% 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_307__5189572_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:17:26.106

% Could-be-implicit typings (10)
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__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 (43)
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_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_ONil_001t__Arrow____Order____Mirabelle____riepwfubkl__Oalt, type,
    nil_Ar277507244le_alt : list_A2130511660le_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_001t__Set__Oset_It__Arrow____Order____Mirabelle____riepwfubkl__Oindi_J, type,
    top_to1799531699e_indi : set_Ar1007576579e_indi).
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__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_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 (138)
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_062a_A_092_060noteq_062_Ab_092_060close_062, axiom,
    ((~ ((a = b))))). % \<open>a \<noteq> 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_PiProf, axiom,
    ((![N : nat]: (member684274596le_alt @ (^[I : arrow_1429744205e_indi]: (if_set550155277le_alt @ (ord_less_nat @ (h @ I) @ N) @ lab @ lba)) @ arrow_1951607831e_Prof)))). % PiProf
thf(fact_6__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_7__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_8__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_9__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_10_dist, axiom,
    ((distin2037765919le_alt @ (cons_A1864255580le_alt @ c @ (cons_A1864255580le_alt @ d @ (cons_A1864255580le_alt @ e @ nil_Ar277507244le_alt)))))). % dist
thf(fact_11__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) => ((![I2 : arrow_1429744205e_indi]: ((member2048039092le_alt @ (produc1494124311le_alt @ A @ B) @ (P @ I2)) = (member2048039092le_alt @ (produc1494124311le_alt @ A2 @ B2) @ (P2 @ I2)))) => ((member2048039092le_alt @ (produc1494124311le_alt @ A @ B) @ (f @ P)) => (member2048039092le_alt @ (produc1494124311le_alt @ A2 @ B2) @ (f @ P2))))))))))))). % "1"
thf(fact_12__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) => ((![I2 : arrow_1429744205e_indi]: ((member2048039092le_alt @ (produc1494124311le_alt @ A @ B) @ (P @ I2)) = (member2048039092le_alt @ (produc1494124311le_alt @ A2 @ B2) @ (P2 @ I2)))) => ((member2048039092le_alt @ (produc1494124311le_alt @ A @ B) @ (f @ P)) = (member2048039092le_alt @ (produc1494124311le_alt @ A2 @ B2) @ (f @ P2))))))))))))). % "2"
thf(fact_13__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) => ((![I2 : arrow_1429744205e_indi]: ((member2048039092le_alt @ (produc1494124311le_alt @ A @ B) @ (P @ I2)) = (member2048039092le_alt @ (produc1494124311le_alt @ B @ A) @ (P2 @ I2)))) => ((member2048039092le_alt @ (produc1494124311le_alt @ A @ B) @ (f @ P)) = (member2048039092le_alt @ (produc1494124311le_alt @ B @ A) @ (f @ P2)))))))))). % "3"
thf(fact_14__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) => ((![I2 : arrow_1429744205e_indi]: ((member2048039092le_alt @ (produc1494124311le_alt @ A @ B) @ (P @ I2)) = (member2048039092le_alt @ (produc1494124311le_alt @ B @ C) @ (P2 @ I2)))) => ((member2048039092le_alt @ (produc1494124311le_alt @ A @ B) @ (f @ P)) = (member2048039092le_alt @ (produc1494124311le_alt @ B @ C) @ (f @ P2)))))))))))). % "4"
