% 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_463__5372450_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:30:20.510

% Could-be-implicit typings (7)
thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_It__Complex__Ocomplex_J_J, type,
    poly_poly_complex : $tType).
thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J, type,
    poly_poly_real : $tType).
thf(ty_n_t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    poly_complex : $tType).
thf(ty_n_t__Polynomial__Opoly_It__Real__Oreal_J, type,
    poly_real : $tType).
thf(ty_n_t__Complex__Ocomplex, type,
    complex : $tType).
thf(ty_n_t__Real__Oreal, type,
    real : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).

% Explicit typings (36)
thf(sy_c_Groups_Oabs__class_Oabs_001t__Complex__Ocomplex, type,
    abs_abs_complex : complex > complex).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Polynomial__Opoly_It__Real__Oreal_J, type,
    abs_abs_poly_real : poly_real > poly_real).
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__Real__Oreal, type,
    minus_minus_real : real > real > real).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal, type,
    one_one_real : real).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Complex__Ocomplex, type,
    uminus1204672759omplex : complex > complex).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    uminus1138659839omplex : poly_complex > poly_complex).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Complex__Ocomplex_J_J, type,
    uminus1762810119omplex : poly_poly_complex > poly_poly_complex).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J, type,
    uminus262047109y_real : poly_poly_real > poly_poly_real).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Polynomial__Opoly_It__Real__Oreal_J, type,
    uminus1613791741y_real : poly_real > poly_real).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal, type,
    uminus_uminus_real : real > real).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Complex__Ocomplex, type,
    zero_zero_complex : complex).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    zero_z1746442943omplex : poly_complex).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Real__Oreal_J, type,
    zero_zero_poly_real : poly_real).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal, type,
    zero_zero_real : real).
thf(sy_c_If_001t__Polynomial__Opoly_It__Real__Oreal_J, type,
    if_poly_real : $o > poly_real > poly_real > poly_real).
thf(sy_c_If_001t__Real__Oreal, type,
    if_real : $o > real > real > real).
thf(sy_c_Orderings_Oord__class_Oless_001t__Polynomial__Opoly_It__Real__Oreal_J, type,
    ord_less_poly_real : poly_real > poly_real > $o).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal, type,
    ord_less_real : real > real > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Polynomial__Opoly_It__Real__Oreal_J, type,
    ord_le1180086932y_real : poly_real > poly_real > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal, type,
    ord_less_eq_real : real > real > $o).
thf(sy_c_Polynomial_Opoly_001t__Complex__Ocomplex, type,
    poly_complex2 : poly_complex > complex > complex).
thf(sy_c_Polynomial_Opoly_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    poly_poly_complex2 : poly_poly_complex > poly_complex > poly_complex).
thf(sy_c_Polynomial_Opoly_001t__Polynomial__Opoly_It__Real__Oreal_J, type,
    poly_poly_real2 : poly_poly_real > poly_real > poly_real).
thf(sy_c_Polynomial_Opoly_001t__Real__Oreal, type,
    poly_real2 : poly_real > real > real).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Complex__Ocomplex, type,
    real_V638595069omplex : complex > real).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Real__Oreal, type,
    real_V646646907m_real : real > real).
thf(sy_v_g____, type,
    g : nat > complex).
thf(sy_v_p, type,
    p : poly_complex).
thf(sy_v_r, type,
    r : real).
thf(sy_v_s____, type,
    s : real).
thf(sy_v_w____, type,
    w : complex).
thf(sy_v_z____, type,
    z : complex).

% Relevant facts (219)
thf(fact_0__092_060open_062_092_060bar_062cmod_A_Ipoly_Ap_Az_J_A_N_A_N_As_092_060bar_062_A_061_A0_092_060close_062, axiom,
    (((abs_abs_real @ (minus_minus_real @ (real_V638595069omplex @ (poly_complex2 @ p @ z)) @ (uminus_uminus_real @ s))) = zero_zero_real))). % \<open>\<bar>cmod (poly p z) - - s\<bar> = 0\<close>
thf(fact_1_poly__minus, axiom,
    ((![P : poly_poly_real, X : poly_real]: ((poly_poly_real2 @ (uminus262047109y_real @ P) @ X) = (uminus1613791741y_real @ (poly_poly_real2 @ P @ X)))))). % poly_minus
thf(fact_2_poly__minus, axiom,
    ((![P : poly_poly_complex, X : poly_complex]: ((poly_poly_complex2 @ (uminus1762810119omplex @ P) @ X) = (uminus1138659839omplex @ (poly_poly_complex2 @ P @ X)))))). % poly_minus
thf(fact_3_poly__minus, axiom,
    ((![P : poly_complex, X : complex]: ((poly_complex2 @ (uminus1138659839omplex @ P) @ X) = (uminus1204672759omplex @ (poly_complex2 @ P @ X)))))). % poly_minus
thf(fact_4_poly__minus, axiom,
    ((![P : poly_real, X : real]: ((poly_real2 @ (uminus1613791741y_real @ P) @ X) = (uminus_uminus_real @ (poly_real2 @ P @ X)))))). % poly_minus
thf(fact_5_norm__minus__cancel, axiom,
    ((![X : complex]: ((real_V638595069omplex @ (uminus1204672759omplex @ X)) = (real_V638595069omplex @ X))))). % norm_minus_cancel
thf(fact_6_norm__minus__cancel, axiom,
    ((![X : real]: ((real_V646646907m_real @ (uminus_uminus_real @ X)) = (real_V646646907m_real @ X))))). % norm_minus_cancel
thf(fact_7__092_060open_0620_A_060_A_092_060bar_062cmod_A_Ipoly_Ap_Az_J_A_N_A_N_As_092_060bar_062_A_092_060Longrightarrow_062_AFalse_092_060close_062, axiom,
    ((~ ((ord_less_real @ zero_zero_real @ (abs_abs_real @ (minus_minus_real @ (real_V638595069omplex @ (poly_complex2 @ p @ z)) @ (uminus_uminus_real @ s)))))))). % \<open>0 < \<bar>cmod (poly p z) - - s\<bar> \<Longrightarrow> False\<close>
