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% 2020-12-16 14:22:40.886

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thf(sy_v_P, type,
    p : list_P1234162913t_char).
thf(sy_v_T____, type,
    t : ty).
thf(sy_v_e, type,
    e2 : exp_list_char).
thf(sy_v_e_H, type,
    e3 : exp_list_char).
thf(sy_v_s, type,
    s : produc515445032on_val).
thf(sy_v_s_H, type,
    s2 : produc515445032on_val).

% Relevant facts (208)
thf(fact_0_refl_Oprems_I2_J, axiom,
    ((typeSa1777264139_sconf @ p @ e @ s2))). % refl.prems(2)
thf(fact_1_local_Owf, axiom,
    ((wf_pro1229751607t_char @ wf_J_mdecl @ p))). % local.wf
thf(fact_2_refl_Oprems_I1_J, axiom,
    ((wTrt @ p @ (hp @ s2) @ e @ e3 @ t))). % refl.prems(1)
thf(fact_3_Red, axiom,
    ((member354538064on_val @ (produc1069526295on_val @ (produc2023458740on_val @ e2 @ s) @ (produc2023458740on_val @ e3 @ s2)) @ (transi111096263on_val @ (red @ p))))). % Red
thf(fact_4_sconf__def, axiom,
    ((typeSa1777264139_sconf = (^[P : list_P1234162913t_char]: (^[E : list_char > option_ty]: (produc99536995_val_o @ (^[H : nat > option2118145160on_val]: (^[L : list_char > option_val]: (((hconf_1244167010t_char @ P @ H)) & ((lconf_485928294t_char @ P @ H @ L @ E))))))))))). % sconf_def
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_old_Oprod_Oinject, axiom,
    ((![A : produc1097884484on_val, B : produc1097884484on_val, A2 : produc1097884484on_val, B2 : produc1097884484on_val]: (((produc1069526295on_val @ A @ B) = (produc1069526295on_val @ A2 @ B2)) = (((A = A2)) & ((B = B2))))))). % old.prod.inject
thf(fact_9_old_Oprod_Oinject, axiom,
    ((![A : exp_list_char, B : produc515445032on_val, A2 : exp_list_char, B2 : produc515445032on_val]: (((produc2023458740on_val @ A @ B) = (produc2023458740on_val @ A2 @ B2)) = (((A = A2)) & ((B = B2))))))). % old.prod.inject
thf(fact_10_old_Oprod_Oinject, axiom,
    ((![A : nat > option2118145160on_val, B : list_char > option_val, A2 : nat > option2118145160on_val, B2 : list_char > option_val]: (((produc52622304on_val @ A @ B) = (produc52622304on_val @ A2 @ B2)) = (((A = A2)) & ((B = B2))))))). % old.prod.inject
thf(fact_11_split__part, axiom,
    ((![P2 : $o, Q : (nat > option2118145160on_val) > (list_char > option_val) > $o]: ((produc99536995_val_o @ (^[A3 : nat > option2118145160on_val]: (^[B3 : list_char > option_val]: ((P2) & ((Q @ A3 @ B3)))))) = (^[Ab : produc515445032on_val]: ((P2) & ((produc99536995_val_o @ Q @ Ab)))))))). % split_part
thf(fact_12_split__part, axiom,
    ((![P2 : $o, Q : produc1097884484on_val > produc1097884484on_val > $o]: ((produc50362220_val_o @ (^[A3 : produc1097884484on_val]: (^[B3 : produc1097884484on_val]: ((P2) & ((Q @ A3 @ B3)))))) = (^[Ab : produc1172664615on_val]: ((P2) & ((produc50362220_val_o @ Q @ Ab)))))))). % split_part
thf(fact_13_split__part, axiom,
    ((![P2 : $o, Q : exp_list_char > produc515445032on_val > $o]: ((produc199145871_val_o @ (^[A3 : exp_list_char]: (^[B3 : produc515445032on_val]: ((P2) & ((Q @ A3 @ B3)))))) = (^[Ab : produc1097884484on_val]: ((P2) & ((produc199145871_val_o @ Q @ Ab)))))))). % split_part
thf(fact_14_case__prod__conv, axiom,
    ((![F : (nat > option2118145160on_val) > (list_char > option_val) > $o, A : nat > option2118145160on_val, B : list_char > option_val]: ((produc99536995_val_o @ F @ (produc52622304on_val @ A @ B)) = (F @ A @ B))))). % case_prod_conv
thf(fact_15_case__prod__conv, axiom,
    ((![F : produc1097884484on_val > produc1097884484on_val > $o, A : produc1097884484on_val, B : produc1097884484on_val]: ((produc50362220_val_o @ F @ (produc1069526295on_val @ A @ B)) = (F @ A @ B))))). % case_prod_conv
thf(fact_16_case__prod__conv, axiom,
    ((![F : exp_list_char > produc515445032on_val > produc1097884484on_val > $o, A : exp_list_char, B : produc515445032on_val]: ((produc1468025474_val_o @ F @ (produc2023458740on_val @ A @ B)) = (F @ A @ B))))). % case_prod_conv
thf(fact_17_case__prod__conv, axiom,
    ((![F : exp_list_char > produc515445032on_val > $o, A : exp_list_char, B : produc515445032on_val]: ((produc199145871_val_o @ F @ (produc2023458740on_val @ A @ B)) = (F @ A @ B))))). % case_prod_conv
thf(fact_18_case__prodI, axiom,
    ((![F : (nat > option2118145160on_val) > (list_char > option_val) > $o, A : nat > option2118145160on_val, B : list_char > option_val]: ((F @ A @ B) => (produc99536995_val_o @ F @ (produc52622304on_val @ A @ B)))))). % case_prodI
thf(fact_19_case__prodI, axiom,
    ((![F : produc1097884484on_val > produc1097884484on_val > $o, A : produc1097884484on_val, B : produc1097884484on_val]: ((F @ A @ B) => (produc50362220_val_o @ F @ (produc1069526295on_val @ A @ B)))))). % case_prodI
thf(fact_20_case__prodI, axiom,
    ((![F : exp_list_char > produc515445032on_val > $o, A : exp_list_char, B : produc515445032on_val]: ((F @ A @ B) => (produc199145871_val_o @ F @ (produc2023458740on_val @ A @ B)))))). % case_prodI
thf(fact_21_case__prodI2, axiom,
    ((![P3 : produc515445032on_val, C : (nat > option2118145160on_val) > (list_char > option_val) > $o]: ((![A4 : nat > option2118145160on_val, B4 : list_char > option_val]: ((P3 = (produc52622304on_val @ A4 @ B4)) => (C @ A4 @ B4))) => (produc99536995_val_o @ C @ P3))))). % case_prodI2
thf(fact_22_case__prodI2, axiom,
    ((![P3 : produc1172664615on_val, C : produc1097884484on_val > produc1097884484on_val > $o]: ((![A4 : produc1097884484on_val, B4 : produc1097884484on_val]: ((P3 = (produc1069526295on_val @ A4 @ B4)) => (C @ A4 @ B4))) => (produc50362220_val_o @ C @ P3))))). % case_prodI2
thf(fact_23_case__prodI2, axiom,
    ((![P3 : produc1097884484on_val, C : exp_list_char > produc515445032on_val > $o]: ((![A4 : exp_list_char, B4 : produc515445032on_val]: ((P3 = (produc2023458740on_val @ A4 @ B4)) => (C @ A4 @ B4))) => (produc199145871_val_o @ C @ P3))))). % case_prodI2
thf(fact_24_surj__pair, axiom,
    ((![P3 : produc1172664615on_val]: (?[X : produc1097884484on_val, Y : produc1097884484on_val]: (P3 = (produc1069526295on_val @ X @ Y)))))). % surj_pair
thf(fact_25_surj__pair, axiom,
    ((![P3 : produc1097884484on_val]: (?[X : exp_list_char, Y : produc515445032on_val]: (P3 = (produc2023458740on_val @ X @ Y)))))). % surj_pair
thf(fact_26_surj__pair, axiom,
    ((![P3 : produc515445032on_val]: (?[X : nat > option2118145160on_val, Y : list_char > option_val]: (P3 = (produc52622304on_val @ X @ Y)))))). % surj_pair
thf(fact_27_case__prodD, axiom,
    ((![F : (nat > option2118145160on_val) > (list_char > option_val) > $o, A : nat > option2118145160on_val, B : list_char > option_val]: ((produc99536995_val_o @ F @ (produc52622304on_val @ A @ B)) => (F @ A @ B))))). % case_prodD
thf(fact_28_case__prodD, axiom,
    ((![F : produc1097884484on_val > produc1097884484on_val > $o, A : produc1097884484on_val, B : produc1097884484on_val]: ((produc50362220_val_o @ F @ (produc1069526295on_val @ A @ B)) => (F @ A @ B))))). % case_prodD
thf(fact_29_case__prodD, axiom,
    ((![F : exp_list_char > produc515445032on_val > $o, A : exp_list_char, B : produc515445032on_val]: ((produc199145871_val_o @ F @ (produc2023458740on_val @ A @ B)) => (F @ A @ B))))). % case_prodD
thf(fact_30_case__prodE, axiom,
    ((![C : (nat > option2118145160on_val) > (list_char > option_val) > $o, P3 : produc515445032on_val]: ((produc99536995_val_o @ C @ P3) => (~ ((![X : nat > option2118145160on_val, Y : list_char > option_val]: ((P3 = (produc52622304on_val @ X @ Y)) => (~ ((C @ X @ Y))))))))))). % case_prodE
thf(fact_31_case__prodE, axiom,
    ((![C : produc1097884484on_val > produc1097884484on_val > $o, P3 : produc1172664615on_val]: ((produc50362220_val_o @ C @ P3) => (~ ((![X : produc1097884484on_val, Y : produc1097884484on_val]: ((P3 = (produc1069526295on_val @ X @ Y)) => (~ ((C @ X @ Y))))))))))). % case_prodE
thf(fact_32_case__prodE, axiom,
    ((![C : exp_list_char > produc515445032on_val > $o, P3 : produc1097884484on_val]: ((produc199145871_val_o @ C @ P3) => (~ ((![X : exp_list_char, Y : produc515445032on_val]: ((P3 = (produc2023458740on_val @ X @ Y)) => (~ ((C @ X @ Y))))))))))). % case_prodE
thf(fact_33_prod__cases, axiom,
    ((![P2 : produc1172664615on_val > $o, P3 : produc1172664615on_val]: ((![A4 : produc1097884484on_val, B4 : produc1097884484on_val]: (P2 @ (produc1069526295on_val @ A4 @ B4))) => (P2 @ P3))))). % prod_cases
thf(fact_34_prod__cases, axiom,
    ((![P2 : produc1097884484on_val > $o, P3 : produc1097884484on_val]: ((![A4 : exp_list_char, B4 : produc515445032on_val]: (P2 @ (produc2023458740on_val @ A4 @ B4))) => (P2 @ P3))))). % prod_cases
thf(fact_35_prod__cases, axiom,
    ((![P2 : produc515445032on_val > $o, P3 : produc515445032on_val]: ((![A4 : nat > option2118145160on_val, B4 : list_char > option_val]: (P2 @ (produc52622304on_val @ A4 @ B4))) => (P2 @ P3))))). % prod_cases
thf(fact_36_Pair__inject, axiom,
    ((![A : produc1097884484on_val, B : produc1097884484on_val, A2 : produc1097884484on_val, B2 : produc1097884484on_val]: (((produc1069526295on_val @ A @ B) = (produc1069526295on_val @ A2 @ B2)) => (~ (((A = A2) => (~ ((B = B2)))))))))). % Pair_inject
