%------------------------------------------------------------------------------
% File     : ITP064^1 : TPTP v8.2.0. Released v7.5.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer GenClock problem prob_727__3251548_1
% Version  : Especial.
% English  :

% Refs     : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
%          : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source   : [Des21]
% Names    : GenClock/prob_727__3251548_1 [Des21]

% Status   : Theorem
% Rating   : 0.30 v8.2.0, 0.23 v8.1.0, 0.27 v7.5.0
% Syntax   : Number of formulae    :  392 ( 188 unt;  33 typ;   0 def)
%            Number of atoms       :  855 ( 341 equ;   0 cnn)
%            Maximal formula atoms :   12 (   2 avg)
%            Number of connectives : 2769 (  26   ~;   6   |;  29   &;2404   @)
%                                         (   0 <=>; 304  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   14 (   6 avg)
%            Number of types       :    3 (   2 usr)
%            Number of type conns  :   73 (  73   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   34 (  31 usr;  15 con; 0-4 aty)
%            Number of variables   :  937 (  65   ^; 868   !;   4   ?; 937   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Sledgehammer 2021-02-23 15:31:14.669
%------------------------------------------------------------------------------
% 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 (31)
thf(sy_c_GenClock__Mirabelle__bsvkzpgbls_OIC,type,
    genClo1160817912lle_IC: nat > nat > real > real ).

thf(sy_c_GenClock__Mirabelle__bsvkzpgbls_O_092_060Lambda_062,type,
    genClo721845095Lambda: real ).

thf(sy_c_GenClock__Mirabelle__bsvkzpgbls_O_092_060beta_062,type,
    genClo1278781456e_beta: real ).

thf(sy_c_GenClock__Mirabelle__bsvkzpgbls_O_092_060theta_062,type,
    genClo1400225944_theta: nat > nat > nat > real ).

thf(sy_c_GenClock__Mirabelle__bsvkzpgbls_Ocorrect,type,
    genClo1015804716orrect: nat > real > $o ).

thf(sy_c_GenClock__Mirabelle__bsvkzpgbls_OokRead1,type,
    genClo293725281kRead1: ( nat > real ) > real > ( nat > $o ) > $o ).

thf(sy_c_GenClock__Mirabelle__bsvkzpgbls_OokRead2,type,
    genClo293725282kRead2: ( nat > real ) > ( nat > real ) > real > ( nat > $o ) > $o ).

thf(sy_c_GenClock__Mirabelle__bsvkzpgbls_Ookmaxsync,type,
    genClo208577157axsync: nat > real > $o ).

thf(sy_c_GenClock__Mirabelle__bsvkzpgbls_Ormax,type,
    genClo1650508560e_rmax: real ).

thf(sy_c_GenClock__Mirabelle__bsvkzpgbls_Ormin,type,
    genClo1651033342e_rmin: real ).

thf(sy_c_GenClock__Mirabelle__bsvkzpgbls_Ote,type,
    genClo1163638703lle_te: nat > nat > real ).

thf(sy_c_Groups_Oabs__class_Oabs_001t__Real__Oreal,type,
    abs_abs_real: real > real ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
    minus_minus_nat: nat > nat > nat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
    minus_minus_real: real > real > real ).

thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
    one_one_nat: nat ).

thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
    one_one_real: real ).

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_If_001t__Nat__Onat,type,
    if_nat: $o > nat > nat > nat ).

thf(sy_c_If_001t__Real__Oreal,type,
    if_real: $o > real > real > real ).

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_Oord__class_Omax_001t__Nat__Onat,type,
    ord_max_nat: nat > nat > nat ).

thf(sy_c_Orderings_Oord__class_Omax_001t__Real__Oreal,type,
    ord_max_real: real > real > real ).

thf(sy_v__092_060delta_062S,type,
    delta_S: real ).

thf(sy_v_i,type,
    i: nat ).

thf(sy_v_l,type,
    l: nat ).

thf(sy_v_m,type,
    m: nat ).

thf(sy_v_p,type,
    p: nat ).

% Relevant facts (353)
thf(fact_0_correct__closed,axiom,
    ! [P: nat,S: real,T: real] :
      ( ( ( ord_less_eq_real @ S @ T )
        & ( genClo1015804716orrect @ P @ T ) )
     => ( genClo1015804716orrect @ P @ S ) ) ).

% correct_closed
thf(fact_1_corr__m,axiom,
    genClo1015804716orrect @ m @ ( genClo1163638703lle_te @ p @ ( plus_plus_nat @ i @ one_one_nat ) ) ).

% corr_m
thf(fact_2_corr__p,axiom,
    genClo1015804716orrect @ p @ ( genClo1163638703lle_te @ p @ ( plus_plus_nat @ i @ one_one_nat ) ) ).

% corr_p
thf(fact_3_tml__le__tp,axiom,
    ord_less_eq_real @ ( ord_max_real @ ( genClo1163638703lle_te @ l @ i ) @ ( genClo1163638703lle_te @ m @ i ) ) @ ( genClo1163638703lle_te @ p @ ( plus_plus_nat @ i @ one_one_nat ) ) ).

% tml_le_tp
thf(fact_4_corr__l,axiom,
    genClo1015804716orrect @ l @ ( genClo1163638703lle_te @ p @ ( plus_plus_nat @ i @ one_one_nat ) ) ).

% corr_l
thf(fact_5_corr__l__tml,axiom,
    genClo1015804716orrect @ l @ ( ord_max_real @ ( genClo1163638703lle_te @ l @ i ) @ ( genClo1163638703lle_te @ m @ i ) ) ).

% corr_l_tml
thf(fact_6_tmi__le__tp,axiom,
    ord_less_eq_real @ ( genClo1163638703lle_te @ m @ i ) @ ( genClo1163638703lle_te @ p @ ( plus_plus_nat @ i @ one_one_nat ) ) ).

% tmi_le_tp
thf(fact_7_tli__le__tp,axiom,
    ord_less_eq_real @ ( genClo1163638703lle_te @ l @ i ) @ ( genClo1163638703lle_te @ p @ ( plus_plus_nat @ i @ one_one_nat ) ) ).

% tli_le_tp
thf(fact_8_max_Oidem,axiom,
    ! [A: real] :
      ( ( ord_max_real @ A @ A )
      = A ) ).

% max.idem
thf(fact_9_max_Oidem,axiom,
    ! [A: nat] :
      ( ( ord_max_nat @ A @ A )
      = A ) ).

% max.idem
thf(fact_10_max_Oleft__idem,axiom,
    ! [A: real,B: real] :
      ( ( ord_max_real @ A @ ( ord_max_real @ A @ B ) )
      = ( ord_max_real @ A @ B ) ) ).

% max.left_idem
thf(fact_11_max_Oleft__idem,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_max_nat @ A @ ( ord_max_nat @ A @ B ) )
      = ( ord_max_nat @ A @ B ) ) ).

