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thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_It__List__Olist_It__String__Ochar_J_J, type,
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    transi100378922on_val : set_Pr1368603847on_val > set_Pr1368603847on_val).
thf(sy_c_Wfrec_Osame__fst_001_062_It__Nat__Onat_Mt__Option__Ooption_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_M_062_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Value__Oval_J_J_J_J_J_001_062_It__List__Olist_It__String__Ochar_J_Mt__Option__Ooption_It__Value__Oval_J_J, type,
    same_f1752972725on_val : ((nat > option2118145160on_val) > $o) > ((nat > option2118145160on_val) > set_Pr1104486497on_val) > set_Pr148564103on_val).
thf(sy_c_Wfrec_Osame__fst_001t__Expr__Oexp_It__List__Olist_It__String__Ochar_J_J_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Option__Ooption_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_M_062_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Value__Oval_J_J_J_J_J_M_062_It__List__Olist_It__String__Ochar_J_Mt__Option__Ooption_It__Value__Oval_J_J_J, type,
    same_f366220041on_val : (exp_list_char > $o) > (exp_list_char > set_Pr148564103on_val) > set_Pr1189492743on_val).
thf(sy_c_Wfrec_Osame__fst_001t__List__Olist_It__String__Ochar_J_001_062_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Value__Oval_J_J, type,
    same_f2003279573on_val : (list_char > $o) > (list_char > set_Pr227219079on_val) > set_Pr2110063281on_val).
thf(sy_c_Wfrec_Osame__fst_001t__Product____Type__Oprod_It__Expr__Oexp_It__List__Olist_It__String__Ochar_J_J_Mt__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Option__Ooption_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_M_062_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Value__Oval_J_J_J_J_J_M_062_It__List__Olist_It__String__Ochar_J_Mt__Option__Ooption_It__Value__Oval_J_J_J_J_001t__Product____Type__Oprod_It__Expr__Oexp_It__List__Olist_It__String__Ochar_J_J_Mt__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Option__Ooption_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_M_062_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Value__Oval_J_J_J_J_J_M_062_It__List__Olist_It__String__Ochar_J_Mt__Option__Ooption_It__Value__Oval_J_J_J_J, type,
    same_f962657004on_val : (produc1097884484on_val > $o) > (produc1097884484on_val > set_Pr1189492743on_val) > set_Pr1368603847on_val).
thf(sy_c_member_001t__List__Olist_It__String__Ochar_J, type,
    member_list_char : list_char > set_list_char > $o).
thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Option__Ooption_It__Value__Oval_J_J_M_062_It__List__Olist_It__String__Ochar_J_Mt__Option__Ooption_It__Value__Oval_J_J_J, type,
    member1177709994on_val : produc1335728385on_val > set_Pr1104486497on_val > $o).
thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Option__Ooption_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_M_062_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Value__Oval_J_J_J_J_J_M_062_It__List__Olist_It__String__Ochar_J_Mt__Option__Ooption_It__Value__Oval_J_J_J, type,
    member652041809on_val : produc515445032on_val > set_Pr723149704on_val > $o).
thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Value__Oval_J_J_M_062_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Value__Oval_J_J_J, type,
    member548035792on_val : produc1731226791on_val > set_Pr227219079on_val > $o).
thf(sy_c_member_001t__Product____Type__Oprod_It__Expr__Oexp_It__List__Olist_It__String__Ochar_J_J_Mt__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Option__Ooption_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_M_062_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Value__Oval_J_J_J_J_J_M_062_It__List__Olist_It__String__Ochar_J_Mt__Option__Ooption_It__Value__Oval_J_J_J_J, type,
    member1888679917on_val : produc1097884484on_val > set_Pr556513572on_val > $o).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_M_062_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Value__Oval_J_J_J, type,
    member1296415951on_val : produc266837688on_val > set_Pr2075231726on_val > $o).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Option__Ooption_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_M_062_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Value__Oval_J_J_J_J_J_M_062_It__List__Olist_It__String__Ochar_J_Mt__Option__Ooption_It__Value__Oval_J_J_J_Mt__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Option__Ooption_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_M_062_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Value__Oval_J_J_J_J_J_M_062_It__List__Olist_It__String__Ochar_J_Mt__Option__Ooption_It__Value__Oval_J_J_J_J, type,
    member2027247312on_val : produc744932519on_val > set_Pr148564103on_val > $o).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Expr__Oexp_It__List__Olist_It__String__Ochar_J_J_Mt__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Option__Ooption_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_M_062_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Value__Oval_J_J_J_J_J_M_062_It__List__Olist_It__String__Ochar_J_Mt__Option__Ooption_It__Value__Oval_J_J_J_J_Mt__Product____Type__Oprod_It__Expr__Oexp_It__List__Olist_It__String__Ochar_J_J_Mt__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Option__Ooption_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_M_062_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Value__Oval_J_J_J_J_J_M_062_It__List__Olist_It__String__Ochar_J_Mt__Option__Ooption_It__Value__Oval_J_J_J_J_J, type,
    member354538064on_val : produc1172664615on_val > set_Pr1189492743on_val > $o).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_M_062_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Value__Oval_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_M_062_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Value__Oval_J_J_J_J, type,
    member197592058on_val : produc1085126737on_val > set_Pr2110063281on_val > $o).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Expr__Oexp_It__List__Olist_It__String__Ochar_J_J_Mt__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Option__Ooption_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_M_062_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Value__Oval_J_J_J_J_J_M_062_It__List__Olist_It__String__Ochar_J_Mt__Option__Ooption_It__Value__Oval_J_J_J_J_Mt__Product____Type__Oprod_It__Expr__Oexp_It__List__Olist_It__String__Ochar_J_J_Mt__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Option__Ooption_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_M_062_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Value__Oval_J_J_J_J_J_M_062_It__List__Olist_It__String__Ochar_J_Mt__Option__Ooption_It__Value__Oval_J_J_J_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Expr__Oexp_It__List__Olist_It__String__Ochar_J_J_Mt__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Option__Ooption_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_M_062_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Value__Oval_J_J_J_J_J_M_062_It__List__Olist_It__String__Ochar_J_Mt__Option__Ooption_It__Value__Oval_J_J_J_J_Mt__Product____Type__Oprod_It__Expr__Oexp_It__List__Olist_It__String__Ochar_J_J_Mt__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Option__Ooption_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_M_062_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Value__Oval_J_J_J_J_J_M_062_It__List__Olist_It__String__Ochar_J_Mt__Option__Ooption_It__Value__Oval_J_J_J_J_J_J, type,
    member208988944on_val : produc1161703655on_val > set_Pr1368603847on_val > $o).
thf(sy_v_P, type,
    p : list_P1234162913t_char).
thf(sy_v_e, type,
    e : exp_list_char).
thf(sy_v_e_H, type,
    e2 : exp_list_char).
thf(sy_v_h, type,
    h : nat > option2118145160on_val).
thf(sy_v_h_H, type,
    h2 : nat > option2118145160on_val).
thf(sy_v_l, type,
    l : list_char > option_val).
thf(sy_v_l_H, type,
    l2 : list_char > option_val).

