% 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_272__5189170_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:17:11.887

% Could-be-implicit typings (8)
thf(ty_n_t__Set__Oset_I_062_I_062_It__Arrow____Order____Mirabelle____riepwfubkl__Oindi_Mt__Set__Oset_It__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_J_J_J, type,
    set_Ar182050865le_alt : $tType).
thf(ty_n_t__Set__Oset_I_062_It__Arrow____Order____Mirabelle____riepwfubkl__Oindi_Mt__Set__Oset_It__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_J_J_J, type,
    set_Ar809243995le_alt : $tType).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_J_J, type,
    set_se2071012361le_alt : $tType).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_J, type,
    set_Pr367596371le_alt : $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 (57)
thf(sy_c_Arrow__Order__Mirabelle__riepwfubkl_OIIA, type,
    arrow_1821794627le_IIA : ((arrow_1429744205e_indi > set_Pr367596371le_alt) > set_Pr367596371le_alt) > $o).
thf(sy_c_Arrow__Order__Mirabelle__riepwfubkl_OLin, type,
    arrow_1848678355le_Lin : set_se2071012361le_alt).
thf(sy_c_Arrow__Order__Mirabelle__riepwfubkl_OProf, type,
    arrow_1951607831e_Prof : set_Ar809243995le_alt).
thf(sy_c_Arrow__Order__Mirabelle__riepwfubkl_Ounanimity, type,
    arrow_52334694nimity : ((arrow_1429744205e_indi > set_Pr367596371le_alt) > set_Pr367596371le_alt) > $o).
thf(sy_c_Finite__Set_Ocard_001t__Arrow____Order____Mirabelle____riepwfubkl__Oindi, type,
    finite927127589e_indi : set_Ar1007576579e_indi > nat).
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_FuncSet_OPi_001_062_It__Arrow____Order____Mirabelle____riepwfubkl__Oindi_Mt__Set__Oset_It__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_J, type,
    pi_Arr479247969le_alt : set_Ar809243995le_alt > ((arrow_1429744205e_indi > set_Pr367596371le_alt) > set_se2071012361le_alt) > set_Ar182050865le_alt).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat, type,
    zero_zero_nat : nat).
thf(sy_c_If_001t__Set__Oset_It__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_J, type,
    if_set550155277le_alt : $o > set_Pr367596371le_alt > set_Pr367596371le_alt > set_Pr367596371le_alt).
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_I_062_I_062_It__Arrow____Order____Mirabelle____riepwfubkl__Oindi_Mt__Set__Oset_It__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_J_J_J, type,
    ord_le1536050821le_alt : set_Ar182050865le_alt > set_Ar182050865le_alt > $o).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_I_062_It__Arrow____Order____Mirabelle____riepwfubkl__Oindi_Mt__Set__Oset_It__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_J_J_J, type,
    ord_le875007495le_alt : set_Ar809243995le_alt > set_Ar809243995le_alt > $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_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_J_J, type,
    ord_le1684089949le_alt : set_se2071012361le_alt > set_se2071012361le_alt > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat, type,
    ord_less_eq_nat : nat > nat > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_I_062_It__Arrow____Order____Mirabelle____riepwfubkl__Oindi_Mt__Set__Oset_It__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_J_J_J, type,
    ord_le1453801617le_alt : set_Ar182050865le_alt > set_Ar182050865le_alt > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Arrow____Order____Mirabelle____riepwfubkl__Oindi_Mt__Set__Oset_It__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_J_J_J, type,
    ord_le22412539le_alt : set_Ar809243995le_alt > set_Ar809243995le_alt > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Arrow____Order____Mirabelle____riepwfubkl__Oindi_J, type,
    ord_le2059613795e_indi : set_Ar1007576579e_indi > set_Ar1007576579e_indi > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J, type,
    ord_less_eq_set_nat : set_nat > set_nat > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_J_J, type,
    ord_le1836905577le_alt : set_se2071012361le_alt > set_se2071012361le_alt > $o).
thf(sy_c_Orderings_Otop__class_Otop_001_062_I_062_I_062_It__Arrow____Order____Mirabelle____riepwfubkl__Oindi_Mt__Set__Oset_It__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_J_J_M_Eo_J, type,
    top_to1892800228_alt_o : ((arrow_1429744205e_indi > set_Pr367596371le_alt) > set_Pr367596371le_alt) > $o).
thf(sy_c_Orderings_Otop__class_Otop_001_062_I_062_It__Arrow____Order____Mirabelle____riepwfubkl__Oindi_Mt__Set__Oset_It__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_J_J_M_Eo_J, type,
    top_to1344581618_alt_o : (arrow_1429744205e_indi > set_Pr367596371le_alt) > $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__Set__Oset_It__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_J_M_Eo_J, type,
    top_to1900088076_alt_o : set_Pr367596371le_alt > $o).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_I_062_It__Arrow____Order____Mirabelle____riepwfubkl__Oindi_Mt__Set__Oset_It__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_J_J_J, type,
    top_to803745505le_alt : set_Ar182050865le_alt).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Arrow____Order____Mirabelle____riepwfubkl__Oindi_Mt__Set__Oset_It__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_J_J_J, type,
    top_to685525675le_alt : set_Ar809243995le_alt).
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__Set__Oset_It__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_J_J, type,
    top_to469035705le_alt : set_se2071012361le_alt).
thf(sy_c_Set_OCollect_001_062_I_062_It__Arrow____Order____Mirabelle____riepwfubkl__Oindi_Mt__Set__Oset_It__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_J_J, type,
    collec1382217680le_alt : (((arrow_1429744205e_indi > set_Pr367596371le_alt) > set_Pr367596371le_alt) > $o) > set_Ar182050865le_alt).
thf(sy_c_Set_OCollect_001_062_It__Arrow____Order____Mirabelle____riepwfubkl__Oindi_Mt__Set__Oset_It__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_J_J, type,
    collec1559089382le_alt : ((arrow_1429744205e_indi > set_Pr367596371le_alt) > $o) > set_Ar809243995le_alt).
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_OCollect_001t__Set__Oset_It__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_J, type,
    collec1399441576le_alt : (set_Pr367596371le_alt > $o) > set_se2071012361le_alt).
thf(sy_c_Set_Oimage_001_062_It__Arrow____Order____Mirabelle____riepwfubkl__Oindi_Mt__Set__Oset_It__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_J_J_001_062_It__Arrow____Order____Mirabelle____riepwfubkl__Oindi_Mt__Set__Oset_It__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_J_J, type,
    image_829063553le_alt : ((arrow_1429744205e_indi > set_Pr367596371le_alt) > arrow_1429744205e_indi > set_Pr367596371le_alt) > set_Ar809243995le_alt > set_Ar809243995le_alt).
thf(sy_c_Set_Oimage_001_062_It__Arrow____Order____Mirabelle____riepwfubkl__Oindi_Mt__Set__Oset_It__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_J_J_001t__Arrow____Order____Mirabelle____riepwfubkl__Oindi, type,
    image_2000124405e_indi : ((arrow_1429744205e_indi > set_Pr367596371le_alt) > arrow_1429744205e_indi) > set_Ar809243995le_alt > set_Ar1007576579e_indi).
thf(sy_c_Set_Oimage_001_062_It__Arrow____Order____Mirabelle____riepwfubkl__Oindi_Mt__Set__Oset_It__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_J_J_001t__Nat__Onat, type,
    image_1037394822lt_nat : ((arrow_1429744205e_indi > set_Pr367596371le_alt) > nat) > set_Ar809243995le_alt > set_nat).
thf(sy_c_Set_Oimage_001_062_It__Arrow____Order____Mirabelle____riepwfubkl__Oindi_Mt__Set__Oset_It__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_J, type,
    image_712506523le_alt : ((arrow_1429744205e_indi > set_Pr367596371le_alt) > set_Pr367596371le_alt) > set_Ar809243995le_alt > set_se2071012361le_alt).
thf(sy_c_Set_Oimage_001t__Arrow____Order____Mirabelle____riepwfubkl__Oindi_001_062_It__Arrow____Order____Mirabelle____riepwfubkl__Oindi_Mt__Set__Oset_It__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_J_J, type,
    image_836174111le_alt : (arrow_1429744205e_indi > arrow_1429744205e_indi > set_Pr367596371le_alt) > set_Ar1007576579e_indi > set_Ar809243995le_alt).
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__Arrow____Order____Mirabelle____riepwfubkl__Oindi_001t__Set__Oset_It__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_J, type,
    image_444707709le_alt : (arrow_1429744205e_indi > set_Pr367596371le_alt) > set_Ar1007576579e_indi > set_se2071012361le_alt).
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__Set__Oset_It__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_J_001_062_It__Arrow____Order____Mirabelle____riepwfubkl__Oindi_Mt__Set__Oset_It__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_J_J, type,
    image_1576623141le_alt : (set_Pr367596371le_alt > arrow_1429744205e_indi > set_Pr367596371le_alt) > set_se2071012361le_alt > set_Ar809243995le_alt).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_J_001t__Arrow____Order____Mirabelle____riepwfubkl__Oindi, type,
    image_220478033e_indi : (set_Pr367596371le_alt > arrow_1429744205e_indi) > set_se2071012361le_alt > set_Ar1007576579e_indi).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_J_001t__Nat__Onat, type,
    image_407642666lt_nat : (set_Pr367596371le_alt > nat) > set_se2071012361le_alt > set_nat).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_J, type,
    image_71834871le_alt : (set_Pr367596371le_alt > set_Pr367596371le_alt) > set_se2071012361le_alt > set_se2071012361le_alt).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat, type,
    set_or562006527an_nat : nat > nat > set_nat).
thf(sy_c_member_001_062_I_062_It__Arrow____Order____Mirabelle____riepwfubkl__Oindi_Mt__Set__Oset_It__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_J_J, type,
    member183760530le_alt : ((arrow_1429744205e_indi > set_Pr367596371le_alt) > set_Pr367596371le_alt) > set_Ar182050865le_alt > $o).
thf(sy_c_member_001_062_It__Arrow____Order____Mirabelle____riepwfubkl__Oindi_Mt__Set__Oset_It__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_J_J, type,
    member684274596le_alt : (arrow_1429744205e_indi > set_Pr367596371le_alt) > set_Ar809243995le_alt > $o).
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_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_J, type,
    member1334244458le_alt : set_Pr367596371le_alt > set_se2071012361le_alt > $o).
thf(sy_v_F, type,
    f : (arrow_1429744205e_indi > set_Pr367596371le_alt) > set_Pr367596371le_alt).
thf(sy_v_Lab____, type,
    lab : set_Pr367596371le_alt).
thf(sy_v_Lba____, type,
    lba : set_Pr367596371le_alt).
thf(sy_v_h____, type,
    h : arrow_1429744205e_indi > nat).

