% 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_390__3254358_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:14:25.819

% Could-be-implicit typings (6)
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 (33)
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_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_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_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_Set_Oinsert_001t__Com__Opname, type,
    insert_pname : pname > set_pname > set_pname).
thf(sy_c_Set_Oinsert_001t__Nat__Onat, type,
    insert_nat : nat > set_nat > set_nat).
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_001tf__a, type,
    member_a : a > set_a > $o).
thf(sy_v_G, type,
    g : set_a).
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_card__insert__disjoint, axiom,
    ((![A : set_pname, X : pname]: ((finite_finite_pname @ A) => ((~ ((member_pname @ X @ A))) => ((finite_card_pname @ (insert_pname @ X @ A)) = (suc @ (finite_card_pname @ A)))))))). % card_insert_disjoint
thf(fact_1_card__insert__disjoint, axiom,
    ((![A : set_a, X : a]: ((finite_finite_a @ A) => ((~ ((member_a @ X @ A))) => ((finite_card_a @ (insert_a @ X @ A)) = (suc @ (finite_card_a @ A)))))))). % card_insert_disjoint
thf(fact_2_card__insert__disjoint, axiom,
    ((![A : set_nat, X : nat]: ((finite_finite_nat @ A) => ((~ ((member_nat @ X @ A))) => ((finite_card_nat @ (insert_nat @ X @ A)) = (suc @ (finite_card_nat @ A)))))))). % card_insert_disjoint
thf(fact_3_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_4_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_5_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_6_card__le__Suc__iff, axiom,
    ((![N : nat, A : set_pname]: ((ord_less_eq_nat @ (suc @ N) @ (finite_card_pname @ A)) = (?[A2 : pname]: (?[B : set_pname]: (((A = (insert_pname @ A2 @ B))) & ((((~ ((member_pname @ A2 @ B)))) & ((((ord_less_eq_nat @ N @ (finite_card_pname @ B))) & ((finite_finite_pname @ B))))))))))))). % card_le_Suc_iff
thf(fact_7_card__le__Suc__iff, axiom,
    ((![N : nat, A : set_a]: ((ord_less_eq_nat @ (suc @ N) @ (finite_card_a @ A)) = (?[A2 : a]: (?[B : set_a]: (((A = (insert_a @ A2 @ B))) & ((((~ ((member_a @ A2 @ B)))) & ((((ord_less_eq_nat @ N @ (finite_card_a @ B))) & ((finite_finite_a @ B))))))))))))). % card_le_Suc_iff
thf(fact_8_card__le__Suc__iff, axiom,
    ((![N : nat, A : set_nat]: ((ord_less_eq_nat @ (suc @ N) @ (finite_card_nat @ A)) = (?[A2 : nat]: (?[B : set_nat]: (((A = (insert_nat @ A2 @ B))) & ((((~ ((member_nat @ A2 @ B)))) & ((((ord_less_eq_nat @ N @ (finite_card_nat @ B))) & ((finite_finite_nat @ B))))))))))))). % card_le_Suc_iff
thf(fact_9_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_10_finite__insert, axiom,
    ((![A3 : pname, A : set_pname]: ((finite_finite_pname @ (insert_pname @ A3 @ A)) = (finite_finite_pname @ A))))). % finite_insert
thf(fact_11_finite__insert, axiom,
    ((![A3 : a, A : set_a]: ((finite_finite_a @ (insert_a @ A3 @ A)) = (finite_finite_a @ A))))). % finite_insert
thf(fact_12_finite__insert, axiom,
    ((![A3 : nat, A : set_nat]: ((finite_finite_nat @ (insert_nat @ A3 @ A)) = (finite_finite_nat @ A))))). % finite_insert
thf(fact_13_image__insert, axiom,
    ((![F : a > nat, A3 : a, B2 : set_a]: ((image_a_nat @ F @ (insert_a @ A3 @ B2)) = (insert_nat @ (F @ A3) @ (image_a_nat @ F @ B2)))))). % image_insert
thf(fact_14_image__insert, axiom,
    ((![F : a > pname, A3 : a, B2 : set_a]: ((image_a_pname @ F @ (insert_a @ A3 @ B2)) = (insert_pname @ (F @ A3) @ (image_a_pname @ F @ B2)))))). % image_insert
thf(fact_15_image__insert, axiom,
    ((![F : nat > a, A3 : nat, B2 : set_nat]: ((image_nat_a @ F @ (insert_nat @ A3 @ B2)) = (insert_a @ (F @ A3) @ (image_nat_a @ F @ B2)))))). % image_insert
thf(fact_16_image__insert, axiom,
    ((![F : nat > nat, A3 : nat, B2 : set_nat]: ((image_nat_nat @ F @ (insert_nat @ A3 @ B2)) = (insert_nat @ (F @ A3) @ (image_nat_nat @ F @ B2)))))). % image_insert
thf(fact_17_image__insert, axiom,
    ((![F : nat > pname, A3 : nat, B2 : set_nat]: ((image_nat_pname @ F @ (insert_nat @ A3 @ B2)) = (insert_pname @ (F @ A3) @ (image_nat_pname @ F @ B2)))))). % image_insert
thf(fact_18_image__insert, axiom,
    ((![F : pname > nat, A3 : pname, B2 : set_pname]: ((image_pname_nat @ F @ (insert_pname @ A3 @ B2)) = (insert_nat @ (F @ A3) @ (image_pname_nat @ F @ B2)))))). % image_insert
thf(fact_19_image__insert, axiom,
    ((![F : pname > pname, A3 : pname, B2 : set_pname]: ((image_pname_pname @ F @ (insert_pname @ A3 @ B2)) = (insert_pname @ (F @ A3) @ (image_pname_pname @ F @ B2)))))). % image_insert
thf(fact_20_image__insert, axiom,
    ((![F : pname > a, A3 : pname, B2 : set_pname]: ((image_pname_a @ F @ (insert_pname @ A3 @ B2)) = (insert_a @ (F @ A3) @ (image_pname_a @ F @ B2)))))). % image_insert
thf(fact_21_image__insert, axiom,
    ((![F : a > a, A3 : a, B2 : set_a]: ((image_a_a @ F @ (insert_a @ A3 @ B2)) = (insert_a @ (F @ A3) @ (image_a_a @ F @ B2)))))). % image_insert
thf(fact_22_insert__image, axiom,
    ((![X : pname, A : set_pname, F : pname > nat]: ((member_pname @ X @ A) => ((insert_nat @ (F @ X) @ (image_pname_nat @ F @ A)) = (image_pname_nat @ F @ A)))))). % insert_image
thf(fact_23_insert__image, axiom,
    ((![X : pname, A : set_pname, F : pname > pname]: ((member_pname @ X @ A) => ((insert_pname @ (F @ X) @ (image_pname_pname @ F @ A)) = (image_pname_pname @ F @ A)))))). % insert_image
thf(fact_24_insert__image, axiom,
    ((![X : a, A : set_a, F : a > nat]: ((member_a @ X @ A) => ((insert_nat @ (F @ X) @ (image_a_nat @ F @ A)) = (image_a_nat @ F @ A)))))). % insert_image
thf(fact_25_insert__image, axiom,
    ((![X : a, A : set_a, F : a > pname]: ((member_a @ X @ A) => ((insert_pname @ (F @ X) @ (image_a_pname @ F @ A)) = (image_a_pname @ F @ A)))))). % insert_image
thf(fact_26_insert__image, axiom,
    ((![X : nat, A : set_nat, F : nat > a]: ((member_nat @ X @ A) => ((insert_a @ (F @ X) @ (image_nat_a @ F @ A)) = (image_nat_a @ F @ A)))))). % insert_image
thf(fact_27_insert__image, axiom,
    ((![X : nat, A : set_nat, F : nat > nat]: ((member_nat @ X @ A) => ((insert_nat @ (F @ X) @ (image_nat_nat @ F @ A)) = (image_nat_nat @ F @ A)))))). % insert_image
thf(fact_28_insert__image, axiom,
    ((![X : nat, A : set_nat, F : nat > pname]: ((member_nat @ X @ A) => ((insert_pname @ (F @ X) @ (image_nat_pname @ F @ A)) = (image_nat_pname @ F @ A)))))). % insert_image
