% 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_293__5189408_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:17:21.107

% Could-be-implicit typings (13)
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__Product____Type__Oprod_I_062_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_M_062_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_M_Eo_J_J_Mt__List__Olist_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_J, type,
    produc1094320767le_alt : $tType).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_Mt__List__Olist_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_J, type,
    produc518943027le_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__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J, type,
    produc16571293le_alt : $tType).
thf(ty_n_t__List__Olist_It__List__Olist_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_J, type,
    list_l1515872562le_alt : $tType).
thf(ty_n_t__List__Olist_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J, type,
    list_A2130511660le_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__Arrow____Order____Mirabelle____riepwfubkl__Oalt, type,
    arrow_1857593510le_alt : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).

% Explicit typings (55)
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_Oabove, type,
    arrow_1726226719_above : set_Pr367596371le_alt > arrow_1857593510le_alt > arrow_1857593510le_alt > set_Pr367596371le_alt).
thf(sy_c_Arrow__Order__Mirabelle__riepwfubkl_Obelow, type,
    arrow_887341619_below : set_Pr367596371le_alt > arrow_1857593510le_alt > arrow_1857593510le_alt > set_Pr367596371le_alt).
thf(sy_c_Arrow__Order__Mirabelle__riepwfubkl_Omkbot, type,
    arrow_843587755_mkbot : set_Pr367596371le_alt > arrow_1857593510le_alt > set_Pr367596371le_alt).
thf(sy_c_Arrow__Order__Mirabelle__riepwfubkl_Omktop, type,
    arrow_992294841_mktop : set_Pr367596371le_alt > arrow_1857593510le_alt > set_Pr367596371le_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_Hilbert__Choice_Oinv__into_001t__Arrow____Order____Mirabelle____riepwfubkl__Oindi_001t__Arrow____Order____Mirabelle____riepwfubkl__Oindi, type,
    hilber700257104e_indi : set_Ar1007576579e_indi > (arrow_1429744205e_indi > arrow_1429744205e_indi) > arrow_1429744205e_indi > arrow_1429744205e_indi).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__Arrow____Order____Mirabelle____riepwfubkl__Oindi_001t__Nat__Onat, type,
    hilber1586975467di_nat : set_Ar1007576579e_indi > (arrow_1429744205e_indi > nat) > nat > arrow_1429744205e_indi).
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_List_Odistinct_001t__Arrow____Order____Mirabelle____riepwfubkl__Oalt, type,
    distin2037765919le_alt : list_A2130511660le_alt > $o).
thf(sy_c_List_Olist_OCons_001t__Arrow____Order____Mirabelle____riepwfubkl__Oalt, type,
    cons_A1864255580le_alt : arrow_1857593510le_alt > list_A2130511660le_alt > list_A2130511660le_alt).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J, type,
    cons_l157309922le_alt : list_A2130511660le_alt > list_l1515872562le_alt > list_l1515872562le_alt).
thf(sy_c_List_Olist_ONil_001t__Arrow____Order____Mirabelle____riepwfubkl__Oalt, type,
    nil_Ar277507244le_alt : list_A2130511660le_alt).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J, type,
    nil_li2110134322le_alt : list_l1515872562le_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_It__Arrow____Order____Mirabelle____riepwfubkl__Oindi_J, type,
    ord_le1187139159e_indi : set_Ar1007576579e_indi > set_Ar1007576579e_indi > $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__Oalt_M_062_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_M_Eo_J_J, type,
    top_to701951606_alt_o : arrow_1857593510le_alt > arrow_1857593510le_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__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_M_Eo_J, type,
    top_to1074038850_alt_o : produc16571293le_alt > $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__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_J, type,
    top_to224369155le_alt : set_Pr367596371le_alt).
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_Product__Type_OPair_001_062_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_M_062_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_M_Eo_J_J_001t__List__Olist_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J, type,
    produc2047839287le_alt : (arrow_1857593510le_alt > arrow_1857593510le_alt > $o) > list_A2130511660le_alt > produc1094320767le_alt).
thf(sy_c_Product__Type_OPair_001t__Arrow____Order____Mirabelle____riepwfubkl__Oalt_001t__Arrow____Order____Mirabelle____riepwfubkl__Oalt, type,
    produc1494124311le_alt : arrow_1857593510le_alt > arrow_1857593510le_alt > produc16571293le_alt).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_001t__List__Olist_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J, type,
    produc961395117le_alt : list_A2130511660le_alt > list_A2130511660le_alt > produc518943027le_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__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J, type,
    collec531981554le_alt : (produc16571293le_alt > $o) > set_Pr367596371le_alt).
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_Wellfounded_Omeasure_001t__Arrow____Order____Mirabelle____riepwfubkl__Oalt, type,
    measur1447705966le_alt : (arrow_1857593510le_alt > nat) > set_Pr367596371le_alt).
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__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J, type,
    member2048039092le_alt : produc16571293le_alt > set_Pr367596371le_alt > $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_P____, type,
    p : arrow_1429744205e_indi > set_Pr367596371le_alt).
thf(sy_v_c____, type,
    c : arrow_1857593510le_alt).
thf(sy_v_d____, type,
    d : arrow_1857593510le_alt).
thf(sy_v_e____, type,
    e : arrow_1857593510le_alt).
thf(sy_v_h____, type,
    h : arrow_1429744205e_indi > nat).
thf(sy_v_n____, type,
    n : nat).

% Relevant facts (149)
thf(fact_0__092_060open_062P_A_092_060in_062_AProf_092_060close_062, axiom,
