% 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_793__5375634_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:33:12.618

% 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 (17)
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_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_Orderings_Oord__class_Oless__eq_001t__Nat__Onat, type,
    ord_less_eq_nat : nat > nat > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal, type,
    ord_less_eq_real : real > real > $o).
thf(sy_c_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_v_a____, type,
    a : complex).
thf(sy_v_k____, type,
    k : nat).
thf(sy_v_m____, type,
    m : real).
thf(sy_v_t____, type,
    t : real).

% Relevant facts (248)
thf(fact_0_kas_I2_J, axiom,
    ((~ ((k = zero_zero_nat))))). % kas(2)
thf(fact_1_t_I1_J, axiom,
    ((ord_less_real @ zero_zero_real @ t))). % t(1)
thf(fact_2_t_I2_J, axiom,
    ((ord_less_real @ t @ one_one_real))). % t(2)
thf(fact_3__092_060open_062_092_060lbrakk_0620_A_092_060le_062_At_059_A0_A_060_Ak_092_060rbrakk_062_A_092_060Longrightarrow_062_At_A_094_Ak_A_060_A1_A_094_Ak_092_060close_062, axiom,
    (((ord_less_eq_real @ zero_zero_real @ t) => ((ord_less_nat @ zero_zero_nat @ k) => (ord_less_real @ (power_power_real @ t @ k) @ (power_power_real @ one_one_real @ k)))))). % \<open>\<lbrakk>0 \<le> t; 0 < k\<rbrakk> \<Longrightarrow> t ^ k < 1 ^ k\<close>
thf(fact_4__092_060open_062_092_060And_062d2_O_A_I0_058_058_063_Ha_J_A_060_Ad2_A_092_060Longrightarrow_062_A_092_060exists_062e_0620_058_058_063_Ha_O_Ae_A_060_A_I1_058_058_063_Ha_J_A_092_060and_062_Ae_A_060_Ad2_092_060close_062, axiom,
    ((![D2 : real]: ((ord_less_real @ zero_zero_real @ D2) => (?[E : real]: ((ord_less_real @ zero_zero_real @ E) & ((ord_less_real @ E @ one_one_real) & (ord_less_real @ E @ D2)))))))). % \<open>\<And>d2. (0::?'a) < d2 \<Longrightarrow> \<exists>e>0::?'a. e < (1::?'a) \<and> e < d2\<close>
thf(fact_5__092_060open_0620_A_060_At_A_094_Ak_092_060close_062, axiom,
    ((ord_less_real @ zero_zero_real @ (power_power_real @ t @ k)))). % \<open>0 < t ^ k\<close>
thf(fact_6_power__one, axiom,
    ((![N : nat]: ((power_power_real @ one_one_real @ N) = one_one_real)))). % power_one
thf(fact_7_power__one, axiom,
    ((![N : nat]: ((power_power_nat @ one_one_nat @ N) = one_one_nat)))). % power_one
thf(fact_8_one__le__power, axiom,
    ((![A : nat, N : nat]: ((ord_less_eq_nat @ one_one_nat @ A) => (ord_less_eq_nat @ one_one_nat @ (power_power_nat @ A @ N)))))). % one_le_power
thf(fact_9_one__le__power, axiom,
    ((![A : real, N : nat]: ((ord_less_eq_real @ one_one_real @ A) => (ord_less_eq_real @ one_one_real @ (power_power_real @ A @ N)))))). % one_le_power
thf(fact_10_power__increasing, axiom,
    ((![N : nat, N2 : nat, A : nat]: ((ord_less_eq_nat @ N @ N2) => ((ord_less_eq_nat @ one_one_nat @ A) => (ord_less_eq_nat @ (power_power_nat @ A @ N) @ (power_power_nat @ A @ N2))))))). % power_increasing
thf(fact_11_power__increasing, axiom,
    ((![N : nat, N2 : nat, A : real]: ((ord_less_eq_nat @ N @ N2) => ((ord_less_eq_real @ one_one_real @ A) => (ord_less_eq_real @ (power_power_real @ A @ N) @ (power_power_real @ A @ N2))))))). % power_increasing
thf(fact_12_order__refl, axiom,
    ((![X : real]: (ord_less_eq_real @ X @ X)))). % order_refl
thf(fact_13_order__refl, axiom,
    ((![X : nat]: (ord_less_eq_nat @ X @ X)))). % order_refl
thf(fact_14_power__increasing__iff, axiom,
    ((![B : real, X : nat, Y : nat]: ((ord_less_real @ one_one_real @ B) => ((ord_less_eq_real @ (power_power_real @ B @ X) @ (power_power_real @ B @ Y)) = (ord_less_eq_nat @ X @ Y)))))). % power_increasing_iff
thf(fact_15_power__increasing__iff, axiom,
    ((![B : nat, X : nat, Y : nat]: ((ord_less_nat @ one_one_nat @ B) => ((ord_less_eq_nat @ (power_power_nat @ B @ X) @ (power_power_nat @ B @ Y)) = (ord_less_eq_nat @ X @ Y)))))). % power_increasing_iff
thf(fact_16_le__numeral__extra_I4_J, axiom,
    ((ord_less_eq_real @ one_one_real @ one_one_real))). % le_numeral_extra(4)
thf(fact_17_le__numeral__extra_I4_J, axiom,
    ((ord_less_eq_nat @ one_one_nat @ one_one_nat))). % le_numeral_extra(4)
thf(fact_18_m_I1_J, axiom,
    ((ord_less_real @ zero_zero_real @ m))). % m(1)
thf(fact_19_power__le__one, axiom,
    ((![A : real, N : nat]: ((ord_less_eq_real @ zero_zero_real @ A) => ((ord_less_eq_real @ A @ one_one_real) => (ord_less_eq_real @ (power_power_real @ A @ N) @ one_one_real)))))). % power_le_one
thf(fact_20_power__le__one, axiom,
    ((![A : nat, N : nat]: ((ord_less_eq_nat @ zero_zero_nat @ A) => ((ord_less_eq_nat @ A @ one_one_nat) => (ord_less_eq_nat @ (power_power_nat @ A @ N) @ one_one_nat)))))). % power_le_one
thf(fact_21_power__decreasing, axiom,
    ((![N : nat, N2 : nat, A : real]: ((ord_less_eq_nat @ N @ N2) => ((ord_less_eq_real @ zero_zero_real @ A) => ((ord_less_eq_real @ A @ one_one_real) => (ord_less_eq_real @ (power_power_real @ A @ N2) @ (power_power_real @ A @ N)))))))). % power_decreasing
thf(fact_22_power__decreasing, axiom,
    ((![N : nat, N2 : nat, A : nat]: ((ord_less_eq_nat @ N @ N2) => ((ord_less_eq_nat @ zero_zero_nat @ A) => ((ord_less_eq_nat @ A @ one_one_nat) => (ord_less_eq_nat @ (power_power_nat @ A @ N2) @ (power_power_nat @ A @ N)))))))). % power_decreasing
thf(fact_23_power__le__imp__le__exp, axiom,
    ((![A : real, M : nat, N : nat]: ((ord_less_real @ one_one_real @ A) => ((ord_less_eq_real @ (power_power_real @ A @ M) @ (power_power_real @ A @ N)) => (ord_less_eq_nat @ M @ N)))))). % power_le_imp_le_exp
thf(fact_24_power__le__imp__le__exp, axiom,
    ((![A : nat, M : nat, N : nat]: ((ord_less_nat @ one_one_nat @ A) => ((ord_less_eq_nat @ (power_power_nat @ A @ M) @ (power_power_nat @ A @ N)) => (ord_less_eq_nat @ M @ N)))))). % power_le_imp_le_exp
thf(fact_25_power__inject__exp, axiom,
