% 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_373__3253916_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:14:07.816

% 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_001t__Com__Opname, type,
    finite_card_pname : set_pname > nat).
thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__Com__Opname_J, type,
    finite1249089560_pname : set_set_pname > nat).
thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_Itf__a_J, type,
    finite_card_set_a : set_set_a > nat).
thf(sy_c_Finite__Set_Ocard_001tf__a, type,
    finite_card_a : set_a > nat).
thf(sy_c_Finite__Set_Ofinite_001t__Com__Opname, type,
    finite_finite_pname : set_pname > $o).
thf(sy_c_Finite__Set_Ofinite_001t__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_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_It__Com__Opname_J_001t__Set__Oset_Itf__a_J, type,
    image_700492503_set_a : (set_pname > set_a) > set_set_pname > set_set_a).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Com__Opname_J_001tf__a, type,
    image_set_pname_a : (set_pname > a) > set_set_pname > set_a).
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_001t__Set__Oset_Itf__a_J, type,
    image_set_a_set_a : (set_a > set_a) > set_set_a > set_set_a).
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_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_finite__Collect__subsets, axiom,
    ((![A : set_set_a]: ((finite_finite_set_a @ A) => (finite1606323175_set_a @ (collect_set_set_a @ (^[B : set_set_a]: (ord_le318720350_set_a @ B @ A)))))))). % finite_Collect_subsets
thf(fact_2_finite__Collect__subsets, axiom,
    ((![A : set_set_pname]: ((finite505202775_pname @ A) => (finite1638948493_pname @ (collec2100311499_pname @ (^[B : set_set_pname]: (ord_le2066558166_pname @ B @ A)))))))). % finite_Collect_subsets
thf(fact_3_finite__Collect__subsets, axiom,
    ((![A : set_pname]: ((finite_finite_pname @ A) => (finite505202775_pname @ (collect_set_pname @ (^[B : set_pname]: (ord_le865024672_pname @ B @ A)))))))). % finite_Collect_subsets
thf(fact_4_finite__Collect__subsets, axiom,
    ((![A : set_a]: ((finite_finite_a @ A) => (finite_finite_set_a @ (collect_set_a @ (^[B : set_a]: (ord_less_eq_set_a @ B @ A)))))))). % finite_Collect_subsets
thf(fact_5_finite__imageI, axiom,
    ((![F : set_pname, H : pname > a]: ((finite_finite_pname @ F) => (finite_finite_a @ (image_pname_a @ H @ F)))))). % finite_imageI
thf(fact_6_finite__imageI, axiom,
    ((![F : set_pname, H : pname > pname]: ((finite_finite_pname @ F) => (finite_finite_pname @ (image_pname_pname @ H @ F)))))). % finite_imageI
thf(fact_7_finite__imageI, axiom,
    ((![F : set_a, H : a > pname]: ((finite_finite_a @ F) => (finite_finite_pname @ (image_a_pname @ H @ F)))))). % finite_imageI
thf(fact_8_finite__imageI, axiom,
    ((![F : set_a, H : a > a]: ((finite_finite_a @ F) => (finite_finite_a @ (image_a_a @ H @ F)))))). % finite_imageI
thf(fact_9_finite__imageI, axiom,
    ((![F : set_pname, H : pname > set_a]: ((finite_finite_pname @ F) => (finite_finite_set_a @ (image_pname_set_a @ H @ F)))))). % finite_imageI
thf(fact_10_finite__imageI, axiom,
    ((![F : set_pname, H : pname > set_pname]: ((finite_finite_pname @ F) => (finite505202775_pname @ (image_747505105_pname @ H @ F)))))). % finite_imageI
thf(fact_11_finite__imageI, axiom,
    ((![F : set_a, H : a > set_a]: ((finite_finite_a @ F) => (finite_finite_set_a @ (image_a_set_a @ H @ F)))))). % finite_imageI
thf(fact_12_finite__imageI, axiom,
    ((![F : set_a, H : a > set_pname]: ((finite_finite_a @ F) => (finite505202775_pname @ (image_a_set_pname @ H @ F)))))). % finite_imageI
thf(fact_13_finite__imageI, axiom,
    ((![F : set_set_a, H : set_a > pname]: ((finite_finite_set_a @ F) => (finite_finite_pname @ (image_set_a_pname @ H @ F)))))). % finite_imageI
thf(fact_14_finite__imageI, axiom,
    ((![F : set_set_a, H : set_a > a]: ((finite_finite_set_a @ F) => (finite_finite_a @ (image_set_a_a @ H @ F)))))). % finite_imageI
thf(fact_15_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 @ (^[X : a]: (((P @ X)) & ((Q @ X)))))))))). % finite_Collect_conjI
thf(fact_16_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 @ (^[X : set_a]: (((P @ X)) & ((Q @ X)))))))))). % finite_Collect_conjI
thf(fact_17_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 @ (^[X : set_pname]: (((P @ X)) & ((Q @ X)))))))))). % finite_Collect_conjI
thf(fact_18_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 @ (^[X : pname]: (((P @ X)) & ((Q @ X)))))))))). % finite_Collect_conjI
thf(fact_19_finite__Collect__disjI, axiom,
    ((![P : a > $o, Q : a > $o]: ((finite_finite_a @ (collect_a @ (^[X : a]: (((P @ X)) | ((Q @ X)))))) = (((finite_finite_a @ (collect_a @ P))) & ((finite_finite_a @ (collect_a @ Q)))))))). % finite_Collect_disjI
thf(fact_20_finite__Collect__disjI, axiom,
    ((![P : set_a > $o, Q : set_a > $o]: ((finite_finite_set_a @ (collect_set_a @ (^[X : set_a]: (((P @ X)) | ((Q @ X)))))) = (((finite_finite_set_a @ (collect_set_a @ P))) & ((finite_finite_set_a @ (collect_set_a @ Q)))))))). % finite_Collect_disjI
thf(fact_21_finite__Collect__disjI, axiom,
    ((![P : set_pname > $o, Q : set_pname > $o]: ((finite505202775_pname @ (collect_set_pname @ (^[X : set_pname]: (((P @ X)) | ((Q @ X)))))) = (((finite505202775_pname @ (collect_set_pname @ P))) & ((finite505202775_pname @ (collect_set_pname @ Q)))))))). % finite_Collect_disjI
thf(fact_22_finite__Collect__disjI, axiom,
    ((![P : pname > $o, Q : pname > $o]: ((finite_finite_pname @ (collect_pname @ (^[X : pname]: (((P @ X)) | ((Q @ X)))))) = (((finite_finite_pname @ (collect_pname @ P))) & ((finite_finite_pname @ (collect_pname @ Q)))))))). % finite_Collect_disjI
thf(fact_23_image__ident, axiom,
    ((![Y : set_pname]: ((image_pname_pname @ (^[X : pname]: X) @ Y) = Y)))). % image_ident
thf(fact_24_image__ident, axiom,
    ((![Y : set_a]: ((image_a_a @ (^[X : a]: X) @ Y) = Y)))). % image_ident
thf(fact_25_card__subset__eq, axiom,
    ((![B2 : set_set_a, A : set_set_a]: ((finite_finite_set_a @ B2) => ((ord_le318720350_set_a @ A @ B2) => (((finite_card_set_a @ A) = (finite_card_set_a @ B2)) => (A = B2))))))). % card_subset_eq
thf(fact_26_card__subset__eq, axiom,
    ((![B2 : set_set_pname, A : set_set_pname]: ((finite505202775_pname @ B2) => ((ord_le2066558166_pname @ A @ B2) => (((finite1249089560_pname @ A) = (finite1249089560_pname @ B2)) => (A = B2))))))). % card_subset_eq
thf(fact_27_card__subset__eq, axiom,
    ((![B2 : set_pname, A : set_pname]: ((finite_finite_pname @ B2) => ((ord_le865024672_pname @ A @ B2) => (((finite_card_pname @ A) = (finite_card_pname @ B2)) => (A = B2))))))). % card_subset_eq
thf(fact_28_card__subset__eq, axiom,
    ((![B2 : set_a, A : set_a]: ((finite_finite_a @ B2) => ((ord_less_eq_set_a @ A @ B2) => (((finite_card_a @ A) = (finite_card_a @ B2)) => (A = B2))))))). % card_subset_eq
thf(fact_29_infinite__arbitrarily__large, axiom,
    ((![A : set_set_a, N : nat]: ((~ ((finite_finite_set_a @ A))) => (?[B3 : set_set_a]: ((finite_finite_set_a @ B3) & (((finite_card_set_a @ B3) = N) & (ord_le318720350_set_a @ B3 @ A)))))))). % infinite_arbitrarily_large
thf(fact_30_infinite__arbitrarily__large, axiom,
    ((![A : set_set_pname, N : nat]: ((~ ((finite505202775_pname @ A))) => (?[B3 : set_set_pname]: ((finite505202775_pname @ B3) & (((finite1249089560_pname @ B3) = N) & (ord_le2066558166_pname @ B3 @ A)))))))). % infinite_arbitrarily_large
thf(fact_31_infinite__arbitrarily__large, axiom,
    ((![A : set_pname, N : nat]: ((~ ((finite_finite_pname @ A))) => (?[B3 : set_pname]: ((finite_finite_pname @ B3) & (((finite_card_pname @ B3) = N) & (ord_le865024672_pname @ B3 @ A)))))))). % infinite_arbitrarily_large
thf(fact_32_infinite__arbitrarily__large, axiom,
    ((![A : set_a, N : nat]: ((~ ((finite_finite_a @ A))) => (?[B3 : set_a]: ((finite_finite_a @ B3) & (((finite_card_a @ B3) = N) & (ord_less_eq_set_a @ B3 @ A)))))))). % infinite_arbitrarily_large
