% 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_24__5366868_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:26:21.411

% Could-be-implicit typings (7)
thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J, type,
    poly_poly_real : $tType).
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__Real__Oreal_J, type,
    poly_real : $tType).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J, type,
    set_real : $tType).
thf(ty_n_t__Polynomial__Opoly_Itf__a_J, type,
    poly_a : $tType).
thf(ty_n_t__Real__Oreal, type,
    real : $tType).
thf(ty_n_tf__a, type,
    a : $tType).

% Explicit typings (20)
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Real__Oreal_J_J, type,
    zero_z1423781445y_real : poly_poly_real).
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_It__Real__Oreal_J, type,
    zero_zero_poly_real : poly_real).
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_001t__Real__Oreal, type,
    zero_zero_real : real).
thf(sy_c_Groups_Ozero__class_Ozero_001tf__a, type,
    zero_zero_a : a).
thf(sy_c_Orderings_Oord__class_Oless_001t__Polynomial__Opoly_It__Real__Oreal_J, type,
    ord_less_poly_real : poly_real > poly_real > $o).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal, type,
    ord_less_real : real > real > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Polynomial__Opoly_It__Real__Oreal_J, type,
    ord_le1180086932y_real : poly_real > poly_real > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal, type,
    ord_less_eq_real : real > real > $o).
thf(sy_c_Polynomial_Opoly_001t__Polynomial__Opoly_It__Real__Oreal_J, type,
    poly_poly_real2 : poly_poly_real > poly_real > poly_real).
thf(sy_c_Polynomial_Opoly_001t__Polynomial__Opoly_Itf__a_J, type,
    poly_poly_a2 : poly_poly_a > poly_a > poly_a).
thf(sy_c_Polynomial_Opoly_001t__Real__Oreal, type,
    poly_real2 : poly_real > real > real).
thf(sy_c_Polynomial_Opoly_001tf__a, type,
    poly_a2 : poly_a > a > a).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Real__Oreal, type,
    real_V646646907m_real : real > real).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001tf__a, type,
    real_V1022479215norm_a : a > real).
thf(sy_c_Set_OCollect_001t__Real__Oreal, type,
    collect_real : (real > $o) > set_real).
thf(sy_c_member_001t__Real__Oreal, type,
    member_real : real > set_real > $o).
thf(sy_v_p, type,
    p : poly_a).
thf(sy_v_r, type,
    r : real).

% Relevant facts (146)
thf(fact_0_norm__le__zero__iff, axiom,
    ((![X : real]: ((ord_less_eq_real @ (real_V646646907m_real @ X) @ zero_zero_real) = (X = zero_zero_real))))). % norm_le_zero_iff
thf(fact_1_norm__le__zero__iff, axiom,
    ((![X : a]: ((ord_less_eq_real @ (real_V1022479215norm_a @ X) @ zero_zero_real) = (X = zero_zero_a))))). % norm_le_zero_iff
thf(fact_2_zero__less__norm__iff, axiom,
    ((![X : real]: ((ord_less_real @ zero_zero_real @ (real_V646646907m_real @ X)) = (~ ((X = zero_zero_real))))))). % zero_less_norm_iff
thf(fact_3_zero__less__norm__iff, axiom,
    ((![X : a]: ((ord_less_real @ zero_zero_real @ (real_V1022479215norm_a @ X)) = (~ ((X = zero_zero_a))))))). % zero_less_norm_iff
thf(fact_4_norm__zero, axiom,
    (((real_V646646907m_real @ zero_zero_real) = zero_zero_real))). % norm_zero
thf(fact_5_norm__zero, axiom,
    (((real_V1022479215norm_a @ zero_zero_a) = zero_zero_real))). % norm_zero
thf(fact_6_norm__eq__zero, axiom,
    ((![X : real]: (((real_V646646907m_real @ X) = zero_zero_real) = (X = zero_zero_real))))). % norm_eq_zero
thf(fact_7_norm__eq__zero, axiom,
    ((![X : a]: (((real_V1022479215norm_a @ X) = zero_zero_real) = (X = zero_zero_a))))). % norm_eq_zero
thf(fact_8_poly__0, axiom,
