% 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_1201__5379386_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:36:29.999

% Could-be-implicit typings (10)
thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_It__Polynomial__Opoly_It__Complex__Ocomplex_J_J_J, type,
    poly_p1267267526omplex : $tType).
thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_It__Complex__Ocomplex_J_J, type,
    poly_poly_complex : $tType).
thf(ty_n_t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    poly_complex : $tType).
thf(ty_n_t__Polynomial__Opoly_It__Nat__Onat_J, type,
    poly_nat : $tType).
thf(ty_n_t__Polynomial__Opoly_It__Int__Oint_J, type,
    poly_int : $tType).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J, type,
    set_real : $tType).
thf(ty_n_t__Complex__Ocomplex, type,
    complex : $tType).
thf(ty_n_t__Real__Oreal, type,
    real : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).
thf(ty_n_t__Int__Oint, type,
    int : $tType).

% Explicit typings (58)
thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Real__Oreal, type,
    archim1031974863r_real : real > int).
thf(sy_c_Factorial__Ring_Ocomm__semiring__1__class_Oprime__elem_001t__Complex__Ocomplex, type,
    factor392545715omplex : complex > $o).
thf(sy_c_Factorial__Ring_Ocomm__semiring__1__class_Oprime__elem_001t__Int__Oint, type,
    factor765094193em_int : int > $o).
thf(sy_c_Factorial__Ring_Ocomm__semiring__1__class_Oprime__elem_001t__Nat__Onat, type,
    factor127820501em_nat : nat > $o).
thf(sy_c_Factorial__Ring_Ocomm__semiring__1__class_Oprime__elem_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    factor477435579omplex : poly_complex > $o).
thf(sy_c_Factorial__Ring_Ocomm__semiring__1__class_Oprime__elem_001t__Polynomial__Opoly_It__Int__Oint_J, type,
    factor907203897ly_int : poly_int > $o).
thf(sy_c_Factorial__Ring_Ocomm__semiring__1__class_Oprime__elem_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    factor745290973ly_nat : poly_nat > $o).
thf(sy_c_Factorial__Ring_Ocomm__semiring__1__class_Oprime__elem_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Complex__Ocomplex_J_J, type,
    factor1360491971omplex : poly_poly_complex > $o).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint, type,
    zero_zero_int : int).
thf(sy_c_If_001t__Nat__Onat, type,
    if_nat : $o > nat > nat > nat).
thf(sy_c_Int_Oring__1__class_Oof__int_001t__Real__Oreal, type,
    ring_1_of_int_real : int > real).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Complex__Ocomplex, type,
    semiri356525583omplex : nat > complex).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint, type,
    semiri2019852685at_int : nat > int).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat, type,
    semiri1382578993at_nat : nat > nat).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    semiri1679838999omplex : nat > poly_complex).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Complex__Ocomplex_J_J, type,
    semiri1266910751omplex : nat > poly_poly_complex).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal, type,
    semiri2110766477t_real : nat > real).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint, type,
    ord_less_eq_int : int > int > $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_Orderings_Oorder__class_OGreatest_001t__Nat__Onat, type,
    order_Greatest_nat : (nat > $o) > nat).
thf(sy_c_Polynomial_Ocontent_001t__Int__Oint, type,
    content_int : poly_int > int).
thf(sy_c_Polynomial_Ocontent_001t__Nat__Onat, type,
    content_nat : poly_nat > nat).
thf(sy_c_Polynomial_Opoly_001t__Complex__Ocomplex, type,
    poly_complex2 : poly_complex > complex > complex).
thf(sy_c_Polynomial_Opoly_001t__Int__Oint, type,
    poly_int2 : poly_int > int > int).
thf(sy_c_Polynomial_Opoly_001t__Nat__Onat, type,
    poly_nat2 : poly_nat > nat > nat).
thf(sy_c_Polynomial_Opoly_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    poly_poly_complex2 : poly_poly_complex > poly_complex > poly_complex).
thf(sy_c_Polynomial_Opoly_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Complex__Ocomplex_J_J, type,
    poly_p282434315omplex : poly_p1267267526omplex > poly_poly_complex > poly_poly_complex).
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__Int__Oint, type,
    power_power_int : int > nat > int).
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__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    power_184595776omplex : poly_complex > nat > poly_complex).
thf(sy_c_Power_Opower__class_Opower_001t__Polynomial__Opoly_It__Int__Oint_J, type,
    power_power_poly_int : poly_int > nat > poly_int).
thf(sy_c_Power_Opower__class_Opower_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    power_power_poly_nat : poly_nat > nat > poly_nat).
thf(sy_c_Power_Opower__class_Opower_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Complex__Ocomplex_J_J, type,
    power_432682568omplex : poly_poly_complex > nat > poly_poly_complex).
thf(sy_c_Power_Opower__class_Opower_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Polynomial__Opoly_It__Complex__Ocomplex_J_J_J, type,
    power_2001192272omplex : poly_p1267267526omplex > nat > poly_p1267267526omplex).
thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal, type,
    power_power_real : real > nat > real).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Complex__Ocomplex, type,
    divide1210191872omplex : complex > complex > complex).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint, type,
    divide_divide_int : int > int > int).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat, type,
    divide_divide_nat : nat > nat > nat).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    divide1187762952omplex : poly_complex > poly_complex > poly_complex).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Polynomial__Opoly_It__Int__Oint_J, type,
    divide1415015302ly_int : poly_int > poly_int > poly_int).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Complex__Ocomplex_J_J, type,
    divide350004240omplex : poly_poly_complex > poly_poly_complex > poly_poly_complex).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal, type,
    divide_divide_real : real > real > real).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Complex__Ocomplex, type,
    dvd_dvd_complex : complex > complex > $o).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint, type,
    dvd_dvd_int : int > int > $o).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat, type,
    dvd_dvd_nat : nat > nat > $o).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    dvd_dvd_poly_complex : poly_complex > poly_complex > $o).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Polynomial__Opoly_It__Int__Oint_J, type,
    dvd_dvd_poly_int : poly_int > poly_int > $o).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    dvd_dvd_poly_nat : poly_nat > poly_nat > $o).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Complex__Ocomplex_J_J, type,
    dvd_dv598755940omplex : poly_poly_complex > poly_poly_complex > $o).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Real__Oreal, type,
    dvd_dvd_real : real > real > $o).
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_n, type,
    n : nat).
thf(sy_v_p, type,
    p : poly_complex).
thf(sy_v_q, type,
    q : poly_complex).
thf(sy_v_r, type,
    r : poly_complex).

% Relevant facts (248)
thf(fact_0__092_060open_062q_A_094_An_A_061_Ar_092_060close_062, axiom,
    (((power_184595776omplex @ q @ n) = r))). % \<open>q ^ n = r\<close>
thf(fact_1__092_060open_062poly_A_Iq_A_094_An_J_A_061_Apoly_Ar_092_060close_062, axiom,