% Relevant facts (183)
thf(fact_0_prod__cases4, axiom,
    ((![Y : produc1172664615on_val]: (~ ((![A : produc1097884484on_val, B : exp_list_char, C : nat > option2118145160on_val, D : list_char > option_val]: (~ ((Y = (produc1069526295on_val @ A @ (produc2023458740on_val @ B @ (produc52622304on_val @ C @ D)))))))))))). % prod_cases4
thf(fact_1_prod__induct4, axiom,
    ((![P : produc1172664615on_val > $o, X : produc1172664615on_val]: ((![A : produc1097884484on_val, B : exp_list_char, C : nat > option2118145160on_val, D : list_char > option_val]: (P @ (produc1069526295on_val @ A @ (produc2023458740on_val @ B @ (produc52622304on_val @ C @ D))))) => (P @ X))))). % prod_induct4
thf(fact_2_sqUn__lem, axiom,
    ((![A2 : option_set_list_char, A3 : option_set_list_char, B2 : option_set_list_char]: ((hyper_1448788648t_char @ A2 @ A3) => (hyper_1448788648t_char @ (hyperUn_list_char @ A2 @ B2) @ (hyperUn_list_char @ A3 @ B2)))))). % sqUn_lem
thf(fact_3_option_Oinject, axiom,
    ((![X2 : set_list_char, Y2 : set_list_char]: (((some_set_list_char @ X2) = (some_set_list_char @ Y2)) = (X2 = Y2))))). % option.inject
thf(fact_4_option_Oinject, axiom,
    ((![X2 : produc266837688on_val, Y2 : produc266837688on_val]: (((some_P1299442195on_val @ X2) = (some_P1299442195on_val @ Y2)) = (X2 = Y2))))). % option.inject
thf(fact_5_prod_Oinject, axiom,
    ((![X1 : produc1097884484on_val, X2 : produc1097884484on_val, Y1 : produc1097884484on_val, Y2 : produc1097884484on_val]: (((produc1069526295on_val @ X1 @ X2) = (produc1069526295on_val @ Y1 @ Y2)) = (((X1 = Y1)) & ((X2 = Y2))))))). % prod.inject
thf(fact_6_prod_Oinject, axiom,
    ((![X1 : exp_list_char, X2 : produc515445032on_val, Y1 : exp_list_char, Y2 : produc515445032on_val]: (((produc2023458740on_val @ X1 @ X2) = (produc2023458740on_val @ Y1 @ Y2)) = (((X1 = Y1)) & ((X2 = Y2))))))). % prod.inject
thf(fact_7_prod_Oinject, axiom,
    ((![X1 : nat > option2118145160on_val, X2 : list_char > option_val, Y1 : nat > option2118145160on_val, Y2 : list_char > option_val]: (((produc52622304on_val @ X1 @ X2) = (produc52622304on_val @ Y1 @ Y2)) = (((X1 = Y1)) & ((X2 = Y2))))))). % prod.inject
thf(fact_8_prod_Oinject, axiom,
    ((![X1 : list_char, X2 : produc1257910179t_char > option_val, Y1 : list_char, Y2 : produc1257910179t_char > option_val]: (((produc904988266on_val @ X1 @ X2) = (produc904988266on_val @ Y1 @ Y2)) = (((X1 = Y1)) & ((X2 = Y2))))))). % prod.inject
thf(fact_9_old_Oprod_Oinject, axiom,
    ((![A4 : produc1097884484on_val, B3 : produc1097884484on_val, A5 : produc1097884484on_val, B4 : produc1097884484on_val]: (((produc1069526295on_val @ A4 @ B3) = (produc1069526295on_val @ A5 @ B4)) = (((A4 = A5)) & ((B3 = B4))))))). % old.prod.inject
thf(fact_10_old_Oprod_Oinject, axiom,
    ((![A4 : exp_list_char, B3 : produc515445032on_val, A5 : exp_list_char, B4 : produc515445032on_val]: (((produc2023458740on_val @ A4 @ B3) = (produc2023458740on_val @ A5 @ B4)) = (((A4 = A5)) & ((B3 = B4))))))). % old.prod.inject
thf(fact_11_old_Oprod_Oinject, axiom,
    ((![A4 : nat > option2118145160on_val, B3 : list_char > option_val, A5 : nat > option2118145160on_val, B4 : list_char > option_val]: (((produc52622304on_val @ A4 @ B3) = (produc52622304on_val @ A5 @ B4)) = (((A4 = A5)) & ((B3 = B4))))))). % old.prod.inject
thf(fact_12_old_Oprod_Oinject, axiom,
    ((![A4 : list_char, B3 : produc1257910179t_char > option_val, A5 : list_char, B4 : produc1257910179t_char > option_val]: (((produc904988266on_val @ A4 @ B3) = (produc904988266on_val @ A5 @ B4)) = (((A4 = A5)) & ((B3 = B4))))))). % old.prod.inject
thf(fact_13_old_Oprod_Oinducts, axiom,
    ((![P : produc1172664615on_val > $o, Prod : produc1172664615on_val]: ((![A : produc1097884484on_val, B : produc1097884484on_val]: (P @ (produc1069526295on_val @ A @ B))) => (P @ Prod))))). % old.prod.inducts
thf(fact_14_old_Oprod_Oinducts, axiom,
    ((![P : produc1097884484on_val > $o, Prod : produc1097884484on_val]: ((![A : exp_list_char, B : produc515445032on_val]: (P @ (produc2023458740on_val @ A @ B))) => (P @ Prod))))). % old.prod.inducts
thf(fact_15_old_Oprod_Oinducts, axiom,
    ((![P : produc515445032on_val > $o, Prod : produc515445032on_val]: ((![A : nat > option2118145160on_val, B : list_char > option_val]: (P @ (produc52622304on_val @ A @ B))) => (P @ Prod))))). % old.prod.inducts
thf(fact_16_old_Oprod_Oinducts, axiom,
    ((![P : produc266837688on_val > $o, Prod : produc266837688on_val]: ((![A : list_char, B : produc1257910179t_char > option_val]: (P @ (produc904988266on_val @ A @ B))) => (P @ Prod))))). % old.prod.inducts
thf(fact_17_old_Oprod_Oexhaust, axiom,
    ((![Y : produc1172664615on_val]: (~ ((![A : produc1097884484on_val, B : produc1097884484on_val]: (~ ((Y = (produc1069526295on_val @ A @ B)))))))))). % old.prod.exhaust
thf(fact_18_old_Oprod_Oexhaust, axiom,
    ((![Y : produc1097884484on_val]: (~ ((![A : exp_list_char, B : produc515445032on_val]: (~ ((Y = (produc2023458740on_val @ A @ B)))))))))). % old.prod.exhaust
thf(fact_19_old_Oprod_Oexhaust, axiom,
    ((![Y : produc515445032on_val]: (~ ((![A : nat > option2118145160on_val, B : list_char > option_val]: (~ ((Y = (produc52622304on_val @ A @ B)))))))))). % old.prod.exhaust
thf(fact_20_old_Oprod_Oexhaust, axiom,
    ((![Y : produc266837688on_val]: (~ ((![A : list_char, B : produc1257910179t_char > option_val]: (~ ((Y = (produc904988266on_val @ A @ B)))))))))). % old.prod.exhaust
thf(fact_21_Pair__inject, axiom,
    ((![A4 : produc1097884484on_val, B3 : produc1097884484on_val, A5 : produc1097884484on_val, B4 : produc1097884484on_val]: (((produc1069526295on_val @ A4 @ B3) = (produc1069526295on_val @ A5 @ B4)) => (~ (((A4 = A5) => (~ ((B3 = B4)))))))))). % Pair_inject
thf(fact_22_Pair__inject, axiom,
    ((![A4 : exp_list_char, B3 : produc515445032on_val, A5 : exp_list_char, B4 : produc515445032on_val]: (((produc2023458740on_val @ A4 @ B3) = (produc2023458740on_val @ A5 @ B4)) => (~ (((A4 = A5) => (~ ((B3 = B4)))))))))). % Pair_inject
thf(fact_23_Pair__inject, axiom,
    ((![A4 : nat > option2118145160on_val, B3 : list_char > option_val, A5 : nat > option2118145160on_val, B4 : list_char > option_val]: (((produc52622304on_val @ A4 @ B3) = (produc52622304on_val @ A5 @ B4)) => (~ (((A4 = A5) => (~ ((B3 = B4)))))))))). % Pair_inject
thf(fact_24_Pair__inject, axiom,
    ((![A4 : list_char, B3 : produc1257910179t_char > option_val, A5 : list_char, B4 : produc1257910179t_char > option_val]: (((produc904988266on_val @ A4 @ B3) = (produc904988266on_val @ A5 @ B4)) => (~ (((A4 = A5) => (~ ((B3 = B4)))))))))). % Pair_inject
thf(fact_25_prod__cases, axiom,
    ((![P : produc1172664615on_val > $o, P2 : produc1172664615on_val]: ((![A : produc1097884484on_val, B : produc1097884484on_val]: (P @ (produc1069526295on_val @ A @ B))) => (P @ P2))))). % prod_cases
thf(fact_26_prod__cases, axiom,
    ((![P : produc1097884484on_val > $o, P2 : produc1097884484on_val]: ((![A : exp_list_char, B : produc515445032on_val]: (P @ (produc2023458740on_val @ A @ B))) => (P @ P2))))). % prod_cases
thf(fact_27_prod__cases, axiom,
    ((![P : produc515445032on_val > $o, P2 : produc515445032on_val]: ((![A : nat > option2118145160on_val, B : list_char > option_val]: (P @ (produc52622304on_val @ A @ B))) => (P @ P2))))). % prod_cases
thf(fact_28_prod__cases, axiom,
    ((![P : produc266837688on_val > $o, P2 : produc266837688on_val]: ((![A : list_char, B : produc1257910179t_char > option_val]: (P @ (produc904988266on_val @ A @ B))) => (P @ P2))))). % prod_cases
thf(fact_29_surj__pair, axiom,
