% 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_146__5369062_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:27:28.819

% Could-be-implicit typings (4)
thf(ty_n_t__Set__Oset_It__Real__Oreal_J, type,
    set_real : $tType).
thf(ty_n_t__Real__Oreal, type,
    real : $tType).
thf(ty_n_t__Num__Onum, type,
    num : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).

% Explicit typings (23)
thf(sy_c_Groups_Oabs__class_Oabs_001t__Real__Oreal, type,
    abs_abs_real : real > real).
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__Nat__Onat, type,
    plus_plus_nat : nat > nat > nat).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Num__Onum, type,
    plus_plus_num : num > num > num).
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__Nat__Onat, type,
    times_times_nat : nat > nat > nat).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Num__Onum, type,
    times_times_num : num > num > num).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal, type,
    times_times_real : real > real > real).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal, type,
    uminus_uminus_real : real > real).
thf(sy_c_Num_Onum_OBit0, type,
    bit0 : num > num).
thf(sy_c_Num_Onum_OOne, type,
    one : num).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat, type,
    numeral_numeral_nat : num > nat).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal, type,
    numeral_numeral_real : num > real).
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__Num__Onum, type,
    ord_less_eq_num : num > num > $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__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_Set_OCollect_001t__Real__Oreal, type,
    collect_real : (real > $o) > set_real).
thf(sy_c_member_001t__Real__Oreal, type,
    member_real : real > set_real > $o).
thf(sy_v_x____, type,
    x : real).
thf(sy_v_y____, type,
    y : real).

% Relevant facts (177)
thf(fact_0__092_060open_062_092_060bar_0622_A_K_Ay_092_060bar_062_092_060_094sup_0622_A_092_060le_062_A1_092_060_094sup_0622_092_060close_062, axiom,
    ((ord_less_eq_real @ (power_power_real @ (abs_abs_real @ (times_times_real @ (numeral_numeral_real @ (bit0 @ one)) @ y)) @ (numeral_numeral_nat @ (bit0 @ one))) @ (power_power_real @ one_one_real @ (numeral_numeral_nat @ (bit0 @ one)))))). % \<open>\<bar>2 * y\<bar>\<^sup>2 \<le> 1\<^sup>2\<close>
thf(fact_1__092_060open_062_092_060bar_0622_A_K_Ax_092_060bar_062_092_060_094sup_0622_A_092_060le_062_A1_092_060_094sup_0622_092_060close_062, axiom,
    ((ord_less_eq_real @ (power_power_real @ (abs_abs_real @ (times_times_real @ (numeral_numeral_real @ (bit0 @ one)) @ x)) @ (numeral_numeral_nat @ (bit0 @ one))) @ (power_power_real @ one_one_real @ (numeral_numeral_nat @ (bit0 @ one)))))). % \<open>\<bar>2 * x\<bar>\<^sup>2 \<le> 1\<^sup>2\<close>
thf(fact_2__092_060open_0622_A_K_Ax_A_092_060le_062_A1_092_060close_062, axiom,
    ((ord_less_eq_real @ (times_times_real @ (numeral_numeral_real @ (bit0 @ one)) @ x) @ one_one_real))). % \<open>2 * x \<le> 1\<close>
thf(fact_3__092_060open_0622_A_K_Ay_A_092_060le_062_A1_092_060close_062, axiom,
    ((ord_less_eq_real @ (times_times_real @ (numeral_numeral_real @ (bit0 @ one)) @ y) @ one_one_real))). % \<open>2 * y \<le> 1\<close>
thf(fact_4__092_060open_062_092_060bar_0622_A_K_Ax_092_060bar_062_A_092_060le_062_A1_092_060close_062, axiom,
    ((ord_less_eq_real @ (abs_abs_real @ (times_times_real @ (numeral_numeral_real @ (bit0 @ one)) @ x)) @ one_one_real))). % \<open>\<bar>2 * x\<bar> \<le> 1\<close>
thf(fact_5__092_060open_062_092_060bar_0622_A_K_Ay_092_060bar_062_A_092_060le_062_A1_092_060close_062, axiom,
    ((ord_less_eq_real @ (abs_abs_real @ (times_times_real @ (numeral_numeral_real @ (bit0 @ one)) @ y)) @ one_one_real))). % \<open>\<bar>2 * y\<bar> \<le> 1\<close>
thf(fact_6_numeral__le__one__iff, axiom,
    ((![N : num]: ((ord_less_eq_real @ (numeral_numeral_real @ N) @ one_one_real) = (ord_less_eq_num @ N @ one))))). % numeral_le_one_iff
thf(fact_7_numeral__le__one__iff, axiom,
    ((![N : num]: ((ord_less_eq_nat @ (numeral_numeral_nat @ N) @ one_one_nat) = (ord_less_eq_num @ N @ one))))). % numeral_le_one_iff
thf(fact_8_numeral__eq__one__iff, axiom,
    ((![N : num]: (((numeral_numeral_real @ N) = one_one_real) = (N = one))))). % numeral_eq_one_iff
thf(fact_9_numeral__eq__one__iff, axiom,
    ((![N : num]: (((numeral_numeral_nat @ N) = one_one_nat) = (N = one))))). % numeral_eq_one_iff
thf(fact_10_one__eq__numeral__iff, axiom,
    ((![N : num]: ((one_one_real = (numeral_numeral_real @ N)) = (one = N))))). % one_eq_numeral_iff
