MEASURABLE_CONTINUOUS_COMPOSE |- (!f. (!g. ((((measurable_on f) UNIV) /\ (((continuous_on g) UNIV) /\ (!k. ((negligible k) ==> (negligible (GSPEC (\GEN%PVAR%3702. (?x. (((SETSPEC GEN%PVAR%3702) ((g x) IN k)) x))))))))) ==> ((measurable_on (f o g)) UNIV))))


CONTINUOUS_ON_COMPLEX_RMUL |- (!f. (!s. (((continuous_on f) s) ==> ((continuous_on (\x. ((complex_mul (f x)) c))) s))))


VECTOR_VARIATION_LIFT_ABS |- (!f. (!s. (((has_bounded_variation_on (\x. (lift (f x)))) s) ==> ((real_le ((vector_variation s) (\x. (lift (real_abs (f x)))))) ((vector_variation s) (\x. (lift (f x))))))))


INCREASING_BOUNDED_REAL_VARIATION |- (!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))))) ==> ((has_bounded_real_variation_on f) (closed_real_interval ((CONS (a , b)) NIL)))))))


TRIANGULATION_INTER_SIMPLEX |- (!tr. (!t. (!t'. (((triangulation tr) /\ ((t IN tr) /\ (t' IN tr))) ==> ((t INTER t') = ((hull convex) ((GSPEC (\GEN%PVAR%2491. (?x. (((SETSPEC GEN%PVAR%2491) ((extreme_point_of x) t)) x)))) INTER (GSPEC (\GEN%PVAR%2492. (?x. (((SETSPEC GEN%PVAR%2492) ((extreme_point_of x) t')) x)))))))))))


Functional_equation.functional_halfbump_x1 |- (halfbump_x1 = (promote1_to_6 halfbump_x))


Nonlinear_lemma.vol3f_lm0 |- (!y1. (!y2. (!y3. (!y4. (!y5. (!y6. ((((real_le ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) y1) /\ (((real_le y2) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) /\ ((real_le y6) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) ==> ((((((vol3f y1) y2) y6) sqrt2) lmfun) = ((((((vol3f_sqrt2_lmplus y1) y2) y3) y4) y5) y6)))))))))


INT_SUB_LNEG |- (!x. (!y. (((int_sub (int_neg x)) y) = (int_neg ((int_add x) y)))))


Lp_ineqs_proofs.lp_cond_trans |- (!L. (!V. (!E. (!g. (!h. (!ds. (((lp_cond (L , (g , h))) (V , E)) ==> ((lp_cond (((split_list_list L) ds) , (g , h))) (V , ((add_diag_list (g , (V , E))) ds))))))))))


AFFINE_INDEPENDENT_STDBASIS |- (~ (affine_dependent ((vec (NUMERAL _0)) INSERT (GSPEC (\GEN%PVAR%1517. (?i. (((SETSPEC GEN%PVAR%1517) (((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV)))) (basis i))))))))


INTEGER_POW |- (!x. (!n. ((integer x) ==> (integer ((real_pow x) n)))))


RECTIFIABLE_PATH_VECTOR_POLYNOMIAL_FUNCTION |- (!p. ((vector_polynomial_function p) ==> (rectifiable_path p)))


PATH_CONNECTED_NEGATIONS |- (!s. ((path_connected s) ==> (path_connected ((IMAGE vector_neg) s))))


HOMOTOPY_EQUIVALENT |- (!s. (!t. (((homotopy_equivalent s) t) = (?f. (?g. (?h. (((continuous_on f) s) /\ ((((IMAGE f) s) SUBSET t) /\ (((continuous_on g) t) /\ ((((IMAGE g) t) SUBSET s) /\ (((continuous_on h) t) /\ ((((IMAGE h) t) SUBSET s) /\ (((((homotopic_with (\x. T)) (s , s)) (g o f)) I) /\ ((((homotopic_with (\x. T)) (t , t)) (f o h)) I))))))))))))))


MAXIMAL_AFFINE_INDEPENDENT_SUBSET_AFFINE |- (!s. (!b. (((affine s) /\ ((b SUBSET s) /\ ((~ (affine_dependent b)) /\ (!b'. (((b SUBSET b') /\ ((b' SUBSET s) /\ (~ (affine_dependent b')))) ==> (b' = b)))))) ==> (((hull affine) b) = s))))


Lp_ineqs_proofs.rhazim3_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 ((rhazim3_fan (V , E)) d)) ((real_add pi) sol0))))))))))


NONNEGATIVE_ABSOLUTELY_REAL_INTEGRABLE |- (!f. (!s. (((!x. ((x IN s) ==> ((real_le (real_of_num (NUMERAL _0))) (f x)))) /\ ((real_integrable_on f) s)) ==> ((absolutely_real_integrable_on f) s))))


LIM_NULL_COMPLEX_POW_EQ |- (!net. (!f. (!n. ((~ (n = (NUMERAL _0))) ==> ((((--> (\x. ((complex_pow (f x)) n))) (Cx (real_of_num (NUMERAL _0)))) net) = (((--> f) (Cx (real_of_num (NUMERAL _0)))) net))))))


Trigonometry1.COS_PERIODIC_N2PI |- (!x. (!n. ((cos ((real_add x) ((real_mul (real_of_num n)) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) pi)))) = (cos x))))


ARC_SUBPATH_EQ |- (!g. (!u. (!v. ((arc (((subpath u) v) g)) = ((path (((subpath u) v) g)) /\ ((~ (u = v)) /\ (!x. (!y. (((x IN (closed_segment ((CONS (u , v)) NIL))) /\ ((y IN (closed_segment ((CONS (u , v)) NIL))) /\ ((g x) = (g y)))) ==> (x = y))))))))))


Tame_lemmas.fully_surrounded_sum_sol |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> (((sum (face_set (hypermap_of_fan (V , E)))) (\f. ((sol (vec (NUMERAL _0))) ((((dartset_leads_into_fan (vec (NUMERAL _0))) V) E) ((IMAGE (ext_dart (V , E))) f))))) = ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) pi))))))


LIFT_IN_INTERVAL |- ((!a. (!b. (!x. (((lift x) IN (closed_interval ((CONS (a , b)) NIL))) = (x IN (closed_real_interval ((CONS ((drop a) , (drop b))) NIL))))))) /\ (!a. (!b. (!x. (((lift x) IN (open_interval (a , b))) = (x IN (open_real_interval ((drop a) , (drop b)))))))))


CONTRACTIBLE_CONVEX_TWEAK_BOUNDARY_POINTS |- (!s. (!t. (((convex s) /\ (((relative_interior s) SUBSET t) /\ (t SUBSET (closure s)))) ==> (contractible t))))


CONIC_RELATIVE_INTERIOR |- (!s. (((conic s) /\ ((vec (NUMERAL _0)) IN (relative_interior s))) ==> (conic (relative_interior s))))


Merge_ineq.nonf_mud_135_x |- (!a. (!b. (!c. (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((real_le (real_of_num (NUMERAL _0))) a) ==> ((((((((((mud_135_x_v1 a) b) c) x1) x2) x3) x4) x5) x6) = ((real_mul (((mu_y a) (sqrt x1)) (sqrt x3))) (sqrt (((((((((delta_135_x ((real_mul a) a)) ((real_mul b) b)) ((real_mul c) c)) x1) x2) x3) x4) x5) x6))))))))))))))


CONTINUOUS_ON_COMPONENTS_FINITE |- (!f. (!s. (((FINITE (components s)) /\ (!c. ((c IN (components s)) ==> ((continuous_on f) c)))) ==> ((continuous_on f) s))))


Seq2.perm_eq_set_of_list |- (!l. (!s. (((FINITE s) /\ ((perm_eq l) (list_of_set s))) ==> ((set_of_list l) = s))))


REAL_SUMMABLE_NEG |- (!x. (!s. (((real_summable s) x) ==> ((real_summable s) (\n. (real_neg (x n)))))))


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


Seq.mem_seq3 |- (!x. (!y1. (!y2. (!y3. (((MEM x) ((CONS y1) ((CONS y2) ((CONS y3) NIL)))) = ((x = y1) \/ ((x = y2) \/ (x = y3))))))))


EDELSTEIN_FIX_ITER |- (!f. (!s. (!n. (((compact s) /\ ((~ (s = EMPTY)) /\ ((((IMAGE f) s) SUBSET s) /\ (!x. (!y. (((x IN s) /\ ((y IN s) /\ (~ (x = y)))) ==> ((real_lt (distance ((((ITER n) f) x) , (((ITER n) f) y)))) (distance (x , y))))))))) ==> (?!x. ((x IN s) /\ ((f x) = x)))))))


ULC_IMP_LOCALLY_CONNECTED |- (!s. ((!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?d. (((real_lt (real_of_num (NUMERAL _0))) d) /\ (!x. (!y. (((x IN s) /\ ((y IN s) /\ ((real_lt (distance (x , y))) d))) ==> (?c. ((x IN c) /\ ((y IN c) /\ ((c SUBSET s) /\ ((connected c) /\ ((bounded c) /\ ((real_le (diameter c)) e)))))))))))))) ==> ((locally connected) s)))


MINIMAL_CONTINUUM |- (!t. (!s. (((t SUBSET s) /\ ((compact s) /\ (connected s))) ==> (?u. ((t SUBSET u) /\ ((u SUBSET s) /\ ((compact u) /\ ((connected u) /\ (!v. (((v SUBSET u) /\ ((t SUBSET v) /\ ((compact v) /\ (connected v)))) ==> (v = u)))))))))))


INSIDE_INSIDE_SUBSET |- (!s. ((inside (inside s)) SUBSET s))


REAL_MEASURABLE_ON_RPOW |- (!f. (!r. (!s. ((((real_measurable_on f) s) /\ ((real_lt (real_of_num (NUMERAL _0))) r)) ==> ((real_measurable_on (\x. ((rpow (f x)) r))) s)))))


CONNECTED_COMPONENT_EMPTY |- (!x. (((connected_component EMPTY) x) = EMPTY))


Lp_ineqs_quads.ineq120 |- (!V. (!E. ((lp_fan (V , E)) ==> (quad_nonlinear_v10 ==> ((!y1. (!y2. (!y3. (!y4. (!y5. (!y6. ((ineq ((CONS (((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 (BIT1 _0))))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))) , (y1 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) h0)))) ((CONS (((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0)))))) , (y2 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) ((CONS (((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0)))))) , (y3 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) ((CONS (((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0) , (y4 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) h0)))) ((CONS (((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0)))))) , (y5 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) ((CONS (((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0)))))) , (y6 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) NIL))))))) ((real_lt ((((((delta_y y1) y2) y3) y4) y5) y6)) (real_of_num (NUMERAL _0)))))))))) ==> ((!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)))))) ==> ((lp_tau (V , E)) ==> (!d. ((d IN ((darts_k (NUMERAL (BIT0 (BIT0 (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_ge ((real_add ((tau_fan (V , E)) ((face (hypermap_of_fan (V , E))) d))) ((real_sub ((real_mul ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT1 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 _0))))))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 _0))))))))))))))))) ((azim_dart (V , E)) d))) ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 _0))))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))))) (real_of_num (NUMERAL _0)))))))))))))


POLYTOPE_AFF_DIM_1 |- (!p. (((polytope p) /\ ((aff_dim p) = (int_of_num (NUMERAL (BIT1 _0))))) = (?a. (?b. ((~ (a = b)) /\ (p = (closed_segment ((CONS (a , b)) NIL))))))))


EXISTS_VECTOR_3 |- ((?v. (P v)) = (?x. (?y. (?z. (P (vector ((CONS x) ((CONS y) ((CONS z) NIL)))))))))


REAL_LE_TRANS_LE |- (!x. (!y. (((real_le x) y) = (!z. (((real_le y) z) ==> ((real_le x) z))))))


CONIC_HULL_EQ |- (!s. ((((hull conic) s) = s) = (conic s)))


AFF_GE_1_1 |- (!x. (!v. (!w. (((DISJOINT (x INSERT EMPTY)) (v INSERT EMPTY)) ==> (((aff_ge (x INSERT EMPTY)) (v INSERT EMPTY)) = (GSPEC (\GEN%PVAR%5098. (?y. (((SETSPEC GEN%PVAR%5098) (?t1. (?t2. (((real_le (real_of_num (NUMERAL _0))) t2) /\ ((((real_add t1) t2) = (real_of_num (NUMERAL (BIT1 _0)))) /\ (y = ((vector_add ((% t1) x)) ((% t2) v)))))))) y)))))))))


IN_RELATIVE_INTERIOR_IN_OPEN_SEGMENT_STRONG |- (!s. (!x. (!a. (((convex s) /\ ((x IN (relative_interior s)) /\ ((a IN ((hull affine) s)) /\ (~ (x = a))))) ==> (?b. ((b IN (relative_interior s)) /\ (x IN (open_segment (a , b)))))))))


HOLOMORPHIC_ON_LMUL |- (!f. (!c. (!s. (((holomorphic_on f) s) ==> ((holomorphic_on (\x. ((complex_mul c) (f x)))) s)))))


SUM_BIJECTION |- (!f. (!p. (!s. (((!x. ((x IN s) ==> ((p x) IN s))) /\ (!y. ((y IN s) ==> (?!x. ((x IN s) /\ ((p x) = y)))))) ==> (((sum s) f) = ((sum s) (f o p)))))))


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


DOUBLE_LEBESGUE_MEASURABLE |- (!f. ((!t. ((lebesgue_measurable t) ==> (lebesgue_measurable (GSPEC (\GEN%PVAR%3690. (?x. (((SETSPEC GEN%PVAR%3690) ((f x) IN t)) x))))))) = (((measurable_on f) UNIV) /\ (!t. ((negligible t) ==> (lebesgue_measurable (GSPEC (\GEN%PVAR%3691. (?x. (((SETSPEC GEN%PVAR%3691) ((f x) IN t)) x))))))))))


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


Lp_ineqs_proofs.yn_hi |- (!E. (!V. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!v. ((v IN V) ==> ((real_le (yn_fan v)) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))))))))))))


OPEN_EXISTS |- (!Q. ((!a. (open (GSPEC (\GEN%PVAR%544. (?x. (((SETSPEC GEN%PVAR%544) ((Q a) x)) x)))))) ==> (open (GSPEC (\GEN%PVAR%545. (?x. (((SETSPEC GEN%PVAR%545) (?a. ((Q a) x))) x)))))))


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


REAL_LEBESGUE_MEASURABLE_COUNTABLE_INTERS_EXPLICIT |- (!s. ((!n. (real_lebesgue_measurable (s n))) ==> (real_lebesgue_measurable (INTERS (GSPEC (\GEN%PVAR%5042. (?n. (((SETSPEC GEN%PVAR%5042) (n IN UNIV)) (s n)))))))))


Tactics.WELLDEFINED_FUNCTION_1P |- (!P. ((?f. (!x. ((P x) ==> ((f (s x)) = (t x))))) = (!x. (!x'. (((P x) /\ ((P x') /\ ((s x) = (s x')))) ==> ((t x) = (t x')))))))


