% 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_747__5375210_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:32:45.219

% 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__Polynomial__Opoly_It__Nat__Onat_J, type,
    poly_nat : $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 (27)
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_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_Ozero__class_Ozero_001t__Complex__Ocomplex, type,
    zero_zero_complex : complex).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat, type,
    zero_zero_nat : nat).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    zero_z1746442943omplex : poly_complex).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    zero_zero_poly_nat : poly_nat).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__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__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__Nat__Onat, type,
    ord_less_eq_nat : nat > nat > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal, type,
    ord_less_eq_real : real > real > $o).
thf(sy_c_Polynomial_Opoly_001t__Complex__Ocomplex, type,
    poly_complex2 : poly_complex > complex > complex).
thf(sy_c_Polynomial_Opoly_001t__Nat__Onat, type,
    poly_nat2 : poly_nat > nat > nat).
thf(sy_c_Polynomial_Opoly_001t__Real__Oreal, type,
    poly_real2 : poly_real > real > real).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Complex__Ocomplex, type,
    real_V638595069omplex : complex > real).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Real__Oreal, type,
    real_V646646907m_real : real > real).
thf(sy_v_a____, type,
    a : complex).
thf(sy_v_c____, type,
    c : complex).
thf(sy_v_k____, type,
    k : nat).
thf(sy_v_p, type,
    p : poly_complex).
thf(sy_v_pa____, type,
    pa : poly_complex).
thf(sy_v_q____, type,
    q : poly_complex).
thf(sy_v_s____, type,
    s : poly_complex).
thf(sy_v_thesis____, type,
    thesis : $o).
thf(sy_v_w____, type,
    w : complex).

% Relevant facts (236)
thf(fact_0_nc, axiom,
    ((~ ((fundam1158420650omplex @ (poly_complex2 @ p)))))). % nc
thf(fact_1__092_060open_062_092_060exists_062m_0620_O_A_092_060forall_062z_O_Acmod_Az_A_092_060le_062_Acmod_Aw_A_092_060longrightarrow_062_Acmod_A_Ipoly_As_Az_J_A_092_060le_062_Am_092_060close_062, axiom,
    ((?[M : real]: ((ord_less_real @ zero_zero_real @ M) & (![Z : complex]: ((ord_less_eq_real @ (real_V638595069omplex @ Z) @ (real_V638595069omplex @ w)) => (ord_less_eq_real @ (real_V638595069omplex @ (poly_complex2 @ s @ Z)) @ M))))))). % \<open>\<exists>m>0. \<forall>z. cmod z \<le> cmod w \<longrightarrow> cmod (poly s z) \<le> m\<close>
thf(fact_2_real__sup__exists, axiom,
    ((![P : real > $o]: ((?[X_1 : real]: (P @ X_1)) => ((?[Z : real]: (![X : real]: ((P @ X) => (ord_less_real @ X @ Z)))) => (?[S : real]: (![Y : real]: ((?[X2 : real]: (((P @ X2)) & ((ord_less_real @ Y @ X2)))) = (ord_less_real @ Y @ S))))))))). % real_sup_exists
thf(fact_3_poly__bound__exists, axiom,
    ((![R : real, P2 : poly_real]: (?[M : real]: ((ord_less_real @ zero_zero_real @ M) & (![Z : real]: ((ord_less_eq_real @ (real_V646646907m_real @ Z) @ R) => (ord_less_eq_real @ (real_V646646907m_real @ (poly_real2 @ P2 @ Z)) @ M)))))))). % poly_bound_exists
thf(fact_4_poly__bound__exists, axiom,
    ((![R : real, P2 : poly_complex]: (?[M : real]: ((ord_less_real @ zero_zero_real @ M) & (![Z : complex]: ((ord_less_eq_real @ (real_V638595069omplex @ Z) @ R) => (ord_less_eq_real @ (real_V638595069omplex @ (poly_complex2 @ P2 @ Z)) @ M)))))))). % poly_bound_exists
thf(fact_5_poly__minimum__modulus, axiom,
    ((![P2 : poly_complex]: (?[Z2 : complex]: (![W : complex]: (ord_less_eq_real @ (real_V638595069omplex @ (poly_complex2 @ P2 @ Z2)) @ (real_V638595069omplex @ (poly_complex2 @ P2 @ W)))))))). % poly_minimum_modulus
thf(fact_6_poly__minimum__modulus__disc, axiom,
    ((![R : real, P2 : poly_complex]: (?[Z2 : complex]: (![W : complex]: ((ord_less_eq_real @ (real_V638595069omplex @ W) @ R) => (ord_less_eq_real @ (real_V638595069omplex @ (poly_complex2 @ P2 @ Z2)) @ (real_V638595069omplex @ (poly_complex2 @ P2 @ W))))))))). % poly_minimum_modulus_disc
thf(fact_7_norm__le__zero__iff, axiom,
    ((![X3 : real]: ((ord_less_eq_real @ (real_V646646907m_real @ X3) @ zero_zero_real) = (X3 = zero_zero_real))))). % norm_le_zero_iff
thf(fact_8_norm__le__zero__iff, axiom,
    ((![X3 : complex]: ((ord_less_eq_real @ (real_V638595069omplex @ X3) @ zero_zero_real) = (X3 = zero_zero_complex))))). % norm_le_zero_iff
thf(fact_9_zero__less__norm__iff, axiom,
    ((![X3 : real]: ((ord_less_real @ zero_zero_real @ (real_V646646907m_real @ X3)) = (~ ((X3 = zero_zero_real))))))). % zero_less_norm_iff
thf(fact_10_zero__less__norm__iff, axiom,
    ((![X3 : complex]: ((ord_less_real @ zero_zero_real @ (real_V638595069omplex @ X3)) = (~ ((X3 = zero_zero_complex))))))). % zero_less_norm_iff
thf(fact_11_norm__zero, axiom,
    (((real_V646646907m_real @ zero_zero_real) = zero_zero_real))). % norm_zero
thf(fact_12_norm__zero, axiom,
    (((real_V638595069omplex @ zero_zero_complex) = zero_zero_real))). % norm_zero
thf(fact_13_norm__eq__zero, axiom,
    ((![X3 : real]: (((real_V646646907m_real @ X3) = zero_zero_real) = (X3 = zero_zero_real))))). % norm_eq_zero
thf(fact_14_norm__eq__zero, axiom,
    ((![X3 : complex]: (((real_V638595069omplex @ X3) = zero_zero_real) = (X3 = zero_zero_complex))))). % norm_eq_zero
thf(fact_15_poly__0, axiom,
    ((![X3 : real]: ((poly_real2 @ zero_zero_poly_real @ X3) = zero_zero_real)))). % poly_0
thf(fact_16_poly__0, axiom,
    ((![X3 : complex]: ((poly_complex2 @ zero_z1746442943omplex @ X3) = zero_zero_complex)))). % poly_0
thf(fact_17_poly__0, axiom,
