% 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_504__5372862_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:30:41.108

% Could-be-implicit typings (7)
thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_Itf__a_J_J, type,
    poly_poly_a : $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__Polynomial__Opoly_Itf__a_J, type,
    poly_a : $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 (50)
thf(sy_c_Groups_Oabs__class_Oabs_001t__Real__Oreal, type,
    abs_abs_real : real > real).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Complex__Ocomplex, type,
    minus_minus_complex : complex > complex > complex).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    minus_174331535omplex : poly_complex > poly_complex > poly_complex).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Polynomial__Opoly_It__Real__Oreal_J, type,
    minus_240770701y_real : poly_real > poly_real > poly_real).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Polynomial__Opoly_Itf__a_J, type,
    minus_minus_poly_a : poly_a > poly_a > poly_a).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal, type,
    minus_minus_real : real > real > real).
thf(sy_c_Groups_Ominus__class_Ominus_001tf__a, type,
    minus_minus_a : a > a > a).
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_Oone__class_Oone_001tf__a, type,
    one_one_a : a).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Complex__Ocomplex, type,
    plus_plus_complex : complex > complex > complex).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    plus_p1547158847omplex : poly_complex > poly_complex > poly_complex).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Polynomial__Opoly_It__Real__Oreal_J, type,
    plus_plus_poly_real : poly_real > poly_real > poly_real).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Polynomial__Opoly_Itf__a_J, type,
    plus_plus_poly_a : poly_a > poly_a > poly_a).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal, type,
    plus_plus_real : real > real > real).
thf(sy_c_Groups_Oplus__class_Oplus_001tf__a, type,
    plus_plus_a : a > a > a).
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__Real__Oreal_J, type,
    times_775122617y_real : poly_real > poly_real > poly_real).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Polynomial__Opoly_Itf__a_J, type,
    times_times_poly_a : poly_a > poly_a > poly_a).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal, type,
    times_times_real : real > real > real).
thf(sy_c_Groups_Otimes__class_Otimes_001tf__a, type,
    times_times_a : a > a > a).
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_Itf__a_J_J, type,
    zero_z2096148049poly_a : poly_poly_a).
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__Polynomial__Opoly_Itf__a_J, type,
    zero_zero_poly_a : poly_a).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal, type,
    zero_zero_real : real).
thf(sy_c_Groups_Ozero__class_Ozero_001tf__a, type,
    zero_zero_a : a).
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_Itf__a_J, type,
    pCons_poly_a : poly_a > poly_poly_a > poly_poly_a).
thf(sy_c_Polynomial_OpCons_001t__Real__Oreal, type,
    pCons_real : real > poly_real > poly_real).
thf(sy_c_Polynomial_OpCons_001tf__a, type,
    pCons_a : a > poly_a > poly_a).
thf(sy_c_Polynomial_Opoly_001t__Complex__Ocomplex, type,
    poly_complex2 : poly_complex > complex > complex).
thf(sy_c_Polynomial_Opoly_001t__Polynomial__Opoly_Itf__a_J, type,
    poly_poly_a2 : poly_poly_a > poly_a > poly_a).
thf(sy_c_Polynomial_Opoly_001t__Real__Oreal, type,
    poly_real2 : poly_real > real > real).
thf(sy_c_Polynomial_Opoly_001tf__a, type,
    poly_a2 : poly_a > a > a).
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_Onorm__class_Onorm_001tf__a, type,
    real_V1022479215norm_a : a > real).
thf(sy_v_aa____, type,
    aa : a).
thf(sy_v_c____, type,
    c : a).
thf(sy_v_cs____, type,
    cs : poly_a).
thf(sy_v_da____, type,
    da : real).
thf(sy_v_p, type,
    p : poly_a).
thf(sy_v_r____, type,
    r : real).
thf(sy_v_z____, type,
    z : a).

% Relevant facts (220)
thf(fact_0_False, axiom,
    ((~ ((cs = zero_zero_poly_a))))). % False
thf(fact_1_th2, axiom,
    ((ord_less_eq_real @ (minus_minus_real @ (real_V1022479215norm_a @ (times_times_a @ z @ (poly_a2 @ (pCons_a @ c @ cs) @ z))) @ (real_V1022479215norm_a @ aa)) @ (real_V1022479215norm_a @ (poly_a2 @ (pCons_a @ aa @ (pCons_a @ c @ cs)) @ z))))). % th2
thf(fact_2_th1, axiom,
    ((ord_less_eq_real @ da @ (minus_minus_real @ (real_V1022479215norm_a @ (times_times_a @ z @ (poly_a2 @ (pCons_a @ c @ cs) @ z))) @ (real_V1022479215norm_a @ aa))))). % th1
thf(fact_3_pCons_Oprems, axiom,
    ((~ (((pCons_a @ c @ cs) = zero_zero_poly_a))))). % pCons.prems
thf(fact_4_r0, axiom,
    ((ord_less_eq_real @ r @ (real_V1022479215norm_a @ z)))). % r0
thf(fact_5_ex, axiom,
    ((~ ((p = zero_zero_poly_a))))). % ex
thf(fact_6_r, axiom,
    ((![Z : a]: ((ord_less_eq_real @ r @ (real_V1022479215norm_a @ Z)) => (ord_less_eq_real @ (plus_plus_real @ da @ (real_V1022479215norm_a @ aa)) @ (real_V1022479215norm_a @ (poly_a2 @ (pCons_a @ c @ cs) @ Z))))))). % r
