% 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_651__5374402_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:31:51.514

% Could-be-implicit typings (9)
thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_It__Complex__Ocomplex_J_J, type,
    poly_poly_complex : $tType).
thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J, type,
    poly_poly_real : $tType).
thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_It__Nat__Onat_J_J, type,
    poly_poly_nat : $tType).
thf(ty_n_t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    poly_complex : $tType).
thf(ty_n_t__Polynomial__Opoly_It__Real__Oreal_J, type,
    poly_real : $tType).
thf(ty_n_t__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 (40)
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_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Opsize_001t__Nat__Onat, type,
    fundam1567013434ze_nat : poly_nat > nat).
thf(sy_c_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Opsize_001t__Real__Oreal, type,
    fundam1947011094e_real : poly_real > nat).
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__Polynomial__Opoly_It__Complex__Ocomplex_J_J, type,
    zero_z1040703943omplex : poly_poly_complex).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Nat__Onat_J_J, type,
    zero_z1059985641ly_nat : poly_poly_nat).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J, type,
    zero_z1423781445y_real : poly_poly_real).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Real__Oreal_J, type,
    zero_zero_poly_real : poly_real).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal, type,
    zero_zero_real : real).
thf(sy_c_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_Oorder_001t__Complex__Ocomplex, type,
    order_complex : complex > poly_complex > nat).
thf(sy_c_Polynomial_Oorder_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    order_poly_complex : poly_complex > poly_poly_complex > nat).
thf(sy_c_Polynomial_Oorder_001t__Polynomial__Opoly_It__Real__Oreal_J, type,
    order_poly_real : poly_real > poly_poly_real > nat).
thf(sy_c_Polynomial_Oorder_001t__Real__Oreal, type,
    order_real : real > poly_real > nat).
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__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    poly_poly_complex2 : poly_poly_complex > poly_complex > poly_complex).
thf(sy_c_Polynomial_Opoly_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    poly_poly_nat2 : poly_poly_nat > poly_nat > poly_nat).
thf(sy_c_Polynomial_Opoly_001t__Polynomial__Opoly_It__Real__Oreal_J, type,
    poly_poly_real2 : poly_poly_real > poly_real > poly_real).
thf(sy_c_Polynomial_Opoly_001t__Real__Oreal, type,
    poly_real2 : poly_real > real > real).
thf(sy_c_Polynomial_Oreflect__poly_001t__Complex__Ocomplex, type,
    reflect_poly_complex : poly_complex > poly_complex).
thf(sy_c_Polynomial_Oreflect__poly_001t__Nat__Onat, type,
    reflect_poly_nat : poly_nat > poly_nat).
thf(sy_c_Polynomial_Oreflect__poly_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    reflec309385472omplex : poly_poly_complex > poly_poly_complex).
thf(sy_c_Polynomial_Oreflect__poly_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    reflec781175074ly_nat : poly_poly_nat > poly_poly_nat).
thf(sy_c_Polynomial_Oreflect__poly_001t__Polynomial__Opoly_It__Real__Oreal_J, type,
    reflec1522834046y_real : poly_poly_real > poly_poly_real).
thf(sy_c_Polynomial_Oreflect__poly_001t__Real__Oreal, type,
    reflect_poly_real : poly_real > poly_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_Transcendental_Oarsinh_001t__Real__Oreal, type,
    arsinh_real : real > real).
thf(sy_c_Transcendental_Oartanh_001t__Real__Oreal, type,
    artanh_real : real > real).
thf(sy_v_c____, type,
    c : complex).
thf(sy_v_p, type,
    p : poly_complex).
thf(sy_v_pa____, type,
    pa : poly_complex).

% Relevant facts (244)
thf(fact_0_True, axiom,
    (((poly_complex2 @ pa @ c) = zero_zero_complex))). % True
thf(fact_1_nc, axiom,
    ((~ ((fundam1158420650omplex @ (poly_complex2 @ p)))))). % nc
thf(fact_2_less_Oprems, axiom,
    ((~ ((fundam1158420650omplex @ (poly_complex2 @ pa)))))). % less.prems
thf(fact_3_poly__0, axiom,
    ((![X : poly_real]: ((poly_poly_real2 @ zero_z1423781445y_real @ X) = zero_zero_poly_real)))). % poly_0
thf(fact_4_poly__0, axiom,
    ((![X : poly_nat]: ((poly_poly_nat2 @ zero_z1059985641ly_nat @ X) = zero_zero_poly_nat)))). % poly_0
thf(fact_5_poly__0, axiom,
    ((![X : poly_complex]: ((poly_poly_complex2 @ zero_z1040703943omplex @ X) = zero_z1746442943omplex)))). % poly_0
thf(fact_6_poly__0, axiom,
    ((![X : complex]: ((poly_complex2 @ zero_z1746442943omplex @ X) = zero_zero_complex)))). % poly_0
thf(fact_7_poly__0, axiom,
    ((![X : nat]: ((poly_nat2 @ zero_zero_poly_nat @ X) = zero_zero_nat)))). % poly_0
thf(fact_8_poly__0, axiom,
    ((![X : real]: ((poly_real2 @ zero_zero_poly_real @ X) = zero_zero_real)))). % poly_0
thf(fact_9_poly__all__0__iff__0, axiom,
    ((![P : poly_poly_real]: ((![X2 : poly_real]: ((poly_poly_real2 @ P @ X2) = zero_zero_poly_real)) = (P = zero_z1423781445y_real))))). % poly_all_0_iff_0
thf(fact_10_poly__all__0__iff__0, axiom,
    ((![P : poly_poly_complex]: ((![X2 : poly_complex]: ((poly_poly_complex2 @ P @ X2) = zero_z1746442943omplex)) = (P = zero_z1040703943omplex))))). % poly_all_0_iff_0
thf(fact_11_poly__all__0__iff__0, axiom,
    ((![P : poly_complex]: ((![X2 : complex]: ((poly_complex2 @ P @ X2) = zero_zero_complex)) = (P = zero_z1746442943omplex))))). % poly_all_0_iff_0
