% 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_389__3254028_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:14:24.595

% 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_It__Nat__Onat_J_J, type,
    set_set_nat : $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_tf__a, type,
    a : $tType).

% Explicit typings (40)
thf(sy_c_Finite__Set_Ocard_001t__Com__Opname, type,
    finite_card_pname : set_pname > nat).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat, type,
    finite_card_nat : set_nat > nat).
thf(sy_c_Finite__Set_Ocard_001tf__a, type,
    finite_card_a : set_a > nat).
thf(sy_c_Finite__Set_Ofinite_001t__Com__Opname, type,
    finite_finite_pname : set_pname > $o).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat, type,
    finite_finite_nat : set_nat > $o).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Com__Opname_J, type,
    finite505202775_pname : set_set_pname > $o).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J, type,
    finite2012248349et_nat : set_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_Nat_OSuc, type,
    suc : nat > nat).
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_It__Nat__Onat_J, type,
    ord_less_eq_set_nat : set_nat > set_nat > $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__Nat__Onat, type,
    collect_nat : (nat > $o) > set_nat).
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__Nat__Onat_J, type,
    collect_set_nat : (set_nat > $o) > set_set_nat).
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__Nat__Onat, type,
    image_pname_nat : (pname > nat) > set_pname > set_nat).
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__Nat__Onat_001t__Com__Opname, type,
    image_nat_pname : (nat > pname) > set_nat > set_pname).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat, type,
    image_nat_nat : (nat > nat) > set_nat > set_nat).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001tf__a, type,
    image_nat_a : (nat > a) > set_nat > 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__Nat__Onat, type,
    image_a_nat : (a > nat) > set_a > set_nat).
thf(sy_c_Set_Oimage_001tf__a_001tf__a, type,
    image_a_a : (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_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_mgt__call, type,
    mgt_call : pname > a).
thf(sy_v_na, type,
    na : nat).
thf(sy_v_pn, type,
    pn : pname).
thf(sy_v_uG, type,
    uG : set_a).

% Relevant facts (247)
thf(fact_0_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_1_diff__Suc__Suc, axiom,
    ((![M : nat, N : nat]: ((minus_minus_nat @ (suc @ M) @ (suc @ N)) = (minus_minus_nat @ M @ N))))). % diff_Suc_Suc
thf(fact_2_Suc__diff__diff, axiom,
    ((![M : nat, N : nat, K : nat]: ((minus_minus_nat @ (minus_minus_nat @ (suc @ M) @ N) @ (suc @ K)) = (minus_minus_nat @ (minus_minus_nat @ M @ N) @ K))))). % Suc_diff_diff
thf(fact_3_Suc__le__mono, axiom,
    ((![N : nat, M : nat]: ((ord_less_eq_nat @ (suc @ N) @ (suc @ M)) = (ord_less_eq_nat @ N @ M))))). % Suc_le_mono
thf(fact_4_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_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 > nat]: ((finite_finite_pname @ F) => (finite_finite_nat @ (image_pname_nat @ 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_a, H : a > nat]: ((finite_finite_a @ F) => (finite_finite_nat @ (image_a_nat @ H @ F)))))). % finite_imageI
thf(fact_10_finite__imageI, axiom,
    ((![F : set_nat, H : nat > pname]: ((finite_finite_nat @ F) => (finite_finite_pname @ (image_nat_pname @ H @ F)))))). % finite_imageI
thf(fact_11_finite__imageI, axiom,
    ((![F : set_nat, H : nat > a]: ((finite_finite_nat @ F) => (finite_finite_a @ (image_nat_a @ H @ F)))))). % finite_imageI
thf(fact_12_finite__imageI, axiom,
    ((![F : set_nat, H : nat > nat]: ((finite_finite_nat @ F) => (finite_finite_nat @ (image_nat_nat @ H @ F)))))). % finite_imageI
thf(fact_13_card__image__le, axiom,
    ((![A : set_pname, F2 : pname > pname]: ((finite_finite_pname @ A) => (ord_less_eq_nat @ (finite_card_pname @ (image_pname_pname @ F2 @ A)) @ (finite_card_pname @ A)))))). % card_image_le
thf(fact_14_card__image__le, axiom,
    ((![A : set_a, F2 : a > pname]: ((finite_finite_a @ A) => (ord_less_eq_nat @ (finite_card_pname @ (image_a_pname @ F2 @ A)) @ (finite_card_a @ A)))))). % card_image_le
thf(fact_15_card__image__le, axiom,
    ((![A : set_nat, F2 : nat > pname]: ((finite_finite_nat @ A) => (ord_less_eq_nat @ (finite_card_pname @ (image_nat_pname @ F2 @ A)) @ (finite_card_nat @ A)))))). % card_image_le
thf(fact_16_card__image__le, axiom,
    ((![A : set_pname, F2 : pname > a]: ((finite_finite_pname @ A) => (ord_less_eq_nat @ (finite_card_a @ (image_pname_a @ F2 @ A)) @ (finite_card_pname @ A)))))). % card_image_le
thf(fact_17_card__image__le, axiom,
    ((![A : set_pname, F2 : pname > nat]: ((finite_finite_pname @ A) => (ord_less_eq_nat @ (finite_card_nat @ (image_pname_nat @ F2 @ A)) @ (finite_card_pname @ A)))))). % card_image_le
thf(fact_18_card__image__le, axiom,
    ((![A : set_a, F2 : a > a]: ((finite_finite_a @ A) => (ord_less_eq_nat @ (finite_card_a @ (image_a_a @ F2 @ A)) @ (finite_card_a @ A)))))). % card_image_le
thf(fact_19_card__image__le, axiom,
    ((![A : set_a, F2 : a > nat]: ((finite_finite_a @ A) => (ord_less_eq_nat @ (finite_card_nat @ (image_a_nat @ F2 @ A)) @ (finite_card_a @ A)))))). % card_image_le
thf(fact_20_card__image__le, axiom,
    ((![A : set_nat, F2 : nat > a]: ((finite_finite_nat @ A) => (ord_less_eq_nat @ (finite_card_a @ (image_nat_a @ F2 @ A)) @ (finite_card_nat @ A)))))). % card_image_le
thf(fact_21_card__image__le, axiom,
    ((![A : set_nat, F2 : nat > nat]: ((finite_finite_nat @ A) => (ord_less_eq_nat @ (finite_card_nat @ (image_nat_nat @ F2 @ A)) @ (finite_card_nat @ A)))))). % card_image_le
thf(fact_22_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_23_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_24_finite__Collect__conjI, axiom,
    ((![P : nat > $o, Q : nat > $o]: (((finite_finite_nat @ (collect_nat @ P)) | (finite_finite_nat @ (collect_nat @ Q))) => (finite_finite_nat @ (collect_nat @ (^[X : nat]: (((P @ X)) & ((Q @ X)))))))))). % finite_Collect_conjI
thf(fact_25_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_26_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_27_finite__Collect__disjI, axiom,
    ((![P : nat > $o, Q : nat > $o]: ((finite_finite_nat @ (collect_nat @ (^[X : nat]: (((P @ X)) | ((Q @ X)))))) = (((finite_finite_nat @ (collect_nat @ P))) & ((finite_finite_nat @ (collect_nat @ Q)))))))). % finite_Collect_disjI
thf(fact_28_image__ident, axiom,
    ((![Y : set_nat]: ((image_nat_nat @ (^[X : nat]: X) @ Y) = Y)))). % image_ident
thf(fact_29_image__ident, axiom,
    ((![Y : set_a]: ((image_a_a @ (^[X : a]: X) @ Y) = Y)))). % image_ident
thf(fact_30_card__le__if__inj__on__rel, axiom,
    ((![B : set_pname, A : set_pname, R : pname > pname > $o]: ((finite_finite_pname @ B) => ((![A2 : pname]: ((member_pname @ A2 @ A) => (?[B2 : pname]: ((member_pname @ B2 @ B) & (R @ A2 @ B2))))) => ((![A1 : pname, A22 : pname, B3 : pname]: ((member_pname @ A1 @ A) => ((member_pname @ A22 @ A) => ((member_pname @ B3 @ B) => ((R @ A1 @ B3) => ((R @ A22 @ B3) => (A1 = A22))))))) => (ord_less_eq_nat @ (finite_card_pname @ A) @ (finite_card_pname @ B)))))))). % card_le_if_inj_on_rel
