% 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_531__5373136_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:30:57.694

% 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__Rat__Orat_J, type,
    poly_rat : $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__Rat__Orat, type,
    rat : $tType).

% Explicit typings (30)
thf(sy_c_Groups_Ozero__class_Ozero_001t__Complex__Ocomplex, type,
    zero_zero_complex : complex).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    zero_z1746442943omplex : poly_complex).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Rat__Orat_J, type,
    zero_zero_poly_rat : poly_rat).
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__Rat__Orat, type,
    zero_zero_rat : rat).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal, type,
    zero_zero_real : real).
thf(sy_c_Orderings_Oord__class_Oless_001t__Rat__Orat, type,
    ord_less_rat : rat > rat > $o).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal, type,
    ord_less_real : real > real > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Rat__Orat_J, type,
    ord_less_eq_o_rat : ($o > rat) > ($o > rat) > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Real__Oreal_J, type,
    ord_less_eq_o_real : ($o > real) > ($o > real) > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Rat__Orat, type,
    ord_less_eq_rat : rat > rat > $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__Rat__Orat, type,
    order_Greatest_rat : (rat > $o) > rat).
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__Rat__Orat_001t__Rat__Orat, type,
    order_802983140at_rat : (rat > rat) > $o).
thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Rat__Orat_001t__Real__Oreal, type,
    order_1851487160t_real : (rat > real) > $o).
thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Real__Oreal_001t__Rat__Orat, type,
    order_765710392al_rat : (real > rat) > $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_OpCons_001t__Complex__Ocomplex, type,
    pCons_complex : complex > poly_complex > poly_complex).
thf(sy_c_Polynomial_OpCons_001t__Rat__Orat, type,
    pCons_rat : rat > poly_rat > poly_rat).
thf(sy_c_Polynomial_OpCons_001t__Real__Oreal, type,
    pCons_real : real > poly_real > poly_real).
thf(sy_c_Polynomial_Opoly_001t__Complex__Ocomplex, type,
    poly_complex2 : poly_complex > complex > complex).
thf(sy_c_Polynomial_Opoly_001t__Rat__Orat, type,
    poly_rat2 : poly_rat > rat > rat).
thf(sy_c_Polynomial_Opoly_001t__Real__Oreal, type,
    poly_real2 : poly_real > real > real).
thf(sy_c_Real_ORatreal, type,
    ratreal : rat > 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).

% Relevant facts (248)
thf(fact_0_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_1_order__refl, axiom,
    ((![X : real]: (ord_less_eq_real @ X @ X)))). % order_refl
thf(fact_2_order__refl, axiom,
    ((![X : rat]: (ord_less_eq_rat @ X @ X)))). % order_refl
thf(fact_3_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_4_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_5_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_6_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_7_order__subst1, axiom,
    ((![A : real, F : rat > real, B : rat, C : rat]: ((ord_less_eq_real @ A @ (F @ B)) => ((ord_less_eq_rat @ B @ C) => ((![X3 : rat, Y : rat]: ((ord_less_eq_rat @ X3 @ Y) => (ord_less_eq_real @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_real @ A @ (F @ C)))))))). % order_subst1
thf(fact_8_order__subst1, axiom,
    ((![A : rat, F : real > rat, B : real, C : real]: ((ord_less_eq_rat @ A @ (F @ B)) => ((ord_less_eq_real @ B @ C) => ((![X3 : real, Y : real]: ((ord_less_eq_real @ X3 @ Y) => (ord_less_eq_rat @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_rat @ A @ (F @ C)))))))). % order_subst1
thf(fact_9_order__subst1, axiom,
    ((![A : rat, F : rat > rat, B : rat, C : rat]: ((ord_less_eq_rat @ A @ (F @ B)) => ((ord_less_eq_rat @ B @ C) => ((![X3 : rat, Y : rat]: ((ord_less_eq_rat @ X3 @ Y) => (ord_less_eq_rat @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_rat @ A @ (F @ C)))))))). % order_subst1
thf(fact_10_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_11_order__subst2, axiom,
    ((![A : real, B : real, F : real > rat, C : rat]: ((ord_less_eq_real @ A @ B) => ((ord_less_eq_rat @ (F @ B) @ C) => ((![X3 : real, Y : real]: ((ord_less_eq_real @ X3 @ Y) => (ord_less_eq_rat @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_rat @ (F @ A) @ C))))))). % order_subst2
thf(fact_12_order__subst2, axiom,
    ((![A : rat, B : rat, F : rat > real, C : real]: ((ord_less_eq_rat @ A @ B) => ((ord_less_eq_real @ (F @ B) @ C) => ((![X3 : rat, Y : rat]: ((ord_less_eq_rat @ X3 @ Y) => (ord_less_eq_real @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_real @ (F @ A) @ C))))))). % order_subst2
thf(fact_13_order__subst2, axiom,
    ((![A : rat, B : rat, F : rat > rat, C : rat]: ((ord_less_eq_rat @ A @ B) => ((ord_less_eq_rat @ (F @ B) @ C) => ((![X3 : rat, Y : rat]: ((ord_less_eq_rat @ X3 @ Y) => (ord_less_eq_rat @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_rat @ (F @ A) @ C))))))). % order_subst2
thf(fact_14_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_15_verit__la__disequality, axiom,
    ((![A : rat, B : rat]: ((A = B) | ((~ ((ord_less_eq_rat @ A @ B))) | (~ ((ord_less_eq_rat @ B @ A)))))))). % verit_la_disequality
thf(fact_16_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_17_ord__eq__le__subst, axiom,
