% 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_465__5372464_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:30:21.889

% Could-be-implicit typings (6)
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 (26)
thf(sy_c_Groups_Oone__class_Oone_001t__Complex__Ocomplex, type,
    one_one_complex : complex).
thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    one_one_poly_complex : poly_complex).
thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_It__Real__Oreal_J, type,
    one_one_poly_real : poly_real).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal, type,
    one_one_real : real).
thf(sy_c_Groups_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__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_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_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_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 (146)
thf(fact_0__092_060open_062cmod_A_Ipoly_Ap_Az_J_A_061_A_N_As_092_060close_062, axiom,
    (((real_V638595069omplex @ (poly_complex2 @ p @ z)) = (uminus_uminus_real @ s)))). % \<open>cmod (poly p z) = - s\<close>
thf(fact_1__092_060open_062_N_As_A_092_060le_062_Acmod_A_Ipoly_Ap_Aw_J_092_060close_062, axiom,
    ((ord_less_eq_real @ (uminus_uminus_real @ s) @ (real_V638595069omplex @ (poly_complex2 @ p @ w))))). % \<open>- s \<le> cmod (poly p w)\<close>
thf(fact_2_wr, axiom,
    ((ord_less_eq_real @ (real_V638595069omplex @ w) @ r))). % wr
thf(fact_3_mth1, axiom,
    ((?[X : real, Z : complex]: ((ord_less_eq_real @ (real_V638595069omplex @ Z) @ r) & ((real_V638595069omplex @ (poly_complex2 @ p @ Z)) = (uminus_uminus_real @ X)))))). % mth1
thf(fact_4_order__refl, axiom,
    ((![X2 : real]: (ord_less_eq_real @ X2 @ X2)))). % order_refl
thf(fact_5_s1m, axiom,
    ((![Z2 : complex]: ((ord_less_eq_real @ (real_V638595069omplex @ Z2) @ r) => (ord_less_eq_real @ (uminus_uminus_real @ s) @ (real_V638595069omplex @ (poly_complex2 @ p @ Z2))))))). % s1m
thf(fact_6_g_I1_J, axiom,
    ((![N : nat]: (ord_less_eq_real @ (real_V638595069omplex @ (g @ N)) @ r)))). % g(1)
thf(fact_7_s, axiom,
    ((![Y : real]: ((?[X3 : real]: (((?[Z3 : complex]: (((ord_less_eq_real @ (real_V638595069omplex @ Z3) @ r)) & (((real_V638595069omplex @ (poly_complex2 @ p @ Z3)) = (uminus_uminus_real @ X3)))))) & ((ord_less_real @ Y @ X3)))) = (ord_less_real @ Y @ s))))). % s
thf(fact_8_complex__mod__minus__le__complex__mod, axiom,
    ((![X2 : complex]: (ord_less_eq_real @ (uminus_uminus_real @ (real_V638595069omplex @ X2)) @ (real_V638595069omplex @ X2))))). % complex_mod_minus_le_complex_mod
thf(fact_9_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_10__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_11_complete__real, axiom,
    ((![S : set_real]: ((?[X4 : real]: (member_real @ X4 @ S)) => ((?[Z4 : real]: (![X : real]: ((member_real @ X @ S) => (ord_less_eq_real @ X @ Z4)))) => (?[Y2 : real]: ((![X4 : real]: ((member_real @ X4 @ S) => (ord_less_eq_real @ X4 @ Y2))) & (![Z4 : real]: ((![X : real]: ((member_real @ X @ S) => (ord_less_eq_real @ X @ Z4))) => (ord_less_eq_real @ Y2 @ Z4)))))))))). % complete_real
thf(fact_12__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,
