% 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_461__5372420_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:30:19.011

% 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__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__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 (46)
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__Polynomial__Opoly_It__Complex__Ocomplex_J_J, type,
    minus_1169194391omplex : poly_poly_complex > poly_poly_complex > poly_poly_complex).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J, type,
    minus_181436949y_real : poly_poly_real > poly_poly_real > poly_poly_real).
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_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__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__Polynomial__Opoly_It__Complex__Ocomplex_J_J, type,
    zero_z1040703943omplex : poly_poly_complex).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J, type,
    zero_z1423781445y_real : poly_poly_real).
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__Nat__Onat, type,
    ord_less_nat : nat > nat > $o).
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__Nat__Onat, type,
    ord_less_eq_nat : nat > nat > $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_Orderings_Oorder__class_Ostrict__mono_001t__Nat__Onat_001t__Nat__Onat, type,
    order_769474267at_nat : (nat > nat) > $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_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_f____, type,
    f : nat > nat).
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 (246)
thf(fact_0__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_1_norm__zero, axiom,
    (((real_V638595069omplex @ zero_zero_complex) = zero_zero_real))). % norm_zero
thf(fact_2_norm__zero, axiom,
    (((real_V646646907m_real @ zero_zero_real) = zero_zero_real))). % norm_zero
thf(fact_3_norm__eq__zero, axiom,
    ((![X : complex]: (((real_V638595069omplex @ X) = zero_zero_real) = (X = zero_zero_complex))))). % norm_eq_zero
thf(fact_4_norm__eq__zero, axiom,
    ((![X : real]: (((real_V646646907m_real @ X) = zero_zero_real) = (X = zero_zero_real))))). % norm_eq_zero
thf(fact_5_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_6_verit__minus__simplify_I3_J, axiom,
    ((![B : poly_complex]: ((minus_174331535omplex @ zero_z1746442943omplex @ B) = (uminus1138659839omplex @ B))))). % verit_minus_simplify(3)
thf(fact_7_verit__minus__simplify_I3_J, axiom,
    ((![B : complex]: ((minus_minus_complex @ zero_zero_complex @ B) = (uminus1204672759omplex @ B))))). % verit_minus_simplify(3)
thf(fact_8_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_9_diff__0, axiom,
    ((![A : poly_real]: ((minus_240770701y_real @ zero_zero_poly_real @ A) = (uminus1613791741y_real @ A))))). % diff_0
thf(fact_10_diff__0, axiom,
    ((![A : poly_complex]: ((minus_174331535omplex @ zero_z1746442943omplex @ A) = (uminus1138659839omplex @ A))))). % diff_0
thf(fact_11_diff__0, axiom,
    ((![A : complex]: ((minus_minus_complex @ zero_zero_complex @ A) = (uminus1204672759omplex @ A))))). % diff_0
thf(fact_12_diff__0, axiom,
    ((![A : real]: ((minus_minus_real @ zero_zero_real @ A) = (uminus_uminus_real @ A))))). % diff_0
thf(fact_13_abs__norm__cancel, axiom,
    ((![A : complex]: ((abs_abs_real @ (real_V638595069omplex @ A)) = (real_V638595069omplex @ A))))). % abs_norm_cancel
thf(fact_14_abs__norm__cancel, axiom,
    ((![A : real]: ((abs_abs_real @ (real_V646646907m_real @ A)) = (real_V646646907m_real @ A))))). % abs_norm_cancel
thf(fact_15_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_16_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_17_poly__minus, axiom,
    ((![P : poly_complex, X : complex]: ((poly_complex2 @ (uminus1138659839omplex @ P) @ X) = (uminus1204672759omplex @ (poly_complex2 @ P @ X)))))). % poly_minus
thf(fact_18_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_19_poly__diff, axiom,
    ((![P : poly_poly_complex, Q : poly_poly_complex, X : poly_complex]: ((poly_poly_complex2 @ (minus_1169194391omplex @ P @ Q) @ X) = (minus_174331535omplex @ (poly_poly_complex2 @ P @ X) @ (poly_poly_complex2 @ Q @ X)))))). % poly_diff
thf(fact_20_poly__diff, axiom,
    ((![P : poly_poly_real, Q : poly_poly_real, X : poly_real]: ((poly_poly_real2 @ (minus_181436949y_real @ P @ Q) @ X) = (minus_240770701y_real @ (poly_poly_real2 @ P @ X) @ (poly_poly_real2 @ Q @ X)))))). % poly_diff
thf(fact_21_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_22_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_23_poly__0, axiom,
