% 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/Arrow_Order/prob_256__5188744_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:17:00.401

% Could-be-implicit typings (5)
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Arrow____Order____Mirabelle____riepwfubkl__Oindi_J_J, type,
    set_se1867414201e_indi : $tType).
thf(ty_n_t__Set__Oset_It__Arrow____Order____Mirabelle____riepwfubkl__Oindi_J, type,
    set_Ar1007576579e_indi : $tType).
thf(ty_n_t__Arrow____Order____Mirabelle____riepwfubkl__Oindi, type,
    arrow_1429744205e_indi : $tType).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J, type,
    set_nat : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).

% Explicit typings (35)
thf(sy_c_Finite__Set_OFpow_001t__Arrow____Order____Mirabelle____riepwfubkl__Oindi, type,
    finite110102887e_indi : set_Ar1007576579e_indi > set_se1867414201e_indi).
thf(sy_c_Finite__Set_Ocard_001t__Arrow____Order____Mirabelle____riepwfubkl__Oindi, type,
    finite927127589e_indi : set_Ar1007576579e_indi > nat).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat, type,
    finite_card_nat : set_nat > nat).
thf(sy_c_Finite__Set_Ofinite_001t__Arrow____Order____Mirabelle____riepwfubkl__Oindi, type,
    finite183240804e_indi : set_Ar1007576579e_indi > $o).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat, type,
    finite_finite_nat : set_nat > $o).
thf(sy_c_Fun_Oinj__on_001t__Arrow____Order____Mirabelle____riepwfubkl__Oindi_001t__Arrow____Order____Mirabelle____riepwfubkl__Oindi, type,
    inj_on1663454827e_indi : (arrow_1429744205e_indi > arrow_1429744205e_indi) > set_Ar1007576579e_indi > $o).
thf(sy_c_Fun_Oinj__on_001t__Arrow____Order____Mirabelle____riepwfubkl__Oindi_001t__Nat__Onat, type,
    inj_on528257168di_nat : (arrow_1429744205e_indi > nat) > set_Ar1007576579e_indi > $o).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Arrow____Order____Mirabelle____riepwfubkl__Oindi, type,
    inj_on1775343632e_indi : (nat > arrow_1429744205e_indi) > set_nat > $o).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat, type,
    inj_on_nat_nat : (nat > nat) > set_nat > $o).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Arrow____Order____Mirabelle____riepwfubkl__Oindi_J_001t__Set__Oset_It__Arrow____Order____Mirabelle____riepwfubkl__Oindi_J, type,
    inj_on5180053e_indi : (set_Ar1007576579e_indi > set_Ar1007576579e_indi) > set_se1867414201e_indi > $o).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Arrow____Order____Mirabelle____riepwfubkl__Oindi_J_001t__Set__Oset_It__Nat__Onat_J, type,
    inj_on384466940et_nat : (set_Ar1007576579e_indi > set_nat) > set_se1867414201e_indi > $o).
thf(sy_c_Fun_Oswap_001t__Arrow____Order____Mirabelle____riepwfubkl__Oindi_001t__Arrow____Order____Mirabelle____riepwfubkl__Oindi, type,
    swap_A1408198358e_indi : arrow_1429744205e_indi > arrow_1429744205e_indi > (arrow_1429744205e_indi > arrow_1429744205e_indi) > arrow_1429744205e_indi > arrow_1429744205e_indi).
thf(sy_c_Fun_Oswap_001t__Arrow____Order____Mirabelle____riepwfubkl__Oindi_001t__Nat__Onat, type,
    swap_A1227367653di_nat : arrow_1429744205e_indi > arrow_1429744205e_indi > (arrow_1429744205e_indi > nat) > arrow_1429744205e_indi > nat).
thf(sy_c_Fun_Othe__inv__into_001t__Arrow____Order____Mirabelle____riepwfubkl__Oindi_001t__Arrow____Order____Mirabelle____riepwfubkl__Oindi, type,
    the_in1730881961e_indi : set_Ar1007576579e_indi > (arrow_1429744205e_indi > arrow_1429744205e_indi) > arrow_1429744205e_indi > arrow_1429744205e_indi).
thf(sy_c_Fun_Othe__inv__into_001t__Arrow____Order____Mirabelle____riepwfubkl__Oindi_001t__Nat__Onat, type,
    the_in1052612818di_nat : set_Ar1007576579e_indi > (arrow_1429744205e_indi > nat) > nat > arrow_1429744205e_indi).
thf(sy_c_Fun_Othe__inv__into_001t__Nat__Onat_001t__Arrow____Order____Mirabelle____riepwfubkl__Oindi, type,
    the_in152215634e_indi : set_nat > (nat > arrow_1429744205e_indi) > arrow_1429744205e_indi > nat).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat, type,
    zero_zero_nat : nat).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat, type,
    ord_less_nat : nat > nat > $o).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Arrow____Order____Mirabelle____riepwfubkl__Oindi_J, type,
    ord_le1187139159e_indi : set_Ar1007576579e_indi > set_Ar1007576579e_indi > $o).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J, type,
    ord_less_set_nat : set_nat > set_nat > $o).
thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Nat__Onat_001t__Nat__Onat, type,
    order_769474267at_nat : (nat > nat) > $o).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Arrow____Order____Mirabelle____riepwfubkl__Oindi_M_Eo_J, type,
    top_to1473733010indi_o : arrow_1429744205e_indi > $o).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Nat__Onat_M_Eo_J, type,
    top_top_nat_o : nat > $o).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Arrow____Order____Mirabelle____riepwfubkl__Oindi_J, type,
    top_to1799531699e_indi : set_Ar1007576579e_indi).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J, type,
    top_top_set_nat : set_nat).
thf(sy_c_Set_OCollect_001t__Arrow____Order____Mirabelle____riepwfubkl__Oindi, type,
    collec1169676194e_indi : (arrow_1429744205e_indi > $o) > set_Ar1007576579e_indi).
thf(sy_c_Set_OCollect_001t__Nat__Onat, type,
    collect_nat : (nat > $o) > set_nat).
thf(sy_c_Set_Oimage_001t__Arrow____Order____Mirabelle____riepwfubkl__Oindi_001t__Arrow____Order____Mirabelle____riepwfubkl__Oindi, type,
    image_688677079e_indi : (arrow_1429744205e_indi > arrow_1429744205e_indi) > set_Ar1007576579e_indi > set_Ar1007576579e_indi).
thf(sy_c_Set_Oimage_001t__Arrow____Order____Mirabelle____riepwfubkl__Oindi_001t__Nat__Onat, type,
    image_555606308di_nat : (arrow_1429744205e_indi > nat) > set_Ar1007576579e_indi > set_nat).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Arrow____Order____Mirabelle____riepwfubkl__Oindi, type,
    image_1802692772e_indi : (nat > arrow_1429744205e_indi) > set_nat > set_Ar1007576579e_indi).
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__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat, type,
    set_or562006527an_nat : nat > nat > set_nat).
thf(sy_c_member_001t__Arrow____Order____Mirabelle____riepwfubkl__Oindi, type,
    member1966420836e_indi : arrow_1429744205e_indi > set_Ar1007576579e_indi > $o).
thf(sy_c_member_001t__Nat__Onat, type,
    member_nat : nat > set_nat > $o).
thf(sy_v_thesis____, type,
    thesis : $o).