thf(fact_12_poly__all__0__iff__0, axiom,
    ((![P : poly_real]: ((![X2 : real]: ((poly_real2 @ P @ X2) = zero_zero_real)) = (P = zero_zero_poly_real))))). % poly_all_0_iff_0
thf(fact_13_c, axiom,
    ((![W : complex]: (ord_less_eq_real @ (real_V638595069omplex @ (poly_complex2 @ pa @ c)) @ (real_V638595069omplex @ (poly_complex2 @ pa @ W)))))). % c
thf(fact_14_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_15_poly__eq__poly__eq__iff, axiom,
    ((![P : poly_real, Q : poly_real]: (((poly_real2 @ P) = (poly_real2 @ Q)) = (P = Q))))). % poly_eq_poly_eq_iff
thf(fact_16_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_17_zero__reorient, axiom,
    ((![X : poly_real]: ((zero_zero_poly_real = X) = (X = zero_zero_poly_real))))). % zero_reorient
thf(fact_18_zero__reorient, axiom,
    ((![X : poly_nat]: ((zero_zero_poly_nat = X) = (X = zero_zero_poly_nat))))). % zero_reorient
thf(fact_19_zero__reorient, axiom,
    ((![X : poly_complex]: ((zero_z1746442943omplex = X) = (X = zero_z1746442943omplex))))). % zero_reorient
thf(fact_20_zero__reorient, axiom,
    ((![X : complex]: ((zero_zero_complex = X) = (X = zero_zero_complex))))). % zero_reorient
thf(fact_21_zero__reorient, axiom,
    ((![X : nat]: ((zero_zero_nat = X) = (X = zero_zero_nat))))). % zero_reorient
thf(fact_22_zero__reorient, axiom,
    ((![X : real]: ((zero_zero_real = X) = (X = zero_zero_real))))). % zero_reorient
thf(fact_23__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_24_arsinh__0, axiom,
    (((arsinh_real @ zero_zero_real) = zero_zero_real))). % arsinh_0
thf(fact_25_artanh__0, axiom,
    (((artanh_real @ zero_zero_real) = zero_zero_real))). % artanh_0
thf(fact_26_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_27_reflect__poly__at__0__eq__0__iff, axiom,
    ((![P : poly_nat]: (((poly_nat2 @ (reflect_poly_nat @ P) @ zero_zero_nat) = zero_zero_nat) = (P = zero_zero_poly_nat))))). % reflect_poly_at_0_eq_0_iff
thf(fact_28_reflect__poly__at__0__eq__0__iff, axiom,
    ((![P : poly_real]: (((poly_real2 @ (reflect_poly_real @ P) @ zero_zero_real) = zero_zero_real) = (P = zero_zero_poly_real))))). % reflect_poly_at_0_eq_0_iff
thf(fact_29_reflect__poly__at__0__eq__0__iff, axiom,
    ((![P : poly_poly_real]: (((poly_poly_real2 @ (reflec1522834046y_real @ P) @ zero_zero_poly_real) = zero_zero_poly_real) = (P = zero_z1423781445y_real))))). % reflect_poly_at_0_eq_0_iff
thf(fact_30_reflect__poly__at__0__eq__0__iff, axiom,
    ((![P : poly_poly_nat]: (((poly_poly_nat2 @ (reflec781175074ly_nat @ P) @ zero_zero_poly_nat) = zero_zero_poly_nat) = (P = zero_z1059985641ly_nat))))). % reflect_poly_at_0_eq_0_iff
thf(fact_31_reflect__poly__at__0__eq__0__iff, axiom,
    ((![P : poly_poly_complex]: (((poly_poly_complex2 @ (reflec309385472omplex @ P) @ zero_z1746442943omplex) = zero_z1746442943omplex) = (P = zero_z1040703943omplex))))). % reflect_poly_at_0_eq_0_iff
thf(fact_32_order__0I, axiom,
    ((![P : poly_complex, A : complex]: ((~ (((poly_complex2 @ P @ A) = zero_zero_complex))) => ((order_complex @ A @ P) = zero_zero_nat))))). % order_0I
thf(fact_33_order__0I, axiom,
    ((![P : poly_real, A : real]: ((~ (((poly_real2 @ P @ A) = zero_zero_real))) => ((order_real @ A @ P) = zero_zero_nat))))). % order_0I
thf(fact_34_order__0I, axiom,
    ((![P : poly_poly_real, A : poly_real]: ((~ (((poly_poly_real2 @ P @ A) = zero_zero_poly_real))) => ((order_poly_real @ A @ P) = zero_zero_nat))))). % order_0I
thf(fact_35_order__0I, axiom,
    ((![P : poly_poly_complex, A : poly_complex]: ((~ (((poly_poly_complex2 @ P @ A) = zero_z1746442943omplex))) => ((order_poly_complex @ A @ P) = zero_zero_nat))))). % order_0I
thf(fact_36_le__zero__eq, axiom,
    ((![N : nat]: ((ord_less_eq_nat @ N @ zero_zero_nat) = (N = zero_zero_nat))))). % le_zero_eq
thf(fact_37_not__gr__zero, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr_zero
thf(fact_38_reflect__poly__0, axiom,
    (((reflect_poly_real @ zero_zero_poly_real) = zero_zero_poly_real))). % reflect_poly_0
thf(fact_39_reflect__poly__0, axiom,
    (((reflect_poly_nat @ zero_zero_poly_nat) = zero_zero_poly_nat))). % reflect_poly_0
thf(fact_40_reflect__poly__0, axiom,
    (((reflect_poly_complex @ zero_z1746442943omplex) = zero_z1746442943omplex))). % reflect_poly_0
thf(fact_41_psize__eq__0__iff, axiom,
    ((![P : poly_real]: (((fundam1947011094e_real @ P) = zero_zero_nat) = (P = zero_zero_poly_real))))). % psize_eq_0_iff
thf(fact_42_psize__eq__0__iff, axiom,
    ((![P : poly_nat]: (((fundam1567013434ze_nat @ P) = zero_zero_nat) = (P = zero_zero_poly_nat))))). % psize_eq_0_iff
thf(fact_43_psize__eq__0__iff, axiom,
    ((![P : poly_complex]: (((fundam1709708056omplex @ P) = zero_zero_nat) = (P = zero_z1746442943omplex))))). % psize_eq_0_iff
thf(fact_44_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_45_gr__implies__not__zero, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not_zero
thf(fact_46_not__less__zero, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less_zero
thf(fact_47_gr__zeroI, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr_zeroI
thf(fact_48_zero__le, axiom,
    ((![X : nat]: (ord_less_eq_nat @ zero_zero_nat @ X)))). % zero_le
thf(fact_49_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_50_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_51_constant__def, axiom,
    ((fundam1158420650omplex = (^[F : complex > complex]: (![X2 : complex]: (![Y : complex]: ((F @ X2) = (F @ Y)))))))). % constant_def
thf(fact_52_order__root, axiom,