    ((![P2 : produc1172664615on_val]: (?[X3 : produc1097884484on_val, Y3 : produc1097884484on_val]: (P2 = (produc1069526295on_val @ X3 @ Y3)))))). % surj_pair
thf(fact_30_surj__pair, axiom,
    ((![P2 : produc1097884484on_val]: (?[X3 : exp_list_char, Y3 : produc515445032on_val]: (P2 = (produc2023458740on_val @ X3 @ Y3)))))). % surj_pair
thf(fact_31_surj__pair, axiom,
    ((![P2 : produc515445032on_val]: (?[X3 : nat > option2118145160on_val, Y3 : list_char > option_val]: (P2 = (produc52622304on_val @ X3 @ Y3)))))). % surj_pair
thf(fact_32_surj__pair, axiom,
    ((![P2 : produc266837688on_val]: (?[X3 : list_char, Y3 : produc1257910179t_char > option_val]: (P2 = (produc904988266on_val @ X3 @ Y3)))))). % surj_pair
thf(fact_33_hyperUn__assoc, axiom,
    ((![A2 : option_set_list_char, B2 : option_set_list_char, C2 : option_set_list_char]: ((hyperUn_list_char @ (hyperUn_list_char @ A2 @ B2) @ C2) = (hyperUn_list_char @ A2 @ (hyperUn_list_char @ B2 @ C2)))))). % hyperUn_assoc
thf(fact_34_prod__induct3, axiom,
    ((![P : produc1172664615on_val > $o, X : produc1172664615on_val]: ((![A : produc1097884484on_val, B : exp_list_char, C : produc515445032on_val]: (P @ (produc1069526295on_val @ A @ (produc2023458740on_val @ B @ C)))) => (P @ X))))). % prod_induct3
thf(fact_35_prod__induct3, axiom,
    ((![P : produc1097884484on_val > $o, X : produc1097884484on_val]: ((![A : exp_list_char, B : nat > option2118145160on_val, C : list_char > option_val]: (P @ (produc2023458740on_val @ A @ (produc52622304on_val @ B @ C)))) => (P @ X))))). % prod_induct3
thf(fact_36_prod__cases3, axiom,
    ((![Y : produc1172664615on_val]: (~ ((![A : produc1097884484on_val, B : exp_list_char, C : produc515445032on_val]: (~ ((Y = (produc1069526295on_val @ A @ (produc2023458740on_val @ B @ C))))))))))). % prod_cases3
thf(fact_37_prod__cases3, axiom,
    ((![Y : produc1097884484on_val]: (~ ((![A : exp_list_char, B : nat > option2118145160on_val, C : list_char > option_val]: (~ ((Y = (produc2023458740on_val @ A @ (produc52622304on_val @ B @ C))))))))))). % prod_cases3
thf(fact_38_domI, axiom,
    ((![M : list_char > option_val, A4 : list_char, B3 : val]: (((M @ A4) = (some_val @ B3)) => (member_list_char @ A4 @ (dom_list_char_val @ M)))))). % domI
thf(fact_39_domI, axiom,
    ((![M : produc1172664615on_val > option_set_list_char, A4 : produc1172664615on_val, B3 : set_list_char]: (((M @ A4) = (some_set_list_char @ B3)) => (member354538064on_val @ A4 @ (dom_Pr1335625324t_char @ M)))))). % domI
thf(fact_40_domI, axiom,
    ((![M : produc1172664615on_val > option2118145160on_val, A4 : produc1172664615on_val, B3 : produc266837688on_val]: (((M @ A4) = (some_P1299442195on_val @ B3)) => (member354538064on_val @ A4 @ (dom_Pr269338893on_val @ M)))))). % domI
thf(fact_41_domD, axiom,
    ((![A4 : list_char, M : list_char > option_val]: ((member_list_char @ A4 @ (dom_list_char_val @ M)) => (?[B : val]: ((M @ A4) = (some_val @ B))))))). % domD
thf(fact_42_domD, axiom,
    ((![A4 : produc1172664615on_val, M : produc1172664615on_val > option_set_list_char]: ((member354538064on_val @ A4 @ (dom_Pr1335625324t_char @ M)) => (?[B : set_list_char]: ((M @ A4) = (some_set_list_char @ B))))))). % domD
thf(fact_43_domD, axiom,
    ((![A4 : produc1172664615on_val, M : produc1172664615on_val > option2118145160on_val]: ((member354538064on_val @ A4 @ (dom_Pr269338893on_val @ M)) => (?[B : produc266837688on_val]: ((M @ A4) = (some_P1299442195on_val @ B))))))). % domD
thf(fact_44_red__lcl__incr, axiom,
    ((![E : exp_list_char, H_0 : nat > option2118145160on_val, L_0 : list_char > option_val, E2 : exp_list_char, H_1 : nat > option2118145160on_val, L_1 : list_char > option_val, P : list_P1234162913t_char]: ((member354538064on_val @ (produc1069526295on_val @ (produc2023458740on_val @ E @ (produc52622304on_val @ H_0 @ L_0)) @ (produc2023458740on_val @ E2 @ (produc52622304on_val @ H_1 @ L_1))) @ (red @ P)) => (ord_le1836582007t_char @ (dom_list_char_val @ L_0) @ (dom_list_char_val @ L_1)))))). % red_lcl_incr
thf(fact_45_same__fstI, axiom,
    ((![P : (nat > option2118145160on_val) > $o, X : nat > option2118145160on_val, Y4 : list_char > option_val, Y : list_char > option_val, R : (nat > option2118145160on_val) > set_Pr1104486497on_val]: ((P @ X) => ((member1177709994on_val @ (produc746553401on_val @ Y4 @ Y) @ (R @ X)) => (member2027247312on_val @ (produc2106877847on_val @ (produc52622304on_val @ X @ Y4) @ (produc52622304on_val @ X @ Y)) @ (same_f1752972725on_val @ P @ R))))))). % same_fstI
thf(fact_46_same__fstI, axiom,
    ((![P : list_char > $o, X : list_char, Y4 : produc1257910179t_char > option_val, Y : produc1257910179t_char > option_val, R : list_char > set_Pr227219079on_val]: ((P @ X) => ((member548035792on_val @ (produc797901719on_val @ Y4 @ Y) @ (R @ X)) => (member197592058on_val @ (produc881885833on_val @ (produc904988266on_val @ X @ Y4) @ (produc904988266on_val @ X @ Y)) @ (same_f2003279573on_val @ P @ R))))))). % same_fstI
thf(fact_47_same__fstI, axiom,
    ((![P : exp_list_char > $o, X : exp_list_char, Y4 : produc515445032on_val, Y : produc515445032on_val, R : exp_list_char > set_Pr148564103on_val]: ((P @ X) => ((member2027247312on_val @ (produc2106877847on_val @ Y4 @ Y) @ (R @ X)) => (member354538064on_val @ (produc1069526295on_val @ (produc2023458740on_val @ X @ Y4) @ (produc2023458740on_val @ X @ Y)) @ (same_f366220041on_val @ P @ R))))))). % same_fstI
thf(fact_48_same__fstI, axiom,
    ((![P : produc1097884484on_val > $o, X : produc1097884484on_val, Y4 : produc1097884484on_val, Y : produc1097884484on_val, R : produc1097884484on_val > set_Pr1189492743on_val]: ((P @ X) => ((member354538064on_val @ (produc1069526295on_val @ Y4 @ Y) @ (R @ X)) => (member208988944on_val @ (produc564964183on_val @ (produc1069526295on_val @ X @ Y4) @ (produc1069526295on_val @ X @ Y)) @ (same_f962657004on_val @ P @ R))))))). % same_fstI
thf(fact_49_red__hext__incr, axiom,
    ((![E : exp_list_char, H : nat > option2118145160on_val, L : list_char > option_val, E2 : exp_list_char, H2 : nat > option2118145160on_val, L2 : list_char > option_val, P : list_P1234162913t_char]: ((member354538064on_val @ (produc1069526295on_val @ (produc2023458740on_val @ E @ (produc52622304on_val @ H @ L)) @ (produc2023458740on_val @ E2 @ (produc52622304on_val @ H2 @ L2))) @ (red @ P)) => (hext @ H @ H2))))). % red_hext_incr
thf(fact_50_diff__lem, axiom,
    ((![A2 : option_set_list_char, A3 : option_set_list_char, B3 : list_char]: ((hyper_1448788648t_char @ A2 @ A3) => (hyper_1448788648t_char @ (hyperDiff1_list_char @ A2 @ B3) @ (hyperDiff1_list_char @ A3 @ B3)))))). % diff_lem
thf(fact_51_ssubst__Pair__rhs, axiom,
    ((![R2 : produc1097884484on_val, S : produc1097884484on_val, R : set_Pr1189492743on_val, S2 : produc1097884484on_val]: ((member354538064on_val @ (produc1069526295on_val @ R2 @ S) @ R) => ((S2 = S) => (member354538064on_val @ (produc1069526295on_val @ R2 @ S2) @ R)))))). % ssubst_Pair_rhs
thf(fact_52_ssubst__Pair__rhs, axiom,
    ((![R2 : exp_list_char, S : produc515445032on_val, R : set_Pr556513572on_val, S2 : produc515445032on_val]: ((member1888679917on_val @ (produc2023458740on_val @ R2 @ S) @ R) => ((S2 = S) => (member1888679917on_val @ (produc2023458740on_val @ R2 @ S2) @ R)))))). % ssubst_Pair_rhs
thf(fact_53_ssubst__Pair__rhs, axiom,
    ((![R2 : nat > option2118145160on_val, S : list_char > option_val, R : set_Pr723149704on_val, S2 : list_char > option_val]: ((member652041809on_val @ (produc52622304on_val @ R2 @ S) @ R) => ((S2 = S) => (member652041809on_val @ (produc52622304on_val @ R2 @ S2) @ R)))))). % ssubst_Pair_rhs