% Relevant facts (192)
thf(fact_0_card__image, axiom,
    ((![F : nat > arrow_1429744205e_indi, A : set_nat]: ((inj_on1775343632e_indi @ F @ A) => ((finite927127589e_indi @ (image_1802692772e_indi @ F @ A)) = (finite_card_nat @ A)))))). % card_image
thf(fact_1_card__image, axiom,
    ((![F : nat > nat, A : set_nat]: ((inj_on_nat_nat @ F @ A) => ((finite_card_nat @ (image_nat_nat @ F @ A)) = (finite_card_nat @ A)))))). % card_image
thf(fact_2_card__image, axiom,
    ((![F : arrow_1429744205e_indi > nat, A : set_Ar1007576579e_indi]: ((inj_on528257168di_nat @ F @ A) => ((finite_card_nat @ (image_555606308di_nat @ F @ A)) = (finite927127589e_indi @ A)))))). % card_image
thf(fact_3_card__image, axiom,
    ((![F : arrow_1429744205e_indi > arrow_1429744205e_indi, A : set_Ar1007576579e_indi]: ((inj_on1663454827e_indi @ F @ A) => ((finite927127589e_indi @ (image_688677079e_indi @ F @ A)) = (finite927127589e_indi @ A)))))). % card_image
thf(fact_4_range__ex1__eq, axiom,
    ((![F : arrow_1429744205e_indi > arrow_1429744205e_indi, B : arrow_1429744205e_indi]: ((inj_on1663454827e_indi @ F @ top_to1799531699e_indi) => ((member1966420836e_indi @ B @ (image_688677079e_indi @ F @ top_to1799531699e_indi)) = (?[X : arrow_1429744205e_indi]: (((B = (F @ X))) & ((![Y : arrow_1429744205e_indi]: (((B = (F @ Y))) => ((Y = X)))))))))))). % range_ex1_eq
thf(fact_5_range__ex1__eq, axiom,
    ((![F : arrow_1429744205e_indi > nat, B : nat]: ((inj_on528257168di_nat @ F @ top_to1799531699e_indi) => ((member_nat @ B @ (image_555606308di_nat @ F @ top_to1799531699e_indi)) = (?[X : arrow_1429744205e_indi]: (((B = (F @ X))) & ((![Y : arrow_1429744205e_indi]: (((B = (F @ Y))) => ((Y = X)))))))))))). % range_ex1_eq
thf(fact_6_inj__image__eq__iff, axiom,
    ((![F : arrow_1429744205e_indi > arrow_1429744205e_indi, A : set_Ar1007576579e_indi, B2 : set_Ar1007576579e_indi]: ((inj_on1663454827e_indi @ F @ top_to1799531699e_indi) => (((image_688677079e_indi @ F @ A) = (image_688677079e_indi @ F @ B2)) = (A = B2)))))). % inj_image_eq_iff
thf(fact_7_inj__image__eq__iff, axiom,
    ((![F : arrow_1429744205e_indi > nat, A : set_Ar1007576579e_indi, B2 : set_Ar1007576579e_indi]: ((inj_on528257168di_nat @ F @ top_to1799531699e_indi) => (((image_555606308di_nat @ F @ A) = (image_555606308di_nat @ F @ B2)) = (A = B2)))))). % inj_image_eq_iff
thf(fact_8_inj__image__mem__iff, axiom,
    ((![F : nat > nat, A2 : nat, A : set_nat]: ((inj_on_nat_nat @ F @ top_top_set_nat) => ((member_nat @ (F @ A2) @ (image_nat_nat @ F @ A)) = (member_nat @ A2 @ A)))))). % inj_image_mem_iff
thf(fact_9_inj__image__mem__iff, axiom,
    ((![F : nat > arrow_1429744205e_indi, A2 : nat, A : set_nat]: ((inj_on1775343632e_indi @ F @ top_top_set_nat) => ((member1966420836e_indi @ (F @ A2) @ (image_1802692772e_indi @ F @ A)) = (member_nat @ A2 @ A)))))). % inj_image_mem_iff
thf(fact_10_inj__image__mem__iff, axiom,
    ((![F : arrow_1429744205e_indi > arrow_1429744205e_indi, A2 : arrow_1429744205e_indi, A : set_Ar1007576579e_indi]: ((inj_on1663454827e_indi @ F @ top_to1799531699e_indi) => ((member1966420836e_indi @ (F @ A2) @ (image_688677079e_indi @ F @ A)) = (member1966420836e_indi @ A2 @ A)))))). % inj_image_mem_iff
thf(fact_11_inj__image__mem__iff, axiom,
    ((![F : arrow_1429744205e_indi > nat, A2 : arrow_1429744205e_indi, A : set_Ar1007576579e_indi]: ((inj_on528257168di_nat @ F @ top_to1799531699e_indi) => ((member_nat @ (F @ A2) @ (image_555606308di_nat @ F @ A)) = (member1966420836e_indi @ A2 @ A)))))). % inj_image_mem_iff
thf(fact_12_UNIV__I, axiom,
    ((![X2 : nat]: (member_nat @ X2 @ top_top_set_nat)))). % UNIV_I
thf(fact_13_UNIV__I, axiom,
    ((![X2 : arrow_1429744205e_indi]: (member1966420836e_indi @ X2 @ top_to1799531699e_indi)))). % UNIV_I
thf(fact_14_iso__tuple__UNIV__I, axiom,
    ((![X2 : nat]: (member_nat @ X2 @ top_top_set_nat)))). % iso_tuple_UNIV_I
thf(fact_15_iso__tuple__UNIV__I, axiom,
    ((![X2 : arrow_1429744205e_indi]: (member1966420836e_indi @ X2 @ top_to1799531699e_indi)))). % iso_tuple_UNIV_I
thf(fact_16_image__eqI, axiom,
    ((![B : nat, F : nat > nat, X2 : nat, A : set_nat]: ((B = (F @ X2)) => ((member_nat @ X2 @ A) => (member_nat @ B @ (image_nat_nat @ F @ A))))))). % image_eqI
thf(fact_17_image__eqI, axiom,
    ((![B : arrow_1429744205e_indi, F : nat > arrow_1429744205e_indi, X2 : nat, A : set_nat]: ((B = (F @ X2)) => ((member_nat @ X2 @ A) => (member1966420836e_indi @ B @ (image_1802692772e_indi @ F @ A))))))). % image_eqI
thf(fact_18_image__eqI, axiom,
    ((![B : arrow_1429744205e_indi, F : arrow_1429744205e_indi > arrow_1429744205e_indi, X2 : arrow_1429744205e_indi, A : set_Ar1007576579e_indi]: ((B = (F @ X2)) => ((member1966420836e_indi @ X2 @ A) => (member1966420836e_indi @ B @ (image_688677079e_indi @ F @ A))))))). % image_eqI
thf(fact_19_image__eqI, axiom,
    ((![B : nat, F : arrow_1429744205e_indi > nat, X2 : arrow_1429744205e_indi, A : set_Ar1007576579e_indi]: ((B = (F @ X2)) => ((member1966420836e_indi @ X2 @ A) => (member_nat @ B @ (image_555606308di_nat @ F @ A))))))). % image_eqI
thf(fact_20_injD, axiom,
    ((![F : arrow_1429744205e_indi > arrow_1429744205e_indi, X2 : arrow_1429744205e_indi, Y2 : arrow_1429744205e_indi]: ((inj_on1663454827e_indi @ F @ top_to1799531699e_indi) => (((F @ X2) = (F @ Y2)) => (X2 = Y2)))))). % injD
thf(fact_21_injD, axiom,
    ((![F : arrow_1429744205e_indi > nat, X2 : arrow_1429744205e_indi, Y2 : arrow_1429744205e_indi]: ((inj_on528257168di_nat @ F @ top_to1799531699e_indi) => (((F @ X2) = (F @ Y2)) => (X2 = Y2)))))). % injD
thf(fact_22_injI, axiom,
    ((![F : arrow_1429744205e_indi > nat]: ((![X3 : arrow_1429744205e_indi, Y3 : arrow_1429744205e_indi]: (((F @ X3) = (F @ Y3)) => (X3 = Y3))) => (inj_on528257168di_nat @ F @ top_to1799531699e_indi))))). % injI
thf(fact_23_injI, axiom,
    ((![F : arrow_1429744205e_indi > arrow_1429744205e_indi]: ((![X3 : arrow_1429744205e_indi, Y3 : arrow_1429744205e_indi]: (((F @ X3) = (F @ Y3)) => (X3 = Y3))) => (inj_on1663454827e_indi @ F @ top_to1799531699e_indi))))). % injI
thf(fact_24_inj__eq, axiom,
    ((![F : arrow_1429744205e_indi > nat, X2 : arrow_1429744205e_indi, Y2 : arrow_1429744205e_indi]: ((inj_on528257168di_nat @ F @ top_to1799531699e_indi) => (((F @ X2) = (F @ Y2)) = (X2 = Y2)))))). % inj_eq
thf(fact_25_inj__eq, axiom,
    ((![F : arrow_1429744205e_indi > arrow_1429744205e_indi, X2 : arrow_1429744205e_indi, Y2 : arrow_1429744205e_indi]: ((inj_on1663454827e_indi @ F @ top_to1799531699e_indi) => (((F @ X2) = (F @ Y2)) = (X2 = Y2)))))). % inj_eq
thf(fact_26_top__set__def, axiom,
    ((top_to1799531699e_indi = (collec1169676194e_indi @ top_to1473733010indi_o)))). % top_set_def
thf(fact_27_zero__reorient, axiom,
    ((![X2 : nat]: ((zero_zero_nat = X2) = (X2 = zero_zero_nat))))). % zero_reorient
