% 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_121__5368110_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:27:14.817

% Could-be-implicit typings (2)
thf(ty_n_t__Set__Oset_It__Real__Oreal_J, type,
    set_real : $tType).
thf(ty_n_t__Real__Oreal, type,
    real : $tType).

% Explicit typings (19)
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001t__Real__Oreal, type,
    condit1201756488e_real : set_real > $o).
thf(sy_c_Lattices_Osup__class_Osup_001t__Real__Oreal, type,
    sup_sup_real : real > real > real).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Real__Oreal_J, type,
    sup_sup_set_real : set_real > set_real > set_real).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Real__Oreal_M_Eo_J, type,
    bot_bot_real_o : real > $o).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J, type,
    bot_bot_set_real : set_real).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal, type,
    ord_less_real : real > real > $o).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Real__Oreal_J, type,
    ord_less_set_real : set_real > set_real > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal, type,
    ord_less_eq_real : real > real > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J, type,
    ord_less_eq_set_real : set_real > set_real > $o).
thf(sy_c_Set_OCollect_001t__Real__Oreal, type,
    collect_real : (real > $o) > set_real).
thf(sy_c_Set_Oinsert_001t__Real__Oreal, type,
    insert_real : real > set_real > set_real).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Real__Oreal, type,
    set_or656347191t_real : real > real > set_real).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Real__Oreal, type,
    set_or2075149659n_real : real > real > set_real).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Real__Oreal, type,
    set_ord_atMost_real : real > set_real).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Real__Oreal, type,
    set_or1638906204t_real : real > real > set_real).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Real__Oreal, type,
    set_or951364608n_real : real > real > set_real).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Real__Oreal, type,
    set_or1211449801n_real : real > set_real).
thf(sy_c_member_001t__Real__Oreal, type,
    member_real : real > set_real > $o).
thf(sy_v_P, type,
    p : real > $o).

% Relevant facts (138)
thf(fact_0_bz, axiom,
    ((?[Z : real]: (![X : real]: ((p @ X) => (ord_less_real @ X @ Z)))))). % bz
thf(fact_1_ex, axiom,
    ((?[X_1 : real]: (p @ X_1)))). % ex
thf(fact_2_bdd__above__empty, axiom,
    ((condit1201756488e_real @ bot_bot_set_real))). % bdd_above_empty
thf(fact_3_bdd__above__Ioo, axiom,
    ((![A : real, B : real]: (condit1201756488e_real @ (set_or951364608n_real @ A @ B))))). % bdd_above_Ioo
thf(fact_4_bdd__above__Iic, axiom,
    ((![B : real]: (condit1201756488e_real @ (set_ord_atMost_real @ B))))). % bdd_above_Iic
thf(fact_5_bdd__above__Ioc, axiom,
    ((![A : real, B : real]: (condit1201756488e_real @ (set_or1638906204t_real @ A @ B))))). % bdd_above_Ioc
thf(fact_6_bdd__aboveI, axiom,
    ((![A2 : set_real, M : real]: ((![X2 : real]: ((member_real @ X2 @ A2) => (ord_less_eq_real @ X2 @ M))) => (condit1201756488e_real @ A2))))). % bdd_aboveI
thf(fact_7_bdd__above__insert, axiom,
    ((![A : real, A2 : set_real]: ((condit1201756488e_real @ (insert_real @ A @ A2)) = (condit1201756488e_real @ A2))))). % bdd_above_insert
thf(fact_8_bdd__above__Un, axiom,
    ((![A2 : set_real, B2 : set_real]: ((condit1201756488e_real @ (sup_sup_set_real @ A2 @ B2)) = (((condit1201756488e_real @ A2)) & ((condit1201756488e_real @ B2))))))). % bdd_above_Un
thf(fact_9_bdd__above__Iio, axiom,
    ((![B : real]: (condit1201756488e_real @ (set_or1211449801n_real @ B))))). % bdd_above_Iio
thf(fact_10_bdd__above__Ico, axiom,
    ((![A : real, B : real]: (condit1201756488e_real @ (set_or2075149659n_real @ A @ B))))). % bdd_above_Ico
thf(fact_11_bdd__above__Icc, axiom,
    ((![A : real, B : real]: (condit1201756488e_real @ (set_or656347191t_real @ A @ B))))). % bdd_above_Icc
