TPTP Problem File: SLH0985^1.p

%------------------------------------------------------------------------------
% File     : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain   : Archive of Formal Proofs
% Problem  :
% Version  : Especial.
% English  :

% Refs     : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source   : [Des23]
% Names    : Frequency_Moments/0087_Frequency_Moment_k/prob_00245_009348__20001030_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1536 ( 531 unt; 396 typ;   0 def)
%            Number of atoms       : 3454 (1121 equ;   0 cnn)
%            Maximal formula atoms :   12 (   3 avg)
%            Number of connectives : 10519 ( 407   ~;  90   |; 323   &;8232   @)
%                                         (   0 <=>;1467  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   20 (   7 avg)
%            Number of types       :   61 (  60 usr)
%            Number of type conns  : 1226 (1226   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  339 ( 336 usr;  34 con; 0-4 aty)
%            Number of variables   : 3589 ( 482   ^;2965   !; 142   ?;3589   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 12:19:14.501
%------------------------------------------------------------------------------
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thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Nat__Onat_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__List__Olist_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
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thf(ty_n_t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
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thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
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thf(ty_n_t__Nat__Onat,type,
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% Explicit typings (336)
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
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thf(sy_c_Finite__Set_Ocard_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Finite__Set_Ocard_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Finite__Set_Ocard_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
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thf(sy_c_Frequency__Moment__k_OM_092_060_094sub_0621,type,
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thf(sy_c_Frequency__Moment__k_Osketch_001t__Nat__Onat,type,
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thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    inj_on6829720656803765732at_nat: ( nat > produc859450856879609959at_nat ) > set_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    inj_on7264072676751113700at_nat: ( nat > produc3843707927480180839at_nat ) > set_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
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thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    inj_on8969904277767023793at_nat: ( product_prod_nat_nat > product_prod_nat_nat ) > set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
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thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    inj_on229527429751349963at_nat: ( produc859450856879609959at_nat > product_prod_nat_nat ) > set_Pr8693737435421807431at_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    inj_on7797493862353638193at_nat: ( produc859450856879609959at_nat > produc859450856879609959at_nat ) > set_Pr8693737435421807431at_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_001t__Nat__Onat,type,
    inj_on7963584518437281414at_nat: ( produc3843707927480180839at_nat > nat ) > set_Pr4329608150637261639at_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    inj_on8430303779476321073at_nat: ( produc3843707927480180839at_nat > produc3843707927480180839at_nat ) > set_Pr4329608150637261639at_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    inj_on104525631246638609at_nat: ( produc3843707927480180839at_nat > set_Pr4329608150637261639at_nat ) > set_Pr4329608150637261639at_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_J_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_J,type,
    inj_on3942866938762682161at_nat: ( produc1319942482725812455at_nat > produc1319942482725812455at_nat ) > set_Pr7459493094073627847at_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    inj_on699426629415580343at_nat: ( set_Pr1261947904930325089at_nat > produc3843707927480180839at_nat ) > set_se7855581050983116737at_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_J,type,
    inj_on1468925692805905105at_nat: ( set_Pr4329608150637261639at_nat > produc1319942482725812455at_nat ) > set_se5479201105039950375at_nat > $o ).

thf(sy_c_Fun__Def_Omax__strict,type,
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thf(sy_c_Fun__Def_Omax__weak,type,
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thf(sy_c_Fun__Def_Omin__strict,type,
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thf(sy_c_Fun__Def_Omin__weak,type,
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thf(sy_c_Fun__Def_Opair__leq,type,
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thf(sy_c_Fun__Def_Opair__less,type,
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thf(sy_c_Fun__Def_Oreduction__pair_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Fun__Def_Orp__inv__image_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
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thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
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thf(sy_c_Groups_Oone__class_Oone_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Nat__Onat_J,type,
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_J,type,
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Nat__Onat_J,type,
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    member8062223511168850704at_nat: produc349518998152878311at_nat > set_Pr553994874890374343at_nat > $o ).

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    member4971006070143632144at_nat: produc5602960702702913767at_nat > set_Pr1386498806471529671at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Nat__Onat_J_J,type,
    member3480088626692287306at_nat: produc2087150632810049441at_nat > set_Pr5070339090747523585at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    member6233145392104960784at_nat: produc6865100024664242407at_nat > set_Pr5975235057897686215at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    member6009668390874684176at_nat: produc1906917888121415911at_nat > set_Pr5564308138774400199at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_J_Mt__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_J_J,type,
    member8215962448328497936at_nat: produc1429535001352425191at_nat > set_Pr1160274906116719815at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    member8277197624267554838et_nat: produc7819656566062154093et_nat > set_Pr5488025237498180813et_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    member8757157785044589968at_nat: produc3843707927480180839at_nat > set_Pr4329608150637261639at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    member5855424355840516880at_nat: produc6487378988399798503at_nat > set_Pr5297940549829899463at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_J,type,
    member1466754251312161552at_nat: produc1319942482725812455at_nat > set_Pr7459493094073627847at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_Mt__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    member4300370950623927696at_nat: produc7299876168958306407at_nat > set_Pr1604112560894399303at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_J_Mt__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_J_J,type,
    member6436561159828024080at_nat: produc1296870326321989863at_nat > set_Pr5540436994911067335at_nat > $o ).

thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    member2643936169264416010at_nat: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat > $o ).

thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    member6273874620674182256at_nat: set_Pr4329608150637261639at_nat > set_se5479201105039950375at_nat > $o ).

thf(sy_v_as,type,
    as: list_nat ).

thf(sy_v_k,type,
    k: nat ).

% Relevant facts (1136)
thf(fact_0__092_060open_062_092_060And_062y_Ax_O_A_092_060lbrakk_062x_A_060_Ay_059_Ay_A_060_Alength_Aas_092_060rbrakk_062_A_092_060Longrightarrow_062_Acount__list_A_Idrop_A_Iy_A_L_A1_J_Aas_J_A_Ias_A_B_Ay_J_A_060_Acount__list_A_Idrop_A_Ix_A_L_A1_J_Aas_J_A_Ias_A_B_Ay_J_092_060close_062,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ ( size_size_list_nat @ as ) )
       => ( ord_less_nat @ ( count_list_nat @ ( drop_nat @ ( plus_plus_nat @ Y @ one_one_nat ) @ as ) @ ( nth_nat @ as @ Y ) ) @ ( count_list_nat @ ( drop_nat @ ( plus_plus_nat @ X @ one_one_nat ) @ as ) @ ( nth_nat @ as @ Y ) ) ) ) ) ).

% \<open>\<And>y x. \<lbrakk>x < y; y < length as\<rbrakk> \<Longrightarrow> count_list (drop (y + 1) as) (as ! y) < count_list (drop (x + 1) as) (as ! y)\<close>
thf(fact_1_assms,axiom,
    as != nil_nat ).

% assms
thf(fact_2_sketch__def,axiom,
    ( freque2132854181332412888ch_nat
    = ( ^ [As: list_nat,I: nat] : ( product_Pair_nat_nat @ ( nth_nat @ As @ I ) @ ( count_list_nat @ ( drop_nat @ ( plus_plus_nat @ I @ one_one_nat ) @ As ) @ ( nth_nat @ As @ I ) ) ) ) ) ).

% sketch_def
thf(fact_3_drop__drop,axiom,
    ! [N: nat,M: nat,Xs: list_nat] :
      ( ( drop_nat @ N @ ( drop_nat @ M @ Xs ) )
      = ( drop_nat @ ( plus_plus_nat @ N @ M ) @ Xs ) ) ).

% drop_drop
thf(fact_4_count__list__ge__2__iff,axiom,
    ! [Y: nat,Z: nat,Xs: list_nat] :
      ( ( ord_less_nat @ Y @ Z )
     => ( ( ord_less_nat @ Z @ ( size_size_list_nat @ Xs ) )
       => ( ( ( nth_nat @ Xs @ Y )
            = ( nth_nat @ Xs @ Z ) )
         => ( ord_less_nat @ one_one_nat @ ( count_list_nat @ Xs @ ( nth_nat @ Xs @ Y ) ) ) ) ) ) ).

% count_list_ge_2_iff
thf(fact_5_nat__add__left__cancel__less,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% nat_add_left_cancel_less
thf(fact_6_add__Pair,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( plus_p9057090461656269880at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( product_Pair_nat_nat @ C @ D ) )
      = ( product_Pair_nat_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ).

% add_Pair
thf(fact_7_add__Pair,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,C: product_prod_nat_nat,D: product_prod_nat_nat] :
      ( ( plus_p4591053195553783070at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( produc6161850002892822231at_nat @ C @ D ) )
      = ( produc6161850002892822231at_nat @ ( plus_p9057090461656269880at_nat @ A @ C ) @ ( plus_p9057090461656269880at_nat @ B @ D ) ) ) ).

% add_Pair
thf(fact_8_add__Pair,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,D: set_Pr1261947904930325089at_nat] :
      ( ( plus_p7352146897519463710at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ ( produc2922128104949294807at_nat @ C @ D ) )
      = ( produc2922128104949294807at_nat @ ( plus_p2407200367410362776at_nat @ A @ C ) @ ( plus_p2407200367410362776at_nat @ B @ D ) ) ) ).

% add_Pair
thf(fact_9_add__Pair,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat,D: set_Pr4329608150637261639at_nat] :
      ( ( plus_p8578679959789082782at_nat @ ( produc9060074326276436823at_nat @ A @ B ) @ ( produc9060074326276436823at_nat @ C @ D ) )
      = ( produc9060074326276436823at_nat @ ( plus_p5270993086169515006at_nat @ A @ C ) @ ( plus_p5270993086169515006at_nat @ B @ D ) ) ) ).

% add_Pair
thf(fact_10_add__Pair,axiom,
    ! [A: nat,B: product_prod_nat_nat,C: nat,D: product_prod_nat_nat] :
      ( ( plus_p6845766728538896175at_nat @ ( produc487386426758144856at_nat @ A @ B ) @ ( produc487386426758144856at_nat @ C @ D ) )
      = ( produc487386426758144856at_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_p9057090461656269880at_nat @ B @ D ) ) ) ).

% add_Pair
thf(fact_11_add__Pair,axiom,
    ! [A: product_prod_nat_nat,B: nat,C: product_prod_nat_nat,D: nat] :
      ( ( plus_p7971253712506197257at_nat @ ( produc6350711070570205562at_nat @ A @ B ) @ ( produc6350711070570205562at_nat @ C @ D ) )
      = ( produc6350711070570205562at_nat @ ( plus_p9057090461656269880at_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ).

% add_Pair
thf(fact_12_add__Pair,axiom,
    ! [A: nat,B: set_Pr1261947904930325089at_nat,C: nat,D: set_Pr1261947904930325089at_nat] :
      ( ( plus_p2083956751892040847at_nat @ ( produc5463602008962177208at_nat @ A @ B ) @ ( produc5463602008962177208at_nat @ C @ D ) )
      = ( produc5463602008962177208at_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_p2407200367410362776at_nat @ B @ D ) ) ) ).

% add_Pair
thf(fact_13_add__Pair,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: nat,C: set_Pr1261947904930325089at_nat,D: nat] :
      ( ( plus_p8359802151116154473at_nat @ ( produc3149970401713881818at_nat @ A @ B ) @ ( produc3149970401713881818at_nat @ C @ D ) )
      = ( produc3149970401713881818at_nat @ ( plus_p2407200367410362776at_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ).

% add_Pair
thf(fact_14_add__Pair,axiom,
    ! [A: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat,C: product_prod_nat_nat,D: set_Pr1261947904930325089at_nat] :
      ( ( plus_p1276725947666404990at_nat @ ( produc5966227314700062775at_nat @ A @ B ) @ ( produc5966227314700062775at_nat @ C @ D ) )
      = ( produc5966227314700062775at_nat @ ( plus_p9057090461656269880at_nat @ A @ C ) @ ( plus_p2407200367410362776at_nat @ B @ D ) ) ) ).

% add_Pair
thf(fact_15_add__Pair,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat,C: set_Pr1261947904930325089at_nat,D: product_prod_nat_nat] :
      ( ( plus_p3735141320629908414at_nat @ ( produc4320700058201173367at_nat @ A @ B ) @ ( produc4320700058201173367at_nat @ C @ D ) )
      = ( produc4320700058201173367at_nat @ ( plus_p2407200367410362776at_nat @ A @ C ) @ ( plus_p9057090461656269880at_nat @ B @ D ) ) ) ).

% add_Pair
thf(fact_16_add__less__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
      = ( ord_less_nat @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_17_add__less__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
      = ( ord_less_nat @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_18_nth__equalityI,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
           => ( ( nth_nat @ Xs @ I2 )
              = ( nth_nat @ Ys @ I2 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_19_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > nat > $o] :
      ( ( ! [I: nat] :
            ( ( ord_less_nat @ I @ K )
           => ? [X2: nat] : ( P @ I @ X2 ) ) )
      = ( ? [Xs2: list_nat] :
            ( ( ( size_size_list_nat @ Xs2 )
              = K )
            & ! [I: nat] :
                ( ( ord_less_nat @ I @ K )
               => ( P @ I @ ( nth_nat @ Xs2 @ I ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_20_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y2: list_nat,Z2: list_nat] : ( Y2 = Z2 ) )
    = ( ^ [Xs2: list_nat,Ys2: list_nat] :
          ( ( ( size_size_list_nat @ Xs2 )
            = ( size_size_list_nat @ Ys2 ) )
          & ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
             => ( ( nth_nat @ Xs2 @ I )
                = ( nth_nat @ Ys2 @ I ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_21_less__add__one,axiom,
    ! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).

% less_add_one
thf(fact_22_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_23_add__right__cancel,axiom,
    ! [B: product_prod_nat_nat,A: product_prod_nat_nat,C: product_prod_nat_nat] :
      ( ( ( plus_p9057090461656269880at_nat @ B @ A )
        = ( plus_p9057090461656269880at_nat @ C @ A ) )
      = ( B = C ) ) ).

% add_right_cancel
thf(fact_24_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_25_add__left__cancel,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,C: product_prod_nat_nat] :
      ( ( ( plus_p9057090461656269880at_nat @ A @ B )
        = ( plus_p9057090461656269880at_nat @ A @ C ) )
      = ( B = C ) ) ).

% add_left_cancel
thf(fact_26_fk__update__2_Ocases,axiom,
    ! [X: produc8642769642335960151at_nat] :
      ~ ! [A2: nat,M2: nat,X3: nat,L: nat] :
          ( X
         != ( produc6385450045882626063at_nat @ A2 @ ( produc487386426758144856at_nat @ M2 @ ( product_Pair_nat_nat @ X3 @ L ) ) ) ) ).

% fk_update_2.cases
thf(fact_27_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [X: produc4471711990508489141at_nat] :
      ~ ! [F: nat > nat > nat,A2: nat,B2: nat,Acc: nat] :
          ( X
         != ( produc3209952032786966637at_nat @ F @ ( produc487386426758144856at_nat @ A2 @ ( product_Pair_nat_nat @ B2 @ Acc ) ) ) ) ).

% fold_atLeastAtMost_nat.cases
thf(fact_28_drop__Nil,axiom,
    ! [N: nat] :
      ( ( drop_nat @ N @ nil_nat )
      = nil_nat ) ).

% drop_Nil
thf(fact_29_one__reorient,axiom,
    ! [X: nat] :
      ( ( one_one_nat = X )
      = ( X = one_one_nat ) ) ).

% one_reorient
thf(fact_30_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_31_add__right__imp__eq,axiom,
    ! [B: product_prod_nat_nat,A: product_prod_nat_nat,C: product_prod_nat_nat] :
      ( ( ( plus_p9057090461656269880at_nat @ B @ A )
        = ( plus_p9057090461656269880at_nat @ C @ A ) )
     => ( B = C ) ) ).

% add_right_imp_eq
thf(fact_32_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_33_add__left__imp__eq,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,C: product_prod_nat_nat] :
      ( ( ( plus_p9057090461656269880at_nat @ A @ B )
        = ( plus_p9057090461656269880at_nat @ A @ C ) )
     => ( B = C ) ) ).

% add_left_imp_eq
thf(fact_34_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_35_add_Oleft__commute,axiom,
    ! [B: produc1319942482725812455at_nat,A: produc1319942482725812455at_nat,C: produc1319942482725812455at_nat] :
      ( ( plus_p8578679959789082782at_nat @ B @ ( plus_p8578679959789082782at_nat @ A @ C ) )
      = ( plus_p8578679959789082782at_nat @ A @ ( plus_p8578679959789082782at_nat @ B @ C ) ) ) ).

% add.left_commute
thf(fact_36_add_Oleft__commute,axiom,
    ! [B: produc3843707927480180839at_nat,A: produc3843707927480180839at_nat,C: produc3843707927480180839at_nat] :
      ( ( plus_p7352146897519463710at_nat @ B @ ( plus_p7352146897519463710at_nat @ A @ C ) )
      = ( plus_p7352146897519463710at_nat @ A @ ( plus_p7352146897519463710at_nat @ B @ C ) ) ) ).

% add.left_commute
thf(fact_37_add_Oleft__commute,axiom,
    ! [B: product_prod_nat_nat,A: product_prod_nat_nat,C: product_prod_nat_nat] :
      ( ( plus_p9057090461656269880at_nat @ B @ ( plus_p9057090461656269880at_nat @ A @ C ) )
      = ( plus_p9057090461656269880at_nat @ A @ ( plus_p9057090461656269880at_nat @ B @ C ) ) ) ).

% add.left_commute
thf(fact_38_add_Oleft__commute,axiom,
    ! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat] :
      ( ( plus_p5270993086169515006at_nat @ B @ ( plus_p5270993086169515006at_nat @ A @ C ) )
      = ( plus_p5270993086169515006at_nat @ A @ ( plus_p5270993086169515006at_nat @ B @ C ) ) ) ).

% add.left_commute
thf(fact_39_add_Oleft__commute,axiom,
    ! [B: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
      ( ( plus_p2407200367410362776at_nat @ B @ ( plus_p2407200367410362776at_nat @ A @ C ) )
      = ( plus_p2407200367410362776at_nat @ A @ ( plus_p2407200367410362776at_nat @ B @ C ) ) ) ).

% add.left_commute
thf(fact_40_add_Ocommute,axiom,
    ( plus_plus_nat
    = ( ^ [A3: nat,B3: nat] : ( plus_plus_nat @ B3 @ A3 ) ) ) ).

% add.commute
thf(fact_41_add_Ocommute,axiom,
    ( plus_p8578679959789082782at_nat
    = ( ^ [A3: produc1319942482725812455at_nat,B3: produc1319942482725812455at_nat] : ( plus_p8578679959789082782at_nat @ B3 @ A3 ) ) ) ).

% add.commute
thf(fact_42_add_Ocommute,axiom,
    ( plus_p7352146897519463710at_nat
    = ( ^ [A3: produc3843707927480180839at_nat,B3: produc3843707927480180839at_nat] : ( plus_p7352146897519463710at_nat @ B3 @ A3 ) ) ) ).

% add.commute
thf(fact_43_add_Ocommute,axiom,
    ( plus_p9057090461656269880at_nat
    = ( ^ [A3: product_prod_nat_nat,B3: product_prod_nat_nat] : ( plus_p9057090461656269880at_nat @ B3 @ A3 ) ) ) ).

% add.commute
thf(fact_44_add_Ocommute,axiom,
    ( plus_p5270993086169515006at_nat
    = ( ^ [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] : ( plus_p5270993086169515006at_nat @ B3 @ A3 ) ) ) ).

% add.commute
thf(fact_45_add_Ocommute,axiom,
    ( plus_p2407200367410362776at_nat
    = ( ^ [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] : ( plus_p2407200367410362776at_nat @ B3 @ A3 ) ) ) ).

% add.commute
thf(fact_46_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_47_add_Oassoc,axiom,
    ! [A: produc1319942482725812455at_nat,B: produc1319942482725812455at_nat,C: produc1319942482725812455at_nat] :
      ( ( plus_p8578679959789082782at_nat @ ( plus_p8578679959789082782at_nat @ A @ B ) @ C )
      = ( plus_p8578679959789082782at_nat @ A @ ( plus_p8578679959789082782at_nat @ B @ C ) ) ) ).

% add.assoc
thf(fact_48_add_Oassoc,axiom,
    ! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat,C: produc3843707927480180839at_nat] :
      ( ( plus_p7352146897519463710at_nat @ ( plus_p7352146897519463710at_nat @ A @ B ) @ C )
      = ( plus_p7352146897519463710at_nat @ A @ ( plus_p7352146897519463710at_nat @ B @ C ) ) ) ).

% add.assoc
thf(fact_49_add_Oassoc,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,C: product_prod_nat_nat] :
      ( ( plus_p9057090461656269880at_nat @ ( plus_p9057090461656269880at_nat @ A @ B ) @ C )
      = ( plus_p9057090461656269880at_nat @ A @ ( plus_p9057090461656269880at_nat @ B @ C ) ) ) ).

% add.assoc
thf(fact_50_add_Oassoc,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat] :
      ( ( plus_p5270993086169515006at_nat @ ( plus_p5270993086169515006at_nat @ A @ B ) @ C )
      = ( plus_p5270993086169515006at_nat @ A @ ( plus_p5270993086169515006at_nat @ B @ C ) ) ) ).

% add.assoc
thf(fact_51_add_Oassoc,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
      ( ( plus_p2407200367410362776at_nat @ ( plus_p2407200367410362776at_nat @ A @ B ) @ C )
      = ( plus_p2407200367410362776at_nat @ A @ ( plus_p2407200367410362776at_nat @ B @ C ) ) ) ).

% add.assoc
thf(fact_52_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_53_group__cancel_Oadd2,axiom,
    ! [B4: produc1319942482725812455at_nat,K: produc1319942482725812455at_nat,B: produc1319942482725812455at_nat,A: produc1319942482725812455at_nat] :
      ( ( B4
        = ( plus_p8578679959789082782at_nat @ K @ B ) )
     => ( ( plus_p8578679959789082782at_nat @ A @ B4 )
        = ( plus_p8578679959789082782at_nat @ K @ ( plus_p8578679959789082782at_nat @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_54_group__cancel_Oadd2,axiom,
    ! [B4: produc3843707927480180839at_nat,K: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat,A: produc3843707927480180839at_nat] :
      ( ( B4
        = ( plus_p7352146897519463710at_nat @ K @ B ) )
     => ( ( plus_p7352146897519463710at_nat @ A @ B4 )
        = ( plus_p7352146897519463710at_nat @ K @ ( plus_p7352146897519463710at_nat @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_55_group__cancel_Oadd2,axiom,
    ! [B4: product_prod_nat_nat,K: product_prod_nat_nat,B: product_prod_nat_nat,A: product_prod_nat_nat] :
      ( ( B4
        = ( plus_p9057090461656269880at_nat @ K @ B ) )
     => ( ( plus_p9057090461656269880at_nat @ A @ B4 )
        = ( plus_p9057090461656269880at_nat @ K @ ( plus_p9057090461656269880at_nat @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_56_group__cancel_Oadd2,axiom,
    ! [B4: set_Pr4329608150637261639at_nat,K: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( B4
        = ( plus_p5270993086169515006at_nat @ K @ B ) )
     => ( ( plus_p5270993086169515006at_nat @ A @ B4 )
        = ( plus_p5270993086169515006at_nat @ K @ ( plus_p5270993086169515006at_nat @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_57_group__cancel_Oadd2,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,K: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( B4
        = ( plus_p2407200367410362776at_nat @ K @ B ) )
     => ( ( plus_p2407200367410362776at_nat @ A @ B4 )
        = ( plus_p2407200367410362776at_nat @ K @ ( plus_p2407200367410362776at_nat @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_58_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_59_group__cancel_Oadd1,axiom,
    ! [A4: produc1319942482725812455at_nat,K: produc1319942482725812455at_nat,A: produc1319942482725812455at_nat,B: produc1319942482725812455at_nat] :
      ( ( A4
        = ( plus_p8578679959789082782at_nat @ K @ A ) )
     => ( ( plus_p8578679959789082782at_nat @ A4 @ B )
        = ( plus_p8578679959789082782at_nat @ K @ ( plus_p8578679959789082782at_nat @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_60_group__cancel_Oadd1,axiom,
    ! [A4: produc3843707927480180839at_nat,K: produc3843707927480180839at_nat,A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
      ( ( A4
        = ( plus_p7352146897519463710at_nat @ K @ A ) )
     => ( ( plus_p7352146897519463710at_nat @ A4 @ B )
        = ( plus_p7352146897519463710at_nat @ K @ ( plus_p7352146897519463710at_nat @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_61_group__cancel_Oadd1,axiom,
    ! [A4: product_prod_nat_nat,K: product_prod_nat_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( ( A4
        = ( plus_p9057090461656269880at_nat @ K @ A ) )
     => ( ( plus_p9057090461656269880at_nat @ A4 @ B )
        = ( plus_p9057090461656269880at_nat @ K @ ( plus_p9057090461656269880at_nat @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_62_group__cancel_Oadd1,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,K: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( A4
        = ( plus_p5270993086169515006at_nat @ K @ A ) )
     => ( ( plus_p5270993086169515006at_nat @ A4 @ B )
        = ( plus_p5270993086169515006at_nat @ K @ ( plus_p5270993086169515006at_nat @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_63_group__cancel_Oadd1,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,K: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( A4
        = ( plus_p2407200367410362776at_nat @ K @ A ) )
     => ( ( plus_p2407200367410362776at_nat @ A4 @ B )
        = ( plus_p2407200367410362776at_nat @ K @ ( plus_p2407200367410362776at_nat @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_64_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I3: nat,J: nat,K: nat,L2: nat] :
      ( ( ( I3 = J )
        & ( K = L2 ) )
     => ( ( plus_plus_nat @ I3 @ K )
        = ( plus_plus_nat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_65_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_66_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: produc1319942482725812455at_nat,B: produc1319942482725812455at_nat,C: produc1319942482725812455at_nat] :
      ( ( plus_p8578679959789082782at_nat @ ( plus_p8578679959789082782at_nat @ A @ B ) @ C )
      = ( plus_p8578679959789082782at_nat @ A @ ( plus_p8578679959789082782at_nat @ B @ C ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_67_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat,C: produc3843707927480180839at_nat] :
      ( ( plus_p7352146897519463710at_nat @ ( plus_p7352146897519463710at_nat @ A @ B ) @ C )
      = ( plus_p7352146897519463710at_nat @ A @ ( plus_p7352146897519463710at_nat @ B @ C ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_68_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,C: product_prod_nat_nat] :
      ( ( plus_p9057090461656269880at_nat @ ( plus_p9057090461656269880at_nat @ A @ B ) @ C )
      = ( plus_p9057090461656269880at_nat @ A @ ( plus_p9057090461656269880at_nat @ B @ C ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_69_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat] :
      ( ( plus_p5270993086169515006at_nat @ ( plus_p5270993086169515006at_nat @ A @ B ) @ C )
      = ( plus_p5270993086169515006at_nat @ A @ ( plus_p5270993086169515006at_nat @ B @ C ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_70_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
      ( ( plus_p2407200367410362776at_nat @ ( plus_p2407200367410362776at_nat @ A @ B ) @ C )
      = ( plus_p2407200367410362776at_nat @ A @ ( plus_p2407200367410362776at_nat @ B @ C ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_71_mem__Collect__eq,axiom,
    ! [A: produc859450856879609959at_nat,P: produc859450856879609959at_nat > $o] :
      ( ( member8206827879206165904at_nat @ A @ ( collec7088162979684241874at_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_72_mem__Collect__eq,axiom,
    ! [A: produc3843707927480180839at_nat,P: produc3843707927480180839at_nat > $o] :
      ( ( member8757157785044589968at_nat @ A @ ( collec6321179662152712658at_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_73_mem__Collect__eq,axiom,
    ! [A: nat,P: nat > $o] :
      ( ( member_nat @ A @ ( collect_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_74_mem__Collect__eq,axiom,
    ! [A: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
      ( ( member8440522571783428010at_nat @ A @ ( collec3392354462482085612at_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_75_Collect__mem__eq,axiom,
    ! [A4: set_Pr8693737435421807431at_nat] :
      ( ( collec7088162979684241874at_nat
        @ ^ [X4: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X4 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_76_Collect__mem__eq,axiom,
    ! [A4: set_Pr4329608150637261639at_nat] :
      ( ( collec6321179662152712658at_nat
        @ ^ [X4: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X4 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_77_Collect__mem__eq,axiom,
    ! [A4: set_nat] :
      ( ( collect_nat
        @ ^ [X4: nat] : ( member_nat @ X4 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_78_Collect__mem__eq,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_79_Collect__cong,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ! [X3: nat] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect_nat @ P )
        = ( collect_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_80_Collect__cong,axiom,
    ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ! [X3: product_prod_nat_nat] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collec3392354462482085612at_nat @ P )
        = ( collec3392354462482085612at_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_81_linorder__neqE__nat,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ ( ord_less_nat @ X @ Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_neqE_nat
thf(fact_82_infinite__descent,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N2: nat] :
          ( ~ ( P @ N2 )
         => ? [M3: nat] :
              ( ( ord_less_nat @ M3 @ N2 )
              & ~ ( P @ M3 ) ) )
     => ( P @ N ) ) ).

% infinite_descent
thf(fact_83_nat__less__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N2: nat] :
          ( ! [M3: nat] :
              ( ( ord_less_nat @ M3 @ N2 )
             => ( P @ M3 ) )
         => ( P @ N2 ) )
     => ( P @ N ) ) ).

% nat_less_induct
thf(fact_84_less__irrefl__nat,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_irrefl_nat
thf(fact_85_less__not__refl3,axiom,
    ! [S: nat,T: nat] :
      ( ( ord_less_nat @ S @ T )
     => ( S != T ) ) ).

% less_not_refl3
thf(fact_86_less__not__refl2,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ N @ M )
     => ( M != N ) ) ).

% less_not_refl2
thf(fact_87_less__not__refl,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_not_refl
thf(fact_88_nat__neq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( M != N )
      = ( ( ord_less_nat @ M @ N )
        | ( ord_less_nat @ N @ M ) ) ) ).

% nat_neq_iff
thf(fact_89_size__neq__size__imp__neq,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( ( size_size_list_nat @ X )
       != ( size_size_list_nat @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_90_neq__if__length__neq,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
       != ( size_size_list_nat @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_91_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_nat] :
      ( ( size_size_list_nat @ Xs3 )
      = N ) ).

% Ex_list_of_length
thf(fact_92_add__less__imp__less__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
     => ( ord_less_nat @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_93_add__less__imp__less__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
     => ( ord_less_nat @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_94_add__strict__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).

% add_strict_right_mono
thf(fact_95_add__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).

% add_strict_left_mono
thf(fact_96_add__strict__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_strict_mono
thf(fact_97_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I3: nat,J: nat,K: nat,L2: nat] :
      ( ( ( ord_less_nat @ I3 @ J )
        & ( K = L2 ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_98_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I3: nat,J: nat,K: nat,L2: nat] :
      ( ( ( I3 = J )
        & ( ord_less_nat @ K @ L2 ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_99_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I3: nat,J: nat,K: nat,L2: nat] :
      ( ( ( ord_less_nat @ I3 @ J )
        & ( ord_less_nat @ K @ L2 ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_100_length__induct,axiom,
    ! [P: list_nat > $o,Xs: list_nat] :
      ( ! [Xs3: list_nat] :
          ( ! [Ys3: list_nat] :
              ( ( ord_less_nat @ ( size_size_list_nat @ Ys3 ) @ ( size_size_list_nat @ Xs3 ) )
             => ( P @ Ys3 ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_101_less__add__eq__less,axiom,
    ! [K: nat,L2: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ K @ L2 )
     => ( ( ( plus_plus_nat @ M @ L2 )
          = ( plus_plus_nat @ K @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% less_add_eq_less
thf(fact_102_trans__less__add2,axiom,
    ! [I3: nat,J: nat,M: nat] :
      ( ( ord_less_nat @ I3 @ J )
     => ( ord_less_nat @ I3 @ ( plus_plus_nat @ M @ J ) ) ) ).

