% 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_305__5370588_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:29:05.775

% 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 (30)
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_Oplus__class_Oplus_001t__Complex__Ocomplex, type,
    plus_plus_complex : complex > complex > complex).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat, type,
    plus_plus_nat : nat > nat > nat).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal, type,
    plus_plus_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_N1____, type,
    n1 : nat).
thf(sy_v_N2____, type,
    n2 : nat).
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_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 (215)
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_that, axiom,
    ((ord_less_real @ zero_zero_real @ e))). % that
thf(fact_3__092_060open_062_092_060And_062n_O_AN1_A_L_AN2_A_092_060le_062_An_A_092_060Longrightarrow_062_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,
    ((![N : nat]: ((ord_less_eq_nat @ (plus_plus_nat @ n1 @ n2) @ N) => (ord_less_real @ (real_V638595069omplex @ (minus_minus_complex @ (s @ (comp_nat_nat_nat @ f @ g @ N)) @ (complex2 @ x @ y))) @ e))))). % \<open>\<And>n. N1 + N2 \<le> n \<Longrightarrow> 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_hs, axiom,
    ((order_769474267at_nat @ (comp_nat_nat_nat @ f @ g)))). % hs
thf(fact_6_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_7_r, axiom,
    ((![N2 : nat]: (ord_less_eq_real @ (real_V638595069omplex @ (s @ N2)) @ r)))). % r
thf(fact_8_comp__apply, axiom,
    ((comp_nat_nat_nat = (^[F : nat > nat]: (^[G : nat > nat]: (^[X2 : nat]: (F @ (G @ X2)))))))). % comp_apply
thf(fact_9_order__refl, axiom,
    ((![X3 : nat]: (ord_less_eq_nat @ X3 @ X3)))). % order_refl
thf(fact_10_order__refl, axiom,
    ((![X3 : real]: (ord_less_eq_real @ X3 @ X3)))). % order_refl
thf(fact_11_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_12_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_13_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_14_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_15_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_16_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_17_add__left__cancel, axiom,
    ((![A : nat, B : nat, C : nat]: (((plus_plus_nat @ A @ B) = (plus_plus_nat @ A @ C)) = (B = C))))). % add_left_cancel
thf(fact_18_add__right__cancel, axiom,
    ((![B : nat, A : nat, C : nat]: (((plus_plus_nat @ B @ A) = (plus_plus_nat @ C @ A)) = (B = C))))). % add_right_cancel
thf(fact_19_rp, axiom,
    ((ord_less_eq_real @ zero_zero_real @ r))). % rp
thf(fact_20_le__zero__eq, axiom,
    ((![N : nat]: ((ord_less_eq_nat @ N @ zero_zero_nat) = (N = zero_zero_nat))))). % le_zero_eq
thf(fact_21_add_Oleft__neutral, axiom,
    ((![A : nat]: ((plus_plus_nat @ zero_zero_nat @ A) = A)))). % add.left_neutral
thf(fact_22_add_Oleft__neutral, axiom,
    ((![A : real]: ((plus_plus_real @ zero_zero_real @ A) = A)))). % add.left_neutral
thf(fact_23_add_Oright__neutral, axiom,
    ((![A : nat]: ((plus_plus_nat @ A @ zero_zero_nat) = A)))). % add.right_neutral
thf(fact_24_add_Oright__neutral, axiom,
    ((![A : real]: ((plus_plus_real @ A @ zero_zero_real) = A)))). % add.right_neutral
thf(fact_25_double__zero, axiom,
    ((![A : real]: (((plus_plus_real @ A @ A) = zero_zero_real) = (A = zero_zero_real))))). % double_zero
thf(fact_26_double__zero__sym, axiom,
    ((![A : real]: ((zero_zero_real = (plus_plus_real @ A @ A)) = (A = zero_zero_real))))). % double_zero_sym
thf(fact_27_add__cancel__left__left, axiom,
    ((![B : nat, A : nat]: (((plus_plus_nat @ B @ A) = A) = (B = zero_zero_nat))))). % add_cancel_left_left
thf(fact_28_add__cancel__left__left, axiom,
    ((![B : real, A : real]: (((plus_plus_real @ B @ A) = A) = (B = zero_zero_real))))). % add_cancel_left_left
thf(fact_29_add__cancel__left__right, axiom,
    ((![A : nat, B : nat]: (((plus_plus_nat @ A @ B) = A) = (B = zero_zero_nat))))). % add_cancel_left_right
thf(fact_30_add__cancel__left__right, axiom,
    ((![A : real, B : real]: (((plus_plus_real @ A @ B) = A) = (B = zero_zero_real))))). % add_cancel_left_right
thf(fact_31_add__cancel__right__left, axiom,
    ((![A : nat, B : nat]: ((A = (plus_plus_nat @ B @ A)) = (B = zero_zero_nat))))). % add_cancel_right_left
thf(fact_32_add__cancel__right__left, axiom,
    ((![A : real, B : real]: ((A = (plus_plus_real @ B @ A)) = (B = zero_zero_real))))). % add_cancel_right_left
thf(fact_33_add__cancel__right__right, axiom,
    ((![A : nat, B : nat]: ((A = (plus_plus_nat @ A @ B)) = (B = zero_zero_nat))))). % add_cancel_right_right
thf(fact_34_add__cancel__right__right, axiom,
    ((![A : real, B : real]: ((A = (plus_plus_real @ A @ B)) = (B = zero_zero_real))))). % add_cancel_right_right
thf(fact_35_add__eq__0__iff__both__eq__0, axiom,
    ((![X3 : nat, Y3 : nat]: (((plus_plus_nat @ X3 @ Y3) = zero_zero_nat) = (((X3 = zero_zero_nat)) & ((Y3 = zero_zero_nat))))))). % add_eq_0_iff_both_eq_0
