% 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_307__5370606_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:29:07.206

% 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 (26)
thf(sy_c_Complex_Ocomplex_OComplex, type,
    complex2 : real > real > complex).
thf(sy_c_Complex_Ocomplex_OIm, type,
    im : complex > real).
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_Oabs__class_Oabs_001t__Real__Oreal, type,
    abs_abs_real : real > real).
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_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).
thf(sy_v_x____, type,
    x : real).
thf(sy_v_y____, type,
    y : real).

% Relevant facts (166)
thf(fact_0_g_I1_J, axiom,
    ((order_769474267at_nat @ g))). % g(1)
thf(fact_1_hs, axiom,
    ((order_769474267at_nat @ (comp_nat_nat_nat @ f @ g)))). % hs
thf(fact_2_f_I1_J, axiom,
    ((order_769474267at_nat @ f))). % f(1)
thf(fact_3__092_060open_062_092_060And_062e_O_A0_A_060_Ae_A_092_060Longrightarrow_062_A_092_060exists_062N_O_A_092_060forall_062n_092_060ge_062N_O_Acmod_A_Is_A_I_If_A_092_060circ_062_Ag_J_An_J_A_N_AComplex_Ax_Ay_J_A_060_Ae_092_060close_062, axiom,
    ((![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 @ (s @ (comp_nat_nat_nat @ f @ g @ N2)) @ (complex2 @ x @ y))) @ E)))))))). % \<open>\<And>e. 0 < e \<Longrightarrow> \<exists>N. \<forall>n\<ge>N. cmod (s ((f \<circ> g) n) - Complex x y) < e\<close>
thf(fact_4_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_5_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_6_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_7_norm__zero, axiom,
    (((real_V638595069omplex @ zero_zero_complex) = zero_zero_real))). % norm_zero
thf(fact_8_norm__zero, axiom,
    (((real_V646646907m_real @ zero_zero_real) = zero_zero_real))). % norm_zero
thf(fact_9_norm__eq__zero, axiom,
    ((![X3 : complex]: (((real_V638595069omplex @ X3) = zero_zero_real) = (X3 = zero_zero_complex))))). % norm_eq_zero
thf(fact_10_norm__eq__zero, axiom,
    ((![X3 : real]: (((real_V646646907m_real @ X3) = zero_zero_real) = (X3 = zero_zero_real))))). % norm_eq_zero
thf(fact_11_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_12_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_13_diff__self, axiom,
    ((![A : complex]: ((minus_minus_complex @ A @ A) = zero_zero_complex)))). % diff_self
thf(fact_14_diff__self, axiom,
    ((![A : real]: ((minus_minus_real @ A @ A) = zero_zero_real)))). % diff_self
thf(fact_15_diff__0__right, axiom,
    ((![A : complex]: ((minus_minus_complex @ A @ zero_zero_complex) = A)))). % diff_0_right
thf(fact_16_diff__0__right, axiom,
    ((![A : real]: ((minus_minus_real @ A @ zero_zero_real) = A)))). % diff_0_right
thf(fact_17_zero__diff, axiom,
    ((![A : nat]: ((minus_minus_nat @ zero_zero_nat @ A) = zero_zero_nat)))). % zero_diff
thf(fact_18_diff__zero, axiom,
    ((![A : complex]: ((minus_minus_complex @ A @ zero_zero_complex) = A)))). % diff_zero
thf(fact_19_diff__zero, axiom,
    ((![A : nat]: ((minus_minus_nat @ A @ zero_zero_nat) = A)))). % diff_zero
thf(fact_20_diff__zero, axiom,