    ((member684274596le_alt @ p @ arrow_1951607831e_Prof))). % \<open>P \<in> Prof\<close>
thf(fact_1__092_060open_062_092_060forall_062P_092_060in_062Prof_O_A_092_060forall_062P_H_092_060in_062Prof_O_A_092_060forall_062a_Ab_O_A_I_092_060forall_062i_O_A_Ia_A_060_092_060_094bsub_062P_Ai_092_060_094esub_062_Ab_J_A_061_A_Ia_A_060_092_060_094bsub_062P_H_Ai_092_060_094esub_062_Ab_J_J_A_092_060longrightarrow_062_A_Ia_A_060_092_060_094bsub_062F_AP_092_060_094esub_062_Ab_J_A_061_A_Ia_A_060_092_060_094bsub_062F_AP_H_092_060_094esub_062_Ab_J_092_060close_062, axiom,
    ((![X : arrow_1429744205e_indi > set_Pr367596371le_alt]: ((member684274596le_alt @ X @ arrow_1951607831e_Prof) => (![Xa : arrow_1429744205e_indi > set_Pr367596371le_alt]: ((member684274596le_alt @ Xa @ arrow_1951607831e_Prof) => (![A : arrow_1857593510le_alt, B : arrow_1857593510le_alt]: ((![I : arrow_1429744205e_indi]: ((member2048039092le_alt @ (produc1494124311le_alt @ A @ B) @ (X @ I)) = (member2048039092le_alt @ (produc1494124311le_alt @ A @ B) @ (Xa @ I)))) => ((member2048039092le_alt @ (produc1494124311le_alt @ A @ B) @ (f @ X)) = (member2048039092le_alt @ (produc1494124311le_alt @ A @ B) @ (f @ Xa))))))))))). % \<open>\<forall>P\<in>Prof. \<forall>P'\<in>Prof. \<forall>a b. (\<forall>i. (a <\<^bsub>P i\<^esub> b) = (a <\<^bsub>P' i\<^esub> b)) \<longrightarrow> (a <\<^bsub>F P\<^esub> b) = (a <\<^bsub>F P'\<^esub> b)\<close>
thf(fact_2_assms_I3_J, axiom,
    ((arrow_1821794627le_IIA @ f))). % assms(3)
thf(fact_3_u, axiom,
    ((arrow_52334694nimity @ f))). % u
thf(fact_4__092_060open_062c_A_092_060noteq_062_Ad_092_060close_062, axiom,
    ((~ ((c = d))))). % \<open>c \<noteq> d\<close>
thf(fact_5__C1_C, axiom,
    ((![A2 : arrow_1857593510le_alt, B2 : arrow_1857593510le_alt, A3 : arrow_1857593510le_alt, B3 : arrow_1857593510le_alt, P : arrow_1429744205e_indi > set_Pr367596371le_alt, P2 : arrow_1429744205e_indi > set_Pr367596371le_alt]: ((~ ((A2 = B2))) => ((~ ((A3 = B3))) => ((~ ((A2 = B3))) => ((~ ((B2 = A3))) => ((member684274596le_alt @ P @ arrow_1951607831e_Prof) => ((member684274596le_alt @ P2 @ arrow_1951607831e_Prof) => ((![I : arrow_1429744205e_indi]: ((member2048039092le_alt @ (produc1494124311le_alt @ A2 @ B2) @ (P @ I)) = (member2048039092le_alt @ (produc1494124311le_alt @ A3 @ B3) @ (P2 @ I)))) => ((member2048039092le_alt @ (produc1494124311le_alt @ A2 @ B2) @ (f @ P)) => (member2048039092le_alt @ (produc1494124311le_alt @ A3 @ B3) @ (f @ P2))))))))))))). % "1"
thf(fact_6__C2_C, axiom,
    ((![A2 : arrow_1857593510le_alt, B2 : arrow_1857593510le_alt, A3 : arrow_1857593510le_alt, B3 : arrow_1857593510le_alt, P : arrow_1429744205e_indi > set_Pr367596371le_alt, P2 : arrow_1429744205e_indi > set_Pr367596371le_alt]: ((~ ((A2 = B2))) => ((~ ((A3 = B3))) => ((~ ((A2 = B3))) => ((~ ((B2 = A3))) => ((member684274596le_alt @ P @ arrow_1951607831e_Prof) => ((member684274596le_alt @ P2 @ arrow_1951607831e_Prof) => ((![I : arrow_1429744205e_indi]: ((member2048039092le_alt @ (produc1494124311le_alt @ A2 @ B2) @ (P @ I)) = (member2048039092le_alt @ (produc1494124311le_alt @ A3 @ B3) @ (P2 @ I)))) => ((member2048039092le_alt @ (produc1494124311le_alt @ A2 @ B2) @ (f @ P)) = (member2048039092le_alt @ (produc1494124311le_alt @ A3 @ B3) @ (f @ P2))))))))))))). % "2"
thf(fact_7__C3_C, axiom,
    ((![A2 : arrow_1857593510le_alt, B2 : arrow_1857593510le_alt, P : arrow_1429744205e_indi > set_Pr367596371le_alt, P2 : arrow_1429744205e_indi > set_Pr367596371le_alt]: ((~ ((A2 = B2))) => ((member684274596le_alt @ P @ arrow_1951607831e_Prof) => ((member684274596le_alt @ P2 @ arrow_1951607831e_Prof) => ((![I : arrow_1429744205e_indi]: ((member2048039092le_alt @ (produc1494124311le_alt @ A2 @ B2) @ (P @ I)) = (member2048039092le_alt @ (produc1494124311le_alt @ B2 @ A2) @ (P2 @ I)))) => ((member2048039092le_alt @ (produc1494124311le_alt @ A2 @ B2) @ (f @ P)) = (member2048039092le_alt @ (produc1494124311le_alt @ B2 @ A2) @ (f @ P2)))))))))). % "3"
thf(fact_8__C4_C, axiom,
    ((![A2 : arrow_1857593510le_alt, B2 : arrow_1857593510le_alt, C : arrow_1857593510le_alt, P : arrow_1429744205e_indi > set_Pr367596371le_alt, P2 : arrow_1429744205e_indi > set_Pr367596371le_alt]: ((~ ((A2 = B2))) => ((~ ((B2 = C))) => ((~ ((A2 = C))) => ((member684274596le_alt @ P @ arrow_1951607831e_Prof) => ((member684274596le_alt @ P2 @ arrow_1951607831e_Prof) => ((![I : arrow_1429744205e_indi]: ((member2048039092le_alt @ (produc1494124311le_alt @ A2 @ B2) @ (P @ I)) = (member2048039092le_alt @ (produc1494124311le_alt @ B2 @ C) @ (P2 @ I)))) => ((member2048039092le_alt @ (produc1494124311le_alt @ A2 @ B2) @ (f @ P)) = (member2048039092le_alt @ (produc1494124311le_alt @ B2 @ C) @ (f @ P2)))))))))))). % "4"
thf(fact_9_pairwise__neutrality, axiom,
    ((![A2 : arrow_1857593510le_alt, B2 : arrow_1857593510le_alt, A3 : arrow_1857593510le_alt, B3 : arrow_1857593510le_alt, P : arrow_1429744205e_indi > set_Pr367596371le_alt, P2 : arrow_1429744205e_indi > set_Pr367596371le_alt]: ((~ ((A2 = B2))) => ((~ ((A3 = B3))) => ((member684274596le_alt @ P @ arrow_1951607831e_Prof) => ((member684274596le_alt @ P2 @ arrow_1951607831e_Prof) => ((![I : arrow_1429744205e_indi]: ((member2048039092le_alt @ (produc1494124311le_alt @ A2 @ B2) @ (P @ I)) = (member2048039092le_alt @ (produc1494124311le_alt @ A3 @ B3) @ (P2 @ I)))) => ((member2048039092le_alt @ (produc1494124311le_alt @ A2 @ B2) @ (f @ P)) = (member2048039092le_alt @ (produc1494124311le_alt @ A3 @ B3) @ (f @ P2))))))))))). % pairwise_neutrality
thf(fact_10__092_060open_062_I_092_060lambda_062i_O_Aif_Ah_Ai_A_060_An_Athen_Amktop_A_IP_Ai_J_Ae_Aelse_Aif_Ah_Ai_A_061_An_Athen_AArrow__Order__Mirabelle__riepwfubkl_Oabove_A_IP_Ai_J_Ac_Ae_Aelse_Amkbot_A_IP_Ai_J_Ae_J_A_092_060in_062_AProf_092_060close_062, axiom,
    ((member684274596le_alt @ (^[I2 : arrow_1429744205e_indi]: (if_set550155277le_alt @ (ord_less_nat @ (h @ I2) @ n) @ (arrow_992294841_mktop @ (p @ I2) @ e) @ (if_set550155277le_alt @ ((h @ I2) = n) @ (arrow_1726226719_above @ (p @ I2) @ c @ e) @ (arrow_843587755_mkbot @ (p @ I2) @ e)))) @ arrow_1951607831e_Prof))). % \<open>(\<lambda>i. if h i < n then mktop (P i) e else if h i = n then Arrow_Order_Mirabelle_riepwfubkl.above (P i) c e else mkbot (P i) e) \<in> Prof\<close>
