% 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/FFT/prob_381__3226240_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:10:51.080

% Could-be-implicit typings (2)
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).
thf(ty_n_tf__a, type,
    a : $tType).

% Explicit typings (17)
thf(sy_c_Fields_Oinverse__class_Oinverse_001tf__a, type,
    inverse_inverse_a : a > a).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat, type,
    minus_minus_nat : nat > nat > nat).
thf(sy_c_Groups_Ominus__class_Ominus_001tf__a, type,
    minus_minus_a : a > a > a).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat, type,
    one_one_nat : nat).
thf(sy_c_Groups_Oone__class_Oone_001tf__a, type,
    one_one_a : a).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat, type,
    times_times_nat : nat > nat > nat).
thf(sy_c_Groups_Otimes__class_Otimes_001tf__a, type,
    times_times_a : a > a > a).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat, type,
    zero_zero_nat : nat).
thf(sy_c_Groups_Ozero__class_Ozero_001tf__a, type,
    zero_zero_a : a).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat, type,
    ord_less_eq_nat : nat > nat > $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_001tf__a, type,
    power_power_a : a > nat > a).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat, type,
    divide_divide_nat : nat > nat > nat).
thf(sy_c_Rings_Odivide__class_Odivide_001tf__a, type,
    divide_divide_a : a > a > a).
thf(sy_v_a, type,
    a2 : a).
thf(sy_v_m, type,
    m : nat).
thf(sy_v_n, type,
    n : nat).

% Relevant facts (135)
thf(fact_0_nz, axiom,
    ((~ ((a2 = zero_zero_a))))). % nz
thf(fact_1_diff__diff__cancel, axiom,
    ((![I : nat, N : nat]: ((ord_less_eq_nat @ I @ N) => ((minus_minus_nat @ N @ (minus_minus_nat @ N @ I)) = I))))). % diff_diff_cancel
thf(fact_2_inverse__divide, axiom,
    ((![A : a, B : a]: ((inverse_inverse_a @ (divide_divide_a @ A @ B)) = (divide_divide_a @ B @ A))))). % inverse_divide
thf(fact_3_inverse__eq__iff__eq, axiom,
    ((![A : a, B : a]: (((inverse_inverse_a @ A) = (inverse_inverse_a @ B)) = (A = B))))). % inverse_eq_iff_eq
thf(fact_4_inverse__inverse__eq, axiom,
    ((![A : a]: ((inverse_inverse_a @ (inverse_inverse_a @ A)) = A)))). % inverse_inverse_eq
thf(fact_5_order__refl, axiom,
    ((![X : nat]: (ord_less_eq_nat @ X @ X)))). % order_refl
thf(fact_6_eq__diff__iff, axiom,
    ((![K : nat, M : nat, N : nat]: ((ord_less_eq_nat @ K @ M) => ((ord_less_eq_nat @ K @ N) => (((minus_minus_nat @ M @ K) = (minus_minus_nat @ N @ K)) = (M = N))))))). % eq_diff_iff
thf(fact_7_le__diff__iff, axiom,
    ((![K : nat, M : nat, N : nat]: ((ord_less_eq_nat @ K @ M) => ((ord_less_eq_nat @ K @ N) => ((ord_less_eq_nat @ (minus_minus_nat @ M @ K) @ (minus_minus_nat @ N @ K)) = (ord_less_eq_nat @ M @ N))))))). % le_diff_iff
thf(fact_8_Nat_Odiff__diff__eq, axiom,
    ((![K : nat, M : nat, N : nat]: ((ord_less_eq_nat @ K @ M) => ((ord_less_eq_nat @ K @ N) => ((minus_minus_nat @ (minus_minus_nat @ M @ K) @ (minus_minus_nat @ N @ K)) = (minus_minus_nat @ M @ N))))))). % Nat.diff_diff_eq