Merge_ineq.nonf_dih4_x_div_sqrtdelta_posbranch |- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((((((dih4_x_div_sqrtdelta_posbranch x1) x2) x3) x4) x5) x6) = ((real_mul ((real_div (sqrt ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) x4))) ((((((delta_x4 x4) x2) x6) x1) x5) x3))) (matan ((real_div ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) ((real_mul x4) ((((((delta_x x1) x2) x3) x4) x5) x6)))) ((real_pow ((((((delta_x4 x4) x2) x6) x1) x5) x3)) (NUMERAL (BIT0 (BIT1 _0))))))))))))))


REALLIM_POSINFINITY_SEQUENTIALLY |- (!f. (!l. ((((---> f) l) at_posinfinity) ==> (((---> (\n. (f (real_of_num n)))) l) sequentially))))


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


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


VECTOR_DERIVATIVE_CONST_AT |- (!c. (!a. (((vector_derivative (\x. c)) (at a)) = (vec (NUMERAL _0)))))


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


Lp_ineqs_proofs.rho_hi |- (!E. (!V. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!v. ((v IN V) ==> ((real_le (rho_fan v)) ((real_add (real_of_num (NUMERAL (BIT1 _0)))) ((real_div sol0) pi)))))))))))


SUMS_OFFSET_REV |- (!f. (!l. (!m. (!n. (((((sums f) l) (from m)) /\ (((NUMERAL _0) < m) /\ (n <= m))) ==> (((sums f) ((vector_add l) ((vsum (n .. (m - (NUMERAL (BIT1 _0))))) f))) (from n)))))))


Ssrnat.maxKn |- (!m. (!n. (((minn n) ((maxn m) n)) = n)))


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


Tactics_jordan.REAL_DOUBLE |- (!x. (((real_add x) x) = ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) x)))


Tame_list.HEIGHTS |- ((heights ((((Graph fs) n) f) h)) = h)


Rnsyjxm.f_fan_pair_ext_in_dart1 |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!d. ((d IN (dart1_of_fan (V , E))) ==> (((f_fan_pair_ext (V , E)) d) IN (dart1_of_fan (V , E))))))))


Gbycpxs.SIGMA_SY_LE1 |- (!s. ((real_le (sigma_sy s)) (real_of_num (NUMERAL (BIT1 _0)))))


CONNECTED_LIMIT_POINTS |- (!s. ((connected s) ==> (connected (GSPEC (\GEN%PVAR%585. (?x. (((SETSPEC GEN%PVAR%585) ((limit_point_of x) s)) x)))))))


HOLOMORPHIC_ON_CCOS |- (!s. ((holomorphic_on ccos) s))


SAME_EIGENVECTORS_MATRIX_INV |- (!A. (!c. (!v. (((transp A) = A) ==> ((((matrix_vector_mul (matrix_inv A)) v) = ((% c) v)) = (((matrix_vector_mul A) v) = ((% (real_inv c)) v)))))))


REAL_VARIATION_REFLECT2 |- (!f. (!s. (((real_variation ((IMAGE real_neg) s)) (\x. (f (real_neg x)))) = ((real_variation s) f))))


CONTINUOUS_MAP_CLOSURES |- (!f. (((continuous_on f) UNIV) = (!s. (((IMAGE f) (closure s)) SUBSET (closure ((IMAGE f) s))))))


Functional_equation.gchi4_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)))))) ==> (((((((gchi4_x x1) x2) x3) x4) x5) x6) = ((((((gchi4_x' x1) x2) x3) x4) x5) x6)))))))))


OPEN_MAP_IFF_LOWER_HEMICONTINUOUS_PREIMAGE |- (!f. (!s. (!t. ((((IMAGE f) s) SUBSET t) ==> ((!u. (((open_in ((subtopology euclidean) s)) u) ==> ((open_in ((subtopology euclidean) t)) ((IMAGE f) u)))) = (!u. (((closed_in ((subtopology euclidean) s)) u) ==> ((closed_in ((subtopology euclidean) t)) (GSPEC (\GEN%PVAR%885. (?y. (((SETSPEC GEN%PVAR%885) ((y IN t) /\ ((GSPEC (\GEN%PVAR%884. (?x. (((SETSPEC GEN%PVAR%884) ((x IN s) /\ ((f x) = y))) x)))) SUBSET u))) y))))))))))))


Localization.HAFL_CIRCLE_FORM_LOCAL_FAN_ALT |- (!V. (!E. (!FF. (!v. (!w. (!HS. (!fv. ((((local_fan (V , (E , FF))) /\ ((v IN V) /\ ((w IN V) /\ ((~ (v = w)) /\ ((!z. (!t. (((z IN (v INSERT (w INSERT EMPTY))) /\ (t IN (V DIFF (z INSERT EMPTY)))) ==> (~ (collinear ((vec (NUMERAL _0)) INSERT (z INSERT (t INSERT EMPTY)))))))) /\ (!x. ((x IN FF) ==> (((aff_gt ((vec (NUMERAL _0)) INSERT EMPTY)) (v INSERT (w INSERT EMPTY))) SUBSET ((wedge_in_fan_gt x) E))))))))) /\ ((HS = (hypermap (HYP ((vec (NUMERAL _0)) , (V , (E UNION ((v INSERT (w INSERT EMPTY)) INSERT EMPTY))))))) /\ (fv = ((face HS) (v , ((rho_node1 FF) v)))))) ==> (local_fan (((v_prime V) fv) , (((e_prime (E UNION ((v INSERT (w INSERT EMPTY)) INSERT EMPTY))) fv) , fv)))))))))))


Planarity.RWXUYZZ |- (!x. (!V. (!E. (!ds. (((FAN (x , (V , E))) /\ ((!v. ((v IN V) ==> ((CARD (((set_of_edge v) V) E)) > (NUMERAL (BIT1 _0))))) /\ ((fan80 (x , (V , E))) /\ (ds IN (face_set (hypermap1_of_fanx (x , (V , E)))))))) ==> ((?s. (!y. ((y IN ds) ==> (s = (((((dart_leads_into x) V) E) (pr2 y)) (pr3 y)))))) /\ (((((dartset_leads_into_fan x) V) E) ds) IN (topological_component_yfan (x , (V , E))))))))))


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


Lp_ineqs_proofs.f_fan_pair_ext_power_in_dart1 |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!k. (!d. ((d IN (dart1_of_fan (V , E))) ==> ((((POWER (f_fan_pair_ext (V , E))) k) d) IN (dart1_of_fan (V , E)))))))))


Ssrnat.distn_add2l |- (!d. (!m. (!n. (((distn (d + m)) (d + n)) = ((distn m) n)))))


Ssrbool.negbNE |- (!b. ((~ (~ b)) ==> b))


LEBESGUE_DENSITY_THEOREM |- (!s. ((lebesgue_measurable s) ==> (?k. ((negligible k) /\ (!x. ((~ (x IN k)) ==> (((---> (\e. ((real_div (measure (s INTER (cball (x , e))))) (measure (cball (x , e)))))) (((COND (x IN s)) (real_of_num (NUMERAL (BIT1 _0)))) (real_of_num (NUMERAL _0)))) ((within (atreal (real_of_num (NUMERAL _0)))) (GSPEC (\GEN%PVAR%5068. (?e. (((SETSPEC GEN%PVAR%5068) ((real_lt (real_of_num (NUMERAL _0))) e)) e))))))))))))


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


REAL_CONTINUOUS_COMPLEX_COMPONENTS_WITHIN |- (!s. (!z. (((real_continuous Re) ((within (at z)) s)) /\ ((real_continuous Im) ((within (at z)) s)))))


INTEGRAL_COMPONENT_LE_AE |- (!f. (!g. (!s. (!k. (!t. ((((NUMERAL (BIT1 _0)) <= k) /\ ((k <= (dimindex UNIV)) /\ ((negligible t) /\ (((integrable_on f) s) /\ (((integrable_on g) s) /\ (!x. ((x IN (s DIFF t)) ==> ((real_le ((f x) $ k)) ((g x) $ k))))))))) ==> ((real_le (((integral s) f) $ k)) (((integral s) g) $ k))))))))


REAL_INTEGRABLE_COMBINE |- (!f. (!a. (!b. (!c. ((((real_le a) c) /\ (((real_le c) b) /\ (((real_integrable_on f) (closed_real_interval ((CONS (a , c)) NIL))) /\ ((real_integrable_on f) (closed_real_interval ((CONS (c , b)) NIL)))))) ==> ((real_integrable_on f) (closed_real_interval ((CONS (a , b)) NIL))))))))


Lp_ineqs_proofs.dart3_sol_fan_eq_sol |- (!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)))) ==> (((sol_fan (V , E)) ((face (hypermap_of_fan (V , E))) d)) = ((((((sol_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)))))))))))


AFF_DIM_BALL |- (!a. (!r. ((aff_dim (ball (a , r))) = (((COND ((real_lt (real_of_num (NUMERAL _0))) r)) (int_of_num (dimindex UNIV))) (int_neg (int_of_num (NUMERAL (BIT1 _0))))))))


INJECTIVE_MAP_OPEN_IFF_CLOSED |- (!f. (!s. (!t. ((((continuous_on f) s) /\ ((((IMAGE f) s) = t) /\ (!x. (!y. (((x IN s) /\ ((y IN s) /\ ((f x) = (f y)))) ==> (x = y)))))) ==> ((!u. (((open_in ((subtopology euclidean) s)) u) ==> ((open_in ((subtopology euclidean) t)) ((IMAGE f) u)))) = (!u. (((closed_in ((subtopology euclidean) s)) u) ==> ((closed_in ((subtopology euclidean) t)) ((IMAGE f) u)))))))))


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


SERIES_DIRICHLET_COMPLEX |- (!f. (!g. (!N. (!k. (!m. (((bounded (GSPEC (\GEN%PVAR%4782. (?n. (((SETSPEC GEN%PVAR%4782) (n IN UNIV)) ((vsum (m .. n)) f)))))) /\ ((!n. (real (g n))) /\ ((!n. ((N <= n) ==> ((real_le (Re (g (n + (NUMERAL (BIT1 _0)))))) (Re (g n))))) /\ (((--> g) (Cx (real_of_num (NUMERAL _0)))) sequentially)))) ==> ((summable (from k)) (\n. ((complex_mul (f n)) (g n))))))))))


CONVEX_AFF_LE |- (!s. (!t. (convex ((aff_le s) t))))


IVT_DECREASING_IM |- (!f. (!a. (!b. (!y. ((((real_le a) b) /\ ((!x. ((((real_le a) x) /\ ((real_le x) b)) ==> ((continuous f) (at (Cx x))))) /\ (((real_le (Im (f (Cx b)))) y) /\ ((real_le y) (Im (f (Cx a))))))) ==> (?x. (((real_le a) x) /\ (((real_le x) b) /\ ((Im (f (Cx x))) = y)))))))))


REAL_MEASURABLE_ON_COMPOSE_CONTINUOUS_OPEN_INTERVAL |- (!f. (!g. (!a. (!b. ((((real_measurable_on f) UNIV) /\ ((!x. ((f x) IN (open_real_interval (a , b)))) /\ ((real_continuous_on g) (open_real_interval (a , b))))) ==> ((real_measurable_on (g o f)) UNIV))))))


LOCALLY_CONVEX |- (!s. (((locally convex) s) = (!x. ((x IN s) ==> (?u. (?v. ((x IN u) /\ ((u SUBSET v) /\ ((v SUBSET s) /\ (((open_in ((subtopology euclidean) s)) u) /\ (convex v)))))))))))


MEASURABLE_ON_MEASURABLE_PREIMAGE_HALFSPACE_COMPONENT_GT |- (!f. (!s. ((measurable s) ==> (((measurable_on f) s) = (!a. (!k. ((((NUMERAL (BIT1 _0)) <= k) /\ (k <= (dimindex UNIV))) ==> (measurable (GSPEC (\GEN%PVAR%3650. (?x. (((SETSPEC GEN%PVAR%3650) ((x IN s) /\ ((real_gt ((f x) $ k)) a))) x))))))))))))


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


NSUM_DEGENERATE |- (!f. (!s. ((~ (FINITE (GSPEC (\GEN%PVAR%171. (?x. (((SETSPEC GEN%PVAR%171) ((x IN s) /\ (~ ((f x) = (NUMERAL _0))))) x)))))) ==> (((nsum s) f) = (NUMERAL _0)))))


QUOTIENT_MAP_IMP_CONTINUOUS_CLOSED |- (!f. (!s. (!t. (((((IMAGE f) s) SUBSET t) /\ (!u. ((u SUBSET t) ==> (((closed_in ((subtopology euclidean) s)) (GSPEC (\GEN%PVAR%645. (?x. (((SETSPEC GEN%PVAR%645) ((x IN s) /\ ((f x) IN u))) x))))) = ((closed_in ((subtopology euclidean) t)) u))))) ==> ((continuous_on f) s)))))


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


INT_LE_NEGL |- (!x. (((int_le (int_neg x)) x) = ((int_le (int_of_num (NUMERAL _0))) x)))


EXISTS_SUBARC_OF_ARC_NOENDS |- (!g. (!a. (!b. (((arc g) /\ ((a IN (path_image g)) /\ ((b IN (path_image g)) /\ ((~ (a = b)) /\ (((a INSERT (b INSERT EMPTY)) INTER ((pathstart g) INSERT ((pathfinish g) INSERT EMPTY))) = EMPTY))))) ==> (?h. ((arc h) /\ (((pathstart h) = a) /\ (((pathfinish h) = b) /\ ((path_image h) SUBSET ((path_image g) DIFF ((pathstart g) INSERT ((pathfinish g) INSERT EMPTY))))))))))))


Tame_list.AQIUNPP2 |- (!L. (!N. (!r. (!x. (((transform_assumption_v2 L) /\ ((good_list L) /\ ((good_list_nodes L) /\ ((((marked_list L) N) r) x)))) ==> ((((marked_list L) (((((ntrans L) N) r) x) (NUMERAL (BIT1 _0)))) (((((rtrans L) N) r) x) (NUMERAL (BIT1 _0)))) x))))))


URYSOHN |- (!s. (!t. (!a. (!b. (((closed s) /\ ((closed t) /\ ((s INTER t) = EMPTY))) ==> (?f. (((continuous_on f) UNIV) /\ ((!x. ((f x) IN (closed_segment ((CONS (a , b)) NIL)))) /\ ((!x. ((x IN s) ==> ((f x) = a))) /\ (!x. ((x IN t) ==> ((f x) = b))))))))))))


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


LINEAR_SUBSPACE_GRAPH |- (!f. ((linear f) = (subspace (GSPEC (\GEN%PVAR%153. (?x. (((SETSPEC GEN%PVAR%153) (x IN UNIV)) ((pastecart x) (f x)))))))))


Lp_ineqs_proofs.rhazim2_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 ((rhazim2_fan (V , E)) d)) ((real_add pi) sol0))))))))))


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


Lp_gen_theory.sum_face_iso |- (!H. (!G. (!g. (((hyp_iso g) (G , H)) ==> (!r. (!f. ((f IN (face_set G)) ==> (((sum ((IMAGE g) f)) r) = ((sum f) (r o g))))))))))


Lp_ineqs_proofs.INJ_pair |- (!h1. (!h2. (!g. ((!x. ((g x) = ((h1 (FST x)) , (h2 (SND x))))) ==> (!s1. (!t1. (!s2. (!t2. (((((INJ h1) s1) t1) /\ (((INJ h2) s2) t2)) ==> (((INJ g) (s1 CROSS s2)) (t1 CROSS t2)))))))))))