    (((poly_complex2 @ (power_184595776omplex @ q @ n)) = (poly_complex2 @ r)))). % \<open>poly (q ^ n) = poly r\<close>
thf(fact_2_h, axiom,
    ((![X : complex]: ((poly_complex2 @ (power_184595776omplex @ q @ n) @ X) = (poly_complex2 @ r @ X))))). % h
thf(fact_3_dvd__power__same, axiom,
    ((![X : poly_nat, Y : poly_nat, N : nat]: ((dvd_dvd_poly_nat @ X @ Y) => (dvd_dvd_poly_nat @ (power_power_poly_nat @ X @ N) @ (power_power_poly_nat @ Y @ N)))))). % dvd_power_same
thf(fact_4_dvd__power__same, axiom,
    ((![X : poly_int, Y : poly_int, N : nat]: ((dvd_dvd_poly_int @ X @ Y) => (dvd_dvd_poly_int @ (power_power_poly_int @ X @ N) @ (power_power_poly_int @ Y @ N)))))). % dvd_power_same
thf(fact_5_dvd__power__same, axiom,
    ((![X : complex, Y : complex, N : nat]: ((dvd_dvd_complex @ X @ Y) => (dvd_dvd_complex @ (power_power_complex @ X @ N) @ (power_power_complex @ Y @ N)))))). % dvd_power_same
thf(fact_6_dvd__power__same, axiom,
    ((![X : poly_poly_complex, Y : poly_poly_complex, N : nat]: ((dvd_dv598755940omplex @ X @ Y) => (dvd_dv598755940omplex @ (power_432682568omplex @ X @ N) @ (power_432682568omplex @ Y @ N)))))). % dvd_power_same
thf(fact_7_dvd__power__same, axiom,
    ((![X : int, Y : int, N : nat]: ((dvd_dvd_int @ X @ Y) => (dvd_dvd_int @ (power_power_int @ X @ N) @ (power_power_int @ Y @ N)))))). % dvd_power_same
thf(fact_8_dvd__power__same, axiom,
    ((![X : nat, Y : nat, N : nat]: ((dvd_dvd_nat @ X @ Y) => (dvd_dvd_nat @ (power_power_nat @ X @ N) @ (power_power_nat @ Y @ N)))))). % dvd_power_same
thf(fact_9_dvd__power__same, axiom,
    ((![X : poly_complex, Y : poly_complex, N : nat]: ((dvd_dvd_poly_complex @ X @ Y) => (dvd_dvd_poly_complex @ (power_184595776omplex @ X @ N) @ (power_184595776omplex @ Y @ N)))))). % dvd_power_same
thf(fact_10_dvd__refl, axiom,
    ((![A : poly_nat]: (dvd_dvd_poly_nat @ A @ A)))). % dvd_refl
thf(fact_11_dvd__refl, axiom,
    ((![A : poly_int]: (dvd_dvd_poly_int @ A @ A)))). % dvd_refl
thf(fact_12_dvd__refl, axiom,
    ((![A : poly_complex]: (dvd_dvd_poly_complex @ A @ A)))). % dvd_refl
thf(fact_13_dvd__refl, axiom,
    ((![A : int]: (dvd_dvd_int @ A @ A)))). % dvd_refl
thf(fact_14_dvd__refl, axiom,
    ((![A : nat]: (dvd_dvd_nat @ A @ A)))). % dvd_refl
thf(fact_15_dvd__trans, axiom,
    ((![A : poly_nat, B : poly_nat, C : poly_nat]: ((dvd_dvd_poly_nat @ A @ B) => ((dvd_dvd_poly_nat @ B @ C) => (dvd_dvd_poly_nat @ A @ C)))))). % dvd_trans
thf(fact_16_dvd__trans, axiom,
    ((![A : poly_int, B : poly_int, C : poly_int]: ((dvd_dvd_poly_int @ A @ B) => ((dvd_dvd_poly_int @ B @ C) => (dvd_dvd_poly_int @ A @ C)))))). % dvd_trans
thf(fact_17_dvd__trans, axiom,
    ((![A : poly_complex, B : poly_complex, C : poly_complex]: ((dvd_dvd_poly_complex @ A @ B) => ((dvd_dvd_poly_complex @ B @ C) => (dvd_dvd_poly_complex @ A @ C)))))). % dvd_trans
thf(fact_18_dvd__trans, axiom,
    ((![A : int, B : int, C : int]: ((dvd_dvd_int @ A @ B) => ((dvd_dvd_int @ B @ C) => (dvd_dvd_int @ A @ C)))))). % dvd_trans
thf(fact_19_dvd__trans, axiom,
    ((![A : nat, B : nat, C : nat]: ((dvd_dvd_nat @ A @ B) => ((dvd_dvd_nat @ B @ C) => (dvd_dvd_nat @ A @ C)))))). % dvd_trans
thf(fact_20_content__dvd__contentI, axiom,
    ((![P : poly_int, Q : poly_int]: ((dvd_dvd_poly_int @ P @ Q) => (dvd_dvd_int @ (content_int @ P) @ (content_int @ Q)))))). % content_dvd_contentI
thf(fact_21_content__dvd__contentI, axiom,
    ((![P : poly_nat, Q : poly_nat]: ((dvd_dvd_poly_nat @ P @ Q) => (dvd_dvd_nat @ (content_nat @ P) @ (content_nat @ Q)))))). % content_dvd_contentI
thf(fact_22_poly__power, axiom,
    ((![P : poly_nat, N : nat, X : nat]: ((poly_nat2 @ (power_power_poly_nat @ P @ N) @ X) = (power_power_nat @ (poly_nat2 @ P @ X) @ N))))). % poly_power
thf(fact_23_poly__power, axiom,
    ((![P : poly_int, N : nat, X : int]: ((poly_int2 @ (power_power_poly_int @ P @ N) @ X) = (power_power_int @ (poly_int2 @ P @ X) @ N))))). % poly_power
thf(fact_24_poly__power, axiom,
    ((![P : poly_p1267267526omplex, N : nat, X : poly_poly_complex]: ((poly_p282434315omplex @ (power_2001192272omplex @ P @ N) @ X) = (power_432682568omplex @ (poly_p282434315omplex @ P @ X) @ N))))). % poly_power
thf(fact_25_poly__power, axiom,
    ((![P : poly_poly_complex, N : nat, X : poly_complex]: ((poly_poly_complex2 @ (power_432682568omplex @ P @ N) @ X) = (power_184595776omplex @ (poly_poly_complex2 @ P @ X) @ N))))). % poly_power
thf(fact_26_poly__power, axiom,
    ((![P : poly_complex, N : nat, X : complex]: ((poly_complex2 @ (power_184595776omplex @ P @ N) @ X) = (power_power_complex @ (poly_complex2 @ P @ X) @ N))))). % poly_power
thf(fact_27_dvd__power__le, axiom,
    ((![X : poly_nat, Y : poly_nat, N : nat, M : nat]: ((dvd_dvd_poly_nat @ X @ Y) => ((ord_less_eq_nat @ N @ M) => (dvd_dvd_poly_nat @ (power_power_poly_nat @ X @ N) @ (power_power_poly_nat @ Y @ M))))))). % dvd_power_le
thf(fact_28_dvd__power__le, axiom,
    ((![X : poly_int, Y : poly_int, N : nat, M : nat]: ((dvd_dvd_poly_int @ X @ Y) => ((ord_less_eq_nat @ N @ M) => (dvd_dvd_poly_int @ (power_power_poly_int @ X @ N) @ (power_power_poly_int @ Y @ M))))))). % dvd_power_le
thf(fact_29_dvd__power__le, axiom,
    ((![X : complex, Y : complex, N : nat, M : nat]: ((dvd_dvd_complex @ X @ Y) => ((ord_less_eq_nat @ N @ M) => (dvd_dvd_complex @ (power_power_complex @ X @ N) @ (power_power_complex @ Y @ M))))))). % dvd_power_le
thf(fact_30_dvd__power__le, axiom,
    ((![X : poly_poly_complex, Y : poly_poly_complex, N : nat, M : nat]: ((dvd_dv598755940omplex @ X @ Y) => ((ord_less_eq_nat @ N @ M) => (dvd_dv598755940omplex @ (power_432682568omplex @ X @ N) @ (power_432682568omplex @ Y @ M))))))). % dvd_power_le
thf(fact_31_dvd__power__le, axiom,
    ((![X : int, Y : int, N : nat, M : nat]: ((dvd_dvd_int @ X @ Y) => ((ord_less_eq_nat @ N @ M) => (dvd_dvd_int @ (power_power_int @ X @ N) @ (power_power_int @ Y @ M))))))). % dvd_power_le
thf(fact_32_dvd__power__le, axiom,
    ((![X : nat, Y : nat, N : nat, M : nat]: ((dvd_dvd_nat @ X @ Y) => ((ord_less_eq_nat @ N @ M) => (dvd_dvd_nat @ (power_power_nat @ X @ N) @ (power_power_nat @ Y @ M))))))). % dvd_power_le
thf(fact_33_dvd__power__le, axiom,
    ((![X : poly_complex, Y : poly_complex, N : nat, M : nat]: ((dvd_dvd_poly_complex @ X @ Y) => ((ord_less_eq_nat @ N @ M) => (dvd_dvd_poly_complex @ (power_184595776omplex @ X @ N) @ (power_184595776omplex @ Y @ M))))))). % dvd_power_le
thf(fact_34_power__le__dvd, axiom,
    ((![A : poly_nat, N : nat, B : poly_nat, M : nat]: ((dvd_dvd_poly_nat @ (power_power_poly_nat @ A @ N) @ B) => ((ord_less_eq_nat @ M @ N) => (dvd_dvd_poly_nat @ (power_power_poly_nat @ A @ M) @ B)))))). % power_le_dvd
thf(fact_35_power__le__dvd, axiom,
    ((![A : poly_int, N : nat, B : poly_int, M : nat]: ((dvd_dvd_poly_int @ (power_power_poly_int @ A @ N) @ B) => ((ord_less_eq_nat @ M @ N) => (dvd_dvd_poly_int @ (power_power_poly_int @ A @ M) @ B)))))). % power_le_dvd
thf(fact_36_power__le__dvd, axiom,
    ((![A : complex, N : nat, B : complex, M : nat]: ((dvd_dvd_complex @ (power_power_complex @ A @ N) @ B) => ((ord_less_eq_nat @ M @ N) => (dvd_dvd_complex @ (power_power_complex @ A @ M) @ B)))))). % power_le_dvd
thf(fact_37_power__le__dvd, axiom,
    ((![A : poly_poly_complex, N : nat, B : poly_poly_complex, M : nat]: ((dvd_dv598755940omplex @ (power_432682568omplex @ A @ N) @ B) => ((ord_less_eq_nat @ M @ N) => (dvd_dv598755940omplex @ (power_432682568omplex @ A @ M) @ B)))))). % power_le_dvd
thf(fact_38_power__le__dvd, axiom,
    ((![A : int, N : nat, B : int, M : nat]: ((dvd_dvd_int @ (power_power_int @ A @ N) @ B) => ((ord_less_eq_nat @ M @ N) => (dvd_dvd_int @ (power_power_int @ A @ M) @ B)))))). % power_le_dvd
thf(fact_39_power__le__dvd, axiom,
