% 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_375__3253928_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:14:09.829

% Could-be-implicit typings (8)
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__Com__Ocom, type,
    com : $tType).
thf(ty_n_tf__a, type,
    a : $tType).

% Explicit typings (36)
thf(sy_c_Com_Obody, type,
    body : pname > option_com).
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_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_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_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_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_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_Set_Ois__empty_001t__Com__Opname, type,
    is_empty_pname : set_pname > $o).
thf(sy_c_Set_Ois__empty_001tf__a, type,
    is_empty_a : set_a > $o).
thf(sy_c_Set_Ois__singleton_001t__Com__Opname, type,
    is_singleton_pname : set_pname > $o).
thf(sy_c_Set_Ois__singleton_001tf__a, type,
    is_singleton_a : set_a > $o).
thf(sy_c_Set_Othe__elem_001t__Com__Opname, type,
    the_elem_pname : set_pname > pname).
thf(sy_c_Set_Othe__elem_001tf__a, type,
    the_elem_a : set_a > a).
thf(sy_c_member_001t__Com__Opname, type,
    member_pname : pname > set_pname > $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_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_1_singleton__conv, axiom,
    ((![A : pname]: ((collect_pname @ (^[X2 : pname]: (X2 = A))) = (insert_pname @ A @ bot_bot_set_pname))))). % singleton_conv
thf(fact_2_singleton__conv, axiom,
    ((![A : a]: ((collect_a @ (^[X2 : a]: (X2 = A))) = (insert_a @ A @ bot_bot_set_a))))). % singleton_conv
thf(fact_3_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_4_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_5_finite__insert, axiom,
    ((![A : a, A2 : set_a]: ((finite_finite_a @ (insert_a @ A @ A2)) = (finite_finite_a @ A2))))). % finite_insert
thf(fact_6_finite__insert, axiom,
    ((![A : pname, A2 : set_pname]: ((finite_finite_pname @ (insert_pname @ A @ A2)) = (finite_finite_pname @ A2))))). % finite_insert
thf(fact_7_singletonI, axiom,
    ((![A : pname]: (member_pname @ A @ (insert_pname @ A @ bot_bot_set_pname))))). % singletonI
thf(fact_8_singletonI, axiom,
    ((![A : a]: (member_a @ A @ (insert_a @ A @ bot_bot_set_a))))). % singletonI
thf(fact_9_image__insert, axiom,
    ((![F : a > pname, A : a, B : set_a]: ((image_a_pname @ F @ (insert_a @ A @ B)) = (insert_pname @ (F @ A) @ (image_a_pname @ F @ B)))))). % image_insert
thf(fact_10_image__insert, axiom,
    ((![F : pname > pname, A : pname, B : set_pname]: ((image_pname_pname @ F @ (insert_pname @ A @ B)) = (insert_pname @ (F @ A) @ (image_pname_pname @ F @ B)))))). % image_insert
thf(fact_11_image__insert, axiom,
    ((![F : pname > a, A : pname, B : set_pname]: ((image_pname_a @ F @ (insert_pname @ A @ B)) = (insert_a @ (F @ A) @ (image_pname_a @ F @ B)))))). % image_insert
thf(fact_12_image__insert, axiom,
    ((![F : a > a, A : a, B : set_a]: ((image_a_a @ F @ (insert_a @ A @ B)) = (insert_a @ (F @ A) @ (image_a_a @ F @ B)))))). % image_insert
thf(fact_13_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_14_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_15_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_16_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_17_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_18_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_19_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_20_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_21_image__empty, axiom,
    ((![F : a > pname]: ((image_a_pname @ F @ bot_bot_set_a) = bot_bot_set_pname)))). % image_empty
thf(fact_22_image__empty, axiom,
    ((![F : pname > pname]: ((image_pname_pname @ F @ bot_bot_set_pname) = bot_bot_set_pname)))). % image_empty
thf(fact_23_image__empty, axiom,
    ((![F : pname > a]: ((image_pname_a @ F @ bot_bot_set_pname) = bot_bot_set_a)))). % image_empty
thf(fact_24_image__empty, axiom,
    ((![F : a > a]: ((image_a_a @ F @ bot_bot_set_a) = bot_bot_set_a)))). % image_empty
thf(fact_25_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_26_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_27_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_28_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_29_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_30_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_31_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_32_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_33_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_34_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_35_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_36_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_37_image__eqI, axiom,
    ((![B2 : pname, F : pname > pname, X3 : pname, A2 : set_pname]: ((B2 = (F @ X3)) => ((member_pname @ X3 @ A2) => (member_pname @ B2 @ (image_pname_pname @ F @ A2))))))). % image_eqI
thf(fact_38_image__eqI, axiom,
    ((![B2 : a, F : pname > a, X3 : pname, A2 : set_pname]: ((B2 = (F @ X3)) => ((member_pname @ X3 @ A2) => (member_a @ B2 @ (image_pname_a @ F @ A2))))))). % image_eqI
thf(fact_39_image__eqI, axiom,
