% 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/Fundamental_Theorem_Algebra/prob_288__5370386_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:28:56.023

% Could-be-implicit typings (2)
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).
thf(ty_n_t__Int__Oint, type,
    int : $tType).

% Explicit typings (20)
thf(sy_c_Fun_Ocomp_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J, type,
    comp_n997826594at_nat : ((nat > nat) > nat > nat) > ((nat > nat) > nat > nat) > (nat > nat) > nat > nat).
thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat, type,
    comp_nat_nat_nat : (nat > nat) > (nat > nat) > nat > nat).
thf(sy_c_Fun_Omap__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat, type,
    map_fu972944474at_nat : (nat > nat) > (nat > nat) > (nat > nat) > nat > nat).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint, type,
    zero_zero_int : int).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat, type,
    zero_zero_nat : nat).
thf(sy_c_If_001t__Nat__Onat, type,
    if_nat : $o > nat > nat > nat).
thf(sy_c_Int_Onat, type,
    nat2 : int > nat).
thf(sy_c_Int_Oring__1__class_Oof__int_001t__Int__Oint, type,
    ring_1_of_int_int : int > int).
thf(sy_c_Nat_OSuc, type,
    suc : nat > nat).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint, type,
    semiri2019852685at_int : nat > int).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat, type,
    semiri1382578993at_nat : nat > nat).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint, type,
    ord_less_int : int > int > $o).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat, type,
    ord_less_nat : nat > nat > $o).
thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Int__Oint_001t__Int__Oint, type,
    order_1320016787nt_int : (int > int) > $o).
thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Int__Oint_001t__Nat__Onat, type,
    order_682743095nt_nat : (int > nat) > $o).
thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Nat__Onat_001t__Int__Oint, type,
    order_1406747959at_int : (nat > int) > $o).
thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Nat__Onat_001t__Nat__Onat, type,
    order_769474267at_nat : (nat > nat) > $o).
thf(sy_c_Topological__Spaces_Otopological__space__class_Oconvergent_001t__Nat__Onat, type,
    topolo768750839nt_nat : (nat > nat) > $o).
thf(sy_v_f____, type,
    f : nat > nat).
thf(sy_v_g____, type,
    g : nat > nat).

% Relevant facts (198)
thf(fact_0_f_I1_J, axiom,
    ((order_769474267at_nat @ f))). % f(1)
thf(fact_1_g_I1_J, axiom,
    ((order_769474267at_nat @ g))). % g(1)
thf(fact_2_comp__apply, axiom,
    ((comp_nat_nat_nat = (^[F : nat > nat]: (^[G : nat > nat]: (^[X : nat]: (F @ (G @ X)))))))). % comp_apply
thf(fact_3_comp__def, axiom,
    ((comp_nat_nat_nat = (^[F : nat > nat]: (^[G : nat > nat]: (^[X : nat]: (F @ (G @ X)))))))). % comp_def
thf(fact_4_comp__assoc, axiom,
    ((![F2 : nat > nat, G2 : nat > nat, H : nat > nat]: ((comp_nat_nat_nat @ (comp_nat_nat_nat @ F2 @ G2) @ H) = (comp_nat_nat_nat @ F2 @ (comp_nat_nat_nat @ G2 @ H)))))). % comp_assoc
thf(fact_5_comp__eq__dest, axiom,
    ((![A : nat > nat, B : nat > nat, C : nat > nat, D : nat > nat, V : nat]: (((comp_nat_nat_nat @ A @ B) = (comp_nat_nat_nat @ C @ D)) => ((A @ (B @ V)) = (C @ (D @ V))))))). % comp_eq_dest
thf(fact_6_comp__eq__elim, axiom,
    ((![A : nat > nat, B : nat > nat, C : nat > nat, D : nat > nat]: (((comp_nat_nat_nat @ A @ B) = (comp_nat_nat_nat @ C @ D)) => (![V2 : nat]: ((A @ (B @ V2)) = (C @ (D @ V2)))))))). % comp_eq_elim
thf(fact_7_comp__cong, axiom,
    ((![F2 : nat > nat, G2 : nat > nat, X2 : nat, F3 : nat > nat, G3 : nat > nat, X3 : nat]: (((F2 @ (G2 @ X2)) = (F3 @ (G3 @ X3))) => ((comp_nat_nat_nat @ F2 @ G2 @ X2) = (comp_nat_nat_nat @ F3 @ G3 @ X3)))))). % comp_cong
thf(fact_8_comp__eq__dest__lhs, axiom,
    ((![A : nat > nat, B : nat > nat, C : nat > nat, V : nat]: (((comp_nat_nat_nat @ A @ B) = C) => ((A @ (B @ V)) = (C @ V)))))). % comp_eq_dest_lhs
thf(fact_9_comp__apply__eq, axiom,
    ((![F2 : nat > nat, G2 : nat > nat, X2 : nat, H : nat > nat, K : nat > nat]: (((F2 @ (G2 @ X2)) = (H @ (K @ X2))) => ((comp_nat_nat_nat @ F2 @ G2 @ X2) = (comp_nat_nat_nat @ H @ K @ X2)))))). % comp_apply_eq
thf(fact_10_fun_Omap__comp, axiom,
    ((![G2 : nat > nat, F2 : nat > nat, V : nat > nat]: ((comp_nat_nat_nat @ G2 @ (comp_nat_nat_nat @ F2 @ V)) = (comp_nat_nat_nat @ (comp_nat_nat_nat @ G2 @ F2) @ V))))). % fun.map_comp
thf(fact_11_type__copy__map__cong0, axiom,
    ((![M : nat > nat, G2 : nat > nat, X2 : nat, N : nat > nat, H : nat > nat, F2 : nat > nat]: (((M @ (G2 @ X2)) = (N @ (H @ X2))) => ((comp_nat_nat_nat @ (comp_nat_nat_nat @ F2 @ M) @ G2 @ X2) = (comp_nat_nat_nat @ (comp_nat_nat_nat @ F2 @ N) @ H @ X2)))))). % type_copy_map_cong0
thf(fact_12_rewriteL__comp__comp, axiom,
    ((![F2 : nat > nat, G2 : nat > nat, L : nat > nat, H : nat > nat]: (((comp_nat_nat_nat @ F2 @ G2) = L) => ((comp_nat_nat_nat @ F2 @ (comp_nat_nat_nat @ G2 @ H)) = (comp_nat_nat_nat @ L @ H)))))). % rewriteL_comp_comp
thf(fact_13_rewriteR__comp__comp, axiom,
