% 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_303__5370572_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:29:04.306

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

% Explicit typings (34)
thf(sy_c_Complex_Ocomplex_OComplex, type,
    complex2 : real > real > complex).
thf(sy_c_Complex_Ocomplex_OIm, type,
    im : complex > real).
thf(sy_c_Complex_Ocomplex_ORe, type,
    re : 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_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_Num_Onum_OBit0, type,
    bit0 : num > num).
thf(sy_c_Num_Onum_OOne, type,
    one : num).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Complex__Ocomplex, type,
    numera632737353omplex : num > complex).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal, type,
    numeral_numeral_real : num > 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_Real__Vector__Spaces_Onorm__class_Onorm_001t__Complex__Ocomplex, type,
    real_V638595069omplex : complex > real).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Real__Oreal, type,
    real_V646646907m_real : real > real).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Complex__Ocomplex, type,
    divide1210191872omplex : complex > complex > complex).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal, type,
    divide_divide_real : 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_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 (197)
thf(fact_0_r, axiom,
    ((![N : nat]: (ord_less_eq_real @ (real_V638595069omplex @ (s @ N)) @ r)))). % r
thf(fact_1_f_I1_J, axiom,
    ((order_769474267at_nat @ f))). % f(1)
thf(fact_2_g_I1_J, axiom,
    ((order_769474267at_nat @ g))). % g(1)
thf(fact_3__092_060open_062cmod_A_Is_A_If_A_Ig_An_J_J_A_N_AComplex_Ax_Ay_J_A_092_060le_062_A_092_060bar_062Re_A_Is_A_If_A_Ig_An_J_J_J_A_N_ARe_A_IComplex_Ax_Ay_J_092_060bar_062_A_L_A_092_060bar_062Im_A_Is_A_If_A_Ig_An_J_J_J_A_N_AIm_A_IComplex_Ax_Ay_J_092_060bar_062_092_060close_062, axiom,
    ((ord_less_eq_real @ (real_V638595069omplex @ (minus_minus_complex @ (s @ (f @ (g @ n))) @ (complex2 @ x @ y))) @ (plus_plus_real @ (abs_abs_real @ (minus_minus_real @ (re @ (s @ (f @ (g @ n)))) @ (re @ (complex2 @ x @ y)))) @ (abs_abs_real @ (minus_minus_real @ (im @ (s @ (f @ (g @ n)))) @ (im @ (complex2 @ x @ y)))))))). % \<open>cmod (s (f (g n)) - Complex x y) \<le> \<bar>Re (s (f (g n))) - Re (Complex x y)\<bar> + \<bar>Im (s (f (g n))) - Im (Complex x y)\<bar>\<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__092_060open_062_092_060bar_062Re_A_Is_A_If_A_Ig_An_J_J_J_A_N_Ax_092_060bar_062_A_L_A_092_060bar_062Im_A_Is_A_If_A_Ig_An_J_J_J_A_N_Ay_092_060bar_062_A_060_Ae_A_P_A2_A_L_Ae_A_P_A2_092_060close_062, axiom,
    ((ord_less_real @ (plus_plus_real @ (abs_abs_real @ (minus_minus_real @ (re @ (s @ (f @ (g @ n)))) @ x)) @ (abs_abs_real @ (minus_minus_real @ (im @ (s @ (f @ (g @ n)))) @ y))) @ (plus_plus_real @ (divide_divide_real @ e @ (numeral_numeral_real @ (bit0 @ one))) @ (divide_divide_real @ e @ (numeral_numeral_real @ (bit0 @ one))))))). % \<open>\<bar>Re (s (f (g n))) - x\<bar> + \<bar>Im (s (f (g n))) - y\<bar> < e / 2 + e / 2\<close>
thf(fact_6_hs, axiom,
    ((order_769474267at_nat @ (comp_nat_nat_nat @ f @ g)))). % hs
thf(fact_7_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_8_comp__apply, axiom,
    ((comp_nat_nat_nat = (^[F : nat > nat]: (^[G : nat > nat]: (^[X2 : nat]: (F @ (G @ X2)))))))). % comp_apply
thf(fact_9_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_10_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_11_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_12_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_13_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_14_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_15_norm__diff__triangle__less, axiom,
    ((![X3 : complex, Y3 : complex, E1 : real, Z2 : complex, E2 : real]: ((ord_less_real @ (real_V638595069omplex @ (minus_minus_complex @ X3 @ Y3)) @ E1) => ((ord_less_real @ (real_V638595069omplex @ (minus_minus_complex @ Y3 @ Z2)) @ E2) => (ord_less_real @ (real_V638595069omplex @ (minus_minus_complex @ X3 @ Z2)) @ (plus_plus_real @ E1 @ E2))))))). % norm_diff_triangle_less
thf(fact_16_norm__diff__triangle__less, axiom,