thf(fact_29_insert__image, axiom,
    ((![X : pname, A : set_pname, F : pname > a]: ((member_pname @ X @ A) => ((insert_a @ (F @ X) @ (image_pname_a @ F @ A)) = (image_pname_a @ F @ A)))))). % insert_image
thf(fact_30_insert__image, axiom,
    ((![X : a, A : set_a, F : a > a]: ((member_a @ X @ A) => ((insert_a @ (F @ X) @ (image_a_a @ F @ A)) = (image_a_a @ F @ A)))))). % insert_image
thf(fact_31_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_32_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_33_finite__imageI, axiom,
    ((![F2 : set_pname, H : pname > nat]: ((finite_finite_pname @ F2) => (finite_finite_nat @ (image_pname_nat @ H @ F2)))))). % finite_imageI
thf(fact_34_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_35_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_36_finite__imageI, axiom,
    ((![F2 : set_a, H : a > nat]: ((finite_finite_a @ F2) => (finite_finite_nat @ (image_a_nat @ H @ F2)))))). % finite_imageI
thf(fact_37_finite__imageI, axiom,
    ((![F2 : set_nat, H : nat > pname]: ((finite_finite_nat @ F2) => (finite_finite_pname @ (image_nat_pname @ H @ F2)))))). % finite_imageI
thf(fact_38_finite__imageI, axiom,
    ((![F2 : set_nat, H : nat > a]: ((finite_finite_nat @ F2) => (finite_finite_a @ (image_nat_a @ H @ F2)))))). % finite_imageI
thf(fact_39_finite__imageI, axiom,
    ((![F2 : set_nat, H : nat > nat]: ((finite_finite_nat @ F2) => (finite_finite_nat @ (image_nat_nat @ H @ F2)))))). % finite_imageI
thf(fact_40_card__insert__le, axiom,
    ((![A : set_pname, X : pname]: ((finite_finite_pname @ A) => (ord_less_eq_nat @ (finite_card_pname @ A) @ (finite_card_pname @ (insert_pname @ X @ A))))))). % card_insert_le
thf(fact_41_card__insert__le, axiom,
    ((![A : set_a, X : a]: ((finite_finite_a @ A) => (ord_less_eq_nat @ (finite_card_a @ A) @ (finite_card_a @ (insert_a @ X @ A))))))). % card_insert_le
thf(fact_42_card__insert__le, axiom,
    ((![A : set_nat, X : nat]: ((finite_finite_nat @ A) => (ord_less_eq_nat @ (finite_card_nat @ A) @ (finite_card_nat @ (insert_nat @ X @ A))))))). % card_insert_le
thf(fact_43_card__insert__if, axiom,
    ((![A : set_pname, X : pname]: ((finite_finite_pname @ A) => (((member_pname @ X @ A) => ((finite_card_pname @ (insert_pname @ X @ A)) = (finite_card_pname @ A))) & ((~ ((member_pname @ X @ A))) => ((finite_card_pname @ (insert_pname @ X @ A)) = (suc @ (finite_card_pname @ A))))))))). % card_insert_if
thf(fact_44_card__insert__if, axiom,
    ((![A : set_a, X : a]: ((finite_finite_a @ A) => (((member_a @ X @ A) => ((finite_card_a @ (insert_a @ X @ A)) = (finite_card_a @ A))) & ((~ ((member_a @ X @ A))) => ((finite_card_a @ (insert_a @ X @ A)) = (suc @ (finite_card_a @ A))))))))). % card_insert_if
thf(fact_45_card__insert__if, axiom,
    ((![A : set_nat, X : nat]: ((finite_finite_nat @ A) => (((member_nat @ X @ A) => ((finite_card_nat @ (insert_nat @ X @ A)) = (finite_card_nat @ A))) & ((~ ((member_nat @ X @ A))) => ((finite_card_nat @ (insert_nat @ X @ A)) = (suc @ (finite_card_nat @ A))))))))). % card_insert_if
thf(fact_46_image__eqI, axiom,
    ((![B3 : pname, F : pname > pname, X : pname, A : set_pname]: ((B3 = (F @ X)) => ((member_pname @ X @ A) => (member_pname @ B3 @ (image_pname_pname @ F @ A))))))). % image_eqI
thf(fact_47_image__eqI, axiom,
    ((![B3 : a, F : pname > a, X : pname, A : set_pname]: ((B3 = (F @ X)) => ((member_pname @ X @ A) => (member_a @ B3 @ (image_pname_a @ F @ A))))))). % image_eqI
thf(fact_48_image__eqI, axiom,
    ((![B3 : nat, F : pname > nat, X : pname, A : set_pname]: ((B3 = (F @ X)) => ((member_pname @ X @ A) => (member_nat @ B3 @ (image_pname_nat @ F @ A))))))). % image_eqI
thf(fact_49_image__eqI, axiom,
    ((![B3 : pname, F : a > pname, X : a, A : set_a]: ((B3 = (F @ X)) => ((member_a @ X @ A) => (member_pname @ B3 @ (image_a_pname @ F @ A))))))). % image_eqI
thf(fact_50_image__eqI, axiom,
    ((![B3 : a, F : a > a, X : a, A : set_a]: ((B3 = (F @ X)) => ((member_a @ X @ A) => (member_a @ B3 @ (image_a_a @ F @ A))))))). % image_eqI
thf(fact_51_image__eqI, axiom,
    ((![B3 : nat, F : a > nat, X : a, A : set_a]: ((B3 = (F @ X)) => ((member_a @ X @ A) => (member_nat @ B3 @ (image_a_nat @ F @ A))))))). % image_eqI
thf(fact_52_image__eqI, axiom,
    ((![B3 : pname, F : nat > pname, X : nat, A : set_nat]: ((B3 = (F @ X)) => ((member_nat @ X @ A) => (member_pname @ B3 @ (image_nat_pname @ F @ A))))))). % image_eqI
thf(fact_53_image__eqI, axiom,
    ((![B3 : a, F : nat > a, X : nat, A : set_nat]: ((B3 = (F @ X)) => ((member_nat @ X @ A) => (member_a @ B3 @ (image_nat_a @ F @ A))))))). % image_eqI
thf(fact_54_image__eqI, axiom,
    ((![B3 : nat, F : nat > nat, X : nat, A : set_nat]: ((B3 = (F @ X)) => ((member_nat @ X @ A) => (member_nat @ B3 @ (image_nat_nat @ F @ A))))))). % image_eqI
thf(fact_55_old_Onat_Oinject, axiom,
    ((![Nat : nat, Nat2 : nat]: (((suc @ Nat) = (suc @ Nat2)) = (Nat = Nat2))))). % old.nat.inject
thf(fact_56_nat_Oinject, axiom,
    ((![X2 : nat, Y2 : nat]: (((suc @ X2) = (suc @ Y2)) = (X2 = Y2))))). % nat.inject
thf(fact_57_insert__absorb2, axiom,
    ((![X : a, A : set_a]: ((insert_a @ X @ (insert_a @ X @ A)) = (insert_a @ X @ A))))). % insert_absorb2
thf(fact_58_insert__absorb2, axiom,
    ((![X : nat, A : set_nat]: ((insert_nat @ X @ (insert_nat @ X @ A)) = (insert_nat @ X @ A))))). % insert_absorb2
thf(fact_59_insert__absorb2, axiom,
    ((![X : pname, A : set_pname]: ((insert_pname @ X @ (insert_pname @ X @ A)) = (insert_pname @ X @ A))))). % insert_absorb2
thf(fact_60_insert__iff, axiom,
    ((![A3 : pname, B3 : pname, A : set_pname]: ((member_pname @ A3 @ (insert_pname @ B3 @ A)) = (((A3 = B3)) | ((member_pname @ A3 @ A))))))). % insert_iff
thf(fact_61_insert__iff, axiom,
    ((![A3 : a, B3 : a, A : set_a]: ((member_a @ A3 @ (insert_a @ B3 @ A)) = (((A3 = B3)) | ((member_a @ A3 @ A))))))). % insert_iff
thf(fact_62_insert__iff, axiom,
    ((![A3 : nat, B3 : nat, A : set_nat]: ((member_nat @ A3 @ (insert_nat @ B3 @ A)) = (((A3 = B3)) | ((member_nat @ A3 @ A))))))). % insert_iff
thf(fact_63_insertCI, axiom,
    ((![A3 : pname, B2 : set_pname, B3 : pname]: (((~ ((member_pname @ A3 @ B2))) => (A3 = B3)) => (member_pname @ A3 @ (insert_pname @ B3 @ B2)))))). % insertCI
thf(fact_64_insertCI, axiom,
    ((![A3 : a, B2 : set_a, B3 : a]: (((~ ((member_a @ A3 @ B2))) => (A3 = B3)) => (member_a @ A3 @ (insert_a @ B3 @ B2)))))). % insertCI
thf(fact_65_insertCI, axiom,