thf(fact_36_zero__eq__add__iff__both__eq__0, axiom,
    ((![X3 : nat, Y3 : nat]: ((zero_zero_nat = (plus_plus_nat @ X3 @ Y3)) = (((X3 = zero_zero_nat)) & ((Y3 = zero_zero_nat))))))). % zero_eq_add_iff_both_eq_0
thf(fact_37_add__le__cancel__right, axiom,
    ((![A : nat, C : nat, B : nat]: ((ord_less_eq_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ C)) = (ord_less_eq_nat @ A @ B))))). % add_le_cancel_right
thf(fact_38_add__le__cancel__right, axiom,
    ((![A : real, C : real, B : real]: ((ord_less_eq_real @ (plus_plus_real @ A @ C) @ (plus_plus_real @ B @ C)) = (ord_less_eq_real @ A @ B))))). % add_le_cancel_right
thf(fact_39_add__le__cancel__left, axiom,
    ((![C : nat, A : nat, B : nat]: ((ord_less_eq_nat @ (plus_plus_nat @ C @ A) @ (plus_plus_nat @ C @ B)) = (ord_less_eq_nat @ A @ B))))). % add_le_cancel_left
thf(fact_40_add__le__cancel__left, axiom,
    ((![C : real, A : real, B : real]: ((ord_less_eq_real @ (plus_plus_real @ C @ A) @ (plus_plus_real @ C @ B)) = (ord_less_eq_real @ A @ B))))). % add_le_cancel_left
thf(fact_41_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_42_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_43_diff__zero, axiom,
    ((![A : real]: ((minus_minus_real @ A @ zero_zero_real) = A)))). % diff_zero
thf(fact_44_diff__zero, axiom,
    ((![A : complex]: ((minus_minus_complex @ A @ zero_zero_complex) = A)))). % diff_zero
thf(fact_45_diff__0__right, axiom,
    ((![A : real]: ((minus_minus_real @ A @ zero_zero_real) = A)))). % diff_0_right
thf(fact_46_diff__0__right, axiom,
    ((![A : complex]: ((minus_minus_complex @ A @ zero_zero_complex) = A)))). % diff_0_right
thf(fact_47_diff__self, axiom,
    ((![A : real]: ((minus_minus_real @ A @ A) = zero_zero_real)))). % diff_self
thf(fact_48_diff__self, axiom,
    ((![A : complex]: ((minus_minus_complex @ A @ A) = zero_zero_complex)))). % diff_self
thf(fact_49_add__less__cancel__right, axiom,
    ((![A : nat, C : nat, B : nat]: ((ord_less_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ C)) = (ord_less_nat @ A @ B))))). % add_less_cancel_right
thf(fact_50_add__less__cancel__right, axiom,
    ((![A : real, C : real, B : real]: ((ord_less_real @ (plus_plus_real @ A @ C) @ (plus_plus_real @ B @ C)) = (ord_less_real @ A @ B))))). % add_less_cancel_right
thf(fact_51_add__less__cancel__left, axiom,
    ((![C : nat, A : nat, B : nat]: ((ord_less_nat @ (plus_plus_nat @ C @ A) @ (plus_plus_nat @ C @ B)) = (ord_less_nat @ A @ B))))). % add_less_cancel_left
thf(fact_52_add__less__cancel__left, axiom,
    ((![C : real, A : real, B : real]: ((ord_less_real @ (plus_plus_real @ C @ A) @ (plus_plus_real @ C @ B)) = (ord_less_real @ A @ B))))). % add_less_cancel_left
thf(fact_53_add__diff__cancel__right_H, axiom,
    ((![A : nat, B : nat]: ((minus_minus_nat @ (plus_plus_nat @ A @ B) @ B) = A)))). % add_diff_cancel_right'
thf(fact_54_add__diff__cancel__right_H, axiom,
    ((![A : complex, B : complex]: ((minus_minus_complex @ (plus_plus_complex @ A @ B) @ B) = A)))). % add_diff_cancel_right'
thf(fact_55_add__diff__cancel__right, axiom,
    ((![A : nat, C : nat, B : nat]: ((minus_minus_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ C)) = (minus_minus_nat @ A @ B))))). % add_diff_cancel_right
thf(fact_56_add__diff__cancel__right, axiom,
    ((![A : complex, C : complex, B : complex]: ((minus_minus_complex @ (plus_plus_complex @ A @ C) @ (plus_plus_complex @ B @ C)) = (minus_minus_complex @ A @ B))))). % add_diff_cancel_right
thf(fact_57_add__diff__cancel__left_H, axiom,
    ((![A : nat, B : nat]: ((minus_minus_nat @ (plus_plus_nat @ A @ B) @ A) = B)))). % add_diff_cancel_left'
thf(fact_58_add__diff__cancel__left_H, axiom,
    ((![A : complex, B : complex]: ((minus_minus_complex @ (plus_plus_complex @ A @ B) @ A) = B)))). % add_diff_cancel_left'
thf(fact_59_add__diff__cancel__left, axiom,
    ((![C : nat, A : nat, B : nat]: ((minus_minus_nat @ (plus_plus_nat @ C @ A) @ (plus_plus_nat @ C @ B)) = (minus_minus_nat @ A @ B))))). % add_diff_cancel_left
thf(fact_60_add__diff__cancel__left, axiom,
    ((![C : complex, A : complex, B : complex]: ((minus_minus_complex @ (plus_plus_complex @ C @ A) @ (plus_plus_complex @ C @ B)) = (minus_minus_complex @ A @ B))))). % add_diff_cancel_left
thf(fact_61_diff__add__cancel, axiom,
    ((![A : complex, B : complex]: ((plus_plus_complex @ (minus_minus_complex @ A @ B) @ B) = A)))). % diff_add_cancel
thf(fact_62_add__diff__cancel, axiom,
    ((![A : complex, B : complex]: ((minus_minus_complex @ (plus_plus_complex @ A @ B) @ B) = A)))). % add_diff_cancel
thf(fact_63_zero__le__double__add__iff__zero__le__single__add, axiom,
    ((![A : real]: ((ord_less_eq_real @ zero_zero_real @ (plus_plus_real @ A @ A)) = (ord_less_eq_real @ zero_zero_real @ A))))). % zero_le_double_add_iff_zero_le_single_add
thf(fact_64_double__add__le__zero__iff__single__add__le__zero, axiom,
    ((![A : real]: ((ord_less_eq_real @ (plus_plus_real @ A @ A) @ zero_zero_real) = (ord_less_eq_real @ A @ zero_zero_real))))). % double_add_le_zero_iff_single_add_le_zero
thf(fact_65_le__add__same__cancel2, axiom,