thf(fact_28_rev__image__eqI, axiom,
    ((![X2 : nat, A : set_nat, B : nat, F : nat > nat]: ((member_nat @ X2 @ A) => ((B = (F @ X2)) => (member_nat @ B @ (image_nat_nat @ F @ A))))))). % rev_image_eqI
thf(fact_29_rev__image__eqI, axiom,
    ((![X2 : nat, A : set_nat, B : arrow_1429744205e_indi, F : nat > arrow_1429744205e_indi]: ((member_nat @ X2 @ A) => ((B = (F @ X2)) => (member1966420836e_indi @ B @ (image_1802692772e_indi @ F @ A))))))). % rev_image_eqI
thf(fact_30_rev__image__eqI, axiom,
    ((![X2 : arrow_1429744205e_indi, A : set_Ar1007576579e_indi, B : nat, F : arrow_1429744205e_indi > nat]: ((member1966420836e_indi @ X2 @ A) => ((B = (F @ X2)) => (member_nat @ B @ (image_555606308di_nat @ F @ A))))))). % rev_image_eqI
thf(fact_31_rev__image__eqI, axiom,
    ((![X2 : arrow_1429744205e_indi, A : set_Ar1007576579e_indi, B : arrow_1429744205e_indi, F : arrow_1429744205e_indi > arrow_1429744205e_indi]: ((member1966420836e_indi @ X2 @ A) => ((B = (F @ X2)) => (member1966420836e_indi @ B @ (image_688677079e_indi @ F @ A))))))). % rev_image_eqI
thf(fact_32_ball__imageD, axiom,
    ((![F : arrow_1429744205e_indi > nat, A : set_Ar1007576579e_indi, P : nat > $o]: ((![X3 : nat]: ((member_nat @ X3 @ (image_555606308di_nat @ F @ A)) => (P @ X3))) => (![X4 : arrow_1429744205e_indi]: ((member1966420836e_indi @ X4 @ A) => (P @ (F @ X4)))))))). % ball_imageD
thf(fact_33_ball__imageD, axiom,
    ((![F : arrow_1429744205e_indi > arrow_1429744205e_indi, A : set_Ar1007576579e_indi, P : arrow_1429744205e_indi > $o]: ((![X3 : arrow_1429744205e_indi]: ((member1966420836e_indi @ X3 @ (image_688677079e_indi @ F @ A)) => (P @ X3))) => (![X4 : arrow_1429744205e_indi]: ((member1966420836e_indi @ X4 @ A) => (P @ (F @ X4)))))))). % ball_imageD
thf(fact_34_image__cong, axiom,
    ((![M : set_Ar1007576579e_indi, N : set_Ar1007576579e_indi, F : arrow_1429744205e_indi > nat, G : arrow_1429744205e_indi > nat]: ((M = N) => ((![X3 : arrow_1429744205e_indi]: ((member1966420836e_indi @ X3 @ N) => ((F @ X3) = (G @ X3)))) => ((image_555606308di_nat @ F @ M) = (image_555606308di_nat @ G @ N))))))). % image_cong
thf(fact_35_image__cong, axiom,
    ((![M : set_Ar1007576579e_indi, N : set_Ar1007576579e_indi, F : arrow_1429744205e_indi > arrow_1429744205e_indi, G : arrow_1429744205e_indi > arrow_1429744205e_indi]: ((M = N) => ((![X3 : arrow_1429744205e_indi]: ((member1966420836e_indi @ X3 @ N) => ((F @ X3) = (G @ X3)))) => ((image_688677079e_indi @ F @ M) = (image_688677079e_indi @ G @ N))))))). % image_cong
thf(fact_36_bex__imageD, axiom,
    ((![F : arrow_1429744205e_indi > nat, A : set_Ar1007576579e_indi, P : nat > $o]: ((?[X4 : nat]: ((member_nat @ X4 @ (image_555606308di_nat @ F @ A)) & (P @ X4))) => (?[X3 : arrow_1429744205e_indi]: ((member1966420836e_indi @ X3 @ A) & (P @ (F @ X3)))))))). % bex_imageD
thf(fact_37_bex__imageD, axiom,
    ((![F : arrow_1429744205e_indi > arrow_1429744205e_indi, A : set_Ar1007576579e_indi, P : arrow_1429744205e_indi > $o]: ((?[X4 : arrow_1429744205e_indi]: ((member1966420836e_indi @ X4 @ (image_688677079e_indi @ F @ A)) & (P @ X4))) => (?[X3 : arrow_1429744205e_indi]: ((member1966420836e_indi @ X3 @ A) & (P @ (F @ X3)))))))). % bex_imageD
thf(fact_38_image__iff, axiom,
    ((![Z : nat, F : arrow_1429744205e_indi > nat, A : set_Ar1007576579e_indi]: ((member_nat @ Z @ (image_555606308di_nat @ F @ A)) = (?[X : arrow_1429744205e_indi]: (((member1966420836e_indi @ X @ A)) & ((Z = (F @ X))))))))). % image_iff
thf(fact_39_image__iff, axiom,
    ((![Z : arrow_1429744205e_indi, F : arrow_1429744205e_indi > arrow_1429744205e_indi, A : set_Ar1007576579e_indi]: ((member1966420836e_indi @ Z @ (image_688677079e_indi @ F @ A)) = (?[X : arrow_1429744205e_indi]: (((member1966420836e_indi @ X @ A)) & ((Z = (F @ X))))))))). % image_iff
thf(fact_40_imageI, axiom,
    ((![X2 : nat, A : set_nat, F : nat > nat]: ((member_nat @ X2 @ A) => (member_nat @ (F @ X2) @ (image_nat_nat @ F @ A)))))). % imageI
thf(fact_41_imageI, axiom,
    ((![X2 : nat, A : set_nat, F : nat > arrow_1429744205e_indi]: ((member_nat @ X2 @ A) => (member1966420836e_indi @ (F @ X2) @ (image_1802692772e_indi @ F @ A)))))). % imageI
thf(fact_42_imageI, axiom,
    ((![X2 : arrow_1429744205e_indi, A : set_Ar1007576579e_indi, F : arrow_1429744205e_indi > nat]: ((member1966420836e_indi @ X2 @ A) => (member_nat @ (F @ X2) @ (image_555606308di_nat @ F @ A)))))). % imageI
thf(fact_43_imageI, axiom,
    ((![X2 : arrow_1429744205e_indi, A : set_Ar1007576579e_indi, F : arrow_1429744205e_indi > arrow_1429744205e_indi]: ((member1966420836e_indi @ X2 @ A) => (member1966420836e_indi @ (F @ X2) @ (image_688677079e_indi @ F @ A)))))). % imageI
thf(fact_44_UNIV__witness, axiom,
    ((?[X3 : nat]: (member_nat @ X3 @ top_top_set_nat)))). % UNIV_witness
thf(fact_45_UNIV__witness, axiom,
    ((?[X3 : arrow_1429744205e_indi]: (member1966420836e_indi @ X3 @ top_to1799531699e_indi)))). % UNIV_witness
thf(fact_46_UNIV__eq__I, axiom,
    ((![A : set_nat]: ((![X3 : nat]: (member_nat @ X3 @ A)) => (top_top_set_nat = A))))). % UNIV_eq_I
thf(fact_47_UNIV__eq__I, axiom,
    ((![A : set_Ar1007576579e_indi]: ((![X3 : arrow_1429744205e_indi]: (member1966420836e_indi @ X3 @ A)) => (top_to1799531699e_indi = A))))). % UNIV_eq_I
thf(fact_48_inj__on__inverseI, axiom,
    ((![A : set_Ar1007576579e_indi, G : nat > arrow_1429744205e_indi, F : arrow_1429744205e_indi > nat]: ((![X3 : arrow_1429744205e_indi]: ((member1966420836e_indi @ X3 @ A) => ((G @ (F @ X3)) = X3))) => (inj_on528257168di_nat @ F @ A))))). % inj_on_inverseI
thf(fact_49_inj__on__inverseI, axiom,
    ((![A : set_Ar1007576579e_indi, G : arrow_1429744205e_indi > arrow_1429744205e_indi, F : arrow_1429744205e_indi > arrow_1429744205e_indi]: ((![X3 : arrow_1429744205e_indi]: ((member1966420836e_indi @ X3 @ A) => ((G @ (F @ X3)) = X3))) => (inj_on1663454827e_indi @ F @ A))))). % inj_on_inverseI
thf(fact_50_inj__on__contraD, axiom,
    ((![F : arrow_1429744205e_indi > nat, A : set_Ar1007576579e_indi, X2 : arrow_1429744205e_indi, Y2 : arrow_1429744205e_indi]: ((inj_on528257168di_nat @ F @ A) => ((~ ((X2 = Y2))) => ((member1966420836e_indi @ X2 @ A) => ((member1966420836e_indi @ Y2 @ A) => (~ (((F @ X2) = (F @ Y2))))))))))). % inj_on_contraD
thf(fact_51_inj__on__contraD, axiom,
    ((![F : arrow_1429744205e_indi > arrow_1429744205e_indi, A : set_Ar1007576579e_indi, X2 : arrow_1429744205e_indi, Y2 : arrow_1429744205e_indi]: ((inj_on1663454827e_indi @ F @ A) => ((~ ((X2 = Y2))) => ((member1966420836e_indi @ X2 @ A) => ((member1966420836e_indi @ Y2 @ A) => (~ (((F @ X2) = (F @ Y2))))))))))). % inj_on_contraD
thf(fact_52_inj__on__eq__iff, axiom,
