% 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_797__5375724_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:33:14.019

% 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 (58)
thf(sy_c_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Oconstant_001t__Complex__Ocomplex_001t__Complex__Ocomplex, type,
    fundam1158420650omplex : (complex > complex) > $o).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Real__Oreal, type,
    abs_abs_real : real > real).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Complex__Ocomplex, type,
    minus_minus_complex : complex > complex > complex).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    minus_174331535omplex : poly_complex > poly_complex > poly_complex).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Polynomial__Opoly_It__Real__Oreal_J, type,
    minus_240770701y_real : poly_real > poly_real > poly_real).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal, type,
    minus_minus_real : real > real > real).
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_Ouminus__class_Ouminus_001t__Complex__Ocomplex, type,
    uminus1204672759omplex : complex > complex).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    uminus1138659839omplex : poly_complex > poly_complex).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Polynomial__Opoly_It__Real__Oreal_J, type,
    uminus1613791741y_real : poly_real > poly_real).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal, type,
    uminus_uminus_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_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 (249)
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_kas_I2_J, axiom,
    ((~ ((k = zero_zero_nat))))). % kas(2)
thf(fact_3_kas_I1_J, axiom,
    ((~ ((a = zero_zero_complex))))). % kas(1)
thf(fact_4_t_I1_J, axiom,
    ((ord_less_real @ zero_zero_real @ t))). % t(1)
thf(fact_5_tw, axiom,
    ((ord_less_eq_real @ (real_V638595069omplex @ (times_times_complex @ (real_V306493662omplex @ t) @ w)) @ (real_V638595069omplex @ w)))). % tw
thf(fact_6_th12, axiom,
    ((ord_less_real @ (plus_plus_real @ (abs_abs_real @ (minus_minus_real @ one_one_real @ (power_power_real @ t @ k))) @ (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))))) @ one_one_real))). % th12
thf(fact_7__092_060open_0621_A_L_A_Icomplex__of__real_At_A_K_Aw_J_A_094_Ak_A_K_A_Ia_A_L_Acomplex__of__real_At_A_K_Aw_A_K_Apoly_As_A_Icomplex__of__real_At_A_K_Aw_J_J_A_061_A1_A_L_Acomplex__of__real_At_A_094_Ak_A_K_A_Iw_A_094_Ak_A_K_Aa_J_A_L_A_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_092_060close_062, axiom,
    (((plus_plus_complex @ one_one_complex @ (times_times_complex @ (power_power_complex @ (times_times_complex @ (real_V306493662omplex @ t) @ w) @ k) @ (plus_plus_complex @ a @ (times_times_complex @ (times_times_complex @ (real_V306493662omplex @ t) @ w) @ (poly_complex2 @ s @ (times_times_complex @ (real_V306493662omplex @ t) @ w)))))) = (plus_plus_complex @ (plus_plus_complex @ one_one_complex @ (times_times_complex @ (power_power_complex @ (real_V306493662omplex @ t) @ k) @ (times_times_complex @ (power_power_complex @ w @ k) @ a))) @ (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))))))). % \<open>1 + (complex_of_real t * w) ^ k * (a + complex_of_real t * w * poly s (complex_of_real t * w)) = 1 + complex_of_real t ^ k * (w ^ k * a) + (complex_of_real t * w) ^ k * (complex_of_real t * w) * poly s (complex_of_real t * w)\<close>
thf(fact_8_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_9_th120, axiom,
    ((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)))). % th120
thf(fact_10_th121, axiom,
    ((ord_less_eq_real @ (power_power_real @ t @ k) @ one_one_real))). % th121
thf(fact_11_w, axiom,
    (((plus_plus_complex @ one_one_complex @ (times_times_complex @ (power_power_complex @ w @ k) @ a)) = zero_zero_complex))). % w
thf(fact_12_th11, axiom,
    ((ord_less_eq_real @ (real_V638595069omplex @ (plus_plus_complex @ one_one_complex @ (times_times_complex @ (power_power_complex @ (times_times_complex @ (real_V306493662omplex @ t) @ w) @ k) @ (plus_plus_complex @ a @ (times_times_complex @ (times_times_complex @ (real_V306493662omplex @ t) @ w) @ (poly_complex2 @ s @ (times_times_complex @ (real_V306493662omplex @ t) @ w))))))) @ (plus_plus_real @ (abs_abs_real @ (minus_minus_real @ one_one_real @ (power_power_real @ t @ k))) @ (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)))))))). % th11
thf(fact_13__092_060open_0621_A_L_Acomplex__of__real_At_A_094_Ak_A_K_A_Iw_A_094_Ak_A_K_Aa_J_A_L_A_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_A_061_Acomplex__of__real_A_I1_A_N_At_A_094_Ak_J_A_L_A_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_092_060close_062, axiom,
    (((plus_plus_complex @ (plus_plus_complex @ one_one_complex @ (times_times_complex @ (power_power_complex @ (real_V306493662omplex @ t) @ k) @ (times_times_complex @ (power_power_complex @ w @ k) @ a))) @ (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)))) = (plus_plus_complex @ (real_V306493662omplex @ (minus_minus_real @ one_one_real @ (power_power_real @ t @ k))) @ (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))))))). % \<open>1 + complex_of_real t ^ k * (w ^ k * a) + (complex_of_real t * w) ^ k * (complex_of_real t * w) * poly s (complex_of_real t * w) = complex_of_real (1 - t ^ k) + (complex_of_real t * w) ^ k * (complex_of_real t * w) * poly s (complex_of_real t * w)\<close>