    ((![G2 : nat > nat, H : nat > nat, R : nat > nat, F2 : nat > nat]: (((comp_nat_nat_nat @ G2 @ H) = R) => ((comp_nat_nat_nat @ (comp_nat_nat_nat @ F2 @ G2) @ H) = (comp_nat_nat_nat @ F2 @ R)))))). % rewriteR_comp_comp
thf(fact_14_rewriteR__comp__comp2, axiom,
    ((![G2 : nat > nat, H : nat > nat, R1 : nat > nat, R2 : nat > nat, F2 : nat > nat, L : nat > nat]: (((comp_nat_nat_nat @ G2 @ H) = (comp_nat_nat_nat @ R1 @ R2)) => (((comp_nat_nat_nat @ F2 @ R1) = L) => ((comp_nat_nat_nat @ (comp_nat_nat_nat @ F2 @ G2) @ H) = (comp_nat_nat_nat @ L @ R2))))))). % rewriteR_comp_comp2
thf(fact_15_rewriteL__comp__comp2, axiom,
    ((![F2 : nat > nat, G2 : nat > nat, L1 : nat > nat, L2 : nat > nat, H : nat > nat, R : nat > nat]: (((comp_nat_nat_nat @ F2 @ G2) = (comp_nat_nat_nat @ L1 @ L2)) => (((comp_nat_nat_nat @ L2 @ H) = R) => ((comp_nat_nat_nat @ F2 @ (comp_nat_nat_nat @ G2 @ H)) = (comp_nat_nat_nat @ L1 @ R))))))). % rewriteL_comp_comp2
thf(fact_16_strict__mono__o, axiom,
    ((![R : nat > nat, S : nat > nat]: ((order_769474267at_nat @ R) => ((order_769474267at_nat @ S) => (order_769474267at_nat @ (comp_nat_nat_nat @ R @ S))))))). % strict_mono_o
thf(fact_17_strict__mono__less, axiom,
    ((![F2 : int > nat, X2 : int, Y : int]: ((order_682743095nt_nat @ F2) => ((ord_less_nat @ (F2 @ X2) @ (F2 @ Y)) = (ord_less_int @ X2 @ Y)))))). % strict_mono_less
thf(fact_18_strict__mono__less, axiom,
    ((![F2 : nat > int, X2 : nat, Y : nat]: ((order_1406747959at_int @ F2) => ((ord_less_int @ (F2 @ X2) @ (F2 @ Y)) = (ord_less_nat @ X2 @ Y)))))). % strict_mono_less
thf(fact_19_strict__mono__less, axiom,
    ((![F2 : int > int, X2 : int, Y : int]: ((order_1320016787nt_int @ F2) => ((ord_less_int @ (F2 @ X2) @ (F2 @ Y)) = (ord_less_int @ X2 @ Y)))))). % strict_mono_less
thf(fact_20_strict__mono__less, axiom,
    ((![F2 : nat > nat, X2 : nat, Y : nat]: ((order_769474267at_nat @ F2) => ((ord_less_nat @ (F2 @ X2) @ (F2 @ Y)) = (ord_less_nat @ X2 @ Y)))))). % strict_mono_less
thf(fact_21_strict__mono__def, axiom,
    ((order_1406747959at_int = (^[F : nat > int]: (![X : nat]: (![Y2 : nat]: (((ord_less_nat @ X @ Y2)) => ((ord_less_int @ (F @ X) @ (F @ Y2)))))))))). % strict_mono_def
thf(fact_22_strict__mono__def, axiom,
    ((order_682743095nt_nat = (^[F : int > nat]: (![X : int]: (![Y2 : int]: (((ord_less_int @ X @ Y2)) => ((ord_less_nat @ (F @ X) @ (F @ Y2)))))))))). % strict_mono_def
thf(fact_23_strict__mono__def, axiom,
    ((order_1320016787nt_int = (^[F : int > int]: (![X : int]: (![Y2 : int]: (((ord_less_int @ X @ Y2)) => ((ord_less_int @ (F @ X) @ (F @ Y2)))))))))). % strict_mono_def
thf(fact_24_strict__mono__def, axiom,
    ((order_769474267at_nat = (^[F : nat > nat]: (![X : nat]: (![Y2 : nat]: (((ord_less_nat @ X @ Y2)) => ((ord_less_nat @ (F @ X) @ (F @ Y2)))))))))). % strict_mono_def
thf(fact_25_strict__monoI, axiom,
    ((![F2 : nat > int]: ((![X4 : nat, Y3 : nat]: ((ord_less_nat @ X4 @ Y3) => (ord_less_int @ (F2 @ X4) @ (F2 @ Y3)))) => (order_1406747959at_int @ F2))))). % strict_monoI
thf(fact_26_strict__monoI, axiom,
    ((![F2 : int > nat]: ((![X4 : int, Y3 : int]: ((ord_less_int @ X4 @ Y3) => (ord_less_nat @ (F2 @ X4) @ (F2 @ Y3)))) => (order_682743095nt_nat @ F2))))). % strict_monoI
thf(fact_27_strict__monoI, axiom,
    ((![F2 : int > int]: ((![X4 : int, Y3 : int]: ((ord_less_int @ X4 @ Y3) => (ord_less_int @ (F2 @ X4) @ (F2 @ Y3)))) => (order_1320016787nt_int @ F2))))). % strict_monoI
thf(fact_28_strict__monoI, axiom,
    ((![F2 : nat > nat]: ((![X4 : nat, Y3 : nat]: ((ord_less_nat @ X4 @ Y3) => (ord_less_nat @ (F2 @ X4) @ (F2 @ Y3)))) => (order_769474267at_nat @ F2))))). % strict_monoI
thf(fact_29_strict__monoD, axiom,
    ((![F2 : nat > int, X2 : nat, Y : nat]: ((order_1406747959at_int @ F2) => ((ord_less_nat @ X2 @ Y) => (ord_less_int @ (F2 @ X2) @ (F2 @ Y))))))). % strict_monoD
thf(fact_30_strict__monoD, axiom,
    ((![F2 : int > nat, X2 : int, Y : int]: ((order_682743095nt_nat @ F2) => ((ord_less_int @ X2 @ Y) => (ord_less_nat @ (F2 @ X2) @ (F2 @ Y))))))). % strict_monoD
thf(fact_31_strict__monoD, axiom,
    ((![F2 : int > int, X2 : int, Y : int]: ((order_1320016787nt_int @ F2) => ((ord_less_int @ X2 @ Y) => (ord_less_int @ (F2 @ X2) @ (F2 @ Y))))))). % strict_monoD
thf(fact_32_strict__monoD, axiom,
    ((![F2 : nat > nat, X2 : nat, Y : nat]: ((order_769474267at_nat @ F2) => ((ord_less_nat @ X2 @ Y) => (ord_less_nat @ (F2 @ X2) @ (F2 @ Y))))))). % strict_monoD
thf(fact_33_strict__mono__eq, axiom,
    ((![F2 : nat > nat, X2 : nat, Y : nat]: ((order_769474267at_nat @ F2) => (((F2 @ X2) = (F2 @ Y)) = (X2 = Y)))))). % strict_mono_eq
thf(fact_34_linorder__neqE__nat, axiom,
    ((![X2 : nat, Y : nat]: ((~ ((X2 = Y))) => ((~ ((ord_less_nat @ X2 @ Y))) => (ord_less_nat @ Y @ X2)))))). % linorder_neqE_nat
thf(fact_35_infinite__descent, axiom,
    ((![P : nat > $o, N2 : nat]: ((![N3 : nat]: ((~ ((P @ N3))) => (?[M2 : nat]: ((ord_less_nat @ M2 @ N3) & (~ ((P @ M2))))))) => (P @ N2))))). % infinite_descent
thf(fact_36_nat__less__induct, axiom,
    ((![P : nat > $o, N2 : nat]: ((![N3 : nat]: ((![M2 : nat]: ((ord_less_nat @ M2 @ N3) => (P @ M2))) => (P @ N3))) => (P @ N2))))). % nat_less_induct
thf(fact_37_ord__eq__less__subst, axiom,