thf(fact_12_bdd__above__mono, axiom,
    ((![B2 : set_real, A2 : set_real]: ((condit1201756488e_real @ B2) => ((ord_less_eq_set_real @ A2 @ B2) => (condit1201756488e_real @ A2)))))). % bdd_above_mono
thf(fact_13_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_14_complete__interval, axiom,
    ((![A : real, B : real, P : real > $o]: ((ord_less_real @ A @ B) => ((P @ A) => ((~ ((P @ B))) => (?[C : real]: ((ord_less_eq_real @ A @ C) & ((ord_less_eq_real @ C @ B) & ((![X : real]: (((ord_less_eq_real @ A @ X) & (ord_less_real @ X @ C)) => (P @ X))) & (![D : real]: ((![X2 : real]: (((ord_less_eq_real @ A @ X2) & (ord_less_real @ X2 @ D)) => (P @ X2))) => (ord_less_eq_real @ D @ C))))))))))))). % complete_interval
thf(fact_15_bdd__above__def, axiom,
    ((condit1201756488e_real = (^[A3 : set_real]: (?[M2 : real]: (![X3 : real]: (((member_real @ X3 @ A3)) => ((ord_less_eq_real @ X3 @ M2))))))))). % bdd_above_def
thf(fact_16_Icc__subset__Iic__iff, axiom,
    ((![L : real, H : real, H2 : real]: ((ord_less_eq_set_real @ (set_or656347191t_real @ L @ H) @ (set_ord_atMost_real @ H2)) = (((~ ((ord_less_eq_real @ L @ H)))) | ((ord_less_eq_real @ H @ H2))))))). % Icc_subset_Iic_iff
thf(fact_17_greaterThanAtMost__empty__iff, axiom,
    ((![K : real, L : real]: (((set_or1638906204t_real @ K @ L) = bot_bot_set_real) = (~ ((ord_less_real @ K @ L))))))). % greaterThanAtMost_empty_iff
thf(fact_18_greaterThanAtMost__empty__iff2, axiom,
    ((![K : real, L : real]: ((bot_bot_set_real = (set_or1638906204t_real @ K @ L)) = (~ ((ord_less_real @ K @ L))))))). % greaterThanAtMost_empty_iff2
thf(fact_19_greaterThanAtMost__empty, axiom,
    ((![L : real, K : real]: ((ord_less_eq_real @ L @ K) => ((set_or1638906204t_real @ K @ L) = bot_bot_set_real))))). % greaterThanAtMost_empty
thf(fact_20_greaterThanLessThan__empty, axiom,
    ((![L : real, K : real]: ((ord_less_eq_real @ L @ K) => ((set_or951364608n_real @ K @ L) = bot_bot_set_real))))). % greaterThanLessThan_empty
thf(fact_21_greaterThanLessThan__empty__iff, axiom,
    ((![A : real, B : real]: (((set_or951364608n_real @ A @ B) = bot_bot_set_real) = (ord_less_eq_real @ B @ A))))). % greaterThanLessThan_empty_iff
thf(fact_22_greaterThanLessThan__empty__iff2, axiom,
    ((![A : real, B : real]: ((bot_bot_set_real = (set_or951364608n_real @ A @ B)) = (ord_less_eq_real @ B @ A))))). % greaterThanLessThan_empty_iff2
thf(fact_23_greaterThanAtMost__iff, axiom,
    ((![I : real, L : real, U : real]: ((member_real @ I @ (set_or1638906204t_real @ L @ U)) = (((ord_less_real @ L @ I)) & ((ord_less_eq_real @ I @ U))))))). % greaterThanAtMost_iff
thf(fact_24_atLeastAtMost__singleton, axiom,
    ((![A : real]: ((set_or656347191t_real @ A @ A) = (insert_real @ A @ bot_bot_set_real))))). % atLeastAtMost_singleton
thf(fact_25_atLeastAtMost__singleton__iff, axiom,
    ((![A : real, B : real, C2 : real]: (((set_or656347191t_real @ A @ B) = (insert_real @ C2 @ bot_bot_set_real)) = (((A = B)) & ((B = C2))))))). % atLeastAtMost_singleton_iff
thf(fact_26_atMost__eq__iff, axiom,
    ((![X4 : real, Y : real]: (((set_ord_atMost_real @ X4) = (set_ord_atMost_real @ Y)) = (X4 = Y))))). % atMost_eq_iff
thf(fact_27_atLeastAtMost__iff, axiom,
    ((![I : real, L : real, U : real]: ((member_real @ I @ (set_or656347191t_real @ L @ U)) = (((ord_less_eq_real @ L @ I)) & ((ord_less_eq_real @ I @ U))))))). % atLeastAtMost_iff
thf(fact_28_Icc__eq__Icc, axiom,
    ((![L : real, H : real, L2 : real, H2 : real]: (((set_or656347191t_real @ L @ H) = (set_or656347191t_real @ L2 @ H2)) = (((((L = L2)) & ((H = H2)))) | ((((~ ((ord_less_eq_real @ L @ H)))) & ((~ ((ord_less_eq_real @ L2 @ H2))))))))))). % Icc_eq_Icc
thf(fact_29_lessThan__iff, axiom,
    ((![I : real, K : real]: ((member_real @ I @ (set_or1211449801n_real @ K)) = (ord_less_real @ I @ K))))). % lessThan_iff
thf(fact_30_atMost__iff, axiom,
    ((![I : real, K : real]: ((member_real @ I @ (set_ord_atMost_real @ K)) = (ord_less_eq_real @ I @ K))))). % atMost_iff
thf(fact_31_greaterThanLessThan__iff, axiom,
