% 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_298__5370464_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:29:01.572

% 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 (24)
thf(sy_c_Complex_Ocomplex_OComplex, type,
    complex2 : real > real > complex).
thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat, type,
    comp_nat_nat_nat : (nat > nat) > (nat > nat) > nat > nat).
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__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_e____, type,
    e : real).
thf(sy_v_f____, type,
    f : nat > nat).
thf(sy_v_g____, type,
    g : nat > nat).
thf(sy_v_n____, type,
    n : nat).
thf(sy_v_r, type,
    r : real).
thf(sy_v_s, type,
    s : nat > complex).
thf(sy_v_x____, type,
    x : real).
thf(sy_v_y____, type,
    y : real).

% Relevant facts (164)
thf(fact_0_f_I1_J, axiom,
    ((order_769474267at_nat @ f))). % f(1)
thf(fact_1_g_I1_J, axiom,
    ((order_769474267at_nat @ g))). % g(1)
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_hs, axiom,
    ((order_769474267at_nat @ (comp_nat_nat_nat @ f @ g)))). % hs
thf(fact_4_complex_Oinject, axiom,
    ((![X1 : real, X22 : real, Y1 : real, Y2 : real]: (((complex2 @ X1 @ X22) = (complex2 @ Y1 @ Y2)) = (((X1 = Y1)) & ((X22 = Y2))))))). % complex.inject
thf(fact_5_r, axiom,
    ((![N : nat]: (ord_less_eq_real @ (real_V638595069omplex @ (s @ N)) @ r)))). % r
thf(fact_6_comp__apply, axiom,
    ((comp_nat_nat_nat = (^[F : nat > nat]: (^[G : nat > nat]: (^[X2 : nat]: (F @ (G @ X2)))))))). % comp_apply
thf(fact_7_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_8_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_9_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_10_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_11_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_12_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_13_complex_Oexhaust, axiom,
    ((![Y3 : complex]: (~ ((![X12 : real, X23 : real]: (~ ((Y3 = (complex2 @ X12 @ X23)))))))))). % complex.exhaust
thf(fact_14_comp__def, axiom,
    ((comp_nat_nat_nat = (^[F : nat > nat]: (^[G : nat > nat]: (^[X2 : nat]: (F @ (G @ X2)))))))). % comp_def
thf(fact_15_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_16_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_17_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_18_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_19_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_20_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_21_complex__diff, axiom,
    ((![A : real, B : real, C : real, D : real]: ((minus_minus_complex @ (complex2 @ A @ B) @ (complex2 @ C @ D)) = (complex2 @ (minus_minus_real @ A @ C) @ (minus_minus_real @ B @ D)))))). % complex_diff
thf(fact_22_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_23_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_24_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_25_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_26_comp__eq__dest__lhs, axiom,
    ((![A : nat > nat, B : nat > nat, C : nat > nat, V : nat]: (((comp_nat_nat_nat @ A @ B) = C) => ((A @ (B @ V)) = (C @ V)))))). % comp_eq_dest_lhs
thf(fact_27_comp__eq__elim, axiom,
    ((![A : nat > nat, B : nat > nat, C : nat > nat, D : nat > nat]: (((comp_nat_nat_nat @ A @ B) = (comp_nat_nat_nat @ C @ D)) => (![V2 : nat]: ((A @ (B @ V2)) = (C @ (D @ V2)))))))). % comp_eq_elim
thf(fact_28_comp__eq__dest, axiom,
    ((![A : nat > nat, B : nat > nat, C : nat > nat, D : nat > nat, V : nat]: (((comp_nat_nat_nat @ A @ B) = (comp_nat_nat_nat @ C @ D)) => ((A @ (B @ V)) = (C @ (D @ V))))))). % comp_eq_dest
