% 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_374__3253922_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:14:08.792

% Could-be-implicit typings (11)
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Com__Opname_J_J_J, type,
    set_set_set_pname : $tType).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J, type,
    set_set_set_a : $tType).
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_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 (50)
thf(sy_c_Com_Obody, type,
    body : pname > option_com).
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__Set__Oset_It__Com__Opname_J, type,
    finite505202775_pname : set_set_pname > $o).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Set__Oset_It__Com__Opname_J_J, type,
    finite1638948493_pname : set_set_set_pname > $o).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Set__Oset_Itf__a_J_J, type,
    finite1606323175_set_a : set_set_set_a > $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_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__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__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_It__Set__Oset_It__Com__Opname_J_J, type,
    ord_le2066558166_pname : set_set_pname > set_set_pname > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__a_J_J, type,
    ord_le318720350_set_a : set_set_a > set_set_a > $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_001t__Set__Oset_It__Com__Opname_J, type,
    collect_set_pname : (set_pname > $o) > set_set_pname).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_It__Com__Opname_J_J, type,
    collec2100311499_pname : (set_set_pname > $o) > set_set_set_pname).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_Itf__a_J_J, type,
    collect_set_set_a : (set_set_a > $o) > set_set_set_a).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__a_J, type,
    collect_set_a : (set_a > $o) > set_set_a).
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_001t__Set__Oset_It__Com__Opname_J, type,
    image_747505105_pname : (pname > set_pname) > set_pname > set_set_pname).
thf(sy_c_Set_Oimage_001t__Com__Opname_001t__Set__Oset_Itf__a_J, type,
    image_pname_set_a : (pname > set_a) > set_pname > set_set_a).
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_001t__Set__Oset_It__Com__Opname_J_001t__Com__Opname, type,
    image_1672683217_pname : (set_pname > pname) > set_set_pname > set_pname).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Com__Opname, type,
    image_set_a_pname : (set_a > pname) > set_set_a > set_pname).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001tf__a, type,
    image_set_a_a : (set_a > a) > set_set_a > 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_001t__Set__Oset_It__Com__Opname_J, type,
    image_a_set_pname : (a > set_pname) > set_a > set_set_pname).
thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_Itf__a_J, type,
    image_a_set_a : (a > set_a) > set_a > set_set_a).
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_001t__Set__Oset_It__Com__Opname_J, type,
    insert_set_pname : set_pname > set_set_pname > set_set_pname).
thf(sy_c_Set_Oinsert_001t__Set__Oset_Itf__a_J, type,
    insert_set_a : set_a > set_set_a > set_set_a).
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__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_G, type,
    g : set_a).
thf(sy_v_P, type,
    p : set_a > set_a > $o).
thf(sy_v_U, type,
    u : set_pname).
thf(sy_v_c, type,
    c : com).
thf(sy_v_mgt, type,
    mgt : com > a).
thf(sy_v_mgt__call, type,
    mgt_call : pname > a).
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_finite__Collect__subsets, axiom,
    ((![A2 : set_set_a]: ((finite_finite_set_a @ A2) => (finite1606323175_set_a @ (collect_set_set_a @ (^[B2 : set_set_a]: (ord_le318720350_set_a @ B2 @ A2)))))))). % finite_Collect_subsets
thf(fact_7_finite__Collect__subsets, axiom,
    ((![A2 : set_set_pname]: ((finite505202775_pname @ A2) => (finite1638948493_pname @ (collec2100311499_pname @ (^[B2 : set_set_pname]: (ord_le2066558166_pname @ B2 @ A2)))))))). % finite_Collect_subsets
thf(fact_8_finite__Collect__subsets, axiom,
    ((![A2 : set_pname]: ((finite_finite_pname @ A2) => (finite505202775_pname @ (collect_set_pname @ (^[B2 : set_pname]: (ord_le865024672_pname @ B2 @ A2)))))))). % finite_Collect_subsets
thf(fact_9_finite__Collect__subsets, axiom,
    ((![A2 : set_a]: ((finite_finite_a @ A2) => (finite_finite_set_a @ (collect_set_a @ (^[B2 : set_a]: (ord_less_eq_set_a @ B2 @ A2)))))))). % finite_Collect_subsets
thf(fact_10_singleton__conv, axiom,
    ((![A : set_a]: ((collect_set_a @ (^[X2 : set_a]: (X2 = A))) = (insert_set_a @ A @ bot_bot_set_set_a))))). % singleton_conv
thf(fact_11_singleton__conv, axiom,
    ((![A : set_pname]: ((collect_set_pname @ (^[X2 : set_pname]: (X2 = A))) = (insert_set_pname @ A @ bot_bo1397849354_pname))))). % singleton_conv
thf(fact_12_singleton__conv, axiom,
    ((![A : pname]: ((collect_pname @ (^[X2 : pname]: (X2 = A))) = (insert_pname @ A @ bot_bot_set_pname))))). % singleton_conv
thf(fact_13_singleton__conv, axiom,
    ((![A : a]: ((collect_a @ (^[X2 : a]: (X2 = A))) = (insert_a @ A @ bot_bot_set_a))))). % singleton_conv
thf(fact_14_singleton__conv2, axiom,
    ((![A : set_a]: ((collect_set_a @ ((^[Y : set_a]: (^[Z : set_a]: (Y = Z))) @ A)) = (insert_set_a @ A @ bot_bot_set_set_a))))). % singleton_conv2
thf(fact_15_singleton__conv2, axiom,
    ((![A : set_pname]: ((collect_set_pname @ ((^[Y : set_pname]: (^[Z : set_pname]: (Y = Z))) @ A)) = (insert_set_pname @ A @ bot_bo1397849354_pname))))). % singleton_conv2
thf(fact_16_singleton__conv2, axiom,
    ((![A : pname]: ((collect_pname @ ((^[Y : pname]: (^[Z : pname]: (Y = Z))) @ A)) = (insert_pname @ A @ bot_bot_set_pname))))). % singleton_conv2
thf(fact_17_singleton__conv2, axiom,
    ((![A : a]: ((collect_a @ ((^[Y : a]: (^[Z : a]: (Y = Z))) @ A)) = (insert_a @ A @ bot_bot_set_a))))). % singleton_conv2
thf(fact_18_insert__subset, axiom,
    ((![X3 : pname, A2 : set_pname, B3 : set_pname]: ((ord_le865024672_pname @ (insert_pname @ X3 @ A2) @ B3) = (((member_pname @ X3 @ B3)) & ((ord_le865024672_pname @ A2 @ B3))))))). % insert_subset
thf(fact_19_insert__subset, axiom,