thf(fact_14__092_060open_062cmod_A_I1_A_L_A_Icomplex__of__real_At_A_K_Aw_J_A_094_Ak_A_K_A_Ia_A_L_Acomplex__of__real_At_A_K_Aw_A_K_Apoly_As_A_Icomplex__of__real_At_A_K_Aw_J_J_J_A_061_Acmod_A_Icomplex__of__real_A_I1_A_N_At_A_094_Ak_J_A_L_A_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_092_060close_062, axiom,
    (((real_V638595069omplex @ (plus_plus_complex @ one_one_complex @ (times_times_complex @ (power_power_complex @ (times_times_complex @ (real_V306493662omplex @ t) @ w) @ k) @ (plus_plus_complex @ a @ (times_times_complex @ (times_times_complex @ (real_V306493662omplex @ t) @ w) @ (poly_complex2 @ s @ (times_times_complex @ (real_V306493662omplex @ t) @ w))))))) = (real_V638595069omplex @ (plus_plus_complex @ (real_V306493662omplex @ (minus_minus_real @ one_one_real @ (power_power_real @ t @ k))) @ (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)))))))). % \<open>cmod (1 + (complex_of_real t * w) ^ k * (a + complex_of_real t * w * poly s (complex_of_real t * w))) = cmod (complex_of_real (1 - t ^ k) + (complex_of_real t * w) ^ k * (complex_of_real t * w) * poly s (complex_of_real t * w))\<close>
thf(fact_15_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_16__092_060open_062cmod_A_Icomplex__of__real_A_I1_A_N_At_A_094_Ak_J_A_L_A_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_Acmod_A_Icomplex__of__real_A_I1_A_N_At_A_094_Ak_J_J_A_L_Acmod_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_092_060close_062, axiom,
    ((ord_less_eq_real @ (real_V638595069omplex @ (plus_plus_complex @ (real_V306493662omplex @ (minus_minus_real @ one_one_real @ (power_power_real @ t @ k))) @ (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))))) @ (plus_plus_real @ (real_V638595069omplex @ (real_V306493662omplex @ (minus_minus_real @ one_one_real @ (power_power_real @ t @ k)))) @ (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)))))))). % \<open>cmod (complex_of_real (1 - t ^ k) + (complex_of_real t * w) ^ k * (complex_of_real t * w) * poly s (complex_of_real t * w)) \<le> cmod (complex_of_real (1 - t ^ k)) + cmod ((complex_of_real t * w) ^ k * (complex_of_real t * w) * poly s (complex_of_real t * w))\<close>
thf(fact_17__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062w_O_A1_A_L_Aw_A_094_Ak_A_K_Aa_A_061_A0_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom,
    ((~ ((![W : complex]: (~ (((plus_plus_complex @ one_one_complex @ (times_times_complex @ (power_power_complex @ W @ k) @ a)) = zero_zero_complex)))))))). % \<open>\<And>thesis. (\<And>w. 1 + w ^ k * a = 0 \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_18_of__real__1, axiom,
    (((real_V1205483228l_real @ one_one_real) = one_one_real))). % of_real_1
thf(fact_19_of__real__1, axiom,
    (((real_V306493662omplex @ one_one_real) = one_one_complex))). % of_real_1
thf(fact_20_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_21_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_22_norm__one, axiom,
    (((real_V638595069omplex @ one_one_complex) = one_one_real))). % norm_one
thf(fact_23_norm__one, axiom,
    (((real_V646646907m_real @ one_one_real) = one_one_real))). % norm_one
thf(fact_24_power__inject__exp, axiom,
    ((![A : real, M : nat, N : nat]: ((ord_less_real @ one_one_real @ A) => (((power_power_real @ A @ M) = (power_power_real @ A @ N)) = (M = N)))))). % power_inject_exp
thf(fact_25_power__inject__exp, axiom,
    ((![A : nat, M : nat, N : nat]: ((ord_less_nat @ one_one_nat @ A) => (((power_power_nat @ A @ M) = (power_power_nat @ A @ N)) = (M = N)))))). % power_inject_exp
thf(fact_26_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_27_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_28_wm1, axiom,
    (((times_times_complex @ (power_power_complex @ w @ k) @ a) = (uminus1204672759omplex @ one_one_complex)))). % wm1
thf(fact_29_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_30_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_31_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_32_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_33_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_34_poly__1, axiom,
    ((![X3 : complex]: ((poly_complex2 @ one_one_poly_complex @ X3) = one_one_complex)))). % poly_1
thf(fact_35_poly__1, axiom,
    ((![X3 : real]: ((poly_real2 @ one_one_poly_real @ X3) = one_one_real)))). % poly_1
thf(fact_36_poly__1, axiom,
    ((![X3 : nat]: ((poly_nat2 @ one_one_poly_nat @ X3) = one_one_nat)))). % poly_1
thf(fact_37_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_38_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_39_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_40_m_I1_J, axiom,
    ((ord_less_real @ zero_zero_real @ m))). % m(1)
thf(fact_41_w0, axiom,
    ((~ ((w = zero_zero_complex))))). % w0
thf(fact_42__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_43_power__one__right, axiom,
    ((![A : complex]: ((power_power_complex @ A @ one_one_nat) = A)))). % power_one_right
thf(fact_44_power__one__right, axiom,
    ((![A : real]: ((power_power_real @ A @ one_one_nat) = A)))). % power_one_right
thf(fact_45_power__one__right, axiom,
    ((![A : nat]: ((power_power_nat @ A @ one_one_nat) = A)))). % power_one_right
thf(fact_46_of__real__eq__iff, axiom,
    ((![X3 : real, Y2 : real]: (((real_V306493662omplex @ X3) = (real_V306493662omplex @ Y2)) = (X3 = Y2))))). % of_real_eq_iff
thf(fact_47__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_48_ath, axiom,
    ((![X3 : real, T : real]: ((ord_less_eq_real @ zero_zero_real @ X3) => ((ord_less_real @ X3 @ T) => ((ord_less_eq_real @ T @ one_one_real) => (ord_less_real @ (plus_plus_real @ (abs_abs_real @ (minus_minus_real @ one_one_real @ T)) @ X3) @ one_one_real))))))). % ath