    ((![X3 : nat]: ((poly_nat2 @ zero_zero_poly_nat @ X3) = zero_zero_nat)))). % poly_0
thf(fact_18_not__gr__zero, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr_zero
thf(fact_19_le__zero__eq, axiom,
    ((![N : nat]: ((ord_less_eq_nat @ N @ zero_zero_nat) = (N = zero_zero_nat))))). % le_zero_eq
thf(fact_20_norm__ge__zero, axiom,
    ((![X3 : complex]: (ord_less_eq_real @ zero_zero_real @ (real_V638595069omplex @ X3))))). % norm_ge_zero
thf(fact_21_norm__ge__zero, axiom,
    ((![X3 : real]: (ord_less_eq_real @ zero_zero_real @ (real_V646646907m_real @ X3))))). % norm_ge_zero
thf(fact_22_norm__not__less__zero, axiom,
    ((![X3 : complex]: (~ ((ord_less_real @ (real_V638595069omplex @ X3) @ zero_zero_real)))))). % norm_not_less_zero
thf(fact_23_norm__not__less__zero, axiom,
    ((![X3 : real]: (~ ((ord_less_real @ (real_V646646907m_real @ X3) @ zero_zero_real)))))). % norm_not_less_zero
thf(fact_24__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_25_order__refl, axiom,
    ((![X3 : real]: (ord_less_eq_real @ X3 @ X3)))). % order_refl
thf(fact_26_kas_I1_J, axiom,
    ((~ ((a = zero_zero_complex))))). % kas(1)
thf(fact_27_kas_I2_J, axiom,
    ((~ ((k = zero_zero_nat))))). % kas(2)
thf(fact_28_less_Oprems, axiom,
    ((~ ((fundam1158420650omplex @ (poly_complex2 @ pa)))))). % less.prems
thf(fact_29_c, axiom,
    ((![W : complex]: (ord_less_eq_real @ (real_V638595069omplex @ (poly_complex2 @ pa @ c)) @ (real_V638595069omplex @ (poly_complex2 @ pa @ W)))))). % c
thf(fact_30_constant__def, axiom,
    ((fundam1158420650omplex = (^[F : complex > complex]: (![X2 : complex]: (![Y2 : complex]: ((F @ X2) = (F @ Y2)))))))). % constant_def
thf(fact_31_poly__IVT__pos, axiom,
    ((![A : real, B : real, P2 : poly_real]: ((ord_less_real @ A @ B) => ((ord_less_real @ (poly_real2 @ P2 @ A) @ zero_zero_real) => ((ord_less_real @ zero_zero_real @ (poly_real2 @ P2 @ B)) => (?[X : real]: ((ord_less_real @ A @ X) & ((ord_less_real @ X @ B) & ((poly_real2 @ P2 @ X) = zero_zero_real)))))))))). % poly_IVT_pos
thf(fact_32_poly__IVT__neg, axiom,
    ((![A : real, B : real, P2 : poly_real]: ((ord_less_real @ A @ B) => ((ord_less_real @ zero_zero_real @ (poly_real2 @ P2 @ A)) => ((ord_less_real @ (poly_real2 @ P2 @ B) @ zero_zero_real) => (?[X : real]: ((ord_less_real @ A @ X) & ((ord_less_real @ X @ B) & ((poly_real2 @ P2 @ X) = zero_zero_real)))))))))). % poly_IVT_neg
thf(fact_33_zero__reorient, axiom,
    ((![X3 : real]: ((zero_zero_real = X3) = (X3 = zero_zero_real))))). % zero_reorient
thf(fact_34_zero__reorient, axiom,
    ((![X3 : complex]: ((zero_zero_complex = X3) = (X3 = zero_zero_complex))))). % zero_reorient
thf(fact_35_zero__reorient, axiom,
    ((![X3 : nat]: ((zero_zero_nat = X3) = (X3 = zero_zero_nat))))). % zero_reorient
thf(fact_36_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_37_dual__order_Oeq__iff, axiom,
    (((^[Y3 : real]: (^[Z3 : real]: (Y3 = Z3))) = (^[A2 : real]: (^[B2 : real]: (((ord_less_eq_real @ B2 @ A2)) & ((ord_less_eq_real @ A2 @ B2)))))))). % dual_order.eq_iff
thf(fact_38_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_39_linorder__wlog, axiom,
    ((![P : real > real > $o, A : real, B : real]: ((![A3 : real, B3 : real]: ((ord_less_eq_real @ A3 @ B3) => (P @ A3 @ B3))) => ((![A3 : real, B3 : real]: ((P @ B3 @ A3) => (P @ A3 @ B3))) => (P @ A @ B)))))). % linorder_wlog
thf(fact_40_dual__order_Orefl, axiom,
    ((![A : real]: (ord_less_eq_real @ A @ A)))). % dual_order.refl
thf(fact_41_order__trans, axiom,
    ((![X3 : real, Y4 : real, Z4 : real]: ((ord_less_eq_real @ X3 @ Y4) => ((ord_less_eq_real @ Y4 @ Z4) => (ord_less_eq_real @ X3 @ Z4)))))). % order_trans
thf(fact_42_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_43_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_44_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_45_order__class_Oorder_Oeq__iff, axiom,
    (((^[Y3 : real]: (^[Z3 : real]: (Y3 = Z3))) = (^[A2 : real]: (^[B2 : real]: (((ord_less_eq_real @ A2 @ B2)) & ((ord_less_eq_real @ B2 @ A2)))))))). % order_class.order.eq_iff
thf(fact_46_antisym__conv, axiom,
    ((![Y4 : real, X3 : real]: ((ord_less_eq_real @ Y4 @ X3) => ((ord_less_eq_real @ X3 @ Y4) = (X3 = Y4)))))). % antisym_conv
thf(fact_47_le__cases3, axiom,
    ((![X3 : real, Y4 : real, Z4 : real]: (((ord_less_eq_real @ X3 @ Y4) => (~ ((ord_less_eq_real @ Y4 @ Z4)))) => (((ord_less_eq_real @ Y4 @ X3) => (~ ((ord_less_eq_real @ X3 @ Z4)))) => (((ord_less_eq_real @ X3 @ Z4) => (~ ((ord_less_eq_real @ Z4 @ Y4)))) => (((ord_less_eq_real @ Z4 @ Y4) => (~ ((ord_less_eq_real @ Y4 @ X3)))) => (((ord_less_eq_real @ Y4 @ Z4) => (~ ((ord_less_eq_real @ Z4 @ X3)))) => (~ (((ord_less_eq_real @ Z4 @ X3) => (~ ((ord_less_eq_real @ X3 @ Y4)))))))))))))). % le_cases3
thf(fact_48_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_49_le__cases, axiom,
    ((![X3 : real, Y4 : real]: ((~ ((ord_less_eq_real @ X3 @ Y4))) => (ord_less_eq_real @ Y4 @ X3))))). % le_cases
thf(fact_50_eq__refl, axiom,
    ((![X3 : real, Y4 : real]: ((X3 = Y4) => (ord_less_eq_real @ X3 @ Y4))))). % eq_refl
thf(fact_51_linear, axiom,
    ((![X3 : real, Y4 : real]: ((ord_less_eq_real @ X3 @ Y4) | (ord_less_eq_real @ Y4 @ X3))))). % linear
thf(fact_52_antisym, axiom,
    ((![X3 : real, Y4 : real]: ((ord_less_eq_real @ X3 @ Y4) => ((ord_less_eq_real @ Y4 @ X3) => (X3 = Y4)))))). % antisym
thf(fact_53_eq__iff, axiom,
    (((^[Y3 : real]: (^[Z3 : real]: (Y3 = Z3))) = (^[X2 : real]: (^[Y2 : real]: (((ord_less_eq_real @ X2 @ Y2)) & ((ord_less_eq_real @ Y2 @ X2)))))))). % eq_iff
thf(fact_54_ord__le__eq__subst, axiom,
    ((![A : real, B : real, F2 : real > real, C2 : real]: ((ord_less_eq_real @ A @ B) => (((F2 @ B) = C2) => ((![X : real, Y5 : real]: ((ord_less_eq_real @ X @ Y5) => (ord_less_eq_real @ (F2 @ X) @ (F2 @ Y5)))) => (ord_less_eq_real @ (F2 @ A) @ C2))))))). % ord_le_eq_subst