    ((![X3 : a, A2 : set_a, B3 : set_a]: ((ord_less_eq_set_a @ (insert_a @ X3 @ A2) @ B3) = (((member_a @ X3 @ B3)) & ((ord_less_eq_set_a @ A2 @ B3))))))). % insert_subset
thf(fact_20_finite__insert, axiom,
    ((![A : set_a, A2 : set_set_a]: ((finite_finite_set_a @ (insert_set_a @ A @ A2)) = (finite_finite_set_a @ A2))))). % finite_insert
thf(fact_21_finite__insert, axiom,
    ((![A : set_pname, A2 : set_set_pname]: ((finite505202775_pname @ (insert_set_pname @ A @ A2)) = (finite505202775_pname @ A2))))). % finite_insert
thf(fact_22_finite__insert, axiom,
    ((![A : a, A2 : set_a]: ((finite_finite_a @ (insert_a @ A @ A2)) = (finite_finite_a @ A2))))). % finite_insert
thf(fact_23_finite__insert, axiom,
    ((![A : pname, A2 : set_pname]: ((finite_finite_pname @ (insert_pname @ A @ A2)) = (finite_finite_pname @ A2))))). % finite_insert
thf(fact_24_singletonI, axiom,
    ((![A : pname]: (member_pname @ A @ (insert_pname @ A @ bot_bot_set_pname))))). % singletonI
thf(fact_25_singletonI, axiom,
    ((![A : a]: (member_a @ A @ (insert_a @ A @ bot_bot_set_a))))). % singletonI
thf(fact_26_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_27_subset__empty, axiom,
    ((![A2 : set_pname]: ((ord_le865024672_pname @ A2 @ bot_bot_set_pname) = (A2 = bot_bot_set_pname))))). % subset_empty
thf(fact_28_empty__subsetI, axiom,
    ((![A2 : set_a]: (ord_less_eq_set_a @ bot_bot_set_a @ A2)))). % empty_subsetI
thf(fact_29_empty__subsetI, axiom,
    ((![A2 : set_pname]: (ord_le865024672_pname @ bot_bot_set_pname @ A2)))). % empty_subsetI
thf(fact_30_image__insert, axiom,
    ((![F : a > a, A : a, B3 : set_a]: ((image_a_a @ F @ (insert_a @ A @ B3)) = (insert_a @ (F @ A) @ (image_a_a @ F @ B3)))))). % image_insert
thf(fact_31_image__insert, axiom,
    ((![F : a > pname, A : a, B3 : set_a]: ((image_a_pname @ F @ (insert_a @ A @ B3)) = (insert_pname @ (F @ A) @ (image_a_pname @ F @ B3)))))). % image_insert
thf(fact_32_image__insert, axiom,
    ((![F : pname > a, A : pname, B3 : set_pname]: ((image_pname_a @ F @ (insert_pname @ A @ B3)) = (insert_a @ (F @ A) @ (image_pname_a @ F @ B3)))))). % image_insert
thf(fact_33_image__insert, axiom,
    ((![F : pname > pname, A : pname, B3 : set_pname]: ((image_pname_pname @ F @ (insert_pname @ A @ B3)) = (insert_pname @ (F @ A) @ (image_pname_pname @ F @ B3)))))). % image_insert
thf(fact_34_insert__image, axiom,
    ((![X3 : pname, A2 : set_pname, F : pname > a]: ((member_pname @ X3 @ A2) => ((insert_a @ (F @ X3) @ (image_pname_a @ F @ A2)) = (image_pname_a @ F @ A2)))))). % insert_image
thf(fact_35_insert__image, axiom,
    ((![X3 : pname, A2 : set_pname, F : pname > pname]: ((member_pname @ X3 @ A2) => ((insert_pname @ (F @ X3) @ (image_pname_pname @ F @ A2)) = (image_pname_pname @ F @ A2)))))). % insert_image
thf(fact_36_insert__image, axiom,
    ((![X3 : a, A2 : set_a, F : a > a]: ((member_a @ X3 @ A2) => ((insert_a @ (F @ X3) @ (image_a_a @ F @ A2)) = (image_a_a @ F @ A2)))))). % insert_image
thf(fact_37_insert__image, axiom,
    ((![X3 : a, A2 : set_a, F : a > pname]: ((member_a @ X3 @ A2) => ((insert_pname @ (F @ X3) @ (image_a_pname @ F @ A2)) = (image_a_pname @ F @ A2)))))). % insert_image
thf(fact_38_image__eqI, axiom,
    ((![B : pname, F : pname > pname, X3 : pname, A2 : set_pname]: ((B = (F @ X3)) => ((member_pname @ X3 @ A2) => (member_pname @ B @ (image_pname_pname @ F @ A2))))))). % image_eqI
thf(fact_39_image__eqI, axiom,
    ((![B : a, F : pname > a, X3 : pname, A2 : set_pname]: ((B = (F @ X3)) => ((member_pname @ X3 @ A2) => (member_a @ B @ (image_pname_a @ F @ A2))))))). % image_eqI
thf(fact_40_image__eqI, axiom,
    ((![B : pname, F : a > pname, X3 : a, A2 : set_a]: ((B = (F @ X3)) => ((member_a @ X3 @ A2) => (member_pname @ B @ (image_a_pname @ F @ A2))))))). % image_eqI
thf(fact_41_image__eqI, axiom,
    ((![B : a, F : a > a, X3 : a, A2 : set_a]: ((B = (F @ X3)) => ((member_a @ X3 @ A2) => (member_a @ B @ (image_a_a @ F @ A2))))))). % image_eqI
thf(fact_42_empty__Collect__eq, axiom,
    ((![P : set_a > $o]: ((bot_bot_set_set_a = (collect_set_a @ P)) = (![X2 : set_a]: (~ ((P @ X2)))))))). % empty_Collect_eq
thf(fact_43_empty__Collect__eq, axiom,
    ((![P : set_pname > $o]: ((bot_bo1397849354_pname = (collect_set_pname @ P)) = (![X2 : set_pname]: (~ ((P @ X2)))))))). % empty_Collect_eq
thf(fact_44_empty__Collect__eq, axiom,
    ((![P : a > $o]: ((bot_bot_set_a = (collect_a @ P)) = (![X2 : a]: (~ ((P @ X2)))))))). % empty_Collect_eq
thf(fact_45_empty__Collect__eq, axiom,
    ((![P : pname > $o]: ((bot_bot_set_pname = (collect_pname @ P)) = (![X2 : pname]: (~ ((P @ X2)))))))). % empty_Collect_eq
thf(fact_46_Collect__empty__eq, axiom,
    ((![P : set_a > $o]: (((collect_set_a @ P) = bot_bot_set_set_a) = (![X2 : set_a]: (~ ((P @ X2)))))))). % Collect_empty_eq
thf(fact_47_Collect__empty__eq, axiom,
    ((![P : set_pname > $o]: (((collect_set_pname @ P) = bot_bo1397849354_pname) = (![X2 : set_pname]: (~ ((P @ X2)))))))). % Collect_empty_eq
thf(fact_48_Collect__empty__eq, axiom,
    ((![P : a > $o]: (((collect_a @ P) = bot_bot_set_a) = (![X2 : a]: (~ ((P @ X2)))))))). % Collect_empty_eq
thf(fact_49_Collect__empty__eq, axiom,
    ((![P : pname > $o]: (((collect_pname @ P) = bot_bot_set_pname) = (![X2 : pname]: (~ ((P @ X2)))))))). % Collect_empty_eq
thf(fact_50_all__not__in__conv, axiom,
    ((![A2 : set_a]: ((![X2 : a]: (~ ((member_a @ X2 @ A2)))) = (A2 = bot_bot_set_a))))). % all_not_in_conv
thf(fact_51_all__not__in__conv, axiom,
    ((![A2 : set_pname]: ((![X2 : pname]: (~ ((member_pname @ X2 @ A2)))) = (A2 = bot_bot_set_pname))))). % all_not_in_conv
thf(fact_52_empty__iff, axiom,
    ((![C : a]: (~ ((member_a @ C @ bot_bot_set_a)))))). % empty_iff
thf(fact_53_empty__iff, axiom,
    ((![C : pname]: (~ ((member_pname @ C @ bot_bot_set_pname)))))). % empty_iff
thf(fact_54_subset__antisym, axiom,
    ((![A2 : set_a, B3 : set_a]: ((ord_less_eq_set_a @ A2 @ B3) => ((ord_less_eq_set_a @ B3 @ A2) => (A2 = B3)))))). % subset_antisym
thf(fact_55_subset__antisym, axiom,
    ((![A2 : set_pname, B3 : set_pname]: ((ord_le865024672_pname @ A2 @ B3) => ((ord_le865024672_pname @ B3 @ A2) => (A2 = B3)))))). % subset_antisym
thf(fact_56_subsetI, axiom,
    ((![A2 : set_a, B3 : set_a]: ((![X : a]: ((member_a @ X @ A2) => (member_a @ X @ B3))) => (ord_less_eq_set_a @ A2 @ B3))))). % subsetI