thf(fact_49_power__one, axiom,
    ((![N : nat]: ((power_power_complex @ one_one_complex @ N) = one_one_complex)))). % power_one
thf(fact_50_power__one, axiom,
    ((![N : nat]: ((power_power_real @ one_one_real @ N) = one_one_real)))). % power_one
thf(fact_51_power__one, axiom,
    ((![N : nat]: ((power_power_nat @ one_one_nat @ N) = one_one_nat)))). % power_one
thf(fact_52_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_53_norm__minus__cancel, axiom,
    ((![X3 : complex]: ((real_V638595069omplex @ (uminus1204672759omplex @ X3)) = (real_V638595069omplex @ X3))))). % norm_minus_cancel
thf(fact_54_norm__minus__cancel, axiom,
    ((![X3 : real]: ((real_V646646907m_real @ (uminus_uminus_real @ X3)) = (real_V646646907m_real @ X3))))). % norm_minus_cancel
thf(fact_55_poly__0, axiom,
    ((![X3 : nat]: ((poly_nat2 @ zero_zero_poly_nat @ X3) = zero_zero_nat)))). % poly_0
thf(fact_56_poly__0, axiom,
    ((![X3 : complex]: ((poly_complex2 @ zero_z1746442943omplex @ X3) = zero_zero_complex)))). % poly_0
thf(fact_57_poly__0, axiom,
    ((![X3 : real]: ((poly_real2 @ zero_zero_poly_real @ X3) = zero_zero_real)))). % poly_0
thf(fact_58_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_59_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_60_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_61_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_62_of__real__minus, axiom,
    ((![X3 : real]: ((real_V306493662omplex @ (uminus_uminus_real @ X3)) = (uminus1204672759omplex @ (real_V306493662omplex @ X3)))))). % of_real_minus
thf(fact_63_minus__of__real__eq__of__real__iff, axiom,
    ((![X3 : real, Y2 : real]: (((uminus1204672759omplex @ (real_V306493662omplex @ X3)) = (real_V306493662omplex @ Y2)) = ((uminus_uminus_real @ X3) = Y2))))). % minus_of_real_eq_of_real_iff
thf(fact_64_of__real__eq__minus__of__real__iff, axiom,
    ((![X3 : real, Y2 : real]: (((real_V306493662omplex @ X3) = (uminus1204672759omplex @ (real_V306493662omplex @ Y2))) = (X3 = (uminus_uminus_real @ Y2)))))). % of_real_eq_minus_of_real_iff
thf(fact_65_poly__diff, axiom,
    ((![P2 : poly_complex, Q : poly_complex, X3 : complex]: ((poly_complex2 @ (minus_174331535omplex @ P2 @ Q) @ X3) = (minus_minus_complex @ (poly_complex2 @ P2 @ X3) @ (poly_complex2 @ Q @ X3)))))). % poly_diff
thf(fact_66_poly__diff, axiom,
    ((![P2 : poly_real, Q : poly_real, X3 : real]: ((poly_real2 @ (minus_240770701y_real @ P2 @ Q) @ X3) = (minus_minus_real @ (poly_real2 @ P2 @ X3) @ (poly_real2 @ Q @ X3)))))). % poly_diff
thf(fact_67_poly__minus, axiom,
    ((![P2 : poly_real, X3 : real]: ((poly_real2 @ (uminus1613791741y_real @ P2) @ X3) = (uminus_uminus_real @ (poly_real2 @ P2 @ X3)))))). % poly_minus
thf(fact_68_poly__minus, axiom,
    ((![P2 : poly_complex, X3 : complex]: ((poly_complex2 @ (uminus1138659839omplex @ P2) @ X3) = (uminus1204672759omplex @ (poly_complex2 @ P2 @ X3)))))). % poly_minus
thf(fact_69_abs__norm__cancel, axiom,
    ((![A : complex]: ((abs_abs_real @ (real_V638595069omplex @ A)) = (real_V638595069omplex @ A))))). % abs_norm_cancel
thf(fact_70_abs__norm__cancel, axiom,
    ((![A : real]: ((abs_abs_real @ (real_V646646907m_real @ A)) = (real_V646646907m_real @ A))))). % abs_norm_cancel
thf(fact_71__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,
    ((~ ((![M2 : real]: ((ord_less_real @ zero_zero_real @ M2) => (~ ((![Z : complex]: ((ord_less_eq_real @ (real_V638595069omplex @ Z) @ (real_V638595069omplex @ w)) => (ord_less_eq_real @ (real_V638595069omplex @ (poly_complex2 @ s @ Z)) @ M2))))))))))). % \<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_72__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,
    ((?[M2 : real]: ((ord_less_real @ zero_zero_real @ M2) & (![Z : complex]: ((ord_less_eq_real @ (real_V638595069omplex @ Z) @ (real_V638595069omplex @ w)) => (ord_less_eq_real @ (real_V638595069omplex @ (poly_complex2 @ s @ Z)) @ M2))))))). % \<open>\<exists>m>0. \<forall>z. cmod z \<le> cmod w \<longrightarrow> cmod (poly s z) \<le> m\<close>
thf(fact_73_less_Oprems, axiom,
    ((~ ((fundam1158420650omplex @ (poly_complex2 @ pa)))))). % less.prems
thf(fact_74__092_060open_062_092_060lbrakk_0620_A_092_060le_062_At_059_A0_A_060_Ak_092_060rbrakk_062_A_092_060Longrightarrow_062_At_A_094_Ak_A_060_A1_A_094_Ak_092_060close_062, axiom,
    (((ord_less_eq_real @ zero_zero_real @ t) => ((ord_less_nat @ zero_zero_nat @ k) => (ord_less_real @ (power_power_real @ t @ k) @ (power_power_real @ one_one_real @ k)))))). % \<open>\<lbrakk>0 \<le> t; 0 < k\<rbrakk> \<Longrightarrow> t ^ k < 1 ^ k\<close>
thf(fact_75_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_76_norm__zero, axiom,
    (((real_V638595069omplex @ zero_zero_complex) = zero_zero_real))). % norm_zero
thf(fact_77_norm__zero, axiom,
    (((real_V646646907m_real @ zero_zero_real) = zero_zero_real))). % norm_zero
thf(fact_78_norm__eq__zero, axiom,
    ((![X3 : complex]: (((real_V638595069omplex @ X3) = zero_zero_real) = (X3 = zero_zero_complex))))). % norm_eq_zero
thf(fact_79_norm__eq__zero, axiom,
    ((![X3 : real]: (((real_V646646907m_real @ X3) = zero_zero_real) = (X3 = zero_zero_real))))). % norm_eq_zero
thf(fact_80_abs__power__minus, axiom,
    ((![A : real, N : nat]: ((abs_abs_real @ (power_power_real @ (uminus_uminus_real @ A) @ N)) = (abs_abs_real @ (power_power_real @ A @ N)))))). % abs_power_minus
thf(fact_81_of__real__0, axiom,
    (((real_V1205483228l_real @ zero_zero_real) = zero_zero_real))). % of_real_0
thf(fact_82_of__real__0, axiom,