thf(fact_15_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) => ((![I2 : arrow_1429744205e_indi]: ((member2048039092le_alt @ (produc1494124311le_alt @ A @ B) @ (P @ I2)) = (member2048039092le_alt @ (produc1494124311le_alt @ A2 @ B2) @ (P2 @ I2)))) => ((member2048039092le_alt @ (produc1494124311le_alt @ A @ B) @ (f @ P)) = (member2048039092le_alt @ (produc1494124311le_alt @ A2 @ B2) @ (f @ P2))))))))))). % pairwise_neutrality
thf(fact_16__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_17__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_18__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_19__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,
    ((![I3 : arrow_1429744205e_indi]: ((((((ord_less_nat @ (h @ I3) @ n)) => ((member2048039092le_alt @ (produc1494124311le_alt @ e @ d) @ (arrow_992294841_mktop @ (p @ I3) @ e))))) & ((((~ ((ord_less_nat @ (h @ I3) @ n)))) => (((((((h @ I3) = n)) => ((member2048039092le_alt @ (produc1494124311le_alt @ e @ d) @ (arrow_1726226719_above @ (p @ I3) @ c @ e))))) & ((((~ (((h @ I3) = n)))) => ((member2048039092le_alt @ (produc1494124311le_alt @ e @ d) @ (arrow_843587755_mkbot @ (p @ I3) @ e)))))))))) = (((((ord_less_nat @ (h @ I3) @ n)) => ((member2048039092le_alt @ (produc1494124311le_alt @ b @ a) @ lab)))) & ((((~ ((ord_less_nat @ (h @ I3) @ 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_20__092_060open_062_092_060lbrakk_062e_A_092_060noteq_062_Ad_059_Ab_A_092_060noteq_062_Aa_059_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_A_092_060in_062_AProf_059_A_I_092_060lambda_062i_O_Aif_Ah_Ai_A_060_An_Athen_ALab_Aelse_ALba_J_A_092_060in_062_AProf_059_A_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_060rbrakk_062_A_092_060Longrightarrow_062_A_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,
    (((~ ((e = d))) => ((~ ((b = a))) => ((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) => ((member684274596le_alt @ (^[I : arrow_1429744205e_indi]: (if_set550155277le_alt @ (ord_less_nat @ (h @ I) @ n) @ lab @ lba)) @ arrow_1951607831e_Prof) => ((![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))))))) => ((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>\<lbrakk>e \<noteq> d; b \<noteq> a; (\<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; (\<lambda>i. if h i < n then Lab else Lba) \<in> Prof; \<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)\<rbrakk> \<Longrightarrow> (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_21__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_22__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_23_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_24_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_25_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_26_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_27__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_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_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_30_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_31__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_32__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,
