% 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_175__5369312_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:27:40.729

% Could-be-implicit typings (3)
thf(ty_n_t__Complex__Ocomplex, type,
    complex : $tType).
thf(ty_n_t__Real__Oreal, type,
    real : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).

% Explicit typings (24)
thf(sy_c_Groups_Oone__class_Oone_001t__Complex__Ocomplex, type,
    one_one_complex : complex).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat, type,
    one_one_nat : nat).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal, type,
    one_one_real : real).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Complex__Ocomplex, type,
    plus_plus_complex : complex > complex > complex).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat, type,
    plus_plus_nat : nat > nat > nat).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal, type,
    plus_plus_real : real > real > real).
thf(sy_c_Groups_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_Ozero__class_Ozero_001t__Complex__Ocomplex, type,
    zero_zero_complex : complex).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat, type,
    zero_zero_nat : nat).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal, type,
    zero_zero_real : real).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat, type,
    ord_less_nat : nat > nat > $o).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal, type,
    ord_less_real : real > real > $o).
thf(sy_c_Power_Opower__class_Opower_001t__Complex__Ocomplex, type,
    power_power_complex : complex > nat > complex).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat, type,
    power_power_nat : nat > nat > nat).
thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal, type,
    power_power_real : real > nat > real).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Complex__Ocomplex, type,
    real_V638595069omplex : complex > real).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Real__Oreal, type,
    real_V646646907m_real : real > real).
thf(sy_v_b, type,
    b : complex).
thf(sy_v_m____, type,
    m : nat).
thf(sy_v_n, type,
    n : nat).
thf(sy_v_na____, type,
    na : nat).
thf(sy_v_thesis____, type,
    thesis : $o).

% Relevant facts (244)
thf(fact_0__092_060open_062m_A_092_060noteq_062_A0_092_060close_062, axiom,
    ((~ ((m = zero_zero_nat))))). % \<open>m \<noteq> 0\<close>
thf(fact_1__092_060open_062m_A_060_An_092_060close_062, axiom,
    ((ord_less_nat @ m @ na))). % \<open>m < n\<close>
thf(fact_2__092_060open_062_092_060lbrakk_062m_A_060_An_059_Am_A_092_060noteq_062_A0_092_060rbrakk_062_A_092_060Longrightarrow_062_A_092_060exists_062z_O_Acmod_A_I1_A_L_Ab_A_K_Az_A_094_Am_J_A_060_A1_092_060close_062, axiom,
    (((ord_less_nat @ m @ na) => ((~ ((m = zero_zero_nat))) => (?[Z : complex]: (ord_less_real @ (real_V638595069omplex @ (plus_plus_complex @ one_one_complex @ (times_times_complex @ b @ (power_power_complex @ Z @ m)))) @ one_one_real)))))). % \<open>\<lbrakk>m < n; m \<noteq> 0\<rbrakk> \<Longrightarrow> \<exists>z. cmod (1 + b * z ^ m) < 1\<close>
thf(fact_3_b, axiom,
    ((~ ((b = zero_zero_complex))))). % b
thf(fact_4_IH, axiom,
    ((![M : nat]: ((ord_less_nat @ M @ na) => ((~ ((M = zero_zero_nat))) => (?[Z : complex]: (ord_less_real @ (real_V638595069omplex @ (plus_plus_complex @ one_one_complex @ (times_times_complex @ b @ (power_power_complex @ Z @ M)))) @ one_one_real))))))). % IH
thf(fact_5_real__sup__exists, axiom,
    ((![P : real > $o]: ((?[X_1 : real]: (P @ X_1)) => ((?[Z2 : real]: (![X : real]: ((P @ X) => (ord_less_real @ X @ Z2)))) => (?[S : real]: (![Y : real]: ((?[X2 : real]: (((P @ X2)) & ((ord_less_real @ Y @ X2)))) = (ord_less_real @ Y @ S))))))))). % real_sup_exists
thf(fact_6_norm__one, axiom,
    (((real_V646646907m_real @ one_one_real) = one_one_real))). % norm_one
thf(fact_7_norm__one, axiom,
    (((real_V638595069omplex @ one_one_complex) = one_one_real))). % norm_one
thf(fact_8_power__inject__exp, axiom,
    ((![A : real, M2 : nat, N : nat]: ((ord_less_real @ one_one_real @ A) => (((power_power_real @ A @ M2) = (power_power_real @ A @ N)) = (M2 = N)))))). % power_inject_exp
thf(fact_9_power__inject__exp, axiom,
    ((![A : nat, M2 : nat, N : nat]: ((ord_less_nat @ one_one_nat @ A) => (((power_power_nat @ A @ M2) = (power_power_nat @ A @ N)) = (M2 = N)))))). % power_inject_exp
thf(fact_10_power__strict__increasing__iff, axiom,
    ((![B : real, X3 : nat, Y2 : nat]: ((ord_less_real @ one_one_real @ B) => ((ord_less_real @ (power_power_real @ B @ X3) @ (power_power_real @ B @ Y2)) = (ord_less_nat @ X3 @ Y2)))))). % power_strict_increasing_iff
thf(fact_11_power__strict__increasing__iff, axiom,
    ((![B : nat, X3 : nat, Y2 : nat]: ((ord_less_nat @ one_one_nat @ B) => ((ord_less_nat @ (power_power_nat @ B @ X3) @ (power_power_nat @ B @ Y2)) = (ord_less_nat @ X3 @ Y2)))))). % power_strict_increasing_iff
thf(fact_12_power__one, axiom,
    ((![N : nat]: ((power_power_real @ one_one_real @ N) = one_one_real)))). % power_one
thf(fact_13_power__one, axiom,
    ((![N : nat]: ((power_power_complex @ one_one_complex @ N) = one_one_complex)))). % power_one
thf(fact_14_power__one, axiom,
    ((![N : nat]: ((power_power_nat @ one_one_nat @ N) = one_one_nat)))). % power_one
thf(fact_15_mult_Oleft__neutral, axiom,
    ((![A : real]: ((times_times_real @ one_one_real @ A) = A)))). % mult.left_neutral
thf(fact_16_mult_Oleft__neutral, axiom,
    ((![A : complex]: ((times_times_complex @ one_one_complex @ A) = A)))). % mult.left_neutral
thf(fact_17_mult_Oleft__neutral, axiom,
    ((![A : nat]: ((times_times_nat @ one_one_nat @ A) = A)))). % mult.left_neutral
thf(fact_18_mult_Oright__neutral, axiom,
    ((![A : real]: ((times_times_real @ A @ one_one_real) = A)))). % mult.right_neutral
