% 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_271__5370180_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:28:47.292

% 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 (53)
thf(sy_c_Complex_Ocomplex_OIm, type,
    im : complex > real).
thf(sy_c_Complex_Ocomplex_ORe, type,
    re : complex > real).
thf(sy_c_Complex_Ocomplex_Ocase__complex_001t__Real__Oreal, type,
    case_complex_real : (real > real > real) > complex > real).
thf(sy_c_Fun_Ocomp_001t__Complex__Ocomplex_001t__Complex__Ocomplex_001t__Complex__Ocomplex, type,
    comp_c130555887omplex : (complex > complex) > (complex > complex) > complex > complex).
thf(sy_c_Fun_Ocomp_001t__Complex__Ocomplex_001t__Complex__Ocomplex_001t__Nat__Onat, type,
    comp_c438056209ex_nat : (complex > complex) > (nat > complex) > nat > complex).
thf(sy_c_Fun_Ocomp_001t__Complex__Ocomplex_001t__Complex__Ocomplex_001t__Real__Oreal, type,
    comp_c595887981x_real : (complex > complex) > (real > complex) > real > complex).
thf(sy_c_Fun_Ocomp_001t__Complex__Ocomplex_001t__Nat__Onat_001t__Nat__Onat, type,
    comp_complex_nat_nat : (complex > nat) > (nat > complex) > nat > nat).
thf(sy_c_Fun_Ocomp_001t__Complex__Ocomplex_001t__Real__Oreal_001t__Complex__Ocomplex, type,
    comp_c317287661omplex : (complex > real) > (complex > complex) > complex > real).
thf(sy_c_Fun_Ocomp_001t__Complex__Ocomplex_001t__Real__Oreal_001t__Nat__Onat, type,
    comp_c1631780367al_nat : (complex > real) > (nat > complex) > nat > real).
thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Complex__Ocomplex_001t__Nat__Onat, type,
    comp_nat_complex_nat : (nat > complex) > (nat > nat) > nat > 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_Fun_Ocomp_001t__Nat__Onat_001t__Real__Oreal_001t__Complex__Ocomplex, type,
    comp_n1816297743omplex : (nat > real) > (complex > nat) > complex > real).
thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Real__Oreal_001t__Nat__Onat, type,
    comp_nat_real_nat : (nat > real) > (nat > nat) > nat > real).
thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Complex__Ocomplex_001t__Complex__Ocomplex, type,
    comp_r667767405omplex : (real > complex) > (complex > real) > complex > complex).
thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Complex__Ocomplex_001t__Nat__Onat, type,
    comp_r2119295887ex_nat : (real > complex) > (nat > real) > nat > complex).
thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Complex__Ocomplex_001t__Real__Oreal, type,
    comp_r701421291x_real : (real > complex) > (real > real) > real > complex).
thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Real__Oreal_001t__Complex__Ocomplex, type,
    comp_r422820971omplex : (real > real) > (complex > real) > complex > real).
thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Real__Oreal_001t__Nat__Onat, type,
    comp_real_real_nat : (real > real) > (nat > real) > nat > real).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Complex__Ocomplex, type,
    abs_abs_complex : complex > complex).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Real__Oreal, type,
    abs_abs_real : real > real).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal, type,
    one_one_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__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_Obounded__bilinear_001t__Complex__Ocomplex_001t__Complex__Ocomplex_001t__Complex__Ocomplex, type,
    real_V1641145724omplex : (complex > complex > complex) > $o).
thf(sy_c_Real__Vector__Spaces_Obounded__bilinear_001t__Complex__Ocomplex_001t__Complex__Ocomplex_001t__Real__Oreal, type,
    real_V179155066x_real : (complex > complex > real) > $o).
thf(sy_c_Real__Vector__Spaces_Obounded__bilinear_001t__Complex__Ocomplex_001t__Real__Oreal_001t__Complex__Ocomplex, type,
    real_V2048038394omplex : (complex > real > complex) > $o).
thf(sy_c_Real__Vector__Spaces_Obounded__bilinear_001t__Complex__Ocomplex_001t__Real__Oreal_001t__Real__Oreal, type,
    real_V735084024l_real : (complex > real > real) > $o).
thf(sy_c_Real__Vector__Spaces_Obounded__bilinear_001t__Real__Oreal_001t__Complex__Ocomplex_001t__Complex__Ocomplex, type,
    real_V251034490omplex : (real > complex > complex) > $o).
thf(sy_c_Real__Vector__Spaces_Obounded__bilinear_001t__Real__Oreal_001t__Complex__Ocomplex_001t__Real__Oreal, type,
    real_V616866680x_real : (real > complex > real) > $o).
thf(sy_c_Real__Vector__Spaces_Obounded__bilinear_001t__Real__Oreal_001t__Real__Oreal_001t__Complex__Ocomplex, type,
    real_V338266360omplex : (real > real > complex) > $o).
thf(sy_c_Real__Vector__Spaces_Obounded__bilinear_001t__Real__Oreal_001t__Real__Oreal_001t__Real__Oreal, type,
    real_V1182092790l_real : (real > real > real) > $o).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Complex__Ocomplex, type,
    real_V638595069omplex : complex > real).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Real__Oreal, type,
    real_V646646907m_real : real > real).
thf(sy_c_Real__Vector__Spaces_Oof__real_001t__Complex__Ocomplex, type,
    real_V306493662omplex : real > complex).
thf(sy_c_Real__Vector__Spaces_Oof__real_001t__Real__Oreal, type,
    real_V1205483228l_real : real > real).
thf(sy_c_Topological__Spaces_Omonoseq_001t__Nat__Onat, type,
    topolo1922093437eq_nat : (nat > nat) > $o).
thf(sy_c_Topological__Spaces_Omonoseq_001t__Real__Oreal, type,
    topolo144289241q_real : (nat > real) > $o).
thf(sy_c_Topological__Spaces_Otopological__space__class_Oconvergent_001t__Complex__Ocomplex, type,
    topolo1054921685omplex : (nat > complex) > $o).
thf(sy_c_Topological__Spaces_Otopological__space__class_Oconvergent_001t__Real__Oreal, type,
    topolo795669587t_real : (nat > real) > $o).
thf(sy_v_f____, type,
    f : nat > nat).
thf(sy_v_g____, type,
    g : nat > nat).
thf(sy_v_r, type,
    r : real).
thf(sy_v_s, type,
    s : nat > complex).

