% 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_770__5375384_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:32:58.242

% Could-be-implicit typings (4)
thf(ty_n_t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    poly_complex : $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 (39)
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__Complex__Ocomplex, type,
    abs_abs_complex : complex > complex).
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__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__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__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__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__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__Real__Oreal, type,
    zero_zero_real : real).
thf(sy_c_If_001t__Real__Oreal, type,
    if_real : $o > real > 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_Power_Opower__class_Opower_001t__Complex__Ocomplex, type,
    power_power_complex : complex > nat > complex).
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 (226)
thf(fact_0_nc, axiom,
    ((~ ((fundam1158420650omplex @ (poly_complex2 @ p)))))). % nc
thf(fact_1_kas_I2_J, axiom,
    ((~ ((k = zero_zero_nat))))). % kas(2)
thf(fact_2_kas_I1_J, axiom,
    ((~ ((a = zero_zero_complex))))). % kas(1)
thf(fact_3__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_4_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_5_t_I2_J, axiom,
    ((ord_less_real @ t @ one_one_real))). % t(2)
thf(fact_6__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_7__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_8__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_9_w, axiom,
    (((plus_plus_complex @ one_one_complex @ (times_times_complex @ (power_power_complex @ w @ k) @ a)) = zero_zero_complex))). % w
thf(fact_10_poly__minimum__modulus, axiom,
    ((![P : poly_complex]: (?[Z2 : complex]: (![W : complex]: (ord_less_eq_real @ (real_V638595069omplex @ (poly_complex2 @ P @ Z2)) @ (real_V638595069omplex @ (poly_complex2 @ P @ W)))))))). % poly_minimum_modulus
thf(fact_11_complex__mod__triangle__sub, axiom,
    ((![W2 : complex, Z3 : complex]: (ord_less_eq_real @ (real_V638595069omplex @ W2) @ (plus_plus_real @ (real_V638595069omplex @ (plus_plus_complex @ W2 @ Z3)) @ (real_V638595069omplex @ Z3)))))). % complex_mod_triangle_sub
thf(fact_12_poly__minimum__modulus__disc, axiom,
    ((![R : real, P : poly_complex]: (?[Z2 : complex]: (![W : complex]: ((ord_less_eq_real @ (real_V638595069omplex @ W) @ R) => (ord_less_eq_real @ (real_V638595069omplex @ (poly_complex2 @ P @ Z2)) @ (real_V638595069omplex @ (poly_complex2 @ P @ W))))))))). % poly_minimum_modulus_disc
thf(fact_13_norm__of__real__add1, axiom,
    ((![X : real]: ((real_V638595069omplex @ (plus_plus_complex @ (real_V306493662omplex @ X) @ one_one_complex)) = (abs_abs_real @ (plus_plus_real @ X @ one_one_real)))))). % norm_of_real_add1
thf(fact_14_norm__of__real__add1, axiom,
    ((![X : real]: ((real_V646646907m_real @ (plus_plus_real @ (real_V1205483228l_real @ X) @ one_one_real)) = (abs_abs_real @ (plus_plus_real @ X @ one_one_real)))))). % norm_of_real_add1
thf(fact_15_norm__of__real, axiom,
    ((![R : real]: ((real_V638595069omplex @ (real_V306493662omplex @ R)) = (abs_abs_real @ R))))). % norm_of_real
thf(fact_16_norm__of__real, axiom,
    ((![R : real]: ((real_V646646907m_real @ (real_V1205483228l_real @ R)) = (abs_abs_real @ R))))). % norm_of_real
thf(fact_17_of__real__diff, axiom,
    ((![X : real, Y : real]: ((real_V1205483228l_real @ (minus_minus_real @ X @ Y)) = (minus_minus_real @ (real_V1205483228l_real @ X) @ (real_V1205483228l_real @ Y)))))). % of_real_diff
thf(fact_18_of__real__diff, axiom,
    ((![X : real, Y : real]: ((real_V306493662omplex @ (minus_minus_real @ X @ Y)) = (minus_minus_complex @ (real_V306493662omplex @ X) @ (real_V306493662omplex @ Y)))))). % of_real_diff
thf(fact_19__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,
    ((~ ((![W3 : complex]: (~ (((plus_plus_complex @ one_one_complex @ (times_times_complex @ (power_power_complex @ W3 @ k) @ a)) = zero_zero_complex)))))))). % \<open>\<And>thesis. (\<And>w. 1 + w ^ k * a = 0 \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_20_of__real__power, axiom,
    ((![X : real, N : nat]: ((real_V1205483228l_real @ (power_power_real @ X @ N)) = (power_power_real @ (real_V1205483228l_real @ X) @ N))))). % of_real_power
thf(fact_21_of__real__power, axiom,
    ((![X : real, N : nat]: ((real_V306493662omplex @ (power_power_real @ X @ N)) = (power_power_complex @ (real_V306493662omplex @ X) @ N))))). % of_real_power
thf(fact_22_of__real__add, axiom,
    ((![X : real, Y : real]: ((real_V1205483228l_real @ (plus_plus_real @ X @ Y)) = (plus_plus_real @ (real_V1205483228l_real @ X) @ (real_V1205483228l_real @ Y)))))). % of_real_add
thf(fact_23_of__real__add, axiom,
    ((![X : real, Y : real]: ((real_V306493662omplex @ (plus_plus_real @ X @ Y)) = (plus_plus_complex @ (real_V306493662omplex @ X) @ (real_V306493662omplex @ Y)))))). % of_real_add
thf(fact_24_of__real__1, axiom,
    (((real_V1205483228l_real @ one_one_real) = one_one_real))). % of_real_1
thf(fact_25_of__real__1, axiom,
    (((real_V306493662omplex @ one_one_real) = one_one_complex))). % of_real_1
thf(fact_26_of__real__eq__1__iff, axiom,
    ((![X : real]: (((real_V1205483228l_real @ X) = one_one_real) = (X = one_one_real))))). % of_real_eq_1_iff
thf(fact_27_of__real__eq__1__iff, axiom,
    ((![X : real]: (((real_V306493662omplex @ X) = one_one_complex) = (X = one_one_real))))). % of_real_eq_1_iff
thf(fact_28_norm__one, axiom,
    (((real_V638595069omplex @ one_one_complex) = one_one_real))). % norm_one
thf(fact_29_norm__one, axiom,
    (((real_V646646907m_real @ one_one_real) = one_one_real))). % norm_one
thf(fact_30_le__add__diff__inverse, axiom,
    ((![B : real, A : real]: ((ord_less_eq_real @ B @ A) => ((plus_plus_real @ B @ (minus_minus_real @ A @ B)) = A))))). % le_add_diff_inverse
thf(fact_31_le__add__diff__inverse2, axiom,
    ((![B : real, A : real]: ((ord_less_eq_real @ B @ A) => ((plus_plus_real @ (minus_minus_real @ A @ B) @ B) = A))))). % le_add_diff_inverse2
