% 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/Hoare/prob_369__3253656_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:14:04.686

% Could-be-implicit typings (10)
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Com__Opname_J_J, type,
    set_set_pname : $tType).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J, type,
    set_set_a : $tType).
thf(ty_n_t__Option__Ooption_It__Com__Ocom_J, type,
    option_com : $tType).
thf(ty_n_t__Set__Oset_It__Com__Opname_J, type,
    set_pname : $tType).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J, type,
    set_nat : $tType).
thf(ty_n_t__Set__Oset_Itf__a_J, type,
    set_a : $tType).
thf(ty_n_t__Com__Opname, type,
    pname : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).
thf(ty_n_t__Com__Ocom, type,
    com : $tType).
thf(ty_n_tf__a, type,
    a : $tType).

% Explicit typings (38)
thf(sy_c_Com_Obody, type,
    body : pname > option_com).
thf(sy_c_Finite__Set_Ocard_001t__Com__Opname, type,
    finite_card_pname : set_pname > nat).
thf(sy_c_Finite__Set_Ocard_001tf__a, type,
    finite_card_a : set_a > nat).
thf(sy_c_Finite__Set_Ofinite_001t__Com__Opname, type,
    finite_finite_pname : set_pname > $o).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat, type,
    finite_finite_nat : set_nat > $o).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Com__Opname_J, type,
    finite505202775_pname : set_set_pname > $o).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_Itf__a_J, type,
    finite_finite_set_a : set_set_a > $o).
thf(sy_c_Finite__Set_Ofinite_001tf__a, type,
    finite_finite_a : set_a > $o).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat, type,
    minus_minus_nat : nat > nat > nat).
thf(sy_c_Option_Ooption_Othe_001t__Com__Ocom, type,
    the_com : option_com > com).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Com__Opname_J, type,
    bot_bot_set_pname : set_pname).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J, type,
    bot_bot_set_nat : set_nat).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Com__Opname_J_J, type,
    bot_bo1397849354_pname : set_set_pname).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_Itf__a_J_J, type,
    bot_bot_set_set_a : set_set_a).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J, type,
    bot_bot_set_a : set_a).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat, type,
    ord_less_eq_nat : nat > nat > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Com__Opname_J, type,
    ord_le865024672_pname : set_pname > set_pname > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J, type,
    ord_less_eq_set_a : set_a > set_a > $o).
thf(sy_c_Set_OCollect_001t__Com__Opname, type,
    collect_pname : (pname > $o) > set_pname).
thf(sy_c_Set_OCollect_001tf__a, type,
    collect_a : (a > $o) > set_a).
thf(sy_c_Set_Oimage_001t__Com__Opname_001t__Com__Opname, type,
    image_pname_pname : (pname > pname) > set_pname > set_pname).
thf(sy_c_Set_Oimage_001t__Com__Opname_001tf__a, type,
    image_pname_a : (pname > a) > set_pname > set_a).
thf(sy_c_Set_Oimage_001tf__a_001t__Com__Opname, type,
    image_a_pname : (a > pname) > set_a > set_pname).
thf(sy_c_Set_Oimage_001tf__a_001tf__a, type,
    image_a_a : (a > a) > set_a > set_a).
thf(sy_c_Set_Oinsert_001t__Com__Opname, type,
    insert_pname : pname > set_pname > set_pname).
thf(sy_c_Set_Oinsert_001tf__a, type,
    insert_a : a > set_a > set_a).
thf(sy_c_member_001t__Com__Opname, type,
    member_pname : pname > set_pname > $o).
thf(sy_c_member_001t__Nat__Onat, type,
    member_nat : nat > set_nat > $o).
thf(sy_c_member_001t__Set__Oset_It__Com__Opname_J, type,
    member_set_pname : set_pname > set_set_pname > $o).
thf(sy_c_member_001t__Set__Oset_Itf__a_J, type,
    member_set_a : set_a > set_set_a > $o).
thf(sy_c_member_001tf__a, type,
    member_a : a > set_a > $o).
thf(sy_v_P, type,
    p : set_a > set_a > $o).
thf(sy_v_U, type,
    u : set_pname).
thf(sy_v_mgt, type,
    mgt : com > a).
thf(sy_v_mgt__call, type,
    mgt_call : pname > a).
thf(sy_v_n, type,
    n : nat).
thf(sy_v_uG, type,
    uG : set_a).
thf(sy_v_wt, type,
    wt : com > $o).

% Relevant facts (248)
thf(fact_0_assms_I1_J, axiom,
    ((![Ts : set_a, G : set_a]: ((ord_less_eq_set_a @ Ts @ G) => (p @ G @ Ts))))). % assms(1)
thf(fact_1_assms_I3_J, axiom,
    ((![C : com, G : set_a]: ((wt @ C) => ((![X : pname]: ((member_pname @ X @ u) => (p @ G @ (insert_a @ (mgt_call @ X) @ bot_bot_set_a)))) => (p @ G @ (insert_a @ (mgt @ C) @ bot_bot_set_a))))))). % assms(3)
thf(fact_2_singleton__insert__inj__eq, axiom,
    ((![B : pname, A : pname, A2 : set_pname]: (((insert_pname @ B @ bot_bot_set_pname) = (insert_pname @ A @ A2)) = (((A = B)) & ((ord_le865024672_pname @ A2 @ (insert_pname @ B @ bot_bot_set_pname)))))))). % singleton_insert_inj_eq
thf(fact_3_singleton__insert__inj__eq, axiom,
    ((![B : a, A : a, A2 : set_a]: (((insert_a @ B @ bot_bot_set_a) = (insert_a @ A @ A2)) = (((A = B)) & ((ord_less_eq_set_a @ A2 @ (insert_a @ B @ bot_bot_set_a)))))))). % singleton_insert_inj_eq
thf(fact_4_singleton__insert__inj__eq_H, axiom,
    ((![A : pname, A2 : set_pname, B : pname]: (((insert_pname @ A @ A2) = (insert_pname @ B @ bot_bot_set_pname)) = (((A = B)) & ((ord_le865024672_pname @ A2 @ (insert_pname @ B @ bot_bot_set_pname)))))))). % singleton_insert_inj_eq'
thf(fact_5_singleton__insert__inj__eq_H, axiom,
    ((![A : a, A2 : set_a, B : a]: (((insert_a @ A @ A2) = (insert_a @ B @ bot_bot_set_a)) = (((A = B)) & ((ord_less_eq_set_a @ A2 @ (insert_a @ B @ bot_bot_set_a)))))))). % singleton_insert_inj_eq'
thf(fact_6_diff__diff__cancel, axiom,
    ((![I : nat, N : nat]: ((ord_less_eq_nat @ I @ N) => ((minus_minus_nat @ N @ (minus_minus_nat @ N @ I)) = I))))). % diff_diff_cancel
thf(fact_7_surj__card__le, axiom,
    ((![A2 : set_pname, B2 : set_pname, F : pname > pname]: ((finite_finite_pname @ A2) => ((ord_le865024672_pname @ B2 @ (image_pname_pname @ F @ A2)) => (ord_less_eq_nat @ (finite_card_pname @ B2) @ (finite_card_pname @ A2))))))). % surj_card_le
thf(fact_8_surj__card__le, axiom,
    ((![A2 : set_a, B2 : set_pname, F : a > pname]: ((finite_finite_a @ A2) => ((ord_le865024672_pname @ B2 @ (image_a_pname @ F @ A2)) => (ord_less_eq_nat @ (finite_card_pname @ B2) @ (finite_card_a @ A2))))))). % surj_card_le
thf(fact_9_surj__card__le, axiom,
    ((![A2 : set_a, B2 : set_a, F : a > a]: ((finite_finite_a @ A2) => ((ord_less_eq_set_a @ B2 @ (image_a_a @ F @ A2)) => (ord_less_eq_nat @ (finite_card_a @ B2) @ (finite_card_a @ A2))))))). % surj_card_le
thf(fact_10_surj__card__le, axiom,
    ((![A2 : set_pname, B2 : set_a, F : pname > a]: ((finite_finite_pname @ A2) => ((ord_less_eq_set_a @ B2 @ (image_pname_a @ F @ A2)) => (ord_less_eq_nat @ (finite_card_a @ B2) @ (finite_card_pname @ A2))))))). % surj_card_le
thf(fact_11_insert__subset, axiom,
    ((![X2 : pname, A2 : set_pname, B2 : set_pname]: ((ord_le865024672_pname @ (insert_pname @ X2 @ A2) @ B2) = (((member_pname @ X2 @ B2)) & ((ord_le865024672_pname @ A2 @ B2))))))). % insert_subset
thf(fact_12_insert__subset, axiom,