    ((![B2 : pname, F : a > pname, X3 : a, A2 : set_a]: ((B2 = (F @ X3)) => ((member_a @ X3 @ A2) => (member_pname @ B2 @ (image_a_pname @ F @ A2))))))). % image_eqI
thf(fact_40_image__eqI, axiom,
    ((![B2 : a, F : a > a, X3 : a, A2 : set_a]: ((B2 = (F @ X3)) => ((member_a @ X3 @ A2) => (member_a @ B2 @ (image_a_a @ F @ A2))))))). % image_eqI
thf(fact_41_empty__Collect__eq, axiom,
    ((![P : a > $o]: ((bot_bot_set_a = (collect_a @ P)) = (![X2 : a]: (~ ((P @ X2)))))))). % empty_Collect_eq
thf(fact_42_empty__Collect__eq, axiom,
    ((![P : pname > $o]: ((bot_bot_set_pname = (collect_pname @ P)) = (![X2 : pname]: (~ ((P @ X2)))))))). % empty_Collect_eq
thf(fact_43_Collect__empty__eq, axiom,
    ((![P : a > $o]: (((collect_a @ P) = bot_bot_set_a) = (![X2 : a]: (~ ((P @ X2)))))))). % Collect_empty_eq
thf(fact_44_Collect__empty__eq, axiom,
    ((![P : pname > $o]: (((collect_pname @ P) = bot_bot_set_pname) = (![X2 : pname]: (~ ((P @ X2)))))))). % Collect_empty_eq
thf(fact_45_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_46_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_47_empty__iff, axiom,
    ((![C : a]: (~ ((member_a @ C @ bot_bot_set_a)))))). % empty_iff
thf(fact_48_empty__iff, axiom,
    ((![C : pname]: (~ ((member_pname @ C @ bot_bot_set_pname)))))). % empty_iff
thf(fact_49_insert__absorb2, axiom,
    ((![X3 : a, A2 : set_a]: ((insert_a @ X3 @ (insert_a @ X3 @ A2)) = (insert_a @ X3 @ A2))))). % insert_absorb2
thf(fact_50_insert__absorb2, axiom,
    ((![X3 : pname, A2 : set_pname]: ((insert_pname @ X3 @ (insert_pname @ X3 @ A2)) = (insert_pname @ X3 @ A2))))). % insert_absorb2
thf(fact_51_insert__iff, axiom,
    ((![A : pname, B2 : pname, A2 : set_pname]: ((member_pname @ A @ (insert_pname @ B2 @ A2)) = (((A = B2)) | ((member_pname @ A @ A2))))))). % insert_iff
thf(fact_52_insert__iff, axiom,
    ((![A : a, B2 : a, A2 : set_a]: ((member_a @ A @ (insert_a @ B2 @ A2)) = (((A = B2)) | ((member_a @ A @ A2))))))). % insert_iff
thf(fact_53_insertCI, axiom,
    ((![A : pname, B : set_pname, B2 : pname]: (((~ ((member_pname @ A @ B))) => (A = B2)) => (member_pname @ A @ (insert_pname @ B2 @ B)))))). % insertCI
thf(fact_54_insertCI, axiom,
    ((![A : a, B : set_a, B2 : a]: (((~ ((member_a @ A @ B))) => (A = B2)) => (member_a @ A @ (insert_a @ B2 @ B)))))). % insertCI
thf(fact_55_image__ident, axiom,
    ((![Y2 : set_a]: ((image_a_a @ (^[X2 : a]: X2) @ Y2) = Y2)))). % image_ident
thf(fact_56_image__ident, axiom,
    ((![Y2 : set_pname]: ((image_pname_pname @ (^[X2 : pname]: X2) @ Y2) = Y2)))). % image_ident
thf(fact_57_rev__image__eqI, axiom,
    ((![X3 : pname, A2 : set_pname, B2 : pname, F : pname > pname]: ((member_pname @ X3 @ A2) => ((B2 = (F @ X3)) => (member_pname @ B2 @ (image_pname_pname @ F @ A2))))))). % rev_image_eqI
thf(fact_58_rev__image__eqI, axiom,
    ((![X3 : pname, A2 : set_pname, B2 : a, F : pname > a]: ((member_pname @ X3 @ A2) => ((B2 = (F @ X3)) => (member_a @ B2 @ (image_pname_a @ F @ A2))))))). % rev_image_eqI
thf(fact_59_rev__image__eqI, axiom,
    ((![X3 : a, A2 : set_a, B2 : pname, F : a > pname]: ((member_a @ X3 @ A2) => ((B2 = (F @ X3)) => (member_pname @ B2 @ (image_a_pname @ F @ A2))))))). % rev_image_eqI
thf(fact_60_rev__image__eqI, axiom,
    ((![X3 : a, A2 : set_a, B2 : a, F : a > a]: ((member_a @ X3 @ A2) => ((B2 = (F @ X3)) => (member_a @ B2 @ (image_a_a @ F @ A2))))))). % rev_image_eqI
thf(fact_61_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_62_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_63_ball__imageD, axiom,
    ((![F : pname > pname, A2 : set_pname, P : pname > $o]: ((![X : pname]: ((member_pname @ X @ (image_pname_pname @ F @ A2)) => (P @ X))) => (![X4 : pname]: ((member_pname @ X4 @ A2) => (P @ (F @ X4)))))))). % ball_imageD
thf(fact_64_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_65_image__cong, axiom,
    ((![M : set_pname, N : set_pname, F : pname > pname, G2 : pname > pname]: ((M = N) => ((![X : pname]: ((member_pname @ X @ N) => ((F @ X) = (G2 @ X)))) => ((image_pname_pname @ F @ M) = (image_pname_pname @ G2 @ N))))))). % image_cong
thf(fact_66_image__cong, axiom,
    ((![M : set_a, N : set_a, F : a > a, G2 : a > a]: ((M = N) => ((![X : a]: ((member_a @ X @ N) => ((F @ X) = (G2 @ X)))) => ((image_a_a @ F @ M) = (image_a_a @ G2 @ N))))))). % image_cong
thf(fact_67_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_68_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_69_bex__imageD, axiom,
    ((![F : pname > pname, A2 : set_pname, P : pname > $o]: ((?[X4 : pname]: ((member_pname @ X4 @ (image_pname_pname @ F @ A2)) & (P @ X4))) => (?[X : pname]: ((member_pname @ X @ A2) & (P @ (F @ X)))))))). % bex_imageD
thf(fact_70_image__iff, axiom,
    ((![Z2 : pname, F : pname > pname, A2 : set_pname]: ((member_pname @ Z2 @ (image_pname_pname @ F @ A2)) = (?[X2 : pname]: (((member_pname @ X2 @ A2)) & ((Z2 = (F @ X2))))))))). % image_iff
thf(fact_71_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_72_image__iff, axiom,
    ((![Z2 : a, F : a > a, A2 : set_a]: ((member_a @ Z2 @ (image_a_a @ F @ A2)) = (?[X2 : a]: (((member_a @ X2 @ A2)) & ((Z2 = (F @ X2))))))))). % image_iff
thf(fact_73_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_74_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_75_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_76_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_77_ex__in__conv, axiom,
