% 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_278__5370302_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:28:53.177

% Could-be-implicit typings (6)
thf(ty_n_t__Filter__Ofilter_It__Complex__Ocomplex_J, type,
    filter_complex : $tType).
thf(ty_n_t__Filter__Ofilter_It__Real__Oreal_J, type,
    filter_real : $tType).
thf(ty_n_t__Filter__Ofilter_It__Nat__Onat_J, type,
    filter_nat : $tType).
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 (48)
thf(sy_c_Complex_Ocomplex_OIm, type,
    im : complex > real).
thf(sy_c_Complex_Ocomplex_ORe, type,
    re : complex > real).
thf(sy_c_Filter_Oat__top_001t__Nat__Onat, type,
    at_top_nat : filter_nat).
thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Complex__Ocomplex, type,
    filter1919943476omplex : (nat > complex) > filter_complex > filter_nat > $o).
thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Nat__Onat, type,
    filterlim_nat_nat : (nat > nat) > filter_nat > filter_nat > $o).
thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Real__Oreal, type,
    filterlim_nat_real : (nat > real) > filter_real > filter_nat > $o).
thf(sy_c_Filter_Ofilterlim_001t__Real__Oreal_001t__Nat__Onat, type,
    filterlim_real_nat : (real > nat) > filter_nat > filter_real > $o).
thf(sy_c_Filter_Ofilterlim_001t__Real__Oreal_001t__Real__Oreal, type,
    filterlim_real_real : (real > real) > filter_real > filter_real > $o).
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__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__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__Real__Oreal, type,
    abs_abs_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_HOL_OUniq_001t__Nat__Onat, type,
    uniq_nat : (nat > $o) > $o).
thf(sy_c_HOL_OUniq_001t__Real__Oreal, type,
    uniq_real : (real > $o) > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Filter__Ofilter_It__Nat__Onat_J, type,
    ord_le1745708096er_nat : filter_nat > filter_nat > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Filter__Ofilter_It__Real__Oreal_J, type,
    ord_le132810396r_real : filter_real > filter_real > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat, type,
    ord_less_eq_nat : nat > nat > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal, type,
    ord_less_eq_real : real > real > $o).
thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Nat__Onat_001t__Nat__Onat, type,
    order_769474267at_nat : (nat > nat) > $o).
thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Nat__Onat_001t__Real__Oreal, type,
    order_952716343t_real : (nat > real) > $o).
thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Real__Oreal_001t__Real__Oreal, type,
    order_1818878995l_real : (real > real) > $o).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Complex__Ocomplex, type,
    real_V638595069omplex : complex > real).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Real__Oreal, type,
    real_V646646907m_real : real > real).
thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Complex__Ocomplex, type,
    real_V1560324349omplex : real > complex > complex).
thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Real__Oreal, type,
    real_V453051771R_real : 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__Nat__Onat, type,
    topolo768750839nt_nat : (nat > nat) > $o).
thf(sy_c_Topological__Spaces_Otopological__space__class_Oconvergent_001t__Real__Oreal, type,
    topolo795669587t_real : (nat > real) > $o).
thf(sy_c_Topological__Spaces_Otopological__space__class_Onhds_001t__Complex__Ocomplex, type,
    topolo155787769omplex : complex > filter_complex).
thf(sy_c_Topological__Spaces_Otopological__space__class_Onhds_001t__Nat__Onat, type,
    topolo1564986139ds_nat : nat > filter_nat).
thf(sy_c_Topological__Spaces_Otopological__space__class_Onhds_001t__Real__Oreal, type,
    topolo1664202871s_real : real > filter_real).
thf(sy_v_f____, type,
    f : nat > nat).
thf(sy_v_g____, type,
    g : nat > nat).
thf(sy_v_r, type,
    r : real).
thf(sy_v_s, type,
    s : nat > complex).
thf(sy_v_thesis____, type,
    thesis : $o).

% Relevant facts (192)
thf(fact_0__092_060open_062_092_060exists_062L_O_A_I_092_060lambda_062n_O_ARe_A_Is_A_If_An_J_J_J_A_092_060longlonglongrightarrow_062_AL_092_060close_062, axiom,
    ((?[L : real]: (filterlim_nat_real @ (^[N : nat]: (re @ (s @ (f @ N)))) @ (topolo1664202871s_real @ L) @ at_top_nat)))). % \<open>\<exists>L. (\<lambda>n. Re (s (f n))) \<longlonglongrightarrow> L\<close>
thf(fact_1_f_I1_J, axiom,
    ((order_769474267at_nat @ f))). % f(1)
thf(fact_2_f_I2_J, axiom,
    ((topolo144289241q_real @ (^[N : nat]: (re @ (s @ (f @ N))))))). % f(2)
thf(fact_3_tendsto__const, axiom,
    ((![K : nat, F : filter_nat]: (filterlim_nat_nat @ (^[X : nat]: K) @ (topolo1564986139ds_nat @ K) @ F)))). % tendsto_const
thf(fact_4_tendsto__const, axiom,
    ((![K : real, F : filter_nat]: (filterlim_nat_real @ (^[X : nat]: K) @ (topolo1664202871s_real @ K) @ F)))). % tendsto_const
thf(fact_5_conv1, axiom,
    ((topolo795669587t_real @ (^[N : nat]: (re @ (s @ (f @ N))))))). % conv1
thf(fact_6_LIMSEQ__const__iff, axiom,
    ((![K : nat, L2 : nat]: ((filterlim_nat_nat @ (^[N : nat]: K) @ (topolo1564986139ds_nat @ L2) @ at_top_nat) = (K = L2))))). % LIMSEQ_const_iff
thf(fact_7_LIMSEQ__const__iff, axiom,
    ((![K : real, L2 : real]: ((filterlim_nat_real @ (^[N : nat]: K) @ (topolo1664202871s_real @ L2) @ at_top_nat) = (K = L2))))). % LIMSEQ_const_iff
thf(fact_8_LIMSEQ__unique, axiom,
