% 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_1143__5378654_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:36:08.300

% Could-be-implicit typings (5)
thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_Itf__a_J_J, type,
    poly_poly_a : $tType).
thf(ty_n_t__Polynomial__Opoly_It__Nat__Onat_J, type,
    poly_nat : $tType).
thf(ty_n_t__Polynomial__Opoly_Itf__a_J, type,
    poly_a : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).
thf(ty_n_tf__a, type,
    a : $tType).

% Explicit typings (45)
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat, type,
    minus_minus_nat : nat > nat > nat).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    minus_minus_poly_nat : poly_nat > poly_nat > poly_nat).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Polynomial__Opoly_It__Polynomial__Opoly_Itf__a_J_J, type,
    minus_154650241poly_a : poly_poly_a > poly_poly_a > poly_poly_a).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Polynomial__Opoly_Itf__a_J, type,
    minus_minus_poly_a : poly_a > poly_a > poly_a).
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_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    one_one_poly_nat : poly_nat).
thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_It__Polynomial__Opoly_Itf__a_J_J, type,
    one_one_poly_poly_a : poly_poly_a).
thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_Itf__a_J, type,
    one_one_poly_a : poly_a).
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_001t__Polynomial__Opoly_Itf__a_J, type,
    times_times_poly_a : poly_a > poly_a > poly_a).
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_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    zero_zero_poly_nat : poly_nat).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_Itf__a_J_J, type,
    zero_z2096148049poly_a : poly_poly_a).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_Itf__a_J, type,
    zero_zero_poly_a : poly_a).
thf(sy_c_Groups_Ozero__class_Ozero_001tf__a, type,
    zero_zero_a : a).
thf(sy_c_Polynomial_Omap__poly_001t__Nat__Onat_001t__Nat__Onat, type,
    map_poly_nat_nat : (nat > nat) > poly_nat > poly_nat).
thf(sy_c_Polynomial_Omap__poly_001t__Nat__Onat_001t__Polynomial__Opoly_Itf__a_J, type,
    map_poly_nat_poly_a : (nat > poly_a) > poly_nat > poly_poly_a).
thf(sy_c_Polynomial_Omap__poly_001t__Nat__Onat_001tf__a, type,
    map_poly_nat_a : (nat > a) > poly_nat > poly_a).
thf(sy_c_Polynomial_Omap__poly_001t__Polynomial__Opoly_Itf__a_J_001t__Nat__Onat, type,
    map_poly_poly_a_nat : (poly_a > nat) > poly_poly_a > poly_nat).
thf(sy_c_Polynomial_Omap__poly_001t__Polynomial__Opoly_Itf__a_J_001t__Polynomial__Opoly_Itf__a_J, type,
    map_po495521320poly_a : (poly_a > poly_a) > poly_poly_a > poly_poly_a).
thf(sy_c_Polynomial_Omap__poly_001t__Polynomial__Opoly_Itf__a_J_001tf__a, type,
    map_poly_poly_a_a : (poly_a > a) > poly_poly_a > poly_a).
thf(sy_c_Polynomial_Omap__poly_001tf__a_001t__Nat__Onat, type,
    map_poly_a_nat : (a > nat) > poly_a > poly_nat).
thf(sy_c_Polynomial_Omap__poly_001tf__a_001t__Polynomial__Opoly_Itf__a_J, type,
    map_poly_a_poly_a : (a > poly_a) > poly_a > poly_poly_a).
thf(sy_c_Polynomial_Omap__poly_001tf__a_001tf__a, type,
    map_poly_a_a : (a > a) > poly_a > poly_a).
thf(sy_c_Polynomial_OpCons_001t__Nat__Onat, type,
    pCons_nat : nat > poly_nat > poly_nat).
thf(sy_c_Polynomial_OpCons_001t__Polynomial__Opoly_Itf__a_J, type,
    pCons_poly_a : poly_a > poly_poly_a > poly_poly_a).
thf(sy_c_Polynomial_OpCons_001tf__a, type,
    pCons_a : a > poly_a > poly_a).
thf(sy_c_Polynomial_Osmult_001t__Nat__Onat, type,
    smult_nat : nat > poly_nat > poly_nat).
thf(sy_c_Polynomial_Osmult_001t__Polynomial__Opoly_Itf__a_J, type,
    smult_poly_a : poly_a > poly_poly_a > poly_poly_a).
thf(sy_c_Polynomial_Osmult_001tf__a, type,
    smult_a : a > poly_a > poly_a).
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__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_Itf__a_J_J, type,
    dvd_dvd_poly_poly_a : poly_poly_a > poly_poly_a > $o).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Polynomial__Opoly_Itf__a_J, type,
    dvd_dvd_poly_a : poly_a > poly_a > $o).
thf(sy_c_Rings_Odvd__class_Odvd_001tf__a, type,
    dvd_dvd_a : a > a > $o).
thf(sy_v_a, type,
    a2 : a).
thf(sy_v_p, type,
    p : poly_a).
thf(sy_v_p_H, type,
    p2 : poly_a).
thf(sy_v_q, type,
    q : poly_a).
thf(sy_v_r, type,
    r : poly_a).
thf(sy_v_t____, type,
    t : poly_a).
thf(sy_v_u____, type,
    u : poly_a).

% Relevant facts (245)
thf(fact_0_t, axiom,
    ((p2 = (times_times_poly_a @ p @ t)))). % t
thf(fact_1_u, axiom,
    ((q = (times_times_poly_a @ p @ u)))). % u
thf(fact_2__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062t_O_Ap_H_A_061_Ap_A_K_At_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom,
    ((~ ((![T : poly_a]: (~ ((p2 = (times_times_poly_a @ p @ T))))))))). % \<open>\<And>thesis. (\<And>t. p' = p * t \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_3__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062u_O_Aq_A_061_Ap_A_K_Au_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom,
    ((~ ((![U : poly_a]: (~ ((q = (times_times_poly_a @ p @ U))))))))). % \<open>\<And>thesis. (\<And>u. q = p * u \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_4_a0, axiom,
    ((~ ((a2 = zero_zero_a))))). % a0
thf(fact_5_qrp_H, axiom,
    (((minus_minus_poly_a @ (smult_a @ a2 @ q) @ p2) = r))). % qrp'
thf(fact_6_pp_H, axiom,
    ((dvd_dvd_poly_a @ p @ p2))). % pp'
thf(fact_7_that, axiom,
    ((dvd_dvd_poly_a @ p @ q))). % that
thf(fact_8_mult__smult__left, axiom,
    ((![A : a, P : poly_a, Q : poly_a]: ((times_times_poly_a @ (smult_a @ A @ P) @ Q) = (smult_a @ A @ (times_times_poly_a @ P @ Q)))))). % mult_smult_left
thf(fact_9_mult__smult__right, axiom,
    ((![P : poly_a, A : a, Q : poly_a]: ((times_times_poly_a @ P @ (smult_a @ A @ Q)) = (smult_a @ A @ (times_times_poly_a @ P @ Q)))))). % mult_smult_right
thf(fact_10_smult__smult, axiom,
    ((![A : a, B : a, P : poly_a]: ((smult_a @ A @ (smult_a @ B @ P)) = (smult_a @ (times_times_a @ A @ B) @ P))))). % smult_smult
thf(fact_11_smult__smult, axiom,
    ((![A : poly_a, B : poly_a, P : poly_poly_a]: ((smult_poly_a @ A @ (smult_poly_a @ B @ P)) = (smult_poly_a @ (times_times_poly_a @ A @ B) @ P))))). % smult_smult
