% 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_78__5185380_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:15:53.277

% Could-be-implicit typings (6)
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_Mt__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_J_J, type,
    set_Pr448479039le_alt : $tType).
thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_Mt__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_J, type,
    produc1556433673le_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__Arrow____Order____Mirabelle____riepwfubkl__Oalt, type,
    arrow_1857593510le_alt : $tType).

% Explicit typings (31)
thf(sy_c_Arrow__Order__Mirabelle__riepwfubkl_OLin, type,
    arrow_1848678355le_Lin : set_se2071012361le_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_Fun__Def_Oin__rel_001t__Arrow____Order____Mirabelle____riepwfubkl__Oalt_001t__Arrow____Order____Mirabelle____riepwfubkl__Oalt, type,
    fun_in1555915436le_alt : set_Pr367596371le_alt > arrow_1857593510le_alt > arrow_1857593510le_alt > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_J, type,
    ord_le1885758387le_alt : set_Pr367596371le_alt > set_Pr367596371le_alt > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_J_J, type,
    ord_le1836905577le_alt : set_se2071012361le_alt > set_se2071012361le_alt > $o).
thf(sy_c_Product__Type_OPair_001t__Arrow____Order____Mirabelle____riepwfubkl__Oalt_001t__Arrow____Order____Mirabelle____riepwfubkl__Oalt, type,
    produc1494124311le_alt : arrow_1857593510le_alt > arrow_1857593510le_alt > produc16571293le_alt).
thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_001t__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J, type,
    produc41061307le_alt : produc16571293le_alt > produc16571293le_alt > produc1556433673le_alt).
thf(sy_c_Product__Type_Oapfst_001t__Arrow____Order____Mirabelle____riepwfubkl__Oalt_001t__Arrow____Order____Mirabelle____riepwfubkl__Oalt_001t__Arrow____Order____Mirabelle____riepwfubkl__Oalt, type,
    produc1245692284le_alt : (arrow_1857593510le_alt > arrow_1857593510le_alt) > produc16571293le_alt > produc16571293le_alt).
thf(sy_c_Product__Type_Oapsnd_001t__Arrow____Order____Mirabelle____riepwfubkl__Oalt_001t__Arrow____Order____Mirabelle____riepwfubkl__Oalt_001t__Arrow____Order____Mirabelle____riepwfubkl__Oalt, type,
    produc1710313402le_alt : (arrow_1857593510le_alt > arrow_1857593510le_alt) > produc16571293le_alt > produc16571293le_alt).
thf(sy_c_Product__Type_Ocurry_001t__Arrow____Order____Mirabelle____riepwfubkl__Oalt_001t__Arrow____Order____Mirabelle____riepwfubkl__Oalt_001_Eo, type,
    produc392997491_alt_o : (produc16571293le_alt > $o) > arrow_1857593510le_alt > arrow_1857593510le_alt > $o).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Arrow____Order____Mirabelle____riepwfubkl__Oalt_001t__Arrow____Order____Mirabelle____riepwfubkl__Oalt_001_Eo, type,
    produc624247124_alt_o : (arrow_1857593510le_alt > arrow_1857593510le_alt > $o) > produc16571293le_alt > $o).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Arrow____Order____Mirabelle____riepwfubkl__Oalt_001t__Arrow____Order____Mirabelle____riepwfubkl__Oalt_001t__Set__Oset_It__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_J, type,
    produc860300173le_alt : (arrow_1857593510le_alt > arrow_1857593510le_alt > set_Pr367596371le_alt) > produc16571293le_alt > set_Pr367596371le_alt).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Arrow____Order____Mirabelle____riepwfubkl__Oalt_001t__Arrow____Order____Mirabelle____riepwfubkl__Oalt_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,
    produc1042938349le_alt : (arrow_1857593510le_alt > arrow_1857593510le_alt > set_se2071012361le_alt) > produc16571293le_alt > set_se2071012361le_alt).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Arrow____Order____Mirabelle____riepwfubkl__Oalt_001t__Arrow____Order____Mirabelle____riepwfubkl__Oalt, type,
    produc2071652395le_alt : produc16571293le_alt > arrow_1857593510le_alt).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Arrow____Order____Mirabelle____riepwfubkl__Oalt_001t__Arrow____Order____Mirabelle____riepwfubkl__Oalt, type,
    produc1272650601le_alt : produc16571293le_alt > arrow_1857593510le_alt).
