% 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_259__5370086_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:28:40.784

% Could-be-implicit typings (4)
thf(ty_n_t__Set__Oset_It__Real__Oreal_J, type,
    set_real : $tType).
thf(ty_n_t__Complex__Ocomplex, type,
    complex : $tType).
thf(ty_n_t__Real__Oreal, type,
    real : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).

% Explicit typings (20)
thf(sy_c_Complex_Ocomplex_ORe, type,
    re : complex > real).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Complex__Ocomplex, type,
    abs_abs_complex : complex > complex).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Real__Oreal, type,
    abs_abs_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__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_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_f____, type,
    f : nat > nat).
thf(sy_v_g____, type,
    g : nat > nat).
thf(sy_v_r, type,
    r : real).
thf(sy_v_s, type,
    s : nat > complex).

% Relevant facts (157)
thf(fact_0_f_I1_J, axiom,
    ((order_769474267at_nat @ f))). % f(1)
thf(fact_1_rp, axiom,
    ((ord_less_eq_real @ zero_zero_real @ r))). % rp
thf(fact_2_real__sup__exists, axiom,
    ((![P : real > $o]: ((?[X_1 : real]: (P @ X_1)) => ((?[Z : real]: (![X : real]: ((P @ X) => (ord_less_real @ X @ Z)))) => (?[S : real]: (![Y : real]: ((?[X2 : real]: (((P @ X2)) & ((ord_less_real @ Y @ X2)))) = (ord_less_real @ Y @ S))))))))). % real_sup_exists
thf(fact_3_zero__less__abs__iff, axiom,
    ((![A : real]: ((ord_less_real @ zero_zero_real @ (abs_abs_real @ A)) = (~ ((A = zero_zero_real))))))). % zero_less_abs_iff
thf(fact_4_abs__of__nonneg, axiom,
    ((![A : real]: ((ord_less_eq_real @ zero_zero_real @ A) => ((abs_abs_real @ A) = A))))). % abs_of_nonneg
thf(fact_5_abs__le__self__iff, axiom,
    ((![A : real]: ((ord_less_eq_real @ (abs_abs_real @ A) @ A) = (ord_less_eq_real @ zero_zero_real @ A))))). % abs_le_self_iff
thf(fact_6_abs__le__zero__iff, axiom,
    ((![A : real]: ((ord_less_eq_real @ (abs_abs_real @ A) @ zero_zero_real) = (A = zero_zero_real))))). % abs_le_zero_iff
thf(fact_7_abs__0, axiom,
    (((abs_abs_real @ zero_zero_real) = zero_zero_real))). % abs_0
thf(fact_8_abs__0, axiom,
    (((abs_abs_complex @ zero_zero_complex) = zero_zero_complex))). % abs_0
thf(fact_9_abs__0__eq, axiom,
    ((![A : real]: ((zero_zero_real = (abs_abs_real @ A)) = (A = zero_zero_real))))). % abs_0_eq
thf(fact_10_abs__eq__0, axiom,
    ((![A : real]: (((abs_abs_real @ A) = zero_zero_real) = (A = zero_zero_real))))). % abs_eq_0
thf(fact_11_abs__zero, axiom,
    (((abs_abs_real @ zero_zero_real) = zero_zero_real))). % abs_zero
thf(fact_12_not__gr__zero, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr_zero
thf(fact_13_le__zero__eq, axiom,
    ((![N : nat]: ((ord_less_eq_nat @ N @ zero_zero_nat) = (N = zero_zero_nat))))). % le_zero_eq
thf(fact_14_dense__eq0__I, axiom,
    ((![X3 : real]: ((![E : real]: ((ord_less_real @ zero_zero_real @ E) => (ord_less_eq_real @ (abs_abs_real @ X3) @ E))) => (X3 = zero_zero_real))))). % dense_eq0_I
thf(fact_15_abs__of__pos, axiom,
    ((![A : real]: ((ord_less_real @ zero_zero_real @ A) => ((abs_abs_real @ A) = A))))). % abs_of_pos
thf(fact_16_g_I1_J, axiom,
    ((order_769474267at_nat @ g))). % g(1)
