% 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_370__3253900_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:14:05.746

% Could-be-implicit typings (9)
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__Set__Oset_It__Com__Opname_J, type,
    set_pname : $tType).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J, type,
    set_nat : $tType).
thf(ty_n_t__Set__Oset_Itf__a_J, type,
    set_a : $tType).
thf(ty_n_t__Com__Opname, type,
    pname : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).
thf(ty_n_t__Com__Ocom, type,
    com : $tType).
thf(ty_n_tf__a, type,
    a : $tType).

% Explicit typings (36)
thf(sy_c_Finite__Set_Ocard_001t__Com__Opname, type,
    finite_card_pname : set_pname > nat).
thf(sy_c_Finite__Set_Ocard_001tf__a, type,
    finite_card_a : set_a > nat).
thf(sy_c_Finite__Set_Ofinite_001t__Com__Opname, type,
    finite_finite_pname : set_pname > $o).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat, type,
    finite_finite_nat : set_nat > $o).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_Itf__a_J, type,
    finite_finite_set_a : set_set_a > $o).
thf(sy_c_Finite__Set_Ofinite_001tf__a, type,
    finite_finite_a : set_a > $o).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat, type,
    minus_minus_nat : nat > nat > nat).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Com__Opname_J, type,
    minus_1937938585_pname : set_pname > set_pname > set_pname).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J, type,
    minus_minus_set_a : set_a > set_a > set_a).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat, type,
    zero_zero_nat : nat).
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__Nat__Onat, type,
    ord_less_eq_nat : nat > nat > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Com__Opname_J, type,
    ord_le865024672_pname : set_pname > set_pname > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J, type,
    ord_less_eq_set_a : set_a > set_a > $o).
thf(sy_c_Set_OCollect_001t__Com__Opname, type,
    collect_pname : (pname > $o) > set_pname).
thf(sy_c_Set_OCollect_001tf__a, type,
    collect_a : (a > $o) > set_a).
thf(sy_c_Set_Oimage_001t__Com__Opname_001t__Com__Opname, type,
    image_pname_pname : (pname > pname) > set_pname > set_pname).
thf(sy_c_Set_Oimage_001t__Com__Opname_001tf__a, type,
    image_pname_a : (pname > a) > set_pname > set_a).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Com__Opname_J_001t__Set__Oset_It__Com__Opname_J, type,
    image_1068293127_pname : (set_pname > set_pname) > set_set_pname > set_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_001tf__a_001t__Com__Opname, type,
    image_a_pname : (a > pname) > set_a > set_pname).
thf(sy_c_Set_Oimage_001tf__a_001tf__a, type,
    image_a_a : (a > a) > set_a > set_a).
thf(sy_c_Set_Oinsert_001t__Com__Opname, type,
    insert_pname : pname > set_pname > set_pname).
thf(sy_c_Set_Oinsert_001tf__a, type,
    insert_a : a > set_a > set_a).
thf(sy_c_member_001t__Com__Opname, type,
    member_pname : pname > set_pname > $o).
thf(sy_c_member_001t__Nat__Onat, type,
    member_nat : nat > set_nat > $o).
thf(sy_c_member_001t__Set__Oset_It__Com__Opname_J, type,
    member_set_pname : set_pname > set_set_pname > $o).
thf(sy_c_member_001t__Set__Oset_Itf__a_J, type,
    member_set_a : set_a > set_set_a > $o).
thf(sy_c_member_001tf__a, type,
    member_a : a > set_a > $o).
thf(sy_v_P, type,
    p : set_a > set_a > $o).
thf(sy_v_U, type,
    u : set_pname).
thf(sy_v_mgt, type,
    mgt : com > a).
thf(sy_v_mgt__call, type,
    mgt_call : pname > a).
thf(sy_v_uG, type,
    uG : set_a).
thf(sy_v_wt, type,
    wt : com > $o).

% Relevant facts (206)
thf(fact_0_assms_I1_J, axiom,
    ((![Ts : set_a, G : set_a]: ((ord_less_eq_set_a @ Ts @ G) => (p @ G @ Ts))))). % assms(1)
thf(fact_1_assms_I3_J, axiom,
    ((![C : com, G : set_a]: ((wt @ C) => ((![X : pname]: ((member_pname @ X @ u) => (p @ G @ (insert_a @ (mgt_call @ X) @ bot_bot_set_a)))) => (p @ G @ (insert_a @ (mgt @ C) @ bot_bot_set_a))))))). % assms(3)
thf(fact_2_card__0__eq, axiom,
    ((![A : set_pname]: ((finite_finite_pname @ A) => (((finite_card_pname @ A) = zero_zero_nat) = (A = bot_bot_set_pname)))))). % card_0_eq
thf(fact_3_card__0__eq, axiom,
    ((![A : set_a]: ((finite_finite_a @ A) => (((finite_card_a @ A) = zero_zero_nat) = (A = bot_bot_set_a)))))). % card_0_eq
thf(fact_4_diff__is__0__eq, axiom,
    ((![M : nat, N : nat]: (((minus_minus_nat @ M @ N) = zero_zero_nat) = (ord_less_eq_nat @ M @ N))))). % diff_is_0_eq
thf(fact_5_diff__is__0__eq_H, axiom,
    ((![M : nat, N : nat]: ((ord_less_eq_nat @ M @ N) => ((minus_minus_nat @ M @ N) = zero_zero_nat))))). % diff_is_0_eq'
thf(fact_6_card_Oinfinite, axiom,
    ((![A : set_a]: ((~ ((finite_finite_a @ A))) => ((finite_card_a @ A) = zero_zero_nat))))). % card.infinite
thf(fact_7_card_Oinfinite, axiom,
    ((![A : set_pname]: ((~ ((finite_finite_pname @ A))) => ((finite_card_pname @ A) = zero_zero_nat))))). % card.infinite
thf(fact_8_card_Oempty, axiom,
    (((finite_card_pname @ bot_bot_set_pname) = zero_zero_nat))). % card.empty
thf(fact_9_card_Oempty, axiom,
    (((finite_card_a @ bot_bot_set_a) = zero_zero_nat))). % card.empty
thf(fact_10_singleton__insert__inj__eq, axiom,
    ((![B : pname, A2 : pname, A : set_pname]: (((insert_pname @ B @ bot_bot_set_pname) = (insert_pname @ A2 @ A)) = (((A2 = B)) & ((ord_le865024672_pname @ A @ (insert_pname @ B @ bot_bot_set_pname)))))))). % singleton_insert_inj_eq