    ((![A : rat, F : real > rat, 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_rat @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_rat @ A @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_18_ord__eq__le__subst, axiom,
    ((![A : real, F : rat > real, B : rat, C : rat]: ((A = (F @ B)) => ((ord_less_eq_rat @ B @ C) => ((![X3 : rat, Y : rat]: ((ord_less_eq_rat @ X3 @ Y) => (ord_less_eq_real @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_real @ A @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_19_ord__eq__le__subst, axiom,
    ((![A : rat, F : rat > rat, B : rat, C : rat]: ((A = (F @ B)) => ((ord_less_eq_rat @ B @ C) => ((![X3 : rat, Y : rat]: ((ord_less_eq_rat @ X3 @ Y) => (ord_less_eq_rat @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_rat @ A @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_20_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_21_ord__le__eq__subst, axiom,
    ((![A : real, B : real, F : real > rat, C : rat]: ((ord_less_eq_real @ A @ B) => (((F @ B) = C) => ((![X3 : real, Y : real]: ((ord_less_eq_real @ X3 @ Y) => (ord_less_eq_rat @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_rat @ (F @ A) @ C))))))). % ord_le_eq_subst
thf(fact_22_ord__le__eq__subst, axiom,
    ((![A : rat, B : rat, F : rat > real, C : real]: ((ord_less_eq_rat @ A @ B) => (((F @ B) = C) => ((![X3 : rat, Y : rat]: ((ord_less_eq_rat @ X3 @ Y) => (ord_less_eq_real @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_real @ (F @ A) @ C))))))). % ord_le_eq_subst
thf(fact_23_ord__le__eq__subst, axiom,
    ((![A : rat, B : rat, F : rat > rat, C : rat]: ((ord_less_eq_rat @ A @ B) => (((F @ B) = C) => ((![X3 : rat, Y : rat]: ((ord_less_eq_rat @ X3 @ Y) => (ord_less_eq_rat @ (F @ X3) @ (F @ Y)))) => (ord_less_eq_rat @ (F @ A) @ C))))))). % ord_le_eq_subst
thf(fact_24_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_25_dual__order_Oantisym, axiom,
    ((![B : rat, A : rat]: ((ord_less_eq_rat @ B @ A) => ((ord_less_eq_rat @ A @ B) => (A = B)))))). % dual_order.antisym
thf(fact_26_dual__order_Oeq__iff, axiom,
    (((^[Y2 : real]: (^[Z3 : real]: (Y2 = Z3))) = (^[A2 : real]: (^[B2 : real]: (((ord_less_eq_real @ B2 @ A2)) & ((ord_less_eq_real @ A2 @ B2)))))))). % dual_order.eq_iff
thf(fact_27_dual__order_Oeq__iff, axiom,
    (((^[Y2 : rat]: (^[Z3 : rat]: (Y2 = Z3))) = (^[A2 : rat]: (^[B2 : rat]: (((ord_less_eq_rat @ B2 @ A2)) & ((ord_less_eq_rat @ A2 @ B2)))))))). % dual_order.eq_iff
thf(fact_28_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_29_dual__order_Otrans, axiom,
    ((![B : rat, A : rat, C : rat]: ((ord_less_eq_rat @ B @ A) => ((ord_less_eq_rat @ C @ B) => (ord_less_eq_rat @ C @ A)))))). % dual_order.trans
thf(fact_30_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_31_linorder__wlog, axiom,
    ((![P2 : rat > rat > $o, A : rat, B : rat]: ((![A3 : rat, B3 : rat]: ((ord_less_eq_rat @ A3 @ B3) => (P2 @ A3 @ B3))) => ((![A3 : rat, B3 : rat]: ((P2 @ B3 @ A3) => (P2 @ A3 @ B3))) => (P2 @ A @ B)))))). % linorder_wlog
thf(fact_32_dual__order_Orefl, axiom,
    ((![A : real]: (ord_less_eq_real @ A @ A)))). % dual_order.refl
thf(fact_33_dual__order_Orefl, axiom,
    ((![A : rat]: (ord_less_eq_rat @ A @ A)))). % dual_order.refl
thf(fact_34_order__trans, axiom,
    ((![X : real, Y3 : real, Z4 : real]: ((ord_less_eq_real @ X @ Y3) => ((ord_less_eq_real @ Y3 @ Z4) => (ord_less_eq_real @ X @ Z4)))))). % order_trans
thf(fact_35_order__trans, axiom,
    ((![X : rat, Y3 : rat, Z4 : rat]: ((ord_less_eq_rat @ X @ Y3) => ((ord_less_eq_rat @ Y3 @ Z4) => (ord_less_eq_rat @ X @ Z4)))))). % order_trans
thf(fact_36_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_37_order__class_Oorder_Oantisym, axiom,
    ((![A : rat, B : rat]: ((ord_less_eq_rat @ A @ B) => ((ord_less_eq_rat @ B @ A) => (A = B)))))). % order_class.order.antisym
thf(fact_38_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_39_ord__le__eq__trans, axiom,
    ((![A : rat, B : rat, C : rat]: ((ord_less_eq_rat @ A @ B) => ((B = C) => (ord_less_eq_rat @ A @ C)))))). % ord_le_eq_trans
thf(fact_40_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_41_ord__eq__le__trans, axiom,
    ((![A : rat, B : rat, C : rat]: ((A = B) => ((ord_less_eq_rat @ B @ C) => (ord_less_eq_rat @ A @ C)))))). % ord_eq_le_trans
thf(fact_42_order__class_Oorder_Oeq__iff, axiom,
    (((^[Y2 : real]: (^[Z3 : real]: (Y2 = Z3))) = (^[A2 : real]: (^[B2 : real]: (((ord_less_eq_real @ A2 @ B2)) & ((ord_less_eq_real @ B2 @ A2)))))))). % order_class.order.eq_iff
thf(fact_43_order__class_Oorder_Oeq__iff, axiom,
    (((^[Y2 : rat]: (^[Z3 : rat]: (Y2 = Z3))) = (^[A2 : rat]: (^[B2 : rat]: (((ord_less_eq_rat @ A2 @ B2)) & ((ord_less_eq_rat @ B2 @ A2)))))))). % order_class.order.eq_iff
thf(fact_44_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_45_antisym__conv, axiom,
    ((![Y3 : rat, X : rat]: ((ord_less_eq_rat @ Y3 @ X) => ((ord_less_eq_rat @ X @ Y3) = (X = Y3)))))). % antisym_conv
thf(fact_46_le__cases3, axiom,
    ((![X : real, Y3 : real, Z4 : real]: (((ord_less_eq_real @ X @ Y3) => (~ ((ord_less_eq_real @ Y3 @ Z4)))) => (((ord_less_eq_real @ Y3 @ X) => (~ ((ord_less_eq_real @ X @ Z4)))) => (((ord_less_eq_real @ X @ Z4) => (~ ((ord_less_eq_real @ Z4 @ Y3)))) => (((ord_less_eq_real @ Z4 @ Y3) => (~ ((ord_less_eq_real @ Y3 @ X)))) => (((ord_less_eq_real @ Y3 @ Z4) => (~ ((ord_less_eq_real @ Z4 @ X)))) => (~ (((ord_less_eq_real @ Z4 @ X) => (~ ((ord_less_eq_real @ X @ Y3)))))))))))))). % le_cases3
thf(fact_47_le__cases3, axiom,