    ((![A : real, M : nat, N : nat]: ((ord_less_real @ one_one_real @ A) => (((power_power_real @ A @ M) = (power_power_real @ A @ N)) = (M = N)))))). % power_inject_exp
thf(fact_26_power__inject__exp, axiom,
    ((![A : nat, M : nat, N : nat]: ((ord_less_nat @ one_one_nat @ A) => (((power_power_nat @ A @ M) = (power_power_nat @ A @ N)) = (M = N)))))). % power_inject_exp
thf(fact_27_kas_I1_J, axiom,
    ((~ ((a = zero_zero_complex))))). % kas(1)
thf(fact_28_power__one__right, axiom,
    ((![A : real]: ((power_power_real @ A @ one_one_nat) = A)))). % power_one_right
thf(fact_29_power__one__right, axiom,
    ((![A : nat]: ((power_power_nat @ A @ one_one_nat) = A)))). % power_one_right
thf(fact_30_le__zero__eq, axiom,
    ((![N : nat]: ((ord_less_eq_nat @ N @ zero_zero_nat) = (N = zero_zero_nat))))). % le_zero_eq
thf(fact_31_not__gr__zero, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr_zero
thf(fact_32_nat__zero__less__power__iff, axiom,
    ((![X : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ (power_power_nat @ X @ N)) = (((ord_less_nat @ zero_zero_nat @ X)) | ((N = zero_zero_nat))))))). % nat_zero_less_power_iff
thf(fact_33_power__strict__increasing__iff, axiom,
    ((![B : real, X : nat, Y : nat]: ((ord_less_real @ one_one_real @ B) => ((ord_less_real @ (power_power_real @ B @ X) @ (power_power_real @ B @ Y)) = (ord_less_nat @ X @ Y)))))). % power_strict_increasing_iff
thf(fact_34_power__strict__increasing__iff, axiom,
    ((![B : nat, X : nat, Y : nat]: ((ord_less_nat @ one_one_nat @ B) => ((ord_less_nat @ (power_power_nat @ B @ X) @ (power_power_nat @ B @ Y)) = (ord_less_nat @ X @ Y)))))). % power_strict_increasing_iff
thf(fact_35_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_36_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_37_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_38_power__strict__decreasing__iff, axiom,
    ((![B : real, M : nat, N : nat]: ((ord_less_real @ zero_zero_real @ B) => ((ord_less_real @ B @ one_one_real) => ((ord_less_real @ (power_power_real @ B @ M) @ (power_power_real @ B @ N)) = (ord_less_nat @ N @ M))))))). % power_strict_decreasing_iff
thf(fact_39_power__strict__decreasing__iff, axiom,
    ((![B : nat, M : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ B) => ((ord_less_nat @ B @ one_one_nat) => ((ord_less_nat @ (power_power_nat @ B @ M) @ (power_power_nat @ B @ N)) = (ord_less_nat @ N @ M))))))). % power_strict_decreasing_iff
thf(fact_40_power__mono__iff, axiom,
    ((![A : real, B : real, N : nat]: ((ord_less_eq_real @ zero_zero_real @ A) => ((ord_less_eq_real @ zero_zero_real @ B) => ((ord_less_nat @ zero_zero_nat @ N) => ((ord_less_eq_real @ (power_power_real @ A @ N) @ (power_power_real @ B @ N)) = (ord_less_eq_real @ A @ B)))))))). % power_mono_iff
thf(fact_41_power__mono__iff, axiom,
    ((![A : nat, B : nat, N : nat]: ((ord_less_eq_nat @ zero_zero_nat @ A) => ((ord_less_eq_nat @ zero_zero_nat @ B) => ((ord_less_nat @ zero_zero_nat @ N) => ((ord_less_eq_nat @ (power_power_nat @ A @ N) @ (power_power_nat @ B @ N)) = (ord_less_eq_nat @ A @ B)))))))). % power_mono_iff
thf(fact_42_power__decreasing__iff, axiom,
    ((![B : real, M : nat, N : nat]: ((ord_less_real @ zero_zero_real @ B) => ((ord_less_real @ B @ one_one_real) => ((ord_less_eq_real @ (power_power_real @ B @ M) @ (power_power_real @ B @ N)) = (ord_less_eq_nat @ N @ M))))))). % power_decreasing_iff
thf(fact_43_power__decreasing__iff, axiom,
    ((![B : nat, M : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ B) => ((ord_less_nat @ B @ one_one_nat) => ((ord_less_eq_nat @ (power_power_nat @ B @ M) @ (power_power_nat @ B @ N)) = (ord_less_eq_nat @ N @ M))))))). % power_decreasing_iff
thf(fact_44_less__numeral__extra_I3_J, axiom,
    ((~ ((ord_less_real @ zero_zero_real @ zero_zero_real))))). % less_numeral_extra(3)
thf(fact_45_less__numeral__extra_I3_J, axiom,
    ((~ ((ord_less_nat @ zero_zero_nat @ zero_zero_nat))))). % less_numeral_extra(3)
thf(fact_46_zero__reorient, axiom,
    ((![X : nat]: ((zero_zero_nat = X) = (X = zero_zero_nat))))). % zero_reorient
thf(fact_47_zero__reorient, axiom,
    ((![X : real]: ((zero_zero_real = X) = (X = zero_zero_real))))). % zero_reorient
thf(fact_48_zero__reorient, axiom,
    ((![X : complex]: ((zero_zero_complex = X) = (X = zero_zero_complex))))). % zero_reorient
thf(fact_49_ord__eq__less__subst, axiom,
    ((![A : real, F : real > real, B : real, C : real]: ((A = (F @ B)) => ((ord_less_real @ B @ C) => ((![X2 : real, Y2 : real]: ((ord_less_real @ X2 @ Y2) => (ord_less_real @ (F @ X2) @ (F @ Y2)))) => (ord_less_real @ A @ (F @ C)))))))). % ord_eq_less_subst
thf(fact_50_ord__eq__less__subst, axiom,
    ((![A : nat, F : real > nat, B : real, C : real]: ((A = (F @ B)) => ((ord_less_real @ B @ C) => ((![X2 : real, Y2 : real]: ((ord_less_real @ X2 @ Y2) => (ord_less_nat @ (F @ X2) @ (F @ Y2)))) => (ord_less_nat @ A @ (F @ C)))))))). % ord_eq_less_subst
thf(fact_51_ord__eq__less__subst, axiom,
    ((![A : real, F : nat > real, B : nat, C : nat]: ((A = (F @ B)) => ((ord_less_nat @ B @ C) => ((![X2 : nat, Y2 : nat]: ((ord_less_nat @ X2 @ Y2) => (ord_less_real @ (F @ X2) @ (F @ Y2)))) => (ord_less_real @ A @ (F @ C)))))))). % ord_eq_less_subst
thf(fact_52_ord__eq__less__subst, axiom,
    ((![A : nat, F : nat > nat, B : nat, C : nat]: ((A = (F @ B)) => ((ord_less_nat @ B @ C) => ((![X2 : nat, Y2 : nat]: ((ord_less_nat @ X2 @ Y2) => (ord_less_nat @ (F @ X2) @ (F @ Y2)))) => (ord_less_nat @ A @ (F @ C)))))))). % ord_eq_less_subst
thf(fact_53_ord__less__eq__subst, axiom,
    ((![A : real, B : real, F : real > real, C : real]: ((ord_less_real @ A @ B) => (((F @ B) = C) => ((![X2 : real, Y2 : real]: ((ord_less_real @ X2 @ Y2) => (ord_less_real @ (F @ X2) @ (F @ Y2)))) => (ord_less_real @ (F @ A) @ C))))))). % ord_less_eq_subst
thf(fact_54_ord__less__eq__subst, axiom,