    ((![A3 : nat, B2 : set_nat, B3 : nat]: (((~ ((member_nat @ A3 @ B2))) => (A3 = B3)) => (member_nat @ A3 @ (insert_nat @ B3 @ B2)))))). % insertCI
thf(fact_66_image__ident, axiom,
    ((![Y : set_nat]: ((image_nat_nat @ (^[X3 : nat]: X3) @ Y) = Y)))). % image_ident
thf(fact_67_image__ident, axiom,
    ((![Y : set_a]: ((image_a_a @ (^[X3 : a]: X3) @ Y) = Y)))). % image_ident
thf(fact_68_finite__Collect__disjI, axiom,
    ((![P : pname > $o, Q : pname > $o]: ((finite_finite_pname @ (collect_pname @ (^[X3 : pname]: (((P @ X3)) | ((Q @ X3)))))) = (((finite_finite_pname @ (collect_pname @ P))) & ((finite_finite_pname @ (collect_pname @ Q)))))))). % finite_Collect_disjI
thf(fact_69_finite__Collect__disjI, axiom,
    ((![P : a > $o, Q : a > $o]: ((finite_finite_a @ (collect_a @ (^[X3 : a]: (((P @ X3)) | ((Q @ X3)))))) = (((finite_finite_a @ (collect_a @ P))) & ((finite_finite_a @ (collect_a @ Q)))))))). % finite_Collect_disjI
thf(fact_70_finite__Collect__disjI, axiom,
    ((![P : nat > $o, Q : nat > $o]: ((finite_finite_nat @ (collect_nat @ (^[X3 : nat]: (((P @ X3)) | ((Q @ X3)))))) = (((finite_finite_nat @ (collect_nat @ P))) & ((finite_finite_nat @ (collect_nat @ Q)))))))). % finite_Collect_disjI
thf(fact_71_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 @ (^[X3 : pname]: (((P @ X3)) & ((Q @ X3)))))))))). % finite_Collect_conjI
thf(fact_72_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 @ (^[X3 : a]: (((P @ X3)) & ((Q @ X3)))))))))). % finite_Collect_conjI
thf(fact_73_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 @ (^[X3 : nat]: (((P @ X3)) & ((Q @ X3)))))))))). % finite_Collect_conjI
thf(fact_74_finite__Collect__le__nat, axiom,
    ((![K : nat]: (finite_finite_nat @ (collect_nat @ (^[N2 : nat]: (ord_less_eq_nat @ N2 @ K))))))). % finite_Collect_le_nat
thf(fact_75_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_76_rev__image__eqI, axiom,
    ((![X : pname, A : set_pname, B3 : pname, F : pname > pname]: ((member_pname @ X @ A) => ((B3 = (F @ X)) => (member_pname @ B3 @ (image_pname_pname @ F @ A))))))). % rev_image_eqI
thf(fact_77_rev__image__eqI, axiom,
    ((![X : pname, A : set_pname, B3 : a, F : pname > a]: ((member_pname @ X @ A) => ((B3 = (F @ X)) => (member_a @ B3 @ (image_pname_a @ F @ A))))))). % rev_image_eqI
thf(fact_78_rev__image__eqI, axiom,
    ((![X : pname, A : set_pname, B3 : nat, F : pname > nat]: ((member_pname @ X @ A) => ((B3 = (F @ X)) => (member_nat @ B3 @ (image_pname_nat @ F @ A))))))). % rev_image_eqI
thf(fact_79_rev__image__eqI, axiom,
    ((![X : a, A : set_a, B3 : pname, F : a > pname]: ((member_a @ X @ A) => ((B3 = (F @ X)) => (member_pname @ B3 @ (image_a_pname @ F @ A))))))). % rev_image_eqI
thf(fact_80_rev__image__eqI, axiom,
    ((![X : a, A : set_a, B3 : a, F : a > a]: ((member_a @ X @ A) => ((B3 = (F @ X)) => (member_a @ B3 @ (image_a_a @ F @ A))))))). % rev_image_eqI
thf(fact_81_rev__image__eqI, axiom,
    ((![X : a, A : set_a, B3 : nat, F : a > nat]: ((member_a @ X @ A) => ((B3 = (F @ X)) => (member_nat @ B3 @ (image_a_nat @ F @ A))))))). % rev_image_eqI
thf(fact_82_rev__image__eqI, axiom,
    ((![X : nat, A : set_nat, B3 : pname, F : nat > pname]: ((member_nat @ X @ A) => ((B3 = (F @ X)) => (member_pname @ B3 @ (image_nat_pname @ F @ A))))))). % rev_image_eqI
thf(fact_83_rev__image__eqI, axiom,
    ((![X : nat, A : set_nat, B3 : a, F : nat > a]: ((member_nat @ X @ A) => ((B3 = (F @ X)) => (member_a @ B3 @ (image_nat_a @ F @ A))))))). % rev_image_eqI
thf(fact_84_rev__image__eqI, axiom,
    ((![X : nat, A : set_nat, B3 : nat, F : nat > nat]: ((member_nat @ X @ A) => ((B3 = (F @ X)) => (member_nat @ B3 @ (image_nat_nat @ F @ A))))))). % rev_image_eqI
thf(fact_85_ball__imageD, axiom,
    ((![F : pname > a, A : set_pname, P : a > $o]: ((![X4 : a]: ((member_a @ X4 @ (image_pname_a @ F @ A)) => (P @ X4))) => (![X5 : pname]: ((member_pname @ X5 @ A) => (P @ (F @ X5)))))))). % ball_imageD
thf(fact_86_ball__imageD, axiom,
    ((![F : nat > nat, A : set_nat, P : nat > $o]: ((![X4 : nat]: ((member_nat @ X4 @ (image_nat_nat @ F @ A)) => (P @ X4))) => (![X5 : nat]: ((member_nat @ X5 @ A) => (P @ (F @ X5)))))))). % ball_imageD
thf(fact_87_ball__imageD, axiom,
    ((![F : nat > a, A : set_nat, P : a > $o]: ((![X4 : a]: ((member_a @ X4 @ (image_nat_a @ F @ A)) => (P @ X4))) => (![X5 : nat]: ((member_nat @ X5 @ A) => (P @ (F @ X5)))))))). % ball_imageD
thf(fact_88_ball__imageD, axiom,
    ((![F : a > pname, A : set_a, P : pname > $o]: ((![X4 : pname]: ((member_pname @ X4 @ (image_a_pname @ F @ A)) => (P @ X4))) => (![X5 : a]: ((member_a @ X5 @ A) => (P @ (F @ X5)))))))). % ball_imageD
thf(fact_89_ball__imageD, axiom,
    ((![F : a > nat, A : set_a, P : nat > $o]: ((![X4 : nat]: ((member_nat @ X4 @ (image_a_nat @ F @ A)) => (P @ X4))) => (![X5 : a]: ((member_a @ X5 @ A) => (P @ (F @ X5)))))))). % ball_imageD
thf(fact_90_ball__imageD, axiom,
    ((![F : a > a, A : set_a, P : a > $o]: ((![X4 : a]: ((member_a @ X4 @ (image_a_a @ F @ A)) => (P @ X4))) => (![X5 : a]: ((member_a @ X5 @ A) => (P @ (F @ X5)))))))). % ball_imageD
thf(fact_91_image__cong, axiom,
    ((![M2 : set_pname, N3 : set_pname, F : pname > a, G : pname > a]: ((M2 = N3) => ((![X4 : pname]: ((member_pname @ X4 @ N3) => ((F @ X4) = (G @ X4)))) => ((image_pname_a @ F @ M2) = (image_pname_a @ G @ N3))))))). % image_cong
thf(fact_92_image__cong, axiom,
    ((![M2 : set_a, N3 : set_a, F : a > pname, G : a > pname]: ((M2 = N3) => ((![X4 : a]: ((member_a @ X4 @ N3) => ((F @ X4) = (G @ X4)))) => ((image_a_pname @ F @ M2) = (image_a_pname @ G @ N3))))))). % image_cong
thf(fact_93_image__cong, axiom,
    ((![M2 : set_a, N3 : set_a, F : a > nat, G : a > nat]: ((M2 = N3) => ((![X4 : a]: ((member_a @ X4 @ N3) => ((F @ X4) = (G @ X4)))) => ((image_a_nat @ F @ M2) = (image_a_nat @ G @ N3))))))). % image_cong
thf(fact_94_image__cong, axiom,
    ((![M2 : set_a, N3 : set_a, F : a > a, G : a > a]: ((M2 = N3) => ((![X4 : a]: ((member_a @ X4 @ N3) => ((F @ X4) = (G @ X4)))) => ((image_a_a @ F @ M2) = (image_a_a @ G @ N3))))))). % image_cong
thf(fact_95_image__cong, axiom,
    ((![M2 : set_nat, N3 : set_nat, F : nat > nat, G : nat > nat]: ((M2 = N3) => ((![X4 : nat]: ((member_nat @ X4 @ N3) => ((F @ X4) = (G @ X4)))) => ((image_nat_nat @ F @ M2) = (image_nat_nat @ G @ N3))))))). % image_cong
thf(fact_96_image__cong, axiom,
    ((![M2 : set_nat, N3 : set_nat, F : nat > a, G : nat > a]: ((M2 = N3) => ((![X4 : nat]: ((member_nat @ X4 @ N3) => ((F @ X4) = (G @ X4)))) => ((image_nat_a @ F @ M2) = (image_nat_a @ G @ N3))))))). % image_cong