thf(fact_9_diff__le__mono, axiom,
    ((![M : nat, N : nat, L : nat]: ((ord_less_eq_nat @ M @ N) => (ord_less_eq_nat @ (minus_minus_nat @ M @ L) @ (minus_minus_nat @ N @ L)))))). % diff_le_mono
thf(fact_10_diff__le__self, axiom,
    ((![M : nat, N : nat]: (ord_less_eq_nat @ (minus_minus_nat @ M @ N) @ M)))). % diff_le_self
thf(fact_11_le__diff__iff_H, axiom,
    ((![A : nat, C : nat, B : nat]: ((ord_less_eq_nat @ A @ C) => ((ord_less_eq_nat @ B @ C) => ((ord_less_eq_nat @ (minus_minus_nat @ C @ A) @ (minus_minus_nat @ C @ B)) = (ord_less_eq_nat @ B @ A))))))). % le_diff_iff'
thf(fact_12_division__ring__divide__zero, axiom,
    ((![A : a]: ((divide_divide_a @ A @ zero_zero_a) = zero_zero_a)))). % division_ring_divide_zero
thf(fact_13_divide__cancel__right, axiom,
    ((![A : a, C : a, B : a]: (((divide_divide_a @ A @ C) = (divide_divide_a @ B @ C)) = (((C = zero_zero_a)) | ((A = B))))))). % divide_cancel_right
thf(fact_14_divide__cancel__left, axiom,
    ((![C : a, A : a, B : a]: (((divide_divide_a @ C @ A) = (divide_divide_a @ C @ B)) = (((C = zero_zero_a)) | ((A = B))))))). % divide_cancel_left
thf(fact_15_divide__eq__0__iff, axiom,
    ((![A : a, B : a]: (((divide_divide_a @ A @ B) = zero_zero_a) = (((A = zero_zero_a)) | ((B = zero_zero_a))))))). % divide_eq_0_iff
thf(fact_16_inverse__nonzero__iff__nonzero, axiom,
    ((![A : a]: (((inverse_inverse_a @ A) = zero_zero_a) = (A = zero_zero_a))))). % inverse_nonzero_iff_nonzero
thf(fact_17_inverse__zero, axiom,
    (((inverse_inverse_a @ zero_zero_a) = zero_zero_a))). % inverse_zero
thf(fact_18_nonzero__imp__inverse__nonzero, axiom,
    ((![A : a]: ((~ ((A = zero_zero_a))) => (~ (((inverse_inverse_a @ A) = zero_zero_a))))))). % nonzero_imp_inverse_nonzero
thf(fact_19_nonzero__inverse__inverse__eq, axiom,
    ((![A : a]: ((~ ((A = zero_zero_a))) => ((inverse_inverse_a @ (inverse_inverse_a @ A)) = A))))). % nonzero_inverse_inverse_eq
thf(fact_20_nonzero__inverse__eq__imp__eq, axiom,
    ((![A : a, B : a]: (((inverse_inverse_a @ A) = (inverse_inverse_a @ B)) => ((~ ((A = zero_zero_a))) => ((~ ((B = zero_zero_a))) => (A = B))))))). % nonzero_inverse_eq_imp_eq
thf(fact_21_inverse__zero__imp__zero, axiom,
    ((![A : a]: (((inverse_inverse_a @ A) = zero_zero_a) => (A = zero_zero_a))))). % inverse_zero_imp_zero
thf(fact_22_field__class_Ofield__inverse__zero, axiom,
    (((inverse_inverse_a @ zero_zero_a) = zero_zero_a))). % field_class.field_inverse_zero
thf(fact_23_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_24_dual__order_Oeq__iff, axiom,
    (((^[Y : nat]: (^[Z : nat]: (Y = Z))) = (^[A2 : nat]: (^[B2 : nat]: (((ord_less_eq_nat @ B2 @ A2)) & ((ord_less_eq_nat @ A2 @ B2)))))))). % dual_order.eq_iff
thf(fact_25_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_26_linorder__wlog, axiom,
    ((![P : nat > nat > $o, A : nat, B : nat]: ((![A3 : nat, B3 : nat]: ((ord_less_eq_nat @ A3 @ B3) => (P @ A3 @ B3))) => ((![A3 : nat, B3 : nat]: ((P @ B3 @ A3) => (P @ A3 @ B3))) => (P @ A @ B)))))). % linorder_wlog
