% 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_258__5370010_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:28:37.979

% Could-be-implicit typings (4)
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 (38)
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_Complex_Oimaginary__unit, type,
    imaginary_unit : complex).
thf(sy_c_Complex_Orcis, type,
    rcis : real > real > complex).
thf(sy_c_Filter_Oat__top_001t__Nat__Onat, type,
    at_top_nat : filter_nat).
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_Ominus__class_Ominus_001t__Complex__Ocomplex, type,
    minus_minus_complex : complex > complex > complex).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat, type,
    minus_minus_nat : nat > nat > nat).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal, type,
    minus_minus_real : real > real > real).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Complex__Ocomplex, type,
    plus_plus_complex : complex > complex > complex).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat, type,
    plus_plus_nat : nat > nat > nat).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal, type,
    plus_plus_real : real > real > real).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Complex__Ocomplex, type,
    times_times_complex : complex > complex > complex).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat, type,
    times_times_nat : nat > nat > nat).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal, type,
    times_times_real : real > real > real).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Complex__Ocomplex, type,
    uminus1204672759omplex : complex > complex).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal, type,
    uminus_uminus_real : real > real).
thf(sy_c_Limits_OBfun_001t__Nat__Onat_001t__Complex__Ocomplex, type,
    bfun_nat_complex : (nat > complex) > filter_nat > $o).
thf(sy_c_Limits_OBfun_001t__Nat__Onat_001t__Real__Oreal, type,
    bfun_nat_real : (nat > real) > filter_nat > $o).
thf(sy_c_Nat_OSuc, type,
    suc : nat > nat).
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_Topological__Spaces_Omonoseq_001t__Real__Oreal, type,
    topolo144289241q_real : (nat > real) > $o).
thf(sy_v_f____, type,
    f : nat > nat).
thf(sy_v_g____, type,
    g : nat > nat).
thf(sy_v_s, type,
    s : nat > complex).

% Relevant facts (248)
thf(fact_0_f_I1_J, axiom,
    ((order_769474267at_nat @ f))). % f(1)
thf(fact_1_f_I2_J, axiom,
    ((topolo144289241q_real @ (^[N : nat]: (re @ (s @ (f @ N))))))). % f(2)
thf(fact_2_Bseq__subseq, axiom,
    ((![F : nat > real, G : nat > nat]: ((bfun_nat_real @ F @ at_top_nat) => (bfun_nat_real @ (^[X : nat]: (F @ (G @ X))) @ at_top_nat))))). % Bseq_subseq
thf(fact_3_Bseq__conv__Bfun, axiom,
    ((![X2 : nat > real]: ((bfun_nat_real @ X2 @ at_top_nat) = (bfun_nat_real @ X2 @ at_top_nat))))). % Bseq_conv_Bfun
thf(fact_4_Bfun__const, axiom,
    ((![C : real, F2 : filter_nat]: (bfun_nat_real @ (^[Uu : nat]: C) @ F2)))). % Bfun_const
thf(fact_5_g_I2_J, axiom,
    ((topolo144289241q_real @ (^[N : nat]: (im @ (s @ (f @ (g @ N)))))))). % g(2)
thf(fact_6__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,
    ((~ ((![F3 : nat > nat]: ((order_769474267at_nat @ F3) => (~ ((topolo144289241q_real @ (^[N : nat]: (re @ (s @ (F3 @ 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_7__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062g_O_A_092_060lbrakk_062strict__mono_Ag_059_Amonoseq_A_I_092_060lambda_062n_O_AIm_A_Is_A_If_A_Ig_An_J_J_J_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom,
    ((~ ((![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_8_Bseq__offset, axiom,
    ((![X2 : nat > real, K : nat]: ((bfun_nat_real @ (^[N : nat]: (X2 @ (plus_plus_nat @ N @ K))) @ at_top_nat) => (bfun_nat_real @ X2 @ at_top_nat))))). % Bseq_offset
thf(fact_9_Bseq__ignore__initial__segment, axiom,
    ((![X2 : nat > real, K : nat]: ((bfun_nat_real @ X2 @ at_top_nat) => (bfun_nat_real @ (^[N : nat]: (X2 @ (plus_plus_nat @ N @ K))) @ at_top_nat))))). % Bseq_ignore_initial_segment
thf(fact_10_Bseq__mult, axiom,
    ((![F : nat > complex, G : nat > complex]: ((bfun_nat_complex @ F @ at_top_nat) => ((bfun_nat_complex @ G @ at_top_nat) => (bfun_nat_complex @ (^[X : nat]: (times_times_complex @ (F @ X) @ (G @ X))) @ at_top_nat)))))). % Bseq_mult
thf(fact_11_Bseq__mult, axiom,
    ((![F : nat > real, G : nat > real]: ((bfun_nat_real @ F @ at_top_nat) => ((bfun_nat_real @ G @ at_top_nat) => (bfun_nat_real @ (^[X : nat]: (times_times_real @ (F @ X) @ (G @ X))) @ at_top_nat)))))). % Bseq_mult
thf(fact_12_Bseq__Suc__iff, axiom,
    ((![F : nat > real]: ((bfun_nat_real @ (^[N : nat]: (F @ (suc @ N))) @ at_top_nat) = (bfun_nat_real @ F @ at_top_nat))))). % Bseq_Suc_iff
thf(fact_13_Bseq__add, axiom,
    ((![F : nat > real, C : real]: ((bfun_nat_real @ F @ at_top_nat) => (bfun_nat_real @ (^[X : nat]: (plus_plus_real @ (F @ X) @ C)) @ at_top_nat))))). % Bseq_add
thf(fact_14_Bseq__add, axiom,
    ((![F : nat > complex, C : complex]: ((bfun_nat_complex @ F @ at_top_nat) => (bfun_nat_complex @ (^[X : nat]: (plus_plus_complex @ (F @ X) @ C)) @ at_top_nat))))). % Bseq_add
thf(fact_15_Bseq__add__iff, axiom,
    ((![F : nat > real, C : real]: ((bfun_nat_real @ (^[X : nat]: (plus_plus_real @ (F @ X) @ C)) @ at_top_nat) = (bfun_nat_real @ F @ at_top_nat))))). % Bseq_add_iff
thf(fact_16_Bseq__add__iff, axiom,
    ((![F : nat > complex, C : complex]: ((bfun_nat_complex @ (^[X : nat]: (plus_plus_complex @ (F @ X) @ C)) @ at_top_nat) = (bfun_nat_complex @ F @ at_top_nat))))). % Bseq_add_iff
thf(fact_17_g_I1_J, axiom,
    ((order_769474267at_nat @ g))). % g(1)
thf(fact_18_add__Suc__right, axiom,
    ((![M : nat, N2 : nat]: ((plus_plus_nat @ M @ (suc @ N2)) = (suc @ (plus_plus_nat @ M @ N2)))))). % add_Suc_right
thf(fact_19_mult__Suc__right, axiom,
    ((![M : nat, N2 : nat]: ((times_times_nat @ M @ (suc @ N2)) = (plus_plus_nat @ M @ (times_times_nat @ M @ N2)))))). % mult_Suc_right
thf(fact_20__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,
    ((?[F3 : nat > nat]: ((order_769474267at_nat @ F3) & (topolo144289241q_real @ (^[N : nat]: (comp_c1631780367al_nat @ re @ s @ (F3 @ N)))))))). % \<open>\<exists>f. strict_mono f \<and> monoseq (\<lambda>n. (Re \<circ> s) (f n))\<close>
thf(fact_21_seq__monosub, axiom,
    ((![S : nat > real]: (?[F3 : nat > nat]: ((order_769474267at_nat @ F3) & (topolo144289241q_real @ (^[N : nat]: (S @ (F3 @ N))))))))). % seq_monosub
