% 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_308__5189586_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:17:27.056

% Could-be-implicit typings (9)
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__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J, type,
    produc16571293le_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 (45)
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_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_Groups_Oone__class_Oone_001t__Nat__Onat, type,
    one_one_nat : nat).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat, type,
    plus_plus_nat : nat > nat > nat).
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_Orderings_Oord__class_Oless_001t__Nat__Onat, type,
    ord_less_nat : nat > nat > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat, type,
    ord_less_eq_nat : nat > nat > $o).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Arrow____Order____Mirabelle____riepwfubkl__Oindi_M_Eo_J, type,
    top_to1473733010indi_o : arrow_1429744205e_indi > $o).
thf(sy_c_Orderings_Otop__class_Otop_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_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_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_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_a____, type,
    a : arrow_1857593510le_alt).
thf(sy_v_b____, type,
    b : arrow_1857593510le_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 (152)
thf(fact_0__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_1__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_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_062a_A_060_092_060_094bsub_062Lab_092_060_094esub_062_Ab_092_060close_062, axiom,
    ((member2048039092le_alt @ (produc1494124311le_alt @ a @ b) @ lab))). % \<open>a <\<^bsub>Lab\<^esub> b\<close>
thf(fact_5__092_060open_062b_A_060_092_060_094bsub_062Lba_092_060_094esub_062_Aa_092_060close_062, axiom,
    ((member2048039092le_alt @ (produc1494124311le_alt @ b @ a) @ lba))). % \<open>b <\<^bsub>Lba\<^esub> a\<close>
thf(fact_6__092_060open_062c_A_092_060noteq_062_Ad_092_060close_062, axiom,
    ((~ ((c = d))))). % \<open>c \<noteq> d\<close>
thf(fact_7__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_8_n_I2_J, axiom,
    ((![M : nat]: ((ord_less_eq_nat @ M @ n) => (member2048039092le_alt @ (produc1494124311le_alt @ b @ a) @ (f @ (^[I : arrow_1429744205e_indi]: (if_set550155277le_alt @ (ord_less_nat @ (h @ I) @ M) @ lab @ lba)))))))). % n(2)
thf(fact_9__092_060open_062_Ie_A_060_092_060_094bsub_062F_A_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_092_060_094esub_062_Ad_J_A_061_A_Ib_A_060_092_060_094bsub_062F_A_I_092_060lambda_062i_O_Aif_Ah_Ai_A_060_An_Athen_ALab_Aelse_ALba_J_092_060_094esub_062_Aa_J_092_060close_062, axiom,
    (((member2048039092le_alt @ (produc1494124311le_alt @ e @ d) @ (f @ (^[I : arrow_1429744205e_indi]: (if_set550155277le_alt @ (ord_less_nat @ (h @ I) @ n) @ (arrow_992294841_mktop @ (p @ I) @ e) @ (if_set550155277le_alt @ ((h @ I) = n) @ (arrow_1726226719_above @ (p @ I) @ c @ e) @ (arrow_843587755_mkbot @ (p @ I) @ e)))))) = (member2048039092le_alt @ (produc1494124311le_alt @ b @ a) @ (f @ (^[I : arrow_1429744205e_indi]: (if_set550155277le_alt @ (ord_less_nat @ (h @ I) @ n) @ lab @ lba))))))). % \<open>(e <\<^bsub>F (\<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)\<^esub> d) = (b <\<^bsub>F (\<lambda>i. if h i < n then Lab else Lba)\<^esub> a)\<close>
thf(fact_10__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,
    ((![I2 : arrow_1429744205e_indi]: ((member2048039092le_alt @ (produc1494124311le_alt @ c @ d) @ (p @ I2)) = (((((ord_less_nat @ (h @ I2) @ n)) => ((member2048039092le_alt @ (produc1494124311le_alt @ c @ d) @ (arrow_992294841_mktop @ (p @ I2) @ e))))) & ((((~ ((ord_less_nat @ (h @ I2) @ n)))) => (((((((h @ I2) = n)) => ((member2048039092le_alt @ (produc1494124311le_alt @ c @ d) @ (arrow_1726226719_above @ (p @ I2) @ c @ e))))) & ((((~ (((h @ I2) = n)))) => ((member2048039092le_alt @ (produc1494124311le_alt @ c @ d) @ (arrow_843587755_mkbot @ (p @ I2) @ 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_11__092_060open_062c_A_060_092_060_094bsub_062F_A_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_092_060_094esub_062_Ae_092_060close_062, axiom,
    ((member2048039092le_alt @ (produc1494124311le_alt @ c @ e) @ (f @ (^[I : arrow_1429744205e_indi]: (if_set550155277le_alt @ (ord_less_nat @ (h @ I) @ n) @ (arrow_992294841_mktop @ (p @ I) @ e) @ (if_set550155277le_alt @ ((h @ I) = n) @ (arrow_1726226719_above @ (p @ I) @ c @ e) @ (arrow_843587755_mkbot @ (p @ I) @ e)))))))). % \<open>c <\<^bsub>F (\<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)\<^esub> e\<close>
thf(fact_12_PW, axiom,
    (((member2048039092le_alt @ (produc1494124311le_alt @ c @ d) @ (f @ p)) = (member2048039092le_alt @ (produc1494124311le_alt @ c @ d) @ (f @ (^[I : arrow_1429744205e_indi]: (if_set550155277le_alt @ (ord_less_nat @ (h @ I) @ n) @ (arrow_992294841_mktop @ (p @ I) @ e) @ (if_set550155277le_alt @ ((h @ I) = n) @ (arrow_1726226719_above @ (p @ I) @ c @ e) @ (arrow_843587755_mkbot @ (p @ I) @ e))))))))). % PW
