COLLINEAR_HYPERPLANE_2

|- (!a. (!b. ((((dimindex UNIV) <= (NUMERAL (BIT0 (BIT1 _0)))) /\ (~ (a = (vec (NUMERAL _0))))) ==> (collinear (GSPEC (\GEN%PVAR%1598. (?x. (((SETSPEC GEN%PVAR%1598) (((dot a) x) = b)) x))))))))


VECTOR_MUL_RZERO

|- (((% c) (vec (NUMERAL _0))) = (vec (NUMERAL _0)))


CONNECTED_COMPONENT_LINEAR_IMAGE

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


REAL_INFSUM_UNIQUE

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


sgn_lt

|- (sgn_lt = (\t. ((real_lt t) (real_of_num (NUMERAL _0)))))


FORALL_4

|- (!P. ((!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) ==> (P i))) = ((P (NUMERAL (BIT1 _0))) /\ ((P (NUMERAL (BIT0 (BIT1 _0)))) /\ ((P (NUMERAL (BIT1 (BIT1 _0)))) /\ (P (NUMERAL (BIT0 (BIT0 (BIT1 _0))))))))))


MBOUNDED_MCBALL

|- (!m. (!c. (!b. ((mbounded m) ((mcball m) (c , b))))))


ORTHOGONAL_SYM

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


LIM_NULL_COMPLEX_RMUL_BOUNDED

|- (!net. (!f. (!g. (!B. (((((--> f) (Cx (real_of_num (NUMERAL _0)))) net) /\ ((eventually (\a. (((f a) = (Cx (real_of_num (NUMERAL _0)))) \/ ((real_le (vector_norm (g a))) B)))) net)) ==> (((--> (\z. ((complex_mul (f z)) (g z)))) (Cx (real_of_num (NUMERAL _0)))) net))))))


CONTINUOUS_ON_SUBSET

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


Functional_equation.proj_y1

|- (!x2. (!x3. (!x4. (!x5. (!x6. (!x1. (((((((proj_y1 x1) x2) x3) x4) x5) x6) = (sqrt x1))))))))


inseg

|- (!m. (!l. (((inseg l) m) = (!x. (!y. ((l (x , y)) = ((m (x , y)) /\ ((fl l) y))))))))


CSQRT_CEXP_CLOG

|- (!z. ((~ (z = (Cx (real_of_num (NUMERAL _0))))) ==> ((csqrt z) = (cexp ((complex_div (clog z)) (Cx (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))))))


Lp_ineqs_def.rhazim2_fan

|- (!V. (!E. (!x. (((rhazim2_fan (V , E)) x) = ((rhazim_fan (V , E)) ((f_fan_pair_ext (V , E)) x))))))


CARD_EQ_CARD

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


Ssrnat.addn1

|- (!n. ((n + (NUMERAL (BIT1 _0))) = (SUC n)))


GDELTA_INTER

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


Miqmcsn.SCS_4M8_13_IS_SCS

|- (is_scs_v39 scs_4M8_13)


LT_NZ

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


CLOSURE_APPROACHABLE

|- (!x. (!s. ((x IN (closure s)) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?y. ((y IN s) /\ ((real_lt (distance (y , x))) e))))))))


IN_MCBALL

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


MATRIX_INV_0

|- ((matrix_inv (mat (NUMERAL _0))) = (mat (NUMERAL _0)))


POSITIVE_DEFINITE_2_DET

|- (!A. ((positive_definite A) = (((transp A) = A) /\ (((real_lt (real_of_num (NUMERAL _0))) ((A $ (NUMERAL (BIT1 _0))) $ (NUMERAL (BIT1 _0)))) /\ (((real_lt (real_of_num (NUMERAL _0))) ((A $ (NUMERAL (BIT0 (BIT1 _0)))) $ (NUMERAL (BIT0 (BIT1 _0))))) /\ ((real_lt (real_of_num (NUMERAL _0))) (det A)))))))


COMPLEX_NORM_CX

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


ABSOLUTELY_INTEGRABLE_CHANGE_OF_VARIABLES_LINEAR

|- (!f. (!g. (!s. ((linear g) ==> (((absolutely_integrable_on (\x. ((% (real_abs (det (matrix g)))) (f (g x))))) s) = ((absolutely_integrable_on f) ((IMAGE g) s)))))))


LIFT_SUB

|- (!x. (!y. ((lift ((real_sub x) y)) = ((vector_sub (lift x)) (lift y)))))


SHIFTPATH_TRANSLATION

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


NORM_GE_SQUARE

|- (!x. (((real_ge (vector_norm x)) a) = (((real_le a) (real_of_num (NUMERAL _0))) \/ ((real_ge ((dot x) x)) ((real_pow a) (NUMERAL (BIT0 (BIT1 _0))))))))


INDEPENDENT_IMP_AFFINE_DEPENDENT_0

|- (!s. ((independent s) ==> (~ (affine_dependent ((vec (NUMERAL _0)) INSERT s)))))


COMPLEX_MUL_CCOS_CSIN

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


RECTIFIABLE_PATH_IMP_PATH

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


REAL_SGN_EQ

|- ((!x. (((real_sgn x) = (real_of_num (NUMERAL _0))) = (x = (real_of_num (NUMERAL _0))))) /\ ((!x. (((real_sgn x) = (real_of_num (NUMERAL (BIT1 _0)))) = ((real_gt x) (real_of_num (NUMERAL _0))))) /\ (!x. (((real_sgn x) = (real_neg (real_of_num (NUMERAL (BIT1 _0))))) = ((real_lt x) (real_of_num (NUMERAL _0)))))))