thf(fact_22_strict__mono__add, axiom,
    ((![K : real]: (order_1818878995l_real @ (^[N : real]: (plus_plus_real @ N @ K)))))). % strict_mono_add
thf(fact_23_strict__mono__add, axiom,
    ((![K : nat]: (order_769474267at_nat @ (^[N : nat]: (plus_plus_nat @ N @ K)))))). % strict_mono_add
thf(fact_24_nat_Oinject, axiom,
    ((![X22 : nat, Y2 : nat]: (((suc @ X22) = (suc @ Y2)) = (X22 = Y2))))). % nat.inject
thf(fact_25_old_Onat_Oinject, axiom,
    ((![Nat : nat, Nat2 : nat]: (((suc @ Nat) = (suc @ Nat2)) = (Nat = Nat2))))). % old.nat.inject
thf(fact_26_add__left__cancel, axiom,
    ((![A : nat, B : nat, C : nat]: (((plus_plus_nat @ A @ B) = (plus_plus_nat @ A @ C)) = (B = C))))). % add_left_cancel
thf(fact_27_add__left__cancel, axiom,
    ((![A : real, B : real, C : real]: (((plus_plus_real @ A @ B) = (plus_plus_real @ A @ C)) = (B = C))))). % add_left_cancel
thf(fact_28_add__left__cancel, axiom,
    ((![A : complex, B : complex, C : complex]: (((plus_plus_complex @ A @ B) = (plus_plus_complex @ A @ C)) = (B = C))))). % add_left_cancel
thf(fact_29_add__right__cancel, axiom,
    ((![B : nat, A : nat, C : nat]: (((plus_plus_nat @ B @ A) = (plus_plus_nat @ C @ A)) = (B = C))))). % add_right_cancel
thf(fact_30_add__right__cancel, axiom,
    ((![B : real, A : real, C : real]: (((plus_plus_real @ B @ A) = (plus_plus_real @ C @ A)) = (B = C))))). % add_right_cancel
thf(fact_31_add__right__cancel, axiom,
    ((![B : complex, A : complex, C : complex]: (((plus_plus_complex @ B @ A) = (plus_plus_complex @ C @ A)) = (B = C))))). % add_right_cancel
thf(fact_32_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_33_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_34_mult_Oleft__commute, axiom,
    ((![B : nat, A : nat, C : nat]: ((times_times_nat @ B @ (times_times_nat @ A @ C)) = (times_times_nat @ A @ (times_times_nat @ B @ C)))))). % mult.left_commute
thf(fact_35_mult_Oleft__commute, axiom,
    ((![B : complex, A : complex, C : complex]: ((times_times_complex @ B @ (times_times_complex @ A @ C)) = (times_times_complex @ A @ (times_times_complex @ B @ C)))))). % mult.left_commute
thf(fact_36_mult_Oleft__commute, axiom,
    ((![B : real, A : real, C : real]: ((times_times_real @ B @ (times_times_real @ A @ C)) = (times_times_real @ A @ (times_times_real @ B @ C)))))). % mult.left_commute
thf(fact_37_mult_Ocommute, axiom,
    ((times_times_nat = (^[A2 : nat]: (^[B2 : nat]: (times_times_nat @ B2 @ A2)))))). % mult.commute
thf(fact_38_mult_Ocommute, axiom,
    ((times_times_complex = (^[A2 : complex]: (^[B2 : complex]: (times_times_complex @ B2 @ A2)))))). % mult.commute
thf(fact_39_mult_Ocommute, axiom,
    ((times_times_real = (^[A2 : real]: (^[B2 : real]: (times_times_real @ B2 @ A2)))))). % mult.commute
thf(fact_40_mult_Oassoc, axiom,
    ((![A : nat, B : nat, C : nat]: ((times_times_nat @ (times_times_nat @ A @ B) @ C) = (times_times_nat @ A @ (times_times_nat @ B @ C)))))). % mult.assoc
thf(fact_41_mult_Oassoc, axiom,
    ((![A : complex, B : complex, C : complex]: ((times_times_complex @ (times_times_complex @ A @ B) @ C) = (times_times_complex @ A @ (times_times_complex @ B @ C)))))). % mult.assoc
thf(fact_42_mult_Oassoc, axiom,
    ((![A : real, B : real, C : real]: ((times_times_real @ (times_times_real @ A @ B) @ C) = (times_times_real @ A @ (times_times_real @ B @ C)))))). % mult.assoc
thf(fact_43_ab__semigroup__mult__class_Omult__ac_I1_J, axiom,
    ((![A : nat, B : nat, C : nat]: ((times_times_nat @ (times_times_nat @ A @ B) @ C) = (times_times_nat @ A @ (times_times_nat @ B @ C)))))). % ab_semigroup_mult_class.mult_ac(1)
thf(fact_44_ab__semigroup__mult__class_Omult__ac_I1_J, axiom,
    ((![A : complex, B : complex, C : complex]: ((times_times_complex @ (times_times_complex @ A @ B) @ C) = (times_times_complex @ A @ (times_times_complex @ B @ C)))))). % ab_semigroup_mult_class.mult_ac(1)
thf(fact_45_ab__semigroup__mult__class_Omult__ac_I1_J, axiom,
    ((![A : real, B : real, C : real]: ((times_times_real @ (times_times_real @ A @ B) @ C) = (times_times_real @ A @ (times_times_real @ B @ C)))))). % ab_semigroup_mult_class.mult_ac(1)
thf(fact_46_add__right__imp__eq, axiom,
    ((![B : nat, A : nat, C : nat]: (((plus_plus_nat @ B @ A) = (plus_plus_nat @ C @ A)) => (B = C))))). % add_right_imp_eq
thf(fact_47_add__right__imp__eq, axiom,
    ((![B : real, A : real, C : real]: (((plus_plus_real @ B @ A) = (plus_plus_real @ C @ A)) => (B = C))))). % add_right_imp_eq
thf(fact_48_add__right__imp__eq, axiom,
    ((![B : complex, A : complex, C : complex]: (((plus_plus_complex @ B @ A) = (plus_plus_complex @ C @ A)) => (B = C))))). % add_right_imp_eq
thf(fact_49_add__left__imp__eq, axiom,
    ((![A : nat, B : nat, C : nat]: (((plus_plus_nat @ A @ B) = (plus_plus_nat @ A @ C)) => (B = C))))). % add_left_imp_eq
thf(fact_50_add__left__imp__eq, axiom,
    ((![A : real, B : real, C : real]: (((plus_plus_real @ A @ B) = (plus_plus_real @ A @ C)) => (B = C))))). % add_left_imp_eq
thf(fact_51_add__left__imp__eq, axiom,
    ((![A : complex, B : complex, C : complex]: (((plus_plus_complex @ A @ B) = (plus_plus_complex @ A @ C)) => (B = C))))). % add_left_imp_eq
thf(fact_52_add_Oleft__commute, axiom,
    ((![B : nat, A : nat, C : nat]: ((plus_plus_nat @ B @ (plus_plus_nat @ A @ C)) = (plus_plus_nat @ A @ (plus_plus_nat @ B @ C)))))). % add.left_commute
thf(fact_53_add_Oleft__commute, axiom,
    ((![B : real, A : real, C : real]: ((plus_plus_real @ B @ (plus_plus_real @ A @ C)) = (plus_plus_real @ A @ (plus_plus_real @ B @ C)))))). % add.left_commute
thf(fact_54_add_Oleft__commute, axiom,
    ((![B : complex, A : complex, C : complex]: ((plus_plus_complex @ B @ (plus_plus_complex @ A @ C)) = (plus_plus_complex @ A @ (plus_plus_complex @ B @ C)))))). % add.left_commute
thf(fact_55_add_Ocommute, axiom,
    ((plus_plus_nat = (^[A2 : nat]: (^[B2 : nat]: (plus_plus_nat @ B2 @ A2)))))). % add.commute
thf(fact_56_add_Ocommute, axiom,
    ((plus_plus_real = (^[A2 : real]: (^[B2 : real]: (plus_plus_real @ B2 @ A2)))))). % add.commute
thf(fact_57_add_Ocommute, axiom,
    ((plus_plus_complex = (^[A2 : complex]: (^[B2 : complex]: (plus_plus_complex @ B2 @ A2)))))). % add.commute
thf(fact_58_add_Oright__cancel, axiom,
    ((![B : real, A : real, C : real]: (((plus_plus_real @ B @ A) = (plus_plus_real @ C @ A)) = (B = C))))). % add.right_cancel