    ((![A : nat, N : nat, B : nat, M : nat]: ((dvd_dvd_nat @ (power_power_nat @ A @ N) @ B) => ((ord_less_eq_nat @ M @ N) => (dvd_dvd_nat @ (power_power_nat @ A @ M) @ B)))))). % power_le_dvd
thf(fact_40_power__le__dvd, axiom,
    ((![A : poly_complex, N : nat, B : poly_complex, M : nat]: ((dvd_dvd_poly_complex @ (power_184595776omplex @ A @ N) @ B) => ((ord_less_eq_nat @ M @ N) => (dvd_dvd_poly_complex @ (power_184595776omplex @ A @ M) @ B)))))). % power_le_dvd
thf(fact_41_le__imp__power__dvd, axiom,
    ((![M : nat, N : nat, A : poly_nat]: ((ord_less_eq_nat @ M @ N) => (dvd_dvd_poly_nat @ (power_power_poly_nat @ A @ M) @ (power_power_poly_nat @ A @ N)))))). % le_imp_power_dvd
thf(fact_42_le__imp__power__dvd, axiom,
    ((![M : nat, N : nat, A : poly_int]: ((ord_less_eq_nat @ M @ N) => (dvd_dvd_poly_int @ (power_power_poly_int @ A @ M) @ (power_power_poly_int @ A @ N)))))). % le_imp_power_dvd
thf(fact_43_le__imp__power__dvd, axiom,
    ((![M : nat, N : nat, A : poly_complex]: ((ord_less_eq_nat @ M @ N) => (dvd_dvd_poly_complex @ (power_184595776omplex @ A @ M) @ (power_184595776omplex @ A @ N)))))). % le_imp_power_dvd
thf(fact_44_le__imp__power__dvd, axiom,
    ((![M : nat, N : nat, A : nat]: ((ord_less_eq_nat @ M @ N) => (dvd_dvd_nat @ (power_power_nat @ A @ M) @ (power_power_nat @ A @ N)))))). % le_imp_power_dvd
thf(fact_45_le__imp__power__dvd, axiom,
    ((![M : nat, N : nat, A : int]: ((ord_less_eq_nat @ M @ N) => (dvd_dvd_int @ (power_power_int @ A @ M) @ (power_power_int @ A @ N)))))). % le_imp_power_dvd
thf(fact_46_le__imp__power__dvd, axiom,
    ((![M : nat, N : nat, A : complex]: ((ord_less_eq_nat @ M @ N) => (dvd_dvd_complex @ (power_power_complex @ A @ M) @ (power_power_complex @ A @ N)))))). % le_imp_power_dvd
thf(fact_47_le__imp__power__dvd, axiom,
    ((![M : nat, N : nat, A : poly_poly_complex]: ((ord_less_eq_nat @ M @ N) => (dvd_dv598755940omplex @ (power_432682568omplex @ A @ M) @ (power_432682568omplex @ A @ N)))))). % le_imp_power_dvd
thf(fact_48_prime__elem__dvd__power, axiom,
    ((![P : poly_nat, X : poly_nat, N : nat]: ((factor745290973ly_nat @ P) => ((dvd_dvd_poly_nat @ P @ (power_power_poly_nat @ X @ N)) => (dvd_dvd_poly_nat @ P @ X)))))). % prime_elem_dvd_power
thf(fact_49_prime__elem__dvd__power, axiom,
    ((![P : poly_int, X : poly_int, N : nat]: ((factor907203897ly_int @ P) => ((dvd_dvd_poly_int @ P @ (power_power_poly_int @ X @ N)) => (dvd_dvd_poly_int @ P @ X)))))). % prime_elem_dvd_power
thf(fact_50_prime__elem__dvd__power, axiom,
    ((![P : poly_complex, X : poly_complex, N : nat]: ((factor477435579omplex @ P) => ((dvd_dvd_poly_complex @ P @ (power_184595776omplex @ X @ N)) => (dvd_dvd_poly_complex @ P @ X)))))). % prime_elem_dvd_power
thf(fact_51_prime__elem__dvd__power, axiom,
    ((![P : nat, X : nat, N : nat]: ((factor127820501em_nat @ P) => ((dvd_dvd_nat @ P @ (power_power_nat @ X @ N)) => (dvd_dvd_nat @ P @ X)))))). % prime_elem_dvd_power
thf(fact_52_prime__elem__dvd__power, axiom,
    ((![P : int, X : int, N : nat]: ((factor765094193em_int @ P) => ((dvd_dvd_int @ P @ (power_power_int @ X @ N)) => (dvd_dvd_int @ P @ X)))))). % prime_elem_dvd_power
thf(fact_53_prime__elem__dvd__power, axiom,
    ((![P : complex, X : complex, N : nat]: ((factor392545715omplex @ P) => ((dvd_dvd_complex @ P @ (power_power_complex @ X @ N)) => (dvd_dvd_complex @ P @ X)))))). % prime_elem_dvd_power
thf(fact_54_prime__elem__dvd__power, axiom,
    ((![P : poly_poly_complex, X : poly_poly_complex, N : nat]: ((factor1360491971omplex @ P) => ((dvd_dv598755940omplex @ P @ (power_432682568omplex @ X @ N)) => (dvd_dv598755940omplex @ P @ X)))))). % prime_elem_dvd_power
thf(fact_55_div__power, axiom,
    ((![B : poly_int, A : poly_int, N : nat]: ((dvd_dvd_poly_int @ B @ A) => ((power_power_poly_int @ (divide1415015302ly_int @ A @ B) @ N) = (divide1415015302ly_int @ (power_power_poly_int @ A @ N) @ (power_power_poly_int @ B @ N))))))). % div_power
thf(fact_56_div__power, axiom,
    ((![B : poly_complex, A : poly_complex, N : nat]: ((dvd_dvd_poly_complex @ B @ A) => ((power_184595776omplex @ (divide1187762952omplex @ A @ B) @ N) = (divide1187762952omplex @ (power_184595776omplex @ A @ N) @ (power_184595776omplex @ B @ N))))))). % div_power
thf(fact_57_div__power, axiom,
    ((![B : poly_poly_complex, A : poly_poly_complex, N : nat]: ((dvd_dv598755940omplex @ B @ A) => ((power_432682568omplex @ (divide350004240omplex @ A @ B) @ N) = (divide350004240omplex @ (power_432682568omplex @ A @ N) @ (power_432682568omplex @ B @ N))))))). % div_power
thf(fact_58_div__power, axiom,
    ((![B : nat, A : nat, N : nat]: ((dvd_dvd_nat @ B @ A) => ((power_power_nat @ (divide_divide_nat @ A @ B) @ N) = (divide_divide_nat @ (power_power_nat @ A @ N) @ (power_power_nat @ B @ N))))))). % div_power
thf(fact_59_div__power, axiom,
    ((![B : int, A : int, N : nat]: ((dvd_dvd_int @ B @ A) => ((power_power_int @ (divide_divide_int @ A @ B) @ N) = (divide_divide_int @ (power_power_int @ A @ N) @ (power_power_int @ B @ N))))))). % div_power
thf(fact_60_div__dvd__div, axiom,
    ((![A : poly_complex, B : poly_complex, C : poly_complex]: ((dvd_dvd_poly_complex @ A @ B) => ((dvd_dvd_poly_complex @ A @ C) => ((dvd_dvd_poly_complex @ (divide1187762952omplex @ B @ A) @ (divide1187762952omplex @ C @ A)) = (dvd_dvd_poly_complex @ B @ C))))))). % div_dvd_div
thf(fact_61_div__dvd__div, axiom,
    ((![A : poly_int, B : poly_int, C : poly_int]: ((dvd_dvd_poly_int @ A @ B) => ((dvd_dvd_poly_int @ A @ C) => ((dvd_dvd_poly_int @ (divide1415015302ly_int @ B @ A) @ (divide1415015302ly_int @ C @ A)) = (dvd_dvd_poly_int @ B @ C))))))). % div_dvd_div
thf(fact_62_div__dvd__div, axiom,
    ((![A : nat, B : nat, C : nat]: ((dvd_dvd_nat @ A @ B) => ((dvd_dvd_nat @ A @ C) => ((dvd_dvd_nat @ (divide_divide_nat @ B @ A) @ (divide_divide_nat @ C @ A)) = (dvd_dvd_nat @ B @ C))))))). % div_dvd_div
thf(fact_63_div__dvd__div, axiom,
    ((![A : int, B : int, C : int]: ((dvd_dvd_int @ A @ B) => ((dvd_dvd_int @ A @ C) => ((dvd_dvd_int @ (divide_divide_int @ B @ A) @ (divide_divide_int @ C @ A)) = (dvd_dvd_int @ B @ C))))))). % div_dvd_div
thf(fact_64_poly__eq__poly__eq__iff, axiom,
    ((![P : poly_complex, Q : poly_complex]: (((poly_complex2 @ P) = (poly_complex2 @ Q)) = (P = Q))))). % poly_eq_poly_eq_iff
thf(fact_65_poly__eq__poly__eq__iff, axiom,
    ((![P : poly_poly_complex, Q : poly_poly_complex]: (((poly_poly_complex2 @ P) = (poly_poly_complex2 @ Q)) = (P = Q))))). % poly_eq_poly_eq_iff
thf(fact_66_power__divide, axiom,
    ((![A : complex, B : complex, N : nat]: ((power_power_complex @ (divide1210191872omplex @ A @ B) @ N) = (divide1210191872omplex @ (power_power_complex @ A @ N) @ (power_power_complex @ B @ N)))))). % power_divide
thf(fact_67_power__divide, axiom,
    ((![A : real, B : real, N : nat]: ((power_power_real @ (divide_divide_real @ A @ B) @ N) = (divide_divide_real @ (power_power_real @ A @ N) @ (power_power_real @ B @ N)))))). % power_divide
thf(fact_68_div__div__div__same, axiom,
    ((![D : poly_complex, B : poly_complex, A : poly_complex]: ((dvd_dvd_poly_complex @ D @ B) => ((dvd_dvd_poly_complex @ B @ A) => ((divide1187762952omplex @ (divide1187762952omplex @ A @ D) @ (divide1187762952omplex @ B @ D)) = (divide1187762952omplex @ A @ B))))))). % div_div_div_same
thf(fact_69_div__div__div__same, axiom,
    ((![D : poly_int, B : poly_int, A : poly_int]: ((dvd_dvd_poly_int @ D @ B) => ((dvd_dvd_poly_int @ B @ A) => ((divide1415015302ly_int @ (divide1415015302ly_int @ A @ D) @ (divide1415015302ly_int @ B @ D)) = (divide1415015302ly_int @ A @ B))))))). % div_div_div_same
thf(fact_70_div__div__div__same, axiom,
