LIEB |- (!f. (!g. (!s. (!t. (((!n. ((absolutely_integrable_on (f n)) s)) /\ (((absolutely_integrable_on g) s) /\ ((negligible t) /\ (!x. ((x IN (s DIFF t)) ==> (((--> (\n. ((f n) x))) (g x)) sequentially)))))) ==> (((--> (\n. ((vector_sub ((integral s) (\x. (lift (vector_norm ((vector_sub ((f n) x)) (g x))))))) ((vector_sub ((integral s) (\x. (lift (vector_norm ((f n) x)))))) ((integral s) (\x. (lift (vector_norm (g x))))))))) (vec (NUMERAL _0))) sequentially))))))


RADON |- (!c. ((affine_dependent c) ==> (?m. (?p. ((m SUBSET c) /\ ((p SUBSET c) /\ (((DISJOINT m) p) /\ (~ ((DISJOINT ((hull convex) m)) ((hull convex) p))))))))))


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


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


REAL_ADD_SIN |- (!x. (!y. (((real_add (sin x)) (sin y)) = ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((real_mul (sin ((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)))))))))))


Seq.reshapeKr |- (!sh. (!s. (((sizel s) <= (sumn sh)) ==> ((flatten ((reshape sh) s)) = s))))


WF_REFL |- (!x. ((WF <<) ==> (~ ((<< x) x))))


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


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


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


Nonlinear_lemma.quartic_has_real_derivative |- (!x. (!c0. (!c1. (!c2. (!c3. (!c4. (((has_real_derivative (\x. ((real_add ((real_mul c0) (real_of_num (NUMERAL (BIT1 _0))))) ((real_add ((real_mul c1) ((real_pow x) (NUMERAL (BIT1 _0))))) ((real_add ((real_mul c2) ((real_pow x) (NUMERAL (BIT0 (BIT1 _0)))))) ((real_add ((real_mul c3) ((real_pow x) (NUMERAL (BIT1 (BIT1 _0)))))) ((real_mul c4) ((real_pow x) (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))))))))) ((real_add ((real_mul c0) (real_of_num (NUMERAL _0)))) ((real_add ((real_mul c1) ((real_mul (real_of_num (NUMERAL (BIT1 _0)))) ((real_mul ((real_pow x) ((NUMERAL (BIT1 _0)) - (NUMERAL (BIT1 _0))))) (real_of_num (NUMERAL (BIT1 _0))))))) ((real_add ((real_mul c2) ((real_mul ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((real_pow x) ((NUMERAL (BIT0 (BIT1 _0))) - (NUMERAL (BIT1 _0)))))) (real_of_num (NUMERAL (BIT1 _0)))))) ((real_add ((real_mul c3) ((real_mul ((real_mul (real_of_num (NUMERAL (BIT1 (BIT1 _0))))) ((real_pow x) ((NUMERAL (BIT1 (BIT1 _0))) - (NUMERAL (BIT1 _0)))))) (real_of_num (NUMERAL (BIT1 _0)))))) ((real_mul c4) ((real_mul ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) ((real_pow x) ((NUMERAL (BIT0 (BIT0 (BIT1 _0)))) - (NUMERAL (BIT1 _0)))))) (real_of_num (NUMERAL (BIT1 _0)))))))))) (atreal x))))))))


LIM_RE_UBOUND |- (!net. (!f. (!l. (!b. (((~ (trivial_limit net)) /\ ((((--> f) l) net) /\ ((eventually (\x. ((real_le (Re (f x))) b))) net))) ==> ((real_le (Re l)) b))))))


CONVEX_HALFSPACE_IM_LE |- (!b. (convex (GSPEC (\GEN%PVAR%4774. (?z. (((SETSPEC GEN%PVAR%4774) ((real_le (Im z)) b)) z))))))


TRACE_MUL_POSITIVE_SEMIDEFINITE_LE |- (!A. (!B. (((positive_semidefinite A) /\ (positive_semidefinite B)) ==> ((real_le (trace ((matrix_mul A) B))) ((real_mul (trace A)) (trace B))))))


REAL_INTEGRABLE_ON_LITTLE_SUBINTERVALS |- (!f. (!a. (!b. ((!x. ((x IN (closed_real_interval ((CONS (a , b)) NIL))) ==> (?d. (((real_lt (real_of_num (NUMERAL _0))) d) /\ (!u. (!v. (((x IN (closed_real_interval ((CONS (u , v)) NIL))) /\ (!y. ((y IN (closed_real_interval ((CONS (u , v)) NIL))) ==> (((real_lt (real_abs ((real_sub y) x))) d) /\ (y IN (closed_real_interval ((CONS (a , b)) NIL))))))) ==> ((real_integrable_on f) (closed_real_interval ((CONS (u , v)) NIL)))))))))) ==> ((real_integrable_on f) (closed_real_interval ((CONS (a , b)) NIL)))))))


