% TIMEFORMAT='%3R'; { time (exec 2>&1; /home/martin/bin/satallax -E /home/martin/.isabelle/contrib/e-2.5-1/x86_64-linux/eprover -p tstp -t 5 /home/martin/judgement-day/tptp-thf/tptp/Fundamental_Theorem_Algebra/prob_791__5375592_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:33:11.135

% Could-be-implicit typings (6)
thf(ty_n_t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    poly_complex : $tType).
thf(ty_n_t__Polynomial__Opoly_It__Real__Oreal_J, type,
    poly_real : $tType).
thf(ty_n_t__Polynomial__Opoly_It__Nat__Onat_J, type,
    poly_nat : $tType).
thf(ty_n_t__Complex__Ocomplex, type,
    complex : $tType).
thf(ty_n_t__Real__Oreal, type,
    real : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).

% Explicit typings (53)
thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Complex__Ocomplex, type,
    invers502456322omplex : complex > complex).
thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Real__Oreal, type,
    inverse_inverse_real : real > real).
thf(sy_c_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Oconstant_001t__Complex__Ocomplex_001t__Complex__Ocomplex, type,
    fundam1158420650omplex : (complex > complex) > $o).
thf(sy_c_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Opsize_001t__Complex__Ocomplex, type,
    fundam1709708056omplex : poly_complex > nat).
thf(sy_c_Groups_Oone__class_Oone_001t__Complex__Ocomplex, type,
    one_one_complex : complex).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat, type,
    one_one_nat : nat).
thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    one_one_poly_complex : poly_complex).
thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    one_one_poly_nat : poly_nat).
thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_It__Real__Oreal_J, type,
    one_one_poly_real : poly_real).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal, type,
    one_one_real : real).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Complex__Ocomplex, type,
    plus_plus_complex : complex > complex > complex).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat, type,
    plus_plus_nat : nat > nat > nat).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    plus_p1547158847omplex : poly_complex > poly_complex > poly_complex).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    plus_plus_poly_nat : poly_nat > poly_nat > poly_nat).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Polynomial__Opoly_It__Real__Oreal_J, type,
    plus_plus_poly_real : poly_real > poly_real > poly_real).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal, type,
    plus_plus_real : real > real > real).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Complex__Ocomplex, type,
    times_times_complex : complex > complex > complex).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat, type,
    times_times_nat : nat > nat > nat).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    times_1246143675omplex : poly_complex > poly_complex > poly_complex).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Polynomial__Opoly_It__Real__Oreal_J, type,
    times_775122617y_real : poly_real > poly_real > poly_real).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal, type,
    times_times_real : real > real > real).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Complex__Ocomplex, type,
    zero_zero_complex : complex).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat, type,
    zero_zero_nat : nat).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    zero_z1746442943omplex : poly_complex).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    zero_zero_poly_nat : poly_nat).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Real__Oreal_J, type,
    zero_zero_poly_real : poly_real).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal, type,
    zero_zero_real : real).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat, type,
    ord_less_nat : nat > nat > $o).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal, type,
    ord_less_real : real > real > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat, type,
    ord_less_eq_nat : nat > nat > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal, type,
    ord_less_eq_real : real > real > $o).
thf(sy_c_Polynomial_Opoly_001t__Complex__Ocomplex, type,
    poly_complex2 : poly_complex > complex > complex).
thf(sy_c_Polynomial_Opoly_001t__Nat__Onat, type,
    poly_nat2 : poly_nat > nat > nat).
thf(sy_c_Polynomial_Opoly_001t__Real__Oreal, type,
    poly_real2 : poly_real > real > real).
thf(sy_c_Power_Opower__class_Opower_001t__Complex__Ocomplex, type,
    power_power_complex : complex > nat > complex).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat, type,
    power_power_nat : nat > nat > nat).
thf(sy_c_Power_Opower__class_Opower_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    power_184595776omplex : poly_complex > nat > poly_complex).
thf(sy_c_Power_Opower__class_Opower_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    power_power_poly_nat : poly_nat > nat > poly_nat).
thf(sy_c_Power_Opower__class_Opower_001t__Polynomial__Opoly_It__Real__Oreal_J, type,
    power_2108872382y_real : poly_real > nat > poly_real).
thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal, type,
    power_power_real : real > nat > real).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Complex__Ocomplex, type,
    real_V638595069omplex : complex > real).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Real__Oreal, type,
    real_V646646907m_real : real > real).
thf(sy_c_Real__Vector__Spaces_Oof__real_001t__Complex__Ocomplex, type,
    real_V306493662omplex : real > complex).
thf(sy_c_Real__Vector__Spaces_Oof__real_001t__Real__Oreal, type,
    real_V1205483228l_real : real > real).
thf(sy_v_a____, type,
    a : complex).
thf(sy_v_c____, type,
    c : complex).
thf(sy_v_k____, type,
    k : nat).
thf(sy_v_m____, type,
    m : real).
thf(sy_v_p, type,
    p : poly_complex).
thf(sy_v_pa____, type,
    pa : poly_complex).
thf(sy_v_s____, type,
    s : poly_complex).
thf(sy_v_t____, type,
    t : real).
thf(sy_v_w____, type,
    w : complex).

% Relevant facts (242)
thf(fact_0_t_I2_J, axiom,
    ((ord_less_real @ t @ one_one_real))). % t(2)
thf(fact_1_nc, axiom,
    ((~ ((fundam1158420650omplex @ (poly_complex2 @ p)))))). % nc
thf(fact_2_m_I2_J, axiom,
    ((![Z : complex]: ((ord_less_eq_real @ (real_V638595069omplex @ Z) @ (real_V638595069omplex @ w)) => (ord_less_eq_real @ (real_V638595069omplex @ (poly_complex2 @ s @ Z)) @ m))))). % m(2)
thf(fact_3_kas_I2_J, axiom,
    ((~ ((k = zero_zero_nat))))). % kas(2)
thf(fact_4_t_I1_J, axiom,
    ((ord_less_real @ zero_zero_real @ t))). % t(1)
thf(fact_5_th30, axiom,
    ((ord_less_real @ (times_times_real @ (power_power_real @ t @ k) @ (times_times_real @ t @ (times_times_real @ (power_power_real @ (real_V638595069omplex @ w) @ (plus_plus_nat @ k @ one_one_nat)) @ m))) @ (times_times_real @ (power_power_real @ t @ k) @ one_one_real)))). % th30
thf(fact_6_tw, axiom,
    ((ord_less_eq_real @ (real_V638595069omplex @ (times_times_complex @ (real_V306493662omplex @ t) @ w)) @ (real_V638595069omplex @ w)))). % tw
thf(fact_7_m_I1_J, axiom,
    ((ord_less_real @ zero_zero_real @ m))). % m(1)
thf(fact_8__092_060open_062cmod_A_I_Icomplex__of__real_At_A_K_Aw_J_A_094_Ak_A_K_A_Icomplex__of__real_At_A_K_Aw_J_A_K_Apoly_As_A_Icomplex__of__real_At_A_K_Aw_J_J_A_092_060le_062_At_A_094_Ak_A_K_A_It_A_K_A_Icmod_Aw_A_094_A_Ik_A_L_A1_J_A_K_Am_J_J_092_060close_062, axiom,