    ((![X3 : real, Y3 : real, E1 : real, Z2 : real, E2 : real]: ((ord_less_real @ (real_V646646907m_real @ (minus_minus_real @ X3 @ Y3)) @ E1) => ((ord_less_real @ (real_V646646907m_real @ (minus_minus_real @ Y3 @ Z2)) @ E2) => (ord_less_real @ (real_V646646907m_real @ (minus_minus_real @ X3 @ Z2)) @ (plus_plus_real @ E1 @ E2))))))). % norm_diff_triangle_less
thf(fact_17__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062N1_AN2_O_A_092_060lbrakk_062_092_060forall_062n_092_060ge_062N1_O_A_092_060bar_062Re_A_Is_A_If_An_J_J_A_N_Ax_092_060bar_062_A_060_Ae_A_P_A2_059_A_092_060forall_062n_092_060ge_062N2_O_A_092_060bar_062Im_A_Is_A_If_A_Ig_An_J_J_J_A_N_Ay_092_060bar_062_A_060_Ae_A_P_A2_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom,
    ((~ (((?[N1 : nat]: (![N : nat]: ((ord_less_eq_nat @ N1 @ N) => (ord_less_real @ (abs_abs_real @ (minus_minus_real @ (re @ (s @ (f @ N))) @ x)) @ (divide_divide_real @ e @ (numeral_numeral_real @ (bit0 @ one))))))) => (![N2 : nat]: (~ ((![N : nat]: ((ord_less_eq_nat @ N2 @ N) => (ord_less_real @ (abs_abs_real @ (minus_minus_real @ (im @ (s @ (f @ (g @ N)))) @ y)) @ (divide_divide_real @ e @ (numeral_numeral_real @ (bit0 @ one)))))))))))))). % \<open>\<And>thesis. (\<And>N1 N2. \<lbrakk>\<forall>n\<ge>N1. \<bar>Re (s (f n)) - x\<bar> < e / 2; \<forall>n\<ge>N2. \<bar>Im (s (f (g n))) - y\<bar> < e / 2\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_18_add__left__cancel, axiom,
    ((![A : real, B : real, C : real]: (((plus_plus_real @ A @ B) = (plus_plus_real @ A @ C)) = (B = C))))). % add_left_cancel
thf(fact_19_add__right__cancel, axiom,
    ((![B : real, A : real, C : real]: (((plus_plus_real @ B @ A) = (plus_plus_real @ C @ A)) = (B = C))))). % add_right_cancel
thf(fact_20_abs__idempotent, axiom,
    ((![A : real]: ((abs_abs_real @ (abs_abs_real @ A)) = (abs_abs_real @ A))))). % abs_idempotent
thf(fact_21_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_22_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_23_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_24_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_25_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_26_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_27_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_28_add__diff__cancel__right_H, axiom,
    ((![A : real, B : real]: ((minus_minus_real @ (plus_plus_real @ A @ B) @ B) = A)))). % add_diff_cancel_right'
thf(fact_29_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_30_add__diff__cancel__right, axiom,
    ((![A : real, C : real, B : real]: ((minus_minus_real @ (plus_plus_real @ A @ C) @ (plus_plus_real @ B @ C)) = (minus_minus_real @ A @ B))))). % add_diff_cancel_right
thf(fact_31_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_32_add__diff__cancel__left_H, axiom,
    ((![A : real, B : real]: ((minus_minus_real @ (plus_plus_real @ A @ B) @ A) = B)))). % add_diff_cancel_left'
thf(fact_33_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_34_add__diff__cancel__left, axiom,
    ((![C : real, A : real, B : real]: ((minus_minus_real @ (plus_plus_real @ C @ A) @ (plus_plus_real @ C @ B)) = (minus_minus_real @ A @ B))))). % add_diff_cancel_left
thf(fact_35_diff__add__cancel, axiom,
    ((![A : complex, B : complex]: ((plus_plus_complex @ (minus_minus_complex @ A @ B) @ B) = A)))). % diff_add_cancel
thf(fact_36_diff__add__cancel, axiom,
    ((![A : real, B : real]: ((plus_plus_real @ (minus_minus_real @ A @ B) @ B) = A)))). % diff_add_cancel
thf(fact_37_add__diff__cancel, axiom,
    ((![A : complex, B : complex]: ((minus_minus_complex @ (plus_plus_complex @ A @ B) @ B) = A)))). % add_diff_cancel
thf(fact_38_add__diff__cancel, axiom,
    ((![A : real, B : real]: ((minus_minus_real @ (plus_plus_real @ A @ B) @ B) = A)))). % add_diff_cancel
thf(fact_39_abs__add__abs, axiom,
    ((![A : real, B : real]: ((abs_abs_real @ (plus_plus_real @ (abs_abs_real @ A) @ (abs_abs_real @ B))) = (plus_plus_real @ (abs_abs_real @ A) @ (abs_abs_real @ B)))))). % abs_add_abs
thf(fact_40_abs__norm__cancel, axiom,
    ((![A : complex]: ((abs_abs_real @ (real_V638595069omplex @ A)) = (real_V638595069omplex @ A))))). % abs_norm_cancel
thf(fact_41_abs__norm__cancel, axiom,
    ((![A : real]: ((abs_abs_real @ (real_V646646907m_real @ A)) = (real_V646646907m_real @ A))))). % abs_norm_cancel
thf(fact_42_norm__numeral, axiom,
