% 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_801__5375748_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:33:16.939

% Could-be-implicit typings (5)
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_t__Nat__Onat, type,
    nat : $tType).

% Explicit typings (39)
thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Complex__Ocomplex, type,
    invers502456322omplex : complex > complex).
thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Real__Oreal, type,
    inverse_inverse_real : real > real).
thf(sy_c_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Oconstant_001t__Complex__Ocomplex_001t__Complex__Ocomplex, type,
    fundam1158420650omplex : (complex > complex) > $o).
thf(sy_c_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Opsize_001t__Complex__Ocomplex, type,
    fundam1709708056omplex : poly_complex > nat).
thf(sy_c_Groups_Oone__class_Oone_001t__Complex__Ocomplex, type,
    one_one_complex : complex).
thf(sy_c_Groups_Oone__class_Oone_001t__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__Nat__Onat, type,
    times_times_nat : nat > nat > nat).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    times_1246143675omplex : poly_complex > poly_complex > poly_complex).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Polynomial__Opoly_It__Real__Oreal_J, type,
    times_775122617y_real : poly_real > poly_real > poly_real).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal, type,
    times_times_real : real > real > real).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Complex__Ocomplex, type,
    zero_zero_complex : complex).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat, type,
    zero_zero_nat : nat).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    zero_z1746442943omplex : poly_complex).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Real__Oreal_J, type,
    zero_zero_poly_real : poly_real).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal, type,
    zero_zero_real : real).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat, type,
    ord_less_nat : nat > nat > $o).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal, type,
    ord_less_real : real > real > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat, type,
    ord_less_eq_nat : nat > nat > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal, type,
    ord_less_eq_real : real > real > $o).
thf(sy_c_Polynomial_Opoly_001t__Complex__Ocomplex, type,
    poly_complex2 : poly_complex > complex > complex).
thf(sy_c_Polynomial_Opoly_001t__Real__Oreal, type,
    poly_real2 : poly_real > real > real).
thf(sy_c_Polynomial_Osmult_001t__Complex__Ocomplex, type,
    smult_complex : complex > poly_complex > poly_complex).
thf(sy_c_Polynomial_Osmult_001t__Real__Oreal, type,
    smult_real : real > poly_real > poly_real).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Complex__Ocomplex, type,
    real_V638595069omplex : complex > real).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Real__Oreal, type,
    real_V646646907m_real : real > real).
thf(sy_c_Real__Vector__Spaces_Oof__real_001t__Complex__Ocomplex, type,
    real_V306493662omplex : real > complex).
thf(sy_c_Real__Vector__Spaces_Oof__real_001t__Real__Oreal, type,
    real_V1205483228l_real : real > real).
thf(sy_v_a____, type,
    a : complex).
thf(sy_v_c____, type,
    c : complex).
thf(sy_v_m____, type,
    m : real).
thf(sy_v_p, type,
    p : poly_complex).
thf(sy_v_pa____, type,
    pa : poly_complex).
thf(sy_v_q____, type,
    q : 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 (197)
thf(fact_0_t_I2_J, axiom,
    ((ord_less_real @ t @ one_one_real))). % t(2)
thf(fact_1_nc, axiom,
    ((~ ((fundam1158420650omplex @ (poly_complex2 @ p)))))). % nc
thf(fact_2__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_3_w0, axiom,
    ((~ ((w = zero_zero_complex))))). % w0
thf(fact_4_a00, axiom,
    ((~ (((poly_complex2 @ q @ zero_zero_complex) = zero_zero_complex))))). % a00
thf(fact_5_kas_I1_J, axiom,
    ((~ ((a = zero_zero_complex))))). % kas(1)
thf(fact_6__092_060open_062_I_092_060And_062x_Ay_O_Apoly_A_Ismult_A_Iinverse_A_Ipoly_Aq_A0_J_J_Aq_J_Ax_A_061_Apoly_A_Ismult_A_Iinverse_A_Ipoly_Aq_A0_J_J_Aq_J_Ay_J_A_092_060Longrightarrow_062_AFalse_092_060close_062, axiom,
    ((~ ((![X : complex, Y : complex]: ((poly_complex2 @ (smult_complex @ (invers502456322omplex @ (poly_complex2 @ q @ zero_zero_complex)) @ q) @ X) = (poly_complex2 @ (smult_complex @ (invers502456322omplex @ (poly_complex2 @ q @ zero_zero_complex)) @ q) @ Y))))))). % \<open>(\<And>x y. poly (smult (inverse (poly q 0)) q) x = poly (smult (inverse (poly q 0)) q) y) \<Longrightarrow> False\<close>