    ((![A2 : set_a]: ((?[X2 : a]: (member_a @ X2 @ A2)) = (~ ((A2 = bot_bot_set_a))))))). % ex_in_conv
thf(fact_78_ex__in__conv, axiom,
    ((![A2 : set_pname]: ((?[X2 : pname]: (member_pname @ X2 @ A2)) = (~ ((A2 = bot_bot_set_pname))))))). % ex_in_conv
thf(fact_79_equals0I, axiom,
    ((![A2 : set_a]: ((![Y3 : a]: (~ ((member_a @ Y3 @ A2)))) => (A2 = bot_bot_set_a))))). % equals0I
thf(fact_80_equals0I, axiom,
    ((![A2 : set_pname]: ((![Y3 : pname]: (~ ((member_pname @ Y3 @ A2)))) => (A2 = bot_bot_set_pname))))). % equals0I
thf(fact_81_equals0D, axiom,
    ((![A2 : set_a, A : a]: ((A2 = bot_bot_set_a) => (~ ((member_a @ A @ A2))))))). % equals0D
thf(fact_82_equals0D, axiom,
    ((![A2 : set_pname, A : pname]: ((A2 = bot_bot_set_pname) => (~ ((member_pname @ A @ A2))))))). % equals0D
thf(fact_83_emptyE, axiom,
    ((![A : a]: (~ ((member_a @ A @ bot_bot_set_a)))))). % emptyE
thf(fact_84_emptyE, axiom,
    ((![A : pname]: (~ ((member_pname @ A @ bot_bot_set_pname)))))). % emptyE
thf(fact_85_mk__disjoint__insert, axiom,
    ((![A : pname, A2 : set_pname]: ((member_pname @ A @ A2) => (?[B3 : set_pname]: ((A2 = (insert_pname @ A @ B3)) & (~ ((member_pname @ A @ B3))))))))). % mk_disjoint_insert
thf(fact_86_mk__disjoint__insert, axiom,
    ((![A : a, A2 : set_a]: ((member_a @ A @ A2) => (?[B3 : set_a]: ((A2 = (insert_a @ A @ B3)) & (~ ((member_a @ A @ B3))))))))). % mk_disjoint_insert
thf(fact_87_insert__commute, axiom,
    ((![X3 : a, Y4 : a, A2 : set_a]: ((insert_a @ X3 @ (insert_a @ Y4 @ A2)) = (insert_a @ Y4 @ (insert_a @ X3 @ A2)))))). % insert_commute
thf(fact_88_insert__commute, axiom,
    ((![X3 : pname, Y4 : pname, A2 : set_pname]: ((insert_pname @ X3 @ (insert_pname @ Y4 @ A2)) = (insert_pname @ Y4 @ (insert_pname @ X3 @ A2)))))). % insert_commute
thf(fact_89_insert__eq__iff, axiom,
    ((![A : pname, A2 : set_pname, B2 : pname, B : set_pname]: ((~ ((member_pname @ A @ A2))) => ((~ ((member_pname @ B2 @ B))) => (((insert_pname @ A @ A2) = (insert_pname @ B2 @ B)) = (((((A = B2)) => ((A2 = B)))) & ((((~ ((A = B2)))) => ((?[C2 : set_pname]: (((A2 = (insert_pname @ B2 @ C2))) & ((((~ ((member_pname @ B2 @ C2)))) & ((((B = (insert_pname @ A @ C2))) & ((~ ((member_pname @ A @ C2)))))))))))))))))))). % insert_eq_iff
thf(fact_90_insert__eq__iff, axiom,
    ((![A : a, A2 : set_a, B2 : a, B : set_a]: ((~ ((member_a @ A @ A2))) => ((~ ((member_a @ B2 @ B))) => (((insert_a @ A @ A2) = (insert_a @ B2 @ B)) = (((((A = B2)) => ((A2 = B)))) & ((((~ ((A = B2)))) => ((?[C2 : set_a]: (((A2 = (insert_a @ B2 @ C2))) & ((((~ ((member_a @ B2 @ C2)))) & ((((B = (insert_a @ A @ C2))) & ((~ ((member_a @ A @ C2)))))))))))))))))))). % insert_eq_iff
thf(fact_91_insert__absorb, axiom,
    ((![A : pname, A2 : set_pname]: ((member_pname @ A @ A2) => ((insert_pname @ A @ A2) = A2))))). % insert_absorb
thf(fact_92_insert__absorb, axiom,
    ((![A : a, A2 : set_a]: ((member_a @ A @ A2) => ((insert_a @ A @ A2) = A2))))). % insert_absorb
thf(fact_93_insert__ident, axiom,
    ((![X3 : pname, A2 : set_pname, B : set_pname]: ((~ ((member_pname @ X3 @ A2))) => ((~ ((member_pname @ X3 @ B))) => (((insert_pname @ X3 @ A2) = (insert_pname @ X3 @ B)) = (A2 = B))))))). % insert_ident
thf(fact_94_insert__ident, axiom,
    ((![X3 : a, A2 : set_a, B : set_a]: ((~ ((member_a @ X3 @ A2))) => ((~ ((member_a @ X3 @ B))) => (((insert_a @ X3 @ A2) = (insert_a @ X3 @ B)) = (A2 = B))))))). % insert_ident
thf(fact_95_Set_Oset__insert, axiom,
    ((![X3 : pname, A2 : set_pname]: ((member_pname @ X3 @ A2) => (~ ((![B3 : set_pname]: ((A2 = (insert_pname @ X3 @ B3)) => (member_pname @ X3 @ B3))))))))). % Set.set_insert
thf(fact_96_Set_Oset__insert, axiom,
    ((![X3 : a, A2 : set_a]: ((member_a @ X3 @ A2) => (~ ((![B3 : set_a]: ((A2 = (insert_a @ X3 @ B3)) => (member_a @ X3 @ B3))))))))). % Set.set_insert
thf(fact_97_insertI2, axiom,
    ((![A : pname, B : set_pname, B2 : pname]: ((member_pname @ A @ B) => (member_pname @ A @ (insert_pname @ B2 @ B)))))). % insertI2
thf(fact_98_insertI2, axiom,
    ((![A : a, B : set_a, B2 : a]: ((member_a @ A @ B) => (member_a @ A @ (insert_a @ B2 @ B)))))). % insertI2
thf(fact_99_insertI1, axiom,
    ((![A : pname, B : set_pname]: (member_pname @ A @ (insert_pname @ A @ B))))). % insertI1
thf(fact_100_insertI1, axiom,
    ((![A : a, B : set_a]: (member_a @ A @ (insert_a @ A @ B))))). % insertI1
thf(fact_101_insertE, axiom,
    ((![A : pname, B2 : pname, A2 : set_pname]: ((member_pname @ A @ (insert_pname @ B2 @ A2)) => ((~ ((A = B2))) => (member_pname @ A @ A2)))))). % insertE
thf(fact_102_insertE, axiom,
    ((![A : a, B2 : a, A2 : set_a]: ((member_a @ A @ (insert_a @ B2 @ A2)) => ((~ ((A = B2))) => (member_a @ A @ A2)))))). % insertE
thf(fact_103_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_104_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_105_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_106_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_107_mem__Collect__eq, axiom,
    ((![A : pname, P : pname > $o]: ((member_pname @ A @ (collect_pname @ P)) = (P @ A))))). % mem_Collect_eq
thf(fact_108_mem__Collect__eq, axiom,
    ((![A : a, P : a > $o]: ((member_a @ A @ (collect_a @ P)) = (P @ A))))). % mem_Collect_eq
thf(fact_109_Collect__mem__eq, axiom,