% Relevant facts (246)
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_g_I2_J, axiom,
    ((topolo144289241q_real @ (^[N : nat]: (im @ (s @ (f @ (g @ N)))))))). % g(2)
thf(fact_3_conv1, axiom,
    ((topolo795669587t_real @ (^[N : nat]: (re @ (s @ (f @ N))))))). % conv1
thf(fact_4_f_I2_J, axiom,
    ((topolo144289241q_real @ (^[N : nat]: (re @ (s @ (f @ N))))))). % f(2)
thf(fact_5_convergent__const, axiom,
    ((![C : complex]: (topolo1054921685omplex @ (^[N : nat]: C))))). % convergent_const
thf(fact_6_convergent__const, axiom,
    ((![C : real]: (topolo795669587t_real @ (^[N : nat]: C))))). % convergent_const
thf(fact_7__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062g_O_A_092_060lbrakk_062strict__mono_Ag_059_Amonoseq_A_I_092_060lambda_062n_O_AIm_A_Is_A_If_A_Ig_An_J_J_J_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom,
    ((~ ((![G : nat > nat]: ((order_769474267at_nat @ G) => (~ ((topolo144289241q_real @ (^[N : nat]: (im @ (s @ (f @ (G @ N)))))))))))))). % \<open>\<And>thesis. (\<And>g. \<lbrakk>strict_mono g; monoseq (\<lambda>n. Im (s (f (g n))))\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_8_convergent__norm, axiom,
    ((![F : nat > complex]: ((topolo1054921685omplex @ F) => (topolo795669587t_real @ (^[N : nat]: (real_V638595069omplex @ (F @ N)))))))). % convergent_norm
thf(fact_9_convergent__norm, axiom,
    ((![F : nat > real]: ((topolo795669587t_real @ F) => (topolo795669587t_real @ (^[N : nat]: (real_V646646907m_real @ (F @ N)))))))). % convergent_norm
thf(fact_10_convergent__of__real, axiom,
    ((![F : nat > real]: ((topolo795669587t_real @ F) => (topolo1054921685omplex @ (^[N : nat]: (real_V306493662omplex @ (F @ N)))))))). % convergent_of_real
thf(fact_11_convergent__of__real, axiom,
    ((![F : nat > real]: ((topolo795669587t_real @ F) => (topolo795669587t_real @ (^[N : nat]: (real_V1205483228l_real @ (F @ N)))))))). % convergent_of_real
thf(fact_12__092_060open_062_092_060exists_062fa_O_Astrict__mono_Afa_A_092_060and_062_Amonoseq_A_I_092_060lambda_062n_O_A_IIm_A_092_060circ_062_As_A_092_060circ_062_Af_J_A_Ifa_An_J_J_092_060close_062, axiom,
    ((?[F2 : nat > nat]: ((order_769474267at_nat @ F2) & (topolo144289241q_real @ (^[N : nat]: (comp_nat_real_nat @ (comp_c1631780367al_nat @ im @ s) @ f @ (F2 @ N)))))))). % \<open>\<exists>fa. strict_mono fa \<and> monoseq (\<lambda>n. (Im \<circ> s \<circ> f) (fa n))\<close>
thf(fact_13_r, axiom,
    ((![N2 : nat]: (ord_less_eq_real @ (real_V638595069omplex @ (s @ N2)) @ r)))). % r
thf(fact_14_Im__def, axiom,
    ((im = (case_complex_real @ (^[X1 : real]: (^[X2 : real]: X2)))))). % Im_def
thf(fact_15__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062f_O_A_092_060lbrakk_062strict__mono_Af_059_Amonoseq_A_I_092_060lambda_062n_O_ARe_A_Is_A_If_An_J_J_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom,
    ((~ ((![F2 : nat > nat]: ((order_769474267at_nat @ F2) => (~ ((topolo144289241q_real @ (^[N : nat]: (re @ (s @ (F2 @ N))))))))))))). % \<open>\<And>thesis. (\<And>f. \<lbrakk>strict_mono f; monoseq (\<lambda>n. Re (s (f n)))\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_16_bounded__bilinear_Oconvergent, axiom,
    ((![Prod : real > real > real, X : nat > real, Y : nat > real]: ((real_V1182092790l_real @ Prod) => ((topolo795669587t_real @ X) => ((topolo795669587t_real @ Y) => (topolo795669587t_real @ (^[N : nat]: (Prod @ (X @ N) @ (Y @ N)))))))))). % bounded_bilinear.convergent
thf(fact_17_bounded__bilinear_Oconvergent, axiom,
    ((![Prod : real > real > complex, X : nat > real, Y : nat > real]: ((real_V338266360omplex @ Prod) => ((topolo795669587t_real @ X) => ((topolo795669587t_real @ Y) => (topolo1054921685omplex @ (^[N : nat]: (Prod @ (X @ N) @ (Y @ N)))))))))). % bounded_bilinear.convergent
thf(fact_18_bounded__bilinear_Oconvergent, axiom,
    ((![Prod : real > complex > real, X : nat > real, Y : nat > complex]: ((real_V616866680x_real @ Prod) => ((topolo795669587t_real @ X) => ((topolo1054921685omplex @ Y) => (topolo795669587t_real @ (^[N : nat]: (Prod @ (X @ N) @ (Y @ N)))))))))). % bounded_bilinear.convergent
thf(fact_19_bounded__bilinear_Oconvergent, axiom,
    ((![Prod : real > complex > complex, X : nat > real, Y : nat > complex]: ((real_V251034490omplex @ Prod) => ((topolo795669587t_real @ X) => ((topolo1054921685omplex @ Y) => (topolo1054921685omplex @ (^[N : nat]: (Prod @ (X @ N) @ (Y @ N)))))))))). % bounded_bilinear.convergent
thf(fact_20_bounded__bilinear_Oconvergent, axiom,
    ((![Prod : complex > real > real, X : nat > complex, Y : nat > real]: ((real_V735084024l_real @ Prod) => ((topolo1054921685omplex @ X) => ((topolo795669587t_real @ Y) => (topolo795669587t_real @ (^[N : nat]: (Prod @ (X @ N) @ (Y @ N)))))))))). % bounded_bilinear.convergent
thf(fact_21_bounded__bilinear_Oconvergent, axiom,
    ((![Prod : complex > real > complex, X : nat > complex, Y : nat > real]: ((real_V2048038394omplex @ Prod) => ((topolo1054921685omplex @ X) => ((topolo795669587t_real @ Y) => (topolo1054921685omplex @ (^[N : nat]: (Prod @ (X @ N) @ (Y @ N)))))))))). % bounded_bilinear.convergent
thf(fact_22_bounded__bilinear_Oconvergent, axiom,
    ((![Prod : complex > complex > real, X : nat > complex, Y : nat > complex]: ((real_V179155066x_real @ Prod) => ((topolo1054921685omplex @ X) => ((topolo1054921685omplex @ Y) => (topolo795669587t_real @ (^[N : nat]: (Prod @ (X @ N) @ (Y @ N)))))))))). % bounded_bilinear.convergent
thf(fact_23_bounded__bilinear_Oconvergent, axiom,
    ((![Prod : complex > complex > complex, X : nat > complex, Y : nat > complex]: ((real_V1641145724omplex @ Prod) => ((topolo1054921685omplex @ X) => ((topolo1054921685omplex @ Y) => (topolo1054921685omplex @ (^[N : nat]: (Prod @ (X @ N) @ (Y @ N)))))))))). % bounded_bilinear.convergent
thf(fact_24_Re__complex__of__real, axiom,
    ((![Z : real]: ((re @ (real_V306493662omplex @ Z)) = Z)))). % Re_complex_of_real
thf(fact_25__092_060open_062_092_060exists_062f_O_Astrict__mono_Af_A_092_060and_062_Amonoseq_A_I_092_060lambda_062n_O_A_IRe_A_092_060circ_062_As_J_A_If_An_J_J_092_060close_062, axiom,
    ((?[F2 : nat > nat]: ((order_769474267at_nat @ F2) & (topolo144289241q_real @ (^[N : nat]: (comp_c1631780367al_nat @ re @ s @ (F2 @ N)))))))). % \<open>\<exists>f. strict_mono f \<and> monoseq (\<lambda>n. (Re \<circ> s) (f n))\<close>
thf(fact_26_Re__def, axiom,
    ((re = (case_complex_real @ (^[X1 : real]: (^[X2 : real]: X1)))))). % Re_def
thf(fact_27_monoI1, axiom,
    ((![X : nat > real]: ((![M : nat, N3 : nat]: ((ord_less_eq_nat @ M @ N3) => (ord_less_eq_real @ (X @ M) @ (X @ N3)))) => (topolo144289241q_real @ X))))). % monoI1
thf(fact_28_monoI1, axiom,