thf(fact_17_abs__idempotent, axiom,
    ((![A : real]: ((abs_abs_real @ (abs_abs_real @ A)) = (abs_abs_real @ A))))). % abs_idempotent
thf(fact_18_abs__abs, axiom,
    ((![A : real]: ((abs_abs_real @ (abs_abs_real @ A)) = (abs_abs_real @ A))))). % abs_abs
thf(fact_19_abs__abs, axiom,
    ((![A : complex]: ((abs_abs_complex @ (abs_abs_complex @ A)) = (abs_abs_complex @ A))))). % abs_abs
thf(fact_20_r, axiom,
    ((![N2 : nat]: (ord_less_eq_real @ (real_V638595069omplex @ (s @ N2)) @ r)))). % r
thf(fact_21_zero__reorient, axiom,
    ((![X3 : real]: ((zero_zero_real = X3) = (X3 = zero_zero_real))))). % zero_reorient
thf(fact_22_zero__reorient, axiom,
    ((![X3 : nat]: ((zero_zero_nat = X3) = (X3 = zero_zero_nat))))). % zero_reorient
thf(fact_23_zero__reorient, axiom,
    ((![X3 : complex]: ((zero_zero_complex = X3) = (X3 = zero_zero_complex))))). % zero_reorient
thf(fact_24_linorder__neqE__linordered__idom, axiom,
    ((![X3 : real, Y2 : real]: ((~ ((X3 = Y2))) => ((~ ((ord_less_real @ X3 @ Y2))) => (ord_less_real @ Y2 @ X3)))))). % linorder_neqE_linordered_idom
thf(fact_25_zero__le, axiom,
    ((![X3 : nat]: (ord_less_eq_nat @ zero_zero_nat @ X3)))). % zero_le
thf(fact_26_zero__less__iff__neq__zero, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ N) = (~ ((N = zero_zero_nat))))))). % zero_less_iff_neq_zero
thf(fact_27_gr__implies__not__zero, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not_zero
thf(fact_28_not__less__zero, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less_zero
thf(fact_29_gr__zeroI, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr_zeroI
thf(fact_30_abs__eq__0__iff, axiom,
    ((![A : real]: (((abs_abs_real @ A) = zero_zero_real) = (A = zero_zero_real))))). % abs_eq_0_iff
thf(fact_31_abs__eq__0__iff, axiom,
    ((![A : complex]: (((abs_abs_complex @ A) = zero_zero_complex) = (A = zero_zero_complex))))). % abs_eq_0_iff
thf(fact_32_abs__ge__self, axiom,
    ((![A : real]: (ord_less_eq_real @ A @ (abs_abs_real @ A))))). % abs_ge_self
thf(fact_33_abs__le__D1, axiom,
    ((![A : real, B : real]: ((ord_less_eq_real @ (abs_abs_real @ A) @ B) => (ord_less_eq_real @ A @ B))))). % abs_le_D1
thf(fact_34_abs__ge__zero, axiom,
    ((![A : real]: (ord_less_eq_real @ zero_zero_real @ (abs_abs_real @ A))))). % abs_ge_zero
thf(fact_35_abs__not__less__zero, axiom,
    ((![A : real]: (~ ((ord_less_real @ (abs_abs_real @ A) @ zero_zero_real)))))). % abs_not_less_zero
thf(fact_36__092_060open_062cmod_A_Is_A0_J_A_092_060le_062_Ar_092_060close_062, axiom,
    ((ord_less_eq_real @ (real_V638595069omplex @ (s @ zero_zero_nat)) @ r))). % \<open>cmod (s 0) \<le> r\<close>
thf(fact_37_order__refl, axiom,
    ((![X3 : real]: (ord_less_eq_real @ X3 @ X3)))). % order_refl
thf(fact_38_strict__mono__less, axiom,
    ((![F : real > real, X3 : real, Y2 : real]: ((order_1818878995l_real @ F) => ((ord_less_real @ (F @ X3) @ (F @ Y2)) = (ord_less_real @ X3 @ Y2)))))). % strict_mono_less
thf(fact_39_strict__mono__less, axiom,
    ((![F : nat > nat, X3 : nat, Y2 : nat]: ((order_769474267at_nat @ F) => ((ord_less_nat @ (F @ X3) @ (F @ Y2)) = (ord_less_nat @ X3 @ Y2)))))). % strict_mono_less
thf(fact_40_strict__mono__def, axiom,