    ((![X : rat, Y3 : rat, Z4 : rat]: (((ord_less_eq_rat @ X @ Y3) => (~ ((ord_less_eq_rat @ Y3 @ Z4)))) => (((ord_less_eq_rat @ Y3 @ X) => (~ ((ord_less_eq_rat @ X @ Z4)))) => (((ord_less_eq_rat @ X @ Z4) => (~ ((ord_less_eq_rat @ Z4 @ Y3)))) => (((ord_less_eq_rat @ Z4 @ Y3) => (~ ((ord_less_eq_rat @ Y3 @ X)))) => (((ord_less_eq_rat @ Y3 @ Z4) => (~ ((ord_less_eq_rat @ Z4 @ X)))) => (~ (((ord_less_eq_rat @ Z4 @ X) => (~ ((ord_less_eq_rat @ X @ Y3)))))))))))))). % le_cases3
thf(fact_48_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_49_order_Otrans, axiom,
    ((![A : rat, B : rat, C : rat]: ((ord_less_eq_rat @ A @ B) => ((ord_less_eq_rat @ B @ C) => (ord_less_eq_rat @ A @ C)))))). % order.trans
thf(fact_50_le__cases, axiom,
    ((![X : real, Y3 : real]: ((~ ((ord_less_eq_real @ X @ Y3))) => (ord_less_eq_real @ Y3 @ X))))). % le_cases
thf(fact_51_le__cases, axiom,
    ((![X : rat, Y3 : rat]: ((~ ((ord_less_eq_rat @ X @ Y3))) => (ord_less_eq_rat @ Y3 @ X))))). % le_cases
thf(fact_52_eq__refl, axiom,
    ((![X : real, Y3 : real]: ((X = Y3) => (ord_less_eq_real @ X @ Y3))))). % eq_refl
thf(fact_53_eq__refl, axiom,
    ((![X : rat, Y3 : rat]: ((X = Y3) => (ord_less_eq_rat @ X @ Y3))))). % eq_refl
thf(fact_54_linear, axiom,
    ((![X : real, Y3 : real]: ((ord_less_eq_real @ X @ Y3) | (ord_less_eq_real @ Y3 @ X))))). % linear
thf(fact_55_linear, axiom,
    ((![X : rat, Y3 : rat]: ((ord_less_eq_rat @ X @ Y3) | (ord_less_eq_rat @ Y3 @ X))))). % linear
thf(fact_56_antisym, axiom,
    ((![X : real, Y3 : real]: ((ord_less_eq_real @ X @ Y3) => ((ord_less_eq_real @ Y3 @ X) => (X = Y3)))))). % antisym
thf(fact_57_antisym, axiom,
    ((![X : rat, Y3 : rat]: ((ord_less_eq_rat @ X @ Y3) => ((ord_less_eq_rat @ Y3 @ X) => (X = Y3)))))). % antisym
thf(fact_58_eq__iff, axiom,
    (((^[Y2 : real]: (^[Z3 : real]: (Y2 = Z3))) = (^[X4 : real]: (^[Y4 : real]: (((ord_less_eq_real @ X4 @ Y4)) & ((ord_less_eq_real @ Y4 @ X4)))))))). % eq_iff
thf(fact_59_eq__iff, axiom,
    (((^[Y2 : rat]: (^[Z3 : rat]: (Y2 = Z3))) = (^[X4 : rat]: (^[Y4 : rat]: (((ord_less_eq_rat @ X4 @ Y4)) & ((ord_less_eq_rat @ Y4 @ X4)))))))). % eq_iff
thf(fact_60_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_61_Greatest__equality, axiom,
    ((![P2 : rat > $o, X : rat]: ((P2 @ X) => ((![Y : rat]: ((P2 @ Y) => (ord_less_eq_rat @ Y @ X))) => ((order_Greatest_rat @ P2) = X)))))). % Greatest_equality
thf(fact_62_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_63_GreatestI2__order, axiom,
    ((![P2 : rat > $o, X : rat, Q2 : rat > $o]: ((P2 @ X) => ((![Y : rat]: ((P2 @ Y) => (ord_less_eq_rat @ Y @ X))) => ((![X3 : rat]: ((P2 @ X3) => ((![Y5 : rat]: ((P2 @ Y5) => (ord_less_eq_rat @ Y5 @ X3))) => (Q2 @ X3)))) => (Q2 @ (order_Greatest_rat @ P2)))))))). % GreatestI2_order
thf(fact_64_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_65_le__rel__bool__arg__iff, axiom,
    ((ord_less_eq_o_rat = (^[X5 : $o > rat]: (^[Y6 : $o > rat]: (((ord_less_eq_rat @ (X5 @ $false) @ (Y6 @ $false))) & ((ord_less_eq_rat @ (X5 @ $true) @ (Y6 @ $true))))))))). % le_rel_bool_arg_iff
thf(fact_66_antimono__def, axiom,
    ((order_537808140l_real = (^[F2 : real > real]: (![X4 : real]: (![Y4 : real]: (((ord_less_eq_real @ X4 @ Y4)) => ((ord_less_eq_real @ (F2 @ Y4) @ (F2 @ X4)))))))))). % antimono_def
thf(fact_67_antimono__def, axiom,
    ((order_765710392al_rat = (^[F2 : real > rat]: (![X4 : real]: (![Y4 : real]: (((ord_less_eq_real @ X4 @ Y4)) => ((ord_less_eq_rat @ (F2 @ Y4) @ (F2 @ X4)))))))))). % antimono_def
thf(fact_68_antimono__def, axiom,
    ((order_1851487160t_real = (^[F2 : rat > real]: (![X4 : rat]: (![Y4 : rat]: (((ord_less_eq_rat @ X4 @ Y4)) => ((ord_less_eq_real @ (F2 @ Y4) @ (F2 @ X4)))))))))). % antimono_def
thf(fact_69_antimono__def, axiom,
    ((order_802983140at_rat = (^[F2 : rat > rat]: (![X4 : rat]: (![Y4 : rat]: (((ord_less_eq_rat @ X4 @ Y4)) => ((ord_less_eq_rat @ (F2 @ Y4) @ (F2 @ X4)))))))))). % antimono_def
thf(fact_70_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_71_antimonoI, axiom,
    ((![F : real > rat]: ((![X3 : real, Y : real]: ((ord_less_eq_real @ X3 @ Y) => (ord_less_eq_rat @ (F @ Y) @ (F @ X3)))) => (order_765710392al_rat @ F))))). % antimonoI
thf(fact_72_antimonoI, axiom,
    ((![F : rat > real]: ((![X3 : rat, Y : rat]: ((ord_less_eq_rat @ X3 @ Y) => (ord_less_eq_real @ (F @ Y) @ (F @ X3)))) => (order_1851487160t_real @ F))))). % antimonoI
thf(fact_73_antimonoI, axiom,
    ((![F : rat > rat]: ((![X3 : rat, Y : rat]: ((ord_less_eq_rat @ X3 @ Y) => (ord_less_eq_rat @ (F @ Y) @ (F @ X3)))) => (order_802983140at_rat @ F))))). % antimonoI
thf(fact_74_antimonoE, axiom,
    ((![F : real > real, X : real, Y3 : real]: ((order_537808140l_real @ F) => ((ord_less_eq_real @ X @ Y3) => (ord_less_eq_real @ (F @ Y3) @ (F @ X))))))). % antimonoE
thf(fact_75_antimonoE, axiom,
    ((![F : real > rat, X : real, Y3 : real]: ((order_765710392al_rat @ F) => ((ord_less_eq_real @ X @ Y3) => (ord_less_eq_rat @ (F @ Y3) @ (F @ X))))))). % antimonoE
thf(fact_76_antimonoE, axiom,
    ((![F : rat > real, X : rat, Y3 : rat]: ((order_1851487160t_real @ F) => ((ord_less_eq_rat @ X @ Y3) => (ord_less_eq_real @ (F @ Y3) @ (F @ X))))))). % antimonoE
thf(fact_77_antimonoE, axiom,
    ((![F : rat > rat, X : rat, Y3 : rat]: ((order_802983140at_rat @ F) => ((ord_less_eq_rat @ X @ Y3) => (ord_less_eq_rat @ (F @ Y3) @ (F @ X))))))). % antimonoE
thf(fact_78_antimonoD, axiom,
    ((![F : real > real, X : real, Y3 : real]: ((order_537808140l_real @ F) => ((ord_less_eq_real @ X @ Y3) => (ord_less_eq_real @ (F @ Y3) @ (F @ X))))))). % antimonoD