thf(fact_11_singleton__insert__inj__eq, axiom,
    ((![B : a, A2 : a, A : set_a]: (((insert_a @ B @ bot_bot_set_a) = (insert_a @ A2 @ A)) = (((A2 = B)) & ((ord_less_eq_set_a @ A @ (insert_a @ B @ bot_bot_set_a)))))))). % singleton_insert_inj_eq
thf(fact_12_singleton__insert__inj__eq_H, axiom,
    ((![A2 : pname, A : set_pname, B : pname]: (((insert_pname @ A2 @ A) = (insert_pname @ B @ bot_bot_set_pname)) = (((A2 = B)) & ((ord_le865024672_pname @ A @ (insert_pname @ B @ bot_bot_set_pname)))))))). % singleton_insert_inj_eq'
thf(fact_13_singleton__insert__inj__eq_H, axiom,
    ((![A2 : a, A : set_a, B : a]: (((insert_a @ A2 @ A) = (insert_a @ B @ bot_bot_set_a)) = (((A2 = B)) & ((ord_less_eq_set_a @ A @ (insert_a @ B @ bot_bot_set_a)))))))). % singleton_insert_inj_eq'
thf(fact_14_diff__diff__cancel, axiom,
    ((![I : nat, N : nat]: ((ord_less_eq_nat @ I @ N) => ((minus_minus_nat @ N @ (minus_minus_nat @ N @ I)) = I))))). % diff_diff_cancel
thf(fact_15_diff__0__eq__0, axiom,
    ((![N : nat]: ((minus_minus_nat @ zero_zero_nat @ N) = zero_zero_nat)))). % diff_0_eq_0
thf(fact_16_diff__self__eq__0, axiom,
    ((![M : nat]: ((minus_minus_nat @ M @ M) = zero_zero_nat)))). % diff_self_eq_0
thf(fact_17_surj__card__le, axiom,
    ((![A : set_pname, B2 : set_pname, F : pname > pname]: ((finite_finite_pname @ A) => ((ord_le865024672_pname @ B2 @ (image_pname_pname @ F @ A)) => (ord_less_eq_nat @ (finite_card_pname @ B2) @ (finite_card_pname @ A))))))). % surj_card_le
thf(fact_18_surj__card__le, axiom,
    ((![A : set_a, B2 : set_pname, F : a > pname]: ((finite_finite_a @ A) => ((ord_le865024672_pname @ B2 @ (image_a_pname @ F @ A)) => (ord_less_eq_nat @ (finite_card_pname @ B2) @ (finite_card_a @ A))))))). % surj_card_le
thf(fact_19_surj__card__le, axiom,
    ((![A : set_pname, B2 : set_a, F : pname > a]: ((finite_finite_pname @ A) => ((ord_less_eq_set_a @ B2 @ (image_pname_a @ F @ A)) => (ord_less_eq_nat @ (finite_card_a @ B2) @ (finite_card_pname @ A))))))). % surj_card_le
thf(fact_20_surj__card__le, axiom,
    ((![A : set_a, B2 : set_a, F : a > a]: ((finite_finite_a @ A) => ((ord_less_eq_set_a @ B2 @ (image_a_a @ F @ A)) => (ord_less_eq_nat @ (finite_card_a @ B2) @ (finite_card_a @ A))))))). % surj_card_le
thf(fact_21_image__eqI, axiom,
    ((![B : a, F : pname > a, X2 : pname, A : set_pname]: ((B = (F @ X2)) => ((member_pname @ X2 @ A) => (member_a @ B @ (image_pname_a @ F @ A))))))). % image_eqI
thf(fact_22_Diff__cancel, axiom,
    ((![A : set_a]: ((minus_minus_set_a @ A @ A) = bot_bot_set_a)))). % Diff_cancel
thf(fact_23_Diff__cancel, axiom,
    ((![A : set_pname]: ((minus_1937938585_pname @ A @ A) = bot_bot_set_pname)))). % Diff_cancel
thf(fact_24_empty__Diff, axiom,
    ((![A : set_a]: ((minus_minus_set_a @ bot_bot_set_a @ A) = bot_bot_set_a)))). % empty_Diff
thf(fact_25_empty__Diff, axiom,
    ((![A : set_pname]: ((minus_1937938585_pname @ bot_bot_set_pname @ A) = bot_bot_set_pname)))). % empty_Diff
thf(fact_26_Diff__empty, axiom,
    ((![A : set_a]: ((minus_minus_set_a @ A @ bot_bot_set_a) = A)))). % Diff_empty
thf(fact_27_Diff__empty, axiom,
    ((![A : set_pname]: ((minus_1937938585_pname @ A @ bot_bot_set_pname) = A)))). % Diff_empty
thf(fact_28_empty__Collect__eq, axiom,
    ((![P : a > $o]: ((bot_bot_set_a = (collect_a @ P)) = (![X3 : a]: (~ ((P @ X3)))))))). % empty_Collect_eq
thf(fact_29_empty__Collect__eq, axiom,
    ((![P : pname > $o]: ((bot_bot_set_pname = (collect_pname @ P)) = (![X3 : pname]: (~ ((P @ X3)))))))). % empty_Collect_eq
thf(fact_30_Collect__empty__eq, axiom,
    ((![P : a > $o]: (((collect_a @ P) = bot_bot_set_a) = (![X3 : a]: (~ ((P @ X3)))))))). % Collect_empty_eq
thf(fact_31_Collect__empty__eq, axiom,
    ((![P : pname > $o]: (((collect_pname @ P) = bot_bot_set_pname) = (![X3 : pname]: (~ ((P @ X3)))))))). % Collect_empty_eq
thf(fact_32_all__not__in__conv, axiom,
    ((![A : set_a]: ((![X3 : a]: (~ ((member_a @ X3 @ A)))) = (A = bot_bot_set_a))))). % all_not_in_conv
thf(fact_33_all__not__in__conv, axiom,
    ((![A : set_pname]: ((![X3 : pname]: (~ ((member_pname @ X3 @ A)))) = (A = bot_bot_set_pname))))). % all_not_in_conv
thf(fact_34_empty__iff, axiom,
    ((![C : a]: (~ ((member_a @ C @ bot_bot_set_a)))))). % empty_iff
thf(fact_35_empty__iff, axiom,
    ((![C : pname]: (~ ((member_pname @ C @ bot_bot_set_pname)))))). % empty_iff
thf(fact_36_finite__Diff2, axiom,
    ((![B2 : set_pname, A : set_pname]: ((finite_finite_pname @ B2) => ((finite_finite_pname @ (minus_1937938585_pname @ A @ B2)) = (finite_finite_pname @ A)))))). % finite_Diff2
thf(fact_37_finite__Diff2, axiom,
    ((![B2 : set_a, A : set_a]: ((finite_finite_a @ B2) => ((finite_finite_a @ (minus_minus_set_a @ A @ B2)) = (finite_finite_a @ A)))))). % finite_Diff2
thf(fact_38_finite__Diff, axiom,
    ((![A : set_pname, B2 : set_pname]: ((finite_finite_pname @ A) => (finite_finite_pname @ (minus_1937938585_pname @ A @ B2)))))). % finite_Diff
thf(fact_39_finite__Diff, axiom,
    ((![A : set_a, B2 : set_a]: ((finite_finite_a @ A) => (finite_finite_a @ (minus_minus_set_a @ A @ B2)))))). % finite_Diff
thf(fact_40_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_41_subsetI, axiom,
    ((![A : set_a, B2 : set_a]: ((![X : a]: ((member_a @ X @ A) => (member_a @ X @ B2))) => (ord_less_eq_set_a @ A @ B2))))). % subsetI
thf(fact_42_insert__Diff1, axiom,
