% 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_300__5370526_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:29:02.977

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

% Explicit typings (23)
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_Groups_Oplus__class_Oplus_001t__Nat__Onat, type,
    plus_plus_nat : nat > nat > nat).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal, type,
    plus_plus_real : real > real > real).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat, type,
    zero_zero_nat : nat).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal, type,
    zero_zero_real : real).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Nat__Onat_J, type,
    ord_less_eq_o_nat : ($o > nat) > ($o > nat) > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Real__Oreal_J, type,
    ord_less_eq_o_real : ($o > real) > ($o > real) > $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__Real__Oreal, type,
    ord_less_eq_real : real > real > $o).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat, type,
    order_Greatest_nat : (nat > $o) > nat).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Real__Oreal, type,
    order_Greatest_real : (real > $o) > real).
thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Nat__Onat_001t__Nat__Onat, type,
    order_769474267at_nat : (nat > nat) > $o).
thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Nat__Onat_001t__Real__Oreal, type,
    order_952716343t_real : (nat > real) > $o).
thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Real__Oreal_001t__Nat__Onat, type,
    order_297469111al_nat : (real > nat) > $o).
thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Real__Oreal_001t__Real__Oreal, type,
    order_1818878995l_real : (real > real) > $o).
thf(sy_c_Topological__Spaces_Omonoseq_001t__Nat__Onat, type,
    topolo1922093437eq_nat : (nat > nat) > $o).
thf(sy_c_Topological__Spaces_Omonoseq_001t__Real__Oreal, type,
    topolo144289241q_real : (nat > real) > $o).
thf(sy_v_N1____, type,
    n1 : nat).
thf(sy_v_N2____, type,
    n2 : nat).
thf(sy_v_f____, type,
    f : nat > nat).
thf(sy_v_g____, type,
    g : nat > nat).
thf(sy_v_n____, type,
    n : nat).
thf(sy_v_r, type,
    r : real).

% Relevant facts (211)
thf(fact_0__092_060open_062n_A_092_060le_062_Ag_An_092_060close_062, axiom,
    ((ord_less_eq_nat @ n @ (g @ n)))). % \<open>n \<le> g n\<close>
thf(fact_1_that, axiom,
    ((ord_less_eq_nat @ (plus_plus_nat @ n1 @ n2) @ n))). % that
thf(fact_2_g_I1_J, axiom,
    ((order_769474267at_nat @ g))). % g(1)
thf(fact_3_order__refl, axiom,
    ((![X : nat]: (ord_less_eq_nat @ X @ X)))). % order_refl
thf(fact_4_order__refl, axiom,
    ((![X : real]: (ord_less_eq_real @ X @ X)))). % order_refl
thf(fact_5_le__refl, axiom,
    ((![N : nat]: (ord_less_eq_nat @ N @ N)))). % le_refl
thf(fact_6_le__trans, axiom,
    ((![I : nat, J : nat, K : nat]: ((ord_less_eq_nat @ I @ J) => ((ord_less_eq_nat @ J @ K) => (ord_less_eq_nat @ I @ K)))))). % le_trans
thf(fact_7_eq__imp__le, axiom,
    ((![M : nat, N : nat]: ((M = N) => (ord_less_eq_nat @ M @ N))))). % eq_imp_le
thf(fact_8_le__antisym, axiom,
    ((![M : nat, N : nat]: ((ord_less_eq_nat @ M @ N) => ((ord_less_eq_nat @ N @ M) => (M = N)))))). % le_antisym
thf(fact_9_nat__le__linear, axiom,
    ((![M : nat, N : nat]: ((ord_less_eq_nat @ M @ N) | (ord_less_eq_nat @ N @ M))))). % nat_le_linear
thf(fact_10_Nat_Oex__has__greatest__nat, axiom,
    ((![P : nat > $o, K : nat, B : nat]: ((P @ K) => ((![Y : nat]: ((P @ Y) => (ord_less_eq_nat @ Y @ B))) => (?[X2 : nat]: ((P @ X2) & (![Y2 : nat]: ((P @ Y2) => (ord_less_eq_nat @ Y2 @ X2)))))))))). % Nat.ex_has_greatest_nat
thf(fact_11_bounded__Max__nat, axiom,
    ((![P : nat > $o, X : nat, M2 : nat]: ((P @ X) => ((![X2 : nat]: ((P @ X2) => (ord_less_eq_nat @ X2 @ M2))) => (~ ((![M3 : nat]: ((P @ M3) => (~ ((![X3 : nat]: ((P @ X3) => (ord_less_eq_nat @ X3 @ M3)))))))))))))). % bounded_Max_nat
thf(fact_12_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_13_f_I1_J, axiom,
    ((order_769474267at_nat @ f))). % f(1)
thf(fact_14_strict__mono__eq, axiom,
    ((![F : nat > nat, X : nat, Y3 : nat]: ((order_769474267at_nat @ F) => (((F @ X) = (F @ Y3)) = (X = Y3)))))). % strict_mono_eq
thf(fact_15_strict__mono__imp__increasing, axiom,
    ((![F : nat > nat, N : nat]: ((order_769474267at_nat @ F) => (ord_less_eq_nat @ N @ (F @ N)))))). % strict_mono_imp_increasing
thf(fact_16_strict__mono__less__eq, axiom,
    ((![F : real > nat, X : real, Y3 : real]: ((order_297469111al_nat @ F) => ((ord_less_eq_nat @ (F @ X) @ (F @ Y3)) = (ord_less_eq_real @ X @ Y3)))))). % strict_mono_less_eq
thf(fact_17_strict__mono__less__eq, axiom,
    ((![F : nat > real, X : nat, Y3 : nat]: ((order_952716343t_real @ F) => ((ord_less_eq_real @ (F @ X) @ (F @ Y3)) = (ord_less_eq_nat @ X @ Y3)))))). % strict_mono_less_eq
thf(fact_18_strict__mono__less__eq, axiom,
    ((![F : real > real, X : real, Y3 : real]: ((order_1818878995l_real @ F) => ((ord_less_eq_real @ (F @ X) @ (F @ Y3)) = (ord_less_eq_real @ X @ Y3)))))). % strict_mono_less_eq