% max.left_idem
thf(fact_12_max_Oright__idem,axiom,
    ! [A: real,B: real] :
      ( ( ord_max_real @ ( ord_max_real @ A @ B ) @ B )
      = ( ord_max_real @ A @ B ) ) ).

% max.right_idem
thf(fact_13_max_Oright__idem,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_max_nat @ ( ord_max_nat @ A @ B ) @ B )
      = ( ord_max_nat @ A @ B ) ) ).

% max.right_idem
thf(fact_14_IC__bound,axiom,
    ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( genClo1160817912lle_IC @ l @ i @ ( ord_max_real @ ( genClo1163638703lle_te @ l @ i ) @ ( genClo1163638703lle_te @ m @ i ) ) ) @ ( genClo1160817912lle_IC @ m @ i @ ( ord_max_real @ ( genClo1163638703lle_te @ l @ i ) @ ( genClo1163638703lle_te @ m @ i ) ) ) ) ) @ delta_S ).

% IC_bound
thf(fact_15_max_Oassoc,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_max_real @ ( ord_max_real @ A @ B ) @ C )
      = ( ord_max_real @ A @ ( ord_max_real @ B @ C ) ) ) ).

% max.assoc
thf(fact_16_max_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_max_nat @ ( ord_max_nat @ A @ B ) @ C )
      = ( ord_max_nat @ A @ ( ord_max_nat @ B @ C ) ) ) ).

% max.assoc
thf(fact_17_max_Ocommute,axiom,
    ( ord_max_real
    = ( ^ [A2: real,B2: real] : ( ord_max_real @ B2 @ A2 ) ) ) ).

% max.commute
thf(fact_18_max_Ocommute,axiom,
    ( ord_max_nat
    = ( ^ [A2: nat,B2: nat] : ( ord_max_nat @ B2 @ A2 ) ) ) ).

% max.commute
thf(fact_19_max_Oleft__commute,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_max_real @ B @ ( ord_max_real @ A @ C ) )
      = ( ord_max_real @ A @ ( ord_max_real @ B @ C ) ) ) ).

% max.left_commute
thf(fact_20_max_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_max_nat @ B @ ( ord_max_nat @ A @ C ) )
      = ( ord_max_nat @ A @ ( ord_max_nat @ B @ C ) ) ) ).

% max.left_commute
thf(fact_21_rte,axiom,
    ! [P2: nat,I: nat] :
      ( ( genClo1015804716orrect @ P2 @ ( genClo1163638703lle_te @ P2 @ ( plus_plus_nat @ I @ one_one_nat ) ) )
     => ( ord_less_eq_real @ ( genClo1163638703lle_te @ P2 @ I ) @ ( genClo1163638703lle_te @ P2 @ ( plus_plus_nat @ I @ one_one_nat ) ) ) ) ).

% rte
thf(fact_22_max_Obounded__iff,axiom,
    ! [B: real,C: real,A: real] :
      ( ( ord_less_eq_real @ ( ord_max_real @ B @ C ) @ A )
      = ( ( ord_less_eq_real @ B @ A )
        & ( ord_less_eq_real @ C @ A ) ) ) ).

% max.bounded_iff
thf(fact_23_max_Obounded__iff,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A )
      = ( ( ord_less_eq_nat @ B @ A )
        & ( ord_less_eq_nat @ C @ A ) ) ) ).

% max.bounded_iff
thf(fact_24_max__absorb1,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ Y @ X )
     => ( ( ord_max_real @ X @ Y )
        = X ) ) ).

% max_absorb1
thf(fact_25_max__absorb1,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( ord_max_nat @ X @ Y )
        = X ) ) ).

% max_absorb1
thf(fact_26_order__refl,axiom,
    ! [X: real] : ( ord_less_eq_real @ X @ X ) ).

% order_refl
thf(fact_27_order__refl,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).

% order_refl
thf(fact_28_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,Y2: real] :
              ( ( ord_less_eq_real @ X2 @ Y2 )
             => ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_29_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,Y2: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y2 )
             => ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_30_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,Y2: real] :
              ( ( ord_less_eq_real @ X2 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_31_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,Y2: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_32_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,Y2: real] :
              ( ( ord_less_eq_real @ X2 @ Y2 )
             => ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_33_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,Y2: real] :
              ( ( ord_less_eq_real @ X2 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_34_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,Y2: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y2 )
             => ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_35_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,Y2: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_36_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,Y2: real] :
              ( ( ord_less_eq_real @ X2 @ Y2 )
             => ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_37_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,Y2: real] :
              ( ( ord_less_eq_real @ X2 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_38_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,Y2: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y2 )
             => ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_39_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,Y2: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_40_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,Y2: real] :
              ( ( ord_less_eq_real @ X2 @ Y2 )
             => ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_41_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,Y2: real] :
              ( ( ord_less_eq_real @ X2 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_42_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,Y2: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y2 )
             => ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_43_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,Y2: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_44_eq__iff,axiom,
    ( ( ^ [Y3: real,Z: real] : Y3 = Z )
    = ( ^ [X3: real,Y4: real] :
          ( ( ord_less_eq_real @ X3 @ Y4 )
          & ( ord_less_eq_real @ Y4 @ X3 ) ) ) ) ).

% eq_iff
thf(fact_45_eq__iff,axiom,
    ( ( ^ [Y3: nat,Z: nat] : Y3 = Z )
    = ( ^ [X3: nat,Y4: nat] :
          ( ( ord_less_eq_nat @ X3 @ Y4 )
          & ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ).

% eq_iff
thf(fact_46_antisym,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ Y )
     => ( ( ord_less_eq_real @ Y @ X )
       => ( X = Y ) ) ) ).

% antisym
thf(fact_47_antisym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ X )
       => ( X = Y ) ) ) ).

% antisym
thf(fact_48_linear,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ Y )
      | ( ord_less_eq_real @ Y @ X ) ) ).

% linear
thf(fact_49_linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
      | ( ord_less_eq_nat @ Y @ X ) ) ).

% linear
thf(fact_50_eq__refl,axiom,
    ! [X: real,Y: real] :
      ( ( X = Y )
     => ( ord_less_eq_real @ X @ Y ) ) ).

% eq_refl
thf(fact_51_eq__refl,axiom,
    ! [X: nat,Y: nat] :
      ( ( X = Y )
     => ( ord_less_eq_nat @ X @ Y ) ) ).

% eq_refl
thf(fact_52_le__cases,axiom,
    ! [X: real,Y: real] :
      ( ~ ( ord_less_eq_real @ X @ Y )
     => ( ord_less_eq_real @ Y @ X ) ) ).

% le_cases
thf(fact_53_le__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_eq_nat @ X @ Y )
     => ( ord_less_eq_nat @ Y @ X ) ) ).