thf(fact_8_verit__minus__simplify_I4_J, axiom,
    ((![B : poly_real]: ((uminus1613791741y_real @ (uminus1613791741y_real @ B)) = B)))). % verit_minus_simplify(4)
thf(fact_9_verit__minus__simplify_I4_J, axiom,
    ((![B : poly_complex]: ((uminus1138659839omplex @ (uminus1138659839omplex @ B)) = B)))). % verit_minus_simplify(4)
thf(fact_10_verit__minus__simplify_I4_J, axiom,
    ((![B : complex]: ((uminus1204672759omplex @ (uminus1204672759omplex @ B)) = B)))). % verit_minus_simplify(4)
thf(fact_11_verit__minus__simplify_I4_J, axiom,
    ((![B : real]: ((uminus_uminus_real @ (uminus_uminus_real @ B)) = B)))). % verit_minus_simplify(4)
thf(fact_12_add_Oinverse__inverse, axiom,
    ((![A : poly_real]: ((uminus1613791741y_real @ (uminus1613791741y_real @ A)) = A)))). % add.inverse_inverse
thf(fact_13_add_Oinverse__inverse, axiom,
    ((![A : poly_complex]: ((uminus1138659839omplex @ (uminus1138659839omplex @ A)) = A)))). % add.inverse_inverse
thf(fact_14_add_Oinverse__inverse, axiom,
    ((![A : complex]: ((uminus1204672759omplex @ (uminus1204672759omplex @ A)) = A)))). % add.inverse_inverse
thf(fact_15_add_Oinverse__inverse, axiom,
    ((![A : real]: ((uminus_uminus_real @ (uminus_uminus_real @ A)) = A)))). % add.inverse_inverse
thf(fact_16_neg__equal__iff__equal, axiom,
    ((![A : poly_real, B : poly_real]: (((uminus1613791741y_real @ A) = (uminus1613791741y_real @ B)) = (A = B))))). % neg_equal_iff_equal
thf(fact_17_neg__equal__iff__equal, axiom,
    ((![A : poly_complex, B : poly_complex]: (((uminus1138659839omplex @ A) = (uminus1138659839omplex @ B)) = (A = B))))). % neg_equal_iff_equal
thf(fact_18_neg__equal__iff__equal, axiom,
    ((![A : complex, B : complex]: (((uminus1204672759omplex @ A) = (uminus1204672759omplex @ B)) = (A = B))))). % neg_equal_iff_equal
thf(fact_19_neg__equal__iff__equal, axiom,
    ((![A : real, B : real]: (((uminus_uminus_real @ A) = (uminus_uminus_real @ B)) = (A = B))))). % neg_equal_iff_equal
thf(fact_20_s1m, axiom,
    ((![Z : complex]: ((ord_less_eq_real @ (real_V638595069omplex @ Z) @ r) => (ord_less_eq_real @ (uminus_uminus_real @ s) @ (real_V638595069omplex @ (poly_complex2 @ p @ Z))))))). % s1m
thf(fact_21_s, axiom,
    ((![Y : real]: ((?[X2 : real]: (((?[Z2 : complex]: (((ord_less_eq_real @ (real_V638595069omplex @ Z2) @ r)) & (((real_V638595069omplex @ (poly_complex2 @ p @ Z2)) = (uminus_uminus_real @ X2)))))) & ((ord_less_real @ Y @ X2)))) = (ord_less_real @ Y @ s))))). % s
thf(fact_22_mth1, axiom,
    ((?[X3 : real, Z3 : complex]: ((ord_less_eq_real @ (real_V638595069omplex @ Z3) @ r) & ((real_V638595069omplex @ (poly_complex2 @ p @ Z3)) = (uminus_uminus_real @ X3)))))). % mth1
thf(fact_23_True, axiom,
    ((ord_less_eq_real @ zero_zero_real @ r))). % True
thf(fact_24_wr, axiom,
    ((ord_less_eq_real @ (real_V638595069omplex @ w) @ r))). % wr
thf(fact_25_abs__idempotent, axiom,
    ((![A : real]: ((abs_abs_real @ (abs_abs_real @ A)) = (abs_abs_real @ A))))). % abs_idempotent
thf(fact_26_g_I1_J, axiom,
    ((![N : nat]: (ord_less_eq_real @ (real_V638595069omplex @ (g @ N)) @ r)))). % g(1)
thf(fact_27_diff__self, axiom,
    ((![A : complex]: ((minus_minus_complex @ A @ A) = zero_zero_complex)))). % diff_self
thf(fact_28_diff__self, axiom,
    ((![A : real]: ((minus_minus_real @ A @ A) = zero_zero_real)))). % diff_self
thf(fact_29_diff__0__right, axiom,
    ((![A : complex]: ((minus_minus_complex @ A @ zero_zero_complex) = A)))). % diff_0_right
thf(fact_30_diff__0__right, axiom,
    ((![A : real]: ((minus_minus_real @ A @ zero_zero_real) = A)))). % diff_0_right
thf(fact_31_diff__zero, axiom,
    ((![A : complex]: ((minus_minus_complex @ A @ zero_zero_complex) = A)))). % diff_zero
thf(fact_32_diff__zero, axiom,
    ((![A : real]: ((minus_minus_real @ A @ zero_zero_real) = A)))). % diff_zero
thf(fact_33_cancel__comm__monoid__add__class_Odiff__cancel, axiom,
    ((![A : complex]: ((minus_minus_complex @ A @ A) = zero_zero_complex)))). % cancel_comm_monoid_add_class.diff_cancel
thf(fact_34_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_35_neg__equal__zero, axiom,
    ((![A : real]: (((uminus_uminus_real @ A) = A) = (A = zero_zero_real))))). % neg_equal_zero
thf(fact_36_neg__equal__zero, axiom,
    ((![A : poly_real]: (((uminus1613791741y_real @ A) = A) = (A = zero_zero_poly_real))))). % neg_equal_zero
thf(fact_37_equal__neg__zero, axiom,
    ((![A : real]: ((A = (uminus_uminus_real @ A)) = (A = zero_zero_real))))). % equal_neg_zero
thf(fact_38_equal__neg__zero, axiom,
    ((![A : poly_real]: ((A = (uminus1613791741y_real @ A)) = (A = zero_zero_poly_real))))). % equal_neg_zero
thf(fact_39_neg__equal__0__iff__equal, axiom,
    ((![A : real]: (((uminus_uminus_real @ A) = zero_zero_real) = (A = zero_zero_real))))). % neg_equal_0_iff_equal
thf(fact_40_neg__equal__0__iff__equal, axiom,
    ((![A : poly_real]: (((uminus1613791741y_real @ A) = zero_zero_poly_real) = (A = zero_zero_poly_real))))). % neg_equal_0_iff_equal
thf(fact_41_neg__equal__0__iff__equal, axiom,
    ((![A : poly_complex]: (((uminus1138659839omplex @ A) = zero_z1746442943omplex) = (A = zero_z1746442943omplex))))). % neg_equal_0_iff_equal
thf(fact_42_neg__equal__0__iff__equal, axiom,
    ((![A : complex]: (((uminus1204672759omplex @ A) = zero_zero_complex) = (A = zero_zero_complex))))). % neg_equal_0_iff_equal
thf(fact_43_neg__0__equal__iff__equal, axiom,
    ((![A : real]: ((zero_zero_real = (uminus_uminus_real @ A)) = (zero_zero_real = A))))). % neg_0_equal_iff_equal
thf(fact_44_neg__0__equal__iff__equal, axiom,
    ((![A : poly_real]: ((zero_zero_poly_real = (uminus1613791741y_real @ A)) = (zero_zero_poly_real = A))))). % neg_0_equal_iff_equal
thf(fact_45_neg__0__equal__iff__equal, axiom,