thf(fact_31_card__le__if__inj__on__rel, axiom,
    ((![B : set_pname, A : set_a, R : a > pname > $o]: ((finite_finite_pname @ B) => ((![A2 : a]: ((member_a @ A2 @ A) => (?[B2 : pname]: ((member_pname @ B2 @ B) & (R @ A2 @ B2))))) => ((![A1 : a, A22 : a, B3 : pname]: ((member_a @ A1 @ A) => ((member_a @ A22 @ A) => ((member_pname @ B3 @ B) => ((R @ A1 @ B3) => ((R @ A22 @ B3) => (A1 = A22))))))) => (ord_less_eq_nat @ (finite_card_a @ A) @ (finite_card_pname @ B)))))))). % card_le_if_inj_on_rel
thf(fact_32_card__le__if__inj__on__rel, axiom,
    ((![B : set_pname, A : set_nat, R : nat > pname > $o]: ((finite_finite_pname @ B) => ((![A2 : nat]: ((member_nat @ A2 @ A) => (?[B2 : pname]: ((member_pname @ B2 @ B) & (R @ A2 @ B2))))) => ((![A1 : nat, A22 : nat, B3 : pname]: ((member_nat @ A1 @ A) => ((member_nat @ A22 @ A) => ((member_pname @ B3 @ B) => ((R @ A1 @ B3) => ((R @ A22 @ B3) => (A1 = A22))))))) => (ord_less_eq_nat @ (finite_card_nat @ A) @ (finite_card_pname @ B)))))))). % card_le_if_inj_on_rel
thf(fact_33_card__le__if__inj__on__rel, axiom,
    ((![B : set_a, A : set_pname, R : pname > a > $o]: ((finite_finite_a @ B) => ((![A2 : pname]: ((member_pname @ A2 @ A) => (?[B2 : a]: ((member_a @ B2 @ B) & (R @ A2 @ B2))))) => ((![A1 : pname, A22 : pname, B3 : a]: ((member_pname @ A1 @ A) => ((member_pname @ A22 @ A) => ((member_a @ B3 @ B) => ((R @ A1 @ B3) => ((R @ A22 @ B3) => (A1 = A22))))))) => (ord_less_eq_nat @ (finite_card_pname @ A) @ (finite_card_a @ B)))))))). % card_le_if_inj_on_rel
thf(fact_34_card__le__if__inj__on__rel, axiom,
    ((![B : set_a, A : set_a, R : a > a > $o]: ((finite_finite_a @ B) => ((![A2 : a]: ((member_a @ A2 @ A) => (?[B2 : a]: ((member_a @ B2 @ B) & (R @ A2 @ B2))))) => ((![A1 : a, A22 : a, B3 : a]: ((member_a @ A1 @ A) => ((member_a @ A22 @ A) => ((member_a @ B3 @ B) => ((R @ A1 @ B3) => ((R @ A22 @ B3) => (A1 = A22))))))) => (ord_less_eq_nat @ (finite_card_a @ A) @ (finite_card_a @ B)))))))). % card_le_if_inj_on_rel
thf(fact_35_card__le__if__inj__on__rel, axiom,
    ((![B : set_a, A : set_nat, R : nat > a > $o]: ((finite_finite_a @ B) => ((![A2 : nat]: ((member_nat @ A2 @ A) => (?[B2 : a]: ((member_a @ B2 @ B) & (R @ A2 @ B2))))) => ((![A1 : nat, A22 : nat, B3 : a]: ((member_nat @ A1 @ A) => ((member_nat @ A22 @ A) => ((member_a @ B3 @ B) => ((R @ A1 @ B3) => ((R @ A22 @ B3) => (A1 = A22))))))) => (ord_less_eq_nat @ (finite_card_nat @ A) @ (finite_card_a @ B)))))))). % card_le_if_inj_on_rel
thf(fact_36_card__le__if__inj__on__rel, axiom,
    ((![B : set_nat, A : set_pname, R : pname > nat > $o]: ((finite_finite_nat @ B) => ((![A2 : pname]: ((member_pname @ A2 @ A) => (?[B2 : nat]: ((member_nat @ B2 @ B) & (R @ A2 @ B2))))) => ((![A1 : pname, A22 : pname, B3 : nat]: ((member_pname @ A1 @ A) => ((member_pname @ A22 @ A) => ((member_nat @ B3 @ B) => ((R @ A1 @ B3) => ((R @ A22 @ B3) => (A1 = A22))))))) => (ord_less_eq_nat @ (finite_card_pname @ A) @ (finite_card_nat @ B)))))))). % card_le_if_inj_on_rel
thf(fact_37_card__le__if__inj__on__rel, axiom,
    ((![B : set_nat, A : set_a, R : a > nat > $o]: ((finite_finite_nat @ B) => ((![A2 : a]: ((member_a @ A2 @ A) => (?[B2 : nat]: ((member_nat @ B2 @ B) & (R @ A2 @ B2))))) => ((![A1 : a, A22 : a, B3 : nat]: ((member_a @ A1 @ A) => ((member_a @ A22 @ A) => ((member_nat @ B3 @ B) => ((R @ A1 @ B3) => ((R @ A22 @ B3) => (A1 = A22))))))) => (ord_less_eq_nat @ (finite_card_a @ A) @ (finite_card_nat @ B)))))))). % card_le_if_inj_on_rel
thf(fact_38_card__le__if__inj__on__rel, axiom,
    ((![B : set_nat, A : set_nat, R : nat > nat > $o]: ((finite_finite_nat @ B) => ((![A2 : nat]: ((member_nat @ A2 @ A) => (?[B2 : nat]: ((member_nat @ B2 @ B) & (R @ A2 @ B2))))) => ((![A1 : nat, A22 : nat, B3 : nat]: ((member_nat @ A1 @ A) => ((member_nat @ A22 @ A) => ((member_nat @ B3 @ B) => ((R @ A1 @ B3) => ((R @ A22 @ B3) => (A1 = A22))))))) => (ord_less_eq_nat @ (finite_card_nat @ A) @ (finite_card_nat @ B)))))))). % card_le_if_inj_on_rel
thf(fact_39_Suc__diff__le, axiom,
    ((![N : nat, M : nat]: ((ord_less_eq_nat @ N @ M) => ((minus_minus_nat @ (suc @ M) @ N) = (suc @ (minus_minus_nat @ M @ N))))))). % Suc_diff_le
thf(fact_40_lift__Suc__mono__le, axiom,
    ((![F2 : nat > nat, N : nat, N2 : nat]: ((![N3 : nat]: (ord_less_eq_nat @ (F2 @ N3) @ (F2 @ (suc @ N3)))) => ((ord_less_eq_nat @ N @ N2) => (ord_less_eq_nat @ (F2 @ N) @ (F2 @ N2))))))). % lift_Suc_mono_le
thf(fact_41_lift__Suc__mono__le, axiom,
    ((![F2 : nat > set_a, N : nat, N2 : nat]: ((![N3 : nat]: (ord_less_eq_set_a @ (F2 @ N3) @ (F2 @ (suc @ N3)))) => ((ord_less_eq_nat @ N @ N2) => (ord_less_eq_set_a @ (F2 @ N) @ (F2 @ N2))))))). % lift_Suc_mono_le
thf(fact_42_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_43_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_44_image__eqI, axiom,
    ((![B4 : nat, F2 : pname > nat, X2 : pname, A : set_pname]: ((B4 = (F2 @ X2)) => ((member_pname @ X2 @ A) => (member_nat @ B4 @ (image_pname_nat @ F2 @ A))))))). % image_eqI
thf(fact_45_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_46_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_47_image__eqI, axiom,
    ((![B4 : nat, F2 : a > nat, X2 : a, A : set_a]: ((B4 = (F2 @ X2)) => ((member_a @ X2 @ A) => (member_nat @ B4 @ (image_a_nat @ F2 @ A))))))). % image_eqI
thf(fact_48_image__eqI, axiom,
    ((![B4 : pname, F2 : nat > pname, X2 : nat, A : set_nat]: ((B4 = (F2 @ X2)) => ((member_nat @ X2 @ A) => (member_pname @ B4 @ (image_nat_pname @ F2 @ A))))))). % image_eqI
thf(fact_49_image__eqI, axiom,
    ((![B4 : a, F2 : nat > a, X2 : nat, A : set_nat]: ((B4 = (F2 @ X2)) => ((member_nat @ X2 @ A) => (member_a @ B4 @ (image_nat_a @ F2 @ A))))))). % image_eqI
thf(fact_50_image__eqI, axiom,
    ((![B4 : nat, F2 : nat > nat, X2 : nat, A : set_nat]: ((B4 = (F2 @ X2)) => ((member_nat @ X2 @ A) => (member_nat @ B4 @ (image_nat_nat @ F2 @ A))))))). % image_eqI
thf(fact_51_old_Onat_Oinject, axiom,
    ((![Nat : nat, Nat2 : nat]: (((suc @ Nat) = (suc @ Nat2)) = (Nat = Nat2))))). % old.nat.inject
thf(fact_52_nat_Oinject, axiom,
    ((![X22 : nat, Y2 : nat]: (((suc @ X22) = (suc @ Y2)) = (X22 = Y2))))). % nat.inject
thf(fact_53_finite__Collect__subsets, axiom,
    ((![A : set_pname]: ((finite_finite_pname @ A) => (finite505202775_pname @ (collect_set_pname @ (^[B5 : set_pname]: (ord_le865024672_pname @ B5 @ A)))))))). % finite_Collect_subsets
thf(fact_54_finite__Collect__subsets, axiom,
    ((![A : set_nat]: ((finite_finite_nat @ A) => (finite2012248349et_nat @ (collect_set_nat @ (^[B5 : set_nat]: (ord_less_eq_set_nat @ B5 @ A)))))))). % finite_Collect_subsets
thf(fact_55_finite__Collect__subsets, axiom,
    ((![A : set_a]: ((finite_finite_a @ A) => (finite_finite_set_a @ (collect_set_a @ (^[B5 : set_a]: (ord_less_eq_set_a @ B5 @ A)))))))). % finite_Collect_subsets
thf(fact_56_finite__Collect__le__nat, axiom,
    ((![K : nat]: (finite_finite_nat @ (collect_nat @ (^[N4 : nat]: (ord_less_eq_nat @ N4 @ K))))))). % finite_Collect_le_nat
thf(fact_57_card__Collect__le__nat, axiom,