    ((![X : poly_real]: ((poly_poly_real2 @ zero_z1423781445y_real @ X) = zero_zero_poly_real)))). % poly_0
thf(fact_9_poly__0, axiom,
    ((![X : poly_a]: ((poly_poly_a2 @ zero_z2096148049poly_a @ X) = zero_zero_poly_a)))). % poly_0
thf(fact_10_poly__0, axiom,
    ((![X : a]: ((poly_a2 @ zero_zero_poly_a @ X) = zero_zero_a)))). % poly_0
thf(fact_11_poly__0, axiom,
    ((![X : real]: ((poly_real2 @ zero_zero_poly_real @ X) = zero_zero_real)))). % poly_0
thf(fact_12_norm__ge__zero, axiom,
    ((![X : real]: (ord_less_eq_real @ zero_zero_real @ (real_V646646907m_real @ X))))). % norm_ge_zero
thf(fact_13_norm__ge__zero, axiom,
    ((![X : a]: (ord_less_eq_real @ zero_zero_real @ (real_V1022479215norm_a @ X))))). % norm_ge_zero
thf(fact_14_norm__not__less__zero, axiom,
    ((![X : real]: (~ ((ord_less_real @ (real_V646646907m_real @ X) @ zero_zero_real)))))). % norm_not_less_zero
thf(fact_15_norm__not__less__zero, axiom,
    ((![X : a]: (~ ((ord_less_real @ (real_V1022479215norm_a @ X) @ zero_zero_real)))))). % norm_not_less_zero
thf(fact_16_order__refl, axiom,
    ((![X : real]: (ord_less_eq_real @ X @ X)))). % order_refl
thf(fact_17_poly__all__0__iff__0, axiom,
    ((![P : poly_real]: ((![X2 : real]: ((poly_real2 @ P @ X2) = zero_zero_real)) = (P = zero_zero_poly_real))))). % poly_all_0_iff_0
thf(fact_18_poly__all__0__iff__0, axiom,
    ((![P : poly_poly_real]: ((![X2 : poly_real]: ((poly_poly_real2 @ P @ X2) = zero_zero_poly_real)) = (P = zero_z1423781445y_real))))). % poly_all_0_iff_0
thf(fact_19_poly__IVT__pos, axiom,
    ((![A : real, B : real, P : poly_real]: ((ord_less_real @ A @ B) => ((ord_less_real @ (poly_real2 @ P @ A) @ zero_zero_real) => ((ord_less_real @ zero_zero_real @ (poly_real2 @ P @ B)) => (?[X3 : real]: ((ord_less_real @ A @ X3) & ((ord_less_real @ X3 @ B) & ((poly_real2 @ P @ X3) = zero_zero_real)))))))))). % poly_IVT_pos
thf(fact_20_poly__IVT__neg, axiom,
    ((![A : real, B : real, P : poly_real]: ((ord_less_real @ A @ B) => ((ord_less_real @ zero_zero_real @ (poly_real2 @ P @ A)) => ((ord_less_real @ (poly_real2 @ P @ B) @ zero_zero_real) => (?[X3 : real]: ((ord_less_real @ A @ X3) & ((ord_less_real @ X3 @ B) & ((poly_real2 @ P @ X3) = zero_zero_real)))))))))). % poly_IVT_neg
thf(fact_21_zero__reorient, axiom,
    ((![X : real]: ((zero_zero_real = X) = (X = zero_zero_real))))). % zero_reorient
thf(fact_22_zero__reorient, axiom,
    ((![X : a]: ((zero_zero_a = X) = (X = zero_zero_a))))). % zero_reorient
thf(fact_23_zero__reorient, axiom,
    ((![X : poly_real]: ((zero_zero_poly_real = X) = (X = zero_zero_poly_real))))). % zero_reorient
thf(fact_24_zero__reorient, axiom,
    ((![X : poly_a]: ((zero_zero_poly_a = X) = (X = zero_zero_poly_a))))). % zero_reorient
thf(fact_25_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_26_dual__order_Oeq__iff, axiom,
    (((^[Y : real]: (^[Z : real]: (Y = Z))) = (^[A2 : real]: (^[B2 : real]: (((ord_less_eq_real @ B2 @ A2)) & ((ord_less_eq_real @ A2 @ B2)))))))). % dual_order.eq_iff
thf(fact_27_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_28_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_29_dual__order_Orefl, axiom,
    ((![A : real]: (ord_less_eq_real @ A @ A)))). % dual_order.refl
thf(fact_30_order__trans, axiom,
    ((![X : real, Y2 : real, Z2 : real]: ((ord_less_eq_real @ X @ Y2) => ((ord_less_eq_real @ Y2 @ Z2) => (ord_less_eq_real @ X @ Z2)))))). % order_trans
thf(fact_31_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_32_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_33_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_34_order__class_Oorder_Oeq__iff, axiom,