thf(fact_79_antimonoD, axiom,
    ((![F : real > rat, X : real, Y3 : real]: ((order_765710392al_rat @ F) => ((ord_less_eq_real @ X @ Y3) => (ord_less_eq_rat @ (F @ Y3) @ (F @ X))))))). % antimonoD
thf(fact_80_antimonoD, axiom,
    ((![F : rat > real, X : rat, Y3 : rat]: ((order_1851487160t_real @ F) => ((ord_less_eq_rat @ X @ Y3) => (ord_less_eq_real @ (F @ Y3) @ (F @ X))))))). % antimonoD
thf(fact_81_antimonoD, axiom,
    ((![F : rat > rat, X : rat, Y3 : rat]: ((order_802983140at_rat @ F) => ((ord_less_eq_rat @ X @ Y3) => (ord_less_eq_rat @ (F @ Y3) @ (F @ X))))))). % antimonoD
thf(fact_82_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_83_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_84_real__less__eq__code, axiom,
    ((![X : rat, Y3 : rat]: ((ord_less_eq_real @ (ratreal @ X) @ (ratreal @ Y3)) = (ord_less_eq_rat @ X @ Y3))))). % real_less_eq_code
thf(fact_85_poly__infinity, axiom,
    ((![P : poly_real, D : real, A : real]: ((~ ((P = zero_zero_poly_real))) => (?[R2 : real]: (![Z2 : real]: ((ord_less_eq_real @ R2 @ (real_V646646907m_real @ Z2)) => (ord_less_eq_real @ D @ (real_V646646907m_real @ (poly_real2 @ (pCons_real @ A @ P) @ Z2)))))))))). % poly_infinity
thf(fact_86_poly__infinity, axiom,
    ((![P : poly_complex, D : real, A : complex]: ((~ ((P = zero_z1746442943omplex))) => (?[R2 : real]: (![Z2 : complex]: ((ord_less_eq_real @ R2 @ (real_V638595069omplex @ Z2)) => (ord_less_eq_real @ D @ (real_V638595069omplex @ (poly_complex2 @ (pCons_complex @ A @ P) @ Z2)))))))))). % poly_infinity
thf(fact_87_pCons__0__0, axiom,
    (((pCons_real @ zero_zero_real @ zero_zero_poly_real) = zero_zero_poly_real))). % pCons_0_0
thf(fact_88_pCons__0__0, axiom,
    (((pCons_rat @ zero_zero_rat @ zero_zero_poly_rat) = zero_zero_poly_rat))). % pCons_0_0
thf(fact_89_pCons__eq__0__iff, axiom,
    ((![A : real, P : poly_real]: (((pCons_real @ A @ P) = zero_zero_poly_real) = (((A = zero_zero_real)) & ((P = zero_zero_poly_real))))))). % pCons_eq_0_iff
thf(fact_90_pCons__eq__0__iff, axiom,
    ((![A : rat, P : poly_rat]: (((pCons_rat @ A @ P) = zero_zero_poly_rat) = (((A = zero_zero_rat)) & ((P = zero_zero_poly_rat))))))). % pCons_eq_0_iff
thf(fact_91_poly__0, axiom,
    ((![X : complex]: ((poly_complex2 @ zero_z1746442943omplex @ X) = zero_zero_complex)))). % poly_0
thf(fact_92_poly__0, axiom,
    ((![X : real]: ((poly_real2 @ zero_zero_poly_real @ X) = zero_zero_real)))). % poly_0
thf(fact_93_poly__0, axiom,
    ((![X : rat]: ((poly_rat2 @ zero_zero_poly_rat @ X) = zero_zero_rat)))). % poly_0
thf(fact_94_mem__Collect__eq, axiom,
    ((![A : real, P2 : real > $o]: ((member_real @ A @ (collect_real @ P2)) = (P2 @ A))))). % mem_Collect_eq
thf(fact_95_Collect__mem__eq, axiom,
    ((![A4 : set_real]: ((collect_real @ (^[X4 : real]: (member_real @ X4 @ A4))) = A4)))). % Collect_mem_eq
thf(fact_96_verit__comp__simplify1_I1_J, axiom,
    ((![A : real]: (~ ((ord_less_real @ A @ A)))))). % verit_comp_simplify1(1)
thf(fact_97_verit__comp__simplify1_I1_J, axiom,
    ((![A : rat]: (~ ((ord_less_rat @ A @ A)))))). % verit_comp_simplify1(1)
thf(fact_98_pCons__induct, axiom,
    ((![P2 : poly_real > $o, P : poly_real]: ((P2 @ zero_zero_poly_real) => ((![A3 : real, P3 : poly_real]: (((~ ((A3 = zero_zero_real))) | (~ ((P3 = zero_zero_poly_real)))) => ((P2 @ P3) => (P2 @ (pCons_real @ A3 @ P3))))) => (P2 @ P)))))). % pCons_induct
thf(fact_99_pCons__induct, axiom,
    ((![P2 : poly_rat > $o, P : poly_rat]: ((P2 @ zero_zero_poly_rat) => ((![A3 : rat, P3 : poly_rat]: (((~ ((A3 = zero_zero_rat))) | (~ ((P3 = zero_zero_poly_rat)))) => ((P2 @ P3) => (P2 @ (pCons_rat @ A3 @ P3))))) => (P2 @ P)))))). % pCons_induct
thf(fact_100_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_101_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_102_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_103_field__lbound__gt__zero, axiom,
    ((![D1 : rat, D2 : rat]: ((ord_less_rat @ zero_zero_rat @ D1) => ((ord_less_rat @ zero_zero_rat @ D2) => (?[E : rat]: ((ord_less_rat @ zero_zero_rat @ E) & ((ord_less_rat @ E @ D1) & (ord_less_rat @ E @ D2))))))))). % field_lbound_gt_zero
thf(fact_104_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_105_ord__eq__less__subst, axiom,
    ((![A : rat, F : real > rat, B : real, C : real]: ((A = (F @ B)) => ((ord_less_real @ B @ C) => ((![X3 : real, Y : real]: ((ord_less_real @ X3 @ Y) => (ord_less_rat @ (F @ X3) @ (F @ Y)))) => (ord_less_rat @ A @ (F @ C)))))))). % ord_eq_less_subst
thf(fact_106_ord__eq__less__subst, axiom,
    ((![A : real, F : rat > real, B : rat, C : rat]: ((A = (F @ B)) => ((ord_less_rat @ B @ C) => ((![X3 : rat, Y : rat]: ((ord_less_rat @ X3 @ Y) => (ord_less_real @ (F @ X3) @ (F @ Y)))) => (ord_less_real @ A @ (F @ C)))))))). % ord_eq_less_subst
thf(fact_107_ord__eq__less__subst, axiom,
    ((![A : rat, F : rat > rat, B : rat, C : rat]: ((A = (F @ B)) => ((ord_less_rat @ B @ C) => ((![X3 : rat, Y : rat]: ((ord_less_rat @ X3 @ Y) => (ord_less_rat @ (F @ X3) @ (F @ Y)))) => (ord_less_rat @ A @ (F @ C)))))))). % ord_eq_less_subst
thf(fact_108_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_109_ord__less__eq__subst, axiom,
    ((![A : real, B : real, F : real > rat, C : rat]: ((ord_less_real @ A @ B) => (((F @ B) = C) => ((![X3 : real, Y : real]: ((ord_less_real @ X3 @ Y) => (ord_less_rat @ (F @ X3) @ (F @ Y)))) => (ord_less_rat @ (F @ A) @ C))))))). % ord_less_eq_subst
thf(fact_110_ord__less__eq__subst, axiom,