IN_COMPONENTS_UNIONS_COMPLEMENT |- (!s. (!c. ((c IN (components s)) ==> ((s DIFF c) = (UNIONS ((components s) DELETE c))))))


Collect_geom2.EQ_IMP_COPLANAR |- (!a. (!b. (!c. (!d. (((a = b) \/ ((a = c) \/ (a = d))) ==> (coplanar_alt (a INSERT (b INSERT (c INSERT (d INSERT EMPTY))))))))))


HOMOTOPIC_INTO_RETRACT |- (!f. (!g. (!s. (!t. (!u. (((((IMAGE f) s) SUBSET t) /\ ((((IMAGE g) s) SUBSET t) /\ (((retract_of t) u) /\ ((((homotopic_with (\x. T)) (s , u)) f) g)))) ==> ((((homotopic_with (\x. T)) (s , t)) f) g)))))))


HAS_BOUNDED_VARIATION_COMPOSE_INJECTIVE |- (!f. (!g. (!s. (((is_interval s) /\ (((continuous_on f) s) /\ ((!x. (!y. (((x IN s) /\ ((y IN s) /\ ((f x) = (f y)))) ==> (x = y)))) /\ ((has_bounded_variation_on g) ((IMAGE f) s))))) ==> ((has_bounded_variation_on (g o f)) s)))))


SUMS_INTERVALS |- ((!a. (!b. (!c. (!d. (((~ ((closed_interval ((CONS (a , b)) NIL)) = EMPTY)) /\ (~ ((closed_interval ((CONS (c , d)) NIL)) = EMPTY))) ==> ((GSPEC (\GEN%PVAR%972. (?x. (?y. (((SETSPEC GEN%PVAR%972) ((x IN (closed_interval ((CONS (a , b)) NIL))) /\ (y IN (closed_interval ((CONS (c , d)) NIL))))) ((vector_add x) y)))))) = (closed_interval ((CONS (((vector_add a) c) , ((vector_add b) d))) NIL)))))))) /\ (!a. (!b. (!c. (!d. (((~ ((open_interval (a , b)) = EMPTY)) /\ (~ ((open_interval (c , d)) = EMPTY))) ==> ((GSPEC (\GEN%PVAR%973. (?x. (?y. (((SETSPEC GEN%PVAR%973) ((x IN (open_interval (a , b))) /\ (y IN (open_interval (c , d))))) ((vector_add x) y)))))) = (open_interval (((vector_add a) c) , ((vector_add b) d))))))))))


Functional_equation.functional_acs_sqrt_x2_d4 |- (acs_sqrt_x2_d4 = (uni (acs , ((scalar6 proj_y2) ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))))


LOCALLY_COMPACT_INTER_CBALLS |- (!s. (((locally compact) s) = (!x. ((x IN s) ==> (?e. (((real_lt (real_of_num (NUMERAL _0))) e) /\ (!d. (((real_le d) e) ==> (closed ((cball (x , d)) INTER s))))))))))


REAL_SERIES_DIFFS |- (!f. (!k. ((((---> f) (real_of_num (NUMERAL _0))) sequentially) ==> (((real_sums (\n. ((real_sub (f n)) (f (n + (NUMERAL (BIT1 _0))))))) (f k)) (from k)))))


REAL_POWER_SERIES_CONV_IMP_ABSCONV |- (!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_abs ((real_mul (a n)) ((real_pow w) n))))))))))


Tame_list.nth_EL |- (!x. (!xs. (!n. (!k. (((n = (NUMERAL _0)) ==> ((((\l. (\n. ((EL n) l))) ((CONS x) xs)) n) = x)) /\ ((n = (SUC k)) ==> ((((\l. (\n. ((EL n) l))) ((CONS x) xs)) n) = (((\l. (\n. ((EL n) l))) xs) k))))))))


NPRODUCT_ONE |- (!s. (((nproduct s) (\n. (NUMERAL (BIT1 _0)))) = (NUMERAL (BIT1 _0))))


SUP_INSERT |- (!x. (!s. ((bounded ((IMAGE lift) s)) ==> ((sup (x INSERT s)) = (((COND (s = EMPTY)) x) ((real_max x) (sup s)))))))


RELPOW_M |- (!m. (!n. (!x. (!y. (((((RELPOW (m + n)) R) x) y) = (?z. (((((RELPOW m) R) x) z) /\ ((((RELPOW n) R) z) y))))))))


REAL_CONTINUOUS_ON_CASES |- (!P. (!f. (!g. (!s. (!t. (((real_closed s) /\ ((real_closed t) /\ (((real_continuous_on f) s) /\ (((real_continuous_on g) t) /\ (!x. ((((x IN s) /\ (~ (P x))) \/ ((x IN t) /\ (P x))) ==> ((f x) = (g x)))))))) ==> ((real_continuous_on (\x. (((COND (P x)) (f x)) (g x)))) (s UNION t))))))))


List_hypermap_iso.fan_list_nodes_iso_inj |- (!L. (!f. (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((good_list L) ==> (((hyp_iso f) ((hypermap_of_fan (V , E)) , (hypermap_of_list L))) ==> ((good_list_nodes L) ==> (!x. (!y. (((x IN V) /\ ((y IN V) /\ (((fan_list_nodes_iso (f , (V , E))) x) = ((fan_list_nodes_iso (f , (V , E))) y)))) ==> (x = y))))))))))))


List_hypermap_iso.fan_list_nodes_iso_surj |- (!f. (!E. (!V. (!L. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((good_list L) ==> (((hyp_iso f) ((hypermap_of_fan (V , E)) , (hypermap_of_list L))) ==> (((SURJ (fan_list_nodes_iso (f , (V , E)))) V) (elements_of_list L)))))))))


REAL_INTEGRABLE_ON_REFL |- (!f. (!a. ((real_integrable_on f) (closed_real_interval ((CONS (a , a)) NIL)))))


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


LIM_LOG_OVER_Z |- (((--> (\z. ((complex_div (clog z)) z))) (Cx (real_of_num (NUMERAL _0)))) at_infinity)


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


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


Fatugpd.dot_normalize |- (!v. (((dot v) (normalize v)) = (vector_norm v)))


REAL_CONVEX_ON_LEFT_SECANT |- (!f. (!s. (((real_convex_on f) s) = (!a. (!b. (!x. (((a IN s) /\ ((b IN s) /\ (x IN (open_real_segment (a , b))))) ==> ((real_le ((real_div ((real_sub (f x)) (f a))) (real_abs ((real_sub x) a)))) ((real_div ((real_sub (f b)) (f a))) (real_abs ((real_sub b) a)))))))))))


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


Lp_gen_theory.hyp_iso_face_set_k |- (!H. (!g. (!G. (((hyp_iso g) (G , H)) ==> (!k. (((face_set_k k) H) = ((IMAGE (IMAGE g)) ((face_set_k k) G))))))))


INT_SGN_NEG |- (!x. ((int_sgn (int_neg x)) = (int_neg (int_sgn x))))


COMPLEX_EQ_NEG2 |- (!x. (!y. (((vector_neg x) = (vector_neg y)) = (x = y))))


INT_SUB_TRIANGLE |- (!a. (!b. (!c. (((int_add ((int_sub a) b)) ((int_sub b) c)) = ((int_sub a) c)))))


FRONTIER_UNIONS_SUBSET |- (!f. ((FINITE f) ==> ((frontier (UNIONS f)) SUBSET (UNIONS (GSPEC (\GEN%PVAR%1826. (?t. (((SETSPEC GEN%PVAR%1826) (t IN f)) (frontier t)))))))))


REAL_LOG_CONVEX_IMP_CONVEX |- (!f. (!s. (((real_log_convex_on f) s) ==> ((real_convex_on f) s))))


Lp_main_estimate.ineq_tau3 |- (!V. (!E. ((lp_fan (V , E)) ==> ((!v1. (!v2. (!v3. ((((real_le (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (vector_norm v1)) /\ (((real_le (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (vector_norm v2)) /\ (((real_le (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (vector_norm v3)) /\ (((real_le (vector_norm v1)) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) /\ (((real_le (vector_norm v2)) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) /\ (((real_le (vector_norm v3)) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) /\ (((real_le (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (distance (v1 , v2))) /\ (((real_le (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (distance (v1 , v3))) /\ (((real_le (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (distance (v2 , v3))) /\ (((real_le (distance (v1 , v2))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) /\ (((real_le (distance (v1 , v3))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) /\ ((real_le (distance (v2 , v3))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0))))))))))))) ==> ((real_le (real_of_num (NUMERAL _0))) (((tau3 v1) v2) v3)))))) ==> (!d. ((d IN ((darts_k (NUMERAL (BIT1 (BIT1 _0)))) (hypermap_of_fan (V , E)))) ==> ((real_le (real_of_num (NUMERAL _0))) ((tau_fan (V , E)) ((face (hypermap_of_fan (V , E))) d)))))))))


REAL_INTEGRAL_NEG |- (!f. (!s. (((real_integrable_on f) s) ==> (((real_integral s) (\x. (real_neg (f x)))) = (real_neg ((real_integral s) f))))))


Lp_ineqs_proofs.RHB |- (!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 ((rhazim_fan (V , E)) d)) ((real_mul ((azim_dart (V , E)) d)) ((real_add (real_of_num (NUMERAL (BIT1 _0)))) ((real_div sol0) pi))))))))))))


PATH_LENGTH_EQ_LINE_SEGMENT |- (!g. (((rectifiable_path g) /\ ((path_length g) = (distance ((pathstart g) , (pathfinish g))))) ==> ((path_image g) = (closed_segment ((CONS ((pathstart g) , (pathfinish g))) NIL)))))


VSUM_CMUL_NUMSEG |- (!f. (!c. (!m. (!n. (((vsum (m .. n)) (\x. ((% c) (f x)))) = ((% c) ((vsum (m .. n)) f)))))))


CONNECTED_OPEN_VECTOR_POLYNOMIAL_CONNECTED |- (!s. (((open s) /\ (connected s)) ==> (!x. (!y. (((x IN s) /\ (y IN s)) ==> (?g. ((vector_polynomial_function g) /\ (((path_image g) SUBSET s) /\ (((pathstart g) = x) /\ ((pathfinish g) = y))))))))))


INT_MAX_ACI |- ((((int_max x) y) = ((int_max y) x)) /\ ((((int_max ((int_max x) y)) z) = ((int_max x) ((int_max y) z))) /\ ((((int_max x) ((int_max y) z)) = ((int_max y) ((int_max x) z))) /\ ((((int_max x) x) = x) /\ (((int_max x) ((int_max x) y)) = ((int_max x) y))))))


Functional_equation.functional_proj_y4 |- (proj_y4 = (rotate4 proj_y1))


COMPACT_IN_IFF_COMPACT |- (!s. (((compact_in s) euclidean) = (compact s)))


INFSUM_NEG |- (!s. (!x. (((summable s) x) ==> (((infsum s) (\n. (vector_neg (x n)))) = (vector_neg ((infsum s) x))))))


KERNEL_MATRIX_INV |- (!A. ((GSPEC (\GEN%PVAR%227. (?x. (((SETSPEC GEN%PVAR%227) (((matrix_vector_mul (matrix_inv A)) x) = (vec (NUMERAL _0)))) x)))) = (GSPEC (\GEN%PVAR%228. (?x. (((SETSPEC GEN%PVAR%228) (((matrix_vector_mul (transp A)) x) = (vec (NUMERAL _0)))) x))))))


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


GATEAUX_DERIVATIVE |- (!f. (!f'. (!x. (!y. ((((has_derivative f) f') (at x)) ==> (((--> (\t. ((% (real_inv (drop t))) ((vector_sub (f ((vector_add x) ((% (drop t)) y)))) (f x))))) (f' y)) (at (vec (NUMERAL _0)))))))))


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


Hypermap_iso.surj_ext |- (!f. (!g. (!s. (!t. ((!x. ((x IN s) ==> ((f x) = (g x)))) ==> ((((SURJ f) s) t) ==> (((SURJ g) s) t)))))))


SYMMETRIX_MATRIX_CONJUGATE |- (!A. (!B. (((transp B) = B) ==> ((transp ((matrix_mul (transp A)) ((matrix_mul B) A))) = ((matrix_mul (transp A)) ((matrix_mul B) A))))))


NSUM_BIJECTION |- (!f. (!p. (!s. (((!x. ((x IN s) ==> ((p x) IN s))) /\ (!y. ((y IN s) ==> (?!x. ((x IN s) /\ ((p x) = y)))))) ==> (((nsum s) f) = ((nsum s) (f o p)))))))


CONTINUOUS_ON_CSQRT |- (!s. ((!z. (((z IN s) /\ ((Im z) = (real_of_num (NUMERAL _0)))) ==> ((real_lt (real_of_num (NUMERAL _0))) (Re z)))) ==> ((continuous_on csqrt) s)))


Tame_list.sub_nth |- (((sub a) n) = ((#!# a) n))


Appendix.scs_unadorned |- (!s. ((is_scs_v39 s) ==> (is_scs_v39 (scs_v39 ((scs_k_v39 s) , ((scs_d_v39 s) , ((scs_a_v39 s) , ((scs_a_v39 s) , ((scs_b_v39 s) , ((scs_b_v39 s) , ((scs_J_v39 s) , (EMPTY , (EMPTY , EMPTY)))))))))))))


EXPLICIT_SUBSET_INTERIOR_CONVEX_HULL |- (!s. (((FINITE s) /\ (((hull affine) s) = UNIV)) ==> ((GSPEC (\GEN%PVAR%1794. (?y. (((SETSPEC GEN%PVAR%1794) (?u. ((!x. ((x IN s) ==> (((real_lt (real_of_num (NUMERAL _0))) (u x)) /\ ((real_lt (u x)) (real_of_num (NUMERAL (BIT1 _0))))))) /\ ((((sum s) u) = (real_of_num (NUMERAL (BIT1 _0)))) /\ (((vsum s) (\x. ((% (u x)) x))) = y))))) y)))) SUBSET (interior ((hull convex) s)))))


CONTINUOUS_INJECTIVE_IMAGE_OPEN_SEGMENT_1 |- (!f. (!a. (!b. ((((continuous_on f) (closed_segment ((CONS (a , b)) NIL))) /\ (!x. (!y. (((x IN (closed_segment ((CONS (a , b)) NIL))) /\ ((y IN (closed_segment ((CONS (a , b)) NIL))) /\ ((f x) = (f y)))) ==> (x = y))))) ==> (((IMAGE f) (open_segment (a , b))) = (open_segment ((f a) , (f b))))))))


HAS_MEASURE_ALMOST_EQ |- (!s. (!s'. (!t. (((negligible t) /\ ((s UNION t) = (s' UNION t))) ==> (((has_measure s) m) = ((has_measure s') m))))))


OPEN_IN_SEGMENT |- (!s. (!a. (!b. ((((open_segment (a , b)) SUBSET s) /\ (s SUBSET ((hull affine) (open_segment (a , b))))) ==> ((open_in ((subtopology euclidean) s)) (open_segment (a , b)))))))