thf(fact_19_strict__mono__less__eq, axiom,
    ((![F : nat > nat, X : nat, Y3 : nat]: ((order_769474267at_nat @ F) => ((ord_less_eq_nat @ (F @ X) @ (F @ Y3)) = (ord_less_eq_nat @ X @ Y3)))))). % strict_mono_less_eq
thf(fact_20_nat__le__iff__add, axiom,
    ((ord_less_eq_nat = (^[M4 : nat]: (^[N2 : nat]: (?[K2 : nat]: (N2 = (plus_plus_nat @ M4 @ K2)))))))). % nat_le_iff_add
thf(fact_21_trans__le__add2, axiom,
    ((![I : nat, J : nat, M : nat]: ((ord_less_eq_nat @ I @ J) => (ord_less_eq_nat @ I @ (plus_plus_nat @ M @ J)))))). % trans_le_add2
thf(fact_22_trans__le__add1, axiom,
    ((![I : nat, J : nat, M : nat]: ((ord_less_eq_nat @ I @ J) => (ord_less_eq_nat @ I @ (plus_plus_nat @ J @ M)))))). % trans_le_add1
thf(fact_23_add__le__mono1, axiom,
    ((![I : nat, J : nat, K : nat]: ((ord_less_eq_nat @ I @ J) => (ord_less_eq_nat @ (plus_plus_nat @ I @ K) @ (plus_plus_nat @ J @ K)))))). % add_le_mono1
thf(fact_24_add__le__mono, 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_le_mono
thf(fact_25_le__Suc__ex, axiom,
    ((![K : nat, L : nat]: ((ord_less_eq_nat @ K @ L) => (?[N3 : nat]: (L = (plus_plus_nat @ K @ N3))))))). % le_Suc_ex
thf(fact_26_add__leD2, axiom,
    ((![M : nat, K : nat, N : nat]: ((ord_less_eq_nat @ (plus_plus_nat @ M @ K) @ N) => (ord_less_eq_nat @ K @ N))))). % add_leD2
thf(fact_27_add__leD1, axiom,
    ((![M : nat, K : nat, N : nat]: ((ord_less_eq_nat @ (plus_plus_nat @ M @ K) @ N) => (ord_less_eq_nat @ M @ N))))). % add_leD1
thf(fact_28_le__add2, axiom,
    ((![N : nat, M : nat]: (ord_less_eq_nat @ N @ (plus_plus_nat @ M @ N))))). % le_add2
thf(fact_29_le__add1, axiom,
    ((![N : nat, M : nat]: (ord_less_eq_nat @ N @ (plus_plus_nat @ N @ M))))). % le_add1
thf(fact_30_add__leE, axiom,
    ((![M : nat, K : nat, N : nat]: ((ord_less_eq_nat @ (plus_plus_nat @ M @ K) @ N) => (~ (((ord_less_eq_nat @ M @ N) => (~ ((ord_less_eq_nat @ K @ N)))))))))). % add_leE
thf(fact_31_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_32_dual__order_Oantisym, axiom,
    ((![B : real, A : real]: ((ord_less_eq_real @ B @ A) => ((ord_less_eq_real @ A @ B) => (A = B)))))). % dual_order.antisym
thf(fact_33_dual__order_Oeq__iff, axiom,
    (((^[Y4 : nat]: (^[Z : nat]: (Y4 = Z))) = (^[A2 : nat]: (^[B2 : nat]: (((ord_less_eq_nat @ B2 @ A2)) & ((ord_less_eq_nat @ A2 @ B2)))))))). % dual_order.eq_iff
thf(fact_34_dual__order_Oeq__iff, axiom,
    (((^[Y4 : real]: (^[Z : real]: (Y4 = Z))) = (^[A2 : real]: (^[B2 : real]: (((ord_less_eq_real @ B2 @ A2)) & ((ord_less_eq_real @ A2 @ B2)))))))). % dual_order.eq_iff
thf(fact_35_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_36_dual__order_Otrans, axiom,
    ((![B : real, A : real, C : real]: ((ord_less_eq_real @ B @ A) => ((ord_less_eq_real @ C @ B) => (ord_less_eq_real @ C @ A)))))). % dual_order.trans
thf(fact_37_linorder__wlog, axiom,
    ((![P : nat > nat > $o, A : nat, B : nat]: ((![A3 : nat, B3 : nat]: ((ord_less_eq_nat @ A3 @ B3) => (P @ A3 @ B3))) => ((![A3 : nat, B3 : nat]: ((P @ B3 @ A3) => (P @ A3 @ B3))) => (P @ A @ B)))))). % linorder_wlog
thf(fact_38_linorder__wlog, axiom,
    ((![P : real > real > $o, A : real, B : real]: ((![A3 : real, B3 : real]: ((ord_less_eq_real @ A3 @ B3) => (P @ A3 @ B3))) => ((![A3 : real, B3 : real]: ((P @ B3 @ A3) => (P @ A3 @ B3))) => (P @ A @ B)))))). % linorder_wlog
thf(fact_39_dual__order_Orefl, axiom,
    ((![A : nat]: (ord_less_eq_nat @ A @ A)))). % dual_order.refl
thf(fact_40_dual__order_Orefl, axiom,
    ((![A : real]: (ord_less_eq_real @ A @ A)))). % dual_order.refl
thf(fact_41_order__trans, axiom,
    ((![X : nat, Y3 : nat, Z2 : nat]: ((ord_less_eq_nat @ X @ Y3) => ((ord_less_eq_nat @ Y3 @ Z2) => (ord_less_eq_nat @ X @ Z2)))))). % order_trans
thf(fact_42_order__trans, axiom,
    ((![X : real, Y3 : real, Z2 : real]: ((ord_less_eq_real @ X @ Y3) => ((ord_less_eq_real @ Y3 @ Z2) => (ord_less_eq_real @ X @ Z2)))))). % order_trans
thf(fact_43_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_44_order__class_Oorder_Oantisym, axiom,
    ((![A : real, B : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_eq_real @ B @ A) => (A = B)))))). % order_class.order.antisym
thf(fact_45_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_46_ord__le__eq__trans, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_eq_real @ A @ B) => ((B = C) => (ord_less_eq_real @ A @ C)))))). % ord_le_eq_trans
thf(fact_47_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_48_ord__eq__le__trans, axiom,
    ((![A : real, B : real, C : real]: ((A = B) => ((ord_less_eq_real @ B @ C) => (ord_less_eq_real @ A @ C)))))). % ord_eq_le_trans
thf(fact_49_order__class_Oorder_Oeq__iff, axiom,
    (((^[Y4 : nat]: (^[Z : nat]: (Y4 = Z))) = (^[A2 : nat]: (^[B2 : nat]: (((ord_less_eq_nat @ A2 @ B2)) & ((ord_less_eq_nat @ B2 @ A2)))))))). % order_class.order.eq_iff
thf(fact_50_order__class_Oorder_Oeq__iff, axiom,
    (((^[Y4 : real]: (^[Z : real]: (Y4 = Z))) = (^[A2 : real]: (^[B2 : real]: (((ord_less_eq_real @ A2 @ B2)) & ((ord_less_eq_real @ B2 @ A2)))))))). % order_class.order.eq_iff