    ((![I3 : arrow_1429744205e_indi]: ((((((ord_less_nat @ (h @ I3) @ n)) => ((member2048039092le_alt @ (produc1494124311le_alt @ c @ e) @ (arrow_992294841_mktop @ (p @ I3) @ e))))) & ((((~ ((ord_less_nat @ (h @ I3) @ n)))) => (((((((h @ I3) = n)) => ((member2048039092le_alt @ (produc1494124311le_alt @ c @ e) @ (arrow_1726226719_above @ (p @ I3) @ c @ e))))) & ((((~ (((h @ I3) = n)))) => ((member2048039092le_alt @ (produc1494124311le_alt @ c @ e) @ (arrow_843587755_mkbot @ (p @ I3) @ e)))))))))) = (((((ord_less_nat @ (h @ I3) @ (plus_plus_nat @ n @ one_one_nat))) => ((member2048039092le_alt @ (produc1494124311le_alt @ a @ b) @ lab)))) & ((((~ ((ord_less_nat @ (h @ I3) @ (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_33__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_34_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_35_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_36_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_37_in__measure, axiom,
    ((![X : arrow_1857593510le_alt, Y : arrow_1857593510le_alt, F : arrow_1857593510le_alt > nat]: ((member2048039092le_alt @ (produc1494124311le_alt @ X @ Y) @ (measur1447705966le_alt @ F)) = (ord_less_nat @ (F @ X) @ (F @ Y)))))). % in_measure
thf(fact_38_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_39_n_I1_J, axiom,
    ((ord_less_nat @ n @ (finite927127589e_indi @ top_to1799531699e_indi)))). % n(1)
thf(fact_40_injh, axiom,
    ((inj_on528257168di_nat @ h @ top_to1799531699e_indi))). % injh
thf(fact_41_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_42_alt3, axiom,
    ((?[A3 : arrow_1857593510le_alt, B3 : arrow_1857593510le_alt, C2 : arrow_1857593510le_alt]: (distin2037765919le_alt @ (cons_A1864255580le_alt @ A3 @ (cons_A1864255580le_alt @ B3 @ (cons_A1864255580le_alt @ C2 @ nil_Ar277507244le_alt))))))). % alt3
thf(fact_43_third__alt, axiom,
    ((![A : arrow_1857593510le_alt, B : arrow_1857593510le_alt]: ((~ ((A = B))) => (?[C2 : arrow_1857593510le_alt]: (distin2037765919le_alt @ (cons_A1864255580le_alt @ A @ (cons_A1864255580le_alt @ B @ (cons_A1864255580le_alt @ C2 @ nil_Ar277507244le_alt))))))))). % third_alt
thf(fact_44_linear__alt, axiom,
    ((?[L2 : set_Pr367596371le_alt]: (member1334244458le_alt @ L2 @ arrow_1848678355le_Lin)))). % linear_alt
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 : 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_47_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_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,
    ((![A4 : set_Pr367596371le_alt]: ((collec531981554le_alt @ (^[X3 : produc16571293le_alt]: (member2048039092le_alt @ X3 @ A4))) = A4)))). % Collect_mem_eq
thf(fact_50_Collect__mem__eq, axiom,
    ((![A4 : set_Ar809243995le_alt]: ((collec1559089382le_alt @ (^[X3 : arrow_1429744205e_indi > set_Pr367596371le_alt]: (member684274596le_alt @ X3 @ A4))) = A4)))). % Collect_mem_eq
thf(fact_51_Collect__mem__eq, axiom,
    ((![A4 : set_se2071012361le_alt]: ((collec1399441576le_alt @ (^[X3 : set_Pr367596371le_alt]: (member1334244458le_alt @ X3 @ A4))) = A4)))). % Collect_mem_eq