thf(fact_11__092_060open_062_092_060forall_062i_O_A_Ic_A_060_092_060_094bsub_062P_Ai_092_060_094esub_062_Ad_J_A_061_A_Ic_A_060_092_060_094bsub_062_Iif_Ah_Ai_A_060_An_Athen_Amktop_A_IP_Ai_J_Ae_Aelse_Aif_Ah_Ai_A_061_An_Athen_AArrow__Order__Mirabelle__riepwfubkl_Oabove_A_IP_Ai_J_Ac_Ae_Aelse_Amkbot_A_IP_Ai_J_Ae_J_092_060_094esub_062_Ad_J_092_060close_062, axiom,
    ((![I3 : arrow_1429744205e_indi]: ((member2048039092le_alt @ (produc1494124311le_alt @ c @ d) @ (p @ I3)) = (((((ord_less_nat @ (h @ I3) @ n)) => ((member2048039092le_alt @ (produc1494124311le_alt @ c @ d) @ (arrow_992294841_mktop @ (p @ I3) @ e))))) & ((((~ ((ord_less_nat @ (h @ I3) @ n)))) => (((((((h @ I3) = n)) => ((member2048039092le_alt @ (produc1494124311le_alt @ c @ d) @ (arrow_1726226719_above @ (p @ I3) @ c @ e))))) & ((((~ (((h @ I3) = n)))) => ((member2048039092le_alt @ (produc1494124311le_alt @ c @ d) @ (arrow_843587755_mkbot @ (p @ I3) @ e)))))))))))))). % \<open>\<forall>i. (c <\<^bsub>P i\<^esub> d) = (c <\<^bsub>(if h i < n then mktop (P i) e else if h i = n then Arrow_Order_Mirabelle_riepwfubkl.above (P i) c e else mkbot (P i) e)\<^esub> d)\<close>
thf(fact_12_in__mkbot, axiom,
    ((![X2 : arrow_1857593510le_alt, Y : arrow_1857593510le_alt, L : set_Pr367596371le_alt, Z : arrow_1857593510le_alt]: ((member2048039092le_alt @ (produc1494124311le_alt @ X2 @ Y) @ (arrow_843587755_mkbot @ L @ Z)) = (((~ ((Y = Z)))) & ((((((X2 = Z)) => ((~ ((X2 = Y)))))) & ((((~ ((X2 = Z)))) => ((member2048039092le_alt @ (produc1494124311le_alt @ X2 @ Y) @ L))))))))))). % in_mkbot
thf(fact_13_in__mktop, axiom,
    ((![X2 : arrow_1857593510le_alt, Y : arrow_1857593510le_alt, L : set_Pr367596371le_alt, Z : arrow_1857593510le_alt]: ((member2048039092le_alt @ (produc1494124311le_alt @ X2 @ Y) @ (arrow_992294841_mktop @ L @ Z)) = (((~ ((X2 = Z)))) & ((((((Y = Z)) => ((~ ((X2 = Y)))))) & ((((~ ((Y = Z)))) => ((member2048039092le_alt @ (produc1494124311le_alt @ X2 @ Y) @ L))))))))))). % in_mktop
thf(fact_14__092_060open_062c_A_060_092_060_094bsub_062P_A_Iinv_Ah_An_J_092_060_094esub_062_Ad_092_060close_062, axiom,
    ((member2048039092le_alt @ (produc1494124311le_alt @ c @ d) @ (p @ (hilber1586975467di_nat @ top_to1799531699e_indi @ h @ n))))). % \<open>c <\<^bsub>P (inv h n)\<^esub> d\<close>
thf(fact_15_prod_Oinject, axiom,
    ((![X1 : arrow_1857593510le_alt, X22 : arrow_1857593510le_alt, Y1 : arrow_1857593510le_alt, Y2 : arrow_1857593510le_alt]: (((produc1494124311le_alt @ X1 @ X22) = (produc1494124311le_alt @ Y1 @ Y2)) = (((X1 = Y1)) & ((X22 = Y2))))))). % prod.inject
thf(fact_16_old_Oprod_Oinject, axiom,
    ((![A2 : arrow_1857593510le_alt, B2 : arrow_1857593510le_alt, A3 : arrow_1857593510le_alt, B3 : arrow_1857593510le_alt]: (((produc1494124311le_alt @ A2 @ B2) = (produc1494124311le_alt @ A3 @ B3)) = (((A2 = A3)) & ((B2 = B3))))))). % old.prod.inject
thf(fact_17_pred__equals__eq2, axiom,
    ((![R : set_Pr367596371le_alt, S : set_Pr367596371le_alt]: (((^[X3 : arrow_1857593510le_alt]: (^[Y3 : arrow_1857593510le_alt]: (member2048039092le_alt @ (produc1494124311le_alt @ X3 @ Y3) @ R))) = (^[X3 : arrow_1857593510le_alt]: (^[Y3 : arrow_1857593510le_alt]: (member2048039092le_alt @ (produc1494124311le_alt @ X3 @ Y3) @ S)))) = (R = S))))). % pred_equals_eq2
thf(fact_18_in__measure, axiom,
    ((![X2 : arrow_1857593510le_alt, Y : arrow_1857593510le_alt, F : arrow_1857593510le_alt > nat]: ((member2048039092le_alt @ (produc1494124311le_alt @ X2 @ Y) @ (measur1447705966le_alt @ F)) = (ord_less_nat @ (F @ X2) @ (F @ Y)))))). % in_measure
thf(fact_19_in__above, axiom,
    ((![A2 : arrow_1857593510le_alt, B2 : arrow_1857593510le_alt, L : set_Pr367596371le_alt, X2 : arrow_1857593510le_alt, Y : arrow_1857593510le_alt]: ((~ ((A2 = B2))) => ((member1334244458le_alt @ L @ arrow_1848678355le_Lin) => ((member2048039092le_alt @ (produc1494124311le_alt @ X2 @ Y) @ (arrow_1726226719_above @ L @ A2 @ B2)) = (((~ ((X2 = Y)))) & ((((((X2 = B2)) => ((member2048039092le_alt @ (produc1494124311le_alt @ A2 @ Y) @ L)))) & ((((~ ((X2 = B2)))) => ((((((Y = B2)) => ((((X2 = A2)) | ((member2048039092le_alt @ (produc1494124311le_alt @ X2 @ A2) @ L)))))) & ((((~ ((Y = B2)))) => ((member2048039092le_alt @ (produc1494124311le_alt @ X2 @ Y) @ L))))))))))))))))). % in_above
thf(fact_20_n_I1_J, axiom,
    ((ord_less_nat @ n @ (finite927127589e_indi @ top_to1799531699e_indi)))). % n(1)
thf(fact_21_dist, axiom,
    ((distin2037765919le_alt @ (cons_A1864255580le_alt @ c @ (cons_A1864255580le_alt @ d @ (cons_A1864255580le_alt @ e @ nil_Ar277507244le_alt)))))). % dist
thf(fact_22_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_23_less__not__refl, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_not_refl
thf(fact_24_less__not__refl2, axiom,
    ((![N : nat, M : nat]: ((ord_less_nat @ N @ M) => (~ ((M = N))))))). % less_not_refl2
thf(fact_25_less__not__refl3, axiom,
    ((![S2 : nat, T : nat]: ((ord_less_nat @ S2 @ T) => (~ ((S2 = T))))))). % less_not_refl3
thf(fact_26__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_27__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_28__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062e_O_Adistinct_A_091c_M_Ad_M_Ae_093_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom,
    ((~ ((![E : arrow_1857593510le_alt]: (~ ((distin2037765919le_alt @ (cons_A1864255580le_alt @ c @ (cons_A1864255580le_alt @ d @ (cons_A1864255580le_alt @ E @ nil_Ar277507244le_alt))))))))))). % \<open>\<And>thesis. (\<And>e. distinct [c, d, e] \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_29_injh, axiom,
    ((inj_on528257168di_nat @ h @ top_to1799531699e_indi))). % injh
thf(fact_30_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_31_alt3, axiom,
    ((?[A4 : arrow_1857593510le_alt, B4 : arrow_1857593510le_alt, C2 : arrow_1857593510le_alt]: (distin2037765919le_alt @ (cons_A1864255580le_alt @ A4 @ (cons_A1864255580le_alt @ B4 @ (cons_A1864255580le_alt @ C2 @ nil_Ar277507244le_alt))))))). % alt3
thf(fact_32_third__alt, axiom,
