% 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_283__5370344_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:28:54.595

% Could-be-implicit typings (8)
thf(ty_n_t__Filter__Ofilter_It__Complex__Ocomplex_J, type,
    filter_complex : $tType).
thf(ty_n_t__Set__Oset_It__Complex__Ocomplex_J, type,
    set_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__Set__Oset_It__Real__Oreal_J, type,
    set_real : $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 (57)
thf(sy_c_Complex_Ocomplex_OComplex, type,
    complex2 : real > real > complex).
thf(sy_c_Complex_Ocomplex_OIm, type,
    im : complex > real).
thf(sy_c_Complex_Ocomplex_ORe, type,
    re : complex > real).
thf(sy_c_Complex_Ocomplex_Ocase__complex_001t__Real__Oreal, type,
    case_complex_real : (real > real > real) > 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_HOL_OThe_001t__Nat__Onat, type,
    the_nat : (nat > $o) > nat).
thf(sy_c_HOL_OThe_001t__Real__Oreal, type,
    the_real : (real > $o) > 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_Oantimono_001t__Nat__Onat_001t__Nat__Onat, type,
    order_1631207636at_nat : (nat > nat) > $o).
thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Nat__Onat_001t__Real__Oreal, type,
    order_106095024t_real : (nat > real) > $o).
thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Real__Oreal_001t__Real__Oreal, type,
    order_537808140l_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_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_Ot2__space__class_OLim_001t__Nat__Onat_001t__Nat__Onat, type,
    topolo606707426at_nat : filter_nat > (nat > nat) > nat).
thf(sy_c_Topological__Spaces_Ot2__space__class_OLim_001t__Nat__Onat_001t__Real__Oreal, type,
    topolo307763390t_real : filter_nat > (nat > real) > real).
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_c_Topological__Spaces_Ouniform__space__class_OCauchy_001t__Complex__Ocomplex, type,
    topolo1714400466omplex : (nat > complex) > $o).
thf(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy_001t__Real__Oreal, type,
    topolo597383376y_real : (nat > real) > $o).
thf(sy_c_Topological__Spaces_Ouniform__space__class_Ocomplete_001t__Complex__Ocomplex, type,
    topolo589572398omplex : set_complex > $o).
thf(sy_c_Topological__Spaces_Ouniform__space__class_Ocomplete_001t__Real__Oreal, type,
    topolo2066423596e_real : set_real > $o).
thf(sy_c_member_001t__Complex__Ocomplex, type,
    member_complex : complex > set_complex > $o).
thf(sy_c_member_001t__Real__Oreal, type,
    member_real : real > set_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).
thf(sy_v_thesis____, type,
    thesis : $o).
thf(sy_v_x____, type,
    x : real).

% Relevant facts (180)
thf(fact_0__092_060open_062_092_060exists_062L_O_A_I_092_060lambda_062n_O_AIm_A_Is_A_If_A_Ig_An_J_J_J_J_A_092_060longlonglongrightarrow_062_AL_092_060close_062, axiom,
    ((?[L : real]: (filterlim_nat_real @ (^[N : nat]: (im @ (s @ (f @ (g @ N))))) @ (topolo1664202871s_real @ L) @ at_top_nat)))). % \<open>\<exists>L. (\<lambda>n. Im (s (f (g n)))) \<longlonglongrightarrow> L\<close>
thf(fact_1_f_I1_J, axiom,
    ((order_769474267at_nat @ f))). % f(1)
thf(fact_2_g_I1_J, axiom,
    ((order_769474267at_nat @ g))). % g(1)
thf(fact_3_g_I2_J, axiom,
    ((topolo144289241q_real @ (^[N : nat]: (im @ (s @ (f @ (g @ N)))))))). % g(2)
thf(fact_4__092_060open_062_I_092_060lambda_062n_O_ARe_A_Is_A_If_An_J_J_J_A_092_060longlonglongrightarrow_062_Ax_092_060close_062, axiom,
    ((filterlim_nat_real @ (^[N : nat]: (re @ (s @ (f @ N)))) @ (topolo1664202871s_real @ x) @ at_top_nat))). % \<open>(\<lambda>n. Re (s (f n))) \<longlonglongrightarrow> x\<close>
thf(fact_5_tendsto__const, axiom,
    ((![K : nat, F : filter_nat]: (filterlim_nat_nat @ (^[X : nat]: K) @ (topolo1564986139ds_nat @ K) @ F)))). % tendsto_const
thf(fact_6_tendsto__const, axiom,
    ((![K : real, F : filter_nat]: (filterlim_nat_real @ (^[X : nat]: K) @ (topolo1664202871s_real @ K) @ F)))). % tendsto_const
thf(fact_7_conv2, axiom,
    ((topolo795669587t_real @ (^[N : nat]: (im @ (s @ (f @ (g @ N)))))))). % conv2
thf(fact_8__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_9__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062x_O_A_I_092_060lambda_062n_O_ARe_A_Is_A_If_An_J_J_J_A_092_060longlonglongrightarrow_062_Ax_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom,
