% TIMEFORMAT='%3R'; { time (exec 2>&1; /home/martin/bin/satallax -E /home/martin/.isabelle/contrib/e-2.5-1/x86_64-linux/eprover -p tstp -t 5 /home/martin/judgement-day/tptp-thf/tptp/FFT/prob_79__3223336_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:09:06.407

% Could-be-implicit typings (3)
thf(ty_n_t__Set__Oset_It__Nat__Onat_J, type,
    set_nat : $tType).
thf(ty_n_t__Num__Onum, type,
    num : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).

% Explicit typings (18)
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat, type,
    times_times_nat : nat > nat > nat).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Num__Onum, type,
    times_times_num : num > num > num).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat, type,
    zero_zero_nat : nat).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Nat__Onat_M_Eo_J, type,
    inf_inf_nat_o : (nat > $o) > (nat > $o) > nat > $o).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J, type,
    inf_inf_set_nat : set_nat > set_nat > set_nat).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J, type,
    sup_sup_set_nat : set_nat > set_nat > set_nat).
thf(sy_c_Nat_OSuc, type,
    suc : nat > nat).
thf(sy_c_Num_Onum_OBit0, type,
    bit0 : num > num).
thf(sy_c_Num_Onum_OOne, type,
    one : num).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat, type,
    numeral_numeral_nat : num > nat).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J, type,
    bot_bot_nat_o : nat > $o).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat, type,
    bot_bot_nat : nat).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J, type,
    bot_bot_set_nat : set_nat).
thf(sy_c_Set_OCollect_001t__Nat__Onat, type,
    collect_nat : (nat > $o) > set_nat).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat, type,
    image_nat_nat : (nat > nat) > set_nat > set_nat).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat, type,
    set_or562006527an_nat : nat > nat > set_nat).
thf(sy_c_member_001t__Nat__Onat, type,
    member_nat : nat > set_nat > $o).
thf(sy_v_n, type,
    n : nat).

% Relevant facts (138)
thf(fact_0_image__Suc__atLeastLessThan, axiom,
    ((![I : nat, J : nat]: ((image_nat_nat @ suc @ (set_or562006527an_nat @ I @ J)) = (set_or562006527an_nat @ (suc @ I) @ (suc @ J)))))). % image_Suc_atLeastLessThan
thf(fact_1_mult__eq__1__iff, axiom,
    ((![M : nat, N : nat]: (((times_times_nat @ M @ N) = (suc @ zero_zero_nat)) = (((M = (suc @ zero_zero_nat))) & ((N = (suc @ zero_zero_nat)))))))). % mult_eq_1_iff
thf(fact_2_one__eq__mult__iff, axiom,
    ((![M : nat, N : nat]: (((suc @ zero_zero_nat) = (times_times_nat @ M @ N)) = (((M = (suc @ zero_zero_nat))) & ((N = (suc @ zero_zero_nat)))))))). % one_eq_mult_iff
thf(fact_3_mult__is__0, axiom,
    ((![M : nat, N : nat]: (((times_times_nat @ M @ N) = zero_zero_nat) = (((M = zero_zero_nat)) | ((N = zero_zero_nat))))))). % mult_is_0
thf(fact_4_mult__0__right, axiom,
    ((![M : nat]: ((times_times_nat @ M @ zero_zero_nat) = zero_zero_nat)))). % mult_0_right
thf(fact_5_mult__cancel1, axiom,
    ((![K : nat, M : nat, N : nat]: (((times_times_nat @ K @ M) = (times_times_nat @ K @ N)) = (((M = N)) | ((K = zero_zero_nat))))))). % mult_cancel1
thf(fact_6_mult__cancel2, axiom,