    ((![A : poly_complex]: ((zero_z1746442943omplex = (uminus1138659839omplex @ A)) = (zero_z1746442943omplex = A))))). % neg_0_equal_iff_equal
thf(fact_46_neg__0__equal__iff__equal, axiom,
    ((![A : complex]: ((zero_zero_complex = (uminus1204672759omplex @ A)) = (zero_zero_complex = A))))). % neg_0_equal_iff_equal
thf(fact_47_add_Oinverse__neutral, axiom,
    (((uminus_uminus_real @ zero_zero_real) = zero_zero_real))). % add.inverse_neutral
thf(fact_48_add_Oinverse__neutral, axiom,
    (((uminus1613791741y_real @ zero_zero_poly_real) = zero_zero_poly_real))). % add.inverse_neutral
thf(fact_49_add_Oinverse__neutral, axiom,
    (((uminus1138659839omplex @ zero_z1746442943omplex) = zero_z1746442943omplex))). % add.inverse_neutral
thf(fact_50_add_Oinverse__neutral, axiom,
    (((uminus1204672759omplex @ zero_zero_complex) = zero_zero_complex))). % add.inverse_neutral
thf(fact_51_neg__le__iff__le, axiom,
    ((![B : poly_real, A : poly_real]: ((ord_le1180086932y_real @ (uminus1613791741y_real @ B) @ (uminus1613791741y_real @ A)) = (ord_le1180086932y_real @ A @ B))))). % neg_le_iff_le
thf(fact_52_neg__le__iff__le, axiom,
    ((![B : real, A : real]: ((ord_less_eq_real @ (uminus_uminus_real @ B) @ (uminus_uminus_real @ A)) = (ord_less_eq_real @ A @ B))))). % neg_le_iff_le
thf(fact_53_neg__less__iff__less, axiom,
    ((![B : real, A : real]: ((ord_less_real @ (uminus_uminus_real @ B) @ (uminus_uminus_real @ A)) = (ord_less_real @ A @ B))))). % neg_less_iff_less
thf(fact_54_neg__less__iff__less, axiom,
    ((![B : poly_real, A : poly_real]: ((ord_less_poly_real @ (uminus1613791741y_real @ B) @ (uminus1613791741y_real @ A)) = (ord_less_poly_real @ A @ B))))). % neg_less_iff_less
thf(fact_55_minus__diff__eq, axiom,
    ((![A : real, B : real]: ((uminus_uminus_real @ (minus_minus_real @ A @ B)) = (minus_minus_real @ B @ A))))). % minus_diff_eq
thf(fact_56_minus__diff__eq, axiom,
    ((![A : poly_real, B : poly_real]: ((uminus1613791741y_real @ (minus_240770701y_real @ A @ B)) = (minus_240770701y_real @ B @ A))))). % minus_diff_eq
thf(fact_57_minus__diff__eq, axiom,
    ((![A : poly_complex, B : poly_complex]: ((uminus1138659839omplex @ (minus_174331535omplex @ A @ B)) = (minus_174331535omplex @ B @ A))))). % minus_diff_eq
thf(fact_58_minus__diff__eq, axiom,
    ((![A : complex, B : complex]: ((uminus1204672759omplex @ (minus_minus_complex @ A @ B)) = (minus_minus_complex @ B @ A))))). % minus_diff_eq
thf(fact_59_abs__0__eq, axiom,
    ((![A : real]: ((zero_zero_real = (abs_abs_real @ A)) = (A = zero_zero_real))))). % abs_0_eq
thf(fact_60_abs__eq__0, axiom,
    ((![A : real]: (((abs_abs_real @ A) = zero_zero_real) = (A = zero_zero_real))))). % abs_eq_0
thf(fact_61_abs__zero, axiom,
    (((abs_abs_real @ zero_zero_real) = zero_zero_real))). % abs_zero
thf(fact_62_abs__minus__cancel, axiom,
    ((![A : real]: ((abs_abs_real @ (uminus_uminus_real @ A)) = (abs_abs_real @ A))))). % abs_minus_cancel
thf(fact_63_abs__minus__cancel, axiom,
    ((![A : poly_real]: ((abs_abs_poly_real @ (uminus1613791741y_real @ A)) = (abs_abs_poly_real @ A))))). % abs_minus_cancel
thf(fact_64_poly__0, axiom,
    ((![X : real]: ((poly_real2 @ zero_zero_poly_real @ X) = zero_zero_real)))). % poly_0
thf(fact_65_poly__0, axiom,
    ((![X : complex]: ((poly_complex2 @ zero_z1746442943omplex @ X) = zero_zero_complex)))). % poly_0
thf(fact_66_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_67_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_68_abs__norm__cancel, axiom,
    ((![A : complex]: ((abs_abs_real @ (real_V638595069omplex @ A)) = (real_V638595069omplex @ A))))). % abs_norm_cancel
thf(fact_69_abs__norm__cancel, axiom,
    ((![A : real]: ((abs_abs_real @ (real_V646646907m_real @ A)) = (real_V646646907m_real @ A))))). % abs_norm_cancel
thf(fact_70_mth2, axiom,
    ((?[Z3 : real]: (![X4 : real]: ((?[Za : complex]: ((ord_less_eq_real @ (real_V638595069omplex @ Za) @ r) & ((real_V638595069omplex @ (poly_complex2 @ p @ Za)) = (uminus_uminus_real @ X4)))) => (ord_less_real @ X4 @ Z3)))))). % mth2
thf(fact_71__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062s_O_A_092_060forall_062y_O_A_I_092_060exists_062x_O_A_I_092_060exists_062z_O_Acmod_Az_A_092_060le_062_Ar_A_092_060and_062_Acmod_A_Ipoly_Ap_Az_J_A_061_A_N_Ax_J_A_092_060and_062_Ay_A_060_Ax_J_A_061_A_Iy_A_060_As_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom,
    ((~ ((![S : real]: (~ ((![Y : real]: ((?[X2 : real]: (((?[Z2 : complex]: (((ord_less_eq_real @ (real_V638595069omplex @ Z2) @ r)) & (((real_V638595069omplex @ (poly_complex2 @ p @ Z2)) = (uminus_uminus_real @ X2)))))) & ((ord_less_real @ Y @ X2)))) = (ord_less_real @ Y @ S)))))))))). % \<open>\<And>thesis. (\<And>s. \<forall>y. (\<exists>x. (\<exists>z. cmod z \<le> r \<and> cmod (poly p z) = - x) \<and> y < x) = (y < s) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_72__092_060open_062_092_060exists_062s_O_A_092_060forall_062y_O_A_I_092_060exists_062x_O_A_I_092_060exists_062z_O_Acmod_Az_A_092_060le_062_Ar_A_092_060and_062_Acmod_A_Ipoly_Ap_Az_J_A_061_A_N_Ax_J_A_092_060and_062_Ay_A_060_Ax_J_A_061_A_Iy_A_060_As_J_092_060close_062, axiom,
    ((?[S : real]: (![Y : real]: ((?[X2 : real]: (((?[Z2 : complex]: (((ord_less_eq_real @ (real_V638595069omplex @ Z2) @ r)) & (((real_V638595069omplex @ (poly_complex2 @ p @ Z2)) = (uminus_uminus_real @ X2)))))) & ((ord_less_real @ Y @ X2)))) = (ord_less_real @ Y @ S)))))). % \<open>\<exists>s. \<forall>y. (\<exists>x. (\<exists>z. cmod z \<le> r \<and> cmod (poly p z) = - x) \<and> y < x) = (y < s)\<close>
thf(fact_73_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_74_diff__gt__0__iff__gt, axiom,
    ((![A : real, B : real]: ((ord_less_real @ zero_zero_real @ (minus_minus_real @ A @ B)) = (ord_less_real @ B @ A))))). % diff_gt_0_iff_gt