    ((![X2 : a, A2 : set_a, B2 : set_a]: ((ord_less_eq_set_a @ (insert_a @ X2 @ A2) @ B2) = (((member_a @ X2 @ B2)) & ((ord_less_eq_set_a @ A2 @ B2))))))). % insert_subset
thf(fact_13_finite__insert, axiom,
    ((![A : a, A2 : set_a]: ((finite_finite_a @ (insert_a @ A @ A2)) = (finite_finite_a @ A2))))). % finite_insert
thf(fact_14_finite__insert, axiom,
    ((![A : pname, A2 : set_pname]: ((finite_finite_pname @ (insert_pname @ A @ A2)) = (finite_finite_pname @ A2))))). % finite_insert
thf(fact_15_singletonI, axiom,
    ((![A : pname]: (member_pname @ A @ (insert_pname @ A @ bot_bot_set_pname))))). % singletonI
thf(fact_16_singletonI, axiom,
    ((![A : a]: (member_a @ A @ (insert_a @ A @ bot_bot_set_a))))). % singletonI
thf(fact_17_subset__empty, axiom,
    ((![A2 : set_pname]: ((ord_le865024672_pname @ A2 @ bot_bot_set_pname) = (A2 = bot_bot_set_pname))))). % subset_empty
thf(fact_18_subset__empty, axiom,
    ((![A2 : set_a]: ((ord_less_eq_set_a @ A2 @ bot_bot_set_a) = (A2 = bot_bot_set_a))))). % subset_empty
thf(fact_19_empty__subsetI, axiom,
    ((![A2 : set_a]: (ord_less_eq_set_a @ bot_bot_set_a @ A2)))). % empty_subsetI
thf(fact_20_empty__subsetI, axiom,
    ((![A2 : set_pname]: (ord_le865024672_pname @ bot_bot_set_pname @ A2)))). % empty_subsetI
thf(fact_21_image__insert, axiom,
    ((![F : a > a, A : a, B2 : set_a]: ((image_a_a @ F @ (insert_a @ A @ B2)) = (insert_a @ (F @ A) @ (image_a_a @ F @ B2)))))). % image_insert
thf(fact_22_image__insert, axiom,
    ((![F : a > pname, A : a, B2 : set_a]: ((image_a_pname @ F @ (insert_a @ A @ B2)) = (insert_pname @ (F @ A) @ (image_a_pname @ F @ B2)))))). % image_insert
thf(fact_23_image__insert, axiom,
    ((![F : pname > a, A : pname, B2 : set_pname]: ((image_pname_a @ F @ (insert_pname @ A @ B2)) = (insert_a @ (F @ A) @ (image_pname_a @ F @ B2)))))). % image_insert
thf(fact_24_image__insert, axiom,
    ((![F : pname > pname, A : pname, B2 : set_pname]: ((image_pname_pname @ F @ (insert_pname @ A @ B2)) = (insert_pname @ (F @ A) @ (image_pname_pname @ F @ B2)))))). % image_insert
thf(fact_25_insert__image, axiom,
    ((![X2 : pname, A2 : set_pname, F : pname > a]: ((member_pname @ X2 @ A2) => ((insert_a @ (F @ X2) @ (image_pname_a @ F @ A2)) = (image_pname_a @ F @ A2)))))). % insert_image
thf(fact_26_insert__image, axiom,
    ((![X2 : pname, A2 : set_pname, F : pname > pname]: ((member_pname @ X2 @ A2) => ((insert_pname @ (F @ X2) @ (image_pname_pname @ F @ A2)) = (image_pname_pname @ F @ A2)))))). % insert_image
thf(fact_27_insert__image, axiom,
    ((![X2 : a, A2 : set_a, F : a > a]: ((member_a @ X2 @ A2) => ((insert_a @ (F @ X2) @ (image_a_a @ F @ A2)) = (image_a_a @ F @ A2)))))). % insert_image
thf(fact_28_insert__image, axiom,
    ((![X2 : a, A2 : set_a, F : a > pname]: ((member_a @ X2 @ A2) => ((insert_pname @ (F @ X2) @ (image_a_pname @ F @ A2)) = (image_a_pname @ F @ A2)))))). % insert_image
thf(fact_29_finite__imageI, axiom,
    ((![F2 : set_pname, H : pname > pname]: ((finite_finite_pname @ F2) => (finite_finite_pname @ (image_pname_pname @ H @ F2)))))). % finite_imageI
thf(fact_30_finite__imageI, axiom,
    ((![F2 : set_pname, H : pname > a]: ((finite_finite_pname @ F2) => (finite_finite_a @ (image_pname_a @ H @ F2)))))). % finite_imageI
thf(fact_31_finite__imageI, axiom,
    ((![F2 : set_a, H : a > pname]: ((finite_finite_a @ F2) => (finite_finite_pname @ (image_a_pname @ H @ F2)))))). % finite_imageI
thf(fact_32_finite__imageI, axiom,
    ((![F2 : set_a, H : a > a]: ((finite_finite_a @ F2) => (finite_finite_a @ (image_a_a @ H @ F2)))))). % finite_imageI
thf(fact_33_image__eqI, axiom,
    ((![B : pname, F : pname > pname, X2 : pname, A2 : set_pname]: ((B = (F @ X2)) => ((member_pname @ X2 @ A2) => (member_pname @ B @ (image_pname_pname @ F @ A2))))))). % image_eqI
thf(fact_34_image__eqI, axiom,
    ((![B : a, F : pname > a, X2 : pname, A2 : set_pname]: ((B = (F @ X2)) => ((member_pname @ X2 @ A2) => (member_a @ B @ (image_pname_a @ F @ A2))))))). % image_eqI
thf(fact_35_image__eqI, axiom,
    ((![B : pname, F : a > pname, X2 : a, A2 : set_a]: ((B = (F @ X2)) => ((member_a @ X2 @ A2) => (member_pname @ B @ (image_a_pname @ F @ A2))))))). % image_eqI
thf(fact_36_image__eqI, axiom,
    ((![B : a, F : a > a, X2 : a, A2 : set_a]: ((B = (F @ X2)) => ((member_a @ X2 @ A2) => (member_a @ B @ (image_a_a @ F @ A2))))))). % image_eqI
thf(fact_37_empty__Collect__eq, axiom,
    ((![P : a > $o]: ((bot_bot_set_a = (collect_a @ P)) = (![X3 : a]: (~ ((P @ X3)))))))). % empty_Collect_eq
thf(fact_38_empty__Collect__eq, axiom,
    ((![P : pname > $o]: ((bot_bot_set_pname = (collect_pname @ P)) = (![X3 : pname]: (~ ((P @ X3)))))))). % empty_Collect_eq
thf(fact_39_Collect__empty__eq, axiom,
    ((![P : a > $o]: (((collect_a @ P) = bot_bot_set_a) = (![X3 : a]: (~ ((P @ X3)))))))). % Collect_empty_eq
thf(fact_40_Collect__empty__eq, axiom,
    ((![P : pname > $o]: (((collect_pname @ P) = bot_bot_set_pname) = (![X3 : pname]: (~ ((P @ X3)))))))). % Collect_empty_eq
thf(fact_41_all__not__in__conv, axiom,
    ((![A2 : set_a]: ((![X3 : a]: (~ ((member_a @ X3 @ A2)))) = (A2 = bot_bot_set_a))))). % all_not_in_conv
thf(fact_42_all__not__in__conv, axiom,
    ((![A2 : set_pname]: ((![X3 : pname]: (~ ((member_pname @ X3 @ A2)))) = (A2 = bot_bot_set_pname))))). % all_not_in_conv
thf(fact_43_empty__iff, axiom,
    ((![C : a]: (~ ((member_a @ C @ bot_bot_set_a)))))). % empty_iff
thf(fact_44_empty__iff, axiom,
    ((![C : pname]: (~ ((member_pname @ C @ bot_bot_set_pname)))))). % empty_iff
thf(fact_45_subset__antisym, axiom,
    ((![A2 : set_a, B2 : set_a]: ((ord_less_eq_set_a @ A2 @ B2) => ((ord_less_eq_set_a @ B2 @ A2) => (A2 = B2)))))). % subset_antisym
thf(fact_46_subset__antisym, axiom,
    ((![A2 : set_pname, B2 : set_pname]: ((ord_le865024672_pname @ A2 @ B2) => ((ord_le865024672_pname @ B2 @ A2) => (A2 = B2)))))). % subset_antisym
thf(fact_47_subsetI, axiom,
    ((![A2 : set_a, B2 : set_a]: ((![X : a]: ((member_a @ X @ A2) => (member_a @ X @ B2))) => (ord_less_eq_set_a @ A2 @ B2))))). % subsetI
thf(fact_48_subsetI, axiom,
    ((![A2 : set_pname, B2 : set_pname]: ((![X : pname]: ((member_pname @ X @ A2) => (member_pname @ X @ B2))) => (ord_le865024672_pname @ A2 @ B2))))). % subsetI
thf(fact_49_insert__absorb2, axiom,
    ((![X2 : a, A2 : set_a]: ((insert_a @ X2 @ (insert_a @ X2 @ A2)) = (insert_a @ X2 @ A2))))). % insert_absorb2
thf(fact_50_insert__absorb2, axiom,
    ((![X2 : pname, A2 : set_pname]: ((insert_pname @ X2 @ (insert_pname @ X2 @ A2)) = (insert_pname @ X2 @ A2))))). % insert_absorb2
thf(fact_51_insert__iff, axiom,
    ((![A : pname, B : pname, A2 : set_pname]: ((member_pname @ A @ (insert_pname @ B @ A2)) = (((A = B)) | ((member_pname @ A @ A2))))))). % insert_iff
thf(fact_52_insert__iff, axiom,
    ((![A : a, B : a, A2 : set_a]: ((member_a @ A @ (insert_a @ B @ A2)) = (((A = B)) | ((member_a @ A @ A2))))))). % insert_iff
thf(fact_53_insertCI, axiom,
    ((![A : pname, B2 : set_pname, B : pname]: (((~ ((member_pname @ A @ B2))) => (A = B)) => (member_pname @ A @ (insert_pname @ B @ B2)))))). % insertCI
thf(fact_54_insertCI, axiom,