% trans_less_add2
thf(fact_103_trans__less__add1,axiom,
    ! [I3: nat,J: nat,M: nat] :
      ( ( ord_less_nat @ I3 @ J )
     => ( ord_less_nat @ I3 @ ( plus_plus_nat @ J @ M ) ) ) ).

% trans_less_add1
thf(fact_104_add__less__mono1,axiom,
    ! [I3: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I3 @ J )
     => ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).

% add_less_mono1
thf(fact_105_not__add__less2,axiom,
    ! [J: nat,I3: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I3 ) @ I3 ) ).

% not_add_less2
thf(fact_106_not__add__less1,axiom,
    ! [I3: nat,J: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ I3 @ J ) @ I3 ) ).

% not_add_less1
thf(fact_107_add__less__mono,axiom,
    ! [I3: nat,J: nat,K: nat,L2: nat] :
      ( ( ord_less_nat @ I3 @ J )
     => ( ( ord_less_nat @ K @ L2 )
       => ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ) ).

% add_less_mono
thf(fact_108_add__lessD1,axiom,
    ! [I3: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I3 @ J ) @ K )
     => ( ord_less_nat @ I3 @ K ) ) ).

% add_lessD1
thf(fact_109_add__mono1,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).

% add_mono1
thf(fact_110_inj__on__convol__ident,axiom,
    ! [F2: nat > nat,X5: set_nat] :
      ( inj_on5538052773655684606at_nat
      @ ^ [X4: nat] : ( product_Pair_nat_nat @ X4 @ ( F2 @ X4 ) )
      @ X5 ) ).

% inj_on_convol_ident
thf(fact_111_inj__on__convol__ident,axiom,
    ! [F2: product_prod_nat_nat > product_prod_nat_nat,X5: set_Pr1261947904930325089at_nat] :
      ( inj_on3865918002419084311at_nat
      @ ^ [X4: product_prod_nat_nat] : ( produc6161850002892822231at_nat @ X4 @ ( F2 @ X4 ) )
      @ X5 ) ).

% inj_on_convol_ident
thf(fact_112_inj__on__convol__ident,axiom,
    ! [F2: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat,X5: set_se7855581050983116737at_nat] :
      ( inj_on699426629415580343at_nat
      @ ^ [X4: set_Pr1261947904930325089at_nat] : ( produc2922128104949294807at_nat @ X4 @ ( F2 @ X4 ) )
      @ X5 ) ).

% inj_on_convol_ident
thf(fact_113_inj__on__convol__ident,axiom,
    ! [F2: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat,X5: set_se5479201105039950375at_nat] :
      ( inj_on1468925692805905105at_nat
      @ ^ [X4: set_Pr4329608150637261639at_nat] : ( produc9060074326276436823at_nat @ X4 @ ( F2 @ X4 ) )
      @ X5 ) ).

% inj_on_convol_ident
thf(fact_114_old_Oprod_Oinject,axiom,
    ! [A: nat,B: nat,A5: nat,B5: nat] :
      ( ( ( product_Pair_nat_nat @ A @ B )
        = ( product_Pair_nat_nat @ A5 @ B5 ) )
      = ( ( A = A5 )
        & ( B = B5 ) ) ) ).

% old.prod.inject
thf(fact_115_old_Oprod_Oinject,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A5: product_prod_nat_nat,B5: product_prod_nat_nat] :
      ( ( ( produc6161850002892822231at_nat @ A @ B )
        = ( produc6161850002892822231at_nat @ A5 @ B5 ) )
      = ( ( A = A5 )
        & ( B = B5 ) ) ) ).

% old.prod.inject
thf(fact_116_old_Oprod_Oinject,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( ( produc2922128104949294807at_nat @ A @ B )
        = ( produc2922128104949294807at_nat @ A5 @ B5 ) )
      = ( ( A = A5 )
        & ( B = B5 ) ) ) ).

% old.prod.inject
thf(fact_117_old_Oprod_Oinject,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A5: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ( ( produc9060074326276436823at_nat @ A @ B )
        = ( produc9060074326276436823at_nat @ A5 @ B5 ) )
      = ( ( A = A5 )
        & ( B = B5 ) ) ) ).

% old.prod.inject
thf(fact_118_prod_Oinject,axiom,
    ! [X1: nat,X22: nat,Y1: nat,Y22: nat] :
      ( ( ( product_Pair_nat_nat @ X1 @ X22 )
        = ( product_Pair_nat_nat @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_119_prod_Oinject,axiom,
    ! [X1: product_prod_nat_nat,X22: product_prod_nat_nat,Y1: product_prod_nat_nat,Y22: product_prod_nat_nat] :
      ( ( ( produc6161850002892822231at_nat @ X1 @ X22 )
        = ( produc6161850002892822231at_nat @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_120_prod_Oinject,axiom,
    ! [X1: set_Pr1261947904930325089at_nat,X22: set_Pr1261947904930325089at_nat,Y1: set_Pr1261947904930325089at_nat,Y22: set_Pr1261947904930325089at_nat] :
      ( ( ( produc2922128104949294807at_nat @ X1 @ X22 )
        = ( produc2922128104949294807at_nat @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_121_prod_Oinject,axiom,
    ! [X1: set_Pr4329608150637261639at_nat,X22: set_Pr4329608150637261639at_nat,Y1: set_Pr4329608150637261639at_nat,Y22: set_Pr4329608150637261639at_nat] :
      ( ( ( produc9060074326276436823at_nat @ X1 @ X22 )
        = ( produc9060074326276436823at_nat @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_122_set__plus__intro,axiom,
    ! [A: produc859450856879609959at_nat,C2: set_Pr8693737435421807431at_nat,B: produc859450856879609959at_nat,D2: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ A @ C2 )
     => ( ( member8206827879206165904at_nat @ B @ D2 )
       => ( member8206827879206165904at_nat @ ( plus_p4591053195553783070at_nat @ A @ B ) @ ( plus_p2778368570229234686at_nat @ C2 @ D2 ) ) ) ) ).

% set_plus_intro
thf(fact_123_set__plus__intro,axiom,
    ! [A: nat,C2: set_nat,B: nat,D2: set_nat] :
      ( ( member_nat @ A @ C2 )
     => ( ( member_nat @ B @ D2 )
       => ( member_nat @ ( plus_plus_nat @ A @ B ) @ ( plus_plus_set_nat @ C2 @ D2 ) ) ) ) ).

% set_plus_intro
thf(fact_124_set__plus__intro,axiom,
    ! [A: produc1319942482725812455at_nat,C2: set_Pr7459493094073627847at_nat,B: produc1319942482725812455at_nat,D2: set_Pr7459493094073627847at_nat] :
      ( ( member1466754251312161552at_nat @ A @ C2 )
     => ( ( member1466754251312161552at_nat @ B @ D2 )
       => ( member1466754251312161552at_nat @ ( plus_p8578679959789082782at_nat @ A @ B ) @ ( plus_p5045890354785974398at_nat @ C2 @ D2 ) ) ) ) ).

% set_plus_intro
thf(fact_125_set__plus__intro,axiom,
    ! [A: produc3843707927480180839at_nat,C2: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat,D2: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ A @ C2 )
     => ( ( member8757157785044589968at_nat @ B @ D2 )
       => ( member8757157785044589968at_nat @ ( plus_p7352146897519463710at_nat @ A @ B ) @ ( plus_p5270993086169515006at_nat @ C2 @ D2 ) ) ) ) ).

% set_plus_intro
thf(fact_126_set__plus__intro,axiom,
    ! [A: product_prod_nat_nat,C2: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat,D2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ A @ C2 )
     => ( ( member8440522571783428010at_nat @ B @ D2 )
       => ( member8440522571783428010at_nat @ ( plus_p9057090461656269880at_nat @ A @ B ) @ ( plus_p2407200367410362776at_nat @ C2 @ D2 ) ) ) ) ).

% set_plus_intro
thf(fact_127_set__plus__intro,axiom,
    ! [A: set_Pr4329608150637261639at_nat,C2: set_se5479201105039950375at_nat,B: set_Pr4329608150637261639at_nat,D2: set_se5479201105039950375at_nat] :
      ( ( member6273874620674182256at_nat @ A @ C2 )
     => ( ( member6273874620674182256at_nat @ B @ D2 )
       => ( member6273874620674182256at_nat @ ( plus_p5270993086169515006at_nat @ A @ B ) @ ( plus_p3918552533073458910at_nat @ C2 @ D2 ) ) ) ) ).

% set_plus_intro
thf(fact_128_set__plus__intro,axiom,
    ! [A: set_Pr1261947904930325089at_nat,C2: set_se7855581050983116737at_nat,B: set_Pr1261947904930325089at_nat,D2: set_se7855581050983116737at_nat] :
      ( ( member2643936169264416010at_nat @ A @ C2 )
     => ( ( member2643936169264416010at_nat @ B @ D2 )
       => ( member2643936169264416010at_nat @ ( plus_p2407200367410362776at_nat @ A @ B ) @ ( plus_p6536625912622263032at_nat @ C2 @ D2 ) ) ) ) ).

% set_plus_intro
thf(fact_129_inj__on__add_H,axiom,
    ! [A: nat,A4: set_nat] :
      ( inj_on_nat_nat
      @ ^ [B3: nat] : ( plus_plus_nat @ B3 @ A )
      @ A4 ) ).

% inj_on_add'
thf(fact_130_inj__on__add_H,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( inj_on8969904277767023793at_nat
      @ ^ [B3: product_prod_nat_nat] : ( plus_p9057090461656269880at_nat @ B3 @ A )
      @ A4 ) ).

% inj_on_add'
thf(fact_131_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_nat,N: nat] :
      ( ( ord_less_nat @ M @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_Pr7617993195940197384at_nat @ ( enumerate_nat @ N @ Xs ) @ M )
        = ( product_Pair_nat_nat @ ( plus_plus_nat @ N @ M ) @ ( nth_nat @ Xs @ M ) ) ) ) ).

% nth_enumerate_eq
thf(fact_132_inj__on__add,axiom,
    ! [A: nat,A4: set_nat] : ( inj_on_nat_nat @ ( plus_plus_nat @ A ) @ A4 ) ).

% inj_on_add
thf(fact_133_inj__on__add,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] : ( inj_on8969904277767023793at_nat @ ( plus_p9057090461656269880at_nat @ A ) @ A4 ) ).

% inj_on_add
thf(fact_134_enumerate__simps_I1_J,axiom,
    ! [N: nat] :
      ( ( enumerate_nat @ N @ nil_nat )
      = nil_Pr5478986624290739719at_nat ) ).

% enumerate_simps(1)
thf(fact_135_length__enumerate,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( size_s5460976970255530739at_nat @ ( enumerate_nat @ N @ Xs ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_enumerate
thf(fact_136_set__plus__elim,axiom,
    ! [X: produc859450856879609959at_nat,A4: set_Pr8693737435421807431at_nat,B4: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ X @ ( plus_p2778368570229234686at_nat @ A4 @ B4 ) )
     => ~ ! [A2: produc859450856879609959at_nat,B2: produc859450856879609959at_nat] :
            ( ( X
              = ( plus_p4591053195553783070at_nat @ A2 @ B2 ) )
           => ( ( member8206827879206165904at_nat @ A2 @ A4 )
             => ~ ( member8206827879206165904at_nat @ B2 @ B4 ) ) ) ) ).

% set_plus_elim
thf(fact_137_set__plus__elim,axiom,
    ! [X: nat,A4: set_nat,B4: set_nat] :
      ( ( member_nat @ X @ ( plus_plus_set_nat @ A4 @ B4 ) )
     => ~ ! [A2: nat,B2: nat] :
            ( ( X
              = ( plus_plus_nat @ A2 @ B2 ) )
           => ( ( member_nat @ A2 @ A4 )
             => ~ ( member_nat @ B2 @ B4 ) ) ) ) ).

% set_plus_elim
thf(fact_138_set__plus__elim,axiom,
    ! [X: produc1319942482725812455at_nat,A4: set_Pr7459493094073627847at_nat,B4: set_Pr7459493094073627847at_nat] :
      ( ( member1466754251312161552at_nat @ X @ ( plus_p5045890354785974398at_nat @ A4 @ B4 ) )
     => ~ ! [A2: produc1319942482725812455at_nat,B2: produc1319942482725812455at_nat] :
            ( ( X
              = ( plus_p8578679959789082782at_nat @ A2 @ B2 ) )
           => ( ( member1466754251312161552at_nat @ A2 @ A4 )
             => ~ ( member1466754251312161552at_nat @ B2 @ B4 ) ) ) ) ).

% set_plus_elim
thf(fact_139_set__plus__elim,axiom,
    ! [X: set_Pr4329608150637261639at_nat,A4: set_se5479201105039950375at_nat,B4: set_se5479201105039950375at_nat] :
      ( ( member6273874620674182256at_nat @ X @ ( plus_p3918552533073458910at_nat @ A4 @ B4 ) )
     => ~ ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
            ( ( X
              = ( plus_p5270993086169515006at_nat @ A2 @ B2 ) )
           => ( ( member6273874620674182256at_nat @ A2 @ A4 )
             => ~ ( member6273874620674182256at_nat @ B2 @ B4 ) ) ) ) ).

% set_plus_elim
thf(fact_140_set__plus__elim,axiom,
    ! [X: set_Pr1261947904930325089at_nat,A4: set_se7855581050983116737at_nat,B4: set_se7855581050983116737at_nat] :
      ( ( member2643936169264416010at_nat @ X @ ( plus_p6536625912622263032at_nat @ A4 @ B4 ) )
     => ~ ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
            ( ( X
              = ( plus_p2407200367410362776at_nat @ A2 @ B2 ) )
           => ( ( member2643936169264416010at_nat @ A2 @ A4 )
             => ~ ( member2643936169264416010at_nat @ B2 @ B4 ) ) ) ) ).

% set_plus_elim
thf(fact_141_set__plus__elim,axiom,
    ! [X: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ X @ ( plus_p5270993086169515006at_nat @ A4 @ B4 ) )
     => ~ ! [A2: produc3843707927480180839at_nat,B2: produc3843707927480180839at_nat] :
            ( ( X
              = ( plus_p7352146897519463710at_nat @ A2 @ B2 ) )
           => ( ( member8757157785044589968at_nat @ A2 @ A4 )
             => ~ ( member8757157785044589968at_nat @ B2 @ B4 ) ) ) ) ).

% set_plus_elim
thf(fact_142_set__plus__elim,axiom,
    ! [X: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ X @ ( plus_p2407200367410362776at_nat @ A4 @ B4 ) )
     => ~ ! [A2: product_prod_nat_nat,B2: product_prod_nat_nat] :
            ( ( X
              = ( plus_p9057090461656269880at_nat @ A2 @ B2 ) )
           => ( ( member8440522571783428010at_nat @ A2 @ A4 )
             => ~ ( member8440522571783428010at_nat @ B2 @ B4 ) ) ) ) ).

% set_plus_elim
thf(fact_143_prod_Oexhaust,axiom,
    ! [Y: product_prod_nat_nat] :
      ~ ! [X12: nat,X23: nat] :
          ( Y
         != ( product_Pair_nat_nat @ X12 @ X23 ) ) ).

% prod.exhaust
thf(fact_144_prod_Oexhaust,axiom,
    ! [Y: produc859450856879609959at_nat] :
      ~ ! [X12: product_prod_nat_nat,X23: product_prod_nat_nat] :
          ( Y
         != ( produc6161850002892822231at_nat @ X12 @ X23 ) ) ).

% prod.exhaust
thf(fact_145_prod_Oexhaust,axiom,
    ! [Y: produc3843707927480180839at_nat] :
      ~ ! [X12: set_Pr1261947904930325089at_nat,X23: set_Pr1261947904930325089at_nat] :
          ( Y
         != ( produc2922128104949294807at_nat @ X12 @ X23 ) ) ).

% prod.exhaust
thf(fact_146_prod_Oexhaust,axiom,
    ! [Y: produc1319942482725812455at_nat] :
      ~ ! [X12: set_Pr4329608150637261639at_nat,X23: set_Pr4329608150637261639at_nat] :
          ( Y
         != ( produc9060074326276436823at_nat @ X12 @ X23 ) ) ).

% prod.exhaust
thf(fact_147_surj__pair,axiom,
    ! [P2: product_prod_nat_nat] :
    ? [X3: nat,Y3: nat] :
      ( P2
      = ( product_Pair_nat_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_148_surj__pair,axiom,
    ! [P2: produc859450856879609959at_nat] :
    ? [X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
      ( P2
      = ( produc6161850002892822231at_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_149_surj__pair,axiom,
    ! [P2: produc3843707927480180839at_nat] :
    ? [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
      ( P2
      = ( produc2922128104949294807at_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_150_surj__pair,axiom,
    ! [P2: produc1319942482725812455at_nat] :
    ? [X3: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] :
      ( P2
      = ( produc9060074326276436823at_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_151_prod__cases,axiom,
    ! [P: product_prod_nat_nat > $o,P2: product_prod_nat_nat] :
      ( ! [A2: nat,B2: nat] : ( P @ ( product_Pair_nat_nat @ A2 @ B2 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_152_prod__cases,axiom,
    ! [P: produc859450856879609959at_nat > $o,P2: produc859450856879609959at_nat] :
      ( ! [A2: product_prod_nat_nat,B2: product_prod_nat_nat] : ( P @ ( produc6161850002892822231at_nat @ A2 @ B2 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_153_prod__cases,axiom,
    ! [P: produc3843707927480180839at_nat > $o,P2: produc3843707927480180839at_nat] :
      ( ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : ( P @ ( produc2922128104949294807at_nat @ A2 @ B2 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_154_prod__cases,axiom,
    ! [P: produc1319942482725812455at_nat > $o,P2: produc1319942482725812455at_nat] :
      ( ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] : ( P @ ( produc9060074326276436823at_nat @ A2 @ B2 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_155_Pair__inject,axiom,
    ! [A: nat,B: nat,A5: nat,B5: nat] :
      ( ( ( product_Pair_nat_nat @ A @ B )
        = ( product_Pair_nat_nat @ A5 @ B5 ) )
     => ~ ( ( A = A5 )
         => ( B != B5 ) ) ) ).

% Pair_inject
thf(fact_156_Pair__inject,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A5: product_prod_nat_nat,B5: product_prod_nat_nat] :
      ( ( ( produc6161850002892822231at_nat @ A @ B )
        = ( produc6161850002892822231at_nat @ A5 @ B5 ) )
     => ~ ( ( A = A5 )
         => ( B != B5 ) ) ) ).

% Pair_inject
thf(fact_157_Pair__inject,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( ( produc2922128104949294807at_nat @ A @ B )
        = ( produc2922128104949294807at_nat @ A5 @ B5 ) )
     => ~ ( ( A = A5 )
         => ( B != B5 ) ) ) ).

% Pair_inject
thf(fact_158_Pair__inject,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A5: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ( ( produc9060074326276436823at_nat @ A @ B )
        = ( produc9060074326276436823at_nat @ A5 @ B5 ) )
     => ~ ( ( A = A5 )
         => ( B != B5 ) ) ) ).

% Pair_inject
thf(fact_159_prod__cases3,axiom,
    ! [Y: produc859450856879609959at_nat] :
      ~ ! [A2: product_prod_nat_nat,B2: nat,C3: nat] :
          ( Y
         != ( produc6161850002892822231at_nat @ A2 @ ( product_Pair_nat_nat @ B2 @ C3 ) ) ) ).

% prod_cases3
thf(fact_160_prod__induct3,axiom,
    ! [P: produc859450856879609959at_nat > $o,X: produc859450856879609959at_nat] :
      ( ! [A2: product_prod_nat_nat,B2: nat,C3: nat] : ( P @ ( produc6161850002892822231at_nat @ A2 @ ( product_Pair_nat_nat @ B2 @ C3 ) ) )
     => ( P @ X ) ) ).

% prod_induct3
thf(fact_161_inj__onD,axiom,
    ! [F2: nat > product_prod_nat_nat,A4: set_nat,X: nat,Y: nat] :
      ( ( inj_on5538052773655684606at_nat @ F2 @ A4 )
     => ( ( ( F2 @ X )
          = ( F2 @ Y ) )
       => ( ( member_nat @ X @ A4 )
         => ( ( member_nat @ Y @ A4 )
           => ( X = Y ) ) ) ) ) ).

% inj_onD
thf(fact_162_inj__onD,axiom,
    ! [F2: nat > nat,A4: set_nat,X: nat,Y: nat] :
      ( ( inj_on_nat_nat @ F2 @ A4 )
     => ( ( ( F2 @ X )
          = ( F2 @ Y ) )
       => ( ( member_nat @ X @ A4 )
         => ( ( member_nat @ Y @ A4 )
           => ( X = Y ) ) ) ) ) ).

% inj_onD
thf(fact_163_inj__onI,axiom,
    ! [A4: set_nat,F2: nat > product_prod_nat_nat] :
      ( ! [X3: nat,Y3: nat] :
          ( ( member_nat @ X3 @ A4 )
         => ( ( member_nat @ Y3 @ A4 )
           => ( ( ( F2 @ X3 )
                = ( F2 @ Y3 ) )
             => ( X3 = Y3 ) ) ) )
     => ( inj_on5538052773655684606at_nat @ F2 @ A4 ) ) ).

% inj_onI
thf(fact_164_inj__onI,axiom,
    ! [A4: set_nat,F2: nat > nat] :
      ( ! [X3: nat,Y3: nat] :
          ( ( member_nat @ X3 @ A4 )
         => ( ( member_nat @ Y3 @ A4 )
           => ( ( ( F2 @ X3 )
                = ( F2 @ Y3 ) )
             => ( X3 = Y3 ) ) ) )
     => ( inj_on_nat_nat @ F2 @ A4 ) ) ).

% inj_onI
thf(fact_165_inj__on__def,axiom,
    ( inj_on5538052773655684606at_nat
    = ( ^ [F3: nat > product_prod_nat_nat,A6: set_nat] :
        ! [X4: nat] :
          ( ( member_nat @ X4 @ A6 )
         => ! [Y4: nat] :
              ( ( member_nat @ Y4 @ A6 )
             => ( ( ( F3 @ X4 )
                  = ( F3 @ Y4 ) )
               => ( X4 = Y4 ) ) ) ) ) ) ).

% inj_on_def
thf(fact_166_inj__on__def,axiom,
    ( inj_on_nat_nat
    = ( ^ [F3: nat > nat,A6: set_nat] :
        ! [X4: nat] :
          ( ( member_nat @ X4 @ A6 )
         => ! [Y4: nat] :
              ( ( member_nat @ Y4 @ A6 )
             => ( ( ( F3 @ X4 )
                  = ( F3 @ Y4 ) )
               => ( X4 = Y4 ) ) ) ) ) ) ).

% inj_on_def
thf(fact_167_inj__on__cong,axiom,
    ! [A4: set_nat,F2: nat > product_prod_nat_nat,G: nat > product_prod_nat_nat] :
      ( ! [A2: nat] :
          ( ( member_nat @ A2 @ A4 )
         => ( ( F2 @ A2 )
            = ( G @ A2 ) ) )
     => ( ( inj_on5538052773655684606at_nat @ F2 @ A4 )
        = ( inj_on5538052773655684606at_nat @ G @ A4 ) ) ) ).

% inj_on_cong
thf(fact_168_inj__on__cong,axiom,
    ! [A4: set_nat,F2: nat > nat,G: nat > nat] :
      ( ! [A2: nat] :
          ( ( member_nat @ A2 @ A4 )
         => ( ( F2 @ A2 )
            = ( G @ A2 ) ) )
     => ( ( inj_on_nat_nat @ F2 @ A4 )
        = ( inj_on_nat_nat @ G @ A4 ) ) ) ).

% inj_on_cong
thf(fact_169_inj__on__eq__iff,axiom,
    ! [F2: nat > product_prod_nat_nat,A4: set_nat,X: nat,Y: nat] :
      ( ( inj_on5538052773655684606at_nat @ F2 @ A4 )
     => ( ( member_nat @ X @ A4 )
       => ( ( member_nat @ Y @ A4 )
         => ( ( ( F2 @ X )
              = ( F2 @ Y ) )
            = ( X = Y ) ) ) ) ) ).

% inj_on_eq_iff
thf(fact_170_inj__on__eq__iff,axiom,
    ! [F2: nat > nat,A4: set_nat,X: nat,Y: nat] :
      ( ( inj_on_nat_nat @ F2 @ A4 )
     => ( ( member_nat @ X @ A4 )
       => ( ( member_nat @ Y @ A4 )
         => ( ( ( F2 @ X )
              = ( F2 @ Y ) )
            = ( X = Y ) ) ) ) ) ).

% inj_on_eq_iff
thf(fact_171_inj__on__contraD,axiom,
    ! [F2: nat > product_prod_nat_nat,A4: set_nat,X: nat,Y: nat] :
      ( ( inj_on5538052773655684606at_nat @ F2 @ A4 )
     => ( ( X != Y )
       => ( ( member_nat @ X @ A4 )
         => ( ( member_nat @ Y @ A4 )
           => ( ( F2 @ X )
             != ( F2 @ Y ) ) ) ) ) ) ).

% inj_on_contraD
thf(fact_172_inj__on__contraD,axiom,
    ! [F2: nat > nat,A4: set_nat,X: nat,Y: nat] :
      ( ( inj_on_nat_nat @ F2 @ A4 )
     => ( ( X != Y )
       => ( ( member_nat @ X @ A4 )
         => ( ( member_nat @ Y @ A4 )
           => ( ( F2 @ X )
             != ( F2 @ Y ) ) ) ) ) ) ).

% inj_on_contraD
thf(fact_173_inj__on__inverseI,axiom,
    ! [A4: set_nat,G: product_prod_nat_nat > nat,F2: nat > product_prod_nat_nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A4 )
         => ( ( G @ ( F2 @ X3 ) )
            = X3 ) )
     => ( inj_on5538052773655684606at_nat @ F2 @ A4 ) ) ).

% inj_on_inverseI
thf(fact_174_inj__on__inverseI,axiom,
    ! [A4: set_nat,G: nat > nat,F2: nat > nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A4 )
         => ( ( G @ ( F2 @ X3 ) )
            = X3 ) )
     => ( inj_on_nat_nat @ F2 @ A4 ) ) ).

% inj_on_inverseI
thf(fact_175_inj__on__id2,axiom,
    ! [A4: set_nat] :
      ( inj_on_nat_nat
      @ ^ [X4: nat] : X4
      @ A4 ) ).

% inj_on_id2
thf(fact_176_linorder__inj__onI_H,axiom,
    ! [A4: set_nat,F2: nat > product_prod_nat_nat] :
      ( ! [I2: nat,J2: nat] :
          ( ( member_nat @ I2 @ A4 )
         => ( ( member_nat @ J2 @ A4 )
           => ( ( ord_less_nat @ I2 @ J2 )
             => ( ( F2 @ I2 )
               != ( F2 @ J2 ) ) ) ) )
     => ( inj_on5538052773655684606at_nat @ F2 @ A4 ) ) ).

% linorder_inj_onI'
thf(fact_177_linorder__inj__onI_H,axiom,
    ! [A4: set_nat,F2: nat > nat] :
      ( ! [I2: nat,J2: nat] :
          ( ( member_nat @ I2 @ A4 )
         => ( ( member_nat @ J2 @ A4 )
           => ( ( ord_less_nat @ I2 @ J2 )
             => ( ( F2 @ I2 )
               != ( F2 @ J2 ) ) ) ) )
     => ( inj_on_nat_nat @ F2 @ A4 ) ) ).

% linorder_inj_onI'
thf(fact_178_rel__simps_I71_J,axiom,
    ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).

% rel_simps(71)
thf(fact_179_nth__zip,axiom,
    ! [I3: nat,Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat] :
      ( ( ord_less_nat @ I3 @ ( size_s5460976970255530739at_nat @ Xs ) )
     => ( ( ord_less_nat @ I3 @ ( size_s5460976970255530739at_nat @ Ys ) )
       => ( ( nth_Pr6744343527793145070at_nat @ ( zip_Pr4664179122662387191at_nat @ Xs @ Ys ) @ I3 )
          = ( produc6161850002892822231at_nat @ ( nth_Pr7617993195940197384at_nat @ Xs @ I3 ) @ ( nth_Pr7617993195940197384at_nat @ Ys @ I3 ) ) ) ) ) ).

% nth_zip
thf(fact_180_nth__zip,axiom,
    ! [I3: nat,Xs: list_s1210847774152347623at_nat,Ys: list_s1210847774152347623at_nat] :
      ( ( ord_less_nat @ I3 @ ( size_s8736152011456118867at_nat @ Xs ) )
     => ( ( ord_less_nat @ I3 @ ( size_s8736152011456118867at_nat @ Ys ) )
       => ( ( nth_Pr1239524528086697198at_nat @ ( zip_se5600341670672612855at_nat @ Xs @ Ys ) @ I3 )
          = ( produc2922128104949294807at_nat @ ( nth_se5679702624988424552at_nat @ Xs @ I3 ) @ ( nth_se5679702624988424552at_nat @ Ys @ I3 ) ) ) ) ) ).

% nth_zip
thf(fact_181_nth__zip,axiom,
    ! [I3: nat,Xs: list_s4242580248156217677at_nat,Ys: list_s4242580248156217677at_nat] :
      ( ( ord_less_nat @ I3 @ ( size_s3186697827981139897at_nat @ Xs ) )
     => ( ( ord_less_nat @ I3 @ ( size_s3186697827981139897at_nat @ Ys ) )
       => ( ( nth_Pr4310225385800671342at_nat @ ( zip_se4605334178636606071at_nat @ Xs @ Ys ) @ I3 )
          = ( produc9060074326276436823at_nat @ ( nth_se3273722028368051150at_nat @ Xs @ I3 ) @ ( nth_se3273722028368051150at_nat @ Ys @ I3 ) ) ) ) ) ).

% nth_zip
thf(fact_182_nth__zip,axiom,
    ! [I3: nat,Xs: list_nat,Ys: list_nat] :
      ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
     => ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys ) )
       => ( ( nth_Pr7617993195940197384at_nat @ ( zip_nat_nat @ Xs @ Ys ) @ I3 )
          = ( product_Pair_nat_nat @ ( nth_nat @ Xs @ I3 ) @ ( nth_nat @ Ys @ I3 ) ) ) ) ) ).

% nth_zip
thf(fact_183_nth__drop,axiom,
    ! [N: nat,Xs: list_nat,I3: nat] :
      ( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_nat @ ( drop_nat @ N @ Xs ) @ I3 )
        = ( nth_nat @ Xs @ ( plus_plus_nat @ N @ I3 ) ) ) ) ).

% nth_drop
thf(fact_184_gen__length__def,axiom,
    ( gen_length_nat
    = ( ^ [N3: nat,Xs2: list_nat] : ( plus_plus_nat @ N3 @ ( size_size_list_nat @ Xs2 ) ) ) ) ).

% gen_length_def
thf(fact_185_inj__on__fixpoints,axiom,
    ! [F2: nat > nat] :
      ( inj_on_nat_nat @ F2
      @ ( collect_nat
        @ ^ [X4: nat] :
            ( ( F2 @ X4 )
            = X4 ) ) ) ).

% inj_on_fixpoints
thf(fact_186_inj__on__fixpoints,axiom,
    ! [F2: product_prod_nat_nat > product_prod_nat_nat] :
      ( inj_on8969904277767023793at_nat @ F2
      @ ( collec3392354462482085612at_nat
        @ ^ [X4: product_prod_nat_nat] :
            ( ( F2 @ X4 )
            = X4 ) ) ) ).

% inj_on_fixpoints
thf(fact_187_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_188_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_189_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_190_Nil__eq__zip__iff,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( nil_Pr5478986624290739719at_nat
        = ( zip_nat_nat @ Xs @ Ys ) )
      = ( ( Xs = nil_nat )
        | ( Ys = nil_nat ) ) ) ).

% Nil_eq_zip_iff
thf(fact_191_zip__eq__Nil__iff,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( zip_nat_nat @ Xs @ Ys )
        = nil_Pr5478986624290739719at_nat )
      = ( ( Xs = nil_nat )
        | ( Ys = nil_nat ) ) ) ).