    ((ord_less_eq_real @ (real_V638595069omplex @ (times_times_complex @ (times_times_complex @ (power_power_complex @ (times_times_complex @ (real_V306493662omplex @ t) @ w) @ k) @ (times_times_complex @ (real_V306493662omplex @ t) @ w)) @ (poly_complex2 @ s @ (times_times_complex @ (real_V306493662omplex @ t) @ w)))) @ (times_times_real @ (power_power_real @ t @ k) @ (times_times_real @ t @ (times_times_real @ (power_power_real @ (real_V638595069omplex @ w) @ (plus_plus_nat @ k @ one_one_nat)) @ m)))))). % \<open>cmod ((complex_of_real t * w) ^ k * (complex_of_real t * w) * poly s (complex_of_real t * w)) \<le> t ^ k * (t * (cmod w ^ (k + 1) * m))\<close>
thf(fact_9_real__sup__exists, axiom,
    ((![P : real > $o]: ((?[X_1 : real]: (P @ X_1)) => ((?[Z : real]: (![X : real]: ((P @ X) => (ord_less_real @ X @ Z)))) => (?[S : real]: (![Y : real]: ((?[X2 : real]: (((P @ X2)) & ((ord_less_real @ Y @ X2)))) = (ord_less_real @ Y @ S))))))))). % real_sup_exists
thf(fact_10__092_060open_062cmod_A_I_Icomplex__of__real_At_A_K_Aw_J_A_094_Ak_A_K_A_Icomplex__of__real_At_A_K_Aw_J_A_K_Apoly_As_A_Icomplex__of__real_At_A_K_Aw_J_J_A_061_At_A_094_Ak_A_K_A_It_A_K_A_Icmod_Aw_A_094_A_Ik_A_L_A1_J_A_K_Acmod_A_Ipoly_As_A_Icomplex__of__real_At_A_K_Aw_J_J_J_J_092_060close_062, axiom,
    (((real_V638595069omplex @ (times_times_complex @ (times_times_complex @ (power_power_complex @ (times_times_complex @ (real_V306493662omplex @ t) @ w) @ k) @ (times_times_complex @ (real_V306493662omplex @ t) @ w)) @ (poly_complex2 @ s @ (times_times_complex @ (real_V306493662omplex @ t) @ w)))) = (times_times_real @ (power_power_real @ t @ k) @ (times_times_real @ t @ (times_times_real @ (power_power_real @ (real_V638595069omplex @ w) @ (plus_plus_nat @ k @ one_one_nat)) @ (real_V638595069omplex @ (poly_complex2 @ s @ (times_times_complex @ (real_V306493662omplex @ t) @ w))))))))). % \<open>cmod ((complex_of_real t * w) ^ k * (complex_of_real t * w) * poly s (complex_of_real t * w)) = t ^ k * (t * (cmod w ^ (k + 1) * cmod (poly s (complex_of_real t * w))))\<close>
thf(fact_11__092_060open_062t_A_K_Acmod_Aw_A_092_060le_062_A1_A_K_Acmod_Aw_092_060close_062, axiom,
    ((ord_less_eq_real @ (times_times_real @ t @ (real_V638595069omplex @ w)) @ (times_times_real @ one_one_real @ (real_V638595069omplex @ w))))). % \<open>t * cmod w \<le> 1 * cmod w\<close>
thf(fact_12_of__real__power, axiom,
    ((![X3 : real, N : nat]: ((real_V1205483228l_real @ (power_power_real @ X3 @ N)) = (power_power_real @ (real_V1205483228l_real @ X3) @ N))))). % of_real_power
thf(fact_13_of__real__power, axiom,
    ((![X3 : real, N : nat]: ((real_V306493662omplex @ (power_power_real @ X3 @ N)) = (power_power_complex @ (real_V306493662omplex @ X3) @ N))))). % of_real_power
thf(fact_14__092_060open_0620_A_060_At_A_094_Ak_092_060close_062, axiom,
    ((ord_less_real @ zero_zero_real @ (power_power_real @ t @ k)))). % \<open>0 < t ^ k\<close>
thf(fact_15_poly__power, axiom,
    ((![P2 : poly_complex, N : nat, X3 : complex]: ((poly_complex2 @ (power_184595776omplex @ P2 @ N) @ X3) = (power_power_complex @ (poly_complex2 @ P2 @ X3) @ N))))). % poly_power
thf(fact_16_poly__power, axiom,
    ((![P2 : poly_real, N : nat, X3 : real]: ((poly_real2 @ (power_2108872382y_real @ P2 @ N) @ X3) = (power_power_real @ (poly_real2 @ P2 @ X3) @ N))))). % poly_power
thf(fact_17_poly__power, axiom,
    ((![P2 : poly_nat, N : nat, X3 : nat]: ((poly_nat2 @ (power_power_poly_nat @ P2 @ N) @ X3) = (power_power_nat @ (poly_nat2 @ P2 @ X3) @ N))))). % poly_power
thf(fact_18_poly__mult, axiom,
    ((![P2 : poly_complex, Q : poly_complex, X3 : complex]: ((poly_complex2 @ (times_1246143675omplex @ P2 @ Q) @ X3) = (times_times_complex @ (poly_complex2 @ P2 @ X3) @ (poly_complex2 @ Q @ X3)))))). % poly_mult
thf(fact_19_poly__mult, axiom,
    ((![P2 : poly_real, Q : poly_real, X3 : real]: ((poly_real2 @ (times_775122617y_real @ P2 @ Q) @ X3) = (times_times_real @ (poly_real2 @ P2 @ X3) @ (poly_real2 @ Q @ X3)))))). % poly_mult
thf(fact_20_of__real__mult, axiom,
    ((![X3 : real, Y2 : real]: ((real_V1205483228l_real @ (times_times_real @ X3 @ Y2)) = (times_times_real @ (real_V1205483228l_real @ X3) @ (real_V1205483228l_real @ Y2)))))). % of_real_mult
thf(fact_21_of__real__mult, axiom,
    ((![X3 : real, Y2 : real]: ((real_V306493662omplex @ (times_times_real @ X3 @ Y2)) = (times_times_complex @ (real_V306493662omplex @ X3) @ (real_V306493662omplex @ Y2)))))). % of_real_mult
thf(fact_22__092_060open_062t_A_K_A_Icmod_Aw_A_094_A_Ik_A_L_A1_J_A_K_Am_J_A_060_A1_092_060close_062, axiom,
    ((ord_less_real @ (times_times_real @ t @ (times_times_real @ (power_power_real @ (real_V638595069omplex @ w) @ (plus_plus_nat @ k @ one_one_nat)) @ m)) @ one_one_real))). % \<open>t * (cmod w ^ (k + 1) * m) < 1\<close>
thf(fact_23__092_060open_062_092_060exists_062m_0620_O_A_092_060forall_062z_O_Acmod_Az_A_092_060le_062_Acmod_Aw_A_092_060longrightarrow_062_Acmod_A_Ipoly_As_Az_J_A_092_060le_062_Am_092_060close_062, axiom,
    ((?[M : real]: ((ord_less_real @ zero_zero_real @ M) & (![Z : complex]: ((ord_less_eq_real @ (real_V638595069omplex @ Z) @ (real_V638595069omplex @ w)) => (ord_less_eq_real @ (real_V638595069omplex @ (poly_complex2 @ s @ Z)) @ M))))))). % \<open>\<exists>m>0. \<forall>z. cmod z \<le> cmod w \<longrightarrow> cmod (poly s z) \<le> m\<close>
thf(fact_24__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062m_O_A_092_060lbrakk_0620_A_060_Am_059_A_092_060forall_062z_O_Acmod_Az_A_092_060le_062_Acmod_Aw_A_092_060longrightarrow_062_Acmod_A_Ipoly_As_Az_J_A_092_060le_062_Am_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom,
    ((~ ((![M : real]: ((ord_less_real @ zero_zero_real @ M) => (~ ((![Z : complex]: ((ord_less_eq_real @ (real_V638595069omplex @ Z) @ (real_V638595069omplex @ w)) => (ord_less_eq_real @ (real_V638595069omplex @ (poly_complex2 @ s @ Z)) @ M))))))))))). % \<open>\<And>thesis. (\<And>m. \<lbrakk>0 < m; \<forall>z. cmod z \<le> cmod w \<longrightarrow> cmod (poly s z) \<le> m\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_25_t_I3_J, axiom,
    ((ord_less_real @ t @ (inverse_inverse_real @ (times_times_real @ (power_power_real @ (real_V638595069omplex @ w) @ (plus_plus_nat @ k @ one_one_nat)) @ m))))). % t(3)
thf(fact_26_norm__power, axiom,
    ((![X3 : real, N : nat]: ((real_V646646907m_real @ (power_power_real @ X3 @ N)) = (power_power_real @ (real_V646646907m_real @ X3) @ N))))). % norm_power
thf(fact_27_norm__power, axiom,
    ((![X3 : complex, N : nat]: ((real_V638595069omplex @ (power_power_complex @ X3 @ N)) = (power_power_real @ (real_V638595069omplex @ X3) @ N))))). % norm_power
thf(fact_28__092_060open_062_092_060And_062d2_O_A_I0_058_058_063_Ha_J_A_060_Ad2_A_092_060Longrightarrow_062_A_092_060exists_062e_0620_058_058_063_Ha_O_Ae_A_060_A_I1_058_058_063_Ha_J_A_092_060and_062_Ae_A_060_Ad2_092_060close_062, axiom,
    ((![D2 : real]: ((ord_less_real @ zero_zero_real @ D2) => (?[E : real]: ((ord_less_real @ zero_zero_real @ E) & ((ord_less_real @ E @ one_one_real) & (ord_less_real @ E @ D2)))))))). % \<open>\<And>d2. (0::?'a) < d2 \<Longrightarrow> \<exists>e>0::?'a. e < (1::?'a) \<and> e < d2\<close>
thf(fact_29_of__real__eq__iff, axiom,
    ((![X3 : real, Y2 : real]: (((real_V306493662omplex @ X3) = (real_V306493662omplex @ Y2)) = (X3 = Y2))))). % of_real_eq_iff
thf(fact_30_poly__0, axiom,
    ((![X3 : nat]: ((poly_nat2 @ zero_zero_poly_nat @ X3) = zero_zero_nat)))). % poly_0
thf(fact_31_poly__0, axiom,
    ((![X3 : real]: ((poly_real2 @ zero_zero_poly_real @ X3) = zero_zero_real)))). % poly_0
thf(fact_32_poly__0, axiom,
    ((![X3 : complex]: ((poly_complex2 @ zero_z1746442943omplex @ X3) = zero_zero_complex)))). % poly_0
thf(fact_33_of__real__add, axiom,
    ((![X3 : real, Y2 : real]: ((real_V1205483228l_real @ (plus_plus_real @ X3 @ Y2)) = (plus_plus_real @ (real_V1205483228l_real @ X3) @ (real_V1205483228l_real @ Y2)))))). % of_real_add
thf(fact_34_of__real__add, axiom,