    ((![A : real, B : real, F : real > nat, C : nat]: ((ord_less_real @ A @ B) => (((F @ B) = C) => ((![X2 : real, Y2 : real]: ((ord_less_real @ X2 @ Y2) => (ord_less_nat @ (F @ X2) @ (F @ Y2)))) => (ord_less_nat @ (F @ A) @ C))))))). % ord_less_eq_subst
thf(fact_55_ord__less__eq__subst, axiom,
    ((![A : nat, B : nat, F : nat > real, C : real]: ((ord_less_nat @ A @ B) => (((F @ B) = C) => ((![X2 : nat, Y2 : nat]: ((ord_less_nat @ X2 @ Y2) => (ord_less_real @ (F @ X2) @ (F @ Y2)))) => (ord_less_real @ (F @ A) @ C))))))). % ord_less_eq_subst
thf(fact_56_ord__less__eq__subst, axiom,
    ((![A : nat, B : nat, F : nat > nat, C : nat]: ((ord_less_nat @ A @ B) => (((F @ B) = C) => ((![X2 : nat, Y2 : nat]: ((ord_less_nat @ X2 @ Y2) => (ord_less_nat @ (F @ X2) @ (F @ Y2)))) => (ord_less_nat @ (F @ A) @ C))))))). % ord_less_eq_subst
thf(fact_57_order__less__subst1, axiom,
    ((![A : real, F : real > real, B : real, C : real]: ((ord_less_real @ A @ (F @ B)) => ((ord_less_real @ B @ C) => ((![X2 : real, Y2 : real]: ((ord_less_real @ X2 @ Y2) => (ord_less_real @ (F @ X2) @ (F @ Y2)))) => (ord_less_real @ A @ (F @ C)))))))). % order_less_subst1
thf(fact_58_order__less__subst1, axiom,
    ((![A : real, F : nat > real, B : nat, C : nat]: ((ord_less_real @ A @ (F @ B)) => ((ord_less_nat @ B @ C) => ((![X2 : nat, Y2 : nat]: ((ord_less_nat @ X2 @ Y2) => (ord_less_real @ (F @ X2) @ (F @ Y2)))) => (ord_less_real @ A @ (F @ C)))))))). % order_less_subst1
thf(fact_59_order__less__subst1, axiom,
    ((![A : nat, F : real > nat, B : real, C : real]: ((ord_less_nat @ A @ (F @ B)) => ((ord_less_real @ B @ C) => ((![X2 : real, Y2 : real]: ((ord_less_real @ X2 @ Y2) => (ord_less_nat @ (F @ X2) @ (F @ Y2)))) => (ord_less_nat @ A @ (F @ C)))))))). % order_less_subst1
thf(fact_60_order__less__subst1, axiom,
    ((![A : nat, F : nat > nat, B : nat, C : nat]: ((ord_less_nat @ A @ (F @ B)) => ((ord_less_nat @ B @ C) => ((![X2 : nat, Y2 : nat]: ((ord_less_nat @ X2 @ Y2) => (ord_less_nat @ (F @ X2) @ (F @ Y2)))) => (ord_less_nat @ A @ (F @ C)))))))). % order_less_subst1
thf(fact_61_order__less__subst2, axiom,
    ((![A : real, B : real, F : real > real, C : real]: ((ord_less_real @ A @ B) => ((ord_less_real @ (F @ B) @ C) => ((![X2 : real, Y2 : real]: ((ord_less_real @ X2 @ Y2) => (ord_less_real @ (F @ X2) @ (F @ Y2)))) => (ord_less_real @ (F @ A) @ C))))))). % order_less_subst2
thf(fact_62_order__less__subst2, axiom,
    ((![A : real, B : real, F : real > nat, C : nat]: ((ord_less_real @ A @ B) => ((ord_less_nat @ (F @ B) @ C) => ((![X2 : real, Y2 : real]: ((ord_less_real @ X2 @ Y2) => (ord_less_nat @ (F @ X2) @ (F @ Y2)))) => (ord_less_nat @ (F @ A) @ C))))))). % order_less_subst2
thf(fact_63_order__less__subst2, axiom,
    ((![A : nat, B : nat, F : nat > real, C : real]: ((ord_less_nat @ A @ B) => ((ord_less_real @ (F @ B) @ C) => ((![X2 : nat, Y2 : nat]: ((ord_less_nat @ X2 @ Y2) => (ord_less_real @ (F @ X2) @ (F @ Y2)))) => (ord_less_real @ (F @ A) @ C))))))). % order_less_subst2
thf(fact_64_order__less__subst2, axiom,
    ((![A : nat, B : nat, F : nat > nat, C : nat]: ((ord_less_nat @ A @ B) => ((ord_less_nat @ (F @ B) @ C) => ((![X2 : nat, Y2 : nat]: ((ord_less_nat @ X2 @ Y2) => (ord_less_nat @ (F @ X2) @ (F @ Y2)))) => (ord_less_nat @ (F @ A) @ C))))))). % order_less_subst2
thf(fact_65_lt__ex, axiom,
    ((![X : real]: (?[Y2 : real]: (ord_less_real @ Y2 @ X))))). % lt_ex
thf(fact_66_gt__ex, axiom,
    ((![X : real]: (?[X_1 : real]: (ord_less_real @ X @ X_1))))). % gt_ex
thf(fact_67_gt__ex, axiom,
    ((![X : nat]: (?[X_1 : nat]: (ord_less_nat @ X @ X_1))))). % gt_ex
thf(fact_68_neqE, axiom,
    ((![X : real, Y : real]: ((~ ((X = Y))) => ((~ ((ord_less_real @ X @ Y))) => (ord_less_real @ Y @ X)))))). % neqE
thf(fact_69_neqE, axiom,
    ((![X : nat, Y : nat]: ((~ ((X = Y))) => ((~ ((ord_less_nat @ X @ Y))) => (ord_less_nat @ Y @ X)))))). % neqE
thf(fact_70_nat__power__less__imp__less, axiom,
    ((![I : nat, M : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ I) => ((ord_less_nat @ (power_power_nat @ I @ M) @ (power_power_nat @ I @ N)) => (ord_less_nat @ M @ N)))))). % nat_power_less_imp_less
thf(fact_71_neq__iff, axiom,
    ((![X : real, Y : real]: ((~ ((X = Y))) = (((ord_less_real @ X @ Y)) | ((ord_less_real @ Y @ X))))))). % neq_iff
thf(fact_72_neq__iff, axiom,
    ((![X : nat, Y : nat]: ((~ ((X = Y))) = (((ord_less_nat @ X @ Y)) | ((ord_less_nat @ Y @ X))))))). % neq_iff
thf(fact_73_order_Oasym, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ B) => (~ ((ord_less_real @ B @ A))))))). % order.asym
thf(fact_74_order_Oasym, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (~ ((ord_less_nat @ B @ A))))))). % order.asym
thf(fact_75_dense, axiom,
    ((![X : real, Y : real]: ((ord_less_real @ X @ Y) => (?[Z : real]: ((ord_less_real @ X @ Z) & (ord_less_real @ Z @ Y))))))). % dense
thf(fact_76_less__imp__neq, axiom,
    ((![X : real, Y : real]: ((ord_less_real @ X @ Y) => (~ ((X = Y))))))). % less_imp_neq
thf(fact_77_less__imp__neq, axiom,
    ((![X : nat, Y : nat]: ((ord_less_nat @ X @ Y) => (~ ((X = Y))))))). % less_imp_neq
thf(fact_78_less__asym, axiom,
    ((![X : real, Y : real]: ((ord_less_real @ X @ Y) => (~ ((ord_less_real @ Y @ X))))))). % less_asym
thf(fact_79_less__asym, axiom,
    ((![X : nat, Y : nat]: ((ord_less_nat @ X @ Y) => (~ ((ord_less_nat @ Y @ X))))))). % less_asym
thf(fact_80_less__asym_H, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ B) => (~ ((ord_less_real @ B @ A))))))). % less_asym'
thf(fact_81_less__asym_H, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (~ ((ord_less_nat @ B @ A))))))). % less_asym'
thf(fact_82_less__trans, axiom,
    ((![X : real, Y : real, Z2 : real]: ((ord_less_real @ X @ Y) => ((ord_less_real @ Y @ Z2) => (ord_less_real @ X @ Z2)))))). % less_trans
thf(fact_83_less__trans, axiom,