thf(fact_32_wm1, axiom,
    (((times_times_complex @ (power_power_complex @ w @ k) @ a) = (uminus1204672759omplex @ one_one_complex)))). % wm1
thf(fact_33__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_34_w0, axiom,
    ((~ ((w = zero_zero_complex))))). % w0
thf(fact_35_abs__abs, axiom,
    ((![A : real]: ((abs_abs_real @ (abs_abs_real @ A)) = (abs_abs_real @ A))))). % abs_abs
thf(fact_36_of__real__eq__iff, axiom,
    ((![X : real, Y : real]: (((real_V306493662omplex @ X) = (real_V306493662omplex @ Y)) = (X = Y))))). % of_real_eq_iff
thf(fact_37_t_I1_J, axiom,
    ((ord_less_real @ zero_zero_real @ t))). % t(1)
thf(fact_38_mult__cancel__right, axiom,
    ((![A : nat, C : nat, B : nat]: (((times_times_nat @ A @ C) = (times_times_nat @ B @ C)) = (((C = zero_zero_nat)) | ((A = B))))))). % mult_cancel_right
thf(fact_39_mult__cancel__right, axiom,
    ((![A : real, C : real, B : real]: (((times_times_real @ A @ C) = (times_times_real @ B @ C)) = (((C = zero_zero_real)) | ((A = B))))))). % mult_cancel_right
thf(fact_40_mult__cancel__right, axiom,
    ((![A : complex, C : complex, B : complex]: (((times_times_complex @ A @ C) = (times_times_complex @ B @ C)) = (((C = zero_zero_complex)) | ((A = B))))))). % mult_cancel_right
thf(fact_41_mult__cancel__left, axiom,
    ((![C : nat, A : nat, B : nat]: (((times_times_nat @ C @ A) = (times_times_nat @ C @ B)) = (((C = zero_zero_nat)) | ((A = B))))))). % mult_cancel_left
thf(fact_42_mult__cancel__left, axiom,
    ((![C : real, A : real, B : real]: (((times_times_real @ C @ A) = (times_times_real @ C @ B)) = (((C = zero_zero_real)) | ((A = B))))))). % mult_cancel_left
thf(fact_43_mult__cancel__left, axiom,
    ((![C : complex, A : complex, B : complex]: (((times_times_complex @ C @ A) = (times_times_complex @ C @ B)) = (((C = zero_zero_complex)) | ((A = B))))))). % mult_cancel_left
thf(fact_44_mult__eq__0__iff, axiom,
    ((![A : nat, B : nat]: (((times_times_nat @ A @ B) = zero_zero_nat) = (((A = zero_zero_nat)) | ((B = zero_zero_nat))))))). % mult_eq_0_iff
thf(fact_45_mult__eq__0__iff, axiom,
    ((![A : real, B : real]: (((times_times_real @ A @ B) = zero_zero_real) = (((A = zero_zero_real)) | ((B = zero_zero_real))))))). % mult_eq_0_iff
thf(fact_46_mult__eq__0__iff, axiom,
    ((![A : complex, B : complex]: (((times_times_complex @ A @ B) = zero_zero_complex) = (((A = zero_zero_complex)) | ((B = zero_zero_complex))))))). % mult_eq_0_iff
thf(fact_47_mult__zero__right, axiom,
    ((![A : nat]: ((times_times_nat @ A @ zero_zero_nat) = zero_zero_nat)))). % mult_zero_right
thf(fact_48_mult__zero__right, axiom,
    ((![A : real]: ((times_times_real @ A @ zero_zero_real) = zero_zero_real)))). % mult_zero_right
thf(fact_49_mult__zero__right, axiom,
    ((![A : complex]: ((times_times_complex @ A @ zero_zero_complex) = zero_zero_complex)))). % mult_zero_right
thf(fact_50_mult__zero__left, axiom,
    ((![A : nat]: ((times_times_nat @ zero_zero_nat @ A) = zero_zero_nat)))). % mult_zero_left
thf(fact_51_mult__zero__left, axiom,
    ((![A : real]: ((times_times_real @ zero_zero_real @ A) = zero_zero_real)))). % mult_zero_left
thf(fact_52_mult__zero__left, axiom,
    ((![A : complex]: ((times_times_complex @ zero_zero_complex @ A) = zero_zero_complex)))). % mult_zero_left
thf(fact_53_mult__minus__right, axiom,
    ((![A : complex, B : complex]: ((times_times_complex @ A @ (uminus1204672759omplex @ B)) = (uminus1204672759omplex @ (times_times_complex @ A @ B)))))). % mult_minus_right
thf(fact_54_minus__mult__minus, axiom,
    ((![A : complex, B : complex]: ((times_times_complex @ (uminus1204672759omplex @ A) @ (uminus1204672759omplex @ B)) = (times_times_complex @ A @ B))))). % minus_mult_minus
thf(fact_55_mult__minus__left, axiom,
    ((![A : complex, B : complex]: ((times_times_complex @ (uminus1204672759omplex @ A) @ B) = (uminus1204672759omplex @ (times_times_complex @ A @ B)))))). % mult_minus_left
thf(fact_56_abs__0, axiom,
    (((abs_abs_complex @ zero_zero_complex) = zero_zero_complex))). % abs_0
thf(fact_57_abs__0, axiom,
    (((abs_abs_real @ zero_zero_real) = zero_zero_real))). % abs_0
thf(fact_58_norm__zero, axiom,
    (((real_V638595069omplex @ zero_zero_complex) = zero_zero_real))). % norm_zero
thf(fact_59_norm__zero, axiom,
    (((real_V646646907m_real @ zero_zero_real) = zero_zero_real))). % norm_zero
thf(fact_60_norm__eq__zero, axiom,
    ((![X : complex]: (((real_V638595069omplex @ X) = zero_zero_real) = (X = zero_zero_complex))))). % norm_eq_zero
thf(fact_61_norm__eq__zero, axiom,
    ((![X : real]: (((real_V646646907m_real @ X) = zero_zero_real) = (X = zero_zero_real))))). % norm_eq_zero
thf(fact_62_abs__mult__self__eq, axiom,
    ((![A : real]: ((times_times_real @ (abs_abs_real @ A) @ (abs_abs_real @ A)) = (times_times_real @ A @ A))))). % abs_mult_self_eq
thf(fact_63_abs__1, axiom,
    (((abs_abs_complex @ one_one_complex) = one_one_complex))). % abs_1
thf(fact_64_abs__1, axiom,
    (((abs_abs_real @ one_one_real) = one_one_real))). % abs_1
thf(fact_65_abs__minus, axiom,
    ((![A : real]: ((abs_abs_real @ (uminus_uminus_real @ A)) = (abs_abs_real @ A))))). % abs_minus
thf(fact_66_abs__minus, axiom,
    ((![A : complex]: ((abs_abs_complex @ (uminus1204672759omplex @ A)) = (abs_abs_complex @ A))))). % abs_minus
thf(fact_67_norm__minus__cancel, axiom,
    ((![X : complex]: ((real_V638595069omplex @ (uminus1204672759omplex @ X)) = (real_V638595069omplex @ X))))). % norm_minus_cancel
thf(fact_68_norm__minus__cancel, axiom,
    ((![X : real]: ((real_V646646907m_real @ (uminus_uminus_real @ X)) = (real_V646646907m_real @ X))))). % norm_minus_cancel
thf(fact_69_of__real__0, axiom,
    (((real_V1205483228l_real @ zero_zero_real) = zero_zero_real))). % of_real_0
thf(fact_70_of__real__0, axiom,
    (((real_V306493662omplex @ zero_zero_real) = zero_zero_complex))). % of_real_0
thf(fact_71_of__real__eq__0__iff, axiom,
    ((![X : real]: (((real_V1205483228l_real @ X) = zero_zero_real) = (X = zero_zero_real))))). % of_real_eq_0_iff
thf(fact_72_of__real__eq__0__iff, axiom,
    ((![X : real]: (((real_V306493662omplex @ X) = zero_zero_complex) = (X = zero_zero_real))))). % of_real_eq_0_iff