    ((![A : nat, B : nat]: ((ord_less_eq_nat @ A @ (plus_plus_nat @ B @ A)) = (ord_less_eq_nat @ zero_zero_nat @ B))))). % le_add_same_cancel2
thf(fact_66_le__add__same__cancel2, axiom,
    ((![A : real, B : real]: ((ord_less_eq_real @ A @ (plus_plus_real @ B @ A)) = (ord_less_eq_real @ zero_zero_real @ B))))). % le_add_same_cancel2
thf(fact_67_le__add__same__cancel1, axiom,
    ((![A : nat, B : nat]: ((ord_less_eq_nat @ A @ (plus_plus_nat @ A @ B)) = (ord_less_eq_nat @ zero_zero_nat @ B))))). % le_add_same_cancel1
thf(fact_68_le__add__same__cancel1, axiom,
    ((![A : real, B : real]: ((ord_less_eq_real @ A @ (plus_plus_real @ A @ B)) = (ord_less_eq_real @ zero_zero_real @ B))))). % le_add_same_cancel1
thf(fact_69_add__le__same__cancel2, axiom,
    ((![A : nat, B : nat]: ((ord_less_eq_nat @ (plus_plus_nat @ A @ B) @ B) = (ord_less_eq_nat @ A @ zero_zero_nat))))). % add_le_same_cancel2
thf(fact_70_add__le__same__cancel2, axiom,
    ((![A : real, B : real]: ((ord_less_eq_real @ (plus_plus_real @ A @ B) @ B) = (ord_less_eq_real @ A @ zero_zero_real))))). % add_le_same_cancel2
thf(fact_71_add__le__same__cancel1, axiom,
    ((![B : nat, A : nat]: ((ord_less_eq_nat @ (plus_plus_nat @ B @ A) @ B) = (ord_less_eq_nat @ A @ zero_zero_nat))))). % add_le_same_cancel1
thf(fact_72_add__le__same__cancel1, axiom,
    ((![B : real, A : real]: ((ord_less_eq_real @ (plus_plus_real @ B @ A) @ B) = (ord_less_eq_real @ A @ zero_zero_real))))). % add_le_same_cancel1
thf(fact_73_zero__less__double__add__iff__zero__less__single__add, axiom,
    ((![A : real]: ((ord_less_real @ zero_zero_real @ (plus_plus_real @ A @ A)) = (ord_less_real @ zero_zero_real @ A))))). % zero_less_double_add_iff_zero_less_single_add
thf(fact_74_double__add__less__zero__iff__single__add__less__zero, axiom,
    ((![A : real]: ((ord_less_real @ (plus_plus_real @ A @ A) @ zero_zero_real) = (ord_less_real @ A @ zero_zero_real))))). % double_add_less_zero_iff_single_add_less_zero
thf(fact_75_less__add__same__cancel2, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ (plus_plus_nat @ B @ A)) = (ord_less_nat @ zero_zero_nat @ B))))). % less_add_same_cancel2
thf(fact_76_less__add__same__cancel2, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ (plus_plus_real @ B @ A)) = (ord_less_real @ zero_zero_real @ B))))). % less_add_same_cancel2
thf(fact_77_less__add__same__cancel1, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ (plus_plus_nat @ A @ B)) = (ord_less_nat @ zero_zero_nat @ B))))). % less_add_same_cancel1
thf(fact_78_less__add__same__cancel1, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ (plus_plus_real @ A @ B)) = (ord_less_real @ zero_zero_real @ B))))). % less_add_same_cancel1
thf(fact_79_add__less__same__cancel2, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ (plus_plus_nat @ A @ B) @ B) = (ord_less_nat @ A @ zero_zero_nat))))). % add_less_same_cancel2
thf(fact_80_add__less__same__cancel2, axiom,
    ((![A : real, B : real]: ((ord_less_real @ (plus_plus_real @ A @ B) @ B) = (ord_less_real @ A @ zero_zero_real))))). % add_less_same_cancel2
thf(fact_81_add__less__same__cancel1, axiom,
    ((![B : nat, A : nat]: ((ord_less_nat @ (plus_plus_nat @ B @ A) @ B) = (ord_less_nat @ A @ zero_zero_nat))))). % add_less_same_cancel1
thf(fact_82_add__less__same__cancel1, axiom,
    ((![B : real, A : real]: ((ord_less_real @ (plus_plus_real @ B @ A) @ B) = (ord_less_real @ A @ zero_zero_real))))). % add_less_same_cancel1
thf(fact_83_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_84_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_85_diff__add__zero, axiom,
    ((![A : nat, B : nat]: ((minus_minus_nat @ A @ (plus_plus_nat @ A @ B)) = zero_zero_nat)))). % diff_add_zero
thf(fact_86_norm__zero, axiom,
    (((real_V646646907m_real @ zero_zero_real) = zero_zero_real))). % norm_zero
thf(fact_87_norm__zero, axiom,
    (((real_V638595069omplex @ zero_zero_complex) = zero_zero_real))). % norm_zero
thf(fact_88_norm__eq__zero, axiom,
    ((![X3 : real]: (((real_V646646907m_real @ X3) = zero_zero_real) = (X3 = zero_zero_real))))). % norm_eq_zero
thf(fact_89_norm__eq__zero, axiom,
    ((![X3 : complex]: (((real_V638595069omplex @ X3) = zero_zero_real) = (X3 = zero_zero_complex))))). % norm_eq_zero
thf(fact_90_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_91_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_92_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_93_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_94_ab__semigroup__add__class_Oadd__ac_I1_J, axiom,
    ((![A : nat, B : nat, C : nat]: ((plus_plus_nat @ (plus_plus_nat @ A @ B) @ C) = (plus_plus_nat @ A @ (plus_plus_nat @ B @ C)))))). % ab_semigroup_add_class.add_ac(1)
thf(fact_95_add__mono__thms__linordered__semiring_I4_J, axiom,
    ((![I : nat, J : nat, K : nat, L : nat]: (((I = J) & (K = L)) => ((plus_plus_nat @ I @ K) = (plus_plus_nat @ J @ L)))))). % add_mono_thms_linordered_semiring(4)
thf(fact_96_zero__reorient, axiom,
    ((![X3 : real]: ((zero_zero_real = X3) = (X3 = zero_zero_real))))). % zero_reorient