thf(fact_37_Pair__inject, axiom,
    ((![A : exp_list_char, B : produc515445032on_val, A2 : exp_list_char, B2 : produc515445032on_val]: (((produc2023458740on_val @ A @ B) = (produc2023458740on_val @ A2 @ B2)) => (~ (((A = A2) => (~ ((B = B2)))))))))). % Pair_inject
thf(fact_38_Pair__inject, axiom,
    ((![A : nat > option2118145160on_val, B : list_char > option_val, A2 : nat > option2118145160on_val, B2 : list_char > option_val]: (((produc52622304on_val @ A @ B) = (produc52622304on_val @ A2 @ B2)) => (~ (((A = A2) => (~ ((B = B2)))))))))). % Pair_inject
thf(fact_39_case__prodE2, axiom,
    ((![Q : $o > $o, P2 : (nat > option2118145160on_val) > (list_char > option_val) > $o, Z : produc515445032on_val]: ((Q @ (produc99536995_val_o @ P2 @ Z)) => (~ ((![X : nat > option2118145160on_val, Y : list_char > option_val]: ((Z = (produc52622304on_val @ X @ Y)) => (~ ((Q @ (P2 @ X @ Y)))))))))))). % case_prodE2
thf(fact_40_case__prodE2, axiom,
    ((![Q : $o > $o, P2 : produc1097884484on_val > produc1097884484on_val > $o, Z : produc1172664615on_val]: ((Q @ (produc50362220_val_o @ P2 @ Z)) => (~ ((![X : produc1097884484on_val, Y : produc1097884484on_val]: ((Z = (produc1069526295on_val @ X @ Y)) => (~ ((Q @ (P2 @ X @ Y)))))))))))). % case_prodE2
thf(fact_41_case__prodE2, axiom,
    ((![Q : (produc1097884484on_val > $o) > $o, P2 : exp_list_char > produc515445032on_val > produc1097884484on_val > $o, Z : produc1097884484on_val]: ((Q @ (produc1468025474_val_o @ P2 @ Z)) => (~ ((![X : exp_list_char, Y : produc515445032on_val]: ((Z = (produc2023458740on_val @ X @ Y)) => (~ ((Q @ (P2 @ X @ Y)))))))))))). % case_prodE2
thf(fact_42_case__prodE2, axiom,
    ((![Q : $o > $o, P2 : exp_list_char > produc515445032on_val > $o, Z : produc1097884484on_val]: ((Q @ (produc199145871_val_o @ P2 @ Z)) => (~ ((![X : exp_list_char, Y : produc515445032on_val]: ((Z = (produc2023458740on_val @ X @ Y)) => (~ ((Q @ (P2 @ X @ Y)))))))))))). % case_prodE2
thf(fact_43_prod__cases3, axiom,
    ((![Y3 : produc1172664615on_val]: (~ ((![A4 : produc1097884484on_val, B4 : exp_list_char, C2 : produc515445032on_val]: (~ ((Y3 = (produc1069526295on_val @ A4 @ (produc2023458740on_val @ B4 @ C2))))))))))). % prod_cases3
thf(fact_44_prod__cases3, axiom,
    ((![Y3 : produc1097884484on_val]: (~ ((![A4 : exp_list_char, B4 : nat > option2118145160on_val, C2 : list_char > option_val]: (~ ((Y3 = (produc2023458740on_val @ A4 @ (produc52622304on_val @ B4 @ C2))))))))))). % prod_cases3
thf(fact_45_prod__cases4, axiom,
    ((![Y3 : produc1172664615on_val]: (~ ((![A4 : produc1097884484on_val, B4 : exp_list_char, C2 : nat > option2118145160on_val, D : list_char > option_val]: (~ ((Y3 = (produc1069526295on_val @ A4 @ (produc2023458740on_val @ B4 @ (produc52622304on_val @ C2 @ D)))))))))))). % prod_cases4
thf(fact_46_pred__equals__eq2, axiom,
    ((![R : set_Pr1189492743on_val, S : set_Pr1189492743on_val]: (((^[X3 : produc1097884484on_val]: (^[Y4 : produc1097884484on_val]: (member354538064on_val @ (produc1069526295on_val @ X3 @ Y4) @ R))) = (^[X3 : produc1097884484on_val]: (^[Y4 : produc1097884484on_val]: (member354538064on_val @ (produc1069526295on_val @ X3 @ Y4) @ S)))) = (R = S))))). % pred_equals_eq2
thf(fact_47_pred__equals__eq2, axiom,
    ((![R : set_Pr556513572on_val, S : set_Pr556513572on_val]: (((^[X3 : exp_list_char]: (^[Y4 : produc515445032on_val]: (member1888679917on_val @ (produc2023458740on_val @ X3 @ Y4) @ R))) = (^[X3 : exp_list_char]: (^[Y4 : produc515445032on_val]: (member1888679917on_val @ (produc2023458740on_val @ X3 @ Y4) @ S)))) = (R = S))))). % pred_equals_eq2
thf(fact_48_pred__equals__eq2, axiom,
    ((![R : set_Pr723149704on_val, S : set_Pr723149704on_val]: (((^[X3 : nat > option2118145160on_val]: (^[Y4 : list_char > option_val]: (member652041809on_val @ (produc52622304on_val @ X3 @ Y4) @ R))) = (^[X3 : nat > option2118145160on_val]: (^[Y4 : list_char > option_val]: (member652041809on_val @ (produc52622304on_val @ X3 @ Y4) @ S)))) = (R = S))))). % pred_equals_eq2
thf(fact_49_prod__induct3, axiom,
    ((![P2 : produc1172664615on_val > $o, X4 : produc1172664615on_val]: ((![A4 : produc1097884484on_val, B4 : exp_list_char, C2 : produc515445032on_val]: (P2 @ (produc1069526295on_val @ A4 @ (produc2023458740on_val @ B4 @ C2)))) => (P2 @ X4))))). % prod_induct3
thf(fact_50_prod__induct3, axiom,
    ((![P2 : produc1097884484on_val > $o, X4 : produc1097884484on_val]: ((![A4 : exp_list_char, B4 : nat > option2118145160on_val, C2 : list_char > option_val]: (P2 @ (produc2023458740on_val @ A4 @ (produc52622304on_val @ B4 @ C2)))) => (P2 @ X4))))). % prod_induct3
thf(fact_51_prod__induct4, axiom,
    ((![P2 : produc1172664615on_val > $o, X4 : produc1172664615on_val]: ((![A4 : produc1097884484on_val, B4 : exp_list_char, C2 : nat > option2118145160on_val, D : list_char > option_val]: (P2 @ (produc1069526295on_val @ A4 @ (produc2023458740on_val @ B4 @ (produc52622304on_val @ C2 @ D))))) => (P2 @ X4))))). % prod_induct4
thf(fact_52_case__prod__eta, axiom,
    ((![F : produc515445032on_val > $o]: ((produc99536995_val_o @ (^[X3 : nat > option2118145160on_val]: (^[Y4 : list_char > option_val]: (F @ (produc52622304on_val @ X3 @ Y4))))) = F)))). % case_prod_eta
thf(fact_53_case__prod__eta, axiom,
    ((![F : produc1172664615on_val > $o]: ((produc50362220_val_o @ (^[X3 : produc1097884484on_val]: (^[Y4 : produc1097884484on_val]: (F @ (produc1069526295on_val @ X3 @ Y4))))) = F)))). % case_prod_eta
thf(fact_54_case__prod__eta, axiom,
    ((![F : produc1097884484on_val > produc1097884484on_val > $o]: ((produc1468025474_val_o @ (^[X3 : exp_list_char]: (^[Y4 : produc515445032on_val]: (F @ (produc2023458740on_val @ X3 @ Y4))))) = F)))). % case_prod_eta
thf(fact_55_case__prod__eta, axiom,
    ((![F : produc1097884484on_val > $o]: ((produc199145871_val_o @ (^[X3 : exp_list_char]: (^[Y4 : produc515445032on_val]: (F @ (produc2023458740on_val @ X3 @ Y4))))) = F)))). % case_prod_eta
thf(fact_56_old_Oprod_Ocase, axiom,
    ((![F : (nat > option2118145160on_val) > (list_char > option_val) > $o, X1 : nat > option2118145160on_val, X2 : list_char > option_val]: ((produc99536995_val_o @ F @ (produc52622304on_val @ X1 @ X2)) = (F @ X1 @ X2))))). % old.prod.case
thf(fact_57_old_Oprod_Ocase, axiom,
    ((![F : produc1097884484on_val > produc1097884484on_val > $o, X1 : produc1097884484on_val, X2 : produc1097884484on_val]: ((produc50362220_val_o @ F @ (produc1069526295on_val @ X1 @ X2)) = (F @ X1 @ X2))))). % old.prod.case
thf(fact_58_old_Oprod_Ocase, axiom,
    ((![F : exp_list_char > produc515445032on_val > produc1097884484on_val > $o, X1 : exp_list_char, X2 : produc515445032on_val]: ((produc1468025474_val_o @ F @ (produc2023458740on_val @ X1 @ X2)) = (F @ X1 @ X2))))). % old.prod.case
thf(fact_59_old_Oprod_Ocase, axiom,
    ((![F : exp_list_char > produc515445032on_val > $o, X1 : exp_list_char, X2 : produc515445032on_val]: ((produc199145871_val_o @ F @ (produc2023458740on_val @ X1 @ X2)) = (F @ X1 @ X2))))). % old.prod.case
thf(fact_60_old_Oprod_Oexhaust, axiom,
    ((![Y3 : produc1172664615on_val]: (~ ((![A4 : produc1097884484on_val, B4 : produc1097884484on_val]: (~ ((Y3 = (produc1069526295on_val @ A4 @ B4)))))))))). % old.prod.exhaust
thf(fact_61_old_Oprod_Oexhaust, axiom,
    ((![Y3 : produc1097884484on_val]: (~ ((![A4 : exp_list_char, B4 : produc515445032on_val]: (~ ((Y3 = (produc2023458740on_val @ A4 @ B4)))))))))). % old.prod.exhaust
thf(fact_62_old_Oprod_Oexhaust, axiom,
    ((![Y3 : produc515445032on_val]: (~ ((![A4 : nat > option2118145160on_val, B4 : list_char > option_val]: (~ ((Y3 = (produc52622304on_val @ A4 @ B4)))))))))). % old.prod.exhaust
thf(fact_63_old_Oprod_Oinducts, axiom,
    ((![P2 : produc1172664615on_val > $o, Prod : produc1172664615on_val]: ((![A4 : produc1097884484on_val, B4 : produc1097884484on_val]: (P2 @ (produc1069526295on_val @ A4 @ B4))) => (P2 @ Prod))))). % old.prod.inducts
thf(fact_64_old_Oprod_Oinducts, axiom,
    ((![P2 : produc1097884484on_val > $o, Prod : produc1097884484on_val]: ((![A4 : exp_list_char, B4 : produc515445032on_val]: (P2 @ (produc2023458740on_val @ A4 @ B4))) => (P2 @ Prod))))). % old.prod.inducts
thf(fact_65_old_Oprod_Oinducts, axiom,
    ((![P2 : produc515445032on_val > $o, Prod : produc515445032on_val]: ((![A4 : nat > option2118145160on_val, B4 : list_char > option_val]: (P2 @ (produc52622304on_val @ A4 @ B4))) => (P2 @ Prod))))). % old.prod.inducts
thf(fact_66_prod_Ocase__distrib, axiom,
    ((![H2 : $o > $o, F : (nat > option2118145160on_val) > (list_char > option_val) > $o, Prod : produc515445032on_val]: ((H2 @ (produc99536995_val_o @ F @ Prod)) = (produc99536995_val_o @ (^[X12 : nat > option2118145160on_val]: (^[X22 : list_char > option_val]: (H2 @ (F @ X12 @ X22)))) @ Prod))))). % prod.case_distrib
thf(fact_67_prod_Ocase__distrib, axiom,