    ((order_1818878995l_real = (^[F2 : real > real]: (![X2 : real]: (![Y3 : real]: (((ord_less_real @ X2 @ Y3)) => ((ord_less_real @ (F2 @ X2) @ (F2 @ Y3)))))))))). % strict_mono_def
thf(fact_41_strict__mono__def, axiom,
    ((order_769474267at_nat = (^[F2 : nat > nat]: (![X2 : nat]: (![Y3 : nat]: (((ord_less_nat @ X2 @ Y3)) => ((ord_less_nat @ (F2 @ X2) @ (F2 @ Y3)))))))))). % strict_mono_def
thf(fact_42_strict__monoI, axiom,
    ((![F : real > real]: ((![X : real, Y4 : real]: ((ord_less_real @ X @ Y4) => (ord_less_real @ (F @ X) @ (F @ Y4)))) => (order_1818878995l_real @ F))))). % strict_monoI
thf(fact_43_strict__monoI, axiom,
    ((![F : nat > nat]: ((![X : nat, Y4 : nat]: ((ord_less_nat @ X @ Y4) => (ord_less_nat @ (F @ X) @ (F @ Y4)))) => (order_769474267at_nat @ F))))). % strict_monoI
thf(fact_44_strict__monoD, axiom,
    ((![F : real > real, X3 : real, Y2 : real]: ((order_1818878995l_real @ F) => ((ord_less_real @ X3 @ Y2) => (ord_less_real @ (F @ X3) @ (F @ Y2))))))). % strict_monoD
thf(fact_45_strict__monoD, axiom,
    ((![F : nat > nat, X3 : nat, Y2 : nat]: ((order_769474267at_nat @ F) => ((ord_less_nat @ X3 @ Y2) => (ord_less_nat @ (F @ X3) @ (F @ Y2))))))). % strict_monoD
thf(fact_46_strict__mono__less__eq, axiom,
    ((![F : real > real, X3 : real, Y2 : real]: ((order_1818878995l_real @ F) => ((ord_less_eq_real @ (F @ X3) @ (F @ Y2)) = (ord_less_eq_real @ X3 @ Y2)))))). % strict_mono_less_eq
thf(fact_47_strict__mono__less__eq, axiom,
    ((![F : nat > nat, X3 : nat, Y2 : nat]: ((order_769474267at_nat @ F) => ((ord_less_eq_nat @ (F @ X3) @ (F @ Y2)) = (ord_less_eq_nat @ X3 @ Y2)))))). % strict_mono_less_eq
thf(fact_48_strict__mono__leD, axiom,
    ((![R : real > real, M : real, N : real]: ((order_1818878995l_real @ R) => ((ord_less_eq_real @ M @ N) => (ord_less_eq_real @ (R @ M) @ (R @ N))))))). % strict_mono_leD
thf(fact_49_strict__mono__leD, axiom,
    ((![R : nat > nat, M : nat, N : nat]: ((order_769474267at_nat @ R) => ((ord_less_eq_nat @ M @ N) => (ord_less_eq_nat @ (R @ M) @ (R @ N))))))). % strict_mono_leD
thf(fact_50_zero__complex_Osimps_I1_J, axiom,
    (((re @ zero_zero_complex) = zero_zero_real))). % zero_complex.simps(1)
thf(fact_51_less__eq__real__def, axiom,
    ((ord_less_eq_real = (^[X2 : real]: (^[Y3 : real]: (((ord_less_real @ X2 @ Y3)) | ((X2 = Y3)))))))). % less_eq_real_def
thf(fact_52_complex__Re__le__cmod, axiom,
    ((![X3 : complex]: (ord_less_eq_real @ (re @ X3) @ (real_V638595069omplex @ X3))))). % complex_Re_le_cmod
thf(fact_53_abs__Re__le__cmod, axiom,
    ((![X3 : complex]: (ord_less_eq_real @ (abs_abs_real @ (re @ X3)) @ (real_V638595069omplex @ X3))))). % abs_Re_le_cmod
thf(fact_54_mem__Collect__eq, axiom,
    ((![A : real, P : real > $o]: ((member_real @ A @ (collect_real @ P)) = (P @ A))))). % mem_Collect_eq
thf(fact_55_Collect__mem__eq, axiom,
    ((![A2 : set_real]: ((collect_real @ (^[X2 : real]: (member_real @ X2 @ A2))) = A2)))). % Collect_mem_eq
thf(fact_56_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) => ((![X : real, Y4 : real]: ((ord_less_eq_real @ X @ Y4) => (ord_less_eq_real @ (F @ X) @ (F @ Y4)))) => (ord_less_eq_real @ A @ (F @ C)))))))). % order_subst1
thf(fact_57_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) => ((![X : real, Y4 : real]: ((ord_less_eq_real @ X @ Y4) => (ord_less_eq_real @ (F @ X) @ (F @ Y4)))) => (ord_less_eq_real @ (F @ A) @ C))))))). % order_subst2