    ((~ ((![X2 : real]: (~ ((filterlim_nat_real @ (^[N : nat]: (re @ (s @ (f @ N)))) @ (topolo1664202871s_real @ X2) @ at_top_nat)))))))). % \<open>\<And>thesis. (\<And>x. (\<lambda>n. Re (s (f n))) \<longlonglongrightarrow> x \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_10_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_11_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_12_LIMSEQ__unique, axiom,
    ((![X3 : nat > real, A : real, B : real]: ((filterlim_nat_real @ X3 @ (topolo1664202871s_real @ A) @ at_top_nat) => ((filterlim_nat_real @ X3 @ (topolo1664202871s_real @ B) @ at_top_nat) => (A = B)))))). % LIMSEQ_unique
thf(fact_13_LIMSEQ__unique, axiom,
    ((![X3 : nat > nat, A : nat, B : nat]: ((filterlim_nat_nat @ X3 @ (topolo1564986139ds_nat @ A) @ at_top_nat) => ((filterlim_nat_nat @ X3 @ (topolo1564986139ds_nat @ B) @ at_top_nat) => (A = B)))))). % LIMSEQ_unique
thf(fact_14_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_15_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_16_tendsto__eq__rhs, axiom,
    ((![F2 : nat > real, X4 : real, F : filter_nat, Y : real]: ((filterlim_nat_real @ F2 @ (topolo1664202871s_real @ X4) @ F) => ((X4 = Y) => (filterlim_nat_real @ F2 @ (topolo1664202871s_real @ Y) @ F)))))). % tendsto_eq_rhs
thf(fact_17_tendsto__eq__rhs, axiom,
    ((![F2 : nat > nat, X4 : nat, F : filter_nat, Y : nat]: ((filterlim_nat_nat @ F2 @ (topolo1564986139ds_nat @ X4) @ F) => ((X4 = Y) => (filterlim_nat_nat @ F2 @ (topolo1564986139ds_nat @ Y) @ F)))))). % tendsto_eq_rhs
thf(fact_18_f_I2_J, axiom,
    ((topolo144289241q_real @ (^[N : nat]: (re @ (s @ (f @ N))))))). % f(2)
thf(fact_19_filterlim__ident, axiom,
    ((![F : filter_nat]: (filterlim_nat_nat @ (^[X : nat]: X) @ F @ F)))). % filterlim_ident
thf(fact_20_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_21_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_22_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_23_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_24__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_25_conv1, axiom,
    ((topolo795669587t_real @ (^[N : nat]: (re @ (s @ (f @ N))))))). % conv1
thf(fact_26__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_27_seq__monosub, axiom,
    ((![S : nat > real]: (?[F4 : nat > nat]: ((order_769474267at_nat @ F4) & (topolo144289241q_real @ (^[N : nat]: (S @ (F4 @ N))))))))). % seq_monosub
thf(fact_28_convergent__const, axiom,
    ((![C : real]: (topolo795669587t_real @ (^[N : nat]: C))))). % convergent_const
thf(fact_29_filterlim__subseq, axiom,
    ((![F2 : nat > nat]: ((order_769474267at_nat @ F2) => (filterlim_nat_nat @ F2 @ at_top_nat @ at_top_nat))))). % filterlim_subseq
thf(fact_30_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_31_convergent__def, axiom,
    ((topolo768750839nt_nat = (^[X5 : nat > nat]: (?[L3 : nat]: (filterlim_nat_nat @ X5 @ (topolo1564986139ds_nat @ L3) @ at_top_nat)))))). % convergent_def
thf(fact_32_convergent__def, axiom,
    ((topolo795669587t_real = (^[X5 : nat > real]: (?[L3 : real]: (filterlim_nat_real @ X5 @ (topolo1664202871s_real @ L3) @ at_top_nat)))))). % convergent_def
thf(fact_33_convergentI, axiom,
    ((![X3 : nat > nat, L4 : nat]: ((filterlim_nat_nat @ X3 @ (topolo1564986139ds_nat @ L4) @ at_top_nat) => (topolo768750839nt_nat @ X3))))). % convergentI
thf(fact_34_convergentI, axiom,
    ((![X3 : nat > real, L4 : real]: ((filterlim_nat_real @ X3 @ (topolo1664202871s_real @ L4) @ at_top_nat) => (topolo795669587t_real @ X3))))). % convergentI
thf(fact_35_convergentD, axiom,
    ((![X3 : nat > nat]: ((topolo768750839nt_nat @ X3) => (?[L : nat]: (filterlim_nat_nat @ X3 @ (topolo1564986139ds_nat @ L) @ at_top_nat)))))). % convergentD
thf(fact_36_convergentD, axiom,
    ((![X3 : nat > real]: ((topolo795669587t_real @ X3) => (?[L : real]: (filterlim_nat_real @ X3 @ (topolo1664202871s_real @ L) @ at_top_nat)))))). % convergentD
thf(fact_37_tendsto__complex__iff, axiom,
    ((![F2 : nat > complex, X4 : complex, F : filter_nat]: ((filter1919943476omplex @ F2 @ (topolo155787769omplex @ X4) @ F) = (((filterlim_nat_real @ (^[X : nat]: (re @ (F2 @ X))) @ (topolo1664202871s_real @ (re @ X4)) @ F)) & ((filterlim_nat_real @ (^[X : nat]: (im @ (F2 @ X))) @ (topolo1664202871s_real @ (im @ X4)) @ F))))))). % tendsto_complex_iff