% zip_eq_Nil_iff
thf(fact_192_drop__eq__Nil2,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( nil_nat
        = ( drop_nat @ N @ Xs ) )
      = ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ).

% drop_eq_Nil2
thf(fact_193_drop__eq__Nil,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( ( drop_nat @ N @ Xs )
        = nil_nat )
      = ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ).

% drop_eq_Nil
thf(fact_194_drop__all,axiom,
    ! [Xs: list_nat,N: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N )
     => ( ( drop_nat @ N @ Xs )
        = nil_nat ) ) ).

% drop_all
thf(fact_195_bounded__Max__nat,axiom,
    ! [P: nat > $o,X: nat,M4: nat] :
      ( ( P @ X )
     => ( ! [X3: nat] :
            ( ( P @ X3 )
           => ( ord_less_eq_nat @ X3 @ M4 ) )
       => ~ ! [M2: nat] :
              ( ( P @ M2 )
             => ~ ! [X6: nat] :
                    ( ( P @ X6 )
                   => ( ord_less_eq_nat @ X6 @ M2 ) ) ) ) ) ).

% bounded_Max_nat
thf(fact_196_Nat_Oex__has__greatest__nat,axiom,
    ! [P: nat > $o,K: nat,B: nat] :
      ( ( P @ K )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B ) )
       => ? [X3: nat] :
            ( ( P @ X3 )
            & ! [Y5: nat] :
                ( ( P @ Y5 )
               => ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_197_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_198_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_199_eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( M = N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% eq_imp_le
thf(fact_200_le__trans,axiom,
    ! [I3: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I3 @ J )
     => ( ( ord_less_eq_nat @ J @ K )
       => ( ord_less_eq_nat @ I3 @ K ) ) ) ).

% le_trans
thf(fact_201_le__refl,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).

% le_refl
thf(fact_202_le__numeral__extra_I4_J,axiom,
    ord_less_eq_nat @ one_one_nat @ one_one_nat ).

% le_numeral_extra(4)
thf(fact_203_drop__zip,axiom,
    ! [N: nat,Xs: list_nat,Ys: list_nat] :
      ( ( drop_P8868858903918902087at_nat @ N @ ( zip_nat_nat @ Xs @ Ys ) )
      = ( zip_nat_nat @ ( drop_nat @ N @ Xs ) @ ( drop_nat @ N @ Ys ) ) ) ).

% drop_zip
thf(fact_204_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_205_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_206_le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [A3: nat,B3: nat] :
        ? [C4: nat] :
          ( B3
          = ( plus_plus_nat @ A3 @ C4 ) ) ) ) ).

% le_iff_add
thf(fact_207_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_208_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_209_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_210_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_211_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I3: nat,J: nat,K: nat,L2: nat] :
      ( ( ( ord_less_eq_nat @ I3 @ J )
        & ( ord_less_eq_nat @ K @ L2 ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_212_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I3: nat,J: nat,K: nat,L2: nat] :
      ( ( ( I3 = J )
        & ( ord_less_eq_nat @ K @ L2 ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_213_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I3: nat,J: nat,K: nat,L2: nat] :
      ( ( ( ord_less_eq_nat @ I3 @ J )
        & ( K = L2 ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_214_less__mono__imp__le__mono,axiom,
    ! [F2: nat > nat,I3: nat,J: nat] :
      ( ! [I2: nat,J2: nat] :
          ( ( ord_less_nat @ I2 @ J2 )
         => ( ord_less_nat @ ( F2 @ I2 ) @ ( F2 @ J2 ) ) )
     => ( ( ord_less_eq_nat @ I3 @ J )
       => ( ord_less_eq_nat @ ( F2 @ I3 ) @ ( F2 @ J ) ) ) ) ).

% less_mono_imp_le_mono
thf(fact_215_le__neq__implies__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( M != N )
       => ( ord_less_nat @ M @ N ) ) ) ).

% le_neq_implies_less
thf(fact_216_less__or__eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ( ord_less_nat @ M @ N )
        | ( M = N ) )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% less_or_eq_imp_le
thf(fact_217_le__eq__less__or__eq,axiom,
    ( ord_less_eq_nat
    = ( ^ [M5: nat,N3: nat] :
          ( ( ord_less_nat @ M5 @ N3 )
          | ( M5 = N3 ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_218_less__imp__le__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% less_imp_le_nat
thf(fact_219_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M5: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M5 @ N3 )
          & ( M5 != N3 ) ) ) ) ).

% nat_less_le
thf(fact_220_nat__le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [M5: nat,N3: nat] :
        ? [K2: nat] :
          ( N3
          = ( plus_plus_nat @ M5 @ K2 ) ) ) ) ).

% nat_le_iff_add
thf(fact_221_trans__le__add2,axiom,
    ! [I3: nat,J: nat,M: nat] :
      ( ( ord_less_eq_nat @ I3 @ J )
     => ( ord_less_eq_nat @ I3 @ ( plus_plus_nat @ M @ J ) ) ) ).

% trans_le_add2
thf(fact_222_trans__le__add1,axiom,
    ! [I3: nat,J: nat,M: nat] :
      ( ( ord_less_eq_nat @ I3 @ J )
     => ( ord_less_eq_nat @ I3 @ ( plus_plus_nat @ J @ M ) ) ) ).

% trans_le_add1
thf(fact_223_add__le__mono1,axiom,
    ! [I3: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I3 @ J )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).

% add_le_mono1
thf(fact_224_add__le__mono,axiom,
    ! [I3: nat,J: nat,K: nat,L2: nat] :
      ( ( ord_less_eq_nat @ I3 @ J )
     => ( ( ord_less_eq_nat @ K @ L2 )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ) ).

% add_le_mono
thf(fact_225_le__Suc__ex,axiom,
    ! [K: nat,L2: nat] :
      ( ( ord_less_eq_nat @ K @ L2 )
     => ? [N2: nat] :
          ( L2
          = ( plus_plus_nat @ K @ N2 ) ) ) ).

% le_Suc_ex
thf(fact_226_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_227_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_228_le__add2,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).

% le_add2
thf(fact_229_le__add1,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).

% le_add1
thf(fact_230_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_231_add__less__le__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_232_add__le__less__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_233_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I3: nat,J: nat,K: nat,L2: nat] :
      ( ( ( ord_less_nat @ I3 @ J )
        & ( ord_less_eq_nat @ K @ L2 ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_234_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I3: nat,J: nat,K: nat,L2: nat] :
      ( ( ( ord_less_eq_nat @ I3 @ J )
        & ( ord_less_nat @ K @ L2 ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_235_linorder__inj__onI,axiom,
    ! [A4: set_nat,F2: nat > product_prod_nat_nat] :
      ( ! [X3: nat,Y3: nat] :
          ( ( ord_less_nat @ X3 @ Y3 )
         => ( ( member_nat @ X3 @ A4 )
           => ( ( member_nat @ Y3 @ A4 )
             => ( ( F2 @ X3 )
               != ( F2 @ Y3 ) ) ) ) )
     => ( ! [X3: nat,Y3: nat] :
            ( ( member_nat @ X3 @ A4 )
           => ( ( member_nat @ Y3 @ A4 )
             => ( ( ord_less_eq_nat @ X3 @ Y3 )
                | ( ord_less_eq_nat @ Y3 @ X3 ) ) ) )
       => ( inj_on5538052773655684606at_nat @ F2 @ A4 ) ) ) ).

% linorder_inj_onI
thf(fact_236_linorder__inj__onI,axiom,
    ! [A4: set_nat,F2: nat > nat] :
      ( ! [X3: nat,Y3: nat] :
          ( ( ord_less_nat @ X3 @ Y3 )
         => ( ( member_nat @ X3 @ A4 )
           => ( ( member_nat @ Y3 @ A4 )
             => ( ( F2 @ X3 )
               != ( F2 @ Y3 ) ) ) ) )
     => ( ! [X3: nat,Y3: nat] :
            ( ( member_nat @ X3 @ A4 )
           => ( ( member_nat @ Y3 @ A4 )
             => ( ( ord_less_eq_nat @ X3 @ Y3 )
                | ( ord_less_eq_nat @ Y3 @ X3 ) ) ) )
       => ( inj_on_nat_nat @ F2 @ A4 ) ) ) ).

% linorder_inj_onI
thf(fact_237_mono__nat__linear__lb,axiom,
    ! [F2: nat > nat,M: nat,K: nat] :
      ( ! [M2: nat,N2: nat] :
          ( ( ord_less_nat @ M2 @ N2 )
         => ( ord_less_nat @ ( F2 @ M2 ) @ ( F2 @ N2 ) ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F2 @ M ) @ K ) @ ( F2 @ ( plus_plus_nat @ M @ K ) ) ) ) ).

% mono_nat_linear_lb
thf(fact_238_count__le__length,axiom,
    ! [Xs: list_nat,X: nat] : ( ord_less_eq_nat @ ( count_list_nat @ Xs @ X ) @ ( size_size_list_nat @ Xs ) ) ).

% count_le_length
thf(fact_239_gen__length__code_I1_J,axiom,
    ! [N: nat] :
      ( ( gen_length_nat @ N @ nil_nat )
      = N ) ).

% gen_length_code(1)
thf(fact_240_gcd_Ocases,axiom,
    ! [X: product_prod_nat_nat] :
      ~ ! [A2: nat,B2: nat] :
          ( X
         != ( product_Pair_nat_nat @ A2 @ B2 ) ) ).

% gcd.cases
thf(fact_241_Pair__le,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,C: product_prod_nat_nat,D: product_prod_nat_nat] :
      ( ( ord_le6722121967309221895at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( produc6161850002892822231at_nat @ C @ D ) )
      = ( ( ord_le8460144461188290721at_nat @ A @ C )
        & ( ord_le8460144461188290721at_nat @ B @ D ) ) ) ).

% Pair_le
thf(fact_242_Pair__le,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,D: set_Pr1261947904930325089at_nat] :
      ( ( ord_le8744046415670163463at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ ( produc2922128104949294807at_nat @ C @ D ) )
      = ( ( ord_le3146513528884898305at_nat @ A @ C )
        & ( ord_le3146513528884898305at_nat @ B @ D ) ) ) ).

% Pair_le
thf(fact_243_Pair__le,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat,D: set_Pr4329608150637261639at_nat] :
      ( ( ord_le6901975178745540743at_nat @ ( produc9060074326276436823at_nat @ A @ B ) @ ( produc9060074326276436823at_nat @ C @ D ) )
      = ( ( ord_le1268244103169919719at_nat @ A @ C )
        & ( ord_le1268244103169919719at_nat @ B @ D ) ) ) ).

% Pair_le
thf(fact_244_Pair__le,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_le8460144461188290721at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( product_Pair_nat_nat @ C @ D ) )
      = ( ( ord_less_eq_nat @ A @ C )
        & ( ord_less_eq_nat @ B @ D ) ) ) ).

% Pair_le
thf(fact_245_k__ge__1,axiom,
    ord_less_eq_nat @ one_one_nat @ k ).

% k_ge_1
thf(fact_246_order_Orefl,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% order.refl
thf(fact_247_order__refl,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).

% order_refl
thf(fact_248_nat__descend__induct,axiom,
    ! [N: nat,P: nat > $o,M: nat] :
      ( ! [K3: nat] :
          ( ( ord_less_nat @ N @ K3 )
         => ( P @ K3 ) )
     => ( ! [K3: nat] :
            ( ( ord_less_eq_nat @ K3 @ N )
           => ( ! [I4: nat] :
                  ( ( ord_less_nat @ K3 @ I4 )
                 => ( P @ I4 ) )
             => ( P @ K3 ) ) )
       => ( P @ M ) ) ) ).

% nat_descend_induct
thf(fact_249_Pair__mono,axiom,
    ! [X: product_prod_nat_nat,X7: product_prod_nat_nat,Y: product_prod_nat_nat,Y6: product_prod_nat_nat] :
      ( ( ord_le8460144461188290721at_nat @ X @ X7 )
     => ( ( ord_le8460144461188290721at_nat @ Y @ Y6 )
       => ( ord_le6722121967309221895at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( produc6161850002892822231at_nat @ X7 @ Y6 ) ) ) ) ).

% Pair_mono
thf(fact_250_Pair__mono,axiom,
    ! [X: set_Pr1261947904930325089at_nat,X7: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Y6: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ X @ X7 )
     => ( ( ord_le3146513528884898305at_nat @ Y @ Y6 )
       => ( ord_le8744046415670163463at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ ( produc2922128104949294807at_nat @ X7 @ Y6 ) ) ) ) ).

% Pair_mono
thf(fact_251_Pair__mono,axiom,
    ! [X: set_Pr4329608150637261639at_nat,X7: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Y6: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ X @ X7 )
     => ( ( ord_le1268244103169919719at_nat @ Y @ Y6 )
       => ( ord_le6901975178745540743at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ ( produc9060074326276436823at_nat @ X7 @ Y6 ) ) ) ) ).

% Pair_mono
thf(fact_252_Pair__mono,axiom,
    ! [X: nat,X7: nat,Y: nat,Y6: nat] :
      ( ( ord_less_eq_nat @ X @ X7 )
     => ( ( ord_less_eq_nat @ Y @ Y6 )
       => ( ord_le8460144461188290721at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( product_Pair_nat_nat @ X7 @ Y6 ) ) ) ) ).

% Pair_mono
thf(fact_253_set__plus__mono2,axiom,
    ! [C2: set_Pr4329608150637261639at_nat,D2: set_Pr4329608150637261639at_nat,E: set_Pr4329608150637261639at_nat,F4: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ C2 @ D2 )
     => ( ( ord_le1268244103169919719at_nat @ E @ F4 )
       => ( ord_le1268244103169919719at_nat @ ( plus_p5270993086169515006at_nat @ C2 @ E ) @ ( plus_p5270993086169515006at_nat @ D2 @ F4 ) ) ) ) ).

% set_plus_mono2
thf(fact_254_set__plus__mono2,axiom,
    ! [C2: set_Pr1261947904930325089at_nat,D2: set_Pr1261947904930325089at_nat,E: set_Pr1261947904930325089at_nat,F4: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ C2 @ D2 )
     => ( ( ord_le3146513528884898305at_nat @ E @ F4 )
       => ( ord_le3146513528884898305at_nat @ ( plus_p2407200367410362776at_nat @ C2 @ E ) @ ( plus_p2407200367410362776at_nat @ D2 @ F4 ) ) ) ) ).

% set_plus_mono2
thf(fact_255_inj__on__subset,axiom,
    ! [F2: nat > product_prod_nat_nat,A4: set_nat,B4: set_nat] :
      ( ( inj_on5538052773655684606at_nat @ F2 @ A4 )
     => ( ( ord_less_eq_set_nat @ B4 @ A4 )
       => ( inj_on5538052773655684606at_nat @ F2 @ B4 ) ) ) ).

% inj_on_subset
thf(fact_256_inj__on__subset,axiom,
    ! [F2: nat > nat,A4: set_nat,B4: set_nat] :
      ( ( inj_on_nat_nat @ F2 @ A4 )
     => ( ( ord_less_eq_set_nat @ B4 @ A4 )
       => ( inj_on_nat_nat @ F2 @ B4 ) ) ) ).

% inj_on_subset
thf(fact_257_subset__inj__on,axiom,
    ! [F2: nat > product_prod_nat_nat,B4: set_nat,A4: set_nat] :
      ( ( inj_on5538052773655684606at_nat @ F2 @ B4 )
     => ( ( ord_less_eq_set_nat @ A4 @ B4 )
       => ( inj_on5538052773655684606at_nat @ F2 @ A4 ) ) ) ).

% subset_inj_on
thf(fact_258_subset__inj__on,axiom,
    ! [F2: nat > nat,B4: set_nat,A4: set_nat] :
      ( ( inj_on_nat_nat @ F2 @ B4 )
     => ( ( ord_less_eq_set_nat @ A4 @ B4 )
       => ( inj_on_nat_nat @ F2 @ A4 ) ) ) ).

% subset_inj_on
thf(fact_259_order__antisym__conv,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( ord_less_eq_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_260_linorder__le__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_eq_nat @ X @ Y )
     => ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_le_cases
thf(fact_261_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F2: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F2 @ B )
          = C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_262_ord__eq__le__subst,axiom,
    ! [A: nat,F2: nat > nat,B: nat,C: nat] :
      ( ( A
        = ( F2 @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_263_linorder__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
      | ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_linear
thf(fact_264_order__eq__refl,axiom,
    ! [X: nat,Y: nat] :
      ( ( X = Y )
     => ( ord_less_eq_nat @ X @ Y ) ) ).

% order_eq_refl
thf(fact_265_order__subst2,axiom,
    ! [A: nat,B: nat,F2: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ ( F2 @ B ) @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_266_order__subst1,axiom,
    ! [A: nat,F2: nat > nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).

% order_subst1
thf(fact_267_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y2: nat,Z2: nat] : ( Y2 = Z2 ) )
    = ( ^ [A3: nat,B3: nat] :
          ( ( ord_less_eq_nat @ A3 @ B3 )
          & ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_268_antisym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_269_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_270_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_271_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y2: nat,Z2: nat] : ( Y2 = Z2 ) )
    = ( ^ [A3: nat,B3: nat] :
          ( ( ord_less_eq_nat @ B3 @ A3 )
          & ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_272_linorder__wlog,axiom,
    ! [P: nat > nat > $o,A: nat,B: nat] :
      ( ! [A2: nat,B2: nat] :
          ( ( ord_less_eq_nat @ A2 @ B2 )
         => ( P @ A2 @ B2 ) )
     => ( ! [A2: nat,B2: nat] :
            ( ( P @ B2 @ A2 )
           => ( P @ A2 @ B2 ) )
       => ( P @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_273_order__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z )
       => ( ord_less_eq_nat @ X @ Z ) ) ) ).

% order_trans
thf(fact_274_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_275_order__antisym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_276_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_277_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_278_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y2: nat,Z2: nat] : ( Y2 = Z2 ) )
    = ( ^ [X4: nat,Y4: nat] :
          ( ( ord_less_eq_nat @ X4 @ Y4 )
          & ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_279_le__cases3,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ( ord_less_eq_nat @ X @ Y )
       => ~ ( ord_less_eq_nat @ Y @ Z ) )
     => ( ( ( ord_less_eq_nat @ Y @ X )
         => ~ ( ord_less_eq_nat @ X @ Z ) )
       => ( ( ( ord_less_eq_nat @ X @ Z )
           => ~ ( ord_less_eq_nat @ Z @ Y ) )
         => ( ( ( ord_less_eq_nat @ Z @ Y )
             => ~ ( ord_less_eq_nat @ Y @ X ) )
           => ( ( ( ord_less_eq_nat @ Y @ Z )
               => ~ ( ord_less_eq_nat @ Z @ X ) )
             => ~ ( ( ord_less_eq_nat @ Z @ X )
                 => ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_280_nle__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_eq_nat @ A @ B ) )
      = ( ( ord_less_eq_nat @ B @ A )
        & ( B != A ) ) ) ).

% nle_le
thf(fact_281_order__trans__rules_I28_J,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order_trans_rules(28)
thf(fact_282_order__trans__rules_I27_J,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( B = C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order_trans_rules(27)
thf(fact_283_order__trans__rules_I20_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order_trans_rules(20)
thf(fact_284_order__trans__rules_I19_J,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ Z )
       => ( ord_less_nat @ X @ Z ) ) ) ).

% order_trans_rules(19)
thf(fact_285_order__trans__rules_I12_J,axiom,
    ! [A: nat,F2: nat > nat,B: nat,C: nat] :
      ( ( A
        = ( F2 @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F2 @ C ) ) ) ) ) ).

% order_trans_rules(12)
thf(fact_286_order__trans__rules_I11_J,axiom,
    ! [A: nat,B: nat,F2: nat > nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ( F2 @ B )
          = C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_nat @ ( F2 @ A ) @ C ) ) ) ) ).

% order_trans_rules(11)
thf(fact_287_order__trans__rules_I2_J,axiom,
    ! [A: nat,F2: nat > nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ ( F2 @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F2 @ C ) ) ) ) ) ).

% order_trans_rules(2)
thf(fact_288_order__trans__rules_I1_J,axiom,
    ! [A: nat,B: nat,F2: nat > nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ ( F2 @ B ) @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_nat @ ( F2 @ A ) @ C ) ) ) ) ).

% order_trans_rules(1)
thf(fact_289_gt__ex,axiom,
    ! [X: nat] :
    ? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).

% gt_ex
thf(fact_290_neqE,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ ( ord_less_nat @ X @ Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% neqE
thf(fact_291_neq__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
      = ( ( ord_less_nat @ X @ Y )
        | ( ord_less_nat @ Y @ X ) ) ) ).

% neq_iff
thf(fact_292_less__imp__neq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_293_less__asym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% less_asym
thf(fact_294_order_Oasym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order.asym
thf(fact_295_less__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
      | ( X = Y )
      | ( ord_less_nat @ Y @ X ) ) ).

% less_linear
thf(fact_296_less__irrefl,axiom,
    ! [X: nat] :
      ~ ( ord_less_nat @ X @ X ) ).

% less_irrefl
thf(fact_297_less__imp__not__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_not_eq
thf(fact_298_less__not__sym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% less_not_sym
thf(fact_299_order_Oirrefl,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% order.irrefl
thf(fact_300_less__induct,axiom,
    ! [P: nat > $o,A: nat] :
      ( ! [X3: nat] :
          ( ! [Y5: nat] :
              ( ( ord_less_nat @ Y5 @ X3 )
             => ( P @ Y5 ) )
         => ( P @ X3 ) )
     => ( P @ A ) ) ).

% less_induct
thf(fact_301_antisym__conv3,axiom,
    ! [Y: nat,X: nat] :
      ( ~ ( ord_less_nat @ Y @ X )
     => ( ( ~ ( ord_less_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_302_less__imp__not__eq2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( Y != X ) ) ).

% less_imp_not_eq2
thf(fact_303_less__imp__triv,axiom,
    ! [X: nat,Y: nat,P: $o] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ X )
       => P ) ) ).

% less_imp_triv
thf(fact_304_linorder__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_305_dual__order_Oasym,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ~ ( ord_less_nat @ A @ B ) ) ).

% dual_order.asym
thf(fact_306_less__imp__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% less_imp_not_less
thf(fact_307_exists__least__iff,axiom,
    ( ( ^ [P3: nat > $o] :
        ? [X8: nat] : ( P3 @ X8 ) )
    = ( ^ [P4: nat > $o] :
        ? [N3: nat] :
          ( ( P4 @ N3 )
          & ! [M5: nat] :
              ( ( ord_less_nat @ M5 @ N3 )
             => ~ ( P4 @ M5 ) ) ) ) ) ).

% exists_least_iff
thf(fact_308_linorder__less__wlog,axiom,
    ! [P: nat > nat > $o,A: nat,B: nat] :
      ( ! [A2: nat,B2: nat] :
          ( ( ord_less_nat @ A2 @ B2 )
         => ( P @ A2 @ B2 ) )
     => ( ! [A2: nat] : ( P @ A2 @ A2 )
       => ( ! [A2: nat,B2: nat] :
              ( ( P @ B2 @ A2 )
             => ( P @ A2 @ B2 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_309_order_Ostrict__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_310_not__less__iff__gr__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X @ Y ) )
      = ( ( ord_less_nat @ Y @ X )
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_311_dual__order_Ostrict__trans,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_less_nat @ C @ B )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_312_order_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_313_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_314_order__trans__rules_I22_J,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z )
       => ( ord_less_nat @ X @ Z ) ) ) ).

% order_trans_rules(22)
thf(fact_315_order__trans__rules_I21_J,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ Z )
       => ( ord_less_nat @ X @ Z ) ) ) ).

% order_trans_rules(21)
thf(fact_316_order__trans__rules_I18_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( A != B )
       => ( ord_less_nat @ A @ B ) ) ) ).

% order_trans_rules(18)
thf(fact_317_order__trans__rules_I17_J,axiom,
    ! [A: nat,B: nat] :
      ( ( A != B )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ord_less_nat @ A @ B ) ) ) ).

% order_trans_rules(17)
thf(fact_318_order__trans__rules_I6_J,axiom,
    ! [A: nat,F2: nat > nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ ( F2 @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F2 @ C ) ) ) ) ) ).

% order_trans_rules(6)
thf(fact_319_order__trans__rules_I5_J,axiom,
    ! [A: nat,B: nat,F2: nat > nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ ( F2 @ B ) @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_nat @ ( F2 @ A ) @ C ) ) ) ) ).

% order_trans_rules(5)
thf(fact_320_order__trans__rules_I4_J,axiom,
    ! [A: nat,F2: nat > nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F2 @ C ) ) ) ) ) ).

% order_trans_rules(4)
thf(fact_321_order__trans__rules_I3_J,axiom,
    ! [A: nat,B: nat,F2: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ ( F2 @ B ) @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_nat @ ( F2 @ A ) @ C ) ) ) ) ).

% order_trans_rules(3)
thf(fact_322_leD,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ~ ( ord_less_nat @ X @ Y ) ) ).

% leD
thf(fact_323_leI,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ord_less_eq_nat @ Y @ X ) ) ).

% leI
thf(fact_324_le__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [X4: nat,Y4: nat] :
          ( ( ord_less_nat @ X4 @ Y4 )
          | ( X4 = Y4 ) ) ) ) ).

% le_less
thf(fact_325_less__le,axiom,
    ( ord_less_nat
    = ( ^ [X4: nat,Y4: nat] :
          ( ( ord_less_eq_nat @ X4 @ Y4 )
          & ( X4 != Y4 ) ) ) ) ).

% less_le
thf(fact_326_nless__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_nat @ A @ B ) )
      = ( ~ ( ord_less_eq_nat @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_327_not__le,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_eq_nat @ X @ Y ) )
      = ( ord_less_nat @ Y @ X ) ) ).

% not_le
thf(fact_328_not__less,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X @ Y ) )
      = ( ord_less_eq_nat @ Y @ X ) ) ).

% not_less
thf(fact_329_antisym__conv1,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_330_antisym__conv2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ~ ( ord_less_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_331_less__imp__le,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ord_less_eq_nat @ X @ Y ) ) ).

% less_imp_le
thf(fact_332_le__less__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
      | ( ord_less_nat @ Y @ X ) ) ).

% le_less_linear
thf(fact_333_le__imp__less__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_nat @ X @ Y )
        | ( X = Y ) ) ) ).

% le_imp_less_or_eq
thf(fact_334_less__le__not__le,axiom,
    ( ord_less_nat
    = ( ^ [X4: nat,Y4: nat] :
          ( ( ord_less_eq_nat @ X4 @ Y4 )
          & ~ ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ).

% less_le_not_le
thf(fact_335_not__le__imp__less,axiom,
    ! [Y: nat,X: nat] :
      ( ~ ( ord_less_eq_nat @ Y @ X )
     => ( ord_less_nat @ X @ Y ) ) ).

% not_le_imp_less
thf(fact_336_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [A3: nat,B3: nat] :
          ( ( ord_less_nat @ A3 @ B3 )
          | ( A3 = B3 ) ) ) ) ).

% order.order_iff_strict
thf(fact_337_order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [A3: nat,B3: nat] :
          ( ( ord_less_eq_nat @ A3 @ B3 )
          & ( A3 != B3 ) ) ) ) ).

% order.strict_iff_order
thf(fact_338_order_Ostrict__trans1,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_339_order_Ostrict__trans2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_340_order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [A3: nat,B3: nat] :
          ( ( ord_less_eq_nat @ A3 @ B3 )
          & ~ ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).

% order.strict_iff_not
thf(fact_341_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [B3: nat,A3: nat] :
          ( ( ord_less_nat @ B3 @ A3 )
          | ( A3 = B3 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_342_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [B3: nat,A3: nat] :
          ( ( ord_less_eq_nat @ B3 @ A3 )
          & ( A3 != B3 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_343_dual__order_Ostrict__trans1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_nat @ C @ B )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_344_dual__order_Ostrict__trans2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C @ B )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_345_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [B3: nat,A3: nat] :
          ( ( ord_less_eq_nat @ B3 @ A3 )
          & ~ ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_346_order_Ostrict__implies__order,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_347_dual__order_Ostrict__implies__order,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ord_less_eq_nat @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_348_le__Nil,axiom,
    ! [X: list_nat] :
      ( ( ord_less_eq_list_nat @ X @ nil_nat )
      = ( X = nil_nat ) ) ).

% le_Nil
thf(fact_349_subsetI,axiom,
    ! [A4: set_Pr8693737435421807431at_nat,B4: set_Pr8693737435421807431at_nat] :
      ( ! [X3: produc859450856879609959at_nat] :
          ( ( member8206827879206165904at_nat @ X3 @ A4 )
         => ( member8206827879206165904at_nat @ X3 @ B4 ) )
     => ( ord_le3000389064537975527at_nat @ A4 @ B4 ) ) ).

% subsetI
thf(fact_350_subsetI,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ! [X3: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X3 @ A4 )
         => ( member8440522571783428010at_nat @ X3 @ B4 ) )
     => ( ord_le3146513528884898305at_nat @ A4 @ B4 ) ) ).

% subsetI
thf(fact_351_subsetI,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ! [X3: produc3843707927480180839at_nat] :
          ( ( member8757157785044589968at_nat @ X3 @ A4 )
         => ( member8757157785044589968at_nat @ X3 @ B4 ) )
     => ( ord_le1268244103169919719at_nat @ A4 @ B4 ) ) ).

% subsetI
thf(fact_352_subsetI,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A4 )
         => ( member_nat @ X3 @ B4 ) )
     => ( ord_less_eq_set_nat @ A4 @ B4 ) ) ).

% subsetI
thf(fact_353_Set_Obasic__monos_I7_J,axiom,
    ! [A4: set_Pr8693737435421807431at_nat,B4: set_Pr8693737435421807431at_nat,X: produc859450856879609959at_nat] :
      ( ( ord_le3000389064537975527at_nat @ A4 @ B4 )
     => ( ( member8206827879206165904at_nat @ X @ A4 )
       => ( member8206827879206165904at_nat @ X @ B4 ) ) ) ).

% Set.basic_monos(7)
thf(fact_354_Set_Obasic__monos_I7_J,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( ( ord_le3146513528884898305at_nat @ A4 @ B4 )
     => ( ( member8440522571783428010at_nat @ X @ A4 )
       => ( member8440522571783428010at_nat @ X @ B4 ) ) ) ).

% Set.basic_monos(7)
thf(fact_355_Set_Obasic__monos_I7_J,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A4 @ B4 )
     => ( ( member8757157785044589968at_nat @ X @ A4 )
       => ( member8757157785044589968at_nat @ X @ B4 ) ) ) ).

% Set.basic_monos(7)
thf(fact_356_Set_Obasic__monos_I7_J,axiom,
    ! [A4: set_nat,B4: set_nat,X: nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B4 )
     => ( ( member_nat @ X @ A4 )
       => ( member_nat @ X @ B4 ) ) ) ).

% Set.basic_monos(7)
thf(fact_357_Set_Obasic__monos_I6_J,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ! [X3: nat] :
          ( ( P @ X3 )
         => ( Q @ X3 ) )
     => ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).

% Set.basic_monos(6)
thf(fact_358_Set_Obasic__monos_I6_J,axiom,
    ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ! [X3: product_prod_nat_nat] :
          ( ( P @ X3 )
         => ( Q @ X3 ) )
     => ( ord_le3146513528884898305at_nat @ ( collec3392354462482085612at_nat @ P ) @ ( collec3392354462482085612at_nat @ Q ) ) ) ).

% Set.basic_monos(6)
thf(fact_359_basic__trans__rules_I31_J,axiom,
    ! [A4: set_Pr8693737435421807431at_nat,B4: set_Pr8693737435421807431at_nat,C: produc859450856879609959at_nat] :
      ( ( ord_le3000389064537975527at_nat @ A4 @ B4 )
     => ( ( member8206827879206165904at_nat @ C @ A4 )
       => ( member8206827879206165904at_nat @ C @ B4 ) ) ) ).