thf(fact_58_ord__eq__le__subst, axiom,
    ((![A : real, F : real > real, B : real, C : real]: ((A = (F @ B)) => ((ord_less_eq_real @ B @ C) => ((![X : real, Y4 : real]: ((ord_less_eq_real @ X @ Y4) => (ord_less_eq_real @ (F @ X) @ (F @ Y4)))) => (ord_less_eq_real @ A @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_59_ord__le__eq__subst, axiom,
    ((![A : real, B : real, F : real > real, C : real]: ((ord_less_eq_real @ A @ B) => (((F @ B) = C) => ((![X : real, Y4 : real]: ((ord_less_eq_real @ X @ Y4) => (ord_less_eq_real @ (F @ X) @ (F @ Y4)))) => (ord_less_eq_real @ (F @ A) @ C))))))). % ord_le_eq_subst
thf(fact_60_eq__iff, axiom,
    (((^[Y5 : real]: (^[Z2 : real]: (Y5 = Z2))) = (^[X2 : real]: (^[Y3 : real]: (((ord_less_eq_real @ X2 @ Y3)) & ((ord_less_eq_real @ Y3 @ X2)))))))). % eq_iff
thf(fact_61_antisym, axiom,
    ((![X3 : real, Y2 : real]: ((ord_less_eq_real @ X3 @ Y2) => ((ord_less_eq_real @ Y2 @ X3) => (X3 = Y2)))))). % antisym
thf(fact_62_linear, axiom,
    ((![X3 : real, Y2 : real]: ((ord_less_eq_real @ X3 @ Y2) | (ord_less_eq_real @ Y2 @ X3))))). % linear
thf(fact_63_eq__refl, axiom,
    ((![X3 : real, Y2 : real]: ((X3 = Y2) => (ord_less_eq_real @ X3 @ Y2))))). % eq_refl
thf(fact_64_le__cases, axiom,
    ((![X3 : real, Y2 : real]: ((~ ((ord_less_eq_real @ X3 @ Y2))) => (ord_less_eq_real @ Y2 @ X3))))). % le_cases
thf(fact_65_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_66_le__cases3, axiom,
    ((![X3 : real, Y2 : real, Z3 : real]: (((ord_less_eq_real @ X3 @ Y2) => (~ ((ord_less_eq_real @ Y2 @ Z3)))) => (((ord_less_eq_real @ Y2 @ X3) => (~ ((ord_less_eq_real @ X3 @ Z3)))) => (((ord_less_eq_real @ X3 @ Z3) => (~ ((ord_less_eq_real @ Z3 @ Y2)))) => (((ord_less_eq_real @ Z3 @ Y2) => (~ ((ord_less_eq_real @ Y2 @ X3)))) => (((ord_less_eq_real @ Y2 @ Z3) => (~ ((ord_less_eq_real @ Z3 @ X3)))) => (~ (((ord_less_eq_real @ Z3 @ X3) => (~ ((ord_less_eq_real @ X3 @ Y2)))))))))))))). % le_cases3
thf(fact_67_antisym__conv, axiom,
    ((![Y2 : real, X3 : real]: ((ord_less_eq_real @ Y2 @ X3) => ((ord_less_eq_real @ X3 @ Y2) = (X3 = Y2)))))). % antisym_conv
thf(fact_68_order__class_Oorder_Oeq__iff, axiom,
    (((^[Y5 : real]: (^[Z2 : real]: (Y5 = Z2))) = (^[A3 : real]: (^[B2 : real]: (((ord_less_eq_real @ A3 @ B2)) & ((ord_less_eq_real @ B2 @ A3)))))))). % order_class.order.eq_iff
thf(fact_69_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_70_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_71_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_72_order__trans, axiom,
    ((![X3 : real, Y2 : real, Z3 : real]: ((ord_less_eq_real @ X3 @ Y2) => ((ord_less_eq_real @ Y2 @ Z3) => (ord_less_eq_real @ X3 @ Z3)))))). % order_trans
thf(fact_73_dual__order_Orefl, axiom,
    ((![A : real]: (ord_less_eq_real @ A @ A)))). % dual_order.refl
thf(fact_74_linorder__wlog, axiom,
    ((![P : real > real > $o, A : real, B : real]: ((![A4 : real, B3 : real]: ((ord_less_eq_real @ A4 @ B3) => (P @ A4 @ B3))) => ((![A4 : real, B3 : real]: ((P @ B3 @ A4) => (P @ A4 @ B3))) => (P @ A @ B)))))). % linorder_wlog
thf(fact_75_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_76_dual__order_Oeq__iff, axiom,