    ((![A : a, B2 : set_a, B : a]: (((~ ((member_a @ A @ B2))) => (A = B)) => (member_a @ A @ (insert_a @ B @ B2)))))). % insertCI
thf(fact_55_image__is__empty, axiom,
    ((![F : a > a, A2 : set_a]: (((image_a_a @ F @ A2) = bot_bot_set_a) = (A2 = bot_bot_set_a))))). % image_is_empty
thf(fact_56_image__is__empty, axiom,
    ((![F : pname > a, A2 : set_pname]: (((image_pname_a @ F @ A2) = bot_bot_set_a) = (A2 = bot_bot_set_pname))))). % image_is_empty
thf(fact_57_image__is__empty, axiom,
    ((![F : a > pname, A2 : set_a]: (((image_a_pname @ F @ A2) = bot_bot_set_pname) = (A2 = bot_bot_set_a))))). % image_is_empty
thf(fact_58_image__is__empty, axiom,
    ((![F : pname > pname, A2 : set_pname]: (((image_pname_pname @ F @ A2) = bot_bot_set_pname) = (A2 = bot_bot_set_pname))))). % image_is_empty
thf(fact_59_empty__is__image, axiom,
    ((![F : a > a, A2 : set_a]: ((bot_bot_set_a = (image_a_a @ F @ A2)) = (A2 = bot_bot_set_a))))). % empty_is_image
thf(fact_60_empty__is__image, axiom,
    ((![F : pname > a, A2 : set_pname]: ((bot_bot_set_a = (image_pname_a @ F @ A2)) = (A2 = bot_bot_set_pname))))). % empty_is_image
thf(fact_61_empty__is__image, axiom,
    ((![F : a > pname, A2 : set_a]: ((bot_bot_set_pname = (image_a_pname @ F @ A2)) = (A2 = bot_bot_set_a))))). % empty_is_image
thf(fact_62_empty__is__image, axiom,
    ((![F : pname > pname, A2 : set_pname]: ((bot_bot_set_pname = (image_pname_pname @ F @ A2)) = (A2 = bot_bot_set_pname))))). % empty_is_image
thf(fact_63_image__empty, axiom,
    ((![F : a > a]: ((image_a_a @ F @ bot_bot_set_a) = bot_bot_set_a)))). % image_empty
thf(fact_64_image__empty, axiom,
    ((![F : a > pname]: ((image_a_pname @ F @ bot_bot_set_a) = bot_bot_set_pname)))). % image_empty
thf(fact_65_image__empty, axiom,
    ((![F : pname > a]: ((image_pname_a @ F @ bot_bot_set_pname) = bot_bot_set_a)))). % image_empty
thf(fact_66_image__empty, axiom,
    ((![F : pname > pname]: ((image_pname_pname @ F @ bot_bot_set_pname) = bot_bot_set_pname)))). % image_empty
thf(fact_67_rev__image__eqI, axiom,
    ((![X2 : pname, A2 : set_pname, B : pname, F : pname > pname]: ((member_pname @ X2 @ A2) => ((B = (F @ X2)) => (member_pname @ B @ (image_pname_pname @ F @ A2))))))). % rev_image_eqI
thf(fact_68_rev__image__eqI, axiom,
    ((![X2 : pname, A2 : set_pname, B : a, F : pname > a]: ((member_pname @ X2 @ A2) => ((B = (F @ X2)) => (member_a @ B @ (image_pname_a @ F @ A2))))))). % rev_image_eqI
thf(fact_69_rev__image__eqI, axiom,
    ((![X2 : a, A2 : set_a, B : pname, F : a > pname]: ((member_a @ X2 @ A2) => ((B = (F @ X2)) => (member_pname @ B @ (image_a_pname @ F @ A2))))))). % rev_image_eqI
thf(fact_70_rev__image__eqI, axiom,
    ((![X2 : a, A2 : set_a, B : a, F : a > a]: ((member_a @ X2 @ A2) => ((B = (F @ X2)) => (member_a @ B @ (image_a_a @ F @ A2))))))). % rev_image_eqI
thf(fact_71_ball__imageD, axiom,
    ((![F : pname > a, A2 : set_pname, P : a > $o]: ((![X : a]: ((member_a @ X @ (image_pname_a @ F @ A2)) => (P @ X))) => (![X4 : pname]: ((member_pname @ X4 @ A2) => (P @ (F @ X4)))))))). % ball_imageD
thf(fact_72_ball__imageD, axiom,
    ((![F : a > a, A2 : set_a, P : a > $o]: ((![X : a]: ((member_a @ X @ (image_a_a @ F @ A2)) => (P @ X))) => (![X4 : a]: ((member_a @ X4 @ A2) => (P @ (F @ X4)))))))). % ball_imageD
thf(fact_73_image__cong, axiom,
    ((![M : set_pname, N2 : set_pname, F : pname > a, G2 : pname > a]: ((M = N2) => ((![X : pname]: ((member_pname @ X @ N2) => ((F @ X) = (G2 @ X)))) => ((image_pname_a @ F @ M) = (image_pname_a @ G2 @ N2))))))). % image_cong
thf(fact_74_image__cong, axiom,
    ((![M : set_a, N2 : set_a, F : a > a, G2 : a > a]: ((M = N2) => ((![X : a]: ((member_a @ X @ N2) => ((F @ X) = (G2 @ X)))) => ((image_a_a @ F @ M) = (image_a_a @ G2 @ N2))))))). % image_cong
thf(fact_75_bex__imageD, axiom,
    ((![F : pname > a, A2 : set_pname, P : a > $o]: ((?[X4 : a]: ((member_a @ X4 @ (image_pname_a @ F @ A2)) & (P @ X4))) => (?[X : pname]: ((member_pname @ X @ A2) & (P @ (F @ X)))))))). % bex_imageD
thf(fact_76_bex__imageD, axiom,
    ((![F : a > a, A2 : set_a, P : a > $o]: ((?[X4 : a]: ((member_a @ X4 @ (image_a_a @ F @ A2)) & (P @ X4))) => (?[X : a]: ((member_a @ X @ A2) & (P @ (F @ X)))))))). % bex_imageD
thf(fact_77_image__iff, axiom,
    ((![Z : a, F : pname > a, A2 : set_pname]: ((member_a @ Z @ (image_pname_a @ F @ A2)) = (?[X3 : pname]: (((member_pname @ X3 @ A2)) & ((Z = (F @ X3))))))))). % image_iff
thf(fact_78_image__iff, axiom,
    ((![Z : a, F : a > a, A2 : set_a]: ((member_a @ Z @ (image_a_a @ F @ A2)) = (?[X3 : a]: (((member_a @ X3 @ A2)) & ((Z = (F @ X3))))))))). % image_iff
thf(fact_79_imageI, axiom,
    ((![X2 : pname, A2 : set_pname, F : pname > pname]: ((member_pname @ X2 @ A2) => (member_pname @ (F @ X2) @ (image_pname_pname @ F @ A2)))))). % imageI
thf(fact_80_imageI, axiom,
    ((![X2 : pname, A2 : set_pname, F : pname > a]: ((member_pname @ X2 @ A2) => (member_a @ (F @ X2) @ (image_pname_a @ F @ A2)))))). % imageI
thf(fact_81_imageI, axiom,
    ((![X2 : a, A2 : set_a, F : a > pname]: ((member_a @ X2 @ A2) => (member_pname @ (F @ X2) @ (image_a_pname @ F @ A2)))))). % imageI
thf(fact_82_imageI, axiom,
    ((![X2 : a, A2 : set_a, F : a > a]: ((member_a @ X2 @ A2) => (member_a @ (F @ X2) @ (image_a_a @ F @ A2)))))). % imageI
thf(fact_83_ex__in__conv, axiom,
    ((![A2 : set_a]: ((?[X3 : a]: (member_a @ X3 @ A2)) = (~ ((A2 = bot_bot_set_a))))))). % ex_in_conv
thf(fact_84_ex__in__conv, axiom,
    ((![A2 : set_pname]: ((?[X3 : pname]: (member_pname @ X3 @ A2)) = (~ ((A2 = bot_bot_set_pname))))))). % ex_in_conv
thf(fact_85_equals0I, axiom,
    ((![A2 : set_a]: ((![Y : a]: (~ ((member_a @ Y @ A2)))) => (A2 = bot_bot_set_a))))). % equals0I
thf(fact_86_equals0I, axiom,
    ((![A2 : set_pname]: ((![Y : pname]: (~ ((member_pname @ Y @ A2)))) => (A2 = bot_bot_set_pname))))). % equals0I
thf(fact_87_equals0D, axiom,
    ((![A2 : set_a, A : a]: ((A2 = bot_bot_set_a) => (~ ((member_a @ A @ A2))))))). % equals0D
thf(fact_88_equals0D, axiom,
    ((![A2 : set_pname, A : pname]: ((A2 = bot_bot_set_pname) => (~ ((member_pname @ A @ A2))))))). % equals0D
thf(fact_89_emptyE, axiom,
    ((![A : a]: (~ ((member_a @ A @ bot_bot_set_a)))))). % emptyE
thf(fact_90_emptyE, axiom,
    ((![A : pname]: (~ ((member_pname @ A @ bot_bot_set_pname)))))). % emptyE
thf(fact_91_Collect__mono__iff, axiom,
    ((![P : a > $o, Q : a > $o]: ((ord_less_eq_set_a @ (collect_a @ P) @ (collect_a @ Q)) = (![X3 : a]: (((P @ X3)) => ((Q @ X3)))))))). % Collect_mono_iff
thf(fact_92_Collect__mono__iff, axiom,
    ((![P : pname > $o, Q : pname > $o]: ((ord_le865024672_pname @ (collect_pname @ P) @ (collect_pname @ Q)) = (![X3 : pname]: (((P @ X3)) => ((Q @ X3)))))))). % Collect_mono_iff
thf(fact_93_set__eq__subset, axiom,
    (((^[Y2 : set_a]: (^[Z2 : set_a]: (Y2 = Z2))) = (^[A3 : set_a]: (^[B3 : set_a]: (((ord_less_eq_set_a @ A3 @ B3)) & ((ord_less_eq_set_a @ B3 @ A3)))))))). % set_eq_subset
thf(fact_94_set__eq__subset, axiom,
    (((^[Y2 : set_pname]: (^[Z2 : set_pname]: (Y2 = Z2))) = (^[A3 : set_pname]: (^[B3 : set_pname]: (((ord_le865024672_pname @ A3 @ B3)) & ((ord_le865024672_pname @ B3 @ A3)))))))). % set_eq_subset