thf(fact_19_mult_Oright__neutral, axiom,
    ((![A : complex]: ((times_times_complex @ A @ one_one_complex) = A)))). % mult.right_neutral
thf(fact_20_mult_Oright__neutral, axiom,
    ((![A : nat]: ((times_times_nat @ A @ one_one_nat) = A)))). % mult.right_neutral
thf(fact_21_add__less__cancel__left, axiom,
    ((![C : real, A : real, B : real]: ((ord_less_real @ (plus_plus_real @ C @ A) @ (plus_plus_real @ C @ B)) = (ord_less_real @ A @ B))))). % add_less_cancel_left
thf(fact_22_add__less__cancel__left, axiom,
    ((![C : nat, A : nat, B : nat]: ((ord_less_nat @ (plus_plus_nat @ C @ A) @ (plus_plus_nat @ C @ B)) = (ord_less_nat @ A @ B))))). % add_less_cancel_left
thf(fact_23_add__less__cancel__right, axiom,
    ((![A : real, C : real, B : real]: ((ord_less_real @ (plus_plus_real @ A @ C) @ (plus_plus_real @ B @ C)) = (ord_less_real @ A @ B))))). % add_less_cancel_right
thf(fact_24_add__less__cancel__right, axiom,
    ((![A : nat, C : nat, B : nat]: ((ord_less_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ C)) = (ord_less_nat @ A @ B))))). % add_less_cancel_right
thf(fact_25_power__gt1__lemma, axiom,
    ((![A : real, N : nat]: ((ord_less_real @ one_one_real @ A) => (ord_less_real @ one_one_real @ (times_times_real @ A @ (power_power_real @ A @ N))))))). % power_gt1_lemma
thf(fact_26_power__gt1__lemma, axiom,
    ((![A : nat, N : nat]: ((ord_less_nat @ one_one_nat @ A) => (ord_less_nat @ one_one_nat @ (times_times_nat @ A @ (power_power_nat @ A @ N))))))). % power_gt1_lemma
thf(fact_27_power__less__power__Suc, axiom,
    ((![A : real, N : nat]: ((ord_less_real @ one_one_real @ A) => (ord_less_real @ (power_power_real @ A @ N) @ (times_times_real @ A @ (power_power_real @ A @ N))))))). % power_less_power_Suc
thf(fact_28_power__less__power__Suc, axiom,
    ((![A : nat, N : nat]: ((ord_less_nat @ one_one_nat @ A) => (ord_less_nat @ (power_power_nat @ A @ N) @ (times_times_nat @ A @ (power_power_nat @ A @ N))))))). % power_less_power_Suc
thf(fact_29_norm__add__less, axiom,
    ((![X3 : complex, R : real, Y2 : complex, S2 : real]: ((ord_less_real @ (real_V638595069omplex @ X3) @ R) => ((ord_less_real @ (real_V638595069omplex @ Y2) @ S2) => (ord_less_real @ (real_V638595069omplex @ (plus_plus_complex @ X3 @ Y2)) @ (plus_plus_real @ R @ S2))))))). % norm_add_less
thf(fact_30_norm__add__less, axiom,
    ((![X3 : real, R : real, Y2 : real, S2 : real]: ((ord_less_real @ (real_V646646907m_real @ X3) @ R) => ((ord_less_real @ (real_V646646907m_real @ Y2) @ S2) => (ord_less_real @ (real_V646646907m_real @ (plus_plus_real @ X3 @ Y2)) @ (plus_plus_real @ R @ S2))))))). % norm_add_less
thf(fact_31_norm__triangle__lt, axiom,
    ((![X3 : complex, Y2 : complex, E : real]: ((ord_less_real @ (plus_plus_real @ (real_V638595069omplex @ X3) @ (real_V638595069omplex @ Y2)) @ E) => (ord_less_real @ (real_V638595069omplex @ (plus_plus_complex @ X3 @ Y2)) @ E))))). % norm_triangle_lt
thf(fact_32_norm__triangle__lt, axiom,
    ((![X3 : real, Y2 : real, E : real]: ((ord_less_real @ (plus_plus_real @ (real_V646646907m_real @ X3) @ (real_V646646907m_real @ Y2)) @ E) => (ord_less_real @ (real_V646646907m_real @ (plus_plus_real @ X3 @ Y2)) @ E))))). % norm_triangle_lt
thf(fact_33_assms_I2_J, axiom,
    ((~ ((n = zero_zero_nat))))). % assms(2)
thf(fact_34_n, axiom,
    ((~ ((na = zero_zero_nat))))). % n
thf(fact_35_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_36_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_37_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_38_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_39_power__one__right, axiom,
    ((![A : complex]: ((power_power_complex @ A @ one_one_nat) = A)))). % power_one_right
thf(fact_40_power__one__right, axiom,
    ((![A : nat]: ((power_power_nat @ A @ one_one_nat) = A)))). % power_one_right
thf(fact_41_power__one__right, axiom,
    ((![A : real]: ((power_power_real @ A @ one_one_nat) = A)))). % power_one_right
thf(fact_42_not__gr__zero, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr_zero
thf(fact_43_zero__eq__add__iff__both__eq__0, axiom,
    ((![X3 : nat, Y2 : nat]: ((zero_zero_nat = (plus_plus_nat @ X3 @ Y2)) = (((X3 = zero_zero_nat)) & ((Y2 = zero_zero_nat))))))). % zero_eq_add_iff_both_eq_0
thf(fact_44_add__eq__0__iff__both__eq__0, axiom,
    ((![X3 : nat, Y2 : nat]: (((plus_plus_nat @ X3 @ Y2) = zero_zero_nat) = (((X3 = zero_zero_nat)) & ((Y2 = zero_zero_nat))))))). % add_eq_0_iff_both_eq_0
thf(fact_45_add__cancel__right__right, axiom,
    ((![A : nat, B : nat]: ((A = (plus_plus_nat @ A @ B)) = (B = zero_zero_nat))))). % add_cancel_right_right
thf(fact_46_add__cancel__right__right, axiom,
    ((![A : complex, B : complex]: ((A = (plus_plus_complex @ A @ B)) = (B = zero_zero_complex))))). % add_cancel_right_right
thf(fact_47_add__cancel__right__left, axiom,
    ((![A : nat, B : nat]: ((A = (plus_plus_nat @ B @ A)) = (B = zero_zero_nat))))). % add_cancel_right_left
thf(fact_48_add__cancel__right__left, axiom,
    ((![A : complex, B : complex]: ((A = (plus_plus_complex @ B @ A)) = (B = zero_zero_complex))))). % add_cancel_right_left
thf(fact_49_add__cancel__left__right, axiom,
    ((![A : nat, B : nat]: (((plus_plus_nat @ A @ B) = A) = (B = zero_zero_nat))))). % add_cancel_left_right
thf(fact_50_add__cancel__left__right, axiom,
    ((![A : complex, B : complex]: (((plus_plus_complex @ A @ B) = A) = (B = zero_zero_complex))))). % add_cancel_left_right
thf(fact_51_add__cancel__left__left, axiom,
    ((![B : nat, A : nat]: (((plus_plus_nat @ B @ A) = A) = (B = zero_zero_nat))))). % add_cancel_left_left
thf(fact_52_add__cancel__left__left, axiom,
    ((![B : complex, A : complex]: (((plus_plus_complex @ B @ A) = A) = (B = zero_zero_complex))))). % add_cancel_left_left