thf(fact_12_smult__smult, axiom,
    ((![A : nat, B : nat, P : poly_nat]: ((smult_nat @ A @ (smult_nat @ B @ P)) = (smult_nat @ (times_times_nat @ A @ B) @ P))))). % smult_smult
thf(fact_13_smult__diff__left, axiom,
    ((![A : a, B : a, P : poly_a]: ((smult_a @ (minus_minus_a @ A @ B) @ P) = (minus_minus_poly_a @ (smult_a @ A @ P) @ (smult_a @ B @ P)))))). % smult_diff_left
thf(fact_14_smult__diff__left, axiom,
    ((![A : poly_a, B : poly_a, P : poly_poly_a]: ((smult_poly_a @ (minus_minus_poly_a @ A @ B) @ P) = (minus_154650241poly_a @ (smult_poly_a @ A @ P) @ (smult_poly_a @ B @ P)))))). % smult_diff_left
thf(fact_15_smult__diff__right, axiom,
    ((![A : a, P : poly_a, Q : poly_a]: ((smult_a @ A @ (minus_minus_poly_a @ P @ Q)) = (minus_minus_poly_a @ (smult_a @ A @ P) @ (smult_a @ A @ Q)))))). % smult_diff_right
thf(fact_16_inf__period_I2_J, axiom,
    ((![P2 : poly_a > $o, D : poly_a, Q2 : poly_a > $o]: ((![X : poly_a, K : poly_a]: ((P2 @ X) = (P2 @ (minus_minus_poly_a @ X @ (times_times_poly_a @ K @ D))))) => ((![X : poly_a, K : poly_a]: ((Q2 @ X) = (Q2 @ (minus_minus_poly_a @ X @ (times_times_poly_a @ K @ D))))) => (![X2 : poly_a, K2 : poly_a]: ((((P2 @ X2)) | ((Q2 @ X2))) = (((P2 @ (minus_minus_poly_a @ X2 @ (times_times_poly_a @ K2 @ D)))) | ((Q2 @ (minus_minus_poly_a @ X2 @ (times_times_poly_a @ K2 @ D)))))))))))). % inf_period(2)
thf(fact_17_inf__period_I1_J, axiom,
    ((![P2 : poly_a > $o, D : poly_a, Q2 : poly_a > $o]: ((![X : poly_a, K : poly_a]: ((P2 @ X) = (P2 @ (minus_minus_poly_a @ X @ (times_times_poly_a @ K @ D))))) => ((![X : poly_a, K : poly_a]: ((Q2 @ X) = (Q2 @ (minus_minus_poly_a @ X @ (times_times_poly_a @ K @ D))))) => (![X2 : poly_a, K2 : poly_a]: ((((P2 @ X2)) & ((Q2 @ X2))) = (((P2 @ (minus_minus_poly_a @ X2 @ (times_times_poly_a @ K2 @ D)))) & ((Q2 @ (minus_minus_poly_a @ X2 @ (times_times_poly_a @ K2 @ D)))))))))))). % inf_period(1)
thf(fact_18_left__diff__distrib, axiom,
    ((![A : poly_a, B : poly_a, C : poly_a]: ((times_times_poly_a @ (minus_minus_poly_a @ A @ B) @ C) = (minus_minus_poly_a @ (times_times_poly_a @ A @ C) @ (times_times_poly_a @ B @ C)))))). % left_diff_distrib
thf(fact_19_right__diff__distrib, axiom,
    ((![A : poly_a, B : poly_a, C : poly_a]: ((times_times_poly_a @ A @ (minus_minus_poly_a @ B @ C)) = (minus_minus_poly_a @ (times_times_poly_a @ A @ B) @ (times_times_poly_a @ A @ C)))))). % right_diff_distrib
thf(fact_20_left__diff__distrib_H, axiom,
    ((![B : poly_a, C : poly_a, A : poly_a]: ((times_times_poly_a @ (minus_minus_poly_a @ B @ C) @ A) = (minus_minus_poly_a @ (times_times_poly_a @ B @ A) @ (times_times_poly_a @ C @ A)))))). % left_diff_distrib'
thf(fact_21_left__diff__distrib_H, axiom,
    ((![B : nat, C : nat, A : nat]: ((times_times_nat @ (minus_minus_nat @ B @ C) @ A) = (minus_minus_nat @ (times_times_nat @ B @ A) @ (times_times_nat @ C @ A)))))). % left_diff_distrib'
thf(fact_22_right__diff__distrib_H, axiom,
    ((![A : poly_a, B : poly_a, C : poly_a]: ((times_times_poly_a @ A @ (minus_minus_poly_a @ B @ C)) = (minus_minus_poly_a @ (times_times_poly_a @ A @ B) @ (times_times_poly_a @ A @ C)))))). % right_diff_distrib'
thf(fact_23_right__diff__distrib_H, axiom,
    ((![A : nat, B : nat, C : nat]: ((times_times_nat @ A @ (minus_minus_nat @ B @ C)) = (minus_minus_nat @ (times_times_nat @ A @ B) @ (times_times_nat @ A @ C)))))). % right_diff_distrib'
thf(fact_24_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_25_mult__cancel__right, axiom,
    ((![A : poly_a, C : poly_a, B : poly_a]: (((times_times_poly_a @ A @ C) = (times_times_poly_a @ B @ C)) = (((C = zero_zero_poly_a)) | ((A = B))))))). % mult_cancel_right
thf(fact_26_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_27_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_28_mult__cancel__left, axiom,
    ((![C : poly_a, A : poly_a, B : poly_a]: (((times_times_poly_a @ C @ A) = (times_times_poly_a @ C @ B)) = (((C = zero_zero_poly_a)) | ((A = B))))))). % mult_cancel_left
thf(fact_29_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_30_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_31_mult__eq__0__iff, axiom,
    ((![A : poly_a, B : poly_a]: (((times_times_poly_a @ A @ B) = zero_zero_poly_a) = (((A = zero_zero_poly_a)) | ((B = zero_zero_poly_a))))))). % mult_eq_0_iff
thf(fact_32_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_33_mult__zero__right, axiom,
    ((![A : a]: ((times_times_a @ A @ zero_zero_a) = zero_zero_a)))). % mult_zero_right
thf(fact_34_mult__zero__right, axiom,
    ((![A : poly_a]: ((times_times_poly_a @ A @ zero_zero_poly_a) = zero_zero_poly_a)))). % mult_zero_right
thf(fact_35_mult__zero__right, axiom,
    ((![A : nat]: ((times_times_nat @ A @ zero_zero_nat) = zero_zero_nat)))). % mult_zero_right
thf(fact_36_mult__zero__left, axiom,
    ((![A : a]: ((times_times_a @ zero_zero_a @ A) = zero_zero_a)))). % mult_zero_left
thf(fact_37_mult__zero__left, axiom,
    ((![A : poly_a]: ((times_times_poly_a @ zero_zero_poly_a @ A) = zero_zero_poly_a)))). % mult_zero_left
thf(fact_38_mult__zero__left, axiom,
    ((![A : nat]: ((times_times_nat @ zero_zero_nat @ A) = zero_zero_nat)))). % mult_zero_left
thf(fact_39_dvd__0__right, axiom,
    ((![A : poly_a]: (dvd_dvd_poly_a @ A @ zero_zero_poly_a)))). % dvd_0_right
thf(fact_40_dvd__0__right, axiom,
    ((![A : a]: (dvd_dvd_a @ A @ zero_zero_a)))). % dvd_0_right
thf(fact_41_dvd__0__right, axiom,
    ((![A : nat]: (dvd_dvd_nat @ A @ zero_zero_nat)))). % dvd_0_right
thf(fact_42_dvd__0__left__iff, axiom,
    ((![A : poly_a]: ((dvd_dvd_poly_a @ zero_zero_poly_a @ A) = (A = zero_zero_poly_a))))). % dvd_0_left_iff
thf(fact_43_dvd__0__left__iff, axiom,
    ((![A : a]: ((dvd_dvd_a @ zero_zero_a @ A) = (A = zero_zero_a))))). % dvd_0_left_iff
thf(fact_44_dvd__0__left__iff, axiom,
    ((![A : nat]: ((dvd_dvd_nat @ zero_zero_nat @ A) = (A = zero_zero_nat))))). % dvd_0_left_iff
thf(fact_45_smult__0__left, axiom,
    ((![P : poly_a]: ((smult_a @ zero_zero_a @ P) = zero_zero_poly_a)))). % smult_0_left
thf(fact_46_smult__0__left, axiom,
    ((![P : poly_nat]: ((smult_nat @ zero_zero_nat @ P) = zero_zero_poly_nat)))). % smult_0_left
thf(fact_47_smult__eq__0__iff, axiom,