    ((![H2 : $o > $o, F : produc1097884484on_val > produc1097884484on_val > $o, Prod : produc1172664615on_val]: ((H2 @ (produc50362220_val_o @ F @ Prod)) = (produc50362220_val_o @ (^[X12 : produc1097884484on_val]: (^[X22 : produc1097884484on_val]: (H2 @ (F @ X12 @ X22)))) @ Prod))))). % prod.case_distrib
thf(fact_68_prod_Ocase__distrib, axiom,
    ((![H2 : (produc1097884484on_val > $o) > produc1097884484on_val > $o, F : exp_list_char > produc515445032on_val > produc1097884484on_val > $o, Prod : produc1097884484on_val]: ((H2 @ (produc1468025474_val_o @ F @ Prod)) = (produc1468025474_val_o @ (^[X12 : exp_list_char]: (^[X22 : produc515445032on_val]: (H2 @ (F @ X12 @ X22)))) @ Prod))))). % prod.case_distrib
thf(fact_69_prod_Ocase__distrib, axiom,
    ((![H2 : (produc1097884484on_val > $o) > $o, F : exp_list_char > produc515445032on_val > produc1097884484on_val > $o, Prod : produc1097884484on_val]: ((H2 @ (produc1468025474_val_o @ F @ Prod)) = (produc199145871_val_o @ (^[X12 : exp_list_char]: (^[X22 : produc515445032on_val]: (H2 @ (F @ X12 @ X22)))) @ Prod))))). % prod.case_distrib
thf(fact_70_prod_Ocase__distrib, axiom,
    ((![H2 : $o > produc1097884484on_val > $o, F : exp_list_char > produc515445032on_val > $o, Prod : produc1097884484on_val]: ((H2 @ (produc199145871_val_o @ F @ Prod)) = (produc1468025474_val_o @ (^[X12 : exp_list_char]: (^[X22 : produc515445032on_val]: (H2 @ (F @ X12 @ X22)))) @ Prod))))). % prod.case_distrib
thf(fact_71_prod_Ocase__distrib, axiom,
    ((![H2 : $o > $o, F : exp_list_char > produc515445032on_val > $o, Prod : produc1097884484on_val]: ((H2 @ (produc199145871_val_o @ F @ Prod)) = (produc199145871_val_o @ (^[X12 : exp_list_char]: (^[X22 : produc515445032on_val]: (H2 @ (F @ X12 @ X22)))) @ Prod))))). % prod.case_distrib
thf(fact_72_prod_Odisc__eq__case, axiom,
    ((![Prod : produc515445032on_val]: (produc99536995_val_o @ (^[Uu : nat > option2118145160on_val]: (^[Uv : list_char > option_val]: $true)) @ Prod)))). % prod.disc_eq_case
thf(fact_73_prod_Odisc__eq__case, axiom,
    ((![Prod : produc1172664615on_val]: (produc50362220_val_o @ (^[Uu : produc1097884484on_val]: (^[Uv : produc1097884484on_val]: $true)) @ Prod)))). % prod.disc_eq_case
thf(fact_74_prod_Odisc__eq__case, axiom,
    ((![Prod : produc1097884484on_val]: (produc199145871_val_o @ (^[Uu : exp_list_char]: (^[Uv : produc515445032on_val]: $true)) @ Prod)))). % prod.disc_eq_case
thf(fact_75_cond__case__prod__eta, axiom,
    ((![F : (nat > option2118145160on_val) > (list_char > option_val) > $o, G : produc515445032on_val > $o]: ((![X : nat > option2118145160on_val, Y : list_char > option_val]: ((F @ X @ Y) = (G @ (produc52622304on_val @ X @ Y)))) => ((produc99536995_val_o @ F) = G))))). % cond_case_prod_eta
thf(fact_76_cond__case__prod__eta, axiom,
    ((![F : produc1097884484on_val > produc1097884484on_val > $o, G : produc1172664615on_val > $o]: ((![X : produc1097884484on_val, Y : produc1097884484on_val]: ((F @ X @ Y) = (G @ (produc1069526295on_val @ X @ Y)))) => ((produc50362220_val_o @ F) = G))))). % cond_case_prod_eta
thf(fact_77_cond__case__prod__eta, axiom,
    ((![F : exp_list_char > produc515445032on_val > produc1097884484on_val > $o, G : produc1097884484on_val > produc1097884484on_val > $o]: ((![X : exp_list_char, Y : produc515445032on_val]: ((F @ X @ Y) = (G @ (produc2023458740on_val @ X @ Y)))) => ((produc1468025474_val_o @ F) = G))))). % cond_case_prod_eta
thf(fact_78_cond__case__prod__eta, axiom,
    ((![F : exp_list_char > produc515445032on_val > $o, G : produc1097884484on_val > $o]: ((![X : exp_list_char, Y : produc515445032on_val]: ((F @ X @ Y) = (G @ (produc2023458740on_val @ X @ Y)))) => ((produc199145871_val_o @ F) = G))))). % cond_case_prod_eta
thf(fact_79_mem__Collect__eq, axiom,
    ((![A : produc1172664615on_val, P2 : produc1172664615on_val > $o]: ((member354538064on_val @ A @ (collec2055896722on_val @ P2)) = (P2 @ A))))). % mem_Collect_eq
thf(fact_80_Collect__mem__eq, axiom,
    ((![A5 : set_Pr1189492743on_val]: ((collec2055896722on_val @ (^[X3 : produc1172664615on_val]: (member354538064on_val @ X3 @ A5))) = A5)))). % Collect_mem_eq
thf(fact_81_Collect__cong, axiom,
    ((![P2 : produc1172664615on_val > $o, Q : produc1172664615on_val > $o]: ((![X : produc1172664615on_val]: ((P2 @ X) = (Q @ X))) => ((collec2055896722on_val @ P2) = (collec2055896722on_val @ Q)))))). % Collect_cong
thf(fact_82_red__preserves__sconf, axiom,
    ((![E2 : exp_list_char, S2 : produc515445032on_val, E3 : exp_list_char, S3 : produc515445032on_val, P2 : list_P1234162913t_char, E4 : list_char > option_ty, T : ty]: ((member354538064on_val @ (produc1069526295on_val @ (produc2023458740on_val @ E2 @ S2) @ (produc2023458740on_val @ E3 @ S3)) @ (red @ P2)) => ((wTrt @ P2 @ (hp @ S2) @ E4 @ E2 @ T) => ((typeSa1777264139_sconf @ P2 @ E4 @ S2) => (typeSa1777264139_sconf @ P2 @ E4 @ S3))))))). % red_preserves_sconf
thf(fact_83_same__fstI, axiom,
    ((![P2 : (nat > option2118145160on_val) > $o, X4 : nat > option2118145160on_val, Y5 : list_char > option_val, Y3 : list_char > option_val, R : (nat > option2118145160on_val) > set_Pr1104486497on_val]: ((P2 @ X4) => ((member1177709994on_val @ (produc746553401on_val @ Y5 @ Y3) @ (R @ X4)) => (member2027247312on_val @ (produc2106877847on_val @ (produc52622304on_val @ X4 @ Y5) @ (produc52622304on_val @ X4 @ Y3)) @ (same_f1752972725on_val @ P2 @ R))))))). % same_fstI
thf(fact_84_same__fstI, axiom,
    ((![P2 : exp_list_char > $o, X4 : exp_list_char, Y5 : produc515445032on_val, Y3 : produc515445032on_val, R : exp_list_char > set_Pr148564103on_val]: ((P2 @ X4) => ((member2027247312on_val @ (produc2106877847on_val @ Y5 @ Y3) @ (R @ X4)) => (member354538064on_val @ (produc1069526295on_val @ (produc2023458740on_val @ X4 @ Y5) @ (produc2023458740on_val @ X4 @ Y3)) @ (same_f366220041on_val @ P2 @ R))))))). % same_fstI
thf(fact_85_same__fstI, axiom,
    ((![P2 : produc1097884484on_val > $o, X4 : produc1097884484on_val, Y5 : produc1097884484on_val, Y3 : produc1097884484on_val, R : produc1097884484on_val > set_Pr1189492743on_val]: ((P2 @ X4) => ((member354538064on_val @ (produc1069526295on_val @ Y5 @ Y3) @ (R @ X4)) => (member208988944on_val @ (produc564964183on_val @ (produc1069526295on_val @ X4 @ Y5) @ (produc1069526295on_val @ X4 @ Y3)) @ (same_f962657004on_val @ P2 @ R))))))). % same_fstI
thf(fact_86_rtrancl__induct2, axiom,
    ((![Ax : produc1097884484on_val, Ay : produc1097884484on_val, Bx : produc1097884484on_val, By : produc1097884484on_val, R2 : set_Pr1368603847on_val, P2 : produc1097884484on_val > produc1097884484on_val > $o]: ((member208988944on_val @ (produc564964183on_val @ (produc1069526295on_val @ Ax @ Ay) @ (produc1069526295on_val @ Bx @ By)) @ (transi100378922on_val @ R2)) => ((P2 @ Ax @ Ay) => ((![A4 : produc1097884484on_val, B4 : produc1097884484on_val, Aa : produc1097884484on_val, Ba : produc1097884484on_val]: ((member208988944on_val @ (produc564964183on_val @ (produc1069526295on_val @ Ax @ Ay) @ (produc1069526295on_val @ A4 @ B4)) @ (transi100378922on_val @ R2)) => ((member208988944on_val @ (produc564964183on_val @ (produc1069526295on_val @ A4 @ B4) @ (produc1069526295on_val @ Aa @ Ba)) @ R2) => ((P2 @ A4 @ B4) => (P2 @ Aa @ Ba))))) => (P2 @ Bx @ By))))))). % rtrancl_induct2
thf(fact_87_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, P2 : (nat > option2118145160on_val) > (list_char > option_val) > $o]: ((member2027247312on_val @ (produc2106877847on_val @ (produc52622304on_val @ Ax @ Ay) @ (produc52622304on_val @ Bx @ By)) @ (transi432893227on_val @ R2)) => ((P2 @ Ax @ Ay) => ((![A4 : nat > option2118145160on_val, B4 : list_char > option_val, Aa : nat > option2118145160on_val, Ba : list_char > option_val]: ((member2027247312on_val @ (produc2106877847on_val @ (produc52622304on_val @ Ax @ Ay) @ (produc52622304on_val @ A4 @ B4)) @ (transi432893227on_val @ R2)) => ((member2027247312on_val @ (produc2106877847on_val @ (produc52622304on_val @ A4 @ B4) @ (produc52622304on_val @ Aa @ Ba)) @ R2) => ((P2 @ A4 @ B4) => (P2 @ Aa @ Ba))))) => (P2 @ Bx @ By))))))). % rtrancl_induct2
thf(fact_88_rtrancl__induct2, axiom,
    ((![Ax : exp_list_char, Ay : produc515445032on_val, Bx : exp_list_char, By : produc515445032on_val, R2 : set_Pr1189492743on_val, P2 : exp_list_char > produc515445032on_val > $o]: ((member354538064on_val @ (produc1069526295on_val @ (produc2023458740on_val @ Ax @ Ay) @ (produc2023458740on_val @ Bx @ By)) @ (transi111096263on_val @ R2)) => ((P2 @ Ax @ Ay) => ((![A4 : exp_list_char, B4 : produc515445032on_val, Aa : exp_list_char, Ba : produc515445032on_val]: ((member354538064on_val @ (produc1069526295on_val @ (produc2023458740on_val @ Ax @ Ay) @ (produc2023458740on_val @ A4 @ B4)) @ (transi111096263on_val @ R2)) => ((member354538064on_val @ (produc1069526295on_val @ (produc2023458740on_val @ A4 @ B4) @ (produc2023458740on_val @ Aa @ Ba)) @ R2) => ((P2 @ A4 @ B4) => (P2 @ Aa @ Ba))))) => (P2 @ Bx @ By))))))). % rtrancl_induct2
thf(fact_89_converse__rtranclE2, axiom,