thf(fact_53_add_Oright__neutral, axiom,
    ((![A : nat]: ((plus_plus_nat @ A @ zero_zero_nat) = A)))). % add.right_neutral
thf(fact_54_add_Oright__neutral, axiom,
    ((![A : complex]: ((plus_plus_complex @ A @ zero_zero_complex) = A)))). % add.right_neutral
thf(fact_55_add_Oleft__neutral, axiom,
    ((![A : nat]: ((plus_plus_nat @ zero_zero_nat @ A) = A)))). % add.left_neutral
thf(fact_56_add_Oleft__neutral, axiom,
    ((![A : complex]: ((plus_plus_complex @ zero_zero_complex @ A) = A)))). % add.left_neutral
thf(fact_57_norm__eq__zero, axiom,
    ((![X3 : complex]: (((real_V638595069omplex @ X3) = zero_zero_real) = (X3 = zero_zero_complex))))). % norm_eq_zero
thf(fact_58_norm__eq__zero, axiom,
    ((![X3 : real]: (((real_V646646907m_real @ X3) = zero_zero_real) = (X3 = zero_zero_real))))). % norm_eq_zero
thf(fact_59_norm__zero, axiom,
    (((real_V638595069omplex @ zero_zero_complex) = zero_zero_real))). % norm_zero
thf(fact_60_norm__zero, axiom,
    (((real_V646646907m_real @ zero_zero_real) = zero_zero_real))). % norm_zero
thf(fact_61_nat__zero__less__power__iff, axiom,
    ((![X3 : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ (power_power_nat @ X3 @ N)) = (((ord_less_nat @ zero_zero_nat @ X3)) | ((N = zero_zero_nat))))))). % nat_zero_less_power_iff
thf(fact_62_zero__less__double__add__iff__zero__less__single__add, axiom,
    ((![A : real]: ((ord_less_real @ zero_zero_real @ (plus_plus_real @ A @ A)) = (ord_less_real @ zero_zero_real @ A))))). % zero_less_double_add_iff_zero_less_single_add
thf(fact_63_double__add__less__zero__iff__single__add__less__zero, axiom,
    ((![A : real]: ((ord_less_real @ (plus_plus_real @ A @ A) @ zero_zero_real) = (ord_less_real @ A @ zero_zero_real))))). % double_add_less_zero_iff_single_add_less_zero
thf(fact_64_less__add__same__cancel2, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ (plus_plus_real @ B @ A)) = (ord_less_real @ zero_zero_real @ B))))). % less_add_same_cancel2
thf(fact_65_less__add__same__cancel2, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ (plus_plus_nat @ B @ A)) = (ord_less_nat @ zero_zero_nat @ B))))). % less_add_same_cancel2
thf(fact_66_less__add__same__cancel1, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ (plus_plus_real @ A @ B)) = (ord_less_real @ zero_zero_real @ B))))). % less_add_same_cancel1
thf(fact_67_less__add__same__cancel1, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ (plus_plus_nat @ A @ B)) = (ord_less_nat @ zero_zero_nat @ B))))). % less_add_same_cancel1
thf(fact_68_add__less__same__cancel2, axiom,
    ((![A : real, B : real]: ((ord_less_real @ (plus_plus_real @ A @ B) @ B) = (ord_less_real @ A @ zero_zero_real))))). % add_less_same_cancel2
thf(fact_69_add__less__same__cancel2, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ (plus_plus_nat @ A @ B) @ B) = (ord_less_nat @ A @ zero_zero_nat))))). % add_less_same_cancel2
thf(fact_70_add__less__same__cancel1, axiom,
    ((![B : real, A : real]: ((ord_less_real @ (plus_plus_real @ B @ A) @ B) = (ord_less_real @ A @ zero_zero_real))))). % add_less_same_cancel1
thf(fact_71_add__less__same__cancel1, axiom,
    ((![B : nat, A : nat]: ((ord_less_nat @ (plus_plus_nat @ B @ A) @ B) = (ord_less_nat @ A @ zero_zero_nat))))). % add_less_same_cancel1
thf(fact_72_zero__less__norm__iff, axiom,
    ((![X3 : complex]: ((ord_less_real @ zero_zero_real @ (real_V638595069omplex @ X3)) = (~ ((X3 = zero_zero_complex))))))). % zero_less_norm_iff
thf(fact_73_zero__less__norm__iff, axiom,
    ((![X3 : real]: ((ord_less_real @ zero_zero_real @ (real_V646646907m_real @ X3)) = (~ ((X3 = zero_zero_real))))))). % zero_less_norm_iff
thf(fact_74_power__eq__0__iff, axiom,
    ((![A : complex, N : nat]: (((power_power_complex @ A @ N) = zero_zero_complex) = (((A = zero_zero_complex)) & ((ord_less_nat @ zero_zero_nat @ N))))))). % power_eq_0_iff
thf(fact_75_power__eq__0__iff, axiom,
    ((![A : nat, N : nat]: (((power_power_nat @ A @ N) = zero_zero_nat) = (((A = zero_zero_nat)) & ((ord_less_nat @ zero_zero_nat @ N))))))). % power_eq_0_iff
thf(fact_76_power__eq__0__iff, axiom,
    ((![A : real, N : nat]: (((power_power_real @ A @ N) = zero_zero_real) = (((A = zero_zero_real)) & ((ord_less_nat @ zero_zero_nat @ N))))))). % power_eq_0_iff
thf(fact_77_power__strict__decreasing__iff, axiom,
    ((![B : real, M2 : nat, N : nat]: ((ord_less_real @ zero_zero_real @ B) => ((ord_less_real @ B @ one_one_real) => ((ord_less_real @ (power_power_real @ B @ M2) @ (power_power_real @ B @ N)) = (ord_less_nat @ N @ M2))))))). % power_strict_decreasing_iff
thf(fact_78_power__strict__decreasing__iff, axiom,
    ((![B : nat, M2 : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ B) => ((ord_less_nat @ B @ one_one_nat) => ((ord_less_nat @ (power_power_nat @ B @ M2) @ (power_power_nat @ B @ N)) = (ord_less_nat @ N @ M2))))))). % power_strict_decreasing_iff
thf(fact_79_zero__reorient, axiom,
    ((![X3 : nat]: ((zero_zero_nat = X3) = (X3 = zero_zero_nat))))). % zero_reorient
thf(fact_80_zero__reorient, axiom,
    ((![X3 : complex]: ((zero_zero_complex = X3) = (X3 = zero_zero_complex))))). % zero_reorient
thf(fact_81_nat__power__less__imp__less, axiom,
    ((![I : nat, M2 : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ I) => ((ord_less_nat @ (power_power_nat @ I @ M2) @ (power_power_nat @ I @ N)) => (ord_less_nat @ M2 @ N)))))). % nat_power_less_imp_less
thf(fact_82_power__mult, axiom,
    ((![A : complex, M2 : nat, N : nat]: ((power_power_complex @ A @ (times_times_nat @ M2 @ N)) = (power_power_complex @ (power_power_complex @ A @ M2) @ N))))). % power_mult
thf(fact_83_power__mult, axiom,
    ((![A : nat, M2 : nat, N : nat]: ((power_power_nat @ A @ (times_times_nat @ M2 @ N)) = (power_power_nat @ (power_power_nat @ A @ M2) @ N))))). % power_mult