thf(sy_c_Product__Type_Oprod_Oswap_001t__Arrow____Order____Mirabelle____riepwfubkl__Oalt_001t__Arrow____Order____Mirabelle____riepwfubkl__Oalt, type,
    produc1794822519le_alt : produc16571293le_alt > produc16571293le_alt).
thf(sy_c_Relation_Oconverse_001t__Arrow____Order____Mirabelle____riepwfubkl__Oalt_001t__Arrow____Order____Mirabelle____riepwfubkl__Oalt, type,
    conver1781819635le_alt : set_Pr367596371le_alt > set_Pr367596371le_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_Set_Oimage_001t__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_001t__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J, type,
    image_1873259469le_alt : (produc16571293le_alt > produc16571293le_alt) > set_Pr367596371le_alt > set_Pr367596371le_alt).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_001t__Set__Oset_It__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_J, type,
    image_715032109le_alt : (produc16571293le_alt > set_Pr367596371le_alt) > set_Pr367596371le_alt > set_se2071012361le_alt).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_J_001t__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J, type,
    image_2105633431le_alt : (set_Pr367596371le_alt > produc16571293le_alt) > set_se2071012361le_alt > set_Pr367596371le_alt).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_J, type,
    image_71834871le_alt : (set_Pr367596371le_alt > set_Pr367596371le_alt) > set_se2071012361le_alt > set_se2071012361le_alt).
thf(sy_c_Wfrec_Osame__fst_001t__Arrow____Order____Mirabelle____riepwfubkl__Oalt_001t__Arrow____Order____Mirabelle____riepwfubkl__Oalt, type,
    same_f554660482le_alt : (arrow_1857593510le_alt > $o) > (arrow_1857593510le_alt > set_Pr367596371le_alt) > set_Pr448479039le_alt).
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__Product____Type__Oprod_It__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_Mt__Product____Type__Oprod_It__Arrow____Order____Mirabelle____riepwfubkl__Oalt_Mt__Arrow____Order____Mirabelle____riepwfubkl__Oalt_J_J, type,
    member2146260128le_alt : produc1556433673le_alt > set_Pr448479039le_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_L, type,
    l : set_Pr367596371le_alt).
thf(sy_v_x, type,
    x : arrow_1857593510le_alt).
thf(sy_v_y, type,
    y : arrow_1857593510le_alt).
thf(sy_v_z, type,
    z : arrow_1857593510le_alt).

% Relevant facts (81)
thf(fact_0_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_1_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_2_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_3_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_4_surj__pair, axiom,
    ((![P : produc16571293le_alt]: (?[X3 : arrow_1857593510le_alt, Y3 : arrow_1857593510le_alt]: (P = (produc1494124311le_alt @ X3 @ Y3)))))). % surj_pair
thf(fact_5_prod__cases, axiom,
    ((![P2 : produc16571293le_alt > $o, P : produc16571293le_alt]: ((![A3 : arrow_1857593510le_alt, B3 : arrow_1857593510le_alt]: (P2 @ (produc1494124311le_alt @ A3 @ B3))) => (P2 @ P))))). % prod_cases
thf(fact_6_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_7_linear__alt, axiom,
    ((?[L2 : set_Pr367596371le_alt]: (member1334244458le_alt @ L2 @ arrow_1848678355le_Lin)))). % linear_alt
thf(fact_8_old_Oprod_Oinducts, axiom,
    ((![P2 : produc16571293le_alt > $o, Prod : produc16571293le_alt]: ((![A3 : arrow_1857593510le_alt, B3 : arrow_1857593510le_alt]: (P2 @ (produc1494124311le_alt @ A3 @ B3))) => (P2 @ Prod))))). % old.prod.inducts