    ((![Xa : produc1097884484on_val, Xb : produc1097884484on_val, Za : produc1097884484on_val, Zb : produc1097884484on_val, R2 : set_Pr1368603847on_val]: ((member208988944on_val @ (produc564964183on_val @ (produc1069526295on_val @ Xa @ Xb) @ (produc1069526295on_val @ Za @ Zb)) @ (transi100378922on_val @ R2)) => ((~ (((produc1069526295on_val @ Xa @ Xb) = (produc1069526295on_val @ Za @ Zb)))) => (~ ((![A4 : produc1097884484on_val, B4 : produc1097884484on_val]: ((member208988944on_val @ (produc564964183on_val @ (produc1069526295on_val @ Xa @ Xb) @ (produc1069526295on_val @ A4 @ B4)) @ R2) => (~ ((member208988944on_val @ (produc564964183on_val @ (produc1069526295on_val @ A4 @ B4) @ (produc1069526295on_val @ Za @ Zb)) @ (transi100378922on_val @ R2))))))))))))). % converse_rtranclE2
thf(fact_90_converse__rtranclE2, axiom,
    ((![Xa : nat > option2118145160on_val, Xb : list_char > option_val, Za : nat > option2118145160on_val, Zb : list_char > option_val, R2 : set_Pr148564103on_val]: ((member2027247312on_val @ (produc2106877847on_val @ (produc52622304on_val @ Xa @ Xb) @ (produc52622304on_val @ Za @ Zb)) @ (transi432893227on_val @ R2)) => ((~ (((produc52622304on_val @ Xa @ Xb) = (produc52622304on_val @ Za @ Zb)))) => (~ ((![A4 : nat > option2118145160on_val, B4 : list_char > option_val]: ((member2027247312on_val @ (produc2106877847on_val @ (produc52622304on_val @ Xa @ Xb) @ (produc52622304on_val @ A4 @ B4)) @ R2) => (~ ((member2027247312on_val @ (produc2106877847on_val @ (produc52622304on_val @ A4 @ B4) @ (produc52622304on_val @ Za @ Zb)) @ (transi432893227on_val @ R2))))))))))))). % converse_rtranclE2
thf(fact_91_converse__rtranclE2, axiom,
    ((![Xa : exp_list_char, Xb : produc515445032on_val, Za : exp_list_char, Zb : produc515445032on_val, R2 : set_Pr1189492743on_val]: ((member354538064on_val @ (produc1069526295on_val @ (produc2023458740on_val @ Xa @ Xb) @ (produc2023458740on_val @ Za @ Zb)) @ (transi111096263on_val @ R2)) => ((~ (((produc2023458740on_val @ Xa @ Xb) = (produc2023458740on_val @ Za @ Zb)))) => (~ ((![A4 : exp_list_char, B4 : produc515445032on_val]: ((member354538064on_val @ (produc1069526295on_val @ (produc2023458740on_val @ Xa @ Xb) @ (produc2023458740on_val @ A4 @ B4)) @ R2) => (~ ((member354538064on_val @ (produc1069526295on_val @ (produc2023458740on_val @ A4 @ B4) @ (produc2023458740on_val @ Za @ Zb)) @ (transi111096263on_val @ R2))))))))))))). % converse_rtranclE2
thf(fact_92_converse__rtrancl__induct2, axiom,
    ((![Ax : produc1097884484on_val, Ay : produc1097884484on_val, Bx : produc1097884484on_val, By : produc1097884484on_val, R2 : set_Pr1368603847on_val, P2 : produc1097884484on_val > produc1097884484on_val > $o]: ((member208988944on_val @ (produc564964183on_val @ (produc1069526295on_val @ Ax @ Ay) @ (produc1069526295on_val @ Bx @ By)) @ (transi100378922on_val @ R2)) => ((P2 @ Bx @ By) => ((![A4 : produc1097884484on_val, B4 : produc1097884484on_val, Aa : produc1097884484on_val, Ba : produc1097884484on_val]: ((member208988944on_val @ (produc564964183on_val @ (produc1069526295on_val @ A4 @ B4) @ (produc1069526295on_val @ Aa @ Ba)) @ R2) => ((member208988944on_val @ (produc564964183on_val @ (produc1069526295on_val @ Aa @ Ba) @ (produc1069526295on_val @ Bx @ By)) @ (transi100378922on_val @ R2)) => ((P2 @ Aa @ Ba) => (P2 @ A4 @ B4))))) => (P2 @ Ax @ Ay))))))). % converse_rtrancl_induct2
thf(fact_93_converse__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, P2 : (nat > option2118145160on_val) > (list_char > option_val) > $o]: ((member2027247312on_val @ (produc2106877847on_val @ (produc52622304on_val @ Ax @ Ay) @ (produc52622304on_val @ Bx @ By)) @ (transi432893227on_val @ R2)) => ((P2 @ Bx @ By) => ((![A4 : nat > option2118145160on_val, B4 : list_char > option_val, Aa : nat > option2118145160on_val, Ba : list_char > option_val]: ((member2027247312on_val @ (produc2106877847on_val @ (produc52622304on_val @ A4 @ B4) @ (produc52622304on_val @ Aa @ Ba)) @ R2) => ((member2027247312on_val @ (produc2106877847on_val @ (produc52622304on_val @ Aa @ Ba) @ (produc52622304on_val @ Bx @ By)) @ (transi432893227on_val @ R2)) => ((P2 @ Aa @ Ba) => (P2 @ A4 @ B4))))) => (P2 @ Ax @ Ay))))))). % converse_rtrancl_induct2
thf(fact_94_converse__rtrancl__induct2, axiom,
    ((![Ax : exp_list_char, Ay : produc515445032on_val, Bx : exp_list_char, By : produc515445032on_val, R2 : set_Pr1189492743on_val, P2 : exp_list_char > produc515445032on_val > $o]: ((member354538064on_val @ (produc1069526295on_val @ (produc2023458740on_val @ Ax @ Ay) @ (produc2023458740on_val @ Bx @ By)) @ (transi111096263on_val @ R2)) => ((P2 @ Bx @ By) => ((![A4 : exp_list_char, B4 : produc515445032on_val, Aa : exp_list_char, Ba : produc515445032on_val]: ((member354538064on_val @ (produc1069526295on_val @ (produc2023458740on_val @ A4 @ B4) @ (produc2023458740on_val @ Aa @ Ba)) @ R2) => ((member354538064on_val @ (produc1069526295on_val @ (produc2023458740on_val @ Aa @ Ba) @ (produc2023458740on_val @ Bx @ By)) @ (transi111096263on_val @ R2)) => ((P2 @ Aa @ Ba) => (P2 @ A4 @ B4))))) => (P2 @ Ax @ Ay))))))). % converse_rtrancl_induct2
thf(fact_95_rtrancl__idemp, axiom,
    ((![R2 : set_Pr1189492743on_val]: ((transi111096263on_val @ (transi111096263on_val @ R2)) = (transi111096263on_val @ R2))))). % rtrancl_idemp
thf(fact_96_r__into__rtrancl, axiom,
    ((![P3 : produc1172664615on_val, R2 : set_Pr1189492743on_val]: ((member354538064on_val @ P3 @ R2) => (member354538064on_val @ P3 @ (transi111096263on_val @ R2)))))). % r_into_rtrancl
thf(fact_97_subject__reduction, axiom,
    ((![P2 : list_P1234162913t_char, E2 : exp_list_char, S2 : produc515445032on_val, E3 : exp_list_char, S3 : produc515445032on_val, E4 : list_char > option_ty, T : ty]: ((wf_pro1229751607t_char @ wf_J_mdecl @ P2) => ((member354538064on_val @ (produc1069526295on_val @ (produc2023458740on_val @ E2 @ S2) @ (produc2023458740on_val @ E3 @ S3)) @ (red @ P2)) => ((typeSa1777264139_sconf @ P2 @ E4 @ S2) => ((wTrt @ P2 @ (hp @ S2) @ E4 @ E2 @ T) => (?[T2 : ty]: ((wTrt @ P2 @ (hp @ S3) @ E4 @ E3 @ T2) & (widen_917054840t_char @ P2 @ T2 @ T)))))))))). % subject_reduction
thf(fact_98_rtranclE, axiom,
    ((![A : produc1097884484on_val, B : produc1097884484on_val, R2 : set_Pr1189492743on_val]: ((member354538064on_val @ (produc1069526295on_val @ A @ B) @ (transi111096263on_val @ R2)) => ((~ ((A = B))) => (~ ((![Y : produc1097884484on_val]: ((member354538064on_val @ (produc1069526295on_val @ A @ Y) @ (transi111096263on_val @ R2)) => (~ ((member354538064on_val @ (produc1069526295on_val @ Y @ B) @ R2)))))))))))). % rtranclE
thf(fact_99_rtrancl_Ocases, axiom,
    ((![A1 : produc1097884484on_val, A22 : produc1097884484on_val, R2 : set_Pr1189492743on_val]: ((member354538064on_val @ (produc1069526295on_val @ A1 @ A22) @ (transi111096263on_val @ R2)) => ((~ ((A22 = A1))) => (~ ((![B4 : produc1097884484on_val]: ((member354538064on_val @ (produc1069526295on_val @ A1 @ B4) @ (transi111096263on_val @ R2)) => (~ ((member354538064on_val @ (produc1069526295on_val @ B4 @ A22) @ R2)))))))))))). % rtrancl.cases
thf(fact_100_mem__case__prodI2, axiom,
    ((![P3 : produc1172664615on_val, Z : produc1172664615on_val, C : produc1097884484on_val > produc1097884484on_val > set_Pr1189492743on_val]: ((![A4 : produc1097884484on_val, B4 : produc1097884484on_val]: ((P3 = (produc1069526295on_val @ A4 @ B4)) => (member354538064on_val @ Z @ (C @ A4 @ B4)))) => (member354538064on_val @ Z @ (produc493625819on_val @ C @ P3)))))). % mem_case_prodI2
thf(fact_101_mem__case__prodI2, axiom,
    ((![P3 : produc1097884484on_val, Z : produc1172664615on_val, C : exp_list_char > produc515445032on_val > set_Pr1189492743on_val]: ((![A4 : exp_list_char, B4 : produc515445032on_val]: ((P3 = (produc2023458740on_val @ A4 @ B4)) => (member354538064on_val @ Z @ (C @ A4 @ B4)))) => (member354538064on_val @ Z @ (produc848217214on_val @ C @ P3)))))). % mem_case_prodI2
thf(fact_102_mem__case__prodI2, axiom,
    ((![P3 : produc515445032on_val, Z : produc1172664615on_val, C : (nat > option2118145160on_val) > (list_char > option_val) > set_Pr1189492743on_val]: ((![A4 : nat > option2118145160on_val, B4 : list_char > option_val]: ((P3 = (produc52622304on_val @ A4 @ B4)) => (member354538064on_val @ Z @ (C @ A4 @ B4)))) => (member354538064on_val @ Z @ (produc1919805778on_val @ C @ P3)))))). % mem_case_prodI2
thf(fact_103_mem__case__prodI, axiom,
    ((![Z : produc1172664615on_val, C : produc1097884484on_val > produc1097884484on_val > set_Pr1189492743on_val, A : produc1097884484on_val, B : produc1097884484on_val]: ((member354538064on_val @ Z @ (C @ A @ B)) => (member354538064on_val @ Z @ (produc493625819on_val @ C @ (produc1069526295on_val @ A @ B))))))). % mem_case_prodI
thf(fact_104_mem__case__prodI, axiom,
    ((![Z : produc1172664615on_val, C : exp_list_char > produc515445032on_val > set_Pr1189492743on_val, A : exp_list_char, B : produc515445032on_val]: ((member354538064on_val @ Z @ (C @ A @ B)) => (member354538064on_val @ Z @ (produc848217214on_val @ C @ (produc2023458740on_val @ A @ B))))))). % mem_case_prodI