    ((![A2 : set_pname]: ((collect_pname @ (^[X2 : pname]: (member_pname @ X2 @ A2))) = A2)))). % Collect_mem_eq
thf(fact_110_Collect__mem__eq, axiom,
    ((![A2 : set_a]: ((collect_a @ (^[X2 : a]: (member_a @ X2 @ A2))) = A2)))). % Collect_mem_eq
thf(fact_111_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_112_image__image, axiom,
    ((![F : a > pname, G2 : pname > a, A2 : set_pname]: ((image_a_pname @ F @ (image_pname_a @ G2 @ A2)) = (image_pname_pname @ (^[X2 : pname]: (F @ (G2 @ X2))) @ A2))))). % image_image
thf(fact_113_image__image, axiom,
    ((![F : pname > a, G2 : a > pname, A2 : set_a]: ((image_pname_a @ F @ (image_a_pname @ G2 @ A2)) = (image_a_a @ (^[X2 : a]: (F @ (G2 @ X2))) @ A2))))). % image_image
thf(fact_114_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_115_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_116_image__image, axiom,
    ((![F : a > a, G2 : a > a, A2 : set_a]: ((image_a_a @ F @ (image_a_a @ G2 @ A2)) = (image_a_a @ (^[X2 : a]: (F @ (G2 @ X2))) @ A2))))). % image_image
thf(fact_117_image__image, axiom,
    ((![F : pname > pname, G2 : pname > pname, A2 : set_pname]: ((image_pname_pname @ F @ (image_pname_pname @ G2 @ A2)) = (image_pname_pname @ (^[X2 : pname]: (F @ (G2 @ X2))) @ A2))))). % image_image
thf(fact_118_imageE, axiom,
    ((![B2 : pname, F : pname > pname, A2 : set_pname]: ((member_pname @ B2 @ (image_pname_pname @ F @ A2)) => (~ ((![X : pname]: ((B2 = (F @ X)) => (~ ((member_pname @ X @ A2))))))))))). % imageE
thf(fact_119_imageE, axiom,
    ((![B2 : pname, F : a > pname, A2 : set_a]: ((member_pname @ B2 @ (image_a_pname @ F @ A2)) => (~ ((![X : a]: ((B2 = (F @ X)) => (~ ((member_a @ X @ A2))))))))))). % imageE
thf(fact_120_imageE, axiom,
    ((![B2 : a, F : pname > a, A2 : set_pname]: ((member_a @ B2 @ (image_pname_a @ F @ A2)) => (~ ((![X : pname]: ((B2 = (F @ X)) => (~ ((member_pname @ X @ A2))))))))))). % imageE
thf(fact_121_imageE, axiom,
    ((![B2 : a, F : a > a, A2 : set_a]: ((member_a @ B2 @ (image_a_a @ F @ A2)) => (~ ((![X : a]: ((B2 = (F @ X)) => (~ ((member_a @ X @ A2))))))))))). % imageE
thf(fact_122_empty__def, axiom,
    ((bot_bot_set_a = (collect_a @ (^[X2 : a]: $false))))). % empty_def
thf(fact_123_empty__def, axiom,
    ((bot_bot_set_pname = (collect_pname @ (^[X2 : pname]: $false))))). % empty_def
thf(fact_124_pigeonhole__infinite__rel, axiom,
    ((![A2 : set_pname, B : set_pname, R : pname > pname > $o]: ((~ ((finite_finite_pname @ A2))) => ((finite_finite_pname @ B) => ((![X : pname]: ((member_pname @ X @ A2) => (?[Xa : pname]: ((member_pname @ Xa @ B) & (R @ X @ Xa))))) => (?[X : pname]: ((member_pname @ X @ B) & (~ ((finite_finite_pname @ (collect_pname @ (^[A3 : pname]: (((member_pname @ A3 @ A2)) & ((R @ A3 @ X)))))))))))))))). % pigeonhole_infinite_rel
thf(fact_125_pigeonhole__infinite__rel, axiom,
    ((![A2 : set_pname, B : set_a, R : pname > a > $o]: ((~ ((finite_finite_pname @ A2))) => ((finite_finite_a @ B) => ((![X : pname]: ((member_pname @ X @ A2) => (?[Xa : a]: ((member_a @ Xa @ B) & (R @ X @ Xa))))) => (?[X : a]: ((member_a @ X @ B) & (~ ((finite_finite_pname @ (collect_pname @ (^[A3 : pname]: (((member_pname @ A3 @ A2)) & ((R @ A3 @ X)))))))))))))))). % pigeonhole_infinite_rel
thf(fact_126_pigeonhole__infinite__rel, axiom,
    ((![A2 : set_a, B : set_pname, R : a > pname > $o]: ((~ ((finite_finite_a @ A2))) => ((finite_finite_pname @ B) => ((![X : a]: ((member_a @ X @ A2) => (?[Xa : pname]: ((member_pname @ Xa @ B) & (R @ X @ Xa))))) => (?[X : pname]: ((member_pname @ X @ B) & (~ ((finite_finite_a @ (collect_a @ (^[A3 : a]: (((member_a @ A3 @ A2)) & ((R @ A3 @ X)))))))))))))))). % pigeonhole_infinite_rel
thf(fact_127_pigeonhole__infinite__rel, axiom,
    ((![A2 : set_a, B : set_a, R : a > a > $o]: ((~ ((finite_finite_a @ A2))) => ((finite_finite_a @ B) => ((![X : a]: ((member_a @ X @ A2) => (?[Xa : a]: ((member_a @ Xa @ B) & (R @ X @ Xa))))) => (?[X : a]: ((member_a @ X @ B) & (~ ((finite_finite_a @ (collect_a @ (^[A3 : a]: (((member_a @ A3 @ A2)) & ((R @ A3 @ X)))))))))))))))). % pigeonhole_infinite_rel
thf(fact_128_not__finite__existsD, axiom,
    ((![P : pname > $o]: ((~ ((finite_finite_pname @ (collect_pname @ P)))) => (?[X_1 : pname]: (P @ X_1)))))). % not_finite_existsD
thf(fact_129_not__finite__existsD, axiom,
    ((![P : a > $o]: ((~ ((finite_finite_a @ (collect_a @ P)))) => (?[X_1 : a]: (P @ X_1)))))). % not_finite_existsD
thf(fact_130_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_131_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_132_insert__compr, axiom,
    ((insert_pname = (^[A3 : pname]: (^[B4 : set_pname]: (collect_pname @ (^[X2 : pname]: (((X2 = A3)) | ((member_pname @ X2 @ B4)))))))))). % insert_compr
thf(fact_133_insert__compr, axiom,
    ((insert_a = (^[A3 : a]: (^[B4 : set_a]: (collect_a @ (^[X2 : a]: (((X2 = A3)) | ((member_a @ X2 @ B4)))))))))). % insert_compr
thf(fact_134_infinite__imp__nonempty, axiom,
    ((![S : set_a]: ((~ ((finite_finite_a @ S))) => (~ ((S = bot_bot_set_a))))))). % infinite_imp_nonempty
thf(fact_135_infinite__imp__nonempty, axiom,
    ((![S : set_pname]: ((~ ((finite_finite_pname @ S))) => (~ ((S = bot_bot_set_pname))))))). % infinite_imp_nonempty
thf(fact_136_finite_OemptyI, axiom,
    ((finite_finite_a @ bot_bot_set_a))). % finite.emptyI
thf(fact_137_finite_OemptyI, axiom,