    ((![X2 : a, B2 : set_a, A : set_a]: ((member_a @ X2 @ B2) => ((minus_minus_set_a @ (insert_a @ X2 @ A) @ B2) = (minus_minus_set_a @ A @ B2)))))). % insert_Diff1
thf(fact_43_Diff__insert0, axiom,
    ((![X2 : a, A : set_a, B2 : set_a]: ((~ ((member_a @ X2 @ A))) => ((minus_minus_set_a @ A @ (insert_a @ X2 @ B2)) = (minus_minus_set_a @ A @ B2)))))). % Diff_insert0
thf(fact_44_insert__absorb2, axiom,
    ((![X2 : a, A : set_a]: ((insert_a @ X2 @ (insert_a @ X2 @ A)) = (insert_a @ X2 @ A))))). % insert_absorb2
thf(fact_45_insert__iff, axiom,
    ((![A2 : a, B : a, A : set_a]: ((member_a @ A2 @ (insert_a @ B @ A)) = (((A2 = B)) | ((member_a @ A2 @ A))))))). % insert_iff
thf(fact_46_insertCI, axiom,
    ((![A2 : a, B2 : set_a, B : a]: (((~ ((member_a @ A2 @ B2))) => (A2 = B)) => (member_a @ A2 @ (insert_a @ B @ B2)))))). % insertCI
thf(fact_47_le__zero__eq, axiom,
    ((![N : nat]: ((ord_less_eq_nat @ N @ zero_zero_nat) = (N = zero_zero_nat))))). % le_zero_eq
thf(fact_48_cancel__comm__monoid__add__class_Odiff__cancel, axiom,
    ((![A2 : nat]: ((minus_minus_nat @ A2 @ A2) = zero_zero_nat)))). % cancel_comm_monoid_add_class.diff_cancel
thf(fact_49_diff__zero, axiom,
    ((![A2 : nat]: ((minus_minus_nat @ A2 @ zero_zero_nat) = A2)))). % diff_zero
thf(fact_50_zero__diff, axiom,
    ((![A2 : nat]: ((minus_minus_nat @ zero_zero_nat @ A2) = zero_zero_nat)))). % zero_diff
thf(fact_51_image__is__empty, axiom,
    ((![F : a > a, A : set_a]: (((image_a_a @ F @ A) = bot_bot_set_a) = (A = bot_bot_set_a))))). % image_is_empty
thf(fact_52_image__is__empty, axiom,
    ((![F : pname > a, A : set_pname]: (((image_pname_a @ F @ A) = bot_bot_set_a) = (A = bot_bot_set_pname))))). % image_is_empty
thf(fact_53_image__is__empty, axiom,
    ((![F : a > pname, A : set_a]: (((image_a_pname @ F @ A) = bot_bot_set_pname) = (A = bot_bot_set_a))))). % image_is_empty
thf(fact_54_image__is__empty, axiom,
    ((![F : pname > pname, A : set_pname]: (((image_pname_pname @ F @ A) = bot_bot_set_pname) = (A = bot_bot_set_pname))))). % image_is_empty
thf(fact_55_empty__is__image, axiom,
    ((![F : a > a, A : set_a]: ((bot_bot_set_a = (image_a_a @ F @ A)) = (A = bot_bot_set_a))))). % empty_is_image
thf(fact_56_empty__is__image, axiom,
    ((![F : pname > a, A : set_pname]: ((bot_bot_set_a = (image_pname_a @ F @ A)) = (A = bot_bot_set_pname))))). % empty_is_image
thf(fact_57_empty__is__image, axiom,
    ((![F : a > pname, A : set_a]: ((bot_bot_set_pname = (image_a_pname @ F @ A)) = (A = bot_bot_set_a))))). % empty_is_image
thf(fact_58_empty__is__image, axiom,
    ((![F : pname > pname, A : set_pname]: ((bot_bot_set_pname = (image_pname_pname @ F @ A)) = (A = bot_bot_set_pname))))). % empty_is_image
thf(fact_59_image__empty, axiom,
    ((![F : a > a]: ((image_a_a @ F @ bot_bot_set_a) = bot_bot_set_a)))). % image_empty
thf(fact_60_image__empty, axiom,
    ((![F : a > pname]: ((image_a_pname @ F @ bot_bot_set_a) = bot_bot_set_pname)))). % image_empty
thf(fact_61_image__empty, axiom,
    ((![F : pname > a]: ((image_pname_a @ F @ bot_bot_set_pname) = bot_bot_set_a)))). % image_empty
thf(fact_62_image__empty, axiom,
    ((![F : pname > pname]: ((image_pname_pname @ F @ bot_bot_set_pname) = bot_bot_set_pname)))). % image_empty
thf(fact_63_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_64_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_65_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_66_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_67_insert__image, axiom,
    ((![X2 : pname, A : set_pname, F : pname > a]: ((member_pname @ X2 @ A) => ((insert_a @ (F @ X2) @ (image_pname_a @ F @ A)) = (image_pname_a @ F @ A)))))). % insert_image
thf(fact_68_image__insert, axiom,
    ((![F : pname > a, A2 : pname, B2 : set_pname]: ((image_pname_a @ F @ (insert_pname @ A2 @ B2)) = (insert_a @ (F @ A2) @ (image_pname_a @ F @ B2)))))). % image_insert
thf(fact_69_image__insert, axiom,
    ((![F : a > a, A2 : a, B2 : set_a]: ((image_a_a @ F @ (insert_a @ A2 @ B2)) = (insert_a @ (F @ A2) @ (image_a_a @ F @ B2)))))). % image_insert
thf(fact_70_Diff__eq__empty__iff, axiom,
    ((![A : set_pname, B2 : set_pname]: (((minus_1937938585_pname @ A @ B2) = bot_bot_set_pname) = (ord_le865024672_pname @ A @ B2))))). % Diff_eq_empty_iff
thf(fact_71_Diff__eq__empty__iff, axiom,
    ((![A : set_a, B2 : set_a]: (((minus_minus_set_a @ A @ B2) = bot_bot_set_a) = (ord_less_eq_set_a @ A @ B2))))). % Diff_eq_empty_iff
thf(fact_72_empty__subsetI, axiom,
    ((![A : set_pname]: (ord_le865024672_pname @ bot_bot_set_pname @ A)))). % empty_subsetI
thf(fact_73_empty__subsetI, axiom,
    ((![A : set_a]: (ord_less_eq_set_a @ bot_bot_set_a @ A)))). % empty_subsetI
thf(fact_74_subset__empty, axiom,
    ((![A : set_pname]: ((ord_le865024672_pname @ A @ bot_bot_set_pname) = (A = bot_bot_set_pname))))). % subset_empty
thf(fact_75_subset__empty, axiom,
    ((![A : set_a]: ((ord_less_eq_set_a @ A @ bot_bot_set_a) = (A = bot_bot_set_a))))). % subset_empty
thf(fact_76_insert__Diff__single, axiom,
    ((![A2 : a, A : set_a]: ((insert_a @ A2 @ (minus_minus_set_a @ A @ (insert_a @ A2 @ bot_bot_set_a))) = (insert_a @ A2 @ A))))). % insert_Diff_single
thf(fact_77_insert__Diff__single, axiom,
    ((![A2 : pname, A : set_pname]: ((insert_pname @ A2 @ (minus_1937938585_pname @ A @ (insert_pname @ A2 @ bot_bot_set_pname))) = (insert_pname @ A2 @ A))))). % insert_Diff_single
thf(fact_78_singletonI, axiom,
    ((![A2 : a]: (member_a @ A2 @ (insert_a @ A2 @ bot_bot_set_a))))). % singletonI
