% 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_707__5374812_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:32:22.234

% Could-be-implicit typings (4)
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__Complex__Ocomplex, type,
    complex : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).

% Explicit typings (27)
thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Complex__Ocomplex, type,
    invers502456322omplex : complex > complex).
thf(sy_c_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Oconstant_001t__Complex__Ocomplex_001t__Complex__Ocomplex, type,
    fundam1158420650omplex : (complex > complex) > $o).
thf(sy_c_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Opsize_001t__Complex__Ocomplex, type,
    fundam1709708056omplex : poly_complex > nat).
thf(sy_c_Groups_Oone__class_Oone_001t__Complex__Ocomplex, type,
    one_one_complex : complex).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat, type,
    one_one_nat : nat).
thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    one_one_poly_complex : poly_complex).
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_Oplus__class_Oplus_001t__Complex__Ocomplex, type,
    plus_plus_complex : complex > complex > complex).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat, type,
    plus_plus_nat : nat > nat > nat).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    plus_p1547158847omplex : poly_complex > poly_complex > poly_complex).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Complex__Ocomplex, type,
    times_times_complex : complex > complex > complex).
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_It__Complex__Ocomplex_J, type,
    times_1246143675omplex : poly_complex > poly_complex > poly_complex).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    times_times_poly_nat : poly_nat > poly_nat > poly_nat).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Complex__Ocomplex, type,
    zero_zero_complex : complex).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat, type,
    zero_zero_nat : nat).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    zero_z1746442943omplex : poly_complex).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Nat__Onat_J, type,
    zero_zero_poly_nat : poly_nat).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat, type,
    ord_less_nat : nat > nat > $o).
thf(sy_c_Polynomial_Opoly_001t__Complex__Ocomplex, type,
    poly_complex2 : poly_complex > complex > complex).
thf(sy_c_Polynomial_Opoly_001t__Nat__Onat, type,
    poly_nat2 : poly_nat > nat > nat).
thf(sy_c_Polynomial_Osmult_001t__Complex__Ocomplex, type,
    smult_complex : complex > poly_complex > poly_complex).
thf(sy_c_Polynomial_Osmult_001t__Nat__Onat, type,
    smult_nat : nat > poly_nat > poly_nat).
thf(sy_v_c____, type,
    c : complex).
thf(sy_v_p, type,
    p : poly_complex).
thf(sy_v_pa____, type,
    pa : poly_complex).
thf(sy_v_q____, type,
    q : poly_complex).

% Relevant facts (170)
thf(fact_0_a00, axiom,
    ((~ (((poly_complex2 @ q @ zero_zero_complex) = zero_zero_complex))))). % a00
thf(fact_1_nc, axiom,
    ((~ ((fundam1158420650omplex @ (poly_complex2 @ p)))))). % nc
thf(fact_2__092_060open_062poly_Aq_A0_A_061_Apoly_A_Ismult_A_Iinverse_A_Ipoly_Aq_A0_J_J_Aq_J_A0_A_K_Apoly_Aq_A0_092_060close_062, axiom,
    (((poly_complex2 @ q @ zero_zero_complex) = (times_times_complex @ (poly_complex2 @ (smult_complex @ (invers502456322omplex @ (poly_complex2 @ q @ zero_zero_complex)) @ q) @ zero_zero_complex) @ (poly_complex2 @ q @ zero_zero_complex))))). % \<open>poly q 0 = poly (smult (inverse (poly q 0)) q) 0 * poly q 0\<close>
thf(fact_3__092_060open_062_I_092_060And_062x_Ay_O_Apoly_A_Ismult_A_Iinverse_A_Ipoly_Aq_A0_J_J_Aq_J_Ax_A_061_Apoly_A_Ismult_A_Iinverse_A_Ipoly_Aq_A0_J_J_Aq_J_Ay_J_A_092_060Longrightarrow_062_AFalse_092_060close_062, axiom,
