% 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_862__5376242_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:33:48.654

% Could-be-implicit typings (7)
thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_It__Complex__Ocomplex_J_J, type,
    poly_poly_complex : $tType).
thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J, type,
    poly_poly_real : $tType).
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__Complex__Ocomplex, type,
    complex : $tType).
thf(ty_n_t__Real__Oreal, type,
    real : $tType).
thf(ty_n_tf__a, type,
    a : $tType).

% Explicit typings (40)
thf(sy_c_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Oconstant_001t__Complex__Ocomplex_001t__Complex__Ocomplex, type,
    fundam1158420650omplex : (complex > complex) > $o).
thf(sy_c_Groups_Oone__class_Oone_001t__Complex__Ocomplex, type,
    one_one_complex : complex).
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__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_Otimes__class_Otimes_001t__Complex__Ocomplex, type,
    times_times_complex : complex > complex > complex).
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__Polynomial__Opoly_It__Complex__Ocomplex_J_J, type,
    times_1460995011omplex : poly_poly_complex > poly_poly_complex > poly_poly_complex).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J, type,
    times_614468161y_real : poly_poly_real > poly_poly_real > poly_poly_real).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Polynomial__Opoly_It__Real__Oreal_J, type,
    times_775122617y_real : poly_real > poly_real > poly_real).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal, type,
    times_times_real : real > real > real).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Complex__Ocomplex, type,
    zero_zero_complex : complex).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    zero_z1746442943omplex : poly_complex).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Complex__Ocomplex_J_J, type,
    zero_z1040703943omplex : poly_poly_complex).
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__Real__Oreal, type,
    ord_less_real : real > real > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal, type,
    ord_less_eq_real : real > real > $o).
thf(sy_c_Polynomial_OpCons_001t__Complex__Ocomplex, type,
    pCons_complex : complex > poly_complex > poly_complex).
thf(sy_c_Polynomial_OpCons_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    pCons_poly_complex : poly_complex > poly_poly_complex > poly_poly_complex).
thf(sy_c_Polynomial_OpCons_001t__Real__Oreal, type,
    pCons_real : real > poly_real > poly_real).
thf(sy_c_Polynomial_Opoly_001t__Complex__Ocomplex, type,
    poly_complex2 : poly_complex > complex > complex).
thf(sy_c_Polynomial_Opoly_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    poly_poly_complex2 : poly_poly_complex > poly_complex > poly_complex).
thf(sy_c_Polynomial_Opoly_001t__Polynomial__Opoly_It__Real__Oreal_J, type,
    poly_poly_real2 : poly_poly_real > poly_real > poly_real).
thf(sy_c_Polynomial_Opoly_001t__Real__Oreal, type,
    poly_real2 : poly_real > real > 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_c_Rings_Odivide__class_Odivide_001t__Complex__Ocomplex, type,
    divide1210191872omplex : complex > complex > complex).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    divide1187762952omplex : poly_complex > poly_complex > poly_complex).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Polynomial__Opoly_It__Real__Oreal_J, type,
    divide1727078534y_real : poly_real > poly_real > poly_real).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal, type,
    divide_divide_real : real > real > real).
thf(sy_v_c____, type,
    c : complex).
thf(sy_v_cs____, type,
    cs : poly_complex).
thf(sy_v_d____, type,
    d : complex).
thf(sy_v_ds____, type,
    ds : poly_complex).
thf(sy_v_m____, type,
    m : real).
thf(sy_v_p, type,
    p : poly_complex).
thf(sy_v_x____, type,
    x : real).

% Relevant facts (219)
thf(fact_0__092_060open_062x_A_K_Am_A_060_Acmod_Ad_092_060close_062, axiom,
    ((ord_less_real @ (times_times_real @ x @ m) @ (real_V638595069omplex @ d)))). % \<open>x * m < cmod d\<close>
thf(fact_1_th0, axiom,
    ((ord_less_eq_real @ (real_V638595069omplex @ (times_times_complex @ (real_V306493662omplex @ x) @ (poly_complex2 @ ds @ (real_V306493662omplex @ x)))) @ (times_times_real @ x @ m)))). % th0
thf(fact_2_False, axiom,
    ((~ ((d = zero_zero_complex))))). % False
thf(fact_3_cth, axiom,
    (((real_V638595069omplex @ (times_times_complex @ (real_V306493662omplex @ x) @ (poly_complex2 @ ds @ (real_V306493662omplex @ x)))) = (real_V638595069omplex @ d)))). % cth
thf(fact_4__092_060open_062cmod_A_Ipoly_Ads_A_Icomplex__of__real_Ax_J_J_A_092_060le_062_Am_092_060close_062, axiom,
    ((ord_less_eq_real @ (real_V638595069omplex @ (poly_complex2 @ ds @ (real_V306493662omplex @ x))) @ m))). % \<open>cmod (poly ds (complex_of_real x)) \<le> m\<close>
thf(fact_5_x_I1_J, axiom,
    ((ord_less_real @ zero_zero_real @ x))). % x(1)
thf(fact_6_x_I3_J, axiom,
    ((ord_less_real @ x @ one_one_real))). % x(3)
thf(fact_7_m_I2_J, axiom,
    ((![Z : a, Za : complex]: ((ord_less_eq_real @ (real_V638595069omplex @ Za) @ one_one_real) => (ord_less_eq_real @ (real_V638595069omplex @ (poly_complex2 @ ds @ Za)) @ m))))). % m(2)
thf(fact_8_x_I2_J, axiom,
    ((ord_less_real @ x @ (divide_divide_real @ (real_V638595069omplex @ d) @ m)))). % x(2)
thf(fact_9_poly__mult, axiom,
    ((![P : poly_poly_real, Q : poly_poly_real, X : poly_real]: ((poly_poly_real2 @ (times_614468161y_real @ P @ Q) @ X) = (times_775122617y_real @ (poly_poly_real2 @ P @ X) @ (poly_poly_real2 @ Q @ X)))))). % poly_mult
thf(fact_10_poly__mult, axiom,
    ((![P : poly_poly_complex, Q : poly_poly_complex, X : poly_complex]: ((poly_poly_complex2 @ (times_1460995011omplex @ P @ Q) @ X) = (times_1246143675omplex @ (poly_poly_complex2 @ P @ X) @ (poly_poly_complex2 @ Q @ X)))))). % poly_mult
thf(fact_11_poly__mult, axiom,
    ((![P : poly_complex, Q : poly_complex, X : complex]: ((poly_complex2 @ (times_1246143675omplex @ P @ Q) @ X) = (times_times_complex @ (poly_complex2 @ P @ X) @ (poly_complex2 @ Q @ X)))))). % poly_mult
thf(fact_12_poly__mult, axiom,
    ((![P : poly_real, Q : poly_real, X : real]: ((poly_real2 @ (times_775122617y_real @ P @ Q) @ X) = (times_times_real @ (poly_real2 @ P @ X) @ (poly_real2 @ Q @ X)))))). % poly_mult
