% 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_68__3223060_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:09:00.359

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

% Explicit typings (16)
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat, type,
    plus_plus_nat : nat > nat > nat).
thf(sy_c_Groups_Oplus__class_Oplus_001tf__a, type,
    plus_plus_a : a > a > a).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Nat__Onat, type,
    groups1842438620at_nat : (nat > nat) > set_nat > nat).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001tf__a, type,
    groups1145913330_nat_a : (nat > a) > set_nat > a).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat, type,
    ord_less_nat : nat > nat > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat, type,
    ord_less_eq_nat : nat > nat > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J, type,
    ord_less_eq_set_nat : set_nat > set_nat > $o).
thf(sy_c_Set_OCollect_001t__Nat__Onat, type,
    collect_nat : (nat > $o) > 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_f, type,
    f : nat > a).
thf(sy_v_g, type,
    g : nat > a).
thf(sy_v_h, type,
    h : nat > a).
thf(sy_v_k, type,
    k : nat).
thf(sy_v_m, type,
    m : nat).
thf(sy_v_n, type,
    n : nat).

% Relevant facts (149)
thf(fact_0_le_I2_J, axiom,
    ((ord_less_eq_nat @ k @ n))). % le(2)
thf(fact_1_le_I1_J, axiom,
    ((ord_less_eq_nat @ m @ k))). % le(1)
thf(fact_2_g, axiom,
    ((![I : nat]: ((ord_less_eq_nat @ m @ I) => ((ord_less_nat @ I @ k) => ((g @ I) = (f @ I))))))). % g
thf(fact_3_h, axiom,
    ((![I : nat]: ((ord_less_eq_nat @ k @ I) => ((ord_less_nat @ I @ n) => ((h @ I) = (f @ I))))))). % h
thf(fact_4_add__left__cancel, axiom,
    ((![A : nat, B : nat, C : nat]: (((plus_plus_nat @ A @ B) = (plus_plus_nat @ A @ C)) = (B = C))))). % add_left_cancel
thf(fact_5_add__right__cancel, axiom,
    ((![B : nat, A : nat, C : nat]: (((plus_plus_nat @ B @ A) = (plus_plus_nat @ C @ A)) = (B = C))))). % add_right_cancel
thf(fact_6_sum_OatLeastLessThan__concat, axiom,
    ((![M : nat, N : nat, P : nat, G : nat > nat]: ((ord_less_eq_nat @ M @ N) => ((ord_less_eq_nat @ N @ P) => ((plus_plus_nat @ (groups1842438620at_nat @ G @ (set_or562006527an_nat @ M @ N)) @ (groups1842438620at_nat @ G @ (set_or562006527an_nat @ N @ P))) = (groups1842438620at_nat @ G @ (set_or562006527an_nat @ M @ P)))))))). % sum.atLeastLessThan_concat
thf(fact_7_sum_OatLeastLessThan__concat, axiom,
    ((![M : nat, N : nat, P : nat, G : nat > a]: ((ord_less_eq_nat @ M @ N) => ((ord_less_eq_nat @ N @ P) => ((plus_plus_a @ (groups1145913330_nat_a @ G @ (set_or562006527an_nat @ M @ N)) @ (groups1145913330_nat_a @ G @ (set_or562006527an_nat @ N @ P))) = (groups1145913330_nat_a @ G @ (set_or562006527an_nat @ M @ P)))))))). % sum.atLeastLessThan_concat
thf(fact_8_sum_Ocong, axiom,
    ((![A2 : set_nat, B2 : set_nat, G : nat > nat, H : nat > nat]: ((A2 = B2) => ((![X : nat]: ((member_nat @ X @ B2) => ((G @ X) = (H @ X)))) => ((groups1842438620at_nat @ G @ A2) = (groups1842438620at_nat @ H @ B2))))))). % sum.cong
thf(fact_9_sum_Ocong, axiom,
    ((![A2 : set_nat, B2 : set_nat, G : nat > a, H : nat > a]: ((A2 = B2) => ((![X : nat]: ((member_nat @ X @ B2) => ((G @ X) = (H @ X)))) => ((groups1145913330_nat_a @ G @ A2) = (groups1145913330_nat_a @ H @ B2))))))). % sum.cong
thf(fact_10_sum_Oeq__general, axiom,
    ((![B2 : set_nat, A2 : set_nat, H : nat > nat, Gamma : nat > nat, Phi : nat > nat]: ((![Y : nat]: ((member_nat @ Y @ B2) => (?[X2 : nat]: (((member_nat @ X2 @ A2) & ((H @ X2) = Y)) & (![Ya : nat]: (((member_nat @ Ya @ A2) & ((H @ Ya) = Y)) => (Ya = X2))))))) => ((![X : nat]: ((member_nat @ X @ A2) => ((member_nat @ (H @ X) @ B2) & ((Gamma @ (H @ X)) = (Phi @ X))))) => ((groups1842438620at_nat @ Phi @ A2) = (groups1842438620at_nat @ Gamma @ B2))))))). % sum.eq_general
thf(fact_11_sum_Oeq__general, axiom,
    ((![B2 : set_nat, A2 : set_nat, H : nat > nat, Gamma : nat > a, Phi : nat > a]: ((![Y : nat]: ((member_nat @ Y @ B2) => (?[X2 : nat]: (((member_nat @ X2 @ A2) & ((H @ X2) = Y)) & (![Ya : nat]: (((member_nat @ Ya @ A2) & ((H @ Ya) = Y)) => (Ya = X2))))))) => ((![X : nat]: ((member_nat @ X @ A2) => ((member_nat @ (H @ X) @ B2) & ((Gamma @ (H @ X)) = (Phi @ X))))) => ((groups1145913330_nat_a @ Phi @ A2) = (groups1145913330_nat_a @ Gamma @ B2))))))). % sum.eq_general
