% 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_404__5371488_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:29:53.507

% Could-be-implicit typings (3)
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 (19)
thf(sy_c_Groups_Ominus__class_Ominus_001t__Complex__Ocomplex, type,
    minus_minus_complex : complex > complex > complex).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat, type,
    minus_minus_nat : nat > nat > nat).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal, type,
    minus_minus_real : real > real > real).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Complex__Ocomplex, type,
    zero_zero_complex : complex).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat, type,
    zero_zero_nat : nat).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal, type,
    zero_zero_real : real).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat, type,
    ord_less_nat : nat > nat > $o).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal, type,
    ord_less_real : real > real > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat, type,
    ord_less_eq_nat : nat > nat > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal, type,
    ord_less_eq_real : real > real > $o).
thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Nat__Onat_001t__Nat__Onat, type,
    order_769474267at_nat : (nat > nat) > $o).
thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Nat__Onat_001t__Real__Oreal, type,
    order_952716343t_real : (nat > real) > $o).
thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Real__Oreal_001t__Nat__Onat, type,
    order_297469111al_nat : (real > nat) > $o).
thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Real__Oreal_001t__Real__Oreal, type,
    order_1818878995l_real : (real > real) > $o).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Complex__Ocomplex, type,
    real_V638595069omplex : complex > real).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Real__Oreal, type,
    real_V646646907m_real : real > real).
thf(sy_v_g____, type,
    g : nat > complex).
thf(sy_v_r, type,
    r : real).
thf(sy_v_thesis____, type,
    thesis : $o).

% Relevant facts (234)
thf(fact_0__092_060open_062_092_060exists_062f_Az_O_Astrict__mono_Af_A_092_060and_062_A_I_092_060forall_062e_0620_O_A_092_060exists_062N_O_A_092_060forall_062n_092_060ge_062N_O_Acmod_A_Ig_A_If_An_J_A_N_Az_J_A_060_Ae_J_092_060close_062, axiom,
    ((?[F : nat > nat, Z : complex]: ((order_769474267at_nat @ F) & (![E : real]: ((ord_less_real @ zero_zero_real @ E) => (?[N : nat]: (![N2 : nat]: ((ord_less_eq_nat @ N @ N2) => (ord_less_real @ (real_V638595069omplex @ (minus_minus_complex @ (g @ (F @ N2)) @ Z)) @ E)))))))))). % \<open>\<exists>f z. strict_mono f \<and> (\<forall>e>0. \<exists>N. \<forall>n\<ge>N. cmod (g (f n) - z) < e)\<close>
thf(fact_1_real__sup__exists, axiom,
    ((![P : real > $o]: ((?[X_1 : real]: (P @ X_1)) => ((?[Z2 : real]: (![X : real]: ((P @ X) => (ord_less_real @ X @ Z2)))) => (?[S : real]: (![Y : real]: ((?[X2 : real]: (((P @ X2)) & ((ord_less_real @ Y @ X2)))) = (ord_less_real @ Y @ S))))))))). % real_sup_exists
thf(fact_2_zero__less__norm__iff, axiom,
    ((![X3 : real]: ((ord_less_real @ zero_zero_real @ (real_V646646907m_real @ X3)) = (~ ((X3 = zero_zero_real))))))). % zero_less_norm_iff
thf(fact_3_zero__less__norm__iff, axiom,
    ((![X3 : complex]: ((ord_less_real @ zero_zero_real @ (real_V638595069omplex @ X3)) = (~ ((X3 = zero_zero_complex))))))). % zero_less_norm_iff
thf(fact_4_norm__zero, axiom,
    (((real_V646646907m_real @ zero_zero_real) = zero_zero_real))). % norm_zero
thf(fact_5_norm__zero, axiom,
    (((real_V638595069omplex @ zero_zero_complex) = zero_zero_real))). % norm_zero
thf(fact_6_norm__eq__zero, axiom,
    ((![X3 : real]: (((real_V646646907m_real @ X3) = zero_zero_real) = (X3 = zero_zero_real))))). % norm_eq_zero
thf(fact_7_norm__eq__zero, axiom,
    ((![X3 : complex]: (((real_V638595069omplex @ X3) = zero_zero_real) = (X3 = zero_zero_complex))))). % norm_eq_zero
thf(fact_8_diff__gt__0__iff__gt, axiom,
    ((![A : real, B : real]: ((ord_less_real @ zero_zero_real @ (minus_minus_real @ A @ B)) = (ord_less_real @ B @ A))))). % diff_gt_0_iff_gt
thf(fact_9_diff__ge__0__iff__ge, axiom,
    ((![A : real, B : real]: ((ord_less_eq_real @ zero_zero_real @ (minus_minus_real @ A @ B)) = (ord_less_eq_real @ B @ A))))). % diff_ge_0_iff_ge
thf(fact_10_diff__self, axiom,
    ((![A : real]: ((minus_minus_real @ A @ A) = zero_zero_real)))). % diff_self
thf(fact_11_diff__self, axiom,
    ((![A : complex]: ((minus_minus_complex @ A @ A) = zero_zero_complex)))). % diff_self
thf(fact_12_diff__0__right, axiom,
    ((![A : real]: ((minus_minus_real @ A @ zero_zero_real) = A)))). % diff_0_right
thf(fact_13_diff__0__right, axiom,
    ((![A : complex]: ((minus_minus_complex @ A @ zero_zero_complex) = A)))). % diff_0_right
thf(fact_14_zero__diff, axiom,
    ((![A : nat]: ((minus_minus_nat @ zero_zero_nat @ A) = zero_zero_nat)))). % zero_diff
thf(fact_15_diff__zero, axiom,
    ((![A : complex]: ((minus_minus_complex @ A @ zero_zero_complex) = A)))). % diff_zero
thf(fact_16_diff__zero, axiom,
    ((![A : nat]: ((minus_minus_nat @ A @ zero_zero_nat) = A)))). % diff_zero
thf(fact_17_diff__zero, axiom,
    ((![A : real]: ((minus_minus_real @ A @ zero_zero_real) = A)))). % diff_zero
thf(fact_18_cancel__comm__monoid__add__class_Odiff__cancel, axiom,
    ((![A : complex]: ((minus_minus_complex @ A @ A) = zero_zero_complex)))). % cancel_comm_monoid_add_class.diff_cancel
thf(fact_19_cancel__comm__monoid__add__class_Odiff__cancel, axiom,