INT_POW_NEG |- (!x. (!n. (((int_pow (int_neg x)) n) = (((COND (EVEN n)) ((int_pow x) n)) (int_neg ((int_pow x) n))))))


CLOSURE_IMAGE_BOUNDED |- (!f. (!s. ((((continuous_on f) (closure s)) /\ (bounded s)) ==> ((closure ((IMAGE f) s)) = ((IMAGE f) (closure s))))))


PATH_COMPOSE_JOIN |- (!f. (!p. (!q. ((f o ((++ p) q)) = ((++ (f o p)) (f o q))))))


REAL_INFSUM_SUB |- (!x. (!y. (!s. ((((real_summable s) x) /\ ((real_summable s) y)) ==> (((real_infsum s) (\i. ((real_sub (x i)) (y i)))) = ((real_sub ((real_infsum s) x)) ((real_infsum s) y)))))))


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


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


Seq2.list_sum_EL_alt |- (!s. (!f. (((list_sum s) f) = ((sum ((NUMERAL (BIT1 _0)) .. (sizel s))) (\i. (f ((EL (i - (NUMERAL (BIT1 _0)))) s)))))))


Functional_equation.functional_proj_y6 |- (proj_y6 = (rotate6 proj_y1))


AFFINE_INDEPENDENT_CONVEX_AFFINE_HULL |- (!s. (!t. (((~ (affine_dependent s)) /\ (t SUBSET s)) ==> (((hull convex) t) = (((hull affine) t) INTER ((hull convex) s))))))


RECTIFIABLE_PATH_SYM |- (!p. (!q. ((((pathfinish p) = (pathstart q)) /\ ((pathfinish q) = (pathstart p))) ==> ((rectifiable_path ((++ p) q)) = (rectifiable_path ((++ q) p))))))


Lp_ineqs_proofs.tau_sum4 |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> ((lp_tau (V , E)) ==> (!f. ((f IN (face_set (hypermap_of_fan (V , E)))) ==> (((CARD f) = (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) ==> (((sum f) (rhazim_fan (V , E))) = ((real_add ((tau_fan (V , E)) f)) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((real_add pi) sol0))))))))))))))


LINEAR_IMP_HAS_BOUNDED_VARIATION |- (!f. (((linear f) /\ (bounded s)) ==> ((has_bounded_variation_on f) s)))


Fnjlbxs.scriptL_eq |- (!S. (!k. ((((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)))))) ==> (!v. ((v IN ((IMAGE (vector_of_list (NUMERAL (BIT1 (BIT1 _0))))) S)) ==> ((scriptL (set_of_list (((list_of_vector (NUMERAL (BIT1 (BIT1 _0)))) k) v))) = ((list_sum (((list_of_vector (NUMERAL (BIT1 (BIT1 _0)))) k) v)) (\v. (lmfun ((real_div (vector_norm v)) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))))))))))))


RCONE_GT_NEG |- (!v0. (!v1. (!a. ((((rcone_gt v0) v1) (real_neg a)) = ((IMAGE (\x. ((vector_sub ((% (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) v0)) x))) (UNIV DIFF (((rcone_ge v0) v1) a)))))))


RE_CACS_BOUND |- (!z. ((real_le (real_abs (Re (cacs z)))) pi))


COMPLEX_NEG_MINUS1 |- (!x. ((vector_neg x) = ((complex_mul (vector_neg (Cx (real_of_num (NUMERAL (BIT1 _0)))))) x)))


AFFINE_NEGATIONS |- (!s. ((affine s) ==> (affine ((IMAGE vector_neg) s))))


DIAMETER_SCALING |- (!a. (!s. ((bounded s) ==> ((diameter ((IMAGE (\x. ((% a) x))) s)) = ((real_mul (real_abs a)) (diameter s))))))


Constants_approx.INTERVAL_SQRT |- ((((interval_arith x) (a , b)) /\ (((real_le ((real_mul c) c)) a) /\ ((real_le b) ((real_mul d) d)))) ==> ((interval_arith (sqrt x)) ((real_abs c) , (real_abs d))))