ABSOLUTELY_SETCONTINUOUS_ON_DIVISION

|- (!f. (!a. (!b. (!d. ((((operative vector_add) f) /\ ((division_of d) (closed_interval ((CONS (a , b)) NIL)))) ==> ((!k. ((k IN d) ==> ((absolutely_setcontinuous_on f) k))) = ((absolutely_setcontinuous_on f) (closed_interval ((CONS (a , b)) NIL)))))))))


Seq.in_nil

|- (!x. (((MEM x) NIL) = F))


CBALL_MIN_INTER

|- (!x. (!d. (!e. ((cball (x , ((real_min d) e))) = ((cball (x , d)) INTER (cball (x , e)))))))


NEGLIGIBLE_TRANSLATION_EQ

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


SQRT_WORKS

|- (!x. (((real_le (real_of_num (NUMERAL _0))) x) ==> (((real_le (real_of_num (NUMERAL _0))) (sqrt x)) /\ (((real_pow (sqrt x)) (NUMERAL (BIT0 (BIT1 _0)))) = x))))


EXISTS_LIFT

|- ((?x. (P x)) = (?x. (P (lift x))))


CONTINUOUS_AT_CTAN

|- (!z. ((~ ((ccos z) = (Cx (real_of_num (NUMERAL _0))))) ==> ((continuous ctan) (at z))))


Ineq.dart_std4

|- (!y1. (!y2. (!y3. (!y4. (!y5. (!y6. (!y7. (!y8. (!y9. ((((((((((dart_std4 y1) y2) y3) y4) y5) y6) y7) y8) y9) = ((CONS (((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0)))))) , (y1 , ((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)))))) , (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 , ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 (BIT1 _0))))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))) ((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)))) ((CONS (((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0)))))) , (y7 , ((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)))))) , (y8 , ((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)))))) , (y9 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) NIL)))))))))))))))))))


REAL_NOT_LE

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


INT_LE_NEG

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


Constants_approx.INTERVAL_INV

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


RE_MUL_CX

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


Ssrnat.mul2n

|- (!m. (((NUMERAL (BIT0 (BIT1 _0))) * m) = (double m)))


Real_ext.REAL_MK_NZ_EXP

|- (!x. (~ ((exp x) = (real_of_num (NUMERAL _0)))))


HOMOTOPIC_WITH_COMPOSE_CONTINUOUS_RIGHT

|- (!p. (!f. (!g. (!h. (!W. (!X. (!Y. ((((((homotopic_with (\f. (p (f o h)))) (X , Y)) f) g) /\ (((continuous_on h) W) /\ (((IMAGE h) W) SUBSET X))) ==> ((((homotopic_with p) (W , Y)) (f o h)) (g o h))))))))))


FRECHET_DERIVATIVE_WORKS

|- (!f. (!net. (((differentiable f) net) = (((has_derivative f) ((frechet_derivative f) net)) net))))


SUM_LMUL

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


Sphere.dih2_x

|- (!x2. (!x1. (!x3. (!x5. (!x4. (!x6. (((((((dih2_x x1) x2) x3) x4) x5) x6) = ((((((dih_x x2) x1) x3) x5) x4) x6))))))))


CX_EXP

|- (!x. ((Cx (exp x)) = (cexp (Cx x))))


Hypermap.node_map_and_darts

|- (!H. ((FINITE (dart H)) /\ ((permutes (node_map H)) (dart H))))


BASIS_COMPONENT

|- (!k. (!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV))) ==> (((basis k) $ i) = (((COND (i = k)) (real_of_num (NUMERAL (BIT1 _0)))) (real_of_num (NUMERAL _0)))))))


COMPLEX_NEG_0

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


REAL_ABS_POW

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


VSUM_SUB

|- (!f. (!g. (!s. ((FINITE s) ==> (((vsum s) (\x. ((vector_sub (f x)) (g x)))) = ((vector_sub ((vsum s) f)) ((vsum s) g)))))))


CLOSED_IMP_LOCALLY_COMPACT

|- (!s. ((closed s) ==> ((locally compact) s)))


LIM_UNION

|- (!f. (!x. (!l. (!s. (!t. (((((--> f) l) ((within (at x)) s)) /\ (((--> f) l) ((within (at x)) t))) ==> (((--> f) l) ((within (at x)) (s UNION t)))))))))


TRANSP_TRANSP

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


MEASURABLE_TRANSLATION_EQ

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


Seq2.list_sum_EL

|- (!s. (!f. ((~ (s = NIL)) ==> (((list_sum s) f) = ((sum ((NUMERAL _0) .. ((sizel s) - (NUMERAL (BIT1 _0))))) (\i. (f ((EL i) s))))))))


Hypermap_and_fan.E_N_F_IN_DART1_OF_FAN

|- (!V. (!E. (!x. (((FAN ((vec (NUMERAL _0)) , (V , E))) /\ (x IN (dart1_of_fan (V , E)))) ==> ((((e_fan_pair (V , E)) x) IN (dart1_of_fan (V , E))) /\ ((((n_fan_pair (V , E)) x) IN (dart1_of_fan (V , E))) /\ (((f_fan_pair (V , E)) x) IN (dart1_of_fan (V , E)))))))))


DIST_SYM

|- (!x. (!y. ((distance (x , y)) = (distance (y , x)))))


REAL_MUL_POS_LE

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


Seq.eqS0

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


FORALL_IN_UNION

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


REAL_INTEGRAL_EQ

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


CLOSED_IN_COMPACT

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


matrix_add

|- (!A. (!B. (((matrix_add A) B) = (lambdai. (lambdaj. ((real_add ((A $ i) $ j)) ((B $ i) $ j)))))))


HAS_SIZE_1_EXISTS

|- (!s. ((s HAS_SIZE (NUMERAL (BIT1 _0))) = (?!x. (x IN s))))


Collect_geom.PER_SET2

|- ((a INSERT (b INSERT EMPTY)) = (b INSERT (a INSERT EMPTY)))