thf(fact_11_one__eq__numeral__iff, axiom,
    ((![N : num]: ((one_one_nat = (numeral_numeral_nat @ N)) = (one = N))))). % one_eq_numeral_iff
thf(fact_12__092_060open_062_N_A1_A_092_060le_062_A2_A_K_Ax_092_060close_062, axiom,
    ((ord_less_eq_real @ (uminus_uminus_real @ one_one_real) @ (times_times_real @ (numeral_numeral_real @ (bit0 @ one)) @ x)))). % \<open>- 1 \<le> 2 * x\<close>
thf(fact_13_four__x__squared, axiom,
    ((![X : real]: ((times_times_real @ (numeral_numeral_real @ (bit0 @ (bit0 @ one))) @ (power_power_real @ X @ (numeral_numeral_nat @ (bit0 @ one)))) = (power_power_real @ (times_times_real @ (numeral_numeral_real @ (bit0 @ one)) @ X) @ (numeral_numeral_nat @ (bit0 @ one))))))). % four_x_squared
thf(fact_14_two__realpow__ge__one, axiom,
    ((![N : nat]: (ord_less_eq_real @ one_one_real @ (power_power_real @ (numeral_numeral_real @ (bit0 @ one)) @ N))))). % two_realpow_ge_one
thf(fact_15_one__power2, axiom,
    (((power_power_real @ one_one_real @ (numeral_numeral_nat @ (bit0 @ one))) = one_one_real))). % one_power2
thf(fact_16_one__power2, axiom,
    (((power_power_nat @ one_one_nat @ (numeral_numeral_nat @ (bit0 @ one))) = one_one_nat))). % one_power2
thf(fact_17_power4__eq__xxxx, axiom,
    ((![X : nat]: ((power_power_nat @ X @ (numeral_numeral_nat @ (bit0 @ (bit0 @ one)))) = (times_times_nat @ (times_times_nat @ (times_times_nat @ X @ X) @ X) @ X))))). % power4_eq_xxxx
thf(fact_18_power4__eq__xxxx, axiom,
    ((![X : real]: ((power_power_real @ X @ (numeral_numeral_nat @ (bit0 @ (bit0 @ one)))) = (times_times_real @ (times_times_real @ (times_times_real @ X @ X) @ X) @ X))))). % power4_eq_xxxx
thf(fact_19_power2__eq__square, axiom,
    ((![A : nat]: ((power_power_nat @ A @ (numeral_numeral_nat @ (bit0 @ one))) = (times_times_nat @ A @ A))))). % power2_eq_square
thf(fact_20_power2__eq__square, axiom,
    ((![A : real]: ((power_power_real @ A @ (numeral_numeral_nat @ (bit0 @ one))) = (times_times_real @ A @ A))))). % power2_eq_square
thf(fact_21_semiring__norm_I85_J, axiom,
    ((![M : num]: (~ (((bit0 @ M) = one)))))). % semiring_norm(85)
thf(fact_22_semiring__norm_I83_J, axiom,
    ((![N : num]: (~ ((one = (bit0 @ N))))))). % semiring_norm(83)
thf(fact_23_power__one, axiom,
    ((![N : nat]: ((power_power_real @ one_one_real @ N) = one_one_real)))). % power_one
thf(fact_24_power__one, axiom,
    ((![N : nat]: ((power_power_nat @ one_one_nat @ N) = one_one_nat)))). % power_one
thf(fact_25_numeral__eq__iff, axiom,
    ((![M : num, N : num]: (((numeral_numeral_real @ M) = (numeral_numeral_real @ N)) = (M = N))))). % numeral_eq_iff
thf(fact_26_numeral__eq__iff, axiom,
    ((![M : num, N : num]: (((numeral_numeral_nat @ M) = (numeral_numeral_nat @ N)) = (M = N))))). % numeral_eq_iff
thf(fact_27_semiring__norm_I13_J, axiom,
    ((![M : num, N : num]: ((times_times_num @ (bit0 @ M) @ (bit0 @ N)) = (bit0 @ (bit0 @ (times_times_num @ M @ N))))))). % semiring_norm(13)
thf(fact_28_semiring__norm_I87_J, axiom,
    ((![M : num, N : num]: (((bit0 @ M) = (bit0 @ N)) = (M = N))))). % semiring_norm(87)
thf(fact_29_semiring__norm_I11_J, axiom,
    ((![M : num]: ((times_times_num @ M @ one) = M)))). % semiring_norm(11)
thf(fact_30_semiring__norm_I12_J, axiom,
    ((![N : num]: ((times_times_num @ one @ N) = N)))). % semiring_norm(12)
thf(fact_31_power__one__right, axiom,
    ((![A : real]: ((power_power_real @ A @ one_one_nat) = A)))). % power_one_right
thf(fact_32_power__one__right, axiom,
    ((![A : nat]: ((power_power_nat @ A @ one_one_nat) = A)))). % power_one_right
thf(fact_33_numeral__times__numeral, axiom,
    ((![M : num, N : num]: ((times_times_real @ (numeral_numeral_real @ M) @ (numeral_numeral_real @ N)) = (numeral_numeral_real @ (times_times_num @ M @ N)))))). % numeral_times_numeral
thf(fact_34_numeral__times__numeral, axiom,
    ((![M : num, N : num]: ((times_times_nat @ (numeral_numeral_nat @ M) @ (numeral_numeral_nat @ N)) = (numeral_numeral_nat @ (times_times_num @ M @ N)))))). % numeral_times_numeral
thf(fact_35_mult__numeral__left__semiring__numeral, axiom,
    ((![V : num, W : num, Z : real]: ((times_times_real @ (numeral_numeral_real @ V) @ (times_times_real @ (numeral_numeral_real @ W) @ Z)) = (times_times_real @ (numeral_numeral_real @ (times_times_num @ V @ W)) @ Z))))). % mult_numeral_left_semiring_numeral
thf(fact_36_mult__numeral__left__semiring__numeral, axiom,
    ((![V : num, W : num, Z : nat]: ((times_times_nat @ (numeral_numeral_nat @ V) @ (times_times_nat @ (numeral_numeral_nat @ W) @ Z)) = (times_times_nat @ (numeral_numeral_nat @ (times_times_num @ V @ W)) @ Z))))). % mult_numeral_left_semiring_numeral