    ((![A : a, P : poly_a]: (((smult_a @ A @ P) = zero_zero_poly_a) = (((A = zero_zero_a)) | ((P = zero_zero_poly_a))))))). % smult_eq_0_iff
thf(fact_48_smult__eq__0__iff, axiom,
    ((![A : nat, P : poly_nat]: (((smult_nat @ A @ P) = zero_zero_poly_nat) = (((A = zero_zero_nat)) | ((P = zero_zero_poly_nat))))))). % smult_eq_0_iff
thf(fact_49_dvd__mult__cancel__left, axiom,
    ((![C : a, A : a, B : a]: ((dvd_dvd_a @ (times_times_a @ C @ A) @ (times_times_a @ C @ B)) = (((C = zero_zero_a)) | ((dvd_dvd_a @ A @ B))))))). % dvd_mult_cancel_left
thf(fact_50_dvd__mult__cancel__left, axiom,
    ((![C : poly_a, A : poly_a, B : poly_a]: ((dvd_dvd_poly_a @ (times_times_poly_a @ C @ A) @ (times_times_poly_a @ C @ B)) = (((C = zero_zero_poly_a)) | ((dvd_dvd_poly_a @ A @ B))))))). % dvd_mult_cancel_left
thf(fact_51_dvd__mult__cancel__right, axiom,
    ((![A : a, C : a, B : a]: ((dvd_dvd_a @ (times_times_a @ A @ C) @ (times_times_a @ B @ C)) = (((C = zero_zero_a)) | ((dvd_dvd_a @ A @ B))))))). % dvd_mult_cancel_right
thf(fact_52_dvd__mult__cancel__right, axiom,
    ((![A : poly_a, C : poly_a, B : poly_a]: ((dvd_dvd_poly_a @ (times_times_poly_a @ A @ C) @ (times_times_poly_a @ B @ C)) = (((C = zero_zero_poly_a)) | ((dvd_dvd_poly_a @ A @ B))))))). % dvd_mult_cancel_right
thf(fact_53_dvd__times__left__cancel__iff, axiom,
    ((![A : poly_a, B : poly_a, C : poly_a]: ((~ ((A = zero_zero_poly_a))) => ((dvd_dvd_poly_a @ (times_times_poly_a @ A @ B) @ (times_times_poly_a @ A @ C)) = (dvd_dvd_poly_a @ B @ C)))))). % dvd_times_left_cancel_iff
thf(fact_54_dvd__times__left__cancel__iff, axiom,
    ((![A : nat, B : nat, C : nat]: ((~ ((A = zero_zero_nat))) => ((dvd_dvd_nat @ (times_times_nat @ A @ B) @ (times_times_nat @ A @ C)) = (dvd_dvd_nat @ B @ C)))))). % dvd_times_left_cancel_iff
thf(fact_55_dvd__times__right__cancel__iff, axiom,
    ((![A : poly_a, B : poly_a, C : poly_a]: ((~ ((A = zero_zero_poly_a))) => ((dvd_dvd_poly_a @ (times_times_poly_a @ B @ A) @ (times_times_poly_a @ C @ A)) = (dvd_dvd_poly_a @ B @ C)))))). % dvd_times_right_cancel_iff
thf(fact_56_dvd__times__right__cancel__iff, axiom,
    ((![A : nat, B : nat, C : nat]: ((~ ((A = zero_zero_nat))) => ((dvd_dvd_nat @ (times_times_nat @ B @ A) @ (times_times_nat @ C @ A)) = (dvd_dvd_nat @ B @ C)))))). % dvd_times_right_cancel_iff
thf(fact_57_smult__dvd, axiom,
    ((![P : poly_a, Q : poly_a, A : a]: ((dvd_dvd_poly_a @ P @ Q) => ((~ ((A = zero_zero_a))) => (dvd_dvd_poly_a @ (smult_a @ A @ P) @ Q)))))). % smult_dvd
thf(fact_58_dvd__smult__iff, axiom,
    ((![A : a, P : poly_a, Q : poly_a]: ((~ ((A = zero_zero_a))) => ((dvd_dvd_poly_a @ P @ (smult_a @ A @ Q)) = (dvd_dvd_poly_a @ P @ Q)))))). % dvd_smult_iff
thf(fact_59_smult__dvd__iff, axiom,
    ((![A : a, P : poly_a, Q : poly_a]: ((dvd_dvd_poly_a @ (smult_a @ A @ P) @ Q) = (((((A = zero_zero_a)) => ((Q = zero_zero_poly_a)))) & ((((~ ((A = zero_zero_a)))) => ((dvd_dvd_poly_a @ P @ Q))))))))). % smult_dvd_iff
thf(fact_60_dvd__smult__cancel, axiom,
    ((![P : poly_a, A : a, Q : poly_a]: ((dvd_dvd_poly_a @ P @ (smult_a @ A @ Q)) => ((~ ((A = zero_zero_a))) => (dvd_dvd_poly_a @ P @ Q)))))). % dvd_smult_cancel
thf(fact_61_dvd__refl, axiom,
    ((![A : poly_a]: (dvd_dvd_poly_a @ A @ A)))). % dvd_refl
thf(fact_62_dvd__refl, axiom,
    ((![A : nat]: (dvd_dvd_nat @ A @ A)))). % dvd_refl
thf(fact_63_dvd__trans, axiom,
    ((![A : poly_a, B : poly_a, C : poly_a]: ((dvd_dvd_poly_a @ A @ B) => ((dvd_dvd_poly_a @ B @ C) => (dvd_dvd_poly_a @ A @ C)))))). % dvd_trans
thf(fact_64_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_65_dvd__0__left, axiom,
    ((![A : poly_a]: ((dvd_dvd_poly_a @ zero_zero_poly_a @ A) => (A = zero_zero_poly_a))))). % dvd_0_left
thf(fact_66_dvd__0__left, axiom,
    ((![A : a]: ((dvd_dvd_a @ zero_zero_a @ A) => (A = zero_zero_a))))). % dvd_0_left
thf(fact_67_dvd__0__left, axiom,
    ((![A : nat]: ((dvd_dvd_nat @ zero_zero_nat @ A) => (A = zero_zero_nat))))). % dvd_0_left
thf(fact_68_dvd__triv__right, axiom,
    ((![A : poly_a, B : poly_a]: (dvd_dvd_poly_a @ A @ (times_times_poly_a @ B @ A))))). % dvd_triv_right
thf(fact_69_dvd__triv__right, axiom,
    ((![A : nat, B : nat]: (dvd_dvd_nat @ A @ (times_times_nat @ B @ A))))). % dvd_triv_right
thf(fact_70_dvd__mult__right, axiom,
    ((![A : poly_a, B : poly_a, C : poly_a]: ((dvd_dvd_poly_a @ (times_times_poly_a @ A @ B) @ C) => (dvd_dvd_poly_a @ B @ C))))). % dvd_mult_right
thf(fact_71_dvd__mult__right, axiom,
    ((![A : nat, B : nat, C : nat]: ((dvd_dvd_nat @ (times_times_nat @ A @ B) @ C) => (dvd_dvd_nat @ B @ C))))). % dvd_mult_right
thf(fact_72_mult__dvd__mono, axiom,
    ((![A : poly_a, B : poly_a, C : poly_a, D2 : poly_a]: ((dvd_dvd_poly_a @ A @ B) => ((dvd_dvd_poly_a @ C @ D2) => (dvd_dvd_poly_a @ (times_times_poly_a @ A @ C) @ (times_times_poly_a @ B @ D2))))))). % mult_dvd_mono
thf(fact_73_mult__dvd__mono, axiom,
    ((![A : nat, B : nat, C : nat, D2 : nat]: ((dvd_dvd_nat @ A @ B) => ((dvd_dvd_nat @ C @ D2) => (dvd_dvd_nat @ (times_times_nat @ A @ C) @ (times_times_nat @ B @ D2))))))). % mult_dvd_mono
thf(fact_74_dvd__triv__left, axiom,
    ((![A : poly_a, B : poly_a]: (dvd_dvd_poly_a @ A @ (times_times_poly_a @ A @ B))))). % dvd_triv_left
thf(fact_75_dvd__triv__left, axiom,
    ((![A : nat, B : nat]: (dvd_dvd_nat @ A @ (times_times_nat @ A @ B))))). % dvd_triv_left
thf(fact_76_dvd__mult__left, axiom,
    ((![A : poly_a, B : poly_a, C : poly_a]: ((dvd_dvd_poly_a @ (times_times_poly_a @ A @ B) @ C) => (dvd_dvd_poly_a @ A @ C))))). % dvd_mult_left
thf(fact_77_dvd__mult__left, axiom,
    ((![A : nat, B : nat, C : nat]: ((dvd_dvd_nat @ (times_times_nat @ A @ B) @ C) => (dvd_dvd_nat @ A @ C))))). % dvd_mult_left
thf(fact_78_dvd__mult2, axiom,
    ((![A : poly_a, B : poly_a, C : poly_a]: ((dvd_dvd_poly_a @ A @ B) => (dvd_dvd_poly_a @ A @ (times_times_poly_a @ B @ C)))))). % dvd_mult2
thf(fact_79_dvd__mult2, axiom,
    ((![A : nat, B : nat, C : nat]: ((dvd_dvd_nat @ A @ B) => (dvd_dvd_nat @ A @ (times_times_nat @ B @ C)))))). % dvd_mult2
thf(fact_80_dvd__mult, axiom,
    ((![A : poly_a, C : poly_a, B : poly_a]: ((dvd_dvd_poly_a @ A @ C) => (dvd_dvd_poly_a @ A @ (times_times_poly_a @ B @ C)))))). % dvd_mult
thf(fact_81_dvd__mult, axiom,