% Relevant facts (247)
thf(fact_0__092_060open_062Lba_A_092_060in_062_ALin_092_060close_062, axiom,
    ((member1334244458le_alt @ lba @ arrow_1848678355le_Lin))). % \<open>Lba \<in> Lin\<close>
thf(fact_1__092_060open_062Lab_A_092_060in_062_ALin_092_060close_062, axiom,
    ((member1334244458le_alt @ lab @ arrow_1848678355le_Lin))). % \<open>Lab \<in> Lin\<close>
thf(fact_2__092_060open_062range_Ah_A_092_060subseteq_062_A_1230_O_O_060N_125_092_060close_062, axiom,
    ((ord_less_eq_set_nat @ (image_555606308di_nat @ h @ top_to1799531699e_indi) @ (set_or562006527an_nat @ zero_zero_nat @ (finite927127589e_indi @ top_to1799531699e_indi))))). % \<open>range h \<subseteq> {0..<N}\<close>
thf(fact_3_surjh, axiom,
    (((image_555606308di_nat @ h @ top_to1799531699e_indi) = (set_or562006527an_nat @ zero_zero_nat @ (finite927127589e_indi @ top_to1799531699e_indi))))). % surjh
thf(fact_4_injh, axiom,
    ((inj_on528257168di_nat @ h @ top_to1799531699e_indi))). % injh
thf(fact_5_PiProf, axiom,
    ((![N : nat]: (member684274596le_alt @ (^[I : arrow_1429744205e_indi]: (if_set550155277le_alt @ (ord_less_nat @ (h @ I) @ N) @ lab @ lba)) @ arrow_1951607831e_Prof)))). % PiProf
thf(fact_6_UNIV__I, axiom,
    ((![X : set_Pr367596371le_alt]: (member1334244458le_alt @ X @ top_to469035705le_alt)))). % UNIV_I
thf(fact_7_UNIV__I, axiom,
    ((![X : arrow_1429744205e_indi > set_Pr367596371le_alt]: (member684274596le_alt @ X @ top_to685525675le_alt)))). % UNIV_I
thf(fact_8_UNIV__I, axiom,
    ((![X : (arrow_1429744205e_indi > set_Pr367596371le_alt) > set_Pr367596371le_alt]: (member183760530le_alt @ X @ top_to803745505le_alt)))). % UNIV_I
thf(fact_9_UNIV__I, axiom,
    ((![X : arrow_1429744205e_indi]: (member1966420836e_indi @ X @ top_to1799531699e_indi)))). % UNIV_I
thf(fact_10_iso__tuple__UNIV__I, axiom,
    ((![X : set_Pr367596371le_alt]: (member1334244458le_alt @ X @ top_to469035705le_alt)))). % iso_tuple_UNIV_I
thf(fact_11_iso__tuple__UNIV__I, axiom,
    ((![X : arrow_1429744205e_indi > set_Pr367596371le_alt]: (member684274596le_alt @ X @ top_to685525675le_alt)))). % iso_tuple_UNIV_I
thf(fact_12_iso__tuple__UNIV__I, axiom,
    ((![X : (arrow_1429744205e_indi > set_Pr367596371le_alt) > set_Pr367596371le_alt]: (member183760530le_alt @ X @ top_to803745505le_alt)))). % iso_tuple_UNIV_I
thf(fact_13_iso__tuple__UNIV__I, axiom,
    ((![X : arrow_1429744205e_indi]: (member1966420836e_indi @ X @ top_to1799531699e_indi)))). % iso_tuple_UNIV_I
thf(fact_14__C0_C, axiom,
    ((![N : nat]: (member1334244458le_alt @ (f @ (^[I : arrow_1429744205e_indi]: (if_set550155277le_alt @ (ord_less_nat @ (h @ I) @ N) @ lab @ lba))) @ arrow_1848678355le_Lin)))). % "0"
thf(fact_15_top_Oextremum__strict, axiom,
    ((![A : set_Ar1007576579e_indi]: (~ ((ord_le1187139159e_indi @ top_to1799531699e_indi @ A)))))). % top.extremum_strict
thf(fact_16_top_Oextremum__strict, axiom,
    ((![A : set_Ar182050865le_alt]: (~ ((ord_le1536050821le_alt @ top_to803745505le_alt @ A)))))). % top.extremum_strict
thf(fact_17_top_Oextremum__strict, axiom,
    ((![A : set_Ar809243995le_alt]: (~ ((ord_le875007495le_alt @ top_to685525675le_alt @ A)))))). % top.extremum_strict
thf(fact_18_top_Oextremum__strict, axiom,
    ((![A : set_se2071012361le_alt]: (~ ((ord_le1684089949le_alt @ top_to469035705le_alt @ A)))))). % top.extremum_strict
thf(fact_19_top_Onot__eq__extremum, axiom,
    ((![A : set_Ar1007576579e_indi]: ((~ ((A = top_to1799531699e_indi))) = (ord_le1187139159e_indi @ A @ top_to1799531699e_indi))))). % top.not_eq_extremum
thf(fact_20_top_Onot__eq__extremum, axiom,
    ((![A : set_Ar182050865le_alt]: ((~ ((A = top_to803745505le_alt))) = (ord_le1536050821le_alt @ A @ top_to803745505le_alt))))). % top.not_eq_extremum
thf(fact_21_top_Onot__eq__extremum, axiom,
    ((![A : set_Ar809243995le_alt]: ((~ ((A = top_to685525675le_alt))) = (ord_le875007495le_alt @ A @ top_to685525675le_alt))))). % top.not_eq_extremum
thf(fact_22_top_Onot__eq__extremum, axiom,
    ((![A : set_se2071012361le_alt]: ((~ ((A = top_to469035705le_alt))) = (ord_le1684089949le_alt @ A @ top_to469035705le_alt))))). % top.not_eq_extremum
thf(fact_23_UNIV__def, axiom,
    ((top_to1799531699e_indi = (collec1169676194e_indi @ (^[X2 : arrow_1429744205e_indi]: $true))))). % UNIV_def
thf(fact_24_UNIV__def, axiom,
    ((top_to803745505le_alt = (collec1382217680le_alt @ (^[X2 : (arrow_1429744205e_indi > set_Pr367596371le_alt) > set_Pr367596371le_alt]: $true))))). % UNIV_def
thf(fact_25_UNIV__def, axiom,
    ((top_to685525675le_alt = (collec1559089382le_alt @ (^[X2 : arrow_1429744205e_indi > set_Pr367596371le_alt]: $true))))). % UNIV_def
thf(fact_26_UNIV__def, axiom,
    ((top_to469035705le_alt = (collec1399441576le_alt @ (^[X2 : set_Pr367596371le_alt]: $true))))). % UNIV_def
thf(fact_27_UNIV__eq__I, axiom,
    ((![A2 : set_Ar1007576579e_indi]: ((![X3 : arrow_1429744205e_indi]: (member1966420836e_indi @ X3 @ A2)) => (top_to1799531699e_indi = A2))))). % UNIV_eq_I
thf(fact_28_UNIV__eq__I, axiom,
    ((![A2 : set_Ar182050865le_alt]: ((![X3 : (arrow_1429744205e_indi > set_Pr367596371le_alt) > set_Pr367596371le_alt]: (member183760530le_alt @ X3 @ A2)) => (top_to803745505le_alt = A2))))). % UNIV_eq_I
thf(fact_29_UNIV__eq__I, axiom,
    ((![A2 : set_Ar809243995le_alt]: ((![X3 : arrow_1429744205e_indi > set_Pr367596371le_alt]: (member684274596le_alt @ X3 @ A2)) => (top_to685525675le_alt = A2))))). % UNIV_eq_I
thf(fact_30_UNIV__eq__I, axiom,
    ((![A2 : set_se2071012361le_alt]: ((![X3 : set_Pr367596371le_alt]: (member1334244458le_alt @ X3 @ A2)) => (top_to469035705le_alt = A2))))). % UNIV_eq_I
thf(fact_31_UNIV__witness, axiom,
    ((?[X3 : arrow_1429744205e_indi]: (member1966420836e_indi @ X3 @ top_to1799531699e_indi)))). % UNIV_witness
thf(fact_32_UNIV__witness, axiom,
    ((?[X3 : (arrow_1429744205e_indi > set_Pr367596371le_alt) > set_Pr367596371le_alt]: (member183760530le_alt @ X3 @ top_to803745505le_alt)))). % UNIV_witness
thf(fact_33_UNIV__witness, axiom,
    ((?[X3 : arrow_1429744205e_indi > set_Pr367596371le_alt]: (member684274596le_alt @ X3 @ top_to685525675le_alt)))). % UNIV_witness
thf(fact_34_UNIV__witness, axiom,
    ((?[X3 : set_Pr367596371le_alt]: (member1334244458le_alt @ X3 @ top_to469035705le_alt)))). % UNIV_witness
thf(fact_35_nat__neq__iff, axiom,
    ((![M : nat, N : nat]: ((~ ((M = N))) = (((ord_less_nat @ M @ N)) | ((ord_less_nat @ N @ M))))))). % nat_neq_iff
thf(fact_36_less__not__refl, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_not_refl
thf(fact_37_u, axiom,
    ((arrow_52334694nimity @ f))). % u
thf(fact_38_assms_I3_J, axiom,
    ((arrow_1821794627le_IIA @ f))). % assms(3)
thf(fact_39_order__refl, axiom,
    ((![X : set_nat]: (ord_less_eq_set_nat @ X @ X)))). % order_refl
thf(fact_40_image__eqI, axiom,
    ((![B : nat, F : arrow_1429744205e_indi > nat, X : arrow_1429744205e_indi, A2 : set_Ar1007576579e_indi]: ((B = (F @ X)) => ((member1966420836e_indi @ X @ A2) => (member_nat @ B @ (image_555606308di_nat @ F @ A2))))))). % image_eqI
thf(fact_41_image__eqI, axiom,
    ((![B : arrow_1429744205e_indi, F : arrow_1429744205e_indi > arrow_1429744205e_indi, X : arrow_1429744205e_indi, A2 : set_Ar1007576579e_indi]: ((B = (F @ X)) => ((member1966420836e_indi @ X @ A2) => (member1966420836e_indi @ B @ (image_688677079e_indi @ F @ A2))))))). % image_eqI
thf(fact_42_image__eqI, axiom,
    ((![B : arrow_1429744205e_indi, F : set_Pr367596371le_alt > arrow_1429744205e_indi, X : set_Pr367596371le_alt, A2 : set_se2071012361le_alt]: ((B = (F @ X)) => ((member1334244458le_alt @ X @ A2) => (member1966420836e_indi @ B @ (image_220478033e_indi @ F @ A2))))))). % image_eqI
thf(fact_43_image__eqI, axiom,
    ((![B : set_Pr367596371le_alt, F : arrow_1429744205e_indi > set_Pr367596371le_alt, X : arrow_1429744205e_indi, A2 : set_Ar1007576579e_indi]: ((B = (F @ X)) => ((member1966420836e_indi @ X @ A2) => (member1334244458le_alt @ B @ (image_444707709le_alt @ F @ A2))))))). % image_eqI
thf(fact_44_image__eqI, axiom,
    ((![B : arrow_1429744205e_indi, F : (arrow_1429744205e_indi > set_Pr367596371le_alt) > arrow_1429744205e_indi, X : arrow_1429744205e_indi > set_Pr367596371le_alt, A2 : set_Ar809243995le_alt]: ((B = (F @ X)) => ((member684274596le_alt @ X @ A2) => (member1966420836e_indi @ B @ (image_2000124405e_indi @ F @ A2))))))). % image_eqI
thf(fact_45_image__eqI, axiom,
    ((![B : arrow_1429744205e_indi > set_Pr367596371le_alt, F : arrow_1429744205e_indi > arrow_1429744205e_indi > set_Pr367596371le_alt, X : arrow_1429744205e_indi, A2 : set_Ar1007576579e_indi]: ((B = (F @ X)) => ((member1966420836e_indi @ X @ A2) => (member684274596le_alt @ B @ (image_836174111le_alt @ F @ A2))))))). % image_eqI