    ((![P : poly_complex, A : complex]: (((poly_complex2 @ P @ A) = zero_zero_complex) = (((P = zero_z1746442943omplex)) | ((~ (((order_complex @ A @ P) = zero_zero_nat))))))))). % order_root
thf(fact_53_order__root, axiom,
    ((![P : poly_real, A : real]: (((poly_real2 @ P @ A) = zero_zero_real) = (((P = zero_zero_poly_real)) | ((~ (((order_real @ A @ P) = zero_zero_nat))))))))). % order_root
thf(fact_54_order__root, axiom,
    ((![P : poly_poly_real, A : poly_real]: (((poly_poly_real2 @ P @ A) = zero_zero_poly_real) = (((P = zero_z1423781445y_real)) | ((~ (((order_poly_real @ A @ P) = zero_zero_nat))))))))). % order_root
thf(fact_55_order__root, axiom,
    ((![P : poly_poly_complex, A : poly_complex]: (((poly_poly_complex2 @ P @ A) = zero_z1746442943omplex) = (((P = zero_z1040703943omplex)) | ((~ (((order_poly_complex @ A @ P) = zero_zero_nat))))))))). % order_root
thf(fact_56_norm__le__zero__iff, axiom,
    ((![X : real]: ((ord_less_eq_real @ (real_V646646907m_real @ X) @ zero_zero_real) = (X = zero_zero_real))))). % norm_le_zero_iff
thf(fact_57_norm__le__zero__iff, axiom,
    ((![X : complex]: ((ord_less_eq_real @ (real_V638595069omplex @ X) @ zero_zero_real) = (X = zero_zero_complex))))). % norm_le_zero_iff
thf(fact_58_neq0__conv, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ N))))). % neq0_conv
thf(fact_59_less__nat__zero__code, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_nat_zero_code
thf(fact_60_bot__nat__0_Onot__eq__extremum, axiom,
    ((![A : nat]: ((~ ((A = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ A))))). % bot_nat_0.not_eq_extremum
thf(fact_61_norm__eq__zero, axiom,
    ((![X : real]: (((real_V646646907m_real @ X) = zero_zero_real) = (X = zero_zero_real))))). % norm_eq_zero
thf(fact_62_norm__eq__zero, axiom,
    ((![X : complex]: (((real_V638595069omplex @ X) = zero_zero_real) = (X = zero_zero_complex))))). % norm_eq_zero
thf(fact_63_norm__zero, axiom,
    (((real_V646646907m_real @ zero_zero_real) = zero_zero_real))). % norm_zero
thf(fact_64_norm__zero, axiom,
    (((real_V638595069omplex @ zero_zero_complex) = zero_zero_real))). % norm_zero
thf(fact_65_order__refl, axiom,
    ((![X : real]: (ord_less_eq_real @ X @ X)))). % order_refl
thf(fact_66_order__refl, axiom,
    ((![X : nat]: (ord_less_eq_nat @ X @ X)))). % order_refl
thf(fact_67_norm__ge__zero, axiom,
    ((![X : complex]: (ord_less_eq_real @ zero_zero_real @ (real_V638595069omplex @ X))))). % norm_ge_zero
thf(fact_68_gr0I, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr0I
thf(fact_69_not__gr0, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr0
thf(fact_70_bot__nat__0_Oextremum, axiom,
    ((![A : nat]: (ord_less_eq_nat @ zero_zero_nat @ A)))). % bot_nat_0.extremum
thf(fact_71_le0, axiom,
    ((![N : nat]: (ord_less_eq_nat @ zero_zero_nat @ N)))). % le0
thf(fact_72_zero__less__norm__iff, axiom,
    ((![X : real]: ((ord_less_real @ zero_zero_real @ (real_V646646907m_real @ X)) = (~ ((X = zero_zero_real))))))). % zero_less_norm_iff
thf(fact_73_zero__less__norm__iff, axiom,
    ((![X : complex]: ((ord_less_real @ zero_zero_real @ (real_V638595069omplex @ X)) = (~ ((X = zero_zero_complex))))))). % zero_less_norm_iff
thf(fact_74_poly__IVT__neg, axiom,
    ((![A : real, B : real, P : poly_real]: ((ord_less_real @ A @ B) => ((ord_less_real @ zero_zero_real @ (poly_real2 @ P @ A)) => ((ord_less_real @ (poly_real2 @ P @ B) @ zero_zero_real) => (?[X3 : real]: ((ord_less_real @ A @ X3) & ((ord_less_real @ X3 @ B) & ((poly_real2 @ P @ X3) = zero_zero_real)))))))))). % poly_IVT_neg
thf(fact_75_poly__IVT__pos, axiom,
    ((![A : real, B : real, P : poly_real]: ((ord_less_real @ A @ B) => ((ord_less_real @ (poly_real2 @ P @ A) @ zero_zero_real) => ((ord_less_real @ zero_zero_real @ (poly_real2 @ P @ B)) => (?[X3 : real]: ((ord_less_real @ A @ X3) & ((ord_less_real @ X3 @ B) & ((poly_real2 @ P @ X3) = zero_zero_real)))))))))). % poly_IVT_pos
thf(fact_76_bot__nat__0_Oextremum__uniqueI, axiom,
    ((![A : nat]: ((ord_less_eq_nat @ A @ zero_zero_nat) => (A = zero_zero_nat))))). % bot_nat_0.extremum_uniqueI
thf(fact_77_bot__nat__0_Oextremum__unique, axiom,
    ((![A : nat]: ((ord_less_eq_nat @ A @ zero_zero_nat) = (A = zero_zero_nat))))). % bot_nat_0.extremum_unique
thf(fact_78_le__0__eq, axiom,
    ((![N : nat]: ((ord_less_eq_nat @ N @ zero_zero_nat) = (N = zero_zero_nat))))). % le_0_eq
thf(fact_79_less__eq__nat_Osimps_I1_J, axiom,
    ((![N : nat]: (ord_less_eq_nat @ zero_zero_nat @ N)))). % less_eq_nat.simps(1)
thf(fact_80_less__mono__imp__le__mono, axiom,
    ((![F2 : nat > nat, I : nat, J : nat]: ((![I2 : nat, J2 : nat]: ((ord_less_nat @ I2 @ J2) => (ord_less_nat @ (F2 @ I2) @ (F2 @ J2)))) => ((ord_less_eq_nat @ I @ J) => (ord_less_eq_nat @ (F2 @ I) @ (F2 @ J))))))). % less_mono_imp_le_mono
thf(fact_81_le__neq__implies__less, axiom,
    ((![M : nat, N : nat]: ((ord_less_eq_nat @ M @ N) => ((~ ((M = N))) => (ord_less_nat @ M @ N)))))). % le_neq_implies_less
thf(fact_82_less__or__eq__imp__le, axiom,
    ((![M : nat, N : nat]: (((ord_less_nat @ M @ N) | (M = N)) => (ord_less_eq_nat @ M @ N))))). % less_or_eq_imp_le
thf(fact_83_le__eq__less__or__eq, axiom,
    ((ord_less_eq_nat = (^[M2 : nat]: (^[N2 : nat]: (((ord_less_nat @ M2 @ N2)) | ((M2 = N2)))))))). % le_eq_less_or_eq
thf(fact_84_less__imp__le__nat, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (ord_less_eq_nat @ M @ N))))). % less_imp_le_nat
thf(fact_85_nat__less__le, axiom,
    ((ord_less_nat = (^[M2 : nat]: (^[N2 : nat]: (((ord_less_eq_nat @ M2 @ N2)) & ((~ ((M2 = N2)))))))))). % nat_less_le
thf(fact_86_norm__not__less__zero, axiom,