thf(fact_75_neg__less__eq__nonneg, axiom,
    ((![A : poly_real]: ((ord_le1180086932y_real @ (uminus1613791741y_real @ A) @ A) = (ord_le1180086932y_real @ zero_zero_poly_real @ A))))). % neg_less_eq_nonneg
thf(fact_76_neg__less__eq__nonneg, axiom,
    ((![A : real]: ((ord_less_eq_real @ (uminus_uminus_real @ A) @ A) = (ord_less_eq_real @ zero_zero_real @ A))))). % neg_less_eq_nonneg
thf(fact_77_less__eq__neg__nonpos, axiom,
    ((![A : poly_real]: ((ord_le1180086932y_real @ A @ (uminus1613791741y_real @ A)) = (ord_le1180086932y_real @ A @ zero_zero_poly_real))))). % less_eq_neg_nonpos
thf(fact_78_less__eq__neg__nonpos, axiom,
    ((![A : real]: ((ord_less_eq_real @ A @ (uminus_uminus_real @ A)) = (ord_less_eq_real @ A @ zero_zero_real))))). % less_eq_neg_nonpos
thf(fact_79_neg__le__0__iff__le, axiom,
    ((![A : poly_real]: ((ord_le1180086932y_real @ (uminus1613791741y_real @ A) @ zero_zero_poly_real) = (ord_le1180086932y_real @ zero_zero_poly_real @ A))))). % neg_le_0_iff_le
thf(fact_80_neg__le__0__iff__le, axiom,
    ((![A : real]: ((ord_less_eq_real @ (uminus_uminus_real @ A) @ zero_zero_real) = (ord_less_eq_real @ zero_zero_real @ A))))). % neg_le_0_iff_le
thf(fact_81_neg__0__le__iff__le, axiom,
    ((![A : poly_real]: ((ord_le1180086932y_real @ zero_zero_poly_real @ (uminus1613791741y_real @ A)) = (ord_le1180086932y_real @ A @ zero_zero_poly_real))))). % neg_0_le_iff_le
thf(fact_82_neg__0__le__iff__le, axiom,
    ((![A : real]: ((ord_less_eq_real @ zero_zero_real @ (uminus_uminus_real @ A)) = (ord_less_eq_real @ A @ zero_zero_real))))). % neg_0_le_iff_le
thf(fact_83_neg__less__0__iff__less, axiom,
    ((![A : real]: ((ord_less_real @ (uminus_uminus_real @ A) @ zero_zero_real) = (ord_less_real @ zero_zero_real @ A))))). % neg_less_0_iff_less
thf(fact_84_neg__less__0__iff__less, axiom,
    ((![A : poly_real]: ((ord_less_poly_real @ (uminus1613791741y_real @ A) @ zero_zero_poly_real) = (ord_less_poly_real @ zero_zero_poly_real @ A))))). % neg_less_0_iff_less
thf(fact_85_neg__0__less__iff__less, axiom,
    ((![A : real]: ((ord_less_real @ zero_zero_real @ (uminus_uminus_real @ A)) = (ord_less_real @ A @ zero_zero_real))))). % neg_0_less_iff_less
thf(fact_86_neg__0__less__iff__less, axiom,
    ((![A : poly_real]: ((ord_less_poly_real @ zero_zero_poly_real @ (uminus1613791741y_real @ A)) = (ord_less_poly_real @ A @ zero_zero_poly_real))))). % neg_0_less_iff_less
thf(fact_87_neg__less__pos, axiom,
    ((![A : real]: ((ord_less_real @ (uminus_uminus_real @ A) @ A) = (ord_less_real @ zero_zero_real @ A))))). % neg_less_pos
thf(fact_88_neg__less__pos, axiom,
    ((![A : poly_real]: ((ord_less_poly_real @ (uminus1613791741y_real @ A) @ A) = (ord_less_poly_real @ zero_zero_poly_real @ A))))). % neg_less_pos
thf(fact_89_less__neg__neg, axiom,
    ((![A : real]: ((ord_less_real @ A @ (uminus_uminus_real @ A)) = (ord_less_real @ A @ zero_zero_real))))). % less_neg_neg
thf(fact_90_less__neg__neg, axiom,
    ((![A : poly_real]: ((ord_less_poly_real @ A @ (uminus1613791741y_real @ A)) = (ord_less_poly_real @ A @ zero_zero_poly_real))))). % less_neg_neg
thf(fact_91_diff__0, axiom,
    ((![A : real]: ((minus_minus_real @ zero_zero_real @ A) = (uminus_uminus_real @ A))))). % diff_0
thf(fact_92_diff__0, axiom,
    ((![A : poly_real]: ((minus_240770701y_real @ zero_zero_poly_real @ A) = (uminus1613791741y_real @ A))))). % diff_0
thf(fact_93_diff__0, axiom,
    ((![A : poly_complex]: ((minus_174331535omplex @ zero_z1746442943omplex @ A) = (uminus1138659839omplex @ A))))). % diff_0
thf(fact_94_diff__0, axiom,
    ((![A : complex]: ((minus_minus_complex @ zero_zero_complex @ A) = (uminus1204672759omplex @ A))))). % diff_0
thf(fact_95_verit__minus__simplify_I3_J, axiom,
    ((![B : real]: ((minus_minus_real @ zero_zero_real @ B) = (uminus_uminus_real @ B))))). % verit_minus_simplify(3)
thf(fact_96_verit__minus__simplify_I3_J, axiom,
    ((![B : poly_real]: ((minus_240770701y_real @ zero_zero_poly_real @ B) = (uminus1613791741y_real @ B))))). % verit_minus_simplify(3)
thf(fact_97_verit__minus__simplify_I3_J, axiom,
    ((![B : poly_complex]: ((minus_174331535omplex @ zero_z1746442943omplex @ B) = (uminus1138659839omplex @ B))))). % verit_minus_simplify(3)
thf(fact_98_verit__minus__simplify_I3_J, axiom,
    ((![B : complex]: ((minus_minus_complex @ zero_zero_complex @ B) = (uminus1204672759omplex @ B))))). % verit_minus_simplify(3)
thf(fact_99_abs__of__nonneg, axiom,
    ((![A : real]: ((ord_less_eq_real @ zero_zero_real @ A) => ((abs_abs_real @ A) = A))))). % abs_of_nonneg
thf(fact_100_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_101_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_102_zero__less__abs__iff, axiom,
    ((![A : real]: ((ord_less_real @ zero_zero_real @ (abs_abs_real @ A)) = (~ ((A = zero_zero_real))))))). % zero_less_abs_iff
thf(fact_103_norm__eq__zero, axiom,
    ((![X : complex]: (((real_V638595069omplex @ X) = zero_zero_real) = (X = zero_zero_complex))))). % norm_eq_zero
thf(fact_104_norm__eq__zero, axiom,
    ((![X : real]: (((real_V646646907m_real @ X) = zero_zero_real) = (X = zero_zero_real))))). % norm_eq_zero
thf(fact_105_norm__zero, axiom,
    (((real_V638595069omplex @ zero_zero_complex) = zero_zero_real))). % norm_zero
thf(fact_106_norm__zero, axiom,
    (((real_V646646907m_real @ zero_zero_real) = zero_zero_real))). % norm_zero
thf(fact_107_abs__of__nonpos, axiom,
    ((![A : poly_real]: ((ord_le1180086932y_real @ A @ zero_zero_poly_real) => ((abs_abs_poly_real @ A) = (uminus1613791741y_real @ A)))))). % abs_of_nonpos
thf(fact_108_abs__of__nonpos, axiom,
    ((![A : real]: ((ord_less_eq_real @ A @ zero_zero_real) => ((abs_abs_real @ A) = (uminus_uminus_real @ A)))))). % abs_of_nonpos
thf(fact_109_zero__less__norm__iff, axiom,