    ((![X : nat, Y : nat, Z2 : nat]: ((ord_less_nat @ X @ Y) => ((ord_less_nat @ Y @ Z2) => (ord_less_nat @ X @ Z2)))))). % less_trans
thf(fact_84_less__linear, axiom,
    ((![X : real, Y : real]: ((ord_less_real @ X @ Y) | ((X = Y) | (ord_less_real @ Y @ X)))))). % less_linear
thf(fact_85_less__linear, axiom,
    ((![X : nat, Y : nat]: ((ord_less_nat @ X @ Y) | ((X = Y) | (ord_less_nat @ Y @ X)))))). % less_linear
thf(fact_86_less__irrefl, axiom,
    ((![X : real]: (~ ((ord_less_real @ X @ X)))))). % less_irrefl
thf(fact_87_less__irrefl, axiom,
    ((![X : nat]: (~ ((ord_less_nat @ X @ X)))))). % less_irrefl
thf(fact_88_ord__eq__less__trans, axiom,
    ((![A : real, B : real, C : real]: ((A = B) => ((ord_less_real @ B @ C) => (ord_less_real @ A @ C)))))). % ord_eq_less_trans
thf(fact_89_ord__eq__less__trans, axiom,
    ((![A : nat, B : nat, C : nat]: ((A = B) => ((ord_less_nat @ B @ C) => (ord_less_nat @ A @ C)))))). % ord_eq_less_trans
thf(fact_90_ord__less__eq__trans, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_real @ A @ B) => ((B = C) => (ord_less_real @ A @ C)))))). % ord_less_eq_trans
thf(fact_91_ord__less__eq__trans, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_nat @ A @ B) => ((B = C) => (ord_less_nat @ A @ C)))))). % ord_less_eq_trans
thf(fact_92_dual__order_Oasym, axiom,
    ((![B : real, A : real]: ((ord_less_real @ B @ A) => (~ ((ord_less_real @ A @ B))))))). % dual_order.asym
thf(fact_93_dual__order_Oasym, axiom,
    ((![B : nat, A : nat]: ((ord_less_nat @ B @ A) => (~ ((ord_less_nat @ A @ B))))))). % dual_order.asym
thf(fact_94_less__imp__not__eq, axiom,
    ((![X : real, Y : real]: ((ord_less_real @ X @ Y) => (~ ((X = Y))))))). % less_imp_not_eq
thf(fact_95_less__imp__not__eq, axiom,
    ((![X : nat, Y : nat]: ((ord_less_nat @ X @ Y) => (~ ((X = Y))))))). % less_imp_not_eq
thf(fact_96_less__not__sym, axiom,
    ((![X : real, Y : real]: ((ord_less_real @ X @ Y) => (~ ((ord_less_real @ Y @ X))))))). % less_not_sym
thf(fact_97_less__not__sym, axiom,
    ((![X : nat, Y : nat]: ((ord_less_nat @ X @ Y) => (~ ((ord_less_nat @ Y @ X))))))). % less_not_sym
thf(fact_98_less__induct, axiom,
    ((![P : nat > $o, A : nat]: ((![X2 : nat]: ((![Y3 : nat]: ((ord_less_nat @ Y3 @ X2) => (P @ Y3))) => (P @ X2))) => (P @ A))))). % less_induct
thf(fact_99_antisym__conv3, axiom,
    ((![Y : real, X : real]: ((~ ((ord_less_real @ Y @ X))) => ((~ ((ord_less_real @ X @ Y))) = (X = Y)))))). % antisym_conv3
thf(fact_100_antisym__conv3, axiom,
    ((![Y : nat, X : nat]: ((~ ((ord_less_nat @ Y @ X))) => ((~ ((ord_less_nat @ X @ Y))) = (X = Y)))))). % antisym_conv3
thf(fact_101_less__imp__not__eq2, axiom,
    ((![X : real, Y : real]: ((ord_less_real @ X @ Y) => (~ ((Y = X))))))). % less_imp_not_eq2
thf(fact_102_less__imp__not__eq2, axiom,
    ((![X : nat, Y : nat]: ((ord_less_nat @ X @ Y) => (~ ((Y = X))))))). % less_imp_not_eq2
thf(fact_103_less__imp__triv, axiom,
    ((![X : real, Y : real, P : $o]: ((ord_less_real @ X @ Y) => ((ord_less_real @ Y @ X) => P))))). % less_imp_triv
thf(fact_104_less__imp__triv, axiom,
    ((![X : nat, Y : nat, P : $o]: ((ord_less_nat @ X @ Y) => ((ord_less_nat @ Y @ X) => P))))). % less_imp_triv
thf(fact_105_linorder__cases, axiom,
    ((![X : real, Y : real]: ((~ ((ord_less_real @ X @ Y))) => ((~ ((X = Y))) => (ord_less_real @ Y @ X)))))). % linorder_cases
thf(fact_106_linorder__cases, axiom,
    ((![X : nat, Y : nat]: ((~ ((ord_less_nat @ X @ Y))) => ((~ ((X = Y))) => (ord_less_nat @ Y @ X)))))). % linorder_cases
thf(fact_107_dual__order_Oirrefl, axiom,
    ((![A : real]: (~ ((ord_less_real @ A @ A)))))). % dual_order.irrefl
thf(fact_108_dual__order_Oirrefl, axiom,
    ((![A : nat]: (~ ((ord_less_nat @ A @ A)))))). % dual_order.irrefl
thf(fact_109_order_Ostrict__trans, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_real @ A @ B) => ((ord_less_real @ B @ C) => (ord_less_real @ A @ C)))))). % order.strict_trans
thf(fact_110_order_Ostrict__trans, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_nat @ A @ B) => ((ord_less_nat @ B @ C) => (ord_less_nat @ A @ C)))))). % order.strict_trans
thf(fact_111_less__imp__not__less, axiom,
    ((![X : real, Y : real]: ((ord_less_real @ X @ Y) => (~ ((ord_less_real @ Y @ X))))))). % less_imp_not_less
thf(fact_112_less__imp__not__less, axiom,
    ((![X : nat, Y : nat]: ((ord_less_nat @ X @ Y) => (~ ((ord_less_nat @ Y @ X))))))). % less_imp_not_less
thf(fact_113_exists__least__iff, axiom,
    (((^[P2 : nat > $o]: (?[X3 : nat]: (P2 @ X3))) = (^[P3 : nat > $o]: (?[N3 : nat]: (((P3 @ N3)) & ((![M2 : nat]: (((ord_less_nat @ M2 @ N3)) => ((~ ((P3 @ M2))))))))))))). % exists_least_iff
thf(fact_114_linorder__less__wlog, axiom,
    ((![P : real > real > $o, A : real, B : real]: ((![A2 : real, B2 : real]: ((ord_less_real @ A2 @ B2) => (P @ A2 @ B2))) => ((![A2 : real]: (P @ A2 @ A2)) => ((![A2 : real, B2 : real]: ((P @ B2 @ A2) => (P @ A2 @ B2))) => (P @ A @ B))))))). % linorder_less_wlog
thf(fact_115_linorder__less__wlog, axiom,
    ((![P : nat > nat > $o, A : nat, B : nat]: ((![A2 : nat, B2 : nat]: ((ord_less_nat @ A2 @ B2) => (P @ A2 @ B2))) => ((![A2 : nat]: (P @ A2 @ A2)) => ((![A2 : nat, B2 : nat]: ((P @ B2 @ A2) => (P @ A2 @ B2))) => (P @ A @ B))))))). % linorder_less_wlog
thf(fact_116_dual__order_Ostrict__trans, axiom,
    ((![B : real, A : real, C : real]: ((ord_less_real @ B @ A) => ((ord_less_real @ C @ B) => (ord_less_real @ C @ A)))))). % dual_order.strict_trans
thf(fact_117_dual__order_Ostrict__trans, axiom,
    ((![B : nat, A : nat, C : nat]: ((ord_less_nat @ B @ A) => ((ord_less_nat @ C @ B) => (ord_less_nat @ C @ A)))))). % dual_order.strict_trans
thf(fact_118_not__less__iff__gr__or__eq, axiom,
    ((![X : real, Y : real]: ((~ ((ord_less_real @ X @ Y))) = (((ord_less_real @ Y @ X)) | ((X = Y))))))). % not_less_iff_gr_or_eq