thf(fact_73_of__real__mult, axiom,
    ((![X : real, Y : real]: ((real_V306493662omplex @ (times_times_real @ X @ Y)) = (times_times_complex @ (real_V306493662omplex @ X) @ (real_V306493662omplex @ Y)))))). % of_real_mult
thf(fact_74_of__real__minus, axiom,
    ((![X : real]: ((real_V306493662omplex @ (uminus_uminus_real @ X)) = (uminus1204672759omplex @ (real_V306493662omplex @ X)))))). % of_real_minus
thf(fact_75_minus__of__real__eq__of__real__iff, axiom,
    ((![X : real, Y : real]: (((uminus1204672759omplex @ (real_V306493662omplex @ X)) = (real_V306493662omplex @ Y)) = ((uminus_uminus_real @ X) = Y))))). % minus_of_real_eq_of_real_iff
thf(fact_76_of__real__eq__minus__of__real__iff, axiom,
    ((![X : real, Y : real]: (((real_V306493662omplex @ X) = (uminus1204672759omplex @ (real_V306493662omplex @ Y))) = (X = (uminus_uminus_real @ Y)))))). % of_real_eq_minus_of_real_iff
thf(fact_77_abs__norm__cancel, axiom,
    ((![A : complex]: ((abs_abs_real @ (real_V638595069omplex @ A)) = (real_V638595069omplex @ A))))). % abs_norm_cancel
thf(fact_78_abs__norm__cancel, axiom,
    ((![A : real]: ((abs_abs_real @ (real_V646646907m_real @ A)) = (real_V646646907m_real @ A))))). % abs_norm_cancel
thf(fact_79_m_I1_J, axiom,
    ((ord_less_real @ zero_zero_real @ m))). % m(1)
thf(fact_80_less_Oprems, axiom,
    ((~ ((fundam1158420650omplex @ (poly_complex2 @ pa)))))). % less.prems
thf(fact_81__092_060open_0621_A_L_Aw_A_094_Ak_A_K_Aa_A_N_A1_A_061_A0_A_N_A1_092_060close_062, axiom,
    (((minus_minus_complex @ (plus_plus_complex @ one_one_complex @ (times_times_complex @ (power_power_complex @ w @ k) @ a)) @ one_one_complex) = (minus_minus_complex @ zero_zero_complex @ one_one_complex)))). % \<open>1 + w ^ k * a - 1 = 0 - 1\<close>
thf(fact_82_mult__cancel__right2, axiom,
    ((![A : real, C : real]: (((times_times_real @ A @ C) = C) = (((C = zero_zero_real)) | ((A = one_one_real))))))). % mult_cancel_right2
thf(fact_83_mult__cancel__right2, axiom,
    ((![A : complex, C : complex]: (((times_times_complex @ A @ C) = C) = (((C = zero_zero_complex)) | ((A = one_one_complex))))))). % mult_cancel_right2
thf(fact_84_mult__cancel__right1, axiom,
    ((![C : real, B : real]: ((C = (times_times_real @ B @ C)) = (((C = zero_zero_real)) | ((B = one_one_real))))))). % mult_cancel_right1
thf(fact_85_mult__cancel__right1, axiom,
    ((![C : complex, B : complex]: ((C = (times_times_complex @ B @ C)) = (((C = zero_zero_complex)) | ((B = one_one_complex))))))). % mult_cancel_right1
thf(fact_86_mult__cancel__left2, axiom,
    ((![C : real, A : real]: (((times_times_real @ C @ A) = C) = (((C = zero_zero_real)) | ((A = one_one_real))))))). % mult_cancel_left2
thf(fact_87_mult__cancel__left2, axiom,
    ((![C : complex, A : complex]: (((times_times_complex @ C @ A) = C) = (((C = zero_zero_complex)) | ((A = one_one_complex))))))). % mult_cancel_left2
thf(fact_88_mult__cancel__left1, axiom,
    ((![C : real, B : real]: ((C = (times_times_real @ C @ B)) = (((C = zero_zero_real)) | ((B = one_one_real))))))). % mult_cancel_left1
thf(fact_89_mult__cancel__left1, axiom,
    ((![C : complex, B : complex]: ((C = (times_times_complex @ C @ B)) = (((C = zero_zero_complex)) | ((B = one_one_complex))))))). % mult_cancel_left1
thf(fact_90_zero__less__norm__iff, axiom,
    ((![X : complex]: ((ord_less_real @ zero_zero_real @ (real_V638595069omplex @ X)) = (~ ((X = zero_zero_complex))))))). % zero_less_norm_iff
thf(fact_91_zero__less__norm__iff, axiom,
    ((![X : real]: ((ord_less_real @ zero_zero_real @ (real_V646646907m_real @ X)) = (~ ((X = zero_zero_real))))))). % zero_less_norm_iff
thf(fact_92_norm__le__zero__iff, axiom,
    ((![X : complex]: ((ord_less_eq_real @ (real_V638595069omplex @ X) @ zero_zero_real) = (X = zero_zero_complex))))). % norm_le_zero_iff
thf(fact_93_norm__le__zero__iff, axiom,
    ((![X : real]: ((ord_less_eq_real @ (real_V646646907m_real @ X) @ zero_zero_real) = (X = zero_zero_real))))). % norm_le_zero_iff
thf(fact_94_abs__if, axiom,
    ((abs_abs_real = (^[A2 : real]: (if_real @ (ord_less_real @ A2 @ zero_zero_real) @ (uminus_uminus_real @ A2) @ A2))))). % abs_if
thf(fact_95_zero__less__one, axiom,
    ((ord_less_nat @ zero_zero_nat @ one_one_nat))). % zero_less_one
thf(fact_96_zero__less__one, axiom,
    ((ord_less_real @ zero_zero_real @ one_one_real))). % zero_less_one
thf(fact_97_abs__less__iff, axiom,
    ((![A : real, B : real]: ((ord_less_real @ (abs_abs_real @ A) @ B) = (((ord_less_real @ A @ B)) & ((ord_less_real @ (uminus_uminus_real @ A) @ B))))))). % abs_less_iff
thf(fact_98_add__less__zeroD, axiom,
    ((![X : real, Y : real]: ((ord_less_real @ (plus_plus_real @ X @ Y) @ zero_zero_real) => ((ord_less_real @ X @ zero_zero_real) | (ord_less_real @ Y @ zero_zero_real)))))). % add_less_zeroD
thf(fact_99_mult__neg__neg, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ zero_zero_real) => ((ord_less_real @ B @ zero_zero_real) => (ord_less_real @ zero_zero_real @ (times_times_real @ A @ B))))))). % mult_neg_neg
thf(fact_100_not__square__less__zero, axiom,
    ((![A : real]: (~ ((ord_less_real @ (times_times_real @ A @ A) @ zero_zero_real)))))). % not_square_less_zero
thf(fact_101_mult__less__0__iff, axiom,
    ((![A : real, B : real]: ((ord_less_real @ (times_times_real @ A @ B) @ zero_zero_real) = (((((ord_less_real @ zero_zero_real @ A)) & ((ord_less_real @ B @ zero_zero_real)))) | ((((ord_less_real @ A @ zero_zero_real)) & ((ord_less_real @ zero_zero_real @ B))))))))). % mult_less_0_iff
thf(fact_102_mult__neg__pos, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ zero_zero_nat) => ((ord_less_nat @ zero_zero_nat @ B) => (ord_less_nat @ (times_times_nat @ A @ B) @ zero_zero_nat)))))). % mult_neg_pos
thf(fact_103_mult__neg__pos, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ zero_zero_real) => ((ord_less_real @ zero_zero_real @ B) => (ord_less_real @ (times_times_real @ A @ B) @ zero_zero_real)))))). % mult_neg_pos
thf(fact_104_mult__pos__neg, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ zero_zero_nat @ A) => ((ord_less_nat @ B @ zero_zero_nat) => (ord_less_nat @ (times_times_nat @ A @ B) @ zero_zero_nat)))))). % mult_pos_neg
