% 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_646__5374362_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:31:48.588

% Could-be-implicit typings (7)
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__Set__Oset_It__Real__Oreal_J, type,
    set_real : $tType).
thf(ty_n_t__Complex__Ocomplex, type,
    complex : $tType).
thf(ty_n_t__Real__Oreal, type,
    real : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).

% Explicit typings (30)
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_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 > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Nat__Onat_J, type,
    ord_less_eq_o_nat : ($o > nat) > ($o > nat) > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Real__Oreal_J, type,
    ord_less_eq_o_real : ($o > real) > ($o > 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_Orderings_Oorder__class_OGreatest_001t__Nat__Onat, type,
    order_Greatest_nat : (nat > $o) > nat).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Real__Oreal, type,
    order_Greatest_real : (real > $o) > real).
thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Nat__Onat_001t__Nat__Onat, type,
    order_1631207636at_nat : (nat > nat) > $o).
thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Nat__Onat_001t__Real__Oreal, type,
    order_106095024t_real : (nat > real) > $o).
thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Real__Oreal_001t__Nat__Onat, type,
    order_1598331440al_nat : (real > nat) > $o).
thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Real__Oreal_001t__Real__Oreal, type,
    order_537808140l_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_c_Set_OCollect_001t__Real__Oreal, type,
    collect_real : (real > $o) > set_real).
thf(sy_c_member_001t__Real__Oreal, type,
    member_real : real > set_real > $o).
thf(sy_v_p, type,
    p : poly_complex).
thf(sy_v_pa____, type,
    pa : poly_complex).
thf(sy_v_thesis____, type,
    thesis : $o).

% Relevant facts (248)
thf(fact_0_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_1_nc, axiom,
    ((~ ((fundam1158420650omplex @ (poly_complex2 @ p)))))). % nc
thf(fact_2_less_Oprems, axiom,
    ((~ ((fundam1158420650omplex @ (poly_complex2 @ pa)))))). % less.prems
thf(fact_3_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_4_order__refl, axiom,
    ((![X : real]: (ord_less_eq_real @ X @ X)))). % order_refl
thf(fact_5_order__refl, axiom,
    ((![X : nat]: (ord_less_eq_nat @ X @ X)))). % order_refl
thf(fact_6_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_7_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_8_complete__real, axiom,
    ((![S : set_real]: ((?[X2 : real]: (member_real @ X2 @ S)) => ((?[Z2 : real]: (![X3 : real]: ((member_real @ X3 @ S) => (ord_less_eq_real @ X3 @ Z2)))) => (?[Y : real]: ((![X2 : real]: ((member_real @ X2 @ S) => (ord_less_eq_real @ X2 @ Y))) & (![Z2 : real]: ((![X3 : real]: ((member_real @ X3 @ S) => (ord_less_eq_real @ X3 @ Z2))) => (ord_less_eq_real @ Y @ Z2)))))))))). % complete_real
thf(fact_9_order__subst1, axiom,
    ((![A : real, F : real > real, B : real, C : real]: ((ord_less_eq_real @ A @ (F @ B)) => ((ord_less_eq_real @ B @ C) => ((![X3 : real, Y : real]: ((ord_less_eq_real @ X3 @ Y) => (ord_less_eq_real @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_real @ A @ (F @ C)))))))). % order_subst1
thf(fact_10_order__subst1, axiom,
    ((![A : real, F : nat > real, B : nat, C : nat]: ((ord_less_eq_real @ A @ (F @ B)) => ((ord_less_eq_nat @ B @ C) => ((![X3 : nat, Y : nat]: ((ord_less_eq_nat @ X3 @ Y) => (ord_less_eq_real @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_real @ A @ (F @ C)))))))). % order_subst1
thf(fact_11_order__subst1, axiom,
    ((![A : nat, F : real > nat, B : real, C : real]: ((ord_less_eq_nat @ A @ (F @ B)) => ((ord_less_eq_real @ B @ C) => ((![X3 : real, Y : real]: ((ord_less_eq_real @ X3 @ Y) => (ord_less_eq_nat @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_nat @ A @ (F @ C)))))))). % order_subst1
thf(fact_12_order__subst1, axiom,
    ((![A : nat, F : nat > nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ (F @ B)) => ((ord_less_eq_nat @ B @ C) => ((![X3 : nat, Y : nat]: ((ord_less_eq_nat @ X3 @ Y) => (ord_less_eq_nat @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_nat @ A @ (F @ C)))))))). % order_subst1
thf(fact_13_order__subst2, axiom,
    ((![A : real, B : real, F : real > real, C : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_eq_real @ (F @ B) @ C) => ((![X3 : real, Y : real]: ((ord_less_eq_real @ X3 @ Y) => (ord_less_eq_real @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_real @ (F @ A) @ C))))))). % order_subst2
thf(fact_14_order__subst2, axiom,
    ((![A : real, B : real, F : real > nat, C : nat]: ((ord_less_eq_real @ A @ B) => ((ord_less_eq_nat @ (F @ B) @ C) => ((![X3 : real, Y : real]: ((ord_less_eq_real @ X3 @ Y) => (ord_less_eq_nat @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_nat @ (F @ A) @ C))))))). % order_subst2
thf(fact_15_order__subst2, axiom,
    ((![A : nat, B : nat, F : nat > real, C : real]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_real @ (F @ B) @ C) => ((![X3 : nat, Y : nat]: ((ord_less_eq_nat @ X3 @ Y) => (ord_less_eq_real @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_real @ (F @ A) @ C))))))). % order_subst2
thf(fact_16_order__subst2, axiom,
    ((![A : nat, B : nat, F : nat > nat, C : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_nat @ (F @ B) @ C) => ((![X3 : nat, Y : nat]: ((ord_less_eq_nat @ X3 @ Y) => (ord_less_eq_nat @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_nat @ (F @ A) @ C))))))). % order_subst2
thf(fact_17_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_18_verit__la__disequality, axiom,
    ((![A : nat, B : nat]: ((A = B) | ((~ ((ord_less_eq_nat @ A @ B))) | (~ ((ord_less_eq_nat @ B @ A)))))))). % verit_la_disequality
thf(fact_19_ord__eq__le__subst, axiom,