thf(fact_12_sum_Oeq__general__inverses, axiom,
    ((![B2 : set_nat, K : nat > nat, A2 : set_nat, H : nat > nat, Gamma : nat > nat, Phi : nat > nat]: ((![Y : nat]: ((member_nat @ Y @ B2) => ((member_nat @ (K @ Y) @ A2) & ((H @ (K @ Y)) = Y)))) => ((![X : nat]: ((member_nat @ X @ A2) => ((member_nat @ (H @ X) @ B2) & (((K @ (H @ X)) = X) & ((Gamma @ (H @ X)) = (Phi @ X)))))) => ((groups1842438620at_nat @ Phi @ A2) = (groups1842438620at_nat @ Gamma @ B2))))))). % sum.eq_general_inverses
thf(fact_13_sum_Oeq__general__inverses, axiom,
    ((![B2 : set_nat, K : nat > nat, A2 : set_nat, H : nat > nat, Gamma : nat > a, Phi : nat > a]: ((![Y : nat]: ((member_nat @ Y @ B2) => ((member_nat @ (K @ Y) @ A2) & ((H @ (K @ Y)) = Y)))) => ((![X : nat]: ((member_nat @ X @ A2) => ((member_nat @ (H @ X) @ B2) & (((K @ (H @ X)) = X) & ((Gamma @ (H @ X)) = (Phi @ X)))))) => ((groups1145913330_nat_a @ Phi @ A2) = (groups1145913330_nat_a @ Gamma @ B2))))))). % sum.eq_general_inverses
thf(fact_14_sum_Oreindex__bij__witness, axiom,
    ((![S : set_nat, I : nat > nat, J : nat > nat, T : set_nat, H : nat > a, G : nat > a]: ((![A3 : nat]: ((member_nat @ A3 @ S) => ((I @ (J @ A3)) = A3))) => ((![A3 : nat]: ((member_nat @ A3 @ S) => (member_nat @ (J @ A3) @ T))) => ((![B3 : nat]: ((member_nat @ B3 @ T) => ((J @ (I @ B3)) = B3))) => ((![B3 : nat]: ((member_nat @ B3 @ T) => (member_nat @ (I @ B3) @ S))) => ((![A3 : nat]: ((member_nat @ A3 @ S) => ((H @ (J @ A3)) = (G @ A3)))) => ((groups1145913330_nat_a @ G @ S) = (groups1145913330_nat_a @ H @ T)))))))))). % sum.reindex_bij_witness
thf(fact_15_sum_Oreindex__bij__witness, axiom,
    ((![S : set_nat, I : nat > nat, J : nat > nat, T : set_nat, H : nat > nat, G : nat > nat]: ((![A3 : nat]: ((member_nat @ A3 @ S) => ((I @ (J @ A3)) = A3))) => ((![A3 : nat]: ((member_nat @ A3 @ S) => (member_nat @ (J @ A3) @ T))) => ((![B3 : nat]: ((member_nat @ B3 @ T) => ((J @ (I @ B3)) = B3))) => ((![B3 : nat]: ((member_nat @ B3 @ T) => (member_nat @ (I @ B3) @ S))) => ((![A3 : nat]: ((member_nat @ A3 @ S) => ((H @ (J @ A3)) = (G @ A3)))) => ((groups1842438620at_nat @ G @ S) = (groups1842438620at_nat @ H @ T)))))))))). % sum.reindex_bij_witness
thf(fact_16_ab__semigroup__add__class_Oadd__ac_I1_J, axiom,
    ((![A : a, B : a, C : a]: ((plus_plus_a @ (plus_plus_a @ A @ B) @ C) = (plus_plus_a @ A @ (plus_plus_a @ B @ C)))))). % ab_semigroup_add_class.add_ac(1)
thf(fact_17_ab__semigroup__add__class_Oadd__ac_I1_J, axiom,
    ((![A : nat, B : nat, C : nat]: ((plus_plus_nat @ (plus_plus_nat @ A @ B) @ C) = (plus_plus_nat @ A @ (plus_plus_nat @ B @ C)))))). % ab_semigroup_add_class.add_ac(1)
thf(fact_18_add__mono__thms__linordered__semiring_I4_J, axiom,
    ((![I : nat, J : nat, K : nat, L : nat]: (((I = J) & (K = L)) => ((plus_plus_nat @ I @ K) = (plus_plus_nat @ J @ L)))))). % add_mono_thms_linordered_semiring(4)
thf(fact_19_group__cancel_Oadd1, axiom,
    ((![A2 : a, K : a, A : a, B : a]: ((A2 = (plus_plus_a @ K @ A)) => ((plus_plus_a @ A2 @ B) = (plus_plus_a @ K @ (plus_plus_a @ A @ B))))))). % group_cancel.add1
thf(fact_20_group__cancel_Oadd1, axiom,
    ((![A2 : nat, K : nat, A : nat, B : nat]: ((A2 = (plus_plus_nat @ K @ A)) => ((plus_plus_nat @ A2 @ B) = (plus_plus_nat @ K @ (plus_plus_nat @ A @ B))))))). % group_cancel.add1
thf(fact_21_group__cancel_Oadd2, axiom,
    ((![B2 : a, K : a, B : a, A : a]: ((B2 = (plus_plus_a @ K @ B)) => ((plus_plus_a @ A @ B2) = (plus_plus_a @ K @ (plus_plus_a @ A @ B))))))). % group_cancel.add2
thf(fact_22_group__cancel_Oadd2, axiom,
    ((![B2 : nat, K : nat, B : nat, A : nat]: ((B2 = (plus_plus_nat @ K @ B)) => ((plus_plus_nat @ A @ B2) = (plus_plus_nat @ K @ (plus_plus_nat @ A @ B))))))). % group_cancel.add2
thf(fact_23_add__le__cancel__right, axiom,
    ((![A : nat, C : nat, B : nat]: ((ord_less_eq_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ C)) = (ord_less_eq_nat @ A @ B))))). % add_le_cancel_right