    ((![I : real, L : real, U : real]: ((member_real @ I @ (set_or951364608n_real @ L @ U)) = (((ord_less_real @ L @ I)) & ((ord_less_real @ I @ U))))))). % greaterThanLessThan_iff
thf(fact_32_atLeastLessThan__iff, axiom,
    ((![I : real, L : real, U : real]: ((member_real @ I @ (set_or2075149659n_real @ L @ U)) = (((ord_less_eq_real @ L @ I)) & ((ord_less_real @ I @ U))))))). % atLeastLessThan_iff
thf(fact_33_atLeastatMost__empty__iff2, axiom,
    ((![A : real, B : real]: ((bot_bot_set_real = (set_or656347191t_real @ A @ B)) = (~ ((ord_less_eq_real @ A @ B))))))). % atLeastatMost_empty_iff2
thf(fact_34_atLeastatMost__empty__iff, axiom,
    ((![A : real, B : real]: (((set_or656347191t_real @ A @ B) = bot_bot_set_real) = (~ ((ord_less_eq_real @ A @ B))))))). % atLeastatMost_empty_iff
thf(fact_35_atLeastatMost__empty, axiom,
    ((![B : real, A : real]: ((ord_less_real @ B @ A) => ((set_or656347191t_real @ A @ B) = bot_bot_set_real))))). % atLeastatMost_empty
thf(fact_36_atLeastatMost__subset__iff, axiom,
    ((![A : real, B : real, C2 : real, D2 : real]: ((ord_less_eq_set_real @ (set_or656347191t_real @ A @ B) @ (set_or656347191t_real @ C2 @ D2)) = (((~ ((ord_less_eq_real @ A @ B)))) | ((((ord_less_eq_real @ C2 @ A)) & ((ord_less_eq_real @ B @ D2))))))))). % atLeastatMost_subset_iff
thf(fact_37_atLeastLessThan__empty, axiom,
    ((![B : real, A : real]: ((ord_less_eq_real @ B @ A) => ((set_or2075149659n_real @ A @ B) = bot_bot_set_real))))). % atLeastLessThan_empty
thf(fact_38_atLeastLessThan__empty__iff2, axiom,
    ((![A : real, B : real]: ((bot_bot_set_real = (set_or2075149659n_real @ A @ B)) = (~ ((ord_less_real @ A @ B))))))). % atLeastLessThan_empty_iff2
thf(fact_39_atLeastLessThan__empty__iff, axiom,
    ((![A : real, B : real]: (((set_or2075149659n_real @ A @ B) = bot_bot_set_real) = (~ ((ord_less_real @ A @ B))))))). % atLeastLessThan_empty_iff
thf(fact_40_ivl__subset, axiom,
    ((![I : real, J : real, M3 : real, N : real]: ((ord_less_eq_set_real @ (set_or2075149659n_real @ I @ J) @ (set_or2075149659n_real @ M3 @ N)) = (((ord_less_eq_real @ J @ I)) | ((((ord_less_eq_real @ M3 @ I)) & ((ord_less_eq_real @ J @ N))))))))). % ivl_subset
thf(fact_41_lessThan__subset__iff, axiom,
    ((![X4 : real, Y : real]: ((ord_less_eq_set_real @ (set_or1211449801n_real @ X4) @ (set_or1211449801n_real @ Y)) = (ord_less_eq_real @ X4 @ Y))))). % lessThan_subset_iff
thf(fact_42_atMost__subset__iff, axiom,
    ((![X4 : real, Y : real]: ((ord_less_eq_set_real @ (set_ord_atMost_real @ X4) @ (set_ord_atMost_real @ Y)) = (ord_less_eq_real @ X4 @ Y))))). % atMost_subset_iff
thf(fact_43_mem__Collect__eq, axiom,
    ((![A : real, P : real > $o]: ((member_real @ A @ (collect_real @ P)) = (P @ A))))). % mem_Collect_eq
thf(fact_44_Collect__mem__eq, axiom,
    ((![A2 : set_real]: ((collect_real @ (^[X3 : real]: (member_real @ X3 @ A2))) = A2)))). % Collect_mem_eq
thf(fact_45_Collect__cong, axiom,
    ((![P : real > $o, Q : real > $o]: ((![X2 : real]: ((P @ X2) = (Q @ X2))) => ((collect_real @ P) = (collect_real @ Q)))))). % Collect_cong
thf(fact_46_atLeastLessThan__eq__iff, axiom,
    ((![A : real, B : real, C2 : real, D2 : real]: ((ord_less_real @ A @ B) => ((ord_less_real @ C2 @ D2) => (((set_or2075149659n_real @ A @ B) = (set_or2075149659n_real @ C2 @ D2)) = (((A = C2)) & ((B = D2))))))))). % atLeastLessThan_eq_iff
thf(fact_47_atLeastLessThan__inj_I1_J, axiom,
    ((![A : real, B : real, C2 : real, D2 : real]: (((set_or2075149659n_real @ A @ B) = (set_or2075149659n_real @ C2 @ D2)) => ((ord_less_real @ A @ B) => ((ord_less_real @ C2 @ D2) => (A = C2))))))). % atLeastLessThan_inj(1)
thf(fact_48_atLeastLessThan__inj_I2_J, axiom,
    ((![A : real, B : real, C2 : real, D2 : real]: (((set_or2075149659n_real @ A @ B) = (set_or2075149659n_real @ C2 @ D2)) => ((ord_less_real @ A @ B) => ((ord_less_real @ C2 @ D2) => (B = D2))))))). % atLeastLessThan_inj(2)
thf(fact_49_lessThan__strict__subset__iff, axiom,