thf(fact_37_neg__numeral__eq__iff, axiom,
    ((![M : num, N : num]: (((uminus_uminus_real @ (numeral_numeral_real @ M)) = (uminus_uminus_real @ (numeral_numeral_real @ N))) = (M = N))))). % neg_numeral_eq_iff
thf(fact_38_num__double, axiom,
    ((![N : num]: ((times_times_num @ (bit0 @ one) @ N) = (bit0 @ N))))). % num_double
thf(fact_39_abs__numeral, axiom,
    ((![N : num]: ((abs_abs_real @ (numeral_numeral_real @ N)) = (numeral_numeral_real @ N))))). % abs_numeral
thf(fact_40_power__mult__numeral, axiom,
    ((![A : real, M : num, N : num]: ((power_power_real @ (power_power_real @ A @ (numeral_numeral_nat @ M)) @ (numeral_numeral_nat @ N)) = (power_power_real @ A @ (numeral_numeral_nat @ (times_times_num @ M @ N))))))). % power_mult_numeral
thf(fact_41_power__mult__numeral, axiom,
    ((![A : nat, M : num, N : num]: ((power_power_nat @ (power_power_nat @ A @ (numeral_numeral_nat @ M)) @ (numeral_numeral_nat @ N)) = (power_power_nat @ A @ (numeral_numeral_nat @ (times_times_num @ M @ N))))))). % power_mult_numeral
thf(fact_42_semiring__norm_I71_J, axiom,
    ((![M : num, N : num]: ((ord_less_eq_num @ (bit0 @ M) @ (bit0 @ N)) = (ord_less_eq_num @ M @ N))))). % semiring_norm(71)
thf(fact_43_semiring__norm_I68_J, axiom,
    ((![N : num]: (ord_less_eq_num @ one @ N)))). % semiring_norm(68)
thf(fact_44__092_060open_062_N_A1_A_092_060le_062_A2_A_K_Ay_092_060close_062, axiom,
    ((ord_less_eq_real @ (uminus_uminus_real @ one_one_real) @ (times_times_real @ (numeral_numeral_real @ (bit0 @ one)) @ y)))). % \<open>- 1 \<le> 2 * y\<close>
thf(fact_45_mult__minus1__right, axiom,
    ((![Z : real]: ((times_times_real @ Z @ (uminus_uminus_real @ one_one_real)) = (uminus_uminus_real @ Z))))). % mult_minus1_right
thf(fact_46_mult__minus1, axiom,
    ((![Z : real]: ((times_times_real @ (uminus_uminus_real @ one_one_real) @ Z) = (uminus_uminus_real @ Z))))). % mult_minus1
thf(fact_47_mult__neg__numeral__simps_I1_J, axiom,
    ((![M : num, N : num]: ((times_times_real @ (uminus_uminus_real @ (numeral_numeral_real @ M)) @ (uminus_uminus_real @ (numeral_numeral_real @ N))) = (numeral_numeral_real @ (times_times_num @ M @ N)))))). % mult_neg_numeral_simps(1)
thf(fact_48_mult__neg__numeral__simps_I2_J, axiom,
    ((![M : num, N : num]: ((times_times_real @ (uminus_uminus_real @ (numeral_numeral_real @ M)) @ (numeral_numeral_real @ N)) = (uminus_uminus_real @ (numeral_numeral_real @ (times_times_num @ M @ N))))))). % mult_neg_numeral_simps(2)
thf(fact_49_mult__neg__numeral__simps_I3_J, axiom,
    ((![M : num, N : num]: ((times_times_real @ (numeral_numeral_real @ M) @ (uminus_uminus_real @ (numeral_numeral_real @ N))) = (uminus_uminus_real @ (numeral_numeral_real @ (times_times_num @ M @ N))))))). % mult_neg_numeral_simps(3)
thf(fact_50_semiring__norm_I170_J, axiom,
    ((![V : num, W : num, Y : real]: ((times_times_real @ (uminus_uminus_real @ (numeral_numeral_real @ V)) @ (times_times_real @ (numeral_numeral_real @ W) @ Y)) = (times_times_real @ (uminus_uminus_real @ (numeral_numeral_real @ (times_times_num @ V @ W))) @ Y))))). % semiring_norm(170)
thf(fact_51_semiring__norm_I171_J, axiom,
    ((![V : num, W : num, Y : real]: ((times_times_real @ (numeral_numeral_real @ V) @ (times_times_real @ (uminus_uminus_real @ (numeral_numeral_real @ W)) @ Y)) = (times_times_real @ (uminus_uminus_real @ (numeral_numeral_real @ (times_times_num @ V @ W))) @ Y))))). % semiring_norm(171)
thf(fact_52_semiring__norm_I172_J, axiom,
    ((![V : num, W : num, Y : real]: ((times_times_real @ (uminus_uminus_real @ (numeral_numeral_real @ V)) @ (times_times_real @ (uminus_uminus_real @ (numeral_numeral_real @ W)) @ Y)) = (times_times_real @ (numeral_numeral_real @ (times_times_num @ V @ W)) @ Y))))). % semiring_norm(172)
thf(fact_53_abs__neg__one, axiom,
    (((abs_abs_real @ (uminus_uminus_real @ one_one_real)) = one_one_real))). % abs_neg_one
thf(fact_54_abs__neg__numeral, axiom,
    ((![N : num]: ((abs_abs_real @ (uminus_uminus_real @ (numeral_numeral_real @ N))) = (numeral_numeral_real @ N))))). % abs_neg_numeral
thf(fact_55_mem__Collect__eq, axiom,
    ((![A : real, P : real > $o]: ((member_real @ A @ (collect_real @ P)) = (P @ A))))). % mem_Collect_eq
thf(fact_56_Collect__mem__eq, axiom,
    ((![A2 : set_real]: ((collect_real @ (^[X2 : real]: (member_real @ X2 @ A2))) = A2)))). % Collect_mem_eq
thf(fact_57_abs__power__minus, axiom,
    ((![A : real, N : nat]: ((abs_abs_real @ (power_power_real @ (uminus_uminus_real @ A) @ N)) = (abs_abs_real @ (power_power_real @ A @ N)))))). % abs_power_minus
thf(fact_58_numeral__le__iff, axiom,