    ((![X2 : nat > nat, A : nat, B : nat]: ((filterlim_nat_nat @ X2 @ (topolo1564986139ds_nat @ A) @ at_top_nat) => ((filterlim_nat_nat @ X2 @ (topolo1564986139ds_nat @ B) @ at_top_nat) => (A = B)))))). % LIMSEQ_unique
thf(fact_9_LIMSEQ__unique, axiom,
    ((![X2 : nat > real, A : real, B : real]: ((filterlim_nat_real @ X2 @ (topolo1664202871s_real @ A) @ at_top_nat) => ((filterlim_nat_real @ X2 @ (topolo1664202871s_real @ B) @ at_top_nat) => (A = B)))))). % LIMSEQ_unique
thf(fact_10_tendsto__cong__limit, axiom,
    ((![F2 : nat > nat, L2 : nat, F : filter_nat, K : nat]: ((filterlim_nat_nat @ F2 @ (topolo1564986139ds_nat @ L2) @ F) => ((K = L2) => (filterlim_nat_nat @ F2 @ (topolo1564986139ds_nat @ K) @ F)))))). % tendsto_cong_limit
thf(fact_11_tendsto__cong__limit, axiom,
    ((![F2 : nat > real, L2 : real, F : filter_nat, K : real]: ((filterlim_nat_real @ F2 @ (topolo1664202871s_real @ L2) @ F) => ((K = L2) => (filterlim_nat_real @ F2 @ (topolo1664202871s_real @ K) @ F)))))). % tendsto_cong_limit
thf(fact_12_tendsto__eq__rhs, axiom,
    ((![F2 : nat > nat, X3 : nat, F : filter_nat, Y : nat]: ((filterlim_nat_nat @ F2 @ (topolo1564986139ds_nat @ X3) @ F) => ((X3 = Y) => (filterlim_nat_nat @ F2 @ (topolo1564986139ds_nat @ Y) @ F)))))). % tendsto_eq_rhs
thf(fact_13_tendsto__eq__rhs, axiom,
    ((![F2 : nat > real, X3 : real, F : filter_nat, Y : real]: ((filterlim_nat_real @ F2 @ (topolo1664202871s_real @ X3) @ F) => ((X3 = Y) => (filterlim_nat_real @ F2 @ (topolo1664202871s_real @ Y) @ F)))))). % tendsto_eq_rhs
thf(fact_14_filterlim__ident, axiom,
    ((![F : filter_nat]: (filterlim_nat_nat @ (^[X : nat]: X) @ F @ F)))). % filterlim_ident
thf(fact_15_filterlim__compose, axiom,
    ((![G : real > real, F3 : filter_real, F22 : filter_real, F2 : nat > real, F1 : filter_nat]: ((filterlim_real_real @ G @ F3 @ F22) => ((filterlim_nat_real @ F2 @ F22 @ F1) => (filterlim_nat_real @ (^[X : nat]: (G @ (F2 @ X))) @ F3 @ F1)))))). % filterlim_compose
thf(fact_16_filterlim__compose, axiom,
    ((![G : real > nat, F3 : filter_nat, F22 : filter_real, F2 : nat > real, F1 : filter_nat]: ((filterlim_real_nat @ G @ F3 @ F22) => ((filterlim_nat_real @ F2 @ F22 @ F1) => (filterlim_nat_nat @ (^[X : nat]: (G @ (F2 @ X))) @ F3 @ F1)))))). % filterlim_compose
thf(fact_17_filterlim__compose, axiom,
    ((![G : nat > real, F3 : filter_real, F22 : filter_nat, F2 : nat > nat, F1 : filter_nat]: ((filterlim_nat_real @ G @ F3 @ F22) => ((filterlim_nat_nat @ F2 @ F22 @ F1) => (filterlim_nat_real @ (^[X : nat]: (G @ (F2 @ X))) @ F3 @ F1)))))). % filterlim_compose
thf(fact_18_filterlim__compose, axiom,
    ((![G : nat > nat, F3 : filter_nat, F22 : filter_nat, F2 : nat > nat, F1 : filter_nat]: ((filterlim_nat_nat @ G @ F3 @ F22) => ((filterlim_nat_nat @ F2 @ F22 @ F1) => (filterlim_nat_nat @ (^[X : nat]: (G @ (F2 @ X))) @ F3 @ F1)))))). % filterlim_compose
thf(fact_19_g_I2_J, axiom,
    ((topolo144289241q_real @ (^[N : nat]: (im @ (s @ (f @ (g @ N)))))))). % g(2)
thf(fact_20_LIMSEQ__Uniq, axiom,
    ((![X2 : nat > real]: (uniq_real @ (^[L3 : real]: (filterlim_nat_real @ X2 @ (topolo1664202871s_real @ L3) @ at_top_nat)))))). % LIMSEQ_Uniq
thf(fact_21_LIMSEQ__Uniq, axiom,
    ((![X2 : nat > nat]: (uniq_nat @ (^[L3 : nat]: (filterlim_nat_nat @ X2 @ (topolo1564986139ds_nat @ L3) @ at_top_nat)))))). % LIMSEQ_Uniq
thf(fact_22_tendsto__norm, axiom,
    ((![F2 : nat > complex, A : complex, F : filter_nat]: ((filter1919943476omplex @ F2 @ (topolo155787769omplex @ A) @ F) => (filterlim_nat_real @ (^[X : nat]: (real_V638595069omplex @ (F2 @ X))) @ (topolo1664202871s_real @ (real_V638595069omplex @ A)) @ F))))). % tendsto_norm
thf(fact_23_tendsto__norm, axiom,
    ((![F2 : nat > real, A : real, F : filter_nat]: ((filterlim_nat_real @ F2 @ (topolo1664202871s_real @ A) @ F) => (filterlim_nat_real @ (^[X : nat]: (real_V646646907m_real @ (F2 @ X))) @ (topolo1664202871s_real @ (real_V646646907m_real @ A)) @ F))))). % tendsto_norm
thf(fact_24_tendsto__scaleR, axiom,
    ((![F2 : nat > real, A : real, F : filter_nat, G : nat > real, B : real]: ((filterlim_nat_real @ F2 @ (topolo1664202871s_real @ A) @ F) => ((filterlim_nat_real @ G @ (topolo1664202871s_real @ B) @ F) => (filterlim_nat_real @ (^[X : nat]: (real_V453051771R_real @ (F2 @ X) @ (G @ X))) @ (topolo1664202871s_real @ (real_V453051771R_real @ A @ B)) @ F)))))). % tendsto_scaleR
thf(fact_25_g_I1_J, axiom,
    ((order_769474267at_nat @ g))). % g(1)
thf(fact_26__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,
    ((~ ((![F4 : nat > nat]: ((order_769474267at_nat @ F4) => (~ ((topolo144289241q_real @ (^[N : nat]: (re @ (s @ (F4 @ 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_27__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,
    ((~ ((![G2 : nat > nat]: ((order_769474267at_nat @ G2) => (~ ((topolo144289241q_real @ (^[N : nat]: (im @ (s @ (f @ (G2 @ 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_28_conv2, axiom,
    ((topolo795669587t_real @ (^[N : nat]: (im @ (s @ (f @ (g @ N)))))))). % conv2
thf(fact_29_convergent__norm, axiom,
    ((![F2 : nat > complex]: ((topolo1054921685omplex @ F2) => (topolo795669587t_real @ (^[N : nat]: (real_V638595069omplex @ (F2 @ N)))))))). % convergent_norm
thf(fact_30_convergent__norm, axiom,
    ((![F2 : nat > real]: ((topolo795669587t_real @ F2) => (topolo795669587t_real @ (^[N : nat]: (real_V646646907m_real @ (F2 @ N)))))))). % convergent_norm
thf(fact_31_seq__monosub, axiom,
    ((![S : nat > real]: (?[F4 : nat > nat]: ((order_769474267at_nat @ F4) & (topolo144289241q_real @ (^[N : nat]: (S @ (F4 @ N))))))))). % seq_monosub