    (((^[Y5 : real]: (^[Z2 : real]: (Y5 = Z2))) = (^[A3 : real]: (^[B2 : real]: (((ord_less_eq_real @ B2 @ A3)) & ((ord_less_eq_real @ A3 @ B2)))))))). % dual_order.eq_iff
thf(fact_77_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_78_ord__eq__less__subst, axiom,
    ((![A : real, F : real > real, B : real, C : real]: ((A = (F @ B)) => ((ord_less_real @ B @ C) => ((![X : real, Y4 : real]: ((ord_less_real @ X @ Y4) => (ord_less_real @ (F @ X) @ (F @ Y4)))) => (ord_less_real @ A @ (F @ C)))))))). % ord_eq_less_subst
thf(fact_79_ord__less__eq__subst, axiom,
    ((![A : real, B : real, F : real > real, C : real]: ((ord_less_real @ A @ B) => (((F @ B) = C) => ((![X : real, Y4 : real]: ((ord_less_real @ X @ Y4) => (ord_less_real @ (F @ X) @ (F @ Y4)))) => (ord_less_real @ (F @ A) @ C))))))). % ord_less_eq_subst
thf(fact_80_order__less__subst1, axiom,
    ((![A : real, F : real > real, B : real, C : real]: ((ord_less_real @ A @ (F @ B)) => ((ord_less_real @ B @ C) => ((![X : real, Y4 : real]: ((ord_less_real @ X @ Y4) => (ord_less_real @ (F @ X) @ (F @ Y4)))) => (ord_less_real @ A @ (F @ C)))))))). % order_less_subst1
thf(fact_81_order__less__subst2, axiom,
    ((![A : real, B : real, F : real > real, C : real]: ((ord_less_real @ A @ B) => ((ord_less_real @ (F @ B) @ C) => ((![X : real, Y4 : real]: ((ord_less_real @ X @ Y4) => (ord_less_real @ (F @ X) @ (F @ Y4)))) => (ord_less_real @ (F @ A) @ C))))))). % order_less_subst2
thf(fact_82_lt__ex, axiom,
    ((![X3 : real]: (?[Y4 : real]: (ord_less_real @ Y4 @ X3))))). % lt_ex
thf(fact_83_gt__ex, axiom,
    ((![X3 : real]: (?[X_12 : real]: (ord_less_real @ X3 @ X_12))))). % gt_ex
thf(fact_84_neqE, axiom,
    ((![X3 : real, Y2 : real]: ((~ ((X3 = Y2))) => ((~ ((ord_less_real @ X3 @ Y2))) => (ord_less_real @ Y2 @ X3)))))). % neqE
thf(fact_85_neq__iff, axiom,
    ((![X3 : real, Y2 : real]: ((~ ((X3 = Y2))) = (((ord_less_real @ X3 @ Y2)) | ((ord_less_real @ Y2 @ X3))))))). % neq_iff
thf(fact_86_order_Oasym, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ B) => (~ ((ord_less_real @ B @ A))))))). % order.asym
thf(fact_87_dense, axiom,
    ((![X3 : real, Y2 : real]: ((ord_less_real @ X3 @ Y2) => (?[Z4 : real]: ((ord_less_real @ X3 @ Z4) & (ord_less_real @ Z4 @ Y2))))))). % dense
thf(fact_88_less__imp__neq, axiom,
    ((![X3 : real, Y2 : real]: ((ord_less_real @ X3 @ Y2) => (~ ((X3 = Y2))))))). % less_imp_neq
thf(fact_89_less__asym, axiom,
    ((![X3 : real, Y2 : real]: ((ord_less_real @ X3 @ Y2) => (~ ((ord_less_real @ Y2 @ X3))))))). % less_asym
thf(fact_90_less__asym_H, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ B) => (~ ((ord_less_real @ B @ A))))))). % less_asym'
thf(fact_91_less__trans, axiom,
    ((![X3 : real, Y2 : real, Z3 : real]: ((ord_less_real @ X3 @ Y2) => ((ord_less_real @ Y2 @ Z3) => (ord_less_real @ X3 @ Z3)))))). % less_trans
thf(fact_92_less__linear, axiom,
    ((![X3 : real, Y2 : real]: ((ord_less_real @ X3 @ Y2) | ((X3 = Y2) | (ord_less_real @ Y2 @ X3)))))). % less_linear
thf(fact_93_less__irrefl, axiom,
    ((![X3 : real]: (~ ((ord_less_real @ X3 @ X3)))))). % less_irrefl
thf(fact_94_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_95_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_96_dual__order_Oasym, axiom,