Float.INT_POW_NEG1

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


Local_lemmas.CONDS_IN_HAFL_LINE

|- ((((real_le (real_of_num (NUMERAL _0))) t) /\ (((vector_sub a) x) = ((% t) ((vector_sub b) x)))) ==> (a IN ((aff_ge (x INSERT EMPTY)) (b INSERT EMPTY))))


SUM_IMAGE

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


LE_1

|- ((!n. ((~ (n = (NUMERAL _0))) ==> ((NUMERAL _0) < n))) /\ ((!n. ((~ (n = (NUMERAL _0))) ==> ((NUMERAL (BIT1 _0)) <= n))) /\ ((!n. (((NUMERAL _0) < n) ==> (~ (n = (NUMERAL _0))))) /\ ((!n. (((NUMERAL _0) < n) ==> ((NUMERAL (BIT1 _0)) <= n))) /\ ((!n. (((NUMERAL (BIT1 _0)) <= n) ==> ((NUMERAL _0) < n))) /\ (!n. (((NUMERAL (BIT1 _0)) <= n) ==> (~ (n = (NUMERAL _0))))))))))


COMPACT_CONVEX_HULL

|- (!s. ((compact s) ==> (compact ((hull convex) s))))


Lp_ineqs_proofs.y1_lo

|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!d. ((d IN (dart_of_fan (V , E))) ==> ((real_le (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (y1_fan d))))))))))


LE_SUC

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


frontier

|- (!s. ((frontier s) = ((closure s) DIFF (interior s))))


PATH_CONNECTED_CONVEX_DIFF_CARD_LT

|- (!u. (!s. (((convex u) /\ ((~ (collinear u)) /\ (s <_c UNIV))) ==> (path_connected (u DIFF s)))))


FINITE_SUBSET

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


o_ASSOC

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


CONVEX_LINEAR_IMAGE

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


Fan.POWER_2

|- (!f. (((POWER f) (NUMERAL (BIT0 (BIT1 _0)))) = (f o f)))


Topology.FINITE_ORBITS_SIGMA_FAN

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


LE_REFL

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


COMPLEX_DIFFERENTIABLE_AT_CLOG

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


MEASURE_NEGLIGIBLE_UNIONS_IMAGE

|- (!f. (!s. (((FINITE s) /\ ((!x. ((x IN s) ==> (measurable (f x)))) /\ (!x. (!y. (((x IN s) /\ ((y IN s) /\ (~ (x = y)))) ==> (negligible ((f x) INTER (f y)))))))) ==> ((measure (UNIONS ((IMAGE f) s))) = ((sum s) (\x. (measure (f x))))))))


Seq.gtS0

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


AR_SING

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


Rnsyjxm.f_surr_localization_convex_local

|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> (!d. ((d IN (dart_of_fan (V , E))) ==> ((LET (\f. (LET_END (convex_local_fan (((v_prime V) f) , (((e_prime E) f) , f)))))) ((face (hypermap_of_fan (V , E))) d))))))))


IM_CLOG_POS_LT

|- (!z. ((~ (z = (Cx (real_of_num (NUMERAL _0))))) ==> ((((real_lt (real_of_num (NUMERAL _0))) (Im (clog z))) /\ ((real_lt (Im (clog z))) pi)) = ((real_lt (real_of_num (NUMERAL _0))) (Im z)))))


CONNECTED_PCROSS

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


HOMEOMORPHISM_SYM

|- (!f. (!g. (!s. (!t. (((homeomorphism (s , t)) (f , g)) = ((homeomorphism (t , s)) (g , f)))))))


RE_CX

|- (!x. ((Re (Cx x)) = x))


CONVEX_HALFSPACE_GT

|- (!a. (!b. (convex (GSPEC (\GEN%PVAR%1441. (?x. (((SETSPEC GEN%PVAR%1441) ((real_gt ((dot a) x)) b)) x)))))))


COMPLEX_DERIVATIVE_UNIQUE_AT

|- (!f. (!z. (!f'. (!f''. (((((has_complex_derivative f) f') (at z)) /\ (((has_complex_derivative f) f'') (at z))) ==> (f' = f''))))))


VECTOR_POLYNOMIAL_FUNCTION_ADD

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


Ssrnat.uphalf0

|- ((uphalf (NUMERAL _0)) = (NUMERAL _0))


POSITIVE_SEMIDEFINITE_2_DET

|- (!A. ((positive_semidefinite A) = (((transp A) = A) /\ (((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_of_num (NUMERAL _0))) (det A)))))))


HOMEOMORPHISM_OF_SUBSETS

|- (!f. (!g. (!s. (!t. (!s'. (!t'. ((((homeomorphism (s , t)) (f , g)) /\ ((s' SUBSET s) /\ ((t' SUBSET t) /\ (((IMAGE f) s') = t')))) ==> ((homeomorphism (s' , t')) (f , g)))))))))


COS_DOUBLE

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


CONIC_EMPTY

|- (conic EMPTY)


LINEPATH_TRANSLATION

|- (!a. (!b. (!c. ((linepath (((vector_add a) b) , ((vector_add a) c))) = ((\x. ((vector_add a) x)) o (linepath (b , c)))))))


CONTRACTIBLE_IMP_SIMPLY_CONNECTED

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


REAL_NEG_SUB

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


INTEGRABLE_INCREASING_PRODUCT

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


Ssrnat.ltnNge

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


CLOSED_CLOSED_PREIMAGES_IMP_CONTINUOUS_ON

|- (!f. (!s. (!t. (((((IMAGE f) s) SUBSET t) /\ ((compact t) /\ ((!y. ((closed_in ((subtopology euclidean) s)) (GSPEC (\GEN%PVAR%1941. (?x. (((SETSPEC GEN%PVAR%1941) ((x IN s) /\ ((f x) = y))) x)))))) /\ (!c. (((closed_in ((subtopology euclidean) s)) c) ==> ((closed_in ((subtopology euclidean) t)) ((IMAGE f) c))))))) ==> ((continuous_on f) s)))))


