% 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_678__5374608_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:32:06.578

% Could-be-implicit typings (5)
thf(ty_n_t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    poly_complex : $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 (23)
thf(sy_c_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Oconstant_001t__Complex__Ocomplex_001t__Complex__Ocomplex, type,
    fundam1158420650omplex : (complex > complex) > $o).
thf(sy_c_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Ooffset__poly_001t__Complex__Ocomplex, type,
    fundam1201687030omplex : poly_complex > complex > poly_complex).
thf(sy_c_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Opsize_001t__Complex__Ocomplex, type,
    fundam1709708056omplex : poly_complex > nat).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Complex__Ocomplex, type,
    plus_plus_complex : complex > complex > complex).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat, type,
    plus_plus_nat : nat > nat > nat).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    plus_p1547158847omplex : poly_complex > poly_complex > poly_complex).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal, type,
    plus_plus_real : real > real > real).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Complex__Ocomplex, type,
    zero_zero_complex : complex).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat, type,
    zero_zero_nat : nat).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    zero_z1746442943omplex : poly_complex).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal, type,
    zero_zero_real : real).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat, type,
    ord_less_nat : nat > nat > $o).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal, type,
    ord_less_real : real > real > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__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_Oreflect__poly_001t__Complex__Ocomplex, type,
    reflect_poly_complex : poly_complex > poly_complex).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Complex__Ocomplex, type,
    real_V638595069omplex : complex > real).
thf(sy_c_Set_OCollect_001t__Real__Oreal, type,
    collect_real : (real > $o) > set_real).
thf(sy_c_member_001t__Real__Oreal, type,
    member_real : real > set_real > $o).
thf(sy_v_c____, type,
    c : complex).
thf(sy_v_p, type,
    p : poly_complex).
thf(sy_v_pa____, type,
    pa : poly_complex).
thf(sy_v_q____, type,
    q : poly_complex).

% Relevant facts (127)
thf(fact_0_c, axiom,
    ((![W : complex]: (ord_less_eq_real @ (real_V638595069omplex @ (poly_complex2 @ pa @ c)) @ (real_V638595069omplex @ (poly_complex2 @ pa @ W)))))). % c
thf(fact_1_nc, axiom,
    ((~ ((fundam1158420650omplex @ (poly_complex2 @ p)))))). % nc
thf(fact_2_False, axiom,
    ((~ (((poly_complex2 @ pa @ c) = zero_zero_complex))))). % False
thf(fact_3_pqc0, axiom,
    (((poly_complex2 @ pa @ c) = (poly_complex2 @ q @ zero_zero_complex)))). % pqc0
thf(fact_4_less_Oprems, axiom,
    ((~ ((fundam1158420650omplex @ (poly_complex2 @ pa)))))). % less.prems
thf(fact_5__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062c_O_A_092_060forall_062w_O_Acmod_A_Ipoly_Ap_Ac_J_A_092_060le_062_Acmod_A_Ipoly_Ap_Aw_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom,
    ((~ ((![C : complex]: (~ ((![W : complex]: (ord_less_eq_real @ (real_V638595069omplex @ (poly_complex2 @ pa @ C)) @ (real_V638595069omplex @ (poly_complex2 @ pa @ W))))))))))). % \<open>\<And>thesis. (\<And>c. \<forall>w. cmod (poly p c) \<le> cmod (poly p w) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_6_q_I1_J, axiom,