thf(fact_24_add__le__cancel__left, axiom,
    ((![C : nat, A : nat, B : nat]: ((ord_less_eq_nat @ (plus_plus_nat @ C @ A) @ (plus_plus_nat @ C @ B)) = (ord_less_eq_nat @ A @ B))))). % add_le_cancel_left
thf(fact_25_add__less__cancel__right, axiom,
    ((![A : nat, C : nat, B : nat]: ((ord_less_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ C)) = (ord_less_nat @ A @ B))))). % add_less_cancel_right
thf(fact_26_add__less__cancel__left, axiom,
    ((![C : nat, A : nat, B : nat]: ((ord_less_nat @ (plus_plus_nat @ C @ A) @ (plus_plus_nat @ C @ B)) = (ord_less_nat @ A @ B))))). % add_less_cancel_left
thf(fact_27_ivl__subset, axiom,
    ((![I : nat, J : nat, M : nat, N : nat]: ((ord_less_eq_set_nat @ (set_or562006527an_nat @ I @ J) @ (set_or562006527an_nat @ M @ N)) = (((ord_less_eq_nat @ J @ I)) | ((((ord_less_eq_nat @ M @ I)) & ((ord_less_eq_nat @ J @ N))))))))). % ivl_subset
thf(fact_28_atLeastLessThan__iff, axiom,
    ((![I : nat, L : nat, U : nat]: ((member_nat @ I @ (set_or562006527an_nat @ L @ U)) = (((ord_less_eq_nat @ L @ I)) & ((ord_less_nat @ I @ U))))))). % atLeastLessThan_iff
thf(fact_29_bounded__Max__nat, axiom,
    ((![P2 : nat > $o, X3 : nat, M2 : nat]: ((P2 @ X3) => ((![X : nat]: ((P2 @ X) => (ord_less_eq_nat @ X @ M2))) => (~ ((![M3 : nat]: ((P2 @ M3) => (~ ((![X2 : nat]: ((P2 @ X2) => (ord_less_eq_nat @ X2 @ M3)))))))))))))). % bounded_Max_nat
thf(fact_30_atLeastLessThan__subset__iff, axiom,
    ((![A : nat, B : nat, C : nat, D : nat]: ((ord_less_eq_set_nat @ (set_or562006527an_nat @ A @ B) @ (set_or562006527an_nat @ C @ D)) => ((ord_less_eq_nat @ B @ A) | ((ord_less_eq_nat @ C @ A) & (ord_less_eq_nat @ B @ D))))))). % atLeastLessThan_subset_iff
thf(fact_31_add__less__le__mono, axiom,
    ((![A : nat, B : nat, C : nat, D : nat]: ((ord_less_nat @ A @ B) => ((ord_less_eq_nat @ C @ D) => (ord_less_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ D))))))). % add_less_le_mono
thf(fact_32_add__le__less__mono, axiom,
    ((![A : nat, B : nat, C : nat, D : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_nat @ C @ D) => (ord_less_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ D))))))). % add_le_less_mono
thf(fact_33_add__mono__thms__linordered__field_I3_J, axiom,
    ((![I : nat, J : nat, K : nat, L : nat]: (((ord_less_nat @ I @ J) & (ord_less_eq_nat @ K @ L)) => (ord_less_nat @ (plus_plus_nat @ I @ K) @ (plus_plus_nat @ J @ L)))))). % add_mono_thms_linordered_field(3)
thf(fact_34_add__mono__thms__linordered__field_I4_J, axiom,
    ((![I : nat, J : nat, K : nat, L : nat]: (((ord_less_eq_nat @ I @ J) & (ord_less_nat @ K @ L)) => (ord_less_nat @ (plus_plus_nat @ I @ K) @ (plus_plus_nat @ J @ L)))))). % add_mono_thms_linordered_field(4)
thf(fact_35_add__less__imp__less__right, axiom,
    ((![A : nat, C : nat, B : nat]: ((ord_less_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ C)) => (ord_less_nat @ A @ B))))). % add_less_imp_less_right
thf(fact_36_add__less__imp__less__left, axiom,
    ((![C : nat, A : nat, B : nat]: ((ord_less_nat @ (plus_plus_nat @ C @ A) @ (plus_plus_nat @ C @ B)) => (ord_less_nat @ A @ B))))). % add_less_imp_less_left
thf(fact_37_add__strict__right__mono, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_nat @ A @ B) => (ord_less_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ C)))))). % add_strict_right_mono
thf(fact_38_add__strict__left__mono, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_nat @ A @ B) => (ord_less_nat @ (plus_plus_nat @ C @ A) @ (plus_plus_nat @ C @ B)))))). % add_strict_left_mono
thf(fact_39_add__strict__mono, axiom,
    ((![A : nat, B : nat, C : nat, D : nat]: ((ord_less_nat @ A @ B) => ((ord_less_nat @ C @ D) => (ord_less_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ D))))))). % add_strict_mono
thf(fact_40_add__mono__thms__linordered__field_I1_J, axiom,
    ((![I : nat, J : nat, K : nat, L : nat]: (((ord_less_nat @ I @ J) & (K = L)) => (ord_less_nat @ (plus_plus_nat @ I @ K) @ (plus_plus_nat @ J @ L)))))). % add_mono_thms_linordered_field(1)
thf(fact_41_add__mono__thms__linordered__field_I2_J, axiom,