    ((![A : rat, B : rat, F : rat > real, C : real]: ((ord_less_rat @ A @ B) => (((F @ B) = C) => ((![X3 : rat, Y : rat]: ((ord_less_rat @ X3 @ Y) => (ord_less_real @ (F @ X3) @ (F @ Y)))) => (ord_less_real @ (F @ A) @ C))))))). % ord_less_eq_subst
thf(fact_111_ord__less__eq__subst, axiom,
    ((![A : rat, B : rat, F : rat > rat, C : rat]: ((ord_less_rat @ A @ B) => (((F @ B) = C) => ((![X3 : rat, Y : rat]: ((ord_less_rat @ X3 @ Y) => (ord_less_rat @ (F @ X3) @ (F @ Y)))) => (ord_less_rat @ (F @ A) @ C))))))). % ord_less_eq_subst
thf(fact_112_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_113_order__less__subst1, axiom,
    ((![A : real, F : rat > real, B : rat, C : rat]: ((ord_less_real @ A @ (F @ B)) => ((ord_less_rat @ B @ C) => ((![X3 : rat, Y : rat]: ((ord_less_rat @ X3 @ Y) => (ord_less_real @ (F @ X3) @ (F @ Y)))) => (ord_less_real @ A @ (F @ C)))))))). % order_less_subst1
thf(fact_114_order__less__subst1, axiom,
    ((![A : rat, F : real > rat, B : real, C : real]: ((ord_less_rat @ A @ (F @ B)) => ((ord_less_real @ B @ C) => ((![X3 : real, Y : real]: ((ord_less_real @ X3 @ Y) => (ord_less_rat @ (F @ X3) @ (F @ Y)))) => (ord_less_rat @ A @ (F @ C)))))))). % order_less_subst1
thf(fact_115_order__less__subst1, axiom,
    ((![A : rat, F : rat > rat, B : rat, C : rat]: ((ord_less_rat @ A @ (F @ B)) => ((ord_less_rat @ B @ C) => ((![X3 : rat, Y : rat]: ((ord_less_rat @ X3 @ Y) => (ord_less_rat @ (F @ X3) @ (F @ Y)))) => (ord_less_rat @ A @ (F @ C)))))))). % order_less_subst1
thf(fact_116_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_117_order__less__subst2, axiom,
    ((![A : real, B : real, F : real > rat, C : rat]: ((ord_less_real @ A @ B) => ((ord_less_rat @ (F @ B) @ C) => ((![X3 : real, Y : real]: ((ord_less_real @ X3 @ Y) => (ord_less_rat @ (F @ X3) @ (F @ Y)))) => (ord_less_rat @ (F @ A) @ C))))))). % order_less_subst2
thf(fact_118_order__less__subst2, axiom,
    ((![A : rat, B : rat, F : rat > real, C : real]: ((ord_less_rat @ A @ B) => ((ord_less_real @ (F @ B) @ C) => ((![X3 : rat, Y : rat]: ((ord_less_rat @ X3 @ Y) => (ord_less_real @ (F @ X3) @ (F @ Y)))) => (ord_less_real @ (F @ A) @ C))))))). % order_less_subst2
thf(fact_119_order__less__subst2, axiom,
    ((![A : rat, B : rat, F : rat > rat, C : rat]: ((ord_less_rat @ A @ B) => ((ord_less_rat @ (F @ B) @ C) => ((![X3 : rat, Y : rat]: ((ord_less_rat @ X3 @ Y) => (ord_less_rat @ (F @ X3) @ (F @ Y)))) => (ord_less_rat @ (F @ A) @ C))))))). % order_less_subst2
thf(fact_120_lt__ex, axiom,
    ((![X : real]: (?[Y : real]: (ord_less_real @ Y @ X))))). % lt_ex
thf(fact_121_lt__ex, axiom,
    ((![X : rat]: (?[Y : rat]: (ord_less_rat @ Y @ X))))). % lt_ex
thf(fact_122_gt__ex, axiom,
    ((![X : real]: (?[X_1 : real]: (ord_less_real @ X @ X_1))))). % gt_ex
thf(fact_123_gt__ex, axiom,
    ((![X : rat]: (?[X_1 : rat]: (ord_less_rat @ X @ X_1))))). % gt_ex
thf(fact_124_neqE, axiom,
    ((![X : real, Y3 : real]: ((~ ((X = Y3))) => ((~ ((ord_less_real @ X @ Y3))) => (ord_less_real @ Y3 @ X)))))). % neqE
thf(fact_125_neqE, axiom,
    ((![X : rat, Y3 : rat]: ((~ ((X = Y3))) => ((~ ((ord_less_rat @ X @ Y3))) => (ord_less_rat @ Y3 @ X)))))). % neqE
thf(fact_126_neq__iff, axiom,
    ((![X : real, Y3 : real]: ((~ ((X = Y3))) = (((ord_less_real @ X @ Y3)) | ((ord_less_real @ Y3 @ X))))))). % neq_iff
thf(fact_127_neq__iff, axiom,
    ((![X : rat, Y3 : rat]: ((~ ((X = Y3))) = (((ord_less_rat @ X @ Y3)) | ((ord_less_rat @ Y3 @ X))))))). % neq_iff
thf(fact_128_order_Oasym, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ B) => (~ ((ord_less_real @ B @ A))))))). % order.asym
thf(fact_129_order_Oasym, axiom,
    ((![A : rat, B : rat]: ((ord_less_rat @ A @ B) => (~ ((ord_less_rat @ B @ A))))))). % order.asym
thf(fact_130_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_131_dense, axiom,
    ((![X : rat, Y3 : rat]: ((ord_less_rat @ X @ Y3) => (?[Z : rat]: ((ord_less_rat @ X @ Z) & (ord_less_rat @ Z @ Y3))))))). % dense
thf(fact_132_less__imp__neq, axiom,
    ((![X : real, Y3 : real]: ((ord_less_real @ X @ Y3) => (~ ((X = Y3))))))). % less_imp_neq
thf(fact_133_less__imp__neq, axiom,
    ((![X : rat, Y3 : rat]: ((ord_less_rat @ X @ Y3) => (~ ((X = Y3))))))). % less_imp_neq
thf(fact_134_less__asym, axiom,
    ((![X : real, Y3 : real]: ((ord_less_real @ X @ Y3) => (~ ((ord_less_real @ Y3 @ X))))))). % less_asym
thf(fact_135_less__asym, axiom,
    ((![X : rat, Y3 : rat]: ((ord_less_rat @ X @ Y3) => (~ ((ord_less_rat @ Y3 @ X))))))). % less_asym
thf(fact_136_less__asym_H, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ B) => (~ ((ord_less_real @ B @ A))))))). % less_asym'
thf(fact_137_less__asym_H, axiom,
    ((![A : rat, B : rat]: ((ord_less_rat @ A @ B) => (~ ((ord_less_rat @ B @ A))))))). % less_asym'
thf(fact_138_less__trans, axiom,
    ((![X : real, Y3 : real, Z4 : real]: ((ord_less_real @ X @ Y3) => ((ord_less_real @ Y3 @ Z4) => (ord_less_real @ X @ Z4)))))). % less_trans
thf(fact_139_less__trans, axiom,
    ((![X : rat, Y3 : rat, Z4 : rat]: ((ord_less_rat @ X @ Y3) => ((ord_less_rat @ Y3 @ Z4) => (ord_less_rat @ X @ Z4)))))). % less_trans
thf(fact_140_less__linear, axiom,
    ((![X : real, Y3 : real]: ((ord_less_real @ X @ Y3) | ((X = Y3) | (ord_less_real @ Y3 @ X)))))). % less_linear
thf(fact_141_less__linear, axiom,
    ((![X : rat, Y3 : rat]: ((ord_less_rat @ X @ Y3) | ((X = Y3) | (ord_less_rat @ Y3 @ X)))))). % less_linear
thf(fact_142_less__irrefl, axiom,
    ((![X : real]: (~ ((ord_less_real @ X @ X)))))). % less_irrefl
thf(fact_143_less__irrefl, axiom,