INT_LT_MUL |- (!x. (!y. ((((int_lt (int_of_num (NUMERAL _0))) x) /\ ((int_lt (int_of_num (NUMERAL _0))) y)) ==> ((int_lt (int_of_num (NUMERAL _0))) ((int_mul x) y)))))


MEASURE_INTERVAL_2_ALT |- ((!a. (!b. ((measure (closed_interval ((CONS (a , b)) NIL))) = (((COND (((real_lt (a $ (NUMERAL (BIT1 _0)))) (b $ (NUMERAL (BIT1 _0)))) /\ ((real_lt (a $ (NUMERAL (BIT0 (BIT1 _0))))) (b $ (NUMERAL (BIT0 (BIT1 _0))))))) ((real_mul ((real_sub (b $ (NUMERAL (BIT1 _0)))) (a $ (NUMERAL (BIT1 _0))))) ((real_sub (b $ (NUMERAL (BIT0 (BIT1 _0))))) (a $ (NUMERAL (BIT0 (BIT1 _0))))))) (real_of_num (NUMERAL _0)))))) /\ (!a. (!b. ((measure (open_interval (a , b))) = (((COND (((real_lt (a $ (NUMERAL (BIT1 _0)))) (b $ (NUMERAL (BIT1 _0)))) /\ ((real_lt (a $ (NUMERAL (BIT0 (BIT1 _0))))) (b $ (NUMERAL (BIT0 (BIT1 _0))))))) ((real_mul ((real_sub (b $ (NUMERAL (BIT1 _0)))) (a $ (NUMERAL (BIT1 _0))))) ((real_sub (b $ (NUMERAL (BIT0 (BIT1 _0))))) (a $ (NUMERAL (BIT0 (BIT1 _0))))))) (real_of_num (NUMERAL _0)))))))


ALL2_MAP2 |- (!l. (!m. ((((ALL2 P) ((MAP f) l)) ((MAP g) m)) = (((ALL2 (\x. (\y. ((P (f x)) (g y))))) l) m))))


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


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


ANGLE_EQ_0_DIST_ABS |- (!A. (!B. (!C. (((angle (A , (B , C))) = (real_of_num (NUMERAL _0))) = ((~ (A = B)) /\ ((~ (C = B)) /\ ((distance (A , C)) = (real_abs ((real_sub (distance (A , B))) (distance (C , B)))))))))))


Pack2.finite_voronoi2 |- (!v. (!S. ((packing S) ==> (FINITE (GSPEC (\GEN%PVAR%5977. (?x. (?y. (((SETSPEC GEN%PVAR%5977) ((y IN (S DELETE v)) /\ ((x = v) /\ (y IN (ball (v , (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0))))))))))) ((bis_le x) y))))))))))


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


Functional_equation.domain6_ineq |- (!c1. (!c2. (!P. (!F1. (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (!a1. (!a2. (!a3. (!a4. (!a5. (!a6. (!b1. (!b2. (!b3. (!b4. (!b5. (!b6. ((((domain6 P) c2) c1) ==> (((ineq ((CONS (a1 , (x1 , b1))) ((CONS (a2 , (x2 , b2))) ((CONS (a3 , (x3 , b3))) ((CONS (a4 , (x4 , b4))) ((CONS (a5 , (x5 , b5))) ((CONS (a6 , (x6 , b6))) NIL))))))) ((((((P x1) x2) x3) x4) x5) x6)) ==> (((ineq ((CONS (a1 , (x1 , b1))) ((CONS (a2 , (x2 , b2))) ((CONS (a3 , (x3 , b3))) ((CONS (a4 , (x4 , b4))) ((CONS (a5 , (x5 , b5))) ((CONS (a6 , (x6 , b6))) NIL))))))) (F1 ((((((c2 x1) x2) x3) x4) x5) x6))) = ((ineq ((CONS (a1 , (x1 , b1))) ((CONS (a2 , (x2 , b2))) ((CONS (a3 , (x3 , b3))) ((CONS (a4 , (x4 , b4))) ((CONS (a5 , (x5 , b5))) ((CONS (a6 , (x6 , b6))) NIL))))))) (F1 ((((((c1 x1) x2) x3) x4) x5) x6))))))))))))))))))))))))))))


HAS_BOUNDED_REAL_VARIATION_ON_MAX |- (!f. (!g. (!s. ((((has_bounded_real_variation_on f) s) /\ ((has_bounded_real_variation_on g) s)) ==> ((has_bounded_real_variation_on (\x. ((real_max (f x)) (g x)))) s)))))