thf(fact_7_mrmq__eq, axiom,
    ((![W : complex]: ((ord_less_real @ (real_V638595069omplex @ (poly_complex2 @ (smult_complex @ (invers502456322omplex @ (poly_complex2 @ q @ zero_zero_complex)) @ q) @ W)) @ one_one_real) = (ord_less_real @ (real_V638595069omplex @ (poly_complex2 @ q @ W)) @ (real_V638595069omplex @ (poly_complex2 @ q @ zero_zero_complex))))))). % mrmq_eq
thf(fact_8__092_060open_062poly_Aq_A0_A_061_Apoly_A_Ismult_A_Iinverse_A_Ipoly_Aq_A0_J_J_Aq_J_A0_A_K_Apoly_Aq_A0_092_060close_062, axiom,
    (((poly_complex2 @ q @ zero_zero_complex) = (times_times_complex @ (poly_complex2 @ (smult_complex @ (invers502456322omplex @ (poly_complex2 @ q @ zero_zero_complex)) @ q) @ zero_zero_complex) @ (poly_complex2 @ q @ zero_zero_complex))))). % \<open>poly q 0 = poly (smult (inverse (poly q 0)) q) 0 * poly q 0\<close>
thf(fact_9_qr, axiom,
    ((![Z : complex]: ((poly_complex2 @ q @ Z) = (times_times_complex @ (poly_complex2 @ (smult_complex @ (invers502456322omplex @ (poly_complex2 @ q @ zero_zero_complex)) @ q) @ Z) @ (poly_complex2 @ q @ zero_zero_complex)))))). % qr
thf(fact_10__092_060open_062cmod_A_Ipoly_A_Ismult_A_Iinverse_A_Ipoly_Aq_A0_J_J_Aq_J_A_Icomplex__of__real_At_A_K_Aw_J_J_A_060_A1_092_060close_062, axiom,
    ((ord_less_real @ (real_V638595069omplex @ (poly_complex2 @ (smult_complex @ (invers502456322omplex @ (poly_complex2 @ q @ zero_zero_complex)) @ q) @ (times_times_complex @ (real_V306493662omplex @ t) @ w))) @ one_one_real))). % \<open>cmod (poly (smult (inverse (poly q 0)) q) (complex_of_real t * w)) < 1\<close>
thf(fact_11__092_060open_062constant_A_Ipoly_Aq_J_A_092_060Longrightarrow_062_AFalse_092_060close_062, axiom,
    ((~ ((fundam1158420650omplex @ (poly_complex2 @ q)))))). % \<open>constant (poly q) \<Longrightarrow> False\<close>
thf(fact_12_qnc, axiom,
    ((~ ((fundam1158420650omplex @ (poly_complex2 @ q)))))). % qnc
thf(fact_13_rnc, axiom,
    ((~ ((fundam1158420650omplex @ (poly_complex2 @ (smult_complex @ (invers502456322omplex @ (poly_complex2 @ q @ zero_zero_complex)) @ q))))))). % rnc
thf(fact_14_real__sup__exists, axiom,
    ((![P : real > $o]: ((?[X_1 : real]: (P @ X_1)) => ((?[Z : real]: (![X : real]: ((P @ X) => (ord_less_real @ X @ Z)))) => (?[S : real]: (![Y2 : real]: ((?[X2 : real]: (((P @ X2)) & ((ord_less_real @ Y2 @ X2)))) = (ord_less_real @ Y2 @ S))))))))). % real_sup_exists
thf(fact_15_lgqr, axiom,
    (((fundam1709708056omplex @ q) = (fundam1709708056omplex @ (smult_complex @ (invers502456322omplex @ (poly_complex2 @ q @ zero_zero_complex)) @ q))))). % lgqr
thf(fact_16_norm__one, axiom,
    (((real_V646646907m_real @ one_one_real) = one_one_real))). % norm_one
thf(fact_17_norm__one, axiom,
    (((real_V638595069omplex @ one_one_complex) = one_one_real))). % norm_one
thf(fact_18_zero__less__norm__iff, axiom,
    ((![X3 : real]: ((ord_less_real @ zero_zero_real @ (real_V646646907m_real @ X3)) = (~ ((X3 = zero_zero_real))))))). % zero_less_norm_iff
thf(fact_19_zero__less__norm__iff, axiom,
    ((![X3 : complex]: ((ord_less_real @ zero_zero_real @ (real_V638595069omplex @ X3)) = (~ ((X3 = zero_zero_complex))))))). % zero_less_norm_iff
thf(fact_20_inverse__less__iff__less, axiom,
    ((![A : real, B : real]: ((ord_less_real @ zero_zero_real @ A) => ((ord_less_real @ zero_zero_real @ B) => ((ord_less_real @ (inverse_inverse_real @ A) @ (inverse_inverse_real @ B)) = (ord_less_real @ B @ A))))))). % inverse_less_iff_less
thf(fact_21_inverse__less__iff__less__neg, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ zero_zero_real) => ((ord_less_real @ B @ zero_zero_real) => ((ord_less_real @ (inverse_inverse_real @ A) @ (inverse_inverse_real @ B)) = (ord_less_real @ B @ A))))))). % inverse_less_iff_less_neg
thf(fact_22_inverse__negative__iff__negative, axiom,
    ((![A : real]: ((ord_less_real @ (inverse_inverse_real @ A) @ zero_zero_real) = (ord_less_real @ A @ zero_zero_real))))). % inverse_negative_iff_negative
thf(fact_23_inverse__positive__iff__positive, axiom,
    ((![A : real]: ((ord_less_real @ zero_zero_real @ (inverse_inverse_real @ A)) = (ord_less_real @ zero_zero_real @ A))))). % inverse_positive_iff_positive
thf(fact_24_r01, axiom,
    (((poly_complex2 @ (smult_complex @ (invers502456322omplex @ (poly_complex2 @ q @ zero_zero_complex)) @ q) @ zero_zero_complex) = one_one_complex))). % r01
thf(fact_25_smult__1__left, axiom,