    ((![B : real, A : real]: ((ord_less_real @ B @ A) => (~ ((ord_less_real @ A @ B))))))). % dual_order.asym
thf(fact_97_less__imp__not__eq, axiom,
    ((![X3 : real, Y2 : real]: ((ord_less_real @ X3 @ Y2) => (~ ((X3 = Y2))))))). % less_imp_not_eq
thf(fact_98_less__not__sym, axiom,
    ((![X3 : real, Y2 : real]: ((ord_less_real @ X3 @ Y2) => (~ ((ord_less_real @ Y2 @ X3))))))). % less_not_sym
thf(fact_99_antisym__conv3, axiom,
    ((![Y2 : real, X3 : real]: ((~ ((ord_less_real @ Y2 @ X3))) => ((~ ((ord_less_real @ X3 @ Y2))) = (X3 = Y2)))))). % antisym_conv3
thf(fact_100_less__imp__not__eq2, axiom,
    ((![X3 : real, Y2 : real]: ((ord_less_real @ X3 @ Y2) => (~ ((Y2 = X3))))))). % less_imp_not_eq2
thf(fact_101_less__imp__triv, axiom,
    ((![X3 : real, Y2 : real, P : $o]: ((ord_less_real @ X3 @ Y2) => ((ord_less_real @ Y2 @ X3) => P))))). % less_imp_triv
thf(fact_102_linorder__cases, axiom,
    ((![X3 : real, Y2 : real]: ((~ ((ord_less_real @ X3 @ Y2))) => ((~ ((X3 = Y2))) => (ord_less_real @ Y2 @ X3)))))). % linorder_cases
thf(fact_103_dual__order_Oirrefl, axiom,
    ((![A : real]: (~ ((ord_less_real @ A @ A)))))). % dual_order.irrefl
thf(fact_104_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_105_less__imp__not__less, axiom,
    ((![X3 : real, Y2 : real]: ((ord_less_real @ X3 @ Y2) => (~ ((ord_less_real @ Y2 @ X3))))))). % less_imp_not_less
thf(fact_106_linorder__less__wlog, axiom,
    ((![P : real > real > $o, A : real, B : real]: ((![A4 : real, B3 : real]: ((ord_less_real @ A4 @ B3) => (P @ A4 @ B3))) => ((![A4 : real]: (P @ A4 @ A4)) => ((![A4 : real, B3 : real]: ((P @ B3 @ A4) => (P @ A4 @ B3))) => (P @ A @ B))))))). % linorder_less_wlog
thf(fact_107_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_108_not__less__iff__gr__or__eq, axiom,
    ((![X3 : real, Y2 : real]: ((~ ((ord_less_real @ X3 @ Y2))) = (((ord_less_real @ Y2 @ X3)) | ((X3 = Y2))))))). % not_less_iff_gr_or_eq
thf(fact_109_order_Ostrict__implies__not__eq, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ B) => (~ ((A = B))))))). % order.strict_implies_not_eq
thf(fact_110_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_111_complete__real, axiom,
    ((![S2 : set_real]: ((?[X4 : real]: (member_real @ X4 @ S2)) => ((?[Z : real]: (![X : real]: ((member_real @ X @ S2) => (ord_less_eq_real @ X @ Z)))) => (?[Y4 : real]: ((![X4 : real]: ((member_real @ X4 @ S2) => (ord_less_eq_real @ X4 @ Y4))) & (![Z : real]: ((![X : real]: ((member_real @ X @ S2) => (ord_less_eq_real @ X @ Z))) => (ord_less_eq_real @ Y4 @ Z)))))))))). % complete_real
thf(fact_112_strict__mono__eq, axiom,
    ((![F : nat > nat, X3 : nat, Y2 : nat]: ((order_769474267at_nat @ F) => (((F @ X3) = (F @ Y2)) = (X3 = Y2)))))). % strict_mono_eq
thf(fact_113_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_114_leD, axiom,
    ((![Y2 : real, X3 : real]: ((ord_less_eq_real @ Y2 @ X3) => (~ ((ord_less_real @ X3 @ Y2))))))). % leD
thf(fact_115_leI, axiom,
    ((![X3 : real, Y2 : real]: ((~ ((ord_less_real @ X3 @ Y2))) => (ord_less_eq_real @ Y2 @ X3))))). % leI
thf(fact_116_le__less, axiom,
    ((ord_less_eq_real = (^[X2 : real]: (^[Y3 : real]: (((ord_less_real @ X2 @ Y3)) | ((X2 = Y3)))))))). % le_less
thf(fact_117_less__le, axiom,