thf(fact_13_of__real__mult, axiom,
    ((![X : real, Y : real]: ((real_V1205483228l_real @ (times_times_real @ X @ Y)) = (times_times_real @ (real_V1205483228l_real @ X) @ (real_V1205483228l_real @ Y)))))). % of_real_mult
thf(fact_14_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_15_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_16_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_17_cx_I1_J, axiom,
    ((~ (((real_V306493662omplex @ x) = zero_zero_complex))))). % cx(1)
thf(fact_18_of__real__eq__iff, axiom,
    ((![X : real, Y : real]: (((real_V306493662omplex @ X) = (real_V306493662omplex @ Y)) = (X = Y))))). % of_real_eq_iff
thf(fact_19_cx_I2_J, axiom,
    ((ord_less_eq_real @ (real_V638595069omplex @ (real_V306493662omplex @ x)) @ one_one_real))). % cx(2)
thf(fact_20_norm__mult, axiom,
    ((![X : real, Y : real]: ((real_V646646907m_real @ (times_times_real @ X @ Y)) = (times_times_real @ (real_V646646907m_real @ X) @ (real_V646646907m_real @ Y)))))). % norm_mult
thf(fact_21_norm__mult, axiom,
    ((![X : complex, Y : complex]: ((real_V638595069omplex @ (times_times_complex @ X @ Y)) = (times_times_real @ (real_V638595069omplex @ X) @ (real_V638595069omplex @ Y)))))). % norm_mult
thf(fact_22_pCons_Oprems, axiom,
    ((![W : complex]: ((~ ((W = zero_zero_complex))) => ((poly_complex2 @ (pCons_complex @ d @ ds) @ W) = zero_zero_complex))))). % pCons.prems
thf(fact_23_m_I1_J, axiom,
    ((ord_less_real @ zero_zero_real @ m))). % m(1)
thf(fact_24_pCons__eq__iff, axiom,
    ((![A : complex, P : poly_complex, B : complex, Q : poly_complex]: (((pCons_complex @ A @ P) = (pCons_complex @ B @ Q)) = (((A = B)) & ((P = Q))))))). % pCons_eq_iff
thf(fact_25_pCons__0__0, axiom,
    (((pCons_complex @ zero_zero_complex @ zero_z1746442943omplex) = zero_z1746442943omplex))). % pCons_0_0
thf(fact_26_pCons__0__0, axiom,
    (((pCons_real @ zero_zero_real @ zero_zero_poly_real) = zero_zero_poly_real))). % pCons_0_0
thf(fact_27_pCons__0__0, axiom,
    (((pCons_poly_complex @ zero_z1746442943omplex @ zero_z1040703943omplex) = zero_z1040703943omplex))). % pCons_0_0
thf(fact_28_pCons__eq__0__iff, axiom,
    ((![A : real, P : poly_real]: (((pCons_real @ A @ P) = zero_zero_poly_real) = (((A = zero_zero_real)) & ((P = zero_zero_poly_real))))))). % pCons_eq_0_iff
thf(fact_29_pCons__eq__0__iff, axiom,
    ((![A : poly_complex, P : poly_poly_complex]: (((pCons_poly_complex @ A @ P) = zero_z1040703943omplex) = (((A = zero_z1746442943omplex)) & ((P = zero_z1040703943omplex))))))). % pCons_eq_0_iff
thf(fact_30_pCons__eq__0__iff, axiom,
    ((![A : complex, P : poly_complex]: (((pCons_complex @ A @ P) = zero_z1746442943omplex) = (((A = zero_zero_complex)) & ((P = zero_z1746442943omplex))))))). % pCons_eq_0_iff
thf(fact_31_dm, axiom,
    ((ord_less_real @ zero_zero_real @ (divide_divide_real @ (real_V638595069omplex @ d) @ m)))). % dm
thf(fact_32__092_060open_062_092_060exists_062m_0620_O_A_092_060forall_062z_O_Acmod_Az_A_092_060le_062_A1_A_092_060longrightarrow_062_Acmod_A_Ipoly_Ads_Az_J_A_092_060le_062_Am_092_060close_062, axiom,
    ((?[M : real]: ((ord_less_real @ zero_zero_real @ M) & (![Z : complex]: ((ord_less_eq_real @ (real_V638595069omplex @ Z) @ one_one_real) => (ord_less_eq_real @ (real_V638595069omplex @ (poly_complex2 @ ds @ Z)) @ M))))))). % \<open>\<exists>m>0. \<forall>z. cmod z \<le> 1 \<longrightarrow> cmod (poly ds z) \<le> m\<close>
thf(fact_33_poly__0, axiom,
    ((![X : real]: ((poly_real2 @ zero_zero_poly_real @ X) = zero_zero_real)))). % poly_0
thf(fact_34_poly__0, axiom,
    ((![X : poly_complex]: ((poly_poly_complex2 @ zero_z1040703943omplex @ X) = zero_z1746442943omplex)))). % poly_0
thf(fact_35_poly__0, axiom,
    ((![X : complex]: ((poly_complex2 @ zero_z1746442943omplex @ X) = zero_zero_complex)))). % poly_0
thf(fact_36_poly__1, axiom,
    ((![X : complex]: ((poly_complex2 @ one_one_poly_complex @ X) = one_one_complex)))). % poly_1
thf(fact_37_poly__1, axiom,
    ((![X : real]: ((poly_real2 @ one_one_poly_real @ X) = one_one_real)))). % poly_1
thf(fact_38__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062x_O_A_092_060lbrakk_0620_A_060_Ax_059_Ax_A_060_Acmod_Ad_A_P_Am_059_Ax_A_060_A1_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom,
    ((~ ((![X2 : real]: ((ord_less_real @ zero_zero_real @ X2) => ((ord_less_real @ X2 @ (divide_divide_real @ (real_V638595069omplex @ d) @ m)) => (~ ((ord_less_real @ X2 @ one_one_real)))))))))). % \<open>\<And>thesis. (\<And>x. \<lbrakk>0 < x; x < cmod d / m; x < 1\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_39__092_060open_062_092_060exists_062e_0620_O_Ae_A_060_Acmod_Ad_A_P_Am_A_092_060and_062_Ae_A_060_A1_092_060close_062, axiom,
    ((?[E : real]: ((ord_less_real @ zero_zero_real @ E) & ((ord_less_real @ E @ (divide_divide_real @ (real_V638595069omplex @ d) @ m)) & (ord_less_real @ E @ one_one_real)))))). % \<open>\<exists>e>0. e < cmod d / m \<and> e < 1\<close>
thf(fact_40__092_060open_062poly_A_IpCons_Ad_Ads_J_A_Icomplex__of__real_Ax_J_A_061_A0_092_060close_062, axiom,
    (((poly_complex2 @ (pCons_complex @ d @ ds) @ (real_V306493662omplex @ x)) = zero_zero_complex))). % \<open>poly (pCons d ds) (complex_of_real x) = 0\<close>
thf(fact_41_norm__zero, axiom,
    (((real_V646646907m_real @ zero_zero_real) = zero_zero_real))). % norm_zero
thf(fact_42_norm__zero, axiom,
    (((real_V638595069omplex @ zero_zero_complex) = zero_zero_real))). % norm_zero
thf(fact_43_norm__eq__zero, axiom,
    ((![X : real]: (((real_V646646907m_real @ X) = zero_zero_real) = (X = zero_zero_real))))). % norm_eq_zero
thf(fact_44_norm__eq__zero, axiom,
    ((![X : complex]: (((real_V638595069omplex @ X) = zero_zero_real) = (X = zero_zero_complex))))). % norm_eq_zero
thf(fact_45_of__real__0, axiom,
    (((real_V1205483228l_real @ zero_zero_real) = zero_zero_real))). % of_real_0
thf(fact_46_of__real__0, axiom,
    (((real_V306493662omplex @ zero_zero_real) = zero_zero_complex))). % of_real_0
thf(fact_47_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_48_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_49_norm__one, axiom,
    (((real_V646646907m_real @ one_one_real) = one_one_real))). % norm_one
thf(fact_50_norm__one, axiom,
    (((real_V638595069omplex @ one_one_complex) = one_one_real))). % norm_one
thf(fact_51_pCons_Ohyps_I2_J, axiom,
    (((![W2 : complex]: ((~ ((W2 = zero_zero_complex))) => ((poly_complex2 @ ds @ W2) = zero_zero_complex))) => (ds = zero_z1746442943omplex)))). % pCons.hyps(2)