thf(fact_32_convergent__const, axiom,
    ((![C : real]: (topolo795669587t_real @ (^[N : nat]: C))))). % convergent_const
thf(fact_33_filterlim__subseq, axiom,
    ((![F2 : nat > nat]: ((order_769474267at_nat @ F2) => (filterlim_nat_nat @ F2 @ at_top_nat @ at_top_nat))))). % filterlim_subseq
thf(fact_34_strict__mono__compose, axiom,
    ((![R : nat > nat, S : nat > nat]: ((order_769474267at_nat @ R) => ((order_769474267at_nat @ S) => (order_769474267at_nat @ (^[X : nat]: (R @ (S @ X))))))))). % strict_mono_compose
thf(fact_35_convergent__def, axiom,
    ((topolo768750839nt_nat = (^[X4 : nat > nat]: (?[L4 : nat]: (filterlim_nat_nat @ X4 @ (topolo1564986139ds_nat @ L4) @ at_top_nat)))))). % convergent_def
thf(fact_36_convergent__def, axiom,
    ((topolo795669587t_real = (^[X4 : nat > real]: (?[L4 : real]: (filterlim_nat_real @ X4 @ (topolo1664202871s_real @ L4) @ at_top_nat)))))). % convergent_def
thf(fact_37_convergentI, axiom,
    ((![X2 : nat > nat, L5 : nat]: ((filterlim_nat_nat @ X2 @ (topolo1564986139ds_nat @ L5) @ at_top_nat) => (topolo768750839nt_nat @ X2))))). % convergentI
thf(fact_38_convergentI, axiom,
    ((![X2 : nat > real, L5 : real]: ((filterlim_nat_real @ X2 @ (topolo1664202871s_real @ L5) @ at_top_nat) => (topolo795669587t_real @ X2))))). % convergentI
thf(fact_39_convergentD, axiom,
    ((![X2 : nat > nat]: ((topolo768750839nt_nat @ X2) => (?[L : nat]: (filterlim_nat_nat @ X2 @ (topolo1564986139ds_nat @ L) @ at_top_nat)))))). % convergentD
thf(fact_40_convergentD, axiom,
    ((![X2 : nat > real]: ((topolo795669587t_real @ X2) => (?[L : real]: (filterlim_nat_real @ X2 @ (topolo1664202871s_real @ L) @ at_top_nat)))))). % convergentD
thf(fact_41_tendsto__complex__iff, axiom,
    ((![F2 : nat > complex, X3 : complex, F : filter_nat]: ((filter1919943476omplex @ F2 @ (topolo155787769omplex @ X3) @ F) = (((filterlim_nat_real @ (^[X : nat]: (re @ (F2 @ X))) @ (topolo1664202871s_real @ (re @ X3)) @ F)) & ((filterlim_nat_real @ (^[X : nat]: (im @ (F2 @ X))) @ (topolo1664202871s_real @ (im @ X3)) @ F))))))). % tendsto_complex_iff
thf(fact_42__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,
    ((?[F4 : nat > nat]: ((order_769474267at_nat @ F4) & (topolo144289241q_real @ (^[N : nat]: (comp_c1631780367al_nat @ re @ s @ (F4 @ N)))))))). % \<open>\<exists>f. strict_mono f \<and> monoseq (\<lambda>n. (Re \<circ> s) (f n))\<close>
thf(fact_43_tendsto__Im, axiom,
    ((![G : nat > complex, A : complex, F : filter_nat]: ((filter1919943476omplex @ G @ (topolo155787769omplex @ A) @ F) => (filterlim_nat_real @ (^[X : nat]: (im @ (G @ X))) @ (topolo1664202871s_real @ (im @ A)) @ F))))). % tendsto_Im
thf(fact_44_tendsto__Re, axiom,
    ((![G : nat > complex, A : complex, F : filter_nat]: ((filter1919943476omplex @ G @ (topolo155787769omplex @ A) @ F) => (filterlim_nat_real @ (^[X : nat]: (re @ (G @ X))) @ (topolo1664202871s_real @ (re @ A)) @ F))))). % tendsto_Re
thf(fact_45_complex_Ocoinduct__strong, axiom,
    ((![R2 : complex > complex > $o, Complex : complex, Complex2 : complex]: ((R2 @ Complex @ Complex2) => ((![Complex3 : complex, Complex4 : complex]: ((R2 @ Complex3 @ Complex4) => (((re @ Complex3) = (re @ Complex4)) & ((im @ Complex3) = (im @ Complex4))))) => (Complex = Complex2)))))). % complex.coinduct_strong
thf(fact_46_complex__eq__iff, axiom,
    (((^[Y2 : complex]: (^[Z : complex]: (Y2 = Z))) = (^[X : complex]: (^[Y3 : complex]: ((((re @ X) = (re @ Y3))) & (((im @ X) = (im @ Y3))))))))). % complex_eq_iff
thf(fact_47_complex_Oexpand, axiom,
    ((![Complex : complex, Complex2 : complex]: ((((re @ Complex) = (re @ Complex2)) & ((im @ Complex) = (im @ Complex2))) => (Complex = Complex2))))). % complex.expand
thf(fact_48_complex__eqI, axiom,
    ((![X3 : complex, Y : complex]: (((re @ X3) = (re @ Y)) => (((im @ X3) = (im @ Y)) => (X3 = Y)))))). % complex_eqI
thf(fact_49_r, axiom,
    ((![N2 : nat]: (ord_less_eq_real @ (real_V638595069omplex @ (s @ N2)) @ r)))). % r
thf(fact_50_monoseq__def, axiom,
    ((topolo144289241q_real = (^[X4 : nat > real]: (((![M : nat]: (![N : nat]: (((ord_less_eq_nat @ M @ N)) => ((ord_less_eq_real @ (X4 @ M) @ (X4 @ N))))))) | ((![M : nat]: (![N : nat]: (((ord_less_eq_nat @ M @ N)) => ((ord_less_eq_real @ (X4 @ N) @ (X4 @ M)))))))))))). % monoseq_def
thf(fact_51_monoI2, axiom,
    ((![X2 : nat > real]: ((![M2 : nat, N3 : nat]: ((ord_less_eq_nat @ M2 @ N3) => (ord_less_eq_real @ (X2 @ N3) @ (X2 @ M2)))) => (topolo144289241q_real @ X2))))). % monoI2
thf(fact_52_monoI1, axiom,
    ((![X2 : nat > real]: ((![M2 : nat, N3 : nat]: ((ord_less_eq_nat @ M2 @ N3) => (ord_less_eq_real @ (X2 @ M2) @ (X2 @ N3)))) => (topolo144289241q_real @ X2))))). % monoI1
thf(fact_53_complex__Re__le__cmod, axiom,
    ((![X3 : complex]: (ord_less_eq_real @ (re @ X3) @ (real_V638595069omplex @ X3))))). % complex_Re_le_cmod
thf(fact_54_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_55_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_56_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_57_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_58_lim__mono, axiom,
    ((![N5 : nat, X2 : nat > real, Y4 : nat > real, X3 : real, Y : real]: ((![N3 : nat]: ((ord_less_eq_nat @ N5 @ N3) => (ord_less_eq_real @ (X2 @ N3) @ (Y4 @ N3)))) => ((filterlim_nat_real @ X2 @ (topolo1664202871s_real @ X3) @ at_top_nat) => ((filterlim_nat_real @ Y4 @ (topolo1664202871s_real @ Y) @ at_top_nat) => (ord_less_eq_real @ X3 @ Y))))))). % lim_mono
thf(fact_59_lim__mono, axiom,