thf(fact_29_comp__assoc, axiom,
    ((![F2 : nat > nat, G2 : nat > nat, H : nat > nat]: ((comp_nat_nat_nat @ (comp_nat_nat_nat @ F2 @ G2) @ H) = (comp_nat_nat_nat @ F2 @ (comp_nat_nat_nat @ G2 @ H)))))). % comp_assoc
thf(fact_30__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_31_rp, axiom,
    ((ord_less_eq_real @ zero_zero_real @ r))). % rp
thf(fact_32_order__refl, axiom,
    ((![X3 : real]: (ord_less_eq_real @ X3 @ X3)))). % order_refl
thf(fact_33_strict__mono__o, axiom,
    ((![R : nat > nat, S2 : nat > nat]: ((order_769474267at_nat @ R) => ((order_769474267at_nat @ S2) => (order_769474267at_nat @ (comp_nat_nat_nat @ R @ S2))))))). % strict_mono_o
thf(fact_34_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_35_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_36_strict__mono__def, axiom,
    ((order_1818878995l_real = (^[F : real > real]: (![X2 : real]: (![Y4 : real]: (((ord_less_real @ X2 @ Y4)) => ((ord_less_real @ (F @ X2) @ (F @ Y4)))))))))). % strict_mono_def
thf(fact_37_strict__mono__def, axiom,
    ((order_769474267at_nat = (^[F : nat > nat]: (![X2 : nat]: (![Y4 : nat]: (((ord_less_nat @ X2 @ Y4)) => ((ord_less_nat @ (F @ X2) @ (F @ Y4)))))))))). % strict_mono_def
thf(fact_38_strict__monoI, axiom,
    ((![F2 : real > real]: ((![X : real, Y5 : real]: ((ord_less_real @ X @ Y5) => (ord_less_real @ (F2 @ X) @ (F2 @ Y5)))) => (order_1818878995l_real @ F2))))). % strict_monoI
thf(fact_39_strict__monoI, axiom,
    ((![F2 : nat > nat]: ((![X : nat, Y5 : nat]: ((ord_less_nat @ X @ Y5) => (ord_less_nat @ (F2 @ X) @ (F2 @ Y5)))) => (order_769474267at_nat @ F2))))). % strict_monoI
thf(fact_40_strict__monoD, axiom,
    ((![F2 : real > real, X3 : real, Y3 : real]: ((order_1818878995l_real @ F2) => ((ord_less_real @ X3 @ Y3) => (ord_less_real @ (F2 @ X3) @ (F2 @ Y3))))))). % strict_monoD
thf(fact_41_strict__monoD, axiom,
    ((![F2 : nat > nat, X3 : nat, Y3 : nat]: ((order_769474267at_nat @ F2) => ((ord_less_nat @ X3 @ Y3) => (ord_less_nat @ (F2 @ X3) @ (F2 @ Y3))))))). % strict_monoD
thf(fact_42_strict__mono__leD, axiom,
    ((![R : real > real, M : real, N2 : real]: ((order_1818878995l_real @ R) => ((ord_less_eq_real @ M @ N2) => (ord_less_eq_real @ (R @ M) @ (R @ N2))))))). % strict_mono_leD
thf(fact_43_strict__mono__leD, axiom,
    ((![R : nat > nat, M : nat, N2 : nat]: ((order_769474267at_nat @ R) => ((ord_less_eq_nat @ M @ N2) => (ord_less_eq_nat @ (R @ M) @ (R @ N2))))))). % strict_mono_leD
thf(fact_44_strict__mono__less__eq, axiom,
    ((![F2 : real > real, X3 : real, Y3 : real]: ((order_1818878995l_real @ F2) => ((ord_less_eq_real @ (F2 @ X3) @ (F2 @ Y3)) = (ord_less_eq_real @ X3 @ Y3)))))). % strict_mono_less_eq
thf(fact_45_strict__mono__less__eq, axiom,
    ((![F2 : nat > nat, X3 : nat, Y3 : nat]: ((order_769474267at_nat @ F2) => ((ord_less_eq_nat @ (F2 @ X3) @ (F2 @ Y3)) = (ord_less_eq_nat @ X3 @ Y3)))))). % strict_mono_less_eq
thf(fact_46_le__zero__eq, axiom,
    ((![N2 : nat]: ((ord_less_eq_nat @ N2 @ zero_zero_nat) = (N2 = zero_zero_nat))))). % le_zero_eq
thf(fact_47_not__gr__zero, axiom,
    ((![N2 : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N2))) = (N2 = zero_zero_nat))))). % not_gr_zero
thf(fact_48_diff__self, axiom,
    ((![A : complex]: ((minus_minus_complex @ A @ A) = zero_zero_complex)))). % diff_self
thf(fact_49_diff__self, axiom,
    ((![A : real]: ((minus_minus_real @ A @ A) = zero_zero_real)))). % diff_self
thf(fact_50_diff__0__right, axiom,