    ((![M : nat, K : nat, N : nat]: (((times_times_nat @ M @ K) = (times_times_nat @ N @ K)) = (((M = N)) | ((K = zero_zero_nat))))))). % mult_cancel2
thf(fact_7_Suc__double__not__eq__double, axiom,
    ((![M : nat, N : nat]: (~ (((suc @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ M)) = (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N))))))). % Suc_double_not_eq_double
thf(fact_8_double__not__eq__Suc__double, axiom,
    ((![M : nat, N : nat]: (~ (((times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ M) = (suc @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N)))))))). % double_not_eq_Suc_double
thf(fact_9_inf__bot__left, axiom,
    ((![X : set_nat]: ((inf_inf_set_nat @ bot_bot_set_nat @ X) = bot_bot_set_nat)))). % inf_bot_left
thf(fact_10_inf__bot__right, axiom,
    ((![X : set_nat]: ((inf_inf_set_nat @ X @ bot_bot_set_nat) = bot_bot_set_nat)))). % inf_bot_right
thf(fact_11_numeral__2__eq__2, axiom,
    (((numeral_numeral_nat @ (bit0 @ one)) = (suc @ (suc @ zero_zero_nat))))). % numeral_2_eq_2
thf(fact_12_numeral__eq__iff, axiom,
    ((![M : num, N : num]: (((numeral_numeral_nat @ M) = (numeral_numeral_nat @ N)) = (M = N))))). % numeral_eq_iff
thf(fact_13_old_Onat_Oinject, axiom,
    ((![Nat : nat, Nat2 : nat]: (((suc @ Nat) = (suc @ Nat2)) = (Nat = Nat2))))). % old.nat.inject
thf(fact_14_nat_Oinject, axiom,
    ((![X2 : nat, Y2 : nat]: (((suc @ X2) = (suc @ Y2)) = (X2 = Y2))))). % nat.inject
thf(fact_15_inf__right__idem, axiom,
    ((![X : set_nat, Y : set_nat]: ((inf_inf_set_nat @ (inf_inf_set_nat @ X @ Y) @ Y) = (inf_inf_set_nat @ X @ Y))))). % inf_right_idem
thf(fact_16_inf_Oright__idem, axiom,
    ((![A : set_nat, B : set_nat]: ((inf_inf_set_nat @ (inf_inf_set_nat @ A @ B) @ B) = (inf_inf_set_nat @ A @ B))))). % inf.right_idem
thf(fact_17_inf__left__idem, axiom,
    ((![X : set_nat, Y : set_nat]: ((inf_inf_set_nat @ X @ (inf_inf_set_nat @ X @ Y)) = (inf_inf_set_nat @ X @ Y))))). % inf_left_idem
thf(fact_18_inf_Oleft__idem, axiom,
    ((![A : set_nat, B : set_nat]: ((inf_inf_set_nat @ A @ (inf_inf_set_nat @ A @ B)) = (inf_inf_set_nat @ A @ B))))). % inf.left_idem
thf(fact_19_inf__idem, axiom,
    ((![X : set_nat]: ((inf_inf_set_nat @ X @ X) = X)))). % inf_idem
thf(fact_20_inf_Oidem, axiom,
    ((![A : set_nat]: ((inf_inf_set_nat @ A @ A) = A)))). % inf.idem
thf(fact_21_numeral__times__numeral, axiom,
    ((![M : num, N : num]: ((times_times_nat @ (numeral_numeral_nat @ M) @ (numeral_numeral_nat @ N)) = (numeral_numeral_nat @ (times_times_num @ M @ N)))))). % numeral_times_numeral
thf(fact_22_mult__numeral__left__semiring__numeral, axiom,
    ((![V : num, W : num, Z : nat]: ((times_times_nat @ (numeral_numeral_nat @ V) @ (times_times_nat @ (numeral_numeral_nat @ W) @ Z)) = (times_times_nat @ (numeral_numeral_nat @ (times_times_num @ V @ W)) @ Z))))). % mult_numeral_left_semiring_numeral
thf(fact_23_bot__nat__def, axiom,
    ((bot_bot_nat = zero_zero_nat))). % bot_nat_def
thf(fact_24_n__not__Suc__n, axiom,
    ((![N : nat]: (~ ((N = (suc @ N))))))). % n_not_Suc_n
thf(fact_25_Suc__inject, axiom,
    ((![X : nat, Y : nat]: (((suc @ X) = (suc @ Y)) => (X = Y))))). % Suc_inject
thf(fact_26_inf__left__commute, axiom,