    ((![A : real, F : real > real, B : real, C : real]: ((A = (F @ B)) => ((ord_less_eq_real @ B @ C) => ((![X3 : real, Y : real]: ((ord_less_eq_real @ X3 @ Y) => (ord_less_eq_real @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_real @ A @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_20_ord__eq__le__subst, axiom,
    ((![A : nat, F : real > nat, B : real, C : real]: ((A = (F @ B)) => ((ord_less_eq_real @ B @ C) => ((![X3 : real, Y : real]: ((ord_less_eq_real @ X3 @ Y) => (ord_less_eq_nat @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_nat @ A @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_21_ord__eq__le__subst, axiom,
    ((![A : real, F : nat > real, B : nat, C : nat]: ((A = (F @ B)) => ((ord_less_eq_nat @ B @ C) => ((![X3 : nat, Y : nat]: ((ord_less_eq_nat @ X3 @ Y) => (ord_less_eq_real @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_real @ A @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_22_ord__eq__le__subst, axiom,
    ((![A : nat, F : nat > nat, B : nat, C : nat]: ((A = (F @ B)) => ((ord_less_eq_nat @ B @ C) => ((![X3 : nat, Y : nat]: ((ord_less_eq_nat @ X3 @ Y) => (ord_less_eq_nat @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_nat @ A @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_23_ord__le__eq__subst, axiom,
    ((![A : real, B : real, F : real > real, C : real]: ((ord_less_eq_real @ A @ B) => (((F @ B) = C) => ((![X3 : real, Y : real]: ((ord_less_eq_real @ X3 @ Y) => (ord_less_eq_real @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_real @ (F @ A) @ C))))))). % ord_le_eq_subst
thf(fact_24_ord__le__eq__subst, axiom,
    ((![A : real, B : real, F : real > nat, C : nat]: ((ord_less_eq_real @ A @ B) => (((F @ B) = C) => ((![X3 : real, Y : real]: ((ord_less_eq_real @ X3 @ Y) => (ord_less_eq_nat @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_nat @ (F @ A) @ C))))))). % ord_le_eq_subst
thf(fact_25_ord__le__eq__subst, axiom,
    ((![A : nat, B : nat, F : nat > real, C : real]: ((ord_less_eq_nat @ A @ B) => (((F @ B) = C) => ((![X3 : nat, Y : nat]: ((ord_less_eq_nat @ X3 @ Y) => (ord_less_eq_real @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_real @ (F @ A) @ C))))))). % ord_le_eq_subst
thf(fact_26_ord__le__eq__subst, axiom,
    ((![A : nat, B : nat, F : nat > nat, C : nat]: ((ord_less_eq_nat @ A @ B) => (((F @ B) = C) => ((![X3 : nat, Y : nat]: ((ord_less_eq_nat @ X3 @ Y) => (ord_less_eq_nat @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_nat @ (F @ A) @ C))))))). % ord_le_eq_subst
thf(fact_27_constant__def, axiom,
    ((fundam1158420650omplex = (^[F2 : complex > complex]: (![X4 : complex]: (![Y2 : complex]: ((F2 @ X4) = (F2 @ Y2)))))))). % constant_def
thf(fact_28_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_29_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_30_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_31_dual__order_Oeq__iff, axiom,
    (((^[Y3 : nat]: (^[Z3 : nat]: (Y3 = Z3))) = (^[A2 : nat]: (^[B2 : nat]: (((ord_less_eq_nat @ B2 @ A2)) & ((ord_less_eq_nat @ A2 @ B2)))))))). % dual_order.eq_iff
thf(fact_32_dual__order_Otrans, axiom,
    ((![B : real, A : real, C : real]: ((ord_less_eq_real @ B @ A) => ((ord_less_eq_real @ C @ B) => (ord_less_eq_real @ C @ A)))))). % dual_order.trans
thf(fact_33_dual__order_Otrans, axiom,
    ((![B : nat, A : nat, C : nat]: ((ord_less_eq_nat @ B @ A) => ((ord_less_eq_nat @ C @ B) => (ord_less_eq_nat @ C @ A)))))). % dual_order.trans
thf(fact_34_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_35_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_36_dual__order_Orefl, axiom,
    ((![A : real]: (ord_less_eq_real @ A @ A)))). % dual_order.refl
thf(fact_37_dual__order_Orefl, axiom,
    ((![A : nat]: (ord_less_eq_nat @ A @ A)))). % dual_order.refl
thf(fact_38_order__trans, axiom,
    ((![X : real, Y4 : real, Z4 : real]: ((ord_less_eq_real @ X @ Y4) => ((ord_less_eq_real @ Y4 @ Z4) => (ord_less_eq_real @ X @ Z4)))))). % order_trans
thf(fact_39_order__trans, axiom,
    ((![X : nat, Y4 : nat, Z4 : nat]: ((ord_less_eq_nat @ X @ Y4) => ((ord_less_eq_nat @ Y4 @ Z4) => (ord_less_eq_nat @ X @ Z4)))))). % order_trans
thf(fact_40_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_41_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_42_ord__le__eq__trans, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_eq_real @ A @ B) => ((B = C) => (ord_less_eq_real @ A @ C)))))). % ord_le_eq_trans
thf(fact_43_ord__le__eq__trans, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ B) => ((B = C) => (ord_less_eq_nat @ A @ C)))))). % ord_le_eq_trans
thf(fact_44_ord__eq__le__trans, axiom,
    ((![A : real, B : real, C : real]: ((A = B) => ((ord_less_eq_real @ B @ C) => (ord_less_eq_real @ A @ C)))))). % ord_eq_le_trans
thf(fact_45_ord__eq__le__trans, axiom,
    ((![A : nat, B : nat, C : nat]: ((A = B) => ((ord_less_eq_nat @ B @ C) => (ord_less_eq_nat @ A @ C)))))). % ord_eq_le_trans
thf(fact_46_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_47_order__class_Oorder_Oeq__iff, axiom,
    (((^[Y3 : nat]: (^[Z3 : nat]: (Y3 = Z3))) = (^[A2 : nat]: (^[B2 : nat]: (((ord_less_eq_nat @ A2 @ B2)) & ((ord_less_eq_nat @ B2 @ A2)))))))). % order_class.order.eq_iff
thf(fact_48_antisym__conv, axiom,
    ((![Y4 : real, X : real]: ((ord_less_eq_real @ Y4 @ X) => ((ord_less_eq_real @ X @ Y4) = (X = Y4)))))). % antisym_conv
thf(fact_49_antisym__conv, axiom,
    ((![Y4 : nat, X : nat]: ((ord_less_eq_nat @ Y4 @ X) => ((ord_less_eq_nat @ X @ Y4) = (X = Y4)))))). % antisym_conv
thf(fact_50_le__cases3, axiom,
    ((![X : real, Y4 : real, Z4 : real]: (((ord_less_eq_real @ X @ Y4) => (~ ((ord_less_eq_real @ Y4 @ Z4)))) => (((ord_less_eq_real @ Y4 @ X) => (~ ((ord_less_eq_real @ X @ Z4)))) => (((ord_less_eq_real @ X @ Z4) => (~ ((ord_less_eq_real @ Z4 @ Y4)))) => (((ord_less_eq_real @ Z4 @ Y4) => (~ ((ord_less_eq_real @ Y4 @ X)))) => (((ord_less_eq_real @ Y4 @ Z4) => (~ ((ord_less_eq_real @ Z4 @ X)))) => (~ (((ord_less_eq_real @ Z4 @ X) => (~ ((ord_less_eq_real @ X @ Y4)))))))))))))). % le_cases3
thf(fact_51_le__cases3, axiom,