    ((![A : nat, C : nat, B : nat]: ((dvd_dvd_nat @ A @ C) => (dvd_dvd_nat @ A @ (times_times_nat @ B @ C)))))). % dvd_mult
thf(fact_82_dvd__def, axiom,
    ((dvd_dvd_poly_a = (^[B2 : poly_a]: (^[A2 : poly_a]: (?[K3 : poly_a]: (A2 = (times_times_poly_a @ B2 @ K3)))))))). % dvd_def
thf(fact_83_dvd__def, axiom,
    ((dvd_dvd_nat = (^[B2 : nat]: (^[A2 : nat]: (?[K3 : nat]: (A2 = (times_times_nat @ B2 @ K3)))))))). % dvd_def
thf(fact_84_dvdI, axiom,
    ((![A : poly_a, B : poly_a, K4 : poly_a]: ((A = (times_times_poly_a @ B @ K4)) => (dvd_dvd_poly_a @ B @ A))))). % dvdI
thf(fact_85_dvdI, axiom,
    ((![A : nat, B : nat, K4 : nat]: ((A = (times_times_nat @ B @ K4)) => (dvd_dvd_nat @ B @ A))))). % dvdI
thf(fact_86_dvdE, axiom,
    ((![B : poly_a, A : poly_a]: ((dvd_dvd_poly_a @ B @ A) => (~ ((![K : poly_a]: (~ ((A = (times_times_poly_a @ B @ K))))))))))). % dvdE
thf(fact_87_dvdE, axiom,
    ((![B : nat, A : nat]: ((dvd_dvd_nat @ B @ A) => (~ ((![K : nat]: (~ ((A = (times_times_nat @ B @ K))))))))))). % dvdE
thf(fact_88_dvd__diff, axiom,
    ((![X3 : poly_a, Y : poly_a, Z : poly_a]: ((dvd_dvd_poly_a @ X3 @ Y) => ((dvd_dvd_poly_a @ X3 @ Z) => (dvd_dvd_poly_a @ X3 @ (minus_minus_poly_a @ Y @ Z))))))). % dvd_diff
thf(fact_89_mult__right__cancel, axiom,
    ((![C : a, A : a, B : a]: ((~ ((C = zero_zero_a))) => (((times_times_a @ A @ C) = (times_times_a @ B @ C)) = (A = B)))))). % mult_right_cancel
thf(fact_90_mult__right__cancel, axiom,
    ((![C : poly_a, A : poly_a, B : poly_a]: ((~ ((C = zero_zero_poly_a))) => (((times_times_poly_a @ A @ C) = (times_times_poly_a @ B @ C)) = (A = B)))))). % mult_right_cancel
thf(fact_91_mult__right__cancel, axiom,
    ((![C : nat, A : nat, B : nat]: ((~ ((C = zero_zero_nat))) => (((times_times_nat @ A @ C) = (times_times_nat @ B @ C)) = (A = B)))))). % mult_right_cancel
thf(fact_92_mult__left__cancel, axiom,
    ((![C : a, A : a, B : a]: ((~ ((C = zero_zero_a))) => (((times_times_a @ C @ A) = (times_times_a @ C @ B)) = (A = B)))))). % mult_left_cancel
thf(fact_93_mult__left__cancel, axiom,
    ((![C : poly_a, A : poly_a, B : poly_a]: ((~ ((C = zero_zero_poly_a))) => (((times_times_poly_a @ C @ A) = (times_times_poly_a @ C @ B)) = (A = B)))))). % mult_left_cancel
thf(fact_94_mult__left__cancel, axiom,
    ((![C : nat, A : nat, B : nat]: ((~ ((C = zero_zero_nat))) => (((times_times_nat @ C @ A) = (times_times_nat @ C @ B)) = (A = B)))))). % mult_left_cancel
thf(fact_95_no__zero__divisors, axiom,
    ((![A : a, B : a]: ((~ ((A = zero_zero_a))) => ((~ ((B = zero_zero_a))) => (~ (((times_times_a @ A @ B) = zero_zero_a)))))))). % no_zero_divisors
thf(fact_96_no__zero__divisors, axiom,
    ((![A : poly_a, B : poly_a]: ((~ ((A = zero_zero_poly_a))) => ((~ ((B = zero_zero_poly_a))) => (~ (((times_times_poly_a @ A @ B) = zero_zero_poly_a)))))))). % no_zero_divisors
thf(fact_97_no__zero__divisors, axiom,
    ((![A : nat, B : nat]: ((~ ((A = zero_zero_nat))) => ((~ ((B = zero_zero_nat))) => (~ (((times_times_nat @ A @ B) = zero_zero_nat)))))))). % no_zero_divisors
thf(fact_98_divisors__zero, axiom,
    ((![A : a, B : a]: (((times_times_a @ A @ B) = zero_zero_a) => ((A = zero_zero_a) | (B = zero_zero_a)))))). % divisors_zero
thf(fact_99_divisors__zero, axiom,
    ((![A : poly_a, B : poly_a]: (((times_times_poly_a @ A @ B) = zero_zero_poly_a) => ((A = zero_zero_poly_a) | (B = zero_zero_poly_a)))))). % divisors_zero
thf(fact_100_divisors__zero, axiom,
    ((![A : nat, B : nat]: (((times_times_nat @ A @ B) = zero_zero_nat) => ((A = zero_zero_nat) | (B = zero_zero_nat)))))). % divisors_zero
thf(fact_101_mult__not__zero, axiom,
    ((![A : a, B : a]: ((~ (((times_times_a @ A @ B) = zero_zero_a))) => ((~ ((A = zero_zero_a))) & (~ ((B = zero_zero_a)))))))). % mult_not_zero
thf(fact_102_mult__not__zero, axiom,
    ((![A : poly_a, B : poly_a]: ((~ (((times_times_poly_a @ A @ B) = zero_zero_poly_a))) => ((~ ((A = zero_zero_poly_a))) & (~ ((B = zero_zero_poly_a)))))))). % mult_not_zero
thf(fact_103_mult__not__zero, axiom,
    ((![A : nat, B : nat]: ((~ (((times_times_nat @ A @ B) = zero_zero_nat))) => ((~ ((A = zero_zero_nat))) & (~ ((B = zero_zero_nat)))))))). % mult_not_zero
thf(fact_104_smult__dvd__cancel, axiom,
    ((![A : a, P : poly_a, Q : poly_a]: ((dvd_dvd_poly_a @ (smult_a @ A @ P) @ Q) => (dvd_dvd_poly_a @ P @ Q))))). % smult_dvd_cancel
thf(fact_105_dvd__smult, axiom,
    ((![P : poly_a, Q : poly_a, A : a]: ((dvd_dvd_poly_a @ P @ Q) => (dvd_dvd_poly_a @ P @ (smult_a @ A @ Q)))))). % dvd_smult
thf(fact_106_poly__cancel__eq__conv, axiom,
    ((![X3 : a, A : a, Y : a, B : a]: ((X3 = zero_zero_a) => ((~ ((A = zero_zero_a))) => ((Y = zero_zero_a) = ((minus_minus_a @ (times_times_a @ A @ Y) @ (times_times_a @ B @ X3)) = zero_zero_a))))))). % poly_cancel_eq_conv
thf(fact_107_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_108_cancel__comm__monoid__add__class_Odiff__cancel, axiom,
    ((![A : poly_a]: ((minus_minus_poly_a @ A @ A) = zero_zero_poly_a)))). % cancel_comm_monoid_add_class.diff_cancel
thf(fact_109_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_110_diff__zero, axiom,
    ((![A : a]: ((minus_minus_a @ A @ zero_zero_a) = A)))). % diff_zero
thf(fact_111_diff__zero, axiom,
    ((![A : poly_a]: ((minus_minus_poly_a @ A @ zero_zero_poly_a) = A)))). % diff_zero
thf(fact_112_diff__zero, axiom,
    ((![A : nat]: ((minus_minus_nat @ A @ zero_zero_nat) = A)))). % diff_zero
thf(fact_113_zero__diff, axiom,
    ((![A : nat]: ((minus_minus_nat @ zero_zero_nat @ A) = zero_zero_nat)))). % zero_diff
thf(fact_114_diff__0__right, axiom,
    ((![A : a]: ((minus_minus_a @ A @ zero_zero_a) = A)))). % diff_0_right
thf(fact_115_diff__0__right, axiom,
    ((![A : poly_a]: ((minus_minus_poly_a @ A @ zero_zero_poly_a) = A)))). % diff_0_right
thf(fact_116_diff__self, axiom,
    ((![A : a]: ((minus_minus_a @ A @ A) = zero_zero_a)))). % diff_self
thf(fact_117_diff__self, axiom,
    ((![A : poly_a]: ((minus_minus_poly_a @ A @ A) = zero_zero_poly_a)))). % diff_self
thf(fact_118_division__decomp, axiom,
    ((![A : nat, B : nat, C : nat]: ((dvd_dvd_nat @ A @ (times_times_nat @ B @ C)) => (?[B3 : nat, C2 : nat]: ((A = (times_times_nat @ B3 @ C2)) & ((dvd_dvd_nat @ B3 @ B) & (dvd_dvd_nat @ C2 @ C)))))))). % division_decomp
thf(fact_119_dvd__productE, axiom,