    ((![I : nat, J : nat, K : nat, L : nat]: (((I = J) & (ord_less_nat @ K @ L)) => (ord_less_nat @ (plus_plus_nat @ I @ K) @ (plus_plus_nat @ J @ L)))))). % add_mono_thms_linordered_field(2)
thf(fact_42_add__mono__thms__linordered__field_I5_J, axiom,
    ((![I : nat, J : nat, K : nat, L : nat]: (((ord_less_nat @ I @ J) & (ord_less_nat @ K @ L)) => (ord_less_nat @ (plus_plus_nat @ I @ K) @ (plus_plus_nat @ J @ L)))))). % add_mono_thms_linordered_field(5)
thf(fact_43_add__le__imp__le__right, axiom,
    ((![A : nat, C : nat, B : nat]: ((ord_less_eq_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ C)) => (ord_less_eq_nat @ A @ B))))). % add_le_imp_le_right
thf(fact_44_add__le__imp__le__left, axiom,
    ((![C : nat, A : nat, B : nat]: ((ord_less_eq_nat @ (plus_plus_nat @ C @ A) @ (plus_plus_nat @ C @ B)) => (ord_less_eq_nat @ A @ B))))). % add_le_imp_le_left
thf(fact_45_le__iff__add, axiom,
    ((ord_less_eq_nat = (^[A4 : nat]: (^[B4 : nat]: (?[C2 : nat]: (B4 = (plus_plus_nat @ A4 @ C2)))))))). % le_iff_add
thf(fact_46_add__right__mono, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ B) => (ord_less_eq_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ C)))))). % add_right_mono
thf(fact_47_less__eqE, axiom,
    ((![A : nat, B : nat]: ((ord_less_eq_nat @ A @ B) => (~ ((![C3 : nat]: (~ ((B = (plus_plus_nat @ A @ C3))))))))))). % less_eqE
thf(fact_48_add__left__mono, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ B) => (ord_less_eq_nat @ (plus_plus_nat @ C @ A) @ (plus_plus_nat @ C @ B)))))). % add_left_mono
thf(fact_49_add__mono, axiom,
    ((![A : nat, B : nat, C : nat, D : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_nat @ C @ D) => (ord_less_eq_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ D))))))). % add_mono
thf(fact_50_add__mono__thms__linordered__semiring_I1_J, axiom,
    ((![I : nat, J : nat, K : nat, L : nat]: (((ord_less_eq_nat @ I @ J) & (ord_less_eq_nat @ K @ L)) => (ord_less_eq_nat @ (plus_plus_nat @ I @ K) @ (plus_plus_nat @ J @ L)))))). % add_mono_thms_linordered_semiring(1)
thf(fact_51_add__mono__thms__linordered__semiring_I2_J, axiom,
    ((![I : nat, J : nat, K : nat, L : nat]: (((I = J) & (ord_less_eq_nat @ K @ L)) => (ord_less_eq_nat @ (plus_plus_nat @ I @ K) @ (plus_plus_nat @ J @ L)))))). % add_mono_thms_linordered_semiring(2)
thf(fact_52_mem__Collect__eq, axiom,
    ((![A : nat, P2 : nat > $o]: ((member_nat @ A @ (collect_nat @ P2)) = (P2 @ A))))). % mem_Collect_eq
thf(fact_53_Collect__mem__eq, axiom,
    ((![A2 : set_nat]: ((collect_nat @ (^[X4 : nat]: (member_nat @ X4 @ A2))) = A2)))). % Collect_mem_eq
thf(fact_54_add__mono__thms__linordered__semiring_I3_J, axiom,
    ((![I : nat, J : nat, K : nat, L : nat]: (((ord_less_eq_nat @ I @ J) & (K = L)) => (ord_less_eq_nat @ (plus_plus_nat @ I @ K) @ (plus_plus_nat @ J @ L)))))). % add_mono_thms_linordered_semiring(3)
thf(fact_55_atLeastLessThan__eq__iff, axiom,
    ((![A : nat, B : nat, C : nat, D : nat]: ((ord_less_nat @ A @ B) => ((ord_less_nat @ C @ D) => (((set_or562006527an_nat @ A @ B) = (set_or562006527an_nat @ C @ D)) = (((A = C)) & ((B = D))))))))). % atLeastLessThan_eq_iff
thf(fact_56_atLeastLessThan__inj_I1_J, axiom,
    ((![A : nat, B : nat, C : nat, D : nat]: (((set_or562006527an_nat @ A @ B) = (set_or562006527an_nat @ C @ D)) => ((ord_less_nat @ A @ B) => ((ord_less_nat @ C @ D) => (A = C))))))). % atLeastLessThan_inj(1)
thf(fact_57_atLeastLessThan__inj_I2_J, axiom,
    ((![A : nat, B : nat, C : nat, D : nat]: (((set_or562006527an_nat @ A @ B) = (set_or562006527an_nat @ C @ D)) => ((ord_less_nat @ A @ B) => ((ord_less_nat @ C @ D) => (B = D))))))). % atLeastLessThan_inj(2)
thf(fact_58_sum_Oivl__cong, axiom,
    ((![A : nat, C : nat, B : nat, D : nat, G : nat > a, H : nat > a]: ((A = C) => ((B = D) => ((![X : nat]: ((ord_less_eq_nat @ C @ X) => ((ord_less_nat @ X @ D) => ((G @ X) = (H @ X))))) => ((groups1145913330_nat_a @ G @ (set_or562006527an_nat @ A @ B)) = (groups1145913330_nat_a @ H @ (set_or562006527an_nat @ C @ D))))))))). % sum.ivl_cong
thf(fact_59_sum_Oivl__cong, axiom,