thf(fact_7__092_060open_062d_A_L_Anorm_Aa_A_092_060le_062_Anorm_A_Ipoly_A_IpCons_Ac_Acs_J_Az_J_092_060close_062, axiom,
    ((ord_less_eq_real @ (plus_plus_real @ da @ (real_V1022479215norm_a @ aa)) @ (real_V1022479215norm_a @ (poly_a2 @ (pCons_a @ c @ cs) @ z))))). % \<open>d + norm a \<le> norm (poly (pCons c cs) z)\<close>
thf(fact_8__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062r_O_A_092_060forall_062z_O_Ar_A_092_060le_062_Anorm_Az_A_092_060longrightarrow_062_Ad_A_L_Anorm_Aa_A_092_060le_062_Anorm_A_Ipoly_A_IpCons_Ac_Acs_J_Az_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom,
    ((~ ((![R : real]: (~ ((![Z : a]: ((ord_less_eq_real @ R @ (real_V1022479215norm_a @ Z)) => (ord_less_eq_real @ (plus_plus_real @ da @ (real_V1022479215norm_a @ aa)) @ (real_V1022479215norm_a @ (poly_a2 @ (pCons_a @ c @ cs) @ Z)))))))))))). % \<open>\<And>thesis. (\<And>r. \<forall>z. r \<le> norm z \<longrightarrow> d + norm a \<le> norm (poly (pCons c cs) z) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_9_pCons_Ohyps_I2_J, axiom,
    ((![D : real, A : a]: ((~ ((cs = zero_zero_poly_a))) => (?[R : real]: (![Z : a]: ((ord_less_eq_real @ R @ (real_V1022479215norm_a @ Z)) => (ord_less_eq_real @ D @ (real_V1022479215norm_a @ (poly_a2 @ (pCons_a @ A @ cs) @ Z)))))))))). % pCons.hyps(2)
thf(fact_10__092_060open_062norm_A_Iz_A_K_Apoly_A_IpCons_Ac_Acs_J_Az_J_A_N_Anorm_Aa_A_092_060le_062_Anorm_A_Iz_A_K_Apoly_A_IpCons_Ac_Acs_J_Az_A_L_Aa_J_092_060close_062, axiom,
    ((ord_less_eq_real @ (minus_minus_real @ (real_V1022479215norm_a @ (times_times_a @ z @ (poly_a2 @ (pCons_a @ c @ cs) @ z))) @ (real_V1022479215norm_a @ aa)) @ (real_V1022479215norm_a @ (plus_plus_a @ (times_times_a @ z @ (poly_a2 @ (pCons_a @ c @ cs) @ z)) @ aa))))). % \<open>norm (z * poly (pCons c cs) z) - norm a \<le> norm (z * poly (pCons c cs) z + a)\<close>
thf(fact_11_z1, axiom,
    ((ord_less_eq_real @ one_one_real @ (real_V1022479215norm_a @ z)))). % z1
thf(fact_12__092_060open_062d_A_L_Anorm_Aa_A_092_060le_062_Anorm_Az_A_K_Anorm_A_Ipoly_A_IpCons_Ac_Acs_J_Az_J_092_060close_062, axiom,
    ((ord_less_eq_real @ (plus_plus_real @ da @ (real_V1022479215norm_a @ aa)) @ (times_times_real @ (real_V1022479215norm_a @ z) @ (real_V1022479215norm_a @ (poly_a2 @ (pCons_a @ c @ cs) @ z)))))). % \<open>d + norm a \<le> norm z * norm (poly (pCons c cs) z)\<close>
thf(fact_13_pCons__eq__iff, axiom,
    ((![A : a, P : poly_a, B : a, Q : poly_a]: (((pCons_a @ A @ P) = (pCons_a @ B @ Q)) = (((A = B)) & ((P = Q))))))). % pCons_eq_iff
thf(fact_14_order__refl, axiom,
    ((![X : real]: (ord_less_eq_real @ X @ X)))). % order_refl
thf(fact_15_th0, axiom,
    ((ord_less_eq_real @ (plus_plus_real @ da @ (real_V1022479215norm_a @ aa)) @ (times_times_real @ one_one_real @ (real_V1022479215norm_a @ (poly_a2 @ (pCons_a @ c @ cs) @ z)))))). % th0
thf(fact_16_pCons_Ohyps_I1_J, axiom,
    (((~ ((c = zero_zero_a))) | (~ ((cs = zero_zero_poly_a)))))). % pCons.hyps(1)
thf(fact_17_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_18_add__pCons, axiom,
    ((![A : real, P : poly_real, B : real, Q : poly_real]: ((plus_plus_poly_real @ (pCons_real @ A @ P) @ (pCons_real @ B @ Q)) = (pCons_real @ (plus_plus_real @ A @ B) @ (plus_plus_poly_real @ P @ Q)))))). % add_pCons
thf(fact_19_add__pCons, axiom,
    ((![A : a, P : poly_a, B : a, Q : poly_a]: ((plus_plus_poly_a @ (pCons_a @ A @ P) @ (pCons_a @ B @ Q)) = (pCons_a @ (plus_plus_a @ A @ B) @ (plus_plus_poly_a @ P @ Q)))))). % add_pCons
thf(fact_20_add__pCons, axiom,
    ((![A : complex, P : poly_complex, B : complex, Q : poly_complex]: ((plus_p1547158847omplex @ (pCons_complex @ A @ P) @ (pCons_complex @ B @ Q)) = (pCons_complex @ (plus_plus_complex @ A @ B) @ (plus_p1547158847omplex @ P @ Q)))))). % add_pCons
thf(fact_21_diff__pCons, axiom,
    ((![A : a, P : poly_a, B : a, Q : poly_a]: ((minus_minus_poly_a @ (pCons_a @ A @ P) @ (pCons_a @ B @ Q)) = (pCons_a @ (minus_minus_a @ A @ B) @ (minus_minus_poly_a @ P @ Q)))))). % diff_pCons
thf(fact_22_diff__pCons, axiom,
    ((![A : real, P : poly_real, B : real, Q : poly_real]: ((minus_240770701y_real @ (pCons_real @ A @ P) @ (pCons_real @ B @ Q)) = (pCons_real @ (minus_minus_real @ A @ B) @ (minus_240770701y_real @ P @ Q)))))). % diff_pCons
thf(fact_23_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_24_poly__mult, axiom,
    ((![P : poly_a, Q : poly_a, X : a]: ((poly_a2 @ (times_times_poly_a @ P @ Q) @ X) = (times_times_a @ (poly_a2 @ P @ X) @ (poly_a2 @ Q @ X)))))). % poly_mult
thf(fact_25_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_26_poly__add, axiom,
    ((![P : poly_real, Q : poly_real, X : real]: ((poly_real2 @ (plus_plus_poly_real @ P @ Q) @ X) = (plus_plus_real @ (poly_real2 @ P @ X) @ (poly_real2 @ Q @ X)))))). % poly_add
thf(fact_27_poly__add, axiom,
    ((![P : poly_a, Q : poly_a, X : a]: ((poly_a2 @ (plus_plus_poly_a @ P @ Q) @ X) = (plus_plus_a @ (poly_a2 @ P @ X) @ (poly_a2 @ Q @ X)))))). % poly_add
thf(fact_28_poly__add, axiom,
    ((![P : poly_complex, Q : poly_complex, X : complex]: ((poly_complex2 @ (plus_p1547158847omplex @ P @ Q) @ X) = (plus_plus_complex @ (poly_complex2 @ P @ X) @ (poly_complex2 @ Q @ X)))))). % poly_add