    ((![X : set_nat, Y : set_nat, Z : set_nat]: ((inf_inf_set_nat @ X @ (inf_inf_set_nat @ Y @ Z)) = (inf_inf_set_nat @ Y @ (inf_inf_set_nat @ X @ Z)))))). % inf_left_commute
thf(fact_27_inf_Oleft__commute, axiom,
    ((![B : set_nat, A : set_nat, C : set_nat]: ((inf_inf_set_nat @ B @ (inf_inf_set_nat @ A @ C)) = (inf_inf_set_nat @ A @ (inf_inf_set_nat @ B @ C)))))). % inf.left_commute
thf(fact_28_inf__commute, axiom,
    ((inf_inf_set_nat = (^[X3 : set_nat]: (^[Y3 : set_nat]: (inf_inf_set_nat @ Y3 @ X3)))))). % inf_commute
thf(fact_29_inf_Ocommute, axiom,
    ((inf_inf_set_nat = (^[A2 : set_nat]: (^[B2 : set_nat]: (inf_inf_set_nat @ B2 @ A2)))))). % inf.commute
thf(fact_30_inf__assoc, axiom,
    ((![X : set_nat, Y : set_nat, Z : set_nat]: ((inf_inf_set_nat @ (inf_inf_set_nat @ X @ Y) @ Z) = (inf_inf_set_nat @ X @ (inf_inf_set_nat @ Y @ Z)))))). % inf_assoc
thf(fact_31_inf_Oassoc, axiom,
    ((![A : set_nat, B : set_nat, C : set_nat]: ((inf_inf_set_nat @ (inf_inf_set_nat @ A @ B) @ C) = (inf_inf_set_nat @ A @ (inf_inf_set_nat @ B @ C)))))). % inf.assoc
thf(fact_32_boolean__algebra__cancel_Oinf2, axiom,
    ((![B3 : set_nat, K : set_nat, B : set_nat, A : set_nat]: ((B3 = (inf_inf_set_nat @ K @ B)) => ((inf_inf_set_nat @ A @ B3) = (inf_inf_set_nat @ K @ (inf_inf_set_nat @ A @ B))))))). % boolean_algebra_cancel.inf2
thf(fact_33_boolean__algebra__cancel_Oinf1, axiom,
    ((![A3 : set_nat, K : set_nat, A : set_nat, B : set_nat]: ((A3 = (inf_inf_set_nat @ K @ A)) => ((inf_inf_set_nat @ A3 @ B) = (inf_inf_set_nat @ K @ (inf_inf_set_nat @ A @ B))))))). % boolean_algebra_cancel.inf1
thf(fact_34_inf__sup__aci_I1_J, axiom,
    ((inf_inf_set_nat = (^[X3 : set_nat]: (^[Y3 : set_nat]: (inf_inf_set_nat @ Y3 @ X3)))))). % inf_sup_aci(1)
thf(fact_35_inf__sup__aci_I2_J, axiom,
    ((![X : set_nat, Y : set_nat, Z : set_nat]: ((inf_inf_set_nat @ (inf_inf_set_nat @ X @ Y) @ Z) = (inf_inf_set_nat @ X @ (inf_inf_set_nat @ Y @ Z)))))). % inf_sup_aci(2)
thf(fact_36_inf__sup__aci_I3_J, axiom,
    ((![X : set_nat, Y : set_nat, Z : set_nat]: ((inf_inf_set_nat @ X @ (inf_inf_set_nat @ Y @ Z)) = (inf_inf_set_nat @ Y @ (inf_inf_set_nat @ X @ Z)))))). % inf_sup_aci(3)
thf(fact_37_inf__sup__aci_I4_J, axiom,
    ((![X : set_nat, Y : set_nat]: ((inf_inf_set_nat @ X @ (inf_inf_set_nat @ X @ Y)) = (inf_inf_set_nat @ X @ Y))))). % inf_sup_aci(4)
thf(fact_38_zero__neq__numeral, axiom,
    ((![N : num]: (~ ((zero_zero_nat = (numeral_numeral_nat @ N))))))). % zero_neq_numeral
thf(fact_39_not0__implies__Suc, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (?[M2 : nat]: (N = (suc @ M2))))))). % not0_implies_Suc
thf(fact_40_mem__Collect__eq, axiom,
    ((![A : nat, P : nat > $o]: ((member_nat @ A @ (collect_nat @ P)) = (P @ A))))). % mem_Collect_eq
thf(fact_41_Collect__mem__eq, axiom,
    ((![A3 : set_nat]: ((collect_nat @ (^[X3 : nat]: (member_nat @ X3 @ A3))) = A3)))). % Collect_mem_eq
thf(fact_42_old_Onat_Oinducts, axiom,
    ((![P : nat > $o, Nat : nat]: ((P @ zero_zero_nat) => ((![Nat3 : nat]: ((P @ Nat3) => (P @ (suc @ Nat3)))) => (P @ Nat)))))). % old.nat.inducts
thf(fact_43_old_Onat_Oexhaust, axiom,
    ((![Y : nat]: ((~ ((Y = zero_zero_nat))) => (~ ((![Nat3 : nat]: (~ ((Y = (suc @ Nat3))))))))))). % old.nat.exhaust
thf(fact_44_Zero__not__Suc, axiom,