thf(fact_46_image__eqI, axiom,
    ((![B : set_Pr367596371le_alt, F : set_Pr367596371le_alt > set_Pr367596371le_alt, X : set_Pr367596371le_alt, A2 : set_se2071012361le_alt]: ((B = (F @ X)) => ((member1334244458le_alt @ X @ A2) => (member1334244458le_alt @ B @ (image_71834871le_alt @ F @ A2))))))). % image_eqI
thf(fact_47_image__eqI, axiom,
    ((![B : arrow_1429744205e_indi > set_Pr367596371le_alt, F : set_Pr367596371le_alt > arrow_1429744205e_indi > set_Pr367596371le_alt, X : set_Pr367596371le_alt, A2 : set_se2071012361le_alt]: ((B = (F @ X)) => ((member1334244458le_alt @ X @ A2) => (member684274596le_alt @ B @ (image_1576623141le_alt @ F @ A2))))))). % image_eqI
thf(fact_48_image__eqI, axiom,
    ((![B : set_Pr367596371le_alt, F : (arrow_1429744205e_indi > set_Pr367596371le_alt) > set_Pr367596371le_alt, X : arrow_1429744205e_indi > set_Pr367596371le_alt, A2 : set_Ar809243995le_alt]: ((B = (F @ X)) => ((member684274596le_alt @ X @ A2) => (member1334244458le_alt @ B @ (image_712506523le_alt @ F @ A2))))))). % image_eqI
thf(fact_49_image__eqI, axiom,
    ((![B : arrow_1429744205e_indi > set_Pr367596371le_alt, F : (arrow_1429744205e_indi > set_Pr367596371le_alt) > arrow_1429744205e_indi > set_Pr367596371le_alt, X : arrow_1429744205e_indi > set_Pr367596371le_alt, A2 : set_Ar809243995le_alt]: ((B = (F @ X)) => ((member684274596le_alt @ X @ A2) => (member684274596le_alt @ B @ (image_829063553le_alt @ F @ A2))))))). % image_eqI
thf(fact_50_subsetI, axiom,
    ((![A2 : set_se2071012361le_alt, B2 : set_se2071012361le_alt]: ((![X3 : set_Pr367596371le_alt]: ((member1334244458le_alt @ X3 @ A2) => (member1334244458le_alt @ X3 @ B2))) => (ord_le1836905577le_alt @ A2 @ B2))))). % subsetI
thf(fact_51_subsetI, axiom,
    ((![A2 : set_Ar809243995le_alt, B2 : set_Ar809243995le_alt]: ((![X3 : arrow_1429744205e_indi > set_Pr367596371le_alt]: ((member684274596le_alt @ X3 @ A2) => (member684274596le_alt @ X3 @ B2))) => (ord_le22412539le_alt @ A2 @ B2))))). % subsetI
thf(fact_52_subsetI, axiom,
    ((![A2 : set_Ar182050865le_alt, B2 : set_Ar182050865le_alt]: ((![X3 : (arrow_1429744205e_indi > set_Pr367596371le_alt) > set_Pr367596371le_alt]: ((member183760530le_alt @ X3 @ A2) => (member183760530le_alt @ X3 @ B2))) => (ord_le1453801617le_alt @ A2 @ B2))))). % subsetI
thf(fact_53_subsetI, axiom,
    ((![A2 : set_Ar1007576579e_indi, B2 : set_Ar1007576579e_indi]: ((![X3 : arrow_1429744205e_indi]: ((member1966420836e_indi @ X3 @ A2) => (member1966420836e_indi @ X3 @ B2))) => (ord_le2059613795e_indi @ A2 @ B2))))). % subsetI
thf(fact_54_subsetI, axiom,
    ((![A2 : set_nat, B2 : set_nat]: ((![X3 : nat]: ((member_nat @ X3 @ A2) => (member_nat @ X3 @ B2))) => (ord_less_eq_set_nat @ A2 @ B2))))). % subsetI
thf(fact_55_psubsetI, axiom,
    ((![A2 : set_nat, B2 : set_nat]: ((ord_less_eq_set_nat @ A2 @ B2) => ((~ ((A2 = B2))) => (ord_less_set_nat @ A2 @ B2)))))). % psubsetI
thf(fact_56_subset__antisym, axiom,
    ((![A2 : set_nat, B2 : set_nat]: ((ord_less_eq_set_nat @ A2 @ B2) => ((ord_less_eq_set_nat @ B2 @ A2) => (A2 = B2)))))). % subset_antisym
thf(fact_57__092_060open_062F_A_I_092_060lambda_062i_O_ALba_J_A_092_060in_062_ALin_092_060close_062, axiom,
    ((member1334244458le_alt @ (f @ (^[I : arrow_1429744205e_indi]: lba)) @ arrow_1848678355le_Lin))). % \<open>F (\<lambda>i. Lba) \<in> Lin\<close>
thf(fact_58_bot__nat__0_Onot__eq__extremum, axiom,
    ((![A : nat]: ((~ ((A = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ A))))). % bot_nat_0.not_eq_extremum
thf(fact_59_less__nat__zero__code, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_nat_zero_code
thf(fact_60_neq0__conv, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ N))))). % neq0_conv
thf(fact_61__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062h_O_A_092_060lbrakk_062inj_Ah_059_Arange_Ah_A_061_A_1230_O_O_060N_125_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom,
    ((~ ((![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))))))))))). % \<open>\<And>thesis. (\<And>h. \<lbrakk>inj h; range h = {0..<N}\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_62_assms_I1_J, axiom,
    ((member183760530le_alt @ f @ (pi_Arr479247969le_alt @ arrow_1951607831e_Prof @ (^[Uu : arrow_1429744205e_indi > set_Pr367596371le_alt]: arrow_1848678355le_Lin))))). % assms(1)
thf(fact_63_imageI, axiom,
    ((![X : arrow_1429744205e_indi, A2 : set_Ar1007576579e_indi, F : arrow_1429744205e_indi > nat]: ((member1966420836e_indi @ X @ A2) => (member_nat @ (F @ X) @ (image_555606308di_nat @ F @ A2)))))). % imageI
thf(fact_64_imageI, axiom,
    ((![X : arrow_1429744205e_indi, A2 : set_Ar1007576579e_indi, F : arrow_1429744205e_indi > arrow_1429744205e_indi]: ((member1966420836e_indi @ X @ A2) => (member1966420836e_indi @ (F @ X) @ (image_688677079e_indi @ F @ A2)))))). % imageI
thf(fact_65_imageI, axiom,
    ((![X : set_Pr367596371le_alt, A2 : set_se2071012361le_alt, F : set_Pr367596371le_alt > arrow_1429744205e_indi]: ((member1334244458le_alt @ X @ A2) => (member1966420836e_indi @ (F @ X) @ (image_220478033e_indi @ F @ A2)))))). % imageI
thf(fact_66_imageI, axiom,
    ((![X : arrow_1429744205e_indi, A2 : set_Ar1007576579e_indi, F : arrow_1429744205e_indi > set_Pr367596371le_alt]: ((member1966420836e_indi @ X @ A2) => (member1334244458le_alt @ (F @ X) @ (image_444707709le_alt @ F @ A2)))))). % imageI
thf(fact_67_imageI, axiom,
    ((![X : arrow_1429744205e_indi > set_Pr367596371le_alt, A2 : set_Ar809243995le_alt, F : (arrow_1429744205e_indi > set_Pr367596371le_alt) > arrow_1429744205e_indi]: ((member684274596le_alt @ X @ A2) => (member1966420836e_indi @ (F @ X) @ (image_2000124405e_indi @ F @ A2)))))). % imageI
thf(fact_68_imageI, axiom,
    ((![X : arrow_1429744205e_indi, A2 : set_Ar1007576579e_indi, F : arrow_1429744205e_indi > arrow_1429744205e_indi > set_Pr367596371le_alt]: ((member1966420836e_indi @ X @ A2) => (member684274596le_alt @ (F @ X) @ (image_836174111le_alt @ F @ A2)))))). % imageI
thf(fact_69_imageI, axiom,
    ((![X : set_Pr367596371le_alt, A2 : set_se2071012361le_alt, F : set_Pr367596371le_alt > set_Pr367596371le_alt]: ((member1334244458le_alt @ X @ A2) => (member1334244458le_alt @ (F @ X) @ (image_71834871le_alt @ F @ A2)))))). % imageI
thf(fact_70_imageI, axiom,
    ((![X : set_Pr367596371le_alt, A2 : set_se2071012361le_alt, F : set_Pr367596371le_alt > arrow_1429744205e_indi > set_Pr367596371le_alt]: ((member1334244458le_alt @ X @ A2) => (member684274596le_alt @ (F @ X) @ (image_1576623141le_alt @ F @ A2)))))). % imageI
thf(fact_71_imageI, axiom,
    ((![X : arrow_1429744205e_indi > set_Pr367596371le_alt, A2 : set_Ar809243995le_alt, F : (arrow_1429744205e_indi > set_Pr367596371le_alt) > set_Pr367596371le_alt]: ((member684274596le_alt @ X @ A2) => (member1334244458le_alt @ (F @ X) @ (image_712506523le_alt @ F @ A2)))))). % imageI
thf(fact_72_imageI, axiom,
    ((![X : arrow_1429744205e_indi > set_Pr367596371le_alt, A2 : set_Ar809243995le_alt, F : (arrow_1429744205e_indi > set_Pr367596371le_alt) > arrow_1429744205e_indi > set_Pr367596371le_alt]: ((member684274596le_alt @ X @ A2) => (member684274596le_alt @ (F @ X) @ (image_829063553le_alt @ F @ A2)))))). % imageI
thf(fact_73_in__mono, axiom,
    ((![A2 : set_se2071012361le_alt, B2 : set_se2071012361le_alt, X : set_Pr367596371le_alt]: ((ord_le1836905577le_alt @ A2 @ B2) => ((member1334244458le_alt @ X @ A2) => (member1334244458le_alt @ X @ B2)))))). % in_mono
thf(fact_74_in__mono, axiom,
    ((![A2 : set_Ar809243995le_alt, B2 : set_Ar809243995le_alt, X : arrow_1429744205e_indi > set_Pr367596371le_alt]: ((ord_le22412539le_alt @ A2 @ B2) => ((member684274596le_alt @ X @ A2) => (member684274596le_alt @ X @ B2)))))). % in_mono
thf(fact_75_in__mono, axiom,
    ((![A2 : set_Ar182050865le_alt, B2 : set_Ar182050865le_alt, X : (arrow_1429744205e_indi > set_Pr367596371le_alt) > set_Pr367596371le_alt]: ((ord_le1453801617le_alt @ A2 @ B2) => ((member183760530le_alt @ X @ A2) => (member183760530le_alt @ X @ B2)))))). % in_mono
thf(fact_76_in__mono, axiom,
    ((![A2 : set_Ar1007576579e_indi, B2 : set_Ar1007576579e_indi, X : arrow_1429744205e_indi]: ((ord_le2059613795e_indi @ A2 @ B2) => ((member1966420836e_indi @ X @ A2) => (member1966420836e_indi @ X @ B2)))))). % in_mono
thf(fact_77_in__mono, axiom,
    ((![A2 : set_nat, B2 : set_nat, X : nat]: ((ord_less_eq_set_nat @ A2 @ B2) => ((member_nat @ X @ A2) => (member_nat @ X @ B2)))))). % in_mono
thf(fact_78_subsetD, axiom,
    ((![A2 : set_se2071012361le_alt, B2 : set_se2071012361le_alt, C : set_Pr367596371le_alt]: ((ord_le1836905577le_alt @ A2 @ B2) => ((member1334244458le_alt @ C @ A2) => (member1334244458le_alt @ C @ B2)))))). % subsetD
thf(fact_79_subsetD, axiom,