HOLOMORPHIC_ON_CSIN |- (!s. ((holomorphic_on csin) s))


LOCALLY_PATH_CONNECTED_RIGHT_INVERTIBLE_IMAGE |- (!f. (!g. (!s. ((((continuous_on f) s) /\ (((continuous_on g) ((IMAGE f) s)) /\ ((((IMAGE g) ((IMAGE f) s)) SUBSET s) /\ ((!x. ((x IN ((IMAGE f) s)) ==> ((f (g x)) = x))) /\ ((locally path_connected) s))))) ==> ((locally path_connected) ((IMAGE f) s))))))


ABSOLUTELY_REAL_INTEGRABLE_SUM |- (!f. (!s. (!t. (((FINITE t) /\ (!a. ((a IN t) ==> ((absolutely_real_integrable_on (f a)) s)))) ==> ((absolutely_real_integrable_on (\x. ((sum t) (\a. ((f a) x))))) s)))))


CPOW_REAL_REAL |- (!w. (!z. (((real w) /\ ((real z) /\ ((real_lt (real_of_num (NUMERAL _0))) (Re w)))) ==> (((cpow w) z) = (Cx (exp ((real_mul (Re z)) (log (Re w)))))))))


EXISTS_CNJ |- ((?z. (P (cnj z))) = (?z. (P z)))


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


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


Bump.MCELL2_CELL_PARAMETERS_EXIST |- (!V. (!ul. (!X. (((packing V) /\ ((saturated V) /\ ((X = ((mcell2 V) ul)) /\ ((ul IN ((barV V) (NUMERAL (BIT1 (BIT1 _0))))) /\ (~ (negligible X)))))) ==> ((FST ((cell_params V) X)) = (NUMERAL (BIT0 (BIT1 _0))))))))


NEGLIGIBLE_AFFINE_HULL_2 |- (!a. (!b. (negligible ((hull affine) (a INSERT (b INSERT EMPTY))))))


Functional_equation.functional_rotate2 |- (!f. ((rotate2 f) = (((((((compose6 f) proj_x2) proj_x3) proj_x1) proj_x5) proj_x6) proj_x4)))


ROOT_EQ_0 |- (!n. (!x. ((((root n) x) = (real_of_num (NUMERAL _0))) = (x = (real_of_num (NUMERAL _0))))))


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


PROPER_LOCALLY_INJECTIVE_OPEN_IMP_HOMEOMORPHISM |- (!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%2105. (?x. (((SETSPEC GEN%PVAR%2105) ((x IN s) /\ ((f x) IN k))) x))))))) /\ ((!u. (((open_in ((subtopology euclidean) s)) u) ==> ((open_in ((subtopology euclidean) t)) ((IMAGE f) u)))) /\ (!x. ((x IN s) ==> (?u. ((x IN u) /\ (((open_in ((subtopology euclidean) s)) u) /\ (!y. (!z. (((y IN u) /\ ((z IN u) /\ ((f y) = (f z)))) ==> (y = z))))))))))))))) ==> (?g. ((homeomorphism (s , t)) (f , g)))))))


REAL_MEASURE_REAL_NEGLIGIBLE_UNION |- (!s. (!t. (((real_measurable s) /\ ((real_measurable t) /\ (real_negligible (s INTER t)))) ==> ((real_measure (s UNION t)) = ((real_add (real_measure s)) (real_measure t))))))


COMPLEX_TAYLOR_MVT |- (!f. (!w. (!z. (!n. ((!i. (!x. (((x IN (closed_segment ((CONS (w , z)) NIL))) /\ (i <= n)) ==> (((has_complex_derivative (f i)) ((f (i + (NUMERAL (BIT1 _0)))) x)) (at x))))) ==> (?u. ((u IN (closed_segment ((CONS (w , z)) NIL))) /\ ((Re ((f (NUMERAL _0)) z)) = (Re ((vector_add ((vsum ((NUMERAL _0) .. n)) (\i. ((complex_mul ((f i) w)) ((complex_div ((complex_pow ((vector_sub z) w)) i)) (Cx (real_of_num (FACT i)))))))) ((complex_mul ((complex_mul ((f (n + (NUMERAL (BIT1 _0)))) u)) ((complex_div ((complex_pow ((vector_sub z) u)) n)) (Cx (real_of_num (FACT n)))))) ((vector_sub z) w))))))))))))