thf(fact_38__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_39_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_40_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_41_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_42_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_43_complex_Oexpand, axiom,
    ((![Complex : complex, Complex2 : complex]: ((((re @ Complex) = (re @ Complex2)) & ((im @ Complex) = (im @ Complex2))) => (Complex = Complex2))))). % complex.expand
thf(fact_44_complex__eqI, axiom,
    ((![X4 : complex, Y : complex]: (((re @ X4) = (re @ Y)) => (((im @ X4) = (im @ Y)) => (X4 = Y)))))). % complex_eqI
thf(fact_45_convergent__LIMSEQ__iff, axiom,
    ((topolo768750839nt_nat = (^[X5 : nat > nat]: (filterlim_nat_nat @ X5 @ (topolo1564986139ds_nat @ (topolo606707426at_nat @ at_top_nat @ X5)) @ at_top_nat))))). % convergent_LIMSEQ_iff
thf(fact_46_convergent__LIMSEQ__iff, axiom,
    ((topolo795669587t_real = (^[X5 : nat > real]: (filterlim_nat_real @ X5 @ (topolo1664202871s_real @ (topolo307763390t_real @ at_top_nat @ X5)) @ at_top_nat))))). % convergent_LIMSEQ_iff
thf(fact_47_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_48_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_49_limI, axiom,
    ((![X3 : nat > real, L4 : real]: ((filterlim_nat_real @ X3 @ (topolo1664202871s_real @ L4) @ at_top_nat) => ((topolo307763390t_real @ at_top_nat @ X3) = L4))))). % limI
thf(fact_50_limI, axiom,
    ((![X3 : nat > nat, L4 : nat]: ((filterlim_nat_nat @ X3 @ (topolo1564986139ds_nat @ L4) @ at_top_nat) => ((topolo606707426at_nat @ at_top_nat @ X3) = L4))))). % limI
thf(fact_51__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_52_comp__apply, axiom,
    ((comp_c1631780367al_nat = (^[F5 : complex > real]: (^[G3 : nat > complex]: (^[X : nat]: (F5 @ (G3 @ X)))))))). % comp_apply
thf(fact_53_comp__apply, axiom,
    ((comp_nat_real_nat = (^[F5 : nat > real]: (^[G3 : nat > nat]: (^[X : nat]: (F5 @ (G3 @ X)))))))). % comp_apply
thf(fact_54_lim__def, axiom,
    ((![X3 : nat > real]: ((topolo307763390t_real @ at_top_nat @ X3) = (the_real @ (^[L3 : real]: (filterlim_nat_real @ X3 @ (topolo1664202871s_real @ L3) @ at_top_nat))))))). % lim_def
thf(fact_55_lim__def, axiom,
    ((![X3 : nat > nat]: ((topolo606707426at_nat @ at_top_nat @ X3) = (the_nat @ (^[L3 : nat]: (filterlim_nat_nat @ X3 @ (topolo1564986139ds_nat @ L3) @ at_top_nat))))))). % lim_def
thf(fact_56_tendsto__Complex, 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) => (filter1919943476omplex @ (^[X : nat]: (complex2 @ (F2 @ X) @ (G @ X))) @ (topolo155787769omplex @ (complex2 @ A @ B)) @ F)))))). % tendsto_Complex
thf(fact_57_LIMSEQ__Uniq, axiom,
    ((![X3 : nat > real]: (uniq_real @ (^[L5 : real]: (filterlim_nat_real @ X3 @ (topolo1664202871s_real @ L5) @ at_top_nat)))))). % LIMSEQ_Uniq
thf(fact_58_LIMSEQ__Uniq, axiom,
    ((![X3 : nat > nat]: (uniq_nat @ (^[L5 : nat]: (filterlim_nat_nat @ X3 @ (topolo1564986139ds_nat @ L5) @ at_top_nat)))))). % LIMSEQ_Uniq
thf(fact_59_K__record__comp, axiom,
    ((![C : real, F2 : nat > complex]: ((comp_c1631780367al_nat @ (^[X : complex]: C) @ F2) = (^[X : nat]: C))))). % K_record_comp
thf(fact_60_K__record__comp, axiom,
    ((![C : real, F2 : nat > nat]: ((comp_nat_real_nat @ (^[X : nat]: C) @ F2) = (^[X : nat]: C))))). % K_record_comp
thf(fact_61_complex_Oinject, axiom,
    ((![X1 : real, X22 : real, Y1 : real, Y22 : real]: (((complex2 @ X1 @ X22) = (complex2 @ Y1 @ Y22)) = (((X1 = Y1)) & ((X22 = Y22))))))). % complex.inject
thf(fact_62_complex__surj, axiom,
    ((![Z2 : complex]: ((complex2 @ (re @ Z2) @ (im @ Z2)) = Z2)))). % complex_surj
thf(fact_63_complex_Ocollapse, axiom,
    ((![Complex : complex]: ((complex2 @ (re @ Complex) @ (im @ Complex)) = Complex)))). % complex.collapse
thf(fact_64_complex_Oexhaust, axiom,
    ((![Y : complex]: (~ ((![X12 : real, X23 : real]: (~ ((Y = (complex2 @ X12 @ X23)))))))))). % complex.exhaust
thf(fact_65_complex_Osel_I1_J, axiom,
    ((![X1 : real, X22 : real]: ((re @ (complex2 @ X1 @ X22)) = X1)))). % complex.sel(1)
thf(fact_66_complex_Osel_I2_J, axiom,
    ((![X1 : real, X22 : real]: ((im @ (complex2 @ X1 @ X22)) = X22)))). % complex.sel(2)
thf(fact_67_complex_Oexhaust__sel, axiom,
    ((![Complex : complex]: (Complex = (complex2 @ (re @ Complex) @ (im @ Complex)))))). % complex.exhaust_sel
thf(fact_68_convergent__subseq__convergent, axiom,
    ((![X3 : nat > real, F2 : nat > nat]: ((topolo795669587t_real @ X3) => ((order_769474267at_nat @ F2) => (topolo795669587t_real @ (comp_nat_real_nat @ X3 @ F2))))))). % convergent_subseq_convergent