    ((?[S2 : real]: (![Y : real]: ((?[X3 : real]: (((?[Z3 : complex]: (((ord_less_eq_real @ (real_V638595069omplex @ Z3) @ r)) & (((real_V638595069omplex @ (poly_complex2 @ p @ Z3)) = (uminus_uminus_real @ X3)))))) & ((ord_less_real @ Y @ X3)))) = (ord_less_real @ Y @ S2)))))). % \<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_13__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,
    ((~ ((![S2 : real]: (~ ((![Y : real]: ((?[X3 : real]: (((?[Z3 : complex]: (((ord_less_eq_real @ (real_V638595069omplex @ Z3) @ r)) & (((real_V638595069omplex @ (poly_complex2 @ p @ Z3)) = (uminus_uminus_real @ X3)))))) & ((ord_less_real @ Y @ X3)))) = (ord_less_real @ Y @ S2)))))))))). % \<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_14_mth2, axiom,
    ((?[Z : 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 @ Z)))))). % mth2
thf(fact_15_True, axiom,
    ((ord_less_eq_real @ zero_zero_real @ r))). % True
thf(fact_16_poly__0, axiom,
    ((![X2 : complex]: ((poly_complex2 @ zero_z1746442943omplex @ X2) = zero_zero_complex)))). % poly_0
thf(fact_17_poly__0, axiom,
    ((![X2 : real]: ((poly_real2 @ zero_zero_poly_real @ X2) = zero_zero_real)))). % poly_0
thf(fact_18_poly__minus, axiom,
    ((![P : poly_complex, X2 : complex]: ((poly_complex2 @ (uminus1138659839omplex @ P) @ X2) = (uminus1204672759omplex @ (poly_complex2 @ P @ X2)))))). % poly_minus
thf(fact_19_poly__minus, axiom,
    ((![P : poly_real, X2 : real]: ((poly_real2 @ (uminus1613791741y_real @ P) @ X2) = (uminus_uminus_real @ (poly_real2 @ P @ X2)))))). % poly_minus
thf(fact_20_field__lbound__gt__zero, axiom,
    ((![D1 : real, D2 : real]: ((ord_less_real @ zero_zero_real @ D1) => ((ord_less_real @ zero_zero_real @ D2) => (?[E : real]: ((ord_less_real @ zero_zero_real @ E) & ((ord_less_real @ E @ D1) & (ord_less_real @ E @ D2))))))))). % field_lbound_gt_zero
thf(fact_21_ord__eq__less__subst, axiom,
    ((![A : real, F : real > real, B : real, C : real]: ((A = (F @ B)) => ((ord_less_real @ B @ C) => ((![X : real, Y2 : real]: ((ord_less_real @ X @ Y2) => (ord_less_real @ (F @ X) @ (F @ Y2)))) => (ord_less_real @ A @ (F @ C)))))))). % ord_eq_less_subst
thf(fact_22_ord__less__eq__subst, axiom,
    ((![A : real, B : real, F : real > real, C : real]: ((ord_less_real @ A @ B) => (((F @ B) = C) => ((![X : real, Y2 : real]: ((ord_less_real @ X @ Y2) => (ord_less_real @ (F @ X) @ (F @ Y2)))) => (ord_less_real @ (F @ A) @ C))))))). % ord_less_eq_subst
thf(fact_23_order__less__subst1, axiom,
    ((![A : real, F : real > real, B : real, C : real]: ((ord_less_real @ A @ (F @ B)) => ((ord_less_real @ B @ C) => ((![X : real, Y2 : real]: ((ord_less_real @ X @ Y2) => (ord_less_real @ (F @ X) @ (F @ Y2)))) => (ord_less_real @ A @ (F @ C)))))))). % order_less_subst1
thf(fact_24_order__less__subst2, axiom,
    ((![A : real, B : real, F : real > real, C : real]: ((ord_less_real @ A @ B) => ((ord_less_real @ (F @ B) @ C) => ((![X : real, Y2 : real]: ((ord_less_real @ X @ Y2) => (ord_less_real @ (F @ X) @ (F @ Y2)))) => (ord_less_real @ (F @ A) @ C))))))). % order_less_subst2
thf(fact_25_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_26_poly__all__0__iff__0, axiom,
    ((![P : poly_real]: ((![X3 : real]: ((poly_real2 @ P @ X3) = zero_zero_real)) = (P = zero_zero_poly_real))))). % poly_all_0_iff_0
thf(fact_27_lt__ex, axiom,
    ((![X2 : real]: (?[Y2 : real]: (ord_less_real @ Y2 @ X2))))). % lt_ex