thf(fact_51_antisym__conv, axiom,
    ((![Y3 : nat, X : nat]: ((ord_less_eq_nat @ Y3 @ X) => ((ord_less_eq_nat @ X @ Y3) = (X = Y3)))))). % antisym_conv
thf(fact_52_antisym__conv, axiom,
    ((![Y3 : real, X : real]: ((ord_less_eq_real @ Y3 @ X) => ((ord_less_eq_real @ X @ Y3) = (X = Y3)))))). % antisym_conv
thf(fact_53_le__cases3, axiom,
    ((![X : nat, Y3 : nat, Z2 : nat]: (((ord_less_eq_nat @ X @ Y3) => (~ ((ord_less_eq_nat @ Y3 @ Z2)))) => (((ord_less_eq_nat @ Y3 @ X) => (~ ((ord_less_eq_nat @ X @ Z2)))) => (((ord_less_eq_nat @ X @ Z2) => (~ ((ord_less_eq_nat @ Z2 @ Y3)))) => (((ord_less_eq_nat @ Z2 @ Y3) => (~ ((ord_less_eq_nat @ Y3 @ X)))) => (((ord_less_eq_nat @ Y3 @ Z2) => (~ ((ord_less_eq_nat @ Z2 @ X)))) => (~ (((ord_less_eq_nat @ Z2 @ X) => (~ ((ord_less_eq_nat @ X @ Y3)))))))))))))). % le_cases3
thf(fact_54_le__cases3, axiom,
    ((![X : real, Y3 : real, Z2 : real]: (((ord_less_eq_real @ X @ Y3) => (~ ((ord_less_eq_real @ Y3 @ Z2)))) => (((ord_less_eq_real @ Y3 @ X) => (~ ((ord_less_eq_real @ X @ Z2)))) => (((ord_less_eq_real @ X @ Z2) => (~ ((ord_less_eq_real @ Z2 @ Y3)))) => (((ord_less_eq_real @ Z2 @ Y3) => (~ ((ord_less_eq_real @ Y3 @ X)))) => (((ord_less_eq_real @ Y3 @ Z2) => (~ ((ord_less_eq_real @ Z2 @ X)))) => (~ (((ord_less_eq_real @ Z2 @ X) => (~ ((ord_less_eq_real @ X @ Y3)))))))))))))). % le_cases3
thf(fact_55_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_56_order_Otrans, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_eq_real @ B @ C) => (ord_less_eq_real @ A @ C)))))). % order.trans
thf(fact_57_le__cases, axiom,
    ((![X : nat, Y3 : nat]: ((~ ((ord_less_eq_nat @ X @ Y3))) => (ord_less_eq_nat @ Y3 @ X))))). % le_cases
thf(fact_58_le__cases, axiom,
    ((![X : real, Y3 : real]: ((~ ((ord_less_eq_real @ X @ Y3))) => (ord_less_eq_real @ Y3 @ X))))). % le_cases
thf(fact_59_eq__refl, axiom,
    ((![X : nat, Y3 : nat]: ((X = Y3) => (ord_less_eq_nat @ X @ Y3))))). % eq_refl
thf(fact_60_eq__refl, axiom,
    ((![X : real, Y3 : real]: ((X = Y3) => (ord_less_eq_real @ X @ Y3))))). % eq_refl
thf(fact_61_linear, axiom,
    ((![X : nat, Y3 : nat]: ((ord_less_eq_nat @ X @ Y3) | (ord_less_eq_nat @ Y3 @ X))))). % linear
thf(fact_62_linear, axiom,
    ((![X : real, Y3 : real]: ((ord_less_eq_real @ X @ Y3) | (ord_less_eq_real @ Y3 @ X))))). % linear
thf(fact_63_antisym, axiom,
    ((![X : nat, Y3 : nat]: ((ord_less_eq_nat @ X @ Y3) => ((ord_less_eq_nat @ Y3 @ X) => (X = Y3)))))). % antisym
thf(fact_64_antisym, axiom,
    ((![X : real, Y3 : real]: ((ord_less_eq_real @ X @ Y3) => ((ord_less_eq_real @ Y3 @ X) => (X = Y3)))))). % antisym
thf(fact_65_eq__iff, axiom,
    (((^[Y4 : nat]: (^[Z : nat]: (Y4 = Z))) = (^[X4 : nat]: (^[Y5 : nat]: (((ord_less_eq_nat @ X4 @ Y5)) & ((ord_less_eq_nat @ Y5 @ X4)))))))). % eq_iff
thf(fact_66_eq__iff, axiom,
    (((^[Y4 : real]: (^[Z : real]: (Y4 = Z))) = (^[X4 : real]: (^[Y5 : real]: (((ord_less_eq_real @ X4 @ Y5)) & ((ord_less_eq_real @ Y5 @ X4)))))))). % eq_iff
thf(fact_67_ord__le__eq__subst, axiom,
    ((![A : nat, B : nat, F : nat > nat, C : nat]: ((ord_less_eq_nat @ A @ B) => (((F @ B) = C) => ((![X2 : nat, Y : nat]: ((ord_less_eq_nat @ X2 @ Y) => (ord_less_eq_nat @ (F @ X2) @ (F @ Y)))) => (ord_less_eq_nat @ (F @ A) @ C))))))). % ord_le_eq_subst
thf(fact_68_ord__le__eq__subst, axiom,
    ((![A : nat, B : nat, F : nat > real, C : real]: ((ord_less_eq_nat @ A @ B) => (((F @ B) = C) => ((![X2 : nat, Y : nat]: ((ord_less_eq_nat @ X2 @ Y) => (ord_less_eq_real @ (F @ X2) @ (F @ Y)))) => (ord_less_eq_real @ (F @ A) @ C))))))). % ord_le_eq_subst
thf(fact_69_ord__le__eq__subst, axiom,
    ((![A : real, B : real, F : real > nat, C : nat]: ((ord_less_eq_real @ A @ B) => (((F @ B) = C) => ((![X2 : real, Y : real]: ((ord_less_eq_real @ X2 @ Y) => (ord_less_eq_nat @ (F @ X2) @ (F @ Y)))) => (ord_less_eq_nat @ (F @ A) @ C))))))). % ord_le_eq_subst
thf(fact_70_ord__le__eq__subst, axiom,
    ((![A : real, B : real, F : real > real, C : real]: ((ord_less_eq_real @ A @ B) => (((F @ B) = C) => ((![X2 : real, Y : real]: ((ord_less_eq_real @ X2 @ Y) => (ord_less_eq_real @ (F @ X2) @ (F @ Y)))) => (ord_less_eq_real @ (F @ A) @ C))))))). % ord_le_eq_subst
thf(fact_71_ord__eq__le__subst, axiom,
    ((![A : nat, F : nat > nat, B : nat, C : nat]: ((A = (F @ B)) => ((ord_less_eq_nat @ B @ C) => ((![X2 : nat, Y : nat]: ((ord_less_eq_nat @ X2 @ Y) => (ord_less_eq_nat @ (F @ X2) @ (F @ Y)))) => (ord_less_eq_nat @ A @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_72_ord__eq__le__subst, axiom,
    ((![A : real, F : nat > real, B : nat, C : nat]: ((A = (F @ B)) => ((ord_less_eq_nat @ B @ C) => ((![X2 : nat, Y : nat]: ((ord_less_eq_nat @ X2 @ Y) => (ord_less_eq_real @ (F @ X2) @ (F @ Y)))) => (ord_less_eq_real @ A @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_73_ord__eq__le__subst, axiom,
    ((![A : nat, F : real > nat, B : real, C : real]: ((A = (F @ B)) => ((ord_less_eq_real @ B @ C) => ((![X2 : real, Y : real]: ((ord_less_eq_real @ X2 @ Y) => (ord_less_eq_nat @ (F @ X2) @ (F @ Y)))) => (ord_less_eq_nat @ A @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_74_ord__eq__le__subst, axiom,