thf(fact_29_poly__1, axiom,
    ((![X : complex]: ((poly_complex2 @ one_one_poly_complex @ X) = one_one_complex)))). % poly_1
thf(fact_30_poly__1, axiom,
    ((![X : real]: ((poly_real2 @ one_one_poly_real @ X) = one_one_real)))). % poly_1
thf(fact_31_poly__diff, axiom,
    ((![P : poly_complex, Q : poly_complex, X : complex]: ((poly_complex2 @ (minus_174331535omplex @ P @ Q) @ X) = (minus_minus_complex @ (poly_complex2 @ P @ X) @ (poly_complex2 @ Q @ X)))))). % poly_diff
thf(fact_32_poly__diff, axiom,
    ((![P : poly_real, Q : poly_real, X : real]: ((poly_real2 @ (minus_240770701y_real @ P @ Q) @ X) = (minus_minus_real @ (poly_real2 @ P @ X) @ (poly_real2 @ Q @ X)))))). % poly_diff
thf(fact_33_pCons__0__0, axiom,
    (((pCons_poly_a @ zero_zero_poly_a @ zero_z2096148049poly_a) = zero_z2096148049poly_a))). % pCons_0_0
thf(fact_34_pCons__0__0, axiom,
    (((pCons_a @ zero_zero_a @ zero_zero_poly_a) = zero_zero_poly_a))). % pCons_0_0
thf(fact_35_pCons__eq__0__iff, axiom,
    ((![A : poly_a, P : poly_poly_a]: (((pCons_poly_a @ A @ P) = zero_z2096148049poly_a) = (((A = zero_zero_poly_a)) & ((P = zero_z2096148049poly_a))))))). % pCons_eq_0_iff
thf(fact_36_pCons__eq__0__iff, axiom,
    ((![A : a, P : poly_a]: (((pCons_a @ A @ P) = zero_zero_poly_a) = (((A = zero_zero_a)) & ((P = zero_zero_poly_a))))))). % pCons_eq_0_iff
thf(fact_37_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_38_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_39_poly__0, axiom,
    ((![X : complex]: ((poly_complex2 @ zero_z1746442943omplex @ X) = zero_zero_complex)))). % poly_0
thf(fact_40_poly__0, axiom,
    ((![X : poly_a]: ((poly_poly_a2 @ zero_z2096148049poly_a @ X) = zero_zero_poly_a)))). % poly_0
thf(fact_41_poly__0, axiom,
    ((![X : a]: ((poly_a2 @ zero_zero_poly_a @ X) = zero_zero_a)))). % poly_0
thf(fact_42_poly__pCons, axiom,
    ((![A : complex, P : poly_complex, X : complex]: ((poly_complex2 @ (pCons_complex @ A @ P) @ X) = (plus_plus_complex @ A @ (times_times_complex @ X @ (poly_complex2 @ P @ X))))))). % poly_pCons
thf(fact_43_poly__pCons, axiom,
    ((![A : a, P : poly_a, X : a]: ((poly_a2 @ (pCons_a @ A @ P) @ X) = (plus_plus_a @ A @ (times_times_a @ X @ (poly_a2 @ P @ X))))))). % poly_pCons
thf(fact_44_poly__pCons, axiom,
    ((![A : real, P : poly_real, X : real]: ((poly_real2 @ (pCons_real @ A @ P) @ X) = (plus_plus_real @ A @ (times_times_real @ X @ (poly_real2 @ P @ X))))))). % poly_pCons
thf(fact_45_that, axiom,
    ((ord_less_eq_real @ (plus_plus_real @ one_one_real @ (abs_abs_real @ r)) @ (real_V1022479215norm_a @ z)))). % that
thf(fact_46_pCons__one, axiom,
    (((pCons_real @ one_one_real @ zero_zero_poly_real) = one_one_poly_real))). % pCons_one
thf(fact_47_pCons__induct, axiom,
    ((![P2 : poly_poly_a > $o, P : poly_poly_a]: ((P2 @ zero_z2096148049poly_a) => ((![A2 : poly_a, P3 : poly_poly_a]: (((~ ((A2 = zero_zero_poly_a))) | (~ ((P3 = zero_z2096148049poly_a)))) => ((P2 @ P3) => (P2 @ (pCons_poly_a @ A2 @ P3))))) => (P2 @ P)))))). % pCons_induct
thf(fact_48_pCons__induct, axiom,
    ((![P2 : poly_a > $o, P : poly_a]: ((P2 @ zero_zero_poly_a) => ((![A2 : a, P3 : poly_a]: (((~ ((A2 = zero_zero_a))) | (~ ((P3 = zero_zero_poly_a)))) => ((P2 @ P3) => (P2 @ (pCons_a @ A2 @ P3))))) => (P2 @ P)))))). % pCons_induct
thf(fact_49_poly__all__0__iff__0, axiom,
    ((![P : poly_complex]: ((![X2 : complex]: ((poly_complex2 @ P @ X2) = zero_zero_complex)) = (P = zero_z1746442943omplex))))). % poly_all_0_iff_0
thf(fact_50_poly__induct2, axiom,
    ((![P2 : poly_a > poly_a > $o, P : poly_a, Q : poly_a]: ((P2 @ zero_zero_poly_a @ zero_zero_poly_a) => ((![A2 : a, P3 : poly_a, B2 : a, Q2 : poly_a]: ((P2 @ P3 @ Q2) => (P2 @ (pCons_a @ A2 @ P3) @ (pCons_a @ B2 @ Q2)))) => (P2 @ P @ Q)))))). % poly_induct2
thf(fact_51_dual__order_Oantisym, axiom,
    ((![B : real, A : real]: ((ord_less_eq_real @ B @ A) => ((ord_less_eq_real @ A @ B) => (A = B)))))). % dual_order.antisym
thf(fact_52_dual__order_Oeq__iff, axiom,
    (((^[Y : real]: (^[Z2 : real]: (Y = Z2))) = (^[A3 : real]: (^[B3 : real]: (((ord_less_eq_real @ B3 @ A3)) & ((ord_less_eq_real @ A3 @ B3)))))))). % dual_order.eq_iff
thf(fact_53_dual__order_Otrans, axiom,
    ((![B : real, A : real, C : real]: ((ord_less_eq_real @ B @ A) => ((ord_less_eq_real @ C @ B) => (ord_less_eq_real @ C @ A)))))). % dual_order.trans
thf(fact_54_linorder__wlog, axiom,
    ((![P2 : real > real > $o, A : real, B : real]: ((![A2 : real, B2 : real]: ((ord_less_eq_real @ A2 @ B2) => (P2 @ A2 @ B2))) => ((![A2 : real, B2 : real]: ((P2 @ B2 @ A2) => (P2 @ A2 @ B2))) => (P2 @ A @ B)))))). % linorder_wlog
thf(fact_55_dual__order_Orefl, axiom,
    ((![A : real]: (ord_less_eq_real @ A @ A)))). % dual_order.refl
thf(fact_56_order__trans, axiom,
    ((![X : real, Y2 : real, Z3 : real]: ((ord_less_eq_real @ X @ Y2) => ((ord_less_eq_real @ Y2 @ Z3) => (ord_less_eq_real @ X @ Z3)))))). % order_trans
thf(fact_57_order__class_Oorder_Oantisym, axiom,
    ((![A : real, B : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_eq_real @ B @ A) => (A = B)))))). % order_class.order.antisym