Functional_equation.functional_vol3f_x_lfun |- (((domain6 (\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))) x6)))))))))) vol3f_x_lfun) ((sub6 ((mul6 (constant6 ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((real_div mm1) pi)))) ((add6 ((mul6 two6) (mk_126 dih_x))) ((add6 ((mul6 two6) (mk_126 dih2_x))) ((add6 ((mul6 two6) (mk_126 dih6_x))) ((add6 (mk_126 dih3_x)) ((add6 (mk_126 dih4_x)) ((sub6 (mk_126 dih5_x)) (constant6 ((real_mul (real_of_num (NUMERAL (BIT1 (BIT1 _0))))) pi)))))))))) ((mul6 (constant6 ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 _0))))))) ((real_div mm2) pi)))) ((add6 (mk_126 ldih_x)) ((add6 (mk_126 ldih2_x)) (mk_126 ldih6_x))))))


BOUNDED_PARTIAL_REAL_SUMS |- (!f. (!k. ((real_bounded (GSPEC (\GEN%PVAR%4868. (?n. (((SETSPEC GEN%PVAR%4868) (n IN UNIV)) ((sum (k .. n)) f)))))) ==> (real_bounded (GSPEC (\GEN%PVAR%4869. (?m. (?n. (((SETSPEC GEN%PVAR%4869) ((m IN UNIV) /\ (n IN UNIV))) ((sum (m .. n)) f))))))))))


TRANS_ALT |- (!R. (!S. (!U. ((!x. (!z. ((?y. (((R x) y) /\ ((S y) z))) ==> ((U x) z)))) = (!x. (!y. (!z. ((((R x) y) /\ ((S y) z)) ==> ((U x) z)))))))))


Ssrnat.maxn_mull |- (!m1. (!m2. (!n. ((((maxn m1) m2) * n) = ((maxn (m1 * n)) (m2 * n))))))


ISO_FUN |- ((((ISO f) f') /\ ((ISO g) g')) ==> ((ISO (\h. (\a'. (g (h (f' a')))))) (\h. (\a. (g' (h (f a)))))))


FRECHET_DERIVATIVE_CONST_AT |- (!c. (!a. (((frechet_derivative (\x. c)) (at a)) = (\h. (vec (NUMERAL _0))))))


COMPLEX_SUB_POW_L1 |- (!x. (!n. (((NUMERAL (BIT1 _0)) <= n) ==> (((vector_sub (Cx (real_of_num (NUMERAL (BIT1 _0))))) ((complex_pow x) n)) = ((complex_mul ((vector_sub (Cx (real_of_num (NUMERAL (BIT1 _0))))) x)) ((vsum ((NUMERAL _0) .. (n - (NUMERAL (BIT1 _0))))) (\i. ((complex_pow x) i))))))))


Trigonometry.LLOYXRK2 |- (!x. ((sin ((real_add ((real_div pi) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) x)) = (cos x)))


REAL_INTEGRABLE_REFLECT |- (!f. (!a. (!b. (((real_integrable_on (\x. (f (real_neg x)))) (closed_real_interval ((CONS ((real_neg b) , (real_neg a))) NIL))) = ((real_integrable_on f) (closed_real_interval ((CONS (a , b)) NIL)))))))


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


COLLINEAR_CONVEX_HULL_COLLINEAR |- (!s. ((collinear ((hull convex) s)) = (collinear s)))


SETDIST_LE_HAUSDIST |- (!s. (!t. (((bounded s) /\ (bounded t)) ==> ((real_le (setdist (s , t))) (hausdist (s , t))))))


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


Sphere.vol_solid_triangle |- (!r. (!v0. (!v1. (!v2. (!v3. ((((((vol_solid_triangle v0) v1) v2) v3) r) = ((LET (\a123. (LET_END ((LET (\a231. (LET_END ((LET (\a312. (LET_END ((real_mul ((real_add a123) ((real_add a231) ((real_sub a312) pi)))) ((real_div ((real_pow r) (NUMERAL (BIT1 (BIT1 _0))))) (real_of_num (NUMERAL (BIT1 (BIT1 _0))))))))) ((((dihV v0) v3) v1) v2))))) ((((dihV v0) v2) v3) v1))))) ((((dihV v0) v1) v2) v3))))))))


REALLIM_AT_POSINFINITY |- (!f. (!l. ((((---> f) l) at_posinfinity) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?b. (!x. (((real_ge x) b) ==> ((real_lt (real_abs ((real_sub (f x)) l))) e)))))))))


FRAC_NEG |- (!x. ((frac (real_neg x)) = (((COND (integer x)) (real_of_num (NUMERAL _0))) ((real_sub (real_of_num (NUMERAL (BIT1 _0)))) (frac x)))))


Lp_ineqs_proofs.ln_hi |- (!E. (!V. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!v. ((v IN V) ==> ((real_le (ln_fan v)) (real_of_num (NUMERAL (BIT1 _0))))))))))))


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


Hypermap_iso.iso_number_of_edges |- (!f. (!G. (!H. (((hyp_iso f) (H , G)) ==> ((number_of_edges G) = (number_of_edges H))))))


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


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


EXISTS_SUBPATH_OF_ARC_NOENDS |- (!g. (!a. (!b. (((arc g) /\ ((a IN (path_image g)) /\ ((b IN (path_image g)) /\ (((a INSERT (b INSERT EMPTY)) INTER ((pathstart g) INSERT ((pathfinish g) INSERT EMPTY))) = EMPTY)))) ==> (?h. ((path h) /\ (((pathstart h) = a) /\ (((pathfinish h) = b) /\ ((path_image h) SUBSET ((path_image g) DIFF ((pathstart g) INSERT ((pathfinish g) INSERT EMPTY))))))))))))


REAL_DIFFERENTIABLE_FROM_COMPLEX_AT |- (!f. (!x. ((((complex_differentiable f) (at (Cx x))) /\ (!z. ((real z) ==> (real (f z))))) ==> ((real_differentiable (Re o (f o Cx))) (atreal x)))))


Lp_ineqs_proofs.dart4_tau_fan_eq_tauq |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> ((lp_tau (V , E)) ==> (!d. ((d IN ((darts_k (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) (hypermap_of_fan (V , E)))) ==> (((tau_fan (V , E)) ((face (hypermap_of_fan (V , E))) d)) = (((((((((tauq (y1_fan d)) (y2_fan d)) ((y3_fan (V , E)) d)) ((y4'_fan (V , E)) d)) ((y5_fan (V , E)) d)) (y6_fan d)) ((y7_fan (V , E)) d)) ((y8_fan (V , E)) d)) ((y9_fan (V , E)) d))))))))))))


YOUNG_INEQUALITY |- (!a. (!b. (!p. (!q. ((((real_le (real_of_num (NUMERAL _0))) a) /\ (((real_le (real_of_num (NUMERAL _0))) b) /\ (((real_lt (real_of_num (NUMERAL _0))) p) /\ (((real_lt (real_of_num (NUMERAL _0))) q) /\ (((real_add (real_inv p)) (real_inv q)) = (real_of_num (NUMERAL (BIT1 _0)))))))) ==> ((real_le ((real_mul a) b)) ((real_add ((real_div ((rpow a) p)) p)) ((real_div ((rpow b) q)) q))))))))


FINITE_COLUMNS |- (!A. (FINITE (columns A)))


Lp_ineqs_proofs.yy10_std |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> ((E = (ESTD V)) ==> (!d. ((d IN (dart_of_fan (V , E))) ==> ((real_le (ye_fan d)) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))))))))))


Nonlinear_lemma.pathL_bound |- (!y. (!a. ((((real_le (FST (pathL a))) y) /\ ((real_le y) (SND (pathL a)))) ==> (((real_le (FST a)) y) /\ ((real_le y) (SND a))))))


Tame_list.list_update_ALT |- (((((list_update NIL) i) v) = NIL) /\ ((((list_update ((CONS x) xs)) i) v) = (((COND (i = (NUMERAL _0))) ((CONS v) xs)) ((LET (\j. (LET_END ((CONS x) (((list_update xs) j) v))))) (PRE i)))))


COMPLEX_DIFFERENTIABLE_POW_AT |- (!f. (!n. (!z. (((complex_differentiable f) (at z)) ==> ((complex_differentiable (\z. ((complex_pow (f z)) n))) (at z))))))


COS_EQ_MINUS1 |- (!x. (((cos x) = (real_neg (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_of_num (NUMERAL (BIT1 _0))))) pi))))))


TAN_ABS_GE_X |- (!x. (((real_lt (real_abs x)) ((real_div pi) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) ==> ((real_le (real_abs x)) (real_abs (tan x)))))


REAL_INFSUM_0 |- (((real_infsum s) (\i. (real_of_num (NUMERAL _0)))) = (real_of_num (NUMERAL _0)))


INT_POW_EQ_ABS |- (!n. (!x. (!y. (((~ (n = (NUMERAL _0))) /\ (((int_pow x) n) = ((int_pow y) n))) ==> ((int_abs x) = (int_abs y))))))


INT_LE_SQUARE |- (!x. ((int_le (int_of_num (NUMERAL _0))) ((int_mul x) x)))


OPEN_SET_IRRATIONAL_COORDINATES |- (!s. (((open s) /\ (~ (s = EMPTY))) ==> (?x. ((x IN s) /\ (!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV))) ==> (~ (rational (x $ i)))))))))


MOD_EXISTS |- (!m. (!n. ((?q. (m = (n * q))) = (((COND (n = (NUMERAL _0))) (m = (NUMERAL _0))) ((m MOD n) = (NUMERAL _0))))))


COBOUNDED_HAS_BOUNDED_COMPONENT |- (!s. ((((NUMERAL (BIT0 (BIT1 _0))) <= (dimindex UNIV)) /\ ((bounded (UNIV DIFF s)) /\ (~ (connected s)))) ==> (?c. ((c IN (components s)) /\ (bounded c)))))


Nonlinear_lemma.delta_x_eq0 |- ((((((((delta_x (real_of_num (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 _0))))))) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 _0))))))) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) = (real_of_num (NUMERAL _0))) /\ (((((((delta_x (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 _0))))))) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 _0))))))) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) = (real_of_num (NUMERAL _0))))


REAL_MEASURABLE_ON_COMPOSE_CONTINUOUS_CLOSED_SET_0 |- (!f. (!g. (!s. (!t. (((real_closed s) /\ (((real_measurable_on f) t) /\ ((!x. ((f x) IN s)) /\ (((real_continuous_on g) s) /\ (((real_of_num (NUMERAL _0)) IN s) /\ ((g (real_of_num (NUMERAL _0))) = (real_of_num (NUMERAL _0)))))))) ==> ((real_measurable_on (g o f)) t))))))


UNIFORMLY_CONTINUOUS_ON_ID |- (!s. ((uniformly_continuous_on (\x. x)) s))


ORTHOGONAL_SUBSPACE_DECOMP |- (!s. (!x. (?! (GABS (\f. (!y. (!z. ((GEQ (f (y , z))) ((y IN (span s)) /\ ((z IN (GSPEC (\GEN%PVAR%185. (?z. (((SETSPEC GEN%PVAR%185) (!x. ((x IN (span s)) ==> ((orthogonal z) x)))) z))))) /\ (x = ((vector_add y) z))))))))))))


Ssrnat.leq_pexp2l |- (!m. (!n1. (!n2. (((NUMERAL _0) < m) ==> ((n1 <= n2) ==> ((m EXP n1) <= (m EXP n2)))))))


Lp_ineqs_proofs.dart3_rhazim3_fan_eq_rhazim3 |- (!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)))) ==> (((rhazim3_fan (V , E)) d) = ((((((rhazim3 (y1_fan d)) (y2_fan d)) ((y3_fan (V , E)) d)) ((y4_fan (V , E)) d)) ((y5_fan (V , E)) d)) (y6_fan d)))))))))))


ORTHOGONAL_TRANSFORMATION_LOWDIM_HORIZONTAL |- (!s. (((dim s) < (dimindex UNIV)) ==> (?f. ((orthogonal_transformation f) /\ (((IMAGE f) s) SUBSET (GSPEC (\GEN%PVAR%457. (?z. (((SETSPEC GEN%PVAR%457) ((z $ (dimindex UNIV)) = (real_of_num (NUMERAL _0)))) z)))))))))


DIV_REFL |- (!n. ((~ (n = (NUMERAL _0))) ==> ((n DIV n) = (NUMERAL (BIT1 _0)))))


Asfutbf.azim_cycle_mono |- (!V. (!E. (!v. (!x. (!x'. (((FAN ((vec (NUMERAL _0)) , (V , E))) /\ ((x IN (((set_of_edge v) V) E)) /\ ((x' IN (((set_of_edge v) V) E)) /\ (((((azim_cycle (((set_of_edge v) V) E)) (vec (NUMERAL _0))) v) x) = ((((azim_cycle (((set_of_edge v) V) E)) (vec (NUMERAL _0))) v) x'))))) ==> (x = x')))))))


Ssrfun.some_inj |- (injective SOME)


SUM_OVER_PERMUTATIONS_NUMSEG |- (!f. (!m. (!n. ((m <= n) ==> (((sum (GSPEC (\GEN%PVAR%297. (?p. (((SETSPEC GEN%PVAR%297) ((permutes p) (m .. n))) p))))) f) = ((sum (m .. n)) (\i. ((sum (GSPEC (\GEN%PVAR%298. (?p. (((SETSPEC GEN%PVAR%298) ((permutes p) ((m + (NUMERAL (BIT1 _0))) .. n))) p))))) (\q. (f ((swap (m , i)) o q)))))))))))


SPHERE_EQ_SING |- (!a. (!r. (!x. (((sphere (a , r)) = (x INSERT EMPTY)) = ((x = a) /\ (r = (real_of_num (NUMERAL _0))))))))


ABSOLUTELY_INTEGRABLE_ON_LINEAR_IMAGE |- (!f. (!g. (!s. ((linear g) ==> (((absolutely_integrable_on f) ((IMAGE g) s)) = (((absolutely_integrable_on (f o g)) s) \/ ((det (matrix g)) = (real_of_num (NUMERAL _0)))))))))


NON_MEASURABLE_SET |- (?s. (~ (lebesgue_measurable s)))


INT_ABS_SIGN2 |- (!x. (!y. (((int_lt (int_abs ((int_sub x) y))) (int_neg y)) ==> ((int_lt x) (int_of_num (NUMERAL _0))))))


Collect_geom2.ZJEWPAP |- (!v1. (!v2. (!v3. (!v4. ((LET (\s. (LET_END ((((CARD s) = (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) /\ (~ (coplanar_alt s))) ==> ((real_le (radV (v1 INSERT (v2 INSERT (v3 INSERT EMPTY))))) (radV s)))))) (v1 INSERT (v2 INSERT (v3 INSERT (v4 INSERT EMPTY)))))))))


URYSOHN_STRONG |- (!s. (!t. (!a. (!b. (((closed s) /\ ((closed t) /\ (((s INTER t) = EMPTY) /\ (~ (a = b))))) ==> (?f. (((continuous_on f) UNIV) /\ ((!x. ((f x) IN (closed_segment ((CONS (a , b)) NIL)))) /\ ((!x. (((f x) = a) = (x IN s))) /\ (!x. (((f x) = b) = (x IN t))))))))))))