    ((![A : real, F : real > real, B : real, C : real]: ((A = (F @ B)) => ((ord_less_eq_real @ B @ C) => ((![X2 : real, Y : real]: ((ord_less_eq_real @ X2 @ Y) => (ord_less_eq_real @ (F @ X2) @ (F @ Y)))) => (ord_less_eq_real @ A @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_75_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) => ((![X2 : nat, Y : nat]: ((ord_less_eq_nat @ X2 @ Y) => (ord_less_eq_nat @ (F @ X2) @ (F @ Y)))) => (ord_less_eq_nat @ (F @ A) @ C))))))). % order_subst2
thf(fact_76_order__subst2, axiom,
    ((![A : nat, B : nat, F : nat > real, C : real]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_real @ (F @ B) @ C) => ((![X2 : nat, Y : nat]: ((ord_less_eq_nat @ X2 @ Y) => (ord_less_eq_real @ (F @ X2) @ (F @ Y)))) => (ord_less_eq_real @ (F @ A) @ C))))))). % order_subst2
thf(fact_77_order__subst2, axiom,
    ((![A : real, B : real, F : real > nat, C : nat]: ((ord_less_eq_real @ A @ B) => ((ord_less_eq_nat @ (F @ B) @ C) => ((![X2 : real, Y : real]: ((ord_less_eq_real @ X2 @ Y) => (ord_less_eq_nat @ (F @ X2) @ (F @ Y)))) => (ord_less_eq_nat @ (F @ A) @ C))))))). % order_subst2
thf(fact_78_order__subst2, axiom,
    ((![A : real, B : real, F : real > real, C : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_eq_real @ (F @ B) @ C) => ((![X2 : real, Y : real]: ((ord_less_eq_real @ X2 @ Y) => (ord_less_eq_real @ (F @ X2) @ (F @ Y)))) => (ord_less_eq_real @ (F @ A) @ C))))))). % order_subst2
thf(fact_79_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) => ((![X2 : nat, Y : nat]: ((ord_less_eq_nat @ X2 @ Y) => (ord_less_eq_nat @ (F @ X2) @ (F @ Y)))) => (ord_less_eq_nat @ A @ (F @ C)))))))). % order_subst1
thf(fact_80_order__subst1, axiom,
    ((![A : nat, F : real > nat, B : real, C : real]: ((ord_less_eq_nat @ A @ (F @ B)) => ((ord_less_eq_real @ B @ C) => ((![X2 : real, Y : real]: ((ord_less_eq_real @ X2 @ Y) => (ord_less_eq_nat @ (F @ X2) @ (F @ Y)))) => (ord_less_eq_nat @ A @ (F @ C)))))))). % order_subst1
thf(fact_81_order__subst1, axiom,
    ((![A : real, F : nat > real, B : nat, C : nat]: ((ord_less_eq_real @ A @ (F @ B)) => ((ord_less_eq_nat @ B @ C) => ((![X2 : nat, Y : nat]: ((ord_less_eq_nat @ X2 @ Y) => (ord_less_eq_real @ (F @ X2) @ (F @ Y)))) => (ord_less_eq_real @ A @ (F @ C)))))))). % order_subst1
thf(fact_82_order__subst1, axiom,
    ((![A : real, F : real > real, B : real, C : real]: ((ord_less_eq_real @ A @ (F @ B)) => ((ord_less_eq_real @ B @ C) => ((![X2 : real, Y : real]: ((ord_less_eq_real @ X2 @ Y) => (ord_less_eq_real @ (F @ X2) @ (F @ Y)))) => (ord_less_eq_real @ A @ (F @ C)))))))). % order_subst1
thf(fact_83_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_84_add__le__cancel__right, axiom,
    ((![A : real, C : real, B : real]: ((ord_less_eq_real @ (plus_plus_real @ A @ C) @ (plus_plus_real @ B @ C)) = (ord_less_eq_real @ A @ B))))). % add_le_cancel_right
thf(fact_85_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_86_add__le__cancel__left, axiom,
    ((![C : real, A : real, B : real]: ((ord_less_eq_real @ (plus_plus_real @ C @ A) @ (plus_plus_real @ C @ B)) = (ord_less_eq_real @ A @ B))))). % add_le_cancel_left
thf(fact_87_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_88_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_89_strict__mono__leD, axiom,
    ((![R : nat > real, M : nat, N : nat]: ((order_952716343t_real @ R) => ((ord_less_eq_nat @ M @ N) => (ord_less_eq_real @ (R @ M) @ (R @ N))))))). % strict_mono_leD
thf(fact_90_strict__mono__leD, axiom,
    ((![R : real > nat, M : real, N : real]: ((order_297469111al_nat @ R) => ((ord_less_eq_real @ M @ N) => (ord_less_eq_nat @ (R @ M) @ (R @ N))))))). % strict_mono_leD
thf(fact_91_strict__mono__leD, axiom,
    ((![R : real > real, M : real, N : real]: ((order_1818878995l_real @ R) => ((ord_less_eq_real @ M @ N) => (ord_less_eq_real @ (R @ M) @ (R @ N))))))). % strict_mono_leD
thf(fact_92_strict__mono__leD, axiom,
    ((![R : nat > nat, M : nat, N : nat]: ((order_769474267at_nat @ R) => ((ord_less_eq_nat @ M @ N) => (ord_less_eq_nat @ (R @ M) @ (R @ N))))))). % strict_mono_leD
thf(fact_93_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_94_add__le__imp__le__right, axiom,
    ((![A : real, C : real, B : real]: ((ord_less_eq_real @ (plus_plus_real @ A @ C) @ (plus_plus_real @ B @ C)) => (ord_less_eq_real @ A @ B))))). % add_le_imp_le_right
thf(fact_95_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_96_add__le__imp__le__left, axiom,
    ((![C : real, A : real, B : real]: ((ord_less_eq_real @ (plus_plus_real @ C @ A) @ (plus_plus_real @ C @ B)) => (ord_less_eq_real @ A @ B))))). % add_le_imp_le_left
thf(fact_97_le__iff__add, axiom,
    ((ord_less_eq_nat = (^[A2 : nat]: (^[B2 : nat]: (?[C2 : nat]: (B2 = (plus_plus_nat @ A2 @ C2)))))))). % le_iff_add
thf(fact_98_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_99_add__right__mono, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_eq_real @ A @ B) => (ord_less_eq_real @ (plus_plus_real @ A @ C) @ (plus_plus_real @ B @ C)))))). % add_right_mono