thf(fact_54_ssubst__Pair__rhs, axiom,
    ((![R2 : list_char, S : produc1257910179t_char > option_val, R : set_Pr2075231726on_val, S2 : produc1257910179t_char > option_val]: ((member1296415951on_val @ (produc904988266on_val @ R2 @ S) @ R) => ((S2 = S) => (member1296415951on_val @ (produc904988266on_val @ R2 @ S2) @ R)))))). % ssubst_Pair_rhs
thf(fact_55_curryI, axiom,
    ((![F : produc1172664615on_val > $o, A4 : produc1097884484on_val, B3 : produc1097884484on_val]: ((F @ (produc1069526295on_val @ A4 @ B3)) => (produc360843917_val_o @ F @ A4 @ B3))))). % curryI
thf(fact_56_curryI, axiom,
    ((![F : produc1097884484on_val > $o, A4 : exp_list_char, B3 : produc515445032on_val]: ((F @ (produc2023458740on_val @ A4 @ B3)) => (produc699483184_val_o @ F @ A4 @ B3))))). % curryI
thf(fact_57_curryI, axiom,
    ((![F : produc515445032on_val > $o, A4 : nat > option2118145160on_val, B3 : list_char > option_val]: ((F @ (produc52622304on_val @ A4 @ B3)) => (produc761261956_val_o @ F @ A4 @ B3))))). % curryI
thf(fact_58_curryI, axiom,
    ((![F : produc266837688on_val > $o, A4 : list_char, B3 : produc1257910179t_char > option_val]: ((F @ (produc904988266on_val @ A4 @ B3)) => (produc988598624_val_o @ F @ A4 @ B3))))). % curryI
thf(fact_59_curryD, axiom,
    ((![F : produc1172664615on_val > $o, A4 : produc1097884484on_val, B3 : produc1097884484on_val]: ((produc360843917_val_o @ F @ A4 @ B3) => (F @ (produc1069526295on_val @ A4 @ B3)))))). % curryD
thf(fact_60_curryD, axiom,
    ((![F : produc1097884484on_val > $o, A4 : exp_list_char, B3 : produc515445032on_val]: ((produc699483184_val_o @ F @ A4 @ B3) => (F @ (produc2023458740on_val @ A4 @ B3)))))). % curryD
thf(fact_61_curryD, axiom,
    ((![F : produc515445032on_val > $o, A4 : nat > option2118145160on_val, B3 : list_char > option_val]: ((produc761261956_val_o @ F @ A4 @ B3) => (F @ (produc52622304on_val @ A4 @ B3)))))). % curryD
thf(fact_62_curryD, axiom,
    ((![F : produc266837688on_val > $o, A4 : list_char, B3 : produc1257910179t_char > option_val]: ((produc988598624_val_o @ F @ A4 @ B3) => (F @ (produc904988266on_val @ A4 @ B3)))))). % curryD
thf(fact_63_curryE, axiom,
    ((![F : produc1172664615on_val > $o, A4 : produc1097884484on_val, B3 : produc1097884484on_val]: ((produc360843917_val_o @ F @ A4 @ B3) => (F @ (produc1069526295on_val @ A4 @ B3)))))). % curryE
thf(fact_64_curryE, axiom,
    ((![F : produc1097884484on_val > $o, A4 : exp_list_char, B3 : produc515445032on_val]: ((produc699483184_val_o @ F @ A4 @ B3) => (F @ (produc2023458740on_val @ A4 @ B3)))))). % curryE
thf(fact_65_curryE, axiom,
    ((![F : produc515445032on_val > $o, A4 : nat > option2118145160on_val, B3 : list_char > option_val]: ((produc761261956_val_o @ F @ A4 @ B3) => (F @ (produc52622304on_val @ A4 @ B3)))))). % curryE
thf(fact_66_curryE, axiom,
    ((![F : produc266837688on_val > $o, A4 : list_char, B3 : produc1257910179t_char > option_val]: ((produc988598624_val_o @ F @ A4 @ B3) => (F @ (produc904988266on_val @ A4 @ B3)))))). % curryE
thf(fact_67_hext__refl, axiom,
    ((![H : nat > option2118145160on_val]: (hext @ H @ H)))). % hext_refl
thf(fact_68_subset__antisym, axiom,
    ((![A2 : set_list_char, B2 : set_list_char]: ((ord_le1836582007t_char @ A2 @ B2) => ((ord_le1836582007t_char @ B2 @ A2) => (A2 = B2)))))). % subset_antisym
thf(fact_69_subsetI, axiom,
    ((![A2 : set_Pr1189492743on_val, B2 : set_Pr1189492743on_val]: ((![X3 : produc1172664615on_val]: ((member354538064on_val @ X3 @ A2) => (member354538064on_val @ X3 @ B2))) => (ord_le837504935on_val @ A2 @ B2))))). % subsetI
thf(fact_70_subsetI, axiom,
    ((![A2 : set_list_char, B2 : set_list_char]: ((![X3 : list_char]: ((member_list_char @ X3 @ A2) => (member_list_char @ X3 @ B2))) => (ord_le1836582007t_char @ A2 @ B2))))). % subsetI
thf(fact_71_order__refl, axiom,
    ((![X : set_list_char]: (ord_le1836582007t_char @ X @ X)))). % order_refl
thf(fact_72_A__fv, axiom,
    ((![E : exp_list_char, A2 : set_list_char]: (((a_list_char @ E) = (some_set_list_char @ A2)) => (ord_le1836582007t_char @ A2 @ (fv @ E)))))). % A_fv
thf(fact_73_red__lcl__add, axiom,
    ((![E : exp_list_char, H : nat > option2118145160on_val, L : list_char > option_val, E2 : exp_list_char, H2 : nat > option2118145160on_val, L2 : list_char > option_val, P : list_P1234162913t_char, L_0 : list_char > option_val]: ((member354538064on_val @ (produc1069526295on_val @ (produc2023458740on_val @ E @ (produc52622304on_val @ H @ L)) @ (produc2023458740on_val @ E2 @ (produc52622304on_val @ H2 @ L2))) @ (red @ P)) => (member354538064on_val @ (produc1069526295on_val @ (produc2023458740on_val @ E @ (produc52622304on_val @ H @ (map_ad1615113112ar_val @ L_0 @ L))) @ (produc2023458740on_val @ E2 @ (produc52622304on_val @ H2 @ (map_ad1615113112ar_val @ L_0 @ L2)))) @ (red @ P)))))). % red_lcl_add
thf(fact_74_hextI, axiom,
    ((![H : nat > option2118145160on_val, H2 : nat > option2118145160on_val]: ((![A : nat, C3 : list_char, Fs : produc1257910179t_char > option_val]: (((H @ A) = (some_P1299442195on_val @ (produc904988266on_val @ C3 @ Fs))) => (?[Fs2 : produc1257910179t_char > option_val]: ((H2 @ A) = (some_P1299442195on_val @ (produc904988266on_val @ C3 @ Fs2)))))) => (hext @ H @ H2))))). % hextI
thf(fact_75_hext__def, axiom,
    ((hext = (^[H3 : nat > option2118145160on_val]: (^[H4 : nat > option2118145160on_val]: (![A6 : nat]: (![C4 : list_char]: (![Fs3 : produc1257910179t_char > option_val]: ((((H3 @ A6) = (some_P1299442195on_val @ (produc904988266on_val @ C4 @ Fs3)))) => ((?[Fs4 : produc1257910179t_char > option_val]: ((H4 @ A6) = (some_P1299442195on_val @ (produc904988266on_val @ C4 @ Fs4)))))))))))))). % hext_def
thf(fact_76_map__add__assoc, axiom,
    ((![M1 : list_char > option_val, M2 : list_char > option_val, M3 : list_char > option_val]: ((map_ad1615113112ar_val @ M1 @ (map_ad1615113112ar_val @ M2 @ M3)) = (map_ad1615113112ar_val @ (map_ad1615113112ar_val @ M1 @ M2) @ M3))))). % map_add_assoc
thf(fact_77_map__add__find__right, axiom,
    ((![N : list_char > option_val, K : list_char, Xx : val, M : list_char > option_val]: (((N @ K) = (some_val @ Xx)) => ((map_ad1615113112ar_val @ M @ N @ K) = (some_val @ Xx)))))). % map_add_find_right
thf(fact_78_map__add__dom__app__simps_I1_J, axiom,
    ((![M : list_char, L22 : list_char > option_val, L1 : list_char > option_val]: ((member_list_char @ M @ (dom_list_char_val @ L22)) => ((map_ad1615113112ar_val @ L1 @ L22 @ M) = (L22 @ M)))))). % map_add_dom_app_simps(1)
thf(fact_79_map__add__dom__app__simps_I2_J, axiom,
    ((![M : list_char, L1 : list_char > option_val, L22 : list_char > option_val]: ((~ ((member_list_char @ M @ (dom_list_char_val @ L1)))) => ((map_ad1615113112ar_val @ L1 @ L22 @ M) = (L22 @ M)))))). % map_add_dom_app_simps(2)
thf(fact_80_mem__Collect__eq, axiom,
    ((![A4 : produc1172664615on_val, P : produc1172664615on_val > $o]: ((member354538064on_val @ A4 @ (collec2055896722on_val @ P)) = (P @ A4))))). % mem_Collect_eq
thf(fact_81_Collect__mem__eq, axiom,
    ((![A2 : set_Pr1189492743on_val]: ((collec2055896722on_val @ (^[X4 : produc1172664615on_val]: (member354538064on_val @ X4 @ A2))) = A2)))). % Collect_mem_eq
thf(fact_82_map__add__dom__app__simps_I3_J, axiom,
    ((![M : list_char, L22 : list_char > option_val, L1 : list_char > option_val]: ((~ ((member_list_char @ M @ (dom_list_char_val @ L22)))) => ((map_ad1615113112ar_val @ L1 @ L22 @ M) = (L1 @ M)))))). % map_add_dom_app_simps(3)
thf(fact_83_order__subst1, axiom,