    ((![M : nat]: (~ ((zero_zero_nat = (suc @ M))))))). % Zero_not_Suc
thf(fact_45_Zero__neq__Suc, axiom,
    ((![M : nat]: (~ ((zero_zero_nat = (suc @ M))))))). % Zero_neq_Suc
thf(fact_46_Suc__neq__Zero, axiom,
    ((![M : nat]: (~ (((suc @ M) = zero_zero_nat)))))). % Suc_neq_Zero
thf(fact_47_zero__induct, axiom,
    ((![P : nat > $o, K : nat]: ((P @ K) => ((![N2 : nat]: ((P @ (suc @ N2)) => (P @ N2))) => (P @ zero_zero_nat)))))). % zero_induct
thf(fact_48_diff__induct, axiom,
    ((![P : nat > nat > $o, M : nat, N : nat]: ((![X4 : nat]: (P @ X4 @ zero_zero_nat)) => ((![Y4 : nat]: (P @ zero_zero_nat @ (suc @ Y4))) => ((![X4 : nat, Y4 : nat]: ((P @ X4 @ Y4) => (P @ (suc @ X4) @ (suc @ Y4)))) => (P @ M @ N))))))). % diff_induct
thf(fact_49_nat__induct, axiom,
    ((![P : nat > $o, N : nat]: ((P @ zero_zero_nat) => ((![N2 : nat]: ((P @ N2) => (P @ (suc @ N2)))) => (P @ N)))))). % nat_induct
thf(fact_50_nat_OdiscI, axiom,
    ((![Nat : nat, X2 : nat]: ((Nat = (suc @ X2)) => (~ ((Nat = zero_zero_nat))))))). % nat.discI
thf(fact_51_old_Onat_Odistinct_I1_J, axiom,
    ((![Nat2 : nat]: (~ ((zero_zero_nat = (suc @ Nat2))))))). % old.nat.distinct(1)
thf(fact_52_old_Onat_Odistinct_I2_J, axiom,
    ((![Nat2 : nat]: (~ (((suc @ Nat2) = zero_zero_nat)))))). % old.nat.distinct(2)
thf(fact_53_nat_Odistinct_I1_J, axiom,
    ((![X2 : nat]: (~ ((zero_zero_nat = (suc @ X2))))))). % nat.distinct(1)
thf(fact_54_mult__0, axiom,
    ((![N : nat]: ((times_times_nat @ zero_zero_nat @ N) = zero_zero_nat)))). % mult_0
thf(fact_55_Suc__mult__cancel1, axiom,
    ((![K : nat, M : nat, N : nat]: (((times_times_nat @ (suc @ K) @ M) = (times_times_nat @ (suc @ K) @ N)) = (M = N))))). % Suc_mult_cancel1
thf(fact_56_mult__numeral__1__right, axiom,
    ((![A : nat]: ((times_times_nat @ A @ (numeral_numeral_nat @ one)) = A)))). % mult_numeral_1_right
thf(fact_57_mult__numeral__1, axiom,
    ((![A : nat]: ((times_times_nat @ (numeral_numeral_nat @ one) @ A) = A)))). % mult_numeral_1
thf(fact_58_ivl__disj__int__two_I3_J, axiom,
    ((![L : nat, M : nat, U : nat]: ((inf_inf_set_nat @ (set_or562006527an_nat @ L @ M) @ (set_or562006527an_nat @ M @ U)) = bot_bot_set_nat)))). % ivl_disj_int_two(3)
thf(fact_59_zero__notin__Suc__image, axiom,
    ((![A3 : set_nat]: (~ ((member_nat @ zero_zero_nat @ (image_nat_nat @ suc @ A3))))))). % zero_notin_Suc_image
thf(fact_60_atLeastLessThan0, axiom,
    ((![M : nat]: ((set_or562006527an_nat @ M @ zero_zero_nat) = bot_bot_set_nat)))). % atLeastLessThan0
thf(fact_61_numeral__1__eq__Suc__0, axiom,
    (((numeral_numeral_nat @ one) = (suc @ zero_zero_nat)))). % numeral_1_eq_Suc_0
thf(fact_62_image__is__empty, axiom,
    ((![F : nat > nat, A3 : set_nat]: (((image_nat_nat @ F @ A3) = bot_bot_set_nat) = (A3 = bot_bot_set_nat))))). % image_is_empty
thf(fact_63_empty__is__image, axiom,
    ((![F : nat > nat, A3 : set_nat]: ((bot_bot_set_nat = (image_nat_nat @ F @ A3)) = (A3 = bot_bot_set_nat))))). % empty_is_image
thf(fact_64_image__empty, axiom,
    ((![F : nat > nat]: ((image_nat_nat @ F @ bot_bot_set_nat) = bot_bot_set_nat)))). % image_empty
thf(fact_65_semiring__norm_I83_J, axiom,
    ((![N : num]: (~ ((one = (bit0 @ N))))))). % semiring_norm(83)
thf(fact_66_semiring__norm_I85_J, axiom,