thf(fact_84_power__mult, axiom,
    ((![A : real, M2 : nat, N : nat]: ((power_power_real @ A @ (times_times_nat @ M2 @ N)) = (power_power_real @ (power_power_real @ A @ M2) @ N))))). % power_mult
thf(fact_85_zero__power, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((power_power_complex @ zero_zero_complex @ N) = zero_zero_complex))))). % zero_power
thf(fact_86_zero__power, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((power_power_nat @ zero_zero_nat @ N) = zero_zero_nat))))). % zero_power
thf(fact_87_zero__power, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((power_power_real @ zero_zero_real @ N) = zero_zero_real))))). % zero_power
thf(fact_88_power__add, axiom,
    ((![A : real, M2 : nat, N : nat]: ((power_power_real @ A @ (plus_plus_nat @ M2 @ N)) = (times_times_real @ (power_power_real @ A @ M2) @ (power_power_real @ A @ N)))))). % power_add
thf(fact_89_power__add, axiom,
    ((![A : complex, M2 : nat, N : nat]: ((power_power_complex @ A @ (plus_plus_nat @ M2 @ N)) = (times_times_complex @ (power_power_complex @ A @ M2) @ (power_power_complex @ A @ N)))))). % power_add
thf(fact_90_power__add, axiom,
    ((![A : nat, M2 : nat, N : nat]: ((power_power_nat @ A @ (plus_plus_nat @ M2 @ N)) = (times_times_nat @ (power_power_nat @ A @ M2) @ (power_power_nat @ A @ N)))))). % power_add
thf(fact_91_zero__less__iff__neq__zero, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ N) = (~ ((N = zero_zero_nat))))))). % zero_less_iff_neq_zero
thf(fact_92_gr__implies__not__zero, axiom,
    ((![M2 : nat, N : nat]: ((ord_less_nat @ M2 @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not_zero
thf(fact_93_not__less__zero, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less_zero
thf(fact_94_gr__zeroI, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr_zeroI
thf(fact_95_add_Ogroup__left__neutral, axiom,
    ((![A : complex]: ((plus_plus_complex @ zero_zero_complex @ A) = A)))). % add.group_left_neutral
thf(fact_96_add_Ocomm__neutral, axiom,
    ((![A : nat]: ((plus_plus_nat @ A @ zero_zero_nat) = A)))). % add.comm_neutral
thf(fact_97_add_Ocomm__neutral, axiom,
    ((![A : complex]: ((plus_plus_complex @ A @ zero_zero_complex) = A)))). % add.comm_neutral
thf(fact_98_comm__monoid__add__class_Oadd__0, axiom,
    ((![A : nat]: ((plus_plus_nat @ zero_zero_nat @ A) = A)))). % comm_monoid_add_class.add_0
thf(fact_99_comm__monoid__add__class_Oadd__0, axiom,
    ((![A : complex]: ((plus_plus_complex @ zero_zero_complex @ A) = A)))). % comm_monoid_add_class.add_0
thf(fact_100_power__not__zero, axiom,
    ((![A : complex, N : nat]: ((~ ((A = zero_zero_complex))) => (~ (((power_power_complex @ A @ N) = zero_zero_complex))))))). % power_not_zero
thf(fact_101_power__not__zero, axiom,
    ((![A : nat, N : nat]: ((~ ((A = zero_zero_nat))) => (~ (((power_power_nat @ A @ N) = zero_zero_nat))))))). % power_not_zero
thf(fact_102_power__not__zero, axiom,
    ((![A : real, N : nat]: ((~ ((A = zero_zero_real))) => (~ (((power_power_real @ A @ N) = zero_zero_real))))))). % power_not_zero
thf(fact_103_power__0__left, axiom,
    ((![N : nat]: (((N = zero_zero_nat) => ((power_power_complex @ zero_zero_complex @ N) = one_one_complex)) & ((~ ((N = zero_zero_nat))) => ((power_power_complex @ zero_zero_complex @ N) = zero_zero_complex)))))). % power_0_left
thf(fact_104_power__0__left, axiom,
    ((![N : nat]: (((N = zero_zero_nat) => ((power_power_nat @ zero_zero_nat @ N) = one_one_nat)) & ((~ ((N = zero_zero_nat))) => ((power_power_nat @ zero_zero_nat @ N) = zero_zero_nat)))))). % power_0_left
thf(fact_105_power__0__left, axiom,
    ((![N : nat]: (((N = zero_zero_nat) => ((power_power_real @ zero_zero_real @ N) = one_one_real)) & ((~ ((N = zero_zero_nat))) => ((power_power_real @ zero_zero_real @ N) = zero_zero_real)))))). % power_0_left
thf(fact_106_power__eq__imp__eq__norm, axiom,
    ((![W : complex, N : nat, Z3 : complex]: (((power_power_complex @ W @ N) = (power_power_complex @ Z3 @ N)) => ((ord_less_nat @ zero_zero_nat @ N) => ((real_V638595069omplex @ W) = (real_V638595069omplex @ Z3))))))). % power_eq_imp_eq_norm
thf(fact_107_power__eq__imp__eq__norm, axiom,
    ((![W : real, N : nat, Z3 : real]: (((power_power_real @ W @ N) = (power_power_real @ Z3 @ N)) => ((ord_less_nat @ zero_zero_nat @ N) => ((real_V646646907m_real @ W) = (real_V646646907m_real @ Z3))))))). % power_eq_imp_eq_norm
thf(fact_108_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_109_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_110_mult_Ocommute, axiom,
    ((times_times_complex = (^[A2 : complex]: (^[B2 : complex]: (times_times_complex @ B2 @ A2)))))). % mult.commute
thf(fact_111_mult_Ocommute, axiom,
    ((times_times_nat = (^[A2 : nat]: (^[B2 : nat]: (times_times_nat @ B2 @ A2)))))). % mult.commute
thf(fact_112_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_113_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_114_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_115_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_116_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_117_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_118_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_119_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_120_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_121_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_122_add_Ocommute, axiom,
    ((plus_plus_complex = (^[A2 : complex]: (^[B2 : complex]: (plus_plus_complex @ B2 @ A2)))))). % add.commute
thf(fact_123_add_Ocommute, axiom,
    ((plus_plus_nat = (^[A2 : nat]: (^[B2 : nat]: (plus_plus_nat @ B2 @ A2)))))). % add.commute