thf(fact_33_finite__surj, axiom,
    ((![A : set_pname, B2 : set_pname, F2 : pname > pname]: ((finite_finite_pname @ A) => ((ord_le865024672_pname @ B2 @ (image_pname_pname @ F2 @ A)) => (finite_finite_pname @ B2)))))). % finite_surj
thf(fact_34_finite__surj, axiom,
    ((![A : set_pname, B2 : set_a, F2 : pname > a]: ((finite_finite_pname @ A) => ((ord_less_eq_set_a @ B2 @ (image_pname_a @ F2 @ A)) => (finite_finite_a @ B2)))))). % finite_surj
thf(fact_35_finite__surj, axiom,
    ((![A : set_a, B2 : set_a, F2 : a > a]: ((finite_finite_a @ A) => ((ord_less_eq_set_a @ B2 @ (image_a_a @ F2 @ A)) => (finite_finite_a @ B2)))))). % finite_surj
thf(fact_36_finite__surj, axiom,
    ((![A : set_a, B2 : set_pname, F2 : a > pname]: ((finite_finite_a @ A) => ((ord_le865024672_pname @ B2 @ (image_a_pname @ F2 @ A)) => (finite_finite_pname @ B2)))))). % finite_surj
thf(fact_37_finite__surj, axiom,
    ((![A : set_pname, B2 : set_set_a, F2 : pname > set_a]: ((finite_finite_pname @ A) => ((ord_le318720350_set_a @ B2 @ (image_pname_set_a @ F2 @ A)) => (finite_finite_set_a @ B2)))))). % finite_surj
thf(fact_38_finite__surj, axiom,
    ((![A : set_pname, B2 : set_set_pname, F2 : pname > set_pname]: ((finite_finite_pname @ A) => ((ord_le2066558166_pname @ B2 @ (image_747505105_pname @ F2 @ A)) => (finite505202775_pname @ B2)))))). % finite_surj
thf(fact_39_finite__surj, axiom,
    ((![A : set_a, B2 : set_set_a, F2 : a > set_a]: ((finite_finite_a @ A) => ((ord_le318720350_set_a @ B2 @ (image_a_set_a @ F2 @ A)) => (finite_finite_set_a @ B2)))))). % finite_surj
thf(fact_40_finite__surj, axiom,
    ((![A : set_a, B2 : set_set_pname, F2 : a > set_pname]: ((finite_finite_a @ A) => ((ord_le2066558166_pname @ B2 @ (image_a_set_pname @ F2 @ A)) => (finite505202775_pname @ B2)))))). % finite_surj
thf(fact_41_finite__surj, axiom,
    ((![A : set_set_a, B2 : set_a, F2 : set_a > a]: ((finite_finite_set_a @ A) => ((ord_less_eq_set_a @ B2 @ (image_set_a_a @ F2 @ A)) => (finite_finite_a @ B2)))))). % finite_surj
thf(fact_42_finite__surj, axiom,
    ((![A : set_set_pname, B2 : set_a, F2 : set_pname > a]: ((finite505202775_pname @ A) => ((ord_less_eq_set_a @ B2 @ (image_set_pname_a @ F2 @ A)) => (finite_finite_a @ B2)))))). % finite_surj
thf(fact_43_finite__subset__image, axiom,
    ((![B2 : set_pname, F2 : pname > pname, A : set_pname]: ((finite_finite_pname @ B2) => ((ord_le865024672_pname @ B2 @ (image_pname_pname @ F2 @ A)) => (?[C : set_pname]: ((ord_le865024672_pname @ C @ A) & ((finite_finite_pname @ C) & (B2 = (image_pname_pname @ F2 @ C)))))))))). % finite_subset_image
thf(fact_44_finite__subset__image, axiom,
    ((![B2 : set_pname, F2 : a > pname, A : set_a]: ((finite_finite_pname @ B2) => ((ord_le865024672_pname @ B2 @ (image_a_pname @ F2 @ A)) => (?[C : set_a]: ((ord_less_eq_set_a @ C @ A) & ((finite_finite_a @ C) & (B2 = (image_a_pname @ F2 @ C)))))))))). % finite_subset_image
thf(fact_45_finite__subset__image, axiom,
    ((![B2 : set_a, F2 : pname > a, A : set_pname]: ((finite_finite_a @ B2) => ((ord_less_eq_set_a @ B2 @ (image_pname_a @ F2 @ A)) => (?[C : set_pname]: ((ord_le865024672_pname @ C @ A) & ((finite_finite_pname @ C) & (B2 = (image_pname_a @ F2 @ C)))))))))). % finite_subset_image
thf(fact_46_finite__subset__image, axiom,
    ((![B2 : set_a, F2 : a > a, A : set_a]: ((finite_finite_a @ B2) => ((ord_less_eq_set_a @ B2 @ (image_a_a @ F2 @ A)) => (?[C : set_a]: ((ord_less_eq_set_a @ C @ A) & ((finite_finite_a @ C) & (B2 = (image_a_a @ F2 @ C)))))))))). % finite_subset_image
thf(fact_47_finite__subset__image, axiom,
    ((![B2 : set_set_a, F2 : a > set_a, A : set_a]: ((finite_finite_set_a @ B2) => ((ord_le318720350_set_a @ B2 @ (image_a_set_a @ F2 @ A)) => (?[C : set_a]: ((ord_less_eq_set_a @ C @ A) & ((finite_finite_a @ C) & (B2 = (image_a_set_a @ F2 @ C)))))))))). % finite_subset_image
thf(fact_48_finite__subset__image, axiom,
    ((![B2 : set_set_pname, F2 : a > set_pname, A : set_a]: ((finite505202775_pname @ B2) => ((ord_le2066558166_pname @ B2 @ (image_a_set_pname @ F2 @ A)) => (?[C : set_a]: ((ord_less_eq_set_a @ C @ A) & ((finite_finite_a @ C) & (B2 = (image_a_set_pname @ F2 @ C)))))))))). % finite_subset_image
thf(fact_49_finite__subset__image, axiom,
    ((![B2 : set_set_a, F2 : pname > set_a, A : set_pname]: ((finite_finite_set_a @ B2) => ((ord_le318720350_set_a @ B2 @ (image_pname_set_a @ F2 @ A)) => (?[C : set_pname]: ((ord_le865024672_pname @ C @ A) & ((finite_finite_pname @ C) & (B2 = (image_pname_set_a @ F2 @ C)))))))))). % finite_subset_image
thf(fact_50_finite__subset__image, axiom,
    ((![B2 : set_set_pname, F2 : pname > set_pname, A : set_pname]: ((finite505202775_pname @ B2) => ((ord_le2066558166_pname @ B2 @ (image_747505105_pname @ F2 @ A)) => (?[C : set_pname]: ((ord_le865024672_pname @ C @ A) & ((finite_finite_pname @ C) & (B2 = (image_747505105_pname @ F2 @ C)))))))))). % finite_subset_image
thf(fact_51_finite__subset__image, axiom,
    ((![B2 : set_a, F2 : set_a > a, A : set_set_a]: ((finite_finite_a @ B2) => ((ord_less_eq_set_a @ B2 @ (image_set_a_a @ F2 @ A)) => (?[C : set_set_a]: ((ord_le318720350_set_a @ C @ A) & ((finite_finite_set_a @ C) & (B2 = (image_set_a_a @ F2 @ C)))))))))). % finite_subset_image
thf(fact_52_finite__subset__image, axiom,
    ((![B2 : set_a, F2 : set_pname > a, A : set_set_pname]: ((finite_finite_a @ B2) => ((ord_less_eq_set_a @ B2 @ (image_set_pname_a @ F2 @ A)) => (?[C : set_set_pname]: ((ord_le2066558166_pname @ C @ A) & ((finite505202775_pname @ C) & (B2 = (image_set_pname_a @ F2 @ C)))))))))). % finite_subset_image
thf(fact_53_ex__finite__subset__image, axiom,
    ((![F2 : a > a, A : set_a, P : set_a > $o]: ((?[B : set_a]: (((finite_finite_a @ B)) & ((((ord_less_eq_set_a @ B @ (image_a_a @ F2 @ A))) & ((P @ B)))))) = (?[B : set_a]: (((finite_finite_a @ B)) & ((((ord_less_eq_set_a @ B @ A)) & ((P @ (image_a_a @ F2 @ B))))))))))). % ex_finite_subset_image
thf(fact_54_ex__finite__subset__image, axiom,
    ((![F2 : pname > a, A : set_pname, P : set_a > $o]: ((?[B : set_a]: (((finite_finite_a @ B)) & ((((ord_less_eq_set_a @ B @ (image_pname_a @ F2 @ A))) & ((P @ B)))))) = (?[B : set_pname]: (((finite_finite_pname @ B)) & ((((ord_le865024672_pname @ B @ A)) & ((P @ (image_pname_a @ F2 @ B))))))))))). % ex_finite_subset_image
thf(fact_55_ex__finite__subset__image, axiom,
    ((![F2 : a > pname, A : set_a, P : set_pname > $o]: ((?[B : set_pname]: (((finite_finite_pname @ B)) & ((((ord_le865024672_pname @ B @ (image_a_pname @ F2 @ A))) & ((P @ B)))))) = (?[B : set_a]: (((finite_finite_a @ B)) & ((((ord_less_eq_set_a @ B @ A)) & ((P @ (image_a_pname @ F2 @ B))))))))))). % ex_finite_subset_image
thf(fact_56_ex__finite__subset__image, axiom,
    ((![F2 : pname > pname, A : set_pname, P : set_pname > $o]: ((?[B : set_pname]: (((finite_finite_pname @ B)) & ((((ord_le865024672_pname @ B @ (image_pname_pname @ F2 @ A))) & ((P @ B)))))) = (?[B : set_pname]: (((finite_finite_pname @ B)) & ((((ord_le865024672_pname @ B @ A)) & ((P @ (image_pname_pname @ F2 @ B))))))))))). % ex_finite_subset_image
thf(fact_57_ex__finite__subset__image, axiom,
    ((![F2 : a > set_a, A : set_a, P : set_set_a > $o]: ((?[B : set_set_a]: (((finite_finite_set_a @ B)) & ((((ord_le318720350_set_a @ B @ (image_a_set_a @ F2 @ A))) & ((P @ B)))))) = (?[B : set_a]: (((finite_finite_a @ B)) & ((((ord_less_eq_set_a @ B @ A)) & ((P @ (image_a_set_a @ F2 @ B))))))))))). % ex_finite_subset_image
thf(fact_58_ex__finite__subset__image, axiom,
    ((![F2 : a > set_pname, A : set_a, P : set_set_pname > $o]: ((?[B : set_set_pname]: (((finite505202775_pname @ B)) & ((((ord_le2066558166_pname @ B @ (image_a_set_pname @ F2 @ A))) & ((P @ B)))))) = (?[B : set_a]: (((finite_finite_a @ B)) & ((((ord_less_eq_set_a @ B @ A)) & ((P @ (image_a_set_pname @ F2 @ B))))))))))). % ex_finite_subset_image
thf(fact_59_ex__finite__subset__image, axiom,