thf(fact_55_ord__eq__le__subst, axiom,
    ((![A : real, F2 : real > real, B : real, C2 : real]: ((A = (F2 @ B)) => ((ord_less_eq_real @ B @ C2) => ((![X : real, Y5 : real]: ((ord_less_eq_real @ X @ Y5) => (ord_less_eq_real @ (F2 @ X) @ (F2 @ Y5)))) => (ord_less_eq_real @ A @ (F2 @ C2)))))))). % ord_eq_le_subst
thf(fact_56_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) => ((![X : real, Y5 : real]: ((ord_less_eq_real @ X @ Y5) => (ord_less_eq_real @ (F2 @ X) @ (F2 @ Y5)))) => (ord_less_eq_real @ (F2 @ A) @ C2))))))). % order_subst2
thf(fact_57_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) => ((![X : real, Y5 : real]: ((ord_less_eq_real @ X @ Y5) => (ord_less_eq_real @ (F2 @ X) @ (F2 @ Y5)))) => (ord_less_eq_real @ A @ (F2 @ C2)))))))). % order_subst1
thf(fact_58_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_59_dual__order_Ostrict__implies__not__eq, axiom,
    ((![B : nat, A : nat]: ((ord_less_nat @ B @ A) => (~ ((A = B))))))). % dual_order.strict_implies_not_eq
thf(fact_60_order_Ostrict__implies__not__eq, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ B) => (~ ((A = B))))))). % order.strict_implies_not_eq
thf(fact_61_order_Ostrict__implies__not__eq, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (~ ((A = B))))))). % order.strict_implies_not_eq
thf(fact_62_not__less__iff__gr__or__eq, axiom,
    ((![X3 : real, Y4 : real]: ((~ ((ord_less_real @ X3 @ Y4))) = (((ord_less_real @ Y4 @ X3)) | ((X3 = Y4))))))). % not_less_iff_gr_or_eq
thf(fact_63_not__less__iff__gr__or__eq, axiom,
    ((![X3 : nat, Y4 : nat]: ((~ ((ord_less_nat @ X3 @ Y4))) = (((ord_less_nat @ Y4 @ X3)) | ((X3 = Y4))))))). % not_less_iff_gr_or_eq
thf(fact_64_dual__order_Ostrict__trans, axiom,
    ((![B : real, A : real, C2 : real]: ((ord_less_real @ B @ A) => ((ord_less_real @ C2 @ B) => (ord_less_real @ C2 @ A)))))). % dual_order.strict_trans
thf(fact_65_dual__order_Ostrict__trans, axiom,
    ((![B : nat, A : nat, C2 : nat]: ((ord_less_nat @ B @ A) => ((ord_less_nat @ C2 @ B) => (ord_less_nat @ C2 @ A)))))). % dual_order.strict_trans
thf(fact_66_linorder__less__wlog, axiom,
    ((![P : real > real > $o, A : real, B : real]: ((![A3 : real, B3 : real]: ((ord_less_real @ A3 @ B3) => (P @ A3 @ B3))) => ((![A3 : real]: (P @ A3 @ A3)) => ((![A3 : real, B3 : real]: ((P @ B3 @ A3) => (P @ A3 @ B3))) => (P @ A @ B))))))). % linorder_less_wlog
thf(fact_67_linorder__less__wlog, axiom,
    ((![P : nat > nat > $o, A : nat, B : nat]: ((![A3 : nat, B3 : nat]: ((ord_less_nat @ A3 @ B3) => (P @ A3 @ B3))) => ((![A3 : nat]: (P @ A3 @ A3)) => ((![A3 : nat, B3 : nat]: ((P @ B3 @ A3) => (P @ A3 @ B3))) => (P @ A @ B))))))). % linorder_less_wlog
thf(fact_68_exists__least__iff, axiom,
    (((^[P3 : nat > $o]: (?[X4 : nat]: (P3 @ X4))) = (^[P4 : nat > $o]: (?[N2 : nat]: (((P4 @ N2)) & ((![M2 : nat]: (((ord_less_nat @ M2 @ N2)) => ((~ ((P4 @ M2))))))))))))). % exists_least_iff
thf(fact_69_less__imp__not__less, axiom,
    ((![X3 : real, Y4 : real]: ((ord_less_real @ X3 @ Y4) => (~ ((ord_less_real @ Y4 @ X3))))))). % less_imp_not_less
thf(fact_70_less__imp__not__less, axiom,
    ((![X3 : nat, Y4 : nat]: ((ord_less_nat @ X3 @ Y4) => (~ ((ord_less_nat @ Y4 @ X3))))))). % less_imp_not_less
thf(fact_71_order_Ostrict__trans, axiom,
    ((![A : real, B : real, C2 : real]: ((ord_less_real @ A @ B) => ((ord_less_real @ B @ C2) => (ord_less_real @ A @ C2)))))). % order.strict_trans
thf(fact_72_order_Ostrict__trans, axiom,
    ((![A : nat, B : nat, C2 : nat]: ((ord_less_nat @ A @ B) => ((ord_less_nat @ B @ C2) => (ord_less_nat @ A @ C2)))))). % order.strict_trans
thf(fact_73_dual__order_Oirrefl, axiom,
    ((![A : real]: (~ ((ord_less_real @ A @ A)))))). % dual_order.irrefl
thf(fact_74_dual__order_Oirrefl, axiom,
    ((![A : nat]: (~ ((ord_less_nat @ A @ A)))))). % dual_order.irrefl
thf(fact_75_linorder__cases, axiom,
    ((![X3 : real, Y4 : real]: ((~ ((ord_less_real @ X3 @ Y4))) => ((~ ((X3 = Y4))) => (ord_less_real @ Y4 @ X3)))))). % linorder_cases
thf(fact_76_linorder__cases, axiom,
    ((![X3 : nat, Y4 : nat]: ((~ ((ord_less_nat @ X3 @ Y4))) => ((~ ((X3 = Y4))) => (ord_less_nat @ Y4 @ X3)))))). % linorder_cases
thf(fact_77_less__imp__triv, axiom,
    ((![X3 : real, Y4 : real, P : $o]: ((ord_less_real @ X3 @ Y4) => ((ord_less_real @ Y4 @ X3) => P))))). % less_imp_triv
thf(fact_78_less__imp__triv, axiom,
    ((![X3 : nat, Y4 : nat, P : $o]: ((ord_less_nat @ X3 @ Y4) => ((ord_less_nat @ Y4 @ X3) => P))))). % less_imp_triv
thf(fact_79_less__imp__not__eq2, axiom,
    ((![X3 : real, Y4 : real]: ((ord_less_real @ X3 @ Y4) => (~ ((Y4 = X3))))))). % less_imp_not_eq2
thf(fact_80_less__imp__not__eq2, axiom,
    ((![X3 : nat, Y4 : nat]: ((ord_less_nat @ X3 @ Y4) => (~ ((Y4 = X3))))))). % less_imp_not_eq2
thf(fact_81_antisym__conv3, axiom,
    ((![Y4 : real, X3 : real]: ((~ ((ord_less_real @ Y4 @ X3))) => ((~ ((ord_less_real @ X3 @ Y4))) = (X3 = Y4)))))). % antisym_conv3
thf(fact_82_antisym__conv3, axiom,
    ((![Y4 : nat, X3 : nat]: ((~ ((ord_less_nat @ Y4 @ X3))) => ((~ ((ord_less_nat @ X3 @ Y4))) = (X3 = Y4)))))). % antisym_conv3
thf(fact_83_less__induct, axiom,
    ((![P : nat > $o, A : nat]: ((![X : nat]: ((![Y : nat]: ((ord_less_nat @ Y @ X) => (P @ Y))) => (P @ X))) => (P @ A))))). % less_induct
thf(fact_84_less__not__sym, axiom,
    ((![X3 : real, Y4 : real]: ((ord_less_real @ X3 @ Y4) => (~ ((ord_less_real @ Y4 @ X3))))))). % less_not_sym
thf(fact_85_less__not__sym, axiom,
    ((![X3 : nat, Y4 : nat]: ((ord_less_nat @ X3 @ Y4) => (~ ((ord_less_nat @ Y4 @ X3))))))). % less_not_sym
thf(fact_86_less__imp__not__eq, axiom,