thf(fact_58_ord__le__eq__trans, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_eq_real @ A @ B) => ((B = C) => (ord_less_eq_real @ A @ C)))))). % ord_le_eq_trans
thf(fact_59_ord__eq__le__trans, axiom,
    ((![A : real, B : real, C : real]: ((A = B) => ((ord_less_eq_real @ B @ C) => (ord_less_eq_real @ A @ C)))))). % ord_eq_le_trans
thf(fact_60_order__class_Oorder_Oeq__iff, axiom,
    (((^[Y : real]: (^[Z2 : real]: (Y = Z2))) = (^[A3 : real]: (^[B3 : real]: (((ord_less_eq_real @ A3 @ B3)) & ((ord_less_eq_real @ B3 @ A3)))))))). % order_class.order.eq_iff
thf(fact_61_antisym__conv, axiom,
    ((![Y2 : real, X : real]: ((ord_less_eq_real @ Y2 @ X) => ((ord_less_eq_real @ X @ Y2) = (X = Y2)))))). % antisym_conv
thf(fact_62_le__cases3, axiom,
    ((![X : real, Y2 : real, Z3 : real]: (((ord_less_eq_real @ X @ Y2) => (~ ((ord_less_eq_real @ Y2 @ Z3)))) => (((ord_less_eq_real @ Y2 @ X) => (~ ((ord_less_eq_real @ X @ Z3)))) => (((ord_less_eq_real @ X @ Z3) => (~ ((ord_less_eq_real @ Z3 @ Y2)))) => (((ord_less_eq_real @ Z3 @ Y2) => (~ ((ord_less_eq_real @ Y2 @ X)))) => (((ord_less_eq_real @ Y2 @ Z3) => (~ ((ord_less_eq_real @ Z3 @ X)))) => (~ (((ord_less_eq_real @ Z3 @ X) => (~ ((ord_less_eq_real @ X @ Y2)))))))))))))). % le_cases3
thf(fact_63_order_Otrans, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_eq_real @ B @ C) => (ord_less_eq_real @ A @ C)))))). % order.trans
thf(fact_64_le__cases, axiom,
    ((![X : real, Y2 : real]: ((~ ((ord_less_eq_real @ X @ Y2))) => (ord_less_eq_real @ Y2 @ X))))). % le_cases
thf(fact_65_eq__refl, axiom,
    ((![X : real, Y2 : real]: ((X = Y2) => (ord_less_eq_real @ X @ Y2))))). % eq_refl
thf(fact_66_linear, axiom,
    ((![X : real, Y2 : real]: ((ord_less_eq_real @ X @ Y2) | (ord_less_eq_real @ Y2 @ X))))). % linear
thf(fact_67_antisym, axiom,
    ((![X : real, Y2 : real]: ((ord_less_eq_real @ X @ Y2) => ((ord_less_eq_real @ Y2 @ X) => (X = Y2)))))). % antisym
thf(fact_68_eq__iff, axiom,
    (((^[Y : real]: (^[Z2 : real]: (Y = Z2))) = (^[X2 : real]: (^[Y3 : real]: (((ord_less_eq_real @ X2 @ Y3)) & ((ord_less_eq_real @ Y3 @ X2)))))))). % eq_iff
thf(fact_69_ord__le__eq__subst, axiom,
    ((![A : real, B : real, F : real > real, C : real]: ((ord_less_eq_real @ A @ B) => (((F @ B) = C) => ((![X3 : real, Y4 : real]: ((ord_less_eq_real @ X3 @ Y4) => (ord_less_eq_real @ (F @ X3) @ (F @ Y4)))) => (ord_less_eq_real @ (F @ A) @ C))))))). % ord_le_eq_subst
thf(fact_70_ord__eq__le__subst, axiom,
    ((![A : real, F : real > real, B : real, C : real]: ((A = (F @ B)) => ((ord_less_eq_real @ B @ C) => ((![X3 : real, Y4 : real]: ((ord_less_eq_real @ X3 @ Y4) => (ord_less_eq_real @ (F @ X3) @ (F @ Y4)))) => (ord_less_eq_real @ A @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_71_order__subst2, axiom,
    ((![A : real, B : real, F : real > real, C : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_eq_real @ (F @ B) @ C) => ((![X3 : real, Y4 : real]: ((ord_less_eq_real @ X3 @ Y4) => (ord_less_eq_real @ (F @ X3) @ (F @ Y4)))) => (ord_less_eq_real @ (F @ A) @ C))))))). % order_subst2
thf(fact_72_order__subst1, axiom,
    ((![A : real, F : real > real, B : real, C : real]: ((ord_less_eq_real @ A @ (F @ B)) => ((ord_less_eq_real @ B @ C) => ((![X3 : real, Y4 : real]: ((ord_less_eq_real @ X3 @ Y4) => (ord_less_eq_real @ (F @ X3) @ (F @ Y4)))) => (ord_less_eq_real @ A @ (F @ C)))))))). % order_subst1
thf(fact_73_pCons__cases, axiom,
    ((![P : poly_a]: (~ ((![A2 : a, Q2 : poly_a]: (~ ((P = (pCons_a @ A2 @ Q2)))))))))). % pCons_cases
thf(fact_74_norm__one, axiom,
    (((real_V1022479215norm_a @ one_one_a) = one_one_real))). % norm_one
thf(fact_75_norm__one, axiom,
    (((real_V646646907m_real @ one_one_real) = one_one_real))). % norm_one
thf(fact_76_norm__one, axiom,
    (((real_V638595069omplex @ one_one_complex) = one_one_real))). % norm_one
thf(fact_77_norm__le__zero__iff, axiom,
    ((![X : a]: ((ord_less_eq_real @ (real_V1022479215norm_a @ X) @ zero_zero_real) = (X = zero_zero_a))))). % norm_le_zero_iff
thf(fact_78_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_79_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_80_le__add__diff__inverse2, axiom,
    ((![B : real, A : real]: ((ord_less_eq_real @ B @ A) => ((plus_plus_real @ (minus_minus_real @ A @ B) @ B) = A))))). % le_add_diff_inverse2
thf(fact_81_le__add__diff__inverse, axiom,
    ((![B : real, A : real]: ((ord_less_eq_real @ B @ A) => ((plus_plus_real @ B @ (minus_minus_real @ A @ B)) = A))))). % le_add_diff_inverse
thf(fact_82_diff__numeral__special_I9_J, axiom,
    (((minus_minus_a @ one_one_a @ one_one_a) = zero_zero_a))). % diff_numeral_special(9)
thf(fact_83_diff__numeral__special_I9_J, axiom,
    (((minus_minus_real @ one_one_real @ one_one_real) = zero_zero_real))). % diff_numeral_special(9)
thf(fact_84_mult__cancel__left1, axiom,
    ((![C : a, B : a]: ((C = (times_times_a @ C @ B)) = (((C = zero_zero_a)) | ((B = one_one_a))))))). % mult_cancel_left1
thf(fact_85_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_86_mult__cancel__left2, axiom,
    ((![C : a, A : a]: (((times_times_a @ C @ A) = C) = (((C = zero_zero_a)) | ((A = one_one_a))))))). % mult_cancel_left2