    ((![F2 : pname > set_a, A : set_pname, P : set_set_a > $o]: ((?[B : set_set_a]: (((finite_finite_set_a @ B)) & ((((ord_le318720350_set_a @ B @ (image_pname_set_a @ F2 @ A))) & ((P @ B)))))) = (?[B : set_pname]: (((finite_finite_pname @ B)) & ((((ord_le865024672_pname @ B @ A)) & ((P @ (image_pname_set_a @ F2 @ B))))))))))). % ex_finite_subset_image
thf(fact_60_ex__finite__subset__image, axiom,
    ((![F2 : pname > set_pname, A : set_pname, P : set_set_pname > $o]: ((?[B : set_set_pname]: (((finite505202775_pname @ B)) & ((((ord_le2066558166_pname @ B @ (image_747505105_pname @ F2 @ A))) & ((P @ B)))))) = (?[B : set_pname]: (((finite_finite_pname @ B)) & ((((ord_le865024672_pname @ B @ A)) & ((P @ (image_747505105_pname @ F2 @ B))))))))))). % ex_finite_subset_image
thf(fact_61_ex__finite__subset__image, axiom,
    ((![F2 : set_a > a, A : set_set_a, P : set_a > $o]: ((?[B : set_a]: (((finite_finite_a @ B)) & ((((ord_less_eq_set_a @ B @ (image_set_a_a @ F2 @ A))) & ((P @ B)))))) = (?[B : set_set_a]: (((finite_finite_set_a @ B)) & ((((ord_le318720350_set_a @ B @ A)) & ((P @ (image_set_a_a @ F2 @ B))))))))))). % ex_finite_subset_image
thf(fact_62_ex__finite__subset__image, axiom,
    ((![F2 : set_pname > a, A : set_set_pname, P : set_a > $o]: ((?[B : set_a]: (((finite_finite_a @ B)) & ((((ord_less_eq_set_a @ B @ (image_set_pname_a @ F2 @ A))) & ((P @ B)))))) = (?[B : set_set_pname]: (((finite505202775_pname @ B)) & ((((ord_le2066558166_pname @ B @ A)) & ((P @ (image_set_pname_a @ F2 @ B))))))))))). % ex_finite_subset_image
thf(fact_63_all__finite__subset__image, axiom,
    ((![F2 : a > a, A : set_a, P : set_a > $o]: ((![B : set_a]: (((((finite_finite_a @ B)) & ((ord_less_eq_set_a @ B @ (image_a_a @ F2 @ A))))) => ((P @ B)))) = (![B : set_a]: (((((finite_finite_a @ B)) & ((ord_less_eq_set_a @ B @ A)))) => ((P @ (image_a_a @ F2 @ B))))))))). % all_finite_subset_image
thf(fact_64_all__finite__subset__image, axiom,
    ((![F2 : pname > a, A : set_pname, P : set_a > $o]: ((![B : set_a]: (((((finite_finite_a @ B)) & ((ord_less_eq_set_a @ B @ (image_pname_a @ F2 @ A))))) => ((P @ B)))) = (![B : set_pname]: (((((finite_finite_pname @ B)) & ((ord_le865024672_pname @ B @ A)))) => ((P @ (image_pname_a @ F2 @ B))))))))). % all_finite_subset_image
thf(fact_65_all__finite__subset__image, axiom,
    ((![F2 : a > pname, A : set_a, P : set_pname > $o]: ((![B : set_pname]: (((((finite_finite_pname @ B)) & ((ord_le865024672_pname @ B @ (image_a_pname @ F2 @ A))))) => ((P @ B)))) = (![B : set_a]: (((((finite_finite_a @ B)) & ((ord_less_eq_set_a @ B @ A)))) => ((P @ (image_a_pname @ F2 @ B))))))))). % all_finite_subset_image
thf(fact_66_all__finite__subset__image, axiom,
    ((![F2 : pname > pname, A : set_pname, P : set_pname > $o]: ((![B : set_pname]: (((((finite_finite_pname @ B)) & ((ord_le865024672_pname @ B @ (image_pname_pname @ F2 @ A))))) => ((P @ B)))) = (![B : set_pname]: (((((finite_finite_pname @ B)) & ((ord_le865024672_pname @ B @ A)))) => ((P @ (image_pname_pname @ F2 @ B))))))))). % all_finite_subset_image
thf(fact_67_all__finite__subset__image, axiom,
    ((![F2 : a > set_a, A : set_a, P : set_set_a > $o]: ((![B : set_set_a]: (((((finite_finite_set_a @ B)) & ((ord_le318720350_set_a @ B @ (image_a_set_a @ F2 @ A))))) => ((P @ B)))) = (![B : set_a]: (((((finite_finite_a @ B)) & ((ord_less_eq_set_a @ B @ A)))) => ((P @ (image_a_set_a @ F2 @ B))))))))). % all_finite_subset_image
thf(fact_68_all__finite__subset__image, axiom,
    ((![F2 : a > set_pname, A : set_a, P : set_set_pname > $o]: ((![B : set_set_pname]: (((((finite505202775_pname @ B)) & ((ord_le2066558166_pname @ B @ (image_a_set_pname @ F2 @ A))))) => ((P @ B)))) = (![B : set_a]: (((((finite_finite_a @ B)) & ((ord_less_eq_set_a @ B @ A)))) => ((P @ (image_a_set_pname @ F2 @ B))))))))). % all_finite_subset_image
thf(fact_69_all__finite__subset__image, axiom,
    ((![F2 : pname > set_a, A : set_pname, P : set_set_a > $o]: ((![B : set_set_a]: (((((finite_finite_set_a @ B)) & ((ord_le318720350_set_a @ B @ (image_pname_set_a @ F2 @ A))))) => ((P @ B)))) = (![B : set_pname]: (((((finite_finite_pname @ B)) & ((ord_le865024672_pname @ B @ A)))) => ((P @ (image_pname_set_a @ F2 @ B))))))))). % all_finite_subset_image
thf(fact_70_all__finite__subset__image, axiom,
    ((![F2 : pname > set_pname, A : set_pname, P : set_set_pname > $o]: ((![B : set_set_pname]: (((((finite505202775_pname @ B)) & ((ord_le2066558166_pname @ B @ (image_747505105_pname @ F2 @ A))))) => ((P @ B)))) = (![B : set_pname]: (((((finite_finite_pname @ B)) & ((ord_le865024672_pname @ B @ A)))) => ((P @ (image_747505105_pname @ F2 @ B))))))))). % all_finite_subset_image
thf(fact_71_all__finite__subset__image, axiom,
    ((![F2 : set_a > a, A : set_set_a, P : set_a > $o]: ((![B : set_a]: (((((finite_finite_a @ B)) & ((ord_less_eq_set_a @ B @ (image_set_a_a @ F2 @ A))))) => ((P @ B)))) = (![B : set_set_a]: (((((finite_finite_set_a @ B)) & ((ord_le318720350_set_a @ B @ A)))) => ((P @ (image_set_a_a @ F2 @ B))))))))). % all_finite_subset_image
thf(fact_72_all__finite__subset__image, axiom,
    ((![F2 : set_pname > a, A : set_set_pname, P : set_a > $o]: ((![B : set_a]: (((((finite_finite_a @ B)) & ((ord_less_eq_set_a @ B @ (image_set_pname_a @ F2 @ A))))) => ((P @ B)))) = (![B : set_set_pname]: (((((finite505202775_pname @ B)) & ((ord_le2066558166_pname @ B @ A)))) => ((P @ (image_set_pname_a @ F2 @ B))))))))). % all_finite_subset_image
thf(fact_73_image__eqI, axiom,
    ((![B4 : a, F2 : a > a, X2 : a, A : set_a]: ((B4 = (F2 @ X2)) => ((member_a @ X2 @ A) => (member_a @ B4 @ (image_a_a @ F2 @ A))))))). % image_eqI
thf(fact_74_image__eqI, axiom,
    ((![B4 : pname, F2 : a > pname, X2 : a, A : set_a]: ((B4 = (F2 @ X2)) => ((member_a @ X2 @ A) => (member_pname @ B4 @ (image_a_pname @ F2 @ A))))))). % image_eqI
thf(fact_75_image__eqI, axiom,
    ((![B4 : a, F2 : pname > a, X2 : pname, A : set_pname]: ((B4 = (F2 @ X2)) => ((member_pname @ X2 @ A) => (member_a @ B4 @ (image_pname_a @ F2 @ A))))))). % image_eqI
thf(fact_76_image__eqI, axiom,
    ((![B4 : pname, F2 : pname > pname, X2 : pname, A : set_pname]: ((B4 = (F2 @ X2)) => ((member_pname @ X2 @ A) => (member_pname @ B4 @ (image_pname_pname @ F2 @ A))))))). % image_eqI
thf(fact_77_subset__antisym, axiom,
    ((![A : set_a, B2 : set_a]: ((ord_less_eq_set_a @ A @ B2) => ((ord_less_eq_set_a @ B2 @ A) => (A = B2)))))). % subset_antisym
thf(fact_78_subset__antisym, axiom,
    ((![A : set_pname, B2 : set_pname]: ((ord_le865024672_pname @ A @ B2) => ((ord_le865024672_pname @ B2 @ A) => (A = B2)))))). % subset_antisym
thf(fact_79_subsetI, axiom,
    ((![A : set_a, B2 : set_a]: ((![X3 : a]: ((member_a @ X3 @ A) => (member_a @ X3 @ B2))) => (ord_less_eq_set_a @ A @ B2))))). % subsetI
thf(fact_80_subsetI, axiom,
    ((![A : set_pname, B2 : set_pname]: ((![X3 : pname]: ((member_pname @ X3 @ A) => (member_pname @ X3 @ B2))) => (ord_le865024672_pname @ A @ B2))))). % subsetI
thf(fact_81_rev__image__eqI, axiom,
    ((![X2 : a, A : set_a, B4 : a, F2 : a > a]: ((member_a @ X2 @ A) => ((B4 = (F2 @ X2)) => (member_a @ B4 @ (image_a_a @ F2 @ A))))))). % rev_image_eqI
thf(fact_82_rev__image__eqI, axiom,
    ((![X2 : a, A : set_a, B4 : pname, F2 : a > pname]: ((member_a @ X2 @ A) => ((B4 = (F2 @ X2)) => (member_pname @ B4 @ (image_a_pname @ F2 @ A))))))). % rev_image_eqI
thf(fact_83_rev__image__eqI, axiom,
    ((![X2 : pname, A : set_pname, B4 : a, F2 : pname > a]: ((member_pname @ X2 @ A) => ((B4 = (F2 @ X2)) => (member_a @ B4 @ (image_pname_a @ F2 @ A))))))). % rev_image_eqI
thf(fact_84_rev__image__eqI, axiom,
    ((![X2 : pname, A : set_pname, B4 : pname, F2 : pname > pname]: ((member_pname @ X2 @ A) => ((B4 = (F2 @ X2)) => (member_pname @ B4 @ (image_pname_pname @ F2 @ A))))))). % rev_image_eqI
thf(fact_85_ball__imageD, axiom,
    ((![F2 : pname > a, A : set_pname, P : a > $o]: ((![X3 : a]: ((member_a @ X3 @ (image_pname_a @ F2 @ A)) => (P @ X3))) => (![X4 : pname]: ((member_pname @ X4 @ A) => (P @ (F2 @ X4)))))))). % ball_imageD