% le_cases
thf(fact_54_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_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_le__cases3,axiom,
    ! [X: real,Y: real,Z2: real] :
      ( ( ( ord_less_eq_real @ X @ Y )
       => ~ ( ord_less_eq_real @ Y @ Z2 ) )
     => ( ( ( ord_less_eq_real @ Y @ X )
         => ~ ( ord_less_eq_real @ X @ Z2 ) )
       => ( ( ( ord_less_eq_real @ X @ Z2 )
           => ~ ( ord_less_eq_real @ Z2 @ Y ) )
         => ( ( ( ord_less_eq_real @ Z2 @ Y )
             => ~ ( ord_less_eq_real @ Y @ X ) )
           => ( ( ( ord_less_eq_real @ Y @ Z2 )
               => ~ ( ord_less_eq_real @ Z2 @ X ) )
             => ~ ( ( ord_less_eq_real @ Z2 @ X )
                 => ~ ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_57_le__cases3,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( ( ( ord_less_eq_nat @ X @ Y )
       => ~ ( ord_less_eq_nat @ Y @ Z2 ) )
     => ( ( ( ord_less_eq_nat @ Y @ X )
         => ~ ( ord_less_eq_nat @ X @ Z2 ) )
       => ( ( ( ord_less_eq_nat @ X @ Z2 )
           => ~ ( ord_less_eq_nat @ Z2 @ Y ) )
         => ( ( ( ord_less_eq_nat @ Z2 @ Y )
             => ~ ( ord_less_eq_nat @ Y @ X ) )
           => ( ( ( ord_less_eq_nat @ Y @ Z2 )
               => ~ ( ord_less_eq_nat @ Z2 @ X ) )
             => ~ ( ( ord_less_eq_nat @ Z2 @ X )
                 => ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_58_antisym__conv,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ Y @ X )
     => ( ( ord_less_eq_real @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv
thf(fact_59_antisym__conv,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( ord_less_eq_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv
thf(fact_60_order__class_Oorder_Oeq__iff,axiom,
    ( ( ^ [Y3: real,Z: real] : Y3 = Z )
    = ( ^ [A2: real,B2: real] :
          ( ( ord_less_eq_real @ A2 @ B2 )
          & ( ord_less_eq_real @ B2 @ A2 ) ) ) ) ).

% order_class.order.eq_iff
thf(fact_61_order__class_Oorder_Oeq__iff,axiom,
    ( ( ^ [Y3: nat,Z: nat] : Y3 = Z )
    = ( ^ [A2: nat,B2: nat] :
          ( ( ord_less_eq_nat @ A2 @ B2 )
          & ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).

% order_class.order.eq_iff
thf(fact_62_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_63_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_64_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_65_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_66_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_67_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_68_order__trans,axiom,
    ! [X: real,Y: real,Z2: real] :
      ( ( ord_less_eq_real @ X @ Y )
     => ( ( ord_less_eq_real @ Y @ Z2 )
       => ( ord_less_eq_real @ X @ Z2 ) ) ) ).

% order_trans
thf(fact_69_order__trans,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z2 )
       => ( ord_less_eq_nat @ X @ Z2 ) ) ) ).

% order_trans
thf(fact_70_dual__order_Orefl,axiom,
    ! [A: real] : ( ord_less_eq_real @ A @ A ) ).

% dual_order.refl
thf(fact_71_dual__order_Orefl,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% dual_order.refl
thf(fact_72_linorder__wlog,axiom,
    ! [P3: real > real > $o,A: real,B: real] :
      ( ! [A3: real,B3: real] :
          ( ( ord_less_eq_real @ A3 @ B3 )
         => ( P3 @ A3 @ B3 ) )
     => ( ! [A3: real,B3: real] :
            ( ( P3 @ B3 @ A3 )
           => ( P3 @ A3 @ B3 ) )
       => ( P3 @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_73_linorder__wlog,axiom,
    ! [P3: nat > nat > $o,A: nat,B: nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( ord_less_eq_nat @ A3 @ B3 )
         => ( P3 @ A3 @ B3 ) )
     => ( ! [A3: nat,B3: nat] :
            ( ( P3 @ B3 @ A3 )
           => ( P3 @ A3 @ B3 ) )
       => ( P3 @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_74_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_75_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_76_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y3: real,Z: real] : Y3 = Z )
    = ( ^ [A2: real,B2: real] :
          ( ( ord_less_eq_real @ B2 @ A2 )
          & ( ord_less_eq_real @ A2 @ B2 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_77_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y3: nat,Z: nat] : Y3 = Z )
    = ( ^ [A2: nat,B2: nat] :
          ( ( ord_less_eq_nat @ B2 @ A2 )
          & ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_78_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_79_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_80_max_OcoboundedI2,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_eq_real @ C @ B )
     => ( ord_less_eq_real @ C @ ( ord_max_real @ A @ B ) ) ) ).

% max.coboundedI2
thf(fact_81_max_OcoboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ord_less_eq_nat @ C @ B )
     => ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).

% max.coboundedI2
thf(fact_82_max_OcoboundedI1,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_eq_real @ C @ A )
     => ( ord_less_eq_real @ C @ ( ord_max_real @ A @ B ) ) ) ).

% max.coboundedI1
thf(fact_83_max_OcoboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ C @ A )
     => ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).

% max.coboundedI1
thf(fact_84_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_real
    = ( ^ [A2: real,B2: real] :
          ( ( ord_max_real @ A2 @ B2 )
          = B2 ) ) ) ).

% max.absorb_iff2
thf(fact_85_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [A2: nat,B2: nat] :
          ( ( ord_max_nat @ A2 @ B2 )
          = B2 ) ) ) ).

% max.absorb_iff2
thf(fact_86_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_real
    = ( ^ [B2: real,A2: real] :
          ( ( ord_max_real @ A2 @ B2 )
          = A2 ) ) ) ).

% max.absorb_iff1
thf(fact_87_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [B2: nat,A2: nat] :
          ( ( ord_max_nat @ A2 @ B2 )
          = A2 ) ) ) ).

% max.absorb_iff1
thf(fact_88_le__max__iff__disj,axiom,
    ! [Z2: real,X: real,Y: real] :
      ( ( ord_less_eq_real @ Z2 @ ( ord_max_real @ X @ Y ) )
      = ( ( ord_less_eq_real @ Z2 @ X )
        | ( ord_less_eq_real @ Z2 @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_89_le__max__iff__disj,axiom,
    ! [Z2: nat,X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ Z2 @ ( ord_max_nat @ X @ Y ) )
      = ( ( ord_less_eq_nat @ Z2 @ X )
        | ( ord_less_eq_nat @ Z2 @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_90_max_Ocobounded2,axiom,
    ! [B: real,A: real] : ( ord_less_eq_real @ B @ ( ord_max_real @ A @ B ) ) ).