thf(fact_28_gt__ex, axiom,
    ((![X2 : real]: (?[X_1 : real]: (ord_less_real @ X2 @ X_1))))). % gt_ex
thf(fact_29_neqE, axiom,
    ((![X2 : real, Y3 : real]: ((~ ((X2 = Y3))) => ((~ ((ord_less_real @ X2 @ Y3))) => (ord_less_real @ Y3 @ X2)))))). % neqE
thf(fact_30_neq__iff, axiom,
    ((![X2 : real, Y3 : real]: ((~ ((X2 = Y3))) = (((ord_less_real @ X2 @ Y3)) | ((ord_less_real @ Y3 @ X2))))))). % neq_iff
thf(fact_31_order_Oasym, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ B) => (~ ((ord_less_real @ B @ A))))))). % order.asym
thf(fact_32_dense, axiom,
    ((![X2 : real, Y3 : real]: ((ord_less_real @ X2 @ Y3) => (?[Z : real]: ((ord_less_real @ X2 @ Z) & (ord_less_real @ Z @ Y3))))))). % dense
thf(fact_33_less__imp__neq, axiom,
    ((![X2 : real, Y3 : real]: ((ord_less_real @ X2 @ Y3) => (~ ((X2 = Y3))))))). % less_imp_neq
thf(fact_34_less__asym, axiom,
    ((![X2 : real, Y3 : real]: ((ord_less_real @ X2 @ Y3) => (~ ((ord_less_real @ Y3 @ X2))))))). % less_asym
thf(fact_35_less__asym_H, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ B) => (~ ((ord_less_real @ B @ A))))))). % less_asym'
thf(fact_36_less__trans, axiom,
    ((![X2 : real, Y3 : real, Z2 : real]: ((ord_less_real @ X2 @ Y3) => ((ord_less_real @ Y3 @ Z2) => (ord_less_real @ X2 @ Z2)))))). % less_trans
thf(fact_37_less__linear, axiom,
    ((![X2 : real, Y3 : real]: ((ord_less_real @ X2 @ Y3) | ((X2 = Y3) | (ord_less_real @ Y3 @ X2)))))). % less_linear
thf(fact_38_less__irrefl, axiom,
    ((![X2 : real]: (~ ((ord_less_real @ X2 @ X2)))))). % less_irrefl
thf(fact_39_ord__eq__less__trans, axiom,
    ((![A : real, B : real, C : real]: ((A = B) => ((ord_less_real @ B @ C) => (ord_less_real @ A @ C)))))). % ord_eq_less_trans
thf(fact_40_ord__less__eq__trans, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_real @ A @ B) => ((B = C) => (ord_less_real @ A @ C)))))). % ord_less_eq_trans
thf(fact_41_dual__order_Oasym, axiom,
    ((![B : real, A : real]: ((ord_less_real @ B @ A) => (~ ((ord_less_real @ A @ B))))))). % dual_order.asym
thf(fact_42_less__imp__not__eq, axiom,
    ((![X2 : real, Y3 : real]: ((ord_less_real @ X2 @ Y3) => (~ ((X2 = Y3))))))). % less_imp_not_eq
thf(fact_43_less__not__sym, axiom,
    ((![X2 : real, Y3 : real]: ((ord_less_real @ X2 @ Y3) => (~ ((ord_less_real @ Y3 @ X2))))))). % less_not_sym
thf(fact_44_antisym__conv3, axiom,
    ((![Y3 : real, X2 : real]: ((~ ((ord_less_real @ Y3 @ X2))) => ((~ ((ord_less_real @ X2 @ Y3))) = (X2 = Y3)))))). % antisym_conv3
thf(fact_45_mem__Collect__eq, axiom,
    ((![A : real, P2 : real > $o]: ((member_real @ A @ (collect_real @ P2)) = (P2 @ A))))). % mem_Collect_eq
thf(fact_46_Collect__mem__eq, axiom,
    ((![A2 : set_real]: ((collect_real @ (^[X3 : real]: (member_real @ X3 @ A2))) = A2)))). % Collect_mem_eq
thf(fact_47_less__imp__not__eq2, axiom,
    ((![X2 : real, Y3 : real]: ((ord_less_real @ X2 @ Y3) => (~ ((Y3 = X2))))))). % less_imp_not_eq2
thf(fact_48_less__imp__triv, axiom,
    ((![X2 : real, Y3 : real, P2 : $o]: ((ord_less_real @ X2 @ Y3) => ((ord_less_real @ Y3 @ X2) => P2))))). % less_imp_triv
thf(fact_49_linorder__cases, axiom,
    ((![X2 : real, Y3 : real]: ((~ ((ord_less_real @ X2 @ Y3))) => ((~ ((X2 = Y3))) => (ord_less_real @ Y3 @ X2)))))). % linorder_cases
thf(fact_50_dual__order_Oirrefl, axiom,