    ((![X3 : real, Y4 : real]: ((ord_less_real @ X3 @ Y4) => (~ ((X3 = Y4))))))). % less_imp_not_eq
thf(fact_87_less__imp__not__eq, axiom,
    ((![X3 : nat, Y4 : nat]: ((ord_less_nat @ X3 @ Y4) => (~ ((X3 = Y4))))))). % less_imp_not_eq
thf(fact_88_dual__order_Oasym, axiom,
    ((![B : real, A : real]: ((ord_less_real @ B @ A) => (~ ((ord_less_real @ A @ B))))))). % dual_order.asym
thf(fact_89_dual__order_Oasym, axiom,
    ((![B : nat, A : nat]: ((ord_less_nat @ B @ A) => (~ ((ord_less_nat @ A @ B))))))). % dual_order.asym
thf(fact_90_ord__less__eq__trans, axiom,
    ((![A : real, B : real, C2 : real]: ((ord_less_real @ A @ B) => ((B = C2) => (ord_less_real @ A @ C2)))))). % ord_less_eq_trans
thf(fact_91_ord__less__eq__trans, axiom,
    ((![A : nat, B : nat, C2 : nat]: ((ord_less_nat @ A @ B) => ((B = C2) => (ord_less_nat @ A @ C2)))))). % ord_less_eq_trans
thf(fact_92_ord__eq__less__trans, axiom,
    ((![A : real, B : real, C2 : real]: ((A = B) => ((ord_less_real @ B @ C2) => (ord_less_real @ A @ C2)))))). % ord_eq_less_trans
thf(fact_93_ord__eq__less__trans, axiom,
    ((![A : nat, B : nat, C2 : nat]: ((A = B) => ((ord_less_nat @ B @ C2) => (ord_less_nat @ A @ C2)))))). % ord_eq_less_trans
thf(fact_94_less__irrefl, axiom,
    ((![X3 : real]: (~ ((ord_less_real @ X3 @ X3)))))). % less_irrefl
thf(fact_95_less__irrefl, axiom,
    ((![X3 : nat]: (~ ((ord_less_nat @ X3 @ X3)))))). % less_irrefl
thf(fact_96_less__linear, axiom,
    ((![X3 : real, Y4 : real]: ((ord_less_real @ X3 @ Y4) | ((X3 = Y4) | (ord_less_real @ Y4 @ X3)))))). % less_linear
thf(fact_97_less__linear, axiom,
    ((![X3 : nat, Y4 : nat]: ((ord_less_nat @ X3 @ Y4) | ((X3 = Y4) | (ord_less_nat @ Y4 @ X3)))))). % less_linear
thf(fact_98_less__trans, axiom,
    ((![X3 : real, Y4 : real, Z4 : real]: ((ord_less_real @ X3 @ Y4) => ((ord_less_real @ Y4 @ Z4) => (ord_less_real @ X3 @ Z4)))))). % less_trans
thf(fact_99_less__trans, axiom,
    ((![X3 : nat, Y4 : nat, Z4 : nat]: ((ord_less_nat @ X3 @ Y4) => ((ord_less_nat @ Y4 @ Z4) => (ord_less_nat @ X3 @ Z4)))))). % less_trans
thf(fact_100_less__asym_H, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ B) => (~ ((ord_less_real @ B @ A))))))). % less_asym'
thf(fact_101_less__asym_H, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (~ ((ord_less_nat @ B @ A))))))). % less_asym'
thf(fact_102_less__asym, axiom,
    ((![X3 : real, Y4 : real]: ((ord_less_real @ X3 @ Y4) => (~ ((ord_less_real @ Y4 @ X3))))))). % less_asym
thf(fact_103_less__asym, axiom,
    ((![X3 : nat, Y4 : nat]: ((ord_less_nat @ X3 @ Y4) => (~ ((ord_less_nat @ Y4 @ X3))))))). % less_asym
thf(fact_104_less__imp__neq, axiom,
    ((![X3 : real, Y4 : real]: ((ord_less_real @ X3 @ Y4) => (~ ((X3 = Y4))))))). % less_imp_neq
thf(fact_105_less__imp__neq, axiom,
    ((![X3 : nat, Y4 : nat]: ((ord_less_nat @ X3 @ Y4) => (~ ((X3 = Y4))))))). % less_imp_neq
thf(fact_106_dense, axiom,
    ((![X3 : real, Y4 : real]: ((ord_less_real @ X3 @ Y4) => (?[Z2 : real]: ((ord_less_real @ X3 @ Z2) & (ord_less_real @ Z2 @ Y4))))))). % dense
thf(fact_107_order_Oasym, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ B) => (~ ((ord_less_real @ B @ A))))))). % order.asym
thf(fact_108_order_Oasym, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (~ ((ord_less_nat @ B @ A))))))). % order.asym
thf(fact_109_neq__iff, axiom,
    ((![X3 : real, Y4 : real]: ((~ ((X3 = Y4))) = (((ord_less_real @ X3 @ Y4)) | ((ord_less_real @ Y4 @ X3))))))). % neq_iff
thf(fact_110_neq__iff, axiom,
    ((![X3 : nat, Y4 : nat]: ((~ ((X3 = Y4))) = (((ord_less_nat @ X3 @ Y4)) | ((ord_less_nat @ Y4 @ X3))))))). % neq_iff
thf(fact_111_neqE, axiom,
    ((![X3 : real, Y4 : real]: ((~ ((X3 = Y4))) => ((~ ((ord_less_real @ X3 @ Y4))) => (ord_less_real @ Y4 @ X3)))))). % neqE
thf(fact_112_neqE, axiom,
    ((![X3 : nat, Y4 : nat]: ((~ ((X3 = Y4))) => ((~ ((ord_less_nat @ X3 @ Y4))) => (ord_less_nat @ Y4 @ X3)))))). % neqE
thf(fact_113_gt__ex, axiom,
    ((![X3 : real]: (?[X_12 : real]: (ord_less_real @ X3 @ X_12))))). % gt_ex
thf(fact_114_gt__ex, axiom,
    ((![X3 : nat]: (?[X_12 : nat]: (ord_less_nat @ X3 @ X_12))))). % gt_ex
thf(fact_115_lt__ex, axiom,
    ((![X3 : real]: (?[Y5 : real]: (ord_less_real @ Y5 @ X3))))). % lt_ex
thf(fact_116_order__less__subst2, axiom,
    ((![A : real, B : real, F2 : real > real, C2 : real]: ((ord_less_real @ A @ B) => ((ord_less_real @ (F2 @ B) @ C2) => ((![X : real, Y5 : real]: ((ord_less_real @ X @ Y5) => (ord_less_real @ (F2 @ X) @ (F2 @ Y5)))) => (ord_less_real @ (F2 @ A) @ C2))))))). % order_less_subst2
thf(fact_117_order__less__subst2, axiom,
    ((![A : real, B : real, F2 : real > nat, C2 : nat]: ((ord_less_real @ A @ B) => ((ord_less_nat @ (F2 @ B) @ C2) => ((![X : real, Y5 : real]: ((ord_less_real @ X @ Y5) => (ord_less_nat @ (F2 @ X) @ (F2 @ Y5)))) => (ord_less_nat @ (F2 @ A) @ C2))))))). % order_less_subst2
thf(fact_118_order__less__subst2, axiom,
    ((![A : nat, B : nat, F2 : nat > real, C2 : real]: ((ord_less_nat @ A @ B) => ((ord_less_real @ (F2 @ B) @ C2) => ((![X : nat, Y5 : nat]: ((ord_less_nat @ X @ Y5) => (ord_less_real @ (F2 @ X) @ (F2 @ Y5)))) => (ord_less_real @ (F2 @ A) @ C2))))))). % order_less_subst2
thf(fact_119_order__less__subst2, axiom,
    ((![A : nat, B : nat, F2 : nat > nat, C2 : nat]: ((ord_less_nat @ A @ B) => ((ord_less_nat @ (F2 @ B) @ C2) => ((![X : nat, Y5 : nat]: ((ord_less_nat @ X @ Y5) => (ord_less_nat @ (F2 @ X) @ (F2 @ Y5)))) => (ord_less_nat @ (F2 @ A) @ C2))))))). % order_less_subst2
thf(fact_120_order__less__subst1, axiom,
    ((![A : real, F2 : real > real, B : real, C2 : real]: ((ord_less_real @ A @ (F2 @ B)) => ((ord_less_real @ B @ C2) => ((![X : real, Y5 : real]: ((ord_less_real @ X @ Y5) => (ord_less_real @ (F2 @ X) @ (F2 @ Y5)))) => (ord_less_real @ A @ (F2 @ C2)))))))). % order_less_subst1