thf(fact_105_mult__pos__neg, axiom,
    ((![A : real, B : real]: ((ord_less_real @ zero_zero_real @ A) => ((ord_less_real @ B @ zero_zero_real) => (ord_less_real @ (times_times_real @ A @ B) @ zero_zero_real)))))). % mult_pos_neg
thf(fact_106_mult__pos__pos, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ zero_zero_nat @ A) => ((ord_less_nat @ zero_zero_nat @ B) => (ord_less_nat @ zero_zero_nat @ (times_times_nat @ A @ B))))))). % mult_pos_pos
thf(fact_107_mult__pos__pos, axiom,
    ((![A : real, B : real]: ((ord_less_real @ zero_zero_real @ A) => ((ord_less_real @ zero_zero_real @ B) => (ord_less_real @ zero_zero_real @ (times_times_real @ A @ B))))))). % mult_pos_pos
thf(fact_108_mult__pos__neg2, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ zero_zero_nat @ A) => ((ord_less_nat @ B @ zero_zero_nat) => (ord_less_nat @ (times_times_nat @ B @ A) @ zero_zero_nat)))))). % mult_pos_neg2
thf(fact_109_mult__pos__neg2, axiom,
    ((![A : real, B : real]: ((ord_less_real @ zero_zero_real @ A) => ((ord_less_real @ B @ zero_zero_real) => (ord_less_real @ (times_times_real @ B @ A) @ zero_zero_real)))))). % mult_pos_neg2
thf(fact_110_zero__less__mult__iff, axiom,
    ((![A : real, B : real]: ((ord_less_real @ zero_zero_real @ (times_times_real @ A @ B)) = (((((ord_less_real @ zero_zero_real @ A)) & ((ord_less_real @ zero_zero_real @ B)))) | ((((ord_less_real @ A @ zero_zero_real)) & ((ord_less_real @ B @ zero_zero_real))))))))). % zero_less_mult_iff
thf(fact_111_linorder__neqE__linordered__idom, axiom,
    ((![X : real, Y : real]: ((~ ((X = Y))) => ((~ ((ord_less_real @ X @ Y))) => (ord_less_real @ Y @ X)))))). % linorder_neqE_linordered_idom
thf(fact_112_not__one__less__zero, axiom,
    ((~ ((ord_less_nat @ one_one_nat @ zero_zero_nat))))). % not_one_less_zero
thf(fact_113_not__one__less__zero, axiom,
    ((~ ((ord_less_real @ one_one_real @ zero_zero_real))))). % not_one_less_zero
thf(fact_114_zero__less__mult__pos, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ zero_zero_nat @ (times_times_nat @ A @ B)) => ((ord_less_nat @ zero_zero_nat @ A) => (ord_less_nat @ zero_zero_nat @ B)))))). % zero_less_mult_pos
thf(fact_115_zero__less__mult__pos, axiom,
    ((![A : real, B : real]: ((ord_less_real @ zero_zero_real @ (times_times_real @ A @ B)) => ((ord_less_real @ zero_zero_real @ A) => (ord_less_real @ zero_zero_real @ B)))))). % zero_less_mult_pos
thf(fact_116_zero__less__mult__pos2, axiom,
    ((![B : nat, A : nat]: ((ord_less_nat @ zero_zero_nat @ (times_times_nat @ B @ A)) => ((ord_less_nat @ zero_zero_nat @ A) => (ord_less_nat @ zero_zero_nat @ B)))))). % zero_less_mult_pos2
thf(fact_117_zero__less__mult__pos2, axiom,
    ((![B : real, A : real]: ((ord_less_real @ zero_zero_real @ (times_times_real @ B @ A)) => ((ord_less_real @ zero_zero_real @ A) => (ord_less_real @ zero_zero_real @ B)))))). % zero_less_mult_pos2
thf(fact_118_mult__less__cancel__left__neg, axiom,
    ((![C : real, A : real, B : real]: ((ord_less_real @ C @ zero_zero_real) => ((ord_less_real @ (times_times_real @ C @ A) @ (times_times_real @ C @ B)) = (ord_less_real @ B @ A)))))). % mult_less_cancel_left_neg
thf(fact_119_mult__less__cancel__left__pos, axiom,
    ((![C : real, A : real, B : real]: ((ord_less_real @ zero_zero_real @ C) => ((ord_less_real @ (times_times_real @ C @ A) @ (times_times_real @ C @ B)) = (ord_less_real @ A @ B)))))). % mult_less_cancel_left_pos
thf(fact_120_mult__strict__left__mono__neg, axiom,
    ((![B : real, A : real, C : real]: ((ord_less_real @ B @ A) => ((ord_less_real @ C @ zero_zero_real) => (ord_less_real @ (times_times_real @ C @ A) @ (times_times_real @ C @ B))))))). % mult_strict_left_mono_neg
thf(fact_121_mult__strict__left__mono, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_nat @ A @ B) => ((ord_less_nat @ zero_zero_nat @ C) => (ord_less_nat @ (times_times_nat @ C @ A) @ (times_times_nat @ C @ B))))))). % mult_strict_left_mono
thf(fact_122_mult__strict__left__mono, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_real @ A @ B) => ((ord_less_real @ zero_zero_real @ C) => (ord_less_real @ (times_times_real @ C @ A) @ (times_times_real @ C @ B))))))). % mult_strict_left_mono
thf(fact_123_mult__less__cancel__left__disj, axiom,
    ((![C : real, A : real, B : real]: ((ord_less_real @ (times_times_real @ C @ A) @ (times_times_real @ C @ B)) = (((((ord_less_real @ zero_zero_real @ C)) & ((ord_less_real @ A @ B)))) | ((((ord_less_real @ C @ zero_zero_real)) & ((ord_less_real @ B @ A))))))))). % mult_less_cancel_left_disj
thf(fact_124_mult__strict__right__mono__neg, axiom,
    ((![B : real, A : real, C : real]: ((ord_less_real @ B @ A) => ((ord_less_real @ C @ zero_zero_real) => (ord_less_real @ (times_times_real @ A @ C) @ (times_times_real @ B @ C))))))). % mult_strict_right_mono_neg
thf(fact_125_mult__strict__right__mono, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_nat @ A @ B) => ((ord_less_nat @ zero_zero_nat @ C) => (ord_less_nat @ (times_times_nat @ A @ C) @ (times_times_nat @ B @ C))))))). % mult_strict_right_mono
thf(fact_126_mult__strict__right__mono, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_real @ A @ B) => ((ord_less_real @ zero_zero_real @ C) => (ord_less_real @ (times_times_real @ A @ C) @ (times_times_real @ B @ C))))))). % mult_strict_right_mono
thf(fact_127_mult__less__cancel__right__disj, axiom,
    ((![A : real, C : real, B : real]: ((ord_less_real @ (times_times_real @ A @ C) @ (times_times_real @ B @ C)) = (((((ord_less_real @ zero_zero_real @ C)) & ((ord_less_real @ A @ B)))) | ((((ord_less_real @ C @ zero_zero_real)) & ((ord_less_real @ B @ A))))))))). % mult_less_cancel_right_disj
thf(fact_128_real__sup__exists, axiom,
    ((![P2 : real > $o]: ((?[X_1 : real]: (P2 @ X_1)) => ((?[Z : real]: (![X2 : real]: ((P2 @ X2) => (ord_less_real @ X2 @ Z)))) => (?[S : real]: (![Y2 : real]: ((?[X3 : real]: (((P2 @ X3)) & ((ord_less_real @ Y2 @ X3)))) = (ord_less_real @ Y2 @ S))))))))). % real_sup_exists