    ((![X : poly_complex]: ((poly_poly_complex2 @ zero_z1040703943omplex @ X) = zero_z1746442943omplex)))). % poly_0
thf(fact_24_poly__0, axiom,
    ((![X : poly_real]: ((poly_poly_real2 @ zero_z1423781445y_real @ X) = zero_zero_poly_real)))). % poly_0
thf(fact_25_poly__0, axiom,
    ((![X : real]: ((poly_real2 @ zero_zero_poly_real @ X) = zero_zero_real)))). % poly_0
thf(fact_26_poly__0, axiom,
    ((![X : complex]: ((poly_complex2 @ zero_z1746442943omplex @ X) = zero_zero_complex)))). % poly_0
thf(fact_27_poly__cont, axiom,
    ((![E : real, Z : complex, P : poly_complex]: ((ord_less_real @ zero_zero_real @ E) => (?[D : real]: ((ord_less_real @ zero_zero_real @ D) & (![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)) @ D)) => (ord_less_real @ (real_V638595069omplex @ (minus_minus_complex @ (poly_complex2 @ P @ W) @ (poly_complex2 @ P @ Z))) @ E))))))))). % poly_cont
thf(fact_28_poly__cont, axiom,
    ((![E : real, Z : real, P : poly_real]: ((ord_less_real @ zero_zero_real @ E) => (?[D : real]: ((ord_less_real @ zero_zero_real @ D) & (![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)) @ D)) => (ord_less_real @ (real_V646646907m_real @ (minus_minus_real @ (poly_real2 @ P @ W) @ (poly_real2 @ P @ Z))) @ E))))))))). % poly_cont
thf(fact_29_norm__minus__cancel, axiom,
    ((![X : complex]: ((real_V638595069omplex @ (uminus1204672759omplex @ X)) = (real_V638595069omplex @ X))))). % norm_minus_cancel
thf(fact_30_norm__minus__cancel, axiom,
    ((![X : real]: ((real_V646646907m_real @ (uminus_uminus_real @ X)) = (real_V646646907m_real @ X))))). % norm_minus_cancel
thf(fact_31_abs__minus, axiom,
    ((![A : real]: ((abs_abs_real @ (uminus_uminus_real @ A)) = (abs_abs_real @ A))))). % abs_minus
thf(fact_32_abs__minus, axiom,
    ((![A : complex]: ((abs_abs_complex @ (uminus1204672759omplex @ A)) = (abs_abs_complex @ A))))). % abs_minus
thf(fact_33_abs__minus, axiom,
    ((![A : poly_real]: ((abs_abs_poly_real @ (uminus1613791741y_real @ A)) = (abs_abs_poly_real @ A))))). % abs_minus
thf(fact_34_abs__minus__cancel, axiom,
    ((![A : real]: ((abs_abs_real @ (uminus_uminus_real @ A)) = (abs_abs_real @ A))))). % abs_minus_cancel
thf(fact_35_abs__minus__cancel, axiom,
    ((![A : poly_real]: ((abs_abs_poly_real @ (uminus1613791741y_real @ A)) = (abs_abs_poly_real @ A))))). % abs_minus_cancel
thf(fact_36_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_37_neg__equal__iff__equal, axiom,
    ((![A : complex, B : complex]: (((uminus1204672759omplex @ A) = (uminus1204672759omplex @ B)) = (A = B))))). % neg_equal_iff_equal
thf(fact_38_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_39_neg__equal__iff__equal, axiom,
    ((![A : poly_complex, B : poly_complex]: (((uminus1138659839omplex @ A) = (uminus1138659839omplex @ B)) = (A = B))))). % neg_equal_iff_equal
thf(fact_40_add_Oinverse__inverse, axiom,
    ((![A : real]: ((uminus_uminus_real @ (uminus_uminus_real @ A)) = A)))). % add.inverse_inverse
thf(fact_41_add_Oinverse__inverse, axiom,
    ((![A : complex]: ((uminus1204672759omplex @ (uminus1204672759omplex @ A)) = A)))). % add.inverse_inverse
thf(fact_42_add_Oinverse__inverse, axiom,
    ((![A : poly_real]: ((uminus1613791741y_real @ (uminus1613791741y_real @ A)) = A)))). % add.inverse_inverse
thf(fact_43_add_Oinverse__inverse, axiom,
    ((![A : poly_complex]: ((uminus1138659839omplex @ (uminus1138659839omplex @ A)) = A)))). % add.inverse_inverse
thf(fact_44_verit__minus__simplify_I4_J, axiom,
    ((![B : real]: ((uminus_uminus_real @ (uminus_uminus_real @ B)) = B)))). % verit_minus_simplify(4)
thf(fact_45_verit__minus__simplify_I4_J, axiom,
    ((![B : complex]: ((uminus1204672759omplex @ (uminus1204672759omplex @ B)) = B)))). % verit_minus_simplify(4)
thf(fact_46_verit__minus__simplify_I4_J, axiom,
    ((![B : poly_real]: ((uminus1613791741y_real @ (uminus1613791741y_real @ B)) = B)))). % verit_minus_simplify(4)
thf(fact_47_verit__minus__simplify_I4_J, axiom,
    ((![B : poly_complex]: ((uminus1138659839omplex @ (uminus1138659839omplex @ B)) = B)))). % verit_minus_simplify(4)
thf(fact_48_abs__idempotent, axiom,
    ((![A : real]: ((abs_abs_real @ (abs_abs_real @ A)) = (abs_abs_real @ A))))). % abs_idempotent
thf(fact_49_abs__abs, axiom,
    ((![A : real]: ((abs_abs_real @ (abs_abs_real @ A)) = (abs_abs_real @ A))))). % abs_abs
thf(fact_50_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_51_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_52_cancel__comm__monoid__add__class_Odiff__cancel, axiom,