thf(fact_97_group__cancel_Oadd1, axiom,
    ((![A2 : nat, K : nat, A : nat, B : nat]: ((A2 = (plus_plus_nat @ K @ A)) => ((plus_plus_nat @ A2 @ B) = (plus_plus_nat @ K @ (plus_plus_nat @ A @ B))))))). % group_cancel.add1
thf(fact_98_group__cancel_Oadd2, axiom,
    ((![B2 : nat, K : nat, B : nat, A : nat]: ((B2 = (plus_plus_nat @ K @ B)) => ((plus_plus_nat @ A @ B2) = (plus_plus_nat @ K @ (plus_plus_nat @ A @ B))))))). % group_cancel.add2
thf(fact_99_comm__monoid__add__class_Oadd__0, axiom,
    ((![A : nat]: ((plus_plus_nat @ zero_zero_nat @ A) = A)))). % comm_monoid_add_class.add_0
thf(fact_100_comm__monoid__add__class_Oadd__0, axiom,
    ((![A : real]: ((plus_plus_real @ zero_zero_real @ A) = A)))). % comm_monoid_add_class.add_0
thf(fact_101_add_Oassoc, axiom,
    ((![A : nat, B : nat, C : nat]: ((plus_plus_nat @ (plus_plus_nat @ A @ B) @ C) = (plus_plus_nat @ A @ (plus_plus_nat @ B @ C)))))). % add.assoc
thf(fact_102_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_103_add_Ocommute, axiom,
    ((plus_plus_nat = (^[A3 : nat]: (^[B3 : nat]: (plus_plus_nat @ B3 @ A3)))))). % add.commute
thf(fact_104_add_Ocomm__neutral, axiom,
    ((![A : nat]: ((plus_plus_nat @ A @ zero_zero_nat) = A)))). % add.comm_neutral
thf(fact_105_add_Ocomm__neutral, axiom,
    ((![A : real]: ((plus_plus_real @ A @ zero_zero_real) = A)))). % add.comm_neutral
thf(fact_106_add_Ogroup__left__neutral, axiom,
    ((![A : real]: ((plus_plus_real @ zero_zero_real @ A) = A)))). % add.group_left_neutral
thf(fact_107_add_Oleft__commute, axiom,
    ((![B : nat, A : nat, C : nat]: ((plus_plus_nat @ B @ (plus_plus_nat @ A @ C)) = (plus_plus_nat @ A @ (plus_plus_nat @ B @ C)))))). % add.left_commute
thf(fact_108_add__left__imp__eq, axiom,
    ((![A : nat, B : nat, C : nat]: (((plus_plus_nat @ A @ B) = (plus_plus_nat @ A @ C)) => (B = C))))). % add_left_imp_eq
thf(fact_109_add__right__imp__eq, axiom,
    ((![B : nat, A : nat, C : nat]: (((plus_plus_nat @ B @ A) = (plus_plus_nat @ C @ A)) => (B = C))))). % add_right_imp_eq
thf(fact_110_norm__add__leD, axiom,
    ((![A : complex, B : complex, C : real]: ((ord_less_eq_real @ (real_V638595069omplex @ (plus_plus_complex @ A @ B)) @ C) => (ord_less_eq_real @ (real_V638595069omplex @ B) @ (plus_plus_real @ (real_V638595069omplex @ A) @ C)))))). % norm_add_leD
thf(fact_111_norm__ge__zero, axiom,
    ((![X3 : complex]: (ord_less_eq_real @ zero_zero_real @ (real_V638595069omplex @ X3))))). % norm_ge_zero
thf(fact_112_norm__diff__ineq, axiom,
    ((![A : complex, B : complex]: (ord_less_eq_real @ (minus_minus_real @ (real_V638595069omplex @ A) @ (real_V638595069omplex @ B)) @ (real_V638595069omplex @ (plus_plus_complex @ A @ B)))))). % norm_diff_ineq
thf(fact_113_norm__triangle__le, axiom,
    ((![X3 : complex, Y3 : complex, E : real]: ((ord_less_eq_real @ (plus_plus_real @ (real_V638595069omplex @ X3) @ (real_V638595069omplex @ Y3)) @ E) => (ord_less_eq_real @ (real_V638595069omplex @ (plus_plus_complex @ X3 @ Y3)) @ E))))). % norm_triangle_le
thf(fact_114_norm__triangle__ineq, axiom,
    ((![X3 : complex, Y3 : complex]: (ord_less_eq_real @ (real_V638595069omplex @ (plus_plus_complex @ X3 @ Y3)) @ (plus_plus_real @ (real_V638595069omplex @ X3) @ (real_V638595069omplex @ Y3)))))). % norm_triangle_ineq
thf(fact_115_norm__triangle__mono, axiom,
    ((![A : complex, R : real, B : complex, S2 : real]: ((ord_less_eq_real @ (real_V638595069omplex @ A) @ R) => ((ord_less_eq_real @ (real_V638595069omplex @ B) @ S2) => (ord_less_eq_real @ (real_V638595069omplex @ (plus_plus_complex @ A @ B)) @ (plus_plus_real @ R @ S2))))))). % norm_triangle_mono
thf(fact_116_add__nonpos__eq__0__iff, axiom,
    ((![X3 : nat, Y3 : nat]: ((ord_less_eq_nat @ X3 @ zero_zero_nat) => ((ord_less_eq_nat @ Y3 @ zero_zero_nat) => (((plus_plus_nat @ X3 @ Y3) = zero_zero_nat) = (((X3 = zero_zero_nat)) & ((Y3 = zero_zero_nat))))))))). % add_nonpos_eq_0_iff
thf(fact_117_add__nonpos__eq__0__iff, axiom,
    ((![X3 : real, Y3 : real]: ((ord_less_eq_real @ X3 @ zero_zero_real) => ((ord_less_eq_real @ Y3 @ zero_zero_real) => (((plus_plus_real @ X3 @ Y3) = zero_zero_real) = (((X3 = zero_zero_real)) & ((Y3 = zero_zero_real))))))))). % add_nonpos_eq_0_iff
thf(fact_118_add__nonneg__eq__0__iff, axiom,
    ((![X3 : nat, Y3 : nat]: ((ord_less_eq_nat @ zero_zero_nat @ X3) => ((ord_less_eq_nat @ zero_zero_nat @ Y3) => (((plus_plus_nat @ X3 @ Y3) = zero_zero_nat) = (((X3 = zero_zero_nat)) & ((Y3 = zero_zero_nat))))))))). % add_nonneg_eq_0_iff
thf(fact_119_add__nonneg__eq__0__iff, axiom,
    ((![X3 : real, Y3 : real]: ((ord_less_eq_real @ zero_zero_real @ X3) => ((ord_less_eq_real @ zero_zero_real @ Y3) => (((plus_plus_real @ X3 @ Y3) = zero_zero_real) = (((X3 = zero_zero_real)) & ((Y3 = zero_zero_real))))))))). % add_nonneg_eq_0_iff
thf(fact_120_add__nonpos__nonpos, axiom,
    ((![A : nat, B : nat]: ((ord_less_eq_nat @ A @ zero_zero_nat) => ((ord_less_eq_nat @ B @ zero_zero_nat) => (ord_less_eq_nat @ (plus_plus_nat @ A @ B) @ zero_zero_nat)))))). % add_nonpos_nonpos
thf(fact_121_add__nonpos__nonpos, axiom,