    ((![W : num]: ((real_V638595069omplex @ (numera632737353omplex @ W)) = (numeral_numeral_real @ W))))). % norm_numeral
thf(fact_43_norm__numeral, axiom,
    ((![W : num]: ((real_V646646907m_real @ (numeral_numeral_real @ W)) = (numeral_numeral_real @ W))))). % norm_numeral
thf(fact_44_complex_Ocollapse, axiom,
    ((![Complex : complex]: ((complex2 @ (re @ Complex) @ (im @ Complex)) = Complex)))). % complex.collapse
thf(fact_45_complex__surj, axiom,
    ((![Z2 : complex]: ((complex2 @ (re @ Z2) @ (im @ Z2)) = Z2)))). % complex_surj
thf(fact_46_N1, axiom,
    ((![N : nat]: ((ord_less_eq_nat @ n1 @ N) => (ord_less_real @ (abs_abs_real @ (minus_minus_real @ (re @ (s @ (f @ N))) @ x)) @ (divide_divide_real @ e @ (numeral_numeral_real @ (bit0 @ one)))))))). % N1
thf(fact_47_nN2, axiom,
    ((ord_less_eq_nat @ n2 @ n))). % nN2
thf(fact_48_N2, axiom,
    ((![N : nat]: ((ord_less_eq_nat @ n2 @ N) => (ord_less_real @ (abs_abs_real @ (minus_minus_real @ (im @ (s @ (f @ (g @ N)))) @ y)) @ (divide_divide_real @ e @ (numeral_numeral_real @ (bit0 @ one)))))))). % N2
thf(fact_49__092_060open_062_092_060exists_062n0_O_A_092_060forall_062n_092_060ge_062n0_O_A_092_060bar_062Re_A_Is_A_If_An_J_J_A_N_Ax_092_060bar_062_A_060_Ae_A_P_A2_092_060close_062, axiom,
    ((?[N0 : nat]: (![N : nat]: ((ord_less_eq_nat @ N0 @ N) => (ord_less_real @ (abs_abs_real @ (minus_minus_real @ (re @ (s @ (f @ N))) @ x)) @ (divide_divide_real @ e @ (numeral_numeral_real @ (bit0 @ one))))))))). % \<open>\<exists>n0. \<forall>n\<ge>n0. \<bar>Re (s (f n)) - x\<bar> < e / 2\<close>
thf(fact_50__092_060open_062_092_060exists_062n0_O_A_092_060forall_062n_092_060ge_062n0_O_A_092_060bar_062Im_A_Is_A_If_A_Ig_An_J_J_J_A_N_Ay_092_060bar_062_A_060_Ae_A_P_A2_092_060close_062, axiom,
    ((?[N0 : nat]: (![N : nat]: ((ord_less_eq_nat @ N0 @ N) => (ord_less_real @ (abs_abs_real @ (minus_minus_real @ (im @ (s @ (f @ (g @ N)))) @ y)) @ (divide_divide_real @ e @ (numeral_numeral_real @ (bit0 @ one))))))))). % \<open>\<exists>n0. \<forall>n\<ge>n0. \<bar>Im (s (f (g n))) - y\<bar> < e / 2\<close>
thf(fact_51_norm__divide__numeral, axiom,
    ((![A : complex, W : num]: ((real_V638595069omplex @ (divide1210191872omplex @ A @ (numera632737353omplex @ W))) = (divide_divide_real @ (real_V638595069omplex @ A) @ (numeral_numeral_real @ W)))))). % norm_divide_numeral
thf(fact_52_norm__divide__numeral, axiom,
    ((![A : real, W : num]: ((real_V646646907m_real @ (divide_divide_real @ A @ (numeral_numeral_real @ W))) = (divide_divide_real @ (real_V646646907m_real @ A) @ (numeral_numeral_real @ W)))))). % norm_divide_numeral
thf(fact_53_nN1, axiom,
    ((ord_less_eq_nat @ n1 @ (g @ n)))). % nN1
thf(fact_54_ab__semigroup__add__class_Oadd__ac_I1_J, axiom,
    ((![A : real, B : real, C : real]: ((plus_plus_real @ (plus_plus_real @ A @ B) @ C) = (plus_plus_real @ A @ (plus_plus_real @ B @ C)))))). % ab_semigroup_add_class.add_ac(1)
thf(fact_55_add__mono__thms__linordered__semiring_I4_J, axiom,
    ((![I : real, J : real, K : real, L : real]: (((I = J) & (K = L)) => ((plus_plus_real @ I @ K) = (plus_plus_real @ J @ L)))))). % add_mono_thms_linordered_semiring(4)
thf(fact_56_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_57_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_58_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_59_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_60_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_61_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_62_complex__eqI, axiom,
    ((![X3 : complex, Y3 : complex]: (((re @ X3) = (re @ Y3)) => (((im @ X3) = (im @ Y3)) => (X3 = Y3)))))). % complex_eqI
thf(fact_63_complex_Oexpand, axiom,
    ((![Complex : complex, Complex2 : complex]: ((((re @ Complex) = (re @ Complex2)) & ((im @ Complex) = (im @ Complex2))) => (Complex = Complex2))))). % complex.expand
thf(fact_64_complex__eq__iff, axiom,
    (((^[Y4 : complex]: (^[Z3 : complex]: (Y4 = Z3))) = (^[X2 : complex]: (^[Y5 : complex]: ((((re @ X2) = (re @ Y5))) & (((im @ X2) = (im @ Y5))))))))). % complex_eq_iff
thf(fact_65_group__cancel_Oadd1, axiom,
    ((![A2 : real, K : real, A : real, B : real]: ((A2 = (plus_plus_real @ K @ A)) => ((plus_plus_real @ A2 @ B) = (plus_plus_real @ K @ (plus_plus_real @ A @ B))))))). % group_cancel.add1