    ((![A : poly_complex]: ((minus_174331535omplex @ A @ A) = zero_z1746442943omplex)))). % cancel_comm_monoid_add_class.diff_cancel
thf(fact_53_cancel__comm__monoid__add__class_Odiff__cancel, axiom,
    ((![A : poly_real]: ((minus_240770701y_real @ A @ A) = zero_zero_poly_real)))). % cancel_comm_monoid_add_class.diff_cancel
thf(fact_54_diff__zero, axiom,
    ((![A : real]: ((minus_minus_real @ A @ zero_zero_real) = A)))). % diff_zero
thf(fact_55_diff__zero, axiom,
    ((![A : complex]: ((minus_minus_complex @ A @ zero_zero_complex) = A)))). % diff_zero
thf(fact_56_diff__zero, axiom,
    ((![A : poly_complex]: ((minus_174331535omplex @ A @ zero_z1746442943omplex) = A)))). % diff_zero
thf(fact_57_diff__zero, axiom,
    ((![A : poly_real]: ((minus_240770701y_real @ A @ zero_zero_poly_real) = A)))). % diff_zero
thf(fact_58_diff__0__right, axiom,
    ((![A : real]: ((minus_minus_real @ A @ zero_zero_real) = A)))). % diff_0_right
thf(fact_59_diff__0__right, axiom,
    ((![A : complex]: ((minus_minus_complex @ A @ zero_zero_complex) = A)))). % diff_0_right
thf(fact_60_diff__0__right, axiom,
    ((![A : poly_complex]: ((minus_174331535omplex @ A @ zero_z1746442943omplex) = A)))). % diff_0_right
thf(fact_61_diff__0__right, axiom,
    ((![A : poly_real]: ((minus_240770701y_real @ A @ zero_zero_poly_real) = A)))). % diff_0_right
thf(fact_62_diff__self, axiom,
    ((![A : real]: ((minus_minus_real @ A @ A) = zero_zero_real)))). % diff_self
thf(fact_63_diff__self, axiom,
    ((![A : complex]: ((minus_minus_complex @ A @ A) = zero_zero_complex)))). % diff_self
thf(fact_64_diff__self, axiom,
    ((![A : poly_complex]: ((minus_174331535omplex @ A @ A) = zero_z1746442943omplex)))). % diff_self
thf(fact_65_diff__self, axiom,
    ((![A : poly_real]: ((minus_240770701y_real @ A @ A) = zero_zero_poly_real)))). % diff_self
thf(fact_66_neg__equal__zero, axiom,
    ((![A : real]: (((uminus_uminus_real @ A) = A) = (A = zero_zero_real))))). % neg_equal_zero
thf(fact_67_neg__equal__zero, axiom,
    ((![A : poly_real]: (((uminus1613791741y_real @ A) = A) = (A = zero_zero_poly_real))))). % neg_equal_zero
thf(fact_68_equal__neg__zero, axiom,
    ((![A : real]: ((A = (uminus_uminus_real @ A)) = (A = zero_zero_real))))). % equal_neg_zero
thf(fact_69_equal__neg__zero, axiom,
    ((![A : poly_real]: ((A = (uminus1613791741y_real @ A)) = (A = zero_zero_poly_real))))). % equal_neg_zero
thf(fact_70_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_71_neg__equal__0__iff__equal, axiom,
    ((![A : complex]: (((uminus1204672759omplex @ A) = zero_zero_complex) = (A = zero_zero_complex))))). % neg_equal_0_iff_equal
thf(fact_72_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_73_neg__equal__0__iff__equal, axiom,
    ((![A : poly_complex]: (((uminus1138659839omplex @ A) = zero_z1746442943omplex) = (A = zero_z1746442943omplex))))). % neg_equal_0_iff_equal
thf(fact_74_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_75_neg__0__equal__iff__equal, axiom,
    ((![A : complex]: ((zero_zero_complex = (uminus1204672759omplex @ A)) = (zero_zero_complex = A))))). % neg_0_equal_iff_equal
thf(fact_76_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_77_neg__0__equal__iff__equal, axiom,
    ((![A : poly_complex]: ((zero_z1746442943omplex = (uminus1138659839omplex @ A)) = (zero_z1746442943omplex = A))))). % neg_0_equal_iff_equal
thf(fact_78_add_Oinverse__neutral, axiom,
    (((uminus_uminus_real @ zero_zero_real) = zero_zero_real))). % add.inverse_neutral
thf(fact_79_add_Oinverse__neutral, axiom,
    (((uminus1204672759omplex @ zero_zero_complex) = zero_zero_complex))). % add.inverse_neutral
thf(fact_80_add_Oinverse__neutral, axiom,
    (((uminus1613791741y_real @ zero_zero_poly_real) = zero_zero_poly_real))). % add.inverse_neutral
thf(fact_81_add_Oinverse__neutral, axiom,
    (((uminus1138659839omplex @ zero_z1746442943omplex) = zero_z1746442943omplex))). % add.inverse_neutral
thf(fact_82_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_83_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_84_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_85_minus__diff__eq, axiom,
    ((![A : complex, B : complex]: ((uminus1204672759omplex @ (minus_minus_complex @ A @ B)) = (minus_minus_complex @ B @ A))))). % minus_diff_eq
thf(fact_86_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_87_minus__diff__eq, axiom,
    ((![A : poly_complex, B : poly_complex]: ((uminus1138659839omplex @ (minus_174331535omplex @ A @ B)) = (minus_174331535omplex @ B @ A))))). % minus_diff_eq