thf(fact_57_subsetI, axiom,
    ((![A2 : set_pname, B3 : set_pname]: ((![X : pname]: ((member_pname @ X @ A2) => (member_pname @ X @ B3))) => (ord_le865024672_pname @ A2 @ B3))))). % subsetI
thf(fact_58_insert__absorb2, axiom,
    ((![X3 : a, A2 : set_a]: ((insert_a @ X3 @ (insert_a @ X3 @ A2)) = (insert_a @ X3 @ A2))))). % insert_absorb2
thf(fact_59_insert__absorb2, axiom,
    ((![X3 : pname, A2 : set_pname]: ((insert_pname @ X3 @ (insert_pname @ X3 @ A2)) = (insert_pname @ X3 @ A2))))). % insert_absorb2
thf(fact_60_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_61_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_62_insertCI, axiom,
    ((![A : pname, B3 : set_pname, B : pname]: (((~ ((member_pname @ A @ B3))) => (A = B)) => (member_pname @ A @ (insert_pname @ B @ B3)))))). % insertCI
thf(fact_63_insertCI, axiom,
    ((![A : a, B3 : set_a, B : a]: (((~ ((member_a @ A @ B3))) => (A = B)) => (member_a @ A @ (insert_a @ B @ B3)))))). % insertCI
thf(fact_64_finite__Collect__disjI, axiom,
    ((![P : pname > $o, Q : pname > $o]: ((finite_finite_pname @ (collect_pname @ (^[X2 : pname]: (((P @ X2)) | ((Q @ X2)))))) = (((finite_finite_pname @ (collect_pname @ P))) & ((finite_finite_pname @ (collect_pname @ Q)))))))). % finite_Collect_disjI
thf(fact_65_finite__Collect__disjI, axiom,
    ((![P : a > $o, Q : a > $o]: ((finite_finite_a @ (collect_a @ (^[X2 : a]: (((P @ X2)) | ((Q @ X2)))))) = (((finite_finite_a @ (collect_a @ P))) & ((finite_finite_a @ (collect_a @ Q)))))))). % finite_Collect_disjI
thf(fact_66_finite__Collect__disjI, axiom,
    ((![P : set_a > $o, Q : set_a > $o]: ((finite_finite_set_a @ (collect_set_a @ (^[X2 : set_a]: (((P @ X2)) | ((Q @ X2)))))) = (((finite_finite_set_a @ (collect_set_a @ P))) & ((finite_finite_set_a @ (collect_set_a @ Q)))))))). % finite_Collect_disjI
thf(fact_67_finite__Collect__disjI, axiom,
    ((![P : set_pname > $o, Q : set_pname > $o]: ((finite505202775_pname @ (collect_set_pname @ (^[X2 : set_pname]: (((P @ X2)) | ((Q @ X2)))))) = (((finite505202775_pname @ (collect_set_pname @ P))) & ((finite505202775_pname @ (collect_set_pname @ Q)))))))). % finite_Collect_disjI
thf(fact_68_finite__Collect__conjI, axiom,
    ((![P : pname > $o, Q : pname > $o]: (((finite_finite_pname @ (collect_pname @ P)) | (finite_finite_pname @ (collect_pname @ Q))) => (finite_finite_pname @ (collect_pname @ (^[X2 : pname]: (((P @ X2)) & ((Q @ X2)))))))))). % finite_Collect_conjI
thf(fact_69_finite__Collect__conjI, axiom,
    ((![P : a > $o, Q : a > $o]: (((finite_finite_a @ (collect_a @ P)) | (finite_finite_a @ (collect_a @ Q))) => (finite_finite_a @ (collect_a @ (^[X2 : a]: (((P @ X2)) & ((Q @ X2)))))))))). % finite_Collect_conjI
thf(fact_70_finite__Collect__conjI, axiom,
    ((![P : set_a > $o, Q : set_a > $o]: (((finite_finite_set_a @ (collect_set_a @ P)) | (finite_finite_set_a @ (collect_set_a @ Q))) => (finite_finite_set_a @ (collect_set_a @ (^[X2 : set_a]: (((P @ X2)) & ((Q @ X2)))))))))). % finite_Collect_conjI
thf(fact_71_finite__Collect__conjI, axiom,
    ((![P : set_pname > $o, Q : set_pname > $o]: (((finite505202775_pname @ (collect_set_pname @ P)) | (finite505202775_pname @ (collect_set_pname @ Q))) => (finite505202775_pname @ (collect_set_pname @ (^[X2 : set_pname]: (((P @ X2)) & ((Q @ X2)))))))))). % finite_Collect_conjI
thf(fact_72_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_73_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_74_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_75_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_76_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_77_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_78_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_79_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_80_image__empty, axiom,
    ((![F : a > a]: ((image_a_a @ F @ bot_bot_set_a) = bot_bot_set_a)))). % image_empty
thf(fact_81_image__empty, axiom,
    ((![F : a > pname]: ((image_a_pname @ F @ bot_bot_set_a) = bot_bot_set_pname)))). % image_empty
thf(fact_82_image__empty, axiom,
    ((![F : pname > a]: ((image_pname_a @ F @ bot_bot_set_pname) = bot_bot_set_a)))). % image_empty
thf(fact_83_image__empty, axiom,
    ((![F : pname > pname]: ((image_pname_pname @ F @ bot_bot_set_pname) = bot_bot_set_pname)))). % image_empty
thf(fact_84_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_85_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_86_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_87_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_88_finite__imageI, axiom,
    ((![F2 : set_pname, H : pname > set_a]: ((finite_finite_pname @ F2) => (finite_finite_set_a @ (image_pname_set_a @ H @ F2)))))). % finite_imageI
thf(fact_89_finite__imageI, axiom,
    ((![F2 : set_pname, H : pname > set_pname]: ((finite_finite_pname @ F2) => (finite505202775_pname @ (image_747505105_pname @ H @ F2)))))). % finite_imageI
thf(fact_90_finite__imageI, axiom,
    ((![F2 : set_a, H : a > set_a]: ((finite_finite_a @ F2) => (finite_finite_set_a @ (image_a_set_a @ H @ F2)))))). % finite_imageI
thf(fact_91_finite__imageI, axiom,
    ((![F2 : set_a, H : a > set_pname]: ((finite_finite_a @ F2) => (finite505202775_pname @ (image_a_set_pname @ H @ F2)))))). % finite_imageI
thf(fact_92_finite__imageI, axiom,
    ((![F2 : set_set_a, H : set_a > pname]: ((finite_finite_set_a @ F2) => (finite_finite_pname @ (image_set_a_pname @ H @ F2)))))). % finite_imageI
thf(fact_93_finite__imageI, axiom,
    ((![F2 : set_set_a, H : set_a > a]: ((finite_finite_set_a @ F2) => (finite_finite_a @ (image_set_a_a @ H @ F2)))))). % finite_imageI
thf(fact_94_rev__image__eqI, axiom,
    ((![X3 : pname, A2 : set_pname, B : pname, F : pname > pname]: ((member_pname @ X3 @ A2) => ((B = (F @ X3)) => (member_pname @ B @ (image_pname_pname @ F @ A2))))))). % rev_image_eqI
thf(fact_95_rev__image__eqI, axiom,
    ((![X3 : pname, A2 : set_pname, B : a, F : pname > a]: ((member_pname @ X3 @ A2) => ((B = (F @ X3)) => (member_a @ B @ (image_pname_a @ F @ A2))))))). % rev_image_eqI
thf(fact_96_rev__image__eqI, axiom,
    ((![X3 : a, A2 : set_a, B : pname, F : a > pname]: ((member_a @ X3 @ A2) => ((B = (F @ X3)) => (member_pname @ B @ (image_a_pname @ F @ A2))))))). % rev_image_eqI