thf(fact_86_ball__imageD, axiom,
    ((![F2 : pname > pname, A : set_pname, P : pname > $o]: ((![X3 : pname]: ((member_pname @ X3 @ (image_pname_pname @ F2 @ A)) => (P @ X3))) => (![X4 : pname]: ((member_pname @ X4 @ A) => (P @ (F2 @ X4)))))))). % ball_imageD
thf(fact_87_ball__imageD, axiom,
    ((![F2 : a > a, A : set_a, P : a > $o]: ((![X3 : a]: ((member_a @ X3 @ (image_a_a @ F2 @ A)) => (P @ X3))) => (![X4 : a]: ((member_a @ X4 @ A) => (P @ (F2 @ X4)))))))). % ball_imageD
thf(fact_88_ball__imageD, axiom,
    ((![F2 : a > pname, A : set_a, P : pname > $o]: ((![X3 : pname]: ((member_pname @ X3 @ (image_a_pname @ F2 @ A)) => (P @ X3))) => (![X4 : a]: ((member_a @ X4 @ A) => (P @ (F2 @ X4)))))))). % ball_imageD
thf(fact_89_image__cong, axiom,
    ((![M : set_a, N2 : set_a, F2 : a > a, G2 : a > a]: ((M = N2) => ((![X3 : a]: ((member_a @ X3 @ N2) => ((F2 @ X3) = (G2 @ X3)))) => ((image_a_a @ F2 @ M) = (image_a_a @ G2 @ N2))))))). % image_cong
thf(fact_90_image__cong, axiom,
    ((![M : set_a, N2 : set_a, F2 : a > pname, G2 : a > pname]: ((M = N2) => ((![X3 : a]: ((member_a @ X3 @ N2) => ((F2 @ X3) = (G2 @ X3)))) => ((image_a_pname @ F2 @ M) = (image_a_pname @ G2 @ N2))))))). % image_cong
thf(fact_91_image__cong, axiom,
    ((![M : set_pname, N2 : set_pname, F2 : pname > a, G2 : pname > a]: ((M = N2) => ((![X3 : pname]: ((member_pname @ X3 @ N2) => ((F2 @ X3) = (G2 @ X3)))) => ((image_pname_a @ F2 @ M) = (image_pname_a @ G2 @ N2))))))). % image_cong
thf(fact_92_image__cong, axiom,
    ((![M : set_pname, N2 : set_pname, F2 : pname > pname, G2 : pname > pname]: ((M = N2) => ((![X3 : pname]: ((member_pname @ X3 @ N2) => ((F2 @ X3) = (G2 @ X3)))) => ((image_pname_pname @ F2 @ M) = (image_pname_pname @ G2 @ N2))))))). % image_cong
thf(fact_93_bex__imageD, axiom,
    ((![F2 : pname > a, A : set_pname, P : a > $o]: ((?[X4 : a]: ((member_a @ X4 @ (image_pname_a @ F2 @ A)) & (P @ X4))) => (?[X3 : pname]: ((member_pname @ X3 @ A) & (P @ (F2 @ X3)))))))). % bex_imageD
thf(fact_94_bex__imageD, axiom,
    ((![F2 : pname > pname, A : set_pname, P : pname > $o]: ((?[X4 : pname]: ((member_pname @ X4 @ (image_pname_pname @ F2 @ A)) & (P @ X4))) => (?[X3 : pname]: ((member_pname @ X3 @ A) & (P @ (F2 @ X3)))))))). % bex_imageD
thf(fact_95_bex__imageD, axiom,
    ((![F2 : a > a, A : set_a, P : a > $o]: ((?[X4 : a]: ((member_a @ X4 @ (image_a_a @ F2 @ A)) & (P @ X4))) => (?[X3 : a]: ((member_a @ X3 @ A) & (P @ (F2 @ X3)))))))). % bex_imageD
thf(fact_96_bex__imageD, axiom,
    ((![F2 : a > pname, A : set_a, P : pname > $o]: ((?[X4 : pname]: ((member_pname @ X4 @ (image_a_pname @ F2 @ A)) & (P @ X4))) => (?[X3 : a]: ((member_a @ X3 @ A) & (P @ (F2 @ X3)))))))). % bex_imageD
thf(fact_97_image__iff, axiom,
    ((![Z : a, F2 : pname > a, A : set_pname]: ((member_a @ Z @ (image_pname_a @ F2 @ A)) = (?[X : pname]: (((member_pname @ X @ A)) & ((Z = (F2 @ X))))))))). % image_iff
thf(fact_98_image__iff, axiom,
    ((![Z : a, F2 : a > a, A : set_a]: ((member_a @ Z @ (image_a_a @ F2 @ A)) = (?[X : a]: (((member_a @ X @ A)) & ((Z = (F2 @ X))))))))). % image_iff
thf(fact_99_image__iff, axiom,
    ((![Z : pname, F2 : pname > pname, A : set_pname]: ((member_pname @ Z @ (image_pname_pname @ F2 @ A)) = (?[X : pname]: (((member_pname @ X @ A)) & ((Z = (F2 @ X))))))))). % image_iff
thf(fact_100_image__iff, axiom,
    ((![Z : pname, F2 : a > pname, A : set_a]: ((member_pname @ Z @ (image_a_pname @ F2 @ A)) = (?[X : a]: (((member_a @ X @ A)) & ((Z = (F2 @ X))))))))). % image_iff
thf(fact_101_imageI, axiom,
    ((![X2 : a, A : set_a, F2 : a > a]: ((member_a @ X2 @ A) => (member_a @ (F2 @ X2) @ (image_a_a @ F2 @ A)))))). % imageI
thf(fact_102_imageI, axiom,
    ((![X2 : a, A : set_a, F2 : a > pname]: ((member_a @ X2 @ A) => (member_pname @ (F2 @ X2) @ (image_a_pname @ F2 @ A)))))). % imageI
thf(fact_103_imageI, axiom,
    ((![X2 : pname, A : set_pname, F2 : pname > a]: ((member_pname @ X2 @ A) => (member_a @ (F2 @ X2) @ (image_pname_a @ F2 @ A)))))). % imageI
thf(fact_104_imageI, axiom,
    ((![X2 : pname, A : set_pname, F2 : pname > pname]: ((member_pname @ X2 @ A) => (member_pname @ (F2 @ X2) @ (image_pname_pname @ F2 @ A)))))). % imageI
thf(fact_105_Collect__mono__iff, axiom,
    ((![P : set_a > $o, Q : set_a > $o]: ((ord_le318720350_set_a @ (collect_set_a @ P) @ (collect_set_a @ Q)) = (![X : set_a]: (((P @ X)) => ((Q @ X)))))))). % Collect_mono_iff
thf(fact_106_Collect__mono__iff, axiom,
    ((![P : set_pname > $o, Q : set_pname > $o]: ((ord_le2066558166_pname @ (collect_set_pname @ P) @ (collect_set_pname @ Q)) = (![X : set_pname]: (((P @ X)) => ((Q @ X)))))))). % Collect_mono_iff
thf(fact_107_Collect__mono__iff, axiom,
    ((![P : a > $o, Q : a > $o]: ((ord_less_eq_set_a @ (collect_a @ P) @ (collect_a @ Q)) = (![X : a]: (((P @ X)) => ((Q @ X)))))))). % Collect_mono_iff
thf(fact_108_Collect__mono__iff, axiom,
    ((![P : pname > $o, Q : pname > $o]: ((ord_le865024672_pname @ (collect_pname @ P) @ (collect_pname @ Q)) = (![X : pname]: (((P @ X)) => ((Q @ X)))))))). % Collect_mono_iff
thf(fact_109_set__eq__subset, axiom,
    (((^[Y2 : set_a]: (^[Z2 : set_a]: (Y2 = Z2))) = (^[A2 : set_a]: (^[B : set_a]: (((ord_less_eq_set_a @ A2 @ B)) & ((ord_less_eq_set_a @ B @ A2)))))))). % set_eq_subset
thf(fact_110_set__eq__subset, axiom,
    (((^[Y2 : set_pname]: (^[Z2 : set_pname]: (Y2 = Z2))) = (^[A2 : set_pname]: (^[B : set_pname]: (((ord_le865024672_pname @ A2 @ B)) & ((ord_le865024672_pname @ B @ A2)))))))). % set_eq_subset
thf(fact_111_subset__trans, axiom,
    ((![A : set_a, B2 : set_a, C2 : set_a]: ((ord_less_eq_set_a @ A @ B2) => ((ord_less_eq_set_a @ B2 @ C2) => (ord_less_eq_set_a @ A @ C2)))))). % subset_trans
thf(fact_112_subset__trans, axiom,
    ((![A : set_pname, B2 : set_pname, C2 : set_pname]: ((ord_le865024672_pname @ A @ B2) => ((ord_le865024672_pname @ B2 @ C2) => (ord_le865024672_pname @ A @ C2)))))). % subset_trans
thf(fact_113_Collect__mono, axiom,
    ((![P : set_a > $o, Q : set_a > $o]: ((![X3 : set_a]: ((P @ X3) => (Q @ X3))) => (ord_le318720350_set_a @ (collect_set_a @ P) @ (collect_set_a @ Q)))))). % Collect_mono
thf(fact_114_Collect__mono, axiom,
    ((![P : set_pname > $o, Q : set_pname > $o]: ((![X3 : set_pname]: ((P @ X3) => (Q @ X3))) => (ord_le2066558166_pname @ (collect_set_pname @ P) @ (collect_set_pname @ Q)))))). % Collect_mono
thf(fact_115_Collect__mono, axiom,
    ((![P : a > $o, Q : a > $o]: ((![X3 : a]: ((P @ X3) => (Q @ X3))) => (ord_less_eq_set_a @ (collect_a @ P) @ (collect_a @ Q)))))). % Collect_mono
thf(fact_116_Collect__mono, axiom,
    ((![P : pname > $o, Q : pname > $o]: ((![X3 : pname]: ((P @ X3) => (Q @ X3))) => (ord_le865024672_pname @ (collect_pname @ P) @ (collect_pname @ Q)))))). % Collect_mono
thf(fact_117_subset__refl, axiom,
    ((![A : set_a]: (ord_less_eq_set_a @ A @ A)))). % subset_refl
thf(fact_118_subset__refl, axiom,
    ((![A : set_pname]: (ord_le865024672_pname @ A @ A)))). % subset_refl
thf(fact_119_subset__iff, axiom,
    ((ord_less_eq_set_a = (^[A2 : set_a]: (^[B : set_a]: (![T : a]: (((member_a @ T @ A2)) => ((member_a @ T @ B))))))))). % subset_iff
thf(fact_120_subset__iff, axiom,
    ((ord_le865024672_pname = (^[A2 : set_pname]: (^[B : set_pname]: (![T : pname]: (((member_pname @ T @ A2)) => ((member_pname @ T @ B))))))))). % subset_iff
thf(fact_121_equalityD2, axiom,
    ((![A : set_a, B2 : set_a]: ((A = B2) => (ord_less_eq_set_a @ B2 @ A))))). % equalityD2
thf(fact_122_equalityD2, axiom,
    ((![A : set_pname, B2 : set_pname]: ((A = B2) => (ord_le865024672_pname @ B2 @ A))))). % equalityD2
thf(fact_123_equalityD1, axiom,
    ((![A : set_a, B2 : set_a]: ((A = B2) => (ord_less_eq_set_a @ A @ B2))))). % equalityD1
thf(fact_124_equalityD1, axiom,
    ((![A : set_pname, B2 : set_pname]: ((A = B2) => (ord_le865024672_pname @ A @ B2))))). % equalityD1
thf(fact_125_subset__eq, axiom,
    ((ord_less_eq_set_a = (^[A2 : set_a]: (^[B : set_a]: (![X : a]: (((member_a @ X @ A2)) => ((member_a @ X @ B))))))))). % subset_eq