thf(fact_88_abs__zero, axiom,
    (((abs_abs_real @ zero_zero_real) = zero_zero_real))). % abs_zero
thf(fact_89_abs__zero, axiom,
    (((abs_abs_poly_real @ zero_zero_poly_real) = zero_zero_poly_real))). % abs_zero
thf(fact_90_abs__eq__0, axiom,
    ((![A : real]: (((abs_abs_real @ A) = zero_zero_real) = (A = zero_zero_real))))). % abs_eq_0
thf(fact_91_abs__eq__0, axiom,
    ((![A : poly_real]: (((abs_abs_poly_real @ A) = zero_zero_poly_real) = (A = zero_zero_poly_real))))). % abs_eq_0
thf(fact_92_abs__0__eq, axiom,
    ((![A : real]: ((zero_zero_real = (abs_abs_real @ A)) = (A = zero_zero_real))))). % abs_0_eq
thf(fact_93_abs__0__eq, axiom,
    ((![A : poly_real]: ((zero_zero_poly_real = (abs_abs_poly_real @ A)) = (A = zero_zero_poly_real))))). % abs_0_eq
thf(fact_94_abs__0, axiom,
    (((abs_abs_real @ zero_zero_real) = zero_zero_real))). % abs_0
thf(fact_95_abs__0, axiom,
    (((abs_abs_complex @ zero_zero_complex) = zero_zero_complex))). % abs_0
thf(fact_96_abs__0, axiom,
    (((abs_abs_poly_real @ zero_zero_poly_real) = zero_zero_poly_real))). % abs_0
thf(fact_97_diff__gt__0__iff__gt, axiom,
    ((![A : poly_real, B : poly_real]: ((ord_less_poly_real @ zero_zero_poly_real @ (minus_240770701y_real @ A @ B)) = (ord_less_poly_real @ B @ A))))). % diff_gt_0_iff_gt
thf(fact_98_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_99_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_100_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_101_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_102_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_103_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_104_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_105_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_106_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_107_zero__less__abs__iff, axiom,
    ((![A : poly_real]: ((ord_less_poly_real @ zero_zero_poly_real @ (abs_abs_poly_real @ A)) = (~ ((A = zero_zero_poly_real))))))). % zero_less_abs_iff
thf(fact_108_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_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_verit__comp__simplify1_I1_J, axiom,
    ((![A : real]: (~ ((ord_less_real @ A @ A)))))). % verit_comp_simplify1(1)
thf(fact_112_linorder__neqE__linordered__idom, axiom,
    ((![X : real, Y : real]: ((~ ((X = Y))) => ((~ ((ord_less_real @ X @ Y))) => (ord_less_real @ Y @ X)))))). % linorder_neqE_linordered_idom
thf(fact_113_mem__Collect__eq, axiom,
    ((![A : real, P2 : real > $o]: ((member_real @ A @ (collect_real @ P2)) = (P2 @ A))))). % mem_Collect_eq
thf(fact_114_Collect__mem__eq, axiom,
    ((![A2 : set_real]: ((collect_real @ (^[X2 : real]: (member_real @ X2 @ A2))) = A2)))). % Collect_mem_eq
thf(fact_115_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_116_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_117_real__norm__def, axiom,
    ((real_V646646907m_real = abs_abs_real))). % real_norm_def
thf(fact_118_real__sup__exists, axiom,
    ((![P2 : real > $o]: ((?[X_1 : real]: (P2 @ X_1)) => ((?[Z2 : real]: (![X3 : real]: ((P2 @ X3) => (ord_less_real @ X3 @ Z2)))) => (?[S : real]: (![Y2 : real]: ((?[X2 : real]: (((P2 @ X2)) & ((ord_less_real @ Y2 @ X2)))) = (ord_less_real @ Y2 @ S))))))))). % real_sup_exists
thf(fact_119_diff__strict__right__mono, axiom,
    ((![A : poly_real, B : poly_real, C : poly_real]: ((ord_less_poly_real @ A @ B) => (ord_less_poly_real @ (minus_240770701y_real @ A @ C) @ (minus_240770701y_real @ B @ C)))))). % diff_strict_right_mono
thf(fact_120_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_121_diff__strict__left__mono, axiom,
    ((![B : poly_real, A : poly_real, C : poly_real]: ((ord_less_poly_real @ B @ A) => (ord_less_poly_real @ (minus_240770701y_real @ C @ A) @ (minus_240770701y_real @ C @ B)))))). % diff_strict_left_mono
thf(fact_122_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_123_diff__eq__diff__less, axiom,
    ((![A : poly_real, B : poly_real, C : poly_real, D2 : poly_real]: (((minus_240770701y_real @ A @ B) = (minus_240770701y_real @ C @ D2)) => ((ord_less_poly_real @ A @ B) = (ord_less_poly_real @ C @ D2)))))). % diff_eq_diff_less
thf(fact_124_diff__eq__diff__less, axiom,
    ((![A : real, B : real, C : real, D2 : real]: (((minus_minus_real @ A @ B) = (minus_minus_real @ C @ D2)) => ((ord_less_real @ A @ B) = (ord_less_real @ C @ D2)))))). % diff_eq_diff_less
thf(fact_125_diff__strict__mono, axiom,