    ((![N : nat]: ((finite_card_nat @ (collect_nat @ (^[I2 : nat]: (ord_less_eq_nat @ I2 @ N)))) = (suc @ N))))). % card_Collect_le_nat
thf(fact_58_all__subset__image, axiom,
    ((![F2 : nat > nat, A : set_nat, P : set_nat > $o]: ((![B5 : set_nat]: (((ord_less_eq_set_nat @ B5 @ (image_nat_nat @ F2 @ A))) => ((P @ B5)))) = (![B5 : set_nat]: (((ord_less_eq_set_nat @ B5 @ A)) => ((P @ (image_nat_nat @ F2 @ B5))))))))). % all_subset_image
thf(fact_59_all__subset__image, axiom,
    ((![F2 : a > pname, A : set_a, P : set_pname > $o]: ((![B5 : set_pname]: (((ord_le865024672_pname @ B5 @ (image_a_pname @ F2 @ A))) => ((P @ B5)))) = (![B5 : set_a]: (((ord_less_eq_set_a @ B5 @ A)) => ((P @ (image_a_pname @ F2 @ B5))))))))). % all_subset_image
thf(fact_60_all__subset__image, axiom,
    ((![F2 : a > nat, A : set_a, P : set_nat > $o]: ((![B5 : set_nat]: (((ord_less_eq_set_nat @ B5 @ (image_a_nat @ F2 @ A))) => ((P @ B5)))) = (![B5 : set_a]: (((ord_less_eq_set_a @ B5 @ A)) => ((P @ (image_a_nat @ F2 @ B5))))))))). % all_subset_image
thf(fact_61_all__subset__image, axiom,
    ((![F2 : pname > a, A : set_pname, P : set_a > $o]: ((![B5 : set_a]: (((ord_less_eq_set_a @ B5 @ (image_pname_a @ F2 @ A))) => ((P @ B5)))) = (![B5 : set_pname]: (((ord_le865024672_pname @ B5 @ A)) => ((P @ (image_pname_a @ F2 @ B5))))))))). % all_subset_image
thf(fact_62_all__subset__image, axiom,
    ((![F2 : nat > a, A : set_nat, P : set_a > $o]: ((![B5 : set_a]: (((ord_less_eq_set_a @ B5 @ (image_nat_a @ F2 @ A))) => ((P @ B5)))) = (![B5 : set_nat]: (((ord_less_eq_set_nat @ B5 @ A)) => ((P @ (image_nat_a @ F2 @ B5))))))))). % all_subset_image
thf(fact_63_all__subset__image, axiom,
    ((![F2 : a > a, A : set_a, P : set_a > $o]: ((![B5 : set_a]: (((ord_less_eq_set_a @ B5 @ (image_a_a @ F2 @ A))) => ((P @ B5)))) = (![B5 : set_a]: (((ord_less_eq_set_a @ B5 @ A)) => ((P @ (image_a_a @ F2 @ B5))))))))). % all_subset_image
thf(fact_64_subset__image__iff, axiom,
    ((![B : set_nat, F2 : nat > nat, A : set_nat]: ((ord_less_eq_set_nat @ B @ (image_nat_nat @ F2 @ A)) = (?[AA : set_nat]: (((ord_less_eq_set_nat @ AA @ A)) & ((B = (image_nat_nat @ F2 @ AA))))))))). % subset_image_iff
thf(fact_65_subset__image__iff, axiom,
    ((![B : set_pname, F2 : a > pname, A : set_a]: ((ord_le865024672_pname @ B @ (image_a_pname @ F2 @ A)) = (?[AA : set_a]: (((ord_less_eq_set_a @ AA @ A)) & ((B = (image_a_pname @ F2 @ AA))))))))). % subset_image_iff
thf(fact_66_subset__image__iff, axiom,
    ((![B : set_nat, F2 : a > nat, A : set_a]: ((ord_less_eq_set_nat @ B @ (image_a_nat @ F2 @ A)) = (?[AA : set_a]: (((ord_less_eq_set_a @ AA @ A)) & ((B = (image_a_nat @ F2 @ AA))))))))). % subset_image_iff
thf(fact_67_subset__image__iff, axiom,
    ((![B : set_a, F2 : pname > a, A : set_pname]: ((ord_less_eq_set_a @ B @ (image_pname_a @ F2 @ A)) = (?[AA : set_pname]: (((ord_le865024672_pname @ AA @ A)) & ((B = (image_pname_a @ F2 @ AA))))))))). % subset_image_iff
thf(fact_68_subset__image__iff, axiom,
    ((![B : set_a, F2 : nat > a, A : set_nat]: ((ord_less_eq_set_a @ B @ (image_nat_a @ F2 @ A)) = (?[AA : set_nat]: (((ord_less_eq_set_nat @ AA @ A)) & ((B = (image_nat_a @ F2 @ AA))))))))). % subset_image_iff
thf(fact_69_subset__image__iff, axiom,
    ((![B : set_a, F2 : a > a, A : set_a]: ((ord_less_eq_set_a @ B @ (image_a_a @ F2 @ A)) = (?[AA : set_a]: (((ord_less_eq_set_a @ AA @ A)) & ((B = (image_a_a @ F2 @ AA))))))))). % subset_image_iff
thf(fact_70_image__subset__iff, axiom,
    ((![F2 : a > pname, A : set_a, B : set_pname]: ((ord_le865024672_pname @ (image_a_pname @ F2 @ A) @ B) = (![X : a]: (((member_a @ X @ A)) => ((member_pname @ (F2 @ X) @ B)))))))). % image_subset_iff
thf(fact_71_image__subset__iff, axiom,
    ((![F2 : nat > nat, A : set_nat, B : set_nat]: ((ord_less_eq_set_nat @ (image_nat_nat @ F2 @ A) @ B) = (![X : nat]: (((member_nat @ X @ A)) => ((member_nat @ (F2 @ X) @ B)))))))). % image_subset_iff
thf(fact_72_image__subset__iff, axiom,
    ((![F2 : a > nat, A : set_a, B : set_nat]: ((ord_less_eq_set_nat @ (image_a_nat @ F2 @ A) @ B) = (![X : a]: (((member_a @ X @ A)) => ((member_nat @ (F2 @ X) @ B)))))))). % image_subset_iff
thf(fact_73_image__subset__iff, axiom,
    ((![F2 : pname > a, A : set_pname, B : set_a]: ((ord_less_eq_set_a @ (image_pname_a @ F2 @ A) @ B) = (![X : pname]: (((member_pname @ X @ A)) => ((member_a @ (F2 @ X) @ B)))))))). % image_subset_iff
thf(fact_74_image__subset__iff, axiom,
    ((![F2 : nat > a, A : set_nat, B : set_a]: ((ord_less_eq_set_a @ (image_nat_a @ F2 @ A) @ B) = (![X : nat]: (((member_nat @ X @ A)) => ((member_a @ (F2 @ X) @ B)))))))). % image_subset_iff
thf(fact_75_image__subset__iff, axiom,
    ((![F2 : a > a, A : set_a, B : set_a]: ((ord_less_eq_set_a @ (image_a_a @ F2 @ A) @ B) = (![X : a]: (((member_a @ X @ A)) => ((member_a @ (F2 @ X) @ B)))))))). % image_subset_iff
thf(fact_76_subset__imageE, axiom,
    ((![B : set_nat, F2 : nat > nat, A : set_nat]: ((ord_less_eq_set_nat @ B @ (image_nat_nat @ F2 @ A)) => (~ ((![C : set_nat]: ((ord_less_eq_set_nat @ C @ A) => (~ ((B = (image_nat_nat @ F2 @ C)))))))))))). % subset_imageE
thf(fact_77_subset__imageE, axiom,
    ((![B : set_pname, F2 : a > pname, A : set_a]: ((ord_le865024672_pname @ B @ (image_a_pname @ F2 @ A)) => (~ ((![C : set_a]: ((ord_less_eq_set_a @ C @ A) => (~ ((B = (image_a_pname @ F2 @ C)))))))))))). % subset_imageE
thf(fact_78_subset__imageE, axiom,
    ((![B : set_nat, F2 : a > nat, A : set_a]: ((ord_less_eq_set_nat @ B @ (image_a_nat @ F2 @ A)) => (~ ((![C : set_a]: ((ord_less_eq_set_a @ C @ A) => (~ ((B = (image_a_nat @ F2 @ C)))))))))))). % subset_imageE
thf(fact_79_subset__imageE, axiom,
    ((![B : set_a, F2 : pname > a, A : set_pname]: ((ord_less_eq_set_a @ B @ (image_pname_a @ F2 @ A)) => (~ ((![C : set_pname]: ((ord_le865024672_pname @ C @ A) => (~ ((B = (image_pname_a @ F2 @ C)))))))))))). % subset_imageE
thf(fact_80_subset__imageE, axiom,
    ((![B : set_a, F2 : nat > a, A : set_nat]: ((ord_less_eq_set_a @ B @ (image_nat_a @ F2 @ A)) => (~ ((![C : set_nat]: ((ord_less_eq_set_nat @ C @ A) => (~ ((B = (image_nat_a @ F2 @ C)))))))))))). % subset_imageE
thf(fact_81_subset__imageE, axiom,
    ((![B : set_a, F2 : a > a, A : set_a]: ((ord_less_eq_set_a @ B @ (image_a_a @ F2 @ A)) => (~ ((![C : set_a]: ((ord_less_eq_set_a @ C @ A) => (~ ((B = (image_a_a @ F2 @ C)))))))))))). % subset_imageE
thf(fact_82_image__subsetI, axiom,
    ((![A : set_pname, F2 : pname > pname, B : set_pname]: ((![X3 : pname]: ((member_pname @ X3 @ A) => (member_pname @ (F2 @ X3) @ B))) => (ord_le865024672_pname @ (image_pname_pname @ F2 @ A) @ B))))). % image_subsetI
thf(fact_83_image__subsetI, axiom,
    ((![A : set_pname, F2 : pname > nat, B : set_nat]: ((![X3 : pname]: ((member_pname @ X3 @ A) => (member_nat @ (F2 @ X3) @ B))) => (ord_less_eq_set_nat @ (image_pname_nat @ F2 @ A) @ B))))). % image_subsetI
thf(fact_84_image__subsetI, axiom,
    ((![A : set_a, F2 : a > pname, B : set_pname]: ((![X3 : a]: ((member_a @ X3 @ A) => (member_pname @ (F2 @ X3) @ B))) => (ord_le865024672_pname @ (image_a_pname @ F2 @ A) @ B))))). % image_subsetI
thf(fact_85_image__subsetI, axiom,
    ((![A : set_a, F2 : a > nat, B : set_nat]: ((![X3 : a]: ((member_a @ X3 @ A) => (member_nat @ (F2 @ X3) @ B))) => (ord_less_eq_set_nat @ (image_a_nat @ F2 @ A) @ B))))). % image_subsetI