thf(fact_97_bex__imageD, axiom,
    ((![F : pname > a, A : set_pname, P : a > $o]: ((?[X5 : a]: ((member_a @ X5 @ (image_pname_a @ F @ A)) & (P @ X5))) => (?[X4 : pname]: ((member_pname @ X4 @ A) & (P @ (F @ X4)))))))). % bex_imageD
thf(fact_98_bex__imageD, axiom,
    ((![F : nat > nat, A : set_nat, P : nat > $o]: ((?[X5 : nat]: ((member_nat @ X5 @ (image_nat_nat @ F @ A)) & (P @ X5))) => (?[X4 : nat]: ((member_nat @ X4 @ A) & (P @ (F @ X4)))))))). % bex_imageD
thf(fact_99_bex__imageD, axiom,
    ((![F : nat > a, A : set_nat, P : a > $o]: ((?[X5 : a]: ((member_a @ X5 @ (image_nat_a @ F @ A)) & (P @ X5))) => (?[X4 : nat]: ((member_nat @ X4 @ A) & (P @ (F @ X4)))))))). % bex_imageD
thf(fact_100_bex__imageD, axiom,
    ((![F : a > pname, A : set_a, P : pname > $o]: ((?[X5 : pname]: ((member_pname @ X5 @ (image_a_pname @ F @ A)) & (P @ X5))) => (?[X4 : a]: ((member_a @ X4 @ A) & (P @ (F @ X4)))))))). % bex_imageD
thf(fact_101_bex__imageD, axiom,
    ((![F : a > nat, A : set_a, P : nat > $o]: ((?[X5 : nat]: ((member_nat @ X5 @ (image_a_nat @ F @ A)) & (P @ X5))) => (?[X4 : a]: ((member_a @ X4 @ A) & (P @ (F @ X4)))))))). % bex_imageD
thf(fact_102_bex__imageD, axiom,
    ((![F : a > a, A : set_a, P : a > $o]: ((?[X5 : a]: ((member_a @ X5 @ (image_a_a @ F @ A)) & (P @ X5))) => (?[X4 : a]: ((member_a @ X4 @ A) & (P @ (F @ X4)))))))). % bex_imageD
thf(fact_103_image__iff, axiom,
    ((![Z : pname, F : a > pname, A : set_a]: ((member_pname @ Z @ (image_a_pname @ F @ A)) = (?[X3 : a]: (((member_a @ X3 @ A)) & ((Z = (F @ X3))))))))). % image_iff
thf(fact_104_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_105_image__iff, axiom,
    ((![Z : a, F : nat > a, A : set_nat]: ((member_a @ Z @ (image_nat_a @ F @ A)) = (?[X3 : nat]: (((member_nat @ X3 @ A)) & ((Z = (F @ X3))))))))). % image_iff
thf(fact_106_image__iff, axiom,
    ((![Z : a, F : a > a, A : set_a]: ((member_a @ Z @ (image_a_a @ F @ A)) = (?[X3 : a]: (((member_a @ X3 @ A)) & ((Z = (F @ X3))))))))). % image_iff
thf(fact_107_image__iff, axiom,
    ((![Z : nat, F : nat > nat, A : set_nat]: ((member_nat @ Z @ (image_nat_nat @ F @ A)) = (?[X3 : nat]: (((member_nat @ X3 @ A)) & ((Z = (F @ X3))))))))). % image_iff
thf(fact_108_image__iff, axiom,
    ((![Z : nat, F : a > nat, A : set_a]: ((member_nat @ Z @ (image_a_nat @ F @ A)) = (?[X3 : a]: (((member_a @ X3 @ A)) & ((Z = (F @ X3))))))))). % image_iff
thf(fact_109_imageI, axiom,
    ((![X : pname, A : set_pname, F : pname > pname]: ((member_pname @ X @ A) => (member_pname @ (F @ X) @ (image_pname_pname @ F @ A)))))). % imageI
thf(fact_110_imageI, axiom,
    ((![X : pname, A : set_pname, F : pname > a]: ((member_pname @ X @ A) => (member_a @ (F @ X) @ (image_pname_a @ F @ A)))))). % imageI
thf(fact_111_imageI, axiom,
    ((![X : pname, A : set_pname, F : pname > nat]: ((member_pname @ X @ A) => (member_nat @ (F @ X) @ (image_pname_nat @ F @ A)))))). % imageI
thf(fact_112_imageI, axiom,
    ((![X : a, A : set_a, F : a > pname]: ((member_a @ X @ A) => (member_pname @ (F @ X) @ (image_a_pname @ F @ A)))))). % imageI
thf(fact_113_imageI, axiom,
    ((![X : a, A : set_a, F : a > a]: ((member_a @ X @ A) => (member_a @ (F @ X) @ (image_a_a @ F @ A)))))). % imageI
thf(fact_114_imageI, axiom,
    ((![X : a, A : set_a, F : a > nat]: ((member_a @ X @ A) => (member_nat @ (F @ X) @ (image_a_nat @ F @ A)))))). % imageI
thf(fact_115_imageI, axiom,
    ((![X : nat, A : set_nat, F : nat > pname]: ((member_nat @ X @ A) => (member_pname @ (F @ X) @ (image_nat_pname @ F @ A)))))). % imageI
thf(fact_116_imageI, axiom,
    ((![X : nat, A : set_nat, F : nat > a]: ((member_nat @ X @ A) => (member_a @ (F @ X) @ (image_nat_a @ F @ A)))))). % imageI
thf(fact_117_imageI, axiom,
    ((![X : nat, A : set_nat, F : nat > nat]: ((member_nat @ X @ A) => (member_nat @ (F @ X) @ (image_nat_nat @ F @ A)))))). % imageI
thf(fact_118_n__not__Suc__n, axiom,
    ((![N : nat]: (~ ((N = (suc @ N))))))). % n_not_Suc_n
thf(fact_119_Suc__inject, axiom,
    ((![X : nat, Y3 : nat]: (((suc @ X) = (suc @ Y3)) => (X = Y3))))). % Suc_inject
thf(fact_120_mk__disjoint__insert, axiom,
    ((![A3 : pname, A : set_pname]: ((member_pname @ A3 @ A) => (?[B4 : set_pname]: ((A = (insert_pname @ A3 @ B4)) & (~ ((member_pname @ A3 @ B4))))))))). % mk_disjoint_insert
thf(fact_121_mk__disjoint__insert, axiom,
    ((![A3 : a, A : set_a]: ((member_a @ A3 @ A) => (?[B4 : set_a]: ((A = (insert_a @ A3 @ B4)) & (~ ((member_a @ A3 @ B4))))))))). % mk_disjoint_insert
thf(fact_122_mk__disjoint__insert, axiom,
    ((![A3 : nat, A : set_nat]: ((member_nat @ A3 @ A) => (?[B4 : set_nat]: ((A = (insert_nat @ A3 @ B4)) & (~ ((member_nat @ A3 @ B4))))))))). % mk_disjoint_insert
thf(fact_123_insert__commute, axiom,
    ((![X : a, Y3 : a, A : set_a]: ((insert_a @ X @ (insert_a @ Y3 @ A)) = (insert_a @ Y3 @ (insert_a @ X @ A)))))). % insert_commute
thf(fact_124_insert__commute, axiom,
    ((![X : nat, Y3 : nat, A : set_nat]: ((insert_nat @ X @ (insert_nat @ Y3 @ A)) = (insert_nat @ Y3 @ (insert_nat @ X @ A)))))). % insert_commute
thf(fact_125_insert__commute, axiom,
    ((![X : pname, Y3 : pname, A : set_pname]: ((insert_pname @ X @ (insert_pname @ Y3 @ A)) = (insert_pname @ Y3 @ (insert_pname @ X @ A)))))). % insert_commute
thf(fact_126_insert__eq__iff, axiom,
    ((![A3 : pname, A : set_pname, B3 : pname, B2 : set_pname]: ((~ ((member_pname @ A3 @ A))) => ((~ ((member_pname @ B3 @ B2))) => (((insert_pname @ A3 @ A) = (insert_pname @ B3 @ B2)) = (((((A3 = B3)) => ((A = B2)))) & ((((~ ((A3 = B3)))) => ((?[C : set_pname]: (((A = (insert_pname @ B3 @ C))) & ((((~ ((member_pname @ B3 @ C)))) & ((((B2 = (insert_pname @ A3 @ C))) & ((~ ((member_pname @ A3 @ C)))))))))))))))))))). % insert_eq_iff
thf(fact_127_insert__eq__iff, axiom,
    ((![A3 : a, A : set_a, B3 : a, B2 : set_a]: ((~ ((member_a @ A3 @ A))) => ((~ ((member_a @ B3 @ B2))) => (((insert_a @ A3 @ A) = (insert_a @ B3 @ B2)) = (((((A3 = B3)) => ((A = B2)))) & ((((~ ((A3 = B3)))) => ((?[C : set_a]: (((A = (insert_a @ B3 @ C))) & ((((~ ((member_a @ B3 @ C)))) & ((((B2 = (insert_a @ A3 @ C))) & ((~ ((member_a @ A3 @ C)))))))))))))))))))). % insert_eq_iff
thf(fact_128_insert__eq__iff, axiom,
    ((![A3 : nat, A : set_nat, B3 : nat, B2 : set_nat]: ((~ ((member_nat @ A3 @ A))) => ((~ ((member_nat @ B3 @ B2))) => (((insert_nat @ A3 @ A) = (insert_nat @ B3 @ B2)) = (((((A3 = B3)) => ((A = B2)))) & ((((~ ((A3 = B3)))) => ((?[C : set_nat]: (((A = (insert_nat @ B3 @ C))) & ((((~ ((member_nat @ B3 @ C)))) & ((((B2 = (insert_nat @ A3 @ C))) & ((~ ((member_nat @ A3 @ C)))))))))))))))))))). % insert_eq_iff