    ((![X : complex]: ((ord_less_real @ zero_zero_real @ (real_V638595069omplex @ X)) = (~ ((X = zero_zero_complex))))))). % zero_less_norm_iff
thf(fact_110_zero__less__norm__iff, axiom,
    ((![X : real]: ((ord_less_real @ zero_zero_real @ (real_V646646907m_real @ X)) = (~ ((X = zero_zero_real))))))). % zero_less_norm_iff
thf(fact_111_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_112_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_113_verit__comp__simplify1_I3_J, axiom,
    ((![B2 : real, A2 : real]: ((~ ((ord_less_eq_real @ B2 @ A2))) = (ord_less_real @ A2 @ B2))))). % verit_comp_simplify1(3)
thf(fact_114_verit__comp__simplify1_I1_J, axiom,
    ((![A : real]: (~ ((ord_less_real @ A @ A)))))). % verit_comp_simplify1(1)
thf(fact_115_abs__if__raw, axiom,
    ((abs_abs_real = (^[A3 : real]: (if_real @ (ord_less_real @ A3 @ zero_zero_real) @ (uminus_uminus_real @ A3) @ A3))))). % abs_if_raw
thf(fact_116_abs__if__raw, axiom,
    ((abs_abs_poly_real = (^[A3 : poly_real]: (if_poly_real @ (ord_less_poly_real @ A3 @ zero_zero_poly_real) @ (uminus1613791741y_real @ A3) @ A3))))). % abs_if_raw
thf(fact_117_dense__eq0__I, axiom,
    ((![X : real]: ((![E : real]: ((ord_less_real @ zero_zero_real @ E) => (ord_less_eq_real @ (abs_abs_real @ X) @ E))) => (X = zero_zero_real))))). % dense_eq0_I
thf(fact_118_zero__reorient, axiom,
    ((![X : real]: ((zero_zero_real = X) = (X = zero_zero_real))))). % zero_reorient
thf(fact_119_zero__reorient, axiom,
    ((![X : complex]: ((zero_zero_complex = X) = (X = zero_zero_complex))))). % zero_reorient
thf(fact_120_poly__IVT__neg, axiom,
    ((![A : real, B : real, P : poly_real]: ((ord_less_real @ A @ B) => ((ord_less_real @ zero_zero_real @ (poly_real2 @ P @ A)) => ((ord_less_real @ (poly_real2 @ P @ B) @ zero_zero_real) => (?[X3 : real]: ((ord_less_real @ A @ X3) & ((ord_less_real @ X3 @ B) & ((poly_real2 @ P @ X3) = zero_zero_real)))))))))). % poly_IVT_neg
thf(fact_121_poly__IVT__pos, axiom,
    ((![A : real, B : real, P : poly_real]: ((ord_less_real @ A @ B) => ((ord_less_real @ (poly_real2 @ P @ A) @ zero_zero_real) => ((ord_less_real @ zero_zero_real @ (poly_real2 @ P @ B)) => (?[X3 : real]: ((ord_less_real @ A @ X3) & ((ord_less_real @ X3 @ B) & ((poly_real2 @ P @ X3) = zero_zero_real)))))))))). % poly_IVT_pos
thf(fact_122_verit__la__disequality, axiom,
    ((![A : real, B : real]: ((A = B) | ((~ ((ord_less_eq_real @ A @ B))) | (~ ((ord_less_eq_real @ B @ A)))))))). % verit_la_disequality
thf(fact_123_real__norm__def, axiom,
    ((real_V646646907m_real = abs_abs_real))). % real_norm_def
thf(fact_124_diff__eq__diff__eq, axiom,
    ((![A : real, B : real, C : real, D : real]: (((minus_minus_real @ A @ B) = (minus_minus_real @ C @ D)) => ((A = B) = (C = D)))))). % diff_eq_diff_eq
thf(fact_125_eq__iff__diff__eq__0, axiom,
    (((^[Y2 : complex]: (^[Z4 : complex]: (Y2 = Z4))) = (^[A3 : complex]: (^[B3 : complex]: ((minus_minus_complex @ A3 @ B3) = zero_zero_complex)))))). % eq_iff_diff_eq_0
thf(fact_126_eq__iff__diff__eq__0, axiom,
    (((^[Y2 : real]: (^[Z4 : real]: (Y2 = Z4))) = (^[A3 : real]: (^[B3 : real]: ((minus_minus_real @ A3 @ B3) = zero_zero_real)))))). % eq_iff_diff_eq_0
thf(fact_127_diff__mono, axiom,
    ((![A : real, B : real, D : real, C : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_eq_real @ D @ C) => (ord_less_eq_real @ (minus_minus_real @ A @ C) @ (minus_minus_real @ B @ D))))))). % diff_mono
thf(fact_128_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_129_abs__of__neg, axiom,
    ((![A : real]: ((ord_less_real @ A @ zero_zero_real) => ((abs_abs_real @ A) = (uminus_uminus_real @ A)))))). % abs_of_neg
thf(fact_130_abs__of__neg, axiom,
    ((![A : poly_real]: ((ord_less_poly_real @ A @ zero_zero_poly_real) => ((abs_abs_poly_real @ A) = (uminus1613791741y_real @ A)))))). % abs_of_neg
thf(fact_131_abs__of__pos, axiom,
    ((![A : real]: ((ord_less_real @ zero_zero_real @ A) => ((abs_abs_real @ A) = A))))). % abs_of_pos
thf(fact_132_diff__left__mono, axiom,
    ((![B : real, A : real, C : real]: ((ord_less_eq_real @ B @ A) => (ord_less_eq_real @ (minus_minus_real @ C @ A) @ (minus_minus_real @ C @ B)))))). % diff_left_mono
thf(fact_133_abs__ge__self, axiom,
    ((![A : real]: (ord_less_eq_real @ A @ (abs_abs_real @ A))))). % abs_ge_self
thf(fact_134_abs__ge__zero, axiom,
    ((![A : real]: (ord_less_eq_real @ zero_zero_real @ (abs_abs_real @ A))))). % abs_ge_zero
thf(fact_135_diff__right__mono, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_eq_real @ A @ B) => (ord_less_eq_real @ (minus_minus_real @ A @ C) @ (minus_minus_real @ B @ C)))))). % diff_right_mono
thf(fact_136_diff__strict__mono, axiom,
    ((![A : real, B : real, D : real, C : real]: ((ord_less_real @ A @ B) => ((ord_less_real @ D @ C) => (ord_less_real @ (minus_minus_real @ A @ C) @ (minus_minus_real @ B @ D))))))). % diff_strict_mono
thf(fact_137_le__iff__diff__le__0, axiom,
    ((ord_less_eq_real = (^[A3 : real]: (^[B3 : real]: (ord_less_eq_real @ (minus_minus_real @ A3 @ B3) @ zero_zero_real)))))). % le_iff_diff_le_0
thf(fact_138_diff__eq__diff__less, axiom,
    ((![A : real, B : real, C : real, D : real]: (((minus_minus_real @ A @ B) = (minus_minus_real @ C @ D)) => ((ord_less_real @ A @ B) = (ord_less_real @ C @ D)))))). % diff_eq_diff_less
thf(fact_139_diff__eq__diff__less__eq, axiom,
    ((![A : real, B : real, C : real, D : real]: (((minus_minus_real @ A @ B) = (minus_minus_real @ C @ D)) => ((ord_less_eq_real @ A @ B) = (ord_less_eq_real @ C @ D)))))). % diff_eq_diff_less_eq
thf(fact_140_less__iff__diff__less__0, axiom,
    ((ord_less_real = (^[A3 : real]: (^[B3 : real]: (ord_less_real @ (minus_minus_real @ A3 @ B3) @ zero_zero_real)))))). % less_iff_diff_less_0
thf(fact_141_cancel__ab__semigroup__add__class_Odiff__right__commute, axiom,