    ((![P2 : poly_real]: ((smult_real @ one_one_real @ P2) = P2)))). % smult_1_left
thf(fact_26_smult__1__left, axiom,
    ((![P2 : poly_complex]: ((smult_complex @ one_one_complex @ P2) = P2)))). % smult_1_left
thf(fact_27_poly__1, axiom,
    ((![X3 : real]: ((poly_real2 @ one_one_poly_real @ X3) = one_one_real)))). % poly_1
thf(fact_28_poly__1, axiom,
    ((![X3 : complex]: ((poly_complex2 @ one_one_poly_complex @ X3) = one_one_complex)))). % poly_1
thf(fact_29_smult__0__left, axiom,
    ((![P2 : poly_complex]: ((smult_complex @ zero_zero_complex @ P2) = zero_z1746442943omplex)))). % smult_0_left
thf(fact_30_smult__0__left, axiom,
    ((![P2 : poly_real]: ((smult_real @ zero_zero_real @ P2) = zero_zero_poly_real)))). % smult_0_left
thf(fact_31_smult__eq__0__iff, axiom,
    ((![A : complex, P2 : poly_complex]: (((smult_complex @ A @ P2) = zero_z1746442943omplex) = (((A = zero_zero_complex)) | ((P2 = zero_z1746442943omplex))))))). % smult_eq_0_iff
thf(fact_32_smult__eq__0__iff, axiom,
    ((![A : real, P2 : poly_real]: (((smult_real @ A @ P2) = zero_zero_poly_real) = (((A = zero_zero_real)) | ((P2 = zero_zero_poly_real))))))). % smult_eq_0_iff
thf(fact_33_t_I1_J, axiom,
    ((ord_less_real @ zero_zero_real @ t))). % t(1)
thf(fact_34_m_I1_J, axiom,
    ((ord_less_real @ zero_zero_real @ m))). % m(1)
thf(fact_35_inverse__inverse__eq, axiom,
    ((![A : complex]: ((invers502456322omplex @ (invers502456322omplex @ A)) = A)))). % inverse_inverse_eq
thf(fact_36_inverse__eq__iff__eq, axiom,
    ((![A : complex, B : complex]: (((invers502456322omplex @ A) = (invers502456322omplex @ B)) = (A = B))))). % inverse_eq_iff_eq
thf(fact_37_of__real__eq__iff, axiom,
    ((![X3 : real, Y3 : real]: (((real_V306493662omplex @ X3) = (real_V306493662omplex @ Y3)) = (X3 = Y3))))). % of_real_eq_iff
thf(fact_38_inverse__nonzero__iff__nonzero, axiom,
    ((![A : real]: (((inverse_inverse_real @ A) = zero_zero_real) = (A = zero_zero_real))))). % inverse_nonzero_iff_nonzero
thf(fact_39_inverse__nonzero__iff__nonzero, axiom,
    ((![A : complex]: (((invers502456322omplex @ A) = zero_zero_complex) = (A = zero_zero_complex))))). % inverse_nonzero_iff_nonzero
thf(fact_40_inverse__zero, axiom,
    (((inverse_inverse_real @ zero_zero_real) = zero_zero_real))). % inverse_zero
thf(fact_41_inverse__zero, axiom,
    (((invers502456322omplex @ zero_zero_complex) = zero_zero_complex))). % inverse_zero
thf(fact_42_inverse__mult__distrib, axiom,
    ((![A : real, B : real]: ((inverse_inverse_real @ (times_times_real @ A @ B)) = (times_times_real @ (inverse_inverse_real @ A) @ (inverse_inverse_real @ B)))))). % inverse_mult_distrib
thf(fact_43_inverse__mult__distrib, axiom,
    ((![A : complex, B : complex]: ((invers502456322omplex @ (times_times_complex @ A @ B)) = (times_times_complex @ (invers502456322omplex @ A) @ (invers502456322omplex @ B)))))). % inverse_mult_distrib
thf(fact_44_inverse__1, axiom,
    (((inverse_inverse_real @ one_one_real) = one_one_real))). % inverse_1
thf(fact_45_inverse__1, axiom,
    (((invers502456322omplex @ one_one_complex) = one_one_complex))). % inverse_1
thf(fact_46_inverse__eq__1__iff, axiom,
    ((![X3 : real]: (((inverse_inverse_real @ X3) = one_one_real) = (X3 = one_one_real))))). % inverse_eq_1_iff
thf(fact_47_inverse__eq__1__iff, axiom,
    ((![X3 : complex]: (((invers502456322omplex @ X3) = one_one_complex) = (X3 = one_one_complex))))). % inverse_eq_1_iff
thf(fact_48_of__real__mult, axiom,
    ((![X3 : real, Y3 : real]: ((real_V1205483228l_real @ (times_times_real @ X3 @ Y3)) = (times_times_real @ (real_V1205483228l_real @ X3) @ (real_V1205483228l_real @ Y3)))))). % of_real_mult
thf(fact_49_of__real__mult, axiom,
    ((![X3 : real, Y3 : real]: ((real_V306493662omplex @ (times_times_real @ X3 @ Y3)) = (times_times_complex @ (real_V306493662omplex @ X3) @ (real_V306493662omplex @ Y3)))))). % of_real_mult
thf(fact_50_poly__mult, axiom,
    ((![P2 : poly_complex, Q : poly_complex, X3 : complex]: ((poly_complex2 @ (times_1246143675omplex @ P2 @ Q) @ X3) = (times_times_complex @ (poly_complex2 @ P2 @ X3) @ (poly_complex2 @ Q @ X3)))))). % poly_mult
thf(fact_51_poly__mult, axiom,
    ((![P2 : poly_real, Q : poly_real, X3 : real]: ((poly_real2 @ (times_775122617y_real @ P2 @ Q) @ X3) = (times_times_real @ (poly_real2 @ P2 @ X3) @ (poly_real2 @ Q @ X3)))))). % poly_mult
thf(fact_52_smult__smult, axiom,
    ((![A : complex, B : complex, P2 : poly_complex]: ((smult_complex @ A @ (smult_complex @ B @ P2)) = (smult_complex @ (times_times_complex @ A @ B) @ P2))))). % smult_smult