thf(fact_105_mem__case__prodI, axiom,
    ((![Z : produc1172664615on_val, C : (nat > option2118145160on_val) > (list_char > option_val) > set_Pr1189492743on_val, A : nat > option2118145160on_val, B : list_char > option_val]: ((member354538064on_val @ Z @ (C @ A @ B)) => (member354538064on_val @ Z @ (produc1919805778on_val @ C @ (produc52622304on_val @ A @ B))))))). % mem_case_prodI
thf(fact_106_case__prodI2_H, axiom,
    ((![P3 : produc1097884484on_val, C : exp_list_char > produc515445032on_val > produc1097884484on_val > $o, X4 : produc1097884484on_val]: ((![A4 : exp_list_char, B4 : produc515445032on_val]: (((produc2023458740on_val @ A4 @ B4) = P3) => (C @ A4 @ B4 @ X4))) => (produc1468025474_val_o @ C @ P3 @ X4))))). % case_prodI2'
thf(fact_107_mem__case__prodE, axiom,
    ((![Z : produc1172664615on_val, C : produc1097884484on_val > produc1097884484on_val > set_Pr1189492743on_val, P3 : produc1172664615on_val]: ((member354538064on_val @ Z @ (produc493625819on_val @ C @ P3)) => (~ ((![X : produc1097884484on_val, Y : produc1097884484on_val]: ((P3 = (produc1069526295on_val @ X @ Y)) => (~ ((member354538064on_val @ Z @ (C @ X @ Y)))))))))))). % mem_case_prodE
thf(fact_108_mem__case__prodE, axiom,
    ((![Z : produc1172664615on_val, C : exp_list_char > produc515445032on_val > set_Pr1189492743on_val, P3 : produc1097884484on_val]: ((member354538064on_val @ Z @ (produc848217214on_val @ C @ P3)) => (~ ((![X : exp_list_char, Y : produc515445032on_val]: ((P3 = (produc2023458740on_val @ X @ Y)) => (~ ((member354538064on_val @ Z @ (C @ X @ Y)))))))))))). % mem_case_prodE
thf(fact_109_mem__case__prodE, axiom,
    ((![Z : produc1172664615on_val, C : (nat > option2118145160on_val) > (list_char > option_val) > set_Pr1189492743on_val, P3 : produc515445032on_val]: ((member354538064on_val @ Z @ (produc1919805778on_val @ C @ P3)) => (~ ((![X : nat > option2118145160on_val, Y : list_char > option_val]: ((P3 = (produc52622304on_val @ X @ Y)) => (~ ((member354538064on_val @ Z @ (C @ X @ Y)))))))))))). % mem_case_prodE
thf(fact_110_case__prodE_H, axiom,
    ((![C : exp_list_char > produc515445032on_val > produc1097884484on_val > $o, P3 : produc1097884484on_val, Z : produc1097884484on_val]: ((produc1468025474_val_o @ C @ P3 @ Z) => (~ ((![X : exp_list_char, Y : produc515445032on_val]: ((P3 = (produc2023458740on_val @ X @ Y)) => (~ ((C @ X @ Y @ Z))))))))))). % case_prodE'
thf(fact_111_case__prodD_H, axiom,
    ((![R : exp_list_char > produc515445032on_val > produc1097884484on_val > $o, A : exp_list_char, B : produc515445032on_val, C : produc1097884484on_val]: ((produc1468025474_val_o @ R @ (produc2023458740on_val @ A @ B) @ C) => (R @ A @ B @ C))))). % case_prodD'
thf(fact_112_same__fst__def, axiom,
    ((same_f1752972725on_val = (^[P : (nat > option2118145160on_val) > $o]: (^[R3 : (nat > option2118145160on_val) > set_Pr1104486497on_val]: (collec1161811218on_val @ (produc271780076_val_o @ (produc1588213706_val_o @ (^[X5 : nat > option2118145160on_val]: (^[Y6 : list_char > option_val]: (produc99536995_val_o @ (^[X3 : nat > option2118145160on_val]: (^[Y4 : list_char > option_val]: (((X5 = X3)) & ((((P @ X3)) & ((member1177709994on_val @ (produc746553401on_val @ Y6 @ Y4) @ (R3 @ X3))))))))))))))))))). % same_fst_def
thf(fact_113_same__fst__def, axiom,
    ((same_f962657004on_val = (^[P : produc1097884484on_val > $o]: (^[R3 : produc1097884484on_val > set_Pr1189492743on_val]: (collec608254290on_val @ (produc669005100_val_o @ (produc791746242_val_o @ (^[X5 : produc1097884484on_val]: (^[Y6 : produc1097884484on_val]: (produc50362220_val_o @ (^[X3 : produc1097884484on_val]: (^[Y4 : produc1097884484on_val]: (((X5 = X3)) & ((((P @ X3)) & ((member354538064on_val @ (produc1069526295on_val @ Y6 @ Y4) @ (R3 @ X3))))))))))))))))))). % same_fst_def
thf(fact_114_same__fst__def, axiom,
    ((same_f366220041on_val = (^[P : exp_list_char > $o]: (^[R3 : exp_list_char > set_Pr148564103on_val]: (collec2055896722on_val @ (produc50362220_val_o @ (produc1468025474_val_o @ (^[X5 : exp_list_char]: (^[Y6 : produc515445032on_val]: (produc199145871_val_o @ (^[X3 : exp_list_char]: (^[Y4 : produc515445032on_val]: (((X5 = X3)) & ((((P @ X3)) & ((member2027247312on_val @ (produc2106877847on_val @ Y6 @ Y4) @ (R3 @ X3))))))))))))))))))). % same_fst_def
thf(fact_115_subject__reduction2, axiom,
    ((![P2 : list_P1234162913t_char, E2 : exp_list_char, H2 : nat > option2118145160on_val, L2 : list_char > option_val, E3 : exp_list_char, H3 : nat > option2118145160on_val, L3 : list_char > option_val, E4 : list_char > option_ty, T : ty]: ((wf_pro1229751607t_char @ wf_J_mdecl @ P2) => ((member354538064on_val @ (produc1069526295on_val @ (produc2023458740on_val @ E2 @ (produc52622304on_val @ H2 @ L2)) @ (produc2023458740on_val @ E3 @ (produc52622304on_val @ H3 @ L3))) @ (red @ P2)) => ((typeSa1777264139_sconf @ P2 @ E4 @ (produc52622304on_val @ H2 @ L2)) => ((wTrt @ P2 @ H2 @ E4 @ E2 @ T) => (?[T2 : ty]: ((wTrt @ P2 @ H3 @ E4 @ E3 @ T2) & (widen_917054840t_char @ P2 @ T2 @ T)))))))))). % subject_reduction2
thf(fact_116_red__preserves__hconf, axiom,
    ((![E2 : exp_list_char, H2 : nat > option2118145160on_val, L2 : list_char > option_val, E3 : exp_list_char, H3 : nat > option2118145160on_val, L3 : list_char > option_val, P2 : list_P1234162913t_char, E4 : list_char > option_ty, T : ty]: ((member354538064on_val @ (produc1069526295on_val @ (produc2023458740on_val @ E2 @ (produc52622304on_val @ H2 @ L2)) @ (produc2023458740on_val @ E3 @ (produc52622304on_val @ H3 @ L3))) @ (red @ P2)) => ((wTrt @ P2 @ H2 @ E4 @ E2 @ T) => ((hconf_1244167010t_char @ P2 @ H2) => (hconf_1244167010t_char @ P2 @ H3))))))). % red_preserves_hconf
thf(fact_117_red__preserves__lconf, axiom,
    ((![E2 : exp_list_char, H2 : nat > option2118145160on_val, L2 : list_char > option_val, E3 : exp_list_char, H3 : nat > option2118145160on_val, L3 : list_char > option_val, P2 : list_P1234162913t_char, E4 : list_char > option_ty, T : ty]: ((member354538064on_val @ (produc1069526295on_val @ (produc2023458740on_val @ E2 @ (produc52622304on_val @ H2 @ L2)) @ (produc2023458740on_val @ E3 @ (produc52622304on_val @ H3 @ L3))) @ (red @ P2)) => ((wTrt @ P2 @ H2 @ E4 @ E2 @ T) => ((lconf_485928294t_char @ P2 @ H2 @ L2 @ E4) => (lconf_485928294t_char @ P2 @ H3 @ L3 @ E4))))))). % red_preserves_lconf
thf(fact_118_converse__rtrancl__into__rtrancl, axiom,
    ((![A : produc1097884484on_val, B : produc1097884484on_val, R2 : set_Pr1189492743on_val, C : produc1097884484on_val]: ((member354538064on_val @ (produc1069526295on_val @ A @ B) @ R2) => ((member354538064on_val @ (produc1069526295on_val @ B @ C) @ (transi111096263on_val @ R2)) => (member354538064on_val @ (produc1069526295on_val @ A @ C) @ (transi111096263on_val @ R2))))))). % converse_rtrancl_into_rtrancl
thf(fact_119_rtrancl_Ortrancl__into__rtrancl, axiom,
    ((![A : produc1097884484on_val, B : produc1097884484on_val, R2 : set_Pr1189492743on_val, C : produc1097884484on_val]: ((member354538064on_val @ (produc1069526295on_val @ A @ B) @ (transi111096263on_val @ R2)) => ((member354538064on_val @ (produc1069526295on_val @ B @ C) @ R2) => (member354538064on_val @ (produc1069526295on_val @ A @ C) @ (transi111096263on_val @ R2))))))). % rtrancl.rtrancl_into_rtrancl
thf(fact_120_converse__rtrancl__induct, axiom,
    ((![A : produc1097884484on_val, B : produc1097884484on_val, R2 : set_Pr1189492743on_val, P2 : produc1097884484on_val > $o]: ((member354538064on_val @ (produc1069526295on_val @ A @ B) @ (transi111096263on_val @ R2)) => ((P2 @ B) => ((![Y : produc1097884484on_val, Z2 : produc1097884484on_val]: ((member354538064on_val @ (produc1069526295on_val @ Y @ Z2) @ R2) => ((member354538064on_val @ (produc1069526295on_val @ Z2 @ B) @ (transi111096263on_val @ R2)) => ((P2 @ Z2) => (P2 @ Y))))) => (P2 @ A))))))). % converse_rtrancl_induct
thf(fact_121_rtrancl_Ortrancl__refl, axiom,
    ((![A : produc1097884484on_val, R2 : set_Pr1189492743on_val]: (member354538064on_val @ (produc1069526295on_val @ A @ A) @ (transi111096263on_val @ R2))))). % rtrancl.rtrancl_refl
thf(fact_122_converse__rtranclE, axiom,
    ((![X4 : produc1097884484on_val, Z : produc1097884484on_val, R2 : set_Pr1189492743on_val]: ((member354538064on_val @ (produc1069526295on_val @ X4 @ Z) @ (transi111096263on_val @ R2)) => ((~ ((X4 = Z))) => (~ ((![Y : produc1097884484on_val]: ((member354538064on_val @ (produc1069526295on_val @ X4 @ Y) @ R2) => (~ ((member354538064on_val @ (produc1069526295on_val @ Y @ Z) @ (transi111096263on_val @ R2))))))))))))). % converse_rtranclE
thf(fact_123_rtrancl_Oinducts, axiom,
    ((![X1 : produc1097884484on_val, X2 : produc1097884484on_val, R2 : set_Pr1189492743on_val, P2 : produc1097884484on_val > produc1097884484on_val > $o]: ((member354538064on_val @ (produc1069526295on_val @ X1 @ X2) @ (transi111096263on_val @ R2)) => ((![A4 : produc1097884484on_val]: (P2 @ A4 @ A4)) => ((![A4 : produc1097884484on_val, B4 : produc1097884484on_val, C2 : produc1097884484on_val]: ((member354538064on_val @ (produc1069526295on_val @ A4 @ B4) @ (transi111096263on_val @ R2)) => ((P2 @ A4 @ B4) => ((member354538064on_val @ (produc1069526295on_val @ B4 @ C2) @ R2) => (P2 @ A4 @ C2))))) => (P2 @ X1 @ X2))))))). % rtrancl.inducts