    ((![X : nat, Y4 : nat, Z4 : nat]: (((ord_less_eq_nat @ X @ Y4) => (~ ((ord_less_eq_nat @ Y4 @ Z4)))) => (((ord_less_eq_nat @ Y4 @ X) => (~ ((ord_less_eq_nat @ X @ Z4)))) => (((ord_less_eq_nat @ X @ Z4) => (~ ((ord_less_eq_nat @ Z4 @ Y4)))) => (((ord_less_eq_nat @ Z4 @ Y4) => (~ ((ord_less_eq_nat @ Y4 @ X)))) => (((ord_less_eq_nat @ Y4 @ Z4) => (~ ((ord_less_eq_nat @ Z4 @ X)))) => (~ (((ord_less_eq_nat @ Z4 @ X) => (~ ((ord_less_eq_nat @ X @ Y4)))))))))))))). % le_cases3
thf(fact_52_order_Otrans, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_eq_real @ B @ C) => (ord_less_eq_real @ A @ C)))))). % order.trans
thf(fact_53_order_Otrans, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_nat @ B @ C) => (ord_less_eq_nat @ A @ C)))))). % order.trans
thf(fact_54_le__cases, axiom,
    ((![X : real, Y4 : real]: ((~ ((ord_less_eq_real @ X @ Y4))) => (ord_less_eq_real @ Y4 @ X))))). % le_cases
thf(fact_55_le__cases, axiom,
    ((![X : nat, Y4 : nat]: ((~ ((ord_less_eq_nat @ X @ Y4))) => (ord_less_eq_nat @ Y4 @ X))))). % le_cases
thf(fact_56_eq__refl, axiom,
    ((![X : real, Y4 : real]: ((X = Y4) => (ord_less_eq_real @ X @ Y4))))). % eq_refl
thf(fact_57_eq__refl, axiom,
    ((![X : nat, Y4 : nat]: ((X = Y4) => (ord_less_eq_nat @ X @ Y4))))). % eq_refl
thf(fact_58_linear, axiom,
    ((![X : real, Y4 : real]: ((ord_less_eq_real @ X @ Y4) | (ord_less_eq_real @ Y4 @ X))))). % linear
thf(fact_59_linear, axiom,
    ((![X : nat, Y4 : nat]: ((ord_less_eq_nat @ X @ Y4) | (ord_less_eq_nat @ Y4 @ X))))). % linear
thf(fact_60_antisym, axiom,
    ((![X : real, Y4 : real]: ((ord_less_eq_real @ X @ Y4) => ((ord_less_eq_real @ Y4 @ X) => (X = Y4)))))). % antisym
thf(fact_61_antisym, axiom,
    ((![X : nat, Y4 : nat]: ((ord_less_eq_nat @ X @ Y4) => ((ord_less_eq_nat @ Y4 @ X) => (X = Y4)))))). % antisym
thf(fact_62_eq__iff, axiom,
    (((^[Y3 : real]: (^[Z3 : real]: (Y3 = Z3))) = (^[X4 : real]: (^[Y2 : real]: (((ord_less_eq_real @ X4 @ Y2)) & ((ord_less_eq_real @ Y2 @ X4)))))))). % eq_iff
thf(fact_63_eq__iff, axiom,
    (((^[Y3 : nat]: (^[Z3 : nat]: (Y3 = Z3))) = (^[X4 : nat]: (^[Y2 : nat]: (((ord_less_eq_nat @ X4 @ Y2)) & ((ord_less_eq_nat @ Y2 @ X4)))))))). % eq_iff
thf(fact_64_Greatest__equality, axiom,
    ((![P2 : real > $o, X : real]: ((P2 @ X) => ((![Y : real]: ((P2 @ Y) => (ord_less_eq_real @ Y @ X))) => ((order_Greatest_real @ P2) = X)))))). % Greatest_equality
thf(fact_65_Greatest__equality, axiom,
    ((![P2 : nat > $o, X : nat]: ((P2 @ X) => ((![Y : nat]: ((P2 @ Y) => (ord_less_eq_nat @ Y @ X))) => ((order_Greatest_nat @ P2) = X)))))). % Greatest_equality
thf(fact_66_GreatestI2__order, axiom,
    ((![P2 : real > $o, X : real, Q2 : real > $o]: ((P2 @ X) => ((![Y : real]: ((P2 @ Y) => (ord_less_eq_real @ Y @ X))) => ((![X3 : real]: ((P2 @ X3) => ((![Y5 : real]: ((P2 @ Y5) => (ord_less_eq_real @ Y5 @ X3))) => (Q2 @ X3)))) => (Q2 @ (order_Greatest_real @ P2)))))))). % GreatestI2_order
thf(fact_67_GreatestI2__order, axiom,
    ((![P2 : nat > $o, X : nat, Q2 : nat > $o]: ((P2 @ X) => ((![Y : nat]: ((P2 @ Y) => (ord_less_eq_nat @ Y @ X))) => ((![X3 : nat]: ((P2 @ X3) => ((![Y5 : nat]: ((P2 @ Y5) => (ord_less_eq_nat @ Y5 @ X3))) => (Q2 @ X3)))) => (Q2 @ (order_Greatest_nat @ P2)))))))). % GreatestI2_order
thf(fact_68_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_69_le__rel__bool__arg__iff, axiom,
    ((ord_less_eq_o_real = (^[X5 : $o > real]: (^[Y6 : $o > real]: (((ord_less_eq_real @ (X5 @ $false) @ (Y6 @ $false))) & ((ord_less_eq_real @ (X5 @ $true) @ (Y6 @ $true))))))))). % le_rel_bool_arg_iff
thf(fact_70_le__rel__bool__arg__iff, axiom,
    ((ord_less_eq_o_nat = (^[X5 : $o > nat]: (^[Y6 : $o > nat]: (((ord_less_eq_nat @ (X5 @ $false) @ (Y6 @ $false))) & ((ord_less_eq_nat @ (X5 @ $true) @ (Y6 @ $true))))))))). % le_rel_bool_arg_iff
thf(fact_71_antimono__def, axiom,
    ((order_537808140l_real = (^[F2 : real > real]: (![X4 : real]: (![Y2 : real]: (((ord_less_eq_real @ X4 @ Y2)) => ((ord_less_eq_real @ (F2 @ Y2) @ (F2 @ X4)))))))))). % antimono_def
thf(fact_72_antimono__def, axiom,
    ((order_1598331440al_nat = (^[F2 : real > nat]: (![X4 : real]: (![Y2 : real]: (((ord_less_eq_real @ X4 @ Y2)) => ((ord_less_eq_nat @ (F2 @ Y2) @ (F2 @ X4)))))))))). % antimono_def
thf(fact_73_antimono__def, axiom,
    ((order_106095024t_real = (^[F2 : nat > real]: (![X4 : nat]: (![Y2 : nat]: (((ord_less_eq_nat @ X4 @ Y2)) => ((ord_less_eq_real @ (F2 @ Y2) @ (F2 @ X4)))))))))). % antimono_def
thf(fact_74_antimono__def, axiom,
    ((order_1631207636at_nat = (^[F2 : nat > nat]: (![X4 : nat]: (![Y2 : nat]: (((ord_less_eq_nat @ X4 @ Y2)) => ((ord_less_eq_nat @ (F2 @ Y2) @ (F2 @ X4)))))))))). % antimono_def
thf(fact_75_antimonoI, axiom,
    ((![F : real > real]: ((![X3 : real, Y : real]: ((ord_less_eq_real @ X3 @ Y) => (ord_less_eq_real @ (F @ Y) @ (F @ X3)))) => (order_537808140l_real @ F))))). % antimonoI
thf(fact_76_antimonoI, axiom,
    ((![F : real > nat]: ((![X3 : real, Y : real]: ((ord_less_eq_real @ X3 @ Y) => (ord_less_eq_nat @ (F @ Y) @ (F @ X3)))) => (order_1598331440al_nat @ F))))). % antimonoI
thf(fact_77_antimonoI, axiom,
    ((![F : nat > real]: ((![X3 : nat, Y : nat]: ((ord_less_eq_nat @ X3 @ Y) => (ord_less_eq_real @ (F @ Y) @ (F @ X3)))) => (order_106095024t_real @ F))))). % antimonoI
thf(fact_78_antimonoI, axiom,
    ((![F : nat > nat]: ((![X3 : nat, Y : nat]: ((ord_less_eq_nat @ X3 @ Y) => (ord_less_eq_nat @ (F @ Y) @ (F @ X3)))) => (order_1631207636at_nat @ F))))). % antimonoI
thf(fact_79_antimonoE, axiom,
    ((![F : real > real, X : real, Y4 : real]: ((order_537808140l_real @ F) => ((ord_less_eq_real @ X @ Y4) => (ord_less_eq_real @ (F @ Y4) @ (F @ X))))))). % antimonoE
thf(fact_80_antimonoE, axiom,
    ((![F : real > nat, X : real, Y4 : real]: ((order_1598331440al_nat @ F) => ((ord_less_eq_real @ X @ Y4) => (ord_less_eq_nat @ (F @ Y4) @ (F @ X))))))). % antimonoE
thf(fact_81_antimonoE, axiom,
    ((![F : nat > real, X : nat, Y4 : nat]: ((order_106095024t_real @ F) => ((ord_less_eq_nat @ X @ Y4) => (ord_less_eq_real @ (F @ Y4) @ (F @ X))))))). % antimonoE
thf(fact_82_antimonoE, axiom,