    (((real_V306493662omplex @ zero_zero_real) = zero_zero_complex))). % of_real_0
thf(fact_83_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_84_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_85_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_86_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_87_of__real__diff, axiom,
    ((![X3 : real, Y2 : real]: ((real_V1205483228l_real @ (minus_minus_real @ X3 @ Y2)) = (minus_minus_real @ (real_V1205483228l_real @ X3) @ (real_V1205483228l_real @ Y2)))))). % of_real_diff
thf(fact_88_of__real__diff, axiom,
    ((![X3 : real, Y2 : real]: ((real_V306493662omplex @ (minus_minus_real @ X3 @ Y2)) = (minus_minus_complex @ (real_V306493662omplex @ X3) @ (real_V306493662omplex @ Y2)))))). % of_real_diff
thf(fact_89_norm__of__real, axiom,
    ((![R : real]: ((real_V638595069omplex @ (real_V306493662omplex @ R)) = (abs_abs_real @ R))))). % norm_of_real
thf(fact_90_norm__of__real, axiom,
    ((![R : real]: ((real_V646646907m_real @ (real_V1205483228l_real @ R)) = (abs_abs_real @ R))))). % norm_of_real
thf(fact_91_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_92_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_93_left__minus__one__mult__self, axiom,
    ((![N : nat, A : real]: ((times_times_real @ (power_power_real @ (uminus_uminus_real @ one_one_real) @ N) @ (times_times_real @ (power_power_real @ (uminus_uminus_real @ one_one_real) @ N) @ A)) = A)))). % left_minus_one_mult_self
thf(fact_94_left__minus__one__mult__self, axiom,
    ((![N : nat, A : complex]: ((times_times_complex @ (power_power_complex @ (uminus1204672759omplex @ one_one_complex) @ N) @ (times_times_complex @ (power_power_complex @ (uminus1204672759omplex @ one_one_complex) @ N) @ A)) = A)))). % left_minus_one_mult_self
thf(fact_95_minus__one__mult__self, axiom,
    ((![N : nat]: ((times_times_real @ (power_power_real @ (uminus_uminus_real @ one_one_real) @ N) @ (power_power_real @ (uminus_uminus_real @ one_one_real) @ N)) = one_one_real)))). % minus_one_mult_self
thf(fact_96_minus__one__mult__self, axiom,
    ((![N : nat]: ((times_times_complex @ (power_power_complex @ (uminus1204672759omplex @ one_one_complex) @ N) @ (power_power_complex @ (uminus1204672759omplex @ one_one_complex) @ N)) = one_one_complex)))). % minus_one_mult_self
thf(fact_97_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_98_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_99_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_100_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_101_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_102_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_103_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_104_power__decreasing__iff, axiom,
    ((![B : nat, M : 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 @ M) @ (power_power_nat @ B @ N)) = (ord_less_eq_nat @ N @ M))))))). % power_decreasing_iff
thf(fact_105_power__decreasing__iff, axiom,
    ((![B : real, M : 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 @ M) @ (power_power_real @ B @ N)) = (ord_less_eq_nat @ N @ M))))))). % power_decreasing_iff
thf(fact_106_power__strict__decreasing__iff, axiom,
    ((![B : real, M : nat, N : nat]: ((ord_less_real @ zero_zero_real @ B) => ((ord_less_real @ B @ one_one_real) => ((ord_less_real @ (power_power_real @ B @ M) @ (power_power_real @ B @ N)) = (ord_less_nat @ N @ M))))))). % power_strict_decreasing_iff
thf(fact_107_power__strict__decreasing__iff, axiom,
    ((![B : nat, M : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ B) => ((ord_less_nat @ B @ one_one_nat) => ((ord_less_nat @ (power_power_nat @ B @ M) @ (power_power_nat @ B @ N)) = (ord_less_nat @ N @ M))))))). % power_strict_decreasing_iff
thf(fact_108_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_109_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_110_zero__less__power__abs__iff, axiom,
    ((![A : real, N : nat]: ((ord_less_real @ zero_zero_real @ (power_power_real @ (abs_abs_real @ A) @ N)) = (((~ ((A = zero_zero_real)))) | ((N = zero_zero_nat))))))). % zero_less_power_abs_iff
thf(fact_111_norm__of__real__add1, axiom,
    ((![X3 : real]: ((real_V638595069omplex @ (plus_plus_complex @ (real_V306493662omplex @ X3) @ one_one_complex)) = (abs_abs_real @ (plus_plus_real @ X3 @ one_one_real)))))). % norm_of_real_add1
thf(fact_112_norm__of__real__add1, axiom,
    ((![X3 : real]: ((real_V646646907m_real @ (plus_plus_real @ (real_V1205483228l_real @ X3) @ one_one_real)) = (abs_abs_real @ (plus_plus_real @ X3 @ one_one_real)))))). % norm_of_real_add1
thf(fact_113_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_114_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_115_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_116_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_117_nat__power__less__imp__less, axiom,
    ((![I : nat, M : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ I) => ((ord_less_nat @ (power_power_nat @ I @ M) @ (power_power_nat @ I @ N)) => (ord_less_nat @ M @ N)))))). % nat_power_less_imp_less
thf(fact_118_real__norm__def, axiom,
    ((real_V646646907m_real = abs_abs_real))). % real_norm_def
thf(fact_119_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_120_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_121_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_122_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_123_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_124_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_125_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_126_zero__le__power__abs, axiom,