thf(fact_52_of__real__1, axiom,
    (((real_V1205483228l_real @ one_one_real) = one_one_real))). % of_real_1
thf(fact_53_of__real__1, axiom,
    (((real_V306493662omplex @ one_one_real) = one_one_complex))). % of_real_1
thf(fact_54_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_55_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_56_of__real__divide, axiom,
    ((![X : real, Y : real]: ((real_V1205483228l_real @ (divide_divide_real @ X @ Y)) = (divide_divide_real @ (real_V1205483228l_real @ X) @ (real_V1205483228l_real @ Y)))))). % of_real_divide
thf(fact_57_of__real__divide, axiom,
    ((![X : real, Y : real]: ((real_V306493662omplex @ (divide_divide_real @ X @ Y)) = (divide1210191872omplex @ (real_V306493662omplex @ X) @ (real_V306493662omplex @ Y)))))). % of_real_divide
thf(fact_58_pCons_Ohyps_I1_J, axiom,
    (((~ ((d = zero_zero_complex))) | (~ ((ds = zero_z1746442943omplex)))))). % pCons.hyps(1)
thf(fact_59_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_60_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_61_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_62_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_63__092_060open_062_092_060forall_062w_O_Aw_A_092_060noteq_062_A0_A_092_060longrightarrow_062_Apoly_Acs_Aw_A_061_A0_092_060close_062, axiom,
    ((![W : complex]: ((~ ((W = zero_zero_complex))) => ((poly_complex2 @ cs @ W) = zero_zero_complex))))). % \<open>\<forall>w. w \<noteq> 0 \<longrightarrow> poly cs w = 0\<close>
thf(fact_64_assms, axiom,
    ((~ ((?[A2 : complex, L : poly_complex]: ((~ ((A2 = zero_zero_complex))) & ((L = zero_z1746442943omplex) & (p = (pCons_complex @ A2 @ L))))))))). % assms
thf(fact_65_poly__IVT, axiom,
    ((![A : real, B : real, P : poly_real]: ((ord_less_real @ A @ B) => ((ord_less_real @ (times_times_real @ (poly_real2 @ P @ A) @ (poly_real2 @ P @ B)) @ zero_zero_real) => (?[X2 : real]: ((ord_less_real @ A @ X2) & ((ord_less_real @ X2 @ B) & ((poly_real2 @ P @ X2) = zero_zero_real))))))))). % poly_IVT
thf(fact_66_pCons__cases, axiom,
    ((![P : poly_complex]: (~ ((![A3 : complex, Q2 : poly_complex]: (~ ((P = (pCons_complex @ A3 @ Q2)))))))))). % pCons_cases
thf(fact_67_pCons__induct, axiom,
    ((![P2 : poly_real > $o, P : poly_real]: ((P2 @ zero_zero_poly_real) => ((![A3 : real, P3 : poly_real]: (((~ ((A3 = zero_zero_real))) | (~ ((P3 = zero_zero_poly_real)))) => ((P2 @ P3) => (P2 @ (pCons_real @ A3 @ P3))))) => (P2 @ P)))))). % pCons_induct
thf(fact_68_pCons__induct, axiom,
    ((![P2 : poly_poly_complex > $o, P : poly_poly_complex]: ((P2 @ zero_z1040703943omplex) => ((![A3 : poly_complex, P3 : poly_poly_complex]: (((~ ((A3 = zero_z1746442943omplex))) | (~ ((P3 = zero_z1040703943omplex)))) => ((P2 @ P3) => (P2 @ (pCons_poly_complex @ A3 @ P3))))) => (P2 @ P)))))). % pCons_induct
thf(fact_69_pCons__induct, axiom,
    ((![P2 : poly_complex > $o, P : poly_complex]: ((P2 @ zero_z1746442943omplex) => ((![A3 : complex, P3 : poly_complex]: (((~ ((A3 = zero_zero_complex))) | (~ ((P3 = zero_z1746442943omplex)))) => ((P2 @ P3) => (P2 @ (pCons_complex @ A3 @ P3))))) => (P2 @ P)))))). % pCons_induct
thf(fact_70_pderiv_Ocases, axiom,
    ((![X : poly_complex]: (~ ((![A3 : complex, P3 : poly_complex]: (~ ((X = (pCons_complex @ A3 @ P3)))))))))). % pderiv.cases
thf(fact_71_poly__IVT__neg, axiom,
    ((![A : real, B : real, P : poly_real]: ((ord_less_real @ A @ B) => ((ord_less_real @ zero_zero_real @ (poly_real2 @ P @ A)) => ((ord_less_real @ (poly_real2 @ P @ B) @ zero_zero_real) => (?[X2 : real]: ((ord_less_real @ A @ X2) & ((ord_less_real @ X2 @ B) & ((poly_real2 @ P @ X2) = zero_zero_real)))))))))). % poly_IVT_neg
thf(fact_72_poly__IVT__pos, axiom,
    ((![A : real, B : real, P : poly_real]: ((ord_less_real @ A @ B) => ((ord_less_real @ (poly_real2 @ P @ A) @ zero_zero_real) => ((ord_less_real @ zero_zero_real @ (poly_real2 @ P @ B)) => (?[X2 : real]: ((ord_less_real @ A @ X2) & ((ord_less_real @ X2 @ B) & ((poly_real2 @ P @ X2) = zero_zero_real)))))))))). % poly_IVT_pos
thf(fact_73_poly__all__0__iff__0, axiom,
    ((![P : poly_complex]: ((![X3 : complex]: ((poly_complex2 @ P @ X3) = zero_zero_complex)) = (P = zero_z1746442943omplex))))). % poly_all_0_iff_0
thf(fact_74_poly__all__0__iff__0, axiom,
    ((![P : poly_real]: ((![X3 : real]: ((poly_real2 @ P @ X3) = zero_zero_real)) = (P = zero_zero_poly_real))))). % poly_all_0_iff_0
thf(fact_75_poly__all__0__iff__0, axiom,
    ((![P : poly_poly_complex]: ((![X3 : poly_complex]: ((poly_poly_complex2 @ P @ X3) = zero_z1746442943omplex)) = (P = zero_z1040703943omplex))))). % poly_all_0_iff_0
thf(fact_76_norm__divide, axiom,
    ((![A : real, B : real]: ((real_V646646907m_real @ (divide_divide_real @ A @ B)) = (divide_divide_real @ (real_V646646907m_real @ A) @ (real_V646646907m_real @ B)))))). % norm_divide
thf(fact_77_norm__divide, axiom,
    ((![A : complex, B : complex]: ((real_V638595069omplex @ (divide1210191872omplex @ A @ B)) = (divide_divide_real @ (real_V638595069omplex @ A) @ (real_V638595069omplex @ B)))))). % norm_divide
thf(fact_78_nonzero__norm__divide, axiom,
    ((![B : real, A : real]: ((~ ((B = zero_zero_real))) => ((real_V646646907m_real @ (divide_divide_real @ A @ B)) = (divide_divide_real @ (real_V646646907m_real @ A) @ (real_V646646907m_real @ B))))))). % nonzero_norm_divide
thf(fact_79_nonzero__norm__divide, axiom,
    ((![B : complex, A : complex]: ((~ ((B = zero_zero_complex))) => ((real_V638595069omplex @ (divide1210191872omplex @ A @ B)) = (divide_divide_real @ (real_V638595069omplex @ A) @ (real_V638595069omplex @ B))))))). % nonzero_norm_divide
thf(fact_80_nonzero__of__real__divide, axiom,
    ((![Y : real, X : real]: ((~ ((Y = zero_zero_real))) => ((real_V1205483228l_real @ (divide_divide_real @ X @ Y)) = (divide_divide_real @ (real_V1205483228l_real @ X) @ (real_V1205483228l_real @ Y))))))). % nonzero_of_real_divide
thf(fact_81_nonzero__of__real__divide, axiom,
    ((![Y : real, X : real]: ((~ ((Y = zero_zero_real))) => ((real_V306493662omplex @ (divide_divide_real @ X @ Y)) = (divide1210191872omplex @ (real_V306493662omplex @ X) @ (real_V306493662omplex @ Y))))))). % nonzero_of_real_divide
thf(fact_82_norm__not__less__zero, axiom,
    ((![X : complex]: (~ ((ord_less_real @ (real_V638595069omplex @ X) @ zero_zero_real)))))). % norm_not_less_zero
thf(fact_83_norm__ge__zero, axiom,