thf(fact_124_rtrancl__induct, axiom,
    ((![A : produc1097884484on_val, B : produc1097884484on_val, R2 : set_Pr1189492743on_val, P2 : produc1097884484on_val > $o]: ((member354538064on_val @ (produc1069526295on_val @ A @ B) @ (transi111096263on_val @ R2)) => ((P2 @ A) => ((![Y : produc1097884484on_val, Z2 : produc1097884484on_val]: ((member354538064on_val @ (produc1069526295on_val @ A @ Y) @ (transi111096263on_val @ R2)) => ((member354538064on_val @ (produc1069526295on_val @ Y @ Z2) @ R2) => ((P2 @ Y) => (P2 @ Z2))))) => (P2 @ B))))))). % rtrancl_induct
thf(fact_125_rtrancl__trans, axiom,
    ((![X4 : produc1097884484on_val, Y3 : produc1097884484on_val, R2 : set_Pr1189492743on_val, Z : produc1097884484on_val]: ((member354538064on_val @ (produc1069526295on_val @ X4 @ Y3) @ (transi111096263on_val @ R2)) => ((member354538064on_val @ (produc1069526295on_val @ Y3 @ Z) @ (transi111096263on_val @ R2)) => (member354538064on_val @ (produc1069526295on_val @ X4 @ Z) @ (transi111096263on_val @ R2))))))). % rtrancl_trans
thf(fact_126_rtrancl_Osimps, axiom,
    ((![A1 : produc1097884484on_val, A22 : produc1097884484on_val, R2 : set_Pr1189492743on_val]: ((member354538064on_val @ (produc1069526295on_val @ A1 @ A22) @ (transi111096263on_val @ R2)) = (((?[A3 : produc1097884484on_val]: (((A1 = A3)) & ((A22 = A3))))) | ((?[A3 : produc1097884484on_val]: (?[B3 : produc1097884484on_val]: (?[C3 : produc1097884484on_val]: (((A1 = A3)) & ((((A22 = C3)) & ((((member354538064on_val @ (produc1069526295on_val @ A3 @ B3) @ (transi111096263on_val @ R2))) & ((member354538064on_val @ (produc1069526295on_val @ B3 @ C3) @ R2)))))))))))))))). % rtrancl.simps
thf(fact_127_converse__rtrancl__induct__red, axiom,
    ((![E2 : exp_list_char, H2 : nat > option2118145160on_val, L2 : list_char > option_val, E3 : exp_list_char, H3 : nat > option2118145160on_val, L3 : list_char > option_val, P2 : 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 @ E2 @ (produc52622304on_val @ H2 @ L2)) @ (produc2023458740on_val @ E3 @ (produc52622304on_val @ H3 @ L3))) @ (transi111096263on_val @ (red @ P2))) => ((![E5 : exp_list_char, H4 : nat > option2118145160on_val, L4 : list_char > option_val]: (R @ E5 @ H4 @ L4 @ E5 @ H4 @ L4)) => ((![E_0 : exp_list_char, H_0 : nat > option2118145160on_val, L_0 : list_char > option_val, E_1 : exp_list_char, H_1 : nat > option2118145160on_val, L_1 : list_char > option_val, E6 : exp_list_char, H5 : nat > option2118145160on_val, L5 : list_char > option_val]: ((member354538064on_val @ (produc1069526295on_val @ (produc2023458740on_val @ E_0 @ (produc52622304on_val @ H_0 @ L_0)) @ (produc2023458740on_val @ E_1 @ (produc52622304on_val @ H_1 @ L_1))) @ (red @ P2)) => ((R @ E_1 @ H_1 @ L_1 @ E6 @ H5 @ L5) => (R @ E_0 @ H_0 @ L_0 @ E6 @ H5 @ L5)))) => (R @ E2 @ H2 @ L2 @ E3 @ H3 @ L3))))))). % converse_rtrancl_induct_red
thf(fact_128_split__cong, axiom,
    ((![Q2 : produc515445032on_val, F : (nat > option2118145160on_val) > (list_char > option_val) > $o, G : (nat > option2118145160on_val) > (list_char > option_val) > $o, P3 : produc515445032on_val]: ((![X : nat > option2118145160on_val, Y : list_char > option_val]: (((produc52622304on_val @ X @ Y) = Q2) => ((F @ X @ Y) = (G @ X @ Y)))) => ((P3 = Q2) => ((produc99536995_val_o @ F @ P3) = (produc99536995_val_o @ G @ Q2))))))). % split_cong
thf(fact_129_split__cong, axiom,
    ((![Q2 : produc1172664615on_val, F : produc1097884484on_val > produc1097884484on_val > $o, G : produc1097884484on_val > produc1097884484on_val > $o, P3 : produc1172664615on_val]: ((![X : produc1097884484on_val, Y : produc1097884484on_val]: (((produc1069526295on_val @ X @ Y) = Q2) => ((F @ X @ Y) = (G @ X @ Y)))) => ((P3 = Q2) => ((produc50362220_val_o @ F @ P3) = (produc50362220_val_o @ G @ Q2))))))). % split_cong
thf(fact_130_split__cong, axiom,
    ((![Q2 : produc1097884484on_val, F : exp_list_char > produc515445032on_val > produc1097884484on_val > $o, G : exp_list_char > produc515445032on_val > produc1097884484on_val > $o, P3 : produc1097884484on_val]: ((![X : exp_list_char, Y : produc515445032on_val]: (((produc2023458740on_val @ X @ Y) = Q2) => ((F @ X @ Y) = (G @ X @ Y)))) => ((P3 = Q2) => ((produc1468025474_val_o @ F @ P3) = (produc1468025474_val_o @ G @ Q2))))))). % split_cong
thf(fact_131_split__cong, axiom,
    ((![Q2 : produc1097884484on_val, F : exp_list_char > produc515445032on_val > $o, G : exp_list_char > produc515445032on_val > $o, P3 : produc1097884484on_val]: ((![X : exp_list_char, Y : produc515445032on_val]: (((produc2023458740on_val @ X @ Y) = Q2) => ((F @ X @ Y) = (G @ X @ Y)))) => ((P3 = Q2) => ((produc199145871_val_o @ F @ P3) = (produc199145871_val_o @ G @ Q2))))))). % split_cong
thf(fact_132_Pair__def, axiom,
    ((produc1069526295on_val = (^[A3 : produc1097884484on_val]: (^[B3 : produc1097884484on_val]: (produc1027460005on_val @ (produc1020551558on_val @ A3 @ B3))))))). % Pair_def
thf(fact_133_Pair__def, axiom,
    ((produc2023458740on_val = (^[A3 : exp_list_char]: (^[B3 : produc515445032on_val]: (produc2033438530on_val @ (produc1641020579on_val @ A3 @ B3))))))). % Pair_def
thf(fact_134_Pair__def, axiom,
    ((produc52622304on_val = (^[A3 : nat > option2118145160on_val]: (^[B3 : list_char > option_val]: (produc1377548910on_val @ (produc239262799on_val @ A3 @ B3))))))). % Pair_def
thf(fact_135_case__prod__app, axiom,
    ((produc1468025474_val_o = (^[F2 : exp_list_char > produc515445032on_val > produc1097884484on_val > $o]: (^[X3 : produc1097884484on_val]: (^[Y4 : produc1097884484on_val]: (produc199145871_val_o @ (^[L : exp_list_char]: (^[R4 : produc515445032on_val]: (F2 @ L @ R4 @ Y4))) @ X3))))))). % case_prod_app
thf(fact_136_internal__case__prod__def, axiom,
    ((produc563955725_val_o = produc99536995_val_o))). % internal_case_prod_def
thf(fact_137_internal__case__prod__def, axiom,
    ((produc1351546646_val_o = produc50362220_val_o))). % internal_case_prod_def
thf(fact_138_internal__case__prod__def, axiom,
    ((produc914566744_val_o = produc1468025474_val_o))). % internal_case_prod_def
thf(fact_139_internal__case__prod__def, axiom,
    ((produc1875489849_val_o = produc199145871_val_o))). % internal_case_prod_def
thf(fact_140_case__prod__Pair__iden, axiom,
    ((![P3 : produc1172664615on_val]: ((produc64793339on_val @ produc1069526295on_val @ P3) = P3)))). % case_prod_Pair_iden
thf(fact_141_case__prod__Pair__iden, axiom,
    ((![P3 : produc1097884484on_val]: ((produc1537230907on_val @ produc2023458740on_val @ P3) = P3)))). % case_prod_Pair_iden
thf(fact_142_case__prod__Pair__iden, axiom,
    ((![P3 : produc515445032on_val]: ((produc2088825459on_val @ produc52622304on_val @ P3) = P3)))). % case_prod_Pair_iden
thf(fact_143_lex__prod__def, axiom,
    ((lex_pr683585528on_val = (^[Ra : set_Pr1465729071on_val]: (^[Rb : set_Pr1104486497on_val]: (collec1161811218on_val @ (produc271780076_val_o @ (produc1588213706_val_o @ (^[A3 : nat > option2118145160on_val]: (^[B3 : list_char > option_val]: (produc99536995_val_o @ (^[A6 : nat > option2118145160on_val]: (^[B5 : list_char > option_val]: (((member78096504on_val @ (produc1392944775on_val @ A3 @ A6) @ Ra)) | ((((A3 = A6)) & ((member1177709994on_val @ (produc746553401on_val @ B3 @ B5) @ Rb)))))))))))))))))). % lex_prod_def
thf(fact_144_lex__prod__def, axiom,
    ((lex_pr186272047on_val = (^[Ra : set_Pr1189492743on_val]: (^[Rb : set_Pr1189492743on_val]: (collec608254290on_val @ (produc669005100_val_o @ (produc791746242_val_o @ (^[A3 : produc1097884484on_val]: (^[B3 : produc1097884484on_val]: (produc50362220_val_o @ (^[A6 : produc1097884484on_val]: (^[B5 : produc1097884484on_val]: (((member354538064on_val @ (produc1069526295on_val @ A3 @ A6) @ Ra)) | ((((A3 = A6)) & ((member354538064on_val @ (produc1069526295on_val @ B3 @ B5) @ Rb)))))))))))))))))). % lex_prod_def
thf(fact_145_lex__prod__def, axiom,
    ((lex_pr1764546508on_val = (^[Ra : set_Pr224372167t_char]: (^[Rb : set_Pr148564103on_val]: (collec2055896722on_val @ (produc50362220_val_o @ (produc1468025474_val_o @ (^[A3 : exp_list_char]: (^[B3 : produc515445032on_val]: (produc199145871_val_o @ (^[A6 : exp_list_char]: (^[B5 : produc515445032on_val]: (((member1699993616t_char @ (produc240508503t_char @ A3 @ A6) @ Ra)) | ((((A3 = A6)) & ((member2027247312on_val @ (produc2106877847on_val @ B3 @ B5) @ Rb)))))))))))))))))). % lex_prod_def
thf(fact_146_Red__lcl__add, axiom,