    ((![A : complex]: ((minus_minus_complex @ A @ zero_zero_complex) = A)))). % diff_0_right
thf(fact_51_diff__0__right, axiom,
    ((![A : real]: ((minus_minus_real @ A @ zero_zero_real) = A)))). % diff_0_right
thf(fact_52_zero__diff, axiom,
    ((![A : nat]: ((minus_minus_nat @ zero_zero_nat @ A) = zero_zero_nat)))). % zero_diff
thf(fact_53_diff__zero, axiom,
    ((![A : nat]: ((minus_minus_nat @ A @ zero_zero_nat) = A)))). % diff_zero
thf(fact_54_diff__zero, axiom,
    ((![A : complex]: ((minus_minus_complex @ A @ zero_zero_complex) = A)))). % diff_zero
thf(fact_55_diff__zero, axiom,
    ((![A : real]: ((minus_minus_real @ A @ zero_zero_real) = A)))). % diff_zero
thf(fact_56_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_57_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_58_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_59_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_60_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_61_norm__eq__zero, axiom,
    ((![X3 : complex]: (((real_V638595069omplex @ X3) = zero_zero_real) = (X3 = zero_zero_complex))))). % norm_eq_zero
thf(fact_62_norm__eq__zero, axiom,
    ((![X3 : real]: (((real_V646646907m_real @ X3) = zero_zero_real) = (X3 = zero_zero_real))))). % norm_eq_zero
thf(fact_63_norm__zero, axiom,
    (((real_V638595069omplex @ zero_zero_complex) = zero_zero_real))). % norm_zero
thf(fact_64_norm__zero, axiom,
    (((real_V646646907m_real @ zero_zero_real) = zero_zero_real))). % norm_zero
thf(fact_65_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_66_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_67_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_68_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_69_zero__reorient, axiom,
    ((![X3 : nat]: ((zero_zero_nat = X3) = (X3 = zero_zero_nat))))). % zero_reorient
thf(fact_70_zero__reorient, axiom,
    ((![X3 : real]: ((zero_zero_real = X3) = (X3 = zero_zero_real))))). % zero_reorient
thf(fact_71_zero__le, axiom,
    ((![X3 : nat]: (ord_less_eq_nat @ zero_zero_nat @ X3)))). % zero_le
thf(fact_72_gr__zeroI, axiom,
    ((![N2 : nat]: ((~ ((N2 = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N2))))). % gr_zeroI
thf(fact_73_not__less__zero, axiom,
    ((![N2 : nat]: (~ ((ord_less_nat @ N2 @ zero_zero_nat)))))). % not_less_zero
thf(fact_74_gr__implies__not__zero, axiom,
    ((![M : nat, N2 : nat]: ((ord_less_nat @ M @ N2) => (~ ((N2 = zero_zero_nat))))))). % gr_implies_not_zero
thf(fact_75_zero__less__iff__neq__zero, axiom,
    ((![N2 : nat]: ((ord_less_nat @ zero_zero_nat @ N2) = (~ ((N2 = zero_zero_nat))))))). % zero_less_iff_neq_zero
thf(fact_76_eq__iff__diff__eq__0, axiom,
    (((^[Y6 : complex]: (^[Z2 : complex]: (Y6 = Z2))) = (^[A2 : complex]: (^[B2 : complex]: ((minus_minus_complex @ A2 @ B2) = zero_zero_complex)))))). % eq_iff_diff_eq_0
thf(fact_77_eq__iff__diff__eq__0, axiom,
    (((^[Y6 : real]: (^[Z2 : real]: (Y6 = Z2))) = (^[A2 : real]: (^[B2 : real]: ((minus_minus_real @ A2 @ B2) = zero_zero_real)))))). % eq_iff_diff_eq_0
thf(fact_78_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_79_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_80_norm__not__less__zero, axiom,
    ((![X3 : complex]: (~ ((ord_less_real @ (real_V638595069omplex @ X3) @ zero_zero_real)))))). % norm_not_less_zero
thf(fact_81_norm__not__less__zero, axiom,
    ((![X3 : real]: (~ ((ord_less_real @ (real_V646646907m_real @ X3) @ zero_zero_real)))))). % norm_not_less_zero
thf(fact_82_norm__ge__zero, axiom,
    ((![X3 : complex]: (ord_less_eq_real @ zero_zero_real @ (real_V638595069omplex @ X3))))). % norm_ge_zero