thf(fact_129_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_nat @ A @ B) => ((ord_less_nat @ zero_zero_nat @ C) => (ord_less_nat @ (times_times_nat @ C @ A) @ (times_times_nat @ C @ B))))))). % linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_130_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_real @ A @ B) => ((ord_less_real @ zero_zero_real @ C) => (ord_less_real @ (times_times_real @ C @ A) @ (times_times_real @ C @ B))))))). % linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_131_minus__mult__commute, axiom,
    ((![A : complex, B : complex]: ((times_times_complex @ (uminus1204672759omplex @ A) @ B) = (times_times_complex @ A @ (uminus1204672759omplex @ B)))))). % minus_mult_commute
thf(fact_132_square__eq__iff, axiom,
    ((![A : complex, B : complex]: (((times_times_complex @ A @ A) = (times_times_complex @ B @ B)) = (((A = B)) | ((A = (uminus1204672759omplex @ B)))))))). % square_eq_iff
thf(fact_133_abs__eq__iff, axiom,
    ((![X : real, Y : real]: (((abs_abs_real @ X) = (abs_abs_real @ Y)) = (((X = Y)) | ((X = (uminus_uminus_real @ Y)))))))). % abs_eq_iff
thf(fact_134_mult__right__cancel, axiom,
    ((![C : nat, A : nat, B : nat]: ((~ ((C = zero_zero_nat))) => (((times_times_nat @ A @ C) = (times_times_nat @ B @ C)) = (A = B)))))). % mult_right_cancel
thf(fact_135_mult__right__cancel, axiom,
    ((![C : real, A : real, B : real]: ((~ ((C = zero_zero_real))) => (((times_times_real @ A @ C) = (times_times_real @ B @ C)) = (A = B)))))). % mult_right_cancel
thf(fact_136_mult__right__cancel, axiom,
    ((![C : complex, A : complex, B : complex]: ((~ ((C = zero_zero_complex))) => (((times_times_complex @ A @ C) = (times_times_complex @ B @ C)) = (A = B)))))). % mult_right_cancel
thf(fact_137_mult__left__cancel, axiom,
    ((![C : nat, A : nat, B : nat]: ((~ ((C = zero_zero_nat))) => (((times_times_nat @ C @ A) = (times_times_nat @ C @ B)) = (A = B)))))). % mult_left_cancel
thf(fact_138_mult__left__cancel, axiom,
    ((![C : real, A : real, B : real]: ((~ ((C = zero_zero_real))) => (((times_times_real @ C @ A) = (times_times_real @ C @ B)) = (A = B)))))). % mult_left_cancel
thf(fact_139_mult__left__cancel, axiom,
    ((![C : complex, A : complex, B : complex]: ((~ ((C = zero_zero_complex))) => (((times_times_complex @ C @ A) = (times_times_complex @ C @ B)) = (A = B)))))). % mult_left_cancel
thf(fact_140_no__zero__divisors, axiom,
    ((![A : nat, B : nat]: ((~ ((A = zero_zero_nat))) => ((~ ((B = zero_zero_nat))) => (~ (((times_times_nat @ A @ B) = zero_zero_nat)))))))). % no_zero_divisors
thf(fact_141_no__zero__divisors, axiom,
    ((![A : real, B : real]: ((~ ((A = zero_zero_real))) => ((~ ((B = zero_zero_real))) => (~ (((times_times_real @ A @ B) = zero_zero_real)))))))). % no_zero_divisors
thf(fact_142_no__zero__divisors, axiom,
    ((![A : complex, B : complex]: ((~ ((A = zero_zero_complex))) => ((~ ((B = zero_zero_complex))) => (~ (((times_times_complex @ A @ B) = zero_zero_complex)))))))). % no_zero_divisors
thf(fact_143_divisors__zero, axiom,
    ((![A : nat, B : nat]: (((times_times_nat @ A @ B) = zero_zero_nat) => ((A = zero_zero_nat) | (B = zero_zero_nat)))))). % divisors_zero
thf(fact_144_divisors__zero, axiom,
    ((![A : real, B : real]: (((times_times_real @ A @ B) = zero_zero_real) => ((A = zero_zero_real) | (B = zero_zero_real)))))). % divisors_zero
thf(fact_145_divisors__zero, axiom,
    ((![A : complex, B : complex]: (((times_times_complex @ A @ B) = zero_zero_complex) => ((A = zero_zero_complex) | (B = zero_zero_complex)))))). % divisors_zero
thf(fact_146_mult__not__zero, axiom,
    ((![A : nat, B : nat]: ((~ (((times_times_nat @ A @ B) = zero_zero_nat))) => ((~ ((A = zero_zero_nat))) & (~ ((B = zero_zero_nat)))))))). % mult_not_zero
thf(fact_147_mult__not__zero, axiom,
    ((![A : real, B : real]: ((~ (((times_times_real @ A @ B) = zero_zero_real))) => ((~ ((A = zero_zero_real))) & (~ ((B = zero_zero_real)))))))). % mult_not_zero
thf(fact_148_mult__not__zero, axiom,
    ((![A : complex, B : complex]: ((~ (((times_times_complex @ A @ B) = zero_zero_complex))) => ((~ ((A = zero_zero_complex))) & (~ ((B = zero_zero_complex)))))))). % mult_not_zero
thf(fact_149_zero__neq__one, axiom,
    ((~ ((zero_zero_nat = one_one_nat))))). % zero_neq_one
thf(fact_150_zero__neq__one, axiom,
    ((~ ((zero_zero_complex = one_one_complex))))). % zero_neq_one
thf(fact_151_zero__neq__one, axiom,
    ((~ ((zero_zero_real = one_one_real))))). % zero_neq_one
thf(fact_152_real__norm__def, axiom,
    ((real_V646646907m_real = abs_abs_real))). % real_norm_def
thf(fact_153_abs__eq__0__iff, axiom,
    ((![A : complex]: (((abs_abs_complex @ A) = zero_zero_complex) = (A = zero_zero_complex))))). % abs_eq_0_iff
thf(fact_154_abs__eq__0__iff, axiom,
    ((![A : real]: (((abs_abs_real @ A) = zero_zero_real) = (A = zero_zero_real))))). % abs_eq_0_iff
thf(fact_155_mult__less__le__imp__less, axiom,
    ((![A : nat, B : nat, C : nat, D : nat]: ((ord_less_nat @ A @ B) => ((ord_less_eq_nat @ C @ D) => ((ord_less_eq_nat @ zero_zero_nat @ A) => ((ord_less_nat @ zero_zero_nat @ C) => (ord_less_nat @ (times_times_nat @ A @ C) @ (times_times_nat @ B @ D))))))))). % mult_less_le_imp_less
thf(fact_156_mult__less__le__imp__less, axiom,
    ((![A : real, B : real, C : real, D : real]: ((ord_less_real @ A @ B) => ((ord_less_eq_real @ C @ D) => ((ord_less_eq_real @ zero_zero_real @ A) => ((ord_less_real @ zero_zero_real @ C) => (ord_less_real @ (times_times_real @ A @ C) @ (times_times_real @ B @ D))))))))). % mult_less_le_imp_less
thf(fact_157_mult__le__less__imp__less, axiom,
    ((![A : nat, B : nat, C : nat, D : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_nat @ C @ D) => ((ord_less_nat @ zero_zero_nat @ A) => ((ord_less_eq_nat @ zero_zero_nat @ C) => (ord_less_nat @ (times_times_nat @ A @ C) @ (times_times_nat @ B @ D))))))))). % mult_le_less_imp_less
thf(fact_158_mult__le__less__imp__less, axiom,
    ((![A : real, B : real, C : real, D : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_real @ C @ D) => ((ord_less_real @ zero_zero_real @ A) => ((ord_less_eq_real @ zero_zero_real @ C) => (ord_less_real @ (times_times_real @ A @ C) @ (times_times_real @ B @ D))))))))). % mult_le_less_imp_less
thf(fact_159_mult__right__le__imp__le, axiom,