    ((![F : nat > nat, X : nat, Y4 : nat]: ((order_1631207636at_nat @ F) => ((ord_less_eq_nat @ X @ Y4) => (ord_less_eq_nat @ (F @ Y4) @ (F @ X))))))). % antimonoE
thf(fact_83_antimonoD, axiom,
    ((![F : real > real, X : real, Y4 : real]: ((order_537808140l_real @ F) => ((ord_less_eq_real @ X @ Y4) => (ord_less_eq_real @ (F @ Y4) @ (F @ X))))))). % antimonoD
thf(fact_84_antimonoD, axiom,
    ((![F : real > nat, X : real, Y4 : real]: ((order_1598331440al_nat @ F) => ((ord_less_eq_real @ X @ Y4) => (ord_less_eq_nat @ (F @ Y4) @ (F @ X))))))). % antimonoD
thf(fact_85_antimonoD, axiom,
    ((![F : nat > real, X : nat, Y4 : nat]: ((order_106095024t_real @ F) => ((ord_less_eq_nat @ X @ Y4) => (ord_less_eq_real @ (F @ Y4) @ (F @ X))))))). % antimonoD
thf(fact_86_antimonoD, axiom,
    ((![F : nat > nat, X : nat, Y4 : nat]: ((order_1631207636at_nat @ F) => ((ord_less_eq_nat @ X @ Y4) => (ord_less_eq_nat @ (F @ Y4) @ (F @ X))))))). % antimonoD
thf(fact_87_poly__bound__exists, axiom,
    ((![R : real, P : poly_real]: (?[M : real]: ((ord_less_real @ zero_zero_real @ M) & (![Z2 : real]: ((ord_less_eq_real @ (real_V646646907m_real @ Z2) @ R) => (ord_less_eq_real @ (real_V646646907m_real @ (poly_real2 @ P @ Z2)) @ M)))))))). % poly_bound_exists
thf(fact_88_poly__bound__exists, axiom,
    ((![R : real, P : poly_complex]: (?[M : real]: ((ord_less_real @ zero_zero_real @ M) & (![Z2 : complex]: ((ord_less_eq_real @ (real_V638595069omplex @ Z2) @ R) => (ord_less_eq_real @ (real_V638595069omplex @ (poly_complex2 @ P @ Z2)) @ M)))))))). % poly_bound_exists
thf(fact_89_poly__0, axiom,
    ((![X : complex]: ((poly_complex2 @ zero_z1746442943omplex @ X) = zero_zero_complex)))). % poly_0
thf(fact_90_poly__0, axiom,
    ((![X : real]: ((poly_real2 @ zero_zero_poly_real @ X) = zero_zero_real)))). % poly_0
thf(fact_91_poly__0, axiom,
    ((![X : nat]: ((poly_nat2 @ zero_zero_poly_nat @ X) = zero_zero_nat)))). % poly_0
thf(fact_92_mem__Collect__eq, axiom,
    ((![A : real, P2 : real > $o]: ((member_real @ A @ (collect_real @ P2)) = (P2 @ A))))). % mem_Collect_eq
thf(fact_93_Collect__mem__eq, axiom,
    ((![A4 : set_real]: ((collect_real @ (^[X4 : real]: (member_real @ X4 @ A4))) = A4)))). % Collect_mem_eq
thf(fact_94_verit__comp__simplify1_I1_J, axiom,
    ((![A : nat]: (~ ((ord_less_nat @ A @ A)))))). % verit_comp_simplify1(1)
thf(fact_95_verit__comp__simplify1_I1_J, axiom,
    ((![A : real]: (~ ((ord_less_real @ A @ A)))))). % verit_comp_simplify1(1)
thf(fact_96_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_97_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_98_field__lbound__gt__zero, axiom,
    ((![D1 : real, D2 : real]: ((ord_less_real @ zero_zero_real @ D1) => ((ord_less_real @ zero_zero_real @ D2) => (?[E : real]: ((ord_less_real @ zero_zero_real @ E) & ((ord_less_real @ E @ D1) & (ord_less_real @ E @ D2))))))))). % field_lbound_gt_zero
thf(fact_99_ord__eq__less__subst, axiom,
    ((![A : nat, F : nat > nat, B : nat, C : nat]: ((A = (F @ B)) => ((ord_less_nat @ B @ C) => ((![X3 : nat, Y : nat]: ((ord_less_nat @ X3 @ Y) => (ord_less_nat @ (F @ X3) @ (F @ Y)))) => (ord_less_nat @ A @ (F @ C)))))))). % ord_eq_less_subst
thf(fact_100_ord__eq__less__subst, axiom,
    ((![A : real, F : nat > real, B : nat, C : nat]: ((A = (F @ B)) => ((ord_less_nat @ B @ C) => ((![X3 : nat, Y : nat]: ((ord_less_nat @ X3 @ Y) => (ord_less_real @ (F @ X3) @ (F @ Y)))) => (ord_less_real @ A @ (F @ C)))))))). % ord_eq_less_subst
thf(fact_101_ord__eq__less__subst, axiom,
    ((![A : nat, F : real > nat, B : real, C : real]: ((A = (F @ B)) => ((ord_less_real @ B @ C) => ((![X3 : real, Y : real]: ((ord_less_real @ X3 @ Y) => (ord_less_nat @ (F @ X3) @ (F @ Y)))) => (ord_less_nat @ A @ (F @ C)))))))). % ord_eq_less_subst
thf(fact_102_ord__eq__less__subst, axiom,
    ((![A : real, F : real > real, B : real, C : real]: ((A = (F @ B)) => ((ord_less_real @ B @ C) => ((![X3 : real, Y : real]: ((ord_less_real @ X3 @ Y) => (ord_less_real @ (F @ X3) @ (F @ Y)))) => (ord_less_real @ A @ (F @ C)))))))). % ord_eq_less_subst
thf(fact_103_ord__less__eq__subst, axiom,
    ((![A : nat, B : nat, F : nat > nat, C : nat]: ((ord_less_nat @ A @ B) => (((F @ B) = C) => ((![X3 : nat, Y : nat]: ((ord_less_nat @ X3 @ Y) => (ord_less_nat @ (F @ X3) @ (F @ Y)))) => (ord_less_nat @ (F @ A) @ C))))))). % ord_less_eq_subst
thf(fact_104_ord__less__eq__subst, axiom,
    ((![A : nat, B : nat, F : nat > real, C : real]: ((ord_less_nat @ A @ B) => (((F @ B) = C) => ((![X3 : nat, Y : nat]: ((ord_less_nat @ X3 @ Y) => (ord_less_real @ (F @ X3) @ (F @ Y)))) => (ord_less_real @ (F @ A) @ C))))))). % ord_less_eq_subst
thf(fact_105_ord__less__eq__subst, axiom,
    ((![A : real, B : real, F : real > nat, C : nat]: ((ord_less_real @ A @ B) => (((F @ B) = C) => ((![X3 : real, Y : real]: ((ord_less_real @ X3 @ Y) => (ord_less_nat @ (F @ X3) @ (F @ Y)))) => (ord_less_nat @ (F @ A) @ C))))))). % ord_less_eq_subst
thf(fact_106_ord__less__eq__subst, axiom,
    ((![A : real, B : real, F : real > real, C : real]: ((ord_less_real @ A @ B) => (((F @ B) = C) => ((![X3 : real, Y : real]: ((ord_less_real @ X3 @ Y) => (ord_less_real @ (F @ X3) @ (F @ Y)))) => (ord_less_real @ (F @ A) @ C))))))). % ord_less_eq_subst
thf(fact_107_order__less__subst1, axiom,
    ((![A : nat, F : nat > nat, B : nat, C : nat]: ((ord_less_nat @ A @ (F @ B)) => ((ord_less_nat @ B @ C) => ((![X3 : nat, Y : nat]: ((ord_less_nat @ X3 @ Y) => (ord_less_nat @ (F @ X3) @ (F @ Y)))) => (ord_less_nat @ A @ (F @ C)))))))). % order_less_subst1
thf(fact_108_order__less__subst1, axiom,
    ((![A : nat, F : real > nat, B : real, C : real]: ((ord_less_nat @ A @ (F @ B)) => ((ord_less_real @ B @ C) => ((![X3 : real, Y : real]: ((ord_less_real @ X3 @ Y) => (ord_less_nat @ (F @ X3) @ (F @ Y)))) => (ord_less_nat @ A @ (F @ C)))))))). % order_less_subst1
thf(fact_109_order__less__subst1, axiom,
    ((![A : real, F : nat > real, B : nat, C : nat]: ((ord_less_real @ A @ (F @ B)) => ((ord_less_nat @ B @ C) => ((![X3 : nat, Y : nat]: ((ord_less_nat @ X3 @ Y) => (ord_less_real @ (F @ X3) @ (F @ Y)))) => (ord_less_real @ A @ (F @ C)))))))). % order_less_subst1