thf(fact_119_not__less__iff__gr__or__eq, axiom,
    ((![X : nat, Y : nat]: ((~ ((ord_less_nat @ X @ Y))) = (((ord_less_nat @ Y @ X)) | ((X = Y))))))). % not_less_iff_gr_or_eq
thf(fact_120_order_Ostrict__implies__not__eq, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ B) => (~ ((A = B))))))). % order.strict_implies_not_eq
thf(fact_121_order_Ostrict__implies__not__eq, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (~ ((A = B))))))). % order.strict_implies_not_eq
thf(fact_122_gr__zeroI, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr_zeroI
thf(fact_123_dual__order_Ostrict__implies__not__eq, axiom,
    ((![B : real, A : real]: ((ord_less_real @ B @ A) => (~ ((A = B))))))). % dual_order.strict_implies_not_eq
thf(fact_124_dual__order_Ostrict__implies__not__eq, axiom,
    ((![B : nat, A : nat]: ((ord_less_nat @ B @ A) => (~ ((A = B))))))). % dual_order.strict_implies_not_eq
thf(fact_125_not__less__zero, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less_zero
thf(fact_126_gr__implies__not__zero, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not_zero
thf(fact_127_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_128_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_129_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_130_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_131_less__numeral__extra_I1_J, axiom,
    ((ord_less_real @ zero_zero_real @ one_one_real))). % less_numeral_extra(1)
thf(fact_132_less__numeral__extra_I1_J, axiom,
    ((ord_less_nat @ zero_zero_nat @ one_one_nat))). % less_numeral_extra(1)
thf(fact_133_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_134_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_135_power__strict__mono, axiom,
    ((![A : real, B : real, N : nat]: ((ord_less_real @ A @ B) => ((ord_less_eq_real @ zero_zero_real @ A) => ((ord_less_nat @ zero_zero_nat @ N) => (ord_less_real @ (power_power_real @ A @ N) @ (power_power_real @ B @ N)))))))). % power_strict_mono
thf(fact_136_power__strict__mono, axiom,
    ((![A : nat, B : nat, N : nat]: ((ord_less_nat @ A @ B) => ((ord_less_eq_nat @ zero_zero_nat @ A) => ((ord_less_nat @ zero_zero_nat @ N) => (ord_less_nat @ (power_power_nat @ A @ N) @ (power_power_nat @ B @ N)))))))). % power_strict_mono
thf(fact_137_real__sup__exists, axiom,
    ((![P : real > $o]: ((?[X_12 : real]: (P @ X_12)) => ((?[Z3 : real]: (![X2 : real]: ((P @ X2) => (ord_less_real @ X2 @ Z3)))) => (?[S : real]: (![Y3 : real]: ((?[X4 : real]: (((P @ X4)) & ((ord_less_real @ Y3 @ X4)))) = (ord_less_real @ Y3 @ S))))))))). % real_sup_exists
thf(fact_138_order_Onot__eq__order__implies__strict, axiom,
    ((![A : real, B : real]: ((~ ((A = B))) => ((ord_less_eq_real @ A @ B) => (ord_less_real @ A @ B)))))). % order.not_eq_order_implies_strict
thf(fact_139_order_Onot__eq__order__implies__strict, axiom,
    ((![A : nat, B : nat]: ((~ ((A = B))) => ((ord_less_eq_nat @ A @ B) => (ord_less_nat @ A @ B)))))). % order.not_eq_order_implies_strict
thf(fact_140_dual__order_Ostrict__implies__order, axiom,
    ((![B : real, A : real]: ((ord_less_real @ B @ A) => (ord_less_eq_real @ B @ A))))). % dual_order.strict_implies_order
thf(fact_141_dual__order_Ostrict__implies__order, axiom,
    ((![B : nat, A : nat]: ((ord_less_nat @ B @ A) => (ord_less_eq_nat @ B @ A))))). % dual_order.strict_implies_order
thf(fact_142_dual__order_Ostrict__iff__order, axiom,
    ((ord_less_real = (^[B3 : real]: (^[A3 : real]: (((ord_less_eq_real @ B3 @ A3)) & ((~ ((A3 = B3)))))))))). % dual_order.strict_iff_order
thf(fact_143_dual__order_Ostrict__iff__order, axiom,
    ((ord_less_nat = (^[B3 : nat]: (^[A3 : nat]: (((ord_less_eq_nat @ B3 @ A3)) & ((~ ((A3 = B3)))))))))). % dual_order.strict_iff_order
thf(fact_144_dual__order_Oorder__iff__strict, axiom,
    ((ord_less_eq_real = (^[B3 : real]: (^[A3 : real]: (((ord_less_real @ B3 @ A3)) | ((A3 = B3)))))))). % dual_order.order_iff_strict
thf(fact_145_dual__order_Oorder__iff__strict, axiom,
    ((ord_less_eq_nat = (^[B3 : nat]: (^[A3 : nat]: (((ord_less_nat @ B3 @ A3)) | ((A3 = B3)))))))). % dual_order.order_iff_strict
thf(fact_146_order_Ostrict__implies__order, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ B) => (ord_less_eq_real @ A @ B))))). % order.strict_implies_order
thf(fact_147_order_Ostrict__implies__order, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (ord_less_eq_nat @ A @ B))))). % order.strict_implies_order
thf(fact_148_dense__le__bounded, axiom,
    ((![X : real, Y : real, Z2 : real]: ((ord_less_real @ X @ Y) => ((![W : real]: ((ord_less_real @ X @ W) => ((ord_less_real @ W @ Y) => (ord_less_eq_real @ W @ Z2)))) => (ord_less_eq_real @ Y @ Z2)))))). % dense_le_bounded
thf(fact_149_dense__ge__bounded, axiom,
    ((![Z2 : real, X : real, Y : real]: ((ord_less_real @ Z2 @ X) => ((![W : real]: ((ord_less_real @ Z2 @ W) => ((ord_less_real @ W @ X) => (ord_less_eq_real @ Y @ W)))) => (ord_less_eq_real @ Y @ Z2)))))). % dense_ge_bounded
thf(fact_150_dual__order_Ostrict__trans2, axiom,
    ((![B : real, A : real, C : real]: ((ord_less_real @ B @ A) => ((ord_less_eq_real @ C @ B) => (ord_less_real @ C @ A)))))). % dual_order.strict_trans2
thf(fact_151_dual__order_Ostrict__trans2, axiom,
    ((![B : nat, A : nat, C : nat]: ((ord_less_nat @ B @ A) => ((ord_less_eq_nat @ C @ B) => (ord_less_nat @ C @ A)))))). % dual_order.strict_trans2
thf(fact_152_dual__order_Ostrict__trans1, axiom,
    ((![B : real, A : real, C : real]: ((ord_less_eq_real @ B @ A) => ((ord_less_real @ C @ B) => (ord_less_real @ C @ A)))))). % dual_order.strict_trans1
thf(fact_153_dual__order_Ostrict__trans1, axiom,
    ((![B : nat, A : nat, C : nat]: ((ord_less_eq_nat @ B @ A) => ((ord_less_nat @ C @ B) => (ord_less_nat @ C @ A)))))). % dual_order.strict_trans1
thf(fact_154_order_Ostrict__iff__order, axiom,