thf(fact_66_group__cancel_Oadd2, axiom,
    ((![B2 : real, K : real, B : real, A : real]: ((B2 = (plus_plus_real @ K @ B)) => ((plus_plus_real @ A @ B2) = (plus_plus_real @ K @ (plus_plus_real @ A @ B))))))). % group_cancel.add2
thf(fact_67_complex_Ocoinduct__strong, axiom,
    ((![R : complex > complex > $o, Complex : complex, Complex2 : complex]: ((R @ Complex @ Complex2) => ((![Complex3 : complex, Complex4 : complex]: ((R @ Complex3 @ Complex4) => (((re @ Complex3) = (re @ Complex4)) & ((im @ Complex3) = (im @ Complex4))))) => (Complex = Complex2)))))). % complex.coinduct_strong
thf(fact_68_real__norm__def, axiom,
    ((real_V646646907m_real = abs_abs_real))). % real_norm_def
thf(fact_69_add_Oassoc, axiom,
    ((![A : real, B : real, C : real]: ((plus_plus_real @ (plus_plus_real @ A @ B) @ C) = (plus_plus_real @ A @ (plus_plus_real @ B @ C)))))). % add.assoc
thf(fact_70_add_Oleft__cancel, axiom,
    ((![A : real, B : real, C : real]: (((plus_plus_real @ A @ B) = (plus_plus_real @ A @ C)) = (B = C))))). % add.left_cancel
thf(fact_71_add_Oright__cancel, axiom,
    ((![B : real, A : real, C : real]: (((plus_plus_real @ B @ A) = (plus_plus_real @ C @ A)) = (B = C))))). % add.right_cancel
thf(fact_72_add_Ocommute, axiom,
    ((plus_plus_real = (^[A3 : real]: (^[B3 : real]: (plus_plus_real @ B3 @ A3)))))). % add.commute
thf(fact_73_add_Oleft__commute, axiom,
    ((![B : real, A : real, C : real]: ((plus_plus_real @ B @ (plus_plus_real @ A @ C)) = (plus_plus_real @ A @ (plus_plus_real @ B @ C)))))). % add.left_commute
thf(fact_74_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_75_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_76_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_77_add__left__imp__eq, axiom,
    ((![A : real, B : real, C : real]: (((plus_plus_real @ A @ B) = (plus_plus_real @ A @ C)) => (B = C))))). % add_left_imp_eq
thf(fact_78_abs__ge__self, axiom,
    ((![A : real]: (ord_less_eq_real @ A @ (abs_abs_real @ A))))). % abs_ge_self
thf(fact_79_add__right__imp__eq, axiom,
    ((![B : real, A : real, C : real]: (((plus_plus_real @ B @ A) = (plus_plus_real @ C @ A)) => (B = C))))). % add_right_imp_eq
thf(fact_80_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_81_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_82_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_83_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_84_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_85_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_86_abs__triangle__ineq, axiom,
    ((![A : real, B : real]: (ord_less_eq_real @ (abs_abs_real @ (plus_plus_real @ A @ B)) @ (plus_plus_real @ (abs_abs_real @ A) @ (abs_abs_real @ B)))))). % abs_triangle_ineq
thf(fact_87_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_88_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_89_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_90_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_91_cmod__le, axiom,
    ((![Z2 : complex]: (ord_less_eq_real @ (real_V638595069omplex @ Z2) @ (plus_plus_real @ (abs_abs_real @ (re @ Z2)) @ (abs_abs_real @ (im @ Z2))))))). % cmod_le
thf(fact_92_cmod__Re__le__iff, axiom,
    ((![X3 : complex, Y3 : complex]: (((im @ X3) = (im @ Y3)) => ((ord_less_eq_real @ (real_V638595069omplex @ X3) @ (real_V638595069omplex @ Y3)) = (ord_less_eq_real @ (abs_abs_real @ (re @ X3)) @ (abs_abs_real @ (re @ Y3)))))))). % cmod_Re_le_iff
thf(fact_93_cmod__Im__le__iff, axiom,
    ((![X3 : complex, Y3 : complex]: (((re @ X3) = (re @ Y3)) => ((ord_less_eq_real @ (real_V638595069omplex @ X3) @ (real_V638595069omplex @ Y3)) = (ord_less_eq_real @ (abs_abs_real @ (im @ X3)) @ (abs_abs_real @ (im @ Y3)))))))). % cmod_Im_le_iff
thf(fact_94_abs__diff__triangle__ineq, axiom,
    ((![A : real, B : real, C : real, D : real]: (ord_less_eq_real @ (abs_abs_real @ (minus_minus_real @ (plus_plus_real @ A @ B) @ (plus_plus_real @ C @ D))) @ (plus_plus_real @ (abs_abs_real @ (minus_minus_real @ A @ C)) @ (abs_abs_real @ (minus_minus_real @ B @ D))))))). % abs_diff_triangle_ineq
thf(fact_95_abs__triangle__ineq4, axiom,
    ((![A : real, B : real]: (ord_less_eq_real @ (abs_abs_real @ (minus_minus_real @ A @ B)) @ (plus_plus_real @ (abs_abs_real @ A) @ (abs_abs_real @ B)))))). % abs_triangle_ineq4