Lp_main_estimate.v_prime_eq |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!d. (!k. ((d IN ((darts_k k) (hypermap_of_fan (V , E)))) ==> (((v_prime V) ((face (hypermap_of_fan (V , E))) d)) = (GSPEC (\GEN%PVAR%8731. (?i. (((SETSPEC GEN%PVAR%8731) (i < k)) (FST (((POWER (f_fan_pair_ext (V , E))) i) d)))))))))))))


COMPLEX_NOT_ROOT_UNITY |- (!n. (((NUMERAL (BIT1 _0)) <= n) ==> (?u. (((vector_norm u) = (real_of_num (NUMERAL (BIT1 _0)))) /\ (~ (((complex_pow u) n) = (Cx (real_of_num (NUMERAL (BIT1 _0))))))))))


POWER_REAL_SERIES_CONV_IMP_ABSCONV_WEAK |- (!a. (!k. (!w. (!z. ((((real_summable k) (\n. ((real_mul (a n)) ((real_pow z) n)))) /\ ((real_lt (real_abs w)) (real_abs z))) ==> ((real_summable k) (\n. ((real_mul (real_abs (a n))) ((real_pow w) n)))))))))


Geomdetail.aff |- (aff = (hull affine))


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


Lp_ineqs_proofs.ln_fan_ly |- (!E. (!V. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!v. ((v IN V) ==> ((ln_fan v) = (ly (vector_norm v)))))))))))


REALLIM_AT_ZERO |- (!f. (!l. (!a. ((((---> f) l) (at a)) = (((---> (\x. (f ((vector_add a) x)))) l) (at (vec (NUMERAL _0))))))))


VECTOR_EXPAND_4 |- (!x. (x = (vector ((CONS (x $ (NUMERAL (BIT1 _0)))) ((CONS (x $ (NUMERAL (BIT0 (BIT1 _0))))) ((CONS (x $ (NUMERAL (BIT1 (BIT1 _0))))) ((CONS (x $ (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) NIL)))))))


Pwssrat.rotate_to_hd |- (((rotate_to ((CONS h) t)) h) = ((CONS h) t))


Seq.eq_all_r |- (!s1. (!s2. ((!x. (((MEM x) s1) = ((MEM x) s2))) ==> (!a. (((all a) s1) = ((all a) s2))))))


SUBMETRIC_UNIV |- (((submetric m) UNIV) = m)


OPEN_NON_PATH_COMPONENT |- (!s. (!x. ((open s) ==> (open (s DIFF ((path_component s) x))))))


HAS_VECTOR_DERIVATIVE_VECTOR_POLYNOMIAL_FUNCTION |- (!p. ((vector_polynomial_function p) ==> (?p'. ((vector_polynomial_function p') /\ (!x. (((has_vector_derivative p) (p' x)) (at x)))))))


Pack2.measure_unions_sum_voronoi |- (!S. (!p. (!r. ((((real_le (real_of_num (NUMERAL _0))) r) /\ ((packing S) /\ (saturated S))) ==> ((measure (UNIONS (GSPEC (\GEN%PVAR%5985. (?w. (?v. (((SETSPEC GEN%PVAR%5985) ((v IN (S INTER (ball (p , ((real_add r) (real_of_num (NUMERAL (BIT1 _0)))))))) /\ (w = S))) ((voronoi_closed w) v)))))))) = ((sum (S INTER (ball (p , ((real_add r) (real_of_num (NUMERAL (BIT1 _0)))))))) (\v. (measure ((voronoi_closed S) v)))))))))


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


INT_SUB_LDISTRIB |- (!x. (!y. (!z. (((int_mul x) ((int_sub y) z)) = ((int_sub ((int_mul x) y)) ((int_mul x) z))))))


INF_INSERT |- (!x. (!s. ((bounded ((IMAGE lift) s)) ==> ((inf (x INSERT s)) = (((COND (s = EMPTY)) x) ((real_min x) (inf s)))))))