    ((![A : nat, C : nat, B : nat]: ((ord_less_eq_nat @ (times_times_nat @ A @ C) @ (times_times_nat @ B @ C)) => ((ord_less_nat @ zero_zero_nat @ C) => (ord_less_eq_nat @ A @ B)))))). % mult_right_le_imp_le
thf(fact_160_mult__right__le__imp__le, axiom,
    ((![A : real, C : real, B : real]: ((ord_less_eq_real @ (times_times_real @ A @ C) @ (times_times_real @ B @ C)) => ((ord_less_real @ zero_zero_real @ C) => (ord_less_eq_real @ A @ B)))))). % mult_right_le_imp_le
thf(fact_161_mult__left__le__imp__le, axiom,
    ((![C : nat, A : nat, B : nat]: ((ord_less_eq_nat @ (times_times_nat @ C @ A) @ (times_times_nat @ C @ B)) => ((ord_less_nat @ zero_zero_nat @ C) => (ord_less_eq_nat @ A @ B)))))). % mult_left_le_imp_le
thf(fact_162_mult__left__le__imp__le, axiom,
    ((![C : real, A : real, B : real]: ((ord_less_eq_real @ (times_times_real @ C @ A) @ (times_times_real @ C @ B)) => ((ord_less_real @ zero_zero_real @ C) => (ord_less_eq_real @ A @ B)))))). % mult_left_le_imp_le
thf(fact_163_mult__le__cancel__left__pos, axiom,
    ((![C : real, A : real, B : real]: ((ord_less_real @ zero_zero_real @ C) => ((ord_less_eq_real @ (times_times_real @ C @ A) @ (times_times_real @ C @ B)) = (ord_less_eq_real @ A @ B)))))). % mult_le_cancel_left_pos
thf(fact_164_mult__le__cancel__left__neg, axiom,
    ((![C : real, A : real, B : real]: ((ord_less_real @ C @ zero_zero_real) => ((ord_less_eq_real @ (times_times_real @ C @ A) @ (times_times_real @ C @ B)) = (ord_less_eq_real @ B @ A)))))). % mult_le_cancel_left_neg
thf(fact_165_mult__less__cancel__right, axiom,
    ((![A : real, C : real, B : real]: ((ord_less_real @ (times_times_real @ A @ C) @ (times_times_real @ B @ C)) = (((((ord_less_eq_real @ zero_zero_real @ C)) => ((ord_less_real @ A @ B)))) & ((((ord_less_eq_real @ C @ zero_zero_real)) => ((ord_less_real @ B @ A))))))))). % mult_less_cancel_right
thf(fact_166_mult__strict__mono_H, axiom,
    ((![A : nat, B : nat, C : nat, D : nat]: ((ord_less_nat @ A @ B) => ((ord_less_nat @ C @ D) => ((ord_less_eq_nat @ zero_zero_nat @ A) => ((ord_less_eq_nat @ zero_zero_nat @ C) => (ord_less_nat @ (times_times_nat @ A @ C) @ (times_times_nat @ B @ D))))))))). % mult_strict_mono'
thf(fact_167_mult__strict__mono_H, axiom,
    ((![A : real, B : real, C : real, D : real]: ((ord_less_real @ A @ B) => ((ord_less_real @ C @ D) => ((ord_less_eq_real @ zero_zero_real @ A) => ((ord_less_eq_real @ zero_zero_real @ C) => (ord_less_real @ (times_times_real @ A @ C) @ (times_times_real @ B @ D))))))))). % mult_strict_mono'
thf(fact_168_mult__right__less__imp__less, axiom,
    ((![A : nat, C : nat, B : nat]: ((ord_less_nat @ (times_times_nat @ A @ C) @ (times_times_nat @ B @ C)) => ((ord_less_eq_nat @ zero_zero_nat @ C) => (ord_less_nat @ A @ B)))))). % mult_right_less_imp_less
thf(fact_169_mult__right__less__imp__less, axiom,
    ((![A : real, C : real, B : real]: ((ord_less_real @ (times_times_real @ A @ C) @ (times_times_real @ B @ C)) => ((ord_less_eq_real @ zero_zero_real @ C) => (ord_less_real @ A @ B)))))). % mult_right_less_imp_less
thf(fact_170_mult__less__cancel__left, axiom,
    ((![C : real, A : real, B : real]: ((ord_less_real @ (times_times_real @ C @ A) @ (times_times_real @ C @ B)) = (((((ord_less_eq_real @ zero_zero_real @ C)) => ((ord_less_real @ A @ B)))) & ((((ord_less_eq_real @ C @ zero_zero_real)) => ((ord_less_real @ B @ A))))))))). % mult_less_cancel_left
thf(fact_171_mult__strict__mono, axiom,
    ((![A : nat, B : nat, C : nat, D : nat]: ((ord_less_nat @ A @ B) => ((ord_less_nat @ C @ D) => ((ord_less_nat @ zero_zero_nat @ B) => ((ord_less_eq_nat @ zero_zero_nat @ C) => (ord_less_nat @ (times_times_nat @ A @ C) @ (times_times_nat @ B @ D))))))))). % mult_strict_mono
thf(fact_172_mult__strict__mono, axiom,
    ((![A : real, B : real, C : real, D : real]: ((ord_less_real @ A @ B) => ((ord_less_real @ C @ D) => ((ord_less_real @ zero_zero_real @ B) => ((ord_less_eq_real @ zero_zero_real @ C) => (ord_less_real @ (times_times_real @ A @ C) @ (times_times_real @ B @ D))))))))). % mult_strict_mono
thf(fact_173_mult__left__less__imp__less, axiom,
    ((![C : nat, A : nat, B : nat]: ((ord_less_nat @ (times_times_nat @ C @ A) @ (times_times_nat @ C @ B)) => ((ord_less_eq_nat @ zero_zero_nat @ C) => (ord_less_nat @ A @ B)))))). % mult_left_less_imp_less
thf(fact_174_mult__left__less__imp__less, axiom,
    ((![C : real, A : real, B : real]: ((ord_less_real @ (times_times_real @ C @ A) @ (times_times_real @ C @ B)) => ((ord_less_eq_real @ zero_zero_real @ C) => (ord_less_real @ A @ B)))))). % mult_left_less_imp_less
thf(fact_175_mult__le__cancel__right, axiom,
    ((![A : real, C : real, B : real]: ((ord_less_eq_real @ (times_times_real @ A @ C) @ (times_times_real @ B @ C)) = (((((ord_less_real @ zero_zero_real @ C)) => ((ord_less_eq_real @ A @ B)))) & ((((ord_less_real @ C @ zero_zero_real)) => ((ord_less_eq_real @ B @ A))))))))). % mult_le_cancel_right
thf(fact_176_mult__le__cancel__left, axiom,
    ((![C : real, A : real, B : real]: ((ord_less_eq_real @ (times_times_real @ C @ A) @ (times_times_real @ C @ B)) = (((((ord_less_real @ zero_zero_real @ C)) => ((ord_less_eq_real @ A @ B)))) & ((((ord_less_real @ C @ zero_zero_real)) => ((ord_less_eq_real @ B @ A))))))))). % mult_le_cancel_left
thf(fact_177_not__sum__squares__lt__zero, axiom,
    ((![X : real, Y : real]: (~ ((ord_less_real @ (plus_plus_real @ (times_times_real @ X @ X) @ (times_times_real @ Y @ Y)) @ zero_zero_real)))))). % not_sum_squares_lt_zero
thf(fact_178_zero__less__two, axiom,
    ((ord_less_nat @ zero_zero_nat @ (plus_plus_nat @ one_one_nat @ one_one_nat)))). % zero_less_two
thf(fact_179_zero__less__two, axiom,
    ((ord_less_real @ zero_zero_real @ (plus_plus_real @ one_one_real @ one_one_real)))). % zero_less_two
thf(fact_180_eq__abs__iff_H, axiom,
    ((![A : real, B : real]: ((A = (abs_abs_real @ B)) = (((ord_less_eq_real @ zero_zero_real @ A)) & ((((B = A)) | ((B = (uminus_uminus_real @ A)))))))))). % eq_abs_iff'
thf(fact_181_abs__eq__iff_H, axiom,
    ((![A : real, B : real]: (((abs_abs_real @ A) = B) = (((ord_less_eq_real @ zero_zero_real @ B)) & ((((A = B)) | ((A = (uminus_uminus_real @ B)))))))))). % abs_eq_iff'