    ((![A : nat]: ((minus_minus_nat @ A @ A) = zero_zero_nat)))). % cancel_comm_monoid_add_class.diff_cancel
thf(fact_20_cancel__comm__monoid__add__class_Odiff__cancel, axiom,
    ((![A : real]: ((minus_minus_real @ A @ A) = zero_zero_real)))). % cancel_comm_monoid_add_class.diff_cancel
thf(fact_21_not__gr__zero, axiom,
    ((![N3 : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N3))) = (N3 = zero_zero_nat))))). % not_gr_zero
thf(fact_22_le__zero__eq, axiom,
    ((![N3 : nat]: ((ord_less_eq_nat @ N3 @ zero_zero_nat) = (N3 = zero_zero_nat))))). % le_zero_eq
thf(fact_23_True, axiom,
    ((ord_less_eq_real @ zero_zero_real @ r))). % True
thf(fact_24_norm__le__zero__iff, axiom,
    ((![X3 : complex]: ((ord_less_eq_real @ (real_V638595069omplex @ X3) @ zero_zero_real) = (X3 = zero_zero_complex))))). % norm_le_zero_iff
thf(fact_25_norm__le__zero__iff, axiom,
    ((![X3 : real]: ((ord_less_eq_real @ (real_V646646907m_real @ X3) @ zero_zero_real) = (X3 = zero_zero_real))))). % norm_le_zero_iff
thf(fact_26_g_I1_J, axiom,
    ((![N2 : nat]: (ord_less_eq_real @ (real_V638595069omplex @ (g @ N2)) @ r)))). % g(1)
thf(fact_27_norm__triangle__ineq2, axiom,
    ((![A : complex, B : complex]: (ord_less_eq_real @ (minus_minus_real @ (real_V638595069omplex @ A) @ (real_V638595069omplex @ B)) @ (real_V638595069omplex @ (minus_minus_complex @ A @ B)))))). % norm_triangle_ineq2
thf(fact_28_norm__triangle__ineq2, axiom,
    ((![A : real, B : real]: (ord_less_eq_real @ (minus_minus_real @ (real_V646646907m_real @ A) @ (real_V646646907m_real @ B)) @ (real_V646646907m_real @ (minus_minus_real @ A @ B)))))). % norm_triangle_ineq2
thf(fact_29_norm__ge__zero, axiom,
    ((![X3 : complex]: (ord_less_eq_real @ zero_zero_real @ (real_V638595069omplex @ X3))))). % norm_ge_zero
thf(fact_30_norm__ge__zero, axiom,
    ((![X3 : real]: (ord_less_eq_real @ zero_zero_real @ (real_V646646907m_real @ X3))))). % norm_ge_zero
thf(fact_31_zero__reorient, axiom,
    ((![X3 : real]: ((zero_zero_real = X3) = (X3 = zero_zero_real))))). % zero_reorient
thf(fact_32_zero__reorient, axiom,
    ((![X3 : nat]: ((zero_zero_nat = X3) = (X3 = zero_zero_nat))))). % zero_reorient
thf(fact_33_zero__reorient, axiom,
    ((![X3 : complex]: ((zero_zero_complex = X3) = (X3 = zero_zero_complex))))). % zero_reorient
thf(fact_34_cancel__ab__semigroup__add__class_Odiff__right__commute, axiom,
    ((![A : complex, C : complex, B : complex]: ((minus_minus_complex @ (minus_minus_complex @ A @ C) @ B) = (minus_minus_complex @ (minus_minus_complex @ A @ B) @ C))))). % cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_35_cancel__ab__semigroup__add__class_Odiff__right__commute, axiom,
    ((![A : nat, C : nat, B : nat]: ((minus_minus_nat @ (minus_minus_nat @ A @ C) @ B) = (minus_minus_nat @ (minus_minus_nat @ A @ B) @ C))))). % cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_36_cancel__ab__semigroup__add__class_Odiff__right__commute, axiom,
    ((![A : real, C : real, B : real]: ((minus_minus_real @ (minus_minus_real @ A @ C) @ B) = (minus_minus_real @ (minus_minus_real @ A @ B) @ C))))). % cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_37_diff__eq__diff__eq, axiom,
    ((![A : complex, B : complex, C : complex, D : complex]: (((minus_minus_complex @ A @ B) = (minus_minus_complex @ C @ D)) => ((A = B) = (C = D)))))). % diff_eq_diff_eq
thf(fact_38_diff__eq__diff__eq, axiom,
    ((![A : real, B : real, C : real, D : real]: (((minus_minus_real @ A @ B) = (minus_minus_real @ C @ D)) => ((A = B) = (C = D)))))). % diff_eq_diff_eq
thf(fact_39_zero__le, axiom,
    ((![X3 : nat]: (ord_less_eq_nat @ zero_zero_nat @ X3)))). % zero_le
thf(fact_40_zero__less__iff__neq__zero, axiom,
    ((![N3 : nat]: ((ord_less_nat @ zero_zero_nat @ N3) = (~ ((N3 = zero_zero_nat))))))). % zero_less_iff_neq_zero
thf(fact_41_gr__implies__not__zero, axiom,
    ((![M : nat, N3 : nat]: ((ord_less_nat @ M @ N3) => (~ ((N3 = zero_zero_nat))))))). % gr_implies_not_zero
thf(fact_42_not__less__zero, axiom,
    ((![N3 : nat]: (~ ((ord_less_nat @ N3 @ zero_zero_nat)))))). % not_less_zero
thf(fact_43_gr__zeroI, axiom,
    ((![N3 : nat]: ((~ ((N3 = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N3))))). % gr_zeroI
thf(fact_44_eq__iff__diff__eq__0, axiom,
    (((^[Y2 : complex]: (^[Z3 : complex]: (Y2 = Z3))) = (^[A2 : complex]: (^[B2 : complex]: ((minus_minus_complex @ A2 @ B2) = zero_zero_complex)))))). % eq_iff_diff_eq_0
thf(fact_45_eq__iff__diff__eq__0, axiom,
    (((^[Y2 : real]: (^[Z3 : real]: (Y2 = Z3))) = (^[A2 : real]: (^[B2 : real]: ((minus_minus_real @ A2 @ B2) = zero_zero_real)))))). % eq_iff_diff_eq_0
thf(fact_46_diff__eq__diff__less__eq, axiom,
    ((![A : real, B : real, C : real, D : real]: (((minus_minus_real @ A @ B) = (minus_minus_real @ C @ D)) => ((ord_less_eq_real @ A @ B) = (ord_less_eq_real @ C @ D)))))). % diff_eq_diff_less_eq
thf(fact_47_diff__right__mono, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_eq_real @ A @ B) => (ord_less_eq_real @ (minus_minus_real @ A @ C) @ (minus_minus_real @ B @ C)))))). % diff_right_mono
thf(fact_48_diff__left__mono, axiom,
    ((![B : real, A : real, C : real]: ((ord_less_eq_real @ B @ A) => (ord_less_eq_real @ (minus_minus_real @ C @ A) @ (minus_minus_real @ C @ B)))))). % diff_left_mono
thf(fact_49_diff__mono, axiom,
    ((![A : real, B : real, D : real, C : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_eq_real @ D @ C) => (ord_less_eq_real @ (minus_minus_real @ A @ C) @ (minus_minus_real @ B @ D))))))). % diff_mono