thf(fact_13_in__mkbot, axiom,
    ((![X : arrow_1857593510le_alt, Y : arrow_1857593510le_alt, L : set_Pr367596371le_alt, Z : arrow_1857593510le_alt]: ((member2048039092le_alt @ (produc1494124311le_alt @ X @ Y) @ (arrow_843587755_mkbot @ L @ Z)) = (((~ ((Y = Z)))) & ((((((X = Z)) => ((~ ((X = Y)))))) & ((((~ ((X = Z)))) => ((member2048039092le_alt @ (produc1494124311le_alt @ X @ Y) @ L))))))))))). % in_mkbot
thf(fact_14_in__mktop, axiom,
    ((![X : arrow_1857593510le_alt, Y : arrow_1857593510le_alt, L : set_Pr367596371le_alt, Z : arrow_1857593510le_alt]: ((member2048039092le_alt @ (produc1494124311le_alt @ X @ Y) @ (arrow_992294841_mktop @ L @ Z)) = (((~ ((X = Z)))) & ((((((Y = Z)) => ((~ ((X = Y)))))) & ((((~ ((Y = Z)))) => ((member2048039092le_alt @ (produc1494124311le_alt @ X @ Y) @ L))))))))))). % in_mktop
thf(fact_15__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 @ (^[I : arrow_1429744205e_indi]: (if_set550155277le_alt @ (ord_less_nat @ (h @ I) @ n) @ (arrow_992294841_mktop @ (p @ I) @ e) @ (if_set550155277le_alt @ ((h @ I) = n) @ (arrow_1726226719_above @ (p @ I) @ c @ e) @ (arrow_843587755_mkbot @ (p @ I) @ 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_16__092_060open_062_092_060forall_062i_O_A_Ie_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_A_061_A_Ib_A_060_092_060_094bsub_062_Iif_Ah_Ai_A_060_An_Athen_ALab_Aelse_ALba_J_092_060_094esub_062_Aa_J_092_060close_062, axiom,
    ((![I2 : arrow_1429744205e_indi]: ((((((ord_less_nat @ (h @ I2) @ n)) => ((member2048039092le_alt @ (produc1494124311le_alt @ e @ d) @ (arrow_992294841_mktop @ (p @ I2) @ e))))) & ((((~ ((ord_less_nat @ (h @ I2) @ n)))) => (((((((h @ I2) = n)) => ((member2048039092le_alt @ (produc1494124311le_alt @ e @ d) @ (arrow_1726226719_above @ (p @ I2) @ c @ e))))) & ((((~ (((h @ I2) = n)))) => ((member2048039092le_alt @ (produc1494124311le_alt @ e @ d) @ (arrow_843587755_mkbot @ (p @ I2) @ e)))))))))) = (((((ord_less_nat @ (h @ I2) @ n)) => ((member2048039092le_alt @ (produc1494124311le_alt @ b @ a) @ lab)))) & ((((~ ((ord_less_nat @ (h @ I2) @ n)))) => ((member2048039092le_alt @ (produc1494124311le_alt @ b @ a) @ lba))))))))). % \<open>\<forall>i. (e <\<^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) = (b <\<^bsub>(if h i < n then Lab else Lba)\<^esub> a)\<close>
thf(fact_17__C1_C, axiom,
    ((![A : arrow_1857593510le_alt, B : arrow_1857593510le_alt, A2 : arrow_1857593510le_alt, B2 : arrow_1857593510le_alt, P : arrow_1429744205e_indi > set_Pr367596371le_alt, P2 : arrow_1429744205e_indi > set_Pr367596371le_alt]: ((~ ((A = B))) => ((~ ((A2 = B2))) => ((~ ((A = B2))) => ((~ ((B = A2))) => ((member684274596le_alt @ P @ arrow_1951607831e_Prof) => ((member684274596le_alt @ P2 @ arrow_1951607831e_Prof) => ((![I3 : arrow_1429744205e_indi]: ((member2048039092le_alt @ (produc1494124311le_alt @ A @ B) @ (P @ I3)) = (member2048039092le_alt @ (produc1494124311le_alt @ A2 @ B2) @ (P2 @ I3)))) => ((member2048039092le_alt @ (produc1494124311le_alt @ A @ B) @ (f @ P)) => (member2048039092le_alt @ (produc1494124311le_alt @ A2 @ B2) @ (f @ P2))))))))))))). % "1"
thf(fact_18__C2_C, axiom,
    ((![A : arrow_1857593510le_alt, B : arrow_1857593510le_alt, A2 : arrow_1857593510le_alt, B2 : arrow_1857593510le_alt, P : arrow_1429744205e_indi > set_Pr367596371le_alt, P2 : arrow_1429744205e_indi > set_Pr367596371le_alt]: ((~ ((A = B))) => ((~ ((A2 = B2))) => ((~ ((A = B2))) => ((~ ((B = A2))) => ((member684274596le_alt @ P @ arrow_1951607831e_Prof) => ((member684274596le_alt @ P2 @ arrow_1951607831e_Prof) => ((![I3 : arrow_1429744205e_indi]: ((member2048039092le_alt @ (produc1494124311le_alt @ A @ B) @ (P @ I3)) = (member2048039092le_alt @ (produc1494124311le_alt @ A2 @ B2) @ (P2 @ I3)))) => ((member2048039092le_alt @ (produc1494124311le_alt @ A @ B) @ (f @ P)) = (member2048039092le_alt @ (produc1494124311le_alt @ A2 @ B2) @ (f @ P2))))))))))))). % "2"
thf(fact_19__C3_C, axiom,
    ((![A : arrow_1857593510le_alt, B : arrow_1857593510le_alt, P : arrow_1429744205e_indi > set_Pr367596371le_alt, P2 : arrow_1429744205e_indi > set_Pr367596371le_alt]: ((~ ((A = B))) => ((member684274596le_alt @ P @ arrow_1951607831e_Prof) => ((member684274596le_alt @ P2 @ arrow_1951607831e_Prof) => ((![I3 : arrow_1429744205e_indi]: ((member2048039092le_alt @ (produc1494124311le_alt @ A @ B) @ (P @ I3)) = (member2048039092le_alt @ (produc1494124311le_alt @ B @ A) @ (P2 @ I3)))) => ((member2048039092le_alt @ (produc1494124311le_alt @ A @ B) @ (f @ P)) = (member2048039092le_alt @ (produc1494124311le_alt @ B @ A) @ (f @ P2)))))))))). % "3"
thf(fact_20__C4_C, axiom,