    (((fundam1709708056omplex @ q) = (fundam1709708056omplex @ pa)))). % q(1)
thf(fact_7__092_060open_062constant_A_Ipoly_Aq_J_A_092_060Longrightarrow_062_AFalse_092_060close_062, axiom,
    ((~ ((fundam1158420650omplex @ (poly_complex2 @ q)))))). % \<open>constant (poly q) \<Longrightarrow> False\<close>
thf(fact_8_qnc, axiom,
    ((~ ((fundam1158420650omplex @ (poly_complex2 @ q)))))). % qnc
thf(fact_9_q_I2_J, axiom,
    ((![X : complex]: ((poly_complex2 @ q @ X) = (poly_complex2 @ pa @ (plus_plus_complex @ c @ X)))))). % q(2)
thf(fact_10_poly__minimum__modulus, axiom,
    ((![P : poly_complex]: (?[Z : complex]: (![W : complex]: (ord_less_eq_real @ (real_V638595069omplex @ (poly_complex2 @ P @ Z)) @ (real_V638595069omplex @ (poly_complex2 @ P @ W)))))))). % poly_minimum_modulus
thf(fact_11_poly__minimum__modulus__disc, axiom,
    ((![R : real, P : poly_complex]: (?[Z : complex]: (![W : complex]: ((ord_less_eq_real @ (real_V638595069omplex @ W) @ R) => (ord_less_eq_real @ (real_V638595069omplex @ (poly_complex2 @ P @ Z)) @ (real_V638595069omplex @ (poly_complex2 @ P @ W))))))))). % poly_minimum_modulus_disc
thf(fact_12_norm__le__zero__iff, axiom,
    ((![X2 : complex]: ((ord_less_eq_real @ (real_V638595069omplex @ X2) @ zero_zero_real) = (X2 = zero_zero_complex))))). % norm_le_zero_iff
thf(fact_13_poly__0, axiom,
    ((![X2 : complex]: ((poly_complex2 @ zero_z1746442943omplex @ X2) = zero_zero_complex)))). % poly_0
thf(fact_14_norm__zero, axiom,
    (((real_V638595069omplex @ zero_zero_complex) = zero_zero_real))). % norm_zero
thf(fact_15_norm__eq__zero, axiom,
    ((![X2 : complex]: (((real_V638595069omplex @ X2) = zero_zero_real) = (X2 = zero_zero_complex))))). % norm_eq_zero
thf(fact_16_order__refl, axiom,
    ((![X2 : real]: (ord_less_eq_real @ X2 @ X2)))). % order_refl
thf(fact_17_poly__all__0__iff__0, axiom,
    ((![P : poly_complex]: ((![X3 : complex]: ((poly_complex2 @ P @ X3) = zero_zero_complex)) = (P = zero_z1746442943omplex))))). % poly_all_0_iff_0
thf(fact_18_le__numeral__extra_I3_J, axiom,
    ((ord_less_eq_real @ zero_zero_real @ zero_zero_real))). % le_numeral_extra(3)
thf(fact_19_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_20_complete__real, axiom,
    ((![S : set_real]: ((?[X : real]: (member_real @ X @ S)) => ((?[Z2 : real]: (![X4 : real]: ((member_real @ X4 @ S) => (ord_less_eq_real @ X4 @ Z2)))) => (?[Y : real]: ((![X : real]: ((member_real @ X @ S) => (ord_less_eq_real @ X @ Y))) & (![Z2 : real]: ((![X4 : real]: ((member_real @ X4 @ S) => (ord_less_eq_real @ X4 @ Z2))) => (ord_less_eq_real @ Y @ Z2)))))))))). % complete_real
thf(fact_21_add__left__cancel, axiom,
    ((![A : complex, B : complex, C2 : complex]: (((plus_plus_complex @ A @ B) = (plus_plus_complex @ A @ C2)) = (B = C2))))). % add_left_cancel
thf(fact_22_add__right__cancel, axiom,
    ((![B : complex, A : complex, C2 : complex]: (((plus_plus_complex @ B @ A) = (plus_plus_complex @ C2 @ A)) = (B = C2))))). % add_right_cancel
thf(fact_23_add__cancel__right__right, axiom,
    ((![A : complex, B : complex]: ((A = (plus_plus_complex @ A @ B)) = (B = zero_zero_complex))))). % add_cancel_right_right
thf(fact_24_add__cancel__right__left, axiom,
    ((![A : complex, B : complex]: ((A = (plus_plus_complex @ B @ A)) = (B = zero_zero_complex))))). % add_cancel_right_left
thf(fact_25_add__cancel__left__right, axiom,
    ((![A : complex, B : complex]: (((plus_plus_complex @ A @ B) = A) = (B = zero_zero_complex))))). % add_cancel_left_right
thf(fact_26_add__cancel__left__left, axiom,
    ((![B : complex, A : complex]: (((plus_plus_complex @ B @ A) = A) = (B = zero_zero_complex))))). % add_cancel_left_left
thf(fact_27_add_Oright__neutral, axiom,