    (((^[Y : real]: (^[Z : real]: (Y = Z))) = (^[A2 : real]: (^[B2 : real]: (((ord_less_eq_real @ A2 @ B2)) & ((ord_less_eq_real @ B2 @ A2)))))))). % order_class.order.eq_iff
thf(fact_35_antisym__conv, axiom,
    ((![Y2 : real, X : real]: ((ord_less_eq_real @ Y2 @ X) => ((ord_less_eq_real @ X @ Y2) = (X = Y2)))))). % antisym_conv
thf(fact_36_le__cases3, axiom,
    ((![X : real, Y2 : real, Z2 : real]: (((ord_less_eq_real @ X @ Y2) => (~ ((ord_less_eq_real @ Y2 @ Z2)))) => (((ord_less_eq_real @ Y2 @ X) => (~ ((ord_less_eq_real @ X @ Z2)))) => (((ord_less_eq_real @ X @ Z2) => (~ ((ord_less_eq_real @ Z2 @ Y2)))) => (((ord_less_eq_real @ Z2 @ Y2) => (~ ((ord_less_eq_real @ Y2 @ X)))) => (((ord_less_eq_real @ Y2 @ Z2) => (~ ((ord_less_eq_real @ Z2 @ X)))) => (~ (((ord_less_eq_real @ Z2 @ X) => (~ ((ord_less_eq_real @ X @ Y2)))))))))))))). % le_cases3
thf(fact_37_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_38_le__cases, axiom,
    ((![X : real, Y2 : real]: ((~ ((ord_less_eq_real @ X @ Y2))) => (ord_less_eq_real @ Y2 @ X))))). % le_cases
thf(fact_39_eq__refl, axiom,
    ((![X : real, Y2 : real]: ((X = Y2) => (ord_less_eq_real @ X @ Y2))))). % eq_refl
thf(fact_40_linear, axiom,
    ((![X : real, Y2 : real]: ((ord_less_eq_real @ X @ Y2) | (ord_less_eq_real @ Y2 @ X))))). % linear
thf(fact_41_antisym, axiom,
    ((![X : real, Y2 : real]: ((ord_less_eq_real @ X @ Y2) => ((ord_less_eq_real @ Y2 @ X) => (X = Y2)))))). % antisym
thf(fact_42_eq__iff, axiom,
    (((^[Y : real]: (^[Z : real]: (Y = Z))) = (^[X2 : real]: (^[Y3 : real]: (((ord_less_eq_real @ X2 @ Y3)) & ((ord_less_eq_real @ Y3 @ X2)))))))). % eq_iff
thf(fact_43_ord__le__eq__subst, axiom,
    ((![A : real, B : real, F : real > real, C : real]: ((ord_less_eq_real @ A @ B) => (((F @ B) = C) => ((![X3 : real, Y4 : real]: ((ord_less_eq_real @ X3 @ Y4) => (ord_less_eq_real @ (F @ X3) @ (F @ Y4)))) => (ord_less_eq_real @ (F @ A) @ C))))))). % ord_le_eq_subst
thf(fact_44_ord__eq__le__subst, axiom,
    ((![A : real, F : real > real, B : real, C : real]: ((A = (F @ B)) => ((ord_less_eq_real @ B @ C) => ((![X3 : real, Y4 : real]: ((ord_less_eq_real @ X3 @ Y4) => (ord_less_eq_real @ (F @ X3) @ (F @ Y4)))) => (ord_less_eq_real @ A @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_45_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) => ((![X3 : real, Y4 : real]: ((ord_less_eq_real @ X3 @ Y4) => (ord_less_eq_real @ (F @ X3) @ (F @ Y4)))) => (ord_less_eq_real @ (F @ A) @ C))))))). % order_subst2
thf(fact_46_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) => ((![X3 : real, Y4 : real]: ((ord_less_eq_real @ X3 @ Y4) => (ord_less_eq_real @ (F @ X3) @ (F @ Y4)))) => (ord_less_eq_real @ A @ (F @ C)))))))). % order_subst1
thf(fact_47_dual__order_Ostrict__implies__not__eq, axiom,
    ((![B : real, A : real]: ((ord_less_real @ B @ A) => (~ ((A = B))))))). % dual_order.strict_implies_not_eq
thf(fact_48_order_Ostrict__implies__not__eq, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ B) => (~ ((A = B))))))). % order.strict_implies_not_eq
thf(fact_49_not__less__iff__gr__or__eq, axiom,
    ((![X : real, Y2 : real]: ((~ ((ord_less_real @ X @ Y2))) = (((ord_less_real @ Y2 @ X)) | ((X = Y2))))))). % not_less_iff_gr_or_eq
thf(fact_50_dual__order_Ostrict__trans, axiom,
    ((![B : real, A : real, C : real]: ((ord_less_real @ B @ A) => ((ord_less_real @ C @ B) => (ord_less_real @ C @ A)))))). % dual_order.strict_trans
thf(fact_51_mem__Collect__eq, axiom,
    ((![A : real, P2 : real > $o]: ((member_real @ A @ (collect_real @ P2)) = (P2 @ A))))). % mem_Collect_eq
thf(fact_52_Collect__mem__eq, axiom,
    ((![A4 : set_real]: ((collect_real @ (^[X2 : real]: (member_real @ X2 @ A4))) = A4)))). % Collect_mem_eq