    ((~ ((![X : complex, Y : complex]: ((poly_complex2 @ (smult_complex @ (invers502456322omplex @ (poly_complex2 @ q @ zero_zero_complex)) @ q) @ X) = (poly_complex2 @ (smult_complex @ (invers502456322omplex @ (poly_complex2 @ q @ zero_zero_complex)) @ q) @ Y))))))). % \<open>(\<And>x y. poly (smult (inverse (poly q 0)) q) x = poly (smult (inverse (poly q 0)) q) y) \<Longrightarrow> False\<close>
thf(fact_4_qr, axiom,
    ((![Z : complex]: ((poly_complex2 @ q @ Z) = (times_times_complex @ (poly_complex2 @ (smult_complex @ (invers502456322omplex @ (poly_complex2 @ q @ zero_zero_complex)) @ q) @ Z) @ (poly_complex2 @ q @ zero_zero_complex)))))). % qr
thf(fact_5__092_060open_062constant_A_Ipoly_Aq_J_A_092_060Longrightarrow_062_AFalse_092_060close_062, axiom,
    ((~ ((fundam1158420650omplex @ (poly_complex2 @ q)))))). % \<open>constant (poly q) \<Longrightarrow> False\<close>
thf(fact_6_qnc, axiom,
    ((~ ((fundam1158420650omplex @ (poly_complex2 @ q)))))). % qnc
thf(fact_7_rnc, axiom,
    ((~ ((fundam1158420650omplex @ (poly_complex2 @ (smult_complex @ (invers502456322omplex @ (poly_complex2 @ q @ zero_zero_complex)) @ q))))))). % rnc
thf(fact_8_lgqr, axiom,
    (((fundam1709708056omplex @ q) = (fundam1709708056omplex @ (smult_complex @ (invers502456322omplex @ (poly_complex2 @ q @ zero_zero_complex)) @ q))))). % lgqr
thf(fact_9_smult__1__left, axiom,
    ((![P : poly_complex]: ((smult_complex @ one_one_complex @ P) = P)))). % smult_1_left
thf(fact_10_smult__1__left, axiom,
    ((![P : poly_nat]: ((smult_nat @ one_one_nat @ P) = P)))). % smult_1_left
thf(fact_11_poly__1, axiom,
    ((![X2 : complex]: ((poly_complex2 @ one_one_poly_complex @ X2) = one_one_complex)))). % poly_1
thf(fact_12_poly__1, axiom,
    ((![X2 : nat]: ((poly_nat2 @ one_one_poly_nat @ X2) = one_one_nat)))). % poly_1
thf(fact_13_smult__0__left, axiom,
    ((![P : poly_complex]: ((smult_complex @ zero_zero_complex @ P) = zero_z1746442943omplex)))). % smult_0_left
thf(fact_14_smult__0__left, axiom,
    ((![P : poly_nat]: ((smult_nat @ zero_zero_nat @ P) = zero_zero_poly_nat)))). % smult_0_left
thf(fact_15_smult__eq__0__iff, axiom,
    ((![A : complex, P : poly_complex]: (((smult_complex @ A @ P) = zero_z1746442943omplex) = (((A = zero_zero_complex)) | ((P = zero_z1746442943omplex))))))). % smult_eq_0_iff
thf(fact_16_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_17_poly__0, axiom,
    ((![X2 : complex]: ((poly_complex2 @ zero_z1746442943omplex @ X2) = zero_zero_complex)))). % poly_0
thf(fact_18_poly__0, axiom,
    ((![X2 : nat]: ((poly_nat2 @ zero_zero_poly_nat @ X2) = zero_zero_nat)))). % poly_0
thf(fact_19_inverse__eq__1__iff, axiom,
    ((![X2 : complex]: (((invers502456322omplex @ X2) = one_one_complex) = (X2 = one_one_complex))))). % inverse_eq_1_iff
thf(fact_20_inverse__1, axiom,
    (((invers502456322omplex @ one_one_complex) = one_one_complex))). % inverse_1
thf(fact_21_inverse__zero, axiom,
    (((invers502456322omplex @ zero_zero_complex) = zero_zero_complex))). % inverse_zero
thf(fact_22_inverse__nonzero__iff__nonzero, axiom,
    ((![A : complex]: (((invers502456322omplex @ A) = zero_zero_complex) = (A = zero_zero_complex))))). % inverse_nonzero_iff_nonzero
thf(fact_23_inverse__eq__iff__eq, axiom,
    ((![A : complex, B : complex]: (((invers502456322omplex @ A) = (invers502456322omplex @ B)) = (A = B))))). % inverse_eq_iff_eq
thf(fact_24_inverse__inverse__eq, axiom,
    ((![A : complex]: ((invers502456322omplex @ (invers502456322omplex @ A)) = A)))). % inverse_inverse_eq
thf(fact_25_inverse__mult__distrib, axiom,
    ((![A : complex, B : complex]: ((invers502456322omplex @ (times_times_complex @ A @ B)) = (times_times_complex @ (invers502456322omplex @ A) @ (invers502456322omplex @ B)))))). % inverse_mult_distrib
thf(fact_26_poly__mult, axiom,
    ((![P : poly_complex, Q : poly_complex, X2 : complex]: ((poly_complex2 @ (times_1246143675omplex @ P @ Q) @ X2) = (times_times_complex @ (poly_complex2 @ P @ X2) @ (poly_complex2 @ Q @ X2)))))). % poly_mult
thf(fact_27_poly__mult, axiom,
    ((![P : poly_nat, Q : poly_nat, X2 : nat]: ((poly_nat2 @ (times_times_poly_nat @ P @ Q) @ X2) = (times_times_nat @ (poly_nat2 @ P @ X2) @ (poly_nat2 @ Q @ X2)))))). % poly_mult