    ((![A4 : set_list_char, F : set_list_char > set_list_char, B3 : set_list_char, C5 : set_list_char]: ((ord_le1836582007t_char @ A4 @ (F @ B3)) => ((ord_le1836582007t_char @ B3 @ C5) => ((![X3 : set_list_char, Y3 : set_list_char]: ((ord_le1836582007t_char @ X3 @ Y3) => (ord_le1836582007t_char @ (F @ X3) @ (F @ Y3)))) => (ord_le1836582007t_char @ A4 @ (F @ C5)))))))). % order_subst1
thf(fact_84_order__subst2, axiom,
    ((![A4 : set_list_char, B3 : set_list_char, F : set_list_char > set_list_char, C5 : set_list_char]: ((ord_le1836582007t_char @ A4 @ B3) => ((ord_le1836582007t_char @ (F @ B3) @ C5) => ((![X3 : set_list_char, Y3 : set_list_char]: ((ord_le1836582007t_char @ X3 @ Y3) => (ord_le1836582007t_char @ (F @ X3) @ (F @ Y3)))) => (ord_le1836582007t_char @ (F @ A4) @ C5))))))). % order_subst2
thf(fact_85_ord__eq__le__subst, axiom,
    ((![A4 : set_list_char, F : set_list_char > set_list_char, B3 : set_list_char, C5 : set_list_char]: ((A4 = (F @ B3)) => ((ord_le1836582007t_char @ B3 @ C5) => ((![X3 : set_list_char, Y3 : set_list_char]: ((ord_le1836582007t_char @ X3 @ Y3) => (ord_le1836582007t_char @ (F @ X3) @ (F @ Y3)))) => (ord_le1836582007t_char @ A4 @ (F @ C5)))))))). % ord_eq_le_subst
thf(fact_86_ord__le__eq__subst, axiom,
    ((![A4 : set_list_char, B3 : set_list_char, F : set_list_char > set_list_char, C5 : set_list_char]: ((ord_le1836582007t_char @ A4 @ B3) => (((F @ B3) = C5) => ((![X3 : set_list_char, Y3 : set_list_char]: ((ord_le1836582007t_char @ X3 @ Y3) => (ord_le1836582007t_char @ (F @ X3) @ (F @ Y3)))) => (ord_le1836582007t_char @ (F @ A4) @ C5))))))). % ord_le_eq_subst
thf(fact_87_eq__iff, axiom,
    (((^[Y5 : set_list_char]: (^[Z : set_list_char]: (Y5 = Z))) = (^[X4 : set_list_char]: (^[Y6 : set_list_char]: (((ord_le1836582007t_char @ X4 @ Y6)) & ((ord_le1836582007t_char @ Y6 @ X4)))))))). % eq_iff
thf(fact_88_antisym, axiom,
    ((![X : set_list_char, Y : set_list_char]: ((ord_le1836582007t_char @ X @ Y) => ((ord_le1836582007t_char @ Y @ X) => (X = Y)))))). % antisym
thf(fact_89_eq__refl, axiom,
    ((![X : set_list_char, Y : set_list_char]: ((X = Y) => (ord_le1836582007t_char @ X @ Y))))). % eq_refl
thf(fact_90_order_Otrans, axiom,
    ((![A4 : set_list_char, B3 : set_list_char, C5 : set_list_char]: ((ord_le1836582007t_char @ A4 @ B3) => ((ord_le1836582007t_char @ B3 @ C5) => (ord_le1836582007t_char @ A4 @ C5)))))). % order.trans
thf(fact_91_antisym__conv, axiom,
    ((![Y : set_list_char, X : set_list_char]: ((ord_le1836582007t_char @ Y @ X) => ((ord_le1836582007t_char @ X @ Y) = (X = Y)))))). % antisym_conv
thf(fact_92_order__class_Oorder_Oeq__iff, axiom,
    (((^[Y5 : set_list_char]: (^[Z : set_list_char]: (Y5 = Z))) = (^[A6 : set_list_char]: (^[B5 : set_list_char]: (((ord_le1836582007t_char @ A6 @ B5)) & ((ord_le1836582007t_char @ B5 @ A6)))))))). % order_class.order.eq_iff
thf(fact_93_ord__eq__le__trans, axiom,
    ((![A4 : set_list_char, B3 : set_list_char, C5 : set_list_char]: ((A4 = B3) => ((ord_le1836582007t_char @ B3 @ C5) => (ord_le1836582007t_char @ A4 @ C5)))))). % ord_eq_le_trans
thf(fact_94_ord__le__eq__trans, axiom,
    ((![A4 : set_list_char, B3 : set_list_char, C5 : set_list_char]: ((ord_le1836582007t_char @ A4 @ B3) => ((B3 = C5) => (ord_le1836582007t_char @ A4 @ C5)))))). % ord_le_eq_trans
thf(fact_95_order__class_Oorder_Oantisym, axiom,
    ((![A4 : set_list_char, B3 : set_list_char]: ((ord_le1836582007t_char @ A4 @ B3) => ((ord_le1836582007t_char @ B3 @ A4) => (A4 = B3)))))). % order_class.order.antisym
thf(fact_96_order__trans, axiom,
    ((![X : set_list_char, Y : set_list_char, Z2 : set_list_char]: ((ord_le1836582007t_char @ X @ Y) => ((ord_le1836582007t_char @ Y @ Z2) => (ord_le1836582007t_char @ X @ Z2)))))). % order_trans
thf(fact_97_dual__order_Orefl, axiom,
    ((![A4 : set_list_char]: (ord_le1836582007t_char @ A4 @ A4)))). % dual_order.refl
thf(fact_98_dual__order_Otrans, axiom,
    ((![B3 : set_list_char, A4 : set_list_char, C5 : set_list_char]: ((ord_le1836582007t_char @ B3 @ A4) => ((ord_le1836582007t_char @ C5 @ B3) => (ord_le1836582007t_char @ C5 @ A4)))))). % dual_order.trans
thf(fact_99_dual__order_Oeq__iff, axiom,
    (((^[Y5 : set_list_char]: (^[Z : set_list_char]: (Y5 = Z))) = (^[A6 : set_list_char]: (^[B5 : set_list_char]: (((ord_le1836582007t_char @ B5 @ A6)) & ((ord_le1836582007t_char @ A6 @ B5)))))))). % dual_order.eq_iff
thf(fact_100_dual__order_Oantisym, axiom,
    ((![B3 : set_list_char, A4 : set_list_char]: ((ord_le1836582007t_char @ B3 @ A4) => ((ord_le1836582007t_char @ A4 @ B3) => (A4 = B3)))))). % dual_order.antisym
thf(fact_101_in__mono, axiom,
    ((![A2 : set_Pr1189492743on_val, B2 : set_Pr1189492743on_val, X : produc1172664615on_val]: ((ord_le837504935on_val @ A2 @ B2) => ((member354538064on_val @ X @ A2) => (member354538064on_val @ X @ B2)))))). % in_mono
thf(fact_102_in__mono, axiom,
    ((![A2 : set_list_char, B2 : set_list_char, X : list_char]: ((ord_le1836582007t_char @ A2 @ B2) => ((member_list_char @ X @ A2) => (member_list_char @ X @ B2)))))). % in_mono
thf(fact_103_subsetD, axiom,
    ((![A2 : set_Pr1189492743on_val, B2 : set_Pr1189492743on_val, C5 : produc1172664615on_val]: ((ord_le837504935on_val @ A2 @ B2) => ((member354538064on_val @ C5 @ A2) => (member354538064on_val @ C5 @ B2)))))). % subsetD
thf(fact_104_subsetD, axiom,
    ((![A2 : set_list_char, B2 : set_list_char, C5 : list_char]: ((ord_le1836582007t_char @ A2 @ B2) => ((member_list_char @ C5 @ A2) => (member_list_char @ C5 @ B2)))))). % subsetD
thf(fact_105_equalityE, axiom,
    ((![A2 : set_list_char, B2 : set_list_char]: ((A2 = B2) => (~ (((ord_le1836582007t_char @ A2 @ B2) => (~ ((ord_le1836582007t_char @ B2 @ A2)))))))))). % equalityE
thf(fact_106_subset__eq, axiom,
    ((ord_le837504935on_val = (^[A7 : set_Pr1189492743on_val]: (^[B6 : set_Pr1189492743on_val]: (![X4 : produc1172664615on_val]: (((member354538064on_val @ X4 @ A7)) => ((member354538064on_val @ X4 @ B6))))))))). % subset_eq
thf(fact_107_subset__eq, axiom,
    ((ord_le1836582007t_char = (^[A7 : set_list_char]: (^[B6 : set_list_char]: (![X4 : list_char]: (((member_list_char @ X4 @ A7)) => ((member_list_char @ X4 @ B6))))))))). % subset_eq
thf(fact_108_equalityD1, axiom,
    ((![A2 : set_list_char, B2 : set_list_char]: ((A2 = B2) => (ord_le1836582007t_char @ A2 @ B2))))). % equalityD1
thf(fact_109_equalityD2, axiom,
    ((![A2 : set_list_char, B2 : set_list_char]: ((A2 = B2) => (ord_le1836582007t_char @ B2 @ A2))))). % equalityD2
thf(fact_110_subset__iff, axiom,
    ((ord_le837504935on_val = (^[A7 : set_Pr1189492743on_val]: (^[B6 : set_Pr1189492743on_val]: (![T : produc1172664615on_val]: (((member354538064on_val @ T @ A7)) => ((member354538064on_val @ T @ B6))))))))). % subset_iff
thf(fact_111_subset__iff, axiom,
    ((ord_le1836582007t_char = (^[A7 : set_list_char]: (^[B6 : set_list_char]: (![T : list_char]: (((member_list_char @ T @ A7)) => ((member_list_char @ T @ B6))))))))). % subset_iff
thf(fact_112_subset__refl, axiom,
    ((![A2 : set_list_char]: (ord_le1836582007t_char @ A2 @ A2)))). % subset_refl
thf(fact_113_Collect__mono, axiom,
    ((![P : list_char > $o, Q : list_char > $o]: ((![X3 : list_char]: ((P @ X3) => (Q @ X3))) => (ord_le1836582007t_char @ (collect_list_char @ P) @ (collect_list_char @ Q)))))). % Collect_mono
thf(fact_114_subset__trans, axiom,
    ((![A2 : set_list_char, B2 : set_list_char, C2 : set_list_char]: ((ord_le1836582007t_char @ A2 @ B2) => ((ord_le1836582007t_char @ B2 @ C2) => (ord_le1836582007t_char @ A2 @ C2)))))). % subset_trans
thf(fact_115_set__eq__subset, axiom,