    ((![A : real]: ((minus_minus_real @ A @ zero_zero_real) = A)))). % diff_zero
thf(fact_21_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_22_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_23_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_24_not__gr__zero, axiom,
    ((![N3 : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N3))) = (N3 = zero_zero_nat))))). % not_gr_zero
thf(fact_25_le__zero__eq, axiom,
    ((![N3 : nat]: ((ord_less_eq_nat @ N3 @ zero_zero_nat) = (N3 = zero_zero_nat))))). % le_zero_eq
thf(fact_26_r, axiom,
    ((![N2 : nat]: (ord_less_eq_real @ (real_V638595069omplex @ (s @ N2)) @ r)))). % r
thf(fact_27_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_28_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_29_rp, axiom,
    ((ord_less_eq_real @ zero_zero_real @ r))). % rp
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_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_32_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_33_norm__ge__zero, axiom,
    ((![X3 : complex]: (ord_less_eq_real @ zero_zero_real @ (real_V638595069omplex @ X3))))). % norm_ge_zero
thf(fact_34_norm__ge__zero, axiom,
    ((![X3 : real]: (ord_less_eq_real @ zero_zero_real @ (real_V646646907m_real @ X3))))). % norm_ge_zero
thf(fact_35_zero__reorient, axiom,
    ((![X3 : real]: ((zero_zero_real = X3) = (X3 = zero_zero_real))))). % zero_reorient
thf(fact_36_zero__reorient, axiom,
    ((![X3 : nat]: ((zero_zero_nat = X3) = (X3 = zero_zero_nat))))). % zero_reorient
thf(fact_37_zero__reorient, axiom,
    ((![X3 : complex]: ((zero_zero_complex = X3) = (X3 = zero_zero_complex))))). % zero_reorient
thf(fact_38_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_39_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_40_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_41_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_42_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_43_zero__le, axiom,
    ((![X3 : nat]: (ord_less_eq_nat @ zero_zero_nat @ X3)))). % zero_le
thf(fact_44_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_45_gr__implies__not__zero, axiom,
    ((![M : nat, N3 : nat]: ((ord_less_nat @ M @ N3) => (~ ((N3 = zero_zero_nat))))))). % gr_implies_not_zero
thf(fact_46_not__less__zero, axiom,
    ((![N3 : nat]: (~ ((ord_less_nat @ N3 @ zero_zero_nat)))))). % not_less_zero
thf(fact_47_gr__zeroI, axiom,
    ((![N3 : nat]: ((~ ((N3 = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N3))))). % gr_zeroI
thf(fact_48_eq__iff__diff__eq__0, axiom,
    (((^[Y2 : complex]: (^[Z2 : complex]: (Y2 = Z2))) = (^[A2 : complex]: (^[B2 : complex]: ((minus_minus_complex @ A2 @ B2) = zero_zero_complex)))))). % eq_iff_diff_eq_0
thf(fact_49_eq__iff__diff__eq__0, axiom,
    (((^[Y2 : real]: (^[Z2 : real]: (Y2 = Z2))) = (^[A2 : real]: (^[B2 : real]: ((minus_minus_real @ A2 @ B2) = zero_zero_real)))))). % eq_iff_diff_eq_0