    ((![A : real]: (~ ((ord_less_real @ A @ A)))))). % dual_order.irrefl
thf(fact_51_order_Ostrict__trans, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_real @ A @ B) => ((ord_less_real @ B @ C) => (ord_less_real @ A @ C)))))). % order.strict_trans
thf(fact_52_less__imp__not__less, axiom,
    ((![X2 : real, Y3 : real]: ((ord_less_real @ X2 @ Y3) => (~ ((ord_less_real @ Y3 @ X2))))))). % less_imp_not_less
thf(fact_53_linorder__less__wlog, axiom,
    ((![P2 : real > real > $o, A : real, B : real]: ((![A3 : real, B2 : real]: ((ord_less_real @ A3 @ B2) => (P2 @ A3 @ B2))) => ((![A3 : real]: (P2 @ A3 @ A3)) => ((![A3 : real, B2 : real]: ((P2 @ B2 @ A3) => (P2 @ A3 @ B2))) => (P2 @ A @ B))))))). % linorder_less_wlog
thf(fact_54_dual__order_Ostrict__trans, axiom,
    ((![B : real, A : real, C : real]: ((ord_less_real @ B @ A) => ((ord_less_real @ C @ B) => (ord_less_real @ C @ A)))))). % dual_order.strict_trans
thf(fact_55_not__less__iff__gr__or__eq, axiom,
    ((![X2 : real, Y3 : real]: ((~ ((ord_less_real @ X2 @ Y3))) = (((ord_less_real @ Y3 @ X2)) | ((X2 = Y3))))))). % not_less_iff_gr_or_eq
thf(fact_56_order_Ostrict__implies__not__eq, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ B) => (~ ((A = B))))))). % order.strict_implies_not_eq
thf(fact_57_dual__order_Ostrict__implies__not__eq, axiom,
    ((![B : real, A : real]: ((ord_less_real @ B @ A) => (~ ((A = B))))))). % dual_order.strict_implies_not_eq
thf(fact_58_real__sup__exists, axiom,
    ((![P2 : real > $o]: ((?[X_12 : real]: (P2 @ X_12)) => ((?[Z4 : real]: (![X : real]: ((P2 @ X) => (ord_less_real @ X @ Z4)))) => (?[S2 : real]: (![Y : real]: ((?[X3 : real]: (((P2 @ X3)) & ((ord_less_real @ Y @ X3)))) = (ord_less_real @ Y @ S2))))))))). % real_sup_exists
thf(fact_59_order_Onot__eq__order__implies__strict, axiom,
    ((![A : real, B : real]: ((~ ((A = B))) => ((ord_less_eq_real @ A @ B) => (ord_less_real @ A @ B)))))). % order.not_eq_order_implies_strict
thf(fact_60_dual__order_Ostrict__implies__order, axiom,
    ((![B : real, A : real]: ((ord_less_real @ B @ A) => (ord_less_eq_real @ B @ A))))). % dual_order.strict_implies_order
thf(fact_61_dual__order_Ostrict__iff__order, axiom,
    ((ord_less_real = (^[B3 : real]: (^[A4 : real]: (((ord_less_eq_real @ B3 @ A4)) & ((~ ((A4 = B3)))))))))). % dual_order.strict_iff_order
thf(fact_62_dual__order_Oorder__iff__strict, axiom,
    ((ord_less_eq_real = (^[B3 : real]: (^[A4 : real]: (((ord_less_real @ B3 @ A4)) | ((A4 = B3)))))))). % dual_order.order_iff_strict
thf(fact_63_order_Ostrict__implies__order, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ B) => (ord_less_eq_real @ A @ B))))). % order.strict_implies_order
thf(fact_64_dense__le__bounded, axiom,
    ((![X2 : real, Y3 : real, Z2 : real]: ((ord_less_real @ X2 @ Y3) => ((![W : real]: ((ord_less_real @ X2 @ W) => ((ord_less_real @ W @ Y3) => (ord_less_eq_real @ W @ Z2)))) => (ord_less_eq_real @ Y3 @ Z2)))))). % dense_le_bounded
thf(fact_65_dense__ge__bounded, axiom,
    ((![Z2 : real, X2 : real, Y3 : real]: ((ord_less_real @ Z2 @ X2) => ((![W : real]: ((ord_less_real @ Z2 @ W) => ((ord_less_real @ W @ X2) => (ord_less_eq_real @ Y3 @ W)))) => (ord_less_eq_real @ Y3 @ Z2)))))). % dense_ge_bounded
thf(fact_66_dual__order_Ostrict__trans2, axiom,