% max.cobounded2
thf(fact_91_max_Ocobounded2,axiom,
    ! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( ord_max_nat @ A @ B ) ) ).

% max.cobounded2
thf(fact_92_max_Ocobounded1,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ A @ ( ord_max_real @ A @ B ) ) ).

% max.cobounded1
thf(fact_93_max_Ocobounded1,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( ord_max_nat @ A @ B ) ) ).

% max.cobounded1
thf(fact_94_max_Oorder__iff,axiom,
    ( ord_less_eq_real
    = ( ^ [B2: real,A2: real] :
          ( A2
          = ( ord_max_real @ A2 @ B2 ) ) ) ) ).

% max.order_iff
thf(fact_95_max_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [B2: nat,A2: nat] :
          ( A2
          = ( ord_max_nat @ A2 @ B2 ) ) ) ) ).

% max.order_iff
thf(fact_96_max_OboundedI,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( ord_less_eq_real @ C @ A )
       => ( ord_less_eq_real @ ( ord_max_real @ B @ C ) @ A ) ) ) ).

% max.boundedI
thf(fact_97_max_OboundedI,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C @ A )
       => ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A ) ) ) ).

% max.boundedI
thf(fact_98_max_OboundedE,axiom,
    ! [B: real,C: real,A: real] :
      ( ( ord_less_eq_real @ ( ord_max_real @ B @ C ) @ A )
     => ~ ( ( ord_less_eq_real @ B @ A )
         => ~ ( ord_less_eq_real @ C @ A ) ) ) ).

% max.boundedE
thf(fact_99_max_OboundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A )
     => ~ ( ( ord_less_eq_nat @ B @ A )
         => ~ ( ord_less_eq_nat @ C @ A ) ) ) ).

% max.boundedE
thf(fact_100_max_Oabsorb2,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_max_real @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_101_max_Oabsorb2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_max_nat @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_102_max_Oabsorb1,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( ord_max_real @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_103_max_Oabsorb1,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_max_nat @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_104_max_OorderI,axiom,
    ! [A: real,B: real] :
      ( ( A
        = ( ord_max_real @ A @ B ) )
     => ( ord_less_eq_real @ B @ A ) ) ).

% max.orderI
thf(fact_105_max_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( ord_max_nat @ A @ B ) )
     => ( ord_less_eq_nat @ B @ A ) ) ).

% max.orderI
thf(fact_106_max_OorderE,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( A
        = ( ord_max_real @ A @ B ) ) ) ).

% max.orderE
thf(fact_107_max_OorderE,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( A
        = ( ord_max_nat @ A @ B ) ) ) ).

% max.orderE
thf(fact_108_max_Omono,axiom,
    ! [C: real,A: real,D: real,B: real] :
      ( ( ord_less_eq_real @ C @ A )
     => ( ( ord_less_eq_real @ D @ B )
       => ( ord_less_eq_real @ ( ord_max_real @ C @ D ) @ ( ord_max_real @ A @ B ) ) ) ) ).

% max.mono
thf(fact_109_max_Omono,axiom,
    ! [C: nat,A: nat,D: nat,B: nat] :
      ( ( ord_less_eq_nat @ C @ A )
     => ( ( ord_less_eq_nat @ D @ B )
       => ( ord_less_eq_nat @ ( ord_max_nat @ C @ D ) @ ( ord_max_nat @ A @ B ) ) ) ) ).

% max.mono
thf(fact_110_max__def,axiom,
    ( ord_max_real
    = ( ^ [A2: real,B2: real] : ( if_real @ ( ord_less_eq_real @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).

% max_def
thf(fact_111_max__def,axiom,
    ( ord_max_nat
    = ( ^ [A2: nat,B2: nat] : ( if_nat @ ( ord_less_eq_nat @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).

% max_def
thf(fact_112_max__absorb2,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ Y )
     => ( ( ord_max_real @ X @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_113_max__absorb2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_max_nat @ X @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_114_okmaxsync__def,axiom,
    ( genClo208577157axsync
    = ( ^ [I2: nat,X3: real] :
        ! [P4: nat,Q: nat] :
          ( ( ( genClo1015804716orrect @ P4 @ ( ord_max_real @ ( genClo1163638703lle_te @ P4 @ I2 ) @ ( genClo1163638703lle_te @ Q @ I2 ) ) )
            & ( genClo1015804716orrect @ Q @ ( ord_max_real @ ( genClo1163638703lle_te @ P4 @ I2 ) @ ( genClo1163638703lle_te @ Q @ I2 ) ) ) )
         => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( genClo1160817912lle_IC @ P4 @ I2 @ ( ord_max_real @ ( genClo1163638703lle_te @ P4 @ I2 ) @ ( genClo1163638703lle_te @ Q @ I2 ) ) ) @ ( genClo1160817912lle_IC @ Q @ I2 @ ( ord_max_real @ ( genClo1163638703lle_te @ P4 @ I2 ) @ ( genClo1163638703lle_te @ Q @ I2 ) ) ) ) ) @ X3 ) ) ) ) ).

% okmaxsync_def
thf(fact_115_le__add__diff__inverse,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
        = A ) ) ).

% le_add_diff_inverse
thf(fact_116_le__add__diff__inverse,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
        = A ) ) ).

% le_add_diff_inverse
thf(fact_117_le__add__diff__inverse2,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
        = A ) ) ).

% le_add_diff_inverse2
thf(fact_118_le__add__diff__inverse2,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
        = A ) ) ).

% le_add_diff_inverse2
thf(fact_119_abs__1,axiom,
    ( ( abs_abs_real @ one_one_real )
    = one_one_real ) ).

% abs_1
thf(fact_120_abs__add__abs,axiom,
    ! [A: real,B: real] :
      ( ( abs_abs_real @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) )
      = ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).

% abs_add_abs
thf(fact_121_add__diff__cancel,axiom,
    ! [A: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_122_diff__add__cancel,axiom,
    ! [A: real,B: real] :
      ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_123_add__diff__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
      = ( minus_minus_real @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_124_add__diff__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
      = ( minus_minus_nat @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_125_add__diff__cancel__left_H,axiom,
    ! [A: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ A )
      = B ) ).

% add_diff_cancel_left'
thf(fact_126_add__diff__cancel__left_H,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
      = B ) ).

% add_diff_cancel_left'
thf(fact_127_add__diff__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
      = ( minus_minus_real @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_128_add__diff__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
      = ( minus_minus_nat @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_129_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_130_add__right__cancel,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ( plus_plus_real @ B @ A )
        = ( plus_plus_real @ C @ A ) )
      = ( B = C ) ) ).

% add_right_cancel
thf(fact_131_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_132_add__left__cancel,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ( plus_plus_real @ A @ B )
        = ( plus_plus_real @ A @ C ) )
      = ( B = C ) ) ).