thf(fact_79_singletonI, axiom,
    ((![A2 : pname]: (member_pname @ A2 @ (insert_pname @ A2 @ bot_bot_set_pname))))). % singletonI
thf(fact_80_finite__Diff__insert, axiom,
    ((![A : set_pname, A2 : pname, B2 : set_pname]: ((finite_finite_pname @ (minus_1937938585_pname @ A @ (insert_pname @ A2 @ B2))) = (finite_finite_pname @ (minus_1937938585_pname @ A @ B2)))))). % finite_Diff_insert
thf(fact_81_finite__Diff__insert, axiom,
    ((![A : set_a, A2 : a, B2 : set_a]: ((finite_finite_a @ (minus_minus_set_a @ A @ (insert_a @ A2 @ B2))) = (finite_finite_a @ (minus_minus_set_a @ A @ B2)))))). % finite_Diff_insert
thf(fact_82_finite__insert, axiom,
    ((![A2 : pname, A : set_pname]: ((finite_finite_pname @ (insert_pname @ A2 @ A)) = (finite_finite_pname @ A))))). % finite_insert
thf(fact_83_finite__insert, axiom,
    ((![A2 : a, A : set_a]: ((finite_finite_a @ (insert_a @ A2 @ A)) = (finite_finite_a @ A))))). % finite_insert
thf(fact_84_insert__subset, axiom,
    ((![X2 : a, A : set_a, B2 : set_a]: ((ord_less_eq_set_a @ (insert_a @ X2 @ A) @ B2) = (((member_a @ X2 @ B2)) & ((ord_less_eq_set_a @ A @ B2))))))). % insert_subset
thf(fact_85_bot__nat__0_Oextremum, axiom,
    ((![A2 : nat]: (ord_less_eq_nat @ zero_zero_nat @ A2)))). % bot_nat_0.extremum
thf(fact_86_le0, axiom,
    ((![N : nat]: (ord_less_eq_nat @ zero_zero_nat @ N)))). % le0
thf(fact_87_Diff__infinite__finite, axiom,
    ((![T : set_pname, S : set_pname]: ((finite_finite_pname @ T) => ((~ ((finite_finite_pname @ S))) => (~ ((finite_finite_pname @ (minus_1937938585_pname @ S @ T))))))))). % Diff_infinite_finite
thf(fact_88_Diff__infinite__finite, axiom,
    ((![T : set_a, S : set_a]: ((finite_finite_a @ T) => ((~ ((finite_finite_a @ S))) => (~ ((finite_finite_a @ (minus_minus_set_a @ S @ T))))))))). % Diff_infinite_finite
thf(fact_89_double__diff, axiom,
    ((![A : set_a, B2 : set_a, C2 : set_a]: ((ord_less_eq_set_a @ A @ B2) => ((ord_less_eq_set_a @ B2 @ C2) => ((minus_minus_set_a @ B2 @ (minus_minus_set_a @ C2 @ A)) = A)))))). % double_diff
thf(fact_90_Diff__subset, axiom,
    ((![A : set_a, B2 : set_a]: (ord_less_eq_set_a @ (minus_minus_set_a @ A @ B2) @ A)))). % Diff_subset
thf(fact_91_Diff__mono, axiom,
    ((![A : set_a, C2 : set_a, D : set_a, B2 : set_a]: ((ord_less_eq_set_a @ A @ C2) => ((ord_less_eq_set_a @ D @ B2) => (ord_less_eq_set_a @ (minus_minus_set_a @ A @ B2) @ (minus_minus_set_a @ C2 @ D))))))). % Diff_mono
thf(fact_92_insert__Diff__if, axiom,
    ((![X2 : a, B2 : set_a, A : set_a]: (((member_a @ X2 @ B2) => ((minus_minus_set_a @ (insert_a @ X2 @ A) @ B2) = (minus_minus_set_a @ A @ B2))) & ((~ ((member_a @ X2 @ B2))) => ((minus_minus_set_a @ (insert_a @ X2 @ A) @ B2) = (insert_a @ X2 @ (minus_minus_set_a @ A @ B2)))))))). % insert_Diff_if
thf(fact_93_in__image__insert__iff, axiom,
    ((![B2 : set_set_a, X2 : a, A : set_a]: ((![C3 : set_a]: ((member_set_a @ C3 @ B2) => (~ ((member_a @ X2 @ C3))))) => ((member_set_a @ A @ (image_set_a_set_a @ (insert_a @ X2) @ B2)) = (((member_a @ X2 @ A)) & ((member_set_a @ (minus_minus_set_a @ A @ (insert_a @ X2 @ bot_bot_set_a)) @ B2)))))))). % in_image_insert_iff
thf(fact_94_in__image__insert__iff, axiom,
    ((![B2 : set_set_pname, X2 : pname, A : set_pname]: ((![C3 : set_pname]: ((member_set_pname @ C3 @ B2) => (~ ((member_pname @ X2 @ C3))))) => ((member_set_pname @ A @ (image_1068293127_pname @ (insert_pname @ X2) @ B2)) = (((member_pname @ X2 @ A)) & ((member_set_pname @ (minus_1937938585_pname @ A @ (insert_pname @ X2 @ bot_bot_set_pname)) @ B2)))))))). % in_image_insert_iff
thf(fact_95_image__diff__subset, axiom,
    ((![F : pname > a, A : set_pname, B2 : set_pname]: (ord_less_eq_set_a @ (minus_minus_set_a @ (image_pname_a @ F @ A) @ (image_pname_a @ F @ B2)) @ (image_pname_a @ F @ (minus_1937938585_pname @ A @ B2)))))). % image_diff_subset
thf(fact_96_Diff__insert__absorb, axiom,
    ((![X2 : a, A : set_a]: ((~ ((member_a @ X2 @ A))) => ((minus_minus_set_a @ (insert_a @ X2 @ A) @ (insert_a @ X2 @ bot_bot_set_a)) = A))))). % Diff_insert_absorb
thf(fact_97_Diff__insert__absorb, axiom,
    ((![X2 : pname, A : set_pname]: ((~ ((member_pname @ X2 @ A))) => ((minus_1937938585_pname @ (insert_pname @ X2 @ A) @ (insert_pname @ X2 @ bot_bot_set_pname)) = A))))). % Diff_insert_absorb
thf(fact_98_Diff__insert2, axiom,
    ((![A : set_a, A2 : a, B2 : set_a]: ((minus_minus_set_a @ A @ (insert_a @ A2 @ B2)) = (minus_minus_set_a @ (minus_minus_set_a @ A @ (insert_a @ A2 @ bot_bot_set_a)) @ B2))))). % Diff_insert2
thf(fact_99_Diff__insert2, axiom,
    ((![A : set_pname, A2 : pname, B2 : set_pname]: ((minus_1937938585_pname @ A @ (insert_pname @ A2 @ B2)) = (minus_1937938585_pname @ (minus_1937938585_pname @ A @ (insert_pname @ A2 @ bot_bot_set_pname)) @ B2))))). % Diff_insert2
thf(fact_100_insert__Diff, axiom,
    ((![A2 : a, A : set_a]: ((member_a @ A2 @ A) => ((insert_a @ A2 @ (minus_minus_set_a @ A @ (insert_a @ A2 @ bot_bot_set_a))) = A))))). % insert_Diff
thf(fact_101_insert__Diff, axiom,
    ((![A2 : pname, A : set_pname]: ((member_pname @ A2 @ A) => ((insert_pname @ A2 @ (minus_1937938585_pname @ A @ (insert_pname @ A2 @ bot_bot_set_pname))) = A))))). % insert_Diff
thf(fact_102_Diff__insert, axiom,
    ((![A : set_a, A2 : a, B2 : set_a]: ((minus_minus_set_a @ A @ (insert_a @ A2 @ B2)) = (minus_minus_set_a @ (minus_minus_set_a @ A @ B2) @ (insert_a @ A2 @ bot_bot_set_a)))))). % Diff_insert