thf(fact_100_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_101_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_102_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_103_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_104_add_Ocommute, axiom,
    ((plus_plus_nat = (^[A2 : nat]: (^[B2 : nat]: (plus_plus_nat @ B2 @ A2)))))). % add.commute
thf(fact_105_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_106_group__cancel_Oadd2, axiom,
    ((![B4 : nat, K : nat, B : nat, A : nat]: ((B4 = (plus_plus_nat @ K @ B)) => ((plus_plus_nat @ A @ B4) = (plus_plus_nat @ K @ (plus_plus_nat @ A @ B))))))). % group_cancel.add2
thf(fact_107_group__cancel_Oadd1, axiom,
    ((![A4 : nat, K : nat, A : nat, B : nat]: ((A4 = (plus_plus_nat @ K @ A)) => ((plus_plus_nat @ A4 @ B) = (plus_plus_nat @ K @ (plus_plus_nat @ A @ B))))))). % group_cancel.add1
thf(fact_108_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_109_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_110_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_111_add__mono__thms__linordered__semiring_I3_J, axiom,
    ((![I : real, J : real, K : real, L : real]: (((ord_less_eq_real @ I @ J) & (K = L)) => (ord_less_eq_real @ (plus_plus_real @ I @ K) @ (plus_plus_real @ J @ L)))))). % add_mono_thms_linordered_semiring(3)
thf(fact_112_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_113_add__mono__thms__linordered__semiring_I2_J, axiom,
    ((![I : real, J : real, K : real, L : real]: (((I = J) & (ord_less_eq_real @ K @ L)) => (ord_less_eq_real @ (plus_plus_real @ I @ K) @ (plus_plus_real @ J @ L)))))). % add_mono_thms_linordered_semiring(2)
thf(fact_114_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_115_add__mono__thms__linordered__semiring_I1_J, axiom,
    ((![I : real, J : real, K : real, L : real]: (((ord_less_eq_real @ I @ J) & (ord_less_eq_real @ K @ L)) => (ord_less_eq_real @ (plus_plus_real @ I @ K) @ (plus_plus_real @ J @ L)))))). % add_mono_thms_linordered_semiring(1)
thf(fact_116_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_117_add__mono, axiom,
    ((![A : real, B : real, C : real, D : real]: ((ord_less_eq_real @ A @ B) => ((ord_less_eq_real @ C @ D) => (ord_less_eq_real @ (plus_plus_real @ A @ C) @ (plus_plus_real @ B @ D))))))). % add_mono
thf(fact_118_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_119_add__left__mono, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_eq_real @ A @ B) => (ord_less_eq_real @ (plus_plus_real @ C @ A) @ (plus_plus_real @ C @ B)))))). % add_left_mono
thf(fact_120_hs, axiom,
    ((order_769474267at_nat @ (comp_nat_nat_nat @ f @ g)))). % hs
thf(fact_121_GreatestI2__order, axiom,
    ((![P : real > $o, X : real, Q : real > $o]: ((P @ X) => ((![Y : real]: ((P @ Y) => (ord_less_eq_real @ Y @ X))) => ((![X2 : real]: ((P @ X2) => ((![Y2 : real]: ((P @ Y2) => (ord_less_eq_real @ Y2 @ X2))) => (Q @ X2)))) => (Q @ (order_Greatest_real @ P)))))))). % GreatestI2_order
thf(fact_122_GreatestI2__order, axiom,
    ((![P : nat > $o, X : nat, Q : nat > $o]: ((P @ X) => ((![Y : nat]: ((P @ Y) => (ord_less_eq_nat @ Y @ X))) => ((![X2 : nat]: ((P @ X2) => ((![Y2 : nat]: ((P @ Y2) => (ord_less_eq_nat @ Y2 @ X2))) => (Q @ X2)))) => (Q @ (order_Greatest_nat @ P)))))))). % GreatestI2_order
thf(fact_123_Greatest__equality, axiom,
    ((![P : real > $o, X : real]: ((P @ X) => ((![Y : real]: ((P @ Y) => (ord_less_eq_real @ Y @ X))) => ((order_Greatest_real @ P) = X)))))). % Greatest_equality
thf(fact_124_Greatest__equality, axiom,
    ((![P : nat > $o, X : nat]: ((P @ X) => ((![Y : nat]: ((P @ Y) => (ord_less_eq_nat @ Y @ X))) => ((order_Greatest_nat @ P) = X)))))). % Greatest_equality
thf(fact_125_monoI1, axiom,
    ((![X5 : nat > nat]: ((![M3 : nat, N3 : nat]: ((ord_less_eq_nat @ M3 @ N3) => (ord_less_eq_nat @ (X5 @ M3) @ (X5 @ N3)))) => (topolo1922093437eq_nat @ X5))))). % monoI1
thf(fact_126_monoI1, axiom,
    ((![X5 : nat > real]: ((![M3 : nat, N3 : nat]: ((ord_less_eq_nat @ M3 @ N3) => (ord_less_eq_real @ (X5 @ M3) @ (X5 @ N3)))) => (topolo144289241q_real @ X5))))). % monoI1
thf(fact_127_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_128_GreatestI__ex__nat, axiom,
    ((![P : nat > $o, B : nat]: ((?[X_1 : nat]: (P @ X_1)) => ((![Y : nat]: ((P @ Y) => (ord_less_eq_nat @ Y @ B))) => (P @ (order_Greatest_nat @ P))))))). % GreatestI_ex_nat
thf(fact_129_Greatest__le__nat, axiom,
    ((![P : nat > $o, K : nat, B : nat]: ((P @ K) => ((![Y : nat]: ((P @ Y) => (ord_less_eq_nat @ Y @ B))) => (ord_less_eq_nat @ K @ (order_Greatest_nat @ P))))))). % Greatest_le_nat
thf(fact_130_GreatestI__nat, axiom,
    ((![P : nat > $o, K : nat, B : nat]: ((P @ K) => ((![Y : nat]: ((P @ Y) => (ord_less_eq_nat @ Y @ B))) => (P @ (order_Greatest_nat @ P))))))). % GreatestI_nat
thf(fact_131_monoseq__def, axiom,
    ((topolo1922093437eq_nat = (^[X6 : nat > nat]: (((![M4 : nat]: (![N2 : nat]: (((ord_less_eq_nat @ M4 @ N2)) => ((ord_less_eq_nat @ (X6 @ M4) @ (X6 @ N2))))))) | ((![M4 : nat]: (![N2 : nat]: (((ord_less_eq_nat @ M4 @ N2)) => ((ord_less_eq_nat @ (X6 @ N2) @ (X6 @ M4)))))))))))). % monoseq_def