    ((![A2 : set_Ar809243995le_alt, B2 : set_Ar809243995le_alt, C : arrow_1429744205e_indi > set_Pr367596371le_alt]: ((ord_le22412539le_alt @ A2 @ B2) => ((member684274596le_alt @ C @ A2) => (member684274596le_alt @ C @ B2)))))). % subsetD
thf(fact_80_subsetD, axiom,
    ((![A2 : set_Ar182050865le_alt, B2 : set_Ar182050865le_alt, C : (arrow_1429744205e_indi > set_Pr367596371le_alt) > set_Pr367596371le_alt]: ((ord_le1453801617le_alt @ A2 @ B2) => ((member183760530le_alt @ C @ A2) => (member183760530le_alt @ C @ B2)))))). % subsetD
thf(fact_81_subsetD, axiom,
    ((![A2 : set_Ar1007576579e_indi, B2 : set_Ar1007576579e_indi, C : arrow_1429744205e_indi]: ((ord_le2059613795e_indi @ A2 @ B2) => ((member1966420836e_indi @ C @ A2) => (member1966420836e_indi @ C @ B2)))))). % subsetD
thf(fact_82_subsetD, axiom,
    ((![A2 : set_nat, B2 : set_nat, C : nat]: ((ord_less_eq_set_nat @ A2 @ B2) => ((member_nat @ C @ A2) => (member_nat @ C @ B2)))))). % subsetD
thf(fact_83_psubsetE, axiom,
    ((![A2 : set_nat, B2 : set_nat]: ((ord_less_set_nat @ A2 @ B2) => (~ (((ord_less_eq_set_nat @ A2 @ B2) => (ord_less_eq_set_nat @ B2 @ A2)))))))). % psubsetE
thf(fact_84_equalityE, axiom,
    ((![A2 : set_nat, B2 : set_nat]: ((A2 = B2) => (~ (((ord_less_eq_set_nat @ A2 @ B2) => (~ ((ord_less_eq_set_nat @ B2 @ A2)))))))))). % equalityE
thf(fact_85_image__iff, axiom,
    ((![Z : nat, F : arrow_1429744205e_indi > nat, A2 : set_Ar1007576579e_indi]: ((member_nat @ Z @ (image_555606308di_nat @ F @ A2)) = (?[X2 : arrow_1429744205e_indi]: (((member1966420836e_indi @ X2 @ A2)) & ((Z = (F @ X2))))))))). % image_iff
thf(fact_86_subset__eq, axiom,
    ((ord_le1836905577le_alt = (^[A3 : set_se2071012361le_alt]: (^[B3 : set_se2071012361le_alt]: (![X2 : set_Pr367596371le_alt]: (((member1334244458le_alt @ X2 @ A3)) => ((member1334244458le_alt @ X2 @ B3))))))))). % subset_eq
thf(fact_87_subset__eq, axiom,
    ((ord_le22412539le_alt = (^[A3 : set_Ar809243995le_alt]: (^[B3 : set_Ar809243995le_alt]: (![X2 : arrow_1429744205e_indi > set_Pr367596371le_alt]: (((member684274596le_alt @ X2 @ A3)) => ((member684274596le_alt @ X2 @ B3))))))))). % subset_eq
thf(fact_88_subset__eq, axiom,
    ((ord_le1453801617le_alt = (^[A3 : set_Ar182050865le_alt]: (^[B3 : set_Ar182050865le_alt]: (![X2 : (arrow_1429744205e_indi > set_Pr367596371le_alt) > set_Pr367596371le_alt]: (((member183760530le_alt @ X2 @ A3)) => ((member183760530le_alt @ X2 @ B3))))))))). % subset_eq
thf(fact_89_subset__eq, axiom,
    ((ord_le2059613795e_indi = (^[A3 : set_Ar1007576579e_indi]: (^[B3 : set_Ar1007576579e_indi]: (![X2 : arrow_1429744205e_indi]: (((member1966420836e_indi @ X2 @ A3)) => ((member1966420836e_indi @ X2 @ B3))))))))). % subset_eq
thf(fact_90_subset__eq, axiom,
    ((ord_less_eq_set_nat = (^[A3 : set_nat]: (^[B3 : set_nat]: (![X2 : nat]: (((member_nat @ X2 @ A3)) => ((member_nat @ X2 @ B3))))))))). % subset_eq
thf(fact_91_bex__imageD, axiom,
    ((![F : arrow_1429744205e_indi > nat, A2 : set_Ar1007576579e_indi, P : nat > $o]: ((?[X4 : nat]: ((member_nat @ X4 @ (image_555606308di_nat @ F @ A2)) & (P @ X4))) => (?[X3 : arrow_1429744205e_indi]: ((member1966420836e_indi @ X3 @ A2) & (P @ (F @ X3)))))))). % bex_imageD
thf(fact_92_equalityD1, axiom,
    ((![A2 : set_nat, B2 : set_nat]: ((A2 = B2) => (ord_less_eq_set_nat @ A2 @ B2))))). % equalityD1
thf(fact_93_equalityD2, axiom,
    ((![A2 : set_nat, B2 : set_nat]: ((A2 = B2) => (ord_less_eq_set_nat @ B2 @ A2))))). % equalityD2
thf(fact_94_image__cong, axiom,
    ((![M2 : set_Ar1007576579e_indi, N2 : set_Ar1007576579e_indi, F : arrow_1429744205e_indi > nat, G : arrow_1429744205e_indi > nat]: ((M2 = N2) => ((![X3 : arrow_1429744205e_indi]: ((member1966420836e_indi @ X3 @ N2) => ((F @ X3) = (G @ X3)))) => ((image_555606308di_nat @ F @ M2) = (image_555606308di_nat @ G @ N2))))))). % image_cong
thf(fact_95_image__mono, axiom,
    ((![A2 : set_Ar1007576579e_indi, B2 : set_Ar1007576579e_indi, F : arrow_1429744205e_indi > nat]: ((ord_le2059613795e_indi @ A2 @ B2) => (ord_less_eq_set_nat @ (image_555606308di_nat @ F @ A2) @ (image_555606308di_nat @ F @ B2)))))). % image_mono
thf(fact_96_image__mono, axiom,
    ((![A2 : set_nat, B2 : set_nat, F : nat > nat]: ((ord_less_eq_set_nat @ A2 @ B2) => (ord_less_eq_set_nat @ (image_nat_nat @ F @ A2) @ (image_nat_nat @ F @ B2)))))). % image_mono
thf(fact_97_psubset__eq, axiom,
    ((ord_less_set_nat = (^[A3 : set_nat]: (^[B3 : set_nat]: (((ord_less_eq_set_nat @ A3 @ B3)) & ((~ ((A3 = B3)))))))))). % psubset_eq
thf(fact_98_subset__iff, axiom,
    ((ord_le1836905577le_alt = (^[A3 : set_se2071012361le_alt]: (^[B3 : set_se2071012361le_alt]: (![T : set_Pr367596371le_alt]: (((member1334244458le_alt @ T @ A3)) => ((member1334244458le_alt @ T @ B3))))))))). % subset_iff
thf(fact_99_subset__iff, axiom,
    ((ord_le22412539le_alt = (^[A3 : set_Ar809243995le_alt]: (^[B3 : set_Ar809243995le_alt]: (![T : arrow_1429744205e_indi > set_Pr367596371le_alt]: (((member684274596le_alt @ T @ A3)) => ((member684274596le_alt @ T @ B3))))))))). % subset_iff
thf(fact_100_subset__iff, axiom,
    ((ord_le1453801617le_alt = (^[A3 : set_Ar182050865le_alt]: (^[B3 : set_Ar182050865le_alt]: (![T : (arrow_1429744205e_indi > set_Pr367596371le_alt) > set_Pr367596371le_alt]: (((member183760530le_alt @ T @ A3)) => ((member183760530le_alt @ T @ B3))))))))). % subset_iff
thf(fact_101_subset__iff, axiom,
    ((ord_le2059613795e_indi = (^[A3 : set_Ar1007576579e_indi]: (^[B3 : set_Ar1007576579e_indi]: (![T : arrow_1429744205e_indi]: (((member1966420836e_indi @ T @ A3)) => ((member1966420836e_indi @ T @ B3))))))))). % subset_iff
thf(fact_102_subset__iff, axiom,
    ((ord_less_eq_set_nat = (^[A3 : set_nat]: (^[B3 : set_nat]: (![T : nat]: (((member_nat @ T @ A3)) => ((member_nat @ T @ B3))))))))). % subset_iff
thf(fact_103_mem__Collect__eq, axiom,
    ((![A : set_Pr367596371le_alt, P : set_Pr367596371le_alt > $o]: ((member1334244458le_alt @ A @ (collec1399441576le_alt @ P)) = (P @ A))))). % mem_Collect_eq
thf(fact_104_mem__Collect__eq, axiom,
    ((![A : arrow_1429744205e_indi > set_Pr367596371le_alt, P : (arrow_1429744205e_indi > set_Pr367596371le_alt) > $o]: ((member684274596le_alt @ A @ (collec1559089382le_alt @ P)) = (P @ A))))). % mem_Collect_eq
thf(fact_105_mem__Collect__eq, axiom,
    ((![A : (arrow_1429744205e_indi > set_Pr367596371le_alt) > set_Pr367596371le_alt, P : ((arrow_1429744205e_indi > set_Pr367596371le_alt) > set_Pr367596371le_alt) > $o]: ((member183760530le_alt @ A @ (collec1382217680le_alt @ P)) = (P @ A))))). % mem_Collect_eq
thf(fact_106_mem__Collect__eq, axiom,
    ((![A : arrow_1429744205e_indi, P : arrow_1429744205e_indi > $o]: ((member1966420836e_indi @ A @ (collec1169676194e_indi @ P)) = (P @ A))))). % mem_Collect_eq
thf(fact_107_Collect__mem__eq, axiom,
    ((![A2 : set_se2071012361le_alt]: ((collec1399441576le_alt @ (^[X2 : set_Pr367596371le_alt]: (member1334244458le_alt @ X2 @ A2))) = A2)))). % Collect_mem_eq
thf(fact_108_Collect__mem__eq, axiom,
    ((![A2 : set_Ar809243995le_alt]: ((collec1559089382le_alt @ (^[X2 : arrow_1429744205e_indi > set_Pr367596371le_alt]: (member684274596le_alt @ X2 @ A2))) = A2)))). % Collect_mem_eq
thf(fact_109_Collect__mem__eq, axiom,
    ((![A2 : set_Ar182050865le_alt]: ((collec1382217680le_alt @ (^[X2 : (arrow_1429744205e_indi > set_Pr367596371le_alt) > set_Pr367596371le_alt]: (member183760530le_alt @ X2 @ A2))) = A2)))). % Collect_mem_eq
thf(fact_110_Collect__mem__eq, axiom,
    ((![A2 : set_Ar1007576579e_indi]: ((collec1169676194e_indi @ (^[X2 : arrow_1429744205e_indi]: (member1966420836e_indi @ X2 @ A2))) = A2)))). % Collect_mem_eq
thf(fact_111_ball__imageD, axiom,
    ((![F : arrow_1429744205e_indi > nat, A2 : set_Ar1007576579e_indi, P : nat > $o]: ((![X3 : nat]: ((member_nat @ X3 @ (image_555606308di_nat @ F @ A2)) => (P @ X3))) => (![X4 : arrow_1429744205e_indi]: ((member1966420836e_indi @ X4 @ A2) => (P @ (F @ X4)))))))). % ball_imageD
thf(fact_112_subset__refl, axiom,
    ((![A2 : set_nat]: (ord_less_eq_set_nat @ A2 @ A2)))). % subset_refl
thf(fact_113_Collect__mono, axiom,
    ((![P : nat > $o, Q : nat > $o]: ((![X3 : nat]: ((P @ X3) => (Q @ X3))) => (ord_less_eq_set_nat @ (collect_nat @ P) @ (collect_nat @ Q)))))). % Collect_mono
thf(fact_114_subset__trans, axiom,
    ((![A2 : set_nat, B2 : set_nat, C2 : set_nat]: ((ord_less_eq_set_nat @ A2 @ B2) => ((ord_less_eq_set_nat @ B2 @ C2) => (ord_less_eq_set_nat @ A2 @ C2)))))). % subset_trans
thf(fact_115_image__subsetI, axiom,
    ((![A2 : set_Ar1007576579e_indi, F : arrow_1429744205e_indi > arrow_1429744205e_indi, B2 : set_Ar1007576579e_indi]: ((![X3 : arrow_1429744205e_indi]: ((member1966420836e_indi @ X3 @ A2) => (member1966420836e_indi @ (F @ X3) @ B2))) => (ord_le2059613795e_indi @ (image_688677079e_indi @ F @ A2) @ B2))))). % image_subsetI