thf(fact_110_order__less__subst1, axiom,
    ((![A : real, F : real > real, B : real, C : real]: ((ord_less_real @ A @ (F @ B)) => ((ord_less_real @ B @ C) => ((![X3 : real, Y : real]: ((ord_less_real @ X3 @ Y) => (ord_less_real @ (F @ X3) @ (F @ Y)))) => (ord_less_real @ A @ (F @ C)))))))). % order_less_subst1
thf(fact_111_order__less__subst2, axiom,
    ((![A : nat, B : nat, F : nat > nat, C : nat]: ((ord_less_nat @ A @ B) => ((ord_less_nat @ (F @ B) @ C) => ((![X3 : nat, Y : nat]: ((ord_less_nat @ X3 @ Y) => (ord_less_nat @ (F @ X3) @ (F @ Y)))) => (ord_less_nat @ (F @ A) @ C))))))). % order_less_subst2
thf(fact_112_order__less__subst2, axiom,
    ((![A : nat, B : nat, F : nat > real, C : real]: ((ord_less_nat @ A @ B) => ((ord_less_real @ (F @ B) @ C) => ((![X3 : nat, Y : nat]: ((ord_less_nat @ X3 @ Y) => (ord_less_real @ (F @ X3) @ (F @ Y)))) => (ord_less_real @ (F @ A) @ C))))))). % order_less_subst2
thf(fact_113_order__less__subst2, axiom,
    ((![A : real, B : real, F : real > nat, C : nat]: ((ord_less_real @ A @ B) => ((ord_less_nat @ (F @ B) @ C) => ((![X3 : real, Y : real]: ((ord_less_real @ X3 @ Y) => (ord_less_nat @ (F @ X3) @ (F @ Y)))) => (ord_less_nat @ (F @ A) @ C))))))). % order_less_subst2
thf(fact_114_order__less__subst2, axiom,
    ((![A : real, B : real, F : real > real, C : real]: ((ord_less_real @ A @ B) => ((ord_less_real @ (F @ B) @ C) => ((![X3 : real, Y : real]: ((ord_less_real @ X3 @ Y) => (ord_less_real @ (F @ X3) @ (F @ Y)))) => (ord_less_real @ (F @ A) @ C))))))). % order_less_subst2
thf(fact_115_lt__ex, axiom,
    ((![X : real]: (?[Y : real]: (ord_less_real @ Y @ X))))). % lt_ex
thf(fact_116_gt__ex, axiom,
    ((![X : nat]: (?[X_1 : nat]: (ord_less_nat @ X @ X_1))))). % gt_ex
thf(fact_117_gt__ex, axiom,
    ((![X : real]: (?[X_1 : real]: (ord_less_real @ X @ X_1))))). % gt_ex
thf(fact_118_neqE, axiom,
    ((![X : nat, Y4 : nat]: ((~ ((X = Y4))) => ((~ ((ord_less_nat @ X @ Y4))) => (ord_less_nat @ Y4 @ X)))))). % neqE
thf(fact_119_neqE, axiom,
    ((![X : real, Y4 : real]: ((~ ((X = Y4))) => ((~ ((ord_less_real @ X @ Y4))) => (ord_less_real @ Y4 @ X)))))). % neqE
thf(fact_120_neq__iff, axiom,
    ((![X : nat, Y4 : nat]: ((~ ((X = Y4))) = (((ord_less_nat @ X @ Y4)) | ((ord_less_nat @ Y4 @ X))))))). % neq_iff
thf(fact_121_neq__iff, axiom,
    ((![X : real, Y4 : real]: ((~ ((X = Y4))) = (((ord_less_real @ X @ Y4)) | ((ord_less_real @ Y4 @ X))))))). % neq_iff
thf(fact_122_order_Oasym, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (~ ((ord_less_nat @ B @ A))))))). % order.asym
thf(fact_123_order_Oasym, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ B) => (~ ((ord_less_real @ B @ A))))))). % order.asym
thf(fact_124_dense, axiom,
    ((![X : real, Y4 : real]: ((ord_less_real @ X @ Y4) => (?[Z : real]: ((ord_less_real @ X @ Z) & (ord_less_real @ Z @ Y4))))))). % dense
thf(fact_125_less__imp__neq, axiom,
    ((![X : nat, Y4 : nat]: ((ord_less_nat @ X @ Y4) => (~ ((X = Y4))))))). % less_imp_neq
thf(fact_126_less__imp__neq, axiom,
    ((![X : real, Y4 : real]: ((ord_less_real @ X @ Y4) => (~ ((X = Y4))))))). % less_imp_neq
thf(fact_127_less__asym, axiom,
    ((![X : nat, Y4 : nat]: ((ord_less_nat @ X @ Y4) => (~ ((ord_less_nat @ Y4 @ X))))))). % less_asym
thf(fact_128_less__asym, axiom,
    ((![X : real, Y4 : real]: ((ord_less_real @ X @ Y4) => (~ ((ord_less_real @ Y4 @ X))))))). % less_asym
thf(fact_129_less__asym_H, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (~ ((ord_less_nat @ B @ A))))))). % less_asym'
thf(fact_130_less__asym_H, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ B) => (~ ((ord_less_real @ B @ A))))))). % less_asym'
thf(fact_131_less__trans, axiom,
    ((![X : nat, Y4 : nat, Z4 : nat]: ((ord_less_nat @ X @ Y4) => ((ord_less_nat @ Y4 @ Z4) => (ord_less_nat @ X @ Z4)))))). % less_trans
thf(fact_132_less__trans, axiom,
    ((![X : real, Y4 : real, Z4 : real]: ((ord_less_real @ X @ Y4) => ((ord_less_real @ Y4 @ Z4) => (ord_less_real @ X @ Z4)))))). % less_trans
thf(fact_133_less__linear, axiom,
    ((![X : nat, Y4 : nat]: ((ord_less_nat @ X @ Y4) | ((X = Y4) | (ord_less_nat @ Y4 @ X)))))). % less_linear
thf(fact_134_less__linear, axiom,
    ((![X : real, Y4 : real]: ((ord_less_real @ X @ Y4) | ((X = Y4) | (ord_less_real @ Y4 @ X)))))). % less_linear
thf(fact_135_less__irrefl, axiom,
    ((![X : nat]: (~ ((ord_less_nat @ X @ X)))))). % less_irrefl
thf(fact_136_less__irrefl, axiom,
    ((![X : real]: (~ ((ord_less_real @ X @ X)))))). % less_irrefl
thf(fact_137_ord__eq__less__trans, axiom,
    ((![A : nat, B : nat, C : nat]: ((A = B) => ((ord_less_nat @ B @ C) => (ord_less_nat @ A @ C)))))). % ord_eq_less_trans
thf(fact_138_ord__eq__less__trans, axiom,
    ((![A : real, B : real, C : real]: ((A = B) => ((ord_less_real @ B @ C) => (ord_less_real @ A @ C)))))). % ord_eq_less_trans
thf(fact_139_ord__less__eq__trans, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_nat @ A @ B) => ((B = C) => (ord_less_nat @ A @ C)))))). % ord_less_eq_trans
thf(fact_140_ord__less__eq__trans, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_real @ A @ B) => ((B = C) => (ord_less_real @ A @ C)))))). % ord_less_eq_trans
thf(fact_141_dual__order_Oasym, axiom,
    ((![B : nat, A : nat]: ((ord_less_nat @ B @ A) => (~ ((ord_less_nat @ A @ B))))))). % dual_order.asym
thf(fact_142_dual__order_Oasym, axiom,
    ((![B : real, A : real]: ((ord_less_real @ B @ A) => (~ ((ord_less_real @ A @ B))))))). % dual_order.asym
thf(fact_143_less__imp__not__eq, axiom,
    ((![X : nat, Y4 : nat]: ((ord_less_nat @ X @ Y4) => (~ ((X = Y4))))))). % less_imp_not_eq
thf(fact_144_less__imp__not__eq, axiom,
    ((![X : real, Y4 : real]: ((ord_less_real @ X @ Y4) => (~ ((X = Y4))))))). % less_imp_not_eq
thf(fact_145_less__not__sym, axiom,
    ((![X : nat, Y4 : nat]: ((ord_less_nat @ X @ Y4) => (~ ((ord_less_nat @ Y4 @ X))))))). % less_not_sym
thf(fact_146_less__not__sym, axiom,
    ((![X : real, Y4 : real]: ((ord_less_real @ X @ Y4) => (~ ((ord_less_real @ Y4 @ X))))))). % less_not_sym