thf(fact_27_dual__order_Orefl, axiom,
    ((![A : nat]: (ord_less_eq_nat @ A @ A)))). % dual_order.refl
thf(fact_28_order__trans, axiom,
    ((![X : nat, Y2 : nat, Z2 : nat]: ((ord_less_eq_nat @ X @ Y2) => ((ord_less_eq_nat @ Y2 @ Z2) => (ord_less_eq_nat @ X @ Z2)))))). % order_trans
thf(fact_29_order__class_Oorder_Oantisym, axiom,
    ((![A : nat, B : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_nat @ B @ A) => (A = B)))))). % order_class.order.antisym
thf(fact_30_ord__le__eq__trans, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ B) => ((B = C) => (ord_less_eq_nat @ A @ C)))))). % ord_le_eq_trans
thf(fact_31_ord__eq__le__trans, axiom,
    ((![A : nat, B : nat, C : nat]: ((A = B) => ((ord_less_eq_nat @ B @ C) => (ord_less_eq_nat @ A @ C)))))). % ord_eq_le_trans
thf(fact_32_order__class_Oorder_Oeq__iff, axiom,
    (((^[Y : nat]: (^[Z : nat]: (Y = Z))) = (^[A2 : nat]: (^[B2 : nat]: (((ord_less_eq_nat @ A2 @ B2)) & ((ord_less_eq_nat @ B2 @ A2)))))))). % order_class.order.eq_iff
thf(fact_33_antisym__conv, axiom,
    ((![Y2 : nat, X : nat]: ((ord_less_eq_nat @ Y2 @ X) => ((ord_less_eq_nat @ X @ Y2) = (X = Y2)))))). % antisym_conv
thf(fact_34_le__cases3, axiom,
    ((![X : nat, Y2 : nat, Z2 : nat]: (((ord_less_eq_nat @ X @ Y2) => (~ ((ord_less_eq_nat @ Y2 @ Z2)))) => (((ord_less_eq_nat @ Y2 @ X) => (~ ((ord_less_eq_nat @ X @ Z2)))) => (((ord_less_eq_nat @ X @ Z2) => (~ ((ord_less_eq_nat @ Z2 @ Y2)))) => (((ord_less_eq_nat @ Z2 @ Y2) => (~ ((ord_less_eq_nat @ Y2 @ X)))) => (((ord_less_eq_nat @ Y2 @ Z2) => (~ ((ord_less_eq_nat @ Z2 @ X)))) => (~ (((ord_less_eq_nat @ Z2 @ X) => (~ ((ord_less_eq_nat @ X @ Y2)))))))))))))). % le_cases3
thf(fact_35_order_Otrans, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_nat @ B @ C) => (ord_less_eq_nat @ A @ C)))))). % order.trans
thf(fact_36_le__cases, axiom,
    ((![X : nat, Y2 : nat]: ((~ ((ord_less_eq_nat @ X @ Y2))) => (ord_less_eq_nat @ Y2 @ X))))). % le_cases
thf(fact_37_eq__refl, axiom,
    ((![X : nat, Y2 : nat]: ((X = Y2) => (ord_less_eq_nat @ X @ Y2))))). % eq_refl
thf(fact_38_linear, axiom,
    ((![X : nat, Y2 : nat]: ((ord_less_eq_nat @ X @ Y2) | (ord_less_eq_nat @ Y2 @ X))))). % linear
thf(fact_39_antisym, axiom,
    ((![X : nat, Y2 : nat]: ((ord_less_eq_nat @ X @ Y2) => ((ord_less_eq_nat @ Y2 @ X) => (X = Y2)))))). % antisym
thf(fact_40_eq__iff, axiom,
    (((^[Y : nat]: (^[Z : nat]: (Y = Z))) = (^[X2 : nat]: (^[Y3 : nat]: (((ord_less_eq_nat @ X2 @ Y3)) & ((ord_less_eq_nat @ Y3 @ X2)))))))). % eq_iff
thf(fact_41_ord__le__eq__subst, axiom,
    ((![A : nat, B : nat, F : nat > nat, C : nat]: ((ord_less_eq_nat @ A @ B) => (((F @ B) = C) => ((![X3 : nat, Y4 : nat]: ((ord_less_eq_nat @ X3 @ Y4) => (ord_less_eq_nat @ (F @ X3) @ (F @ Y4)))) => (ord_less_eq_nat @ (F @ A) @ C))))))). % ord_le_eq_subst
thf(fact_42_ord__eq__le__subst, axiom,
    ((![A : nat, F : nat > nat, B : nat, C : nat]: ((A = (F @ B)) => ((ord_less_eq_nat @ B @ C) => ((![X3 : nat, Y4 : nat]: ((ord_less_eq_nat @ X3 @ Y4) => (ord_less_eq_nat @ (F @ X3) @ (F @ Y4)))) => (ord_less_eq_nat @ A @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_43_order__subst2, axiom,