    ((![X : rat]: (~ ((ord_less_rat @ X @ X)))))). % less_irrefl
thf(fact_144_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_145_ord__eq__less__trans, axiom,
    ((![A : rat, B : rat, C : rat]: ((A = B) => ((ord_less_rat @ B @ C) => (ord_less_rat @ A @ C)))))). % ord_eq_less_trans
thf(fact_146_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_147_ord__less__eq__trans, axiom,
    ((![A : rat, B : rat, C : rat]: ((ord_less_rat @ A @ B) => ((B = C) => (ord_less_rat @ A @ C)))))). % ord_less_eq_trans
thf(fact_148_dual__order_Oasym, axiom,
    ((![B : real, A : real]: ((ord_less_real @ B @ A) => (~ ((ord_less_real @ A @ B))))))). % dual_order.asym
thf(fact_149_dual__order_Oasym, axiom,
    ((![B : rat, A : rat]: ((ord_less_rat @ B @ A) => (~ ((ord_less_rat @ A @ B))))))). % dual_order.asym
thf(fact_150_less__imp__not__eq, axiom,
    ((![X : real, Y3 : real]: ((ord_less_real @ X @ Y3) => (~ ((X = Y3))))))). % less_imp_not_eq
thf(fact_151_less__imp__not__eq, axiom,
    ((![X : rat, Y3 : rat]: ((ord_less_rat @ X @ Y3) => (~ ((X = Y3))))))). % less_imp_not_eq
thf(fact_152_less__not__sym, axiom,
    ((![X : real, Y3 : real]: ((ord_less_real @ X @ Y3) => (~ ((ord_less_real @ Y3 @ X))))))). % less_not_sym
thf(fact_153_less__not__sym, axiom,
    ((![X : rat, Y3 : rat]: ((ord_less_rat @ X @ Y3) => (~ ((ord_less_rat @ Y3 @ X))))))). % less_not_sym
thf(fact_154_antisym__conv3, axiom,
    ((![Y3 : real, X : real]: ((~ ((ord_less_real @ Y3 @ X))) => ((~ ((ord_less_real @ X @ Y3))) = (X = Y3)))))). % antisym_conv3
thf(fact_155_antisym__conv3, axiom,
    ((![Y3 : rat, X : rat]: ((~ ((ord_less_rat @ Y3 @ X))) => ((~ ((ord_less_rat @ X @ Y3))) = (X = Y3)))))). % antisym_conv3
thf(fact_156_less__imp__not__eq2, axiom,
    ((![X : real, Y3 : real]: ((ord_less_real @ X @ Y3) => (~ ((Y3 = X))))))). % less_imp_not_eq2
thf(fact_157_less__imp__not__eq2, axiom,
    ((![X : rat, Y3 : rat]: ((ord_less_rat @ X @ Y3) => (~ ((Y3 = X))))))). % less_imp_not_eq2
thf(fact_158_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_159_less__imp__triv, axiom,
    ((![X : rat, Y3 : rat, P2 : $o]: ((ord_less_rat @ X @ Y3) => ((ord_less_rat @ Y3 @ X) => P2))))). % less_imp_triv
thf(fact_160_linorder__cases, axiom,
    ((![X : real, Y3 : real]: ((~ ((ord_less_real @ X @ Y3))) => ((~ ((X = Y3))) => (ord_less_real @ Y3 @ X)))))). % linorder_cases
thf(fact_161_linorder__cases, axiom,
    ((![X : rat, Y3 : rat]: ((~ ((ord_less_rat @ X @ Y3))) => ((~ ((X = Y3))) => (ord_less_rat @ Y3 @ X)))))). % linorder_cases
thf(fact_162_dual__order_Oirrefl, axiom,
    ((![A : real]: (~ ((ord_less_real @ A @ A)))))). % dual_order.irrefl
thf(fact_163_dual__order_Oirrefl, axiom,
    ((![A : rat]: (~ ((ord_less_rat @ A @ A)))))). % dual_order.irrefl
thf(fact_164_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_165_order_Ostrict__trans, axiom,
    ((![A : rat, B : rat, C : rat]: ((ord_less_rat @ A @ B) => ((ord_less_rat @ B @ C) => (ord_less_rat @ A @ C)))))). % order.strict_trans
thf(fact_166_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_167_less__imp__not__less, axiom,
    ((![X : rat, Y3 : rat]: ((ord_less_rat @ X @ Y3) => (~ ((ord_less_rat @ Y3 @ X))))))). % less_imp_not_less
thf(fact_168_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_169_linorder__less__wlog, axiom,
    ((![P2 : rat > rat > $o, A : rat, B : rat]: ((![A3 : rat, B3 : rat]: ((ord_less_rat @ A3 @ B3) => (P2 @ A3 @ B3))) => ((![A3 : rat]: (P2 @ A3 @ A3)) => ((![A3 : rat, B3 : rat]: ((P2 @ B3 @ A3) => (P2 @ A3 @ B3))) => (P2 @ A @ B))))))). % linorder_less_wlog
thf(fact_170_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_171_dual__order_Ostrict__trans, axiom,
    ((![B : rat, A : rat, C : rat]: ((ord_less_rat @ B @ A) => ((ord_less_rat @ C @ B) => (ord_less_rat @ C @ A)))))). % dual_order.strict_trans
thf(fact_172_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_173_not__less__iff__gr__or__eq, axiom,
    ((![X : rat, Y3 : rat]: ((~ ((ord_less_rat @ X @ Y3))) = (((ord_less_rat @ Y3 @ X)) | ((X = Y3))))))). % not_less_iff_gr_or_eq
thf(fact_174_order_Ostrict__implies__not__eq, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ B) => (~ ((A = B))))))). % order.strict_implies_not_eq
thf(fact_175_order_Ostrict__implies__not__eq, axiom,
    ((![A : rat, B : rat]: ((ord_less_rat @ A @ B) => (~ ((A = B))))))). % order.strict_implies_not_eq
thf(fact_176_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_177_dual__order_Ostrict__implies__not__eq, axiom,
    ((![B : rat, A : rat]: ((ord_less_rat @ B @ A) => (~ ((A = B))))))). % dual_order.strict_implies_not_eq
thf(fact_178_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_179_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_180_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_181_poly__all__0__iff__0, axiom,
    ((![P : poly_rat]: ((![X4 : rat]: ((poly_rat2 @ P @ X4) = zero_zero_rat)) = (P = zero_zero_poly_rat))))). % poly_all_0_iff_0
thf(fact_182_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_183_verit__comp__simplify1_I3_J, axiom,
    ((![B4 : rat, A5 : rat]: ((~ ((ord_less_eq_rat @ B4 @ A5))) = (ord_less_rat @ A5 @ B4))))). % verit_comp_simplify1(3)
thf(fact_184_leD, axiom,
    ((![Y3 : real, X : real]: ((ord_less_eq_real @ Y3 @ X) => (~ ((ord_less_real @ X @ Y3))))))). % leD
thf(fact_185_leD, axiom,
    ((![Y3 : rat, X : rat]: ((ord_less_eq_rat @ Y3 @ X) => (~ ((ord_less_rat @ X @ Y3))))))). % leD
thf(fact_186_leI, axiom,
    ((![X : real, Y3 : real]: ((~ ((ord_less_real @ X @ Y3))) => (ord_less_eq_real @ Y3 @ X))))). % leI