    ((ord_less_real = (^[A3 : real]: (^[B3 : real]: (((ord_less_eq_real @ A3 @ B3)) & ((~ ((A3 = B3)))))))))). % order.strict_iff_order
thf(fact_155_order_Ostrict__iff__order, axiom,
    ((ord_less_nat = (^[A3 : nat]: (^[B3 : nat]: (((ord_less_eq_nat @ A3 @ B3)) & ((~ ((A3 = B3)))))))))). % order.strict_iff_order
thf(fact_156_order_Oorder__iff__strict, axiom,
    ((ord_less_eq_real = (^[A3 : real]: (^[B3 : real]: (((ord_less_real @ A3 @ B3)) | ((A3 = B3)))))))). % order.order_iff_strict
thf(fact_157_order_Oorder__iff__strict, axiom,
    ((ord_less_eq_nat = (^[A3 : nat]: (^[B3 : nat]: (((ord_less_nat @ A3 @ B3)) | ((A3 = B3)))))))). % order.order_iff_strict
thf(fact_158_order_Ostrict__trans2, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_real @ A @ B) => ((ord_less_eq_real @ B @ C) => (ord_less_real @ A @ C)))))). % order.strict_trans2
thf(fact_159_order_Ostrict__trans2, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_nat @ A @ B) => ((ord_less_eq_nat @ B @ C) => (ord_less_nat @ A @ C)))))). % order.strict_trans2
thf(fact_160_order_Ostrict__trans1, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_real @ B @ C) => (ord_less_real @ A @ C)))))). % order.strict_trans1
thf(fact_161_order_Ostrict__trans1, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_nat @ B @ C) => (ord_less_nat @ A @ C)))))). % order.strict_trans1
thf(fact_162_not__le__imp__less, axiom,
    ((![Y : real, X : real]: ((~ ((ord_less_eq_real @ Y @ X))) => (ord_less_real @ X @ Y))))). % not_le_imp_less
thf(fact_163_not__le__imp__less, axiom,
    ((![Y : nat, X : nat]: ((~ ((ord_less_eq_nat @ Y @ X))) => (ord_less_nat @ X @ Y))))). % not_le_imp_less
thf(fact_164_less__le__not__le, axiom,
    ((ord_less_real = (^[X4 : real]: (^[Y4 : real]: (((ord_less_eq_real @ X4 @ Y4)) & ((~ ((ord_less_eq_real @ Y4 @ X4)))))))))). % less_le_not_le
thf(fact_165_less__le__not__le, axiom,
    ((ord_less_nat = (^[X4 : nat]: (^[Y4 : nat]: (((ord_less_eq_nat @ X4 @ Y4)) & ((~ ((ord_less_eq_nat @ Y4 @ X4)))))))))). % less_le_not_le
thf(fact_166_le__imp__less__or__eq, axiom,
    ((![X : real, Y : real]: ((ord_less_eq_real @ X @ Y) => ((ord_less_real @ X @ Y) | (X = Y)))))). % le_imp_less_or_eq
thf(fact_167_le__imp__less__or__eq, axiom,
    ((![X : nat, Y : nat]: ((ord_less_eq_nat @ X @ Y) => ((ord_less_nat @ X @ Y) | (X = Y)))))). % le_imp_less_or_eq
thf(fact_168_le__less__linear, axiom,
    ((![X : real, Y : real]: ((ord_less_eq_real @ X @ Y) | (ord_less_real @ Y @ X))))). % le_less_linear
thf(fact_169_le__less__linear, axiom,
    ((![X : nat, Y : nat]: ((ord_less_eq_nat @ X @ Y) | (ord_less_nat @ Y @ X))))). % le_less_linear
thf(fact_170_dense__le, axiom,
    ((![Y : real, Z2 : real]: ((![X2 : real]: ((ord_less_real @ X2 @ Y) => (ord_less_eq_real @ X2 @ Z2))) => (ord_less_eq_real @ Y @ Z2))))). % dense_le
thf(fact_171_dense__ge, axiom,
    ((![Z2 : real, Y : real]: ((![X2 : real]: ((ord_less_real @ Z2 @ X2) => (ord_less_eq_real @ Y @ X2))) => (ord_less_eq_real @ Y @ Z2))))). % dense_ge
thf(fact_172_less__le__trans, axiom,
    ((![X : real, Y : real, Z2 : real]: ((ord_less_real @ X @ Y) => ((ord_less_eq_real @ Y @ Z2) => (ord_less_real @ X @ Z2)))))). % less_le_trans
thf(fact_173_less__le__trans, axiom,
    ((![X : nat, Y : nat, Z2 : nat]: ((ord_less_nat @ X @ Y) => ((ord_less_eq_nat @ Y @ Z2) => (ord_less_nat @ X @ Z2)))))). % less_le_trans
thf(fact_174_le__less__trans, axiom,
    ((![X : real, Y : real, Z2 : real]: ((ord_less_eq_real @ X @ Y) => ((ord_less_real @ Y @ Z2) => (ord_less_real @ X @ Z2)))))). % le_less_trans
thf(fact_175_le__less__trans, axiom,
    ((![X : nat, Y : nat, Z2 : nat]: ((ord_less_eq_nat @ X @ Y) => ((ord_less_nat @ Y @ Z2) => (ord_less_nat @ X @ Z2)))))). % le_less_trans
thf(fact_176_less__imp__le, axiom,
    ((![X : real, Y : real]: ((ord_less_real @ X @ Y) => (ord_less_eq_real @ X @ Y))))). % less_imp_le
thf(fact_177_less__imp__le, axiom,
    ((![X : nat, Y : nat]: ((ord_less_nat @ X @ Y) => (ord_less_eq_nat @ X @ Y))))). % less_imp_le
thf(fact_178_antisym__conv2, axiom,
    ((![X : real, Y : real]: ((ord_less_eq_real @ X @ Y) => ((~ ((ord_less_real @ X @ Y))) = (X = Y)))))). % antisym_conv2
thf(fact_179_antisym__conv2, axiom,
    ((![X : nat, Y : nat]: ((ord_less_eq_nat @ X @ Y) => ((~ ((ord_less_nat @ X @ Y))) = (X = Y)))))). % antisym_conv2
thf(fact_180_antisym__conv1, axiom,
    ((![X : real, Y : real]: ((~ ((ord_less_real @ X @ Y))) => ((ord_less_eq_real @ X @ Y) = (X = Y)))))). % antisym_conv1
thf(fact_181_antisym__conv1, axiom,
    ((![X : nat, Y : nat]: ((~ ((ord_less_nat @ X @ Y))) => ((ord_less_eq_nat @ X @ Y) = (X = Y)))))). % antisym_conv1
thf(fact_182_le__neq__trans, axiom,
    ((![A : real, B : real]: ((ord_less_eq_real @ A @ B) => ((~ ((A = B))) => (ord_less_real @ A @ B)))))). % le_neq_trans
thf(fact_183_le__neq__trans, axiom,
    ((![A : nat, B : nat]: ((ord_less_eq_nat @ A @ B) => ((~ ((A = B))) => (ord_less_nat @ A @ B)))))). % le_neq_trans
thf(fact_184_not__less, axiom,
    ((![X : real, Y : real]: ((~ ((ord_less_real @ X @ Y))) = (ord_less_eq_real @ Y @ X))))). % not_less
thf(fact_185_not__less, axiom,
    ((![X : nat, Y : nat]: ((~ ((ord_less_nat @ X @ Y))) = (ord_less_eq_nat @ Y @ X))))). % not_less
thf(fact_186_not__le, axiom,
    ((![X : real, Y : real]: ((~ ((ord_less_eq_real @ X @ Y))) = (ord_less_real @ Y @ X))))). % not_le
thf(fact_187_not__le, axiom,
    ((![X : nat, Y : nat]: ((~ ((ord_less_eq_nat @ X @ Y))) = (ord_less_nat @ Y @ X))))). % not_le
thf(fact_188_order__less__le__subst2, axiom,
    ((![A : real, B : real, F : real > real, C : real]: ((ord_less_real @ A @ B) => ((ord_less_eq_real @ (F @ B) @ C) => ((![X2 : real, Y2 : real]: ((ord_less_real @ X2 @ Y2) => (ord_less_real @ (F @ X2) @ (F @ Y2)))) => (ord_less_real @ (F @ A) @ C))))))). % order_less_le_subst2
thf(fact_189_order__less__le__subst2, axiom,