thf(fact_132_monoseq__def, axiom,
    ((topolo144289241q_real = (^[X6 : nat > real]: (((![M4 : nat]: (![N2 : nat]: (((ord_less_eq_nat @ M4 @ N2)) => ((ord_less_eq_real @ (X6 @ M4) @ (X6 @ N2))))))) | ((![M4 : nat]: (![N2 : nat]: (((ord_less_eq_nat @ M4 @ N2)) => ((ord_less_eq_real @ (X6 @ N2) @ (X6 @ M4)))))))))))). % monoseq_def
thf(fact_133_monoI2, axiom,
    ((![X5 : nat > nat]: ((![M3 : nat, N3 : nat]: ((ord_less_eq_nat @ M3 @ N3) => (ord_less_eq_nat @ (X5 @ N3) @ (X5 @ M3)))) => (topolo1922093437eq_nat @ X5))))). % monoI2
thf(fact_134_monoI2, axiom,
    ((![X5 : nat > real]: ((![M3 : nat, N3 : nat]: ((ord_less_eq_nat @ M3 @ N3) => (ord_less_eq_real @ (X5 @ N3) @ (X5 @ M3)))) => (topolo144289241q_real @ X5))))). % monoI2
thf(fact_135_comp__apply, axiom,
    ((comp_nat_nat_nat = (^[F2 : nat > nat]: (^[G : nat > nat]: (^[X4 : nat]: (F2 @ (G @ X4)))))))). % comp_apply
thf(fact_136_comp__def, axiom,
    ((comp_nat_nat_nat = (^[F2 : nat > nat]: (^[G : nat > nat]: (^[X4 : nat]: (F2 @ (G @ X4)))))))). % comp_def
thf(fact_137_comp__assoc, axiom,
    ((![F : nat > nat, G2 : nat > nat, H : nat > nat]: ((comp_nat_nat_nat @ (comp_nat_nat_nat @ F @ G2) @ H) = (comp_nat_nat_nat @ F @ (comp_nat_nat_nat @ G2 @ H)))))). % comp_assoc
thf(fact_138_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_139_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_140_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_141_rewriteR__comp__comp2, axiom,
    ((![G2 : nat > nat, H : nat > nat, R1 : nat > nat, R2 : nat > nat, F : nat > nat, L : nat > nat]: (((comp_nat_nat_nat @ G2 @ H) = (comp_nat_nat_nat @ R1 @ R2)) => (((comp_nat_nat_nat @ F @ R1) = L) => ((comp_nat_nat_nat @ (comp_nat_nat_nat @ F @ G2) @ H) = (comp_nat_nat_nat @ L @ R2))))))). % rewriteR_comp_comp2
thf(fact_142_rewriteL__comp__comp2, axiom,
    ((![F : nat > nat, G2 : nat > nat, L1 : nat > nat, L2 : nat > nat, H : nat > nat, R : nat > nat]: (((comp_nat_nat_nat @ F @ G2) = (comp_nat_nat_nat @ L1 @ L2)) => (((comp_nat_nat_nat @ L2 @ H) = R) => ((comp_nat_nat_nat @ F @ (comp_nat_nat_nat @ G2 @ H)) = (comp_nat_nat_nat @ L1 @ R))))))). % rewriteL_comp_comp2
thf(fact_143_rewriteR__comp__comp, axiom,
    ((![G2 : nat > nat, H : nat > nat, R : nat > nat, F : nat > nat]: (((comp_nat_nat_nat @ G2 @ H) = R) => ((comp_nat_nat_nat @ (comp_nat_nat_nat @ F @ G2) @ H) = (comp_nat_nat_nat @ F @ R)))))). % rewriteR_comp_comp
thf(fact_144_rewriteL__comp__comp, axiom,
    ((![F : nat > nat, G2 : nat > nat, L : nat > nat, H : nat > nat]: (((comp_nat_nat_nat @ F @ G2) = L) => ((comp_nat_nat_nat @ F @ (comp_nat_nat_nat @ G2 @ H)) = (comp_nat_nat_nat @ L @ H)))))). % rewriteL_comp_comp
thf(fact_145_comp__cong, axiom,
    ((![F : nat > nat, G2 : nat > nat, X : nat, F3 : nat > nat, G3 : nat > nat, X7 : nat]: (((F @ (G2 @ X)) = (F3 @ (G3 @ X7))) => ((comp_nat_nat_nat @ F @ G2 @ X) = (comp_nat_nat_nat @ F3 @ G3 @ X7)))))). % comp_cong
thf(fact_146_comp__apply__eq, axiom,
    ((![F : nat > nat, G2 : nat > nat, X : nat, H : nat > nat, K : nat > nat]: (((F @ (G2 @ X)) = (H @ (K @ X))) => ((comp_nat_nat_nat @ F @ G2 @ X) = (comp_nat_nat_nat @ H @ K @ X)))))). % comp_apply_eq
thf(fact_147_fun_Omap__comp, axiom,
    ((![G2 : nat > nat, F : nat > nat, V : nat > nat]: ((comp_nat_nat_nat @ G2 @ (comp_nat_nat_nat @ F @ V)) = (comp_nat_nat_nat @ (comp_nat_nat_nat @ G2 @ F) @ V))))). % fun.map_comp
thf(fact_148_type__copy__map__cong0, axiom,
    ((![M2 : nat > nat, G2 : nat > nat, X : nat, N4 : nat > nat, H : nat > nat, F : nat > nat]: (((M2 @ (G2 @ X)) = (N4 @ (H @ X))) => ((comp_nat_nat_nat @ (comp_nat_nat_nat @ F @ M2) @ G2 @ X) = (comp_nat_nat_nat @ (comp_nat_nat_nat @ F @ N4) @ H @ X)))))). % type_copy_map_cong0
thf(fact_149_verit__la__disequality, axiom,
    ((![A : nat, B : nat]: ((A = B) | ((~ ((ord_less_eq_nat @ A @ B))) | (~ ((ord_less_eq_nat @ B @ A)))))))). % verit_la_disequality
thf(fact_150_verit__la__disequality, axiom,
    ((![A : real, B : real]: ((A = B) | ((~ ((ord_less_eq_real @ A @ B))) | (~ ((ord_less_eq_real @ B @ A)))))))). % verit_la_disequality
thf(fact_151_le__rel__bool__arg__iff, axiom,
    ((ord_less_eq_o_nat = (^[X6 : $o > nat]: (^[Y6 : $o > nat]: (((ord_less_eq_nat @ (X6 @ $false) @ (Y6 @ $false))) & ((ord_less_eq_nat @ (X6 @ $true) @ (Y6 @ $true))))))))). % le_rel_bool_arg_iff
thf(fact_152_le__rel__bool__arg__iff, axiom,
    ((ord_less_eq_o_real = (^[X6 : $o > real]: (^[Y6 : $o > real]: (((ord_less_eq_real @ (X6 @ $false) @ (Y6 @ $false))) & ((ord_less_eq_real @ (X6 @ $true) @ (Y6 @ $true))))))))). % le_rel_bool_arg_iff
thf(fact_153_add__le__same__cancel1, axiom,
    ((![B : nat, A : nat]: ((ord_less_eq_nat @ (plus_plus_nat @ B @ A) @ B) = (ord_less_eq_nat @ A @ zero_zero_nat))))). % add_le_same_cancel1
thf(fact_154_add__le__same__cancel1, axiom,
    ((![B : real, A : real]: ((ord_less_eq_real @ (plus_plus_real @ B @ A) @ B) = (ord_less_eq_real @ A @ zero_zero_real))))). % add_le_same_cancel1