thf(fact_53_smult__smult, axiom,
    ((![A : real, B : real, P2 : poly_real]: ((smult_real @ A @ (smult_real @ B @ P2)) = (smult_real @ (times_times_real @ A @ B) @ P2))))). % smult_smult
thf(fact_54_of__real__inverse, axiom,
    ((![X3 : real]: ((real_V306493662omplex @ (inverse_inverse_real @ X3)) = (invers502456322omplex @ (real_V306493662omplex @ X3)))))). % of_real_inverse
thf(fact_55_smult__0__right, axiom,
    ((![A : complex]: ((smult_complex @ A @ zero_z1746442943omplex) = zero_z1746442943omplex)))). % smult_0_right
thf(fact_56_psize__eq__0__iff, axiom,
    ((![P2 : poly_complex]: (((fundam1709708056omplex @ P2) = zero_zero_nat) = (P2 = zero_z1746442943omplex))))). % psize_eq_0_iff
thf(fact_57_less_Oprems, axiom,
    ((~ ((fundam1158420650omplex @ (poly_complex2 @ pa)))))). % less.prems
thf(fact_58_q_I1_J, axiom,
    (((fundam1709708056omplex @ q) = (fundam1709708056omplex @ pa)))). % q(1)
thf(fact_59_norm__eq__zero, axiom,
    ((![X3 : real]: (((real_V646646907m_real @ X3) = zero_zero_real) = (X3 = zero_zero_real))))). % norm_eq_zero
thf(fact_60_norm__eq__zero, axiom,
    ((![X3 : complex]: (((real_V638595069omplex @ X3) = zero_zero_real) = (X3 = zero_zero_complex))))). % norm_eq_zero
thf(fact_61_norm__zero, axiom,
    (((real_V646646907m_real @ zero_zero_real) = zero_zero_real))). % norm_zero
thf(fact_62_norm__zero, axiom,
    (((real_V638595069omplex @ zero_zero_complex) = zero_zero_real))). % norm_zero
thf(fact_63_of__real__eq__0__iff, axiom,
    ((![X3 : real]: (((real_V1205483228l_real @ X3) = zero_zero_real) = (X3 = zero_zero_real))))). % of_real_eq_0_iff
thf(fact_64_of__real__eq__0__iff, axiom,
    ((![X3 : real]: (((real_V306493662omplex @ X3) = zero_zero_complex) = (X3 = zero_zero_real))))). % of_real_eq_0_iff
thf(fact_65_of__real__0, axiom,
    (((real_V1205483228l_real @ zero_zero_real) = zero_zero_real))). % of_real_0
thf(fact_66_of__real__0, axiom,
    (((real_V306493662omplex @ zero_zero_real) = zero_zero_complex))). % of_real_0
thf(fact_67_of__real__eq__1__iff, axiom,
    ((![X3 : real]: (((real_V1205483228l_real @ X3) = one_one_real) = (X3 = one_one_real))))). % of_real_eq_1_iff
thf(fact_68_of__real__eq__1__iff, axiom,
    ((![X3 : real]: (((real_V306493662omplex @ X3) = one_one_complex) = (X3 = one_one_real))))). % of_real_eq_1_iff
thf(fact_69_of__real__1, axiom,
    (((real_V1205483228l_real @ one_one_real) = one_one_real))). % of_real_1
thf(fact_70_of__real__1, axiom,
    (((real_V306493662omplex @ one_one_real) = one_one_complex))). % of_real_1
thf(fact_71_poly__0, axiom,
    ((![X3 : complex]: ((poly_complex2 @ zero_z1746442943omplex @ X3) = zero_zero_complex)))). % poly_0
thf(fact_72_poly__0, axiom,
    ((![X3 : real]: ((poly_real2 @ zero_zero_poly_real @ X3) = zero_zero_real)))). % poly_0
thf(fact_73_poly__smult, axiom,
    ((![A : complex, P2 : poly_complex, X3 : complex]: ((poly_complex2 @ (smult_complex @ A @ P2) @ X3) = (times_times_complex @ A @ (poly_complex2 @ P2 @ X3)))))). % poly_smult
thf(fact_74_poly__smult, axiom,
    ((![A : real, P2 : poly_real, X3 : real]: ((poly_real2 @ (smult_real @ A @ P2) @ X3) = (times_times_real @ A @ (poly_real2 @ P2 @ X3)))))). % poly_smult
thf(fact_75_right__inverse, axiom,
    ((![A : real]: ((~ ((A = zero_zero_real))) => ((times_times_real @ A @ (inverse_inverse_real @ A)) = one_one_real))))). % right_inverse
thf(fact_76_right__inverse, axiom,
    ((![A : complex]: ((~ ((A = zero_zero_complex))) => ((times_times_complex @ A @ (invers502456322omplex @ A)) = one_one_complex))))). % right_inverse
thf(fact_77_left__inverse, axiom,
    ((![A : real]: ((~ ((A = zero_zero_real))) => ((times_times_real @ (inverse_inverse_real @ A) @ A) = one_one_real))))). % left_inverse
thf(fact_78_left__inverse, axiom,
    ((![A : complex]: ((~ ((A = zero_zero_complex))) => ((times_times_complex @ (invers502456322omplex @ A) @ A) = one_one_complex))))). % left_inverse
thf(fact_79_poly__IVT__neg, axiom,
    ((![A : real, B : real, P2 : poly_real]: ((ord_less_real @ A @ B) => ((ord_less_real @ zero_zero_real @ (poly_real2 @ P2 @ A)) => ((ord_less_real @ (poly_real2 @ P2 @ B) @ zero_zero_real) => (?[X : real]: ((ord_less_real @ A @ X) & ((ord_less_real @ X @ B) & ((poly_real2 @ P2 @ X) = zero_zero_real)))))))))). % poly_IVT_neg
thf(fact_80_poly__IVT__pos, axiom,