% add_left_cancel
thf(fact_133_abs__idempotent,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( abs_abs_real @ A ) )
      = ( abs_abs_real @ A ) ) ).

% abs_idempotent
thf(fact_134_abs__abs,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( abs_abs_real @ A ) )
      = ( abs_abs_real @ A ) ) ).

% abs_abs
thf(fact_135_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_136_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_137_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_138_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_139_add__diff__cancel__right_H,axiom,
    ! [A: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel_right'
thf(fact_140_add__diff__cancel__right_H,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel_right'
thf(fact_141_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_142_add__right__imp__eq,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ( plus_plus_real @ B @ A )
        = ( plus_plus_real @ C @ A ) )
     => ( B = C ) ) ).

% add_right_imp_eq
thf(fact_143_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_144_add__left__imp__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ( plus_plus_real @ A @ B )
        = ( plus_plus_real @ A @ C ) )
     => ( B = C ) ) ).

% add_left_imp_eq
thf(fact_145_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_146_add_Oleft__commute,axiom,
    ! [B: real,A: real,C: real] :
      ( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C ) )
      = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).

% add.left_commute
thf(fact_147_add_Ocommute,axiom,
    ( plus_plus_nat
    = ( ^ [A2: nat,B2: nat] : ( plus_plus_nat @ B2 @ A2 ) ) ) ).

% add.commute
thf(fact_148_add_Ocommute,axiom,
    ( plus_plus_real
    = ( ^ [A2: real,B2: real] : ( plus_plus_real @ B2 @ A2 ) ) ) ).

% add.commute
thf(fact_149_add_Oright__cancel,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ( plus_plus_real @ B @ A )
        = ( plus_plus_real @ C @ A ) )
      = ( B = C ) ) ).

% add.right_cancel
thf(fact_150_add_Oleft__cancel,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ( plus_plus_real @ A @ B )
        = ( plus_plus_real @ A @ C ) )
      = ( B = C ) ) ).

% add.left_cancel
thf(fact_151_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_152_add_Oassoc,axiom,
    ! [A: real,B: real,C: real] :
      ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
      = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).

% add.assoc
thf(fact_153_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_154_group__cancel_Oadd2,axiom,
    ! [B4: real,K: real,B: real,A: real] :
      ( ( B4
        = ( plus_plus_real @ K @ B ) )
     => ( ( plus_plus_real @ A @ B4 )
        = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_155_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_156_group__cancel_Oadd1,axiom,
    ! [A4: real,K: real,A: real,B: real] :
      ( ( A4
        = ( plus_plus_real @ K @ A ) )
     => ( ( plus_plus_real @ A4 @ B )
        = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_157_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_158_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( ( plus_plus_real @ I @ K )
        = ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_159_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_160_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: real,B: real,C: real] :
      ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
      = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_161_one__reorient,axiom,
    ! [X: nat] :
      ( ( one_one_nat = X )
      = ( X = one_one_nat ) ) ).

% one_reorient
thf(fact_162_diff__right__commute,axiom,
    ! [A: real,C: real,B: real] :
      ( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B )
      = ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C ) ) ).

% diff_right_commute
thf(fact_163_diff__right__commute,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
      = ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).

% diff_right_commute
thf(fact_164_diff__eq__diff__eq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ( minus_minus_real @ A @ B )
        = ( minus_minus_real @ C @ D ) )
     => ( ( A = B )
        = ( C = D ) ) ) ).

% diff_eq_diff_eq
thf(fact_165_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_166_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_167_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_168_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_169_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_170_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_171_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_172_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_173_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_174_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_175_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_176_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_177_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_178_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_179_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_180_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_181_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_182_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_183_diff__eq__diff__less__eq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ( minus_minus_real @ A @ B )
        = ( minus_minus_real @ C @ D ) )
     => ( ( ord_less_eq_real @ A @ B )
        = ( ord_less_eq_real @ C @ D ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_184_diff__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).

% diff_right_mono
thf(fact_185_diff__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).

% diff_left_mono
thf(fact_186_diff__mono,axiom,
    ! [A: real,B: real,D: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ D @ C )
       => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).

% diff_mono
thf(fact_187_add__implies__diff,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ( plus_plus_real @ C @ B )
        = A )
     => ( C
        = ( minus_minus_real @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_188_add__implies__diff,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ( plus_plus_nat @ C @ B )
        = A )
     => ( C
        = ( minus_minus_nat @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_189_diff__diff__add,axiom,
    ! [A: real,B: real,C: real] :
      ( ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C )
      = ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).

% diff_diff_add
thf(fact_190_diff__diff__add,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
      = ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).

% diff_diff_add
thf(fact_191_diff__add__eq__diff__diff__swap,axiom,
    ! [A: real,B: real,C: real] :
      ( ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) )
      = ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_192_diff__add__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ C )
      = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).

% diff_add_eq
thf(fact_193_diff__diff__eq2,axiom,
    ! [A: real,B: real,C: real] :
      ( ( minus_minus_real @ A @ ( minus_minus_real @ B @ C ) )
      = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).

% diff_diff_eq2
thf(fact_194_add__diff__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( plus_plus_real @ A @ ( minus_minus_real @ B @ C ) )
      = ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).

% add_diff_eq
thf(fact_195_eq__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( ( A
        = ( minus_minus_real @ C @ B ) )
      = ( ( plus_plus_real @ A @ B )
        = C ) ) ).

% eq_diff_eq
thf(fact_196_diff__eq__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ( minus_minus_real @ A @ B )
        = C )
      = ( A
        = ( plus_plus_real @ C @ B ) ) ) ).

% diff_eq_eq
thf(fact_197_group__cancel_Osub1,axiom,
    ! [A4: real,K: real,A: real,B: real] :
      ( ( A4
        = ( plus_plus_real @ K @ A ) )
     => ( ( minus_minus_real @ A4 @ B )
        = ( plus_plus_real @ K @ ( minus_minus_real @ A @ B ) ) ) ) ).

% group_cancel.sub1
thf(fact_198_abs__ge__self,axiom,
    ! [A: real] : ( ord_less_eq_real @ A @ ( abs_abs_real @ A ) ) ).

% abs_ge_self
thf(fact_199_abs__le__D1,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
     => ( ord_less_eq_real @ A @ B ) ) ).

% abs_le_D1
thf(fact_200_abs__one,axiom,
    ( ( abs_abs_real @ one_one_real )
    = one_one_real ) ).

% abs_one
thf(fact_201_abs__minus__commute,axiom,
    ! [A: real,B: real] :
      ( ( abs_abs_real @ ( minus_minus_real @ A @ B ) )
      = ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).

% abs_minus_commute
thf(fact_202_max__add__distrib__left,axiom,
    ! [X: real,Y: real,Z2: real] :
      ( ( plus_plus_real @ ( ord_max_real @ X @ Y ) @ Z2 )
      = ( ord_max_real @ ( plus_plus_real @ X @ Z2 ) @ ( plus_plus_real @ Y @ Z2 ) ) ) ).