    ((![N5 : nat, X2 : nat > nat, Y4 : nat > nat, X3 : nat, Y : nat]: ((![N3 : nat]: ((ord_less_eq_nat @ N5 @ N3) => (ord_less_eq_nat @ (X2 @ N3) @ (Y4 @ N3)))) => ((filterlim_nat_nat @ X2 @ (topolo1564986139ds_nat @ X3) @ at_top_nat) => ((filterlim_nat_nat @ Y4 @ (topolo1564986139ds_nat @ Y) @ at_top_nat) => (ord_less_eq_nat @ X3 @ Y))))))). % lim_mono
thf(fact_60_LIMSEQ__le, axiom,
    ((![X2 : nat > real, X3 : real, Y4 : nat > real, Y : real]: ((filterlim_nat_real @ X2 @ (topolo1664202871s_real @ X3) @ at_top_nat) => ((filterlim_nat_real @ Y4 @ (topolo1664202871s_real @ Y) @ at_top_nat) => ((?[N6 : nat]: (![N3 : nat]: ((ord_less_eq_nat @ N6 @ N3) => (ord_less_eq_real @ (X2 @ N3) @ (Y4 @ N3))))) => (ord_less_eq_real @ X3 @ Y))))))). % LIMSEQ_le
thf(fact_61_LIMSEQ__le, axiom,
    ((![X2 : nat > nat, X3 : nat, Y4 : nat > nat, Y : nat]: ((filterlim_nat_nat @ X2 @ (topolo1564986139ds_nat @ X3) @ at_top_nat) => ((filterlim_nat_nat @ Y4 @ (topolo1564986139ds_nat @ Y) @ at_top_nat) => ((?[N6 : nat]: (![N3 : nat]: ((ord_less_eq_nat @ N6 @ N3) => (ord_less_eq_nat @ (X2 @ N3) @ (Y4 @ N3))))) => (ord_less_eq_nat @ X3 @ Y))))))). % LIMSEQ_le
thf(fact_62_Lim__bounded, axiom,
    ((![F2 : nat > real, L2 : real, M4 : nat, C2 : real]: ((filterlim_nat_real @ F2 @ (topolo1664202871s_real @ L2) @ at_top_nat) => ((![N3 : nat]: ((ord_less_eq_nat @ M4 @ N3) => (ord_less_eq_real @ (F2 @ N3) @ C2))) => (ord_less_eq_real @ L2 @ C2)))))). % Lim_bounded
thf(fact_63_Lim__bounded, axiom,
    ((![F2 : nat > nat, L2 : nat, M4 : nat, C2 : nat]: ((filterlim_nat_nat @ F2 @ (topolo1564986139ds_nat @ L2) @ at_top_nat) => ((![N3 : nat]: ((ord_less_eq_nat @ M4 @ N3) => (ord_less_eq_nat @ (F2 @ N3) @ C2))) => (ord_less_eq_nat @ L2 @ C2)))))). % Lim_bounded
thf(fact_64_Lim__bounded2, axiom,
    ((![F2 : nat > real, L2 : real, N5 : nat, C2 : real]: ((filterlim_nat_real @ F2 @ (topolo1664202871s_real @ L2) @ at_top_nat) => ((![N3 : nat]: ((ord_less_eq_nat @ N5 @ N3) => (ord_less_eq_real @ C2 @ (F2 @ N3)))) => (ord_less_eq_real @ C2 @ L2)))))). % Lim_bounded2
thf(fact_65_Lim__bounded2, axiom,
    ((![F2 : nat > nat, L2 : nat, N5 : nat, C2 : nat]: ((filterlim_nat_nat @ F2 @ (topolo1564986139ds_nat @ L2) @ at_top_nat) => ((![N3 : nat]: ((ord_less_eq_nat @ N5 @ N3) => (ord_less_eq_nat @ C2 @ (F2 @ N3)))) => (ord_less_eq_nat @ C2 @ L2)))))). % Lim_bounded2
thf(fact_66_LIMSEQ__le__const, axiom,
    ((![X2 : nat > real, X3 : real, A : real]: ((filterlim_nat_real @ X2 @ (topolo1664202871s_real @ X3) @ at_top_nat) => ((?[N6 : nat]: (![N3 : nat]: ((ord_less_eq_nat @ N6 @ N3) => (ord_less_eq_real @ A @ (X2 @ N3))))) => (ord_less_eq_real @ A @ X3)))))). % LIMSEQ_le_const
thf(fact_67_LIMSEQ__le__const, axiom,
    ((![X2 : nat > nat, X3 : nat, A : nat]: ((filterlim_nat_nat @ X2 @ (topolo1564986139ds_nat @ X3) @ at_top_nat) => ((?[N6 : nat]: (![N3 : nat]: ((ord_less_eq_nat @ N6 @ N3) => (ord_less_eq_nat @ A @ (X2 @ N3))))) => (ord_less_eq_nat @ A @ X3)))))). % LIMSEQ_le_const
thf(fact_68_LIMSEQ__le__const2, axiom,
    ((![X2 : nat > real, X3 : real, A : real]: ((filterlim_nat_real @ X2 @ (topolo1664202871s_real @ X3) @ at_top_nat) => ((?[N6 : nat]: (![N3 : nat]: ((ord_less_eq_nat @ N6 @ N3) => (ord_less_eq_real @ (X2 @ N3) @ A)))) => (ord_less_eq_real @ X3 @ A)))))). % LIMSEQ_le_const2
thf(fact_69_LIMSEQ__le__const2, axiom,
    ((![X2 : nat > nat, X3 : nat, A : nat]: ((filterlim_nat_nat @ X2 @ (topolo1564986139ds_nat @ X3) @ at_top_nat) => ((?[N6 : nat]: (![N3 : nat]: ((ord_less_eq_nat @ N6 @ N3) => (ord_less_eq_nat @ (X2 @ N3) @ A)))) => (ord_less_eq_nat @ X3 @ A)))))). % LIMSEQ_le_const2
thf(fact_70_monoseq__le, axiom,
    ((![A : nat > real, X3 : real]: ((topolo144289241q_real @ A) => ((filterlim_nat_real @ A @ (topolo1664202871s_real @ X3) @ at_top_nat) => (((![N2 : nat]: (ord_less_eq_real @ (A @ N2) @ X3)) & (![M5 : nat, N2 : nat]: ((ord_less_eq_nat @ M5 @ N2) => (ord_less_eq_real @ (A @ M5) @ (A @ N2))))) | ((![N2 : nat]: (ord_less_eq_real @ X3 @ (A @ N2))) & (![M5 : nat, N2 : nat]: ((ord_less_eq_nat @ M5 @ N2) => (ord_less_eq_real @ (A @ N2) @ (A @ M5))))))))))). % monoseq_le
thf(fact_71_monoseq__le, axiom,
    ((![A : nat > nat, X3 : nat]: ((topolo1922093437eq_nat @ A) => ((filterlim_nat_nat @ A @ (topolo1564986139ds_nat @ X3) @ at_top_nat) => (((![N2 : nat]: (ord_less_eq_nat @ (A @ N2) @ X3)) & (![M5 : nat, N2 : nat]: ((ord_less_eq_nat @ M5 @ N2) => (ord_less_eq_nat @ (A @ M5) @ (A @ N2))))) | ((![N2 : nat]: (ord_less_eq_nat @ X3 @ (A @ N2))) & (![M5 : nat, N2 : nat]: ((ord_less_eq_nat @ M5 @ N2) => (ord_less_eq_nat @ (A @ N2) @ (A @ M5))))))))))). % monoseq_le
thf(fact_72__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_73__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,
    ((?[F4 : nat > nat]: ((order_769474267at_nat @ F4) & (topolo144289241q_real @ (^[N : nat]: (comp_nat_real_nat @ (comp_c1631780367al_nat @ im @ s) @ f @ (F4 @ N)))))))). % \<open>\<exists>fa. strict_mono fa \<and> monoseq (\<lambda>n. (Im \<circ> s \<circ> f) (fa n))\<close>
thf(fact_74_rp, axiom,
    ((ord_less_eq_real @ zero_zero_real @ r))). % rp
thf(fact_75_comp__apply, axiom,
    ((comp_c1631780367al_nat = (^[F5 : complex > real]: (^[G3 : nat > complex]: (^[X : nat]: (F5 @ (G3 @ X)))))))). % comp_apply
thf(fact_76_comp__apply, axiom,
    ((comp_nat_real_nat = (^[F5 : nat > real]: (^[G3 : nat > nat]: (^[X : nat]: (F5 @ (G3 @ X)))))))). % comp_apply
thf(fact_77_order__refl, axiom,
    ((![X3 : real]: (ord_less_eq_real @ X3 @ X3)))). % order_refl
thf(fact_78_strict__mono__less__eq, axiom,