    ((![D : nat, B : nat, A : nat]: ((dvd_dvd_nat @ D @ B) => ((dvd_dvd_nat @ B @ A) => ((divide_divide_nat @ (divide_divide_nat @ A @ D) @ (divide_divide_nat @ B @ D)) = (divide_divide_nat @ A @ B))))))). % div_div_div_same
thf(fact_71_div__div__div__same, axiom,
    ((![D : int, B : int, A : int]: ((dvd_dvd_int @ D @ B) => ((dvd_dvd_int @ B @ A) => ((divide_divide_int @ (divide_divide_int @ A @ D) @ (divide_divide_int @ B @ D)) = (divide_divide_int @ A @ B))))))). % div_div_div_same
thf(fact_72_dvd__div__eq__cancel, axiom,
    ((![A : poly_complex, C : poly_complex, B : poly_complex]: (((divide1187762952omplex @ A @ C) = (divide1187762952omplex @ B @ C)) => ((dvd_dvd_poly_complex @ C @ A) => ((dvd_dvd_poly_complex @ C @ B) => (A = B))))))). % dvd_div_eq_cancel
thf(fact_73_dvd__div__eq__cancel, axiom,
    ((![A : poly_int, C : poly_int, B : poly_int]: (((divide1415015302ly_int @ A @ C) = (divide1415015302ly_int @ B @ C)) => ((dvd_dvd_poly_int @ C @ A) => ((dvd_dvd_poly_int @ C @ B) => (A = B))))))). % dvd_div_eq_cancel
thf(fact_74_dvd__div__eq__cancel, axiom,
    ((![A : nat, C : nat, B : nat]: (((divide_divide_nat @ A @ C) = (divide_divide_nat @ B @ C)) => ((dvd_dvd_nat @ C @ A) => ((dvd_dvd_nat @ C @ B) => (A = B))))))). % dvd_div_eq_cancel
thf(fact_75_dvd__div__eq__cancel, axiom,
    ((![A : int, C : int, B : int]: (((divide_divide_int @ A @ C) = (divide_divide_int @ B @ C)) => ((dvd_dvd_int @ C @ A) => ((dvd_dvd_int @ C @ B) => (A = B))))))). % dvd_div_eq_cancel
thf(fact_76_dvd__div__eq__cancel, axiom,
    ((![A : real, C : real, B : real]: (((divide_divide_real @ A @ C) = (divide_divide_real @ B @ C)) => ((dvd_dvd_real @ C @ A) => ((dvd_dvd_real @ C @ B) => (A = B))))))). % dvd_div_eq_cancel
thf(fact_77_dvd__div__eq__iff, axiom,
    ((![C : poly_complex, A : poly_complex, B : poly_complex]: ((dvd_dvd_poly_complex @ C @ A) => ((dvd_dvd_poly_complex @ C @ B) => (((divide1187762952omplex @ A @ C) = (divide1187762952omplex @ B @ C)) = (A = B))))))). % dvd_div_eq_iff
thf(fact_78_dvd__div__eq__iff, axiom,
    ((![C : poly_int, A : poly_int, B : poly_int]: ((dvd_dvd_poly_int @ C @ A) => ((dvd_dvd_poly_int @ C @ B) => (((divide1415015302ly_int @ A @ C) = (divide1415015302ly_int @ B @ C)) = (A = B))))))). % dvd_div_eq_iff
thf(fact_79_dvd__div__eq__iff, axiom,
    ((![C : nat, A : nat, B : nat]: ((dvd_dvd_nat @ C @ A) => ((dvd_dvd_nat @ C @ B) => (((divide_divide_nat @ A @ C) = (divide_divide_nat @ B @ C)) = (A = B))))))). % dvd_div_eq_iff
thf(fact_80_dvd__div__eq__iff, axiom,
    ((![C : int, A : int, B : int]: ((dvd_dvd_int @ C @ A) => ((dvd_dvd_int @ C @ B) => (((divide_divide_int @ A @ C) = (divide_divide_int @ B @ C)) = (A = B))))))). % dvd_div_eq_iff
thf(fact_81_dvd__div__eq__iff, axiom,
    ((![C : real, A : real, B : real]: ((dvd_dvd_real @ C @ A) => ((dvd_dvd_real @ C @ B) => (((divide_divide_real @ A @ C) = (divide_divide_real @ B @ C)) = (A = B))))))). % dvd_div_eq_iff
thf(fact_82_order__refl, axiom,
    ((![X : nat]: (ord_less_eq_nat @ X @ X)))). % order_refl
thf(fact_83_order__refl, axiom,
    ((![X : int]: (ord_less_eq_int @ X @ X)))). % order_refl
thf(fact_84_order__refl, axiom,
    ((![X : real]: (ord_less_eq_real @ X @ X)))). % order_refl
thf(fact_85_of__nat__le__of__nat__power__cancel__iff, axiom,
    ((![B : nat, W : nat, X : nat]: ((ord_less_eq_nat @ (power_power_nat @ (semiri1382578993at_nat @ B) @ W) @ (semiri1382578993at_nat @ X)) = (ord_less_eq_nat @ (power_power_nat @ B @ W) @ X))))). % of_nat_le_of_nat_power_cancel_iff
thf(fact_86_of__nat__le__of__nat__power__cancel__iff, axiom,
    ((![B : nat, W : nat, X : nat]: ((ord_less_eq_int @ (power_power_int @ (semiri2019852685at_int @ B) @ W) @ (semiri2019852685at_int @ X)) = (ord_less_eq_nat @ (power_power_nat @ B @ W) @ X))))). % of_nat_le_of_nat_power_cancel_iff
thf(fact_87_of__nat__le__of__nat__power__cancel__iff, axiom,
    ((![B : nat, W : nat, X : nat]: ((ord_less_eq_real @ (power_power_real @ (semiri2110766477t_real @ B) @ W) @ (semiri2110766477t_real @ X)) = (ord_less_eq_nat @ (power_power_nat @ B @ W) @ X))))). % of_nat_le_of_nat_power_cancel_iff
thf(fact_88_of__nat__power__le__of__nat__cancel__iff, axiom,
    ((![X : nat, B : nat, W : nat]: ((ord_less_eq_nat @ (semiri1382578993at_nat @ X) @ (power_power_nat @ (semiri1382578993at_nat @ B) @ W)) = (ord_less_eq_nat @ X @ (power_power_nat @ B @ W)))))). % of_nat_power_le_of_nat_cancel_iff
thf(fact_89_of__nat__power__le__of__nat__cancel__iff, axiom,
    ((![X : nat, B : nat, W : nat]: ((ord_less_eq_int @ (semiri2019852685at_int @ X) @ (power_power_int @ (semiri2019852685at_int @ B) @ W)) = (ord_less_eq_nat @ X @ (power_power_nat @ B @ W)))))). % of_nat_power_le_of_nat_cancel_iff
thf(fact_90_of__nat__power__le__of__nat__cancel__iff, axiom,
    ((![X : nat, B : nat, W : nat]: ((ord_less_eq_real @ (semiri2110766477t_real @ X) @ (power_power_real @ (semiri2110766477t_real @ B) @ W)) = (ord_less_eq_nat @ X @ (power_power_nat @ B @ W)))))). % of_nat_power_le_of_nat_cancel_iff
thf(fact_91_le__refl, axiom,
    ((![N : nat]: (ord_less_eq_nat @ N @ N)))). % le_refl
thf(fact_92_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_93_eq__imp__le, axiom,
    ((![M : nat, N : nat]: ((M = N) => (ord_less_eq_nat @ M @ N))))). % eq_imp_le
thf(fact_94_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_95_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_96_Nat_Oex__has__greatest__nat, axiom,
    ((![P2 : nat > $o, K : nat, B : nat]: ((P2 @ K) => ((![Y2 : nat]: ((P2 @ Y2) => (ord_less_eq_nat @ Y2 @ B))) => (?[X2 : nat]: ((P2 @ X2) & (![Y3 : nat]: ((P2 @ Y3) => (ord_less_eq_nat @ Y3 @ X2)))))))))). % Nat.ex_has_greatest_nat
thf(fact_97_bounded__Max__nat, axiom,
    ((![P2 : nat > $o, X : nat, M2 : nat]: ((P2 @ X) => ((![X2 : nat]: ((P2 @ X2) => (ord_less_eq_nat @ X2 @ M2))) => (~ ((![M3 : nat]: ((P2 @ M3) => (~ ((![X3 : nat]: ((P2 @ X3) => (ord_less_eq_nat @ X3 @ M3)))))))))))))). % bounded_Max_nat
thf(fact_98_of__nat__eq__iff, axiom,
    ((![M : nat, N : nat]: (((semiri2019852685at_int @ M) = (semiri2019852685at_int @ N)) = (M = N))))). % of_nat_eq_iff
thf(fact_99_of__nat__eq__iff, axiom,
    ((![M : nat, N : nat]: (((semiri2110766477t_real @ M) = (semiri2110766477t_real @ N)) = (M = N))))). % of_nat_eq_iff
thf(fact_100_of__nat__le__iff, axiom,
    ((![M : nat, N : nat]: ((ord_less_eq_nat @ (semiri1382578993at_nat @ M) @ (semiri1382578993at_nat @ N)) = (ord_less_eq_nat @ M @ N))))). % of_nat_le_iff
thf(fact_101_of__nat__le__iff, axiom,
    ((![M : nat, N : nat]: ((ord_less_eq_int @ (semiri2019852685at_int @ M) @ (semiri2019852685at_int @ N)) = (ord_less_eq_nat @ M @ N))))). % of_nat_le_iff
thf(fact_102_of__nat__le__iff, axiom,
    ((![M : nat, N : nat]: ((ord_less_eq_real @ (semiri2110766477t_real @ M) @ (semiri2110766477t_real @ N)) = (ord_less_eq_nat @ M @ N))))). % of_nat_le_iff
thf(fact_103_of__nat__power, axiom,
    ((![M : nat, N : nat]: ((semiri1679838999omplex @ (power_power_nat @ M @ N)) = (power_184595776omplex @ (semiri1679838999omplex @ M) @ N))))). % of_nat_power
thf(fact_104_of__nat__power, axiom,
    ((![M : nat, N : nat]: ((semiri1382578993at_nat @ (power_power_nat @ M @ N)) = (power_power_nat @ (semiri1382578993at_nat @ M) @ N))))). % of_nat_power
thf(fact_105_of__nat__power, axiom,
    ((![M : nat, N : nat]: ((semiri356525583omplex @ (power_power_nat @ M @ N)) = (power_power_complex @ (semiri356525583omplex @ M) @ N))))). % of_nat_power
thf(fact_106_of__nat__power, axiom,
    ((![M : nat, N : nat]: ((semiri1266910751omplex @ (power_power_nat @ M @ N)) = (power_432682568omplex @ (semiri1266910751omplex @ M) @ N))))). % of_nat_power
thf(fact_107_of__nat__power, axiom,