thf(fact_83_norm__ge__zero, axiom,
    ((![X3 : real]: (ord_less_eq_real @ zero_zero_real @ (real_V646646907m_real @ X3))))). % norm_ge_zero
thf(fact_84_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_85_dual__order_Oeq__iff, axiom,
    (((^[Y6 : real]: (^[Z2 : real]: (Y6 = Z2))) = (^[A2 : real]: (^[B2 : real]: (((ord_less_eq_real @ B2 @ A2)) & ((ord_less_eq_real @ A2 @ B2)))))))). % dual_order.eq_iff
thf(fact_86_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_87_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_88_dual__order_Orefl, axiom,
    ((![A : real]: (ord_less_eq_real @ A @ A)))). % dual_order.refl
thf(fact_89_order__trans, axiom,
    ((![X3 : real, Y3 : real, Z3 : real]: ((ord_less_eq_real @ X3 @ Y3) => ((ord_less_eq_real @ Y3 @ Z3) => (ord_less_eq_real @ X3 @ Z3)))))). % order_trans
thf(fact_90_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_91_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_92_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_93_order__class_Oorder_Oeq__iff, axiom,
    (((^[Y6 : real]: (^[Z2 : real]: (Y6 = Z2))) = (^[A2 : real]: (^[B2 : real]: (((ord_less_eq_real @ A2 @ B2)) & ((ord_less_eq_real @ B2 @ A2)))))))). % order_class.order.eq_iff
thf(fact_94_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_95_le__cases3, axiom,
    ((![X3 : real, Y3 : real, Z3 : real]: (((ord_less_eq_real @ X3 @ Y3) => (~ ((ord_less_eq_real @ Y3 @ Z3)))) => (((ord_less_eq_real @ Y3 @ X3) => (~ ((ord_less_eq_real @ X3 @ Z3)))) => (((ord_less_eq_real @ X3 @ Z3) => (~ ((ord_less_eq_real @ Z3 @ Y3)))) => (((ord_less_eq_real @ Z3 @ Y3) => (~ ((ord_less_eq_real @ Y3 @ X3)))) => (((ord_less_eq_real @ Y3 @ Z3) => (~ ((ord_less_eq_real @ Z3 @ X3)))) => (~ (((ord_less_eq_real @ Z3 @ X3) => (~ ((ord_less_eq_real @ X3 @ Y3)))))))))))))). % le_cases3
thf(fact_96_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_97_le__cases, axiom,
    ((![X3 : real, Y3 : real]: ((~ ((ord_less_eq_real @ X3 @ Y3))) => (ord_less_eq_real @ Y3 @ X3))))). % le_cases
thf(fact_98_eq__refl, axiom,
    ((![X3 : real, Y3 : real]: ((X3 = Y3) => (ord_less_eq_real @ X3 @ Y3))))). % eq_refl
thf(fact_99_linear, axiom,
    ((![X3 : real, Y3 : real]: ((ord_less_eq_real @ X3 @ Y3) | (ord_less_eq_real @ Y3 @ X3))))). % linear
thf(fact_100_antisym, axiom,
    ((![X3 : real, Y3 : real]: ((ord_less_eq_real @ X3 @ Y3) => ((ord_less_eq_real @ Y3 @ X3) => (X3 = Y3)))))). % antisym
thf(fact_101_eq__iff, axiom,
    (((^[Y6 : real]: (^[Z2 : real]: (Y6 = Z2))) = (^[X2 : real]: (^[Y4 : real]: (((ord_less_eq_real @ X2 @ Y4)) & ((ord_less_eq_real @ Y4 @ X2)))))))). % eq_iff
thf(fact_102_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_103_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_104_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_105_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_106_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_107_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_108_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_109_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_110_lt__ex, axiom,
    ((![X3 : real]: (?[Y5 : real]: (ord_less_real @ Y5 @ X3))))). % lt_ex
thf(fact_111_gt__ex, axiom,
    ((![X3 : real]: (?[X_12 : real]: (ord_less_real @ X3 @ X_12))))). % gt_ex
thf(fact_112_neqE, axiom,
    ((![X3 : real, Y3 : real]: ((~ ((X3 = Y3))) => ((~ ((ord_less_real @ X3 @ Y3))) => (ord_less_real @ Y3 @ X3)))))). % neqE
thf(fact_113_neq__iff, axiom,
    ((![X3 : real, Y3 : real]: ((~ ((X3 = Y3))) = (((ord_less_real @ X3 @ Y3)) | ((ord_less_real @ Y3 @ X3))))))). % neq_iff