% max_add_distrib_left
thf(fact_203_max__add__distrib__left,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( ( plus_plus_nat @ ( ord_max_nat @ X @ Y ) @ Z2 )
      = ( ord_max_nat @ ( plus_plus_nat @ X @ Z2 ) @ ( plus_plus_nat @ Y @ Z2 ) ) ) ).

% max_add_distrib_left
thf(fact_204_max__add__distrib__right,axiom,
    ! [X: real,Y: real,Z2: real] :
      ( ( plus_plus_real @ X @ ( ord_max_real @ Y @ Z2 ) )
      = ( ord_max_real @ ( plus_plus_real @ X @ Y ) @ ( plus_plus_real @ X @ Z2 ) ) ) ).

% max_add_distrib_right
thf(fact_205_max__add__distrib__right,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( ( plus_plus_nat @ X @ ( ord_max_nat @ Y @ Z2 ) )
      = ( ord_max_nat @ ( plus_plus_nat @ X @ Y ) @ ( plus_plus_nat @ X @ Z2 ) ) ) ).

% max_add_distrib_right
thf(fact_206_max__diff__distrib__left,axiom,
    ! [X: real,Y: real,Z2: real] :
      ( ( minus_minus_real @ ( ord_max_real @ X @ Y ) @ Z2 )
      = ( ord_max_real @ ( minus_minus_real @ X @ Z2 ) @ ( minus_minus_real @ Y @ Z2 ) ) ) ).

% max_diff_distrib_left
thf(fact_207_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ( ( minus_minus_nat @ B @ A )
            = C )
          = ( B
            = ( plus_plus_nat @ C @ A ) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_208_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
        = B ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_209_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_210_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
        = ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_211_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
        = ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_212_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
        = ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_213_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_214_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_215_le__add__diff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).

% le_add_diff
thf(fact_216_add__le__add__imp__diff__le,axiom,
    ! [I: real,K: real,N: real,J: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
     => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
       => ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
         => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
           => ( ord_less_eq_real @ ( minus_minus_real @ N @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_217_add__le__add__imp__diff__le,axiom,
    ! [I: nat,K: nat,N: nat,J: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
     => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
       => ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
         => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
           => ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_218_diff__add,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
        = B ) ) ).

% diff_add
thf(fact_219_add__le__imp__le__diff,axiom,
    ! [I: real,K: real,N: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
     => ( ord_less_eq_real @ I @ ( minus_minus_real @ N @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_220_add__le__imp__le__diff,axiom,
    ! [I: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
     => ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_221_le__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ A @ ( minus_minus_real @ C @ B ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).

% le_diff_eq
thf(fact_222_diff__le__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ ( minus_minus_real @ A @ B ) @ C )
      = ( ord_less_eq_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).

% diff_le_eq
thf(fact_223_abs__triangle__ineq,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ A @ B ) ) @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).

% abs_triangle_ineq
thf(fact_224_abs__triangle__ineq2__sym,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).

% abs_triangle_ineq2_sym
thf(fact_225_abs__triangle__ineq3,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) ) ).

% abs_triangle_ineq3
thf(fact_226_abs__triangle__ineq2,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) ) ).

% abs_triangle_ineq2
thf(fact_227_abs__diff__triangle__ineq,axiom,
    ! [A: real,B: real,C: real,D: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ ( minus_minus_real @ A @ C ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ D ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_228_abs__triangle__ineq4,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).

% abs_triangle_ineq4
thf(fact_229_abs__diff__le__iff,axiom,
    ! [X: real,A: real,R: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ X @ A ) ) @ R )
      = ( ( ord_less_eq_real @ ( minus_minus_real @ A @ R ) @ X )
        & ( ord_less_eq_real @ X @ ( plus_plus_real @ A @ R ) ) ) ) ).

% abs_diff_le_iff
thf(fact_230_rts1,axiom,
    ! [P: nat,T: real,I3: nat] :
      ( ( ( genClo1015804716orrect @ P @ T )
        & ( ord_less_eq_real @ ( genClo1163638703lle_te @ P @ ( plus_plus_nat @ I3 @ one_one_nat ) ) @ T ) )
     => ( ord_less_eq_real @ genClo1651033342e_rmin @ ( minus_minus_real @ T @ ( genClo1163638703lle_te @ P @ I3 ) ) ) ) ).

% rts1
thf(fact_231_rts1d,axiom,
    ! [P2: nat,I: nat] :
      ( ( genClo1015804716orrect @ P2 @ ( genClo1163638703lle_te @ P2 @ ( plus_plus_nat @ I @ one_one_nat ) ) )
     => ( ord_less_eq_real @ genClo1651033342e_rmin @ ( minus_minus_real @ ( genClo1163638703lle_te @ P2 @ ( plus_plus_nat @ I @ one_one_nat ) ) @ ( genClo1163638703lle_te @ P2 @ I ) ) ) ) ).

% rts1d
thf(fact_232_rts0,axiom,
    ! [P: nat,T: real,I3: nat] :
      ( ( ( genClo1015804716orrect @ P @ T )
        & ( ord_less_eq_real @ T @ ( genClo1163638703lle_te @ P @ ( plus_plus_nat @ I3 @ one_one_nat ) ) ) )
     => ( ord_less_eq_real @ ( minus_minus_real @ T @ ( genClo1163638703lle_te @ P @ I3 ) ) @ genClo1650508560e_rmax ) ) ).

% rts0
thf(fact_233_rts0d,axiom,
    ! [P2: nat,I: nat] :
      ( ( genClo1015804716orrect @ P2 @ ( genClo1163638703lle_te @ P2 @ ( plus_plus_nat @ I @ one_one_nat ) ) )
     => ( ord_less_eq_real @ ( minus_minus_real @ ( genClo1163638703lle_te @ P2 @ ( plus_plus_nat @ I @ one_one_nat ) ) @ ( genClo1163638703lle_te @ P2 @ I ) ) @ genClo1650508560e_rmax ) ) ).

% rts0d
thf(fact_234_okRead2__def,axiom,
    ( genClo293725282kRead2
    = ( ^ [F2: nat > real,G: nat > real,X3: real,Ppred: nat > $o] :
        ! [P4: nat] :
          ( ( Ppred @ P4 )
         => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( F2 @ P4 ) @ ( G @ P4 ) ) ) @ X3 ) ) ) ) ).

% okRead2_def
thf(fact_235_sin__bound__lemma,axiom,
    ! [X: real,Y: real,U: real,V: real] :
      ( ( X = Y )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ U ) @ V )
       => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ X @ U ) @ Y ) ) @ V ) ) ) ).