    ((![B : real, A : real, C : real]: ((ord_less_real @ B @ A) => ((ord_less_eq_real @ C @ B) => (ord_less_real @ C @ A)))))). % dual_order.strict_trans2
thf(fact_67_dual__order_Ostrict__trans1, axiom,
    ((![B : real, A : real, C : real]: ((ord_less_eq_real @ B @ A) => ((ord_less_real @ C @ B) => (ord_less_real @ C @ A)))))). % dual_order.strict_trans1
thf(fact_68_order_Ostrict__iff__order, axiom,
    ((ord_less_real = (^[A4 : real]: (^[B3 : real]: (((ord_less_eq_real @ A4 @ B3)) & ((~ ((A4 = B3)))))))))). % order.strict_iff_order
thf(fact_69_order_Oorder__iff__strict, axiom,
    ((ord_less_eq_real = (^[A4 : real]: (^[B3 : real]: (((ord_less_real @ A4 @ B3)) | ((A4 = B3)))))))). % order.order_iff_strict
thf(fact_70_order_Ostrict__trans2, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_real @ A @ B) => ((ord_less_eq_real @ B @ C) => (ord_less_real @ A @ C)))))). % order.strict_trans2
thf(fact_71_order_Ostrict__trans1, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_real @ B @ C) => (ord_less_real @ A @ C)))))). % order.strict_trans1
thf(fact_72_not__le__imp__less, axiom,
    ((![Y3 : real, X2 : real]: ((~ ((ord_less_eq_real @ Y3 @ X2))) => (ord_less_real @ X2 @ Y3))))). % not_le_imp_less
thf(fact_73_less__le__not__le, axiom,
    ((ord_less_real = (^[X3 : real]: (^[Y4 : real]: (((ord_less_eq_real @ X3 @ Y4)) & ((~ ((ord_less_eq_real @ Y4 @ X3)))))))))). % less_le_not_le
thf(fact_74_le__imp__less__or__eq, axiom,
    ((![X2 : real, Y3 : real]: ((ord_less_eq_real @ X2 @ Y3) => ((ord_less_real @ X2 @ Y3) | (X2 = Y3)))))). % le_imp_less_or_eq
thf(fact_75_le__less__linear, axiom,
    ((![X2 : real, Y3 : real]: ((ord_less_eq_real @ X2 @ Y3) | (ord_less_real @ Y3 @ X2))))). % le_less_linear
thf(fact_76_dense__le, axiom,
    ((![Y3 : real, Z2 : real]: ((![X : real]: ((ord_less_real @ X @ Y3) => (ord_less_eq_real @ X @ Z2))) => (ord_less_eq_real @ Y3 @ Z2))))). % dense_le
thf(fact_77_dense__ge, axiom,
    ((![Z2 : real, Y3 : real]: ((![X : real]: ((ord_less_real @ Z2 @ X) => (ord_less_eq_real @ Y3 @ X))) => (ord_less_eq_real @ Y3 @ Z2))))). % dense_ge
thf(fact_78_less__le__trans, axiom,
    ((![X2 : real, Y3 : real, Z2 : real]: ((ord_less_real @ X2 @ Y3) => ((ord_less_eq_real @ Y3 @ Z2) => (ord_less_real @ X2 @ Z2)))))). % less_le_trans
thf(fact_79_le__less__trans, axiom,
    ((![X2 : real, Y3 : real, Z2 : real]: ((ord_less_eq_real @ X2 @ Y3) => ((ord_less_real @ Y3 @ Z2) => (ord_less_real @ X2 @ Z2)))))). % le_less_trans
thf(fact_80_less__imp__le, axiom,
    ((![X2 : real, Y3 : real]: ((ord_less_real @ X2 @ Y3) => (ord_less_eq_real @ X2 @ Y3))))). % less_imp_le
thf(fact_81_antisym__conv2, axiom,
    ((![X2 : real, Y3 : real]: ((ord_less_eq_real @ X2 @ Y3) => ((~ ((ord_less_real @ X2 @ Y3))) = (X2 = Y3)))))). % antisym_conv2
thf(fact_82_antisym__conv1, axiom,
    ((![X2 : real, Y3 : real]: ((~ ((ord_less_real @ X2 @ Y3))) => ((ord_less_eq_real @ X2 @ Y3) = (X2 = Y3)))))). % antisym_conv1
thf(fact_83_le__neq__trans, axiom,
    ((![A : real, B : real]: ((ord_less_eq_real @ A @ B) => ((~ ((A = B))) => (ord_less_real @ A @ B)))))). % le_neq_trans
thf(fact_84_not__less, axiom,
    ((![X2 : real, Y3 : real]: ((~ ((ord_less_real @ X2 @ Y3))) = (ord_less_eq_real @ Y3 @ X2))))). % not_less
thf(fact_85_not__le, axiom,
    ((![X2 : real, Y3 : real]: ((~ ((ord_less_eq_real @ X2 @ Y3))) = (ord_less_real @ Y3 @ X2))))). % not_le