    ((![A : nat, B : nat, F : nat > nat, C : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_nat @ (F @ B) @ C) => ((![X3 : nat, Y4 : nat]: ((ord_less_eq_nat @ X3 @ Y4) => (ord_less_eq_nat @ (F @ X3) @ (F @ Y4)))) => (ord_less_eq_nat @ (F @ A) @ C))))))). % order_subst2
thf(fact_44_order__subst1, axiom,
    ((![A : nat, F : nat > nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ (F @ B)) => ((ord_less_eq_nat @ B @ C) => ((![X3 : nat, Y4 : nat]: ((ord_less_eq_nat @ X3 @ Y4) => (ord_less_eq_nat @ (F @ X3) @ (F @ Y4)))) => (ord_less_eq_nat @ A @ (F @ C)))))))). % order_subst1
thf(fact_45_Nat_Oex__has__greatest__nat, axiom,
    ((![P : nat > $o, K : nat, B : nat]: ((P @ K) => ((![Y4 : nat]: ((P @ Y4) => (ord_less_eq_nat @ Y4 @ B))) => (?[X3 : nat]: ((P @ X3) & (![Y5 : nat]: ((P @ Y5) => (ord_less_eq_nat @ Y5 @ X3)))))))))). % Nat.ex_has_greatest_nat
thf(fact_46_nat__le__linear, axiom,
    ((![M : nat, N : nat]: ((ord_less_eq_nat @ M @ N) | (ord_less_eq_nat @ N @ M))))). % nat_le_linear
thf(fact_47_le__antisym, axiom,
    ((![M : nat, N : nat]: ((ord_less_eq_nat @ M @ N) => ((ord_less_eq_nat @ N @ M) => (M = N)))))). % le_antisym
thf(fact_48_eq__imp__le, axiom,
    ((![M : nat, N : nat]: ((M = N) => (ord_less_eq_nat @ M @ N))))). % eq_imp_le
thf(fact_49_le__trans, axiom,
    ((![I : nat, J : nat, K : nat]: ((ord_less_eq_nat @ I @ J) => ((ord_less_eq_nat @ J @ K) => (ord_less_eq_nat @ I @ K)))))). % le_trans
thf(fact_50_le__refl, axiom,
    ((![N : nat]: (ord_less_eq_nat @ N @ N)))). % le_refl
thf(fact_51_diff__commute, axiom,
    ((![I : nat, J : nat, K : nat]: ((minus_minus_nat @ (minus_minus_nat @ I @ J) @ K) = (minus_minus_nat @ (minus_minus_nat @ I @ K) @ J))))). % diff_commute
thf(fact_52_inverse__eq__imp__eq, axiom,
    ((![A : a, B : a]: (((inverse_inverse_a @ A) = (inverse_inverse_a @ B)) => (A = B))))). % inverse_eq_imp_eq
thf(fact_53_diff__divide__distrib, axiom,
    ((![A : a, B : a, C : a]: ((divide_divide_a @ (minus_minus_a @ A @ B) @ C) = (minus_minus_a @ (divide_divide_a @ A @ C) @ (divide_divide_a @ B @ C)))))). % diff_divide_distrib
thf(fact_54_diff__le__mono2, axiom,
    ((![M : nat, N : nat, L : nat]: ((ord_less_eq_nat @ M @ N) => (ord_less_eq_nat @ (minus_minus_nat @ L @ N) @ (minus_minus_nat @ L @ M)))))). % diff_le_mono2
thf(fact_55_power__diff, axiom,
    ((![A : nat, N : nat, M : nat]: ((~ ((A = zero_zero_nat))) => ((ord_less_eq_nat @ N @ M) => ((power_power_nat @ A @ (minus_minus_nat @ M @ N)) = (divide_divide_nat @ (power_power_nat @ A @ M) @ (power_power_nat @ A @ N)))))))). % power_diff
thf(fact_56_power__diff, axiom,
    ((![A : a, N : nat, M : nat]: ((~ ((A = zero_zero_a))) => ((ord_less_eq_nat @ N @ M) => ((power_power_a @ A @ (minus_minus_nat @ M @ N)) = (divide_divide_a @ (power_power_a @ A @ M) @ (power_power_a @ A @ N)))))))). % power_diff
thf(fact_57_bits__div__by__0, axiom,
    ((![A : nat]: ((divide_divide_nat @ A @ zero_zero_nat) = zero_zero_nat)))). % bits_div_by_0