thf(fact_9_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_10_ssubst__Pair__rhs, axiom,
    ((![R : arrow_1857593510le_alt, S : arrow_1857593510le_alt, R2 : set_Pr367596371le_alt, S2 : arrow_1857593510le_alt]: ((member2048039092le_alt @ (produc1494124311le_alt @ R @ S) @ R2) => ((S2 = S) => (member2048039092le_alt @ (produc1494124311le_alt @ R @ S2) @ R2)))))). % ssubst_Pair_rhs
thf(fact_11_curryI, axiom,
    ((![F : produc16571293le_alt > $o, A : arrow_1857593510le_alt, B : arrow_1857593510le_alt]: ((F @ (produc1494124311le_alt @ A @ B)) => (produc392997491_alt_o @ F @ A @ B))))). % curryI
thf(fact_12_swap__simp, axiom,
    ((![X : arrow_1857593510le_alt, Y : arrow_1857593510le_alt]: ((produc1794822519le_alt @ (produc1494124311le_alt @ X @ Y)) = (produc1494124311le_alt @ Y @ X))))). % swap_simp
thf(fact_13_apsnd__conv, axiom,
    ((![F : arrow_1857593510le_alt > arrow_1857593510le_alt, X : arrow_1857593510le_alt, Y : arrow_1857593510le_alt]: ((produc1710313402le_alt @ F @ (produc1494124311le_alt @ X @ Y)) = (produc1494124311le_alt @ X @ (F @ Y)))))). % apsnd_conv
thf(fact_14_apfst__conv, axiom,
    ((![F : arrow_1857593510le_alt > arrow_1857593510le_alt, X : arrow_1857593510le_alt, Y : arrow_1857593510le_alt]: ((produc1245692284le_alt @ F @ (produc1494124311le_alt @ X @ Y)) = (produc1494124311le_alt @ (F @ X) @ Y))))). % apfst_conv
thf(fact_15_curryD, axiom,
    ((![F : produc16571293le_alt > $o, A : arrow_1857593510le_alt, B : arrow_1857593510le_alt]: ((produc392997491_alt_o @ F @ A @ B) => (F @ (produc1494124311le_alt @ A @ B)))))). % curryD
thf(fact_16_curryE, axiom,
    ((![F : produc16571293le_alt > $o, A : arrow_1857593510le_alt, B : arrow_1857593510le_alt]: ((produc392997491_alt_o @ F @ A @ B) => (F @ (produc1494124311le_alt @ A @ B)))))). % curryE
thf(fact_17_pair__in__swap__image, axiom,
    ((![Y : arrow_1857593510le_alt, X : arrow_1857593510le_alt, A4 : set_Pr367596371le_alt]: ((member2048039092le_alt @ (produc1494124311le_alt @ Y @ X) @ (image_1873259469le_alt @ produc1794822519le_alt @ A4)) = (member2048039092le_alt @ (produc1494124311le_alt @ X @ Y) @ A4))))). % pair_in_swap_image
thf(fact_18_converse__in__Lin, axiom,
    ((![L : set_Pr367596371le_alt]: ((member1334244458le_alt @ (conver1781819635le_alt @ L) @ arrow_1848678355le_Lin) = (member1334244458le_alt @ L @ arrow_1848678355le_Lin))))). % converse_in_Lin
thf(fact_19_same__fstI, axiom,
    ((![P2 : arrow_1857593510le_alt > $o, X : arrow_1857593510le_alt, Y4 : arrow_1857593510le_alt, Y : arrow_1857593510le_alt, R2 : arrow_1857593510le_alt > set_Pr367596371le_alt]: ((P2 @ X) => ((member2048039092le_alt @ (produc1494124311le_alt @ Y4 @ Y) @ (R2 @ X)) => (member2146260128le_alt @ (produc41061307le_alt @ (produc1494124311le_alt @ X @ Y4) @ (produc1494124311le_alt @ X @ Y)) @ (same_f554660482le_alt @ P2 @ R2))))))). % same_fstI
thf(fact_20_mem__Collect__eq, axiom,
    ((![A : produc16571293le_alt, P2 : produc16571293le_alt > $o]: ((member2048039092le_alt @ A @ (collec531981554le_alt @ P2)) = (P2 @ A))))). % mem_Collect_eq
thf(fact_21_mem__Collect__eq, axiom,
    ((![A : set_Pr367596371le_alt, P2 : set_Pr367596371le_alt > $o]: ((member1334244458le_alt @ A @ (collec1399441576le_alt @ P2)) = (P2 @ A))))). % mem_Collect_eq