    ((![A : real, C : real, B : real]: ((minus_minus_real @ (minus_minus_real @ A @ C) @ B) = (minus_minus_real @ (minus_minus_real @ A @ B) @ C))))). % cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_142_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_143_abs__minus__le__zero, axiom,
    ((![A : poly_real]: (ord_le1180086932y_real @ (uminus1613791741y_real @ (abs_abs_poly_real @ A)) @ zero_zero_poly_real)))). % abs_minus_le_zero
thf(fact_144_abs__minus__le__zero, axiom,
    ((![A : real]: (ord_less_eq_real @ (uminus_uminus_real @ (abs_abs_real @ A)) @ zero_zero_real)))). % abs_minus_le_zero
thf(fact_145_abs__not__less__zero, axiom,
    ((![A : real]: (~ ((ord_less_real @ (abs_abs_real @ A) @ zero_zero_real)))))). % abs_not_less_zero
thf(fact_146_diff__strict__left__mono, axiom,
    ((![B : real, A : real, C : real]: ((ord_less_real @ B @ A) => (ord_less_real @ (minus_minus_real @ C @ A) @ (minus_minus_real @ C @ B)))))). % diff_strict_left_mono
thf(fact_147_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_148_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_149_diff__strict__right__mono, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_real @ A @ B) => (ord_less_real @ (minus_minus_real @ A @ C) @ (minus_minus_real @ B @ C)))))). % diff_strict_right_mono
thf(fact_150_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_151_norm__triangle__ineq3, axiom,
    ((![A : complex, B : complex]: (ord_less_eq_real @ (abs_abs_real @ (minus_minus_real @ (real_V638595069omplex @ A) @ (real_V638595069omplex @ B))) @ (real_V638595069omplex @ (minus_minus_complex @ A @ B)))))). % norm_triangle_ineq3
thf(fact_152_norm__triangle__ineq3, axiom,
    ((![A : real, B : real]: (ord_less_eq_real @ (abs_abs_real @ (minus_minus_real @ (real_V646646907m_real @ A) @ (real_V646646907m_real @ B))) @ (real_V646646907m_real @ (minus_minus_real @ A @ B)))))). % norm_triangle_ineq3
thf(fact_153_real__sup__exists, axiom,
    ((![P2 : real > $o]: ((?[X_1 : real]: (P2 @ X_1)) => ((?[Z5 : real]: (![X3 : real]: ((P2 @ X3) => (ord_less_real @ X3 @ Z5)))) => (?[S : real]: (![Y : real]: ((?[X2 : real]: (((P2 @ X2)) & ((ord_less_real @ Y @ X2)))) = (ord_less_real @ Y @ S))))))))). % real_sup_exists
thf(fact_154_norm__triangle__ineq2, axiom,
    ((![A : complex, B : complex]: (ord_less_eq_real @ (minus_minus_real @ (real_V638595069omplex @ A) @ (real_V638595069omplex @ B)) @ (real_V638595069omplex @ (minus_minus_complex @ A @ B)))))). % norm_triangle_ineq2
thf(fact_155_norm__triangle__ineq2, axiom,
    ((![A : real, B : real]: (ord_less_eq_real @ (minus_minus_real @ (real_V646646907m_real @ A) @ (real_V646646907m_real @ B)) @ (real_V646646907m_real @ (minus_minus_real @ A @ B)))))). % norm_triangle_ineq2
thf(fact_156_abs__ge__minus__self, axiom,
    ((![A : poly_real]: (ord_le1180086932y_real @ (uminus1613791741y_real @ A) @ (abs_abs_poly_real @ A))))). % abs_ge_minus_self
thf(fact_157_abs__ge__minus__self, axiom,
    ((![A : real]: (ord_less_eq_real @ (uminus_uminus_real @ A) @ (abs_abs_real @ A))))). % abs_ge_minus_self
thf(fact_158_abs__le__iff, axiom,
    ((![A : poly_real, B : poly_real]: ((ord_le1180086932y_real @ (abs_abs_poly_real @ A) @ B) = (((ord_le1180086932y_real @ A @ B)) & ((ord_le1180086932y_real @ (uminus1613791741y_real @ A) @ B))))))). % abs_le_iff
thf(fact_159_abs__le__iff, axiom,
    ((![A : real, B : real]: ((ord_less_eq_real @ (abs_abs_real @ A) @ B) = (((ord_less_eq_real @ A @ B)) & ((ord_less_eq_real @ (uminus_uminus_real @ A) @ B))))))). % abs_le_iff
thf(fact_160_abs__le__D2, axiom,
    ((![A : poly_real, B : poly_real]: ((ord_le1180086932y_real @ (abs_abs_poly_real @ A) @ B) => (ord_le1180086932y_real @ (uminus1613791741y_real @ A) @ B))))). % abs_le_D2
thf(fact_161_abs__le__D2, axiom,
    ((![A : real, B : real]: ((ord_less_eq_real @ (abs_abs_real @ A) @ B) => (ord_less_eq_real @ (uminus_uminus_real @ A) @ B))))). % abs_le_D2
thf(fact_162_abs__leI, axiom,
    ((![A : poly_real, B : poly_real]: ((ord_le1180086932y_real @ A @ B) => ((ord_le1180086932y_real @ (uminus1613791741y_real @ A) @ B) => (ord_le1180086932y_real @ (abs_abs_poly_real @ A) @ B)))))). % abs_leI
thf(fact_163_abs__leI, axiom,
    ((![A : real, B : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_eq_real @ (uminus_uminus_real @ A) @ B) => (ord_less_eq_real @ (abs_abs_real @ A) @ B)))))). % abs_leI
thf(fact_164_norm__not__less__zero, axiom,
    ((![X : complex]: (~ ((ord_less_real @ (real_V638595069omplex @ X) @ zero_zero_real)))))). % norm_not_less_zero
thf(fact_165_norm__not__less__zero, axiom,
    ((![X : real]: (~ ((ord_less_real @ (real_V646646907m_real @ X) @ zero_zero_real)))))). % norm_not_less_zero
thf(fact_166_norm__ge__zero, axiom,
    ((![X : complex]: (ord_less_eq_real @ zero_zero_real @ (real_V638595069omplex @ X))))). % norm_ge_zero
thf(fact_167_norm__ge__zero, axiom,
    ((![X : real]: (ord_less_eq_real @ zero_zero_real @ (real_V646646907m_real @ X))))). % norm_ge_zero
thf(fact_168_poly__all__0__iff__0, axiom,
    ((![P : poly_real]: ((![X2 : real]: ((poly_real2 @ P @ X2) = zero_zero_real)) = (P = zero_zero_poly_real))))). % poly_all_0_iff_0
thf(fact_169_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_170_minus__less__iff, axiom,
    ((![A : real, B : real]: ((ord_less_real @ (uminus_uminus_real @ A) @ B) = (ord_less_real @ (uminus_uminus_real @ B) @ A))))). % minus_less_iff
thf(fact_171_minus__less__iff, axiom,
    ((![A : poly_real, B : poly_real]: ((ord_less_poly_real @ (uminus1613791741y_real @ A) @ B) = (ord_less_poly_real @ (uminus1613791741y_real @ B) @ A))))). % minus_less_iff
thf(fact_172_less__minus__iff, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ (uminus_uminus_real @ B)) = (ord_less_real @ B @ (uminus_uminus_real @ A)))))). % less_minus_iff
thf(fact_173_less__minus__iff, axiom,