    ((ord_less_real = (^[X2 : real]: (^[Y3 : real]: (((ord_less_eq_real @ X2 @ Y3)) & ((~ ((X2 = Y3)))))))))). % less_le
thf(fact_118_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) => ((![X : real, Y4 : real]: ((ord_less_real @ X @ Y4) => (ord_less_real @ (F @ X) @ (F @ Y4)))) => (ord_less_real @ A @ (F @ C)))))))). % order_le_less_subst1
thf(fact_119_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) => ((![X : real, Y4 : real]: ((ord_less_eq_real @ X @ Y4) => (ord_less_eq_real @ (F @ X) @ (F @ Y4)))) => (ord_less_real @ (F @ A) @ C))))))). % order_le_less_subst2
thf(fact_120_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) => ((![X : real, Y4 : real]: ((ord_less_eq_real @ X @ Y4) => (ord_less_eq_real @ (F @ X) @ (F @ Y4)))) => (ord_less_real @ A @ (F @ C)))))))). % order_less_le_subst1
thf(fact_121_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) => ((![X : real, Y4 : real]: ((ord_less_real @ X @ Y4) => (ord_less_real @ (F @ X) @ (F @ Y4)))) => (ord_less_real @ (F @ A) @ C))))))). % order_less_le_subst2
thf(fact_122_not__le, axiom,
    ((![X3 : real, Y2 : real]: ((~ ((ord_less_eq_real @ X3 @ Y2))) = (ord_less_real @ Y2 @ X3))))). % not_le
thf(fact_123_not__less, axiom,
    ((![X3 : real, Y2 : real]: ((~ ((ord_less_real @ X3 @ Y2))) = (ord_less_eq_real @ Y2 @ X3))))). % not_less
thf(fact_124_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_125_antisym__conv1, axiom,
    ((![X3 : real, Y2 : real]: ((~ ((ord_less_real @ X3 @ Y2))) => ((ord_less_eq_real @ X3 @ Y2) = (X3 = Y2)))))). % antisym_conv1
thf(fact_126_antisym__conv2, axiom,
    ((![X3 : real, Y2 : real]: ((ord_less_eq_real @ X3 @ Y2) => ((~ ((ord_less_real @ X3 @ Y2))) = (X3 = Y2)))))). % antisym_conv2
thf(fact_127_less__imp__le, axiom,
    ((![X3 : real, Y2 : real]: ((ord_less_real @ X3 @ Y2) => (ord_less_eq_real @ X3 @ Y2))))). % less_imp_le
thf(fact_128_le__less__trans, axiom,
    ((![X3 : real, Y2 : real, Z3 : real]: ((ord_less_eq_real @ X3 @ Y2) => ((ord_less_real @ Y2 @ Z3) => (ord_less_real @ X3 @ Z3)))))). % le_less_trans
thf(fact_129_less__le__trans, axiom,
    ((![X3 : real, Y2 : real, Z3 : real]: ((ord_less_real @ X3 @ Y2) => ((ord_less_eq_real @ Y2 @ Z3) => (ord_less_real @ X3 @ Z3)))))). % less_le_trans
thf(fact_130_dense__ge, axiom,
    ((![Z3 : real, Y2 : real]: ((![X : real]: ((ord_less_real @ Z3 @ X) => (ord_less_eq_real @ Y2 @ X))) => (ord_less_eq_real @ Y2 @ Z3))))). % dense_ge
thf(fact_131_dense__le, axiom,
    ((![Y2 : real, Z3 : real]: ((![X : real]: ((ord_less_real @ X @ Y2) => (ord_less_eq_real @ X @ Z3))) => (ord_less_eq_real @ Y2 @ Z3))))). % dense_le
thf(fact_132_le__less__linear, axiom,
    ((![X3 : real, Y2 : real]: ((ord_less_eq_real @ X3 @ Y2) | (ord_less_real @ Y2 @ X3))))). % le_less_linear
thf(fact_133_le__imp__less__or__eq, axiom,
    ((![X3 : real, Y2 : real]: ((ord_less_eq_real @ X3 @ Y2) => ((ord_less_real @ X3 @ Y2) | (X3 = Y2)))))). % le_imp_less_or_eq
thf(fact_134_less__le__not__le, axiom,
    ((ord_less_real = (^[X2 : real]: (^[Y3 : real]: (((ord_less_eq_real @ X2 @ Y3)) & ((~ ((ord_less_eq_real @ Y3 @ X2)))))))))). % less_le_not_le
thf(fact_135_not__le__imp__less, axiom,
    ((![Y2 : real, X3 : real]: ((~ ((ord_less_eq_real @ Y2 @ X3))) => (ord_less_real @ X3 @ Y2))))). % not_le_imp_less