    ((![M : nat, N : nat]: ((semiri2019852685at_int @ (power_power_nat @ M @ N)) = (power_power_int @ (semiri2019852685at_int @ M) @ N))))). % of_nat_power
thf(fact_108_of__nat__power, axiom,
    ((![M : nat, N : nat]: ((semiri2110766477t_real @ (power_power_nat @ M @ N)) = (power_power_real @ (semiri2110766477t_real @ M) @ N))))). % of_nat_power
thf(fact_109_of__nat__eq__of__nat__power__cancel__iff, axiom,
    ((![B : nat, W : nat, X : nat]: (((power_184595776omplex @ (semiri1679838999omplex @ B) @ W) = (semiri1679838999omplex @ X)) = ((power_power_nat @ B @ W) = X))))). % of_nat_eq_of_nat_power_cancel_iff
thf(fact_110_of__nat__eq__of__nat__power__cancel__iff, axiom,
    ((![B : nat, W : nat, X : nat]: (((power_power_nat @ (semiri1382578993at_nat @ B) @ W) = (semiri1382578993at_nat @ X)) = ((power_power_nat @ B @ W) = X))))). % of_nat_eq_of_nat_power_cancel_iff
thf(fact_111_of__nat__eq__of__nat__power__cancel__iff, axiom,
    ((![B : nat, W : nat, X : nat]: (((power_power_complex @ (semiri356525583omplex @ B) @ W) = (semiri356525583omplex @ X)) = ((power_power_nat @ B @ W) = X))))). % of_nat_eq_of_nat_power_cancel_iff
thf(fact_112_of__nat__eq__of__nat__power__cancel__iff, axiom,
    ((![B : nat, W : nat, X : nat]: (((power_432682568omplex @ (semiri1266910751omplex @ B) @ W) = (semiri1266910751omplex @ X)) = ((power_power_nat @ B @ W) = X))))). % of_nat_eq_of_nat_power_cancel_iff
thf(fact_113_of__nat__eq__of__nat__power__cancel__iff, axiom,
    ((![B : nat, W : nat, X : nat]: (((power_power_int @ (semiri2019852685at_int @ B) @ W) = (semiri2019852685at_int @ X)) = ((power_power_nat @ B @ W) = X))))). % of_nat_eq_of_nat_power_cancel_iff
thf(fact_114_of__nat__eq__of__nat__power__cancel__iff, axiom,
    ((![B : nat, W : nat, X : nat]: (((power_power_real @ (semiri2110766477t_real @ B) @ W) = (semiri2110766477t_real @ X)) = ((power_power_nat @ B @ W) = X))))). % of_nat_eq_of_nat_power_cancel_iff
thf(fact_115_of__nat__power__eq__of__nat__cancel__iff, axiom,
    ((![X : nat, B : nat, W : nat]: (((semiri1679838999omplex @ X) = (power_184595776omplex @ (semiri1679838999omplex @ B) @ W)) = (X = (power_power_nat @ B @ W)))))). % of_nat_power_eq_of_nat_cancel_iff
thf(fact_116_of__nat__power__eq__of__nat__cancel__iff, axiom,
    ((![X : nat, B : nat, W : nat]: (((semiri1382578993at_nat @ X) = (power_power_nat @ (semiri1382578993at_nat @ B) @ W)) = (X = (power_power_nat @ B @ W)))))). % of_nat_power_eq_of_nat_cancel_iff
thf(fact_117_of__nat__power__eq__of__nat__cancel__iff, axiom,
    ((![X : nat, B : nat, W : nat]: (((semiri356525583omplex @ X) = (power_power_complex @ (semiri356525583omplex @ B) @ W)) = (X = (power_power_nat @ B @ W)))))). % of_nat_power_eq_of_nat_cancel_iff
thf(fact_118_of__nat__power__eq__of__nat__cancel__iff, axiom,
    ((![X : nat, B : nat, W : nat]: (((semiri1266910751omplex @ X) = (power_432682568omplex @ (semiri1266910751omplex @ B) @ W)) = (X = (power_power_nat @ B @ W)))))). % of_nat_power_eq_of_nat_cancel_iff
thf(fact_119_of__nat__power__eq__of__nat__cancel__iff, axiom,
    ((![X : nat, B : nat, W : nat]: (((semiri2019852685at_int @ X) = (power_power_int @ (semiri2019852685at_int @ B) @ W)) = (X = (power_power_nat @ B @ W)))))). % of_nat_power_eq_of_nat_cancel_iff
thf(fact_120_of__nat__power__eq__of__nat__cancel__iff, axiom,
    ((![X : nat, B : nat, W : nat]: (((semiri2110766477t_real @ X) = (power_power_real @ (semiri2110766477t_real @ B) @ W)) = (X = (power_power_nat @ B @ W)))))). % of_nat_power_eq_of_nat_cancel_iff
thf(fact_121_of__nat__mono, axiom,
    ((![I : nat, J : nat]: ((ord_less_eq_nat @ I @ J) => (ord_less_eq_nat @ (semiri1382578993at_nat @ I) @ (semiri1382578993at_nat @ J)))))). % of_nat_mono
thf(fact_122_of__nat__mono, axiom,
    ((![I : nat, J : nat]: ((ord_less_eq_nat @ I @ J) => (ord_less_eq_int @ (semiri2019852685at_int @ I) @ (semiri2019852685at_int @ J)))))). % of_nat_mono
thf(fact_123_of__nat__mono, axiom,
    ((![I : nat, J : nat]: ((ord_less_eq_nat @ I @ J) => (ord_less_eq_real @ (semiri2110766477t_real @ I) @ (semiri2110766477t_real @ J)))))). % of_nat_mono
thf(fact_124_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_125_dual__order_Oantisym, axiom,
    ((![B : int, A : int]: ((ord_less_eq_int @ B @ A) => ((ord_less_eq_int @ A @ B) => (A = B)))))). % dual_order.antisym
thf(fact_126_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_127_dual__order_Oeq__iff, axiom,
    (((^[Y4 : nat]: (^[Z : nat]: (Y4 = Z))) = (^[A2 : nat]: (^[B2 : nat]: (((ord_less_eq_nat @ B2 @ A2)) & ((ord_less_eq_nat @ A2 @ B2)))))))). % dual_order.eq_iff
thf(fact_128_dual__order_Oeq__iff, axiom,
    (((^[Y4 : int]: (^[Z : int]: (Y4 = Z))) = (^[A2 : int]: (^[B2 : int]: (((ord_less_eq_int @ B2 @ A2)) & ((ord_less_eq_int @ A2 @ B2)))))))). % dual_order.eq_iff
thf(fact_129_dual__order_Oeq__iff, axiom,
    (((^[Y4 : real]: (^[Z : real]: (Y4 = Z))) = (^[A2 : real]: (^[B2 : real]: (((ord_less_eq_real @ B2 @ A2)) & ((ord_less_eq_real @ A2 @ B2)))))))). % dual_order.eq_iff
thf(fact_130_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_131_dual__order_Otrans, axiom,
    ((![B : int, A : int, C : int]: ((ord_less_eq_int @ B @ A) => ((ord_less_eq_int @ C @ B) => (ord_less_eq_int @ C @ A)))))). % dual_order.trans
thf(fact_132_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_133_linorder__wlog, axiom,
    ((![P2 : nat > nat > $o, A : nat, B : nat]: ((![A3 : nat, B3 : nat]: ((ord_less_eq_nat @ A3 @ B3) => (P2 @ A3 @ B3))) => ((![A3 : nat, B3 : nat]: ((P2 @ B3 @ A3) => (P2 @ A3 @ B3))) => (P2 @ A @ B)))))). % linorder_wlog
thf(fact_134_linorder__wlog, axiom,
    ((![P2 : int > int > $o, A : int, B : int]: ((![A3 : int, B3 : int]: ((ord_less_eq_int @ A3 @ B3) => (P2 @ A3 @ B3))) => ((![A3 : int, B3 : int]: ((P2 @ B3 @ A3) => (P2 @ A3 @ B3))) => (P2 @ A @ B)))))). % linorder_wlog
thf(fact_135_linorder__wlog, axiom,
    ((![P2 : real > real > $o, A : real, B : real]: ((![A3 : real, B3 : real]: ((ord_less_eq_real @ A3 @ B3) => (P2 @ A3 @ B3))) => ((![A3 : real, B3 : real]: ((P2 @ B3 @ A3) => (P2 @ A3 @ B3))) => (P2 @ A @ B)))))). % linorder_wlog
thf(fact_136_dual__order_Orefl, axiom,
    ((![A : nat]: (ord_less_eq_nat @ A @ A)))). % dual_order.refl
thf(fact_137_dual__order_Orefl, axiom,
    ((![A : int]: (ord_less_eq_int @ A @ A)))). % dual_order.refl
thf(fact_138_dual__order_Orefl, axiom,
    ((![A : real]: (ord_less_eq_real @ A @ A)))). % dual_order.refl
thf(fact_139_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
thf(fact_140_order__trans, axiom,
    ((![X : int, Y : int, Z2 : int]: ((ord_less_eq_int @ X @ Y) => ((ord_less_eq_int @ Y @ Z2) => (ord_less_eq_int @ X @ Z2)))))). % order_trans
thf(fact_141_order__trans, axiom,
    ((![X : real, Y : real, Z2 : real]: ((ord_less_eq_real @ X @ Y) => ((ord_less_eq_real @ Y @ Z2) => (ord_less_eq_real @ X @ Z2)))))). % order_trans
thf(fact_142_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_143_order__class_Oorder_Oantisym, axiom,
    ((![A : int, B : int]: ((ord_less_eq_int @ A @ B) => ((ord_less_eq_int @ B @ A) => (A = B)))))). % order_class.order.antisym
thf(fact_144_order__class_Oorder_Oantisym, axiom,
    ((![A : real, B : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_eq_real @ B @ A) => (A = B)))))). % order_class.order.antisym
thf(fact_145_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_146_ord__le__eq__trans, axiom,
    ((![A : int, B : int, C : int]: ((ord_less_eq_int @ A @ B) => ((B = C) => (ord_less_eq_int @ A @ C)))))). % ord_le_eq_trans
thf(fact_147_ord__le__eq__trans, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_eq_real @ A @ B) => ((B = C) => (ord_less_eq_real @ A @ C)))))). % ord_le_eq_trans