thf(fact_58_bits__div__0, axiom,
    ((![A : nat]: ((divide_divide_nat @ zero_zero_nat @ A) = zero_zero_nat)))). % bits_div_0
thf(fact_59_div__by__0, axiom,
    ((![A : nat]: ((divide_divide_nat @ A @ zero_zero_nat) = zero_zero_nat)))). % div_by_0
thf(fact_60_div__by__0, axiom,
    ((![A : a]: ((divide_divide_a @ A @ zero_zero_a) = zero_zero_a)))). % div_by_0
thf(fact_61_div__0, axiom,
    ((![A : nat]: ((divide_divide_nat @ zero_zero_nat @ A) = zero_zero_nat)))). % div_0
thf(fact_62_div__0, axiom,
    ((![A : a]: ((divide_divide_a @ zero_zero_a @ A) = zero_zero_a)))). % div_0
thf(fact_63_cancel__comm__monoid__add__class_Odiff__cancel, axiom,
    ((![A : a]: ((minus_minus_a @ A @ A) = zero_zero_a)))). % cancel_comm_monoid_add_class.diff_cancel
thf(fact_64_cancel__comm__monoid__add__class_Odiff__cancel, axiom,
    ((![A : nat]: ((minus_minus_nat @ A @ A) = zero_zero_nat)))). % cancel_comm_monoid_add_class.diff_cancel
thf(fact_65_diff__zero, axiom,
    ((![A : a]: ((minus_minus_a @ A @ zero_zero_a) = A)))). % diff_zero
thf(fact_66_diff__zero, axiom,
    ((![A : nat]: ((minus_minus_nat @ A @ zero_zero_nat) = A)))). % diff_zero
thf(fact_67_zero__diff, axiom,
    ((![A : nat]: ((minus_minus_nat @ zero_zero_nat @ A) = zero_zero_nat)))). % zero_diff
thf(fact_68_diff__0__right, axiom,
    ((![A : a]: ((minus_minus_a @ A @ zero_zero_a) = A)))). % diff_0_right
thf(fact_69_le0, axiom,
    ((![N : nat]: (ord_less_eq_nat @ zero_zero_nat @ N)))). % le0
thf(fact_70_bot__nat__0_Oextremum, axiom,
    ((![A : nat]: (ord_less_eq_nat @ zero_zero_nat @ A)))). % bot_nat_0.extremum
thf(fact_71_diff__0__eq__0, axiom,
    ((![N : nat]: ((minus_minus_nat @ zero_zero_nat @ N) = zero_zero_nat)))). % diff_0_eq_0
thf(fact_72_diff__self__eq__0, axiom,
    ((![M : nat]: ((minus_minus_nat @ M @ M) = zero_zero_nat)))). % diff_self_eq_0
thf(fact_73_le__zero__eq, axiom,
    ((![N : nat]: ((ord_less_eq_nat @ N @ zero_zero_nat) = (N = zero_zero_nat))))). % le_zero_eq
thf(fact_74_diff__self, axiom,
    ((![A : a]: ((minus_minus_a @ A @ A) = zero_zero_a)))). % diff_self
thf(fact_75_diff__is__0__eq, axiom,
    ((![M : nat, N : nat]: (((minus_minus_nat @ M @ N) = zero_zero_nat) = (ord_less_eq_nat @ M @ N))))). % diff_is_0_eq
thf(fact_76_diff__is__0__eq_H, axiom,
    ((![M : nat, N : nat]: ((ord_less_eq_nat @ M @ N) => ((minus_minus_nat @ M @ N) = zero_zero_nat))))). % diff_is_0_eq'
thf(fact_77_less__eq__nat_Osimps_I1_J, axiom,
    ((![N : nat]: (ord_less_eq_nat @ zero_zero_nat @ N)))). % less_eq_nat.simps(1)
thf(fact_78_le__0__eq, axiom,
    ((![N : nat]: ((ord_less_eq_nat @ N @ zero_zero_nat) = (N = zero_zero_nat))))). % le_0_eq
thf(fact_79_bot__nat__0_Oextremum__unique, axiom,
    ((![A : nat]: ((ord_less_eq_nat @ A @ zero_zero_nat) = (A = zero_zero_nat))))). % bot_nat_0.extremum_unique
thf(fact_80_bot__nat__0_Oextremum__uniqueI, axiom,
    ((![A : nat]: ((ord_less_eq_nat @ A @ zero_zero_nat) => (A = zero_zero_nat))))). % bot_nat_0.extremum_uniqueI
thf(fact_81_diffs0__imp__equal, axiom,