thf(fact_22_Collect__mem__eq, axiom,
    ((![A4 : set_Pr367596371le_alt]: ((collec531981554le_alt @ (^[X4 : produc16571293le_alt]: (member2048039092le_alt @ X4 @ A4))) = A4)))). % Collect_mem_eq
thf(fact_23_Collect__mem__eq, axiom,
    ((![A4 : set_se2071012361le_alt]: ((collec1399441576le_alt @ (^[X4 : set_Pr367596371le_alt]: (member1334244458le_alt @ X4 @ A4))) = A4)))). % Collect_mem_eq
thf(fact_24_prod_Ocollapse, axiom,
    ((![Prod : produc16571293le_alt]: ((produc1494124311le_alt @ (produc2071652395le_alt @ Prod) @ (produc1272650601le_alt @ Prod)) = Prod)))). % prod.collapse
thf(fact_25_surjective__pairing, axiom,
    ((![T : produc16571293le_alt]: (T = (produc1494124311le_alt @ (produc2071652395le_alt @ T) @ (produc1272650601le_alt @ T)))))). % surjective_pairing
thf(fact_26_Product__Type_OCollect__case__prodD, axiom,
    ((![X : produc16571293le_alt, A4 : arrow_1857593510le_alt > arrow_1857593510le_alt > $o]: ((member2048039092le_alt @ X @ (collec531981554le_alt @ (produc624247124_alt_o @ A4))) => (A4 @ (produc2071652395le_alt @ X) @ (produc1272650601le_alt @ X)))))). % Product_Type.Collect_case_prodD
thf(fact_27_prod_Oexhaust__sel, axiom,
    ((![Prod : produc16571293le_alt]: (Prod = (produc1494124311le_alt @ (produc2071652395le_alt @ Prod) @ (produc1272650601le_alt @ Prod)))))). % prod.exhaust_sel
thf(fact_28_prod_Oswap__def, axiom,
    ((produc1794822519le_alt = (^[P3 : produc16571293le_alt]: (produc1494124311le_alt @ (produc1272650601le_alt @ P3) @ (produc2071652395le_alt @ P3)))))). % prod.swap_def
thf(fact_29_snd__conv, axiom,
    ((![X1 : arrow_1857593510le_alt, X2 : arrow_1857593510le_alt]: ((produc1272650601le_alt @ (produc1494124311le_alt @ X1 @ X2)) = X2)))). % snd_conv
thf(fact_30_snd__eqD, axiom,
    ((![X : arrow_1857593510le_alt, Y : arrow_1857593510le_alt, A : arrow_1857593510le_alt]: (((produc1272650601le_alt @ (produc1494124311le_alt @ X @ Y)) = A) => (Y = A))))). % snd_eqD
thf(fact_31_fst__conv, axiom,
    ((![X1 : arrow_1857593510le_alt, X2 : arrow_1857593510le_alt]: ((produc2071652395le_alt @ (produc1494124311le_alt @ X1 @ X2)) = X1)))). % fst_conv
thf(fact_32_fst__eqD, axiom,
    ((![X : arrow_1857593510le_alt, Y : arrow_1857593510le_alt, A : arrow_1857593510le_alt]: (((produc2071652395le_alt @ (produc1494124311le_alt @ X @ Y)) = A) => (X = A))))). % fst_eqD
thf(fact_33_mem__case__prodE, axiom,
    ((![Z : produc16571293le_alt, C : arrow_1857593510le_alt > arrow_1857593510le_alt > set_Pr367596371le_alt, P : produc16571293le_alt]: ((member2048039092le_alt @ Z @ (produc860300173le_alt @ C @ P)) => (~ ((![X3 : arrow_1857593510le_alt, Y3 : arrow_1857593510le_alt]: ((P = (produc1494124311le_alt @ X3 @ Y3)) => (~ ((member2048039092le_alt @ Z @ (C @ X3 @ Y3)))))))))))). % mem_case_prodE
thf(fact_34_mem__case__prodE, axiom,
    ((![Z : set_Pr367596371le_alt, C : arrow_1857593510le_alt > arrow_1857593510le_alt > set_se2071012361le_alt, P : produc16571293le_alt]: ((member1334244458le_alt @ Z @ (produc1042938349le_alt @ C @ P)) => (~ ((![X3 : arrow_1857593510le_alt, Y3 : arrow_1857593510le_alt]: ((P = (produc1494124311le_alt @ X3 @ Y3)) => (~ ((member1334244458le_alt @ Z @ (C @ X3 @ Y3)))))))))))). % mem_case_prodE
thf(fact_35_converse__iff, axiom,