    ((![A : poly_real, B : poly_real]: ((ord_less_poly_real @ A @ (uminus1613791741y_real @ B)) = (ord_less_poly_real @ B @ (uminus1613791741y_real @ A)))))). % less_minus_iff
thf(fact_174_verit__negate__coefficient_I2_J, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ B) => (ord_less_real @ (uminus_uminus_real @ B) @ (uminus_uminus_real @ A)))))). % verit_negate_coefficient(2)
thf(fact_175_verit__negate__coefficient_I2_J, axiom,
    ((![A : poly_real, B : poly_real]: ((ord_less_poly_real @ A @ B) => (ord_less_poly_real @ (uminus1613791741y_real @ B) @ (uminus1613791741y_real @ A)))))). % verit_negate_coefficient(2)
thf(fact_176_le__imp__neg__le, axiom,
    ((![A : poly_real, B : poly_real]: ((ord_le1180086932y_real @ A @ B) => (ord_le1180086932y_real @ (uminus1613791741y_real @ B) @ (uminus1613791741y_real @ A)))))). % le_imp_neg_le
thf(fact_177_le__imp__neg__le, axiom,
    ((![A : real, B : real]: ((ord_less_eq_real @ A @ B) => (ord_less_eq_real @ (uminus_uminus_real @ B) @ (uminus_uminus_real @ A)))))). % le_imp_neg_le
thf(fact_178_minus__le__iff, axiom,
    ((![A : poly_real, B : poly_real]: ((ord_le1180086932y_real @ (uminus1613791741y_real @ A) @ B) = (ord_le1180086932y_real @ (uminus1613791741y_real @ B) @ A))))). % minus_le_iff
thf(fact_179_minus__le__iff, axiom,
    ((![A : real, B : real]: ((ord_less_eq_real @ (uminus_uminus_real @ A) @ B) = (ord_less_eq_real @ (uminus_uminus_real @ B) @ A))))). % minus_le_iff
thf(fact_180_le__minus__iff, axiom,
    ((![A : poly_real, B : poly_real]: ((ord_le1180086932y_real @ A @ (uminus1613791741y_real @ B)) = (ord_le1180086932y_real @ B @ (uminus1613791741y_real @ A)))))). % le_minus_iff
thf(fact_181_le__minus__iff, axiom,
    ((![A : real, B : real]: ((ord_less_eq_real @ A @ (uminus_uminus_real @ B)) = (ord_less_eq_real @ B @ (uminus_uminus_real @ A)))))). % le_minus_iff
thf(fact_182_poly__bound__exists, axiom,
    ((![R : real, P : poly_complex]: (?[M : real]: ((ord_less_real @ zero_zero_real @ M) & (![Z5 : complex]: ((ord_less_eq_real @ (real_V638595069omplex @ Z5) @ R) => (ord_less_eq_real @ (real_V638595069omplex @ (poly_complex2 @ P @ Z5)) @ M)))))))). % poly_bound_exists
thf(fact_183_poly__bound__exists, axiom,
    ((![R : real, P : poly_real]: (?[M : real]: ((ord_less_real @ zero_zero_real @ M) & (![Z5 : real]: ((ord_less_eq_real @ (real_V646646907m_real @ Z5) @ R) => (ord_less_eq_real @ (real_V646646907m_real @ (poly_real2 @ P @ Z5)) @ M)))))))). % poly_bound_exists
thf(fact_184_poly__cont, axiom,
    ((![E2 : real, Z : complex, P : poly_complex]: ((ord_less_real @ zero_zero_real @ E2) => (?[D2 : real]: ((ord_less_real @ zero_zero_real @ D2) & (![W : complex]: (((ord_less_real @ zero_zero_real @ (real_V638595069omplex @ (minus_minus_complex @ W @ Z))) & (ord_less_real @ (real_V638595069omplex @ (minus_minus_complex @ W @ Z)) @ D2)) => (ord_less_real @ (real_V638595069omplex @ (minus_minus_complex @ (poly_complex2 @ P @ W) @ (poly_complex2 @ P @ Z))) @ E2))))))))). % poly_cont
thf(fact_185_poly__cont, axiom,
    ((![E2 : real, Z : real, P : poly_real]: ((ord_less_real @ zero_zero_real @ E2) => (?[D2 : real]: ((ord_less_real @ zero_zero_real @ D2) & (![W : real]: (((ord_less_real @ zero_zero_real @ (real_V646646907m_real @ (minus_minus_real @ W @ Z))) & (ord_less_real @ (real_V646646907m_real @ (minus_minus_real @ W @ Z)) @ D2)) => (ord_less_real @ (real_V646646907m_real @ (minus_minus_real @ (poly_real2 @ P @ W) @ (poly_real2 @ P @ Z))) @ E2))))))))). % poly_cont
thf(fact_186_minus__diff__commute, axiom,
    ((![B : real, A : real]: ((minus_minus_real @ (uminus_uminus_real @ B) @ A) = (minus_minus_real @ (uminus_uminus_real @ A) @ B))))). % minus_diff_commute
thf(fact_187_minus__diff__commute, axiom,
    ((![B : poly_real, A : poly_real]: ((minus_240770701y_real @ (uminus1613791741y_real @ B) @ A) = (minus_240770701y_real @ (uminus1613791741y_real @ A) @ B))))). % minus_diff_commute
thf(fact_188_minus__diff__commute, axiom,
    ((![B : poly_complex, A : poly_complex]: ((minus_174331535omplex @ (uminus1138659839omplex @ B) @ A) = (minus_174331535omplex @ (uminus1138659839omplex @ A) @ B))))). % minus_diff_commute
thf(fact_189_minus__diff__commute, axiom,
    ((![B : complex, A : complex]: ((minus_minus_complex @ (uminus1204672759omplex @ B) @ A) = (minus_minus_complex @ (uminus1204672759omplex @ A) @ B))))). % minus_diff_commute
thf(fact_190_norm__minus__commute, axiom,
    ((![A : complex, B : complex]: ((real_V638595069omplex @ (minus_minus_complex @ A @ B)) = (real_V638595069omplex @ (minus_minus_complex @ B @ A)))))). % norm_minus_commute
thf(fact_191_norm__minus__commute, axiom,
    ((![A : real, B : real]: ((real_V646646907m_real @ (minus_minus_real @ A @ B)) = (real_V646646907m_real @ (minus_minus_real @ B @ A)))))). % norm_minus_commute
thf(fact_192_minus__equation__iff, axiom,
    ((![A : real, B : real]: (((uminus_uminus_real @ A) = B) = ((uminus_uminus_real @ B) = A))))). % minus_equation_iff
thf(fact_193_minus__equation__iff, axiom,
    ((![A : poly_real, B : poly_real]: (((uminus1613791741y_real @ A) = B) = ((uminus1613791741y_real @ B) = A))))). % minus_equation_iff
thf(fact_194_minus__equation__iff, axiom,
    ((![A : poly_complex, B : poly_complex]: (((uminus1138659839omplex @ A) = B) = ((uminus1138659839omplex @ B) = A))))). % minus_equation_iff
thf(fact_195_minus__equation__iff, axiom,
    ((![A : complex, B : complex]: (((uminus1204672759omplex @ A) = B) = ((uminus1204672759omplex @ B) = A))))). % minus_equation_iff
thf(fact_196_equation__minus__iff, axiom,
    ((![A : real, B : real]: ((A = (uminus_uminus_real @ B)) = (B = (uminus_uminus_real @ A)))))). % equation_minus_iff
thf(fact_197_equation__minus__iff, axiom,
    ((![A : poly_real, B : poly_real]: ((A = (uminus1613791741y_real @ B)) = (B = (uminus1613791741y_real @ A)))))). % equation_minus_iff