thf(fact_28_smult__smult, axiom,
    ((![A : complex, B : complex, P : poly_complex]: ((smult_complex @ A @ (smult_complex @ B @ P)) = (smult_complex @ (times_times_complex @ A @ B) @ P))))). % smult_smult
thf(fact_29_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_30_smult__0__right, axiom,
    ((![A : complex]: ((smult_complex @ A @ zero_z1746442943omplex) = zero_z1746442943omplex)))). % smult_0_right
thf(fact_31_smult__0__right, axiom,
    ((![A : nat]: ((smult_nat @ A @ zero_zero_poly_nat) = zero_zero_poly_nat)))). % smult_0_right
thf(fact_32_psize__eq__0__iff, axiom,
    ((![P : poly_complex]: (((fundam1709708056omplex @ P) = zero_zero_nat) = (P = zero_z1746442943omplex))))). % psize_eq_0_iff
thf(fact_33_less_Oprems, axiom,
    ((~ ((fundam1158420650omplex @ (poly_complex2 @ pa)))))). % less.prems
thf(fact_34_q_I1_J, axiom,
    (((fundam1709708056omplex @ q) = (fundam1709708056omplex @ pa)))). % q(1)
thf(fact_35_poly__smult, axiom,
    ((![A : complex, P : poly_complex, X2 : complex]: ((poly_complex2 @ (smult_complex @ A @ P) @ X2) = (times_times_complex @ A @ (poly_complex2 @ P @ X2)))))). % poly_smult
thf(fact_36_poly__smult, axiom,
    ((![A : nat, P : poly_nat, X2 : nat]: ((poly_nat2 @ (smult_nat @ A @ P) @ X2) = (times_times_nat @ A @ (poly_nat2 @ P @ X2)))))). % poly_smult
thf(fact_37_right__inverse, axiom,
    ((![A : complex]: ((~ ((A = zero_zero_complex))) => ((times_times_complex @ A @ (invers502456322omplex @ A)) = one_one_complex))))). % right_inverse
thf(fact_38_left__inverse, axiom,
    ((![A : complex]: ((~ ((A = zero_zero_complex))) => ((times_times_complex @ (invers502456322omplex @ A) @ A) = one_one_complex))))). % left_inverse
thf(fact_39_constant__def, axiom,
    ((fundam1158420650omplex = (^[F : complex > complex]: (![X3 : complex]: (![Y2 : complex]: ((F @ X3) = (F @ Y2)))))))). % constant_def
thf(fact_40_mult__commute__imp__mult__inverse__commute, axiom,
    ((![Y3 : complex, X2 : complex]: (((times_times_complex @ Y3 @ X2) = (times_times_complex @ X2 @ Y3)) => ((times_times_complex @ (invers502456322omplex @ Y3) @ X2) = (times_times_complex @ X2 @ (invers502456322omplex @ Y3))))))). % mult_commute_imp_mult_inverse_commute
thf(fact_41_inverse__eq__imp__eq, axiom,
    ((![A : complex, B : complex]: (((invers502456322omplex @ A) = (invers502456322omplex @ B)) => (A = B))))). % inverse_eq_imp_eq
thf(fact_42_nonzero__inverse__mult__distrib, axiom,
    ((![A : complex, B : complex]: ((~ ((A = zero_zero_complex))) => ((~ ((B = zero_zero_complex))) => ((invers502456322omplex @ (times_times_complex @ A @ B)) = (times_times_complex @ (invers502456322omplex @ B) @ (invers502456322omplex @ A)))))))). % nonzero_inverse_mult_distrib
thf(fact_43_inverse__unique, axiom,
    ((![A : complex, B : complex]: (((times_times_complex @ A @ B) = one_one_complex) => ((invers502456322omplex @ A) = B))))). % inverse_unique
thf(fact_44_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_45_field__class_Ofield__inverse, axiom,
    ((![A : complex]: ((~ ((A = zero_zero_complex))) => ((times_times_complex @ (invers502456322omplex @ A) @ A) = one_one_complex))))). % field_class.field_inverse
thf(fact_46_nonzero__imp__inverse__nonzero, axiom,
    ((![A : complex]: ((~ ((A = zero_zero_complex))) => (~ (((invers502456322omplex @ A) = zero_zero_complex))))))). % nonzero_imp_inverse_nonzero
thf(fact_47_nonzero__inverse__inverse__eq, axiom,
    ((![A : complex]: ((~ ((A = zero_zero_complex))) => ((invers502456322omplex @ (invers502456322omplex @ A)) = A))))). % nonzero_inverse_inverse_eq
thf(fact_48_nonzero__inverse__eq__imp__eq, axiom,
    ((![A : complex, B : complex]: (((invers502456322omplex @ A) = (invers502456322omplex @ B)) => ((~ ((A = zero_zero_complex))) => ((~ ((B = zero_zero_complex))) => (A = B))))))). % nonzero_inverse_eq_imp_eq
thf(fact_49_inverse__zero__imp__zero, axiom,
    ((![A : complex]: (((invers502456322omplex @ A) = zero_zero_complex) => (A = zero_zero_complex))))). % inverse_zero_imp_zero