    ((![A : arrow_1857593510le_alt, B : arrow_1857593510le_alt, R : set_Pr367596371le_alt]: ((member2048039092le_alt @ (produc1494124311le_alt @ A @ B) @ (conver1781819635le_alt @ R)) = (member2048039092le_alt @ (produc1494124311le_alt @ B @ A) @ R))))). % converse_iff
thf(fact_36_BNF__Greatest__Fixpoint_Osubst__Pair, axiom,
    ((![P2 : arrow_1857593510le_alt > arrow_1857593510le_alt > $o, X : arrow_1857593510le_alt, Y : arrow_1857593510le_alt, A : produc16571293le_alt]: ((P2 @ X @ Y) => ((A = (produc1494124311le_alt @ X @ Y)) => (P2 @ (produc2071652395le_alt @ A) @ (produc1272650601le_alt @ A))))))). % BNF_Greatest_Fixpoint.subst_Pair
thf(fact_37_conjI__realizer, axiom,
    ((![P2 : arrow_1857593510le_alt > $o, P : arrow_1857593510le_alt, Q : arrow_1857593510le_alt > $o, Q2 : arrow_1857593510le_alt]: ((P2 @ P) => ((Q @ Q2) => ((P2 @ (produc2071652395le_alt @ (produc1494124311le_alt @ P @ Q2))) & (Q @ (produc1272650601le_alt @ (produc1494124311le_alt @ P @ Q2))))))))). % conjI_realizer
thf(fact_38_exI__realizer, axiom,
    ((![P2 : arrow_1857593510le_alt > arrow_1857593510le_alt > $o, Y : arrow_1857593510le_alt, X : arrow_1857593510le_alt]: ((P2 @ Y @ X) => (P2 @ (produc1272650601le_alt @ (produc1494124311le_alt @ X @ Y)) @ (produc2071652395le_alt @ (produc1494124311le_alt @ X @ Y))))))). % exI_realizer
thf(fact_39_converse__inject, axiom,
    ((![R : set_Pr367596371le_alt, S : set_Pr367596371le_alt]: (((conver1781819635le_alt @ R) = (conver1781819635le_alt @ S)) = (R = S))))). % converse_inject
thf(fact_40_converse__converse, axiom,
    ((![R : set_Pr367596371le_alt]: ((conver1781819635le_alt @ (conver1781819635le_alt @ R)) = R)))). % converse_converse
thf(fact_41_converse_Oinducts, axiom,
    ((![X1 : arrow_1857593510le_alt, X2 : arrow_1857593510le_alt, R : set_Pr367596371le_alt, P2 : arrow_1857593510le_alt > arrow_1857593510le_alt > $o]: ((member2048039092le_alt @ (produc1494124311le_alt @ X1 @ X2) @ (conver1781819635le_alt @ R)) => ((![A3 : arrow_1857593510le_alt, B3 : arrow_1857593510le_alt]: ((member2048039092le_alt @ (produc1494124311le_alt @ A3 @ B3) @ R) => (P2 @ B3 @ A3))) => (P2 @ X1 @ X2)))))). % converse.inducts
thf(fact_42_converse_Ointros, axiom,
    ((![A : arrow_1857593510le_alt, B : arrow_1857593510le_alt, R : set_Pr367596371le_alt]: ((member2048039092le_alt @ (produc1494124311le_alt @ A @ B) @ R) => (member2048039092le_alt @ (produc1494124311le_alt @ B @ A) @ (conver1781819635le_alt @ R)))))). % converse.intros
thf(fact_43_converse_Osimps, axiom,
    ((![A1 : arrow_1857593510le_alt, A22 : arrow_1857593510le_alt, R : set_Pr367596371le_alt]: ((member2048039092le_alt @ (produc1494124311le_alt @ A1 @ A22) @ (conver1781819635le_alt @ R)) = (?[A5 : arrow_1857593510le_alt]: (?[B4 : arrow_1857593510le_alt]: (((A1 = B4)) & ((((A22 = A5)) & ((member2048039092le_alt @ (produc1494124311le_alt @ A5 @ B4) @ R))))))))))). % converse.simps
thf(fact_44_converse_Ocases, axiom,
    ((![A1 : arrow_1857593510le_alt, A22 : arrow_1857593510le_alt, R : set_Pr367596371le_alt]: ((member2048039092le_alt @ (produc1494124311le_alt @ A1 @ A22) @ (conver1781819635le_alt @ R)) => (member2048039092le_alt @ (produc1494124311le_alt @ A22 @ A1) @ R))))). % converse.cases
thf(fact_45_converseE, axiom,
    ((![Yx : produc16571293le_alt, R : set_Pr367596371le_alt]: ((member2048039092le_alt @ Yx @ (conver1781819635le_alt @ R)) => (~ ((![X3 : arrow_1857593510le_alt, Y3 : arrow_1857593510le_alt]: ((Yx = (produc1494124311le_alt @ Y3 @ X3)) => (~ ((member2048039092le_alt @ (produc1494124311le_alt @ X3 @ Y3) @ R))))))))))). % converseE