thf(fact_50_diff__strict__right__mono, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_real @ A @ B) => (ord_less_real @ (minus_minus_real @ A @ C) @ (minus_minus_real @ B @ C)))))). % diff_strict_right_mono
thf(fact_51_diff__strict__left__mono, axiom,
    ((![B : real, A : real, C : real]: ((ord_less_real @ B @ A) => (ord_less_real @ (minus_minus_real @ C @ A) @ (minus_minus_real @ C @ B)))))). % diff_strict_left_mono
thf(fact_52_diff__eq__diff__less, axiom,
    ((![A : real, B : real, C : real, D : real]: (((minus_minus_real @ A @ B) = (minus_minus_real @ C @ D)) => ((ord_less_real @ A @ B) = (ord_less_real @ C @ D)))))). % diff_eq_diff_less
thf(fact_53_diff__strict__mono, axiom,
    ((![A : real, B : real, D : real, C : real]: ((ord_less_real @ A @ B) => ((ord_less_real @ D @ C) => (ord_less_real @ (minus_minus_real @ A @ C) @ (minus_minus_real @ B @ D))))))). % diff_strict_mono
thf(fact_54_norm__minus__commute, axiom,
    ((![A : complex, B : complex]: ((real_V638595069omplex @ (minus_minus_complex @ A @ B)) = (real_V638595069omplex @ (minus_minus_complex @ B @ A)))))). % norm_minus_commute
thf(fact_55_norm__minus__commute, axiom,
    ((![A : real, B : real]: ((real_V646646907m_real @ (minus_minus_real @ A @ B)) = (real_V646646907m_real @ (minus_minus_real @ B @ A)))))). % norm_minus_commute
thf(fact_56_Bolzano__Weierstrass__complex__disc, axiom,
    ((![S2 : nat > complex, R : real]: ((![N4 : nat]: (ord_less_eq_real @ (real_V638595069omplex @ (S2 @ N4)) @ R)) => (?[F : nat > nat, Z : complex]: ((order_769474267at_nat @ F) & (![E : real]: ((ord_less_real @ zero_zero_real @ E) => (?[N : nat]: (![N2 : nat]: ((ord_less_eq_nat @ N @ N2) => (ord_less_real @ (real_V638595069omplex @ (minus_minus_complex @ (S2 @ (F @ N2)) @ Z)) @ E)))))))))))). % Bolzano_Weierstrass_complex_disc
thf(fact_57_le__iff__diff__le__0, axiom,
    ((ord_less_eq_real = (^[A2 : real]: (^[B2 : real]: (ord_less_eq_real @ (minus_minus_real @ A2 @ B2) @ zero_zero_real)))))). % le_iff_diff_le_0
thf(fact_58_less__iff__diff__less__0, axiom,
    ((ord_less_real = (^[A2 : real]: (^[B2 : real]: (ord_less_real @ (minus_minus_real @ A2 @ B2) @ zero_zero_real)))))). % less_iff_diff_less_0
thf(fact_59_norm__not__less__zero, axiom,
    ((![X3 : complex]: (~ ((ord_less_real @ (real_V638595069omplex @ X3) @ zero_zero_real)))))). % norm_not_less_zero
thf(fact_60_norm__not__less__zero, axiom,
    ((![X3 : real]: (~ ((ord_less_real @ (real_V646646907m_real @ X3) @ zero_zero_real)))))). % norm_not_less_zero
thf(fact_61_diff__is__0__eq, axiom,
    ((![M : nat, N3 : nat]: (((minus_minus_nat @ M @ N3) = zero_zero_nat) = (ord_less_eq_nat @ M @ N3))))). % diff_is_0_eq
thf(fact_62_diff__is__0__eq_H, axiom,
    ((![M : nat, N3 : nat]: ((ord_less_eq_nat @ M @ N3) => ((minus_minus_nat @ M @ N3) = zero_zero_nat))))). % diff_is_0_eq'
thf(fact_63_diff__diff__cancel, axiom,
    ((![I : nat, N3 : nat]: ((ord_less_eq_nat @ I @ N3) => ((minus_minus_nat @ N3 @ (minus_minus_nat @ N3 @ I)) = I))))). % diff_diff_cancel
thf(fact_64_bot__nat__0_Oextremum, axiom,
    ((![A : nat]: (ord_less_eq_nat @ zero_zero_nat @ A)))). % bot_nat_0.extremum
thf(fact_65_le0, axiom,
    ((![N3 : nat]: (ord_less_eq_nat @ zero_zero_nat @ N3)))). % le0
thf(fact_66_order__refl, axiom,
    ((![X3 : nat]: (ord_less_eq_nat @ X3 @ X3)))). % order_refl
thf(fact_67_order__refl, axiom,
    ((![X3 : real]: (ord_less_eq_real @ X3 @ X3)))). % order_refl
thf(fact_68_strict__mono__imp__increasing, axiom,
    ((![F2 : nat > nat, N3 : nat]: ((order_769474267at_nat @ F2) => (ord_less_eq_nat @ N3 @ (F2 @ N3)))))). % strict_mono_imp_increasing
thf(fact_69_strict__mono__less, axiom,
    ((![F2 : real > real, X3 : real, Y3 : real]: ((order_1818878995l_real @ F2) => ((ord_less_real @ (F2 @ X3) @ (F2 @ Y3)) = (ord_less_real @ X3 @ Y3)))))). % strict_mono_less
thf(fact_70_strict__mono__less, axiom,
    ((![F2 : nat > real, X3 : nat, Y3 : nat]: ((order_952716343t_real @ F2) => ((ord_less_real @ (F2 @ X3) @ (F2 @ Y3)) = (ord_less_nat @ X3 @ Y3)))))). % strict_mono_less
thf(fact_71_strict__mono__less, axiom,
    ((![F2 : real > nat, X3 : real, Y3 : real]: ((order_297469111al_nat @ F2) => ((ord_less_nat @ (F2 @ X3) @ (F2 @ Y3)) = (ord_less_real @ X3 @ Y3)))))). % strict_mono_less
thf(fact_72_strict__mono__less, axiom,
    ((![F2 : nat > nat, X3 : nat, Y3 : nat]: ((order_769474267at_nat @ F2) => ((ord_less_nat @ (F2 @ X3) @ (F2 @ Y3)) = (ord_less_nat @ X3 @ Y3)))))). % strict_mono_less
thf(fact_73_strict__mono__def, axiom,
    ((order_1818878995l_real = (^[F3 : real > real]: (![X2 : real]: (![Y4 : real]: (((ord_less_real @ X2 @ Y4)) => ((ord_less_real @ (F3 @ X2) @ (F3 @ Y4)))))))))). % strict_mono_def
thf(fact_74_strict__mono__def, axiom,
    ((order_297469111al_nat = (^[F3 : real > nat]: (![X2 : real]: (![Y4 : real]: (((ord_less_real @ X2 @ Y4)) => ((ord_less_nat @ (F3 @ X2) @ (F3 @ Y4)))))))))). % strict_mono_def
thf(fact_75_strict__mono__def, axiom,
    ((order_952716343t_real = (^[F3 : nat > real]: (![X2 : nat]: (![Y4 : nat]: (((ord_less_nat @ X2 @ Y4)) => ((ord_less_real @ (F3 @ X2) @ (F3 @ Y4)))))))))). % strict_mono_def
thf(fact_76_strict__mono__def, axiom,
    ((order_769474267at_nat = (^[F3 : nat > nat]: (![X2 : nat]: (![Y4 : nat]: (((ord_less_nat @ X2 @ Y4)) => ((ord_less_nat @ (F3 @ X2) @ (F3 @ Y4)))))))))). % strict_mono_def
thf(fact_77_neq0__conv, axiom,
    ((![N3 : nat]: ((~ ((N3 = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ N3))))). % neq0_conv
thf(fact_78_less__nat__zero__code, axiom,
    ((![N3 : nat]: (~ ((ord_less_nat @ N3 @ zero_zero_nat)))))). % less_nat_zero_code