    ((![A : complex]: ((plus_plus_complex @ A @ zero_zero_complex) = A)))). % add.right_neutral
thf(fact_28_add_Oleft__neutral, axiom,
    ((![A : complex]: ((plus_plus_complex @ zero_zero_complex @ A) = A)))). % add.left_neutral
thf(fact_29_add__le__cancel__right, axiom,
    ((![A : real, C2 : real, B : real]: ((ord_less_eq_real @ (plus_plus_real @ A @ C2) @ (plus_plus_real @ B @ C2)) = (ord_less_eq_real @ A @ B))))). % add_le_cancel_right
thf(fact_30_add__le__cancel__left, axiom,
    ((![C2 : real, A : real, B : real]: ((ord_less_eq_real @ (plus_plus_real @ C2 @ A) @ (plus_plus_real @ C2 @ B)) = (ord_less_eq_real @ A @ B))))). % add_le_cancel_left
thf(fact_31__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062q_O_A_092_060lbrakk_062psize_Aq_A_061_Apsize_Ap_059_A_092_060forall_062x_O_Apoly_Aq_Ax_A_061_Apoly_Ap_A_Ic_A_L_Ax_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom,
    ((~ ((![Q2 : poly_complex]: (((fundam1709708056omplex @ Q2) = (fundam1709708056omplex @ pa)) => (~ ((![X : complex]: ((poly_complex2 @ Q2 @ X) = (poly_complex2 @ pa @ (plus_plus_complex @ c @ X)))))))))))). % \<open>\<And>thesis. (\<And>q. \<lbrakk>psize q = psize p; \<forall>x. poly q x = poly p (c + x)\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_32__092_060open_062_092_060exists_062q_O_Apsize_Aq_A_061_Apsize_Ap_A_092_060and_062_A_I_092_060forall_062x_O_Apoly_Aq_Ax_A_061_Apoly_Ap_A_Ic_A_L_Ax_J_J_092_060close_062, axiom,
    ((?[Q2 : poly_complex]: (((fundam1709708056omplex @ Q2) = (fundam1709708056omplex @ pa)) & (![X : complex]: ((poly_complex2 @ Q2 @ X) = (poly_complex2 @ pa @ (plus_plus_complex @ c @ X)))))))). % \<open>\<exists>q. psize q = psize p \<and> (\<forall>x. poly q x = poly p (c + x))\<close>
thf(fact_33_poly__add, axiom,
    ((![P : poly_complex, Q : poly_complex, X2 : complex]: ((poly_complex2 @ (plus_p1547158847omplex @ P @ Q) @ X2) = (plus_plus_complex @ (poly_complex2 @ P @ X2) @ (poly_complex2 @ Q @ X2)))))). % poly_add
thf(fact_34_psize__eq__0__iff, axiom,
    ((![P : poly_complex]: (((fundam1709708056omplex @ P) = zero_zero_nat) = (P = zero_z1746442943omplex))))). % psize_eq_0_iff
thf(fact_35_zero__le__double__add__iff__zero__le__single__add, axiom,
    ((![A : real]: ((ord_less_eq_real @ zero_zero_real @ (plus_plus_real @ A @ A)) = (ord_less_eq_real @ zero_zero_real @ A))))). % zero_le_double_add_iff_zero_le_single_add
thf(fact_36_double__add__le__zero__iff__single__add__le__zero, axiom,
    ((![A : real]: ((ord_less_eq_real @ (plus_plus_real @ A @ A) @ zero_zero_real) = (ord_less_eq_real @ A @ zero_zero_real))))). % double_add_le_zero_iff_single_add_le_zero
thf(fact_37_le__add__same__cancel2, axiom,
    ((![A : real, B : real]: ((ord_less_eq_real @ A @ (plus_plus_real @ B @ A)) = (ord_less_eq_real @ zero_zero_real @ B))))). % le_add_same_cancel2
thf(fact_38_mem__Collect__eq, axiom,
    ((![A : real, P2 : real > $o]: ((member_real @ A @ (collect_real @ P2)) = (P2 @ A))))). % mem_Collect_eq
thf(fact_39_Collect__mem__eq, axiom,
    ((![A2 : set_real]: ((collect_real @ (^[X3 : real]: (member_real @ X3 @ A2))) = A2)))). % Collect_mem_eq
thf(fact_40_le__add__same__cancel1, axiom,
    ((![A : real, B : real]: ((ord_less_eq_real @ A @ (plus_plus_real @ A @ B)) = (ord_less_eq_real @ zero_zero_real @ B))))). % le_add_same_cancel1
thf(fact_41_add__le__same__cancel2, axiom,
    ((![A : real, B : real]: ((ord_less_eq_real @ (plus_plus_real @ A @ B) @ B) = (ord_less_eq_real @ A @ zero_zero_real))))). % add_le_same_cancel2
thf(fact_42_add__le__same__cancel1, axiom,
    ((![B : real, A : real]: ((ord_less_eq_real @ (plus_plus_real @ B @ A) @ B) = (ord_less_eq_real @ A @ zero_zero_real))))). % add_le_same_cancel1