    ((![A : arrow_1857593510le_alt, B : arrow_1857593510le_alt, C : arrow_1857593510le_alt, P : arrow_1429744205e_indi > set_Pr367596371le_alt, P2 : arrow_1429744205e_indi > set_Pr367596371le_alt]: ((~ ((A = B))) => ((~ ((B = C))) => ((~ ((A = C))) => ((member684274596le_alt @ P @ arrow_1951607831e_Prof) => ((member684274596le_alt @ P2 @ arrow_1951607831e_Prof) => ((![I3 : arrow_1429744205e_indi]: ((member2048039092le_alt @ (produc1494124311le_alt @ A @ B) @ (P @ I3)) = (member2048039092le_alt @ (produc1494124311le_alt @ B @ C) @ (P2 @ I3)))) => ((member2048039092le_alt @ (produc1494124311le_alt @ A @ B) @ (f @ P)) = (member2048039092le_alt @ (produc1494124311le_alt @ B @ C) @ (f @ P2)))))))))))). % "4"
thf(fact_21_pairwise__neutrality, axiom,
    ((![A : arrow_1857593510le_alt, B : arrow_1857593510le_alt, A2 : arrow_1857593510le_alt, B2 : arrow_1857593510le_alt, P : arrow_1429744205e_indi > set_Pr367596371le_alt, P2 : arrow_1429744205e_indi > set_Pr367596371le_alt]: ((~ ((A = B))) => ((~ ((A2 = B2))) => ((member684274596le_alt @ P @ arrow_1951607831e_Prof) => ((member684274596le_alt @ P2 @ arrow_1951607831e_Prof) => ((![I3 : arrow_1429744205e_indi]: ((member2048039092le_alt @ (produc1494124311le_alt @ A @ B) @ (P @ I3)) = (member2048039092le_alt @ (produc1494124311le_alt @ A2 @ B2) @ (P2 @ I3)))) => ((member2048039092le_alt @ (produc1494124311le_alt @ A @ B) @ (f @ P)) = (member2048039092le_alt @ (produc1494124311le_alt @ A2 @ B2) @ (f @ P2))))))))))). % pairwise_neutrality
thf(fact_22__092_060open_062_Ia_M_Ab_J_A_092_060notin_062_ALba_092_060close_062, axiom,
    ((~ ((member2048039092le_alt @ (produc1494124311le_alt @ a @ b) @ lba))))). % \<open>(a, b) \<notin> Lba\<close>
thf(fact_23__092_060open_062_Ib_M_Aa_J_A_092_060notin_062_ALab_092_060close_062, axiom,
    ((~ ((member2048039092le_alt @ (produc1494124311le_alt @ b @ a) @ lab))))). % \<open>(b, a) \<notin> Lab\<close>
thf(fact_24__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_25_prod_Oinject, axiom,
    ((![X1 : arrow_1857593510le_alt, X2 : arrow_1857593510le_alt, Y1 : arrow_1857593510le_alt, Y2 : arrow_1857593510le_alt]: (((produc1494124311le_alt @ X1 @ X2) = (produc1494124311le_alt @ Y1 @ Y2)) = (((X1 = Y1)) & ((X2 = Y2))))))). % prod.inject
thf(fact_26_old_Oprod_Oinject, axiom,
    ((![A : arrow_1857593510le_alt, B : arrow_1857593510le_alt, A2 : arrow_1857593510le_alt, B2 : arrow_1857593510le_alt]: (((produc1494124311le_alt @ A @ B) = (produc1494124311le_alt @ A2 @ B2)) = (((A = A2)) & ((B = B2))))))). % old.prod.inject
thf(fact_27__092_060open_062a_A_092_060noteq_062_Ab_092_060close_062, axiom,
    ((~ ((a = b))))). % \<open>a \<noteq> b\<close>
thf(fact_28__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062Lba_O_A_092_060lbrakk_062b_A_060_092_060_094bsub_062Lba_092_060_094esub_062_Aa_059_ALba_A_092_060in_062_ALin_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom,
    ((~ ((![Lba : set_Pr367596371le_alt]: ((member2048039092le_alt @ (produc1494124311le_alt @ b @ a) @ Lba) => (~ ((member1334244458le_alt @ Lba @ arrow_1848678355le_Lin))))))))). % \<open>\<And>thesis. (\<And>Lba. \<lbrakk>b <\<^bsub>Lba\<^esub> a; Lba \<in> Lin\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_29__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062Lab_O_A_092_060lbrakk_062a_A_060_092_060_094bsub_062Lab_092_060_094esub_062_Ab_059_ALab_A_092_060in_062_ALin_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom,
    ((~ ((![Lab : set_Pr367596371le_alt]: ((member2048039092le_alt @ (produc1494124311le_alt @ a @ b) @ Lab) => (~ ((member1334244458le_alt @ Lab @ arrow_1848678355le_Lin))))))))). % \<open>\<And>thesis. (\<And>Lab. \<lbrakk>a <\<^bsub>Lab\<^esub> b; Lab \<in> Lin\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_30_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_31_n_I1_J, axiom,
    ((ord_less_nat @ n @ (finite927127589e_indi @ top_to1799531699e_indi)))). % n(1)
thf(fact_32_injh, axiom,
    ((inj_on528257168di_nat @ h @ top_to1799531699e_indi))). % injh
thf(fact_33_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_34_linear__alt, axiom,
    ((?[L2 : set_Pr367596371le_alt]: (member1334244458le_alt @ L2 @ arrow_1848678355le_Lin)))). % linear_alt
thf(fact_35_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_36_unanimity__def, axiom,
    ((arrow_52334694nimity = (^[F : (arrow_1429744205e_indi > set_Pr367596371le_alt) > set_Pr367596371le_alt]: (![X3 : arrow_1429744205e_indi > set_Pr367596371le_alt]: (((member684274596le_alt @ X3 @ arrow_1951607831e_Prof)) => ((![A3 : arrow_1857593510le_alt]: (![B3 : arrow_1857593510le_alt]: (((![I : arrow_1429744205e_indi]: (member2048039092le_alt @ (produc1494124311le_alt @ A3 @ B3) @ (X3 @ I)))) => ((member2048039092le_alt @ (produc1494124311le_alt @ A3 @ B3) @ (F @ X3))))))))))))). % unanimity_def
thf(fact_37_IIA__def, axiom,
    ((arrow_1821794627le_IIA = (^[F : (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)) => ((![A3 : arrow_1857593510le_alt]: (![B3 : arrow_1857593510le_alt]: (((![I : arrow_1429744205e_indi]: ((member2048039092le_alt @ (produc1494124311le_alt @ A3 @ B3) @ (X3 @ I)) = (member2048039092le_alt @ (produc1494124311le_alt @ A3 @ B3) @ (Y3 @ I))))) => (((member2048039092le_alt @ (produc1494124311le_alt @ A3 @ B3) @ (F @ X3)) = (member2048039092le_alt @ (produc1494124311le_alt @ A3 @ B3) @ (F @ Y3))))))))))))))))). % IIA_def
thf(fact_38_notin__Lin__iff, axiom,