FINITE_INSERT

|- (!s. (!x. ((FINITE (x INSERT s)) = (FINITE s))))


CONTINUOUS_COMPLEX_MUL

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


CARD_MUL_C

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


ANALYTIC_ON_SUBSET

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


POLYTOPE_TRANSLATION_EQ

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


MATRIX_VECTOR_MUL_RZERO

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


Ardbzye.K_SCS_3T3

|- ((scs_k_v39 scs_3T3) = (NUMERAL (BIT1 (BIT1 _0))))


Fan.subset_d_fan

|- (!x. (!V. (!E. (((d1_fan (x , (V , E))) SUBSET (d_fan (x , (V , E)))) /\ ((d20_fan (x , (V , E))) SUBSET (d_fan (x , (V , E))))))))


UNIFORMLY_CONTINUOUS_ON_COMPOSE

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


HAS_MEASURE_UNION_NEGLIGIBLE_EQ

|- (!s. (!t. (!m. ((negligible t) ==> (((has_measure (s UNION t)) m) = ((has_measure s) m))))))


Hypermap_iso.res_inv_left

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


ARG_CNJ

|- (!z. ((Arg (cnj z)) = (((COND ((real z) /\ ((real_le (real_of_num (NUMERAL _0))) (Re z)))) (Arg z)) ((real_sub ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) pi)) (Arg z)))))


INTEGRAL_SPIKE_SET

|- (!f. (!g. (!s. (!t. ((negligible ((s DIFF t) UNION (t DIFF s))) ==> (((integral s) f) = ((integral t) f)))))))


HAS_MEASURE_DISJOINT_UNIONS

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


FORALL_LIFT

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


CONTINUOUS_MUL

|- (!net. (!f. (!c. ((((continuous (lift o c)) net) /\ ((continuous f) net)) ==> ((continuous (\x. ((% (c x)) (f x)))) net)))))


Real_ext.REAL_SV_EXP_0

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


NSUM_LMUL

|- (!f. (!c. (!s. (((nsum s) (\x. (c * (f x)))) = (c * ((nsum s) f))))))


HAS_BOUNDED_VARIATION_ON_VMUL

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


IM_NEG

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


CARD_LE_SUBSET

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


SPHERE_LINEAR_IMAGE

|- (!f. (!x. (!r. (((linear f) /\ ((!y. (?x. ((f x) = y))) /\ (!x. ((vector_norm (f x)) = (vector_norm x))))) ==> ((sphere ((f x) , r)) = ((IMAGE f) (sphere (x , r))))))))


RANK_TRANSP

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


HAS_REAL_DERIVATIVE_CONST

|- (!c. (!net. (((has_real_derivative (\x. c)) (real_of_num (NUMERAL _0))) net)))


Fan.INVERSE1_SIGMA_FAN

|- (!x. (!V. (!E. (!v. ((FAN (x , (V , E))) ==> ((!w. (((v INSERT (w INSERT EMPTY)) IN E) ==> ((v INSERT ((((((inverse1_sigma_fan x) V) E) v) w) INSERT EMPTY)) IN E))) /\ ((!w. (((v INSERT (w INSERT EMPTY)) IN E) ==> ((((((sigma_fan x) V) E) v) (((((inverse1_sigma_fan x) V) E) v) w)) = w))) /\ (!w. (((v INSERT (w INSERT EMPTY)) IN E) ==> ((((((inverse1_sigma_fan x) V) E) v) (((((sigma_fan x) V) E) v) w)) = w))))))))))


SC_INC

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


CPRODUCT_CLAUSES

|- ((!f. (((cproduct EMPTY) f) = (Cx (real_of_num (NUMERAL (BIT1 _0)))))) /\ (!x. (!f. (!s. ((FINITE s) ==> (((cproduct (x INSERT s)) f) = (((COND (x IN s)) ((cproduct s) f)) ((complex_mul (f x)) ((cproduct s) f)))))))))


DOT_LNEG

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


ARC_IMP_SIMPLE_PATH

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


Lp_ineqs_def.ye_fan

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


HAS_REAL_MEASURE_REAL_NEGLIGIBLE_UNION

|- (!s1. (!s2. (!m1. (!m2. ((((has_real_measure s1) m1) /\ (((has_real_measure s2) m2) /\ (real_negligible (s1 INTER s2)))) ==> ((has_real_measure (s1 UNION s2)) ((real_add m1) m2)))))))


Pack1.saturated

|- (!S. ((saturated S) = (!x. (?y. ((y IN S) /\ ((real_lt (distance (x , y))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))))))


MATRIX_TRANSP_MULTIPLE_INV_LEFT

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


SELF_ADJOINT_ORTHOGONAL_EIGENVECTORS

|- (!f. (!v. (!w. (!a. (!b. (((linear f) /\ (((adjoint f) = f) /\ (((f v) = ((% a) v)) /\ (((f w) = ((% b) w)) /\ (~ (a = b)))))) ==> ((orthogonal v) w)))))))


Seq.lastI

|- (!x. (!s. (((CONS x) s) = ((rcons ((belast x) s)) ((last x) s)))))


REAL_CONTINUOUS_WITHIN_ACS_STRONG

|- (!x. ((real_continuous acs) ((within (atreal x)) (GSPEC (\GEN%PVAR%4911. (?t. (((SETSPEC GEN%PVAR%4911) ((real_le (real_abs t)) (real_of_num (NUMERAL (BIT1 _0))))) t)))))))