    ((![A : nat, F2 : nat > nat, B : nat, C : nat]: ((A = (F2 @ B)) => ((ord_less_nat @ B @ C) => ((![X4 : nat, Y3 : nat]: ((ord_less_nat @ X4 @ Y3) => (ord_less_nat @ (F2 @ X4) @ (F2 @ Y3)))) => (ord_less_nat @ A @ (F2 @ C)))))))). % ord_eq_less_subst
thf(fact_38_ord__eq__less__subst, axiom,
    ((![A : int, F2 : nat > int, B : nat, C : nat]: ((A = (F2 @ B)) => ((ord_less_nat @ B @ C) => ((![X4 : nat, Y3 : nat]: ((ord_less_nat @ X4 @ Y3) => (ord_less_int @ (F2 @ X4) @ (F2 @ Y3)))) => (ord_less_int @ A @ (F2 @ C)))))))). % ord_eq_less_subst
thf(fact_39_ord__eq__less__subst, axiom,
    ((![A : nat, F2 : int > nat, B : int, C : int]: ((A = (F2 @ B)) => ((ord_less_int @ B @ C) => ((![X4 : int, Y3 : int]: ((ord_less_int @ X4 @ Y3) => (ord_less_nat @ (F2 @ X4) @ (F2 @ Y3)))) => (ord_less_nat @ A @ (F2 @ C)))))))). % ord_eq_less_subst
thf(fact_40_ord__eq__less__subst, axiom,
    ((![A : int, F2 : int > int, B : int, C : int]: ((A = (F2 @ B)) => ((ord_less_int @ B @ C) => ((![X4 : int, Y3 : int]: ((ord_less_int @ X4 @ Y3) => (ord_less_int @ (F2 @ X4) @ (F2 @ Y3)))) => (ord_less_int @ A @ (F2 @ C)))))))). % ord_eq_less_subst
thf(fact_41_ord__less__eq__subst, axiom,
    ((![A : nat, B : nat, F2 : nat > nat, C : nat]: ((ord_less_nat @ A @ B) => (((F2 @ B) = C) => ((![X4 : nat, Y3 : nat]: ((ord_less_nat @ X4 @ Y3) => (ord_less_nat @ (F2 @ X4) @ (F2 @ Y3)))) => (ord_less_nat @ (F2 @ A) @ C))))))). % ord_less_eq_subst
thf(fact_42_ord__less__eq__subst, axiom,
    ((![A : nat, B : nat, F2 : nat > int, C : int]: ((ord_less_nat @ A @ B) => (((F2 @ B) = C) => ((![X4 : nat, Y3 : nat]: ((ord_less_nat @ X4 @ Y3) => (ord_less_int @ (F2 @ X4) @ (F2 @ Y3)))) => (ord_less_int @ (F2 @ A) @ C))))))). % ord_less_eq_subst
thf(fact_43_ord__less__eq__subst, axiom,
    ((![A : int, B : int, F2 : int > nat, C : nat]: ((ord_less_int @ A @ B) => (((F2 @ B) = C) => ((![X4 : int, Y3 : int]: ((ord_less_int @ X4 @ Y3) => (ord_less_nat @ (F2 @ X4) @ (F2 @ Y3)))) => (ord_less_nat @ (F2 @ A) @ C))))))). % ord_less_eq_subst
thf(fact_44_ord__less__eq__subst, axiom,
    ((![A : int, B : int, F2 : int > int, C : int]: ((ord_less_int @ A @ B) => (((F2 @ B) = C) => ((![X4 : int, Y3 : int]: ((ord_less_int @ X4 @ Y3) => (ord_less_int @ (F2 @ X4) @ (F2 @ Y3)))) => (ord_less_int @ (F2 @ A) @ C))))))). % ord_less_eq_subst
thf(fact_45_order__less__subst1, axiom,
    ((![A : nat, F2 : nat > nat, B : nat, C : nat]: ((ord_less_nat @ A @ (F2 @ B)) => ((ord_less_nat @ B @ C) => ((![X4 : nat, Y3 : nat]: ((ord_less_nat @ X4 @ Y3) => (ord_less_nat @ (F2 @ X4) @ (F2 @ Y3)))) => (ord_less_nat @ A @ (F2 @ C)))))))). % order_less_subst1
thf(fact_46_order__less__subst1, axiom,
    ((![A : nat, F2 : int > nat, B : int, C : int]: ((ord_less_nat @ A @ (F2 @ B)) => ((ord_less_int @ B @ C) => ((![X4 : int, Y3 : int]: ((ord_less_int @ X4 @ Y3) => (ord_less_nat @ (F2 @ X4) @ (F2 @ Y3)))) => (ord_less_nat @ A @ (F2 @ C)))))))). % order_less_subst1
thf(fact_47_order__less__subst1, axiom,
    ((![A : int, F2 : nat > int, B : nat, C : nat]: ((ord_less_int @ A @ (F2 @ B)) => ((ord_less_nat @ B @ C) => ((![X4 : nat, Y3 : nat]: ((ord_less_nat @ X4 @ Y3) => (ord_less_int @ (F2 @ X4) @ (F2 @ Y3)))) => (ord_less_int @ A @ (F2 @ C)))))))). % order_less_subst1
thf(fact_48_order__less__subst1, axiom,
    ((![A : int, F2 : int > int, B : int, C : int]: ((ord_less_int @ A @ (F2 @ B)) => ((ord_less_int @ B @ C) => ((![X4 : int, Y3 : int]: ((ord_less_int @ X4 @ Y3) => (ord_less_int @ (F2 @ X4) @ (F2 @ Y3)))) => (ord_less_int @ A @ (F2 @ C)))))))). % order_less_subst1
thf(fact_49_order__less__subst2, axiom,
    ((![A : nat, B : nat, F2 : nat > nat, C : nat]: ((ord_less_nat @ A @ B) => ((ord_less_nat @ (F2 @ B) @ C) => ((![X4 : nat, Y3 : nat]: ((ord_less_nat @ X4 @ Y3) => (ord_less_nat @ (F2 @ X4) @ (F2 @ Y3)))) => (ord_less_nat @ (F2 @ A) @ C))))))). % order_less_subst2
thf(fact_50_order__less__subst2, axiom,
    ((![A : nat, B : nat, F2 : nat > int, C : int]: ((ord_less_nat @ A @ B) => ((ord_less_int @ (F2 @ B) @ C) => ((![X4 : nat, Y3 : nat]: ((ord_less_nat @ X4 @ Y3) => (ord_less_int @ (F2 @ X4) @ (F2 @ Y3)))) => (ord_less_int @ (F2 @ A) @ C))))))). % order_less_subst2
thf(fact_51_order__less__subst2, axiom,
    ((![A : int, B : int, F2 : int > nat, C : nat]: ((ord_less_int @ A @ B) => ((ord_less_nat @ (F2 @ B) @ C) => ((![X4 : int, Y3 : int]: ((ord_less_int @ X4 @ Y3) => (ord_less_nat @ (F2 @ X4) @ (F2 @ Y3)))) => (ord_less_nat @ (F2 @ A) @ C))))))). % order_less_subst2
thf(fact_52_order__less__subst2, axiom,
    ((![A : int, B : int, F2 : int > int, C : int]: ((ord_less_int @ A @ B) => ((ord_less_int @ (F2 @ B) @ C) => ((![X4 : int, Y3 : int]: ((ord_less_int @ X4 @ Y3) => (ord_less_int @ (F2 @ X4) @ (F2 @ Y3)))) => (ord_less_int @ (F2 @ A) @ C))))))). % order_less_subst2
thf(fact_53_lt__ex, axiom,
    ((![X2 : int]: (?[Y3 : int]: (ord_less_int @ Y3 @ X2))))). % lt_ex
thf(fact_54_gt__ex, axiom,
    ((![X2 : nat]: (?[X_1 : nat]: (ord_less_nat @ X2 @ X_1))))). % gt_ex
thf(fact_55_gt__ex, axiom,
    ((![X2 : int]: (?[X_1 : int]: (ord_less_int @ X2 @ X_1))))). % gt_ex
thf(fact_56_neqE, axiom,