    (((^[Y5 : set_list_char]: (^[Z : set_list_char]: (Y5 = Z))) = (^[A7 : set_list_char]: (^[B6 : set_list_char]: (((ord_le1836582007t_char @ A7 @ B6)) & ((ord_le1836582007t_char @ B6 @ A7)))))))). % set_eq_subset
thf(fact_116_Collect__mono__iff, axiom,
    ((![P : list_char > $o, Q : list_char > $o]: ((ord_le1836582007t_char @ (collect_list_char @ P) @ (collect_list_char @ Q)) = (![X4 : list_char]: (((P @ X4)) => ((Q @ X4)))))))). % Collect_mono_iff
thf(fact_117_hext__trans, axiom,
    ((![H : nat > option2118145160on_val, H2 : nat > option2118145160on_val, H5 : nat > option2118145160on_val]: ((hext @ H @ H2) => ((hext @ H2 @ H5) => (hext @ H @ H5)))))). % hext_trans
thf(fact_118_hext__objD, axiom,
    ((![H : nat > option2118145160on_val, H2 : nat > option2118145160on_val, A4 : nat, C2 : list_char, Fs5 : produc1257910179t_char > option_val]: ((hext @ H @ H2) => (((H @ A4) = (some_P1299442195on_val @ (produc904988266on_val @ C2 @ Fs5))) => (?[Fs6 : produc1257910179t_char > option_val]: ((H2 @ A4) = (some_P1299442195on_val @ (produc904988266on_val @ C2 @ Fs6))))))))). % hext_objD
thf(fact_119_Red__dom__lcl, axiom,
    ((![E : exp_list_char, H : nat > option2118145160on_val, L : list_char > option_val, E2 : exp_list_char, H2 : nat > option2118145160on_val, L2 : list_char > option_val, P : list_P1234162913t_char]: ((member354538064on_val @ (produc1069526295on_val @ (produc2023458740on_val @ E @ (produc52622304on_val @ H @ L)) @ (produc2023458740on_val @ E2 @ (produc52622304on_val @ H2 @ L2))) @ (red @ P)) => (ord_le1836582007t_char @ (dom_list_char_val @ L2) @ (sup_su1099679555t_char @ (dom_list_char_val @ L) @ (fv @ E))))))). % Red_dom_lcl
thf(fact_120_Red__lcl__add, axiom,
    ((![E : exp_list_char, H : nat > option2118145160on_val, L : list_char > option_val, E2 : exp_list_char, H2 : nat > option2118145160on_val, L2 : list_char > option_val, P : list_P1234162913t_char, L_0 : list_char > option_val]: ((member354538064on_val @ (produc1069526295on_val @ (produc2023458740on_val @ E @ (produc52622304on_val @ H @ L)) @ (produc2023458740on_val @ E2 @ (produc52622304on_val @ H2 @ L2))) @ (transi111096263on_val @ (red @ P))) => (member354538064on_val @ (produc1069526295on_val @ (produc2023458740on_val @ E @ (produc52622304on_val @ H @ (map_ad1615113112ar_val @ L_0 @ L))) @ (produc2023458740on_val @ E2 @ (produc52622304on_val @ H2 @ (map_ad1615113112ar_val @ L_0 @ L2)))) @ (transi111096263on_val @ (red @ P))))))). % Red_lcl_add
thf(fact_121_relChain__def, axiom,
    ((bNF_Ca1733382455t_char = (^[R3 : set_Pr1189492743on_val]: (^[As : produc1097884484on_val > set_list_char]: (![I : produc1097884484on_val]: (![J : produc1097884484on_val]: (((member354538064on_val @ (produc1069526295on_val @ I @ J) @ R3)) => ((ord_le1836582007t_char @ (As @ I) @ (As @ J))))))))))). % relChain_def
thf(fact_122_converse__rtrancl__induct__red, axiom,
    ((![E : exp_list_char, H : nat > option2118145160on_val, L : list_char > option_val, E2 : exp_list_char, H2 : nat > option2118145160on_val, L2 : list_char > option_val, P : list_P1234162913t_char, R : exp_list_char > (nat > option2118145160on_val) > (list_char > option_val) > exp_list_char > (nat > option2118145160on_val) > (list_char > option_val) > $o]: ((member354538064on_val @ (produc1069526295on_val @ (produc2023458740on_val @ E @ (produc52622304on_val @ H @ L)) @ (produc2023458740on_val @ E2 @ (produc52622304on_val @ H2 @ L2))) @ (transi111096263on_val @ (red @ P))) => ((![E3 : exp_list_char, H6 : nat > option2118145160on_val, L3 : list_char > option_val]: (R @ E3 @ H6 @ L3 @ E3 @ H6 @ L3)) => ((![E_0 : exp_list_char, H_02 : nat > option2118145160on_val, L_02 : list_char > option_val, E_1 : exp_list_char, H_12 : nat > option2118145160on_val, L_12 : list_char > option_val, E4 : exp_list_char, H7 : nat > option2118145160on_val, L4 : list_char > option_val]: ((member354538064on_val @ (produc1069526295on_val @ (produc2023458740on_val @ E_0 @ (produc52622304on_val @ H_02 @ L_02)) @ (produc2023458740on_val @ E_1 @ (produc52622304on_val @ H_12 @ L_12))) @ (red @ P)) => ((R @ E_1 @ H_12 @ L_12 @ E4 @ H7 @ L4) => (R @ E_0 @ H_02 @ L_02 @ E4 @ H7 @ L4)))) => (R @ E @ H @ L @ E2 @ H2 @ L2))))))). % converse_rtrancl_induct_red
thf(fact_123_GreatestI2__order, axiom,
    ((![P : set_list_char > $o, X : set_list_char, Q : set_list_char > $o]: ((P @ X) => ((![Y3 : set_list_char]: ((P @ Y3) => (ord_le1836582007t_char @ Y3 @ X))) => ((![X3 : set_list_char]: ((P @ X3) => ((![Y7 : set_list_char]: ((P @ Y7) => (ord_le1836582007t_char @ Y7 @ X3))) => (Q @ X3)))) => (Q @ (order_1708226430t_char @ P)))))))). % GreatestI2_order
thf(fact_124_UnCI, axiom,
    ((![C5 : produc1172664615on_val, B2 : set_Pr1189492743on_val, A2 : set_Pr1189492743on_val]: (((~ ((member354538064on_val @ C5 @ B2))) => (member354538064on_val @ C5 @ A2)) => (member354538064on_val @ C5 @ (sup_su886787547on_val @ A2 @ B2)))))). % UnCI
thf(fact_125_UnCI, axiom,
    ((![C5 : list_char, B2 : set_list_char, A2 : set_list_char]: (((~ ((member_list_char @ C5 @ B2))) => (member_list_char @ C5 @ A2)) => (member_list_char @ C5 @ (sup_su1099679555t_char @ A2 @ B2)))))). % UnCI
thf(fact_126_Un__iff, axiom,
    ((![C5 : produc1172664615on_val, A2 : set_Pr1189492743on_val, B2 : set_Pr1189492743on_val]: ((member354538064on_val @ C5 @ (sup_su886787547on_val @ A2 @ B2)) = (((member354538064on_val @ C5 @ A2)) | ((member354538064on_val @ C5 @ B2))))))). % Un_iff
thf(fact_127_Un__iff, axiom,
    ((![C5 : list_char, A2 : set_list_char, B2 : set_list_char]: ((member_list_char @ C5 @ (sup_su1099679555t_char @ A2 @ B2)) = (((member_list_char @ C5 @ A2)) | ((member_list_char @ C5 @ B2))))))). % Un_iff
thf(fact_128_Un__subset__iff, axiom,
    ((![A2 : set_list_char, B2 : set_list_char, C2 : set_list_char]: ((ord_le1836582007t_char @ (sup_su1099679555t_char @ A2 @ B2) @ C2) = (((ord_le1836582007t_char @ A2 @ C2)) & ((ord_le1836582007t_char @ B2 @ C2))))))). % Un_subset_iff
thf(fact_129_dom__map__add, axiom,
    ((![M : list_char > option_val, N : list_char > option_val]: ((dom_list_char_val @ (map_ad1615113112ar_val @ M @ N)) = (sup_su1099679555t_char @ (dom_list_char_val @ N) @ (dom_list_char_val @ M)))))). % dom_map_add
thf(fact_130_UnE, axiom,
    ((![C5 : produc1172664615on_val, A2 : set_Pr1189492743on_val, B2 : set_Pr1189492743on_val]: ((member354538064on_val @ C5 @ (sup_su886787547on_val @ A2 @ B2)) => ((~ ((member354538064on_val @ C5 @ A2))) => (member354538064on_val @ C5 @ B2)))))). % UnE
thf(fact_131_UnE, axiom,
    ((![C5 : list_char, A2 : set_list_char, B2 : set_list_char]: ((member_list_char @ C5 @ (sup_su1099679555t_char @ A2 @ B2)) => ((~ ((member_list_char @ C5 @ A2))) => (member_list_char @ C5 @ B2)))))). % UnE
thf(fact_132_UnI1, axiom,
    ((![C5 : produc1172664615on_val, A2 : set_Pr1189492743on_val, B2 : set_Pr1189492743on_val]: ((member354538064on_val @ C5 @ A2) => (member354538064on_val @ C5 @ (sup_su886787547on_val @ A2 @ B2)))))). % UnI1
thf(fact_133_UnI1, axiom,
    ((![C5 : list_char, A2 : set_list_char, B2 : set_list_char]: ((member_list_char @ C5 @ A2) => (member_list_char @ C5 @ (sup_su1099679555t_char @ A2 @ B2)))))). % UnI1
thf(fact_134_UnI2, axiom,
    ((![C5 : produc1172664615on_val, B2 : set_Pr1189492743on_val, A2 : set_Pr1189492743on_val]: ((member354538064on_val @ C5 @ B2) => (member354538064on_val @ C5 @ (sup_su886787547on_val @ A2 @ B2)))))). % UnI2
thf(fact_135_UnI2, axiom,
    ((![C5 : list_char, B2 : set_list_char, A2 : set_list_char]: ((member_list_char @ C5 @ B2) => (member_list_char @ C5 @ (sup_su1099679555t_char @ A2 @ B2)))))). % UnI2
thf(fact_136_bex__Un, axiom,