thf(fact_43_ab__semigroup__add__class_Oadd__ac_I1_J, axiom,
    ((![A : complex, B : complex, C2 : complex]: ((plus_plus_complex @ (plus_plus_complex @ A @ B) @ C2) = (plus_plus_complex @ A @ (plus_plus_complex @ B @ C2)))))). % ab_semigroup_add_class.add_ac(1)
thf(fact_44_is__num__normalize_I1_J, axiom,
    ((![A : complex, B : complex, C2 : complex]: ((plus_plus_complex @ (plus_plus_complex @ A @ B) @ C2) = (plus_plus_complex @ A @ (plus_plus_complex @ B @ C2)))))). % is_num_normalize(1)
thf(fact_45_group__cancel_Oadd1, axiom,
    ((![A2 : complex, K : complex, A : complex, B : complex]: ((A2 = (plus_plus_complex @ K @ A)) => ((plus_plus_complex @ A2 @ B) = (plus_plus_complex @ K @ (plus_plus_complex @ A @ B))))))). % group_cancel.add1
thf(fact_46_group__cancel_Oadd2, axiom,
    ((![B2 : complex, K : complex, B : complex, A : complex]: ((B2 = (plus_plus_complex @ K @ B)) => ((plus_plus_complex @ A @ B2) = (plus_plus_complex @ K @ (plus_plus_complex @ A @ B))))))). % group_cancel.add2
thf(fact_47_add_Oassoc, axiom,
    ((![A : complex, B : complex, C2 : complex]: ((plus_plus_complex @ (plus_plus_complex @ A @ B) @ C2) = (plus_plus_complex @ A @ (plus_plus_complex @ B @ C2)))))). % add.assoc
thf(fact_48_add_Oleft__cancel, axiom,
    ((![A : complex, B : complex, C2 : complex]: (((plus_plus_complex @ A @ B) = (plus_plus_complex @ A @ C2)) = (B = C2))))). % add.left_cancel
thf(fact_49_add_Oright__cancel, axiom,
    ((![B : complex, A : complex, C2 : complex]: (((plus_plus_complex @ B @ A) = (plus_plus_complex @ C2 @ A)) = (B = C2))))). % add.right_cancel
thf(fact_50_add_Ocommute, axiom,
    ((plus_plus_complex = (^[A3 : complex]: (^[B3 : complex]: (plus_plus_complex @ B3 @ A3)))))). % add.commute
thf(fact_51_add_Oleft__commute, axiom,
    ((![B : complex, A : complex, C2 : complex]: ((plus_plus_complex @ B @ (plus_plus_complex @ A @ C2)) = (plus_plus_complex @ A @ (plus_plus_complex @ B @ C2)))))). % add.left_commute
thf(fact_52_add__left__imp__eq, axiom,
    ((![A : complex, B : complex, C2 : complex]: (((plus_plus_complex @ A @ B) = (plus_plus_complex @ A @ C2)) => (B = C2))))). % add_left_imp_eq
thf(fact_53_add__right__imp__eq, axiom,
    ((![B : complex, A : complex, C2 : complex]: (((plus_plus_complex @ B @ A) = (plus_plus_complex @ C2 @ A)) => (B = C2))))). % add_right_imp_eq
thf(fact_54_add_Ogroup__left__neutral, axiom,
    ((![A : complex]: ((plus_plus_complex @ zero_zero_complex @ A) = A)))). % add.group_left_neutral
thf(fact_55_add_Ocomm__neutral, axiom,
    ((![A : complex]: ((plus_plus_complex @ A @ zero_zero_complex) = A)))). % add.comm_neutral
thf(fact_56_comm__monoid__add__class_Oadd__0, axiom,
    ((![A : complex]: ((plus_plus_complex @ zero_zero_complex @ A) = A)))). % comm_monoid_add_class.add_0
thf(fact_57_add__le__imp__le__right, axiom,
    ((![A : real, C2 : real, B : real]: ((ord_less_eq_real @ (plus_plus_real @ A @ C2) @ (plus_plus_real @ B @ C2)) => (ord_less_eq_real @ A @ B))))). % add_le_imp_le_right
thf(fact_58_add__le__imp__le__left, axiom,
    ((![C2 : real, A : real, B : real]: ((ord_less_eq_real @ (plus_plus_real @ C2 @ A) @ (plus_plus_real @ C2 @ B)) => (ord_less_eq_real @ A @ B))))). % add_le_imp_le_left
thf(fact_59_add__right__mono, axiom,
    ((![A : real, B : real, C2 : real]: ((ord_less_eq_real @ A @ B) => (ord_less_eq_real @ (plus_plus_real @ A @ C2) @ (plus_plus_real @ B @ C2)))))). % add_right_mono
thf(fact_60_add__left__mono, axiom,
    ((![A : real, B : real, C2 : real]: ((ord_less_eq_real @ A @ B) => (ord_less_eq_real @ (plus_plus_real @ C2 @ A) @ (plus_plus_real @ C2 @ B)))))). % add_left_mono
thf(fact_61_add__mono, axiom,