thf(fact_69_comp__def, axiom,
    ((comp_c1631780367al_nat = (^[F5 : complex > real]: (^[G3 : nat > complex]: (^[X : nat]: (F5 @ (G3 @ X)))))))). % comp_def
thf(fact_70_comp__def, axiom,
    ((comp_nat_real_nat = (^[F5 : nat > real]: (^[G3 : nat > nat]: (^[X : nat]: (F5 @ (G3 @ X)))))))). % comp_def
thf(fact_71_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_72_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_73_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_74_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_75_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_76_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_77_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_78_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_79_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_80_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_81_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_82_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_83_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_84_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_85_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_86_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_87_fun_Omap__comp, axiom,
    ((![G : real > real, F2 : complex > real, V : nat > complex]: ((comp_real_real_nat @ G @ (comp_c1631780367al_nat @ F2 @ V)) = (comp_c1631780367al_nat @ (comp_r422820971omplex @ G @ F2) @ V))))). % fun.map_comp
thf(fact_88_fun_Omap__comp, axiom,
    ((![G : real > real, F2 : nat > real, V : nat > nat]: ((comp_real_real_nat @ G @ (comp_nat_real_nat @ F2 @ V)) = (comp_nat_real_nat @ (comp_real_real_nat @ G @ F2) @ V))))). % fun.map_comp
thf(fact_89_fun_Omap__comp, axiom,
    ((![G : complex > real, F2 : complex > complex, V : nat > complex]: ((comp_c1631780367al_nat @ G @ (comp_c438056209ex_nat @ F2 @ V)) = (comp_c1631780367al_nat @ (comp_c317287661omplex @ G @ F2) @ V))))). % fun.map_comp
thf(fact_90_fun_Omap__comp, axiom,
    ((![G : complex > real, F2 : nat > complex, V : nat > nat]: ((comp_c1631780367al_nat @ G @ (comp_nat_complex_nat @ F2 @ V)) = (comp_nat_real_nat @ (comp_c1631780367al_nat @ G @ F2) @ V))))). % fun.map_comp
thf(fact_91_fun_Omap__comp, axiom,
    ((![G : nat > real, F2 : complex > nat, V : nat > complex]: ((comp_nat_real_nat @ G @ (comp_complex_nat_nat @ F2 @ V)) = (comp_c1631780367al_nat @ (comp_n1816297743omplex @ G @ F2) @ V))))). % fun.map_comp
thf(fact_92_fun_Omap__comp, axiom,
    ((![G : nat > real, F2 : nat > nat, V : nat > nat]: ((comp_nat_real_nat @ G @ (comp_nat_nat_nat @ F2 @ V)) = (comp_nat_real_nat @ (comp_nat_real_nat @ G @ F2) @ V))))). % fun.map_comp
thf(fact_93_Lim__def, axiom,
    ((topolo307763390t_real = (^[A2 : filter_nat]: (^[F5 : nat > real]: (the_real @ (^[L5 : real]: (filterlim_nat_real @ F5 @ (topolo1664202871s_real @ L5) @ A2)))))))). % Lim_def
thf(fact_94_Lim__def, axiom,
    ((topolo606707426at_nat = (^[A2 : filter_nat]: (^[F5 : nat > nat]: (the_nat @ (^[L5 : nat]: (filterlim_nat_nat @ F5 @ (topolo1564986139ds_nat @ L5) @ A2)))))))). % Lim_def
thf(fact_95_LIMSEQ__subseq__LIMSEQ, axiom,
    ((![X3 : nat > real, L4 : real, F2 : nat > nat]: ((filterlim_nat_real @ X3 @ (topolo1664202871s_real @ L4) @ at_top_nat) => ((order_769474267at_nat @ F2) => (filterlim_nat_real @ (comp_nat_real_nat @ X3 @ F2) @ (topolo1664202871s_real @ L4) @ at_top_nat)))))). % LIMSEQ_subseq_LIMSEQ
thf(fact_96_LIMSEQ__subseq__LIMSEQ, axiom,
    ((![X3 : nat > nat, L4 : nat, F2 : nat > nat]: ((filterlim_nat_nat @ X3 @ (topolo1564986139ds_nat @ L4) @ at_top_nat) => ((order_769474267at_nat @ F2) => (filterlim_nat_nat @ (comp_nat_nat_nat @ X3 @ F2) @ (topolo1564986139ds_nat @ L4) @ at_top_nat)))))). % LIMSEQ_subseq_LIMSEQ
thf(fact_97_complex_Osplit__sel, axiom,
    ((![P : real > $o, F2 : real > real > real, Complex : complex]: ((P @ (case_complex_real @ F2 @ Complex)) = (((Complex = (complex2 @ (re @ Complex) @ (im @ Complex)))) => ((P @ (F2 @ (re @ Complex) @ (im @ Complex))))))))). % complex.split_sel
thf(fact_98_complex_Osplit__sel__asm, axiom,
    ((![P : real > $o, F2 : real > real > real, Complex : complex]: ((P @ (case_complex_real @ F2 @ Complex)) = (~ ((((Complex = (complex2 @ (re @ Complex) @ (im @ Complex)))) & ((~ ((P @ (F2 @ (re @ Complex) @ (im @ Complex))))))))))))). % complex.split_sel_asm
thf(fact_99_complex_Ocase__distrib, axiom,