thf(fact_52_Collect__mem__eq, axiom,
    ((![A4 : set_Ar182050865le_alt]: ((collec1382217680le_alt @ (^[X3 : (arrow_1429744205e_indi > set_Pr367596371le_alt) > set_Pr367596371le_alt]: (member183760530le_alt @ X3 @ A4))) = A4)))). % Collect_mem_eq
thf(fact_53_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_54_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_55_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_56_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_57_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_58_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_59_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_60_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]: (![B4 : arrow_1857593510le_alt]: (((![I : arrow_1429744205e_indi]: (member2048039092le_alt @ (produc1494124311le_alt @ A5 @ B4) @ (X3 @ I)))) => ((member2048039092le_alt @ (produc1494124311le_alt @ A5 @ B4) @ (F2 @ X3))))))))))))). % unanimity_def
thf(fact_61_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]: (![B4 : arrow_1857593510le_alt]: (((![I : arrow_1429744205e_indi]: ((member2048039092le_alt @ (produc1494124311le_alt @ A5 @ B4) @ (X3 @ I)) = (member2048039092le_alt @ (produc1494124311le_alt @ A5 @ B4) @ (Y3 @ I))))) => (((member2048039092le_alt @ (produc1494124311le_alt @ A5 @ B4) @ (F2 @ X3)) = (member2048039092le_alt @ (produc1494124311le_alt @ A5 @ B4) @ (F2 @ Y3))))))))))))))))). % IIA_def
thf(fact_62_old_Oprod_Oinducts, axiom,
    ((![P : produc16571293le_alt > $o, Prod : produc16571293le_alt]: ((![A3 : arrow_1857593510le_alt, B3 : arrow_1857593510le_alt]: (P @ (produc1494124311le_alt @ A3 @ B3))) => (P @ Prod))))). % old.prod.inducts
thf(fact_63_old_Oprod_Oexhaust, axiom,
    ((![Y : produc16571293le_alt]: (~ ((![A3 : arrow_1857593510le_alt, B3 : arrow_1857593510le_alt]: (~ ((Y = (produc1494124311le_alt @ A3 @ B3)))))))))). % old.prod.exhaust
thf(fact_64_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_65_prod__cases, axiom,
    ((![P : produc16571293le_alt > $o, P4 : produc16571293le_alt]: ((![A3 : arrow_1857593510le_alt, B3 : arrow_1857593510le_alt]: (P @ (produc1494124311le_alt @ A3 @ B3))) => (P @ P4))))). % prod_cases
thf(fact_66_surj__pair, axiom,
    ((![P4 : produc16571293le_alt]: (?[X4 : arrow_1857593510le_alt, Y4 : arrow_1857593510le_alt]: (P4 = (produc1494124311le_alt @ X4 @ Y4)))))). % surj_pair
thf(fact_67__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_68__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_69_inv__identity, axiom,
    (((hilber700257104e_indi @ top_to1799531699e_indi @ (^[A5 : arrow_1429744205e_indi]: A5)) = (^[A5 : arrow_1429744205e_indi]: A5)))). % inv_identity
thf(fact_70_nat__add__left__cancel__le, axiom,
    ((![K : nat, M2 : nat, N : nat]: ((ord_less_eq_nat @ (plus_plus_nat @ K @ M2) @ (plus_plus_nat @ K @ N)) = (ord_less_eq_nat @ M2 @ N))))). % nat_add_left_cancel_le
thf(fact_71_nat__add__left__cancel__less, axiom,
    ((![K : nat, M2 : nat, N : nat]: ((ord_less_nat @ (plus_plus_nat @ K @ M2) @ (plus_plus_nat @ K @ N)) = (ord_less_nat @ M2 @ N))))). % nat_add_left_cancel_less
thf(fact_72_add__less__cancel__right, axiom,