    ((![A : real, B : real, P2 : poly_real]: ((ord_less_real @ A @ B) => ((ord_less_real @ (poly_real2 @ P2 @ A) @ zero_zero_real) => ((ord_less_real @ zero_zero_real @ (poly_real2 @ P2 @ B)) => (?[X : real]: ((ord_less_real @ A @ X) & ((ord_less_real @ X @ B) & ((poly_real2 @ P2 @ X) = zero_zero_real)))))))))). % poly_IVT_pos
thf(fact_81_norm__mult, axiom,
    ((![X3 : real, Y3 : real]: ((real_V646646907m_real @ (times_times_real @ X3 @ Y3)) = (times_times_real @ (real_V646646907m_real @ X3) @ (real_V646646907m_real @ Y3)))))). % norm_mult
thf(fact_82_norm__mult, axiom,
    ((![X3 : complex, Y3 : complex]: ((real_V638595069omplex @ (times_times_complex @ X3 @ Y3)) = (times_times_real @ (real_V638595069omplex @ X3) @ (real_V638595069omplex @ Y3)))))). % norm_mult
thf(fact_83_nonzero__of__real__inverse, axiom,
    ((![X3 : real]: ((~ ((X3 = zero_zero_real))) => ((real_V306493662omplex @ (inverse_inverse_real @ X3)) = (invers502456322omplex @ (real_V306493662omplex @ X3))))))). % nonzero_of_real_inverse
thf(fact_84_constant__def, axiom,
    ((fundam1158420650omplex = (^[F : complex > complex]: (![X2 : complex]: (![Y4 : complex]: ((F @ X2) = (F @ Y4)))))))). % constant_def
thf(fact_85_norm__mult__less, axiom,
    ((![X3 : real, R : real, Y3 : real, S2 : real]: ((ord_less_real @ (real_V646646907m_real @ X3) @ R) => ((ord_less_real @ (real_V646646907m_real @ Y3) @ S2) => (ord_less_real @ (real_V646646907m_real @ (times_times_real @ X3 @ Y3)) @ (times_times_real @ R @ S2))))))). % norm_mult_less
thf(fact_86_norm__mult__less, axiom,
    ((![X3 : complex, R : real, Y3 : complex, S2 : real]: ((ord_less_real @ (real_V638595069omplex @ X3) @ R) => ((ord_less_real @ (real_V638595069omplex @ Y3) @ S2) => (ord_less_real @ (real_V638595069omplex @ (times_times_complex @ X3 @ Y3)) @ (times_times_real @ R @ S2))))))). % norm_mult_less
thf(fact_87_norm__not__less__zero, axiom,
    ((![X3 : complex]: (~ ((ord_less_real @ (real_V638595069omplex @ X3) @ zero_zero_real)))))). % norm_not_less_zero
thf(fact_88_mult__commute__imp__mult__inverse__commute, axiom,
    ((![Y3 : real, X3 : real]: (((times_times_real @ Y3 @ X3) = (times_times_real @ X3 @ Y3)) => ((times_times_real @ (inverse_inverse_real @ Y3) @ X3) = (times_times_real @ X3 @ (inverse_inverse_real @ Y3))))))). % mult_commute_imp_mult_inverse_commute
thf(fact_89_mult__commute__imp__mult__inverse__commute, axiom,
    ((![Y3 : complex, X3 : complex]: (((times_times_complex @ Y3 @ X3) = (times_times_complex @ X3 @ Y3)) => ((times_times_complex @ (invers502456322omplex @ Y3) @ X3) = (times_times_complex @ X3 @ (invers502456322omplex @ Y3))))))). % mult_commute_imp_mult_inverse_commute
thf(fact_90_linordered__field__no__ub, axiom,
    ((![X4 : real]: (?[X_12 : real]: (ord_less_real @ X4 @ X_12))))). % linordered_field_no_ub
thf(fact_91_linordered__field__no__lb, axiom,
    ((![X4 : real]: (?[Y : real]: (ord_less_real @ Y @ X4))))). % linordered_field_no_lb
thf(fact_92_inverse__eq__imp__eq, axiom,
    ((![A : complex, B : complex]: (((invers502456322omplex @ A) = (invers502456322omplex @ B)) => (A = B))))). % inverse_eq_imp_eq
thf(fact_93_nonzero__inverse__mult__distrib, axiom,
    ((![A : real, B : real]: ((~ ((A = zero_zero_real))) => ((~ ((B = zero_zero_real))) => ((inverse_inverse_real @ (times_times_real @ A @ B)) = (times_times_real @ (inverse_inverse_real @ B) @ (inverse_inverse_real @ A)))))))). % nonzero_inverse_mult_distrib
thf(fact_94_nonzero__inverse__mult__distrib, axiom,
    ((![A : complex, B : complex]: ((~ ((A = zero_zero_complex))) => ((~ ((B = zero_zero_complex))) => ((invers502456322omplex @ (times_times_complex @ A @ B)) = (times_times_complex @ (invers502456322omplex @ B) @ (invers502456322omplex @ A)))))))). % nonzero_inverse_mult_distrib
thf(fact_95_inverse__unique, axiom,
    ((![A : real, B : real]: (((times_times_real @ A @ B) = one_one_real) => ((inverse_inverse_real @ A) = B))))). % inverse_unique
thf(fact_96_inverse__unique, axiom,
    ((![A : complex, B : complex]: (((times_times_complex @ A @ B) = one_one_complex) => ((invers502456322omplex @ A) = B))))). % inverse_unique
thf(fact_97_poly__eq__poly__eq__iff, axiom,
    ((![P2 : poly_complex, Q : poly_complex]: (((poly_complex2 @ P2) = (poly_complex2 @ Q)) = (P2 = Q))))). % poly_eq_poly_eq_iff
thf(fact_98_poly__eq__poly__eq__iff, axiom,
    ((![P2 : poly_real, Q : poly_real]: (((poly_real2 @ P2) = (poly_real2 @ Q)) = (P2 = Q))))). % poly_eq_poly_eq_iff
thf(fact_99_field__class_Ofield__inverse, axiom,
    ((![A : real]: ((~ ((A = zero_zero_real))) => ((times_times_real @ (inverse_inverse_real @ A) @ A) = one_one_real))))). % field_class.field_inverse