thf(fact_59_add_Oright__cancel, axiom,
    ((![B : complex, A : complex, C : complex]: (((plus_plus_complex @ B @ A) = (plus_plus_complex @ C @ A)) = (B = C))))). % add.right_cancel
thf(fact_60_add_Oleft__cancel, axiom,
    ((![A : real, B : real, C : real]: (((plus_plus_real @ A @ B) = (plus_plus_real @ A @ C)) = (B = C))))). % add.left_cancel
thf(fact_61_add_Oleft__cancel, axiom,
    ((![A : complex, B : complex, C : complex]: (((plus_plus_complex @ A @ B) = (plus_plus_complex @ A @ C)) = (B = C))))). % add.left_cancel
thf(fact_62_add_Oassoc, axiom,
    ((![A : nat, B : nat, C : nat]: ((plus_plus_nat @ (plus_plus_nat @ A @ B) @ C) = (plus_plus_nat @ A @ (plus_plus_nat @ B @ C)))))). % add.assoc
thf(fact_63_add_Oassoc, axiom,
    ((![A : real, B : real, C : real]: ((plus_plus_real @ (plus_plus_real @ A @ B) @ C) = (plus_plus_real @ A @ (plus_plus_real @ B @ C)))))). % add.assoc
thf(fact_64_add_Oassoc, axiom,
    ((![A : complex, B : complex, C : complex]: ((plus_plus_complex @ (plus_plus_complex @ A @ B) @ C) = (plus_plus_complex @ A @ (plus_plus_complex @ B @ C)))))). % add.assoc
thf(fact_65_group__cancel_Oadd2, axiom,
    ((![B3 : nat, K : nat, B : nat, A : nat]: ((B3 = (plus_plus_nat @ K @ B)) => ((plus_plus_nat @ A @ B3) = (plus_plus_nat @ K @ (plus_plus_nat @ A @ B))))))). % group_cancel.add2
thf(fact_66_group__cancel_Oadd2, axiom,
    ((![B3 : real, K : real, B : real, A : real]: ((B3 = (plus_plus_real @ K @ B)) => ((plus_plus_real @ A @ B3) = (plus_plus_real @ K @ (plus_plus_real @ A @ B))))))). % group_cancel.add2
thf(fact_67_group__cancel_Oadd2, axiom,
    ((![B3 : complex, K : complex, B : complex, A : complex]: ((B3 = (plus_plus_complex @ K @ B)) => ((plus_plus_complex @ A @ B3) = (plus_plus_complex @ K @ (plus_plus_complex @ A @ B))))))). % group_cancel.add2
thf(fact_68_group__cancel_Oadd1, axiom,
    ((![A3 : nat, K : nat, A : nat, B : nat]: ((A3 = (plus_plus_nat @ K @ A)) => ((plus_plus_nat @ A3 @ B) = (plus_plus_nat @ K @ (plus_plus_nat @ A @ B))))))). % group_cancel.add1
thf(fact_69_group__cancel_Oadd1, axiom,
    ((![A3 : real, K : real, A : real, B : real]: ((A3 = (plus_plus_real @ K @ A)) => ((plus_plus_real @ A3 @ B) = (plus_plus_real @ K @ (plus_plus_real @ A @ B))))))). % group_cancel.add1
thf(fact_70_group__cancel_Oadd1, axiom,
    ((![A3 : complex, K : complex, A : complex, B : complex]: ((A3 = (plus_plus_complex @ K @ A)) => ((plus_plus_complex @ A3 @ B) = (plus_plus_complex @ K @ (plus_plus_complex @ A @ B))))))). % group_cancel.add1
thf(fact_71_add__mono__thms__linordered__semiring_I4_J, axiom,
    ((![I : nat, J : nat, K : nat, L : nat]: (((I = J) & (K = L)) => ((plus_plus_nat @ I @ K) = (plus_plus_nat @ J @ L)))))). % add_mono_thms_linordered_semiring(4)
thf(fact_72_add__mono__thms__linordered__semiring_I4_J, axiom,
    ((![I : real, J : real, K : real, L : real]: (((I = J) & (K = L)) => ((plus_plus_real @ I @ K) = (plus_plus_real @ J @ L)))))). % add_mono_thms_linordered_semiring(4)
thf(fact_73_ab__semigroup__add__class_Oadd__ac_I1_J, axiom,
    ((![A : nat, B : nat, C : nat]: ((plus_plus_nat @ (plus_plus_nat @ A @ B) @ C) = (plus_plus_nat @ A @ (plus_plus_nat @ B @ C)))))). % ab_semigroup_add_class.add_ac(1)
thf(fact_74_ab__semigroup__add__class_Oadd__ac_I1_J, axiom,
    ((![A : real, B : real, C : real]: ((plus_plus_real @ (plus_plus_real @ A @ B) @ C) = (plus_plus_real @ A @ (plus_plus_real @ B @ C)))))). % ab_semigroup_add_class.add_ac(1)
thf(fact_75_ab__semigroup__add__class_Oadd__ac_I1_J, axiom,
    ((![A : complex, B : complex, C : complex]: ((plus_plus_complex @ (plus_plus_complex @ A @ B) @ C) = (plus_plus_complex @ A @ (plus_plus_complex @ B @ C)))))). % ab_semigroup_add_class.add_ac(1)
thf(fact_76_Suc__mult__cancel1, axiom,
    ((![K : nat, M : nat, N2 : nat]: (((times_times_nat @ (suc @ K) @ M) = (times_times_nat @ (suc @ K) @ N2)) = (M = N2))))). % Suc_mult_cancel1
thf(fact_77_n__not__Suc__n, axiom,
    ((![N2 : nat]: (~ ((N2 = (suc @ N2))))))). % n_not_Suc_n
thf(fact_78_Suc__inject, axiom,
    ((![X3 : nat, Y : nat]: (((suc @ X3) = (suc @ Y)) => (X3 = Y))))). % Suc_inject
thf(fact_79_add__mult__distrib2, axiom,
    ((![K : nat, M : nat, N2 : nat]: ((times_times_nat @ K @ (plus_plus_nat @ M @ N2)) = (plus_plus_nat @ (times_times_nat @ K @ M) @ (times_times_nat @ K @ N2)))))). % add_mult_distrib2
thf(fact_80_add__mult__distrib, axiom,
    ((![M : nat, N2 : nat, K : nat]: ((times_times_nat @ (plus_plus_nat @ M @ N2) @ K) = (plus_plus_nat @ (times_times_nat @ M @ K) @ (times_times_nat @ N2 @ K)))))). % add_mult_distrib
thf(fact_81_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_82_mult__Suc, axiom,
    ((![M : nat, N2 : nat]: ((times_times_nat @ (suc @ M) @ N2) = (plus_plus_nat @ N2 @ (times_times_nat @ M @ N2)))))). % mult_Suc
thf(fact_83_add__Suc, axiom,
    ((![M : nat, N2 : nat]: ((plus_plus_nat @ (suc @ M) @ N2) = (suc @ (plus_plus_nat @ M @ N2)))))). % add_Suc
thf(fact_84_nat__arith_Osuc1, axiom,
    ((![A3 : nat, K : nat, A : nat]: ((A3 = (plus_plus_nat @ K @ A)) => ((suc @ A3) = (plus_plus_nat @ K @ (suc @ A))))))). % nat_arith.suc1
thf(fact_85_add__Suc__shift, axiom,
    ((![M : nat, N2 : nat]: ((plus_plus_nat @ (suc @ M) @ N2) = (plus_plus_nat @ M @ (suc @ N2)))))). % add_Suc_shift
thf(fact_86__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,
    ((?[F3 : nat > nat]: ((order_769474267at_nat @ F3) & (topolo144289241q_real @ (^[N : nat]: (comp_nat_real_nat @ (comp_c1631780367al_nat @ im @ s) @ f @ (F3 @ N)))))))). % \<open>\<exists>fa. strict_mono fa \<and> monoseq (\<lambda>n. (Im \<circ> s \<circ> f) (fa n))\<close>
thf(fact_87_comp__apply, axiom,
    ((comp_c1631780367al_nat = (^[F4 : complex > real]: (^[G3 : nat > complex]: (^[X : nat]: (F4 @ (G3 @ X)))))))). % comp_apply
thf(fact_88_comp__apply, axiom,
    ((comp_nat_real_nat = (^[F4 : nat > real]: (^[G3 : nat > nat]: (^[X : nat]: (F4 @ (G3 @ X)))))))). % comp_apply
thf(fact_89_complex__eqI, axiom,
    ((![X3 : complex, Y : complex]: (((re @ X3) = (re @ Y)) => (((im @ X3) = (im @ Y)) => (X3 = Y)))))). % complex_eqI
thf(fact_90_complex_Oexpand, axiom,
    ((![Complex : complex, Complex2 : complex]: ((((re @ Complex) = (re @ Complex2)) & ((im @ Complex) = (im @ Complex2))) => (Complex = Complex2))))). % complex.expand
thf(fact_91_complex__eq__iff, axiom,