    ((![X3 : real, Y2 : real]: ((real_V306493662omplex @ (plus_plus_real @ X3 @ Y2)) = (plus_plus_complex @ (real_V306493662omplex @ X3) @ (real_V306493662omplex @ Y2)))))). % of_real_add
thf(fact_35_poly__add, axiom,
    ((![P2 : poly_nat, Q : poly_nat, X3 : nat]: ((poly_nat2 @ (plus_plus_poly_nat @ P2 @ Q) @ X3) = (plus_plus_nat @ (poly_nat2 @ P2 @ X3) @ (poly_nat2 @ Q @ X3)))))). % poly_add
thf(fact_36_poly__add, axiom,
    ((![P2 : poly_complex, Q : poly_complex, X3 : complex]: ((poly_complex2 @ (plus_p1547158847omplex @ P2 @ Q) @ X3) = (plus_plus_complex @ (poly_complex2 @ P2 @ X3) @ (poly_complex2 @ Q @ X3)))))). % poly_add
thf(fact_37_poly__add, axiom,
    ((![P2 : poly_real, Q : poly_real, X3 : real]: ((poly_real2 @ (plus_plus_poly_real @ P2 @ Q) @ X3) = (plus_plus_real @ (poly_real2 @ P2 @ X3) @ (poly_real2 @ Q @ X3)))))). % poly_add
thf(fact_38_poly__1, axiom,
    ((![X3 : real]: ((poly_real2 @ one_one_poly_real @ X3) = one_one_real)))). % poly_1
thf(fact_39_poly__1, axiom,
    ((![X3 : nat]: ((poly_nat2 @ one_one_poly_nat @ X3) = one_one_nat)))). % poly_1
thf(fact_40_poly__1, axiom,
    ((![X3 : complex]: ((poly_complex2 @ one_one_poly_complex @ X3) = one_one_complex)))). % poly_1
thf(fact_41_less_Oprems, axiom,
    ((~ ((fundam1158420650omplex @ (poly_complex2 @ pa)))))). % less.prems
thf(fact_42_norm__zero, axiom,
    (((real_V646646907m_real @ zero_zero_real) = zero_zero_real))). % norm_zero
thf(fact_43_norm__zero, axiom,
    (((real_V638595069omplex @ zero_zero_complex) = zero_zero_real))). % norm_zero
thf(fact_44_norm__eq__zero, axiom,
    ((![X3 : real]: (((real_V646646907m_real @ X3) = zero_zero_real) = (X3 = zero_zero_real))))). % norm_eq_zero
thf(fact_45_norm__eq__zero, axiom,
    ((![X3 : complex]: (((real_V638595069omplex @ X3) = zero_zero_real) = (X3 = zero_zero_complex))))). % norm_eq_zero
thf(fact_46_of__real__0, axiom,
    (((real_V1205483228l_real @ zero_zero_real) = zero_zero_real))). % of_real_0
thf(fact_47_of__real__0, axiom,
    (((real_V306493662omplex @ zero_zero_real) = zero_zero_complex))). % of_real_0
thf(fact_48_of__real__eq__0__iff, axiom,
    ((![X3 : real]: (((real_V1205483228l_real @ X3) = zero_zero_real) = (X3 = zero_zero_real))))). % of_real_eq_0_iff
thf(fact_49_of__real__eq__0__iff, axiom,
    ((![X3 : real]: (((real_V306493662omplex @ X3) = zero_zero_complex) = (X3 = zero_zero_real))))). % of_real_eq_0_iff
thf(fact_50_norm__one, axiom,
    (((real_V646646907m_real @ one_one_real) = one_one_real))). % norm_one
thf(fact_51_norm__one, axiom,
    (((real_V638595069omplex @ one_one_complex) = one_one_real))). % norm_one
thf(fact_52_of__real__eq__1__iff, axiom,
    ((![X3 : real]: (((real_V1205483228l_real @ X3) = one_one_real) = (X3 = one_one_real))))). % of_real_eq_1_iff
thf(fact_53_of__real__eq__1__iff, axiom,
    ((![X3 : real]: (((real_V306493662omplex @ X3) = one_one_complex) = (X3 = one_one_real))))). % of_real_eq_1_iff
thf(fact_54_of__real__1, axiom,
    (((real_V1205483228l_real @ one_one_real) = one_one_real))). % of_real_1
thf(fact_55_of__real__1, axiom,
    (((real_V306493662omplex @ one_one_real) = one_one_complex))). % of_real_1
thf(fact_56_of__real__inverse, axiom,
    ((![X3 : real]: ((real_V306493662omplex @ (inverse_inverse_real @ X3)) = (invers502456322omplex @ (real_V306493662omplex @ X3)))))). % of_real_inverse
thf(fact_57_of__real__inverse, axiom,
    ((![X3 : real]: ((real_V1205483228l_real @ (inverse_inverse_real @ X3)) = (inverse_inverse_real @ (real_V1205483228l_real @ X3)))))). % of_real_inverse
thf(fact_58_inv0, axiom,
    ((ord_less_real @ zero_zero_real @ (inverse_inverse_real @ (times_times_real @ (power_power_real @ (real_V638595069omplex @ w) @ (plus_plus_nat @ k @ one_one_nat)) @ m))))). % inv0
thf(fact_59__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062t_O_A_092_060lbrakk_0620_A_060_At_059_At_A_060_A1_059_At_A_060_Ainverse_A_Icmod_Aw_A_094_A_Ik_A_L_A1_J_A_K_Am_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom,
    ((~ ((![T : real]: ((ord_less_real @ zero_zero_real @ T) => ((ord_less_real @ T @ one_one_real) => (~ ((ord_less_real @ T @ (inverse_inverse_real @ (times_times_real @ (power_power_real @ (real_V638595069omplex @ w) @ (plus_plus_nat @ k @ one_one_nat)) @ m)))))))))))). % \<open>\<And>thesis. (\<And>t. \<lbrakk>0 < t; t < 1; t < inverse (cmod w ^ (k + 1) * m)\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_60_zero__less__norm__iff, axiom,
    ((![X3 : real]: ((ord_less_real @ zero_zero_real @ (real_V646646907m_real @ X3)) = (~ ((X3 = zero_zero_real))))))). % zero_less_norm_iff
thf(fact_61_zero__less__norm__iff, axiom,
    ((![X3 : complex]: ((ord_less_real @ zero_zero_real @ (real_V638595069omplex @ X3)) = (~ ((X3 = zero_zero_complex))))))). % zero_less_norm_iff
thf(fact_62_norm__le__zero__iff, axiom,
    ((![X3 : real]: ((ord_less_eq_real @ (real_V646646907m_real @ X3) @ zero_zero_real) = (X3 = zero_zero_real))))). % norm_le_zero_iff
thf(fact_63_norm__le__zero__iff, axiom,
    ((![X3 : complex]: ((ord_less_eq_real @ (real_V638595069omplex @ X3) @ zero_zero_real) = (X3 = zero_zero_complex))))). % norm_le_zero_iff
thf(fact_64_poly__IVT, axiom,
    ((![A : real, B : real, P2 : poly_real]: ((ord_less_real @ A @ B) => ((ord_less_real @ (times_times_real @ (poly_real2 @ P2 @ A) @ (poly_real2 @ P2 @ B)) @ zero_zero_real) => (?[X : real]: ((ord_less_real @ A @ X) & ((ord_less_real @ X @ B) & ((poly_real2 @ P2 @ X) = zero_zero_real))))))))). % poly_IVT
thf(fact_65_poly__IVT__neg, axiom,
    ((![A : real, B : real, P2 : poly_real]: ((ord_less_real @ A @ B) => ((ord_less_real @ zero_zero_real @ (poly_real2 @ P2 @ A)) => ((ord_less_real @ (poly_real2 @ P2 @ B) @ zero_zero_real) => (?[X : real]: ((ord_less_real @ A @ X) & ((ord_less_real @ X @ B) & ((poly_real2 @ P2 @ X) = zero_zero_real)))))))))). % poly_IVT_neg
thf(fact_66_poly__IVT__pos, axiom,
    ((![A : real, B : real, P2 : poly_real]: ((ord_less_real @ A @ B) => ((ord_less_real @ (poly_real2 @ P2 @ A) @ zero_zero_real) => ((ord_less_real @ zero_zero_real @ (poly_real2 @ P2 @ B)) => (?[X : real]: ((ord_less_real @ A @ X) & ((ord_less_real @ X @ B) & ((poly_real2 @ P2 @ X) = zero_zero_real)))))))))). % poly_IVT_pos
thf(fact_67_poly__all__0__iff__0, axiom,
    ((![P2 : poly_real]: ((![X2 : real]: ((poly_real2 @ P2 @ X2) = zero_zero_real)) = (P2 = zero_zero_poly_real))))). % poly_all_0_iff_0
thf(fact_68_poly__all__0__iff__0, axiom,
    ((![P2 : poly_complex]: ((![X2 : complex]: ((poly_complex2 @ P2 @ X2) = zero_zero_complex)) = (P2 = zero_z1746442943omplex))))). % poly_all_0_iff_0
thf(fact_69_norm__inverse, axiom,
    ((![A : real]: ((real_V646646907m_real @ (inverse_inverse_real @ A)) = (inverse_inverse_real @ (real_V646646907m_real @ A)))))). % norm_inverse
thf(fact_70_norm__inverse, axiom,
    ((![A : complex]: ((real_V638595069omplex @ (invers502456322omplex @ A)) = (inverse_inverse_real @ (real_V638595069omplex @ A)))))). % norm_inverse
thf(fact_71_nonzero__norm__inverse, axiom,
    ((![A : real]: ((~ ((A = zero_zero_real))) => ((real_V646646907m_real @ (inverse_inverse_real @ A)) = (inverse_inverse_real @ (real_V646646907m_real @ A))))))). % nonzero_norm_inverse
thf(fact_72_nonzero__norm__inverse, axiom,
    ((![A : complex]: ((~ ((A = zero_zero_complex))) => ((real_V638595069omplex @ (invers502456322omplex @ A)) = (inverse_inverse_real @ (real_V638595069omplex @ A))))))). % nonzero_norm_inverse
thf(fact_73_norm__inverse__le__norm, axiom,
    ((![R : real, X3 : real]: ((ord_less_eq_real @ R @ (real_V646646907m_real @ X3)) => ((ord_less_real @ zero_zero_real @ R) => (ord_less_eq_real @ (real_V646646907m_real @ (inverse_inverse_real @ X3)) @ (inverse_inverse_real @ R))))))). % norm_inverse_le_norm
thf(fact_74_norm__inverse__le__norm, axiom,