    ((![A : real, B : real, C2 : real, D : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_eq_real @ C2 @ D) => (ord_less_eq_real @ (plus_plus_real @ A @ C2) @ (plus_plus_real @ B @ D))))))). % add_mono
thf(fact_62_add__mono__thms__linordered__semiring_I1_J, axiom,
    ((![I : real, J : real, K : real, L : real]: (((ord_less_eq_real @ I @ J) & (ord_less_eq_real @ K @ L)) => (ord_less_eq_real @ (plus_plus_real @ I @ K) @ (plus_plus_real @ J @ L)))))). % add_mono_thms_linordered_semiring(1)
thf(fact_63_add__mono__thms__linordered__semiring_I2_J, axiom,
    ((![I : real, J : real, K : real, L : real]: (((I = J) & (ord_less_eq_real @ K @ L)) => (ord_less_eq_real @ (plus_plus_real @ I @ K) @ (plus_plus_real @ J @ L)))))). % add_mono_thms_linordered_semiring(2)
thf(fact_64_add__mono__thms__linordered__semiring_I3_J, axiom,
    ((![I : real, J : real, K : real, L : real]: (((ord_less_eq_real @ I @ J) & (K = L)) => (ord_less_eq_real @ (plus_plus_real @ I @ K) @ (plus_plus_real @ J @ L)))))). % add_mono_thms_linordered_semiring(3)
thf(fact_65_norm__triangle__mono, axiom,
    ((![A : complex, R : real, B : complex, S2 : real]: ((ord_less_eq_real @ (real_V638595069omplex @ A) @ R) => ((ord_less_eq_real @ (real_V638595069omplex @ B) @ S2) => (ord_less_eq_real @ (real_V638595069omplex @ (plus_plus_complex @ A @ B)) @ (plus_plus_real @ R @ S2))))))). % norm_triangle_mono
thf(fact_66_norm__triangle__ineq, axiom,
    ((![X2 : complex, Y2 : complex]: (ord_less_eq_real @ (real_V638595069omplex @ (plus_plus_complex @ X2 @ Y2)) @ (plus_plus_real @ (real_V638595069omplex @ X2) @ (real_V638595069omplex @ Y2)))))). % norm_triangle_ineq
thf(fact_67_norm__triangle__le, axiom,
    ((![X2 : complex, Y2 : complex, E : real]: ((ord_less_eq_real @ (plus_plus_real @ (real_V638595069omplex @ X2) @ (real_V638595069omplex @ Y2)) @ E) => (ord_less_eq_real @ (real_V638595069omplex @ (plus_plus_complex @ X2 @ Y2)) @ E))))). % norm_triangle_le
thf(fact_68_norm__add__leD, axiom,
    ((![A : complex, B : complex, C2 : real]: ((ord_less_eq_real @ (real_V638595069omplex @ (plus_plus_complex @ A @ B)) @ C2) => (ord_less_eq_real @ (real_V638595069omplex @ B) @ (plus_plus_real @ (real_V638595069omplex @ A) @ C2)))))). % norm_add_leD
thf(fact_69_poly__offset, axiom,
    ((![P : poly_complex, A : complex]: (?[Q2 : poly_complex]: (((fundam1709708056omplex @ Q2) = (fundam1709708056omplex @ P)) & (![X : complex]: ((poly_complex2 @ Q2 @ X) = (poly_complex2 @ P @ (plus_plus_complex @ A @ X))))))))). % poly_offset
thf(fact_70_constant__def, axiom,
    ((fundam1158420650omplex = (^[F : complex > complex]: (![X3 : complex]: (![Y3 : complex]: ((F @ X3) = (F @ Y3)))))))). % constant_def
thf(fact_71_add__nonpos__eq__0__iff, axiom,
    ((![X2 : real, Y2 : real]: ((ord_less_eq_real @ X2 @ zero_zero_real) => ((ord_less_eq_real @ Y2 @ zero_zero_real) => (((plus_plus_real @ X2 @ Y2) = zero_zero_real) = (((X2 = zero_zero_real)) & ((Y2 = zero_zero_real))))))))). % add_nonpos_eq_0_iff
thf(fact_72_add__nonneg__eq__0__iff, axiom,
    ((![X2 : real, Y2 : real]: ((ord_less_eq_real @ zero_zero_real @ X2) => ((ord_less_eq_real @ zero_zero_real @ Y2) => (((plus_plus_real @ X2 @ Y2) = zero_zero_real) = (((X2 = zero_zero_real)) & ((Y2 = zero_zero_real))))))))). % add_nonneg_eq_0_iff
thf(fact_73_add__nonpos__nonpos, axiom,
    ((![A : real, B : real]: ((ord_less_eq_real @ A @ zero_zero_real) => ((ord_less_eq_real @ B @ zero_zero_real) => (ord_less_eq_real @ (plus_plus_real @ A @ B) @ zero_zero_real)))))). % add_nonpos_nonpos
thf(fact_74_add__nonneg__nonneg, axiom,
    ((![A : real, B : real]: ((ord_less_eq_real @ zero_zero_real @ A) => ((ord_less_eq_real @ zero_zero_real @ B) => (ord_less_eq_real @ zero_zero_real @ (plus_plus_real @ A @ B))))))). % add_nonneg_nonneg