    (((^[Y3 : complex]: (^[Z : complex]: (Y3 = Z))) = (^[X : complex]: (^[Y4 : complex]: ((((re @ X) = (re @ Y4))) & (((im @ X) = (im @ Y4))))))))). % complex_eq_iff
thf(fact_92_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_93_times__complex_Osimps_I2_J, axiom,
    ((![X3 : complex, Y : complex]: ((im @ (times_times_complex @ X3 @ Y)) = (plus_plus_real @ (times_times_real @ (re @ X3) @ (im @ Y)) @ (times_times_real @ (im @ X3) @ (re @ Y))))))). % times_complex.simps(2)
thf(fact_94_left__add__mult__distrib, axiom,
    ((![I : nat, U : nat, J : nat, K : nat]: ((plus_plus_nat @ (times_times_nat @ I @ U) @ (plus_plus_nat @ (times_times_nat @ J @ U) @ K)) = (plus_plus_nat @ (times_times_nat @ (plus_plus_nat @ I @ J) @ U) @ K))))). % left_add_mult_distrib
thf(fact_95_comp__def, axiom,
    ((comp_c1631780367al_nat = (^[F4 : complex > real]: (^[G3 : nat > complex]: (^[X : nat]: (F4 @ (G3 @ X)))))))). % comp_def
thf(fact_96_comp__def, axiom,
    ((comp_nat_real_nat = (^[F4 : nat > real]: (^[G3 : nat > nat]: (^[X : nat]: (F4 @ (G3 @ X)))))))). % comp_def
thf(fact_97_comp__assoc, axiom,
    ((![F : real > real, G : complex > real, H : nat > complex]: ((comp_c1631780367al_nat @ (comp_r422820971omplex @ F @ G) @ H) = (comp_real_real_nat @ F @ (comp_c1631780367al_nat @ G @ H)))))). % comp_assoc
thf(fact_98_comp__assoc, axiom,
    ((![F : complex > real, G : complex > complex, H : nat > complex]: ((comp_c1631780367al_nat @ (comp_c317287661omplex @ F @ G) @ H) = (comp_c1631780367al_nat @ F @ (comp_c438056209ex_nat @ G @ H)))))). % comp_assoc
thf(fact_99_comp__assoc, axiom,
    ((![F : nat > real, G : complex > nat, H : nat > complex]: ((comp_c1631780367al_nat @ (comp_n1816297743omplex @ F @ G) @ H) = (comp_nat_real_nat @ F @ (comp_complex_nat_nat @ G @ H)))))). % comp_assoc
thf(fact_100_comp__assoc, axiom,
    ((![F : real > real, G : nat > real, H : nat > nat]: ((comp_nat_real_nat @ (comp_real_real_nat @ F @ G) @ H) = (comp_real_real_nat @ F @ (comp_nat_real_nat @ G @ H)))))). % comp_assoc
thf(fact_101_comp__assoc, axiom,
    ((![F : complex > real, G : nat > complex, H : nat > nat]: ((comp_nat_real_nat @ (comp_c1631780367al_nat @ F @ G) @ H) = (comp_c1631780367al_nat @ F @ (comp_nat_complex_nat @ G @ H)))))). % comp_assoc
thf(fact_102_comp__assoc, axiom,
    ((![F : nat > real, G : nat > nat, H : nat > nat]: ((comp_nat_real_nat @ (comp_nat_real_nat @ F @ G) @ H) = (comp_nat_real_nat @ F @ (comp_nat_nat_nat @ G @ H)))))). % comp_assoc
thf(fact_103_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_104_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_105_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_106_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_107_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_108_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_109_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_110_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_111_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_112_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_113_plus__complex_Osimps_I1_J, axiom,
    ((![X3 : complex, Y : complex]: ((re @ (plus_plus_complex @ X3 @ Y)) = (plus_plus_real @ (re @ X3) @ (re @ Y)))))). % plus_complex.simps(1)
thf(fact_114_plus__complex_Osimps_I2_J, axiom,
    ((![X3 : complex, Y : complex]: ((im @ (plus_plus_complex @ X3 @ Y)) = (plus_plus_real @ (im @ X3) @ (im @ Y)))))). % plus_complex.simps(2)
thf(fact_115_ring__class_Oring__distribs_I2_J, axiom,
    ((![A : complex, B : complex, C : complex]: ((times_times_complex @ (plus_plus_complex @ A @ B) @ C) = (plus_plus_complex @ (times_times_complex @ A @ C) @ (times_times_complex @ B @ C)))))). % ring_class.ring_distribs(2)
thf(fact_116_ring__class_Oring__distribs_I2_J, axiom,
    ((![A : real, B : real, C : real]: ((times_times_real @ (plus_plus_real @ A @ B) @ C) = (plus_plus_real @ (times_times_real @ A @ C) @ (times_times_real @ B @ C)))))). % ring_class.ring_distribs(2)
thf(fact_117_ring__class_Oring__distribs_I1_J, axiom,
    ((![A : complex, B : complex, C : complex]: ((times_times_complex @ A @ (plus_plus_complex @ B @ C)) = (plus_plus_complex @ (times_times_complex @ A @ B) @ (times_times_complex @ A @ C)))))). % ring_class.ring_distribs(1)
thf(fact_118_ring__class_Oring__distribs_I1_J, axiom,
    ((![A : real, B : real, C : real]: ((times_times_real @ A @ (plus_plus_real @ B @ C)) = (plus_plus_real @ (times_times_real @ A @ B) @ (times_times_real @ A @ C)))))). % ring_class.ring_distribs(1)
thf(fact_119_comm__semiring__class_Odistrib, axiom,
    ((![A : nat, B : nat, C : nat]: ((times_times_nat @ (plus_plus_nat @ A @ B) @ C) = (plus_plus_nat @ (times_times_nat @ A @ C) @ (times_times_nat @ B @ C)))))). % comm_semiring_class.distrib
thf(fact_120_comm__semiring__class_Odistrib, axiom,
    ((![A : complex, B : complex, C : complex]: ((times_times_complex @ (plus_plus_complex @ A @ B) @ C) = (plus_plus_complex @ (times_times_complex @ A @ C) @ (times_times_complex @ B @ C)))))). % comm_semiring_class.distrib
thf(fact_121_comm__semiring__class_Odistrib, axiom,
    ((![A : real, B : real, C : real]: ((times_times_real @ (plus_plus_real @ A @ B) @ C) = (plus_plus_real @ (times_times_real @ A @ C) @ (times_times_real @ B @ C)))))). % comm_semiring_class.distrib
thf(fact_122_distrib__left, axiom,
    ((![A : nat, B : nat, C : nat]: ((times_times_nat @ A @ (plus_plus_nat @ B @ C)) = (plus_plus_nat @ (times_times_nat @ A @ B) @ (times_times_nat @ A @ C)))))). % distrib_left
thf(fact_123_distrib__left, axiom,
    ((![A : complex, B : complex, C : complex]: ((times_times_complex @ A @ (plus_plus_complex @ B @ C)) = (plus_plus_complex @ (times_times_complex @ A @ B) @ (times_times_complex @ A @ C)))))). % distrib_left
thf(fact_124_distrib__left, axiom,
    ((![A : real, B : real, C : real]: ((times_times_real @ A @ (plus_plus_real @ B @ C)) = (plus_plus_real @ (times_times_real @ A @ B) @ (times_times_real @ A @ C)))))). % distrib_left
thf(fact_125_distrib__right, axiom,
    ((![A : nat, B : nat, C : nat]: ((times_times_nat @ (plus_plus_nat @ A @ B) @ C) = (plus_plus_nat @ (times_times_nat @ A @ C) @ (times_times_nat @ B @ C)))))). % distrib_right
thf(fact_126_distrib__right, axiom,
    ((![A : complex, B : complex, C : complex]: ((times_times_complex @ (plus_plus_complex @ A @ B) @ C) = (plus_plus_complex @ (times_times_complex @ A @ C) @ (times_times_complex @ B @ C)))))). % distrib_right
thf(fact_127_distrib__right, axiom,