thf(fact_129_insert__absorb, axiom,
    ((![A3 : pname, A : set_pname]: ((member_pname @ A3 @ A) => ((insert_pname @ A3 @ A) = A))))). % insert_absorb
thf(fact_130_insert__absorb, axiom,
    ((![A3 : a, A : set_a]: ((member_a @ A3 @ A) => ((insert_a @ A3 @ A) = A))))). % insert_absorb
thf(fact_131_insert__absorb, axiom,
    ((![A3 : nat, A : set_nat]: ((member_nat @ A3 @ A) => ((insert_nat @ A3 @ A) = A))))). % insert_absorb
thf(fact_132_insert__ident, axiom,
    ((![X : pname, A : set_pname, B2 : set_pname]: ((~ ((member_pname @ X @ A))) => ((~ ((member_pname @ X @ B2))) => (((insert_pname @ X @ A) = (insert_pname @ X @ B2)) = (A = B2))))))). % insert_ident
thf(fact_133_insert__ident, axiom,
    ((![X : a, A : set_a, B2 : set_a]: ((~ ((member_a @ X @ A))) => ((~ ((member_a @ X @ B2))) => (((insert_a @ X @ A) = (insert_a @ X @ B2)) = (A = B2))))))). % insert_ident
thf(fact_134_insert__ident, axiom,
    ((![X : nat, A : set_nat, B2 : set_nat]: ((~ ((member_nat @ X @ A))) => ((~ ((member_nat @ X @ B2))) => (((insert_nat @ X @ A) = (insert_nat @ X @ B2)) = (A = B2))))))). % insert_ident
thf(fact_135_Set_Oset__insert, axiom,
    ((![X : pname, A : set_pname]: ((member_pname @ X @ A) => (~ ((![B4 : set_pname]: ((A = (insert_pname @ X @ B4)) => (member_pname @ X @ B4))))))))). % Set.set_insert
thf(fact_136_Set_Oset__insert, axiom,
    ((![X : a, A : set_a]: ((member_a @ X @ A) => (~ ((![B4 : set_a]: ((A = (insert_a @ X @ B4)) => (member_a @ X @ B4))))))))). % Set.set_insert
thf(fact_137_Set_Oset__insert, axiom,
    ((![X : nat, A : set_nat]: ((member_nat @ X @ A) => (~ ((![B4 : set_nat]: ((A = (insert_nat @ X @ B4)) => (member_nat @ X @ B4))))))))). % Set.set_insert
thf(fact_138_insertI2, axiom,
    ((![A3 : pname, B2 : set_pname, B3 : pname]: ((member_pname @ A3 @ B2) => (member_pname @ A3 @ (insert_pname @ B3 @ B2)))))). % insertI2
thf(fact_139_insertI2, axiom,
    ((![A3 : a, B2 : set_a, B3 : a]: ((member_a @ A3 @ B2) => (member_a @ A3 @ (insert_a @ B3 @ B2)))))). % insertI2
thf(fact_140_insertI2, axiom,
    ((![A3 : nat, B2 : set_nat, B3 : nat]: ((member_nat @ A3 @ B2) => (member_nat @ A3 @ (insert_nat @ B3 @ B2)))))). % insertI2
thf(fact_141_insertI1, axiom,
    ((![A3 : pname, B2 : set_pname]: (member_pname @ A3 @ (insert_pname @ A3 @ B2))))). % insertI1
thf(fact_142_insertI1, axiom,
    ((![A3 : a, B2 : set_a]: (member_a @ A3 @ (insert_a @ A3 @ B2))))). % insertI1
thf(fact_143_insertI1, axiom,
    ((![A3 : nat, B2 : set_nat]: (member_nat @ A3 @ (insert_nat @ A3 @ B2))))). % insertI1
thf(fact_144_insertE, axiom,
    ((![A3 : pname, B3 : pname, A : set_pname]: ((member_pname @ A3 @ (insert_pname @ B3 @ A)) => ((~ ((A3 = B3))) => (member_pname @ A3 @ A)))))). % insertE
thf(fact_145_insertE, axiom,
    ((![A3 : a, B3 : a, A : set_a]: ((member_a @ A3 @ (insert_a @ B3 @ A)) => ((~ ((A3 = B3))) => (member_a @ A3 @ A)))))). % insertE
thf(fact_146_insertE, axiom,
    ((![A3 : nat, B3 : nat, A : set_nat]: ((member_nat @ A3 @ (insert_nat @ B3 @ A)) => ((~ ((A3 = B3))) => (member_nat @ A3 @ A)))))). % insertE
thf(fact_147_Nat_Oex__has__greatest__nat, axiom,
    ((![P : nat > $o, K : nat, B3 : nat]: ((P @ K) => ((![Y4 : nat]: ((P @ Y4) => (ord_less_eq_nat @ Y4 @ B3))) => (?[X4 : nat]: ((P @ X4) & (![Y5 : nat]: ((P @ Y5) => (ord_less_eq_nat @ Y5 @ X4)))))))))). % Nat.ex_has_greatest_nat
thf(fact_148_mem__Collect__eq, axiom,
    ((![A3 : pname, P : pname > $o]: ((member_pname @ A3 @ (collect_pname @ P)) = (P @ A3))))). % mem_Collect_eq
thf(fact_149_mem__Collect__eq, axiom,
    ((![A3 : a, P : a > $o]: ((member_a @ A3 @ (collect_a @ P)) = (P @ A3))))). % mem_Collect_eq
thf(fact_150_mem__Collect__eq, axiom,
    ((![A3 : nat, P : nat > $o]: ((member_nat @ A3 @ (collect_nat @ P)) = (P @ A3))))). % mem_Collect_eq
thf(fact_151_Collect__mem__eq, axiom,
    ((![A : set_pname]: ((collect_pname @ (^[X3 : pname]: (member_pname @ X3 @ A))) = A)))). % Collect_mem_eq
thf(fact_152_Collect__mem__eq, axiom,
    ((![A : set_a]: ((collect_a @ (^[X3 : a]: (member_a @ X3 @ A))) = A)))). % Collect_mem_eq
thf(fact_153_Collect__mem__eq, axiom,
    ((![A : set_nat]: ((collect_nat @ (^[X3 : nat]: (member_nat @ X3 @ A))) = A)))). % Collect_mem_eq
thf(fact_154_Collect__cong, axiom,
    ((![P : nat > $o, Q : nat > $o]: ((![X4 : nat]: ((P @ X4) = (Q @ X4))) => ((collect_nat @ P) = (collect_nat @ Q)))))). % Collect_cong
thf(fact_155_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_156_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_157_eq__imp__le, axiom,
    ((![M : nat, N : nat]: ((M = N) => (ord_less_eq_nat @ M @ N))))). % eq_imp_le
thf(fact_158_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_159_le__refl, axiom,
    ((![N : nat]: (ord_less_eq_nat @ N @ N)))). % le_refl
thf(fact_160_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_161_Compr__image__eq, axiom,
    ((![F : pname > pname, A : set_pname, P : pname > $o]: ((collect_pname @ (^[X3 : pname]: (((member_pname @ X3 @ (image_pname_pname @ F @ A))) & ((P @ X3))))) = (image_pname_pname @ F @ (collect_pname @ (^[X3 : pname]: (((member_pname @ X3 @ A)) & ((P @ (F @ X3))))))))))). % Compr_image_eq
thf(fact_162_Compr__image__eq, axiom,
    ((![F : a > pname, A : set_a, P : pname > $o]: ((collect_pname @ (^[X3 : pname]: (((member_pname @ X3 @ (image_a_pname @ F @ A))) & ((P @ X3))))) = (image_a_pname @ F @ (collect_a @ (^[X3 : a]: (((member_a @ X3 @ A)) & ((P @ (F @ X3))))))))))). % Compr_image_eq
thf(fact_163_Compr__image__eq, axiom,
    ((![F : pname > a, A : set_pname, P : a > $o]: ((collect_a @ (^[X3 : a]: (((member_a @ X3 @ (image_pname_a @ F @ A))) & ((P @ X3))))) = (image_pname_a @ F @ (collect_pname @ (^[X3 : pname]: (((member_pname @ X3 @ A)) & ((P @ (F @ X3))))))))))). % Compr_image_eq
thf(fact_164_Compr__image__eq, axiom,
    ((![F : a > a, A : set_a, P : a > $o]: ((collect_a @ (^[X3 : a]: (((member_a @ X3 @ (image_a_a @ F @ A))) & ((P @ X3))))) = (image_a_a @ F @ (collect_a @ (^[X3 : a]: (((member_a @ X3 @ A)) & ((P @ (F @ X3))))))))))). % Compr_image_eq
thf(fact_165_Compr__image__eq, axiom,
    ((![F : nat > pname, A : set_nat, P : pname > $o]: ((collect_pname @ (^[X3 : pname]: (((member_pname @ X3 @ (image_nat_pname @ F @ A))) & ((P @ X3))))) = (image_nat_pname @ F @ (collect_nat @ (^[X3 : nat]: (((member_nat @ X3 @ A)) & ((P @ (F @ X3))))))))))). % Compr_image_eq