    ((![M : num, N : num]: ((ord_less_eq_real @ (numeral_numeral_real @ M) @ (numeral_numeral_real @ N)) = (ord_less_eq_num @ M @ N))))). % numeral_le_iff
thf(fact_59_numeral__le__iff, axiom,
    ((![M : num, N : num]: ((ord_less_eq_nat @ (numeral_numeral_nat @ M) @ (numeral_numeral_nat @ N)) = (ord_less_eq_num @ M @ N))))). % numeral_le_iff
thf(fact_60_semiring__norm_I69_J, axiom,
    ((![M : num]: (~ ((ord_less_eq_num @ (bit0 @ M) @ one)))))). % semiring_norm(69)
thf(fact_61_left__minus__one__mult__self, axiom,
    ((![N : nat, A : real]: ((times_times_real @ (power_power_real @ (uminus_uminus_real @ one_one_real) @ N) @ (times_times_real @ (power_power_real @ (uminus_uminus_real @ one_one_real) @ N) @ A)) = A)))). % left_minus_one_mult_self
thf(fact_62_minus__one__mult__self, axiom,
    ((![N : nat]: ((times_times_real @ (power_power_real @ (uminus_uminus_real @ one_one_real) @ N) @ (power_power_real @ (uminus_uminus_real @ one_one_real) @ N)) = one_one_real)))). % minus_one_mult_self
thf(fact_63_numeral__eq__neg__one__iff, axiom,
    ((![N : num]: (((uminus_uminus_real @ (numeral_numeral_real @ N)) = (uminus_uminus_real @ one_one_real)) = (N = one))))). % numeral_eq_neg_one_iff
thf(fact_64_neg__one__eq__numeral__iff, axiom,
    ((![N : num]: (((uminus_uminus_real @ one_one_real) = (uminus_uminus_real @ (numeral_numeral_real @ N))) = (N = one))))). % neg_one_eq_numeral_iff
thf(fact_65_neg__numeral__le__iff, axiom,
    ((![M : num, N : num]: ((ord_less_eq_real @ (uminus_uminus_real @ (numeral_numeral_real @ M)) @ (uminus_uminus_real @ (numeral_numeral_real @ N))) = (ord_less_eq_num @ N @ M))))). % neg_numeral_le_iff
thf(fact_66_not__neg__one__le__neg__numeral__iff, axiom,
    ((![M : num]: ((~ ((ord_less_eq_real @ (uminus_uminus_real @ one_one_real) @ (uminus_uminus_real @ (numeral_numeral_real @ M))))) = (~ ((M = one))))))). % not_neg_one_le_neg_numeral_iff
thf(fact_67_Power_Oring__1__class_Opower__minus__even, axiom,
    ((![A : real, N : nat]: ((power_power_real @ (uminus_uminus_real @ A) @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N)) = (power_power_real @ A @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N)))))). % Power.ring_1_class.power_minus_even
thf(fact_68_power2__minus, axiom,
    ((![A : real]: ((power_power_real @ (uminus_uminus_real @ A) @ (numeral_numeral_nat @ (bit0 @ one))) = (power_power_real @ A @ (numeral_numeral_nat @ (bit0 @ one))))))). % power2_minus
thf(fact_69_power2__abs, axiom,
    ((![A : real]: ((power_power_real @ (abs_abs_real @ A) @ (numeral_numeral_nat @ (bit0 @ one))) = (power_power_real @ A @ (numeral_numeral_nat @ (bit0 @ one))))))). % power2_abs
thf(fact_70_abs__power2, axiom,
    ((![A : real]: ((abs_abs_real @ (power_power_real @ A @ (numeral_numeral_nat @ (bit0 @ one)))) = (power_power_real @ A @ (numeral_numeral_nat @ (bit0 @ one))))))). % abs_power2
thf(fact_71_power__minus1__even, axiom,
    ((![N : nat]: ((power_power_real @ (uminus_uminus_real @ one_one_real) @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N)) = one_one_real)))). % power_minus1_even
thf(fact_72_one__neq__neg__one, axiom,
    ((~ ((one_one_real = (uminus_uminus_real @ one_one_real)))))). % one_neq_neg_one
thf(fact_73_numeral__neq__neg__numeral, axiom,
    ((![M : num, N : num]: (~ (((numeral_numeral_real @ M) = (uminus_uminus_real @ (numeral_numeral_real @ N)))))))). % numeral_neq_neg_numeral
thf(fact_74_neg__numeral__neq__numeral, axiom,
    ((![M : num, N : num]: (~ (((uminus_uminus_real @ (numeral_numeral_real @ M)) = (numeral_numeral_real @ N))))))). % neg_numeral_neq_numeral
thf(fact_75_le__num__One__iff, axiom,
    ((![X : num]: ((ord_less_eq_num @ X @ one) = (X = one))))). % le_num_One_iff
thf(fact_76_power__mult, axiom,
    ((![A : real, M : nat, N : nat]: ((power_power_real @ A @ (times_times_nat @ M @ N)) = (power_power_real @ (power_power_real @ A @ M) @ N))))). % power_mult
thf(fact_77_power__mult, axiom,
    ((![A : nat, M : nat, N : nat]: ((power_power_nat @ A @ (times_times_nat @ M @ N)) = (power_power_nat @ (power_power_nat @ A @ M) @ N))))). % power_mult
thf(fact_78_power2__nat__le__imp__le, axiom,
    ((![M : nat, N : nat]: ((ord_less_eq_nat @ (power_power_nat @ M @ (numeral_numeral_nat @ (bit0 @ one))) @ N) => (ord_less_eq_nat @ M @ N))))). % power2_nat_le_imp_le
thf(fact_79_power2__nat__le__eq__le, axiom,
    ((![M : nat, N : nat]: ((ord_less_eq_nat @ (power_power_nat @ M @ (numeral_numeral_nat @ (bit0 @ one))) @ (power_power_nat @ N @ (numeral_numeral_nat @ (bit0 @ one)))) = (ord_less_eq_nat @ M @ N))))). % power2_nat_le_eq_le
thf(fact_80_self__le__ge2__pow, axiom,