    ((![A2 : arrow_1857593510le_alt, B2 : arrow_1857593510le_alt]: ((~ ((A2 = B2))) => (?[C2 : arrow_1857593510le_alt]: (distin2037765919le_alt @ (cons_A1864255580le_alt @ A2 @ (cons_A1864255580le_alt @ B2 @ (cons_A1864255580le_alt @ C2 @ nil_Ar277507244le_alt))))))))). % third_alt
thf(fact_33_linear__alt, axiom,
    ((?[L2 : set_Pr367596371le_alt]: (member1334244458le_alt @ L2 @ arrow_1848678355le_Lin)))). % linear_alt
thf(fact_34_const__Lin__Prof, axiom,
    ((![L : set_Pr367596371le_alt]: ((member1334244458le_alt @ L @ arrow_1848678355le_Lin) => (member684274596le_alt @ (^[P3 : arrow_1429744205e_indi]: L) @ arrow_1951607831e_Prof))))). % const_Lin_Prof
thf(fact_35_notin__Lin__iff, axiom,
    ((![L : set_Pr367596371le_alt, X2 : arrow_1857593510le_alt, Y : arrow_1857593510le_alt]: ((member1334244458le_alt @ L @ arrow_1848678355le_Lin) => ((~ ((X2 = Y))) => ((~ ((member2048039092le_alt @ (produc1494124311le_alt @ X2 @ Y) @ L))) = (member2048039092le_alt @ (produc1494124311le_alt @ Y @ X2) @ L))))))). % notin_Lin_iff
thf(fact_36_complete__Lin, axiom,
    ((![A2 : arrow_1857593510le_alt, B2 : arrow_1857593510le_alt]: ((~ ((A2 = B2))) => (?[X4 : set_Pr367596371le_alt]: ((member1334244458le_alt @ X4 @ arrow_1848678355le_Lin) & (member2048039092le_alt @ (produc1494124311le_alt @ A2 @ B2) @ X4))))))). % complete_Lin
thf(fact_37_Lin__irrefl, axiom,
    ((![L : set_Pr367596371le_alt, A2 : arrow_1857593510le_alt, B2 : arrow_1857593510le_alt]: ((member1334244458le_alt @ L @ arrow_1848678355le_Lin) => ((member2048039092le_alt @ (produc1494124311le_alt @ A2 @ B2) @ L) => (~ ((member2048039092le_alt @ (produc1494124311le_alt @ B2 @ A2) @ L)))))))). % Lin_irrefl
thf(fact_38_mktop__Lin, axiom,
    ((![L : set_Pr367596371le_alt, X2 : arrow_1857593510le_alt]: ((member1334244458le_alt @ L @ arrow_1848678355le_Lin) => (member1334244458le_alt @ (arrow_992294841_mktop @ L @ X2) @ arrow_1848678355le_Lin))))). % mktop_Lin
thf(fact_39_mkbot__Lin, axiom,
    ((![L : set_Pr367596371le_alt, X2 : arrow_1857593510le_alt]: ((member1334244458le_alt @ L @ arrow_1848678355le_Lin) => (member1334244458le_alt @ (arrow_843587755_mkbot @ L @ X2) @ arrow_1848678355le_Lin))))). % mkbot_Lin
thf(fact_40_above__Lin, axiom,
    ((![X2 : arrow_1857593510le_alt, Y : arrow_1857593510le_alt, L : set_Pr367596371le_alt]: ((~ ((X2 = Y))) => ((member1334244458le_alt @ L @ arrow_1848678355le_Lin) => (member1334244458le_alt @ (arrow_1726226719_above @ L @ X2 @ Y) @ arrow_1848678355le_Lin)))))). % above_Lin
thf(fact_41_unanimity__def, axiom,
    ((arrow_52334694nimity = (^[F2 : (arrow_1429744205e_indi > set_Pr367596371le_alt) > set_Pr367596371le_alt]: (![X3 : arrow_1429744205e_indi > set_Pr367596371le_alt]: (((member684274596le_alt @ X3 @ arrow_1951607831e_Prof)) => ((![A5 : arrow_1857593510le_alt]: (![B5 : arrow_1857593510le_alt]: (((![I2 : arrow_1429744205e_indi]: (member2048039092le_alt @ (produc1494124311le_alt @ A5 @ B5) @ (X3 @ I2)))) => ((member2048039092le_alt @ (produc1494124311le_alt @ A5 @ B5) @ (F2 @ X3))))))))))))). % unanimity_def
thf(fact_42_IIA__def, axiom,
    ((arrow_1821794627le_IIA = (^[F2 : (arrow_1429744205e_indi > set_Pr367596371le_alt) > set_Pr367596371le_alt]: (![X3 : arrow_1429744205e_indi > set_Pr367596371le_alt]: (((member684274596le_alt @ X3 @ arrow_1951607831e_Prof)) => ((![Y3 : arrow_1429744205e_indi > set_Pr367596371le_alt]: (((member684274596le_alt @ Y3 @ arrow_1951607831e_Prof)) => ((![A5 : arrow_1857593510le_alt]: (![B5 : arrow_1857593510le_alt]: (((![I2 : arrow_1429744205e_indi]: ((member2048039092le_alt @ (produc1494124311le_alt @ A5 @ B5) @ (X3 @ I2)) = (member2048039092le_alt @ (produc1494124311le_alt @ A5 @ B5) @ (Y3 @ I2))))) => (((member2048039092le_alt @ (produc1494124311le_alt @ A5 @ B5) @ (F2 @ X3)) = (member2048039092le_alt @ (produc1494124311le_alt @ A5 @ B5) @ (F2 @ Y3))))))))))))))))). % IIA_def
thf(fact_43_mem__Collect__eq, axiom,
    ((![A2 : produc16571293le_alt, P : produc16571293le_alt > $o]: ((member2048039092le_alt @ A2 @ (collec531981554le_alt @ P)) = (P @ A2))))). % mem_Collect_eq
thf(fact_44_mem__Collect__eq, axiom,
    ((![A2 : arrow_1429744205e_indi > set_Pr367596371le_alt, P : (arrow_1429744205e_indi > set_Pr367596371le_alt) > $o]: ((member684274596le_alt @ A2 @ (collec1559089382le_alt @ P)) = (P @ A2))))). % mem_Collect_eq
thf(fact_45_mem__Collect__eq, axiom,
    ((![A2 : set_Pr367596371le_alt, P : set_Pr367596371le_alt > $o]: ((member1334244458le_alt @ A2 @ (collec1399441576le_alt @ P)) = (P @ A2))))). % mem_Collect_eq
thf(fact_46_mem__Collect__eq, axiom,
    ((![A2 : (arrow_1429744205e_indi > set_Pr367596371le_alt) > set_Pr367596371le_alt, P : ((arrow_1429744205e_indi > set_Pr367596371le_alt) > set_Pr367596371le_alt) > $o]: ((member183760530le_alt @ A2 @ (collec1382217680le_alt @ P)) = (P @ A2))))). % mem_Collect_eq
thf(fact_47_Collect__mem__eq, axiom,
    ((![A6 : set_Pr367596371le_alt]: ((collec531981554le_alt @ (^[X3 : produc16571293le_alt]: (member2048039092le_alt @ X3 @ A6))) = A6)))). % Collect_mem_eq
thf(fact_48_Collect__mem__eq, axiom,
    ((![A6 : set_Ar809243995le_alt]: ((collec1559089382le_alt @ (^[X3 : arrow_1429744205e_indi > set_Pr367596371le_alt]: (member684274596le_alt @ X3 @ A6))) = A6)))). % Collect_mem_eq
thf(fact_49_Collect__mem__eq, axiom,
    ((![A6 : set_se2071012361le_alt]: ((collec1399441576le_alt @ (^[X3 : set_Pr367596371le_alt]: (member1334244458le_alt @ X3 @ A6))) = A6)))). % Collect_mem_eq
thf(fact_50_Collect__mem__eq, axiom,
    ((![A6 : set_Ar182050865le_alt]: ((collec1382217680le_alt @ (^[X3 : (arrow_1429744205e_indi > set_Pr367596371le_alt) > set_Pr367596371le_alt]: (member183760530le_alt @ X3 @ A6))) = A6)))). % Collect_mem_eq
thf(fact_51_old_Oprod_Oinducts, axiom,
    ((![P : produc16571293le_alt > $o, Prod : produc16571293le_alt]: ((![A4 : arrow_1857593510le_alt, B4 : arrow_1857593510le_alt]: (P @ (produc1494124311le_alt @ A4 @ B4))) => (P @ Prod))))). % old.prod.inducts
thf(fact_52_old_Oprod_Oexhaust, axiom,
    ((![Y : produc16571293le_alt]: (~ ((![A4 : arrow_1857593510le_alt, B4 : arrow_1857593510le_alt]: (~ ((Y = (produc1494124311le_alt @ A4 @ B4)))))))))). % old.prod.exhaust
thf(fact_53_Pair__inject, axiom,