thf(fact_50_field__class_Ofield__inverse__zero, axiom,
    (((invers502456322omplex @ zero_zero_complex) = zero_zero_complex))). % field_class.field_inverse_zero
thf(fact_51_poly__all__0__iff__0, axiom,
    ((![P : poly_complex]: ((![X3 : complex]: ((poly_complex2 @ P @ X3) = zero_zero_complex)) = (P = zero_z1746442943omplex))))). % poly_all_0_iff_0
thf(fact_52_mult__cancel__right2, axiom,
    ((![A : complex, C : complex]: (((times_times_complex @ A @ C) = C) = (((C = zero_zero_complex)) | ((A = one_one_complex))))))). % mult_cancel_right2
thf(fact_53_mult__cancel__right1, axiom,
    ((![C : complex, B : complex]: ((C = (times_times_complex @ B @ C)) = (((C = zero_zero_complex)) | ((B = one_one_complex))))))). % mult_cancel_right1
thf(fact_54_mult__cancel__left2, axiom,
    ((![C : complex, A : complex]: (((times_times_complex @ C @ A) = C) = (((C = zero_zero_complex)) | ((A = one_one_complex))))))). % mult_cancel_left2
thf(fact_55_mult__cancel__left1, axiom,
    ((![C : complex, B : complex]: ((C = (times_times_complex @ C @ B)) = (((C = zero_zero_complex)) | ((B = one_one_complex))))))). % mult_cancel_left1
thf(fact_56_mult_Oright__neutral, axiom,
    ((![A : complex]: ((times_times_complex @ A @ one_one_complex) = A)))). % mult.right_neutral
thf(fact_57_mult_Oright__neutral, axiom,
    ((![A : nat]: ((times_times_nat @ A @ one_one_nat) = A)))). % mult.right_neutral
thf(fact_58_mult_Oleft__neutral, axiom,
    ((![A : complex]: ((times_times_complex @ one_one_complex @ A) = A)))). % mult.left_neutral
thf(fact_59_mult_Oleft__neutral, axiom,
    ((![A : nat]: ((times_times_nat @ one_one_nat @ A) = A)))). % mult.left_neutral
thf(fact_60_mult__cancel__right, axiom,
    ((![A : complex, C : complex, B : complex]: (((times_times_complex @ A @ C) = (times_times_complex @ B @ C)) = (((C = zero_zero_complex)) | ((A = B))))))). % mult_cancel_right
thf(fact_61_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_62_mult__cancel__left, axiom,
    ((![C : complex, A : complex, B : complex]: (((times_times_complex @ C @ A) = (times_times_complex @ C @ B)) = (((C = zero_zero_complex)) | ((A = B))))))). % mult_cancel_left
thf(fact_63_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_64_mult__eq__0__iff, axiom,
    ((![A : complex, B : complex]: (((times_times_complex @ A @ B) = zero_zero_complex) = (((A = zero_zero_complex)) | ((B = zero_zero_complex))))))). % mult_eq_0_iff
thf(fact_65_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_66_False, axiom,
    ((~ (((poly_complex2 @ pa @ c) = zero_zero_complex))))). % False
thf(fact_67_mult__zero__left, axiom,
    ((![A : complex]: ((times_times_complex @ zero_zero_complex @ A) = zero_zero_complex)))). % mult_zero_left
thf(fact_68_mult__zero__left, axiom,
    ((![A : nat]: ((times_times_nat @ zero_zero_nat @ A) = zero_zero_nat)))). % mult_zero_left
thf(fact_69_mult__zero__right, axiom,
    ((![A : complex]: ((times_times_complex @ A @ zero_zero_complex) = zero_zero_complex)))). % mult_zero_right
thf(fact_70_mult__zero__right, axiom,
    ((![A : nat]: ((times_times_nat @ A @ zero_zero_nat) = zero_zero_nat)))). % mult_zero_right
thf(fact_71_mult__smult__left, axiom,
    ((![A : complex, P : poly_complex, Q : poly_complex]: ((times_1246143675omplex @ (smult_complex @ A @ P) @ Q) = (smult_complex @ A @ (times_1246143675omplex @ P @ Q)))))). % mult_smult_left
thf(fact_72_mult__smult__left, axiom,
    ((![A : nat, P : poly_nat, Q : poly_nat]: ((times_times_poly_nat @ (smult_nat @ A @ P) @ Q) = (smult_nat @ A @ (times_times_poly_nat @ P @ Q)))))). % mult_smult_left
thf(fact_73_mult__smult__right, axiom,
    ((![P : poly_complex, A : complex, Q : poly_complex]: ((times_1246143675omplex @ P @ (smult_complex @ A @ Q)) = (smult_complex @ A @ (times_1246143675omplex @ P @ Q)))))). % mult_smult_right
thf(fact_74_mult__smult__right, axiom,
    ((![P : poly_nat, A : nat, Q : poly_nat]: ((times_times_poly_nat @ P @ (smult_nat @ A @ Q)) = (smult_nat @ A @ (times_times_poly_nat @ P @ Q)))))). % mult_smult_right