    ((![X : complex]: (~ ((ord_less_real @ (real_V638595069omplex @ X) @ zero_zero_real)))))). % norm_not_less_zero
thf(fact_87_ex__least__nat__le, axiom,
    ((![P2 : nat > $o, N : nat]: ((P2 @ N) => ((~ ((P2 @ zero_zero_nat))) => (?[K : nat]: ((ord_less_eq_nat @ K @ N) & ((![I3 : nat]: ((ord_less_nat @ I3 @ K) => (~ ((P2 @ I3))))) & (P2 @ K))))))))). % ex_least_nat_le
thf(fact_88_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_89_dual__order_Oantisym, axiom,
    ((![B : nat, A : nat]: ((ord_less_eq_nat @ B @ A) => ((ord_less_eq_nat @ A @ B) => (A = B)))))). % dual_order.antisym
thf(fact_90_dual__order_Oeq__iff, axiom,
    (((^[Y2 : real]: (^[Z2 : real]: (Y2 = Z2))) = (^[A2 : real]: (^[B2 : real]: (((ord_less_eq_real @ B2 @ A2)) & ((ord_less_eq_real @ A2 @ B2)))))))). % dual_order.eq_iff
thf(fact_91_dual__order_Oeq__iff, axiom,
    (((^[Y2 : nat]: (^[Z2 : nat]: (Y2 = Z2))) = (^[A2 : nat]: (^[B2 : nat]: (((ord_less_eq_nat @ B2 @ A2)) & ((ord_less_eq_nat @ A2 @ B2)))))))). % dual_order.eq_iff
thf(fact_92_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_93_dual__order_Otrans, axiom,
    ((![B : nat, A : nat, C2 : nat]: ((ord_less_eq_nat @ B @ A) => ((ord_less_eq_nat @ C2 @ B) => (ord_less_eq_nat @ C2 @ A)))))). % dual_order.trans
thf(fact_94_linorder__wlog, axiom,
    ((![P2 : real > real > $o, A : real, B : real]: ((![A3 : real, B3 : real]: ((ord_less_eq_real @ A3 @ B3) => (P2 @ A3 @ B3))) => ((![A3 : real, B3 : real]: ((P2 @ B3 @ A3) => (P2 @ A3 @ B3))) => (P2 @ A @ B)))))). % linorder_wlog
thf(fact_95_linorder__wlog, axiom,
    ((![P2 : nat > nat > $o, A : nat, B : nat]: ((![A3 : nat, B3 : nat]: ((ord_less_eq_nat @ A3 @ B3) => (P2 @ A3 @ B3))) => ((![A3 : nat, B3 : nat]: ((P2 @ B3 @ A3) => (P2 @ A3 @ B3))) => (P2 @ A @ B)))))). % linorder_wlog
thf(fact_96_dual__order_Orefl, axiom,
    ((![A : real]: (ord_less_eq_real @ A @ A)))). % dual_order.refl
thf(fact_97_dual__order_Orefl, axiom,
    ((![A : nat]: (ord_less_eq_nat @ A @ A)))). % dual_order.refl
thf(fact_98_order__trans, axiom,
    ((![X : real, Y3 : real, Z3 : real]: ((ord_less_eq_real @ X @ Y3) => ((ord_less_eq_real @ Y3 @ Z3) => (ord_less_eq_real @ X @ Z3)))))). % order_trans
thf(fact_99_order__trans, axiom,
    ((![X : nat, Y3 : nat, Z3 : nat]: ((ord_less_eq_nat @ X @ Y3) => ((ord_less_eq_nat @ Y3 @ Z3) => (ord_less_eq_nat @ X @ Z3)))))). % order_trans
thf(fact_100_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_101_order__class_Oorder_Oantisym, axiom,
    ((![A : nat, B : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_nat @ B @ A) => (A = B)))))). % order_class.order.antisym
thf(fact_102_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_103_ord__le__eq__trans, axiom,
    ((![A : nat, B : nat, C2 : nat]: ((ord_less_eq_nat @ A @ B) => ((B = C2) => (ord_less_eq_nat @ A @ C2)))))). % ord_le_eq_trans
thf(fact_104_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_105_ord__eq__le__trans, axiom,
    ((![A : nat, B : nat, C2 : nat]: ((A = B) => ((ord_less_eq_nat @ B @ C2) => (ord_less_eq_nat @ A @ C2)))))). % ord_eq_le_trans
thf(fact_106_order__class_Oorder_Oeq__iff, axiom,
    (((^[Y2 : real]: (^[Z2 : real]: (Y2 = Z2))) = (^[A2 : real]: (^[B2 : real]: (((ord_less_eq_real @ A2 @ B2)) & ((ord_less_eq_real @ B2 @ A2)))))))). % order_class.order.eq_iff
thf(fact_107_order__class_Oorder_Oeq__iff, axiom,
    (((^[Y2 : nat]: (^[Z2 : nat]: (Y2 = Z2))) = (^[A2 : nat]: (^[B2 : nat]: (((ord_less_eq_nat @ A2 @ B2)) & ((ord_less_eq_nat @ B2 @ A2)))))))). % order_class.order.eq_iff
thf(fact_108_antisym__conv, axiom,
    ((![Y3 : real, X : real]: ((ord_less_eq_real @ Y3 @ X) => ((ord_less_eq_real @ X @ Y3) = (X = Y3)))))). % antisym_conv
thf(fact_109_antisym__conv, axiom,
    ((![Y3 : nat, X : nat]: ((ord_less_eq_nat @ Y3 @ X) => ((ord_less_eq_nat @ X @ Y3) = (X = Y3)))))). % antisym_conv
thf(fact_110_le__cases3, axiom,
    ((![X : real, Y3 : real, Z3 : real]: (((ord_less_eq_real @ X @ Y3) => (~ ((ord_less_eq_real @ Y3 @ Z3)))) => (((ord_less_eq_real @ Y3 @ X) => (~ ((ord_less_eq_real @ X @ Z3)))) => (((ord_less_eq_real @ X @ Z3) => (~ ((ord_less_eq_real @ Z3 @ Y3)))) => (((ord_less_eq_real @ Z3 @ Y3) => (~ ((ord_less_eq_real @ Y3 @ X)))) => (((ord_less_eq_real @ Y3 @ Z3) => (~ ((ord_less_eq_real @ Z3 @ X)))) => (~ (((ord_less_eq_real @ Z3 @ X) => (~ ((ord_less_eq_real @ X @ Y3)))))))))))))). % le_cases3
thf(fact_111_le__cases3, axiom,
    ((![X : nat, Y3 : nat, Z3 : nat]: (((ord_less_eq_nat @ X @ Y3) => (~ ((ord_less_eq_nat @ Y3 @ Z3)))) => (((ord_less_eq_nat @ Y3 @ X) => (~ ((ord_less_eq_nat @ X @ Z3)))) => (((ord_less_eq_nat @ X @ Z3) => (~ ((ord_less_eq_nat @ Z3 @ Y3)))) => (((ord_less_eq_nat @ Z3 @ Y3) => (~ ((ord_less_eq_nat @ Y3 @ X)))) => (((ord_less_eq_nat @ Y3 @ Z3) => (~ ((ord_less_eq_nat @ Z3 @ X)))) => (~ (((ord_less_eq_nat @ Z3 @ X) => (~ ((ord_less_eq_nat @ X @ Y3)))))))))))))). % le_cases3
thf(fact_112_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_113_order_Otrans, axiom,
    ((![A : nat, B : nat, C2 : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_nat @ B @ C2) => (ord_less_eq_nat @ A @ C2)))))). % order.trans
thf(fact_114_le__cases, axiom,
    ((![X : real, Y3 : real]: ((~ ((ord_less_eq_real @ X @ Y3))) => (ord_less_eq_real @ Y3 @ X))))). % le_cases
thf(fact_115_le__cases, axiom,
    ((![X : nat, Y3 : nat]: ((~ ((ord_less_eq_nat @ X @ Y3))) => (ord_less_eq_nat @ Y3 @ X))))). % le_cases
thf(fact_116_eq__refl, axiom,
    ((![X : real, Y3 : real]: ((X = Y3) => (ord_less_eq_real @ X @ Y3))))). % eq_refl
thf(fact_117_eq__refl, axiom,