thf(fact_95_subset__trans, axiom,
    ((![A2 : set_a, B2 : set_a, C2 : set_a]: ((ord_less_eq_set_a @ A2 @ B2) => ((ord_less_eq_set_a @ B2 @ C2) => (ord_less_eq_set_a @ A2 @ C2)))))). % subset_trans
thf(fact_96_subset__trans, axiom,
    ((![A2 : set_pname, B2 : set_pname, C2 : set_pname]: ((ord_le865024672_pname @ A2 @ B2) => ((ord_le865024672_pname @ B2 @ C2) => (ord_le865024672_pname @ A2 @ C2)))))). % subset_trans
thf(fact_97_Collect__mono, axiom,
    ((![P : a > $o, Q : a > $o]: ((![X : a]: ((P @ X) => (Q @ X))) => (ord_less_eq_set_a @ (collect_a @ P) @ (collect_a @ Q)))))). % Collect_mono
thf(fact_98_Collect__mono, axiom,
    ((![P : pname > $o, Q : pname > $o]: ((![X : pname]: ((P @ X) => (Q @ X))) => (ord_le865024672_pname @ (collect_pname @ P) @ (collect_pname @ Q)))))). % Collect_mono
thf(fact_99_subset__refl, axiom,
    ((![A2 : set_a]: (ord_less_eq_set_a @ A2 @ A2)))). % subset_refl
thf(fact_100_subset__refl, axiom,
    ((![A2 : set_pname]: (ord_le865024672_pname @ A2 @ A2)))). % subset_refl
thf(fact_101_mem__Collect__eq, axiom,
    ((![A : pname, P : pname > $o]: ((member_pname @ A @ (collect_pname @ P)) = (P @ A))))). % mem_Collect_eq
thf(fact_102_mem__Collect__eq, axiom,
    ((![A : a, P : a > $o]: ((member_a @ A @ (collect_a @ P)) = (P @ A))))). % mem_Collect_eq
thf(fact_103_Collect__mem__eq, axiom,
    ((![A2 : set_pname]: ((collect_pname @ (^[X3 : pname]: (member_pname @ X3 @ A2))) = A2)))). % Collect_mem_eq
thf(fact_104_Collect__mem__eq, axiom,
    ((![A2 : set_a]: ((collect_a @ (^[X3 : a]: (member_a @ X3 @ A2))) = A2)))). % Collect_mem_eq
thf(fact_105_subset__iff, axiom,
    ((ord_less_eq_set_a = (^[A3 : set_a]: (^[B3 : set_a]: (![T : a]: (((member_a @ T @ A3)) => ((member_a @ T @ B3))))))))). % subset_iff
thf(fact_106_subset__iff, axiom,
    ((ord_le865024672_pname = (^[A3 : set_pname]: (^[B3 : set_pname]: (![T : pname]: (((member_pname @ T @ A3)) => ((member_pname @ T @ B3))))))))). % subset_iff
thf(fact_107_equalityD2, axiom,
    ((![A2 : set_a, B2 : set_a]: ((A2 = B2) => (ord_less_eq_set_a @ B2 @ A2))))). % equalityD2
thf(fact_108_equalityD2, axiom,
    ((![A2 : set_pname, B2 : set_pname]: ((A2 = B2) => (ord_le865024672_pname @ B2 @ A2))))). % equalityD2
thf(fact_109_equalityD1, axiom,
    ((![A2 : set_a, B2 : set_a]: ((A2 = B2) => (ord_less_eq_set_a @ A2 @ B2))))). % equalityD1
thf(fact_110_equalityD1, axiom,
    ((![A2 : set_pname, B2 : set_pname]: ((A2 = B2) => (ord_le865024672_pname @ A2 @ B2))))). % equalityD1
thf(fact_111_subset__eq, axiom,
    ((ord_less_eq_set_a = (^[A3 : set_a]: (^[B3 : set_a]: (![X3 : a]: (((member_a @ X3 @ A3)) => ((member_a @ X3 @ B3))))))))). % subset_eq
thf(fact_112_subset__eq, axiom,
    ((ord_le865024672_pname = (^[A3 : set_pname]: (^[B3 : set_pname]: (![X3 : pname]: (((member_pname @ X3 @ A3)) => ((member_pname @ X3 @ B3))))))))). % subset_eq
thf(fact_113_equalityE, axiom,
    ((![A2 : set_a, B2 : set_a]: ((A2 = B2) => (~ (((ord_less_eq_set_a @ A2 @ B2) => (~ ((ord_less_eq_set_a @ B2 @ A2)))))))))). % equalityE
thf(fact_114_equalityE, axiom,
    ((![A2 : set_pname, B2 : set_pname]: ((A2 = B2) => (~ (((ord_le865024672_pname @ A2 @ B2) => (~ ((ord_le865024672_pname @ B2 @ A2)))))))))). % equalityE
thf(fact_115_subsetD, axiom,
    ((![A2 : set_a, B2 : set_a, C : a]: ((ord_less_eq_set_a @ A2 @ B2) => ((member_a @ C @ A2) => (member_a @ C @ B2)))))). % subsetD
thf(fact_116_subsetD, axiom,
    ((![A2 : set_pname, B2 : set_pname, C : pname]: ((ord_le865024672_pname @ A2 @ B2) => ((member_pname @ C @ A2) => (member_pname @ C @ B2)))))). % subsetD
thf(fact_117_in__mono, axiom,
    ((![A2 : set_a, B2 : set_a, X2 : a]: ((ord_less_eq_set_a @ A2 @ B2) => ((member_a @ X2 @ A2) => (member_a @ X2 @ B2)))))). % in_mono
thf(fact_118_in__mono, axiom,
    ((![A2 : set_pname, B2 : set_pname, X2 : pname]: ((ord_le865024672_pname @ A2 @ B2) => ((member_pname @ X2 @ A2) => (member_pname @ X2 @ B2)))))). % in_mono
thf(fact_119_mk__disjoint__insert, axiom,
    ((![A : pname, A2 : set_pname]: ((member_pname @ A @ A2) => (?[B4 : set_pname]: ((A2 = (insert_pname @ A @ B4)) & (~ ((member_pname @ A @ B4))))))))). % mk_disjoint_insert
thf(fact_120_mk__disjoint__insert, axiom,
    ((![A : a, A2 : set_a]: ((member_a @ A @ A2) => (?[B4 : set_a]: ((A2 = (insert_a @ A @ B4)) & (~ ((member_a @ A @ B4))))))))). % mk_disjoint_insert
thf(fact_121_insert__commute, axiom,
    ((![X2 : a, Y3 : a, A2 : set_a]: ((insert_a @ X2 @ (insert_a @ Y3 @ A2)) = (insert_a @ Y3 @ (insert_a @ X2 @ A2)))))). % insert_commute
thf(fact_122_insert__commute, axiom,
    ((![X2 : pname, Y3 : pname, A2 : set_pname]: ((insert_pname @ X2 @ (insert_pname @ Y3 @ A2)) = (insert_pname @ Y3 @ (insert_pname @ X2 @ A2)))))). % insert_commute
thf(fact_123_insert__eq__iff, axiom,
    ((![A : pname, A2 : set_pname, B : pname, B2 : set_pname]: ((~ ((member_pname @ A @ A2))) => ((~ ((member_pname @ B @ B2))) => (((insert_pname @ A @ A2) = (insert_pname @ B @ B2)) = (((((A = B)) => ((A2 = B2)))) & ((((~ ((A = B)))) => ((?[C3 : set_pname]: (((A2 = (insert_pname @ B @ C3))) & ((((~ ((member_pname @ B @ C3)))) & ((((B2 = (insert_pname @ A @ C3))) & ((~ ((member_pname @ A @ C3)))))))))))))))))))). % insert_eq_iff
thf(fact_124_insert__eq__iff, axiom,
    ((![A : a, A2 : set_a, B : a, B2 : set_a]: ((~ ((member_a @ A @ A2))) => ((~ ((member_a @ B @ B2))) => (((insert_a @ A @ A2) = (insert_a @ B @ B2)) = (((((A = B)) => ((A2 = B2)))) & ((((~ ((A = B)))) => ((?[C3 : set_a]: (((A2 = (insert_a @ B @ C3))) & ((((~ ((member_a @ B @ C3)))) & ((((B2 = (insert_a @ A @ C3))) & ((~ ((member_a @ A @ C3)))))))))))))))))))). % insert_eq_iff
thf(fact_125_insert__absorb, axiom,
    ((![A : pname, A2 : set_pname]: ((member_pname @ A @ A2) => ((insert_pname @ A @ A2) = A2))))). % insert_absorb
thf(fact_126_insert__absorb, axiom,
    ((![A : a, A2 : set_a]: ((member_a @ A @ A2) => ((insert_a @ A @ A2) = A2))))). % insert_absorb
thf(fact_127_insert__ident, axiom,
    ((![X2 : pname, A2 : set_pname, B2 : set_pname]: ((~ ((member_pname @ X2 @ A2))) => ((~ ((member_pname @ X2 @ B2))) => (((insert_pname @ X2 @ A2) = (insert_pname @ X2 @ B2)) = (A2 = B2))))))). % insert_ident
thf(fact_128_insert__ident, axiom,
    ((![X2 : a, A2 : set_a, B2 : set_a]: ((~ ((member_a @ X2 @ A2))) => ((~ ((member_a @ X2 @ B2))) => (((insert_a @ X2 @ A2) = (insert_a @ X2 @ B2)) = (A2 = B2))))))). % insert_ident
thf(fact_129_Set_Oset__insert, axiom,
    ((![X2 : pname, A2 : set_pname]: ((member_pname @ X2 @ A2) => (~ ((![B4 : set_pname]: ((A2 = (insert_pname @ X2 @ B4)) => (member_pname @ X2 @ B4))))))))). % Set.set_insert
thf(fact_130_Set_Oset__insert, axiom,
    ((![X2 : a, A2 : set_a]: ((member_a @ X2 @ A2) => (~ ((![B4 : set_a]: ((A2 = (insert_a @ X2 @ B4)) => (member_a @ X2 @ B4))))))))). % Set.set_insert
thf(fact_131_insertI2, axiom,
    ((![A : pname, B2 : set_pname, B : pname]: ((member_pname @ A @ B2) => (member_pname @ A @ (insert_pname @ B @ B2)))))). % insertI2
thf(fact_132_insertI2, axiom,
    ((![A : a, B2 : set_a, B : a]: ((member_a @ A @ B2) => (member_a @ A @ (insert_a @ B @ B2)))))). % insertI2
thf(fact_133_insertI1, axiom,