thf(fact_96_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_97_norm__diff__ineq, axiom,
    ((![A : real, B : real]: (ord_less_eq_real @ (minus_minus_real @ (real_V646646907m_real @ A) @ (real_V646646907m_real @ B)) @ (real_V646646907m_real @ (plus_plus_real @ A @ B)))))). % norm_diff_ineq
thf(fact_98_norm__triangle__mono, axiom,
    ((![A : complex, R2 : real, B : complex, S2 : real]: ((ord_less_eq_real @ (real_V638595069omplex @ A) @ R2) => ((ord_less_eq_real @ (real_V638595069omplex @ B) @ S2) => (ord_less_eq_real @ (real_V638595069omplex @ (plus_plus_complex @ A @ B)) @ (plus_plus_real @ R2 @ S2))))))). % norm_triangle_mono
thf(fact_99_norm__triangle__mono, axiom,
    ((![A : real, R2 : real, B : real, S2 : real]: ((ord_less_eq_real @ (real_V646646907m_real @ A) @ R2) => ((ord_less_eq_real @ (real_V646646907m_real @ B) @ S2) => (ord_less_eq_real @ (real_V646646907m_real @ (plus_plus_real @ A @ B)) @ (plus_plus_real @ R2 @ S2))))))). % norm_triangle_mono
thf(fact_100_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_101_norm__triangle__ineq, axiom,
    ((![X3 : real, Y3 : real]: (ord_less_eq_real @ (real_V646646907m_real @ (plus_plus_real @ X3 @ Y3)) @ (plus_plus_real @ (real_V646646907m_real @ X3) @ (real_V646646907m_real @ Y3)))))). % norm_triangle_ineq
thf(fact_102_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_103_norm__triangle__le, axiom,
    ((![X3 : real, Y3 : real, E : real]: ((ord_less_eq_real @ (plus_plus_real @ (real_V646646907m_real @ X3) @ (real_V646646907m_real @ Y3)) @ E) => (ord_less_eq_real @ (real_V646646907m_real @ (plus_plus_real @ X3 @ Y3)) @ E))))). % norm_triangle_le
thf(fact_104_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_105_norm__add__leD, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_eq_real @ (real_V646646907m_real @ (plus_plus_real @ A @ B)) @ C) => (ord_less_eq_real @ (real_V646646907m_real @ B) @ (plus_plus_real @ (real_V646646907m_real @ A) @ C)))))). % norm_add_leD
thf(fact_106_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_107_abs__Re__le__cmod, axiom,
    ((![X3 : complex]: (ord_less_eq_real @ (abs_abs_real @ (re @ X3)) @ (real_V638595069omplex @ X3))))). % abs_Re_le_cmod
thf(fact_108_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_109_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_110_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
thf(fact_111_add__less__le__mono, axiom,
    ((![A : real, B : real, C : real, D : real]: ((ord_less_real @ A @ B) => ((ord_less_eq_real @ C @ D) => (ord_less_real @ (plus_plus_real @ A @ C) @ (plus_plus_real @ B @ D))))))). % add_less_le_mono
thf(fact_112_add__less__le__mono, axiom,
    ((![A : nat, B : nat, C : nat, D : nat]: ((ord_less_nat @ A @ B) => ((ord_less_eq_nat @ C @ D) => (ord_less_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ D))))))). % add_less_le_mono
thf(fact_113_add__le__less__mono, axiom,
    ((![A : real, B : real, C : real, D : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_real @ C @ D) => (ord_less_real @ (plus_plus_real @ A @ C) @ (plus_plus_real @ B @ D))))))). % add_le_less_mono
thf(fact_114_add__le__less__mono, axiom,
    ((![A : nat, B : nat, C : nat, D : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_nat @ C @ D) => (ord_less_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ D))))))). % add_le_less_mono
thf(fact_115_add__mono__thms__linordered__field_I3_J, axiom,
    ((![I : real, J : real, K : real, L : real]: (((ord_less_real @ I @ J) & (ord_less_eq_real @ K @ L)) => (ord_less_real @ (plus_plus_real @ I @ K) @ (plus_plus_real @ J @ L)))))). % add_mono_thms_linordered_field(3)
thf(fact_116_add__mono__thms__linordered__field_I3_J, axiom,
    ((![I : nat, J : nat, K : nat, L : nat]: (((ord_less_nat @ I @ J) & (ord_less_eq_nat @ K @ L)) => (ord_less_nat @ (plus_plus_nat @ I @ K) @ (plus_plus_nat @ J @ L)))))). % add_mono_thms_linordered_field(3)
thf(fact_117_add__mono__thms__linordered__field_I4_J, axiom,
    ((![I : real, J : real, K : real, L : real]: (((ord_less_eq_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(4)
thf(fact_118_add__mono__thms__linordered__field_I4_J, axiom,
    ((![I : nat, J : nat, K : nat, L : nat]: (((ord_less_eq_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(4)