    ((![F : arrow_1429744205e_indi > nat, A : set_Ar1007576579e_indi, X2 : arrow_1429744205e_indi, Y2 : arrow_1429744205e_indi]: ((inj_on528257168di_nat @ F @ A) => ((member1966420836e_indi @ X2 @ A) => ((member1966420836e_indi @ Y2 @ A) => (((F @ X2) = (F @ Y2)) = (X2 = Y2)))))))). % inj_on_eq_iff
thf(fact_53_inj__on__eq__iff, axiom,
    ((![F : arrow_1429744205e_indi > arrow_1429744205e_indi, A : set_Ar1007576579e_indi, X2 : arrow_1429744205e_indi, Y2 : arrow_1429744205e_indi]: ((inj_on1663454827e_indi @ F @ A) => ((member1966420836e_indi @ X2 @ A) => ((member1966420836e_indi @ Y2 @ A) => (((F @ X2) = (F @ Y2)) = (X2 = Y2)))))))). % inj_on_eq_iff
thf(fact_54_inj__on__cong, axiom,
    ((![A : set_Ar1007576579e_indi, F : arrow_1429744205e_indi > nat, G : arrow_1429744205e_indi > nat]: ((![A3 : arrow_1429744205e_indi]: ((member1966420836e_indi @ A3 @ A) => ((F @ A3) = (G @ A3)))) => ((inj_on528257168di_nat @ F @ A) = (inj_on528257168di_nat @ G @ A)))))). % inj_on_cong
thf(fact_55_inj__on__cong, axiom,
    ((![A : set_Ar1007576579e_indi, F : arrow_1429744205e_indi > arrow_1429744205e_indi, G : arrow_1429744205e_indi > arrow_1429744205e_indi]: ((![A3 : arrow_1429744205e_indi]: ((member1966420836e_indi @ A3 @ A) => ((F @ A3) = (G @ A3)))) => ((inj_on1663454827e_indi @ F @ A) = (inj_on1663454827e_indi @ G @ A)))))). % inj_on_cong
thf(fact_56_inj__on__def, axiom,
    ((inj_on528257168di_nat = (^[F2 : arrow_1429744205e_indi > nat]: (^[A4 : set_Ar1007576579e_indi]: (![X : arrow_1429744205e_indi]: (((member1966420836e_indi @ X @ A4)) => ((![Y : arrow_1429744205e_indi]: (((member1966420836e_indi @ Y @ A4)) => (((((F2 @ X) = (F2 @ Y))) => ((X = Y)))))))))))))). % inj_on_def
thf(fact_57_inj__on__def, axiom,
    ((inj_on1663454827e_indi = (^[F2 : arrow_1429744205e_indi > arrow_1429744205e_indi]: (^[A4 : set_Ar1007576579e_indi]: (![X : arrow_1429744205e_indi]: (((member1966420836e_indi @ X @ A4)) => ((![Y : arrow_1429744205e_indi]: (((member1966420836e_indi @ Y @ A4)) => (((((F2 @ X) = (F2 @ Y))) => ((X = Y)))))))))))))). % inj_on_def
thf(fact_58_inj__onI, axiom,
    ((![A : set_Ar1007576579e_indi, F : arrow_1429744205e_indi > nat]: ((![X3 : arrow_1429744205e_indi, Y3 : arrow_1429744205e_indi]: ((member1966420836e_indi @ X3 @ A) => ((member1966420836e_indi @ Y3 @ A) => (((F @ X3) = (F @ Y3)) => (X3 = Y3))))) => (inj_on528257168di_nat @ F @ A))))). % inj_onI
thf(fact_59_inj__onI, axiom,
    ((![A : set_Ar1007576579e_indi, F : arrow_1429744205e_indi > arrow_1429744205e_indi]: ((![X3 : arrow_1429744205e_indi, Y3 : arrow_1429744205e_indi]: ((member1966420836e_indi @ X3 @ A) => ((member1966420836e_indi @ Y3 @ A) => (((F @ X3) = (F @ Y3)) => (X3 = Y3))))) => (inj_on1663454827e_indi @ F @ A))))). % inj_onI
thf(fact_60_inj__onD, axiom,
    ((![F : arrow_1429744205e_indi > nat, A : set_Ar1007576579e_indi, X2 : arrow_1429744205e_indi, Y2 : arrow_1429744205e_indi]: ((inj_on528257168di_nat @ F @ A) => (((F @ X2) = (F @ Y2)) => ((member1966420836e_indi @ X2 @ A) => ((member1966420836e_indi @ Y2 @ A) => (X2 = Y2)))))))). % inj_onD
thf(fact_61_inj__onD, axiom,
    ((![F : arrow_1429744205e_indi > arrow_1429744205e_indi, A : set_Ar1007576579e_indi, X2 : arrow_1429744205e_indi, Y2 : arrow_1429744205e_indi]: ((inj_on1663454827e_indi @ F @ A) => (((F @ X2) = (F @ Y2)) => ((member1966420836e_indi @ X2 @ A) => ((member1966420836e_indi @ Y2 @ A) => (X2 = Y2)))))))). % inj_onD
thf(fact_62_range__eqI, axiom,
    ((![B : nat, F : arrow_1429744205e_indi > nat, X2 : arrow_1429744205e_indi]: ((B = (F @ X2)) => (member_nat @ B @ (image_555606308di_nat @ F @ top_to1799531699e_indi)))))). % range_eqI
thf(fact_63_range__eqI, axiom,
    ((![B : arrow_1429744205e_indi, F : arrow_1429744205e_indi > arrow_1429744205e_indi, X2 : arrow_1429744205e_indi]: ((B = (F @ X2)) => (member1966420836e_indi @ B @ (image_688677079e_indi @ F @ top_to1799531699e_indi)))))). % range_eqI
thf(fact_64_surj__def, axiom,
    ((![F : arrow_1429744205e_indi > nat]: (((image_555606308di_nat @ F @ top_to1799531699e_indi) = top_top_set_nat) = (![Y : nat]: (?[X : arrow_1429744205e_indi]: (Y = (F @ X)))))))). % surj_def
thf(fact_65_surj__def, axiom,
    ((![F : arrow_1429744205e_indi > arrow_1429744205e_indi]: (((image_688677079e_indi @ F @ top_to1799531699e_indi) = top_to1799531699e_indi) = (![Y : arrow_1429744205e_indi]: (?[X : arrow_1429744205e_indi]: (Y = (F @ X)))))))). % surj_def
thf(fact_66_rangeI, axiom,
    ((![F : arrow_1429744205e_indi > nat, X2 : arrow_1429744205e_indi]: (member_nat @ (F @ X2) @ (image_555606308di_nat @ F @ top_to1799531699e_indi))))). % rangeI
thf(fact_67_rangeI, axiom,
    ((![F : arrow_1429744205e_indi > arrow_1429744205e_indi, X2 : arrow_1429744205e_indi]: (member1966420836e_indi @ (F @ X2) @ (image_688677079e_indi @ F @ top_to1799531699e_indi))))). % rangeI
thf(fact_68_surjI, axiom,
    ((![G : arrow_1429744205e_indi > nat, F : nat > arrow_1429744205e_indi]: ((![X3 : nat]: ((G @ (F @ X3)) = X3)) => ((image_555606308di_nat @ G @ top_to1799531699e_indi) = top_top_set_nat))))). % surjI
thf(fact_69_surjI, axiom,
    ((![G : arrow_1429744205e_indi > arrow_1429744205e_indi, F : arrow_1429744205e_indi > arrow_1429744205e_indi]: ((![X3 : arrow_1429744205e_indi]: ((G @ (F @ X3)) = X3)) => ((image_688677079e_indi @ G @ top_to1799531699e_indi) = top_to1799531699e_indi))))). % surjI
thf(fact_70_surjE, axiom,
    ((![F : arrow_1429744205e_indi > nat, Y2 : nat]: (((image_555606308di_nat @ F @ top_to1799531699e_indi) = top_top_set_nat) => (~ ((![X3 : arrow_1429744205e_indi]: (~ ((Y2 = (F @ X3))))))))))). % surjE
thf(fact_71_surjE, axiom,
    ((![F : arrow_1429744205e_indi > arrow_1429744205e_indi, Y2 : arrow_1429744205e_indi]: (((image_688677079e_indi @ F @ top_to1799531699e_indi) = top_to1799531699e_indi) => (~ ((![X3 : arrow_1429744205e_indi]: (~ ((Y2 = (F @ X3))))))))))). % surjE
thf(fact_72_surjD, axiom,
    ((![F : arrow_1429744205e_indi > nat, Y2 : nat]: (((image_555606308di_nat @ F @ top_to1799531699e_indi) = top_top_set_nat) => (?[X3 : arrow_1429744205e_indi]: (Y2 = (F @ X3))))))). % surjD
thf(fact_73_surjD, axiom,
    ((![F : arrow_1429744205e_indi > arrow_1429744205e_indi, Y2 : arrow_1429744205e_indi]: (((image_688677079e_indi @ F @ top_to1799531699e_indi) = top_to1799531699e_indi) => (?[X3 : arrow_1429744205e_indi]: (Y2 = (F @ X3))))))). % surjD
thf(fact_74_inj__on__image__iff, axiom,
    ((![A : set_Ar1007576579e_indi, G : arrow_1429744205e_indi > nat, F : arrow_1429744205e_indi > arrow_1429744205e_indi]: ((![X3 : arrow_1429744205e_indi]: ((member1966420836e_indi @ X3 @ A) => (![Xa : arrow_1429744205e_indi]: ((member1966420836e_indi @ Xa @ A) => (((G @ (F @ X3)) = (G @ (F @ Xa))) = ((G @ X3) = (G @ Xa))))))) => ((inj_on1663454827e_indi @ F @ A) => ((inj_on528257168di_nat @ G @ (image_688677079e_indi @ F @ A)) = (inj_on528257168di_nat @ G @ A))))))). % inj_on_image_iff
thf(fact_75_inj__on__image__iff, axiom,