    ((![A : pname, B2 : set_pname]: (member_pname @ A @ (insert_pname @ A @ B2))))). % insertI1
thf(fact_134_insertI1, axiom,
    ((![A : a, B2 : set_a]: (member_a @ A @ (insert_a @ A @ B2))))). % insertI1
thf(fact_135_insertE, axiom,
    ((![A : pname, B : pname, A2 : set_pname]: ((member_pname @ A @ (insert_pname @ B @ A2)) => ((~ ((A = B))) => (member_pname @ A @ A2)))))). % insertE
thf(fact_136_insertE, axiom,
    ((![A : a, B : a, A2 : set_a]: ((member_a @ A @ (insert_a @ B @ A2)) => ((~ ((A = B))) => (member_a @ A @ A2)))))). % insertE
thf(fact_137_Nat_Oex__has__greatest__nat, axiom,
    ((![P : nat > $o, K : nat, B : nat]: ((P @ K) => ((![Y : nat]: ((P @ Y) => (ord_less_eq_nat @ Y @ B))) => (?[X : nat]: ((P @ X) & (![Y4 : nat]: ((P @ Y4) => (ord_less_eq_nat @ Y4 @ X)))))))))). % Nat.ex_has_greatest_nat
thf(fact_138_nat__le__linear, axiom,
    ((![M2 : nat, N : nat]: ((ord_less_eq_nat @ M2 @ N) | (ord_less_eq_nat @ N @ M2))))). % nat_le_linear
thf(fact_139_le__antisym, axiom,
    ((![M2 : nat, N : nat]: ((ord_less_eq_nat @ M2 @ N) => ((ord_less_eq_nat @ N @ M2) => (M2 = N)))))). % le_antisym
thf(fact_140_eq__imp__le, axiom,
    ((![M2 : nat, N : nat]: ((M2 = N) => (ord_less_eq_nat @ M2 @ N))))). % eq_imp_le
thf(fact_141_le__trans, axiom,
    ((![I : nat, J : nat, K : nat]: ((ord_less_eq_nat @ I @ J) => ((ord_less_eq_nat @ J @ K) => (ord_less_eq_nat @ I @ K)))))). % le_trans
thf(fact_142_le__refl, axiom,
    ((![N : nat]: (ord_less_eq_nat @ N @ N)))). % le_refl
thf(fact_143_diff__commute, axiom,
    ((![I : nat, J : nat, K : nat]: ((minus_minus_nat @ (minus_minus_nat @ I @ J) @ K) = (minus_minus_nat @ (minus_minus_nat @ I @ K) @ J))))). % diff_commute
thf(fact_144_finite__has__minimal2, axiom,
    ((![A2 : set_set_a, A : set_a]: ((finite_finite_set_a @ A2) => ((member_set_a @ A @ A2) => (?[X : set_a]: ((member_set_a @ X @ A2) & ((ord_less_eq_set_a @ X @ A) & (![Xa : set_a]: ((member_set_a @ Xa @ A2) => ((ord_less_eq_set_a @ Xa @ X) => (X = Xa)))))))))))). % finite_has_minimal2
thf(fact_145_finite__has__minimal2, axiom,
    ((![A2 : set_nat, A : nat]: ((finite_finite_nat @ A2) => ((member_nat @ A @ A2) => (?[X : nat]: ((member_nat @ X @ A2) & ((ord_less_eq_nat @ X @ A) & (![Xa : nat]: ((member_nat @ Xa @ A2) => ((ord_less_eq_nat @ Xa @ X) => (X = Xa)))))))))))). % finite_has_minimal2
thf(fact_146_finite__has__minimal2, axiom,
    ((![A2 : set_set_pname, A : set_pname]: ((finite505202775_pname @ A2) => ((member_set_pname @ A @ A2) => (?[X : set_pname]: ((member_set_pname @ X @ A2) & ((ord_le865024672_pname @ X @ A) & (![Xa : set_pname]: ((member_set_pname @ Xa @ A2) => ((ord_le865024672_pname @ Xa @ X) => (X = Xa)))))))))))). % finite_has_minimal2
thf(fact_147_finite__has__maximal2, axiom,
    ((![A2 : set_set_a, A : set_a]: ((finite_finite_set_a @ A2) => ((member_set_a @ A @ A2) => (?[X : set_a]: ((member_set_a @ X @ A2) & ((ord_less_eq_set_a @ A @ X) & (![Xa : set_a]: ((member_set_a @ Xa @ A2) => ((ord_less_eq_set_a @ X @ Xa) => (X = Xa)))))))))))). % finite_has_maximal2
thf(fact_148_finite__has__maximal2, axiom,
    ((![A2 : set_nat, A : nat]: ((finite_finite_nat @ A2) => ((member_nat @ A @ A2) => (?[X : nat]: ((member_nat @ X @ A2) & ((ord_less_eq_nat @ A @ X) & (![Xa : nat]: ((member_nat @ Xa @ A2) => ((ord_less_eq_nat @ X @ Xa) => (X = Xa)))))))))))). % finite_has_maximal2
thf(fact_149_finite__has__maximal2, axiom,
    ((![A2 : set_set_pname, A : set_pname]: ((finite505202775_pname @ A2) => ((member_set_pname @ A @ A2) => (?[X : set_pname]: ((member_set_pname @ X @ A2) & ((ord_le865024672_pname @ A @ X) & (![Xa : set_pname]: ((member_set_pname @ Xa @ A2) => ((ord_le865024672_pname @ X @ Xa) => (X = Xa)))))))))))). % finite_has_maximal2
thf(fact_150_infinite__imp__nonempty, axiom,
    ((![S : set_a]: ((~ ((finite_finite_a @ S))) => (~ ((S = bot_bot_set_a))))))). % infinite_imp_nonempty
thf(fact_151_infinite__imp__nonempty, axiom,
    ((![S : set_pname]: ((~ ((finite_finite_pname @ S))) => (~ ((S = bot_bot_set_pname))))))). % infinite_imp_nonempty
thf(fact_152_finite_OemptyI, axiom,
    ((finite_finite_a @ bot_bot_set_a))). % finite.emptyI
thf(fact_153_finite_OemptyI, axiom,
    ((finite_finite_pname @ bot_bot_set_pname))). % finite.emptyI
thf(fact_154_all__subset__image, axiom,
    ((![F : a > a, A2 : set_a, P : set_a > $o]: ((![B3 : set_a]: (((ord_less_eq_set_a @ B3 @ (image_a_a @ F @ A2))) => ((P @ B3)))) = (![B3 : set_a]: (((ord_less_eq_set_a @ B3 @ A2)) => ((P @ (image_a_a @ F @ B3))))))))). % all_subset_image
thf(fact_155_all__subset__image, axiom,
    ((![F : pname > a, A2 : set_pname, P : set_a > $o]: ((![B3 : set_a]: (((ord_less_eq_set_a @ B3 @ (image_pname_a @ F @ A2))) => ((P @ B3)))) = (![B3 : set_pname]: (((ord_le865024672_pname @ B3 @ A2)) => ((P @ (image_pname_a @ F @ B3))))))))). % all_subset_image
thf(fact_156_all__subset__image, axiom,
    ((![F : a > pname, A2 : set_a, P : set_pname > $o]: ((![B3 : set_pname]: (((ord_le865024672_pname @ B3 @ (image_a_pname @ F @ A2))) => ((P @ B3)))) = (![B3 : set_a]: (((ord_less_eq_set_a @ B3 @ A2)) => ((P @ (image_a_pname @ F @ B3))))))))). % all_subset_image
thf(fact_157_all__subset__image, axiom,
    ((![F : pname > pname, A2 : set_pname, P : set_pname > $o]: ((![B3 : set_pname]: (((ord_le865024672_pname @ B3 @ (image_pname_pname @ F @ A2))) => ((P @ B3)))) = (![B3 : set_pname]: (((ord_le865024672_pname @ B3 @ A2)) => ((P @ (image_pname_pname @ F @ B3))))))))). % all_subset_image
thf(fact_158_subset__image__iff, axiom,
    ((![B2 : set_a, F : a > a, A2 : set_a]: ((ord_less_eq_set_a @ B2 @ (image_a_a @ F @ A2)) = (?[AA : set_a]: (((ord_less_eq_set_a @ AA @ A2)) & ((B2 = (image_a_a @ F @ AA))))))))). % subset_image_iff
thf(fact_159_subset__image__iff, axiom,
    ((![B2 : set_a, F : pname > a, A2 : set_pname]: ((ord_less_eq_set_a @ B2 @ (image_pname_a @ F @ A2)) = (?[AA : set_pname]: (((ord_le865024672_pname @ AA @ A2)) & ((B2 = (image_pname_a @ F @ AA))))))))). % subset_image_iff
thf(fact_160_subset__image__iff, axiom,
    ((![B2 : set_pname, F : a > pname, A2 : set_a]: ((ord_le865024672_pname @ B2 @ (image_a_pname @ F @ A2)) = (?[AA : set_a]: (((ord_less_eq_set_a @ AA @ A2)) & ((B2 = (image_a_pname @ F @ AA))))))))). % subset_image_iff
thf(fact_161_subset__image__iff, axiom,
    ((![B2 : set_pname, F : pname > pname, A2 : set_pname]: ((ord_le865024672_pname @ B2 @ (image_pname_pname @ F @ A2)) = (?[AA : set_pname]: (((ord_le865024672_pname @ AA @ A2)) & ((B2 = (image_pname_pname @ F @ AA))))))))). % subset_image_iff
thf(fact_162_image__subset__iff, axiom,
    ((![F : pname > a, A2 : set_pname, B2 : set_a]: ((ord_less_eq_set_a @ (image_pname_a @ F @ A2) @ B2) = (![X3 : pname]: (((member_pname @ X3 @ A2)) => ((member_a @ (F @ X3) @ B2)))))))). % image_subset_iff
thf(fact_163_image__subset__iff, axiom,
    ((![F : a > a, A2 : set_a, B2 : set_a]: ((ord_less_eq_set_a @ (image_a_a @ F @ A2) @ B2) = (![X3 : a]: (((member_a @ X3 @ A2)) => ((member_a @ (F @ X3) @ B2)))))))). % image_subset_iff
thf(fact_164_subset__imageE, axiom,