thf(fact_147_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_148_antisym__conv3, axiom,
    ((![Y4 : nat, X : nat]: ((~ ((ord_less_nat @ Y4 @ X))) => ((~ ((ord_less_nat @ X @ Y4))) = (X = Y4)))))). % antisym_conv3
thf(fact_149_antisym__conv3, axiom,
    ((![Y4 : real, X : real]: ((~ ((ord_less_real @ Y4 @ X))) => ((~ ((ord_less_real @ X @ Y4))) = (X = Y4)))))). % antisym_conv3
thf(fact_150_less__imp__not__eq2, axiom,
    ((![X : nat, Y4 : nat]: ((ord_less_nat @ X @ Y4) => (~ ((Y4 = X))))))). % less_imp_not_eq2
thf(fact_151_less__imp__not__eq2, axiom,
    ((![X : real, Y4 : real]: ((ord_less_real @ X @ Y4) => (~ ((Y4 = X))))))). % less_imp_not_eq2
thf(fact_152_less__imp__triv, axiom,
    ((![X : nat, Y4 : nat, P2 : $o]: ((ord_less_nat @ X @ Y4) => ((ord_less_nat @ Y4 @ X) => P2))))). % less_imp_triv
thf(fact_153_less__imp__triv, axiom,
    ((![X : real, Y4 : real, P2 : $o]: ((ord_less_real @ X @ Y4) => ((ord_less_real @ Y4 @ X) => P2))))). % less_imp_triv
thf(fact_154_linorder__cases, axiom,
    ((![X : nat, Y4 : nat]: ((~ ((ord_less_nat @ X @ Y4))) => ((~ ((X = Y4))) => (ord_less_nat @ Y4 @ X)))))). % linorder_cases
thf(fact_155_linorder__cases, axiom,
    ((![X : real, Y4 : real]: ((~ ((ord_less_real @ X @ Y4))) => ((~ ((X = Y4))) => (ord_less_real @ Y4 @ X)))))). % linorder_cases
thf(fact_156_dual__order_Oirrefl, axiom,
    ((![A : nat]: (~ ((ord_less_nat @ A @ A)))))). % dual_order.irrefl
thf(fact_157_dual__order_Oirrefl, axiom,
    ((![A : real]: (~ ((ord_less_real @ A @ A)))))). % dual_order.irrefl
thf(fact_158_order_Ostrict__trans, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_nat @ A @ B) => ((ord_less_nat @ B @ C) => (ord_less_nat @ A @ C)))))). % order.strict_trans
thf(fact_159_order_Ostrict__trans, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_real @ A @ B) => ((ord_less_real @ B @ C) => (ord_less_real @ A @ C)))))). % order.strict_trans
thf(fact_160_less__imp__not__less, axiom,
    ((![X : nat, Y4 : nat]: ((ord_less_nat @ X @ Y4) => (~ ((ord_less_nat @ Y4 @ X))))))). % less_imp_not_less
thf(fact_161_less__imp__not__less, axiom,
    ((![X : real, Y4 : real]: ((ord_less_real @ X @ Y4) => (~ ((ord_less_real @ Y4 @ X))))))). % less_imp_not_less
thf(fact_162_exists__least__iff, axiom,
    (((^[P3 : nat > $o]: (?[X6 : nat]: (P3 @ X6))) = (^[P4 : nat > $o]: (?[N : nat]: (((P4 @ N)) & ((![M2 : nat]: (((ord_less_nat @ M2 @ N)) => ((~ ((P4 @ M2))))))))))))). % exists_least_iff
thf(fact_163_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_164_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_165_dual__order_Ostrict__trans, axiom,
    ((![B : nat, A : nat, C : nat]: ((ord_less_nat @ B @ A) => ((ord_less_nat @ C @ B) => (ord_less_nat @ C @ A)))))). % dual_order.strict_trans
thf(fact_166_dual__order_Ostrict__trans, axiom,
    ((![B : real, A : real, C : real]: ((ord_less_real @ B @ A) => ((ord_less_real @ C @ B) => (ord_less_real @ C @ A)))))). % dual_order.strict_trans
thf(fact_167_not__less__iff__gr__or__eq, axiom,
    ((![X : nat, Y4 : nat]: ((~ ((ord_less_nat @ X @ Y4))) = (((ord_less_nat @ Y4 @ X)) | ((X = Y4))))))). % not_less_iff_gr_or_eq
thf(fact_168_not__less__iff__gr__or__eq, axiom,
    ((![X : real, Y4 : real]: ((~ ((ord_less_real @ X @ Y4))) = (((ord_less_real @ Y4 @ X)) | ((X = Y4))))))). % not_less_iff_gr_or_eq
thf(fact_169_order_Ostrict__implies__not__eq, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (~ ((A = B))))))). % order.strict_implies_not_eq
thf(fact_170_order_Ostrict__implies__not__eq, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ B) => (~ ((A = B))))))). % order.strict_implies_not_eq
thf(fact_171_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_172_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_173_real__sup__exists, axiom,
    ((![P2 : real > $o]: ((?[X_12 : real]: (P2 @ X_12)) => ((?[Z2 : real]: (![X3 : real]: ((P2 @ X3) => (ord_less_real @ X3 @ Z2)))) => (?[S2 : real]: (![Y5 : real]: ((?[X4 : real]: (((P2 @ X4)) & ((ord_less_real @ Y5 @ X4)))) = (ord_less_real @ Y5 @ S2))))))))). % real_sup_exists
thf(fact_174_poly__all__0__iff__0, axiom,
    ((![P : poly_complex]: ((![X4 : complex]: ((poly_complex2 @ P @ X4) = zero_zero_complex)) = (P = zero_z1746442943omplex))))). % poly_all_0_iff_0
thf(fact_175_poly__all__0__iff__0, axiom,
    ((![P : poly_real]: ((![X4 : real]: ((poly_real2 @ P @ X4) = zero_zero_real)) = (P = zero_zero_poly_real))))). % poly_all_0_iff_0
thf(fact_176_verit__comp__simplify1_I3_J, axiom,
    ((![B4 : real, A5 : real]: ((~ ((ord_less_eq_real @ B4 @ A5))) = (ord_less_real @ A5 @ B4))))). % verit_comp_simplify1(3)
thf(fact_177_verit__comp__simplify1_I3_J, axiom,
    ((![B4 : nat, A5 : nat]: ((~ ((ord_less_eq_nat @ B4 @ A5))) = (ord_less_nat @ A5 @ B4))))). % verit_comp_simplify1(3)
thf(fact_178_leD, axiom,
    ((![Y4 : real, X : real]: ((ord_less_eq_real @ Y4 @ X) => (~ ((ord_less_real @ X @ Y4))))))). % leD
thf(fact_179_leD, axiom,
    ((![Y4 : nat, X : nat]: ((ord_less_eq_nat @ Y4 @ X) => (~ ((ord_less_nat @ X @ Y4))))))). % leD
thf(fact_180_leI, axiom,
    ((![X : real, Y4 : real]: ((~ ((ord_less_real @ X @ Y4))) => (ord_less_eq_real @ Y4 @ X))))). % leI
thf(fact_181_leI, axiom,
    ((![X : nat, Y4 : nat]: ((~ ((ord_less_nat @ X @ Y4))) => (ord_less_eq_nat @ Y4 @ X))))). % leI
thf(fact_182_le__less, axiom,
    ((ord_less_eq_real = (^[X4 : real]: (^[Y2 : real]: (((ord_less_real @ X4 @ Y2)) | ((X4 = Y2)))))))). % le_less
thf(fact_183_le__less, axiom,
    ((ord_less_eq_nat = (^[X4 : nat]: (^[Y2 : nat]: (((ord_less_nat @ X4 @ Y2)) | ((X4 = Y2)))))))). % le_less
thf(fact_184_less__le, axiom,
    ((ord_less_real = (^[X4 : real]: (^[Y2 : real]: (((ord_less_eq_real @ X4 @ Y2)) & ((~ ((X4 = Y2)))))))))). % less_le
thf(fact_185_less__le, axiom,
    ((ord_less_nat = (^[X4 : nat]: (^[Y2 : nat]: (((ord_less_eq_nat @ X4 @ Y2)) & ((~ ((X4 = Y2)))))))))). % less_le
thf(fact_186_order__le__less__subst1, axiom,