thf(fact_116_image__subsetI, axiom,
    ((![A2 : set_Ar1007576579e_indi, F : arrow_1429744205e_indi > nat, B2 : set_nat]: ((![X3 : arrow_1429744205e_indi]: ((member1966420836e_indi @ X3 @ A2) => (member_nat @ (F @ X3) @ B2))) => (ord_less_eq_set_nat @ (image_555606308di_nat @ F @ A2) @ B2))))). % image_subsetI
thf(fact_117_image__subsetI, axiom,
    ((![A2 : set_se2071012361le_alt, F : set_Pr367596371le_alt > arrow_1429744205e_indi, B2 : set_Ar1007576579e_indi]: ((![X3 : set_Pr367596371le_alt]: ((member1334244458le_alt @ X3 @ A2) => (member1966420836e_indi @ (F @ X3) @ B2))) => (ord_le2059613795e_indi @ (image_220478033e_indi @ F @ A2) @ B2))))). % image_subsetI
thf(fact_118_image__subsetI, axiom,
    ((![A2 : set_Ar1007576579e_indi, F : arrow_1429744205e_indi > set_Pr367596371le_alt, B2 : set_se2071012361le_alt]: ((![X3 : arrow_1429744205e_indi]: ((member1966420836e_indi @ X3 @ A2) => (member1334244458le_alt @ (F @ X3) @ B2))) => (ord_le1836905577le_alt @ (image_444707709le_alt @ F @ A2) @ B2))))). % image_subsetI
thf(fact_119_image__subsetI, axiom,
    ((![A2 : set_se2071012361le_alt, F : set_Pr367596371le_alt > nat, B2 : set_nat]: ((![X3 : set_Pr367596371le_alt]: ((member1334244458le_alt @ X3 @ A2) => (member_nat @ (F @ X3) @ B2))) => (ord_less_eq_set_nat @ (image_407642666lt_nat @ F @ A2) @ B2))))). % image_subsetI
thf(fact_120_image__subsetI, axiom,
    ((![A2 : set_Ar809243995le_alt, F : (arrow_1429744205e_indi > set_Pr367596371le_alt) > arrow_1429744205e_indi, B2 : set_Ar1007576579e_indi]: ((![X3 : arrow_1429744205e_indi > set_Pr367596371le_alt]: ((member684274596le_alt @ X3 @ A2) => (member1966420836e_indi @ (F @ X3) @ B2))) => (ord_le2059613795e_indi @ (image_2000124405e_indi @ F @ A2) @ B2))))). % image_subsetI
thf(fact_121_image__subsetI, axiom,
    ((![A2 : set_Ar1007576579e_indi, F : arrow_1429744205e_indi > arrow_1429744205e_indi > set_Pr367596371le_alt, B2 : set_Ar809243995le_alt]: ((![X3 : arrow_1429744205e_indi]: ((member1966420836e_indi @ X3 @ A2) => (member684274596le_alt @ (F @ X3) @ B2))) => (ord_le22412539le_alt @ (image_836174111le_alt @ F @ A2) @ B2))))). % image_subsetI
thf(fact_122_image__subsetI, axiom,
    ((![A2 : set_Ar809243995le_alt, F : (arrow_1429744205e_indi > set_Pr367596371le_alt) > nat, B2 : set_nat]: ((![X3 : arrow_1429744205e_indi > set_Pr367596371le_alt]: ((member684274596le_alt @ X3 @ A2) => (member_nat @ (F @ X3) @ B2))) => (ord_less_eq_set_nat @ (image_1037394822lt_nat @ F @ A2) @ B2))))). % image_subsetI
thf(fact_123_image__subsetI, axiom,
    ((![A2 : set_se2071012361le_alt, F : set_Pr367596371le_alt > set_Pr367596371le_alt, B2 : set_se2071012361le_alt]: ((![X3 : set_Pr367596371le_alt]: ((member1334244458le_alt @ X3 @ A2) => (member1334244458le_alt @ (F @ X3) @ B2))) => (ord_le1836905577le_alt @ (image_71834871le_alt @ F @ A2) @ B2))))). % image_subsetI
thf(fact_124_image__subsetI, axiom,
    ((![A2 : set_se2071012361le_alt, F : set_Pr367596371le_alt > arrow_1429744205e_indi > set_Pr367596371le_alt, B2 : set_Ar809243995le_alt]: ((![X3 : set_Pr367596371le_alt]: ((member1334244458le_alt @ X3 @ A2) => (member684274596le_alt @ (F @ X3) @ B2))) => (ord_le22412539le_alt @ (image_1576623141le_alt @ F @ A2) @ B2))))). % image_subsetI
thf(fact_125_rev__image__eqI, axiom,
    ((![X : arrow_1429744205e_indi, A2 : set_Ar1007576579e_indi, B : nat, F : arrow_1429744205e_indi > nat]: ((member1966420836e_indi @ X @ A2) => ((B = (F @ X)) => (member_nat @ B @ (image_555606308di_nat @ F @ A2))))))). % rev_image_eqI
thf(fact_126_rev__image__eqI, axiom,
    ((![X : arrow_1429744205e_indi, A2 : set_Ar1007576579e_indi, B : arrow_1429744205e_indi, F : arrow_1429744205e_indi > arrow_1429744205e_indi]: ((member1966420836e_indi @ X @ A2) => ((B = (F @ X)) => (member1966420836e_indi @ B @ (image_688677079e_indi @ F @ A2))))))). % rev_image_eqI
thf(fact_127_rev__image__eqI, axiom,
    ((![X : set_Pr367596371le_alt, A2 : set_se2071012361le_alt, B : arrow_1429744205e_indi, F : set_Pr367596371le_alt > arrow_1429744205e_indi]: ((member1334244458le_alt @ X @ A2) => ((B = (F @ X)) => (member1966420836e_indi @ B @ (image_220478033e_indi @ F @ A2))))))). % rev_image_eqI
thf(fact_128_rev__image__eqI, axiom,
    ((![X : arrow_1429744205e_indi, A2 : set_Ar1007576579e_indi, B : set_Pr367596371le_alt, F : arrow_1429744205e_indi > set_Pr367596371le_alt]: ((member1966420836e_indi @ X @ A2) => ((B = (F @ X)) => (member1334244458le_alt @ B @ (image_444707709le_alt @ F @ A2))))))). % rev_image_eqI
thf(fact_129_rev__image__eqI, axiom,
    ((![X : arrow_1429744205e_indi > set_Pr367596371le_alt, A2 : set_Ar809243995le_alt, B : arrow_1429744205e_indi, F : (arrow_1429744205e_indi > set_Pr367596371le_alt) > arrow_1429744205e_indi]: ((member684274596le_alt @ X @ A2) => ((B = (F @ X)) => (member1966420836e_indi @ B @ (image_2000124405e_indi @ F @ A2))))))). % rev_image_eqI
thf(fact_130_rev__image__eqI, axiom,
    ((![X : arrow_1429744205e_indi, A2 : set_Ar1007576579e_indi, B : arrow_1429744205e_indi > set_Pr367596371le_alt, F : arrow_1429744205e_indi > arrow_1429744205e_indi > set_Pr367596371le_alt]: ((member1966420836e_indi @ X @ A2) => ((B = (F @ X)) => (member684274596le_alt @ B @ (image_836174111le_alt @ F @ A2))))))). % rev_image_eqI
thf(fact_131_rev__image__eqI, axiom,
    ((![X : set_Pr367596371le_alt, A2 : set_se2071012361le_alt, B : set_Pr367596371le_alt, F : set_Pr367596371le_alt > set_Pr367596371le_alt]: ((member1334244458le_alt @ X @ A2) => ((B = (F @ X)) => (member1334244458le_alt @ B @ (image_71834871le_alt @ F @ A2))))))). % rev_image_eqI
thf(fact_132_rev__image__eqI, axiom,
    ((![X : set_Pr367596371le_alt, A2 : set_se2071012361le_alt, B : arrow_1429744205e_indi > set_Pr367596371le_alt, F : set_Pr367596371le_alt > arrow_1429744205e_indi > set_Pr367596371le_alt]: ((member1334244458le_alt @ X @ A2) => ((B = (F @ X)) => (member684274596le_alt @ B @ (image_1576623141le_alt @ F @ A2))))))). % rev_image_eqI
thf(fact_133_rev__image__eqI, axiom,
    ((![X : arrow_1429744205e_indi > set_Pr367596371le_alt, A2 : set_Ar809243995le_alt, B : set_Pr367596371le_alt, F : (arrow_1429744205e_indi > set_Pr367596371le_alt) > set_Pr367596371le_alt]: ((member684274596le_alt @ X @ A2) => ((B = (F @ X)) => (member1334244458le_alt @ B @ (image_712506523le_alt @ F @ A2))))))). % rev_image_eqI
thf(fact_134_rev__image__eqI, axiom,
    ((![X : arrow_1429744205e_indi > set_Pr367596371le_alt, A2 : set_Ar809243995le_alt, B : arrow_1429744205e_indi > set_Pr367596371le_alt, F : (arrow_1429744205e_indi > set_Pr367596371le_alt) > arrow_1429744205e_indi > set_Pr367596371le_alt]: ((member684274596le_alt @ X @ A2) => ((B = (F @ X)) => (member684274596le_alt @ B @ (image_829063553le_alt @ F @ A2))))))). % rev_image_eqI
thf(fact_135_set__eq__subset, axiom,
    (((^[Y : set_nat]: (^[Z2 : set_nat]: (Y = Z2))) = (^[A3 : set_nat]: (^[B3 : set_nat]: (((ord_less_eq_set_nat @ A3 @ B3)) & ((ord_less_eq_set_nat @ B3 @ A3)))))))). % set_eq_subset
thf(fact_136_subset__imageE, axiom,
    ((![B2 : set_nat, F : arrow_1429744205e_indi > nat, A2 : set_Ar1007576579e_indi]: ((ord_less_eq_set_nat @ B2 @ (image_555606308di_nat @ F @ A2)) => (~ ((![C3 : set_Ar1007576579e_indi]: ((ord_le2059613795e_indi @ C3 @ A2) => (~ ((B2 = (image_555606308di_nat @ F @ C3)))))))))))). % subset_imageE
thf(fact_137_subset__imageE, axiom,
    ((![B2 : set_nat, F : nat > nat, A2 : set_nat]: ((ord_less_eq_set_nat @ B2 @ (image_nat_nat @ F @ A2)) => (~ ((![C3 : set_nat]: ((ord_less_eq_set_nat @ C3 @ A2) => (~ ((B2 = (image_nat_nat @ F @ C3)))))))))))). % subset_imageE
thf(fact_138_Collect__mono__iff, axiom,
    ((![P : nat > $o, Q : nat > $o]: ((ord_less_eq_set_nat @ (collect_nat @ P) @ (collect_nat @ Q)) = (![X2 : nat]: (((P @ X2)) => ((Q @ X2)))))))). % Collect_mono_iff
thf(fact_139_image__subset__iff, axiom,
    ((![F : arrow_1429744205e_indi > nat, A2 : set_Ar1007576579e_indi, B2 : set_nat]: ((ord_less_eq_set_nat @ (image_555606308di_nat @ F @ A2) @ B2) = (![X2 : arrow_1429744205e_indi]: (((member1966420836e_indi @ X2 @ A2)) => ((member_nat @ (F @ X2) @ B2)))))))). % image_subset_iff
thf(fact_140_subset__image__iff, axiom,
    ((![B2 : set_nat, F : arrow_1429744205e_indi > nat, A2 : set_Ar1007576579e_indi]: ((ord_less_eq_set_nat @ B2 @ (image_555606308di_nat @ F @ A2)) = (?[AA : set_Ar1007576579e_indi]: (((ord_le2059613795e_indi @ AA @ A2)) & ((B2 = (image_555606308di_nat @ F @ AA))))))))). % subset_image_iff
thf(fact_141_subset__image__iff, axiom,