    ((![A : nat, B : nat, F : nat > real, C : real]: ((ord_less_nat @ A @ B) => ((ord_less_eq_real @ (F @ B) @ C) => ((![X2 : nat, Y2 : nat]: ((ord_less_nat @ X2 @ Y2) => (ord_less_real @ (F @ X2) @ (F @ Y2)))) => (ord_less_real @ (F @ A) @ C))))))). % order_less_le_subst2
thf(fact_190_order__less__le__subst2, axiom,
    ((![A : real, B : real, F : real > nat, C : nat]: ((ord_less_real @ A @ B) => ((ord_less_eq_nat @ (F @ B) @ C) => ((![X2 : real, Y2 : real]: ((ord_less_real @ X2 @ Y2) => (ord_less_nat @ (F @ X2) @ (F @ Y2)))) => (ord_less_nat @ (F @ A) @ C))))))). % order_less_le_subst2
thf(fact_191_order__less__le__subst2, axiom,
    ((![A : nat, B : nat, F : nat > nat, C : nat]: ((ord_less_nat @ A @ B) => ((ord_less_eq_nat @ (F @ B) @ C) => ((![X2 : nat, Y2 : nat]: ((ord_less_nat @ X2 @ Y2) => (ord_less_nat @ (F @ X2) @ (F @ Y2)))) => (ord_less_nat @ (F @ A) @ C))))))). % order_less_le_subst2
thf(fact_192_order__less__le__subst1, axiom,
    ((![A : real, F : real > real, B : real, C : real]: ((ord_less_real @ A @ (F @ B)) => ((ord_less_eq_real @ B @ C) => ((![X2 : real, Y2 : real]: ((ord_less_eq_real @ X2 @ Y2) => (ord_less_eq_real @ (F @ X2) @ (F @ Y2)))) => (ord_less_real @ A @ (F @ C)))))))). % order_less_le_subst1
thf(fact_193_order__less__le__subst1, axiom,
    ((![A : nat, F : real > nat, B : real, C : real]: ((ord_less_nat @ A @ (F @ B)) => ((ord_less_eq_real @ B @ C) => ((![X2 : real, Y2 : real]: ((ord_less_eq_real @ X2 @ Y2) => (ord_less_eq_nat @ (F @ X2) @ (F @ Y2)))) => (ord_less_nat @ A @ (F @ C)))))))). % order_less_le_subst1
thf(fact_194_order__less__le__subst1, axiom,
    ((![A : real, F : nat > real, B : nat, C : nat]: ((ord_less_real @ A @ (F @ B)) => ((ord_less_eq_nat @ B @ C) => ((![X2 : nat, Y2 : nat]: ((ord_less_eq_nat @ X2 @ Y2) => (ord_less_eq_real @ (F @ X2) @ (F @ Y2)))) => (ord_less_real @ A @ (F @ C)))))))). % order_less_le_subst1
thf(fact_195_order__less__le__subst1, axiom,
    ((![A : nat, F : nat > nat, B : nat, C : nat]: ((ord_less_nat @ A @ (F @ B)) => ((ord_less_eq_nat @ B @ C) => ((![X2 : nat, Y2 : nat]: ((ord_less_eq_nat @ X2 @ Y2) => (ord_less_eq_nat @ (F @ X2) @ (F @ Y2)))) => (ord_less_nat @ A @ (F @ C)))))))). % order_less_le_subst1
thf(fact_196_order__le__less__subst2, axiom,
    ((![A : real, B : real, F : real > real, C : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_real @ (F @ B) @ C) => ((![X2 : real, Y2 : real]: ((ord_less_eq_real @ X2 @ Y2) => (ord_less_eq_real @ (F @ X2) @ (F @ Y2)))) => (ord_less_real @ (F @ A) @ C))))))). % order_le_less_subst2
thf(fact_197_order__le__less__subst2, axiom,
    ((![A : real, B : real, F : real > nat, C : nat]: ((ord_less_eq_real @ A @ B) => ((ord_less_nat @ (F @ B) @ C) => ((![X2 : real, Y2 : real]: ((ord_less_eq_real @ X2 @ Y2) => (ord_less_eq_nat @ (F @ X2) @ (F @ Y2)))) => (ord_less_nat @ (F @ A) @ C))))))). % order_le_less_subst2
thf(fact_198_order__le__less__subst2, axiom,
    ((![A : nat, B : nat, F : nat > real, C : real]: ((ord_less_eq_nat @ A @ B) => ((ord_less_real @ (F @ B) @ C) => ((![X2 : nat, Y2 : nat]: ((ord_less_eq_nat @ X2 @ Y2) => (ord_less_eq_real @ (F @ X2) @ (F @ Y2)))) => (ord_less_real @ (F @ A) @ C))))))). % order_le_less_subst2
thf(fact_199_order__le__less__subst2, axiom,
    ((![A : nat, B : nat, F : nat > nat, C : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_nat @ (F @ B) @ C) => ((![X2 : nat, Y2 : nat]: ((ord_less_eq_nat @ X2 @ Y2) => (ord_less_eq_nat @ (F @ X2) @ (F @ Y2)))) => (ord_less_nat @ (F @ A) @ C))))))). % order_le_less_subst2
thf(fact_200_order__le__less__subst1, axiom,
    ((![A : real, F : real > real, B : real, C : real]: ((ord_less_eq_real @ A @ (F @ B)) => ((ord_less_real @ B @ C) => ((![X2 : real, Y2 : real]: ((ord_less_real @ X2 @ Y2) => (ord_less_real @ (F @ X2) @ (F @ Y2)))) => (ord_less_real @ A @ (F @ C)))))))). % order_le_less_subst1
thf(fact_201_order__le__less__subst1, axiom,
    ((![A : real, F : nat > real, B : nat, C : nat]: ((ord_less_eq_real @ A @ (F @ B)) => ((ord_less_nat @ B @ C) => ((![X2 : nat, Y2 : nat]: ((ord_less_nat @ X2 @ Y2) => (ord_less_real @ (F @ X2) @ (F @ Y2)))) => (ord_less_real @ A @ (F @ C)))))))). % order_le_less_subst1
thf(fact_202_order__le__less__subst1, axiom,
    ((![A : nat, F : real > nat, B : real, C : real]: ((ord_less_eq_nat @ A @ (F @ B)) => ((ord_less_real @ B @ C) => ((![X2 : real, Y2 : real]: ((ord_less_real @ X2 @ Y2) => (ord_less_nat @ (F @ X2) @ (F @ Y2)))) => (ord_less_nat @ A @ (F @ C)))))))). % order_le_less_subst1
thf(fact_203_order__le__less__subst1, axiom,
    ((![A : nat, F : nat > nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ (F @ B)) => ((ord_less_nat @ B @ C) => ((![X2 : nat, Y2 : nat]: ((ord_less_nat @ X2 @ Y2) => (ord_less_nat @ (F @ X2) @ (F @ Y2)))) => (ord_less_nat @ A @ (F @ C)))))))). % order_le_less_subst1
thf(fact_204_less__le, axiom,
    ((ord_less_real = (^[X4 : real]: (^[Y4 : real]: (((ord_less_eq_real @ X4 @ Y4)) & ((~ ((X4 = Y4)))))))))). % less_le
thf(fact_205_less__le, axiom,
    ((ord_less_nat = (^[X4 : nat]: (^[Y4 : nat]: (((ord_less_eq_nat @ X4 @ Y4)) & ((~ ((X4 = Y4)))))))))). % less_le
thf(fact_206_le__less, axiom,
    ((ord_less_eq_real = (^[X4 : real]: (^[Y4 : real]: (((ord_less_real @ X4 @ Y4)) | ((X4 = Y4)))))))). % le_less
thf(fact_207_le__less, axiom,
    ((ord_less_eq_nat = (^[X4 : nat]: (^[Y4 : nat]: (((ord_less_nat @ X4 @ Y4)) | ((X4 = Y4)))))))). % le_less
thf(fact_208_leI, axiom,
    ((![X : real, Y : real]: ((~ ((ord_less_real @ X @ Y))) => (ord_less_eq_real @ Y @ X))))). % leI
thf(fact_209_leI, axiom,
    ((![X : nat, Y : nat]: ((~ ((ord_less_nat @ X @ Y))) => (ord_less_eq_nat @ Y @ X))))). % leI
thf(fact_210_leD, axiom,
    ((![Y : real, X : real]: ((ord_less_eq_real @ Y @ X) => (~ ((ord_less_real @ X @ Y))))))). % leD
thf(fact_211_leD, axiom,