% sin_bound_lemma
thf(fact_236_eq__diff__eq_H,axiom,
    ! [X: real,Y: real,Z2: real] :
      ( ( X
        = ( minus_minus_real @ Y @ Z2 ) )
      = ( Y
        = ( plus_plus_real @ X @ Z2 ) ) ) ).

% eq_diff_eq'
thf(fact_237_beta__bound2,axiom,
    ! [P2: nat,I: nat,Q2: nat] :
      ( ( genClo1015804716orrect @ P2 @ ( genClo1163638703lle_te @ P2 @ ( plus_plus_nat @ I @ one_one_nat ) ) )
     => ( ( genClo1015804716orrect @ Q2 @ ( genClo1163638703lle_te @ Q2 @ I ) )
       => ( ord_less_eq_real @ ( minus_minus_real @ ( genClo1163638703lle_te @ P2 @ ( plus_plus_nat @ I @ one_one_nat ) ) @ ( genClo1163638703lle_te @ Q2 @ I ) ) @ ( plus_plus_real @ genClo1650508560e_rmax @ genClo1278781456e_beta ) ) ) ) ).

% beta_bound2
thf(fact_238_rts2b,axiom,
    ! [P: nat,Q3: nat,I3: nat] :
      ( ( ( genClo1015804716orrect @ P @ ( genClo1163638703lle_te @ P @ I3 ) )
        & ( genClo1015804716orrect @ Q3 @ ( genClo1163638703lle_te @ Q3 @ I3 ) ) )
     => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( genClo1163638703lle_te @ P @ I3 ) @ ( genClo1163638703lle_te @ Q3 @ I3 ) ) ) @ genClo1278781456e_beta ) ) ).

% rts2b
thf(fact_239_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_240_diff__diff__cancel,axiom,
    ! [I: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ N )
     => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
        = I ) ) ).

% diff_diff_cancel
thf(fact_241_diff__diff__left,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
      = ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).

% diff_diff_left
thf(fact_242_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).

% Nat.diff_diff_right
thf(fact_243_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
        = ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).

% Nat.add_diff_assoc2
thf(fact_244_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).

% Nat.add_diff_assoc
thf(fact_245_Nat_Oex__has__greatest__nat,axiom,
    ! [P3: nat > $o,K: nat,B: nat] :
      ( ( P3 @ K )
     => ( ! [Y2: nat] :
            ( ( P3 @ Y2 )
           => ( ord_less_eq_nat @ Y2 @ B ) )
       => ? [X2: nat] :
            ( ( P3 @ X2 )
            & ! [Y5: nat] :
                ( ( P3 @ Y5 )
               => ( ord_less_eq_nat @ Y5 @ X2 ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_246_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_247_diff__le__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).

% diff_le_mono2
thf(fact_248_le__diff__iff_H,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ C )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
          = ( ord_less_eq_nat @ B @ A ) ) ) ) ).

% le_diff_iff'
thf(fact_249_diff__le__self,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).

% diff_le_self
thf(fact_250_diff__le__mono,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).

% diff_le_mono
thf(fact_251_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( minus_minus_nat @ M @ N ) ) ) ) ).

% Nat.diff_diff_eq
thf(fact_252_le__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( ord_less_eq_nat @ M @ N ) ) ) ) ).

% le_diff_iff
thf(fact_253_eq__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ( minus_minus_nat @ M @ K )
            = ( minus_minus_nat @ N @ K ) )
          = ( M = N ) ) ) ) ).

% eq_diff_iff
thf(fact_254_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_255_eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( M = N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% eq_imp_le
thf(fact_256_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_257_le__refl,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).

% le_refl
thf(fact_258_Nat_Ole__imp__diff__is__add,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ( minus_minus_nat @ J @ I )
          = K )
        = ( J
          = ( plus_plus_nat @ K @ I ) ) ) ) ).

% Nat.le_imp_diff_is_add
thf(fact_259_nat__minus__add__max,axiom,
    ! [N: nat,M: nat] :
      ( ( plus_plus_nat @ ( minus_minus_nat @ N @ M ) @ M )
      = ( ord_max_nat @ N @ M ) ) ).

% nat_minus_add_max
thf(fact_260_diff__add__inverse2,axiom,
    ! [M: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
      = M ) ).

% diff_add_inverse2
thf(fact_261_diff__add__inverse,axiom,
    ! [N: nat,M: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
      = M ) ).

% diff_add_inverse
thf(fact_262_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
        = ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).

% Nat.diff_add_assoc2
thf(fact_263_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
        = ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).

% Nat.diff_add_assoc
thf(fact_264_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).

% Nat.le_diff_conv2
thf(fact_265_le__diff__conv,axiom,
    ! [J: nat,K: nat,I: nat] :
      ( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
      = ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).

% le_diff_conv
thf(fact_266_diff__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
      = ( minus_minus_nat @ M @ N ) ) ).

% diff_cancel2
thf(fact_267_Nat_Odiff__cancel,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
      = ( minus_minus_nat @ M @ N ) ) ).

% Nat.diff_cancel
thf(fact_268_nat__add__max__right,axiom,
    ! [M: nat,N: nat,Q2: nat] :
      ( ( plus_plus_nat @ M @ ( ord_max_nat @ N @ Q2 ) )
      = ( ord_max_nat @ ( plus_plus_nat @ M @ N ) @ ( plus_plus_nat @ M @ Q2 ) ) ) ).

% nat_add_max_right
thf(fact_269_nat__add__max__left,axiom,
    ! [M: nat,N: nat,Q2: nat] :
      ( ( plus_plus_nat @ ( ord_max_nat @ M @ N ) @ Q2 )
      = ( ord_max_nat @ ( plus_plus_nat @ M @ Q2 ) @ ( plus_plus_nat @ N @ Q2 ) ) ) ).

% nat_add_max_left
thf(fact_270_nat__le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [M2: nat,N2: nat] :
        ? [K2: nat] :
          ( N2
          = ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).

% nat_le_iff_add
thf(fact_271_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_272_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_273_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_274_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_275_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_276_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_277_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_278_le__add2,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).

% le_add2
thf(fact_279_le__add1,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).

% le_add1
thf(fact_280_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_281_nonoverlap,axiom,
    ord_less_eq_real @ genClo1278781456e_beta @ genClo1651033342e_rmin ).

% nonoverlap
thf(fact_282_rts2a,axiom,
    ! [P: nat,Q3: nat,T: real,I3: nat] :
      ( ( ( genClo1015804716orrect @ P @ T )
        & ( genClo1015804716orrect @ Q3 @ T )
        & ( ord_less_eq_real @ ( plus_plus_real @ ( genClo1163638703lle_te @ Q3 @ I3 ) @ genClo1278781456e_beta ) @ T ) )
     => ( ord_less_eq_real @ ( genClo1163638703lle_te @ P @ I3 ) @ T ) ) ).