    ((![B2 : set_nat, F : nat > nat, A2 : set_nat]: ((ord_less_eq_set_nat @ B2 @ (image_nat_nat @ F @ A2)) = (?[AA : set_nat]: (((ord_less_eq_set_nat @ AA @ A2)) & ((B2 = (image_nat_nat @ F @ AA))))))))). % subset_image_iff
thf(fact_142_order__subst1, axiom,
    ((![A : set_nat, F : set_nat > set_nat, B : set_nat, C : set_nat]: ((ord_less_eq_set_nat @ A @ (F @ B)) => ((ord_less_eq_set_nat @ B @ C) => ((![X3 : set_nat, Y2 : set_nat]: ((ord_less_eq_set_nat @ X3 @ Y2) => (ord_less_eq_set_nat @ (F @ X3) @ (F @ Y2)))) => (ord_less_eq_set_nat @ A @ (F @ C)))))))). % order_subst1
thf(fact_143_order__subst2, axiom,
    ((![A : set_nat, B : set_nat, F : set_nat > set_nat, C : set_nat]: ((ord_less_eq_set_nat @ A @ B) => ((ord_less_eq_set_nat @ (F @ B) @ C) => ((![X3 : set_nat, Y2 : set_nat]: ((ord_less_eq_set_nat @ X3 @ Y2) => (ord_less_eq_set_nat @ (F @ X3) @ (F @ Y2)))) => (ord_less_eq_set_nat @ (F @ A) @ C))))))). % order_subst2
thf(fact_144_psubset__imp__subset, axiom,
    ((![A2 : set_nat, B2 : set_nat]: ((ord_less_set_nat @ A2 @ B2) => (ord_less_eq_set_nat @ A2 @ B2))))). % psubset_imp_subset
thf(fact_145_psubset__subset__trans, axiom,
    ((![A2 : set_nat, B2 : set_nat, C2 : set_nat]: ((ord_less_set_nat @ A2 @ B2) => ((ord_less_eq_set_nat @ B2 @ C2) => (ord_less_set_nat @ A2 @ C2)))))). % psubset_subset_trans
thf(fact_146_subset__not__subset__eq, axiom,
    ((ord_less_set_nat = (^[A3 : set_nat]: (^[B3 : set_nat]: (((ord_less_eq_set_nat @ A3 @ B3)) & ((~ ((ord_less_eq_set_nat @ B3 @ A3)))))))))). % subset_not_subset_eq
thf(fact_147_subset__psubset__trans, axiom,
    ((![A2 : set_nat, B2 : set_nat, C2 : set_nat]: ((ord_less_eq_set_nat @ A2 @ B2) => ((ord_less_set_nat @ B2 @ C2) => (ord_less_set_nat @ A2 @ C2)))))). % subset_psubset_trans
thf(fact_148_ord__eq__le__subst, axiom,
    ((![A : set_nat, F : set_nat > set_nat, B : set_nat, C : set_nat]: ((A = (F @ B)) => ((ord_less_eq_set_nat @ B @ C) => ((![X3 : set_nat, Y2 : set_nat]: ((ord_less_eq_set_nat @ X3 @ Y2) => (ord_less_eq_set_nat @ (F @ X3) @ (F @ Y2)))) => (ord_less_eq_set_nat @ A @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_149_ord__le__eq__subst, axiom,
    ((![A : set_nat, B : set_nat, F : set_nat > set_nat, C : set_nat]: ((ord_less_eq_set_nat @ A @ B) => (((F @ B) = C) => ((![X3 : set_nat, Y2 : set_nat]: ((ord_less_eq_set_nat @ X3 @ Y2) => (ord_less_eq_set_nat @ (F @ X3) @ (F @ Y2)))) => (ord_less_eq_set_nat @ (F @ A) @ C))))))). % ord_le_eq_subst
thf(fact_150_subset__iff__psubset__eq, axiom,
    ((ord_less_eq_set_nat = (^[A3 : set_nat]: (^[B3 : set_nat]: (((ord_less_set_nat @ A3 @ B3)) | ((A3 = B3)))))))). % subset_iff_psubset_eq
thf(fact_151_eq__iff, axiom,
    (((^[Y : set_nat]: (^[Z2 : set_nat]: (Y = Z2))) = (^[X2 : set_nat]: (^[Y3 : set_nat]: (((ord_less_eq_set_nat @ X2 @ Y3)) & ((ord_less_eq_set_nat @ Y3 @ X2)))))))). % eq_iff
thf(fact_152_antisym, axiom,
    ((![X : set_nat, Y4 : set_nat]: ((ord_less_eq_set_nat @ X @ Y4) => ((ord_less_eq_set_nat @ Y4 @ X) => (X = Y4)))))). % antisym
thf(fact_153_eq__refl, axiom,
    ((![X : set_nat, Y4 : set_nat]: ((X = Y4) => (ord_less_eq_set_nat @ X @ Y4))))). % eq_refl
thf(fact_154_order_Otrans, axiom,
    ((![A : set_nat, B : set_nat, C : set_nat]: ((ord_less_eq_set_nat @ A @ B) => ((ord_less_eq_set_nat @ B @ C) => (ord_less_eq_set_nat @ A @ C)))))). % order.trans
thf(fact_155_antisym__conv, axiom,
    ((![Y4 : set_nat, X : set_nat]: ((ord_less_eq_set_nat @ Y4 @ X) => ((ord_less_eq_set_nat @ X @ Y4) = (X = Y4)))))). % antisym_conv
thf(fact_156_order__class_Oorder_Oeq__iff, axiom,
    (((^[Y : set_nat]: (^[Z2 : set_nat]: (Y = Z2))) = (^[A4 : set_nat]: (^[B4 : set_nat]: (((ord_less_eq_set_nat @ A4 @ B4)) & ((ord_less_eq_set_nat @ B4 @ A4)))))))). % order_class.order.eq_iff
thf(fact_157_ord__eq__le__trans, axiom,
    ((![A : set_nat, B : set_nat, C : set_nat]: ((A = B) => ((ord_less_eq_set_nat @ B @ C) => (ord_less_eq_set_nat @ A @ C)))))). % ord_eq_le_trans
thf(fact_158_ord__le__eq__trans, axiom,
    ((![A : set_nat, B : set_nat, C : set_nat]: ((ord_less_eq_set_nat @ A @ B) => ((B = C) => (ord_less_eq_set_nat @ A @ C)))))). % ord_le_eq_trans
thf(fact_159_order__class_Oorder_Oantisym, axiom,
    ((![A : set_nat, B : set_nat]: ((ord_less_eq_set_nat @ A @ B) => ((ord_less_eq_set_nat @ B @ A) => (A = B)))))). % order_class.order.antisym
thf(fact_160_order__trans, axiom,
    ((![X : set_nat, Y4 : set_nat, Z : set_nat]: ((ord_less_eq_set_nat @ X @ Y4) => ((ord_less_eq_set_nat @ Y4 @ Z) => (ord_less_eq_set_nat @ X @ Z)))))). % order_trans
thf(fact_161_dual__order_Orefl, axiom,
    ((![A : set_nat]: (ord_less_eq_set_nat @ A @ A)))). % dual_order.refl
thf(fact_162_dual__order_Otrans, axiom,
    ((![B : set_nat, A : set_nat, C : set_nat]: ((ord_less_eq_set_nat @ B @ A) => ((ord_less_eq_set_nat @ C @ B) => (ord_less_eq_set_nat @ C @ A)))))). % dual_order.trans
thf(fact_163_dual__order_Oeq__iff, axiom,
    (((^[Y : set_nat]: (^[Z2 : set_nat]: (Y = Z2))) = (^[A4 : set_nat]: (^[B4 : set_nat]: (((ord_less_eq_set_nat @ B4 @ A4)) & ((ord_less_eq_set_nat @ A4 @ B4)))))))). % dual_order.eq_iff
thf(fact_164_dual__order_Oantisym, axiom,
    ((![B : set_nat, A : set_nat]: ((ord_less_eq_set_nat @ B @ A) => ((ord_less_eq_set_nat @ A @ B) => (A = B)))))). % dual_order.antisym
thf(fact_165_linear__alt, axiom,
    ((?[L : set_Pr367596371le_alt]: (member1334244458le_alt @ L @ arrow_1848678355le_Lin)))). % linear_alt
thf(fact_166_const__Lin__Prof, axiom,
    ((![L2 : set_Pr367596371le_alt]: ((member1334244458le_alt @ L2 @ arrow_1848678355le_Lin) => (member684274596le_alt @ (^[P2 : arrow_1429744205e_indi]: L2) @ arrow_1951607831e_Prof))))). % const_Lin_Prof
thf(fact_167_Compr__image__eq, axiom,
    ((![F : arrow_1429744205e_indi > nat, A2 : set_Ar1007576579e_indi, P : nat > $o]: ((collect_nat @ (^[X2 : nat]: (((member_nat @ X2 @ (image_555606308di_nat @ F @ A2))) & ((P @ X2))))) = (image_555606308di_nat @ F @ (collec1169676194e_indi @ (^[X2 : arrow_1429744205e_indi]: (((member1966420836e_indi @ X2 @ A2)) & ((P @ (F @ X2))))))))))). % Compr_image_eq
thf(fact_168_Compr__image__eq, axiom,
    ((![F : arrow_1429744205e_indi > arrow_1429744205e_indi, A2 : set_Ar1007576579e_indi, P : arrow_1429744205e_indi > $o]: ((collec1169676194e_indi @ (^[X2 : arrow_1429744205e_indi]: (((member1966420836e_indi @ X2 @ (image_688677079e_indi @ F @ A2))) & ((P @ X2))))) = (image_688677079e_indi @ F @ (collec1169676194e_indi @ (^[X2 : arrow_1429744205e_indi]: (((member1966420836e_indi @ X2 @ A2)) & ((P @ (F @ X2))))))))))). % Compr_image_eq
thf(fact_169_Compr__image__eq, axiom,
    ((![F : arrow_1429744205e_indi > set_Pr367596371le_alt, A2 : set_Ar1007576579e_indi, P : set_Pr367596371le_alt > $o]: ((collec1399441576le_alt @ (^[X2 : set_Pr367596371le_alt]: (((member1334244458le_alt @ X2 @ (image_444707709le_alt @ F @ A2))) & ((P @ X2))))) = (image_444707709le_alt @ F @ (collec1169676194e_indi @ (^[X2 : arrow_1429744205e_indi]: (((member1966420836e_indi @ X2 @ A2)) & ((P @ (F @ X2))))))))))). % Compr_image_eq
thf(fact_170_Compr__image__eq, axiom,
    ((![F : set_Pr367596371le_alt > arrow_1429744205e_indi, A2 : set_se2071012361le_alt, P : arrow_1429744205e_indi > $o]: ((collec1169676194e_indi @ (^[X2 : arrow_1429744205e_indi]: (((member1966420836e_indi @ X2 @ (image_220478033e_indi @ F @ A2))) & ((P @ X2))))) = (image_220478033e_indi @ F @ (collec1399441576le_alt @ (^[X2 : set_Pr367596371le_alt]: (((member1334244458le_alt @ X2 @ A2)) & ((P @ (F @ X2))))))))))). % Compr_image_eq
thf(fact_171_Compr__image__eq, axiom,
    ((![F : arrow_1429744205e_indi > arrow_1429744205e_indi > set_Pr367596371le_alt, A2 : set_Ar1007576579e_indi, P : (arrow_1429744205e_indi > set_Pr367596371le_alt) > $o]: ((collec1559089382le_alt @ (^[X2 : arrow_1429744205e_indi > set_Pr367596371le_alt]: (((member684274596le_alt @ X2 @ (image_836174111le_alt @ F @ A2))) & ((P @ X2))))) = (image_836174111le_alt @ F @ (collec1169676194e_indi @ (^[X2 : arrow_1429744205e_indi]: (((member1966420836e_indi @ X2 @ A2)) & ((P @ (F @ X2))))))))))). % Compr_image_eq
thf(fact_172_Compr__image__eq, axiom,
    ((![F : (arrow_1429744205e_indi > set_Pr367596371le_alt) > arrow_1429744205e_indi, A2 : set_Ar809243995le_alt, P : arrow_1429744205e_indi > $o]: ((collec1169676194e_indi @ (^[X2 : arrow_1429744205e_indi]: (((member1966420836e_indi @ X2 @ (image_2000124405e_indi @ F @ A2))) & ((P @ X2))))) = (image_2000124405e_indi @ F @ (collec1559089382le_alt @ (^[X2 : arrow_1429744205e_indi > set_Pr367596371le_alt]: (((member684274596le_alt @ X2 @ A2)) & ((P @ (F @ X2))))))))))). % Compr_image_eq
thf(fact_173_Compr__image__eq, axiom,