thf(fact_75_zero__reorient, axiom,
    ((![X2 : complex]: ((zero_zero_complex = X2) = (X2 = zero_zero_complex))))). % zero_reorient
thf(fact_76_zero__reorient, axiom,
    ((![X2 : nat]: ((zero_zero_nat = X2) = (X2 = zero_zero_nat))))). % zero_reorient
thf(fact_77_mult_Oleft__commute, axiom,
    ((![B : complex, A : complex, C : complex]: ((times_times_complex @ B @ (times_times_complex @ A @ C)) = (times_times_complex @ A @ (times_times_complex @ B @ C)))))). % mult.left_commute
thf(fact_78_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_79_mult_Ocommute, axiom,
    ((times_times_complex = (^[A2 : complex]: (^[B2 : complex]: (times_times_complex @ B2 @ A2)))))). % mult.commute
thf(fact_80_mult_Ocommute, axiom,
    ((times_times_nat = (^[A2 : nat]: (^[B2 : nat]: (times_times_nat @ B2 @ A2)))))). % mult.commute
thf(fact_81_mult_Oassoc, axiom,
    ((![A : complex, B : complex, C : complex]: ((times_times_complex @ (times_times_complex @ A @ B) @ C) = (times_times_complex @ A @ (times_times_complex @ B @ C)))))). % mult.assoc
thf(fact_82_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_83_ab__semigroup__mult__class_Omult__ac_I1_J, axiom,
    ((![A : complex, B : complex, C : complex]: ((times_times_complex @ (times_times_complex @ A @ B) @ C) = (times_times_complex @ A @ (times_times_complex @ B @ C)))))). % ab_semigroup_mult_class.mult_ac(1)
thf(fact_84_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_85_one__reorient, axiom,
    ((![X2 : complex]: ((one_one_complex = X2) = (X2 = one_one_complex))))). % one_reorient
thf(fact_86_one__reorient, axiom,
    ((![X2 : nat]: ((one_one_nat = X2) = (X2 = one_one_nat))))). % one_reorient
thf(fact_87_mult__not__zero, axiom,
    ((![A : complex, B : complex]: ((~ (((times_times_complex @ A @ B) = zero_zero_complex))) => ((~ ((A = zero_zero_complex))) & (~ ((B = zero_zero_complex)))))))). % mult_not_zero
thf(fact_88_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_89_divisors__zero, axiom,
    ((![A : complex, B : complex]: (((times_times_complex @ A @ B) = zero_zero_complex) => ((A = zero_zero_complex) | (B = zero_zero_complex)))))). % divisors_zero
thf(fact_90_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_91_no__zero__divisors, axiom,
    ((![A : complex, B : complex]: ((~ ((A = zero_zero_complex))) => ((~ ((B = zero_zero_complex))) => (~ (((times_times_complex @ A @ B) = zero_zero_complex)))))))). % no_zero_divisors
thf(fact_92_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_93_mult__left__cancel, axiom,
    ((![C : complex, A : complex, B : complex]: ((~ ((C = zero_zero_complex))) => (((times_times_complex @ C @ A) = (times_times_complex @ 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_mult__right__cancel, axiom,
    ((![C : complex, A : complex, B : complex]: ((~ ((C = zero_zero_complex))) => (((times_times_complex @ A @ C) = (times_times_complex @ B @ C)) = (A = B)))))). % mult_right_cancel
thf(fact_96_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_97_zero__neq__one, axiom,
    ((~ ((zero_zero_complex = one_one_complex))))). % zero_neq_one
thf(fact_98_zero__neq__one, axiom,
    ((~ ((zero_zero_nat = one_one_nat))))). % zero_neq_one
thf(fact_99_comm__monoid__mult__class_Omult__1, axiom,
    ((![A : complex]: ((times_times_complex @ one_one_complex @ A) = A)))). % comm_monoid_mult_class.mult_1
thf(fact_100_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_101_mult_Ocomm__neutral, axiom,
    ((![A : complex]: ((times_times_complex @ A @ one_one_complex) = A)))). % mult.comm_neutral
thf(fact_102_mult_Ocomm__neutral, axiom,
    ((![A : nat]: ((times_times_nat @ A @ one_one_nat) = A)))). % mult.comm_neutral
thf(fact_103_pqc0, axiom,
    (((poly_complex2 @ pa @ c) = (poly_complex2 @ q @ zero_zero_complex)))). % pqc0
thf(fact_104_less_Ohyps, axiom,
    ((![P : poly_complex]: ((ord_less_nat @ (fundam1709708056omplex @ P) @ (fundam1709708056omplex @ pa)) => ((~ ((fundam1158420650omplex @ (poly_complex2 @ P)))) => (?[Z2 : complex]: ((poly_complex2 @ P @ Z2) = zero_zero_complex))))))). % less.hyps