    ((finite_finite_pname @ bot_bot_set_pname))). % finite.emptyI
thf(fact_138_singleton__inject, axiom,
    ((![A : a, B2 : a]: (((insert_a @ A @ bot_bot_set_a) = (insert_a @ B2 @ bot_bot_set_a)) => (A = B2))))). % singleton_inject
thf(fact_139_singleton__inject, axiom,
    ((![A : pname, B2 : pname]: (((insert_pname @ A @ bot_bot_set_pname) = (insert_pname @ B2 @ bot_bot_set_pname)) => (A = B2))))). % singleton_inject
thf(fact_140_insert__not__empty, axiom,
    ((![A : a, A2 : set_a]: (~ (((insert_a @ A @ A2) = bot_bot_set_a)))))). % insert_not_empty
thf(fact_141_insert__not__empty, axiom,
    ((![A : pname, A2 : set_pname]: (~ (((insert_pname @ A @ A2) = bot_bot_set_pname)))))). % insert_not_empty
thf(fact_142_doubleton__eq__iff, axiom,
    ((![A : a, B2 : a, C : a, D : a]: (((insert_a @ A @ (insert_a @ B2 @ bot_bot_set_a)) = (insert_a @ C @ (insert_a @ D @ bot_bot_set_a))) = (((((A = C)) & ((B2 = D)))) | ((((A = D)) & ((B2 = C))))))))). % doubleton_eq_iff
thf(fact_143_doubleton__eq__iff, axiom,
    ((![A : pname, B2 : pname, C : pname, D : pname]: (((insert_pname @ A @ (insert_pname @ B2 @ bot_bot_set_pname)) = (insert_pname @ C @ (insert_pname @ D @ bot_bot_set_pname))) = (((((A = C)) & ((B2 = D)))) | ((((A = D)) & ((B2 = C))))))))). % doubleton_eq_iff
thf(fact_144_singleton__iff, axiom,
    ((![B2 : a, A : a]: ((member_a @ B2 @ (insert_a @ A @ bot_bot_set_a)) = (B2 = A))))). % singleton_iff
thf(fact_145_singleton__iff, axiom,
    ((![B2 : pname, A : pname]: ((member_pname @ B2 @ (insert_pname @ A @ bot_bot_set_pname)) = (B2 = A))))). % singleton_iff
thf(fact_146_singletonD, axiom,
    ((![B2 : a, A : a]: ((member_a @ B2 @ (insert_a @ A @ bot_bot_set_a)) => (B2 = A))))). % singletonD
thf(fact_147_singletonD, axiom,
    ((![B2 : pname, A : pname]: ((member_pname @ B2 @ (insert_pname @ A @ bot_bot_set_pname)) => (B2 = A))))). % singletonD
thf(fact_148_finite_OinsertI, axiom,
    ((![A2 : set_pname, A : pname]: ((finite_finite_pname @ A2) => (finite_finite_pname @ (insert_pname @ A @ A2)))))). % finite.insertI
thf(fact_149_finite_OinsertI, axiom,
    ((![A2 : set_a, A : a]: ((finite_finite_a @ A2) => (finite_finite_a @ (insert_a @ A @ A2)))))). % finite.insertI
thf(fact_150_pigeonhole__infinite, axiom,
    ((![A2 : set_pname, F : pname > pname]: ((~ ((finite_finite_pname @ A2))) => ((finite_finite_pname @ (image_pname_pname @ F @ A2)) => (?[X : pname]: ((member_pname @ X @ A2) & (~ ((finite_finite_pname @ (collect_pname @ (^[A3 : pname]: (((member_pname @ A3 @ A2)) & (((F @ A3) = (F @ X)))))))))))))))). % pigeonhole_infinite
thf(fact_151_pigeonhole__infinite, axiom,
    ((![A2 : set_pname, F : pname > a]: ((~ ((finite_finite_pname @ A2))) => ((finite_finite_a @ (image_pname_a @ F @ A2)) => (?[X : pname]: ((member_pname @ X @ A2) & (~ ((finite_finite_pname @ (collect_pname @ (^[A3 : pname]: (((member_pname @ A3 @ A2)) & (((F @ A3) = (F @ X)))))))))))))))). % pigeonhole_infinite
thf(fact_152_pigeonhole__infinite, axiom,
    ((![A2 : set_a, F : a > pname]: ((~ ((finite_finite_a @ A2))) => ((finite_finite_pname @ (image_a_pname @ F @ A2)) => (?[X : a]: ((member_a @ X @ A2) & (~ ((finite_finite_a @ (collect_a @ (^[A3 : a]: (((member_a @ A3 @ A2)) & (((F @ A3) = (F @ X)))))))))))))))). % pigeonhole_infinite
thf(fact_153_pigeonhole__infinite, axiom,
    ((![A2 : set_a, F : a > a]: ((~ ((finite_finite_a @ A2))) => ((finite_finite_a @ (image_a_a @ F @ A2)) => (?[X : a]: ((member_a @ X @ A2) & (~ ((finite_finite_a @ (collect_a @ (^[A3 : a]: (((member_a @ A3 @ A2)) & (((F @ A3) = (F @ X)))))))))))))))). % pigeonhole_infinite
thf(fact_154_Collect__conv__if2, axiom,
    ((![P : a > $o, A : a]: (((P @ A) => ((collect_a @ (^[X2 : a]: (((A = X2)) & ((P @ X2))))) = (insert_a @ A @ bot_bot_set_a))) & ((~ ((P @ A))) => ((collect_a @ (^[X2 : a]: (((A = X2)) & ((P @ X2))))) = bot_bot_set_a)))))). % Collect_conv_if2
thf(fact_155_Collect__conv__if2, axiom,
    ((![P : pname > $o, A : pname]: (((P @ A) => ((collect_pname @ (^[X2 : pname]: (((A = X2)) & ((P @ X2))))) = (insert_pname @ A @ bot_bot_set_pname))) & ((~ ((P @ A))) => ((collect_pname @ (^[X2 : pname]: (((A = X2)) & ((P @ X2))))) = bot_bot_set_pname)))))). % Collect_conv_if2
thf(fact_156_Collect__conv__if, axiom,
    ((![P : a > $o, A : a]: (((P @ A) => ((collect_a @ (^[X2 : a]: (((X2 = A)) & ((P @ X2))))) = (insert_a @ A @ bot_bot_set_a))) & ((~ ((P @ A))) => ((collect_a @ (^[X2 : a]: (((X2 = A)) & ((P @ X2))))) = bot_bot_set_a)))))). % Collect_conv_if
thf(fact_157_Collect__conv__if, axiom,
    ((![P : pname > $o, A : pname]: (((P @ A) => ((collect_pname @ (^[X2 : pname]: (((X2 = A)) & ((P @ X2))))) = (insert_pname @ A @ bot_bot_set_pname))) & ((~ ((P @ A))) => ((collect_pname @ (^[X2 : pname]: (((X2 = A)) & ((P @ X2))))) = bot_bot_set_pname)))))). % Collect_conv_if
thf(fact_158_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_159_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_160_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_161_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_162_finite_Oinducts, axiom,
    ((![X3 : set_a, P : set_a > $o]: ((finite_finite_a @ X3) => ((P @ bot_bot_set_a) => ((![A4 : set_a, A5 : a]: ((finite_finite_a @ A4) => ((P @ A4) => (P @ (insert_a @ A5 @ A4))))) => (P @ X3))))))). % finite.inducts