    ((![L : set_Pr367596371le_alt, X : arrow_1857593510le_alt, Y : arrow_1857593510le_alt]: ((member1334244458le_alt @ L @ arrow_1848678355le_Lin) => ((~ ((X = Y))) => ((~ ((member2048039092le_alt @ (produc1494124311le_alt @ X @ Y) @ L))) = (member2048039092le_alt @ (produc1494124311le_alt @ Y @ X) @ L))))))). % notin_Lin_iff
thf(fact_39_complete__Lin, axiom,
    ((![A : arrow_1857593510le_alt, B : arrow_1857593510le_alt]: ((~ ((A = B))) => (?[X4 : set_Pr367596371le_alt]: ((member1334244458le_alt @ X4 @ arrow_1848678355le_Lin) & (member2048039092le_alt @ (produc1494124311le_alt @ A @ B) @ X4))))))). % complete_Lin
thf(fact_40_Lin__irrefl, axiom,
    ((![L : set_Pr367596371le_alt, A : arrow_1857593510le_alt, B : arrow_1857593510le_alt]: ((member1334244458le_alt @ L @ arrow_1848678355le_Lin) => ((member2048039092le_alt @ (produc1494124311le_alt @ A @ B) @ L) => (~ ((member2048039092le_alt @ (produc1494124311le_alt @ B @ A) @ L)))))))). % Lin_irrefl
thf(fact_41_mktop__Lin, axiom,
    ((![L : set_Pr367596371le_alt, X : arrow_1857593510le_alt]: ((member1334244458le_alt @ L @ arrow_1848678355le_Lin) => (member1334244458le_alt @ (arrow_992294841_mktop @ L @ X) @ arrow_1848678355le_Lin))))). % mktop_Lin
thf(fact_42_mkbot__Lin, axiom,
    ((![L : set_Pr367596371le_alt, X : arrow_1857593510le_alt]: ((member1334244458le_alt @ L @ arrow_1848678355le_Lin) => (member1334244458le_alt @ (arrow_843587755_mkbot @ L @ X) @ arrow_1848678355le_Lin))))). % mkbot_Lin
thf(fact_43_above__Lin, axiom,
    ((![X : arrow_1857593510le_alt, Y : arrow_1857593510le_alt, L : set_Pr367596371le_alt]: ((~ ((X = Y))) => ((member1334244458le_alt @ L @ arrow_1848678355le_Lin) => (member1334244458le_alt @ (arrow_1726226719_above @ L @ X @ Y) @ arrow_1848678355le_Lin)))))). % above_Lin
thf(fact_44_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_45_mem__Collect__eq, axiom,
    ((![A : produc16571293le_alt, P : produc16571293le_alt > $o]: ((member2048039092le_alt @ A @ (collec531981554le_alt @ P)) = (P @ A))))). % mem_Collect_eq
thf(fact_46_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_47_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_48_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_49_Collect__mem__eq, axiom,
    ((![A5 : set_Pr367596371le_alt]: ((collec531981554le_alt @ (^[X3 : produc16571293le_alt]: (member2048039092le_alt @ X3 @ A5))) = A5)))). % Collect_mem_eq
thf(fact_50_Collect__mem__eq, axiom,
    ((![A5 : set_se2071012361le_alt]: ((collec1399441576le_alt @ (^[X3 : set_Pr367596371le_alt]: (member1334244458le_alt @ X3 @ A5))) = A5)))). % Collect_mem_eq
thf(fact_51_Collect__mem__eq, axiom,
    ((![A5 : set_Ar809243995le_alt]: ((collec1559089382le_alt @ (^[X3 : arrow_1429744205e_indi > set_Pr367596371le_alt]: (member684274596le_alt @ X3 @ A5))) = A5)))). % Collect_mem_eq
thf(fact_52_Collect__mem__eq, axiom,
    ((![A5 : set_Ar182050865le_alt]: ((collec1382217680le_alt @ (^[X3 : (arrow_1429744205e_indi > set_Pr367596371le_alt) > set_Pr367596371le_alt]: (member183760530le_alt @ X3 @ A5))) = A5)))). % Collect_mem_eq
thf(fact_53_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_54_Pair__inject, axiom,
    ((![A : arrow_1857593510le_alt, B : arrow_1857593510le_alt, A2 : arrow_1857593510le_alt, B2 : arrow_1857593510le_alt]: (((produc1494124311le_alt @ A @ B) = (produc1494124311le_alt @ A2 @ B2)) => (~ (((A = A2) => (~ ((B = B2)))))))))). % Pair_inject
thf(fact_55_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_56_surj__pair, axiom,
    ((![P4 : produc16571293le_alt]: (?[X4 : arrow_1857593510le_alt, Y4 : arrow_1857593510le_alt]: (P4 = (produc1494124311le_alt @ X4 @ Y4)))))). % surj_pair
thf(fact_57_in__above, axiom,
    ((![A : arrow_1857593510le_alt, B : arrow_1857593510le_alt, L : set_Pr367596371le_alt, X : arrow_1857593510le_alt, Y : arrow_1857593510le_alt]: ((~ ((A = B))) => ((member1334244458le_alt @ L @ arrow_1848678355le_Lin) => ((member2048039092le_alt @ (produc1494124311le_alt @ X @ Y) @ (arrow_1726226719_above @ L @ A @ B)) = (((~ ((X = Y)))) & ((((((X = B)) => ((member2048039092le_alt @ (produc1494124311le_alt @ A @ Y) @ L)))) & ((((~ ((X = B)))) => ((((((Y = B)) => ((((X = A)) | ((member2048039092le_alt @ (produc1494124311le_alt @ X @ A) @ L)))))) & ((((~ ((Y = B)))) => ((member2048039092le_alt @ (produc1494124311le_alt @ X @ Y) @ L))))))))))))))))). % in_above
thf(fact_58_inv__identity, axiom,
    (((hilber700257104e_indi @ top_to1799531699e_indi @ (^[A3 : arrow_1429744205e_indi]: A3)) = (^[A3 : arrow_1429744205e_indi]: A3)))). % inv_identity
thf(fact_59_n_I3_J, axiom,
    ((member2048039092le_alt @ (produc1494124311le_alt @ a @ b) @ (f @ (^[I : arrow_1429744205e_indi]: (if_set550155277le_alt @ (ord_less_nat @ (h @ I) @ (plus_plus_nat @ n @ one_one_nat)) @ lab @ lba)))))). % n(3)
thf(fact_60__092_060open_062_Ic_A_060_092_060_094bsub_062F_A_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_092_060_094esub_062_Ae_J_A_061_A_Ia_A_060_092_060_094bsub_062F_A_I_092_060lambda_062i_O_Aif_Ah_Ai_A_060_An_A_L_A1_Athen_ALab_Aelse_ALba_J_092_060_094esub_062_Ab_J_092_060close_062, axiom,