    ((![A : real, B : real]: ((ord_less_eq_real @ A @ zero_zero_real) => ((ord_less_eq_real @ B @ zero_zero_real) => (ord_less_eq_real @ (plus_plus_real @ A @ B) @ zero_zero_real)))))). % add_nonpos_nonpos
thf(fact_122_add__nonneg__nonneg, axiom,
    ((![A : nat, B : nat]: ((ord_less_eq_nat @ zero_zero_nat @ A) => ((ord_less_eq_nat @ zero_zero_nat @ B) => (ord_less_eq_nat @ zero_zero_nat @ (plus_plus_nat @ A @ B))))))). % add_nonneg_nonneg
thf(fact_123_add__nonneg__nonneg, axiom,
    ((![A : real, B : real]: ((ord_less_eq_real @ zero_zero_real @ A) => ((ord_less_eq_real @ zero_zero_real @ B) => (ord_less_eq_real @ zero_zero_real @ (plus_plus_real @ A @ B))))))). % add_nonneg_nonneg
thf(fact_124_add__increasing2, axiom,
    ((![C : nat, B : nat, A : nat]: ((ord_less_eq_nat @ zero_zero_nat @ C) => ((ord_less_eq_nat @ B @ A) => (ord_less_eq_nat @ B @ (plus_plus_nat @ A @ C))))))). % add_increasing2
thf(fact_125_add__increasing2, axiom,
    ((![C : real, B : real, A : real]: ((ord_less_eq_real @ zero_zero_real @ C) => ((ord_less_eq_real @ B @ A) => (ord_less_eq_real @ B @ (plus_plus_real @ A @ C))))))). % add_increasing2
thf(fact_126_add__decreasing2, axiom,
    ((![C : nat, A : nat, B : nat]: ((ord_less_eq_nat @ C @ zero_zero_nat) => ((ord_less_eq_nat @ A @ B) => (ord_less_eq_nat @ (plus_plus_nat @ A @ C) @ B)))))). % add_decreasing2
thf(fact_127_add__decreasing2, axiom,
    ((![C : real, A : real, B : real]: ((ord_less_eq_real @ C @ zero_zero_real) => ((ord_less_eq_real @ A @ B) => (ord_less_eq_real @ (plus_plus_real @ A @ C) @ B)))))). % add_decreasing2
thf(fact_128_add__increasing, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_eq_nat @ zero_zero_nat @ A) => ((ord_less_eq_nat @ B @ C) => (ord_less_eq_nat @ B @ (plus_plus_nat @ A @ C))))))). % add_increasing
thf(fact_129_add__increasing, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_eq_real @ zero_zero_real @ A) => ((ord_less_eq_real @ B @ C) => (ord_less_eq_real @ B @ (plus_plus_real @ A @ C))))))). % add_increasing
thf(fact_130_add__decreasing, axiom,
    ((![A : nat, C : nat, B : nat]: ((ord_less_eq_nat @ A @ zero_zero_nat) => ((ord_less_eq_nat @ C @ B) => (ord_less_eq_nat @ (plus_plus_nat @ A @ C) @ B)))))). % add_decreasing
thf(fact_131_add__decreasing, axiom,
    ((![A : real, C : real, B : real]: ((ord_less_eq_real @ A @ zero_zero_real) => ((ord_less_eq_real @ C @ B) => (ord_less_eq_real @ (plus_plus_real @ A @ C) @ B)))))). % add_decreasing
thf(fact_132_pos__add__strict, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_nat @ zero_zero_nat @ A) => ((ord_less_nat @ B @ C) => (ord_less_nat @ B @ (plus_plus_nat @ A @ C))))))). % pos_add_strict
thf(fact_133_pos__add__strict, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_real @ zero_zero_real @ A) => ((ord_less_real @ B @ C) => (ord_less_real @ B @ (plus_plus_real @ A @ C))))))). % pos_add_strict
thf(fact_134_canonically__ordered__monoid__add__class_OlessE, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (~ ((![C2 : nat]: ((B = (plus_plus_nat @ A @ C2)) => (C2 = zero_zero_nat))))))))). % canonically_ordered_monoid_add_class.lessE
thf(fact_135_add__pos__pos, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ zero_zero_nat @ A) => ((ord_less_nat @ zero_zero_nat @ B) => (ord_less_nat @ zero_zero_nat @ (plus_plus_nat @ A @ B))))))). % add_pos_pos
thf(fact_136_add__pos__pos, axiom,
    ((![A : real, B : real]: ((ord_less_real @ zero_zero_real @ A) => ((ord_less_real @ zero_zero_real @ B) => (ord_less_real @ zero_zero_real @ (plus_plus_real @ A @ B))))))). % add_pos_pos
thf(fact_137_add__neg__neg, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ zero_zero_nat) => ((ord_less_nat @ B @ zero_zero_nat) => (ord_less_nat @ (plus_plus_nat @ A @ B) @ zero_zero_nat)))))). % add_neg_neg
thf(fact_138_add__neg__neg, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ zero_zero_real) => ((ord_less_real @ B @ zero_zero_real) => (ord_less_real @ (plus_plus_real @ A @ B) @ zero_zero_real)))))). % add_neg_neg
thf(fact_139_zero__le, axiom,
    ((![X3 : nat]: (ord_less_eq_nat @ zero_zero_nat @ X3)))). % zero_le
thf(fact_140_norm__diff__triangle__ineq, axiom,
    ((![A : complex, B : complex, C : complex, D : complex]: (ord_less_eq_real @ (real_V638595069omplex @ (minus_minus_complex @ (plus_plus_complex @ A @ B) @ (plus_plus_complex @ C @ D))) @ (plus_plus_real @ (real_V638595069omplex @ (minus_minus_complex @ A @ C)) @ (real_V638595069omplex @ (minus_minus_complex @ B @ D))))))). % norm_diff_triangle_ineq
thf(fact_141_eq__iff__diff__eq__0, axiom,
    (((^[Y4 : real]: (^[Z2 : real]: (Y4 = Z2))) = (^[A3 : real]: (^[B3 : real]: ((minus_minus_real @ A3 @ B3) = zero_zero_real)))))). % eq_iff_diff_eq_0
thf(fact_142_eq__iff__diff__eq__0, axiom,
    (((^[Y4 : complex]: (^[Z2 : complex]: (Y4 = Z2))) = (^[A3 : complex]: (^[B3 : complex]: ((minus_minus_complex @ A3 @ B3) = zero_zero_complex)))))). % eq_iff_diff_eq_0
thf(fact_143_strict__mono__less__eq, axiom,