    ((![B2 : set_a, F : a > a, A2 : set_a]: ((ord_less_eq_set_a @ B2 @ (image_a_a @ F @ A2)) => (~ ((![C4 : set_a]: ((ord_less_eq_set_a @ C4 @ A2) => (~ ((B2 = (image_a_a @ F @ C4)))))))))))). % subset_imageE
thf(fact_165_subset__imageE, axiom,
    ((![B2 : set_a, F : pname > a, A2 : set_pname]: ((ord_less_eq_set_a @ B2 @ (image_pname_a @ F @ A2)) => (~ ((![C4 : set_pname]: ((ord_le865024672_pname @ C4 @ A2) => (~ ((B2 = (image_pname_a @ F @ C4)))))))))))). % subset_imageE
thf(fact_166_subset__imageE, axiom,
    ((![B2 : set_pname, F : a > pname, A2 : set_a]: ((ord_le865024672_pname @ B2 @ (image_a_pname @ F @ A2)) => (~ ((![C4 : set_a]: ((ord_less_eq_set_a @ C4 @ A2) => (~ ((B2 = (image_a_pname @ F @ C4)))))))))))). % subset_imageE
thf(fact_167_subset__imageE, axiom,
    ((![B2 : set_pname, F : pname > pname, A2 : set_pname]: ((ord_le865024672_pname @ B2 @ (image_pname_pname @ F @ A2)) => (~ ((![C4 : set_pname]: ((ord_le865024672_pname @ C4 @ A2) => (~ ((B2 = (image_pname_pname @ F @ C4)))))))))))). % subset_imageE
thf(fact_168_image__subsetI, axiom,
    ((![A2 : set_pname, F : pname > a, B2 : set_a]: ((![X : pname]: ((member_pname @ X @ A2) => (member_a @ (F @ X) @ B2))) => (ord_less_eq_set_a @ (image_pname_a @ F @ A2) @ B2))))). % image_subsetI
thf(fact_169_image__subsetI, axiom,
    ((![A2 : set_a, F : a > a, B2 : set_a]: ((![X : a]: ((member_a @ X @ A2) => (member_a @ (F @ X) @ B2))) => (ord_less_eq_set_a @ (image_a_a @ F @ A2) @ B2))))). % image_subsetI
thf(fact_170_image__subsetI, axiom,
    ((![A2 : set_pname, F : pname > pname, B2 : set_pname]: ((![X : pname]: ((member_pname @ X @ A2) => (member_pname @ (F @ X) @ B2))) => (ord_le865024672_pname @ (image_pname_pname @ F @ A2) @ B2))))). % image_subsetI
thf(fact_171_image__subsetI, axiom,
    ((![A2 : set_a, F : a > pname, B2 : set_pname]: ((![X : a]: ((member_a @ X @ A2) => (member_pname @ (F @ X) @ B2))) => (ord_le865024672_pname @ (image_a_pname @ F @ A2) @ B2))))). % image_subsetI
thf(fact_172_image__mono, axiom,
    ((![A2 : set_a, B2 : set_a, F : a > a]: ((ord_less_eq_set_a @ A2 @ B2) => (ord_less_eq_set_a @ (image_a_a @ F @ A2) @ (image_a_a @ F @ B2)))))). % image_mono
thf(fact_173_image__mono, axiom,
    ((![A2 : set_a, B2 : set_a, F : a > pname]: ((ord_less_eq_set_a @ A2 @ B2) => (ord_le865024672_pname @ (image_a_pname @ F @ A2) @ (image_a_pname @ F @ B2)))))). % image_mono
thf(fact_174_image__mono, axiom,
    ((![A2 : set_pname, B2 : set_pname, F : pname > a]: ((ord_le865024672_pname @ A2 @ B2) => (ord_less_eq_set_a @ (image_pname_a @ F @ A2) @ (image_pname_a @ F @ B2)))))). % image_mono
thf(fact_175_image__mono, axiom,
    ((![A2 : set_pname, B2 : set_pname, F : pname > pname]: ((ord_le865024672_pname @ A2 @ B2) => (ord_le865024672_pname @ (image_pname_pname @ F @ A2) @ (image_pname_pname @ F @ B2)))))). % image_mono
thf(fact_176_singleton__inject, axiom,
    ((![A : a, B : a]: (((insert_a @ A @ bot_bot_set_a) = (insert_a @ B @ bot_bot_set_a)) => (A = B))))). % singleton_inject
thf(fact_177_singleton__inject, axiom,
    ((![A : pname, B : pname]: (((insert_pname @ A @ bot_bot_set_pname) = (insert_pname @ B @ bot_bot_set_pname)) => (A = B))))). % singleton_inject
thf(fact_178_insert__not__empty, axiom,
    ((![A : a, A2 : set_a]: (~ (((insert_a @ A @ A2) = bot_bot_set_a)))))). % insert_not_empty
thf(fact_179_insert__not__empty, axiom,
    ((![A : pname, A2 : set_pname]: (~ (((insert_pname @ A @ A2) = bot_bot_set_pname)))))). % insert_not_empty
thf(fact_180_doubleton__eq__iff, axiom,
    ((![A : a, B : a, C : a, D : a]: (((insert_a @ A @ (insert_a @ B @ bot_bot_set_a)) = (insert_a @ C @ (insert_a @ D @ bot_bot_set_a))) = (((((A = C)) & ((B = D)))) | ((((A = D)) & ((B = C))))))))). % doubleton_eq_iff
thf(fact_181_doubleton__eq__iff, axiom,
    ((![A : pname, B : pname, C : pname, D : pname]: (((insert_pname @ A @ (insert_pname @ B @ bot_bot_set_pname)) = (insert_pname @ C @ (insert_pname @ D @ bot_bot_set_pname))) = (((((A = C)) & ((B = D)))) | ((((A = D)) & ((B = C))))))))). % doubleton_eq_iff
thf(fact_182_singleton__iff, axiom,
    ((![B : a, A : a]: ((member_a @ B @ (insert_a @ A @ bot_bot_set_a)) = (B = A))))). % singleton_iff
thf(fact_183_singleton__iff, axiom,
    ((![B : pname, A : pname]: ((member_pname @ B @ (insert_pname @ A @ bot_bot_set_pname)) = (B = A))))). % singleton_iff
thf(fact_184_singletonD, axiom,
    ((![B : a, A : a]: ((member_a @ B @ (insert_a @ A @ bot_bot_set_a)) => (B = A))))). % singletonD
thf(fact_185_singletonD, axiom,
    ((![B : pname, A : pname]: ((member_pname @ B @ (insert_pname @ A @ bot_bot_set_pname)) => (B = A))))). % singletonD
thf(fact_186_rev__finite__subset, axiom,
    ((![B2 : set_a, A2 : set_a]: ((finite_finite_a @ B2) => ((ord_less_eq_set_a @ A2 @ B2) => (finite_finite_a @ A2)))))). % rev_finite_subset
thf(fact_187_rev__finite__subset, axiom,
    ((![B2 : set_pname, A2 : set_pname]: ((finite_finite_pname @ B2) => ((ord_le865024672_pname @ A2 @ B2) => (finite_finite_pname @ A2)))))). % rev_finite_subset
thf(fact_188_infinite__super, axiom,
    ((![S : set_a, T2 : set_a]: ((ord_less_eq_set_a @ S @ T2) => ((~ ((finite_finite_a @ S))) => (~ ((finite_finite_a @ T2)))))))). % infinite_super
thf(fact_189_infinite__super, axiom,
    ((![S : set_pname, T2 : set_pname]: ((ord_le865024672_pname @ S @ T2) => ((~ ((finite_finite_pname @ S))) => (~ ((finite_finite_pname @ T2)))))))). % infinite_super
thf(fact_190_finite__subset, axiom,
    ((![A2 : set_a, B2 : set_a]: ((ord_less_eq_set_a @ A2 @ B2) => ((finite_finite_a @ B2) => (finite_finite_a @ A2)))))). % finite_subset
thf(fact_191_finite__subset, axiom,
    ((![A2 : set_pname, B2 : set_pname]: ((ord_le865024672_pname @ A2 @ B2) => ((finite_finite_pname @ B2) => (finite_finite_pname @ A2)))))). % finite_subset
thf(fact_192_finite_OinsertI, axiom,
    ((![A2 : set_pname, A : pname]: ((finite_finite_pname @ A2) => (finite_finite_pname @ (insert_pname @ A @ A2)))))). % finite.insertI
thf(fact_193_finite_OinsertI, axiom,
    ((![A2 : set_a, A : a]: ((finite_finite_a @ A2) => (finite_finite_a @ (insert_a @ A @ A2)))))). % finite.insertI
thf(fact_194_subset__insertI2, axiom,
    ((![A2 : set_a, B2 : set_a, B : a]: ((ord_less_eq_set_a @ A2 @ B2) => (ord_less_eq_set_a @ A2 @ (insert_a @ B @ B2)))))). % subset_insertI2
thf(fact_195_subset__insertI2, axiom,
    ((![A2 : set_pname, B2 : set_pname, B : pname]: ((ord_le865024672_pname @ A2 @ B2) => (ord_le865024672_pname @ A2 @ (insert_pname @ B @ B2)))))). % subset_insertI2
thf(fact_196_subset__insertI, axiom,
    ((![B2 : set_a, A : a]: (ord_less_eq_set_a @ B2 @ (insert_a @ A @ B2))))). % subset_insertI
thf(fact_197_subset__insertI, axiom,
    ((![B2 : set_pname, A : pname]: (ord_le865024672_pname @ B2 @ (insert_pname @ A @ B2))))). % subset_insertI
thf(fact_198_subset__insert, axiom,
    ((![X2 : a, A2 : set_a, B2 : set_a]: ((~ ((member_a @ X2 @ A2))) => ((ord_less_eq_set_a @ A2 @ (insert_a @ X2 @ B2)) = (ord_less_eq_set_a @ A2 @ B2)))))). % subset_insert
thf(fact_199_subset__insert, axiom,
    ((![X2 : pname, A2 : set_pname, B2 : set_pname]: ((~ ((member_pname @ X2 @ A2))) => ((ord_le865024672_pname @ A2 @ (insert_pname @ X2 @ B2)) = (ord_le865024672_pname @ A2 @ B2)))))). % subset_insert