    ((![A : real, F : nat > real, B : nat, C : nat]: ((ord_less_eq_real @ A @ (F @ B)) => ((ord_less_nat @ B @ C) => ((![X3 : nat, Y : nat]: ((ord_less_nat @ X3 @ Y) => (ord_less_real @ (F @ X3) @ (F @ Y)))) => (ord_less_real @ A @ (F @ C)))))))). % order_le_less_subst1
thf(fact_187_order__le__less__subst1, axiom,
    ((![A : real, F : real > real, B : real, C : real]: ((ord_less_eq_real @ A @ (F @ B)) => ((ord_less_real @ B @ C) => ((![X3 : real, Y : real]: ((ord_less_real @ X3 @ Y) => (ord_less_real @ (F @ X3) @ (F @ Y)))) => (ord_less_real @ A @ (F @ C)))))))). % order_le_less_subst1
thf(fact_188_order__le__less__subst1, axiom,
    ((![A : nat, F : nat > nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ (F @ B)) => ((ord_less_nat @ B @ C) => ((![X3 : nat, Y : nat]: ((ord_less_nat @ X3 @ Y) => (ord_less_nat @ (F @ X3) @ (F @ Y)))) => (ord_less_nat @ A @ (F @ C)))))))). % order_le_less_subst1
thf(fact_189_order__le__less__subst1, axiom,
    ((![A : nat, F : real > nat, B : real, C : real]: ((ord_less_eq_nat @ A @ (F @ B)) => ((ord_less_real @ B @ C) => ((![X3 : real, Y : real]: ((ord_less_real @ X3 @ Y) => (ord_less_nat @ (F @ X3) @ (F @ Y)))) => (ord_less_nat @ A @ (F @ C)))))))). % order_le_less_subst1
thf(fact_190_order__le__less__subst2, axiom,
    ((![A : real, B : real, F : real > real, C : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_real @ (F @ B) @ C) => ((![X3 : real, Y : real]: ((ord_less_eq_real @ X3 @ Y) => (ord_less_eq_real @ (F @ X3) @ (F @ Y)))) => (ord_less_real @ (F @ A) @ C))))))). % order_le_less_subst2
thf(fact_191_order__le__less__subst2, axiom,
    ((![A : real, B : real, F : real > nat, C : nat]: ((ord_less_eq_real @ A @ B) => ((ord_less_nat @ (F @ B) @ C) => ((![X3 : real, Y : real]: ((ord_less_eq_real @ X3 @ Y) => (ord_less_eq_nat @ (F @ X3) @ (F @ Y)))) => (ord_less_nat @ (F @ A) @ C))))))). % order_le_less_subst2
thf(fact_192_order__le__less__subst2, axiom,
    ((![A : nat, B : nat, F : nat > real, C : real]: ((ord_less_eq_nat @ A @ B) => ((ord_less_real @ (F @ B) @ C) => ((![X3 : nat, Y : nat]: ((ord_less_eq_nat @ X3 @ Y) => (ord_less_eq_real @ (F @ X3) @ (F @ Y)))) => (ord_less_real @ (F @ A) @ C))))))). % order_le_less_subst2
thf(fact_193_order__le__less__subst2, axiom,
    ((![A : nat, B : nat, F : nat > nat, C : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_nat @ (F @ B) @ C) => ((![X3 : nat, Y : nat]: ((ord_less_eq_nat @ X3 @ Y) => (ord_less_eq_nat @ (F @ X3) @ (F @ Y)))) => (ord_less_nat @ (F @ A) @ C))))))). % order_le_less_subst2
thf(fact_194_order__less__le__subst1, axiom,
    ((![A : real, F : real > real, B : real, C : real]: ((ord_less_real @ A @ (F @ B)) => ((ord_less_eq_real @ B @ C) => ((![X3 : real, Y : real]: ((ord_less_eq_real @ X3 @ Y) => (ord_less_eq_real @ (F @ X3) @ (F @ Y)))) => (ord_less_real @ A @ (F @ C)))))))). % order_less_le_subst1
thf(fact_195_order__less__le__subst1, axiom,
    ((![A : nat, F : real > nat, B : real, C : real]: ((ord_less_nat @ A @ (F @ B)) => ((ord_less_eq_real @ B @ C) => ((![X3 : real, Y : real]: ((ord_less_eq_real @ X3 @ Y) => (ord_less_eq_nat @ (F @ X3) @ (F @ Y)))) => (ord_less_nat @ A @ (F @ C)))))))). % order_less_le_subst1
thf(fact_196_order__less__le__subst1, axiom,
    ((![A : real, F : nat > real, B : nat, C : nat]: ((ord_less_real @ A @ (F @ B)) => ((ord_less_eq_nat @ B @ C) => ((![X3 : nat, Y : nat]: ((ord_less_eq_nat @ X3 @ Y) => (ord_less_eq_real @ (F @ X3) @ (F @ Y)))) => (ord_less_real @ A @ (F @ C)))))))). % order_less_le_subst1
thf(fact_197_order__less__le__subst1, axiom,
    ((![A : nat, F : nat > nat, B : nat, C : nat]: ((ord_less_nat @ A @ (F @ B)) => ((ord_less_eq_nat @ B @ C) => ((![X3 : nat, Y : nat]: ((ord_less_eq_nat @ X3 @ Y) => (ord_less_eq_nat @ (F @ X3) @ (F @ Y)))) => (ord_less_nat @ A @ (F @ C)))))))). % order_less_le_subst1
thf(fact_198_order__less__le__subst2, axiom,
    ((![A : nat, B : nat, F : nat > real, C : real]: ((ord_less_nat @ A @ B) => ((ord_less_eq_real @ (F @ B) @ C) => ((![X3 : nat, Y : nat]: ((ord_less_nat @ X3 @ Y) => (ord_less_real @ (F @ X3) @ (F @ Y)))) => (ord_less_real @ (F @ A) @ C))))))). % order_less_le_subst2
thf(fact_199_order__less__le__subst2, axiom,
    ((![A : real, B : real, F : real > real, C : real]: ((ord_less_real @ A @ B) => ((ord_less_eq_real @ (F @ B) @ C) => ((![X3 : real, Y : real]: ((ord_less_real @ X3 @ Y) => (ord_less_real @ (F @ X3) @ (F @ Y)))) => (ord_less_real @ (F @ A) @ C))))))). % order_less_le_subst2
thf(fact_200_order__less__le__subst2, axiom,
    ((![A : nat, B : nat, F : nat > nat, C : nat]: ((ord_less_nat @ A @ B) => ((ord_less_eq_nat @ (F @ B) @ C) => ((![X3 : nat, Y : nat]: ((ord_less_nat @ X3 @ Y) => (ord_less_nat @ (F @ X3) @ (F @ Y)))) => (ord_less_nat @ (F @ A) @ C))))))). % order_less_le_subst2
thf(fact_201_order__less__le__subst2, axiom,
    ((![A : real, B : real, F : real > nat, C : nat]: ((ord_less_real @ A @ B) => ((ord_less_eq_nat @ (F @ B) @ C) => ((![X3 : real, Y : real]: ((ord_less_real @ X3 @ Y) => (ord_less_nat @ (F @ X3) @ (F @ Y)))) => (ord_less_nat @ (F @ A) @ C))))))). % order_less_le_subst2
thf(fact_202_not__le, axiom,
    ((![X : real, Y4 : real]: ((~ ((ord_less_eq_real @ X @ Y4))) = (ord_less_real @ Y4 @ X))))). % not_le
thf(fact_203_not__le, axiom,
    ((![X : nat, Y4 : nat]: ((~ ((ord_less_eq_nat @ X @ Y4))) = (ord_less_nat @ Y4 @ X))))). % not_le
thf(fact_204_not__less, axiom,
    ((![X : real, Y4 : real]: ((~ ((ord_less_real @ X @ Y4))) = (ord_less_eq_real @ Y4 @ X))))). % not_less
thf(fact_205_not__less, axiom,
    ((![X : nat, Y4 : nat]: ((~ ((ord_less_nat @ X @ Y4))) = (ord_less_eq_nat @ Y4 @ X))))). % not_less
thf(fact_206_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_207_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_208_antisym__conv1, axiom,
    ((![X : real, Y4 : real]: ((~ ((ord_less_real @ X @ Y4))) => ((ord_less_eq_real @ X @ Y4) = (X = Y4)))))). % antisym_conv1
thf(fact_209_antisym__conv1, axiom,
    ((![X : nat, Y4 : nat]: ((~ ((ord_less_nat @ X @ Y4))) => ((ord_less_eq_nat @ X @ Y4) = (X = Y4)))))). % antisym_conv1
thf(fact_210_antisym__conv2, axiom,