thf(fact_86_order__less__le__subst2, axiom,
    ((![A : real, B : real, F : real > real, C : real]: ((ord_less_real @ A @ B) => ((ord_less_eq_real @ (F @ B) @ C) => ((![X : real, Y2 : real]: ((ord_less_real @ X @ Y2) => (ord_less_real @ (F @ X) @ (F @ Y2)))) => (ord_less_real @ (F @ A) @ C))))))). % order_less_le_subst2
thf(fact_87_order__less__le__subst1, axiom,
    ((![A : real, F : real > real, B : real, C : real]: ((ord_less_real @ A @ (F @ B)) => ((ord_less_eq_real @ B @ C) => ((![X : real, Y2 : real]: ((ord_less_eq_real @ X @ Y2) => (ord_less_eq_real @ (F @ X) @ (F @ Y2)))) => (ord_less_real @ A @ (F @ C)))))))). % order_less_le_subst1
thf(fact_88_order__le__less__subst2, axiom,
    ((![A : real, B : real, F : real > real, C : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_real @ (F @ B) @ C) => ((![X : real, Y2 : real]: ((ord_less_eq_real @ X @ Y2) => (ord_less_eq_real @ (F @ X) @ (F @ Y2)))) => (ord_less_real @ (F @ A) @ C))))))). % order_le_less_subst2
thf(fact_89_order__le__less__subst1, axiom,
    ((![A : real, F : real > real, B : real, C : real]: ((ord_less_eq_real @ A @ (F @ B)) => ((ord_less_real @ B @ C) => ((![X : real, Y2 : real]: ((ord_less_real @ X @ Y2) => (ord_less_real @ (F @ X) @ (F @ Y2)))) => (ord_less_real @ A @ (F @ C)))))))). % order_le_less_subst1
thf(fact_90_less__le, axiom,
    ((ord_less_real = (^[X3 : real]: (^[Y4 : real]: (((ord_less_eq_real @ X3 @ Y4)) & ((~ ((X3 = Y4)))))))))). % less_le
thf(fact_91_le__less, axiom,
    ((ord_less_eq_real = (^[X3 : real]: (^[Y4 : real]: (((ord_less_real @ X3 @ Y4)) | ((X3 = Y4)))))))). % le_less
thf(fact_92_leI, axiom,
    ((![X2 : real, Y3 : real]: ((~ ((ord_less_real @ X2 @ Y3))) => (ord_less_eq_real @ Y3 @ X2))))). % leI
thf(fact_93_leD, axiom,
    ((![Y3 : real, X2 : real]: ((ord_less_eq_real @ Y3 @ X2) => (~ ((ord_less_real @ X2 @ Y3))))))). % leD
thf(fact_94_less__eq__real__def, axiom,
    ((ord_less_eq_real = (^[X3 : real]: (^[Y4 : real]: (((ord_less_real @ X3 @ Y4)) | ((X3 = Y4)))))))). % less_eq_real_def
thf(fact_95_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_96_dual__order_Oeq__iff, axiom,
    (((^[Y5 : real]: (^[Z5 : real]: (Y5 = Z5))) = (^[A4 : real]: (^[B3 : real]: (((ord_less_eq_real @ B3 @ A4)) & ((ord_less_eq_real @ A4 @ B3)))))))). % dual_order.eq_iff
thf(fact_97_dual__order_Otrans, axiom,
    ((![B : real, A : real, C : real]: ((ord_less_eq_real @ B @ A) => ((ord_less_eq_real @ C @ B) => (ord_less_eq_real @ C @ A)))))). % dual_order.trans
thf(fact_98_linorder__wlog, axiom,
    ((![P2 : real > real > $o, A : real, B : real]: ((![A3 : real, B2 : real]: ((ord_less_eq_real @ A3 @ B2) => (P2 @ A3 @ B2))) => ((![A3 : real, B2 : real]: ((P2 @ B2 @ A3) => (P2 @ A3 @ B2))) => (P2 @ A @ B)))))). % linorder_wlog
thf(fact_99_dual__order_Orefl, axiom,
    ((![A : real]: (ord_less_eq_real @ A @ A)))). % dual_order.refl
thf(fact_100_order__trans, axiom,
    ((![X2 : real, Y3 : real, Z2 : real]: ((ord_less_eq_real @ X2 @ Y3) => ((ord_less_eq_real @ Y3 @ Z2) => (ord_less_eq_real @ X2 @ Z2)))))). % order_trans
thf(fact_101_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_102_ord__le__eq__trans, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_eq_real @ A @ B) => ((B = C) => (ord_less_eq_real @ A @ C)))))). % ord_le_eq_trans