    ((![A : nat, C : nat, B : nat, D : nat, G : nat > nat, H : nat > nat]: ((A = C) => ((B = D) => ((![X : nat]: ((ord_less_eq_nat @ C @ X) => ((ord_less_nat @ X @ D) => ((G @ X) = (H @ X))))) => ((groups1842438620at_nat @ G @ (set_or562006527an_nat @ A @ B)) = (groups1842438620at_nat @ H @ (set_or562006527an_nat @ C @ D))))))))). % sum.ivl_cong
thf(fact_60_add__right__imp__eq, axiom,
    ((![B : nat, A : nat, C : nat]: (((plus_plus_nat @ B @ A) = (plus_plus_nat @ C @ A)) => (B = C))))). % add_right_imp_eq
thf(fact_61_add__left__imp__eq, axiom,
    ((![A : nat, B : nat, C : nat]: (((plus_plus_nat @ A @ B) = (plus_plus_nat @ A @ C)) => (B = C))))). % add_left_imp_eq
thf(fact_62_add_Oleft__commute, axiom,
    ((![B : a, A : a, C : a]: ((plus_plus_a @ B @ (plus_plus_a @ A @ C)) = (plus_plus_a @ A @ (plus_plus_a @ B @ C)))))). % add.left_commute
thf(fact_63_add_Oleft__commute, axiom,
    ((![B : nat, A : nat, C : nat]: ((plus_plus_nat @ B @ (plus_plus_nat @ A @ C)) = (plus_plus_nat @ A @ (plus_plus_nat @ B @ C)))))). % add.left_commute
thf(fact_64_add_Ocommute, axiom,
    ((plus_plus_a = (^[A4 : a]: (^[B4 : a]: (plus_plus_a @ B4 @ A4)))))). % add.commute
thf(fact_65_add_Ocommute, axiom,
    ((plus_plus_nat = (^[A4 : nat]: (^[B4 : nat]: (plus_plus_nat @ B4 @ A4)))))). % add.commute
thf(fact_66_add_Oassoc, axiom,
    ((![A : a, B : a, C : a]: ((plus_plus_a @ (plus_plus_a @ A @ B) @ C) = (plus_plus_a @ A @ (plus_plus_a @ B @ C)))))). % add.assoc
thf(fact_67_add_Oassoc, axiom,
    ((![A : nat, B : nat, C : nat]: ((plus_plus_nat @ (plus_plus_nat @ A @ B) @ C) = (plus_plus_nat @ A @ (plus_plus_nat @ B @ C)))))). % add.assoc
thf(fact_68_nat__add__left__cancel__le, axiom,
    ((![K : nat, M : nat, N : nat]: ((ord_less_eq_nat @ (plus_plus_nat @ K @ M) @ (plus_plus_nat @ K @ N)) = (ord_less_eq_nat @ M @ N))))). % nat_add_left_cancel_le
thf(fact_69_nat__add__left__cancel__less, axiom,
    ((![K : nat, M : nat, N : nat]: ((ord_less_nat @ (plus_plus_nat @ K @ M) @ (plus_plus_nat @ K @ N)) = (ord_less_nat @ M @ N))))). % nat_add_left_cancel_less
thf(fact_70_order__refl, axiom,
    ((![X3 : nat]: (ord_less_eq_nat @ X3 @ X3)))). % order_refl
thf(fact_71_nat__less__le, axiom,
    ((ord_less_nat = (^[M4 : nat]: (^[N2 : nat]: (((ord_less_eq_nat @ M4 @ N2)) & ((~ ((M4 = N2)))))))))). % nat_less_le
thf(fact_72_less__imp__le__nat, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (ord_less_eq_nat @ M @ N))))). % less_imp_le_nat
thf(fact_73_le__eq__less__or__eq, axiom,
    ((ord_less_eq_nat = (^[M4 : nat]: (^[N2 : nat]: (((ord_less_nat @ M4 @ N2)) | ((M4 = N2)))))))). % le_eq_less_or_eq
thf(fact_74_less__or__eq__imp__le, axiom,
    ((![M : nat, N : nat]: (((ord_less_nat @ M @ N) | (M = N)) => (ord_less_eq_nat @ M @ N))))). % less_or_eq_imp_le
thf(fact_75_mono__nat__linear__lb, axiom,
    ((![F : nat > nat, M : nat, K : nat]: ((![M3 : nat, N3 : nat]: ((ord_less_nat @ M3 @ N3) => (ord_less_nat @ (F @ M3) @ (F @ N3)))) => (ord_less_eq_nat @ (plus_plus_nat @ (F @ M) @ K) @ (F @ (plus_plus_nat @ M @ K))))))). % mono_nat_linear_lb
thf(fact_76_le__neq__implies__less, axiom,
    ((![M : nat, N : nat]: ((ord_less_eq_nat @ M @ N) => ((~ ((M = N))) => (ord_less_nat @ M @ N)))))). % le_neq_implies_less
thf(fact_77_dual__order_Oantisym, axiom,
    ((![B : nat, A : nat]: ((ord_less_eq_nat @ B @ A) => ((ord_less_eq_nat @ A @ B) => (A = B)))))). % dual_order.antisym
thf(fact_78_dual__order_Oeq__iff, axiom,
    (((^[Y2 : nat]: (^[Z : nat]: (Y2 = Z))) = (^[A4 : nat]: (^[B4 : nat]: (((ord_less_eq_nat @ B4 @ A4)) & ((ord_less_eq_nat @ A4 @ B4)))))))). % dual_order.eq_iff
thf(fact_79_dual__order_Otrans, axiom,
    ((![B : nat, A : nat, C : nat]: ((ord_less_eq_nat @ B @ A) => ((ord_less_eq_nat @ C @ B) => (ord_less_eq_nat @ C @ A)))))). % dual_order.trans
thf(fact_80_linorder__wlog, axiom,
    ((![P2 : nat > nat > $o, A : nat, B : nat]: ((![A3 : nat, B3 : nat]: ((ord_less_eq_nat @ A3 @ B3) => (P2 @ A3 @ B3))) => ((![A3 : nat, B3 : nat]: ((P2 @ B3 @ A3) => (P2 @ A3 @ B3))) => (P2 @ A @ B)))))). % linorder_wlog
thf(fact_81_dual__order_Orefl, axiom,
    ((![A : nat]: (ord_less_eq_nat @ A @ A)))). % dual_order.refl
thf(fact_82_order__trans, axiom,
    ((![X3 : nat, Y3 : nat, Z2 : nat]: ((ord_less_eq_nat @ X3 @ Y3) => ((ord_less_eq_nat @ Y3 @ Z2) => (ord_less_eq_nat @ X3 @ Z2)))))). % order_trans