thf(fact_75_add__increasing2, axiom,
    ((![C2 : real, B : real, A : real]: ((ord_less_eq_real @ zero_zero_real @ C2) => ((ord_less_eq_real @ B @ A) => (ord_less_eq_real @ B @ (plus_plus_real @ A @ C2))))))). % add_increasing2
thf(fact_76_add__decreasing2, axiom,
    ((![C2 : real, A : real, B : real]: ((ord_less_eq_real @ C2 @ zero_zero_real) => ((ord_less_eq_real @ A @ B) => (ord_less_eq_real @ (plus_plus_real @ A @ C2) @ B)))))). % add_decreasing2
thf(fact_77_add__increasing, axiom,
    ((![A : real, B : real, C2 : real]: ((ord_less_eq_real @ zero_zero_real @ A) => ((ord_less_eq_real @ B @ C2) => (ord_less_eq_real @ B @ (plus_plus_real @ A @ C2))))))). % add_increasing
thf(fact_78_add__decreasing, axiom,
    ((![A : real, C2 : real, B : real]: ((ord_less_eq_real @ A @ zero_zero_real) => ((ord_less_eq_real @ C2 @ B) => (ord_less_eq_real @ (plus_plus_real @ A @ C2) @ B)))))). % add_decreasing
thf(fact_79_norm__ge__zero, axiom,
    ((![X2 : complex]: (ord_less_eq_real @ zero_zero_real @ (real_V638595069omplex @ X2))))). % norm_ge_zero
thf(fact_80_zero__reorient, axiom,
    ((![X2 : complex]: ((zero_zero_complex = X2) = (X2 = zero_zero_complex))))). % zero_reorient
thf(fact_81_dual__order_Oantisym, axiom,
    ((![B : real, A : real]: ((ord_less_eq_real @ B @ A) => ((ord_less_eq_real @ A @ B) => (A = B)))))). % dual_order.antisym
thf(fact_82_dual__order_Oeq__iff, axiom,
    (((^[Y4 : real]: (^[Z3 : real]: (Y4 = Z3))) = (^[A3 : real]: (^[B3 : real]: (((ord_less_eq_real @ B3 @ A3)) & ((ord_less_eq_real @ A3 @ B3)))))))). % dual_order.eq_iff
thf(fact_83_dual__order_Otrans, axiom,
    ((![B : real, A : real, C2 : real]: ((ord_less_eq_real @ B @ A) => ((ord_less_eq_real @ C2 @ B) => (ord_less_eq_real @ C2 @ A)))))). % dual_order.trans
thf(fact_84_linorder__wlog, axiom,
    ((![P2 : real > real > $o, A : real, B : real]: ((![A4 : real, B4 : real]: ((ord_less_eq_real @ A4 @ B4) => (P2 @ A4 @ B4))) => ((![A4 : real, B4 : real]: ((P2 @ B4 @ A4) => (P2 @ A4 @ B4))) => (P2 @ A @ B)))))). % linorder_wlog
thf(fact_85_dual__order_Orefl, axiom,
    ((![A : real]: (ord_less_eq_real @ A @ A)))). % dual_order.refl
thf(fact_86_order__trans, axiom,
    ((![X2 : real, Y2 : real, Z4 : real]: ((ord_less_eq_real @ X2 @ Y2) => ((ord_less_eq_real @ Y2 @ Z4) => (ord_less_eq_real @ X2 @ Z4)))))). % order_trans
thf(fact_87_order__class_Oorder_Oantisym, axiom,
    ((![A : real, B : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_eq_real @ B @ A) => (A = B)))))). % order_class.order.antisym
thf(fact_88_ord__le__eq__trans, axiom,
    ((![A : real, B : real, C2 : real]: ((ord_less_eq_real @ A @ B) => ((B = C2) => (ord_less_eq_real @ A @ C2)))))). % ord_le_eq_trans
thf(fact_89_ord__eq__le__trans, axiom,
    ((![A : real, B : real, C2 : real]: ((A = B) => ((ord_less_eq_real @ B @ C2) => (ord_less_eq_real @ A @ C2)))))). % ord_eq_le_trans
thf(fact_90_order__class_Oorder_Oeq__iff, axiom,
    (((^[Y4 : real]: (^[Z3 : real]: (Y4 = Z3))) = (^[A3 : real]: (^[B3 : real]: (((ord_less_eq_real @ A3 @ B3)) & ((ord_less_eq_real @ B3 @ A3)))))))). % order_class.order.eq_iff
thf(fact_91_antisym__conv, axiom,
    ((![Y2 : real, X2 : real]: ((ord_less_eq_real @ Y2 @ X2) => ((ord_less_eq_real @ X2 @ Y2) = (X2 = Y2)))))). % antisym_conv
thf(fact_92_le__cases3, axiom,
    ((![X2 : real, Y2 : real, Z4 : real]: (((ord_less_eq_real @ X2 @ Y2) => (~ ((ord_less_eq_real @ Y2 @ Z4)))) => (((ord_less_eq_real @ Y2 @ X2) => (~ ((ord_less_eq_real @ X2 @ Z4)))) => (((ord_less_eq_real @ X2 @ Z4) => (~ ((ord_less_eq_real @ Z4 @ Y2)))) => (((ord_less_eq_real @ Z4 @ Y2) => (~ ((ord_less_eq_real @ Y2 @ X2)))) => (((ord_less_eq_real @ Y2 @ Z4) => (~ ((ord_less_eq_real @ Z4 @ X2)))) => (~ (((ord_less_eq_real @ Z4 @ X2) => (~ ((ord_less_eq_real @ X2 @ Y2)))))))))))))). % le_cases3