    ((![F : set_Pr367596371le_alt > set_Pr367596371le_alt, A2 : set_se2071012361le_alt, P : set_Pr367596371le_alt > $o]: ((collec1399441576le_alt @ (^[X2 : set_Pr367596371le_alt]: (((member1334244458le_alt @ X2 @ (image_71834871le_alt @ F @ A2))) & ((P @ X2))))) = (image_71834871le_alt @ F @ (collec1399441576le_alt @ (^[X2 : set_Pr367596371le_alt]: (((member1334244458le_alt @ X2 @ A2)) & ((P @ (F @ X2))))))))))). % Compr_image_eq
thf(fact_174_Compr__image__eq, axiom,
    ((![F : (arrow_1429744205e_indi > set_Pr367596371le_alt) > set_Pr367596371le_alt, A2 : set_Ar809243995le_alt, P : set_Pr367596371le_alt > $o]: ((collec1399441576le_alt @ (^[X2 : set_Pr367596371le_alt]: (((member1334244458le_alt @ X2 @ (image_712506523le_alt @ F @ A2))) & ((P @ X2))))) = (image_712506523le_alt @ F @ (collec1559089382le_alt @ (^[X2 : arrow_1429744205e_indi > set_Pr367596371le_alt]: (((member684274596le_alt @ X2 @ A2)) & ((P @ (F @ X2))))))))))). % Compr_image_eq
thf(fact_175_Compr__image__eq, axiom,
    ((![F : set_Pr367596371le_alt > arrow_1429744205e_indi > set_Pr367596371le_alt, A2 : set_se2071012361le_alt, P : (arrow_1429744205e_indi > set_Pr367596371le_alt) > $o]: ((collec1559089382le_alt @ (^[X2 : arrow_1429744205e_indi > set_Pr367596371le_alt]: (((member684274596le_alt @ X2 @ (image_1576623141le_alt @ F @ A2))) & ((P @ X2))))) = (image_1576623141le_alt @ F @ (collec1399441576le_alt @ (^[X2 : set_Pr367596371le_alt]: (((member1334244458le_alt @ X2 @ A2)) & ((P @ (F @ X2))))))))))). % Compr_image_eq
thf(fact_176_Compr__image__eq, axiom,
    ((![F : (arrow_1429744205e_indi > set_Pr367596371le_alt) > arrow_1429744205e_indi > set_Pr367596371le_alt, A2 : set_Ar809243995le_alt, P : (arrow_1429744205e_indi > set_Pr367596371le_alt) > $o]: ((collec1559089382le_alt @ (^[X2 : arrow_1429744205e_indi > set_Pr367596371le_alt]: (((member684274596le_alt @ X2 @ (image_829063553le_alt @ F @ A2))) & ((P @ X2))))) = (image_829063553le_alt @ F @ (collec1559089382le_alt @ (^[X2 : arrow_1429744205e_indi > set_Pr367596371le_alt]: (((member684274596le_alt @ X2 @ A2)) & ((P @ (F @ X2))))))))))). % Compr_image_eq
thf(fact_177_image__image, axiom,
    ((![F : nat > nat, G : arrow_1429744205e_indi > nat, A2 : set_Ar1007576579e_indi]: ((image_nat_nat @ F @ (image_555606308di_nat @ G @ A2)) = (image_555606308di_nat @ (^[X2 : arrow_1429744205e_indi]: (F @ (G @ X2))) @ A2))))). % image_image
thf(fact_178_image__image, axiom,
    ((![F : arrow_1429744205e_indi > nat, G : arrow_1429744205e_indi > arrow_1429744205e_indi, A2 : set_Ar1007576579e_indi]: ((image_555606308di_nat @ F @ (image_688677079e_indi @ G @ A2)) = (image_555606308di_nat @ (^[X2 : arrow_1429744205e_indi]: (F @ (G @ X2))) @ A2))))). % image_image
thf(fact_179_imageE, axiom,
    ((![B : nat, F : arrow_1429744205e_indi > nat, A2 : set_Ar1007576579e_indi]: ((member_nat @ B @ (image_555606308di_nat @ F @ A2)) => (~ ((![X3 : arrow_1429744205e_indi]: ((B = (F @ X3)) => (~ ((member1966420836e_indi @ X3 @ A2))))))))))). % imageE
thf(fact_180_imageE, axiom,
    ((![B : arrow_1429744205e_indi, F : arrow_1429744205e_indi > arrow_1429744205e_indi, A2 : set_Ar1007576579e_indi]: ((member1966420836e_indi @ B @ (image_688677079e_indi @ F @ A2)) => (~ ((![X3 : arrow_1429744205e_indi]: ((B = (F @ X3)) => (~ ((member1966420836e_indi @ X3 @ A2))))))))))). % imageE
thf(fact_181_imageE, axiom,
    ((![B : set_Pr367596371le_alt, F : arrow_1429744205e_indi > set_Pr367596371le_alt, A2 : set_Ar1007576579e_indi]: ((member1334244458le_alt @ B @ (image_444707709le_alt @ F @ A2)) => (~ ((![X3 : arrow_1429744205e_indi]: ((B = (F @ X3)) => (~ ((member1966420836e_indi @ X3 @ A2))))))))))). % imageE
thf(fact_182_imageE, axiom,
    ((![B : arrow_1429744205e_indi, F : set_Pr367596371le_alt > arrow_1429744205e_indi, A2 : set_se2071012361le_alt]: ((member1966420836e_indi @ B @ (image_220478033e_indi @ F @ A2)) => (~ ((![X3 : set_Pr367596371le_alt]: ((B = (F @ X3)) => (~ ((member1334244458le_alt @ X3 @ A2))))))))))). % imageE
thf(fact_183_imageE, axiom,
    ((![B : arrow_1429744205e_indi > set_Pr367596371le_alt, F : arrow_1429744205e_indi > arrow_1429744205e_indi > set_Pr367596371le_alt, A2 : set_Ar1007576579e_indi]: ((member684274596le_alt @ B @ (image_836174111le_alt @ F @ A2)) => (~ ((![X3 : arrow_1429744205e_indi]: ((B = (F @ X3)) => (~ ((member1966420836e_indi @ X3 @ A2))))))))))). % imageE
thf(fact_184_imageE, axiom,
    ((![B : arrow_1429744205e_indi, F : (arrow_1429744205e_indi > set_Pr367596371le_alt) > arrow_1429744205e_indi, A2 : set_Ar809243995le_alt]: ((member1966420836e_indi @ B @ (image_2000124405e_indi @ F @ A2)) => (~ ((![X3 : arrow_1429744205e_indi > set_Pr367596371le_alt]: ((B = (F @ X3)) => (~ ((member684274596le_alt @ X3 @ A2))))))))))). % imageE
thf(fact_185_imageE, axiom,
    ((![B : set_Pr367596371le_alt, F : set_Pr367596371le_alt > set_Pr367596371le_alt, A2 : set_se2071012361le_alt]: ((member1334244458le_alt @ B @ (image_71834871le_alt @ F @ A2)) => (~ ((![X3 : set_Pr367596371le_alt]: ((B = (F @ X3)) => (~ ((member1334244458le_alt @ X3 @ A2))))))))))). % imageE
thf(fact_186_imageE, axiom,
    ((![B : set_Pr367596371le_alt, F : (arrow_1429744205e_indi > set_Pr367596371le_alt) > set_Pr367596371le_alt, A2 : set_Ar809243995le_alt]: ((member1334244458le_alt @ B @ (image_712506523le_alt @ F @ A2)) => (~ ((![X3 : arrow_1429744205e_indi > set_Pr367596371le_alt]: ((B = (F @ X3)) => (~ ((member684274596le_alt @ X3 @ A2))))))))))). % imageE
thf(fact_187_imageE, axiom,
    ((![B : arrow_1429744205e_indi > set_Pr367596371le_alt, F : set_Pr367596371le_alt > arrow_1429744205e_indi > set_Pr367596371le_alt, A2 : set_se2071012361le_alt]: ((member684274596le_alt @ B @ (image_1576623141le_alt @ F @ A2)) => (~ ((![X3 : set_Pr367596371le_alt]: ((B = (F @ X3)) => (~ ((member1334244458le_alt @ X3 @ A2))))))))))). % imageE
thf(fact_188_imageE, axiom,
    ((![B : arrow_1429744205e_indi > set_Pr367596371le_alt, F : (arrow_1429744205e_indi > set_Pr367596371le_alt) > arrow_1429744205e_indi > set_Pr367596371le_alt, A2 : set_Ar809243995le_alt]: ((member684274596le_alt @ B @ (image_829063553le_alt @ F @ A2)) => (~ ((![X3 : arrow_1429744205e_indi > set_Pr367596371le_alt]: ((B = (F @ X3)) => (~ ((member684274596le_alt @ X3 @ A2))))))))))). % imageE
thf(fact_189_top__set__def, axiom,
    ((top_to1799531699e_indi = (collec1169676194e_indi @ top_to1473733010indi_o)))). % top_set_def
thf(fact_190_top__set__def, axiom,
    ((top_to803745505le_alt = (collec1382217680le_alt @ top_to1892800228_alt_o)))). % top_set_def
thf(fact_191_top__set__def, axiom,
    ((top_to685525675le_alt = (collec1559089382le_alt @ top_to1344581618_alt_o)))). % top_set_def
thf(fact_192_top__set__def, axiom,
    ((top_to469035705le_alt = (collec1399441576le_alt @ top_to1900088076_alt_o)))). % top_set_def
thf(fact_193_Collect__subset, axiom,
    ((![A2 : set_se2071012361le_alt, P : set_Pr367596371le_alt > $o]: (ord_le1836905577le_alt @ (collec1399441576le_alt @ (^[X2 : set_Pr367596371le_alt]: (((member1334244458le_alt @ X2 @ A2)) & ((P @ X2))))) @ A2)))). % Collect_subset
thf(fact_194_Collect__subset, axiom,
    ((![A2 : set_Ar809243995le_alt, P : (arrow_1429744205e_indi > set_Pr367596371le_alt) > $o]: (ord_le22412539le_alt @ (collec1559089382le_alt @ (^[X2 : arrow_1429744205e_indi > set_Pr367596371le_alt]: (((member684274596le_alt @ X2 @ A2)) & ((P @ X2))))) @ A2)))). % Collect_subset
thf(fact_195_Collect__subset, axiom,
    ((![A2 : set_Ar182050865le_alt, P : ((arrow_1429744205e_indi > set_Pr367596371le_alt) > set_Pr367596371le_alt) > $o]: (ord_le1453801617le_alt @ (collec1382217680le_alt @ (^[X2 : (arrow_1429744205e_indi > set_Pr367596371le_alt) > set_Pr367596371le_alt]: (((member183760530le_alt @ X2 @ A2)) & ((P @ X2))))) @ A2)))). % Collect_subset
thf(fact_196_Collect__subset, axiom,
    ((![A2 : set_Ar1007576579e_indi, P : arrow_1429744205e_indi > $o]: (ord_le2059613795e_indi @ (collec1169676194e_indi @ (^[X2 : arrow_1429744205e_indi]: (((member1966420836e_indi @ X2 @ A2)) & ((P @ X2))))) @ A2)))). % Collect_subset
thf(fact_197_Collect__subset, axiom,
    ((![A2 : set_nat, P : nat > $o]: (ord_less_eq_set_nat @ (collect_nat @ (^[X2 : nat]: (((member_nat @ X2 @ A2)) & ((P @ X2))))) @ A2)))). % Collect_subset
thf(fact_198_order_Onot__eq__order__implies__strict, axiom,
    ((![A : nat, B : nat]: ((~ ((A = B))) => ((ord_less_eq_nat @ A @ B) => (ord_less_nat @ A @ B)))))). % order.not_eq_order_implies_strict
thf(fact_199_order_Onot__eq__order__implies__strict, axiom,
    ((![A : set_nat, B : set_nat]: ((~ ((A = B))) => ((ord_less_eq_set_nat @ A @ B) => (ord_less_set_nat @ A @ B)))))). % order.not_eq_order_implies_strict
thf(fact_200_dual__order_Ostrict__implies__order, axiom,
    ((![B : nat, A : nat]: ((ord_less_nat @ B @ A) => (ord_less_eq_nat @ B @ A))))). % dual_order.strict_implies_order