thf(fact_83_order__class_Oorder_Oantisym, axiom,
    ((![A : nat, B : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_nat @ B @ A) => (A = B)))))). % order_class.order.antisym
thf(fact_84_ord__le__eq__trans, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ B) => ((B = C) => (ord_less_eq_nat @ A @ C)))))). % ord_le_eq_trans
thf(fact_85_ord__eq__le__trans, axiom,
    ((![A : nat, B : nat, C : nat]: ((A = B) => ((ord_less_eq_nat @ B @ C) => (ord_less_eq_nat @ A @ C)))))). % ord_eq_le_trans
thf(fact_86_order__class_Oorder_Oeq__iff, axiom,
    (((^[Y2 : nat]: (^[Z : nat]: (Y2 = Z))) = (^[A4 : nat]: (^[B4 : nat]: (((ord_less_eq_nat @ A4 @ B4)) & ((ord_less_eq_nat @ B4 @ A4)))))))). % order_class.order.eq_iff
thf(fact_87_antisym__conv, axiom,
    ((![Y3 : nat, X3 : nat]: ((ord_less_eq_nat @ Y3 @ X3) => ((ord_less_eq_nat @ X3 @ Y3) = (X3 = Y3)))))). % antisym_conv
thf(fact_88_le__cases3, axiom,
    ((![X3 : nat, Y3 : nat, Z2 : nat]: (((ord_less_eq_nat @ X3 @ Y3) => (~ ((ord_less_eq_nat @ Y3 @ Z2)))) => (((ord_less_eq_nat @ Y3 @ X3) => (~ ((ord_less_eq_nat @ X3 @ Z2)))) => (((ord_less_eq_nat @ X3 @ Z2) => (~ ((ord_less_eq_nat @ Z2 @ Y3)))) => (((ord_less_eq_nat @ Z2 @ Y3) => (~ ((ord_less_eq_nat @ Y3 @ X3)))) => (((ord_less_eq_nat @ Y3 @ Z2) => (~ ((ord_less_eq_nat @ Z2 @ X3)))) => (~ (((ord_less_eq_nat @ Z2 @ X3) => (~ ((ord_less_eq_nat @ X3 @ Y3)))))))))))))). % le_cases3
thf(fact_89_order_Otrans, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_nat @ B @ C) => (ord_less_eq_nat @ A @ C)))))). % order.trans
thf(fact_90_le__cases, axiom,
    ((![X3 : nat, Y3 : nat]: ((~ ((ord_less_eq_nat @ X3 @ Y3))) => (ord_less_eq_nat @ Y3 @ X3))))). % le_cases
thf(fact_91_eq__refl, axiom,
    ((![X3 : nat, Y3 : nat]: ((X3 = Y3) => (ord_less_eq_nat @ X3 @ Y3))))). % eq_refl
thf(fact_92_linear, axiom,
    ((![X3 : nat, Y3 : nat]: ((ord_less_eq_nat @ X3 @ Y3) | (ord_less_eq_nat @ Y3 @ X3))))). % linear
thf(fact_93_antisym, axiom,
    ((![X3 : nat, Y3 : nat]: ((ord_less_eq_nat @ X3 @ Y3) => ((ord_less_eq_nat @ Y3 @ X3) => (X3 = Y3)))))). % antisym
thf(fact_94_eq__iff, axiom,
    (((^[Y2 : nat]: (^[Z : nat]: (Y2 = Z))) = (^[X4 : nat]: (^[Y4 : nat]: (((ord_less_eq_nat @ X4 @ Y4)) & ((ord_less_eq_nat @ Y4 @ X4)))))))). % eq_iff
thf(fact_95_ord__le__eq__subst, axiom,
    ((![A : nat, B : nat, F : nat > nat, C : nat]: ((ord_less_eq_nat @ A @ B) => (((F @ B) = C) => ((![X : nat, Y : nat]: ((ord_less_eq_nat @ X @ Y) => (ord_less_eq_nat @ (F @ X) @ (F @ Y)))) => (ord_less_eq_nat @ (F @ A) @ C))))))). % ord_le_eq_subst
thf(fact_96_ord__eq__le__subst, axiom,
    ((![A : nat, F : nat > nat, B : nat, C : nat]: ((A = (F @ B)) => ((ord_less_eq_nat @ B @ C) => ((![X : nat, Y : nat]: ((ord_less_eq_nat @ X @ Y) => (ord_less_eq_nat @ (F @ X) @ (F @ Y)))) => (ord_less_eq_nat @ A @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_97_order__subst2, axiom,
    ((![A : nat, B : nat, F : nat > nat, C : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_nat @ (F @ B) @ C) => ((![X : nat, Y : nat]: ((ord_less_eq_nat @ X @ Y) => (ord_less_eq_nat @ (F @ X) @ (F @ Y)))) => (ord_less_eq_nat @ (F @ A) @ C))))))). % order_subst2
thf(fact_98_order__subst1, axiom,
    ((![A : nat, F : nat > nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ (F @ B)) => ((ord_less_eq_nat @ B @ C) => ((![X : nat, Y : nat]: ((ord_less_eq_nat @ X @ Y) => (ord_less_eq_nat @ (F @ X) @ (F @ Y)))) => (ord_less_eq_nat @ A @ (F @ C)))))))). % order_subst1
thf(fact_99_dual__order_Ostrict__implies__not__eq, axiom,
    ((![B : nat, A : nat]: ((ord_less_nat @ B @ A) => (~ ((A = B))))))). % dual_order.strict_implies_not_eq
thf(fact_100_order_Ostrict__implies__not__eq, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (~ ((A = B))))))). % order.strict_implies_not_eq
thf(fact_101_not__less__iff__gr__or__eq, axiom,
    ((![X3 : nat, Y3 : nat]: ((~ ((ord_less_nat @ X3 @ Y3))) = (((ord_less_nat @ Y3 @ X3)) | ((X3 = Y3))))))). % not_less_iff_gr_or_eq
thf(fact_102_dual__order_Ostrict__trans, axiom,
