% 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_552__5373354_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:31:07.909

% Could-be-implicit typings (8)
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__Complex__Ocomplex_J, type,
    poly_complex : $tType).
thf(ty_n_t__Polynomial__Opoly_It__Real__Oreal_J, type,
    poly_real : $tType).
thf(ty_n_t__Polynomial__Opoly_It__Nat__Onat_J, type,
    poly_nat : $tType).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J, type,
    set_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 (59)
thf(sy_c_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Opsize_001t__Complex__Ocomplex, type,
    fundam1709708056omplex : poly_complex > nat).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Complex__Ocomplex, type,
    abs_abs_complex : complex > complex).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Real__Oreal, type,
    abs_abs_real : real > real).
thf(sy_c_Groups_Oone__class_Oone_001t__Complex__Ocomplex, type,
    one_one_complex : complex).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat, type,
    one_one_nat : nat).
thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    one_one_poly_complex : poly_complex).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal, type,
    one_one_real : real).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Complex__Ocomplex, type,
    zero_zero_complex : complex).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat, type,
    zero_zero_nat : nat).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    zero_z1746442943omplex : poly_complex).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    zero_zero_poly_nat : poly_nat).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__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_Num_Oneg__numeral__class_Odbl__inc_001t__Complex__Ocomplex, type,
    neg_nu484426047omplex : complex > complex).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    neg_nu1217972807omplex : poly_complex > poly_complex).
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_Odegree_001t__Complex__Ocomplex, type,
    degree_complex : poly_complex > nat).
thf(sy_c_Polynomial_Odegree_001t__Nat__Onat, type,
    degree_nat : poly_nat > nat).
thf(sy_c_Polynomial_Odegree_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    degree_poly_complex : poly_poly_complex > nat).
thf(sy_c_Polynomial_Ois__zero_001t__Complex__Ocomplex, type,
    is_zero_complex : poly_complex > $o).
thf(sy_c_Polynomial_Omap__poly_001t__Complex__Ocomplex_001t__Complex__Ocomplex, type,
    map_po1637612550omplex : (complex > complex) > poly_complex > poly_complex).
thf(sy_c_Polynomial_Omap__poly_001t__Complex__Ocomplex_001t__Nat__Onat, type,
    map_poly_complex_nat : (complex > nat) > poly_complex > poly_nat).
thf(sy_c_Polynomial_Omap__poly_001t__Complex__Ocomplex_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    map_po366287630omplex : (complex > poly_complex) > poly_complex > poly_poly_complex).
thf(sy_c_Polynomial_Omap__poly_001t__Nat__Onat_001t__Complex__Ocomplex, type,
    map_poly_nat_complex : (nat > complex) > poly_nat > poly_complex).
thf(sy_c_Polynomial_Omap__poly_001t__Nat__Onat_001t__Nat__Onat, type,
    map_poly_nat_nat : (nat > nat) > poly_nat > poly_nat).
thf(sy_c_Polynomial_Omap__poly_001t__Nat__Onat_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    map_po2099259440omplex : (nat > poly_complex) > poly_nat > poly_poly_complex).
thf(sy_c_Polynomial_Omap__poly_001t__Polynomial__Opoly_It__Complex__Ocomplex_J_001t__Complex__Ocomplex, type,
    map_po944112910omplex : (poly_complex > complex) > poly_poly_complex > poly_complex).
thf(sy_c_Polynomial_Omap__poly_001t__Polynomial__Opoly_It__Complex__Ocomplex_J_001t__Nat__Onat, type,
    map_po1895962672ex_nat : (poly_complex > nat) > poly_poly_complex > poly_nat).
thf(sy_c_Polynomial_Omap__poly_001t__Polynomial__Opoly_It__Complex__Ocomplex_J_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    map_po1985147926omplex : (poly_complex > poly_complex) > poly_poly_complex > poly_poly_complex).
thf(sy_c_Polynomial_OpCons_001t__Complex__Ocomplex, type,
    pCons_complex : complex > poly_complex > poly_complex).
thf(sy_c_Polynomial_OpCons_001t__Nat__Onat, type,
    pCons_nat : nat > poly_nat > poly_nat).
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_Opcompose_001t__Complex__Ocomplex, type,
    pcompose_complex : poly_complex > poly_complex > poly_complex).
thf(sy_c_Polynomial_Opcompose_001t__Nat__Onat, type,
    pcompose_nat : poly_nat > poly_nat > poly_nat).
thf(sy_c_Polynomial_Opcompose_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    pcompo1411605209omplex : poly_poly_complex > poly_poly_complex > poly_poly_complex).
thf(sy_c_Polynomial_Opoly_001t__Complex__Ocomplex, type,
    poly_complex2 : poly_complex > complex > complex).
thf(sy_c_Polynomial_Opoly_001t__Nat__Onat, type,
    poly_nat2 : poly_nat > nat > nat).
thf(sy_c_Polynomial_Opoly_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    poly_poly_complex2 : poly_poly_complex > poly_complex > poly_complex).
thf(sy_c_Polynomial_Opoly_001t__Real__Oreal, type,
    poly_real2 : poly_real > real > real).
thf(sy_c_Polynomial_Opoly_Ocoeff_001t__Complex__Ocomplex, type,
    coeff_complex : poly_complex > nat > complex).
thf(sy_c_Polynomial_Opoly_Ocoeff_001t__Nat__Onat, type,
    coeff_nat : poly_nat > nat > nat).
thf(sy_c_Polynomial_Opoly_Ocoeff_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    coeff_poly_complex : poly_poly_complex > nat > poly_complex).
thf(sy_c_Polynomial_Opoly__cutoff_001t__Complex__Ocomplex, type,
    poly_cutoff_complex : nat > poly_complex > poly_complex).
thf(sy_c_Polynomial_Opoly__shift_001t__Complex__Ocomplex, type,
    poly_shift_complex : nat > poly_complex > poly_complex).
thf(sy_c_Polynomial_Oreflect__poly_001t__Complex__Ocomplex, type,
    reflect_poly_complex : poly_complex > poly_complex).
thf(sy_c_Polynomial_Oreflect__poly_001t__Nat__Onat, type,
    reflect_poly_nat : poly_nat > poly_nat).
thf(sy_c_Polynomial_Oreflect__poly_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    reflec309385472omplex : poly_poly_complex > poly_poly_complex).
thf(sy_c_Polynomial_Osynthetic__div_001t__Complex__Ocomplex, type,
    synthe151143547omplex : poly_complex > complex > poly_complex).
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_Set_OCollect_001t__Real__Oreal, type,
    collect_real : (real > $o) > set_real).
thf(sy_c_member_001t__Real__Oreal, type,
    member_real : real > set_real > $o).
thf(sy_v_c____, type,
    c : complex).
thf(sy_v_cs____, type,
    cs : poly_complex).
thf(sy_v_r____, type,
    r : real).
thf(sy_v_v____, type,
    v : complex).