    ((![A : nat, C : nat, B : nat]: ((ord_less_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ C)) = (ord_less_nat @ A @ B))))). % add_less_cancel_right
thf(fact_73_add__less__cancel__left, axiom,
    ((![C : nat, A : nat, B : nat]: ((ord_less_nat @ (plus_plus_nat @ C @ A) @ (plus_plus_nat @ C @ B)) = (ord_less_nat @ A @ B))))). % add_less_cancel_left
thf(fact_74_add__right__cancel, axiom,
    ((![B : nat, A : nat, C : nat]: (((plus_plus_nat @ B @ A) = (plus_plus_nat @ C @ A)) = (B = C))))). % add_right_cancel
thf(fact_75_add__left__cancel, axiom,
    ((![A : nat, B : nat, C : nat]: (((plus_plus_nat @ A @ B) = (plus_plus_nat @ A @ C)) = (B = C))))). % add_left_cancel
thf(fact_76_add__le__cancel__right, axiom,
    ((![A : nat, C : nat, B : nat]: ((ord_less_eq_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ C)) = (ord_less_eq_nat @ A @ B))))). % add_le_cancel_right
thf(fact_77_add__le__cancel__left, axiom,
    ((![C : nat, A : nat, B : nat]: ((ord_less_eq_nat @ (plus_plus_nat @ C @ A) @ (plus_plus_nat @ C @ B)) = (ord_less_eq_nat @ A @ B))))). % add_le_cancel_left
thf(fact_78_inv__into__f__f, axiom,
    ((![F : arrow_1429744205e_indi > nat, A4 : set_Ar1007576579e_indi, X : arrow_1429744205e_indi]: ((inj_on528257168di_nat @ F @ A4) => ((member1966420836e_indi @ X @ A4) => ((hilber1586975467di_nat @ A4 @ F @ (F @ X)) = X)))))). % inv_into_f_f
thf(fact_79_inv__into__f__eq, axiom,
    ((![F : arrow_1429744205e_indi > nat, A4 : set_Ar1007576579e_indi, X : arrow_1429744205e_indi, Y : nat]: ((inj_on528257168di_nat @ F @ A4) => ((member1966420836e_indi @ X @ A4) => (((F @ X) = Y) => ((hilber1586975467di_nat @ A4 @ F @ Y) = X))))))). % inv_into_f_eq
thf(fact_80_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_81_inv__f__eq, axiom,
    ((![F : arrow_1429744205e_indi > nat, X : arrow_1429744205e_indi, Y : nat]: ((inj_on528257168di_nat @ F @ top_to1799531699e_indi) => (((F @ X) = Y) => ((hilber1586975467di_nat @ top_to1799531699e_indi @ F @ Y) = X)))))). % inv_f_eq
thf(fact_82_inv__f__f, axiom,
    ((![F : arrow_1429744205e_indi > nat, X : arrow_1429744205e_indi]: ((inj_on528257168di_nat @ F @ top_to1799531699e_indi) => ((hilber1586975467di_nat @ top_to1799531699e_indi @ F @ (F @ X)) = X))))). % inv_f_f
thf(fact_83_add__right__imp__eq, axiom,
    ((![B : nat, A : nat, C : nat]: (((plus_plus_nat @ B @ A) = (plus_plus_nat @ C @ A)) => (B = C))))). % add_right_imp_eq
thf(fact_84_add__left__imp__eq, axiom,
    ((![A : nat, B : nat, C : nat]: (((plus_plus_nat @ A @ B) = (plus_plus_nat @ A @ C)) => (B = C))))). % add_left_imp_eq
thf(fact_85_add_Oleft__commute, axiom,
    ((![B : nat, A : nat, C : nat]: ((plus_plus_nat @ B @ (plus_plus_nat @ A @ C)) = (plus_plus_nat @ A @ (plus_plus_nat @ B @ C)))))). % add.left_commute
thf(fact_86_add_Ocommute, axiom,
    ((plus_plus_nat = (^[A5 : nat]: (^[B4 : nat]: (plus_plus_nat @ B4 @ A5)))))). % add.commute
thf(fact_87_add_Oassoc, axiom,
    ((![A : nat, B : nat, C : nat]: ((plus_plus_nat @ (plus_plus_nat @ A @ B) @ C) = (plus_plus_nat @ A @ (plus_plus_nat @ B @ C)))))). % add.assoc
thf(fact_88_group__cancel_Oadd2, axiom,