    ((![X : nat, Y3 : nat]: ((X = Y3) => (ord_less_eq_nat @ X @ Y3))))). % eq_refl
thf(fact_118_linear, axiom,
    ((![X : real, Y3 : real]: ((ord_less_eq_real @ X @ Y3) | (ord_less_eq_real @ Y3 @ X))))). % linear
thf(fact_119_linear, axiom,
    ((![X : nat, Y3 : nat]: ((ord_less_eq_nat @ X @ Y3) | (ord_less_eq_nat @ Y3 @ X))))). % linear
thf(fact_120_antisym, axiom,
    ((![X : real, Y3 : real]: ((ord_less_eq_real @ X @ Y3) => ((ord_less_eq_real @ Y3 @ X) => (X = Y3)))))). % antisym
thf(fact_121_antisym, axiom,
    ((![X : nat, Y3 : nat]: ((ord_less_eq_nat @ X @ Y3) => ((ord_less_eq_nat @ Y3 @ X) => (X = Y3)))))). % antisym
thf(fact_122_eq__iff, axiom,
    (((^[Y2 : real]: (^[Z2 : real]: (Y2 = Z2))) = (^[X2 : real]: (^[Y : real]: (((ord_less_eq_real @ X2 @ Y)) & ((ord_less_eq_real @ Y @ X2)))))))). % eq_iff
thf(fact_123_eq__iff, axiom,
    (((^[Y2 : nat]: (^[Z2 : nat]: (Y2 = Z2))) = (^[X2 : nat]: (^[Y : nat]: (((ord_less_eq_nat @ X2 @ Y)) & ((ord_less_eq_nat @ Y @ X2)))))))). % eq_iff
thf(fact_124_ord__le__eq__subst, axiom,
    ((![A : real, B : real, F2 : real > real, C2 : real]: ((ord_less_eq_real @ A @ B) => (((F2 @ B) = C2) => ((![X3 : real, Y4 : real]: ((ord_less_eq_real @ X3 @ Y4) => (ord_less_eq_real @ (F2 @ X3) @ (F2 @ Y4)))) => (ord_less_eq_real @ (F2 @ A) @ C2))))))). % ord_le_eq_subst
thf(fact_125_ord__le__eq__subst, axiom,
    ((![A : real, B : real, F2 : real > nat, C2 : nat]: ((ord_less_eq_real @ A @ B) => (((F2 @ B) = C2) => ((![X3 : real, Y4 : real]: ((ord_less_eq_real @ X3 @ Y4) => (ord_less_eq_nat @ (F2 @ X3) @ (F2 @ Y4)))) => (ord_less_eq_nat @ (F2 @ A) @ C2))))))). % ord_le_eq_subst
thf(fact_126_ord__le__eq__subst, axiom,
    ((![A : nat, B : nat, F2 : nat > real, C2 : real]: ((ord_less_eq_nat @ A @ B) => (((F2 @ B) = C2) => ((![X3 : nat, Y4 : nat]: ((ord_less_eq_nat @ X3 @ Y4) => (ord_less_eq_real @ (F2 @ X3) @ (F2 @ Y4)))) => (ord_less_eq_real @ (F2 @ A) @ C2))))))). % ord_le_eq_subst
thf(fact_127_ord__le__eq__subst, axiom,
    ((![A : nat, B : nat, F2 : nat > nat, C2 : nat]: ((ord_less_eq_nat @ A @ B) => (((F2 @ B) = C2) => ((![X3 : nat, Y4 : nat]: ((ord_less_eq_nat @ X3 @ Y4) => (ord_less_eq_nat @ (F2 @ X3) @ (F2 @ Y4)))) => (ord_less_eq_nat @ (F2 @ A) @ C2))))))). % ord_le_eq_subst
thf(fact_128_ord__eq__le__subst, axiom,
    ((![A : real, F2 : real > real, B : real, C2 : real]: ((A = (F2 @ B)) => ((ord_less_eq_real @ B @ C2) => ((![X3 : real, Y4 : real]: ((ord_less_eq_real @ X3 @ Y4) => (ord_less_eq_real @ (F2 @ X3) @ (F2 @ Y4)))) => (ord_less_eq_real @ A @ (F2 @ C2)))))))). % ord_eq_le_subst
thf(fact_129_ord__eq__le__subst, axiom,
    ((![A : nat, F2 : real > nat, B : real, C2 : real]: ((A = (F2 @ B)) => ((ord_less_eq_real @ B @ C2) => ((![X3 : real, Y4 : real]: ((ord_less_eq_real @ X3 @ Y4) => (ord_less_eq_nat @ (F2 @ X3) @ (F2 @ Y4)))) => (ord_less_eq_nat @ A @ (F2 @ C2)))))))). % ord_eq_le_subst
thf(fact_130_ord__eq__le__subst, axiom,
    ((![A : real, F2 : nat > real, B : nat, C2 : nat]: ((A = (F2 @ B)) => ((ord_less_eq_nat @ B @ C2) => ((![X3 : nat, Y4 : nat]: ((ord_less_eq_nat @ X3 @ Y4) => (ord_less_eq_real @ (F2 @ X3) @ (F2 @ Y4)))) => (ord_less_eq_real @ A @ (F2 @ C2)))))))). % ord_eq_le_subst
thf(fact_131_ord__eq__le__subst, axiom,
    ((![A : nat, F2 : nat > nat, B : nat, C2 : nat]: ((A = (F2 @ B)) => ((ord_less_eq_nat @ B @ C2) => ((![X3 : nat, Y4 : nat]: ((ord_less_eq_nat @ X3 @ Y4) => (ord_less_eq_nat @ (F2 @ X3) @ (F2 @ Y4)))) => (ord_less_eq_nat @ A @ (F2 @ C2)))))))). % ord_eq_le_subst
thf(fact_132_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) => ((![X3 : real, Y4 : real]: ((ord_less_eq_real @ X3 @ Y4) => (ord_less_eq_real @ (F2 @ X3) @ (F2 @ Y4)))) => (ord_less_eq_real @ (F2 @ A) @ C2))))))). % order_subst2
thf(fact_133_order__subst2, axiom,
    ((![A : real, B : real, F2 : real > nat, C2 : nat]: ((ord_less_eq_real @ A @ B) => ((ord_less_eq_nat @ (F2 @ B) @ C2) => ((![X3 : real, Y4 : real]: ((ord_less_eq_real @ X3 @ Y4) => (ord_less_eq_nat @ (F2 @ X3) @ (F2 @ Y4)))) => (ord_less_eq_nat @ (F2 @ A) @ C2))))))). % order_subst2
thf(fact_134_order__subst2, axiom,
    ((![A : nat, B : nat, F2 : nat > real, C2 : real]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_real @ (F2 @ B) @ C2) => ((![X3 : nat, Y4 : nat]: ((ord_less_eq_nat @ X3 @ Y4) => (ord_less_eq_real @ (F2 @ X3) @ (F2 @ Y4)))) => (ord_less_eq_real @ (F2 @ A) @ C2))))))). % order_subst2
thf(fact_135_order__subst2, axiom,
    ((![A : nat, B : nat, F2 : nat > nat, C2 : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_nat @ (F2 @ B) @ C2) => ((![X3 : nat, Y4 : nat]: ((ord_less_eq_nat @ X3 @ Y4) => (ord_less_eq_nat @ (F2 @ X3) @ (F2 @ Y4)))) => (ord_less_eq_nat @ (F2 @ A) @ C2))))))). % order_subst2
thf(fact_136_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) => ((![X3 : real, Y4 : real]: ((ord_less_eq_real @ X3 @ Y4) => (ord_less_eq_real @ (F2 @ X3) @ (F2 @ Y4)))) => (ord_less_eq_real @ A @ (F2 @ C2)))))))). % order_subst1
thf(fact_137_order__subst1, axiom,
    ((![A : real, F2 : nat > real, B : nat, C2 : nat]: ((ord_less_eq_real @ A @ (F2 @ B)) => ((ord_less_eq_nat @ B @ C2) => ((![X3 : nat, Y4 : nat]: ((ord_less_eq_nat @ X3 @ Y4) => (ord_less_eq_real @ (F2 @ X3) @ (F2 @ Y4)))) => (ord_less_eq_real @ A @ (F2 @ C2)))))))). % order_subst1
thf(fact_138_order__subst1, axiom,
    ((![A : nat, F2 : real > nat, B : real, C2 : real]: ((ord_less_eq_nat @ A @ (F2 @ B)) => ((ord_less_eq_real @ B @ C2) => ((![X3 : real, Y4 : real]: ((ord_less_eq_real @ X3 @ Y4) => (ord_less_eq_nat @ (F2 @ X3) @ (F2 @ Y4)))) => (ord_less_eq_nat @ A @ (F2 @ C2)))))))). % order_subst1