thf(fact_46_converseD, axiom,
    ((![A : arrow_1857593510le_alt, B : arrow_1857593510le_alt, R : set_Pr367596371le_alt]: ((member2048039092le_alt @ (produc1494124311le_alt @ A @ B) @ (conver1781819635le_alt @ R)) => (member2048039092le_alt @ (produc1494124311le_alt @ B @ A) @ R))))). % converseD
thf(fact_47_image__eqI, axiom,
    ((![B : produc16571293le_alt, F : produc16571293le_alt > produc16571293le_alt, X : produc16571293le_alt, A4 : set_Pr367596371le_alt]: ((B = (F @ X)) => ((member2048039092le_alt @ X @ A4) => (member2048039092le_alt @ B @ (image_1873259469le_alt @ F @ A4))))))). % image_eqI
thf(fact_48_image__eqI, axiom,
    ((![B : set_Pr367596371le_alt, F : produc16571293le_alt > set_Pr367596371le_alt, X : produc16571293le_alt, A4 : set_Pr367596371le_alt]: ((B = (F @ X)) => ((member2048039092le_alt @ X @ A4) => (member1334244458le_alt @ B @ (image_715032109le_alt @ F @ A4))))))). % image_eqI
thf(fact_49_image__eqI, axiom,
    ((![B : produc16571293le_alt, F : set_Pr367596371le_alt > produc16571293le_alt, X : set_Pr367596371le_alt, A4 : set_se2071012361le_alt]: ((B = (F @ X)) => ((member1334244458le_alt @ X @ A4) => (member2048039092le_alt @ B @ (image_2105633431le_alt @ F @ A4))))))). % image_eqI
thf(fact_50_image__eqI, axiom,
    ((![B : set_Pr367596371le_alt, F : set_Pr367596371le_alt > set_Pr367596371le_alt, X : set_Pr367596371le_alt, A4 : set_se2071012361le_alt]: ((B = (F @ X)) => ((member1334244458le_alt @ X @ A4) => (member1334244458le_alt @ B @ (image_71834871le_alt @ F @ A4))))))). % image_eqI
thf(fact_51_eq__snd__iff, axiom,
    ((![B : arrow_1857593510le_alt, P : produc16571293le_alt]: ((B = (produc1272650601le_alt @ P)) = (?[A5 : arrow_1857593510le_alt]: (P = (produc1494124311le_alt @ A5 @ B))))))). % eq_snd_iff
thf(fact_52_sndI, axiom,
    ((![X : produc16571293le_alt, Y : arrow_1857593510le_alt, Z : arrow_1857593510le_alt]: ((X = (produc1494124311le_alt @ Y @ Z)) => ((produc1272650601le_alt @ X) = Z))))). % sndI
thf(fact_53_eq__fst__iff, axiom,
    ((![A : arrow_1857593510le_alt, P : produc16571293le_alt]: ((A = (produc2071652395le_alt @ P)) = (?[B4 : arrow_1857593510le_alt]: (P = (produc1494124311le_alt @ A @ B4))))))). % eq_fst_iff
thf(fact_54_imageI, axiom,
    ((![X : produc16571293le_alt, A4 : set_Pr367596371le_alt, F : produc16571293le_alt > produc16571293le_alt]: ((member2048039092le_alt @ X @ A4) => (member2048039092le_alt @ (F @ X) @ (image_1873259469le_alt @ F @ A4)))))). % imageI
thf(fact_55_imageI, axiom,
    ((![X : produc16571293le_alt, A4 : set_Pr367596371le_alt, F : produc16571293le_alt > set_Pr367596371le_alt]: ((member2048039092le_alt @ X @ A4) => (member1334244458le_alt @ (F @ X) @ (image_715032109le_alt @ F @ A4)))))). % imageI
thf(fact_56_imageI, axiom,
    ((![X : set_Pr367596371le_alt, A4 : set_se2071012361le_alt, F : set_Pr367596371le_alt > produc16571293le_alt]: ((member1334244458le_alt @ X @ A4) => (member2048039092le_alt @ (F @ X) @ (image_2105633431le_alt @ F @ A4)))))). % imageI
thf(fact_57_imageI, axiom,
    ((![X : set_Pr367596371le_alt, A4 : set_se2071012361le_alt, F : set_Pr367596371le_alt > set_Pr367596371le_alt]: ((member1334244458le_alt @ X @ A4) => (member1334244458le_alt @ (F @ X) @ (image_71834871le_alt @ F @ A4)))))). % imageI