% Relevant facts (223)
thf(fact_0__092_060open_062_092_060And_062z_O_Ar_A_092_060le_062_Acmod_Az_A_092_060or_062_Acmod_Az_A_092_060le_062_A_092_060bar_062r_092_060bar_062_092_060close_062, axiom,
    ((![Z : complex]: ((ord_less_eq_real @ r @ (real_V638595069omplex @ Z)) | (ord_less_eq_real @ (real_V638595069omplex @ Z) @ (abs_abs_real @ r)))))). % \<open>\<And>z. r \<le> cmod z \<or> cmod z \<le> \<bar>r\<bar>\<close>
thf(fact_1_pCons_Ohyps_I2_J, axiom,
    ((?[Z2 : complex]: (![W : complex]: (ord_less_eq_real @ (real_V638595069omplex @ (poly_complex2 @ cs @ Z2)) @ (real_V638595069omplex @ (poly_complex2 @ cs @ W))))))). % pCons.hyps(2)
thf(fact_2__092_060open_062_092_060And_062z_O_Ar_A_092_060le_062_Acmod_Az_A_092_060Longrightarrow_062_Acmod_A_Ipoly_A_IpCons_Ac_Acs_J_Av_J_A_092_060le_062_Acmod_A_Ipoly_A_IpCons_Ac_Acs_J_Az_J_092_060close_062, axiom,
    ((![Z : complex]: ((ord_less_eq_real @ r @ (real_V638595069omplex @ Z)) => (ord_less_eq_real @ (real_V638595069omplex @ (poly_complex2 @ (pCons_complex @ c @ cs) @ v)) @ (real_V638595069omplex @ (poly_complex2 @ (pCons_complex @ c @ cs) @ Z))))))). % \<open>\<And>z. r \<le> cmod z \<Longrightarrow> cmod (poly (pCons c cs) v) \<le> cmod (poly (pCons c cs) z)\<close>
thf(fact_3_ath, axiom,
    ((![R : real, Z : complex]: ((ord_less_eq_real @ R @ (real_V638595069omplex @ Z)) | (ord_less_eq_real @ (real_V638595069omplex @ Z) @ (abs_abs_real @ R)))))). % ath
thf(fact_4_v, axiom,
    ((![W2 : complex]: ((ord_less_eq_real @ (real_V638595069omplex @ W2) @ (abs_abs_real @ r)) => (ord_less_eq_real @ (real_V638595069omplex @ (poly_complex2 @ (pCons_complex @ c @ cs) @ v)) @ (real_V638595069omplex @ (poly_complex2 @ (pCons_complex @ c @ cs) @ W2))))))). % v
thf(fact_5_False, axiom,
    ((~ ((cs = zero_z1746442943omplex))))). % False
thf(fact_6__092_060open_062_092_060exists_062z_O_A_092_060forall_062w_O_Acmod_Aw_A_092_060le_062_A_092_060bar_062r_092_060bar_062_A_092_060longrightarrow_062_Acmod_A_Ipoly_A_IpCons_Ac_Acs_J_Az_J_A_092_060le_062_Acmod_A_Ipoly_A_IpCons_Ac_Acs_J_Aw_J_092_060close_062, axiom,
    ((?[Z2 : complex]: (![W : complex]: ((ord_less_eq_real @ (real_V638595069omplex @ W) @ (abs_abs_real @ r)) => (ord_less_eq_real @ (real_V638595069omplex @ (poly_complex2 @ (pCons_complex @ c @ cs) @ Z2)) @ (real_V638595069omplex @ (poly_complex2 @ (pCons_complex @ c @ cs) @ W)))))))). % \<open>\<exists>z. \<forall>w. cmod w \<le> \<bar>r\<bar> \<longrightarrow> cmod (poly (pCons c cs) z) \<le> cmod (poly (pCons c cs) w)\<close>
thf(fact_7__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062v_O_A_I_092_060And_062w_O_Acmod_Aw_A_092_060le_062_A_092_060bar_062r_092_060bar_062_A_092_060Longrightarrow_062_Acmod_A_Ipoly_A_IpCons_Ac_Acs_J_Av_J_A_092_060le_062_Acmod_A_Ipoly_A_IpCons_Ac_Acs_J_Aw_J_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom,
    ((~ ((![V : complex]: (~ ((![W : complex]: ((ord_less_eq_real @ (real_V638595069omplex @ W) @ (abs_abs_real @ r)) => (ord_less_eq_real @ (real_V638595069omplex @ (poly_complex2 @ (pCons_complex @ c @ cs) @ V)) @ (real_V638595069omplex @ (poly_complex2 @ (pCons_complex @ c @ cs) @ W)))))))))))). % \<open>\<And>thesis. (\<And>v. (\<And>w. cmod w \<le> \<bar>r\<bar> \<Longrightarrow> cmod (poly (pCons c cs) v) \<le> cmod (poly (pCons c cs) w)) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_8_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_9__092_060open_062_092_060exists_062r_O_A_092_060forall_062z_O_Ar_A_092_060le_062_Acmod_Az_A_092_060longrightarrow_062_Acmod_A_Ipoly_A_IpCons_Ac_Acs_J_A0_J_A_092_060le_062_Acmod_A_Ipoly_A_IpCons_Ac_Acs_J_Az_J_092_060close_062, axiom,
    ((?[R2 : real]: (![Z3 : complex]: ((ord_less_eq_real @ R2 @ (real_V638595069omplex @ Z3)) => (ord_less_eq_real @ (real_V638595069omplex @ (poly_complex2 @ (pCons_complex @ c @ cs) @ zero_zero_complex)) @ (real_V638595069omplex @ (poly_complex2 @ (pCons_complex @ c @ cs) @ Z3)))))))). % \<open>\<exists>r. \<forall>z. r \<le> cmod z \<longrightarrow> cmod (poly (pCons c cs) 0) \<le> cmod (poly (pCons c cs) z)\<close>
thf(fact_10__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062r_O_A_I_092_060And_062z_O_Ar_A_092_060le_062_Acmod_Az_A_092_060Longrightarrow_062_Acmod_A_Ipoly_A_IpCons_Ac_Acs_J_A0_J_A_092_060le_062_Acmod_A_Ipoly_A_IpCons_Ac_Acs_J_Az_J_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom,
    ((~ ((![R2 : real]: (~ ((![Z3 : complex]: ((ord_less_eq_real @ R2 @ (real_V638595069omplex @ Z3)) => (ord_less_eq_real @ (real_V638595069omplex @ (poly_complex2 @ (pCons_complex @ c @ cs) @ zero_zero_complex)) @ (real_V638595069omplex @ (poly_complex2 @ (pCons_complex @ c @ cs) @ Z3)))))))))))). % \<open>\<And>thesis. (\<And>r. (\<And>z. r \<le> cmod z \<Longrightarrow> cmod (poly (pCons c cs) 0) \<le> cmod (poly (pCons c cs) z)) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_11_r, axiom,
    ((![Z : complex]: ((ord_less_eq_real @ r @ (real_V638595069omplex @ Z)) => (ord_less_eq_real @ (real_V638595069omplex @ (poly_complex2 @ (pCons_complex @ c @ cs) @ zero_zero_complex)) @ (real_V638595069omplex @ (poly_complex2 @ (pCons_complex @ c @ cs) @ Z))))))). % r
thf(fact_12_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_13_order__refl, axiom,
    ((![X : real]: (ord_less_eq_real @ X @ X)))). % order_refl
thf(fact_14_order__refl, axiom,
    ((![X : nat]: (ord_less_eq_nat @ X @ X)))). % order_refl
thf(fact_15_pCons_Ohyps_I1_J, axiom,
    (((~ ((c = zero_zero_complex))) | (~ ((cs = zero_z1746442943omplex)))))). % pCons.hyps(1)
thf(fact_16_poly__infinity, axiom,
    ((![P : poly_complex, D : real, A : complex]: ((~ ((P = zero_z1746442943omplex))) => (?[R2 : real]: (![Z3 : complex]: ((ord_less_eq_real @ R2 @ (real_V638595069omplex @ Z3)) => (ord_less_eq_real @ D @ (real_V638595069omplex @ (poly_complex2 @ (pCons_complex @ A @ P) @ Z3)))))))))). % poly_infinity
thf(fact_17_poly__infinity, axiom,
    ((![P : poly_real, D : real, A : real]: ((~ ((P = zero_zero_poly_real))) => (?[R2 : real]: (![Z3 : real]: ((ord_less_eq_real @ R2 @ (real_V646646907m_real @ Z3)) => (ord_less_eq_real @ D @ (real_V646646907m_real @ (poly_real2 @ (pCons_real @ A @ P) @ Z3)))))))))). % poly_infinity
thf(fact_18_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_19_pCons__cases, axiom,