    ((![X2 : nat, Y : nat]: ((~ ((X2 = Y))) => ((~ ((ord_less_nat @ X2 @ Y))) => (ord_less_nat @ Y @ X2)))))). % neqE
thf(fact_57_neqE, axiom,
    ((![X2 : int, Y : int]: ((~ ((X2 = Y))) => ((~ ((ord_less_int @ X2 @ Y))) => (ord_less_int @ Y @ X2)))))). % neqE
thf(fact_58_neq__iff, axiom,
    ((![X2 : nat, Y : nat]: ((~ ((X2 = Y))) = (((ord_less_nat @ X2 @ Y)) | ((ord_less_nat @ Y @ X2))))))). % neq_iff
thf(fact_59_neq__iff, axiom,
    ((![X2 : int, Y : int]: ((~ ((X2 = Y))) = (((ord_less_int @ X2 @ Y)) | ((ord_less_int @ Y @ X2))))))). % neq_iff
thf(fact_60_order_Oasym, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (~ ((ord_less_nat @ B @ A))))))). % order.asym
thf(fact_61_order_Oasym, axiom,
    ((![A : int, B : int]: ((ord_less_int @ A @ B) => (~ ((ord_less_int @ B @ A))))))). % order.asym
thf(fact_62_less__imp__neq, axiom,
    ((![X2 : nat, Y : nat]: ((ord_less_nat @ X2 @ Y) => (~ ((X2 = Y))))))). % less_imp_neq
thf(fact_63_less__imp__neq, axiom,
    ((![X2 : int, Y : int]: ((ord_less_int @ X2 @ Y) => (~ ((X2 = Y))))))). % less_imp_neq
thf(fact_64_less__asym, axiom,
    ((![X2 : nat, Y : nat]: ((ord_less_nat @ X2 @ Y) => (~ ((ord_less_nat @ Y @ X2))))))). % less_asym
thf(fact_65_less__asym, axiom,
    ((![X2 : int, Y : int]: ((ord_less_int @ X2 @ Y) => (~ ((ord_less_int @ Y @ X2))))))). % less_asym
thf(fact_66_less__asym_H, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (~ ((ord_less_nat @ B @ A))))))). % less_asym'
thf(fact_67_less__asym_H, axiom,
    ((![A : int, B : int]: ((ord_less_int @ A @ B) => (~ ((ord_less_int @ B @ A))))))). % less_asym'
thf(fact_68_less__trans, axiom,
    ((![X2 : nat, Y : nat, Z : nat]: ((ord_less_nat @ X2 @ Y) => ((ord_less_nat @ Y @ Z) => (ord_less_nat @ X2 @ Z)))))). % less_trans
thf(fact_69_less__trans, axiom,
    ((![X2 : int, Y : int, Z : int]: ((ord_less_int @ X2 @ Y) => ((ord_less_int @ Y @ Z) => (ord_less_int @ X2 @ Z)))))). % less_trans
thf(fact_70_less__linear, axiom,
    ((![X2 : nat, Y : nat]: ((ord_less_nat @ X2 @ Y) | ((X2 = Y) | (ord_less_nat @ Y @ X2)))))). % less_linear
thf(fact_71_less__linear, axiom,
    ((![X2 : int, Y : int]: ((ord_less_int @ X2 @ Y) | ((X2 = Y) | (ord_less_int @ Y @ X2)))))). % less_linear
thf(fact_72_less__irrefl, axiom,
    ((![X2 : nat]: (~ ((ord_less_nat @ X2 @ X2)))))). % less_irrefl
thf(fact_73_less__irrefl, axiom,
    ((![X2 : int]: (~ ((ord_less_int @ X2 @ X2)))))). % less_irrefl
thf(fact_74_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_75_ord__eq__less__trans, axiom,
    ((![A : int, B : int, C : int]: ((A = B) => ((ord_less_int @ B @ C) => (ord_less_int @ A @ C)))))). % ord_eq_less_trans
thf(fact_76_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_77_ord__less__eq__trans, axiom,
    ((![A : int, B : int, C : int]: ((ord_less_int @ A @ B) => ((B = C) => (ord_less_int @ A @ C)))))). % ord_less_eq_trans
thf(fact_78_dual__order_Oasym, axiom,
    ((![B : nat, A : nat]: ((ord_less_nat @ B @ A) => (~ ((ord_less_nat @ A @ B))))))). % dual_order.asym
thf(fact_79_dual__order_Oasym, axiom,
    ((![B : int, A : int]: ((ord_less_int @ B @ A) => (~ ((ord_less_int @ A @ B))))))). % dual_order.asym
thf(fact_80_less__imp__not__eq, axiom,
    ((![X2 : nat, Y : nat]: ((ord_less_nat @ X2 @ Y) => (~ ((X2 = Y))))))). % less_imp_not_eq
thf(fact_81_less__imp__not__eq, axiom,
    ((![X2 : int, Y : int]: ((ord_less_int @ X2 @ Y) => (~ ((X2 = Y))))))). % less_imp_not_eq
thf(fact_82_less__not__sym, axiom,
    ((![X2 : nat, Y : nat]: ((ord_less_nat @ X2 @ Y) => (~ ((ord_less_nat @ Y @ X2))))))). % less_not_sym
thf(fact_83_less__not__sym, axiom,
    ((![X2 : int, Y : int]: ((ord_less_int @ X2 @ Y) => (~ ((ord_less_int @ Y @ X2))))))). % less_not_sym
thf(fact_84_less__induct, axiom,
    ((![P : nat > $o, A : nat]: ((![X4 : nat]: ((![Y4 : nat]: ((ord_less_nat @ Y4 @ X4) => (P @ Y4))) => (P @ X4))) => (P @ A))))). % less_induct
thf(fact_85_antisym__conv3, axiom,
    ((![Y : nat, X2 : nat]: ((~ ((ord_less_nat @ Y @ X2))) => ((~ ((ord_less_nat @ X2 @ Y))) = (X2 = Y)))))). % antisym_conv3
thf(fact_86_antisym__conv3, axiom,
    ((![Y : int, X2 : int]: ((~ ((ord_less_int @ Y @ X2))) => ((~ ((ord_less_int @ X2 @ Y))) = (X2 = Y)))))). % antisym_conv3
thf(fact_87_less__imp__not__eq2, axiom,
    ((![X2 : nat, Y : nat]: ((ord_less_nat @ X2 @ Y) => (~ ((Y = X2))))))). % less_imp_not_eq2
thf(fact_88_less__imp__not__eq2, axiom,
    ((![X2 : int, Y : int]: ((ord_less_int @ X2 @ Y) => (~ ((Y = X2))))))). % less_imp_not_eq2
thf(fact_89_less__imp__triv, axiom,
    ((![X2 : nat, Y : nat, P : $o]: ((ord_less_nat @ X2 @ Y) => ((ord_less_nat @ Y @ X2) => P))))). % less_imp_triv
thf(fact_90_less__imp__triv, axiom,
    ((![X2 : int, Y : int, P : $o]: ((ord_less_int @ X2 @ Y) => ((ord_less_int @ Y @ X2) => P))))). % less_imp_triv
thf(fact_91_linorder__cases, axiom,
    ((![X2 : nat, Y : nat]: ((~ ((ord_less_nat @ X2 @ Y))) => ((~ ((X2 = Y))) => (ord_less_nat @ Y @ X2)))))). % linorder_cases
thf(fact_92_linorder__cases, axiom,
    ((![X2 : int, Y : int]: ((~ ((ord_less_int @ X2 @ Y))) => ((~ ((X2 = Y))) => (ord_less_int @ Y @ X2)))))). % linorder_cases
thf(fact_93_dual__order_Oirrefl, axiom,
    ((![A : nat]: (~ ((ord_less_nat @ A @ A)))))). % dual_order.irrefl