    ((![A : real, N : nat]: (ord_less_eq_real @ zero_zero_real @ (power_power_real @ (abs_abs_real @ A) @ N))))). % zero_le_power_abs
thf(fact_127_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_128_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_129_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_130_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_131_norm__ge__zero, axiom,
    ((![X3 : complex]: (ord_less_eq_real @ zero_zero_real @ (real_V638595069omplex @ X3))))). % norm_ge_zero
thf(fact_132_norm__ge__zero, axiom,
    ((![X3 : real]: (ord_less_eq_real @ zero_zero_real @ (real_V646646907m_real @ X3))))). % norm_ge_zero
thf(fact_133_norm__triangle__ineq3, axiom,
    ((![A : complex, B : complex]: (ord_less_eq_real @ (abs_abs_real @ (minus_minus_real @ (real_V638595069omplex @ A) @ (real_V638595069omplex @ B))) @ (real_V638595069omplex @ (minus_minus_complex @ A @ B)))))). % norm_triangle_ineq3
thf(fact_134_norm__triangle__ineq3, axiom,
    ((![A : real, B : real]: (ord_less_eq_real @ (abs_abs_real @ (minus_minus_real @ (real_V646646907m_real @ A) @ (real_V646646907m_real @ B))) @ (real_V646646907m_real @ (minus_minus_real @ A @ B)))))). % norm_triangle_ineq3
thf(fact_135_power__decreasing, axiom,
    ((![N : nat, N2 : nat, A : nat]: ((ord_less_eq_nat @ N @ N2) => ((ord_less_eq_nat @ zero_zero_nat @ A) => ((ord_less_eq_nat @ A @ one_one_nat) => (ord_less_eq_nat @ (power_power_nat @ A @ N2) @ (power_power_nat @ A @ N)))))))). % power_decreasing
thf(fact_136_power__decreasing, axiom,
    ((![N : nat, N2 : nat, A : real]: ((ord_less_eq_nat @ N @ N2) => ((ord_less_eq_real @ zero_zero_real @ A) => ((ord_less_eq_real @ A @ one_one_real) => (ord_less_eq_real @ (power_power_real @ A @ N2) @ (power_power_real @ A @ N)))))))). % power_decreasing
thf(fact_137_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_138_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_139_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_140_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_141_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_142_power__strict__mono, axiom,
    ((![A : nat, B : nat, N : nat]: ((ord_less_nat @ A @ B) => ((ord_less_eq_nat @ zero_zero_nat @ A) => ((ord_less_nat @ zero_zero_nat @ N) => (ord_less_nat @ (power_power_nat @ A @ N) @ (power_power_nat @ B @ N)))))))). % power_strict_mono
thf(fact_143_power__strict__mono, axiom,
    ((![A : real, B : real, N : nat]: ((ord_less_real @ A @ B) => ((ord_less_eq_real @ zero_zero_real @ A) => ((ord_less_nat @ zero_zero_nat @ N) => (ord_less_real @ (power_power_real @ A @ N) @ (power_power_real @ B @ N)))))))). % power_strict_mono
thf(fact_144_power__0__left, axiom,
    ((![N : nat]: (((N = zero_zero_nat) => ((power_power_complex @ zero_zero_complex @ N) = one_one_complex)) & ((~ ((N = zero_zero_nat))) => ((power_power_complex @ zero_zero_complex @ N) = zero_zero_complex)))))). % power_0_left
thf(fact_145_power__0__left, axiom,
    ((![N : nat]: (((N = zero_zero_nat) => ((power_power_real @ zero_zero_real @ N) = one_one_real)) & ((~ ((N = zero_zero_nat))) => ((power_power_real @ zero_zero_real @ N) = zero_zero_real)))))). % power_0_left
thf(fact_146_power__0__left, axiom,
    ((![N : nat]: (((N = zero_zero_nat) => ((power_power_nat @ zero_zero_nat @ N) = one_one_nat)) & ((~ ((N = zero_zero_nat))) => ((power_power_nat @ zero_zero_nat @ N) = zero_zero_nat)))))). % power_0_left
thf(fact_147_power__eq__imp__eq__norm, axiom,
    ((![W2 : complex, N : nat, Z2 : complex]: (((power_power_complex @ W2 @ N) = (power_power_complex @ Z2 @ N)) => ((ord_less_nat @ zero_zero_nat @ N) => ((real_V638595069omplex @ W2) = (real_V638595069omplex @ Z2))))))). % power_eq_imp_eq_norm
thf(fact_148_power__eq__imp__eq__norm, axiom,
    ((![W2 : real, N : nat, Z2 : real]: (((power_power_real @ W2 @ N) = (power_power_real @ Z2 @ N)) => ((ord_less_nat @ zero_zero_nat @ N) => ((real_V646646907m_real @ W2) = (real_V646646907m_real @ Z2))))))). % power_eq_imp_eq_norm
thf(fact_149_norm__triangle__ineq2, axiom,
    ((![A : complex, B : complex]: (ord_less_eq_real @ (minus_minus_real @ (real_V638595069omplex @ A) @ (real_V638595069omplex @ B)) @ (real_V638595069omplex @ (minus_minus_complex @ A @ B)))))). % norm_triangle_ineq2
thf(fact_150_norm__triangle__ineq2, axiom,
    ((![A : real, B : real]: (ord_less_eq_real @ (minus_minus_real @ (real_V646646907m_real @ A) @ (real_V646646907m_real @ B)) @ (real_V646646907m_real @ (minus_minus_real @ A @ B)))))). % norm_triangle_ineq2
thf(fact_151_norm__triangle__le__diff, 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 @ (minus_minus_complex @ X3 @ Y2)) @ E2))))). % norm_triangle_le_diff
thf(fact_152_norm__triangle__le__diff, 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 @ (minus_minus_real @ X3 @ Y2)) @ E2))))). % norm_triangle_le_diff
thf(fact_153_norm__diff__triangle__le, axiom,
    ((![X3 : complex, Y2 : complex, E1 : real, Z2 : complex, E22 : real]: ((ord_less_eq_real @ (real_V638595069omplex @ (minus_minus_complex @ X3 @ Y2)) @ E1) => ((ord_less_eq_real @ (real_V638595069omplex @ (minus_minus_complex @ Y2 @ Z2)) @ E22) => (ord_less_eq_real @ (real_V638595069omplex @ (minus_minus_complex @ X3 @ Z2)) @ (plus_plus_real @ E1 @ E22))))))). % norm_diff_triangle_le
thf(fact_154_norm__diff__triangle__le, axiom,
    ((![X3 : real, Y2 : real, E1 : real, Z2 : real, E22 : real]: ((ord_less_eq_real @ (real_V646646907m_real @ (minus_minus_real @ X3 @ Y2)) @ E1) => ((ord_less_eq_real @ (real_V646646907m_real @ (minus_minus_real @ Y2 @ Z2)) @ E22) => (ord_less_eq_real @ (real_V646646907m_real @ (minus_minus_real @ X3 @ Z2)) @ (plus_plus_real @ E1 @ E22))))))). % norm_diff_triangle_le