    ((![E2 : exp_list_char, H2 : nat > option2118145160on_val, L2 : list_char > option_val, E3 : exp_list_char, H3 : nat > option2118145160on_val, L3 : list_char > option_val, P2 : list_P1234162913t_char, L_02 : list_char > option_val]: ((member354538064on_val @ (produc1069526295on_val @ (produc2023458740on_val @ E2 @ (produc52622304on_val @ H2 @ L2)) @ (produc2023458740on_val @ E3 @ (produc52622304on_val @ H3 @ L3))) @ (transi111096263on_val @ (red @ P2))) => (member354538064on_val @ (produc1069526295on_val @ (produc2023458740on_val @ E2 @ (produc52622304on_val @ H2 @ (map_ad1615113112ar_val @ L_02 @ L2))) @ (produc2023458740on_val @ E3 @ (produc52622304on_val @ H3 @ (map_ad1615113112ar_val @ L_02 @ L3)))) @ (transi111096263on_val @ (red @ P2))))))). % Red_lcl_add
thf(fact_147_red__hext__incr, axiom,
    ((![E2 : exp_list_char, H2 : nat > option2118145160on_val, L2 : list_char > option_val, E3 : exp_list_char, H3 : nat > option2118145160on_val, L3 : list_char > option_val, P2 : list_P1234162913t_char]: ((member354538064on_val @ (produc1069526295on_val @ (produc2023458740on_val @ E2 @ (produc52622304on_val @ H2 @ L2)) @ (produc2023458740on_val @ E3 @ (produc52622304on_val @ H3 @ L3))) @ (red @ P2)) => (hext @ H2 @ H3))))). % red_hext_incr
thf(fact_148_red__lcl__add, axiom,
    ((![E2 : exp_list_char, H2 : nat > option2118145160on_val, L2 : list_char > option_val, E3 : exp_list_char, H3 : nat > option2118145160on_val, L3 : list_char > option_val, P2 : list_P1234162913t_char, L_02 : list_char > option_val]: ((member354538064on_val @ (produc1069526295on_val @ (produc2023458740on_val @ E2 @ (produc52622304on_val @ H2 @ L2)) @ (produc2023458740on_val @ E3 @ (produc52622304on_val @ H3 @ L3))) @ (red @ P2)) => (member354538064on_val @ (produc1069526295on_val @ (produc2023458740on_val @ E2 @ (produc52622304on_val @ H2 @ (map_ad1615113112ar_val @ L_02 @ L2))) @ (produc2023458740on_val @ E3 @ (produc52622304on_val @ H3 @ (map_ad1615113112ar_val @ L_02 @ L3)))) @ (red @ P2)))))). % red_lcl_add
thf(fact_149_in__lex__prod, axiom,
    ((![A : produc1097884484on_val, B : produc1097884484on_val, A2 : produc1097884484on_val, B2 : produc1097884484on_val, R2 : set_Pr1189492743on_val, S2 : set_Pr1189492743on_val]: ((member208988944on_val @ (produc564964183on_val @ (produc1069526295on_val @ A @ B) @ (produc1069526295on_val @ A2 @ B2)) @ (lex_pr186272047on_val @ R2 @ S2)) = (((member354538064on_val @ (produc1069526295on_val @ A @ A2) @ R2)) | ((((A = A2)) & ((member354538064on_val @ (produc1069526295on_val @ B @ B2) @ S2))))))))). % in_lex_prod
thf(fact_150_in__lex__prod, axiom,
    ((![A : nat > option2118145160on_val, B : list_char > option_val, A2 : nat > option2118145160on_val, B2 : list_char > option_val, R2 : set_Pr1465729071on_val, S2 : set_Pr1104486497on_val]: ((member2027247312on_val @ (produc2106877847on_val @ (produc52622304on_val @ A @ B) @ (produc52622304on_val @ A2 @ B2)) @ (lex_pr683585528on_val @ R2 @ S2)) = (((member78096504on_val @ (produc1392944775on_val @ A @ A2) @ R2)) | ((((A = A2)) & ((member1177709994on_val @ (produc746553401on_val @ B @ B2) @ S2))))))))). % in_lex_prod
thf(fact_151_in__lex__prod, axiom,
    ((![A : exp_list_char, B : produc515445032on_val, A2 : exp_list_char, B2 : produc515445032on_val, R2 : set_Pr224372167t_char, S2 : set_Pr148564103on_val]: ((member354538064on_val @ (produc1069526295on_val @ (produc2023458740on_val @ A @ B) @ (produc2023458740on_val @ A2 @ B2)) @ (lex_pr1764546508on_val @ R2 @ S2)) = (((member1699993616t_char @ (produc240508503t_char @ A @ A2) @ R2)) | ((((A = A2)) & ((member2027247312on_val @ (produc2106877847on_val @ B @ B2) @ S2))))))))). % in_lex_prod
thf(fact_152_hext__refl, axiom,
    ((![H2 : nat > option2118145160on_val]: (hext @ H2 @ H2)))). % hext_refl
thf(fact_153_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_154_lconf__hext, axiom,
    ((![P2 : list_P1234162913t_char, H2 : nat > option2118145160on_val, L2 : list_char > option_val, E4 : list_char > option_ty, H3 : nat > option2118145160on_val]: ((lconf_485928294t_char @ P2 @ H2 @ L2 @ E4) => ((hext @ H2 @ H3) => (lconf_485928294t_char @ P2 @ H3 @ L2 @ E4)))))). % lconf_hext
thf(fact_155_hext__trans, axiom,
    ((![H2 : nat > option2118145160on_val, H3 : nat > option2118145160on_val, H6 : nat > option2118145160on_val]: ((hext @ H2 @ H3) => ((hext @ H3 @ H6) => (hext @ H2 @ H6)))))). % hext_trans
thf(fact_156_WTrt__hext__mono, axiom,
    ((![P2 : list_P1234162913t_char, H2 : nat > option2118145160on_val, E4 : list_char > option_ty, E2 : exp_list_char, T : ty, H3 : nat > option2118145160on_val]: ((wTrt @ P2 @ H2 @ E4 @ E2 @ T) => ((hext @ H2 @ H3) => (wTrt @ P2 @ H3 @ E4 @ E2 @ T)))))). % WTrt_hext_mono
thf(fact_157_inv__image__def, axiom,
    ((inv_im130641106on_val = (^[R4 : set_Pr1189492743on_val]: (^[F2 : produc1097884484on_val > produc1097884484on_val]: (collec2055896722on_val @ (produc50362220_val_o @ (^[X3 : produc1097884484on_val]: (^[Y4 : produc1097884484on_val]: (member354538064on_val @ (produc1069526295on_val @ (F2 @ X3) @ (F2 @ Y4)) @ R4)))))))))). % inv_image_def
thf(fact_158_natural_Oinject, axiom,
    ((![Natural : code_natural, Natural2 : code_natural]: (((code_Suc @ Natural) = (code_Suc @ Natural2)) = (Natural = Natural2))))). % natural.inject
thf(fact_159_in__inv__image, axiom,
    ((![X4 : produc1097884484on_val, Y3 : produc1097884484on_val, R2 : set_Pr1189492743on_val, F : produc1097884484on_val > produc1097884484on_val]: ((member354538064on_val @ (produc1069526295on_val @ X4 @ Y3) @ (inv_im130641106on_val @ R2 @ F)) = (member354538064on_val @ (produc1069526295on_val @ (F @ X4) @ (F @ Y3)) @ R2))))). % in_inv_image
thf(fact_160_widen__refl, axiom,
    ((![P2 : list_P1234162913t_char, T : ty]: (widen_917054840t_char @ P2 @ T @ T)))). % widen_refl
thf(fact_161_widen__trans, axiom,
    ((![P2 : list_P1234162913t_char, S : ty, U : ty, T : ty]: ((widen_917054840t_char @ P2 @ S @ U) => ((widen_917054840t_char @ P2 @ U @ T) => (widen_917054840t_char @ P2 @ S @ T)))))). % widen_trans
thf(fact_162_redp__red__eq, axiom,
    ((redp = (^[P : list_P1234162913t_char]: (^[X3 : exp_list_char]: (^[Y4 : produc515445032on_val]: (^[Z3 : exp_list_char]: (^[Aa2 : produc515445032on_val]: (member354538064on_val @ (produc1069526295on_val @ (produc2023458740on_val @ X3 @ Y4) @ (produc2023458740on_val @ Z3 @ Aa2)) @ (red @ P)))))))))). % redp_red_eq
thf(fact_163_The__split__eq, axiom,
    ((![X4 : nat > option2118145160on_val, Y3 : list_char > option_val]: ((the_Pr958633775on_val @ (produc99536995_val_o @ (^[X5 : nat > option2118145160on_val]: (^[Y6 : list_char > option_val]: (((X4 = X5)) & ((Y3 = Y6))))))) = (produc52622304on_val @ X4 @ Y3))))). % The_split_eq
thf(fact_164_The__split__eq, axiom,
    ((![X4 : produc1097884484on_val, Y3 : produc1097884484on_val]: ((the_Pr779199278on_val @ (produc50362220_val_o @ (^[X5 : produc1097884484on_val]: (^[Y6 : produc1097884484on_val]: (((X4 = X5)) & ((Y3 = Y6))))))) = (produc1069526295on_val @ X4 @ Y3))))). % The_split_eq
thf(fact_165_The__split__eq, axiom,
    ((![X4 : exp_list_char, Y3 : produc515445032on_val]: ((the_Pr506350027on_val @ (produc199145871_val_o @ (^[X5 : exp_list_char]: (^[Y6 : produc515445032on_val]: (((X4 = X5)) & ((Y3 = Y6))))))) = (produc2023458740on_val @ X4 @ Y3))))). % The_split_eq
thf(fact_166_red__def, axiom,
    ((red = (^[P : list_P1234162913t_char]: (collec2055896722on_val @ (produc50362220_val_o @ (produc1468025474_val_o @ (^[X3 : exp_list_char]: (^[Y4 : produc515445032on_val]: (produc199145871_val_o @ (redp @ P @ X3 @ Y4))))))))))). % red_def
thf(fact_167_curry__case__prod, axiom,
    ((![F : (nat > option2118145160on_val) > (list_char > option_val) > $o]: ((produc761261956_val_o @ (produc99536995_val_o @ F)) = F)))). % curry_case_prod
thf(fact_168_curry__case__prod, axiom,
    ((![F : produc1097884484on_val > produc1097884484on_val > $o]: ((produc360843917_val_o @ (produc50362220_val_o @ F)) = F)))). % curry_case_prod
thf(fact_169_curry__case__prod, axiom,
    ((![F : exp_list_char > produc515445032on_val > produc1097884484on_val > $o]: ((produc1143790113_val_o @ (produc1468025474_val_o @ F)) = F)))). % curry_case_prod
thf(fact_170_curry__case__prod, axiom,
    ((![F : exp_list_char > produc515445032on_val > $o]: ((produc699483184_val_o @ (produc199145871_val_o @ F)) = F)))). % curry_case_prod
thf(fact_171_case__prod__curry, axiom,
    ((![F : produc515445032on_val > $o]: ((produc99536995_val_o @ (produc761261956_val_o @ F)) = F)))). % case_prod_curry
thf(fact_172_case__prod__curry, axiom,
    ((![F : produc1172664615on_val > $o]: ((produc50362220_val_o @ (produc360843917_val_o @ F)) = F)))). % case_prod_curry
thf(fact_173_case__prod__curry, axiom,
    ((![F : produc1097884484on_val > produc1097884484on_val > $o]: ((produc1468025474_val_o @ (produc1143790113_val_o @ F)) = F)))). % case_prod_curry
thf(fact_174_case__prod__curry, axiom,
    ((![F : produc1097884484on_val > $o]: ((produc199145871_val_o @ (produc699483184_val_o @ F)) = F)))). % case_prod_curry
thf(fact_175_curryI, axiom,
    ((![F : produc1172664615on_val > $o, A : produc1097884484on_val, B : produc1097884484on_val]: ((F @ (produc1069526295on_val @ A @ B)) => (produc360843917_val_o @ F @ A @ B))))). % curryI
thf(fact_176_curryI, axiom,