    ((![A : real, B : real, C : real]: ((times_times_real @ (plus_plus_real @ A @ B) @ C) = (plus_plus_real @ (times_times_real @ A @ C) @ (times_times_real @ B @ C)))))). % distrib_right
thf(fact_128_combine__common__factor, axiom,
    ((![A : nat, E : nat, B : nat, C : nat]: ((plus_plus_nat @ (times_times_nat @ A @ E) @ (plus_plus_nat @ (times_times_nat @ B @ E) @ C)) = (plus_plus_nat @ (times_times_nat @ (plus_plus_nat @ A @ B) @ E) @ C))))). % combine_common_factor
thf(fact_129_combine__common__factor, axiom,
    ((![A : complex, E : complex, B : complex, C : complex]: ((plus_plus_complex @ (times_times_complex @ A @ E) @ (plus_plus_complex @ (times_times_complex @ B @ E) @ C)) = (plus_plus_complex @ (times_times_complex @ (plus_plus_complex @ A @ B) @ E) @ C))))). % combine_common_factor
thf(fact_130_combine__common__factor, axiom,
    ((![A : real, E : real, B : real, C : real]: ((plus_plus_real @ (times_times_real @ A @ E) @ (plus_plus_real @ (times_times_real @ B @ E) @ C)) = (plus_plus_real @ (times_times_real @ (plus_plus_real @ A @ B) @ E) @ C))))). % combine_common_factor
thf(fact_131_crossproduct__noteq, axiom,
    ((![A : nat, B : nat, C : nat, D : nat]: ((((~ ((A = B)))) & ((~ ((C = D))))) = (~ (((plus_plus_nat @ (times_times_nat @ A @ C) @ (times_times_nat @ B @ D)) = (plus_plus_nat @ (times_times_nat @ A @ D) @ (times_times_nat @ B @ C))))))))). % crossproduct_noteq
thf(fact_132_crossproduct__noteq, axiom,
    ((![A : complex, B : complex, C : complex, D : complex]: ((((~ ((A = B)))) & ((~ ((C = D))))) = (~ (((plus_plus_complex @ (times_times_complex @ A @ C) @ (times_times_complex @ B @ D)) = (plus_plus_complex @ (times_times_complex @ A @ D) @ (times_times_complex @ B @ C))))))))). % crossproduct_noteq
thf(fact_133_crossproduct__noteq, axiom,
    ((![A : real, B : real, C : real, D : real]: ((((~ ((A = B)))) & ((~ ((C = D))))) = (~ (((plus_plus_real @ (times_times_real @ A @ C) @ (times_times_real @ B @ D)) = (plus_plus_real @ (times_times_real @ A @ D) @ (times_times_real @ B @ C))))))))). % crossproduct_noteq
thf(fact_134_crossproduct__eq, axiom,
    ((![W : nat, Y : nat, X3 : nat, Z2 : nat]: (((plus_plus_nat @ (times_times_nat @ W @ Y) @ (times_times_nat @ X3 @ Z2)) = (plus_plus_nat @ (times_times_nat @ W @ Z2) @ (times_times_nat @ X3 @ Y))) = (((W = X3)) | ((Y = Z2))))))). % crossproduct_eq
thf(fact_135_crossproduct__eq, axiom,
    ((![W : complex, Y : complex, X3 : complex, Z2 : complex]: (((plus_plus_complex @ (times_times_complex @ W @ Y) @ (times_times_complex @ X3 @ Z2)) = (plus_plus_complex @ (times_times_complex @ W @ Z2) @ (times_times_complex @ X3 @ Y))) = (((W = X3)) | ((Y = Z2))))))). % crossproduct_eq
thf(fact_136_crossproduct__eq, axiom,
    ((![W : real, Y : real, X3 : real, Z2 : real]: (((plus_plus_real @ (times_times_real @ W @ Y) @ (times_times_real @ X3 @ Z2)) = (plus_plus_real @ (times_times_real @ W @ Z2) @ (times_times_real @ X3 @ Y))) = (((W = X3)) | ((Y = Z2))))))). % crossproduct_eq
thf(fact_137_K__record__comp, axiom,
    ((![C : real, F : nat > complex]: ((comp_c1631780367al_nat @ (^[X : complex]: C) @ F) = (^[X : nat]: C))))). % K_record_comp
thf(fact_138_K__record__comp, axiom,
    ((![C : real, F : nat > nat]: ((comp_nat_real_nat @ (^[X : nat]: C) @ F) = (^[X : nat]: C))))). % K_record_comp
thf(fact_139_fun_Omap__comp, axiom,
    ((![G : real > real, F : complex > real, V : nat > complex]: ((comp_real_real_nat @ G @ (comp_c1631780367al_nat @ F @ V)) = (comp_c1631780367al_nat @ (comp_r422820971omplex @ G @ F) @ V))))). % fun.map_comp
thf(fact_140_fun_Omap__comp, axiom,
    ((![G : real > real, F : nat > real, V : nat > nat]: ((comp_real_real_nat @ G @ (comp_nat_real_nat @ F @ V)) = (comp_nat_real_nat @ (comp_real_real_nat @ G @ F) @ V))))). % fun.map_comp
thf(fact_141_fun_Omap__comp, axiom,
    ((![G : complex > real, F : complex > complex, V : nat > complex]: ((comp_c1631780367al_nat @ G @ (comp_c438056209ex_nat @ F @ V)) = (comp_c1631780367al_nat @ (comp_c317287661omplex @ G @ F) @ V))))). % fun.map_comp
thf(fact_142_fun_Omap__comp, axiom,
    ((![G : complex > real, F : nat > complex, V : nat > nat]: ((comp_c1631780367al_nat @ G @ (comp_nat_complex_nat @ F @ V)) = (comp_nat_real_nat @ (comp_c1631780367al_nat @ G @ F) @ V))))). % fun.map_comp
thf(fact_143_fun_Omap__comp, axiom,
    ((![G : nat > real, F : complex > nat, V : nat > complex]: ((comp_nat_real_nat @ G @ (comp_complex_nat_nat @ F @ V)) = (comp_c1631780367al_nat @ (comp_n1816297743omplex @ G @ F) @ V))))). % fun.map_comp
thf(fact_144_fun_Omap__comp, axiom,
    ((![G : nat > real, F : nat > nat, V : nat > nat]: ((comp_nat_real_nat @ G @ (comp_nat_nat_nat @ F @ V)) = (comp_nat_real_nat @ (comp_nat_real_nat @ G @ F) @ V))))). % fun.map_comp
thf(fact_145_rcis__mult, axiom,
    ((![R1 : real, A : real, R22 : real, B : real]: ((times_times_complex @ (rcis @ R1 @ A) @ (rcis @ R22 @ B)) = (rcis @ (times_times_real @ R1 @ R22) @ (plus_plus_real @ A @ B)))))). % rcis_mult
thf(fact_146_Im__i__times, axiom,
    ((![Z2 : complex]: ((im @ (times_times_complex @ imaginary_unit @ Z2)) = (re @ Z2))))). % Im_i_times
thf(fact_147_mult__commute__abs, axiom,
    ((![C : nat]: ((^[X : nat]: (times_times_nat @ X @ C)) = (times_times_nat @ C))))). % mult_commute_abs
thf(fact_148_mult__commute__abs, axiom,
    ((![C : complex]: ((^[X : complex]: (times_times_complex @ X @ C)) = (times_times_complex @ C))))). % mult_commute_abs
thf(fact_149_mult__commute__abs, axiom,
    ((![C : real]: ((^[X : real]: (times_times_real @ X @ C)) = (times_times_real @ C))))). % mult_commute_abs
thf(fact_150_plus__complex_Ocode, axiom,
    ((plus_plus_complex = (^[X : complex]: (^[Y4 : complex]: (complex2 @ (plus_plus_real @ (re @ X) @ (re @ Y4)) @ (plus_plus_real @ (im @ X) @ (im @ Y4)))))))). % plus_complex.code
thf(fact_151_complex_Oinject, axiom,
    ((![X1 : real, X22 : real, Y1 : real, Y2 : real]: (((complex2 @ X1 @ X22) = (complex2 @ Y1 @ Y2)) = (((X1 = Y1)) & ((X22 = Y2))))))). % complex.inject
thf(fact_152_complex_Ocollapse, axiom,
    ((![Complex : complex]: ((complex2 @ (re @ Complex) @ (im @ Complex)) = Complex)))). % complex.collapse
thf(fact_153_complex__surj, axiom,
    ((![Z2 : complex]: ((complex2 @ (re @ Z2) @ (im @ Z2)) = Z2)))). % complex_surj
thf(fact_154_complex_Osel_I1_J, axiom,
    ((![X1 : real, X22 : real]: ((re @ (complex2 @ X1 @ X22)) = X1)))). % complex.sel(1)