    ((![P : poly_complex]: (~ ((![A2 : complex, Q2 : poly_complex]: (~ ((P = (pCons_complex @ A2 @ Q2)))))))))). % pCons_cases
thf(fact_20_pderiv_Ocases, axiom,
    ((![X : poly_complex]: (~ ((![A2 : complex, P2 : poly_complex]: (~ ((X = (pCons_complex @ A2 @ P2)))))))))). % pderiv.cases
thf(fact_21_complete__real, axiom,
    ((![S : set_real]: ((?[X2 : real]: (member_real @ X2 @ S)) => ((?[Z3 : real]: (![X3 : real]: ((member_real @ X3 @ S) => (ord_less_eq_real @ X3 @ Z3)))) => (?[Y : real]: ((![X2 : real]: ((member_real @ X2 @ S) => (ord_less_eq_real @ X2 @ Y))) & (![Z3 : real]: ((![X3 : real]: ((member_real @ X3 @ S) => (ord_less_eq_real @ X3 @ Z3))) => (ord_less_eq_real @ Y @ Z3)))))))))). % complete_real
thf(fact_22_pCons__0__0, axiom,
    (((pCons_poly_complex @ zero_z1746442943omplex @ zero_z1040703943omplex) = zero_z1040703943omplex))). % pCons_0_0
thf(fact_23_pCons__0__0, axiom,
    (((pCons_complex @ zero_zero_complex @ zero_z1746442943omplex) = zero_z1746442943omplex))). % pCons_0_0
thf(fact_24_pCons__0__0, axiom,
    (((pCons_nat @ zero_zero_nat @ zero_zero_poly_nat) = zero_zero_poly_nat))). % pCons_0_0
thf(fact_25_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_26_pCons__eq__0__iff, axiom,
    ((![A : nat, P : poly_nat]: (((pCons_nat @ A @ P) = zero_zero_poly_nat) = (((A = zero_zero_nat)) & ((P = zero_zero_poly_nat))))))). % pCons_eq_0_iff
thf(fact_27_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_28_poly__0, axiom,
    ((![X : poly_complex]: ((poly_poly_complex2 @ zero_z1040703943omplex @ X) = zero_z1746442943omplex)))). % poly_0
thf(fact_29_poly__0, axiom,
    ((![X : nat]: ((poly_nat2 @ zero_zero_poly_nat @ X) = zero_zero_nat)))). % poly_0
thf(fact_30_poly__0, axiom,
    ((![X : complex]: ((poly_complex2 @ zero_z1746442943omplex @ X) = zero_zero_complex)))). % poly_0
thf(fact_31_pCons__induct, axiom,
    ((![P3 : poly_poly_complex > $o, P : poly_poly_complex]: ((P3 @ zero_z1040703943omplex) => ((![A2 : poly_complex, P2 : poly_poly_complex]: (((~ ((A2 = zero_z1746442943omplex))) | (~ ((P2 = zero_z1040703943omplex)))) => ((P3 @ P2) => (P3 @ (pCons_poly_complex @ A2 @ P2))))) => (P3 @ P)))))). % pCons_induct
thf(fact_32_pCons__induct, axiom,
    ((![P3 : poly_nat > $o, P : poly_nat]: ((P3 @ zero_zero_poly_nat) => ((![A2 : nat, P2 : poly_nat]: (((~ ((A2 = zero_zero_nat))) | (~ ((P2 = zero_zero_poly_nat)))) => ((P3 @ P2) => (P3 @ (pCons_nat @ A2 @ P2))))) => (P3 @ P)))))). % pCons_induct
thf(fact_33_pCons__induct, axiom,
    ((![P3 : poly_complex > $o, P : poly_complex]: ((P3 @ zero_z1746442943omplex) => ((![A2 : complex, P2 : poly_complex]: (((~ ((A2 = zero_zero_complex))) | (~ ((P2 = zero_z1746442943omplex)))) => ((P3 @ P2) => (P3 @ (pCons_complex @ A2 @ P2))))) => (P3 @ P)))))). % pCons_induct
thf(fact_34_poly__all__0__iff__0, axiom,
    ((![P : poly_poly_complex]: ((![X4 : poly_complex]: ((poly_poly_complex2 @ P @ X4) = zero_z1746442943omplex)) = (P = zero_z1040703943omplex))))). % poly_all_0_iff_0
thf(fact_35_poly__all__0__iff__0, axiom,
    ((![P : poly_complex]: ((![X4 : complex]: ((poly_complex2 @ P @ X4) = zero_zero_complex)) = (P = zero_z1746442943omplex))))). % poly_all_0_iff_0
thf(fact_36_pderiv_Oinduct, axiom,
    ((![P3 : poly_complex > $o, A0 : poly_complex]: ((![A2 : complex, P2 : poly_complex]: (((~ ((P2 = zero_z1746442943omplex))) => (P3 @ P2)) => (P3 @ (pCons_complex @ A2 @ P2)))) => (P3 @ A0))))). % pderiv.induct
thf(fact_37_poly__induct2, axiom,
    ((![P3 : poly_complex > poly_complex > $o, P : poly_complex, Q : poly_complex]: ((P3 @ zero_z1746442943omplex @ zero_z1746442943omplex) => ((![A2 : complex, P2 : poly_complex, B2 : complex, Q2 : poly_complex]: ((P3 @ P2 @ Q2) => (P3 @ (pCons_complex @ A2 @ P2) @ (pCons_complex @ B2 @ Q2)))) => (P3 @ P @ Q)))))). % poly_induct2
thf(fact_38_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_39_dual__order_Oantisym, axiom,
    ((![B : nat, A : nat]: ((ord_less_eq_nat @ B @ A) => ((ord_less_eq_nat @ A @ B) => (A = B)))))). % dual_order.antisym
thf(fact_40_dual__order_Oeq__iff, axiom,
    (((^[Y2 : real]: (^[Z4 : real]: (Y2 = Z4))) = (^[A3 : real]: (^[B3 : real]: (((ord_less_eq_real @ B3 @ A3)) & ((ord_less_eq_real @ A3 @ B3)))))))). % dual_order.eq_iff
thf(fact_41_dual__order_Oeq__iff, axiom,
    (((^[Y2 : nat]: (^[Z4 : nat]: (Y2 = Z4))) = (^[A3 : nat]: (^[B3 : nat]: (((ord_less_eq_nat @ B3 @ A3)) & ((ord_less_eq_nat @ A3 @ B3)))))))). % dual_order.eq_iff
thf(fact_42_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_43_dual__order_Otrans, axiom,
    ((![B : nat, A : nat, C : nat]: ((ord_less_eq_nat @ B @ A) => ((ord_less_eq_nat @ C @ B) => (ord_less_eq_nat @ C @ A)))))). % dual_order.trans
thf(fact_44_linorder__wlog, axiom,
    ((![P3 : real > real > $o, A : real, B : real]: ((![A2 : real, B2 : real]: ((ord_less_eq_real @ A2 @ B2) => (P3 @ A2 @ B2))) => ((![A2 : real, B2 : real]: ((P3 @ B2 @ A2) => (P3 @ A2 @ B2))) => (P3 @ A @ B)))))). % linorder_wlog
thf(fact_45_linorder__wlog, axiom,
    ((![P3 : nat > nat > $o, A : nat, B : nat]: ((![A2 : nat, B2 : nat]: ((ord_less_eq_nat @ A2 @ B2) => (P3 @ A2 @ B2))) => ((![A2 : nat, B2 : nat]: ((P3 @ B2 @ A2) => (P3 @ A2 @ B2))) => (P3 @ A @ B)))))). % linorder_wlog
thf(fact_46_dual__order_Orefl, axiom,
    ((![A : real]: (ord_less_eq_real @ A @ A)))). % dual_order.refl
thf(fact_47_dual__order_Orefl, axiom,
    ((![A : nat]: (ord_less_eq_nat @ A @ A)))). % dual_order.refl
thf(fact_48_order__trans, axiom,
    ((![X : real, Y3 : real, Z : real]: ((ord_less_eq_real @ X @ Y3) => ((ord_less_eq_real @ Y3 @ Z) => (ord_less_eq_real @ X @ Z)))))). % order_trans
thf(fact_49_order__trans, axiom,
    ((![X : nat, Y3 : nat, Z : nat]: ((ord_less_eq_nat @ X @ Y3) => ((ord_less_eq_nat @ Y3 @ Z) => (ord_less_eq_nat @ X @ Z)))))). % order_trans
thf(fact_50_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_51_order__class_Oorder_Oantisym, axiom,
    ((![A : nat, B : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_nat @ B @ A) => (A = B)))))). % order_class.order.antisym
thf(fact_52_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_53_ord__le__eq__trans, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ B) => ((B = C) => (ord_less_eq_nat @ A @ C)))))). % ord_le_eq_trans
thf(fact_54_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_55_ord__eq__le__trans, axiom,