STEINHAUS

|- (!s. (((measurable s) /\ ((real_lt (real_of_num (NUMERAL _0))) (measure s))) ==> (?d. (((real_lt (real_of_num (NUMERAL _0))) d) /\ ((ball ((vec (NUMERAL _0)) , d)) SUBSET (GSPEC (\GEN%PVAR%3461. (?x. (?y. (((SETSPEC GEN%PVAR%3461) ((x IN s) /\ (y IN s))) ((vector_sub x) y)))))))))))


INTERIOR_MAXIMAL

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


RELATIVE_INTERIOR_INJECTIVE_LINEAR_IMAGE

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


ANALYTIC_ON_MUL

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


ORTHOGONAL_CLAUSES

|- ((!a. ((orthogonal a) (vec (NUMERAL _0)))) /\ ((!a. (!x. (!c. (((orthogonal a) x) ==> ((orthogonal a) ((% c) x)))))) /\ ((!a. (!x. (((orthogonal a) x) ==> ((orthogonal a) (vector_neg x))))) /\ ((!a. (!x. (!y. ((((orthogonal a) x) /\ ((orthogonal a) y)) ==> ((orthogonal a) ((vector_add x) y)))))) /\ ((!a. (!x. (!y. ((((orthogonal a) x) /\ ((orthogonal a) y)) ==> ((orthogonal a) ((vector_sub x) y)))))) /\ ((!a. ((orthogonal (vec (NUMERAL _0))) a)) /\ ((!a. (!x. (!c. (((orthogonal x) a) ==> ((orthogonal ((% c) x)) a))))) /\ ((!a. (!x. (((orthogonal x) a) ==> ((orthogonal (vector_neg x)) a)))) /\ ((!a. (!x. (!y. ((((orthogonal x) a) /\ ((orthogonal y) a)) ==> ((orthogonal ((vector_add x) y)) a))))) /\ (!a. (!x. (!y. ((((orthogonal x) a) /\ ((orthogonal y) a)) ==> ((orthogonal ((vector_sub x) y)) a))))))))))))))


SQRT_WORKS_GEN

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


REAL_EQ_LCANCEL_IMP

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


Nkezbfc_local.slicef

|- (!E. (!FF. (!w. (!v. (((((slicef E) FF) v) w) = ((GSPEC (\GEN%PVAR%7012. (?f. (((SETSPEC GEN%PVAR%7012) (?u. ((u IN (((((slicev E) FF) v) w) DELETE w)) /\ (f = (u , ((rho_node1 FF) u)))))) f)))) UNION ((w , v) INSERT EMPTY)))))))


SUP

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


REALLIM_LMUL

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


aff_le_def

|- (aff_le = (affsign sgn_le))


LIMPT_OF_UNIV

|- (!x. ((limit_point_of x) UNIV))


MATRIX_MUL_ASSOC

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


CCOS_EQ_MINUS1

|- (!z. (((ccos z) = (vector_neg (Cx (real_of_num (NUMERAL (BIT1 _0)))))) = (?n. ((integer n) /\ (z = (Cx ((real_mul ((real_add ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) n)) (real_of_num (NUMERAL (BIT1 _0))))) pi)))))))


RAT_LEMMA4

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


CONIC_IMP_PATH_CONNECTED

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


Ssrbool.if_same

|- (!b. (!vT. ((((COND b) vT) vT) = vT)))


Conforming.SIGMA_FAN_OF_FANADD_AT_POINT6

|- (!x. (!V. (!E. (!v. (!u. (!w. (!w1. (((FAN (x , (V , E))) /\ ((FAN (x , (V , E1))) /\ ((fan80 (x , (V , E))) /\ (((v INSERT (u INSERT EMPTY)) IN E) /\ (((u INSERT (w INSERT EMPTY)) IN E) /\ ((~ ((w INSERT (v INSERT EMPTY)) IN E)) /\ ((w1 = (((((inverse1_sigma_fan x) V) E) w) u)) /\ (((w INSERT (w1 INSERT EMPTY)) IN E) /\ (((((((sigma_fan x) V) E) u) w) = v) /\ ((!v. ((v IN V) ==> ((CARD (((set_of_edge v) V) E)) > (NUMERAL (BIT1 _0))))) /\ ((E UNION ((v INSERT (w INSERT EMPTY)) INSERT EMPTY)) = E1))))))))))) ==> ((((((sigma_fan x) V) E1) w) w1) = v)))))))))


ORTHOGONAL_SUBSPACE_DECOMP_EXISTS

|- (!s. (!x. (?y. (?z. ((y IN (span s)) /\ ((!w. ((w IN (span s)) ==> ((orthogonal z) w))) /\ (x = ((vector_add y) z))))))))


DROP_CMUL

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


AFFSIGN_INJECTIVE_LINEAR_IMAGE

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


DOT_LZERO

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


SUBSET_INTERIOR

|- (!s. (!t. ((s SUBSET t) ==> ((interior s) SUBSET (interior t)))))


LIM_NULL_NORM

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


Hypermap_iso.hypermap_of_fan_rep_alt

|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (((dart (hypermap1_of_fanx ((vec (NUMERAL _0)) , (V , E)))) = (d_fan ((vec (NUMERAL _0)) , (V , E)))) /\ (((edge_map (hypermap1_of_fanx ((vec (NUMERAL _0)) , (V , E)))) = ((res (((e_fan (vec (NUMERAL _0))) V) E)) (d1_fan ((vec (NUMERAL _0)) , (V , E))))) /\ (((node_map (hypermap1_of_fanx ((vec (NUMERAL _0)) , (V , E)))) = ((res (((n_fan (vec (NUMERAL _0))) V) E)) (d1_fan ((vec (NUMERAL _0)) , (V , E))))) /\ ((face_map (hypermap1_of_fanx ((vec (NUMERAL _0)) , (V , E)))) = ((res (((f1_fan (vec (NUMERAL _0))) V) E)) (d1_fan ((vec (NUMERAL _0)) , (V , E)))))))))))