thf(fact_86_image__subsetI, axiom,
    ((![A : set_nat, F2 : nat > pname, B : set_pname]: ((![X3 : nat]: ((member_nat @ X3 @ A) => (member_pname @ (F2 @ X3) @ B))) => (ord_le865024672_pname @ (image_nat_pname @ F2 @ A) @ B))))). % image_subsetI
thf(fact_87_image__subsetI, axiom,
    ((![A : set_nat, F2 : nat > nat, B : set_nat]: ((![X3 : nat]: ((member_nat @ X3 @ A) => (member_nat @ (F2 @ X3) @ B))) => (ord_less_eq_set_nat @ (image_nat_nat @ F2 @ A) @ B))))). % image_subsetI
thf(fact_88_image__subsetI, axiom,
    ((![A : set_pname, F2 : pname > a, B : set_a]: ((![X3 : pname]: ((member_pname @ X3 @ A) => (member_a @ (F2 @ X3) @ B))) => (ord_less_eq_set_a @ (image_pname_a @ F2 @ A) @ B))))). % image_subsetI
thf(fact_89_image__subsetI, axiom,
    ((![A : set_a, F2 : a > a, B : set_a]: ((![X3 : a]: ((member_a @ X3 @ A) => (member_a @ (F2 @ X3) @ B))) => (ord_less_eq_set_a @ (image_a_a @ F2 @ A) @ B))))). % image_subsetI
thf(fact_90_image__subsetI, axiom,
    ((![A : set_nat, F2 : nat > a, B : set_a]: ((![X3 : nat]: ((member_nat @ X3 @ A) => (member_a @ (F2 @ X3) @ B))) => (ord_less_eq_set_a @ (image_nat_a @ F2 @ A) @ B))))). % image_subsetI
thf(fact_91_image__mono, axiom,
    ((![A : set_nat, B : set_nat, F2 : nat > nat]: ((ord_less_eq_set_nat @ A @ B) => (ord_less_eq_set_nat @ (image_nat_nat @ F2 @ A) @ (image_nat_nat @ F2 @ B)))))). % image_mono
thf(fact_92_image__mono, axiom,
    ((![A : set_pname, B : set_pname, F2 : pname > a]: ((ord_le865024672_pname @ A @ B) => (ord_less_eq_set_a @ (image_pname_a @ F2 @ A) @ (image_pname_a @ F2 @ B)))))). % image_mono
thf(fact_93_image__mono, axiom,
    ((![A : set_nat, B : set_nat, F2 : nat > a]: ((ord_less_eq_set_nat @ A @ B) => (ord_less_eq_set_a @ (image_nat_a @ F2 @ A) @ (image_nat_a @ F2 @ B)))))). % image_mono
thf(fact_94_image__mono, axiom,
    ((![A : set_a, B : set_a, F2 : a > pname]: ((ord_less_eq_set_a @ A @ B) => (ord_le865024672_pname @ (image_a_pname @ F2 @ A) @ (image_a_pname @ F2 @ B)))))). % image_mono
thf(fact_95_image__mono, axiom,
    ((![A : set_a, B : set_a, F2 : a > nat]: ((ord_less_eq_set_a @ A @ B) => (ord_less_eq_set_nat @ (image_a_nat @ F2 @ A) @ (image_a_nat @ F2 @ B)))))). % image_mono
thf(fact_96_image__mono, axiom,
    ((![A : set_a, B : set_a, F2 : a > a]: ((ord_less_eq_set_a @ A @ B) => (ord_less_eq_set_a @ (image_a_a @ F2 @ A) @ (image_a_a @ F2 @ B)))))). % image_mono
thf(fact_97_rev__finite__subset, axiom,
    ((![B : set_pname, A : set_pname]: ((finite_finite_pname @ B) => ((ord_le865024672_pname @ A @ B) => (finite_finite_pname @ A)))))). % rev_finite_subset
thf(fact_98_rev__finite__subset, axiom,
    ((![B : set_nat, A : set_nat]: ((finite_finite_nat @ B) => ((ord_less_eq_set_nat @ A @ B) => (finite_finite_nat @ A)))))). % rev_finite_subset
thf(fact_99_rev__finite__subset, axiom,
    ((![B : set_a, A : set_a]: ((finite_finite_a @ B) => ((ord_less_eq_set_a @ A @ B) => (finite_finite_a @ A)))))). % rev_finite_subset
thf(fact_100_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_101_infinite__super, axiom,
    ((![S : set_nat, T : set_nat]: ((ord_less_eq_set_nat @ S @ T) => ((~ ((finite_finite_nat @ S))) => (~ ((finite_finite_nat @ T)))))))). % infinite_super
thf(fact_102_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_103_finite__subset, axiom,
    ((![A : set_pname, B : set_pname]: ((ord_le865024672_pname @ A @ B) => ((finite_finite_pname @ B) => (finite_finite_pname @ A)))))). % finite_subset
thf(fact_104_finite__subset, axiom,
    ((![A : set_nat, B : set_nat]: ((ord_less_eq_set_nat @ A @ B) => ((finite_finite_nat @ B) => (finite_finite_nat @ A)))))). % finite_subset
thf(fact_105_finite__subset, axiom,
    ((![A : set_a, B : set_a]: ((ord_less_eq_set_a @ A @ B) => ((finite_finite_a @ B) => (finite_finite_a @ A)))))). % finite_subset
thf(fact_106_finite__nat__set__iff__bounded__le, axiom,
    ((finite_finite_nat = (^[N5 : set_nat]: (?[M2 : nat]: (![X : nat]: (((member_nat @ X @ N5)) => ((ord_less_eq_nat @ X @ M2))))))))). % finite_nat_set_iff_bounded_le
thf(fact_107_finite__less__ub, axiom,
    ((![F2 : nat > nat, U : nat]: ((![N3 : nat]: (ord_less_eq_nat @ N3 @ (F2 @ N3))) => (finite_finite_nat @ (collect_nat @ (^[N4 : nat]: (ord_less_eq_nat @ (F2 @ N4) @ U)))))))). % finite_less_ub
thf(fact_108_all__finite__subset__image, axiom,
    ((![F2 : pname > pname, A : set_pname, P : set_pname > $o]: ((![B5 : set_pname]: (((((finite_finite_pname @ B5)) & ((ord_le865024672_pname @ B5 @ (image_pname_pname @ F2 @ A))))) => ((P @ B5)))) = (![B5 : set_pname]: (((((finite_finite_pname @ B5)) & ((ord_le865024672_pname @ B5 @ A)))) => ((P @ (image_pname_pname @ F2 @ B5))))))))). % all_finite_subset_image
thf(fact_109_all__finite__subset__image, axiom,
    ((![F2 : nat > pname, A : set_nat, P : set_pname > $o]: ((![B5 : set_pname]: (((((finite_finite_pname @ B5)) & ((ord_le865024672_pname @ B5 @ (image_nat_pname @ F2 @ A))))) => ((P @ B5)))) = (![B5 : set_nat]: (((((finite_finite_nat @ B5)) & ((ord_less_eq_set_nat @ B5 @ A)))) => ((P @ (image_nat_pname @ F2 @ B5))))))))). % all_finite_subset_image
thf(fact_110_all__finite__subset__image, axiom,
    ((![F2 : pname > nat, A : set_pname, P : set_nat > $o]: ((![B5 : set_nat]: (((((finite_finite_nat @ B5)) & ((ord_less_eq_set_nat @ B5 @ (image_pname_nat @ F2 @ A))))) => ((P @ B5)))) = (![B5 : set_pname]: (((((finite_finite_pname @ B5)) & ((ord_le865024672_pname @ B5 @ A)))) => ((P @ (image_pname_nat @ F2 @ B5))))))))). % all_finite_subset_image
thf(fact_111_all__finite__subset__image, axiom,
    ((![F2 : nat > nat, A : set_nat, P : set_nat > $o]: ((![B5 : set_nat]: (((((finite_finite_nat @ B5)) & ((ord_less_eq_set_nat @ B5 @ (image_nat_nat @ F2 @ A))))) => ((P @ B5)))) = (![B5 : set_nat]: (((((finite_finite_nat @ B5)) & ((ord_less_eq_set_nat @ B5 @ A)))) => ((P @ (image_nat_nat @ F2 @ B5))))))))). % all_finite_subset_image
thf(fact_112_all__finite__subset__image, axiom,
    ((![F2 : a > pname, A : set_a, P : set_pname > $o]: ((![B5 : set_pname]: (((((finite_finite_pname @ B5)) & ((ord_le865024672_pname @ B5 @ (image_a_pname @ F2 @ A))))) => ((P @ B5)))) = (![B5 : set_a]: (((((finite_finite_a @ B5)) & ((ord_less_eq_set_a @ B5 @ A)))) => ((P @ (image_a_pname @ F2 @ B5))))))))). % all_finite_subset_image
thf(fact_113_all__finite__subset__image, axiom,
    ((![F2 : a > nat, A : set_a, P : set_nat > $o]: ((![B5 : set_nat]: (((((finite_finite_nat @ B5)) & ((ord_less_eq_set_nat @ B5 @ (image_a_nat @ F2 @ A))))) => ((P @ B5)))) = (![B5 : set_a]: (((((finite_finite_a @ B5)) & ((ord_less_eq_set_a @ B5 @ A)))) => ((P @ (image_a_nat @ F2 @ B5))))))))). % all_finite_subset_image
thf(fact_114_all__finite__subset__image, axiom,
    ((![F2 : pname > a, A : set_pname, P : set_a > $o]: ((![B5 : set_a]: (((((finite_finite_a @ B5)) & ((ord_less_eq_set_a @ B5 @ (image_pname_a @ F2 @ A))))) => ((P @ B5)))) = (![B5 : set_pname]: (((((finite_finite_pname @ B5)) & ((ord_le865024672_pname @ B5 @ A)))) => ((P @ (image_pname_a @ F2 @ B5))))))))). % all_finite_subset_image
thf(fact_115_all__finite__subset__image, axiom,