    ((![K : nat, M : nat]: ((ord_less_eq_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ K) => (ord_less_eq_nat @ M @ (power_power_nat @ K @ M)))))). % self_le_ge2_pow
thf(fact_81_le__minus__one__simps_I2_J, axiom,
    ((ord_less_eq_real @ (uminus_uminus_real @ one_one_real) @ one_one_real))). % le_minus_one_simps(2)
thf(fact_82_le__minus__one__simps_I4_J, axiom,
    ((~ ((ord_less_eq_real @ one_one_real @ (uminus_uminus_real @ one_one_real)))))). % le_minus_one_simps(4)
thf(fact_83_not__numeral__le__neg__numeral, axiom,
    ((![M : num, N : num]: (~ ((ord_less_eq_real @ (numeral_numeral_real @ M) @ (uminus_uminus_real @ (numeral_numeral_real @ N)))))))). % not_numeral_le_neg_numeral
thf(fact_84_neg__numeral__le__numeral, axiom,
    ((![M : num, N : num]: (ord_less_eq_real @ (uminus_uminus_real @ (numeral_numeral_real @ M)) @ (numeral_numeral_real @ N))))). % neg_numeral_le_numeral
thf(fact_85_numeral__times__minus__swap, axiom,
    ((![W : num, X : real]: ((times_times_real @ (numeral_numeral_real @ W) @ (uminus_uminus_real @ X)) = (times_times_real @ X @ (uminus_uminus_real @ (numeral_numeral_real @ W))))))). % numeral_times_minus_swap
thf(fact_86_one__neq__neg__numeral, axiom,
    ((![N : num]: (~ ((one_one_real = (uminus_uminus_real @ (numeral_numeral_real @ N)))))))). % one_neq_neg_numeral
thf(fact_87_numeral__neq__neg__one, axiom,
    ((![N : num]: (~ (((numeral_numeral_real @ N) = (uminus_uminus_real @ one_one_real))))))). % numeral_neq_neg_one
thf(fact_88_real__minus__mult__self__le, axiom,
    ((![U : real, X : real]: (ord_less_eq_real @ (uminus_uminus_real @ (times_times_real @ U @ U)) @ (times_times_real @ X @ X))))). % real_minus_mult_self_le
thf(fact_89_numerals_I1_J, axiom,
    (((numeral_numeral_nat @ one) = one_one_nat))). % numerals(1)
thf(fact_90_power__abs, axiom,
    ((![A : real, N : nat]: ((abs_abs_real @ (power_power_real @ A @ N)) = (power_power_real @ (abs_abs_real @ A) @ N))))). % power_abs
thf(fact_91_not__one__le__neg__numeral, axiom,
    ((![M : num]: (~ ((ord_less_eq_real @ one_one_real @ (uminus_uminus_real @ (numeral_numeral_real @ M)))))))). % not_one_le_neg_numeral
thf(fact_92_not__numeral__le__neg__one, axiom,
    ((![M : num]: (~ ((ord_less_eq_real @ (numeral_numeral_real @ M) @ (uminus_uminus_real @ one_one_real))))))). % not_numeral_le_neg_one
thf(fact_93_neg__numeral__le__neg__one, axiom,
    ((![M : num]: (ord_less_eq_real @ (uminus_uminus_real @ (numeral_numeral_real @ M)) @ (uminus_uminus_real @ one_one_real))))). % neg_numeral_le_neg_one
thf(fact_94_neg__one__le__numeral, axiom,
    ((![M : num]: (ord_less_eq_real @ (uminus_uminus_real @ one_one_real) @ (numeral_numeral_real @ M))))). % neg_one_le_numeral
thf(fact_95_neg__numeral__le__one, axiom,
    ((![M : num]: (ord_less_eq_real @ (uminus_uminus_real @ (numeral_numeral_real @ M)) @ one_one_real)))). % neg_numeral_le_one
thf(fact_96_mult__1s__ring__1_I1_J, axiom,
    ((![B : real]: ((times_times_real @ (uminus_uminus_real @ (numeral_numeral_real @ one)) @ B) = (uminus_uminus_real @ B))))). % mult_1s_ring_1(1)
thf(fact_97_mult__1s__ring__1_I2_J, axiom,
    ((![B : real]: ((times_times_real @ B @ (uminus_uminus_real @ (numeral_numeral_real @ one))) = (uminus_uminus_real @ B))))). % mult_1s_ring_1(2)
thf(fact_98_power__minus, axiom,
    ((![A : real, N : nat]: ((power_power_real @ (uminus_uminus_real @ A) @ N) = (times_times_real @ (power_power_real @ (uminus_uminus_real @ one_one_real) @ N) @ (power_power_real @ A @ N)))))). % power_minus
thf(fact_99_uminus__numeral__One, axiom,
    (((uminus_uminus_real @ (numeral_numeral_real @ one)) = (uminus_uminus_real @ one_one_real)))). % uminus_numeral_One
thf(fact_100_power__minus__Bit0, axiom,
    ((![X : real, K : num]: ((power_power_real @ (uminus_uminus_real @ X) @ (numeral_numeral_nat @ (bit0 @ K))) = (power_power_real @ X @ (numeral_numeral_nat @ (bit0 @ K))))))). % power_minus_Bit0
thf(fact_101_power2__eq__iff, axiom,
    ((![X : real, Y : real]: (((power_power_real @ X @ (numeral_numeral_nat @ (bit0 @ one))) = (power_power_real @ Y @ (numeral_numeral_nat @ (bit0 @ one)))) = (((X = Y)) | ((X = (uminus_uminus_real @ Y)))))))). % power2_eq_iff
thf(fact_102_power__even__eq, axiom,
    ((![A : real, N : nat]: ((power_power_real @ A @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N)) = (power_power_real @ (power_power_real @ A @ N) @ (numeral_numeral_nat @ (bit0 @ one))))))). % power_even_eq
thf(fact_103_power__even__eq, axiom,
    ((![A : nat, N : nat]: ((power_power_nat @ A @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N)) = (power_power_nat @ (power_power_nat @ A @ N) @ (numeral_numeral_nat @ (bit0 @ one))))))). % power_even_eq