CLOPEN

|- (!s. (((closed s) /\ (open s)) = ((s = EMPTY) \/ (s = UNIV))))


ORTHOGONAL_TRANSFORMATION_INTO_SUBSPACE

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


HAS_COMPLEX_DERIVATIVE_TRANSFORM_WITHIN_OPEN

|- (!f. (!g. (!f'. (!s. (!z. (((open s) /\ ((z IN s) /\ ((!w. ((w IN s) ==> ((f w) = (g w)))) /\ (((has_complex_derivative f) f') (at z))))) ==> (((has_complex_derivative g) f') (at z))))))))


COMPLEX_MUL_CSIN_CSIN

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


FORALL_IN_IMAGE

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


column

|- (!A. (!j. (((column j) A) = (lambdai. ((A $ i) $ j)))))


VECTOR_ADD_LID

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


INTER_UNIV

|- ((!s. ((UNIV INTER s) = s)) /\ (!s. ((s INTER UNIV) = s)))


cpow

|- (!z. (!w. (((cpow w) z) = (((COND (w = (Cx (real_of_num (NUMERAL _0))))) (Cx (real_of_num (NUMERAL _0)))) (cexp ((complex_mul z) (clog w)))))))


HOMOTOPIC_ON_EMPTY

|- (!t. (!f. (!g. ((((homotopic_with (\x. T)) (EMPTY , t)) f) g))))


INCREASING_RIGHT_LIMIT_1

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


ARITH_EXP

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


ENR_IMP_ANR

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


Dih2k_hypermap.vecmat

|- (!f. (!j. (((vecmat j) f) = (lambdai. (f $ ((j * (dimindex UNIV)) + i))))))


Lp_ineqs_proofs.add_diag

|- (!V. (!v. (!w. (!E. (((add_diag (V , E)) (v , w)) = (((COND ((v , w) IN (dart1_of_fan (V , E)))) ((w INSERT ((((((sigma_fan (vec (NUMERAL _0))) V) E) v) w) INSERT EMPTY)) INSERT E)) E))))))


Seq.perm_eq_mem

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


NEUTRAL_ADD

|- ((neutral +) = (NUMERAL _0))


Hypermap.number_of_nodes

|- (!H. ((number_of_nodes H) = (CARD (node_set H))))


Pack2.BOUNDED_VORONOI_CLOSED

|- (!s. (!v. ((saturated s) ==> (bounded ((voronoi_closed s) v)))))


Hypermap.FINITE_HYPERMAP_ORBITS

|- (!H. ((FINITE (edge_set H)) /\ ((FINITE (node_set H)) /\ (FINITE (face_set H)))))


Hypermap.is_node_nondegenerate

|- (!H. ((is_node_nondegenerate H) = (!x. ((x IN (dart H)) ==> (~ (((node_map H) x) = x))))))


FRECHET_DERIVATIVE_UNIQUE_WITHIN_CLOSED_INTERVAL

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


OPEN_COMPONENTS

|- (!u. (!s. (((open u) /\ (s IN (components u))) ==> (open s))))


QUOTIENT_MAP_FROM_COMPOSITION

|- (!f. (!g. (!s. (!t. (!u. ((((continuous_on f) s) /\ ((((IMAGE f) s) SUBSET t) /\ (((continuous_on g) t) /\ ((((IMAGE g) t) SUBSET u) /\ (!v. ((v SUBSET u) ==> (((open_in ((subtopology euclidean) s)) (GSPEC (\GEN%PVAR%667. (?x. (((SETSPEC GEN%PVAR%667) ((x IN s) /\ (((g o f) x) IN v))) x))))) = ((open_in ((subtopology euclidean) u)) v)))))))) ==> (!v. ((v SUBSET u) ==> (((open_in ((subtopology euclidean) t)) (GSPEC (\GEN%PVAR%668. (?x. (((SETSPEC GEN%PVAR%668) ((x IN t) /\ ((g x) IN v))) x))))) = ((open_in ((subtopology euclidean) u)) v))))))))))


OUTSIDE_TRANSLATION

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


OPEN_IMP_LOCALLY_COMPACT

|- (!s. ((open s) ==> ((locally compact) s)))


TC_CASES

|- (!R. (!a0. (!a1. ((((TC R) a0) a1) = (((R a0) a1) \/ (?y. ((((TC R) a0) y) /\ (((TC R) y) a1))))))))


LIM_ZERO_NEGINFINITY

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


NEGLIGIBLE_ON_INTERVALS

|- (!s. ((negligible s) = (!a. (!b. (negligible (s INTER (closed_interval ((CONS (a , b)) NIL))))))))


DIFFERENTIABLE_NEG

|- (!f. (!net. (((differentiable f) net) ==> ((differentiable (\z. (vector_neg (f z)))) net))))


INTERIOR_SUBSET_RELATIVE_INTERIOR

|- (!s. ((interior s) SUBSET (relative_interior s)))


REAL_CONVEX_ON

|- (!f. (!s. (((real_convex_on f) s) = ((convex_on (f o drop)) ((IMAGE lift) s)))))


Fan.pr2

|- (pr2 = (GABS (\f. (!x. (!v. (!w. (!w1. ((GEQ (f (x , (v , (w , w1))))) v))))))))


SING_SUBSET

|- (!s. (!x. (((x INSERT EMPTY) SUBSET s) = (x IN s))))