    ((![A : set_Ar1007576579e_indi, G : arrow_1429744205e_indi > arrow_1429744205e_indi, F : arrow_1429744205e_indi > arrow_1429744205e_indi]: ((![X3 : arrow_1429744205e_indi]: ((member1966420836e_indi @ X3 @ A) => (![Xa : arrow_1429744205e_indi]: ((member1966420836e_indi @ Xa @ A) => (((G @ (F @ X3)) = (G @ (F @ Xa))) = ((G @ X3) = (G @ Xa))))))) => ((inj_on1663454827e_indi @ F @ A) => ((inj_on1663454827e_indi @ G @ (image_688677079e_indi @ F @ A)) = (inj_on1663454827e_indi @ G @ A))))))). % inj_on_image_iff
thf(fact_76_inj__def, axiom,
    ((![F : arrow_1429744205e_indi > nat]: ((inj_on528257168di_nat @ F @ top_to1799531699e_indi) = (![X : arrow_1429744205e_indi]: (![Y : arrow_1429744205e_indi]: ((((F @ X) = (F @ Y))) => ((X = Y))))))))). % inj_def
thf(fact_77_inj__def, axiom,
    ((![F : arrow_1429744205e_indi > arrow_1429744205e_indi]: ((inj_on1663454827e_indi @ F @ top_to1799531699e_indi) = (![X : arrow_1429744205e_indi]: (![Y : arrow_1429744205e_indi]: ((((F @ X) = (F @ Y))) => ((X = Y))))))))). % inj_def
thf(fact_78_the__inv__into__onto, axiom,
    ((![F : nat > arrow_1429744205e_indi, A : set_nat]: ((inj_on1775343632e_indi @ F @ A) => ((image_555606308di_nat @ (the_in152215634e_indi @ A @ F) @ (image_1802692772e_indi @ F @ A)) = A))))). % the_inv_into_onto
thf(fact_79_the__inv__into__onto, axiom,
    ((![F : arrow_1429744205e_indi > nat, A : set_Ar1007576579e_indi]: ((inj_on528257168di_nat @ F @ A) => ((image_1802692772e_indi @ (the_in1052612818di_nat @ A @ F) @ (image_555606308di_nat @ F @ A)) = A))))). % the_inv_into_onto
thf(fact_80_the__inv__into__onto, axiom,
    ((![F : arrow_1429744205e_indi > arrow_1429744205e_indi, A : set_Ar1007576579e_indi]: ((inj_on1663454827e_indi @ F @ A) => ((image_688677079e_indi @ (the_in1730881961e_indi @ A @ F) @ (image_688677079e_indi @ F @ A)) = A))))). % the_inv_into_onto
thf(fact_81_inj__on__image__Fpow, axiom,
    ((![F : arrow_1429744205e_indi > nat, A : set_Ar1007576579e_indi]: ((inj_on528257168di_nat @ F @ A) => (inj_on384466940et_nat @ (image_555606308di_nat @ F) @ (finite110102887e_indi @ A)))))). % inj_on_image_Fpow
thf(fact_82_inj__on__image__Fpow, axiom,
    ((![F : arrow_1429744205e_indi > arrow_1429744205e_indi, A : set_Ar1007576579e_indi]: ((inj_on1663454827e_indi @ F @ A) => (inj_on5180053e_indi @ (image_688677079e_indi @ F) @ (finite110102887e_indi @ A)))))). % inj_on_image_Fpow
thf(fact_83_surj__swap__iff, axiom,
    ((![A2 : arrow_1429744205e_indi, B : arrow_1429744205e_indi, F : arrow_1429744205e_indi > nat]: (((image_555606308di_nat @ (swap_A1227367653di_nat @ A2 @ B @ F) @ top_to1799531699e_indi) = top_top_set_nat) = ((image_555606308di_nat @ F @ top_to1799531699e_indi) = top_top_set_nat))))). % surj_swap_iff
thf(fact_84_surj__swap__iff, axiom,
    ((![A2 : arrow_1429744205e_indi, B : arrow_1429744205e_indi, F : arrow_1429744205e_indi > arrow_1429744205e_indi]: (((image_688677079e_indi @ (swap_A1408198358e_indi @ A2 @ B @ F) @ top_to1799531699e_indi) = top_to1799531699e_indi) = ((image_688677079e_indi @ F @ top_to1799531699e_indi) = top_to1799531699e_indi))))). % surj_swap_iff
thf(fact_85_top__empty__eq, axiom,
    ((top_top_nat_o = (^[X : nat]: (member_nat @ X @ top_top_set_nat))))). % top_empty_eq
thf(fact_86_top__empty__eq, axiom,
    ((top_to1473733010indi_o = (^[X : arrow_1429744205e_indi]: (member1966420836e_indi @ X @ top_to1799531699e_indi))))). % top_empty_eq
thf(fact_87_zero__natural_Orsp, axiom,
    ((zero_zero_nat = zero_zero_nat))). % zero_natural.rsp
thf(fact_88_Sup_OSUP__cong, axiom,
    ((![A : set_Ar1007576579e_indi, B2 : set_Ar1007576579e_indi, C : arrow_1429744205e_indi > nat, D : arrow_1429744205e_indi > nat, Sup : set_nat > nat]: ((A = B2) => ((![X3 : arrow_1429744205e_indi]: ((member1966420836e_indi @ X3 @ B2) => ((C @ X3) = (D @ X3)))) => ((Sup @ (image_555606308di_nat @ C @ A)) = (Sup @ (image_555606308di_nat @ D @ B2)))))))). % Sup.SUP_cong
thf(fact_89_Sup_OSUP__cong, axiom,
    ((![A : set_Ar1007576579e_indi, B2 : set_Ar1007576579e_indi, C : arrow_1429744205e_indi > arrow_1429744205e_indi, D : arrow_1429744205e_indi > arrow_1429744205e_indi, Sup : set_Ar1007576579e_indi > arrow_1429744205e_indi]: ((A = B2) => ((![X3 : arrow_1429744205e_indi]: ((member1966420836e_indi @ X3 @ B2) => ((C @ X3) = (D @ X3)))) => ((Sup @ (image_688677079e_indi @ C @ A)) = (Sup @ (image_688677079e_indi @ D @ B2)))))))). % Sup.SUP_cong
thf(fact_90_mem__Collect__eq, axiom,
    ((![A2 : nat, P : nat > $o]: ((member_nat @ A2 @ (collect_nat @ P)) = (P @ A2))))). % mem_Collect_eq
thf(fact_91_mem__Collect__eq, axiom,
    ((![A2 : arrow_1429744205e_indi, P : arrow_1429744205e_indi > $o]: ((member1966420836e_indi @ A2 @ (collec1169676194e_indi @ P)) = (P @ A2))))). % mem_Collect_eq
thf(fact_92_Collect__mem__eq, axiom,
    ((![A : set_nat]: ((collect_nat @ (^[X : nat]: (member_nat @ X @ A))) = A)))). % Collect_mem_eq
thf(fact_93_Collect__mem__eq, axiom,
    ((![A : set_Ar1007576579e_indi]: ((collec1169676194e_indi @ (^[X : arrow_1429744205e_indi]: (member1966420836e_indi @ X @ A))) = A)))). % Collect_mem_eq
thf(fact_94_Inf_OINF__cong, axiom,
    ((![A : set_Ar1007576579e_indi, B2 : set_Ar1007576579e_indi, C : arrow_1429744205e_indi > nat, D : arrow_1429744205e_indi > nat, Inf : set_nat > nat]: ((A = B2) => ((![X3 : arrow_1429744205e_indi]: ((member1966420836e_indi @ X3 @ B2) => ((C @ X3) = (D @ X3)))) => ((Inf @ (image_555606308di_nat @ C @ A)) = (Inf @ (image_555606308di_nat @ D @ B2)))))))). % Inf.INF_cong
thf(fact_95_Inf_OINF__cong, axiom,
    ((![A : set_Ar1007576579e_indi, B2 : set_Ar1007576579e_indi, C : arrow_1429744205e_indi > arrow_1429744205e_indi, D : arrow_1429744205e_indi > arrow_1429744205e_indi, Inf : set_Ar1007576579e_indi > arrow_1429744205e_indi]: ((A = B2) => ((![X3 : arrow_1429744205e_indi]: ((member1966420836e_indi @ X3 @ B2) => ((C @ X3) = (D @ X3)))) => ((Inf @ (image_688677079e_indi @ C @ A)) = (Inf @ (image_688677079e_indi @ D @ B2)))))))). % Inf.INF_cong
thf(fact_96_pigeonhole, axiom,
    ((![F : arrow_1429744205e_indi > arrow_1429744205e_indi, A : set_Ar1007576579e_indi]: ((ord_less_nat @ (finite927127589e_indi @ (image_688677079e_indi @ F @ A)) @ (finite927127589e_indi @ A)) => (~ ((inj_on1663454827e_indi @ F @ A))))))). % pigeonhole
thf(fact_97_pigeonhole, axiom,
    ((![F : nat > arrow_1429744205e_indi, A : set_nat]: ((ord_less_nat @ (finite927127589e_indi @ (image_1802692772e_indi @ F @ A)) @ (finite_card_nat @ A)) => (~ ((inj_on1775343632e_indi @ F @ A))))))). % pigeonhole
thf(fact_98_pigeonhole, axiom,
    ((![F : arrow_1429744205e_indi > nat, A : set_Ar1007576579e_indi]: ((ord_less_nat @ (finite_card_nat @ (image_555606308di_nat @ F @ A)) @ (finite927127589e_indi @ A)) => (~ ((inj_on528257168di_nat @ F @ A))))))). % pigeonhole