thf(fact_119_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_nat @ A @ B) => (((minus_minus_nat @ B @ A) = C) = (B = (plus_plus_nat @ C @ A)))))))). % ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_120_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse, axiom,
    ((![A : nat, B : nat]: ((ord_less_eq_nat @ A @ B) => ((plus_plus_nat @ A @ (minus_minus_nat @ B @ A)) = B))))). % ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_121_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ B) => ((minus_minus_nat @ C @ (minus_minus_nat @ B @ A)) = (minus_minus_nat @ (plus_plus_nat @ C @ A) @ B)))))). % ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_122_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ B) => ((minus_minus_nat @ (plus_plus_nat @ B @ C) @ A) = (plus_plus_nat @ (minus_minus_nat @ B @ A) @ C)))))). % ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_123_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ B) => ((plus_plus_nat @ (minus_minus_nat @ B @ A) @ C) = (minus_minus_nat @ (plus_plus_nat @ B @ C) @ A)))))). % ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_124_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ B) => ((minus_minus_nat @ (plus_plus_nat @ C @ B) @ A) = (plus_plus_nat @ C @ (minus_minus_nat @ B @ A))))))). % ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_125_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ B) => ((plus_plus_nat @ C @ (minus_minus_nat @ B @ A)) = (minus_minus_nat @ (plus_plus_nat @ C @ B) @ A)))))). % ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_126_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_nat @ C @ (minus_minus_nat @ B @ A)) = (ord_less_eq_nat @ (plus_plus_nat @ C @ A) @ B)))))). % ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_127_le__add__diff, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ B) => (ord_less_eq_nat @ C @ (minus_minus_nat @ (plus_plus_nat @ B @ C) @ A)))))). % le_add_diff
thf(fact_128_ordered__cancel__comm__monoid__diff__class_Odiff__add, axiom,
    ((![A : nat, B : nat]: ((ord_less_eq_nat @ A @ B) => ((plus_plus_nat @ (minus_minus_nat @ B @ A) @ A) = B))))). % ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_129_le__diff__eq, axiom,
    ((![A : real, C : real, B : real]: ((ord_less_eq_real @ A @ (minus_minus_real @ C @ B)) = (ord_less_eq_real @ (plus_plus_real @ A @ B) @ C))))). % le_diff_eq
thf(fact_130_diff__le__eq, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_eq_real @ (minus_minus_real @ A @ B) @ C) = (ord_less_eq_real @ A @ (plus_plus_real @ C @ B)))))). % diff_le_eq
thf(fact_131_norm__divide, axiom,
    ((![A : complex, B : complex]: ((real_V638595069omplex @ (divide1210191872omplex @ A @ B)) = (divide_divide_real @ (real_V638595069omplex @ A) @ (real_V638595069omplex @ B)))))). % norm_divide
thf(fact_132_norm__divide, axiom,
    ((![A : real, B : real]: ((real_V646646907m_real @ (divide_divide_real @ A @ B)) = (divide_divide_real @ (real_V646646907m_real @ A) @ (real_V646646907m_real @ B)))))). % norm_divide
thf(fact_133_complex_Oexhaust__sel, axiom,
    ((![Complex : complex]: (Complex = (complex2 @ (re @ Complex) @ (im @ Complex)))))). % complex.exhaust_sel
thf(fact_134_complex__Re__le__cmod, axiom,
    ((![X3 : complex]: (ord_less_eq_real @ (re @ X3) @ (real_V638595069omplex @ X3))))). % complex_Re_le_cmod
thf(fact_135_minus__complex_Osimps_I2_J, axiom,
    ((![X3 : complex, Y3 : complex]: ((im @ (minus_minus_complex @ X3 @ Y3)) = (minus_minus_real @ (im @ X3) @ (im @ Y3)))))). % minus_complex.simps(2)
thf(fact_136_minus__complex_Osimps_I1_J, axiom,
    ((![X3 : complex, Y3 : complex]: ((re @ (minus_minus_complex @ X3 @ Y3)) = (minus_minus_real @ (re @ X3) @ (re @ Y3)))))). % minus_complex.simps(1)
thf(fact_137_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_138_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_139_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_140_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_141_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_142_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_143_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_144_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_145_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_146_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_147_add__implies__diff, axiom,