    ((![A : nat, B : nat, C : nat]: ((A = B) => ((ord_less_eq_nat @ B @ C) => (ord_less_eq_nat @ A @ C)))))). % ord_eq_le_trans
thf(fact_56_order__class_Oorder_Oeq__iff, axiom,
    (((^[Y2 : real]: (^[Z4 : real]: (Y2 = Z4))) = (^[A3 : real]: (^[B3 : real]: (((ord_less_eq_real @ A3 @ B3)) & ((ord_less_eq_real @ B3 @ A3)))))))). % order_class.order.eq_iff
thf(fact_57_order__class_Oorder_Oeq__iff, axiom,
    (((^[Y2 : nat]: (^[Z4 : nat]: (Y2 = Z4))) = (^[A3 : nat]: (^[B3 : nat]: (((ord_less_eq_nat @ A3 @ B3)) & ((ord_less_eq_nat @ B3 @ A3)))))))). % order_class.order.eq_iff
thf(fact_58_antisym__conv, axiom,
    ((![Y3 : real, X : real]: ((ord_less_eq_real @ Y3 @ X) => ((ord_less_eq_real @ X @ Y3) = (X = Y3)))))). % antisym_conv
thf(fact_59_antisym__conv, axiom,
    ((![Y3 : nat, X : nat]: ((ord_less_eq_nat @ Y3 @ X) => ((ord_less_eq_nat @ X @ Y3) = (X = Y3)))))). % antisym_conv
thf(fact_60_le__cases3, axiom,
    ((![X : real, Y3 : real, Z : real]: (((ord_less_eq_real @ X @ Y3) => (~ ((ord_less_eq_real @ Y3 @ Z)))) => (((ord_less_eq_real @ Y3 @ X) => (~ ((ord_less_eq_real @ X @ Z)))) => (((ord_less_eq_real @ X @ Z) => (~ ((ord_less_eq_real @ Z @ Y3)))) => (((ord_less_eq_real @ Z @ Y3) => (~ ((ord_less_eq_real @ Y3 @ X)))) => (((ord_less_eq_real @ Y3 @ Z) => (~ ((ord_less_eq_real @ Z @ X)))) => (~ (((ord_less_eq_real @ Z @ X) => (~ ((ord_less_eq_real @ X @ Y3)))))))))))))). % le_cases3
thf(fact_61_le__cases3, axiom,
    ((![X : nat, Y3 : nat, Z : nat]: (((ord_less_eq_nat @ X @ Y3) => (~ ((ord_less_eq_nat @ Y3 @ Z)))) => (((ord_less_eq_nat @ Y3 @ X) => (~ ((ord_less_eq_nat @ X @ Z)))) => (((ord_less_eq_nat @ X @ Z) => (~ ((ord_less_eq_nat @ Z @ Y3)))) => (((ord_less_eq_nat @ Z @ Y3) => (~ ((ord_less_eq_nat @ Y3 @ X)))) => (((ord_less_eq_nat @ Y3 @ Z) => (~ ((ord_less_eq_nat @ Z @ X)))) => (~ (((ord_less_eq_nat @ Z @ X) => (~ ((ord_less_eq_nat @ X @ Y3)))))))))))))). % le_cases3
thf(fact_62_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_63_order_Otrans, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_nat @ B @ C) => (ord_less_eq_nat @ A @ C)))))). % order.trans
thf(fact_64_le__cases, axiom,
    ((![X : real, Y3 : real]: ((~ ((ord_less_eq_real @ X @ Y3))) => (ord_less_eq_real @ Y3 @ X))))). % le_cases
thf(fact_65_le__cases, axiom,
    ((![X : nat, Y3 : nat]: ((~ ((ord_less_eq_nat @ X @ Y3))) => (ord_less_eq_nat @ Y3 @ X))))). % le_cases
thf(fact_66_eq__refl, axiom,
    ((![X : real, Y3 : real]: ((X = Y3) => (ord_less_eq_real @ X @ Y3))))). % eq_refl
thf(fact_67_eq__refl, axiom,
    ((![X : nat, Y3 : nat]: ((X = Y3) => (ord_less_eq_nat @ X @ Y3))))). % eq_refl
thf(fact_68_linear, axiom,
    ((![X : real, Y3 : real]: ((ord_less_eq_real @ X @ Y3) | (ord_less_eq_real @ Y3 @ X))))). % linear
thf(fact_69_linear, axiom,
    ((![X : nat, Y3 : nat]: ((ord_less_eq_nat @ X @ Y3) | (ord_less_eq_nat @ Y3 @ X))))). % linear
thf(fact_70_antisym, axiom,
    ((![X : real, Y3 : real]: ((ord_less_eq_real @ X @ Y3) => ((ord_less_eq_real @ Y3 @ X) => (X = Y3)))))). % antisym
thf(fact_71_antisym, axiom,
    ((![X : nat, Y3 : nat]: ((ord_less_eq_nat @ X @ Y3) => ((ord_less_eq_nat @ Y3 @ X) => (X = Y3)))))). % antisym
thf(fact_72_mem__Collect__eq, axiom,
    ((![A : real, P3 : real > $o]: ((member_real @ A @ (collect_real @ P3)) = (P3 @ A))))). % mem_Collect_eq
thf(fact_73_Collect__mem__eq, axiom,
    ((![A4 : set_real]: ((collect_real @ (^[X4 : real]: (member_real @ X4 @ A4))) = A4)))). % Collect_mem_eq
thf(fact_74_eq__iff, axiom,
    (((^[Y2 : real]: (^[Z4 : real]: (Y2 = Z4))) = (^[X4 : real]: (^[Y4 : real]: (((ord_less_eq_real @ X4 @ Y4)) & ((ord_less_eq_real @ Y4 @ X4)))))))). % eq_iff
thf(fact_75_eq__iff, axiom,
    (((^[Y2 : nat]: (^[Z4 : nat]: (Y2 = Z4))) = (^[X4 : nat]: (^[Y4 : nat]: (((ord_less_eq_nat @ X4 @ Y4)) & ((ord_less_eq_nat @ Y4 @ X4)))))))). % eq_iff
thf(fact_76_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, Y : real]: ((ord_less_eq_real @ X3 @ Y) => (ord_less_eq_real @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_real @ (F @ A) @ C))))))). % ord_le_eq_subst
thf(fact_77_ord__le__eq__subst, axiom,
    ((![A : real, B : real, F : real > nat, C : nat]: ((ord_less_eq_real @ A @ B) => (((F @ B) = C) => ((![X3 : real, Y : real]: ((ord_less_eq_real @ X3 @ Y) => (ord_less_eq_nat @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_nat @ (F @ A) @ C))))))). % ord_le_eq_subst
thf(fact_78_ord__le__eq__subst, axiom,
    ((![A : nat, B : nat, F : nat > real, C : real]: ((ord_less_eq_nat @ A @ B) => (((F @ B) = C) => ((![X3 : nat, Y : nat]: ((ord_less_eq_nat @ X3 @ Y) => (ord_less_eq_real @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_real @ (F @ A) @ C))))))). % ord_le_eq_subst
thf(fact_79_ord__le__eq__subst, axiom,
    ((![A : nat, B : nat, F : nat > nat, C : nat]: ((ord_less_eq_nat @ A @ B) => (((F @ B) = C) => ((![X3 : nat, Y : nat]: ((ord_less_eq_nat @ X3 @ Y) => (ord_less_eq_nat @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_nat @ (F @ A) @ C))))))). % ord_le_eq_subst
thf(fact_80_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, Y : real]: ((ord_less_eq_real @ X3 @ Y) => (ord_less_eq_real @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_real @ A @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_81_ord__eq__le__subst, axiom,
    ((![A : nat, F : real > nat, B : real, C : real]: ((A = (F @ B)) => ((ord_less_eq_real @ B @ C) => ((![X3 : real, Y : real]: ((ord_less_eq_real @ X3 @ Y) => (ord_less_eq_nat @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_nat @ A @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_82_ord__eq__le__subst, axiom,
    ((![A : real, F : nat > real, B : nat, C : nat]: ((A = (F @ B)) => ((ord_less_eq_nat @ B @ C) => ((![X3 : nat, Y : nat]: ((ord_less_eq_nat @ X3 @ Y) => (ord_less_eq_real @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_real @ A @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_83_ord__eq__le__subst, axiom,
    ((![A : nat, F : nat > nat, B : nat, C : nat]: ((A = (F @ B)) => ((ord_less_eq_nat @ B @ C) => ((![X3 : nat, Y : nat]: ((ord_less_eq_nat @ X3 @ Y) => (ord_less_eq_nat @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_nat @ A @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_84_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, Y : real]: ((ord_less_eq_real @ X3 @ Y) => (ord_less_eq_real @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_real @ (F @ A) @ C))))))). % order_subst2