thf(fact_99_pigeonhole, axiom,
    ((![F : nat > nat, A : set_nat]: ((ord_less_nat @ (finite_card_nat @ (image_nat_nat @ F @ A)) @ (finite_card_nat @ A)) => (~ ((inj_on_nat_nat @ F @ A))))))). % pigeonhole
thf(fact_100_not__gr__zero, axiom,
    ((![N2 : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N2))) = (N2 = zero_zero_nat))))). % not_gr_zero
thf(fact_101_neq0__conv, axiom,
    ((![N2 : nat]: ((~ ((N2 = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ N2))))). % neq0_conv
thf(fact_102_less__nat__zero__code, axiom,
    ((![N2 : nat]: (~ ((ord_less_nat @ N2 @ zero_zero_nat)))))). % less_nat_zero_code
thf(fact_103_bot__nat__0_Onot__eq__extremum, axiom,
    ((![A2 : nat]: ((~ ((A2 = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ A2))))). % bot_nat_0.not_eq_extremum
thf(fact_104_swap__image__eq, axiom,
    ((![A2 : arrow_1429744205e_indi, A : set_Ar1007576579e_indi, B : arrow_1429744205e_indi, F : arrow_1429744205e_indi > nat]: ((member1966420836e_indi @ A2 @ A) => ((member1966420836e_indi @ B @ A) => ((image_555606308di_nat @ (swap_A1227367653di_nat @ A2 @ B @ F) @ A) = (image_555606308di_nat @ F @ A))))))). % swap_image_eq
thf(fact_105_swap__image__eq, axiom,
    ((![A2 : arrow_1429744205e_indi, A : set_Ar1007576579e_indi, B : arrow_1429744205e_indi, F : arrow_1429744205e_indi > arrow_1429744205e_indi]: ((member1966420836e_indi @ A2 @ A) => ((member1966420836e_indi @ B @ A) => ((image_688677079e_indi @ (swap_A1408198358e_indi @ A2 @ B @ F) @ A) = (image_688677079e_indi @ F @ A))))))). % swap_image_eq
thf(fact_106_inj__on__swap__iff, axiom,
    ((![A2 : arrow_1429744205e_indi, A : set_Ar1007576579e_indi, B : arrow_1429744205e_indi, F : arrow_1429744205e_indi > nat]: ((member1966420836e_indi @ A2 @ A) => ((member1966420836e_indi @ B @ A) => ((inj_on528257168di_nat @ (swap_A1227367653di_nat @ A2 @ B @ F) @ A) = (inj_on528257168di_nat @ F @ A))))))). % inj_on_swap_iff
thf(fact_107_inj__on__swap__iff, axiom,
    ((![A2 : arrow_1429744205e_indi, A : set_Ar1007576579e_indi, B : arrow_1429744205e_indi, F : arrow_1429744205e_indi > arrow_1429744205e_indi]: ((member1966420836e_indi @ A2 @ A) => ((member1966420836e_indi @ B @ A) => ((inj_on1663454827e_indi @ (swap_A1408198358e_indi @ A2 @ B @ F) @ A) = (inj_on1663454827e_indi @ F @ A))))))). % inj_on_swap_iff
thf(fact_108_ord__eq__less__subst, axiom,
    ((![A2 : nat, F : nat > nat, B : nat, C2 : nat]: ((A2 = (F @ B)) => ((ord_less_nat @ B @ C2) => ((![X3 : nat, Y3 : nat]: ((ord_less_nat @ X3 @ Y3) => (ord_less_nat @ (F @ X3) @ (F @ Y3)))) => (ord_less_nat @ A2 @ (F @ C2)))))))). % ord_eq_less_subst
thf(fact_109_ord__less__eq__subst, axiom,
    ((![A2 : nat, B : nat, F : nat > nat, C2 : nat]: ((ord_less_nat @ A2 @ B) => (((F @ B) = C2) => ((![X3 : nat, Y3 : nat]: ((ord_less_nat @ X3 @ Y3) => (ord_less_nat @ (F @ X3) @ (F @ Y3)))) => (ord_less_nat @ (F @ A2) @ C2))))))). % ord_less_eq_subst
thf(fact_110_order__less__subst1, axiom,
    ((![A2 : nat, F : nat > nat, B : nat, C2 : nat]: ((ord_less_nat @ A2 @ (F @ B)) => ((ord_less_nat @ B @ C2) => ((![X3 : nat, Y3 : nat]: ((ord_less_nat @ X3 @ Y3) => (ord_less_nat @ (F @ X3) @ (F @ Y3)))) => (ord_less_nat @ A2 @ (F @ C2)))))))). % order_less_subst1
thf(fact_111_order__less__subst2, axiom,
    ((![A2 : nat, B : nat, F : nat > nat, C2 : nat]: ((ord_less_nat @ A2 @ B) => ((ord_less_nat @ (F @ B) @ C2) => ((![X3 : nat, Y3 : nat]: ((ord_less_nat @ X3 @ Y3) => (ord_less_nat @ (F @ X3) @ (F @ Y3)))) => (ord_less_nat @ (F @ A2) @ C2))))))). % order_less_subst2
thf(fact_112_gt__ex, axiom,
    ((![X2 : nat]: (?[X_1 : nat]: (ord_less_nat @ X2 @ X_1))))). % gt_ex
thf(fact_113_neqE, axiom,
    ((![X2 : nat, Y2 : nat]: ((~ ((X2 = Y2))) => ((~ ((ord_less_nat @ X2 @ Y2))) => (ord_less_nat @ Y2 @ X2)))))). % neqE
thf(fact_114_neq__iff, axiom,
    ((![X2 : nat, Y2 : nat]: ((~ ((X2 = Y2))) = (((ord_less_nat @ X2 @ Y2)) | ((ord_less_nat @ Y2 @ X2))))))). % neq_iff
thf(fact_115_order_Oasym, axiom,
    ((![A2 : nat, B : nat]: ((ord_less_nat @ A2 @ B) => (~ ((ord_less_nat @ B @ A2))))))). % order.asym
thf(fact_116_less__imp__neq, axiom,
    ((![X2 : nat, Y2 : nat]: ((ord_less_nat @ X2 @ Y2) => (~ ((X2 = Y2))))))). % less_imp_neq
thf(fact_117_strict__monoD, axiom,
    ((![F : nat > nat, X2 : nat, Y2 : nat]: ((order_769474267at_nat @ F) => ((ord_less_nat @ X2 @ Y2) => (ord_less_nat @ (F @ X2) @ (F @ Y2))))))). % strict_monoD
thf(fact_118_strict__monoI, axiom,
    ((![F : nat > nat]: ((![X3 : nat, Y3 : nat]: ((ord_less_nat @ X3 @ Y3) => (ord_less_nat @ (F @ X3) @ (F @ Y3)))) => (order_769474267at_nat @ F))))). % strict_monoI
thf(fact_119_less__asym, axiom,
    ((![X2 : nat, Y2 : nat]: ((ord_less_nat @ X2 @ Y2) => (~ ((ord_less_nat @ Y2 @ X2))))))). % less_asym
thf(fact_120_less__asym_H, axiom,
    ((![A2 : nat, B : nat]: ((ord_less_nat @ A2 @ B) => (~ ((ord_less_nat @ B @ A2))))))). % less_asym'
thf(fact_121_less__trans, axiom,
    ((![X2 : nat, Y2 : nat, Z : nat]: ((ord_less_nat @ X2 @ Y2) => ((ord_less_nat @ Y2 @ Z) => (ord_less_nat @ X2 @ Z)))))). % less_trans
thf(fact_122_less__linear, axiom,
    ((![X2 : nat, Y2 : nat]: ((ord_less_nat @ X2 @ Y2) | ((X2 = Y2) | (ord_less_nat @ Y2 @ X2)))))). % less_linear
thf(fact_123_less__irrefl, axiom,
    ((![X2 : nat]: (~ ((ord_less_nat @ X2 @ X2)))))). % less_irrefl
thf(fact_124_ord__eq__less__trans, axiom,
    ((![A2 : nat, B : nat, C2 : nat]: ((A2 = B) => ((ord_less_nat @ B @ C2) => (ord_less_nat @ A2 @ C2)))))). % ord_eq_less_trans
thf(fact_125_ord__less__eq__trans, axiom,
    ((![A2 : nat, B : nat, C2 : nat]: ((ord_less_nat @ A2 @ B) => ((B = C2) => (ord_less_nat @ A2 @ C2)))))). % ord_less_eq_trans
thf(fact_126_dual__order_Oasym, axiom,
    ((![B : nat, A2 : nat]: ((ord_less_nat @ B @ A2) => (~ ((ord_less_nat @ A2 @ B))))))). % dual_order.asym
thf(fact_127_less__imp__not__eq, axiom,
    ((![X2 : nat, Y2 : nat]: ((ord_less_nat @ X2 @ Y2) => (~ ((X2 = Y2))))))). % less_imp_not_eq
thf(fact_128_strict__mono__def, axiom,
    ((order_769474267at_nat = (^[F2 : nat > nat]: (![X : nat]: (![Y : nat]: (((ord_less_nat @ X @ Y)) => ((ord_less_nat @ (F2 @ X) @ (F2 @ Y)))))))))). % strict_mono_def
thf(fact_129_less__not__sym, axiom,
    ((![X2 : nat, Y2 : nat]: ((ord_less_nat @ X2 @ Y2) => (~ ((ord_less_nat @ Y2 @ X2))))))). % less_not_sym
thf(fact_130_less__induct, axiom,
    ((![P : nat > $o, A2 : nat]: ((![X3 : nat]: ((![Y4 : nat]: ((ord_less_nat @ Y4 @ X3) => (P @ Y4))) => (P @ X3))) => (P @ A2))))). % less_induct
thf(fact_131_antisym__conv3, axiom,
    ((![Y2 : nat, X2 : nat]: ((~ ((ord_less_nat @ Y2 @ X2))) => ((~ ((ord_less_nat @ X2 @ Y2))) = (X2 = Y2)))))). % antisym_conv3
thf(fact_132_less__imp__not__eq2, axiom,
    ((![X2 : nat, Y2 : nat]: ((ord_less_nat @ X2 @ Y2) => (~ ((Y2 = X2))))))). % less_imp_not_eq2