    ((![H : real > real, F2 : real > real > real, Complex : complex]: ((H @ (case_complex_real @ F2 @ Complex)) = (case_complex_real @ (^[X13 : real]: (^[X24 : real]: (H @ (F2 @ X13 @ X24)))) @ Complex))))). % complex.case_distrib
thf(fact_100_complex_Ocase, axiom,
    ((![F2 : real > real > real, X1 : real, X22 : real]: ((case_complex_real @ F2 @ (complex2 @ X1 @ X22)) = (F2 @ X1 @ X22))))). % complex.case
thf(fact_101_Re__def, axiom,
    ((re = (case_complex_real @ (^[X13 : real]: (^[X24 : real]: X13)))))). % Re_def
thf(fact_102_Im__def, axiom,
    ((im = (case_complex_real @ (^[X13 : real]: (^[X24 : real]: X24)))))). % Im_def
thf(fact_103_complex_Ocase__eq__if, axiom,
    ((case_complex_real = (^[F5 : real > real > real]: (^[Complex5 : complex]: (F5 @ (re @ Complex5) @ (im @ Complex5))))))). % complex.case_eq_if
thf(fact_104_Cauchy__converges__subseq, axiom,
    ((![U : nat > complex, R : nat > nat, L2 : complex]: ((topolo1714400466omplex @ U) => ((order_769474267at_nat @ R) => ((filter1919943476omplex @ (comp_nat_complex_nat @ U @ R) @ (topolo155787769omplex @ L2) @ at_top_nat) => (filter1919943476omplex @ U @ (topolo155787769omplex @ L2) @ at_top_nat))))))). % Cauchy_converges_subseq
thf(fact_105_Cauchy__converges__subseq, axiom,
    ((![U : nat > real, R : nat > nat, L2 : real]: ((topolo597383376y_real @ U) => ((order_769474267at_nat @ R) => ((filterlim_nat_real @ (comp_nat_real_nat @ U @ R) @ (topolo1664202871s_real @ L2) @ at_top_nat) => (filterlim_nat_real @ U @ (topolo1664202871s_real @ L2) @ at_top_nat))))))). % Cauchy_converges_subseq
thf(fact_106_strict__mono__eq, axiom,
    ((![F2 : nat > nat, X4 : nat, Y : nat]: ((order_769474267at_nat @ F2) => (((F2 @ X4) = (F2 @ Y)) = (X4 = Y)))))). % strict_mono_eq
thf(fact_107_lim__le, axiom,
    ((![F2 : nat > real, X4 : real]: ((topolo795669587t_real @ F2) => ((![N2 : nat]: (ord_less_eq_real @ (F2 @ N2) @ X4)) => (ord_less_eq_real @ (topolo307763390t_real @ at_top_nat @ F2) @ X4)))))). % lim_le
thf(fact_108_order__refl, axiom,
    ((![X4 : real]: (ord_less_eq_real @ X4 @ X4)))). % order_refl
thf(fact_109_real__Cauchy__convergent, axiom,
    ((![X3 : nat > real]: ((topolo597383376y_real @ X3) => (topolo795669587t_real @ X3))))). % real_Cauchy_convergent
thf(fact_110_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_111_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_112_monoseq__def, axiom,
    ((topolo144289241q_real = (^[X5 : nat > real]: (((![M : nat]: (![N : nat]: (((ord_less_eq_nat @ M @ N)) => ((ord_less_eq_real @ (X5 @ M) @ (X5 @ N))))))) | ((![M : nat]: (![N : nat]: (((ord_less_eq_nat @ M @ N)) => ((ord_less_eq_real @ (X5 @ N) @ (X5 @ M)))))))))))). % monoseq_def
thf(fact_113_monoI2, axiom,
    ((![X3 : nat > real]: ((![M2 : nat, N2 : nat]: ((ord_less_eq_nat @ M2 @ N2) => (ord_less_eq_real @ (X3 @ N2) @ (X3 @ M2)))) => (topolo144289241q_real @ X3))))). % monoI2
thf(fact_114_monoI1, axiom,
    ((![X3 : nat > real]: ((![M2 : nat, N2 : nat]: ((ord_less_eq_nat @ M2 @ N2) => (ord_less_eq_real @ (X3 @ M2) @ (X3 @ N2)))) => (topolo144289241q_real @ X3))))). % monoI1
thf(fact_115_strict__mono__less__eq, axiom,
    ((![F2 : real > real, X4 : real, Y : real]: ((order_1818878995l_real @ F2) => ((ord_less_eq_real @ (F2 @ X4) @ (F2 @ Y)) = (ord_less_eq_real @ X4 @ Y)))))). % strict_mono_less_eq
thf(fact_116_strict__mono__less__eq, axiom,
    ((![F2 : nat > nat, X4 : nat, Y : nat]: ((order_769474267at_nat @ F2) => ((ord_less_eq_nat @ (F2 @ X4) @ (F2 @ Y)) = (ord_less_eq_nat @ X4 @ Y)))))). % strict_mono_less_eq
thf(fact_117_Cauchy__convergent, axiom,
    ((![X3 : nat > complex]: ((topolo1714400466omplex @ X3) => (topolo1054921685omplex @ X3))))). % Cauchy_convergent
thf(fact_118_Cauchy__convergent, axiom,
    ((![X3 : nat > real]: ((topolo597383376y_real @ X3) => (topolo795669587t_real @ X3))))). % Cauchy_convergent
thf(fact_119_Cauchy__convergent__iff, axiom,
    ((topolo1714400466omplex = topolo1054921685omplex))). % Cauchy_convergent_iff
thf(fact_120_Cauchy__convergent__iff, axiom,
    ((topolo597383376y_real = topolo795669587t_real))). % Cauchy_convergent_iff
thf(fact_121_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) => ((![X2 : real, Y4 : real]: ((ord_less_eq_real @ X2 @ Y4) => (ord_less_eq_real @ (F2 @ X2) @ (F2 @ Y4)))) => (ord_less_eq_real @ A @ (F2 @ C)))))))). % order_subst1