thf(fact_94_dual__order_Oirrefl, axiom,
    ((![A : int]: (~ ((ord_less_int @ A @ A)))))). % dual_order.irrefl
thf(fact_95_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_96_order_Ostrict__trans, axiom,
    ((![A : int, B : int, C : int]: ((ord_less_int @ A @ B) => ((ord_less_int @ B @ C) => (ord_less_int @ A @ C)))))). % order.strict_trans
thf(fact_97_less__imp__not__less, axiom,
    ((![X2 : nat, Y : nat]: ((ord_less_nat @ X2 @ Y) => (~ ((ord_less_nat @ Y @ X2))))))). % less_imp_not_less
thf(fact_98_less__imp__not__less, axiom,
    ((![X2 : int, Y : int]: ((ord_less_int @ X2 @ Y) => (~ ((ord_less_int @ Y @ X2))))))). % less_imp_not_less
thf(fact_99_exists__least__iff, axiom,
    (((^[P2 : nat > $o]: (?[X5 : nat]: (P2 @ X5))) = (^[P3 : nat > $o]: (?[N4 : nat]: (((P3 @ N4)) & ((![M3 : nat]: (((ord_less_nat @ M3 @ N4)) => ((~ ((P3 @ M3))))))))))))). % exists_least_iff
thf(fact_100_linorder__less__wlog, axiom,
    ((![P : nat > nat > $o, A : nat, B : nat]: ((![A2 : nat, B2 : nat]: ((ord_less_nat @ A2 @ B2) => (P @ A2 @ B2))) => ((![A2 : nat]: (P @ A2 @ A2)) => ((![A2 : nat, B2 : nat]: ((P @ B2 @ A2) => (P @ A2 @ B2))) => (P @ A @ B))))))). % linorder_less_wlog
thf(fact_101_linorder__less__wlog, axiom,
    ((![P : int > int > $o, A : int, B : int]: ((![A2 : int, B2 : int]: ((ord_less_int @ A2 @ B2) => (P @ A2 @ B2))) => ((![A2 : int]: (P @ A2 @ A2)) => ((![A2 : int, B2 : int]: ((P @ B2 @ A2) => (P @ A2 @ B2))) => (P @ A @ B))))))). % linorder_less_wlog
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_dual__order_Ostrict__trans, axiom,
    ((![B : int, A : int, C : int]: ((ord_less_int @ B @ A) => ((ord_less_int @ C @ B) => (ord_less_int @ C @ A)))))). % dual_order.strict_trans
thf(fact_104_not__less__iff__gr__or__eq, axiom,
    ((![X2 : nat, Y : nat]: ((~ ((ord_less_nat @ X2 @ Y))) = (((ord_less_nat @ Y @ X2)) | ((X2 = Y))))))). % not_less_iff_gr_or_eq
thf(fact_105_not__less__iff__gr__or__eq, axiom,
    ((![X2 : int, Y : int]: ((~ ((ord_less_int @ X2 @ Y))) = (((ord_less_int @ Y @ X2)) | ((X2 = Y))))))). % not_less_iff_gr_or_eq
thf(fact_106_order_Ostrict__implies__not__eq, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (~ ((A = B))))))). % order.strict_implies_not_eq
thf(fact_107_order_Ostrict__implies__not__eq, axiom,
    ((![A : int, B : int]: ((ord_less_int @ A @ B) => (~ ((A = B))))))). % order.strict_implies_not_eq
thf(fact_108_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_109_dual__order_Ostrict__implies__not__eq, axiom,
    ((![B : int, A : int]: ((ord_less_int @ B @ A) => (~ ((A = B))))))). % dual_order.strict_implies_not_eq
thf(fact_110_nat__neq__iff, axiom,
    ((![M4 : nat, N2 : nat]: ((~ ((M4 = N2))) = (((ord_less_nat @ M4 @ N2)) | ((ord_less_nat @ N2 @ M4))))))). % nat_neq_iff
thf(fact_111_less__not__refl, axiom,
    ((![N2 : nat]: (~ ((ord_less_nat @ N2 @ N2)))))). % less_not_refl
thf(fact_112_less__not__refl2, axiom,
    ((![N2 : nat, M4 : nat]: ((ord_less_nat @ N2 @ M4) => (~ ((M4 = N2))))))). % less_not_refl2
thf(fact_113_less__not__refl3, axiom,
    ((![S : nat, T : nat]: ((ord_less_nat @ S @ T) => (~ ((S = T))))))). % less_not_refl3
thf(fact_114_less__irrefl__nat, axiom,
    ((![N2 : nat]: (~ ((ord_less_nat @ N2 @ N2)))))). % less_irrefl_nat
thf(fact_115_minf_I7_J, axiom,
    ((![T : nat]: (?[Z2 : nat]: (![X6 : nat]: ((ord_less_nat @ X6 @ Z2) => (~ ((ord_less_nat @ T @ X6))))))))). % minf(7)
thf(fact_116_minf_I7_J, axiom,
    ((![T : int]: (?[Z2 : int]: (![X6 : int]: ((ord_less_int @ X6 @ Z2) => (~ ((ord_less_int @ T @ X6))))))))). % minf(7)
thf(fact_117_minf_I5_J, axiom,
    ((![T : nat]: (?[Z2 : nat]: (![X6 : nat]: ((ord_less_nat @ X6 @ Z2) => (ord_less_nat @ X6 @ T))))))). % minf(5)
thf(fact_118_minf_I5_J, axiom,
    ((![T : int]: (?[Z2 : int]: (![X6 : int]: ((ord_less_int @ X6 @ Z2) => (ord_less_int @ X6 @ T))))))). % minf(5)
thf(fact_119_minf_I4_J, axiom,
    ((![T : nat]: (?[Z2 : nat]: (![X6 : nat]: ((ord_less_nat @ X6 @ Z2) => (~ ((X6 = T))))))))). % minf(4)
thf(fact_120_minf_I4_J, axiom,
    ((![T : int]: (?[Z2 : int]: (![X6 : int]: ((ord_less_int @ X6 @ Z2) => (~ ((X6 = T))))))))). % minf(4)
thf(fact_121_minf_I3_J, axiom,
    ((![T : nat]: (?[Z2 : nat]: (![X6 : nat]: ((ord_less_nat @ X6 @ Z2) => (~ ((X6 = T))))))))). % minf(3)
thf(fact_122_minf_I3_J, axiom,
    ((![T : int]: (?[Z2 : int]: (![X6 : int]: ((ord_less_int @ X6 @ Z2) => (~ ((X6 = T))))))))). % minf(3)
thf(fact_123_minf_I2_J, axiom,
    ((![P : nat > $o, P4 : nat > $o, Q : nat > $o, Q2 : nat > $o]: ((?[Z3 : nat]: (![X4 : nat]: ((ord_less_nat @ X4 @ Z3) => ((P @ X4) = (P4 @ X4))))) => ((?[Z3 : nat]: (![X4 : nat]: ((ord_less_nat @ X4 @ Z3) => ((Q @ X4) = (Q2 @ X4))))) => (?[Z2 : nat]: (![X6 : nat]: ((ord_less_nat @ X6 @ Z2) => ((((P @ X6)) | ((Q @ X6))) = (((P4 @ X6)) | ((Q2 @ X6)))))))))))). % minf(2)
thf(fact_124_minf_I2_J, axiom,
    ((![P : int > $o, P4 : int > $o, Q : int > $o, Q2 : int > $o]: ((?[Z3 : int]: (![X4 : int]: ((ord_less_int @ X4 @ Z3) => ((P @ X4) = (P4 @ X4))))) => ((?[Z3 : int]: (![X4 : int]: ((ord_less_int @ X4 @ Z3) => ((Q @ X4) = (Q2 @ X4))))) => (?[Z2 : int]: (![X6 : int]: ((ord_less_int @ X6 @ Z2) => ((((P @ X6)) | ((Q @ X6))) = (((P4 @ X6)) | ((Q2 @ X6)))))))))))). % minf(2)