thf(fact_133_less__imp__triv, axiom,
    ((![X2 : nat, Y2 : nat, P : $o]: ((ord_less_nat @ X2 @ Y2) => ((ord_less_nat @ Y2 @ X2) => P))))). % less_imp_triv
thf(fact_134_linorder__cases, axiom,
    ((![X2 : nat, Y2 : nat]: ((~ ((ord_less_nat @ X2 @ Y2))) => ((~ ((X2 = Y2))) => (ord_less_nat @ Y2 @ X2)))))). % linorder_cases
thf(fact_135_dual__order_Oirrefl, axiom,
    ((![A2 : nat]: (~ ((ord_less_nat @ A2 @ A2)))))). % dual_order.irrefl
thf(fact_136_order_Ostrict__trans, axiom,
    ((![A2 : nat, B : nat, C2 : nat]: ((ord_less_nat @ A2 @ B) => ((ord_less_nat @ B @ C2) => (ord_less_nat @ A2 @ C2)))))). % order.strict_trans
thf(fact_137_strict__mono__less, axiom,
    ((![F : nat > nat, X2 : nat, Y2 : nat]: ((order_769474267at_nat @ F) => ((ord_less_nat @ (F @ X2) @ (F @ Y2)) = (ord_less_nat @ X2 @ Y2)))))). % strict_mono_less
thf(fact_138_less__imp__not__less, axiom,
    ((![X2 : nat, Y2 : nat]: ((ord_less_nat @ X2 @ Y2) => (~ ((ord_less_nat @ Y2 @ X2))))))). % less_imp_not_less
thf(fact_139_exists__least__iff, axiom,
    (((^[P2 : nat > $o]: (?[X5 : nat]: (P2 @ X5))) = (^[P3 : nat > $o]: (?[N3 : nat]: (((P3 @ N3)) & ((![M2 : nat]: (((ord_less_nat @ M2 @ N3)) => ((~ ((P3 @ M2))))))))))))). % exists_least_iff
thf(fact_140_linorder__less__wlog, axiom,
    ((![P : nat > nat > $o, A2 : nat, B : nat]: ((![A3 : nat, B3 : nat]: ((ord_less_nat @ A3 @ B3) => (P @ A3 @ B3))) => ((![A3 : nat]: (P @ A3 @ A3)) => ((![A3 : nat, B3 : nat]: ((P @ B3 @ A3) => (P @ A3 @ B3))) => (P @ A2 @ B))))))). % linorder_less_wlog
thf(fact_141_dual__order_Ostrict__trans, axiom,
    ((![B : nat, A2 : nat, C2 : nat]: ((ord_less_nat @ B @ A2) => ((ord_less_nat @ C2 @ B) => (ord_less_nat @ C2 @ A2)))))). % dual_order.strict_trans
thf(fact_142_not__less__iff__gr__or__eq, axiom,
    ((![X2 : nat, Y2 : nat]: ((~ ((ord_less_nat @ X2 @ Y2))) = (((ord_less_nat @ Y2 @ X2)) | ((X2 = Y2))))))). % not_less_iff_gr_or_eq
thf(fact_143_order_Ostrict__implies__not__eq, axiom,
    ((![A2 : nat, B : nat]: ((ord_less_nat @ A2 @ B) => (~ ((A2 = B))))))). % order.strict_implies_not_eq
thf(fact_144_dual__order_Ostrict__implies__not__eq, axiom,
    ((![B : nat, A2 : nat]: ((ord_less_nat @ B @ A2) => (~ ((A2 = B))))))). % dual_order.strict_implies_not_eq
thf(fact_145_linorder__neqE__nat, axiom,
    ((![X2 : nat, Y2 : nat]: ((~ ((X2 = Y2))) => ((~ ((ord_less_nat @ X2 @ Y2))) => (ord_less_nat @ Y2 @ X2)))))). % linorder_neqE_nat
thf(fact_146_infinite__descent, axiom,
    ((![P : nat > $o, N2 : nat]: ((![N4 : nat]: ((~ ((P @ N4))) => (?[M3 : nat]: ((ord_less_nat @ M3 @ N4) & (~ ((P @ M3))))))) => (P @ N2))))). % infinite_descent
thf(fact_147_nat__less__induct, axiom,
    ((![P : nat > $o, N2 : nat]: ((![N4 : nat]: ((![M3 : nat]: ((ord_less_nat @ M3 @ N4) => (P @ M3))) => (P @ N4))) => (P @ N2))))). % nat_less_induct
thf(fact_148_less__irrefl__nat, axiom,
    ((![N2 : nat]: (~ ((ord_less_nat @ N2 @ N2)))))). % less_irrefl_nat
thf(fact_149_less__not__refl3, axiom,
    ((![S : nat, T : nat]: ((ord_less_nat @ S @ T) => (~ ((S = T))))))). % less_not_refl3
thf(fact_150_less__not__refl2, axiom,
    ((![N2 : nat, M4 : nat]: ((ord_less_nat @ N2 @ M4) => (~ ((M4 = N2))))))). % less_not_refl2
thf(fact_151_less__not__refl, axiom,
    ((![N2 : nat]: (~ ((ord_less_nat @ N2 @ N2)))))). % less_not_refl
thf(fact_152_nat__neq__iff, axiom,
    ((![M4 : nat, N2 : nat]: ((~ ((M4 = N2))) = (((ord_less_nat @ M4 @ N2)) | ((ord_less_nat @ N2 @ M4))))))). % nat_neq_iff
thf(fact_153_gr0I, axiom,
    ((![N2 : nat]: ((~ ((N2 = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N2))))). % gr0I
thf(fact_154_not__gr0, axiom,
    ((![N2 : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N2))) = (N2 = zero_zero_nat))))). % not_gr0
thf(fact_155_not__less0, axiom,
    ((![N2 : nat]: (~ ((ord_less_nat @ N2 @ zero_zero_nat)))))). % not_less0
thf(fact_156_less__zeroE, axiom,
    ((![N2 : nat]: (~ ((ord_less_nat @ N2 @ zero_zero_nat)))))). % less_zeroE
thf(fact_157_gr__implies__not0, axiom,
    ((![M4 : nat, N2 : nat]: ((ord_less_nat @ M4 @ N2) => (~ ((N2 = zero_zero_nat))))))). % gr_implies_not0
thf(fact_158_infinite__descent0, axiom,
    ((![P : nat > $o, N2 : nat]: ((P @ zero_zero_nat) => ((![N4 : nat]: ((ord_less_nat @ zero_zero_nat @ N4) => ((~ ((P @ N4))) => (?[M3 : nat]: ((ord_less_nat @ M3 @ N4) & (~ ((P @ M3)))))))) => (P @ N2)))))). % infinite_descent0
thf(fact_159_bot__nat__0_Oextremum__strict, axiom,
    ((![A2 : nat]: (~ ((ord_less_nat @ A2 @ zero_zero_nat)))))). % bot_nat_0.extremum_strict
thf(fact_160_gr__zeroI, axiom,
    ((![N2 : nat]: ((~ ((N2 = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N2))))). % gr_zeroI
thf(fact_161_not__less__zero, axiom,
    ((![N2 : nat]: (~ ((ord_less_nat @ N2 @ zero_zero_nat)))))). % not_less_zero
thf(fact_162_gr__implies__not__zero, axiom,
    ((![M4 : nat, N2 : nat]: ((ord_less_nat @ M4 @ N2) => (~ ((N2 = zero_zero_nat))))))). % gr_implies_not_zero
thf(fact_163_zero__less__iff__neq__zero, axiom,
    ((![N2 : nat]: ((ord_less_nat @ zero_zero_nat @ N2) = (~ ((N2 = zero_zero_nat))))))). % zero_less_iff_neq_zero
thf(fact_164_top_Oextremum__strict, axiom,
    ((![A2 : set_Ar1007576579e_indi]: (~ ((ord_le1187139159e_indi @ top_to1799531699e_indi @ A2)))))). % top.extremum_strict
thf(fact_165_top_Onot__eq__extremum, axiom,
    ((![A2 : set_Ar1007576579e_indi]: ((~ ((A2 = top_to1799531699e_indi))) = (ord_le1187139159e_indi @ A2 @ top_to1799531699e_indi))))). % top.not_eq_extremum
thf(fact_166_inj__on__imp__inj__on__swap, axiom,
    ((![F : arrow_1429744205e_indi > nat, A : set_Ar1007576579e_indi, A2 : arrow_1429744205e_indi, B : arrow_1429744205e_indi]: ((inj_on528257168di_nat @ F @ A) => ((member1966420836e_indi @ A2 @ A) => ((member1966420836e_indi @ B @ A) => (inj_on528257168di_nat @ (swap_A1227367653di_nat @ A2 @ B @ F) @ A))))))). % inj_on_imp_inj_on_swap
thf(fact_167_inj__on__imp__inj__on__swap, axiom,
    ((![F : arrow_1429744205e_indi > arrow_1429744205e_indi, A : set_Ar1007576579e_indi, A2 : arrow_1429744205e_indi, B : arrow_1429744205e_indi]: ((inj_on1663454827e_indi @ F @ A) => ((member1966420836e_indi @ A2 @ A) => ((member1966420836e_indi @ B @ A) => (inj_on1663454827e_indi @ (swap_A1408198358e_indi @ A2 @ B @ F) @ A))))))). % inj_on_imp_inj_on_swap
thf(fact_168_the__inv__into__f__f, axiom,
    ((![F : arrow_1429744205e_indi > nat, A : set_Ar1007576579e_indi, X2 : arrow_1429744205e_indi]: ((inj_on528257168di_nat @ F @ A) => ((member1966420836e_indi @ X2 @ A) => ((the_in1052612818di_nat @ A @ F @ (F @ X2)) = X2)))))). % the_inv_into_f_f
thf(fact_169_the__inv__into__f__f, axiom,
    ((![F : arrow_1429744205e_indi > arrow_1429744205e_indi, A : set_Ar1007576579e_indi, X2 : arrow_1429744205e_indi]: ((inj_on1663454827e_indi @ F @ A) => ((member1966420836e_indi @ X2 @ A) => ((the_in1730881961e_indi @ A @ F @ (F @ X2)) = X2)))))). % the_inv_into_f_f