    (((member2048039092le_alt @ (produc1494124311le_alt @ c @ e) @ (f @ (^[I : arrow_1429744205e_indi]: (if_set550155277le_alt @ (ord_less_nat @ (h @ I) @ n) @ (arrow_992294841_mktop @ (p @ I) @ e) @ (if_set550155277le_alt @ ((h @ I) = n) @ (arrow_1726226719_above @ (p @ I) @ c @ e) @ (arrow_843587755_mkbot @ (p @ I) @ e)))))) = (member2048039092le_alt @ (produc1494124311le_alt @ a @ b) @ (f @ (^[I : arrow_1429744205e_indi]: (if_set550155277le_alt @ (ord_less_nat @ (h @ I) @ (plus_plus_nat @ n @ one_one_nat)) @ lab @ lba))))))). % \<open>(c <\<^bsub>F (\<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)\<^esub> e) = (a <\<^bsub>F (\<lambda>i. if h i < n + 1 then Lab else Lba)\<^esub> b)\<close>
thf(fact_61__092_060open_062_092_060forall_062i_O_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_Ae_J_A_061_A_Ia_A_060_092_060_094bsub_062_Iif_Ah_Ai_A_060_An_A_L_A1_Athen_ALab_Aelse_ALba_J_092_060_094esub_062_Ab_J_092_060close_062, axiom,
    ((![I2 : arrow_1429744205e_indi]: ((((((ord_less_nat @ (h @ I2) @ n)) => ((member2048039092le_alt @ (produc1494124311le_alt @ c @ e) @ (arrow_992294841_mktop @ (p @ I2) @ e))))) & ((((~ ((ord_less_nat @ (h @ I2) @ n)))) => (((((((h @ I2) = n)) => ((member2048039092le_alt @ (produc1494124311le_alt @ c @ e) @ (arrow_1726226719_above @ (p @ I2) @ c @ e))))) & ((((~ (((h @ I2) = n)))) => ((member2048039092le_alt @ (produc1494124311le_alt @ c @ e) @ (arrow_843587755_mkbot @ (p @ I2) @ e)))))))))) = (((((ord_less_nat @ (h @ I2) @ (plus_plus_nat @ n @ one_one_nat))) => ((member2048039092le_alt @ (produc1494124311le_alt @ a @ b) @ lab)))) & ((((~ ((ord_less_nat @ (h @ I2) @ (plus_plus_nat @ n @ one_one_nat))))) => ((member2048039092le_alt @ (produc1494124311le_alt @ a @ b) @ lba))))))))). % \<open>\<forall>i. (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> e) = (a <\<^bsub>(if h i < n + 1 then Lab else Lba)\<^esub> b)\<close>
thf(fact_62_UNIV__I, axiom,
    ((![X : produc16571293le_alt]: (member2048039092le_alt @ X @ top_to224369155le_alt)))). % UNIV_I
thf(fact_63_UNIV__I, axiom,
    ((![X : set_Pr367596371le_alt]: (member1334244458le_alt @ X @ top_to469035705le_alt)))). % UNIV_I
thf(fact_64_UNIV__I, axiom,
    ((![X : arrow_1429744205e_indi > set_Pr367596371le_alt]: (member684274596le_alt @ X @ top_to685525675le_alt)))). % UNIV_I
thf(fact_65_UNIV__I, axiom,
    ((![X : (arrow_1429744205e_indi > set_Pr367596371le_alt) > set_Pr367596371le_alt]: (member183760530le_alt @ X @ top_to803745505le_alt)))). % UNIV_I
thf(fact_66_UNIV__I, axiom,
    ((![X : arrow_1429744205e_indi]: (member1966420836e_indi @ X @ top_to1799531699e_indi)))). % UNIV_I
thf(fact_67_iso__tuple__UNIV__I, axiom,
    ((![X : produc16571293le_alt]: (member2048039092le_alt @ X @ top_to224369155le_alt)))). % iso_tuple_UNIV_I
thf(fact_68_iso__tuple__UNIV__I, axiom,
    ((![X : set_Pr367596371le_alt]: (member1334244458le_alt @ X @ top_to469035705le_alt)))). % iso_tuple_UNIV_I
thf(fact_69_iso__tuple__UNIV__I, axiom,
    ((![X : arrow_1429744205e_indi > set_Pr367596371le_alt]: (member684274596le_alt @ X @ top_to685525675le_alt)))). % iso_tuple_UNIV_I
thf(fact_70_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_71_iso__tuple__UNIV__I, axiom,
    ((![X : arrow_1429744205e_indi]: (member1966420836e_indi @ X @ top_to1799531699e_indi)))). % iso_tuple_UNIV_I
thf(fact_72_order__refl, axiom,
    ((![X : nat]: (ord_less_eq_nat @ X @ X)))). % order_refl
thf(fact_73_inv__into__f__f, axiom,
    ((![F2 : arrow_1429744205e_indi > nat, A5 : set_Ar1007576579e_indi, X : arrow_1429744205e_indi]: ((inj_on528257168di_nat @ F2 @ A5) => ((member1966420836e_indi @ X @ A5) => ((hilber1586975467di_nat @ A5 @ F2 @ (F2 @ X)) = X)))))). % inv_into_f_f
thf(fact_74__092_060open_062_092_060exists_062n_060N_O_A_I_092_060forall_062m_092_060le_062n_O_Ab_A_060_092_060_094bsub_062F_A_I_092_060lambda_062i_O_Aif_Ah_Ai_A_060_Am_Athen_ALab_Aelse_ALba_J_092_060_094esub_062_Aa_J_A_092_060and_062_Aa_A_060_092_060_094bsub_062F_A_I_092_060lambda_062i_O_Aif_Ah_Ai_A_060_An_A_L_A1_Athen_ALab_Aelse_ALba_J_092_060_094esub_062_Ab_092_060close_062, axiom,
    ((?[N2 : nat]: ((ord_less_nat @ N2 @ (finite927127589e_indi @ top_to1799531699e_indi)) & ((![M : nat]: ((ord_less_eq_nat @ M @ N2) => (member2048039092le_alt @ (produc1494124311le_alt @ b @ a) @ (f @ (^[I : arrow_1429744205e_indi]: (if_set550155277le_alt @ (ord_less_nat @ (h @ I) @ M) @ lab @ lba)))))) & (member2048039092le_alt @ (produc1494124311le_alt @ a @ b) @ (f @ (^[I : arrow_1429744205e_indi]: (if_set550155277le_alt @ (ord_less_nat @ (h @ I) @ (plus_plus_nat @ N2 @ one_one_nat)) @ lab @ lba))))))))). % \<open>\<exists>n<N. (\<forall>m\<le>n. b <\<^bsub>F (\<lambda>i. if h i < m then Lab else Lba)\<^esub> a) \<and> a <\<^bsub>F (\<lambda>i. if h i < n + 1 then Lab else Lba)\<^esub> b\<close>
thf(fact_75__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062n_O_A_092_060lbrakk_062n_A_060_AN_059_A_092_060forall_062m_092_060le_062n_O_Ab_A_060_092_060_094bsub_062F_A_I_092_060lambda_062i_O_Aif_Ah_Ai_A_060_Am_Athen_ALab_Aelse_ALba_J_092_060_094esub_062_Aa_059_Aa_A_060_092_060_094bsub_062F_A_I_092_060lambda_062i_O_Aif_Ah_Ai_A_060_An_A_L_A1_Athen_ALab_Aelse_ALba_J_092_060_094esub_062_Ab_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom,