thf(fact_85_order__subst2, axiom,
    ((![A : real, B : real, F : real > nat, C : nat]: ((ord_less_eq_real @ A @ B) => ((ord_less_eq_nat @ (F @ B) @ C) => ((![X3 : real, Y : real]: ((ord_less_eq_real @ X3 @ Y) => (ord_less_eq_nat @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_nat @ (F @ A) @ C))))))). % order_subst2
thf(fact_86_order__subst2, axiom,
    ((![A : nat, B : nat, F : nat > real, C : real]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_real @ (F @ B) @ C) => ((![X3 : nat, Y : nat]: ((ord_less_eq_nat @ X3 @ Y) => (ord_less_eq_real @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_real @ (F @ A) @ C))))))). % order_subst2
thf(fact_87_order__subst2, axiom,
    ((![A : nat, B : nat, F : nat > nat, C : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_nat @ (F @ B) @ C) => ((![X3 : nat, Y : nat]: ((ord_less_eq_nat @ X3 @ Y) => (ord_less_eq_nat @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_nat @ (F @ A) @ C))))))). % order_subst2
thf(fact_88_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, Y : real]: ((ord_less_eq_real @ X3 @ Y) => (ord_less_eq_real @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_real @ A @ (F @ C)))))))). % order_subst1
thf(fact_89_order__subst1, axiom,
    ((![A : real, F : nat > real, B : nat, C : nat]: ((ord_less_eq_real @ A @ (F @ B)) => ((ord_less_eq_nat @ B @ C) => ((![X3 : nat, Y : nat]: ((ord_less_eq_nat @ X3 @ Y) => (ord_less_eq_real @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_real @ A @ (F @ C)))))))). % order_subst1
thf(fact_90_order__subst1, axiom,
    ((![A : nat, F : real > nat, B : real, C : real]: ((ord_less_eq_nat @ A @ (F @ B)) => ((ord_less_eq_real @ B @ C) => ((![X3 : real, Y : real]: ((ord_less_eq_real @ X3 @ Y) => (ord_less_eq_nat @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_nat @ A @ (F @ C)))))))). % order_subst1
thf(fact_91_order__subst1, axiom,
    ((![A : nat, F : nat > nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ (F @ B)) => ((ord_less_eq_nat @ B @ C) => ((![X3 : nat, Y : nat]: ((ord_less_eq_nat @ X3 @ Y) => (ord_less_eq_nat @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_nat @ A @ (F @ C)))))))). % order_subst1
thf(fact_92_norm__le__zero__iff, axiom,
    ((![X : complex]: ((ord_less_eq_real @ (real_V638595069omplex @ X) @ zero_zero_real) = (X = zero_zero_complex))))). % norm_le_zero_iff
thf(fact_93_norm__le__zero__iff, axiom,
    ((![X : real]: ((ord_less_eq_real @ (real_V646646907m_real @ X) @ zero_zero_real) = (X = zero_zero_real))))). % norm_le_zero_iff
thf(fact_94_abs__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_95_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_96_abs__of__nonneg, axiom,
    ((![A : real]: ((ord_less_eq_real @ zero_zero_real @ A) => ((abs_abs_real @ A) = A))))). % abs_of_nonneg
thf(fact_97_abs__norm__cancel, axiom,
    ((![A : complex]: ((abs_abs_real @ (real_V638595069omplex @ A)) = (real_V638595069omplex @ A))))). % abs_norm_cancel
thf(fact_98_abs__norm__cancel, axiom,
    ((![A : real]: ((abs_abs_real @ (real_V646646907m_real @ A)) = (real_V646646907m_real @ A))))). % abs_norm_cancel
thf(fact_99_norm__zero, axiom,
    (((real_V638595069omplex @ zero_zero_complex) = zero_zero_real))). % norm_zero
thf(fact_100_norm__zero, axiom,
    (((real_V646646907m_real @ zero_zero_real) = zero_zero_real))). % norm_zero
thf(fact_101_norm__eq__zero, axiom,
    ((![X : complex]: (((real_V638595069omplex @ X) = zero_zero_real) = (X = zero_zero_complex))))). % norm_eq_zero
thf(fact_102_norm__eq__zero, axiom,
    ((![X : real]: (((real_V646646907m_real @ X) = zero_zero_real) = (X = zero_zero_real))))). % norm_eq_zero
thf(fact_103_abs__0, axiom,
    (((abs_abs_real @ zero_zero_real) = zero_zero_real))). % abs_0
thf(fact_104_abs__0, axiom,
    (((abs_abs_complex @ zero_zero_complex) = zero_zero_complex))). % abs_0
thf(fact_105_abs__0__eq, axiom,
    ((![A : real]: ((zero_zero_real = (abs_abs_real @ A)) = (A = zero_zero_real))))). % abs_0_eq
thf(fact_106_abs__idempotent, axiom,
    ((![A : real]: ((abs_abs_real @ (abs_abs_real @ A)) = (abs_abs_real @ A))))). % abs_idempotent
thf(fact_107_abs__abs, axiom,
    ((![A : real]: ((abs_abs_real @ (abs_abs_real @ A)) = (abs_abs_real @ A))))). % abs_abs
thf(fact_108_le__zero__eq, axiom,
    ((![N : nat]: ((ord_less_eq_nat @ N @ zero_zero_nat) = (N = zero_zero_nat))))). % le_zero_eq
thf(fact_109_abs__zero, axiom,
    (((abs_abs_real @ zero_zero_real) = zero_zero_real))). % abs_zero
thf(fact_110_abs__eq__0, axiom,
    ((![A : real]: (((abs_abs_real @ A) = zero_zero_real) = (A = zero_zero_real))))). % abs_eq_0
thf(fact_111_norm__ge__zero, axiom,
    ((![X : complex]: (ord_less_eq_real @ zero_zero_real @ (real_V638595069omplex @ X))))). % norm_ge_zero
thf(fact_112_norm__ge__zero, axiom,
    ((![X : real]: (ord_less_eq_real @ zero_zero_real @ (real_V646646907m_real @ X))))). % norm_ge_zero
thf(fact_113_zero__reorient, axiom,
    ((![X : poly_complex]: ((zero_z1746442943omplex = X) = (X = zero_z1746442943omplex))))). % zero_reorient
thf(fact_114_zero__reorient, axiom,
    ((![X : complex]: ((zero_zero_complex = X) = (X = zero_zero_complex))))). % zero_reorient
thf(fact_115_zero__reorient, axiom,
    ((![X : nat]: ((zero_zero_nat = X) = (X = zero_zero_nat))))). % zero_reorient
thf(fact_116_real__norm__def, axiom,
    ((real_V646646907m_real = abs_abs_real))). % real_norm_def
thf(fact_117_zero__le, axiom,
    ((![X : nat]: (ord_less_eq_nat @ zero_zero_nat @ X)))). % zero_le
thf(fact_118_abs__eq__0__iff, axiom,
    ((![A : real]: (((abs_abs_real @ A) = zero_zero_real) = (A = zero_zero_real))))). % abs_eq_0_iff
thf(fact_119_abs__eq__0__iff, axiom,
    ((![A : complex]: (((abs_abs_complex @ A) = zero_zero_complex) = (A = zero_zero_complex))))). % abs_eq_0_iff
thf(fact_120_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_121_abs__ge__self, axiom,
    ((![A : real]: (ord_less_eq_real @ A @ (abs_abs_real @ A))))). % abs_ge_self
thf(fact_122_abs__ge__zero, axiom,
    ((![A : real]: (ord_less_eq_real @ zero_zero_real @ (abs_abs_real @ A))))). % abs_ge_zero
thf(fact_123_le__numeral__extra_I3_J, axiom,
    ((ord_less_eq_real @ zero_zero_real @ zero_zero_real))). % le_numeral_extra(3)