thf(fact_79_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_80_diff__0__eq__0, axiom,
    ((![N3 : nat]: ((minus_minus_nat @ zero_zero_nat @ N3) = zero_zero_nat)))). % diff_0_eq_0
thf(fact_81_diff__self__eq__0, axiom,
    ((![M : nat]: ((minus_minus_nat @ M @ M) = zero_zero_nat)))). % diff_self_eq_0
thf(fact_82_zero__less__diff, axiom,
    ((![N3 : nat, M : nat]: ((ord_less_nat @ zero_zero_nat @ (minus_minus_nat @ N3 @ M)) = (ord_less_nat @ M @ N3))))). % zero_less_diff
thf(fact_83_gr0I, axiom,
    ((![N3 : nat]: ((~ ((N3 = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N3))))). % gr0I
thf(fact_84_not__gr0, axiom,
    ((![N3 : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N3))) = (N3 = zero_zero_nat))))). % not_gr0
thf(fact_85_diff__less, axiom,
    ((![N3 : nat, M : nat]: ((ord_less_nat @ zero_zero_nat @ N3) => ((ord_less_nat @ zero_zero_nat @ M) => (ord_less_nat @ (minus_minus_nat @ M @ N3) @ M)))))). % diff_less
thf(fact_86_not__less0, axiom,
    ((![N3 : nat]: (~ ((ord_less_nat @ N3 @ zero_zero_nat)))))). % not_less0
thf(fact_87_less__zeroE, axiom,
    ((![N3 : nat]: (~ ((ord_less_nat @ N3 @ zero_zero_nat)))))). % less_zeroE
thf(fact_88_diff__commute, axiom,
    ((![I : nat, J : nat, K : nat]: ((minus_minus_nat @ (minus_minus_nat @ I @ J) @ K) = (minus_minus_nat @ (minus_minus_nat @ I @ K) @ J))))). % diff_commute
thf(fact_89_diff__less__mono2, axiom,
    ((![M : nat, N3 : nat, L : nat]: ((ord_less_nat @ M @ N3) => ((ord_less_nat @ M @ L) => (ord_less_nat @ (minus_minus_nat @ L @ N3) @ (minus_minus_nat @ L @ M))))))). % diff_less_mono2
thf(fact_90_gr__implies__not0, axiom,
    ((![M : nat, N3 : nat]: ((ord_less_nat @ M @ N3) => (~ ((N3 = zero_zero_nat))))))). % gr_implies_not0
thf(fact_91_diffs0__imp__equal, axiom,
    ((![M : nat, N3 : nat]: (((minus_minus_nat @ M @ N3) = zero_zero_nat) => (((minus_minus_nat @ N3 @ M) = zero_zero_nat) => (M = N3)))))). % diffs0_imp_equal
thf(fact_92_minus__nat_Odiff__0, axiom,
    ((![M : nat]: ((minus_minus_nat @ M @ zero_zero_nat) = M)))). % minus_nat.diff_0
thf(fact_93_infinite__descent0, axiom,
    ((![P : nat > $o, N3 : nat]: ((P @ zero_zero_nat) => ((![N4 : nat]: ((ord_less_nat @ zero_zero_nat @ N4) => ((~ ((P @ N4))) => (?[M2 : nat]: ((ord_less_nat @ M2 @ N4) & (~ ((P @ M2)))))))) => (P @ N3)))))). % infinite_descent0
thf(fact_94_less__imp__diff__less, axiom,
    ((![J : nat, K : nat, N3 : nat]: ((ord_less_nat @ J @ K) => (ord_less_nat @ (minus_minus_nat @ J @ N3) @ K))))). % less_imp_diff_less
thf(fact_95_bot__nat__0_Oextremum__strict, axiom,
    ((![A : nat]: (~ ((ord_less_nat @ A @ zero_zero_nat)))))). % bot_nat_0.extremum_strict
thf(fact_96_order__subst1, axiom,
    ((![A : nat, F2 : nat > nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ (F2 @ B)) => ((ord_less_eq_nat @ B @ C) => ((![X : nat, Y5 : nat]: ((ord_less_eq_nat @ X @ Y5) => (ord_less_eq_nat @ (F2 @ X) @ (F2 @ Y5)))) => (ord_less_eq_nat @ A @ (F2 @ C)))))))). % order_subst1
thf(fact_97_order__subst1, axiom,
    ((![A : nat, F2 : real > nat, B : real, C : real]: ((ord_less_eq_nat @ A @ (F2 @ B)) => ((ord_less_eq_real @ B @ C) => ((![X : real, Y5 : real]: ((ord_less_eq_real @ X @ Y5) => (ord_less_eq_nat @ (F2 @ X) @ (F2 @ Y5)))) => (ord_less_eq_nat @ A @ (F2 @ C)))))))). % order_subst1
thf(fact_98_order__subst1, axiom,
    ((![A : real, F2 : nat > real, B : nat, C : nat]: ((ord_less_eq_real @ A @ (F2 @ B)) => ((ord_less_eq_nat @ B @ C) => ((![X : nat, Y5 : nat]: ((ord_less_eq_nat @ X @ Y5) => (ord_less_eq_real @ (F2 @ X) @ (F2 @ Y5)))) => (ord_less_eq_real @ A @ (F2 @ C)))))))). % order_subst1
thf(fact_99_order__subst1, axiom,
    ((![A : real, F2 : real > real, B : real, C : real]: ((ord_less_eq_real @ A @ (F2 @ B)) => ((ord_less_eq_real @ B @ C) => ((![X : real, Y5 : real]: ((ord_less_eq_real @ X @ Y5) => (ord_less_eq_real @ (F2 @ X) @ (F2 @ Y5)))) => (ord_less_eq_real @ A @ (F2 @ C)))))))). % order_subst1
thf(fact_100_order__subst2, axiom,
    ((![A : nat, B : nat, F2 : nat > nat, C : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_nat @ (F2 @ B) @ C) => ((![X : nat, Y5 : nat]: ((ord_less_eq_nat @ X @ Y5) => (ord_less_eq_nat @ (F2 @ X) @ (F2 @ Y5)))) => (ord_less_eq_nat @ (F2 @ A) @ C))))))). % order_subst2
thf(fact_101_order__subst2, axiom,
    ((![A : nat, B : nat, F2 : nat > real, C : real]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_real @ (F2 @ B) @ C) => ((![X : nat, Y5 : nat]: ((ord_less_eq_nat @ X @ Y5) => (ord_less_eq_real @ (F2 @ X) @ (F2 @ Y5)))) => (ord_less_eq_real @ (F2 @ A) @ C))))))). % order_subst2
thf(fact_102_order__subst2, axiom,
    ((![A : real, B : real, F2 : real > nat, C : nat]: ((ord_less_eq_real @ A @ B) => ((ord_less_eq_nat @ (F2 @ B) @ C) => ((![X : real, Y5 : real]: ((ord_less_eq_real @ X @ Y5) => (ord_less_eq_nat @ (F2 @ X) @ (F2 @ Y5)))) => (ord_less_eq_nat @ (F2 @ A) @ C))))))). % order_subst2
thf(fact_103_order__subst2, axiom,
    ((![A : real, B : real, F2 : real > real, C : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_eq_real @ (F2 @ B) @ C) => ((![X : real, Y5 : real]: ((ord_less_eq_real @ X @ Y5) => (ord_less_eq_real @ (F2 @ X) @ (F2 @ Y5)))) => (ord_less_eq_real @ (F2 @ A) @ C))))))). % order_subst2
thf(fact_104_ord__eq__le__subst, axiom,
    ((![A : nat, F2 : nat > nat, B : nat, C : nat]: ((A = (F2 @ B)) => ((ord_less_eq_nat @ B @ C) => ((![X : nat, Y5 : nat]: ((ord_less_eq_nat @ X @ Y5) => (ord_less_eq_nat @ (F2 @ X) @ (F2 @ Y5)))) => (ord_less_eq_nat @ A @ (F2 @ C)))))))). % ord_eq_le_subst
thf(fact_105_ord__eq__le__subst, axiom,