    ((~ ((![N2 : nat]: ((ord_less_nat @ N2 @ (finite927127589e_indi @ top_to1799531699e_indi)) => ((![M : nat]: ((ord_less_eq_nat @ M @ N2) => (member2048039092le_alt @ (produc1494124311le_alt @ b @ a) @ (f @ (^[I : arrow_1429744205e_indi]: (if_set550155277le_alt @ (ord_less_nat @ (h @ I) @ M) @ lab @ lba)))))) => (~ ((member2048039092le_alt @ (produc1494124311le_alt @ a @ b) @ (f @ (^[I : arrow_1429744205e_indi]: (if_set550155277le_alt @ (ord_less_nat @ (h @ I) @ (plus_plus_nat @ N2 @ one_one_nat)) @ lab @ lba))))))))))))). % \<open>\<And>thesis. (\<And>n. \<lbrakk>n < N; \<forall>m\<le>n. b <\<^bsub>F (\<lambda>i. if h i < m then Lab else Lba)\<^esub> a; a <\<^bsub>F (\<lambda>i. if h i < n + 1 then Lab else Lba)\<^esub> b\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
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,
    ((![F2 : arrow_1429744205e_indi > nat, A5 : set_Ar1007576579e_indi, X : arrow_1429744205e_indi, Y : nat]: ((inj_on528257168di_nat @ F2 @ A5) => ((member1966420836e_indi @ X @ A5) => (((F2 @ X) = Y) => ((hilber1586975467di_nat @ A5 @ F2 @ Y) = X))))))). % inv_into_f_eq
thf(fact_78_inj__imp__inv__eq, axiom,
    ((![F2 : arrow_1429744205e_indi > nat, G : nat > arrow_1429744205e_indi]: ((inj_on528257168di_nat @ F2 @ top_to1799531699e_indi) => ((![X4 : nat]: ((F2 @ (G @ X4)) = X4)) => ((hilber1586975467di_nat @ top_to1799531699e_indi @ F2) = G)))))). % inj_imp_inv_eq
thf(fact_79_inv__f__eq, axiom,
    ((![F2 : arrow_1429744205e_indi > nat, X : arrow_1429744205e_indi, Y : nat]: ((inj_on528257168di_nat @ F2 @ top_to1799531699e_indi) => (((F2 @ X) = Y) => ((hilber1586975467di_nat @ top_to1799531699e_indi @ F2 @ Y) = X)))))). % inv_f_eq
thf(fact_80_inv__f__f, axiom,
    ((![F2 : arrow_1429744205e_indi > nat, X : arrow_1429744205e_indi]: ((inj_on528257168di_nat @ F2 @ top_to1799531699e_indi) => ((hilber1586975467di_nat @ top_to1799531699e_indi @ F2 @ (F2 @ X)) = X))))). % inv_f_f
thf(fact_81_dual__order_Oantisym, axiom,
    ((![B : nat, A : nat]: ((ord_less_eq_nat @ B @ A) => ((ord_less_eq_nat @ A @ B) => (A = B)))))). % dual_order.antisym
thf(fact_82_dual__order_Oeq__iff, axiom,
    (((^[Y5 : nat]: (^[Z2 : nat]: (Y5 = Z2))) = (^[A3 : nat]: (^[B3 : nat]: (((ord_less_eq_nat @ B3 @ A3)) & ((ord_less_eq_nat @ A3 @ B3)))))))). % dual_order.eq_iff
thf(fact_83_dual__order_Otrans, axiom,
    ((![B : nat, A : nat, C : nat]: ((ord_less_eq_nat @ B @ A) => ((ord_less_eq_nat @ C @ B) => (ord_less_eq_nat @ C @ A)))))). % dual_order.trans
thf(fact_84_linorder__wlog, axiom,
    ((![P : nat > nat > $o, A : nat, B : nat]: ((![A4 : nat, B4 : nat]: ((ord_less_eq_nat @ A4 @ B4) => (P @ A4 @ B4))) => ((![A4 : nat, B4 : nat]: ((P @ B4 @ A4) => (P @ A4 @ B4))) => (P @ A @ B)))))). % linorder_wlog
thf(fact_85_dual__order_Orefl, axiom,
    ((![A : nat]: (ord_less_eq_nat @ A @ A)))). % dual_order.refl
thf(fact_86_order__trans, axiom,
    ((![X : nat, Y : nat, Z : nat]: ((ord_less_eq_nat @ X @ Y) => ((ord_less_eq_nat @ Y @ Z) => (ord_less_eq_nat @ X @ Z)))))). % order_trans
thf(fact_87_order__class_Oorder_Oantisym, axiom,
    ((![A : nat, B : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_nat @ B @ A) => (A = B)))))). % order_class.order.antisym
thf(fact_88_ord__le__eq__trans, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ B) => ((B = C) => (ord_less_eq_nat @ A @ C)))))). % ord_le_eq_trans
thf(fact_89_ord__eq__le__trans, axiom,
    ((![A : nat, B : nat, C : nat]: ((A = B) => ((ord_less_eq_nat @ B @ C) => (ord_less_eq_nat @ A @ C)))))). % ord_eq_le_trans
thf(fact_90_order__class_Oorder_Oeq__iff, axiom,
    (((^[Y5 : nat]: (^[Z2 : nat]: (Y5 = Z2))) = (^[A3 : nat]: (^[B3 : nat]: (((ord_less_eq_nat @ A3 @ B3)) & ((ord_less_eq_nat @ B3 @ A3)))))))). % order_class.order.eq_iff
thf(fact_91_antisym__conv, axiom,
    ((![Y : nat, X : nat]: ((ord_less_eq_nat @ Y @ X) => ((ord_less_eq_nat @ X @ Y) = (X = Y)))))). % antisym_conv
thf(fact_92_le__cases3, axiom,
    ((![X : nat, Y : nat, Z : nat]: (((ord_less_eq_nat @ X @ Y) => (~ ((ord_less_eq_nat @ Y @ Z)))) => (((ord_less_eq_nat @ Y @ X) => (~ ((ord_less_eq_nat @ X @ Z)))) => (((ord_less_eq_nat @ X @ Z) => (~ ((ord_less_eq_nat @ Z @ Y)))) => (((ord_less_eq_nat @ Z @ Y) => (~ ((ord_less_eq_nat @ Y @ X)))) => (((ord_less_eq_nat @ Y @ Z) => (~ ((ord_less_eq_nat @ Z @ X)))) => (~ (((ord_less_eq_nat @ Z @ X) => (~ ((ord_less_eq_nat @ X @ Y)))))))))))))). % le_cases3
thf(fact_93_order_Otrans, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_nat @ B @ C) => (ord_less_eq_nat @ A @ C)))))). % order.trans
thf(fact_94_le__cases, axiom,
    ((![X : nat, Y : nat]: ((~ ((ord_less_eq_nat @ X @ Y))) => (ord_less_eq_nat @ Y @ X))))). % le_cases
thf(fact_95_eq__refl, axiom,
    ((![X : nat, Y : nat]: ((X = Y) => (ord_less_eq_nat @ X @ Y))))). % eq_refl
thf(fact_96_linear, axiom,