thf(fact_122_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) => ((![X2 : real, Y4 : real]: ((ord_less_eq_real @ X2 @ Y4) => (ord_less_eq_real @ (F2 @ X2) @ (F2 @ Y4)))) => (ord_less_eq_real @ (F2 @ A) @ C))))))). % order_subst2
thf(fact_123_ord__eq__le__subst, axiom,
    ((![A : real, F2 : real > real, B : real, C : real]: ((A = (F2 @ B)) => ((ord_less_eq_real @ B @ C) => ((![X2 : real, Y4 : real]: ((ord_less_eq_real @ X2 @ Y4) => (ord_less_eq_real @ (F2 @ X2) @ (F2 @ Y4)))) => (ord_less_eq_real @ A @ (F2 @ C)))))))). % ord_eq_le_subst
thf(fact_124_ord__le__eq__subst, axiom,
    ((![A : real, B : real, F2 : real > real, C : real]: ((ord_less_eq_real @ A @ B) => (((F2 @ B) = C) => ((![X2 : real, Y4 : real]: ((ord_less_eq_real @ X2 @ Y4) => (ord_less_eq_real @ (F2 @ X2) @ (F2 @ Y4)))) => (ord_less_eq_real @ (F2 @ A) @ C))))))). % ord_le_eq_subst
thf(fact_125_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_126_antisym, axiom,
    ((![X4 : real, Y : real]: ((ord_less_eq_real @ X4 @ Y) => ((ord_less_eq_real @ Y @ X4) => (X4 = Y)))))). % antisym
thf(fact_127_linear, axiom,
    ((![X4 : real, Y : real]: ((ord_less_eq_real @ X4 @ Y) | (ord_less_eq_real @ Y @ X4))))). % linear
thf(fact_128_eq__refl, axiom,
    ((![X4 : real, Y : real]: ((X4 = Y) => (ord_less_eq_real @ X4 @ Y))))). % eq_refl
thf(fact_129_le__cases, axiom,
    ((![X4 : real, Y : real]: ((~ ((ord_less_eq_real @ X4 @ Y))) => (ord_less_eq_real @ Y @ X4))))). % le_cases
thf(fact_130_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_131_le__cases3, axiom,
    ((![X4 : real, Y : real, Z2 : real]: (((ord_less_eq_real @ X4 @ Y) => (~ ((ord_less_eq_real @ Y @ Z2)))) => (((ord_less_eq_real @ Y @ X4) => (~ ((ord_less_eq_real @ X4 @ Z2)))) => (((ord_less_eq_real @ X4 @ Z2) => (~ ((ord_less_eq_real @ Z2 @ Y)))) => (((ord_less_eq_real @ Z2 @ Y) => (~ ((ord_less_eq_real @ Y @ X4)))) => (((ord_less_eq_real @ Y @ Z2) => (~ ((ord_less_eq_real @ Z2 @ X4)))) => (~ (((ord_less_eq_real @ Z2 @ X4) => (~ ((ord_less_eq_real @ X4 @ Y)))))))))))))). % le_cases3
thf(fact_132_antisym__conv, axiom,
    ((![Y : real, X4 : real]: ((ord_less_eq_real @ Y @ X4) => ((ord_less_eq_real @ X4 @ Y) = (X4 = Y)))))). % antisym_conv
thf(fact_133_order__class_Oorder_Oeq__iff, axiom,
    (((^[Y2 : real]: (^[Z : real]: (Y2 = Z))) = (^[A3 : real]: (^[B2 : real]: (((ord_less_eq_real @ A3 @ B2)) & ((ord_less_eq_real @ B2 @ A3)))))))). % order_class.order.eq_iff
thf(fact_134_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_135_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_136_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_137_order__trans, axiom,
    ((![X4 : real, Y : real, Z2 : real]: ((ord_less_eq_real @ X4 @ Y) => ((ord_less_eq_real @ Y @ Z2) => (ord_less_eq_real @ X4 @ Z2)))))). % order_trans
thf(fact_138_dual__order_Orefl, axiom,
    ((![A : real]: (ord_less_eq_real @ A @ A)))). % dual_order.refl
thf(fact_139_linorder__wlog, axiom,
    ((![P : real > real > $o, A : real, B : real]: ((![A4 : real, B3 : real]: ((ord_less_eq_real @ A4 @ B3) => (P @ A4 @ B3))) => ((![A4 : real, B3 : real]: ((P @ B3 @ A4) => (P @ A4 @ B3))) => (P @ A @ B)))))). % linorder_wlog
thf(fact_140_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_141_dual__order_Oeq__iff, axiom,
    (((^[Y2 : real]: (^[Z : real]: (Y2 = Z))) = (^[A3 : real]: (^[B2 : real]: (((ord_less_eq_real @ B2 @ A3)) & ((ord_less_eq_real @ A3 @ B2)))))))). % dual_order.eq_iff
thf(fact_142_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_143_Cauchy__Im, axiom,
    ((![X3 : nat > complex]: ((topolo1714400466omplex @ X3) => (topolo597383376y_real @ (^[N : nat]: (im @ (X3 @ N)))))))). % Cauchy_Im
thf(fact_144_Cauchy__Re, axiom,
    ((![X3 : nat > complex]: ((topolo1714400466omplex @ X3) => (topolo597383376y_real @ (^[N : nat]: (re @ (X3 @ N)))))))). % Cauchy_Re
thf(fact_145_strict__mono__leD, axiom,
    ((![R : real > real, M3 : real, N3 : real]: ((order_1818878995l_real @ R) => ((ord_less_eq_real @ M3 @ N3) => (ord_less_eq_real @ (R @ M3) @ (R @ N3))))))). % strict_mono_leD
thf(fact_146_strict__mono__leD, axiom,
    ((![R : nat > nat, M3 : nat, N3 : nat]: ((order_769474267at_nat @ R) => ((ord_less_eq_nat @ M3 @ N3) => (ord_less_eq_nat @ (R @ M3) @ (R @ N3))))))). % strict_mono_leD
thf(fact_147_convergent__Cauchy, axiom,
    ((![X3 : nat > complex]: ((topolo1054921685omplex @ X3) => (topolo1714400466omplex @ X3))))). % convergent_Cauchy
thf(fact_148_convergent__Cauchy, axiom,