thf(fact_201_dual__order_Ostrict__implies__order, axiom,
    ((![B : set_nat, A : set_nat]: ((ord_less_set_nat @ B @ A) => (ord_less_eq_set_nat @ B @ A))))). % dual_order.strict_implies_order
thf(fact_202_dual__order_Ostrict__iff__order, axiom,
    ((ord_less_nat = (^[B4 : nat]: (^[A4 : nat]: (((ord_less_eq_nat @ B4 @ A4)) & ((~ ((A4 = B4)))))))))). % dual_order.strict_iff_order
thf(fact_203_dual__order_Ostrict__iff__order, axiom,
    ((ord_less_set_nat = (^[B4 : set_nat]: (^[A4 : set_nat]: (((ord_less_eq_set_nat @ B4 @ A4)) & ((~ ((A4 = B4)))))))))). % dual_order.strict_iff_order
thf(fact_204_dual__order_Oorder__iff__strict, axiom,
    ((ord_less_eq_nat = (^[B4 : nat]: (^[A4 : nat]: (((ord_less_nat @ B4 @ A4)) | ((A4 = B4)))))))). % dual_order.order_iff_strict
thf(fact_205_dual__order_Oorder__iff__strict, axiom,
    ((ord_less_eq_set_nat = (^[B4 : set_nat]: (^[A4 : set_nat]: (((ord_less_set_nat @ B4 @ A4)) | ((A4 = B4)))))))). % dual_order.order_iff_strict
thf(fact_206_order_Ostrict__implies__order, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (ord_less_eq_nat @ A @ B))))). % order.strict_implies_order
thf(fact_207_order_Ostrict__implies__order, axiom,
    ((![A : set_nat, B : set_nat]: ((ord_less_set_nat @ A @ B) => (ord_less_eq_set_nat @ A @ B))))). % order.strict_implies_order
thf(fact_208_dual__order_Ostrict__trans2, axiom,
    ((![B : nat, A : nat, C : nat]: ((ord_less_nat @ B @ A) => ((ord_less_eq_nat @ C @ B) => (ord_less_nat @ C @ A)))))). % dual_order.strict_trans2
thf(fact_209_dual__order_Ostrict__trans2, axiom,
    ((![B : set_nat, A : set_nat, C : set_nat]: ((ord_less_set_nat @ B @ A) => ((ord_less_eq_set_nat @ C @ B) => (ord_less_set_nat @ C @ A)))))). % dual_order.strict_trans2
thf(fact_210_dual__order_Ostrict__trans1, axiom,
    ((![B : nat, A : nat, C : nat]: ((ord_less_eq_nat @ B @ A) => ((ord_less_nat @ C @ B) => (ord_less_nat @ C @ A)))))). % dual_order.strict_trans1
thf(fact_211_dual__order_Ostrict__trans1, axiom,
    ((![B : set_nat, A : set_nat, C : set_nat]: ((ord_less_eq_set_nat @ B @ A) => ((ord_less_set_nat @ C @ B) => (ord_less_set_nat @ C @ A)))))). % dual_order.strict_trans1
thf(fact_212_order_Ostrict__iff__order, axiom,
    ((ord_less_nat = (^[A4 : nat]: (^[B4 : nat]: (((ord_less_eq_nat @ A4 @ B4)) & ((~ ((A4 = B4)))))))))). % order.strict_iff_order
thf(fact_213_order_Ostrict__iff__order, axiom,
    ((ord_less_set_nat = (^[A4 : set_nat]: (^[B4 : set_nat]: (((ord_less_eq_set_nat @ A4 @ B4)) & ((~ ((A4 = B4)))))))))). % order.strict_iff_order
thf(fact_214_order_Oorder__iff__strict, axiom,
    ((ord_less_eq_nat = (^[A4 : nat]: (^[B4 : nat]: (((ord_less_nat @ A4 @ B4)) | ((A4 = B4)))))))). % order.order_iff_strict
thf(fact_215_order_Oorder__iff__strict, axiom,
    ((ord_less_eq_set_nat = (^[A4 : set_nat]: (^[B4 : set_nat]: (((ord_less_set_nat @ A4 @ B4)) | ((A4 = B4)))))))). % order.order_iff_strict
thf(fact_216_order_Ostrict__trans2, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_nat @ A @ B) => ((ord_less_eq_nat @ B @ C) => (ord_less_nat @ A @ C)))))). % order.strict_trans2
thf(fact_217_order_Ostrict__trans2, axiom,
    ((![A : set_nat, B : set_nat, C : set_nat]: ((ord_less_set_nat @ A @ B) => ((ord_less_eq_set_nat @ B @ C) => (ord_less_set_nat @ A @ C)))))). % order.strict_trans2
thf(fact_218_order_Ostrict__trans1, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_nat @ B @ C) => (ord_less_nat @ A @ C)))))). % order.strict_trans1
thf(fact_219_order_Ostrict__trans1, axiom,
    ((![A : set_nat, B : set_nat, C : set_nat]: ((ord_less_eq_set_nat @ A @ B) => ((ord_less_set_nat @ B @ C) => (ord_less_set_nat @ A @ C)))))). % order.strict_trans1
thf(fact_220_not__le__imp__less, axiom,
    ((![Y4 : nat, X : nat]: ((~ ((ord_less_eq_nat @ Y4 @ X))) => (ord_less_nat @ X @ Y4))))). % not_le_imp_less
thf(fact_221_less__le__not__le, axiom,
    ((ord_less_nat = (^[X2 : nat]: (^[Y3 : nat]: (((ord_less_eq_nat @ X2 @ Y3)) & ((~ ((ord_less_eq_nat @ Y3 @ X2)))))))))). % less_le_not_le
thf(fact_222_less__le__not__le, axiom,
    ((ord_less_set_nat = (^[X2 : set_nat]: (^[Y3 : set_nat]: (((ord_less_eq_set_nat @ X2 @ Y3)) & ((~ ((ord_less_eq_set_nat @ Y3 @ X2)))))))))). % less_le_not_le
thf(fact_223_le__imp__less__or__eq, axiom,
    ((![X : nat, Y4 : nat]: ((ord_less_eq_nat @ X @ Y4) => ((ord_less_nat @ X @ Y4) | (X = Y4)))))). % le_imp_less_or_eq
thf(fact_224_le__imp__less__or__eq, axiom,
    ((![X : set_nat, Y4 : set_nat]: ((ord_less_eq_set_nat @ X @ Y4) => ((ord_less_set_nat @ X @ Y4) | (X = Y4)))))). % le_imp_less_or_eq
thf(fact_225_le__less__linear, axiom,
    ((![X : nat, Y4 : nat]: ((ord_less_eq_nat @ X @ Y4) | (ord_less_nat @ Y4 @ X))))). % le_less_linear
thf(fact_226_less__le__trans, axiom,
    ((![X : nat, Y4 : nat, Z : nat]: ((ord_less_nat @ X @ Y4) => ((ord_less_eq_nat @ Y4 @ Z) => (ord_less_nat @ X @ Z)))))). % less_le_trans
thf(fact_227_less__le__trans, axiom,
    ((![X : set_nat, Y4 : set_nat, Z : set_nat]: ((ord_less_set_nat @ X @ Y4) => ((ord_less_eq_set_nat @ Y4 @ Z) => (ord_less_set_nat @ X @ Z)))))). % less_le_trans
thf(fact_228_le__less__trans, axiom,
    ((![X : nat, Y4 : nat, Z : nat]: ((ord_less_eq_nat @ X @ Y4) => ((ord_less_nat @ Y4 @ Z) => (ord_less_nat @ X @ Z)))))). % le_less_trans
thf(fact_229_le__less__trans, axiom,
    ((![X : set_nat, Y4 : set_nat, Z : set_nat]: ((ord_less_eq_set_nat @ X @ Y4) => ((ord_less_set_nat @ Y4 @ Z) => (ord_less_set_nat @ X @ Z)))))). % le_less_trans
thf(fact_230_less__imp__le, axiom,
    ((![X : nat, Y4 : nat]: ((ord_less_nat @ X @ Y4) => (ord_less_eq_nat @ X @ Y4))))). % less_imp_le
thf(fact_231_less__imp__le, axiom,
    ((![X : set_nat, Y4 : set_nat]: ((ord_less_set_nat @ X @ Y4) => (ord_less_eq_set_nat @ X @ Y4))))). % less_imp_le
thf(fact_232_antisym__conv2, axiom,
    ((![X : nat, Y4 : nat]: ((ord_less_eq_nat @ X @ Y4) => ((~ ((ord_less_nat @ X @ Y4))) = (X = Y4)))))). % antisym_conv2
thf(fact_233_antisym__conv2, axiom,
    ((![X : set_nat, Y4 : set_nat]: ((ord_less_eq_set_nat @ X @ Y4) => ((~ ((ord_less_set_nat @ X @ Y4))) = (X = Y4)))))). % antisym_conv2
thf(fact_234_antisym__conv1, axiom,
    ((![X : nat, Y4 : nat]: ((~ ((ord_less_nat @ X @ Y4))) => ((ord_less_eq_nat @ X @ Y4) = (X = Y4)))))). % antisym_conv1
thf(fact_235_antisym__conv1, axiom,
    ((![X : set_nat, Y4 : set_nat]: ((~ ((ord_less_set_nat @ X @ Y4))) => ((ord_less_eq_set_nat @ X @ Y4) = (X = Y4)))))). % antisym_conv1
thf(fact_236_le__neq__trans, axiom,
    ((![A : nat, B : nat]: ((ord_less_eq_nat @ A @ B) => ((~ ((A = B))) => (ord_less_nat @ A @ B)))))). % le_neq_trans
thf(fact_237_le__neq__trans, axiom,
    ((![A : set_nat, B : set_nat]: ((ord_less_eq_set_nat @ A @ B) => ((~ ((A = B))) => (ord_less_set_nat @ A @ B)))))). % le_neq_trans
thf(fact_238_not__less, axiom,
    ((![X : nat, Y4 : nat]: ((~ ((ord_less_nat @ X @ Y4))) = (ord_less_eq_nat @ Y4 @ X))))). % not_less
thf(fact_239_not__le, axiom,
    ((![X : nat, Y4 : nat]: ((~ ((ord_less_eq_nat @ X @ Y4))) = (ord_less_nat @ Y4 @ X))))). % not_le
thf(fact_240_bot__nat__0_Oextremum__strict, axiom,
    ((![A : nat]: (~ ((ord_less_nat @ A @ zero_zero_nat)))))). % bot_nat_0.extremum_strict
thf(fact_241_infinite__descent0, axiom,
    ((![P : nat > $o, N : nat]: ((P @ zero_zero_nat) => ((![N3 : nat]: ((ord_less_nat @ zero_zero_nat @ N3) => ((~ ((P @ N3))) => (?[M3 : nat]: ((ord_less_nat @ M3 @ N3) & (~ ((P @ M3)))))))) => (P @ N)))))). % infinite_descent0
thf(fact_242_gr__implies__not0, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not0
thf(fact_243_less__zeroE, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_zeroE
thf(fact_244_not__less0, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less0
thf(fact_245_not__gr0, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr0
thf(fact_246_gr0I, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr0I

% Helper facts (3)
thf(help_If_3_1_If_001t__Set__Oset_It__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_J_T, axiom,
    ((![P : $o]: ((P = $true) | (P = $false))))).
thf(help_If_2_1_If_001t__Set__Oset_It__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_J_T, axiom,
    ((![X : set_Pr367596371le_alt, Y4 : set_Pr367596371le_alt]: ((if_set550155277le_alt @ $false @ X @ Y4) = Y4)))).
thf(help_If_1_1_If_001t__Set__Oset_It__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_J_T, axiom,
    ((![X : set_Pr367596371le_alt, Y4 : set_Pr367596371le_alt]: ((if_set550155277le_alt @ $true @ X @ Y4) = X)))).

% Conjectures (1)
thf(conj_0, conjecture,
    (((^[I : arrow_1429744205e_indi]: (if_set550155277le_alt @ (ord_less_nat @ (h @ I) @ (finite927127589e_indi @ top_to1799531699e_indi)) @ lab @ lba)) = (^[P2 : arrow_1429744205e_indi]: lab)))).