    ((![M3 : real, N : real]: ((ord_less_set_real @ (set_or1211449801n_real @ M3) @ (set_or1211449801n_real @ N)) = (ord_less_real @ M3 @ N))))). % lessThan_strict_subset_iff
thf(fact_50_lessThan__non__empty, axiom,
    ((![X4 : real]: (~ (((set_or1211449801n_real @ X4) = bot_bot_set_real)))))). % lessThan_non_empty
thf(fact_51_not__empty__eq__Iic__eq__empty, axiom,
    ((![H : real]: (~ ((bot_bot_set_real = (set_ord_atMost_real @ H))))))). % not_empty_eq_Iic_eq_empty
thf(fact_52_Ioc__inj, axiom,
    ((![A : real, B : real, C2 : real, D2 : real]: (((set_or1638906204t_real @ A @ B) = (set_or1638906204t_real @ C2 @ D2)) = (((((ord_less_eq_real @ B @ A)) & ((ord_less_eq_real @ D2 @ C2)))) | ((((A = C2)) & ((B = D2))))))))). % Ioc_inj
thf(fact_53_not__Iic__eq__Icc, axiom,
    ((![H2 : real, L : real, H : real]: (~ (((set_ord_atMost_real @ H2) = (set_or656347191t_real @ L @ H))))))). % not_Iic_eq_Icc
thf(fact_54_atLeastatMost__psubset__iff, axiom,
    ((![A : real, B : real, C2 : real, D2 : real]: ((ord_less_set_real @ (set_or656347191t_real @ A @ B) @ (set_or656347191t_real @ C2 @ D2)) = (((((~ ((ord_less_eq_real @ A @ B)))) | ((((ord_less_eq_real @ C2 @ A)) & ((((ord_less_eq_real @ B @ D2)) & ((((ord_less_real @ C2 @ A)) | ((ord_less_real @ B @ D2)))))))))) & ((ord_less_eq_real @ C2 @ D2))))))). % atLeastatMost_psubset_iff
thf(fact_55_atLeastLessThan__subset__iff, axiom,
    ((![A : real, B : real, C2 : real, D2 : real]: ((ord_less_eq_set_real @ (set_or2075149659n_real @ A @ B) @ (set_or2075149659n_real @ C2 @ D2)) => ((ord_less_eq_real @ B @ A) | ((ord_less_eq_real @ C2 @ A) & (ord_less_eq_real @ B @ D2))))))). % atLeastLessThan_subset_iff
thf(fact_56_ivl__disj__un__two__touch_I4_J, axiom,
    ((![L : real, M3 : real, U : real]: ((ord_less_eq_real @ L @ M3) => ((ord_less_eq_real @ M3 @ U) => ((sup_sup_set_real @ (set_or656347191t_real @ L @ M3) @ (set_or656347191t_real @ M3 @ U)) = (set_or656347191t_real @ L @ U))))))). % ivl_disj_un_two_touch(4)
thf(fact_57_atLeastAtMost__singleton_H, axiom,
    ((![A : real, B : real]: ((A = B) => ((set_or656347191t_real @ A @ B) = (insert_real @ A @ bot_bot_set_real)))))). % atLeastAtMost_singleton'
thf(fact_58_ivl__disj__un__two_I3_J, axiom,
    ((![L : real, M3 : real, U : real]: ((ord_less_eq_real @ L @ M3) => ((ord_less_eq_real @ M3 @ U) => ((sup_sup_set_real @ (set_or2075149659n_real @ L @ M3) @ (set_or2075149659n_real @ M3 @ U)) = (set_or2075149659n_real @ L @ U))))))). % ivl_disj_un_two(3)
thf(fact_59_Ioc__subset__iff, axiom,
    ((![A : real, B : real, C2 : real, D2 : real]: ((ord_less_eq_set_real @ (set_or1638906204t_real @ A @ B) @ (set_or1638906204t_real @ C2 @ D2)) = (((ord_less_eq_real @ B @ A)) | ((((ord_less_eq_real @ C2 @ A)) & ((ord_less_eq_real @ B @ D2))))))))). % Ioc_subset_iff
thf(fact_60_not__Iic__le__Icc, axiom,
    ((![H : real, L2 : real, H2 : real]: (~ ((ord_less_eq_set_real @ (set_ord_atMost_real @ H) @ (set_or656347191t_real @ L2 @ H2))))))). % not_Iic_le_Icc
thf(fact_61_ivl__disj__un__two_I6_J, axiom,
    ((![L : real, M3 : real, U : real]: ((ord_less_eq_real @ L @ M3) => ((ord_less_eq_real @ M3 @ U) => ((sup_sup_set_real @ (set_or1638906204t_real @ L @ M3) @ (set_or1638906204t_real @ M3 @ U)) = (set_or1638906204t_real @ L @ U))))))). % ivl_disj_un_two(6)
thf(fact_62_greaterThanLessThan__subseteq__greaterThanLessThan, axiom,
    ((![A : real, B : real, C2 : real, D2 : real]: ((ord_less_eq_set_real @ (set_or951364608n_real @ A @ B) @ (set_or951364608n_real @ C2 @ D2)) = (((ord_less_real @ A @ B)) => ((((ord_less_eq_real @ C2 @ A)) & ((ord_less_eq_real @ B @ D2))))))))). % greaterThanLessThan_subseteq_greaterThanLessThan
thf(fact_63_ivl__disj__un__two_I7_J, axiom,
    ((![L : real, M3 : real, U : real]: ((ord_less_eq_real @ L @ M3) => ((ord_less_eq_real @ M3 @ U) => ((sup_sup_set_real @ (set_or2075149659n_real @ L @ M3) @ (set_or656347191t_real @ M3 @ U)) = (set_or656347191t_real @ L @ U))))))). % ivl_disj_un_two(7)
thf(fact_64_Iic__subset__Iio__iff, axiom,
    ((![A : real, B : real]: ((ord_less_eq_set_real @ (set_ord_atMost_real @ A) @ (set_or1211449801n_real @ B)) = (ord_less_real @ A @ B))))). % Iic_subset_Iio_iff