thf(fact_124_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_125_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_126_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_127_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_128_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_129_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_130_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_131_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_132_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_133_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_134_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_135_one__reorient, axiom,
    ((![X3 : complex]: ((one_one_complex = X3) = (X3 = one_one_complex))))). % one_reorient
thf(fact_136_one__reorient, axiom,
    ((![X3 : real]: ((one_one_real = X3) = (X3 = one_one_real))))). % one_reorient
thf(fact_137_one__reorient, axiom,
    ((![X3 : nat]: ((one_one_nat = X3) = (X3 = one_one_nat))))). % one_reorient
thf(fact_138_pos__add__strict, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_real @ zero_zero_real @ A) => ((ord_less_real @ B @ C) => (ord_less_real @ B @ (plus_plus_real @ A @ C))))))). % pos_add_strict
thf(fact_139_pos__add__strict, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_nat @ zero_zero_nat @ A) => ((ord_less_nat @ B @ C) => (ord_less_nat @ B @ (plus_plus_nat @ A @ C))))))). % pos_add_strict
thf(fact_140_canonically__ordered__monoid__add__class_OlessE, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (~ ((![C2 : nat]: ((B = (plus_plus_nat @ A @ C2)) => (C2 = zero_zero_nat))))))))). % canonically_ordered_monoid_add_class.lessE
thf(fact_141_add__pos__pos, axiom,
    ((![A : real, B : real]: ((ord_less_real @ zero_zero_real @ A) => ((ord_less_real @ zero_zero_real @ B) => (ord_less_real @ zero_zero_real @ (plus_plus_real @ A @ B))))))). % add_pos_pos
thf(fact_142_add__pos__pos, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ zero_zero_nat @ A) => ((ord_less_nat @ zero_zero_nat @ B) => (ord_less_nat @ zero_zero_nat @ (plus_plus_nat @ A @ B))))))). % add_pos_pos
thf(fact_143_add__neg__neg, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ zero_zero_real) => ((ord_less_real @ B @ zero_zero_real) => (ord_less_real @ (plus_plus_real @ A @ B) @ zero_zero_real)))))). % add_neg_neg
thf(fact_144_add__neg__neg, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ zero_zero_nat) => ((ord_less_nat @ B @ zero_zero_nat) => (ord_less_nat @ (plus_plus_nat @ A @ B) @ zero_zero_nat)))))). % add_neg_neg
thf(fact_145_zero__less__power, axiom,
    ((![A : real, N : nat]: ((ord_less_real @ zero_zero_real @ A) => (ord_less_real @ zero_zero_real @ (power_power_real @ A @ N)))))). % zero_less_power
thf(fact_146_zero__less__power, axiom,
    ((![A : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ A) => (ord_less_nat @ zero_zero_nat @ (power_power_nat @ A @ N)))))). % zero_less_power
thf(fact_147_power__0, axiom,
    ((![A : complex]: ((power_power_complex @ A @ zero_zero_nat) = one_one_complex)))). % power_0
thf(fact_148_power__0, axiom,
    ((![A : nat]: ((power_power_nat @ A @ zero_zero_nat) = one_one_nat)))). % power_0
thf(fact_149_power__0, axiom,
    ((![A : real]: ((power_power_real @ A @ zero_zero_nat) = one_one_real)))). % power_0
thf(fact_150_sum__squares__gt__zero__iff, axiom,
    ((![X3 : real, Y2 : real]: ((ord_less_real @ zero_zero_real @ (plus_plus_real @ (times_times_real @ X3 @ X3) @ (times_times_real @ Y2 @ Y2))) = (((~ ((X3 = zero_zero_real)))) | ((~ ((Y2 = zero_zero_real))))))))). % sum_squares_gt_zero_iff
thf(fact_151_power__strict__decreasing, axiom,
    ((![N : nat, N2 : nat, A : real]: ((ord_less_nat @ N @ N2) => ((ord_less_real @ zero_zero_real @ A) => ((ord_less_real @ A @ one_one_real) => (ord_less_real @ (power_power_real @ A @ N2) @ (power_power_real @ A @ N)))))))). % power_strict_decreasing
thf(fact_152_power__strict__decreasing, axiom,
    ((![N : nat, N2 : nat, A : nat]: ((ord_less_nat @ N @ N2) => ((ord_less_nat @ zero_zero_nat @ A) => ((ord_less_nat @ A @ one_one_nat) => (ord_less_nat @ (power_power_nat @ A @ N2) @ (power_power_nat @ A @ N)))))))). % power_strict_decreasing
thf(fact_153_one__less__power, axiom,
    ((![A : real, N : nat]: ((ord_less_real @ one_one_real @ A) => ((ord_less_nat @ zero_zero_nat @ N) => (ord_less_real @ one_one_real @ (power_power_real @ A @ N))))))). % one_less_power
thf(fact_154_one__less__power, axiom,
    ((![A : nat, N : nat]: ((ord_less_nat @ one_one_nat @ A) => ((ord_less_nat @ zero_zero_nat @ N) => (ord_less_nat @ one_one_nat @ (power_power_nat @ A @ N))))))). % one_less_power
thf(fact_155_add__less__imp__less__right, axiom,
    ((![A : real, C : real, B : real]: ((ord_less_real @ (plus_plus_real @ A @ C) @ (plus_plus_real @ B @ C)) => (ord_less_real @ A @ B))))). % add_less_imp_less_right
thf(fact_156_add__less__imp__less__right, axiom,
    ((![A : nat, C : nat, B : nat]: ((ord_less_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ C)) => (ord_less_nat @ A @ B))))). % add_less_imp_less_right
thf(fact_157_add__less__imp__less__left, axiom,
    ((![C : real, A : real, B : real]: ((ord_less_real @ (plus_plus_real @ C @ A) @ (plus_plus_real @ C @ B)) => (ord_less_real @ A @ B))))). % add_less_imp_less_left
thf(fact_158_add__less__imp__less__left, axiom,
    ((![C : nat, A : nat, B : nat]: ((ord_less_nat @ (plus_plus_nat @ C @ A) @ (plus_plus_nat @ C @ B)) => (ord_less_nat @ A @ B))))). % add_less_imp_less_left
thf(fact_159_add__strict__right__mono, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_real @ A @ B) => (ord_less_real @ (plus_plus_real @ A @ C) @ (plus_plus_real @ B @ C)))))). % add_strict_right_mono
thf(fact_160_add__strict__right__mono, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_nat @ A @ B) => (ord_less_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ C)))))). % add_strict_right_mono