    ((![A2 : arrow_1857593510le_alt, B2 : arrow_1857593510le_alt, A3 : arrow_1857593510le_alt, B3 : arrow_1857593510le_alt]: (((produc1494124311le_alt @ A2 @ B2) = (produc1494124311le_alt @ A3 @ B3)) => (~ (((A2 = A3) => (~ ((B2 = B3)))))))))). % Pair_inject
thf(fact_54_prod__cases, axiom,
    ((![P : produc16571293le_alt > $o, P4 : produc16571293le_alt]: ((![A4 : arrow_1857593510le_alt, B4 : arrow_1857593510le_alt]: (P @ (produc1494124311le_alt @ A4 @ B4))) => (P @ P4))))). % prod_cases
thf(fact_55_surj__pair, axiom,
    ((![P4 : produc16571293le_alt]: (?[X4 : arrow_1857593510le_alt, Y4 : arrow_1857593510le_alt]: (P4 = (produc1494124311le_alt @ X4 @ Y4)))))). % surj_pair
thf(fact_56_linorder__neqE__nat, axiom,
    ((![X2 : nat, Y : nat]: ((~ ((X2 = Y))) => ((~ ((ord_less_nat @ X2 @ Y))) => (ord_less_nat @ Y @ X2)))))). % linorder_neqE_nat
thf(fact_57_infinite__descent, axiom,
    ((![P : nat > $o, N : nat]: ((![N2 : nat]: ((~ ((P @ N2))) => (?[M2 : nat]: ((ord_less_nat @ M2 @ N2) & (~ ((P @ M2))))))) => (P @ N))))). % infinite_descent
thf(fact_58_nat__less__induct, axiom,
    ((![P : nat > $o, N : nat]: ((![N2 : nat]: ((![M2 : nat]: ((ord_less_nat @ M2 @ N2) => (P @ M2))) => (P @ N2))) => (P @ N))))). % nat_less_induct
thf(fact_59_less__irrefl__nat, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_irrefl_nat
thf(fact_60_inv__identity, axiom,
    (((hilber700257104e_indi @ top_to1799531699e_indi @ (^[A5 : arrow_1429744205e_indi]: A5)) = (^[A5 : arrow_1429744205e_indi]: A5)))). % inv_identity
thf(fact_61_distinct__singleton, axiom,
    ((![X2 : arrow_1857593510le_alt]: (distin2037765919le_alt @ (cons_A1864255580le_alt @ X2 @ nil_Ar277507244le_alt))))). % distinct_singleton
thf(fact_62_UNIV__I, axiom,
    ((![X2 : produc16571293le_alt]: (member2048039092le_alt @ X2 @ top_to224369155le_alt)))). % UNIV_I
thf(fact_63_UNIV__I, axiom,
    ((![X2 : arrow_1429744205e_indi > set_Pr367596371le_alt]: (member684274596le_alt @ X2 @ top_to685525675le_alt)))). % UNIV_I
thf(fact_64_UNIV__I, axiom,
    ((![X2 : set_Pr367596371le_alt]: (member1334244458le_alt @ X2 @ top_to469035705le_alt)))). % UNIV_I
thf(fact_65_UNIV__I, axiom,
    ((![X2 : (arrow_1429744205e_indi > set_Pr367596371le_alt) > set_Pr367596371le_alt]: (member183760530le_alt @ X2 @ top_to803745505le_alt)))). % UNIV_I
thf(fact_66_UNIV__I, axiom,
    ((![X2 : arrow_1429744205e_indi]: (member1966420836e_indi @ X2 @ top_to1799531699e_indi)))). % UNIV_I
thf(fact_67_iso__tuple__UNIV__I, axiom,
    ((![X2 : produc16571293le_alt]: (member2048039092le_alt @ X2 @ top_to224369155le_alt)))). % iso_tuple_UNIV_I
thf(fact_68_iso__tuple__UNIV__I, axiom,
    ((![X2 : arrow_1429744205e_indi > set_Pr367596371le_alt]: (member684274596le_alt @ X2 @ top_to685525675le_alt)))). % iso_tuple_UNIV_I
thf(fact_69_iso__tuple__UNIV__I, axiom,
    ((![X2 : set_Pr367596371le_alt]: (member1334244458le_alt @ X2 @ top_to469035705le_alt)))). % iso_tuple_UNIV_I
thf(fact_70_iso__tuple__UNIV__I, axiom,
    ((![X2 : (arrow_1429744205e_indi > set_Pr367596371le_alt) > set_Pr367596371le_alt]: (member183760530le_alt @ X2 @ top_to803745505le_alt)))). % iso_tuple_UNIV_I
thf(fact_71_iso__tuple__UNIV__I, axiom,
    ((![X2 : arrow_1429744205e_indi]: (member1966420836e_indi @ X2 @ top_to1799531699e_indi)))). % iso_tuple_UNIV_I
thf(fact_72_list_Oinject, axiom,
    ((![X21 : arrow_1857593510le_alt, X222 : list_A2130511660le_alt, Y21 : arrow_1857593510le_alt, Y22 : list_A2130511660le_alt]: (((cons_A1864255580le_alt @ X21 @ X222) = (cons_A1864255580le_alt @ Y21 @ Y22)) = (((X21 = Y21)) & ((X222 = Y22))))))). % list.inject
thf(fact_73_in__below, axiom,
    ((![A2 : arrow_1857593510le_alt, B2 : arrow_1857593510le_alt, L : set_Pr367596371le_alt, X2 : arrow_1857593510le_alt, Y : arrow_1857593510le_alt]: ((~ ((A2 = B2))) => ((member1334244458le_alt @ L @ arrow_1848678355le_Lin) => ((member2048039092le_alt @ (produc1494124311le_alt @ X2 @ Y) @ (arrow_887341619_below @ L @ A2 @ B2)) = (((~ ((X2 = Y)))) & ((((((Y = A2)) => ((member2048039092le_alt @ (produc1494124311le_alt @ X2 @ B2) @ L)))) & ((((~ ((Y = A2)))) => ((((((X2 = A2)) => ((((Y = B2)) | ((member2048039092le_alt @ (produc1494124311le_alt @ B2 @ Y) @ L)))))) & ((((~ ((X2 = A2)))) => ((member2048039092le_alt @ (produc1494124311le_alt @ X2 @ Y) @ L))))))))))))))))). % in_below
thf(fact_74_PiProf, axiom,
    ((![N : nat]: (member684274596le_alt @ (^[I2 : arrow_1429744205e_indi]: (if_set550155277le_alt @ (ord_less_nat @ (h @ I2) @ N) @ lab @ lba)) @ arrow_1951607831e_Prof)))). % PiProf
thf(fact_75_inv__into__f__f, axiom,
    ((![F : arrow_1429744205e_indi > nat, A6 : set_Ar1007576579e_indi, X2 : arrow_1429744205e_indi]: ((inj_on528257168di_nat @ F @ A6) => ((member1966420836e_indi @ X2 @ A6) => ((hilber1586975467di_nat @ A6 @ F @ (F @ X2)) = X2)))))). % inv_into_f_f
thf(fact_76_top__set__def, axiom,
    ((top_to1799531699e_indi = (collec1169676194e_indi @ top_to1473733010indi_o)))). % top_set_def
thf(fact_77_inv__into__f__eq, axiom,
    ((![F : arrow_1429744205e_indi > nat, A6 : set_Ar1007576579e_indi, X2 : arrow_1429744205e_indi, Y : nat]: ((inj_on528257168di_nat @ F @ A6) => ((member1966420836e_indi @ X2 @ A6) => (((F @ X2) = Y) => ((hilber1586975467di_nat @ A6 @ F @ Y) = X2))))))). % inv_into_f_eq
thf(fact_78_top__empty__eq2, axiom,
    ((top_to701951606_alt_o = (^[X3 : arrow_1857593510le_alt]: (^[Y3 : arrow_1857593510le_alt]: (member2048039092le_alt @ (produc1494124311le_alt @ X3 @ Y3) @ top_to224369155le_alt)))))). % top_empty_eq2
thf(fact_79_top__empty__eq, axiom,
    ((top_to1074038850_alt_o = (^[X3 : produc16571293le_alt]: (member2048039092le_alt @ X3 @ top_to224369155le_alt))))). % top_empty_eq
thf(fact_80_top__empty__eq, axiom,
    ((top_to1344581618_alt_o = (^[X3 : arrow_1429744205e_indi > set_Pr367596371le_alt]: (member684274596le_alt @ X3 @ top_to685525675le_alt))))). % top_empty_eq
thf(fact_81_top__empty__eq, axiom,
    ((top_to1900088076_alt_o = (^[X3 : set_Pr367596371le_alt]: (member1334244458le_alt @ X3 @ top_to469035705le_alt))))). % top_empty_eq
thf(fact_82_top__empty__eq, axiom,
    ((top_to1892800228_alt_o = (^[X3 : (arrow_1429744205e_indi > set_Pr367596371le_alt) > set_Pr367596371le_alt]: (member183760530le_alt @ X3 @ top_to803745505le_alt))))). % top_empty_eq