    ((![M : num]: (~ (((bit0 @ M) = one)))))). % semiring_norm(85)
thf(fact_67_mult__cancel__right, axiom,
    ((![A : nat, C : nat, B : nat]: (((times_times_nat @ A @ C) = (times_times_nat @ B @ C)) = (((C = zero_zero_nat)) | ((A = B))))))). % mult_cancel_right
thf(fact_68_mult__cancel__left, axiom,
    ((![C : nat, A : nat, B : nat]: (((times_times_nat @ C @ A) = (times_times_nat @ C @ B)) = (((C = zero_zero_nat)) | ((A = B))))))). % mult_cancel_left
thf(fact_69_mult__eq__0__iff, axiom,
    ((![A : nat, B : nat]: (((times_times_nat @ A @ B) = zero_zero_nat) = (((A = zero_zero_nat)) | ((B = zero_zero_nat))))))). % mult_eq_0_iff
thf(fact_70_mult__zero__right, axiom,
    ((![A : nat]: ((times_times_nat @ A @ zero_zero_nat) = zero_zero_nat)))). % mult_zero_right
thf(fact_71_mult__zero__left, axiom,
    ((![A : nat]: ((times_times_nat @ zero_zero_nat @ A) = zero_zero_nat)))). % mult_zero_left
thf(fact_72_semiring__norm_I87_J, axiom,
    ((![M : num, N : num]: (((bit0 @ M) = (bit0 @ N)) = (M = N))))). % semiring_norm(87)
thf(fact_73_image__eqI, axiom,
    ((![B : nat, F : nat > nat, X : nat, A3 : set_nat]: ((B = (F @ X)) => ((member_nat @ X @ A3) => (member_nat @ B @ (image_nat_nat @ F @ A3))))))). % image_eqI
thf(fact_74_empty__iff, axiom,
    ((![C : nat]: (~ ((member_nat @ C @ bot_bot_set_nat)))))). % empty_iff
thf(fact_75_all__not__in__conv, axiom,
    ((![A3 : set_nat]: ((![X3 : nat]: (~ ((member_nat @ X3 @ A3)))) = (A3 = bot_bot_set_nat))))). % all_not_in_conv
thf(fact_76_Collect__empty__eq, axiom,
    ((![P : nat > $o]: (((collect_nat @ P) = bot_bot_set_nat) = (![X3 : nat]: (~ ((P @ X3)))))))). % Collect_empty_eq
thf(fact_77_empty__Collect__eq, axiom,
    ((![P : nat > $o]: ((bot_bot_set_nat = (collect_nat @ P)) = (![X3 : nat]: (~ ((P @ X3)))))))). % empty_Collect_eq
thf(fact_78_IntI, axiom,
    ((![C : nat, A3 : set_nat, B3 : set_nat]: ((member_nat @ C @ A3) => ((member_nat @ C @ B3) => (member_nat @ C @ (inf_inf_set_nat @ A3 @ B3))))))). % IntI
thf(fact_79_Int__iff, axiom,
    ((![C : nat, A3 : set_nat, B3 : set_nat]: ((member_nat @ C @ (inf_inf_set_nat @ A3 @ B3)) = (((member_nat @ C @ A3)) & ((member_nat @ C @ B3))))))). % Int_iff
thf(fact_80_image__ident, axiom,
    ((![Y5 : set_nat]: ((image_nat_nat @ (^[X3 : nat]: X3) @ Y5) = Y5)))). % image_ident
thf(fact_81_semiring__norm_I13_J, axiom,
    ((![M : num, N : num]: ((times_times_num @ (bit0 @ M) @ (bit0 @ N)) = (bit0 @ (bit0 @ (times_times_num @ M @ N))))))). % semiring_norm(13)
thf(fact_82_semiring__norm_I12_J, axiom,
    ((![N : num]: ((times_times_num @ one @ N) = N)))). % semiring_norm(12)
thf(fact_83_semiring__norm_I11_J, axiom,
    ((![M : num]: ((times_times_num @ M @ one) = M)))). % semiring_norm(11)
thf(fact_84_num__double, axiom,
    ((![N : num]: ((times_times_num @ (bit0 @ one) @ N) = (bit0 @ N))))). % num_double
thf(fact_85_imageI, axiom,
    ((![X : nat, A3 : set_nat, F : nat > nat]: ((member_nat @ X @ A3) => (member_nat @ (F @ X) @ (image_nat_nat @ F @ A3)))))). % imageI
thf(fact_86_image__iff, axiom,
    ((![Z : nat, F : nat > nat, A3 : set_nat]: ((member_nat @ Z @ (image_nat_nat @ F @ A3)) = (?[X3 : nat]: (((member_nat @ X3 @ A3)) & ((Z = (F @ X3))))))))). % image_iff
thf(fact_87_bex__imageD, axiom,