    ((![X : complex]: (ord_less_eq_real @ zero_zero_real @ (real_V638595069omplex @ X))))). % norm_ge_zero
thf(fact_84_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_85_poly__bound__exists, axiom,
    ((![R : real, P : poly_real]: (?[M : real]: ((ord_less_real @ zero_zero_real @ M) & (![Z : real]: ((ord_less_eq_real @ (real_V646646907m_real @ Z) @ R) => (ord_less_eq_real @ (real_V646646907m_real @ (poly_real2 @ P @ Z)) @ M)))))))). % poly_bound_exists
thf(fact_86_poly__bound__exists, axiom,
    ((![R : real, P : poly_complex]: (?[M : real]: ((ord_less_real @ zero_zero_real @ M) & (![Z : complex]: ((ord_less_eq_real @ (real_V638595069omplex @ Z) @ R) => (ord_less_eq_real @ (real_V638595069omplex @ (poly_complex2 @ P @ Z)) @ M)))))))). % poly_bound_exists
thf(fact_87_norm__mult__less, axiom,
    ((![X : real, R : real, Y : real, S2 : real]: ((ord_less_real @ (real_V646646907m_real @ X) @ R) => ((ord_less_real @ (real_V646646907m_real @ Y) @ S2) => (ord_less_real @ (real_V646646907m_real @ (times_times_real @ X @ Y)) @ (times_times_real @ R @ S2))))))). % norm_mult_less
thf(fact_88_norm__mult__less, axiom,
    ((![X : complex, R : real, Y : complex, S2 : real]: ((ord_less_real @ (real_V638595069omplex @ X) @ R) => ((ord_less_real @ (real_V638595069omplex @ Y) @ S2) => (ord_less_real @ (real_V638595069omplex @ (times_times_complex @ X @ Y)) @ (times_times_real @ R @ S2))))))). % norm_mult_less
thf(fact_89_norm__mult__ineq, axiom,
    ((![X : real, Y : real]: (ord_less_eq_real @ (real_V646646907m_real @ (times_times_real @ X @ Y)) @ (times_times_real @ (real_V646646907m_real @ X) @ (real_V646646907m_real @ Y)))))). % norm_mult_ineq
thf(fact_90_norm__mult__ineq, axiom,
    ((![X : complex, Y : complex]: (ord_less_eq_real @ (real_V638595069omplex @ (times_times_complex @ X @ Y)) @ (times_times_real @ (real_V638595069omplex @ X) @ (real_V638595069omplex @ Y)))))). % norm_mult_ineq
thf(fact_91_poly__eq__poly__eq__iff, axiom,
    ((![P : poly_complex, Q : poly_complex]: (((poly_complex2 @ P) = (poly_complex2 @ Q)) = (P = Q))))). % poly_eq_poly_eq_iff
thf(fact_92_poly__eq__poly__eq__iff, axiom,
    ((![P : poly_real, Q : poly_real]: (((poly_real2 @ P) = (poly_real2 @ Q)) = (P = Q))))). % poly_eq_poly_eq_iff
thf(fact_93_divide__le__eq__1__neg, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ zero_zero_real) => ((ord_less_eq_real @ (divide_divide_real @ B @ A) @ one_one_real) = (ord_less_eq_real @ A @ B)))))). % divide_le_eq_1_neg
thf(fact_94_divide__le__eq__1__pos, axiom,
    ((![A : real, B : real]: ((ord_less_real @ zero_zero_real @ A) => ((ord_less_eq_real @ (divide_divide_real @ B @ A) @ one_one_real) = (ord_less_eq_real @ B @ A)))))). % divide_le_eq_1_pos
thf(fact_95_le__divide__eq__1__neg, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ zero_zero_real) => ((ord_less_eq_real @ one_one_real @ (divide_divide_real @ B @ A)) = (ord_less_eq_real @ B @ A)))))). % le_divide_eq_1_neg
thf(fact_96_le__divide__eq__1__pos, axiom,
    ((![A : real, B : real]: ((ord_less_real @ zero_zero_real @ A) => ((ord_less_eq_real @ one_one_real @ (divide_divide_real @ B @ A)) = (ord_less_eq_real @ A @ B)))))). % le_divide_eq_1_pos
thf(fact_97_nonzero__divide__mult__cancel__left, axiom,
    ((![A : complex, B : complex]: ((~ ((A = zero_zero_complex))) => ((divide1210191872omplex @ A @ (times_times_complex @ A @ B)) = (divide1210191872omplex @ one_one_complex @ B)))))). % nonzero_divide_mult_cancel_left
thf(fact_98_nonzero__divide__mult__cancel__left, axiom,
    ((![A : real, B : real]: ((~ ((A = zero_zero_real))) => ((divide_divide_real @ A @ (times_times_real @ A @ B)) = (divide_divide_real @ one_one_real @ B)))))). % nonzero_divide_mult_cancel_left
thf(fact_99_nonzero__divide__mult__cancel__right, axiom,
    ((![B : complex, A : complex]: ((~ ((B = zero_zero_complex))) => ((divide1210191872omplex @ B @ (times_times_complex @ A @ B)) = (divide1210191872omplex @ one_one_complex @ A)))))). % nonzero_divide_mult_cancel_right
thf(fact_100_nonzero__divide__mult__cancel__right, axiom,
    ((![B : real, A : real]: ((~ ((B = zero_zero_real))) => ((divide_divide_real @ B @ (times_times_real @ A @ B)) = (divide_divide_real @ one_one_real @ A)))))). % nonzero_divide_mult_cancel_right
thf(fact_101_divide__less__0__1__iff, axiom,
    ((![A : real]: ((ord_less_real @ (divide_divide_real @ one_one_real @ A) @ zero_zero_real) = (ord_less_real @ A @ zero_zero_real))))). % divide_less_0_1_iff
thf(fact_102_divide__less__eq__1__neg, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ zero_zero_real) => ((ord_less_real @ (divide_divide_real @ B @ A) @ one_one_real) = (ord_less_real @ A @ B)))))). % divide_less_eq_1_neg
thf(fact_103_divide__less__eq__1__pos, axiom,
    ((![A : real, B : real]: ((ord_less_real @ zero_zero_real @ A) => ((ord_less_real @ (divide_divide_real @ B @ A) @ one_one_real) = (ord_less_real @ B @ A)))))). % divide_less_eq_1_pos
thf(fact_104_division__ring__divide__zero, axiom,
    ((![A : complex]: ((divide1210191872omplex @ A @ zero_zero_complex) = zero_zero_complex)))). % division_ring_divide_zero
thf(fact_105_division__ring__divide__zero, axiom,
    ((![A : real]: ((divide_divide_real @ A @ zero_zero_real) = zero_zero_real)))). % division_ring_divide_zero
thf(fact_106_divide__cancel__right, axiom,
    ((![A : complex, C : complex, B : complex]: (((divide1210191872omplex @ A @ C) = (divide1210191872omplex @ B @ C)) = (((C = zero_zero_complex)) | ((A = B))))))). % divide_cancel_right
thf(fact_107_divide__cancel__right, axiom,
    ((![A : real, C : real, B : real]: (((divide_divide_real @ A @ C) = (divide_divide_real @ B @ C)) = (((C = zero_zero_real)) | ((A = B))))))). % divide_cancel_right
thf(fact_108_divide__cancel__left, axiom,
    ((![C : complex, A : complex, B : complex]: (((divide1210191872omplex @ C @ A) = (divide1210191872omplex @ C @ B)) = (((C = zero_zero_complex)) | ((A = B))))))). % divide_cancel_left
thf(fact_109_divide__cancel__left, axiom,
    ((![C : real, A : real, B : real]: (((divide_divide_real @ C @ A) = (divide_divide_real @ C @ B)) = (((C = zero_zero_real)) | ((A = B))))))). % divide_cancel_left
thf(fact_110_divide__eq__0__iff, axiom,
    ((![A : complex, B : complex]: (((divide1210191872omplex @ A @ B) = zero_zero_complex) = (((A = zero_zero_complex)) | ((B = zero_zero_complex))))))). % divide_eq_0_iff
thf(fact_111_divide__eq__0__iff, axiom,
    ((![A : real, B : real]: (((divide_divide_real @ A @ B) = zero_zero_real) = (((A = zero_zero_real)) | ((B = zero_zero_real))))))). % divide_eq_0_iff