thf(fact_53_linorder__less__wlog, axiom,
    ((![P2 : real > real > $o, A : real, B : real]: ((![A3 : real, B3 : real]: ((ord_less_real @ A3 @ B3) => (P2 @ A3 @ B3))) => ((![A3 : real]: (P2 @ A3 @ A3)) => ((![A3 : real, B3 : real]: ((P2 @ B3 @ A3) => (P2 @ A3 @ B3))) => (P2 @ A @ B))))))). % linorder_less_wlog
thf(fact_54_less__imp__not__less, axiom,
    ((![X : real, Y2 : real]: ((ord_less_real @ X @ Y2) => (~ ((ord_less_real @ Y2 @ X))))))). % less_imp_not_less
thf(fact_55_order_Ostrict__trans, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_real @ A @ B) => ((ord_less_real @ B @ C) => (ord_less_real @ A @ C)))))). % order.strict_trans
thf(fact_56_dual__order_Oirrefl, axiom,
    ((![A : real]: (~ ((ord_less_real @ A @ A)))))). % dual_order.irrefl
thf(fact_57_linorder__cases, axiom,
    ((![X : real, Y2 : real]: ((~ ((ord_less_real @ X @ Y2))) => ((~ ((X = Y2))) => (ord_less_real @ Y2 @ X)))))). % linorder_cases
thf(fact_58_less__imp__triv, axiom,
    ((![X : real, Y2 : real, P2 : $o]: ((ord_less_real @ X @ Y2) => ((ord_less_real @ Y2 @ X) => P2))))). % less_imp_triv
thf(fact_59_less__imp__not__eq2, axiom,
    ((![X : real, Y2 : real]: ((ord_less_real @ X @ Y2) => (~ ((Y2 = X))))))). % less_imp_not_eq2
thf(fact_60_antisym__conv3, axiom,
    ((![Y2 : real, X : real]: ((~ ((ord_less_real @ Y2 @ X))) => ((~ ((ord_less_real @ X @ Y2))) = (X = Y2)))))). % antisym_conv3
thf(fact_61_less__not__sym, axiom,
    ((![X : real, Y2 : real]: ((ord_less_real @ X @ Y2) => (~ ((ord_less_real @ Y2 @ X))))))). % less_not_sym
thf(fact_62_less__imp__not__eq, axiom,
    ((![X : real, Y2 : real]: ((ord_less_real @ X @ Y2) => (~ ((X = Y2))))))). % less_imp_not_eq
thf(fact_63_dual__order_Oasym, axiom,
    ((![B : real, A : real]: ((ord_less_real @ B @ A) => (~ ((ord_less_real @ A @ B))))))). % dual_order.asym
thf(fact_64_ord__less__eq__trans, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_real @ A @ B) => ((B = C) => (ord_less_real @ A @ C)))))). % ord_less_eq_trans
thf(fact_65_ord__eq__less__trans, axiom,
    ((![A : real, B : real, C : real]: ((A = B) => ((ord_less_real @ B @ C) => (ord_less_real @ A @ C)))))). % ord_eq_less_trans
thf(fact_66_less__irrefl, axiom,
    ((![X : real]: (~ ((ord_less_real @ X @ X)))))). % less_irrefl
thf(fact_67_less__linear, axiom,
    ((![X : real, Y2 : real]: ((ord_less_real @ X @ Y2) | ((X = Y2) | (ord_less_real @ Y2 @ X)))))). % less_linear
thf(fact_68_less__trans, axiom,
    ((![X : real, Y2 : real, Z2 : real]: ((ord_less_real @ X @ Y2) => ((ord_less_real @ Y2 @ Z2) => (ord_less_real @ X @ Z2)))))). % less_trans
thf(fact_69_less__asym_H, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ B) => (~ ((ord_less_real @ B @ A))))))). % less_asym'
thf(fact_70_less__asym, axiom,
    ((![X : real, Y2 : real]: ((ord_less_real @ X @ Y2) => (~ ((ord_less_real @ Y2 @ X))))))). % less_asym
thf(fact_71_less__imp__neq, axiom,
    ((![X : real, Y2 : real]: ((ord_less_real @ X @ Y2) => (~ ((X = Y2))))))). % less_imp_neq
thf(fact_72_dense, axiom,
    ((![X : real, Y2 : real]: ((ord_less_real @ X @ Y2) => (?[Z3 : real]: ((ord_less_real @ X @ Z3) & (ord_less_real @ Z3 @ Y2))))))). % dense
thf(fact_73_order_Oasym, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ B) => (~ ((ord_less_real @ B @ A))))))). % order.asym
thf(fact_74_neq__iff, axiom,
    ((![X : real, Y2 : real]: ((~ ((X = Y2))) = (((ord_less_real @ X @ Y2)) | ((ord_less_real @ Y2 @ X))))))). % neq_iff
thf(fact_75_neqE, axiom,
    ((![X : real, Y2 : real]: ((~ ((X = Y2))) => ((~ ((ord_less_real @ X @ Y2))) => (ord_less_real @ Y2 @ X)))))). % neqE
thf(fact_76_gt__ex, axiom,