    ((![B : nat, A : nat, C : nat]: ((ord_less_nat @ B @ A) => ((ord_less_nat @ C @ B) => (ord_less_nat @ C @ A)))))). % dual_order.strict_trans
thf(fact_103_linorder__less__wlog, axiom,
    ((![P2 : nat > nat > $o, A : nat, B : nat]: ((![A3 : nat, B3 : nat]: ((ord_less_nat @ A3 @ B3) => (P2 @ A3 @ B3))) => ((![A3 : nat]: (P2 @ A3 @ A3)) => ((![A3 : nat, B3 : nat]: ((P2 @ B3 @ A3) => (P2 @ A3 @ B3))) => (P2 @ A @ B))))))). % linorder_less_wlog
thf(fact_104_exists__least__iff, axiom,
    (((^[P3 : nat > $o]: (?[X5 : nat]: (P3 @ X5))) = (^[P4 : nat > $o]: (?[N2 : nat]: (((P4 @ N2)) & ((![M4 : nat]: (((ord_less_nat @ M4 @ N2)) => ((~ ((P4 @ M4))))))))))))). % exists_least_iff
thf(fact_105_less__imp__not__less, axiom,
    ((![X3 : nat, Y3 : nat]: ((ord_less_nat @ X3 @ Y3) => (~ ((ord_less_nat @ Y3 @ X3))))))). % less_imp_not_less
thf(fact_106_order_Ostrict__trans, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_nat @ A @ B) => ((ord_less_nat @ B @ C) => (ord_less_nat @ A @ C)))))). % order.strict_trans
thf(fact_107_dual__order_Oirrefl, axiom,
    ((![A : nat]: (~ ((ord_less_nat @ A @ A)))))). % dual_order.irrefl
thf(fact_108_linorder__cases, axiom,
    ((![X3 : nat, Y3 : nat]: ((~ ((ord_less_nat @ X3 @ Y3))) => ((~ ((X3 = Y3))) => (ord_less_nat @ Y3 @ X3)))))). % linorder_cases
thf(fact_109_less__imp__triv, axiom,
    ((![X3 : nat, Y3 : nat, P2 : $o]: ((ord_less_nat @ X3 @ Y3) => ((ord_less_nat @ Y3 @ X3) => P2))))). % less_imp_triv
thf(fact_110_less__imp__not__eq2, axiom,
    ((![X3 : nat, Y3 : nat]: ((ord_less_nat @ X3 @ Y3) => (~ ((Y3 = X3))))))). % less_imp_not_eq2
thf(fact_111_antisym__conv3, axiom,
    ((![Y3 : nat, X3 : nat]: ((~ ((ord_less_nat @ Y3 @ X3))) => ((~ ((ord_less_nat @ X3 @ Y3))) = (X3 = Y3)))))). % antisym_conv3
thf(fact_112_less__induct, axiom,
    ((![P2 : nat > $o, A : nat]: ((![X : nat]: ((![Y5 : nat]: ((ord_less_nat @ Y5 @ X) => (P2 @ Y5))) => (P2 @ X))) => (P2 @ A))))). % less_induct
thf(fact_113_less__not__sym, axiom,
    ((![X3 : nat, Y3 : nat]: ((ord_less_nat @ X3 @ Y3) => (~ ((ord_less_nat @ Y3 @ X3))))))). % less_not_sym
thf(fact_114_less__imp__not__eq, axiom,
    ((![X3 : nat, Y3 : nat]: ((ord_less_nat @ X3 @ Y3) => (~ ((X3 = Y3))))))). % less_imp_not_eq
thf(fact_115_dual__order_Oasym, axiom,
    ((![B : nat, A : nat]: ((ord_less_nat @ B @ A) => (~ ((ord_less_nat @ A @ B))))))). % dual_order.asym
thf(fact_116_ord__less__eq__trans, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_nat @ A @ B) => ((B = C) => (ord_less_nat @ A @ C)))))). % ord_less_eq_trans
thf(fact_117_ord__eq__less__trans, axiom,
    ((![A : nat, B : nat, C : nat]: ((A = B) => ((ord_less_nat @ B @ C) => (ord_less_nat @ A @ C)))))). % ord_eq_less_trans
thf(fact_118_less__irrefl, axiom,
    ((![X3 : nat]: (~ ((ord_less_nat @ X3 @ X3)))))). % less_irrefl
thf(fact_119_less__linear, axiom,
    ((![X3 : nat, Y3 : nat]: ((ord_less_nat @ X3 @ Y3) | ((X3 = Y3) | (ord_less_nat @ Y3 @ X3)))))). % less_linear
thf(fact_120_less__trans, axiom,
    ((![X3 : nat, Y3 : nat, Z2 : nat]: ((ord_less_nat @ X3 @ Y3) => ((ord_less_nat @ Y3 @ Z2) => (ord_less_nat @ X3 @ Z2)))))). % less_trans
thf(fact_121_less__asym_H, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (~ ((ord_less_nat @ B @ A))))))). % less_asym'
thf(fact_122_less__asym, axiom,
    ((![X3 : nat, Y3 : nat]: ((ord_less_nat @ X3 @ Y3) => (~ ((ord_less_nat @ Y3 @ X3))))))). % less_asym
thf(fact_123_less__imp__neq, axiom,
    ((![X3 : nat, Y3 : nat]: ((ord_less_nat @ X3 @ Y3) => (~ ((X3 = Y3))))))). % less_imp_neq
thf(fact_124_order_Oasym, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (~ ((ord_less_nat @ B @ A))))))). % order.asym
thf(fact_125_neq__iff, axiom,
    ((![X3 : nat, Y3 : nat]: ((~ ((X3 = Y3))) = (((ord_less_nat @ X3 @ Y3)) | ((ord_less_nat @ Y3 @ X3))))))). % neq_iff
thf(fact_126_neqE, axiom,
    ((![X3 : nat, Y3 : nat]: ((~ ((X3 = Y3))) => ((~ ((ord_less_nat @ X3 @ Y3))) => (ord_less_nat @ Y3 @ X3)))))). % neqE
thf(fact_127_gt__ex, axiom,
    ((![X3 : nat]: (?[X_1 : nat]: (ord_less_nat @ X3 @ X_1))))). % gt_ex
thf(fact_128_order__less__subst2, axiom,