thf(fact_87_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_88_mult__cancel__right1, axiom,
    ((![C : a, B : a]: ((C = (times_times_a @ B @ C)) = (((C = zero_zero_a)) | ((B = one_one_a))))))). % mult_cancel_right1
thf(fact_89_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_90_add__right__cancel, axiom,
    ((![B : real, A : real, C : real]: (((plus_plus_real @ B @ A) = (plus_plus_real @ C @ A)) = (B = C))))). % add_right_cancel
thf(fact_91_add__right__cancel, axiom,
    ((![B : a, A : a, C : a]: (((plus_plus_a @ B @ A) = (plus_plus_a @ C @ A)) = (B = C))))). % add_right_cancel
thf(fact_92_add__right__cancel, axiom,
    ((![B : complex, A : complex, C : complex]: (((plus_plus_complex @ B @ A) = (plus_plus_complex @ C @ A)) = (B = C))))). % add_right_cancel
thf(fact_93_add__left__cancel, axiom,
    ((![A : real, B : real, C : real]: (((plus_plus_real @ A @ B) = (plus_plus_real @ A @ C)) = (B = C))))). % add_left_cancel
thf(fact_94_add__left__cancel, axiom,
    ((![A : a, B : a, C : a]: (((plus_plus_a @ A @ B) = (plus_plus_a @ A @ C)) = (B = C))))). % add_left_cancel
thf(fact_95_add__left__cancel, axiom,
    ((![A : complex, B : complex, C : complex]: (((plus_plus_complex @ A @ B) = (plus_plus_complex @ A @ C)) = (B = C))))). % add_left_cancel
thf(fact_96_abs__idempotent, axiom,
    ((![A : real]: ((abs_abs_real @ (abs_abs_real @ A)) = (abs_abs_real @ A))))). % abs_idempotent
thf(fact_97_abs__abs, axiom,
    ((![A : real]: ((abs_abs_real @ (abs_abs_real @ A)) = (abs_abs_real @ A))))). % abs_abs
thf(fact_98_mult__cancel__right, axiom,
    ((![A : a, C : a, B : a]: (((times_times_a @ A @ C) = (times_times_a @ B @ C)) = (((C = zero_zero_a)) | ((A = B))))))). % mult_cancel_right
thf(fact_99_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_100_mult__cancel__left, axiom,
    ((![C : a, A : a, B : a]: (((times_times_a @ C @ A) = (times_times_a @ C @ B)) = (((C = zero_zero_a)) | ((A = B))))))). % mult_cancel_left
thf(fact_101_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_102_mult__eq__0__iff, axiom,
    ((![A : poly_a, B : poly_a]: (((times_times_poly_a @ A @ B) = zero_zero_poly_a) = (((A = zero_zero_poly_a)) | ((B = zero_zero_poly_a))))))). % mult_eq_0_iff
thf(fact_103_mult__eq__0__iff, axiom,
    ((![A : a, B : a]: (((times_times_a @ A @ B) = zero_zero_a) = (((A = zero_zero_a)) | ((B = zero_zero_a))))))). % mult_eq_0_iff
thf(fact_104_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_105_mult__zero__right, axiom,
    ((![A : poly_a]: ((times_times_poly_a @ A @ zero_zero_poly_a) = zero_zero_poly_a)))). % mult_zero_right
thf(fact_106_mult__zero__right, axiom,
    ((![A : a]: ((times_times_a @ A @ zero_zero_a) = zero_zero_a)))). % mult_zero_right
thf(fact_107_mult__zero__right, axiom,
    ((![A : real]: ((times_times_real @ A @ zero_zero_real) = zero_zero_real)))). % mult_zero_right
thf(fact_108_mult__zero__left, axiom,
    ((![A : poly_a]: ((times_times_poly_a @ zero_zero_poly_a @ A) = zero_zero_poly_a)))). % mult_zero_left
thf(fact_109_mult__zero__left, axiom,
    ((![A : a]: ((times_times_a @ zero_zero_a @ A) = zero_zero_a)))). % mult_zero_left
thf(fact_110_mult__zero__left, axiom,
    ((![A : real]: ((times_times_real @ zero_zero_real @ A) = zero_zero_real)))). % mult_zero_left
thf(fact_111_add__cancel__right__right, axiom,
    ((![A : real, B : real]: ((A = (plus_plus_real @ A @ B)) = (B = zero_zero_real))))). % add_cancel_right_right
thf(fact_112_add__cancel__right__right, axiom,
    ((![A : complex, B : complex]: ((A = (plus_plus_complex @ A @ B)) = (B = zero_zero_complex))))). % add_cancel_right_right
thf(fact_113_add__cancel__right__right, axiom,
    ((![A : poly_a, B : poly_a]: ((A = (plus_plus_poly_a @ A @ B)) = (B = zero_zero_poly_a))))). % add_cancel_right_right
thf(fact_114_add__cancel__right__right, axiom,
    ((![A : a, B : a]: ((A = (plus_plus_a @ A @ B)) = (B = zero_zero_a))))). % add_cancel_right_right
thf(fact_115_add__cancel__right__left, axiom,
    ((![A : real, B : real]: ((A = (plus_plus_real @ B @ A)) = (B = zero_zero_real))))). % add_cancel_right_left
thf(fact_116_add__cancel__right__left, axiom,
    ((![A : complex, B : complex]: ((A = (plus_plus_complex @ B @ A)) = (B = zero_zero_complex))))). % add_cancel_right_left
thf(fact_117_add__cancel__right__left, axiom,
    ((![A : poly_a, B : poly_a]: ((A = (plus_plus_poly_a @ B @ A)) = (B = zero_zero_poly_a))))). % add_cancel_right_left
thf(fact_118_add__cancel__right__left, axiom,
    ((![A : a, B : a]: ((A = (plus_plus_a @ B @ A)) = (B = zero_zero_a))))). % add_cancel_right_left
thf(fact_119_add__cancel__left__right, axiom,
    ((![A : real, B : real]: (((plus_plus_real @ A @ B) = A) = (B = zero_zero_real))))). % add_cancel_left_right
thf(fact_120_add__cancel__left__right, axiom,
    ((![A : complex, B : complex]: (((plus_plus_complex @ A @ B) = A) = (B = zero_zero_complex))))). % add_cancel_left_right
thf(fact_121_add__cancel__left__right, axiom,
    ((![A : poly_a, B : poly_a]: (((plus_plus_poly_a @ A @ B) = A) = (B = zero_zero_poly_a))))). % add_cancel_left_right
thf(fact_122_add__cancel__left__right, axiom,
    ((![A : a, B : a]: (((plus_plus_a @ A @ B) = A) = (B = zero_zero_a))))). % add_cancel_left_right