thf(fact_97_rev__image__eqI, axiom,
    ((![X3 : a, A2 : set_a, B : a, F : a > a]: ((member_a @ X3 @ A2) => ((B = (F @ X3)) => (member_a @ B @ (image_a_a @ F @ A2))))))). % rev_image_eqI
thf(fact_98_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_99_image__cong, axiom,
    ((![M : set_pname, N : set_pname, F : pname > a, G2 : pname > a]: ((M = N) => ((![X : pname]: ((member_pname @ X @ N) => ((F @ X) = (G2 @ X)))) => ((image_pname_a @ F @ M) = (image_pname_a @ G2 @ N))))))). % image_cong
thf(fact_100_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_101_image__iff, axiom,
    ((![Z2 : a, F : pname > a, A2 : set_pname]: ((member_a @ Z2 @ (image_pname_a @ F @ A2)) = (?[X2 : pname]: (((member_pname @ X2 @ A2)) & ((Z2 = (F @ X2))))))))). % image_iff
thf(fact_102_imageI, axiom,
    ((![X3 : pname, A2 : set_pname, F : pname > pname]: ((member_pname @ X3 @ A2) => (member_pname @ (F @ X3) @ (image_pname_pname @ F @ A2)))))). % imageI
thf(fact_103_imageI, axiom,
    ((![X3 : pname, A2 : set_pname, F : pname > a]: ((member_pname @ X3 @ A2) => (member_a @ (F @ X3) @ (image_pname_a @ F @ A2)))))). % imageI
thf(fact_104_imageI, axiom,
    ((![X3 : a, A2 : set_a, F : a > pname]: ((member_a @ X3 @ A2) => (member_pname @ (F @ X3) @ (image_a_pname @ F @ A2)))))). % imageI
thf(fact_105_imageI, axiom,
    ((![X3 : a, A2 : set_a, F : a > a]: ((member_a @ X3 @ A2) => (member_a @ (F @ X3) @ (image_a_a @ F @ A2)))))). % imageI
thf(fact_106_ex__in__conv, axiom,
    ((![A2 : set_a]: ((?[X2 : a]: (member_a @ X2 @ A2)) = (~ ((A2 = bot_bot_set_a))))))). % ex_in_conv
thf(fact_107_ex__in__conv, axiom,
    ((![A2 : set_pname]: ((?[X2 : pname]: (member_pname @ X2 @ A2)) = (~ ((A2 = bot_bot_set_pname))))))). % ex_in_conv
thf(fact_108_equals0I, axiom,
    ((![A2 : set_a]: ((![Y2 : a]: (~ ((member_a @ Y2 @ A2)))) => (A2 = bot_bot_set_a))))). % equals0I
thf(fact_109_equals0I, axiom,
    ((![A2 : set_pname]: ((![Y2 : pname]: (~ ((member_pname @ Y2 @ A2)))) => (A2 = bot_bot_set_pname))))). % equals0I
thf(fact_110_equals0D, axiom,
    ((![A2 : set_a, A : a]: ((A2 = bot_bot_set_a) => (~ ((member_a @ A @ A2))))))). % equals0D
thf(fact_111_equals0D, axiom,
    ((![A2 : set_pname, A : pname]: ((A2 = bot_bot_set_pname) => (~ ((member_pname @ A @ A2))))))). % equals0D
thf(fact_112_emptyE, axiom,
    ((![A : a]: (~ ((member_a @ A @ bot_bot_set_a)))))). % emptyE
thf(fact_113_emptyE, axiom,
    ((![A : pname]: (~ ((member_pname @ A @ bot_bot_set_pname)))))). % emptyE
thf(fact_114_Collect__mono__iff, axiom,
    ((![P : set_a > $o, Q : set_a > $o]: ((ord_le318720350_set_a @ (collect_set_a @ P) @ (collect_set_a @ Q)) = (![X2 : set_a]: (((P @ X2)) => ((Q @ X2)))))))). % Collect_mono_iff
thf(fact_115_Collect__mono__iff, axiom,
    ((![P : set_pname > $o, Q : set_pname > $o]: ((ord_le2066558166_pname @ (collect_set_pname @ P) @ (collect_set_pname @ Q)) = (![X2 : set_pname]: (((P @ X2)) => ((Q @ X2)))))))). % Collect_mono_iff
thf(fact_116_Collect__mono__iff, axiom,
    ((![P : a > $o, Q : a > $o]: ((ord_less_eq_set_a @ (collect_a @ P) @ (collect_a @ Q)) = (![X2 : a]: (((P @ X2)) => ((Q @ X2)))))))). % Collect_mono_iff
thf(fact_117_Collect__mono__iff, axiom,
    ((![P : pname > $o, Q : pname > $o]: ((ord_le865024672_pname @ (collect_pname @ P) @ (collect_pname @ Q)) = (![X2 : pname]: (((P @ X2)) => ((Q @ X2)))))))). % Collect_mono_iff
thf(fact_118_mem__Collect__eq, axiom,
    ((![A : pname, P : pname > $o]: ((member_pname @ A @ (collect_pname @ P)) = (P @ A))))). % mem_Collect_eq
thf(fact_119_mem__Collect__eq, axiom,
    ((![A : set_a, P : set_a > $o]: ((member_set_a @ A @ (collect_set_a @ P)) = (P @ A))))). % mem_Collect_eq
thf(fact_120_mem__Collect__eq, axiom,
    ((![A : set_pname, P : set_pname > $o]: ((member_set_pname @ A @ (collect_set_pname @ P)) = (P @ A))))). % mem_Collect_eq
thf(fact_121_mem__Collect__eq, axiom,
    ((![A : a, P : a > $o]: ((member_a @ A @ (collect_a @ P)) = (P @ A))))). % mem_Collect_eq
thf(fact_122_Collect__mem__eq, axiom,
    ((![A2 : set_pname]: ((collect_pname @ (^[X2 : pname]: (member_pname @ X2 @ A2))) = A2)))). % Collect_mem_eq
thf(fact_123_Collect__mem__eq, axiom,
    ((![A2 : set_set_a]: ((collect_set_a @ (^[X2 : set_a]: (member_set_a @ X2 @ A2))) = A2)))). % Collect_mem_eq
thf(fact_124_Collect__mem__eq, axiom,
    ((![A2 : set_set_pname]: ((collect_set_pname @ (^[X2 : set_pname]: (member_set_pname @ X2 @ A2))) = A2)))). % Collect_mem_eq
thf(fact_125_Collect__mem__eq, axiom,
    ((![A2 : set_a]: ((collect_a @ (^[X2 : a]: (member_a @ X2 @ A2))) = A2)))). % Collect_mem_eq
thf(fact_126_Collect__cong, axiom,
    ((![P : set_a > $o, Q : set_a > $o]: ((![X : set_a]: ((P @ X) = (Q @ X))) => ((collect_set_a @ P) = (collect_set_a @ Q)))))). % Collect_cong
thf(fact_127_Collect__cong, axiom,
    ((![P : set_pname > $o, Q : set_pname > $o]: ((![X : set_pname]: ((P @ X) = (Q @ X))) => ((collect_set_pname @ P) = (collect_set_pname @ Q)))))). % Collect_cong
thf(fact_128_Collect__cong, axiom,
    ((![P : a > $o, Q : a > $o]: ((![X : a]: ((P @ X) = (Q @ X))) => ((collect_a @ P) = (collect_a @ Q)))))). % Collect_cong
thf(fact_129_set__eq__subset, axiom,
    (((^[Y : set_a]: (^[Z : set_a]: (Y = Z))) = (^[A3 : set_a]: (^[B2 : set_a]: (((ord_less_eq_set_a @ A3 @ B2)) & ((ord_less_eq_set_a @ B2 @ A3)))))))). % set_eq_subset
thf(fact_130_set__eq__subset, axiom,
    (((^[Y : set_pname]: (^[Z : set_pname]: (Y = Z))) = (^[A3 : set_pname]: (^[B2 : set_pname]: (((ord_le865024672_pname @ A3 @ B2)) & ((ord_le865024672_pname @ B2 @ A3)))))))). % set_eq_subset
thf(fact_131_subset__trans, axiom,
    ((![A2 : set_a, B3 : set_a, C2 : set_a]: ((ord_less_eq_set_a @ A2 @ B3) => ((ord_less_eq_set_a @ B3 @ C2) => (ord_less_eq_set_a @ A2 @ C2)))))). % subset_trans
thf(fact_132_subset__trans, axiom,
    ((![A2 : set_pname, B3 : set_pname, C2 : set_pname]: ((ord_le865024672_pname @ A2 @ B3) => ((ord_le865024672_pname @ B3 @ C2) => (ord_le865024672_pname @ A2 @ C2)))))). % subset_trans
thf(fact_133_Collect__mono, axiom,
    ((![P : set_a > $o, Q : set_a > $o]: ((![X : set_a]: ((P @ X) => (Q @ X))) => (ord_le318720350_set_a @ (collect_set_a @ P) @ (collect_set_a @ Q)))))). % Collect_mono
thf(fact_134_Collect__mono, axiom,