    ((![A : real, F2 : nat > real, B : nat, C : nat]: ((A = (F2 @ B)) => ((ord_less_eq_nat @ B @ C) => ((![X : nat, Y5 : nat]: ((ord_less_eq_nat @ X @ Y5) => (ord_less_eq_real @ (F2 @ X) @ (F2 @ Y5)))) => (ord_less_eq_real @ A @ (F2 @ C)))))))). % ord_eq_le_subst
thf(fact_106_ord__eq__le__subst, axiom,
    ((![A : nat, F2 : real > nat, B : real, C : real]: ((A = (F2 @ B)) => ((ord_less_eq_real @ B @ C) => ((![X : real, Y5 : real]: ((ord_less_eq_real @ X @ Y5) => (ord_less_eq_nat @ (F2 @ X) @ (F2 @ Y5)))) => (ord_less_eq_nat @ A @ (F2 @ C)))))))). % ord_eq_le_subst
thf(fact_107_ord__eq__le__subst, axiom,
    ((![A : real, F2 : real > real, B : real, C : real]: ((A = (F2 @ B)) => ((ord_less_eq_real @ B @ C) => ((![X : real, Y5 : real]: ((ord_less_eq_real @ X @ Y5) => (ord_less_eq_real @ (F2 @ X) @ (F2 @ Y5)))) => (ord_less_eq_real @ A @ (F2 @ C)))))))). % ord_eq_le_subst
thf(fact_108_ord__le__eq__subst, axiom,
    ((![A : nat, B : nat, F2 : nat > nat, C : nat]: ((ord_less_eq_nat @ A @ B) => (((F2 @ B) = C) => ((![X : nat, Y5 : nat]: ((ord_less_eq_nat @ X @ Y5) => (ord_less_eq_nat @ (F2 @ X) @ (F2 @ Y5)))) => (ord_less_eq_nat @ (F2 @ A) @ C))))))). % ord_le_eq_subst
thf(fact_109_ord__le__eq__subst, axiom,
    ((![A : nat, B : nat, F2 : nat > real, C : real]: ((ord_less_eq_nat @ A @ B) => (((F2 @ B) = C) => ((![X : nat, Y5 : nat]: ((ord_less_eq_nat @ X @ Y5) => (ord_less_eq_real @ (F2 @ X) @ (F2 @ Y5)))) => (ord_less_eq_real @ (F2 @ A) @ C))))))). % ord_le_eq_subst
thf(fact_110_ord__le__eq__subst, axiom,
    ((![A : real, B : real, F2 : real > nat, C : nat]: ((ord_less_eq_real @ A @ B) => (((F2 @ B) = C) => ((![X : real, Y5 : real]: ((ord_less_eq_real @ X @ Y5) => (ord_less_eq_nat @ (F2 @ X) @ (F2 @ Y5)))) => (ord_less_eq_nat @ (F2 @ A) @ C))))))). % ord_le_eq_subst
thf(fact_111_ord__le__eq__subst, axiom,
    ((![A : real, B : real, F2 : real > real, C : real]: ((ord_less_eq_real @ A @ B) => (((F2 @ B) = C) => ((![X : real, Y5 : real]: ((ord_less_eq_real @ X @ Y5) => (ord_less_eq_real @ (F2 @ X) @ (F2 @ Y5)))) => (ord_less_eq_real @ (F2 @ A) @ C))))))). % ord_le_eq_subst
thf(fact_112_eq__iff, axiom,
    (((^[Y2 : nat]: (^[Z3 : nat]: (Y2 = Z3))) = (^[X2 : nat]: (^[Y4 : nat]: (((ord_less_eq_nat @ X2 @ Y4)) & ((ord_less_eq_nat @ Y4 @ X2)))))))). % eq_iff
thf(fact_113_eq__iff, axiom,
    (((^[Y2 : real]: (^[Z3 : real]: (Y2 = Z3))) = (^[X2 : real]: (^[Y4 : real]: (((ord_less_eq_real @ X2 @ Y4)) & ((ord_less_eq_real @ Y4 @ X2)))))))). % eq_iff
thf(fact_114_antisym, axiom,
    ((![X3 : nat, Y3 : nat]: ((ord_less_eq_nat @ X3 @ Y3) => ((ord_less_eq_nat @ Y3 @ X3) => (X3 = Y3)))))). % antisym
thf(fact_115_antisym, axiom,
    ((![X3 : real, Y3 : real]: ((ord_less_eq_real @ X3 @ Y3) => ((ord_less_eq_real @ Y3 @ X3) => (X3 = Y3)))))). % antisym
thf(fact_116_linear, axiom,
    ((![X3 : nat, Y3 : nat]: ((ord_less_eq_nat @ X3 @ Y3) | (ord_less_eq_nat @ Y3 @ X3))))). % linear
thf(fact_117_linear, axiom,
    ((![X3 : real, Y3 : real]: ((ord_less_eq_real @ X3 @ Y3) | (ord_less_eq_real @ Y3 @ X3))))). % linear
thf(fact_118_eq__refl, axiom,
    ((![X3 : nat, Y3 : nat]: ((X3 = Y3) => (ord_less_eq_nat @ X3 @ Y3))))). % eq_refl
thf(fact_119_eq__refl, axiom,
    ((![X3 : real, Y3 : real]: ((X3 = Y3) => (ord_less_eq_real @ X3 @ Y3))))). % eq_refl
thf(fact_120_le__cases, axiom,
    ((![X3 : nat, Y3 : nat]: ((~ ((ord_less_eq_nat @ X3 @ Y3))) => (ord_less_eq_nat @ Y3 @ X3))))). % le_cases
thf(fact_121_le__cases, axiom,
    ((![X3 : real, Y3 : real]: ((~ ((ord_less_eq_real @ X3 @ Y3))) => (ord_less_eq_real @ Y3 @ X3))))). % le_cases
thf(fact_122_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_123_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_124_le__cases3, axiom,
    ((![X3 : nat, Y3 : nat, Z4 : nat]: (((ord_less_eq_nat @ X3 @ Y3) => (~ ((ord_less_eq_nat @ Y3 @ Z4)))) => (((ord_less_eq_nat @ Y3 @ X3) => (~ ((ord_less_eq_nat @ X3 @ Z4)))) => (((ord_less_eq_nat @ X3 @ Z4) => (~ ((ord_less_eq_nat @ Z4 @ Y3)))) => (((ord_less_eq_nat @ Z4 @ Y3) => (~ ((ord_less_eq_nat @ Y3 @ X3)))) => (((ord_less_eq_nat @ Y3 @ Z4) => (~ ((ord_less_eq_nat @ Z4 @ X3)))) => (~ (((ord_less_eq_nat @ Z4 @ X3) => (~ ((ord_less_eq_nat @ X3 @ Y3)))))))))))))). % le_cases3
thf(fact_125_le__cases3, axiom,
    ((![X3 : real, Y3 : real, Z4 : real]: (((ord_less_eq_real @ X3 @ Y3) => (~ ((ord_less_eq_real @ Y3 @ Z4)))) => (((ord_less_eq_real @ Y3 @ X3) => (~ ((ord_less_eq_real @ X3 @ Z4)))) => (((ord_less_eq_real @ X3 @ Z4) => (~ ((ord_less_eq_real @ Z4 @ Y3)))) => (((ord_less_eq_real @ Z4 @ Y3) => (~ ((ord_less_eq_real @ Y3 @ X3)))) => (((ord_less_eq_real @ Y3 @ Z4) => (~ ((ord_less_eq_real @ Z4 @ X3)))) => (~ (((ord_less_eq_real @ Z4 @ X3) => (~ ((ord_less_eq_real @ X3 @ Y3)))))))))))))). % le_cases3
thf(fact_126_antisym__conv, axiom,
    ((![Y3 : nat, X3 : nat]: ((ord_less_eq_nat @ Y3 @ X3) => ((ord_less_eq_nat @ X3 @ Y3) = (X3 = Y3)))))). % antisym_conv
thf(fact_127_antisym__conv, axiom,
    ((![Y3 : real, X3 : real]: ((ord_less_eq_real @ Y3 @ X3) => ((ord_less_eq_real @ X3 @ Y3) = (X3 = Y3)))))). % antisym_conv
thf(fact_128_order__class_Oorder_Oeq__iff, axiom,
    (((^[Y2 : nat]: (^[Z3 : nat]: (Y2 = Z3))) = (^[A2 : nat]: (^[B2 : nat]: (((ord_less_eq_nat @ A2 @ B2)) & ((ord_less_eq_nat @ B2 @ A2)))))))). % order_class.order.eq_iff
thf(fact_129_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_130_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_131_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_132_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_133_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_134_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_135_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_136_order__trans, axiom,