INT_BOUNDS_LT |- (!x. (!k. ((((int_lt (int_neg k)) x) /\ ((int_lt x) k)) = ((int_lt (int_abs x)) k))))


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


CEXP_MUL_CPOW |- (!w. (!z. ((((real_lt (real_neg pi)) (Im w)) /\ ((real_le (Im w)) pi)) ==> ((cexp ((complex_mul w) z)) = ((cpow (cexp w)) z)))))


Lp_gen_theory.hyp_iso_comm_special |- (!H. (!g. (!G. (((hyp_iso g) (G , H)) ==> (!d. ((d IN (dart G)) ==> ((((face_map H) (g d)) = (g ((face_map G) d))) /\ ((((node_map H) (g d)) = (g ((node_map G) d))) /\ ((((edge_map H) (g d)) = (g ((edge_map G) d))) /\ (!k. ((((POWER (face_map H)) k) (g d)) = (g (((POWER (face_map G)) k) d)))))))))))))


COLLINEAR_STANDARD_HYPERPLANE_2 |- (!k. (!b. (((dimindex UNIV) <= (NUMERAL (BIT0 (BIT1 _0)))) ==> (collinear (GSPEC (\GEN%PVAR%1599. (?x. (((SETSPEC GEN%PVAR%1599) ((x $ k) = b)) x))))))))


List_hypermap_iso.set_of_edge_subset |- (!V. (!E. (!E'. (!x. ((E SUBSET E') ==> ((((set_of_edge x) V) E) SUBSET (((set_of_edge x) V) E')))))))


AFF_DIM_HALFSPACE_LT |- (!a. (!b. ((aff_dim (GSPEC (\GEN%PVAR%1548. (?x. (((SETSPEC GEN%PVAR%1548) ((real_lt ((dot a) x)) b)) x))))) = (((COND ((a = (vec (NUMERAL _0))) /\ ((real_le b) (real_of_num (NUMERAL _0))))) (int_neg (int_of_num (NUMERAL (BIT1 _0))))) (int_of_num (dimindex UNIV))))))


NSUM_INJECTION |- (!f. (!p. (!s. (((FINITE s) /\ ((!x. ((x IN s) ==> ((p x) IN s))) /\ (!x. (!y. (((x IN s) /\ ((y IN s) /\ ((p x) = (p y)))) ==> (x = y)))))) ==> (((nsum s) (f o p)) = ((nsum s) f))))))


Nonlinear_lemma.sqrt8_nn |- ((real_le (real_of_num (NUMERAL _0))) sqrt8)


NSUM_LE_GEN |- (!f. (!g. (!s. (((!x. ((x IN s) ==> ((f x) <= (g x)))) /\ (FINITE (GSPEC (\GEN%PVAR%188. (?x. (((SETSPEC GEN%PVAR%188) ((x IN s) /\ (~ ((g x) = (NUMERAL _0))))) x)))))) ==> (((nsum s) f) <= ((nsum s) g))))))


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


HAS_REAL_INTEGRAL_ALT |- (!f. (!s. (!i. ((((has_real_integral f) i) s) = ((!a. (!b. ((real_integrable_on (\x. (((COND (x IN s)) (f x)) (real_of_num (NUMERAL _0))))) (closed_real_interval ((CONS (a , b)) NIL))))) /\ (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?B. (((real_lt (real_of_num (NUMERAL _0))) B) /\ (!a. (!b. (((open_real_interval ((real_neg B) , B)) SUBSET (closed_real_interval ((CONS (a , b)) NIL))) ==> ((real_lt (real_abs ((real_sub ((real_integral (closed_real_interval ((CONS (a , b)) NIL))) (\x. (((COND (x IN s)) (f x)) (real_of_num (NUMERAL _0)))))) i))) e)))))))))))))


MEASURABLE_INSERT |- (!x. (!s. ((measurable (x INSERT s)) = (measurable s))))


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


Hypermap_and_fan.FAN_PAIR_FIXED_POINT |- (!V. (!x. ((x IN (GSPEC (\GEN%PVAR%5632. (?v. (((SETSPEC GEN%PVAR%5632) ((v IN V) /\ ((((set_of_edge v) V) E) = EMPTY))) (v , v)))))) ==> ((((n_fan_pair_ext (V , E)) x) = x) /\ (((f_fan_pair (V , E)) x) = x)))))


INFSUM_CMUL |- (!s. (!x. (!c. (((summable s) x) ==> (((infsum s) (\n. ((% c) (x n)))) = ((% c) ((infsum s) x)))))))


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


LENGTH_MAP2 |- (!f. (!l. (!m. (((LENGTH l) = (LENGTH m)) ==> ((LENGTH (((MAP2 f) l) m)) = (LENGTH m))))))


EVENTUALLY_INV1_LT |- (!e. (((eventually (\n. ((real_lt (real_inv ((real_add (real_of_num n)) (real_of_num (NUMERAL (BIT1 _0)))))) e))) sequentially) = ((real_lt (real_of_num (NUMERAL _0))) e)))


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


ABSOLUTE_RETRACTION_CONVEX_CLOSED_RELATIVE |- (!s. (!t. (((convex s) /\ ((closed s) /\ ((~ (s = EMPTY)) /\ (s SUBSET t)))) ==> (?r. (((retraction (t , s)) r) /\ (!x. ((x IN (((hull affine) s) DIFF (relative_interior s))) ==> ((r x) IN (relative_frontier s)))))))))


Merge_ineq.nonf_ldih6_x_div_sqrtdelta_posbranch |- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((((((ldih6_x_div_sqrtdelta_posbranch x1) x2) x3) x4) x5) x6) = ((real_mul (lfun ((real_div (sqrt x6)) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) ((real_mul ((real_div (sqrt ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) x6))) ((((((delta_x4 x6) x1) x5) x3) x4) x2))) (matan ((real_div ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) ((real_mul x6) ((((((delta_x x1) x2) x3) x4) x5) x6)))) ((real_pow ((((((delta_x4 x6) x1) x5) x3) x4) x2)) (NUMERAL (BIT0 (BIT1 _0)))))))))))))))


MATRIX_CMUL_SUB_LDISTRIB |- (!A. (!B. (!c. (((%% c) ((matrix_sub A) B)) = ((matrix_sub ((%% c) A)) ((%% c) B))))))


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


CLOG_MUL_II |- (!z. ((~ (z = (Cx (real_of_num (NUMERAL _0))))) ==> ((clog ((complex_mul ii) z)) = (((COND (((real_le (real_of_num (NUMERAL _0))) (Re z)) \/ ((real_lt (Im z)) (real_of_num (NUMERAL _0))))) ((vector_add (clog z)) ((complex_mul ii) (Cx ((real_div pi) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))))) ((vector_sub (clog z)) ((complex_mul ii) (Cx ((real_mul (real_of_num (NUMERAL (BIT1 (BIT1 _0))))) ((real_div pi) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))))))))))


Lp_ineqs_quads.ineq120_std |- (!V. (!E. ((lp_fan (V , E)) ==> (quad_nonlinear_v10 ==> ((!y1. (!y2. (!y3. (!y4. (!y5. (!y6. ((ineq ((CONS (((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 (BIT1 _0))))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))) , (y1 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) h0)))) ((CONS (((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0)))))) , (y2 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) ((CONS (((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0)))))) , (y3 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) ((CONS (((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0) , (y4 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) h0)))) ((CONS (((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0)))))) , (y5 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) ((CONS (((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0)))))) , (y6 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) NIL))))))) ((real_lt ((((((delta_y y1) y2) y3) y4) y5) y6)) (real_of_num (NUMERAL _0)))))))))) ==> ((!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)))))) ==> ((lp_tau (V , E)) ==> ((E = (ESTD V)) ==> (!d. ((d IN ((darts_k (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) (hypermap_of_fan (V , E)))) ==> ((real_ge ((real_add ((tau_fan (V , E)) ((face (hypermap_of_fan (V , E))) d))) ((real_sub ((real_mul ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT1 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 _0))))))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 _0))))))))))))))))) ((azim_dart (V , E)) d))) ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 _0))))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))))) (real_of_num (NUMERAL _0)))))))))))))


Functional_equation.functional_proj_x1 |- (proj_x1 = (promote1_to_6 I))


Lp_ineqs_proofs.y8_list_def |- (!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)))) ==> ((ALL (\d. (((y8_list (g , (V , E))) d) = ((ye_list g) ((inverse (f_list_ext L)) ((inverse (f_list_ext L)) d)))))) (list_of_darts L)))))))))


CRAMER_MATRIX_LEFT_INVERSE |- (!A. (!A'. ((((matrix_mul A') A) = (mat (NUMERAL (BIT1 _0)))) = ((~ ((det A) = (real_of_num (NUMERAL _0)))) /\ (A' = (lambdak. (lambdal. ((real_div (det (lambdai. (lambdaj. (((COND (j = l)) (((COND (i = k)) (real_of_num (NUMERAL (BIT1 _0)))) (real_of_num (NUMERAL _0)))) ((A $ j) $ i)))))) (det A)))))))))


SUBSET_HYPERPLANES |- (!a. (!b. (!a'. (!b'. (((GSPEC (\GEN%PVAR%442. (?x. (((SETSPEC GEN%PVAR%442) (((dot a) x) = b)) x)))) SUBSET (GSPEC (\GEN%PVAR%443. (?x. (((SETSPEC GEN%PVAR%443) (((dot a') x) = b')) x))))) = (((GSPEC (\GEN%PVAR%444. (?x. (((SETSPEC GEN%PVAR%444) (((dot a) x) = b)) x)))) = EMPTY) \/ (((GSPEC (\GEN%PVAR%445. (?x. (((SETSPEC GEN%PVAR%445) (((dot a') x) = b')) x)))) = UNIV) \/ ((GSPEC (\GEN%PVAR%446. (?x. (((SETSPEC GEN%PVAR%446) (((dot a) x) = b)) x)))) = (GSPEC (\GEN%PVAR%447. (?x. (((SETSPEC GEN%PVAR%447) (((dot a') x) = b')) x))))))))))))


REAL_SERIES_LE |- (!f. (!g. (!s. (!y. (!z. (((((real_sums f) y) s) /\ ((((real_sums g) z) s) /\ (!i. ((i IN s) ==> ((real_le (f i)) (g i)))))) ==> ((real_le y) z)))))))


HAS_BOUNDED_REAL_VARIATION_TRANSLATION_EQ_INTERVAL |- (!a. (!f. (!u. (!v. (((has_bounded_real_variation_on (\x. (f ((real_add a) x)))) (closed_real_interval ((CONS (u , v)) NIL))) = ((has_bounded_real_variation_on f) (closed_real_interval ((CONS (((real_add a) u) , ((real_add a) v))) NIL))))))))


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


Merge_ineq.nonf_edge2_126_x |- (!d. (!x4'. (!x5'. (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. ((((((((((edge2_126_x d) x4') x5') x1) x2) x3) x4) x5) x6) = ((((((edge2_flatD_x1 d) x1) x2) x6) x4') x5')))))))))))


INT_ABS_NZ |- (!x. ((~ (x = (int_of_num (NUMERAL _0)))) = ((int_lt (int_of_num (NUMERAL _0))) (int_abs x))))


CONTENT_POS_LT_1 |- (!a. (!b. (((real_lt (drop a)) (drop b)) ==> ((real_lt (real_of_num (NUMERAL _0))) (content (closed_interval ((CONS (a , b)) NIL)))))))


HAS_BOUNDED_VARIATION_ON_COMPLEX_MUL |- (!f. (!g. (!s. ((((has_bounded_variation_on f) s) /\ (((has_bounded_variation_on g) s) /\ (is_interval s))) ==> ((has_bounded_variation_on (\x. ((complex_mul (f x)) (g x)))) s)))))


NPRODUCT_EQ_NUMSEG |- (!f. (!g. (!m. (!n. ((!i. (((m <= i) /\ (i <= n)) ==> ((f i) = (g i)))) ==> (((nproduct (m .. n)) f) = ((nproduct (m .. n)) g)))))))


Tame_list.INSERT_alt |- ((a INSERT B) = (GSPEC (\GEN%PVAR%8608. (?x. (((SETSPEC GEN%PVAR%8608) ((x = a) \/ (x IN B))) x)))))


Hxhytij.HXHYTIJ |- (!s. (!vv. (!ww. (((is_scs_v39 s) /\ (((BBprime2_v39 s) vv) /\ ((BBs_v39 s) ww))) ==> (((real_lt ((taustar_v39 s) vv)) ((taustar_v39 s) ww)) \/ (((BBindex_v39 s) vv) <= ((BBindex_v39 s) ww)))))))