    ((![M : nat, N : nat]: (((minus_minus_nat @ M @ N) = zero_zero_nat) => (((minus_minus_nat @ N @ M) = zero_zero_nat) => (M = N)))))). % diffs0_imp_equal
thf(fact_82_minus__nat_Odiff__0, axiom,
    ((![M : nat]: ((minus_minus_nat @ M @ zero_zero_nat) = M)))). % minus_nat.diff_0
thf(fact_83_zero__reorient, axiom,
    ((![X : a]: ((zero_zero_a = X) = (X = zero_zero_a))))). % zero_reorient
thf(fact_84_zero__reorient, axiom,
    ((![X : nat]: ((zero_zero_nat = X) = (X = zero_zero_nat))))). % zero_reorient
thf(fact_85_diff__right__commute, axiom,
    ((![A : nat, C : nat, B : nat]: ((minus_minus_nat @ (minus_minus_nat @ A @ C) @ B) = (minus_minus_nat @ (minus_minus_nat @ A @ B) @ C))))). % diff_right_commute
thf(fact_86_zero__le, axiom,
    ((![X : nat]: (ord_less_eq_nat @ zero_zero_nat @ X)))). % zero_le
thf(fact_87_eq__iff__diff__eq__0, axiom,
    (((^[Y : a]: (^[Z : a]: (Y = Z))) = (^[A2 : a]: (^[B2 : a]: ((minus_minus_a @ A2 @ B2) = zero_zero_a)))))). % eq_iff_diff_eq_0
thf(fact_88_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_89_power__not__zero, axiom,
    ((![A : a, N : nat]: ((~ ((A = zero_zero_a))) => (~ (((power_power_a @ A @ N) = zero_zero_a))))))). % power_not_zero
thf(fact_90_power__divide, axiom,
    ((![A : a, B : a, N : nat]: ((power_power_a @ (divide_divide_a @ A @ B) @ N) = (divide_divide_a @ (power_power_a @ A @ N) @ (power_power_a @ B @ N)))))). % power_divide
thf(fact_91_power__inverse, axiom,
    ((![A : a, N : nat]: ((power_power_a @ (inverse_inverse_a @ A) @ N) = (inverse_inverse_a @ (power_power_a @ A @ N)))))). % power_inverse
thf(fact_92_power__mono, axiom,
    ((![A : nat, B : nat, N : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_nat @ zero_zero_nat @ A) => (ord_less_eq_nat @ (power_power_nat @ A @ N) @ (power_power_nat @ B @ N))))))). % power_mono
thf(fact_93_zero__le__power, axiom,
    ((![A : nat, N : nat]: ((ord_less_eq_nat @ zero_zero_nat @ A) => (ord_less_eq_nat @ zero_zero_nat @ (power_power_nat @ A @ N)))))). % zero_le_power
thf(fact_94_le__numeral__extra_I3_J, axiom,
    ((ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat))). % le_numeral_extra(3)
thf(fact_95_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_96_power__one__right, axiom,
    ((![A : a]: ((power_power_a @ A @ one_one_nat) = A)))). % power_one_right
thf(fact_97_mult__is__0, axiom,
    ((![M : nat, N : nat]: (((times_times_nat @ M @ N) = zero_zero_nat) = (((M = zero_zero_nat)) | ((N = zero_zero_nat))))))). % mult_is_0
thf(fact_98_mult__0__right, axiom,
    ((![M : nat]: ((times_times_nat @ M @ zero_zero_nat) = zero_zero_nat)))). % mult_0_right
thf(fact_99_mult__cancel1, axiom,
    ((![K : nat, M : nat, N : nat]: (((times_times_nat @ K @ M) = (times_times_nat @ K @ N)) = (((M = N)) | ((K = zero_zero_nat))))))). % mult_cancel1
thf(fact_100_mult__cancel2, axiom,
    ((![M : nat, K : nat, N : nat]: (((times_times_nat @ M @ K) = (times_times_nat @ N @ K)) = (((M = N)) | ((K = zero_zero_nat))))))). % mult_cancel2
thf(fact_101_mult__cancel__right, axiom,