    ((![X : real]: (?[X_1 : real]: (ord_less_real @ X @ X_1))))). % gt_ex
thf(fact_77_lt__ex, axiom,
    ((![X : real]: (?[Y4 : real]: (ord_less_real @ Y4 @ X))))). % lt_ex
thf(fact_78_order__less__subst2, axiom,
    ((![A : real, B : real, F : real > real, C : real]: ((ord_less_real @ A @ B) => ((ord_less_real @ (F @ B) @ C) => ((![X3 : real, Y4 : real]: ((ord_less_real @ X3 @ Y4) => (ord_less_real @ (F @ X3) @ (F @ Y4)))) => (ord_less_real @ (F @ A) @ C))))))). % order_less_subst2
thf(fact_79_order__less__subst1, axiom,
    ((![A : real, F : real > real, B : real, C : real]: ((ord_less_real @ A @ (F @ B)) => ((ord_less_real @ B @ C) => ((![X3 : real, Y4 : real]: ((ord_less_real @ X3 @ Y4) => (ord_less_real @ (F @ X3) @ (F @ Y4)))) => (ord_less_real @ A @ (F @ C)))))))). % order_less_subst1
thf(fact_80_ord__less__eq__subst, axiom,
    ((![A : real, B : real, F : real > real, C : real]: ((ord_less_real @ A @ B) => (((F @ B) = C) => ((![X3 : real, Y4 : real]: ((ord_less_real @ X3 @ Y4) => (ord_less_real @ (F @ X3) @ (F @ Y4)))) => (ord_less_real @ (F @ A) @ C))))))). % ord_less_eq_subst
thf(fact_81_ord__eq__less__subst, axiom,
    ((![A : real, F : real > real, B : real, C : real]: ((A = (F @ B)) => ((ord_less_real @ B @ C) => ((![X3 : real, Y4 : real]: ((ord_less_real @ X3 @ Y4) => (ord_less_real @ (F @ X3) @ (F @ Y4)))) => (ord_less_real @ A @ (F @ C)))))))). % ord_eq_less_subst
thf(fact_82_poly__eq__poly__eq__iff, axiom,
    ((![P : poly_real, Q : poly_real]: (((poly_real2 @ P) = (poly_real2 @ Q)) = (P = Q))))). % poly_eq_poly_eq_iff
thf(fact_83_order_Onot__eq__order__implies__strict, axiom,
    ((![A : real, B : real]: ((~ ((A = B))) => ((ord_less_eq_real @ A @ B) => (ord_less_real @ A @ B)))))). % order.not_eq_order_implies_strict
thf(fact_84_dual__order_Ostrict__implies__order, axiom,
    ((![B : real, A : real]: ((ord_less_real @ B @ A) => (ord_less_eq_real @ B @ A))))). % dual_order.strict_implies_order
thf(fact_85_dual__order_Ostrict__iff__order, axiom,
    ((ord_less_real = (^[B2 : real]: (^[A2 : real]: (((ord_less_eq_real @ B2 @ A2)) & ((~ ((A2 = B2)))))))))). % dual_order.strict_iff_order
thf(fact_86_dual__order_Oorder__iff__strict, axiom,
    ((ord_less_eq_real = (^[B2 : real]: (^[A2 : real]: (((ord_less_real @ B2 @ A2)) | ((A2 = B2)))))))). % dual_order.order_iff_strict
thf(fact_87_order_Ostrict__implies__order, axiom,
    ((![A : real, B : real]: ((ord_less_real @ A @ B) => (ord_less_eq_real @ A @ B))))). % order.strict_implies_order
thf(fact_88_dense__le__bounded, axiom,
    ((![X : real, Y2 : real, Z2 : real]: ((ord_less_real @ X @ Y2) => ((![W : real]: ((ord_less_real @ X @ W) => ((ord_less_real @ W @ Y2) => (ord_less_eq_real @ W @ Z2)))) => (ord_less_eq_real @ Y2 @ Z2)))))). % dense_le_bounded
thf(fact_89_dense__ge__bounded, axiom,
    ((![Z2 : real, X : real, Y2 : real]: ((ord_less_real @ Z2 @ X) => ((![W : real]: ((ord_less_real @ Z2 @ W) => ((ord_less_real @ W @ X) => (ord_less_eq_real @ Y2 @ W)))) => (ord_less_eq_real @ Y2 @ Z2)))))). % dense_ge_bounded
thf(fact_90_dual__order_Ostrict__trans2, axiom,
    ((![B : real, A : real, C : real]: ((ord_less_real @ B @ A) => ((ord_less_eq_real @ C @ B) => (ord_less_real @ C @ A)))))). % dual_order.strict_trans2
thf(fact_91_dual__order_Ostrict__trans1, axiom,
    ((![B : real, A : real, C : real]: ((ord_less_eq_real @ B @ A) => ((ord_less_real @ C @ B) => (ord_less_real @ C @ A)))))). % dual_order.strict_trans1
thf(fact_92_order_Ostrict__iff__order, axiom,
    ((ord_less_real = (^[A2 : real]: (^[B2 : real]: (((ord_less_eq_real @ A2 @ B2)) & ((~ ((A2 = B2)))))))))). % order.strict_iff_order
thf(fact_93_order_Oorder__iff__strict, axiom,