    ((![F2 : real > nat, X3 : real, Y3 : real]: ((order_297469111al_nat @ F2) => ((ord_less_eq_nat @ (F2 @ X3) @ (F2 @ Y3)) = (ord_less_eq_real @ X3 @ Y3)))))). % strict_mono_less_eq
thf(fact_144_strict__mono__less__eq, axiom,
    ((![F2 : nat > real, X3 : nat, Y3 : nat]: ((order_952716343t_real @ F2) => ((ord_less_eq_real @ (F2 @ X3) @ (F2 @ Y3)) = (ord_less_eq_nat @ X3 @ Y3)))))). % strict_mono_less_eq
thf(fact_145_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_146_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_147_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_148_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_149_strict__mono__def, axiom,
    ((order_1818878995l_real = (^[F : real > real]: (![X2 : real]: (![Y5 : real]: (((ord_less_real @ X2 @ Y5)) => ((ord_less_real @ (F @ X2) @ (F @ Y5)))))))))). % strict_mono_def
thf(fact_150_strict__mono__def, axiom,
    ((order_769474267at_nat = (^[F : nat > nat]: (![X2 : nat]: (![Y5 : nat]: (((ord_less_nat @ X2 @ Y5)) => ((ord_less_nat @ (F @ X2) @ (F @ Y5)))))))))). % strict_mono_def
thf(fact_151_strict__monoI, axiom,
    ((![F2 : real > real]: ((![X : real, Y6 : real]: ((ord_less_real @ X @ Y6) => (ord_less_real @ (F2 @ X) @ (F2 @ Y6)))) => (order_1818878995l_real @ F2))))). % strict_monoI
thf(fact_152_strict__monoI, axiom,
    ((![F2 : nat > nat]: ((![X : nat, Y6 : nat]: ((ord_less_nat @ X @ Y6) => (ord_less_nat @ (F2 @ X) @ (F2 @ Y6)))) => (order_769474267at_nat @ F2))))). % strict_monoI
thf(fact_153_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_154_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_155_add__strict__increasing2, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_eq_nat @ zero_zero_nat @ A) => ((ord_less_nat @ B @ C) => (ord_less_nat @ B @ (plus_plus_nat @ A @ C))))))). % add_strict_increasing2
thf(fact_156_add__strict__increasing2, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_eq_real @ zero_zero_real @ A) => ((ord_less_real @ B @ C) => (ord_less_real @ B @ (plus_plus_real @ A @ C))))))). % add_strict_increasing2
thf(fact_157_add__strict__increasing, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_nat @ zero_zero_nat @ A) => ((ord_less_eq_nat @ B @ C) => (ord_less_nat @ B @ (plus_plus_nat @ A @ C))))))). % add_strict_increasing
thf(fact_158_add__strict__increasing, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_real @ zero_zero_real @ A) => ((ord_less_eq_real @ B @ C) => (ord_less_real @ B @ (plus_plus_real @ A @ C))))))). % add_strict_increasing
thf(fact_159_add__pos__nonneg, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ zero_zero_nat @ A) => ((ord_less_eq_nat @ zero_zero_nat @ B) => (ord_less_nat @ zero_zero_nat @ (plus_plus_nat @ A @ B))))))). % add_pos_nonneg
thf(fact_160_add__pos__nonneg, axiom,
    ((![A : real, B : real]: ((ord_less_real @ zero_zero_real @ A) => ((ord_less_eq_real @ zero_zero_real @ B) => (ord_less_real @ zero_zero_real @ (plus_plus_real @ A @ B))))))). % add_pos_nonneg
thf(fact_161_add__nonpos__neg, axiom,
    ((![A : nat, B : nat]: ((ord_less_eq_nat @ A @ zero_zero_nat) => ((ord_less_nat @ B @ zero_zero_nat) => (ord_less_nat @ (plus_plus_nat @ A @ B) @ zero_zero_nat)))))). % add_nonpos_neg
thf(fact_162_add__nonpos__neg, axiom,
    ((![A : real, B : real]: ((ord_less_eq_real @ A @ zero_zero_real) => ((ord_less_real @ B @ zero_zero_real) => (ord_less_real @ (plus_plus_real @ A @ B) @ zero_zero_real)))))). % add_nonpos_neg
thf(fact_163_add__nonneg__pos, axiom,
    ((![A : nat, B : nat]: ((ord_less_eq_nat @ zero_zero_nat @ A) => ((ord_less_nat @ zero_zero_nat @ B) => (ord_less_nat @ zero_zero_nat @ (plus_plus_nat @ A @ B))))))). % add_nonneg_pos
thf(fact_164_add__nonneg__pos, axiom,
    ((![A : real, B : real]: ((ord_less_eq_real @ zero_zero_real @ A) => ((ord_less_real @ zero_zero_real @ B) => (ord_less_real @ zero_zero_real @ (plus_plus_real @ A @ B))))))). % add_nonneg_pos
thf(fact_165_add__neg__nonpos, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ zero_zero_nat) => ((ord_less_eq_nat @ B @ zero_zero_nat) => (ord_less_nat @ (plus_plus_nat @ A @ B) @ zero_zero_nat)))))). % add_neg_nonpos
thf(fact_166_add__neg__nonpos, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ zero_zero_real) => ((ord_less_eq_real @ B @ zero_zero_real) => (ord_less_real @ (plus_plus_real @ A @ B) @ zero_zero_real)))))). % add_neg_nonpos
thf(fact_167_norm__triangle__lt, axiom,
    ((![X3 : complex, Y3 : complex, E : real]: ((ord_less_real @ (plus_plus_real @ (real_V638595069omplex @ X3) @ (real_V638595069omplex @ Y3)) @ E) => (ord_less_real @ (real_V638595069omplex @ (plus_plus_complex @ X3 @ Y3)) @ E))))). % norm_triangle_lt
thf(fact_168_norm__add__less, axiom,
    ((![X3 : complex, R : real, Y3 : complex, S2 : real]: ((ord_less_real @ (real_V638595069omplex @ X3) @ R) => ((ord_less_real @ (real_V638595069omplex @ Y3) @ S2) => (ord_less_real @ (real_V638595069omplex @ (plus_plus_complex @ X3 @ Y3)) @ (plus_plus_real @ R @ S2))))))). % norm_add_less
thf(fact_169_le__iff__diff__le__0, axiom,
    ((ord_less_eq_real = (^[A3 : real]: (^[B3 : real]: (ord_less_eq_real @ (minus_minus_real @ A3 @ B3) @ zero_zero_real)))))). % le_iff_diff_le_0