thf(fact_148_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_149_ord__eq__le__trans, axiom,
    ((![A : int, B : int, C : int]: ((A = B) => ((ord_less_eq_int @ B @ C) => (ord_less_eq_int @ A @ C)))))). % ord_eq_le_trans
thf(fact_150_ord__eq__le__trans, axiom,
    ((![A : real, B : real, C : real]: ((A = B) => ((ord_less_eq_real @ B @ C) => (ord_less_eq_real @ A @ C)))))). % ord_eq_le_trans
thf(fact_151_order__class_Oorder_Oeq__iff, axiom,
    (((^[Y4 : nat]: (^[Z : nat]: (Y4 = Z))) = (^[A2 : nat]: (^[B2 : nat]: (((ord_less_eq_nat @ A2 @ B2)) & ((ord_less_eq_nat @ B2 @ A2)))))))). % order_class.order.eq_iff
thf(fact_152_order__class_Oorder_Oeq__iff, axiom,
    (((^[Y4 : int]: (^[Z : int]: (Y4 = Z))) = (^[A2 : int]: (^[B2 : int]: (((ord_less_eq_int @ A2 @ B2)) & ((ord_less_eq_int @ B2 @ A2)))))))). % order_class.order.eq_iff
thf(fact_153_order__class_Oorder_Oeq__iff, axiom,
    (((^[Y4 : real]: (^[Z : real]: (Y4 = Z))) = (^[A2 : real]: (^[B2 : real]: (((ord_less_eq_real @ A2 @ B2)) & ((ord_less_eq_real @ B2 @ A2)))))))). % order_class.order.eq_iff
thf(fact_154_mem__Collect__eq, axiom,
    ((![A : real, P2 : real > $o]: ((member_real @ A @ (collect_real @ P2)) = (P2 @ A))))). % mem_Collect_eq
thf(fact_155_Collect__mem__eq, axiom,
    ((![A4 : set_real]: ((collect_real @ (^[X4 : real]: (member_real @ X4 @ A4))) = A4)))). % Collect_mem_eq
thf(fact_156_antisym__conv, axiom,
    ((![Y : nat, X : nat]: ((ord_less_eq_nat @ Y @ X) => ((ord_less_eq_nat @ X @ Y) = (X = Y)))))). % antisym_conv
thf(fact_157_antisym__conv, axiom,
    ((![Y : int, X : int]: ((ord_less_eq_int @ Y @ X) => ((ord_less_eq_int @ X @ Y) = (X = Y)))))). % antisym_conv
thf(fact_158_antisym__conv, axiom,
    ((![Y : real, X : real]: ((ord_less_eq_real @ Y @ X) => ((ord_less_eq_real @ X @ Y) = (X = Y)))))). % antisym_conv
thf(fact_159_le__cases3, axiom,
    ((![X : nat, Y : nat, Z2 : nat]: (((ord_less_eq_nat @ X @ Y) => (~ ((ord_less_eq_nat @ Y @ Z2)))) => (((ord_less_eq_nat @ Y @ X) => (~ ((ord_less_eq_nat @ X @ Z2)))) => (((ord_less_eq_nat @ X @ Z2) => (~ ((ord_less_eq_nat @ Z2 @ Y)))) => (((ord_less_eq_nat @ Z2 @ Y) => (~ ((ord_less_eq_nat @ Y @ X)))) => (((ord_less_eq_nat @ Y @ Z2) => (~ ((ord_less_eq_nat @ Z2 @ X)))) => (~ (((ord_less_eq_nat @ Z2 @ X) => (~ ((ord_less_eq_nat @ X @ Y)))))))))))))). % le_cases3
thf(fact_160_le__cases3, axiom,
    ((![X : int, Y : int, Z2 : int]: (((ord_less_eq_int @ X @ Y) => (~ ((ord_less_eq_int @ Y @ Z2)))) => (((ord_less_eq_int @ Y @ X) => (~ ((ord_less_eq_int @ X @ Z2)))) => (((ord_less_eq_int @ X @ Z2) => (~ ((ord_less_eq_int @ Z2 @ Y)))) => (((ord_less_eq_int @ Z2 @ Y) => (~ ((ord_less_eq_int @ Y @ X)))) => (((ord_less_eq_int @ Y @ Z2) => (~ ((ord_less_eq_int @ Z2 @ X)))) => (~ (((ord_less_eq_int @ Z2 @ X) => (~ ((ord_less_eq_int @ X @ Y)))))))))))))). % le_cases3
thf(fact_161_le__cases3, axiom,
    ((![X : real, Y : real, Z2 : real]: (((ord_less_eq_real @ X @ Y) => (~ ((ord_less_eq_real @ Y @ Z2)))) => (((ord_less_eq_real @ Y @ X) => (~ ((ord_less_eq_real @ X @ Z2)))) => (((ord_less_eq_real @ X @ Z2) => (~ ((ord_less_eq_real @ Z2 @ Y)))) => (((ord_less_eq_real @ Z2 @ Y) => (~ ((ord_less_eq_real @ Y @ X)))) => (((ord_less_eq_real @ Y @ Z2) => (~ ((ord_less_eq_real @ Z2 @ X)))) => (~ (((ord_less_eq_real @ Z2 @ X) => (~ ((ord_less_eq_real @ X @ Y)))))))))))))). % le_cases3
thf(fact_162_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_163_order_Otrans, axiom,
    ((![A : int, B : int, C : int]: ((ord_less_eq_int @ A @ B) => ((ord_less_eq_int @ B @ C) => (ord_less_eq_int @ A @ C)))))). % order.trans
thf(fact_164_order_Otrans, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_eq_real @ B @ C) => (ord_less_eq_real @ A @ C)))))). % order.trans
thf(fact_165_le__cases, axiom,
    ((![X : nat, Y : nat]: ((~ ((ord_less_eq_nat @ X @ Y))) => (ord_less_eq_nat @ Y @ X))))). % le_cases
thf(fact_166_le__cases, axiom,
    ((![X : int, Y : int]: ((~ ((ord_less_eq_int @ X @ Y))) => (ord_less_eq_int @ Y @ X))))). % le_cases
thf(fact_167_le__cases, axiom,
    ((![X : real, Y : real]: ((~ ((ord_less_eq_real @ X @ Y))) => (ord_less_eq_real @ Y @ X))))). % le_cases
thf(fact_168_eq__refl, axiom,
    ((![X : nat, Y : nat]: ((X = Y) => (ord_less_eq_nat @ X @ Y))))). % eq_refl
thf(fact_169_eq__refl, axiom,
    ((![X : int, Y : int]: ((X = Y) => (ord_less_eq_int @ X @ Y))))). % eq_refl
thf(fact_170_eq__refl, axiom,
    ((![X : real, Y : real]: ((X = Y) => (ord_less_eq_real @ X @ Y))))). % eq_refl
thf(fact_171_linear, axiom,
    ((![X : nat, Y : nat]: ((ord_less_eq_nat @ X @ Y) | (ord_less_eq_nat @ Y @ X))))). % linear
thf(fact_172_linear, axiom,
    ((![X : int, Y : int]: ((ord_less_eq_int @ X @ Y) | (ord_less_eq_int @ Y @ X))))). % linear
thf(fact_173_linear, axiom,
    ((![X : real, Y : real]: ((ord_less_eq_real @ X @ Y) | (ord_less_eq_real @ Y @ X))))). % linear
thf(fact_174_antisym, axiom,
    ((![X : nat, Y : nat]: ((ord_less_eq_nat @ X @ Y) => ((ord_less_eq_nat @ Y @ X) => (X = Y)))))). % antisym
thf(fact_175_antisym, axiom,
    ((![X : int, Y : int]: ((ord_less_eq_int @ X @ Y) => ((ord_less_eq_int @ Y @ X) => (X = Y)))))). % antisym
thf(fact_176_antisym, axiom,
    ((![X : real, Y : real]: ((ord_less_eq_real @ X @ Y) => ((ord_less_eq_real @ Y @ X) => (X = Y)))))). % antisym
thf(fact_177_eq__iff, axiom,
    (((^[Y4 : nat]: (^[Z : nat]: (Y4 = Z))) = (^[X4 : nat]: (^[Y5 : nat]: (((ord_less_eq_nat @ X4 @ Y5)) & ((ord_less_eq_nat @ Y5 @ X4)))))))). % eq_iff
thf(fact_178_eq__iff, axiom,
    (((^[Y4 : int]: (^[Z : int]: (Y4 = Z))) = (^[X4 : int]: (^[Y5 : int]: (((ord_less_eq_int @ X4 @ Y5)) & ((ord_less_eq_int @ Y5 @ X4)))))))). % eq_iff
thf(fact_179_eq__iff, axiom,
    (((^[Y4 : real]: (^[Z : real]: (Y4 = Z))) = (^[X4 : real]: (^[Y5 : real]: (((ord_less_eq_real @ X4 @ Y5)) & ((ord_less_eq_real @ Y5 @ X4)))))))). % eq_iff
thf(fact_180_ord__le__eq__subst, axiom,
    ((![A : nat, B : nat, F : nat > nat, C : nat]: ((ord_less_eq_nat @ A @ B) => (((F @ B) = C) => ((![X2 : nat, Y2 : nat]: ((ord_less_eq_nat @ X2 @ Y2) => (ord_less_eq_nat @ (F @ X2) @ (F @ Y2)))) => (ord_less_eq_nat @ (F @ A) @ C))))))). % ord_le_eq_subst
thf(fact_181_ord__le__eq__subst, axiom,
    ((![A : nat, B : nat, F : nat > int, C : int]: ((ord_less_eq_nat @ A @ B) => (((F @ B) = C) => ((![X2 : nat, Y2 : nat]: ((ord_less_eq_nat @ X2 @ Y2) => (ord_less_eq_int @ (F @ X2) @ (F @ Y2)))) => (ord_less_eq_int @ (F @ A) @ C))))))). % ord_le_eq_subst
thf(fact_182_ord__le__eq__subst, axiom,
    ((![A : nat, B : nat, F : nat > real, C : real]: ((ord_less_eq_nat @ A @ B) => (((F @ B) = C) => ((![X2 : nat, Y2 : nat]: ((ord_less_eq_nat @ X2 @ Y2) => (ord_less_eq_real @ (F @ X2) @ (F @ Y2)))) => (ord_less_eq_real @ (F @ A) @ C))))))). % ord_le_eq_subst
thf(fact_183_ord__le__eq__subst, axiom,
    ((![A : int, B : int, F : int > nat, C : nat]: ((ord_less_eq_int @ A @ B) => (((F @ B) = C) => ((![X2 : int, Y2 : int]: ((ord_less_eq_int @ X2 @ Y2) => (ord_less_eq_nat @ (F @ X2) @ (F @ Y2)))) => (ord_less_eq_nat @ (F @ A) @ C))))))). % ord_le_eq_subst
thf(fact_184_ord__le__eq__subst, axiom,
    ((![A : int, B : int, F : int > int, C : int]: ((ord_less_eq_int @ A @ B) => (((F @ B) = C) => ((![X2 : int, Y2 : int]: ((ord_less_eq_int @ X2 @ Y2) => (ord_less_eq_int @ (F @ X2) @ (F @ Y2)))) => (ord_less_eq_int @ (F @ A) @ C))))))). % ord_le_eq_subst
thf(fact_185_ord__le__eq__subst, axiom,
    ((![A : int, B : int, F : int > real, C : real]: ((ord_less_eq_int @ A @ B) => (((F @ B) = C) => ((![X2 : int, Y2 : int]: ((ord_less_eq_int @ X2 @ Y2) => (ord_less_eq_real @ (F @ X2) @ (F @ Y2)))) => (ord_less_eq_real @ (F @ A) @ C))))))). % ord_le_eq_subst
thf(fact_186_ord__le__eq__subst, axiom,