thf(fact_114_order_Oasym, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ B) => (~ ((ord_less_real @ B @ A))))))). % order.asym
thf(fact_115_dense, axiom,
    ((![X3 : real, Y3 : real]: ((ord_less_real @ X3 @ Y3) => (?[Z4 : real]: ((ord_less_real @ X3 @ Z4) & (ord_less_real @ Z4 @ Y3))))))). % dense
thf(fact_116_less__imp__neq, axiom,
    ((![X3 : real, Y3 : real]: ((ord_less_real @ X3 @ Y3) => (~ ((X3 = Y3))))))). % less_imp_neq
thf(fact_117_less__asym, axiom,
    ((![X3 : real, Y3 : real]: ((ord_less_real @ X3 @ Y3) => (~ ((ord_less_real @ Y3 @ X3))))))). % less_asym
thf(fact_118_less__asym_H, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ B) => (~ ((ord_less_real @ B @ A))))))). % less_asym'
thf(fact_119_less__trans, axiom,
    ((![X3 : real, Y3 : real, Z3 : real]: ((ord_less_real @ X3 @ Y3) => ((ord_less_real @ Y3 @ Z3) => (ord_less_real @ X3 @ Z3)))))). % less_trans
thf(fact_120_less__linear, axiom,
    ((![X3 : real, Y3 : real]: ((ord_less_real @ X3 @ Y3) | ((X3 = Y3) | (ord_less_real @ Y3 @ X3)))))). % less_linear
thf(fact_121_less__irrefl, axiom,
    ((![X3 : real]: (~ ((ord_less_real @ X3 @ X3)))))). % less_irrefl
thf(fact_122_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_123_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_124_dual__order_Oasym, axiom,
    ((![B : real, A : real]: ((ord_less_real @ B @ A) => (~ ((ord_less_real @ A @ B))))))). % dual_order.asym
thf(fact_125_less__imp__not__eq, axiom,
    ((![X3 : real, Y3 : real]: ((ord_less_real @ X3 @ Y3) => (~ ((X3 = Y3))))))). % less_imp_not_eq
thf(fact_126_less__not__sym, axiom,
    ((![X3 : real, Y3 : real]: ((ord_less_real @ X3 @ Y3) => (~ ((ord_less_real @ Y3 @ X3))))))). % less_not_sym
thf(fact_127_antisym__conv3, axiom,
    ((![Y3 : real, X3 : real]: ((~ ((ord_less_real @ Y3 @ X3))) => ((~ ((ord_less_real @ X3 @ Y3))) = (X3 = Y3)))))). % antisym_conv3
thf(fact_128_less__imp__not__eq2, axiom,
    ((![X3 : real, Y3 : real]: ((ord_less_real @ X3 @ Y3) => (~ ((Y3 = X3))))))). % less_imp_not_eq2
thf(fact_129_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_130_linorder__cases, axiom,
    ((![X3 : real, Y3 : real]: ((~ ((ord_less_real @ X3 @ Y3))) => ((~ ((X3 = Y3))) => (ord_less_real @ Y3 @ X3)))))). % linorder_cases
thf(fact_131_dual__order_Oirrefl, axiom,
    ((![A : real]: (~ ((ord_less_real @ A @ A)))))). % dual_order.irrefl
thf(fact_132_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_133_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_134_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_135_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_136_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_137_order_Ostrict__implies__not__eq, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ B) => (~ ((A = B))))))). % order.strict_implies_not_eq
thf(fact_138_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_139_strict__mono__eq, axiom,
    ((![F2 : nat > nat, X3 : nat, Y3 : nat]: ((order_769474267at_nat @ F2) => (((F2 @ X3) = (F2 @ Y3)) = (X3 = Y3)))))). % strict_mono_eq
thf(fact_140_leD, axiom,
    ((![Y3 : real, X3 : real]: ((ord_less_eq_real @ Y3 @ X3) => (~ ((ord_less_real @ X3 @ Y3))))))). % leD
thf(fact_141_leI, axiom,
    ((![X3 : real, Y3 : real]: ((~ ((ord_less_real @ X3 @ Y3))) => (ord_less_eq_real @ Y3 @ X3))))). % leI
thf(fact_142_le__less, axiom,
    ((ord_less_eq_real = (^[X2 : real]: (^[Y4 : real]: (((ord_less_real @ X2 @ Y4)) | ((X2 = Y4)))))))). % le_less
thf(fact_143_less__le, axiom,