thf(fact_170_the__inv__into__f__eq, axiom,
    ((![F : arrow_1429744205e_indi > nat, A : set_Ar1007576579e_indi, X2 : arrow_1429744205e_indi, Y2 : nat]: ((inj_on528257168di_nat @ F @ A) => (((F @ X2) = Y2) => ((member1966420836e_indi @ X2 @ A) => ((the_in1052612818di_nat @ A @ F @ Y2) = X2))))))). % the_inv_into_f_eq
thf(fact_171_the__inv__into__f__eq, axiom,
    ((![F : arrow_1429744205e_indi > arrow_1429744205e_indi, A : set_Ar1007576579e_indi, X2 : arrow_1429744205e_indi, Y2 : arrow_1429744205e_indi]: ((inj_on1663454827e_indi @ F @ A) => (((F @ X2) = Y2) => ((member1966420836e_indi @ X2 @ A) => ((the_in1730881961e_indi @ A @ F @ Y2) = X2))))))). % the_inv_into_f_eq
thf(fact_172_surj__imp__surj__swap, axiom,
    ((![F : arrow_1429744205e_indi > nat, A2 : arrow_1429744205e_indi, B : arrow_1429744205e_indi]: (((image_555606308di_nat @ F @ top_to1799531699e_indi) = top_top_set_nat) => ((image_555606308di_nat @ (swap_A1227367653di_nat @ A2 @ B @ F) @ top_to1799531699e_indi) = top_top_set_nat))))). % surj_imp_surj_swap
thf(fact_173_surj__imp__surj__swap, axiom,
    ((![F : arrow_1429744205e_indi > arrow_1429744205e_indi, A2 : arrow_1429744205e_indi, B : arrow_1429744205e_indi]: (((image_688677079e_indi @ F @ top_to1799531699e_indi) = top_to1799531699e_indi) => ((image_688677079e_indi @ (swap_A1408198358e_indi @ A2 @ B @ F) @ top_to1799531699e_indi) = top_to1799531699e_indi))))). % surj_imp_surj_swap
thf(fact_174_f__the__inv__into__f, axiom,
    ((![F : arrow_1429744205e_indi > nat, A : set_Ar1007576579e_indi, Y2 : nat]: ((inj_on528257168di_nat @ F @ A) => ((member_nat @ Y2 @ (image_555606308di_nat @ F @ A)) => ((F @ (the_in1052612818di_nat @ A @ F @ Y2)) = Y2)))))). % f_the_inv_into_f
thf(fact_175_f__the__inv__into__f, axiom,
    ((![F : arrow_1429744205e_indi > arrow_1429744205e_indi, A : set_Ar1007576579e_indi, Y2 : arrow_1429744205e_indi]: ((inj_on1663454827e_indi @ F @ A) => ((member1966420836e_indi @ Y2 @ (image_688677079e_indi @ F @ A)) => ((F @ (the_in1730881961e_indi @ A @ F @ Y2)) = Y2)))))). % f_the_inv_into_f
thf(fact_176_inj__on__the__inv__into, axiom,
    ((![F : nat > arrow_1429744205e_indi, A : set_nat]: ((inj_on1775343632e_indi @ F @ A) => (inj_on528257168di_nat @ (the_in152215634e_indi @ A @ F) @ (image_1802692772e_indi @ F @ A)))))). % inj_on_the_inv_into
thf(fact_177_inj__on__the__inv__into, axiom,
    ((![F : arrow_1429744205e_indi > nat, A : set_Ar1007576579e_indi]: ((inj_on528257168di_nat @ F @ A) => (inj_on1775343632e_indi @ (the_in1052612818di_nat @ A @ F) @ (image_555606308di_nat @ F @ A)))))). % inj_on_the_inv_into
thf(fact_178_inj__on__the__inv__into, axiom,
    ((![F : arrow_1429744205e_indi > arrow_1429744205e_indi, A : set_Ar1007576579e_indi]: ((inj_on1663454827e_indi @ F @ A) => (inj_on1663454827e_indi @ (the_in1730881961e_indi @ A @ F) @ (image_688677079e_indi @ F @ A)))))). % inj_on_the_inv_into
thf(fact_179_the__inv__f__f, axiom,
    ((![F : arrow_1429744205e_indi > nat, X2 : arrow_1429744205e_indi]: ((inj_on528257168di_nat @ F @ top_to1799531699e_indi) => ((the_in1052612818di_nat @ top_to1799531699e_indi @ F @ (F @ X2)) = X2))))). % the_inv_f_f
thf(fact_180_the__inv__f__f, axiom,
    ((![F : arrow_1429744205e_indi > arrow_1429744205e_indi, X2 : arrow_1429744205e_indi]: ((inj_on1663454827e_indi @ F @ top_to1799531699e_indi) => ((the_in1730881961e_indi @ top_to1799531699e_indi @ F @ (F @ X2)) = X2))))). % the_inv_f_f
thf(fact_181_ex__nat__less__eq, axiom,
    ((![N2 : nat, P : nat > $o]: ((?[M2 : nat]: (((ord_less_nat @ M2 @ N2)) & ((P @ M2)))) = (?[X : nat]: (((member_nat @ X @ (set_or562006527an_nat @ zero_zero_nat @ N2))) & ((P @ X)))))))). % ex_nat_less_eq
thf(fact_182_all__nat__less__eq, axiom,
    ((![N2 : nat, P : nat > $o]: ((![M2 : nat]: (((ord_less_nat @ M2 @ N2)) => ((P @ M2)))) = (![X : nat]: (((member_nat @ X @ (set_or562006527an_nat @ zero_zero_nat @ N2))) => ((P @ X)))))))). % all_nat_less_eq
thf(fact_183_atLeastLessThan__inj_I2_J, axiom,
    ((![A2 : nat, B : nat, C2 : nat, D2 : nat]: (((set_or562006527an_nat @ A2 @ B) = (set_or562006527an_nat @ C2 @ D2)) => ((ord_less_nat @ A2 @ B) => ((ord_less_nat @ C2 @ D2) => (B = D2))))))). % atLeastLessThan_inj(2)
thf(fact_184_atLeastLessThan__inj_I1_J, axiom,
    ((![A2 : nat, B : nat, C2 : nat, D2 : nat]: (((set_or562006527an_nat @ A2 @ B) = (set_or562006527an_nat @ C2 @ D2)) => ((ord_less_nat @ A2 @ B) => ((ord_less_nat @ C2 @ D2) => (A2 = C2))))))). % atLeastLessThan_inj(1)
thf(fact_185_atLeastLessThan__eq__iff, axiom,
    ((![A2 : nat, B : nat, C2 : nat, D2 : nat]: ((ord_less_nat @ A2 @ B) => ((ord_less_nat @ C2 @ D2) => (((set_or562006527an_nat @ A2 @ B) = (set_or562006527an_nat @ C2 @ D2)) = (((A2 = C2)) & ((B = D2))))))))). % atLeastLessThan_eq_iff
thf(fact_186_less__numeral__extra_I3_J, axiom,
    ((~ ((ord_less_nat @ zero_zero_nat @ zero_zero_nat))))). % less_numeral_extra(3)
thf(fact_187_inj__on__strict__subset, axiom,
    ((![F : arrow_1429744205e_indi > nat, B2 : set_Ar1007576579e_indi, A : set_Ar1007576579e_indi]: ((inj_on528257168di_nat @ F @ B2) => ((ord_le1187139159e_indi @ A @ B2) => (ord_less_set_nat @ (image_555606308di_nat @ F @ A) @ (image_555606308di_nat @ F @ B2))))))). % inj_on_strict_subset
thf(fact_188_inj__on__strict__subset, axiom,
    ((![F : arrow_1429744205e_indi > arrow_1429744205e_indi, B2 : set_Ar1007576579e_indi, A : set_Ar1007576579e_indi]: ((inj_on1663454827e_indi @ F @ B2) => ((ord_le1187139159e_indi @ A @ B2) => (ord_le1187139159e_indi @ (image_688677079e_indi @ F @ A) @ (image_688677079e_indi @ F @ B2))))))). % inj_on_strict_subset
thf(fact_189_card__range__greater__zero, axiom,
    ((![F : arrow_1429744205e_indi > arrow_1429744205e_indi]: ((finite183240804e_indi @ (image_688677079e_indi @ F @ top_to1799531699e_indi)) => (ord_less_nat @ zero_zero_nat @ (finite927127589e_indi @ (image_688677079e_indi @ F @ top_to1799531699e_indi))))))). % card_range_greater_zero
thf(fact_190_card__range__greater__zero, axiom,
    ((![F : arrow_1429744205e_indi > nat]: ((finite_finite_nat @ (image_555606308di_nat @ F @ top_to1799531699e_indi)) => (ord_less_nat @ zero_zero_nat @ (finite_card_nat @ (image_555606308di_nat @ F @ top_to1799531699e_indi))))))). % card_range_greater_zero
thf(fact_191_finite__atLeastLessThan, axiom,
    ((![L : nat, U : nat]: (finite_finite_nat @ (set_or562006527an_nat @ L @ U))))). % finite_atLeastLessThan

% Conjectures (2)
thf(conj_0, hypothesis,
    ((![H : arrow_1429744205e_indi > nat]: ((inj_on528257168di_nat @ H @ top_to1799531699e_indi) => (((image_555606308di_nat @ H @ top_to1799531699e_indi) = (set_or562006527an_nat @ zero_zero_nat @ (finite927127589e_indi @ top_to1799531699e_indi))) => thesis))))).
thf(conj_1, conjecture,
    (thesis)).