thf(fact_187_leI, axiom,
    ((![X : rat, Y3 : rat]: ((~ ((ord_less_rat @ X @ Y3))) => (ord_less_eq_rat @ Y3 @ X))))). % leI
thf(fact_188_le__less, axiom,
    ((ord_less_eq_real = (^[X4 : real]: (^[Y4 : real]: (((ord_less_real @ X4 @ Y4)) | ((X4 = Y4)))))))). % le_less
thf(fact_189_le__less, axiom,
    ((ord_less_eq_rat = (^[X4 : rat]: (^[Y4 : rat]: (((ord_less_rat @ X4 @ Y4)) | ((X4 = Y4)))))))). % le_less
thf(fact_190_less__le, axiom,
    ((ord_less_real = (^[X4 : real]: (^[Y4 : real]: (((ord_less_eq_real @ X4 @ Y4)) & ((~ ((X4 = Y4)))))))))). % less_le
thf(fact_191_less__le, axiom,
    ((ord_less_rat = (^[X4 : rat]: (^[Y4 : rat]: (((ord_less_eq_rat @ X4 @ Y4)) & ((~ ((X4 = Y4)))))))))). % less_le
thf(fact_192_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_193_order__le__less__subst1, axiom,
    ((![A : real, F : rat > real, B : rat, C : rat]: ((ord_less_eq_real @ A @ (F @ B)) => ((ord_less_rat @ B @ C) => ((![X3 : rat, Y : rat]: ((ord_less_rat @ X3 @ Y) => (ord_less_real @ (F @ X3) @ (F @ Y)))) => (ord_less_real @ A @ (F @ C)))))))). % order_le_less_subst1
thf(fact_194_order__le__less__subst1, axiom,
    ((![A : rat, F : real > rat, B : real, C : real]: ((ord_less_eq_rat @ A @ (F @ B)) => ((ord_less_real @ B @ C) => ((![X3 : real, Y : real]: ((ord_less_real @ X3 @ Y) => (ord_less_rat @ (F @ X3) @ (F @ Y)))) => (ord_less_rat @ A @ (F @ C)))))))). % order_le_less_subst1
thf(fact_195_order__le__less__subst1, axiom,
    ((![A : rat, F : rat > rat, B : rat, C : rat]: ((ord_less_eq_rat @ A @ (F @ B)) => ((ord_less_rat @ B @ C) => ((![X3 : rat, Y : rat]: ((ord_less_rat @ X3 @ Y) => (ord_less_rat @ (F @ X3) @ (F @ Y)))) => (ord_less_rat @ A @ (F @ C)))))))). % order_le_less_subst1
thf(fact_196_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_197_order__le__less__subst2, axiom,
    ((![A : real, B : real, F : real > rat, C : rat]: ((ord_less_eq_real @ A @ B) => ((ord_less_rat @ (F @ B) @ C) => ((![X3 : real, Y : real]: ((ord_less_eq_real @ X3 @ Y) => (ord_less_eq_rat @ (F @ X3) @ (F @ Y)))) => (ord_less_rat @ (F @ A) @ C))))))). % order_le_less_subst2
thf(fact_198_order__le__less__subst2, axiom,
    ((![A : rat, B : rat, F : rat > real, C : real]: ((ord_less_eq_rat @ A @ B) => ((ord_less_real @ (F @ B) @ C) => ((![X3 : rat, Y : rat]: ((ord_less_eq_rat @ X3 @ Y) => (ord_less_eq_real @ (F @ X3) @ (F @ Y)))) => (ord_less_real @ (F @ A) @ C))))))). % order_le_less_subst2
thf(fact_199_order__le__less__subst2, axiom,
    ((![A : rat, B : rat, F : rat > rat, C : rat]: ((ord_less_eq_rat @ A @ B) => ((ord_less_rat @ (F @ B) @ C) => ((![X3 : rat, Y : rat]: ((ord_less_eq_rat @ X3 @ Y) => (ord_less_eq_rat @ (F @ X3) @ (F @ Y)))) => (ord_less_rat @ (F @ A) @ C))))))). % order_le_less_subst2
thf(fact_200_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_201_order__less__le__subst1, axiom,
    ((![A : rat, F : real > rat, B : real, C : real]: ((ord_less_rat @ A @ (F @ B)) => ((ord_less_eq_real @ B @ C) => ((![X3 : real, Y : real]: ((ord_less_eq_real @ X3 @ Y) => (ord_less_eq_rat @ (F @ X3) @ (F @ Y)))) => (ord_less_rat @ A @ (F @ C)))))))). % order_less_le_subst1
thf(fact_202_order__less__le__subst1, axiom,
    ((![A : real, F : rat > real, B : rat, C : rat]: ((ord_less_real @ A @ (F @ B)) => ((ord_less_eq_rat @ B @ C) => ((![X3 : rat, Y : rat]: ((ord_less_eq_rat @ X3 @ Y) => (ord_less_eq_real @ (F @ X3) @ (F @ Y)))) => (ord_less_real @ A @ (F @ C)))))))). % order_less_le_subst1
thf(fact_203_order__less__le__subst1, axiom,
    ((![A : rat, F : rat > rat, B : rat, C : rat]: ((ord_less_rat @ A @ (F @ B)) => ((ord_less_eq_rat @ B @ C) => ((![X3 : rat, Y : rat]: ((ord_less_eq_rat @ X3 @ Y) => (ord_less_eq_rat @ (F @ X3) @ (F @ Y)))) => (ord_less_rat @ A @ (F @ C)))))))). % order_less_le_subst1
thf(fact_204_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_205_order__less__le__subst2, axiom,
    ((![A : rat, B : rat, F : rat > real, C : real]: ((ord_less_rat @ A @ B) => ((ord_less_eq_real @ (F @ B) @ C) => ((![X3 : rat, Y : rat]: ((ord_less_rat @ X3 @ Y) => (ord_less_real @ (F @ X3) @ (F @ Y)))) => (ord_less_real @ (F @ A) @ C))))))). % order_less_le_subst2
thf(fact_206_order__less__le__subst2, axiom,
    ((![A : real, B : real, F : real > rat, C : rat]: ((ord_less_real @ A @ B) => ((ord_less_eq_rat @ (F @ B) @ C) => ((![X3 : real, Y : real]: ((ord_less_real @ X3 @ Y) => (ord_less_rat @ (F @ X3) @ (F @ Y)))) => (ord_less_rat @ (F @ A) @ C))))))). % order_less_le_subst2
thf(fact_207_order__less__le__subst2, axiom,
    ((![A : rat, B : rat, F : rat > rat, C : rat]: ((ord_less_rat @ A @ B) => ((ord_less_eq_rat @ (F @ B) @ C) => ((![X3 : rat, Y : rat]: ((ord_less_rat @ X3 @ Y) => (ord_less_rat @ (F @ X3) @ (F @ Y)))) => (ord_less_rat @ (F @ A) @ C))))))). % order_less_le_subst2
thf(fact_208_not__le, axiom,
    ((![X : real, Y3 : real]: ((~ ((ord_less_eq_real @ X @ Y3))) = (ord_less_real @ Y3 @ X))))). % not_le
thf(fact_209_not__le, axiom,
    ((![X : rat, Y3 : rat]: ((~ ((ord_less_eq_rat @ X @ Y3))) = (ord_less_rat @ Y3 @ X))))). % not_le
thf(fact_210_not__less, axiom,
    ((![X : real, Y3 : real]: ((~ ((ord_less_real @ X @ Y3))) = (ord_less_eq_real @ Y3 @ X))))). % not_less
thf(fact_211_not__less, axiom,
    ((![X : rat, Y3 : rat]: ((~ ((ord_less_rat @ X @ Y3))) = (ord_less_eq_rat @ Y3 @ X))))). % not_less
thf(fact_212_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_213_le__neq__trans, axiom,
    ((![A : rat, B : rat]: ((ord_less_eq_rat @ A @ B) => ((~ ((A = B))) => (ord_less_rat @ A @ B)))))). % le_neq_trans