    ((![F2 : real > real, X3 : real, Y : real]: ((order_1818878995l_real @ F2) => ((ord_less_eq_real @ (F2 @ X3) @ (F2 @ Y)) = (ord_less_eq_real @ X3 @ Y)))))). % strict_mono_less_eq
thf(fact_79_strict__mono__less__eq, axiom,
    ((![F2 : nat > nat, X3 : nat, Y : nat]: ((order_769474267at_nat @ F2) => ((ord_less_eq_nat @ (F2 @ X3) @ (F2 @ Y)) = (ord_less_eq_nat @ X3 @ Y)))))). % strict_mono_less_eq
thf(fact_80_monoseq__convergent, axiom,
    ((![X2 : nat > real, B2 : real]: ((topolo144289241q_real @ X2) => ((![I : nat]: (ord_less_eq_real @ (abs_abs_real @ (X2 @ I)) @ B2)) => (~ ((![L : real]: (~ ((filterlim_nat_real @ X2 @ (topolo1664202871s_real @ L) @ at_top_nat))))))))))). % monoseq_convergent
thf(fact_81_filterlim__mono, axiom,
    ((![F2 : nat > real, F22 : filter_real, F1 : filter_nat, F23 : filter_real, F12 : filter_nat]: ((filterlim_nat_real @ F2 @ F22 @ F1) => ((ord_le132810396r_real @ F22 @ F23) => ((ord_le1745708096er_nat @ F12 @ F1) => (filterlim_nat_real @ F2 @ F23 @ F12))))))). % filterlim_mono
thf(fact_82_filterlim__mono, axiom,
    ((![F2 : nat > nat, F22 : filter_nat, F1 : filter_nat, F23 : filter_nat, F12 : filter_nat]: ((filterlim_nat_nat @ F2 @ F22 @ F1) => ((ord_le1745708096er_nat @ F22 @ F23) => ((ord_le1745708096er_nat @ F12 @ F1) => (filterlim_nat_nat @ F2 @ F23 @ F12))))))). % filterlim_mono
thf(fact_83_tendsto__rabs__zero__cancel, axiom,
    ((![F2 : nat > real, F : filter_nat]: ((filterlim_nat_real @ (^[X : nat]: (abs_abs_real @ (F2 @ X))) @ (topolo1664202871s_real @ zero_zero_real) @ F) => (filterlim_nat_real @ F2 @ (topolo1664202871s_real @ zero_zero_real) @ F))))). % tendsto_rabs_zero_cancel
thf(fact_84_tendsto__rabs__zero__iff, axiom,
    ((![F2 : nat > real, F : filter_nat]: ((filterlim_nat_real @ (^[X : nat]: (abs_abs_real @ (F2 @ X))) @ (topolo1664202871s_real @ zero_zero_real) @ F) = (filterlim_nat_real @ F2 @ (topolo1664202871s_real @ zero_zero_real) @ F))))). % tendsto_rabs_zero_iff
thf(fact_85_tendsto__rabs__zero, axiom,
    ((![F2 : nat > real, F : filter_nat]: ((filterlim_nat_real @ F2 @ (topolo1664202871s_real @ zero_zero_real) @ F) => (filterlim_nat_real @ (^[X : nat]: (abs_abs_real @ (F2 @ X))) @ (topolo1664202871s_real @ zero_zero_real) @ F))))). % tendsto_rabs_zero
thf(fact_86_Im__eq__0, axiom,
    ((![Z2 : complex]: (((abs_abs_real @ (re @ Z2)) = (real_V638595069omplex @ Z2)) => ((im @ Z2) = zero_zero_real))))). % Im_eq_0
thf(fact_87_cmod__eq__Im, axiom,
    ((![Z2 : complex]: (((re @ Z2) = zero_zero_real) => ((real_V638595069omplex @ Z2) = (abs_abs_real @ (im @ Z2))))))). % cmod_eq_Im
thf(fact_88_cmod__eq__Re, axiom,
    ((![Z2 : complex]: (((im @ Z2) = zero_zero_real) => ((real_V638595069omplex @ Z2) = (abs_abs_real @ (re @ Z2))))))). % cmod_eq_Re
thf(fact_89_tendsto__mono, axiom,
    ((![F : filter_nat, F6 : filter_nat, F2 : nat > real, L2 : real]: ((ord_le1745708096er_nat @ F @ F6) => ((filterlim_nat_real @ F2 @ (topolo1664202871s_real @ L2) @ F6) => (filterlim_nat_real @ F2 @ (topolo1664202871s_real @ L2) @ F)))))). % tendsto_mono
thf(fact_90_tendsto__mono, axiom,
    ((![F : filter_nat, F6 : filter_nat, F2 : nat > nat, L2 : nat]: ((ord_le1745708096er_nat @ F @ F6) => ((filterlim_nat_nat @ F2 @ (topolo1564986139ds_nat @ L2) @ F6) => (filterlim_nat_nat @ F2 @ (topolo1564986139ds_nat @ L2) @ F)))))). % tendsto_mono
thf(fact_91_convergent__subseq__convergent, axiom,
    ((![X2 : nat > real, F2 : nat > nat]: ((topolo795669587t_real @ X2) => ((order_769474267at_nat @ F2) => (topolo795669587t_real @ (comp_nat_real_nat @ X2 @ F2))))))). % convergent_subseq_convergent
thf(fact_92_tendsto__rabs, axiom,
    ((![F2 : nat > real, L2 : real, F : filter_nat]: ((filterlim_nat_real @ F2 @ (topolo1664202871s_real @ L2) @ F) => (filterlim_nat_real @ (^[X : nat]: (abs_abs_real @ (F2 @ X))) @ (topolo1664202871s_real @ (abs_abs_real @ L2)) @ F))))). % tendsto_rabs
thf(fact_93_tendsto__norm__zero__cancel, axiom,
    ((![F2 : nat > complex, F : filter_nat]: ((filterlim_nat_real @ (^[X : nat]: (real_V638595069omplex @ (F2 @ X))) @ (topolo1664202871s_real @ zero_zero_real) @ F) => (filter1919943476omplex @ F2 @ (topolo155787769omplex @ zero_zero_complex) @ F))))). % tendsto_norm_zero_cancel
thf(fact_94_tendsto__norm__zero__cancel, axiom,
    ((![F2 : nat > real, F : filter_nat]: ((filterlim_nat_real @ (^[X : nat]: (real_V646646907m_real @ (F2 @ X))) @ (topolo1664202871s_real @ zero_zero_real) @ F) => (filterlim_nat_real @ F2 @ (topolo1664202871s_real @ zero_zero_real) @ F))))). % tendsto_norm_zero_cancel
thf(fact_95_tendsto__norm__zero__iff, axiom,
    ((![F2 : nat > complex, F : filter_nat]: ((filterlim_nat_real @ (^[X : nat]: (real_V638595069omplex @ (F2 @ X))) @ (topolo1664202871s_real @ zero_zero_real) @ F) = (filter1919943476omplex @ F2 @ (topolo155787769omplex @ zero_zero_complex) @ F))))). % tendsto_norm_zero_iff
thf(fact_96_tendsto__norm__zero__iff, axiom,
    ((![F2 : nat > real, F : filter_nat]: ((filterlim_nat_real @ (^[X : nat]: (real_V646646907m_real @ (F2 @ X))) @ (topolo1664202871s_real @ zero_zero_real) @ F) = (filterlim_nat_real @ F2 @ (topolo1664202871s_real @ zero_zero_real) @ F))))). % tendsto_norm_zero_iff
thf(fact_97_tendsto__norm__zero, axiom,
    ((![F2 : nat > complex, F : filter_nat]: ((filter1919943476omplex @ F2 @ (topolo155787769omplex @ zero_zero_complex) @ F) => (filterlim_nat_real @ (^[X : nat]: (real_V638595069omplex @ (F2 @ X))) @ (topolo1664202871s_real @ zero_zero_real) @ F))))). % tendsto_norm_zero
thf(fact_98_tendsto__norm__zero, axiom,