    ((![R : real, X3 : complex]: ((ord_less_eq_real @ R @ (real_V638595069omplex @ X3)) => ((ord_less_real @ zero_zero_real @ R) => (ord_less_eq_real @ (real_V638595069omplex @ (invers502456322omplex @ X3)) @ (inverse_inverse_real @ R))))))). % norm_inverse_le_norm
thf(fact_75_nonzero__of__real__inverse, axiom,
    ((![X3 : real]: ((~ ((X3 = zero_zero_real))) => ((real_V306493662omplex @ (inverse_inverse_real @ X3)) = (invers502456322omplex @ (real_V306493662omplex @ X3))))))). % nonzero_of_real_inverse
thf(fact_76_nonzero__of__real__inverse, axiom,
    ((![X3 : real]: ((~ ((X3 = zero_zero_real))) => ((real_V1205483228l_real @ (inverse_inverse_real @ X3)) = (inverse_inverse_real @ (real_V1205483228l_real @ X3))))))). % nonzero_of_real_inverse
thf(fact_77_norm__triangle__mono, axiom,
    ((![A : real, R : real, B : real, S2 : real]: ((ord_less_eq_real @ (real_V646646907m_real @ A) @ R) => ((ord_less_eq_real @ (real_V646646907m_real @ B) @ S2) => (ord_less_eq_real @ (real_V646646907m_real @ (plus_plus_real @ A @ B)) @ (plus_plus_real @ R @ S2))))))). % norm_triangle_mono
thf(fact_78_norm__triangle__mono, axiom,
    ((![A : complex, R : real, B : complex, S2 : real]: ((ord_less_eq_real @ (real_V638595069omplex @ A) @ R) => ((ord_less_eq_real @ (real_V638595069omplex @ B) @ S2) => (ord_less_eq_real @ (real_V638595069omplex @ (plus_plus_complex @ A @ B)) @ (plus_plus_real @ R @ S2))))))). % norm_triangle_mono
thf(fact_79_norm__triangle__ineq, axiom,
    ((![X3 : real, Y2 : real]: (ord_less_eq_real @ (real_V646646907m_real @ (plus_plus_real @ X3 @ Y2)) @ (plus_plus_real @ (real_V646646907m_real @ X3) @ (real_V646646907m_real @ Y2)))))). % norm_triangle_ineq
thf(fact_80_norm__triangle__ineq, axiom,
    ((![X3 : complex, Y2 : complex]: (ord_less_eq_real @ (real_V638595069omplex @ (plus_plus_complex @ X3 @ Y2)) @ (plus_plus_real @ (real_V638595069omplex @ X3) @ (real_V638595069omplex @ Y2)))))). % norm_triangle_ineq
thf(fact_81_norm__triangle__le, axiom,
    ((![X3 : real, Y2 : real, E2 : real]: ((ord_less_eq_real @ (plus_plus_real @ (real_V646646907m_real @ X3) @ (real_V646646907m_real @ Y2)) @ E2) => (ord_less_eq_real @ (real_V646646907m_real @ (plus_plus_real @ X3 @ Y2)) @ E2))))). % norm_triangle_le
thf(fact_82_norm__triangle__le, axiom,
    ((![X3 : complex, Y2 : complex, E2 : real]: ((ord_less_eq_real @ (plus_plus_real @ (real_V638595069omplex @ X3) @ (real_V638595069omplex @ Y2)) @ E2) => (ord_less_eq_real @ (real_V638595069omplex @ (plus_plus_complex @ X3 @ Y2)) @ E2))))). % norm_triangle_le
thf(fact_83_norm__add__leD, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_eq_real @ (real_V646646907m_real @ (plus_plus_real @ A @ B)) @ C) => (ord_less_eq_real @ (real_V646646907m_real @ B) @ (plus_plus_real @ (real_V646646907m_real @ A) @ C)))))). % norm_add_leD
thf(fact_84_norm__add__leD, axiom,
    ((![A : complex, B : complex, C : real]: ((ord_less_eq_real @ (real_V638595069omplex @ (plus_plus_complex @ A @ B)) @ C) => (ord_less_eq_real @ (real_V638595069omplex @ B) @ (plus_plus_real @ (real_V638595069omplex @ A) @ C)))))). % norm_add_leD
thf(fact_85_norm__ge__zero, axiom,
    ((![X3 : complex]: (ord_less_eq_real @ zero_zero_real @ (real_V638595069omplex @ X3))))). % norm_ge_zero
thf(fact_86_constant__def, axiom,
    ((fundam1158420650omplex = (^[F : complex > complex]: (![X2 : complex]: (![Y3 : complex]: ((F @ X2) = (F @ Y3)))))))). % constant_def
thf(fact_87_power__eq__1__iff, axiom,
    ((![W : real, N : nat]: (((power_power_real @ W @ N) = one_one_real) => (((real_V646646907m_real @ W) = one_one_real) | (N = zero_zero_nat)))))). % power_eq_1_iff
thf(fact_88_power__eq__1__iff, axiom,
    ((![W : complex, N : nat]: (((power_power_complex @ W @ N) = one_one_complex) => (((real_V638595069omplex @ W) = one_one_real) | (N = zero_zero_nat)))))). % power_eq_1_iff
thf(fact_89_norm__triangle__lt, axiom,
    ((![X3 : real, Y2 : real, E2 : real]: ((ord_less_real @ (plus_plus_real @ (real_V646646907m_real @ X3) @ (real_V646646907m_real @ Y2)) @ E2) => (ord_less_real @ (real_V646646907m_real @ (plus_plus_real @ X3 @ Y2)) @ E2))))). % norm_triangle_lt
thf(fact_90_norm__triangle__lt, axiom,
    ((![X3 : complex, Y2 : complex, E2 : real]: ((ord_less_real @ (plus_plus_real @ (real_V638595069omplex @ X3) @ (real_V638595069omplex @ Y2)) @ E2) => (ord_less_real @ (real_V638595069omplex @ (plus_plus_complex @ X3 @ Y2)) @ E2))))). % norm_triangle_lt
thf(fact_91_norm__add__less, axiom,
    ((![X3 : real, R : real, Y2 : real, S2 : real]: ((ord_less_real @ (real_V646646907m_real @ X3) @ R) => ((ord_less_real @ (real_V646646907m_real @ Y2) @ S2) => (ord_less_real @ (real_V646646907m_real @ (plus_plus_real @ X3 @ Y2)) @ (plus_plus_real @ R @ S2))))))). % norm_add_less
thf(fact_92_norm__add__less, axiom,
    ((![X3 : complex, R : real, Y2 : complex, S2 : real]: ((ord_less_real @ (real_V638595069omplex @ X3) @ R) => ((ord_less_real @ (real_V638595069omplex @ Y2) @ S2) => (ord_less_real @ (real_V638595069omplex @ (plus_plus_complex @ X3 @ Y2)) @ (plus_plus_real @ R @ S2))))))). % norm_add_less
thf(fact_93_norm__not__less__zero, axiom,
    ((![X3 : complex]: (~ ((ord_less_real @ (real_V638595069omplex @ X3) @ zero_zero_real)))))). % norm_not_less_zero
thf(fact_94_poly__bound__exists, axiom,
    ((![R : real, P2 : poly_real]: (?[M : real]: ((ord_less_real @ zero_zero_real @ M) & (![Z : real]: ((ord_less_eq_real @ (real_V646646907m_real @ Z) @ R) => (ord_less_eq_real @ (real_V646646907m_real @ (poly_real2 @ P2 @ Z)) @ M)))))))). % poly_bound_exists
thf(fact_95_poly__bound__exists, axiom,
    ((![R : real, P2 : poly_complex]: (?[M : real]: ((ord_less_real @ zero_zero_real @ M) & (![Z : complex]: ((ord_less_eq_real @ (real_V638595069omplex @ Z) @ R) => (ord_less_eq_real @ (real_V638595069omplex @ (poly_complex2 @ P2 @ Z)) @ M)))))))). % poly_bound_exists
thf(fact_96_norm__mult__ineq, axiom,
    ((![X3 : real, Y2 : real]: (ord_less_eq_real @ (real_V646646907m_real @ (times_times_real @ X3 @ Y2)) @ (times_times_real @ (real_V646646907m_real @ X3) @ (real_V646646907m_real @ Y2)))))). % norm_mult_ineq
thf(fact_97_norm__mult__ineq, axiom,
    ((![X3 : complex, Y2 : complex]: (ord_less_eq_real @ (real_V638595069omplex @ (times_times_complex @ X3 @ Y2)) @ (times_times_real @ (real_V638595069omplex @ X3) @ (real_V638595069omplex @ Y2)))))). % norm_mult_ineq
thf(fact_98_poly__eq__poly__eq__iff, axiom,
    ((![P2 : poly_complex, Q : poly_complex]: (((poly_complex2 @ P2) = (poly_complex2 @ Q)) = (P2 = Q))))). % poly_eq_poly_eq_iff
thf(fact_99_poly__eq__poly__eq__iff, axiom,
    ((![P2 : poly_real, Q : poly_real]: (((poly_real2 @ P2) = (poly_real2 @ Q)) = (P2 = Q))))). % poly_eq_poly_eq_iff
thf(fact_100_norm__less__p1, axiom,
    ((![X3 : real]: (ord_less_real @ (real_V646646907m_real @ X3) @ (real_V646646907m_real @ (plus_plus_real @ (real_V1205483228l_real @ (real_V646646907m_real @ X3)) @ one_one_real)))))). % norm_less_p1
thf(fact_101_norm__less__p1, axiom,
    ((![X3 : complex]: (ord_less_real @ (real_V638595069omplex @ X3) @ (real_V638595069omplex @ (plus_plus_complex @ (real_V306493662omplex @ (real_V638595069omplex @ X3)) @ one_one_complex)))))). % norm_less_p1
thf(fact_102_norm__power__ineq, axiom,
    ((![X3 : real, N : nat]: (ord_less_eq_real @ (real_V646646907m_real @ (power_power_real @ X3 @ N)) @ (power_power_real @ (real_V646646907m_real @ X3) @ N))))). % norm_power_ineq
thf(fact_103_norm__power__ineq, axiom,
    ((![X3 : complex, N : nat]: (ord_less_eq_real @ (real_V638595069omplex @ (power_power_complex @ X3 @ N)) @ (power_power_real @ (real_V638595069omplex @ X3) @ N))))). % norm_power_ineq
thf(fact_104_norm__mult, axiom,
    ((![X3 : real, Y2 : real]: ((real_V646646907m_real @ (times_times_real @ X3 @ Y2)) = (times_times_real @ (real_V646646907m_real @ X3) @ (real_V646646907m_real @ Y2)))))). % norm_mult