    ((![X : nat > nat]: ((![M : nat, N3 : nat]: ((ord_less_eq_nat @ M @ N3) => (ord_less_eq_nat @ (X @ M) @ (X @ N3)))) => (topolo1922093437eq_nat @ X))))). % monoI1
thf(fact_29_monoI2, axiom,
    ((![X : nat > real]: ((![M : nat, N3 : nat]: ((ord_less_eq_nat @ M @ N3) => (ord_less_eq_real @ (X @ N3) @ (X @ M)))) => (topolo144289241q_real @ X))))). % monoI2
thf(fact_30_monoI2, axiom,
    ((![X : nat > nat]: ((![M : nat, N3 : nat]: ((ord_less_eq_nat @ M @ N3) => (ord_less_eq_nat @ (X @ N3) @ (X @ M)))) => (topolo1922093437eq_nat @ X))))). % monoI2
thf(fact_31_complex_Ocase__eq__if, axiom,
    ((case_complex_real = (^[F3 : real > real > real]: (^[Complex : complex]: (F3 @ (re @ Complex) @ (im @ Complex))))))). % complex.case_eq_if
thf(fact_32_complex__Re__le__cmod, axiom,
    ((![X3 : complex]: (ord_less_eq_real @ (re @ X3) @ (real_V638595069omplex @ X3))))). % complex_Re_le_cmod
thf(fact_33_complex_Ocase__distrib, axiom,
    ((![H : real > real, F : real > real > real, Complex2 : complex]: ((H @ (case_complex_real @ F @ Complex2)) = (case_complex_real @ (^[X1 : real]: (^[X2 : real]: (H @ (F @ X1 @ X2)))) @ Complex2))))). % complex.case_distrib
thf(fact_34_monoseq__def, axiom,
    ((topolo144289241q_real = (^[X4 : nat > real]: (((![M2 : nat]: (![N : nat]: (((ord_less_eq_nat @ M2 @ N)) => ((ord_less_eq_real @ (X4 @ M2) @ (X4 @ N))))))) | ((![M2 : nat]: (![N : nat]: (((ord_less_eq_nat @ M2 @ N)) => ((ord_less_eq_real @ (X4 @ N) @ (X4 @ M2)))))))))))). % monoseq_def
thf(fact_35_monoseq__def, axiom,
    ((topolo1922093437eq_nat = (^[X4 : nat > nat]: (((![M2 : nat]: (![N : nat]: (((ord_less_eq_nat @ M2 @ N)) => ((ord_less_eq_nat @ (X4 @ M2) @ (X4 @ N))))))) | ((![M2 : nat]: (![N : nat]: (((ord_less_eq_nat @ M2 @ N)) => ((ord_less_eq_nat @ (X4 @ N) @ (X4 @ M2)))))))))))). % monoseq_def
thf(fact_36_seq__monosub, axiom,
    ((![S : nat > real]: (?[F2 : nat > nat]: ((order_769474267at_nat @ F2) & (topolo144289241q_real @ (^[N : nat]: (S @ (F2 @ N))))))))). % seq_monosub
thf(fact_37_strict__mono__o, axiom,
    ((![R : nat > real, S : nat > nat]: ((order_952716343t_real @ R) => ((order_769474267at_nat @ S) => (order_952716343t_real @ (comp_nat_real_nat @ R @ S))))))). % strict_mono_o
thf(fact_38_strict__mono__o, axiom,
    ((![R : nat > nat, S : nat > nat]: ((order_769474267at_nat @ R) => ((order_769474267at_nat @ S) => (order_769474267at_nat @ (comp_nat_nat_nat @ R @ S))))))). % strict_mono_o
thf(fact_39_strict__mono__leD, axiom,
    ((![R : real > real, M3 : real, N4 : real]: ((order_1818878995l_real @ R) => ((ord_less_eq_real @ M3 @ N4) => (ord_less_eq_real @ (R @ M3) @ (R @ N4))))))). % strict_mono_leD
thf(fact_40_strict__mono__leD, axiom,
    ((![R : real > nat, M3 : real, N4 : real]: ((order_297469111al_nat @ R) => ((ord_less_eq_real @ M3 @ N4) => (ord_less_eq_nat @ (R @ M3) @ (R @ N4))))))). % strict_mono_leD
thf(fact_41_strict__mono__leD, axiom,
    ((![R : nat > real, M3 : nat, N4 : nat]: ((order_952716343t_real @ R) => ((ord_less_eq_nat @ M3 @ N4) => (ord_less_eq_real @ (R @ M3) @ (R @ N4))))))). % strict_mono_leD
thf(fact_42_strict__mono__leD, axiom,
    ((![R : nat > nat, M3 : nat, N4 : nat]: ((order_769474267at_nat @ R) => ((ord_less_eq_nat @ M3 @ N4) => (ord_less_eq_nat @ (R @ M3) @ (R @ N4))))))). % strict_mono_leD
thf(fact_43_strict__mono__compose, axiom,
    ((![R : nat > nat, S : nat > nat]: ((order_769474267at_nat @ R) => ((order_769474267at_nat @ S) => (order_769474267at_nat @ (^[X5 : nat]: (R @ (S @ X5))))))))). % strict_mono_compose
thf(fact_44_convergent__subseq__convergent, axiom,
    ((![X : nat > real, F : nat > nat]: ((topolo795669587t_real @ X) => ((order_769474267at_nat @ F) => (topolo795669587t_real @ (comp_nat_real_nat @ X @ F))))))). % convergent_subseq_convergent
thf(fact_45_convergent__subseq__convergent, axiom,
    ((![X : nat > complex, F : nat > nat]: ((topolo1054921685omplex @ X) => ((order_769474267at_nat @ F) => (topolo1054921685omplex @ (comp_nat_complex_nat @ X @ F))))))). % convergent_subseq_convergent
thf(fact_46_complex_Ocoinduct__strong, axiom,
    ((![R2 : complex > complex > $o, Complex2 : complex, Complex3 : complex]: ((R2 @ Complex2 @ Complex3) => ((![Complex4 : complex, Complex5 : complex]: ((R2 @ Complex4 @ Complex5) => (((re @ Complex4) = (re @ Complex5)) & ((im @ Complex4) = (im @ Complex5))))) => (Complex2 = Complex3)))))). % complex.coinduct_strong
thf(fact_47_complex__eq__iff, axiom,
    (((^[Y2 : complex]: (^[Z2 : complex]: (Y2 = Z2))) = (^[X5 : complex]: (^[Y3 : complex]: ((((re @ X5) = (re @ Y3))) & (((im @ X5) = (im @ Y3))))))))). % complex_eq_iff
thf(fact_48_complex_Oexpand, axiom,
    ((![Complex2 : complex, Complex3 : complex]: ((((re @ Complex2) = (re @ Complex3)) & ((im @ Complex2) = (im @ Complex3))) => (Complex2 = Complex3))))). % complex.expand
thf(fact_49_complex__eqI, axiom,
    ((![X3 : complex, Y4 : complex]: (((re @ X3) = (re @ Y4)) => (((im @ X3) = (im @ Y4)) => (X3 = Y4)))))). % complex_eqI
thf(fact_50__092_060open_062cmod_A_Is_A0_J_A_092_060le_062_Ar_092_060close_062, axiom,
    ((ord_less_eq_real @ (real_V638595069omplex @ (s @ zero_zero_nat)) @ r))). % \<open>cmod (s 0) \<le> r\<close>
thf(fact_51_rp, axiom,
    ((ord_less_eq_real @ zero_zero_real @ r))). % rp
thf(fact_52_of__real__eq__iff, axiom,
    ((![X3 : real, Y4 : real]: (((real_V306493662omplex @ X3) = (real_V306493662omplex @ Y4)) = (X3 = Y4))))). % of_real_eq_iff
thf(fact_53_of__real__eq__iff, axiom,
    ((![X3 : real, Y4 : real]: (((real_V1205483228l_real @ X3) = (real_V1205483228l_real @ Y4)) = (X3 = Y4))))). % of_real_eq_iff
thf(fact_54_comp__apply, axiom,
    ((comp_nat_real_nat = (^[F3 : nat > real]: (^[G2 : nat > nat]: (^[X5 : nat]: (F3 @ (G2 @ X5)))))))). % comp_apply
thf(fact_55_comp__apply, axiom,
    ((comp_c1631780367al_nat = (^[F3 : complex > real]: (^[G2 : nat > complex]: (^[X5 : nat]: (F3 @ (G2 @ X5)))))))). % comp_apply
thf(fact_56_comp__apply, axiom,
    ((comp_r667767405omplex = (^[F3 : real > complex]: (^[G2 : complex > real]: (^[X5 : complex]: (F3 @ (G2 @ X5)))))))). % comp_apply
thf(fact_57_order__refl, axiom,
    ((![X3 : real]: (ord_less_eq_real @ X3 @ X3)))). % order_refl
thf(fact_58_order__refl, axiom,
    ((![X3 : nat]: (ord_less_eq_nat @ X3 @ X3)))). % order_refl
thf(fact_59_strict__mono__less__eq, axiom,
    ((![F : real > real, X3 : real, Y4 : real]: ((order_1818878995l_real @ F) => ((ord_less_eq_real @ (F @ X3) @ (F @ Y4)) = (ord_less_eq_real @ X3 @ Y4)))))). % strict_mono_less_eq
thf(fact_60_strict__mono__less__eq, axiom,
    ((![F : nat > real, X3 : nat, Y4 : nat]: ((order_952716343t_real @ F) => ((ord_less_eq_real @ (F @ X3) @ (F @ Y4)) = (ord_less_eq_nat @ X3 @ Y4)))))). % strict_mono_less_eq