thf(fact_58_rev__image__eqI, axiom,
    ((![X : produc16571293le_alt, A4 : set_Pr367596371le_alt, B : produc16571293le_alt, F : produc16571293le_alt > produc16571293le_alt]: ((member2048039092le_alt @ X @ A4) => ((B = (F @ X)) => (member2048039092le_alt @ B @ (image_1873259469le_alt @ F @ A4))))))). % rev_image_eqI
thf(fact_59_rev__image__eqI, axiom,
    ((![X : produc16571293le_alt, A4 : set_Pr367596371le_alt, B : set_Pr367596371le_alt, F : produc16571293le_alt > set_Pr367596371le_alt]: ((member2048039092le_alt @ X @ A4) => ((B = (F @ X)) => (member1334244458le_alt @ B @ (image_715032109le_alt @ F @ A4))))))). % rev_image_eqI
thf(fact_60_rev__image__eqI, axiom,
    ((![X : set_Pr367596371le_alt, A4 : set_se2071012361le_alt, B : produc16571293le_alt, F : set_Pr367596371le_alt > produc16571293le_alt]: ((member1334244458le_alt @ X @ A4) => ((B = (F @ X)) => (member2048039092le_alt @ B @ (image_2105633431le_alt @ F @ A4))))))). % rev_image_eqI
thf(fact_61_rev__image__eqI, axiom,
    ((![X : set_Pr367596371le_alt, A4 : set_se2071012361le_alt, B : set_Pr367596371le_alt, F : set_Pr367596371le_alt > set_Pr367596371le_alt]: ((member1334244458le_alt @ X @ A4) => ((B = (F @ X)) => (member1334244458le_alt @ B @ (image_71834871le_alt @ F @ A4))))))). % rev_image_eqI
thf(fact_62_fstI, axiom,
    ((![X : produc16571293le_alt, Y : arrow_1857593510le_alt, Z : arrow_1857593510le_alt]: ((X = (produc1494124311le_alt @ Y @ Z)) => ((produc2071652395le_alt @ X) = Y))))). % fstI
thf(fact_63_subsetI, axiom,
    ((![A4 : set_Pr367596371le_alt, B5 : set_Pr367596371le_alt]: ((![X3 : produc16571293le_alt]: ((member2048039092le_alt @ X3 @ A4) => (member2048039092le_alt @ X3 @ B5))) => (ord_le1885758387le_alt @ A4 @ B5))))). % subsetI
thf(fact_64_subsetI, axiom,
    ((![A4 : set_se2071012361le_alt, B5 : set_se2071012361le_alt]: ((![X3 : set_Pr367596371le_alt]: ((member1334244458le_alt @ X3 @ A4) => (member1334244458le_alt @ X3 @ B5))) => (ord_le1836905577le_alt @ A4 @ B5))))). % subsetI
thf(fact_65_converse__mono, axiom,
    ((![R : set_Pr367596371le_alt, S : set_Pr367596371le_alt]: ((ord_le1885758387le_alt @ (conver1781819635le_alt @ R) @ (conver1781819635le_alt @ S)) = (ord_le1885758387le_alt @ R @ S))))). % converse_mono
thf(fact_66_in__mono, axiom,
    ((![A4 : set_Pr367596371le_alt, B5 : set_Pr367596371le_alt, X : produc16571293le_alt]: ((ord_le1885758387le_alt @ A4 @ B5) => ((member2048039092le_alt @ X @ A4) => (member2048039092le_alt @ X @ B5)))))). % in_mono
thf(fact_67_in__mono, axiom,
    ((![A4 : set_se2071012361le_alt, B5 : set_se2071012361le_alt, X : set_Pr367596371le_alt]: ((ord_le1836905577le_alt @ A4 @ B5) => ((member1334244458le_alt @ X @ A4) => (member1334244458le_alt @ X @ B5)))))). % in_mono
thf(fact_68_subsetD, axiom,
    ((![A4 : set_Pr367596371le_alt, B5 : set_Pr367596371le_alt, C : produc16571293le_alt]: ((ord_le1885758387le_alt @ A4 @ B5) => ((member2048039092le_alt @ C @ A4) => (member2048039092le_alt @ C @ B5)))))). % subsetD
thf(fact_69_subsetD, axiom,
    ((![A4 : set_se2071012361le_alt, B5 : set_se2071012361le_alt, C : set_Pr367596371le_alt]: ((ord_le1836905577le_alt @ A4 @ B5) => ((member1334244458le_alt @ C @ A4) => (member1334244458le_alt @ C @ B5)))))). % subsetD
thf(fact_70_subset__eq, axiom,