thf(fact_155_norm__triangle__ineq4, axiom,
    ((![A : complex, B : complex]: (ord_less_eq_real @ (real_V638595069omplex @ (minus_minus_complex @ A @ B)) @ (plus_plus_real @ (real_V638595069omplex @ A) @ (real_V638595069omplex @ B)))))). % norm_triangle_ineq4
thf(fact_156_norm__triangle__ineq4, axiom,
    ((![A : real, B : real]: (ord_less_eq_real @ (real_V646646907m_real @ (minus_minus_real @ A @ B)) @ (plus_plus_real @ (real_V646646907m_real @ A) @ (real_V646646907m_real @ B)))))). % norm_triangle_ineq4
thf(fact_157_norm__triangle__sub, axiom,
    ((![X3 : complex, Y2 : complex]: (ord_less_eq_real @ (real_V638595069omplex @ X3) @ (plus_plus_real @ (real_V638595069omplex @ Y2) @ (real_V638595069omplex @ (minus_minus_complex @ X3 @ Y2))))))). % norm_triangle_sub
thf(fact_158_norm__triangle__sub, axiom,
    ((![X3 : real, Y2 : real]: (ord_less_eq_real @ (real_V646646907m_real @ X3) @ (plus_plus_real @ (real_V646646907m_real @ Y2) @ (real_V646646907m_real @ (minus_minus_real @ X3 @ Y2))))))). % norm_triangle_sub
thf(fact_159_constant__def, axiom,
    ((fundam1158420650omplex = (^[F : complex > complex]: (![X2 : complex]: (![Y3 : complex]: ((F @ X2) = (F @ Y3)))))))). % constant_def
thf(fact_160_norm__uminus__minus, axiom,
    ((![X3 : complex, Y2 : complex]: ((real_V638595069omplex @ (minus_minus_complex @ (uminus1204672759omplex @ X3) @ Y2)) = (real_V638595069omplex @ (plus_plus_complex @ X3 @ Y2)))))). % norm_uminus_minus
thf(fact_161_norm__uminus__minus, axiom,
    ((![X3 : real, Y2 : real]: ((real_V646646907m_real @ (minus_minus_real @ (uminus_uminus_real @ X3) @ Y2)) = (real_V646646907m_real @ (plus_plus_real @ X3 @ Y2)))))). % norm_uminus_minus
thf(fact_162_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_163_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_164_sum__squares__le__zero__iff, axiom,
    ((![X3 : real, Y2 : real]: ((ord_less_eq_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_le_zero_iff
thf(fact_165_power__less__imp__less__base, axiom,
    ((![A : nat, N : nat, B : nat]: ((ord_less_nat @ (power_power_nat @ A @ N) @ (power_power_nat @ B @ N)) => ((ord_less_eq_nat @ zero_zero_nat @ B) => (ord_less_nat @ A @ B)))))). % power_less_imp_less_base
thf(fact_166_power__less__imp__less__base, axiom,
    ((![A : real, N : nat, B : real]: ((ord_less_real @ (power_power_real @ A @ N) @ (power_power_real @ B @ N)) => ((ord_less_eq_real @ zero_zero_real @ B) => (ord_less_real @ A @ B)))))). % power_less_imp_less_base
thf(fact_167_power__le__one, axiom,
    ((![A : nat, N : nat]: ((ord_less_eq_nat @ zero_zero_nat @ A) => ((ord_less_eq_nat @ A @ one_one_nat) => (ord_less_eq_nat @ (power_power_nat @ A @ N) @ one_one_nat)))))). % power_le_one
thf(fact_168_power__le__one, axiom,
    ((![A : real, N : nat]: ((ord_less_eq_real @ zero_zero_real @ A) => ((ord_less_eq_real @ A @ one_one_real) => (ord_less_eq_real @ (power_power_real @ A @ N) @ one_one_real)))))). % power_le_one
thf(fact_169_power__add, axiom,
    ((![A : nat, M : nat, N : nat]: ((power_power_nat @ A @ (plus_plus_nat @ M @ N)) = (times_times_nat @ (power_power_nat @ A @ M) @ (power_power_nat @ A @ N)))))). % power_add
thf(fact_170_power__add, axiom,
    ((![A : complex, M : nat, N : nat]: ((power_power_complex @ A @ (plus_plus_nat @ M @ N)) = (times_times_complex @ (power_power_complex @ A @ M) @ (power_power_complex @ A @ N)))))). % power_add
thf(fact_171_power__add, axiom,
    ((![A : real, M : nat, N : nat]: ((power_power_real @ A @ (plus_plus_nat @ M @ N)) = (times_times_real @ (power_power_real @ A @ M) @ (power_power_real @ A @ N)))))). % power_add
thf(fact_172_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_173_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_174_norm__of__real__diff, axiom,
    ((![B : real, A : real]: (ord_less_eq_real @ (real_V638595069omplex @ (minus_minus_complex @ (real_V306493662omplex @ B) @ (real_V306493662omplex @ A))) @ (abs_abs_real @ (minus_minus_real @ B @ A)))))). % norm_of_real_diff
thf(fact_175_norm__of__real__diff, axiom,
    ((![B : real, A : real]: (ord_less_eq_real @ (real_V646646907m_real @ (minus_minus_real @ (real_V1205483228l_real @ B) @ (real_V1205483228l_real @ A))) @ (abs_abs_real @ (minus_minus_real @ B @ A)))))). % norm_of_real_diff
thf(fact_176_power__0, axiom,
    ((![A : complex]: ((power_power_complex @ A @ zero_zero_nat) = one_one_complex)))). % power_0
thf(fact_177_power__0, axiom,
    ((![A : real]: ((power_power_real @ A @ zero_zero_nat) = one_one_real)))). % power_0
thf(fact_178_power__0, axiom,
    ((![A : nat]: ((power_power_nat @ A @ zero_zero_nat) = one_one_nat)))). % power_0
thf(fact_179_norm__not__less__zero, axiom,
    ((![X3 : complex]: (~ ((ord_less_real @ (real_V638595069omplex @ X3) @ zero_zero_real)))))). % norm_not_less_zero
thf(fact_180_norm__not__less__zero, axiom,
    ((![X3 : real]: (~ ((ord_less_real @ (real_V646646907m_real @ X3) @ zero_zero_real)))))). % norm_not_less_zero
thf(fact_181_power__abs, axiom,
    ((![A : real, N : nat]: ((abs_abs_real @ (power_power_real @ A @ N)) = (power_power_real @ (abs_abs_real @ A) @ N))))). % power_abs
thf(fact_182_norm__minus__commute, axiom,
    ((![A : complex, B : complex]: ((real_V638595069omplex @ (minus_minus_complex @ A @ B)) = (real_V638595069omplex @ (minus_minus_complex @ B @ A)))))). % norm_minus_commute
thf(fact_183_norm__minus__commute, axiom,