thf(fact_61_strict__mono__less__eq, axiom,
    ((![F : real > nat, X3 : real, Y4 : real]: ((order_297469111al_nat @ F) => ((ord_less_eq_nat @ (F @ X3) @ (F @ Y4)) = (ord_less_eq_real @ X3 @ Y4)))))). % strict_mono_less_eq
thf(fact_62_strict__mono__less__eq, axiom,
    ((![F : nat > nat, X3 : nat, Y4 : nat]: ((order_769474267at_nat @ F) => ((ord_less_eq_nat @ (F @ X3) @ (F @ Y4)) = (ord_less_eq_nat @ X3 @ Y4)))))). % strict_mono_less_eq
thf(fact_63_K__record__comp, axiom,
    ((![C : real, F : nat > nat]: ((comp_nat_real_nat @ (^[X5 : nat]: C) @ F) = (^[X5 : nat]: C))))). % K_record_comp
thf(fact_64_K__record__comp, axiom,
    ((![C : real, F : nat > complex]: ((comp_c1631780367al_nat @ (^[X5 : complex]: C) @ F) = (^[X5 : nat]: C))))). % K_record_comp
thf(fact_65_K__record__comp, axiom,
    ((![C : complex, F : complex > real]: ((comp_r667767405omplex @ (^[X5 : real]: C) @ F) = (^[X5 : complex]: C))))). % K_record_comp
thf(fact_66_cmod__Im__le__iff, axiom,
    ((![X3 : complex, Y4 : complex]: (((re @ X3) = (re @ Y4)) => ((ord_less_eq_real @ (real_V638595069omplex @ X3) @ (real_V638595069omplex @ Y4)) = (ord_less_eq_real @ (abs_abs_real @ (im @ X3)) @ (abs_abs_real @ (im @ Y4)))))))). % cmod_Im_le_iff
thf(fact_67_abs__norm__cancel, axiom,
    ((![A : complex]: ((abs_abs_real @ (real_V638595069omplex @ A)) = (real_V638595069omplex @ A))))). % abs_norm_cancel
thf(fact_68_abs__norm__cancel, axiom,
    ((![A : real]: ((abs_abs_real @ (real_V646646907m_real @ A)) = (real_V646646907m_real @ A))))). % abs_norm_cancel
thf(fact_69_norm__zero, axiom,
    (((real_V638595069omplex @ zero_zero_complex) = zero_zero_real))). % norm_zero
thf(fact_70_norm__zero, axiom,
    (((real_V646646907m_real @ zero_zero_real) = zero_zero_real))). % norm_zero
thf(fact_71_norm__eq__zero, axiom,
    ((![X3 : complex]: (((real_V638595069omplex @ X3) = zero_zero_real) = (X3 = zero_zero_complex))))). % norm_eq_zero
thf(fact_72_norm__eq__zero, axiom,
    ((![X3 : real]: (((real_V646646907m_real @ X3) = zero_zero_real) = (X3 = zero_zero_real))))). % norm_eq_zero
thf(fact_73_of__real__0, axiom,
    (((real_V306493662omplex @ zero_zero_real) = zero_zero_complex))). % of_real_0
thf(fact_74_of__real__0, axiom,
    (((real_V1205483228l_real @ zero_zero_real) = zero_zero_real))). % of_real_0
thf(fact_75_of__real__eq__0__iff, axiom,
    ((![X3 : real]: (((real_V306493662omplex @ X3) = zero_zero_complex) = (X3 = zero_zero_real))))). % of_real_eq_0_iff
thf(fact_76_of__real__eq__0__iff, axiom,
    ((![X3 : real]: (((real_V1205483228l_real @ X3) = zero_zero_real) = (X3 = zero_zero_real))))). % of_real_eq_0_iff
thf(fact_77_norm__of__real, axiom,
    ((![R : real]: ((real_V638595069omplex @ (real_V306493662omplex @ R)) = (abs_abs_real @ R))))). % norm_of_real
thf(fact_78_norm__of__real, axiom,
    ((![R : real]: ((real_V646646907m_real @ (real_V1205483228l_real @ R)) = (abs_abs_real @ R))))). % norm_of_real
thf(fact_79_Im__complex__of__real, axiom,
    ((![Z : real]: ((im @ (real_V306493662omplex @ Z)) = zero_zero_real)))). % Im_complex_of_real
thf(fact_80_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_81_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_82_real__norm__def, axiom,
    ((real_V646646907m_real = abs_abs_real))). % real_norm_def
thf(fact_83_norm__ge__zero, axiom,
    ((![X3 : complex]: (ord_less_eq_real @ zero_zero_real @ (real_V638595069omplex @ X3))))). % norm_ge_zero
thf(fact_84_norm__ge__zero, axiom,
    ((![X3 : real]: (ord_less_eq_real @ zero_zero_real @ (real_V646646907m_real @ X3))))). % norm_ge_zero
thf(fact_85_cmod__eq__Re, axiom,
    ((![Z : complex]: (((im @ Z) = zero_zero_real) => ((real_V638595069omplex @ Z) = (abs_abs_real @ (re @ Z))))))). % cmod_eq_Re
thf(fact_86_cmod__eq__Im, axiom,
    ((![Z : complex]: (((re @ Z) = zero_zero_real) => ((real_V638595069omplex @ Z) = (abs_abs_real @ (im @ Z))))))). % cmod_eq_Im
thf(fact_87_Im__eq__0, axiom,
    ((![Z : complex]: (((abs_abs_real @ (re @ Z)) = (real_V638595069omplex @ Z)) => ((im @ Z) = zero_zero_real))))). % Im_eq_0
thf(fact_88_dual__order_Oantisym, axiom,
    ((![B : real, A : real]: ((ord_less_eq_real @ B @ A) => ((ord_less_eq_real @ A @ B) => (A = B)))))). % dual_order.antisym
thf(fact_89_dual__order_Oantisym, axiom,
    ((![B : nat, A : nat]: ((ord_less_eq_nat @ B @ A) => ((ord_less_eq_nat @ A @ B) => (A = B)))))). % dual_order.antisym
thf(fact_90_dual__order_Oeq__iff, axiom,
    (((^[Y2 : real]: (^[Z2 : real]: (Y2 = Z2))) = (^[A2 : real]: (^[B2 : real]: (((ord_less_eq_real @ B2 @ A2)) & ((ord_less_eq_real @ A2 @ B2)))))))). % dual_order.eq_iff
thf(fact_91_dual__order_Oeq__iff, axiom,
    (((^[Y2 : nat]: (^[Z2 : nat]: (Y2 = Z2))) = (^[A2 : nat]: (^[B2 : nat]: (((ord_less_eq_nat @ B2 @ A2)) & ((ord_less_eq_nat @ A2 @ B2)))))))). % dual_order.eq_iff
thf(fact_92_dual__order_Otrans, axiom,
    ((![B : real, A : real, C : real]: ((ord_less_eq_real @ B @ A) => ((ord_less_eq_real @ C @ B) => (ord_less_eq_real @ C @ A)))))). % dual_order.trans
thf(fact_93_dual__order_Otrans, axiom,
    ((![B : nat, A : nat, C : nat]: ((ord_less_eq_nat @ B @ A) => ((ord_less_eq_nat @ C @ B) => (ord_less_eq_nat @ C @ A)))))). % dual_order.trans
thf(fact_94_linorder__wlog, axiom,
    ((![P : real > real > $o, A : real, B : real]: ((![A3 : real, B3 : real]: ((ord_less_eq_real @ A3 @ B3) => (P @ A3 @ B3))) => ((![A3 : real, B3 : real]: ((P @ B3 @ A3) => (P @ A3 @ B3))) => (P @ A @ B)))))). % linorder_wlog
thf(fact_95_linorder__wlog, axiom,
    ((![P : nat > nat > $o, A : nat, B : nat]: ((![A3 : nat, B3 : nat]: ((ord_less_eq_nat @ A3 @ B3) => (P @ A3 @ B3))) => ((![A3 : nat, B3 : nat]: ((P @ B3 @ A3) => (P @ A3 @ B3))) => (P @ A @ B)))))). % linorder_wlog
thf(fact_96_dual__order_Orefl, axiom,
    ((![A : real]: (ord_less_eq_real @ A @ A)))). % dual_order.refl
thf(fact_97_dual__order_Orefl, axiom,
    ((![A : nat]: (ord_less_eq_nat @ A @ A)))). % dual_order.refl
thf(fact_98_order__trans, axiom,
    ((![X3 : real, Y4 : real, Z : real]: ((ord_less_eq_real @ X3 @ Y4) => ((ord_less_eq_real @ Y4 @ Z) => (ord_less_eq_real @ X3 @ Z)))))). % order_trans
thf(fact_99_order__trans, axiom,
    ((![X3 : nat, Y4 : nat, Z : nat]: ((ord_less_eq_nat @ X3 @ Y4) => ((ord_less_eq_nat @ Y4 @ Z) => (ord_less_eq_nat @ X3 @ Z)))))). % order_trans
thf(fact_100_order__class_Oorder_Oantisym, axiom,
    ((![A : real, B : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_eq_real @ B @ A) => (A = B)))))). % order_class.order.antisym
thf(fact_101_order__class_Oorder_Oantisym, axiom,
    ((![A : nat, B : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_nat @ B @ A) => (A = B)))))). % order_class.order.antisym