thf(fact_136_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_137_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_138_order_Oorder__iff__strict, axiom,
    ((ord_less_eq_real = (^[A3 : real]: (^[B2 : real]: (((ord_less_real @ A3 @ B2)) | ((A3 = B2)))))))). % order.order_iff_strict
thf(fact_139_order_Ostrict__iff__order, axiom,
    ((ord_less_real = (^[A3 : real]: (^[B2 : real]: (((ord_less_eq_real @ A3 @ B2)) & ((~ ((A3 = B2)))))))))). % order.strict_iff_order
thf(fact_140_dual__order_Ostrict__trans1, axiom,
    ((![B : real, A : real, C : real]: ((ord_less_eq_real @ B @ A) => ((ord_less_real @ C @ B) => (ord_less_real @ C @ A)))))). % dual_order.strict_trans1
thf(fact_141_dual__order_Ostrict__trans2, axiom,
    ((![B : real, A : real, C : real]: ((ord_less_real @ B @ A) => ((ord_less_eq_real @ C @ B) => (ord_less_real @ C @ A)))))). % dual_order.strict_trans2
thf(fact_142_dense__ge__bounded, axiom,
    ((![Z3 : real, X3 : real, Y2 : real]: ((ord_less_real @ Z3 @ X3) => ((![W : real]: ((ord_less_real @ Z3 @ W) => ((ord_less_real @ W @ X3) => (ord_less_eq_real @ Y2 @ W)))) => (ord_less_eq_real @ Y2 @ Z3)))))). % dense_ge_bounded
thf(fact_143_dense__le__bounded, axiom,
    ((![X3 : real, Y2 : real, Z3 : real]: ((ord_less_real @ X3 @ Y2) => ((![W : real]: ((ord_less_real @ X3 @ W) => ((ord_less_real @ W @ Y2) => (ord_less_eq_real @ W @ Z3)))) => (ord_less_eq_real @ Y2 @ Z3)))))). % dense_le_bounded
thf(fact_144_order_Ostrict__implies__order, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ B) => (ord_less_eq_real @ A @ B))))). % order.strict_implies_order
thf(fact_145_dual__order_Oorder__iff__strict, axiom,
    ((ord_less_eq_real = (^[B2 : real]: (^[A3 : real]: (((ord_less_real @ B2 @ A3)) | ((A3 = B2)))))))). % dual_order.order_iff_strict
thf(fact_146_dual__order_Ostrict__iff__order, axiom,
    ((ord_less_real = (^[B2 : real]: (^[A3 : real]: (((ord_less_eq_real @ B2 @ A3)) & ((~ ((A3 = B2)))))))))). % dual_order.strict_iff_order
thf(fact_147_dual__order_Ostrict__implies__order, axiom,
    ((![B : real, A : real]: ((ord_less_real @ B @ A) => (ord_less_eq_real @ B @ A))))). % dual_order.strict_implies_order
thf(fact_148_order_Onot__eq__order__implies__strict, axiom,
    ((![A : real, B : real]: ((~ ((A = B))) => ((ord_less_eq_real @ A @ B) => (ord_less_real @ A @ B)))))). % order.not_eq_order_implies_strict
thf(fact_149_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_150_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_151_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_152_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_153_norm__zero, axiom,
    (((real_V646646907m_real @ zero_zero_real) = zero_zero_real))). % norm_zero
thf(fact_154_norm__zero, axiom,
    (((real_V638595069omplex @ zero_zero_complex) = zero_zero_real))). % norm_zero
thf(fact_155_norm__eq__zero, axiom,
    ((![X3 : real]: (((real_V646646907m_real @ X3) = zero_zero_real) = (X3 = zero_zero_real))))). % norm_eq_zero
thf(fact_156_norm__eq__zero, axiom,
    ((![X3 : complex]: (((real_V638595069omplex @ X3) = zero_zero_real) = (X3 = zero_zero_complex))))). % norm_eq_zero

% Conjectures (1)
thf(conj_0, conjecture,
    ((?[K : real]: ((ord_less_real @ zero_zero_real @ K) & (![N3 : nat]: (ord_less_eq_real @ (abs_abs_real @ (re @ (s @ (f @ N3)))) @ K)))))).