    ((![F2 : nat > real, F : filter_nat]: ((filterlim_nat_real @ F2 @ (topolo1664202871s_real @ zero_zero_real) @ F) => (filterlim_nat_real @ (^[X : nat]: (real_V646646907m_real @ (F2 @ X))) @ (topolo1664202871s_real @ zero_zero_real) @ F))))). % tendsto_norm_zero
thf(fact_99_order__subst1, axiom,
    ((![A : real, F2 : real > real, B : real, C : real]: ((ord_less_eq_real @ A @ (F2 @ B)) => ((ord_less_eq_real @ B @ C) => ((![X5 : real, Y5 : real]: ((ord_less_eq_real @ X5 @ Y5) => (ord_less_eq_real @ (F2 @ X5) @ (F2 @ Y5)))) => (ord_less_eq_real @ A @ (F2 @ C)))))))). % order_subst1
thf(fact_100_order__subst2, axiom,
    ((![A : real, B : real, F2 : real > real, C : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_eq_real @ (F2 @ B) @ C) => ((![X5 : real, Y5 : real]: ((ord_less_eq_real @ X5 @ Y5) => (ord_less_eq_real @ (F2 @ X5) @ (F2 @ Y5)))) => (ord_less_eq_real @ (F2 @ A) @ C))))))). % order_subst2
thf(fact_101_ord__eq__le__subst, axiom,
    ((![A : real, F2 : real > real, B : real, C : real]: ((A = (F2 @ B)) => ((ord_less_eq_real @ B @ C) => ((![X5 : real, Y5 : real]: ((ord_less_eq_real @ X5 @ Y5) => (ord_less_eq_real @ (F2 @ X5) @ (F2 @ Y5)))) => (ord_less_eq_real @ A @ (F2 @ C)))))))). % ord_eq_le_subst
thf(fact_102_ord__le__eq__subst, axiom,
    ((![A : real, B : real, F2 : real > real, C : real]: ((ord_less_eq_real @ A @ B) => (((F2 @ B) = C) => ((![X5 : real, Y5 : real]: ((ord_less_eq_real @ X5 @ Y5) => (ord_less_eq_real @ (F2 @ X5) @ (F2 @ Y5)))) => (ord_less_eq_real @ (F2 @ A) @ C))))))). % ord_le_eq_subst
thf(fact_103_eq__iff, axiom,
    (((^[Y2 : real]: (^[Z : real]: (Y2 = Z))) = (^[X : real]: (^[Y3 : real]: (((ord_less_eq_real @ X @ Y3)) & ((ord_less_eq_real @ Y3 @ X)))))))). % eq_iff
thf(fact_104_antisym, axiom,
    ((![X3 : real, Y : real]: ((ord_less_eq_real @ X3 @ Y) => ((ord_less_eq_real @ Y @ X3) => (X3 = Y)))))). % antisym
thf(fact_105_linear, axiom,
    ((![X3 : real, Y : real]: ((ord_less_eq_real @ X3 @ Y) | (ord_less_eq_real @ Y @ X3))))). % linear
thf(fact_106_eq__refl, axiom,
    ((![X3 : real, Y : real]: ((X3 = Y) => (ord_less_eq_real @ X3 @ Y))))). % eq_refl
thf(fact_107_le__cases, axiom,
    ((![X3 : real, Y : real]: ((~ ((ord_less_eq_real @ X3 @ Y))) => (ord_less_eq_real @ Y @ X3))))). % le_cases
thf(fact_108_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_109_le__cases3, axiom,
    ((![X3 : real, Y : real, Z2 : real]: (((ord_less_eq_real @ X3 @ Y) => (~ ((ord_less_eq_real @ Y @ Z2)))) => (((ord_less_eq_real @ Y @ X3) => (~ ((ord_less_eq_real @ X3 @ Z2)))) => (((ord_less_eq_real @ X3 @ Z2) => (~ ((ord_less_eq_real @ Z2 @ Y)))) => (((ord_less_eq_real @ Z2 @ Y) => (~ ((ord_less_eq_real @ Y @ X3)))) => (((ord_less_eq_real @ Y @ Z2) => (~ ((ord_less_eq_real @ Z2 @ X3)))) => (~ (((ord_less_eq_real @ Z2 @ X3) => (~ ((ord_less_eq_real @ X3 @ Y)))))))))))))). % le_cases3
thf(fact_110_antisym__conv, axiom,
    ((![Y : real, X3 : real]: ((ord_less_eq_real @ Y @ X3) => ((ord_less_eq_real @ X3 @ Y) = (X3 = Y)))))). % antisym_conv
thf(fact_111_order__class_Oorder_Oeq__iff, axiom,
    (((^[Y2 : real]: (^[Z : real]: (Y2 = Z))) = (^[A2 : real]: (^[B3 : real]: (((ord_less_eq_real @ A2 @ B3)) & ((ord_less_eq_real @ B3 @ A2)))))))). % order_class.order.eq_iff
thf(fact_112_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_113_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_114_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_115_order__trans, axiom,
    ((![X3 : real, Y : real, Z2 : real]: ((ord_less_eq_real @ X3 @ Y) => ((ord_less_eq_real @ Y @ Z2) => (ord_less_eq_real @ X3 @ Z2)))))). % order_trans
thf(fact_116_dual__order_Orefl, axiom,
    ((![A : real]: (ord_less_eq_real @ A @ A)))). % dual_order.refl
thf(fact_117_linorder__wlog, axiom,
    ((![P : real > real > $o, A : real, B : real]: ((![A3 : real, B4 : real]: ((ord_less_eq_real @ A3 @ B4) => (P @ A3 @ B4))) => ((![A3 : real, B4 : real]: ((P @ B4 @ A3) => (P @ A3 @ B4))) => (P @ A @ B)))))). % linorder_wlog
thf(fact_118_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_119_dual__order_Oeq__iff, axiom,
    (((^[Y2 : real]: (^[Z : real]: (Y2 = Z))) = (^[A2 : real]: (^[B3 : real]: (((ord_less_eq_real @ B3 @ A2)) & ((ord_less_eq_real @ A2 @ B3)))))))). % dual_order.eq_iff
thf(fact_120_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_121_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_122_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_123_comp__def, axiom,
    ((comp_c1631780367al_nat = (^[F5 : complex > real]: (^[G3 : nat > complex]: (^[X : nat]: (F5 @ (G3 @ X)))))))). % comp_def
thf(fact_124_comp__def, axiom,
    ((comp_nat_real_nat = (^[F5 : nat > real]: (^[G3 : nat > nat]: (^[X : nat]: (F5 @ (G3 @ X)))))))). % comp_def
thf(fact_125_comp__assoc, axiom,
    ((![F2 : real > real, G : complex > real, H : nat > complex]: ((comp_c1631780367al_nat @ (comp_r422820971omplex @ F2 @ G) @ H) = (comp_real_real_nat @ F2 @ (comp_c1631780367al_nat @ G @ H)))))). % comp_assoc
thf(fact_126_comp__assoc, axiom,
    ((![F2 : complex > real, G : complex > complex, H : nat > complex]: ((comp_c1631780367al_nat @ (comp_c317287661omplex @ F2 @ G) @ H) = (comp_c1631780367al_nat @ F2 @ (comp_c438056209ex_nat @ G @ H)))))). % comp_assoc
thf(fact_127_comp__assoc, axiom,
    ((![F2 : nat > real, G : complex > nat, H : nat > complex]: ((comp_c1631780367al_nat @ (comp_n1816297743omplex @ F2 @ G) @ H) = (comp_nat_real_nat @ F2 @ (comp_complex_nat_nat @ G @ H)))))). % comp_assoc
thf(fact_128_comp__assoc, axiom,