    ((![A : real, B : real, F : real > nat, C : nat]: ((ord_less_eq_real @ A @ B) => (((F @ B) = C) => ((![X2 : real, Y2 : real]: ((ord_less_eq_real @ X2 @ Y2) => (ord_less_eq_nat @ (F @ X2) @ (F @ Y2)))) => (ord_less_eq_nat @ (F @ A) @ C))))))). % ord_le_eq_subst
thf(fact_187_ord__le__eq__subst, axiom,
    ((![A : real, B : real, F : real > int, C : int]: ((ord_less_eq_real @ A @ B) => (((F @ B) = C) => ((![X2 : real, Y2 : real]: ((ord_less_eq_real @ X2 @ Y2) => (ord_less_eq_int @ (F @ X2) @ (F @ Y2)))) => (ord_less_eq_int @ (F @ A) @ C))))))). % ord_le_eq_subst
thf(fact_188_ord__le__eq__subst, axiom,
    ((![A : real, B : real, F : real > real, C : real]: ((ord_less_eq_real @ A @ B) => (((F @ B) = C) => ((![X2 : real, Y2 : real]: ((ord_less_eq_real @ X2 @ Y2) => (ord_less_eq_real @ (F @ X2) @ (F @ Y2)))) => (ord_less_eq_real @ (F @ A) @ C))))))). % ord_le_eq_subst
thf(fact_189_ord__eq__le__subst, axiom,
    ((![A : nat, F : nat > nat, B : nat, C : 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_eq_nat @ A @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_190_ord__eq__le__subst, axiom,
    ((![A : int, F : nat > int, B : nat, C : nat]: ((A = (F @ B)) => ((ord_less_eq_nat @ B @ C) => ((![X2 : nat, Y2 : nat]: ((ord_less_eq_nat @ X2 @ Y2) => (ord_less_eq_int @ (F @ X2) @ (F @ Y2)))) => (ord_less_eq_int @ A @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_191_ord__eq__le__subst, axiom,
    ((![A : real, F : nat > real, B : nat, C : nat]: ((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_eq_real @ A @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_192_ord__eq__le__subst, axiom,
    ((![A : nat, F : int > nat, B : int, C : int]: ((A = (F @ B)) => ((ord_less_eq_int @ B @ C) => ((![X2 : int, Y2 : int]: ((ord_less_eq_int @ X2 @ Y2) => (ord_less_eq_nat @ (F @ X2) @ (F @ Y2)))) => (ord_less_eq_nat @ A @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_193_ord__eq__le__subst, axiom,
    ((![A : int, F : int > int, B : int, C : int]: ((A = (F @ B)) => ((ord_less_eq_int @ B @ C) => ((![X2 : int, Y2 : int]: ((ord_less_eq_int @ X2 @ Y2) => (ord_less_eq_int @ (F @ X2) @ (F @ Y2)))) => (ord_less_eq_int @ A @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_194_ord__eq__le__subst, axiom,
    ((![A : real, F : int > real, B : int, C : int]: ((A = (F @ B)) => ((ord_less_eq_int @ B @ C) => ((![X2 : int, Y2 : int]: ((ord_less_eq_int @ X2 @ Y2) => (ord_less_eq_real @ (F @ X2) @ (F @ Y2)))) => (ord_less_eq_real @ A @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_195_ord__eq__le__subst, axiom,
    ((![A : nat, F : real > nat, B : real, C : real]: ((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_eq_nat @ A @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_196_ord__eq__le__subst, axiom,
    ((![A : int, F : real > int, B : real, C : real]: ((A = (F @ B)) => ((ord_less_eq_real @ B @ C) => ((![X2 : real, Y2 : real]: ((ord_less_eq_real @ X2 @ Y2) => (ord_less_eq_int @ (F @ X2) @ (F @ Y2)))) => (ord_less_eq_int @ A @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_197_ord__eq__le__subst, axiom,
    ((![A : real, F : real > real, B : real, C : 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_eq_real @ A @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_198_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) => ((![X2 : nat, Y2 : nat]: ((ord_less_eq_nat @ X2 @ Y2) => (ord_less_eq_nat @ (F @ X2) @ (F @ Y2)))) => (ord_less_eq_nat @ (F @ A) @ C))))))). % order_subst2
thf(fact_199_order__subst2, axiom,
    ((![A : nat, B : nat, F : nat > int, C : int]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_int @ (F @ B) @ C) => ((![X2 : nat, Y2 : nat]: ((ord_less_eq_nat @ X2 @ Y2) => (ord_less_eq_int @ (F @ X2) @ (F @ Y2)))) => (ord_less_eq_int @ (F @ A) @ C))))))). % order_subst2
thf(fact_200_order__subst2, axiom,
    ((![A : nat, B : nat, F : nat > real, C : real]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_real @ (F @ B) @ C) => ((![X2 : nat, Y2 : nat]: ((ord_less_eq_nat @ X2 @ Y2) => (ord_less_eq_real @ (F @ X2) @ (F @ Y2)))) => (ord_less_eq_real @ (F @ A) @ C))))))). % order_subst2
thf(fact_201_order__subst2, axiom,
    ((![A : int, B : int, F : int > nat, C : nat]: ((ord_less_eq_int @ A @ B) => ((ord_less_eq_nat @ (F @ B) @ C) => ((![X2 : int, Y2 : int]: ((ord_less_eq_int @ X2 @ Y2) => (ord_less_eq_nat @ (F @ X2) @ (F @ Y2)))) => (ord_less_eq_nat @ (F @ A) @ C))))))). % order_subst2
thf(fact_202_order__subst2, axiom,
    ((![A : int, B : int, F : int > int, C : int]: ((ord_less_eq_int @ A @ B) => ((ord_less_eq_int @ (F @ B) @ C) => ((![X2 : int, Y2 : int]: ((ord_less_eq_int @ X2 @ Y2) => (ord_less_eq_int @ (F @ X2) @ (F @ Y2)))) => (ord_less_eq_int @ (F @ A) @ C))))))). % order_subst2
thf(fact_203_order__subst2, axiom,
    ((![A : int, B : int, F : int > real, C : real]: ((ord_less_eq_int @ A @ B) => ((ord_less_eq_real @ (F @ B) @ C) => ((![X2 : int, Y2 : int]: ((ord_less_eq_int @ X2 @ Y2) => (ord_less_eq_real @ (F @ X2) @ (F @ Y2)))) => (ord_less_eq_real @ (F @ A) @ C))))))). % order_subst2
thf(fact_204_order__subst2, axiom,
    ((![A : real, B : real, F : real > nat, C : nat]: ((ord_less_eq_real @ A @ B) => ((ord_less_eq_nat @ (F @ B) @ C) => ((![X2 : real, Y2 : real]: ((ord_less_eq_real @ X2 @ Y2) => (ord_less_eq_nat @ (F @ X2) @ (F @ Y2)))) => (ord_less_eq_nat @ (F @ A) @ C))))))). % order_subst2
thf(fact_205_order__subst2, axiom,
    ((![A : real, B : real, F : real > int, C : int]: ((ord_less_eq_real @ A @ B) => ((ord_less_eq_int @ (F @ B) @ C) => ((![X2 : real, Y2 : real]: ((ord_less_eq_real @ X2 @ Y2) => (ord_less_eq_int @ (F @ X2) @ (F @ Y2)))) => (ord_less_eq_int @ (F @ A) @ C))))))). % order_subst2
thf(fact_206_order__subst2, axiom,
    ((![A : real, B : real, F : real > real, C : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_eq_real @ (F @ B) @ C) => ((![X2 : real, Y2 : real]: ((ord_less_eq_real @ X2 @ Y2) => (ord_less_eq_real @ (F @ X2) @ (F @ Y2)))) => (ord_less_eq_real @ (F @ A) @ C))))))). % order_subst2
thf(fact_207_order__subst1, axiom,
    ((![A : nat, F : real > nat, B : real, C : real]: ((ord_less_eq_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_eq_nat @ A @ (F @ C)))))))). % order_subst1
thf(fact_208_order__subst1, axiom,
    ((![A : int, F : nat > int, B : nat, C : nat]: ((ord_less_eq_int @ A @ (F @ B)) => ((ord_less_eq_nat @ B @ C) => ((![X2 : nat, Y2 : nat]: ((ord_less_eq_nat @ X2 @ Y2) => (ord_less_eq_int @ (F @ X2) @ (F @ Y2)))) => (ord_less_eq_int @ A @ (F @ C)))))))). % order_subst1
thf(fact_209_order__subst1, axiom,
    ((![A : int, F : int > int, B : int, C : int]: ((ord_less_eq_int @ A @ (F @ B)) => ((ord_less_eq_int @ B @ C) => ((![X2 : int, Y2 : int]: ((ord_less_eq_int @ X2 @ Y2) => (ord_less_eq_int @ (F @ X2) @ (F @ Y2)))) => (ord_less_eq_int @ A @ (F @ C)))))))). % order_subst1
thf(fact_210_order__subst1, axiom,
    ((![A : int, F : real > int, B : real, C : real]: ((ord_less_eq_int @ A @ (F @ B)) => ((ord_less_eq_real @ B @ C) => ((![X2 : real, Y2 : real]: ((ord_less_eq_real @ X2 @ Y2) => (ord_less_eq_int @ (F @ X2) @ (F @ Y2)))) => (ord_less_eq_int @ A @ (F @ C)))))))). % order_subst1
thf(fact_211_order__subst1, axiom,
    ((![A : real, F : nat > real, B : nat, C : nat]: ((ord_less_eq_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_eq_real @ A @ (F @ C)))))))). % order_subst1
thf(fact_212_order__subst1, axiom,
    ((![A : real, F : int > real, B : int, C : int]: ((ord_less_eq_real @ A @ (F @ B)) => ((ord_less_eq_int @ B @ C) => ((![X2 : int, Y2 : int]: ((ord_less_eq_int @ X2 @ Y2) => (ord_less_eq_real @ (F @ X2) @ (F @ Y2)))) => (ord_less_eq_real @ A @ (F @ C)))))))). % order_subst1
thf(fact_213_order__subst1, axiom,