thf(fact_93_order_Otrans, axiom,
    ((![A : real, B : real, C2 : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_eq_real @ B @ C2) => (ord_less_eq_real @ A @ C2)))))). % order.trans
thf(fact_94_le__cases, axiom,
    ((![X2 : real, Y2 : real]: ((~ ((ord_less_eq_real @ X2 @ Y2))) => (ord_less_eq_real @ Y2 @ X2))))). % le_cases
thf(fact_95_eq__refl, axiom,
    ((![X2 : real, Y2 : real]: ((X2 = Y2) => (ord_less_eq_real @ X2 @ Y2))))). % eq_refl
thf(fact_96_linear, axiom,
    ((![X2 : real, Y2 : real]: ((ord_less_eq_real @ X2 @ Y2) | (ord_less_eq_real @ Y2 @ X2))))). % linear
thf(fact_97_antisym, axiom,
    ((![X2 : real, Y2 : real]: ((ord_less_eq_real @ X2 @ Y2) => ((ord_less_eq_real @ Y2 @ X2) => (X2 = Y2)))))). % antisym
thf(fact_98_eq__iff, axiom,
    (((^[Y4 : real]: (^[Z3 : real]: (Y4 = Z3))) = (^[X3 : real]: (^[Y3 : real]: (((ord_less_eq_real @ X3 @ Y3)) & ((ord_less_eq_real @ Y3 @ X3)))))))). % eq_iff
thf(fact_99_ord__le__eq__subst, axiom,
    ((![A : real, B : real, F2 : real > real, C2 : real]: ((ord_less_eq_real @ A @ B) => (((F2 @ B) = C2) => ((![X4 : real, Y : real]: ((ord_less_eq_real @ X4 @ Y) => (ord_less_eq_real @ (F2 @ X4) @ (F2 @ Y)))) => (ord_less_eq_real @ (F2 @ A) @ C2))))))). % ord_le_eq_subst
thf(fact_100_ord__eq__le__subst, axiom,
    ((![A : real, F2 : real > real, B : real, C2 : real]: ((A = (F2 @ B)) => ((ord_less_eq_real @ B @ C2) => ((![X4 : real, Y : real]: ((ord_less_eq_real @ X4 @ Y) => (ord_less_eq_real @ (F2 @ X4) @ (F2 @ Y)))) => (ord_less_eq_real @ A @ (F2 @ C2)))))))). % ord_eq_le_subst
thf(fact_101_order__subst2, axiom,
    ((![A : real, B : real, F2 : real > real, C2 : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_eq_real @ (F2 @ B) @ C2) => ((![X4 : real, Y : real]: ((ord_less_eq_real @ X4 @ Y) => (ord_less_eq_real @ (F2 @ X4) @ (F2 @ Y)))) => (ord_less_eq_real @ (F2 @ A) @ C2))))))). % order_subst2
thf(fact_102_order__subst1, axiom,
    ((![A : real, F2 : real > real, B : real, C2 : real]: ((ord_less_eq_real @ A @ (F2 @ B)) => ((ord_less_eq_real @ B @ C2) => ((![X4 : real, Y : real]: ((ord_less_eq_real @ X4 @ Y) => (ord_less_eq_real @ (F2 @ X4) @ (F2 @ Y)))) => (ord_less_eq_real @ A @ (F2 @ C2)))))))). % order_subst1
thf(fact_103_less_Ohyps, axiom,
    ((![P : poly_complex]: ((ord_less_nat @ (fundam1709708056omplex @ P) @ (fundam1709708056omplex @ pa)) => ((~ ((fundam1158420650omplex @ (poly_complex2 @ P)))) => (?[Z : complex]: ((poly_complex2 @ P @ Z) = zero_zero_complex))))))). % less.hyps
thf(fact_104_add__0__iff, axiom,
    ((![B : complex, A : complex]: ((B = (plus_plus_complex @ B @ A)) = (A = zero_zero_complex))))). % add_0_iff
thf(fact_105_verit__sum__simplify, axiom,
    ((![A : complex]: ((plus_plus_complex @ A @ zero_zero_complex) = A)))). % verit_sum_simplify
thf(fact_106_reflect__poly__at__0__eq__0__iff, axiom,
    ((![P : poly_complex]: (((poly_complex2 @ (reflect_poly_complex @ P) @ zero_zero_complex) = zero_zero_complex) = (P = zero_z1746442943omplex))))). % reflect_poly_at_0_eq_0_iff
thf(fact_107_poly__bound__exists, axiom,
    ((![R : real, P : poly_complex]: (?[M : real]: ((ord_less_real @ zero_zero_real @ M) & (![Z2 : complex]: ((ord_less_eq_real @ (real_V638595069omplex @ Z2) @ R) => (ord_less_eq_real @ (real_V638595069omplex @ (poly_complex2 @ P @ Z2)) @ M)))))))). % poly_bound_exists
thf(fact_108_poly__offset__poly, axiom,
    ((![P : poly_complex, H : complex, X2 : complex]: ((poly_complex2 @ (fundam1201687030omplex @ P @ H) @ X2) = (poly_complex2 @ P @ (plus_plus_complex @ H @ X2)))))). % poly_offset_poly