    ((![F : produc1097884484on_val > $o, A : exp_list_char, B : produc515445032on_val]: ((F @ (produc2023458740on_val @ A @ B)) => (produc699483184_val_o @ F @ A @ B))))). % curryI
thf(fact_177_curryI, axiom,
    ((![F : produc515445032on_val > $o, A : nat > option2118145160on_val, B : list_char > option_val]: ((F @ (produc52622304on_val @ A @ B)) => (produc761261956_val_o @ F @ A @ B))))). % curryI
thf(fact_178_curryE, axiom,
    ((![F : produc1172664615on_val > $o, A : produc1097884484on_val, B : produc1097884484on_val]: ((produc360843917_val_o @ F @ A @ B) => (F @ (produc1069526295on_val @ A @ B)))))). % curryE
thf(fact_179_curryE, axiom,
    ((![F : produc1097884484on_val > $o, A : exp_list_char, B : produc515445032on_val]: ((produc699483184_val_o @ F @ A @ B) => (F @ (produc2023458740on_val @ A @ B)))))). % curryE
thf(fact_180_curryE, axiom,
    ((![F : produc515445032on_val > $o, A : nat > option2118145160on_val, B : list_char > option_val]: ((produc761261956_val_o @ F @ A @ B) => (F @ (produc52622304on_val @ A @ B)))))). % curryE
thf(fact_181_curryD, axiom,
    ((![F : produc1172664615on_val > $o, A : produc1097884484on_val, B : produc1097884484on_val]: ((produc360843917_val_o @ F @ A @ B) => (F @ (produc1069526295on_val @ A @ B)))))). % curryD
thf(fact_182_curryD, axiom,
    ((![F : produc1097884484on_val > $o, A : exp_list_char, B : produc515445032on_val]: ((produc699483184_val_o @ F @ A @ B) => (F @ (produc2023458740on_val @ A @ B)))))). % curryD
thf(fact_183_curryD, axiom,
    ((![F : produc515445032on_val > $o, A : nat > option2118145160on_val, B : list_char > option_val]: ((produc761261956_val_o @ F @ A @ B) => (F @ (produc52622304on_val @ A @ B)))))). % curryD
thf(fact_184_case__swap, axiom,
    ((![F : (nat > option2118145160on_val) > (list_char > option_val) > $o, P3 : produc515445032on_val]: ((produc656384227_val_o @ (^[Y4 : list_char > option_val]: (^[X3 : nat > option2118145160on_val]: (F @ X3 @ Y4))) @ (produc1496715648on_val @ P3)) = (produc99536995_val_o @ F @ P3))))). % case_swap
thf(fact_185_case__swap, axiom,
    ((![F : exp_list_char > produc515445032on_val > produc1097884484on_val > $o, P3 : produc1097884484on_val]: ((produc1614918792_val_o @ (^[Y4 : produc515445032on_val]: (^[X3 : exp_list_char]: (F @ X3 @ Y4))) @ (produc916018516on_val @ P3)) = (produc1468025474_val_o @ F @ P3))))). % case_swap
thf(fact_186_case__swap, axiom,
    ((![F : exp_list_char > produc515445032on_val > $o, P3 : produc1097884484on_val]: ((produc804340233char_o @ (^[Y4 : produc515445032on_val]: (^[X3 : exp_list_char]: (F @ X3 @ Y4))) @ (produc916018516on_val @ P3)) = (produc199145871_val_o @ F @ P3))))). % case_swap
thf(fact_187_case__swap, axiom,
    ((![F : (list_char > option_val) > (nat > option2118145160on_val) > $o, P3 : produc761789224on_val]: ((produc99536995_val_o @ (^[Y4 : nat > option2118145160on_val]: (^[X3 : list_char > option_val]: (F @ X3 @ Y4))) @ (produc1513644288on_val @ P3)) = (produc656384227_val_o @ F @ P3))))). % case_swap
thf(fact_188_case__swap, axiom,
    ((![F : produc1097884484on_val > produc1097884484on_val > $o, P3 : produc1172664615on_val]: ((produc50362220_val_o @ (^[Y4 : produc1097884484on_val]: (^[X3 : produc1097884484on_val]: (F @ X3 @ Y4))) @ (produc976177847on_val @ P3)) = (produc50362220_val_o @ F @ P3))))). % case_swap
thf(fact_189_case__swap, axiom,
    ((![F : produc515445032on_val > exp_list_char > produc1097884484on_val > $o, P3 : produc1232408010t_char]: ((produc1468025474_val_o @ (^[Y4 : exp_list_char]: (^[X3 : produc515445032on_val]: (F @ X3 @ Y4))) @ (produc1741453146t_char @ P3)) = (produc1614918792_val_o @ F @ P3))))). % case_swap
thf(fact_190_case__swap, axiom,
    ((![F : produc515445032on_val > exp_list_char > $o, P3 : produc1232408010t_char]: ((produc199145871_val_o @ (^[Y4 : exp_list_char]: (^[X3 : produc515445032on_val]: (F @ X3 @ Y4))) @ (produc1741453146t_char @ P3)) = (produc804340233char_o @ F @ P3))))). % case_swap
thf(fact_191_swap__simp, axiom,
    ((![X4 : produc515445032on_val, Y3 : exp_list_char]: ((produc1741453146t_char @ (produc701409722t_char @ X4 @ Y3)) = (produc2023458740on_val @ Y3 @ X4))))). % swap_simp
thf(fact_192_swap__simp, axiom,
    ((![X4 : list_char > option_val, Y3 : nat > option2118145160on_val]: ((produc1513644288on_val @ (produc69550944on_val @ X4 @ Y3)) = (produc52622304on_val @ Y3 @ X4))))). % swap_simp
thf(fact_193_swap__simp, axiom,
    ((![X4 : produc1097884484on_val, Y3 : produc1097884484on_val]: ((produc976177847on_val @ (produc1069526295on_val @ X4 @ Y3)) = (produc1069526295on_val @ Y3 @ X4))))). % swap_simp
thf(fact_194_swap__simp, axiom,
    ((![X4 : exp_list_char, Y3 : produc515445032on_val]: ((produc916018516on_val @ (produc2023458740on_val @ X4 @ Y3)) = (produc701409722t_char @ Y3 @ X4))))). % swap_simp
thf(fact_195_swap__simp, axiom,
    ((![X4 : nat > option2118145160on_val, Y3 : list_char > option_val]: ((produc1496715648on_val @ (produc52622304on_val @ X4 @ Y3)) = (produc69550944on_val @ Y3 @ X4))))). % swap_simp
thf(fact_196_ssubst__Pair__rhs, axiom,
    ((![R2 : produc1097884484on_val, S2 : produc1097884484on_val, R : set_Pr1189492743on_val, S3 : produc1097884484on_val]: ((member354538064on_val @ (produc1069526295on_val @ R2 @ S2) @ R) => ((S3 = S2) => (member354538064on_val @ (produc1069526295on_val @ R2 @ S3) @ R)))))). % ssubst_Pair_rhs
thf(fact_197_ssubst__Pair__rhs, axiom,
    ((![R2 : exp_list_char, S2 : produc515445032on_val, R : set_Pr556513572on_val, S3 : produc515445032on_val]: ((member1888679917on_val @ (produc2023458740on_val @ R2 @ S2) @ R) => ((S3 = S2) => (member1888679917on_val @ (produc2023458740on_val @ R2 @ S3) @ R)))))). % ssubst_Pair_rhs
thf(fact_198_ssubst__Pair__rhs, axiom,
    ((![R2 : nat > option2118145160on_val, S2 : list_char > option_val, R : set_Pr723149704on_val, S3 : list_char > option_val]: ((member652041809on_val @ (produc52622304on_val @ R2 @ S2) @ R) => ((S3 = S2) => (member652041809on_val @ (produc52622304on_val @ R2 @ S3) @ R)))))). % ssubst_Pair_rhs
thf(fact_199_Range__Collect__case__prod, axiom,
    ((![P2 : (nat > option2118145160on_val) > (list_char > option_val) > $o]: ((range_229948606on_val @ (collec879567507on_val @ (produc99536995_val_o @ P2))) = (collec530568741on_val @ (^[Y4 : list_char > option_val]: (?[X3 : nat > option2118145160on_val]: (P2 @ X3 @ Y4)))))))). % Range_Collect_case_prod
thf(fact_200_Range__Collect__case__prod, axiom,
    ((![P2 : produc1097884484on_val > produc1097884484on_val > $o]: ((range_647695861on_val @ (collec2055896722on_val @ (produc50362220_val_o @ P2))) = (collec252351279on_val @ (^[Y4 : produc1097884484on_val]: (?[X3 : produc1097884484on_val]: (P2 @ X3 @ Y4)))))))). % Range_Collect_case_prod
thf(fact_201_Range__Collect__case__prod, axiom,
    ((![P2 : exp_list_char > produc515445032on_val > $o]: ((range_1652189586on_val @ (collec252351279on_val @ (produc199145871_val_o @ P2))) = (collec879567507on_val @ (^[Y4 : produc515445032on_val]: (?[X3 : exp_list_char]: (P2 @ X3 @ Y4)))))))). % Range_Collect_case_prod
thf(fact_202_Domain__Collect__case__prod, axiom,
    ((![P2 : (nat > option2118145160on_val) > (list_char > option_val) > $o]: ((domain1507236181on_val @ (collec879567507on_val @ (produc99536995_val_o @ P2))) = (collec1235631948on_val @ (^[X3 : nat > option2118145160on_val]: (?[X6 : list_char > option_val]: (P2 @ X3 @ X6)))))))). % Domain_Collect_case_prod
thf(fact_203_Domain__Collect__case__prod, axiom,
    ((![P2 : produc1097884484on_val > produc1097884484on_val > $o]: ((domain1815074444on_val @ (collec2055896722on_val @ (produc50362220_val_o @ P2))) = (collec252351279on_val @ (^[X3 : produc1097884484on_val]: (?[X6 : produc1097884484on_val]: (P2 @ X3 @ X6)))))))). % Domain_Collect_case_prod
thf(fact_204_Domain__Collect__case__prod, axiom,
    ((![P2 : exp_list_char > produc515445032on_val > $o]: ((domain327294633on_val @ (collec252351279on_val @ (produc199145871_val_o @ P2))) = (collec1062857398t_char @ (^[X3 : exp_list_char]: (?[X6 : produc515445032on_val]: (P2 @ X3 @ X6)))))))). % Domain_Collect_case_prod
thf(fact_205_Domain__unfold, axiom,
    ((domain1815074444on_val = (^[R4 : set_Pr1189492743on_val]: (collec252351279on_val @ (^[X3 : produc1097884484on_val]: (?[Y4 : produc1097884484on_val]: (member354538064on_val @ (produc1069526295on_val @ X3 @ Y4) @ R4)))))))). % Domain_unfold
thf(fact_206_Domain__unfold, axiom,
    ((domain327294633on_val = (^[R4 : set_Pr556513572on_val]: (collec1062857398t_char @ (^[X3 : exp_list_char]: (?[Y4 : produc515445032on_val]: (member1888679917on_val @ (produc2023458740on_val @ X3 @ Y4) @ R4)))))))). % Domain_unfold
thf(fact_207_Domain__unfold, axiom,
    ((domain1507236181on_val = (^[R4 : set_Pr723149704on_val]: (collec1235631948on_val @ (^[X3 : nat > option2118145160on_val]: (?[Y4 : list_char > option_val]: (member652041809on_val @ (produc52622304on_val @ X3 @ Y4) @ R4)))))))). % Domain_unfold

% Conjectures (1)
thf(conj_0, conjecture,
    ((typeSa1777264139_sconf @ p @ e @ s2))).