thf(fact_123_add__cancel__left__left, axiom,
    ((![B : real, A : real]: (((plus_plus_real @ B @ A) = A) = (B = zero_zero_real))))). % add_cancel_left_left
thf(fact_124_add__cancel__left__left, axiom,
    ((![B : complex, A : complex]: (((plus_plus_complex @ B @ A) = A) = (B = zero_zero_complex))))). % add_cancel_left_left
thf(fact_125_add__cancel__left__left, axiom,
    ((![B : poly_a, A : poly_a]: (((plus_plus_poly_a @ B @ A) = A) = (B = zero_zero_poly_a))))). % add_cancel_left_left
thf(fact_126_add__cancel__left__left, axiom,
    ((![B : a, A : a]: (((plus_plus_a @ B @ A) = A) = (B = zero_zero_a))))). % add_cancel_left_left
thf(fact_127_double__zero__sym, axiom,
    ((![A : real]: ((zero_zero_real = (plus_plus_real @ A @ A)) = (A = zero_zero_real))))). % double_zero_sym
thf(fact_128_double__zero, axiom,
    ((![A : real]: (((plus_plus_real @ A @ A) = zero_zero_real) = (A = zero_zero_real))))). % double_zero
thf(fact_129_add_Oright__neutral, axiom,
    ((![A : real]: ((plus_plus_real @ A @ zero_zero_real) = A)))). % add.right_neutral
thf(fact_130_add_Oright__neutral, axiom,
    ((![A : complex]: ((plus_plus_complex @ A @ zero_zero_complex) = A)))). % add.right_neutral
thf(fact_131_add_Oright__neutral, axiom,
    ((![A : poly_a]: ((plus_plus_poly_a @ A @ zero_zero_poly_a) = A)))). % add.right_neutral
thf(fact_132_add_Oright__neutral, axiom,
    ((![A : a]: ((plus_plus_a @ A @ zero_zero_a) = A)))). % add.right_neutral
thf(fact_133_add_Oleft__neutral, axiom,
    ((![A : real]: ((plus_plus_real @ zero_zero_real @ A) = A)))). % add.left_neutral
thf(fact_134_add_Oleft__neutral, axiom,
    ((![A : complex]: ((plus_plus_complex @ zero_zero_complex @ A) = A)))). % add.left_neutral
thf(fact_135_add_Oleft__neutral, axiom,
    ((![A : poly_a]: ((plus_plus_poly_a @ zero_zero_poly_a @ A) = A)))). % add.left_neutral
thf(fact_136_add_Oleft__neutral, axiom,
    ((![A : a]: ((plus_plus_a @ zero_zero_a @ A) = A)))). % add.left_neutral
thf(fact_137_add__le__cancel__left, axiom,
    ((![C : real, A : real, B : real]: ((ord_less_eq_real @ (plus_plus_real @ C @ A) @ (plus_plus_real @ C @ B)) = (ord_less_eq_real @ A @ B))))). % add_le_cancel_left
thf(fact_138_add__le__cancel__right, axiom,
    ((![A : real, C : real, B : real]: ((ord_less_eq_real @ (plus_plus_real @ A @ C) @ (plus_plus_real @ B @ C)) = (ord_less_eq_real @ A @ B))))). % add_le_cancel_right
thf(fact_139_cancel__comm__monoid__add__class_Odiff__cancel, axiom,
    ((![A : poly_a]: ((minus_minus_poly_a @ A @ A) = zero_zero_poly_a)))). % cancel_comm_monoid_add_class.diff_cancel
thf(fact_140_cancel__comm__monoid__add__class_Odiff__cancel, axiom,
    ((![A : a]: ((minus_minus_a @ A @ A) = zero_zero_a)))). % cancel_comm_monoid_add_class.diff_cancel
thf(fact_141_cancel__comm__monoid__add__class_Odiff__cancel, axiom,
    ((![A : real]: ((minus_minus_real @ A @ A) = zero_zero_real)))). % cancel_comm_monoid_add_class.diff_cancel
thf(fact_142_diff__zero, axiom,
    ((![A : poly_a]: ((minus_minus_poly_a @ A @ zero_zero_poly_a) = A)))). % diff_zero
thf(fact_143_diff__zero, axiom,
    ((![A : a]: ((minus_minus_a @ A @ zero_zero_a) = A)))). % diff_zero
thf(fact_144_diff__zero, axiom,
    ((![A : real]: ((minus_minus_real @ A @ zero_zero_real) = A)))). % diff_zero
thf(fact_145_diff__0__right, axiom,
    ((![A : poly_a]: ((minus_minus_poly_a @ A @ zero_zero_poly_a) = A)))). % diff_0_right
thf(fact_146_diff__0__right, axiom,
    ((![A : a]: ((minus_minus_a @ A @ zero_zero_a) = A)))). % diff_0_right
thf(fact_147_diff__0__right, axiom,
    ((![A : real]: ((minus_minus_real @ A @ zero_zero_real) = A)))). % diff_0_right
thf(fact_148_diff__self, axiom,
    ((![A : poly_a]: ((minus_minus_poly_a @ A @ A) = zero_zero_poly_a)))). % diff_self
thf(fact_149_diff__self, axiom,
    ((![A : a]: ((minus_minus_a @ A @ A) = zero_zero_a)))). % diff_self
thf(fact_150_diff__self, axiom,
    ((![A : real]: ((minus_minus_real @ A @ A) = zero_zero_real)))). % diff_self
thf(fact_151_mult_Oright__neutral, axiom,
    ((![A : a]: ((times_times_a @ A @ one_one_a) = A)))). % mult.right_neutral
thf(fact_152_mult_Oright__neutral, axiom,
    ((![A : real]: ((times_times_real @ A @ one_one_real) = A)))). % mult.right_neutral
thf(fact_153_mult_Oleft__neutral, axiom,
    ((![A : a]: ((times_times_a @ one_one_a @ A) = A)))). % mult.left_neutral
thf(fact_154_mult_Oleft__neutral, axiom,
    ((![A : real]: ((times_times_real @ one_one_real @ A) = A)))). % mult.left_neutral
thf(fact_155_add__diff__cancel__right_H, axiom,
    ((![A : a, B : a]: ((minus_minus_a @ (plus_plus_a @ A @ B) @ B) = A)))). % add_diff_cancel_right'
thf(fact_156_add__diff__cancel__right_H, axiom,
    ((![A : complex, B : complex]: ((minus_minus_complex @ (plus_plus_complex @ A @ B) @ B) = A)))). % add_diff_cancel_right'
thf(fact_157_add__diff__cancel__right_H, axiom,
    ((![A : real, B : real]: ((minus_minus_real @ (plus_plus_real @ A @ B) @ B) = A)))). % add_diff_cancel_right'
thf(fact_158_add__diff__cancel__right, axiom,
    ((![A : a, C : a, B : a]: ((minus_minus_a @ (plus_plus_a @ A @ C) @ (plus_plus_a @ B @ C)) = (minus_minus_a @ A @ B))))). % add_diff_cancel_right