% basic_trans_rules(31)
thf(fact_360_basic__trans__rules_I31_J,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,C: product_prod_nat_nat] :
      ( ( ord_le3146513528884898305at_nat @ A4 @ B4 )
     => ( ( member8440522571783428010at_nat @ C @ A4 )
       => ( member8440522571783428010at_nat @ C @ B4 ) ) ) ).

% basic_trans_rules(31)
thf(fact_361_basic__trans__rules_I31_J,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,C: produc3843707927480180839at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A4 @ B4 )
     => ( ( member8757157785044589968at_nat @ C @ A4 )
       => ( member8757157785044589968at_nat @ C @ B4 ) ) ) ).

% basic_trans_rules(31)
thf(fact_362_basic__trans__rules_I31_J,axiom,
    ! [A4: set_nat,B4: set_nat,C: nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B4 )
     => ( ( member_nat @ C @ A4 )
       => ( member_nat @ C @ B4 ) ) ) ).

% basic_trans_rules(31)
thf(fact_363_subset__eq,axiom,
    ( ord_le3000389064537975527at_nat
    = ( ^ [A6: set_Pr8693737435421807431at_nat,B6: set_Pr8693737435421807431at_nat] :
        ! [X4: produc859450856879609959at_nat] :
          ( ( member8206827879206165904at_nat @ X4 @ A6 )
         => ( member8206827879206165904at_nat @ X4 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_364_subset__eq,axiom,
    ( ord_le3146513528884898305at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
        ! [X4: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X4 @ A6 )
         => ( member8440522571783428010at_nat @ X4 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_365_subset__eq,axiom,
    ( ord_le1268244103169919719at_nat
    = ( ^ [A6: set_Pr4329608150637261639at_nat,B6: set_Pr4329608150637261639at_nat] :
        ! [X4: produc3843707927480180839at_nat] :
          ( ( member8757157785044589968at_nat @ X4 @ A6 )
         => ( member8757157785044589968at_nat @ X4 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_366_subset__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
        ! [X4: nat] :
          ( ( member_nat @ X4 @ A6 )
         => ( member_nat @ X4 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_367_subset__iff,axiom,
    ( ord_le3000389064537975527at_nat
    = ( ^ [A6: set_Pr8693737435421807431at_nat,B6: set_Pr8693737435421807431at_nat] :
        ! [T2: produc859450856879609959at_nat] :
          ( ( member8206827879206165904at_nat @ T2 @ A6 )
         => ( member8206827879206165904at_nat @ T2 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_368_subset__iff,axiom,
    ( ord_le3146513528884898305at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
        ! [T2: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ T2 @ A6 )
         => ( member8440522571783428010at_nat @ T2 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_369_subset__iff,axiom,
    ( ord_le1268244103169919719at_nat
    = ( ^ [A6: set_Pr4329608150637261639at_nat,B6: set_Pr4329608150637261639at_nat] :
        ! [T2: produc3843707927480180839at_nat] :
          ( ( member8757157785044589968at_nat @ T2 @ A6 )
         => ( member8757157785044589968at_nat @ T2 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_370_subset__iff,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
        ! [T2: nat] :
          ( ( member_nat @ T2 @ A6 )
         => ( member_nat @ T2 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_371_Collect__subset,axiom,
    ! [A4: set_Pr8693737435421807431at_nat,P: produc859450856879609959at_nat > $o] :
      ( ord_le3000389064537975527at_nat
      @ ( collec7088162979684241874at_nat
        @ ^ [X4: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X4 @ A4 )
            & ( P @ X4 ) ) )
      @ A4 ) ).

% Collect_subset
thf(fact_372_Collect__subset,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,P: produc3843707927480180839at_nat > $o] :
      ( ord_le1268244103169919719at_nat
      @ ( collec6321179662152712658at_nat
        @ ^ [X4: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X4 @ A4 )
            & ( P @ X4 ) ) )
      @ A4 ) ).

% Collect_subset
thf(fact_373_Collect__subset,axiom,
    ! [A4: set_nat,P: nat > $o] :
      ( ord_less_eq_set_nat
      @ ( collect_nat
        @ ^ [X4: nat] :
            ( ( member_nat @ X4 @ A4 )
            & ( P @ X4 ) ) )
      @ A4 ) ).

% Collect_subset
thf(fact_374_Collect__subset,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
      ( ord_le3146513528884898305at_nat
      @ ( collec3392354462482085612at_nat
        @ ^ [X4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X4 @ A4 )
            & ( P @ X4 ) ) )
      @ A4 ) ).

% Collect_subset
thf(fact_375_less__eq__set__def,axiom,
    ( ord_le3000389064537975527at_nat
    = ( ^ [A6: set_Pr8693737435421807431at_nat,B6: set_Pr8693737435421807431at_nat] :
          ( ord_le7858099551454983350_nat_o
          @ ^ [X4: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X4 @ A6 )
          @ ^ [X4: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X4 @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_376_less__eq__set__def,axiom,
    ( ord_le3146513528884898305at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
          ( ord_le704812498762024988_nat_o
          @ ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ A6 )
          @ ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_377_less__eq__set__def,axiom,
    ( ord_le1268244103169919719at_nat
    = ( ^ [A6: set_Pr4329608150637261639at_nat,B6: set_Pr4329608150637261639at_nat] :
          ( ord_le1999416642142172342_nat_o
          @ ^ [X4: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X4 @ A6 )
          @ ^ [X4: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X4 @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_378_less__eq__set__def,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
          ( ord_less_eq_nat_o
          @ ^ [X4: nat] : ( member_nat @ X4 @ A6 )
          @ ^ [X4: nat] : ( member_nat @ X4 @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_379_Collect__mono__iff,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
      = ( ! [X4: nat] :
            ( ( P @ X4 )
           => ( Q @ X4 ) ) ) ) ).

% Collect_mono_iff
thf(fact_380_Collect__mono__iff,axiom,
    ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ( ord_le3146513528884898305at_nat @ ( collec3392354462482085612at_nat @ P ) @ ( collec3392354462482085612at_nat @ Q ) )
      = ( ! [X4: product_prod_nat_nat] :
            ( ( P @ X4 )
           => ( Q @ X4 ) ) ) ) ).

% Collect_mono_iff
thf(fact_381_less__list__code_I1_J,axiom,
    ! [Xs: list_nat] :
      ~ ( ord_less_list_nat @ Xs @ nil_nat ) ).

% less_list_code(1)
thf(fact_382_not__less__Nil,axiom,
    ! [X: list_nat] :
      ~ ( ord_less_list_nat @ X @ nil_nat ) ).

% not_less_Nil
thf(fact_383_Nil__le__Cons,axiom,
    ! [X: list_nat] : ( ord_less_eq_list_nat @ nil_nat @ X ) ).

% Nil_le_Cons
thf(fact_384_less__eq__list__code_I2_J,axiom,
    ! [Xs: list_nat] : ( ord_less_eq_list_nat @ nil_nat @ Xs ) ).

% less_eq_list_code(2)
thf(fact_385_complete__interval,axiom,
    ! [A: nat,B: nat,P: nat > $o] :
      ( ( ord_less_nat @ A @ B )
     => ( ( P @ A )
       => ( ~ ( P @ B )
         => ? [C3: nat] :
              ( ( ord_less_eq_nat @ A @ C3 )
              & ( ord_less_eq_nat @ C3 @ B )
              & ! [X6: nat] :
                  ( ( ( ord_less_eq_nat @ A @ X6 )
                    & ( ord_less_nat @ X6 @ C3 ) )
                 => ( P @ X6 ) )
              & ! [D3: nat] :
                  ( ! [X3: nat] :
                      ( ( ( ord_less_eq_nat @ A @ X3 )
                        & ( ord_less_nat @ X3 @ D3 ) )
                     => ( P @ X3 ) )
                 => ( ord_less_eq_nat @ D3 @ C3 ) ) ) ) ) ) ).

% complete_interval
thf(fact_386_verit__comp__simplify1_I3_J,axiom,
    ! [B5: nat,A5: nat] :
      ( ( ~ ( ord_less_eq_nat @ B5 @ A5 ) )
      = ( ord_less_nat @ A5 @ B5 ) ) ).

% verit_comp_simplify1(3)
thf(fact_387_psubsetD,axiom,
    ! [A4: set_Pr8693737435421807431at_nat,B4: set_Pr8693737435421807431at_nat,C: produc859450856879609959at_nat] :
      ( ( ord_le6428140832669894131at_nat @ A4 @ B4 )
     => ( ( member8206827879206165904at_nat @ C @ A4 )
       => ( member8206827879206165904at_nat @ C @ B4 ) ) ) ).

% psubsetD
thf(fact_388_psubsetD,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,C: product_prod_nat_nat] :
      ( ( ord_le7866589430770878221at_nat @ A4 @ B4 )
     => ( ( member8440522571783428010at_nat @ C @ A4 )
       => ( member8440522571783428010at_nat @ C @ B4 ) ) ) ).

% psubsetD
thf(fact_389_psubsetD,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,C: produc3843707927480180839at_nat] :
      ( ( ord_le2604355607129572851at_nat @ A4 @ B4 )
     => ( ( member8757157785044589968at_nat @ C @ A4 )
       => ( member8757157785044589968at_nat @ C @ B4 ) ) ) ).

% psubsetD
thf(fact_390_psubsetD,axiom,
    ! [A4: set_nat,B4: set_nat,C: nat] :
      ( ( ord_less_set_nat @ A4 @ B4 )
     => ( ( member_nat @ C @ A4 )
       => ( member_nat @ C @ B4 ) ) ) ).

% psubsetD
thf(fact_391_less__set__def,axiom,
    ( ord_le6428140832669894131at_nat
    = ( ^ [A6: set_Pr8693737435421807431at_nat,B6: set_Pr8693737435421807431at_nat] :
          ( ord_le7432393201274343594_nat_o
          @ ^ [X4: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X4 @ A6 )
          @ ^ [X4: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X4 @ B6 ) ) ) ) ).

% less_set_def
thf(fact_392_less__set__def,axiom,
    ( ord_le7866589430770878221at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
          ( ord_le549003669493604880_nat_o
          @ ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ A6 )
          @ ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ B6 ) ) ) ) ).

% less_set_def
thf(fact_393_less__set__def,axiom,
    ( ord_le2604355607129572851at_nat
    = ( ^ [A6: set_Pr4329608150637261639at_nat,B6: set_Pr4329608150637261639at_nat] :
          ( ord_le8739539344792662186_nat_o
          @ ^ [X4: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X4 @ A6 )
          @ ^ [X4: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X4 @ B6 ) ) ) ) ).

% less_set_def
thf(fact_394_less__set__def,axiom,
    ( ord_less_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
          ( ord_less_nat_o
          @ ^ [X4: nat] : ( member_nat @ X4 @ A6 )
          @ ^ [X4: nat] : ( member_nat @ X4 @ B6 ) ) ) ) ).

% less_set_def
thf(fact_395_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_396_verit__comp__simplify1_I2_J,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_397_verit__comp__simplify_I1_J,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% verit_comp_simplify(1)
thf(fact_398_pred__subset__eq,axiom,
    ! [R: set_Pr8693737435421807431at_nat,S2: set_Pr8693737435421807431at_nat] :
      ( ( ord_le7858099551454983350_nat_o
        @ ^ [X4: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X4 @ R )
        @ ^ [X4: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X4 @ S2 ) )
      = ( ord_le3000389064537975527at_nat @ R @ S2 ) ) ).

% pred_subset_eq
thf(fact_399_pred__subset__eq,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le704812498762024988_nat_o
        @ ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ R )
        @ ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ S2 ) )
      = ( ord_le3146513528884898305at_nat @ R @ S2 ) ) ).

% pred_subset_eq
thf(fact_400_pred__subset__eq,axiom,
    ! [R: set_Pr4329608150637261639at_nat,S2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1999416642142172342_nat_o
        @ ^ [X4: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X4 @ R )
        @ ^ [X4: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X4 @ S2 ) )
      = ( ord_le1268244103169919719at_nat @ R @ S2 ) ) ).

% pred_subset_eq
thf(fact_401_pred__subset__eq,axiom,
    ! [R: set_nat,S2: set_nat] :
      ( ( ord_less_eq_nat_o
        @ ^ [X4: nat] : ( member_nat @ X4 @ R )
        @ ^ [X4: nat] : ( member_nat @ X4 @ S2 ) )
      = ( ord_less_eq_set_nat @ R @ S2 ) ) ).

% pred_subset_eq
thf(fact_402_arg__min__inj__eq,axiom,
    ! [F2: nat > product_prod_nat_nat,P: nat > $o,A: nat] :
      ( ( inj_on5538052773655684606at_nat @ F2 @ ( collect_nat @ P ) )
     => ( ( P @ A )
       => ( ! [Y3: nat] :
              ( ( P @ Y3 )
             => ( ord_le8460144461188290721at_nat @ ( F2 @ A ) @ ( F2 @ Y3 ) ) )
         => ( ( lattic7534068341178078613at_nat @ F2 @ P )
            = A ) ) ) ) ).

% arg_min_inj_eq
thf(fact_403_arg__min__inj__eq,axiom,
    ! [F2: nat > nat,P: nat > $o,A: nat] :
      ( ( inj_on_nat_nat @ F2 @ ( collect_nat @ P ) )
     => ( ( P @ A )
       => ( ! [Y3: nat] :
              ( ( P @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ A ) @ ( F2 @ Y3 ) ) )
         => ( ( lattic8739620818006775868at_nat @ F2 @ P )
            = A ) ) ) ) ).

% arg_min_inj_eq
thf(fact_404_arg__min__inj__eq,axiom,
    ! [F2: product_prod_nat_nat > nat,P: product_prod_nat_nat > $o,A: product_prod_nat_nat] :
      ( ( inj_on2178005380612969504at_nat @ F2 @ ( collec3392354462482085612at_nat @ P ) )
     => ( ( P @ A )
       => ( ! [Y3: product_prod_nat_nat] :
              ( ( P @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ A ) @ ( F2 @ Y3 ) ) )
         => ( ( lattic4174020948135363511at_nat @ F2 @ P )
            = A ) ) ) ) ).

% arg_min_inj_eq
thf(fact_405_pair__lessI2,axiom,
    ! [A: nat,B: nat,S: nat,T: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ S @ T )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_less ) ) ) ).

% pair_lessI2
thf(fact_406_fold__atLeastAtMost__nat_Opelims,axiom,
    ! [X: nat > nat > nat,Xa: nat,Xb: nat,Xc: nat,Y: nat] :
      ( ( ( set_fo2584398358068434914at_nat @ X @ Xa @ Xb @ Xc )
        = Y )
     => ( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ X @ ( produc487386426758144856at_nat @ Xa @ ( product_Pair_nat_nat @ Xb @ Xc ) ) ) )
       => ~ ( ( ( ( ord_less_nat @ Xb @ Xa )
               => ( Y = Xc ) )
              & ( ~ ( ord_less_nat @ Xb @ Xa )
               => ( Y
                  = ( set_fo2584398358068434914at_nat @ X @ ( plus_plus_nat @ Xa @ one_one_nat ) @ Xb @ ( X @ Xa @ Xc ) ) ) ) )
           => ~ ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ X @ ( produc487386426758144856at_nat @ Xa @ ( product_Pair_nat_nat @ Xb @ Xc ) ) ) ) ) ) ) ).

% fold_atLeastAtMost_nat.pelims
thf(fact_407_fold__atLeastAtMost__nat_Opsimps,axiom,
    ! [F2: nat > nat > nat,A: nat,B: nat,Acc2: nat] :
      ( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ F2 @ ( produc487386426758144856at_nat @ A @ ( product_Pair_nat_nat @ B @ Acc2 ) ) ) )
     => ( ( ( ord_less_nat @ B @ A )
         => ( ( set_fo2584398358068434914at_nat @ F2 @ A @ B @ Acc2 )
            = Acc2 ) )
        & ( ~ ( ord_less_nat @ B @ A )
         => ( ( set_fo2584398358068434914at_nat @ F2 @ A @ B @ Acc2 )
            = ( set_fo2584398358068434914at_nat @ F2 @ ( plus_plus_nat @ A @ one_one_nat ) @ B @ ( F2 @ A @ Acc2 ) ) ) ) ) ) ).

% fold_atLeastAtMost_nat.psimps
thf(fact_408_pair__less__iff1,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( product_Pair_nat_nat @ X @ Z ) ) @ fun_pair_less )
      = ( ord_less_nat @ Y @ Z ) ) ).

% pair_less_iff1
thf(fact_409_subrelI,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ! [X3: nat,Y3: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R2 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ S ) )
     => ( ord_le3146513528884898305at_nat @ R2 @ S ) ) ).

% subrelI
thf(fact_410_subrelI,axiom,
    ! [R2: set_Pr8693737435421807431at_nat,S: set_Pr8693737435421807431at_nat] :
      ( ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
          ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ R2 )
         => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ S ) )
     => ( ord_le3000389064537975527at_nat @ R2 @ S ) ) ).

% subrelI
thf(fact_411_subrelI,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ! [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
          ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ R2 )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ S ) )
     => ( ord_le1268244103169919719at_nat @ R2 @ S ) ) ).

% subrelI
thf(fact_412_subrelI,axiom,
    ! [R2: set_Pr7459493094073627847at_nat,S: set_Pr7459493094073627847at_nat] :
      ( ! [X3: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] :
          ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ Y3 ) @ R2 )
         => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ Y3 ) @ S ) )
     => ( ord_le5997549366648089703at_nat @ R2 @ S ) ) ).

% subrelI
thf(fact_413_pair__lessI1,axiom,
    ! [A: nat,B: nat,S: nat,T: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_less ) ) ).

% pair_lessI1
thf(fact_414_fold__atLeastAtMost__nat_Opinduct,axiom,
    ! [A0: nat > nat > nat,A1: nat,A22: nat,A32: nat,P: ( nat > nat > nat ) > nat > nat > nat > $o] :
      ( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ A0 @ ( produc487386426758144856at_nat @ A1 @ ( product_Pair_nat_nat @ A22 @ A32 ) ) ) )
     => ( ! [F: nat > nat > nat,A2: nat,B2: nat,Acc: nat] :
            ( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ F @ ( produc487386426758144856at_nat @ A2 @ ( product_Pair_nat_nat @ B2 @ Acc ) ) ) )
           => ( ( ~ ( ord_less_nat @ B2 @ A2 )
               => ( P @ F @ ( plus_plus_nat @ A2 @ one_one_nat ) @ B2 @ ( F @ A2 @ Acc ) ) )
             => ( P @ F @ A2 @ B2 @ Acc ) ) )
       => ( P @ A0 @ A1 @ A22 @ A32 ) ) ) ).

% fold_atLeastAtMost_nat.pinduct
thf(fact_415_pred__equals__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ( ( ^ [X4: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ R ) )
        = ( ^ [X4: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ S2 ) ) )
      = ( R = S2 ) ) ).

% pred_equals_eq2
thf(fact_416_pred__equals__eq2,axiom,
    ! [R: set_Pr8693737435421807431at_nat,S2: set_Pr8693737435421807431at_nat] :
      ( ( ( ^ [X4: product_prod_nat_nat,Y4: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y4 ) @ R ) )
        = ( ^ [X4: product_prod_nat_nat,Y4: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y4 ) @ S2 ) ) )
      = ( R = S2 ) ) ).

% pred_equals_eq2
thf(fact_417_pred__equals__eq2,axiom,
    ! [R: set_Pr4329608150637261639at_nat,S2: set_Pr4329608150637261639at_nat] :
      ( ( ( ^ [X4: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y4 ) @ R ) )
        = ( ^ [X4: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y4 ) @ S2 ) ) )
      = ( R = S2 ) ) ).

% pred_equals_eq2
thf(fact_418_pred__equals__eq2,axiom,
    ! [R: set_Pr7459493094073627847at_nat,S2: set_Pr7459493094073627847at_nat] :
      ( ( ( ^ [X4: set_Pr4329608150637261639at_nat,Y4: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X4 @ Y4 ) @ R ) )
        = ( ^ [X4: set_Pr4329608150637261639at_nat,Y4: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X4 @ Y4 ) @ S2 ) ) )
      = ( R = S2 ) ) ).

% pred_equals_eq2
thf(fact_419_pred__subset__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le2646555220125990790_nat_o
        @ ^ [X4: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ R )
        @ ^ [X4: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ S2 ) )
      = ( ord_le3146513528884898305at_nat @ R @ S2 ) ) ).

% pred_subset_eq2
thf(fact_420_pred__subset__eq2,axiom,
    ! [R: set_Pr8693737435421807431at_nat,S2: set_Pr8693737435421807431at_nat] :
      ( ( ord_le5604493270027003598_nat_o
        @ ^ [X4: product_prod_nat_nat,Y4: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y4 ) @ R )
        @ ^ [X4: product_prod_nat_nat,Y4: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y4 ) @ S2 ) )
      = ( ord_le3000389064537975527at_nat @ R @ S2 ) ) ).

% pred_subset_eq2
thf(fact_421_pred__subset__eq2,axiom,
    ! [R: set_Pr4329608150637261639at_nat,S2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le3935385432712749774_nat_o
        @ ^ [X4: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y4 ) @ R )
        @ ^ [X4: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y4 ) @ S2 ) )
      = ( ord_le1268244103169919719at_nat @ R @ S2 ) ) ).

% pred_subset_eq2
thf(fact_422_pred__subset__eq2,axiom,
    ! [R: set_Pr7459493094073627847at_nat,S2: set_Pr7459493094073627847at_nat] :
      ( ( ord_le3072208448688395470_nat_o
        @ ^ [X4: set_Pr4329608150637261639at_nat,Y4: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X4 @ Y4 ) @ R )
        @ ^ [X4: set_Pr4329608150637261639at_nat,Y4: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X4 @ Y4 ) @ S2 ) )
      = ( ord_le5997549366648089703at_nat @ R @ S2 ) ) ).

% pred_subset_eq2
thf(fact_423_pair__leqI2,axiom,
    ! [A: nat,B: nat,S: nat,T: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ S @ T )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_leq ) ) ) ).

% pair_leqI2
thf(fact_424_pair__leqI1,axiom,
    ! [A: nat,B: nat,S: nat,T: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_leq ) ) ).

% pair_leqI1
thf(fact_425_in__measure,axiom,
    ! [X: nat,Y: nat,F2: nat > nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( measure_nat @ F2 ) )
      = ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) ) ).

% in_measure
thf(fact_426_in__measure,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,F2: product_prod_nat_nat > nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( measur8038558561449204169at_nat @ F2 ) )
      = ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) ) ).

% in_measure
thf(fact_427_in__measure,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,F2: set_Pr1261947904930325089at_nat > nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ ( measur1827424007717751593at_nat @ F2 ) )
      = ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) ) ).

% in_measure
thf(fact_428_in__measure,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,F2: set_Pr4329608150637261639at_nat > nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ ( measur4922264856574889999at_nat @ F2 ) )
      = ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) ) ).

% in_measure
thf(fact_429_rotate1__length01,axiom,
    ! [Xs: list_nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
     => ( ( rotate1_nat @ Xs )
        = Xs ) ) ).

% rotate1_length01
thf(fact_430_less__than__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ less_than )
      = ( ord_less_nat @ X @ Y ) ) ).

% less_than_iff
thf(fact_431_in__lex__prod,axiom,
    ! [A: nat,B: nat,A5: nat,B5: nat,R2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( product_Pair_nat_nat @ A5 @ B5 ) ) @ ( lex_prod_nat_nat @ R2 @ S ) )
      = ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A5 ) @ R2 )
        | ( ( A = A5 )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ B5 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_432_in__lex__prod,axiom,
    ! [A: nat,B: product_prod_nat_nat,A5: nat,B5: product_prod_nat_nat,R2: set_Pr1261947904930325089at_nat,S: set_Pr8693737435421807431at_nat] :
      ( ( member3048279782668807382at_nat @ ( produc2653010282673554021at_nat @ ( produc487386426758144856at_nat @ A @ B ) @ ( produc487386426758144856at_nat @ A5 @ B5 ) ) @ ( lex_pr8029265285556086080at_nat @ R2 @ S ) )
      = ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A5 ) @ R2 )
        | ( ( A = A5 )
          & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B @ B5 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_433_in__lex__prod,axiom,
    ! [A: product_prod_nat_nat,B: nat,A5: product_prod_nat_nat,B5: nat,R2: set_Pr8693737435421807431at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( member1995966531042493578at_nat @ ( produc7904928797850150681at_nat @ ( produc6350711070570205562at_nat @ A @ B ) @ ( produc6350711070570205562at_nat @ A5 @ B5 ) ) @ ( lex_pr4669217892513370978at_nat @ R2 @ S ) )
      = ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ A5 ) @ R2 )
        | ( ( A = A5 )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ B5 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_434_in__lex__prod,axiom,
    ! [A: nat,B: set_Pr1261947904930325089at_nat,A5: nat,B5: set_Pr1261947904930325089at_nat,R2: set_Pr1261947904930325089at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ( member4107301959146914710at_nat @ ( produc8824474337961625893at_nat @ ( produc5463602008962177208at_nat @ A @ B ) @ ( produc5463602008962177208at_nat @ A5 @ B5 ) ) @ ( lex_pr4265550801952275104at_nat @ R2 @ S ) )
      = ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A5 ) @ R2 )
        | ( ( A = A5 )
          & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B @ B5 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_435_in__lex__prod,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: nat,A5: set_Pr1261947904930325089at_nat,B5: nat,R2: set_Pr4329608150637261639at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( member3480088626692287306at_nat @ ( produc3585720979125124569at_nat @ ( produc3149970401713881818at_nat @ A @ B ) @ ( produc3149970401713881818at_nat @ A5 @ B5 ) ) @ ( lex_pr1951919194703979714at_nat @ R2 @ S ) )
      = ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ A5 ) @ R2 )
        | ( ( A = A5 )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ B5 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_436_in__lex__prod,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A5: product_prod_nat_nat,B5: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat,S: set_Pr8693737435421807431at_nat] :
      ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( produc6161850002892822231at_nat @ A5 @ B5 ) ) @ ( lex_pr8801849515957261039at_nat @ R2 @ S ) )
      = ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ A5 ) @ R2 )
        | ( ( A = A5 )
          & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B @ B5 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_437_in__lex__prod,axiom,
    ! [A: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat,A5: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat,R2: set_Pr8693737435421807431at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ( member4971006070143632144at_nat @ ( produc3335200116822430551at_nat @ ( produc5966227314700062775at_nat @ A @ B ) @ ( produc5966227314700062775at_nat @ A5 @ B5 ) ) @ ( lex_pr864846108736269391at_nat @ R2 @ S ) )
      = ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ A5 ) @ R2 )
        | ( ( A = A5 )
          & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B @ B5 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_438_in__lex__prod,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat,B5: product_prod_nat_nat,R2: set_Pr4329608150637261639at_nat,S: set_Pr8693737435421807431at_nat] :
      ( ( member6233145392104960784at_nat @ ( produc9001802813318374231at_nat @ ( produc4320700058201173367at_nat @ A @ B ) @ ( produc4320700058201173367at_nat @ A5 @ B5 ) ) @ ( lex_pr8442690889092155791at_nat @ R2 @ S ) )
      = ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ A5 ) @ R2 )
        | ( ( A = A5 )
          & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B @ B5 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_439_in__lex__prod,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ ( produc2922128104949294807at_nat @ A5 @ B5 ) ) @ ( lex_pr8000699672184251119at_nat @ R2 @ S ) )
      = ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ A5 ) @ R2 )
        | ( ( A = A5 )
          & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B @ B5 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_440_in__lex__prod,axiom,
    ! [A: nat,B: set_Pr4329608150637261639at_nat,A5: nat,B5: set_Pr4329608150637261639at_nat,R2: set_Pr1261947904930325089at_nat,S: set_Pr7459493094073627847at_nat] :
      ( ( member4210486617425595874at_nat @ ( produc908947232176754289at_nat @ ( produc6484155927498994462at_nat @ A @ B ) @ ( produc6484155927498994462at_nat @ A5 @ B5 ) ) @ ( lex_pr1368066972704168198at_nat @ R2 @ S ) )
      = ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A5 ) @ R2 )
        | ( ( A = A5 )
          & ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ B @ B5 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_441_rotate1__is__Nil__conv,axiom,
    ! [Xs: list_nat] :
      ( ( ( rotate1_nat @ Xs )
        = nil_nat )
      = ( Xs = nil_nat ) ) ).

% rotate1_is_Nil_conv
thf(fact_442_length__rotate1,axiom,
    ! [Xs: list_nat] :
      ( ( size_size_list_nat @ ( rotate1_nat @ Xs ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_rotate1
thf(fact_443_pair__less__def,axiom,
    ( fun_pair_less
    = ( lex_prod_nat_nat @ less_than @ less_than ) ) ).

% pair_less_def
thf(fact_444_rotate1_Osimps_I1_J,axiom,
    ( ( rotate1_nat @ nil_nat )
    = nil_nat ) ).

% rotate1.simps(1)
thf(fact_445_rotate__length01,axiom,
    ! [Xs: list_nat,N: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
     => ( ( rotate_nat @ N @ Xs )
        = Xs ) ) ).

% rotate_length01
thf(fact_446_count__list__card,axiom,
    ( count_list_nat
    = ( ^ [Xs2: list_nat,X4: nat] :
          ( finite_card_nat
          @ ( collect_nat
            @ ^ [K2: nat] :
                ( ( ord_less_nat @ K2 @ ( size_size_list_nat @ Xs2 ) )
                & ( ( nth_nat @ Xs2 @ K2 )
                  = X4 ) ) ) ) ) ) ).

% count_list_card
thf(fact_447_rotate__is__Nil__conv,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( ( rotate_nat @ N @ Xs )
        = nil_nat )
      = ( Xs = nil_nat ) ) ).

% rotate_is_Nil_conv
thf(fact_448_length__rotate,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( size_size_list_nat @ ( rotate_nat @ N @ Xs ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_rotate
thf(fact_449_ex__card,axiom,
    ! [N: nat,A4: set_nat] :
      ( ( ord_less_eq_nat @ N @ ( finite_card_nat @ A4 ) )
     => ? [S3: set_nat] :
          ( ( ord_less_eq_set_nat @ S3 @ A4 )
          & ( ( finite_card_nat @ S3 )
            = N ) ) ) ).

% ex_card
thf(fact_450_ex__card,axiom,
    ! [N: nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( ord_less_eq_nat @ N @ ( finite711546835091564841at_nat @ A4 ) )
     => ? [S3: set_Pr1261947904930325089at_nat] :
          ( ( ord_le3146513528884898305at_nat @ S3 @ A4 )
          & ( ( finite711546835091564841at_nat @ S3 )
            = N ) ) ) ).

% ex_card
thf(fact_451_card__Collect__less__nat,axiom,
    ! [N: nat] :
      ( ( finite_card_nat
        @ ( collect_nat
          @ ^ [I: nat] : ( ord_less_nat @ I @ N ) ) )
      = N ) ).

% card_Collect_less_nat
thf(fact_452_nth__rotate,axiom,
    ! [N: nat,Xs: list_nat,M: nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_nat @ ( rotate_nat @ M @ Xs ) @ N )
        = ( nth_nat @ Xs @ ( modulo_modulo_nat @ ( plus_plus_nat @ M @ N ) @ ( size_size_list_nat @ Xs ) ) ) ) ) ).