thf(fact_198_equation__minus__iff, axiom,
    ((![A : poly_complex, B : poly_complex]: ((A = (uminus1138659839omplex @ B)) = (B = (uminus1138659839omplex @ A)))))). % equation_minus_iff
thf(fact_199_equation__minus__iff, axiom,
    ((![A : complex, B : complex]: ((A = (uminus1204672759omplex @ B)) = (B = (uminus1204672759omplex @ A)))))). % equation_minus_iff
thf(fact_200_verit__negate__coefficient_I3_J, axiom,
    ((![A : real, B : real]: ((A = B) => ((uminus_uminus_real @ A) = (uminus_uminus_real @ B)))))). % verit_negate_coefficient(3)
thf(fact_201_verit__negate__coefficient_I3_J, axiom,
    ((![A : poly_real, B : poly_real]: ((A = B) => ((uminus1613791741y_real @ A) = (uminus1613791741y_real @ B)))))). % verit_negate_coefficient(3)
thf(fact_202_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_203_poly__eq__poly__eq__iff, axiom,
    ((![P : poly_real, Q : poly_real]: (((poly_real2 @ P) = (poly_real2 @ Q)) = (P = Q))))). % poly_eq_poly_eq_iff
thf(fact_204__092_060open_062_092_060And_062z_Ax_O_A_092_060lbrakk_062cmod_Az_A_092_060le_062_Ar_059_Acmod_A_Ipoly_Ap_Az_J_A_061_A_N_Ax_059_A_092_060not_062_Ax_A_060_A1_092_060rbrakk_062_A_092_060Longrightarrow_062_AFalse_092_060close_062, axiom,
    ((![Z : complex, X : real]: ((ord_less_eq_real @ (real_V638595069omplex @ Z) @ r) => (((real_V638595069omplex @ (poly_complex2 @ p @ Z)) = (uminus_uminus_real @ X)) => (ord_less_real @ X @ one_one_real)))))). % \<open>\<And>z x. \<lbrakk>cmod z \<le> r; cmod (poly p z) = - x; \<not> x < 1\<rbrakk> \<Longrightarrow> False\<close>
thf(fact_205__092_060open_062cmod_A0_A_092_060le_062_Ar_A_092_060and_062_Acmod_A_Ipoly_Ap_A0_J_A_061_A_N_A_I_N_Acmod_A_Ipoly_Ap_A0_J_J_092_060close_062, axiom,
    (((ord_less_eq_real @ (real_V638595069omplex @ zero_zero_complex) @ r) & ((real_V638595069omplex @ (poly_complex2 @ p @ zero_zero_complex)) = (uminus_uminus_real @ (uminus_uminus_real @ (real_V638595069omplex @ (poly_complex2 @ p @ zero_zero_complex)))))))). % \<open>cmod 0 \<le> r \<and> cmod (poly p 0) = - (- cmod (poly p 0))\<close>
thf(fact_206_lemma__interval, axiom,
    ((![A : real, X : real, B : real]: ((ord_less_real @ A @ X) => ((ord_less_real @ X @ B) => (?[D2 : real]: ((ord_less_real @ zero_zero_real @ D2) & (![Y : real]: ((ord_less_real @ (abs_abs_real @ (minus_minus_real @ X @ Y)) @ D2) => ((ord_less_eq_real @ A @ Y) & (ord_less_eq_real @ Y @ B))))))))))). % lemma_interval
thf(fact_207_abs__minus, axiom,
    ((![A : real]: ((abs_abs_real @ (uminus_uminus_real @ A)) = (abs_abs_real @ A))))). % abs_minus
thf(fact_208_abs__minus, axiom,
    ((![A : poly_real]: ((abs_abs_poly_real @ (uminus1613791741y_real @ A)) = (abs_abs_poly_real @ A))))). % abs_minus
thf(fact_209_abs__minus, axiom,
    ((![A : complex]: ((abs_abs_complex @ (uminus1204672759omplex @ A)) = (abs_abs_complex @ A))))). % abs_minus
thf(fact_210_abs__0, axiom,
    (((abs_abs_real @ zero_zero_real) = zero_zero_real))). % abs_0
thf(fact_211_abs__0, axiom,
    (((abs_abs_complex @ zero_zero_complex) = zero_zero_complex))). % abs_0
thf(fact_212_lemma__interval__lt, axiom,
    ((![A : real, X : real, B : real]: ((ord_less_real @ A @ X) => ((ord_less_real @ X @ B) => (?[D2 : real]: ((ord_less_real @ zero_zero_real @ D2) & (![Y : real]: ((ord_less_real @ (abs_abs_real @ (minus_minus_real @ X @ Y)) @ D2) => ((ord_less_real @ A @ Y) & (ord_less_real @ Y @ B))))))))))). % lemma_interval_lt
thf(fact_213_abs__real__def, axiom,
    ((abs_abs_real = (^[A3 : real]: (if_real @ (ord_less_real @ A3 @ zero_zero_real) @ (uminus_uminus_real @ A3) @ A3))))). % abs_real_def
thf(fact_214_Bolzano, axiom,
    ((![A : real, B : real, P2 : real > real > $o]: ((ord_less_eq_real @ A @ B) => ((![A4 : real, B4 : real, C2 : real]: ((P2 @ A4 @ B4) => ((P2 @ B4 @ C2) => ((ord_less_eq_real @ A4 @ B4) => ((ord_less_eq_real @ B4 @ C2) => (P2 @ A4 @ C2)))))) => ((![X3 : real]: ((ord_less_eq_real @ A @ X3) => ((ord_less_eq_real @ X3 @ B) => (?[D3 : real]: ((ord_less_real @ zero_zero_real @ D3) & (![A4 : real, B4 : real]: (((ord_less_eq_real @ A4 @ X3) & ((ord_less_eq_real @ X3 @ B4) & (ord_less_real @ (minus_minus_real @ B4 @ A4) @ D3))) => (P2 @ A4 @ B4)))))))) => (P2 @ A @ B))))))). % Bolzano
thf(fact_215_abs__if, axiom,
    ((abs_abs_real = (^[A3 : real]: (if_real @ (ord_less_real @ A3 @ zero_zero_real) @ (uminus_uminus_real @ A3) @ A3))))). % abs_if
thf(fact_216_abs__if, axiom,
    ((abs_abs_poly_real = (^[A3 : poly_real]: (if_poly_real @ (ord_less_poly_real @ A3 @ zero_zero_poly_real) @ (uminus1613791741y_real @ A3) @ A3))))). % abs_if
thf(fact_217_abs__eq__iff_H, axiom,
    ((![A : poly_real, B : poly_real]: (((abs_abs_poly_real @ A) = B) = (((ord_le1180086932y_real @ zero_zero_poly_real @ B)) & ((((A = B)) | ((A = (uminus1613791741y_real @ B)))))))))). % abs_eq_iff'
thf(fact_218_abs__eq__iff_H, axiom,
    ((![A : real, B : real]: (((abs_abs_real @ A) = B) = (((ord_less_eq_real @ zero_zero_real @ B)) & ((((A = B)) | ((A = (uminus_uminus_real @ B)))))))))). % abs_eq_iff'

% Helper facts (5)
thf(help_If_2_1_If_001t__Real__Oreal_T, axiom,
    ((![X : real, Y3 : real]: ((if_real @ $false @ X @ Y3) = Y3)))).
thf(help_If_1_1_If_001t__Real__Oreal_T, axiom,
    ((![X : real, Y3 : real]: ((if_real @ $true @ X @ Y3) = X)))).
thf(help_If_3_1_If_001t__Polynomial__Opoly_It__Real__Oreal_J_T, axiom,
    ((![P2 : $o]: ((P2 = $true) | (P2 = $false))))).
thf(help_If_2_1_If_001t__Polynomial__Opoly_It__Real__Oreal_J_T, axiom,
    ((![X : poly_real, Y3 : poly_real]: ((if_poly_real @ $false @ X @ Y3) = Y3)))).
thf(help_If_1_1_If_001t__Polynomial__Opoly_It__Real__Oreal_J_T, axiom,
    ((![X : poly_real, Y3 : poly_real]: ((if_poly_real @ $true @ X @ Y3) = X)))).

% Conjectures (1)
thf(conj_0, conjecture,
    (((real_V638595069omplex @ (poly_complex2 @ p @ z)) = (uminus_uminus_real @ s)))).