thf(fact_166_Compr__image__eq, axiom,
    ((![F : nat > a, A : set_nat, P : a > $o]: ((collect_a @ (^[X3 : a]: (((member_a @ X3 @ (image_nat_a @ F @ A))) & ((P @ X3))))) = (image_nat_a @ F @ (collect_nat @ (^[X3 : nat]: (((member_nat @ X3 @ A)) & ((P @ (F @ X3))))))))))). % Compr_image_eq
thf(fact_167_Compr__image__eq, axiom,
    ((![F : pname > nat, A : set_pname, P : nat > $o]: ((collect_nat @ (^[X3 : nat]: (((member_nat @ X3 @ (image_pname_nat @ F @ A))) & ((P @ X3))))) = (image_pname_nat @ F @ (collect_pname @ (^[X3 : pname]: (((member_pname @ X3 @ A)) & ((P @ (F @ X3))))))))))). % Compr_image_eq
thf(fact_168_Compr__image__eq, axiom,
    ((![F : a > nat, A : set_a, P : nat > $o]: ((collect_nat @ (^[X3 : nat]: (((member_nat @ X3 @ (image_a_nat @ F @ A))) & ((P @ X3))))) = (image_a_nat @ F @ (collect_a @ (^[X3 : a]: (((member_a @ X3 @ A)) & ((P @ (F @ X3))))))))))). % Compr_image_eq
thf(fact_169_Compr__image__eq, axiom,
    ((![F : nat > nat, A : set_nat, P : nat > $o]: ((collect_nat @ (^[X3 : nat]: (((member_nat @ X3 @ (image_nat_nat @ F @ A))) & ((P @ X3))))) = (image_nat_nat @ F @ (collect_nat @ (^[X3 : nat]: (((member_nat @ X3 @ A)) & ((P @ (F @ X3))))))))))). % Compr_image_eq
thf(fact_170_image__image, axiom,
    ((![F : pname > pname, G : a > pname, A : set_a]: ((image_pname_pname @ F @ (image_a_pname @ G @ A)) = (image_a_pname @ (^[X3 : a]: (F @ (G @ X3))) @ A))))). % image_image
thf(fact_171_image__image, axiom,
    ((![F : pname > nat, G : a > pname, A : set_a]: ((image_pname_nat @ F @ (image_a_pname @ G @ A)) = (image_a_nat @ (^[X3 : a]: (F @ (G @ X3))) @ A))))). % image_image
thf(fact_172_image__image, axiom,
    ((![F : nat > pname, G : a > nat, A : set_a]: ((image_nat_pname @ F @ (image_a_nat @ G @ A)) = (image_a_pname @ (^[X3 : a]: (F @ (G @ X3))) @ A))))). % image_image
thf(fact_173_image__image, axiom,
    ((![F : pname > a, G : pname > pname, A : set_pname]: ((image_pname_a @ F @ (image_pname_pname @ G @ A)) = (image_pname_a @ (^[X3 : pname]: (F @ (G @ X3))) @ A))))). % image_image
thf(fact_174_image__image, axiom,
    ((![F : pname > a, G : nat > pname, A : set_nat]: ((image_pname_a @ F @ (image_nat_pname @ G @ A)) = (image_nat_a @ (^[X3 : nat]: (F @ (G @ X3))) @ A))))). % image_image
thf(fact_175_image__image, axiom,
    ((![F : pname > a, G : a > pname, A : set_a]: ((image_pname_a @ F @ (image_a_pname @ G @ A)) = (image_a_a @ (^[X3 : a]: (F @ (G @ X3))) @ A))))). % image_image
thf(fact_176_image__image, axiom,
    ((![F : nat > nat, G : nat > nat, A : set_nat]: ((image_nat_nat @ F @ (image_nat_nat @ G @ A)) = (image_nat_nat @ (^[X3 : nat]: (F @ (G @ X3))) @ A))))). % image_image
thf(fact_177_image__image, axiom,
    ((![F : nat > nat, G : a > nat, A : set_a]: ((image_nat_nat @ F @ (image_a_nat @ G @ A)) = (image_a_nat @ (^[X3 : a]: (F @ (G @ X3))) @ A))))). % image_image
thf(fact_178_image__image, axiom,
    ((![F : nat > a, G : pname > nat, A : set_pname]: ((image_nat_a @ F @ (image_pname_nat @ G @ A)) = (image_pname_a @ (^[X3 : pname]: (F @ (G @ X3))) @ A))))). % image_image
thf(fact_179_image__image, axiom,
    ((![F : nat > a, G : nat > nat, A : set_nat]: ((image_nat_a @ F @ (image_nat_nat @ G @ A)) = (image_nat_a @ (^[X3 : nat]: (F @ (G @ X3))) @ A))))). % image_image
thf(fact_180_imageE, axiom,
    ((![B3 : pname, F : pname > pname, A : set_pname]: ((member_pname @ B3 @ (image_pname_pname @ F @ A)) => (~ ((![X4 : pname]: ((B3 = (F @ X4)) => (~ ((member_pname @ X4 @ A))))))))))). % imageE
thf(fact_181_imageE, axiom,
    ((![B3 : pname, F : a > pname, A : set_a]: ((member_pname @ B3 @ (image_a_pname @ F @ A)) => (~ ((![X4 : a]: ((B3 = (F @ X4)) => (~ ((member_a @ X4 @ A))))))))))). % imageE
thf(fact_182_imageE, axiom,
    ((![B3 : pname, F : nat > pname, A : set_nat]: ((member_pname @ B3 @ (image_nat_pname @ F @ A)) => (~ ((![X4 : nat]: ((B3 = (F @ X4)) => (~ ((member_nat @ X4 @ A))))))))))). % imageE
thf(fact_183_imageE, axiom,
    ((![B3 : a, F : pname > a, A : set_pname]: ((member_a @ B3 @ (image_pname_a @ F @ A)) => (~ ((![X4 : pname]: ((B3 = (F @ X4)) => (~ ((member_pname @ X4 @ A))))))))))). % imageE
thf(fact_184_imageE, axiom,
    ((![B3 : a, F : a > a, A : set_a]: ((member_a @ B3 @ (image_a_a @ F @ A)) => (~ ((![X4 : a]: ((B3 = (F @ X4)) => (~ ((member_a @ X4 @ A))))))))))). % imageE
thf(fact_185_imageE, axiom,
    ((![B3 : a, F : nat > a, A : set_nat]: ((member_a @ B3 @ (image_nat_a @ F @ A)) => (~ ((![X4 : nat]: ((B3 = (F @ X4)) => (~ ((member_nat @ X4 @ A))))))))))). % imageE
thf(fact_186_imageE, axiom,
    ((![B3 : nat, F : pname > nat, A : set_pname]: ((member_nat @ B3 @ (image_pname_nat @ F @ A)) => (~ ((![X4 : pname]: ((B3 = (F @ X4)) => (~ ((member_pname @ X4 @ A))))))))))). % imageE
thf(fact_187_imageE, axiom,
    ((![B3 : nat, F : a > nat, A : set_a]: ((member_nat @ B3 @ (image_a_nat @ F @ A)) => (~ ((![X4 : a]: ((B3 = (F @ X4)) => (~ ((member_a @ X4 @ A))))))))))). % imageE
thf(fact_188_imageE, axiom,
    ((![B3 : nat, F : nat > nat, A : set_nat]: ((member_nat @ B3 @ (image_nat_nat @ F @ A)) => (~ ((![X4 : nat]: ((B3 = (F @ X4)) => (~ ((member_nat @ X4 @ A))))))))))). % imageE
thf(fact_189_pigeonhole__infinite__rel, axiom,
    ((![A : set_pname, B2 : set_pname, R : pname > pname > $o]: ((~ ((finite_finite_pname @ A))) => ((finite_finite_pname @ B2) => ((![X4 : pname]: ((member_pname @ X4 @ A) => (?[Xa : pname]: ((member_pname @ Xa @ B2) & (R @ X4 @ Xa))))) => (?[X4 : pname]: ((member_pname @ X4 @ B2) & (~ ((finite_finite_pname @ (collect_pname @ (^[A2 : pname]: (((member_pname @ A2 @ A)) & ((R @ A2 @ X4)))))))))))))))). % pigeonhole_infinite_rel
thf(fact_190_pigeonhole__infinite__rel, axiom,
    ((![A : set_pname, B2 : set_a, R : pname > a > $o]: ((~ ((finite_finite_pname @ A))) => ((finite_finite_a @ B2) => ((![X4 : pname]: ((member_pname @ X4 @ A) => (?[Xa : a]: ((member_a @ Xa @ B2) & (R @ X4 @ Xa))))) => (?[X4 : a]: ((member_a @ X4 @ B2) & (~ ((finite_finite_pname @ (collect_pname @ (^[A2 : pname]: (((member_pname @ A2 @ A)) & ((R @ A2 @ X4)))))))))))))))). % pigeonhole_infinite_rel
thf(fact_191_pigeonhole__infinite__rel, axiom,
    ((![A : set_pname, B2 : set_nat, R : pname > nat > $o]: ((~ ((finite_finite_pname @ A))) => ((finite_finite_nat @ B2) => ((![X4 : pname]: ((member_pname @ X4 @ A) => (?[Xa : nat]: ((member_nat @ Xa @ B2) & (R @ X4 @ Xa))))) => (?[X4 : nat]: ((member_nat @ X4 @ B2) & (~ ((finite_finite_pname @ (collect_pname @ (^[A2 : pname]: (((member_pname @ A2 @ A)) & ((R @ A2 @ X4)))))))))))))))). % pigeonhole_infinite_rel
