) (at x)))))) ==> ((((jacobian f) (at x)) $ k) = (vec (NUMERAL _0))))))))


IN_EPIGRAPH |- (!x. (!y. ((((pastecart x) (lift y)) IN ((epigraph s) f)) = ((x IN s) /\ ((real_le (f x)) y)))))


HAS_INTEGRAL_SPIKE_INTERIOR_EQ |- (!f. (!g. (!a. (!b. (!y. ((!x. ((x IN (open_interval (a , b))) ==> ((g x) = (f x)))) ==> ((((has_integral f) y) (closed_interval ((CONS (a , b)) NIL))) = (((has_integral g) y) (closed_interval ((CONS (a , b)) NIL))))))))))


Trigonometry2.WHEN_K_DIFF0_ARCV |- ((~ (k = (real_of_num (NUMERAL _0)))) ==> ((((arcV (vec (NUMERAL _0))) u) v) = (((arcV (vec (NUMERAL _0))) u) ((% (real_abs k)) v))))


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


INT_ABS_SUB |- (!x. (!y. ((int_abs ((int_sub x) y)) = (int_abs ((int_sub y) x)))))


UNION_FRONTIER |- (!s. (!t. (((frontier s) UNION (frontier t)) = ((frontier (s UNION t)) UNION ((frontier (s INTER t)) UNION ((frontier s) INTER (frontier t)))))))


REAL_CONVEX_ON_CONTINUOUS |- (!f. (!s. (((real_open s) /\ ((real_convex_on f) s)) ==> ((real_continuous_on f) s))))


Ssrnat.max0n |- (!n. (((maxn (NUMERAL _0)) n) = n))


REAL_SECOND_MEAN_VALUE_THEOREM_BONNET |- (!f. (!g. (!a. (!b. (((~ ((closed_real_interval ((CONS (a , b)) NIL)) = EMPTY)) /\ (((real_integrable_on f) (closed_real_interval ((CONS (a , b)) NIL))) /\ ((!x. ((x IN (closed_real_interval ((CONS (a , b)) NIL))) ==> ((real_le (real_of_num (NUMERAL _0))) (g x)))) /\ (!x. (!y. (((x IN (closed_real_interval ((CONS (a , b)) NIL))) /\ ((y IN (closed_real_interval ((CONS (a , b)) NIL))) /\ ((real_le x) y))) ==> ((real_le (g x)) (g y)))))))) ==> (?c. ((c IN (closed_real_interval ((CONS (a , b)) NIL))) /\ (((real_integral (closed_real_interval ((CONS (a , b)) NIL))) (\x. ((real_mul (g x)) (f x)))) = ((real_mul (g b)) ((real_integral (closed_real_interval ((CONS (c , b)) NIL))) f))))))))))


INT_MUL_AC |- ((((int_mul m) n) = ((int_mul n) m)) /\ ((((int_mul ((int_mul m) n)) p) = ((int_mul m) ((int_mul n) p))) /\ (((int_mul m) ((int_mul n) p)) = ((int_mul n) ((int_mul m) p)))))


Conforming.NEGLIGIBLE_AFF_3_INTER_BALL |- (!x. (!v. (!u. (!r. (negligible ((aff (x INSERT (v INSERT (u INSERT EMPTY)))) INTER ((normball x) r)))))))


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


REAL_COMPACT_IMP_CLOSED |- (!s. ((real_compact s) ==> (real_closed s)))


ARZELA_ASCOLI_LIPSCHITZ |- (!f. (!s. (!t. (!b. (((compact s) /\ ((bounded t) /\ ((!n. (!x. ((x IN s) ==> (~ ((((IMAGE (f n)) s) INTER t) = EMPTY))))) /\ (!n. (!x. (!y. (((x IN s) /\ (y IN s)) ==> ((real_le (vector_norm ((vector_sub ((f n) x)) ((f n) y)))) ((real_mul b) (vector_norm ((vector_sub x) y))))))))))) ==> (?g. ((!x. (!y. (((x IN s) /\ (y IN s)) ==> ((real_le (vector_norm ((vector_sub (g x)) (g y)))) ((real_mul b) (vector_norm ((vector_sub x) y))))))) /\ (?r. ((!m. (!n. ((m < n) ==> ((r m) < (r n))))) /\ (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?N. (!n. (!x. (((n >= N) /\ (x IN s)) ==> ((real_lt (vector_norm ((vector_sub ((f (r n)) x)) (g x)))) e))))))))))))))))


REALLIM_CONST_EQ |- (!net. (!c. (!d. ((((---> (\x. c)) d) net) = ((trivial_limit net) \/ (c = d))))))


MOD_MOD_EXP_MIN |- (!x. (!p. (!m. (!n. ((~ (p = (NUMERAL _0))) ==> (((x MOD (p EXP m)) MOD (p EXP n)) = (x MOD (p EXP ((MIN m) n)))))))))