thf(fact_126_subset__eq, axiom,
    ((ord_le865024672_pname = (^[A2 : set_pname]: (^[B : set_pname]: (![X : pname]: (((member_pname @ X @ A2)) => ((member_pname @ X @ B))))))))). % subset_eq
thf(fact_127_equalityE, axiom,
    ((![A : set_a, B2 : set_a]: ((A = B2) => (~ (((ord_less_eq_set_a @ A @ B2) => (~ ((ord_less_eq_set_a @ B2 @ A)))))))))). % equalityE
thf(fact_128_equalityE, axiom,
    ((![A : set_pname, B2 : set_pname]: ((A = B2) => (~ (((ord_le865024672_pname @ A @ B2) => (~ ((ord_le865024672_pname @ B2 @ A)))))))))). % equalityE
thf(fact_129_subsetD, axiom,
    ((![A : set_a, B2 : set_a, C3 : a]: ((ord_less_eq_set_a @ A @ B2) => ((member_a @ C3 @ A) => (member_a @ C3 @ B2)))))). % subsetD
thf(fact_130_subsetD, axiom,
    ((![A : set_pname, B2 : set_pname, C3 : pname]: ((ord_le865024672_pname @ A @ B2) => ((member_pname @ C3 @ A) => (member_pname @ C3 @ B2)))))). % subsetD
thf(fact_131_in__mono, axiom,
    ((![A : set_a, B2 : set_a, X2 : a]: ((ord_less_eq_set_a @ A @ B2) => ((member_a @ X2 @ A) => (member_a @ X2 @ B2)))))). % in_mono
thf(fact_132_in__mono, axiom,
    ((![A : set_pname, B2 : set_pname, X2 : pname]: ((ord_le865024672_pname @ A @ B2) => ((member_pname @ X2 @ A) => (member_pname @ X2 @ B2)))))). % in_mono
thf(fact_133_Compr__image__eq, axiom,
    ((![F2 : a > a, A : set_a, P : a > $o]: ((collect_a @ (^[X : a]: (((member_a @ X @ (image_a_a @ F2 @ A))) & ((P @ X))))) = (image_a_a @ F2 @ (collect_a @ (^[X : a]: (((member_a @ X @ A)) & ((P @ (F2 @ X))))))))))). % Compr_image_eq
thf(fact_134_Compr__image__eq, axiom,
    ((![F2 : pname > a, A : set_pname, P : a > $o]: ((collect_a @ (^[X : a]: (((member_a @ X @ (image_pname_a @ F2 @ A))) & ((P @ X))))) = (image_pname_a @ F2 @ (collect_pname @ (^[X : pname]: (((member_pname @ X @ A)) & ((P @ (F2 @ X))))))))))). % Compr_image_eq
thf(fact_135_Compr__image__eq, axiom,
    ((![F2 : a > pname, A : set_a, P : pname > $o]: ((collect_pname @ (^[X : pname]: (((member_pname @ X @ (image_a_pname @ F2 @ A))) & ((P @ X))))) = (image_a_pname @ F2 @ (collect_a @ (^[X : a]: (((member_a @ X @ A)) & ((P @ (F2 @ X))))))))))). % Compr_image_eq
thf(fact_136_Compr__image__eq, axiom,
    ((![F2 : pname > pname, A : set_pname, P : pname > $o]: ((collect_pname @ (^[X : pname]: (((member_pname @ X @ (image_pname_pname @ F2 @ A))) & ((P @ X))))) = (image_pname_pname @ F2 @ (collect_pname @ (^[X : pname]: (((member_pname @ X @ A)) & ((P @ (F2 @ X))))))))))). % Compr_image_eq
thf(fact_137_Compr__image__eq, axiom,
    ((![F2 : pname > set_a, A : set_pname, P : set_a > $o]: ((collect_set_a @ (^[X : set_a]: (((member_set_a @ X @ (image_pname_set_a @ F2 @ A))) & ((P @ X))))) = (image_pname_set_a @ F2 @ (collect_pname @ (^[X : pname]: (((member_pname @ X @ A)) & ((P @ (F2 @ X))))))))))). % Compr_image_eq
thf(fact_138_Compr__image__eq, axiom,
    ((![F2 : pname > set_pname, A : set_pname, P : set_pname > $o]: ((collect_set_pname @ (^[X : set_pname]: (((member_set_pname @ X @ (image_747505105_pname @ F2 @ A))) & ((P @ X))))) = (image_747505105_pname @ F2 @ (collect_pname @ (^[X : pname]: (((member_pname @ X @ A)) & ((P @ (F2 @ X))))))))))). % Compr_image_eq
thf(fact_139_Compr__image__eq, axiom,
    ((![F2 : set_a > pname, A : set_set_a, P : pname > $o]: ((collect_pname @ (^[X : pname]: (((member_pname @ X @ (image_set_a_pname @ F2 @ A))) & ((P @ X))))) = (image_set_a_pname @ F2 @ (collect_set_a @ (^[X : set_a]: (((member_set_a @ X @ A)) & ((P @ (F2 @ X))))))))))). % Compr_image_eq
thf(fact_140_Compr__image__eq, axiom,
    ((![F2 : set_pname > pname, A : set_set_pname, P : pname > $o]: ((collect_pname @ (^[X : pname]: (((member_pname @ X @ (image_1672683217_pname @ F2 @ A))) & ((P @ X))))) = (image_1672683217_pname @ F2 @ (collect_set_pname @ (^[X : set_pname]: (((member_set_pname @ X @ A)) & ((P @ (F2 @ X))))))))))). % Compr_image_eq
thf(fact_141_Compr__image__eq, axiom,
    ((![F2 : set_a > set_a, A : set_set_a, P : set_a > $o]: ((collect_set_a @ (^[X : set_a]: (((member_set_a @ X @ (image_set_a_set_a @ F2 @ A))) & ((P @ X))))) = (image_set_a_set_a @ F2 @ (collect_set_a @ (^[X : set_a]: (((member_set_a @ X @ A)) & ((P @ (F2 @ X))))))))))). % Compr_image_eq
thf(fact_142_Compr__image__eq, axiom,
    ((![F2 : set_pname > set_a, A : set_set_pname, P : set_a > $o]: ((collect_set_a @ (^[X : set_a]: (((member_set_a @ X @ (image_700492503_set_a @ F2 @ A))) & ((P @ X))))) = (image_700492503_set_a @ F2 @ (collect_set_pname @ (^[X : set_pname]: (((member_set_pname @ X @ A)) & ((P @ (F2 @ X))))))))))). % Compr_image_eq
thf(fact_143_image__image, axiom,
    ((![F2 : pname > a, G2 : pname > pname, A : set_pname]: ((image_pname_a @ F2 @ (image_pname_pname @ G2 @ A)) = (image_pname_a @ (^[X : pname]: (F2 @ (G2 @ X))) @ A))))). % image_image
thf(fact_144_image__image, axiom,
    ((![F2 : pname > a, G2 : a > pname, A : set_a]: ((image_pname_a @ F2 @ (image_a_pname @ G2 @ A)) = (image_a_a @ (^[X : a]: (F2 @ (G2 @ X))) @ A))))). % image_image
thf(fact_145_image__image, axiom,
    ((![F2 : pname > pname, G2 : pname > pname, A : set_pname]: ((image_pname_pname @ F2 @ (image_pname_pname @ G2 @ A)) = (image_pname_pname @ (^[X : pname]: (F2 @ (G2 @ X))) @ A))))). % image_image
thf(fact_146_image__image, axiom,
    ((![F2 : pname > pname, G2 : a > pname, A : set_a]: ((image_pname_pname @ F2 @ (image_a_pname @ G2 @ A)) = (image_a_pname @ (^[X : a]: (F2 @ (G2 @ X))) @ A))))). % image_image
thf(fact_147_image__image, axiom,
    ((![F2 : a > a, G2 : pname > a, A : set_pname]: ((image_a_a @ F2 @ (image_pname_a @ G2 @ A)) = (image_pname_a @ (^[X : pname]: (F2 @ (G2 @ X))) @ A))))). % image_image
thf(fact_148_image__image, axiom,
    ((![F2 : a > a, G2 : a > a, A : set_a]: ((image_a_a @ F2 @ (image_a_a @ G2 @ A)) = (image_a_a @ (^[X : a]: (F2 @ (G2 @ X))) @ A))))). % image_image
thf(fact_149_image__image, axiom,
    ((![F2 : a > pname, G2 : pname > a, A : set_pname]: ((image_a_pname @ F2 @ (image_pname_a @ G2 @ A)) = (image_pname_pname @ (^[X : pname]: (F2 @ (G2 @ X))) @ A))))). % image_image
thf(fact_150_image__image, axiom,
    ((![F2 : a > pname, G2 : a > a, A : set_a]: ((image_a_pname @ F2 @ (image_a_a @ G2 @ A)) = (image_a_pname @ (^[X : a]: (F2 @ (G2 @ X))) @ A))))). % image_image
thf(fact_151_imageE, axiom,
    ((![B4 : a, F2 : a > a, A : set_a]: ((member_a @ B4 @ (image_a_a @ F2 @ A)) => (~ ((![X3 : a]: ((B4 = (F2 @ X3)) => (~ ((member_a @ X3 @ A))))))))))). % imageE
thf(fact_152_imageE, axiom,
    ((![B4 : a, F2 : pname > a, A : set_pname]: ((member_a @ B4 @ (image_pname_a @ F2 @ A)) => (~ ((![X3 : pname]: ((B4 = (F2 @ X3)) => (~ ((member_pname @ X3 @ A))))))))))). % imageE
thf(fact_153_imageE, axiom,
    ((![B4 : pname, F2 : a > pname, A : set_a]: ((member_pname @ B4 @ (image_a_pname @ F2 @ A)) => (~ ((![X3 : a]: ((B4 = (F2 @ X3)) => (~ ((member_a @ X3 @ A))))))))))). % imageE
thf(fact_154_imageE, axiom,
    ((![B4 : pname, F2 : pname > pname, A : set_pname]: ((member_pname @ B4 @ (image_pname_pname @ F2 @ A)) => (~ ((![X3 : pname]: ((B4 = (F2 @ X3)) => (~ ((member_pname @ X3 @ A))))))))))). % imageE
thf(fact_155_pigeonhole__infinite__rel, axiom,
    ((![A : set_pname, B2 : set_pname, R : pname > pname > $o]: ((~ ((finite_finite_pname @ A))) => ((finite_finite_pname @ B2) => ((![X3 : pname]: ((member_pname @ X3 @ A) => (?[Xa : pname]: ((member_pname @ Xa @ B2) & (R @ X3 @ Xa))))) => (?[X3 : pname]: ((member_pname @ X3 @ B2) & (~ ((finite_finite_pname @ (collect_pname @ (^[A3 : pname]: (((member_pname @ A3 @ A)) & ((R @ A3 @ X3)))))))))))))))). % pigeonhole_infinite_rel
thf(fact_156_pigeonhole__infinite__rel, axiom,
    ((![A : set_pname, B2 : set_a, R : pname > a > $o]: ((~ ((finite_finite_pname @ A))) => ((finite_finite_a @ B2) => ((![X3 : pname]: ((member_pname @ X3 @ A) => (?[Xa : a]: ((member_a @ Xa @ B2) & (R @ X3 @ Xa))))) => (?[X3 : a]: ((member_a @ X3 @ B2) & (~ ((finite_finite_pname @ (collect_pname @ (^[A3 : pname]: (((member_pname @ A3 @ A)) & ((R @ A3 @ X3)))))))))))))))). % pigeonhole_infinite_rel