    ((ord_less_real = (^[X2 : real]: (^[Y4 : real]: (((ord_less_eq_real @ X2 @ Y4)) & ((~ ((X2 = Y4)))))))))). % less_le
thf(fact_144_order__le__less__subst1, axiom,
    ((![A : real, F2 : real > real, B : real, C : real]: ((ord_less_eq_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_le_less_subst1
thf(fact_145_order__le__less__subst2, axiom,
    ((![A : real, B : real, F2 : real > real, C : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_real @ (F2 @ B) @ C) => ((![X : real, Y5 : real]: ((ord_less_eq_real @ X @ Y5) => (ord_less_eq_real @ (F2 @ X) @ (F2 @ Y5)))) => (ord_less_real @ (F2 @ A) @ C))))))). % order_le_less_subst2
thf(fact_146_order__less__le__subst1, axiom,
    ((![A : real, F2 : real > real, B : real, C : real]: ((ord_less_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_real @ A @ (F2 @ C)))))))). % order_less_le_subst1
thf(fact_147_order__less__le__subst2, axiom,
    ((![A : real, B : real, F2 : real > real, C : real]: ((ord_less_real @ A @ B) => ((ord_less_eq_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_le_subst2
thf(fact_148_not__le, axiom,
    ((![X3 : real, Y3 : real]: ((~ ((ord_less_eq_real @ X3 @ Y3))) = (ord_less_real @ Y3 @ X3))))). % not_le
thf(fact_149_not__less, axiom,
    ((![X3 : real, Y3 : real]: ((~ ((ord_less_real @ X3 @ Y3))) = (ord_less_eq_real @ Y3 @ X3))))). % not_less
thf(fact_150_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_151_antisym__conv1, axiom,
    ((![X3 : real, Y3 : real]: ((~ ((ord_less_real @ X3 @ Y3))) => ((ord_less_eq_real @ X3 @ Y3) = (X3 = Y3)))))). % antisym_conv1
thf(fact_152_antisym__conv2, axiom,
    ((![X3 : real, Y3 : real]: ((ord_less_eq_real @ X3 @ Y3) => ((~ ((ord_less_real @ X3 @ Y3))) = (X3 = Y3)))))). % antisym_conv2
thf(fact_153_less__imp__le, axiom,
    ((![X3 : real, Y3 : real]: ((ord_less_real @ X3 @ Y3) => (ord_less_eq_real @ X3 @ Y3))))). % less_imp_le
thf(fact_154_le__less__trans, axiom,
    ((![X3 : real, Y3 : real, Z3 : real]: ((ord_less_eq_real @ X3 @ Y3) => ((ord_less_real @ Y3 @ Z3) => (ord_less_real @ X3 @ Z3)))))). % le_less_trans
thf(fact_155_less__le__trans, axiom,
    ((![X3 : real, Y3 : real, Z3 : real]: ((ord_less_real @ X3 @ Y3) => ((ord_less_eq_real @ Y3 @ Z3) => (ord_less_real @ X3 @ Z3)))))). % less_le_trans
thf(fact_156_dense__ge, axiom,
    ((![Z3 : real, Y3 : real]: ((![X : real]: ((ord_less_real @ Z3 @ X) => (ord_less_eq_real @ Y3 @ X))) => (ord_less_eq_real @ Y3 @ Z3))))). % dense_ge
thf(fact_157_dense__le, axiom,
    ((![Y3 : real, Z3 : real]: ((![X : real]: ((ord_less_real @ X @ Y3) => (ord_less_eq_real @ X @ Z3))) => (ord_less_eq_real @ Y3 @ Z3))))). % dense_le
thf(fact_158_le__less__linear, axiom,
    ((![X3 : real, Y3 : real]: ((ord_less_eq_real @ X3 @ Y3) | (ord_less_real @ Y3 @ X3))))). % le_less_linear
thf(fact_159_le__imp__less__or__eq, axiom,
    ((![X3 : real, Y3 : real]: ((ord_less_eq_real @ X3 @ Y3) => ((ord_less_real @ X3 @ Y3) | (X3 = Y3)))))). % le_imp_less_or_eq
thf(fact_160_less__le__not__le, axiom,
    ((ord_less_real = (^[X2 : real]: (^[Y4 : real]: (((ord_less_eq_real @ X2 @ Y4)) & ((~ ((ord_less_eq_real @ Y4 @ X2)))))))))). % less_le_not_le
thf(fact_161_not__le__imp__less, axiom,
    ((![Y3 : real, X3 : real]: ((~ ((ord_less_eq_real @ Y3 @ X3))) => (ord_less_real @ X3 @ Y3))))). % not_le_imp_less
thf(fact_162_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_163_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

% Conjectures (1)
thf(conj_0, conjecture,
    ((ord_less_real @ (real_V638595069omplex @ (minus_minus_complex @ (s @ (comp_nat_nat_nat @ f @ g @ n)) @ (complex2 @ x @ y))) @ e))).