    ((ord_less_eq_real = (^[A2 : real]: (^[B2 : real]: (((ord_less_real @ A2 @ B2)) | ((A2 = B2)))))))). % order.order_iff_strict
thf(fact_94_order_Ostrict__trans2, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_real @ A @ B) => ((ord_less_eq_real @ B @ C) => (ord_less_real @ A @ C)))))). % order.strict_trans2
thf(fact_95_order_Ostrict__trans1, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_real @ B @ C) => (ord_less_real @ A @ C)))))). % order.strict_trans1
thf(fact_96_not__le__imp__less, axiom,
    ((![Y2 : real, X : real]: ((~ ((ord_less_eq_real @ Y2 @ X))) => (ord_less_real @ X @ Y2))))). % not_le_imp_less
thf(fact_97_less__le__not__le, axiom,
    ((ord_less_real = (^[X2 : real]: (^[Y3 : real]: (((ord_less_eq_real @ X2 @ Y3)) & ((~ ((ord_less_eq_real @ Y3 @ X2)))))))))). % less_le_not_le
thf(fact_98_le__imp__less__or__eq, axiom,
    ((![X : real, Y2 : real]: ((ord_less_eq_real @ X @ Y2) => ((ord_less_real @ X @ Y2) | (X = Y2)))))). % le_imp_less_or_eq
thf(fact_99_le__less__linear, axiom,
    ((![X : real, Y2 : real]: ((ord_less_eq_real @ X @ Y2) | (ord_less_real @ Y2 @ X))))). % le_less_linear
thf(fact_100_dense__le, axiom,
    ((![Y2 : real, Z2 : real]: ((![X3 : real]: ((ord_less_real @ X3 @ Y2) => (ord_less_eq_real @ X3 @ Z2))) => (ord_less_eq_real @ Y2 @ Z2))))). % dense_le
thf(fact_101_dense__ge, axiom,
    ((![Z2 : real, Y2 : real]: ((![X3 : real]: ((ord_less_real @ Z2 @ X3) => (ord_less_eq_real @ Y2 @ X3))) => (ord_less_eq_real @ Y2 @ Z2))))). % dense_ge
thf(fact_102_less__le__trans, axiom,
    ((![X : real, Y2 : real, Z2 : real]: ((ord_less_real @ X @ Y2) => ((ord_less_eq_real @ Y2 @ Z2) => (ord_less_real @ X @ Z2)))))). % less_le_trans
thf(fact_103_le__less__trans, axiom,
    ((![X : real, Y2 : real, Z2 : real]: ((ord_less_eq_real @ X @ Y2) => ((ord_less_real @ Y2 @ Z2) => (ord_less_real @ X @ Z2)))))). % le_less_trans
thf(fact_104_less__imp__le, axiom,
    ((![X : real, Y2 : real]: ((ord_less_real @ X @ Y2) => (ord_less_eq_real @ X @ Y2))))). % less_imp_le
thf(fact_105_antisym__conv2, axiom,
    ((![X : real, Y2 : real]: ((ord_less_eq_real @ X @ Y2) => ((~ ((ord_less_real @ X @ Y2))) = (X = Y2)))))). % antisym_conv2
thf(fact_106_antisym__conv1, axiom,
    ((![X : real, Y2 : real]: ((~ ((ord_less_real @ X @ Y2))) => ((ord_less_eq_real @ X @ Y2) = (X = Y2)))))). % antisym_conv1
thf(fact_107_le__neq__trans, axiom,
    ((![A : real, B : real]: ((ord_less_eq_real @ A @ B) => ((~ ((A = B))) => (ord_less_real @ A @ B)))))). % le_neq_trans
thf(fact_108_not__less, axiom,
    ((![X : real, Y2 : real]: ((~ ((ord_less_real @ X @ Y2))) = (ord_less_eq_real @ Y2 @ X))))). % not_less
thf(fact_109_not__le, axiom,
    ((![X : real, Y2 : real]: ((~ ((ord_less_eq_real @ X @ Y2))) = (ord_less_real @ Y2 @ X))))). % not_le
thf(fact_110_order__less__le__subst2, axiom,
    ((![A : real, B : real, F : real > real, C : real]: ((ord_less_real @ A @ B) => ((ord_less_eq_real @ (F @ B) @ C) => ((![X3 : real, Y4 : real]: ((ord_less_real @ X3 @ Y4) => (ord_less_real @ (F @ X3) @ (F @ Y4)))) => (ord_less_real @ (F @ A) @ C))))))). % order_less_le_subst2
thf(fact_111_order__less__le__subst1, axiom,
    ((![A : real, F : real > real, B : real, C : real]: ((ord_less_real @ A @ (F @ B)) => ((ord_less_eq_real @ B @ C) => ((![X3 : real, Y4 : real]: ((ord_less_eq_real @ X3 @ Y4) => (ord_less_eq_real @ (F @ X3) @ (F @ Y4)))) => (ord_less_real @ A @ (F @ C)))))))). % order_less_le_subst1
thf(fact_112_order__le__less__subst2, axiom,
    ((![A : real, B : real, F : real > real, C : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_real @ (F @ B) @ C) => ((![X3 : real, Y4 : real]: ((ord_less_eq_real @ X3 @ Y4) => (ord_less_eq_real @ (F @ X3) @ (F @ Y4)))) => (ord_less_real @ (F @ A) @ C))))))). % order_le_less_subst2