thf(fact_50_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_51_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_52_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_53_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_54_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_55_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_56_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_57_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_58_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_59_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_60_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_61_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_62_norm__not__less__zero, axiom,
    ((![X3 : complex]: (~ ((ord_less_real @ (real_V638595069omplex @ X3) @ zero_zero_real)))))). % norm_not_less_zero
thf(fact_63_norm__not__less__zero, axiom,
    ((![X3 : real]: (~ ((ord_less_real @ (real_V646646907m_real @ X3) @ zero_zero_real)))))). % norm_not_less_zero
thf(fact_64_complex_Oinject, axiom,
    ((![X1 : real, X22 : real, Y1 : real, Y22 : real]: (((complex2 @ X1 @ X22) = (complex2 @ Y1 @ Y22)) = (((X1 = Y1)) & ((X22 = Y22))))))). % complex.inject
thf(fact_65_comp__apply, axiom,
    ((comp_nat_nat_nat = (^[F : nat > nat]: (^[G : nat > nat]: (^[X2 : nat]: (F @ (G @ X2)))))))). % comp_apply
thf(fact_66_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_67_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_68_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_69_bot__nat__0_Oextremum, axiom,
    ((![A : nat]: (ord_less_eq_nat @ zero_zero_nat @ A)))). % bot_nat_0.extremum
thf(fact_70_le0, axiom,
    ((![N3 : nat]: (ord_less_eq_nat @ zero_zero_nat @ N3)))). % le0
thf(fact_71_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_72_neq0__conv, axiom,
    ((![N3 : nat]: ((~ ((N3 = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ N3))))). % neq0_conv
thf(fact_73_less__nat__zero__code, axiom,
    ((![N3 : nat]: (~ ((ord_less_nat @ N3 @ zero_zero_nat)))))). % less_nat_zero_code
thf(fact_74_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_75_diff__0__eq__0, axiom,
    ((![N3 : nat]: ((minus_minus_nat @ zero_zero_nat @ N3) = zero_zero_nat)))). % diff_0_eq_0
thf(fact_76_diff__self__eq__0, axiom,
    ((![M : nat]: ((minus_minus_nat @ M @ M) = zero_zero_nat)))). % diff_self_eq_0
thf(fact_77_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_78_gr0I, axiom,
    ((![N3 : nat]: ((~ ((N3 = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N3))))). % gr0I
thf(fact_79_not__gr0, axiom,
    ((![N3 : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N3))) = (N3 = zero_zero_nat))))). % not_gr0
thf(fact_80_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_81_not__less0, axiom,
    ((![N3 : nat]: (~ ((ord_less_nat @ N3 @ zero_zero_nat)))))). % not_less0
thf(fact_82_less__zeroE, axiom,
    ((![N3 : nat]: (~ ((ord_less_nat @ N3 @ zero_zero_nat)))))). % less_zeroE
thf(fact_83_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_84_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_85_gr__implies__not0, axiom,
    ((![M : nat, N3 : nat]: ((ord_less_nat @ M @ N3) => (~ ((N3 = zero_zero_nat))))))). % gr_implies_not0
thf(fact_86_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_87_minus__nat_Odiff__0, axiom,