thf(fact_65_ivl__disj__un__one_I2_J, axiom,
    ((![L : real, U : real]: ((ord_less_eq_real @ L @ U) => ((sup_sup_set_real @ (set_or1211449801n_real @ L) @ (set_or2075149659n_real @ L @ U)) = (set_or1211449801n_real @ U)))))). % ivl_disj_un_one(2)
thf(fact_66_ivl__disj__un__two_I8_J, axiom,
    ((![L : real, M3 : real, U : real]: ((ord_less_eq_real @ L @ M3) => ((ord_less_eq_real @ M3 @ U) => ((sup_sup_set_real @ (set_or656347191t_real @ L @ M3) @ (set_or1638906204t_real @ M3 @ U)) = (set_or656347191t_real @ L @ U))))))). % ivl_disj_un_two(8)
thf(fact_67_ivl__disj__un__one_I3_J, axiom,
    ((![L : real, U : real]: ((ord_less_eq_real @ L @ U) => ((sup_sup_set_real @ (set_ord_atMost_real @ L) @ (set_or1638906204t_real @ L @ U)) = (set_ord_atMost_real @ U)))))). % ivl_disj_un_one(3)
thf(fact_68_atLeastLessThan__subseteq__atLeastAtMost__iff, axiom,
    ((![A : real, B : real, C2 : real, D2 : real]: ((ord_less_eq_set_real @ (set_or2075149659n_real @ A @ B) @ (set_or656347191t_real @ C2 @ D2)) = (((ord_less_real @ A @ B)) => ((((ord_less_eq_real @ C2 @ A)) & ((ord_less_eq_real @ B @ D2))))))))). % atLeastLessThan_subseteq_atLeastAtMost_iff
thf(fact_69_atLeastAtMost__subseteq__atLeastLessThan__iff, axiom,
    ((![A : real, B : real, C2 : real, D2 : real]: ((ord_less_eq_set_real @ (set_or656347191t_real @ A @ B) @ (set_or2075149659n_real @ C2 @ D2)) = (((ord_less_eq_real @ A @ B)) => ((((ord_less_eq_real @ C2 @ A)) & ((ord_less_real @ B @ D2))))))))). % atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_70_ivl__disj__un__two__touch_I2_J, axiom,
    ((![L : real, M3 : real, U : real]: ((ord_less_eq_real @ L @ M3) => ((ord_less_real @ M3 @ U) => ((sup_sup_set_real @ (set_or656347191t_real @ L @ M3) @ (set_or2075149659n_real @ M3 @ U)) = (set_or2075149659n_real @ L @ U))))))). % ivl_disj_un_two_touch(2)
thf(fact_71_greaterThanLessThan__subseteq__atLeastAtMost__iff, axiom,
    ((![A : real, B : real, C2 : real, D2 : real]: ((ord_less_eq_set_real @ (set_or951364608n_real @ A @ B) @ (set_or656347191t_real @ C2 @ D2)) = (((ord_less_real @ A @ B)) => ((((ord_less_eq_real @ C2 @ A)) & ((ord_less_eq_real @ B @ D2))))))))). % greaterThanLessThan_subseteq_atLeastAtMost_iff
thf(fact_72_greaterThanAtMost__subseteq__atLeastAtMost__iff, axiom,
    ((![A : real, B : real, C2 : real, D2 : real]: ((ord_less_eq_set_real @ (set_or1638906204t_real @ A @ B) @ (set_or656347191t_real @ C2 @ D2)) = (((ord_less_real @ A @ B)) => ((((ord_less_eq_real @ C2 @ A)) & ((ord_less_eq_real @ B @ D2))))))))). % greaterThanAtMost_subseteq_atLeastAtMost_iff
thf(fact_73_greaterThanLessThan__subseteq__atLeastLessThan__iff, axiom,
    ((![A : real, B : real, C2 : real, D2 : real]: ((ord_less_eq_set_real @ (set_or951364608n_real @ A @ B) @ (set_or2075149659n_real @ C2 @ D2)) = (((ord_less_real @ A @ B)) => ((((ord_less_eq_real @ C2 @ A)) & ((ord_less_eq_real @ B @ D2))))))))). % greaterThanLessThan_subseteq_atLeastLessThan_iff
thf(fact_74_greaterThanAtMost__subseteq__atLeastLessThan__iff, axiom,
    ((![A : real, B : real, C2 : real, D2 : real]: ((ord_less_eq_set_real @ (set_or1638906204t_real @ A @ B) @ (set_or2075149659n_real @ C2 @ D2)) = (((ord_less_real @ A @ B)) => ((((ord_less_eq_real @ C2 @ A)) & ((ord_less_real @ B @ D2))))))))). % greaterThanAtMost_subseteq_atLeastLessThan_iff
thf(fact_75_ivl__disj__un__two__touch_I3_J, axiom,
    ((![L : real, M3 : real, U : real]: ((ord_less_real @ L @ M3) => ((ord_less_eq_real @ M3 @ U) => ((sup_sup_set_real @ (set_or1638906204t_real @ L @ M3) @ (set_or656347191t_real @ M3 @ U)) = (set_or1638906204t_real @ L @ U))))))). % ivl_disj_un_two_touch(3)
thf(fact_76_ivl__disj__un__two_I1_J, axiom,
    ((![L : real, M3 : real, U : real]: ((ord_less_real @ L @ M3) => ((ord_less_eq_real @ M3 @ U) => ((sup_sup_set_real @ (set_or951364608n_real @ L @ M3) @ (set_or2075149659n_real @ M3 @ U)) = (set_or951364608n_real @ L @ U))))))). % ivl_disj_un_two(1)
thf(fact_77_ivl__disj__un__one_I4_J, axiom,