thf(fact_83_top__empty__eq, axiom,
    ((top_to1473733010indi_o = (^[X3 : arrow_1429744205e_indi]: (member1966420836e_indi @ X3 @ top_to1799531699e_indi))))). % top_empty_eq
thf(fact_84_inj__imp__inv__eq, axiom,
    ((![F : arrow_1429744205e_indi > nat, G : nat > arrow_1429744205e_indi]: ((inj_on528257168di_nat @ F @ top_to1799531699e_indi) => ((![X4 : nat]: ((F @ (G @ X4)) = X4)) => ((hilber1586975467di_nat @ top_to1799531699e_indi @ F) = G)))))). % inj_imp_inv_eq
thf(fact_85_inv__f__eq, axiom,
    ((![F : arrow_1429744205e_indi > nat, X2 : arrow_1429744205e_indi, Y : nat]: ((inj_on528257168di_nat @ F @ top_to1799531699e_indi) => (((F @ X2) = Y) => ((hilber1586975467di_nat @ top_to1799531699e_indi @ F @ Y) = X2)))))). % inv_f_eq
thf(fact_86_inv__f__f, axiom,
    ((![F : arrow_1429744205e_indi > nat, X2 : arrow_1429744205e_indi]: ((inj_on528257168di_nat @ F @ top_to1799531699e_indi) => ((hilber1586975467di_nat @ top_to1799531699e_indi @ F @ (F @ X2)) = X2))))). % inv_f_f
thf(fact_87_ord__eq__less__subst, axiom,
    ((![A2 : nat, F : nat > nat, B2 : nat, C : nat]: ((A2 = (F @ B2)) => ((ord_less_nat @ B2 @ C) => ((![X4 : nat, Y4 : nat]: ((ord_less_nat @ X4 @ Y4) => (ord_less_nat @ (F @ X4) @ (F @ Y4)))) => (ord_less_nat @ A2 @ (F @ C)))))))). % ord_eq_less_subst
thf(fact_88_ord__less__eq__subst, axiom,
    ((![A2 : nat, B2 : nat, F : nat > nat, C : nat]: ((ord_less_nat @ A2 @ B2) => (((F @ B2) = C) => ((![X4 : nat, Y4 : nat]: ((ord_less_nat @ X4 @ Y4) => (ord_less_nat @ (F @ X4) @ (F @ Y4)))) => (ord_less_nat @ (F @ A2) @ C))))))). % ord_less_eq_subst
thf(fact_89_order__less__subst1, axiom,
    ((![A2 : nat, F : nat > nat, B2 : nat, C : nat]: ((ord_less_nat @ A2 @ (F @ B2)) => ((ord_less_nat @ B2 @ C) => ((![X4 : nat, Y4 : nat]: ((ord_less_nat @ X4 @ Y4) => (ord_less_nat @ (F @ X4) @ (F @ Y4)))) => (ord_less_nat @ A2 @ (F @ C)))))))). % order_less_subst1
thf(fact_90_order__less__subst2, axiom,
    ((![A2 : nat, B2 : nat, F : nat > nat, C : nat]: ((ord_less_nat @ A2 @ B2) => ((ord_less_nat @ (F @ B2) @ C) => ((![X4 : nat, Y4 : nat]: ((ord_less_nat @ X4 @ Y4) => (ord_less_nat @ (F @ X4) @ (F @ Y4)))) => (ord_less_nat @ (F @ A2) @ C))))))). % order_less_subst2
thf(fact_91_gt__ex, axiom,
    ((![X2 : nat]: (?[X_1 : nat]: (ord_less_nat @ X2 @ X_1))))). % gt_ex
thf(fact_92_neqE, axiom,
    ((![X2 : nat, Y : nat]: ((~ ((X2 = Y))) => ((~ ((ord_less_nat @ X2 @ Y))) => (ord_less_nat @ Y @ X2)))))). % neqE
thf(fact_93_neq__iff, axiom,
    ((![X2 : nat, Y : nat]: ((~ ((X2 = Y))) = (((ord_less_nat @ X2 @ Y)) | ((ord_less_nat @ Y @ X2))))))). % neq_iff
thf(fact_94_order_Oasym, axiom,
    ((![A2 : nat, B2 : nat]: ((ord_less_nat @ A2 @ B2) => (~ ((ord_less_nat @ B2 @ A2))))))). % order.asym
thf(fact_95_less__imp__neq, axiom,
    ((![X2 : nat, Y : nat]: ((ord_less_nat @ X2 @ Y) => (~ ((X2 = Y))))))). % less_imp_neq
thf(fact_96_less__asym, axiom,
    ((![X2 : nat, Y : nat]: ((ord_less_nat @ X2 @ Y) => (~ ((ord_less_nat @ Y @ X2))))))). % less_asym
thf(fact_97_less__asym_H, axiom,
    ((![A2 : nat, B2 : nat]: ((ord_less_nat @ A2 @ B2) => (~ ((ord_less_nat @ B2 @ A2))))))). % less_asym'
thf(fact_98_less__trans, axiom,
    ((![X2 : nat, Y : nat, Z : nat]: ((ord_less_nat @ X2 @ Y) => ((ord_less_nat @ Y @ Z) => (ord_less_nat @ X2 @ Z)))))). % less_trans
thf(fact_99_less__linear, axiom,
    ((![X2 : nat, Y : nat]: ((ord_less_nat @ X2 @ Y) | ((X2 = Y) | (ord_less_nat @ Y @ X2)))))). % less_linear
thf(fact_100_less__irrefl, axiom,
    ((![X2 : nat]: (~ ((ord_less_nat @ X2 @ X2)))))). % less_irrefl
thf(fact_101_ord__eq__less__trans, axiom,
    ((![A2 : nat, B2 : nat, C : nat]: ((A2 = B2) => ((ord_less_nat @ B2 @ C) => (ord_less_nat @ A2 @ C)))))). % ord_eq_less_trans
thf(fact_102_ord__less__eq__trans, axiom,
    ((![A2 : nat, B2 : nat, C : nat]: ((ord_less_nat @ A2 @ B2) => ((B2 = C) => (ord_less_nat @ A2 @ C)))))). % ord_less_eq_trans
thf(fact_103_dual__order_Oasym, axiom,
    ((![B2 : nat, A2 : nat]: ((ord_less_nat @ B2 @ A2) => (~ ((ord_less_nat @ A2 @ B2))))))). % dual_order.asym
thf(fact_104_less__imp__not__eq, axiom,
    ((![X2 : nat, Y : nat]: ((ord_less_nat @ X2 @ Y) => (~ ((X2 = Y))))))). % less_imp_not_eq
thf(fact_105_less__not__sym, axiom,
    ((![X2 : nat, Y : nat]: ((ord_less_nat @ X2 @ Y) => (~ ((ord_less_nat @ Y @ X2))))))). % less_not_sym
thf(fact_106_less__induct, axiom,
    ((![P : nat > $o, A2 : nat]: ((![X4 : nat]: ((![Y5 : nat]: ((ord_less_nat @ Y5 @ X4) => (P @ Y5))) => (P @ X4))) => (P @ A2))))). % less_induct
thf(fact_107_antisym__conv3, axiom,
    ((![Y : nat, X2 : nat]: ((~ ((ord_less_nat @ Y @ X2))) => ((~ ((ord_less_nat @ X2 @ Y))) = (X2 = Y)))))). % antisym_conv3
thf(fact_108_less__imp__not__eq2, axiom,
    ((![X2 : nat, Y : nat]: ((ord_less_nat @ X2 @ Y) => (~ ((Y = X2))))))). % less_imp_not_eq2
thf(fact_109_less__imp__triv, axiom,
    ((![X2 : nat, Y : nat, P : $o]: ((ord_less_nat @ X2 @ Y) => ((ord_less_nat @ Y @ X2) => P))))). % less_imp_triv
thf(fact_110_linorder__cases, axiom,
    ((![X2 : nat, Y : nat]: ((~ ((ord_less_nat @ X2 @ Y))) => ((~ ((X2 = Y))) => (ord_less_nat @ Y @ X2)))))). % linorder_cases
thf(fact_111_dual__order_Oirrefl, axiom,
    ((![A2 : nat]: (~ ((ord_less_nat @ A2 @ A2)))))). % dual_order.irrefl
thf(fact_112_order_Ostrict__trans, axiom,
    ((![A2 : nat, B2 : nat, C : nat]: ((ord_less_nat @ A2 @ B2) => ((ord_less_nat @ B2 @ C) => (ord_less_nat @ A2 @ C)))))). % order.strict_trans
thf(fact_113_less__imp__not__less, axiom,
    ((![X2 : nat, Y : nat]: ((ord_less_nat @ X2 @ Y) => (~ ((ord_less_nat @ Y @ X2))))))). % less_imp_not_less
thf(fact_114_exists__least__iff, axiom,
    (((^[P5 : nat > $o]: (?[X5 : nat]: (P5 @ X5))) = (^[P6 : nat > $o]: (?[N3 : nat]: (((P6 @ N3)) & ((![M3 : nat]: (((ord_less_nat @ M3 @ N3)) => ((~ ((P6 @ M3))))))))))))). % exists_least_iff
thf(fact_115_linorder__less__wlog, axiom,
    ((![P : nat > nat > $o, A2 : nat, B2 : nat]: ((![A4 : nat, B4 : nat]: ((ord_less_nat @ A4 @ B4) => (P @ A4 @ B4))) => ((![A4 : nat]: (P @ A4 @ A4)) => ((![A4 : nat, B4 : nat]: ((P @ B4 @ A4) => (P @ A4 @ B4))) => (P @ A2 @ B2))))))). % linorder_less_wlog