thf(fact_112_times__divide__eq__left, axiom,
    ((![B : complex, C : complex, A : complex]: ((times_times_complex @ (divide1210191872omplex @ B @ C) @ A) = (divide1210191872omplex @ (times_times_complex @ B @ A) @ C))))). % times_divide_eq_left
thf(fact_113_times__divide__eq__left, axiom,
    ((![B : real, C : real, A : real]: ((times_times_real @ (divide_divide_real @ B @ C) @ A) = (divide_divide_real @ (times_times_real @ B @ A) @ C))))). % times_divide_eq_left
thf(fact_114_divide__divide__eq__left, axiom,
    ((![A : complex, B : complex, C : complex]: ((divide1210191872omplex @ (divide1210191872omplex @ A @ B) @ C) = (divide1210191872omplex @ A @ (times_times_complex @ B @ C)))))). % divide_divide_eq_left
thf(fact_115_divide__divide__eq__left, axiom,
    ((![A : real, B : real, C : real]: ((divide_divide_real @ (divide_divide_real @ A @ B) @ C) = (divide_divide_real @ A @ (times_times_real @ B @ C)))))). % divide_divide_eq_left
thf(fact_116_divide__divide__eq__right, axiom,
    ((![A : complex, B : complex, C : complex]: ((divide1210191872omplex @ A @ (divide1210191872omplex @ B @ C)) = (divide1210191872omplex @ (times_times_complex @ A @ C) @ B))))). % divide_divide_eq_right
thf(fact_117_divide__divide__eq__right, axiom,
    ((![A : real, B : real, C : real]: ((divide_divide_real @ A @ (divide_divide_real @ B @ C)) = (divide_divide_real @ (times_times_real @ A @ C) @ B))))). % divide_divide_eq_right
thf(fact_118_times__divide__eq__right, axiom,
    ((![A : complex, B : complex, C : complex]: ((times_times_complex @ A @ (divide1210191872omplex @ B @ C)) = (divide1210191872omplex @ (times_times_complex @ A @ B) @ C))))). % times_divide_eq_right
thf(fact_119_times__divide__eq__right, axiom,
    ((![A : real, B : real, C : real]: ((times_times_real @ A @ (divide_divide_real @ B @ C)) = (divide_divide_real @ (times_times_real @ A @ B) @ C))))). % times_divide_eq_right
thf(fact_120_nonzero__mult__divide__mult__cancel__right2, axiom,
    ((![C : complex, A : complex, B : complex]: ((~ ((C = zero_zero_complex))) => ((divide1210191872omplex @ (times_times_complex @ A @ C) @ (times_times_complex @ C @ B)) = (divide1210191872omplex @ A @ B)))))). % nonzero_mult_divide_mult_cancel_right2
thf(fact_121_nonzero__mult__divide__mult__cancel__right2, axiom,
    ((![C : real, A : real, B : real]: ((~ ((C = zero_zero_real))) => ((divide_divide_real @ (times_times_real @ A @ C) @ (times_times_real @ C @ B)) = (divide_divide_real @ A @ B)))))). % nonzero_mult_divide_mult_cancel_right2
thf(fact_122_nonzero__mult__divide__mult__cancel__right, axiom,
    ((![C : complex, A : complex, B : complex]: ((~ ((C = zero_zero_complex))) => ((divide1210191872omplex @ (times_times_complex @ A @ C) @ (times_times_complex @ B @ C)) = (divide1210191872omplex @ A @ B)))))). % nonzero_mult_divide_mult_cancel_right
thf(fact_123_nonzero__mult__divide__mult__cancel__right, axiom,
    ((![C : real, A : real, B : real]: ((~ ((C = zero_zero_real))) => ((divide_divide_real @ (times_times_real @ A @ C) @ (times_times_real @ B @ C)) = (divide_divide_real @ A @ B)))))). % nonzero_mult_divide_mult_cancel_right
thf(fact_124_nonzero__mult__divide__mult__cancel__left2, axiom,
    ((![C : complex, A : complex, B : complex]: ((~ ((C = zero_zero_complex))) => ((divide1210191872omplex @ (times_times_complex @ C @ A) @ (times_times_complex @ B @ C)) = (divide1210191872omplex @ A @ B)))))). % nonzero_mult_divide_mult_cancel_left2
thf(fact_125_nonzero__mult__divide__mult__cancel__left2, axiom,
    ((![C : real, A : real, B : real]: ((~ ((C = zero_zero_real))) => ((divide_divide_real @ (times_times_real @ C @ A) @ (times_times_real @ B @ C)) = (divide_divide_real @ A @ B)))))). % nonzero_mult_divide_mult_cancel_left2
thf(fact_126_nonzero__mult__divide__mult__cancel__left, axiom,
    ((![C : complex, A : complex, B : complex]: ((~ ((C = zero_zero_complex))) => ((divide1210191872omplex @ (times_times_complex @ C @ A) @ (times_times_complex @ C @ B)) = (divide1210191872omplex @ A @ B)))))). % nonzero_mult_divide_mult_cancel_left
thf(fact_127_nonzero__mult__divide__mult__cancel__left, axiom,
    ((![C : real, A : real, B : real]: ((~ ((C = zero_zero_real))) => ((divide_divide_real @ (times_times_real @ C @ A) @ (times_times_real @ C @ B)) = (divide_divide_real @ A @ B)))))). % nonzero_mult_divide_mult_cancel_left
thf(fact_128_mult__divide__mult__cancel__left__if, axiom,
    ((![C : complex, A : complex, B : complex]: (((C = zero_zero_complex) => ((divide1210191872omplex @ (times_times_complex @ C @ A) @ (times_times_complex @ C @ B)) = zero_zero_complex)) & ((~ ((C = zero_zero_complex))) => ((divide1210191872omplex @ (times_times_complex @ C @ A) @ (times_times_complex @ C @ B)) = (divide1210191872omplex @ A @ B))))))). % mult_divide_mult_cancel_left_if
thf(fact_129_mult__divide__mult__cancel__left__if, axiom,
    ((![C : real, A : real, B : real]: (((C = zero_zero_real) => ((divide_divide_real @ (times_times_real @ C @ A) @ (times_times_real @ C @ B)) = zero_zero_real)) & ((~ ((C = zero_zero_real))) => ((divide_divide_real @ (times_times_real @ C @ A) @ (times_times_real @ C @ B)) = (divide_divide_real @ A @ B))))))). % mult_divide_mult_cancel_left_if
thf(fact_130_zero__eq__1__divide__iff, axiom,
    ((![A : real]: ((zero_zero_real = (divide_divide_real @ one_one_real @ A)) = (A = zero_zero_real))))). % zero_eq_1_divide_iff
thf(fact_131_one__divide__eq__0__iff, axiom,
    ((![A : real]: (((divide_divide_real @ one_one_real @ A) = zero_zero_real) = (A = zero_zero_real))))). % one_divide_eq_0_iff
thf(fact_132_eq__divide__eq__1, axiom,
    ((![B : real, A : real]: ((one_one_real = (divide_divide_real @ B @ A)) = (((~ ((A = zero_zero_real)))) & ((A = B))))))). % eq_divide_eq_1
thf(fact_133_divide__eq__eq__1, axiom,
    ((![B : real, A : real]: (((divide_divide_real @ B @ A) = one_one_real) = (((~ ((A = zero_zero_real)))) & ((A = B))))))). % divide_eq_eq_1
thf(fact_134_divide__self__if, axiom,
    ((![A : complex]: (((A = zero_zero_complex) => ((divide1210191872omplex @ A @ A) = zero_zero_complex)) & ((~ ((A = zero_zero_complex))) => ((divide1210191872omplex @ A @ A) = one_one_complex)))))). % divide_self_if
thf(fact_135_divide__self__if, axiom,
    ((![A : real]: (((A = zero_zero_real) => ((divide_divide_real @ A @ A) = zero_zero_real)) & ((~ ((A = zero_zero_real))) => ((divide_divide_real @ A @ A) = one_one_real)))))). % divide_self_if