thf(fact_121_order__less__subst1, axiom,
    ((![A : real, F2 : nat > real, B : nat, C2 : nat]: ((ord_less_real @ A @ (F2 @ B)) => ((ord_less_nat @ B @ C2) => ((![X : nat, Y5 : nat]: ((ord_less_nat @ X @ Y5) => (ord_less_real @ (F2 @ X) @ (F2 @ Y5)))) => (ord_less_real @ A @ (F2 @ C2)))))))). % order_less_subst1
thf(fact_122_order__less__subst1, axiom,
    ((![A : nat, F2 : real > nat, B : real, C2 : real]: ((ord_less_nat @ A @ (F2 @ B)) => ((ord_less_real @ B @ C2) => ((![X : real, Y5 : real]: ((ord_less_real @ X @ Y5) => (ord_less_nat @ (F2 @ X) @ (F2 @ Y5)))) => (ord_less_nat @ A @ (F2 @ C2)))))))). % order_less_subst1
thf(fact_123_order__less__subst1, axiom,
    ((![A : nat, F2 : nat > nat, B : nat, C2 : nat]: ((ord_less_nat @ A @ (F2 @ B)) => ((ord_less_nat @ B @ C2) => ((![X : nat, Y5 : nat]: ((ord_less_nat @ X @ Y5) => (ord_less_nat @ (F2 @ X) @ (F2 @ Y5)))) => (ord_less_nat @ A @ (F2 @ C2)))))))). % order_less_subst1
thf(fact_124_ord__less__eq__subst, axiom,
    ((![A : real, B : real, F2 : real > real, C2 : real]: ((ord_less_real @ A @ B) => (((F2 @ B) = C2) => ((![X : real, Y5 : real]: ((ord_less_real @ X @ Y5) => (ord_less_real @ (F2 @ X) @ (F2 @ Y5)))) => (ord_less_real @ (F2 @ A) @ C2))))))). % ord_less_eq_subst
thf(fact_125_ord__less__eq__subst, axiom,
    ((![A : real, B : real, F2 : real > nat, C2 : nat]: ((ord_less_real @ A @ B) => (((F2 @ B) = C2) => ((![X : real, Y5 : real]: ((ord_less_real @ X @ Y5) => (ord_less_nat @ (F2 @ X) @ (F2 @ Y5)))) => (ord_less_nat @ (F2 @ A) @ C2))))))). % ord_less_eq_subst
thf(fact_126_ord__less__eq__subst, axiom,
    ((![A : nat, B : nat, F2 : nat > real, C2 : real]: ((ord_less_nat @ A @ B) => (((F2 @ B) = C2) => ((![X : nat, Y5 : nat]: ((ord_less_nat @ X @ Y5) => (ord_less_real @ (F2 @ X) @ (F2 @ Y5)))) => (ord_less_real @ (F2 @ A) @ C2))))))). % ord_less_eq_subst
thf(fact_127_ord__less__eq__subst, axiom,
    ((![A : nat, B : nat, F2 : nat > nat, C2 : nat]: ((ord_less_nat @ A @ B) => (((F2 @ B) = C2) => ((![X : nat, Y5 : nat]: ((ord_less_nat @ X @ Y5) => (ord_less_nat @ (F2 @ X) @ (F2 @ Y5)))) => (ord_less_nat @ (F2 @ A) @ C2))))))). % ord_less_eq_subst
thf(fact_128_ord__eq__less__subst, axiom,
    ((![A : real, F2 : real > real, B : real, C2 : real]: ((A = (F2 @ B)) => ((ord_less_real @ B @ C2) => ((![X : real, Y5 : real]: ((ord_less_real @ X @ Y5) => (ord_less_real @ (F2 @ X) @ (F2 @ Y5)))) => (ord_less_real @ A @ (F2 @ C2)))))))). % ord_eq_less_subst
thf(fact_129_ord__eq__less__subst, axiom,
    ((![A : nat, F2 : real > nat, B : real, C2 : real]: ((A = (F2 @ B)) => ((ord_less_real @ B @ C2) => ((![X : real, Y5 : real]: ((ord_less_real @ X @ Y5) => (ord_less_nat @ (F2 @ X) @ (F2 @ Y5)))) => (ord_less_nat @ A @ (F2 @ C2)))))))). % ord_eq_less_subst
thf(fact_130_ord__eq__less__subst, axiom,
    ((![A : real, F2 : nat > real, B : nat, C2 : nat]: ((A = (F2 @ B)) => ((ord_less_nat @ B @ C2) => ((![X : nat, Y5 : nat]: ((ord_less_nat @ X @ Y5) => (ord_less_real @ (F2 @ X) @ (F2 @ Y5)))) => (ord_less_real @ A @ (F2 @ C2)))))))). % ord_eq_less_subst
thf(fact_131_ord__eq__less__subst, axiom,
    ((![A : nat, F2 : nat > nat, B : nat, C2 : nat]: ((A = (F2 @ B)) => ((ord_less_nat @ B @ C2) => ((![X : nat, Y5 : nat]: ((ord_less_nat @ X @ Y5) => (ord_less_nat @ (F2 @ X) @ (F2 @ Y5)))) => (ord_less_nat @ A @ (F2 @ C2)))))))). % ord_eq_less_subst
thf(fact_132_poly__eq__poly__eq__iff, axiom,
    ((![P2 : poly_complex, Q : poly_complex]: (((poly_complex2 @ P2) = (poly_complex2 @ Q)) = (P2 = Q))))). % poly_eq_poly_eq_iff
thf(fact_133_poly__eq__poly__eq__iff, axiom,
    ((![P2 : poly_real, Q : poly_real]: (((poly_real2 @ P2) = (poly_real2 @ Q)) = (P2 = Q))))). % poly_eq_poly_eq_iff
thf(fact_134_zero__le, axiom,
    ((![X3 : nat]: (ord_less_eq_nat @ zero_zero_nat @ X3)))). % zero_le
thf(fact_135_zero__less__iff__neq__zero, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ N) = (~ ((N = zero_zero_nat))))))). % zero_less_iff_neq_zero
thf(fact_136_gr__implies__not__zero, axiom,
    ((![M3 : nat, N : nat]: ((ord_less_nat @ M3 @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not_zero
thf(fact_137_not__less__zero, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less_zero
thf(fact_138_gr__zeroI, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr_zeroI
thf(fact_139_order_Onot__eq__order__implies__strict, axiom,
    ((![A : nat, B : nat]: ((~ ((A = B))) => ((ord_less_eq_nat @ A @ B) => (ord_less_nat @ A @ B)))))). % order.not_eq_order_implies_strict
thf(fact_140_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_141_dual__order_Ostrict__implies__order, axiom,
    ((![B : nat, A : nat]: ((ord_less_nat @ B @ A) => (ord_less_eq_nat @ B @ A))))). % dual_order.strict_implies_order
thf(fact_142_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_143_dual__order_Ostrict__iff__order, axiom,
    ((ord_less_nat = (^[B2 : nat]: (^[A2 : nat]: (((ord_less_eq_nat @ B2 @ A2)) & ((~ ((A2 = B2)))))))))). % dual_order.strict_iff_order
thf(fact_144_dual__order_Ostrict__iff__order, axiom,
    ((ord_less_real = (^[B2 : real]: (^[A2 : real]: (((ord_less_eq_real @ B2 @ A2)) & ((~ ((A2 = B2)))))))))). % dual_order.strict_iff_order
thf(fact_145_dual__order_Oorder__iff__strict, axiom,
    ((ord_less_eq_nat = (^[B2 : nat]: (^[A2 : nat]: (((ord_less_nat @ B2 @ A2)) | ((A2 = B2)))))))). % dual_order.order_iff_strict
thf(fact_146_dual__order_Oorder__iff__strict, axiom,
    ((ord_less_eq_real = (^[B2 : real]: (^[A2 : real]: (((ord_less_real @ B2 @ A2)) | ((A2 = B2)))))))). % dual_order.order_iff_strict
thf(fact_147_order_Ostrict__implies__order, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (ord_less_eq_nat @ A @ B))))). % order.strict_implies_order