    ((![A2 : set_list_char, B2 : set_list_char, P : list_char > $o]: ((?[X4 : list_char]: (((member_list_char @ X4 @ (sup_su1099679555t_char @ A2 @ B2))) & ((P @ X4)))) = (((?[X4 : list_char]: (((member_list_char @ X4 @ A2)) & ((P @ X4))))) | ((?[X4 : list_char]: (((member_list_char @ X4 @ B2)) & ((P @ X4)))))))))). % bex_Un
thf(fact_137_ball__Un, axiom,
    ((![A2 : set_list_char, B2 : set_list_char, P : list_char > $o]: ((![X4 : list_char]: (((member_list_char @ X4 @ (sup_su1099679555t_char @ A2 @ B2))) => ((P @ X4)))) = (((![X4 : list_char]: (((member_list_char @ X4 @ A2)) => ((P @ X4))))) & ((![X4 : list_char]: (((member_list_char @ X4 @ B2)) => ((P @ X4)))))))))). % ball_Un
thf(fact_138_Un__assoc, axiom,
    ((![A2 : set_list_char, B2 : set_list_char, C2 : set_list_char]: ((sup_su1099679555t_char @ (sup_su1099679555t_char @ A2 @ B2) @ C2) = (sup_su1099679555t_char @ A2 @ (sup_su1099679555t_char @ B2 @ C2)))))). % Un_assoc
thf(fact_139_Un__absorb, axiom,
    ((![A2 : set_list_char]: ((sup_su1099679555t_char @ A2 @ A2) = A2)))). % Un_absorb
thf(fact_140_Un__commute, axiom,
    ((sup_su1099679555t_char = (^[A7 : set_list_char]: (^[B6 : set_list_char]: (sup_su1099679555t_char @ B6 @ A7)))))). % Un_commute
thf(fact_141_Un__left__absorb, axiom,
    ((![A2 : set_list_char, B2 : set_list_char]: ((sup_su1099679555t_char @ A2 @ (sup_su1099679555t_char @ A2 @ B2)) = (sup_su1099679555t_char @ A2 @ B2))))). % Un_left_absorb
thf(fact_142_Un__left__commute, axiom,
    ((![A2 : set_list_char, B2 : set_list_char, C2 : set_list_char]: ((sup_su1099679555t_char @ A2 @ (sup_su1099679555t_char @ B2 @ C2)) = (sup_su1099679555t_char @ B2 @ (sup_su1099679555t_char @ A2 @ C2)))))). % Un_left_commute
thf(fact_143_Un__mono, axiom,
    ((![A2 : set_list_char, C2 : set_list_char, B2 : set_list_char, D2 : set_list_char]: ((ord_le1836582007t_char @ A2 @ C2) => ((ord_le1836582007t_char @ B2 @ D2) => (ord_le1836582007t_char @ (sup_su1099679555t_char @ A2 @ B2) @ (sup_su1099679555t_char @ C2 @ D2))))))). % Un_mono
thf(fact_144_Un__least, axiom,
    ((![A2 : set_list_char, C2 : set_list_char, B2 : set_list_char]: ((ord_le1836582007t_char @ A2 @ C2) => ((ord_le1836582007t_char @ B2 @ C2) => (ord_le1836582007t_char @ (sup_su1099679555t_char @ A2 @ B2) @ C2)))))). % Un_least
thf(fact_145_Un__upper1, axiom,
    ((![A2 : set_list_char, B2 : set_list_char]: (ord_le1836582007t_char @ A2 @ (sup_su1099679555t_char @ A2 @ B2))))). % Un_upper1
thf(fact_146_Un__upper2, axiom,
    ((![B2 : set_list_char, A2 : set_list_char]: (ord_le1836582007t_char @ B2 @ (sup_su1099679555t_char @ A2 @ B2))))). % Un_upper2
thf(fact_147_Un__absorb1, axiom,
    ((![A2 : set_list_char, B2 : set_list_char]: ((ord_le1836582007t_char @ A2 @ B2) => ((sup_su1099679555t_char @ A2 @ B2) = B2))))). % Un_absorb1
thf(fact_148_Un__absorb2, axiom,
    ((![B2 : set_list_char, A2 : set_list_char]: ((ord_le1836582007t_char @ B2 @ A2) => ((sup_su1099679555t_char @ A2 @ B2) = A2))))). % Un_absorb2
thf(fact_149_subset__UnE, axiom,
    ((![C2 : set_list_char, A2 : set_list_char, B2 : set_list_char]: ((ord_le1836582007t_char @ C2 @ (sup_su1099679555t_char @ A2 @ B2)) => (~ ((![A8 : set_list_char]: ((ord_le1836582007t_char @ A8 @ A2) => (![B7 : set_list_char]: ((ord_le1836582007t_char @ B7 @ B2) => (~ ((C2 = (sup_su1099679555t_char @ A8 @ B7)))))))))))))). % subset_UnE
thf(fact_150_subset__Un__eq, axiom,
    ((ord_le1836582007t_char = (^[A7 : set_list_char]: (^[B6 : set_list_char]: ((sup_su1099679555t_char @ A7 @ B6) = B6)))))). % subset_Un_eq
thf(fact_151_Greatest__equality, axiom,
    ((![P : set_list_char > $o, X : set_list_char]: ((P @ X) => ((![Y3 : set_list_char]: ((P @ Y3) => (ord_le1836582007t_char @ Y3 @ X))) => ((order_1708226430t_char @ P) = X)))))). % Greatest_equality
thf(fact_152_sup_Obounded__iff, axiom,
    ((![B3 : set_list_char, C5 : set_list_char, A4 : set_list_char]: ((ord_le1836582007t_char @ (sup_su1099679555t_char @ B3 @ C5) @ A4) = (((ord_le1836582007t_char @ B3 @ A4)) & ((ord_le1836582007t_char @ C5 @ A4))))))). % sup.bounded_iff
thf(fact_153_le__sup__iff, axiom,
    ((![X : set_list_char, Y : set_list_char, Z2 : set_list_char]: ((ord_le1836582007t_char @ (sup_su1099679555t_char @ X @ Y) @ Z2) = (((ord_le1836582007t_char @ X @ Z2)) & ((ord_le1836582007t_char @ Y @ Z2))))))). % le_sup_iff
thf(fact_154_rtrancl__induct2, axiom,
    ((![Ax : produc1097884484on_val, Ay : produc1097884484on_val, Bx : produc1097884484on_val, By : produc1097884484on_val, R2 : set_Pr1368603847on_val, P : produc1097884484on_val > produc1097884484on_val > $o]: ((member208988944on_val @ (produc564964183on_val @ (produc1069526295on_val @ Ax @ Ay) @ (produc1069526295on_val @ Bx @ By)) @ (transi100378922on_val @ R2)) => ((P @ Ax @ Ay) => ((![A : produc1097884484on_val, B : produc1097884484on_val, Aa : produc1097884484on_val, Ba : produc1097884484on_val]: ((member208988944on_val @ (produc564964183on_val @ (produc1069526295on_val @ Ax @ Ay) @ (produc1069526295on_val @ A @ B)) @ (transi100378922on_val @ R2)) => ((member208988944on_val @ (produc564964183on_val @ (produc1069526295on_val @ A @ B) @ (produc1069526295on_val @ Aa @ Ba)) @ R2) => ((P @ A @ B) => (P @ Aa @ Ba))))) => (P @ Bx @ By))))))). % rtrancl_induct2
thf(fact_155_rtrancl__induct2, axiom,
    ((![Ax : nat > option2118145160on_val, Ay : list_char > option_val, Bx : nat > option2118145160on_val, By : list_char > option_val, R2 : set_Pr148564103on_val, P : (nat > option2118145160on_val) > (list_char > option_val) > $o]: ((member2027247312on_val @ (produc2106877847on_val @ (produc52622304on_val @ Ax @ Ay) @ (produc52622304on_val @ Bx @ By)) @ (transi432893227on_val @ R2)) => ((P @ Ax @ Ay) => ((![A : nat > option2118145160on_val, B : list_char > option_val, Aa : nat > option2118145160on_val, Ba : list_char > option_val]: ((member2027247312on_val @ (produc2106877847on_val @ (produc52622304on_val @ Ax @ Ay) @ (produc52622304on_val @ A @ B)) @ (transi432893227on_val @ R2)) => ((member2027247312on_val @ (produc2106877847on_val @ (produc52622304on_val @ A @ B) @ (produc52622304on_val @ Aa @ Ba)) @ R2) => ((P @ A @ B) => (P @ Aa @ Ba))))) => (P @ Bx @ By))))))). % rtrancl_induct2
thf(fact_156_rtrancl__induct2, axiom,
    ((![Ax : list_char, Ay : produc1257910179t_char > option_val, Bx : list_char, By : produc1257910179t_char > option_val, R2 : set_Pr2110063281on_val, P : list_char > (produc1257910179t_char > option_val) > $o]: ((member197592058on_val @ (produc881885833on_val @ (produc904988266on_val @ Ax @ Ay) @ (produc904988266on_val @ Bx @ By)) @ (transi2071064181on_val @ R2)) => ((P @ Ax @ Ay) => ((![A : list_char, B : produc1257910179t_char > option_val, Aa : list_char, Ba : produc1257910179t_char > option_val]: ((member197592058on_val @ (produc881885833on_val @ (produc904988266on_val @ Ax @ Ay) @ (produc904988266on_val @ A @ B)) @ (transi2071064181on_val @ R2)) => ((member197592058on_val @ (produc881885833on_val @ (produc904988266on_val @ A @ B) @ (produc904988266on_val @ Aa @ Ba)) @ R2) => ((P @ A @ B) => (P @ Aa @ Ba))))) => (P @ Bx @ By))))))). % rtrancl_induct2
thf(fact_157_rtrancl__induct2, axiom,
    ((![Ax : exp_list_char, Ay : produc515445032on_val, Bx : exp_list_char, By : produc515445032on_val, R2 : set_Pr1189492743on_val, P : exp_list_char > produc515445032on_val > $o]: ((member354538064on_val @ (produc1069526295on_val @ (produc2023458740on_val @ Ax @ Ay) @ (produc2023458740on_val @ Bx @ By)) @ (transi111096263on_val @ R2)) => ((P @ Ax @ Ay) => ((![A : exp_list_char, B : produc515445032on_val, Aa : exp_list_char, Ba : produc515445032on_val]: ((member354538064on_val @ (produc1069526295on_val @ (produc2023458740on_val @ Ax @ Ay) @ (produc2023458740on_val @ A @ B)) @ (transi111096263on_val @ R2)) => ((member354538064on_val @ (produc1069526295on_val @ (produc2023458740on_val @ A @ B) @ (produc2023458740on_val @ Aa @ Ba)) @ R2) => ((P @ A @ B) => (P @ Aa @ Ba))))) => (P @ Bx @ By))))))). % rtrancl_induct2