thf(fact_214_antisym__conv1, axiom,
    ((![X : real, Y3 : real]: ((~ ((ord_less_real @ X @ Y3))) => ((ord_less_eq_real @ X @ Y3) = (X = Y3)))))). % antisym_conv1
thf(fact_215_antisym__conv1, axiom,
    ((![X : rat, Y3 : rat]: ((~ ((ord_less_rat @ X @ Y3))) => ((ord_less_eq_rat @ X @ Y3) = (X = Y3)))))). % antisym_conv1
thf(fact_216_antisym__conv2, axiom,
    ((![X : real, Y3 : real]: ((ord_less_eq_real @ X @ Y3) => ((~ ((ord_less_real @ X @ Y3))) = (X = Y3)))))). % antisym_conv2
thf(fact_217_antisym__conv2, axiom,
    ((![X : rat, Y3 : rat]: ((ord_less_eq_rat @ X @ Y3) => ((~ ((ord_less_rat @ X @ Y3))) = (X = Y3)))))). % antisym_conv2
thf(fact_218_less__imp__le, axiom,
    ((![X : real, Y3 : real]: ((ord_less_real @ X @ Y3) => (ord_less_eq_real @ X @ Y3))))). % less_imp_le
thf(fact_219_less__imp__le, axiom,
    ((![X : rat, Y3 : rat]: ((ord_less_rat @ X @ Y3) => (ord_less_eq_rat @ X @ Y3))))). % less_imp_le
thf(fact_220_le__less__trans, axiom,
    ((![X : real, Y3 : real, Z4 : real]: ((ord_less_eq_real @ X @ Y3) => ((ord_less_real @ Y3 @ Z4) => (ord_less_real @ X @ Z4)))))). % le_less_trans
thf(fact_221_le__less__trans, axiom,
    ((![X : rat, Y3 : rat, Z4 : rat]: ((ord_less_eq_rat @ X @ Y3) => ((ord_less_rat @ Y3 @ Z4) => (ord_less_rat @ X @ Z4)))))). % le_less_trans
thf(fact_222_less__le__trans, axiom,
    ((![X : real, Y3 : real, Z4 : real]: ((ord_less_real @ X @ Y3) => ((ord_less_eq_real @ Y3 @ Z4) => (ord_less_real @ X @ Z4)))))). % less_le_trans
thf(fact_223_less__le__trans, axiom,
    ((![X : rat, Y3 : rat, Z4 : rat]: ((ord_less_rat @ X @ Y3) => ((ord_less_eq_rat @ Y3 @ Z4) => (ord_less_rat @ X @ Z4)))))). % less_le_trans
thf(fact_224_dense__ge, axiom,
    ((![Z4 : real, Y3 : real]: ((![X3 : real]: ((ord_less_real @ Z4 @ X3) => (ord_less_eq_real @ Y3 @ X3))) => (ord_less_eq_real @ Y3 @ Z4))))). % dense_ge
thf(fact_225_dense__ge, axiom,
    ((![Z4 : rat, Y3 : rat]: ((![X3 : rat]: ((ord_less_rat @ Z4 @ X3) => (ord_less_eq_rat @ Y3 @ X3))) => (ord_less_eq_rat @ Y3 @ Z4))))). % dense_ge
thf(fact_226_dense__le, axiom,
    ((![Y3 : real, Z4 : real]: ((![X3 : real]: ((ord_less_real @ X3 @ Y3) => (ord_less_eq_real @ X3 @ Z4))) => (ord_less_eq_real @ Y3 @ Z4))))). % dense_le
thf(fact_227_dense__le, axiom,
    ((![Y3 : rat, Z4 : rat]: ((![X3 : rat]: ((ord_less_rat @ X3 @ Y3) => (ord_less_eq_rat @ X3 @ Z4))) => (ord_less_eq_rat @ Y3 @ Z4))))). % dense_le
thf(fact_228_le__less__linear, axiom,
    ((![X : real, Y3 : real]: ((ord_less_eq_real @ X @ Y3) | (ord_less_real @ Y3 @ X))))). % le_less_linear
thf(fact_229_le__less__linear, axiom,
    ((![X : rat, Y3 : rat]: ((ord_less_eq_rat @ X @ Y3) | (ord_less_rat @ Y3 @ X))))). % le_less_linear
thf(fact_230_le__imp__less__or__eq, axiom,
    ((![X : real, Y3 : real]: ((ord_less_eq_real @ X @ Y3) => ((ord_less_real @ X @ Y3) | (X = Y3)))))). % le_imp_less_or_eq
thf(fact_231_le__imp__less__or__eq, axiom,
    ((![X : rat, Y3 : rat]: ((ord_less_eq_rat @ X @ Y3) => ((ord_less_rat @ X @ Y3) | (X = Y3)))))). % le_imp_less_or_eq
thf(fact_232_less__le__not__le, axiom,
    ((ord_less_real = (^[X4 : real]: (^[Y4 : real]: (((ord_less_eq_real @ X4 @ Y4)) & ((~ ((ord_less_eq_real @ Y4 @ X4)))))))))). % less_le_not_le
thf(fact_233_less__le__not__le, axiom,
    ((ord_less_rat = (^[X4 : rat]: (^[Y4 : rat]: (((ord_less_eq_rat @ X4 @ Y4)) & ((~ ((ord_less_eq_rat @ Y4 @ X4)))))))))). % less_le_not_le
thf(fact_234_not__le__imp__less, axiom,
    ((![Y3 : real, X : real]: ((~ ((ord_less_eq_real @ Y3 @ X))) => (ord_less_real @ X @ Y3))))). % not_le_imp_less
thf(fact_235_not__le__imp__less, axiom,
    ((![Y3 : rat, X : rat]: ((~ ((ord_less_eq_rat @ Y3 @ X))) => (ord_less_rat @ X @ Y3))))). % not_le_imp_less
thf(fact_236_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_237_order_Ostrict__trans1, axiom,
    ((![A : rat, B : rat, C : rat]: ((ord_less_eq_rat @ A @ B) => ((ord_less_rat @ B @ C) => (ord_less_rat @ A @ C)))))). % order.strict_trans1
thf(fact_238_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_239_order_Ostrict__trans2, axiom,
    ((![A : rat, B : rat, C : rat]: ((ord_less_rat @ A @ B) => ((ord_less_eq_rat @ B @ C) => (ord_less_rat @ A @ C)))))). % order.strict_trans2
thf(fact_240_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_241_order_Oorder__iff__strict, axiom,
    ((ord_less_eq_rat = (^[A2 : rat]: (^[B2 : rat]: (((ord_less_rat @ A2 @ B2)) | ((A2 = B2)))))))). % order.order_iff_strict
thf(fact_242_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_243_order_Ostrict__iff__order, axiom,
    ((ord_less_rat = (^[A2 : rat]: (^[B2 : rat]: (((ord_less_eq_rat @ A2 @ B2)) & ((~ ((A2 = B2)))))))))). % order.strict_iff_order
thf(fact_244_dual__order_Ostrict__trans1, axiom,
    ((![B : rat, A : rat, C : rat]: ((ord_less_eq_rat @ B @ A) => ((ord_less_rat @ C @ B) => (ord_less_rat @ C @ A)))))). % dual_order.strict_trans1
thf(fact_245_less__eq__real__def, axiom,
    ((ord_less_eq_real = (^[X4 : real]: (^[Y4 : real]: (((ord_less_real @ X4 @ Y4)) | ((X4 = Y4)))))))). % less_eq_real_def
thf(fact_246_real__less__code, axiom,
    ((![X : rat, Y3 : rat]: ((ord_less_real @ (ratreal @ X) @ (ratreal @ Y3)) = (ord_less_rat @ X @ Y3))))). % real_less_code
thf(fact_247_zero__real__code, axiom,
    ((zero_zero_real = (ratreal @ zero_zero_rat)))). % zero_real_code

% Conjectures (1)
thf(conj_0, conjecture,
    ((?[Z2 : complex]: (![W2 : complex]: (ord_less_eq_real @ (real_V638595069omplex @ (poly_complex2 @ p @ Z2)) @ (real_V638595069omplex @ (poly_complex2 @ p @ W2))))))).