thf(fact_104_complete__real, axiom,
    ((![S : set_real]: ((?[X3 : real]: (member_real @ X3 @ S)) => ((?[Z2 : real]: (![X4 : real]: ((member_real @ X4 @ S) => (ord_less_eq_real @ X4 @ Z2)))) => (?[Y2 : real]: ((![X3 : real]: ((member_real @ X3 @ S) => (ord_less_eq_real @ X3 @ Y2))) & (![Z2 : real]: ((![X4 : real]: ((member_real @ X4 @ S) => (ord_less_eq_real @ X4 @ Z2))) => (ord_less_eq_real @ Y2 @ Z2)))))))))). % complete_real
thf(fact_105_minus__power__mult__self, axiom,
    ((![A : real, N : nat]: ((times_times_real @ (power_power_real @ (uminus_uminus_real @ A) @ N) @ (power_power_real @ (uminus_uminus_real @ A) @ N)) = (power_power_real @ A @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N)))))). % minus_power_mult_self
thf(fact_106_power2__eq__1__iff, axiom,
    ((![A : real]: (((power_power_real @ A @ (numeral_numeral_nat @ (bit0 @ one))) = one_one_real) = (((A = one_one_real)) | ((A = (uminus_uminus_real @ one_one_real)))))))). % power2_eq_1_iff
thf(fact_107_realpow__square__minus__le, axiom,
    ((![U : real, X : real]: (ord_less_eq_real @ (uminus_uminus_real @ (power_power_real @ U @ (numeral_numeral_nat @ (bit0 @ one)))) @ (power_power_real @ X @ (numeral_numeral_nat @ (bit0 @ one))))))). % realpow_square_minus_le
thf(fact_108_square__le__1, axiom,
    ((![X : real]: ((ord_less_eq_real @ (uminus_uminus_real @ one_one_real) @ X) => ((ord_less_eq_real @ X @ one_one_real) => (ord_less_eq_real @ (power_power_real @ X @ (numeral_numeral_nat @ (bit0 @ one))) @ one_one_real)))))). % square_le_1
thf(fact_109_le__numeral__extra_I4_J, axiom,
    ((ord_less_eq_real @ one_one_real @ one_one_real))). % le_numeral_extra(4)
thf(fact_110_le__numeral__extra_I4_J, axiom,
    ((ord_less_eq_nat @ one_one_nat @ one_one_nat))). % le_numeral_extra(4)
thf(fact_111_power__commuting__commutes, axiom,
    ((![X : nat, Y : nat, N : nat]: (((times_times_nat @ X @ Y) = (times_times_nat @ Y @ X)) => ((times_times_nat @ (power_power_nat @ X @ N) @ Y) = (times_times_nat @ Y @ (power_power_nat @ X @ N))))))). % power_commuting_commutes
thf(fact_112_power__commuting__commutes, axiom,
    ((![X : real, Y : real, N : nat]: (((times_times_real @ X @ Y) = (times_times_real @ Y @ X)) => ((times_times_real @ (power_power_real @ X @ N) @ Y) = (times_times_real @ Y @ (power_power_real @ X @ N))))))). % power_commuting_commutes
thf(fact_113_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_114_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_115_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_116_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_117_abs__le__square__iff, axiom,
    ((![X : real, Y : real]: ((ord_less_eq_real @ (abs_abs_real @ X) @ (abs_abs_real @ Y)) = (ord_less_eq_real @ (power_power_real @ X @ (numeral_numeral_nat @ (bit0 @ one))) @ (power_power_real @ Y @ (numeral_numeral_nat @ (bit0 @ one)))))))). % abs_le_square_iff
thf(fact_118_abs__square__eq__1, axiom,
    ((![X : real]: (((power_power_real @ X @ (numeral_numeral_nat @ (bit0 @ one))) = one_one_real) = ((abs_abs_real @ X) = one_one_real))))). % abs_square_eq_1
thf(fact_119_one__le__numeral, axiom,
    ((![N : num]: (ord_less_eq_real @ one_one_real @ (numeral_numeral_real @ N))))). % one_le_numeral
thf(fact_120_one__le__numeral, axiom,
    ((![N : num]: (ord_less_eq_nat @ one_one_nat @ (numeral_numeral_nat @ N))))). % one_le_numeral
thf(fact_121_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_122_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_123_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_124_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_125_mult__numeral__1__right, axiom,
    ((![A : real]: ((times_times_real @ A @ (numeral_numeral_real @ one)) = A)))). % mult_numeral_1_right
thf(fact_126_mult__numeral__1__right, axiom,
    ((![A : nat]: ((times_times_nat @ A @ (numeral_numeral_nat @ one)) = A)))). % mult_numeral_1_right
thf(fact_127_mult__numeral__1, axiom,
    ((![A : real]: ((times_times_real @ (numeral_numeral_real @ one) @ A) = A)))). % mult_numeral_1
thf(fact_128_mult__numeral__1, axiom,
    ((![A : nat]: ((times_times_nat @ (numeral_numeral_nat @ one) @ A) = A)))). % mult_numeral_1
thf(fact_129_left__right__inverse__power, axiom,
    ((![X : nat, Y : nat, N : nat]: (((times_times_nat @ X @ Y) = one_one_nat) => ((times_times_nat @ (power_power_nat @ X @ N) @ (power_power_nat @ Y @ N)) = one_one_nat))))). % left_right_inverse_power
thf(fact_130_left__right__inverse__power, axiom,
    ((![X : real, Y : real, N : nat]: (((times_times_real @ X @ Y) = one_one_real) => ((times_times_real @ (power_power_real @ X @ N) @ (power_power_real @ Y @ N)) = one_one_real))))). % left_right_inverse_power