thf(fact_155_complex_Osel_I2_J, axiom,
    ((![X1 : real, X22 : real]: ((im @ (complex2 @ X1 @ X22)) = X22)))). % complex.sel(2)
thf(fact_156_rcis__Ex, axiom,
    ((![Z2 : complex]: (?[R3 : real, A4 : real]: (Z2 = (rcis @ R3 @ A4)))))). % rcis_Ex
thf(fact_157_complex_Oexhaust, axiom,
    ((![Y : complex]: (~ ((![X12 : real, X23 : real]: (~ ((Y = (complex2 @ X12 @ X23)))))))))). % complex.exhaust
thf(fact_158_complex_Oexhaust__sel, axiom,
    ((![Complex : complex]: (Complex = (complex2 @ (re @ Complex) @ (im @ Complex)))))). % complex.exhaust_sel
thf(fact_159_complex__add, axiom,
    ((![A : real, B : real, C : real, D : real]: ((plus_plus_complex @ (complex2 @ A @ B) @ (complex2 @ C @ D)) = (complex2 @ (plus_plus_real @ A @ C) @ (plus_plus_real @ B @ D)))))). % complex_add
thf(fact_160_times__complex_Ocode, axiom,
    ((times_times_complex = (^[X : complex]: (^[Y4 : complex]: (complex2 @ (minus_minus_real @ (times_times_real @ (re @ X) @ (re @ Y4)) @ (times_times_real @ (im @ X) @ (im @ Y4))) @ (plus_plus_real @ (times_times_real @ (re @ X) @ (im @ Y4)) @ (times_times_real @ (im @ X) @ (re @ Y4))))))))). % times_complex.code
thf(fact_161_complex_Osplit__sel, axiom,
    ((![P : real > $o, F : real > real > real, Complex : complex]: ((P @ (case_complex_real @ F @ Complex)) = (((Complex = (complex2 @ (re @ Complex) @ (im @ Complex)))) => ((P @ (F @ (re @ Complex) @ (im @ Complex))))))))). % complex.split_sel
thf(fact_162_complex_Osplit__sel__asm, axiom,
    ((![P : real > $o, F : real > real > real, Complex : complex]: ((P @ (case_complex_real @ F @ Complex)) = (~ ((((Complex = (complex2 @ (re @ Complex) @ (im @ Complex)))) & ((~ ((P @ (F @ (re @ Complex) @ (im @ Complex))))))))))))). % complex.split_sel_asm
thf(fact_163_Re__i__times, axiom,
    ((![Z2 : complex]: ((re @ (times_times_complex @ imaginary_unit @ Z2)) = (uminus_uminus_real @ (im @ Z2)))))). % Re_i_times
thf(fact_164_neg__equal__iff__equal, axiom,
    ((![A : real, B : real]: (((uminus_uminus_real @ A) = (uminus_uminus_real @ B)) = (A = B))))). % neg_equal_iff_equal
thf(fact_165_add_Oinverse__inverse, axiom,
    ((![A : real]: ((uminus_uminus_real @ (uminus_uminus_real @ A)) = A)))). % add.inverse_inverse
thf(fact_166_add__diff__cancel__right_H, axiom,
    ((![A : nat, B : nat]: ((minus_minus_nat @ (plus_plus_nat @ A @ B) @ B) = A)))). % add_diff_cancel_right'
thf(fact_167_add__diff__cancel__right_H, axiom,
    ((![A : complex, B : complex]: ((minus_minus_complex @ (plus_plus_complex @ A @ B) @ B) = A)))). % add_diff_cancel_right'
thf(fact_168_add__diff__cancel__right_H, axiom,
    ((![A : real, B : real]: ((minus_minus_real @ (plus_plus_real @ A @ B) @ B) = A)))). % add_diff_cancel_right'
thf(fact_169_add__diff__cancel__right, axiom,
    ((![A : nat, C : nat, B : nat]: ((minus_minus_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ C)) = (minus_minus_nat @ A @ B))))). % add_diff_cancel_right
thf(fact_170_add__diff__cancel__right, axiom,
    ((![A : complex, C : complex, B : complex]: ((minus_minus_complex @ (plus_plus_complex @ A @ C) @ (plus_plus_complex @ B @ C)) = (minus_minus_complex @ A @ B))))). % add_diff_cancel_right
thf(fact_171_add__diff__cancel__right, axiom,
    ((![A : real, C : real, B : real]: ((minus_minus_real @ (plus_plus_real @ A @ C) @ (plus_plus_real @ B @ C)) = (minus_minus_real @ A @ B))))). % add_diff_cancel_right
thf(fact_172_add__diff__cancel__left_H, axiom,
    ((![A : nat, B : nat]: ((minus_minus_nat @ (plus_plus_nat @ A @ B) @ A) = B)))). % add_diff_cancel_left'
thf(fact_173_add__diff__cancel__left_H, axiom,
    ((![A : complex, B : complex]: ((minus_minus_complex @ (plus_plus_complex @ A @ B) @ A) = B)))). % add_diff_cancel_left'
thf(fact_174_add__diff__cancel__left_H, axiom,
    ((![A : real, B : real]: ((minus_minus_real @ (plus_plus_real @ A @ B) @ A) = B)))). % add_diff_cancel_left'
thf(fact_175_add__diff__cancel__left, axiom,
    ((![C : nat, A : nat, B : nat]: ((minus_minus_nat @ (plus_plus_nat @ C @ A) @ (plus_plus_nat @ C @ B)) = (minus_minus_nat @ A @ B))))). % add_diff_cancel_left
thf(fact_176_add__diff__cancel__left, axiom,
    ((![C : complex, A : complex, B : complex]: ((minus_minus_complex @ (plus_plus_complex @ C @ A) @ (plus_plus_complex @ C @ B)) = (minus_minus_complex @ A @ B))))). % add_diff_cancel_left
thf(fact_177_add__diff__cancel__left, axiom,
    ((![C : real, A : real, B : real]: ((minus_minus_real @ (plus_plus_real @ C @ A) @ (plus_plus_real @ C @ B)) = (minus_minus_real @ A @ B))))). % add_diff_cancel_left
thf(fact_178_diff__add__cancel, axiom,
    ((![A : complex, B : complex]: ((plus_plus_complex @ (minus_minus_complex @ A @ B) @ B) = A)))). % diff_add_cancel
thf(fact_179_diff__add__cancel, axiom,
    ((![A : real, B : real]: ((plus_plus_real @ (minus_minus_real @ A @ B) @ B) = A)))). % diff_add_cancel
thf(fact_180_add__diff__cancel, axiom,
    ((![A : complex, B : complex]: ((minus_minus_complex @ (plus_plus_complex @ A @ B) @ B) = A)))). % add_diff_cancel
thf(fact_181_add__diff__cancel, axiom,
    ((![A : real, B : real]: ((minus_minus_real @ (plus_plus_real @ A @ B) @ B) = A)))). % add_diff_cancel
thf(fact_182_mult__minus__right, axiom,
    ((![A : complex, B : complex]: ((times_times_complex @ A @ (uminus1204672759omplex @ B)) = (uminus1204672759omplex @ (times_times_complex @ A @ B)))))). % mult_minus_right
thf(fact_183_mult__minus__right, axiom,
    ((![A : real, B : real]: ((times_times_real @ A @ (uminus_uminus_real @ B)) = (uminus_uminus_real @ (times_times_real @ A @ B)))))). % mult_minus_right
thf(fact_184_minus__mult__minus, axiom,
    ((![A : complex, B : complex]: ((times_times_complex @ (uminus1204672759omplex @ A) @ (uminus1204672759omplex @ B)) = (times_times_complex @ A @ B))))). % minus_mult_minus
thf(fact_185_minus__mult__minus, axiom,
    ((![A : real, B : real]: ((times_times_real @ (uminus_uminus_real @ A) @ (uminus_uminus_real @ B)) = (times_times_real @ A @ B))))). % minus_mult_minus
thf(fact_186_mult__minus__left, axiom,
    ((![A : complex, B : complex]: ((times_times_complex @ (uminus1204672759omplex @ A) @ B) = (uminus1204672759omplex @ (times_times_complex @ A @ B)))))). % mult_minus_left
thf(fact_187_mult__minus__left, axiom,
    ((![A : real, B : real]: ((times_times_real @ (uminus_uminus_real @ A) @ B) = (uminus_uminus_real @ (times_times_real @ A @ B)))))). % mult_minus_left
thf(fact_188_minus__add__distrib, axiom,