thf(fact_103_Diff__insert, axiom,
    ((![A : set_pname, A2 : pname, B2 : set_pname]: ((minus_1937938585_pname @ A @ (insert_pname @ A2 @ B2)) = (minus_1937938585_pname @ (minus_1937938585_pname @ A @ B2) @ (insert_pname @ A2 @ bot_bot_set_pname)))))). % Diff_insert
thf(fact_104_subset__Diff__insert, axiom,
    ((![A : set_a, B2 : set_a, X2 : a, C2 : set_a]: ((ord_less_eq_set_a @ A @ (minus_minus_set_a @ B2 @ (insert_a @ X2 @ C2))) = (((ord_less_eq_set_a @ A @ (minus_minus_set_a @ B2 @ C2))) & ((~ ((member_a @ X2 @ A))))))))). % subset_Diff_insert
thf(fact_105_finite__empty__induct, axiom,
    ((![A : set_a, P : set_a > $o]: ((finite_finite_a @ A) => ((P @ A) => ((![A3 : a, A4 : set_a]: ((finite_finite_a @ A4) => ((member_a @ A3 @ A4) => ((P @ A4) => (P @ (minus_minus_set_a @ A4 @ (insert_a @ A3 @ bot_bot_set_a))))))) => (P @ bot_bot_set_a))))))). % finite_empty_induct
thf(fact_106_finite__empty__induct, axiom,
    ((![A : set_pname, P : set_pname > $o]: ((finite_finite_pname @ A) => ((P @ A) => ((![A3 : pname, A4 : set_pname]: ((finite_finite_pname @ A4) => ((member_pname @ A3 @ A4) => ((P @ A4) => (P @ (minus_1937938585_pname @ A4 @ (insert_pname @ A3 @ bot_bot_set_pname))))))) => (P @ bot_bot_set_pname))))))). % finite_empty_induct
thf(fact_107_infinite__coinduct, axiom,
    ((![X4 : set_a > $o, A : set_a]: ((X4 @ A) => ((![A4 : set_a]: ((X4 @ A4) => (?[X5 : a]: ((member_a @ X5 @ A4) & ((X4 @ (minus_minus_set_a @ A4 @ (insert_a @ X5 @ bot_bot_set_a))) | (~ ((finite_finite_a @ (minus_minus_set_a @ A4 @ (insert_a @ X5 @ bot_bot_set_a)))))))))) => (~ ((finite_finite_a @ A)))))))). % infinite_coinduct
thf(fact_108_infinite__coinduct, axiom,
    ((![X4 : set_pname > $o, A : set_pname]: ((X4 @ A) => ((![A4 : set_pname]: ((X4 @ A4) => (?[X5 : pname]: ((member_pname @ X5 @ A4) & ((X4 @ (minus_1937938585_pname @ A4 @ (insert_pname @ X5 @ bot_bot_set_pname))) | (~ ((finite_finite_pname @ (minus_1937938585_pname @ A4 @ (insert_pname @ X5 @ bot_bot_set_pname)))))))))) => (~ ((finite_finite_pname @ A)))))))). % infinite_coinduct
thf(fact_109_infinite__remove, axiom,
    ((![S : set_a, A2 : a]: ((~ ((finite_finite_a @ S))) => (~ ((finite_finite_a @ (minus_minus_set_a @ S @ (insert_a @ A2 @ bot_bot_set_a))))))))). % infinite_remove
thf(fact_110_infinite__remove, axiom,
    ((![S : set_pname, A2 : pname]: ((~ ((finite_finite_pname @ S))) => (~ ((finite_finite_pname @ (minus_1937938585_pname @ S @ (insert_pname @ A2 @ bot_bot_set_pname))))))))). % infinite_remove
thf(fact_111_Diff__single__insert, axiom,
    ((![A : set_pname, X2 : pname, B2 : set_pname]: ((ord_le865024672_pname @ (minus_1937938585_pname @ A @ (insert_pname @ X2 @ bot_bot_set_pname)) @ B2) => (ord_le865024672_pname @ A @ (insert_pname @ X2 @ B2)))))). % Diff_single_insert
thf(fact_112_Diff__single__insert, axiom,
    ((![A : set_a, X2 : a, B2 : set_a]: ((ord_less_eq_set_a @ (minus_minus_set_a @ A @ (insert_a @ X2 @ bot_bot_set_a)) @ B2) => (ord_less_eq_set_a @ A @ (insert_a @ X2 @ B2)))))). % Diff_single_insert
thf(fact_113_subset__insert__iff, axiom,
    ((![A : set_pname, X2 : pname, B2 : set_pname]: ((ord_le865024672_pname @ A @ (insert_pname @ X2 @ B2)) = (((((member_pname @ X2 @ A)) => ((ord_le865024672_pname @ (minus_1937938585_pname @ A @ (insert_pname @ X2 @ bot_bot_set_pname)) @ B2)))) & ((((~ ((member_pname @ X2 @ A)))) => ((ord_le865024672_pname @ A @ B2))))))))). % subset_insert_iff
thf(fact_114_subset__insert__iff, axiom,
    ((![A : set_a, X2 : a, B2 : set_a]: ((ord_less_eq_set_a @ A @ (insert_a @ X2 @ B2)) = (((((member_a @ X2 @ A)) => ((ord_less_eq_set_a @ (minus_minus_set_a @ A @ (insert_a @ X2 @ bot_bot_set_a)) @ B2)))) & ((((~ ((member_a @ X2 @ A)))) => ((ord_less_eq_set_a @ A @ B2))))))))). % subset_insert_iff
thf(fact_115_card__le__sym__Diff, axiom,
    ((![A : set_pname, B2 : set_pname]: ((finite_finite_pname @ A) => ((finite_finite_pname @ B2) => ((ord_less_eq_nat @ (finite_card_pname @ A) @ (finite_card_pname @ B2)) => (ord_less_eq_nat @ (finite_card_pname @ (minus_1937938585_pname @ A @ B2)) @ (finite_card_pname @ (minus_1937938585_pname @ B2 @ A))))))))). % card_le_sym_Diff
thf(fact_116_card__le__sym__Diff, axiom,
    ((![A : set_a, B2 : set_a]: ((finite_finite_a @ A) => ((finite_finite_a @ B2) => ((ord_less_eq_nat @ (finite_card_a @ A) @ (finite_card_a @ B2)) => (ord_less_eq_nat @ (finite_card_a @ (minus_minus_set_a @ A @ B2)) @ (finite_card_a @ (minus_minus_set_a @ B2 @ A))))))))). % card_le_sym_Diff
thf(fact_117_finite__remove__induct, axiom,
    ((![B2 : set_pname, P : set_pname > $o]: ((finite_finite_pname @ B2) => ((P @ bot_bot_set_pname) => ((![A4 : set_pname]: ((finite_finite_pname @ A4) => ((~ ((A4 = bot_bot_set_pname))) => ((ord_le865024672_pname @ A4 @ B2) => ((![X5 : pname]: ((member_pname @ X5 @ A4) => (P @ (minus_1937938585_pname @ A4 @ (insert_pname @ X5 @ bot_bot_set_pname))))) => (P @ A4)))))) => (P @ B2))))))). % finite_remove_induct
thf(fact_118_finite__remove__induct, axiom,
    ((![B2 : set_a, P : set_a > $o]: ((finite_finite_a @ B2) => ((P @ bot_bot_set_a) => ((![A4 : set_a]: ((finite_finite_a @ A4) => ((~ ((A4 = bot_bot_set_a))) => ((ord_less_eq_set_a @ A4 @ B2) => ((![X5 : a]: ((member_a @ X5 @ A4) => (P @ (minus_minus_set_a @ A4 @ (insert_a @ X5 @ bot_bot_set_a))))) => (P @ A4)))))) => (P @ B2))))))). % finite_remove_induct
thf(fact_119_remove__induct, axiom,