thf(fact_100_field__class_Ofield__inverse, axiom,
    ((![A : complex]: ((~ ((A = zero_zero_complex))) => ((times_times_complex @ (invers502456322omplex @ A) @ A) = one_one_complex))))). % field_class.field_inverse
thf(fact_101_nonzero__imp__inverse__nonzero, axiom,
    ((![A : real]: ((~ ((A = zero_zero_real))) => (~ (((inverse_inverse_real @ A) = zero_zero_real))))))). % nonzero_imp_inverse_nonzero
thf(fact_102_nonzero__imp__inverse__nonzero, axiom,
    ((![A : complex]: ((~ ((A = zero_zero_complex))) => (~ (((invers502456322omplex @ A) = zero_zero_complex))))))). % nonzero_imp_inverse_nonzero
thf(fact_103_nonzero__inverse__inverse__eq, axiom,
    ((![A : real]: ((~ ((A = zero_zero_real))) => ((inverse_inverse_real @ (inverse_inverse_real @ A)) = A))))). % nonzero_inverse_inverse_eq
thf(fact_104_nonzero__inverse__inverse__eq, axiom,
    ((![A : complex]: ((~ ((A = zero_zero_complex))) => ((invers502456322omplex @ (invers502456322omplex @ A)) = A))))). % nonzero_inverse_inverse_eq
thf(fact_105_nonzero__inverse__eq__imp__eq, axiom,
    ((![A : real, B : real]: (((inverse_inverse_real @ A) = (inverse_inverse_real @ B)) => ((~ ((A = zero_zero_real))) => ((~ ((B = zero_zero_real))) => (A = B))))))). % nonzero_inverse_eq_imp_eq
thf(fact_106_nonzero__inverse__eq__imp__eq, axiom,
    ((![A : complex, B : complex]: (((invers502456322omplex @ A) = (invers502456322omplex @ B)) => ((~ ((A = zero_zero_complex))) => ((~ ((B = zero_zero_complex))) => (A = B))))))). % nonzero_inverse_eq_imp_eq
thf(fact_107_inverse__zero__imp__zero, axiom,
    ((![A : real]: (((inverse_inverse_real @ A) = zero_zero_real) => (A = zero_zero_real))))). % inverse_zero_imp_zero
thf(fact_108_inverse__zero__imp__zero, axiom,
    ((![A : complex]: (((invers502456322omplex @ A) = zero_zero_complex) => (A = zero_zero_complex))))). % inverse_zero_imp_zero
thf(fact_109_field__class_Ofield__inverse__zero, axiom,
    (((inverse_inverse_real @ zero_zero_real) = zero_zero_real))). % field_class.field_inverse_zero
thf(fact_110_field__class_Ofield__inverse__zero, axiom,
    (((invers502456322omplex @ zero_zero_complex) = zero_zero_complex))). % field_class.field_inverse_zero
thf(fact_111_poly__all__0__iff__0, axiom,
    ((![P2 : poly_complex]: ((![X2 : complex]: ((poly_complex2 @ P2 @ X2) = zero_zero_complex)) = (P2 = zero_z1746442943omplex))))). % poly_all_0_iff_0
thf(fact_112_poly__all__0__iff__0, axiom,
    ((![P2 : poly_real]: ((![X2 : real]: ((poly_real2 @ P2 @ X2) = zero_zero_real)) = (P2 = zero_zero_poly_real))))). % poly_all_0_iff_0
thf(fact_113_norm__inverse, axiom,
    ((![A : complex]: ((real_V638595069omplex @ (invers502456322omplex @ A)) = (inverse_inverse_real @ (real_V638595069omplex @ A)))))). % norm_inverse
thf(fact_114_positive__imp__inverse__positive, axiom,
    ((![A : real]: ((ord_less_real @ zero_zero_real @ A) => (ord_less_real @ zero_zero_real @ (inverse_inverse_real @ A)))))). % positive_imp_inverse_positive
thf(fact_115_negative__imp__inverse__negative, axiom,
    ((![A : real]: ((ord_less_real @ A @ zero_zero_real) => (ord_less_real @ (inverse_inverse_real @ A) @ zero_zero_real))))). % negative_imp_inverse_negative
thf(fact_116_inverse__positive__imp__positive, axiom,
    ((![A : real]: ((ord_less_real @ zero_zero_real @ (inverse_inverse_real @ A)) => ((~ ((A = zero_zero_real))) => (ord_less_real @ zero_zero_real @ A)))))). % inverse_positive_imp_positive
thf(fact_117_inverse__negative__imp__negative, axiom,
    ((![A : real]: ((ord_less_real @ (inverse_inverse_real @ A) @ zero_zero_real) => ((~ ((A = zero_zero_real))) => (ord_less_real @ A @ zero_zero_real)))))). % inverse_negative_imp_negative
thf(fact_118_less__imp__inverse__less__neg, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ B) => ((ord_less_real @ B @ zero_zero_real) => (ord_less_real @ (inverse_inverse_real @ B) @ (inverse_inverse_real @ A))))))). % less_imp_inverse_less_neg
thf(fact_119_inverse__less__imp__less__neg, axiom,
    ((![A : real, B : real]: ((ord_less_real @ (inverse_inverse_real @ A) @ (inverse_inverse_real @ B)) => ((ord_less_real @ B @ zero_zero_real) => (ord_less_real @ B @ A)))))). % inverse_less_imp_less_neg
thf(fact_120_less__imp__inverse__less, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ B) => ((ord_less_real @ zero_zero_real @ A) => (ord_less_real @ (inverse_inverse_real @ B) @ (inverse_inverse_real @ A))))))). % less_imp_inverse_less
thf(fact_121_inverse__less__imp__less, axiom,