thf(fact_116_dual__order_Ostrict__trans, axiom,
    ((![B2 : nat, A2 : nat, C : nat]: ((ord_less_nat @ B2 @ A2) => ((ord_less_nat @ C @ B2) => (ord_less_nat @ C @ A2)))))). % dual_order.strict_trans
thf(fact_117_not__less__iff__gr__or__eq, axiom,
    ((![X2 : nat, Y : nat]: ((~ ((ord_less_nat @ X2 @ Y))) = (((ord_less_nat @ Y @ X2)) | ((X2 = Y))))))). % not_less_iff_gr_or_eq
thf(fact_118_order_Ostrict__implies__not__eq, axiom,
    ((![A2 : nat, B2 : nat]: ((ord_less_nat @ A2 @ B2) => (~ ((A2 = B2))))))). % order.strict_implies_not_eq
thf(fact_119_dual__order_Ostrict__implies__not__eq, axiom,
    ((![B2 : nat, A2 : nat]: ((ord_less_nat @ B2 @ A2) => (~ ((A2 = B2))))))). % dual_order.strict_implies_not_eq
thf(fact_120_not__Cons__self2, axiom,
    ((![X2 : arrow_1857593510le_alt, Xs : list_A2130511660le_alt]: (~ (((cons_A1864255580le_alt @ X2 @ Xs) = Xs)))))). % not_Cons_self2
thf(fact_121_UNIV__witness, axiom,
    ((?[X4 : produc16571293le_alt]: (member2048039092le_alt @ X4 @ top_to224369155le_alt)))). % UNIV_witness
thf(fact_122_UNIV__witness, axiom,
    ((?[X4 : arrow_1429744205e_indi > set_Pr367596371le_alt]: (member684274596le_alt @ X4 @ top_to685525675le_alt)))). % UNIV_witness
thf(fact_123_UNIV__witness, axiom,
    ((?[X4 : set_Pr367596371le_alt]: (member1334244458le_alt @ X4 @ top_to469035705le_alt)))). % UNIV_witness
thf(fact_124_UNIV__witness, axiom,
    ((?[X4 : (arrow_1429744205e_indi > set_Pr367596371le_alt) > set_Pr367596371le_alt]: (member183760530le_alt @ X4 @ top_to803745505le_alt)))). % UNIV_witness
thf(fact_125_UNIV__witness, axiom,
    ((?[X4 : arrow_1429744205e_indi]: (member1966420836e_indi @ X4 @ top_to1799531699e_indi)))). % UNIV_witness
thf(fact_126_UNIV__eq__I, axiom,
    ((![A6 : set_Pr367596371le_alt]: ((![X4 : produc16571293le_alt]: (member2048039092le_alt @ X4 @ A6)) => (top_to224369155le_alt = A6))))). % UNIV_eq_I
thf(fact_127_UNIV__eq__I, axiom,
    ((![A6 : set_Ar809243995le_alt]: ((![X4 : arrow_1429744205e_indi > set_Pr367596371le_alt]: (member684274596le_alt @ X4 @ A6)) => (top_to685525675le_alt = A6))))). % UNIV_eq_I
thf(fact_128_UNIV__eq__I, axiom,
    ((![A6 : set_se2071012361le_alt]: ((![X4 : set_Pr367596371le_alt]: (member1334244458le_alt @ X4 @ A6)) => (top_to469035705le_alt = A6))))). % UNIV_eq_I
thf(fact_129_UNIV__eq__I, axiom,
    ((![A6 : set_Ar182050865le_alt]: ((![X4 : (arrow_1429744205e_indi > set_Pr367596371le_alt) > set_Pr367596371le_alt]: (member183760530le_alt @ X4 @ A6)) => (top_to803745505le_alt = A6))))). % UNIV_eq_I
thf(fact_130_UNIV__eq__I, axiom,
    ((![A6 : set_Ar1007576579e_indi]: ((![X4 : arrow_1429744205e_indi]: (member1966420836e_indi @ X4 @ A6)) => (top_to1799531699e_indi = A6))))). % UNIV_eq_I
thf(fact_131_below__Lin, axiom,
    ((![X2 : arrow_1857593510le_alt, Y : arrow_1857593510le_alt, L : set_Pr367596371le_alt]: ((~ ((X2 = Y))) => ((member1334244458le_alt @ L @ arrow_1848678355le_Lin) => (member1334244458le_alt @ (arrow_887341619_below @ L @ X2 @ Y) @ arrow_1848678355le_Lin)))))). % below_Lin
thf(fact_132_UNIV__def, axiom,
    ((top_to1799531699e_indi = (collec1169676194e_indi @ (^[X3 : arrow_1429744205e_indi]: $true))))). % UNIV_def
thf(fact_133_top_Oextremum__strict, axiom,
    ((![A2 : set_Ar1007576579e_indi]: (~ ((ord_le1187139159e_indi @ top_to1799531699e_indi @ A2)))))). % top.extremum_strict
thf(fact_134_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_135_map__tailrec__rev_Oinduct, axiom,
    ((![P : (arrow_1857593510le_alt > arrow_1857593510le_alt) > list_A2130511660le_alt > list_A2130511660le_alt > $o, A0 : arrow_1857593510le_alt > arrow_1857593510le_alt, A1 : list_A2130511660le_alt, A22 : list_A2130511660le_alt]: ((![F3 : arrow_1857593510le_alt > arrow_1857593510le_alt, X_1 : list_A2130511660le_alt]: (P @ F3 @ nil_Ar277507244le_alt @ X_1)) => ((![F3 : arrow_1857593510le_alt > arrow_1857593510le_alt, A4 : arrow_1857593510le_alt, As : list_A2130511660le_alt, Bs : list_A2130511660le_alt]: ((P @ F3 @ As @ (cons_A1864255580le_alt @ (F3 @ A4) @ Bs)) => (P @ F3 @ (cons_A1864255580le_alt @ A4 @ As) @ Bs))) => (P @ A0 @ A1 @ A22)))))). % map_tailrec_rev.induct
thf(fact_136_list__nonempty__induct, axiom,
    ((![Xs : list_A2130511660le_alt, P : list_A2130511660le_alt > $o]: ((~ ((Xs = nil_Ar277507244le_alt))) => ((![X4 : arrow_1857593510le_alt]: (P @ (cons_A1864255580le_alt @ X4 @ nil_Ar277507244le_alt))) => ((![X4 : arrow_1857593510le_alt, Xs2 : list_A2130511660le_alt]: ((~ ((Xs2 = nil_Ar277507244le_alt))) => ((P @ Xs2) => (P @ (cons_A1864255580le_alt @ X4 @ Xs2))))) => (P @ Xs))))))). % list_nonempty_induct
thf(fact_137_successively_Oinduct, axiom,
    ((![P : (arrow_1857593510le_alt > arrow_1857593510le_alt > $o) > list_A2130511660le_alt > $o, A0 : arrow_1857593510le_alt > arrow_1857593510le_alt > $o, A1 : list_A2130511660le_alt]: ((![P7 : arrow_1857593510le_alt > arrow_1857593510le_alt > $o]: (P @ P7 @ nil_Ar277507244le_alt)) => ((![P7 : arrow_1857593510le_alt > arrow_1857593510le_alt > $o, X4 : arrow_1857593510le_alt]: (P @ P7 @ (cons_A1864255580le_alt @ X4 @ nil_Ar277507244le_alt))) => ((![P7 : arrow_1857593510le_alt > arrow_1857593510le_alt > $o, X4 : arrow_1857593510le_alt, Y4 : arrow_1857593510le_alt, Xs2 : list_A2130511660le_alt]: ((P @ P7 @ (cons_A1864255580le_alt @ Y4 @ Xs2)) => (P @ P7 @ (cons_A1864255580le_alt @ X4 @ (cons_A1864255580le_alt @ Y4 @ Xs2))))) => (P @ A0 @ A1))))))). % successively.induct
thf(fact_138_successively_Ocases, axiom,
    ((![X2 : produc1094320767le_alt]: ((![P7 : arrow_1857593510le_alt > arrow_1857593510le_alt > $o]: (~ ((X2 = (produc2047839287le_alt @ P7 @ nil_Ar277507244le_alt))))) => ((![P7 : arrow_1857593510le_alt > arrow_1857593510le_alt > $o, X4 : arrow_1857593510le_alt]: (~ ((X2 = (produc2047839287le_alt @ P7 @ (cons_A1864255580le_alt @ X4 @ nil_Ar277507244le_alt)))))) => (~ ((![P7 : arrow_1857593510le_alt > arrow_1857593510le_alt > $o, X4 : arrow_1857593510le_alt, Y4 : arrow_1857593510le_alt, Xs2 : list_A2130511660le_alt]: (~ ((X2 = (produc2047839287le_alt @ P7 @ (cons_A1864255580le_alt @ X4 @ (cons_A1864255580le_alt @ Y4 @ Xs2)))))))))))))). % successively.cases