thf(fact_170_less__iff__diff__less__0, axiom,
    ((ord_less_real = (^[A3 : real]: (^[B3 : real]: (ord_less_real @ (minus_minus_real @ A3 @ B3) @ zero_zero_real)))))). % less_iff_diff_less_0
thf(fact_171_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_172_norm__not__less__zero, axiom,
    ((![X3 : complex]: (~ ((ord_less_real @ (real_V638595069omplex @ X3) @ zero_zero_real)))))). % norm_not_less_zero
thf(fact_173_add__le__imp__le__right, axiom,
    ((![A : nat, C : nat, B : nat]: ((ord_less_eq_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ C)) => (ord_less_eq_nat @ A @ B))))). % add_le_imp_le_right
thf(fact_174_add__le__imp__le__right, axiom,
    ((![A : real, C : real, B : real]: ((ord_less_eq_real @ (plus_plus_real @ A @ C) @ (plus_plus_real @ B @ C)) => (ord_less_eq_real @ A @ B))))). % add_le_imp_le_right
thf(fact_175_add__le__imp__le__left, axiom,
    ((![C : nat, A : nat, B : nat]: ((ord_less_eq_nat @ (plus_plus_nat @ C @ A) @ (plus_plus_nat @ C @ B)) => (ord_less_eq_nat @ A @ B))))). % add_le_imp_le_left
thf(fact_176_add__le__imp__le__left, axiom,
    ((![C : real, A : real, B : real]: ((ord_less_eq_real @ (plus_plus_real @ C @ A) @ (plus_plus_real @ C @ B)) => (ord_less_eq_real @ A @ B))))). % add_le_imp_le_left
thf(fact_177_le__iff__add, axiom,
    ((ord_less_eq_nat = (^[A3 : nat]: (^[B3 : nat]: (?[C3 : nat]: (B3 = (plus_plus_nat @ A3 @ C3)))))))). % le_iff_add
thf(fact_178_add__right__mono, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ B) => (ord_less_eq_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ C)))))). % add_right_mono
thf(fact_179_add__right__mono, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_eq_real @ A @ B) => (ord_less_eq_real @ (plus_plus_real @ A @ C) @ (plus_plus_real @ B @ C)))))). % add_right_mono
thf(fact_180_less__eqE, axiom,
    ((![A : nat, B : nat]: ((ord_less_eq_nat @ A @ B) => (~ ((![C2 : nat]: (~ ((B = (plus_plus_nat @ A @ C2))))))))))). % less_eqE
thf(fact_181_add__left__mono, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ B) => (ord_less_eq_nat @ (plus_plus_nat @ C @ A) @ (plus_plus_nat @ C @ B)))))). % add_left_mono
thf(fact_182_add__left__mono, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_eq_real @ A @ B) => (ord_less_eq_real @ (plus_plus_real @ C @ A) @ (plus_plus_real @ C @ B)))))). % add_left_mono
thf(fact_183_add__mono, axiom,
    ((![A : nat, B : nat, C : nat, D : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_nat @ C @ D) => (ord_less_eq_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ D))))))). % add_mono
thf(fact_184_add__mono, axiom,
    ((![A : real, B : real, C : real, D : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_eq_real @ C @ D) => (ord_less_eq_real @ (plus_plus_real @ A @ C) @ (plus_plus_real @ B @ D))))))). % add_mono
thf(fact_185_add__mono__thms__linordered__semiring_I1_J, axiom,
    ((![I : nat, J : nat, K : nat, L : nat]: (((ord_less_eq_nat @ I @ J) & (ord_less_eq_nat @ K @ L)) => (ord_less_eq_nat @ (plus_plus_nat @ I @ K) @ (plus_plus_nat @ J @ L)))))). % add_mono_thms_linordered_semiring(1)
thf(fact_186_add__mono__thms__linordered__semiring_I1_J, axiom,
    ((![I : real, J : real, K : real, L : real]: (((ord_less_eq_real @ I @ J) & (ord_less_eq_real @ K @ L)) => (ord_less_eq_real @ (plus_plus_real @ I @ K) @ (plus_plus_real @ J @ L)))))). % add_mono_thms_linordered_semiring(1)
thf(fact_187_add__mono__thms__linordered__semiring_I2_J, axiom,
    ((![I : nat, J : nat, K : nat, L : nat]: (((I = J) & (ord_less_eq_nat @ K @ L)) => (ord_less_eq_nat @ (plus_plus_nat @ I @ K) @ (plus_plus_nat @ J @ L)))))). % add_mono_thms_linordered_semiring(2)
thf(fact_188_add__mono__thms__linordered__semiring_I2_J, axiom,
    ((![I : real, J : real, K : real, L : real]: (((I = J) & (ord_less_eq_real @ K @ L)) => (ord_less_eq_real @ (plus_plus_real @ I @ K) @ (plus_plus_real @ J @ L)))))). % add_mono_thms_linordered_semiring(2)
thf(fact_189_add__mono__thms__linordered__semiring_I3_J, axiom,
    ((![I : nat, J : nat, K : nat, L : nat]: (((ord_less_eq_nat @ I @ J) & (K = L)) => (ord_less_eq_nat @ (plus_plus_nat @ I @ K) @ (plus_plus_nat @ J @ L)))))). % add_mono_thms_linordered_semiring(3)
thf(fact_190_add__mono__thms__linordered__semiring_I3_J, axiom,
    ((![I : real, J : real, K : real, L : real]: (((ord_less_eq_real @ I @ J) & (K = L)) => (ord_less_eq_real @ (plus_plus_real @ I @ K) @ (plus_plus_real @ J @ L)))))). % add_mono_thms_linordered_semiring(3)
thf(fact_191_add__less__imp__less__right, axiom,
    ((![A : nat, C : nat, B : nat]: ((ord_less_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ C)) => (ord_less_nat @ A @ B))))). % add_less_imp_less_right
thf(fact_192_add__less__imp__less__right, axiom,
    ((![A : real, C : real, B : real]: ((ord_less_real @ (plus_plus_real @ A @ C) @ (plus_plus_real @ B @ C)) => (ord_less_real @ A @ B))))). % add_less_imp_less_right
thf(fact_193_add__less__imp__less__left, axiom,
    ((![C : nat, A : nat, B : nat]: ((ord_less_nat @ (plus_plus_nat @ C @ A) @ (plus_plus_nat @ C @ B)) => (ord_less_nat @ A @ B))))). % add_less_imp_less_left