thf(fact_105_norm__mult, axiom,
    ((![X3 : complex, Y2 : complex]: ((real_V638595069omplex @ (times_times_complex @ X3 @ Y2)) = (times_times_real @ (real_V638595069omplex @ X3) @ (real_V638595069omplex @ Y2)))))). % norm_mult
thf(fact_106_norm__mult__less, axiom,
    ((![X3 : real, R : real, Y2 : real, S2 : real]: ((ord_less_real @ (real_V646646907m_real @ X3) @ R) => ((ord_less_real @ (real_V646646907m_real @ Y2) @ S2) => (ord_less_real @ (real_V646646907m_real @ (times_times_real @ X3 @ Y2)) @ (times_times_real @ R @ S2))))))). % norm_mult_less
thf(fact_107_norm__mult__less, axiom,
    ((![X3 : complex, R : real, Y2 : complex, S2 : real]: ((ord_less_real @ (real_V638595069omplex @ X3) @ R) => ((ord_less_real @ (real_V638595069omplex @ Y2) @ S2) => (ord_less_real @ (real_V638595069omplex @ (times_times_complex @ X3 @ Y2)) @ (times_times_real @ R @ S2))))))). % norm_mult_less
thf(fact_108_poly__minimum__modulus__disc, axiom,
    ((![R : real, P2 : poly_complex]: (?[Z2 : complex]: (![W2 : complex]: ((ord_less_eq_real @ (real_V638595069omplex @ W2) @ R) => (ord_less_eq_real @ (real_V638595069omplex @ (poly_complex2 @ P2 @ Z2)) @ (real_V638595069omplex @ (poly_complex2 @ P2 @ W2))))))))). % poly_minimum_modulus_disc
thf(fact_109_poly__minimum__modulus, axiom,
    ((![P2 : poly_complex]: (?[Z2 : complex]: (![W2 : complex]: (ord_less_eq_real @ (real_V638595069omplex @ (poly_complex2 @ P2 @ Z2)) @ (real_V638595069omplex @ (poly_complex2 @ P2 @ W2)))))))). % poly_minimum_modulus
thf(fact_110_power__decreasing__iff, axiom,
    ((![B : nat, M2 : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ B) => ((ord_less_nat @ B @ one_one_nat) => ((ord_less_eq_nat @ (power_power_nat @ B @ M2) @ (power_power_nat @ B @ N)) = (ord_less_eq_nat @ N @ M2))))))). % power_decreasing_iff
thf(fact_111_power__decreasing__iff, axiom,
    ((![B : real, M2 : nat, N : nat]: ((ord_less_real @ zero_zero_real @ B) => ((ord_less_real @ B @ one_one_real) => ((ord_less_eq_real @ (power_power_real @ B @ M2) @ (power_power_real @ B @ N)) = (ord_less_eq_nat @ N @ M2))))))). % power_decreasing_iff
thf(fact_112_left__inverse, axiom,
    ((![A : complex]: ((~ ((A = zero_zero_complex))) => ((times_times_complex @ (invers502456322omplex @ A) @ A) = one_one_complex))))). % left_inverse
thf(fact_113_left__inverse, axiom,
    ((![A : real]: ((~ ((A = zero_zero_real))) => ((times_times_real @ (inverse_inverse_real @ A) @ A) = one_one_real))))). % left_inverse
thf(fact_114_right__inverse, axiom,
    ((![A : complex]: ((~ ((A = zero_zero_complex))) => ((times_times_complex @ A @ (invers502456322omplex @ A)) = one_one_complex))))). % right_inverse
thf(fact_115_right__inverse, axiom,
    ((![A : real]: ((~ ((A = zero_zero_real))) => ((times_times_real @ A @ (inverse_inverse_real @ A)) = one_one_real))))). % right_inverse
thf(fact_116_inverse__le__iff__le, axiom,
    ((![A : real, B : real]: ((ord_less_real @ zero_zero_real @ A) => ((ord_less_real @ zero_zero_real @ B) => ((ord_less_eq_real @ (inverse_inverse_real @ A) @ (inverse_inverse_real @ B)) = (ord_less_eq_real @ B @ A))))))). % inverse_le_iff_le
thf(fact_117_inverse__le__iff__le__neg, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ zero_zero_real) => ((ord_less_real @ B @ zero_zero_real) => ((ord_less_eq_real @ (inverse_inverse_real @ A) @ (inverse_inverse_real @ B)) = (ord_less_eq_real @ B @ A))))))). % inverse_le_iff_le_neg
thf(fact_118_power__increasing__iff, axiom,
    ((![B : nat, X3 : nat, Y2 : nat]: ((ord_less_nat @ one_one_nat @ B) => ((ord_less_eq_nat @ (power_power_nat @ B @ X3) @ (power_power_nat @ B @ Y2)) = (ord_less_eq_nat @ X3 @ Y2)))))). % power_increasing_iff
thf(fact_119_power__increasing__iff, axiom,
    ((![B : real, X3 : nat, Y2 : nat]: ((ord_less_real @ one_one_real @ B) => ((ord_less_eq_real @ (power_power_real @ B @ X3) @ (power_power_real @ B @ Y2)) = (ord_less_eq_nat @ X3 @ Y2)))))). % power_increasing_iff
thf(fact_120_power__strict__decreasing__iff, axiom,
    ((![B : real, M2 : nat, N : nat]: ((ord_less_real @ zero_zero_real @ B) => ((ord_less_real @ B @ one_one_real) => ((ord_less_real @ (power_power_real @ B @ M2) @ (power_power_real @ B @ N)) = (ord_less_nat @ N @ M2))))))). % power_strict_decreasing_iff
thf(fact_121_power__strict__decreasing__iff, axiom,
    ((![B : nat, M2 : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ B) => ((ord_less_nat @ B @ one_one_nat) => ((ord_less_nat @ (power_power_nat @ B @ M2) @ (power_power_nat @ B @ N)) = (ord_less_nat @ N @ M2))))))). % power_strict_decreasing_iff
thf(fact_122_not__real__square__gt__zero, axiom,
    ((![X3 : real]: ((~ ((ord_less_real @ zero_zero_real @ (times_times_real @ X3 @ X3)))) = (X3 = zero_zero_real))))). % not_real_square_gt_zero
thf(fact_123_inverse__less__iff__less, axiom,
    ((![A : real, B : real]: ((ord_less_real @ zero_zero_real @ A) => ((ord_less_real @ zero_zero_real @ B) => ((ord_less_real @ (inverse_inverse_real @ A) @ (inverse_inverse_real @ B)) = (ord_less_real @ B @ A))))))). % inverse_less_iff_less
thf(fact_124_kas_I1_J, axiom,
    ((~ ((a = zero_zero_complex))))). % kas(1)
thf(fact_125_w0, axiom,
    ((~ ((w = zero_zero_complex))))). % w0
thf(fact_126_inverse__inverse__eq, axiom,
    ((![A : real]: ((inverse_inverse_real @ (inverse_inverse_real @ A)) = A)))). % inverse_inverse_eq
thf(fact_127_inverse__eq__iff__eq, axiom,
    ((![A : real, B : real]: (((inverse_inverse_real @ A) = (inverse_inverse_real @ B)) = (A = B))))). % inverse_eq_iff_eq
thf(fact_128__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062c_O_A_092_060forall_062w_O_Acmod_A_Ipoly_Ap_Ac_J_A_092_060le_062_Acmod_A_Ipoly_Ap_Aw_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom,
    ((~ ((![C2 : complex]: (~ ((![W2 : complex]: (ord_less_eq_real @ (real_V638595069omplex @ (poly_complex2 @ pa @ C2)) @ (real_V638595069omplex @ (poly_complex2 @ pa @ W2))))))))))). % \<open>\<And>thesis. (\<And>c. \<forall>w. cmod (poly p c) \<le> cmod (poly p w) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_129_c, axiom,
    ((![W2 : complex]: (ord_less_eq_real @ (real_V638595069omplex @ (poly_complex2 @ pa @ c)) @ (real_V638595069omplex @ (poly_complex2 @ pa @ W2)))))). % c
thf(fact_130_power__one, axiom,
    ((![N : nat]: ((power_power_complex @ one_one_complex @ N) = one_one_complex)))). % power_one
thf(fact_131_power__one, axiom,
    ((![N : nat]: ((power_power_real @ one_one_real @ N) = one_one_real)))). % power_one
thf(fact_132_power__one, axiom,
    ((![N : nat]: ((power_power_nat @ one_one_nat @ N) = one_one_nat)))). % power_one
thf(fact_133_inverse__nonzero__iff__nonzero, axiom,
    ((![A : complex]: (((invers502456322omplex @ A) = zero_zero_complex) = (A = zero_zero_complex))))). % inverse_nonzero_iff_nonzero
thf(fact_134_inverse__nonzero__iff__nonzero, axiom,
    ((![A : real]: (((inverse_inverse_real @ A) = zero_zero_real) = (A = zero_zero_real))))). % inverse_nonzero_iff_nonzero
thf(fact_135_inverse__zero, axiom,
    (((invers502456322omplex @ zero_zero_complex) = zero_zero_complex))). % inverse_zero
thf(fact_136_inverse__zero, axiom,
    (((inverse_inverse_real @ zero_zero_real) = zero_zero_real))). % inverse_zero
thf(fact_137_nat__zero__less__power__iff, axiom,
    ((![X3 : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ (power_power_nat @ X3 @ N)) = (((ord_less_nat @ zero_zero_nat @ X3)) | ((N = zero_zero_nat))))))). % nat_zero_less_power_iff
thf(fact_138_inverse__mult__distrib, axiom,
    ((![A : complex, B : complex]: ((invers502456322omplex @ (times_times_complex @ A @ B)) = (times_times_complex @ (invers502456322omplex @ A) @ (invers502456322omplex @ B)))))). % inverse_mult_distrib
thf(fact_139_inverse__mult__distrib, axiom,