    ((![C : real, B : real, A : real]: (((plus_plus_real @ C @ B) = A) => (C = (minus_minus_real @ A @ B)))))). % add_implies_diff
thf(fact_148_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_149_diff__diff__add, axiom,
    ((![A : real, B : real, C : real]: ((minus_minus_real @ (minus_minus_real @ A @ B) @ C) = (minus_minus_real @ A @ (plus_plus_real @ B @ C)))))). % diff_diff_add
thf(fact_150_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_151_diff__add__eq__diff__diff__swap, axiom,
    ((![A : real, B : real, C : real]: ((minus_minus_real @ A @ (plus_plus_real @ B @ C)) = (minus_minus_real @ (minus_minus_real @ A @ C) @ B))))). % diff_add_eq_diff_diff_swap
thf(fact_152_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_153_diff__add__eq, axiom,
    ((![A : real, B : real, C : real]: ((plus_plus_real @ (minus_minus_real @ A @ B) @ C) = (minus_minus_real @ (plus_plus_real @ A @ C) @ B))))). % diff_add_eq
thf(fact_154_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_155_diff__diff__eq2, axiom,
    ((![A : real, B : real, C : real]: ((minus_minus_real @ A @ (minus_minus_real @ B @ C)) = (minus_minus_real @ (plus_plus_real @ A @ C) @ B))))). % diff_diff_eq2
thf(fact_156_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
thf(fact_157_add__diff__eq, axiom,
    ((![A : real, B : real, C : real]: ((plus_plus_real @ A @ (minus_minus_real @ B @ C)) = (minus_minus_real @ (plus_plus_real @ A @ B) @ C))))). % add_diff_eq
thf(fact_158_eq__diff__eq, axiom,
    ((![A : complex, C : complex, B : complex]: ((A = (minus_minus_complex @ C @ B)) = ((plus_plus_complex @ A @ B) = C))))). % eq_diff_eq
thf(fact_159_eq__diff__eq, axiom,
    ((![A : real, C : real, B : real]: ((A = (minus_minus_real @ C @ B)) = ((plus_plus_real @ A @ B) = C))))). % eq_diff_eq
thf(fact_160_diff__eq__eq, axiom,
    ((![A : complex, B : complex, C : complex]: (((minus_minus_complex @ A @ B) = C) = (A = (plus_plus_complex @ C @ B)))))). % diff_eq_eq
thf(fact_161_diff__eq__eq, axiom,
    ((![A : real, B : real, C : real]: (((minus_minus_real @ A @ B) = C) = (A = (plus_plus_real @ C @ B)))))). % diff_eq_eq
thf(fact_162_group__cancel_Osub1, axiom,
    ((![A2 : complex, K : complex, A : complex, B : complex]: ((A2 = (plus_plus_complex @ K @ A)) => ((minus_minus_complex @ A2 @ B) = (plus_plus_complex @ K @ (minus_minus_complex @ A @ B))))))). % group_cancel.sub1
thf(fact_163_group__cancel_Osub1, axiom,
    ((![A2 : real, K : real, A : real, B : real]: ((A2 = (plus_plus_real @ K @ A)) => ((minus_minus_real @ A2 @ B) = (plus_plus_real @ K @ (minus_minus_real @ A @ B))))))). % group_cancel.sub1
thf(fact_164_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_165_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_166_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_167_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_168_metric__bound__lemma, axiom,
    ((![X3 : complex, Y3 : complex]: (ord_less_eq_real @ (real_V638595069omplex @ (minus_minus_complex @ X3 @ Y3)) @ (plus_plus_real @ (abs_abs_real @ (minus_minus_real @ (re @ X3) @ (re @ Y3))) @ (abs_abs_real @ (minus_minus_real @ (im @ X3) @ (im @ Y3)))))))). % metric_bound_lemma
thf(fact_169_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_170_norm__diff__triangle__ineq, axiom,
    ((![A : real, B : real, C : real, D : real]: (ord_less_eq_real @ (real_V646646907m_real @ (minus_minus_real @ (plus_plus_real @ A @ B) @ (plus_plus_real @ C @ D))) @ (plus_plus_real @ (real_V646646907m_real @ (minus_minus_real @ A @ C)) @ (real_V646646907m_real @ (minus_minus_real @ B @ D))))))). % norm_diff_triangle_ineq
thf(fact_171_complex_Osel_I2_J, axiom,
    ((![X1 : real, X22 : real]: ((im @ (complex2 @ X1 @ X22)) = X22)))). % complex.sel(2)
thf(fact_172_complex_Osel_I1_J, axiom,
    ((![X1 : real, X22 : real]: ((re @ (complex2 @ X1 @ X22)) = X1)))). % complex.sel(1)
thf(fact_173_minus__complex_Ocode, axiom,
    ((minus_minus_complex = (^[X2 : complex]: (^[Y5 : complex]: (complex2 @ (minus_minus_real @ (re @ X2) @ (re @ Y5)) @ (minus_minus_real @ (im @ X2) @ (im @ Y5)))))))). % minus_complex.code
thf(fact_174_norm__triangle__ineq3, axiom,
    ((![A : complex, B : complex]: (ord_less_eq_real @ (abs_abs_real @ (minus_minus_real @ (real_V638595069omplex @ A) @ (real_V638595069omplex @ B))) @ (real_V638595069omplex @ (minus_minus_complex @ A @ B)))))). % norm_triangle_ineq3