    ((![P : set_pname > $o, B2 : set_pname]: ((P @ bot_bot_set_pname) => (((~ ((finite_finite_pname @ B2))) => (P @ B2)) => ((![A4 : set_pname]: ((finite_finite_pname @ A4) => ((~ ((A4 = bot_bot_set_pname))) => ((ord_le865024672_pname @ A4 @ B2) => ((![X5 : pname]: ((member_pname @ X5 @ A4) => (P @ (minus_1937938585_pname @ A4 @ (insert_pname @ X5 @ bot_bot_set_pname))))) => (P @ A4)))))) => (P @ B2))))))). % remove_induct
thf(fact_120_remove__induct, axiom,
    ((![P : set_a > $o, B2 : set_a]: ((P @ bot_bot_set_a) => (((~ ((finite_finite_a @ B2))) => (P @ B2)) => ((![A4 : set_a]: ((finite_finite_a @ A4) => ((~ ((A4 = bot_bot_set_a))) => ((ord_less_eq_set_a @ A4 @ B2) => ((![X5 : a]: ((member_a @ X5 @ A4) => (P @ (minus_minus_set_a @ A4 @ (insert_a @ X5 @ bot_bot_set_a))))) => (P @ A4)))))) => (P @ B2))))))). % remove_induct
thf(fact_121_card__Diff__subset, axiom,
    ((![B2 : set_pname, A : set_pname]: ((finite_finite_pname @ B2) => ((ord_le865024672_pname @ B2 @ A) => ((finite_card_pname @ (minus_1937938585_pname @ A @ B2)) = (minus_minus_nat @ (finite_card_pname @ A) @ (finite_card_pname @ B2)))))))). % card_Diff_subset
thf(fact_122_card__Diff__subset, axiom,
    ((![B2 : set_a, A : set_a]: ((finite_finite_a @ B2) => ((ord_less_eq_set_a @ B2 @ A) => ((finite_card_a @ (minus_minus_set_a @ A @ B2)) = (minus_minus_nat @ (finite_card_a @ A) @ (finite_card_a @ B2)))))))). % card_Diff_subset
thf(fact_123_diff__card__le__card__Diff, axiom,
    ((![B2 : set_pname, A : set_pname]: ((finite_finite_pname @ B2) => (ord_less_eq_nat @ (minus_minus_nat @ (finite_card_pname @ A) @ (finite_card_pname @ B2)) @ (finite_card_pname @ (minus_1937938585_pname @ A @ B2))))))). % diff_card_le_card_Diff
thf(fact_124_diff__card__le__card__Diff, axiom,
    ((![B2 : set_a, A : set_a]: ((finite_finite_a @ B2) => (ord_less_eq_nat @ (minus_minus_nat @ (finite_card_a @ A) @ (finite_card_a @ B2)) @ (finite_card_a @ (minus_minus_set_a @ A @ B2))))))). % diff_card_le_card_Diff
thf(fact_125_zero__reorient, axiom,
    ((![X2 : nat]: ((zero_zero_nat = X2) = (X2 = zero_zero_nat))))). % zero_reorient
thf(fact_126_rev__image__eqI, axiom,
    ((![X2 : pname, A : set_pname, B : a, F : pname > a]: ((member_pname @ X2 @ A) => ((B = (F @ X2)) => (member_a @ B @ (image_pname_a @ F @ A))))))). % rev_image_eqI
thf(fact_127_ball__imageD, axiom,
    ((![F : pname > a, A : set_pname, P : a > $o]: ((![X : a]: ((member_a @ X @ (image_pname_a @ F @ A)) => (P @ X))) => (![X5 : pname]: ((member_pname @ X5 @ A) => (P @ (F @ X5)))))))). % ball_imageD
thf(fact_128_image__cong, axiom,
    ((![M2 : set_pname, N2 : set_pname, F : pname > a, G2 : pname > a]: ((M2 = N2) => ((![X : pname]: ((member_pname @ X @ N2) => ((F @ X) = (G2 @ X)))) => ((image_pname_a @ F @ M2) = (image_pname_a @ G2 @ N2))))))). % image_cong
thf(fact_129_bex__imageD, axiom,
    ((![F : pname > a, A : set_pname, P : a > $o]: ((?[X5 : a]: ((member_a @ X5 @ (image_pname_a @ F @ A)) & (P @ X5))) => (?[X : pname]: ((member_pname @ X @ A) & (P @ (F @ X)))))))). % bex_imageD
thf(fact_130_image__iff, axiom,
    ((![Z : a, F : pname > a, A : set_pname]: ((member_a @ Z @ (image_pname_a @ F @ A)) = (?[X3 : pname]: (((member_pname @ X3 @ A)) & ((Z = (F @ X3))))))))). % image_iff
thf(fact_131_imageI, axiom,
    ((![X2 : pname, A : set_pname, F : pname > a]: ((member_pname @ X2 @ A) => (member_a @ (F @ X2) @ (image_pname_a @ F @ A)))))). % imageI
thf(fact_132_diff__right__commute, axiom,
    ((![A2 : nat, C : nat, B : nat]: ((minus_minus_nat @ (minus_minus_nat @ A2 @ C) @ B) = (minus_minus_nat @ (minus_minus_nat @ A2 @ B) @ C))))). % diff_right_commute
thf(fact_133_ex__in__conv, axiom,
    ((![A : set_a]: ((?[X3 : a]: (member_a @ X3 @ A)) = (~ ((A = bot_bot_set_a))))))). % ex_in_conv
thf(fact_134_ex__in__conv, axiom,
    ((![A : set_pname]: ((?[X3 : pname]: (member_pname @ X3 @ A)) = (~ ((A = bot_bot_set_pname))))))). % ex_in_conv
thf(fact_135_equals0I, axiom,
    ((![A : set_a]: ((![Y : a]: (~ ((member_a @ Y @ A)))) => (A = bot_bot_set_a))))). % equals0I
thf(fact_136_equals0I, axiom,
    ((![A : set_pname]: ((![Y : pname]: (~ ((member_pname @ Y @ A)))) => (A = bot_bot_set_pname))))). % equals0I
thf(fact_137_equals0D, axiom,
    ((![A : set_a, A2 : a]: ((A = bot_bot_set_a) => (~ ((member_a @ A2 @ A))))))). % equals0D
thf(fact_138_equals0D, axiom,
    ((![A : set_pname, A2 : pname]: ((A = bot_bot_set_pname) => (~ ((member_pname @ A2 @ A))))))). % equals0D
thf(fact_139_emptyE, axiom,
    ((![A2 : a]: (~ ((member_a @ A2 @ bot_bot_set_a)))))). % emptyE
thf(fact_140_emptyE, axiom,
    ((![A2 : pname]: (~ ((member_pname @ A2 @ bot_bot_set_pname)))))). % emptyE
thf(fact_141_Collect__mono__iff, axiom,
    ((![P : a > $o, Q : a > $o]: ((ord_less_eq_set_a @ (collect_a @ P) @ (collect_a @ Q)) = (![X3 : a]: (((P @ X3)) => ((Q @ X3)))))))). % Collect_mono_iff
thf(fact_142_set__eq__subset, axiom,
    (((^[Y2 : set_a]: (^[Z2 : set_a]: (Y2 = Z2))) = (^[A5 : set_a]: (^[B3 : set_a]: (((ord_less_eq_set_a @ A5 @ B3)) & ((ord_less_eq_set_a @ B3 @ A5)))))))). % set_eq_subset
thf(fact_143_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_144_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_145_subset__refl, axiom,
    ((![A : set_a]: (ord_less_eq_set_a @ A @ A)))). % subset_refl
thf(fact_146_subset__iff, axiom,
    ((ord_less_eq_set_a = (^[A5 : set_a]: (^[B3 : set_a]: (![T2 : a]: (((member_a @ T2 @ A5)) => ((member_a @ T2 @ B3))))))))). % subset_iff
thf(fact_147_equalityD2, axiom,