REAL_MIN_MIN

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


Nonlin_def.sol_euler_x_div_sqrtdelta

|- (!x4. (!x5. (!x3. (!x1. (!x2. (!x6. (((((((sol_euler_x_div_sqrtdelta x1) x2) x3) x4) x5) x6) = ((LET (\a. (LET_END ((real_div (matan ((real_div ((((((delta_x x1) x2) x3) x4) x5) x6)) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) ((real_pow a) (NUMERAL (BIT0 (BIT1 _0)))))))) a)))) ((real_add (sqrt ((real_mul x1) ((real_mul x2) x3)))) ((real_add ((real_mul (sqrt x1)) ((real_div ((real_add x2) ((real_sub x3) x4))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) ((real_add ((real_mul (sqrt x2)) ((real_div ((real_add x1) ((real_sub x3) x5))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) ((real_mul (sqrt x3)) ((real_div ((real_add x1) ((real_sub x2) x6))) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))))))))))))))


INJECTIVE_ON_PREIMAGE

|- (!f. (!s. (!u. ((!t. (!t'. (((t SUBSET u) /\ ((t' SUBSET u) /\ ((GSPEC (\GEN%PVAR%142. (?x. (((SETSPEC GEN%PVAR%142) ((x IN s) /\ ((f x) IN t))) x)))) = (GSPEC (\GEN%PVAR%143. (?x. (((SETSPEC GEN%PVAR%143) ((x IN s) /\ ((f x) IN t'))) x))))))) ==> (t = t')))) = (u SUBSET ((IMAGE f) s))))))


Functional_equation.proj_y3

|- (!x1. (!x2. (!x4. (!x5. (!x6. (!x3. (((((((proj_y3 x1) x2) x3) x4) x5) x6) = (sqrt x3))))))))


MEASURABLE_ON_TRANSLATION

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


RELATIVE_INTERIOR_EQ_EMPTY

|- (!s. ((convex s) ==> (((relative_interior s) = EMPTY) = (s = EMPTY))))


CLOSED_HALFSPACE_IM_GE

|- (!b. (closed (GSPEC (\GEN%PVAR%4756. (?z. (((SETSPEC GEN%PVAR%4756) ((real_ge (Im z)) b)) z))))))


Add_triangle.f_fan_pair3

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


ABSOLUTELY_INTEGRABLE_SPIKE

|- (!f. (!g. (!s. (!t. (((negligible s) /\ (!x. ((x IN (t DIFF s)) ==> ((g x) = (f x))))) ==> (((absolutely_integrable_on f) t) ==> ((absolutely_integrable_on g) t)))))))


PERMUTES_EMPTY

|- (!p. (((permutes p) EMPTY) = (p = I)))


ARITH_EVEN

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


nsum

|- (nsum = (iterate +))


List_hypermap.mem_list_of_darts

|- (!d. (!L. (((MEM d) (list_of_darts L)) = (?l. (((MEM l) L) /\ ((MEM d) (list_pairs l)))))))


Planarity.PROPERTIES_TRIANGLE_FAN

|- (!x. (!V. (!E. (!v. (!u. (!w. (((FAN (x , (V , E))) /\ (((v INSERT (u INSERT EMPTY)) IN E) /\ (((u INSERT (w INSERT EMPTY)) IN E) /\ (((w INSERT (v INSERT EMPTY)) IN E) /\ (((((((sigma_fan x) V) E) u) w) = v) /\ ((!v. ((v IN V) ==> ((CARD (((set_of_edge v) V) E)) > (NUMERAL (BIT1 _0))))) /\ (fan80 (x , (V , E))))))))) ==> (((((((sigma_fan x) V) E) v) u) = w) /\ ((((((sigma_fan x) V) E) w) v) = u)))))))))


ARC_TRANSLATION_EQ

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


Ssrnat.sqrn_add

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


STARLIKE_IMP_PATH_CONNECTED

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


Hypermap.COM_POWER

|- (!n. (!f. (((POWER f) (SUC n)) = (f o ((POWER f) n)))))


REAL_SUB_REFL

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


Seq.size_drop

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


IMAGE_ID

|- (!s. (((IMAGE (\x. x)) s) = s))


REAL_ROOT_POW

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


HOMOTOPIC_WITH_COMPOSE

|- (!P. (!Q. (!R. (!f. (!f'. (!g. (!g'. (!s. (!t. (!u. (((!f. (!g. ((((continuous_on f) s) /\ ((((IMAGE f) s) SUBSET t) /\ ((P f) /\ (((continuous_on g) t) /\ ((((IMAGE g) t) SUBSET u) /\ (Q g)))))) ==> (R (g o f))))) /\ (((((homotopic_with P) (s , t)) f) f') /\ ((((homotopic_with Q) (t , u)) g) g'))) ==> ((((homotopic_with R) (s , u)) (g o f)) (g' o f')))))))))))))


COMPLEX_ADD_SYM

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


ETA_AX

|- (!t. ((\x. (t x)) = t))


HOMEOMORPHIC_COMPACTNESS

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


ANR_OPEN_IN

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


HOMOTOPY_EQUIVALENT_EMPTY

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


REAL_SECOND_MEAN_VALUE_THEOREM_FULL

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


HAS_REAL_DERIVATIVE_WITHIN_SUBSET

|- (!f. (!s. (!t. (!x. (((((has_real_derivative f) f') ((within (atreal x)) s)) /\ (t SUBSET s)) ==> (((has_real_derivative f) f') ((within (atreal x)) t)))))))


RPOW_ZERO

|- (!y. (((rpow (real_of_num (NUMERAL _0))) y) = (((COND (y = (real_of_num (NUMERAL _0)))) (real_of_num (NUMERAL (BIT1 _0)))) (real_of_num (NUMERAL _0)))))


Lp_main_estimate.v_prime_subset

|- (!V. (!X. (!f. ((V SUBSET X) ==> (((v_prime V) f) SUBSET X)))))