thf(fact_136_divide__self, axiom,
    ((![A : complex]: ((~ ((A = zero_zero_complex))) => ((divide1210191872omplex @ A @ A) = one_one_complex))))). % divide_self
thf(fact_137_divide__self, axiom,
    ((![A : real]: ((~ ((A = zero_zero_real))) => ((divide_divide_real @ A @ A) = one_one_real))))). % divide_self
thf(fact_138_one__eq__divide__iff, axiom,
    ((![A : complex, B : complex]: ((one_one_complex = (divide1210191872omplex @ A @ B)) = (((~ ((B = zero_zero_complex)))) & ((A = B))))))). % one_eq_divide_iff
thf(fact_139_one__eq__divide__iff, axiom,
    ((![A : real, B : real]: ((one_one_real = (divide_divide_real @ A @ B)) = (((~ ((B = zero_zero_real)))) & ((A = B))))))). % one_eq_divide_iff
thf(fact_140_divide__eq__1__iff, axiom,
    ((![A : complex, B : complex]: (((divide1210191872omplex @ A @ B) = one_one_complex) = (((~ ((B = zero_zero_complex)))) & ((A = B))))))). % divide_eq_1_iff
thf(fact_141_divide__eq__1__iff, axiom,
    ((![A : real, B : real]: (((divide_divide_real @ A @ B) = one_one_real) = (((~ ((B = zero_zero_real)))) & ((A = B))))))). % divide_eq_1_iff
thf(fact_142_nc, axiom,
    ((fundam1158420650omplex @ (poly_complex2 @ (pCons_complex @ c @ cs))))). % nc
thf(fact_143_zero__le__divide__1__iff, axiom,
    ((![A : real]: ((ord_less_eq_real @ zero_zero_real @ (divide_divide_real @ one_one_real @ A)) = (ord_less_eq_real @ zero_zero_real @ A))))). % zero_le_divide_1_iff
thf(fact_144_divide__le__0__1__iff, axiom,
    ((![A : real]: ((ord_less_eq_real @ (divide_divide_real @ one_one_real @ A) @ zero_zero_real) = (ord_less_eq_real @ A @ zero_zero_real))))). % divide_le_0_1_iff
thf(fact_145_zero__less__divide__1__iff, axiom,
    ((![A : real]: ((ord_less_real @ zero_zero_real @ (divide_divide_real @ one_one_real @ A)) = (ord_less_real @ zero_zero_real @ A))))). % zero_less_divide_1_iff
thf(fact_146_less__divide__eq__1__pos, axiom,
    ((![A : real, B : real]: ((ord_less_real @ zero_zero_real @ A) => ((ord_less_real @ one_one_real @ (divide_divide_real @ B @ A)) = (ord_less_real @ A @ B)))))). % less_divide_eq_1_pos
thf(fact_147_less__divide__eq__1__neg, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ zero_zero_real) => ((ord_less_real @ one_one_real @ (divide_divide_real @ B @ A)) = (ord_less_real @ B @ A)))))). % less_divide_eq_1_neg
thf(fact_148_one__poly__eq__simps_I1_J, axiom,
    ((one_one_poly_real = (pCons_real @ one_one_real @ zero_zero_poly_real)))). % one_poly_eq_simps(1)
thf(fact_149_one__poly__eq__simps_I1_J, axiom,
    ((one_one_poly_complex = (pCons_complex @ one_one_complex @ zero_z1746442943omplex)))). % one_poly_eq_simps(1)
thf(fact_150_one__poly__eq__simps_I2_J, axiom,
    (((pCons_real @ one_one_real @ zero_zero_poly_real) = one_one_poly_real))). % one_poly_eq_simps(2)
thf(fact_151_one__poly__eq__simps_I2_J, axiom,
    (((pCons_complex @ one_one_complex @ zero_z1746442943omplex) = one_one_poly_complex))). % one_poly_eq_simps(2)
thf(fact_152__092_060open_062_092_060And_062y_O_Apoly_A_IpCons_Ac_Acs_J_A0_A_061_Apoly_A_IpCons_Ac_Acs_J_Ay_092_060close_062, axiom,
    ((![Y : complex]: ((poly_complex2 @ (pCons_complex @ c @ cs) @ zero_zero_complex) = (poly_complex2 @ (pCons_complex @ c @ cs) @ Y))))). % \<open>\<And>y. poly (pCons c cs) 0 = poly (pCons c cs) y\<close>
thf(fact_153_divide__poly, axiom,
    ((![G : poly_real, F : poly_real]: ((~ ((G = zero_zero_poly_real))) => ((divide1727078534y_real @ (times_775122617y_real @ F @ G) @ G) = F))))). % divide_poly
thf(fact_154_divide__poly, axiom,
    ((![G : poly_complex, F : poly_complex]: ((~ ((G = zero_z1746442943omplex))) => ((divide1187762952omplex @ (times_1246143675omplex @ F @ G) @ G) = F))))). % divide_poly
thf(fact_155_divide__poly__0, axiom,
    ((![F : poly_complex]: ((divide1187762952omplex @ F @ zero_z1746442943omplex) = zero_z1746442943omplex)))). % divide_poly_0
thf(fact_156_poly__div__mult__right, axiom,
    ((![X : poly_real, Y : poly_real, Z3 : poly_real]: ((divide1727078534y_real @ X @ (times_775122617y_real @ Y @ Z3)) = (divide1727078534y_real @ (divide1727078534y_real @ X @ Y) @ Z3))))). % poly_div_mult_right
thf(fact_157_poly__div__mult__right, axiom,
    ((![X : poly_complex, Y : poly_complex, Z3 : poly_complex]: ((divide1187762952omplex @ X @ (times_1246143675omplex @ Y @ Z3)) = (divide1187762952omplex @ (divide1187762952omplex @ X @ Y) @ Z3))))). % poly_div_mult_right
thf(fact_158_pderiv_Oinduct, axiom,
    ((![P2 : poly_complex > $o, A0 : poly_complex]: ((![A3 : complex, P3 : poly_complex]: (((~ ((P3 = zero_z1746442943omplex))) => (P2 @ P3)) => (P2 @ (pCons_complex @ A3 @ P3)))) => (P2 @ A0))))). % pderiv.induct
thf(fact_159_poly__induct2, axiom,
    ((![P2 : poly_complex > poly_complex > $o, P : poly_complex, Q : poly_complex]: ((P2 @ zero_z1746442943omplex @ zero_z1746442943omplex) => ((![A3 : complex, P3 : poly_complex, B2 : complex, Q2 : poly_complex]: ((P2 @ P3 @ Q2) => (P2 @ (pCons_complex @ A3 @ P3) @ (pCons_complex @ B2 @ Q2)))) => (P2 @ P @ Q)))))). % poly_induct2
thf(fact_160_mult__poly__0__right, axiom,
    ((![P : poly_real]: ((times_775122617y_real @ P @ zero_zero_poly_real) = zero_zero_poly_real)))). % mult_poly_0_right
thf(fact_161_mult__poly__0__right, axiom,
    ((![P : poly_complex]: ((times_1246143675omplex @ P @ zero_z1746442943omplex) = zero_z1746442943omplex)))). % mult_poly_0_right
thf(fact_162_mult__poly__0__left, axiom,
    ((![Q : poly_real]: ((times_775122617y_real @ zero_zero_poly_real @ Q) = zero_zero_poly_real)))). % mult_poly_0_left
thf(fact_163_mult__poly__0__left, axiom,
    ((![Q : poly_complex]: ((times_1246143675omplex @ zero_z1746442943omplex @ Q) = zero_z1746442943omplex)))). % mult_poly_0_left
thf(fact_164_pCons__one, axiom,
    (((pCons_real @ one_one_real @ zero_zero_poly_real) = one_one_poly_real))). % pCons_one
thf(fact_165_pCons__one, axiom,
    (((pCons_complex @ one_one_complex @ zero_z1746442943omplex) = one_one_poly_complex))). % pCons_one
thf(fact_166_linordered__field__no__ub, axiom,
    ((![X4 : real]: (?[X_12 : real]: (ord_less_real @ X4 @ X_12))))). % linordered_field_no_ub
thf(fact_167_linordered__field__no__lb, axiom,
    ((![X4 : real]: (?[Y3 : real]: (ord_less_real @ Y3 @ X4))))). % linordered_field_no_lb
thf(fact_168_poly__infinity, axiom,
    ((![P : poly_real, D : real, A : real]: ((~ ((P = zero_zero_poly_real))) => (?[R2 : real]: (![Z : real]: ((ord_less_eq_real @ R2 @ (real_V646646907m_real @ Z)) => (ord_less_eq_real @ D @ (real_V646646907m_real @ (poly_real2 @ (pCons_real @ A @ P) @ Z)))))))))). % poly_infinity