    ((![M : nat]: ((minus_minus_nat @ M @ zero_zero_nat) = M)))). % minus_nat.diff_0
thf(fact_88_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_89_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_90_bot__nat__0_Oextremum__strict, axiom,
    ((![A : nat]: (~ ((ord_less_nat @ A @ zero_zero_nat)))))). % bot_nat_0.extremum_strict
thf(fact_91_le__refl, axiom,
    ((![N3 : nat]: (ord_less_eq_nat @ N3 @ N3)))). % le_refl
thf(fact_92_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_93_eq__imp__le, axiom,
    ((![M : nat, N3 : nat]: ((M = N3) => (ord_less_eq_nat @ M @ N3))))). % eq_imp_le
thf(fact_94_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_95_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_96_Nat_Oex__has__greatest__nat, axiom,
    ((![P : nat > $o, K : nat, B : nat]: ((P @ K) => ((![Y3 : nat]: ((P @ Y3) => (ord_less_eq_nat @ Y3 @ B))) => (?[X : nat]: ((P @ X) & (![Y : nat]: ((P @ Y) => (ord_less_eq_nat @ Y @ X)))))))))). % Nat.ex_has_greatest_nat
thf(fact_97_less__eq__nat_Osimps_I1_J, axiom,
    ((![N3 : nat]: (ord_less_eq_nat @ zero_zero_nat @ N3)))). % less_eq_nat.simps(1)
thf(fact_98_le__0__eq, axiom,
    ((![N3 : nat]: ((ord_less_eq_nat @ N3 @ zero_zero_nat) = (N3 = zero_zero_nat))))). % le_0_eq
thf(fact_99_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_100_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_101_nat__less__le, axiom,
    ((ord_less_nat = (^[M3 : nat]: (^[N5 : nat]: (((ord_less_eq_nat @ M3 @ N5)) & ((~ ((M3 = N5)))))))))). % nat_less_le
thf(fact_102_less__imp__le__nat, axiom,
    ((![M : nat, N3 : nat]: ((ord_less_nat @ M @ N3) => (ord_less_eq_nat @ M @ N3))))). % less_imp_le_nat
thf(fact_103_le__eq__less__or__eq, axiom,
    ((ord_less_eq_nat = (^[M3 : nat]: (^[N5 : nat]: (((ord_less_nat @ M3 @ N5)) | ((M3 = N5)))))))). % le_eq_less_or_eq
thf(fact_104_less__or__eq__imp__le, axiom,
    ((![M : nat, N3 : nat]: (((ord_less_nat @ M @ N3) | (M = N3)) => (ord_less_eq_nat @ M @ N3))))). % less_or_eq_imp_le
thf(fact_105_le__neq__implies__less, axiom,
    ((![M : nat, N3 : nat]: ((ord_less_eq_nat @ M @ N3) => ((~ ((M = N3))) => (ord_less_nat @ M @ N3)))))). % le_neq_implies_less
thf(fact_106_less__mono__imp__le__mono, axiom,
    ((![F2 : nat > nat, I : nat, J : nat]: ((![I2 : nat, J2 : nat]: ((ord_less_nat @ I2 @ J2) => (ord_less_nat @ (F2 @ I2) @ (F2 @ J2)))) => ((ord_less_eq_nat @ I @ J) => (ord_less_eq_nat @ (F2 @ I) @ (F2 @ J))))))). % less_mono_imp_le_mono
thf(fact_107_eq__diff__iff, axiom,
    ((![K : nat, M : nat, N3 : nat]: ((ord_less_eq_nat @ K @ M) => ((ord_less_eq_nat @ K @ N3) => (((minus_minus_nat @ M @ K) = (minus_minus_nat @ N3 @ K)) = (M = N3))))))). % eq_diff_iff
thf(fact_108_le__diff__iff, axiom,
    ((![K : nat, M : nat, N3 : nat]: ((ord_less_eq_nat @ K @ M) => ((ord_less_eq_nat @ K @ N3) => ((ord_less_eq_nat @ (minus_minus_nat @ M @ K) @ (minus_minus_nat @ N3 @ K)) = (ord_less_eq_nat @ M @ N3))))))). % le_diff_iff
thf(fact_109_Nat_Odiff__diff__eq, axiom,