    ((![F2 : real > real, G : nat > real, H : nat > nat]: ((comp_nat_real_nat @ (comp_real_real_nat @ F2 @ G) @ H) = (comp_real_real_nat @ F2 @ (comp_nat_real_nat @ G @ H)))))). % comp_assoc
thf(fact_129_comp__assoc, axiom,
    ((![F2 : complex > real, G : nat > complex, H : nat > nat]: ((comp_nat_real_nat @ (comp_c1631780367al_nat @ F2 @ G) @ H) = (comp_c1631780367al_nat @ F2 @ (comp_nat_complex_nat @ G @ H)))))). % comp_assoc
thf(fact_130_comp__assoc, axiom,
    ((![F2 : nat > real, G : nat > nat, H : nat > nat]: ((comp_nat_real_nat @ (comp_nat_real_nat @ F2 @ G) @ H) = (comp_nat_real_nat @ F2 @ (comp_nat_nat_nat @ G @ H)))))). % comp_assoc
thf(fact_131_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_132_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_133_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_134_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_135_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_136_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_137_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_138_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_139_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_140_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_141_strict__mono__eq, axiom,
    ((![F2 : nat > nat, X3 : nat, Y : nat]: ((order_769474267at_nat @ F2) => (((F2 @ X3) = (F2 @ Y)) = (X3 = Y)))))). % strict_mono_eq
thf(fact_142_LIMSEQ__subseq__LIMSEQ, axiom,
    ((![X2 : nat > real, L5 : real, F2 : nat > nat]: ((filterlim_nat_real @ X2 @ (topolo1664202871s_real @ L5) @ at_top_nat) => ((order_769474267at_nat @ F2) => (filterlim_nat_real @ (comp_nat_real_nat @ X2 @ F2) @ (topolo1664202871s_real @ L5) @ at_top_nat)))))). % LIMSEQ_subseq_LIMSEQ
thf(fact_143_LIMSEQ__subseq__LIMSEQ, axiom,
    ((![X2 : nat > nat, L5 : nat, F2 : nat > nat]: ((filterlim_nat_nat @ X2 @ (topolo1564986139ds_nat @ L5) @ at_top_nat) => ((order_769474267at_nat @ F2) => (filterlim_nat_nat @ (comp_nat_nat_nat @ X2 @ F2) @ (topolo1564986139ds_nat @ L5) @ at_top_nat)))))). % LIMSEQ_subseq_LIMSEQ
thf(fact_144_cmod__Im__le__iff, axiom,
    ((![X3 : complex, Y : complex]: (((re @ X3) = (re @ Y)) => ((ord_less_eq_real @ (real_V638595069omplex @ X3) @ (real_V638595069omplex @ Y)) = (ord_less_eq_real @ (abs_abs_real @ (im @ X3)) @ (abs_abs_real @ (im @ Y)))))))). % cmod_Im_le_iff
thf(fact_145_cmod__Re__le__iff, axiom,
    ((![X3 : complex, Y : complex]: (((im @ X3) = (im @ Y)) => ((ord_less_eq_real @ (real_V638595069omplex @ X3) @ (real_V638595069omplex @ Y)) = (ord_less_eq_real @ (abs_abs_real @ (re @ X3)) @ (abs_abs_real @ (re @ Y)))))))). % cmod_Re_le_iff
thf(fact_146_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_147_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_148_scale__zero__left, axiom,
    ((![X3 : real]: ((real_V453051771R_real @ zero_zero_real @ X3) = zero_zero_real)))). % scale_zero_left
thf(fact_149_scale__zero__left, axiom,
    ((![X3 : complex]: ((real_V1560324349omplex @ zero_zero_real @ X3) = zero_zero_complex)))). % scale_zero_left
thf(fact_150_scale__eq__0__iff, axiom,
    ((![A : real, X3 : real]: (((real_V453051771R_real @ A @ X3) = zero_zero_real) = (((A = zero_zero_real)) | ((X3 = zero_zero_real))))))). % scale_eq_0_iff
thf(fact_151_scale__eq__0__iff, axiom,
    ((![A : real, X3 : complex]: (((real_V1560324349omplex @ A @ X3) = zero_zero_complex) = (((A = zero_zero_real)) | ((X3 = zero_zero_complex))))))). % scale_eq_0_iff
thf(fact_152_norm__zero, axiom,
    (((real_V638595069omplex @ zero_zero_complex) = zero_zero_real))). % norm_zero
thf(fact_153_norm__zero, axiom,
    (((real_V646646907m_real @ zero_zero_real) = zero_zero_real))). % norm_zero
thf(fact_154_norm__eq__zero, axiom,
    ((![X3 : complex]: (((real_V638595069omplex @ X3) = zero_zero_real) = (X3 = zero_zero_complex))))). % norm_eq_zero
thf(fact_155_norm__eq__zero, axiom,
    ((![X3 : real]: (((real_V646646907m_real @ X3) = zero_zero_real) = (X3 = zero_zero_real))))). % norm_eq_zero
thf(fact_156_scale__zero__right, axiom,
    ((![A : real]: ((real_V453051771R_real @ A @ zero_zero_real) = zero_zero_real)))). % scale_zero_right
thf(fact_157_scale__zero__right, axiom,
    ((![A : real]: ((real_V1560324349omplex @ A @ zero_zero_complex) = zero_zero_complex)))). % scale_zero_right
thf(fact_158_scale__cancel__right, axiom,
    ((![A : real, X3 : real, B : real]: (((real_V453051771R_real @ A @ X3) = (real_V453051771R_real @ B @ X3)) = (((A = B)) | ((X3 = zero_zero_real))))))). % scale_cancel_right
thf(fact_159_scale__cancel__right, axiom,
    ((![A : real, X3 : complex, B : real]: (((real_V1560324349omplex @ A @ X3) = (real_V1560324349omplex @ B @ X3)) = (((A = B)) | ((X3 = zero_zero_complex))))))). % scale_cancel_right
thf(fact_160_abs__norm__cancel, axiom,
    ((![A : complex]: ((abs_abs_real @ (real_V638595069omplex @ A)) = (real_V638595069omplex @ A))))). % abs_norm_cancel
thf(fact_161_abs__norm__cancel, axiom,
    ((![A : real]: ((abs_abs_real @ (real_V646646907m_real @ A)) = (real_V646646907m_real @ A))))). % abs_norm_cancel
thf(fact_162_zero__complex_Osimps_I1_J, axiom,
    (((re @ zero_zero_complex) = zero_zero_real))). % zero_complex.simps(1)
thf(fact_163_zero__complex_Osimps_I2_J, axiom,
    (((im @ zero_zero_complex) = zero_zero_real))). % zero_complex.simps(2)
thf(fact_164_real__norm__def, axiom,
    ((real_V646646907m_real = abs_abs_real))). % real_norm_def
thf(fact_165_scale__right__imp__eq, axiom,
    ((![X3 : real, A : real, B : real]: ((~ ((X3 = zero_zero_real))) => (((real_V453051771R_real @ A @ X3) = (real_V453051771R_real @ B @ X3)) => (A = B)))))). % scale_right_imp_eq