    ((![F2 : nat > a, A : set_nat, P : set_a > $o]: ((![B5 : set_a]: (((((finite_finite_a @ B5)) & ((ord_less_eq_set_a @ B5 @ (image_nat_a @ F2 @ A))))) => ((P @ B5)))) = (![B5 : set_nat]: (((((finite_finite_nat @ B5)) & ((ord_less_eq_set_nat @ B5 @ A)))) => ((P @ (image_nat_a @ F2 @ B5))))))))). % all_finite_subset_image
thf(fact_116_all__finite__subset__image, axiom,
    ((![F2 : a > a, A : set_a, P : set_a > $o]: ((![B5 : set_a]: (((((finite_finite_a @ B5)) & ((ord_less_eq_set_a @ B5 @ (image_a_a @ F2 @ A))))) => ((P @ B5)))) = (![B5 : set_a]: (((((finite_finite_a @ B5)) & ((ord_less_eq_set_a @ B5 @ A)))) => ((P @ (image_a_a @ F2 @ B5))))))))). % all_finite_subset_image
thf(fact_117_ex__finite__subset__image, axiom,
    ((![F2 : pname > pname, A : set_pname, P : set_pname > $o]: ((?[B5 : set_pname]: (((finite_finite_pname @ B5)) & ((((ord_le865024672_pname @ B5 @ (image_pname_pname @ F2 @ A))) & ((P @ B5)))))) = (?[B5 : set_pname]: (((finite_finite_pname @ B5)) & ((((ord_le865024672_pname @ B5 @ A)) & ((P @ (image_pname_pname @ F2 @ B5))))))))))). % ex_finite_subset_image
thf(fact_118_ex__finite__subset__image, axiom,
    ((![F2 : nat > pname, A : set_nat, P : set_pname > $o]: ((?[B5 : set_pname]: (((finite_finite_pname @ B5)) & ((((ord_le865024672_pname @ B5 @ (image_nat_pname @ F2 @ A))) & ((P @ B5)))))) = (?[B5 : set_nat]: (((finite_finite_nat @ B5)) & ((((ord_less_eq_set_nat @ B5 @ A)) & ((P @ (image_nat_pname @ F2 @ B5))))))))))). % ex_finite_subset_image
thf(fact_119_ex__finite__subset__image, axiom,
    ((![F2 : pname > nat, A : set_pname, P : set_nat > $o]: ((?[B5 : set_nat]: (((finite_finite_nat @ B5)) & ((((ord_less_eq_set_nat @ B5 @ (image_pname_nat @ F2 @ A))) & ((P @ B5)))))) = (?[B5 : set_pname]: (((finite_finite_pname @ B5)) & ((((ord_le865024672_pname @ B5 @ A)) & ((P @ (image_pname_nat @ F2 @ B5))))))))))). % ex_finite_subset_image
thf(fact_120_ex__finite__subset__image, axiom,
    ((![F2 : nat > nat, A : set_nat, P : set_nat > $o]: ((?[B5 : set_nat]: (((finite_finite_nat @ B5)) & ((((ord_less_eq_set_nat @ B5 @ (image_nat_nat @ F2 @ A))) & ((P @ B5)))))) = (?[B5 : set_nat]: (((finite_finite_nat @ B5)) & ((((ord_less_eq_set_nat @ B5 @ A)) & ((P @ (image_nat_nat @ F2 @ B5))))))))))). % ex_finite_subset_image
thf(fact_121_ex__finite__subset__image, axiom,
    ((![F2 : a > pname, A : set_a, P : set_pname > $o]: ((?[B5 : set_pname]: (((finite_finite_pname @ B5)) & ((((ord_le865024672_pname @ B5 @ (image_a_pname @ F2 @ A))) & ((P @ B5)))))) = (?[B5 : set_a]: (((finite_finite_a @ B5)) & ((((ord_less_eq_set_a @ B5 @ A)) & ((P @ (image_a_pname @ F2 @ B5))))))))))). % ex_finite_subset_image
thf(fact_122_ex__finite__subset__image, axiom,
    ((![F2 : a > nat, A : set_a, P : set_nat > $o]: ((?[B5 : set_nat]: (((finite_finite_nat @ B5)) & ((((ord_less_eq_set_nat @ B5 @ (image_a_nat @ F2 @ A))) & ((P @ B5)))))) = (?[B5 : set_a]: (((finite_finite_a @ B5)) & ((((ord_less_eq_set_a @ B5 @ A)) & ((P @ (image_a_nat @ F2 @ B5))))))))))). % ex_finite_subset_image
thf(fact_123_ex__finite__subset__image, axiom,
    ((![F2 : pname > a, A : set_pname, P : set_a > $o]: ((?[B5 : set_a]: (((finite_finite_a @ B5)) & ((((ord_less_eq_set_a @ B5 @ (image_pname_a @ F2 @ A))) & ((P @ B5)))))) = (?[B5 : set_pname]: (((finite_finite_pname @ B5)) & ((((ord_le865024672_pname @ B5 @ A)) & ((P @ (image_pname_a @ F2 @ B5))))))))))). % ex_finite_subset_image
thf(fact_124_ex__finite__subset__image, axiom,
    ((![F2 : nat > a, A : set_nat, P : set_a > $o]: ((?[B5 : set_a]: (((finite_finite_a @ B5)) & ((((ord_less_eq_set_a @ B5 @ (image_nat_a @ F2 @ A))) & ((P @ B5)))))) = (?[B5 : set_nat]: (((finite_finite_nat @ B5)) & ((((ord_less_eq_set_nat @ B5 @ A)) & ((P @ (image_nat_a @ F2 @ B5))))))))))). % ex_finite_subset_image
thf(fact_125_ex__finite__subset__image, axiom,
    ((![F2 : a > a, A : set_a, P : set_a > $o]: ((?[B5 : set_a]: (((finite_finite_a @ B5)) & ((((ord_less_eq_set_a @ B5 @ (image_a_a @ F2 @ A))) & ((P @ B5)))))) = (?[B5 : set_a]: (((finite_finite_a @ B5)) & ((((ord_less_eq_set_a @ B5 @ A)) & ((P @ (image_a_a @ F2 @ B5))))))))))). % ex_finite_subset_image
thf(fact_126_finite__subset__image, axiom,
    ((![B : set_pname, F2 : pname > pname, A : set_pname]: ((finite_finite_pname @ B) => ((ord_le865024672_pname @ B @ (image_pname_pname @ F2 @ A)) => (?[C : set_pname]: ((ord_le865024672_pname @ C @ A) & ((finite_finite_pname @ C) & (B = (image_pname_pname @ F2 @ C)))))))))). % finite_subset_image
thf(fact_127_finite__subset__image, axiom,
    ((![B : set_pname, F2 : nat > pname, A : set_nat]: ((finite_finite_pname @ B) => ((ord_le865024672_pname @ B @ (image_nat_pname @ F2 @ A)) => (?[C : set_nat]: ((ord_less_eq_set_nat @ C @ A) & ((finite_finite_nat @ C) & (B = (image_nat_pname @ F2 @ C)))))))))). % finite_subset_image
thf(fact_128_finite__subset__image, axiom,
    ((![B : set_nat, F2 : pname > nat, A : set_pname]: ((finite_finite_nat @ B) => ((ord_less_eq_set_nat @ B @ (image_pname_nat @ F2 @ A)) => (?[C : set_pname]: ((ord_le865024672_pname @ C @ A) & ((finite_finite_pname @ C) & (B = (image_pname_nat @ F2 @ C)))))))))). % finite_subset_image
thf(fact_129_finite__subset__image, axiom,
    ((![B : set_nat, F2 : nat > nat, A : set_nat]: ((finite_finite_nat @ B) => ((ord_less_eq_set_nat @ B @ (image_nat_nat @ F2 @ A)) => (?[C : set_nat]: ((ord_less_eq_set_nat @ C @ A) & ((finite_finite_nat @ C) & (B = (image_nat_nat @ F2 @ C)))))))))). % finite_subset_image
thf(fact_130_finite__subset__image, axiom,
    ((![B : set_pname, F2 : a > pname, A : set_a]: ((finite_finite_pname @ B) => ((ord_le865024672_pname @ B @ (image_a_pname @ F2 @ A)) => (?[C : set_a]: ((ord_less_eq_set_a @ C @ A) & ((finite_finite_a @ C) & (B = (image_a_pname @ F2 @ C)))))))))). % finite_subset_image
thf(fact_131_finite__subset__image, axiom,
    ((![B : set_nat, F2 : a > nat, A : set_a]: ((finite_finite_nat @ B) => ((ord_less_eq_set_nat @ B @ (image_a_nat @ F2 @ A)) => (?[C : set_a]: ((ord_less_eq_set_a @ C @ A) & ((finite_finite_a @ C) & (B = (image_a_nat @ F2 @ C)))))))))). % finite_subset_image
thf(fact_132_finite__subset__image, axiom,
    ((![B : set_a, F2 : pname > a, A : set_pname]: ((finite_finite_a @ B) => ((ord_less_eq_set_a @ B @ (image_pname_a @ F2 @ A)) => (?[C : set_pname]: ((ord_le865024672_pname @ C @ A) & ((finite_finite_pname @ C) & (B = (image_pname_a @ F2 @ C)))))))))). % finite_subset_image
thf(fact_133_finite__subset__image, axiom,
    ((![B : set_a, F2 : nat > a, A : set_nat]: ((finite_finite_a @ B) => ((ord_less_eq_set_a @ B @ (image_nat_a @ F2 @ A)) => (?[C : set_nat]: ((ord_less_eq_set_nat @ C @ A) & ((finite_finite_nat @ C) & (B = (image_nat_a @ F2 @ C)))))))))). % finite_subset_image
thf(fact_134_finite__subset__image, axiom,
    ((![B : set_a, F2 : a > a, A : set_a]: ((finite_finite_a @ B) => ((ord_less_eq_set_a @ B @ (image_a_a @ F2 @ A)) => (?[C : set_a]: ((ord_less_eq_set_a @ C @ A) & ((finite_finite_a @ C) & (B = (image_a_a @ F2 @ C)))))))))). % finite_subset_image
thf(fact_135_finite__surj, axiom,
    ((![A : set_pname, B : set_pname, F2 : pname > pname]: ((finite_finite_pname @ A) => ((ord_le865024672_pname @ B @ (image_pname_pname @ F2 @ A)) => (finite_finite_pname @ B)))))). % finite_surj
thf(fact_136_finite__surj, axiom,