thf(fact_125_minf_I1_J, axiom,
    ((![P : nat > $o, P4 : nat > $o, Q : nat > $o, Q2 : nat > $o]: ((?[Z3 : nat]: (![X4 : nat]: ((ord_less_nat @ X4 @ Z3) => ((P @ X4) = (P4 @ X4))))) => ((?[Z3 : nat]: (![X4 : nat]: ((ord_less_nat @ X4 @ Z3) => ((Q @ X4) = (Q2 @ X4))))) => (?[Z2 : nat]: (![X6 : nat]: ((ord_less_nat @ X6 @ Z2) => ((((P @ X6)) & ((Q @ X6))) = (((P4 @ X6)) & ((Q2 @ X6)))))))))))). % minf(1)
thf(fact_126_minf_I1_J, axiom,
    ((![P : int > $o, P4 : int > $o, Q : int > $o, Q2 : int > $o]: ((?[Z3 : int]: (![X4 : int]: ((ord_less_int @ X4 @ Z3) => ((P @ X4) = (P4 @ X4))))) => ((?[Z3 : int]: (![X4 : int]: ((ord_less_int @ X4 @ Z3) => ((Q @ X4) = (Q2 @ X4))))) => (?[Z2 : int]: (![X6 : int]: ((ord_less_int @ X6 @ Z2) => ((((P @ X6)) & ((Q @ X6))) = (((P4 @ X6)) & ((Q2 @ X6)))))))))))). % minf(1)
thf(fact_127_pinf_I1_J, axiom,
    ((![P : nat > $o, P4 : nat > $o, Q : nat > $o, Q2 : nat > $o]: ((?[Z3 : nat]: (![X4 : nat]: ((ord_less_nat @ Z3 @ X4) => ((P @ X4) = (P4 @ X4))))) => ((?[Z3 : nat]: (![X4 : nat]: ((ord_less_nat @ Z3 @ X4) => ((Q @ X4) = (Q2 @ X4))))) => (?[Z2 : nat]: (![X6 : nat]: ((ord_less_nat @ Z2 @ X6) => ((((P @ X6)) & ((Q @ X6))) = (((P4 @ X6)) & ((Q2 @ X6)))))))))))). % pinf(1)
thf(fact_128_pinf_I1_J, axiom,
    ((![P : int > $o, P4 : int > $o, Q : int > $o, Q2 : int > $o]: ((?[Z3 : int]: (![X4 : int]: ((ord_less_int @ Z3 @ X4) => ((P @ X4) = (P4 @ X4))))) => ((?[Z3 : int]: (![X4 : int]: ((ord_less_int @ Z3 @ X4) => ((Q @ X4) = (Q2 @ X4))))) => (?[Z2 : int]: (![X6 : int]: ((ord_less_int @ Z2 @ X6) => ((((P @ X6)) & ((Q @ X6))) = (((P4 @ X6)) & ((Q2 @ X6)))))))))))). % pinf(1)
thf(fact_129_pinf_I2_J, axiom,
    ((![P : nat > $o, P4 : nat > $o, Q : nat > $o, Q2 : nat > $o]: ((?[Z3 : nat]: (![X4 : nat]: ((ord_less_nat @ Z3 @ X4) => ((P @ X4) = (P4 @ X4))))) => ((?[Z3 : nat]: (![X4 : nat]: ((ord_less_nat @ Z3 @ X4) => ((Q @ X4) = (Q2 @ X4))))) => (?[Z2 : nat]: (![X6 : nat]: ((ord_less_nat @ Z2 @ X6) => ((((P @ X6)) | ((Q @ X6))) = (((P4 @ X6)) | ((Q2 @ X6)))))))))))). % pinf(2)
thf(fact_130_pinf_I2_J, axiom,
    ((![P : int > $o, P4 : int > $o, Q : int > $o, Q2 : int > $o]: ((?[Z3 : int]: (![X4 : int]: ((ord_less_int @ Z3 @ X4) => ((P @ X4) = (P4 @ X4))))) => ((?[Z3 : int]: (![X4 : int]: ((ord_less_int @ Z3 @ X4) => ((Q @ X4) = (Q2 @ X4))))) => (?[Z2 : int]: (![X6 : int]: ((ord_less_int @ Z2 @ X6) => ((((P @ X6)) | ((Q @ X6))) = (((P4 @ X6)) | ((Q2 @ X6)))))))))))). % pinf(2)
thf(fact_131_pinf_I3_J, axiom,
    ((![T : nat]: (?[Z2 : nat]: (![X6 : nat]: ((ord_less_nat @ Z2 @ X6) => (~ ((X6 = T))))))))). % pinf(3)
thf(fact_132_pinf_I3_J, axiom,
    ((![T : int]: (?[Z2 : int]: (![X6 : int]: ((ord_less_int @ Z2 @ X6) => (~ ((X6 = T))))))))). % pinf(3)
thf(fact_133_pinf_I4_J, axiom,
    ((![T : nat]: (?[Z2 : nat]: (![X6 : nat]: ((ord_less_nat @ Z2 @ X6) => (~ ((X6 = T))))))))). % pinf(4)
thf(fact_134_pinf_I4_J, axiom,
    ((![T : int]: (?[Z2 : int]: (![X6 : int]: ((ord_less_int @ Z2 @ X6) => (~ ((X6 = T))))))))). % pinf(4)
thf(fact_135_pinf_I5_J, axiom,
    ((![T : nat]: (?[Z2 : nat]: (![X6 : nat]: ((ord_less_nat @ Z2 @ X6) => (~ ((ord_less_nat @ X6 @ T))))))))). % pinf(5)
thf(fact_136_pinf_I5_J, axiom,
    ((![T : int]: (?[Z2 : int]: (![X6 : int]: ((ord_less_int @ Z2 @ X6) => (~ ((ord_less_int @ X6 @ T))))))))). % pinf(5)
thf(fact_137_pinf_I7_J, axiom,
    ((![T : nat]: (?[Z2 : nat]: (![X6 : nat]: ((ord_less_nat @ Z2 @ X6) => (ord_less_nat @ T @ X6))))))). % pinf(7)
thf(fact_138_pinf_I7_J, axiom,
    ((![T : int]: (?[Z2 : int]: (![X6 : int]: ((ord_less_int @ Z2 @ X6) => (ord_less_int @ T @ X6))))))). % pinf(7)
thf(fact_139_linorder__neqE__linordered__idom, axiom,
    ((![X2 : int, Y : int]: ((~ ((X2 = Y))) => ((~ ((ord_less_int @ X2 @ Y))) => (ord_less_int @ Y @ X2)))))). % linorder_neqE_linordered_idom
thf(fact_140_verit__comp__simplify1_I1_J, axiom,
    ((![A : nat]: (~ ((ord_less_nat @ A @ A)))))). % verit_comp_simplify1(1)
thf(fact_141_verit__comp__simplify1_I1_J, axiom,
    ((![A : int]: (~ ((ord_less_int @ A @ A)))))). % verit_comp_simplify1(1)
thf(fact_142_convergent__subseq__convergent, axiom,
    ((![X7 : nat > nat, F2 : nat > nat]: ((topolo768750839nt_nat @ X7) => ((order_769474267at_nat @ F2) => (topolo768750839nt_nat @ (comp_nat_nat_nat @ X7 @ F2))))))). % convergent_subseq_convergent
thf(fact_143_of__nat__less__iff, axiom,
    ((![M4 : nat, N2 : nat]: ((ord_less_nat @ (semiri1382578993at_nat @ M4) @ (semiri1382578993at_nat @ N2)) = (ord_less_nat @ M4 @ N2))))). % of_nat_less_iff
thf(fact_144_of__nat__less__iff, axiom,
    ((![M4 : nat, N2 : nat]: ((ord_less_int @ (semiri2019852685at_int @ M4) @ (semiri2019852685at_int @ N2)) = (ord_less_nat @ M4 @ N2))))). % of_nat_less_iff
thf(fact_145_of__nat__eq__iff, axiom,
    ((![M4 : nat, N2 : nat]: (((semiri2019852685at_int @ M4) = (semiri2019852685at_int @ N2)) = (M4 = N2))))). % of_nat_eq_iff
thf(fact_146_nat__int__comparison_I2_J, axiom,
    ((ord_less_nat = (^[A3 : nat]: (^[B3 : nat]: (ord_less_int @ (semiri2019852685at_int @ A3) @ (semiri2019852685at_int @ B3))))))). % nat_int_comparison(2)
thf(fact_147_less__imp__of__nat__less, axiom,