    ((![P : set_pname > $o, Q : set_pname > $o]: ((![X : set_pname]: ((P @ X) => (Q @ X))) => (ord_le2066558166_pname @ (collect_set_pname @ P) @ (collect_set_pname @ Q)))))). % Collect_mono
thf(fact_135_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_136_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_137_subset__refl, axiom,
    ((![A2 : set_a]: (ord_less_eq_set_a @ A2 @ A2)))). % subset_refl
thf(fact_138_subset__refl, axiom,
    ((![A2 : set_pname]: (ord_le865024672_pname @ A2 @ A2)))). % subset_refl
thf(fact_139_subset__iff, axiom,
    ((ord_less_eq_set_a = (^[A3 : set_a]: (^[B2 : set_a]: (![T : a]: (((member_a @ T @ A3)) => ((member_a @ T @ B2))))))))). % subset_iff
thf(fact_140_subset__iff, axiom,
    ((ord_le865024672_pname = (^[A3 : set_pname]: (^[B2 : set_pname]: (![T : pname]: (((member_pname @ T @ A3)) => ((member_pname @ T @ B2))))))))). % subset_iff
thf(fact_141_equalityD2, axiom,
    ((![A2 : set_a, B3 : set_a]: ((A2 = B3) => (ord_less_eq_set_a @ B3 @ A2))))). % equalityD2
thf(fact_142_equalityD2, axiom,
    ((![A2 : set_pname, B3 : set_pname]: ((A2 = B3) => (ord_le865024672_pname @ B3 @ A2))))). % equalityD2
thf(fact_143_equalityD1, axiom,
    ((![A2 : set_a, B3 : set_a]: ((A2 = B3) => (ord_less_eq_set_a @ A2 @ B3))))). % equalityD1
thf(fact_144_equalityD1, axiom,
    ((![A2 : set_pname, B3 : set_pname]: ((A2 = B3) => (ord_le865024672_pname @ A2 @ B3))))). % equalityD1
thf(fact_145_subset__eq, axiom,
    ((ord_less_eq_set_a = (^[A3 : set_a]: (^[B2 : set_a]: (![X2 : a]: (((member_a @ X2 @ A3)) => ((member_a @ X2 @ B2))))))))). % subset_eq
thf(fact_146_subset__eq, axiom,
    ((ord_le865024672_pname = (^[A3 : set_pname]: (^[B2 : set_pname]: (![X2 : pname]: (((member_pname @ X2 @ A3)) => ((member_pname @ X2 @ B2))))))))). % subset_eq
thf(fact_147_equalityE, axiom,
    ((![A2 : set_a, B3 : set_a]: ((A2 = B3) => (~ (((ord_less_eq_set_a @ A2 @ B3) => (~ ((ord_less_eq_set_a @ B3 @ A2)))))))))). % equalityE
thf(fact_148_equalityE, axiom,
    ((![A2 : set_pname, B3 : set_pname]: ((A2 = B3) => (~ (((ord_le865024672_pname @ A2 @ B3) => (~ ((ord_le865024672_pname @ B3 @ A2)))))))))). % equalityE
thf(fact_149_subsetD, axiom,
    ((![A2 : set_a, B3 : set_a, C : a]: ((ord_less_eq_set_a @ A2 @ B3) => ((member_a @ C @ A2) => (member_a @ C @ B3)))))). % subsetD
thf(fact_150_subsetD, axiom,
    ((![A2 : set_pname, B3 : set_pname, C : pname]: ((ord_le865024672_pname @ A2 @ B3) => ((member_pname @ C @ A2) => (member_pname @ C @ B3)))))). % subsetD
thf(fact_151_in__mono, axiom,
    ((![A2 : set_a, B3 : set_a, X3 : a]: ((ord_less_eq_set_a @ A2 @ B3) => ((member_a @ X3 @ A2) => (member_a @ X3 @ B3)))))). % in_mono
thf(fact_152_in__mono, axiom,
    ((![A2 : set_pname, B3 : set_pname, X3 : pname]: ((ord_le865024672_pname @ A2 @ B3) => ((member_pname @ X3 @ A2) => (member_pname @ X3 @ B3)))))). % in_mono
thf(fact_153_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_154_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_155_insert__commute, axiom,
    ((![X3 : a, Y3 : a, A2 : set_a]: ((insert_a @ X3 @ (insert_a @ Y3 @ A2)) = (insert_a @ Y3 @ (insert_a @ X3 @ A2)))))). % insert_commute
thf(fact_156_insert__commute, axiom,
    ((![X3 : pname, Y3 : pname, A2 : set_pname]: ((insert_pname @ X3 @ (insert_pname @ Y3 @ A2)) = (insert_pname @ Y3 @ (insert_pname @ X3 @ A2)))))). % insert_commute
thf(fact_157_insert__eq__iff, axiom,
    ((![A : pname, A2 : set_pname, B : pname, B3 : set_pname]: ((~ ((member_pname @ A @ A2))) => ((~ ((member_pname @ B @ B3))) => (((insert_pname @ A @ A2) = (insert_pname @ B @ B3)) = (((((A = B)) => ((A2 = B3)))) & ((((~ ((A = B)))) => ((?[C3 : set_pname]: (((A2 = (insert_pname @ B @ C3))) & ((((~ ((member_pname @ B @ C3)))) & ((((B3 = (insert_pname @ A @ C3))) & ((~ ((member_pname @ A @ C3)))))))))))))))))))). % insert_eq_iff
thf(fact_158_insert__eq__iff, axiom,
    ((![A : a, A2 : set_a, B : a, B3 : set_a]: ((~ ((member_a @ A @ A2))) => ((~ ((member_a @ B @ B3))) => (((insert_a @ A @ A2) = (insert_a @ B @ B3)) = (((((A = B)) => ((A2 = B3)))) & ((((~ ((A = B)))) => ((?[C3 : set_a]: (((A2 = (insert_a @ B @ C3))) & ((((~ ((member_a @ B @ C3)))) & ((((B3 = (insert_a @ A @ C3))) & ((~ ((member_a @ A @ C3)))))))))))))))))))). % insert_eq_iff
thf(fact_159_insert__absorb, axiom,
    ((![A : pname, A2 : set_pname]: ((member_pname @ A @ A2) => ((insert_pname @ A @ A2) = A2))))). % insert_absorb
thf(fact_160_insert__absorb, axiom,
    ((![A : a, A2 : set_a]: ((member_a @ A @ A2) => ((insert_a @ A @ A2) = A2))))). % insert_absorb
thf(fact_161_insert__ident, axiom,
    ((![X3 : pname, A2 : set_pname, B3 : set_pname]: ((~ ((member_pname @ X3 @ A2))) => ((~ ((member_pname @ X3 @ B3))) => (((insert_pname @ X3 @ A2) = (insert_pname @ X3 @ B3)) = (A2 = B3))))))). % insert_ident
thf(fact_162_insert__ident, axiom,
    ((![X3 : a, A2 : set_a, B3 : set_a]: ((~ ((member_a @ X3 @ A2))) => ((~ ((member_a @ X3 @ B3))) => (((insert_a @ X3 @ A2) = (insert_a @ X3 @ B3)) = (A2 = B3))))))). % insert_ident
thf(fact_163_Set_Oset__insert, axiom,
    ((![X3 : pname, A2 : set_pname]: ((member_pname @ X3 @ A2) => (~ ((![B4 : set_pname]: ((A2 = (insert_pname @ X3 @ B4)) => (member_pname @ X3 @ B4))))))))). % Set.set_insert
thf(fact_164_Set_Oset__insert, axiom,
    ((![X3 : a, A2 : set_a]: ((member_a @ X3 @ A2) => (~ ((![B4 : set_a]: ((A2 = (insert_a @ X3 @ B4)) => (member_a @ X3 @ B4))))))))). % Set.set_insert
thf(fact_165_insertI2, axiom,
    ((![A : pname, B3 : set_pname, B : pname]: ((member_pname @ A @ B3) => (member_pname @ A @ (insert_pname @ B @ B3)))))). % insertI2
thf(fact_166_insertI2, axiom,
    ((![A : a, B3 : set_a, B : a]: ((member_a @ A @ B3) => (member_a @ A @ (insert_a @ B @ B3)))))). % insertI2
thf(fact_167_insertI1, axiom,
    ((![A : pname, B3 : set_pname]: (member_pname @ A @ (insert_pname @ A @ B3))))). % insertI1
thf(fact_168_insertI1, axiom,
    ((![A : a, B3 : set_a]: (member_a @ A @ (insert_a @ A @ B3))))). % insertI1
thf(fact_169_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_170_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_171_Compr__image__eq, axiom,
    ((![F : pname > pname, A2 : set_pname, P : pname > $o]: ((collect_pname @ (^[X2 : pname]: (((member_pname @ X2 @ (image_pname_pname @ F @ A2))) & ((P @ X2))))) = (image_pname_pname @ F @ (collect_pname @ (^[X2 : pname]: (((member_pname @ X2 @ A2)) & ((P @ (F @ X2))))))))))). % Compr_image_eq