thf(fact_102_ord__le__eq__trans, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_eq_real @ A @ B) => ((B = C) => (ord_less_eq_real @ A @ C)))))). % ord_le_eq_trans
thf(fact_103_ord__le__eq__trans, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ B) => ((B = C) => (ord_less_eq_nat @ A @ C)))))). % ord_le_eq_trans
thf(fact_104_ord__eq__le__trans, axiom,
    ((![A : real, B : real, C : real]: ((A = B) => ((ord_less_eq_real @ B @ C) => (ord_less_eq_real @ A @ C)))))). % ord_eq_le_trans
thf(fact_105_ord__eq__le__trans, axiom,
    ((![A : nat, B : nat, C : nat]: ((A = B) => ((ord_less_eq_nat @ B @ C) => (ord_less_eq_nat @ A @ C)))))). % ord_eq_le_trans
thf(fact_106_order__class_Oorder_Oeq__iff, axiom,
    (((^[Y2 : real]: (^[Z2 : real]: (Y2 = Z2))) = (^[A2 : real]: (^[B2 : real]: (((ord_less_eq_real @ A2 @ B2)) & ((ord_less_eq_real @ B2 @ A2)))))))). % order_class.order.eq_iff
thf(fact_107_order__class_Oorder_Oeq__iff, axiom,
    (((^[Y2 : nat]: (^[Z2 : nat]: (Y2 = Z2))) = (^[A2 : nat]: (^[B2 : nat]: (((ord_less_eq_nat @ A2 @ B2)) & ((ord_less_eq_nat @ B2 @ A2)))))))). % order_class.order.eq_iff
thf(fact_108_antisym__conv, axiom,
    ((![Y4 : real, X3 : real]: ((ord_less_eq_real @ Y4 @ X3) => ((ord_less_eq_real @ X3 @ Y4) = (X3 = Y4)))))). % antisym_conv
thf(fact_109_antisym__conv, axiom,
    ((![Y4 : nat, X3 : nat]: ((ord_less_eq_nat @ Y4 @ X3) => ((ord_less_eq_nat @ X3 @ Y4) = (X3 = Y4)))))). % antisym_conv
thf(fact_110_le__cases3, axiom,
    ((![X3 : real, Y4 : real, Z : real]: (((ord_less_eq_real @ X3 @ Y4) => (~ ((ord_less_eq_real @ Y4 @ Z)))) => (((ord_less_eq_real @ Y4 @ X3) => (~ ((ord_less_eq_real @ X3 @ Z)))) => (((ord_less_eq_real @ X3 @ Z) => (~ ((ord_less_eq_real @ Z @ Y4)))) => (((ord_less_eq_real @ Z @ Y4) => (~ ((ord_less_eq_real @ Y4 @ X3)))) => (((ord_less_eq_real @ Y4 @ Z) => (~ ((ord_less_eq_real @ Z @ X3)))) => (~ (((ord_less_eq_real @ Z @ X3) => (~ ((ord_less_eq_real @ X3 @ Y4)))))))))))))). % le_cases3
thf(fact_111_le__cases3, axiom,
    ((![X3 : nat, Y4 : nat, Z : nat]: (((ord_less_eq_nat @ X3 @ Y4) => (~ ((ord_less_eq_nat @ Y4 @ Z)))) => (((ord_less_eq_nat @ Y4 @ X3) => (~ ((ord_less_eq_nat @ X3 @ Z)))) => (((ord_less_eq_nat @ X3 @ Z) => (~ ((ord_less_eq_nat @ Z @ Y4)))) => (((ord_less_eq_nat @ Z @ Y4) => (~ ((ord_less_eq_nat @ Y4 @ X3)))) => (((ord_less_eq_nat @ Y4 @ Z) => (~ ((ord_less_eq_nat @ Z @ X3)))) => (~ (((ord_less_eq_nat @ Z @ X3) => (~ ((ord_less_eq_nat @ X3 @ Y4)))))))))))))). % le_cases3
thf(fact_112_order_Otrans, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_eq_real @ B @ C) => (ord_less_eq_real @ A @ C)))))). % order.trans
thf(fact_113_order_Otrans, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_nat @ B @ C) => (ord_less_eq_nat @ A @ C)))))). % order.trans
thf(fact_114_le__cases, axiom,
    ((![X3 : real, Y4 : real]: ((~ ((ord_less_eq_real @ X3 @ Y4))) => (ord_less_eq_real @ Y4 @ X3))))). % le_cases
thf(fact_115_le__cases, axiom,
    ((![X3 : nat, Y4 : nat]: ((~ ((ord_less_eq_nat @ X3 @ Y4))) => (ord_less_eq_nat @ Y4 @ X3))))). % le_cases
thf(fact_116_eq__refl, axiom,
    ((![X3 : real, Y4 : real]: ((X3 = Y4) => (ord_less_eq_real @ X3 @ Y4))))). % eq_refl
thf(fact_117_eq__refl, axiom,
    ((![X3 : nat, Y4 : nat]: ((X3 = Y4) => (ord_less_eq_nat @ X3 @ Y4))))). % eq_refl
thf(fact_118_linear, axiom,
    ((![X3 : real, Y4 : real]: ((ord_less_eq_real @ X3 @ Y4) | (ord_less_eq_real @ Y4 @ X3))))). % linear
thf(fact_119_linear, axiom,
    ((![X3 : nat, Y4 : nat]: ((ord_less_eq_nat @ X3 @ Y4) | (ord_less_eq_nat @ Y4 @ X3))))). % linear
thf(fact_120_antisym, axiom,
    ((![X3 : real, Y4 : real]: ((ord_less_eq_real @ X3 @ Y4) => ((ord_less_eq_real @ Y4 @ X3) => (X3 = Y4)))))). % antisym
thf(fact_121_antisym, axiom,
    ((![X3 : nat, Y4 : nat]: ((ord_less_eq_nat @ X3 @ Y4) => ((ord_less_eq_nat @ Y4 @ X3) => (X3 = Y4)))))). % antisym
thf(fact_122_eq__iff, axiom,
    (((^[Y2 : real]: (^[Z2 : real]: (Y2 = Z2))) = (^[X5 : real]: (^[Y3 : real]: (((ord_less_eq_real @ X5 @ Y3)) & ((ord_less_eq_real @ Y3 @ X5)))))))). % eq_iff
thf(fact_123_eq__iff, axiom,
    (((^[Y2 : nat]: (^[Z2 : nat]: (Y2 = Z2))) = (^[X5 : nat]: (^[Y3 : nat]: (((ord_less_eq_nat @ X5 @ Y3)) & ((ord_less_eq_nat @ Y3 @ X5)))))))). % eq_iff
thf(fact_124_ord__le__eq__subst, axiom,
    ((![A : real, B : real, F : real > real, C : real]: ((ord_less_eq_real @ A @ B) => (((F @ B) = C) => ((![X6 : real, Y5 : real]: ((ord_less_eq_real @ X6 @ Y5) => (ord_less_eq_real @ (F @ X6) @ (F @ Y5)))) => (ord_less_eq_real @ (F @ A) @ C))))))). % ord_le_eq_subst
thf(fact_125_ord__le__eq__subst, axiom,
    ((![A : real, B : real, F : real > nat, C : nat]: ((ord_less_eq_real @ A @ B) => (((F @ B) = C) => ((![X6 : real, Y5 : real]: ((ord_less_eq_real @ X6 @ Y5) => (ord_less_eq_nat @ (F @ X6) @ (F @ Y5)))) => (ord_less_eq_nat @ (F @ A) @ C))))))). % ord_le_eq_subst
thf(fact_126_ord__le__eq__subst, axiom,
    ((![A : nat, B : nat, F : nat > real, C : real]: ((ord_less_eq_nat @ A @ B) => (((F @ B) = C) => ((![X6 : nat, Y5 : nat]: ((ord_less_eq_nat @ X6 @ Y5) => (ord_less_eq_real @ (F @ X6) @ (F @ Y5)))) => (ord_less_eq_real @ (F @ A) @ C))))))). % ord_le_eq_subst
thf(fact_127_ord__le__eq__subst, axiom,
    ((![A : nat, B : nat, F : nat > nat, C : nat]: ((ord_less_eq_nat @ A @ B) => (((F @ B) = C) => ((![X6 : nat, Y5 : nat]: ((ord_less_eq_nat @ X6 @ Y5) => (ord_less_eq_nat @ (F @ X6) @ (F @ Y5)))) => (ord_less_eq_nat @ (F @ A) @ C))))))). % ord_le_eq_subst
thf(fact_128_ord__eq__le__subst, axiom,
    ((![A : real, F : real > real, B : real, C : real]: ((A = (F @ B)) => ((ord_less_eq_real @ B @ C) => ((![X6 : real, Y5 : real]: ((ord_less_eq_real @ X6 @ Y5) => (ord_less_eq_real @ (F @ X6) @ (F @ Y5)))) => (ord_less_eq_real @ A @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_129_ord__eq__le__subst, axiom,
    ((![A : nat, F : real > nat, B : real, C : real]: ((A = (F @ B)) => ((ord_less_eq_real @ B @ C) => ((![X6 : real, Y5 : real]: ((ord_less_eq_real @ X6 @ Y5) => (ord_less_eq_nat @ (F @ X6) @ (F @ Y5)))) => (ord_less_eq_nat @ A @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_130_ord__eq__le__subst, axiom,
    ((![A : real, F : nat > real, B : nat, C : nat]: ((A = (F @ B)) => ((ord_less_eq_nat @ B @ C) => ((![X6 : nat, Y5 : nat]: ((ord_less_eq_nat @ X6 @ Y5) => (ord_less_eq_real @ (F @ X6) @ (F @ Y5)))) => (ord_less_eq_real @ A @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_131_ord__eq__le__subst, axiom,
    ((![A : nat, F : nat > nat, B : nat, C : nat]: ((A = (F @ B)) => ((ord_less_eq_nat @ B @ C) => ((![X6 : nat, Y5 : nat]: ((ord_less_eq_nat @ X6 @ Y5) => (ord_less_eq_nat @ (F @ X6) @ (F @ Y5)))) => (ord_less_eq_nat @ A @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_132_order__subst2, axiom,