    ((![M4 : nat, N2 : nat]: ((ord_less_nat @ M4 @ N2) => (ord_less_nat @ (semiri1382578993at_nat @ M4) @ (semiri1382578993at_nat @ N2)))))). % less_imp_of_nat_less
thf(fact_148_less__imp__of__nat__less, axiom,
    ((![M4 : nat, N2 : nat]: ((ord_less_nat @ M4 @ N2) => (ord_less_int @ (semiri2019852685at_int @ M4) @ (semiri2019852685at_int @ N2)))))). % less_imp_of_nat_less
thf(fact_149_of__nat__less__imp__less, axiom,
    ((![M4 : nat, N2 : nat]: ((ord_less_nat @ (semiri1382578993at_nat @ M4) @ (semiri1382578993at_nat @ N2)) => (ord_less_nat @ M4 @ N2))))). % of_nat_less_imp_less
thf(fact_150_of__nat__less__imp__less, axiom,
    ((![M4 : nat, N2 : nat]: ((ord_less_int @ (semiri2019852685at_int @ M4) @ (semiri2019852685at_int @ N2)) => (ord_less_nat @ M4 @ N2))))). % of_nat_less_imp_less
thf(fact_151_nat__int__comparison_I1_J, axiom,
    (((^[Y5 : nat]: (^[Z4 : nat]: (Y5 = Z4))) = (^[A3 : nat]: (^[B3 : nat]: ((semiri2019852685at_int @ A3) = (semiri2019852685at_int @ B3))))))). % nat_int_comparison(1)
thf(fact_152_int__if, axiom,
    ((![P : $o, A : nat, B : nat]: ((P => ((semiri2019852685at_int @ (if_nat @ P @ A @ B)) = (semiri2019852685at_int @ A))) & ((~ (P)) => ((semiri2019852685at_int @ (if_nat @ P @ A @ B)) = (semiri2019852685at_int @ B))))))). % int_if
thf(fact_153_map__fun_Ocomp, axiom,
    ((![F2 : nat > nat, G2 : nat > nat, H : nat > nat, I : nat > nat]: ((comp_n997826594at_nat @ (map_fu972944474at_nat @ F2 @ G2) @ (map_fu972944474at_nat @ H @ I)) = (map_fu972944474at_nat @ (comp_nat_nat_nat @ H @ F2) @ (comp_nat_nat_nat @ G2 @ I)))))). % map_fun.comp
thf(fact_154_of__nat__less__of__int__iff, axiom,
    ((![N2 : nat, X2 : int]: ((ord_less_int @ (semiri2019852685at_int @ N2) @ (ring_1_of_int_int @ X2)) = (ord_less_int @ (semiri2019852685at_int @ N2) @ X2))))). % of_nat_less_of_int_iff
thf(fact_155_strict__mono__Suc__iff, axiom,
    ((order_1406747959at_int = (^[F : nat > int]: (![N4 : nat]: (ord_less_int @ (F @ N4) @ (F @ (suc @ N4)))))))). % strict_mono_Suc_iff
thf(fact_156_strict__mono__Suc__iff, axiom,
    ((order_769474267at_nat = (^[F : nat > nat]: (![N4 : nat]: (ord_less_nat @ (F @ N4) @ (F @ (suc @ N4)))))))). % strict_mono_Suc_iff
thf(fact_157_nat_Oinject, axiom,
    ((![X22 : nat, Y22 : nat]: (((suc @ X22) = (suc @ Y22)) = (X22 = Y22))))). % nat.inject
thf(fact_158_old_Onat_Oinject, axiom,
    ((![Nat : nat, Nat2 : nat]: (((suc @ Nat) = (suc @ Nat2)) = (Nat = Nat2))))). % old.nat.inject
thf(fact_159_lessI, axiom,
    ((![N2 : nat]: (ord_less_nat @ N2 @ (suc @ N2))))). % lessI
thf(fact_160_Suc__mono, axiom,
    ((![M4 : nat, N2 : nat]: ((ord_less_nat @ M4 @ N2) => (ord_less_nat @ (suc @ M4) @ (suc @ N2)))))). % Suc_mono
thf(fact_161_Suc__less__eq, axiom,
    ((![M4 : nat, N2 : nat]: ((ord_less_nat @ (suc @ M4) @ (suc @ N2)) = (ord_less_nat @ M4 @ N2))))). % Suc_less_eq
thf(fact_162_of__int__less__iff, axiom,
    ((![W : int, Z : int]: ((ord_less_int @ (ring_1_of_int_int @ W) @ (ring_1_of_int_int @ Z)) = (ord_less_int @ W @ Z))))). % of_int_less_iff
thf(fact_163_Nat_OlessE, axiom,
    ((![I : nat, K : nat]: ((ord_less_nat @ I @ K) => ((~ ((K = (suc @ I)))) => (~ ((![J : nat]: ((ord_less_nat @ I @ J) => (~ ((K = (suc @ J))))))))))))). % Nat.lessE
thf(fact_164_Suc__lessD, axiom,
    ((![M4 : nat, N2 : nat]: ((ord_less_nat @ (suc @ M4) @ N2) => (ord_less_nat @ M4 @ N2))))). % Suc_lessD
thf(fact_165_Suc__lessE, axiom,
    ((![I : nat, K : nat]: ((ord_less_nat @ (suc @ I) @ K) => (~ ((![J : nat]: ((ord_less_nat @ I @ J) => (~ ((K = (suc @ J)))))))))))). % Suc_lessE
thf(fact_166_Suc__lessI, axiom,
    ((![M4 : nat, N2 : nat]: ((ord_less_nat @ M4 @ N2) => ((~ (((suc @ M4) = N2))) => (ord_less_nat @ (suc @ M4) @ N2)))))). % Suc_lessI
thf(fact_167_less__SucE, axiom,
    ((![M4 : nat, N2 : nat]: ((ord_less_nat @ M4 @ (suc @ N2)) => ((~ ((ord_less_nat @ M4 @ N2))) => (M4 = N2)))))). % less_SucE
thf(fact_168_less__SucI, axiom,
    ((![M4 : nat, N2 : nat]: ((ord_less_nat @ M4 @ N2) => (ord_less_nat @ M4 @ (suc @ N2)))))). % less_SucI
thf(fact_169_Ex__less__Suc, axiom,
    ((![N2 : nat, P : nat > $o]: ((?[I2 : nat]: (((ord_less_nat @ I2 @ (suc @ N2))) & ((P @ I2)))) = (((P @ N2)) | ((?[I2 : nat]: (((ord_less_nat @ I2 @ N2)) & ((P @ I2)))))))))). % Ex_less_Suc
thf(fact_170_less__Suc__eq, axiom,
    ((![M4 : nat, N2 : nat]: ((ord_less_nat @ M4 @ (suc @ N2)) = (((ord_less_nat @ M4 @ N2)) | ((M4 = N2))))))). % less_Suc_eq
thf(fact_171_not__less__eq, axiom,
    ((![M4 : nat, N2 : nat]: ((~ ((ord_less_nat @ M4 @ N2))) = (ord_less_nat @ N2 @ (suc @ M4)))))). % not_less_eq
thf(fact_172_All__less__Suc, axiom,
    ((![N2 : nat, P : nat > $o]: ((![I2 : nat]: (((ord_less_nat @ I2 @ (suc @ N2))) => ((P @ I2)))) = (((P @ N2)) & ((![I2 : nat]: (((ord_less_nat @ I2 @ N2)) => ((P @ I2)))))))))). % All_less_Suc
thf(fact_173_Suc__less__eq2, axiom,
    ((![N2 : nat, M4 : nat]: ((ord_less_nat @ (suc @ N2) @ M4) = (?[M5 : nat]: (((M4 = (suc @ M5))) & ((ord_less_nat @ N2 @ M5)))))))). % Suc_less_eq2
thf(fact_174_less__antisym, axiom,
    ((![N2 : nat, M4 : nat]: ((~ ((ord_less_nat @ N2 @ M4))) => ((ord_less_nat @ N2 @ (suc @ M4)) => (M4 = N2)))))). % less_antisym