thf(fact_155_add__le__same__cancel2, axiom,
    ((![A : nat, B : nat]: ((ord_less_eq_nat @ (plus_plus_nat @ A @ B) @ B) = (ord_less_eq_nat @ A @ zero_zero_nat))))). % add_le_same_cancel2
thf(fact_156_add__le__same__cancel2, axiom,
    ((![A : real, B : real]: ((ord_less_eq_real @ (plus_plus_real @ A @ B) @ B) = (ord_less_eq_real @ A @ zero_zero_real))))). % add_le_same_cancel2
thf(fact_157_rp, axiom,
    ((ord_less_eq_real @ zero_zero_real @ r))). % rp
thf(fact_158_bot__nat__0_Oextremum, axiom,
    ((![A : nat]: (ord_less_eq_nat @ zero_zero_nat @ A)))). % bot_nat_0.extremum
thf(fact_159_le0, axiom,
    ((![N : nat]: (ord_less_eq_nat @ zero_zero_nat @ N)))). % le0
thf(fact_160_Nat_Oadd__0__right, axiom,
    ((![M : nat]: ((plus_plus_nat @ M @ zero_zero_nat) = M)))). % Nat.add_0_right
thf(fact_161_add__is__0, axiom,
    ((![M : nat, N : nat]: (((plus_plus_nat @ M @ N) = zero_zero_nat) = (((M = zero_zero_nat)) & ((N = zero_zero_nat))))))). % add_is_0
thf(fact_162_le__zero__eq, axiom,
    ((![N : nat]: ((ord_less_eq_nat @ N @ zero_zero_nat) = (N = zero_zero_nat))))). % le_zero_eq
thf(fact_163_zero__eq__add__iff__both__eq__0, axiom,
    ((![X : nat, Y3 : nat]: ((zero_zero_nat = (plus_plus_nat @ X @ Y3)) = (((X = zero_zero_nat)) & ((Y3 = zero_zero_nat))))))). % zero_eq_add_iff_both_eq_0
thf(fact_164_add__eq__0__iff__both__eq__0, axiom,
    ((![X : nat, Y3 : nat]: (((plus_plus_nat @ X @ Y3) = zero_zero_nat) = (((X = zero_zero_nat)) & ((Y3 = zero_zero_nat))))))). % add_eq_0_iff_both_eq_0
thf(fact_165_add__cancel__right__right, axiom,
    ((![A : real, B : real]: ((A = (plus_plus_real @ A @ B)) = (B = zero_zero_real))))). % add_cancel_right_right
thf(fact_166_add__cancel__right__right, axiom,
    ((![A : nat, B : nat]: ((A = (plus_plus_nat @ A @ B)) = (B = zero_zero_nat))))). % add_cancel_right_right
thf(fact_167_add__cancel__right__left, axiom,
    ((![A : real, B : real]: ((A = (plus_plus_real @ B @ A)) = (B = zero_zero_real))))). % add_cancel_right_left
thf(fact_168_add__cancel__right__left, axiom,
    ((![A : nat, B : nat]: ((A = (plus_plus_nat @ B @ A)) = (B = zero_zero_nat))))). % add_cancel_right_left
thf(fact_169_add__cancel__left__right, axiom,
    ((![A : real, B : real]: (((plus_plus_real @ A @ B) = A) = (B = zero_zero_real))))). % add_cancel_left_right
thf(fact_170_add__cancel__left__right, axiom,
    ((![A : nat, B : nat]: (((plus_plus_nat @ A @ B) = A) = (B = zero_zero_nat))))). % add_cancel_left_right
thf(fact_171_add__cancel__left__left, axiom,
    ((![B : real, A : real]: (((plus_plus_real @ B @ A) = A) = (B = zero_zero_real))))). % add_cancel_left_left
thf(fact_172_add__cancel__left__left, axiom,
    ((![B : nat, A : nat]: (((plus_plus_nat @ B @ A) = A) = (B = zero_zero_nat))))). % add_cancel_left_left
thf(fact_173_double__zero__sym, axiom,
    ((![A : real]: ((zero_zero_real = (plus_plus_real @ A @ A)) = (A = zero_zero_real))))). % double_zero_sym
thf(fact_174_double__zero, axiom,
    ((![A : real]: (((plus_plus_real @ A @ A) = zero_zero_real) = (A = zero_zero_real))))). % double_zero
thf(fact_175_add_Oright__neutral, axiom,
    ((![A : real]: ((plus_plus_real @ A @ zero_zero_real) = A)))). % add.right_neutral
thf(fact_176_add_Oright__neutral, axiom,
    ((![A : nat]: ((plus_plus_nat @ A @ zero_zero_nat) = A)))). % add.right_neutral
thf(fact_177_add_Oleft__neutral, axiom,
    ((![A : real]: ((plus_plus_real @ zero_zero_real @ A) = A)))). % add.left_neutral
thf(fact_178_add_Oleft__neutral, axiom,
    ((![A : nat]: ((plus_plus_nat @ zero_zero_nat @ A) = A)))). % add.left_neutral
thf(fact_179_zero__le__double__add__iff__zero__le__single__add, axiom,
    ((![A : real]: ((ord_less_eq_real @ zero_zero_real @ (plus_plus_real @ A @ A)) = (ord_less_eq_real @ zero_zero_real @ A))))). % zero_le_double_add_iff_zero_le_single_add
thf(fact_180_double__add__le__zero__iff__single__add__le__zero, axiom,
    ((![A : real]: ((ord_less_eq_real @ (plus_plus_real @ A @ A) @ zero_zero_real) = (ord_less_eq_real @ A @ zero_zero_real))))). % double_add_le_zero_iff_single_add_le_zero
thf(fact_181_le__add__same__cancel2, axiom,
    ((![A : nat, B : nat]: ((ord_less_eq_nat @ A @ (plus_plus_nat @ B @ A)) = (ord_less_eq_nat @ zero_zero_nat @ B))))). % le_add_same_cancel2
thf(fact_182_le__add__same__cancel2, axiom,
    ((![A : real, B : real]: ((ord_less_eq_real @ A @ (plus_plus_real @ B @ A)) = (ord_less_eq_real @ zero_zero_real @ B))))). % le_add_same_cancel2
thf(fact_183_le__add__same__cancel1, axiom,
    ((![A : nat, B : nat]: ((ord_less_eq_nat @ A @ (plus_plus_nat @ A @ B)) = (ord_less_eq_nat @ zero_zero_nat @ B))))). % le_add_same_cancel1
thf(fact_184_le__add__same__cancel1, axiom,
    ((![A : real, B : real]: ((ord_less_eq_real @ A @ (plus_plus_real @ A @ B)) = (ord_less_eq_real @ zero_zero_real @ B))))). % le_add_same_cancel1
thf(fact_185_le__numeral__extra_I3_J, axiom,
    ((ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat))). % le_numeral_extra(3)
thf(fact_186_le__numeral__extra_I3_J, axiom,
    ((ord_less_eq_real @ zero_zero_real @ zero_zero_real))). % le_numeral_extra(3)