    ((![P : nat, A : nat, B : nat]: ((dvd_dvd_nat @ P @ (times_times_nat @ A @ B)) => (~ ((![X : nat, Y2 : nat]: ((P = (times_times_nat @ X @ Y2)) => ((dvd_dvd_nat @ X @ A) => (~ ((dvd_dvd_nat @ Y2 @ B)))))))))))). % dvd_productE
thf(fact_120_smult__0__right, axiom,
    ((![A : a]: ((smult_a @ A @ zero_zero_poly_a) = zero_zero_poly_a)))). % smult_0_right
thf(fact_121_mult__poly__0__left, axiom,
    ((![Q : poly_a]: ((times_times_poly_a @ zero_zero_poly_a @ Q) = zero_zero_poly_a)))). % mult_poly_0_left
thf(fact_122_mult__poly__0__right, axiom,
    ((![P : poly_a]: ((times_times_poly_a @ P @ zero_zero_poly_a) = zero_zero_poly_a)))). % mult_poly_0_right
thf(fact_123_zero__reorient, axiom,
    ((![X3 : a]: ((zero_zero_a = X3) = (X3 = zero_zero_a))))). % zero_reorient
thf(fact_124_zero__reorient, axiom,
    ((![X3 : nat]: ((zero_zero_nat = X3) = (X3 = zero_zero_nat))))). % zero_reorient
thf(fact_125_ab__semigroup__mult__class_Omult__ac_I1_J, axiom,
    ((![A : poly_a, B : poly_a, C : poly_a]: ((times_times_poly_a @ (times_times_poly_a @ A @ B) @ C) = (times_times_poly_a @ A @ (times_times_poly_a @ B @ C)))))). % ab_semigroup_mult_class.mult_ac(1)
thf(fact_126_ab__semigroup__mult__class_Omult__ac_I1_J, axiom,
    ((![A : nat, B : nat, C : nat]: ((times_times_nat @ (times_times_nat @ A @ B) @ C) = (times_times_nat @ A @ (times_times_nat @ B @ C)))))). % ab_semigroup_mult_class.mult_ac(1)
thf(fact_127_mult_Oassoc, axiom,
    ((![A : poly_a, B : poly_a, C : poly_a]: ((times_times_poly_a @ (times_times_poly_a @ A @ B) @ C) = (times_times_poly_a @ A @ (times_times_poly_a @ B @ C)))))). % mult.assoc
thf(fact_128_mult_Oassoc, axiom,
    ((![A : nat, B : nat, C : nat]: ((times_times_nat @ (times_times_nat @ A @ B) @ C) = (times_times_nat @ A @ (times_times_nat @ B @ C)))))). % mult.assoc
thf(fact_129_mult_Ocommute, axiom,
    ((times_times_poly_a = (^[A2 : poly_a]: (^[B2 : poly_a]: (times_times_poly_a @ B2 @ A2)))))). % mult.commute
thf(fact_130_mult_Ocommute, axiom,
    ((times_times_nat = (^[A2 : nat]: (^[B2 : nat]: (times_times_nat @ B2 @ A2)))))). % mult.commute
thf(fact_131_mult_Oleft__commute, axiom,
    ((![B : poly_a, A : poly_a, C : poly_a]: ((times_times_poly_a @ B @ (times_times_poly_a @ A @ C)) = (times_times_poly_a @ A @ (times_times_poly_a @ B @ C)))))). % mult.left_commute
thf(fact_132_mult_Oleft__commute, axiom,
    ((![B : nat, A : nat, C : nat]: ((times_times_nat @ B @ (times_times_nat @ A @ C)) = (times_times_nat @ A @ (times_times_nat @ B @ C)))))). % mult.left_commute
thf(fact_133_diff__eq__diff__eq, axiom,
    ((![A : poly_a, B : poly_a, C : poly_a, D2 : poly_a]: (((minus_minus_poly_a @ A @ B) = (minus_minus_poly_a @ C @ D2)) => ((A = B) = (C = D2)))))). % diff_eq_diff_eq
thf(fact_134_cancel__ab__semigroup__add__class_Odiff__right__commute, axiom,
    ((![A : poly_a, C : poly_a, B : poly_a]: ((minus_minus_poly_a @ (minus_minus_poly_a @ A @ C) @ B) = (minus_minus_poly_a @ (minus_minus_poly_a @ A @ B) @ C))))). % cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_135_cancel__ab__semigroup__add__class_Odiff__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))))). % cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_136_eq__iff__diff__eq__0, axiom,
    (((^[Y3 : a]: (^[Z2 : a]: (Y3 = Z2))) = (^[A2 : a]: (^[B2 : a]: ((minus_minus_a @ A2 @ B2) = zero_zero_a)))))). % eq_iff_diff_eq_0
thf(fact_137_eq__iff__diff__eq__0, axiom,
    (((^[Y3 : poly_a]: (^[Z2 : poly_a]: (Y3 = Z2))) = (^[A2 : poly_a]: (^[B2 : poly_a]: ((minus_minus_poly_a @ A2 @ B2) = zero_zero_poly_a)))))). % eq_iff_diff_eq_0
thf(fact_138_dvd__field__iff, axiom,
    ((dvd_dvd_a = (^[A2 : a]: (^[B2 : a]: (((A2 = zero_zero_a)) => ((B2 = zero_zero_a)))))))). % dvd_field_iff
thf(fact_139_map__poly__smult, axiom,
    ((![F : a > a, C : a, P : poly_a]: (((F @ zero_zero_a) = zero_zero_a) => ((![C3 : a, X : a]: ((F @ (times_times_a @ C3 @ X)) = (times_times_a @ (F @ C3) @ (F @ X)))) => ((map_poly_a_a @ F @ (smult_a @ C @ P)) = (smult_a @ (F @ C) @ (map_poly_a_a @ F @ P)))))))). % map_poly_smult
thf(fact_140_map__poly__smult, axiom,
    ((![F : a > poly_a, C : a, P : poly_a]: (((F @ zero_zero_a) = zero_zero_poly_a) => ((![C3 : a, X : a]: ((F @ (times_times_a @ C3 @ X)) = (times_times_poly_a @ (F @ C3) @ (F @ X)))) => ((map_poly_a_poly_a @ F @ (smult_a @ C @ P)) = (smult_poly_a @ (F @ C) @ (map_poly_a_poly_a @ F @ P)))))))). % map_poly_smult
thf(fact_141_map__poly__smult, axiom,
    ((![F : a > nat, C : a, P : poly_a]: (((F @ zero_zero_a) = zero_zero_nat) => ((![C3 : a, X : a]: ((F @ (times_times_a @ C3 @ X)) = (times_times_nat @ (F @ C3) @ (F @ X)))) => ((map_poly_a_nat @ F @ (smult_a @ C @ P)) = (smult_nat @ (F @ C) @ (map_poly_a_nat @ F @ P)))))))). % map_poly_smult
thf(fact_142_map__poly__smult, axiom,
    ((![F : poly_a > a, C : poly_a, P : poly_poly_a]: (((F @ zero_zero_poly_a) = zero_zero_a) => ((![C3 : poly_a, X : poly_a]: ((F @ (times_times_poly_a @ C3 @ X)) = (times_times_a @ (F @ C3) @ (F @ X)))) => ((map_poly_poly_a_a @ F @ (smult_poly_a @ C @ P)) = (smult_a @ (F @ C) @ (map_poly_poly_a_a @ F @ P)))))))). % map_poly_smult
thf(fact_143_map__poly__smult, axiom,
    ((![F : poly_a > poly_a, C : poly_a, P : poly_poly_a]: (((F @ zero_zero_poly_a) = zero_zero_poly_a) => ((![C3 : poly_a, X : poly_a]: ((F @ (times_times_poly_a @ C3 @ X)) = (times_times_poly_a @ (F @ C3) @ (F @ X)))) => ((map_po495521320poly_a @ F @ (smult_poly_a @ C @ P)) = (smult_poly_a @ (F @ C) @ (map_po495521320poly_a @ F @ P)))))))). % map_poly_smult
thf(fact_144_map__poly__smult, axiom,
    ((![F : poly_a > nat, C : poly_a, P : poly_poly_a]: (((F @ zero_zero_poly_a) = zero_zero_nat) => ((![C3 : poly_a, X : poly_a]: ((F @ (times_times_poly_a @ C3 @ X)) = (times_times_nat @ (F @ C3) @ (F @ X)))) => ((map_poly_poly_a_nat @ F @ (smult_poly_a @ C @ P)) = (smult_nat @ (F @ C) @ (map_poly_poly_a_nat @ F @ P)))))))). % map_poly_smult
thf(fact_145_map__poly__smult, axiom,
    ((![F : nat > a, C : nat, P : poly_nat]: (((F @ zero_zero_nat) = zero_zero_a) => ((![C3 : nat, X : nat]: ((F @ (times_times_nat @ C3 @ X)) = (times_times_a @ (F @ C3) @ (F @ X)))) => ((map_poly_nat_a @ F @ (smult_nat @ C @ P)) = (smult_a @ (F @ C) @ (map_poly_nat_a @ F @ P)))))))). % map_poly_smult
thf(fact_146_map__poly__smult, axiom,