thf(fact_175_Suc__less__SucD, axiom,
    ((![M4 : nat, N2 : nat]: ((ord_less_nat @ (suc @ M4) @ (suc @ N2)) => (ord_less_nat @ M4 @ N2))))). % Suc_less_SucD
thf(fact_176_less__trans__Suc, axiom,
    ((![I : nat, J2 : nat, K : nat]: ((ord_less_nat @ I @ J2) => ((ord_less_nat @ J2 @ K) => (ord_less_nat @ (suc @ I) @ K)))))). % less_trans_Suc
thf(fact_177_less__Suc__induct, axiom,
    ((![I : nat, J2 : nat, P : nat > nat > $o]: ((ord_less_nat @ I @ J2) => ((![I3 : nat]: (P @ I3 @ (suc @ I3))) => ((![I3 : nat, J : nat, K2 : nat]: ((ord_less_nat @ I3 @ J) => ((ord_less_nat @ J @ K2) => ((P @ I3 @ J) => ((P @ J @ K2) => (P @ I3 @ K2)))))) => (P @ I @ J2))))))). % less_Suc_induct
thf(fact_178_strict__inc__induct, axiom,
    ((![I : nat, J2 : nat, P : nat > $o]: ((ord_less_nat @ I @ J2) => ((![I3 : nat]: ((J2 = (suc @ I3)) => (P @ I3))) => ((![I3 : nat]: ((ord_less_nat @ I3 @ J2) => ((P @ (suc @ I3)) => (P @ I3)))) => (P @ I))))))). % strict_inc_induct
thf(fact_179_not__less__less__Suc__eq, axiom,
    ((![N2 : nat, M4 : nat]: ((~ ((ord_less_nat @ N2 @ M4))) => ((ord_less_nat @ N2 @ (suc @ M4)) = (N2 = M4)))))). % not_less_less_Suc_eq
thf(fact_180_Suc__inject, axiom,
    ((![X2 : nat, Y : nat]: (((suc @ X2) = (suc @ Y)) => (X2 = Y))))). % Suc_inject
thf(fact_181_n__not__Suc__n, axiom,
    ((![N2 : nat]: (~ ((N2 = (suc @ N2))))))). % n_not_Suc_n
thf(fact_182_map__fun_Ocompositionality, axiom,
    ((![F2 : nat > nat, G2 : nat > nat, H : nat > nat, I : nat > nat, Fun : nat > nat]: ((map_fu972944474at_nat @ F2 @ G2 @ (map_fu972944474at_nat @ H @ I @ Fun)) = (map_fu972944474at_nat @ (comp_nat_nat_nat @ H @ F2) @ (comp_nat_nat_nat @ G2 @ I) @ Fun))))). % map_fun.compositionality
thf(fact_183_map__fun__def, axiom,
    ((map_fu972944474at_nat = (^[F : nat > nat]: (^[G : nat > nat]: (^[H2 : nat > nat]: (comp_nat_nat_nat @ (comp_nat_nat_nat @ G @ H2) @ F))))))). % map_fun_def
thf(fact_184_lift__Suc__mono__less, axiom,
    ((![F2 : nat > nat, N2 : nat, N5 : nat]: ((![N3 : nat]: (ord_less_nat @ (F2 @ N3) @ (F2 @ (suc @ N3)))) => ((ord_less_nat @ N2 @ N5) => (ord_less_nat @ (F2 @ N2) @ (F2 @ N5))))))). % lift_Suc_mono_less
thf(fact_185_lift__Suc__mono__less, axiom,
    ((![F2 : nat > int, N2 : nat, N5 : nat]: ((![N3 : nat]: (ord_less_int @ (F2 @ N3) @ (F2 @ (suc @ N3)))) => ((ord_less_nat @ N2 @ N5) => (ord_less_int @ (F2 @ N2) @ (F2 @ N5))))))). % lift_Suc_mono_less
thf(fact_186_lift__Suc__mono__less__iff, axiom,
    ((![F2 : nat > nat, N2 : nat, M4 : nat]: ((![N3 : nat]: (ord_less_nat @ (F2 @ N3) @ (F2 @ (suc @ N3)))) => ((ord_less_nat @ (F2 @ N2) @ (F2 @ M4)) = (ord_less_nat @ N2 @ M4)))))). % lift_Suc_mono_less_iff
thf(fact_187_lift__Suc__mono__less__iff, axiom,
    ((![F2 : nat > int, N2 : nat, M4 : nat]: ((![N3 : nat]: (ord_less_int @ (F2 @ N3) @ (F2 @ (suc @ N3)))) => ((ord_less_int @ (F2 @ N2) @ (F2 @ M4)) = (ord_less_nat @ N2 @ M4)))))). % lift_Suc_mono_less_iff
thf(fact_188_card_Ocomp__fun__commute, axiom,
    (((comp_nat_nat_nat @ suc @ suc) = (comp_nat_nat_nat @ suc @ suc)))). % card.comp_fun_commute
thf(fact_189_zless__nat__eq__int__zless, axiom,
    ((![M4 : nat, Z : int]: ((ord_less_nat @ M4 @ (nat2 @ Z)) = (ord_less_int @ (semiri2019852685at_int @ M4) @ Z))))). % zless_nat_eq_int_zless
thf(fact_190_zless__nat__conj, axiom,
    ((![W : int, Z : int]: ((ord_less_nat @ (nat2 @ W) @ (nat2 @ Z)) = (((ord_less_int @ zero_zero_int @ Z)) & ((ord_less_int @ W @ Z))))))). % zless_nat_conj
thf(fact_191_nat__mono__iff, axiom,
    ((![Z : int, W : int]: ((ord_less_int @ zero_zero_int @ Z) => ((ord_less_nat @ (nat2 @ W) @ (nat2 @ Z)) = (ord_less_int @ W @ Z)))))). % nat_mono_iff
thf(fact_192_of__nat__0, axiom,
    (((semiri2019852685at_int @ zero_zero_nat) = zero_zero_int))). % of_nat_0
thf(fact_193_of__nat__0__eq__iff, axiom,
    ((![N2 : nat]: ((zero_zero_int = (semiri2019852685at_int @ N2)) = (zero_zero_nat = N2))))). % of_nat_0_eq_iff
thf(fact_194_of__nat__eq__0__iff, axiom,
    ((![M4 : nat]: (((semiri2019852685at_int @ M4) = zero_zero_int) = (M4 = zero_zero_nat))))). % of_nat_eq_0_iff
thf(fact_195_of__nat__0__less__iff, axiom,
    ((![N2 : nat]: ((ord_less_nat @ zero_zero_nat @ (semiri1382578993at_nat @ N2)) = (ord_less_nat @ zero_zero_nat @ N2))))). % of_nat_0_less_iff
thf(fact_196_of__nat__0__less__iff, axiom,
    ((![N2 : nat]: ((ord_less_int @ zero_zero_int @ (semiri2019852685at_int @ N2)) = (ord_less_nat @ zero_zero_nat @ N2))))). % of_nat_0_less_iff
thf(fact_197_of__int__0__less__iff, axiom,
    ((![Z : int]: ((ord_less_int @ zero_zero_int @ (ring_1_of_int_int @ Z)) = (ord_less_int @ zero_zero_int @ Z))))). % of_int_0_less_iff

% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T, axiom,
    ((![P : $o]: ((P = $true) | (P = $false))))).
thf(help_If_2_1_If_001t__Nat__Onat_T, axiom,
    ((![X2 : nat, Y : nat]: ((if_nat @ $false @ X2 @ Y) = Y)))).
thf(help_If_1_1_If_001t__Nat__Onat_T, axiom,
    ((![X2 : nat, Y : nat]: ((if_nat @ $true @ X2 @ Y) = X2)))).

% Conjectures (1)
thf(conj_0, conjecture,
    ((![X4 : nat, Y3 : nat]: ((~ ((ord_less_nat @ X4 @ Y3))) | (ord_less_nat @ (comp_nat_nat_nat @ f @ g @ X4) @ (comp_nat_nat_nat @ f @ g @ Y3)))))).