    ((![A : real, B : real]: ((inverse_inverse_real @ (times_times_real @ A @ B)) = (times_times_real @ (inverse_inverse_real @ A) @ (inverse_inverse_real @ B)))))). % inverse_mult_distrib
thf(fact_140_inverse__1, axiom,
    (((invers502456322omplex @ one_one_complex) = one_one_complex))). % inverse_1
thf(fact_141_inverse__1, axiom,
    (((inverse_inverse_real @ one_one_real) = one_one_real))). % inverse_1
thf(fact_142_inverse__eq__1__iff, axiom,
    ((![X3 : complex]: (((invers502456322omplex @ X3) = one_one_complex) = (X3 = one_one_complex))))). % inverse_eq_1_iff
thf(fact_143_inverse__eq__1__iff, axiom,
    ((![X3 : real]: (((inverse_inverse_real @ X3) = one_one_real) = (X3 = one_one_real))))). % inverse_eq_1_iff
thf(fact_144_power__one__right, axiom,
    ((![A : complex]: ((power_power_complex @ A @ one_one_nat) = A)))). % power_one_right
thf(fact_145_power__one__right, axiom,
    ((![A : real]: ((power_power_real @ A @ one_one_nat) = A)))). % power_one_right
thf(fact_146_power__one__right, axiom,
    ((![A : nat]: ((power_power_nat @ A @ one_one_nat) = A)))). % power_one_right
thf(fact_147_kn, axiom,
    ((~ (((fundam1709708056omplex @ pa) = (plus_plus_nat @ k @ one_one_nat)))))). % kn
thf(fact_148_sum__squares__eq__zero__iff, axiom,
    ((![X3 : real, Y2 : real]: (((plus_plus_real @ (times_times_real @ X3 @ X3) @ (times_times_real @ Y2 @ Y2)) = zero_zero_real) = (((X3 = zero_zero_real)) & ((Y2 = zero_zero_real))))))). % sum_squares_eq_zero_iff
thf(fact_149_power__inject__exp, axiom,
    ((![A : real, M2 : nat, N : nat]: ((ord_less_real @ one_one_real @ A) => (((power_power_real @ A @ M2) = (power_power_real @ A @ N)) = (M2 = N)))))). % power_inject_exp
thf(fact_150_power__inject__exp, axiom,
    ((![A : nat, M2 : nat, N : nat]: ((ord_less_nat @ one_one_nat @ A) => (((power_power_nat @ A @ M2) = (power_power_nat @ A @ N)) = (M2 = N)))))). % power_inject_exp
thf(fact_151_inverse__nonpositive__iff__nonpositive, axiom,
    ((![A : real]: ((ord_less_eq_real @ (inverse_inverse_real @ A) @ zero_zero_real) = (ord_less_eq_real @ A @ zero_zero_real))))). % inverse_nonpositive_iff_nonpositive
thf(fact_152_inverse__nonnegative__iff__nonnegative, axiom,
    ((![A : real]: ((ord_less_eq_real @ zero_zero_real @ (inverse_inverse_real @ A)) = (ord_less_eq_real @ zero_zero_real @ A))))). % inverse_nonnegative_iff_nonnegative
thf(fact_153_inverse__positive__iff__positive, axiom,
    ((![A : real]: ((ord_less_real @ zero_zero_real @ (inverse_inverse_real @ A)) = (ord_less_real @ zero_zero_real @ A))))). % inverse_positive_iff_positive
thf(fact_154_inverse__negative__iff__negative, axiom,
    ((![A : real]: ((ord_less_real @ (inverse_inverse_real @ A) @ zero_zero_real) = (ord_less_real @ A @ zero_zero_real))))). % inverse_negative_iff_negative
thf(fact_155_inverse__less__iff__less__neg, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ zero_zero_real) => ((ord_less_real @ B @ zero_zero_real) => ((ord_less_real @ (inverse_inverse_real @ A) @ (inverse_inverse_real @ B)) = (ord_less_real @ B @ A))))))). % inverse_less_iff_less_neg
thf(fact_156_power__strict__increasing__iff, axiom,
    ((![B : real, X3 : nat, Y2 : nat]: ((ord_less_real @ one_one_real @ B) => ((ord_less_real @ (power_power_real @ B @ X3) @ (power_power_real @ B @ Y2)) = (ord_less_nat @ X3 @ Y2)))))). % power_strict_increasing_iff
thf(fact_157_power__strict__increasing__iff, axiom,
    ((![B : nat, X3 : nat, Y2 : nat]: ((ord_less_nat @ one_one_nat @ B) => ((ord_less_nat @ (power_power_nat @ B @ X3) @ (power_power_nat @ B @ Y2)) = (ord_less_nat @ X3 @ Y2)))))). % power_strict_increasing_iff
thf(fact_158_power__eq__0__iff, axiom,
    ((![A : complex, N : nat]: (((power_power_complex @ A @ N) = zero_zero_complex) = (((A = zero_zero_complex)) & ((ord_less_nat @ zero_zero_nat @ N))))))). % power_eq_0_iff
thf(fact_159_power__eq__0__iff, axiom,
    ((![A : real, N : nat]: (((power_power_real @ A @ N) = zero_zero_real) = (((A = zero_zero_real)) & ((ord_less_nat @ zero_zero_nat @ N))))))). % power_eq_0_iff
thf(fact_160_power__eq__0__iff, axiom,
    ((![A : nat, N : nat]: (((power_power_nat @ A @ N) = zero_zero_nat) = (((A = zero_zero_nat)) & ((ord_less_nat @ zero_zero_nat @ N))))))). % power_eq_0_iff
thf(fact_161_w, axiom,
    (((plus_plus_complex @ one_one_complex @ (times_times_complex @ (power_power_complex @ w @ k) @ a)) = zero_zero_complex))). % w
thf(fact_162_power__mono__iff, axiom,
    ((![A : nat, B : nat, N : nat]: ((ord_less_eq_nat @ zero_zero_nat @ A) => ((ord_less_eq_nat @ zero_zero_nat @ B) => ((ord_less_nat @ zero_zero_nat @ N) => ((ord_less_eq_nat @ (power_power_nat @ A @ N) @ (power_power_nat @ B @ N)) = (ord_less_eq_nat @ A @ B)))))))). % power_mono_iff
thf(fact_163_power__mono__iff, axiom,
    ((![A : real, B : real, N : nat]: ((ord_less_eq_real @ zero_zero_real @ A) => ((ord_less_eq_real @ zero_zero_real @ B) => ((ord_less_nat @ zero_zero_nat @ N) => ((ord_less_eq_real @ (power_power_real @ A @ N) @ (power_power_real @ B @ N)) = (ord_less_eq_real @ A @ B)))))))). % power_mono_iff
thf(fact_164_nat__power__less__imp__less, axiom,
    ((![I : nat, M2 : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ I) => ((ord_less_nat @ (power_power_nat @ I @ M2) @ (power_power_nat @ I @ N)) => (ord_less_nat @ M2 @ N)))))). % nat_power_less_imp_less
thf(fact_165_complex__mod__triangle__sub, axiom,
    ((![W : complex, Z3 : complex]: (ord_less_eq_real @ (real_V638595069omplex @ W) @ (plus_plus_real @ (real_V638595069omplex @ (plus_plus_complex @ W @ Z3)) @ (real_V638595069omplex @ Z3)))))). % complex_mod_triangle_sub
thf(fact_166_zero__power, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((power_power_complex @ zero_zero_complex @ N) = zero_zero_complex))))). % zero_power
thf(fact_167_zero__power, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((power_power_real @ zero_zero_real @ N) = zero_zero_real))))). % zero_power
thf(fact_168_zero__power, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((power_power_nat @ zero_zero_nat @ N) = zero_zero_nat))))). % zero_power
thf(fact_169_reduce__poly__simple, axiom,
    ((![B : complex, N : nat]: ((~ ((B = zero_zero_complex))) => ((~ ((N = zero_zero_nat))) => (?[Z2 : complex]: (ord_less_real @ (real_V638595069omplex @ (plus_plus_complex @ one_one_complex @ (times_times_complex @ B @ (power_power_complex @ Z2 @ N)))) @ one_one_real))))))). % reduce_poly_simple
thf(fact_170_power__eq__imp__eq__base, axiom,
    ((![A : nat, N : nat, B : nat]: (((power_power_nat @ A @ N) = (power_power_nat @ B @ N)) => ((ord_less_eq_nat @ zero_zero_nat @ A) => ((ord_less_eq_nat @ zero_zero_nat @ B) => ((ord_less_nat @ zero_zero_nat @ N) => (A = B)))))))). % power_eq_imp_eq_base
thf(fact_171_power__eq__imp__eq__base, axiom,
    ((![A : real, N : nat, B : real]: (((power_power_real @ A @ N) = (power_power_real @ B @ N)) => ((ord_less_eq_real @ zero_zero_real @ A) => ((ord_less_eq_real @ zero_zero_real @ B) => ((ord_less_nat @ zero_zero_nat @ N) => (A = B)))))))). % power_eq_imp_eq_base
thf(fact_172_power__eq__iff__eq__base, axiom,
    ((![N : nat, A : nat, B : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((ord_less_eq_nat @ zero_zero_nat @ A) => ((ord_less_eq_nat @ zero_zero_nat @ B) => (((power_power_nat @ A @ N) = (power_power_nat @ B @ N)) = (A = B)))))))). % power_eq_iff_eq_base
thf(fact_173_power__eq__iff__eq__base, axiom,
    ((![N : nat, A : real, B : real]: ((ord_less_nat @ zero_zero_nat @ N) => ((ord_less_eq_real @ zero_zero_real @ A) => ((ord_less_eq_real @ zero_zero_real @ B) => (((power_power_real @ A @ N) = (power_power_real @ B @ N)) = (A = B)))))))). % power_eq_iff_eq_base
thf(fact_174_self__le__power, axiom,
    ((![A : nat, N : nat]: ((ord_less_eq_nat @ one_one_nat @ A) => ((ord_less_nat @ zero_zero_nat @ N) => (ord_less_eq_nat @ A @ (power_power_nat @ A @ N))))))). % self_le_power