    ((![A : poly_real, B : poly_real, D2 : poly_real, C : poly_real]: ((ord_less_poly_real @ A @ B) => ((ord_less_poly_real @ D2 @ C) => (ord_less_poly_real @ (minus_240770701y_real @ A @ C) @ (minus_240770701y_real @ B @ D2))))))). % diff_strict_mono
thf(fact_126_diff__strict__mono, axiom,
    ((![A : real, B : real, D2 : real, C : real]: ((ord_less_real @ A @ B) => ((ord_less_real @ D2 @ C) => (ord_less_real @ (minus_minus_real @ A @ C) @ (minus_minus_real @ B @ D2))))))). % diff_strict_mono
thf(fact_127_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_128_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_129_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_130_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_131_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_132_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_133_zero__reorient, axiom,
    ((![X : real]: ((zero_zero_real = X) = (X = zero_zero_real))))). % zero_reorient
thf(fact_134_zero__reorient, axiom,
    ((![X : complex]: ((zero_zero_complex = X) = (X = zero_zero_complex))))). % zero_reorient
thf(fact_135_zero__reorient, axiom,
    ((![X : poly_complex]: ((zero_z1746442943omplex = X) = (X = zero_z1746442943omplex))))). % zero_reorient
thf(fact_136_zero__reorient, axiom,
    ((![X : poly_real]: ((zero_zero_poly_real = X) = (X = zero_zero_poly_real))))). % zero_reorient
thf(fact_137_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_138_cancel__ab__semigroup__add__class_Odiff__right__commute, axiom,
    ((![A : complex, C : complex, B : complex]: ((minus_minus_complex @ (minus_minus_complex @ A @ C) @ B) = (minus_minus_complex @ (minus_minus_complex @ A @ B) @ C))))). % cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_139_cancel__ab__semigroup__add__class_Odiff__right__commute, axiom,
    ((![A : poly_complex, C : poly_complex, B : poly_complex]: ((minus_174331535omplex @ (minus_174331535omplex @ A @ C) @ B) = (minus_174331535omplex @ (minus_174331535omplex @ A @ B) @ C))))). % cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_140_cancel__ab__semigroup__add__class_Odiff__right__commute, axiom,
    ((![A : poly_real, C : poly_real, B : poly_real]: ((minus_240770701y_real @ (minus_240770701y_real @ A @ C) @ B) = (minus_240770701y_real @ (minus_240770701y_real @ A @ B) @ C))))). % cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_141_diff__eq__diff__eq, axiom,
    ((![A : real, B : real, C : real, D2 : real]: (((minus_minus_real @ A @ B) = (minus_minus_real @ C @ D2)) => ((A = B) = (C = D2)))))). % diff_eq_diff_eq
thf(fact_142_diff__eq__diff__eq, axiom,
    ((![A : complex, B : complex, C : complex, D2 : complex]: (((minus_minus_complex @ A @ B) = (minus_minus_complex @ C @ D2)) => ((A = B) = (C = D2)))))). % diff_eq_diff_eq
thf(fact_143_diff__eq__diff__eq, axiom,
    ((![A : poly_complex, B : poly_complex, C : poly_complex, D2 : poly_complex]: (((minus_174331535omplex @ A @ B) = (minus_174331535omplex @ C @ D2)) => ((A = B) = (C = D2)))))). % diff_eq_diff_eq
thf(fact_144_diff__eq__diff__eq, axiom,
    ((![A : poly_real, B : poly_real, C : poly_real, D2 : poly_real]: (((minus_240770701y_real @ A @ B) = (minus_240770701y_real @ C @ D2)) => ((A = B) = (C = D2)))))). % diff_eq_diff_eq
thf(fact_145_minus__equation__iff, axiom,
    ((![A : real, B : real]: (((uminus_uminus_real @ A) = B) = ((uminus_uminus_real @ B) = A))))). % minus_equation_iff
thf(fact_146_minus__equation__iff, axiom,
    ((![A : complex, B : complex]: (((uminus1204672759omplex @ A) = B) = ((uminus1204672759omplex @ B) = A))))). % minus_equation_iff
thf(fact_147_minus__equation__iff, axiom,
    ((![A : poly_real, B : poly_real]: (((uminus1613791741y_real @ A) = B) = ((uminus1613791741y_real @ B) = A))))). % minus_equation_iff
thf(fact_148_minus__equation__iff, axiom,
    ((![A : poly_complex, B : poly_complex]: (((uminus1138659839omplex @ A) = B) = ((uminus1138659839omplex @ B) = A))))). % minus_equation_iff
thf(fact_149_equation__minus__iff, axiom,
    ((![A : real, B : real]: ((A = (uminus_uminus_real @ B)) = (B = (uminus_uminus_real @ A)))))). % equation_minus_iff
thf(fact_150_equation__minus__iff, axiom,
    ((![A : complex, B : complex]: ((A = (uminus1204672759omplex @ B)) = (B = (uminus1204672759omplex @ A)))))). % equation_minus_iff
thf(fact_151_equation__minus__iff, axiom,
    ((![A : poly_real, B : poly_real]: ((A = (uminus1613791741y_real @ B)) = (B = (uminus1613791741y_real @ A)))))). % equation_minus_iff
thf(fact_152_equation__minus__iff, axiom,
    ((![A : poly_complex, B : poly_complex]: ((A = (uminus1138659839omplex @ B)) = (B = (uminus1138659839omplex @ A)))))). % equation_minus_iff