CONIC_POSITIVE_ORTHANT |- (conic (GSPEC (\GEN%PVAR%1467. (?x. (((SETSPEC GEN%PVAR%1467) (!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV))) ==> ((real_le (real_of_num (NUMERAL _0))) (x $ i))))) x)))))


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


FRONTIER_UNION |- (!s. (!t. ((((closure s) INTER (closure t)) = EMPTY) ==> ((frontier (s UNION t)) = ((frontier s) UNION (frontier t))))))


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


Ssrnat.distn_eq1 |- (!m. (!n. ((((distn m) n) = (NUMERAL (BIT1 _0))) = (((COND (m < n)) ((SUC m) = n)) (m = (SUC n))))))


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


Seq.size_behead |- (!s. ((sizel (behead s)) = ((sizel s) - (NUMERAL (BIT1 _0)))))


LIM_CSIN_OVER_X |- (((--> (\z. ((complex_div (csin z)) z))) (Cx (real_of_num (NUMERAL (BIT1 _0))))) (at (Cx (real_of_num (NUMERAL _0)))))


Functional_equation.functional_proj_x3 |- (proj_x3 = (rotate3 proj_x1))


Ssrbool.addbI |- (!x. (!y1. (!y2. ((((XOR x) y1) = ((XOR x) y2)) ==> (y1 = y2)))))


CONVEX_ON_IMP_MIDPOINT_CONVEX |- (!f. (!s. (!x. (!y. ((((convex_on f) s) /\ ((x IN s) /\ (y IN s))) ==> ((real_le (f (midpoint (x , y)))) ((real_div ((real_add (f x)) (f y))) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))))))))


CARD_EQ_COMPACT_SETS |- ((GSPEC (\GEN%PVAR%1256. (?s. (((SETSPEC GEN%PVAR%1256) (compact s)) s)))) =_c UNIV)


HAUSDIST_ALT |- (!s. (!t. (((bounded s) /\ ((bounded t) /\ ((~ (s = EMPTY)) /\ (~ (t = EMPTY))))) ==> ((hausdist (s , t)) = (sup (GSPEC (\GEN%PVAR%1203. (?x. (((SETSPEC GEN%PVAR%1203) (x IN UNIV)) (real_abs ((real_sub (setdist ((x INSERT EMPTY) , s))) (setdist ((x INSERT EMPTY) , t)))))))))))))


RELATIVE_BOUNDARY_RETRACT_OF_PUNCTURED_AFFINE_HULL |- (!s. (!a. (((convex s) /\ ((compact s) /\ (a IN (relative_interior s)))) ==> ((retract_of (s DIFF (relative_interior s))) (((hull affine) s) DELETE a)))))


FCCOVERABLE_INTERMEDIATE_CLOSURE |- (!s. (!t. (((s SUBSET t) /\ ((t SUBSET (closure s)) /\ (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?c. ((FINITE c) /\ (((UNIONS c) = s) /\ (!t. ((t IN c) ==> ((connected t) /\ ((bounded t) /\ ((real_le (diameter t)) e)))))))))))) ==> (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?c. ((FINITE c) /\ (((UNIONS c) = t) /\ (!t. ((t IN c) ==> ((connected t) /\ ((bounded t) /\ ((real_le (diameter t)) e)))))))))))))


DENSE_LIMIT_POINTS |- (!x. (((GSPEC (\GEN%PVAR%583. (?x. (((SETSPEC GEN%PVAR%583) ((limit_point_of x) s)) x)))) = UNIV) = ((closure s) = UNIV)))


Rogers.QXSKIIT |- (!vf. (!b. (((FINITE ((IMAGE vf) UNIV)) /\ ((~ (affine_dependent ((IMAGE vf) UNIV))) /\ (!i. (!j. (((vf i) = (vf j)) ==> ((b i) = (b j))))))) ==> (?!p. ((p IN ((hull affine) ((IMAGE vf) UNIV))) /\ (!i. (!j. (((dot p) ((vector_sub (vf i)) (vf j))) = ((real_sub (b i)) (b j))))))))))


SECOND_MEAN_VALUE_THEOREM_GEN |- (!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))) /\ (((integral (closed_interval ((CONS (a , b)) NIL))) (\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)))))))))))))


EXTREME_POINT_OF_CONIC |- (!s. (!x. (((conic s) /\ ((extreme_point_of x) s)) ==> (x = (vec (NUMERAL _0))))))


REAL_DIFFERENTIABLE_WITHIN_SQRT |- (!s. (!x. (((real_lt (real_of_num (NUMERAL _0))) x) ==> ((real_differentiable sqrt) ((within (atreal x)) s)))))


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


Seq.count_predUI |- (!a1. (!a2. (!s. ((((count ((predU a1) a2)) s) + ((count ((predI a1) a2)) s)) = (((count a1) s) + ((count a2) s))))))


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


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