    ((![X3 : nat, Y3 : nat, Z4 : nat]: ((ord_less_eq_nat @ X3 @ Y3) => ((ord_less_eq_nat @ Y3 @ Z4) => (ord_less_eq_nat @ X3 @ Z4)))))). % order_trans
thf(fact_137_order__trans, axiom,
    ((![X3 : real, Y3 : real, Z4 : real]: ((ord_less_eq_real @ X3 @ Y3) => ((ord_less_eq_real @ Y3 @ Z4) => (ord_less_eq_real @ X3 @ Z4)))))). % order_trans
thf(fact_138_dual__order_Orefl, axiom,
    ((![A : nat]: (ord_less_eq_nat @ A @ A)))). % dual_order.refl
thf(fact_139_dual__order_Orefl, axiom,
    ((![A : real]: (ord_less_eq_real @ A @ A)))). % dual_order.refl
thf(fact_140_linorder__wlog, axiom,
    ((![P : nat > nat > $o, A : nat, B : nat]: ((![A3 : nat, B3 : nat]: ((ord_less_eq_nat @ A3 @ B3) => (P @ A3 @ B3))) => ((![A3 : nat, B3 : nat]: ((P @ B3 @ A3) => (P @ A3 @ B3))) => (P @ A @ B)))))). % linorder_wlog
thf(fact_141_linorder__wlog, axiom,
    ((![P : real > real > $o, A : real, B : real]: ((![A3 : real, B3 : real]: ((ord_less_eq_real @ A3 @ B3) => (P @ A3 @ B3))) => ((![A3 : real, B3 : real]: ((P @ B3 @ A3) => (P @ A3 @ B3))) => (P @ A @ B)))))). % linorder_wlog
thf(fact_142_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_143_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_144_dual__order_Oeq__iff, axiom,
    (((^[Y2 : nat]: (^[Z3 : nat]: (Y2 = Z3))) = (^[A2 : nat]: (^[B2 : nat]: (((ord_less_eq_nat @ B2 @ A2)) & ((ord_less_eq_nat @ A2 @ B2)))))))). % dual_order.eq_iff
thf(fact_145_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_146_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_147_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_148_ord__eq__less__subst, axiom,
    ((![A : real, F2 : real > real, B : real, C : real]: ((A = (F2 @ B)) => ((ord_less_real @ B @ C) => ((![X : real, Y5 : real]: ((ord_less_real @ X @ Y5) => (ord_less_real @ (F2 @ X) @ (F2 @ Y5)))) => (ord_less_real @ A @ (F2 @ C)))))))). % ord_eq_less_subst
thf(fact_149_ord__eq__less__subst, axiom,
    ((![A : nat, F2 : real > nat, B : real, C : real]: ((A = (F2 @ B)) => ((ord_less_real @ B @ C) => ((![X : real, Y5 : real]: ((ord_less_real @ X @ Y5) => (ord_less_nat @ (F2 @ X) @ (F2 @ Y5)))) => (ord_less_nat @ A @ (F2 @ C)))))))). % ord_eq_less_subst
thf(fact_150_ord__eq__less__subst, axiom,
    ((![A : real, F2 : nat > real, B : nat, C : nat]: ((A = (F2 @ B)) => ((ord_less_nat @ B @ C) => ((![X : nat, Y5 : nat]: ((ord_less_nat @ X @ Y5) => (ord_less_real @ (F2 @ X) @ (F2 @ Y5)))) => (ord_less_real @ A @ (F2 @ C)))))))). % ord_eq_less_subst
thf(fact_151_ord__eq__less__subst, axiom,
    ((![A : nat, F2 : nat > nat, B : nat, C : nat]: ((A = (F2 @ B)) => ((ord_less_nat @ B @ C) => ((![X : nat, Y5 : nat]: ((ord_less_nat @ X @ Y5) => (ord_less_nat @ (F2 @ X) @ (F2 @ Y5)))) => (ord_less_nat @ A @ (F2 @ C)))))))). % ord_eq_less_subst
thf(fact_152_ord__less__eq__subst, axiom,
    ((![A : real, B : real, F2 : real > real, C : real]: ((ord_less_real @ A @ B) => (((F2 @ B) = C) => ((![X : real, Y5 : real]: ((ord_less_real @ X @ Y5) => (ord_less_real @ (F2 @ X) @ (F2 @ Y5)))) => (ord_less_real @ (F2 @ A) @ C))))))). % ord_less_eq_subst
thf(fact_153_ord__less__eq__subst, axiom,
    ((![A : real, B : real, F2 : real > nat, C : nat]: ((ord_less_real @ A @ B) => (((F2 @ B) = C) => ((![X : real, Y5 : real]: ((ord_less_real @ X @ Y5) => (ord_less_nat @ (F2 @ X) @ (F2 @ Y5)))) => (ord_less_nat @ (F2 @ A) @ C))))))). % ord_less_eq_subst
thf(fact_154_ord__less__eq__subst, axiom,
    ((![A : nat, B : nat, F2 : nat > real, C : real]: ((ord_less_nat @ A @ B) => (((F2 @ B) = C) => ((![X : nat, Y5 : nat]: ((ord_less_nat @ X @ Y5) => (ord_less_real @ (F2 @ X) @ (F2 @ Y5)))) => (ord_less_real @ (F2 @ A) @ C))))))). % ord_less_eq_subst
thf(fact_155_ord__less__eq__subst, axiom,
    ((![A : nat, B : nat, F2 : nat > nat, C : nat]: ((ord_less_nat @ A @ B) => (((F2 @ B) = C) => ((![X : nat, Y5 : nat]: ((ord_less_nat @ X @ Y5) => (ord_less_nat @ (F2 @ X) @ (F2 @ Y5)))) => (ord_less_nat @ (F2 @ A) @ C))))))). % ord_less_eq_subst
thf(fact_156_order__less__subst1, axiom,
    ((![A : real, F2 : real > real, B : real, C : real]: ((ord_less_real @ A @ (F2 @ B)) => ((ord_less_real @ B @ C) => ((![X : real, Y5 : real]: ((ord_less_real @ X @ Y5) => (ord_less_real @ (F2 @ X) @ (F2 @ Y5)))) => (ord_less_real @ A @ (F2 @ C)))))))). % order_less_subst1
thf(fact_157_order__less__subst1, axiom,
    ((![A : real, F2 : nat > real, B : nat, C : nat]: ((ord_less_real @ A @ (F2 @ B)) => ((ord_less_nat @ B @ C) => ((![X : nat, Y5 : nat]: ((ord_less_nat @ X @ Y5) => (ord_less_real @ (F2 @ X) @ (F2 @ Y5)))) => (ord_less_real @ A @ (F2 @ C)))))))). % order_less_subst1
thf(fact_158_order__less__subst1, axiom,
    ((![A : nat, F2 : real > nat, B : real, C : real]: ((ord_less_nat @ A @ (F2 @ B)) => ((ord_less_real @ B @ C) => ((![X : real, Y5 : real]: ((ord_less_real @ X @ Y5) => (ord_less_nat @ (F2 @ X) @ (F2 @ Y5)))) => (ord_less_nat @ A @ (F2 @ C)))))))). % order_less_subst1
thf(fact_159_order__less__subst1, axiom,
    ((![A : nat, F2 : nat > nat, B : nat, C : nat]: ((ord_less_nat @ A @ (F2 @ B)) => ((ord_less_nat @ B @ C) => ((![X : nat, Y5 : nat]: ((ord_less_nat @ X @ Y5) => (ord_less_nat @ (F2 @ X) @ (F2 @ Y5)))) => (ord_less_nat @ A @ (F2 @ C)))))))). % order_less_subst1
thf(fact_160_order__less__subst2, axiom,
    ((![A : real, B : real, F2 : real > real, C : real]: ((ord_less_real @ A @ B) => ((ord_less_real @ (F2 @ B) @ C) => ((![X : real, Y5 : real]: ((ord_less_real @ X @ Y5) => (ord_less_real @ (F2 @ X) @ (F2 @ Y5)))) => (ord_less_real @ (F2 @ A) @ C))))))). % order_less_subst2