    ((![K : nat, M : nat, N3 : nat]: ((ord_less_eq_nat @ K @ M) => ((ord_less_eq_nat @ K @ N3) => ((minus_minus_nat @ (minus_minus_nat @ M @ K) @ (minus_minus_nat @ N3 @ K)) = (minus_minus_nat @ M @ N3))))))). % Nat.diff_diff_eq
thf(fact_110_diff__le__mono, axiom,
    ((![M : nat, N3 : nat, L : nat]: ((ord_less_eq_nat @ M @ N3) => (ord_less_eq_nat @ (minus_minus_nat @ M @ L) @ (minus_minus_nat @ N3 @ L)))))). % diff_le_mono
thf(fact_111_diff__le__self, axiom,
    ((![M : nat, N3 : nat]: (ord_less_eq_nat @ (minus_minus_nat @ M @ N3) @ M)))). % diff_le_self
thf(fact_112_le__diff__iff_H, axiom,
    ((![A : nat, C : nat, B : nat]: ((ord_less_eq_nat @ A @ C) => ((ord_less_eq_nat @ B @ C) => ((ord_less_eq_nat @ (minus_minus_nat @ C @ A) @ (minus_minus_nat @ C @ B)) = (ord_less_eq_nat @ B @ A))))))). % le_diff_iff'
thf(fact_113_diff__le__mono2, axiom,
    ((![M : nat, N3 : nat, L : nat]: ((ord_less_eq_nat @ M @ N3) => (ord_less_eq_nat @ (minus_minus_nat @ L @ N3) @ (minus_minus_nat @ L @ M)))))). % diff_le_mono2
thf(fact_114_ex__least__nat__le, axiom,
    ((![P : nat > $o, N3 : nat]: ((P @ N3) => ((~ ((P @ zero_zero_nat))) => (?[K2 : nat]: ((ord_less_eq_nat @ K2 @ N3) & ((![I3 : nat]: ((ord_less_nat @ I3 @ K2) => (~ ((P @ I3))))) & (P @ K2))))))))). % ex_least_nat_le
thf(fact_115_diff__less__mono, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_nat @ A @ B) => ((ord_less_eq_nat @ C @ A) => (ord_less_nat @ (minus_minus_nat @ A @ C) @ (minus_minus_nat @ B @ C))))))). % diff_less_mono
thf(fact_116_less__diff__iff, axiom,
    ((![K : nat, M : nat, N3 : nat]: ((ord_less_eq_nat @ K @ M) => ((ord_less_eq_nat @ K @ N3) => ((ord_less_nat @ (minus_minus_nat @ M @ K) @ (minus_minus_nat @ N3 @ K)) = (ord_less_nat @ M @ N3))))))). % less_diff_iff
thf(fact_117_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_118_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_119_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_120_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_121_comp__def, axiom,
    ((comp_nat_nat_nat = (^[F : nat > nat]: (^[G : nat > nat]: (^[X2 : nat]: (F @ (G @ X2)))))))). % comp_def
thf(fact_122_complex_Oexhaust, axiom,
    ((![Y4 : complex]: (~ ((![X12 : real, X23 : real]: (~ ((Y4 = (complex2 @ X12 @ X23)))))))))). % complex.exhaust
thf(fact_123_Complex__eq__0, axiom,
    ((![A : real, B : real]: (((complex2 @ A @ B) = zero_zero_complex) = (((A = zero_zero_real)) & ((B = zero_zero_real))))))). % Complex_eq_0
thf(fact_124_zero__complex_Ocode, axiom,
    ((zero_zero_complex = (complex2 @ zero_zero_real @ zero_zero_real)))). % zero_complex.code
thf(fact_125_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_126_Bolzano, axiom,
    ((![A : real, B : real, P : real > real > $o]: ((ord_less_eq_real @ A @ B) => ((![A3 : real, B3 : real, C2 : real]: ((P @ A3 @ B3) => ((P @ B3 @ C2) => ((ord_less_eq_real @ A3 @ B3) => ((ord_less_eq_real @ B3 @ C2) => (P @ A3 @ C2)))))) => ((![X : real]: ((ord_less_eq_real @ A @ X) => ((ord_less_eq_real @ X @ B) => (?[D2 : real]: ((ord_less_real @ zero_zero_real @ D2) & (![A3 : real, B3 : real]: (((ord_less_eq_real @ A3 @ X) & ((ord_less_eq_real @ X @ B3) & (ord_less_real @ (minus_minus_real @ B3 @ A3) @ D2))) => (P @ A3 @ B3)))))))) => (P @ A @ B))))))). % Bolzano