    ((![F : nat > nat, A3 : set_nat, P : nat > $o]: ((?[X5 : nat]: ((member_nat @ X5 @ (image_nat_nat @ F @ A3)) & (P @ X5))) => (?[X4 : nat]: ((member_nat @ X4 @ A3) & (P @ (F @ X4)))))))). % bex_imageD
thf(fact_88_image__cong, axiom,
    ((![M3 : set_nat, N3 : set_nat, F : nat > nat, G : nat > nat]: ((M3 = N3) => ((![X4 : nat]: ((member_nat @ X4 @ N3) => ((F @ X4) = (G @ X4)))) => ((image_nat_nat @ F @ M3) = (image_nat_nat @ G @ N3))))))). % image_cong
thf(fact_89_ball__imageD, axiom,
    ((![F : nat > nat, A3 : set_nat, P : nat > $o]: ((![X4 : nat]: ((member_nat @ X4 @ (image_nat_nat @ F @ A3)) => (P @ X4))) => (![X5 : nat]: ((member_nat @ X5 @ A3) => (P @ (F @ X5)))))))). % ball_imageD
thf(fact_90_rev__image__eqI, axiom,
    ((![X : nat, A3 : set_nat, B : nat, F : nat > nat]: ((member_nat @ X @ A3) => ((B = (F @ X)) => (member_nat @ B @ (image_nat_nat @ F @ A3))))))). % rev_image_eqI
thf(fact_91_emptyE, axiom,
    ((![A : nat]: (~ ((member_nat @ A @ bot_bot_set_nat)))))). % emptyE
thf(fact_92_equals0D, axiom,
    ((![A3 : set_nat, A : nat]: ((A3 = bot_bot_set_nat) => (~ ((member_nat @ A @ A3))))))). % equals0D
thf(fact_93_equals0I, axiom,
    ((![A3 : set_nat]: ((![Y4 : nat]: (~ ((member_nat @ Y4 @ A3)))) => (A3 = bot_bot_set_nat))))). % equals0I
thf(fact_94_ex__in__conv, axiom,
    ((![A3 : set_nat]: ((?[X3 : nat]: (member_nat @ X3 @ A3)) = (~ ((A3 = bot_bot_set_nat))))))). % ex_in_conv
thf(fact_95_bot__set__def, axiom,
    ((bot_bot_set_nat = (collect_nat @ bot_bot_nat_o)))). % bot_set_def
thf(fact_96_IntE, axiom,
    ((![C : nat, A3 : set_nat, B3 : set_nat]: ((member_nat @ C @ (inf_inf_set_nat @ A3 @ B3)) => (~ (((member_nat @ C @ A3) => (~ ((member_nat @ C @ B3)))))))))). % IntE
thf(fact_97_IntD1, axiom,
    ((![C : nat, A3 : set_nat, B3 : set_nat]: ((member_nat @ C @ (inf_inf_set_nat @ A3 @ B3)) => (member_nat @ C @ A3))))). % IntD1
thf(fact_98_IntD2, axiom,
    ((![C : nat, A3 : set_nat, B3 : set_nat]: ((member_nat @ C @ (inf_inf_set_nat @ A3 @ B3)) => (member_nat @ C @ B3))))). % IntD2
thf(fact_99_Int__assoc, axiom,
    ((![A3 : set_nat, B3 : set_nat, C2 : set_nat]: ((inf_inf_set_nat @ (inf_inf_set_nat @ A3 @ B3) @ C2) = (inf_inf_set_nat @ A3 @ (inf_inf_set_nat @ B3 @ C2)))))). % Int_assoc
thf(fact_100_Int__absorb, axiom,
    ((![A3 : set_nat]: ((inf_inf_set_nat @ A3 @ A3) = A3)))). % Int_absorb
thf(fact_101_Int__commute, axiom,
    ((inf_inf_set_nat = (^[A4 : set_nat]: (^[B4 : set_nat]: (inf_inf_set_nat @ B4 @ A4)))))). % Int_commute
thf(fact_102_Int__left__absorb, axiom,
    ((![A3 : set_nat, B3 : set_nat]: ((inf_inf_set_nat @ A3 @ (inf_inf_set_nat @ A3 @ B3)) = (inf_inf_set_nat @ A3 @ B3))))). % Int_left_absorb
thf(fact_103_Int__left__commute, axiom,
    ((![A3 : set_nat, B3 : set_nat, C2 : set_nat]: ((inf_inf_set_nat @ A3 @ (inf_inf_set_nat @ B3 @ C2)) = (inf_inf_set_nat @ B3 @ (inf_inf_set_nat @ A3 @ C2)))))). % Int_left_commute
thf(fact_104_imageE, axiom,
    ((![B : nat, F : nat > nat, A3 : set_nat]: ((member_nat @ B @ (image_nat_nat @ F @ A3)) => (~ ((![X4 : nat]: ((B = (F @ X4)) => (~ ((member_nat @ X4 @ A3))))))))))). % imageE
thf(fact_105_image__image, axiom,
    ((![F : nat > nat, G : nat > nat, A3 : set_nat]: ((image_nat_nat @ F @ (image_nat_nat @ G @ A3)) = (image_nat_nat @ (^[X3 : nat]: (F @ (G @ X3))) @ A3))))). % image_image