    ((![A : set_pname, B : set_nat, F2 : pname > nat]: ((finite_finite_pname @ A) => ((ord_less_eq_set_nat @ B @ (image_pname_nat @ F2 @ A)) => (finite_finite_nat @ B)))))). % finite_surj
thf(fact_137_finite__surj, axiom,
    ((![A : set_a, B : set_pname, F2 : a > pname]: ((finite_finite_a @ A) => ((ord_le865024672_pname @ B @ (image_a_pname @ F2 @ A)) => (finite_finite_pname @ B)))))). % finite_surj
thf(fact_138_finite__surj, axiom,
    ((![A : set_a, B : set_nat, F2 : a > nat]: ((finite_finite_a @ A) => ((ord_less_eq_set_nat @ B @ (image_a_nat @ F2 @ A)) => (finite_finite_nat @ B)))))). % finite_surj
thf(fact_139_finite__surj, axiom,
    ((![A : set_nat, B : set_pname, F2 : nat > pname]: ((finite_finite_nat @ A) => ((ord_le865024672_pname @ B @ (image_nat_pname @ F2 @ A)) => (finite_finite_pname @ B)))))). % finite_surj
thf(fact_140_finite__surj, axiom,
    ((![A : set_nat, B : set_nat, F2 : nat > nat]: ((finite_finite_nat @ A) => ((ord_less_eq_set_nat @ B @ (image_nat_nat @ F2 @ A)) => (finite_finite_nat @ B)))))). % finite_surj
thf(fact_141_finite__surj, axiom,
    ((![A : set_pname, B : set_a, F2 : pname > a]: ((finite_finite_pname @ A) => ((ord_less_eq_set_a @ B @ (image_pname_a @ F2 @ A)) => (finite_finite_a @ B)))))). % finite_surj
thf(fact_142_finite__surj, axiom,
    ((![A : set_a, B : set_a, F2 : a > a]: ((finite_finite_a @ A) => ((ord_less_eq_set_a @ B @ (image_a_a @ F2 @ A)) => (finite_finite_a @ B)))))). % finite_surj
thf(fact_143_finite__surj, axiom,
    ((![A : set_nat, B : set_a, F2 : nat > a]: ((finite_finite_nat @ A) => ((ord_less_eq_set_a @ B @ (image_nat_a @ F2 @ A)) => (finite_finite_a @ B)))))). % finite_surj
thf(fact_144_infinite__arbitrarily__large, axiom,
    ((![A : set_pname, N : nat]: ((~ ((finite_finite_pname @ A))) => (?[B6 : set_pname]: ((finite_finite_pname @ B6) & (((finite_card_pname @ B6) = N) & (ord_le865024672_pname @ B6 @ A)))))))). % infinite_arbitrarily_large
thf(fact_145_infinite__arbitrarily__large, axiom,
    ((![A : set_nat, N : nat]: ((~ ((finite_finite_nat @ A))) => (?[B6 : set_nat]: ((finite_finite_nat @ B6) & (((finite_card_nat @ B6) = N) & (ord_less_eq_set_nat @ B6 @ A)))))))). % infinite_arbitrarily_large
thf(fact_146_infinite__arbitrarily__large, axiom,
    ((![A : set_a, N : nat]: ((~ ((finite_finite_a @ A))) => (?[B6 : set_a]: ((finite_finite_a @ B6) & (((finite_card_a @ B6) = N) & (ord_less_eq_set_a @ B6 @ A)))))))). % infinite_arbitrarily_large
thf(fact_147_card__subset__eq, axiom,
    ((![B : set_pname, A : set_pname]: ((finite_finite_pname @ B) => ((ord_le865024672_pname @ A @ B) => (((finite_card_pname @ A) = (finite_card_pname @ B)) => (A = B))))))). % card_subset_eq
thf(fact_148_card__subset__eq, axiom,
    ((![B : set_nat, A : set_nat]: ((finite_finite_nat @ B) => ((ord_less_eq_set_nat @ A @ B) => (((finite_card_nat @ A) = (finite_card_nat @ B)) => (A = B))))))). % card_subset_eq
thf(fact_149_card__subset__eq, axiom,
    ((![B : set_a, A : set_a]: ((finite_finite_a @ B) => ((ord_less_eq_set_a @ A @ B) => (((finite_card_a @ A) = (finite_card_a @ B)) => (A = B))))))). % card_subset_eq
thf(fact_150_finite__if__finite__subsets__card__bdd, axiom,
    ((![F : set_pname, C2 : nat]: ((![G : set_pname]: ((ord_le865024672_pname @ G @ F) => ((finite_finite_pname @ G) => (ord_less_eq_nat @ (finite_card_pname @ G) @ C2)))) => ((finite_finite_pname @ F) & (ord_less_eq_nat @ (finite_card_pname @ F) @ C2)))))). % finite_if_finite_subsets_card_bdd
thf(fact_151_finite__if__finite__subsets__card__bdd, axiom,
    ((![F : set_nat, C2 : nat]: ((![G : set_nat]: ((ord_less_eq_set_nat @ G @ F) => ((finite_finite_nat @ G) => (ord_less_eq_nat @ (finite_card_nat @ G) @ C2)))) => ((finite_finite_nat @ F) & (ord_less_eq_nat @ (finite_card_nat @ F) @ C2)))))). % finite_if_finite_subsets_card_bdd
thf(fact_152_finite__if__finite__subsets__card__bdd, axiom,
    ((![F : set_a, C2 : nat]: ((![G : set_a]: ((ord_less_eq_set_a @ G @ F) => ((finite_finite_a @ G) => (ord_less_eq_nat @ (finite_card_a @ G) @ C2)))) => ((finite_finite_a @ F) & (ord_less_eq_nat @ (finite_card_a @ F) @ C2)))))). % finite_if_finite_subsets_card_bdd
thf(fact_153_obtain__subset__with__card__n, axiom,
    ((![N : nat, S : set_pname]: ((ord_less_eq_nat @ N @ (finite_card_pname @ S)) => (~ ((![T2 : set_pname]: ((ord_le865024672_pname @ T2 @ S) => (((finite_card_pname @ T2) = N) => (~ ((finite_finite_pname @ T2)))))))))))). % obtain_subset_with_card_n
thf(fact_154_obtain__subset__with__card__n, axiom,
    ((![N : nat, S : set_nat]: ((ord_less_eq_nat @ N @ (finite_card_nat @ S)) => (~ ((![T2 : set_nat]: ((ord_less_eq_set_nat @ T2 @ S) => (((finite_card_nat @ T2) = N) => (~ ((finite_finite_nat @ T2)))))))))))). % obtain_subset_with_card_n
thf(fact_155_obtain__subset__with__card__n, axiom,
    ((![N : nat, S : set_a]: ((ord_less_eq_nat @ N @ (finite_card_a @ S)) => (~ ((![T2 : set_a]: ((ord_less_eq_set_a @ T2 @ S) => (((finite_card_a @ T2) = N) => (~ ((finite_finite_a @ T2)))))))))))). % obtain_subset_with_card_n
thf(fact_156_card__seteq, axiom,
    ((![B : set_pname, A : set_pname]: ((finite_finite_pname @ B) => ((ord_le865024672_pname @ A @ B) => ((ord_less_eq_nat @ (finite_card_pname @ B) @ (finite_card_pname @ A)) => (A = B))))))). % card_seteq
thf(fact_157_card__seteq, axiom,
    ((![B : set_nat, A : set_nat]: ((finite_finite_nat @ B) => ((ord_less_eq_set_nat @ A @ B) => ((ord_less_eq_nat @ (finite_card_nat @ B) @ (finite_card_nat @ A)) => (A = B))))))). % card_seteq
thf(fact_158_card__seteq, axiom,
    ((![B : set_a, A : set_a]: ((finite_finite_a @ B) => ((ord_less_eq_set_a @ A @ B) => ((ord_less_eq_nat @ (finite_card_a @ B) @ (finite_card_a @ A)) => (A = B))))))). % card_seteq
thf(fact_159_card__mono, axiom,
    ((![B : set_pname, A : set_pname]: ((finite_finite_pname @ B) => ((ord_le865024672_pname @ A @ B) => (ord_less_eq_nat @ (finite_card_pname @ A) @ (finite_card_pname @ B))))))). % card_mono
thf(fact_160_card__mono, axiom,
    ((![B : set_nat, A : set_nat]: ((finite_finite_nat @ B) => ((ord_less_eq_set_nat @ A @ B) => (ord_less_eq_nat @ (finite_card_nat @ A) @ (finite_card_nat @ B))))))). % card_mono
thf(fact_161_card__mono, axiom,
    ((![B : set_a, A : set_a]: ((finite_finite_a @ B) => ((ord_less_eq_set_a @ A @ B) => (ord_less_eq_nat @ (finite_card_a @ A) @ (finite_card_a @ B))))))). % card_mono
thf(fact_162_surj__card__le, axiom,
    ((![A : set_pname, B : set_nat, F2 : pname > nat]: ((finite_finite_pname @ A) => ((ord_less_eq_set_nat @ B @ (image_pname_nat @ F2 @ A)) => (ord_less_eq_nat @ (finite_card_nat @ B) @ (finite_card_pname @ A))))))). % surj_card_le
thf(fact_163_surj__card__le, axiom,
    ((![A : set_pname, B : set_pname, F2 : pname > pname]: ((finite_finite_pname @ A) => ((ord_le865024672_pname @ B @ (image_pname_pname @ F2 @ A)) => (ord_less_eq_nat @ (finite_card_pname @ B) @ (finite_card_pname @ A))))))). % surj_card_le
thf(fact_164_surj__card__le, axiom,
    ((![A : set_a, B : set_nat, F2 : a > nat]: ((finite_finite_a @ A) => ((ord_less_eq_set_nat @ B @ (image_a_nat @ F2 @ A)) => (ord_less_eq_nat @ (finite_card_nat @ B) @ (finite_card_a @ A))))))). % surj_card_le
thf(fact_165_surj__card__le, axiom,