thf(fact_148_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_149_dense__le__bounded, axiom,
    ((![X3 : real, Y4 : real, Z4 : real]: ((ord_less_real @ X3 @ Y4) => ((![W2 : real]: ((ord_less_real @ X3 @ W2) => ((ord_less_real @ W2 @ Y4) => (ord_less_eq_real @ W2 @ Z4)))) => (ord_less_eq_real @ Y4 @ Z4)))))). % dense_le_bounded
thf(fact_150_dense__ge__bounded, axiom,
    ((![Z4 : real, X3 : real, Y4 : real]: ((ord_less_real @ Z4 @ X3) => ((![W2 : real]: ((ord_less_real @ Z4 @ W2) => ((ord_less_real @ W2 @ X3) => (ord_less_eq_real @ Y4 @ W2)))) => (ord_less_eq_real @ Y4 @ Z4)))))). % dense_ge_bounded
thf(fact_151_dual__order_Ostrict__trans2, axiom,
    ((![B : nat, A : nat, C2 : nat]: ((ord_less_nat @ B @ A) => ((ord_less_eq_nat @ C2 @ B) => (ord_less_nat @ C2 @ A)))))). % dual_order.strict_trans2
thf(fact_152_dual__order_Ostrict__trans2, axiom,
    ((![B : real, A : real, C2 : real]: ((ord_less_real @ B @ A) => ((ord_less_eq_real @ C2 @ B) => (ord_less_real @ C2 @ A)))))). % dual_order.strict_trans2
thf(fact_153_dual__order_Ostrict__trans1, axiom,
    ((![B : nat, A : nat, C2 : nat]: ((ord_less_eq_nat @ B @ A) => ((ord_less_nat @ C2 @ B) => (ord_less_nat @ C2 @ A)))))). % dual_order.strict_trans1
thf(fact_154_dual__order_Ostrict__trans1, axiom,
    ((![B : real, A : real, C2 : real]: ((ord_less_eq_real @ B @ A) => ((ord_less_real @ C2 @ B) => (ord_less_real @ C2 @ A)))))). % dual_order.strict_trans1
thf(fact_155_order_Ostrict__iff__order, axiom,
    ((ord_less_nat = (^[A2 : nat]: (^[B2 : nat]: (((ord_less_eq_nat @ A2 @ B2)) & ((~ ((A2 = B2)))))))))). % order.strict_iff_order
thf(fact_156_order_Ostrict__iff__order, axiom,
    ((ord_less_real = (^[A2 : real]: (^[B2 : real]: (((ord_less_eq_real @ A2 @ B2)) & ((~ ((A2 = B2)))))))))). % order.strict_iff_order
thf(fact_157_order_Oorder__iff__strict, axiom,
    ((ord_less_eq_nat = (^[A2 : nat]: (^[B2 : nat]: (((ord_less_nat @ A2 @ B2)) | ((A2 = B2)))))))). % order.order_iff_strict
thf(fact_158_order_Oorder__iff__strict, axiom,
    ((ord_less_eq_real = (^[A2 : real]: (^[B2 : real]: (((ord_less_real @ A2 @ B2)) | ((A2 = B2)))))))). % order.order_iff_strict
thf(fact_159_order_Ostrict__trans2, axiom,
    ((![A : nat, B : nat, C2 : nat]: ((ord_less_nat @ A @ B) => ((ord_less_eq_nat @ B @ C2) => (ord_less_nat @ A @ C2)))))). % order.strict_trans2
thf(fact_160_order_Ostrict__trans2, axiom,
    ((![A : real, B : real, C2 : real]: ((ord_less_real @ A @ B) => ((ord_less_eq_real @ B @ C2) => (ord_less_real @ A @ C2)))))). % order.strict_trans2
thf(fact_161_order_Ostrict__trans1, axiom,
    ((![A : nat, B : nat, C2 : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_nat @ B @ C2) => (ord_less_nat @ A @ C2)))))). % order.strict_trans1
thf(fact_162_order_Ostrict__trans1, axiom,
    ((![A : real, B : real, C2 : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_real @ B @ C2) => (ord_less_real @ A @ C2)))))). % order.strict_trans1
thf(fact_163_not__le__imp__less, axiom,
    ((![Y4 : nat, X3 : nat]: ((~ ((ord_less_eq_nat @ Y4 @ X3))) => (ord_less_nat @ X3 @ Y4))))). % not_le_imp_less
thf(fact_164_not__le__imp__less, axiom,
    ((![Y4 : real, X3 : real]: ((~ ((ord_less_eq_real @ Y4 @ X3))) => (ord_less_real @ X3 @ Y4))))). % not_le_imp_less
thf(fact_165_less__le__not__le, axiom,
    ((ord_less_nat = (^[X2 : nat]: (^[Y2 : nat]: (((ord_less_eq_nat @ X2 @ Y2)) & ((~ ((ord_less_eq_nat @ Y2 @ X2)))))))))). % less_le_not_le
thf(fact_166_less__le__not__le, axiom,
    ((ord_less_real = (^[X2 : real]: (^[Y2 : real]: (((ord_less_eq_real @ X2 @ Y2)) & ((~ ((ord_less_eq_real @ Y2 @ X2)))))))))). % less_le_not_le
thf(fact_167_le__imp__less__or__eq, axiom,
    ((![X3 : nat, Y4 : nat]: ((ord_less_eq_nat @ X3 @ Y4) => ((ord_less_nat @ X3 @ Y4) | (X3 = Y4)))))). % le_imp_less_or_eq
thf(fact_168_le__imp__less__or__eq, axiom,
    ((![X3 : real, Y4 : real]: ((ord_less_eq_real @ X3 @ Y4) => ((ord_less_real @ X3 @ Y4) | (X3 = Y4)))))). % le_imp_less_or_eq
thf(fact_169_le__less__linear, axiom,
    ((![X3 : nat, Y4 : nat]: ((ord_less_eq_nat @ X3 @ Y4) | (ord_less_nat @ Y4 @ X3))))). % le_less_linear
thf(fact_170_le__less__linear, axiom,
    ((![X3 : real, Y4 : real]: ((ord_less_eq_real @ X3 @ Y4) | (ord_less_real @ Y4 @ X3))))). % le_less_linear
thf(fact_171_dense__le, axiom,
    ((![Y4 : real, Z4 : real]: ((![X : real]: ((ord_less_real @ X @ Y4) => (ord_less_eq_real @ X @ Z4))) => (ord_less_eq_real @ Y4 @ Z4))))). % dense_le
thf(fact_172_dense__ge, axiom,
    ((![Z4 : real, Y4 : real]: ((![X : real]: ((ord_less_real @ Z4 @ X) => (ord_less_eq_real @ Y4 @ X))) => (ord_less_eq_real @ Y4 @ Z4))))). % dense_ge
thf(fact_173_less__le__trans, axiom,
    ((![X3 : nat, Y4 : nat, Z4 : nat]: ((ord_less_nat @ X3 @ Y4) => ((ord_less_eq_nat @ Y4 @ Z4) => (ord_less_nat @ X3 @ Z4)))))). % less_le_trans
thf(fact_174_less__le__trans, axiom,
    ((![X3 : real, Y4 : real, Z4 : real]: ((ord_less_real @ X3 @ Y4) => ((ord_less_eq_real @ Y4 @ Z4) => (ord_less_real @ X3 @ Z4)))))). % less_le_trans
thf(fact_175_le__less__trans, axiom,
    ((![X3 : nat, Y4 : nat, Z4 : nat]: ((ord_less_eq_nat @ X3 @ Y4) => ((ord_less_nat @ Y4 @ Z4) => (ord_less_nat @ X3 @ Z4)))))). % le_less_trans
thf(fact_176_le__less__trans, axiom,
    ((![X3 : real, Y4 : real, Z4 : real]: ((ord_less_eq_real @ X3 @ Y4) => ((ord_less_real @ Y4 @ Z4) => (ord_less_real @ X3 @ Z4)))))). % le_less_trans
thf(fact_177_less__imp__le, axiom,
    ((![X3 : nat, Y4 : nat]: ((ord_less_nat @ X3 @ Y4) => (ord_less_eq_nat @ X3 @ Y4))))). % less_imp_le
thf(fact_178_less__imp__le, axiom,
    ((![X3 : real, Y4 : real]: ((ord_less_real @ X3 @ Y4) => (ord_less_eq_real @ X3 @ Y4))))). % less_imp_le
thf(fact_179_antisym__conv2, axiom,