    ((![A : complex, B : complex]: ((uminus1204672759omplex @ (plus_plus_complex @ A @ B)) = (plus_plus_complex @ (uminus1204672759omplex @ A) @ (uminus1204672759omplex @ B)))))). % minus_add_distrib
thf(fact_189_minus__add__distrib, axiom,
    ((![A : real, B : real]: ((uminus_uminus_real @ (plus_plus_real @ A @ B)) = (plus_plus_real @ (uminus_uminus_real @ A) @ (uminus_uminus_real @ B)))))). % minus_add_distrib
thf(fact_190_minus__add__cancel, axiom,
    ((![A : complex, B : complex]: ((plus_plus_complex @ (uminus1204672759omplex @ A) @ (plus_plus_complex @ A @ B)) = B)))). % minus_add_cancel
thf(fact_191_minus__add__cancel, axiom,
    ((![A : real, B : real]: ((plus_plus_real @ (uminus_uminus_real @ A) @ (plus_plus_real @ A @ B)) = B)))). % minus_add_cancel
thf(fact_192_add__minus__cancel, axiom,
    ((![A : complex, B : complex]: ((plus_plus_complex @ A @ (plus_plus_complex @ (uminus1204672759omplex @ A) @ B)) = B)))). % add_minus_cancel
thf(fact_193_add__minus__cancel, axiom,
    ((![A : real, B : real]: ((plus_plus_real @ A @ (plus_plus_real @ (uminus_uminus_real @ A) @ B)) = B)))). % add_minus_cancel
thf(fact_194_minus__diff__eq, axiom,
    ((![A : real, B : real]: ((uminus_uminus_real @ (minus_minus_real @ A @ B)) = (minus_minus_real @ B @ A))))). % minus_diff_eq
thf(fact_195_diff__minus__eq__add, axiom,
    ((![A : complex, B : complex]: ((minus_minus_complex @ A @ (uminus1204672759omplex @ B)) = (plus_plus_complex @ A @ B))))). % diff_minus_eq_add
thf(fact_196_diff__minus__eq__add, axiom,
    ((![A : real, B : real]: ((minus_minus_real @ A @ (uminus_uminus_real @ B)) = (plus_plus_real @ A @ B))))). % diff_minus_eq_add
thf(fact_197_uminus__add__conv__diff, axiom,
    ((![A : complex, B : complex]: ((plus_plus_complex @ (uminus1204672759omplex @ A) @ B) = (minus_minus_complex @ B @ A))))). % uminus_add_conv_diff
thf(fact_198_uminus__add__conv__diff, axiom,
    ((![A : real, B : real]: ((plus_plus_real @ (uminus_uminus_real @ A) @ B) = (minus_minus_real @ B @ A))))). % uminus_add_conv_diff
thf(fact_199_group__cancel_Osub2, axiom,
    ((![B3 : complex, K : complex, B : complex, A : complex]: ((B3 = (plus_plus_complex @ K @ B)) => ((minus_minus_complex @ A @ B3) = (plus_plus_complex @ (uminus1204672759omplex @ K) @ (minus_minus_complex @ A @ B))))))). % group_cancel.sub2
thf(fact_200_group__cancel_Osub2, axiom,
    ((![B3 : real, K : real, B : real, A : real]: ((B3 = (plus_plus_real @ K @ B)) => ((minus_minus_real @ A @ B3) = (plus_plus_real @ (uminus_uminus_real @ K) @ (minus_minus_real @ A @ B))))))). % group_cancel.sub2
thf(fact_201_diff__conv__add__uminus, axiom,
    ((minus_minus_complex = (^[A2 : complex]: (^[B2 : complex]: (plus_plus_complex @ A2 @ (uminus1204672759omplex @ B2))))))). % diff_conv_add_uminus
thf(fact_202_diff__conv__add__uminus, axiom,
    ((minus_minus_real = (^[A2 : real]: (^[B2 : real]: (plus_plus_real @ A2 @ (uminus_uminus_real @ B2))))))). % diff_conv_add_uminus
thf(fact_203_ab__group__add__class_Oab__diff__conv__add__uminus, axiom,
    ((minus_minus_complex = (^[A2 : complex]: (^[B2 : complex]: (plus_plus_complex @ A2 @ (uminus1204672759omplex @ B2))))))). % ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_204_ab__group__add__class_Oab__diff__conv__add__uminus, axiom,
    ((minus_minus_real = (^[A2 : real]: (^[B2 : real]: (plus_plus_real @ A2 @ (uminus_uminus_real @ B2))))))). % ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_205_right__diff__distrib_H, axiom,
    ((![A : nat, B : nat, C : nat]: ((times_times_nat @ A @ (minus_minus_nat @ B @ C)) = (minus_minus_nat @ (times_times_nat @ A @ B) @ (times_times_nat @ A @ C)))))). % right_diff_distrib'
thf(fact_206_right__diff__distrib_H, axiom,
    ((![A : complex, B : complex, C : complex]: ((times_times_complex @ A @ (minus_minus_complex @ B @ C)) = (minus_minus_complex @ (times_times_complex @ A @ B) @ (times_times_complex @ A @ C)))))). % right_diff_distrib'
thf(fact_207_right__diff__distrib_H, axiom,
    ((![A : real, B : real, C : real]: ((times_times_real @ A @ (minus_minus_real @ B @ C)) = (minus_minus_real @ (times_times_real @ A @ B) @ (times_times_real @ A @ C)))))). % right_diff_distrib'
thf(fact_208_left__diff__distrib_H, axiom,
    ((![B : nat, C : nat, A : nat]: ((times_times_nat @ (minus_minus_nat @ B @ C) @ A) = (minus_minus_nat @ (times_times_nat @ B @ A) @ (times_times_nat @ C @ A)))))). % left_diff_distrib'
thf(fact_209_left__diff__distrib_H, axiom,
    ((![B : complex, C : complex, A : complex]: ((times_times_complex @ (minus_minus_complex @ B @ C) @ A) = (minus_minus_complex @ (times_times_complex @ B @ A) @ (times_times_complex @ C @ A)))))). % left_diff_distrib'
thf(fact_210_left__diff__distrib_H, axiom,
    ((![B : real, C : real, A : real]: ((times_times_real @ (minus_minus_real @ B @ C) @ A) = (minus_minus_real @ (times_times_real @ B @ A) @ (times_times_real @ C @ A)))))). % left_diff_distrib'
thf(fact_211_right__diff__distrib, axiom,
    ((![A : complex, B : complex, C : complex]: ((times_times_complex @ A @ (minus_minus_complex @ B @ C)) = (minus_minus_complex @ (times_times_complex @ A @ B) @ (times_times_complex @ A @ C)))))). % right_diff_distrib
thf(fact_212_right__diff__distrib, axiom,
    ((![A : real, B : real, C : real]: ((times_times_real @ A @ (minus_minus_real @ B @ C)) = (minus_minus_real @ (times_times_real @ A @ B) @ (times_times_real @ A @ C)))))). % right_diff_distrib
thf(fact_213_left__diff__distrib, axiom,
    ((![A : complex, B : complex, C : complex]: ((times_times_complex @ (minus_minus_complex @ A @ B) @ C) = (minus_minus_complex @ (times_times_complex @ A @ C) @ (times_times_complex @ B @ C)))))). % left_diff_distrib
thf(fact_214_left__diff__distrib, axiom,
    ((![A : real, B : real, C : real]: ((times_times_real @ (minus_minus_real @ A @ B) @ C) = (minus_minus_real @ (times_times_real @ A @ C) @ (times_times_real @ B @ C)))))). % left_diff_distrib
thf(fact_215_minus__mult__commute, axiom,
    ((![A : complex, B : complex]: ((times_times_complex @ (uminus1204672759omplex @ A) @ B) = (times_times_complex @ A @ (uminus1204672759omplex @ B)))))). % minus_mult_commute
thf(fact_216_minus__mult__commute, axiom,
    ((![A : real, B : real]: ((times_times_real @ (uminus_uminus_real @ A) @ B) = (times_times_real @ A @ (uminus_uminus_real @ B)))))). % minus_mult_commute
thf(fact_217_square__eq__iff, axiom,