    ((![L : real, U : real]: ((ord_less_eq_real @ L @ U) => ((sup_sup_set_real @ (set_or1211449801n_real @ L) @ (set_or656347191t_real @ L @ U)) = (set_ord_atMost_real @ U)))))). % ivl_disj_un_one(4)
thf(fact_78_ivl__disj__un__singleton_I2_J, axiom,
    ((![U : real]: ((sup_sup_set_real @ (set_or1211449801n_real @ U) @ (insert_real @ U @ bot_bot_set_real)) = (set_ord_atMost_real @ U))))). % ivl_disj_un_singleton(2)
thf(fact_79_greaterThanLessThan__subseteq__greaterThanAtMost__iff, axiom,
    ((![A : real, B : real, C2 : real, D2 : real]: ((ord_less_eq_set_real @ (set_or951364608n_real @ A @ B) @ (set_or1638906204t_real @ C2 @ D2)) = (((ord_less_real @ A @ B)) => ((((ord_less_eq_real @ C2 @ A)) & ((ord_less_eq_real @ B @ D2))))))))). % greaterThanLessThan_subseteq_greaterThanAtMost_iff
thf(fact_80_ivl__disj__un__two_I2_J, axiom,
    ((![L : real, M3 : real, U : real]: ((ord_less_eq_real @ L @ M3) => ((ord_less_real @ M3 @ U) => ((sup_sup_set_real @ (set_or1638906204t_real @ L @ M3) @ (set_or951364608n_real @ M3 @ U)) = (set_or951364608n_real @ L @ U))))))). % ivl_disj_un_two(2)
thf(fact_81_ivl__disj__un__one_I1_J, axiom,
    ((![L : real, U : real]: ((ord_less_real @ L @ U) => ((sup_sup_set_real @ (set_ord_atMost_real @ L) @ (set_or951364608n_real @ L @ U)) = (set_or1211449801n_real @ U)))))). % ivl_disj_un_one(1)
thf(fact_82_ivl__disj__un__two__touch_I1_J, axiom,
    ((![L : real, M3 : real, U : real]: ((ord_less_real @ L @ M3) => ((ord_less_real @ M3 @ U) => ((sup_sup_set_real @ (set_or1638906204t_real @ L @ M3) @ (set_or2075149659n_real @ M3 @ U)) = (set_or951364608n_real @ L @ U))))))). % ivl_disj_un_two_touch(1)
thf(fact_83_ivl__disj__un__singleton_I6_J, axiom,
    ((![L : real, U : real]: ((ord_less_eq_real @ L @ U) => ((sup_sup_set_real @ (set_or2075149659n_real @ L @ U) @ (insert_real @ U @ bot_bot_set_real)) = (set_or656347191t_real @ L @ U)))))). % ivl_disj_un_singleton(6)
thf(fact_84_ivl__disj__un__singleton_I5_J, axiom,
    ((![L : real, U : real]: ((ord_less_eq_real @ L @ U) => ((sup_sup_set_real @ (insert_real @ L @ bot_bot_set_real) @ (set_or1638906204t_real @ L @ U)) = (set_or656347191t_real @ L @ U)))))). % ivl_disj_un_singleton(5)
thf(fact_85_ivl__disj__un__two_I4_J, axiom,
    ((![L : real, M3 : real, U : real]: ((ord_less_eq_real @ L @ M3) => ((ord_less_real @ M3 @ U) => ((sup_sup_set_real @ (set_or656347191t_real @ L @ M3) @ (set_or951364608n_real @ M3 @ U)) = (set_or2075149659n_real @ L @ U))))))). % ivl_disj_un_two(4)
thf(fact_86_ivl__disj__un__singleton_I3_J, axiom,
    ((![L : real, U : real]: ((ord_less_real @ L @ U) => ((sup_sup_set_real @ (insert_real @ L @ bot_bot_set_real) @ (set_or951364608n_real @ L @ U)) = (set_or2075149659n_real @ L @ U)))))). % ivl_disj_un_singleton(3)
thf(fact_87_ivl__disj__un__two_I5_J, axiom,
    ((![L : real, M3 : real, U : real]: ((ord_less_real @ L @ M3) => ((ord_less_eq_real @ M3 @ U) => ((sup_sup_set_real @ (set_or951364608n_real @ L @ M3) @ (set_or656347191t_real @ M3 @ U)) = (set_or1638906204t_real @ L @ U))))))). % ivl_disj_un_two(5)
thf(fact_88_ivl__disj__un__singleton_I4_J, axiom,
    ((![L : real, U : real]: ((ord_less_real @ L @ U) => ((sup_sup_set_real @ (set_or951364608n_real @ L @ U) @ (insert_real @ U @ bot_bot_set_real)) = (set_or1638906204t_real @ L @ U)))))). % ivl_disj_un_singleton(4)
thf(fact_89_singleton__insert__inj__eq_H, axiom,
    ((![A : real, A2 : set_real, B : real]: (((insert_real @ A @ A2) = (insert_real @ B @ bot_bot_set_real)) = (((A = B)) & ((ord_less_eq_set_real @ A2 @ (insert_real @ B @ bot_bot_set_real)))))))). % singleton_insert_inj_eq'
thf(fact_90_singleton__insert__inj__eq, axiom,
    ((![B : real, A : real, A2 : set_real]: (((insert_real @ B @ bot_bot_set_real) = (insert_real @ A @ A2)) = (((A = B)) & ((ord_less_eq_set_real @ A2 @ (insert_real @ B @ bot_bot_set_real)))))))). % singleton_insert_inj_eq
thf(fact_91_Un__insert__right, axiom,
    ((![A2 : set_real, A : real, B2 : set_real]: ((sup_sup_set_real @ A2 @ (insert_real @ A @ B2)) = (insert_real @ A @ (sup_sup_set_real @ A2 @ B2)))))). % Un_insert_right