    ((![B5 : nat, K : nat, B : nat, A : nat]: ((B5 = (plus_plus_nat @ K @ B)) => ((plus_plus_nat @ A @ B5) = (plus_plus_nat @ K @ (plus_plus_nat @ A @ B))))))). % group_cancel.add2
thf(fact_89_group__cancel_Oadd1, axiom,
    ((![A4 : nat, K : nat, A : nat, B : nat]: ((A4 = (plus_plus_nat @ K @ A)) => ((plus_plus_nat @ A4 @ B) = (plus_plus_nat @ K @ (plus_plus_nat @ A @ B))))))). % group_cancel.add1
thf(fact_90_add__mono__thms__linordered__semiring_I4_J, axiom,
    ((![I4 : nat, J : nat, K : nat, L3 : nat]: (((I4 = J) & (K = L3)) => ((plus_plus_nat @ I4 @ K) = (plus_plus_nat @ J @ L3)))))). % add_mono_thms_linordered_semiring(4)
thf(fact_91_ab__semigroup__add__class_Oadd__ac_I1_J, axiom,
    ((![A : nat, B : nat, C : nat]: ((plus_plus_nat @ (plus_plus_nat @ A @ B) @ C) = (plus_plus_nat @ A @ (plus_plus_nat @ B @ C)))))). % ab_semigroup_add_class.add_ac(1)
thf(fact_92_one__reorient, axiom,
    ((![X : nat]: ((one_one_nat = X) = (X = one_one_nat))))). % one_reorient
thf(fact_93_nat__neq__iff, axiom,
    ((![M2 : nat, N : nat]: ((~ ((M2 = N))) = (((ord_less_nat @ M2 @ N)) | ((ord_less_nat @ N @ M2))))))). % nat_neq_iff
thf(fact_94_less__not__refl, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_not_refl
thf(fact_95_less__not__refl2, axiom,
    ((![N : nat, M2 : nat]: ((ord_less_nat @ N @ M2) => (~ ((M2 = N))))))). % less_not_refl2
thf(fact_96_less__not__refl3, axiom,
    ((![S2 : nat, T : nat]: ((ord_less_nat @ S2 @ T) => (~ ((S2 = T))))))). % less_not_refl3
thf(fact_97_less__irrefl__nat, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_irrefl_nat
thf(fact_98_nat__less__induct, axiom,
    ((![P : nat > $o, N : nat]: ((![N2 : nat]: ((![M : nat]: ((ord_less_nat @ M @ N2) => (P @ M))) => (P @ N2))) => (P @ N))))). % nat_less_induct
thf(fact_99_infinite__descent, axiom,
    ((![P : nat > $o, N : nat]: ((![N2 : nat]: ((~ ((P @ N2))) => (?[M : nat]: ((ord_less_nat @ M @ N2) & (~ ((P @ M))))))) => (P @ N))))). % infinite_descent
thf(fact_100_linorder__neqE__nat, axiom,
    ((![X : nat, Y : nat]: ((~ ((X = Y))) => ((~ ((ord_less_nat @ X @ Y))) => (ord_less_nat @ Y @ X)))))). % linorder_neqE_nat
thf(fact_101_Nat_Oex__has__greatest__nat, axiom,
    ((![P : nat > $o, K : nat, B : nat]: ((P @ K) => ((![Y4 : nat]: ((P @ Y4) => (ord_less_eq_nat @ Y4 @ B))) => (?[X4 : nat]: ((P @ X4) & (![Y5 : nat]: ((P @ Y5) => (ord_less_eq_nat @ Y5 @ X4)))))))))). % Nat.ex_has_greatest_nat
thf(fact_102_nat__le__linear, axiom,
    ((![M2 : nat, N : nat]: ((ord_less_eq_nat @ M2 @ N) | (ord_less_eq_nat @ N @ M2))))). % nat_le_linear
thf(fact_103_le__antisym, axiom,
    ((![M2 : nat, N : nat]: ((ord_less_eq_nat @ M2 @ N) => ((ord_less_eq_nat @ N @ M2) => (M2 = N)))))). % le_antisym
thf(fact_104_eq__imp__le, axiom,
    ((![M2 : nat, N : nat]: ((M2 = N) => (ord_less_eq_nat @ M2 @ N))))). % eq_imp_le
thf(fact_105_le__trans, axiom,
    ((![I4 : nat, J : nat, K : nat]: ((ord_less_eq_nat @ I4 @ J) => ((ord_less_eq_nat @ J @ K) => (ord_less_eq_nat @ I4 @ K)))))). % le_trans
thf(fact_106_le__refl, axiom,
    ((![N : nat]: (ord_less_eq_nat @ N @ N)))). % le_refl
thf(fact_107_add__le__imp__le__right, axiom,
    ((![A : nat, C : nat, B : nat]: ((ord_less_eq_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ C)) => (ord_less_eq_nat @ A @ B))))). % add_le_imp_le_right