thf(fact_157_pigeonhole__infinite__rel, axiom,
    ((![A : set_a, B2 : set_pname, R : a > pname > $o]: ((~ ((finite_finite_a @ A))) => ((finite_finite_pname @ B2) => ((![X3 : a]: ((member_a @ X3 @ A) => (?[Xa : pname]: ((member_pname @ Xa @ B2) & (R @ X3 @ Xa))))) => (?[X3 : pname]: ((member_pname @ X3 @ B2) & (~ ((finite_finite_a @ (collect_a @ (^[A3 : a]: (((member_a @ A3 @ A)) & ((R @ A3 @ X3)))))))))))))))). % pigeonhole_infinite_rel
thf(fact_158_pigeonhole__infinite__rel, axiom,
    ((![A : set_a, B2 : set_a, R : a > a > $o]: ((~ ((finite_finite_a @ A))) => ((finite_finite_a @ B2) => ((![X3 : a]: ((member_a @ X3 @ A) => (?[Xa : a]: ((member_a @ Xa @ B2) & (R @ X3 @ Xa))))) => (?[X3 : a]: ((member_a @ X3 @ B2) & (~ ((finite_finite_a @ (collect_a @ (^[A3 : a]: (((member_a @ A3 @ A)) & ((R @ A3 @ X3)))))))))))))))). % pigeonhole_infinite_rel
thf(fact_159_pigeonhole__infinite__rel, axiom,
    ((![A : set_pname, B2 : set_set_a, R : pname > set_a > $o]: ((~ ((finite_finite_pname @ A))) => ((finite_finite_set_a @ B2) => ((![X3 : pname]: ((member_pname @ X3 @ A) => (?[Xa : set_a]: ((member_set_a @ Xa @ B2) & (R @ X3 @ Xa))))) => (?[X3 : set_a]: ((member_set_a @ X3 @ B2) & (~ ((finite_finite_pname @ (collect_pname @ (^[A3 : pname]: (((member_pname @ A3 @ A)) & ((R @ A3 @ X3)))))))))))))))). % pigeonhole_infinite_rel
thf(fact_160_pigeonhole__infinite__rel, axiom,
    ((![A : set_pname, B2 : set_set_pname, R : pname > set_pname > $o]: ((~ ((finite_finite_pname @ A))) => ((finite505202775_pname @ B2) => ((![X3 : pname]: ((member_pname @ X3 @ A) => (?[Xa : set_pname]: ((member_set_pname @ Xa @ B2) & (R @ X3 @ Xa))))) => (?[X3 : set_pname]: ((member_set_pname @ X3 @ B2) & (~ ((finite_finite_pname @ (collect_pname @ (^[A3 : pname]: (((member_pname @ A3 @ A)) & ((R @ A3 @ X3)))))))))))))))). % pigeonhole_infinite_rel
thf(fact_161_pigeonhole__infinite__rel, axiom,
    ((![A : set_a, B2 : set_set_a, R : a > set_a > $o]: ((~ ((finite_finite_a @ A))) => ((finite_finite_set_a @ B2) => ((![X3 : a]: ((member_a @ X3 @ A) => (?[Xa : set_a]: ((member_set_a @ Xa @ B2) & (R @ X3 @ Xa))))) => (?[X3 : set_a]: ((member_set_a @ X3 @ B2) & (~ ((finite_finite_a @ (collect_a @ (^[A3 : a]: (((member_a @ A3 @ A)) & ((R @ A3 @ X3)))))))))))))))). % pigeonhole_infinite_rel
thf(fact_162_pigeonhole__infinite__rel, axiom,
    ((![A : set_a, B2 : set_set_pname, R : a > set_pname > $o]: ((~ ((finite_finite_a @ A))) => ((finite505202775_pname @ B2) => ((![X3 : a]: ((member_a @ X3 @ A) => (?[Xa : set_pname]: ((member_set_pname @ Xa @ B2) & (R @ X3 @ Xa))))) => (?[X3 : set_pname]: ((member_set_pname @ X3 @ B2) & (~ ((finite_finite_a @ (collect_a @ (^[A3 : a]: (((member_a @ A3 @ A)) & ((R @ A3 @ X3)))))))))))))))). % pigeonhole_infinite_rel
thf(fact_163_pigeonhole__infinite__rel, axiom,
    ((![A : set_set_a, B2 : set_pname, R : set_a > pname > $o]: ((~ ((finite_finite_set_a @ A))) => ((finite_finite_pname @ B2) => ((![X3 : set_a]: ((member_set_a @ X3 @ A) => (?[Xa : pname]: ((member_pname @ Xa @ B2) & (R @ X3 @ Xa))))) => (?[X3 : pname]: ((member_pname @ X3 @ B2) & (~ ((finite_finite_set_a @ (collect_set_a @ (^[A3 : set_a]: (((member_set_a @ A3 @ A)) & ((R @ A3 @ X3)))))))))))))))). % pigeonhole_infinite_rel
thf(fact_164_pigeonhole__infinite__rel, axiom,
    ((![A : set_set_a, B2 : set_a, R : set_a > a > $o]: ((~ ((finite_finite_set_a @ A))) => ((finite_finite_a @ B2) => ((![X3 : set_a]: ((member_set_a @ X3 @ A) => (?[Xa : a]: ((member_a @ Xa @ B2) & (R @ X3 @ Xa))))) => (?[X3 : a]: ((member_a @ X3 @ B2) & (~ ((finite_finite_set_a @ (collect_set_a @ (^[A3 : set_a]: (((member_set_a @ A3 @ A)) & ((R @ A3 @ X3)))))))))))))))). % pigeonhole_infinite_rel
thf(fact_165_not__finite__existsD, axiom,
    ((![P : pname > $o]: ((~ ((finite_finite_pname @ (collect_pname @ P)))) => (?[X_1 : pname]: (P @ X_1)))))). % not_finite_existsD
thf(fact_166_not__finite__existsD, axiom,
    ((![P : a > $o]: ((~ ((finite_finite_a @ (collect_a @ P)))) => (?[X_1 : a]: (P @ X_1)))))). % not_finite_existsD
thf(fact_167_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_168_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_169_Collect__subset, axiom,
    ((![A : set_set_a, P : set_a > $o]: (ord_le318720350_set_a @ (collect_set_a @ (^[X : set_a]: (((member_set_a @ X @ A)) & ((P @ X))))) @ A)))). % Collect_subset
thf(fact_170_Collect__subset, axiom,
    ((![A : set_set_pname, P : set_pname > $o]: (ord_le2066558166_pname @ (collect_set_pname @ (^[X : set_pname]: (((member_set_pname @ X @ A)) & ((P @ X))))) @ A)))). % Collect_subset
thf(fact_171_Collect__subset, axiom,
    ((![A : set_a, P : a > $o]: (ord_less_eq_set_a @ (collect_a @ (^[X : a]: (((member_a @ X @ A)) & ((P @ X))))) @ A)))). % Collect_subset
thf(fact_172_Collect__subset, axiom,
    ((![A : set_pname, P : pname > $o]: (ord_le865024672_pname @ (collect_pname @ (^[X : pname]: (((member_pname @ X @ A)) & ((P @ X))))) @ A)))). % Collect_subset
thf(fact_173_finite__has__minimal2, axiom,
    ((![A : set_set_a, A4 : set_a]: ((finite_finite_set_a @ A) => ((member_set_a @ A4 @ A) => (?[X3 : set_a]: ((member_set_a @ X3 @ A) & ((ord_less_eq_set_a @ X3 @ A4) & (![Xa : set_a]: ((member_set_a @ Xa @ A) => ((ord_less_eq_set_a @ Xa @ X3) => (X3 = Xa)))))))))))). % finite_has_minimal2
thf(fact_174_finite__has__minimal2, axiom,
    ((![A : set_set_pname, A4 : set_pname]: ((finite505202775_pname @ A) => ((member_set_pname @ A4 @ A) => (?[X3 : set_pname]: ((member_set_pname @ X3 @ A) & ((ord_le865024672_pname @ X3 @ A4) & (![Xa : set_pname]: ((member_set_pname @ Xa @ A) => ((ord_le865024672_pname @ Xa @ X3) => (X3 = Xa)))))))))))). % finite_has_minimal2
thf(fact_175_finite__has__maximal2, axiom,
    ((![A : set_set_a, A4 : set_a]: ((finite_finite_set_a @ A) => ((member_set_a @ A4 @ A) => (?[X3 : set_a]: ((member_set_a @ X3 @ A) & ((ord_less_eq_set_a @ A4 @ X3) & (![Xa : set_a]: ((member_set_a @ Xa @ A) => ((ord_less_eq_set_a @ X3 @ Xa) => (X3 = Xa)))))))))))). % finite_has_maximal2
thf(fact_176_finite__has__maximal2, axiom,
    ((![A : set_set_pname, A4 : set_pname]: ((finite505202775_pname @ A) => ((member_set_pname @ A4 @ A) => (?[X3 : set_pname]: ((member_set_pname @ X3 @ A) & ((ord_le865024672_pname @ A4 @ X3) & (![Xa : set_pname]: ((member_set_pname @ Xa @ A) => ((ord_le865024672_pname @ X3 @ Xa) => (X3 = Xa)))))))))))). % finite_has_maximal2
thf(fact_177_all__subset__image, axiom,
    ((![F2 : a > a, A : set_a, P : set_a > $o]: ((![B : set_a]: (((ord_less_eq_set_a @ B @ (image_a_a @ F2 @ A))) => ((P @ B)))) = (![B : set_a]: (((ord_less_eq_set_a @ B @ A)) => ((P @ (image_a_a @ F2 @ B))))))))). % all_subset_image
thf(fact_178_all__subset__image, axiom,
    ((![F2 : pname > a, A : set_pname, P : set_a > $o]: ((![B : set_a]: (((ord_less_eq_set_a @ B @ (image_pname_a @ F2 @ A))) => ((P @ B)))) = (![B : set_pname]: (((ord_le865024672_pname @ B @ A)) => ((P @ (image_pname_a @ F2 @ B))))))))). % all_subset_image
thf(fact_179_all__subset__image, axiom,
    ((![F2 : a > pname, A : set_a, P : set_pname > $o]: ((![B : set_pname]: (((ord_le865024672_pname @ B @ (image_a_pname @ F2 @ A))) => ((P @ B)))) = (![B : set_a]: (((ord_less_eq_set_a @ B @ A)) => ((P @ (image_a_pname @ F2 @ B))))))))). % all_subset_image
thf(fact_180_all__subset__image, axiom,
    ((![F2 : pname > pname, A : set_pname, P : set_pname > $o]: ((![B : set_pname]: (((ord_le865024672_pname @ B @ (image_pname_pname @ F2 @ A))) => ((P @ B)))) = (![B : set_pname]: (((ord_le865024672_pname @ B @ A)) => ((P @ (image_pname_pname @ F2 @ B))))))))). % all_subset_image
thf(fact_181_mem__Collect__eq, axiom,
    ((![A4 : set_a, P : set_a > $o]: ((member_set_a @ A4 @ (collect_set_a @ P)) = (P @ A4))))). % mem_Collect_eq
thf(fact_182_mem__Collect__eq, axiom,
    ((![A4 : set_pname, P : set_pname > $o]: ((member_set_pname @ A4 @ (collect_set_pname @ P)) = (P @ A4))))). % mem_Collect_eq
thf(fact_183_mem__Collect__eq, axiom,