thf(fact_161_order__less__subst2, axiom,
    ((![A : real, B : real, F2 : real > nat, C : nat]: ((ord_less_real @ A @ B) => ((ord_less_nat @ (F2 @ B) @ C) => ((![X : real, Y5 : real]: ((ord_less_real @ X @ Y5) => (ord_less_nat @ (F2 @ X) @ (F2 @ Y5)))) => (ord_less_nat @ (F2 @ A) @ C))))))). % order_less_subst2
thf(fact_162_order__less__subst2, axiom,
    ((![A : nat, B : nat, F2 : nat > real, C : real]: ((ord_less_nat @ A @ B) => ((ord_less_real @ (F2 @ B) @ C) => ((![X : nat, Y5 : nat]: ((ord_less_nat @ X @ Y5) => (ord_less_real @ (F2 @ X) @ (F2 @ Y5)))) => (ord_less_real @ (F2 @ A) @ C))))))). % order_less_subst2
thf(fact_163_order__less__subst2, axiom,
    ((![A : nat, B : nat, F2 : nat > nat, C : nat]: ((ord_less_nat @ A @ B) => ((ord_less_nat @ (F2 @ B) @ C) => ((![X : nat, Y5 : nat]: ((ord_less_nat @ X @ Y5) => (ord_less_nat @ (F2 @ X) @ (F2 @ Y5)))) => (ord_less_nat @ (F2 @ A) @ C))))))). % order_less_subst2
thf(fact_164_lt__ex, axiom,
    ((![X3 : real]: (?[Y5 : real]: (ord_less_real @ Y5 @ X3))))). % lt_ex
thf(fact_165_gt__ex, axiom,
    ((![X3 : real]: (?[X_12 : real]: (ord_less_real @ X3 @ X_12))))). % gt_ex
thf(fact_166_gt__ex, axiom,
    ((![X3 : nat]: (?[X_12 : nat]: (ord_less_nat @ X3 @ X_12))))). % gt_ex
thf(fact_167_neqE, axiom,
    ((![X3 : real, Y3 : real]: ((~ ((X3 = Y3))) => ((~ ((ord_less_real @ X3 @ Y3))) => (ord_less_real @ Y3 @ X3)))))). % neqE
thf(fact_168_neqE, axiom,
    ((![X3 : nat, Y3 : nat]: ((~ ((X3 = Y3))) => ((~ ((ord_less_nat @ X3 @ Y3))) => (ord_less_nat @ Y3 @ X3)))))). % neqE
thf(fact_169_neq__iff, axiom,
    ((![X3 : real, Y3 : real]: ((~ ((X3 = Y3))) = (((ord_less_real @ X3 @ Y3)) | ((ord_less_real @ Y3 @ X3))))))). % neq_iff
thf(fact_170_neq__iff, axiom,
    ((![X3 : nat, Y3 : nat]: ((~ ((X3 = Y3))) = (((ord_less_nat @ X3 @ Y3)) | ((ord_less_nat @ Y3 @ X3))))))). % neq_iff
thf(fact_171_order_Oasym, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ B) => (~ ((ord_less_real @ B @ A))))))). % order.asym
thf(fact_172_order_Oasym, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (~ ((ord_less_nat @ B @ A))))))). % order.asym
thf(fact_173_dense, axiom,
    ((![X3 : real, Y3 : real]: ((ord_less_real @ X3 @ Y3) => (?[Z : real]: ((ord_less_real @ X3 @ Z) & (ord_less_real @ Z @ Y3))))))). % dense
thf(fact_174_less__imp__neq, axiom,
    ((![X3 : real, Y3 : real]: ((ord_less_real @ X3 @ Y3) => (~ ((X3 = Y3))))))). % less_imp_neq
thf(fact_175_less__imp__neq, axiom,
    ((![X3 : nat, Y3 : nat]: ((ord_less_nat @ X3 @ Y3) => (~ ((X3 = Y3))))))). % less_imp_neq
thf(fact_176_less__asym, axiom,
    ((![X3 : real, Y3 : real]: ((ord_less_real @ X3 @ Y3) => (~ ((ord_less_real @ Y3 @ X3))))))). % less_asym
thf(fact_177_less__asym, axiom,
    ((![X3 : nat, Y3 : nat]: ((ord_less_nat @ X3 @ Y3) => (~ ((ord_less_nat @ Y3 @ X3))))))). % less_asym
thf(fact_178_less__asym_H, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ B) => (~ ((ord_less_real @ B @ A))))))). % less_asym'
thf(fact_179_less__asym_H, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (~ ((ord_less_nat @ B @ A))))))). % less_asym'
thf(fact_180_less__trans, axiom,
    ((![X3 : real, Y3 : real, Z4 : real]: ((ord_less_real @ X3 @ Y3) => ((ord_less_real @ Y3 @ Z4) => (ord_less_real @ X3 @ Z4)))))). % less_trans
thf(fact_181_less__trans, axiom,
    ((![X3 : nat, Y3 : nat, Z4 : nat]: ((ord_less_nat @ X3 @ Y3) => ((ord_less_nat @ Y3 @ Z4) => (ord_less_nat @ X3 @ Z4)))))). % less_trans
thf(fact_182_less__linear, axiom,
    ((![X3 : real, Y3 : real]: ((ord_less_real @ X3 @ Y3) | ((X3 = Y3) | (ord_less_real @ Y3 @ X3)))))). % less_linear
thf(fact_183_less__linear, axiom,
    ((![X3 : nat, Y3 : nat]: ((ord_less_nat @ X3 @ Y3) | ((X3 = Y3) | (ord_less_nat @ Y3 @ X3)))))). % less_linear
thf(fact_184_less__irrefl, axiom,
    ((![X3 : real]: (~ ((ord_less_real @ X3 @ X3)))))). % less_irrefl
thf(fact_185_less__irrefl, axiom,
    ((![X3 : nat]: (~ ((ord_less_nat @ X3 @ X3)))))). % less_irrefl
thf(fact_186_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_187_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_188_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_189_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_190_dual__order_Oasym, axiom,
    ((![B : real, A : real]: ((ord_less_real @ B @ A) => (~ ((ord_less_real @ A @ B))))))). % dual_order.asym
thf(fact_191_dual__order_Oasym, axiom,
    ((![B : nat, A : nat]: ((ord_less_nat @ B @ A) => (~ ((ord_less_nat @ A @ B))))))). % dual_order.asym
thf(fact_192_less__imp__not__eq, axiom,
    ((![X3 : real, Y3 : real]: ((ord_less_real @ X3 @ Y3) => (~ ((X3 = Y3))))))). % less_imp_not_eq
thf(fact_193_less__imp__not__eq, axiom,
    ((![X3 : nat, Y3 : nat]: ((ord_less_nat @ X3 @ Y3) => (~ ((X3 = Y3))))))). % less_imp_not_eq
thf(fact_194_less__not__sym, axiom,
    ((![X3 : real, Y3 : real]: ((ord_less_real @ X3 @ Y3) => (~ ((ord_less_real @ Y3 @ X3))))))). % less_not_sym
thf(fact_195_less__not__sym, axiom,
    ((![X3 : nat, Y3 : nat]: ((ord_less_nat @ X3 @ Y3) => (~ ((ord_less_nat @ Y3 @ X3))))))). % less_not_sym
thf(fact_196_less__induct, axiom,
    ((![P : nat > $o, A : nat]: ((![X : nat]: ((![Y : nat]: ((ord_less_nat @ Y @ X) => (P @ Y))) => (P @ X))) => (P @ A))))). % less_induct
thf(fact_197_antisym__conv3, axiom,
    ((![Y3 : real, X3 : real]: ((~ ((ord_less_real @ Y3 @ X3))) => ((~ ((ord_less_real @ X3 @ Y3))) = (X3 = Y3)))))). % antisym_conv3
thf(fact_198_antisym__conv3, axiom,
    ((![Y3 : nat, X3 : nat]: ((~ ((ord_less_nat @ Y3 @ X3))) => ((~ ((ord_less_nat @ X3 @ Y3))) = (X3 = Y3)))))). % antisym_conv3