thf(fact_139_remdups__adj_Oinduct, axiom,
    ((![P : list_A2130511660le_alt > $o, A0 : list_A2130511660le_alt]: ((P @ nil_Ar277507244le_alt) => ((![X4 : arrow_1857593510le_alt]: (P @ (cons_A1864255580le_alt @ X4 @ nil_Ar277507244le_alt))) => ((![X4 : arrow_1857593510le_alt, Y4 : arrow_1857593510le_alt, Xs2 : list_A2130511660le_alt]: (((X4 = Y4) => (P @ (cons_A1864255580le_alt @ X4 @ Xs2))) => (((~ ((X4 = Y4))) => (P @ (cons_A1864255580le_alt @ Y4 @ Xs2))) => (P @ (cons_A1864255580le_alt @ X4 @ (cons_A1864255580le_alt @ Y4 @ Xs2)))))) => (P @ A0))))))). % remdups_adj.induct
thf(fact_140_sorted__wrt_Oinduct, axiom,
    ((![P : (arrow_1857593510le_alt > arrow_1857593510le_alt > $o) > list_A2130511660le_alt > $o, A0 : arrow_1857593510le_alt > arrow_1857593510le_alt > $o, A1 : list_A2130511660le_alt]: ((![P7 : arrow_1857593510le_alt > arrow_1857593510le_alt > $o]: (P @ P7 @ nil_Ar277507244le_alt)) => ((![P7 : arrow_1857593510le_alt > arrow_1857593510le_alt > $o, X4 : arrow_1857593510le_alt, Ys : list_A2130511660le_alt]: ((P @ P7 @ Ys) => (P @ P7 @ (cons_A1864255580le_alt @ X4 @ Ys)))) => (P @ A0 @ A1)))))). % sorted_wrt.induct
thf(fact_141_remdups__adj_Ocases, axiom,
    ((![X2 : list_A2130511660le_alt]: ((~ ((X2 = nil_Ar277507244le_alt))) => ((![X4 : arrow_1857593510le_alt]: (~ ((X2 = (cons_A1864255580le_alt @ X4 @ nil_Ar277507244le_alt))))) => (~ ((![X4 : arrow_1857593510le_alt, Y4 : arrow_1857593510le_alt, Xs2 : list_A2130511660le_alt]: (~ ((X2 = (cons_A1864255580le_alt @ X4 @ (cons_A1864255580le_alt @ Y4 @ Xs2))))))))))))). % remdups_adj.cases
thf(fact_142_sorted__wrt_Ocases, axiom,
    ((![X2 : produc1094320767le_alt]: ((![P7 : arrow_1857593510le_alt > arrow_1857593510le_alt > $o]: (~ ((X2 = (produc2047839287le_alt @ P7 @ nil_Ar277507244le_alt))))) => (~ ((![P7 : arrow_1857593510le_alt > arrow_1857593510le_alt > $o, X4 : arrow_1857593510le_alt, Ys : list_A2130511660le_alt]: (~ ((X2 = (produc2047839287le_alt @ P7 @ (cons_A1864255580le_alt @ X4 @ Ys)))))))))))). % sorted_wrt.cases
thf(fact_143_transpose_Ocases, axiom,
    ((![X2 : list_l1515872562le_alt]: ((~ ((X2 = nil_li2110134322le_alt))) => ((![Xss : list_l1515872562le_alt]: (~ ((X2 = (cons_l157309922le_alt @ nil_Ar277507244le_alt @ Xss))))) => (~ ((![X4 : arrow_1857593510le_alt, Xs2 : list_A2130511660le_alt, Xss : list_l1515872562le_alt]: (~ ((X2 = (cons_l157309922le_alt @ (cons_A1864255580le_alt @ X4 @ Xs2) @ Xss)))))))))))). % transpose.cases
thf(fact_144_shuffles_Oinduct, axiom,
    ((![P : list_A2130511660le_alt > list_A2130511660le_alt > $o, A0 : list_A2130511660le_alt, A1 : list_A2130511660le_alt]: ((![X_1 : list_A2130511660le_alt]: (P @ nil_Ar277507244le_alt @ X_1)) => ((![Xs2 : list_A2130511660le_alt]: (P @ Xs2 @ nil_Ar277507244le_alt)) => ((![X4 : arrow_1857593510le_alt, Xs2 : list_A2130511660le_alt, Y4 : arrow_1857593510le_alt, Ys : list_A2130511660le_alt]: ((P @ Xs2 @ (cons_A1864255580le_alt @ Y4 @ Ys)) => ((P @ (cons_A1864255580le_alt @ X4 @ Xs2) @ Ys) => (P @ (cons_A1864255580le_alt @ X4 @ Xs2) @ (cons_A1864255580le_alt @ Y4 @ Ys))))) => (P @ A0 @ A1))))))). % shuffles.induct
thf(fact_145_shuffles_Ocases, axiom,
    ((![X2 : produc518943027le_alt]: ((![Ys : list_A2130511660le_alt]: (~ ((X2 = (produc961395117le_alt @ nil_Ar277507244le_alt @ Ys))))) => ((![Xs2 : list_A2130511660le_alt]: (~ ((X2 = (produc961395117le_alt @ Xs2 @ nil_Ar277507244le_alt))))) => (~ ((![X4 : arrow_1857593510le_alt, Xs2 : list_A2130511660le_alt, Y4 : arrow_1857593510le_alt, Ys : list_A2130511660le_alt]: (~ ((X2 = (produc961395117le_alt @ (cons_A1864255580le_alt @ X4 @ Xs2) @ (cons_A1864255580le_alt @ Y4 @ Ys))))))))))))). % shuffles.cases
thf(fact_146_induct__list012, axiom,
    ((![P : list_A2130511660le_alt > $o, Xs : list_A2130511660le_alt]: ((P @ nil_Ar277507244le_alt) => ((![X4 : arrow_1857593510le_alt]: (P @ (cons_A1864255580le_alt @ X4 @ nil_Ar277507244le_alt))) => ((![X4 : arrow_1857593510le_alt, Y4 : arrow_1857593510le_alt, Zs : list_A2130511660le_alt]: ((P @ Zs) => ((P @ (cons_A1864255580le_alt @ Y4 @ Zs)) => (P @ (cons_A1864255580le_alt @ X4 @ (cons_A1864255580le_alt @ Y4 @ Zs)))))) => (P @ Xs))))))). % induct_list012
thf(fact_147_splice_Oinduct, axiom,
    ((![P : list_A2130511660le_alt > list_A2130511660le_alt > $o, A0 : list_A2130511660le_alt, A1 : list_A2130511660le_alt]: ((![X_1 : list_A2130511660le_alt]: (P @ nil_Ar277507244le_alt @ X_1)) => ((![X4 : arrow_1857593510le_alt, Xs2 : list_A2130511660le_alt, Ys : list_A2130511660le_alt]: ((P @ Ys @ Xs2) => (P @ (cons_A1864255580le_alt @ X4 @ Xs2) @ Ys))) => (P @ A0 @ A1)))))). % splice.induct
thf(fact_148_list__induct2_H, axiom,
    ((![P : list_A2130511660le_alt > list_A2130511660le_alt > $o, Xs : list_A2130511660le_alt, Ys2 : list_A2130511660le_alt]: ((P @ nil_Ar277507244le_alt @ nil_Ar277507244le_alt) => ((![X4 : arrow_1857593510le_alt, Xs2 : list_A2130511660le_alt]: (P @ (cons_A1864255580le_alt @ X4 @ Xs2) @ nil_Ar277507244le_alt)) => ((![Y4 : arrow_1857593510le_alt, Ys : list_A2130511660le_alt]: (P @ nil_Ar277507244le_alt @ (cons_A1864255580le_alt @ Y4 @ Ys))) => ((![X4 : arrow_1857593510le_alt, Xs2 : list_A2130511660le_alt, Y4 : arrow_1857593510le_alt, Ys : list_A2130511660le_alt]: ((P @ Xs2 @ Ys) => (P @ (cons_A1864255580le_alt @ X4 @ Xs2) @ (cons_A1864255580le_alt @ Y4 @ Ys)))) => (P @ Xs @ Ys2)))))))). % list_induct2'

% 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,
    ((![X2 : set_Pr367596371le_alt, Y : set_Pr367596371le_alt]: ((if_set550155277le_alt @ $false @ X2 @ Y) = Y)))).
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,
    ((![X2 : set_Pr367596371le_alt, Y : set_Pr367596371le_alt]: ((if_set550155277le_alt @ $true @ X2 @ Y) = X2)))).

% Conjectures (1)
thf(conj_0, conjecture,
    ((~ (((member2048039092le_alt @ (produc1494124311le_alt @ c @ d) @ (f @ p)) = (~ ((member2048039092le_alt @ (produc1494124311le_alt @ c @ d) @ (f @ (^[I2 : arrow_1429744205e_indi]: (if_set550155277le_alt @ (ord_less_nat @ (h @ I2) @ n) @ (arrow_992294841_mktop @ (p @ I2) @ e) @ (if_set550155277le_alt @ ((h @ I2) = n) @ (arrow_1726226719_above @ (p @ I2) @ c @ e) @ (arrow_843587755_mkbot @ (p @ I2) @ e))))))))))))).