thf(fact_161_add__strict__left__mono, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_real @ A @ B) => (ord_less_real @ (plus_plus_real @ C @ A) @ (plus_plus_real @ C @ B)))))). % add_strict_left_mono
thf(fact_162_add__strict__left__mono, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_nat @ A @ B) => (ord_less_nat @ (plus_plus_nat @ C @ A) @ (plus_plus_nat @ C @ B)))))). % add_strict_left_mono
thf(fact_163_add__strict__mono, axiom,
    ((![A : real, B : real, C : real, D : real]: ((ord_less_real @ A @ B) => ((ord_less_real @ C @ D) => (ord_less_real @ (plus_plus_real @ A @ C) @ (plus_plus_real @ B @ D))))))). % add_strict_mono
thf(fact_164_add__strict__mono, axiom,
    ((![A : nat, B : nat, C : nat, D : nat]: ((ord_less_nat @ A @ B) => ((ord_less_nat @ C @ D) => (ord_less_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ D))))))). % add_strict_mono
thf(fact_165_add__mono__thms__linordered__field_I1_J, axiom,
    ((![I : real, J : real, K : real, L : real]: (((ord_less_real @ I @ J) & (K = L)) => (ord_less_real @ (plus_plus_real @ I @ K) @ (plus_plus_real @ J @ L)))))). % add_mono_thms_linordered_field(1)
thf(fact_166_add__mono__thms__linordered__field_I1_J, axiom,
    ((![I : nat, J : nat, K : nat, L : nat]: (((ord_less_nat @ I @ J) & (K = L)) => (ord_less_nat @ (plus_plus_nat @ I @ K) @ (plus_plus_nat @ J @ L)))))). % add_mono_thms_linordered_field(1)
thf(fact_167_add__mono__thms__linordered__field_I2_J, axiom,
    ((![I : real, J : real, K : real, L : real]: (((I = J) & (ord_less_real @ K @ L)) => (ord_less_real @ (plus_plus_real @ I @ K) @ (plus_plus_real @ J @ L)))))). % add_mono_thms_linordered_field(2)
thf(fact_168_add__mono__thms__linordered__field_I2_J, axiom,
    ((![I : nat, J : nat, K : nat, L : nat]: (((I = J) & (ord_less_nat @ K @ L)) => (ord_less_nat @ (plus_plus_nat @ I @ K) @ (plus_plus_nat @ J @ L)))))). % add_mono_thms_linordered_field(2)
thf(fact_169_add__mono__thms__linordered__field_I5_J, axiom,
    ((![I : real, J : real, K : real, L : real]: (((ord_less_real @ I @ J) & (ord_less_real @ K @ L)) => (ord_less_real @ (plus_plus_real @ I @ K) @ (plus_plus_real @ J @ L)))))). % add_mono_thms_linordered_field(5)
thf(fact_170_add__mono__thms__linordered__field_I5_J, axiom,
    ((![I : nat, J : nat, K : nat, L : nat]: (((ord_less_nat @ I @ J) & (ord_less_nat @ K @ L)) => (ord_less_nat @ (plus_plus_nat @ I @ K) @ (plus_plus_nat @ J @ L)))))). % add_mono_thms_linordered_field(5)
thf(fact_171_mult_Ocomm__neutral, axiom,
    ((![A : real]: ((times_times_real @ A @ one_one_real) = A)))). % mult.comm_neutral
thf(fact_172_mult_Ocomm__neutral, axiom,
    ((![A : complex]: ((times_times_complex @ A @ one_one_complex) = A)))). % mult.comm_neutral
thf(fact_173_mult_Ocomm__neutral, axiom,
    ((![A : nat]: ((times_times_nat @ A @ one_one_nat) = A)))). % mult.comm_neutral
thf(fact_174_comm__monoid__mult__class_Omult__1, axiom,
    ((![A : real]: ((times_times_real @ one_one_real @ A) = A)))). % comm_monoid_mult_class.mult_1
thf(fact_175_comm__monoid__mult__class_Omult__1, axiom,
    ((![A : complex]: ((times_times_complex @ one_one_complex @ A) = A)))). % comm_monoid_mult_class.mult_1
thf(fact_176_comm__monoid__mult__class_Omult__1, axiom,
    ((![A : nat]: ((times_times_nat @ one_one_nat @ A) = A)))). % comm_monoid_mult_class.mult_1
thf(fact_177_power__commuting__commutes, axiom,
    ((![X3 : real, Y2 : real, N : nat]: (((times_times_real @ X3 @ Y2) = (times_times_real @ Y2 @ X3)) => ((times_times_real @ (power_power_real @ X3 @ N) @ Y2) = (times_times_real @ Y2 @ (power_power_real @ X3 @ N))))))). % power_commuting_commutes
thf(fact_178_power__commuting__commutes, axiom,
    ((![X3 : complex, Y2 : complex, N : nat]: (((times_times_complex @ X3 @ Y2) = (times_times_complex @ Y2 @ X3)) => ((times_times_complex @ (power_power_complex @ X3 @ N) @ Y2) = (times_times_complex @ Y2 @ (power_power_complex @ X3 @ N))))))). % power_commuting_commutes
thf(fact_179_power__commuting__commutes, axiom,
    ((![X3 : nat, Y2 : nat, N : nat]: (((times_times_nat @ X3 @ Y2) = (times_times_nat @ Y2 @ X3)) => ((times_times_nat @ (power_power_nat @ X3 @ N) @ Y2) = (times_times_nat @ Y2 @ (power_power_nat @ X3 @ N))))))). % power_commuting_commutes
thf(fact_180_power__mult__distrib, axiom,
    ((![A : real, B : real, N : nat]: ((power_power_real @ (times_times_real @ A @ B) @ N) = (times_times_real @ (power_power_real @ A @ N) @ (power_power_real @ B @ N)))))). % power_mult_distrib
thf(fact_181_power__mult__distrib, axiom,
    ((![A : complex, B : complex, N : nat]: ((power_power_complex @ (times_times_complex @ A @ B) @ N) = (times_times_complex @ (power_power_complex @ A @ N) @ (power_power_complex @ B @ N)))))). % power_mult_distrib
thf(fact_182_power__mult__distrib, axiom,
    ((![A : nat, B : nat, N : nat]: ((power_power_nat @ (times_times_nat @ A @ B) @ N) = (times_times_nat @ (power_power_nat @ A @ N) @ (power_power_nat @ B @ N)))))). % power_mult_distrib
thf(fact_183_power__commutes, axiom,
    ((![A : real, N : nat]: ((times_times_real @ (power_power_real @ A @ N) @ A) = (times_times_real @ A @ (power_power_real @ A @ N)))))). % power_commutes
thf(fact_184_power__commutes, axiom,
    ((![A : complex, N : nat]: ((times_times_complex @ (power_power_complex @ A @ N) @ A) = (times_times_complex @ A @ (power_power_complex @ A @ N)))))). % power_commutes
thf(fact_185_power__commutes, axiom,
    ((![A : nat, N : nat]: ((times_times_nat @ (power_power_nat @ A @ N) @ A) = (times_times_nat @ A @ (power_power_nat @ A @ N)))))). % power_commutes
thf(fact_186_norm__mult, axiom,
    ((![X3 : complex, Y2 : complex]: ((real_V638595069omplex @ (times_times_complex @ X3 @ Y2)) = (times_times_real @ (real_V638595069omplex @ X3) @ (real_V638595069omplex @ Y2)))))). % norm_mult