    ((![A : real, B : real]: ((real_V646646907m_real @ (minus_minus_real @ A @ B)) = (real_V646646907m_real @ (minus_minus_real @ B @ A)))))). % norm_minus_commute
thf(fact_184_power__le__imp__le__exp, axiom,
    ((![A : nat, M : nat, N : nat]: ((ord_less_nat @ one_one_nat @ A) => ((ord_less_eq_nat @ (power_power_nat @ A @ M) @ (power_power_nat @ A @ N)) => (ord_less_eq_nat @ M @ N)))))). % power_le_imp_le_exp
thf(fact_185_power__le__imp__le__exp, axiom,
    ((![A : real, M : nat, N : nat]: ((ord_less_real @ one_one_real @ A) => ((ord_less_eq_real @ (power_power_real @ A @ M) @ (power_power_real @ A @ N)) => (ord_less_eq_nat @ M @ N)))))). % power_le_imp_le_exp
thf(fact_186_power__minus, axiom,
    ((![A : real, N : nat]: ((power_power_real @ (uminus_uminus_real @ A) @ N) = (times_times_real @ (power_power_real @ (uminus_uminus_real @ one_one_real) @ N) @ (power_power_real @ A @ N)))))). % power_minus
thf(fact_187_power__minus, axiom,
    ((![A : complex, N : nat]: ((power_power_complex @ (uminus1204672759omplex @ A) @ N) = (times_times_complex @ (power_power_complex @ (uminus1204672759omplex @ one_one_complex) @ N) @ (power_power_complex @ A @ N)))))). % power_minus
thf(fact_188_sum__squares__gt__zero__iff, axiom,
    ((![X3 : real, Y2 : real]: ((ord_less_real @ zero_zero_real @ (plus_plus_real @ (times_times_real @ X3 @ X3) @ (times_times_real @ Y2 @ Y2))) = (((~ ((X3 = zero_zero_real)))) | ((~ ((Y2 = zero_zero_real))))))))). % sum_squares_gt_zero_iff
thf(fact_189_power__strict__decreasing, axiom,
    ((![N : nat, N2 : nat, A : real]: ((ord_less_nat @ N @ N2) => ((ord_less_real @ zero_zero_real @ A) => ((ord_less_real @ A @ one_one_real) => (ord_less_real @ (power_power_real @ A @ N2) @ (power_power_real @ A @ N)))))))). % power_strict_decreasing
thf(fact_190_power__strict__decreasing, axiom,
    ((![N : nat, N2 : nat, A : nat]: ((ord_less_nat @ N @ N2) => ((ord_less_nat @ zero_zero_nat @ A) => ((ord_less_nat @ A @ one_one_nat) => (ord_less_nat @ (power_power_nat @ A @ N2) @ (power_power_nat @ A @ N)))))))). % power_strict_decreasing
thf(fact_191_one__less__power, axiom,
    ((![A : real, N : nat]: ((ord_less_real @ one_one_real @ A) => ((ord_less_nat @ zero_zero_nat @ N) => (ord_less_real @ one_one_real @ (power_power_real @ A @ N))))))). % one_less_power
thf(fact_192_one__less__power, axiom,
    ((![A : nat, N : nat]: ((ord_less_nat @ one_one_nat @ A) => ((ord_less_nat @ zero_zero_nat @ N) => (ord_less_nat @ one_one_nat @ (power_power_nat @ A @ N))))))). % one_less_power
thf(fact_193_norm__diff__triangle__ineq, axiom,
    ((![A : complex, B : complex, C : complex, D : complex]: (ord_less_eq_real @ (real_V638595069omplex @ (minus_minus_complex @ (plus_plus_complex @ A @ B) @ (plus_plus_complex @ C @ D))) @ (plus_plus_real @ (real_V638595069omplex @ (minus_minus_complex @ A @ C)) @ (real_V638595069omplex @ (minus_minus_complex @ B @ D))))))). % norm_diff_triangle_ineq
thf(fact_194_norm__diff__triangle__ineq, axiom,
    ((![A : real, B : real, C : real, D : real]: (ord_less_eq_real @ (real_V646646907m_real @ (minus_minus_real @ (plus_plus_real @ A @ B) @ (plus_plus_real @ C @ D))) @ (plus_plus_real @ (real_V646646907m_real @ (minus_minus_real @ A @ C)) @ (real_V646646907m_real @ (minus_minus_real @ B @ D))))))). % norm_diff_triangle_ineq
thf(fact_195_norm__diff__ineq, axiom,
    ((![A : complex, B : complex]: (ord_less_eq_real @ (minus_minus_real @ (real_V638595069omplex @ A) @ (real_V638595069omplex @ B)) @ (real_V638595069omplex @ (plus_plus_complex @ A @ B)))))). % norm_diff_ineq
thf(fact_196_norm__diff__ineq, axiom,
    ((![A : real, B : real]: (ord_less_eq_real @ (minus_minus_real @ (real_V646646907m_real @ A) @ (real_V646646907m_real @ B)) @ (real_V646646907m_real @ (plus_plus_real @ A @ B)))))). % norm_diff_ineq
thf(fact_197_norm__diff__triangle__less, axiom,
    ((![X3 : complex, Y2 : complex, E1 : real, Z2 : complex, E22 : real]: ((ord_less_real @ (real_V638595069omplex @ (minus_minus_complex @ X3 @ Y2)) @ E1) => ((ord_less_real @ (real_V638595069omplex @ (minus_minus_complex @ Y2 @ Z2)) @ E22) => (ord_less_real @ (real_V638595069omplex @ (minus_minus_complex @ X3 @ Z2)) @ (plus_plus_real @ E1 @ E22))))))). % norm_diff_triangle_less
thf(fact_198_norm__diff__triangle__less, axiom,
    ((![X3 : real, Y2 : real, E1 : real, Z2 : real, E22 : real]: ((ord_less_real @ (real_V646646907m_real @ (minus_minus_real @ X3 @ Y2)) @ E1) => ((ord_less_real @ (real_V646646907m_real @ (minus_minus_real @ Y2 @ Z2)) @ E22) => (ord_less_real @ (real_V646646907m_real @ (minus_minus_real @ X3 @ Z2)) @ (plus_plus_real @ E1 @ E22))))))). % norm_diff_triangle_less
thf(fact_199_poly__cont, axiom,
    ((![E2 : real, Z2 : complex, P2 : poly_complex]: ((ord_less_real @ zero_zero_real @ E2) => (?[D3 : real]: ((ord_less_real @ zero_zero_real @ D3) & (![W3 : complex]: (((ord_less_real @ zero_zero_real @ (real_V638595069omplex @ (minus_minus_complex @ W3 @ Z2))) & (ord_less_real @ (real_V638595069omplex @ (minus_minus_complex @ W3 @ Z2)) @ D3)) => (ord_less_real @ (real_V638595069omplex @ (minus_minus_complex @ (poly_complex2 @ P2 @ W3) @ (poly_complex2 @ P2 @ Z2))) @ E2))))))))). % poly_cont
thf(fact_200_poly__cont, axiom,
    ((![E2 : real, Z2 : real, P2 : poly_real]: ((ord_less_real @ zero_zero_real @ E2) => (?[D3 : real]: ((ord_less_real @ zero_zero_real @ D3) & (![W3 : real]: (((ord_less_real @ zero_zero_real @ (real_V646646907m_real @ (minus_minus_real @ W3 @ Z2))) & (ord_less_real @ (real_V646646907m_real @ (minus_minus_real @ W3 @ Z2)) @ D3)) => (ord_less_real @ (real_V646646907m_real @ (minus_minus_real @ (poly_real2 @ P2 @ W3) @ (poly_real2 @ P2 @ Z2))) @ E2))))))))). % poly_cont