    ((![A : set_a, B : set_pname, F2 : a > pname]: ((finite_finite_a @ A) => ((ord_le865024672_pname @ B @ (image_a_pname @ F2 @ A)) => (ord_less_eq_nat @ (finite_card_pname @ B) @ (finite_card_a @ A))))))). % surj_card_le
thf(fact_166_surj__card__le, axiom,
    ((![A : set_nat, B : set_nat, F2 : nat > nat]: ((finite_finite_nat @ A) => ((ord_less_eq_set_nat @ B @ (image_nat_nat @ F2 @ A)) => (ord_less_eq_nat @ (finite_card_nat @ B) @ (finite_card_nat @ A))))))). % surj_card_le
thf(fact_167_surj__card__le, axiom,
    ((![A : set_nat, B : set_pname, F2 : nat > pname]: ((finite_finite_nat @ A) => ((ord_le865024672_pname @ B @ (image_nat_pname @ F2 @ A)) => (ord_less_eq_nat @ (finite_card_pname @ B) @ (finite_card_nat @ A))))))). % surj_card_le
thf(fact_168_surj__card__le, axiom,
    ((![A : set_pname, B : set_a, F2 : pname > a]: ((finite_finite_pname @ A) => ((ord_less_eq_set_a @ B @ (image_pname_a @ F2 @ A)) => (ord_less_eq_nat @ (finite_card_a @ B) @ (finite_card_pname @ A))))))). % surj_card_le
thf(fact_169_surj__card__le, axiom,
    ((![A : set_a, B : set_a, F2 : a > a]: ((finite_finite_a @ A) => ((ord_less_eq_set_a @ B @ (image_a_a @ F2 @ A)) => (ord_less_eq_nat @ (finite_card_a @ B) @ (finite_card_a @ A))))))). % surj_card_le
thf(fact_170_surj__card__le, axiom,
    ((![A : set_nat, B : set_a, F2 : nat > a]: ((finite_finite_nat @ A) => ((ord_less_eq_set_a @ B @ (image_nat_a @ F2 @ A)) => (ord_less_eq_nat @ (finite_card_a @ B) @ (finite_card_nat @ A))))))). % surj_card_le
thf(fact_171_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_172_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_173_rev__image__eqI, axiom,
    ((![X2 : pname, A : set_pname, B4 : nat, F2 : pname > nat]: ((member_pname @ X2 @ A) => ((B4 = (F2 @ X2)) => (member_nat @ B4 @ (image_pname_nat @ F2 @ A))))))). % rev_image_eqI
thf(fact_174_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_175_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_176_rev__image__eqI, axiom,
    ((![X2 : a, A : set_a, B4 : nat, F2 : a > nat]: ((member_a @ X2 @ A) => ((B4 = (F2 @ X2)) => (member_nat @ B4 @ (image_a_nat @ F2 @ A))))))). % rev_image_eqI
thf(fact_177_rev__image__eqI, axiom,
    ((![X2 : nat, A : set_nat, B4 : pname, F2 : nat > pname]: ((member_nat @ X2 @ A) => ((B4 = (F2 @ X2)) => (member_pname @ B4 @ (image_nat_pname @ F2 @ A))))))). % rev_image_eqI
thf(fact_178_rev__image__eqI, axiom,
    ((![X2 : nat, A : set_nat, B4 : a, F2 : nat > a]: ((member_nat @ X2 @ A) => ((B4 = (F2 @ X2)) => (member_a @ B4 @ (image_nat_a @ F2 @ A))))))). % rev_image_eqI
thf(fact_179_rev__image__eqI, axiom,
    ((![X2 : nat, A : set_nat, B4 : nat, F2 : nat > nat]: ((member_nat @ X2 @ A) => ((B4 = (F2 @ X2)) => (member_nat @ B4 @ (image_nat_nat @ F2 @ A))))))). % rev_image_eqI
thf(fact_180_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_181_ball__imageD, axiom,
    ((![F2 : nat > nat, A : set_nat, P : nat > $o]: ((![X3 : nat]: ((member_nat @ X3 @ (image_nat_nat @ F2 @ A)) => (P @ X3))) => (![X4 : nat]: ((member_nat @ X4 @ A) => (P @ (F2 @ X4)))))))). % ball_imageD
thf(fact_182_ball__imageD, axiom,
    ((![F2 : nat > a, A : set_nat, P : a > $o]: ((![X3 : a]: ((member_a @ X3 @ (image_nat_a @ F2 @ A)) => (P @ X3))) => (![X4 : nat]: ((member_nat @ X4 @ A) => (P @ (F2 @ X4)))))))). % ball_imageD
thf(fact_183_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_184_ball__imageD, axiom,
    ((![F2 : a > nat, A : set_a, P : nat > $o]: ((![X3 : nat]: ((member_nat @ X3 @ (image_a_nat @ F2 @ A)) => (P @ X3))) => (![X4 : a]: ((member_a @ X4 @ A) => (P @ (F2 @ X4)))))))). % ball_imageD
thf(fact_185_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_186_image__cong, axiom,
    ((![M3 : set_pname, N6 : set_pname, F2 : pname > a, G2 : pname > a]: ((M3 = N6) => ((![X3 : pname]: ((member_pname @ X3 @ N6) => ((F2 @ X3) = (G2 @ X3)))) => ((image_pname_a @ F2 @ M3) = (image_pname_a @ G2 @ N6))))))). % image_cong
thf(fact_187_image__cong, axiom,
    ((![M3 : set_a, N6 : set_a, F2 : a > pname, G2 : a > pname]: ((M3 = N6) => ((![X3 : a]: ((member_a @ X3 @ N6) => ((F2 @ X3) = (G2 @ X3)))) => ((image_a_pname @ F2 @ M3) = (image_a_pname @ G2 @ N6))))))). % image_cong
thf(fact_188_image__cong, axiom,
    ((![M3 : set_a, N6 : set_a, F2 : a > nat, G2 : a > nat]: ((M3 = N6) => ((![X3 : a]: ((member_a @ X3 @ N6) => ((F2 @ X3) = (G2 @ X3)))) => ((image_a_nat @ F2 @ M3) = (image_a_nat @ G2 @ N6))))))). % image_cong
thf(fact_189_image__cong, axiom,
    ((![M3 : set_a, N6 : set_a, F2 : a > a, G2 : a > a]: ((M3 = N6) => ((![X3 : a]: ((member_a @ X3 @ N6) => ((F2 @ X3) = (G2 @ X3)))) => ((image_a_a @ F2 @ M3) = (image_a_a @ G2 @ N6))))))). % image_cong
thf(fact_190_image__cong, axiom,
    ((![M3 : set_nat, N6 : set_nat, F2 : nat > nat, G2 : nat > nat]: ((M3 = N6) => ((![X3 : nat]: ((member_nat @ X3 @ N6) => ((F2 @ X3) = (G2 @ X3)))) => ((image_nat_nat @ F2 @ M3) = (image_nat_nat @ G2 @ N6))))))). % image_cong
thf(fact_191_image__cong, axiom,
    ((![M3 : set_nat, N6 : set_nat, F2 : nat > a, G2 : nat > a]: ((M3 = N6) => ((![X3 : nat]: ((member_nat @ X3 @ N6) => ((F2 @ X3) = (G2 @ X3)))) => ((image_nat_a @ F2 @ M3) = (image_nat_a @ G2 @ N6))))))). % image_cong
thf(fact_192_mem__Collect__eq, axiom,
    ((![A3 : nat, P : nat > $o]: ((member_nat @ A3 @ (collect_nat @ P)) = (P @ A3))))). % mem_Collect_eq
thf(fact_193_mem__Collect__eq, axiom,
    ((![A3 : a, P : a > $o]: ((member_a @ A3 @ (collect_a @ P)) = (P @ A3))))). % mem_Collect_eq
thf(fact_194_mem__Collect__eq, axiom,
    ((![A3 : pname, P : pname > $o]: ((member_pname @ A3 @ (collect_pname @ P)) = (P @ A3))))). % mem_Collect_eq
thf(fact_195_Collect__mem__eq, axiom,
    ((![A : set_nat]: ((collect_nat @ (^[X : nat]: (member_nat @ X @ A))) = A)))). % Collect_mem_eq
thf(fact_196_Collect__mem__eq, axiom,
    ((![A : set_a]: ((collect_a @ (^[X : a]: (member_a @ X @ A))) = A)))). % Collect_mem_eq
thf(fact_197_Collect__mem__eq, axiom,
    ((![A : set_pname]: ((collect_pname @ (^[X : pname]: (member_pname @ X @ A))) = A)))). % Collect_mem_eq
thf(fact_198_Collect__cong, axiom,
    ((![P : nat > $o, Q : nat > $o]: ((![X3 : nat]: ((P @ X3) = (Q @ X3))) => ((collect_nat @ P) = (collect_nat @ Q)))))). % Collect_cong
thf(fact_199_Collect__cong, axiom,
    ((![P : a > $o, Q : a > $o]: ((![X3 : a]: ((P @ X3) = (Q @ X3))) => ((collect_a @ P) = (collect_a @ Q)))))). % Collect_cong
thf(fact_200_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_201_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_202_bex__imageD, axiom,
    ((![F2 : nat > nat, A : set_nat, P : nat > $o]: ((?[X4 : nat]: ((member_nat @ X4 @ (image_nat_nat @ F2 @ A)) & (P @ X4))) => (?[X3 : nat]: ((member_nat @ X3 @ A) & (P @ (F2 @ X3)))))))). % bex_imageD
thf(fact_203_bex__imageD, axiom,
    ((![F2 : nat > a, A : set_nat, P : a > $o]: ((?[X4 : a]: ((member_a @ X4 @ (image_nat_a @ F2 @ A)) & (P @ X4))) => (?[X3 : nat]: ((member_nat @ X3 @ A) & (P @ (F2 @ X3)))))))). % bex_imageD
thf(fact_204_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_205_bex__imageD, axiom,
    ((![F2 : a > nat, A : set_a, P : nat > $o]: ((?[X4 : nat]: ((member_nat @ X4 @ (image_a_nat @ F2 @ A)) & (P @ X4))) => (?[X3 : a]: ((member_a @ X3 @ A) & (P @ (F2 @ X3)))))))). % bex_imageD