thf(fact_192_pigeonhole__infinite__rel, axiom,
    ((![A : set_a, B2 : set_pname, R : a > pname > $o]: ((~ ((finite_finite_a @ A))) => ((finite_finite_pname @ B2) => ((![X4 : a]: ((member_a @ X4 @ A) => (?[Xa : pname]: ((member_pname @ Xa @ B2) & (R @ X4 @ Xa))))) => (?[X4 : pname]: ((member_pname @ X4 @ B2) & (~ ((finite_finite_a @ (collect_a @ (^[A2 : a]: (((member_a @ A2 @ A)) & ((R @ A2 @ X4)))))))))))))))). % pigeonhole_infinite_rel
thf(fact_193_pigeonhole__infinite__rel, axiom,
    ((![A : set_a, B2 : set_a, R : a > a > $o]: ((~ ((finite_finite_a @ A))) => ((finite_finite_a @ B2) => ((![X4 : a]: ((member_a @ X4 @ A) => (?[Xa : a]: ((member_a @ Xa @ B2) & (R @ X4 @ Xa))))) => (?[X4 : a]: ((member_a @ X4 @ B2) & (~ ((finite_finite_a @ (collect_a @ (^[A2 : a]: (((member_a @ A2 @ A)) & ((R @ A2 @ X4)))))))))))))))). % pigeonhole_infinite_rel
thf(fact_194_pigeonhole__infinite__rel, axiom,
    ((![A : set_a, B2 : set_nat, R : a > nat > $o]: ((~ ((finite_finite_a @ A))) => ((finite_finite_nat @ B2) => ((![X4 : a]: ((member_a @ X4 @ A) => (?[Xa : nat]: ((member_nat @ Xa @ B2) & (R @ X4 @ Xa))))) => (?[X4 : nat]: ((member_nat @ X4 @ B2) & (~ ((finite_finite_a @ (collect_a @ (^[A2 : a]: (((member_a @ A2 @ A)) & ((R @ A2 @ X4)))))))))))))))). % pigeonhole_infinite_rel
thf(fact_195_pigeonhole__infinite__rel, axiom,
    ((![A : set_nat, B2 : set_pname, R : nat > pname > $o]: ((~ ((finite_finite_nat @ A))) => ((finite_finite_pname @ B2) => ((![X4 : nat]: ((member_nat @ X4 @ A) => (?[Xa : pname]: ((member_pname @ Xa @ B2) & (R @ X4 @ Xa))))) => (?[X4 : pname]: ((member_pname @ X4 @ B2) & (~ ((finite_finite_nat @ (collect_nat @ (^[A2 : nat]: (((member_nat @ A2 @ A)) & ((R @ A2 @ X4)))))))))))))))). % pigeonhole_infinite_rel
thf(fact_196_pigeonhole__infinite__rel, axiom,
    ((![A : set_nat, B2 : set_a, R : nat > a > $o]: ((~ ((finite_finite_nat @ A))) => ((finite_finite_a @ B2) => ((![X4 : nat]: ((member_nat @ X4 @ A) => (?[Xa : a]: ((member_a @ Xa @ B2) & (R @ X4 @ Xa))))) => (?[X4 : a]: ((member_a @ X4 @ B2) & (~ ((finite_finite_nat @ (collect_nat @ (^[A2 : nat]: (((member_nat @ A2 @ A)) & ((R @ A2 @ X4)))))))))))))))). % pigeonhole_infinite_rel
thf(fact_197_pigeonhole__infinite__rel, axiom,
    ((![A : set_nat, B2 : set_nat, R : nat > nat > $o]: ((~ ((finite_finite_nat @ A))) => ((finite_finite_nat @ B2) => ((![X4 : nat]: ((member_nat @ X4 @ A) => (?[Xa : nat]: ((member_nat @ Xa @ B2) & (R @ X4 @ Xa))))) => (?[X4 : nat]: ((member_nat @ X4 @ B2) & (~ ((finite_finite_nat @ (collect_nat @ (^[A2 : nat]: (((member_nat @ A2 @ A)) & ((R @ A2 @ X4)))))))))))))))). % pigeonhole_infinite_rel
thf(fact_198_not__finite__existsD, axiom,
    ((![P : pname > $o]: ((~ ((finite_finite_pname @ (collect_pname @ P)))) => (?[X_1 : pname]: (P @ X_1)))))). % not_finite_existsD
thf(fact_199_not__finite__existsD, axiom,
    ((![P : a > $o]: ((~ ((finite_finite_a @ (collect_a @ P)))) => (?[X_1 : a]: (P @ X_1)))))). % not_finite_existsD
thf(fact_200_not__finite__existsD, axiom,
    ((![P : nat > $o]: ((~ ((finite_finite_nat @ (collect_nat @ P)))) => (?[X_1 : nat]: (P @ X_1)))))). % not_finite_existsD
thf(fact_201_insert__Collect, axiom,
    ((![A3 : a, P : a > $o]: ((insert_a @ A3 @ (collect_a @ P)) = (collect_a @ (^[U : a]: (((~ ((U = A3)))) => ((P @ U))))))))). % insert_Collect
thf(fact_202_insert__Collect, axiom,
    ((![A3 : pname, P : pname > $o]: ((insert_pname @ A3 @ (collect_pname @ P)) = (collect_pname @ (^[U : pname]: (((~ ((U = A3)))) => ((P @ U))))))))). % insert_Collect
thf(fact_203_insert__Collect, axiom,
    ((![A3 : nat, P : nat > $o]: ((insert_nat @ A3 @ (collect_nat @ P)) = (collect_nat @ (^[U : nat]: (((~ ((U = A3)))) => ((P @ U))))))))). % insert_Collect
thf(fact_204_insert__compr, axiom,
    ((insert_pname = (^[A2 : pname]: (^[B : set_pname]: (collect_pname @ (^[X3 : pname]: (((X3 = A2)) | ((member_pname @ X3 @ B)))))))))). % insert_compr
thf(fact_205_insert__compr, axiom,
    ((insert_a = (^[A2 : a]: (^[B : set_a]: (collect_a @ (^[X3 : a]: (((X3 = A2)) | ((member_a @ X3 @ B)))))))))). % insert_compr
thf(fact_206_insert__compr, axiom,
    ((insert_nat = (^[A2 : nat]: (^[B : set_nat]: (collect_nat @ (^[X3 : nat]: (((X3 = A2)) | ((member_nat @ X3 @ B)))))))))). % insert_compr
thf(fact_207_finite__has__minimal2, axiom,
    ((![A : set_nat, A3 : nat]: ((finite_finite_nat @ A) => ((member_nat @ A3 @ A) => (?[X4 : nat]: ((member_nat @ X4 @ A) & ((ord_less_eq_nat @ X4 @ A3) & (![Xa : nat]: ((member_nat @ Xa @ A) => ((ord_less_eq_nat @ Xa @ X4) => (X4 = Xa)))))))))))). % finite_has_minimal2
thf(fact_208_finite__has__maximal2, axiom,
    ((![A : set_nat, A3 : nat]: ((finite_finite_nat @ A) => ((member_nat @ A3 @ A) => (?[X4 : nat]: ((member_nat @ X4 @ A) & ((ord_less_eq_nat @ A3 @ X4) & (![Xa : nat]: ((member_nat @ Xa @ A) => ((ord_less_eq_nat @ X4 @ Xa) => (X4 = Xa)))))))))))). % finite_has_maximal2
thf(fact_209_finite_OinsertI, axiom,
    ((![A : set_pname, A3 : pname]: ((finite_finite_pname @ A) => (finite_finite_pname @ (insert_pname @ A3 @ A)))))). % finite.insertI
thf(fact_210_finite_OinsertI, axiom,
    ((![A : set_a, A3 : a]: ((finite_finite_a @ A) => (finite_finite_a @ (insert_a @ A3 @ A)))))). % finite.insertI
thf(fact_211_finite_OinsertI, axiom,
    ((![A : set_nat, A3 : nat]: ((finite_finite_nat @ A) => (finite_finite_nat @ (insert_nat @ A3 @ A)))))). % finite.insertI
thf(fact_212_transitive__stepwise__le, axiom,
    ((![M : nat, N : nat, R : nat > nat > $o]: ((ord_less_eq_nat @ M @ N) => ((![X4 : nat]: (R @ X4 @ X4)) => ((![X4 : nat, Y4 : nat, Z2 : nat]: ((R @ X4 @ Y4) => ((R @ Y4 @ Z2) => (R @ X4 @ Z2)))) => ((![N4 : nat]: (R @ N4 @ (suc @ N4))) => (R @ M @ N)))))))). % transitive_stepwise_le
thf(fact_213_nat__induct__at__least, axiom,
    ((![M : nat, N : nat, P : nat > $o]: ((ord_less_eq_nat @ M @ N) => ((P @ M) => ((![N4 : nat]: ((ord_less_eq_nat @ M @ N4) => ((P @ N4) => (P @ (suc @ N4))))) => (P @ N))))))). % nat_induct_at_least
thf(fact_214_full__nat__induct, axiom,
    ((![P : nat > $o, N : nat]: ((![N4 : nat]: ((![M3 : nat]: ((ord_less_eq_nat @ (suc @ M3) @ N4) => (P @ M3))) => (P @ N4))) => (P @ N))))). % full_nat_induct
thf(fact_215_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_216_Suc__n__not__le__n, axiom,
    ((![N : nat]: (~ ((ord_less_eq_nat @ (suc @ N) @ N)))))). % Suc_n_not_le_n