thf(fact_139_order__subst1, axiom,
    ((![A : nat, F2 : nat > nat, B : nat, C2 : nat]: ((ord_less_eq_nat @ A @ (F2 @ B)) => ((ord_less_eq_nat @ B @ C2) => ((![X3 : nat, Y4 : nat]: ((ord_less_eq_nat @ X3 @ Y4) => (ord_less_eq_nat @ (F2 @ X3) @ (F2 @ Y4)))) => (ord_less_eq_nat @ A @ (F2 @ C2)))))))). % order_subst1
thf(fact_140_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_141_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_142_order_Ostrict__implies__not__eq, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (~ ((A = B))))))). % order.strict_implies_not_eq
thf(fact_143_order_Ostrict__implies__not__eq, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ B) => (~ ((A = B))))))). % order.strict_implies_not_eq
thf(fact_144_not__less__iff__gr__or__eq, axiom,
    ((![X : nat, Y3 : nat]: ((~ ((ord_less_nat @ X @ Y3))) = (((ord_less_nat @ Y3 @ X)) | ((X = Y3))))))). % not_less_iff_gr_or_eq
thf(fact_145_not__less__iff__gr__or__eq, axiom,
    ((![X : real, Y3 : real]: ((~ ((ord_less_real @ X @ Y3))) = (((ord_less_real @ Y3 @ X)) | ((X = Y3))))))). % not_less_iff_gr_or_eq
thf(fact_146_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_147_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_148_linorder__less__wlog, axiom,
    ((![P2 : nat > nat > $o, A : nat, B : nat]: ((![A3 : nat, B3 : nat]: ((ord_less_nat @ A3 @ B3) => (P2 @ A3 @ B3))) => ((![A3 : nat]: (P2 @ A3 @ A3)) => ((![A3 : nat, B3 : nat]: ((P2 @ B3 @ A3) => (P2 @ A3 @ B3))) => (P2 @ A @ B))))))). % linorder_less_wlog
thf(fact_149_linorder__less__wlog, axiom,
    ((![P2 : real > real > $o, A : real, B : real]: ((![A3 : real, B3 : real]: ((ord_less_real @ A3 @ B3) => (P2 @ A3 @ B3))) => ((![A3 : real]: (P2 @ A3 @ A3)) => ((![A3 : real, B3 : real]: ((P2 @ B3 @ A3) => (P2 @ A3 @ B3))) => (P2 @ A @ B))))))). % linorder_less_wlog
thf(fact_150_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_151_less__imp__not__less, axiom,
    ((![X : nat, Y3 : nat]: ((ord_less_nat @ X @ Y3) => (~ ((ord_less_nat @ Y3 @ X))))))). % less_imp_not_less
thf(fact_152_less__imp__not__less, axiom,
    ((![X : real, Y3 : real]: ((ord_less_real @ X @ Y3) => (~ ((ord_less_real @ Y3 @ X))))))). % less_imp_not_less
thf(fact_153_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_154_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_155_dual__order_Oirrefl, axiom,
    ((![A : nat]: (~ ((ord_less_nat @ A @ A)))))). % dual_order.irrefl
thf(fact_156_dual__order_Oirrefl, axiom,
    ((![A : real]: (~ ((ord_less_real @ A @ A)))))). % dual_order.irrefl
thf(fact_157_linorder__cases, axiom,
    ((![X : nat, Y3 : nat]: ((~ ((ord_less_nat @ X @ Y3))) => ((~ ((X = Y3))) => (ord_less_nat @ Y3 @ X)))))). % linorder_cases
thf(fact_158_linorder__cases, axiom,
    ((![X : real, Y3 : real]: ((~ ((ord_less_real @ X @ Y3))) => ((~ ((X = Y3))) => (ord_less_real @ Y3 @ X)))))). % linorder_cases
thf(fact_159_less__imp__triv, axiom,
    ((![X : nat, Y3 : nat, P2 : $o]: ((ord_less_nat @ X @ Y3) => ((ord_less_nat @ Y3 @ X) => P2))))). % less_imp_triv
thf(fact_160_less__imp__triv, axiom,
    ((![X : real, Y3 : real, P2 : $o]: ((ord_less_real @ X @ Y3) => ((ord_less_real @ Y3 @ X) => P2))))). % less_imp_triv
thf(fact_161_less__imp__not__eq2, axiom,
    ((![X : nat, Y3 : nat]: ((ord_less_nat @ X @ Y3) => (~ ((Y3 = X))))))). % less_imp_not_eq2
thf(fact_162_less__imp__not__eq2, axiom,
    ((![X : real, Y3 : real]: ((ord_less_real @ X @ Y3) => (~ ((Y3 = X))))))). % less_imp_not_eq2
thf(fact_163_antisym__conv3, axiom,
    ((![Y3 : nat, X : nat]: ((~ ((ord_less_nat @ Y3 @ X))) => ((~ ((ord_less_nat @ X @ Y3))) = (X = Y3)))))). % antisym_conv3
thf(fact_164_antisym__conv3, axiom,
    ((![Y3 : real, X : real]: ((~ ((ord_less_real @ Y3 @ X))) => ((~ ((ord_less_real @ X @ Y3))) = (X = Y3)))))). % antisym_conv3
thf(fact_165_less__induct, axiom,
    ((![P2 : nat > $o, A : nat]: ((![X3 : nat]: ((![Y5 : nat]: ((ord_less_nat @ Y5 @ X3) => (P2 @ Y5))) => (P2 @ X3))) => (P2 @ A))))). % less_induct
thf(fact_166_less__not__sym, axiom,
    ((![X : nat, Y3 : nat]: ((ord_less_nat @ X @ Y3) => (~ ((ord_less_nat @ Y3 @ X))))))). % less_not_sym
thf(fact_167_less__not__sym, axiom,
    ((![X : real, Y3 : real]: ((ord_less_real @ X @ Y3) => (~ ((ord_less_real @ Y3 @ X))))))). % less_not_sym
thf(fact_168_less__imp__not__eq, axiom,
    ((![X : nat, Y3 : nat]: ((ord_less_nat @ X @ Y3) => (~ ((X = Y3))))))). % less_imp_not_eq
thf(fact_169_less__imp__not__eq, axiom,
    ((![X : real, Y3 : real]: ((ord_less_real @ X @ Y3) => (~ ((X = Y3))))))). % less_imp_not_eq
thf(fact_170_dual__order_Oasym, axiom,
    ((![B : nat, A : nat]: ((ord_less_nat @ B @ A) => (~ ((ord_less_nat @ A @ B))))))). % dual_order.asym
thf(fact_171_dual__order_Oasym, axiom,
    ((![B : real, A : real]: ((ord_less_real @ B @ A) => (~ ((ord_less_real @ A @ B))))))). % dual_order.asym
thf(fact_172_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_173_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_174_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_175_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_176_less__irrefl, axiom,
    ((![X : nat]: (~ ((ord_less_nat @ X @ X)))))). % less_irrefl
thf(fact_177_less__irrefl, axiom,
    ((![X : real]: (~ ((ord_less_real @ X @ X)))))). % less_irrefl
thf(fact_178_less__linear, axiom,
    ((![X : nat, Y3 : nat]: ((ord_less_nat @ X @ Y3) | ((X = Y3) | (ord_less_nat @ Y3 @ X)))))). % less_linear
thf(fact_179_less__linear, axiom,
    ((![X : real, Y3 : real]: ((ord_less_real @ X @ Y3) | ((X = Y3) | (ord_less_real @ Y3 @ X)))))). % less_linear