% nth_rotate
thf(fact_453_subset__Collect__iff,axiom,
    ! [B4: set_Pr8693737435421807431at_nat,A4: set_Pr8693737435421807431at_nat,P: produc859450856879609959at_nat > $o] :
      ( ( ord_le3000389064537975527at_nat @ B4 @ A4 )
     => ( ( ord_le3000389064537975527at_nat @ B4
          @ ( collec7088162979684241874at_nat
            @ ^ [X4: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ X4 @ A4 )
                & ( P @ X4 ) ) ) )
        = ( ! [X4: produc859450856879609959at_nat] :
              ( ( member8206827879206165904at_nat @ X4 @ B4 )
             => ( P @ X4 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_454_subset__Collect__iff,axiom,
    ! [B4: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat,P: produc3843707927480180839at_nat > $o] :
      ( ( ord_le1268244103169919719at_nat @ B4 @ A4 )
     => ( ( ord_le1268244103169919719at_nat @ B4
          @ ( collec6321179662152712658at_nat
            @ ^ [X4: produc3843707927480180839at_nat] :
                ( ( member8757157785044589968at_nat @ X4 @ A4 )
                & ( P @ X4 ) ) ) )
        = ( ! [X4: produc3843707927480180839at_nat] :
              ( ( member8757157785044589968at_nat @ X4 @ B4 )
             => ( P @ X4 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_455_subset__Collect__iff,axiom,
    ! [B4: set_nat,A4: set_nat,P: nat > $o] :
      ( ( ord_less_eq_set_nat @ B4 @ A4 )
     => ( ( ord_less_eq_set_nat @ B4
          @ ( collect_nat
            @ ^ [X4: nat] :
                ( ( member_nat @ X4 @ A4 )
                & ( P @ X4 ) ) ) )
        = ( ! [X4: nat] :
              ( ( member_nat @ X4 @ B4 )
             => ( P @ X4 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_456_subset__Collect__iff,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
      ( ( ord_le3146513528884898305at_nat @ B4 @ A4 )
     => ( ( ord_le3146513528884898305at_nat @ B4
          @ ( collec3392354462482085612at_nat
            @ ^ [X4: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X4 @ A4 )
                & ( P @ X4 ) ) ) )
        = ( ! [X4: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X4 @ B4 )
             => ( P @ X4 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_457_rotate__conv__mod,axiom,
    ( rotate_nat
    = ( ^ [N3: nat,Xs2: list_nat] : ( rotate_nat @ ( modulo_modulo_nat @ N3 @ ( size_size_list_nat @ Xs2 ) ) @ Xs2 ) ) ) ).

% rotate_conv_mod
thf(fact_458_subset__CollectI,axiom,
    ! [B4: set_Pr8693737435421807431at_nat,A4: set_Pr8693737435421807431at_nat,Q: produc859450856879609959at_nat > $o,P: produc859450856879609959at_nat > $o] :
      ( ( ord_le3000389064537975527at_nat @ B4 @ A4 )
     => ( ! [X3: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X3 @ B4 )
           => ( ( Q @ X3 )
             => ( P @ X3 ) ) )
       => ( ord_le3000389064537975527at_nat
          @ ( collec7088162979684241874at_nat
            @ ^ [X4: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ X4 @ B4 )
                & ( Q @ X4 ) ) )
          @ ( collec7088162979684241874at_nat
            @ ^ [X4: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ X4 @ A4 )
                & ( P @ X4 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_459_subset__CollectI,axiom,
    ! [B4: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat,Q: produc3843707927480180839at_nat > $o,P: produc3843707927480180839at_nat > $o] :
      ( ( ord_le1268244103169919719at_nat @ B4 @ A4 )
     => ( ! [X3: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X3 @ B4 )
           => ( ( Q @ X3 )
             => ( P @ X3 ) ) )
       => ( ord_le1268244103169919719at_nat
          @ ( collec6321179662152712658at_nat
            @ ^ [X4: produc3843707927480180839at_nat] :
                ( ( member8757157785044589968at_nat @ X4 @ B4 )
                & ( Q @ X4 ) ) )
          @ ( collec6321179662152712658at_nat
            @ ^ [X4: produc3843707927480180839at_nat] :
                ( ( member8757157785044589968at_nat @ X4 @ A4 )
                & ( P @ X4 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_460_subset__CollectI,axiom,
    ! [B4: set_nat,A4: set_nat,Q: nat > $o,P: nat > $o] :
      ( ( ord_less_eq_set_nat @ B4 @ A4 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ B4 )
           => ( ( Q @ X3 )
             => ( P @ X3 ) ) )
       => ( ord_less_eq_set_nat
          @ ( collect_nat
            @ ^ [X4: nat] :
                ( ( member_nat @ X4 @ B4 )
                & ( Q @ X4 ) ) )
          @ ( collect_nat
            @ ^ [X4: nat] :
                ( ( member_nat @ X4 @ A4 )
                & ( P @ X4 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_461_subset__CollectI,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,Q: product_prod_nat_nat > $o,P: product_prod_nat_nat > $o] :
      ( ( ord_le3146513528884898305at_nat @ B4 @ A4 )
     => ( ! [X3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ B4 )
           => ( ( Q @ X3 )
             => ( P @ X3 ) ) )
       => ( ord_le3146513528884898305at_nat
          @ ( collec3392354462482085612at_nat
            @ ^ [X4: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X4 @ B4 )
                & ( Q @ X4 ) ) )
          @ ( collec3392354462482085612at_nat
            @ ^ [X4: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X4 @ A4 )
                & ( P @ X4 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_462_card__space,axiom,
    ( ( as != nil_nat )
   => ( ( finite711546835091564841at_nat @ ( frequency_Moment_M_1 @ as ) )
      = ( size_size_list_nat @ as ) ) ) ).

% card_space
thf(fact_463_mod__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( modulo_modulo_nat @ M @ N )
        = M ) ) ).

% mod_less
thf(fact_464_mod__add__self1,axiom,
    ! [B: nat,A: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ B @ A ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_add_self1
thf(fact_465_mod__add__self2,axiom,
    ! [A: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_add_self2
thf(fact_466_mod__add__eq,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).

% mod_add_eq
thf(fact_467_mod__add__cong,axiom,
    ! [A: nat,C: nat,A5: nat,B: nat,B5: nat] :
      ( ( ( modulo_modulo_nat @ A @ C )
        = ( modulo_modulo_nat @ A5 @ C ) )
     => ( ( ( modulo_modulo_nat @ B @ C )
          = ( modulo_modulo_nat @ B5 @ C ) )
       => ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C )
          = ( modulo_modulo_nat @ ( plus_plus_nat @ A5 @ B5 ) @ C ) ) ) ) ).

% mod_add_cong
thf(fact_468_mod__add__left__eq,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ B ) @ C )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).

% mod_add_left_eq
thf(fact_469_mod__add__right__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).

% mod_add_right_eq
thf(fact_470_mod__less__eq__dividend,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ M ) ).

% mod_less_eq_dividend
thf(fact_471_non__empty__space,axiom,
    ( ( as != nil_nat )
   => ( ( frequency_Moment_M_1 @ as )
     != bot_bo2099793752762293965at_nat ) ) ).

% non_empty_space
thf(fact_472_M_092_060_094sub_0621__def,axiom,
    ( ( frequency_Moment_M_1 @ as )
    = ( collec3392354462482085612at_nat
      @ ( produc6081775807080527818_nat_o
        @ ^ [U: nat,V: nat] : ( ord_less_nat @ V @ ( count_list_nat @ as @ U ) ) ) ) ) ).

% M\<^sub>1_def
thf(fact_473_fin__space,axiom,
    ( ( as != nil_nat )
   => ( finite6177210948735845034at_nat @ ( frequency_Moment_M_1 @ as ) ) ) ).

% fin_space
thf(fact_474_nth__rotate1,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_nat @ ( rotate1_nat @ Xs ) @ N )
        = ( nth_nat @ Xs @ ( modulo_modulo_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs ) ) ) ) ) ).

% nth_rotate1
thf(fact_475_empty__Collect__eq,axiom,
    ! [P: nat > $o] :
      ( ( bot_bot_set_nat
        = ( collect_nat @ P ) )
      = ( ! [X4: nat] :
            ~ ( P @ X4 ) ) ) ).

% empty_Collect_eq
thf(fact_476_empty__Collect__eq,axiom,
    ! [P: product_prod_nat_nat > $o] :
      ( ( bot_bo2099793752762293965at_nat
        = ( collec3392354462482085612at_nat @ P ) )
      = ( ! [X4: product_prod_nat_nat] :
            ~ ( P @ X4 ) ) ) ).

% empty_Collect_eq
thf(fact_477_empty__Collect__eq,axiom,
    ! [P: produc3843707927480180839at_nat > $o] :
      ( ( bot_bo228742789529271731at_nat
        = ( collec6321179662152712658at_nat @ P ) )
      = ( ! [X4: produc3843707927480180839at_nat] :
            ~ ( P @ X4 ) ) ) ).

% empty_Collect_eq
thf(fact_478_Collect__empty__eq,axiom,
    ! [P: nat > $o] :
      ( ( ( collect_nat @ P )
        = bot_bot_set_nat )
      = ( ! [X4: nat] :
            ~ ( P @ X4 ) ) ) ).

% Collect_empty_eq
thf(fact_479_Collect__empty__eq,axiom,
    ! [P: product_prod_nat_nat > $o] :
      ( ( ( collec3392354462482085612at_nat @ P )
        = bot_bo2099793752762293965at_nat )
      = ( ! [X4: product_prod_nat_nat] :
            ~ ( P @ X4 ) ) ) ).

% Collect_empty_eq
thf(fact_480_Collect__empty__eq,axiom,
    ! [P: produc3843707927480180839at_nat > $o] :
      ( ( ( collec6321179662152712658at_nat @ P )
        = bot_bo228742789529271731at_nat )
      = ( ! [X4: produc3843707927480180839at_nat] :
            ~ ( P @ X4 ) ) ) ).

% Collect_empty_eq
thf(fact_481_all__not__in__conv,axiom,
    ! [A4: set_Pr8693737435421807431at_nat] :
      ( ( ! [X4: produc859450856879609959at_nat] :
            ~ ( member8206827879206165904at_nat @ X4 @ A4 ) )
      = ( A4 = bot_bo5327735625951526323at_nat ) ) ).

% all_not_in_conv
thf(fact_482_all__not__in__conv,axiom,
    ! [A4: set_nat] :
      ( ( ! [X4: nat] :
            ~ ( member_nat @ X4 @ A4 ) )
      = ( A4 = bot_bot_set_nat ) ) ).

% all_not_in_conv
thf(fact_483_all__not__in__conv,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ( ! [X4: product_prod_nat_nat] :
            ~ ( member8440522571783428010at_nat @ X4 @ A4 ) )
      = ( A4 = bot_bo2099793752762293965at_nat ) ) ).

% all_not_in_conv
thf(fact_484_all__not__in__conv,axiom,
    ! [A4: set_Pr4329608150637261639at_nat] :
      ( ( ! [X4: produc3843707927480180839at_nat] :
            ~ ( member8757157785044589968at_nat @ X4 @ A4 ) )
      = ( A4 = bot_bo228742789529271731at_nat ) ) ).

% all_not_in_conv
thf(fact_485_empty__iff,axiom,
    ! [C: produc859450856879609959at_nat] :
      ~ ( member8206827879206165904at_nat @ C @ bot_bo5327735625951526323at_nat ) ).

% empty_iff
thf(fact_486_empty__iff,axiom,
    ! [C: nat] :
      ~ ( member_nat @ C @ bot_bot_set_nat ) ).

% empty_iff
thf(fact_487_empty__iff,axiom,
    ! [C: product_prod_nat_nat] :
      ~ ( member8440522571783428010at_nat @ C @ bot_bo2099793752762293965at_nat ) ).

% empty_iff
thf(fact_488_empty__iff,axiom,
    ! [C: produc3843707927480180839at_nat] :
      ~ ( member8757157785044589968at_nat @ C @ bot_bo228742789529271731at_nat ) ).

% empty_iff
thf(fact_489_old_Onat_Oinject,axiom,
    ! [Nat: nat,Nat2: nat] :
      ( ( ( suc @ Nat )
        = ( suc @ Nat2 ) )
      = ( Nat = Nat2 ) ) ).

% old.nat.inject
thf(fact_490_nat_Oinject,axiom,
    ! [X22: nat,Y22: nat] :
      ( ( ( suc @ X22 )
        = ( suc @ Y22 ) )
      = ( X22 = Y22 ) ) ).

% nat.inject
thf(fact_491_finite__Collect__disjI,axiom,
    ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ( finite6177210948735845034at_nat
        @ ( collec3392354462482085612at_nat
          @ ^ [X4: product_prod_nat_nat] :
              ( ( P @ X4 )
              | ( Q @ X4 ) ) ) )
      = ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P ) )
        & ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_492_finite__Collect__disjI,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X4: nat] :
              ( ( P @ X4 )
              | ( Q @ X4 ) ) ) )
      = ( ( finite_finite_nat @ ( collect_nat @ P ) )
        & ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_493_finite__Collect__conjI,axiom,
    ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P ) )
        | ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ Q ) ) )
     => ( finite6177210948735845034at_nat
        @ ( collec3392354462482085612at_nat
          @ ^ [X4: product_prod_nat_nat] :
              ( ( P @ X4 )
              & ( Q @ X4 ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_494_finite__Collect__conjI,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( ( finite_finite_nat @ ( collect_nat @ P ) )
        | ( finite_finite_nat @ ( collect_nat @ Q ) ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X4: nat] :
              ( ( P @ X4 )
              & ( Q @ X4 ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_495_split__part,axiom,
    ! [P: $o,Q: nat > nat > $o] :
      ( ( produc6081775807080527818_nat_o
        @ ^ [A3: nat,B3: nat] :
            ( P
            & ( Q @ A3 @ B3 ) ) )
      = ( ^ [Ab: product_prod_nat_nat] :
            ( P
            & ( produc6081775807080527818_nat_o @ Q @ Ab ) ) ) ) ).

% split_part
thf(fact_496_subset__empty,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A4 @ bot_bo2099793752762293965at_nat )
      = ( A4 = bot_bo2099793752762293965at_nat ) ) ).

% subset_empty
thf(fact_497_subset__empty,axiom,
    ! [A4: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A4 @ bot_bo228742789529271731at_nat )
      = ( A4 = bot_bo228742789529271731at_nat ) ) ).

% subset_empty
thf(fact_498_empty__subsetI,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ bot_bo2099793752762293965at_nat @ A4 ) ).

% empty_subsetI
thf(fact_499_empty__subsetI,axiom,
    ! [A4: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ bot_bo228742789529271731at_nat @ A4 ) ).

% empty_subsetI
thf(fact_500_lessI,axiom,
    ! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).

% lessI
thf(fact_501_Suc__mono,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).

% Suc_mono
thf(fact_502_Suc__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% Suc_less_eq
thf(fact_503_Suc__le__mono,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
      = ( ord_less_eq_nat @ N @ M ) ) ).

% Suc_le_mono
thf(fact_504_inj__on__empty,axiom,
    ! [F2: nat > product_prod_nat_nat] : ( inj_on5538052773655684606at_nat @ F2 @ bot_bot_set_nat ) ).

% inj_on_empty
thf(fact_505_inj__on__empty,axiom,
    ! [F2: nat > nat] : ( inj_on_nat_nat @ F2 @ bot_bot_set_nat ) ).

% inj_on_empty
thf(fact_506_add__Suc__right,axiom,
    ! [M: nat,N: nat] :
      ( ( plus_plus_nat @ M @ ( suc @ N ) )
      = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).

% add_Suc_right
thf(fact_507_sumset__empty_I1_J,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ( plus_p2407200367410362776at_nat @ A4 @ bot_bo2099793752762293965at_nat )
      = bot_bo2099793752762293965at_nat ) ).

% sumset_empty(1)
thf(fact_508_sumset__empty_I1_J,axiom,
    ! [A4: set_Pr4329608150637261639at_nat] :
      ( ( plus_p5270993086169515006at_nat @ A4 @ bot_bo228742789529271731at_nat )
      = bot_bo228742789529271731at_nat ) ).

% sumset_empty(1)
thf(fact_509_sumset__empty_I2_J,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ( plus_p2407200367410362776at_nat @ bot_bo2099793752762293965at_nat @ A4 )
      = bot_bo2099793752762293965at_nat ) ).

% sumset_empty(2)
thf(fact_510_sumset__empty_I2_J,axiom,
    ! [A4: set_Pr4329608150637261639at_nat] :
      ( ( plus_p5270993086169515006at_nat @ bot_bo228742789529271731at_nat @ A4 )
      = bot_bo228742789529271731at_nat ) ).

% sumset_empty(2)
thf(fact_511_finite__Collect__subsets,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A4 )
     => ( finite9047747110432174090at_nat
        @ ( collec5514110066124741708at_nat
          @ ^ [B6: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ B6 @ A4 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_512_finite__Collect__subsets,axiom,
    ! [A4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [B6: set_nat] : ( ord_less_eq_set_nat @ B6 @ A4 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_513_case__prod__conv,axiom,
    ! [F2: nat > nat > $o,A: nat,B: nat] :
      ( ( produc6081775807080527818_nat_o @ F2 @ ( product_Pair_nat_nat @ A @ B ) )
      = ( F2 @ A @ B ) ) ).

% case_prod_conv
thf(fact_514_case__prodI2,axiom,
    ! [P2: produc859450856879609959at_nat,C: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ! [A2: product_prod_nat_nat,B2: product_prod_nat_nat] :
          ( ( P2
            = ( produc6161850002892822231at_nat @ A2 @ B2 ) )
         => ( C @ A2 @ B2 ) )
     => ( produc6590410687421337004_nat_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_515_case__prodI2,axiom,
    ! [P2: produc3843707927480180839at_nat,C: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o] :
      ( ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
          ( ( P2
            = ( produc2922128104949294807at_nat @ A2 @ B2 ) )
         => ( C @ A2 @ B2 ) )
     => ( produc410239310623530412_nat_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_516_case__prodI2,axiom,
    ! [P2: produc1319942482725812455at_nat,C: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o] :
      ( ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
          ( ( P2
            = ( produc9060074326276436823at_nat @ A2 @ B2 ) )
         => ( C @ A2 @ B2 ) )
     => ( produc9020218426428501292_nat_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_517_case__prodI2,axiom,
    ! [P2: product_prod_nat_nat,C: nat > nat > $o] :
      ( ! [A2: nat,B2: nat] :
          ( ( P2
            = ( product_Pair_nat_nat @ A2 @ B2 ) )
         => ( C @ A2 @ B2 ) )
     => ( produc6081775807080527818_nat_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_518_case__prodI,axiom,
    ! [F2: product_prod_nat_nat > product_prod_nat_nat > $o,A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( ( F2 @ A @ B )
     => ( produc6590410687421337004_nat_o @ F2 @ ( produc6161850002892822231at_nat @ A @ B ) ) ) ).

% case_prodI
thf(fact_519_case__prodI,axiom,
    ! [F2: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( F2 @ A @ B )
     => ( produc410239310623530412_nat_o @ F2 @ ( produc2922128104949294807at_nat @ A @ B ) ) ) ).

% case_prodI
thf(fact_520_case__prodI,axiom,
    ! [F2: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( F2 @ A @ B )
     => ( produc9020218426428501292_nat_o @ F2 @ ( produc9060074326276436823at_nat @ A @ B ) ) ) ).

% case_prodI
thf(fact_521_case__prodI,axiom,
    ! [F2: nat > nat > $o,A: nat,B: nat] :
      ( ( F2 @ A @ B )
     => ( produc6081775807080527818_nat_o @ F2 @ ( product_Pair_nat_nat @ A @ B ) ) ) ).

% case_prodI
thf(fact_522_card__Collect__le__nat,axiom,
    ! [N: nat] :
      ( ( finite_card_nat
        @ ( collect_nat
          @ ^ [I: nat] : ( ord_less_eq_nat @ I @ N ) ) )
      = ( suc @ N ) ) ).

% card_Collect_le_nat
thf(fact_523_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_nat,F2: nat > nat] :
      ( ( finite_finite_nat @ S2 )
     => ( ( S2 != bot_bot_set_nat )
       => ~ ? [X6: nat] :
              ( ( member_nat @ X6 @ S2 )
              & ( ord_less_nat @ ( F2 @ X6 ) @ ( F2 @ ( lattic7446932960582359483at_nat @ F2 @ S2 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_524_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_Pr1261947904930325089at_nat,F2: product_prod_nat_nat > nat] :
      ( ( finite6177210948735845034at_nat @ S2 )
     => ( ( S2 != bot_bo2099793752762293965at_nat )
       => ~ ? [X6: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X6 @ S2 )
              & ( ord_less_nat @ ( F2 @ X6 ) @ ( F2 @ ( lattic4984276347100956536at_nat @ F2 @ S2 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_525_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_Pr4329608150637261639at_nat,F2: produc3843707927480180839at_nat > nat] :
      ( ( finite4343798906461161616at_nat @ S2 )
     => ( ( S2 != bot_bo228742789529271731at_nat )
       => ~ ? [X6: produc3843707927480180839at_nat] :
              ( ( member8757157785044589968at_nat @ X6 @ S2 )
              & ( ord_less_nat @ ( F2 @ X6 ) @ ( F2 @ ( lattic3831113108725116382at_nat @ F2 @ S2 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_526_arg__min__least,axiom,
    ! [S2: set_Pr8693737435421807431at_nat,Y: produc859450856879609959at_nat,F2: produc859450856879609959at_nat > nat] :
      ( ( finite4392333629123659920at_nat @ S2 )
     => ( ( S2 != bot_bo5327735625951526323at_nat )
       => ( ( member8206827879206165904at_nat @ Y @ S2 )
         => ( ord_less_eq_nat @ ( F2 @ ( lattic390166758595302878at_nat @ F2 @ S2 ) ) @ ( F2 @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_527_arg__min__least,axiom,
    ! [S2: set_nat,Y: nat,F2: nat > nat] :
      ( ( finite_finite_nat @ S2 )
     => ( ( S2 != bot_bot_set_nat )
       => ( ( member_nat @ Y @ S2 )
         => ( ord_less_eq_nat @ ( F2 @ ( lattic7446932960582359483at_nat @ F2 @ S2 ) ) @ ( F2 @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_528_arg__min__least,axiom,
    ! [S2: set_Pr1261947904930325089at_nat,Y: product_prod_nat_nat,F2: product_prod_nat_nat > nat] :
      ( ( finite6177210948735845034at_nat @ S2 )
     => ( ( S2 != bot_bo2099793752762293965at_nat )
       => ( ( member8440522571783428010at_nat @ Y @ S2 )
         => ( ord_less_eq_nat @ ( F2 @ ( lattic4984276347100956536at_nat @ F2 @ S2 ) ) @ ( F2 @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_529_arg__min__least,axiom,
    ! [S2: set_Pr4329608150637261639at_nat,Y: produc3843707927480180839at_nat,F2: produc3843707927480180839at_nat > nat] :
      ( ( finite4343798906461161616at_nat @ S2 )
     => ( ( S2 != bot_bo228742789529271731at_nat )
       => ( ( member8757157785044589968at_nat @ Y @ S2 )
         => ( ord_less_eq_nat @ ( F2 @ ( lattic3831113108725116382at_nat @ F2 @ S2 ) ) @ ( F2 @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_530_transitive__stepwise__le,axiom,
    ! [M: nat,N: nat,R: nat > nat > $o] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ! [X3: nat] : ( R @ X3 @ X3 )
       => ( ! [X3: nat,Y3: nat,Z3: nat] :
              ( ( R @ X3 @ Y3 )
             => ( ( R @ Y3 @ Z3 )
               => ( R @ X3 @ Z3 ) ) )
         => ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
           => ( R @ M @ N ) ) ) ) ) ).

% transitive_stepwise_le
thf(fact_531_nat__induct__at__least,axiom,
    ! [M: nat,N: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( P @ M )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ M @ N2 )
             => ( ( P @ N2 )
               => ( P @ ( suc @ N2 ) ) ) )
         => ( P @ N ) ) ) ) ).

% nat_induct_at_least
thf(fact_532_full__nat__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N2: nat] :
          ( ! [M3: nat] :
              ( ( ord_less_eq_nat @ ( suc @ M3 ) @ N2 )
             => ( P @ M3 ) )
         => ( P @ N2 ) )
     => ( P @ N ) ) ).

% full_nat_induct
thf(fact_533_not__less__eq__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ~ ( ord_less_eq_nat @ M @ N ) )
      = ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).

% not_less_eq_eq
thf(fact_534_Suc__n__not__le__n,axiom,
    ! [N: nat] :
      ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).

% Suc_n_not_le_n
thf(fact_535_le__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
      = ( ( ord_less_eq_nat @ M @ N )
        | ( M
          = ( suc @ N ) ) ) ) ).

% le_Suc_eq
thf(fact_536_Suc__le__D,axiom,
    ! [N: nat,M6: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
     => ? [M2: nat] :
          ( M6
          = ( suc @ M2 ) ) ) ).

% Suc_le_D
thf(fact_537_le__SucI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).

% le_SucI
thf(fact_538_le__SucE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ~ ( ord_less_eq_nat @ M @ N )
       => ( M
          = ( suc @ N ) ) ) ) ).

% le_SucE
thf(fact_539_Suc__leD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% Suc_leD
thf(fact_540_le__bot,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ bot_bo2099793752762293965at_nat )
     => ( A = bot_bo2099793752762293965at_nat ) ) ).

% le_bot
thf(fact_541_le__bot,axiom,
    ! [A: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A @ bot_bo228742789529271731at_nat )
     => ( A = bot_bo228742789529271731at_nat ) ) ).

% le_bot
thf(fact_542_le__bot,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ bot_bot_nat )
     => ( A = bot_bot_nat ) ) ).

% le_bot
thf(fact_543_bot__least,axiom,
    ! [A: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ bot_bo2099793752762293965at_nat @ A ) ).

% bot_least
thf(fact_544_bot__least,axiom,
    ! [A: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ bot_bo228742789529271731at_nat @ A ) ).

% bot_least
thf(fact_545_bot__least,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).

% bot_least
thf(fact_546_bot__unique,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ bot_bo2099793752762293965at_nat )
      = ( A = bot_bo2099793752762293965at_nat ) ) ).

% bot_unique
thf(fact_547_bot__unique,axiom,
    ! [A: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A @ bot_bo228742789529271731at_nat )
      = ( A = bot_bo228742789529271731at_nat ) ) ).

% bot_unique
thf(fact_548_bot__unique,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ bot_bot_nat )
      = ( A = bot_bot_nat ) ) ).

% bot_unique
thf(fact_549_less__by__empty,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( A4 = bot_bo2099793752762293965at_nat )
     => ( ord_le3146513528884898305at_nat @ A4 @ B4 ) ) ).

% less_by_empty
thf(fact_550_less__by__empty,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( A4 = bot_bo228742789529271731at_nat )
     => ( ord_le1268244103169919719at_nat @ A4 @ B4 ) ) ).

% less_by_empty
thf(fact_551_not__psubset__empty,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ~ ( ord_le7866589430770878221at_nat @ A4 @ bot_bo2099793752762293965at_nat ) ).

% not_psubset_empty
thf(fact_552_not__psubset__empty,axiom,
    ! [A4: set_Pr4329608150637261639at_nat] :
      ~ ( ord_le2604355607129572851at_nat @ A4 @ bot_bo228742789529271731at_nat ) ).

% not_psubset_empty
thf(fact_553_finite__set__plus,axiom,
    ! [S: set_nat,T: set_nat] :
      ( ( finite_finite_nat @ S )
     => ( ( finite_finite_nat @ T )
       => ( finite_finite_nat @ ( plus_plus_set_nat @ S @ T ) ) ) ) ).

% finite_set_plus
thf(fact_554_finite__set__plus,axiom,
    ! [S: set_Pr4329608150637261639at_nat,T: set_Pr4329608150637261639at_nat] :
      ( ( finite4343798906461161616at_nat @ S )
     => ( ( finite4343798906461161616at_nat @ T )
       => ( finite4343798906461161616at_nat @ ( plus_p5270993086169515006at_nat @ S @ T ) ) ) ) ).

% finite_set_plus
thf(fact_555_finite__set__plus,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ S )
     => ( ( finite6177210948735845034at_nat @ T )
       => ( finite6177210948735845034at_nat @ ( plus_p2407200367410362776at_nat @ S @ T ) ) ) ) ).

% finite_set_plus
thf(fact_556_plus__nat_Osimps_I2_J,axiom,
    ! [M: nat,N: nat] :
      ( ( plus_plus_nat @ ( suc @ M ) @ N )
      = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).

% plus_nat.simps(2)
thf(fact_557_nat__arith_Osuc1,axiom,
    ! [A4: nat,K: nat,A: nat] :
      ( ( A4
        = ( plus_plus_nat @ K @ A ) )
     => ( ( suc @ A4 )
        = ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).

% nat_arith.suc1
thf(fact_558_add__Suc__shift,axiom,
    ! [M: nat,N: nat] :
      ( ( plus_plus_nat @ ( suc @ M ) @ N )
      = ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).

% add_Suc_shift
thf(fact_559_inj__Suc,axiom,
    ! [N4: set_nat] : ( inj_on_nat_nat @ suc @ N4 ) ).

% inj_Suc
thf(fact_560_prod_Ocase__distrib,axiom,
    ! [H: $o > $o,F2: nat > nat > $o,Prod: product_prod_nat_nat] :
      ( ( H @ ( produc6081775807080527818_nat_o @ F2 @ Prod ) )
      = ( produc6081775807080527818_nat_o
        @ ^ [X13: nat,X24: nat] : ( H @ ( F2 @ X13 @ X24 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_561_n__not__Suc__n,axiom,
    ! [N: nat] :
      ( N
     != ( suc @ N ) ) ).

% n_not_Suc_n
thf(fact_562_Suc__inject,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( suc @ X )
        = ( suc @ Y ) )
     => ( X = Y ) ) ).

% Suc_inject
thf(fact_563_prod_Odisc__eq__case,axiom,
    ! [Prod: product_prod_nat_nat] :
      ( produc6081775807080527818_nat_o
      @ ^ [Uu: nat,Uv: nat] : $true
      @ Prod ) ).

% prod.disc_eq_case
thf(fact_564_ex__in__conv,axiom,
    ! [A4: set_Pr8693737435421807431at_nat] :
      ( ( ? [X4: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X4 @ A4 ) )
      = ( A4 != bot_bo5327735625951526323at_nat ) ) ).

% ex_in_conv
thf(fact_565_ex__in__conv,axiom,
    ! [A4: set_nat] :
      ( ( ? [X4: nat] : ( member_nat @ X4 @ A4 ) )
      = ( A4 != bot_bot_set_nat ) ) ).

% ex_in_conv
thf(fact_566_ex__in__conv,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ( ? [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ A4 ) )
      = ( A4 != bot_bo2099793752762293965at_nat ) ) ).

% ex_in_conv
thf(fact_567_ex__in__conv,axiom,
    ! [A4: set_Pr4329608150637261639at_nat] :
      ( ( ? [X4: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X4 @ A4 ) )
      = ( A4 != bot_bo228742789529271731at_nat ) ) ).

% ex_in_conv
thf(fact_568_Set_Oempty__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat
      @ ^ [X4: nat] : $false ) ) ).

% Set.empty_def
thf(fact_569_Set_Oempty__def,axiom,
    ( bot_bo2099793752762293965at_nat
    = ( collec3392354462482085612at_nat
      @ ^ [X4: product_prod_nat_nat] : $false ) ) ).

% Set.empty_def
thf(fact_570_Set_Oempty__def,axiom,
    ( bot_bo228742789529271731at_nat
    = ( collec6321179662152712658at_nat
      @ ^ [X4: produc3843707927480180839at_nat] : $false ) ) ).

% Set.empty_def
thf(fact_571_equals0I,axiom,
    ! [A4: set_Pr8693737435421807431at_nat] :
      ( ! [Y3: produc859450856879609959at_nat] :
          ~ ( member8206827879206165904at_nat @ Y3 @ A4 )
     => ( A4 = bot_bo5327735625951526323at_nat ) ) ).

% equals0I
thf(fact_572_equals0I,axiom,
    ! [A4: set_nat] :
      ( ! [Y3: nat] :
          ~ ( member_nat @ Y3 @ A4 )
     => ( A4 = bot_bot_set_nat ) ) ).

% equals0I
thf(fact_573_equals0I,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ! [Y3: product_prod_nat_nat] :
          ~ ( member8440522571783428010at_nat @ Y3 @ A4 )
     => ( A4 = bot_bo2099793752762293965at_nat ) ) ).