thf(fact_206_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_207_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_208_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_209_image__iff, axiom,
    ((![Z : a, F2 : nat > a, A : set_nat]: ((member_a @ Z @ (image_nat_a @ F2 @ A)) = (?[X : nat]: (((member_nat @ X @ A)) & ((Z = (F2 @ X))))))))). % image_iff
thf(fact_210_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_211_image__iff, axiom,
    ((![Z : nat, F2 : nat > nat, A : set_nat]: ((member_nat @ Z @ (image_nat_nat @ F2 @ A)) = (?[X : nat]: (((member_nat @ X @ A)) & ((Z = (F2 @ X))))))))). % image_iff
thf(fact_212_image__iff, axiom,
    ((![Z : nat, F2 : a > nat, A : set_a]: ((member_nat @ Z @ (image_a_nat @ F2 @ A)) = (?[X : a]: (((member_a @ X @ A)) & ((Z = (F2 @ X))))))))). % image_iff
thf(fact_213_imageI, axiom,
    ((![X2 : a, A : set_a, F2 : a > nat]: ((member_a @ X2 @ A) => (member_nat @ (F2 @ X2) @ (image_a_nat @ F2 @ A)))))). % imageI
thf(fact_214_imageI, axiom,
    ((![X2 : nat, A : set_nat, F2 : nat > pname]: ((member_nat @ X2 @ A) => (member_pname @ (F2 @ X2) @ (image_nat_pname @ F2 @ A)))))). % imageI
thf(fact_215_imageI, axiom,
    ((![X2 : nat, A : set_nat, F2 : nat > a]: ((member_nat @ X2 @ A) => (member_a @ (F2 @ X2) @ (image_nat_a @ F2 @ A)))))). % imageI
thf(fact_216_imageI, axiom,
    ((![X2 : nat, A : set_nat, F2 : nat > nat]: ((member_nat @ X2 @ A) => (member_nat @ (F2 @ X2) @ (image_nat_nat @ F2 @ A)))))). % imageI
thf(fact_217_n__not__Suc__n, axiom,
    ((![N : nat]: (~ ((N = (suc @ N))))))). % n_not_Suc_n
thf(fact_218_Suc__inject, axiom,
    ((![X2 : nat, Y3 : nat]: (((suc @ X2) = (suc @ Y3)) => (X2 = Y3))))). % Suc_inject
thf(fact_219_bounded__Max__nat, axiom,
    ((![P : nat > $o, X2 : nat, M3 : nat]: ((P @ X2) => ((![X3 : nat]: ((P @ X3) => (ord_less_eq_nat @ X3 @ M3))) => (~ ((![M4 : nat]: ((P @ M4) => (~ ((![X4 : nat]: ((P @ X4) => (ord_less_eq_nat @ X4 @ M4)))))))))))))). % bounded_Max_nat
thf(fact_220_Nat_Oex__has__greatest__nat, axiom,
    ((![P : nat > $o, K : nat, B4 : nat]: ((P @ K) => ((![Y4 : nat]: ((P @ Y4) => (ord_less_eq_nat @ Y4 @ B4))) => (?[X3 : nat]: ((P @ X3) & (![Y5 : nat]: ((P @ Y5) => (ord_less_eq_nat @ Y5 @ X3)))))))))). % Nat.ex_has_greatest_nat
thf(fact_221_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_222_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_223_eq__imp__le, axiom,
    ((![M : nat, N : nat]: ((M = N) => (ord_less_eq_nat @ M @ N))))). % eq_imp_le
thf(fact_224_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_225_le__refl, axiom,
    ((![N : nat]: (ord_less_eq_nat @ N @ N)))). % le_refl
thf(fact_226_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_227_transitive__stepwise__le, axiom,
    ((![M : nat, N : nat, R2 : nat > nat > $o]: ((ord_less_eq_nat @ M @ N) => ((![X3 : nat]: (R2 @ X3 @ X3)) => ((![X3 : nat, Y4 : nat, Z2 : nat]: ((R2 @ X3 @ Y4) => ((R2 @ Y4 @ Z2) => (R2 @ X3 @ Z2)))) => ((![N3 : nat]: (R2 @ N3 @ (suc @ N3))) => (R2 @ M @ N)))))))). % transitive_stepwise_le
thf(fact_228_nat__induct__at__least, axiom,
    ((![M : nat, N : nat, P : nat > $o]: ((ord_less_eq_nat @ M @ N) => ((P @ M) => ((![N3 : nat]: ((ord_less_eq_nat @ M @ N3) => ((P @ N3) => (P @ (suc @ N3))))) => (P @ N))))))). % nat_induct_at_least
thf(fact_229_full__nat__induct, axiom,
    ((![P : nat > $o, N : nat]: ((![N3 : nat]: ((![M5 : nat]: ((ord_less_eq_nat @ (suc @ M5) @ N3) => (P @ M5))) => (P @ N3))) => (P @ N))))). % full_nat_induct
thf(fact_230_not__less__eq__eq, axiom,
    ((![M : nat, N : nat]: ((~ ((ord_less_eq_nat @ M @ N))) = (ord_less_eq_nat @ (suc @ N) @ M))))). % not_less_eq_eq
thf(fact_231_Suc__n__not__le__n, axiom,
    ((![N : nat]: (~ ((ord_less_eq_nat @ (suc @ N) @ N)))))). % Suc_n_not_le_n
thf(fact_232_le__Suc__eq, axiom,
    ((![M : nat, N : nat]: ((ord_less_eq_nat @ M @ (suc @ N)) = (((ord_less_eq_nat @ M @ N)) | ((M = (suc @ N)))))))). % le_Suc_eq
thf(fact_233_Suc__le__D, axiom,
    ((![N : nat, M6 : nat]: ((ord_less_eq_nat @ (suc @ N) @ M6) => (?[M4 : nat]: (M6 = (suc @ M4))))))). % Suc_le_D
thf(fact_234_le__SucI, axiom,
    ((![M : nat, N : nat]: ((ord_less_eq_nat @ M @ N) => (ord_less_eq_nat @ M @ (suc @ N)))))). % le_SucI
thf(fact_235_le__SucE, axiom,
    ((![M : nat, N : nat]: ((ord_less_eq_nat @ M @ (suc @ N)) => ((~ ((ord_less_eq_nat @ M @ N))) => (M = (suc @ N))))))). % le_SucE
thf(fact_236_Suc__leD, axiom,
    ((![M : nat, N : nat]: ((ord_less_eq_nat @ (suc @ M) @ N) => (ord_less_eq_nat @ M @ N))))). % Suc_leD
thf(fact_237_zero__induct__lemma, axiom,
    ((![P : nat > $o, K : nat, I : nat]: ((P @ K) => ((![N3 : nat]: ((P @ (suc @ N3)) => (P @ N3))) => (P @ (minus_minus_nat @ K @ I))))))). % zero_induct_lemma
thf(fact_238_diff__le__mono2, axiom,
    ((![M : nat, N : nat, L : nat]: ((ord_less_eq_nat @ M @ N) => (ord_less_eq_nat @ (minus_minus_nat @ L @ N) @ (minus_minus_nat @ L @ M)))))). % diff_le_mono2
thf(fact_239_le__diff__iff_H, axiom,
    ((![A3 : nat, C3 : nat, B4 : nat]: ((ord_less_eq_nat @ A3 @ C3) => ((ord_less_eq_nat @ B4 @ C3) => ((ord_less_eq_nat @ (minus_minus_nat @ C3 @ A3) @ (minus_minus_nat @ C3 @ B4)) = (ord_less_eq_nat @ B4 @ A3))))))). % le_diff_iff'
thf(fact_240_diff__le__self, axiom,
    ((![M : nat, N : nat]: (ord_less_eq_nat @ (minus_minus_nat @ M @ N) @ M)))). % diff_le_self
thf(fact_241_diff__le__mono, axiom,
    ((![M : nat, N : nat, L : nat]: ((ord_less_eq_nat @ M @ N) => (ord_less_eq_nat @ (minus_minus_nat @ M @ L) @ (minus_minus_nat @ N @ L)))))). % diff_le_mono
thf(fact_242_Nat_Odiff__diff__eq, axiom,
    ((![K : nat, M : nat, N : nat]: ((ord_less_eq_nat @ K @ M) => ((ord_less_eq_nat @ K @ N) => ((minus_minus_nat @ (minus_minus_nat @ M @ K) @ (minus_minus_nat @ N @ K)) = (minus_minus_nat @ M @ N))))))). % Nat.diff_diff_eq
thf(fact_243_le__diff__iff, axiom,
    ((![K : nat, M : nat, N : nat]: ((ord_less_eq_nat @ K @ M) => ((ord_less_eq_nat @ K @ N) => ((ord_less_eq_nat @ (minus_minus_nat @ M @ K) @ (minus_minus_nat @ N @ K)) = (ord_less_eq_nat @ M @ N))))))). % le_diff_iff
thf(fact_244_eq__diff__iff, axiom,
    ((![K : nat, M : nat, N : nat]: ((ord_less_eq_nat @ K @ M) => ((ord_less_eq_nat @ K @ N) => (((minus_minus_nat @ M @ K) = (minus_minus_nat @ N @ K)) = (M = N))))))). % eq_diff_iff
thf(fact_245_assms_I1_J, axiom,
    ((![Ts : set_a, G3 : set_a]: ((ord_less_eq_set_a @ Ts @ G3) => (p @ G3 @ Ts))))). % assms(1)
thf(fact_246_Suc__le__D__lemma, axiom,
    ((![N : nat, M6 : nat, P : nat > $o]: ((ord_less_eq_nat @ (suc @ N) @ M6) => ((![M4 : nat]: ((ord_less_eq_nat @ N @ M4) => (P @ (suc @ M4)))) => (P @ M6)))))). % Suc_le_D_lemma

% Conjectures (7)
thf(conj_0, hypothesis,
    ((finite_finite_pname @ u))).
thf(conj_1, hypothesis,
    ((uG = (image_pname_a @ mgt_call @ u)))).
thf(conj_2, hypothesis,
    ((ord_less_eq_nat @ (suc @ na) @ (finite_card_a @ (image_pname_a @ mgt_call @ u))))).
thf(conj_3, hypothesis,
    (((finite_card_a @ g) = (minus_minus_nat @ (finite_card_a @ (image_pname_a @ mgt_call @ u)) @ (suc @ na))))).
thf(conj_4, hypothesis,
    ((member_pname @ pn @ u))).
thf(conj_5, hypothesis,
    ((~ ((member_a @ (mgt_call @ pn) @ g))))).
thf(conj_6, conjecture,
    ((finite_finite_a @ (image_pname_a @ mgt_call @ u)))).