    ((![A : real, B : real]: ((ord_less_real @ (inverse_inverse_real @ A) @ (inverse_inverse_real @ B)) => ((ord_less_real @ zero_zero_real @ A) => (ord_less_real @ B @ A)))))). % inverse_less_imp_less
thf(fact_122_nonzero__norm__inverse, axiom,
    ((![A : real]: ((~ ((A = zero_zero_real))) => ((real_V646646907m_real @ (inverse_inverse_real @ A)) = (inverse_inverse_real @ (real_V646646907m_real @ A))))))). % nonzero_norm_inverse
thf(fact_123_nonzero__norm__inverse, axiom,
    ((![A : complex]: ((~ ((A = zero_zero_complex))) => ((real_V638595069omplex @ (invers502456322omplex @ A)) = (inverse_inverse_real @ (real_V638595069omplex @ A))))))). % nonzero_norm_inverse
thf(fact_124_one__less__inverse__iff, axiom,
    ((![X3 : real]: ((ord_less_real @ one_one_real @ (inverse_inverse_real @ X3)) = (((ord_less_real @ zero_zero_real @ X3)) & ((ord_less_real @ X3 @ one_one_real))))))). % one_less_inverse_iff
thf(fact_125_one__less__inverse, axiom,
    ((![A : real]: ((ord_less_real @ zero_zero_real @ A) => ((ord_less_real @ A @ one_one_real) => (ord_less_real @ one_one_real @ (inverse_inverse_real @ A))))))). % one_less_inverse
thf(fact_126_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_127_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_128_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_129_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_130_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_131_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_132_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_133_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_134_tw, axiom,
    ((ord_less_eq_real @ (real_V638595069omplex @ (times_times_complex @ (real_V306493662omplex @ t) @ w)) @ (real_V638595069omplex @ w)))). % tw
thf(fact_135_mult_Oright__neutral, axiom,
    ((![A : complex]: ((times_times_complex @ A @ one_one_complex) = A)))). % mult.right_neutral
thf(fact_136_mult_Oright__neutral, axiom,
    ((![A : real]: ((times_times_real @ A @ one_one_real) = A)))). % mult.right_neutral
thf(fact_137_mult_Oleft__neutral, axiom,
    ((![A : complex]: ((times_times_complex @ one_one_complex @ A) = A)))). % mult.left_neutral
thf(fact_138_mult_Oleft__neutral, axiom,
    ((![A : real]: ((times_times_real @ one_one_real @ A) = A)))). % mult.left_neutral
thf(fact_139_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_140_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_141__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062c_O_A_092_060forall_062w_O_Acmod_A_Ipoly_Ap_Ac_J_A_092_060le_062_Acmod_A_Ipoly_Ap_Aw_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom,
    ((~ ((![C2 : complex]: (~ ((![W2 : complex]: (ord_less_eq_real @ (real_V638595069omplex @ (poly_complex2 @ pa @ C2)) @ (real_V638595069omplex @ (poly_complex2 @ pa @ W2))))))))))). % \<open>\<And>thesis. (\<And>c. \<forall>w. cmod (poly p c) \<le> cmod (poly p w) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_142_c, axiom,
    ((![W2 : complex]: (ord_less_eq_real @ (real_V638595069omplex @ (poly_complex2 @ pa @ c)) @ (real_V638595069omplex @ (poly_complex2 @ pa @ W2)))))). % c
thf(fact_143_less_Ohyps, axiom,
    ((![P2 : poly_complex]: ((ord_less_nat @ (fundam1709708056omplex @ P2) @ (fundam1709708056omplex @ pa)) => ((~ ((fundam1158420650omplex @ (poly_complex2 @ P2)))) => (?[Z2 : complex]: ((poly_complex2 @ P2 @ Z2) = zero_zero_complex))))))). % less.hyps
thf(fact_144__092_060open_062t_A_K_Acmod_Aw_A_092_060le_062_A1_A_K_Acmod_Aw_092_060close_062, axiom,
    ((ord_less_eq_real @ (times_times_real @ t @ (real_V638595069omplex @ w)) @ (times_times_real @ one_one_real @ (real_V638595069omplex @ w))))). % \<open>t * cmod w \<le> 1 * cmod w\<close>
thf(fact_145_not__gr__zero, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr_zero
thf(fact_146_mult__zero__left, axiom,
    ((![A : complex]: ((times_times_complex @ zero_zero_complex @ A) = zero_zero_complex)))). % mult_zero_left
thf(fact_147_mult__zero__left, axiom,
    ((![A : real]: ((times_times_real @ zero_zero_real @ A) = zero_zero_real)))). % mult_zero_left
thf(fact_148_mult__zero__right, axiom,
    ((![A : complex]: ((times_times_complex @ A @ zero_zero_complex) = zero_zero_complex)))). % mult_zero_right
thf(fact_149_mult__zero__right, axiom,
    ((![A : real]: ((times_times_real @ A @ zero_zero_real) = zero_zero_real)))). % mult_zero_right
thf(fact_150_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_151_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_152_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_153_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_154_cq0, axiom,