thf(fact_163_finite_Oinducts, axiom,
    ((![X3 : set_pname, P : set_pname > $o]: ((finite_finite_pname @ X3) => ((P @ bot_bot_set_pname) => ((![A4 : set_pname, A5 : pname]: ((finite_finite_pname @ A4) => ((P @ A4) => (P @ (insert_pname @ A5 @ A4))))) => (P @ X3))))))). % finite.inducts
thf(fact_164_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_165_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_166_finite_Osimps, axiom,
    ((finite_finite_a = (^[A3 : set_a]: (((A3 = bot_bot_set_a)) | ((?[A6 : set_a]: (?[B5 : a]: (((A3 = (insert_a @ B5 @ A6))) & ((finite_finite_a @ A6))))))))))). % finite.simps
thf(fact_167_finite_Osimps, axiom,
    ((finite_finite_pname = (^[A3 : set_pname]: (((A3 = bot_bot_set_pname)) | ((?[A6 : set_pname]: (?[B5 : pname]: (((A3 = (insert_pname @ B5 @ A6))) & ((finite_finite_pname @ A6))))))))))). % finite.simps
thf(fact_168_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_169_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_170_image__constant__conv, axiom,
    ((![A2 : set_a, C : a]: (((A2 = bot_bot_set_a) => ((image_a_a @ (^[X2 : a]: C) @ A2) = bot_bot_set_a)) & ((~ ((A2 = bot_bot_set_a))) => ((image_a_a @ (^[X2 : a]: C) @ A2) = (insert_a @ C @ bot_bot_set_a))))))). % image_constant_conv
thf(fact_171_image__constant__conv, axiom,
    ((![A2 : set_a, C : pname]: (((A2 = bot_bot_set_a) => ((image_a_pname @ (^[X2 : a]: C) @ A2) = bot_bot_set_pname)) & ((~ ((A2 = bot_bot_set_a))) => ((image_a_pname @ (^[X2 : a]: C) @ A2) = (insert_pname @ C @ bot_bot_set_pname))))))). % image_constant_conv
thf(fact_172_image__constant__conv, axiom,
    ((![A2 : set_pname, C : a]: (((A2 = bot_bot_set_pname) => ((image_pname_a @ (^[X2 : pname]: C) @ A2) = bot_bot_set_a)) & ((~ ((A2 = bot_bot_set_pname))) => ((image_pname_a @ (^[X2 : pname]: C) @ A2) = (insert_a @ C @ bot_bot_set_a))))))). % image_constant_conv
thf(fact_173_image__constant__conv, axiom,
    ((![A2 : set_pname, C : pname]: (((A2 = bot_bot_set_pname) => ((image_pname_pname @ (^[X2 : pname]: C) @ A2) = bot_bot_set_pname)) & ((~ ((A2 = bot_bot_set_pname))) => ((image_pname_pname @ (^[X2 : pname]: C) @ A2) = (insert_pname @ C @ bot_bot_set_pname))))))). % image_constant_conv
thf(fact_174_image__constant, axiom,
    ((![X3 : pname, A2 : set_pname, C : a]: ((member_pname @ X3 @ A2) => ((image_pname_a @ (^[X2 : pname]: C) @ A2) = (insert_a @ C @ bot_bot_set_a)))))). % image_constant
thf(fact_175_image__constant, axiom,
    ((![X3 : a, A2 : set_a, C : a]: ((member_a @ X3 @ A2) => ((image_a_a @ (^[X2 : a]: C) @ A2) = (insert_a @ C @ bot_bot_set_a)))))). % image_constant
thf(fact_176_image__constant, axiom,
    ((![X3 : pname, A2 : set_pname, C : pname]: ((member_pname @ X3 @ A2) => ((image_pname_pname @ (^[X2 : pname]: C) @ A2) = (insert_pname @ C @ bot_bot_set_pname)))))). % image_constant
thf(fact_177_image__constant, axiom,
    ((![X3 : a, A2 : set_a, C : pname]: ((member_a @ X3 @ A2) => ((image_a_pname @ (^[X2 : a]: C) @ A2) = (insert_pname @ C @ bot_bot_set_pname)))))). % image_constant
thf(fact_178_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_179_the__elem__eq, axiom,
    ((![X3 : a]: ((the_elem_a @ (insert_a @ X3 @ bot_bot_set_a)) = X3)))). % the_elem_eq
thf(fact_180_the__elem__eq, axiom,
    ((![X3 : pname]: ((the_elem_pname @ (insert_pname @ X3 @ bot_bot_set_pname)) = X3)))). % the_elem_eq
thf(fact_181_is__singletonI, axiom,
    ((![X3 : a]: (is_singleton_a @ (insert_a @ X3 @ bot_bot_set_a))))). % is_singletonI
thf(fact_182_is__singletonI, axiom,
    ((![X3 : pname]: (is_singleton_pname @ (insert_pname @ X3 @ bot_bot_set_pname))))). % is_singletonI
thf(fact_183_assms_I1_J, axiom,
    ((![Ts : set_a, G : set_a]: ((ord_less_eq_set_a @ Ts @ G) => (p @ G @ Ts))))). % assms(1)
thf(fact_184_assms_I4_J, axiom,
    ((![Pn : pname]: ((member_pname @ Pn @ u) => (wt @ (the_com @ (body @ Pn))))))). % assms(4)
thf(fact_185_the__elem__image__unique, axiom,
    ((![A2 : set_a, F : a > a, X3 : a]: ((~ ((A2 = bot_bot_set_a))) => ((![Y3 : a]: ((member_a @ Y3 @ A2) => ((F @ Y3) = (F @ X3)))) => ((the_elem_a @ (image_a_a @ F @ A2)) = (F @ X3))))))). % the_elem_image_unique
thf(fact_186_the__elem__image__unique, axiom,
    ((![A2 : set_pname, F : pname > a, X3 : pname]: ((~ ((A2 = bot_bot_set_pname))) => ((![Y3 : pname]: ((member_pname @ Y3 @ A2) => ((F @ Y3) = (F @ X3)))) => ((the_elem_a @ (image_pname_a @ F @ A2)) = (F @ X3))))))). % the_elem_image_unique
thf(fact_187_the__elem__image__unique, axiom,
    ((![A2 : set_pname, F : pname > pname, X3 : pname]: ((~ ((A2 = bot_bot_set_pname))) => ((![Y3 : pname]: ((member_pname @ Y3 @ A2) => ((F @ Y3) = (F @ X3)))) => ((the_elem_pname @ (image_pname_pname @ F @ A2)) = (F @ X3))))))). % the_elem_image_unique
thf(fact_188_Set_Ois__empty__def, axiom,
    ((is_empty_a = (^[A6 : set_a]: (A6 = bot_bot_set_a))))). % Set.is_empty_def
thf(fact_189_Set_Ois__empty__def, axiom,
    ((is_empty_pname = (^[A6 : set_pname]: (A6 = bot_bot_set_pname))))). % Set.is_empty_def
thf(fact_190_is__singleton__def, axiom,
    ((is_singleton_a = (^[A6 : set_a]: (?[X2 : a]: (A6 = (insert_a @ X2 @ bot_bot_set_a))))))). % is_singleton_def
thf(fact_191_is__singleton__def, axiom,
    ((is_singleton_pname = (^[A6 : set_pname]: (?[X2 : pname]: (A6 = (insert_pname @ X2 @ bot_bot_set_pname))))))). % is_singleton_def