thf(fact_187_bot__nat__0_Oextremum__uniqueI, axiom,
    ((![A : nat]: ((ord_less_eq_nat @ A @ zero_zero_nat) => (A = zero_zero_nat))))). % bot_nat_0.extremum_uniqueI
thf(fact_188_bot__nat__0_Oextremum__unique, axiom,
    ((![A : nat]: ((ord_less_eq_nat @ A @ zero_zero_nat) = (A = zero_zero_nat))))). % bot_nat_0.extremum_unique
thf(fact_189_le__0__eq, axiom,
    ((![N : nat]: ((ord_less_eq_nat @ N @ zero_zero_nat) = (N = zero_zero_nat))))). % le_0_eq
thf(fact_190_less__eq__nat_Osimps_I1_J, axiom,
    ((![N : nat]: (ord_less_eq_nat @ zero_zero_nat @ N)))). % less_eq_nat.simps(1)
thf(fact_191_zero__le, axiom,
    ((![X : nat]: (ord_less_eq_nat @ zero_zero_nat @ X)))). % zero_le
thf(fact_192_add__decreasing, axiom,
    ((![A : nat, C : nat, B : nat]: ((ord_less_eq_nat @ A @ zero_zero_nat) => ((ord_less_eq_nat @ C @ B) => (ord_less_eq_nat @ (plus_plus_nat @ A @ C) @ B)))))). % add_decreasing
thf(fact_193_add__decreasing, axiom,
    ((![A : real, C : real, B : real]: ((ord_less_eq_real @ A @ zero_zero_real) => ((ord_less_eq_real @ C @ B) => (ord_less_eq_real @ (plus_plus_real @ A @ C) @ B)))))). % add_decreasing
thf(fact_194_add__increasing, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_eq_nat @ zero_zero_nat @ A) => ((ord_less_eq_nat @ B @ C) => (ord_less_eq_nat @ B @ (plus_plus_nat @ A @ C))))))). % add_increasing
thf(fact_195_add__increasing, axiom,
    ((![A : real, B : real, C : real]: ((ord_less_eq_real @ zero_zero_real @ A) => ((ord_less_eq_real @ B @ C) => (ord_less_eq_real @ B @ (plus_plus_real @ A @ C))))))). % add_increasing
thf(fact_196_add__decreasing2, axiom,
    ((![C : nat, A : nat, B : nat]: ((ord_less_eq_nat @ C @ zero_zero_nat) => ((ord_less_eq_nat @ A @ B) => (ord_less_eq_nat @ (plus_plus_nat @ A @ C) @ B)))))). % add_decreasing2
thf(fact_197_add__decreasing2, axiom,
    ((![C : real, A : real, B : real]: ((ord_less_eq_real @ C @ zero_zero_real) => ((ord_less_eq_real @ A @ B) => (ord_less_eq_real @ (plus_plus_real @ A @ C) @ B)))))). % add_decreasing2
thf(fact_198_add__increasing2, axiom,
    ((![C : nat, B : nat, A : nat]: ((ord_less_eq_nat @ zero_zero_nat @ C) => ((ord_less_eq_nat @ B @ A) => (ord_less_eq_nat @ B @ (plus_plus_nat @ A @ C))))))). % add_increasing2
thf(fact_199_add__increasing2, axiom,
    ((![C : real, B : real, A : real]: ((ord_less_eq_real @ zero_zero_real @ C) => ((ord_less_eq_real @ B @ A) => (ord_less_eq_real @ B @ (plus_plus_real @ A @ C))))))). % add_increasing2
thf(fact_200_add__nonneg__nonneg, axiom,
    ((![A : nat, B : nat]: ((ord_less_eq_nat @ zero_zero_nat @ A) => ((ord_less_eq_nat @ zero_zero_nat @ B) => (ord_less_eq_nat @ zero_zero_nat @ (plus_plus_nat @ A @ B))))))). % add_nonneg_nonneg
thf(fact_201_add__nonneg__nonneg, axiom,
    ((![A : real, B : real]: ((ord_less_eq_real @ zero_zero_real @ A) => ((ord_less_eq_real @ zero_zero_real @ B) => (ord_less_eq_real @ zero_zero_real @ (plus_plus_real @ A @ B))))))). % add_nonneg_nonneg
thf(fact_202_add__nonpos__nonpos, axiom,
    ((![A : nat, B : nat]: ((ord_less_eq_nat @ A @ zero_zero_nat) => ((ord_less_eq_nat @ B @ zero_zero_nat) => (ord_less_eq_nat @ (plus_plus_nat @ A @ B) @ zero_zero_nat)))))). % add_nonpos_nonpos
thf(fact_203_add__nonpos__nonpos, axiom,
    ((![A : real, B : real]: ((ord_less_eq_real @ A @ zero_zero_real) => ((ord_less_eq_real @ B @ zero_zero_real) => (ord_less_eq_real @ (plus_plus_real @ A @ B) @ zero_zero_real)))))). % add_nonpos_nonpos
thf(fact_204_add__nonneg__eq__0__iff, axiom,
    ((![X : nat, Y3 : nat]: ((ord_less_eq_nat @ zero_zero_nat @ X) => ((ord_less_eq_nat @ zero_zero_nat @ Y3) => (((plus_plus_nat @ X @ Y3) = zero_zero_nat) = (((X = zero_zero_nat)) & ((Y3 = zero_zero_nat))))))))). % add_nonneg_eq_0_iff
thf(fact_205_add__nonneg__eq__0__iff, axiom,
    ((![X : real, Y3 : real]: ((ord_less_eq_real @ zero_zero_real @ X) => ((ord_less_eq_real @ zero_zero_real @ Y3) => (((plus_plus_real @ X @ Y3) = zero_zero_real) = (((X = zero_zero_real)) & ((Y3 = zero_zero_real))))))))). % add_nonneg_eq_0_iff
thf(fact_206_add__nonpos__eq__0__iff, axiom,
    ((![X : nat, Y3 : nat]: ((ord_less_eq_nat @ X @ zero_zero_nat) => ((ord_less_eq_nat @ Y3 @ zero_zero_nat) => (((plus_plus_nat @ X @ Y3) = zero_zero_nat) = (((X = zero_zero_nat)) & ((Y3 = zero_zero_nat))))))))). % add_nonpos_eq_0_iff
thf(fact_207_add__nonpos__eq__0__iff, axiom,
    ((![X : real, Y3 : real]: ((ord_less_eq_real @ X @ zero_zero_real) => ((ord_less_eq_real @ Y3 @ zero_zero_real) => (((plus_plus_real @ X @ Y3) = zero_zero_real) = (((X = zero_zero_real)) & ((Y3 = zero_zero_real))))))))). % add_nonpos_eq_0_iff
thf(fact_208_zero__reorient, axiom,
    ((![X : real]: ((zero_zero_real = X) = (X = zero_zero_real))))). % zero_reorient
thf(fact_209_zero__reorient, axiom,
    ((![X : nat]: ((zero_zero_nat = X) = (X = zero_zero_nat))))). % zero_reorient
thf(fact_210_add_Ogroup__left__neutral, axiom,
    ((![A : real]: ((plus_plus_real @ zero_zero_real @ A) = A)))). % add.group_left_neutral

% Conjectures (1)
thf(conj_0, conjecture,
    ((ord_less_eq_nat @ n1 @ (g @ n)))).