thf(fact_180_less__trans, axiom,
    ((![X : nat, Y3 : nat, Z3 : nat]: ((ord_less_nat @ X @ Y3) => ((ord_less_nat @ Y3 @ Z3) => (ord_less_nat @ X @ Z3)))))). % less_trans
thf(fact_181_less__trans, axiom,
    ((![X : real, Y3 : real, Z3 : real]: ((ord_less_real @ X @ Y3) => ((ord_less_real @ Y3 @ Z3) => (ord_less_real @ X @ Z3)))))). % less_trans
thf(fact_182_less__asym_H, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (~ ((ord_less_nat @ B @ A))))))). % less_asym'
thf(fact_183_less__asym_H, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ B) => (~ ((ord_less_real @ B @ A))))))). % less_asym'
thf(fact_184_less__asym, axiom,
    ((![X : nat, Y3 : nat]: ((ord_less_nat @ X @ Y3) => (~ ((ord_less_nat @ Y3 @ X))))))). % less_asym
thf(fact_185_less__asym, axiom,
    ((![X : real, Y3 : real]: ((ord_less_real @ X @ Y3) => (~ ((ord_less_real @ Y3 @ X))))))). % less_asym
thf(fact_186_less__imp__neq, axiom,
    ((![X : nat, Y3 : nat]: ((ord_less_nat @ X @ Y3) => (~ ((X = Y3))))))). % less_imp_neq
thf(fact_187_less__imp__neq, axiom,
    ((![X : real, Y3 : real]: ((ord_less_real @ X @ Y3) => (~ ((X = Y3))))))). % less_imp_neq
thf(fact_188_dense, axiom,
    ((![X : real, Y3 : real]: ((ord_less_real @ X @ Y3) => (?[Z : real]: ((ord_less_real @ X @ Z) & (ord_less_real @ Z @ Y3))))))). % dense
thf(fact_189_order_Oasym, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (~ ((ord_less_nat @ B @ A))))))). % order.asym
thf(fact_190_order_Oasym, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ B) => (~ ((ord_less_real @ B @ A))))))). % order.asym
thf(fact_191_neq__iff, axiom,
    ((![X : nat, Y3 : nat]: ((~ ((X = Y3))) = (((ord_less_nat @ X @ Y3)) | ((ord_less_nat @ Y3 @ X))))))). % neq_iff
thf(fact_192_neq__iff, axiom,
    ((![X : real, Y3 : real]: ((~ ((X = Y3))) = (((ord_less_real @ X @ Y3)) | ((ord_less_real @ Y3 @ X))))))). % neq_iff
thf(fact_193_neqE, axiom,
    ((![X : nat, Y3 : nat]: ((~ ((X = Y3))) => ((~ ((ord_less_nat @ X @ Y3))) => (ord_less_nat @ Y3 @ X)))))). % neqE
thf(fact_194_neqE, axiom,
    ((![X : real, Y3 : real]: ((~ ((X = Y3))) => ((~ ((ord_less_real @ X @ Y3))) => (ord_less_real @ Y3 @ X)))))). % neqE
thf(fact_195_gt__ex, axiom,
    ((![X : nat]: (?[X_1 : nat]: (ord_less_nat @ X @ X_1))))). % gt_ex
thf(fact_196_gt__ex, axiom,
    ((![X : real]: (?[X_1 : real]: (ord_less_real @ X @ X_1))))). % gt_ex
thf(fact_197_lt__ex, axiom,
    ((![X : real]: (?[Y4 : real]: (ord_less_real @ Y4 @ X))))). % lt_ex
thf(fact_198_order__less__subst2, axiom,
    ((![A : nat, B : nat, F2 : nat > nat, C2 : nat]: ((ord_less_nat @ A @ B) => ((ord_less_nat @ (F2 @ B) @ C2) => ((![X3 : nat, Y4 : nat]: ((ord_less_nat @ X3 @ Y4) => (ord_less_nat @ (F2 @ X3) @ (F2 @ Y4)))) => (ord_less_nat @ (F2 @ A) @ C2))))))). % order_less_subst2
thf(fact_199_order__less__subst2, axiom,
    ((![A : nat, B : nat, F2 : nat > real, C2 : real]: ((ord_less_nat @ A @ B) => ((ord_less_real @ (F2 @ B) @ C2) => ((![X3 : nat, Y4 : nat]: ((ord_less_nat @ X3 @ Y4) => (ord_less_real @ (F2 @ X3) @ (F2 @ Y4)))) => (ord_less_real @ (F2 @ A) @ C2))))))). % order_less_subst2
thf(fact_200_order__less__subst2, axiom,
    ((![A : real, B : real, F2 : real > nat, C2 : nat]: ((ord_less_real @ A @ B) => ((ord_less_nat @ (F2 @ B) @ C2) => ((![X3 : real, Y4 : real]: ((ord_less_real @ X3 @ Y4) => (ord_less_nat @ (F2 @ X3) @ (F2 @ Y4)))) => (ord_less_nat @ (F2 @ A) @ C2))))))). % order_less_subst2
thf(fact_201_order__less__subst2, axiom,
    ((![A : real, B : real, F2 : real > real, C2 : real]: ((ord_less_real @ A @ B) => ((ord_less_real @ (F2 @ B) @ C2) => ((![X3 : real, Y4 : real]: ((ord_less_real @ X3 @ Y4) => (ord_less_real @ (F2 @ X3) @ (F2 @ Y4)))) => (ord_less_real @ (F2 @ A) @ C2))))))). % order_less_subst2
thf(fact_202_order__less__subst1, axiom,
    ((![A : nat, F2 : nat > nat, B : nat, C2 : nat]: ((ord_less_nat @ A @ (F2 @ B)) => ((ord_less_nat @ B @ C2) => ((![X3 : nat, Y4 : nat]: ((ord_less_nat @ X3 @ Y4) => (ord_less_nat @ (F2 @ X3) @ (F2 @ Y4)))) => (ord_less_nat @ A @ (F2 @ C2)))))))). % order_less_subst1
thf(fact_203_order__less__subst1, axiom,
    ((![A : nat, F2 : real > nat, B : real, C2 : real]: ((ord_less_nat @ A @ (F2 @ B)) => ((ord_less_real @ B @ C2) => ((![X3 : real, Y4 : real]: ((ord_less_real @ X3 @ Y4) => (ord_less_nat @ (F2 @ X3) @ (F2 @ Y4)))) => (ord_less_nat @ A @ (F2 @ C2)))))))). % order_less_subst1
thf(fact_204_order__less__subst1, axiom,
    ((![A : real, F2 : nat > real, B : nat, C2 : nat]: ((ord_less_real @ A @ (F2 @ B)) => ((ord_less_nat @ B @ C2) => ((![X3 : nat, Y4 : nat]: ((ord_less_nat @ X3 @ Y4) => (ord_less_real @ (F2 @ X3) @ (F2 @ Y4)))) => (ord_less_real @ A @ (F2 @ C2)))))))). % order_less_subst1
thf(fact_205_order__less__subst1, axiom,
    ((![A : real, F2 : real > real, B : real, C2 : real]: ((ord_less_real @ A @ (F2 @ B)) => ((ord_less_real @ B @ C2) => ((![X3 : real, Y4 : real]: ((ord_less_real @ X3 @ Y4) => (ord_less_real @ (F2 @ X3) @ (F2 @ Y4)))) => (ord_less_real @ A @ (F2 @ C2)))))))). % order_less_subst1
thf(fact_206_ord__less__eq__subst, axiom,
    ((![A : nat, B : nat, F2 : nat > nat, C2 : nat]: ((ord_less_nat @ A @ B) => (((F2 @ B) = C2) => ((![X3 : nat, Y4 : nat]: ((ord_less_nat @ X3 @ Y4) => (ord_less_nat @ (F2 @ X3) @ (F2 @ Y4)))) => (ord_less_nat @ (F2 @ A) @ C2))))))). % ord_less_eq_subst
thf(fact_207_ord__less__eq__subst, axiom,
    ((![A : nat, B : nat, F2 : nat > real, C2 : real]: ((ord_less_nat @ A @ B) => (((F2 @ B) = C2) => ((![X3 : nat, Y4 : nat]: ((ord_less_nat @ X3 @ Y4) => (ord_less_real @ (F2 @ X3) @ (F2 @ Y4)))) => (ord_less_real @ (F2 @ A) @ C2))))))). % ord_less_eq_subst