    ((![X3 : nat > real]: ((topolo795669587t_real @ X3) => (topolo597383376y_real @ X3))))). % convergent_Cauchy
thf(fact_149_LIMSEQ__le__const2, axiom,
    ((![X3 : nat > real, X4 : real, A : real]: ((filterlim_nat_real @ X3 @ (topolo1664202871s_real @ X4) @ at_top_nat) => ((?[N4 : nat]: (![N2 : nat]: ((ord_less_eq_nat @ N4 @ N2) => (ord_less_eq_real @ (X3 @ N2) @ A)))) => (ord_less_eq_real @ X4 @ A)))))). % LIMSEQ_le_const2
thf(fact_150_LIMSEQ__le__const2, axiom,
    ((![X3 : nat > nat, X4 : nat, A : nat]: ((filterlim_nat_nat @ X3 @ (topolo1564986139ds_nat @ X4) @ at_top_nat) => ((?[N4 : nat]: (![N2 : nat]: ((ord_less_eq_nat @ N4 @ N2) => (ord_less_eq_nat @ (X3 @ N2) @ A)))) => (ord_less_eq_nat @ X4 @ A)))))). % LIMSEQ_le_const2
thf(fact_151_LIMSEQ__le__const, axiom,
    ((![X3 : nat > real, X4 : real, A : real]: ((filterlim_nat_real @ X3 @ (topolo1664202871s_real @ X4) @ at_top_nat) => ((?[N4 : nat]: (![N2 : nat]: ((ord_less_eq_nat @ N4 @ N2) => (ord_less_eq_real @ A @ (X3 @ N2))))) => (ord_less_eq_real @ A @ X4)))))). % LIMSEQ_le_const
thf(fact_152_LIMSEQ__le__const, axiom,
    ((![X3 : nat > nat, X4 : nat, A : nat]: ((filterlim_nat_nat @ X3 @ (topolo1564986139ds_nat @ X4) @ at_top_nat) => ((?[N4 : nat]: (![N2 : nat]: ((ord_less_eq_nat @ N4 @ N2) => (ord_less_eq_nat @ A @ (X3 @ N2))))) => (ord_less_eq_nat @ A @ X4)))))). % LIMSEQ_le_const
thf(fact_153_Lim__bounded2, axiom,
    ((![F2 : nat > real, L2 : real, N5 : nat, C2 : real]: ((filterlim_nat_real @ F2 @ (topolo1664202871s_real @ L2) @ at_top_nat) => ((![N2 : nat]: ((ord_less_eq_nat @ N5 @ N2) => (ord_less_eq_real @ C2 @ (F2 @ N2)))) => (ord_less_eq_real @ C2 @ L2)))))). % Lim_bounded2
thf(fact_154_Lim__bounded2, axiom,
    ((![F2 : nat > nat, L2 : nat, N5 : nat, C2 : nat]: ((filterlim_nat_nat @ F2 @ (topolo1564986139ds_nat @ L2) @ at_top_nat) => ((![N2 : nat]: ((ord_less_eq_nat @ N5 @ N2) => (ord_less_eq_nat @ C2 @ (F2 @ N2)))) => (ord_less_eq_nat @ C2 @ L2)))))). % Lim_bounded2
thf(fact_155_Lim__bounded, axiom,
    ((![F2 : nat > real, L2 : real, M4 : nat, C2 : real]: ((filterlim_nat_real @ F2 @ (topolo1664202871s_real @ L2) @ at_top_nat) => ((![N2 : nat]: ((ord_less_eq_nat @ M4 @ N2) => (ord_less_eq_real @ (F2 @ N2) @ C2))) => (ord_less_eq_real @ L2 @ C2)))))). % Lim_bounded
thf(fact_156_Lim__bounded, axiom,
    ((![F2 : nat > nat, L2 : nat, M4 : nat, C2 : nat]: ((filterlim_nat_nat @ F2 @ (topolo1564986139ds_nat @ L2) @ at_top_nat) => ((![N2 : nat]: ((ord_less_eq_nat @ M4 @ N2) => (ord_less_eq_nat @ (F2 @ N2) @ C2))) => (ord_less_eq_nat @ L2 @ C2)))))). % Lim_bounded
thf(fact_157_LIMSEQ__le, axiom,
    ((![X3 : nat > real, X4 : real, Y5 : nat > real, Y : real]: ((filterlim_nat_real @ X3 @ (topolo1664202871s_real @ X4) @ at_top_nat) => ((filterlim_nat_real @ Y5 @ (topolo1664202871s_real @ Y) @ at_top_nat) => ((?[N4 : nat]: (![N2 : nat]: ((ord_less_eq_nat @ N4 @ N2) => (ord_less_eq_real @ (X3 @ N2) @ (Y5 @ N2))))) => (ord_less_eq_real @ X4 @ Y))))))). % LIMSEQ_le
thf(fact_158_LIMSEQ__le, axiom,
    ((![X3 : nat > nat, X4 : nat, Y5 : nat > nat, Y : nat]: ((filterlim_nat_nat @ X3 @ (topolo1564986139ds_nat @ X4) @ at_top_nat) => ((filterlim_nat_nat @ Y5 @ (topolo1564986139ds_nat @ Y) @ at_top_nat) => ((?[N4 : nat]: (![N2 : nat]: ((ord_less_eq_nat @ N4 @ N2) => (ord_less_eq_nat @ (X3 @ N2) @ (Y5 @ N2))))) => (ord_less_eq_nat @ X4 @ Y))))))). % LIMSEQ_le
thf(fact_159_lim__mono, axiom,
    ((![N5 : nat, X3 : nat > real, Y5 : nat > real, X4 : real, Y : real]: ((![N2 : nat]: ((ord_less_eq_nat @ N5 @ N2) => (ord_less_eq_real @ (X3 @ N2) @ (Y5 @ N2)))) => ((filterlim_nat_real @ X3 @ (topolo1664202871s_real @ X4) @ at_top_nat) => ((filterlim_nat_real @ Y5 @ (topolo1664202871s_real @ Y) @ at_top_nat) => (ord_less_eq_real @ X4 @ Y))))))). % lim_mono
thf(fact_160_lim__mono, axiom,
    ((![N5 : nat, X3 : nat > nat, Y5 : nat > nat, X4 : nat, Y : nat]: ((![N2 : nat]: ((ord_less_eq_nat @ N5 @ N2) => (ord_less_eq_nat @ (X3 @ N2) @ (Y5 @ N2)))) => ((filterlim_nat_nat @ X3 @ (topolo1564986139ds_nat @ X4) @ at_top_nat) => ((filterlim_nat_nat @ Y5 @ (topolo1564986139ds_nat @ Y) @ at_top_nat) => (ord_less_eq_nat @ X4 @ Y))))))). % lim_mono
thf(fact_161_monoseq__le, axiom,