thf(fact_182_constant__def, axiom,
    ((fundam1158420650omplex = (^[F : complex > complex]: (![X3 : complex]: (![Y3 : complex]: ((F @ X3) = (F @ Y3)))))))). % constant_def
thf(fact_183_mult__less__cancel__right2, axiom,
    ((![A : real, C : real]: ((ord_less_real @ (times_times_real @ A @ C) @ C) = (((((ord_less_eq_real @ zero_zero_real @ C)) => ((ord_less_real @ A @ one_one_real)))) & ((((ord_less_eq_real @ C @ zero_zero_real)) => ((ord_less_real @ one_one_real @ A))))))))). % mult_less_cancel_right2
thf(fact_184_mult__less__cancel__right1, axiom,
    ((![C : real, B : real]: ((ord_less_real @ C @ (times_times_real @ B @ C)) = (((((ord_less_eq_real @ zero_zero_real @ C)) => ((ord_less_real @ one_one_real @ B)))) & ((((ord_less_eq_real @ C @ zero_zero_real)) => ((ord_less_real @ B @ one_one_real))))))))). % mult_less_cancel_right1
thf(fact_185_mult__less__cancel__left2, axiom,
    ((![C : real, A : real]: ((ord_less_real @ (times_times_real @ C @ A) @ C) = (((((ord_less_eq_real @ zero_zero_real @ C)) => ((ord_less_real @ A @ one_one_real)))) & ((((ord_less_eq_real @ C @ zero_zero_real)) => ((ord_less_real @ one_one_real @ A))))))))). % mult_less_cancel_left2
thf(fact_186_mult__less__cancel__left1, axiom,
    ((![C : real, B : real]: ((ord_less_real @ C @ (times_times_real @ C @ B)) = (((((ord_less_eq_real @ zero_zero_real @ C)) => ((ord_less_real @ one_one_real @ B)))) & ((((ord_less_eq_real @ C @ zero_zero_real)) => ((ord_less_real @ B @ one_one_real))))))))). % mult_less_cancel_left1
thf(fact_187_mult__le__cancel__right2, axiom,
    ((![A : real, C : real]: ((ord_less_eq_real @ (times_times_real @ A @ C) @ C) = (((((ord_less_real @ zero_zero_real @ C)) => ((ord_less_eq_real @ A @ one_one_real)))) & ((((ord_less_real @ C @ zero_zero_real)) => ((ord_less_eq_real @ one_one_real @ A))))))))). % mult_le_cancel_right2
thf(fact_188_mult__le__cancel__right1, axiom,
    ((![C : real, B : real]: ((ord_less_eq_real @ C @ (times_times_real @ B @ C)) = (((((ord_less_real @ zero_zero_real @ C)) => ((ord_less_eq_real @ one_one_real @ B)))) & ((((ord_less_real @ C @ zero_zero_real)) => ((ord_less_eq_real @ B @ one_one_real))))))))). % mult_le_cancel_right1
thf(fact_189_mult__le__cancel__left2, axiom,
    ((![C : real, A : real]: ((ord_less_eq_real @ (times_times_real @ C @ A) @ C) = (((((ord_less_real @ zero_zero_real @ C)) => ((ord_less_eq_real @ A @ one_one_real)))) & ((((ord_less_real @ C @ zero_zero_real)) => ((ord_less_eq_real @ one_one_real @ A))))))))). % mult_le_cancel_left2
thf(fact_190_mult__le__cancel__left1, axiom,
    ((![C : real, B : real]: ((ord_less_eq_real @ C @ (times_times_real @ C @ B)) = (((((ord_less_real @ zero_zero_real @ C)) => ((ord_less_eq_real @ one_one_real @ B)))) & ((((ord_less_real @ C @ zero_zero_real)) => ((ord_less_eq_real @ B @ one_one_real))))))))). % mult_le_cancel_left1
thf(fact_191_abs__add__one__gt__zero, axiom,
    ((![X : real]: (ord_less_real @ zero_zero_real @ (plus_plus_real @ one_one_real @ (abs_abs_real @ X)))))). % abs_add_one_gt_zero
thf(fact_192_square__eq__1__iff, axiom,
    ((![X : real]: (((times_times_real @ X @ X) = one_one_real) = (((X = one_one_real)) | ((X = (uminus_uminus_real @ one_one_real)))))))). % square_eq_1_iff
thf(fact_193_square__eq__1__iff, axiom,
    ((![X : complex]: (((times_times_complex @ X @ X) = one_one_complex) = (((X = one_one_complex)) | ((X = (uminus1204672759omplex @ one_one_complex)))))))). % square_eq_1_iff
thf(fact_194_less__1__mult, axiom,
    ((![M : real, N : real]: ((ord_less_real @ one_one_real @ M) => ((ord_less_real @ one_one_real @ N) => (ord_less_real @ one_one_real @ (times_times_real @ M @ N))))))). % less_1_mult
thf(fact_195_add__mono1, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ B) => (ord_less_real @ (plus_plus_real @ A @ one_one_real) @ (plus_plus_real @ B @ one_one_real)))))). % add_mono1
thf(fact_196_less__add__one, axiom,
    ((![A : real]: (ord_less_real @ A @ (plus_plus_real @ A @ one_one_real))))). % less_add_one
thf(fact_197_linordered__semidom__class_Oadd__diff__inverse, axiom,
    ((![A : real, B : real]: ((~ ((ord_less_real @ A @ B))) => ((plus_plus_real @ B @ (minus_minus_real @ A @ B)) = A))))). % linordered_semidom_class.add_diff_inverse
thf(fact_198_abs__mult__less, axiom,
    ((![A : real, C : real, B : real, D : real]: ((ord_less_real @ (abs_abs_real @ A) @ C) => ((ord_less_real @ (abs_abs_real @ B) @ D) => (ord_less_real @ (times_times_real @ (abs_abs_real @ A) @ (abs_abs_real @ B)) @ (times_times_real @ C @ D))))))). % abs_mult_less
thf(fact_199_ordered__comm__semiring__class_Ocomm__mult__left__mono, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_nat @ zero_zero_nat @ C) => (ord_less_eq_nat @ (times_times_nat @ C @ A) @ (times_times_nat @ C @ B))))))). % ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_200_ordered__comm__semiring__class_Ocomm__mult__left__mono, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_eq_real @ zero_zero_real @ C) => (ord_less_eq_real @ (times_times_real @ C @ A) @ (times_times_real @ C @ B))))))). % ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_201_zero__le__mult__iff, axiom,
    ((![A : real, B : real]: ((ord_less_eq_real @ zero_zero_real @ (times_times_real @ A @ B)) = (((((ord_less_eq_real @ zero_zero_real @ A)) & ((ord_less_eq_real @ zero_zero_real @ B)))) | ((((ord_less_eq_real @ A @ zero_zero_real)) & ((ord_less_eq_real @ B @ zero_zero_real))))))))). % zero_le_mult_iff
thf(fact_202_mult__nonneg__nonpos2, axiom,
    ((![A : nat, B : nat]: ((ord_less_eq_nat @ zero_zero_nat @ A) => ((ord_less_eq_nat @ B @ zero_zero_nat) => (ord_less_eq_nat @ (times_times_nat @ B @ A) @ zero_zero_nat)))))). % mult_nonneg_nonpos2
thf(fact_203_mult__nonneg__nonpos2, axiom,
    ((![A : real, B : real]: ((ord_less_eq_real @ zero_zero_real @ A) => ((ord_less_eq_real @ B @ zero_zero_real) => (ord_less_eq_real @ (times_times_real @ B @ A) @ zero_zero_real)))))). % mult_nonneg_nonpos2
thf(fact_204_mult__nonpos__nonneg, axiom,
    ((![A : nat, B : nat]: ((ord_less_eq_nat @ A @ zero_zero_nat) => ((ord_less_eq_nat @ zero_zero_nat @ B) => (ord_less_eq_nat @ (times_times_nat @ A @ B) @ zero_zero_nat)))))). % mult_nonpos_nonneg