thf(fact_192_is__singletonE, axiom,
    ((![A2 : set_a]: ((is_singleton_a @ A2) => (~ ((![X : a]: (~ ((A2 = (insert_a @ X @ bot_bot_set_a))))))))))). % is_singletonE
thf(fact_193_is__singletonE, axiom,
    ((![A2 : set_pname]: ((is_singleton_pname @ A2) => (~ ((![X : pname]: (~ ((A2 = (insert_pname @ X @ bot_bot_set_pname))))))))))). % is_singletonE
thf(fact_194_order__refl, axiom,
    ((![X3 : set_a]: (ord_less_eq_set_a @ X3 @ X3)))). % order_refl
thf(fact_195_subsetI, axiom,
    ((![A2 : set_pname, B : set_pname]: ((![X : pname]: ((member_pname @ X @ A2) => (member_pname @ X @ B))) => (ord_le865024672_pname @ A2 @ B))))). % subsetI
thf(fact_196_subsetI, axiom,
    ((![A2 : set_a, B : set_a]: ((![X : a]: ((member_a @ X @ A2) => (member_a @ X @ B))) => (ord_less_eq_set_a @ A2 @ B))))). % subsetI
thf(fact_197_subset__antisym, axiom,
    ((![A2 : set_a, B : set_a]: ((ord_less_eq_set_a @ A2 @ B) => ((ord_less_eq_set_a @ B @ A2) => (A2 = B)))))). % subset_antisym
thf(fact_198_subset__empty, axiom,
    ((![A2 : set_pname]: ((ord_le865024672_pname @ A2 @ bot_bot_set_pname) = (A2 = bot_bot_set_pname))))). % subset_empty
thf(fact_199_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_200_empty__subsetI, axiom,
    ((![A2 : set_pname]: (ord_le865024672_pname @ bot_bot_set_pname @ A2)))). % empty_subsetI
thf(fact_201_empty__subsetI, axiom,
    ((![A2 : set_a]: (ord_less_eq_set_a @ bot_bot_set_a @ A2)))). % empty_subsetI
thf(fact_202_insert__subset, axiom,
    ((![X3 : pname, A2 : set_pname, B : set_pname]: ((ord_le865024672_pname @ (insert_pname @ X3 @ A2) @ B) = (((member_pname @ X3 @ B)) & ((ord_le865024672_pname @ A2 @ B))))))). % insert_subset
thf(fact_203_insert__subset, axiom,
    ((![X3 : a, A2 : set_a, B : set_a]: ((ord_less_eq_set_a @ (insert_a @ X3 @ A2) @ B) = (((member_a @ X3 @ B)) & ((ord_less_eq_set_a @ A2 @ B))))))). % insert_subset
thf(fact_204_finite__Collect__subsets, axiom,
    ((![A2 : set_pname]: ((finite_finite_pname @ A2) => (finite505202775_pname @ (collect_set_pname @ (^[B4 : set_pname]: (ord_le865024672_pname @ B4 @ A2)))))))). % finite_Collect_subsets
thf(fact_205_finite__Collect__subsets, axiom,
    ((![A2 : set_a]: ((finite_finite_a @ A2) => (finite_finite_set_a @ (collect_set_a @ (^[B4 : set_a]: (ord_less_eq_set_a @ B4 @ A2)))))))). % finite_Collect_subsets
thf(fact_206_singleton__insert__inj__eq, axiom,
    ((![B2 : pname, A : pname, A2 : set_pname]: (((insert_pname @ B2 @ bot_bot_set_pname) = (insert_pname @ A @ A2)) = (((A = B2)) & ((ord_le865024672_pname @ A2 @ (insert_pname @ B2 @ bot_bot_set_pname)))))))). % singleton_insert_inj_eq
thf(fact_207_singleton__insert__inj__eq, axiom,
    ((![B2 : a, A : a, A2 : set_a]: (((insert_a @ B2 @ bot_bot_set_a) = (insert_a @ A @ A2)) = (((A = B2)) & ((ord_less_eq_set_a @ A2 @ (insert_a @ B2 @ bot_bot_set_a)))))))). % singleton_insert_inj_eq
thf(fact_208_singleton__insert__inj__eq_H, axiom,
    ((![A : pname, A2 : set_pname, B2 : pname]: (((insert_pname @ A @ A2) = (insert_pname @ B2 @ bot_bot_set_pname)) = (((A = B2)) & ((ord_le865024672_pname @ A2 @ (insert_pname @ B2 @ bot_bot_set_pname)))))))). % singleton_insert_inj_eq'
thf(fact_209_singleton__insert__inj__eq_H, axiom,
    ((![A : a, A2 : set_a, B2 : a]: (((insert_a @ A @ A2) = (insert_a @ B2 @ bot_bot_set_a)) = (((A = B2)) & ((ord_less_eq_set_a @ A2 @ (insert_a @ B2 @ bot_bot_set_a)))))))). % singleton_insert_inj_eq'
thf(fact_210_in__mono, axiom,
    ((![A2 : set_pname, B : set_pname, X3 : pname]: ((ord_le865024672_pname @ A2 @ B) => ((member_pname @ X3 @ A2) => (member_pname @ X3 @ B)))))). % in_mono
thf(fact_211_in__mono, axiom,
    ((![A2 : set_a, B : set_a, X3 : a]: ((ord_less_eq_set_a @ A2 @ B) => ((member_a @ X3 @ A2) => (member_a @ X3 @ B)))))). % in_mono
thf(fact_212_subsetD, axiom,
    ((![A2 : set_pname, B : set_pname, C : pname]: ((ord_le865024672_pname @ A2 @ B) => ((member_pname @ C @ A2) => (member_pname @ C @ B)))))). % subsetD
thf(fact_213_subsetD, axiom,
    ((![A2 : set_a, B : set_a, C : a]: ((ord_less_eq_set_a @ A2 @ B) => ((member_a @ C @ A2) => (member_a @ C @ B)))))). % subsetD
thf(fact_214_equalityE, axiom,
    ((![A2 : set_a, B : set_a]: ((A2 = B) => (~ (((ord_less_eq_set_a @ A2 @ B) => (~ ((ord_less_eq_set_a @ B @ A2)))))))))). % equalityE
thf(fact_215_subset__eq, axiom,
    ((ord_le865024672_pname = (^[A6 : set_pname]: (^[B4 : set_pname]: (![X2 : pname]: (((member_pname @ X2 @ A6)) => ((member_pname @ X2 @ B4))))))))). % subset_eq
thf(fact_216_subset__eq, axiom,
    ((ord_less_eq_set_a = (^[A6 : set_a]: (^[B4 : set_a]: (![X2 : a]: (((member_a @ X2 @ A6)) => ((member_a @ X2 @ B4))))))))). % subset_eq
thf(fact_217_equalityD1, axiom,
    ((![A2 : set_a, B : set_a]: ((A2 = B) => (ord_less_eq_set_a @ A2 @ B))))). % equalityD1
thf(fact_218_equalityD2, axiom,
    ((![A2 : set_a, B : set_a]: ((A2 = B) => (ord_less_eq_set_a @ B @ A2))))). % equalityD2
thf(fact_219_subset__iff, axiom,
    ((ord_le865024672_pname = (^[A6 : set_pname]: (^[B4 : set_pname]: (![T : pname]: (((member_pname @ T @ A6)) => ((member_pname @ T @ B4))))))))). % subset_iff
thf(fact_220_subset__iff, axiom,
    ((ord_less_eq_set_a = (^[A6 : set_a]: (^[B4 : set_a]: (![T : a]: (((member_a @ T @ A6)) => ((member_a @ T @ B4))))))))). % subset_iff
thf(fact_221_subset__refl, axiom,
    ((![A2 : set_a]: (ord_less_eq_set_a @ A2 @ A2)))). % subset_refl