thf(fact_131_numeral__One, axiom,
    (((numeral_numeral_real @ one) = one_one_real))). % numeral_One
thf(fact_132_numeral__One, axiom,
    (((numeral_numeral_nat @ one) = one_one_nat))). % numeral_One
thf(fact_133_abs__square__le__1, axiom,
    ((![X : real]: ((ord_less_eq_real @ (power_power_real @ X @ (numeral_numeral_nat @ (bit0 @ one))) @ one_one_real) = (ord_less_eq_real @ (abs_abs_real @ X) @ one_one_real))))). % abs_square_le_1
thf(fact_134_xy, axiom,
    (((plus_plus_real @ (power_power_real @ x @ (numeral_numeral_nat @ (bit0 @ one))) @ (power_power_real @ y @ (numeral_numeral_nat @ (bit0 @ one)))) = one_one_real))). % xy
thf(fact_135_abs__minus, axiom,
    ((![A : real]: ((abs_abs_real @ (uminus_uminus_real @ A)) = (abs_abs_real @ A))))). % abs_minus
thf(fact_136_abs__minus__cancel, axiom,
    ((![A : real]: ((abs_abs_real @ (uminus_uminus_real @ A)) = (abs_abs_real @ A))))). % abs_minus_cancel
thf(fact_137_abs__1, axiom,
    (((abs_abs_real @ one_one_real) = one_one_real))). % abs_1
thf(fact_138_abs__mult__self__eq, axiom,
    ((![A : real]: ((times_times_real @ (abs_abs_real @ A) @ (abs_abs_real @ A)) = (times_times_real @ A @ A))))). % abs_mult_self_eq
thf(fact_139_mult__minus__right, axiom,
    ((![A : real, B : real]: ((times_times_real @ A @ (uminus_uminus_real @ B)) = (uminus_uminus_real @ (times_times_real @ A @ B)))))). % mult_minus_right
thf(fact_140_minus__mult__minus, axiom,
    ((![A : real, B : real]: ((times_times_real @ (uminus_uminus_real @ A) @ (uminus_uminus_real @ B)) = (times_times_real @ A @ B))))). % minus_mult_minus
thf(fact_141_mult__minus__left, axiom,
    ((![A : real, B : real]: ((times_times_real @ (uminus_uminus_real @ A) @ B) = (uminus_uminus_real @ (times_times_real @ A @ B)))))). % mult_minus_left
thf(fact_142_add__right__cancel, axiom,
    ((![B : real, A : real, C : real]: (((plus_plus_real @ B @ A) = (plus_plus_real @ C @ A)) = (B = C))))). % add_right_cancel
thf(fact_143_add__left__cancel, axiom,
    ((![A : real, B : real, C : real]: (((plus_plus_real @ A @ B) = (plus_plus_real @ A @ C)) = (B = C))))). % add_left_cancel
thf(fact_144_neg__equal__iff__equal, axiom,
    ((![A : real, B : real]: (((uminus_uminus_real @ A) = (uminus_uminus_real @ B)) = (A = B))))). % neg_equal_iff_equal
thf(fact_145_add_Oinverse__inverse, axiom,
    ((![A : real]: ((uminus_uminus_real @ (uminus_uminus_real @ A)) = A)))). % add.inverse_inverse
thf(fact_146_abs__idempotent, axiom,
    ((![A : real]: ((abs_abs_real @ (abs_abs_real @ A)) = (abs_abs_real @ A))))). % abs_idempotent
thf(fact_147_abs__abs, axiom,
    ((![A : real]: ((abs_abs_real @ (abs_abs_real @ A)) = (abs_abs_real @ A))))). % abs_abs
thf(fact_148_add__le__cancel__left, axiom,
    ((![C : real, A : real, B : real]: ((ord_less_eq_real @ (plus_plus_real @ C @ A) @ (plus_plus_real @ C @ B)) = (ord_less_eq_real @ A @ B))))). % add_le_cancel_left
thf(fact_149_add__le__cancel__left, axiom,
    ((![C : nat, A : nat, B : nat]: ((ord_less_eq_nat @ (plus_plus_nat @ C @ A) @ (plus_plus_nat @ C @ B)) = (ord_less_eq_nat @ A @ B))))). % add_le_cancel_left
thf(fact_150_add__le__cancel__right, axiom,
    ((![A : real, C : real, B : real]: ((ord_less_eq_real @ (plus_plus_real @ A @ C) @ (plus_plus_real @ B @ C)) = (ord_less_eq_real @ A @ B))))). % add_le_cancel_right
thf(fact_151_add__le__cancel__right, axiom,
    ((![A : nat, C : nat, B : nat]: ((ord_less_eq_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ C)) = (ord_less_eq_nat @ A @ B))))). % add_le_cancel_right
thf(fact_152_mult_Oleft__neutral, axiom,
    ((![A : nat]: ((times_times_nat @ one_one_nat @ A) = A)))). % mult.left_neutral
thf(fact_153_mult_Oleft__neutral, axiom,
    ((![A : real]: ((times_times_real @ one_one_real @ A) = A)))). % mult.left_neutral
thf(fact_154_mult_Oright__neutral, axiom,
    ((![A : nat]: ((times_times_nat @ A @ one_one_nat) = A)))). % mult.right_neutral
thf(fact_155_mult_Oright__neutral, axiom,
    ((![A : real]: ((times_times_real @ A @ one_one_real) = A)))). % mult.right_neutral
thf(fact_156_neg__le__iff__le, axiom,
    ((![B : real, A : real]: ((ord_less_eq_real @ (uminus_uminus_real @ B) @ (uminus_uminus_real @ A)) = (ord_less_eq_real @ A @ B))))). % neg_le_iff_le
thf(fact_157_add__numeral__left, axiom,
    ((![V : num, W : num, Z : real]: ((plus_plus_real @ (numeral_numeral_real @ V) @ (plus_plus_real @ (numeral_numeral_real @ W) @ Z)) = (plus_plus_real @ (numeral_numeral_real @ (plus_plus_num @ V @ W)) @ Z))))). % add_numeral_left
thf(fact_158_add__numeral__left, axiom,