thf(fact_175_norm__triangle__ineq3, axiom,
    ((![A : real, B : real]: (ord_less_eq_real @ (abs_abs_real @ (minus_minus_real @ (real_V646646907m_real @ A) @ (real_V646646907m_real @ B))) @ (real_V646646907m_real @ (minus_minus_real @ A @ B)))))). % norm_triangle_ineq3
thf(fact_176_norm__triangle__le__diff, 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 @ (minus_minus_complex @ X3 @ Y3)) @ E))))). % norm_triangle_le_diff
thf(fact_177_norm__triangle__le__diff, axiom,
    ((![X3 : real, Y3 : real, E : real]: ((ord_less_eq_real @ (plus_plus_real @ (real_V646646907m_real @ X3) @ (real_V646646907m_real @ Y3)) @ E) => (ord_less_eq_real @ (real_V646646907m_real @ (minus_minus_real @ X3 @ Y3)) @ E))))). % norm_triangle_le_diff
thf(fact_178_norm__diff__triangle__le, axiom,
    ((![X3 : complex, Y3 : complex, E1 : real, Z2 : complex, E2 : real]: ((ord_less_eq_real @ (real_V638595069omplex @ (minus_minus_complex @ X3 @ Y3)) @ E1) => ((ord_less_eq_real @ (real_V638595069omplex @ (minus_minus_complex @ Y3 @ Z2)) @ E2) => (ord_less_eq_real @ (real_V638595069omplex @ (minus_minus_complex @ X3 @ Z2)) @ (plus_plus_real @ E1 @ E2))))))). % norm_diff_triangle_le
thf(fact_179_norm__diff__triangle__le, axiom,
    ((![X3 : real, Y3 : real, E1 : real, Z2 : real, E2 : real]: ((ord_less_eq_real @ (real_V646646907m_real @ (minus_minus_real @ X3 @ Y3)) @ E1) => ((ord_less_eq_real @ (real_V646646907m_real @ (minus_minus_real @ Y3 @ Z2)) @ E2) => (ord_less_eq_real @ (real_V646646907m_real @ (minus_minus_real @ X3 @ Z2)) @ (plus_plus_real @ E1 @ E2))))))). % norm_diff_triangle_le
thf(fact_180_norm__triangle__ineq4, axiom,
    ((![A : complex, B : complex]: (ord_less_eq_real @ (real_V638595069omplex @ (minus_minus_complex @ A @ B)) @ (plus_plus_real @ (real_V638595069omplex @ A) @ (real_V638595069omplex @ B)))))). % norm_triangle_ineq4
thf(fact_181_norm__triangle__ineq4, axiom,
    ((![A : real, B : real]: (ord_less_eq_real @ (real_V646646907m_real @ (minus_minus_real @ A @ B)) @ (plus_plus_real @ (real_V646646907m_real @ A) @ (real_V646646907m_real @ B)))))). % norm_triangle_ineq4
thf(fact_182_norm__triangle__sub, axiom,
    ((![X3 : complex, Y3 : complex]: (ord_less_eq_real @ (real_V638595069omplex @ X3) @ (plus_plus_real @ (real_V638595069omplex @ Y3) @ (real_V638595069omplex @ (minus_minus_complex @ X3 @ Y3))))))). % norm_triangle_sub
thf(fact_183_norm__triangle__sub, axiom,
    ((![X3 : real, Y3 : real]: (ord_less_eq_real @ (real_V646646907m_real @ X3) @ (plus_plus_real @ (real_V646646907m_real @ Y3) @ (real_V646646907m_real @ (minus_minus_real @ X3 @ Y3))))))). % norm_triangle_sub
thf(fact_184_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_185_norm__triangle__lt, axiom,
    ((![X3 : real, Y3 : real, E : real]: ((ord_less_real @ (plus_plus_real @ (real_V646646907m_real @ X3) @ (real_V646646907m_real @ Y3)) @ E) => (ord_less_real @ (real_V646646907m_real @ (plus_plus_real @ X3 @ Y3)) @ E))))). % norm_triangle_lt
thf(fact_186_norm__add__less, axiom,
    ((![X3 : complex, R2 : real, Y3 : complex, S2 : real]: ((ord_less_real @ (real_V638595069omplex @ X3) @ R2) => ((ord_less_real @ (real_V638595069omplex @ Y3) @ S2) => (ord_less_real @ (real_V638595069omplex @ (plus_plus_complex @ X3 @ Y3)) @ (plus_plus_real @ R2 @ S2))))))). % norm_add_less
thf(fact_187_norm__add__less, axiom,
    ((![X3 : real, R2 : real, Y3 : real, S2 : real]: ((ord_less_real @ (real_V646646907m_real @ X3) @ R2) => ((ord_less_real @ (real_V646646907m_real @ Y3) @ S2) => (ord_less_real @ (real_V646646907m_real @ (plus_plus_real @ X3 @ Y3)) @ (plus_plus_real @ R2 @ S2))))))). % norm_add_less
thf(fact_188_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_189_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_190_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_191_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_192_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_193_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_194_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_195_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_196_comp__def, axiom,
    ((comp_nat_nat_nat = (^[F : nat > nat]: (^[G : nat > nat]: (^[X2 : nat]: (F @ (G @ X2)))))))). % comp_def

% 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))).