thf(fact_106_Compr__image__eq, axiom,
    ((![F : nat > nat, A3 : set_nat, P : nat > $o]: ((collect_nat @ (^[X3 : nat]: (((member_nat @ X3 @ (image_nat_nat @ F @ A3))) & ((P @ X3))))) = (image_nat_nat @ F @ (collect_nat @ (^[X3 : nat]: (((member_nat @ X3 @ A3)) & ((P @ (F @ X3))))))))))). % Compr_image_eq
thf(fact_107_empty__def, axiom,
    ((bot_bot_set_nat = (collect_nat @ (^[X3 : nat]: $false))))). % empty_def
thf(fact_108_Int__def, axiom,
    ((inf_inf_set_nat = (^[A4 : set_nat]: (^[B4 : set_nat]: (collect_nat @ (^[X3 : nat]: (((member_nat @ X3 @ A4)) & ((member_nat @ X3 @ B4)))))))))). % Int_def
thf(fact_109_Int__Collect, axiom,
    ((![X : nat, A3 : set_nat, P : nat > $o]: ((member_nat @ X @ (inf_inf_set_nat @ A3 @ (collect_nat @ P))) = (((member_nat @ X @ A3)) & ((P @ X))))))). % Int_Collect
thf(fact_110_Collect__conj__eq, axiom,
    ((![P : nat > $o, Q : nat > $o]: ((collect_nat @ (^[X3 : nat]: (((P @ X3)) & ((Q @ X3))))) = (inf_inf_set_nat @ (collect_nat @ P) @ (collect_nat @ Q)))))). % Collect_conj_eq
thf(fact_111_inf__set__def, axiom,
    ((inf_inf_set_nat = (^[A4 : set_nat]: (^[B4 : set_nat]: (collect_nat @ (inf_inf_nat_o @ (^[X3 : nat]: (member_nat @ X3 @ A4)) @ (^[X3 : nat]: (member_nat @ X3 @ B4))))))))). % inf_set_def
thf(fact_112_mult__not__zero, axiom,
    ((![A : nat, B : nat]: ((~ (((times_times_nat @ A @ B) = zero_zero_nat))) => ((~ ((A = zero_zero_nat))) & (~ ((B = zero_zero_nat)))))))). % mult_not_zero
thf(fact_113_divisors__zero, axiom,
    ((![A : nat, B : nat]: (((times_times_nat @ A @ B) = zero_zero_nat) => ((A = zero_zero_nat) | (B = zero_zero_nat)))))). % divisors_zero
thf(fact_114_no__zero__divisors, axiom,
    ((![A : nat, B : nat]: ((~ ((A = zero_zero_nat))) => ((~ ((B = zero_zero_nat))) => (~ (((times_times_nat @ A @ B) = zero_zero_nat)))))))). % no_zero_divisors
thf(fact_115_mult__left__cancel, axiom,
    ((![C : nat, A : nat, B : nat]: ((~ ((C = zero_zero_nat))) => (((times_times_nat @ C @ A) = (times_times_nat @ C @ B)) = (A = B)))))). % mult_left_cancel
thf(fact_116_mult__right__cancel, axiom,
    ((![C : nat, A : nat, B : nat]: ((~ ((C = zero_zero_nat))) => (((times_times_nat @ A @ C) = (times_times_nat @ B @ C)) = (A = B)))))). % mult_right_cancel
thf(fact_117_Int__emptyI, axiom,
    ((![A3 : set_nat, B3 : set_nat]: ((![X4 : nat]: ((member_nat @ X4 @ A3) => (~ ((member_nat @ X4 @ B3))))) => ((inf_inf_set_nat @ A3 @ B3) = bot_bot_set_nat))))). % Int_emptyI
thf(fact_118_disjoint__iff, axiom,
    ((![A3 : set_nat, B3 : set_nat]: (((inf_inf_set_nat @ A3 @ B3) = bot_bot_set_nat) = (![X3 : nat]: (((member_nat @ X3 @ A3)) => ((~ ((member_nat @ X3 @ B3)))))))))). % disjoint_iff
thf(fact_119_Int__empty__left, axiom,
    ((![B3 : set_nat]: ((inf_inf_set_nat @ bot_bot_set_nat @ B3) = bot_bot_set_nat)))). % Int_empty_left
thf(fact_120_Int__empty__right, axiom,
    ((![A3 : set_nat]: ((inf_inf_set_nat @ A3 @ bot_bot_set_nat) = bot_bot_set_nat)))). % Int_empty_right
thf(fact_121_disjoint__iff__not__equal, axiom,
    ((![A3 : set_nat, B3 : set_nat]: (((inf_inf_set_nat @ A3 @ B3) = bot_bot_set_nat) = (![X3 : nat]: (((member_nat @ X3 @ A3)) => ((![Y3 : nat]: (((member_nat @ Y3 @ B3)) => ((~ ((X3 = Y3))))))))))))). % disjoint_iff_not_equal