    ((![A : a, C : a, B : a]: (((times_times_a @ A @ C) = (times_times_a @ B @ C)) = (((C = zero_zero_a)) | ((A = B))))))). % mult_cancel_right
thf(fact_102_mult__cancel__right, axiom,
    ((![A : nat, C : nat, B : nat]: (((times_times_nat @ A @ C) = (times_times_nat @ B @ C)) = (((C = zero_zero_nat)) | ((A = B))))))). % mult_cancel_right
thf(fact_103_mult__cancel__left, axiom,
    ((![C : a, A : a, B : a]: (((times_times_a @ C @ A) = (times_times_a @ C @ B)) = (((C = zero_zero_a)) | ((A = B))))))). % mult_cancel_left
thf(fact_104_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_105_mult__eq__0__iff, axiom,
    ((![A : a, B : a]: (((times_times_a @ A @ B) = zero_zero_a) = (((A = zero_zero_a)) | ((B = zero_zero_a))))))). % mult_eq_0_iff
thf(fact_106_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_107_mult__zero__right, axiom,
    ((![A : a]: ((times_times_a @ A @ zero_zero_a) = zero_zero_a)))). % mult_zero_right
thf(fact_108_mult__zero__right, axiom,
    ((![A : nat]: ((times_times_nat @ A @ zero_zero_nat) = zero_zero_nat)))). % mult_zero_right
thf(fact_109_mult__zero__left, axiom,
    ((![A : a]: ((times_times_a @ zero_zero_a @ A) = zero_zero_a)))). % mult_zero_left
thf(fact_110_mult__zero__left, axiom,
    ((![A : nat]: ((times_times_nat @ zero_zero_nat @ A) = zero_zero_nat)))). % mult_zero_left
thf(fact_111_mult_Oright__neutral, axiom,
    ((![A : nat]: ((times_times_nat @ A @ one_one_nat) = A)))). % mult.right_neutral
thf(fact_112_mult_Oleft__neutral, axiom,
    ((![A : nat]: ((times_times_nat @ one_one_nat @ A) = A)))). % mult.left_neutral
thf(fact_113_times__divide__eq__left, axiom,
    ((![B : a, C : a, A : a]: ((times_times_a @ (divide_divide_a @ B @ C) @ A) = (divide_divide_a @ (times_times_a @ B @ A) @ C))))). % times_divide_eq_left
thf(fact_114_times__divide__eq__right, axiom,
    ((![A : a, B : a, C : a]: ((times_times_a @ A @ (divide_divide_a @ B @ C)) = (divide_divide_a @ (times_times_a @ A @ B) @ C))))). % times_divide_eq_right
thf(fact_115_div__by__1, axiom,
    ((![A : a]: ((divide_divide_a @ A @ one_one_a) = A)))). % div_by_1
thf(fact_116_power__one, axiom,
    ((![N : nat]: ((power_power_a @ one_one_a @ N) = one_one_a)))). % power_one
thf(fact_117_inverse__mult__distrib, axiom,
    ((![A : a, B : a]: ((inverse_inverse_a @ (times_times_a @ A @ B)) = (times_times_a @ (inverse_inverse_a @ A) @ (inverse_inverse_a @ B)))))). % inverse_mult_distrib
thf(fact_118_inverse__eq__1__iff, axiom,
    ((![X : a]: (((inverse_inverse_a @ X) = one_one_a) = (X = one_one_a))))). % inverse_eq_1_iff
thf(fact_119_inverse__1, axiom,
    (((inverse_inverse_a @ one_one_a) = one_one_a))). % inverse_1
thf(fact_120_mult__cancel__left1, axiom,
    ((![C : a, B : a]: ((C = (times_times_a @ C @ B)) = (((C = zero_zero_a)) | ((B = one_one_a))))))). % mult_cancel_left1
thf(fact_121_mult__cancel__left2, axiom,
    ((![C : a, A : a]: (((times_times_a @ C @ A) = C) = (((C = zero_zero_a)) | ((A = one_one_a))))))). % mult_cancel_left2
thf(fact_122_mult__cancel__right1, axiom,