thf(fact_158_sup_Oidem, axiom,
    ((![A4 : set_list_char]: ((sup_su1099679555t_char @ A4 @ A4) = A4)))). % sup.idem
thf(fact_159_sup__idem, axiom,
    ((![X : set_list_char]: ((sup_su1099679555t_char @ X @ X) = X)))). % sup_idem
thf(fact_160_sup_Oleft__idem, axiom,
    ((![A4 : set_list_char, B3 : set_list_char]: ((sup_su1099679555t_char @ A4 @ (sup_su1099679555t_char @ A4 @ B3)) = (sup_su1099679555t_char @ A4 @ B3))))). % sup.left_idem
thf(fact_161_sup__left__idem, axiom,
    ((![X : set_list_char, Y : set_list_char]: ((sup_su1099679555t_char @ X @ (sup_su1099679555t_char @ X @ Y)) = (sup_su1099679555t_char @ X @ Y))))). % sup_left_idem
thf(fact_162_sup_Oright__idem, axiom,
    ((![A4 : set_list_char, B3 : set_list_char]: ((sup_su1099679555t_char @ (sup_su1099679555t_char @ A4 @ B3) @ B3) = (sup_su1099679555t_char @ A4 @ B3))))). % sup.right_idem
thf(fact_163_rtrancl__idemp, axiom,
    ((![R2 : set_Pr1189492743on_val]: ((transi111096263on_val @ (transi111096263on_val @ R2)) = (transi111096263on_val @ R2))))). % rtrancl_idemp
thf(fact_164_r__into__rtrancl, axiom,
    ((![P2 : produc1172664615on_val, R2 : set_Pr1189492743on_val]: ((member354538064on_val @ P2 @ R2) => (member354538064on_val @ P2 @ (transi111096263on_val @ R2)))))). % r_into_rtrancl
thf(fact_165_rtrancl__Un__separatorE, axiom,
    ((![A4 : produc1097884484on_val, B3 : produc1097884484on_val, P : set_Pr1189492743on_val, Q : set_Pr1189492743on_val]: ((member354538064on_val @ (produc1069526295on_val @ A4 @ B3) @ (transi111096263on_val @ (sup_su886787547on_val @ P @ Q))) => ((![X3 : produc1097884484on_val, Y3 : produc1097884484on_val]: ((member354538064on_val @ (produc1069526295on_val @ A4 @ X3) @ (transi111096263on_val @ P)) => ((member354538064on_val @ (produc1069526295on_val @ X3 @ Y3) @ Q) => (X3 = Y3)))) => (member354538064on_val @ (produc1069526295on_val @ A4 @ B3) @ (transi111096263on_val @ P))))))). % rtrancl_Un_separatorE
thf(fact_166_rtrancl__Un__separator__converseE, axiom,
    ((![A4 : produc1097884484on_val, B3 : produc1097884484on_val, P : set_Pr1189492743on_val, Q : set_Pr1189492743on_val]: ((member354538064on_val @ (produc1069526295on_val @ A4 @ B3) @ (transi111096263on_val @ (sup_su886787547on_val @ P @ Q))) => ((![X3 : produc1097884484on_val, Y3 : produc1097884484on_val]: ((member354538064on_val @ (produc1069526295on_val @ X3 @ B3) @ (transi111096263on_val @ P)) => ((member354538064on_val @ (produc1069526295on_val @ Y3 @ X3) @ Q) => (Y3 = X3)))) => (member354538064on_val @ (produc1069526295on_val @ A4 @ B3) @ (transi111096263on_val @ P))))))). % rtrancl_Un_separator_converseE
thf(fact_167_rtrancl__Un__subset, axiom,
    ((![R : set_Pr1189492743on_val, S3 : set_Pr1189492743on_val]: (ord_le837504935on_val @ (sup_su886787547on_val @ (transi111096263on_val @ R) @ (transi111096263on_val @ S3)) @ (transi111096263on_val @ (sup_su886787547on_val @ R @ S3)))))). % rtrancl_Un_subset
thf(fact_168_rtrancl__Un__rtrancl, axiom,
    ((![R : set_Pr1189492743on_val, S3 : set_Pr1189492743on_val]: ((transi111096263on_val @ (sup_su886787547on_val @ (transi111096263on_val @ R) @ (transi111096263on_val @ S3))) = (transi111096263on_val @ (sup_su886787547on_val @ R @ S3)))))). % rtrancl_Un_rtrancl
thf(fact_169_in__rtrancl__UnI, axiom,
    ((![X : produc1172664615on_val, R : set_Pr1189492743on_val, S3 : set_Pr1189492743on_val]: (((member354538064on_val @ X @ (transi111096263on_val @ R)) | (member354538064on_val @ X @ (transi111096263on_val @ S3))) => (member354538064on_val @ X @ (transi111096263on_val @ (sup_su886787547on_val @ R @ S3))))))). % in_rtrancl_UnI
thf(fact_170_inf__sup__aci_I8_J, axiom,
    ((![X : set_list_char, Y : set_list_char]: ((sup_su1099679555t_char @ X @ (sup_su1099679555t_char @ X @ Y)) = (sup_su1099679555t_char @ X @ Y))))). % inf_sup_aci(8)
thf(fact_171_inf__sup__aci_I7_J, axiom,
    ((![X : set_list_char, Y : set_list_char, Z2 : set_list_char]: ((sup_su1099679555t_char @ X @ (sup_su1099679555t_char @ Y @ Z2)) = (sup_su1099679555t_char @ Y @ (sup_su1099679555t_char @ X @ Z2)))))). % inf_sup_aci(7)
thf(fact_172_inf__sup__aci_I6_J, axiom,
    ((![X : set_list_char, Y : set_list_char, Z2 : set_list_char]: ((sup_su1099679555t_char @ (sup_su1099679555t_char @ X @ Y) @ Z2) = (sup_su1099679555t_char @ X @ (sup_su1099679555t_char @ Y @ Z2)))))). % inf_sup_aci(6)
thf(fact_173_inf__sup__aci_I5_J, axiom,
    ((sup_su1099679555t_char = (^[X4 : set_list_char]: (^[Y6 : set_list_char]: (sup_su1099679555t_char @ Y6 @ X4)))))). % inf_sup_aci(5)
thf(fact_174_boolean__algebra__cancel_Osup1, axiom,
    ((![A2 : set_list_char, K : set_list_char, A4 : set_list_char, B3 : set_list_char]: ((A2 = (sup_su1099679555t_char @ K @ A4)) => ((sup_su1099679555t_char @ A2 @ B3) = (sup_su1099679555t_char @ K @ (sup_su1099679555t_char @ A4 @ B3))))))). % boolean_algebra_cancel.sup1
thf(fact_175_boolean__algebra__cancel_Osup2, axiom,
    ((![B2 : set_list_char, K : set_list_char, B3 : set_list_char, A4 : set_list_char]: ((B2 = (sup_su1099679555t_char @ K @ B3)) => ((sup_su1099679555t_char @ A4 @ B2) = (sup_su1099679555t_char @ K @ (sup_su1099679555t_char @ A4 @ B3))))))). % boolean_algebra_cancel.sup2
thf(fact_176_sup_Oassoc, axiom,
    ((![A4 : set_list_char, B3 : set_list_char, C5 : set_list_char]: ((sup_su1099679555t_char @ (sup_su1099679555t_char @ A4 @ B3) @ C5) = (sup_su1099679555t_char @ A4 @ (sup_su1099679555t_char @ B3 @ C5)))))). % sup.assoc
thf(fact_177_sup__assoc, axiom,
    ((![X : set_list_char, Y : set_list_char, Z2 : set_list_char]: ((sup_su1099679555t_char @ (sup_su1099679555t_char @ X @ Y) @ Z2) = (sup_su1099679555t_char @ X @ (sup_su1099679555t_char @ Y @ Z2)))))). % sup_assoc
thf(fact_178_sup_Ocommute, axiom,
    ((sup_su1099679555t_char = (^[A6 : set_list_char]: (^[B5 : set_list_char]: (sup_su1099679555t_char @ B5 @ A6)))))). % sup.commute
thf(fact_179_sup__commute, axiom,
    ((sup_su1099679555t_char = (^[X4 : set_list_char]: (^[Y6 : set_list_char]: (sup_su1099679555t_char @ Y6 @ X4)))))). % sup_commute
thf(fact_180_sup_Oleft__commute, axiom,
    ((![B3 : set_list_char, A4 : set_list_char, C5 : set_list_char]: ((sup_su1099679555t_char @ B3 @ (sup_su1099679555t_char @ A4 @ C5)) = (sup_su1099679555t_char @ A4 @ (sup_su1099679555t_char @ B3 @ C5)))))). % sup.left_commute
thf(fact_181_sup__left__commute, axiom,
    ((![X : set_list_char, Y : set_list_char, Z2 : set_list_char]: ((sup_su1099679555t_char @ X @ (sup_su1099679555t_char @ Y @ Z2)) = (sup_su1099679555t_char @ Y @ (sup_su1099679555t_char @ X @ Z2)))))). % sup_left_commute
thf(fact_182_rtrancl__mono, axiom,
    ((![R2 : set_Pr1189492743on_val, S : set_Pr1189492743on_val]: ((ord_le837504935on_val @ R2 @ S) => (ord_le837504935on_val @ (transi111096263on_val @ R2) @ (transi111096263on_val @ S)))))). % rtrancl_mono

% Conjectures (2)
thf(conj_0, hypothesis,
    ((member354538064on_val @ (produc1069526295on_val @ (produc2023458740on_val @ e @ (produc52622304on_val @ h @ l)) @ (produc2023458740on_val @ e2 @ (produc52622304on_val @ h2 @ l2))) @ (red @ p)))).
thf(conj_1, conjecture,
    ((hyper_1448788648t_char @ (hyperUn_list_char @ (some_set_list_char @ (dom_list_char_val @ l)) @ (a_list_char @ e)) @ (hyperUn_list_char @ (some_set_list_char @ (dom_list_char_val @ l2)) @ (a_list_char @ e2))))).