    ((![A : real, B : real, F : real > real, C : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_eq_real @ (F @ B) @ C) => ((![X6 : real, Y5 : real]: ((ord_less_eq_real @ X6 @ Y5) => (ord_less_eq_real @ (F @ X6) @ (F @ Y5)))) => (ord_less_eq_real @ (F @ A) @ C))))))). % order_subst2
thf(fact_133_order__subst2, axiom,
    ((![A : real, B : real, F : real > nat, C : nat]: ((ord_less_eq_real @ A @ B) => ((ord_less_eq_nat @ (F @ B) @ C) => ((![X6 : real, Y5 : real]: ((ord_less_eq_real @ X6 @ Y5) => (ord_less_eq_nat @ (F @ X6) @ (F @ Y5)))) => (ord_less_eq_nat @ (F @ A) @ C))))))). % order_subst2
thf(fact_134_order__subst2, axiom,
    ((![A : nat, B : nat, F : nat > real, C : real]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_real @ (F @ B) @ C) => ((![X6 : nat, Y5 : nat]: ((ord_less_eq_nat @ X6 @ Y5) => (ord_less_eq_real @ (F @ X6) @ (F @ Y5)))) => (ord_less_eq_real @ (F @ A) @ C))))))). % order_subst2
thf(fact_135_order__subst2, axiom,
    ((![A : nat, B : nat, F : nat > nat, C : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_nat @ (F @ B) @ C) => ((![X6 : nat, Y5 : nat]: ((ord_less_eq_nat @ X6 @ Y5) => (ord_less_eq_nat @ (F @ X6) @ (F @ Y5)))) => (ord_less_eq_nat @ (F @ A) @ C))))))). % order_subst2
thf(fact_136_order__subst1, axiom,
    ((![A : real, F : real > real, B : real, C : real]: ((ord_less_eq_real @ A @ (F @ B)) => ((ord_less_eq_real @ B @ C) => ((![X6 : real, Y5 : real]: ((ord_less_eq_real @ X6 @ Y5) => (ord_less_eq_real @ (F @ X6) @ (F @ Y5)))) => (ord_less_eq_real @ A @ (F @ C)))))))). % order_subst1
thf(fact_137_order__subst1, axiom,
    ((![A : real, F : nat > real, B : nat, C : nat]: ((ord_less_eq_real @ A @ (F @ B)) => ((ord_less_eq_nat @ B @ C) => ((![X6 : nat, Y5 : nat]: ((ord_less_eq_nat @ X6 @ Y5) => (ord_less_eq_real @ (F @ X6) @ (F @ Y5)))) => (ord_less_eq_real @ A @ (F @ C)))))))). % order_subst1
thf(fact_138_order__subst1, axiom,
    ((![A : nat, F : real > nat, B : real, C : real]: ((ord_less_eq_nat @ A @ (F @ B)) => ((ord_less_eq_real @ B @ C) => ((![X6 : real, Y5 : real]: ((ord_less_eq_real @ X6 @ Y5) => (ord_less_eq_nat @ (F @ X6) @ (F @ Y5)))) => (ord_less_eq_nat @ A @ (F @ C)))))))). % order_subst1
thf(fact_139_order__subst1, axiom,
    ((![A : nat, F : nat > nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ (F @ B)) => ((ord_less_eq_nat @ B @ C) => ((![X6 : nat, Y5 : nat]: ((ord_less_eq_nat @ X6 @ Y5) => (ord_less_eq_nat @ (F @ X6) @ (F @ Y5)))) => (ord_less_eq_nat @ A @ (F @ C)))))))). % order_subst1
thf(fact_140_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_141_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_142_fun_Omap__comp, axiom,
    ((![G3 : real > real, F : nat > real, V : nat > nat]: ((comp_real_real_nat @ G3 @ (comp_nat_real_nat @ F @ V)) = (comp_nat_real_nat @ (comp_real_real_nat @ G3 @ F) @ V))))). % fun.map_comp
thf(fact_143_fun_Omap__comp, axiom,
    ((![G3 : real > complex, F : complex > real, V : nat > complex]: ((comp_r2119295887ex_nat @ G3 @ (comp_c1631780367al_nat @ F @ V)) = (comp_c438056209ex_nat @ (comp_r667767405omplex @ G3 @ F) @ V))))). % fun.map_comp
thf(fact_144_fun_Omap__comp, axiom,
    ((![G3 : real > real, F : complex > real, V : nat > complex]: ((comp_real_real_nat @ G3 @ (comp_c1631780367al_nat @ F @ V)) = (comp_c1631780367al_nat @ (comp_r422820971omplex @ G3 @ F) @ V))))). % fun.map_comp
thf(fact_145_fun_Omap__comp, axiom,
    ((![G3 : complex > complex, F : real > complex, V : complex > real]: ((comp_c130555887omplex @ G3 @ (comp_r667767405omplex @ F @ V)) = (comp_r667767405omplex @ (comp_c595887981x_real @ G3 @ F) @ V))))). % fun.map_comp
thf(fact_146_fun_Omap__comp, axiom,
    ((![G3 : nat > real, F : nat > nat, V : nat > nat]: ((comp_nat_real_nat @ G3 @ (comp_nat_nat_nat @ F @ V)) = (comp_nat_real_nat @ (comp_nat_real_nat @ G3 @ F) @ V))))). % fun.map_comp
thf(fact_147_fun_Omap__comp, axiom,
    ((![G3 : nat > real, F : complex > nat, V : nat > complex]: ((comp_nat_real_nat @ G3 @ (comp_complex_nat_nat @ F @ V)) = (comp_c1631780367al_nat @ (comp_n1816297743omplex @ G3 @ F) @ V))))). % fun.map_comp
thf(fact_148_fun_Omap__comp, axiom,
    ((![G3 : complex > real, F : nat > complex, V : nat > nat]: ((comp_c1631780367al_nat @ G3 @ (comp_nat_complex_nat @ F @ V)) = (comp_nat_real_nat @ (comp_c1631780367al_nat @ G3 @ F) @ V))))). % fun.map_comp
thf(fact_149_fun_Omap__comp, axiom,
    ((![G3 : complex > real, F : complex > complex, V : nat > complex]: ((comp_c1631780367al_nat @ G3 @ (comp_c438056209ex_nat @ F @ V)) = (comp_c1631780367al_nat @ (comp_c317287661omplex @ G3 @ F) @ V))))). % fun.map_comp
thf(fact_150_fun_Omap__comp, axiom,
    ((![G3 : real > complex, F : complex > real, V : complex > complex]: ((comp_r667767405omplex @ G3 @ (comp_c317287661omplex @ F @ V)) = (comp_c130555887omplex @ (comp_r667767405omplex @ G3 @ F) @ V))))). % fun.map_comp
thf(fact_151_fun_Omap__comp, axiom,
    ((![G3 : real > complex, F : real > real, V : complex > real]: ((comp_r667767405omplex @ G3 @ (comp_r422820971omplex @ F @ V)) = (comp_r667767405omplex @ (comp_r701421291x_real @ G3 @ F) @ V))))). % fun.map_comp
thf(fact_152_comp__eq__dest__lhs, axiom,
    ((![A : nat > real, B : nat > nat, C : nat > real, V : nat]: (((comp_nat_real_nat @ A @ B) = C) => ((A @ (B @ V)) = (C @ V)))))). % comp_eq_dest_lhs
thf(fact_153_comp__eq__dest__lhs, axiom,
    ((![A : complex > real, B : nat > complex, C : nat > real, V : nat]: (((comp_c1631780367al_nat @ A @ B) = C) => ((A @ (B @ V)) = (C @ V)))))). % comp_eq_dest_lhs
thf(fact_154_comp__eq__dest__lhs, axiom,
    ((![A : real > complex, B : complex > real, C : complex > complex, V : complex]: (((comp_r667767405omplex @ A @ B) = C) => ((A @ (B @ V)) = (C @ V)))))). % comp_eq_dest_lhs
thf(fact_155_comp__eq__elim, axiom,
    ((![A : nat > real, B : nat > nat, C : nat > real, D : nat > nat]: (((comp_nat_real_nat @ A @ B) = (comp_nat_real_nat @ C @ D)) => (![V2 : nat]: ((A @ (B @ V2)) = (C @ (D @ V2)))))))). % comp_eq_elim
thf(fact_156_comp__eq__elim, axiom,
    ((![A : nat > real, B : nat > nat, C : complex > real, D : nat > complex]: (((comp_nat_real_nat @ A @ B) = (comp_c1631780367al_nat @ C @ D)) => (![V2 : nat]: ((A @ (B @ V2)) = (C @ (D @ V2)))))))). % comp_eq_elim
thf(fact_157_comp__eq__elim, axiom,
    ((![A : complex > real, B : nat > complex, C : nat > real, D : nat > nat]: (((comp_c1631780367al_nat @ A @ B) = (comp_nat_real_nat @ C @ D)) => (![V2 : nat]: ((A @ (B @ V2)) = (C @ (D @ V2)))))))). % comp_eq_elim
thf(fact_158_comp__eq__elim, axiom,
    ((![A : complex > real, B : nat > complex, C : complex > real, D : nat > complex]: (((comp_c1631780367al_nat @ A @ B) = (comp_c1631780367al_nat @ C @ D)) => (![V2 : nat]: ((A @ (B @ V2)) = (C @ (D @ V2)))))))). % comp_eq_elim