    ((![F : nat > poly_a, C : nat, P : poly_nat]: (((F @ zero_zero_nat) = zero_zero_poly_a) => ((![C3 : nat, X : nat]: ((F @ (times_times_nat @ C3 @ X)) = (times_times_poly_a @ (F @ C3) @ (F @ X)))) => ((map_poly_nat_poly_a @ F @ (smult_nat @ C @ P)) = (smult_poly_a @ (F @ C) @ (map_poly_nat_poly_a @ F @ P)))))))). % map_poly_smult
thf(fact_147_map__poly__smult, axiom,
    ((![F : nat > nat, C : nat, P : poly_nat]: (((F @ zero_zero_nat) = zero_zero_nat) => ((![C3 : nat, X : nat]: ((F @ (times_times_nat @ C3 @ X)) = (times_times_nat @ (F @ C3) @ (F @ X)))) => ((map_poly_nat_nat @ F @ (smult_nat @ C @ P)) = (smult_nat @ (F @ C) @ (map_poly_nat_nat @ F @ P)))))))). % map_poly_smult
thf(fact_148_const__poly__dvd__const__poly__iff, axiom,
    ((![A : poly_a, B : poly_a]: ((dvd_dvd_poly_poly_a @ (pCons_poly_a @ A @ zero_z2096148049poly_a) @ (pCons_poly_a @ B @ zero_z2096148049poly_a)) = (dvd_dvd_poly_a @ A @ B))))). % const_poly_dvd_const_poly_iff
thf(fact_149_const__poly__dvd__const__poly__iff, axiom,
    ((![A : nat, B : nat]: ((dvd_dvd_poly_nat @ (pCons_nat @ A @ zero_zero_poly_nat) @ (pCons_nat @ B @ zero_zero_poly_nat)) = (dvd_dvd_nat @ A @ B))))). % const_poly_dvd_const_poly_iff
thf(fact_150_const__poly__dvd__const__poly__iff, axiom,
    ((![A : a, B : a]: ((dvd_dvd_poly_a @ (pCons_a @ A @ zero_zero_poly_a) @ (pCons_a @ B @ zero_zero_poly_a)) = (dvd_dvd_a @ A @ B))))). % const_poly_dvd_const_poly_iff
thf(fact_151_mult_Oright__neutral, axiom,
    ((![A : poly_a]: ((times_times_poly_a @ A @ one_one_poly_a) = A)))). % mult.right_neutral
thf(fact_152_mult_Oright__neutral, axiom,
    ((![A : nat]: ((times_times_nat @ A @ one_one_nat) = A)))). % mult.right_neutral
thf(fact_153_mult_Oleft__neutral, axiom,
    ((![A : poly_a]: ((times_times_poly_a @ one_one_poly_a @ A) = A)))). % mult.left_neutral
thf(fact_154_mult_Oleft__neutral, axiom,
    ((![A : nat]: ((times_times_nat @ one_one_nat @ A) = A)))). % mult.left_neutral
thf(fact_155_smult__1__left, axiom,
    ((![P : poly_a]: ((smult_a @ one_one_a @ P) = P)))). % smult_1_left
thf(fact_156_smult__1__left, axiom,
    ((![P : poly_nat]: ((smult_nat @ one_one_nat @ P) = P)))). % smult_1_left
thf(fact_157_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_158_mult__cancel__right2, axiom,
    ((![A : poly_a, C : poly_a]: (((times_times_poly_a @ A @ C) = C) = (((C = zero_zero_poly_a)) | ((A = one_one_poly_a))))))). % mult_cancel_right2
thf(fact_159_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_160_mult__cancel__right1, axiom,
    ((![C : poly_a, B : poly_a]: ((C = (times_times_poly_a @ B @ C)) = (((C = zero_zero_poly_a)) | ((B = one_one_poly_a))))))). % mult_cancel_right1
thf(fact_161_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_162_mult__cancel__left2, axiom,
    ((![C : poly_a, A : poly_a]: (((times_times_poly_a @ C @ A) = C) = (((C = zero_zero_poly_a)) | ((A = one_one_poly_a))))))). % mult_cancel_left2
thf(fact_163_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_164_mult__cancel__left1, axiom,
    ((![C : poly_a, B : poly_a]: ((C = (times_times_poly_a @ C @ B)) = (((C = zero_zero_poly_a)) | ((B = one_one_poly_a))))))). % mult_cancel_left1
thf(fact_165_unit__prod, axiom,
    ((![A : poly_a, B : poly_a]: ((dvd_dvd_poly_a @ A @ one_one_poly_a) => ((dvd_dvd_poly_a @ B @ one_one_poly_a) => (dvd_dvd_poly_a @ (times_times_poly_a @ A @ B) @ one_one_poly_a)))))). % unit_prod
thf(fact_166_unit__prod, axiom,
    ((![A : nat, B : nat]: ((dvd_dvd_nat @ A @ one_one_nat) => ((dvd_dvd_nat @ B @ one_one_nat) => (dvd_dvd_nat @ (times_times_nat @ A @ B) @ one_one_nat)))))). % unit_prod
thf(fact_167_pCons__eq__0__iff, axiom,
    ((![A : a, P : poly_a]: (((pCons_a @ A @ P) = zero_zero_poly_a) = (((A = zero_zero_a)) & ((P = zero_zero_poly_a))))))). % pCons_eq_0_iff
thf(fact_168_pCons__eq__0__iff, axiom,
    ((![A : nat, P : poly_nat]: (((pCons_nat @ A @ P) = zero_zero_poly_nat) = (((A = zero_zero_nat)) & ((P = zero_zero_poly_nat))))))). % pCons_eq_0_iff
thf(fact_169_pCons__0__0, axiom,
    (((pCons_a @ zero_zero_a @ zero_zero_poly_a) = zero_zero_poly_a))). % pCons_0_0
thf(fact_170_pCons__0__0, axiom,
    (((pCons_nat @ zero_zero_nat @ zero_zero_poly_nat) = zero_zero_poly_nat))). % pCons_0_0
thf(fact_171_smult__pCons, axiom,
    ((![A : a, B : a, P : poly_a]: ((smult_a @ A @ (pCons_a @ B @ P)) = (pCons_a @ (times_times_a @ A @ B) @ (smult_a @ A @ P)))))). % smult_pCons
thf(fact_172_smult__pCons, axiom,
    ((![A : poly_a, B : poly_a, P : poly_poly_a]: ((smult_poly_a @ A @ (pCons_poly_a @ B @ P)) = (pCons_poly_a @ (times_times_poly_a @ A @ B) @ (smult_poly_a @ A @ P)))))). % smult_pCons
thf(fact_173_smult__pCons, axiom,
    ((![A : nat, B : nat, P : poly_nat]: ((smult_nat @ A @ (pCons_nat @ B @ P)) = (pCons_nat @ (times_times_nat @ A @ B) @ (smult_nat @ A @ P)))))). % smult_pCons
thf(fact_174_diff__pCons, axiom,
    ((![A : poly_a, P : poly_poly_a, B : poly_a, Q : poly_poly_a]: ((minus_154650241poly_a @ (pCons_poly_a @ A @ P) @ (pCons_poly_a @ B @ Q)) = (pCons_poly_a @ (minus_minus_poly_a @ A @ B) @ (minus_154650241poly_a @ P @ Q)))))). % diff_pCons
thf(fact_175_diff__pCons, axiom,
    ((![A : nat, P : poly_nat, B : nat, Q : poly_nat]: ((minus_minus_poly_nat @ (pCons_nat @ A @ P) @ (pCons_nat @ B @ Q)) = (pCons_nat @ (minus_minus_nat @ A @ B) @ (minus_minus_poly_nat @ P @ Q)))))). % diff_pCons
thf(fact_176_diff__pCons, axiom,
    ((![A : a, P : poly_a, B : a, Q : poly_a]: ((minus_minus_poly_a @ (pCons_a @ A @ P) @ (pCons_a @ B @ Q)) = (pCons_a @ (minus_minus_a @ A @ B) @ (minus_minus_poly_a @ P @ Q)))))). % diff_pCons
thf(fact_177_map__poly__1_H, axiom,
    ((![F : nat > nat]: (((F @ one_one_nat) = one_one_nat) => ((map_poly_nat_nat @ F @ one_one_poly_nat) = one_one_poly_nat))))). % map_poly_1'
thf(fact_178_one__poly__eq__simps_I1_J, axiom,
    ((one_one_poly_nat = (pCons_nat @ one_one_nat @ zero_zero_poly_nat)))). % one_poly_eq_simps(1)
thf(fact_179_one__poly__eq__simps_I2_J, axiom,
    (((pCons_nat @ one_one_nat @ zero_zero_poly_nat) = one_one_poly_nat))). % one_poly_eq_simps(2)
thf(fact_180_smult__one, axiom,
    ((![C : a]: ((smult_a @ C @ one_one_poly_a) = (pCons_a @ C @ zero_zero_poly_a))))). % smult_one
thf(fact_181_is__unit__triv, axiom,