thf(fact_159_add__diff__cancel__right, axiom,
    ((![A : complex, C : complex, B : complex]: ((minus_minus_complex @ (plus_plus_complex @ A @ C) @ (plus_plus_complex @ B @ C)) = (minus_minus_complex @ A @ B))))). % add_diff_cancel_right
thf(fact_160_add__diff__cancel__right, axiom,
    ((![A : real, C : real, B : real]: ((minus_minus_real @ (plus_plus_real @ A @ C) @ (plus_plus_real @ B @ C)) = (minus_minus_real @ A @ B))))). % add_diff_cancel_right
thf(fact_161_add__diff__cancel__left_H, axiom,
    ((![A : a, B : a]: ((minus_minus_a @ (plus_plus_a @ A @ B) @ A) = B)))). % add_diff_cancel_left'
thf(fact_162_add__diff__cancel__left_H, axiom,
    ((![A : complex, B : complex]: ((minus_minus_complex @ (plus_plus_complex @ A @ B) @ A) = B)))). % add_diff_cancel_left'
thf(fact_163_add__diff__cancel__left_H, axiom,
    ((![A : real, B : real]: ((minus_minus_real @ (plus_plus_real @ A @ B) @ A) = B)))). % add_diff_cancel_left'
thf(fact_164_add__diff__cancel__left, axiom,
    ((![C : a, A : a, B : a]: ((minus_minus_a @ (plus_plus_a @ C @ A) @ (plus_plus_a @ C @ B)) = (minus_minus_a @ A @ B))))). % add_diff_cancel_left
thf(fact_165_add__diff__cancel__left, axiom,
    ((![C : complex, A : complex, B : complex]: ((minus_minus_complex @ (plus_plus_complex @ C @ A) @ (plus_plus_complex @ C @ B)) = (minus_minus_complex @ A @ B))))). % add_diff_cancel_left
thf(fact_166_add__diff__cancel__left, axiom,
    ((![C : real, A : real, B : real]: ((minus_minus_real @ (plus_plus_real @ C @ A) @ (plus_plus_real @ C @ B)) = (minus_minus_real @ A @ B))))). % add_diff_cancel_left
thf(fact_167_diff__add__cancel, axiom,
    ((![A : a, B : a]: ((plus_plus_a @ (minus_minus_a @ A @ B) @ B) = A)))). % diff_add_cancel
thf(fact_168_diff__add__cancel, axiom,
    ((![A : complex, B : complex]: ((plus_plus_complex @ (minus_minus_complex @ A @ B) @ B) = A)))). % diff_add_cancel
thf(fact_169_diff__add__cancel, axiom,
    ((![A : real, B : real]: ((plus_plus_real @ (minus_minus_real @ A @ B) @ B) = A)))). % diff_add_cancel
thf(fact_170_add__diff__cancel, axiom,
    ((![A : a, B : a]: ((minus_minus_a @ (plus_plus_a @ A @ B) @ B) = A)))). % add_diff_cancel
thf(fact_171_add__diff__cancel, axiom,
    ((![A : complex, B : complex]: ((minus_minus_complex @ (plus_plus_complex @ A @ B) @ B) = A)))). % add_diff_cancel
thf(fact_172_add__diff__cancel, axiom,
    ((![A : real, B : real]: ((minus_minus_real @ (plus_plus_real @ A @ B) @ B) = A)))). % add_diff_cancel
thf(fact_173_abs__zero, axiom,
    (((abs_abs_real @ zero_zero_real) = zero_zero_real))). % abs_zero
thf(fact_174_abs__eq__0, axiom,
    ((![A : real]: (((abs_abs_real @ A) = zero_zero_real) = (A = zero_zero_real))))). % abs_eq_0
thf(fact_175_abs__0__eq, axiom,
    ((![A : real]: ((zero_zero_real = (abs_abs_real @ A)) = (A = zero_zero_real))))). % abs_0_eq
thf(fact_176_abs__0, axiom,
    (((abs_abs_real @ zero_zero_real) = zero_zero_real))). % abs_0
thf(fact_177_abs__mult__self__eq, axiom,
    ((![A : real]: ((times_times_real @ (abs_abs_real @ A) @ (abs_abs_real @ A)) = (times_times_real @ A @ A))))). % abs_mult_self_eq
thf(fact_178_abs__add__abs, axiom,
    ((![A : real, B : real]: ((abs_abs_real @ (plus_plus_real @ (abs_abs_real @ A) @ (abs_abs_real @ B))) = (plus_plus_real @ (abs_abs_real @ A) @ (abs_abs_real @ B)))))). % abs_add_abs
thf(fact_179_abs__1, axiom,
    (((abs_abs_real @ one_one_real) = one_one_real))). % abs_1
thf(fact_180_abs__norm__cancel, axiom,
    ((![A : a]: ((abs_abs_real @ (real_V1022479215norm_a @ A)) = (real_V1022479215norm_a @ A))))). % abs_norm_cancel
thf(fact_181_abs__norm__cancel, axiom,
    ((![A : real]: ((abs_abs_real @ (real_V646646907m_real @ A)) = (real_V646646907m_real @ A))))). % abs_norm_cancel
thf(fact_182_abs__norm__cancel, axiom,
    ((![A : complex]: ((abs_abs_real @ (real_V638595069omplex @ A)) = (real_V638595069omplex @ A))))). % abs_norm_cancel
thf(fact_183_zero__le__double__add__iff__zero__le__single__add, axiom,
    ((![A : real]: ((ord_less_eq_real @ zero_zero_real @ (plus_plus_real @ A @ A)) = (ord_less_eq_real @ zero_zero_real @ A))))). % zero_le_double_add_iff_zero_le_single_add
thf(fact_184_double__add__le__zero__iff__single__add__le__zero, axiom,
    ((![A : real]: ((ord_less_eq_real @ (plus_plus_real @ A @ A) @ zero_zero_real) = (ord_less_eq_real @ A @ zero_zero_real))))). % double_add_le_zero_iff_single_add_le_zero
thf(fact_185_le__add__same__cancel2, axiom,
    ((![A : real, B : real]: ((ord_less_eq_real @ A @ (plus_plus_real @ B @ A)) = (ord_less_eq_real @ zero_zero_real @ B))))). % le_add_same_cancel2
thf(fact_186_le__add__same__cancel1, axiom,
    ((![A : real, B : real]: ((ord_less_eq_real @ A @ (plus_plus_real @ A @ B)) = (ord_less_eq_real @ zero_zero_real @ B))))). % le_add_same_cancel1
thf(fact_187_add__le__same__cancel2, axiom,
    ((![A : real, B : real]: ((ord_less_eq_real @ (plus_plus_real @ A @ B) @ B) = (ord_less_eq_real @ A @ zero_zero_real))))). % add_le_same_cancel2
thf(fact_188_add__le__same__cancel1, axiom,
    ((![B : real, A : real]: ((ord_less_eq_real @ (plus_plus_real @ B @ A) @ B) = (ord_less_eq_real @ A @ zero_zero_real))))). % add_le_same_cancel1
thf(fact_189_diff__ge__0__iff__ge, axiom,