thf(fact_159_comp__eq__elim, axiom,
    ((![A : real > complex, B : complex > real, C : real > complex, D : complex > real]: (((comp_r667767405omplex @ A @ B) = (comp_r667767405omplex @ C @ D)) => (![V2 : complex]: ((A @ (B @ V2)) = (C @ (D @ V2)))))))). % comp_eq_elim
thf(fact_160_comp__eq__dest, axiom,
    ((![A : nat > real, B : nat > nat, C : nat > real, D : nat > nat, V : nat]: (((comp_nat_real_nat @ A @ B) = (comp_nat_real_nat @ C @ D)) => ((A @ (B @ V)) = (C @ (D @ V))))))). % comp_eq_dest
thf(fact_161_comp__eq__dest, axiom,
    ((![A : nat > real, B : nat > nat, C : complex > real, D : nat > complex, V : nat]: (((comp_nat_real_nat @ A @ B) = (comp_c1631780367al_nat @ C @ D)) => ((A @ (B @ V)) = (C @ (D @ V))))))). % comp_eq_dest
thf(fact_162_comp__eq__dest, axiom,
    ((![A : complex > real, B : nat > complex, C : nat > real, D : nat > nat, V : nat]: (((comp_c1631780367al_nat @ A @ B) = (comp_nat_real_nat @ C @ D)) => ((A @ (B @ V)) = (C @ (D @ V))))))). % comp_eq_dest
thf(fact_163_comp__eq__dest, axiom,
    ((![A : complex > real, B : nat > complex, C : complex > real, D : nat > complex, V : nat]: (((comp_c1631780367al_nat @ A @ B) = (comp_c1631780367al_nat @ C @ D)) => ((A @ (B @ V)) = (C @ (D @ V))))))). % comp_eq_dest
thf(fact_164_comp__eq__dest, axiom,
    ((![A : real > complex, B : complex > real, C : real > complex, D : complex > real, V : complex]: (((comp_r667767405omplex @ A @ B) = (comp_r667767405omplex @ C @ D)) => ((A @ (B @ V)) = (C @ (D @ V))))))). % comp_eq_dest
thf(fact_165_comp__assoc, axiom,
    ((![F : real > complex, G3 : complex > real, H : nat > complex]: ((comp_c438056209ex_nat @ (comp_r667767405omplex @ F @ G3) @ H) = (comp_r2119295887ex_nat @ F @ (comp_c1631780367al_nat @ G3 @ H)))))). % comp_assoc
thf(fact_166_comp__assoc, axiom,
    ((![F : real > complex, G3 : complex > real, H : complex > complex]: ((comp_c130555887omplex @ (comp_r667767405omplex @ F @ G3) @ H) = (comp_r667767405omplex @ F @ (comp_c317287661omplex @ G3 @ H)))))). % comp_assoc
thf(fact_167_comp__assoc, axiom,
    ((![F : real > real, G3 : nat > real, H : nat > nat]: ((comp_nat_real_nat @ (comp_real_real_nat @ F @ G3) @ H) = (comp_real_real_nat @ F @ (comp_nat_real_nat @ G3 @ H)))))). % comp_assoc
thf(fact_168_comp__assoc, axiom,
    ((![F : nat > real, G3 : nat > nat, H : nat > nat]: ((comp_nat_real_nat @ (comp_nat_real_nat @ F @ G3) @ H) = (comp_nat_real_nat @ F @ (comp_nat_nat_nat @ G3 @ H)))))). % comp_assoc
thf(fact_169_comp__assoc, axiom,
    ((![F : complex > real, G3 : nat > complex, H : nat > nat]: ((comp_nat_real_nat @ (comp_c1631780367al_nat @ F @ G3) @ H) = (comp_c1631780367al_nat @ F @ (comp_nat_complex_nat @ G3 @ H)))))). % comp_assoc
thf(fact_170_comp__assoc, axiom,
    ((![F : real > real, G3 : complex > real, H : nat > complex]: ((comp_c1631780367al_nat @ (comp_r422820971omplex @ F @ G3) @ H) = (comp_real_real_nat @ F @ (comp_c1631780367al_nat @ G3 @ H)))))). % comp_assoc
thf(fact_171_comp__assoc, axiom,
    ((![F : nat > real, G3 : complex > nat, H : nat > complex]: ((comp_c1631780367al_nat @ (comp_n1816297743omplex @ F @ G3) @ H) = (comp_nat_real_nat @ F @ (comp_complex_nat_nat @ G3 @ H)))))). % comp_assoc
thf(fact_172_comp__assoc, axiom,
    ((![F : complex > real, G3 : complex > complex, H : nat > complex]: ((comp_c1631780367al_nat @ (comp_c317287661omplex @ F @ G3) @ H) = (comp_c1631780367al_nat @ F @ (comp_c438056209ex_nat @ G3 @ H)))))). % comp_assoc
thf(fact_173_comp__assoc, axiom,
    ((![F : complex > complex, G3 : real > complex, H : complex > real]: ((comp_r667767405omplex @ (comp_c595887981x_real @ F @ G3) @ H) = (comp_c130555887omplex @ F @ (comp_r667767405omplex @ G3 @ H)))))). % comp_assoc
thf(fact_174_comp__assoc, axiom,
    ((![F : real > complex, G3 : real > real, H : complex > real]: ((comp_r667767405omplex @ (comp_r701421291x_real @ F @ G3) @ H) = (comp_r667767405omplex @ F @ (comp_r422820971omplex @ G3 @ H)))))). % comp_assoc
thf(fact_175_comp__def, axiom,
    ((comp_nat_real_nat = (^[F3 : nat > real]: (^[G2 : nat > nat]: (^[X5 : nat]: (F3 @ (G2 @ X5)))))))). % comp_def
thf(fact_176_comp__def, axiom,
    ((comp_c1631780367al_nat = (^[F3 : complex > real]: (^[G2 : nat > complex]: (^[X5 : nat]: (F3 @ (G2 @ X5)))))))). % comp_def
thf(fact_177_comp__def, axiom,
    ((comp_r667767405omplex = (^[F3 : real > complex]: (^[G2 : complex > real]: (^[X5 : complex]: (F3 @ (G2 @ X5)))))))). % comp_def
thf(fact_178_strict__mono__eq, axiom,
    ((![F : nat > nat, X3 : nat, Y4 : nat]: ((order_769474267at_nat @ F) => (((F @ X3) = (F @ Y4)) = (X3 = Y4)))))). % strict_mono_eq
thf(fact_179_cmod__Re__le__iff, axiom,
    ((![X3 : complex, Y4 : complex]: (((im @ X3) = (im @ Y4)) => ((ord_less_eq_real @ (real_V638595069omplex @ X3) @ (real_V638595069omplex @ Y4)) = (ord_less_eq_real @ (abs_abs_real @ (re @ X3)) @ (abs_abs_real @ (re @ Y4)))))))). % cmod_Re_le_iff
thf(fact_180_abs__of__nonneg, axiom,
    ((![A : real]: ((ord_less_eq_real @ zero_zero_real @ A) => ((abs_abs_real @ A) = A))))). % abs_of_nonneg
thf(fact_181_abs__le__self__iff, axiom,
    ((![A : real]: ((ord_less_eq_real @ (abs_abs_real @ A) @ A) = (ord_less_eq_real @ zero_zero_real @ A))))). % abs_le_self_iff
thf(fact_182_abs__le__zero__iff, axiom,
    ((![A : real]: ((ord_less_eq_real @ (abs_abs_real @ A) @ zero_zero_real) = (A = zero_zero_real))))). % abs_le_zero_iff
thf(fact_183_le0, axiom,
    ((![N4 : nat]: (ord_less_eq_nat @ zero_zero_nat @ N4)))). % le0
thf(fact_184_bot__nat__0_Oextremum, axiom,
    ((![A : nat]: (ord_less_eq_nat @ zero_zero_nat @ A)))). % bot_nat_0.extremum
thf(fact_185_abs__zero, axiom,
    (((abs_abs_real @ zero_zero_real) = zero_zero_real))). % abs_zero
thf(fact_186_abs__idempotent, axiom,
    ((![A : real]: ((abs_abs_real @ (abs_abs_real @ A)) = (abs_abs_real @ A))))). % abs_idempotent
thf(fact_187_le__zero__eq, axiom,
    ((![N4 : nat]: ((ord_less_eq_nat @ N4 @ zero_zero_nat) = (N4 = zero_zero_nat))))). % le_zero_eq
thf(fact_188_abs__0__eq, axiom,
    ((![A : real]: ((zero_zero_real = (abs_abs_real @ A)) = (A = zero_zero_real))))). % abs_0_eq
thf(fact_189_abs__eq__0, axiom,
    ((![A : real]: (((abs_abs_real @ A) = zero_zero_real) = (A = zero_zero_real))))). % abs_eq_0
thf(fact_190_abs__complex__def, axiom,
    ((abs_abs_complex = (comp_r667767405omplex @ real_V306493662omplex @ real_V638595069omplex)))). % abs_complex_def
thf(fact_191_zero__complex_Osimps_I1_J, axiom,
    (((re @ zero_zero_complex) = zero_zero_real))). % zero_complex.simps(1)
thf(fact_192_zero__complex_Osimps_I2_J, axiom,
    (((im @ zero_zero_complex) = zero_zero_real))). % zero_complex.simps(2)