    ((![X : nat, Y : nat]: ((ord_less_eq_nat @ X @ Y) | (ord_less_eq_nat @ Y @ X))))). % linear
thf(fact_97_antisym, axiom,
    ((![X : nat, Y : nat]: ((ord_less_eq_nat @ X @ Y) => ((ord_less_eq_nat @ Y @ X) => (X = Y)))))). % antisym
thf(fact_98_eq__iff, axiom,
    (((^[Y5 : nat]: (^[Z2 : nat]: (Y5 = Z2))) = (^[X3 : nat]: (^[Y3 : nat]: (((ord_less_eq_nat @ X3 @ Y3)) & ((ord_less_eq_nat @ Y3 @ X3)))))))). % eq_iff
thf(fact_99_ord__le__eq__subst, axiom,
    ((![A : nat, B : nat, F2 : nat > nat, C : nat]: ((ord_less_eq_nat @ A @ B) => (((F2 @ B) = C) => ((![X4 : nat, Y4 : nat]: ((ord_less_eq_nat @ X4 @ Y4) => (ord_less_eq_nat @ (F2 @ X4) @ (F2 @ Y4)))) => (ord_less_eq_nat @ (F2 @ A) @ C))))))). % ord_le_eq_subst
thf(fact_100_ord__eq__le__subst, axiom,
    ((![A : nat, F2 : nat > nat, B : nat, C : nat]: ((A = (F2 @ B)) => ((ord_less_eq_nat @ B @ C) => ((![X4 : nat, Y4 : nat]: ((ord_less_eq_nat @ X4 @ Y4) => (ord_less_eq_nat @ (F2 @ X4) @ (F2 @ Y4)))) => (ord_less_eq_nat @ A @ (F2 @ C)))))))). % ord_eq_le_subst
thf(fact_101_order__subst2, axiom,
    ((![A : nat, B : nat, F2 : nat > nat, C : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_nat @ (F2 @ B) @ C) => ((![X4 : nat, Y4 : nat]: ((ord_less_eq_nat @ X4 @ Y4) => (ord_less_eq_nat @ (F2 @ X4) @ (F2 @ Y4)))) => (ord_less_eq_nat @ (F2 @ A) @ C))))))). % order_subst2
thf(fact_102_order__subst1, axiom,
    ((![A : nat, F2 : nat > nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ (F2 @ B)) => ((ord_less_eq_nat @ B @ C) => ((![X4 : nat, Y4 : nat]: ((ord_less_eq_nat @ X4 @ Y4) => (ord_less_eq_nat @ (F2 @ X4) @ (F2 @ Y4)))) => (ord_less_eq_nat @ A @ (F2 @ C)))))))). % order_subst1
thf(fact_103_ord__eq__less__subst, axiom,
    ((![A : nat, F2 : nat > nat, B : nat, C : nat]: ((A = (F2 @ B)) => ((ord_less_nat @ B @ C) => ((![X4 : nat, Y4 : nat]: ((ord_less_nat @ X4 @ Y4) => (ord_less_nat @ (F2 @ X4) @ (F2 @ Y4)))) => (ord_less_nat @ A @ (F2 @ C)))))))). % ord_eq_less_subst
thf(fact_104_ord__less__eq__subst, axiom,
    ((![A : nat, B : nat, F2 : nat > nat, C : nat]: ((ord_less_nat @ A @ B) => (((F2 @ B) = C) => ((![X4 : nat, Y4 : nat]: ((ord_less_nat @ X4 @ Y4) => (ord_less_nat @ (F2 @ X4) @ (F2 @ Y4)))) => (ord_less_nat @ (F2 @ A) @ C))))))). % ord_less_eq_subst
thf(fact_105_order__less__subst1, axiom,
    ((![A : nat, F2 : nat > nat, B : nat, C : nat]: ((ord_less_nat @ A @ (F2 @ B)) => ((ord_less_nat @ B @ C) => ((![X4 : nat, Y4 : nat]: ((ord_less_nat @ X4 @ Y4) => (ord_less_nat @ (F2 @ X4) @ (F2 @ Y4)))) => (ord_less_nat @ A @ (F2 @ C)))))))). % order_less_subst1
thf(fact_106_order__less__subst2, axiom,
    ((![A : nat, B : nat, F2 : nat > nat, C : nat]: ((ord_less_nat @ A @ B) => ((ord_less_nat @ (F2 @ B) @ C) => ((![X4 : nat, Y4 : nat]: ((ord_less_nat @ X4 @ Y4) => (ord_less_nat @ (F2 @ X4) @ (F2 @ Y4)))) => (ord_less_nat @ (F2 @ A) @ C))))))). % order_less_subst2
thf(fact_107_gt__ex, axiom,
    ((![X : nat]: (?[X_1 : nat]: (ord_less_nat @ X @ X_1))))). % gt_ex
thf(fact_108_neqE, axiom,
    ((![X : nat, Y : nat]: ((~ ((X = Y))) => ((~ ((ord_less_nat @ X @ Y))) => (ord_less_nat @ Y @ X)))))). % neqE
thf(fact_109_neq__iff, axiom,
    ((![X : nat, Y : nat]: ((~ ((X = Y))) = (((ord_less_nat @ X @ Y)) | ((ord_less_nat @ Y @ X))))))). % neq_iff
thf(fact_110_order_Oasym, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (~ ((ord_less_nat @ B @ A))))))). % order.asym
thf(fact_111_less__imp__neq, axiom,
    ((![X : nat, Y : nat]: ((ord_less_nat @ X @ Y) => (~ ((X = Y))))))). % less_imp_neq
thf(fact_112_less__asym, axiom,
    ((![X : nat, Y : nat]: ((ord_less_nat @ X @ Y) => (~ ((ord_less_nat @ Y @ X))))))). % less_asym
thf(fact_113_less__asym_H, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (~ ((ord_less_nat @ B @ A))))))). % less_asym'
thf(fact_114_less__trans, axiom,
    ((![X : nat, Y : nat, Z : nat]: ((ord_less_nat @ X @ Y) => ((ord_less_nat @ Y @ Z) => (ord_less_nat @ X @ Z)))))). % less_trans
thf(fact_115_less__linear, axiom,
    ((![X : nat, Y : nat]: ((ord_less_nat @ X @ Y) | ((X = Y) | (ord_less_nat @ Y @ X)))))). % less_linear
thf(fact_116_less__irrefl, axiom,
    ((![X : nat]: (~ ((ord_less_nat @ X @ X)))))). % less_irrefl
thf(fact_117_ord__eq__less__trans, axiom,
    ((![A : nat, B : nat, C : nat]: ((A = B) => ((ord_less_nat @ B @ C) => (ord_less_nat @ A @ C)))))). % ord_eq_less_trans
thf(fact_118_ord__less__eq__trans, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_nat @ A @ B) => ((B = C) => (ord_less_nat @ A @ C)))))). % ord_less_eq_trans
thf(fact_119_dual__order_Oasym, axiom,
    ((![B : nat, A : nat]: ((ord_less_nat @ B @ A) => (~ ((ord_less_nat @ A @ B))))))). % dual_order.asym
thf(fact_120_less__imp__not__eq, axiom,
    ((![X : nat, Y : nat]: ((ord_less_nat @ X @ Y) => (~ ((X = Y))))))). % less_imp_not_eq
thf(fact_121_less__not__sym, axiom,
    ((![X : nat, Y : nat]: ((ord_less_nat @ X @ Y) => (~ ((ord_less_nat @ Y @ X))))))). % less_not_sym
thf(fact_122_less__induct, axiom,
    ((![P : nat > $o, A : nat]: ((![X4 : nat]: ((![Y6 : nat]: ((ord_less_nat @ Y6 @ X4) => (P @ Y6))) => (P @ X4))) => (P @ A))))). % less_induct