thf(fact_194_add__less__imp__less__left, axiom,
    ((![C : real, A : real, B : real]: ((ord_less_real @ (plus_plus_real @ C @ A) @ (plus_plus_real @ C @ B)) => (ord_less_real @ A @ B))))). % add_less_imp_less_left
thf(fact_195_add__strict__right__mono, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_nat @ A @ B) => (ord_less_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ C)))))). % add_strict_right_mono
thf(fact_196_add__strict__right__mono, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_real @ A @ B) => (ord_less_real @ (plus_plus_real @ A @ C) @ (plus_plus_real @ B @ C)))))). % add_strict_right_mono
thf(fact_197_add__strict__left__mono, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_nat @ A @ B) => (ord_less_nat @ (plus_plus_nat @ C @ A) @ (plus_plus_nat @ C @ B)))))). % add_strict_left_mono
thf(fact_198_add__strict__left__mono, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_real @ A @ B) => (ord_less_real @ (plus_plus_real @ C @ A) @ (plus_plus_real @ C @ B)))))). % add_strict_left_mono
thf(fact_199_add__strict__mono, axiom,
    ((![A : nat, B : nat, C : nat, D : nat]: ((ord_less_nat @ A @ B) => ((ord_less_nat @ C @ D) => (ord_less_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ D))))))). % add_strict_mono
thf(fact_200_add__strict__mono, axiom,
    ((![A : real, B : real, C : real, D : real]: ((ord_less_real @ A @ B) => ((ord_less_real @ C @ D) => (ord_less_real @ (plus_plus_real @ A @ C) @ (plus_plus_real @ B @ D))))))). % add_strict_mono
thf(fact_201_add__mono__thms__linordered__field_I1_J, axiom,
    ((![I : nat, J : nat, K : nat, L : nat]: (((ord_less_nat @ I @ J) & (K = L)) => (ord_less_nat @ (plus_plus_nat @ I @ K) @ (plus_plus_nat @ J @ L)))))). % add_mono_thms_linordered_field(1)
thf(fact_202_add__mono__thms__linordered__field_I1_J, axiom,
    ((![I : real, J : real, K : real, L : real]: (((ord_less_real @ I @ J) & (K = L)) => (ord_less_real @ (plus_plus_real @ I @ K) @ (plus_plus_real @ J @ L)))))). % add_mono_thms_linordered_field(1)
thf(fact_203_add__mono__thms__linordered__field_I2_J, axiom,
    ((![I : nat, J : nat, K : nat, L : nat]: (((I = J) & (ord_less_nat @ K @ L)) => (ord_less_nat @ (plus_plus_nat @ I @ K) @ (plus_plus_nat @ J @ L)))))). % add_mono_thms_linordered_field(2)
thf(fact_204_add__mono__thms__linordered__field_I2_J, axiom,
    ((![I : real, J : real, K : real, L : real]: (((I = J) & (ord_less_real @ K @ L)) => (ord_less_real @ (plus_plus_real @ I @ K) @ (plus_plus_real @ J @ L)))))). % add_mono_thms_linordered_field(2)
thf(fact_205_add__mono__thms__linordered__field_I5_J, axiom,
    ((![I : nat, J : nat, K : nat, L : nat]: (((ord_less_nat @ I @ J) & (ord_less_nat @ K @ L)) => (ord_less_nat @ (plus_plus_nat @ I @ K) @ (plus_plus_nat @ J @ L)))))). % add_mono_thms_linordered_field(5)
thf(fact_206_add__mono__thms__linordered__field_I5_J, axiom,
    ((![I : real, J : real, K : real, L : real]: (((ord_less_real @ I @ J) & (ord_less_real @ K @ L)) => (ord_less_real @ (plus_plus_real @ I @ K) @ (plus_plus_real @ J @ L)))))). % add_mono_thms_linordered_field(5)
thf(fact_207_add__implies__diff, axiom,
    ((![C : nat, B : nat, A : nat]: (((plus_plus_nat @ C @ B) = A) => (C = (minus_minus_nat @ A @ B)))))). % add_implies_diff
thf(fact_208_add__implies__diff, axiom,
    ((![C : complex, B : complex, A : complex]: (((plus_plus_complex @ C @ B) = A) => (C = (minus_minus_complex @ A @ B)))))). % add_implies_diff
thf(fact_209_diff__diff__add, axiom,
    ((![A : nat, B : nat, C : nat]: ((minus_minus_nat @ (minus_minus_nat @ A @ B) @ C) = (minus_minus_nat @ A @ (plus_plus_nat @ B @ C)))))). % diff_diff_add
thf(fact_210_diff__diff__add, axiom,
    ((![A : complex, B : complex, C : complex]: ((minus_minus_complex @ (minus_minus_complex @ A @ B) @ C) = (minus_minus_complex @ A @ (plus_plus_complex @ B @ C)))))). % diff_diff_add
thf(fact_211_diff__add__eq__diff__diff__swap, axiom,
    ((![A : complex, B : complex, C : complex]: ((minus_minus_complex @ A @ (plus_plus_complex @ B @ C)) = (minus_minus_complex @ (minus_minus_complex @ A @ C) @ B))))). % diff_add_eq_diff_diff_swap
thf(fact_212_diff__add__eq, axiom,
    ((![A : complex, B : complex, C : complex]: ((plus_plus_complex @ (minus_minus_complex @ A @ B) @ C) = (minus_minus_complex @ (plus_plus_complex @ A @ C) @ B))))). % diff_add_eq
thf(fact_213_diff__diff__eq2, axiom,
    ((![A : complex, B : complex, C : complex]: ((minus_minus_complex @ A @ (minus_minus_complex @ B @ C)) = (minus_minus_complex @ (plus_plus_complex @ A @ C) @ B))))). % diff_diff_eq2
thf(fact_214_add__diff__eq, axiom,
    ((![A : complex, B : complex, C : complex]: ((plus_plus_complex @ A @ (minus_minus_complex @ B @ C)) = (minus_minus_complex @ (plus_plus_complex @ A @ B) @ C))))). % add_diff_eq

% Conjectures (1)
thf(conj_0, conjecture,
    ((?[N3 : nat]: (![N4 : nat]: ((~ ((ord_less_eq_nat @ N3 @ N4))) | (ord_less_real @ (real_V638595069omplex @ (minus_minus_complex @ (s @ (comp_nat_nat_nat @ f @ g @ N4)) @ (complex2 @ x @ y))) @ e)))))).