thf(fact_124_le__numeral__extra_I3_J, axiom,
    ((ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat))). % le_numeral_extra(3)
thf(fact_125_psize__eq__0__iff, axiom,
    ((![P : poly_complex]: (((fundam1709708056omplex @ P) = zero_zero_nat) = (P = zero_z1746442943omplex))))). % psize_eq_0_iff
thf(fact_126_synthetic__div__pCons, axiom,
    ((![A : complex, P : poly_complex, C : complex]: ((synthe151143547omplex @ (pCons_complex @ A @ P) @ C) = (pCons_complex @ (poly_complex2 @ P @ C) @ (synthe151143547omplex @ P @ C)))))). % synthetic_div_pCons
thf(fact_127_is__zero__null, axiom,
    ((is_zero_complex = (^[P4 : poly_complex]: (P4 = zero_z1746442943omplex))))). % is_zero_null
thf(fact_128_poly__cutoff__0, axiom,
    ((![N : nat]: ((poly_cutoff_complex @ N @ zero_z1746442943omplex) = zero_z1746442943omplex)))). % poly_cutoff_0
thf(fact_129_synthetic__div__0, axiom,
    ((![C : complex]: ((synthe151143547omplex @ zero_z1746442943omplex @ C) = zero_z1746442943omplex)))). % synthetic_div_0
thf(fact_130_le0, axiom,
    ((![N : nat]: (ord_less_eq_nat @ zero_zero_nat @ N)))). % le0
thf(fact_131_bot__nat__0_Oextremum, axiom,
    ((![A : nat]: (ord_less_eq_nat @ zero_zero_nat @ A)))). % bot_nat_0.extremum
thf(fact_132_reflect__poly__at__0__eq__0__iff, axiom,
    ((![P : poly_poly_complex]: (((poly_poly_complex2 @ (reflec309385472omplex @ P) @ zero_z1746442943omplex) = zero_z1746442943omplex) = (P = zero_z1040703943omplex))))). % reflect_poly_at_0_eq_0_iff
thf(fact_133_reflect__poly__at__0__eq__0__iff, axiom,
    ((![P : poly_complex]: (((poly_complex2 @ (reflect_poly_complex @ P) @ zero_zero_complex) = zero_zero_complex) = (P = zero_z1746442943omplex))))). % reflect_poly_at_0_eq_0_iff
thf(fact_134_reflect__poly__at__0__eq__0__iff, axiom,
    ((![P : poly_nat]: (((poly_nat2 @ (reflect_poly_nat @ P) @ zero_zero_nat) = zero_zero_nat) = (P = zero_zero_poly_nat))))). % reflect_poly_at_0_eq_0_iff
thf(fact_135_poly__cutoff__1, axiom,
    ((![N : nat]: (((N = zero_zero_nat) => ((poly_cutoff_complex @ N @ one_one_poly_complex) = zero_z1746442943omplex)) & ((~ ((N = zero_zero_nat))) => ((poly_cutoff_complex @ N @ one_one_poly_complex) = one_one_poly_complex)))))). % poly_cutoff_1
thf(fact_136_abs__1, axiom,
    (((abs_abs_real @ one_one_real) = one_one_real))). % abs_1
thf(fact_137_norm__one, axiom,
    (((real_V638595069omplex @ one_one_complex) = one_one_real))). % norm_one
thf(fact_138_norm__one, axiom,
    (((real_V646646907m_real @ one_one_real) = one_one_real))). % norm_one
thf(fact_139_reflect__poly__0, axiom,
    (((reflect_poly_complex @ zero_z1746442943omplex) = zero_z1746442943omplex))). % reflect_poly_0
thf(fact_140_poly__1, axiom,
    ((![X : complex]: ((poly_complex2 @ one_one_poly_complex @ X) = one_one_complex)))). % poly_1
thf(fact_141_reflect__poly__const, axiom,
    ((![A : complex]: ((reflect_poly_complex @ (pCons_complex @ A @ zero_z1746442943omplex)) = (pCons_complex @ A @ zero_z1746442943omplex))))). % reflect_poly_const
thf(fact_142_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_143_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_144_zero__neq__one, axiom,
    ((~ ((zero_z1746442943omplex = one_one_poly_complex))))). % zero_neq_one
thf(fact_145_zero__neq__one, axiom,
    ((~ ((zero_zero_complex = one_one_complex))))). % zero_neq_one
thf(fact_146_zero__neq__one, axiom,
    ((~ ((zero_zero_nat = one_one_nat))))). % zero_neq_one
thf(fact_147_pCons__one, axiom,
    (((pCons_complex @ one_one_complex @ zero_z1746442943omplex) = one_one_poly_complex))). % pCons_one
thf(fact_148_le__numeral__extra_I4_J, axiom,
    ((ord_less_eq_real @ one_one_real @ one_one_real))). % le_numeral_extra(4)
thf(fact_149_le__numeral__extra_I4_J, axiom,
    ((ord_less_eq_nat @ one_one_nat @ one_one_nat))). % le_numeral_extra(4)
thf(fact_150_abs__one, axiom,
    (((abs_abs_real @ one_one_real) = one_one_real))). % abs_one
thf(fact_151_le__refl, axiom,
    ((![N : nat]: (ord_less_eq_nat @ N @ N)))). % le_refl
thf(fact_152_le__trans, axiom,
    ((![I : nat, J : nat, K : nat]: ((ord_less_eq_nat @ I @ J) => ((ord_less_eq_nat @ J @ K) => (ord_less_eq_nat @ I @ K)))))). % le_trans
thf(fact_153_eq__imp__le, axiom,
    ((![M : nat, N : nat]: ((M = N) => (ord_less_eq_nat @ M @ N))))). % eq_imp_le
thf(fact_154_le__antisym, axiom,
    ((![M : nat, N : nat]: ((ord_less_eq_nat @ M @ N) => ((ord_less_eq_nat @ N @ M) => (M = N)))))). % le_antisym
thf(fact_155_nat__le__linear, axiom,
    ((![M : nat, N : nat]: ((ord_less_eq_nat @ M @ N) | (ord_less_eq_nat @ N @ M))))). % nat_le_linear
thf(fact_156_Nat_Oex__has__greatest__nat, axiom,
    ((![P3 : nat > $o, K : nat, B : nat]: ((P3 @ K) => ((![Y : nat]: ((P3 @ Y) => (ord_less_eq_nat @ Y @ B))) => (?[X3 : nat]: ((P3 @ X3) & (![Y5 : nat]: ((P3 @ Y5) => (ord_less_eq_nat @ Y5 @ X3)))))))))). % Nat.ex_has_greatest_nat
thf(fact_157_not__one__le__zero, axiom,
    ((~ ((ord_less_eq_real @ one_one_real @ zero_zero_real))))). % not_one_le_zero
thf(fact_158_not__one__le__zero, axiom,
    ((~ ((ord_less_eq_nat @ one_one_nat @ zero_zero_nat))))). % not_one_le_zero
thf(fact_159_zero__le__one, axiom,
    ((ord_less_eq_real @ zero_zero_real @ one_one_real))). % zero_le_one
thf(fact_160_zero__le__one, axiom,
    ((ord_less_eq_nat @ zero_zero_nat @ one_one_nat))). % zero_le_one
thf(fact_161_bot__nat__0_Oextremum__uniqueI, axiom,
    ((![A : nat]: ((ord_less_eq_nat @ A @ zero_zero_nat) => (A = zero_zero_nat))))). % bot_nat_0.extremum_uniqueI
thf(fact_162_bot__nat__0_Oextremum__unique, axiom,
    ((![A : nat]: ((ord_less_eq_nat @ A @ zero_zero_nat) = (A = zero_zero_nat))))). % bot_nat_0.extremum_unique
thf(fact_163_le__0__eq, axiom,
    ((![N : nat]: ((ord_less_eq_nat @ N @ zero_zero_nat) = (N = zero_zero_nat))))). % le_0_eq
thf(fact_164_less__eq__nat_Osimps_I1_J, axiom,
    ((![N : nat]: (ord_less_eq_nat @ zero_zero_nat @ N)))). % less_eq_nat.simps(1)
thf(fact_165_poly__shift__1, axiom,
    ((![N : nat]: (((N = zero_zero_nat) => ((poly_shift_complex @ N @ one_one_poly_complex) = one_one_poly_complex)) & ((~ ((N = zero_zero_nat))) => ((poly_shift_complex @ N @ one_one_poly_complex) = zero_z1746442943omplex)))))). % poly_shift_1
thf(fact_166_dbl__inc__simps_I2_J, axiom,
    (((neg_nu1217972807omplex @ zero_z1746442943omplex) = one_one_poly_complex))). % dbl_inc_simps(2)