thf(fact_169_poly__infinity, axiom,
    ((![P : poly_complex, D : real, A : complex]: ((~ ((P = zero_z1746442943omplex))) => (?[R2 : real]: (![Z : complex]: ((ord_less_eq_real @ R2 @ (real_V638595069omplex @ Z)) => (ord_less_eq_real @ D @ (real_V638595069omplex @ (poly_complex2 @ (pCons_complex @ A @ P) @ Z)))))))))). % poly_infinity
thf(fact_170_times__divide__times__eq, axiom,
    ((![X : complex, Y : complex, Z3 : complex, W3 : complex]: ((times_times_complex @ (divide1210191872omplex @ X @ Y) @ (divide1210191872omplex @ Z3 @ W3)) = (divide1210191872omplex @ (times_times_complex @ X @ Z3) @ (times_times_complex @ Y @ W3)))))). % times_divide_times_eq
thf(fact_171_times__divide__times__eq, axiom,
    ((![X : real, Y : real, Z3 : real, W3 : real]: ((times_times_real @ (divide_divide_real @ X @ Y) @ (divide_divide_real @ Z3 @ W3)) = (divide_divide_real @ (times_times_real @ X @ Z3) @ (times_times_real @ Y @ W3)))))). % times_divide_times_eq
thf(fact_172_divide__divide__times__eq, axiom,
    ((![X : complex, Y : complex, Z3 : complex, W3 : complex]: ((divide1210191872omplex @ (divide1210191872omplex @ X @ Y) @ (divide1210191872omplex @ Z3 @ W3)) = (divide1210191872omplex @ (times_times_complex @ X @ W3) @ (times_times_complex @ Y @ Z3)))))). % divide_divide_times_eq
thf(fact_173_divide__divide__times__eq, axiom,
    ((![X : real, Y : real, Z3 : real, W3 : real]: ((divide_divide_real @ (divide_divide_real @ X @ Y) @ (divide_divide_real @ Z3 @ W3)) = (divide_divide_real @ (times_times_real @ X @ W3) @ (times_times_real @ Y @ Z3)))))). % divide_divide_times_eq
thf(fact_174_divide__divide__eq__left_H, axiom,
    ((![A : complex, B : complex, C : complex]: ((divide1210191872omplex @ (divide1210191872omplex @ A @ B) @ C) = (divide1210191872omplex @ A @ (times_times_complex @ C @ B)))))). % divide_divide_eq_left'
thf(fact_175_divide__divide__eq__left_H, axiom,
    ((![A : real, B : real, C : real]: ((divide_divide_real @ (divide_divide_real @ A @ B) @ C) = (divide_divide_real @ A @ (times_times_real @ C @ B)))))). % divide_divide_eq_left'
thf(fact_176_divide__right__mono__neg, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_eq_real @ C @ zero_zero_real) => (ord_less_eq_real @ (divide_divide_real @ B @ C) @ (divide_divide_real @ A @ C))))))). % divide_right_mono_neg
thf(fact_177_divide__nonpos__nonpos, axiom,
    ((![X : real, Y : real]: ((ord_less_eq_real @ X @ zero_zero_real) => ((ord_less_eq_real @ Y @ zero_zero_real) => (ord_less_eq_real @ zero_zero_real @ (divide_divide_real @ X @ Y))))))). % divide_nonpos_nonpos
thf(fact_178_divide__nonpos__nonneg, axiom,
    ((![X : real, Y : real]: ((ord_less_eq_real @ X @ zero_zero_real) => ((ord_less_eq_real @ zero_zero_real @ Y) => (ord_less_eq_real @ (divide_divide_real @ X @ Y) @ zero_zero_real)))))). % divide_nonpos_nonneg
thf(fact_179_divide__nonneg__nonpos, axiom,
    ((![X : real, Y : real]: ((ord_less_eq_real @ zero_zero_real @ X) => ((ord_less_eq_real @ Y @ zero_zero_real) => (ord_less_eq_real @ (divide_divide_real @ X @ Y) @ zero_zero_real)))))). % divide_nonneg_nonpos
thf(fact_180_divide__nonneg__nonneg, axiom,
    ((![X : real, Y : real]: ((ord_less_eq_real @ zero_zero_real @ X) => ((ord_less_eq_real @ zero_zero_real @ Y) => (ord_less_eq_real @ zero_zero_real @ (divide_divide_real @ X @ Y))))))). % divide_nonneg_nonneg
thf(fact_181_zero__le__divide__iff, axiom,
    ((![A : real, B : real]: ((ord_less_eq_real @ zero_zero_real @ (divide_divide_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_divide_iff
thf(fact_182_divide__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 @ (divide_divide_real @ A @ C) @ (divide_divide_real @ B @ C))))))). % divide_right_mono
thf(fact_183_divide__le__0__iff, axiom,
    ((![A : real, B : real]: ((ord_less_eq_real @ (divide_divide_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))))))))). % divide_le_0_iff
thf(fact_184_divide__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 @ (divide_divide_real @ A @ C) @ (divide_divide_real @ B @ C))))))). % divide_strict_right_mono_neg
thf(fact_185_divide__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 @ (divide_divide_real @ A @ C) @ (divide_divide_real @ B @ C))))))). % divide_strict_right_mono
thf(fact_186_zero__less__divide__iff, axiom,
    ((![A : real, B : real]: ((ord_less_real @ zero_zero_real @ (divide_divide_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_divide_iff
thf(fact_187_divide__less__cancel, axiom,
    ((![A : real, C : real, B : real]: ((ord_less_real @ (divide_divide_real @ A @ C) @ (divide_divide_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)))) & ((~ ((C = zero_zero_real))))))))))). % divide_less_cancel
thf(fact_188_divide__less__0__iff, axiom,
    ((![A : real, B : real]: ((ord_less_real @ (divide_divide_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))))))))). % divide_less_0_iff
thf(fact_189_divide__pos__pos, axiom,
    ((![X : real, Y : real]: ((ord_less_real @ zero_zero_real @ X) => ((ord_less_real @ zero_zero_real @ Y) => (ord_less_real @ zero_zero_real @ (divide_divide_real @ X @ Y))))))). % divide_pos_pos
thf(fact_190_divide__pos__neg, axiom,
    ((![X : real, Y : real]: ((ord_less_real @ zero_zero_real @ X) => ((ord_less_real @ Y @ zero_zero_real) => (ord_less_real @ (divide_divide_real @ X @ Y) @ zero_zero_real)))))). % divide_pos_neg
thf(fact_191_divide__neg__pos, axiom,
    ((![X : real, Y : real]: ((ord_less_real @ X @ zero_zero_real) => ((ord_less_real @ zero_zero_real @ Y) => (ord_less_real @ (divide_divide_real @ X @ Y) @ zero_zero_real)))))). % divide_neg_pos
thf(fact_192_divide__neg__neg, axiom,
    ((![X : real, Y : real]: ((ord_less_real @ X @ zero_zero_real) => ((ord_less_real @ Y @ zero_zero_real) => (ord_less_real @ zero_zero_real @ (divide_divide_real @ X @ Y))))))). % divide_neg_neg
thf(fact_193_nonzero__eq__divide__eq, axiom,
    ((![C : complex, A : complex, B : complex]: ((~ ((C = zero_zero_complex))) => ((A = (divide1210191872omplex @ B @ C)) = ((times_times_complex @ A @ C) = B)))))). % nonzero_eq_divide_eq
thf(fact_194_nonzero__eq__divide__eq, axiom,
    ((![C : real, A : real, B : real]: ((~ ((C = zero_zero_real))) => ((A = (divide_divide_real @ B @ C)) = ((times_times_real @ A @ C) = B)))))). % nonzero_eq_divide_eq
thf(fact_195_nonzero__divide__eq__eq, axiom,