thf(fact_153_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_154_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_155_less__iff__diff__less__0, axiom,
    ((ord_less_poly_real = (^[A3 : poly_real]: (^[B2 : poly_real]: (ord_less_poly_real @ (minus_240770701y_real @ A3 @ B2) @ zero_zero_poly_real)))))). % less_iff_diff_less_0
thf(fact_156_less__iff__diff__less__0, axiom,
    ((ord_less_real = (^[A3 : real]: (^[B2 : real]: (ord_less_real @ (minus_minus_real @ A3 @ B2) @ zero_zero_real)))))). % less_iff_diff_less_0
thf(fact_157_abs__not__less__zero, axiom,
    ((![A : poly_real]: (~ ((ord_less_poly_real @ (abs_abs_poly_real @ A) @ zero_zero_poly_real)))))). % abs_not_less_zero
thf(fact_158_abs__not__less__zero, axiom,
    ((![A : real]: (~ ((ord_less_real @ (abs_abs_real @ A) @ zero_zero_real)))))). % abs_not_less_zero
thf(fact_159_abs__of__pos, axiom,
    ((![A : poly_real]: ((ord_less_poly_real @ zero_zero_poly_real @ A) => ((abs_abs_poly_real @ A) = A))))). % abs_of_pos
thf(fact_160_abs__of__pos, axiom,
    ((![A : real]: ((ord_less_real @ zero_zero_real @ A) => ((abs_abs_real @ A) = A))))). % abs_of_pos
thf(fact_161_abs__less__iff, axiom,
    ((![A : real, B : real]: ((ord_less_real @ (abs_abs_real @ A) @ B) = (((ord_less_real @ A @ B)) & ((ord_less_real @ (uminus_uminus_real @ A) @ B))))))). % abs_less_iff
thf(fact_162_abs__less__iff, axiom,
    ((![A : poly_real, B : poly_real]: ((ord_less_poly_real @ (abs_abs_poly_real @ A) @ B) = (((ord_less_poly_real @ A @ B)) & ((ord_less_poly_real @ (uminus1613791741y_real @ A) @ B))))))). % abs_less_iff
thf(fact_163_norm__not__less__zero, axiom,
    ((![X : complex]: (~ ((ord_less_real @ (real_V638595069omplex @ X) @ zero_zero_real)))))). % norm_not_less_zero
thf(fact_164_norm__not__less__zero, axiom,
    ((![X : real]: (~ ((ord_less_real @ (real_V646646907m_real @ X) @ zero_zero_real)))))). % norm_not_less_zero
thf(fact_165_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_166_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_167_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_168_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_169_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_170_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_171_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_172_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_173_eq__iff__diff__eq__0, axiom,
    (((^[Y3 : real]: (^[Z3 : real]: (Y3 = Z3))) = (^[A3 : real]: (^[B2 : real]: ((minus_minus_real @ A3 @ B2) = zero_zero_real)))))). % eq_iff_diff_eq_0
thf(fact_174_eq__iff__diff__eq__0, axiom,
    (((^[Y3 : complex]: (^[Z3 : complex]: (Y3 = Z3))) = (^[A3 : complex]: (^[B2 : complex]: ((minus_minus_complex @ A3 @ B2) = zero_zero_complex)))))). % eq_iff_diff_eq_0
thf(fact_175_eq__iff__diff__eq__0, axiom,
    (((^[Y3 : poly_complex]: (^[Z3 : poly_complex]: (Y3 = Z3))) = (^[A3 : poly_complex]: (^[B2 : poly_complex]: ((minus_174331535omplex @ A3 @ B2) = zero_z1746442943omplex)))))). % eq_iff_diff_eq_0
thf(fact_176_eq__iff__diff__eq__0, axiom,
    (((^[Y3 : poly_real]: (^[Z3 : poly_real]: (Y3 = Z3))) = (^[A3 : poly_real]: (^[B2 : poly_real]: ((minus_240770701y_real @ A3 @ B2) = zero_zero_poly_real)))))). % eq_iff_diff_eq_0
thf(fact_177_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_178_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_179_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_180_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_181_abs__eq__0__iff, axiom,
    ((![A : real]: (((abs_abs_real @ A) = zero_zero_real) = (A = zero_zero_real))))). % abs_eq_0_iff
thf(fact_182_abs__eq__0__iff, axiom,
    ((![A : complex]: (((abs_abs_complex @ A) = zero_zero_complex) = (A = zero_zero_complex))))). % abs_eq_0_iff
thf(fact_183_abs__eq__0__iff, axiom,
    ((![A : poly_real]: (((abs_abs_poly_real @ A) = zero_zero_poly_real) = (A = zero_zero_poly_real))))). % abs_eq_0_iff
thf(fact_184_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_185_abs__minus__commute, axiom,
    ((![A : poly_real, B : poly_real]: ((abs_abs_poly_real @ (minus_240770701y_real @ A @ B)) = (abs_abs_poly_real @ (minus_240770701y_real @ B @ A)))))). % abs_minus_commute
thf(fact_186_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_187_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_188_abs__eq__iff, axiom,
    ((![X : real, Y : real]: (((abs_abs_real @ X) = (abs_abs_real @ Y)) = (((X = Y)) | ((X = (uminus_uminus_real @ Y)))))))). % abs_eq_iff