    ((![X3 : nat, Y4 : nat]: ((ord_less_eq_nat @ X3 @ Y4) => ((~ ((ord_less_nat @ X3 @ Y4))) = (X3 = Y4)))))). % antisym_conv2
thf(fact_180_antisym__conv2, axiom,
    ((![X3 : real, Y4 : real]: ((ord_less_eq_real @ X3 @ Y4) => ((~ ((ord_less_real @ X3 @ Y4))) = (X3 = Y4)))))). % antisym_conv2
thf(fact_181_antisym__conv1, axiom,
    ((![X3 : nat, Y4 : nat]: ((~ ((ord_less_nat @ X3 @ Y4))) => ((ord_less_eq_nat @ X3 @ Y4) = (X3 = Y4)))))). % antisym_conv1
thf(fact_182_antisym__conv1, axiom,
    ((![X3 : real, Y4 : real]: ((~ ((ord_less_real @ X3 @ Y4))) => ((ord_less_eq_real @ X3 @ Y4) = (X3 = Y4)))))). % antisym_conv1
thf(fact_183_le__neq__trans, axiom,
    ((![A : nat, B : nat]: ((ord_less_eq_nat @ A @ B) => ((~ ((A = B))) => (ord_less_nat @ A @ B)))))). % le_neq_trans
thf(fact_184_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_185_not__less, axiom,
    ((![X3 : nat, Y4 : nat]: ((~ ((ord_less_nat @ X3 @ Y4))) = (ord_less_eq_nat @ Y4 @ X3))))). % not_less
thf(fact_186_not__less, axiom,
    ((![X3 : real, Y4 : real]: ((~ ((ord_less_real @ X3 @ Y4))) = (ord_less_eq_real @ Y4 @ X3))))). % not_less
thf(fact_187_not__le, axiom,
    ((![X3 : nat, Y4 : nat]: ((~ ((ord_less_eq_nat @ X3 @ Y4))) = (ord_less_nat @ Y4 @ X3))))). % not_le
thf(fact_188_not__le, axiom,
    ((![X3 : real, Y4 : real]: ((~ ((ord_less_eq_real @ X3 @ Y4))) = (ord_less_real @ Y4 @ X3))))). % not_le
thf(fact_189_order__less__le__subst2, axiom,
    ((![A : real, B : real, F2 : real > nat, C2 : nat]: ((ord_less_real @ A @ B) => ((ord_less_eq_nat @ (F2 @ B) @ C2) => ((![X : real, Y5 : real]: ((ord_less_real @ X @ Y5) => (ord_less_nat @ (F2 @ X) @ (F2 @ Y5)))) => (ord_less_nat @ (F2 @ A) @ C2))))))). % order_less_le_subst2
thf(fact_190_order__less__le__subst2, axiom,
    ((![A : nat, B : nat, F2 : nat > nat, C2 : nat]: ((ord_less_nat @ A @ B) => ((ord_less_eq_nat @ (F2 @ B) @ C2) => ((![X : nat, Y5 : nat]: ((ord_less_nat @ X @ Y5) => (ord_less_nat @ (F2 @ X) @ (F2 @ Y5)))) => (ord_less_nat @ (F2 @ A) @ C2))))))). % order_less_le_subst2
thf(fact_191_order__less__le__subst2, axiom,
    ((![A : real, B : real, F2 : real > real, C2 : real]: ((ord_less_real @ A @ B) => ((ord_less_eq_real @ (F2 @ B) @ C2) => ((![X : real, Y5 : real]: ((ord_less_real @ X @ Y5) => (ord_less_real @ (F2 @ X) @ (F2 @ Y5)))) => (ord_less_real @ (F2 @ A) @ C2))))))). % order_less_le_subst2
thf(fact_192_order__less__le__subst2, axiom,
    ((![A : nat, B : nat, F2 : nat > real, C2 : real]: ((ord_less_nat @ A @ B) => ((ord_less_eq_real @ (F2 @ B) @ C2) => ((![X : nat, Y5 : nat]: ((ord_less_nat @ X @ Y5) => (ord_less_real @ (F2 @ X) @ (F2 @ Y5)))) => (ord_less_real @ (F2 @ A) @ C2))))))). % order_less_le_subst2
thf(fact_193_order__less__le__subst1, axiom,
    ((![A : nat, F2 : real > nat, B : real, C2 : real]: ((ord_less_nat @ A @ (F2 @ B)) => ((ord_less_eq_real @ B @ C2) => ((![X : real, Y5 : real]: ((ord_less_eq_real @ X @ Y5) => (ord_less_eq_nat @ (F2 @ X) @ (F2 @ Y5)))) => (ord_less_nat @ A @ (F2 @ C2)))))))). % order_less_le_subst1
thf(fact_194_order__less__le__subst1, axiom,
    ((![A : real, F2 : real > real, B : real, C2 : real]: ((ord_less_real @ A @ (F2 @ B)) => ((ord_less_eq_real @ B @ C2) => ((![X : real, Y5 : real]: ((ord_less_eq_real @ X @ Y5) => (ord_less_eq_real @ (F2 @ X) @ (F2 @ Y5)))) => (ord_less_real @ A @ (F2 @ C2)))))))). % order_less_le_subst1
thf(fact_195_order__le__less__subst2, axiom,
    ((![A : real, B : real, F2 : real > nat, C2 : nat]: ((ord_less_eq_real @ A @ B) => ((ord_less_nat @ (F2 @ B) @ C2) => ((![X : real, Y5 : real]: ((ord_less_eq_real @ X @ Y5) => (ord_less_eq_nat @ (F2 @ X) @ (F2 @ Y5)))) => (ord_less_nat @ (F2 @ A) @ C2))))))). % order_le_less_subst2
thf(fact_196_order__le__less__subst2, axiom,
    ((![A : real, B : real, F2 : real > real, C2 : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_real @ (F2 @ B) @ C2) => ((![X : real, Y5 : real]: ((ord_less_eq_real @ X @ Y5) => (ord_less_eq_real @ (F2 @ X) @ (F2 @ Y5)))) => (ord_less_real @ (F2 @ A) @ C2))))))). % order_le_less_subst2
thf(fact_197_order__le__less__subst1, axiom,
    ((![A : nat, F2 : real > nat, B : real, C2 : real]: ((ord_less_eq_nat @ A @ (F2 @ B)) => ((ord_less_real @ B @ C2) => ((![X : real, Y5 : real]: ((ord_less_real @ X @ Y5) => (ord_less_nat @ (F2 @ X) @ (F2 @ Y5)))) => (ord_less_nat @ A @ (F2 @ C2)))))))). % order_le_less_subst1
thf(fact_198_order__le__less__subst1, axiom,
    ((![A : nat, F2 : nat > nat, B : nat, C2 : nat]: ((ord_less_eq_nat @ A @ (F2 @ B)) => ((ord_less_nat @ B @ C2) => ((![X : nat, Y5 : nat]: ((ord_less_nat @ X @ Y5) => (ord_less_nat @ (F2 @ X) @ (F2 @ Y5)))) => (ord_less_nat @ A @ (F2 @ C2)))))))). % order_le_less_subst1
thf(fact_199_order__le__less__subst1, axiom,
    ((![A : real, F2 : real > real, B : real, C2 : real]: ((ord_less_eq_real @ A @ (F2 @ B)) => ((ord_less_real @ B @ C2) => ((![X : real, Y5 : real]: ((ord_less_real @ X @ Y5) => (ord_less_real @ (F2 @ X) @ (F2 @ Y5)))) => (ord_less_real @ A @ (F2 @ C2)))))))). % order_le_less_subst1
thf(fact_200_order__le__less__subst1, axiom,
    ((![A : real, F2 : nat > real, B : nat, C2 : nat]: ((ord_less_eq_real @ A @ (F2 @ B)) => ((ord_less_nat @ B @ C2) => ((![X : nat, Y5 : nat]: ((ord_less_nat @ X @ Y5) => (ord_less_real @ (F2 @ X) @ (F2 @ Y5)))) => (ord_less_real @ A @ (F2 @ C2)))))))). % order_le_less_subst1
thf(fact_201_less__le, axiom,
    ((ord_less_nat = (^[X2 : nat]: (^[Y2 : nat]: (((ord_less_eq_nat @ X2 @ Y2)) & ((~ ((X2 = Y2)))))))))). % less_le
thf(fact_202_less__le, axiom,
    ((ord_less_real = (^[X2 : real]: (^[Y2 : real]: (((ord_less_eq_real @ X2 @ Y2)) & ((~ ((X2 = Y2)))))))))). % less_le
thf(fact_203_le__less, axiom,
    ((ord_less_eq_nat = (^[X2 : nat]: (^[Y2 : nat]: (((ord_less_nat @ X2 @ Y2)) | ((X2 = Y2)))))))). % le_less