    ((![A4 : pname, P : pname > $o]: ((member_pname @ A4 @ (collect_pname @ P)) = (P @ A4))))). % mem_Collect_eq
thf(fact_184_Collect__mem__eq, axiom,
    ((![A : set_set_a]: ((collect_set_a @ (^[X : set_a]: (member_set_a @ X @ A))) = A)))). % Collect_mem_eq
thf(fact_185_Collect__mem__eq, axiom,
    ((![A : set_set_pname]: ((collect_set_pname @ (^[X : set_pname]: (member_set_pname @ X @ A))) = A)))). % Collect_mem_eq
thf(fact_186_Collect__mem__eq, axiom,
    ((![A : set_pname]: ((collect_pname @ (^[X : pname]: (member_pname @ X @ A))) = A)))). % Collect_mem_eq
thf(fact_187_Collect__cong, axiom,
    ((![P : set_a > $o, Q : set_a > $o]: ((![X3 : set_a]: ((P @ X3) = (Q @ X3))) => ((collect_set_a @ P) = (collect_set_a @ Q)))))). % Collect_cong
thf(fact_188_Collect__cong, axiom,
    ((![P : set_pname > $o, Q : set_pname > $o]: ((![X3 : set_pname]: ((P @ X3) = (Q @ X3))) => ((collect_set_pname @ P) = (collect_set_pname @ Q)))))). % Collect_cong
thf(fact_189_Collect__cong, axiom,
    ((![P : pname > $o, Q : pname > $o]: ((![X3 : pname]: ((P @ X3) = (Q @ X3))) => ((collect_pname @ P) = (collect_pname @ Q)))))). % Collect_cong
thf(fact_190_subset__image__iff, axiom,
    ((![B2 : set_a, F2 : a > a, A : set_a]: ((ord_less_eq_set_a @ B2 @ (image_a_a @ F2 @ A)) = (?[AA : set_a]: (((ord_less_eq_set_a @ AA @ A)) & ((B2 = (image_a_a @ F2 @ AA))))))))). % subset_image_iff
thf(fact_191_subset__image__iff, axiom,
    ((![B2 : set_a, F2 : pname > a, A : set_pname]: ((ord_less_eq_set_a @ B2 @ (image_pname_a @ F2 @ A)) = (?[AA : set_pname]: (((ord_le865024672_pname @ AA @ A)) & ((B2 = (image_pname_a @ F2 @ AA))))))))). % subset_image_iff
thf(fact_192_subset__image__iff, axiom,
    ((![B2 : set_pname, F2 : a > pname, A : set_a]: ((ord_le865024672_pname @ B2 @ (image_a_pname @ F2 @ A)) = (?[AA : set_a]: (((ord_less_eq_set_a @ AA @ A)) & ((B2 = (image_a_pname @ F2 @ AA))))))))). % subset_image_iff
thf(fact_193_subset__image__iff, axiom,
    ((![B2 : set_pname, F2 : pname > pname, A : set_pname]: ((ord_le865024672_pname @ B2 @ (image_pname_pname @ F2 @ A)) = (?[AA : set_pname]: (((ord_le865024672_pname @ AA @ A)) & ((B2 = (image_pname_pname @ F2 @ AA))))))))). % subset_image_iff
thf(fact_194_image__subset__iff, axiom,
    ((![F2 : pname > a, A : set_pname, B2 : set_a]: ((ord_less_eq_set_a @ (image_pname_a @ F2 @ A) @ B2) = (![X : pname]: (((member_pname @ X @ A)) => ((member_a @ (F2 @ X) @ B2)))))))). % image_subset_iff
thf(fact_195_image__subset__iff, axiom,
    ((![F2 : a > a, A : set_a, B2 : set_a]: ((ord_less_eq_set_a @ (image_a_a @ F2 @ A) @ B2) = (![X : a]: (((member_a @ X @ A)) => ((member_a @ (F2 @ X) @ B2)))))))). % image_subset_iff
thf(fact_196_image__subset__iff, axiom,
    ((![F2 : pname > pname, A : set_pname, B2 : set_pname]: ((ord_le865024672_pname @ (image_pname_pname @ F2 @ A) @ B2) = (![X : pname]: (((member_pname @ X @ A)) => ((member_pname @ (F2 @ X) @ B2)))))))). % image_subset_iff
thf(fact_197_image__subset__iff, axiom,
    ((![F2 : a > pname, A : set_a, B2 : set_pname]: ((ord_le865024672_pname @ (image_a_pname @ F2 @ A) @ B2) = (![X : a]: (((member_a @ X @ A)) => ((member_pname @ (F2 @ X) @ B2)))))))). % image_subset_iff
thf(fact_198_subset__imageE, axiom,
    ((![B2 : set_a, F2 : a > a, A : set_a]: ((ord_less_eq_set_a @ B2 @ (image_a_a @ F2 @ A)) => (~ ((![C : set_a]: ((ord_less_eq_set_a @ C @ A) => (~ ((B2 = (image_a_a @ F2 @ C)))))))))))). % subset_imageE
thf(fact_199_subset__imageE, axiom,
    ((![B2 : set_a, F2 : pname > a, A : set_pname]: ((ord_less_eq_set_a @ B2 @ (image_pname_a @ F2 @ A)) => (~ ((![C : set_pname]: ((ord_le865024672_pname @ C @ A) => (~ ((B2 = (image_pname_a @ F2 @ C)))))))))))). % subset_imageE
thf(fact_200_subset__imageE, axiom,
    ((![B2 : set_pname, F2 : a > pname, A : set_a]: ((ord_le865024672_pname @ B2 @ (image_a_pname @ F2 @ A)) => (~ ((![C : set_a]: ((ord_less_eq_set_a @ C @ A) => (~ ((B2 = (image_a_pname @ F2 @ C)))))))))))). % subset_imageE
thf(fact_201_subset__imageE, axiom,
    ((![B2 : set_pname, F2 : pname > pname, A : set_pname]: ((ord_le865024672_pname @ B2 @ (image_pname_pname @ F2 @ A)) => (~ ((![C : set_pname]: ((ord_le865024672_pname @ C @ A) => (~ ((B2 = (image_pname_pname @ F2 @ C)))))))))))). % subset_imageE
thf(fact_202_image__subsetI, axiom,
    ((![A : set_a, F2 : a > a, B2 : set_a]: ((![X3 : a]: ((member_a @ X3 @ A) => (member_a @ (F2 @ X3) @ B2))) => (ord_less_eq_set_a @ (image_a_a @ F2 @ A) @ B2))))). % image_subsetI
thf(fact_203_image__subsetI, axiom,
    ((![A : set_pname, F2 : pname > a, B2 : set_a]: ((![X3 : pname]: ((member_pname @ X3 @ A) => (member_a @ (F2 @ X3) @ B2))) => (ord_less_eq_set_a @ (image_pname_a @ F2 @ A) @ B2))))). % image_subsetI
thf(fact_204_image__subsetI, axiom,
    ((![A : set_a, F2 : a > pname, B2 : set_pname]: ((![X3 : a]: ((member_a @ X3 @ A) => (member_pname @ (F2 @ X3) @ B2))) => (ord_le865024672_pname @ (image_a_pname @ F2 @ A) @ B2))))). % image_subsetI
thf(fact_205_image__subsetI, axiom,
    ((![A : set_pname, F2 : pname > pname, B2 : set_pname]: ((![X3 : pname]: ((member_pname @ X3 @ A) => (member_pname @ (F2 @ X3) @ B2))) => (ord_le865024672_pname @ (image_pname_pname @ F2 @ A) @ B2))))). % image_subsetI
thf(fact_206_image__mono, axiom,
    ((![A : set_a, B2 : set_a, F2 : a > a]: ((ord_less_eq_set_a @ A @ B2) => (ord_less_eq_set_a @ (image_a_a @ F2 @ A) @ (image_a_a @ F2 @ B2)))))). % image_mono
thf(fact_207_image__mono, axiom,
    ((![A : set_a, B2 : set_a, F2 : a > pname]: ((ord_less_eq_set_a @ A @ B2) => (ord_le865024672_pname @ (image_a_pname @ F2 @ A) @ (image_a_pname @ F2 @ B2)))))). % image_mono
thf(fact_208_image__mono, axiom,
    ((![A : set_pname, B2 : set_pname, F2 : pname > a]: ((ord_le865024672_pname @ A @ B2) => (ord_less_eq_set_a @ (image_pname_a @ F2 @ A) @ (image_pname_a @ F2 @ B2)))))). % image_mono
thf(fact_209_image__mono, axiom,
    ((![A : set_pname, B2 : set_pname, F2 : pname > pname]: ((ord_le865024672_pname @ A @ B2) => (ord_le865024672_pname @ (image_pname_pname @ F2 @ A) @ (image_pname_pname @ F2 @ B2)))))). % image_mono
thf(fact_210_rev__finite__subset, axiom,
    ((![B2 : set_set_a, A : set_set_a]: ((finite_finite_set_a @ B2) => ((ord_le318720350_set_a @ A @ B2) => (finite_finite_set_a @ A)))))). % rev_finite_subset
thf(fact_211_rev__finite__subset, axiom,
    ((![B2 : set_set_pname, A : set_set_pname]: ((finite505202775_pname @ B2) => ((ord_le2066558166_pname @ A @ B2) => (finite505202775_pname @ A)))))). % rev_finite_subset
thf(fact_212_rev__finite__subset, axiom,
    ((![B2 : set_a, A : set_a]: ((finite_finite_a @ B2) => ((ord_less_eq_set_a @ A @ B2) => (finite_finite_a @ A)))))). % rev_finite_subset
thf(fact_213_rev__finite__subset, axiom,
    ((![B2 : set_pname, A : set_pname]: ((finite_finite_pname @ B2) => ((ord_le865024672_pname @ A @ B2) => (finite_finite_pname @ A)))))). % rev_finite_subset
thf(fact_214_infinite__super, axiom,
    ((![S : set_set_a, T2 : set_set_a]: ((ord_le318720350_set_a @ S @ T2) => ((~ ((finite_finite_set_a @ S))) => (~ ((finite_finite_set_a @ T2)))))))). % infinite_super
thf(fact_215_infinite__super, axiom,
    ((![S : set_set_pname, T2 : set_set_pname]: ((ord_le2066558166_pname @ S @ T2) => ((~ ((finite505202775_pname @ S))) => (~ ((finite505202775_pname @ T2)))))))). % infinite_super
thf(fact_216_infinite__super, axiom,
    ((![S : set_a, T2 : set_a]: ((ord_less_eq_set_a @ S @ T2) => ((~ ((finite_finite_a @ S))) => (~ ((finite_finite_a @ T2)))))))). % infinite_super
thf(fact_217_infinite__super, axiom,
    ((![S : set_pname, T2 : set_pname]: ((ord_le865024672_pname @ S @ T2) => ((~ ((finite_finite_pname @ S))) => (~ ((finite_finite_pname @ T2)))))))). % infinite_super
thf(fact_218_finite__subset, axiom,
    ((![A : set_set_a, B2 : set_set_a]: ((ord_le318720350_set_a @ A @ B2) => ((finite_finite_set_a @ B2) => (finite_finite_set_a @ A)))))). % finite_subset
thf(fact_219_finite__subset, axiom,