thf(fact_189_abs__eq__iff, axiom,
    ((![X : poly_real, Y : poly_real]: (((abs_abs_poly_real @ X) = (abs_abs_poly_real @ Y)) = (((X = Y)) | ((X = (uminus1613791741y_real @ Y)))))))). % abs_eq_iff
thf(fact_190_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_191_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_192_poly__all__0__iff__0, axiom,
    ((![P : poly_poly_complex]: ((![X2 : poly_complex]: ((poly_poly_complex2 @ P @ X2) = zero_z1746442943omplex)) = (P = zero_z1040703943omplex))))). % poly_all_0_iff_0
thf(fact_193_poly__all__0__iff__0, axiom,
    ((![P : poly_poly_real]: ((![X2 : poly_real]: ((poly_poly_real2 @ P @ X2) = zero_zero_poly_real)) = (P = zero_z1423781445y_real))))). % poly_all_0_iff_0
thf(fact_194_lemma__interval__lt, axiom,
    ((![A : real, X : real, B : real]: ((ord_less_real @ A @ X) => ((ord_less_real @ X @ B) => (?[D : real]: ((ord_less_real @ zero_zero_real @ D) & (![Y2 : real]: ((ord_less_real @ (abs_abs_real @ (minus_minus_real @ X @ Y2)) @ D) => ((ord_less_real @ A @ Y2) & (ord_less_real @ Y2 @ B))))))))))). % lemma_interval_lt
thf(fact_195_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_196_s, axiom,
    ((![Y2 : real]: ((?[X2 : real]: (((?[Z4 : complex]: (((ord_less_eq_real @ (real_V638595069omplex @ Z4) @ r)) & (((real_V638595069omplex @ (poly_complex2 @ p @ Z4)) = (uminus_uminus_real @ X2)))))) & ((ord_less_real @ Y2 @ X2)))) = (ord_less_real @ Y2 @ s))))). % s
thf(fact_197_fz_I2_J, axiom,
    ((![E2 : real]: ((ord_less_real @ zero_zero_real @ E2) => (?[N : nat]: (![N2 : nat]: ((ord_less_eq_nat @ N @ N2) => (ord_less_real @ (real_V638595069omplex @ (minus_minus_complex @ (g @ (f @ N2)) @ z)) @ E2)))))))). % fz(2)
thf(fact_198_minus__diff__minus, axiom,
    ((![A : real, B : real]: ((minus_minus_real @ (uminus_uminus_real @ A) @ (uminus_uminus_real @ B)) = (uminus_uminus_real @ (minus_minus_real @ A @ B)))))). % minus_diff_minus
thf(fact_199_minus__diff__minus, axiom,
    ((![A : complex, B : complex]: ((minus_minus_complex @ (uminus1204672759omplex @ A) @ (uminus1204672759omplex @ B)) = (uminus1204672759omplex @ (minus_minus_complex @ A @ B)))))). % minus_diff_minus
thf(fact_200_minus__diff__minus, axiom,
    ((![A : poly_real, B : poly_real]: ((minus_240770701y_real @ (uminus1613791741y_real @ A) @ (uminus1613791741y_real @ B)) = (uminus1613791741y_real @ (minus_240770701y_real @ A @ B)))))). % minus_diff_minus
thf(fact_201_minus__diff__minus, axiom,
    ((![A : poly_complex, B : poly_complex]: ((minus_174331535omplex @ (uminus1138659839omplex @ A) @ (uminus1138659839omplex @ B)) = (uminus1138659839omplex @ (minus_174331535omplex @ A @ B)))))). % minus_diff_minus
thf(fact_202_True, axiom,
    ((ord_less_eq_real @ zero_zero_real @ r))). % True
thf(fact_203_wr, axiom,
    ((ord_less_eq_real @ (real_V638595069omplex @ w) @ r))). % wr
thf(fact_204_g_I1_J, axiom,
    ((![N2 : nat]: (ord_less_eq_real @ (real_V638595069omplex @ (g @ N2)) @ r)))). % g(1)
thf(fact_205_le__zero__eq, axiom,
    ((![N3 : nat]: ((ord_less_eq_nat @ N3 @ zero_zero_nat) = (N3 = zero_zero_nat))))). % le_zero_eq
thf(fact_206_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_207_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_208_mth1, axiom,
    ((?[X3 : real, Z5 : complex]: ((ord_less_eq_real @ (real_V638595069omplex @ Z5) @ r) & ((real_V638595069omplex @ (poly_complex2 @ p @ Z5)) = (uminus_uminus_real @ X3)))))). % mth1
thf(fact_209__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_210__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]: (![Y2 : real]: ((?[X2 : real]: (((?[Z4 : complex]: (((ord_less_eq_real @ (real_V638595069omplex @ Z4) @ r)) & (((real_V638595069omplex @ (poly_complex2 @ p @ Z4)) = (uminus_uminus_real @ X2)))))) & ((ord_less_real @ Y2 @ X2)))) = (ord_less_real @ Y2 @ 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_211__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]: (~ ((![Y2 : real]: ((?[X2 : real]: (((?[Z4 : complex]: (((ord_less_eq_real @ (real_V638595069omplex @ Z4) @ r)) & (((real_V638595069omplex @ (poly_complex2 @ p @ Z4)) = (uminus_uminus_real @ X2)))))) & ((ord_less_real @ Y2 @ X2)))) = (ord_less_real @ Y2 @ 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_212_mth2, axiom,
    ((?[Z5 : 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 @ Z5)))))). % mth2
thf(fact_213_fz_I1_J, axiom,
    ((order_769474267at_nat @ f))). % fz(1)
thf(fact_214_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_215_diff__ge__0__iff__ge, axiom,