    ((![Y : nat, X : nat]: ((ord_less_eq_nat @ Y @ X) => (~ ((ord_less_nat @ X @ Y))))))). % leD
thf(fact_212_less__numeral__extra_I4_J, axiom,
    ((~ ((ord_less_real @ one_one_real @ one_one_real))))). % less_numeral_extra(4)
thf(fact_213_less__numeral__extra_I4_J, axiom,
    ((~ ((ord_less_nat @ one_one_nat @ one_one_nat))))). % less_numeral_extra(4)
thf(fact_214_zero__le, axiom,
    ((![X : nat]: (ord_less_eq_nat @ zero_zero_nat @ X)))). % zero_le
thf(fact_215_le__numeral__extra_I3_J, axiom,
    ((ord_less_eq_real @ zero_zero_real @ zero_zero_real))). % le_numeral_extra(3)
thf(fact_216_le__numeral__extra_I3_J, axiom,
    ((ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat))). % le_numeral_extra(3)
thf(fact_217_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_218_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_219_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_220_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_221_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_222_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_223_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_224_power__eq__imp__eq__base, axiom,
    ((![A : real, N : nat, B : real]: (((power_power_real @ A @ N) = (power_power_real @ B @ N)) => ((ord_less_eq_real @ zero_zero_real @ A) => ((ord_less_eq_real @ zero_zero_real @ B) => ((ord_less_nat @ zero_zero_nat @ N) => (A = B)))))))). % power_eq_imp_eq_base
thf(fact_225_power__eq__imp__eq__base, axiom,
    ((![A : nat, N : nat, B : nat]: (((power_power_nat @ A @ N) = (power_power_nat @ B @ N)) => ((ord_less_eq_nat @ zero_zero_nat @ A) => ((ord_less_eq_nat @ zero_zero_nat @ B) => ((ord_less_nat @ zero_zero_nat @ N) => (A = B)))))))). % power_eq_imp_eq_base
thf(fact_226_power__eq__iff__eq__base, axiom,
    ((![N : nat, A : real, B : real]: ((ord_less_nat @ zero_zero_nat @ N) => ((ord_less_eq_real @ zero_zero_real @ A) => ((ord_less_eq_real @ zero_zero_real @ B) => (((power_power_real @ A @ N) = (power_power_real @ B @ N)) = (A = B)))))))). % power_eq_iff_eq_base
thf(fact_227_power__eq__iff__eq__base, axiom,
    ((![N : nat, A : nat, B : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((ord_less_eq_nat @ zero_zero_nat @ A) => ((ord_less_eq_nat @ zero_zero_nat @ B) => (((power_power_nat @ A @ N) = (power_power_nat @ B @ N)) = (A = B)))))))). % power_eq_iff_eq_base
thf(fact_228_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_229_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_230_power__less__imp__less__exp, axiom,
    ((![A : real, M : nat, N : nat]: ((ord_less_real @ one_one_real @ A) => ((ord_less_real @ (power_power_real @ A @ M) @ (power_power_real @ A @ N)) => (ord_less_nat @ M @ N)))))). % power_less_imp_less_exp
thf(fact_231_power__less__imp__less__exp, axiom,
    ((![A : nat, M : nat, N : nat]: ((ord_less_nat @ one_one_nat @ A) => ((ord_less_nat @ (power_power_nat @ A @ M) @ (power_power_nat @ A @ N)) => (ord_less_nat @ M @ N)))))). % power_less_imp_less_exp
thf(fact_232_power__less__imp__less__base, axiom,
    ((![A : real, N : nat, B : real]: ((ord_less_real @ (power_power_real @ A @ N) @ (power_power_real @ B @ N)) => ((ord_less_eq_real @ zero_zero_real @ B) => (ord_less_real @ A @ B)))))). % power_less_imp_less_base
thf(fact_233_power__less__imp__less__base, axiom,
    ((![A : nat, N : nat, B : nat]: ((ord_less_nat @ (power_power_nat @ A @ N) @ (power_power_nat @ B @ N)) => ((ord_less_eq_nat @ zero_zero_nat @ B) => (ord_less_nat @ A @ B)))))). % power_less_imp_less_base
thf(fact_234_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_235_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_236_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_237_dual__order_Oantisym, axiom,
    ((![B : real, A : real]: ((ord_less_eq_real @ B @ A) => ((ord_less_eq_real @ A @ B) => (A = B)))))). % dual_order.antisym
thf(fact_238_dual__order_Oantisym, axiom,
    ((![B : nat, A : nat]: ((ord_less_eq_nat @ B @ A) => ((ord_less_eq_nat @ A @ B) => (A = B)))))). % dual_order.antisym
thf(fact_239_dual__order_Oeq__iff, axiom,
    (((^[Y5 : real]: (^[Z4 : real]: (Y5 = Z4))) = (^[A3 : real]: (^[B3 : real]: (((ord_less_eq_real @ B3 @ A3)) & ((ord_less_eq_real @ A3 @ B3)))))))). % dual_order.eq_iff
thf(fact_240_dual__order_Oeq__iff, axiom,
    (((^[Y5 : nat]: (^[Z4 : nat]: (Y5 = Z4))) = (^[A3 : nat]: (^[B3 : nat]: (((ord_less_eq_nat @ B3 @ A3)) & ((ord_less_eq_nat @ A3 @ B3)))))))). % dual_order.eq_iff
thf(fact_241_dual__order_Otrans, axiom,
    ((![B : real, A : real, C : real]: ((ord_less_eq_real @ B @ A) => ((ord_less_eq_real @ C @ B) => (ord_less_eq_real @ C @ A)))))). % dual_order.trans
thf(fact_242_dual__order_Otrans, axiom,
    ((![B : nat, A : nat, C : nat]: ((ord_less_eq_nat @ B @ A) => ((ord_less_eq_nat @ C @ B) => (ord_less_eq_nat @ C @ A)))))). % dual_order.trans
thf(fact_243_linorder__wlog, axiom,
    ((![P : real > real > $o, A : real, B : real]: ((![A2 : real, B2 : real]: ((ord_less_eq_real @ A2 @ B2) => (P @ A2 @ B2))) => ((![A2 : real, B2 : real]: ((P @ B2 @ A2) => (P @ A2 @ B2))) => (P @ A @ B)))))). % linorder_wlog
thf(fact_244_linorder__wlog, axiom,
    ((![P : nat > nat > $o, A : nat, B : nat]: ((![A2 : nat, B2 : nat]: ((ord_less_eq_nat @ A2 @ B2) => (P @ A2 @ B2))) => ((![A2 : nat, B2 : nat]: ((P @ B2 @ A2) => (P @ A2 @ B2))) => (P @ A @ B)))))). % linorder_wlog
thf(fact_245_dual__order_Orefl, axiom,
    ((![A : real]: (ord_less_eq_real @ A @ A)))). % dual_order.refl
thf(fact_246_dual__order_Orefl, axiom,
    ((![A : nat]: (ord_less_eq_nat @ A @ A)))). % dual_order.refl
thf(fact_247_order__trans, axiom,
    ((![X : nat, Y : nat, Z2 : nat]: ((ord_less_eq_nat @ X @ Y) => ((ord_less_eq_nat @ Y @ Z2) => (ord_less_eq_nat @ X @ Z2)))))). % order_trans

% Conjectures (1)
thf(conj_0, conjecture,
    ((ord_less_eq_real @ (power_power_real @ t @ k) @ one_one_real))).