thf(fact_167_dbl__inc__simps_I2_J, axiom,
    (((neg_nu484426047omplex @ zero_zero_complex) = one_one_complex))). % dbl_inc_simps(2)
thf(fact_168_pcompose__idR, axiom,
    ((![P : poly_poly_complex]: ((pcompo1411605209omplex @ P @ (pCons_poly_complex @ zero_z1746442943omplex @ (pCons_poly_complex @ one_one_poly_complex @ zero_z1040703943omplex))) = P)))). % pcompose_idR
thf(fact_169_pcompose__idR, axiom,
    ((![P : poly_complex]: ((pcompose_complex @ P @ (pCons_complex @ zero_zero_complex @ (pCons_complex @ one_one_complex @ zero_z1746442943omplex))) = P)))). % pcompose_idR
thf(fact_170_pcompose__idR, axiom,
    ((![P : poly_nat]: ((pcompose_nat @ P @ (pCons_nat @ zero_zero_nat @ (pCons_nat @ one_one_nat @ zero_zero_poly_nat))) = P)))). % pcompose_idR
thf(fact_171_pcompose__0, axiom,
    ((![Q : poly_complex]: ((pcompose_complex @ zero_z1746442943omplex @ Q) = zero_z1746442943omplex)))). % pcompose_0
thf(fact_172_poly__shift__0, axiom,
    ((![N : nat]: ((poly_shift_complex @ N @ zero_z1746442943omplex) = zero_z1746442943omplex)))). % poly_shift_0
thf(fact_173_pcompose__const, axiom,
    ((![A : complex, Q : poly_complex]: ((pcompose_complex @ (pCons_complex @ A @ zero_z1746442943omplex) @ Q) = (pCons_complex @ A @ zero_z1746442943omplex))))). % pcompose_const
thf(fact_174_poly__pcompose, axiom,
    ((![P : poly_complex, Q : poly_complex, X : complex]: ((poly_complex2 @ (pcompose_complex @ P @ Q) @ X) = (poly_complex2 @ P @ (poly_complex2 @ Q @ X)))))). % poly_pcompose
thf(fact_175_coeff__0__reflect__poly__0__iff, axiom,
    ((![P : poly_poly_complex]: (((coeff_poly_complex @ (reflec309385472omplex @ P) @ zero_zero_nat) = zero_z1746442943omplex) = (P = zero_z1040703943omplex))))). % coeff_0_reflect_poly_0_iff
thf(fact_176_coeff__0__reflect__poly__0__iff, axiom,
    ((![P : poly_complex]: (((coeff_complex @ (reflect_poly_complex @ P) @ zero_zero_nat) = zero_zero_complex) = (P = zero_z1746442943omplex))))). % coeff_0_reflect_poly_0_iff
thf(fact_177_coeff__0__reflect__poly__0__iff, axiom,
    ((![P : poly_nat]: (((coeff_nat @ (reflect_poly_nat @ P) @ zero_zero_nat) = zero_zero_nat) = (P = zero_zero_poly_nat))))). % coeff_0_reflect_poly_0_iff
thf(fact_178_map__poly__0, axiom,
    ((![F : complex > complex]: ((map_po1637612550omplex @ F @ zero_z1746442943omplex) = zero_z1746442943omplex)))). % map_poly_0
thf(fact_179_coeff__0, axiom,
    ((![N : nat]: ((coeff_poly_complex @ zero_z1040703943omplex @ N) = zero_z1746442943omplex)))). % coeff_0
thf(fact_180_coeff__0, axiom,
    ((![N : nat]: ((coeff_nat @ zero_zero_poly_nat @ N) = zero_zero_nat)))). % coeff_0
thf(fact_181_coeff__0, axiom,
    ((![N : nat]: ((coeff_complex @ zero_z1746442943omplex @ N) = zero_zero_complex)))). % coeff_0
thf(fact_182_coeff__pCons__0, axiom,
    ((![A : complex, P : poly_complex]: ((coeff_complex @ (pCons_complex @ A @ P) @ zero_zero_nat) = A)))). % coeff_pCons_0
thf(fact_183_reflect__poly__reflect__poly, axiom,
    ((![P : poly_poly_complex]: ((~ (((coeff_poly_complex @ P @ zero_zero_nat) = zero_z1746442943omplex))) => ((reflec309385472omplex @ (reflec309385472omplex @ P)) = P))))). % reflect_poly_reflect_poly
thf(fact_184_reflect__poly__reflect__poly, axiom,
    ((![P : poly_complex]: ((~ (((coeff_complex @ P @ zero_zero_nat) = zero_zero_complex))) => ((reflect_poly_complex @ (reflect_poly_complex @ P)) = P))))). % reflect_poly_reflect_poly
thf(fact_185_reflect__poly__reflect__poly, axiom,
    ((![P : poly_nat]: ((~ (((coeff_nat @ P @ zero_zero_nat) = zero_zero_nat))) => ((reflect_poly_nat @ (reflect_poly_nat @ P)) = P))))). % reflect_poly_reflect_poly
thf(fact_186_zero__poly_Orep__eq, axiom,
    (((coeff_poly_complex @ zero_z1040703943omplex) = (^[Uu : nat]: zero_z1746442943omplex)))). % zero_poly.rep_eq
thf(fact_187_zero__poly_Orep__eq, axiom,
    (((coeff_nat @ zero_zero_poly_nat) = (^[Uu : nat]: zero_zero_nat)))). % zero_poly.rep_eq
thf(fact_188_zero__poly_Orep__eq, axiom,
    (((coeff_complex @ zero_z1746442943omplex) = (^[Uu : nat]: zero_zero_complex)))). % zero_poly.rep_eq
thf(fact_189_coeff__map__poly, axiom,
    ((![F : poly_complex > poly_complex, P : poly_poly_complex, N : nat]: (((F @ zero_z1746442943omplex) = zero_z1746442943omplex) => ((coeff_poly_complex @ (map_po1985147926omplex @ F @ P) @ N) = (F @ (coeff_poly_complex @ P @ N))))))). % coeff_map_poly
thf(fact_190_coeff__map__poly, axiom,
    ((![F : poly_complex > complex, P : poly_poly_complex, N : nat]: (((F @ zero_z1746442943omplex) = zero_zero_complex) => ((coeff_complex @ (map_po944112910omplex @ F @ P) @ N) = (F @ (coeff_poly_complex @ P @ N))))))). % coeff_map_poly
thf(fact_191_coeff__map__poly, axiom,
    ((![F : poly_complex > nat, P : poly_poly_complex, N : nat]: (((F @ zero_z1746442943omplex) = zero_zero_nat) => ((coeff_nat @ (map_po1895962672ex_nat @ F @ P) @ N) = (F @ (coeff_poly_complex @ P @ N))))))). % coeff_map_poly
thf(fact_192_coeff__map__poly, axiom,
    ((![F : complex > poly_complex, P : poly_complex, N : nat]: (((F @ zero_zero_complex) = zero_z1746442943omplex) => ((coeff_poly_complex @ (map_po366287630omplex @ F @ P) @ N) = (F @ (coeff_complex @ P @ N))))))). % coeff_map_poly
thf(fact_193_coeff__map__poly, axiom,
    ((![F : complex > complex, P : poly_complex, N : nat]: (((F @ zero_zero_complex) = zero_zero_complex) => ((coeff_complex @ (map_po1637612550omplex @ F @ P) @ N) = (F @ (coeff_complex @ P @ N))))))). % coeff_map_poly
thf(fact_194_coeff__map__poly, axiom,
    ((![F : complex > nat, P : poly_complex, N : nat]: (((F @ zero_zero_complex) = zero_zero_nat) => ((coeff_nat @ (map_poly_complex_nat @ F @ P) @ N) = (F @ (coeff_complex @ P @ N))))))). % coeff_map_poly
thf(fact_195_coeff__map__poly, axiom,
    ((![F : nat > poly_complex, P : poly_nat, N : nat]: (((F @ zero_zero_nat) = zero_z1746442943omplex) => ((coeff_poly_complex @ (map_po2099259440omplex @ F @ P) @ N) = (F @ (coeff_nat @ P @ N))))))). % coeff_map_poly
thf(fact_196_coeff__map__poly, axiom,
    ((![F : nat > complex, P : poly_nat, N : nat]: (((F @ zero_zero_nat) = zero_zero_complex) => ((coeff_complex @ (map_poly_nat_complex @ F @ P) @ N) = (F @ (coeff_nat @ P @ N))))))). % coeff_map_poly
thf(fact_197_coeff__map__poly, axiom,