    ((![A : real, F : real > real, B : real, C : real]: ((ord_less_eq_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_eq_real @ A @ (F @ C)))))))). % order_subst1
thf(fact_214_nat__int__comparison_I3_J, axiom,
    ((ord_less_eq_nat = (^[A2 : nat]: (^[B2 : nat]: (ord_less_eq_int @ (semiri2019852685at_int @ A2) @ (semiri2019852685at_int @ B2))))))). % nat_int_comparison(3)
thf(fact_215_zle__int, axiom,
    ((![M : nat, N : nat]: ((ord_less_eq_int @ (semiri2019852685at_int @ M) @ (semiri2019852685at_int @ N)) = (ord_less_eq_nat @ M @ N))))). % zle_int
thf(fact_216_div__le__dividend, axiom,
    ((![M : nat, N : nat]: (ord_less_eq_nat @ (divide_divide_nat @ M @ N) @ M)))). % div_le_dividend
thf(fact_217_int__dvd__int__iff, axiom,
    ((![M : nat, N : nat]: ((dvd_dvd_int @ (semiri2019852685at_int @ M) @ (semiri2019852685at_int @ N)) = (dvd_dvd_nat @ M @ N))))). % int_dvd_int_iff
thf(fact_218_dvd__antisym, axiom,
    ((![M : nat, N : nat]: ((dvd_dvd_nat @ M @ N) => ((dvd_dvd_nat @ N @ M) => (M = N)))))). % dvd_antisym
thf(fact_219_verit__la__generic, axiom,
    ((![A : int, X : int]: ((ord_less_eq_int @ A @ X) | ((A = X) | (ord_less_eq_int @ X @ A)))))). % verit_la_generic
thf(fact_220_int__int__eq, axiom,
    ((![M : nat, N : nat]: (((semiri2019852685at_int @ M) = (semiri2019852685at_int @ N)) = (M = N))))). % int_int_eq
thf(fact_221_int__if, axiom,
    ((![P2 : $o, A : nat, B : nat]: ((P2 => ((semiri2019852685at_int @ (if_nat @ P2 @ A @ B)) = (semiri2019852685at_int @ A))) & ((~ (P2)) => ((semiri2019852685at_int @ (if_nat @ P2 @ A @ B)) = (semiri2019852685at_int @ B))))))). % int_if
thf(fact_222_nat__int__comparison_I1_J, axiom,
    (((^[Y4 : nat]: (^[Z : nat]: (Y4 = Z))) = (^[A2 : nat]: (^[B2 : nat]: ((semiri2019852685at_int @ A2) = (semiri2019852685at_int @ B2))))))). % nat_int_comparison(1)
thf(fact_223_GreatestI__nat, axiom,
    ((![P2 : nat > $o, K : nat, B : nat]: ((P2 @ K) => ((![Y2 : nat]: ((P2 @ Y2) => (ord_less_eq_nat @ Y2 @ B))) => (P2 @ (order_Greatest_nat @ P2))))))). % GreatestI_nat
thf(fact_224_Greatest__le__nat, axiom,
    ((![P2 : nat > $o, K : nat, B : nat]: ((P2 @ K) => ((![Y2 : nat]: ((P2 @ Y2) => (ord_less_eq_nat @ Y2 @ B))) => (ord_less_eq_nat @ K @ (order_Greatest_nat @ P2))))))). % Greatest_le_nat
thf(fact_225_GreatestI__ex__nat, axiom,
    ((![P2 : nat > $o, B : nat]: ((?[X_1 : nat]: (P2 @ X_1)) => ((![Y2 : nat]: ((P2 @ Y2) => (ord_less_eq_nat @ Y2 @ B))) => (P2 @ (order_Greatest_nat @ P2))))))). % GreatestI_ex_nat
thf(fact_226_div__le__mono, axiom,
    ((![M : nat, N : nat, K : nat]: ((ord_less_eq_nat @ M @ N) => (ord_less_eq_nat @ (divide_divide_nat @ M @ K) @ (divide_divide_nat @ N @ K)))))). % div_le_mono
thf(fact_227_zdiv__int, axiom,
    ((![A : nat, B : nat]: ((semiri2019852685at_int @ (divide_divide_nat @ A @ B)) = (divide_divide_int @ (semiri2019852685at_int @ A) @ (semiri2019852685at_int @ B)))))). % zdiv_int
thf(fact_228_real__of__nat__div, axiom,
    ((![D : nat, N : nat]: ((dvd_dvd_nat @ D @ N) => ((semiri2110766477t_real @ (divide_divide_nat @ N @ D)) = (divide_divide_real @ (semiri2110766477t_real @ N) @ (semiri2110766477t_real @ D))))))). % real_of_nat_div
thf(fact_229_real__of__nat__div4, axiom,
    ((![N : nat, X : nat]: (ord_less_eq_real @ (semiri2110766477t_real @ (divide_divide_nat @ N @ X)) @ (divide_divide_real @ (semiri2110766477t_real @ N) @ (semiri2110766477t_real @ X)))))). % real_of_nat_div4
thf(fact_230_gcd__nat_Oasym, axiom,
    ((![A : nat, B : nat]: (((dvd_dvd_nat @ A @ B) & (~ ((A = B)))) => (~ (((dvd_dvd_nat @ B @ A) & (~ ((B = A)))))))))). % gcd_nat.asym
thf(fact_231_gcd__nat_Orefl, axiom,
    ((![A : nat]: (dvd_dvd_nat @ A @ A)))). % gcd_nat.refl
thf(fact_232_gcd__nat_Otrans, axiom,
    ((![A : nat, B : nat, C : nat]: ((dvd_dvd_nat @ A @ B) => ((dvd_dvd_nat @ B @ C) => (dvd_dvd_nat @ A @ C)))))). % gcd_nat.trans
thf(fact_233_complete__real, axiom,
    ((![S : set_real]: ((?[X3 : real]: (member_real @ X3 @ S)) => ((?[Z3 : real]: (![X2 : real]: ((member_real @ X2 @ S) => (ord_less_eq_real @ X2 @ Z3)))) => (?[Y2 : real]: ((![X3 : real]: ((member_real @ X3 @ S) => (ord_less_eq_real @ X3 @ Y2))) & (![Z3 : real]: ((![X2 : real]: ((member_real @ X2 @ S) => (ord_less_eq_real @ X2 @ Z3))) => (ord_less_eq_real @ Y2 @ Z3)))))))))). % complete_real
thf(fact_234_gcd__nat_Onot__eq__order__implies__strict, axiom,
    ((![A : nat, B : nat]: ((~ ((A = B))) => ((dvd_dvd_nat @ A @ B) => ((dvd_dvd_nat @ A @ B) & (~ ((A = B))))))))). % gcd_nat.not_eq_order_implies_strict
thf(fact_235_gcd__nat_Ostrict__implies__not__eq, axiom,
    ((![A : nat, B : nat]: (((dvd_dvd_nat @ A @ B) & (~ ((A = B)))) => (~ ((A = B))))))). % gcd_nat.strict_implies_not_eq
thf(fact_236_gcd__nat_Ostrict__implies__order, axiom,
    ((![A : nat, B : nat]: (((dvd_dvd_nat @ A @ B) & (~ ((A = B)))) => (dvd_dvd_nat @ A @ B))))). % gcd_nat.strict_implies_order
thf(fact_237_gcd__nat_Ostrict__iff__order, axiom,
    ((![A : nat, B : nat]: ((((dvd_dvd_nat @ A @ B)) & ((~ ((A = B))))) = (((dvd_dvd_nat @ A @ B)) & ((~ ((A = B))))))))). % gcd_nat.strict_iff_order
thf(fact_238_gcd__nat_Oorder__iff__strict, axiom,
    ((dvd_dvd_nat = (^[A2 : nat]: (^[B2 : nat]: (((((dvd_dvd_nat @ A2 @ B2)) & ((~ ((A2 = B2)))))) | ((A2 = B2)))))))). % gcd_nat.order_iff_strict
thf(fact_239_gcd__nat_Ostrict__trans2, axiom,
    ((![A : nat, B : nat, C : nat]: (((dvd_dvd_nat @ A @ B) & (~ ((A = B)))) => ((dvd_dvd_nat @ B @ C) => ((dvd_dvd_nat @ A @ C) & (~ ((A = C))))))))). % gcd_nat.strict_trans2
thf(fact_240_gcd__nat_Ostrict__trans1, axiom,
    ((![A : nat, B : nat, C : nat]: ((dvd_dvd_nat @ A @ B) => (((dvd_dvd_nat @ B @ C) & (~ ((B = C)))) => ((dvd_dvd_nat @ A @ C) & (~ ((A = C))))))))). % gcd_nat.strict_trans1
thf(fact_241_gcd__nat_Ostrict__trans, axiom,
    ((![A : nat, B : nat, C : nat]: (((dvd_dvd_nat @ A @ B) & (~ ((A = B)))) => (((dvd_dvd_nat @ B @ C) & (~ ((B = C)))) => ((dvd_dvd_nat @ A @ C) & (~ ((A = C))))))))). % gcd_nat.strict_trans
thf(fact_242_gcd__nat_Oantisym, axiom,
    ((![A : nat, B : nat]: ((dvd_dvd_nat @ A @ B) => ((dvd_dvd_nat @ B @ A) => (A = B)))))). % gcd_nat.antisym
thf(fact_243_gcd__nat_Oirrefl, axiom,
    ((![A : nat]: (~ (((dvd_dvd_nat @ A @ A) & (~ ((A = A))))))))). % gcd_nat.irrefl
thf(fact_244_gcd__nat_Oeq__iff, axiom,
    (((^[Y4 : nat]: (^[Z : nat]: (Y4 = Z))) = (^[A2 : nat]: (^[B2 : nat]: (((dvd_dvd_nat @ A2 @ B2)) & ((dvd_dvd_nat @ B2 @ A2)))))))). % gcd_nat.eq_iff
thf(fact_245_real__of__int__div4, axiom,
    ((![N : int, X : int]: (ord_less_eq_real @ (ring_1_of_int_real @ (divide_divide_int @ N @ X)) @ (divide_divide_real @ (ring_1_of_int_real @ N) @ (ring_1_of_int_real @ X)))))). % real_of_int_div4
thf(fact_246_real__of__int__div, axiom,
    ((![D : int, N : int]: ((dvd_dvd_int @ D @ N) => ((ring_1_of_int_real @ (divide_divide_int @ N @ D)) = (divide_divide_real @ (ring_1_of_int_real @ N) @ (ring_1_of_int_real @ D))))))). % real_of_int_div
thf(fact_247_floor__divide__real__eq__div, axiom,
    ((![B : int, A : real]: ((ord_less_eq_int @ zero_zero_int @ B) => ((archim1031974863r_real @ (divide_divide_real @ A @ (ring_1_of_int_real @ B))) = (divide_divide_int @ (archim1031974863r_real @ A) @ B)))))). % floor_divide_real_eq_div

% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T, axiom,
    ((![P2 : $o]: ((P2 = $true) | (P2 = $false))))).
thf(help_If_2_1_If_001t__Nat__Onat_T, axiom,
    ((![X : nat, Y : nat]: ((if_nat @ $false @ X @ Y) = Y)))).
thf(help_If_1_1_If_001t__Nat__Onat_T, axiom,
    ((![X : nat, Y : nat]: ((if_nat @ $true @ X @ Y) = X)))).

% Conjectures (1)
thf(conj_0, conjecture,
    (((dvd_dvd_poly_complex @ p @ (power_184595776omplex @ q @ n)) = (dvd_dvd_poly_complex @ p @ r)))).