    ((![A : a]: ((~ ((A = zero_zero_a))) => (dvd_dvd_poly_a @ (pCons_a @ A @ zero_zero_poly_a) @ one_one_poly_a))))). % is_unit_triv
thf(fact_182_is__unit__pCons__iff, axiom,
    ((![A : a, P : poly_a]: ((dvd_dvd_poly_a @ (pCons_a @ A @ P) @ one_one_poly_a) = (((P = zero_zero_poly_a)) & ((~ ((A = zero_zero_a))))))))). % is_unit_pCons_iff
thf(fact_183_one__dvd, axiom,
    ((![A : poly_a]: (dvd_dvd_poly_a @ one_one_poly_a @ A)))). % one_dvd
thf(fact_184_one__dvd, axiom,
    ((![A : nat]: (dvd_dvd_nat @ one_one_nat @ A)))). % one_dvd
thf(fact_185_unit__imp__dvd, axiom,
    ((![B : poly_a, A : poly_a]: ((dvd_dvd_poly_a @ B @ one_one_poly_a) => (dvd_dvd_poly_a @ B @ A))))). % unit_imp_dvd
thf(fact_186_unit__imp__dvd, axiom,
    ((![B : nat, A : nat]: ((dvd_dvd_nat @ B @ one_one_nat) => (dvd_dvd_nat @ B @ A))))). % unit_imp_dvd
thf(fact_187_dvd__unit__imp__unit, axiom,
    ((![A : poly_a, B : poly_a]: ((dvd_dvd_poly_a @ A @ B) => ((dvd_dvd_poly_a @ B @ one_one_poly_a) => (dvd_dvd_poly_a @ A @ one_one_poly_a)))))). % dvd_unit_imp_unit
thf(fact_188_dvd__unit__imp__unit, axiom,
    ((![A : nat, B : nat]: ((dvd_dvd_nat @ A @ B) => ((dvd_dvd_nat @ B @ one_one_nat) => (dvd_dvd_nat @ A @ one_one_nat)))))). % dvd_unit_imp_unit
thf(fact_189_zero__neq__one, axiom,
    ((~ ((zero_zero_a = one_one_a))))). % zero_neq_one
thf(fact_190_zero__neq__one, axiom,
    ((~ ((zero_zero_nat = one_one_nat))))). % zero_neq_one
thf(fact_191_one__reorient, axiom,
    ((![X3 : nat]: ((one_one_nat = X3) = (X3 = one_one_nat))))). % one_reorient
thf(fact_192_comm__monoid__mult__class_Omult__1, axiom,
    ((![A : poly_a]: ((times_times_poly_a @ one_one_poly_a @ A) = A)))). % comm_monoid_mult_class.mult_1
thf(fact_193_comm__monoid__mult__class_Omult__1, axiom,
    ((![A : nat]: ((times_times_nat @ one_one_nat @ A) = A)))). % comm_monoid_mult_class.mult_1
thf(fact_194_mult_Ocomm__neutral, axiom,
    ((![A : poly_a]: ((times_times_poly_a @ A @ one_one_poly_a) = A)))). % mult.comm_neutral
thf(fact_195_mult_Ocomm__neutral, axiom,
    ((![A : nat]: ((times_times_nat @ A @ one_one_nat) = A)))). % mult.comm_neutral
thf(fact_196_is__unit__const__poly__iff, axiom,
    ((![C : poly_a]: ((dvd_dvd_poly_poly_a @ (pCons_poly_a @ C @ zero_z2096148049poly_a) @ one_one_poly_poly_a) = (dvd_dvd_poly_a @ C @ one_one_poly_a))))). % is_unit_const_poly_iff
thf(fact_197_is__unit__const__poly__iff, axiom,
    ((![C : a]: ((dvd_dvd_poly_a @ (pCons_a @ C @ zero_zero_poly_a) @ one_one_poly_a) = (dvd_dvd_a @ C @ one_one_a))))). % is_unit_const_poly_iff
thf(fact_198_is__unit__const__poly__iff, axiom,
    ((![C : nat]: ((dvd_dvd_poly_nat @ (pCons_nat @ C @ zero_zero_poly_nat) @ one_one_poly_nat) = (dvd_dvd_nat @ C @ one_one_nat))))). % is_unit_const_poly_iff
thf(fact_199_gcd__nat_Oextremum__uniqueI, axiom,
    ((![A : nat]: ((dvd_dvd_nat @ zero_zero_nat @ A) => (A = zero_zero_nat))))). % gcd_nat.extremum_uniqueI
thf(fact_200_is__unit__poly__iff, axiom,
    ((![P : poly_poly_a]: ((dvd_dvd_poly_poly_a @ P @ one_one_poly_poly_a) = (?[C4 : poly_a]: (((P = (pCons_poly_a @ C4 @ zero_z2096148049poly_a))) & ((dvd_dvd_poly_a @ C4 @ one_one_poly_a)))))))). % is_unit_poly_iff
thf(fact_201_is__unit__poly__iff, axiom,
    ((![P : poly_a]: ((dvd_dvd_poly_a @ P @ one_one_poly_a) = (?[C4 : a]: (((P = (pCons_a @ C4 @ zero_zero_poly_a))) & ((dvd_dvd_a @ C4 @ one_one_a)))))))). % is_unit_poly_iff
thf(fact_202_is__unit__poly__iff, axiom,
    ((![P : poly_nat]: ((dvd_dvd_poly_nat @ P @ one_one_poly_nat) = (?[C4 : nat]: (((P = (pCons_nat @ C4 @ zero_zero_poly_nat))) & ((dvd_dvd_nat @ C4 @ one_one_nat)))))))). % is_unit_poly_iff
thf(fact_203_gcd__nat_Onot__eq__extremum, axiom,
    ((![A : nat]: ((~ ((A = zero_zero_nat))) = (((dvd_dvd_nat @ A @ zero_zero_nat)) & ((~ ((A = zero_zero_nat))))))))). % gcd_nat.not_eq_extremum
thf(fact_204_gcd__nat_Oextremum__unique, axiom,
    ((![A : nat]: ((dvd_dvd_nat @ zero_zero_nat @ A) = (A = zero_zero_nat))))). % gcd_nat.extremum_unique
thf(fact_205_gcd__nat_Oextremum__strict, axiom,
    ((![A : nat]: (~ (((dvd_dvd_nat @ zero_zero_nat @ A) & (~ ((zero_zero_nat = A))))))))). % gcd_nat.extremum_strict
thf(fact_206_map__poly__pCons, axiom,
    ((![F : a > a, C : a, P : poly_a]: (((F @ zero_zero_a) = zero_zero_a) => ((map_poly_a_a @ F @ (pCons_a @ C @ P)) = (pCons_a @ (F @ C) @ (map_poly_a_a @ F @ P))))))). % map_poly_pCons
thf(fact_207_map__poly__pCons, axiom,
    ((![F : a > nat, C : a, P : poly_a]: (((F @ zero_zero_a) = zero_zero_nat) => ((map_poly_a_nat @ F @ (pCons_a @ C @ P)) = (pCons_nat @ (F @ C) @ (map_poly_a_nat @ F @ P))))))). % map_poly_pCons
thf(fact_208_map__poly__pCons, axiom,
    ((![F : nat > a, C : nat, P : poly_nat]: (((F @ zero_zero_nat) = zero_zero_a) => ((map_poly_nat_a @ F @ (pCons_nat @ C @ P)) = (pCons_a @ (F @ C) @ (map_poly_nat_a @ F @ P))))))). % map_poly_pCons
thf(fact_209_map__poly__pCons, axiom,
    ((![F : nat > nat, C : nat, P : poly_nat]: (((F @ zero_zero_nat) = zero_zero_nat) => ((map_poly_nat_nat @ F @ (pCons_nat @ C @ P)) = (pCons_nat @ (F @ C) @ (map_poly_nat_nat @ F @ P))))))). % map_poly_pCons
thf(fact_210_is__unit__polyE, axiom,
    ((![P : poly_poly_a]: ((dvd_dvd_poly_poly_a @ P @ one_one_poly_poly_a) => (~ ((![C3 : poly_a]: ((P = (pCons_poly_a @ C3 @ zero_z2096148049poly_a)) => (~ ((dvd_dvd_poly_a @ C3 @ one_one_poly_a))))))))))). % is_unit_polyE
thf(fact_211_is__unit__polyE, axiom,
    ((![P : poly_a]: ((dvd_dvd_poly_a @ P @ one_one_poly_a) => (~ ((![C3 : a]: ((P = (pCons_a @ C3 @ zero_zero_poly_a)) => (~ ((dvd_dvd_a @ C3 @ one_one_a))))))))))). % is_unit_polyE
thf(fact_212_is__unit__polyE, axiom,
    ((![P : poly_nat]: ((dvd_dvd_poly_nat @ P @ one_one_poly_nat) => (~ ((![C3 : nat]: ((P = (pCons_nat @ C3 @ zero_zero_poly_nat)) => (~ ((dvd_dvd_nat @ C3 @ one_one_nat))))))))))). % is_unit_polyE
thf(fact_213_pCons__one, axiom,
    (((pCons_nat @ one_one_nat @ zero_zero_poly_nat) = one_one_poly_nat))). % pCons_one
thf(fact_214_gcd__nat_Oextremum, axiom,
    ((![A : nat]: (dvd_dvd_nat @ A @ zero_zero_nat)))). % gcd_nat.extremum
thf(fact_215_is__unit__smult__iff, axiom,
    ((![C : poly_a, P : poly_poly_a]: ((dvd_dvd_poly_poly_a @ (smult_poly_a @ C @ P) @ one_one_poly_poly_a) = (((dvd_dvd_poly_a @ C @ one_one_poly_a)) & ((dvd_dvd_poly_poly_a @ P @ one_one_poly_poly_a))))))). % is_unit_smult_iff