    ((![A : real, B : real]: ((ord_less_eq_real @ zero_zero_real @ (minus_minus_real @ A @ B)) = (ord_less_eq_real @ B @ A))))). % diff_ge_0_iff_ge
thf(fact_190_mult__cancel__right2, axiom,
    ((![A : a, C : a]: (((times_times_a @ A @ C) = C) = (((C = zero_zero_a)) | ((A = one_one_a))))))). % mult_cancel_right2
thf(fact_191_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_192_abs__le__zero__iff, axiom,
    ((![A : real]: ((ord_less_eq_real @ (abs_abs_real @ A) @ zero_zero_real) = (A = zero_zero_real))))). % abs_le_zero_iff
thf(fact_193_abs__le__self__iff, axiom,
    ((![A : real]: ((ord_less_eq_real @ (abs_abs_real @ A) @ A) = (ord_less_eq_real @ zero_zero_real @ A))))). % abs_le_self_iff
thf(fact_194_abs__of__nonneg, axiom,
    ((![A : real]: ((ord_less_eq_real @ zero_zero_real @ A) => ((abs_abs_real @ A) = A))))). % abs_of_nonneg
thf(fact_195_norm__eq__zero, axiom,
    ((![X : a]: (((real_V1022479215norm_a @ X) = zero_zero_real) = (X = zero_zero_a))))). % norm_eq_zero
thf(fact_196_norm__eq__zero, axiom,
    ((![X : real]: (((real_V646646907m_real @ X) = zero_zero_real) = (X = zero_zero_real))))). % norm_eq_zero
thf(fact_197_norm__eq__zero, axiom,
    ((![X : complex]: (((real_V638595069omplex @ X) = zero_zero_real) = (X = zero_zero_complex))))). % norm_eq_zero
thf(fact_198_norm__zero, axiom,
    (((real_V1022479215norm_a @ zero_zero_a) = zero_zero_real))). % norm_zero
thf(fact_199_norm__zero, axiom,
    (((real_V646646907m_real @ zero_zero_real) = zero_zero_real))). % norm_zero
thf(fact_200_norm__zero, axiom,
    (((real_V638595069omplex @ zero_zero_complex) = zero_zero_real))). % norm_zero
thf(fact_201_abs__mult, axiom,
    ((![A : real, B : real]: ((abs_abs_real @ (times_times_real @ A @ B)) = (times_times_real @ (abs_abs_real @ A) @ (abs_abs_real @ B)))))). % abs_mult
thf(fact_202_abs__one, axiom,
    (((abs_abs_real @ one_one_real) = one_one_real))). % abs_one
thf(fact_203_abs__eq__0__iff, axiom,
    ((![A : real]: (((abs_abs_real @ A) = zero_zero_real) = (A = zero_zero_real))))). % abs_eq_0_iff
thf(fact_204_abs__minus__commute, axiom,
    ((![A : real, B : real]: ((abs_abs_real @ (minus_minus_real @ A @ B)) = (abs_abs_real @ (minus_minus_real @ B @ A)))))). % abs_minus_commute
thf(fact_205_abs__ge__self, axiom,
    ((![A : real]: (ord_less_eq_real @ A @ (abs_abs_real @ A))))). % abs_ge_self
thf(fact_206_abs__le__D1, axiom,
    ((![A : real, B : real]: ((ord_less_eq_real @ (abs_abs_real @ A) @ B) => (ord_less_eq_real @ A @ B))))). % abs_le_D1
thf(fact_207_real__norm__def, axiom,
    ((real_V646646907m_real = abs_abs_real))). % real_norm_def
thf(fact_208_complex__mod__triangle__sub, axiom,
    ((![W : complex, Z3 : complex]: (ord_less_eq_real @ (real_V638595069omplex @ W) @ (plus_plus_real @ (real_V638595069omplex @ (plus_plus_complex @ W @ Z3)) @ (real_V638595069omplex @ Z3)))))). % complex_mod_triangle_sub
thf(fact_209_poly__minimum__modulus__disc, axiom,
    ((![R2 : real, P : poly_complex]: (?[Z4 : complex]: (![W2 : complex]: ((ord_less_eq_real @ (real_V638595069omplex @ W2) @ R2) => (ord_less_eq_real @ (real_V638595069omplex @ (poly_complex2 @ P @ Z4)) @ (real_V638595069omplex @ (poly_complex2 @ P @ W2))))))))). % poly_minimum_modulus_disc
thf(fact_210_abs__ge__zero, axiom,
    ((![A : real]: (ord_less_eq_real @ zero_zero_real @ (abs_abs_real @ A))))). % abs_ge_zero
thf(fact_211_abs__triangle__ineq, axiom,
    ((![A : real, B : real]: (ord_less_eq_real @ (abs_abs_real @ (plus_plus_real @ A @ B)) @ (plus_plus_real @ (abs_abs_real @ A) @ (abs_abs_real @ B)))))). % abs_triangle_ineq
thf(fact_212_abs__triangle__ineq2, axiom,
    ((![A : real, B : real]: (ord_less_eq_real @ (minus_minus_real @ (abs_abs_real @ A) @ (abs_abs_real @ B)) @ (abs_abs_real @ (minus_minus_real @ A @ B)))))). % abs_triangle_ineq2
thf(fact_213_abs__triangle__ineq3, axiom,
    ((![A : real, B : real]: (ord_less_eq_real @ (abs_abs_real @ (minus_minus_real @ (abs_abs_real @ A) @ (abs_abs_real @ B))) @ (abs_abs_real @ (minus_minus_real @ A @ B)))))). % abs_triangle_ineq3
thf(fact_214_abs__triangle__ineq2__sym, axiom,
    ((![A : real, B : real]: (ord_less_eq_real @ (minus_minus_real @ (abs_abs_real @ A) @ (abs_abs_real @ B)) @ (abs_abs_real @ (minus_minus_real @ B @ A)))))). % abs_triangle_ineq2_sym
thf(fact_215_mult__poly__0__right, axiom,
    ((![P : poly_a]: ((times_times_poly_a @ P @ zero_zero_poly_a) = zero_zero_poly_a)))). % mult_poly_0_right
thf(fact_216_mult__poly__0__left, axiom,
    ((![Q : poly_a]: ((times_times_poly_a @ zero_zero_poly_a @ Q) = zero_zero_poly_a)))). % mult_poly_0_left
thf(fact_217_norm__ge__zero, axiom,
    ((![X : a]: (ord_less_eq_real @ zero_zero_real @ (real_V1022479215norm_a @ X))))). % norm_ge_zero
thf(fact_218_norm__ge__zero, axiom,
    ((![X : real]: (ord_less_eq_real @ zero_zero_real @ (real_V646646907m_real @ X))))). % norm_ge_zero
thf(fact_219_norm__ge__zero, axiom,
    ((![X : complex]: (ord_less_eq_real @ zero_zero_real @ (real_V638595069omplex @ X))))). % norm_ge_zero

% Conjectures (1)
thf(conj_0, conjecture,
    ((ord_less_eq_real @ da @ (real_V1022479215norm_a @ (poly_a2 @ (pCons_a @ aa @ (pCons_a @ c @ cs)) @ z))))).