    ((![A : nat, B : nat, F : nat > nat, C : nat]: ((ord_less_nat @ A @ B) => ((ord_less_nat @ (F @ B) @ C) => ((![X : nat, Y : nat]: ((ord_less_nat @ X @ Y) => (ord_less_nat @ (F @ X) @ (F @ Y)))) => (ord_less_nat @ (F @ A) @ C))))))). % order_less_subst2
thf(fact_129_order__less__subst1, axiom,
    ((![A : nat, F : nat > nat, B : nat, C : nat]: ((ord_less_nat @ A @ (F @ B)) => ((ord_less_nat @ B @ C) => ((![X : nat, Y : nat]: ((ord_less_nat @ X @ Y) => (ord_less_nat @ (F @ X) @ (F @ Y)))) => (ord_less_nat @ A @ (F @ C)))))))). % order_less_subst1
thf(fact_130_ord__less__eq__subst, axiom,
    ((![A : nat, B : nat, F : nat > nat, C : nat]: ((ord_less_nat @ A @ B) => (((F @ B) = C) => ((![X : nat, Y : nat]: ((ord_less_nat @ X @ Y) => (ord_less_nat @ (F @ X) @ (F @ Y)))) => (ord_less_nat @ (F @ A) @ C))))))). % ord_less_eq_subst
thf(fact_131_ord__eq__less__subst, axiom,
    ((![A : nat, F : nat > nat, B : nat, C : nat]: ((A = (F @ B)) => ((ord_less_nat @ B @ C) => ((![X : nat, Y : nat]: ((ord_less_nat @ X @ Y) => (ord_less_nat @ (F @ X) @ (F @ Y)))) => (ord_less_nat @ A @ (F @ C)))))))). % ord_eq_less_subst
thf(fact_132_linorder__neqE__nat, axiom,
    ((![X3 : nat, Y3 : nat]: ((~ ((X3 = Y3))) => ((~ ((ord_less_nat @ X3 @ Y3))) => (ord_less_nat @ Y3 @ X3)))))). % linorder_neqE_nat
thf(fact_133_less__add__eq__less, axiom,
    ((![K : nat, L : nat, M : nat, N : nat]: ((ord_less_nat @ K @ L) => (((plus_plus_nat @ M @ L) = (plus_plus_nat @ K @ N)) => (ord_less_nat @ M @ N)))))). % less_add_eq_less
thf(fact_134_infinite__descent, axiom,
    ((![P2 : nat > $o, N : nat]: ((![N3 : nat]: ((~ ((P2 @ N3))) => (?[M5 : nat]: ((ord_less_nat @ M5 @ N3) & (~ ((P2 @ M5))))))) => (P2 @ N))))). % infinite_descent
thf(fact_135_trans__less__add2, axiom,
    ((![I : nat, J : nat, M : nat]: ((ord_less_nat @ I @ J) => (ord_less_nat @ I @ (plus_plus_nat @ M @ J)))))). % trans_less_add2
thf(fact_136_trans__less__add1, axiom,
    ((![I : nat, J : nat, M : nat]: ((ord_less_nat @ I @ J) => (ord_less_nat @ I @ (plus_plus_nat @ J @ M)))))). % trans_less_add1
thf(fact_137_nat__less__induct, axiom,
    ((![P2 : nat > $o, N : nat]: ((![N3 : nat]: ((![M5 : nat]: ((ord_less_nat @ M5 @ N3) => (P2 @ M5))) => (P2 @ N3))) => (P2 @ N))))). % nat_less_induct
thf(fact_138_less__irrefl__nat, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_irrefl_nat
thf(fact_139_less__not__refl3, axiom,
    ((![S2 : nat, T2 : nat]: ((ord_less_nat @ S2 @ T2) => (~ ((S2 = T2))))))). % less_not_refl3
thf(fact_140_less__not__refl2, axiom,
    ((![N : nat, M : nat]: ((ord_less_nat @ N @ M) => (~ ((M = N))))))). % less_not_refl2
thf(fact_141_add__less__mono1, axiom,
    ((![I : nat, J : nat, K : nat]: ((ord_less_nat @ I @ J) => (ord_less_nat @ (plus_plus_nat @ I @ K) @ (plus_plus_nat @ J @ K)))))). % add_less_mono1
thf(fact_142_not__add__less2, axiom,
    ((![J : nat, I : nat]: (~ ((ord_less_nat @ (plus_plus_nat @ J @ I) @ I)))))). % not_add_less2
thf(fact_143_not__add__less1, axiom,
    ((![I : nat, J : nat]: (~ ((ord_less_nat @ (plus_plus_nat @ I @ J) @ I)))))). % not_add_less1
thf(fact_144_less__not__refl, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_not_refl
thf(fact_145_add__less__mono, axiom,
    ((![I : nat, J : nat, K : nat, L : nat]: ((ord_less_nat @ I @ J) => ((ord_less_nat @ K @ L) => (ord_less_nat @ (plus_plus_nat @ I @ K) @ (plus_plus_nat @ J @ L))))))). % add_less_mono
thf(fact_146_nat__neq__iff, axiom,
    ((![M : nat, N : nat]: ((~ ((M = N))) = (((ord_less_nat @ M @ N)) | ((ord_less_nat @ N @ M))))))). % nat_neq_iff
thf(fact_147_add__lessD1, axiom,
    ((![I : nat, J : nat, K : nat]: ((ord_less_nat @ (plus_plus_nat @ I @ J) @ K) => (ord_less_nat @ I @ K))))). % add_lessD1
thf(fact_148_Nat_Oex__has__greatest__nat, axiom,
    ((![P2 : nat > $o, K : nat, B : nat]: ((P2 @ K) => ((![Y : nat]: ((P2 @ Y) => (ord_less_eq_nat @ Y @ B))) => (?[X : nat]: ((P2 @ X) & (![Y5 : nat]: ((P2 @ Y5) => (ord_less_eq_nat @ Y5 @ X)))))))))). % Nat.ex_has_greatest_nat

% Conjectures (1)
thf(conj_0, conjecture,
    (((plus_plus_a @ (groups1145913330_nat_a @ g @ (set_or562006527an_nat @ m @ k)) @ (groups1145913330_nat_a @ h @ (set_or562006527an_nat @ k @ n))) = (groups1145913330_nat_a @ f @ (set_or562006527an_nat @ m @ n))))).