    ((![V : num, W : num, Z : nat]: ((plus_plus_nat @ (numeral_numeral_nat @ V) @ (plus_plus_nat @ (numeral_numeral_nat @ W) @ Z)) = (plus_plus_nat @ (numeral_numeral_nat @ (plus_plus_num @ V @ W)) @ Z))))). % add_numeral_left
thf(fact_159_numeral__plus__numeral, axiom,
    ((![M : num, N : num]: ((plus_plus_real @ (numeral_numeral_real @ M) @ (numeral_numeral_real @ N)) = (numeral_numeral_real @ (plus_plus_num @ M @ N)))))). % numeral_plus_numeral
thf(fact_160_numeral__plus__numeral, axiom,
    ((![M : num, N : num]: ((plus_plus_nat @ (numeral_numeral_nat @ M) @ (numeral_numeral_nat @ N)) = (numeral_numeral_nat @ (plus_plus_num @ M @ N)))))). % numeral_plus_numeral
thf(fact_161_minus__add__distrib, axiom,
    ((![A : real, B : real]: ((uminus_uminus_real @ (plus_plus_real @ A @ B)) = (plus_plus_real @ (uminus_uminus_real @ A) @ (uminus_uminus_real @ B)))))). % minus_add_distrib
thf(fact_162_minus__add__cancel, axiom,
    ((![A : real, B : real]: ((plus_plus_real @ (uminus_uminus_real @ A) @ (plus_plus_real @ A @ B)) = B)))). % minus_add_cancel
thf(fact_163_add__minus__cancel, axiom,
    ((![A : real, B : real]: ((plus_plus_real @ A @ (plus_plus_real @ (uminus_uminus_real @ A) @ B)) = B)))). % add_minus_cancel
thf(fact_164_abs__add__abs, axiom,
    ((![A : real, B : real]: ((abs_abs_real @ (plus_plus_real @ (abs_abs_real @ A) @ (abs_abs_real @ B))) = (plus_plus_real @ (abs_abs_real @ A) @ (abs_abs_real @ B)))))). % abs_add_abs
thf(fact_165_distrib__left__numeral, axiom,
    ((![V : num, B : real, C : real]: ((times_times_real @ (numeral_numeral_real @ V) @ (plus_plus_real @ B @ C)) = (plus_plus_real @ (times_times_real @ (numeral_numeral_real @ V) @ B) @ (times_times_real @ (numeral_numeral_real @ V) @ C)))))). % distrib_left_numeral
thf(fact_166_distrib__left__numeral, axiom,
    ((![V : num, B : nat, C : nat]: ((times_times_nat @ (numeral_numeral_nat @ V) @ (plus_plus_nat @ B @ C)) = (plus_plus_nat @ (times_times_nat @ (numeral_numeral_nat @ V) @ B) @ (times_times_nat @ (numeral_numeral_nat @ V) @ C)))))). % distrib_left_numeral
thf(fact_167_distrib__right__numeral, axiom,
    ((![A : real, B : real, V : num]: ((times_times_real @ (plus_plus_real @ A @ B) @ (numeral_numeral_real @ V)) = (plus_plus_real @ (times_times_real @ A @ (numeral_numeral_real @ V)) @ (times_times_real @ B @ (numeral_numeral_real @ V))))))). % distrib_right_numeral
thf(fact_168_distrib__right__numeral, axiom,
    ((![A : nat, B : nat, V : num]: ((times_times_nat @ (plus_plus_nat @ A @ B) @ (numeral_numeral_nat @ V)) = (plus_plus_nat @ (times_times_nat @ A @ (numeral_numeral_nat @ V)) @ (times_times_nat @ B @ (numeral_numeral_nat @ V))))))). % distrib_right_numeral
thf(fact_169_semiring__norm_I168_J, axiom,
    ((![V : num, W : num, Y : real]: ((plus_plus_real @ (uminus_uminus_real @ (numeral_numeral_real @ V)) @ (plus_plus_real @ (uminus_uminus_real @ (numeral_numeral_real @ W)) @ Y)) = (plus_plus_real @ (uminus_uminus_real @ (numeral_numeral_real @ (plus_plus_num @ V @ W))) @ Y))))). % semiring_norm(168)
thf(fact_170_add__neg__numeral__simps_I3_J, axiom,
    ((![M : num, N : num]: ((plus_plus_real @ (uminus_uminus_real @ (numeral_numeral_real @ M)) @ (uminus_uminus_real @ (numeral_numeral_real @ N))) = (uminus_uminus_real @ (plus_plus_real @ (numeral_numeral_real @ M) @ (numeral_numeral_real @ N))))))). % add_neg_numeral_simps(3)
thf(fact_171_numeral__plus__one, axiom,
    ((![N : num]: ((plus_plus_real @ (numeral_numeral_real @ N) @ one_one_real) = (numeral_numeral_real @ (plus_plus_num @ N @ one)))))). % numeral_plus_one
thf(fact_172_numeral__plus__one, axiom,
    ((![N : num]: ((plus_plus_nat @ (numeral_numeral_nat @ N) @ one_one_nat) = (numeral_numeral_nat @ (plus_plus_num @ N @ one)))))). % numeral_plus_one
thf(fact_173_one__plus__numeral, axiom,
    ((![N : num]: ((plus_plus_real @ one_one_real @ (numeral_numeral_real @ N)) = (numeral_numeral_real @ (plus_plus_num @ one @ N)))))). % one_plus_numeral
thf(fact_174_one__plus__numeral, axiom,
    ((![N : num]: ((plus_plus_nat @ one_one_nat @ (numeral_numeral_nat @ N)) = (numeral_numeral_nat @ (plus_plus_num @ one @ N)))))). % one_plus_numeral
thf(fact_175_one__add__one, axiom,
    (((plus_plus_real @ one_one_real @ one_one_real) = (numeral_numeral_real @ (bit0 @ one))))). % one_add_one
thf(fact_176_one__add__one, axiom,
    (((plus_plus_nat @ one_one_nat @ one_one_nat) = (numeral_numeral_nat @ (bit0 @ one))))). % one_add_one

% Conjectures (1)
thf(conj_0, conjecture,
    ((ord_less_eq_real @ (times_times_real @ (numeral_numeral_real @ (bit0 @ (bit0 @ one))) @ (power_power_real @ x @ (numeral_numeral_nat @ (bit0 @ one)))) @ one_one_real))).