    ((![A : nat > real, X4 : real]: ((topolo144289241q_real @ A) => ((filterlim_nat_real @ A @ (topolo1664202871s_real @ X4) @ at_top_nat) => (((![N6 : nat]: (ord_less_eq_real @ (A @ N6) @ X4)) & (![M5 : nat, N6 : nat]: ((ord_less_eq_nat @ M5 @ N6) => (ord_less_eq_real @ (A @ M5) @ (A @ N6))))) | ((![N6 : nat]: (ord_less_eq_real @ X4 @ (A @ N6))) & (![M5 : nat, N6 : nat]: ((ord_less_eq_nat @ M5 @ N6) => (ord_less_eq_real @ (A @ N6) @ (A @ M5))))))))))). % monoseq_le
thf(fact_162_monoseq__le, axiom,
    ((![A : nat > nat, X4 : nat]: ((topolo1922093437eq_nat @ A) => ((filterlim_nat_nat @ A @ (topolo1564986139ds_nat @ X4) @ at_top_nat) => (((![N6 : nat]: (ord_less_eq_nat @ (A @ N6) @ X4)) & (![M5 : nat, N6 : nat]: ((ord_less_eq_nat @ M5 @ N6) => (ord_less_eq_nat @ (A @ M5) @ (A @ N6))))) | ((![N6 : nat]: (ord_less_eq_nat @ X4 @ (A @ N6))) & (![M5 : nat, N6 : nat]: ((ord_less_eq_nat @ M5 @ N6) => (ord_less_eq_nat @ (A @ N6) @ (A @ M5))))))))))). % monoseq_le
thf(fact_163_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_164_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_165_Cauchy__subseq__Cauchy, axiom,
    ((![X3 : nat > real, F2 : nat > nat]: ((topolo597383376y_real @ X3) => ((order_769474267at_nat @ F2) => (topolo597383376y_real @ (comp_nat_real_nat @ X3 @ F2))))))). % Cauchy_subseq_Cauchy
thf(fact_166_Cauchy__subseq__Cauchy, axiom,
    ((![X3 : nat > complex, F2 : nat > nat]: ((topolo1714400466omplex @ X3) => ((order_769474267at_nat @ F2) => (topolo1714400466omplex @ (comp_nat_complex_nat @ X3 @ F2))))))). % Cauchy_subseq_Cauchy
thf(fact_167_LIMSEQ__imp__Cauchy, axiom,
    ((![X3 : nat > complex, X4 : complex]: ((filter1919943476omplex @ X3 @ (topolo155787769omplex @ X4) @ at_top_nat) => (topolo1714400466omplex @ X3))))). % LIMSEQ_imp_Cauchy
thf(fact_168_LIMSEQ__imp__Cauchy, axiom,
    ((![X3 : nat > real, X4 : real]: ((filterlim_nat_real @ X3 @ (topolo1664202871s_real @ X4) @ at_top_nat) => (topolo597383376y_real @ X3))))). % LIMSEQ_imp_Cauchy
thf(fact_169_complete__def, axiom,
    ((topolo589572398omplex = (^[S2 : set_complex]: (![F5 : nat > complex]: (((((![N : nat]: (member_complex @ (F5 @ N) @ S2))) & ((topolo1714400466omplex @ F5)))) => ((?[X : complex]: (((member_complex @ X @ S2)) & ((filter1919943476omplex @ F5 @ (topolo155787769omplex @ X) @ at_top_nat))))))))))). % complete_def
thf(fact_170_complete__def, axiom,
    ((topolo2066423596e_real = (^[S2 : set_real]: (![F5 : nat > real]: (((((![N : nat]: (member_real @ (F5 @ N) @ S2))) & ((topolo597383376y_real @ F5)))) => ((?[X : real]: (((member_real @ X @ S2)) & ((filterlim_nat_real @ F5 @ (topolo1664202871s_real @ X) @ at_top_nat))))))))))). % complete_def
thf(fact_171_r, axiom,
    ((![N6 : nat]: (ord_less_eq_real @ (real_V638595069omplex @ (s @ N6)) @ r)))). % r
thf(fact_172_decseq__ge, axiom,
    ((![X3 : nat > real, L4 : real, N3 : nat]: ((order_106095024t_real @ X3) => ((filterlim_nat_real @ X3 @ (topolo1664202871s_real @ L4) @ at_top_nat) => (ord_less_eq_real @ L4 @ (X3 @ N3))))))). % decseq_ge
thf(fact_173_decseq__ge, axiom,
    ((![X3 : nat > nat, L4 : nat, N3 : nat]: ((order_1631207636at_nat @ X3) => ((filterlim_nat_nat @ X3 @ (topolo1564986139ds_nat @ L4) @ at_top_nat) => (ord_less_eq_nat @ L4 @ (X3 @ N3))))))). % decseq_ge
thf(fact_174_complex__Re__le__cmod, axiom,
    ((![X4 : complex]: (ord_less_eq_real @ (re @ X4) @ (real_V638595069omplex @ X4))))). % complex_Re_le_cmod
thf(fact_175_decseq__def, axiom,
    ((order_106095024t_real = (^[X5 : nat > real]: (![M : nat]: (![N : nat]: (((ord_less_eq_nat @ M @ N)) => ((ord_less_eq_real @ (X5 @ N) @ (X5 @ M)))))))))). % decseq_def
thf(fact_176_decseqD, axiom,
    ((![F2 : nat > real, I : nat, J : nat]: ((order_106095024t_real @ F2) => ((ord_less_eq_nat @ I @ J) => (ord_less_eq_real @ (F2 @ J) @ (F2 @ I))))))). % decseqD
thf(fact_177_decseq__imp__monoseq, axiom,
    ((![X3 : nat > real]: ((order_106095024t_real @ X3) => (topolo144289241q_real @ X3))))). % decseq_imp_monoseq
thf(fact_178_antimono__def, axiom,
    ((order_537808140l_real = (^[F5 : real > real]: (![X : real]: (![Y3 : real]: (((ord_less_eq_real @ X @ Y3)) => ((ord_less_eq_real @ (F5 @ Y3) @ (F5 @ X)))))))))). % antimono_def
thf(fact_179_antimonoI, axiom,
    ((![F2 : real > real]: ((![X2 : real, Y4 : real]: ((ord_less_eq_real @ X2 @ Y4) => (ord_less_eq_real @ (F2 @ Y4) @ (F2 @ X2)))) => (order_537808140l_real @ F2))))). % antimonoI

% Conjectures (2)
thf(conj_0, hypothesis,
    ((![Y6 : real]: ((filterlim_nat_real @ (^[N : nat]: (im @ (s @ (f @ (g @ N))))) @ (topolo1664202871s_real @ Y6) @ at_top_nat) => thesis)))).
thf(conj_1, conjecture,
    (thesis)).