    ((![C : complex, B : complex, A : complex]: ((~ ((C = zero_zero_complex))) => (((divide1210191872omplex @ B @ C) = A) = (B = (times_times_complex @ A @ C))))))). % nonzero_divide_eq_eq
thf(fact_196_nonzero__divide__eq__eq, axiom,
    ((![C : real, B : real, A : real]: ((~ ((C = zero_zero_real))) => (((divide_divide_real @ B @ C) = A) = (B = (times_times_real @ A @ C))))))). % nonzero_divide_eq_eq
thf(fact_197_eq__divide__imp, axiom,
    ((![C : complex, A : complex, B : complex]: ((~ ((C = zero_zero_complex))) => (((times_times_complex @ A @ C) = B) => (A = (divide1210191872omplex @ B @ C))))))). % eq_divide_imp
thf(fact_198_eq__divide__imp, axiom,
    ((![C : real, A : real, B : real]: ((~ ((C = zero_zero_real))) => (((times_times_real @ A @ C) = B) => (A = (divide_divide_real @ B @ C))))))). % eq_divide_imp
thf(fact_199_divide__eq__imp, axiom,
    ((![C : complex, B : complex, A : complex]: ((~ ((C = zero_zero_complex))) => ((B = (times_times_complex @ A @ C)) => ((divide1210191872omplex @ B @ C) = A)))))). % divide_eq_imp
thf(fact_200_divide__eq__imp, axiom,
    ((![C : real, B : real, A : real]: ((~ ((C = zero_zero_real))) => ((B = (times_times_real @ A @ C)) => ((divide_divide_real @ B @ C) = A)))))). % divide_eq_imp
thf(fact_201_eq__divide__eq, axiom,
    ((![A : complex, B : complex, C : complex]: ((A = (divide1210191872omplex @ B @ C)) = (((((~ ((C = zero_zero_complex)))) => (((times_times_complex @ A @ C) = B)))) & ((((C = zero_zero_complex)) => ((A = zero_zero_complex))))))))). % eq_divide_eq
thf(fact_202_eq__divide__eq, axiom,
    ((![A : real, B : real, C : real]: ((A = (divide_divide_real @ B @ C)) = (((((~ ((C = zero_zero_real)))) => (((times_times_real @ A @ C) = B)))) & ((((C = zero_zero_real)) => ((A = zero_zero_real))))))))). % eq_divide_eq
thf(fact_203_divide__eq__eq, axiom,
    ((![B : complex, C : complex, A : complex]: (((divide1210191872omplex @ B @ C) = A) = (((((~ ((C = zero_zero_complex)))) => ((B = (times_times_complex @ A @ C))))) & ((((C = zero_zero_complex)) => ((A = zero_zero_complex))))))))). % divide_eq_eq
thf(fact_204_divide__eq__eq, axiom,
    ((![B : real, C : real, A : real]: (((divide_divide_real @ B @ C) = A) = (((((~ ((C = zero_zero_real)))) => ((B = (times_times_real @ A @ C))))) & ((((C = zero_zero_real)) => ((A = zero_zero_real))))))))). % divide_eq_eq
thf(fact_205_frac__eq__eq, axiom,
    ((![Y : complex, Z3 : complex, X : complex, W3 : complex]: ((~ ((Y = zero_zero_complex))) => ((~ ((Z3 = zero_zero_complex))) => (((divide1210191872omplex @ X @ Y) = (divide1210191872omplex @ W3 @ Z3)) = ((times_times_complex @ X @ Z3) = (times_times_complex @ W3 @ Y)))))))). % frac_eq_eq
thf(fact_206_frac__eq__eq, axiom,
    ((![Y : real, Z3 : real, X : real, W3 : real]: ((~ ((Y = zero_zero_real))) => ((~ ((Z3 = zero_zero_real))) => (((divide_divide_real @ X @ Y) = (divide_divide_real @ W3 @ Z3)) = ((times_times_real @ X @ Z3) = (times_times_real @ W3 @ Y)))))))). % frac_eq_eq
thf(fact_207_right__inverse__eq, axiom,
    ((![B : complex, A : complex]: ((~ ((B = zero_zero_complex))) => (((divide1210191872omplex @ A @ B) = one_one_complex) = (A = B)))))). % right_inverse_eq
thf(fact_208_right__inverse__eq, axiom,
    ((![B : real, A : real]: ((~ ((B = zero_zero_real))) => (((divide_divide_real @ A @ B) = one_one_real) = (A = B)))))). % right_inverse_eq
thf(fact_209_divide__nonpos__pos, axiom,
    ((![X : real, Y : real]: ((ord_less_eq_real @ X @ zero_zero_real) => ((ord_less_real @ zero_zero_real @ Y) => (ord_less_eq_real @ (divide_divide_real @ X @ Y) @ zero_zero_real)))))). % divide_nonpos_pos
thf(fact_210_divide__nonpos__neg, axiom,
    ((![X : real, Y : real]: ((ord_less_eq_real @ X @ zero_zero_real) => ((ord_less_real @ Y @ zero_zero_real) => (ord_less_eq_real @ zero_zero_real @ (divide_divide_real @ X @ Y))))))). % divide_nonpos_neg
thf(fact_211_divide__nonneg__pos, axiom,
    ((![X : real, Y : real]: ((ord_less_eq_real @ zero_zero_real @ X) => ((ord_less_real @ zero_zero_real @ Y) => (ord_less_eq_real @ zero_zero_real @ (divide_divide_real @ X @ Y))))))). % divide_nonneg_pos
thf(fact_212_divide__nonneg__neg, axiom,
    ((![X : real, Y : real]: ((ord_less_eq_real @ zero_zero_real @ X) => ((ord_less_real @ Y @ zero_zero_real) => (ord_less_eq_real @ (divide_divide_real @ X @ Y) @ zero_zero_real)))))). % divide_nonneg_neg
thf(fact_213_divide__le__cancel, axiom,
    ((![A : real, C : real, B : real]: ((ord_less_eq_real @ (divide_divide_real @ A @ C) @ (divide_divide_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))))))))). % divide_le_cancel
thf(fact_214_frac__less2, axiom,
    ((![X : real, Y : real, W3 : real, Z3 : real]: ((ord_less_real @ zero_zero_real @ X) => ((ord_less_eq_real @ X @ Y) => ((ord_less_real @ zero_zero_real @ W3) => ((ord_less_real @ W3 @ Z3) => (ord_less_real @ (divide_divide_real @ X @ Z3) @ (divide_divide_real @ Y @ W3))))))))). % frac_less2
thf(fact_215_frac__less, axiom,
    ((![X : real, Y : real, W3 : real, Z3 : real]: ((ord_less_eq_real @ zero_zero_real @ X) => ((ord_less_real @ X @ Y) => ((ord_less_real @ zero_zero_real @ W3) => ((ord_less_eq_real @ W3 @ Z3) => (ord_less_real @ (divide_divide_real @ X @ Z3) @ (divide_divide_real @ Y @ W3))))))))). % frac_less
thf(fact_216_frac__le, axiom,
    ((![Y : real, X : real, W3 : real, Z3 : real]: ((ord_less_eq_real @ zero_zero_real @ Y) => ((ord_less_eq_real @ X @ Y) => ((ord_less_real @ zero_zero_real @ W3) => ((ord_less_eq_real @ W3 @ Z3) => (ord_less_eq_real @ (divide_divide_real @ X @ Z3) @ (divide_divide_real @ Y @ W3))))))))). % frac_le
thf(fact_217_divide__strict__left__mono__neg, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_real @ A @ B) => ((ord_less_real @ C @ zero_zero_real) => ((ord_less_real @ zero_zero_real @ (times_times_real @ A @ B)) => (ord_less_real @ (divide_divide_real @ C @ A) @ (divide_divide_real @ C @ B)))))))). % divide_strict_left_mono_neg
thf(fact_218_divide__strict__left__mono, axiom,
    ((![B : real, A : real, C : real]: ((ord_less_real @ B @ A) => ((ord_less_real @ zero_zero_real @ C) => ((ord_less_real @ zero_zero_real @ (times_times_real @ A @ B)) => (ord_less_real @ (divide_divide_real @ C @ A) @ (divide_divide_real @ C @ B)))))))). % divide_strict_left_mono

% Conjectures (1)
thf(conj_0, conjecture,
    ((~ (((real_V638595069omplex @ (times_times_complex @ (real_V306493662omplex @ x) @ (poly_complex2 @ ds @ (real_V306493662omplex @ x)))) = (real_V638595069omplex @ d)))))).