thf(fact_200_insert__mono, axiom,
    ((![C2 : set_a, D2 : set_a, A : a]: ((ord_less_eq_set_a @ C2 @ D2) => (ord_less_eq_set_a @ (insert_a @ A @ C2) @ (insert_a @ A @ D2)))))). % insert_mono
thf(fact_201_insert__mono, axiom,
    ((![C2 : set_pname, D2 : set_pname, A : pname]: ((ord_le865024672_pname @ C2 @ D2) => (ord_le865024672_pname @ (insert_pname @ A @ C2) @ (insert_pname @ A @ D2)))))). % insert_mono
thf(fact_202_diff__le__mono2, axiom,
    ((![M2 : nat, N : nat, L : nat]: ((ord_less_eq_nat @ M2 @ N) => (ord_less_eq_nat @ (minus_minus_nat @ L @ N) @ (minus_minus_nat @ L @ M2)))))). % diff_le_mono2
thf(fact_203_le__diff__iff_H, axiom,
    ((![A : nat, C : nat, B : nat]: ((ord_less_eq_nat @ A @ C) => ((ord_less_eq_nat @ B @ C) => ((ord_less_eq_nat @ (minus_minus_nat @ C @ A) @ (minus_minus_nat @ C @ B)) = (ord_less_eq_nat @ B @ A))))))). % le_diff_iff'
thf(fact_204_diff__le__self, axiom,
    ((![M2 : nat, N : nat]: (ord_less_eq_nat @ (minus_minus_nat @ M2 @ N) @ M2)))). % diff_le_self
thf(fact_205_diff__le__mono, axiom,
    ((![M2 : nat, N : nat, L : nat]: ((ord_less_eq_nat @ M2 @ N) => (ord_less_eq_nat @ (minus_minus_nat @ M2 @ L) @ (minus_minus_nat @ N @ L)))))). % diff_le_mono
thf(fact_206_Nat_Odiff__diff__eq, axiom,
    ((![K : nat, M2 : nat, N : nat]: ((ord_less_eq_nat @ K @ M2) => ((ord_less_eq_nat @ K @ N) => ((minus_minus_nat @ (minus_minus_nat @ M2 @ K) @ (minus_minus_nat @ N @ K)) = (minus_minus_nat @ M2 @ N))))))). % Nat.diff_diff_eq
thf(fact_207_le__diff__iff, axiom,
    ((![K : nat, M2 : nat, N : nat]: ((ord_less_eq_nat @ K @ M2) => ((ord_less_eq_nat @ K @ N) => ((ord_less_eq_nat @ (minus_minus_nat @ M2 @ K) @ (minus_minus_nat @ N @ K)) = (ord_less_eq_nat @ M2 @ N))))))). % le_diff_iff
thf(fact_208_eq__diff__iff, axiom,
    ((![K : nat, M2 : nat, N : nat]: ((ord_less_eq_nat @ K @ M2) => ((ord_less_eq_nat @ K @ N) => (((minus_minus_nat @ M2 @ K) = (minus_minus_nat @ N @ K)) = (M2 = N))))))). % eq_diff_iff
thf(fact_209_finite__has__minimal, axiom,
    ((![A2 : set_set_a]: ((finite_finite_set_a @ A2) => ((~ ((A2 = bot_bot_set_set_a))) => (?[X : set_a]: ((member_set_a @ X @ A2) & (![Xa : set_a]: ((member_set_a @ Xa @ A2) => ((ord_less_eq_set_a @ Xa @ X) => (X = Xa))))))))))). % finite_has_minimal
thf(fact_210_finite__has__minimal, axiom,
    ((![A2 : set_nat]: ((finite_finite_nat @ A2) => ((~ ((A2 = bot_bot_set_nat))) => (?[X : nat]: ((member_nat @ X @ A2) & (![Xa : nat]: ((member_nat @ Xa @ A2) => ((ord_less_eq_nat @ Xa @ X) => (X = Xa))))))))))). % finite_has_minimal
thf(fact_211_finite__has__minimal, axiom,
    ((![A2 : set_set_pname]: ((finite505202775_pname @ A2) => ((~ ((A2 = bot_bo1397849354_pname))) => (?[X : set_pname]: ((member_set_pname @ X @ A2) & (![Xa : set_pname]: ((member_set_pname @ Xa @ A2) => ((ord_le865024672_pname @ Xa @ X) => (X = Xa))))))))))). % finite_has_minimal
thf(fact_212_finite__has__maximal, axiom,
    ((![A2 : set_set_a]: ((finite_finite_set_a @ A2) => ((~ ((A2 = bot_bot_set_set_a))) => (?[X : set_a]: ((member_set_a @ X @ A2) & (![Xa : set_a]: ((member_set_a @ Xa @ A2) => ((ord_less_eq_set_a @ X @ Xa) => (X = Xa))))))))))). % finite_has_maximal
thf(fact_213_finite__has__maximal, axiom,
    ((![A2 : set_nat]: ((finite_finite_nat @ A2) => ((~ ((A2 = bot_bot_set_nat))) => (?[X : nat]: ((member_nat @ X @ A2) & (![Xa : nat]: ((member_nat @ Xa @ A2) => ((ord_less_eq_nat @ X @ Xa) => (X = Xa))))))))))). % finite_has_maximal
thf(fact_214_finite__has__maximal, axiom,
    ((![A2 : set_set_pname]: ((finite505202775_pname @ A2) => ((~ ((A2 = bot_bo1397849354_pname))) => (?[X : set_pname]: ((member_set_pname @ X @ A2) & (![Xa : set_pname]: ((member_set_pname @ Xa @ A2) => ((ord_le865024672_pname @ X @ Xa) => (X = Xa))))))))))). % finite_has_maximal
thf(fact_215_all__finite__subset__image, axiom,
    ((![F : a > a, A2 : set_a, P : set_a > $o]: ((![B3 : set_a]: (((((finite_finite_a @ B3)) & ((ord_less_eq_set_a @ B3 @ (image_a_a @ F @ A2))))) => ((P @ B3)))) = (![B3 : set_a]: (((((finite_finite_a @ B3)) & ((ord_less_eq_set_a @ B3 @ A2)))) => ((P @ (image_a_a @ F @ B3))))))))). % all_finite_subset_image
thf(fact_216_all__finite__subset__image, axiom,
    ((![F : pname > a, A2 : set_pname, P : set_a > $o]: ((![B3 : set_a]: (((((finite_finite_a @ B3)) & ((ord_less_eq_set_a @ B3 @ (image_pname_a @ F @ A2))))) => ((P @ B3)))) = (![B3 : set_pname]: (((((finite_finite_pname @ B3)) & ((ord_le865024672_pname @ B3 @ A2)))) => ((P @ (image_pname_a @ F @ B3))))))))). % all_finite_subset_image
thf(fact_217_all__finite__subset__image, axiom,
    ((![F : a > pname, A2 : set_a, P : set_pname > $o]: ((![B3 : set_pname]: (((((finite_finite_pname @ B3)) & ((ord_le865024672_pname @ B3 @ (image_a_pname @ F @ A2))))) => ((P @ B3)))) = (![B3 : set_a]: (((((finite_finite_a @ B3)) & ((ord_less_eq_set_a @ B3 @ A2)))) => ((P @ (image_a_pname @ F @ B3))))))))). % all_finite_subset_image
thf(fact_218_all__finite__subset__image, axiom,
    ((![F : pname > pname, A2 : set_pname, P : set_pname > $o]: ((![B3 : set_pname]: (((((finite_finite_pname @ B3)) & ((ord_le865024672_pname @ B3 @ (image_pname_pname @ F @ A2))))) => ((P @ B3)))) = (![B3 : set_pname]: (((((finite_finite_pname @ B3)) & ((ord_le865024672_pname @ B3 @ A2)))) => ((P @ (image_pname_pname @ F @ B3))))))))). % all_finite_subset_image
thf(fact_219_ex__finite__subset__image, axiom,
    ((![F : a > a, A2 : set_a, P : set_a > $o]: ((?[B3 : set_a]: (((finite_finite_a @ B3)) & ((((ord_less_eq_set_a @ B3 @ (image_a_a @ F @ A2))) & ((P @ B3)))))) = (?[B3 : set_a]: (((finite_finite_a @ B3)) & ((((ord_less_eq_set_a @ B3 @ A2)) & ((P @ (image_a_a @ F @ B3))))))))))). % ex_finite_subset_image
thf(fact_220_ex__finite__subset__image, axiom,
    ((![F : pname > a, A2 : set_pname, P : set_a > $o]: ((?[B3 : set_a]: (((finite_finite_a @ B3)) & ((((ord_less_eq_set_a @ B3 @ (image_pname_a @ F @ A2))) & ((P @ B3)))))) = (?[B3 : set_pname]: (((finite_finite_pname @ B3)) & ((((ord_le865024672_pname @ B3 @ A2)) & ((P @ (image_pname_a @ F @ B3))))))))))). % ex_finite_subset_image
thf(fact_221_ex__finite__subset__image, axiom,
    ((![F : a > pname, A2 : set_a, P : set_pname > $o]: ((?[B3 : set_pname]: (((finite_finite_pname @ B3)) & ((((ord_le865024672_pname @ B3 @ (image_a_pname @ F @ A2))) & ((P @ B3)))))) = (?[B3 : set_a]: (((finite_finite_a @ B3)) & ((((ord_less_eq_set_a @ B3 @ A2)) & ((P @ (image_a_pname @ F @ B3))))))))))). % ex_finite_subset_image
thf(fact_222_ex__finite__subset__image, axiom,
    ((![F : pname > pname, A2 : set_pname, P : set_pname > $o]: ((?[B3 : set_pname]: (((finite_finite_pname @ B3)) & ((((ord_le865024672_pname @ B3 @ (image_pname_pname @ F @ A2))) & ((P @ B3)))))) = (?[B3 : set_pname]: (((finite_finite_pname @ B3)) & ((((ord_le865024672_pname @ B3 @ A2)) & ((P @ (image_pname_pname @ F @ B3))))))))))). % ex_finite_subset_image
thf(fact_223_finite__subset__image, axiom,
    ((![B2 : set_a, F : a > a, A2 : set_a]: ((finite_finite_a @ B2) => ((ord_less_eq_set_a @ B2 @ (image_a_a @ F @ A2)) => (?[C4 : set_a]: ((ord_less_eq_set_a @ C4 @ A2) & ((finite_finite_a @ C4) & (B2 = (image_a_a @ F @ C4)))))))))). % finite_subset_image