Functional_equation.delta_all_atn |- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. ((((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))))))) ==> ((((((((rhazim_x x1) x2) x3) x4) x5) x6) = ((((((rhazimatn_x x1) x2) x3) x4) x5) x6)) /\ ((((((((rhazim2_x x1) x2) x3) x4) x5) x6) = ((((((rhazim2atn_x x1) x2) x3) x4) x5) x6)) /\ ((((((((rhazim3_x x1) x2) x3) x4) x5) x6) = ((((((rhazim3atn_x x1) x2) x3) x4) x5) x6)) /\ ((((((((ldih_x x1) x2) x3) x4) x5) x6) = ((((((ldihatn_x x1) x2) x3) x4) x5) x6)) /\ ((((((((ldih2_x x1) x2) x3) x4) x5) x6) = ((((((ldih2atn_x x1) x2) x3) x4) x5) x6)) /\ ((((((((ldih3_x x1) x2) x3) x4) x5) x6) = ((((((ldih3atn_x x1) x2) x3) x4) x5) x6)) /\ ((((((((ldih6_x x1) x2) x3) x4) x5) x6) = ((((((ldih6atn_x x1) x2) x3) x4) x5) x6)) /\ ((((((((gchi1_x x1) x2) x3) x4) x5) x6) = ((real_mul (gchi (sqrt x1))) ((((((dihatn_x x1) x2) x3) x4) x5) x6))) /\ ((((((((gchi2_x x1) x2) x3) x4) x5) x6) = ((real_mul (gchi (sqrt x2))) ((((((dih2atn_x x1) x2) x3) x4) x5) x6))) /\ ((((((((gchi3_x x1) x2) x3) x4) x5) x6) = ((real_mul (gchi (sqrt x3))) ((((((dih3atn_x x1) x2) x3) x4) x5) x6))) /\ ((((((((gchi4_x x1) x2) x3) x4) x5) x6) = ((real_mul (gchi (sqrt x4))) ((((((dih4atn_x x1) x2) x3) x4) x5) x6))) /\ ((((((((gchi5_x x1) x2) x3) x4) x5) x6) = ((real_mul (gchi (sqrt x5))) ((((((dih5atn_x x1) x2) x3) x4) x5) x6))) /\ ((((((((gchi6_x x1) x2) x3) x4) x5) x6) = ((real_mul (gchi (sqrt x6))) ((((((dih6atn_x x1) x2) x3) x4) x5) x6))) /\ ((((((((dih_x x1) x2) x3) x4) x5) x6) = ((((((dihatn_x x1) x2) x3) x4) x5) x6)) /\ ((((((((dih2_x x1) x2) x3) x4) x5) x6) = ((((((dih2atn_x x1) x2) x3) x4) x5) x6)) /\ ((((((((dih3_x x1) x2) x3) x4) x5) x6) = ((((((dih3atn_x x1) x2) x3) x4) x5) x6)) /\ ((((((((dih4_x x1) x2) x3) x4) x5) x6) = ((((((dih4atn_x x1) x2) x3) x4) x5) x6)) /\ ((((((((dih5_x x1) x2) x3) x4) x5) x6) = ((((((dih5atn_x x1) x2) x3) x4) x5) x6)) /\ ((((((((dih6_x x1) x2) x3) x4) x5) x6) = ((((((dih6atn_x x1) x2) x3) x4) x5) x6)) /\ ((((((((eta2_126 x1) x2) x3) x4) x5) x6) = ((real_pow (((eta_x x1) x2) x6)) (NUMERAL (BIT0 (BIT1 _0))))) /\ ((((((((eta2_135 x1) x2) x3) x4) x5) x6) = ((real_pow (((eta_x x1) x3) x5)) (NUMERAL (BIT0 (BIT1 _0))))) /\ (((((((eta2_456 x1) x2) x3) x4) x5) x6) = ((real_pow (((eta_x x4) x5) x6)) (NUMERAL (BIT0 (BIT1 _0)))))))))))))))))))))))))))))))))


REAL_CONTINUOUS_AT_AZIM |- (!v. (!w. (!w1. (!w2. ((~ (coplanar (v INSERT (w INSERT (w1 INSERT (w2 INSERT EMPTY)))))) ==> ((real_continuous (((azim v) w) w1)) (at w2)))))))


BARYCENTRE_NOT_IN_SET |- (!s. (((~ (affine_dependent s)) /\ (~ (?a. (s = (a INSERT EMPTY))))) ==> (~ ((barycentre s) IN s))))


ISOMETRY_SUBSET_SUBSPACE |- (!s. (!t. (((subspace s) /\ ((subspace t) /\ ((dim s) <= (dim t)))) ==> (?f. ((linear f) /\ ((((IMAGE f) s) SUBSET t) /\ (!x. ((x IN s) ==> ((vector_norm (f x)) = (vector_norm x))))))))))


EXISTS_SUBPATH_OF_PATH |- (!g. (!a. (!b. (((path g) /\ ((a IN (path_image g)) /\ (b IN (path_image g)))) ==> (?h. ((path h) /\ (((pathstart h) = a) /\ (((pathfinish h) = b) /\ ((path_image h) SUBSET (path_image g))))))))))