    ((![C : a, B : a]: ((C = (times_times_a @ B @ C)) = (((C = zero_zero_a)) | ((B = one_one_a))))))). % mult_cancel_right1
thf(fact_123_mult__cancel__right2, axiom,
    ((![A : a, C : a]: (((times_times_a @ A @ C) = C) = (((C = zero_zero_a)) | ((A = one_one_a))))))). % mult_cancel_right2
thf(fact_124_diff__numeral__special_I9_J, axiom,
    (((minus_minus_a @ one_one_a @ one_one_a) = zero_zero_a))). % diff_numeral_special(9)
thf(fact_125_mult__divide__mult__cancel__left__if, axiom,
    ((![C : a, A : a, B : a]: (((C = zero_zero_a) => ((divide_divide_a @ (times_times_a @ C @ A) @ (times_times_a @ C @ B)) = zero_zero_a)) & ((~ ((C = zero_zero_a))) => ((divide_divide_a @ (times_times_a @ C @ A) @ (times_times_a @ C @ B)) = (divide_divide_a @ A @ B))))))). % mult_divide_mult_cancel_left_if
thf(fact_126_nonzero__mult__divide__mult__cancel__left, axiom,
    ((![C : a, A : a, B : a]: ((~ ((C = zero_zero_a))) => ((divide_divide_a @ (times_times_a @ C @ A) @ (times_times_a @ C @ B)) = (divide_divide_a @ A @ B)))))). % nonzero_mult_divide_mult_cancel_left
thf(fact_127_nonzero__mult__divide__mult__cancel__left2, axiom,
    ((![C : a, A : a, B : a]: ((~ ((C = zero_zero_a))) => ((divide_divide_a @ (times_times_a @ C @ A) @ (times_times_a @ B @ C)) = (divide_divide_a @ A @ B)))))). % nonzero_mult_divide_mult_cancel_left2
thf(fact_128_nonzero__mult__divide__mult__cancel__right, axiom,
    ((![C : a, A : a, B : a]: ((~ ((C = zero_zero_a))) => ((divide_divide_a @ (times_times_a @ A @ C) @ (times_times_a @ B @ C)) = (divide_divide_a @ A @ B)))))). % nonzero_mult_divide_mult_cancel_right
thf(fact_129_nonzero__mult__divide__mult__cancel__right2, axiom,
    ((![C : a, A : a, B : a]: ((~ ((C = zero_zero_a))) => ((divide_divide_a @ (times_times_a @ A @ C) @ (times_times_a @ C @ B)) = (divide_divide_a @ A @ B)))))). % nonzero_mult_divide_mult_cancel_right2
thf(fact_130_nonzero__mult__div__cancel__left, axiom,
    ((![A : nat, B : nat]: ((~ ((A = zero_zero_nat))) => ((divide_divide_nat @ (times_times_nat @ A @ B) @ A) = B))))). % nonzero_mult_div_cancel_left
thf(fact_131_nonzero__mult__div__cancel__left, axiom,
    ((![A : a, B : a]: ((~ ((A = zero_zero_a))) => ((divide_divide_a @ (times_times_a @ A @ B) @ A) = B))))). % nonzero_mult_div_cancel_left
thf(fact_132_nonzero__mult__div__cancel__right, axiom,
    ((![B : nat, A : nat]: ((~ ((B = zero_zero_nat))) => ((divide_divide_nat @ (times_times_nat @ A @ B) @ B) = A))))). % nonzero_mult_div_cancel_right
thf(fact_133_nonzero__mult__div__cancel__right, axiom,
    ((![B : a, A : a]: ((~ ((B = zero_zero_a))) => ((divide_divide_a @ (times_times_a @ A @ B) @ B) = A))))). % nonzero_mult_div_cancel_right
thf(fact_134_divide__eq__1__iff, axiom,
    ((![A : a, B : a]: (((divide_divide_a @ A @ B) = one_one_a) = (((~ ((B = zero_zero_a)))) & ((A = B))))))). % divide_eq_1_iff

% Conjectures (2)
thf(conj_0, hypothesis,
    ((ord_less_eq_nat @ m @ n))).
thf(conj_1, conjecture,
    (((power_power_a @ (inverse_inverse_a @ a2) @ (minus_minus_nat @ n @ m)) = (divide_divide_a @ (power_power_a @ a2 @ m) @ (power_power_a @ a2 @ n))))).