thf(fact_105_q_I2_J, axiom,
    ((![X4 : complex]: ((poly_complex2 @ q @ X4) = (poly_complex2 @ pa @ (plus_plus_complex @ c @ X4)))))). % q(2)
thf(fact_106_mult__is__0, axiom,
    ((![M : nat, N : nat]: (((times_times_nat @ M @ N) = zero_zero_nat) = (((M = zero_zero_nat)) | ((N = zero_zero_nat))))))). % mult_is_0
thf(fact_107_add__right__cancel, axiom,
    ((![B : complex, A : complex, C : complex]: (((plus_plus_complex @ B @ A) = (plus_plus_complex @ C @ A)) = (B = C))))). % add_right_cancel
thf(fact_108_add__left__cancel, axiom,
    ((![A : complex, B : complex, C : complex]: (((plus_plus_complex @ A @ B) = (plus_plus_complex @ A @ C)) = (B = C))))). % add_left_cancel
thf(fact_109_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
thf(fact_110_nat__mult__eq__1__iff, axiom,
    ((![M : nat, N : nat]: (((times_times_nat @ M @ N) = one_one_nat) = (((M = one_one_nat)) & ((N = one_one_nat))))))). % nat_mult_eq_1_iff
thf(fact_111_not__gr__zero, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr_zero
thf(fact_112_zero__eq__add__iff__both__eq__0, axiom,
    ((![X2 : nat, Y3 : nat]: ((zero_zero_nat = (plus_plus_nat @ X2 @ Y3)) = (((X2 = zero_zero_nat)) & ((Y3 = zero_zero_nat))))))). % zero_eq_add_iff_both_eq_0
thf(fact_113_add__eq__0__iff__both__eq__0, axiom,
    ((![X2 : nat, Y3 : nat]: (((plus_plus_nat @ X2 @ Y3) = zero_zero_nat) = (((X2 = zero_zero_nat)) & ((Y3 = zero_zero_nat))))))). % add_eq_0_iff_both_eq_0
thf(fact_114_add__cancel__right__right, axiom,
    ((![A : complex, B : complex]: ((A = (plus_plus_complex @ A @ B)) = (B = zero_zero_complex))))). % add_cancel_right_right
thf(fact_115_add__cancel__right__right, axiom,
    ((![A : nat, B : nat]: ((A = (plus_plus_nat @ A @ B)) = (B = zero_zero_nat))))). % add_cancel_right_right
thf(fact_116_add__cancel__right__left, axiom,
    ((![A : complex, B : complex]: ((A = (plus_plus_complex @ B @ A)) = (B = zero_zero_complex))))). % add_cancel_right_left
thf(fact_117_add__cancel__right__left, axiom,
    ((![A : nat, B : nat]: ((A = (plus_plus_nat @ B @ A)) = (B = zero_zero_nat))))). % add_cancel_right_left
thf(fact_118_add__cancel__left__right, axiom,
    ((![A : complex, B : complex]: (((plus_plus_complex @ A @ B) = A) = (B = zero_zero_complex))))). % add_cancel_left_right
thf(fact_119_add__cancel__left__right, axiom,
    ((![A : nat, B : nat]: (((plus_plus_nat @ A @ B) = A) = (B = zero_zero_nat))))). % add_cancel_left_right
thf(fact_120_add__cancel__left__left, axiom,
    ((![B : complex, A : complex]: (((plus_plus_complex @ B @ A) = A) = (B = zero_zero_complex))))). % add_cancel_left_left
thf(fact_121_add__cancel__left__left, axiom,
    ((![B : nat, A : nat]: (((plus_plus_nat @ B @ A) = A) = (B = zero_zero_nat))))). % add_cancel_left_left
thf(fact_122_add_Oright__neutral, axiom,
    ((![A : complex]: ((plus_plus_complex @ A @ zero_zero_complex) = A)))). % add.right_neutral
thf(fact_123_add_Oright__neutral, axiom,
    ((![A : nat]: ((plus_plus_nat @ A @ zero_zero_nat) = A)))). % add.right_neutral
thf(fact_124_add_Oleft__neutral, axiom,
    ((![A : complex]: ((plus_plus_complex @ zero_zero_complex @ A) = A)))). % add.left_neutral
thf(fact_125_add_Oleft__neutral, axiom,
    ((![A : nat]: ((plus_plus_nat @ zero_zero_nat @ A) = A)))). % add.left_neutral
thf(fact_126_add__less__cancel__right, axiom,
    ((![A : nat, C : nat, B : nat]: ((ord_less_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ C)) = (ord_less_nat @ A @ B))))). % add_less_cancel_right
thf(fact_127_add__less__cancel__left, axiom,
    ((![C : nat, A : nat, B : nat]: ((ord_less_nat @ (plus_plus_nat @ C @ A) @ (plus_plus_nat @ C @ B)) = (ord_less_nat @ A @ B))))). % add_less_cancel_left
thf(fact_128__092_060open_062_092_060exists_062q_O_Apsize_Aq_A_061_Apsize_Ap_A_092_060and_062_A_I_092_060forall_062x_O_Apoly_Aq_Ax_A_061_Apoly_Ap_A_Ic_A_L_Ax_J_J_092_060close_062, axiom,