thf(fact_222_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_223_subset__trans, axiom,
    ((![A2 : set_a, B : set_a, C3 : set_a]: ((ord_less_eq_set_a @ A2 @ B) => ((ord_less_eq_set_a @ B @ C3) => (ord_less_eq_set_a @ A2 @ C3)))))). % subset_trans
thf(fact_224_set__eq__subset, axiom,
    (((^[Y : set_a]: (^[Z : set_a]: (Y = Z))) = (^[A6 : set_a]: (^[B4 : set_a]: (((ord_less_eq_set_a @ A6 @ B4)) & ((ord_less_eq_set_a @ B4 @ A6)))))))). % set_eq_subset
thf(fact_225_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_226_dual__order_Oantisym, axiom,
    ((![B2 : set_a, A : set_a]: ((ord_less_eq_set_a @ B2 @ A) => ((ord_less_eq_set_a @ A @ B2) => (A = B2)))))). % dual_order.antisym
thf(fact_227_dual__order_Oeq__iff, axiom,
    (((^[Y : set_a]: (^[Z : set_a]: (Y = Z))) = (^[A3 : set_a]: (^[B5 : set_a]: (((ord_less_eq_set_a @ B5 @ A3)) & ((ord_less_eq_set_a @ A3 @ B5)))))))). % dual_order.eq_iff
thf(fact_228_dual__order_Otrans, axiom,
    ((![B2 : set_a, A : set_a, C : set_a]: ((ord_less_eq_set_a @ B2 @ A) => ((ord_less_eq_set_a @ C @ B2) => (ord_less_eq_set_a @ C @ A)))))). % dual_order.trans
thf(fact_229_dual__order_Orefl, axiom,
    ((![A : set_a]: (ord_less_eq_set_a @ A @ A)))). % dual_order.refl
thf(fact_230_order__trans, axiom,
    ((![X3 : set_a, Y4 : set_a, Z2 : set_a]: ((ord_less_eq_set_a @ X3 @ Y4) => ((ord_less_eq_set_a @ Y4 @ Z2) => (ord_less_eq_set_a @ X3 @ Z2)))))). % order_trans
thf(fact_231_order__class_Oorder_Oantisym, axiom,
    ((![A : set_a, B2 : set_a]: ((ord_less_eq_set_a @ A @ B2) => ((ord_less_eq_set_a @ B2 @ A) => (A = B2)))))). % order_class.order.antisym
thf(fact_232_ord__le__eq__trans, axiom,
    ((![A : set_a, B2 : set_a, C : set_a]: ((ord_less_eq_set_a @ A @ B2) => ((B2 = C) => (ord_less_eq_set_a @ A @ C)))))). % ord_le_eq_trans
thf(fact_233_ord__eq__le__trans, axiom,
    ((![A : set_a, B2 : set_a, C : set_a]: ((A = B2) => ((ord_less_eq_set_a @ B2 @ C) => (ord_less_eq_set_a @ A @ C)))))). % ord_eq_le_trans
thf(fact_234_order__class_Oorder_Oeq__iff, axiom,
    (((^[Y : set_a]: (^[Z : set_a]: (Y = Z))) = (^[A3 : set_a]: (^[B5 : set_a]: (((ord_less_eq_set_a @ A3 @ B5)) & ((ord_less_eq_set_a @ B5 @ A3)))))))). % order_class.order.eq_iff
thf(fact_235_antisym__conv, axiom,
    ((![Y4 : set_a, X3 : set_a]: ((ord_less_eq_set_a @ Y4 @ X3) => ((ord_less_eq_set_a @ X3 @ Y4) = (X3 = Y4)))))). % antisym_conv
thf(fact_236_order_Otrans, axiom,
    ((![A : set_a, B2 : set_a, C : set_a]: ((ord_less_eq_set_a @ A @ B2) => ((ord_less_eq_set_a @ B2 @ C) => (ord_less_eq_set_a @ A @ C)))))). % order.trans
thf(fact_237_eq__refl, axiom,
    ((![X3 : set_a, Y4 : set_a]: ((X3 = Y4) => (ord_less_eq_set_a @ X3 @ Y4))))). % eq_refl
thf(fact_238_antisym, axiom,
    ((![X3 : set_a, Y4 : set_a]: ((ord_less_eq_set_a @ X3 @ Y4) => ((ord_less_eq_set_a @ Y4 @ X3) => (X3 = Y4)))))). % antisym
thf(fact_239_eq__iff, axiom,
    (((^[Y : set_a]: (^[Z : set_a]: (Y = Z))) = (^[X2 : set_a]: (^[Y5 : set_a]: (((ord_less_eq_set_a @ X2 @ Y5)) & ((ord_less_eq_set_a @ Y5 @ X2)))))))). % eq_iff
thf(fact_240_ord__le__eq__subst, axiom,
    ((![A : set_a, B2 : set_a, F : set_a > set_a, C : set_a]: ((ord_less_eq_set_a @ A @ B2) => (((F @ B2) = C) => ((![X : set_a, Y3 : set_a]: ((ord_less_eq_set_a @ X @ Y3) => (ord_less_eq_set_a @ (F @ X) @ (F @ Y3)))) => (ord_less_eq_set_a @ (F @ A) @ C))))))). % ord_le_eq_subst
thf(fact_241_ord__eq__le__subst, axiom,
    ((![A : set_a, F : set_a > set_a, B2 : set_a, C : set_a]: ((A = (F @ B2)) => ((ord_less_eq_set_a @ B2 @ C) => ((![X : set_a, Y3 : set_a]: ((ord_less_eq_set_a @ X @ Y3) => (ord_less_eq_set_a @ (F @ X) @ (F @ Y3)))) => (ord_less_eq_set_a @ A @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_242_order__subst2, axiom,
    ((![A : set_a, B2 : set_a, F : set_a > set_a, C : set_a]: ((ord_less_eq_set_a @ A @ B2) => ((ord_less_eq_set_a @ (F @ B2) @ C) => ((![X : set_a, Y3 : set_a]: ((ord_less_eq_set_a @ X @ Y3) => (ord_less_eq_set_a @ (F @ X) @ (F @ Y3)))) => (ord_less_eq_set_a @ (F @ A) @ C))))))). % order_subst2
thf(fact_243_order__subst1, axiom,
    ((![A : set_a, F : set_a > set_a, B2 : set_a, C : set_a]: ((ord_less_eq_set_a @ A @ (F @ B2)) => ((ord_less_eq_set_a @ B2 @ C) => ((![X : set_a, Y3 : set_a]: ((ord_less_eq_set_a @ X @ Y3) => (ord_less_eq_set_a @ (F @ X) @ (F @ Y3)))) => (ord_less_eq_set_a @ A @ (F @ C)))))))). % order_subst1
thf(fact_244_bot_Oextremum, axiom,
    ((![A : set_pname]: (ord_le865024672_pname @ bot_bot_set_pname @ A)))). % bot.extremum
thf(fact_245_bot_Oextremum, axiom,
    ((![A : set_a]: (ord_less_eq_set_a @ bot_bot_set_a @ A)))). % bot.extremum
thf(fact_246_bot_Oextremum__unique, axiom,
    ((![A : set_pname]: ((ord_le865024672_pname @ A @ bot_bot_set_pname) = (A = bot_bot_set_pname))))). % bot.extremum_unique
thf(fact_247_bot_Oextremum__unique, axiom,
    ((![A : set_a]: ((ord_less_eq_set_a @ A @ bot_bot_set_a) = (A = bot_bot_set_a))))). % bot.extremum_unique

% Conjectures (5)
thf(conj_0, hypothesis,
    ((finite_finite_pname @ u))).
thf(conj_1, hypothesis,
    ((uG = (image_pname_a @ mgt_call @ u)))).
thf(conj_2, hypothesis,
    ((wt @ c))).
thf(conj_3, hypothesis,
    ((g = (image_pname_a @ mgt_call @ u)))).
thf(conj_4, conjecture,
    ((p @ (image_pname_a @ mgt_call @ u) @ (insert_a @ (mgt @ c) @ bot_bot_set_a)))).