Functional_equation.functional_cayleyR6u |- ((LET (\x1. (LET_END ((LET (\x2. (LET_END ((LET (\x3. (LET_END ((LET (\x4. (LET_END ((LET (\x5. (LET_END ((LET (\x6. (LET_END ((LET (\four. (LET_END ((LET (\t01. (LET_END ((LET (\two. (LET_END ((LET (\mone. (LET_END (cayleyR6u = ((add6 ((mul6 mone) ((mul6 x1) ((mul6 x1) ((mul6 x4) x4))))) ((add6 ((sub6 ((mul6 two) ((mul6 x1) ((mul6 x6) ((mul6 x4) x4))))) ((mul6 x6) ((mul6 x6) ((mul6 x4) x4))))) ((add6 ((sub6 ((sub6 ((mul6 two) ((mul6 x2) ((mul6 x1) ((mul6 x4) x5))))) ((mul6 two) ((mul6 x2) ((mul6 x6) ((mul6 x4) x5)))))) ((mul6 two) ((mul6 x1) ((mul6 x6) ((mul6 x4) x5)))))) ((add6 ((sub6 ((mul6 two) ((mul6 x6) ((mul6 x6) ((mul6 x4) x5))))) ((mul6 x2) ((mul6 x2) ((mul6 x5) x5))))) ((add6 ((sub6 ((sub6 ((mul6 two) ((mul6 x2) ((mul6 x6) ((mul6 x5) x5))))) ((mul6 x6) ((mul6 x6) ((mul6 x5) x5))))) ((mul6 two) ((mul6 x2) ((mul6 x1) ((mul6 x4) four)))))) ((add6 ((mul6 two) ((mul6 x1) ((mul6 x1) ((mul6 x4) four))))) ((add6 ((sub6 ((mul6 two) ((mul6 x2) ((mul6 x6) ((mul6 x4) four))))) ((mul6 two) ((mul6 x1) ((mul6 x6) ((mul6 x4) four)))))) ((add6 ((sub6 ((sub6 ((mul6 two) ((mul6 x2) ((mul6 x2) ((mul6 x5) four))))) ((mul6 two) ((mul6 x2) ((mul6 x1) ((mul6 x5) four)))))) ((mul6 two) ((mul6 x2) ((mul6 x6) ((mul6 x5) four)))))) ((add6 ((sub6 ((mul6 two) ((mul6 x1) ((mul6 x6) ((mul6 x5) four))))) ((mul6 x2) ((mul6 x2) ((mul6 four) four))))) ((add6 ((sub6 ((mul6 two) ((mul6 x2) ((mul6 x1) ((mul6 four) four))))) ((mul6 x1) ((mul6 x1) ((mul6 four) four))))) ((add6 ((sub6 ((sub6 ((mul6 two) ((mul6 x3) ((mul6 x1) ((mul6 x4) four))))) ((mul6 two) ((mul6 x3) ((mul6 x6) ((mul6 x4) four)))))) ((mul6 two) ((mul6 x1) ((mul6 x6) ((mul6 x4) four)))))) ((add6 ((mul6 two) ((mul6 x6) ((mul6 x6) ((mul6 x4) four))))) ((add6 ((sub6 ((sub6 ((mul6 two) ((mul6 x3) ((mul6 x2) ((mul6 x5) four))))) ((mul6 two) ((mul6 x3) ((mul6 x6) ((mul6 x5) four)))))) ((mul6 two) ((mul6 x2) ((mul6 x6) ((mul6 x5) four)))))) ((add6 ((mul6 two) ((mul6 x6) ((mul6 x6) ((mul6 x5) four))))) ((add6 ((sub6 ((sub6 ((mul6 four) ((mul6 x6) ((mul6 x4) ((mul6 x5) four))))) ((mul6 two) ((mul6 x3) ((mul6 x2) ((mul6 four) four)))))) ((mul6 two) ((mul6 x3) ((mul6 x1) ((mul6 four) four)))))) ((add6 ((mul6 four) ((mul6 x2) ((mul6 x1) ((mul6 four) four))))) ((add6 ((sub6 ((sub6 ((sub6 ((sub6 ((sub6 ((sub6 ((mul6 four) ((mul6 x3) ((mul6 x6) ((mul6 four) four))))) ((mul6 two) ((mul6 x2) ((mul6 x6) ((mul6 four) four)))))) ((mul6 two) ((mul6 x1) ((mul6 x6) ((mul6 four) four)))))) ((mul6 two) ((mul6 x1) ((mul6 x4) ((mul6 four) four)))))) ((mul6 two) ((mul6 x6) ((mul6 x4) ((mul6 four) four)))))) ((mul6 two) ((mul6 x2) ((mul6 x5) ((mul6 four) four)))))) ((mul6 two) ((mul6 x6) ((mul6 x5) ((mul6 four) four)))))) ((add6 ((mul6 two) ((mul6 x2) ((mul6 four) ((mul6 four) four))))) ((add6 ((sub6 ((mul6 two) ((mul6 x1) ((mul6 four) ((mul6 four) four))))) ((mul6 x3) ((mul6 x3) ((mul6 four) four))))) ((add6 ((sub6 ((mul6 two) ((mul6 x3) ((mul6 x6) ((mul6 four) four))))) ((mul6 x6) ((mul6 x6) ((mul6 four) four))))) ((add6 ((mul6 two) ((mul6 x3) ((mul6 four) ((mul6 four) four))))) ((add6 ((sub6 ((sub6 ((mul6 two) ((mul6 x6) ((mul6 four) ((mul6 four) four))))) ((mul6 four) ((mul6 four) ((mul6 four) four))))) ((mul6 two) ((mul6 x3) ((mul6 x1) ((mul6 x4) ((mul6 t01) t01))))))) ((add6 ((mul6 two) ((mul6 x1) ((mul6 x1) ((mul6 x4) ((mul6 t01) t01)))))) ((add6 ((sub6 ((sub6 ((mul6 two) ((mul6 x3) ((mul6 x6) ((mul6 x4) ((mul6 t01) t01)))))) ((mul6 two) ((mul6 x1) ((mul6 x6) ((mul6 x4) ((mul6 t01) t01))))))) ((mul6 two) ((mul6 x3) ((mul6 x2) ((mul6 x5) ((mul6 t01) t01))))))) ((add6 ((sub6 ((sub6 ((mul6 four) ((mul6 x3) ((mul6 x1) ((mul6 x5) ((mul6 t01) t01)))))) ((mul6 two) ((mul6 x2) ((mul6 x1) ((mul6 x5) ((mul6 t01) t01))))))) ((mul6 two) ((mul6 x3) ((mul6 x6) ((mul6 x5) ((mul6 t01) t01))))))) ((add6 ((sub6 ((sub6 ((sub6 ((mul6 four) ((mul6 x2) ((mul6 x6) ((mul6 x5) ((mul6 t01) t01)))))) ((mul6 two) ((mul6 x1) ((mul6 x6) ((mul6 x5) ((mul6 t01) t01))))))) ((mul6 two) ((mul6 x1) ((mul6 x4) ((mul6 x5) ((mul6 t01) t01))))))) ((mul6 two) ((mul6 x6) ((mul6 x4) ((mul6 x5) ((mul6 t01) t01))))))) ((add6 ((mul6 two) ((mul6 x2) ((mul6 x5) ((mul6 x5) ((mul6 t01) t01)))))) ((add6 ((mul6 two) ((mul6 x6) ((mul6 x5) ((mul6 x5) ((mul6 t01) t01)))))) ((add6 ((sub6 ((sub6 ((mul6 two) ((mul6 x3) ((mul6 x2) ((mul6 four) ((mul6 t01) t01)))))) ((mul6 two) ((mul6 x3) ((mul6 x1) ((mul6 four) ((mul6 t01) t01))))))) ((mul6 two) ((mul6 x2) ((mul6 x1) ((mul6 four) ((mul6 t01) t01))))))) ((add6 ((mul6 two) ((mul6 x1) ((mul6 x1) ((mul6 four) ((mul6 t01) t01)))))) ((add6 ((sub6 ((sub6 ((mul6 four) ((mul6 x1) ((mul6 x4) ((mul6 four) ((mul6 t01) t01)))))) ((mul6 two) ((mul6 x2) ((mul6 x5) ((mul6 four) ((mul6 t01) t01))))))) ((mul6 two) ((mul6 x1) ((mul6 x5) ((mul6 four) ((mul6 t01) t01))))))) ((add6 ((sub6 ((sub6 ((mul6 two) ((mul6 x3) ((mul6 x3) ((mul6 four) ((mul6 t01) t01)))))) ((mul6 two) ((mul6 x3) ((mul6 x1) ((mul6 four) ((mul6 t01) t01))))))) ((mul6 two) ((mul6 x3) ((mul6 x6) ((mul6 four) ((mul6 t01) t01))))))) ((add6 ((sub6 ((sub6 ((sub6 ((sub6 ((mul6 two) ((mul6 x1) ((mul6 x6) ((mul6 four) ((mul6 t01) t01)))))) ((mul6 two) ((mul6 x3) ((mul6 x5) ((mul6 four) ((mul6 t01) t01))))))) ((mul6 two) ((mul6 x6) ((mul6 x5) ((mul6 four) ((mul6 t01) t01))))))) ((mul6 two) ((mul6 x3) ((mul6 four) ((mul6 four) ((mul6 t01) t01))))))) ((mul6 two) ((mul6 x1) ((mul6 four) ((mul6 four) ((mul6 t01) t01))))))) ((add6 ((sub6 ((mul6 two) ((mul6 x5) ((mul6 four) ((mul6 four) ((mul6 t01) t01)))))) ((mul6 x3) ((mul6 x3) ((mul6 ((mul6 t01) t01)) ((mul6 t01) t01)))))) ((add6 ((sub6 ((mul6 two) ((mul6 x3) ((mul6 x1) ((mul6 ((mul6 t01) t01)) ((mul6 t01) t01)))))) ((mul6 x1) ((mul6 x1) ((mul6 ((mul6 t01) t01)) ((mul6 t01) t01)))))) ((add6 ((mul6 two) ((mul6 x3) ((mul6 x5) ((mul6 ((mul6 t01) t01)) ((mul6 t01) t01)))))) ((add6 ((sub6 ((mul6 two) ((mul6 x1) ((mul6 x5) ((mul6 ((mul6 t01) t01)) ((mul6 t01) t01)))))) ((mul6 x5) ((mul6 x5) ((mul6 ((mul6 t01) t01)) ((mul6 t01) t01)))))) ((add6 ((sub6 ((sub6 ((sub6 ((sub6 ((mul6 four) ((mul6 x3) ((mul6 x2) ((mul6 x4) ((mul6 t01) t01)))))) ((mul6 two) ((mul6 x3) ((mul6 x1) ((mul6 x4) ((mul6 t01) t01))))))) ((mul6 two) ((mul6 x2) ((mul6 x1) ((mul6 x4) ((mul6 t01) t01))))))) ((mul6 two) ((mul6 x3) ((mul6 x6) ((mul6 x4) ((mul6 t01) t01))))))) ((mul6 two) ((mul6 x2) ((mul6 x6) ((mul6 x4) ((mul6 t01) t01))))))) ((add6 ((mul6 four) ((mul6 x1) ((mul6 x6) ((mul6 x4) ((mul6 t01) t01)))))) ((add6 ((mul6 two) ((mul6 x1) ((mul6 x4) ((mul6 x4) ((mul6 t01) t01)))))) ((add6 ((sub6 ((mul6 two) ((mul6 x6) ((mul6 x4) ((mul6 x4) ((mul6 t01) t01)))))) ((mul6 two) ((mul6 x3) ((mul6 x2) ((mul6 x5) ((mul6 t01) t01))))))) ((add6 ((mul6 two) ((mul6 x2) ((mul6 x2) ((mul6 x5) ((mul6 t01) t01)))))) ((add6 ((sub6 ((sub6 ((sub6 ((sub6 ((mul6 two) ((mul6 x3) ((mul6 x6) ((mul6 x5) ((mul6 t01) t01)))))) ((mul6 two) ((mul6 x2) ((mul6 x6) ((mul6 x5) ((mul6 t01) t01))))))) ((mul6 two) ((mul6 x2) ((mul6 x4) ((mul6 x5) ((mul6 t01) t01))))))) ((mul6 two) ((mul6 x6) ((mul6 x4) ((mul6 x5) ((mul6 t01) t01))))))) ((mul6 two) ((mul6 x3) ((mul6 x2) ((mul6 four) ((mul6 t01) t01))))))) ((add6 ((mul6 two) ((mul6 x2) ((mul6 x2) ((mul6 four) ((mul6 t01) t01)))))) ((add6 ((sub6 ((sub6 ((sub6 ((mul6 two) ((mul6 x3) ((mul6 x1) ((mul6 four) ((mul6 t01) t01)))))) ((mul6 two) ((mul6 x2) ((mul6 x1) ((mul6 four) ((mul6 t01) t01))))))) ((mul6 two) ((mul6 x2) ((mul6 x4) ((mul6 four) ((mul6 t01) t01))))))) ((mul6 two) ((mul6 x1) ((mul6 x4) ((mul6 four) ((mul6 t01) t01))))))) ((add6 ((mul6 four) ((mul6 x2) ((mul6 x5) ((mul6 four) ((mul6 t01) t01)))))) ((add6 ((sub6 ((sub6 ((mul6 two) ((mul6 x3) ((mul6 x3) ((mul6 four) ((mul6 t01) t01)))))) ((mul6 two) ((mul6 x3) ((mul6 x2) ((mul6 four) ((mul6 t01) t01))))))) ((mul6 two) ((mul6 x3) ((mul6 x6) ((mul6 four) ((mul6 t01) t01))))))) ((add6 ((sub6 ((sub6 ((sub6 ((sub6 ((mul6 two) ((mul6 x2) ((mul6 x6) ((mul6 four) ((mul6 t01) t01)))))) ((mul6 two) ((mul6 x3) ((mul6 x4) ((mul6 four) ((mul6 t01) t01))))))) ((mul6 two) ((mul6 x6) ((mul6 x4) ((mul6 four) ((mul6 t01) t01))))))) ((mul6 two) ((mul6 x3) ((mul6 four) ((mul6 four) ((mul6 t01) t01))))))) ((mul6 two) ((mul6 x2) ((mul6 four) ((mul6 four) ((mul6 t01) t01))))))) ((add6 ((mul6 two) ((mul6 x4) ((mul6 four) ((mul6 four) ((mul6 t01) t01)))))) ((add6 ((sub6 ((sub6 ((mul6 two) ((mul6 x3) ((mul6 x3) ((mul6 ((mul6 t01) t01)) ((mul6 t01) t01)))))) ((mul6 two) ((mul6 x3) ((mul6 x2) ((mul6 ((mul6 t01) t01)) ((mul6 t01) t01))))))) ((mul6 two) ((mul6 x3) ((mul6 x1) ((mul6 ((mul6 t01) t01)) ((mul6 t01) t01))))))) ((add6 ((sub6 ((sub6 ((sub6 ((sub6 ((mul6 two) ((mul6 x2) ((mul6 x1) ((mul6 ((mul6 t01) t01)) ((mul6 t01) t01)))))) ((mul6 two) ((mul6 x3) ((mul6 x4) ((mul6 ((mul6 t01) t01)) ((mul6 t01) t01))))))) ((mul6 two) ((mul6 x1) ((mul6 x4) ((mul6 ((mul6 t01) t01)) ((mul6 t01) t01))))))) ((mul6 two) ((mul6 x3) ((mul6 x5) ((mul6 ((mul6 t01) t01)) ((mul6 t01) t01))))))) ((mul6 two) ((mul6 x2) ((mul6 x5) ((mul6 ((mul6 t01) t01)) ((mul6 t01) t01))))))) ((add6 ((mul6 two) ((mul6 x4) ((mul6 x5) ((mul6 ((mul6 t01) t01)) ((mul6 t01) t01)))))) ((add6 ((sub6 ((mul6 four) ((mul6 x3) ((mul6 four) ((mul6 ((mul6 t01) t01)) ((mul6 t01) t01)))))) ((mul6 x3) ((mul6 x3) ((mul6 ((mul6 t01) t01)) ((mul6 t01) t01)))))) ((add6 ((sub6 ((mul6 two) ((mul6 x3) ((mul6 x2) ((mul6 ((mul6 t01) t01)) ((mul6 t01) t01)))))) ((mul6 x2) ((mul6 x2) ((mul6 ((mul6 t01) t01)) ((mul6 t01) t01)))))) ((add6 ((mul6 two) ((mul6 x3) ((mul6 x4) ((mul6 ((mul6 t01) t01)) ((mul6 t01) t01)))))) ((sub6 ((mul6 two) ((mul6 x2) ((mul6 x4) ((mul6 ((mul6 t01) t01)) ((mul6 t01) t01)))))) ((mul6 x4) ((mul6 x4) ((mul6 ((mul6 t01) t01)) ((mul6 t01) t01)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) (constant6 (real_neg (real_of_num (NUMERAL (BIT1 _0))))))))) (constant6 (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))))) (constant6 ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 _0))))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))))) (constant6 (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))))))) proj_x6)))) proj_x5)))) proj_x4)))) proj_x3)))) proj_x2)))) proj_x1)


Seq2.list_sum_undup |- (!s. (!f. (((list_sum s) f) = ((list_sum (undup s)) (\x. ((real_mul (real_of_num ((count (pred1 x)) s))) (f x)))))))


PRODUCT_CLAUSES_RIGHT |- (!f. (!m. (!n. ((((NUMERAL _0) < n) /\ (m <= n)) ==> (((product (m .. n)) f) = ((real_mul ((product (m .. (n - (NUMERAL (BIT1 _0))))) f)) (f n)))))))


REAL_CONVEX_ON_EXP |- (!s. ((real_convex_on exp) s))


HOMEOMORPHIC_GRAPH |- (!f. (!s. (((continuous_on f) s) ==> ((homeomorphic (GSPEC (\GEN%PVAR%1035. (?x. (((SETSPEC GEN%PVAR%1035) (x IN s)) ((pastecart x) (f x))))))) s))))