    ((![A : complex, B : complex]: (((times_times_complex @ A @ A) = (times_times_complex @ B @ B)) = (((A = B)) | ((A = (uminus1204672759omplex @ B)))))))). % square_eq_iff
thf(fact_218_square__eq__iff, axiom,
    ((![A : real, B : real]: (((times_times_real @ A @ A) = (times_times_real @ B @ B)) = (((A = B)) | ((A = (uminus_uminus_real @ B)))))))). % square_eq_iff
thf(fact_219_group__cancel_Osub1, axiom,
    ((![A3 : complex, K : complex, A : complex, B : complex]: ((A3 = (plus_plus_complex @ K @ A)) => ((minus_minus_complex @ A3 @ B) = (plus_plus_complex @ K @ (minus_minus_complex @ A @ B))))))). % group_cancel.sub1
thf(fact_220_group__cancel_Osub1, axiom,
    ((![A3 : real, K : real, A : real, B : real]: ((A3 = (plus_plus_real @ K @ A)) => ((minus_minus_real @ A3 @ B) = (plus_plus_real @ K @ (minus_minus_real @ A @ B))))))). % group_cancel.sub1
thf(fact_221_diff__eq__eq, axiom,
    ((![A : complex, B : complex, C : complex]: (((minus_minus_complex @ A @ B) = C) = (A = (plus_plus_complex @ C @ B)))))). % diff_eq_eq
thf(fact_222_diff__eq__eq, axiom,
    ((![A : real, B : real, C : real]: (((minus_minus_real @ A @ B) = C) = (A = (plus_plus_real @ C @ B)))))). % diff_eq_eq
thf(fact_223_eq__diff__eq, axiom,
    ((![A : complex, C : complex, B : complex]: ((A = (minus_minus_complex @ C @ B)) = ((plus_plus_complex @ A @ B) = C))))). % eq_diff_eq
thf(fact_224_eq__diff__eq, axiom,
    ((![A : real, C : real, B : real]: ((A = (minus_minus_real @ C @ B)) = ((plus_plus_real @ A @ B) = C))))). % eq_diff_eq
thf(fact_225_add__diff__eq, axiom,
    ((![A : complex, B : complex, C : complex]: ((plus_plus_complex @ A @ (minus_minus_complex @ B @ C)) = (minus_minus_complex @ (plus_plus_complex @ A @ B) @ C))))). % add_diff_eq
thf(fact_226_add__diff__eq, axiom,
    ((![A : real, B : real, C : real]: ((plus_plus_real @ A @ (minus_minus_real @ B @ C)) = (minus_minus_real @ (plus_plus_real @ A @ B) @ C))))). % add_diff_eq
thf(fact_227_diff__diff__eq2, axiom,
    ((![A : complex, B : complex, C : complex]: ((minus_minus_complex @ A @ (minus_minus_complex @ B @ C)) = (minus_minus_complex @ (plus_plus_complex @ A @ C) @ B))))). % diff_diff_eq2
thf(fact_228_diff__diff__eq2, axiom,
    ((![A : real, B : real, C : real]: ((minus_minus_real @ A @ (minus_minus_real @ B @ C)) = (minus_minus_real @ (plus_plus_real @ A @ C) @ B))))). % diff_diff_eq2
thf(fact_229_diff__add__eq, axiom,
    ((![A : complex, B : complex, C : complex]: ((plus_plus_complex @ (minus_minus_complex @ A @ B) @ C) = (minus_minus_complex @ (plus_plus_complex @ A @ C) @ B))))). % diff_add_eq
thf(fact_230_diff__add__eq, axiom,
    ((![A : real, B : real, C : real]: ((plus_plus_real @ (minus_minus_real @ A @ B) @ C) = (minus_minus_real @ (plus_plus_real @ A @ C) @ B))))). % diff_add_eq
thf(fact_231_diff__add__eq__diff__diff__swap, axiom,
    ((![A : complex, B : complex, C : complex]: ((minus_minus_complex @ A @ (plus_plus_complex @ B @ C)) = (minus_minus_complex @ (minus_minus_complex @ A @ C) @ B))))). % diff_add_eq_diff_diff_swap
thf(fact_232_diff__add__eq__diff__diff__swap, axiom,
    ((![A : real, B : real, C : real]: ((minus_minus_real @ A @ (plus_plus_real @ B @ C)) = (minus_minus_real @ (minus_minus_real @ A @ C) @ B))))). % diff_add_eq_diff_diff_swap
thf(fact_233_diff__diff__add, axiom,
    ((![A : nat, B : nat, C : nat]: ((minus_minus_nat @ (minus_minus_nat @ A @ B) @ C) = (minus_minus_nat @ A @ (plus_plus_nat @ B @ C)))))). % diff_diff_add
thf(fact_234_diff__diff__add, axiom,
    ((![A : complex, B : complex, C : complex]: ((minus_minus_complex @ (minus_minus_complex @ A @ B) @ C) = (minus_minus_complex @ A @ (plus_plus_complex @ B @ C)))))). % diff_diff_add
thf(fact_235_diff__diff__add, axiom,
    ((![A : real, B : real, C : real]: ((minus_minus_real @ (minus_minus_real @ A @ B) @ C) = (minus_minus_real @ A @ (plus_plus_real @ B @ C)))))). % diff_diff_add
thf(fact_236_add__implies__diff, axiom,
    ((![C : nat, B : nat, A : nat]: (((plus_plus_nat @ C @ B) = A) => (C = (minus_minus_nat @ A @ B)))))). % add_implies_diff
thf(fact_237_add__implies__diff, axiom,
    ((![C : complex, B : complex, A : complex]: (((plus_plus_complex @ C @ B) = A) => (C = (minus_minus_complex @ A @ B)))))). % add_implies_diff
thf(fact_238_add__implies__diff, axiom,
    ((![C : real, B : real, A : real]: (((plus_plus_real @ C @ B) = A) => (C = (minus_minus_real @ A @ B)))))). % add_implies_diff
thf(fact_239_group__cancel_Oneg1, axiom,
    ((![A3 : complex, K : complex, A : complex]: ((A3 = (plus_plus_complex @ K @ A)) => ((uminus1204672759omplex @ A3) = (plus_plus_complex @ (uminus1204672759omplex @ K) @ (uminus1204672759omplex @ A))))))). % group_cancel.neg1
thf(fact_240_group__cancel_Oneg1, axiom,
    ((![A3 : real, K : real, A : real]: ((A3 = (plus_plus_real @ K @ A)) => ((uminus_uminus_real @ A3) = (plus_plus_real @ (uminus_uminus_real @ K) @ (uminus_uminus_real @ A))))))). % group_cancel.neg1
thf(fact_241_add_Oinverse__distrib__swap, axiom,
    ((![A : complex, B : complex]: ((uminus1204672759omplex @ (plus_plus_complex @ A @ B)) = (plus_plus_complex @ (uminus1204672759omplex @ B) @ (uminus1204672759omplex @ A)))))). % add.inverse_distrib_swap
thf(fact_242_add_Oinverse__distrib__swap, axiom,
    ((![A : real, B : real]: ((uminus_uminus_real @ (plus_plus_real @ A @ B)) = (plus_plus_real @ (uminus_uminus_real @ B) @ (uminus_uminus_real @ A)))))). % add.inverse_distrib_swap
thf(fact_243_Re__def, axiom,
    ((re = (case_complex_real @ (^[X13 : real]: (^[X24 : real]: X13)))))). % Re_def
thf(fact_244_Im__def, axiom,
    ((im = (case_complex_real @ (^[X13 : real]: (^[X24 : real]: X24)))))). % Im_def
thf(fact_245_i__mult__Complex, axiom,
    ((![A : real, B : real]: ((times_times_complex @ imaginary_unit @ (complex2 @ A @ B)) = (complex2 @ (uminus_uminus_real @ B) @ A))))). % i_mult_Complex
thf(fact_246_Complex__mult__i, axiom,
    ((![A : real, B : real]: ((times_times_complex @ (complex2 @ A @ B) @ imaginary_unit) = (complex2 @ (uminus_uminus_real @ B) @ A))))). % Complex_mult_i
thf(fact_247_times__complex_Osimps_I1_J, axiom,
    ((![X3 : complex, Y : complex]: ((re @ (times_times_complex @ X3 @ Y)) = (minus_minus_real @ (times_times_real @ (re @ X3) @ (re @ Y)) @ (times_times_real @ (im @ X3) @ (im @ Y))))))). % times_complex.simps(1)

% Conjectures (1)
thf(conj_0, conjecture,
    ((bfun_nat_real @ (^[N : nat]: (re @ (s @ (f @ N)))) @ at_top_nat))).