    ((?[Q2 : poly_complex]: (((fundam1709708056omplex @ Q2) = (fundam1709708056omplex @ pa)) & (![X4 : complex]: ((poly_complex2 @ Q2 @ X4) = (poly_complex2 @ pa @ (plus_plus_complex @ c @ X4)))))))). % \<open>\<exists>q. psize q = psize p \<and> (\<forall>x. poly q x = poly p (c + x))\<close>
thf(fact_129__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062q_O_A_092_060lbrakk_062psize_Aq_A_061_Apsize_Ap_059_A_092_060forall_062x_O_Apoly_Aq_Ax_A_061_Apoly_Ap_A_Ic_A_L_Ax_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom,
    ((~ ((![Q2 : poly_complex]: (((fundam1709708056omplex @ Q2) = (fundam1709708056omplex @ pa)) => (~ ((![X4 : complex]: ((poly_complex2 @ Q2 @ X4) = (poly_complex2 @ pa @ (plus_plus_complex @ c @ X4)))))))))))). % \<open>\<And>thesis. (\<And>q. \<lbrakk>psize q = psize p; \<forall>x. poly q x = poly p (c + x)\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_130_less__one, axiom,
    ((![N : nat]: ((ord_less_nat @ N @ one_one_nat) = (N = zero_zero_nat))))). % less_one
thf(fact_131_neq0__conv, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ N))))). % neq0_conv
thf(fact_132_less__nat__zero__code, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_nat_zero_code
thf(fact_133_bot__nat__0_Onot__eq__extremum, axiom,
    ((![A : nat]: ((~ ((A = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ A))))). % bot_nat_0.not_eq_extremum
thf(fact_134_mult__cancel2, axiom,
    ((![M : nat, K : nat, N : nat]: (((times_times_nat @ M @ K) = (times_times_nat @ N @ K)) = (((M = N)) | ((K = zero_zero_nat))))))). % mult_cancel2
thf(fact_135_mult__cancel1, axiom,
    ((![K : nat, M : nat, N : nat]: (((times_times_nat @ K @ M) = (times_times_nat @ K @ N)) = (((M = N)) | ((K = zero_zero_nat))))))). % mult_cancel1
thf(fact_136_mult__0__right, axiom,
    ((![M : nat]: ((times_times_nat @ M @ zero_zero_nat) = zero_zero_nat)))). % mult_0_right
thf(fact_137_poly__add, axiom,
    ((![P : poly_complex, Q : poly_complex, X2 : complex]: ((poly_complex2 @ (plus_p1547158847omplex @ P @ Q) @ X2) = (plus_plus_complex @ (poly_complex2 @ P @ X2) @ (poly_complex2 @ Q @ X2)))))). % poly_add
thf(fact_138_add__less__same__cancel1, axiom,
    ((![B : nat, A : nat]: ((ord_less_nat @ (plus_plus_nat @ B @ A) @ B) = (ord_less_nat @ A @ zero_zero_nat))))). % add_less_same_cancel1
thf(fact_139_add__less__same__cancel2, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ (plus_plus_nat @ A @ B) @ B) = (ord_less_nat @ A @ zero_zero_nat))))). % add_less_same_cancel2
thf(fact_140_less__add__same__cancel1, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ (plus_plus_nat @ A @ B)) = (ord_less_nat @ zero_zero_nat @ B))))). % less_add_same_cancel1
thf(fact_141_less__add__same__cancel2, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ (plus_plus_nat @ B @ A)) = (ord_less_nat @ zero_zero_nat @ B))))). % less_add_same_cancel2
thf(fact_142_mult__less__cancel2, axiom,
    ((![M : nat, K : nat, N : nat]: ((ord_less_nat @ (times_times_nat @ M @ K) @ (times_times_nat @ N @ K)) = (((ord_less_nat @ zero_zero_nat @ K)) & ((ord_less_nat @ M @ N))))))). % mult_less_cancel2
thf(fact_143_nat__0__less__mult__iff, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ (times_times_nat @ M @ N)) = (((ord_less_nat @ zero_zero_nat @ M)) & ((ord_less_nat @ zero_zero_nat @ N))))))). % nat_0_less_mult_iff
thf(fact_144_gr0I, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr0I
thf(fact_145_not__gr0, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr0
thf(fact_146_not__less0, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less0
thf(fact_147_less__zeroE, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_zeroE
thf(fact_148_nat__mult__1, axiom,
    ((![N : nat]: ((times_times_nat @ one_one_nat @ N) = N)))). % nat_mult_1
thf(fact_149_gr__implies__not0, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not0
thf(fact_150_mult__less__mono1, axiom,
    ((![I : nat, J : nat, K : nat]: ((ord_less_nat @ I @ J) => ((ord_less_nat @ zero_zero_nat @ K) => (ord_less_nat @ (times_times_nat @ I @ K) @ (times_times_nat @ J @ K))))))). % mult_less_mono1