    ((![A : set_set_pname, B2 : set_set_pname]: ((ord_le2066558166_pname @ A @ B2) => ((finite505202775_pname @ B2) => (finite505202775_pname @ A)))))). % finite_subset
thf(fact_220_finite__subset, axiom,
    ((![A : set_a, B2 : set_a]: ((ord_less_eq_set_a @ A @ B2) => ((finite_finite_a @ B2) => (finite_finite_a @ A)))))). % finite_subset
thf(fact_221_finite__subset, axiom,
    ((![A : set_pname, B2 : set_pname]: ((ord_le865024672_pname @ A @ B2) => ((finite_finite_pname @ B2) => (finite_finite_pname @ A)))))). % finite_subset
thf(fact_222_pigeonhole__infinite, axiom,
    ((![A : set_pname, F2 : pname > pname]: ((~ ((finite_finite_pname @ A))) => ((finite_finite_pname @ (image_pname_pname @ F2 @ A)) => (?[X3 : pname]: ((member_pname @ X3 @ A) & (~ ((finite_finite_pname @ (collect_pname @ (^[A3 : pname]: (((member_pname @ A3 @ A)) & (((F2 @ A3) = (F2 @ X3)))))))))))))))). % pigeonhole_infinite
thf(fact_223_pigeonhole__infinite, axiom,
    ((![A : set_pname, F2 : pname > a]: ((~ ((finite_finite_pname @ A))) => ((finite_finite_a @ (image_pname_a @ F2 @ A)) => (?[X3 : pname]: ((member_pname @ X3 @ A) & (~ ((finite_finite_pname @ (collect_pname @ (^[A3 : pname]: (((member_pname @ A3 @ A)) & (((F2 @ A3) = (F2 @ X3)))))))))))))))). % pigeonhole_infinite
thf(fact_224_pigeonhole__infinite, axiom,
    ((![A : set_a, F2 : a > pname]: ((~ ((finite_finite_a @ A))) => ((finite_finite_pname @ (image_a_pname @ F2 @ A)) => (?[X3 : a]: ((member_a @ X3 @ A) & (~ ((finite_finite_a @ (collect_a @ (^[A3 : a]: (((member_a @ A3 @ A)) & (((F2 @ A3) = (F2 @ X3)))))))))))))))). % pigeonhole_infinite
thf(fact_225_pigeonhole__infinite, axiom,
    ((![A : set_a, F2 : a > a]: ((~ ((finite_finite_a @ A))) => ((finite_finite_a @ (image_a_a @ F2 @ A)) => (?[X3 : a]: ((member_a @ X3 @ A) & (~ ((finite_finite_a @ (collect_a @ (^[A3 : a]: (((member_a @ A3 @ A)) & (((F2 @ A3) = (F2 @ X3)))))))))))))))). % pigeonhole_infinite
thf(fact_226_pigeonhole__infinite, axiom,
    ((![A : set_pname, F2 : pname > set_a]: ((~ ((finite_finite_pname @ A))) => ((finite_finite_set_a @ (image_pname_set_a @ F2 @ A)) => (?[X3 : pname]: ((member_pname @ X3 @ A) & (~ ((finite_finite_pname @ (collect_pname @ (^[A3 : pname]: (((member_pname @ A3 @ A)) & (((F2 @ A3) = (F2 @ X3)))))))))))))))). % pigeonhole_infinite
thf(fact_227_pigeonhole__infinite, axiom,
    ((![A : set_pname, F2 : pname > set_pname]: ((~ ((finite_finite_pname @ A))) => ((finite505202775_pname @ (image_747505105_pname @ F2 @ A)) => (?[X3 : pname]: ((member_pname @ X3 @ A) & (~ ((finite_finite_pname @ (collect_pname @ (^[A3 : pname]: (((member_pname @ A3 @ A)) & (((F2 @ A3) = (F2 @ X3)))))))))))))))). % pigeonhole_infinite
thf(fact_228_pigeonhole__infinite, axiom,
    ((![A : set_a, F2 : a > set_a]: ((~ ((finite_finite_a @ A))) => ((finite_finite_set_a @ (image_a_set_a @ F2 @ A)) => (?[X3 : a]: ((member_a @ X3 @ A) & (~ ((finite_finite_a @ (collect_a @ (^[A3 : a]: (((member_a @ A3 @ A)) & (((F2 @ A3) = (F2 @ X3)))))))))))))))). % pigeonhole_infinite
thf(fact_229_pigeonhole__infinite, axiom,
    ((![A : set_a, F2 : a > set_pname]: ((~ ((finite_finite_a @ A))) => ((finite505202775_pname @ (image_a_set_pname @ F2 @ A)) => (?[X3 : a]: ((member_a @ X3 @ A) & (~ ((finite_finite_a @ (collect_a @ (^[A3 : a]: (((member_a @ A3 @ A)) & (((F2 @ A3) = (F2 @ X3)))))))))))))))). % pigeonhole_infinite
thf(fact_230_pigeonhole__infinite, axiom,
    ((![A : set_set_a, F2 : set_a > pname]: ((~ ((finite_finite_set_a @ A))) => ((finite_finite_pname @ (image_set_a_pname @ F2 @ A)) => (?[X3 : set_a]: ((member_set_a @ X3 @ A) & (~ ((finite_finite_set_a @ (collect_set_a @ (^[A3 : set_a]: (((member_set_a @ A3 @ A)) & (((F2 @ A3) = (F2 @ X3)))))))))))))))). % pigeonhole_infinite
thf(fact_231_pigeonhole__infinite, axiom,
    ((![A : set_set_a, F2 : set_a > a]: ((~ ((finite_finite_set_a @ A))) => ((finite_finite_a @ (image_set_a_a @ F2 @ A)) => (?[X3 : set_a]: ((member_set_a @ X3 @ A) & (~ ((finite_finite_set_a @ (collect_set_a @ (^[A3 : set_a]: (((member_set_a @ A3 @ A)) & (((F2 @ A3) = (F2 @ X3)))))))))))))))). % pigeonhole_infinite
thf(fact_232_order__refl, axiom,
    ((![X2 : set_a]: (ord_less_eq_set_a @ X2 @ X2)))). % order_refl
thf(fact_233_order__refl, axiom,
    ((![X2 : set_pname]: (ord_le865024672_pname @ X2 @ X2)))). % order_refl
thf(fact_234_image__Collect__subsetI, axiom,
    ((![P : a > $o, F2 : a > a, B2 : set_a]: ((![X3 : a]: ((P @ X3) => (member_a @ (F2 @ X3) @ B2))) => (ord_less_eq_set_a @ (image_a_a @ F2 @ (collect_a @ P)) @ B2))))). % image_Collect_subsetI
thf(fact_235_image__Collect__subsetI, axiom,
    ((![P : set_a > $o, F2 : set_a > a, B2 : set_a]: ((![X3 : set_a]: ((P @ X3) => (member_a @ (F2 @ X3) @ B2))) => (ord_less_eq_set_a @ (image_set_a_a @ F2 @ (collect_set_a @ P)) @ B2))))). % image_Collect_subsetI
thf(fact_236_image__Collect__subsetI, axiom,
    ((![P : set_pname > $o, F2 : set_pname > a, B2 : set_a]: ((![X3 : set_pname]: ((P @ X3) => (member_a @ (F2 @ X3) @ B2))) => (ord_less_eq_set_a @ (image_set_pname_a @ F2 @ (collect_set_pname @ P)) @ B2))))). % image_Collect_subsetI
thf(fact_237_image__Collect__subsetI, axiom,
    ((![P : pname > $o, F2 : pname > a, B2 : set_a]: ((![X3 : pname]: ((P @ X3) => (member_a @ (F2 @ X3) @ B2))) => (ord_less_eq_set_a @ (image_pname_a @ F2 @ (collect_pname @ P)) @ B2))))). % image_Collect_subsetI
thf(fact_238_image__Collect__subsetI, axiom,
    ((![P : a > $o, F2 : a > pname, B2 : set_pname]: ((![X3 : a]: ((P @ X3) => (member_pname @ (F2 @ X3) @ B2))) => (ord_le865024672_pname @ (image_a_pname @ F2 @ (collect_a @ P)) @ B2))))). % image_Collect_subsetI
thf(fact_239_image__Collect__subsetI, axiom,
    ((![P : set_a > $o, F2 : set_a > pname, B2 : set_pname]: ((![X3 : set_a]: ((P @ X3) => (member_pname @ (F2 @ X3) @ B2))) => (ord_le865024672_pname @ (image_set_a_pname @ F2 @ (collect_set_a @ P)) @ B2))))). % image_Collect_subsetI
thf(fact_240_image__Collect__subsetI, axiom,
    ((![P : set_pname > $o, F2 : set_pname > pname, B2 : set_pname]: ((![X3 : set_pname]: ((P @ X3) => (member_pname @ (F2 @ X3) @ B2))) => (ord_le865024672_pname @ (image_1672683217_pname @ F2 @ (collect_set_pname @ P)) @ B2))))). % image_Collect_subsetI
thf(fact_241_image__Collect__subsetI, axiom,
    ((![P : pname > $o, F2 : pname > pname, B2 : set_pname]: ((![X3 : pname]: ((P @ X3) => (member_pname @ (F2 @ X3) @ B2))) => (ord_le865024672_pname @ (image_pname_pname @ F2 @ (collect_pname @ P)) @ B2))))). % image_Collect_subsetI
thf(fact_242_assms_I3_J, axiom,
    ((![C3 : com, G : set_a]: ((wt @ C3) => ((![X3 : pname]: ((member_pname @ X3 @ u) => (p @ G @ (insert_a @ (mgt_call @ X3) @ bot_bot_set_a)))) => (p @ G @ (insert_a @ (mgt @ C3) @ bot_bot_set_a))))))). % assms(3)
thf(fact_243_assms_I4_J, axiom,
    ((![Pn : pname]: ((member_pname @ Pn @ u) => (wt @ (the_com @ (body @ Pn))))))). % assms(4)
thf(fact_244_subset__Collect__iff, axiom,
    ((![B2 : set_set_pname, A : set_set_pname, P : set_pname > $o]: ((ord_le2066558166_pname @ B2 @ A) => ((ord_le2066558166_pname @ B2 @ (collect_set_pname @ (^[X : set_pname]: (((member_set_pname @ X @ A)) & ((P @ X)))))) = (![X : set_pname]: (((member_set_pname @ X @ B2)) => ((P @ X))))))))). % subset_Collect_iff
thf(fact_245_subset__Collect__iff, axiom,
    ((![B2 : set_a, A : set_a, P : a > $o]: ((ord_less_eq_set_a @ B2 @ A) => ((ord_less_eq_set_a @ B2 @ (collect_a @ (^[X : a]: (((member_a @ X @ A)) & ((P @ X)))))) = (![X : a]: (((member_a @ X @ B2)) => ((P @ X))))))))). % subset_Collect_iff
thf(fact_246_subset__Collect__iff, axiom,
    ((![B2 : set_pname, A : set_pname, P : pname > $o]: ((ord_le865024672_pname @ B2 @ A) => ((ord_le865024672_pname @ B2 @ (collect_pname @ (^[X : pname]: (((member_pname @ X @ A)) & ((P @ X)))))) = (![X : pname]: (((member_pname @ X @ B2)) => ((P @ X))))))))). % subset_Collect_iff
thf(fact_247_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)

% Conjectures (6)
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, conjecture,
    ((g = (image_pname_a @ mgt_call @ u)))).