thf(fact_103_ord__eq__le__trans, axiom,
    ((![A : real, B : real, C : real]: ((A = B) => ((ord_less_eq_real @ B @ C) => (ord_less_eq_real @ A @ C)))))). % ord_eq_le_trans
thf(fact_104_order__class_Oorder_Oeq__iff, axiom,
    (((^[Y5 : real]: (^[Z5 : real]: (Y5 = Z5))) = (^[A4 : real]: (^[B3 : real]: (((ord_less_eq_real @ A4 @ B3)) & ((ord_less_eq_real @ B3 @ A4)))))))). % order_class.order.eq_iff
thf(fact_105_antisym__conv, axiom,
    ((![Y3 : real, X2 : real]: ((ord_less_eq_real @ Y3 @ X2) => ((ord_less_eq_real @ X2 @ Y3) = (X2 = Y3)))))). % antisym_conv
thf(fact_106_le__cases3, axiom,
    ((![X2 : real, Y3 : real, Z2 : real]: (((ord_less_eq_real @ X2 @ Y3) => (~ ((ord_less_eq_real @ Y3 @ Z2)))) => (((ord_less_eq_real @ Y3 @ X2) => (~ ((ord_less_eq_real @ X2 @ Z2)))) => (((ord_less_eq_real @ X2 @ Z2) => (~ ((ord_less_eq_real @ Z2 @ Y3)))) => (((ord_less_eq_real @ Z2 @ Y3) => (~ ((ord_less_eq_real @ Y3 @ X2)))) => (((ord_less_eq_real @ Y3 @ Z2) => (~ ((ord_less_eq_real @ Z2 @ X2)))) => (~ (((ord_less_eq_real @ Z2 @ X2) => (~ ((ord_less_eq_real @ X2 @ Y3)))))))))))))). % le_cases3
thf(fact_107_order_Otrans, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_eq_real @ B @ C) => (ord_less_eq_real @ A @ C)))))). % order.trans
thf(fact_108_le__cases, axiom,
    ((![X2 : real, Y3 : real]: ((~ ((ord_less_eq_real @ X2 @ Y3))) => (ord_less_eq_real @ Y3 @ X2))))). % le_cases
thf(fact_109_eq__refl, axiom,
    ((![X2 : real, Y3 : real]: ((X2 = Y3) => (ord_less_eq_real @ X2 @ Y3))))). % eq_refl
thf(fact_110_linear, axiom,
    ((![X2 : real, Y3 : real]: ((ord_less_eq_real @ X2 @ Y3) | (ord_less_eq_real @ Y3 @ X2))))). % linear
thf(fact_111_antisym, axiom,
    ((![X2 : real, Y3 : real]: ((ord_less_eq_real @ X2 @ Y3) => ((ord_less_eq_real @ Y3 @ X2) => (X2 = Y3)))))). % antisym
thf(fact_112_eq__iff, axiom,
    (((^[Y5 : real]: (^[Z5 : real]: (Y5 = Z5))) = (^[X3 : real]: (^[Y4 : real]: (((ord_less_eq_real @ X3 @ Y4)) & ((ord_less_eq_real @ Y4 @ X3)))))))). % eq_iff
thf(fact_113_ord__le__eq__subst, axiom,
    ((![A : real, B : real, F : real > real, C : real]: ((ord_less_eq_real @ A @ B) => (((F @ B) = C) => ((![X : real, Y2 : real]: ((ord_less_eq_real @ X @ Y2) => (ord_less_eq_real @ (F @ X) @ (F @ Y2)))) => (ord_less_eq_real @ (F @ A) @ C))))))). % ord_le_eq_subst
thf(fact_114_ord__eq__le__subst, axiom,
    ((![A : real, F : real > real, B : real, C : real]: ((A = (F @ B)) => ((ord_less_eq_real @ B @ C) => ((![X : real, Y2 : real]: ((ord_less_eq_real @ X @ Y2) => (ord_less_eq_real @ (F @ X) @ (F @ Y2)))) => (ord_less_eq_real @ A @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_115_order__subst2, axiom,
    ((![A : real, B : real, F : real > real, C : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_eq_real @ (F @ B) @ C) => ((![X : real, Y2 : real]: ((ord_less_eq_real @ X @ Y2) => (ord_less_eq_real @ (F @ X) @ (F @ Y2)))) => (ord_less_eq_real @ (F @ A) @ C))))))). % order_subst2
thf(fact_116_order__subst1, axiom,
    ((![A : real, F : real > real, B : real, C : real]: ((ord_less_eq_real @ A @ (F @ B)) => ((ord_less_eq_real @ B @ C) => ((![X : real, Y2 : real]: ((ord_less_eq_real @ X @ Y2) => (ord_less_eq_real @ (F @ X) @ (F @ Y2)))) => (ord_less_eq_real @ A @ (F @ C)))))))). % order_subst1
thf(fact_117__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,