thf(fact_113_order__le__less__subst1, axiom,
    ((![A : real, F : real > real, B : real, C : real]: ((ord_less_eq_real @ A @ (F @ B)) => ((ord_less_real @ B @ C) => ((![X3 : real, Y4 : real]: ((ord_less_real @ X3 @ Y4) => (ord_less_real @ (F @ X3) @ (F @ Y4)))) => (ord_less_real @ A @ (F @ C)))))))). % order_le_less_subst1
thf(fact_114_less__le, axiom,
    ((ord_less_real = (^[X2 : real]: (^[Y3 : real]: (((ord_less_eq_real @ X2 @ Y3)) & ((~ ((X2 = Y3)))))))))). % less_le
thf(fact_115_le__less, axiom,
    ((ord_less_eq_real = (^[X2 : real]: (^[Y3 : real]: (((ord_less_real @ X2 @ Y3)) | ((X2 = Y3)))))))). % le_less
thf(fact_116_leI, axiom,
    ((![X : real, Y2 : real]: ((~ ((ord_less_real @ X @ Y2))) => (ord_less_eq_real @ Y2 @ X))))). % leI
thf(fact_117_leD, axiom,
    ((![Y2 : real, X : real]: ((ord_less_eq_real @ Y2 @ X) => (~ ((ord_less_real @ X @ Y2))))))). % leD
thf(fact_118_less__eq__real__def, axiom,
    ((ord_less_eq_real = (^[X2 : real]: (^[Y3 : real]: (((ord_less_real @ X2 @ Y3)) | ((X2 = Y3)))))))). % less_eq_real_def
thf(fact_119_complete__interval, axiom,
    ((![A : real, B : real, P2 : real > $o]: ((ord_less_real @ A @ B) => ((P2 @ A) => ((~ ((P2 @ B))) => (?[C2 : real]: ((ord_less_eq_real @ A @ C2) & ((ord_less_eq_real @ C2 @ B) & ((![X4 : real]: (((ord_less_eq_real @ A @ X4) & (ord_less_real @ X4 @ C2)) => (P2 @ X4))) & (![D : real]: ((![X3 : real]: (((ord_less_eq_real @ A @ X3) & (ord_less_real @ X3 @ D)) => (P2 @ X3))) => (ord_less_eq_real @ D @ C2))))))))))))). % complete_interval
thf(fact_120_verit__comp__simplify1_I3_J, axiom,
    ((![B4 : real, A5 : real]: ((~ ((ord_less_eq_real @ B4 @ A5))) = (ord_less_real @ A5 @ B4))))). % verit_comp_simplify1(3)
thf(fact_121_pinf_I6_J, axiom,
    ((![T : real]: (?[Z3 : real]: (![X4 : real]: ((ord_less_real @ Z3 @ X4) => (~ ((ord_less_eq_real @ X4 @ T))))))))). % pinf(6)
thf(fact_122_pinf_I8_J, axiom,
    ((![T : real]: (?[Z3 : real]: (![X4 : real]: ((ord_less_real @ Z3 @ X4) => (ord_less_eq_real @ T @ X4))))))). % pinf(8)
thf(fact_123_minf_I6_J, axiom,
    ((![T : real]: (?[Z3 : real]: (![X4 : real]: ((ord_less_real @ X4 @ Z3) => (ord_less_eq_real @ X4 @ T))))))). % minf(6)
thf(fact_124_minf_I8_J, axiom,
    ((![T : real]: (?[Z3 : real]: (![X4 : real]: ((ord_less_real @ X4 @ Z3) => (~ ((ord_less_eq_real @ T @ X4))))))))). % minf(8)
thf(fact_125_field__lbound__gt__zero, axiom,
    ((![D1 : real, D2 : real]: ((ord_less_real @ zero_zero_real @ D1) => ((ord_less_real @ zero_zero_real @ D2) => (?[E : real]: ((ord_less_real @ zero_zero_real @ E) & ((ord_less_real @ E @ D1) & (ord_less_real @ E @ D2))))))))). % field_lbound_gt_zero
thf(fact_126_less__numeral__extra_I3_J, axiom,
    ((~ ((ord_less_poly_real @ zero_zero_poly_real @ zero_zero_poly_real))))). % less_numeral_extra(3)
thf(fact_127_less__numeral__extra_I3_J, axiom,
    ((~ ((ord_less_real @ zero_zero_real @ zero_zero_real))))). % less_numeral_extra(3)
thf(fact_128_le__numeral__extra_I3_J, axiom,
    ((ord_le1180086932y_real @ zero_zero_poly_real @ zero_zero_poly_real))). % le_numeral_extra(3)
thf(fact_129_le__numeral__extra_I3_J, axiom,
    ((ord_less_eq_real @ zero_zero_real @ zero_zero_real))). % le_numeral_extra(3)
thf(fact_130_verit__la__disequality, axiom,
    ((![A : real, B : real]: ((A = B) | ((~ ((ord_less_eq_real @ A @ B))) | (~ ((ord_less_eq_real @ B @ A)))))))). % verit_la_disequality
thf(fact_131_minf_I7_J, axiom,
    ((![T : real]: (?[Z3 : real]: (![X4 : real]: ((ord_less_real @ X4 @ Z3) => (~ ((ord_less_real @ T @ X4))))))))). % minf(7)
thf(fact_132_minf_I5_J, axiom,