thf(fact_127_y, axiom,
    ((![R : real]: ((ord_less_real @ zero_zero_real @ R) => (?[N0 : nat]: (![N2 : nat]: ((ord_less_eq_nat @ N0 @ N2) => (ord_less_real @ (abs_abs_real @ (minus_minus_real @ (im @ (s @ (f @ (g @ N2)))) @ y)) @ R)))))))). % y
thf(fact_128_order__refl, axiom,
    ((![X3 : nat]: (ord_less_eq_nat @ X3 @ X3)))). % order_refl
thf(fact_129_order__refl, axiom,
    ((![X3 : real]: (ord_less_eq_real @ X3 @ X3)))). % order_refl
thf(fact_130_strict__mono__o, axiom,
    ((![R2 : nat > nat, S2 : nat > nat]: ((order_769474267at_nat @ R2) => ((order_769474267at_nat @ S2) => (order_769474267at_nat @ (comp_nat_nat_nat @ R2 @ S2))))))). % strict_mono_o
thf(fact_131_strict__monoD, axiom,
    ((![F2 : real > real, X3 : real, Y4 : real]: ((order_1818878995l_real @ F2) => ((ord_less_real @ X3 @ Y4) => (ord_less_real @ (F2 @ X3) @ (F2 @ Y4))))))). % strict_monoD
thf(fact_132_strict__monoD, axiom,
    ((![F2 : real > nat, X3 : real, Y4 : real]: ((order_297469111al_nat @ F2) => ((ord_less_real @ X3 @ Y4) => (ord_less_nat @ (F2 @ X3) @ (F2 @ Y4))))))). % strict_monoD
thf(fact_133_strict__monoD, axiom,
    ((![F2 : nat > real, X3 : nat, Y4 : nat]: ((order_952716343t_real @ F2) => ((ord_less_nat @ X3 @ Y4) => (ord_less_real @ (F2 @ X3) @ (F2 @ Y4))))))). % strict_monoD
thf(fact_134_strict__monoD, axiom,
    ((![F2 : nat > nat, X3 : nat, Y4 : nat]: ((order_769474267at_nat @ F2) => ((ord_less_nat @ X3 @ Y4) => (ord_less_nat @ (F2 @ X3) @ (F2 @ Y4))))))). % strict_monoD
thf(fact_135_abs__idempotent, axiom,
    ((![A : real]: ((abs_abs_real @ (abs_abs_real @ A)) = (abs_abs_real @ A))))). % abs_idempotent
thf(fact_136_abs__zero, axiom,
    (((abs_abs_real @ zero_zero_real) = zero_zero_real))). % abs_zero
thf(fact_137_abs__eq__0, axiom,
    ((![A : real]: (((abs_abs_real @ A) = zero_zero_real) = (A = zero_zero_real))))). % abs_eq_0
thf(fact_138_abs__0__eq, axiom,
    ((![A : real]: ((zero_zero_real = (abs_abs_real @ A)) = (A = zero_zero_real))))). % abs_0_eq
thf(fact_139_abs__norm__cancel, axiom,
    ((![A : complex]: ((abs_abs_real @ (real_V638595069omplex @ A)) = (real_V638595069omplex @ A))))). % abs_norm_cancel
thf(fact_140_abs__norm__cancel, axiom,
    ((![A : real]: ((abs_abs_real @ (real_V646646907m_real @ A)) = (real_V646646907m_real @ A))))). % abs_norm_cancel
thf(fact_141_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_142_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_143_abs__of__nonneg, axiom,
    ((![A : real]: ((ord_less_eq_real @ zero_zero_real @ A) => ((abs_abs_real @ A) = A))))). % abs_of_nonneg
thf(fact_144_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_145_zero__complex_Osimps_I2_J, axiom,
    (((im @ zero_zero_complex) = zero_zero_real))). % zero_complex.simps(2)
thf(fact_146_nat__neq__iff, axiom,
    ((![M : nat, N3 : nat]: ((~ ((M = N3))) = (((ord_less_nat @ M @ N3)) | ((ord_less_nat @ N3 @ M))))))). % nat_neq_iff
thf(fact_147_less__not__refl, axiom,