thf(fact_92_Un__insert__left, axiom,
    ((![A : real, B2 : set_real, C3 : set_real]: ((sup_sup_set_real @ (insert_real @ A @ B2) @ C3) = (insert_real @ A @ (sup_sup_set_real @ B2 @ C3)))))). % Un_insert_left
thf(fact_93_Un__subset__iff, axiom,
    ((![A2 : set_real, B2 : set_real, C3 : set_real]: ((ord_less_eq_set_real @ (sup_sup_set_real @ A2 @ B2) @ C3) = (((ord_less_eq_set_real @ A2 @ C3)) & ((ord_less_eq_set_real @ B2 @ C3))))))). % Un_subset_iff
thf(fact_94_Un__empty, axiom,
    ((![A2 : set_real, B2 : set_real]: (((sup_sup_set_real @ A2 @ B2) = bot_bot_set_real) = (((A2 = bot_bot_set_real)) & ((B2 = bot_bot_set_real))))))). % Un_empty
thf(fact_95_insert__subset, axiom,
    ((![X4 : real, A2 : set_real, B2 : set_real]: ((ord_less_eq_set_real @ (insert_real @ X4 @ A2) @ B2) = (((member_real @ X4 @ B2)) & ((ord_less_eq_set_real @ A2 @ B2))))))). % insert_subset
thf(fact_96_sup__bot_Oright__neutral, axiom,
    ((![A : set_real]: ((sup_sup_set_real @ A @ bot_bot_set_real) = A)))). % sup_bot.right_neutral
thf(fact_97_empty__iff, axiom,
    ((![C2 : real]: (~ ((member_real @ C2 @ bot_bot_set_real)))))). % empty_iff
thf(fact_98_all__not__in__conv, axiom,
    ((![A2 : set_real]: ((![X3 : real]: (~ ((member_real @ X3 @ A2)))) = (A2 = bot_bot_set_real))))). % all_not_in_conv
thf(fact_99_Collect__empty__eq, axiom,
    ((![P : real > $o]: (((collect_real @ P) = bot_bot_set_real) = (![X3 : real]: (~ ((P @ X3)))))))). % Collect_empty_eq
thf(fact_100_empty__Collect__eq, axiom,
    ((![P : real > $o]: ((bot_bot_set_real = (collect_real @ P)) = (![X3 : real]: (~ ((P @ X3)))))))). % empty_Collect_eq
thf(fact_101_subsetI, axiom,
    ((![A2 : set_real, B2 : set_real]: ((![X2 : real]: ((member_real @ X2 @ A2) => (member_real @ X2 @ B2))) => (ord_less_eq_set_real @ A2 @ B2))))). % subsetI
thf(fact_102_insertCI, axiom,
    ((![A : real, B2 : set_real, B : real]: (((~ ((member_real @ A @ B2))) => (A = B)) => (member_real @ A @ (insert_real @ B @ B2)))))). % insertCI
thf(fact_103_insert__iff, axiom,
    ((![A : real, B : real, A2 : set_real]: ((member_real @ A @ (insert_real @ B @ A2)) = (((A = B)) | ((member_real @ A @ A2))))))). % insert_iff
thf(fact_104_insert__absorb2, axiom,
    ((![X4 : real, A2 : set_real]: ((insert_real @ X4 @ (insert_real @ X4 @ A2)) = (insert_real @ X4 @ A2))))). % insert_absorb2
thf(fact_105_sup_Oidem, axiom,
    ((![A : set_real]: ((sup_sup_set_real @ A @ A) = A)))). % sup.idem
thf(fact_106_sup__idem, axiom,
    ((![X4 : set_real]: ((sup_sup_set_real @ X4 @ X4) = X4)))). % sup_idem
thf(fact_107_sup_Oleft__idem, axiom,
    ((![A : set_real, B : set_real]: ((sup_sup_set_real @ A @ (sup_sup_set_real @ A @ B)) = (sup_sup_set_real @ A @ B))))). % sup.left_idem
thf(fact_108_sup__left__idem, axiom,
    ((![X4 : set_real, Y : set_real]: ((sup_sup_set_real @ X4 @ (sup_sup_set_real @ X4 @ Y)) = (sup_sup_set_real @ X4 @ Y))))). % sup_left_idem
thf(fact_109_sup_Oright__idem, axiom,
    ((![A : set_real, B : set_real]: ((sup_sup_set_real @ (sup_sup_set_real @ A @ B) @ B) = (sup_sup_set_real @ A @ B))))). % sup.right_idem
thf(fact_110_UnCI, axiom,
    ((![C2 : real, B2 : set_real, A2 : set_real]: (((~ ((member_real @ C2 @ B2))) => (member_real @ C2 @ A2)) => (member_real @ C2 @ (sup_sup_set_real @ A2 @ B2)))))). % UnCI
thf(fact_111_Un__iff, axiom,
    ((![C2 : real, A2 : set_real, B2 : set_real]: ((member_real @ C2 @ (sup_sup_set_real @ A2 @ B2)) = (((member_real @ C2 @ A2)) | ((member_real @ C2 @ B2))))))). % Un_iff
thf(fact_112_le__sup__iff, axiom,
    ((![X4 : set_real, Y : set_real, Z2 : set_real]: ((ord_less_eq_set_real @ (sup_sup_set_real @ X4 @ Y) @ Z2) = (((ord_less_eq_set_real @ X4 @ Z2)) & ((ord_less_eq_set_real @ Y @ Z2))))))). % le_sup_iff
thf(fact_113_le__sup__iff, axiom,
    ((![X4 : real, Y : real, Z2 : real]: ((ord_less_eq_real @ (sup_sup_real @ X4 @ Y) @ Z2) = (((ord_less_eq_real @ X4 @ Z2)) & ((ord_less_eq_real @ Y @ Z2))))))). % le_sup_iff
thf(fact_114_sup_Obounded__iff, axiom,