thf(fact_193_zero__reorient, axiom,
    ((![X3 : nat]: ((zero_zero_nat = X3) = (X3 = zero_zero_nat))))). % zero_reorient
thf(fact_194_zero__reorient, axiom,
    ((![X3 : real]: ((zero_zero_real = X3) = (X3 = zero_zero_real))))). % zero_reorient
thf(fact_195_zero__reorient, axiom,
    ((![X3 : complex]: ((zero_zero_complex = X3) = (X3 = zero_zero_complex))))). % zero_reorient
thf(fact_196_Nat_Oex__has__greatest__nat, axiom,
    ((![P : nat > $o, K : nat, B : nat]: ((P @ K) => ((![Y5 : nat]: ((P @ Y5) => (ord_less_eq_nat @ Y5 @ B))) => (?[X6 : nat]: ((P @ X6) & (![Y6 : nat]: ((P @ Y6) => (ord_less_eq_nat @ Y6 @ X6)))))))))). % Nat.ex_has_greatest_nat
thf(fact_197_nat__le__linear, axiom,
    ((![M3 : nat, N4 : nat]: ((ord_less_eq_nat @ M3 @ N4) | (ord_less_eq_nat @ N4 @ M3))))). % nat_le_linear
thf(fact_198_le__antisym, axiom,
    ((![M3 : nat, N4 : nat]: ((ord_less_eq_nat @ M3 @ N4) => ((ord_less_eq_nat @ N4 @ M3) => (M3 = N4)))))). % le_antisym
thf(fact_199_eq__imp__le, axiom,
    ((![M3 : nat, N4 : nat]: ((M3 = N4) => (ord_less_eq_nat @ M3 @ N4))))). % eq_imp_le
thf(fact_200_le__trans, axiom,
    ((![I : nat, J : nat, K : nat]: ((ord_less_eq_nat @ I @ J) => ((ord_less_eq_nat @ J @ K) => (ord_less_eq_nat @ I @ K)))))). % le_trans
thf(fact_201_le__refl, axiom,
    ((![N4 : nat]: (ord_less_eq_nat @ N4 @ N4)))). % le_refl
thf(fact_202_zero__le, axiom,
    ((![X3 : nat]: (ord_less_eq_nat @ zero_zero_nat @ X3)))). % zero_le
thf(fact_203_abs__ge__self, axiom,
    ((![A : real]: (ord_less_eq_real @ A @ (abs_abs_real @ A))))). % abs_ge_self
thf(fact_204_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_205_bot__nat__0_Oextremum__uniqueI, axiom,
    ((![A : nat]: ((ord_less_eq_nat @ A @ zero_zero_nat) => (A = zero_zero_nat))))). % bot_nat_0.extremum_uniqueI
thf(fact_206_bot__nat__0_Oextremum__unique, axiom,
    ((![A : nat]: ((ord_less_eq_nat @ A @ zero_zero_nat) = (A = zero_zero_nat))))). % bot_nat_0.extremum_unique
thf(fact_207_le__0__eq, axiom,
    ((![N4 : nat]: ((ord_less_eq_nat @ N4 @ zero_zero_nat) = (N4 = zero_zero_nat))))). % le_0_eq
thf(fact_208_less__eq__nat_Osimps_I1_J, axiom,
    ((![N4 : nat]: (ord_less_eq_nat @ zero_zero_nat @ N4)))). % less_eq_nat.simps(1)
thf(fact_209_strict__mono__imp__increasing, axiom,
    ((![F : nat > nat, N4 : nat]: ((order_769474267at_nat @ F) => (ord_less_eq_nat @ N4 @ (F @ N4)))))). % strict_mono_imp_increasing
thf(fact_210_abs__ge__zero, axiom,
    ((![A : real]: (ord_less_eq_real @ zero_zero_real @ (abs_abs_real @ A))))). % abs_ge_zero
thf(fact_211_abs__0, axiom,
    (((abs_abs_real @ zero_zero_real) = zero_zero_real))). % abs_0
thf(fact_212_abs__0, axiom,
    (((abs_abs_complex @ zero_zero_complex) = zero_zero_complex))). % abs_0
thf(fact_213_abs__abs, axiom,
    ((![A : real]: ((abs_abs_real @ (abs_abs_real @ A)) = (abs_abs_real @ A))))). % abs_abs
thf(fact_214_abs__abs, axiom,
    ((![A : complex]: ((abs_abs_complex @ (abs_abs_complex @ A)) = (abs_abs_complex @ A))))). % abs_abs
thf(fact_215_abs__eq__0__iff, axiom,
    ((![A : real]: (((abs_abs_real @ A) = zero_zero_real) = (A = zero_zero_real))))). % abs_eq_0_iff
thf(fact_216_abs__eq__0__iff, axiom,
    ((![A : complex]: (((abs_abs_complex @ A) = zero_zero_complex) = (A = zero_zero_complex))))). % abs_eq_0_iff
thf(fact_217_le__numeral__extra_I3_J, axiom,
    ((ord_less_eq_real @ zero_zero_real @ zero_zero_real))). % le_numeral_extra(3)
thf(fact_218_le__numeral__extra_I3_J, axiom,
    ((ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat))). % le_numeral_extra(3)
thf(fact_219_th, axiom,
    ((![N2 : nat]: (ord_less_eq_real @ (abs_abs_real @ (im @ (s @ N2))) @ (plus_plus_real @ r @ one_one_real))))). % th
thf(fact_220_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_221_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_222_zero__eq__add__iff__both__eq__0, axiom,
    ((![X3 : nat, Y4 : nat]: ((zero_zero_nat = (plus_plus_nat @ X3 @ Y4)) = (((X3 = zero_zero_nat)) & ((Y4 = zero_zero_nat))))))). % zero_eq_add_iff_both_eq_0
thf(fact_223_add__eq__0__iff__both__eq__0, axiom,
    ((![X3 : nat, Y4 : nat]: (((plus_plus_nat @ X3 @ Y4) = zero_zero_nat) = (((X3 = zero_zero_nat)) & ((Y4 = zero_zero_nat))))))). % add_eq_0_iff_both_eq_0
thf(fact_224_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_225_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_226_add__cancel__right__right, axiom,
    ((![A : complex, B : complex]: ((A = (plus_plus_complex @ A @ B)) = (B = zero_zero_complex))))). % add_cancel_right_right
thf(fact_227_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_228_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_229_add__cancel__right__left, axiom,
    ((![A : complex, B : complex]: ((A = (plus_plus_complex @ B @ A)) = (B = zero_zero_complex))))). % add_cancel_right_left
thf(fact_230_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_231_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_232_add__cancel__left__right, axiom,
    ((![A : complex, B : complex]: (((plus_plus_complex @ A @ B) = A) = (B = zero_zero_complex))))). % add_cancel_left_right
thf(fact_233_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_234_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_235_add__cancel__left__left, axiom,
    ((![B : complex, A : complex]: (((plus_plus_complex @ B @ A) = A) = (B = zero_zero_complex))))). % add_cancel_left_left
thf(fact_236_double__zero__sym, axiom,
    ((![A : real]: ((zero_zero_real = (plus_plus_real @ A @ A)) = (A = zero_zero_real))))). % double_zero_sym
thf(fact_237_double__zero, axiom,
    ((![A : real]: (((plus_plus_real @ A @ A) = zero_zero_real) = (A = zero_zero_real))))). % double_zero
thf(fact_238_add_Oright__neutral, axiom,
    ((![A : nat]: ((plus_plus_nat @ A @ zero_zero_nat) = A)))). % add.right_neutral
thf(fact_239_add_Oright__neutral, axiom,
    ((![A : real]: ((plus_plus_real @ A @ zero_zero_real) = A)))). % add.right_neutral
thf(fact_240_add_Oright__neutral, axiom,
    ((![A : complex]: ((plus_plus_complex @ A @ zero_zero_complex) = A)))). % add.right_neutral
thf(fact_241_add_Oleft__neutral, axiom,
    ((![A : nat]: ((plus_plus_nat @ zero_zero_nat @ A) = A)))). % add.left_neutral
thf(fact_242_add_Oleft__neutral, axiom,
    ((![A : real]: ((plus_plus_real @ zero_zero_real @ A) = A)))). % add.left_neutral
thf(fact_243_add_Oleft__neutral, axiom,
    ((![A : complex]: ((plus_plus_complex @ zero_zero_complex @ A) = A)))). % add.left_neutral
thf(fact_244_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_245_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

% Conjectures (1)
thf(conj_0, conjecture,
    ((topolo795669587t_real @ (^[N : nat]: (im @ (s @ (f @ (g @ N)))))))).