Constants_approx.INTERVAL_INV_NEG |- ((((interval_arith x) (a , b)) /\ ((real_lt b) (real_of_num (NUMERAL _0)))) ==> ((interval_arith (real_inv x)) ((real_inv b) , (real_inv a))))


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


JOINABLE_COMPONENTS_EQ |- (!s. (!t. (!c1. (!c2. (((connected t) /\ ((t SUBSET s) /\ ((c1 IN (components s)) /\ ((c2 IN (components s)) /\ ((~ ((c1 INTER t) = EMPTY)) /\ (~ ((c2 INTER t) = EMPTY))))))) ==> (c1 = c2))))))


REALLIM_ZERO_NEGINFINITY |- (!f. (!l. ((((---> (\x. (f ((real_div (real_of_num (NUMERAL (BIT1 _0)))) x)))) l) (atreal (real_of_num (NUMERAL _0)))) ==> (((---> f) l) at_neginfinity))))


Miqmcsn.J_SCS_3M1_prime |- ((((scs_J_v39 ((scs_prop_equ_v39 scs_3M1_prime) i)) i1) j) = F)


MEASURABLE_INTER_HALFSPACE_GE |- (!s. (!a. (!i. ((measurable s) ==> (measurable (s INTER (GSPEC (\GEN%PVAR%3589. (?x. (((SETSPEC GEN%PVAR%3589) ((real_ge (x $ i)) a)) x))))))))))


CONIC_HALFSPACE_GE |- (!a. (conic (GSPEC (\GEN%PVAR%1465. (?x. (((SETSPEC GEN%PVAR%1465) ((real_ge ((dot a) x)) (real_of_num (NUMERAL _0)))) x))))))


REAL_MEASURE_DIFF_SUBSET |- (!s. (!t. (((real_measurable s) /\ ((real_measurable t) /\ (t SUBSET s))) ==> ((real_measure (s DIFF t)) = ((real_sub (real_measure s)) (real_measure t))))))


MEASURABLE_ON_MEASURABLE_PREIMAGE_OPEN_INTERVAL |- (!f. (!s. ((measurable s) ==> (((measurable_on f) s) = (!a. (!b. (measurable (GSPEC (\GEN%PVAR%3653. (?x. (((SETSPEC GEN%PVAR%3653) ((x IN s) /\ ((f x) IN (open_interval (a , b))))) x)))))))))))


MBOUNDED_MBALL |- (!m. (!c. (!b. ((mbounded m) ((mball m) (c , b))))))


PATH_CONNECTED_PUNCTURED_BALL |- (!a. (!r. (((NUMERAL (BIT0 (BIT1 _0))) <= (dimindex UNIV)) ==> (path_connected ((ball (a , r)) DELETE a)))))


BOUNDED_RECTIFIABLE_PATH_IMAGE |- (!g. ((rectifiable_path g) ==> (bounded (path_image g))))


Functional_equation.functional_acs_sqrt_x1_d4 |- (acs_sqrt_x1_d4 = (uni (acs , ((scalar6 proj_y1) ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))))


REAL_DERIVATIVE_POS_RIGHT_MAXIMUM |- (!f. (!f'. (!a. (!b. (!e. ((((real_lt a) b) /\ (((real_lt (real_of_num (NUMERAL _0))) e) /\ ((((has_real_derivative f) f') ((within (atreal b)) (closed_real_interval ((CONS (a , b)) NIL)))) /\ (!x. (((x IN (closed_real_interval ((CONS (a , b)) NIL))) /\ ((real_lt (real_abs ((real_sub x) b))) e)) ==> ((real_le (f x)) (f b))))))) ==> ((real_le (real_of_num (NUMERAL _0))) f')))))))


TAGGED_DIVISION_OF_ANOTHER |- (!p. (!s. (!s'. ((((tagged_partial_division_of p) s') /\ ((!t. (!k. (((t , k) IN p) ==> (k SUBSET s)))) /\ (!x. ((x IN s) ==> (?t. (?k. (((t , k) IN p) /\ (x IN k)))))))) ==> ((tagged_division_of p) s)))))


NSUM_ADD_GEN |- (!f. (!g. (!s. (((FINITE (GSPEC (\GEN%PVAR%172. (?x. (((SETSPEC GEN%PVAR%172) ((x IN s) /\ (~ ((f x) = (NUMERAL _0))))) x))))) /\ (FINITE (GSPEC (\GEN%PVAR%173. (?x. (((SETSPEC GEN%PVAR%173) ((x IN s) /\ (~ ((g x) = (NUMERAL _0))))) x)))))) ==> (((nsum s) (\x. ((f x) + (g x)))) = (((nsum s) f) + ((nsum s) g)))))))