    ((![F : nat > nat, P : poly_nat, N : nat]: (((F @ zero_zero_nat) = zero_zero_nat) => ((coeff_nat @ (map_poly_nat_nat @ F @ P) @ N) = (F @ (coeff_nat @ P @ N))))))). % coeff_map_poly
thf(fact_198_map__poly__pCons, axiom,
    ((![F : poly_complex > poly_complex, C : poly_complex, P : poly_poly_complex]: (((F @ zero_z1746442943omplex) = zero_z1746442943omplex) => ((map_po1985147926omplex @ F @ (pCons_poly_complex @ C @ P)) = (pCons_poly_complex @ (F @ C) @ (map_po1985147926omplex @ F @ P))))))). % map_poly_pCons
thf(fact_199_map__poly__pCons, axiom,
    ((![F : poly_complex > complex, C : poly_complex, P : poly_poly_complex]: (((F @ zero_z1746442943omplex) = zero_zero_complex) => ((map_po944112910omplex @ F @ (pCons_poly_complex @ C @ P)) = (pCons_complex @ (F @ C) @ (map_po944112910omplex @ F @ P))))))). % map_poly_pCons
thf(fact_200_map__poly__pCons, axiom,
    ((![F : poly_complex > nat, C : poly_complex, P : poly_poly_complex]: (((F @ zero_z1746442943omplex) = zero_zero_nat) => ((map_po1895962672ex_nat @ F @ (pCons_poly_complex @ C @ P)) = (pCons_nat @ (F @ C) @ (map_po1895962672ex_nat @ F @ P))))))). % map_poly_pCons
thf(fact_201_map__poly__pCons, axiom,
    ((![F : complex > poly_complex, C : complex, P : poly_complex]: (((F @ zero_zero_complex) = zero_z1746442943omplex) => ((map_po366287630omplex @ F @ (pCons_complex @ C @ P)) = (pCons_poly_complex @ (F @ C) @ (map_po366287630omplex @ F @ P))))))). % map_poly_pCons
thf(fact_202_map__poly__pCons, axiom,
    ((![F : complex > complex, C : complex, P : poly_complex]: (((F @ zero_zero_complex) = zero_zero_complex) => ((map_po1637612550omplex @ F @ (pCons_complex @ C @ P)) = (pCons_complex @ (F @ C) @ (map_po1637612550omplex @ F @ P))))))). % map_poly_pCons
thf(fact_203_map__poly__pCons, axiom,
    ((![F : complex > nat, C : complex, P : poly_complex]: (((F @ zero_zero_complex) = zero_zero_nat) => ((map_poly_complex_nat @ F @ (pCons_complex @ C @ P)) = (pCons_nat @ (F @ C) @ (map_poly_complex_nat @ F @ P))))))). % map_poly_pCons
thf(fact_204_map__poly__pCons, axiom,
    ((![F : nat > poly_complex, C : nat, P : poly_nat]: (((F @ zero_zero_nat) = zero_z1746442943omplex) => ((map_po2099259440omplex @ F @ (pCons_nat @ C @ P)) = (pCons_poly_complex @ (F @ C) @ (map_po2099259440omplex @ F @ P))))))). % map_poly_pCons
thf(fact_205_map__poly__pCons, axiom,
    ((![F : nat > complex, C : nat, P : poly_nat]: (((F @ zero_zero_nat) = zero_zero_complex) => ((map_poly_nat_complex @ F @ (pCons_nat @ C @ P)) = (pCons_complex @ (F @ C) @ (map_poly_nat_complex @ F @ P))))))). % map_poly_pCons
thf(fact_206_map__poly__pCons, axiom,
    ((![F : nat > nat, C : nat, P : poly_nat]: (((F @ zero_zero_nat) = zero_zero_nat) => ((map_poly_nat_nat @ F @ (pCons_nat @ C @ P)) = (pCons_nat @ (F @ C) @ (map_poly_nat_nat @ F @ P))))))). % map_poly_pCons
thf(fact_207_poly__0__coeff__0, axiom,
    ((![P : poly_poly_complex]: ((poly_poly_complex2 @ P @ zero_z1746442943omplex) = (coeff_poly_complex @ P @ zero_zero_nat))))). % poly_0_coeff_0
thf(fact_208_poly__0__coeff__0, axiom,
    ((![P : poly_complex]: ((poly_complex2 @ P @ zero_zero_complex) = (coeff_complex @ P @ zero_zero_nat))))). % poly_0_coeff_0
thf(fact_209_poly__0__coeff__0, axiom,
    ((![P : poly_nat]: ((poly_nat2 @ P @ zero_zero_nat) = (coeff_nat @ P @ zero_zero_nat))))). % poly_0_coeff_0
thf(fact_210_pcompose__0_H, axiom,
    ((![P : poly_complex]: ((pcompose_complex @ P @ zero_z1746442943omplex) = (pCons_complex @ (coeff_complex @ P @ zero_zero_nat) @ zero_z1746442943omplex))))). % pcompose_0'
thf(fact_211_poly__reflect__poly__0, axiom,
    ((![P : poly_poly_complex]: ((poly_poly_complex2 @ (reflec309385472omplex @ P) @ zero_z1746442943omplex) = (coeff_poly_complex @ P @ (degree_poly_complex @ P)))))). % poly_reflect_poly_0
thf(fact_212_poly__reflect__poly__0, axiom,
    ((![P : poly_complex]: ((poly_complex2 @ (reflect_poly_complex @ P) @ zero_zero_complex) = (coeff_complex @ P @ (degree_complex @ P)))))). % poly_reflect_poly_0
thf(fact_213_poly__reflect__poly__0, axiom,
    ((![P : poly_nat]: ((poly_nat2 @ (reflect_poly_nat @ P) @ zero_zero_nat) = (coeff_nat @ P @ (degree_nat @ P)))))). % poly_reflect_poly_0
thf(fact_214_degree__reflect__poly__eq, axiom,
    ((![P : poly_poly_complex]: ((~ (((coeff_poly_complex @ P @ zero_zero_nat) = zero_z1746442943omplex))) => ((degree_poly_complex @ (reflec309385472omplex @ P)) = (degree_poly_complex @ P)))))). % degree_reflect_poly_eq
thf(fact_215_degree__reflect__poly__eq, axiom,
    ((![P : poly_complex]: ((~ (((coeff_complex @ P @ zero_zero_nat) = zero_zero_complex))) => ((degree_complex @ (reflect_poly_complex @ P)) = (degree_complex @ P)))))). % degree_reflect_poly_eq
thf(fact_216_degree__reflect__poly__eq, axiom,
    ((![P : poly_nat]: ((~ (((coeff_nat @ P @ zero_zero_nat) = zero_zero_nat))) => ((degree_nat @ (reflect_poly_nat @ P)) = (degree_nat @ P)))))). % degree_reflect_poly_eq
thf(fact_217_degree__0, axiom,
    (((degree_complex @ zero_z1746442943omplex) = zero_zero_nat))). % degree_0
thf(fact_218_leading__coeff__0__iff, axiom,
    ((![P : poly_poly_complex]: (((coeff_poly_complex @ P @ (degree_poly_complex @ P)) = zero_z1746442943omplex) = (P = zero_z1040703943omplex))))). % leading_coeff_0_iff
thf(fact_219_leading__coeff__0__iff, axiom,
    ((![P : poly_complex]: (((coeff_complex @ P @ (degree_complex @ P)) = zero_zero_complex) = (P = zero_z1746442943omplex))))). % leading_coeff_0_iff
thf(fact_220_leading__coeff__0__iff, axiom,
    ((![P : poly_nat]: (((coeff_nat @ P @ (degree_nat @ P)) = zero_zero_nat) = (P = zero_zero_poly_nat))))). % leading_coeff_0_iff
thf(fact_221_lead__coeff__pCons_I2_J, axiom,
    ((![P : poly_complex, A : complex]: ((P = zero_z1746442943omplex) => ((coeff_complex @ (pCons_complex @ A @ P) @ (degree_complex @ (pCons_complex @ A @ P))) = A))))). % lead_coeff_pCons(2)
thf(fact_222_lead__coeff__pCons_I1_J, axiom,
    ((![P : poly_complex, A : complex]: ((~ ((P = zero_z1746442943omplex))) => ((coeff_complex @ (pCons_complex @ A @ P) @ (degree_complex @ (pCons_complex @ A @ P))) = (coeff_complex @ P @ (degree_complex @ P))))))). % lead_coeff_pCons(1)

% Conjectures (1)
thf(conj_0, conjecture,
    ((?[Z3 : complex]: (![W3 : complex]: (ord_less_eq_real @ (real_V638595069omplex @ (poly_complex2 @ (pCons_complex @ c @ cs) @ Z3)) @ (real_V638595069omplex @ (poly_complex2 @ (pCons_complex @ c @ cs) @ W3))))))).