    ((![B : set_real, C2 : set_real, A : set_real]: ((ord_less_eq_set_real @ (sup_sup_set_real @ B @ C2) @ A) = (((ord_less_eq_set_real @ B @ A)) & ((ord_less_eq_set_real @ C2 @ A))))))). % sup.bounded_iff
thf(fact_115_sup_Obounded__iff, axiom,
    ((![B : real, C2 : real, A : real]: ((ord_less_eq_real @ (sup_sup_real @ B @ C2) @ A) = (((ord_less_eq_real @ B @ A)) & ((ord_less_eq_real @ C2 @ A))))))). % sup.bounded_iff
thf(fact_116_subset__empty, axiom,
    ((![A2 : set_real]: ((ord_less_eq_set_real @ A2 @ bot_bot_set_real) = (A2 = bot_bot_set_real))))). % subset_empty
thf(fact_117_empty__subsetI, axiom,
    ((![A2 : set_real]: (ord_less_eq_set_real @ bot_bot_set_real @ A2)))). % empty_subsetI
thf(fact_118_singletonI, axiom,
    ((![A : real]: (member_real @ A @ (insert_real @ A @ bot_bot_set_real))))). % singletonI
thf(fact_119_sup__bot__left, axiom,
    ((![X4 : set_real]: ((sup_sup_set_real @ bot_bot_set_real @ X4) = X4)))). % sup_bot_left
thf(fact_120_sup__bot__right, axiom,
    ((![X4 : set_real]: ((sup_sup_set_real @ X4 @ bot_bot_set_real) = X4)))). % sup_bot_right
thf(fact_121_bot__eq__sup__iff, axiom,
    ((![X4 : set_real, Y : set_real]: ((bot_bot_set_real = (sup_sup_set_real @ X4 @ Y)) = (((X4 = bot_bot_set_real)) & ((Y = bot_bot_set_real))))))). % bot_eq_sup_iff
thf(fact_122_sup__eq__bot__iff, axiom,
    ((![X4 : set_real, Y : set_real]: (((sup_sup_set_real @ X4 @ Y) = bot_bot_set_real) = (((X4 = bot_bot_set_real)) & ((Y = bot_bot_set_real))))))). % sup_eq_bot_iff
thf(fact_123_sup__bot_Oeq__neutr__iff, axiom,
    ((![A : set_real, B : set_real]: (((sup_sup_set_real @ A @ B) = bot_bot_set_real) = (((A = bot_bot_set_real)) & ((B = bot_bot_set_real))))))). % sup_bot.eq_neutr_iff
thf(fact_124_sup__bot_Oleft__neutral, axiom,
    ((![A : set_real]: ((sup_sup_set_real @ bot_bot_set_real @ A) = A)))). % sup_bot.left_neutral
thf(fact_125_sup__bot_Oneutr__eq__iff, axiom,
    ((![A : set_real, B : set_real]: ((bot_bot_set_real = (sup_sup_set_real @ A @ B)) = (((A = bot_bot_set_real)) & ((B = bot_bot_set_real))))))). % sup_bot.neutr_eq_iff
thf(fact_126_psubsetD, axiom,
    ((![A2 : set_real, B2 : set_real, C2 : real]: ((ord_less_set_real @ A2 @ B2) => ((member_real @ C2 @ A2) => (member_real @ C2 @ B2)))))). % psubsetD
thf(fact_127_bot__set__def, axiom,
    ((bot_bot_set_real = (collect_real @ bot_bot_real_o)))). % bot_set_def
thf(fact_128_emptyE, axiom,
    ((![A : real]: (~ ((member_real @ A @ bot_bot_set_real)))))). % emptyE
thf(fact_129_equals0D, axiom,
    ((![A2 : set_real, A : real]: ((A2 = bot_bot_set_real) => (~ ((member_real @ A @ A2))))))). % equals0D
thf(fact_130_equals0I, axiom,
    ((![A2 : set_real]: ((![Y2 : real]: (~ ((member_real @ Y2 @ A2)))) => (A2 = bot_bot_set_real))))). % equals0I
thf(fact_131_ex__in__conv, axiom,
    ((![A2 : set_real]: ((?[X3 : real]: (member_real @ X3 @ A2)) = (~ ((A2 = bot_bot_set_real))))))). % ex_in_conv
thf(fact_132_not__psubset__empty, axiom,
    ((![A2 : set_real]: (~ ((ord_less_set_real @ A2 @ bot_bot_set_real)))))). % not_psubset_empty
thf(fact_133_in__mono, axiom,
    ((![A2 : set_real, B2 : set_real, X4 : real]: ((ord_less_eq_set_real @ A2 @ B2) => ((member_real @ X4 @ A2) => (member_real @ X4 @ B2)))))). % in_mono
thf(fact_134_subsetD, axiom,
    ((![A2 : set_real, B2 : set_real, C2 : real]: ((ord_less_eq_set_real @ A2 @ B2) => ((member_real @ C2 @ A2) => (member_real @ C2 @ B2)))))). % subsetD
thf(fact_135_subset__eq, axiom,
    ((ord_less_eq_set_real = (^[A3 : set_real]: (^[B4 : set_real]: (![X3 : real]: (((member_real @ X3 @ A3)) => ((member_real @ X3 @ B4))))))))). % subset_eq
thf(fact_136_subset__iff, axiom,
    ((ord_less_eq_set_real = (^[A3 : set_real]: (^[B4 : set_real]: (![T : real]: (((member_real @ T @ A3)) => ((member_real @ T @ B4))))))))). % subset_iff
thf(fact_137_Collect__mono, axiom,
    ((![P : real > $o, Q : real > $o]: ((![X2 : real]: ((P @ X2) => (Q @ X2))) => (ord_less_eq_set_real @ (collect_real @ P) @ (collect_real @ Q)))))). % Collect_mono

% Conjectures (1)
thf(conj_0, conjecture,
    ((condit1201756488e_real @ (collect_real @ p)))).