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


ORDER_EXISTENCE_FINITE |- (!R. (!f. (!z. (((FINITE (GSPEC (\GEN%PVAR%77. (?n. (((SETSPEC GEN%PVAR%77) (n IN UNIV)) (R (((ITER n) f) z))))))) /\ (((R z) z) /\ ((!x. (!y. (((R x) y) ==> ((R y) x)))) /\ ((!x. (!y. (!z. ((((R x) y) /\ ((R y) z)) ==> ((R x) z))))) /\ (!x. (!y. (((R (f x)) (f y)) = ((R x) y)))))))) ==> (?d. (((NUMERAL _0) < d) /\ (!n. (((R (((ITER n) f) z)) z) = ((num_divides d) n)))))))))


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


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


NSUM_CONST_NUMSEG |- (!c. (!m. (!n. (((nsum (m .. n)) (\n. c)) = (((n + (NUMERAL (BIT1 _0))) - m) * c)))))


Deformation.COMPACT_SPHERE_0 |- (!a. (compact (GSPEC (\GEN%PVAR%7989. (?x. (((SETSPEC GEN%PVAR%7989) ((vector_norm x) = a)) x))))))


CONTINUOUS_ON_CPOW_RIGHT |- (!w. (!f. (!s. (((continuous_on f) s) ==> ((continuous_on (\z. ((cpow w) (f z)))) s)))))


CONTINUOUS_WITHIN_CEXP |- (!s. (!z. ((continuous cexp) ((within (at z)) s))))


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


INJECTIVE_PREIMAGE |- (!f. ((!t. (!t'. (((GSPEC (\GEN%PVAR%145. (?x. (((SETSPEC GEN%PVAR%145) ((f x) IN t)) x)))) = (GSPEC (\GEN%PVAR%146. (?x. (((SETSPEC GEN%PVAR%146) ((f x) IN t')) x))))) ==> (t = t')))) = (((IMAGE f) UNIV) = UNIV)))


ISO_REFL |- ((ISO (\x. x)) (\x. x))


HOMOTOPIC_LINEAR_POSITIVE_SEMIDEFINITE_MAPS |- (!f. (!g. (((((homotopic_with (\f. ((linear f) /\ (positive_semidefinite (matrix f))))) (UNIV , UNIV)) f) g) = ((linear f) /\ ((linear g) /\ ((positive_semidefinite (matrix f)) /\ (positive_semidefinite (matrix g))))))))


CANTOR_BENDIXSON |- (!s. ((closed s) ==> (?t. (?u. ((closed t) /\ ((!x. ((x IN t) ==> ((limit_point_of x) t))) /\ ((COUNTABLE u) /\ (s = (t UNION u)))))))))


INT_EQ_ADD_RCANCEL |- (!x. (!y. (!z. ((((int_add x) z) = ((int_add y) z)) = (x = y)))))


COVERING_SPACE_LIFT |- (!p. (!c. (!s. (!f. (!u. ((((covering_space (c , p)) s) /\ ((simply_connected u) /\ (((locally path_connected) u) /\ (((continuous_on f) u) /\ (((IMAGE f) u) SUBSET s))))) ==> (?g. (((continuous_on g) u) /\ ((((IMAGE g) u) SUBSET c) /\ (!y. ((y IN u) ==> ((p (g y)) = (f y)))))))))))))


TENDSTO_LIM |- (!net. (!f. (!l. (((~ (trivial_limit net)) /\ (((--> f) l) net)) ==> (((lim net) f) = l)))))


CEXP_VSUM |- (!f. (!s. ((FINITE s) ==> ((cexp ((vsum s) f)) = ((cproduct s) (\x. (cexp (f x))))))))


AFF_LE_EQ_AFFINE_HULL |- (!s. (!t. (((aff_le s) EMPTY) = ((hull affine) s))))


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


Topology.FINITE_SERIES |- (!n. (!f. (FINITE (GSPEC (\GEN%PVAR%5487. (?i. (((SETSPEC GEN%PVAR%5487) (i < n)) (f i))))))))


FORALL_INTEGER |- (!P. (((!n. (P (real_of_num n))) /\ (!x. ((P x) ==> (P (real_neg x))))) ==> (!x. ((integer x) ==> (P x)))))


MEASURABLE_ON_MEASURABLE_PREIMAGE_CLOSED_EQ |- (!f. (!s. ((((measurable_on f) s) /\ (measurable s)) = (!t. ((closed t) ==> (measurable (GSPEC (\GEN%PVAR%3659. (?x. (((SETSPEC GEN%PVAR%3659) ((x IN s) /\ ((f x) IN t))) x))))))))))


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


HOMEOMORPHIC_STANDARD_HYPERPLANE_HYPERPLANE |- (!a. (!b. (!k. (!c. (((~ (a = (vec (NUMERAL _0)))) /\ (((NUMERAL (BIT1 _0)) <= k) /\ (k <= (dimindex UNIV)))) ==> ((homeomorphic (GSPEC (\GEN%PVAR%1060. (?x. (((SETSPEC GEN%PVAR%1060) ((x $ k) = c)) x))))) (GSPEC (\GEN%PVAR%1061. (?x. (((SETSPEC GEN%PVAR%1061) (((dot a) x) = b)) x))))))))))


CARD_FACES_OF_SIMPLEX |- (!n. (!s. (((simplex n) s) ==> ((CARD (GSPEC (\GEN%PVAR%2481. (?f. (((SETSPEC GEN%PVAR%2481) ((face_of f) s)) f))))) = ((NUMERAL (BIT0 (BIT1 _0))) EXP (num_of_int ((int_add n) (int_of_num (NUMERAL (BIT1 _0))))))))))


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


Localization.slicef |- (!E. (!FF. (!w. (!v. (((((slicef E) FF) v) w) = ((GSPEC (\GEN%PVAR%7541. (?f. (((SETSPEC GEN%PVAR%7541) (?u. ((u IN (((((slicev E) FF) v) w) DELETE w)) /\ (f = (u , ((rho_node1 FF) u)))))) f)))) UNION ((w , v) INSERT EMPTY)))))))


AFF_DIM_EQ_INTER_HYPERPLANE |- (!s. (!t. (((affine s) /\ ((affine t) /\ ((t SUBSET s) /\ (((int_add (aff_dim t)) (int_of_num (NUMERAL (BIT1 _0)))) = (aff_dim s))))) ==> (?a. (?b. ((~ (a = (vec (NUMERAL _0)))) /\ (((GSPEC (\GEN%PVAR%1574. (?x. (((SETSPEC GEN%PVAR%1574) (((dot a) x) = b)) x)))) INTER s) = t)))))))


INFSUM_LINEAR |- (!f. (!h. (!s. ((((summable s) f) /\ (linear h)) ==> (((infsum s) (\n. (h (f n)))) = (h ((infsum s) f)))))))


Hypermap_iso.disjoint_diff |- (!s. (!t. ((DISJOINT s) (t DIFF s))))


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


LIMPT_PCROSS |- (!s. (!t. (!x. (!y. ((((limit_point_of x) s) /\ ((limit_point_of y) t)) ==> ((limit_point_of ((pastecart x) y)) (s PCROSS t)))))))


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


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


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


REAL_CARD_INTSEG_INT |- (!a. (!b. (((integer a) /\ (integer b)) ==> ((real_of_num (CARD (GSPEC (\GEN%PVAR%86. (?x. (((SETSPEC GEN%PVAR%86) ((integer x) /\ (((real_le a) x) /\ ((real_le x) b)))) x)))))) = (((COND ((real_lt b) a)) (real_of_num (NUMERAL _0))) ((real_add ((real_sub b) a)) (real_of_num (NUMERAL (BIT1 _0)))))))))


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


SYMMETRIC_MATRIX_ORTHOGONAL_EIGENVECTORS |- (!A. (!v. (!w. (!a. (!b. ((((transp A) = A) /\ ((((matrix_vector_mul A) v) = ((% a) v)) /\ ((((matrix_vector_mul A) w) = ((% b) w)) /\ (~ (a = b))))) ==> ((orthogonal v) w)))))))


REAL_INTEGRABLE_ON_EMPTY |- (!f. ((real_integrable_on f) EMPTY))


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


AR_IMP_PATH_CONNECTED |- (!s. ((AR s) ==> (path_connected s)))


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


SUMMABLE_IMP_SUMS_BOUNDED |- (!f. (!k. (((summable (from k)) f) ==> (bounded (GSPEC (\GEN%PVAR%1120. (?n. (((SETSPEC GEN%PVAR%1120) (n IN UNIV)) ((vsum (k .. n)) f)))))))))


RETRACT_OF_HOMOTOPICALLY_TRIVIAL_NULL |- (!s. (!t. (!u. ((((retract_of t) s) /\ (!f. ((((continuous_on f) u) /\ (((IMAGE f) u) SUBSET s)) ==> (?c. ((((homotopic_with (\x. T)) (u , s)) f) (\x. c)))))) ==> (!f. ((((continuous_on f) u) /\ (((IMAGE f) u) SUBSET t)) ==> (?c. ((((homotopic_with (\x. T)) (u , t)) f) (\x. c)))))))))


RELATIVE_FRONTIER_RELATIVE_INTERIOR |- (!s. ((convex s) ==> ((relative_frontier (relative_interior s)) = (relative_frontier s))))


DIST_TRIANGLE_ADD_HALF |- (!x. (!x'. (!y. (!y'. ((((real_lt (distance (x , x'))) ((real_div e) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) /\ ((real_lt (distance (y , y'))) ((real_div e) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) ==> ((real_lt (distance (((vector_add x) y) , ((vector_add x') y')))) e))))))


ABSOLUTELY_CONTINUOUS_INDEFINITE_INTEGRAL_LEFT |- (!f. (!a. (!b. (((absolutely_integrable_on f) (closed_interval ((CONS (a , b)) NIL))) ==> ((absolutely_continuous_on (\x. ((integral (closed_interval ((CONS (x , b)) NIL))) f))) (closed_interval ((CONS (a , b)) NIL)))))))


ORTHOGONAL_MATRIX_ORTHOGONAL_EIGENVECTORS |- (!A. (!v. (!w. (!a. (!b. (((orthogonal_matrix A) /\ ((((matrix_vector_mul A) v) = ((% a) v)) /\ ((((matrix_vector_mul A) w) = ((% b) w)) /\ (~ (a = b))))) ==> ((orthogonal v) w)))))))


REAL_CONTINUOUS_ON_ACS |- (!s. ((!x. ((x IN s) ==> ((real_le (real_abs x)) (real_of_num (NUMERAL (BIT1 _0)))))) ==> ((real_continuous_on acs) s)))


COMPLEX_DERIVATIVE_SUB_AT |- (!f. (!g. (!z. ((((analytic_on f) (z INSERT EMPTY)) /\ ((analytic_on g) (z INSERT EMPTY))) ==> (((complex_derivative (\w. ((vector_sub (f w)) (g w)))) z) = ((vector_sub ((complex_derivative f) z)) ((complex_derivative g) z)))))))


MEASURE_INTERVAL_4_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_lt (a $ (NUMERAL (BIT1 (BIT1 _0))))) (b $ (NUMERAL (BIT1 (BIT1 _0))))) /\ ((real_lt (a $ (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) (b $ (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))))))) ((real_mul ((real_sub (b $ (NUMERAL (BIT1 _0)))) (a $ (NUMERAL (BIT1 _0))))) ((real_mul ((real_sub (b $ (NUMERAL (BIT0 (BIT1 _0))))) (a $ (NUMERAL (BIT0 (BIT1 _0)))))) ((real_mul ((real_sub (b $ (NUMERAL (BIT1 (BIT1 _0))))) (a $ (NUMERAL (BIT1 (BIT1 _0)))))) ((real_sub (b $ (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) (a $ (NUMERAL (BIT0 (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_lt (a $ (NUMERAL (BIT1 (BIT1 _0))))) (b $ (NUMERAL (BIT1 (BIT1 _0))))) /\ ((real_lt (a $ (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) (b $ (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))))))) ((real_mul ((real_sub (b $ (NUMERAL (BIT1 _0)))) (a $ (NUMERAL (BIT1 _0))))) ((real_mul ((real_sub (b $ (NUMERAL (BIT0 (BIT1 _0))))) (a $ (NUMERAL (BIT0 (BIT1 _0)))))) ((real_mul ((real_sub (b $ (NUMERAL (BIT1 (BIT1 _0))))) (a $ (NUMERAL (BIT1 (BIT1 _0)))))) ((real_sub (b $ (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) (a $ (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))))))) (real_of_num (NUMERAL _0)))))))


HAS_REAL_MEASURE_UNION_REAL_NEGLIGIBLE_EQ |- (!s. (!t. (!m. ((real_negligible t) ==> (((has_real_measure (s UNION t)) m) = ((has_real_measure s) m))))))


INDEPENDENT_EQ_DIM_EQ_CARD |- (!s. ((independent s) = ((FINITE s) /\ ((dim s) = (CARD s)))))


NONNEGATIVE_INTEGER |- (!x. (((integer x) /\ ((real_le (real_of_num (NUMERAL _0))) x)) = (?n. (x = (real_of_num n)))))


ANR_INTERIOR |- (!s. (ANR (interior s)))


CLOSED_INTERVAL_RIGHT |- (!a. (closed (GSPEC (\GEN%PVAR%762. (?x. (((SETSPEC GEN%PVAR%762) (!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV))) ==> ((real_le (a $ i)) (x $ i))))) x))))))


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


RESTRICTION_EQ |- (!s. (!f. (!x. (!y. (((x IN s) /\ ((f x) = y)) ==> ((((RESTRICTION s) f) x) = y))))))


Functional_equation.functional_halfbump_x4 |- (halfbump_x4 = (rotate4 halfbump_x1))


POSITIVE_SEMIDEFINITE_2 |- (!A. ((positive_semidefinite A) = ((((A $ (NUMERAL (BIT0 (BIT1 _0)))) $ (NUMERAL (BIT1 _0))) = ((A $ (NUMERAL (BIT1 _0))) $ (NUMERAL (BIT0 (BIT1 _0))))) /\ (((real_le (real_of_num (NUMERAL _0))) ((A $ (NUMERAL (BIT1 _0))) $ (NUMERAL (BIT1 _0)))) /\ (((real_le (real_of_num (NUMERAL _0))) ((A $ (NUMERAL (BIT0 (BIT1 _0)))) $ (NUMERAL (BIT0 (BIT1 _0))))) /\ ((real_le ((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)))))))))))


Lp_ineqs_proofs.tau_sum3 |- (!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 (BIT1 (BIT1 _0)))) ==> (((sum f) (rhazim_fan (V , E))) = ((real_add ((tau_fan (V , E)) f)) ((real_add pi) sol0)))))))))))))


GDELTA_UNIONS |- (!g. (((FINITE g) /\ (!u. ((u IN g) ==> (gdelta u)))) ==> (gdelta (UNIONS g))))