thf(fact_175_self__le__power, axiom,
    ((![A : real, N : nat]: ((ord_less_eq_real @ one_one_real @ A) => ((ord_less_nat @ zero_zero_nat @ N) => (ord_less_eq_real @ A @ (power_power_real @ A @ N))))))). % self_le_power
thf(fact_176_realpow__pos__nth, axiom,
    ((![N : nat, A : real]: ((ord_less_nat @ zero_zero_nat @ N) => ((ord_less_real @ zero_zero_real @ A) => (?[R2 : real]: ((ord_less_real @ zero_zero_real @ R2) & ((power_power_real @ R2 @ N) = A)))))))). % realpow_pos_nth
thf(fact_177_realpow__pos__nth__unique, axiom,
    ((![N : nat, A : real]: ((ord_less_nat @ zero_zero_nat @ N) => ((ord_less_real @ zero_zero_real @ A) => (?[X : real]: (((ord_less_real @ zero_zero_real @ X) & ((power_power_real @ X @ N) = A)) & (![Y : real]: (((ord_less_real @ zero_zero_real @ Y) & ((power_power_real @ Y @ N) = A)) => (Y = X)))))))))). % realpow_pos_nth_unique
thf(fact_178_linordered__field__no__lb, axiom,
    ((![X4 : real]: (?[Y4 : real]: (ord_less_real @ Y4 @ X4))))). % linordered_field_no_lb
thf(fact_179_linordered__field__no__ub, axiom,
    ((![X4 : real]: (?[X_12 : real]: (ord_less_real @ X4 @ X_12))))). % linordered_field_no_ub
thf(fact_180_power__eq__imp__eq__norm, axiom,
    ((![W : real, N : nat, Z3 : real]: (((power_power_real @ W @ N) = (power_power_real @ Z3 @ N)) => ((ord_less_nat @ zero_zero_nat @ N) => ((real_V646646907m_real @ W) = (real_V646646907m_real @ Z3))))))). % power_eq_imp_eq_norm
thf(fact_181_power__eq__imp__eq__norm, axiom,
    ((![W : complex, N : nat, Z3 : complex]: (((power_power_complex @ W @ N) = (power_power_complex @ Z3 @ N)) => ((ord_less_nat @ zero_zero_nat @ N) => ((real_V638595069omplex @ W) = (real_V638595069omplex @ Z3))))))). % power_eq_imp_eq_norm
thf(fact_182_power__mult, axiom,
    ((![A : complex, M2 : nat, N : nat]: ((power_power_complex @ A @ (times_times_nat @ M2 @ N)) = (power_power_complex @ (power_power_complex @ A @ M2) @ N))))). % power_mult
thf(fact_183_power__mult, axiom,
    ((![A : real, M2 : nat, N : nat]: ((power_power_real @ A @ (times_times_nat @ M2 @ N)) = (power_power_real @ (power_power_real @ A @ M2) @ N))))). % power_mult
thf(fact_184_power__mult, axiom,
    ((![A : nat, M2 : nat, N : nat]: ((power_power_nat @ A @ (times_times_nat @ M2 @ N)) = (power_power_nat @ (power_power_nat @ A @ M2) @ N))))). % power_mult
thf(fact_185_inverse__eq__imp__eq, axiom,
    ((![A : real, B : real]: (((inverse_inverse_real @ A) = (inverse_inverse_real @ B)) => (A = B))))). % inverse_eq_imp_eq
thf(fact_186_power__not__zero, axiom,
    ((![A : complex, N : nat]: ((~ ((A = zero_zero_complex))) => (~ (((power_power_complex @ A @ N) = zero_zero_complex))))))). % power_not_zero
thf(fact_187_power__not__zero, axiom,
    ((![A : real, N : nat]: ((~ ((A = zero_zero_real))) => (~ (((power_power_real @ A @ N) = zero_zero_real))))))). % power_not_zero
thf(fact_188_power__not__zero, axiom,
    ((![A : nat, N : nat]: ((~ ((A = zero_zero_nat))) => (~ (((power_power_nat @ A @ N) = zero_zero_nat))))))). % power_not_zero
thf(fact_189_power__commutes, axiom,
    ((![A : nat, N : nat]: ((times_times_nat @ (power_power_nat @ A @ N) @ A) = (times_times_nat @ A @ (power_power_nat @ A @ N)))))). % power_commutes
thf(fact_190_power__commutes, axiom,
    ((![A : complex, N : nat]: ((times_times_complex @ (power_power_complex @ A @ N) @ A) = (times_times_complex @ A @ (power_power_complex @ A @ N)))))). % power_commutes
thf(fact_191_power__commutes, axiom,
    ((![A : real, N : nat]: ((times_times_real @ (power_power_real @ A @ N) @ A) = (times_times_real @ A @ (power_power_real @ A @ N)))))). % power_commutes
thf(fact_192_power__mult__distrib, axiom,
    ((![A : nat, B : nat, N : nat]: ((power_power_nat @ (times_times_nat @ A @ B) @ N) = (times_times_nat @ (power_power_nat @ A @ N) @ (power_power_nat @ B @ N)))))). % power_mult_distrib
thf(fact_193_power__mult__distrib, axiom,
    ((![A : complex, B : complex, N : nat]: ((power_power_complex @ (times_times_complex @ A @ B) @ N) = (times_times_complex @ (power_power_complex @ A @ N) @ (power_power_complex @ B @ N)))))). % power_mult_distrib
thf(fact_194_power__mult__distrib, axiom,
    ((![A : real, B : real, N : nat]: ((power_power_real @ (times_times_real @ A @ B) @ N) = (times_times_real @ (power_power_real @ A @ N) @ (power_power_real @ B @ N)))))). % power_mult_distrib
thf(fact_195_power__commuting__commutes, axiom,
    ((![X3 : nat, Y2 : nat, N : nat]: (((times_times_nat @ X3 @ Y2) = (times_times_nat @ Y2 @ X3)) => ((times_times_nat @ (power_power_nat @ X3 @ N) @ Y2) = (times_times_nat @ Y2 @ (power_power_nat @ X3 @ N))))))). % power_commuting_commutes
thf(fact_196_power__commuting__commutes, axiom,
    ((![X3 : complex, Y2 : complex, N : nat]: (((times_times_complex @ X3 @ Y2) = (times_times_complex @ Y2 @ X3)) => ((times_times_complex @ (power_power_complex @ X3 @ N) @ Y2) = (times_times_complex @ Y2 @ (power_power_complex @ X3 @ N))))))). % power_commuting_commutes
thf(fact_197_power__commuting__commutes, axiom,
    ((![X3 : real, Y2 : real, N : nat]: (((times_times_real @ X3 @ Y2) = (times_times_real @ Y2 @ X3)) => ((times_times_real @ (power_power_real @ X3 @ N) @ Y2) = (times_times_real @ Y2 @ (power_power_real @ X3 @ N))))))). % power_commuting_commutes
thf(fact_198_nonzero__imp__inverse__nonzero, axiom,
    ((![A : complex]: ((~ ((A = zero_zero_complex))) => (~ (((invers502456322omplex @ A) = zero_zero_complex))))))). % nonzero_imp_inverse_nonzero
thf(fact_199_nonzero__imp__inverse__nonzero, axiom,
    ((![A : real]: ((~ ((A = zero_zero_real))) => (~ (((inverse_inverse_real @ A) = zero_zero_real))))))). % nonzero_imp_inverse_nonzero
thf(fact_200_nonzero__inverse__inverse__eq, axiom,
    ((![A : complex]: ((~ ((A = zero_zero_complex))) => ((invers502456322omplex @ (invers502456322omplex @ A)) = A))))). % nonzero_inverse_inverse_eq
thf(fact_201_nonzero__inverse__inverse__eq, axiom,
    ((![A : real]: ((~ ((A = zero_zero_real))) => ((inverse_inverse_real @ (inverse_inverse_real @ A)) = A))))). % nonzero_inverse_inverse_eq
thf(fact_202_nonzero__inverse__eq__imp__eq, axiom,
    ((![A : complex, B : complex]: (((invers502456322omplex @ A) = (invers502456322omplex @ B)) => ((~ ((A = zero_zero_complex))) => ((~ ((B = zero_zero_complex))) => (A = B))))))). % nonzero_inverse_eq_imp_eq
thf(fact_203_nonzero__inverse__eq__imp__eq, axiom,
    ((![A : real, B : real]: (((inverse_inverse_real @ A) = (inverse_inverse_real @ B)) => ((~ ((A = zero_zero_real))) => ((~ ((B = zero_zero_real))) => (A = B))))))). % nonzero_inverse_eq_imp_eq
thf(fact_204_inverse__zero__imp__zero, axiom,
    ((![A : complex]: (((invers502456322omplex @ A) = zero_zero_complex) => (A = zero_zero_complex))))). % inverse_zero_imp_zero
thf(fact_205_inverse__zero__imp__zero, axiom,
    ((![A : real]: (((inverse_inverse_real @ A) = zero_zero_real) => (A = zero_zero_real))))). % inverse_zero_imp_zero
thf(fact_206_field__class_Ofield__inverse__zero, axiom,
    (((invers502456322omplex @ zero_zero_complex) = zero_zero_complex))). % field_class.field_inverse_zero
thf(fact_207_field__class_Ofield__inverse__zero, axiom,
    (((inverse_inverse_real @ zero_zero_real) = zero_zero_real))). % field_class.field_inverse_zero
thf(fact_208_mult__commute__imp__mult__inverse__commute, axiom,
    ((![Y2 : complex, X3 : complex]: (((times_times_complex @ Y2 @ X3) = (times_times_complex @ X3 @ Y2)) => ((times_times_complex @ (invers502456322omplex @ Y2) @ X3) = (times_times_complex @ X3 @ (invers502456322omplex @ Y2))))))). % mult_commute_imp_mult_inverse_commute
thf(fact_209_mult__commute__imp__mult__inverse__commute, axiom,
    ((![Y2 : real, X3 : real]: (((times_times_real @ Y2 @ X3) = (times_times_real @ X3 @ Y2)) => ((times_times_real @ (inverse_inverse_real @ Y2) @ X3) = (times_times_real @ X3 @ (inverse_inverse_real @ Y2))))))). % mult_commute_imp_mult_inverse_commute