thf(fact_108_add__le__imp__le__left, axiom,
    ((![C : nat, A : nat, B : nat]: ((ord_less_eq_nat @ (plus_plus_nat @ C @ A) @ (plus_plus_nat @ C @ B)) => (ord_less_eq_nat @ A @ B))))). % add_le_imp_le_left
thf(fact_109_le__iff__add, axiom,
    ((ord_less_eq_nat = (^[A5 : nat]: (^[B4 : nat]: (?[C3 : nat]: (B4 = (plus_plus_nat @ A5 @ C3)))))))). % le_iff_add
thf(fact_110_add__right__mono, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ B) => (ord_less_eq_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ C)))))). % add_right_mono
thf(fact_111_less__eqE, axiom,
    ((![A : nat, B : nat]: ((ord_less_eq_nat @ A @ B) => (~ ((![C2 : nat]: (~ ((B = (plus_plus_nat @ A @ C2))))))))))). % less_eqE
thf(fact_112_add__left__mono, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ B) => (ord_less_eq_nat @ (plus_plus_nat @ C @ A) @ (plus_plus_nat @ C @ B)))))). % add_left_mono
thf(fact_113_add__mono, axiom,
    ((![A : nat, B : nat, C : nat, D : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_nat @ C @ D) => (ord_less_eq_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ D))))))). % add_mono
thf(fact_114_add__mono__thms__linordered__semiring_I1_J, axiom,
    ((![I4 : nat, J : nat, K : nat, L3 : nat]: (((ord_less_eq_nat @ I4 @ J) & (ord_less_eq_nat @ K @ L3)) => (ord_less_eq_nat @ (plus_plus_nat @ I4 @ K) @ (plus_plus_nat @ J @ L3)))))). % add_mono_thms_linordered_semiring(1)
thf(fact_115_add__mono__thms__linordered__semiring_I2_J, axiom,
    ((![I4 : nat, J : nat, K : nat, L3 : nat]: (((I4 = J) & (ord_less_eq_nat @ K @ L3)) => (ord_less_eq_nat @ (plus_plus_nat @ I4 @ K) @ (plus_plus_nat @ J @ L3)))))). % add_mono_thms_linordered_semiring(2)
thf(fact_116_add__mono__thms__linordered__semiring_I3_J, axiom,
    ((![I4 : nat, J : nat, K : nat, L3 : nat]: (((ord_less_eq_nat @ I4 @ J) & (K = L3)) => (ord_less_eq_nat @ (plus_plus_nat @ I4 @ K) @ (plus_plus_nat @ J @ L3)))))). % add_mono_thms_linordered_semiring(3)
thf(fact_117_add__mono__thms__linordered__field_I5_J, axiom,
    ((![I4 : nat, J : nat, K : nat, L3 : nat]: (((ord_less_nat @ I4 @ J) & (ord_less_nat @ K @ L3)) => (ord_less_nat @ (plus_plus_nat @ I4 @ K) @ (plus_plus_nat @ J @ L3)))))). % add_mono_thms_linordered_field(5)
thf(fact_118_add__mono__thms__linordered__field_I2_J, axiom,
    ((![I4 : nat, J : nat, K : nat, L3 : nat]: (((I4 = J) & (ord_less_nat @ K @ L3)) => (ord_less_nat @ (plus_plus_nat @ I4 @ K) @ (plus_plus_nat @ J @ L3)))))). % add_mono_thms_linordered_field(2)
thf(fact_119_add__mono__thms__linordered__field_I1_J, axiom,
    ((![I4 : nat, J : nat, K : nat, L3 : nat]: (((ord_less_nat @ I4 @ J) & (K = L3)) => (ord_less_nat @ (plus_plus_nat @ I4 @ K) @ (plus_plus_nat @ J @ L3)))))). % add_mono_thms_linordered_field(1)
thf(fact_120_add__strict__mono, axiom,
    ((![A : nat, B : nat, C : nat, D : nat]: ((ord_less_nat @ A @ B) => ((ord_less_nat @ C @ D) => (ord_less_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ D))))))). % add_strict_mono
thf(fact_121_add__strict__left__mono, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_nat @ A @ B) => (ord_less_nat @ (plus_plus_nat @ C @ A) @ (plus_plus_nat @ C @ B)))))). % add_strict_left_mono