thf(fact_204_le__less, axiom,
    ((ord_less_eq_real = (^[X2 : real]: (^[Y2 : real]: (((ord_less_real @ X2 @ Y2)) | ((X2 = Y2)))))))). % le_less
thf(fact_205_leI, axiom,
    ((![X3 : nat, Y4 : nat]: ((~ ((ord_less_nat @ X3 @ Y4))) => (ord_less_eq_nat @ Y4 @ X3))))). % leI
thf(fact_206_leI, axiom,
    ((![X3 : real, Y4 : real]: ((~ ((ord_less_real @ X3 @ Y4))) => (ord_less_eq_real @ Y4 @ X3))))). % leI
thf(fact_207_leD, axiom,
    ((![Y4 : nat, X3 : nat]: ((ord_less_eq_nat @ Y4 @ X3) => (~ ((ord_less_nat @ X3 @ Y4))))))). % leD
thf(fact_208_leD, axiom,
    ((![Y4 : real, X3 : real]: ((ord_less_eq_real @ Y4 @ X3) => (~ ((ord_less_real @ X3 @ Y4))))))). % leD
thf(fact_209_poly__all__0__iff__0, axiom,
    ((![P2 : poly_real]: ((![X2 : real]: ((poly_real2 @ P2 @ X2) = zero_zero_real)) = (P2 = zero_zero_poly_real))))). % poly_all_0_iff_0
thf(fact_210_poly__all__0__iff__0, axiom,
    ((![P2 : poly_complex]: ((![X2 : complex]: ((poly_complex2 @ P2 @ X2) = zero_zero_complex)) = (P2 = zero_z1746442943omplex))))). % poly_all_0_iff_0
thf(fact_211_less_Ohyps, axiom,
    ((![P2 : poly_complex]: ((ord_less_nat @ (fundam1709708056omplex @ P2) @ (fundam1709708056omplex @ pa)) => ((~ ((fundam1158420650omplex @ (poly_complex2 @ P2)))) => (?[Z2 : complex]: ((poly_complex2 @ P2 @ Z2) = zero_zero_complex))))))). % less.hyps
thf(fact_212_cq0, axiom,
    ((![W : complex]: (ord_less_eq_real @ (real_V638595069omplex @ (poly_complex2 @ q @ zero_zero_complex)) @ (real_V638595069omplex @ (poly_complex2 @ pa @ W)))))). % cq0
thf(fact_213_less__eq__real__def, axiom,
    ((ord_less_eq_real = (^[X2 : real]: (^[Y2 : real]: (((ord_less_real @ X2 @ Y2)) | ((X2 = Y2)))))))). % less_eq_real_def
thf(fact_214_complete__interval, axiom,
    ((![A : nat, B : nat, P : nat > $o]: ((ord_less_nat @ A @ B) => ((P @ A) => ((~ ((P @ B))) => (?[C : nat]: ((ord_less_eq_nat @ A @ C) & ((ord_less_eq_nat @ C @ B) & ((![X5 : nat]: (((ord_less_eq_nat @ A @ X5) & (ord_less_nat @ X5 @ C)) => (P @ X5))) & (![D : nat]: ((![X : nat]: (((ord_less_eq_nat @ A @ X) & (ord_less_nat @ X @ D)) => (P @ X))) => (ord_less_eq_nat @ D @ C))))))))))))). % complete_interval
thf(fact_215_complete__interval, axiom,
    ((![A : real, B : real, P : real > $o]: ((ord_less_real @ A @ B) => ((P @ A) => ((~ ((P @ B))) => (?[C : real]: ((ord_less_eq_real @ A @ C) & ((ord_less_eq_real @ C @ B) & ((![X5 : real]: (((ord_less_eq_real @ A @ X5) & (ord_less_real @ X5 @ C)) => (P @ X5))) & (![D : real]: ((![X : real]: (((ord_less_eq_real @ A @ X) & (ord_less_real @ X @ D)) => (P @ X))) => (ord_less_eq_real @ D @ C))))))))))))). % complete_interval
thf(fact_216_verit__comp__simplify1_I3_J, axiom,
    ((![B4 : nat, A4 : nat]: ((~ ((ord_less_eq_nat @ B4 @ A4))) = (ord_less_nat @ A4 @ B4))))). % verit_comp_simplify1(3)
thf(fact_217_verit__comp__simplify1_I3_J, axiom,
    ((![B4 : real, A4 : real]: ((~ ((ord_less_eq_real @ B4 @ A4))) = (ord_less_real @ A4 @ B4))))). % verit_comp_simplify1(3)
thf(fact_218_pinf_I6_J, axiom,
    ((![T : nat]: (?[Z2 : nat]: (![X5 : nat]: ((ord_less_nat @ Z2 @ X5) => (~ ((ord_less_eq_nat @ X5 @ T))))))))). % pinf(6)
thf(fact_219_pinf_I6_J, axiom,
    ((![T : real]: (?[Z2 : real]: (![X5 : real]: ((ord_less_real @ Z2 @ X5) => (~ ((ord_less_eq_real @ X5 @ T))))))))). % pinf(6)
thf(fact_220_pinf_I8_J, axiom,
    ((![T : nat]: (?[Z2 : nat]: (![X5 : nat]: ((ord_less_nat @ Z2 @ X5) => (ord_less_eq_nat @ T @ X5))))))). % pinf(8)
thf(fact_221_pinf_I8_J, axiom,
    ((![T : real]: (?[Z2 : real]: (![X5 : real]: ((ord_less_real @ Z2 @ X5) => (ord_less_eq_real @ T @ X5))))))). % pinf(8)
thf(fact_222_minf_I6_J, axiom,
    ((![T : nat]: (?[Z2 : nat]: (![X5 : nat]: ((ord_less_nat @ X5 @ Z2) => (ord_less_eq_nat @ X5 @ T))))))). % minf(6)
thf(fact_223_minf_I6_J, axiom,
    ((![T : real]: (?[Z2 : real]: (![X5 : real]: ((ord_less_real @ X5 @ Z2) => (ord_less_eq_real @ X5 @ T))))))). % minf(6)
thf(fact_224_minf_I8_J, axiom,
    ((![T : nat]: (?[Z2 : nat]: (![X5 : nat]: ((ord_less_nat @ X5 @ Z2) => (~ ((ord_less_eq_nat @ T @ X5))))))))). % minf(8)
thf(fact_225_minf_I8_J, axiom,
    ((![T : real]: (?[Z2 : real]: (![X5 : real]: ((ord_less_real @ X5 @ Z2) => (~ ((ord_less_eq_real @ T @ X5))))))))). % minf(8)
thf(fact_226_q_I1_J, axiom,
    (((fundam1709708056omplex @ q) = (fundam1709708056omplex @ pa)))). % q(1)
thf(fact_227_a00, axiom,
    ((~ (((poly_complex2 @ q @ zero_zero_complex) = zero_zero_complex))))). % a00
thf(fact_228__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_229_qnc, axiom,
    ((~ ((fundam1158420650omplex @ (poly_complex2 @ q)))))). % qnc
thf(fact_230_pqc0, axiom,
    (((poly_complex2 @ pa @ c) = (poly_complex2 @ q @ zero_zero_complex)))). % pqc0
thf(fact_231_psize__eq__0__iff, axiom,
    ((![P2 : poly_complex]: (((fundam1709708056omplex @ P2) = zero_zero_nat) = (P2 = zero_z1746442943omplex))))). % psize_eq_0_iff
thf(fact_232_q_I2_J, axiom,
    ((![X5 : complex]: ((poly_complex2 @ q @ X5) = (poly_complex2 @ pa @ (plus_plus_complex @ c @ X5)))))). % q(2)
thf(fact_233_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_234_minf_I7_J, axiom,
    ((![T : real]: (?[Z2 : real]: (![X5 : real]: ((ord_less_real @ X5 @ Z2) => (~ ((ord_less_real @ T @ X5))))))))). % minf(7)
thf(fact_235_minf_I7_J, axiom,
    ((![T : nat]: (?[Z2 : nat]: (![X5 : nat]: ((ord_less_nat @ X5 @ Z2) => (~ ((ord_less_nat @ T @ X5))))))))). % minf(7)

% Conjectures (2)
thf(conj_0, hypothesis,
    ((![M4 : real]: ((ord_less_real @ zero_zero_real @ M4) => ((![Z2 : complex]: ((ord_less_eq_real @ (real_V638595069omplex @ Z2) @ (real_V638595069omplex @ w)) => (ord_less_eq_real @ (real_V638595069omplex @ (poly_complex2 @ s @ Z2)) @ M4))) => thesis))))).
thf(conj_1, conjecture,
    (thesis)).