% equals0I
thf(fact_574_equals0I,axiom,
    ! [A4: set_Pr4329608150637261639at_nat] :
      ( ! [Y3: produc3843707927480180839at_nat] :
          ~ ( member8757157785044589968at_nat @ Y3 @ A4 )
     => ( A4 = bot_bo228742789529271731at_nat ) ) ).

% equals0I
thf(fact_575_equals0D,axiom,
    ! [A4: set_Pr8693737435421807431at_nat,A: produc859450856879609959at_nat] :
      ( ( A4 = bot_bo5327735625951526323at_nat )
     => ~ ( member8206827879206165904at_nat @ A @ A4 ) ) ).

% equals0D
thf(fact_576_equals0D,axiom,
    ! [A4: set_nat,A: nat] :
      ( ( A4 = bot_bot_set_nat )
     => ~ ( member_nat @ A @ A4 ) ) ).

% equals0D
thf(fact_577_equals0D,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( ( A4 = bot_bo2099793752762293965at_nat )
     => ~ ( member8440522571783428010at_nat @ A @ A4 ) ) ).

% equals0D
thf(fact_578_equals0D,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
      ( ( A4 = bot_bo228742789529271731at_nat )
     => ~ ( member8757157785044589968at_nat @ A @ A4 ) ) ).

% equals0D
thf(fact_579_emptyE,axiom,
    ! [A: produc859450856879609959at_nat] :
      ~ ( member8206827879206165904at_nat @ A @ bot_bo5327735625951526323at_nat ) ).

% emptyE
thf(fact_580_emptyE,axiom,
    ! [A: nat] :
      ~ ( member_nat @ A @ bot_bot_set_nat ) ).

% emptyE
thf(fact_581_emptyE,axiom,
    ! [A: product_prod_nat_nat] :
      ~ ( member8440522571783428010at_nat @ A @ bot_bo2099793752762293965at_nat ) ).

% emptyE
thf(fact_582_emptyE,axiom,
    ! [A: produc3843707927480180839at_nat] :
      ~ ( member8757157785044589968at_nat @ A @ bot_bo228742789529271731at_nat ) ).

% emptyE
thf(fact_583_not__less__less__Suc__eq,axiom,
    ! [N: nat,M: nat] :
      ( ~ ( ord_less_nat @ N @ M )
     => ( ( ord_less_nat @ N @ ( suc @ M ) )
        = ( N = M ) ) ) ).

% not_less_less_Suc_eq
thf(fact_584_strict__inc__induct,axiom,
    ! [I3: nat,J: nat,P: nat > $o] :
      ( ( ord_less_nat @ I3 @ J )
     => ( ! [I2: nat] :
            ( ( J
              = ( suc @ I2 ) )
           => ( P @ I2 ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ J )
             => ( ( P @ ( suc @ I2 ) )
               => ( P @ I2 ) ) )
         => ( P @ I3 ) ) ) ) ).

% strict_inc_induct
thf(fact_585_less__Suc__induct,axiom,
    ! [I3: nat,J: nat,P: nat > nat > $o] :
      ( ( ord_less_nat @ I3 @ J )
     => ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
       => ( ! [I2: nat,J2: nat,K3: nat] :
              ( ( ord_less_nat @ I2 @ J2 )
             => ( ( ord_less_nat @ J2 @ K3 )
               => ( ( P @ I2 @ J2 )
                 => ( ( P @ J2 @ K3 )
                   => ( P @ I2 @ K3 ) ) ) ) )
         => ( P @ I3 @ J ) ) ) ) ).

% less_Suc_induct
thf(fact_586_less__trans__Suc,axiom,
    ! [I3: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I3 @ J )
     => ( ( ord_less_nat @ J @ K )
       => ( ord_less_nat @ ( suc @ I3 ) @ K ) ) ) ).

% less_trans_Suc
thf(fact_587_Suc__less__SucD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
     => ( ord_less_nat @ M @ N ) ) ).

% Suc_less_SucD
thf(fact_588_less__antisym,axiom,
    ! [N: nat,M: nat] :
      ( ~ ( ord_less_nat @ N @ M )
     => ( ( ord_less_nat @ N @ ( suc @ M ) )
       => ( M = N ) ) ) ).

% less_antisym
thf(fact_589_Suc__less__eq2,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ ( suc @ N ) @ M )
      = ( ? [M7: nat] :
            ( ( M
              = ( suc @ M7 ) )
            & ( ord_less_nat @ N @ M7 ) ) ) ) ).

% Suc_less_eq2
thf(fact_590_All__less__Suc,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( suc @ N ) )
           => ( P @ I ) ) )
      = ( ( P @ N )
        & ! [I: nat] :
            ( ( ord_less_nat @ I @ N )
           => ( P @ I ) ) ) ) ).

% All_less_Suc
thf(fact_591_not__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ~ ( ord_less_nat @ M @ N ) )
      = ( ord_less_nat @ N @ ( suc @ M ) ) ) ).

% not_less_eq
thf(fact_592_less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
      = ( ( ord_less_nat @ M @ N )
        | ( M = N ) ) ) ).

% less_Suc_eq
thf(fact_593_Ex__less__Suc,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [I: nat] :
            ( ( ord_less_nat @ I @ ( suc @ N ) )
            & ( P @ I ) ) )
      = ( ( P @ N )
        | ? [I: nat] :
            ( ( ord_less_nat @ I @ N )
            & ( P @ I ) ) ) ) ).

% Ex_less_Suc
thf(fact_594_less__SucI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).

% less_SucI
thf(fact_595_less__SucE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
     => ( ~ ( ord_less_nat @ M @ N )
       => ( M = N ) ) ) ).

% less_SucE
thf(fact_596_Suc__lessI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( ( suc @ M )
         != N )
       => ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).

% Suc_lessI
thf(fact_597_Suc__lessE,axiom,
    ! [I3: nat,K: nat] :
      ( ( ord_less_nat @ ( suc @ I3 ) @ K )
     => ~ ! [J2: nat] :
            ( ( ord_less_nat @ I3 @ J2 )
           => ( K
             != ( suc @ J2 ) ) ) ) ).

% Suc_lessE
thf(fact_598_Suc__lessD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ N )
     => ( ord_less_nat @ M @ N ) ) ).

% Suc_lessD
thf(fact_599_Nat_OlessE,axiom,
    ! [I3: nat,K: nat] :
      ( ( ord_less_nat @ I3 @ K )
     => ( ( K
         != ( suc @ I3 ) )
       => ~ ! [J2: nat] :
              ( ( ord_less_nat @ I3 @ J2 )
             => ( K
               != ( suc @ J2 ) ) ) ) ) ).

% Nat.lessE
thf(fact_600_bot_Onot__eq__extremum,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( A != bot_bo2099793752762293965at_nat )
      = ( ord_le7866589430770878221at_nat @ bot_bo2099793752762293965at_nat @ A ) ) ).

% bot.not_eq_extremum
thf(fact_601_bot_Onot__eq__extremum,axiom,
    ! [A: set_Pr4329608150637261639at_nat] :
      ( ( A != bot_bo228742789529271731at_nat )
      = ( ord_le2604355607129572851at_nat @ bot_bo228742789529271731at_nat @ A ) ) ).

% bot.not_eq_extremum
thf(fact_602_bot_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != bot_bot_nat )
      = ( ord_less_nat @ bot_bot_nat @ A ) ) ).

% bot.not_eq_extremum
thf(fact_603_bot_Oextremum__strict,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ~ ( ord_le7866589430770878221at_nat @ A @ bot_bo2099793752762293965at_nat ) ).

% bot.extremum_strict
thf(fact_604_bot_Oextremum__strict,axiom,
    ! [A: set_Pr4329608150637261639at_nat] :
      ~ ( ord_le2604355607129572851at_nat @ A @ bot_bo228742789529271731at_nat ) ).

% bot.extremum_strict
thf(fact_605_bot_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ bot_bot_nat ) ).

% bot.extremum_strict
thf(fact_606_ex__min__if__finite,axiom,
    ! [S2: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ S2 )
     => ( ( S2 != bot_bo2099793752762293965at_nat )
       => ? [X3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ S2 )
            & ~ ? [Xa2: product_prod_nat_nat] :
                  ( ( member8440522571783428010at_nat @ Xa2 @ S2 )
                  & ( ord_le1203424502768444845at_nat @ Xa2 @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_607_ex__min__if__finite,axiom,
    ! [S2: set_Pr4329608150637261639at_nat] :
      ( ( finite4343798906461161616at_nat @ S2 )
     => ( ( S2 != bot_bo228742789529271731at_nat )
       => ? [X3: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X3 @ S2 )
            & ~ ? [Xa2: produc3843707927480180839at_nat] :
                  ( ( member8757157785044589968at_nat @ Xa2 @ S2 )
                  & ( ord_le8864659147655897875at_nat @ Xa2 @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_608_ex__min__if__finite,axiom,
    ! [S2: set_nat] :
      ( ( finite_finite_nat @ S2 )
     => ( ( S2 != bot_bot_set_nat )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ S2 )
            & ~ ? [Xa2: nat] :
                  ( ( member_nat @ Xa2 @ S2 )
                  & ( ord_less_nat @ Xa2 @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_609_infinite__growing,axiom,
    ! [X5: set_nat] :
      ( ( X5 != bot_bot_set_nat )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ X5 )
           => ? [Xa2: nat] :
                ( ( member_nat @ Xa2 @ X5 )
                & ( ord_less_nat @ X3 @ Xa2 ) ) )
       => ~ ( finite_finite_nat @ X5 ) ) ) ).

% infinite_growing
thf(fact_610_case__prodD,axiom,
    ! [F2: product_prod_nat_nat > product_prod_nat_nat > $o,A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( ( produc6590410687421337004_nat_o @ F2 @ ( produc6161850002892822231at_nat @ A @ B ) )
     => ( F2 @ A @ B ) ) ).

% case_prodD
thf(fact_611_case__prodD,axiom,
    ! [F2: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( produc410239310623530412_nat_o @ F2 @ ( produc2922128104949294807at_nat @ A @ B ) )
     => ( F2 @ A @ B ) ) ).

% case_prodD
thf(fact_612_case__prodD,axiom,
    ! [F2: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( produc9020218426428501292_nat_o @ F2 @ ( produc9060074326276436823at_nat @ A @ B ) )
     => ( F2 @ A @ B ) ) ).

% case_prodD
thf(fact_613_case__prodD,axiom,
    ! [F2: nat > nat > $o,A: nat,B: nat] :
      ( ( produc6081775807080527818_nat_o @ F2 @ ( product_Pair_nat_nat @ A @ B ) )
     => ( F2 @ A @ B ) ) ).

% case_prodD
thf(fact_614_case__prodE,axiom,
    ! [C: product_prod_nat_nat > product_prod_nat_nat > $o,P2: produc859450856879609959at_nat] :
      ( ( produc6590410687421337004_nat_o @ C @ P2 )
     => ~ ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
            ( ( P2
              = ( produc6161850002892822231at_nat @ X3 @ Y3 ) )
           => ~ ( C @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_615_case__prodE,axiom,
    ! [C: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o,P2: produc3843707927480180839at_nat] :
      ( ( produc410239310623530412_nat_o @ C @ P2 )
     => ~ ! [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
            ( ( P2
              = ( produc2922128104949294807at_nat @ X3 @ Y3 ) )
           => ~ ( C @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_616_case__prodE,axiom,
    ! [C: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o,P2: produc1319942482725812455at_nat] :
      ( ( produc9020218426428501292_nat_o @ C @ P2 )
     => ~ ! [X3: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] :
            ( ( P2
              = ( produc9060074326276436823at_nat @ X3 @ Y3 ) )
           => ~ ( C @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_617_case__prodE,axiom,
    ! [C: nat > nat > $o,P2: product_prod_nat_nat] :
      ( ( produc6081775807080527818_nat_o @ C @ P2 )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P2
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( C @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_618_Pair__bot__bot,axiom,
    ( ( product_Pair_nat_nat @ bot_bot_nat @ bot_bot_nat )
    = bot_bo2769642828321324397at_nat ) ).

% Pair_bot_bot
thf(fact_619_Pair__bot__bot,axiom,
    ( ( produc6161850002892822231at_nat @ bot_bo2769642828321324397at_nat @ bot_bo2769642828321324397at_nat )
    = bot_bo7480491830437098195at_nat ) ).

% Pair_bot_bot
thf(fact_620_Pair__bot__bot,axiom,
    ( ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ bot_bo2099793752762293965at_nat )
    = bot_bo4893629854819490515at_nat ) ).

% Pair_bot_bot
thf(fact_621_Pair__bot__bot,axiom,
    ( ( produc8551506612059717309at_nat @ bot_bo2099793752762293965at_nat @ bot_bo228742789529271731at_nat )
    = bot_bo5829401510961248825at_nat ) ).

% Pair_bot_bot
thf(fact_622_Pair__bot__bot,axiom,
    ( ( produc6644067809570830961at_nat @ bot_bo228742789529271731at_nat @ bot_bo2099793752762293965at_nat )
    = bot_bo3011160614121672429at_nat ) ).

% Pair_bot_bot
thf(fact_623_Pair__bot__bot,axiom,
    ( ( produc9060074326276436823at_nat @ bot_bo228742789529271731at_nat @ bot_bo228742789529271731at_nat )
    = bot_bo2226985256079609683at_nat ) ).

% Pair_bot_bot
thf(fact_624_split__cong,axiom,
    ! [Q2: product_prod_nat_nat,F2: nat > nat > $o,G: nat > nat > $o,P2: product_prod_nat_nat] :
      ( ! [X3: nat,Y3: nat] :
          ( ( ( product_Pair_nat_nat @ X3 @ Y3 )
            = Q2 )
         => ( ( F2 @ X3 @ Y3 )
            = ( G @ X3 @ Y3 ) ) )
     => ( ( P2 = Q2 )
       => ( ( produc6081775807080527818_nat_o @ F2 @ P2 )
          = ( produc6081775807080527818_nat_o @ G @ Q2 ) ) ) ) ).

% split_cong
thf(fact_625_prod_Osimps_I2_J,axiom,
    ! [F2: nat > nat > $o,X1: nat,X22: nat] :
      ( ( produc6081775807080527818_nat_o @ F2 @ ( product_Pair_nat_nat @ X1 @ X22 ) )
      = ( F2 @ X1 @ X22 ) ) ).

% prod.simps(2)
thf(fact_626_cond__case__prod__eta,axiom,
    ! [F2: nat > nat > $o,G: product_prod_nat_nat > $o] :
      ( ! [X3: nat,Y3: nat] :
          ( ( F2 @ X3 @ Y3 )
          = ( G @ ( product_Pair_nat_nat @ X3 @ Y3 ) ) )
     => ( ( produc6081775807080527818_nat_o @ F2 )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_627_case__prod__eta,axiom,
    ! [F2: product_prod_nat_nat > $o] :
      ( ( produc6081775807080527818_nat_o
        @ ^ [X4: nat,Y4: nat] : ( F2 @ ( product_Pair_nat_nat @ X4 @ Y4 ) ) )
      = F2 ) ).

% case_prod_eta
thf(fact_628_case__prodE2,axiom,
    ! [Q: $o > $o,P: nat > nat > $o,Z: product_prod_nat_nat] :
      ( ( Q @ ( produc6081775807080527818_nat_o @ P @ Z ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( Z
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( Q @ ( P @ X3 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_629_finite__psubset__induct,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
      ( ( finite6177210948735845034at_nat @ A4 )
     => ( ! [A7: set_Pr1261947904930325089at_nat] :
            ( ( finite6177210948735845034at_nat @ A7 )
           => ( ! [B7: set_Pr1261947904930325089at_nat] :
                  ( ( ord_le7866589430770878221at_nat @ B7 @ A7 )
                 => ( P @ B7 ) )
             => ( P @ A7 ) ) )
       => ( P @ A4 ) ) ) ).

% finite_psubset_induct
thf(fact_630_finite__psubset__induct,axiom,
    ! [A4: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A4 )
     => ( ! [A7: set_nat] :
            ( ( finite_finite_nat @ A7 )
           => ( ! [B7: set_nat] :
                  ( ( ord_less_set_nat @ B7 @ A7 )
                 => ( P @ B7 ) )
             => ( P @ A7 ) ) )
       => ( P @ A4 ) ) ) ).

% finite_psubset_induct
thf(fact_631_pigeonhole__infinite__rel,axiom,
    ! [A4: set_Pr8693737435421807431at_nat,B4: set_Pr1261947904930325089at_nat,R: produc859450856879609959at_nat > product_prod_nat_nat > $o] :
      ( ~ ( finite4392333629123659920at_nat @ A4 )
     => ( ( finite6177210948735845034at_nat @ B4 )
       => ( ! [X3: produc859450856879609959at_nat] :
              ( ( member8206827879206165904at_nat @ X3 @ A4 )
             => ? [Xa2: product_prod_nat_nat] :
                  ( ( member8440522571783428010at_nat @ Xa2 @ B4 )
                  & ( R @ X3 @ Xa2 ) ) )
         => ? [X3: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X3 @ B4 )
              & ~ ( finite4392333629123659920at_nat
                  @ ( collec7088162979684241874at_nat
                    @ ^ [A3: produc859450856879609959at_nat] :
                        ( ( member8206827879206165904at_nat @ A3 @ A4 )
                        & ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_632_pigeonhole__infinite__rel,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr1261947904930325089at_nat,R: produc3843707927480180839at_nat > product_prod_nat_nat > $o] :
      ( ~ ( finite4343798906461161616at_nat @ A4 )
     => ( ( finite6177210948735845034at_nat @ B4 )
       => ( ! [X3: produc3843707927480180839at_nat] :
              ( ( member8757157785044589968at_nat @ X3 @ A4 )
             => ? [Xa2: product_prod_nat_nat] :
                  ( ( member8440522571783428010at_nat @ Xa2 @ B4 )
                  & ( R @ X3 @ Xa2 ) ) )
         => ? [X3: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X3 @ B4 )
              & ~ ( finite4343798906461161616at_nat
                  @ ( collec6321179662152712658at_nat
                    @ ^ [A3: produc3843707927480180839at_nat] :
                        ( ( member8757157785044589968at_nat @ A3 @ A4 )
                        & ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_633_pigeonhole__infinite__rel,axiom,
    ! [A4: set_Pr8693737435421807431at_nat,B4: set_nat,R: produc859450856879609959at_nat > nat > $o] :
      ( ~ ( finite4392333629123659920at_nat @ A4 )
     => ( ( finite_finite_nat @ B4 )
       => ( ! [X3: produc859450856879609959at_nat] :
              ( ( member8206827879206165904at_nat @ X3 @ A4 )
             => ? [Xa2: nat] :
                  ( ( member_nat @ Xa2 @ B4 )
                  & ( R @ X3 @ Xa2 ) ) )
         => ? [X3: nat] :
              ( ( member_nat @ X3 @ B4 )
              & ~ ( finite4392333629123659920at_nat
                  @ ( collec7088162979684241874at_nat
                    @ ^ [A3: produc859450856879609959at_nat] :
                        ( ( member8206827879206165904at_nat @ A3 @ A4 )
                        & ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_634_pigeonhole__infinite__rel,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_nat,R: produc3843707927480180839at_nat > nat > $o] :
      ( ~ ( finite4343798906461161616at_nat @ A4 )
     => ( ( finite_finite_nat @ B4 )
       => ( ! [X3: produc3843707927480180839at_nat] :
              ( ( member8757157785044589968at_nat @ X3 @ A4 )
             => ? [Xa2: nat] :
                  ( ( member_nat @ Xa2 @ B4 )
                  & ( R @ X3 @ Xa2 ) ) )
         => ? [X3: nat] :
              ( ( member_nat @ X3 @ B4 )
              & ~ ( finite4343798906461161616at_nat
                  @ ( collec6321179662152712658at_nat
                    @ ^ [A3: produc3843707927480180839at_nat] :
                        ( ( member8757157785044589968at_nat @ A3 @ A4 )
                        & ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_635_pigeonhole__infinite__rel,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,R: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ~ ( finite6177210948735845034at_nat @ A4 )
     => ( ( finite6177210948735845034at_nat @ B4 )
       => ( ! [X3: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X3 @ A4 )
             => ? [Xa2: product_prod_nat_nat] :
                  ( ( member8440522571783428010at_nat @ Xa2 @ B4 )
                  & ( R @ X3 @ Xa2 ) ) )
         => ? [X3: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X3 @ B4 )
              & ~ ( finite6177210948735845034at_nat
                  @ ( collec3392354462482085612at_nat
                    @ ^ [A3: product_prod_nat_nat] :
                        ( ( member8440522571783428010at_nat @ A3 @ A4 )
                        & ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_636_pigeonhole__infinite__rel,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_nat,R: product_prod_nat_nat > nat > $o] :
      ( ~ ( finite6177210948735845034at_nat @ A4 )
     => ( ( finite_finite_nat @ B4 )
       => ( ! [X3: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X3 @ A4 )
             => ? [Xa2: nat] :
                  ( ( member_nat @ Xa2 @ B4 )
                  & ( R @ X3 @ Xa2 ) ) )
         => ? [X3: nat] :
              ( ( member_nat @ X3 @ B4 )
              & ~ ( finite6177210948735845034at_nat
                  @ ( collec3392354462482085612at_nat
                    @ ^ [A3: product_prod_nat_nat] :
                        ( ( member8440522571783428010at_nat @ A3 @ A4 )
                        & ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_637_pigeonhole__infinite__rel,axiom,
    ! [A4: set_nat,B4: set_Pr1261947904930325089at_nat,R: nat > product_prod_nat_nat > $o] :
      ( ~ ( finite_finite_nat @ A4 )
     => ( ( finite6177210948735845034at_nat @ B4 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ A4 )
             => ? [Xa2: product_prod_nat_nat] :
                  ( ( member8440522571783428010at_nat @ Xa2 @ B4 )
                  & ( R @ X3 @ Xa2 ) ) )
         => ? [X3: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X3 @ B4 )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A3: nat] :
                        ( ( member_nat @ A3 @ A4 )
                        & ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_638_pigeonhole__infinite__rel,axiom,
    ! [A4: set_nat,B4: set_nat,R: nat > nat > $o] :
      ( ~ ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B4 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ A4 )
             => ? [Xa2: nat] :
                  ( ( member_nat @ Xa2 @ B4 )
                  & ( R @ X3 @ Xa2 ) ) )
         => ? [X3: nat] :
              ( ( member_nat @ X3 @ B4 )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A3: nat] :
                        ( ( member_nat @ A3 @ A4 )
                        & ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_639_not__finite__existsD,axiom,
    ! [P: product_prod_nat_nat > $o] :
      ( ~ ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P ) )
     => ? [X_1: product_prod_nat_nat] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_640_not__finite__existsD,axiom,
    ! [P: nat > $o] :
      ( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
     => ? [X_1: nat] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_641_finite__has__maximal,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A4 )
     => ( ( A4 != bot_bo2099793752762293965at_nat )
       => ? [X3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ A4 )
            & ! [Xa2: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ Xa2 @ A4 )
               => ( ( ord_le8460144461188290721at_nat @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_642_finite__has__maximal,axiom,
    ! [A4: set_Pr4329608150637261639at_nat] :
      ( ( finite4343798906461161616at_nat @ A4 )
     => ( ( A4 != bot_bo228742789529271731at_nat )
       => ? [X3: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X3 @ A4 )
            & ! [Xa2: produc3843707927480180839at_nat] :
                ( ( member8757157785044589968at_nat @ Xa2 @ A4 )
               => ( ( ord_le8744046415670163463at_nat @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_643_finite__has__maximal,axiom,
    ! [A4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A4 )
            & ! [Xa2: nat] :
                ( ( member_nat @ Xa2 @ A4 )
               => ( ( ord_less_eq_nat @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_644_finite__has__minimal,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A4 )
     => ( ( A4 != bot_bo2099793752762293965at_nat )
       => ? [X3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ A4 )
            & ! [Xa2: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ Xa2 @ A4 )
               => ( ( ord_le8460144461188290721at_nat @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_645_finite__has__minimal,axiom,
    ! [A4: set_Pr4329608150637261639at_nat] :
      ( ( finite4343798906461161616at_nat @ A4 )
     => ( ( A4 != bot_bo228742789529271731at_nat )
       => ? [X3: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X3 @ A4 )
            & ! [Xa2: produc3843707927480180839at_nat] :
                ( ( member8757157785044589968at_nat @ Xa2 @ A4 )
               => ( ( ord_le8744046415670163463at_nat @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_646_finite__has__minimal,axiom,
    ! [A4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A4 )
            & ! [Xa2: nat] :
                ( ( member_nat @ Xa2 @ A4 )
               => ( ( ord_less_eq_nat @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_647_finite__has__maximal2,axiom,
    ! [A4: set_Pr8693737435421807431at_nat,A: produc859450856879609959at_nat] :
      ( ( finite4392333629123659920at_nat @ A4 )
     => ( ( member8206827879206165904at_nat @ A @ A4 )
       => ? [X3: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X3 @ A4 )
            & ( ord_le6722121967309221895at_nat @ A @ X3 )
            & ! [Xa2: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ Xa2 @ A4 )
               => ( ( ord_le6722121967309221895at_nat @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_648_finite__has__maximal2,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
      ( ( finite4343798906461161616at_nat @ A4 )
     => ( ( member8757157785044589968at_nat @ A @ A4 )
       => ? [X3: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X3 @ A4 )
            & ( ord_le8744046415670163463at_nat @ A @ X3 )
            & ! [Xa2: produc3843707927480180839at_nat] :
                ( ( member8757157785044589968at_nat @ Xa2 @ A4 )
               => ( ( ord_le8744046415670163463at_nat @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_649_finite__has__maximal2,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( ( finite6177210948735845034at_nat @ A4 )
     => ( ( member8440522571783428010at_nat @ A @ A4 )
       => ? [X3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ A4 )
            & ( ord_le8460144461188290721at_nat @ A @ X3 )
            & ! [Xa2: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ Xa2 @ A4 )
               => ( ( ord_le8460144461188290721at_nat @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_650_finite__has__maximal2,axiom,
    ! [A4: set_nat,A: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( member_nat @ A @ A4 )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A4 )
            & ( ord_less_eq_nat @ A @ X3 )
            & ! [Xa2: nat] :
                ( ( member_nat @ Xa2 @ A4 )
               => ( ( ord_less_eq_nat @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_651_finite__has__minimal2,axiom,
    ! [A4: set_Pr8693737435421807431at_nat,A: produc859450856879609959at_nat] :
      ( ( finite4392333629123659920at_nat @ A4 )
     => ( ( member8206827879206165904at_nat @ A @ A4 )
       => ? [X3: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X3 @ A4 )
            & ( ord_le6722121967309221895at_nat @ X3 @ A )
            & ! [Xa2: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ Xa2 @ A4 )
               => ( ( ord_le6722121967309221895at_nat @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_652_finite__has__minimal2,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
      ( ( finite4343798906461161616at_nat @ A4 )
     => ( ( member8757157785044589968at_nat @ A @ A4 )
       => ? [X3: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X3 @ A4 )
            & ( ord_le8744046415670163463at_nat @ X3 @ A )
            & ! [Xa2: produc3843707927480180839at_nat] :
                ( ( member8757157785044589968at_nat @ Xa2 @ A4 )
               => ( ( ord_le8744046415670163463at_nat @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_653_finite__has__minimal2,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( ( finite6177210948735845034at_nat @ A4 )
     => ( ( member8440522571783428010at_nat @ A @ A4 )
       => ? [X3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ A4 )
            & ( ord_le8460144461188290721at_nat @ X3 @ A )
            & ! [Xa2: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ Xa2 @ A4 )
               => ( ( ord_le8460144461188290721at_nat @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_654_finite__has__minimal2,axiom,
    ! [A4: set_nat,A: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( member_nat @ A @ A4 )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A4 )
            & ( ord_less_eq_nat @ X3 @ A )
            & ! [Xa2: nat] :
                ( ( member_nat @ Xa2 @ A4 )
               => ( ( ord_less_eq_nat @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_655_finite__subset,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A4 @ B4 )
     => ( ( finite6177210948735845034at_nat @ B4 )
       => ( finite6177210948735845034at_nat @ A4 ) ) ) ).

% finite_subset
thf(fact_656_finite__subset,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B4 )
     => ( ( finite_finite_nat @ B4 )
       => ( finite_finite_nat @ A4 ) ) ) ).

% finite_subset
thf(fact_657_infinite__super,axiom,
    ! [S2: set_Pr1261947904930325089at_nat,T3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ S2 @ T3 )
     => ( ~ ( finite6177210948735845034at_nat @ S2 )
       => ~ ( finite6177210948735845034at_nat @ T3 ) ) ) ).

% infinite_super
thf(fact_658_infinite__super,axiom,
    ! [S2: set_nat,T3: set_nat] :
      ( ( ord_less_eq_set_nat @ S2 @ T3 )
     => ( ~ ( finite_finite_nat @ S2 )
       => ~ ( finite_finite_nat @ T3 ) ) ) ).

% infinite_super
thf(fact_659_rev__finite__subset,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ B4 )
     => ( ( ord_le3146513528884898305at_nat @ A4 @ B4 )
       => ( finite6177210948735845034at_nat @ A4 ) ) ) ).

% rev_finite_subset
thf(fact_660_rev__finite__subset,axiom,
    ! [B4: set_nat,A4: set_nat] :
      ( ( finite_finite_nat @ B4 )
     => ( ( ord_less_eq_set_nat @ A4 @ B4 )
       => ( finite_finite_nat @ A4 ) ) ) ).

% rev_finite_subset
thf(fact_661_finite__inverse__image__gen,axiom,
    ! [A4: set_nat,F2: nat > nat,D2: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( inj_on_nat_nat @ F2 @ D2 )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [J3: nat] :
                ( ( member_nat @ J3 @ D2 )
                & ( member_nat @ ( F2 @ J3 ) @ A4 ) ) ) ) ) ) ).

% finite_inverse_image_gen
thf(fact_662_finite__inverse__image__gen,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,F2: nat > product_prod_nat_nat,D2: set_nat] :
      ( ( finite6177210948735845034at_nat @ A4 )
     => ( ( inj_on5538052773655684606at_nat @ F2 @ D2 )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [J3: nat] :
                ( ( member_nat @ J3 @ D2 )
                & ( member8440522571783428010at_nat @ ( F2 @ J3 ) @ A4 ) ) ) ) ) ) ).

% finite_inverse_image_gen
thf(fact_663_finite__inverse__image__gen,axiom,
    ! [A4: set_nat,F2: product_prod_nat_nat > nat,D2: set_Pr1261947904930325089at_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( inj_on2178005380612969504at_nat @ F2 @ D2 )
       => ( finite6177210948735845034at_nat
          @ ( collec3392354462482085612at_nat
            @ ^ [J3: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ J3 @ D2 )
                & ( member_nat @ ( F2 @ J3 ) @ A4 ) ) ) ) ) ) ).

% finite_inverse_image_gen
thf(fact_664_finite__inverse__image__gen,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,F2: product_prod_nat_nat > product_prod_nat_nat,D2: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A4 )
     => ( ( inj_on8969904277767023793at_nat @ F2 @ D2 )
       => ( finite6177210948735845034at_nat
          @ ( collec3392354462482085612at_nat
            @ ^ [J3: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ J3 @ D2 )
                & ( member8440522571783428010at_nat @ ( F2 @ J3 ) @ A4 ) ) ) ) ) ) ).

% finite_inverse_image_gen
thf(fact_665_finite__inverse__image__gen,axiom,
    ! [A4: set_Pr8693737435421807431at_nat,F2: nat > produc859450856879609959at_nat,D2: set_nat] :
      ( ( finite4392333629123659920at_nat @ A4 )
     => ( ( inj_on6829720656803765732at_nat @ F2 @ D2 )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [J3: nat] :
                ( ( member_nat @ J3 @ D2 )
                & ( member8206827879206165904at_nat @ ( F2 @ J3 ) @ A4 ) ) ) ) ) ) ).