thf(fact_172_Compr__image__eq, axiom,
    ((![F : a > pname, A2 : set_a, P : pname > $o]: ((collect_pname @ (^[X2 : pname]: (((member_pname @ X2 @ (image_a_pname @ F @ A2))) & ((P @ X2))))) = (image_a_pname @ F @ (collect_a @ (^[X2 : a]: (((member_a @ X2 @ A2)) & ((P @ (F @ X2))))))))))). % Compr_image_eq
thf(fact_173_Compr__image__eq, axiom,
    ((![F : pname > a, A2 : set_pname, P : a > $o]: ((collect_a @ (^[X2 : a]: (((member_a @ X2 @ (image_pname_a @ F @ A2))) & ((P @ X2))))) = (image_pname_a @ F @ (collect_pname @ (^[X2 : pname]: (((member_pname @ X2 @ A2)) & ((P @ (F @ X2))))))))))). % Compr_image_eq
thf(fact_174_Compr__image__eq, axiom,
    ((![F : a > a, A2 : set_a, P : a > $o]: ((collect_a @ (^[X2 : a]: (((member_a @ X2 @ (image_a_a @ F @ A2))) & ((P @ X2))))) = (image_a_a @ F @ (collect_a @ (^[X2 : a]: (((member_a @ X2 @ A2)) & ((P @ (F @ X2))))))))))). % Compr_image_eq
thf(fact_175_Compr__image__eq, axiom,
    ((![F : set_a > pname, A2 : set_set_a, P : pname > $o]: ((collect_pname @ (^[X2 : pname]: (((member_pname @ X2 @ (image_set_a_pname @ F @ A2))) & ((P @ X2))))) = (image_set_a_pname @ F @ (collect_set_a @ (^[X2 : set_a]: (((member_set_a @ X2 @ A2)) & ((P @ (F @ X2))))))))))). % Compr_image_eq
thf(fact_176_Compr__image__eq, axiom,
    ((![F : set_pname > pname, A2 : set_set_pname, P : pname > $o]: ((collect_pname @ (^[X2 : pname]: (((member_pname @ X2 @ (image_1672683217_pname @ F @ A2))) & ((P @ X2))))) = (image_1672683217_pname @ F @ (collect_set_pname @ (^[X2 : set_pname]: (((member_set_pname @ X2 @ A2)) & ((P @ (F @ X2))))))))))). % Compr_image_eq
thf(fact_177_Compr__image__eq, axiom,
    ((![F : pname > set_a, A2 : set_pname, P : set_a > $o]: ((collect_set_a @ (^[X2 : set_a]: (((member_set_a @ X2 @ (image_pname_set_a @ F @ A2))) & ((P @ X2))))) = (image_pname_set_a @ F @ (collect_pname @ (^[X2 : pname]: (((member_pname @ X2 @ A2)) & ((P @ (F @ X2))))))))))). % Compr_image_eq
thf(fact_178_Compr__image__eq, axiom,
    ((![F : a > set_a, A2 : set_a, P : set_a > $o]: ((collect_set_a @ (^[X2 : set_a]: (((member_set_a @ X2 @ (image_a_set_a @ F @ A2))) & ((P @ X2))))) = (image_a_set_a @ F @ (collect_a @ (^[X2 : a]: (((member_a @ X2 @ A2)) & ((P @ (F @ X2))))))))))). % Compr_image_eq
thf(fact_179_Compr__image__eq, axiom,
    ((![F : pname > set_pname, A2 : set_pname, P : set_pname > $o]: ((collect_set_pname @ (^[X2 : set_pname]: (((member_set_pname @ X2 @ (image_747505105_pname @ F @ A2))) & ((P @ X2))))) = (image_747505105_pname @ F @ (collect_pname @ (^[X2 : pname]: (((member_pname @ X2 @ A2)) & ((P @ (F @ X2))))))))))). % Compr_image_eq
thf(fact_180_Compr__image__eq, axiom,
    ((![F : a > set_pname, A2 : set_a, P : set_pname > $o]: ((collect_set_pname @ (^[X2 : set_pname]: (((member_set_pname @ X2 @ (image_a_set_pname @ F @ A2))) & ((P @ X2))))) = (image_a_set_pname @ F @ (collect_a @ (^[X2 : a]: (((member_a @ X2 @ A2)) & ((P @ (F @ X2))))))))))). % Compr_image_eq
thf(fact_181_image__image, axiom,
    ((![F : a > a, G2 : pname > a, A2 : set_pname]: ((image_a_a @ F @ (image_pname_a @ G2 @ A2)) = (image_pname_a @ (^[X2 : pname]: (F @ (G2 @ X2))) @ A2))))). % image_image
thf(fact_182_image__image, axiom,
    ((![F : pname > a, G2 : pname > pname, A2 : set_pname]: ((image_pname_a @ F @ (image_pname_pname @ G2 @ A2)) = (image_pname_a @ (^[X2 : pname]: (F @ (G2 @ X2))) @ A2))))). % image_image
thf(fact_183_imageE, axiom,
    ((![B : pname, F : pname > pname, A2 : set_pname]: ((member_pname @ B @ (image_pname_pname @ F @ A2)) => (~ ((![X : pname]: ((B = (F @ X)) => (~ ((member_pname @ X @ A2))))))))))). % imageE
thf(fact_184_imageE, axiom,
    ((![B : pname, F : a > pname, A2 : set_a]: ((member_pname @ B @ (image_a_pname @ F @ A2)) => (~ ((![X : a]: ((B = (F @ X)) => (~ ((member_a @ X @ A2))))))))))). % imageE
thf(fact_185_imageE, axiom,
    ((![B : a, F : pname > a, A2 : set_pname]: ((member_a @ B @ (image_pname_a @ F @ A2)) => (~ ((![X : pname]: ((B = (F @ X)) => (~ ((member_pname @ X @ A2))))))))))). % imageE
thf(fact_186_imageE, axiom,
    ((![B : a, F : a > a, A2 : set_a]: ((member_a @ B @ (image_a_a @ F @ A2)) => (~ ((![X : a]: ((B = (F @ X)) => (~ ((member_a @ X @ A2))))))))))). % imageE
thf(fact_187_empty__def, axiom,
    ((bot_bot_set_set_a = (collect_set_a @ (^[X2 : set_a]: $false))))). % empty_def
thf(fact_188_empty__def, axiom,
    ((bot_bo1397849354_pname = (collect_set_pname @ (^[X2 : set_pname]: $false))))). % empty_def
thf(fact_189_empty__def, axiom,
    ((bot_bot_set_a = (collect_a @ (^[X2 : a]: $false))))). % empty_def
thf(fact_190_empty__def, axiom,
    ((bot_bot_set_pname = (collect_pname @ (^[X2 : pname]: $false))))). % empty_def
thf(fact_191_pigeonhole__infinite__rel, axiom,
    ((![A2 : set_pname, B3 : set_pname, R : pname > pname > $o]: ((~ ((finite_finite_pname @ A2))) => ((finite_finite_pname @ B3) => ((![X : pname]: ((member_pname @ X @ A2) => (?[Xa : pname]: ((member_pname @ Xa @ B3) & (R @ X @ Xa))))) => (?[X : pname]: ((member_pname @ X @ B3) & (~ ((finite_finite_pname @ (collect_pname @ (^[A4 : pname]: (((member_pname @ A4 @ A2)) & ((R @ A4 @ X)))))))))))))))). % pigeonhole_infinite_rel
thf(fact_192_pigeonhole__infinite__rel, axiom,
    ((![A2 : set_pname, B3 : set_a, R : pname > a > $o]: ((~ ((finite_finite_pname @ A2))) => ((finite_finite_a @ B3) => ((![X : pname]: ((member_pname @ X @ A2) => (?[Xa : a]: ((member_a @ Xa @ B3) & (R @ X @ Xa))))) => (?[X : a]: ((member_a @ X @ B3) & (~ ((finite_finite_pname @ (collect_pname @ (^[A4 : pname]: (((member_pname @ A4 @ A2)) & ((R @ A4 @ X)))))))))))))))). % pigeonhole_infinite_rel
thf(fact_193_pigeonhole__infinite__rel, axiom,
    ((![A2 : set_a, B3 : set_pname, R : a > pname > $o]: ((~ ((finite_finite_a @ A2))) => ((finite_finite_pname @ B3) => ((![X : a]: ((member_a @ X @ A2) => (?[Xa : pname]: ((member_pname @ Xa @ B3) & (R @ X @ Xa))))) => (?[X : pname]: ((member_pname @ X @ B3) & (~ ((finite_finite_a @ (collect_a @ (^[A4 : a]: (((member_a @ A4 @ A2)) & ((R @ A4 @ X)))))))))))))))). % pigeonhole_infinite_rel
thf(fact_194_pigeonhole__infinite__rel, axiom,