    ((![X : real, Y4 : real]: ((ord_less_eq_real @ X @ Y4) => ((~ ((ord_less_real @ X @ Y4))) = (X = Y4)))))). % antisym_conv2
thf(fact_211_antisym__conv2, axiom,
    ((![X : nat, Y4 : nat]: ((ord_less_eq_nat @ X @ Y4) => ((~ ((ord_less_nat @ X @ Y4))) = (X = Y4)))))). % antisym_conv2
thf(fact_212_less__imp__le, axiom,
    ((![X : real, Y4 : real]: ((ord_less_real @ X @ Y4) => (ord_less_eq_real @ X @ Y4))))). % less_imp_le
thf(fact_213_less__imp__le, axiom,
    ((![X : nat, Y4 : nat]: ((ord_less_nat @ X @ Y4) => (ord_less_eq_nat @ X @ Y4))))). % less_imp_le
thf(fact_214_le__less__trans, axiom,
    ((![X : real, Y4 : real, Z4 : real]: ((ord_less_eq_real @ X @ Y4) => ((ord_less_real @ Y4 @ Z4) => (ord_less_real @ X @ Z4)))))). % le_less_trans
thf(fact_215_le__less__trans, axiom,
    ((![X : nat, Y4 : nat, Z4 : nat]: ((ord_less_eq_nat @ X @ Y4) => ((ord_less_nat @ Y4 @ Z4) => (ord_less_nat @ X @ Z4)))))). % le_less_trans
thf(fact_216_less__le__trans, axiom,
    ((![X : real, Y4 : real, Z4 : real]: ((ord_less_real @ X @ Y4) => ((ord_less_eq_real @ Y4 @ Z4) => (ord_less_real @ X @ Z4)))))). % less_le_trans
thf(fact_217_less__le__trans, axiom,
    ((![X : nat, Y4 : nat, Z4 : nat]: ((ord_less_nat @ X @ Y4) => ((ord_less_eq_nat @ Y4 @ Z4) => (ord_less_nat @ X @ Z4)))))). % less_le_trans
thf(fact_218_dense__ge, axiom,
    ((![Z4 : real, Y4 : real]: ((![X3 : real]: ((ord_less_real @ Z4 @ X3) => (ord_less_eq_real @ Y4 @ X3))) => (ord_less_eq_real @ Y4 @ Z4))))). % dense_ge
thf(fact_219_dense__le, axiom,
    ((![Y4 : real, Z4 : real]: ((![X3 : real]: ((ord_less_real @ X3 @ Y4) => (ord_less_eq_real @ X3 @ Z4))) => (ord_less_eq_real @ Y4 @ Z4))))). % dense_le
thf(fact_220_le__less__linear, axiom,
    ((![X : real, Y4 : real]: ((ord_less_eq_real @ X @ Y4) | (ord_less_real @ Y4 @ X))))). % le_less_linear
thf(fact_221_le__less__linear, axiom,
    ((![X : nat, Y4 : nat]: ((ord_less_eq_nat @ X @ Y4) | (ord_less_nat @ Y4 @ X))))). % le_less_linear
thf(fact_222_le__imp__less__or__eq, axiom,
    ((![X : real, Y4 : real]: ((ord_less_eq_real @ X @ Y4) => ((ord_less_real @ X @ Y4) | (X = Y4)))))). % le_imp_less_or_eq
thf(fact_223_le__imp__less__or__eq, axiom,
    ((![X : nat, Y4 : nat]: ((ord_less_eq_nat @ X @ Y4) => ((ord_less_nat @ X @ Y4) | (X = Y4)))))). % le_imp_less_or_eq
thf(fact_224_less__le__not__le, axiom,
    ((ord_less_real = (^[X4 : real]: (^[Y2 : real]: (((ord_less_eq_real @ X4 @ Y2)) & ((~ ((ord_less_eq_real @ Y2 @ X4)))))))))). % less_le_not_le
thf(fact_225_less__le__not__le, axiom,
    ((ord_less_nat = (^[X4 : nat]: (^[Y2 : nat]: (((ord_less_eq_nat @ X4 @ Y2)) & ((~ ((ord_less_eq_nat @ Y2 @ X4)))))))))). % less_le_not_le
thf(fact_226_not__le__imp__less, axiom,
    ((![Y4 : real, X : real]: ((~ ((ord_less_eq_real @ Y4 @ X))) => (ord_less_real @ X @ Y4))))). % not_le_imp_less
thf(fact_227_not__le__imp__less, axiom,
    ((![Y4 : nat, X : nat]: ((~ ((ord_less_eq_nat @ Y4 @ X))) => (ord_less_nat @ X @ Y4))))). % not_le_imp_less
thf(fact_228_order_Ostrict__trans1, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_real @ B @ C) => (ord_less_real @ A @ C)))))). % order.strict_trans1
thf(fact_229_order_Ostrict__trans1, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_nat @ B @ C) => (ord_less_nat @ A @ C)))))). % order.strict_trans1
thf(fact_230_order_Ostrict__trans2, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_real @ A @ B) => ((ord_less_eq_real @ B @ C) => (ord_less_real @ A @ C)))))). % order.strict_trans2
thf(fact_231_order_Ostrict__trans2, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_nat @ A @ B) => ((ord_less_eq_nat @ B @ C) => (ord_less_nat @ A @ C)))))). % order.strict_trans2
thf(fact_232_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_233_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_234_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_235_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_236_dual__order_Ostrict__trans1, axiom,
    ((![B : real, A : real, C : real]: ((ord_less_eq_real @ B @ A) => ((ord_less_real @ C @ B) => (ord_less_real @ C @ A)))))). % dual_order.strict_trans1
thf(fact_237_dual__order_Ostrict__trans1, axiom,
    ((![B : nat, A : nat, C : nat]: ((ord_less_eq_nat @ B @ A) => ((ord_less_nat @ C @ B) => (ord_less_nat @ C @ A)))))). % dual_order.strict_trans1
thf(fact_238_dual__order_Ostrict__trans2, axiom,
    ((![B : real, A : real, C : real]: ((ord_less_real @ B @ A) => ((ord_less_eq_real @ C @ B) => (ord_less_real @ C @ A)))))). % dual_order.strict_trans2
thf(fact_239_dual__order_Ostrict__trans2, axiom,
    ((![B : nat, A : nat, C : nat]: ((ord_less_nat @ B @ A) => ((ord_less_eq_nat @ C @ B) => (ord_less_nat @ C @ A)))))). % dual_order.strict_trans2
thf(fact_240_dense__ge__bounded, axiom,
    ((![Z4 : real, X : real, Y4 : real]: ((ord_less_real @ Z4 @ X) => ((![W2 : real]: ((ord_less_real @ Z4 @ W2) => ((ord_less_real @ W2 @ X) => (ord_less_eq_real @ Y4 @ W2)))) => (ord_less_eq_real @ Y4 @ Z4)))))). % dense_ge_bounded
thf(fact_241_less__eq__real__def, axiom,
    ((ord_less_eq_real = (^[X4 : real]: (^[Y2 : real]: (((ord_less_real @ X4 @ Y2)) | ((X4 = Y2)))))))). % less_eq_real_def
thf(fact_242_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_243_less__nat__zero__code, axiom,
    ((![N2 : nat]: (~ ((ord_less_nat @ N2 @ zero_zero_nat)))))). % less_nat_zero_code
thf(fact_244_le0, axiom,
    ((![N2 : nat]: (ord_less_eq_nat @ zero_zero_nat @ N2)))). % le0
thf(fact_245_bot__nat__0_Oextremum, axiom,
    ((![A : nat]: (ord_less_eq_nat @ zero_zero_nat @ A)))). % bot_nat_0.extremum
thf(fact_246_neq0__conv, axiom,
    ((![N2 : nat]: ((~ ((N2 = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ N2))))). % neq0_conv
thf(fact_247_less__eq__nat_Osimps_I1_J, axiom,
    ((![N2 : nat]: (ord_less_eq_nat @ zero_zero_nat @ N2)))). % less_eq_nat.simps(1)

% Conjectures (2)
thf(conj_0, hypothesis,
    ((![C2 : complex]: ((![W2 : complex]: (ord_less_eq_real @ (real_V638595069omplex @ (poly_complex2 @ pa @ C2)) @ (real_V638595069omplex @ (poly_complex2 @ pa @ W2)))) => thesis)))).
thf(conj_1, conjecture,
    (thesis)).