    ((ord_le1885758387le_alt = (^[A6 : set_Pr367596371le_alt]: (^[B6 : set_Pr367596371le_alt]: (![X4 : produc16571293le_alt]: (((member2048039092le_alt @ X4 @ A6)) => ((member2048039092le_alt @ X4 @ B6))))))))). % subset_eq
thf(fact_71_subset__eq, axiom,
    ((ord_le1836905577le_alt = (^[A6 : set_se2071012361le_alt]: (^[B6 : set_se2071012361le_alt]: (![X4 : set_Pr367596371le_alt]: (((member1334244458le_alt @ X4 @ A6)) => ((member1334244458le_alt @ X4 @ B6))))))))). % subset_eq
thf(fact_72_subset__iff, axiom,
    ((ord_le1885758387le_alt = (^[A6 : set_Pr367596371le_alt]: (^[B6 : set_Pr367596371le_alt]: (![T2 : produc16571293le_alt]: (((member2048039092le_alt @ T2 @ A6)) => ((member2048039092le_alt @ T2 @ B6))))))))). % subset_iff
thf(fact_73_subset__iff, axiom,
    ((ord_le1836905577le_alt = (^[A6 : set_se2071012361le_alt]: (^[B6 : set_se2071012361le_alt]: (![T2 : set_Pr367596371le_alt]: (((member1334244458le_alt @ T2 @ A6)) => ((member1334244458le_alt @ T2 @ B6))))))))). % subset_iff
thf(fact_74_image__subsetI, axiom,
    ((![A4 : set_Pr367596371le_alt, F : produc16571293le_alt > produc16571293le_alt, B5 : set_Pr367596371le_alt]: ((![X3 : produc16571293le_alt]: ((member2048039092le_alt @ X3 @ A4) => (member2048039092le_alt @ (F @ X3) @ B5))) => (ord_le1885758387le_alt @ (image_1873259469le_alt @ F @ A4) @ B5))))). % image_subsetI
thf(fact_75_image__subsetI, axiom,
    ((![A4 : set_Pr367596371le_alt, F : produc16571293le_alt > set_Pr367596371le_alt, B5 : set_se2071012361le_alt]: ((![X3 : produc16571293le_alt]: ((member2048039092le_alt @ X3 @ A4) => (member1334244458le_alt @ (F @ X3) @ B5))) => (ord_le1836905577le_alt @ (image_715032109le_alt @ F @ A4) @ B5))))). % image_subsetI
thf(fact_76_image__subsetI, axiom,
    ((![A4 : set_se2071012361le_alt, F : set_Pr367596371le_alt > produc16571293le_alt, B5 : set_Pr367596371le_alt]: ((![X3 : set_Pr367596371le_alt]: ((member1334244458le_alt @ X3 @ A4) => (member2048039092le_alt @ (F @ X3) @ B5))) => (ord_le1885758387le_alt @ (image_2105633431le_alt @ F @ A4) @ B5))))). % image_subsetI
thf(fact_77_image__subsetI, axiom,
    ((![A4 : set_se2071012361le_alt, F : set_Pr367596371le_alt > set_Pr367596371le_alt, B5 : set_se2071012361le_alt]: ((![X3 : set_Pr367596371le_alt]: ((member1334244458le_alt @ X3 @ A4) => (member1334244458le_alt @ (F @ X3) @ B5))) => (ord_le1836905577le_alt @ (image_71834871le_alt @ F @ A4) @ B5))))). % image_subsetI
thf(fact_78_converse__subset__swap, axiom,
    ((![R : set_Pr367596371le_alt, S : set_Pr367596371le_alt]: ((ord_le1885758387le_alt @ R @ (conver1781819635le_alt @ S)) = (ord_le1885758387le_alt @ (conver1781819635le_alt @ R) @ S))))). % converse_subset_swap
thf(fact_79_subrelI, axiom,
    ((![R : set_Pr367596371le_alt, S : set_Pr367596371le_alt]: ((![X3 : arrow_1857593510le_alt, Y3 : arrow_1857593510le_alt]: ((member2048039092le_alt @ (produc1494124311le_alt @ X3 @ Y3) @ R) => (member2048039092le_alt @ (produc1494124311le_alt @ X3 @ Y3) @ S))) => (ord_le1885758387le_alt @ R @ S))))). % subrelI
thf(fact_80_in__rel__def, axiom,
    ((fun_in1555915436le_alt = (^[R3 : set_Pr367596371le_alt]: (^[X4 : arrow_1857593510le_alt]: (^[Y5 : arrow_1857593510le_alt]: (member2048039092le_alt @ (produc1494124311le_alt @ X4 @ Y5) @ R3))))))). % in_rel_def

% Conjectures (1)
thf(conj_0, conjecture,
    ((~ (((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)))))))))))))).