thf(fact_187_norm__mult, axiom,
    ((![X3 : real, Y2 : real]: ((real_V646646907m_real @ (times_times_real @ X3 @ Y2)) = (times_times_real @ (real_V646646907m_real @ X3) @ (real_V646646907m_real @ Y2)))))). % norm_mult
thf(fact_188_norm__power, axiom,
    ((![X3 : complex, N : nat]: ((real_V638595069omplex @ (power_power_complex @ X3 @ N)) = (power_power_real @ (real_V638595069omplex @ X3) @ N))))). % norm_power
thf(fact_189_norm__power, axiom,
    ((![X3 : real, N : nat]: ((real_V646646907m_real @ (power_power_real @ X3 @ N)) = (power_power_real @ (real_V646646907m_real @ X3) @ N))))). % norm_power
thf(fact_190_power__Suc__less, axiom,
    ((![A : real, N : nat]: ((ord_less_real @ zero_zero_real @ A) => ((ord_less_real @ A @ one_one_real) => (ord_less_real @ (times_times_real @ A @ (power_power_real @ A @ N)) @ (power_power_real @ A @ N))))))). % power_Suc_less
thf(fact_191_power__Suc__less, axiom,
    ((![A : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ A) => ((ord_less_nat @ A @ one_one_nat) => (ord_less_nat @ (times_times_nat @ A @ (power_power_nat @ A @ N)) @ (power_power_nat @ A @ N))))))). % power_Suc_less
thf(fact_192_power__eq__1__iff, axiom,
    ((![W : complex, N : nat]: (((power_power_complex @ W @ N) = one_one_complex) => (((real_V638595069omplex @ W) = one_one_real) | (N = zero_zero_nat)))))). % power_eq_1_iff
thf(fact_193_power__eq__1__iff, axiom,
    ((![W : real, N : nat]: (((power_power_real @ W @ N) = one_one_real) => (((real_V646646907m_real @ W) = one_one_real) | (N = zero_zero_nat)))))). % power_eq_1_iff
thf(fact_194_power__strict__increasing, axiom,
    ((![N : nat, N2 : nat, A : real]: ((ord_less_nat @ N @ N2) => ((ord_less_real @ one_one_real @ A) => (ord_less_real @ (power_power_real @ A @ N) @ (power_power_real @ A @ N2))))))). % power_strict_increasing
thf(fact_195_power__strict__increasing, axiom,
    ((![N : nat, N2 : nat, A : nat]: ((ord_less_nat @ N @ N2) => ((ord_less_nat @ one_one_nat @ A) => (ord_less_nat @ (power_power_nat @ A @ N) @ (power_power_nat @ A @ N2))))))). % power_strict_increasing
thf(fact_196_power__less__imp__less__exp, axiom,
    ((![A : real, M2 : nat, N : nat]: ((ord_less_real @ one_one_real @ A) => ((ord_less_real @ (power_power_real @ A @ M2) @ (power_power_real @ A @ N)) => (ord_less_nat @ M2 @ N)))))). % power_less_imp_less_exp
thf(fact_197_power__less__imp__less__exp, axiom,
    ((![A : nat, M2 : nat, N : nat]: ((ord_less_nat @ one_one_nat @ A) => ((ord_less_nat @ (power_power_nat @ A @ M2) @ (power_power_nat @ A @ N)) => (ord_less_nat @ M2 @ N)))))). % power_less_imp_less_exp
thf(fact_198_left__right__inverse__power, axiom,
    ((![X3 : real, Y2 : real, N : nat]: (((times_times_real @ X3 @ Y2) = one_one_real) => ((times_times_real @ (power_power_real @ X3 @ N) @ (power_power_real @ Y2 @ N)) = one_one_real))))). % left_right_inverse_power
thf(fact_199_left__right__inverse__power, axiom,
    ((![X3 : complex, Y2 : complex, N : nat]: (((times_times_complex @ X3 @ Y2) = one_one_complex) => ((times_times_complex @ (power_power_complex @ X3 @ N) @ (power_power_complex @ Y2 @ N)) = one_one_complex))))). % left_right_inverse_power
thf(fact_200_left__right__inverse__power, axiom,
    ((![X3 : nat, Y2 : nat, N : nat]: (((times_times_nat @ X3 @ Y2) = one_one_nat) => ((times_times_nat @ (power_power_nat @ X3 @ N) @ (power_power_nat @ Y2 @ N)) = one_one_nat))))). % left_right_inverse_power
thf(fact_201_norm__mult__less, axiom,
    ((![X3 : complex, R : real, Y2 : complex, S2 : real]: ((ord_less_real @ (real_V638595069omplex @ X3) @ R) => ((ord_less_real @ (real_V638595069omplex @ Y2) @ S2) => (ord_less_real @ (real_V638595069omplex @ (times_times_complex @ X3 @ Y2)) @ (times_times_real @ R @ S2))))))). % norm_mult_less
thf(fact_202_norm__mult__less, axiom,
    ((![X3 : real, R : real, Y2 : real, S2 : real]: ((ord_less_real @ (real_V646646907m_real @ X3) @ R) => ((ord_less_real @ (real_V646646907m_real @ Y2) @ S2) => (ord_less_real @ (real_V646646907m_real @ (times_times_real @ X3 @ Y2)) @ (times_times_real @ R @ S2))))))). % norm_mult_less
thf(fact_203_mult__cancel__right2, axiom,
    ((![A : real, C : real]: (((times_times_real @ A @ C) = C) = (((C = zero_zero_real)) | ((A = one_one_real))))))). % mult_cancel_right2
thf(fact_204_mult__cancel__right2, axiom,
    ((![A : complex, C : complex]: (((times_times_complex @ A @ C) = C) = (((C = zero_zero_complex)) | ((A = one_one_complex))))))). % mult_cancel_right2
thf(fact_205_mult__cancel__right1, axiom,
    ((![C : real, B : real]: ((C = (times_times_real @ B @ C)) = (((C = zero_zero_real)) | ((B = one_one_real))))))). % mult_cancel_right1
thf(fact_206_mult__cancel__right1, axiom,
    ((![C : complex, B : complex]: ((C = (times_times_complex @ B @ C)) = (((C = zero_zero_complex)) | ((B = one_one_complex))))))). % mult_cancel_right1
thf(fact_207_mult__cancel__left2, axiom,
    ((![C : real, A : real]: (((times_times_real @ C @ A) = C) = (((C = zero_zero_real)) | ((A = one_one_real))))))). % mult_cancel_left2
thf(fact_208_mult__cancel__left2, axiom,
    ((![C : complex, A : complex]: (((times_times_complex @ C @ A) = C) = (((C = zero_zero_complex)) | ((A = one_one_complex))))))). % mult_cancel_left2
thf(fact_209_mult__cancel__left1, axiom,
    ((![C : real, B : real]: ((C = (times_times_real @ C @ B)) = (((C = zero_zero_real)) | ((B = one_one_real))))))). % mult_cancel_left1
thf(fact_210_mult__cancel__left1, axiom,
    ((![C : complex, B : complex]: ((C = (times_times_complex @ C @ B)) = (((C = zero_zero_complex)) | ((B = one_one_complex))))))). % mult_cancel_left1
thf(fact_211_add__gr__0, axiom,