HOMEOMORPHISM_IMP_QUOTIENT_MAP |- (!f. (!g. (!s. (!t. (((homeomorphism (s , t)) (f , g)) ==> (!u. ((u SUBSET t) ==> (((open_in ((subtopology euclidean) s)) (GSPEC (\GEN%PVAR%1030. (?x. (((SETSPEC GEN%PVAR%1030) ((x IN s) /\ ((f x) IN u))) x))))) = ((open_in ((subtopology euclidean) t)) u)))))))))


Dih2k_hypermap.VECMAT_VEC |- (!i. (!n. ((i < (dimindex UNIV)) ==> (((vecmat i) (vec n)) = (vec n)))))


Vasyyau.A_EQ2h0_IMP_B |- (((!i. ((((((scs_a_v39 s) i) (i + (NUMERAL (BIT1 _0)))) = (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) /\ ((((scs_b_v39 s) i) (i + (NUMERAL (BIT1 _0)))) = ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0))) \/ (((((scs_a_v39 s) i) (i + (NUMERAL (BIT1 _0)))) = ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) /\ ((((scs_b_v39 s) i) (i + (NUMERAL (BIT1 _0)))) = cstab)))) /\ ((((scs_a_v39 s) p) (p + (NUMERAL (BIT1 _0)))) = ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0))) ==> ((((scs_b_v39 s) p) (p + (NUMERAL (BIT1 _0)))) = cstab))


SIN_VECTOR_ANGLE_RMUL |- (!a. (!x. (!y. ((sin ((vector_angle x) ((% a) y))) = (((COND (a = (real_of_num (NUMERAL _0)))) (real_of_num (NUMERAL (BIT1 _0)))) (sin ((vector_angle x) y)))))))


CONIC_HULLS_EQ_IMP_SPANS_EQ |- (!s. (!t. ((((hull conic) s) = ((hull conic) t)) ==> ((span s) = (span t)))))


COUNTABLE_ISOLATED_SET |- (!s. ((!a. ((a IN s) ==> (~ ((limit_point_of a) s)))) ==> (COUNTABLE s)))


UNIFORM_LIM_COMPLEX_DIV |- (!net. (!P. (!f. (!g. (!l. (!m. (!b1. (!b2. ((((eventually (\x. (!n. ((P n) ==> ((real_le (vector_norm (l n))) b1))))) net) /\ (((real_lt (real_of_num (NUMERAL _0))) b2) /\ (((eventually (\x. (!n. ((P n) ==> ((real_le b2) (vector_norm (m n))))))) net) /\ ((!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> ((eventually (\x. (!n. ((P n) ==> ((real_lt (vector_norm ((vector_sub ((f n) x)) (l n)))) e))))) net))) /\ (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> ((eventually (\x. (!n. ((P n) ==> ((real_lt (vector_norm ((vector_sub ((g n) x)) (m n)))) e))))) net))))))) ==> (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> ((eventually (\x. (!n. ((P n) ==> ((real_lt (vector_norm ((vector_sub ((complex_div ((f n) x)) ((g n) x))) ((complex_div (l n)) (m n))))) e))))) net))))))))))))


Functional_equation.functional_dih2_x |- (dih2_x = (rotate2 dih_x))


MATRIX_CMUL_RZERO |- (!c. (((%% c) (mat (NUMERAL _0))) = (mat (NUMERAL _0))))


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


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


INT_EQ_MUL_RCANCEL |- (!x. (!y. (!z. ((((int_mul x) z) = ((int_mul y) z)) = ((x = y) \/ (z = (int_of_num (NUMERAL _0))))))))


CARD_EQ_OPEN_IN_AFFINE |- (!u. (!s. (((affine u) /\ ((~ ((aff_dim u) = (int_of_num (NUMERAL _0)))) /\ (((open_in ((subtopology euclidean) u)) s) /\ (~ (s = EMPTY))))) ==> (s =_c UNIV))))


More_list_hypermap.contravening_dist_not_dart |- (!V. (!h. (!L. (!g. ((contravening V) ==> ((good_list L) ==> (((hyp_iso g) ((hypermap_of_list L) , (hypermap_of_fan (V , (ESTD V))))) ==> ((((BIJ h) (elements_of_list L)) V) ==> ((g = (\d. ((h (FST d)) , (h (SND d))))) ==> (!s. ((((ALL (\d. (((MEM (FST d)) (list_of_elements L)) /\ (((MEM (SND d)) (list_of_elements L)) /\ (((FST d) = (SND d)) = F))))) s) /\ ((ALL (\d. (((MEM d) (list_of_darts L)) = F))) s)) ==> ((ALL (\d. ((real_le ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) (distance (g d))))) s))))))))))))


RETRACTION_ARC |- (!p. ((arc p) ==> (?f. (((continuous_on f) UNIV) /\ ((((IMAGE f) UNIV) SUBSET (path_image p)) /\ (!x. ((x IN (path_image p)) ==> ((f x) = x))))))))


INVERTIBLE_MAT |- (!a. ((invertible (mat a)) = (~ (a = (NUMERAL _0)))))


OPEN_MAP_CLOSED_SUPERSET_PREIMAGE_EQ |- (!f. (!s. (!t. ((((IMAGE f) s) SUBSET t) ==> ((!k. (((open_in ((subtopology euclidean) s)) k) ==> ((open_in ((subtopology euclidean) t)) ((IMAGE f) k)))) = (!u. (!w. ((((closed_in ((subtopology euclidean) s)) u) /\ ((w SUBSET t) /\ ((GSPEC (\GEN%PVAR%699. (?x. (((SETSPEC GEN%PVAR%699) ((x IN s) /\ ((f x) IN w))) x)))) SUBSET u))) ==> (?v. (((closed_in ((subtopology euclidean) t)) v) /\ ((w SUBSET v) /\ ((GSPEC (\GEN%PVAR%700. (?x. (((SETSPEC GEN%PVAR%700) ((x IN s) /\ ((f x) IN v))) x)))) SUBSET u))))))))))))


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


Optimize.EXPAND_1var_lemma |- (!f. (!a. (!b. ((!y. ((ineq ((CONS (a , (y , b))) NIL)) (f y))) = (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. ((ineq ((CONS (a , (x1 , b))) ((CONS ((real_of_num (NUMERAL (BIT1 _0))) , (x2 , (real_of_num (NUMERAL (BIT1 _0)))))) ((CONS ((real_of_num (NUMERAL (BIT1 _0))) , (x3 , (real_of_num (NUMERAL (BIT1 _0)))))) ((CONS ((real_of_num (NUMERAL (BIT1 _0))) , (x4 , (real_of_num (NUMERAL (BIT1 _0)))))) ((CONS ((real_of_num (NUMERAL (BIT1 _0))) , (x5 , (real_of_num (NUMERAL (BIT1 _0)))))) ((CONS ((real_of_num (NUMERAL (BIT1 _0))) , (x6 , (real_of_num (NUMERAL (BIT1 _0)))))) NIL))))))) (f x1))))))))))))


INVERTIBLE_EIGENVALUES |- (!A. ((invertible A) = (!c. (!v. (((((matrix_vector_mul A) v) = ((% c) v)) /\ (~ (v = (vec (NUMERAL _0))))) ==> (~ (c = (real_of_num (NUMERAL _0)))))))))


AR_IMP_ABSOLUTE_RETRACT_UNIV |- (!s. (!s'. (((AR s) /\ (((homeomorphic s) s') /\ (closed s'))) ==> ((retract_of s') UNIV))))


INT_POW_NZ |- (!x. (!n. ((~ (x = (int_of_num (NUMERAL _0)))) ==> (~ (((int_pow x) n) = (int_of_num (NUMERAL _0)))))))


COLUMN_0 |- (!i. (((column i) (mat (NUMERAL _0))) = (vec (NUMERAL _0))))


SKOLEM_THM_GEN |- (!P. (!s. ((!x. ((P x) ==> (?y. ((R x) y)))) = (?f. (!x. ((P x) ==> ((R x) (f x))))))))


CONTINUOUS_AT_CX_RE |- (!z. ((continuous (Cx o Re)) (at z)))


Local_lemmas.AFF_GE_EQ |- ((((real_lt (real_of_num (NUMERAL _0))) t) /\ (((vector_sub a) x) = ((% t) ((vector_sub b) x)))) ==> (((aff_ge (x INSERT EMPTY)) (a INSERT EMPTY)) = ((aff_ge (x INSERT EMPTY)) (b INSERT EMPTY))))


COMPLEX_POW_EQ_1 |- (!z. (!n. ((((complex_pow z) n) = (Cx (real_of_num (NUMERAL (BIT1 _0))))) ==> (((vector_norm z) = (real_of_num (NUMERAL (BIT1 _0)))) \/ (n = (NUMERAL _0))))))


INT_LTE_ADD2 |- (!w. (!x. (!y. (!z. ((((int_lt w) x) /\ ((int_le y) z)) ==> ((int_lt ((int_add w) y)) ((int_add x) z)))))))


INT_ABS_BETWEEN1 |- (!x. (!y. (!z. ((((int_lt x) z) /\ ((int_lt (int_abs ((int_sub y) x))) ((int_sub z) x))) ==> ((int_lt y) z)))))


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


CNJ_CPRODUCT |- (!f. (!s. ((FINITE s) ==> ((cnj ((cproduct s) f)) = ((cproduct s) (\i. (cnj (f i))))))))


REAL_CONTINUOUS_WITHIN_ACS |- (!s. (!x. (((real_lt (real_abs x)) (real_of_num (NUMERAL (BIT1 _0)))) ==> ((real_continuous acs) ((within (atreal x)) s)))))


REALLIM_1_OVER_N_OFFSET |- (!a. (((---> (\n. (real_inv ((real_add (real_of_num n)) a)))) (real_of_num (NUMERAL _0))) sequentially))


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


CONIC_CAP_DEGENERATE |- (!v0. (!r. (!a. (((((conic_cap v0) v0) r) a) = EMPTY))))


HOLOMORPHIC_ON_EMPTY |- (!f. ((holomorphic_on f) EMPTY))


REAL_COMPLEX_INTEGRAL |- (!f. (!s. ((((integrable_on f) s) /\ (!x. ((x IN s) ==> (real (f x))))) ==> (real ((integral s) f)))))


CONVEX_HULL_SUBSET_SPAN |- (!s. (((hull convex) s) SUBSET (span s)))


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


Geomdetail.CONV_EM |- ((conv EMPTY) = EMPTY)


INT_ABS_BETWEEN2 |- (!x0. (!x. (!y0. (!y. ((((int_lt x0) y0) /\ (((int_lt ((int_mul (int_of_num (NUMERAL (BIT0 (BIT1 _0))))) (int_abs ((int_sub x) x0)))) ((int_sub y0) x0)) /\ ((int_lt ((int_mul (int_of_num (NUMERAL (BIT0 (BIT1 _0))))) (int_abs ((int_sub y) y0)))) ((int_sub y0) x0)))) ==> ((int_lt x) y))))))


Pqcsxwg.PQCSXWG2 |- (!v0. (!v1. (!v2. (!v3. (!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) /\ ((~ (collinear (v0 INSERT (v1 INSERT (v2 INSERT EMPTY))))) /\ ((x1 = ((real_pow (distance (v1 , v0))) (NUMERAL (BIT0 (BIT1 _0))))) /\ ((x2 = ((real_pow (distance (v2 , v0))) (NUMERAL (BIT0 (BIT1 _0))))) /\ ((x6 = ((real_pow (distance (v1 , v2))) (NUMERAL (BIT0 (BIT1 _0))))) /\ (((real_lt (real_of_num (NUMERAL _0))) ((((((delta_x x1) x2) x3) x4) x5) x6)) /\ (v3 = (((((((((mk_simplex1 v0) v1) v2) x1) x2) x3) x4) x5) x6))))))))))))) ==> ((continuous (\q. (((((((((mk_simplex1 v0) v1) v2) x1) x2) x3) x4) q) x6))) (atreal x5)))))))))))))


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


SEPARATING_HYPERPLANE_POLYHEDRA |- (!s. (!t. (((polyhedron s) /\ ((polyhedron t) /\ ((~ (s = EMPTY)) /\ ((~ (t = EMPTY)) /\ ((DISJOINT s) t))))) ==> (?a. (?b. ((~ (a = (vec (NUMERAL _0)))) /\ ((!x. ((x IN s) ==> ((real_lt ((dot a) x)) b))) /\ (!x. ((x IN t) ==> ((real_gt ((dot a) x)) b))))))))))


ABSOLUTELY_SETCONTINUOUS_ON_LIFT_ABS |- (!f. (!s. (((absolutely_setcontinuous_on (\x. (lift (real_abs (f x))))) s) = ((absolutely_setcontinuous_on (\x. (lift (f x)))) s))))


INT_SGN_POW_2 |- (!x. ((int_sgn ((int_pow x) (NUMERAL (BIT0 (BIT1 _0))))) = (int_sgn (int_abs x))))


BOUNDED_EQ_TOTALLY_BOUNDED |- (!s. ((bounded s) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?k. ((FINITE k) /\ ((k SUBSET s) /\ (s SUBSET (UNIONS ((IMAGE (\x. (ball (x , e)))) k))))))))))


BOUNDED_SUMS_IMAGES |- (!f. (!t. (!s. (((FINITE s) /\ (!a. ((a IN s) ==> (bounded (GSPEC (\GEN%PVAR%591. (?x. (((SETSPEC GEN%PVAR%591) (x IN t)) ((f x) a))))))))) ==> (bounded (GSPEC (\GEN%PVAR%592. (?x. (((SETSPEC GEN%PVAR%592) (x IN t)) ((vsum s) (f x)))))))))))


ANR_SIMPLICIAL_COMPLEX |- (!c. ((simplicial_complex c) ==> (ANR (UNIONS c))))


DIM_OPEN_IN |- (!s. (!t. (((~ (s = EMPTY)) /\ (((open_in ((subtopology euclidean) t)) s) /\ (subspace t))) ==> ((dim s) = (dim t)))))


ABSOLUTELY_INTEGRABLE_CONVOLUTION_LINF_L1 |- (!bop. (!f. (!g. (!x. (((bilinear bop) /\ (((measurable_on f) UNIV) /\ ((bounded ((IMAGE f) UNIV)) /\ ((absolutely_integrable_on g) UNIV)))) ==> ((absolutely_integrable_on (\y. ((bop (f ((vector_sub x) y))) (g y)))) UNIV))))))


REAL_MEASURABLE_ON_MEASURABLE_SUBSET |- (!f. (!s. (!t. (((s SUBSET t) /\ (((real_measurable_on f) t) /\ (real_measurable s))) ==> ((real_measurable_on f) s)))))


OPEN_IN_CLOSED_IN |- (!s. ((s SUBSET (topspace top)) ==> (((open_in top) s) = ((closed_in top) ((topspace top) DIFF s)))))


REAL_BEPPO_LEVI_INCREASING |- (!f. (!s. (((!k. ((real_integrable_on (f k)) s)) /\ ((!k. (!x. ((x IN s) ==> ((real_le ((f k) x)) ((f (SUC k)) x))))) /\ (real_bounded (GSPEC (\GEN%PVAR%4926. (?k. (((SETSPEC GEN%PVAR%4926) (k IN UNIV)) ((real_integral s) (f k))))))))) ==> (?g. (?k. ((real_negligible k) /\ (!x. ((x IN (s DIFF k)) ==> (((---> (\k. ((f k) x))) (g x)) sequentially)))))))))