% finite_inverse_image_gen
thf(fact_666_finite__inverse__image__gen,axiom,
    ! [A4: set_nat,F2: produc859450856879609959at_nat > nat,D2: set_Pr8693737435421807431at_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( inj_on490725545064285830at_nat @ F2 @ D2 )
       => ( finite4392333629123659920at_nat
          @ ( collec7088162979684241874at_nat
            @ ^ [J3: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ J3 @ D2 )
                & ( member_nat @ ( F2 @ J3 ) @ A4 ) ) ) ) ) ) ).

% finite_inverse_image_gen
thf(fact_667_finite__inverse__image__gen,axiom,
    ! [A4: set_Pr8693737435421807431at_nat,F2: product_prod_nat_nat > produc859450856879609959at_nat,D2: set_Pr1261947904930325089at_nat] :
      ( ( finite4392333629123659920at_nat @ A4 )
     => ( ( inj_on3865918002419084311at_nat @ F2 @ D2 )
       => ( finite6177210948735845034at_nat
          @ ( collec3392354462482085612at_nat
            @ ^ [J3: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ J3 @ D2 )
                & ( member8206827879206165904at_nat @ ( F2 @ J3 ) @ A4 ) ) ) ) ) ) ).

% finite_inverse_image_gen
thf(fact_668_finite__inverse__image__gen,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,F2: nat > produc3843707927480180839at_nat,D2: set_nat] :
      ( ( finite4343798906461161616at_nat @ A4 )
     => ( ( inj_on7264072676751113700at_nat @ F2 @ D2 )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [J3: nat] :
                ( ( member_nat @ J3 @ D2 )
                & ( member8757157785044589968at_nat @ ( F2 @ J3 ) @ A4 ) ) ) ) ) ) ).

% finite_inverse_image_gen
thf(fact_669_finite__inverse__image__gen,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,F2: produc859450856879609959at_nat > product_prod_nat_nat,D2: set_Pr8693737435421807431at_nat] :
      ( ( finite6177210948735845034at_nat @ A4 )
     => ( ( inj_on229527429751349963at_nat @ F2 @ D2 )
       => ( finite4392333629123659920at_nat
          @ ( collec7088162979684241874at_nat
            @ ^ [J3: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ J3 @ D2 )
                & ( member8440522571783428010at_nat @ ( F2 @ J3 ) @ A4 ) ) ) ) ) ) ).

% finite_inverse_image_gen
thf(fact_670_finite__inverse__image__gen,axiom,
    ! [A4: set_nat,F2: produc3843707927480180839at_nat > nat,D2: set_Pr4329608150637261639at_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( inj_on7963584518437281414at_nat @ F2 @ D2 )
       => ( finite4343798906461161616at_nat
          @ ( collec6321179662152712658at_nat
            @ ^ [J3: produc3843707927480180839at_nat] :
                ( ( member8757157785044589968at_nat @ J3 @ D2 )
                & ( member_nat @ ( F2 @ J3 ) @ A4 ) ) ) ) ) ) ).

% finite_inverse_image_gen
thf(fact_671_internal__case__prod__def,axiom,
    produc9139807813499881076_nat_o = produc6081775807080527818_nat_o ).

% internal_case_prod_def
thf(fact_672_Collect__case__prod__mono,axiom,
    ! [A4: nat > nat > $o,B4: nat > nat > $o] :
      ( ( ord_le2646555220125990790_nat_o @ A4 @ B4 )
     => ( ord_le3146513528884898305at_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ A4 ) ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ B4 ) ) ) ) ).

% Collect_case_prod_mono
thf(fact_673_lift__Suc__antimono__le,axiom,
    ! [F2: nat > nat,N: nat,N5: nat] :
      ( ! [N2: nat] : ( ord_less_eq_nat @ ( F2 @ ( suc @ N2 ) ) @ ( F2 @ N2 ) )
     => ( ( ord_less_eq_nat @ N @ N5 )
       => ( ord_less_eq_nat @ ( F2 @ N5 ) @ ( F2 @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_674_lift__Suc__mono__le,axiom,
    ! [F2: nat > nat,N: nat,N5: nat] :
      ( ! [N2: nat] : ( ord_less_eq_nat @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ N @ N5 )
       => ( ord_less_eq_nat @ ( F2 @ N ) @ ( F2 @ N5 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_675_lift__Suc__mono__less__iff,axiom,
    ! [F2: nat > nat,N: nat,M: nat] :
      ( ! [N2: nat] : ( ord_less_nat @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
     => ( ( ord_less_nat @ ( F2 @ N ) @ ( F2 @ M ) )
        = ( ord_less_nat @ N @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_676_lift__Suc__mono__less,axiom,
    ! [F2: nat > nat,N: nat,N5: nat] :
      ( ! [N2: nat] : ( ord_less_nat @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
     => ( ( ord_less_nat @ N @ N5 )
       => ( ord_less_nat @ ( F2 @ N ) @ ( F2 @ N5 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_677_card__subset__eq,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ B4 )
     => ( ( ord_le3146513528884898305at_nat @ A4 @ B4 )
       => ( ( ( finite711546835091564841at_nat @ A4 )
            = ( finite711546835091564841at_nat @ B4 ) )
         => ( A4 = B4 ) ) ) ) ).

% card_subset_eq
thf(fact_678_card__subset__eq,axiom,
    ! [B4: set_nat,A4: set_nat] :
      ( ( finite_finite_nat @ B4 )
     => ( ( ord_less_eq_set_nat @ A4 @ B4 )
       => ( ( ( finite_card_nat @ A4 )
            = ( finite_card_nat @ B4 ) )
         => ( A4 = B4 ) ) ) ) ).

% card_subset_eq
thf(fact_679_infinite__arbitrarily__large,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,N: nat] :
      ( ~ ( finite6177210948735845034at_nat @ A4 )
     => ? [B8: set_Pr1261947904930325089at_nat] :
          ( ( finite6177210948735845034at_nat @ B8 )
          & ( ( finite711546835091564841at_nat @ B8 )
            = N )
          & ( ord_le3146513528884898305at_nat @ B8 @ A4 ) ) ) ).

% infinite_arbitrarily_large
thf(fact_680_infinite__arbitrarily__large,axiom,
    ! [A4: set_nat,N: nat] :
      ( ~ ( finite_finite_nat @ A4 )
     => ? [B8: set_nat] :
          ( ( finite_finite_nat @ B8 )
          & ( ( finite_card_nat @ B8 )
            = N )
          & ( ord_less_eq_set_nat @ B8 @ A4 ) ) ) ).

% infinite_arbitrarily_large
thf(fact_681_le__imp__less__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).

% le_imp_less_Suc
thf(fact_682_less__eq__Suc__le,axiom,
    ( ord_less_nat
    = ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).

% less_eq_Suc_le
thf(fact_683_less__Suc__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% less_Suc_eq_le
thf(fact_684_le__less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( ord_less_nat @ N @ ( suc @ M ) )
        = ( N = M ) ) ) ).

% le_less_Suc_eq
thf(fact_685_Suc__le__lessD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
     => ( ord_less_nat @ M @ N ) ) ).

% Suc_le_lessD
thf(fact_686_inc__induct,axiom,
    ! [I3: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I3 @ J )
     => ( ( P @ J )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ I3 @ N2 )
             => ( ( ord_less_nat @ N2 @ J )
               => ( ( P @ ( suc @ N2 ) )
                 => ( P @ N2 ) ) ) )
         => ( P @ I3 ) ) ) ) ).

% inc_induct
thf(fact_687_dec__induct,axiom,
    ! [I3: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I3 @ J )
     => ( ( P @ I3 )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ I3 @ N2 )
             => ( ( ord_less_nat @ N2 @ J )
               => ( ( P @ N2 )
                 => ( P @ ( suc @ N2 ) ) ) ) )
         => ( P @ J ) ) ) ) ).

% dec_induct
thf(fact_688_Suc__le__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
      = ( ord_less_nat @ M @ N ) ) ).

% Suc_le_eq
thf(fact_689_Suc__leI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).

% Suc_leI
thf(fact_690_less__imp__Suc__add,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ? [K3: nat] :
          ( N
          = ( suc @ ( plus_plus_nat @ M @ K3 ) ) ) ) ).

% less_imp_Suc_add
thf(fact_691_less__iff__Suc__add,axiom,
    ( ord_less_nat
    = ( ^ [M5: nat,N3: nat] :
        ? [K2: nat] :
          ( N3
          = ( suc @ ( plus_plus_nat @ M5 @ K2 ) ) ) ) ) ).

% less_iff_Suc_add
thf(fact_692_less__add__Suc2,axiom,
    ! [I3: nat,M: nat] : ( ord_less_nat @ I3 @ ( suc @ ( plus_plus_nat @ M @ I3 ) ) ) ).

% less_add_Suc2
thf(fact_693_less__add__Suc1,axiom,
    ! [I3: nat,M: nat] : ( ord_less_nat @ I3 @ ( suc @ ( plus_plus_nat @ I3 @ M ) ) ) ).

% less_add_Suc1
thf(fact_694_less__natE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ~ ! [Q3: nat] :
            ( N
           != ( suc @ ( plus_plus_nat @ M @ Q3 ) ) ) ) ).

% less_natE
thf(fact_695_semiring__norm_I174_J,axiom,
    ( suc
    = ( ^ [N3: nat] : ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ).

% semiring_norm(174)
thf(fact_696_Suc__eq__plus1__left,axiom,
    ( suc
    = ( plus_plus_nat @ one_one_nat ) ) ).

% Suc_eq_plus1_left
thf(fact_697_plus__1__eq__Suc,axiom,
    ( ( plus_plus_nat @ one_one_nat )
    = suc ) ).

% plus_1_eq_Suc
thf(fact_698_card__le__if__inj__on__rel,axiom,
    ! [B4: set_nat,A4: set_nat,R2: nat > nat > $o] :
      ( ( finite_finite_nat @ B4 )
     => ( ! [A2: nat] :
            ( ( member_nat @ A2 @ A4 )
           => ? [B9: nat] :
                ( ( member_nat @ B9 @ B4 )
                & ( R2 @ A2 @ B9 ) ) )
       => ( ! [A12: nat,A23: nat,B2: nat] :
              ( ( member_nat @ A12 @ A4 )
             => ( ( member_nat @ A23 @ A4 )
               => ( ( member_nat @ B2 @ B4 )
                 => ( ( R2 @ A12 @ B2 )
                   => ( ( R2 @ A23 @ B2 )
                     => ( A12 = A23 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B4 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_699_card__le__if__inj__on__rel,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,A4: set_nat,R2: nat > product_prod_nat_nat > $o] :
      ( ( finite6177210948735845034at_nat @ B4 )
     => ( ! [A2: nat] :
            ( ( member_nat @ A2 @ A4 )
           => ? [B9: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ B9 @ B4 )
                & ( R2 @ A2 @ B9 ) ) )
       => ( ! [A12: nat,A23: nat,B2: product_prod_nat_nat] :
              ( ( member_nat @ A12 @ A4 )
             => ( ( member_nat @ A23 @ A4 )
               => ( ( member8440522571783428010at_nat @ B2 @ B4 )
                 => ( ( R2 @ A12 @ B2 )
                   => ( ( R2 @ A23 @ B2 )
                     => ( A12 = A23 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite711546835091564841at_nat @ B4 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_700_card__le__if__inj__on__rel,axiom,
    ! [B4: set_nat,A4: set_Pr1261947904930325089at_nat,R2: product_prod_nat_nat > nat > $o] :
      ( ( finite_finite_nat @ B4 )
     => ( ! [A2: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ A2 @ A4 )
           => ? [B9: nat] :
                ( ( member_nat @ B9 @ B4 )
                & ( R2 @ A2 @ B9 ) ) )
       => ( ! [A12: product_prod_nat_nat,A23: product_prod_nat_nat,B2: nat] :
              ( ( member8440522571783428010at_nat @ A12 @ A4 )
             => ( ( member8440522571783428010at_nat @ A23 @ A4 )
               => ( ( member_nat @ B2 @ B4 )
                 => ( ( R2 @ A12 @ B2 )
                   => ( ( R2 @ A23 @ B2 )
                     => ( A12 = A23 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ A4 ) @ ( finite_card_nat @ B4 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_701_card__le__if__inj__on__rel,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,R2: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ( finite6177210948735845034at_nat @ B4 )
     => ( ! [A2: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ A2 @ A4 )
           => ? [B9: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ B9 @ B4 )
                & ( R2 @ A2 @ B9 ) ) )
       => ( ! [A12: product_prod_nat_nat,A23: product_prod_nat_nat,B2: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ A12 @ A4 )
             => ( ( member8440522571783428010at_nat @ A23 @ A4 )
               => ( ( member8440522571783428010at_nat @ B2 @ B4 )
                 => ( ( R2 @ A12 @ B2 )
                   => ( ( R2 @ A23 @ B2 )
                     => ( A12 = A23 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ A4 ) @ ( finite711546835091564841at_nat @ B4 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_702_card__le__if__inj__on__rel,axiom,
    ! [B4: set_Pr8693737435421807431at_nat,A4: set_nat,R2: nat > produc859450856879609959at_nat > $o] :
      ( ( finite4392333629123659920at_nat @ B4 )
     => ( ! [A2: nat] :
            ( ( member_nat @ A2 @ A4 )
           => ? [B9: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ B9 @ B4 )
                & ( R2 @ A2 @ B9 ) ) )
       => ( ! [A12: nat,A23: nat,B2: produc859450856879609959at_nat] :
              ( ( member_nat @ A12 @ A4 )
             => ( ( member_nat @ A23 @ A4 )
               => ( ( member8206827879206165904at_nat @ B2 @ B4 )
                 => ( ( R2 @ A12 @ B2 )
                   => ( ( R2 @ A23 @ B2 )
                     => ( A12 = A23 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite1207074278014112911at_nat @ B4 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_703_card__le__if__inj__on__rel,axiom,
    ! [B4: set_nat,A4: set_Pr8693737435421807431at_nat,R2: produc859450856879609959at_nat > nat > $o] :
      ( ( finite_finite_nat @ B4 )
     => ( ! [A2: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ A2 @ A4 )
           => ? [B9: nat] :
                ( ( member_nat @ B9 @ B4 )
                & ( R2 @ A2 @ B9 ) ) )
       => ( ! [A12: produc859450856879609959at_nat,A23: produc859450856879609959at_nat,B2: nat] :
              ( ( member8206827879206165904at_nat @ A12 @ A4 )
             => ( ( member8206827879206165904at_nat @ A23 @ A4 )
               => ( ( member_nat @ B2 @ B4 )
                 => ( ( R2 @ A12 @ B2 )
                   => ( ( R2 @ A23 @ B2 )
                     => ( A12 = A23 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite1207074278014112911at_nat @ A4 ) @ ( finite_card_nat @ B4 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_704_card__le__if__inj__on__rel,axiom,
    ! [B4: set_Pr4329608150637261639at_nat,A4: set_nat,R2: nat > produc3843707927480180839at_nat > $o] :
      ( ( finite4343798906461161616at_nat @ B4 )
     => ( ! [A2: nat] :
            ( ( member_nat @ A2 @ A4 )
           => ? [B9: produc3843707927480180839at_nat] :
                ( ( member8757157785044589968at_nat @ B9 @ B4 )
                & ( R2 @ A2 @ B9 ) ) )
       => ( ! [A12: nat,A23: nat,B2: produc3843707927480180839at_nat] :
              ( ( member_nat @ A12 @ A4 )
             => ( ( member_nat @ A23 @ A4 )
               => ( ( member8757157785044589968at_nat @ B2 @ B4 )
                 => ( ( R2 @ A12 @ B2 )
                   => ( ( R2 @ A23 @ B2 )
                     => ( A12 = A23 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite3771342082235030671at_nat @ B4 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_705_card__le__if__inj__on__rel,axiom,
    ! [B4: set_Pr8693737435421807431at_nat,A4: set_Pr1261947904930325089at_nat,R2: product_prod_nat_nat > produc859450856879609959at_nat > $o] :
      ( ( finite4392333629123659920at_nat @ B4 )
     => ( ! [A2: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ A2 @ A4 )
           => ? [B9: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ B9 @ B4 )
                & ( R2 @ A2 @ B9 ) ) )
       => ( ! [A12: product_prod_nat_nat,A23: product_prod_nat_nat,B2: produc859450856879609959at_nat] :
              ( ( member8440522571783428010at_nat @ A12 @ A4 )
             => ( ( member8440522571783428010at_nat @ A23 @ A4 )
               => ( ( member8206827879206165904at_nat @ B2 @ B4 )
                 => ( ( R2 @ A12 @ B2 )
                   => ( ( R2 @ A23 @ B2 )
                     => ( A12 = A23 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ A4 ) @ ( finite1207074278014112911at_nat @ B4 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_706_card__le__if__inj__on__rel,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,A4: set_Pr8693737435421807431at_nat,R2: produc859450856879609959at_nat > product_prod_nat_nat > $o] :
      ( ( finite6177210948735845034at_nat @ B4 )
     => ( ! [A2: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ A2 @ A4 )
           => ? [B9: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ B9 @ B4 )
                & ( R2 @ A2 @ B9 ) ) )
       => ( ! [A12: produc859450856879609959at_nat,A23: produc859450856879609959at_nat,B2: product_prod_nat_nat] :
              ( ( member8206827879206165904at_nat @ A12 @ A4 )
             => ( ( member8206827879206165904at_nat @ A23 @ A4 )
               => ( ( member8440522571783428010at_nat @ B2 @ B4 )
                 => ( ( R2 @ A12 @ B2 )
                   => ( ( R2 @ A23 @ B2 )
                     => ( A12 = A23 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite1207074278014112911at_nat @ A4 ) @ ( finite711546835091564841at_nat @ B4 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_707_card__le__if__inj__on__rel,axiom,
    ! [B4: set_nat,A4: set_Pr4329608150637261639at_nat,R2: produc3843707927480180839at_nat > nat > $o] :
      ( ( finite_finite_nat @ B4 )
     => ( ! [A2: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ A2 @ A4 )
           => ? [B9: nat] :
                ( ( member_nat @ B9 @ B4 )
                & ( R2 @ A2 @ B9 ) ) )
       => ( ! [A12: produc3843707927480180839at_nat,A23: produc3843707927480180839at_nat,B2: nat] :
              ( ( member8757157785044589968at_nat @ A12 @ A4 )
             => ( ( member8757157785044589968at_nat @ A23 @ A4 )
               => ( ( member_nat @ B2 @ B4 )
                 => ( ( R2 @ A12 @ B2 )
                   => ( ( R2 @ A23 @ B2 )
                     => ( A12 = A23 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite3771342082235030671at_nat @ A4 ) @ ( finite_card_nat @ B4 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_708_mod__induct,axiom,
    ! [P: nat > $o,N: nat,P2: nat,M: nat] :
      ( ( P @ N )
     => ( ( ord_less_nat @ N @ P2 )
       => ( ( ord_less_nat @ M @ P2 )
         => ( ! [N2: nat] :
                ( ( ord_less_nat @ N2 @ P2 )
               => ( ( P @ N2 )
                 => ( P @ ( modulo_modulo_nat @ ( suc @ N2 ) @ P2 ) ) ) )
           => ( P @ M ) ) ) ) ) ).

% mod_induct
thf(fact_709_mod__Suc__le__divisor,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ ( suc @ N ) ) @ N ) ).

% mod_Suc_le_divisor
thf(fact_710_finite__if__finite__subsets__card__bdd,axiom,
    ! [F4: set_Pr1261947904930325089at_nat,C2: nat] :
      ( ! [G2: set_Pr1261947904930325089at_nat] :
          ( ( ord_le3146513528884898305at_nat @ G2 @ F4 )
         => ( ( finite6177210948735845034at_nat @ G2 )
           => ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ G2 ) @ C2 ) ) )
     => ( ( finite6177210948735845034at_nat @ F4 )
        & ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ F4 ) @ C2 ) ) ) ).

% finite_if_finite_subsets_card_bdd
thf(fact_711_finite__if__finite__subsets__card__bdd,axiom,
    ! [F4: set_nat,C2: nat] :
      ( ! [G2: set_nat] :
          ( ( ord_less_eq_set_nat @ G2 @ F4 )
         => ( ( finite_finite_nat @ G2 )
           => ( ord_less_eq_nat @ ( finite_card_nat @ G2 ) @ C2 ) ) )
     => ( ( finite_finite_nat @ F4 )
        & ( ord_less_eq_nat @ ( finite_card_nat @ F4 ) @ C2 ) ) ) ).

% finite_if_finite_subsets_card_bdd
thf(fact_712_obtain__subset__with__card__n,axiom,
    ! [N: nat,S2: set_Pr1261947904930325089at_nat] :
      ( ( ord_less_eq_nat @ N @ ( finite711546835091564841at_nat @ S2 ) )
     => ~ ! [T4: set_Pr1261947904930325089at_nat] :
            ( ( ord_le3146513528884898305at_nat @ T4 @ S2 )
           => ( ( ( finite711546835091564841at_nat @ T4 )
                = N )
             => ~ ( finite6177210948735845034at_nat @ T4 ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_713_obtain__subset__with__card__n,axiom,
    ! [N: nat,S2: set_nat] :
      ( ( ord_less_eq_nat @ N @ ( finite_card_nat @ S2 ) )
     => ~ ! [T4: set_nat] :
            ( ( ord_less_eq_set_nat @ T4 @ S2 )
           => ( ( ( finite_card_nat @ T4 )
                = N )
             => ~ ( finite_finite_nat @ T4 ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_714_exists__subset__between,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,N: nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ A4 ) @ N )
     => ( ( ord_less_eq_nat @ N @ ( finite711546835091564841at_nat @ C2 ) )
       => ( ( ord_le3146513528884898305at_nat @ A4 @ C2 )
         => ( ( finite6177210948735845034at_nat @ C2 )
           => ? [B8: set_Pr1261947904930325089at_nat] :
                ( ( ord_le3146513528884898305at_nat @ A4 @ B8 )
                & ( ord_le3146513528884898305at_nat @ B8 @ C2 )
                & ( ( finite711546835091564841at_nat @ B8 )
                  = N ) ) ) ) ) ) ).

% exists_subset_between
thf(fact_715_exists__subset__between,axiom,
    ! [A4: set_nat,N: nat,C2: set_nat] :
      ( ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ N )
     => ( ( ord_less_eq_nat @ N @ ( finite_card_nat @ C2 ) )
       => ( ( ord_less_eq_set_nat @ A4 @ C2 )
         => ( ( finite_finite_nat @ C2 )
           => ? [B8: set_nat] :
                ( ( ord_less_eq_set_nat @ A4 @ B8 )
                & ( ord_less_eq_set_nat @ B8 @ C2 )
                & ( ( finite_card_nat @ B8 )
                  = N ) ) ) ) ) ) ).

% exists_subset_between
thf(fact_716_card__seteq,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ B4 )
     => ( ( ord_le3146513528884898305at_nat @ A4 @ B4 )
       => ( ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ B4 ) @ ( finite711546835091564841at_nat @ A4 ) )
         => ( A4 = B4 ) ) ) ) ).

% card_seteq
thf(fact_717_card__seteq,axiom,
    ! [B4: set_nat,A4: set_nat] :
      ( ( finite_finite_nat @ B4 )
     => ( ( ord_less_eq_set_nat @ A4 @ B4 )
       => ( ( ord_less_eq_nat @ ( finite_card_nat @ B4 ) @ ( finite_card_nat @ A4 ) )
         => ( A4 = B4 ) ) ) ) ).

% card_seteq
thf(fact_718_card__mono,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ B4 )
     => ( ( ord_le3146513528884898305at_nat @ A4 @ B4 )
       => ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ A4 ) @ ( finite711546835091564841at_nat @ B4 ) ) ) ) ).

% card_mono
thf(fact_719_card__mono,axiom,
    ! [B4: set_nat,A4: set_nat] :
      ( ( finite_finite_nat @ B4 )
     => ( ( ord_less_eq_set_nat @ A4 @ B4 )
       => ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B4 ) ) ) ) ).

% card_mono
thf(fact_720_card__gr__1__iff,axiom,
    ! [S2: set_Pr8693737435421807431at_nat,X: produc859450856879609959at_nat,Y: produc859450856879609959at_nat] :
      ( ( finite4392333629123659920at_nat @ S2 )
     => ( ( member8206827879206165904at_nat @ X @ S2 )
       => ( ( member8206827879206165904at_nat @ Y @ S2 )
         => ( ( X != Y )
           => ( ord_less_nat @ one_one_nat @ ( finite1207074278014112911at_nat @ S2 ) ) ) ) ) ) ).

% card_gr_1_iff
thf(fact_721_card__gr__1__iff,axiom,
    ! [S2: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( ( finite4343798906461161616at_nat @ S2 )
     => ( ( member8757157785044589968at_nat @ X @ S2 )
       => ( ( member8757157785044589968at_nat @ Y @ S2 )
         => ( ( X != Y )
           => ( ord_less_nat @ one_one_nat @ ( finite3771342082235030671at_nat @ S2 ) ) ) ) ) ) ).

% card_gr_1_iff
thf(fact_722_card__gr__1__iff,axiom,
    ! [S2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( finite6177210948735845034at_nat @ S2 )
     => ( ( member8440522571783428010at_nat @ X @ S2 )
       => ( ( member8440522571783428010at_nat @ Y @ S2 )
         => ( ( X != Y )
           => ( ord_less_nat @ one_one_nat @ ( finite711546835091564841at_nat @ S2 ) ) ) ) ) ) ).

% card_gr_1_iff
thf(fact_723_card__gr__1__iff,axiom,
    ! [S2: set_nat,X: nat,Y: nat] :
      ( ( finite_finite_nat @ S2 )
     => ( ( member_nat @ X @ S2 )
       => ( ( member_nat @ Y @ S2 )
         => ( ( X != Y )
           => ( ord_less_nat @ one_one_nat @ ( finite_card_nat @ S2 ) ) ) ) ) ) ).

% card_gr_1_iff
thf(fact_724_psubset__card__mono,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ B4 )
     => ( ( ord_le7866589430770878221at_nat @ A4 @ B4 )
       => ( ord_less_nat @ ( finite711546835091564841at_nat @ A4 ) @ ( finite711546835091564841at_nat @ B4 ) ) ) ) ).

% psubset_card_mono
thf(fact_725_psubset__card__mono,axiom,
    ! [B4: set_nat,A4: set_nat] :
      ( ( finite_finite_nat @ B4 )
     => ( ( ord_less_set_nat @ A4 @ B4 )
       => ( ord_less_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B4 ) ) ) ) ).

% psubset_card_mono
thf(fact_726_card__psubset,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ B4 )
     => ( ( ord_le3146513528884898305at_nat @ A4 @ B4 )
       => ( ( ord_less_nat @ ( finite711546835091564841at_nat @ A4 ) @ ( finite711546835091564841at_nat @ B4 ) )
         => ( ord_le7866589430770878221at_nat @ A4 @ B4 ) ) ) ) ).

% card_psubset
thf(fact_727_card__psubset,axiom,
    ! [B4: set_nat,A4: set_nat] :
      ( ( finite_finite_nat @ B4 )
     => ( ( ord_less_eq_set_nat @ A4 @ B4 )
       => ( ( ord_less_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B4 ) )
         => ( ord_less_set_nat @ A4 @ B4 ) ) ) ) ).

% card_psubset
thf(fact_728_in__finite__psubset,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A4 @ B4 ) @ finite_psubset_nat )
      = ( ( ord_less_set_nat @ A4 @ B4 )
        & ( finite_finite_nat @ B4 ) ) ) ).

% in_finite_psubset
thf(fact_729_in__finite__psubset,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A4 @ B4 ) @ finite469560695537375940at_nat )
      = ( ( ord_le7866589430770878221at_nat @ A4 @ B4 )
        & ( finite6177210948735845034at_nat @ B4 ) ) ) ).

% in_finite_psubset
thf(fact_730_in__finite__psubset,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A4 @ B4 ) @ finite4695646753290404266at_nat )
      = ( ( ord_le2604355607129572851at_nat @ A4 @ B4 )
        & ( finite4343798906461161616at_nat @ B4 ) ) ) ).

% in_finite_psubset
thf(fact_731_wmax__emptyI,axiom,
    ! [X5: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ X5 )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ X5 ) @ fun_max_weak ) ) ).

% wmax_emptyI
thf(fact_732_pred__nat__def,axiom,
    ( pred_nat
    = ( collec3392354462482085612at_nat
      @ ( produc6081775807080527818_nat_o
        @ ^ [M5: nat,N3: nat] :
            ( N3
            = ( suc @ M5 ) ) ) ) ) ).