thf(fact_224_finite__subset__image, axiom,
    ((![B2 : set_a, F : pname > a, A2 : set_pname]: ((finite_finite_a @ B2) => ((ord_less_eq_set_a @ B2 @ (image_pname_a @ F @ A2)) => (?[C4 : set_pname]: ((ord_le865024672_pname @ C4 @ A2) & ((finite_finite_pname @ C4) & (B2 = (image_pname_a @ F @ C4)))))))))). % finite_subset_image
thf(fact_225_finite__subset__image, axiom,
    ((![B2 : set_pname, F : a > pname, A2 : set_a]: ((finite_finite_pname @ B2) => ((ord_le865024672_pname @ B2 @ (image_a_pname @ F @ A2)) => (?[C4 : set_a]: ((ord_less_eq_set_a @ C4 @ A2) & ((finite_finite_a @ C4) & (B2 = (image_a_pname @ F @ C4)))))))))). % finite_subset_image
thf(fact_226_finite__subset__image, axiom,
    ((![B2 : set_pname, F : pname > pname, A2 : set_pname]: ((finite_finite_pname @ B2) => ((ord_le865024672_pname @ B2 @ (image_pname_pname @ F @ A2)) => (?[C4 : set_pname]: ((ord_le865024672_pname @ C4 @ A2) & ((finite_finite_pname @ C4) & (B2 = (image_pname_pname @ F @ C4)))))))))). % finite_subset_image
thf(fact_227_finite__surj, axiom,
    ((![A2 : set_pname, B2 : set_a, F : pname > a]: ((finite_finite_pname @ A2) => ((ord_less_eq_set_a @ B2 @ (image_pname_a @ F @ A2)) => (finite_finite_a @ B2)))))). % finite_surj
thf(fact_228_finite__surj, axiom,
    ((![A2 : set_a, B2 : set_a, F : a > a]: ((finite_finite_a @ A2) => ((ord_less_eq_set_a @ B2 @ (image_a_a @ F @ A2)) => (finite_finite_a @ B2)))))). % finite_surj
thf(fact_229_finite__surj, axiom,
    ((![A2 : set_pname, B2 : set_pname, F : pname > pname]: ((finite_finite_pname @ A2) => ((ord_le865024672_pname @ B2 @ (image_pname_pname @ F @ A2)) => (finite_finite_pname @ B2)))))). % finite_surj
thf(fact_230_finite__surj, axiom,
    ((![A2 : set_a, B2 : set_pname, F : a > pname]: ((finite_finite_a @ A2) => ((ord_le865024672_pname @ B2 @ (image_a_pname @ F @ A2)) => (finite_finite_pname @ B2)))))). % finite_surj
thf(fact_231_infinite__finite__induct, axiom,
    ((![P : set_a > $o, A2 : set_a]: ((![A4 : set_a]: ((~ ((finite_finite_a @ A4))) => (P @ A4))) => ((P @ bot_bot_set_a) => ((![X : a, F3 : set_a]: ((finite_finite_a @ F3) => ((~ ((member_a @ X @ F3))) => ((P @ F3) => (P @ (insert_a @ X @ F3)))))) => (P @ A2))))))). % infinite_finite_induct
thf(fact_232_infinite__finite__induct, axiom,
    ((![P : set_pname > $o, A2 : set_pname]: ((![A4 : set_pname]: ((~ ((finite_finite_pname @ A4))) => (P @ A4))) => ((P @ bot_bot_set_pname) => ((![X : pname, F3 : set_pname]: ((finite_finite_pname @ F3) => ((~ ((member_pname @ X @ F3))) => ((P @ F3) => (P @ (insert_pname @ X @ F3)))))) => (P @ A2))))))). % infinite_finite_induct
thf(fact_233_finite__ne__induct, axiom,
    ((![F2 : set_a, P : set_a > $o]: ((finite_finite_a @ F2) => ((~ ((F2 = bot_bot_set_a))) => ((![X : a]: (P @ (insert_a @ X @ bot_bot_set_a))) => ((![X : a, F3 : set_a]: ((finite_finite_a @ F3) => ((~ ((F3 = bot_bot_set_a))) => ((~ ((member_a @ X @ F3))) => ((P @ F3) => (P @ (insert_a @ X @ F3))))))) => (P @ F2)))))))). % finite_ne_induct
thf(fact_234_finite__ne__induct, axiom,
    ((![F2 : set_pname, P : set_pname > $o]: ((finite_finite_pname @ F2) => ((~ ((F2 = bot_bot_set_pname))) => ((![X : pname]: (P @ (insert_pname @ X @ bot_bot_set_pname))) => ((![X : pname, F3 : set_pname]: ((finite_finite_pname @ F3) => ((~ ((F3 = bot_bot_set_pname))) => ((~ ((member_pname @ X @ F3))) => ((P @ F3) => (P @ (insert_pname @ X @ F3))))))) => (P @ F2)))))))). % finite_ne_induct
thf(fact_235_finite_Oinducts, axiom,
    ((![X2 : set_a, P : set_a > $o]: ((finite_finite_a @ X2) => ((P @ bot_bot_set_a) => ((![A4 : set_a, A5 : a]: ((finite_finite_a @ A4) => ((P @ A4) => (P @ (insert_a @ A5 @ A4))))) => (P @ X2))))))). % finite.inducts
thf(fact_236_finite_Oinducts, axiom,
    ((![X2 : set_pname, P : set_pname > $o]: ((finite_finite_pname @ X2) => ((P @ bot_bot_set_pname) => ((![A4 : set_pname, A5 : pname]: ((finite_finite_pname @ A4) => ((P @ A4) => (P @ (insert_pname @ A5 @ A4))))) => (P @ X2))))))). % finite.inducts
thf(fact_237_finite__induct, axiom,
    ((![F2 : set_a, P : set_a > $o]: ((finite_finite_a @ F2) => ((P @ bot_bot_set_a) => ((![X : a, F3 : set_a]: ((finite_finite_a @ F3) => ((~ ((member_a @ X @ F3))) => ((P @ F3) => (P @ (insert_a @ X @ F3)))))) => (P @ F2))))))). % finite_induct
thf(fact_238_finite__induct, axiom,
    ((![F2 : set_pname, P : set_pname > $o]: ((finite_finite_pname @ F2) => ((P @ bot_bot_set_pname) => ((![X : pname, F3 : set_pname]: ((finite_finite_pname @ F3) => ((~ ((member_pname @ X @ F3))) => ((P @ F3) => (P @ (insert_pname @ X @ F3)))))) => (P @ F2))))))). % finite_induct
thf(fact_239_finite_Osimps, axiom,
    ((finite_finite_a = (^[A6 : set_a]: (((A6 = bot_bot_set_a)) | ((?[A3 : set_a]: (?[B5 : a]: (((A6 = (insert_a @ B5 @ A3))) & ((finite_finite_a @ A3))))))))))). % finite.simps
thf(fact_240_finite_Osimps, axiom,
    ((finite_finite_pname = (^[A6 : set_pname]: (((A6 = bot_bot_set_pname)) | ((?[A3 : set_pname]: (?[B5 : pname]: (((A6 = (insert_pname @ B5 @ A3))) & ((finite_finite_pname @ A3))))))))))). % finite.simps
thf(fact_241_finite_Ocases, axiom,
    ((![A : set_a]: ((finite_finite_a @ A) => ((~ ((A = bot_bot_set_a))) => (~ ((![A4 : set_a]: ((?[A5 : a]: (A = (insert_a @ A5 @ A4))) => (~ ((finite_finite_a @ A4)))))))))))). % finite.cases
thf(fact_242_finite_Ocases, axiom,
    ((![A : set_pname]: ((finite_finite_pname @ A) => ((~ ((A = bot_bot_set_pname))) => (~ ((![A4 : set_pname]: ((?[A5 : pname]: (A = (insert_pname @ A5 @ A4))) => (~ ((finite_finite_pname @ A4)))))))))))). % finite.cases
thf(fact_243_subset__singleton__iff, axiom,
    ((![X5 : set_a, A : a]: ((ord_less_eq_set_a @ X5 @ (insert_a @ A @ bot_bot_set_a)) = (((X5 = bot_bot_set_a)) | ((X5 = (insert_a @ A @ bot_bot_set_a)))))))). % subset_singleton_iff
thf(fact_244_subset__singleton__iff, axiom,
    ((![X5 : set_pname, A : pname]: ((ord_le865024672_pname @ X5 @ (insert_pname @ A @ bot_bot_set_pname)) = (((X5 = bot_bot_set_pname)) | ((X5 = (insert_pname @ A @ bot_bot_set_pname)))))))). % subset_singleton_iff
thf(fact_245_subset__singletonD, axiom,
    ((![A2 : set_pname, X2 : pname]: ((ord_le865024672_pname @ A2 @ (insert_pname @ X2 @ bot_bot_set_pname)) => ((A2 = bot_bot_set_pname) | (A2 = (insert_pname @ X2 @ bot_bot_set_pname))))))). % subset_singletonD
thf(fact_246_assms_I2_J, axiom,
    ((![Pn : pname, G : set_a]: ((p @ (insert_a @ (mgt_call @ Pn) @ G) @ (insert_a @ (mgt @ (the_com @ (body @ Pn))) @ bot_bot_set_a)) => (p @ G @ (insert_a @ (mgt_call @ Pn) @ bot_bot_set_a)))))). % assms(2)
thf(fact_247_assms_I4_J, axiom,
    ((![Pn : pname]: ((member_pname @ Pn @ u) => (wt @ (the_com @ (body @ Pn))))))). % assms(4)

% Conjectures (3)
thf(conj_0, hypothesis,
    ((finite_finite_pname @ u))).
thf(conj_1, hypothesis,
    ((uG = (image_pname_a @ mgt_call @ u)))).
thf(conj_2, conjecture,
    ((![G3 : set_a]: ((~ ((ord_less_eq_set_a @ G3 @ uG))) | ((~ ((ord_less_eq_nat @ n @ (finite_card_a @ uG)))) | ((~ (((finite_card_a @ G3) = (minus_minus_nat @ (finite_card_a @ uG) @ n)))) | (![C5 : com]: ((~ ((wt @ C5))) | (p @ G3 @ (insert_a @ (mgt @ C5) @ bot_bot_set_a)))))))))).