    ((![Z2 : complex, X2 : real]: ((ord_less_eq_real @ (real_V638595069omplex @ Z2) @ r) => (((real_V638595069omplex @ (poly_complex2 @ p @ Z2)) = (uminus_uminus_real @ X2)) => (ord_less_real @ X2 @ 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_118_norm__le__zero__iff, axiom,
    ((![X2 : real]: ((ord_less_eq_real @ (real_V646646907m_real @ X2) @ zero_zero_real) = (X2 = zero_zero_real))))). % norm_le_zero_iff
thf(fact_119_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_120_zero__less__norm__iff, axiom,
    ((![X2 : real]: ((ord_less_real @ zero_zero_real @ (real_V646646907m_real @ X2)) = (~ ((X2 = zero_zero_real))))))). % zero_less_norm_iff
thf(fact_121_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_122_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_123_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_124_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_125_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_126_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_127_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_128_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_129_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_130_add_Oinverse__inverse, axiom,
    ((![A : real]: ((uminus_uminus_real @ (uminus_uminus_real @ A)) = A)))). % add.inverse_inverse
thf(fact_131_neg__equal__zero, axiom,
    ((![A : real]: (((uminus_uminus_real @ A) = A) = (A = zero_zero_real))))). % neg_equal_zero
thf(fact_132_equal__neg__zero, axiom,
    ((![A : real]: ((A = (uminus_uminus_real @ A)) = (A = zero_zero_real))))). % equal_neg_zero
thf(fact_133_neg__equal__0__iff__equal, axiom,
    ((![A : complex]: (((uminus1204672759omplex @ A) = zero_zero_complex) = (A = zero_zero_complex))))). % neg_equal_0_iff_equal
thf(fact_134_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_135_neg__0__equal__iff__equal, axiom,
    ((![A : complex]: ((zero_zero_complex = (uminus1204672759omplex @ A)) = (zero_zero_complex = A))))). % neg_0_equal_iff_equal
thf(fact_136_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_137_add_Oinverse__neutral, axiom,
    (((uminus1204672759omplex @ zero_zero_complex) = zero_zero_complex))). % add.inverse_neutral
thf(fact_138_add_Oinverse__neutral, axiom,
    (((uminus_uminus_real @ zero_zero_real) = zero_zero_real))). % add.inverse_neutral
thf(fact_139_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_140_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_141_norm__minus__cancel, axiom,
    ((![X2 : real]: ((real_V646646907m_real @ (uminus_uminus_real @ X2)) = (real_V646646907m_real @ X2))))). % norm_minus_cancel
thf(fact_142_norm__minus__cancel, axiom,
    ((![X2 : complex]: ((real_V638595069omplex @ (uminus1204672759omplex @ X2)) = (real_V638595069omplex @ X2))))). % norm_minus_cancel
thf(fact_143_poly__1, axiom,
    ((![X2 : complex]: ((poly_complex2 @ one_one_poly_complex @ X2) = one_one_complex)))). % poly_1
thf(fact_144_poly__1, axiom,
    ((![X2 : real]: ((poly_real2 @ one_one_poly_real @ X2) = one_one_real)))). % poly_1
thf(fact_145_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

% Conjectures (1)
thf(conj_0, conjecture,
    ((ord_less_eq_real @ (real_V638595069omplex @ (poly_complex2 @ p @ z)) @ (real_V638595069omplex @ (poly_complex2 @ p @ w))))).