    ((![A : poly_real, B : poly_real]: ((ord_le1180086932y_real @ zero_zero_poly_real @ (minus_240770701y_real @ A @ B)) = (ord_le1180086932y_real @ B @ A))))). % diff_ge_0_iff_ge
thf(fact_216_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_217_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_218_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_219_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_220_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_221_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_222_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_223_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_224_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_225_abs__of__nonneg, axiom,
    ((![A : poly_real]: ((ord_le1180086932y_real @ zero_zero_poly_real @ A) => ((abs_abs_poly_real @ A) = A))))). % abs_of_nonneg
thf(fact_226_abs__of__nonneg, axiom,
    ((![A : real]: ((ord_less_eq_real @ zero_zero_real @ A) => ((abs_abs_real @ A) = A))))). % abs_of_nonneg
thf(fact_227_abs__le__self__iff, axiom,
    ((![A : poly_real]: ((ord_le1180086932y_real @ (abs_abs_poly_real @ A) @ A) = (ord_le1180086932y_real @ zero_zero_poly_real @ A))))). % abs_le_self_iff
thf(fact_228_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_229_abs__le__zero__iff, axiom,
    ((![A : poly_real]: ((ord_le1180086932y_real @ (abs_abs_poly_real @ A) @ zero_zero_poly_real) = (A = zero_zero_poly_real))))). % abs_le_zero_iff
thf(fact_230_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_231_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_232_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_233_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_234_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_235_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_236_verit__la__disequality, axiom,
    ((![A : nat, B : nat]: ((A = B) | ((~ ((ord_less_eq_nat @ A @ B))) | (~ ((ord_less_eq_nat @ B @ A)))))))). % verit_la_disequality
thf(fact_237_complete__real, axiom,
    ((![S2 : set_real]: ((?[X4 : real]: (member_real @ X4 @ S2)) => ((?[Z2 : real]: (![X3 : real]: ((member_real @ X3 @ S2) => (ord_less_eq_real @ X3 @ Z2)))) => (?[Y4 : real]: ((![X4 : real]: ((member_real @ X4 @ S2) => (ord_less_eq_real @ X4 @ Y4))) & (![Z2 : real]: ((![X3 : real]: ((member_real @ X3 @ S2) => (ord_less_eq_real @ X3 @ Z2))) => (ord_less_eq_real @ Y4 @ Z2)))))))))). % complete_real
thf(fact_238_less__eq__real__def, axiom,
    ((ord_less_eq_real = (^[X2 : real]: (^[Y5 : real]: (((ord_less_real @ X2 @ Y5)) | ((X2 = Y5)))))))). % less_eq_real_def
thf(fact_239_zero__le, axiom,
    ((![X : nat]: (ord_less_eq_nat @ zero_zero_nat @ X)))). % zero_le
thf(fact_240_verit__comp__simplify1_I3_J, axiom,
    ((![B3 : real, A4 : real]: ((~ ((ord_less_eq_real @ B3 @ A4))) = (ord_less_real @ A4 @ B3))))). % verit_comp_simplify1(3)
thf(fact_241_verit__comp__simplify1_I3_J, axiom,
    ((![B3 : nat, A4 : nat]: ((~ ((ord_less_eq_nat @ B3 @ A4))) = (ord_less_nat @ A4 @ B3))))). % verit_comp_simplify1(3)
thf(fact_242_diff__mono, axiom,
    ((![A : poly_real, B : poly_real, D2 : poly_real, C : poly_real]: ((ord_le1180086932y_real @ A @ B) => ((ord_le1180086932y_real @ D2 @ C) => (ord_le1180086932y_real @ (minus_240770701y_real @ A @ C) @ (minus_240770701y_real @ B @ D2))))))). % diff_mono
thf(fact_243_diff__mono, axiom,
    ((![A : real, B : real, D2 : real, C : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_eq_real @ D2 @ C) => (ord_less_eq_real @ (minus_minus_real @ A @ C) @ (minus_minus_real @ B @ D2))))))). % diff_mono
thf(fact_244_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_245_lemma__interval, axiom,
    ((![A : real, X : real, B : real]: ((ord_less_real @ A @ X) => ((ord_less_real @ X @ B) => (?[D : real]: ((ord_less_real @ zero_zero_real @ D) & (![Y2 : real]: ((ord_less_real @ (abs_abs_real @ (minus_minus_real @ X @ Y2)) @ D) => ((ord_less_eq_real @ A @ Y2) & (ord_less_eq_real @ Y2 @ B))))))))))). % lemma_interval

% Helper facts (5)
thf(help_If_2_1_If_001t__Real__Oreal_T, axiom,
    ((![X : real, Y : real]: ((if_real @ $false @ X @ Y) = Y)))).
thf(help_If_1_1_If_001t__Real__Oreal_T, axiom,
    ((![X : real, Y : real]: ((if_real @ $true @ X @ Y) = 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, Y : poly_real]: ((if_poly_real @ $false @ X @ Y) = Y)))).
thf(help_If_1_1_If_001t__Polynomial__Opoly_It__Real__Oreal_J_T, axiom,
    ((![X : poly_real, Y : poly_real]: ((if_poly_real @ $true @ X @ Y) = X)))).

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