Hypermap.lemma_in_subset

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


Hypermap.iterate_map_valuation

|- (!p. (!n. (!x. ((p (((POWER p) n) x)) = (((POWER p) (SUC n)) x)))))


AR_FROM_UNION_AND_INTER

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


Hexagons.LE_sqrt8_2h0

|- ((real_le ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((DECIMAL (NUMERAL (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))))) sqrt8)


PROPER_LOCAL_HOMEOMORPHISM_IMP_COVERING_MAP

|- (!p. (!c. (!s. (((((IMAGE p) c) = s) /\ ((!k. (((k SUBSET s) /\ (compact k)) ==> (compact (GSPEC (\GEN%PVAR%2100. (?x. (((SETSPEC GEN%PVAR%2100) ((x IN c) /\ ((p x) IN k))) x))))))) /\ (!x. ((x IN c) ==> (?t. (?u. (?q. ((x IN t) /\ (((open_in ((subtopology euclidean) c)) t) /\ (((open_in ((subtopology euclidean) s)) u) /\ ((homeomorphism (t , u)) (p , q)))))))))))) ==> ((covering_space (c , p)) s)))))


ARITH_EQ

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


SUBTOPOLOGY_UNIV

|- (!top. (((subtopology top) UNIV) = top))


ANALYTIC_ON_ADD

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


Nonlin_def.edge2_126_x

|- (!d. (!x4. (!x5. ((((edge2_126_x d) x4) x5) = (((((((compose6 edge2_flatD_x1) (constant6 d)) proj_x1) proj_x2) proj_x6) (constant6 x4)) (constant6 x5))))))


GE

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


LOCALLY_NEGLIGIBLE_ALT

|- (!s. ((negligible s) = (!x. ((x IN s) ==> (?u. (((open_in ((subtopology euclidean) s)) u) /\ ((x IN u) /\ (negligible u))))))))


LINEAR_INJECTIVE_IMP_SURJECTIVE

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


CX_TAN

|- (!x. ((Cx (tan x)) = (ctan (Cx x))))


FORALL_IN_INSERT

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


ABSOLUTELY_INTEGRABLE_BOUNDED_MEASURABLE_PRODUCT

|- (!op. (!f. (!g. (!s. (((bilinear op) /\ (((measurable_on f) s) /\ ((bounded ((IMAGE f) s)) /\ ((absolutely_integrable_on g) s)))) ==> ((absolutely_integrable_on (\x. ((op (f x)) (g x)))) s))))))


CONTRACTIBLE_TRANSLATION

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


REAL_LT_NEG2

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


Sphere.quadratic_root_plus

|- (!b. (!c. (!a. ((quadratic_root_plus (a , (b , c))) = ((real_div ((real_add (real_neg b)) (sqrt ((real_sub ((real_pow b) (NUMERAL (BIT0 (BIT1 _0))))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) ((real_mul a) c)))))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) a))))))


NEUTRAL_REAL_ADD

|- ((neutral real_add) = (real_of_num (NUMERAL _0)))


COMPLEX_STONE_WEIERSTRASS_ALT

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


LOCALLY_PATH_CONNECTED_QUOTIENT_IMAGE

|- (!f. (!s. (((!t. ((t SUBSET ((IMAGE f) s)) ==> (((open_in ((subtopology euclidean) s)) (GSPEC (\GEN%PVAR%1904. (?x. (((SETSPEC GEN%PVAR%1904) ((x IN s) /\ ((f x) IN t))) x))))) = ((open_in ((subtopology euclidean) ((IMAGE f) s))) t)))) /\ ((locally path_connected) s)) ==> ((locally path_connected) ((IMAGE f) s)))))


RTC_INDUCT

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


Kepler_spec.sum_spec

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


PCROSS

|- (!s. (!t. ((s PCROSS t) = (GSPEC (\GEN%PVAR%239. (?x. (?y. (((SETSPEC GEN%PVAR%239) ((x IN s) /\ (y IN t))) ((pastecart x) y)))))))))


Fan.extension_sigma_fan

|- (!x. (!V. (!E. (!v. (!u. ((((((extension_sigma_fan x) V) E) v) u) = (((COND (~ (u IN (((set_of_edge v) V) E)))) u) (((((sigma_fan x) V) E) v) u))))))))


IS_INTERVAL_PATH_CONNECTED_1

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


HAS_REAL_INTEGRAL_REFLECT

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


UNIFORM_LIM_COMPLEX_INV

|- (!net. (!P. (!f. (!l. (!b. (((!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))) /\ (((real_lt (real_of_num (NUMERAL _0))) b) /\ ((eventually (\x. (!n. ((P n) ==> ((real_le b) (vector_norm (l n))))))) net))) ==> (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> ((eventually (\x. (!n. ((P n) ==> ((real_lt (vector_norm ((vector_sub (complex_inv ((f n) x))) (complex_inv (l n))))) e))))) net)))))))))


DIVMOD_UNIQ

|- (!m. (!n. (!q. (!r. (((m = ((q * n) + r)) /\ (r < n)) ==> (((m DIV n) = q) /\ ((m MOD n) = r)))))))


CONNECTED_BALL

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


BILINEAR_UNIFORMLY_CONTINUOUS_ON_COMPOSE

|- (!f. (!g. (!h. (!s. ((((uniformly_continuous_on f) s) /\ (((uniformly_continuous_on g) s) /\ ((bilinear h) /\ ((bounded ((IMAGE f) s)) /\ (bounded ((IMAGE g) s)))))) ==> ((uniformly_continuous_on (\x. ((h (f x)) (g x)))) s))))))


REAL_DIFFERENTIABLE_AT_SQRT

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


Misc_defs_and_lemmas.NUM_COUNTABLE

|- (COUNTABLE UNIV)