    ((![W2 : complex]: (ord_less_eq_real @ (real_V638595069omplex @ (poly_complex2 @ q @ zero_zero_complex)) @ (real_V638595069omplex @ (poly_complex2 @ pa @ W2)))))). % cq0
thf(fact_155_mult__smult__left, axiom,
    ((![A : complex, P2 : poly_complex, Q : poly_complex]: ((times_1246143675omplex @ (smult_complex @ A @ P2) @ Q) = (smult_complex @ A @ (times_1246143675omplex @ P2 @ Q)))))). % mult_smult_left
thf(fact_156_mult__smult__right, axiom,
    ((![P2 : poly_complex, A : complex, Q : poly_complex]: ((times_1246143675omplex @ P2 @ (smult_complex @ A @ Q)) = (smult_complex @ A @ (times_1246143675omplex @ P2 @ Q)))))). % mult_smult_right
thf(fact_157_inverse__nonnegative__iff__nonnegative, axiom,
    ((![A : real]: ((ord_less_eq_real @ zero_zero_real @ (inverse_inverse_real @ A)) = (ord_less_eq_real @ zero_zero_real @ A))))). % inverse_nonnegative_iff_nonnegative
thf(fact_158_inverse__nonpositive__iff__nonpositive, axiom,
    ((![A : real]: ((ord_less_eq_real @ (inverse_inverse_real @ A) @ zero_zero_real) = (ord_less_eq_real @ A @ zero_zero_real))))). % inverse_nonpositive_iff_nonpositive
thf(fact_159_inverse__le__iff__le, axiom,
    ((![A : real, B : real]: ((ord_less_real @ zero_zero_real @ A) => ((ord_less_real @ zero_zero_real @ B) => ((ord_less_eq_real @ (inverse_inverse_real @ A) @ (inverse_inverse_real @ B)) = (ord_less_eq_real @ B @ A))))))). % inverse_le_iff_le
thf(fact_160_inverse__le__iff__le__neg, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ zero_zero_real) => ((ord_less_real @ B @ zero_zero_real) => ((ord_less_eq_real @ (inverse_inverse_real @ A) @ (inverse_inverse_real @ B)) = (ord_less_eq_real @ B @ A))))))). % inverse_le_iff_le_neg
thf(fact_161_norm__le__zero__iff, axiom,
    ((![X3 : real]: ((ord_less_eq_real @ (real_V646646907m_real @ X3) @ zero_zero_real) = (X3 = zero_zero_real))))). % norm_le_zero_iff
thf(fact_162_norm__le__zero__iff, axiom,
    ((![X3 : complex]: ((ord_less_eq_real @ (real_V638595069omplex @ X3) @ zero_zero_real) = (X3 = zero_zero_complex))))). % norm_le_zero_iff
thf(fact_163_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_164_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
thf(fact_165_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_166_zero__le__square, axiom,
    ((![A : real]: (ord_less_eq_real @ zero_zero_real @ (times_times_real @ A @ A))))). % zero_le_square
thf(fact_167_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_168_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_169_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_170_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_171_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_172_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_173_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_174_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_175_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_176_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_177_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_178_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_179_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_180_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_181_zero__le__one, axiom,
    ((ord_less_eq_real @ zero_zero_real @ one_one_real))). % zero_le_one
thf(fact_182_not__one__le__zero, axiom,
    ((~ ((ord_less_eq_real @ one_one_real @ zero_zero_real))))). % not_one_le_zero
thf(fact_183_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_184_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_185_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_186_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_187_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_188_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_189_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_190_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_191_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_192_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_193_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_194_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_195_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_196_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

% Conjectures (1)
thf(conj_0, conjecture,
    ((?[W2 : complex]: (ord_less_real @ (real_V638595069omplex @ (poly_complex2 @ (smult_complex @ (invers502456322omplex @ (poly_complex2 @ q @ zero_zero_complex)) @ q) @ W2)) @ one_one_real)))).