    ((![A : set_a, B2 : set_a]: ((A = B2) => (ord_less_eq_set_a @ B2 @ A))))). % equalityD2
thf(fact_148_equalityD1, axiom,
    ((![A : set_a, B2 : set_a]: ((A = B2) => (ord_less_eq_set_a @ A @ B2))))). % equalityD1
thf(fact_149_subset__eq, axiom,
    ((ord_less_eq_set_a = (^[A5 : set_a]: (^[B3 : set_a]: (![X3 : a]: (((member_a @ X3 @ A5)) => ((member_a @ X3 @ B3))))))))). % subset_eq
thf(fact_150_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_151_subsetD, axiom,
    ((![A : set_a, B2 : set_a, C : a]: ((ord_less_eq_set_a @ A @ B2) => ((member_a @ C @ A) => (member_a @ C @ B2)))))). % subsetD
thf(fact_152_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_153_mk__disjoint__insert, axiom,
    ((![A2 : a, A : set_a]: ((member_a @ A2 @ A) => (?[B4 : set_a]: ((A = (insert_a @ A2 @ B4)) & (~ ((member_a @ A2 @ B4))))))))). % mk_disjoint_insert
thf(fact_154_insert__commute, axiom,
    ((![X2 : a, Y3 : a, A : set_a]: ((insert_a @ X2 @ (insert_a @ Y3 @ A)) = (insert_a @ Y3 @ (insert_a @ X2 @ A)))))). % insert_commute
thf(fact_155_insert__eq__iff, axiom,
    ((![A2 : a, A : set_a, B : a, B2 : set_a]: ((~ ((member_a @ A2 @ A))) => ((~ ((member_a @ B @ B2))) => (((insert_a @ A2 @ A) = (insert_a @ B @ B2)) = (((((A2 = B)) => ((A = B2)))) & ((((~ ((A2 = B)))) => ((?[C4 : set_a]: (((A = (insert_a @ B @ C4))) & ((((~ ((member_a @ B @ C4)))) & ((((B2 = (insert_a @ A2 @ C4))) & ((~ ((member_a @ A2 @ C4)))))))))))))))))))). % insert_eq_iff
thf(fact_156_insert__absorb, axiom,
    ((![A2 : a, A : set_a]: ((member_a @ A2 @ A) => ((insert_a @ A2 @ A) = A))))). % insert_absorb
thf(fact_157_insert__ident, axiom,
    ((![X2 : a, A : set_a, B2 : set_a]: ((~ ((member_a @ X2 @ A))) => ((~ ((member_a @ X2 @ B2))) => (((insert_a @ X2 @ A) = (insert_a @ X2 @ B2)) = (A = B2))))))). % insert_ident
thf(fact_158_Set_Oset__insert, axiom,
    ((![X2 : a, A : set_a]: ((member_a @ X2 @ A) => (~ ((![B4 : set_a]: ((A = (insert_a @ X2 @ B4)) => (member_a @ X2 @ B4))))))))). % Set.set_insert
thf(fact_159_insertI2, axiom,
    ((![A2 : a, B2 : set_a, B : a]: ((member_a @ A2 @ B2) => (member_a @ A2 @ (insert_a @ B @ B2)))))). % insertI2
thf(fact_160_insertI1, axiom,
    ((![A2 : a, B2 : set_a]: (member_a @ A2 @ (insert_a @ A2 @ B2))))). % insertI1
thf(fact_161_insertE, axiom,
    ((![A2 : a, B : a, A : set_a]: ((member_a @ A2 @ (insert_a @ B @ A)) => ((~ ((A2 = B))) => (member_a @ A2 @ A)))))). % insertE
thf(fact_162_Nat_Oex__has__greatest__nat, axiom,
    ((![P : nat > $o, K : nat, B : nat]: ((P @ K) => ((![Y : nat]: ((P @ Y) => (ord_less_eq_nat @ Y @ B))) => (?[X : nat]: ((P @ X) & (![Y4 : nat]: ((P @ Y4) => (ord_less_eq_nat @ Y4 @ X)))))))))). % Nat.ex_has_greatest_nat
thf(fact_163_nat__le__linear, axiom,
    ((![M : nat, N : nat]: ((ord_less_eq_nat @ M @ N) | (ord_less_eq_nat @ N @ M))))). % nat_le_linear
thf(fact_164_le__antisym, axiom,
    ((![M : nat, N : nat]: ((ord_less_eq_nat @ M @ N) => ((ord_less_eq_nat @ N @ M) => (M = N)))))). % le_antisym
thf(fact_165_eq__imp__le, axiom,
    ((![M : nat, N : nat]: ((M = N) => (ord_less_eq_nat @ M @ N))))). % eq_imp_le
thf(fact_166_le__trans, axiom,
    ((![I : nat, J : nat, K : nat]: ((ord_less_eq_nat @ I @ J) => ((ord_less_eq_nat @ J @ K) => (ord_less_eq_nat @ I @ K)))))). % le_trans
thf(fact_167_le__refl, axiom,
    ((![N : nat]: (ord_less_eq_nat @ N @ N)))). % le_refl
thf(fact_168_diff__commute, axiom,
    ((![I : nat, J : nat, K : nat]: ((minus_minus_nat @ (minus_minus_nat @ I @ J) @ K) = (minus_minus_nat @ (minus_minus_nat @ I @ K) @ J))))). % diff_commute
thf(fact_169_card__Diff1__le, axiom,
    ((![A : set_a, X2 : a]: ((finite_finite_a @ A) => (ord_less_eq_nat @ (finite_card_a @ (minus_minus_set_a @ A @ (insert_a @ X2 @ bot_bot_set_a))) @ (finite_card_a @ A)))))). % card_Diff1_le
thf(fact_170_card__Diff1__le, axiom,
    ((![A : set_pname, X2 : pname]: ((finite_finite_pname @ A) => (ord_less_eq_nat @ (finite_card_pname @ (minus_1937938585_pname @ A @ (insert_pname @ X2 @ bot_bot_set_pname))) @ (finite_card_pname @ A)))))). % card_Diff1_le
thf(fact_171_zero__le, axiom,
    ((![X2 : nat]: (ord_less_eq_nat @ zero_zero_nat @ X2)))). % zero_le
thf(fact_172_finite__has__minimal2, axiom,
    ((![A : set_set_a, A2 : set_a]: ((finite_finite_set_a @ A) => ((member_set_a @ A2 @ A) => (?[X : set_a]: ((member_set_a @ X @ A) & ((ord_less_eq_set_a @ X @ A2) & (![Xa : set_a]: ((member_set_a @ Xa @ A) => ((ord_less_eq_set_a @ Xa @ X) => (X = Xa)))))))))))). % finite_has_minimal2
thf(fact_173_finite__has__minimal2, axiom,
    ((![A : set_nat, A2 : nat]: ((finite_finite_nat @ A) => ((member_nat @ A2 @ A) => (?[X : nat]: ((member_nat @ X @ A) & ((ord_less_eq_nat @ X @ A2) & (![Xa : nat]: ((member_nat @ Xa @ A) => ((ord_less_eq_nat @ Xa @ X) => (X = Xa)))))))))))). % finite_has_minimal2
thf(fact_174_finite__has__maximal2, axiom,
    ((![A : set_set_a, A2 : set_a]: ((finite_finite_set_a @ A) => ((member_set_a @ A2 @ A) => (?[X : set_a]: ((member_set_a @ X @ A) & ((ord_less_eq_set_a @ A2 @ X) & (![Xa : set_a]: ((member_set_a @ Xa @ A) => ((ord_less_eq_set_a @ X @ Xa) => (X = Xa)))))))))))). % finite_has_maximal2
thf(fact_175_finite__has__maximal2, axiom,
    ((![A : set_nat, A2 : nat]: ((finite_finite_nat @ A) => ((member_nat @ A2 @ A) => (?[X : nat]: ((member_nat @ X @ A) & ((ord_less_eq_nat @ A2 @ X) & (![Xa : nat]: ((member_nat @ Xa @ A) => ((ord_less_eq_nat @ X @ Xa) => (X = Xa)))))))))))). % finite_has_maximal2
thf(fact_176_infinite__imp__nonempty, axiom,