    ((![A2 : set_a, B3 : set_a, R : a > a > $o]: ((~ ((finite_finite_a @ A2))) => ((finite_finite_a @ B3) => ((![X : a]: ((member_a @ X @ A2) => (?[Xa : a]: ((member_a @ Xa @ B3) & (R @ X @ Xa))))) => (?[X : a]: ((member_a @ X @ B3) & (~ ((finite_finite_a @ (collect_a @ (^[A4 : a]: (((member_a @ A4 @ A2)) & ((R @ A4 @ X)))))))))))))))). % pigeonhole_infinite_rel
thf(fact_195_pigeonhole__infinite__rel, axiom,
    ((![A2 : set_pname, B3 : set_set_a, R : pname > set_a > $o]: ((~ ((finite_finite_pname @ A2))) => ((finite_finite_set_a @ B3) => ((![X : pname]: ((member_pname @ X @ A2) => (?[Xa : set_a]: ((member_set_a @ Xa @ B3) & (R @ X @ Xa))))) => (?[X : set_a]: ((member_set_a @ X @ B3) & (~ ((finite_finite_pname @ (collect_pname @ (^[A4 : pname]: (((member_pname @ A4 @ A2)) & ((R @ A4 @ X)))))))))))))))). % pigeonhole_infinite_rel
thf(fact_196_pigeonhole__infinite__rel, axiom,
    ((![A2 : set_pname, B3 : set_set_pname, R : pname > set_pname > $o]: ((~ ((finite_finite_pname @ A2))) => ((finite505202775_pname @ B3) => ((![X : pname]: ((member_pname @ X @ A2) => (?[Xa : set_pname]: ((member_set_pname @ Xa @ B3) & (R @ X @ Xa))))) => (?[X : set_pname]: ((member_set_pname @ X @ B3) & (~ ((finite_finite_pname @ (collect_pname @ (^[A4 : pname]: (((member_pname @ A4 @ A2)) & ((R @ A4 @ X)))))))))))))))). % pigeonhole_infinite_rel
thf(fact_197_pigeonhole__infinite__rel, axiom,
    ((![A2 : set_a, B3 : set_set_a, R : a > set_a > $o]: ((~ ((finite_finite_a @ A2))) => ((finite_finite_set_a @ B3) => ((![X : a]: ((member_a @ X @ A2) => (?[Xa : set_a]: ((member_set_a @ Xa @ B3) & (R @ X @ Xa))))) => (?[X : set_a]: ((member_set_a @ X @ B3) & (~ ((finite_finite_a @ (collect_a @ (^[A4 : a]: (((member_a @ A4 @ A2)) & ((R @ A4 @ X)))))))))))))))). % pigeonhole_infinite_rel
thf(fact_198_pigeonhole__infinite__rel, axiom,
    ((![A2 : set_a, B3 : set_set_pname, R : a > set_pname > $o]: ((~ ((finite_finite_a @ A2))) => ((finite505202775_pname @ B3) => ((![X : a]: ((member_a @ X @ A2) => (?[Xa : set_pname]: ((member_set_pname @ Xa @ B3) & (R @ X @ Xa))))) => (?[X : set_pname]: ((member_set_pname @ X @ B3) & (~ ((finite_finite_a @ (collect_a @ (^[A4 : a]: (((member_a @ A4 @ A2)) & ((R @ A4 @ X)))))))))))))))). % pigeonhole_infinite_rel
thf(fact_199_pigeonhole__infinite__rel, axiom,
    ((![A2 : set_set_a, B3 : set_pname, R : set_a > pname > $o]: ((~ ((finite_finite_set_a @ A2))) => ((finite_finite_pname @ B3) => ((![X : set_a]: ((member_set_a @ X @ A2) => (?[Xa : pname]: ((member_pname @ Xa @ B3) & (R @ X @ Xa))))) => (?[X : pname]: ((member_pname @ X @ B3) & (~ ((finite_finite_set_a @ (collect_set_a @ (^[A4 : set_a]: (((member_set_a @ A4 @ A2)) & ((R @ A4 @ X)))))))))))))))). % pigeonhole_infinite_rel
thf(fact_200_pigeonhole__infinite__rel, axiom,
    ((![A2 : set_set_a, B3 : set_a, R : set_a > a > $o]: ((~ ((finite_finite_set_a @ A2))) => ((finite_finite_a @ B3) => ((![X : set_a]: ((member_set_a @ X @ A2) => (?[Xa : a]: ((member_a @ Xa @ B3) & (R @ X @ Xa))))) => (?[X : a]: ((member_a @ X @ B3) & (~ ((finite_finite_set_a @ (collect_set_a @ (^[A4 : set_a]: (((member_set_a @ A4 @ A2)) & ((R @ A4 @ X)))))))))))))))). % pigeonhole_infinite_rel
thf(fact_201_not__finite__existsD, axiom,
    ((![P : pname > $o]: ((~ ((finite_finite_pname @ (collect_pname @ P)))) => (?[X_1 : pname]: (P @ X_1)))))). % not_finite_existsD
thf(fact_202_not__finite__existsD, axiom,
    ((![P : a > $o]: ((~ ((finite_finite_a @ (collect_a @ P)))) => (?[X_1 : a]: (P @ X_1)))))). % not_finite_existsD
thf(fact_203_not__finite__existsD, axiom,
    ((![P : set_a > $o]: ((~ ((finite_finite_set_a @ (collect_set_a @ P)))) => (?[X_1 : set_a]: (P @ X_1)))))). % not_finite_existsD
thf(fact_204_not__finite__existsD, axiom,
    ((![P : set_pname > $o]: ((~ ((finite505202775_pname @ (collect_set_pname @ P)))) => (?[X_1 : set_pname]: (P @ X_1)))))). % not_finite_existsD
thf(fact_205_Collect__subset, axiom,
    ((![A2 : set_set_a, P : set_a > $o]: (ord_le318720350_set_a @ (collect_set_a @ (^[X2 : set_a]: (((member_set_a @ X2 @ A2)) & ((P @ X2))))) @ A2)))). % Collect_subset
thf(fact_206_Collect__subset, axiom,
    ((![A2 : set_set_pname, P : set_pname > $o]: (ord_le2066558166_pname @ (collect_set_pname @ (^[X2 : set_pname]: (((member_set_pname @ X2 @ A2)) & ((P @ X2))))) @ A2)))). % Collect_subset
thf(fact_207_Collect__subset, axiom,
    ((![A2 : set_a, P : a > $o]: (ord_less_eq_set_a @ (collect_a @ (^[X2 : a]: (((member_a @ X2 @ A2)) & ((P @ X2))))) @ A2)))). % Collect_subset
thf(fact_208_Collect__subset, axiom,
    ((![A2 : set_pname, P : pname > $o]: (ord_le865024672_pname @ (collect_pname @ (^[X2 : pname]: (((member_pname @ X2 @ A2)) & ((P @ X2))))) @ A2)))). % Collect_subset
thf(fact_209_insert__Collect, axiom,
    ((![A : pname, P : pname > $o]: ((insert_pname @ A @ (collect_pname @ P)) = (collect_pname @ (^[U : pname]: (((~ ((U = A)))) => ((P @ U))))))))). % insert_Collect
thf(fact_210_insert__Collect, axiom,
    ((![A : set_a, P : set_a > $o]: ((insert_set_a @ A @ (collect_set_a @ P)) = (collect_set_a @ (^[U : set_a]: (((~ ((U = A)))) => ((P @ U))))))))). % insert_Collect
thf(fact_211_insert__Collect, axiom,
    ((![A : set_pname, P : set_pname > $o]: ((insert_set_pname @ A @ (collect_set_pname @ P)) = (collect_set_pname @ (^[U : set_pname]: (((~ ((U = A)))) => ((P @ U))))))))). % insert_Collect
thf(fact_212_insert__Collect, axiom,
    ((![A : a, P : a > $o]: ((insert_a @ A @ (collect_a @ P)) = (collect_a @ (^[U : a]: (((~ ((U = A)))) => ((P @ U))))))))). % insert_Collect
thf(fact_213_insert__compr, axiom,
    ((insert_pname = (^[A4 : pname]: (^[B2 : set_pname]: (collect_pname @ (^[X2 : pname]: (((X2 = A4)) | ((member_pname @ X2 @ B2)))))))))). % insert_compr
thf(fact_214_insert__compr, axiom,
    ((insert_set_a = (^[A4 : set_a]: (^[B2 : set_set_a]: (collect_set_a @ (^[X2 : set_a]: (((X2 = A4)) | ((member_set_a @ X2 @ B2)))))))))). % insert_compr
thf(fact_215_insert__compr, axiom,
    ((insert_set_pname = (^[A4 : set_pname]: (^[B2 : set_set_pname]: (collect_set_pname @ (^[X2 : set_pname]: (((X2 = A4)) | ((member_set_pname @ X2 @ B2)))))))))). % insert_compr
thf(fact_216_insert__compr, axiom,
    ((insert_a = (^[A4 : a]: (^[B2 : set_a]: (collect_a @ (^[X2 : a]: (((X2 = A4)) | ((member_a @ X2 @ B2)))))))))). % insert_compr