thf(fact_151_mult__less__mono2, axiom,
    ((![I : nat, J : nat, K : nat]: ((ord_less_nat @ I @ J) => ((ord_less_nat @ zero_zero_nat @ K) => (ord_less_nat @ (times_times_nat @ K @ I) @ (times_times_nat @ K @ J))))))). % mult_less_mono2
thf(fact_152_nat__mult__1__right, axiom,
    ((![N : nat]: ((times_times_nat @ N @ one_one_nat) = N)))). % nat_mult_1_right
thf(fact_153_infinite__descent0, axiom,
    ((![P2 : nat > $o, N : nat]: ((P2 @ zero_zero_nat) => ((![N2 : nat]: ((ord_less_nat @ zero_zero_nat @ N2) => ((~ ((P2 @ N2))) => (?[M2 : nat]: ((ord_less_nat @ M2 @ N2) & (~ ((P2 @ M2)))))))) => (P2 @ N)))))). % infinite_descent0
thf(fact_154_bot__nat__0_Oextremum__strict, axiom,
    ((![A : nat]: (~ ((ord_less_nat @ A @ zero_zero_nat)))))). % bot_nat_0.extremum_strict
thf(fact_155_less__add__one, axiom,
    ((![A : nat]: (ord_less_nat @ A @ (plus_plus_nat @ A @ one_one_nat))))). % less_add_one
thf(fact_156_add__neg__neg, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ zero_zero_nat) => ((ord_less_nat @ B @ zero_zero_nat) => (ord_less_nat @ (plus_plus_nat @ A @ B) @ zero_zero_nat)))))). % add_neg_neg
thf(fact_157_add__pos__pos, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ zero_zero_nat @ A) => ((ord_less_nat @ zero_zero_nat @ B) => (ord_less_nat @ zero_zero_nat @ (plus_plus_nat @ A @ B))))))). % add_pos_pos
thf(fact_158_canonically__ordered__monoid__add__class_OlessE, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (~ ((![C2 : nat]: ((B = (plus_plus_nat @ A @ C2)) => (C2 = zero_zero_nat))))))))). % canonically_ordered_monoid_add_class.lessE
thf(fact_159_add__mono1, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (ord_less_nat @ (plus_plus_nat @ A @ one_one_nat) @ (plus_plus_nat @ B @ one_one_nat)))))). % add_mono1
thf(fact_160_pos__add__strict, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_nat @ zero_zero_nat @ A) => ((ord_less_nat @ B @ C) => (ord_less_nat @ B @ (plus_plus_nat @ A @ C))))))). % pos_add_strict
thf(fact_161_add__less__imp__less__right, axiom,
    ((![A : nat, C : nat, B : nat]: ((ord_less_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ C)) => (ord_less_nat @ A @ B))))). % add_less_imp_less_right
thf(fact_162_add__less__imp__less__left, axiom,
    ((![C : nat, A : nat, B : nat]: ((ord_less_nat @ (plus_plus_nat @ C @ A) @ (plus_plus_nat @ C @ B)) => (ord_less_nat @ A @ B))))). % add_less_imp_less_left
thf(fact_163_add__strict__right__mono, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_nat @ A @ B) => (ord_less_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ C)))))). % add_strict_right_mono
thf(fact_164_add__strict__left__mono, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_nat @ A @ B) => (ord_less_nat @ (plus_plus_nat @ C @ A) @ (plus_plus_nat @ C @ B)))))). % add_strict_left_mono
thf(fact_165_add__strict__mono, axiom,
    ((![A : nat, B : nat, C : nat, D : nat]: ((ord_less_nat @ A @ B) => ((ord_less_nat @ C @ D) => (ord_less_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ D))))))). % add_strict_mono
thf(fact_166_add__right__imp__eq, axiom,
    ((![B : complex, A : complex, C : complex]: (((plus_plus_complex @ B @ A) = (plus_plus_complex @ C @ A)) => (B = C))))). % add_right_imp_eq
thf(fact_167_add__left__imp__eq, axiom,
    ((![A : complex, B : complex, C : complex]: (((plus_plus_complex @ A @ B) = (plus_plus_complex @ A @ C)) => (B = C))))). % add_left_imp_eq
thf(fact_168_add_Oleft__commute, axiom,
    ((![B : complex, A : complex, C : complex]: ((plus_plus_complex @ B @ (plus_plus_complex @ A @ C)) = (plus_plus_complex @ A @ (plus_plus_complex @ B @ C)))))). % add.left_commute
thf(fact_169_add_Ocommute, axiom,
    ((plus_plus_complex = (^[A2 : complex]: (^[B2 : complex]: (plus_plus_complex @ B2 @ A2)))))). % add.commute

% Conjectures (1)
thf(conj_0, conjecture,
    (((poly_complex2 @ (smult_complex @ (invers502456322omplex @ (poly_complex2 @ q @ zero_zero_complex)) @ q) @ zero_zero_complex) = one_one_complex))).