% rts2a
thf(fact_283_readerror,axiom,
    ! [P: nat,Q3: nat,I3: nat] :
      ( ( ( genClo1015804716orrect @ P @ ( genClo1163638703lle_te @ P @ ( plus_plus_nat @ I3 @ one_one_nat ) ) )
        & ( genClo1015804716orrect @ Q3 @ ( genClo1163638703lle_te @ P @ ( plus_plus_nat @ I3 @ one_one_nat ) ) ) )
     => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( genClo1400225944_theta @ P @ ( plus_plus_nat @ I3 @ one_one_nat ) @ Q3 ) @ ( genClo1160817912lle_IC @ Q3 @ I3 @ ( genClo1163638703lle_te @ P @ ( plus_plus_nat @ I3 @ one_one_nat ) ) ) ) ) @ genClo721845095Lambda ) ) ).

% readerror
thf(fact_284_okRead1__def,axiom,
    ( genClo293725281kRead1
    = ( ^ [F2: nat > real,X3: real,Ppred: nat > $o] :
        ! [L2: nat,M2: nat] :
          ( ( ( Ppred @ L2 )
            & ( Ppred @ M2 ) )
         => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( F2 @ L2 ) @ ( F2 @ M2 ) ) ) @ X3 ) ) ) ) ).

% okRead1_def
thf(fact_285_beta__bound1,axiom,
    ! [P2: nat,I: nat,Q2: nat] :
      ( ( genClo1015804716orrect @ P2 @ ( genClo1163638703lle_te @ P2 @ ( plus_plus_nat @ I @ one_one_nat ) ) )
     => ( ( genClo1015804716orrect @ Q2 @ ( genClo1163638703lle_te @ P2 @ ( plus_plus_nat @ I @ one_one_nat ) ) )
       => ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( genClo1163638703lle_te @ P2 @ ( plus_plus_nat @ I @ one_one_nat ) ) @ ( genClo1163638703lle_te @ Q2 @ I ) ) ) ) ) ).

% beta_bound1
thf(fact_286_le__zero__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_zero_eq
thf(fact_287_zero__eq__add__iff__both__eq__0,axiom,
    ! [X: nat,Y: nat] :
      ( ( zero_zero_nat
        = ( plus_plus_nat @ X @ Y ) )
      = ( ( X = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
thf(fact_288_add__eq__0__iff__both__eq__0,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( plus_plus_nat @ X @ Y )
        = zero_zero_nat )
      = ( ( X = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
thf(fact_289_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_290_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_291_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_292_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_293_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_294_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_295_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_296_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_297_double__zero__sym,axiom,
    ! [A: real] :
      ( ( zero_zero_real
        = ( plus_plus_real @ A @ A ) )
      = ( A = zero_zero_real ) ) ).

% double_zero_sym
thf(fact_298_double__zero,axiom,
    ! [A: real] :
      ( ( ( plus_plus_real @ A @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% double_zero
thf(fact_299_add_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.right_neutral
thf(fact_300_add_Oright__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% add.right_neutral
thf(fact_301_add_Oleft__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A )
      = A ) ).

% add.left_neutral
thf(fact_302_add_Oleft__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ zero_zero_real @ A )
      = A ) ).

% add.left_neutral
thf(fact_303_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ A )
      = zero_zero_real ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_304_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ A @ A )
      = zero_zero_nat ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_305_diff__zero,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ zero_zero_real )
      = A ) ).

% diff_zero
thf(fact_306_diff__zero,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ A @ zero_zero_nat )
      = A ) ).

% diff_zero
thf(fact_307_zero__diff,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% zero_diff
thf(fact_308_diff__0__right,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ zero_zero_real )
      = A ) ).

% diff_0_right
thf(fact_309_diff__self,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ A )
      = zero_zero_real ) ).

% diff_self
thf(fact_310_abs__zero,axiom,
    ( ( abs_abs_real @ zero_zero_real )
    = zero_zero_real ) ).

% abs_zero
thf(fact_311_abs__eq__0,axiom,
    ! [A: real] :
      ( ( ( abs_abs_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% abs_eq_0
thf(fact_312_abs__0__eq,axiom,
    ! [A: real] :
      ( ( zero_zero_real
        = ( abs_abs_real @ A ) )
      = ( A = zero_zero_real ) ) ).

% abs_0_eq
thf(fact_313_abs__0,axiom,
    ( ( abs_abs_real @ zero_zero_real )
    = zero_zero_real ) ).

% abs_0
thf(fact_314_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_315_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_316_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_317_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_318_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_319_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_320_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_321_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_322_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_323_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_324_diff__ge__0__iff__ge,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
      = ( ord_less_eq_real @ B @ A ) ) ).

% diff_ge_0_iff_ge
thf(fact_325_diff__add__zero,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = zero_zero_nat ) ).

% diff_add_zero
thf(fact_326_abs__of__nonneg,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( abs_abs_real @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_327_abs__le__self__iff,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ A )
      = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).

% abs_le_self_iff
thf(fact_328_abs__le__zero__iff,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% abs_le_zero_iff
thf(fact_329_zero__le,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).

% zero_le
thf(fact_330_diff__commute,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
      = ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).

% diff_commute
thf(fact_331_zero__reorient,axiom,
    ! [X: real] :
      ( ( zero_zero_real = X )
      = ( X = zero_zero_real ) ) ).

% zero_reorient
thf(fact_332_abs__eq__0__iff,axiom,
    ! [A: real] :
      ( ( ( abs_abs_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% abs_eq_0_iff
thf(fact_333_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y3: real,Z: real] : Y3 = Z )
    = ( ^ [A2: real,B2: real] :
          ( ( minus_minus_real @ A2 @ B2 )
          = zero_zero_real ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_334_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
thf(fact_335_zero__neq__one,axiom,
    zero_zero_real != one_one_real ).

% zero_neq_one
thf(fact_336_add_Ogroup__left__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ zero_zero_real @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_337_add_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.comm_neutral
thf(fact_338_add_Ocomm__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% add.comm_neutral
thf(fact_339_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_340_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ zero_zero_real @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_341_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_342_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_343_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_344_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_345_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_346_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_347_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_348_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_349_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_350_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_351_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_352_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

% Helper facts (5)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( if_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( if_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_3_1_If_001t__Real__Oreal_T,axiom,
    ! [P3: $o] :
      ( ( P3 = $true )
      | ( P3 = $false ) ) ).

thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
    ! [X: real,Y: real] :
      ( ( if_real @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
    ! [X: real,Y: real] :
      ( ( if_real @ $true @ X @ Y )
      = X ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    genClo1015804716orrect @ m @ ( ord_max_real @ ( genClo1163638703lle_te @ l @ i ) @ ( genClo1163638703lle_te @ m @ i ) ) ).

%------------------------------------------------------------------------------