thf(fact_201_poly__bound__exists, axiom,
    ((![R : real, P2 : poly_complex]: (?[M2 : real]: ((ord_less_real @ zero_zero_real @ M2) & (![Z : complex]: ((ord_less_eq_real @ (real_V638595069omplex @ Z) @ R) => (ord_less_eq_real @ (real_V638595069omplex @ (poly_complex2 @ P2 @ Z)) @ M2)))))))). % poly_bound_exists
thf(fact_202_poly__bound__exists, axiom,
    ((![R : real, P2 : poly_real]: (?[M2 : real]: ((ord_less_real @ zero_zero_real @ M2) & (![Z : real]: ((ord_less_eq_real @ (real_V646646907m_real @ Z) @ R) => (ord_less_eq_real @ (real_V646646907m_real @ (poly_real2 @ P2 @ Z)) @ M2)))))))). % poly_bound_exists
thf(fact_203_power__Suc__less, axiom,
    ((![A : real, N : nat]: ((ord_less_real @ zero_zero_real @ A) => ((ord_less_real @ A @ one_one_real) => (ord_less_real @ (times_times_real @ A @ (power_power_real @ A @ N)) @ (power_power_real @ A @ N))))))). % power_Suc_less
thf(fact_204_power__Suc__less, axiom,
    ((![A : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ A) => ((ord_less_nat @ A @ one_one_nat) => (ord_less_nat @ (times_times_nat @ A @ (power_power_nat @ A @ N)) @ (power_power_nat @ A @ N))))))). % power_Suc_less
thf(fact_205_power__eq__1__iff, axiom,
    ((![W2 : complex, N : nat]: (((power_power_complex @ W2 @ N) = one_one_complex) => (((real_V638595069omplex @ W2) = one_one_real) | (N = zero_zero_nat)))))). % power_eq_1_iff
thf(fact_206_power__eq__1__iff, axiom,
    ((![W2 : real, N : nat]: (((power_power_real @ W2 @ N) = one_one_real) => (((real_V646646907m_real @ W2) = one_one_real) | (N = zero_zero_nat)))))). % power_eq_1_iff
thf(fact_207_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_208_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_209_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_210_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_211_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_212_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_213_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_214_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_215_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_216_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_217_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_218_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_219_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_220_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_221_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_222_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_223_reduce__poly__simple, axiom,
    ((![B : complex, N : nat]: ((~ ((B = zero_zero_complex))) => ((~ ((N = zero_zero_nat))) => (?[Z3 : complex]: (ord_less_real @ (real_V638595069omplex @ (plus_plus_complex @ one_one_complex @ (times_times_complex @ B @ (power_power_complex @ Z3 @ N)))) @ one_one_real))))))). % reduce_poly_simple
thf(fact_224_complex__mod__triangle__sub, axiom,
    ((![W2 : complex, Z2 : complex]: (ord_less_eq_real @ (real_V638595069omplex @ W2) @ (plus_plus_real @ (real_V638595069omplex @ (plus_plus_complex @ W2 @ Z2)) @ (real_V638595069omplex @ Z2)))))). % complex_mod_triangle_sub
thf(fact_225_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_226_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_227_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_228_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_229_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_230_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_231_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_232_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_233_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_234_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_235_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_236_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_237_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_238_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_239_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_240_power__less__imp__less__exp, axiom,
    ((![A : real, M : nat, N : nat]: ((ord_less_real @ one_one_real @ A) => ((ord_less_real @ (power_power_real @ A @ M) @ (power_power_real @ A @ N)) => (ord_less_nat @ M @ N)))))). % power_less_imp_less_exp
thf(fact_241_power__less__imp__less__exp, axiom,
    ((![A : nat, M : nat, N : nat]: ((ord_less_nat @ one_one_nat @ A) => ((ord_less_nat @ (power_power_nat @ A @ M) @ (power_power_nat @ A @ N)) => (ord_less_nat @ M @ N)))))). % power_less_imp_less_exp
thf(fact_242_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_243_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_244_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_245_poly__minimum__modulus__disc, axiom,
    ((![R : real, P2 : poly_complex]: (?[Z3 : complex]: (![W3 : complex]: ((ord_less_eq_real @ (real_V638595069omplex @ W3) @ R) => (ord_less_eq_real @ (real_V638595069omplex @ (poly_complex2 @ P2 @ Z3)) @ (real_V638595069omplex @ (poly_complex2 @ P2 @ W3))))))))). % poly_minimum_modulus_disc
thf(fact_246_poly__minimum__modulus, axiom,
    ((![P2 : poly_complex]: (?[Z3 : complex]: (![W3 : complex]: (ord_less_eq_real @ (real_V638595069omplex @ (poly_complex2 @ P2 @ Z3)) @ (real_V638595069omplex @ (poly_complex2 @ P2 @ W3)))))))). % poly_minimum_modulus
thf(fact_247__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_248__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>

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