thf(fact_205_mult__nonpos__nonneg, axiom,
    ((![A : real, B : real]: ((ord_less_eq_real @ A @ zero_zero_real) => ((ord_less_eq_real @ zero_zero_real @ B) => (ord_less_eq_real @ (times_times_real @ A @ B) @ zero_zero_real)))))). % mult_nonpos_nonneg
thf(fact_206_mult__nonneg__nonpos, axiom,
    ((![A : nat, B : nat]: ((ord_less_eq_nat @ zero_zero_nat @ A) => ((ord_less_eq_nat @ B @ zero_zero_nat) => (ord_less_eq_nat @ (times_times_nat @ A @ B) @ zero_zero_nat)))))). % mult_nonneg_nonpos
thf(fact_207_mult__nonneg__nonpos, axiom,
    ((![A : real, B : real]: ((ord_less_eq_real @ zero_zero_real @ A) => ((ord_less_eq_real @ B @ zero_zero_real) => (ord_less_eq_real @ (times_times_real @ A @ B) @ zero_zero_real)))))). % mult_nonneg_nonpos
thf(fact_208_mult__nonneg__nonneg, axiom,
    ((![A : nat, B : nat]: ((ord_less_eq_nat @ zero_zero_nat @ A) => ((ord_less_eq_nat @ zero_zero_nat @ B) => (ord_less_eq_nat @ zero_zero_nat @ (times_times_nat @ A @ B))))))). % mult_nonneg_nonneg
thf(fact_209_mult__nonneg__nonneg, axiom,
    ((![A : real, B : real]: ((ord_less_eq_real @ zero_zero_real @ A) => ((ord_less_eq_real @ zero_zero_real @ B) => (ord_less_eq_real @ zero_zero_real @ (times_times_real @ A @ B))))))). % mult_nonneg_nonneg
thf(fact_210_split__mult__neg__le, axiom,
    ((![A : nat, B : nat]: ((((ord_less_eq_nat @ zero_zero_nat @ A) & (ord_less_eq_nat @ B @ zero_zero_nat)) | ((ord_less_eq_nat @ A @ zero_zero_nat) & (ord_less_eq_nat @ zero_zero_nat @ B))) => (ord_less_eq_nat @ (times_times_nat @ A @ B) @ zero_zero_nat))))). % split_mult_neg_le
thf(fact_211_split__mult__neg__le, axiom,
    ((![A : real, B : real]: ((((ord_less_eq_real @ zero_zero_real @ A) & (ord_less_eq_real @ B @ zero_zero_real)) | ((ord_less_eq_real @ A @ zero_zero_real) & (ord_less_eq_real @ zero_zero_real @ B))) => (ord_less_eq_real @ (times_times_real @ A @ B) @ zero_zero_real))))). % split_mult_neg_le
thf(fact_212_mult__le__0__iff, axiom,
    ((![A : real, B : real]: ((ord_less_eq_real @ (times_times_real @ A @ B) @ zero_zero_real) = (((((ord_less_eq_real @ zero_zero_real @ A)) & ((ord_less_eq_real @ B @ zero_zero_real)))) | ((((ord_less_eq_real @ A @ zero_zero_real)) & ((ord_less_eq_real @ zero_zero_real @ B))))))))). % mult_le_0_iff
thf(fact_213_mult__right__mono, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_nat @ zero_zero_nat @ C) => (ord_less_eq_nat @ (times_times_nat @ A @ C) @ (times_times_nat @ B @ C))))))). % mult_right_mono
thf(fact_214_mult__right__mono, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_eq_real @ zero_zero_real @ C) => (ord_less_eq_real @ (times_times_real @ A @ C) @ (times_times_real @ B @ C))))))). % mult_right_mono
thf(fact_215_mult__right__mono__neg, axiom,
    ((![B : real, A : real, C : real]: ((ord_less_eq_real @ B @ A) => ((ord_less_eq_real @ C @ zero_zero_real) => (ord_less_eq_real @ (times_times_real @ A @ C) @ (times_times_real @ B @ C))))))). % mult_right_mono_neg
thf(fact_216_mult__left__mono, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_nat @ zero_zero_nat @ C) => (ord_less_eq_nat @ (times_times_nat @ C @ A) @ (times_times_nat @ C @ B))))))). % mult_left_mono
thf(fact_217_mult__left__mono, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_eq_real @ zero_zero_real @ C) => (ord_less_eq_real @ (times_times_real @ C @ A) @ (times_times_real @ C @ B))))))). % mult_left_mono
thf(fact_218_mult__nonpos__nonpos, axiom,
    ((![A : real, B : real]: ((ord_less_eq_real @ A @ zero_zero_real) => ((ord_less_eq_real @ B @ zero_zero_real) => (ord_less_eq_real @ zero_zero_real @ (times_times_real @ A @ B))))))). % mult_nonpos_nonpos
thf(fact_219_mult__left__mono__neg, axiom,
    ((![B : real, A : real, C : real]: ((ord_less_eq_real @ B @ A) => ((ord_less_eq_real @ C @ zero_zero_real) => (ord_less_eq_real @ (times_times_real @ C @ A) @ (times_times_real @ C @ B))))))). % mult_left_mono_neg
thf(fact_220_split__mult__pos__le, axiom,
    ((![A : real, B : real]: ((((ord_less_eq_real @ zero_zero_real @ A) & (ord_less_eq_real @ zero_zero_real @ B)) | ((ord_less_eq_real @ A @ zero_zero_real) & (ord_less_eq_real @ B @ zero_zero_real))) => (ord_less_eq_real @ zero_zero_real @ (times_times_real @ A @ B)))))). % split_mult_pos_le
thf(fact_221_zero__le__square, axiom,
    ((![A : real]: (ord_less_eq_real @ zero_zero_real @ (times_times_real @ A @ A))))). % zero_le_square
thf(fact_222_mult__mono_H, axiom,
    ((![A : nat, B : nat, C : nat, D : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_nat @ C @ D) => ((ord_less_eq_nat @ zero_zero_nat @ A) => ((ord_less_eq_nat @ zero_zero_nat @ C) => (ord_less_eq_nat @ (times_times_nat @ A @ C) @ (times_times_nat @ B @ D))))))))). % mult_mono'
thf(fact_223_mult__mono_H, axiom,
    ((![A : real, B : real, C : real, D : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_eq_real @ C @ D) => ((ord_less_eq_real @ zero_zero_real @ A) => ((ord_less_eq_real @ zero_zero_real @ C) => (ord_less_eq_real @ (times_times_real @ A @ C) @ (times_times_real @ B @ D))))))))). % mult_mono'
thf(fact_224_mult__mono, axiom,
    ((![A : nat, B : nat, C : nat, D : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_nat @ C @ D) => ((ord_less_eq_nat @ zero_zero_nat @ B) => ((ord_less_eq_nat @ zero_zero_nat @ C) => (ord_less_eq_nat @ (times_times_nat @ A @ C) @ (times_times_nat @ B @ D))))))))). % mult_mono
thf(fact_225_mult__mono, axiom,
    ((![A : real, B : real, C : real, D : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_eq_real @ C @ D) => ((ord_less_eq_real @ zero_zero_real @ B) => ((ord_less_eq_real @ zero_zero_real @ C) => (ord_less_eq_real @ (times_times_real @ A @ C) @ (times_times_real @ B @ D))))))))). % mult_mono

% Helper facts (3)
thf(help_If_3_1_If_001t__Real__Oreal_T, axiom,
    ((![P2 : $o]: ((P2 = $true) | (P2 = $false))))).
thf(help_If_2_1_If_001t__Real__Oreal_T, axiom,
    ((![X : real, Y : real]: ((if_real @ $false @ X @ Y) = Y)))).
thf(help_If_1_1_If_001t__Real__Oreal_T, axiom,
    ((![X : real, Y : real]: ((if_real @ $true @ X @ Y) = X)))).

% Conjectures (1)
thf(conj_0, conjecture,
    ((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)))))))).