thf(fact_216_is__unit__smult__iff, axiom,
    ((![C : a, P : poly_a]: ((dvd_dvd_poly_a @ (smult_a @ C @ P) @ one_one_poly_a) = (((dvd_dvd_a @ C @ one_one_a)) & ((dvd_dvd_poly_a @ P @ one_one_poly_a))))))). % is_unit_smult_iff
thf(fact_217_is__unit__smult__iff, axiom,
    ((![C : nat, P : poly_nat]: ((dvd_dvd_poly_nat @ (smult_nat @ C @ P) @ one_one_poly_nat) = (((dvd_dvd_nat @ C @ one_one_nat)) & ((dvd_dvd_poly_nat @ P @ one_one_poly_nat))))))). % is_unit_smult_iff
thf(fact_218_pCons__induct, axiom,
    ((![P2 : poly_a > $o, P : poly_a]: ((P2 @ zero_zero_poly_a) => ((![A3 : a, P3 : poly_a]: (((~ ((A3 = zero_zero_a))) | (~ ((P3 = zero_zero_poly_a)))) => ((P2 @ P3) => (P2 @ (pCons_a @ A3 @ P3))))) => (P2 @ P)))))). % pCons_induct
thf(fact_219_pCons__induct, axiom,
    ((![P2 : poly_nat > $o, P : poly_nat]: ((P2 @ zero_zero_poly_nat) => ((![A3 : nat, P3 : poly_nat]: (((~ ((A3 = zero_zero_nat))) | (~ ((P3 = zero_zero_poly_nat)))) => ((P2 @ P3) => (P2 @ (pCons_nat @ A3 @ P3))))) => (P2 @ P)))))). % pCons_induct
thf(fact_220_poly__divides__pad__rule, axiom,
    ((![P : poly_a, Q : poly_a]: ((dvd_dvd_poly_a @ P @ Q) => (dvd_dvd_poly_a @ P @ (pCons_a @ zero_zero_a @ Q)))))). % poly_divides_pad_rule
thf(fact_221_synthetic__div__unique__lemma, axiom,
    ((![C : a, P : poly_a, A : a]: (((smult_a @ C @ P) = (pCons_a @ A @ P)) => (P = zero_zero_poly_a))))). % synthetic_div_unique_lemma
thf(fact_222_not__is__unit__0, axiom,
    ((~ ((dvd_dvd_poly_a @ zero_zero_poly_a @ one_one_poly_a))))). % not_is_unit_0
thf(fact_223_not__is__unit__0, axiom,
    ((~ ((dvd_dvd_nat @ zero_zero_nat @ one_one_nat))))). % not_is_unit_0
thf(fact_224_unit__mult__right__cancel, axiom,
    ((![A : poly_a, B : poly_a, C : poly_a]: ((dvd_dvd_poly_a @ A @ one_one_poly_a) => (((times_times_poly_a @ B @ A) = (times_times_poly_a @ C @ A)) = (B = C)))))). % unit_mult_right_cancel
thf(fact_225_unit__mult__right__cancel, axiom,
    ((![A : nat, B : nat, C : nat]: ((dvd_dvd_nat @ A @ one_one_nat) => (((times_times_nat @ B @ A) = (times_times_nat @ C @ A)) = (B = C)))))). % unit_mult_right_cancel
thf(fact_226_unit__mult__left__cancel, axiom,
    ((![A : poly_a, B : poly_a, C : poly_a]: ((dvd_dvd_poly_a @ A @ one_one_poly_a) => (((times_times_poly_a @ A @ B) = (times_times_poly_a @ A @ C)) = (B = C)))))). % unit_mult_left_cancel
thf(fact_227_unit__mult__left__cancel, axiom,
    ((![A : nat, B : nat, C : nat]: ((dvd_dvd_nat @ A @ one_one_nat) => (((times_times_nat @ A @ B) = (times_times_nat @ A @ C)) = (B = C)))))). % unit_mult_left_cancel
thf(fact_228_mult__unit__dvd__iff_H, axiom,
    ((![A : poly_a, B : poly_a, C : poly_a]: ((dvd_dvd_poly_a @ A @ one_one_poly_a) => ((dvd_dvd_poly_a @ (times_times_poly_a @ A @ B) @ C) = (dvd_dvd_poly_a @ B @ C)))))). % mult_unit_dvd_iff'
thf(fact_229_mult__unit__dvd__iff_H, axiom,
    ((![A : nat, B : nat, C : nat]: ((dvd_dvd_nat @ A @ one_one_nat) => ((dvd_dvd_nat @ (times_times_nat @ A @ B) @ C) = (dvd_dvd_nat @ B @ C)))))). % mult_unit_dvd_iff'
thf(fact_230_dvd__mult__unit__iff_H, axiom,
    ((![B : poly_a, A : poly_a, C : poly_a]: ((dvd_dvd_poly_a @ B @ one_one_poly_a) => ((dvd_dvd_poly_a @ A @ (times_times_poly_a @ B @ C)) = (dvd_dvd_poly_a @ A @ C)))))). % dvd_mult_unit_iff'
thf(fact_231_dvd__mult__unit__iff_H, axiom,
    ((![B : nat, A : nat, C : nat]: ((dvd_dvd_nat @ B @ one_one_nat) => ((dvd_dvd_nat @ A @ (times_times_nat @ B @ C)) = (dvd_dvd_nat @ A @ C)))))). % dvd_mult_unit_iff'
thf(fact_232_mult__unit__dvd__iff, axiom,
    ((![B : poly_a, A : poly_a, C : poly_a]: ((dvd_dvd_poly_a @ B @ one_one_poly_a) => ((dvd_dvd_poly_a @ (times_times_poly_a @ A @ B) @ C) = (dvd_dvd_poly_a @ A @ C)))))). % mult_unit_dvd_iff
thf(fact_233_mult__unit__dvd__iff, axiom,
    ((![B : nat, A : nat, C : nat]: ((dvd_dvd_nat @ B @ one_one_nat) => ((dvd_dvd_nat @ (times_times_nat @ A @ B) @ C) = (dvd_dvd_nat @ A @ C)))))). % mult_unit_dvd_iff
thf(fact_234_dvd__mult__unit__iff, axiom,
    ((![B : poly_a, A : poly_a, C : poly_a]: ((dvd_dvd_poly_a @ B @ one_one_poly_a) => ((dvd_dvd_poly_a @ A @ (times_times_poly_a @ C @ B)) = (dvd_dvd_poly_a @ A @ C)))))). % dvd_mult_unit_iff
thf(fact_235_dvd__mult__unit__iff, axiom,
    ((![B : nat, A : nat, C : nat]: ((dvd_dvd_nat @ B @ one_one_nat) => ((dvd_dvd_nat @ A @ (times_times_nat @ C @ B)) = (dvd_dvd_nat @ A @ C)))))). % dvd_mult_unit_iff
thf(fact_236_is__unit__mult__iff, axiom,
    ((![A : poly_a, B : poly_a]: ((dvd_dvd_poly_a @ (times_times_poly_a @ A @ B) @ one_one_poly_a) = (((dvd_dvd_poly_a @ A @ one_one_poly_a)) & ((dvd_dvd_poly_a @ B @ one_one_poly_a))))))). % is_unit_mult_iff
thf(fact_237_is__unit__mult__iff, axiom,
    ((![A : nat, B : nat]: ((dvd_dvd_nat @ (times_times_nat @ A @ B) @ one_one_nat) = (((dvd_dvd_nat @ A @ one_one_nat)) & ((dvd_dvd_nat @ B @ one_one_nat))))))). % is_unit_mult_iff
thf(fact_238_unit__dvdE, axiom,
    ((![A : poly_a, B : poly_a]: ((dvd_dvd_poly_a @ A @ one_one_poly_a) => (~ (((~ ((A = zero_zero_poly_a))) => (![C3 : poly_a]: (~ ((B = (times_times_poly_a @ A @ C3)))))))))))). % unit_dvdE
thf(fact_239_unit__dvdE, axiom,
    ((![A : nat, B : nat]: ((dvd_dvd_nat @ A @ one_one_nat) => (~ (((~ ((A = zero_zero_nat))) => (![C3 : nat]: (~ ((B = (times_times_nat @ A @ C3)))))))))))). % unit_dvdE
thf(fact_240_diff__numeral__special_I9_J, axiom,
    (((minus_minus_a @ one_one_a @ one_one_a) = zero_zero_a))). % diff_numeral_special(9)
thf(fact_241_diff__numeral__special_I9_J, axiom,
    (((minus_minus_poly_a @ one_one_poly_a @ one_one_poly_a) = zero_zero_poly_a))). % diff_numeral_special(9)
thf(fact_242_nat__mult__dvd__cancel__disj, axiom,
    ((![K4 : nat, M : nat, N : nat]: ((dvd_dvd_nat @ (times_times_nat @ K4 @ M) @ (times_times_nat @ K4 @ N)) = (((K4 = zero_zero_nat)) | ((dvd_dvd_nat @ M @ N))))))). % nat_mult_dvd_cancel_disj
thf(fact_243_diff__self__eq__0, axiom,
    ((![M : nat]: ((minus_minus_nat @ M @ M) = zero_zero_nat)))). % diff_self_eq_0
thf(fact_244_nat__1__eq__mult__iff, axiom,
    ((![M : nat, N : nat]: ((one_one_nat = (times_times_nat @ M @ N)) = (((M = one_one_nat)) & ((N = one_one_nat))))))). % nat_1_eq_mult_iff

% Conjectures (1)
thf(conj_0, conjecture,
    ((r = (times_times_poly_a @ p @ (minus_minus_poly_a @ (smult_a @ a2 @ u) @ t))))).