    ((![N3 : nat]: (~ ((ord_less_nat @ N3 @ N3)))))). % less_not_refl
thf(fact_148_less__not__refl2, axiom,
    ((![N3 : nat, M : nat]: ((ord_less_nat @ N3 @ M) => (~ ((M = N3))))))). % less_not_refl2
thf(fact_149_less__not__refl3, axiom,
    ((![S2 : nat, T : nat]: ((ord_less_nat @ S2 @ T) => (~ ((S2 = T))))))). % less_not_refl3
thf(fact_150_less__irrefl__nat, axiom,
    ((![N3 : nat]: (~ ((ord_less_nat @ N3 @ N3)))))). % less_irrefl_nat
thf(fact_151_nat__less__induct, axiom,
    ((![P : nat > $o, N3 : nat]: ((![N4 : nat]: ((![M2 : nat]: ((ord_less_nat @ M2 @ N4) => (P @ M2))) => (P @ N4))) => (P @ N3))))). % nat_less_induct
thf(fact_152_infinite__descent, axiom,
    ((![P : nat > $o, N3 : nat]: ((![N4 : nat]: ((~ ((P @ N4))) => (?[M2 : nat]: ((ord_less_nat @ M2 @ N4) & (~ ((P @ M2))))))) => (P @ N3))))). % infinite_descent
thf(fact_153_linorder__neqE__nat, axiom,
    ((![X3 : nat, Y4 : nat]: ((~ ((X3 = Y4))) => ((~ ((ord_less_nat @ X3 @ Y4))) => (ord_less_nat @ Y4 @ X3)))))). % linorder_neqE_nat
thf(fact_154_abs__Im__le__cmod, axiom,
    ((![X3 : complex]: (ord_less_eq_real @ (abs_abs_real @ (im @ X3)) @ (real_V638595069omplex @ X3))))). % abs_Im_le_cmod
thf(fact_155_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_156_abs__ge__self, axiom,
    ((![A : real]: (ord_less_eq_real @ A @ (abs_abs_real @ A))))). % abs_ge_self
thf(fact_157_abs__minus__commute, axiom,
    ((![A : real, B : real]: ((abs_abs_real @ (minus_minus_real @ A @ B)) = (abs_abs_real @ (minus_minus_real @ B @ A)))))). % abs_minus_commute
thf(fact_158_real__norm__def, axiom,
    ((real_V646646907m_real = abs_abs_real))). % real_norm_def
thf(fact_159_complex_Osel_I2_J, axiom,
    ((![X1 : real, X22 : real]: ((im @ (complex2 @ X1 @ X22)) = X22)))). % complex.sel(2)
thf(fact_160_abs__ge__zero, axiom,
    ((![A : real]: (ord_less_eq_real @ zero_zero_real @ (abs_abs_real @ A))))). % abs_ge_zero
thf(fact_161_abs__not__less__zero, axiom,
    ((![A : real]: (~ ((ord_less_real @ (abs_abs_real @ A) @ zero_zero_real)))))). % abs_not_less_zero
thf(fact_162_abs__of__pos, axiom,
    ((![A : real]: ((ord_less_real @ zero_zero_real @ A) => ((abs_abs_real @ A) = A))))). % abs_of_pos
thf(fact_163_abs__triangle__ineq2__sym, axiom,
    ((![A : real, B : real]: (ord_less_eq_real @ (minus_minus_real @ (abs_abs_real @ A) @ (abs_abs_real @ B)) @ (abs_abs_real @ (minus_minus_real @ B @ A)))))). % abs_triangle_ineq2_sym
thf(fact_164_abs__triangle__ineq3, axiom,
    ((![A : real, B : real]: (ord_less_eq_real @ (abs_abs_real @ (minus_minus_real @ (abs_abs_real @ A) @ (abs_abs_real @ B))) @ (abs_abs_real @ (minus_minus_real @ A @ B)))))). % abs_triangle_ineq3
thf(fact_165_abs__triangle__ineq2, axiom,
    ((![A : real, B : real]: (ord_less_eq_real @ (minus_minus_real @ (abs_abs_real @ A) @ (abs_abs_real @ B)) @ (abs_abs_real @ (minus_minus_real @ A @ B)))))). % abs_triangle_ineq2

% Conjectures (1)
thf(conj_0, conjecture,
    ((?[F3 : nat > nat, Z : complex]: ((order_769474267at_nat @ F3) & (![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 @ (s @ (F3 @ N4)) @ Z)) @ E2)))))))))).