thf(fact_199_less__imp__not__eq2, axiom,
    ((![X3 : real, Y3 : real]: ((ord_less_real @ X3 @ Y3) => (~ ((Y3 = X3))))))). % less_imp_not_eq2
thf(fact_200_less__imp__not__eq2, axiom,
    ((![X3 : nat, Y3 : nat]: ((ord_less_nat @ X3 @ Y3) => (~ ((Y3 = X3))))))). % less_imp_not_eq2
thf(fact_201_less__imp__triv, axiom,
    ((![X3 : real, Y3 : real, P : $o]: ((ord_less_real @ X3 @ Y3) => ((ord_less_real @ Y3 @ X3) => P))))). % less_imp_triv
thf(fact_202_less__imp__triv, axiom,
    ((![X3 : nat, Y3 : nat, P : $o]: ((ord_less_nat @ X3 @ Y3) => ((ord_less_nat @ Y3 @ X3) => P))))). % less_imp_triv
thf(fact_203_linorder__cases, axiom,
    ((![X3 : real, Y3 : real]: ((~ ((ord_less_real @ X3 @ Y3))) => ((~ ((X3 = Y3))) => (ord_less_real @ Y3 @ X3)))))). % linorder_cases
thf(fact_204_linorder__cases, axiom,
    ((![X3 : nat, Y3 : nat]: ((~ ((ord_less_nat @ X3 @ Y3))) => ((~ ((X3 = Y3))) => (ord_less_nat @ Y3 @ X3)))))). % linorder_cases
thf(fact_205_dual__order_Oirrefl, axiom,
    ((![A : real]: (~ ((ord_less_real @ A @ A)))))). % dual_order.irrefl
thf(fact_206_dual__order_Oirrefl, axiom,
    ((![A : nat]: (~ ((ord_less_nat @ A @ A)))))). % dual_order.irrefl
thf(fact_207_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_208_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_209_less__imp__not__less, axiom,
    ((![X3 : real, Y3 : real]: ((ord_less_real @ X3 @ Y3) => (~ ((ord_less_real @ Y3 @ X3))))))). % less_imp_not_less
thf(fact_210_less__imp__not__less, axiom,
    ((![X3 : nat, Y3 : nat]: ((ord_less_nat @ X3 @ Y3) => (~ ((ord_less_nat @ Y3 @ X3))))))). % less_imp_not_less
thf(fact_211_exists__least__iff, axiom,
    (((^[P2 : nat > $o]: (?[X4 : nat]: (P2 @ X4))) = (^[P3 : nat > $o]: (?[N5 : nat]: (((P3 @ N5)) & ((![M3 : nat]: (((ord_less_nat @ M3 @ N5)) => ((~ ((P3 @ M3))))))))))))). % exists_least_iff
thf(fact_212_linorder__less__wlog, axiom,
    ((![P : real > real > $o, A : real, B : real]: ((![A3 : real, B3 : real]: ((ord_less_real @ A3 @ B3) => (P @ A3 @ B3))) => ((![A3 : real]: (P @ A3 @ A3)) => ((![A3 : real, B3 : real]: ((P @ B3 @ A3) => (P @ A3 @ B3))) => (P @ A @ B))))))). % linorder_less_wlog
thf(fact_213_linorder__less__wlog, axiom,
    ((![P : nat > nat > $o, A : nat, B : nat]: ((![A3 : nat, B3 : nat]: ((ord_less_nat @ A3 @ B3) => (P @ A3 @ B3))) => ((![A3 : nat]: (P @ A3 @ A3)) => ((![A3 : nat, B3 : nat]: ((P @ B3 @ A3) => (P @ A3 @ B3))) => (P @ A @ B))))))). % linorder_less_wlog
thf(fact_214_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_215_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_216_not__less__iff__gr__or__eq, axiom,
    ((![X3 : real, Y3 : real]: ((~ ((ord_less_real @ X3 @ Y3))) = (((ord_less_real @ Y3 @ X3)) | ((X3 = Y3))))))). % not_less_iff_gr_or_eq
thf(fact_217_not__less__iff__gr__or__eq, axiom,
    ((![X3 : nat, Y3 : nat]: ((~ ((ord_less_nat @ X3 @ Y3))) = (((ord_less_nat @ Y3 @ X3)) | ((X3 = Y3))))))). % not_less_iff_gr_or_eq
thf(fact_218_order_Ostrict__implies__not__eq, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ B) => (~ ((A = B))))))). % order.strict_implies_not_eq
thf(fact_219_order_Ostrict__implies__not__eq, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (~ ((A = B))))))). % order.strict_implies_not_eq
thf(fact_220_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_221_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_222_le__refl, axiom,
    ((![N3 : nat]: (ord_less_eq_nat @ N3 @ N3)))). % le_refl
thf(fact_223_le__trans, axiom,
    ((![I : nat, J : nat, K : nat]: ((ord_less_eq_nat @ I @ J) => ((ord_less_eq_nat @ J @ K) => (ord_less_eq_nat @ I @ K)))))). % le_trans
thf(fact_224_eq__imp__le, axiom,
    ((![M : nat, N3 : nat]: ((M = N3) => (ord_less_eq_nat @ M @ N3))))). % eq_imp_le
thf(fact_225_le__antisym, axiom,
    ((![M : nat, N3 : nat]: ((ord_less_eq_nat @ M @ N3) => ((ord_less_eq_nat @ N3 @ M) => (M = N3)))))). % le_antisym
thf(fact_226_nat__le__linear, axiom,
    ((![M : nat, N3 : nat]: ((ord_less_eq_nat @ M @ N3) | (ord_less_eq_nat @ N3 @ M))))). % nat_le_linear
thf(fact_227_Nat_Oex__has__greatest__nat, axiom,
    ((![P : nat > $o, K : nat, B : nat]: ((P @ K) => ((![Y5 : nat]: ((P @ Y5) => (ord_less_eq_nat @ Y5 @ B))) => (?[X : nat]: ((P @ X) & (![Y : nat]: ((P @ Y) => (ord_less_eq_nat @ Y @ X)))))))))). % Nat.ex_has_greatest_nat
thf(fact_228_less__eq__nat_Osimps_I1_J, axiom,
    ((![N3 : nat]: (ord_less_eq_nat @ zero_zero_nat @ N3)))). % less_eq_nat.simps(1)
thf(fact_229_le__0__eq, axiom,
    ((![N3 : nat]: ((ord_less_eq_nat @ N3 @ zero_zero_nat) = (N3 = zero_zero_nat))))). % le_0_eq
thf(fact_230_bot__nat__0_Oextremum__unique, axiom,
    ((![A : nat]: ((ord_less_eq_nat @ A @ zero_zero_nat) = (A = zero_zero_nat))))). % bot_nat_0.extremum_unique
thf(fact_231_bot__nat__0_Oextremum__uniqueI, axiom,
    ((![A : nat]: ((ord_less_eq_nat @ A @ zero_zero_nat) => (A = zero_zero_nat))))). % bot_nat_0.extremum_uniqueI
thf(fact_232_nat__less__le, axiom,
    ((ord_less_nat = (^[M3 : nat]: (^[N5 : nat]: (((ord_less_eq_nat @ M3 @ N5)) & ((~ ((M3 = N5)))))))))). % nat_less_le
thf(fact_233_less__imp__le__nat, axiom,
    ((![M : nat, N3 : nat]: ((ord_less_nat @ M @ N3) => (ord_less_eq_nat @ M @ N3))))). % less_imp_le_nat

% Conjectures (2)
thf(conj_0, hypothesis,
    ((![F4 : nat > nat, Z2 : complex]: ((order_769474267at_nat @ F4) => ((![E2 : real]: ((ord_less_real @ zero_zero_real @ E2) => (?[N6 : nat]: (![N4 : nat]: ((ord_less_eq_nat @ N6 @ N4) => (ord_less_real @ (real_V638595069omplex @ (minus_minus_complex @ (g @ (F4 @ N4)) @ Z2)) @ E2)))))) => thesis))))).
thf(conj_1, conjecture,
    (thesis)).