thf(fact_208_ord__less__eq__subst, axiom,
    ((![A : real, B : real, F2 : real > nat, C2 : nat]: ((ord_less_real @ A @ B) => (((F2 @ B) = C2) => ((![X3 : real, Y4 : real]: ((ord_less_real @ X3 @ Y4) => (ord_less_nat @ (F2 @ X3) @ (F2 @ Y4)))) => (ord_less_nat @ (F2 @ A) @ C2))))))). % ord_less_eq_subst
thf(fact_209_ord__less__eq__subst, axiom,
    ((![A : real, B : real, F2 : real > real, C2 : real]: ((ord_less_real @ A @ B) => (((F2 @ B) = C2) => ((![X3 : real, Y4 : real]: ((ord_less_real @ X3 @ Y4) => (ord_less_real @ (F2 @ X3) @ (F2 @ Y4)))) => (ord_less_real @ (F2 @ A) @ C2))))))). % ord_less_eq_subst
thf(fact_210_ord__eq__less__subst, axiom,
    ((![A : nat, F2 : nat > nat, B : nat, C2 : nat]: ((A = (F2 @ B)) => ((ord_less_nat @ B @ C2) => ((![X3 : nat, Y4 : nat]: ((ord_less_nat @ X3 @ Y4) => (ord_less_nat @ (F2 @ X3) @ (F2 @ Y4)))) => (ord_less_nat @ A @ (F2 @ C2)))))))). % ord_eq_less_subst
thf(fact_211_ord__eq__less__subst, axiom,
    ((![A : real, F2 : nat > real, B : nat, C2 : nat]: ((A = (F2 @ B)) => ((ord_less_nat @ B @ C2) => ((![X3 : nat, Y4 : nat]: ((ord_less_nat @ X3 @ Y4) => (ord_less_real @ (F2 @ X3) @ (F2 @ Y4)))) => (ord_less_real @ A @ (F2 @ C2)))))))). % ord_eq_less_subst
thf(fact_212_ord__eq__less__subst, axiom,
    ((![A : nat, F2 : real > nat, B : real, C2 : real]: ((A = (F2 @ B)) => ((ord_less_real @ B @ C2) => ((![X3 : real, Y4 : real]: ((ord_less_real @ X3 @ Y4) => (ord_less_nat @ (F2 @ X3) @ (F2 @ Y4)))) => (ord_less_nat @ A @ (F2 @ C2)))))))). % ord_eq_less_subst
thf(fact_213_ord__eq__less__subst, axiom,
    ((![A : real, F2 : real > real, B : real, C2 : real]: ((A = (F2 @ B)) => ((ord_less_real @ B @ C2) => ((![X3 : real, Y4 : real]: ((ord_less_real @ X3 @ Y4) => (ord_less_real @ (F2 @ X3) @ (F2 @ Y4)))) => (ord_less_real @ A @ (F2 @ C2)))))))). % ord_eq_less_subst
thf(fact_214_linorder__neqE__nat, axiom,
    ((![X : nat, Y3 : nat]: ((~ ((X = Y3))) => ((~ ((ord_less_nat @ X @ Y3))) => (ord_less_nat @ Y3 @ X)))))). % linorder_neqE_nat
thf(fact_215_infinite__descent, axiom,
    ((![P2 : nat > $o, N : nat]: ((![N3 : nat]: ((~ ((P2 @ N3))) => (?[M3 : nat]: ((ord_less_nat @ M3 @ N3) & (~ ((P2 @ M3))))))) => (P2 @ N))))). % infinite_descent
thf(fact_216_nat__less__induct, axiom,
    ((![P2 : nat > $o, N : nat]: ((![N3 : nat]: ((![M3 : nat]: ((ord_less_nat @ M3 @ N3) => (P2 @ M3))) => (P2 @ N3))) => (P2 @ N))))). % nat_less_induct
thf(fact_217_less__irrefl__nat, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_irrefl_nat
thf(fact_218_less__not__refl3, axiom,
    ((![S : nat, T : nat]: ((ord_less_nat @ S @ T) => (~ ((S = T))))))). % less_not_refl3
thf(fact_219_less__not__refl2, axiom,
    ((![N : nat, M : nat]: ((ord_less_nat @ N @ M) => (~ ((M = N))))))). % less_not_refl2
thf(fact_220_less__not__refl, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_not_refl
thf(fact_221_nat__neq__iff, axiom,
    ((![M : nat, N : nat]: ((~ ((M = N))) = (((ord_less_nat @ M @ N)) | ((ord_less_nat @ N @ M))))))). % nat_neq_iff
thf(fact_222_poly__bound__exists, axiom,
    ((![R : real, P : poly_real]: (?[M4 : real]: ((ord_less_real @ zero_zero_real @ M4) & (![Z4 : real]: ((ord_less_eq_real @ (real_V646646907m_real @ Z4) @ R) => (ord_less_eq_real @ (real_V646646907m_real @ (poly_real2 @ P @ Z4)) @ M4)))))))). % poly_bound_exists
thf(fact_223_poly__bound__exists, axiom,
    ((![R : real, P : poly_complex]: (?[M4 : real]: ((ord_less_real @ zero_zero_real @ M4) & (![Z4 : complex]: ((ord_less_eq_real @ (real_V638595069omplex @ Z4) @ R) => (ord_less_eq_real @ (real_V638595069omplex @ (poly_complex2 @ P @ Z4)) @ M4)))))))). % poly_bound_exists
thf(fact_224_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_225_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_226_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_227_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_228_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_229_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_230_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_231_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_232_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_233_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_234_dense__le__bounded, axiom,
    ((![X : real, Y3 : real, Z3 : real]: ((ord_less_real @ X @ Y3) => ((![W2 : real]: ((ord_less_real @ X @ W2) => ((ord_less_real @ W2 @ Y3) => (ord_less_eq_real @ W2 @ Z3)))) => (ord_less_eq_real @ Y3 @ Z3)))))). % dense_le_bounded
thf(fact_235_dense__ge__bounded, axiom,
    ((![Z3 : real, X : real, Y3 : real]: ((ord_less_real @ Z3 @ X) => ((![W2 : real]: ((ord_less_real @ Z3 @ W2) => ((ord_less_real @ W2 @ X) => (ord_less_eq_real @ Y3 @ W2)))) => (ord_less_eq_real @ Y3 @ Z3)))))). % dense_ge_bounded
thf(fact_236_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_237_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_238_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_239_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_240_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_241_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_242_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_243_order_Oorder__iff__strict, axiom,
    ((ord_less_eq_nat = (^[A2 : nat]: (^[B2 : nat]: (((ord_less_nat @ A2 @ B2)) | ((A2 = B2)))))))). % order.order_iff_strict

% Conjectures (1)
thf(conj_0, conjecture,
    ((?[Z4 : complex]: ((poly_complex2 @ pa @ Z4) = zero_zero_complex)))).