thf(fact_122_nat__mult__eq__cancel__disj, axiom,
    ((![K : nat, M : nat, N : nat]: (((times_times_nat @ K @ M) = (times_times_nat @ K @ N)) = (((K = zero_zero_nat)) | ((M = N))))))). % nat_mult_eq_cancel_disj
thf(fact_123_lambda__zero, axiom,
    (((^[H : nat]: zero_zero_nat) = (times_times_nat @ zero_zero_nat)))). % lambda_zero
thf(fact_124_ivl__splice__Un, axiom,
    ((![N : nat]: ((set_or562006527an_nat @ zero_zero_nat @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N)) = (sup_sup_set_nat @ (image_nat_nat @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one))) @ (set_or562006527an_nat @ zero_zero_nat @ N)) @ (image_nat_nat @ (^[I2 : nat]: (suc @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ I2))) @ (set_or562006527an_nat @ zero_zero_nat @ N))))))). % ivl_splice_Un
thf(fact_125_verit__eq__simplify_I8_J, axiom,
    ((![X2 : num, Y2 : num]: (((bit0 @ X2) = (bit0 @ Y2)) = (X2 = Y2))))). % verit_eq_simplify(8)
thf(fact_126_sup_Oright__idem, axiom,
    ((![A : set_nat, B : set_nat]: ((sup_sup_set_nat @ (sup_sup_set_nat @ A @ B) @ B) = (sup_sup_set_nat @ A @ B))))). % sup.right_idem
thf(fact_127_sup__left__idem, axiom,
    ((![X : set_nat, Y : set_nat]: ((sup_sup_set_nat @ X @ (sup_sup_set_nat @ X @ Y)) = (sup_sup_set_nat @ X @ Y))))). % sup_left_idem
thf(fact_128_sup_Oleft__idem, axiom,
    ((![A : set_nat, B : set_nat]: ((sup_sup_set_nat @ A @ (sup_sup_set_nat @ A @ B)) = (sup_sup_set_nat @ A @ B))))). % sup.left_idem
thf(fact_129_sup__idem, axiom,
    ((![X : set_nat]: ((sup_sup_set_nat @ X @ X) = X)))). % sup_idem
thf(fact_130_sup_Oidem, axiom,
    ((![A : set_nat]: ((sup_sup_set_nat @ A @ A) = A)))). % sup.idem
thf(fact_131_Un__iff, axiom,
    ((![C : nat, A3 : set_nat, B3 : set_nat]: ((member_nat @ C @ (sup_sup_set_nat @ A3 @ B3)) = (((member_nat @ C @ A3)) | ((member_nat @ C @ B3))))))). % Un_iff
thf(fact_132_UnCI, axiom,
    ((![C : nat, B3 : set_nat, A3 : set_nat]: (((~ ((member_nat @ C @ B3))) => (member_nat @ C @ A3)) => (member_nat @ C @ (sup_sup_set_nat @ A3 @ B3)))))). % UnCI
thf(fact_133_sup__bot__left, axiom,
    ((![X : set_nat]: ((sup_sup_set_nat @ bot_bot_set_nat @ X) = X)))). % sup_bot_left
thf(fact_134_sup__bot__right, axiom,
    ((![X : set_nat]: ((sup_sup_set_nat @ X @ bot_bot_set_nat) = X)))). % sup_bot_right
thf(fact_135_bot__eq__sup__iff, axiom,
    ((![X : set_nat, Y : set_nat]: ((bot_bot_set_nat = (sup_sup_set_nat @ X @ Y)) = (((X = bot_bot_set_nat)) & ((Y = bot_bot_set_nat))))))). % bot_eq_sup_iff
thf(fact_136_sup__eq__bot__iff, axiom,
    ((![X : set_nat, Y : set_nat]: (((sup_sup_set_nat @ X @ Y) = bot_bot_set_nat) = (((X = bot_bot_set_nat)) & ((Y = bot_bot_set_nat))))))). % sup_eq_bot_iff
thf(fact_137_sup__bot_Oeq__neutr__iff, axiom,
    ((![A : set_nat, B : set_nat]: (((sup_sup_set_nat @ A @ B) = bot_bot_set_nat) = (((A = bot_bot_set_nat)) & ((B = bot_bot_set_nat))))))). % sup_bot.eq_neutr_iff

% Conjectures (1)
thf(conj_0, conjecture,
    (((inf_inf_set_nat @ (image_nat_nat @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one))) @ (set_or562006527an_nat @ zero_zero_nat @ n)) @ (image_nat_nat @ (^[I2 : nat]: (suc @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ I2))) @ (set_or562006527an_nat @ zero_zero_nat @ n))) = bot_bot_set_nat))).