thf(fact_210_power__inverse, axiom,
    ((![A : complex, N : nat]: ((power_power_complex @ (invers502456322omplex @ A) @ N) = (invers502456322omplex @ (power_power_complex @ A @ N)))))). % power_inverse
thf(fact_211_power__inverse, axiom,
    ((![A : real, N : nat]: ((power_power_real @ (inverse_inverse_real @ A) @ N) = (inverse_inverse_real @ (power_power_real @ A @ N)))))). % power_inverse
thf(fact_212_power__mono, axiom,
    ((![A : nat, B : nat, N : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_nat @ zero_zero_nat @ A) => (ord_less_eq_nat @ (power_power_nat @ A @ N) @ (power_power_nat @ B @ N))))))). % power_mono
thf(fact_213_power__mono, axiom,
    ((![A : real, B : real, N : nat]: ((ord_less_eq_real @ A @ B) => ((ord_less_eq_real @ zero_zero_real @ A) => (ord_less_eq_real @ (power_power_real @ A @ N) @ (power_power_real @ B @ N))))))). % power_mono
thf(fact_214_zero__le__power, axiom,
    ((![A : nat, N : nat]: ((ord_less_eq_nat @ zero_zero_nat @ A) => (ord_less_eq_nat @ zero_zero_nat @ (power_power_nat @ A @ N)))))). % zero_le_power
thf(fact_215_zero__le__power, axiom,
    ((![A : real, N : nat]: ((ord_less_eq_real @ zero_zero_real @ A) => (ord_less_eq_real @ zero_zero_real @ (power_power_real @ A @ N)))))). % zero_le_power
thf(fact_216_zero__less__power, axiom,
    ((![A : real, N : nat]: ((ord_less_real @ zero_zero_real @ A) => (ord_less_real @ zero_zero_real @ (power_power_real @ A @ N)))))). % zero_less_power
thf(fact_217_zero__less__power, axiom,
    ((![A : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ A) => (ord_less_nat @ zero_zero_nat @ (power_power_nat @ A @ N)))))). % zero_less_power
thf(fact_218_one__le__power, axiom,
    ((![A : nat, N : nat]: ((ord_less_eq_nat @ one_one_nat @ A) => (ord_less_eq_nat @ one_one_nat @ (power_power_nat @ A @ N)))))). % one_le_power
thf(fact_219_one__le__power, axiom,
    ((![A : real, N : nat]: ((ord_less_eq_real @ one_one_real @ A) => (ord_less_eq_real @ one_one_real @ (power_power_real @ A @ N)))))). % one_le_power
thf(fact_220_power__increasing, axiom,
    ((![N : nat, N2 : nat, A : nat]: ((ord_less_eq_nat @ N @ N2) => ((ord_less_eq_nat @ one_one_nat @ A) => (ord_less_eq_nat @ (power_power_nat @ A @ N) @ (power_power_nat @ A @ N2))))))). % power_increasing
thf(fact_221_power__increasing, axiom,
    ((![N : nat, N2 : nat, A : real]: ((ord_less_eq_nat @ N @ N2) => ((ord_less_eq_real @ one_one_real @ A) => (ord_less_eq_real @ (power_power_real @ A @ N) @ (power_power_real @ A @ N2))))))). % power_increasing
thf(fact_222_power__less__imp__less__exp, axiom,
    ((![A : real, M2 : nat, N : nat]: ((ord_less_real @ one_one_real @ A) => ((ord_less_real @ (power_power_real @ A @ M2) @ (power_power_real @ A @ N)) => (ord_less_nat @ M2 @ N)))))). % power_less_imp_less_exp
thf(fact_223_power__less__imp__less__exp, axiom,
    ((![A : nat, M2 : nat, N : nat]: ((ord_less_nat @ one_one_nat @ A) => ((ord_less_nat @ (power_power_nat @ A @ M2) @ (power_power_nat @ A @ N)) => (ord_less_nat @ M2 @ N)))))). % power_less_imp_less_exp
thf(fact_224_power__strict__increasing, axiom,
    ((![N : nat, N2 : nat, A : real]: ((ord_less_nat @ N @ N2) => ((ord_less_real @ one_one_real @ A) => (ord_less_real @ (power_power_real @ A @ N) @ (power_power_real @ A @ N2))))))). % power_strict_increasing
thf(fact_225_power__strict__increasing, axiom,
    ((![N : nat, N2 : nat, A : nat]: ((ord_less_nat @ N @ N2) => ((ord_less_nat @ one_one_nat @ A) => (ord_less_nat @ (power_power_nat @ A @ N) @ (power_power_nat @ A @ N2))))))). % power_strict_increasing
thf(fact_226_left__right__inverse__power, axiom,
    ((![X3 : nat, Y2 : nat, N : nat]: (((times_times_nat @ X3 @ Y2) = one_one_nat) => ((times_times_nat @ (power_power_nat @ X3 @ N) @ (power_power_nat @ Y2 @ N)) = one_one_nat))))). % left_right_inverse_power
thf(fact_227_left__right__inverse__power, axiom,
    ((![X3 : complex, Y2 : complex, N : nat]: (((times_times_complex @ X3 @ Y2) = one_one_complex) => ((times_times_complex @ (power_power_complex @ X3 @ N) @ (power_power_complex @ Y2 @ N)) = one_one_complex))))). % left_right_inverse_power
thf(fact_228_left__right__inverse__power, axiom,
    ((![X3 : real, Y2 : real, N : nat]: (((times_times_real @ X3 @ Y2) = one_one_real) => ((times_times_real @ (power_power_real @ X3 @ N) @ (power_power_real @ Y2 @ N)) = one_one_real))))). % left_right_inverse_power
thf(fact_229_positive__imp__inverse__positive, axiom,
    ((![A : real]: ((ord_less_real @ zero_zero_real @ A) => (ord_less_real @ zero_zero_real @ (inverse_inverse_real @ A)))))). % positive_imp_inverse_positive
thf(fact_230_negative__imp__inverse__negative, axiom,
    ((![A : real]: ((ord_less_real @ A @ zero_zero_real) => (ord_less_real @ (inverse_inverse_real @ A) @ zero_zero_real))))). % negative_imp_inverse_negative
thf(fact_231_inverse__positive__imp__positive, axiom,
    ((![A : real]: ((ord_less_real @ zero_zero_real @ (inverse_inverse_real @ A)) => ((~ ((A = zero_zero_real))) => (ord_less_real @ zero_zero_real @ A)))))). % inverse_positive_imp_positive
thf(fact_232_inverse__negative__imp__negative, axiom,
    ((![A : real]: ((ord_less_real @ (inverse_inverse_real @ A) @ zero_zero_real) => ((~ ((A = zero_zero_real))) => (ord_less_real @ A @ zero_zero_real)))))). % inverse_negative_imp_negative
thf(fact_233_less__imp__inverse__less__neg, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ B) => ((ord_less_real @ B @ zero_zero_real) => (ord_less_real @ (inverse_inverse_real @ B) @ (inverse_inverse_real @ A))))))). % less_imp_inverse_less_neg
thf(fact_234_inverse__less__imp__less__neg, axiom,
    ((![A : real, B : real]: ((ord_less_real @ (inverse_inverse_real @ A) @ (inverse_inverse_real @ B)) => ((ord_less_real @ B @ zero_zero_real) => (ord_less_real @ B @ A)))))). % inverse_less_imp_less_neg
thf(fact_235_less__imp__inverse__less, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ B) => ((ord_less_real @ zero_zero_real @ A) => (ord_less_real @ (inverse_inverse_real @ B) @ (inverse_inverse_real @ A))))))). % less_imp_inverse_less
thf(fact_236_inverse__less__imp__less, axiom,
    ((![A : real, B : real]: ((ord_less_real @ (inverse_inverse_real @ A) @ (inverse_inverse_real @ B)) => ((ord_less_real @ zero_zero_real @ A) => (ord_less_real @ B @ A)))))). % inverse_less_imp_less
thf(fact_237_power__0, axiom,
    ((![A : complex]: ((power_power_complex @ A @ zero_zero_nat) = one_one_complex)))). % power_0
thf(fact_238_power__0, axiom,
    ((![A : real]: ((power_power_real @ A @ zero_zero_nat) = one_one_real)))). % power_0
thf(fact_239_power__0, axiom,
    ((![A : nat]: ((power_power_nat @ A @ zero_zero_nat) = one_one_nat)))). % power_0
thf(fact_240_nonzero__inverse__mult__distrib, axiom,
    ((![A : complex, B : complex]: ((~ ((A = zero_zero_complex))) => ((~ ((B = zero_zero_complex))) => ((invers502456322omplex @ (times_times_complex @ A @ B)) = (times_times_complex @ (invers502456322omplex @ B) @ (invers502456322omplex @ A)))))))). % nonzero_inverse_mult_distrib
thf(fact_241_nonzero__inverse__mult__distrib, axiom,
    ((![A : real, B : real]: ((~ ((A = zero_zero_real))) => ((~ ((B = zero_zero_real))) => ((inverse_inverse_real @ (times_times_real @ A @ B)) = (times_times_real @ (inverse_inverse_real @ B) @ (inverse_inverse_real @ A)))))))). % nonzero_inverse_mult_distrib

% Conjectures (1)
thf(conj_0, conjecture,
    ((ord_less_real @ (real_V638595069omplex @ (times_times_complex @ (times_times_complex @ (power_power_complex @ (times_times_complex @ (real_V306493662omplex @ t) @ w) @ k) @ (times_times_complex @ (real_V306493662omplex @ t) @ w)) @ (poly_complex2 @ s @ (times_times_complex @ (real_V306493662omplex @ t) @ w)))) @ (power_power_real @ t @ k)))).