    ((![S : set_a]: ((~ ((finite_finite_a @ S))) => (~ ((S = bot_bot_set_a))))))). % infinite_imp_nonempty
thf(fact_177_infinite__imp__nonempty, axiom,
    ((![S : set_pname]: ((~ ((finite_finite_pname @ S))) => (~ ((S = bot_bot_set_pname))))))). % infinite_imp_nonempty
thf(fact_178_finite_OemptyI, axiom,
    ((finite_finite_a @ bot_bot_set_a))). % finite.emptyI
thf(fact_179_finite_OemptyI, axiom,
    ((finite_finite_pname @ bot_bot_set_pname))). % finite.emptyI
thf(fact_180_all__subset__image, axiom,
    ((![F : pname > a, A : set_pname, P : set_a > $o]: ((![B3 : set_a]: (((ord_less_eq_set_a @ B3 @ (image_pname_a @ F @ A))) => ((P @ B3)))) = (![B3 : set_pname]: (((ord_le865024672_pname @ B3 @ A)) => ((P @ (image_pname_a @ F @ B3))))))))). % all_subset_image
thf(fact_181_all__subset__image, axiom,
    ((![F : a > a, A : set_a, P : set_a > $o]: ((![B3 : set_a]: (((ord_less_eq_set_a @ B3 @ (image_a_a @ F @ A))) => ((P @ B3)))) = (![B3 : set_a]: (((ord_less_eq_set_a @ B3 @ A)) => ((P @ (image_a_a @ F @ B3))))))))). % all_subset_image
thf(fact_182_subset__image__iff, axiom,
    ((![B2 : set_a, F : pname > a, A : set_pname]: ((ord_less_eq_set_a @ B2 @ (image_pname_a @ F @ A)) = (?[AA : set_pname]: (((ord_le865024672_pname @ AA @ A)) & ((B2 = (image_pname_a @ F @ AA))))))))). % subset_image_iff
thf(fact_183_subset__image__iff, axiom,
    ((![B2 : set_a, F : a > a, A : set_a]: ((ord_less_eq_set_a @ B2 @ (image_a_a @ F @ A)) = (?[AA : set_a]: (((ord_less_eq_set_a @ AA @ A)) & ((B2 = (image_a_a @ F @ AA))))))))). % subset_image_iff
thf(fact_184_image__subset__iff, axiom,
    ((![F : pname > a, A : set_pname, B2 : set_a]: ((ord_less_eq_set_a @ (image_pname_a @ F @ A) @ B2) = (![X3 : pname]: (((member_pname @ X3 @ A)) => ((member_a @ (F @ X3) @ B2)))))))). % image_subset_iff
thf(fact_185_subset__imageE, axiom,
    ((![B2 : set_a, F : pname > a, A : set_pname]: ((ord_less_eq_set_a @ B2 @ (image_pname_a @ F @ A)) => (~ ((![C3 : set_pname]: ((ord_le865024672_pname @ C3 @ A) => (~ ((B2 = (image_pname_a @ F @ C3)))))))))))). % subset_imageE
thf(fact_186_subset__imageE, axiom,
    ((![B2 : set_a, F : a > a, A : set_a]: ((ord_less_eq_set_a @ B2 @ (image_a_a @ F @ A)) => (~ ((![C3 : set_a]: ((ord_less_eq_set_a @ C3 @ A) => (~ ((B2 = (image_a_a @ F @ C3)))))))))))). % subset_imageE
thf(fact_187_image__subsetI, axiom,
    ((![A : set_pname, F : pname > a, B2 : set_a]: ((![X : pname]: ((member_pname @ X @ A) => (member_a @ (F @ X) @ B2))) => (ord_less_eq_set_a @ (image_pname_a @ F @ A) @ B2))))). % image_subsetI
thf(fact_188_image__mono, axiom,
    ((![A : set_pname, B2 : set_pname, F : pname > a]: ((ord_le865024672_pname @ A @ B2) => (ord_less_eq_set_a @ (image_pname_a @ F @ A) @ (image_pname_a @ F @ B2)))))). % image_mono
thf(fact_189_image__mono, axiom,
    ((![A : set_a, B2 : set_a, F : a > a]: ((ord_less_eq_set_a @ A @ B2) => (ord_less_eq_set_a @ (image_a_a @ F @ A) @ (image_a_a @ F @ B2)))))). % image_mono
thf(fact_190_singleton__inject, axiom,
    ((![A2 : a, B : a]: (((insert_a @ A2 @ bot_bot_set_a) = (insert_a @ B @ bot_bot_set_a)) => (A2 = B))))). % singleton_inject
thf(fact_191_singleton__inject, axiom,
    ((![A2 : pname, B : pname]: (((insert_pname @ A2 @ bot_bot_set_pname) = (insert_pname @ B @ bot_bot_set_pname)) => (A2 = B))))). % singleton_inject
thf(fact_192_insert__not__empty, axiom,
    ((![A2 : a, A : set_a]: (~ (((insert_a @ A2 @ A) = bot_bot_set_a)))))). % insert_not_empty
thf(fact_193_insert__not__empty, axiom,
    ((![A2 : pname, A : set_pname]: (~ (((insert_pname @ A2 @ A) = bot_bot_set_pname)))))). % insert_not_empty
thf(fact_194_doubleton__eq__iff, axiom,
    ((![A2 : a, B : a, C : a, D2 : a]: (((insert_a @ A2 @ (insert_a @ B @ bot_bot_set_a)) = (insert_a @ C @ (insert_a @ D2 @ bot_bot_set_a))) = (((((A2 = C)) & ((B = D2)))) | ((((A2 = D2)) & ((B = C))))))))). % doubleton_eq_iff
thf(fact_195_doubleton__eq__iff, axiom,
    ((![A2 : pname, B : pname, C : pname, D2 : pname]: (((insert_pname @ A2 @ (insert_pname @ B @ bot_bot_set_pname)) = (insert_pname @ C @ (insert_pname @ D2 @ bot_bot_set_pname))) = (((((A2 = C)) & ((B = D2)))) | ((((A2 = D2)) & ((B = C))))))))). % doubleton_eq_iff
thf(fact_196_singleton__iff, axiom,
    ((![B : a, A2 : a]: ((member_a @ B @ (insert_a @ A2 @ bot_bot_set_a)) = (B = A2))))). % singleton_iff
thf(fact_197_singleton__iff, axiom,
    ((![B : pname, A2 : pname]: ((member_pname @ B @ (insert_pname @ A2 @ bot_bot_set_pname)) = (B = A2))))). % singleton_iff
thf(fact_198_singletonD, axiom,
    ((![B : a, A2 : a]: ((member_a @ B @ (insert_a @ A2 @ bot_bot_set_a)) => (B = A2))))). % singletonD
thf(fact_199_singletonD, axiom,
    ((![B : pname, A2 : pname]: ((member_pname @ B @ (insert_pname @ A2 @ bot_bot_set_pname)) => (B = A2))))). % singletonD
thf(fact_200_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_201_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_202_infinite__super, axiom,
    ((![S : set_pname, T : set_pname]: ((ord_le865024672_pname @ S @ T) => ((~ ((finite_finite_pname @ S))) => (~ ((finite_finite_pname @ T)))))))). % infinite_super
thf(fact_203_infinite__super, axiom,
    ((![S : set_a, T : set_a]: ((ord_less_eq_set_a @ S @ T) => ((~ ((finite_finite_a @ S))) => (~ ((finite_finite_a @ T)))))))). % infinite_super
thf(fact_204_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_205_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

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