Imjxphr.DEFORMATION_DIST_LE_V3_DEFOR_A |- (((is_scs_v39 s) /\ (((MMs_v39 s) w) /\ (((scs_k_v39 s) = k) /\ (((NUMERAL (BIT1 (BIT1 _0))) < k) /\ (((w l) = v2) /\ (((w (SUC l)) = v1) /\ ((((real_pow (vector_norm v1)) (NUMERAL (BIT0 (BIT1 _0)))) = x1) /\ ((((real_pow (vector_norm v2)) (NUMERAL (BIT0 (BIT1 _0)))) = x2) /\ ((((real_pow (vector_norm ((vector_sub v1) v2))) (NUMERAL (BIT0 (BIT1 _0)))) = x6) /\ ((a = (real_neg (real_of_num (NUMERAL (BIT1 _0))))) /\ (!i. (((~ ((l MOD k) = (i MOD k))) /\ (~ ((l MOD k) = ((SUC i) MOD k)))) ==> ((real_lt (((scs_a_v39 s) l) i)) (distance (v2 , (w i)))))))))))))))) ==> (!i. (((~ ((l MOD k) = (i MOD k))) /\ (~ ((l MOD k) = ((SUC i) MOD k)))) ==> (?e. (((real_lt (real_of_num (NUMERAL _0))) e) /\ (!t. ((((real_lt (real_of_num (NUMERAL _0))) t) /\ ((real_lt t) e)) ==> ((real_lt (((scs_a_v39 s) l) i)) (distance (((((((((v3_defor_v1 a) v1) v2) x1) x2) x6) x6) ((real_sub x2) t)) , (w i)))))))))))


CBALL_INTERVAL_0 |- (!e. ((cball ((vec (NUMERAL _0)) , e)) = (closed_interval ((CONS ((vector_neg (lift e)) , (lift e))) NIL))))


ABSOLUTELY_CONTINUOUS_ON_DIVISION |- (!f. (!a. (!b. (!d. (((division_of d) (closed_interval ((CONS (a , b)) NIL))) ==> ((!k. ((k IN d) ==> ((absolutely_continuous_on f) k))) = ((absolutely_continuous_on f) (closed_interval ((CONS (a , b)) NIL)))))))))


Merge_ineq.eta135_x_bd |- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (!c. ((~ ((real_lt ((real_add ((real_mul ((((((unit6 x1) x2) x3) x4) x5) x6)) c)) ((real_mul ((real_pow (((eta_x x1) x3) x5)) (NUMERAL (BIT0 (BIT1 _0))))) (real_neg (real_of_num (NUMERAL (BIT1 _0))))))) (real_of_num (NUMERAL _0)))) ==> (((real_lt c) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ==> ((real_lt ((real_pow (((eta_x x1) x3) x5)) (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))))))))))


ABSOLUTELY_CONTINUOUS_ON_TRANSLATION |- (!f. (!s. (!a. (((absolutely_continuous_on (\x. ((vector_add a) (f x)))) s) = ((absolutely_continuous_on f) s)))))


RE_CSQRT |- (!z. ((real_le (real_of_num (NUMERAL _0))) (Re (csqrt z))))


RSC_INC_RC |- (!R. (!x. (!y. ((((RC R) x) y) ==> (((RSC R) x) y)))))


RE_COMPLEX_DIV_LE_0 |- (!a. (!b. (((real_le (Re ((complex_div a) b))) (real_of_num (NUMERAL _0))) = ((real_le (Re ((complex_mul a) (cnj b)))) (real_of_num (NUMERAL _0))))))


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


ABSOLUTELY_CONTINUOUS_DIFFERENTIABLE_BV |- (!f. (!a. (!b. ((((differentiable_on f) (closed_interval ((CONS (a , b)) NIL))) /\ ((has_bounded_variation_on f) (closed_interval ((CONS (a , b)) NIL)))) ==> ((absolutely_continuous_on f) (closed_interval ((CONS (a , b)) NIL)))))))


SUBSPACE_ORTHOGONAL_TO_VECTORS |- (!s. (subspace (GSPEC (\GEN%PVAR%163. (?y. (((SETSPEC GEN%PVAR%163) (!x. ((x IN s) ==> ((orthogonal x) y)))) y))))))