thf(fact_166_scale__right__imp__eq, axiom,
    ((![X3 : complex, A : real, B : real]: ((~ ((X3 = zero_zero_complex))) => (((real_V1560324349omplex @ A @ X3) = (real_V1560324349omplex @ B @ X3)) => (A = B)))))). % scale_right_imp_eq
thf(fact_167_norm__ge__zero, axiom,
    ((![X3 : complex]: (ord_less_eq_real @ zero_zero_real @ (real_V638595069omplex @ X3))))). % norm_ge_zero
thf(fact_168_norm__ge__zero, axiom,
    ((![X3 : real]: (ord_less_eq_real @ zero_zero_real @ (real_V646646907m_real @ X3))))). % norm_ge_zero
thf(fact_169_scaleR__right__mono__neg, axiom,
    ((![B : real, A : real, C : real]: ((ord_less_eq_real @ B @ A) => ((ord_less_eq_real @ C @ zero_zero_real) => (ord_less_eq_real @ (real_V453051771R_real @ A @ C) @ (real_V453051771R_real @ B @ C))))))). % scaleR_right_mono_neg
thf(fact_170_scaleR__right__mono, axiom,
    ((![A : real, B : real, X3 : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_eq_real @ zero_zero_real @ X3) => (ord_less_eq_real @ (real_V453051771R_real @ A @ X3) @ (real_V453051771R_real @ B @ X3))))))). % scaleR_right_mono
thf(fact_171_scaleR__left__mono__neg, axiom,
    ((![B : real, A : real, C : real]: ((ord_less_eq_real @ B @ A) => ((ord_less_eq_real @ C @ zero_zero_real) => (ord_less_eq_real @ (real_V453051771R_real @ C @ A) @ (real_V453051771R_real @ C @ B))))))). % scaleR_left_mono_neg
thf(fact_172_scaleR__left__mono, axiom,
    ((![X3 : real, Y : real, A : real]: ((ord_less_eq_real @ X3 @ Y) => ((ord_less_eq_real @ zero_zero_real @ A) => (ord_less_eq_real @ (real_V453051771R_real @ A @ X3) @ (real_V453051771R_real @ A @ Y))))))). % scaleR_left_mono
thf(fact_173_split__scaleR__neg__le, axiom,
    ((![A : real, X3 : real]: ((((ord_less_eq_real @ zero_zero_real @ A) & (ord_less_eq_real @ X3 @ zero_zero_real)) | ((ord_less_eq_real @ A @ zero_zero_real) & (ord_less_eq_real @ zero_zero_real @ X3))) => (ord_less_eq_real @ (real_V453051771R_real @ A @ X3) @ zero_zero_real))))). % split_scaleR_neg_le
thf(fact_174_split__scaleR__pos__le, axiom,
    ((![A : real, B : real]: ((((ord_less_eq_real @ zero_zero_real @ A) & (ord_less_eq_real @ zero_zero_real @ B)) | ((ord_less_eq_real @ A @ zero_zero_real) & (ord_less_eq_real @ B @ zero_zero_real))) => (ord_less_eq_real @ zero_zero_real @ (real_V453051771R_real @ A @ B)))))). % split_scaleR_pos_le
thf(fact_175_scaleR__nonneg__nonneg, axiom,
    ((![A : real, X3 : real]: ((ord_less_eq_real @ zero_zero_real @ A) => ((ord_less_eq_real @ zero_zero_real @ X3) => (ord_less_eq_real @ zero_zero_real @ (real_V453051771R_real @ A @ X3))))))). % scaleR_nonneg_nonneg
thf(fact_176_scaleR__nonneg__nonpos, axiom,
    ((![A : real, X3 : real]: ((ord_less_eq_real @ zero_zero_real @ A) => ((ord_less_eq_real @ X3 @ zero_zero_real) => (ord_less_eq_real @ (real_V453051771R_real @ A @ X3) @ zero_zero_real)))))). % scaleR_nonneg_nonpos
thf(fact_177_scaleR__nonpos__nonneg, axiom,
    ((![A : real, X3 : real]: ((ord_less_eq_real @ A @ zero_zero_real) => ((ord_less_eq_real @ zero_zero_real @ X3) => (ord_less_eq_real @ (real_V453051771R_real @ A @ X3) @ zero_zero_real)))))). % scaleR_nonpos_nonneg
thf(fact_178_scaleR__nonpos__nonpos, axiom,
    ((![A : real, B : real]: ((ord_less_eq_real @ A @ zero_zero_real) => ((ord_less_eq_real @ B @ zero_zero_real) => (ord_less_eq_real @ zero_zero_real @ (real_V453051771R_real @ A @ B))))))). % scaleR_nonpos_nonpos
thf(fact_179_scaleR__mono, axiom,
    ((![A : real, B : real, X3 : real, Y : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_eq_real @ X3 @ Y) => ((ord_less_eq_real @ zero_zero_real @ B) => ((ord_less_eq_real @ zero_zero_real @ X3) => (ord_less_eq_real @ (real_V453051771R_real @ A @ X3) @ (real_V453051771R_real @ B @ Y))))))))). % scaleR_mono
thf(fact_180_scaleR__mono_H, 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 @ zero_zero_real @ A) => ((ord_less_eq_real @ zero_zero_real @ C) => (ord_less_eq_real @ (real_V453051771R_real @ A @ C) @ (real_V453051771R_real @ B @ D))))))))). % scaleR_mono'
thf(fact_181_abs__of__nonneg, axiom,
    ((![A : real]: ((ord_less_eq_real @ zero_zero_real @ A) => ((abs_abs_real @ A) = A))))). % abs_of_nonneg
thf(fact_182_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_183_abs__idempotent, axiom,
    ((![A : real]: ((abs_abs_real @ (abs_abs_real @ A)) = (abs_abs_real @ A))))). % abs_idempotent
thf(fact_184_le__zero__eq, axiom,
    ((![N4 : nat]: ((ord_less_eq_nat @ N4 @ zero_zero_nat) = (N4 = zero_zero_nat))))). % le_zero_eq
thf(fact_185_abs__0__eq, axiom,
    ((![A : real]: ((zero_zero_real = (abs_abs_real @ A)) = (A = zero_zero_real))))). % abs_0_eq
thf(fact_186_abs__eq__0, axiom,
    ((![A : real]: (((abs_abs_real @ A) = zero_zero_real) = (A = zero_zero_real))))). % abs_eq_0
thf(fact_187_abs__zero, axiom,
    (((abs_abs_real @ zero_zero_real) = zero_zero_real))). % abs_zero
thf(fact_188_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_189_zero__reorient, axiom,
    ((![X3 : nat]: ((zero_zero_nat = X3) = (X3 = zero_zero_nat))))). % zero_reorient
thf(fact_190_zero__reorient, axiom,
    ((![X3 : real]: ((zero_zero_real = X3) = (X3 = zero_zero_real))))). % zero_reorient
thf(fact_191_zero__reorient, axiom,
    ((![X3 : complex]: ((zero_zero_complex = X3) = (X3 = zero_zero_complex))))). % zero_reorient

% Conjectures (2)
thf(conj_0, hypothesis,
    ((![X6 : real]: ((filterlim_nat_real @ (^[N : nat]: (re @ (s @ (f @ N)))) @ (topolo1664202871s_real @ X6) @ at_top_nat) => thesis)))).
thf(conj_1, conjecture,
    (thesis)).