thf(fact_109_not__gr__zero, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr_zero
thf(fact_110_add__less__cancel__right, axiom,
    ((![A : nat, C2 : nat, B : nat]: ((ord_less_nat @ (plus_plus_nat @ A @ C2) @ (plus_plus_nat @ B @ C2)) = (ord_less_nat @ A @ B))))). % add_less_cancel_right
thf(fact_111_add__less__cancel__left, axiom,
    ((![C2 : nat, A : nat, B : nat]: ((ord_less_nat @ (plus_plus_nat @ C2 @ A) @ (plus_plus_nat @ C2 @ B)) = (ord_less_nat @ A @ B))))). % add_less_cancel_left
thf(fact_112_less__add__same__cancel2, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ (plus_plus_nat @ B @ A)) = (ord_less_nat @ zero_zero_nat @ B))))). % less_add_same_cancel2
thf(fact_113_less__add__same__cancel1, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ (plus_plus_nat @ A @ B)) = (ord_less_nat @ zero_zero_nat @ B))))). % less_add_same_cancel1
thf(fact_114_add__less__same__cancel2, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ (plus_plus_nat @ A @ B) @ B) = (ord_less_nat @ A @ zero_zero_nat))))). % add_less_same_cancel2
thf(fact_115_add__less__same__cancel1, axiom,
    ((![B : nat, A : nat]: ((ord_less_nat @ (plus_plus_nat @ B @ A) @ B) = (ord_less_nat @ A @ zero_zero_nat))))). % add_less_same_cancel1
thf(fact_116_zero__less__norm__iff, axiom,
    ((![X2 : complex]: ((ord_less_real @ zero_zero_real @ (real_V638595069omplex @ X2)) = (~ ((X2 = zero_zero_complex))))))). % zero_less_norm_iff
thf(fact_117_ord__eq__less__subst, axiom,
    ((![A : nat, F2 : nat > nat, B : nat, C2 : nat]: ((A = (F2 @ B)) => ((ord_less_nat @ B @ C2) => ((![X4 : nat, Y : nat]: ((ord_less_nat @ X4 @ Y) => (ord_less_nat @ (F2 @ X4) @ (F2 @ Y)))) => (ord_less_nat @ A @ (F2 @ C2)))))))). % ord_eq_less_subst
thf(fact_118_ord__less__eq__subst, axiom,
    ((![A : nat, B : nat, F2 : nat > nat, C2 : nat]: ((ord_less_nat @ A @ B) => (((F2 @ B) = C2) => ((![X4 : nat, Y : nat]: ((ord_less_nat @ X4 @ Y) => (ord_less_nat @ (F2 @ X4) @ (F2 @ Y)))) => (ord_less_nat @ (F2 @ A) @ C2))))))). % ord_less_eq_subst
thf(fact_119_order__less__subst1, axiom,
    ((![A : nat, F2 : nat > nat, B : nat, C2 : nat]: ((ord_less_nat @ A @ (F2 @ B)) => ((ord_less_nat @ B @ C2) => ((![X4 : nat, Y : nat]: ((ord_less_nat @ X4 @ Y) => (ord_less_nat @ (F2 @ X4) @ (F2 @ Y)))) => (ord_less_nat @ A @ (F2 @ C2)))))))). % order_less_subst1
thf(fact_120_order__less__subst2, axiom,
    ((![A : nat, B : nat, F2 : nat > nat, C2 : nat]: ((ord_less_nat @ A @ B) => ((ord_less_nat @ (F2 @ B) @ C2) => ((![X4 : nat, Y : nat]: ((ord_less_nat @ X4 @ Y) => (ord_less_nat @ (F2 @ X4) @ (F2 @ Y)))) => (ord_less_nat @ (F2 @ A) @ C2))))))). % order_less_subst2
thf(fact_121_gt__ex, axiom,
    ((![X2 : nat]: (?[X_1 : nat]: (ord_less_nat @ X2 @ X_1))))). % gt_ex
thf(fact_122_neqE, axiom,
    ((![X2 : nat, Y2 : nat]: ((~ ((X2 = Y2))) => ((~ ((ord_less_nat @ X2 @ Y2))) => (ord_less_nat @ Y2 @ X2)))))). % neqE
thf(fact_123_neq__iff, axiom,
    ((![X2 : nat, Y2 : nat]: ((~ ((X2 = Y2))) = (((ord_less_nat @ X2 @ Y2)) | ((ord_less_nat @ Y2 @ X2))))))). % neq_iff
thf(fact_124_order_Oasym, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (~ ((ord_less_nat @ B @ A))))))). % order.asym
thf(fact_125_less__imp__neq, axiom,
    ((![X2 : nat, Y2 : nat]: ((ord_less_nat @ X2 @ Y2) => (~ ((X2 = Y2))))))). % less_imp_neq
thf(fact_126_less__asym, axiom,
    ((![X2 : nat, Y2 : nat]: ((ord_less_nat @ X2 @ Y2) => (~ ((ord_less_nat @ Y2 @ X2))))))). % less_asym

% Conjectures (1)
thf(conj_0, conjecture,
    ((![W2 : complex]: (ord_less_eq_real @ (real_V638595069omplex @ (poly_complex2 @ q @ zero_zero_complex)) @ (real_V638595069omplex @ (poly_complex2 @ pa @ W2)))))).