thf(fact_217_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_218_Suc__le__D, axiom,
    ((![N : nat, M4 : nat]: ((ord_less_eq_nat @ (suc @ N) @ M4) => (?[M5 : nat]: (M4 = (suc @ M5))))))). % Suc_le_D
thf(fact_219_le__SucI, axiom,
    ((![M : nat, N : nat]: ((ord_less_eq_nat @ M @ N) => (ord_less_eq_nat @ M @ (suc @ N)))))). % le_SucI
thf(fact_220_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_221_Suc__leD, axiom,
    ((![M : nat, N : nat]: ((ord_less_eq_nat @ (suc @ M) @ N) => (ord_less_eq_nat @ M @ N))))). % Suc_leD
thf(fact_222_zero__induct__lemma, axiom,
    ((![P : nat > $o, K : nat, I : nat]: ((P @ K) => ((![N4 : nat]: ((P @ (suc @ N4)) => (P @ N4))) => (P @ (minus_minus_nat @ K @ I))))))). % zero_induct_lemma
thf(fact_223_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_224_le__diff__iff_H, axiom,
    ((![A3 : nat, C2 : nat, B3 : nat]: ((ord_less_eq_nat @ A3 @ C2) => ((ord_less_eq_nat @ B3 @ C2) => ((ord_less_eq_nat @ (minus_minus_nat @ C2 @ A3) @ (minus_minus_nat @ C2 @ B3)) = (ord_less_eq_nat @ B3 @ A3))))))). % le_diff_iff'
thf(fact_225_diff__le__self, axiom,
    ((![M : nat, N : nat]: (ord_less_eq_nat @ (minus_minus_nat @ M @ N) @ M)))). % diff_le_self
thf(fact_226_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_227_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_228_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_229_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_230_pigeonhole__infinite, axiom,
    ((![A : set_pname, F : pname > pname]: ((~ ((finite_finite_pname @ A))) => ((finite_finite_pname @ (image_pname_pname @ F @ A)) => (?[X4 : pname]: ((member_pname @ X4 @ A) & (~ ((finite_finite_pname @ (collect_pname @ (^[A2 : pname]: (((member_pname @ A2 @ A)) & (((F @ A2) = (F @ X4)))))))))))))))). % pigeonhole_infinite
thf(fact_231_pigeonhole__infinite, axiom,
    ((![A : set_pname, F : pname > a]: ((~ ((finite_finite_pname @ A))) => ((finite_finite_a @ (image_pname_a @ F @ A)) => (?[X4 : pname]: ((member_pname @ X4 @ A) & (~ ((finite_finite_pname @ (collect_pname @ (^[A2 : pname]: (((member_pname @ A2 @ A)) & (((F @ A2) = (F @ X4)))))))))))))))). % pigeonhole_infinite
thf(fact_232_pigeonhole__infinite, axiom,
    ((![A : set_pname, F : pname > nat]: ((~ ((finite_finite_pname @ A))) => ((finite_finite_nat @ (image_pname_nat @ F @ A)) => (?[X4 : pname]: ((member_pname @ X4 @ A) & (~ ((finite_finite_pname @ (collect_pname @ (^[A2 : pname]: (((member_pname @ A2 @ A)) & (((F @ A2) = (F @ X4)))))))))))))))). % pigeonhole_infinite
thf(fact_233_pigeonhole__infinite, axiom,
    ((![A : set_a, F : a > pname]: ((~ ((finite_finite_a @ A))) => ((finite_finite_pname @ (image_a_pname @ F @ A)) => (?[X4 : a]: ((member_a @ X4 @ A) & (~ ((finite_finite_a @ (collect_a @ (^[A2 : a]: (((member_a @ A2 @ A)) & (((F @ A2) = (F @ X4)))))))))))))))). % pigeonhole_infinite
thf(fact_234_pigeonhole__infinite, axiom,
    ((![A : set_a, F : a > a]: ((~ ((finite_finite_a @ A))) => ((finite_finite_a @ (image_a_a @ F @ A)) => (?[X4 : a]: ((member_a @ X4 @ A) & (~ ((finite_finite_a @ (collect_a @ (^[A2 : a]: (((member_a @ A2 @ A)) & (((F @ A2) = (F @ X4)))))))))))))))). % pigeonhole_infinite
thf(fact_235_pigeonhole__infinite, axiom,
    ((![A : set_a, F : a > nat]: ((~ ((finite_finite_a @ A))) => ((finite_finite_nat @ (image_a_nat @ F @ A)) => (?[X4 : a]: ((member_a @ X4 @ A) & (~ ((finite_finite_a @ (collect_a @ (^[A2 : a]: (((member_a @ A2 @ A)) & (((F @ A2) = (F @ X4)))))))))))))))). % pigeonhole_infinite
thf(fact_236_pigeonhole__infinite, axiom,
    ((![A : set_nat, F : nat > pname]: ((~ ((finite_finite_nat @ A))) => ((finite_finite_pname @ (image_nat_pname @ F @ A)) => (?[X4 : nat]: ((member_nat @ X4 @ A) & (~ ((finite_finite_nat @ (collect_nat @ (^[A2 : nat]: (((member_nat @ A2 @ A)) & (((F @ A2) = (F @ X4)))))))))))))))). % pigeonhole_infinite
thf(fact_237_pigeonhole__infinite, axiom,
    ((![A : set_nat, F : nat > a]: ((~ ((finite_finite_nat @ A))) => ((finite_finite_a @ (image_nat_a @ F @ A)) => (?[X4 : nat]: ((member_nat @ X4 @ A) & (~ ((finite_finite_nat @ (collect_nat @ (^[A2 : nat]: (((member_nat @ A2 @ A)) & (((F @ A2) = (F @ X4)))))))))))))))). % pigeonhole_infinite
thf(fact_238_pigeonhole__infinite, axiom,
    ((![A : set_nat, F : nat > nat]: ((~ ((finite_finite_nat @ A))) => ((finite_finite_nat @ (image_nat_nat @ F @ A)) => (?[X4 : nat]: ((member_nat @ X4 @ A) & (~ ((finite_finite_nat @ (collect_nat @ (^[A2 : nat]: (((member_nat @ A2 @ A)) & (((F @ A2) = (F @ X4)))))))))))))))). % pigeonhole_infinite
thf(fact_239_lift__Suc__antimono__le, axiom,
    ((![F : nat > nat, N : nat, N5 : nat]: ((![N4 : nat]: (ord_less_eq_nat @ (F @ (suc @ N4)) @ (F @ N4))) => ((ord_less_eq_nat @ N @ N5) => (ord_less_eq_nat @ (F @ N5) @ (F @ N))))))). % lift_Suc_antimono_le
thf(fact_240_lift__Suc__mono__le, axiom,
    ((![F : nat > nat, N : nat, N5 : nat]: ((![N4 : nat]: (ord_less_eq_nat @ (F @ N4) @ (F @ (suc @ N4)))) => ((ord_less_eq_nat @ N @ N5) => (ord_less_eq_nat @ (F @ N) @ (F @ N5))))))). % lift_Suc_mono_le
thf(fact_241_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_242_card__image__le, axiom,
    ((![A : set_nat, F : nat > a]: ((finite_finite_nat @ A) => (ord_less_eq_nat @ (finite_card_a @ (image_nat_a @ F @ A)) @ (finite_card_nat @ A)))))). % card_image_le
thf(fact_243_card__image__le, axiom,
    ((![A : set_nat, F : nat > nat]: ((finite_finite_nat @ A) => (ord_less_eq_nat @ (finite_card_nat @ (image_nat_nat @ F @ A)) @ (finite_card_nat @ A)))))). % card_image_le
thf(fact_244_card__image__le, axiom,
    ((![A : set_nat, F : nat > pname]: ((finite_finite_nat @ A) => (ord_less_eq_nat @ (finite_card_pname @ (image_nat_pname @ F @ A)) @ (finite_card_nat @ A)))))). % card_image_le
thf(fact_245_Suc__le__D__lemma, axiom,
    ((![N : nat, M4 : nat, P : nat > $o]: ((ord_less_eq_nat @ (suc @ N) @ M4) => ((![M5 : nat]: ((ord_less_eq_nat @ N @ M5) => (P @ (suc @ M5)))) => (P @ M4)))))). % Suc_le_D_lemma
thf(fact_246_finite__less__ub, axiom,
    ((![F : nat > nat, U2 : nat]: ((![N4 : nat]: (ord_less_eq_nat @ N4 @ (F @ N4))) => (finite_finite_nat @ (collect_nat @ (^[N2 : nat]: (ord_less_eq_nat @ (F @ N2) @ U2)))))))). % finite_less_ub

% Conjectures (8)
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, hypothesis,
    ((finite_finite_a @ g))).
thf(conj_7, conjecture,
    (((finite_card_a @ (insert_a @ (mgt_call @ pn) @ g)) = (minus_minus_nat @ (finite_card_a @ (image_pname_a @ mgt_call @ u)) @ na)))).