thf(fact_123_antisym__conv3, axiom,
    ((![Y : nat, X : nat]: ((~ ((ord_less_nat @ Y @ X))) => ((~ ((ord_less_nat @ X @ Y))) = (X = Y)))))). % antisym_conv3
thf(fact_124_less__imp__not__eq2, axiom,
    ((![X : nat, Y : nat]: ((ord_less_nat @ X @ Y) => (~ ((Y = X))))))). % less_imp_not_eq2
thf(fact_125_less__imp__triv, axiom,
    ((![X : nat, Y : nat, P : $o]: ((ord_less_nat @ X @ Y) => ((ord_less_nat @ Y @ X) => P))))). % less_imp_triv
thf(fact_126_linorder__cases, axiom,
    ((![X : nat, Y : nat]: ((~ ((ord_less_nat @ X @ Y))) => ((~ ((X = Y))) => (ord_less_nat @ Y @ X)))))). % linorder_cases
thf(fact_127_dual__order_Oirrefl, axiom,
    ((![A : nat]: (~ ((ord_less_nat @ A @ A)))))). % dual_order.irrefl
thf(fact_128_order_Ostrict__trans, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_nat @ A @ B) => ((ord_less_nat @ B @ C) => (ord_less_nat @ A @ C)))))). % order.strict_trans
thf(fact_129_less__imp__not__less, axiom,
    ((![X : nat, Y : nat]: ((ord_less_nat @ X @ Y) => (~ ((ord_less_nat @ Y @ X))))))). % less_imp_not_less
thf(fact_130_exists__least__iff, axiom,
    (((^[P5 : nat > $o]: (?[X5 : nat]: (P5 @ X5))) = (^[P6 : nat > $o]: (?[N3 : nat]: (((P6 @ N3)) & ((![M2 : nat]: (((ord_less_nat @ M2 @ N3)) => ((~ ((P6 @ M2))))))))))))). % exists_least_iff
thf(fact_131_linorder__less__wlog, axiom,
    ((![P : nat > nat > $o, A : nat, B : 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 @ A @ B))))))). % linorder_less_wlog
thf(fact_132_dual__order_Ostrict__trans, axiom,
    ((![B : nat, A : nat, C : nat]: ((ord_less_nat @ B @ A) => ((ord_less_nat @ C @ B) => (ord_less_nat @ C @ A)))))). % dual_order.strict_trans
thf(fact_133_not__less__iff__gr__or__eq, axiom,
    ((![X : nat, Y : nat]: ((~ ((ord_less_nat @ X @ Y))) = (((ord_less_nat @ Y @ X)) | ((X = Y))))))). % not_less_iff_gr_or_eq
thf(fact_134_order_Ostrict__implies__not__eq, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (~ ((A = B))))))). % order.strict_implies_not_eq
thf(fact_135_dual__order_Ostrict__implies__not__eq, axiom,
    ((![B : nat, A : nat]: ((ord_less_nat @ B @ A) => (~ ((A = B))))))). % dual_order.strict_implies_not_eq
thf(fact_136_UNIV__witness, axiom,
    ((?[X4 : produc16571293le_alt]: (member2048039092le_alt @ X4 @ top_to224369155le_alt)))). % UNIV_witness
thf(fact_137_UNIV__witness, axiom,
    ((?[X4 : set_Pr367596371le_alt]: (member1334244458le_alt @ X4 @ top_to469035705le_alt)))). % UNIV_witness
thf(fact_138_UNIV__witness, axiom,
    ((?[X4 : arrow_1429744205e_indi > set_Pr367596371le_alt]: (member684274596le_alt @ X4 @ top_to685525675le_alt)))). % UNIV_witness
thf(fact_139_UNIV__witness, axiom,
    ((?[X4 : (arrow_1429744205e_indi > set_Pr367596371le_alt) > set_Pr367596371le_alt]: (member183760530le_alt @ X4 @ top_to803745505le_alt)))). % UNIV_witness
thf(fact_140_UNIV__witness, axiom,
    ((?[X4 : arrow_1429744205e_indi]: (member1966420836e_indi @ X4 @ top_to1799531699e_indi)))). % UNIV_witness
thf(fact_141_UNIV__eq__I, axiom,
    ((![A5 : set_Pr367596371le_alt]: ((![X4 : produc16571293le_alt]: (member2048039092le_alt @ X4 @ A5)) => (top_to224369155le_alt = A5))))). % UNIV_eq_I
thf(fact_142_UNIV__eq__I, axiom,
    ((![A5 : set_se2071012361le_alt]: ((![X4 : set_Pr367596371le_alt]: (member1334244458le_alt @ X4 @ A5)) => (top_to469035705le_alt = A5))))). % UNIV_eq_I
thf(fact_143_UNIV__eq__I, axiom,
    ((![A5 : set_Ar809243995le_alt]: ((![X4 : arrow_1429744205e_indi > set_Pr367596371le_alt]: (member684274596le_alt @ X4 @ A5)) => (top_to685525675le_alt = A5))))). % UNIV_eq_I
thf(fact_144_UNIV__eq__I, axiom,
    ((![A5 : set_Ar182050865le_alt]: ((![X4 : (arrow_1429744205e_indi > set_Pr367596371le_alt) > set_Pr367596371le_alt]: (member183760530le_alt @ X4 @ A5)) => (top_to803745505le_alt = A5))))). % UNIV_eq_I
thf(fact_145_UNIV__eq__I, axiom,
    ((![A5 : set_Ar1007576579e_indi]: ((![X4 : arrow_1429744205e_indi]: (member1966420836e_indi @ X4 @ A5)) => (top_to1799531699e_indi = A5))))). % UNIV_eq_I
thf(fact_146_UNIV__def, axiom,
    ((top_to1799531699e_indi = (collec1169676194e_indi @ (^[X3 : arrow_1429744205e_indi]: $true))))). % UNIV_def
thf(fact_147_leD, axiom,
    ((![Y : nat, X : nat]: ((ord_less_eq_nat @ Y @ X) => (~ ((ord_less_nat @ X @ Y))))))). % leD
thf(fact_148_leI, axiom,
    ((![X : nat, Y : nat]: ((~ ((ord_less_nat @ X @ Y))) => (ord_less_eq_nat @ Y @ X))))). % leI
thf(fact_149_le__less, axiom,
    ((ord_less_eq_nat = (^[X3 : nat]: (^[Y3 : nat]: (((ord_less_nat @ X3 @ Y3)) | ((X3 = Y3)))))))). % le_less
thf(fact_150_less__le, axiom,
    ((ord_less_nat = (^[X3 : nat]: (^[Y3 : nat]: (((ord_less_eq_nat @ X3 @ Y3)) & ((~ ((X3 = Y3)))))))))). % less_le
thf(fact_151_order__le__less__subst1, axiom,
    ((![A : nat, F2 : nat > nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ (F2 @ B)) => ((ord_less_nat @ B @ C) => ((![X4 : nat, Y4 : nat]: ((ord_less_nat @ X4 @ Y4) => (ord_less_nat @ (F2 @ X4) @ (F2 @ Y4)))) => (ord_less_nat @ A @ (F2 @ C)))))))). % order_le_less_subst1

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

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