    ((![M2 : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ (plus_plus_nat @ M2 @ N)) = (((ord_less_nat @ zero_zero_nat @ M2)) | ((ord_less_nat @ zero_zero_nat @ N))))))). % add_gr_0
thf(fact_212_less__one, axiom,
    ((![N : nat]: ((ord_less_nat @ N @ one_one_nat) = (N = zero_zero_nat))))). % less_one
thf(fact_213_neq0__conv, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ N))))). % neq0_conv
thf(fact_214_mult__less__cancel2, axiom,
    ((![M2 : nat, K : nat, N : nat]: ((ord_less_nat @ (times_times_nat @ M2 @ K) @ (times_times_nat @ N @ K)) = (((ord_less_nat @ zero_zero_nat @ K)) & ((ord_less_nat @ M2 @ N))))))). % mult_less_cancel2
thf(fact_215_less__nat__zero__code, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_nat_zero_code
thf(fact_216_mult__zero__left, axiom,
    ((![A : complex]: ((times_times_complex @ zero_zero_complex @ A) = zero_zero_complex)))). % mult_zero_left
thf(fact_217_mult__zero__left, axiom,
    ((![A : nat]: ((times_times_nat @ zero_zero_nat @ A) = zero_zero_nat)))). % mult_zero_left
thf(fact_218_mult__zero__right, axiom,
    ((![A : complex]: ((times_times_complex @ A @ zero_zero_complex) = zero_zero_complex)))). % mult_zero_right
thf(fact_219_mult__zero__right, axiom,
    ((![A : nat]: ((times_times_nat @ A @ zero_zero_nat) = zero_zero_nat)))). % mult_zero_right
thf(fact_220_mult__eq__0__iff, axiom,
    ((![A : complex, B : complex]: (((times_times_complex @ A @ B) = zero_zero_complex) = (((A = zero_zero_complex)) | ((B = zero_zero_complex))))))). % mult_eq_0_iff
thf(fact_221_mult__eq__0__iff, axiom,
    ((![A : nat, B : nat]: (((times_times_nat @ A @ B) = zero_zero_nat) = (((A = zero_zero_nat)) | ((B = zero_zero_nat))))))). % mult_eq_0_iff
thf(fact_222_mult__cancel__left, axiom,
    ((![C : complex, A : complex, B : complex]: (((times_times_complex @ C @ A) = (times_times_complex @ C @ B)) = (((C = zero_zero_complex)) | ((A = B))))))). % mult_cancel_left
thf(fact_223_mult__cancel__left, axiom,
    ((![C : nat, A : nat, B : nat]: (((times_times_nat @ C @ A) = (times_times_nat @ C @ B)) = (((C = zero_zero_nat)) | ((A = B))))))). % mult_cancel_left
thf(fact_224_bot__nat__0_Onot__eq__extremum, axiom,
    ((![A : nat]: ((~ ((A = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ A))))). % bot_nat_0.not_eq_extremum
thf(fact_225_Nat_Oadd__0__right, axiom,
    ((![M2 : nat]: ((plus_plus_nat @ M2 @ zero_zero_nat) = M2)))). % Nat.add_0_right
thf(fact_226_add__is__0, axiom,
    ((![M2 : nat, N : nat]: (((plus_plus_nat @ M2 @ N) = zero_zero_nat) = (((M2 = zero_zero_nat)) & ((N = zero_zero_nat))))))). % add_is_0
thf(fact_227_nat__add__left__cancel__less, axiom,
    ((![K : nat, M2 : nat, N : nat]: ((ord_less_nat @ (plus_plus_nat @ K @ M2) @ (plus_plus_nat @ K @ N)) = (ord_less_nat @ M2 @ N))))). % nat_add_left_cancel_less
thf(fact_228_mult__cancel2, axiom,
    ((![M2 : nat, K : nat, N : nat]: (((times_times_nat @ M2 @ K) = (times_times_nat @ N @ K)) = (((M2 = N)) | ((K = zero_zero_nat))))))). % mult_cancel2
thf(fact_229_mult__cancel1, axiom,
    ((![K : nat, M2 : nat, N : nat]: (((times_times_nat @ K @ M2) = (times_times_nat @ K @ N)) = (((M2 = N)) | ((K = zero_zero_nat))))))). % mult_cancel1
thf(fact_230_mult__0__right, axiom,
    ((![M2 : nat]: ((times_times_nat @ M2 @ zero_zero_nat) = zero_zero_nat)))). % mult_0_right
thf(fact_231_mult__is__0, axiom,
    ((![M2 : nat, N : nat]: (((times_times_nat @ M2 @ N) = zero_zero_nat) = (((M2 = zero_zero_nat)) | ((N = zero_zero_nat))))))). % mult_is_0
thf(fact_232_nat__1__eq__mult__iff, axiom,
    ((![M2 : nat, N : nat]: ((one_one_nat = (times_times_nat @ M2 @ N)) = (((M2 = one_one_nat)) & ((N = one_one_nat))))))). % nat_1_eq_mult_iff
thf(fact_233_nat__mult__eq__1__iff, axiom,
    ((![M2 : nat, N : nat]: (((times_times_nat @ M2 @ N) = one_one_nat) = (((M2 = one_one_nat)) & ((N = one_one_nat))))))). % nat_mult_eq_1_iff
thf(fact_234_nat__0__less__mult__iff, axiom,
    ((![M2 : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ (times_times_nat @ M2 @ N)) = (((ord_less_nat @ zero_zero_nat @ M2)) & ((ord_less_nat @ zero_zero_nat @ N))))))). % nat_0_less_mult_iff
thf(fact_235_nat__mult__1, axiom,
    ((![N : nat]: ((times_times_nat @ one_one_nat @ N) = N)))). % nat_mult_1
thf(fact_236_add__mult__distrib, axiom,
    ((![M2 : nat, N : nat, K : nat]: ((times_times_nat @ (plus_plus_nat @ M2 @ N) @ K) = (plus_plus_nat @ (times_times_nat @ M2 @ K) @ (times_times_nat @ N @ K)))))). % add_mult_distrib
thf(fact_237_nat__mult__1__right, axiom,
    ((![N : nat]: ((times_times_nat @ N @ one_one_nat) = N)))). % nat_mult_1_right
thf(fact_238_add__mult__distrib2, axiom,
    ((![K : nat, M2 : nat, N : nat]: ((times_times_nat @ K @ (plus_plus_nat @ M2 @ N)) = (plus_plus_nat @ (times_times_nat @ K @ M2) @ (times_times_nat @ K @ N)))))). % add_mult_distrib2
thf(fact_239_mult__eq__self__implies__10, axiom,
    ((![M2 : nat, N : nat]: ((M2 = (times_times_nat @ M2 @ N)) => ((N = one_one_nat) | (M2 = zero_zero_nat)))))). % mult_eq_self_implies_10
thf(fact_240_mult__0, axiom,
    ((![N : nat]: ((times_times_nat @ zero_zero_nat @ N) = zero_zero_nat)))). % mult_0
thf(fact_241_add__eq__self__zero, axiom,
    ((![M2 : nat, N : nat]: (((plus_plus_nat @ M2 @ N) = M2) => (N = zero_zero_nat))))). % add_eq_self_zero
thf(fact_242_plus__nat_Oadd__0, axiom,
    ((![N : nat]: ((plus_plus_nat @ zero_zero_nat @ N) = N)))). % plus_nat.add_0
thf(fact_243_linorder__neqE__nat, axiom,
    ((![X3 : nat, Y2 : nat]: ((~ ((X3 = Y2))) => ((~ ((ord_less_nat @ X3 @ Y2))) => (ord_less_nat @ Y2 @ X3)))))). % linorder_neqE_nat

% Conjectures (2)
thf(conj_0, hypothesis,
    ((![Z2 : complex]: ((ord_less_real @ (real_V638595069omplex @ (plus_plus_complex @ one_one_complex @ (times_times_complex @ b @ (power_power_complex @ Z2 @ m)))) @ one_one_real) => thesis)))).
thf(conj_1, conjecture,
    (thesis)).