thf(fact_217_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_218_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_219_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_220_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_221_infinite__imp__nonempty, axiom,
    ((![S : set_set_a]: ((~ ((finite_finite_set_a @ S))) => (~ ((S = bot_bot_set_set_a))))))). % infinite_imp_nonempty
thf(fact_222_infinite__imp__nonempty, axiom,
    ((![S : set_set_pname]: ((~ ((finite505202775_pname @ S))) => (~ ((S = bot_bo1397849354_pname))))))). % infinite_imp_nonempty
thf(fact_223_infinite__imp__nonempty, axiom,
    ((![S : set_a]: ((~ ((finite_finite_a @ S))) => (~ ((S = bot_bot_set_a))))))). % infinite_imp_nonempty
thf(fact_224_infinite__imp__nonempty, axiom,
    ((![S : set_pname]: ((~ ((finite_finite_pname @ S))) => (~ ((S = bot_bot_set_pname))))))). % infinite_imp_nonempty
thf(fact_225_finite_OemptyI, axiom,
    ((finite_finite_set_a @ bot_bot_set_set_a))). % finite.emptyI
thf(fact_226_finite_OemptyI, axiom,
    ((finite505202775_pname @ bot_bo1397849354_pname))). % finite.emptyI
thf(fact_227_finite_OemptyI, axiom,
    ((finite_finite_a @ bot_bot_set_a))). % finite.emptyI
thf(fact_228_finite_OemptyI, axiom,
    ((finite_finite_pname @ bot_bot_set_pname))). % finite.emptyI
thf(fact_229_all__subset__image, axiom,
    ((![F : a > a, A2 : set_a, P : set_a > $o]: ((![B2 : set_a]: (((ord_less_eq_set_a @ B2 @ (image_a_a @ F @ A2))) => ((P @ B2)))) = (![B2 : set_a]: (((ord_less_eq_set_a @ B2 @ A2)) => ((P @ (image_a_a @ F @ B2))))))))). % all_subset_image
thf(fact_230_all__subset__image, axiom,
    ((![F : pname > a, A2 : set_pname, P : set_a > $o]: ((![B2 : set_a]: (((ord_less_eq_set_a @ B2 @ (image_pname_a @ F @ A2))) => ((P @ B2)))) = (![B2 : set_pname]: (((ord_le865024672_pname @ B2 @ A2)) => ((P @ (image_pname_a @ F @ B2))))))))). % all_subset_image
thf(fact_231_all__subset__image, axiom,
    ((![F : a > pname, A2 : set_a, P : set_pname > $o]: ((![B2 : set_pname]: (((ord_le865024672_pname @ B2 @ (image_a_pname @ F @ A2))) => ((P @ B2)))) = (![B2 : set_a]: (((ord_less_eq_set_a @ B2 @ A2)) => ((P @ (image_a_pname @ F @ B2))))))))). % all_subset_image
thf(fact_232_all__subset__image, axiom,
    ((![F : pname > pname, A2 : set_pname, P : set_pname > $o]: ((![B2 : set_pname]: (((ord_le865024672_pname @ B2 @ (image_pname_pname @ F @ A2))) => ((P @ B2)))) = (![B2 : set_pname]: (((ord_le865024672_pname @ B2 @ A2)) => ((P @ (image_pname_pname @ F @ B2))))))))). % all_subset_image
thf(fact_233_subset__image__iff, axiom,
    ((![B3 : set_a, F : a > a, A2 : set_a]: ((ord_less_eq_set_a @ B3 @ (image_a_a @ F @ A2)) = (?[AA : set_a]: (((ord_less_eq_set_a @ AA @ A2)) & ((B3 = (image_a_a @ F @ AA))))))))). % subset_image_iff
thf(fact_234_subset__image__iff, axiom,
    ((![B3 : set_a, F : pname > a, A2 : set_pname]: ((ord_less_eq_set_a @ B3 @ (image_pname_a @ F @ A2)) = (?[AA : set_pname]: (((ord_le865024672_pname @ AA @ A2)) & ((B3 = (image_pname_a @ F @ AA))))))))). % subset_image_iff
thf(fact_235_subset__image__iff, axiom,
    ((![B3 : set_pname, F : a > pname, A2 : set_a]: ((ord_le865024672_pname @ B3 @ (image_a_pname @ F @ A2)) = (?[AA : set_a]: (((ord_less_eq_set_a @ AA @ A2)) & ((B3 = (image_a_pname @ F @ AA))))))))). % subset_image_iff
thf(fact_236_subset__image__iff, axiom,
    ((![B3 : set_pname, F : pname > pname, A2 : set_pname]: ((ord_le865024672_pname @ B3 @ (image_pname_pname @ F @ A2)) = (?[AA : set_pname]: (((ord_le865024672_pname @ AA @ A2)) & ((B3 = (image_pname_pname @ F @ AA))))))))). % subset_image_iff
thf(fact_237_image__subset__iff, axiom,
    ((![F : pname > a, A2 : set_pname, B3 : set_a]: ((ord_less_eq_set_a @ (image_pname_a @ F @ A2) @ B3) = (![X2 : pname]: (((member_pname @ X2 @ A2)) => ((member_a @ (F @ X2) @ B3)))))))). % image_subset_iff
thf(fact_238_subset__imageE, axiom,
    ((![B3 : set_a, F : a > a, A2 : set_a]: ((ord_less_eq_set_a @ B3 @ (image_a_a @ F @ A2)) => (~ ((![C4 : set_a]: ((ord_less_eq_set_a @ C4 @ A2) => (~ ((B3 = (image_a_a @ F @ C4)))))))))))). % subset_imageE
thf(fact_239_subset__imageE, axiom,
    ((![B3 : set_a, F : pname > a, A2 : set_pname]: ((ord_less_eq_set_a @ B3 @ (image_pname_a @ F @ A2)) => (~ ((![C4 : set_pname]: ((ord_le865024672_pname @ C4 @ A2) => (~ ((B3 = (image_pname_a @ F @ C4)))))))))))). % subset_imageE
thf(fact_240_subset__imageE, axiom,
    ((![B3 : set_pname, F : a > pname, A2 : set_a]: ((ord_le865024672_pname @ B3 @ (image_a_pname @ F @ A2)) => (~ ((![C4 : set_a]: ((ord_less_eq_set_a @ C4 @ A2) => (~ ((B3 = (image_a_pname @ F @ C4)))))))))))). % subset_imageE
thf(fact_241_subset__imageE, axiom,
    ((![B3 : set_pname, F : pname > pname, A2 : set_pname]: ((ord_le865024672_pname @ B3 @ (image_pname_pname @ F @ A2)) => (~ ((![C4 : set_pname]: ((ord_le865024672_pname @ C4 @ A2) => (~ ((B3 = (image_pname_pname @ F @ C4)))))))))))). % subset_imageE
thf(fact_242_image__subsetI, axiom,
    ((![A2 : set_pname, F : pname > a, B3 : set_a]: ((![X : pname]: ((member_pname @ X @ A2) => (member_a @ (F @ X) @ B3))) => (ord_less_eq_set_a @ (image_pname_a @ F @ A2) @ B3))))). % image_subsetI
thf(fact_243_image__subsetI, axiom,
    ((![A2 : set_a, F : a > a, B3 : set_a]: ((![X : a]: ((member_a @ X @ A2) => (member_a @ (F @ X) @ B3))) => (ord_less_eq_set_a @ (image_a_a @ F @ A2) @ B3))))). % image_subsetI
thf(fact_244_image__subsetI, axiom,
    ((![A2 : set_pname, F : pname > pname, B3 : set_pname]: ((![X : pname]: ((member_pname @ X @ A2) => (member_pname @ (F @ X) @ B3))) => (ord_le865024672_pname @ (image_pname_pname @ F @ A2) @ B3))))). % image_subsetI
thf(fact_245_image__subsetI, axiom,
    ((![A2 : set_a, F : a > pname, B3 : set_pname]: ((![X : a]: ((member_a @ X @ A2) => (member_pname @ (F @ X) @ B3))) => (ord_le865024672_pname @ (image_a_pname @ F @ A2) @ B3))))). % image_subsetI
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 (7)
thf(conj_0, hypothesis,
    ((finite_finite_pname @ u))).
thf(conj_1, hypothesis,
    ((uG = (image_pname_a @ mgt_call @ u)))).
thf(conj_2, hypothesis,
    ((ord_less_eq_set_a @ g @ (image_pname_a @ mgt_call @ u)))).
thf(conj_3, hypothesis,
    (((finite_card_a @ g) = (finite_card_a @ (image_pname_a @ mgt_call @ u))))).
thf(conj_4, hypothesis,
    ((wt @ c))).
thf(conj_5, hypothesis,
    ((g = (image_pname_a @ mgt_call @ u)))).
thf(conj_6, conjecture,
    ((p @ g @ (insert_a @ (mgt @ c) @ bot_bot_set_a)))).