    ((![T : real]: (?[Z3 : real]: (![X4 : real]: ((ord_less_real @ X4 @ Z3) => (ord_less_real @ X4 @ T))))))). % minf(5)
thf(fact_133_minf_I4_J, axiom,
    ((![T : real]: (?[Z3 : real]: (![X4 : real]: ((ord_less_real @ X4 @ Z3) => (~ ((X4 = T))))))))). % minf(4)
thf(fact_134_minf_I3_J, axiom,
    ((![T : real]: (?[Z3 : real]: (![X4 : real]: ((ord_less_real @ X4 @ Z3) => (~ ((X4 = T))))))))). % minf(3)
thf(fact_135_minf_I2_J, axiom,
    ((![P2 : real > $o, P3 : real > $o, Q2 : real > $o, Q3 : real > $o]: ((?[Z4 : real]: (![X3 : real]: ((ord_less_real @ X3 @ Z4) => ((P2 @ X3) = (P3 @ X3))))) => ((?[Z4 : real]: (![X3 : real]: ((ord_less_real @ X3 @ Z4) => ((Q2 @ X3) = (Q3 @ X3))))) => (?[Z3 : real]: (![X4 : real]: ((ord_less_real @ X4 @ Z3) => ((((P2 @ X4)) | ((Q2 @ X4))) = (((P3 @ X4)) | ((Q3 @ X4)))))))))))). % minf(2)
thf(fact_136_minf_I1_J, axiom,
    ((![P2 : real > $o, P3 : real > $o, Q2 : real > $o, Q3 : real > $o]: ((?[Z4 : real]: (![X3 : real]: ((ord_less_real @ X3 @ Z4) => ((P2 @ X3) = (P3 @ X3))))) => ((?[Z4 : real]: (![X3 : real]: ((ord_less_real @ X3 @ Z4) => ((Q2 @ X3) = (Q3 @ X3))))) => (?[Z3 : real]: (![X4 : real]: ((ord_less_real @ X4 @ Z3) => ((((P2 @ X4)) & ((Q2 @ X4))) = (((P3 @ X4)) & ((Q3 @ X4)))))))))))). % minf(1)
thf(fact_137_pinf_I7_J, axiom,
    ((![T : real]: (?[Z3 : real]: (![X4 : real]: ((ord_less_real @ Z3 @ X4) => (ord_less_real @ T @ X4))))))). % pinf(7)
thf(fact_138_pinf_I5_J, axiom,
    ((![T : real]: (?[Z3 : real]: (![X4 : real]: ((ord_less_real @ Z3 @ X4) => (~ ((ord_less_real @ X4 @ T))))))))). % pinf(5)
thf(fact_139_pinf_I4_J, axiom,
    ((![T : real]: (?[Z3 : real]: (![X4 : real]: ((ord_less_real @ Z3 @ X4) => (~ ((X4 = T))))))))). % pinf(4)
thf(fact_140_pinf_I3_J, axiom,
    ((![T : real]: (?[Z3 : real]: (![X4 : real]: ((ord_less_real @ Z3 @ X4) => (~ ((X4 = T))))))))). % pinf(3)
thf(fact_141_pinf_I2_J, axiom,
    ((![P2 : real > $o, P3 : real > $o, Q2 : real > $o, Q3 : real > $o]: ((?[Z4 : real]: (![X3 : real]: ((ord_less_real @ Z4 @ X3) => ((P2 @ X3) = (P3 @ X3))))) => ((?[Z4 : real]: (![X3 : real]: ((ord_less_real @ Z4 @ X3) => ((Q2 @ X3) = (Q3 @ X3))))) => (?[Z3 : real]: (![X4 : real]: ((ord_less_real @ Z3 @ X4) => ((((P2 @ X4)) | ((Q2 @ X4))) = (((P3 @ X4)) | ((Q3 @ X4)))))))))))). % pinf(2)
thf(fact_142_pinf_I1_J, axiom,
    ((![P2 : real > $o, P3 : real > $o, Q2 : real > $o, Q3 : real > $o]: ((?[Z4 : real]: (![X3 : real]: ((ord_less_real @ Z4 @ X3) => ((P2 @ X3) = (P3 @ X3))))) => ((?[Z4 : real]: (![X3 : real]: ((ord_less_real @ Z4 @ X3) => ((Q2 @ X3) = (Q3 @ X3))))) => (?[Z3 : real]: (![X4 : real]: ((ord_less_real @ Z3 @ X4) => ((((P2 @ X4)) & ((Q2 @ X4))) = (((P3 @ X4)) & ((Q3 @ X4)))))))))))). % pinf(1)
thf(fact_143_verit__comp__simplify1_I1_J, axiom,
    ((![A : real]: (~ ((ord_less_real @ A @ A)))))). % verit_comp_simplify1(1)
thf(fact_144_ex__gt__or__lt, axiom,
    ((![A : real]: (?[B3 : real]: ((ord_less_real @ A @ B3) | (ord_less_real @ B3 @ A)))))). % ex_gt_or_lt
thf(fact_145_complete__real, axiom,
    ((![S : set_real]: ((?[X4 : real]: (member_real @ X4 @ S)) => ((?[Z4 : real]: (![X3 : real]: ((member_real @ X3 @ S) => (ord_less_eq_real @ X3 @ Z4)))) => (?[Y4 : real]: ((![X4 : real]: ((member_real @ X4 @ S) => (ord_less_eq_real @ X4 @ Y4))) & (![Z4 : real]: ((![X3 : real]: ((member_real @ X3 @ S) => (ord_less_eq_real @ X3 @ Z4))) => (ord_less_eq_real @ Y4 @ Z4)))))))))). % complete_real

% Conjectures (1)
thf(conj_0, conjecture,
    ((?[M : real]: ((ord_less_real @ zero_zero_real @ M) & (![Z3 : a]: ((~ ((ord_less_eq_real @ (real_V1022479215norm_a @ Z3) @ r))) | (ord_less_eq_real @ (real_V1022479215norm_a @ (poly_a2 @ p @ Z3)) @ M))))))).