thf(fact_122_add__strict__right__mono, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_nat @ A @ B) => (ord_less_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ C)))))). % add_strict_right_mono
thf(fact_123_add__less__imp__less__left, axiom,
    ((![C : nat, A : nat, B : nat]: ((ord_less_nat @ (plus_plus_nat @ C @ A) @ (plus_plus_nat @ C @ B)) => (ord_less_nat @ A @ B))))). % add_less_imp_less_left
thf(fact_124_add__less__imp__less__right, axiom,
    ((![A : nat, C : nat, B : nat]: ((ord_less_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ C)) => (ord_less_nat @ A @ B))))). % add_less_imp_less_right
thf(fact_125_nat__less__le, axiom,
    ((ord_less_nat = (^[M3 : nat]: (^[N3 : nat]: (((ord_less_eq_nat @ M3 @ N3)) & ((~ ((M3 = N3)))))))))). % nat_less_le
thf(fact_126_less__imp__le__nat, axiom,
    ((![M2 : nat, N : nat]: ((ord_less_nat @ M2 @ N) => (ord_less_eq_nat @ M2 @ N))))). % less_imp_le_nat
thf(fact_127_le__eq__less__or__eq, axiom,
    ((ord_less_eq_nat = (^[M3 : nat]: (^[N3 : nat]: (((ord_less_nat @ M3 @ N3)) | ((M3 = N3)))))))). % le_eq_less_or_eq
thf(fact_128_less__or__eq__imp__le, axiom,
    ((![M2 : nat, N : nat]: (((ord_less_nat @ M2 @ N) | (M2 = N)) => (ord_less_eq_nat @ M2 @ N))))). % less_or_eq_imp_le
thf(fact_129_le__neq__implies__less, axiom,
    ((![M2 : nat, N : nat]: ((ord_less_eq_nat @ M2 @ N) => ((~ ((M2 = N))) => (ord_less_nat @ M2 @ N)))))). % le_neq_implies_less
thf(fact_130_less__mono__imp__le__mono, axiom,
    ((![F : nat > nat, I4 : nat, J : nat]: ((![I2 : nat, J2 : nat]: ((ord_less_nat @ I2 @ J2) => (ord_less_nat @ (F @ I2) @ (F @ J2)))) => ((ord_less_eq_nat @ I4 @ J) => (ord_less_eq_nat @ (F @ I4) @ (F @ J))))))). % less_mono_imp_le_mono
thf(fact_131_add__lessD1, axiom,
    ((![I4 : nat, J : nat, K : nat]: ((ord_less_nat @ (plus_plus_nat @ I4 @ J) @ K) => (ord_less_nat @ I4 @ K))))). % add_lessD1
thf(fact_132_add__less__mono, axiom,
    ((![I4 : nat, J : nat, K : nat, L3 : nat]: ((ord_less_nat @ I4 @ J) => ((ord_less_nat @ K @ L3) => (ord_less_nat @ (plus_plus_nat @ I4 @ K) @ (plus_plus_nat @ J @ L3))))))). % add_less_mono
thf(fact_133_not__add__less1, axiom,
    ((![I4 : nat, J : nat]: (~ ((ord_less_nat @ (plus_plus_nat @ I4 @ J) @ I4)))))). % not_add_less1
thf(fact_134_not__add__less2, axiom,
    ((![J : nat, I4 : nat]: (~ ((ord_less_nat @ (plus_plus_nat @ J @ I4) @ I4)))))). % not_add_less2
thf(fact_135_add__less__mono1, axiom,
    ((![I4 : nat, J : nat, K : nat]: ((ord_less_nat @ I4 @ J) => (ord_less_nat @ (plus_plus_nat @ I4 @ K) @ (plus_plus_nat @ J @ K)))))). % add_less_mono1
thf(fact_136_trans__less__add1, axiom,
    ((![I4 : nat, J : nat, M2 : nat]: ((ord_less_nat @ I4 @ J) => (ord_less_nat @ I4 @ (plus_plus_nat @ J @ M2)))))). % trans_less_add1
thf(fact_137_trans__less__add2, axiom,
    ((![I4 : nat, J : nat, M2 : nat]: ((ord_less_nat @ I4 @ J) => (ord_less_nat @ I4 @ (plus_plus_nat @ M2 @ J)))))). % trans_less_add2

% 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)))))) = (~ ((member2048039092le_alt @ (produc1494124311le_alt @ b @ a) @ (f @ (^[I : arrow_1429744205e_indi]: (if_set550155277le_alt @ (ord_less_nat @ (h @ I) @ n) @ lab @ lba))))))))))).