% pred_nat_def
thf(fact_733_mem__case__prodI,axiom,
    ! [Z: nat,C: nat > nat > set_nat,A: nat,B: nat] :
      ( ( member_nat @ Z @ ( C @ A @ B ) )
     => ( member_nat @ Z @ ( produc6189476227299908564et_nat @ C @ ( product_Pair_nat_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_734_mem__case__prodI,axiom,
    ! [Z: product_prod_nat_nat,C: nat > nat > set_Pr1261947904930325089at_nat,A: nat,B: nat] :
      ( ( member8440522571783428010at_nat @ Z @ ( C @ A @ B ) )
     => ( member8440522571783428010at_nat @ Z @ ( produc8197505143624133779at_nat @ C @ ( product_Pair_nat_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_735_mem__case__prodI,axiom,
    ! [Z: nat,C: product_prod_nat_nat > product_prod_nat_nat > set_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( ( member_nat @ Z @ ( C @ A @ B ) )
     => ( member_nat @ Z @ ( produc1370685125915216498et_nat @ C @ ( produc6161850002892822231at_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_736_mem__case__prodI,axiom,
    ! [Z: produc859450856879609959at_nat,C: nat > nat > set_Pr8693737435421807431at_nat,A: nat,B: nat] :
      ( ( member8206827879206165904at_nat @ Z @ ( C @ A @ B ) )
     => ( member8206827879206165904at_nat @ Z @ ( produc2423807293333670521at_nat @ C @ ( product_Pair_nat_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_737_mem__case__prodI,axiom,
    ! [Z: product_prod_nat_nat,C: product_prod_nat_nat > product_prod_nat_nat > set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ Z @ ( C @ A @ B ) )
     => ( member8440522571783428010at_nat @ Z @ ( produc3380114247675707509at_nat @ C @ ( produc6161850002892822231at_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_738_mem__case__prodI,axiom,
    ! [Z: nat,C: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( member_nat @ Z @ ( C @ A @ B ) )
     => ( member_nat @ Z @ ( produc5395831842199599730et_nat @ C @ ( produc2922128104949294807at_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_739_mem__case__prodI,axiom,
    ! [Z: produc3843707927480180839at_nat,C: nat > nat > set_Pr4329608150637261639at_nat,A: nat,B: nat] :
      ( ( member8757157785044589968at_nat @ Z @ ( C @ A @ B ) )
     => ( member8757157785044589968at_nat @ Z @ ( produc3705604114177432185at_nat @ C @ ( product_Pair_nat_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_740_mem__case__prodI,axiom,
    ! [Z: product_prod_nat_nat,C: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ Z @ ( C @ A @ B ) )
     => ( member8440522571783428010at_nat @ Z @ ( produc1433696742876942453at_nat @ C @ ( produc2922128104949294807at_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_741_mem__case__prodI,axiom,
    ! [Z: produc859450856879609959at_nat,C: product_prod_nat_nat > product_prod_nat_nat > set_Pr8693737435421807431at_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( ( member8206827879206165904at_nat @ Z @ ( C @ A @ B ) )
     => ( member8206827879206165904at_nat @ Z @ ( produc773927175282099035at_nat @ C @ ( produc6161850002892822231at_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_742_mem__case__prodI,axiom,
    ! [Z: produc3843707927480180839at_nat,C: product_prod_nat_nat > product_prod_nat_nat > set_Pr4329608150637261639at_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( ( member8757157785044589968at_nat @ Z @ ( C @ A @ B ) )
     => ( member8757157785044589968at_nat @ Z @ ( produc1422205692209202011at_nat @ C @ ( produc6161850002892822231at_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_743_mem__case__prodI2,axiom,
    ! [P2: product_prod_nat_nat,Z: nat,C: nat > nat > set_nat] :
      ( ! [A2: nat,B2: nat] :
          ( ( P2
            = ( product_Pair_nat_nat @ A2 @ B2 ) )
         => ( member_nat @ Z @ ( C @ A2 @ B2 ) ) )
     => ( member_nat @ Z @ ( produc6189476227299908564et_nat @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_744_mem__case__prodI2,axiom,
    ! [P2: product_prod_nat_nat,Z: product_prod_nat_nat,C: nat > nat > set_Pr1261947904930325089at_nat] :
      ( ! [A2: nat,B2: nat] :
          ( ( P2
            = ( product_Pair_nat_nat @ A2 @ B2 ) )
         => ( member8440522571783428010at_nat @ Z @ ( C @ A2 @ B2 ) ) )
     => ( member8440522571783428010at_nat @ Z @ ( produc8197505143624133779at_nat @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_745_mem__case__prodI2,axiom,
    ! [P2: produc859450856879609959at_nat,Z: nat,C: product_prod_nat_nat > product_prod_nat_nat > set_nat] :
      ( ! [A2: product_prod_nat_nat,B2: product_prod_nat_nat] :
          ( ( P2
            = ( produc6161850002892822231at_nat @ A2 @ B2 ) )
         => ( member_nat @ Z @ ( C @ A2 @ B2 ) ) )
     => ( member_nat @ Z @ ( produc1370685125915216498et_nat @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_746_mem__case__prodI2,axiom,
    ! [P2: product_prod_nat_nat,Z: produc859450856879609959at_nat,C: nat > nat > set_Pr8693737435421807431at_nat] :
      ( ! [A2: nat,B2: nat] :
          ( ( P2
            = ( product_Pair_nat_nat @ A2 @ B2 ) )
         => ( member8206827879206165904at_nat @ Z @ ( C @ A2 @ B2 ) ) )
     => ( member8206827879206165904at_nat @ Z @ ( produc2423807293333670521at_nat @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_747_mem__case__prodI2,axiom,
    ! [P2: produc859450856879609959at_nat,Z: product_prod_nat_nat,C: product_prod_nat_nat > product_prod_nat_nat > set_Pr1261947904930325089at_nat] :
      ( ! [A2: product_prod_nat_nat,B2: product_prod_nat_nat] :
          ( ( P2
            = ( produc6161850002892822231at_nat @ A2 @ B2 ) )
         => ( member8440522571783428010at_nat @ Z @ ( C @ A2 @ B2 ) ) )
     => ( member8440522571783428010at_nat @ Z @ ( produc3380114247675707509at_nat @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_748_mem__case__prodI2,axiom,
    ! [P2: produc3843707927480180839at_nat,Z: nat,C: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_nat] :
      ( ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
          ( ( P2
            = ( produc2922128104949294807at_nat @ A2 @ B2 ) )
         => ( member_nat @ Z @ ( C @ A2 @ B2 ) ) )
     => ( member_nat @ Z @ ( produc5395831842199599730et_nat @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_749_mem__case__prodI2,axiom,
    ! [P2: product_prod_nat_nat,Z: produc3843707927480180839at_nat,C: nat > nat > set_Pr4329608150637261639at_nat] :
      ( ! [A2: nat,B2: nat] :
          ( ( P2
            = ( product_Pair_nat_nat @ A2 @ B2 ) )
         => ( member8757157785044589968at_nat @ Z @ ( C @ A2 @ B2 ) ) )
     => ( member8757157785044589968at_nat @ Z @ ( produc3705604114177432185at_nat @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_750_mem__case__prodI2,axiom,
    ! [P2: produc3843707927480180839at_nat,Z: product_prod_nat_nat,C: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat] :
      ( ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
          ( ( P2
            = ( produc2922128104949294807at_nat @ A2 @ B2 ) )
         => ( member8440522571783428010at_nat @ Z @ ( C @ A2 @ B2 ) ) )
     => ( member8440522571783428010at_nat @ Z @ ( produc1433696742876942453at_nat @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_751_mem__case__prodI2,axiom,
    ! [P2: produc859450856879609959at_nat,Z: produc859450856879609959at_nat,C: product_prod_nat_nat > product_prod_nat_nat > set_Pr8693737435421807431at_nat] :
      ( ! [A2: product_prod_nat_nat,B2: product_prod_nat_nat] :
          ( ( P2
            = ( produc6161850002892822231at_nat @ A2 @ B2 ) )
         => ( member8206827879206165904at_nat @ Z @ ( C @ A2 @ B2 ) ) )
     => ( member8206827879206165904at_nat @ Z @ ( produc773927175282099035at_nat @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_752_mem__case__prodI2,axiom,
    ! [P2: produc859450856879609959at_nat,Z: produc3843707927480180839at_nat,C: product_prod_nat_nat > product_prod_nat_nat > set_Pr4329608150637261639at_nat] :
      ( ! [A2: product_prod_nat_nat,B2: product_prod_nat_nat] :
          ( ( P2
            = ( produc6161850002892822231at_nat @ A2 @ B2 ) )
         => ( member8757157785044589968at_nat @ Z @ ( C @ A2 @ B2 ) ) )
     => ( member8757157785044589968at_nat @ Z @ ( produc1422205692209202011at_nat @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_753_finite__Collect__less__nat,axiom,
    ! [K: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N3: nat] : ( ord_less_nat @ N3 @ K ) ) ) ).

% finite_Collect_less_nat
thf(fact_754_finite__Collect__le__nat,axiom,
    ! [K: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N3: nat] : ( ord_less_eq_nat @ N3 @ K ) ) ) ).

% finite_Collect_le_nat
thf(fact_755_finite__nat__set__iff__bounded__le,axiom,
    ( finite_finite_nat
    = ( ^ [N6: set_nat] :
        ? [M5: nat] :
        ! [X4: nat] :
          ( ( member_nat @ X4 @ N6 )
         => ( ord_less_eq_nat @ X4 @ M5 ) ) ) ) ).

% finite_nat_set_iff_bounded_le
thf(fact_756_bot__set__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat @ bot_bot_nat_o ) ) ).

% bot_set_def
thf(fact_757_bot__set__def,axiom,
    ( bot_bo2099793752762293965at_nat
    = ( collec3392354462482085612at_nat @ bot_bo482883023278783056_nat_o ) ) ).

% bot_set_def
thf(fact_758_bot__set__def,axiom,
    ( bot_bo228742789529271731at_nat
    = ( collec6321179662152712658at_nat @ bot_bo6829958733157050602_nat_o ) ) ).

% bot_set_def
thf(fact_759_bot__list__def,axiom,
    bot_bot_list_nat = nil_nat ).

% bot_list_def
thf(fact_760_finite__nat__set__iff__bounded,axiom,
    ( finite_finite_nat
    = ( ^ [N6: set_nat] :
        ? [M5: nat] :
        ! [X4: nat] :
          ( ( member_nat @ X4 @ N6 )
         => ( ord_less_nat @ X4 @ M5 ) ) ) ) ).

% finite_nat_set_iff_bounded
thf(fact_761_bounded__nat__set__is__finite,axiom,
    ! [N4: set_nat,N: nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ N4 )
         => ( ord_less_nat @ X3 @ N ) )
     => ( finite_finite_nat @ N4 ) ) ).

% bounded_nat_set_is_finite
thf(fact_762_mem__case__prodE,axiom,
    ! [Z: nat,C: nat > nat > set_nat,P2: product_prod_nat_nat] :
      ( ( member_nat @ Z @ ( produc6189476227299908564et_nat @ C @ P2 ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P2
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( member_nat @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_763_mem__case__prodE,axiom,
    ! [Z: product_prod_nat_nat,C: nat > nat > set_Pr1261947904930325089at_nat,P2: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ Z @ ( produc8197505143624133779at_nat @ C @ P2 ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P2
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( member8440522571783428010at_nat @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_764_mem__case__prodE,axiom,
    ! [Z: nat,C: product_prod_nat_nat > product_prod_nat_nat > set_nat,P2: produc859450856879609959at_nat] :
      ( ( member_nat @ Z @ ( produc1370685125915216498et_nat @ C @ P2 ) )
     => ~ ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
            ( ( P2
              = ( produc6161850002892822231at_nat @ X3 @ Y3 ) )
           => ~ ( member_nat @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_765_mem__case__prodE,axiom,
    ! [Z: produc859450856879609959at_nat,C: nat > nat > set_Pr8693737435421807431at_nat,P2: product_prod_nat_nat] :
      ( ( member8206827879206165904at_nat @ Z @ ( produc2423807293333670521at_nat @ C @ P2 ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P2
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( member8206827879206165904at_nat @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_766_mem__case__prodE,axiom,
    ! [Z: product_prod_nat_nat,C: product_prod_nat_nat > product_prod_nat_nat > set_Pr1261947904930325089at_nat,P2: produc859450856879609959at_nat] :
      ( ( member8440522571783428010at_nat @ Z @ ( produc3380114247675707509at_nat @ C @ P2 ) )
     => ~ ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
            ( ( P2
              = ( produc6161850002892822231at_nat @ X3 @ Y3 ) )
           => ~ ( member8440522571783428010at_nat @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_767_mem__case__prodE,axiom,
    ! [Z: nat,C: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_nat,P2: produc3843707927480180839at_nat] :
      ( ( member_nat @ Z @ ( produc5395831842199599730et_nat @ C @ P2 ) )
     => ~ ! [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
            ( ( P2
              = ( produc2922128104949294807at_nat @ X3 @ Y3 ) )
           => ~ ( member_nat @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_768_mem__case__prodE,axiom,
    ! [Z: produc3843707927480180839at_nat,C: nat > nat > set_Pr4329608150637261639at_nat,P2: product_prod_nat_nat] :
      ( ( member8757157785044589968at_nat @ Z @ ( produc3705604114177432185at_nat @ C @ P2 ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P2
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( member8757157785044589968at_nat @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_769_mem__case__prodE,axiom,
    ! [Z: product_prod_nat_nat,C: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat,P2: produc3843707927480180839at_nat] :
      ( ( member8440522571783428010at_nat @ Z @ ( produc1433696742876942453at_nat @ C @ P2 ) )
     => ~ ! [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
            ( ( P2
              = ( produc2922128104949294807at_nat @ X3 @ Y3 ) )
           => ~ ( member8440522571783428010at_nat @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_770_mem__case__prodE,axiom,
    ! [Z: produc859450856879609959at_nat,C: product_prod_nat_nat > product_prod_nat_nat > set_Pr8693737435421807431at_nat,P2: produc859450856879609959at_nat] :
      ( ( member8206827879206165904at_nat @ Z @ ( produc773927175282099035at_nat @ C @ P2 ) )
     => ~ ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
            ( ( P2
              = ( produc6161850002892822231at_nat @ X3 @ Y3 ) )
           => ~ ( member8206827879206165904at_nat @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_771_mem__case__prodE,axiom,
    ! [Z: produc3843707927480180839at_nat,C: product_prod_nat_nat > product_prod_nat_nat > set_Pr4329608150637261639at_nat,P2: produc859450856879609959at_nat] :
      ( ( member8757157785044589968at_nat @ Z @ ( produc1422205692209202011at_nat @ C @ P2 ) )
     => ~ ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
            ( ( P2
              = ( produc6161850002892822231at_nat @ X3 @ Y3 ) )
           => ~ ( member8757157785044589968at_nat @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_772_case__prod__Pair__iden,axiom,
    ! [P2: product_prod_nat_nat] :
      ( ( produc2626176000494625587at_nat @ product_Pair_nat_nat @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_773_case__prod__Pair__iden,axiom,
    ! [P2: produc859450856879609959at_nat] :
      ( ( produc6744312701629110395at_nat @ produc6161850002892822231at_nat @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_774_case__prod__Pair__iden,axiom,
    ! [P2: produc3843707927480180839at_nat] :
      ( ( produc7548845685434208379at_nat @ produc2922128104949294807at_nat @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_775_case__prod__Pair__iden,axiom,
    ! [P2: produc1319942482725812455at_nat] :
      ( ( produc5456455677105495163at_nat @ produc9060074326276436823at_nat @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_776_swap__inj__on,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( inj_on8969904277767023793at_nat
      @ ( produc2626176000494625587at_nat
        @ ^ [I: nat,J3: nat] : ( product_Pair_nat_nat @ J3 @ I ) )
      @ A4 ) ).

% swap_inj_on
thf(fact_777_swap__inj__on,axiom,
    ! [A4: set_Pr8693737435421807431at_nat] :
      ( inj_on7797493862353638193at_nat
      @ ( produc6744312701629110395at_nat
        @ ^ [I: product_prod_nat_nat,J3: product_prod_nat_nat] : ( produc6161850002892822231at_nat @ J3 @ I ) )
      @ A4 ) ).

% swap_inj_on
thf(fact_778_swap__inj__on,axiom,
    ! [A4: set_Pr4329608150637261639at_nat] :
      ( inj_on8430303779476321073at_nat
      @ ( produc7548845685434208379at_nat
        @ ^ [I: set_Pr1261947904930325089at_nat,J3: set_Pr1261947904930325089at_nat] : ( produc2922128104949294807at_nat @ J3 @ I ) )
      @ A4 ) ).

% swap_inj_on
thf(fact_779_swap__inj__on,axiom,
    ! [A4: set_Pr7459493094073627847at_nat] :
      ( inj_on3942866938762682161at_nat
      @ ( produc5456455677105495163at_nat
        @ ^ [I: set_Pr4329608150637261639at_nat,J3: set_Pr4329608150637261639at_nat] : ( produc9060074326276436823at_nat @ J3 @ I ) )
      @ A4 ) ).

% swap_inj_on
thf(fact_780_finite__M__bounded__by__nat,axiom,
    ! [P: nat > $o,I3: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [K2: nat] :
            ( ( P @ K2 )
            & ( ord_less_nat @ K2 @ I3 ) ) ) ) ).

% finite_M_bounded_by_nat
thf(fact_781_finite__less__ub,axiom,
    ! [F2: nat > nat,U2: nat] :
      ( ! [N2: nat] : ( ord_less_eq_nat @ N2 @ ( F2 @ N2 ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [N3: nat] : ( ord_less_eq_nat @ ( F2 @ N3 ) @ U2 ) ) ) ) ).

% finite_less_ub
thf(fact_782_finite__psubset__def,axiom,
    ( finite469560695537375940at_nat
    = ( collec6321179662152712658at_nat
      @ ( produc410239310623530412_nat_o
        @ ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
            ( ( ord_le7866589430770878221at_nat @ A6 @ B6 )
            & ( finite6177210948735845034at_nat @ B6 ) ) ) ) ) ).

% finite_psubset_def
thf(fact_783_finite__psubset__def,axiom,
    ( finite_psubset_nat
    = ( collec6662362479098859352et_nat
      @ ( produc6247414631856714078_nat_o
        @ ^ [A6: set_nat,B6: set_nat] :
            ( ( ord_less_set_nat @ A6 @ B6 )
            & ( finite_finite_nat @ B6 ) ) ) ) ) ).

% finite_psubset_def
thf(fact_784_finite__maxlen,axiom,
    ! [M4: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ M4 )
     => ? [N2: nat] :
        ! [X6: list_nat] :
          ( ( member_list_nat @ X6 @ M4 )
         => ( ord_less_nat @ ( size_size_list_nat @ X6 ) @ N2 ) ) ) ).

% finite_maxlen
thf(fact_785_bot__empty__eq2,axiom,
    ( bot_bo4898103413517107610_nat_o
    = ( ^ [X4: product_prod_nat_nat,Y4: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y4 ) @ bot_bo5327735625951526323at_nat ) ) ) ).

% bot_empty_eq2
thf(fact_786_bot__empty__eq2,axiom,
    ( bot_bo3364206721330744218_nat_o
    = ( ^ [X4: set_Pr4329608150637261639at_nat,Y4: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X4 @ Y4 ) @ bot_bo4948859079157340979at_nat ) ) ) ).

% bot_empty_eq2
thf(fact_787_bot__empty__eq2,axiom,
    ( bot_bot_nat_nat_o
    = ( ^ [X4: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ bot_bo2099793752762293965at_nat ) ) ) ).

% bot_empty_eq2
thf(fact_788_bot__empty__eq2,axiom,
    ( bot_bo394778441745866138_nat_o
    = ( ^ [X4: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y4 ) @ bot_bo228742789529271731at_nat ) ) ) ).

% bot_empty_eq2
thf(fact_789_lex__prod__def,axiom,
    ( lex_prod_nat_nat
    = ( ^ [Ra: set_Pr1261947904930325089at_nat,Rb: set_Pr1261947904930325089at_nat] :
          ( collec7088162979684241874at_nat
          @ ( produc6590410687421337004_nat_o
            @ ( produc8739625826339149834_nat_o
              @ ^ [A3: nat,B3: nat] :
                  ( produc6081775807080527818_nat_o
                  @ ^ [A8: nat,B10: nat] :
                      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ A8 ) @ Ra )
                      | ( ( A3 = A8 )
                        & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B3 @ B10 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_790_lex__prod__def,axiom,
    ( lex_pr8029265285556086080at_nat
    = ( ^ [Ra: set_Pr1261947904930325089at_nat,Rb: set_Pr8693737435421807431at_nat] :
          ( collec3253044014973990424at_nat
          @ ( produc2869316064735525022_nat_o
            @ ( produc7480869177107369060_nat_o
              @ ^ [A3: nat,B3: product_prod_nat_nat] :
                  ( produc5864757623865647827_nat_o
                  @ ^ [A8: nat,B10: product_prod_nat_nat] :
                      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ A8 ) @ Ra )
                      | ( ( A3 = A8 )
                        & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B3 @ B10 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_791_lex__prod__def,axiom,
    ( lex_pr4669217892513370978at_nat
    = ( ^ [Ra: set_Pr8693737435421807431at_nat,Rb: set_Pr1261947904930325089at_nat] :
          ( collec2200730763347676620at_nat
          @ ( produc3600351907215576810_nat_o
            @ ( produc6198819664464039784_nat_o
              @ ^ [A3: product_prod_nat_nat,B3: nat] :
                  ( produc8758269395934548017_nat_o
                  @ ^ [A8: product_prod_nat_nat,B10: nat] :
                      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A3 @ A8 ) @ Ra )
                      | ( ( A3 = A8 )
                        & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B3 @ B10 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_792_lex__prod__def,axiom,
    ( lex_pr4265550801952275104at_nat
    = ( ^ [Ra: set_Pr1261947904930325089at_nat,Rb: set_Pr4329608150637261639at_nat] :
          ( collec2088441199205367000at_nat
          @ ( produc7830147906768200158_nat_o
            @ ( produc1908958340173245860_nat_o
              @ ^ [A3: nat,B3: set_Pr1261947904930325089at_nat] :
                  ( produc7997411485876045171_nat_o
                  @ ^ [A8: nat,B10: set_Pr1261947904930325089at_nat] :
                      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ A8 ) @ Ra )
                      | ( ( A3 = A8 )
                        & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B3 @ B10 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_793_lex__prod__def,axiom,
    ( lex_pr1951919194703979714at_nat
    = ( ^ [Ra: set_Pr4329608150637261639at_nat,Rb: set_Pr1261947904930325089at_nat] :
          ( collec1461227866750739596at_nat
          @ ( produc8701595406162815018_nat_o
            @ ( produc7110675705651593384_nat_o
              @ ^ [A3: set_Pr1261947904930325089at_nat,B3: nat] :
                  ( produc3595121349294487249_nat_o
                  @ ^ [A8: set_Pr1261947904930325089at_nat,B10: nat] :
                      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A3 @ A8 ) @ Ra )
                      | ( ( A3 = A8 )
                        & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B3 @ B10 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_794_lex__prod__def,axiom,
    ( lex_pr8801849515957261039at_nat
    = ( ^ [Ra: set_Pr8693737435421807431at_nat,Rb: set_Pr8693737435421807431at_nat] :
          ( collec7334067512558549330at_nat
          @ ( produc3603361296207914284_nat_o
            @ ( produc5077327844045112642_nat_o
              @ ^ [A3: product_prod_nat_nat,B3: product_prod_nat_nat] :
                  ( produc6590410687421337004_nat_o
                  @ ^ [A8: product_prod_nat_nat,B10: product_prod_nat_nat] :
                      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A3 @ A8 ) @ Ra )
                      | ( ( A3 = A8 )
                        & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B3 @ B10 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_795_lex__prod__def,axiom,
    ( lex_pr864846108736269391at_nat
    = ( ^ [Ra: set_Pr8693737435421807431at_nat,Rb: set_Pr4329608150637261639at_nat] :
          ( collec1712228713741422930at_nat
          @ ( produc3676778756977593644_nat_o
            @ ( produc8821334275656303938_nat_o
              @ ^ [A3: product_prod_nat_nat,B3: set_Pr1261947904930325089at_nat] :
                  ( produc7695416289158663244_nat_o
                  @ ^ [A8: product_prod_nat_nat,B10: set_Pr1261947904930325089at_nat] :
                      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A3 @ A8 ) @ Ra )
                      | ( ( A3 = A8 )
                        & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B3 @ B10 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_796_lex__prod__def,axiom,
    ( lex_pr8442690889092155791at_nat
    = ( ^ [Ra: set_Pr4329608150637261639at_nat,Rb: set_Pr8693737435421807431at_nat] :
          ( collec2974368035702751570at_nat
          @ ( produc4641403137511177516_nat_o
            @ ( produc3500598050016679234_nat_o
              @ ^ [A3: set_Pr1261947904930325089at_nat,B3: product_prod_nat_nat] :
                  ( produc743548188471107340_nat_o
                  @ ^ [A8: set_Pr1261947904930325089at_nat,B10: product_prod_nat_nat] :
                      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A3 @ A8 ) @ Ra )
                      | ( ( A3 = A8 )
                        & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B3 @ B10 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_797_lex__prod__def,axiom,
    ( lex_pr8000699672184251119at_nat
    = ( ^ [Ra: set_Pr4329608150637261639at_nat,Rb: set_Pr4329608150637261639at_nat] :
          ( collec223960383693904210at_nat
          @ ( produc1561160019221259564_nat_o
            @ ( produc1971064617671185730_nat_o
              @ ^ [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
                  ( produc410239310623530412_nat_o
                  @ ^ [A8: set_Pr1261947904930325089at_nat,B10: set_Pr1261947904930325089at_nat] :
                      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A3 @ A8 ) @ Ra )
                      | ( ( A3 = A8 )
                        & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B3 @ B10 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_798_lex__prod__def,axiom,
    ( lex_pr1368066972704168198at_nat
    = ( ^ [Ra: set_Pr1261947904930325089at_nat,Rb: set_Pr7459493094073627847at_nat] :
          ( collec600673703569510692at_nat
          @ ( produc7652991722741833106_nat_o
            @ ( produc2208240802940919896_nat_o
              @ ^ [A3: nat,B3: set_Pr4329608150637261639at_nat] :
                  ( produc735371275256815501_nat_o
                  @ ^ [A8: nat,B10: set_Pr4329608150637261639at_nat] :
                      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ A8 ) @ Ra )
                      | ( ( A3 = A8 )
                        & ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ B3 @ B10 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_799_smax__emptyI,axiom,
    ! [Y7: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ Y7 )
     => ( ( Y7 != bot_bo2099793752762293965at_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ Y7 ) @ fun_max_strict ) ) ) ).

% smax_emptyI
thf(fact_800_same__fst__def,axiom,
    ( same_fst_nat_nat
    = ( ^ [P4: nat > $o,R3: nat > set_Pr1261947904930325089at_nat] :
          ( collec7088162979684241874at_nat
          @ ( produc6590410687421337004_nat_o
            @ ( produc8739625826339149834_nat_o
              @ ^ [X9: nat,Y8: nat] :
                  ( produc6081775807080527818_nat_o
                  @ ^ [X4: nat,Y4: nat] :
                      ( ( X9 = X4 )
                      & ( P4 @ X4 )
                      & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y8 @ Y4 ) @ ( R3 @ X4 ) ) ) ) ) ) ) ) ) ).

% same_fst_def
thf(fact_801_infinite__nat__iff__unbounded__le,axiom,
    ! [S2: set_nat] :
      ( ( ~ ( finite_finite_nat @ S2 ) )
      = ( ! [M5: nat] :
          ? [N3: nat] :
            ( ( ord_less_eq_nat @ M5 @ N3 )
            & ( member_nat @ N3 @ S2 ) ) ) ) ).

% infinite_nat_iff_unbounded_le
thf(fact_802_infinite__nat__iff__unbounded,axiom,
    ! [S2: set_nat] :
      ( ( ~ ( finite_finite_nat @ S2 ) )
      = ( ! [M5: nat] :
          ? [N3: nat] :
            ( ( ord_less_nat @ M5 @ N3 )
            & ( member_nat @ N3 @ S2 ) ) ) ) ).

% infinite_nat_iff_unbounded
thf(fact_803_same__fstI,axiom,
    ! [P: nat > $o,X: nat,Y6: nat,Y: nat,R: nat > set_Pr1261947904930325089at_nat] :
      ( ( P @ X )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y6 @ Y ) @ ( R @ X ) )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ X @ Y6 ) @ ( product_Pair_nat_nat @ X @ Y ) ) @ ( same_fst_nat_nat @ P @ R ) ) ) ) ).

% same_fstI
thf(fact_804_same__fstI,axiom,
    ! [P: product_prod_nat_nat > $o,X: product_prod_nat_nat,Y6: product_prod_nat_nat,Y: product_prod_nat_nat,R: product_prod_nat_nat > set_Pr8693737435421807431at_nat] :
      ( ( P @ X )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y6 @ Y ) @ ( R @ X ) )
       => ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ ( produc6161850002892822231at_nat @ X @ Y6 ) @ ( produc6161850002892822231at_nat @ X @ Y ) ) @ ( same_f4956014544515070124at_nat @ P @ R ) ) ) ) ).

% same_fstI
thf(fact_805_same__fstI,axiom,
    ! [P: set_Pr1261947904930325089at_nat > $o,X: set_Pr1261947904930325089at_nat,Y6: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,R: set_Pr1261947904930325089at_nat > set_Pr4329608150637261639at_nat] :
      ( ( P @ X )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y6 @ Y ) @ ( R @ X ) )
       => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ ( produc2922128104949294807at_nat @ X @ Y6 ) @ ( produc2922128104949294807at_nat @ X @ Y ) ) @ ( same_f1460074399991292076at_nat @ P @ R ) ) ) ) ).

% same_fstI
thf(fact_806_same__fstI,axiom,
    ! [P: set_Pr4329608150637261639at_nat > $o,X: set_Pr4329608150637261639at_nat,Y6: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,R: set_Pr4329608150637261639at_nat > set_Pr7459493094073627847at_nat] :
      ( ( P @ X )
     => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y6 @ Y ) @ ( R @ X ) )
       => ( member8215962448328497936at_nat @ ( produc2220582066590712663at_nat @ ( produc9060074326276436823at_nat @ X @ Y6 ) @ ( produc9060074326276436823at_nat @ X @ Y ) ) @ ( same_f4504843259822650924at_nat @ P @ R ) ) ) ) ).

% same_fstI
thf(fact_807_unbounded__k__infinite,axiom,
    ! [K: nat,S2: set_nat] :
      ( ! [M2: nat] :
          ( ( ord_less_nat @ K @ M2 )
         => ? [N7: nat] :
              ( ( ord_less_nat @ M2 @ N7 )
              & ( member_nat @ N7 @ S2 ) ) )
     => ~ ( finite_finite_nat @ S2 ) ) ).

% unbounded_k_infinite
thf(fact_808_wmin__emptyI,axiom,
    ! [X5: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X5 @ bot_bo2099793752762293965at_nat ) @ fun_min_weak ) ).

% wmin_emptyI
thf(fact_809_smin__emptyI,axiom,
    ! [X5: set_Pr1261947904930325089at_nat] :
      ( ( X5 != bot_bo2099793752762293965at_nat )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X5 @ bot_bo2099793752762293965at_nat ) @ fun_min_strict ) ) ).

% smin_emptyI
thf(fact_810_subset__emptyI,axiom,
    ! [A4: set_Pr8693737435421807431at_nat] :
      ( ! [X3: produc859450856879609959at_nat] :
          ~ ( member8206827879206165904at_nat @ X3 @ A4 )
     => ( ord_le3000389064537975527at_nat @ A4 @ bot_bo5327735625951526323at_nat ) ) ).

% subset_emptyI
thf(fact_811_subset__emptyI,axiom,
    ! [A4: set_nat] :
      ( ! [X3: nat] :
          ~ ( member_nat @ X3 @ A4 )
     => ( ord_less_eq_set_nat @ A4 @ bot_bot_set_nat ) ) ).

% subset_emptyI
thf(fact_812_subset__emptyI,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ! [X3: product_prod_nat_nat] :
          ~ ( member8440522571783428010at_nat @ X3 @ A4 )
     => ( ord_le3146513528884898305at_nat @ A4 @ bot_bo2099793752762293965at_nat ) ) ).

% subset_emptyI
thf(fact_813_subset__emptyI,axiom,
    ! [A4: set_Pr4329608150637261639at_nat] :
      ( ! [X3: produc3843707927480180839at_nat] :
          ~ ( member8757157785044589968at_nat @ X3 @ A4 )
     => ( ord_le1268244103169919719at_nat @ A4 @ bot_bo228742789529271731at_nat ) ) ).

% subset_emptyI
thf(fact_814_ssubst__Pair__rhs,axiom,
    ! [R2: nat,S: nat,R: set_Pr1261947904930325089at_nat,S4: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R2 @ S ) @ R )
     => ( ( S4 = S )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R2 @ S4 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_815_ssubst__Pair__rhs,axiom,
    ! [R2: product_prod_nat_nat,S: product_prod_nat_nat,R: set_Pr8693737435421807431at_nat,S4: product_prod_nat_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ R2 @ S ) @ R )
     => ( ( S4 = S )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ R2 @ S4 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_816_ssubst__Pair__rhs,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat,S4: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ S ) @ R )
     => ( ( S4 = S )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ S4 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_817_ssubst__Pair__rhs,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat,R: set_Pr7459493094073627847at_nat,S4: set_Pr4329608150637261639at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R2 @ S ) @ R )
     => ( ( S4 = S )
       => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R2 @ S4 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_818_prop__restrict,axiom,
    ! [X: produc859450856879609959at_nat,Z4: set_Pr8693737435421807431at_nat,X5: set_Pr8693737435421807431at_nat,P: produc859450856879609959at_nat > $o] :
      ( ( member8206827879206165904at_nat @ X @ Z4 )
     => ( ( ord_le3000389064537975527at_nat @ Z4
          @ ( collec7088162979684241874at_nat
            @ ^ [X4: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ X4 @ X5 )
                & ( P @ X4 ) ) ) )
       => ( P @ X ) ) ) ).

% prop_restrict
thf(fact_819_prop__restrict,axiom,
    ! [X: produc3843707927480180839at_nat,Z4: set_Pr4329608150637261639at_nat,X5: set_Pr4329608150637261639at_nat,P: produc3843707927480180839at_nat > $o] :
      ( ( member8757157785044589968at_nat @ X @ Z4 )
     => ( ( ord_le1268244103169919719at_nat @ Z4
          @ ( collec6321179662152712658at_nat
            @ ^ [X4: produc3843707927480180839at_nat] :
                ( ( member8757157785044589968at_nat @ X4 @ X5 )
                & ( P @ X4 ) ) ) )
       => ( P @ X ) ) ) ).

% prop_restrict
thf(fact_820_prop__restrict,axiom,
    ! [X: nat,Z4: set_nat,X5: set_nat,P: nat > $o] :
      ( ( member_nat @ X @ Z4 )
     => ( ( ord_less_eq_set_nat @ Z4
          @ ( collect_nat
            @ ^ [X4: nat] :
                ( ( member_nat @ X4 @ X5 )
                & ( P @ X4 ) ) ) )
       => ( P @ X ) ) ) ).

% prop_restrict
thf(fact_821_prop__restrict,axiom,
    ! [X: product_prod_nat_nat,Z4: set_Pr1261947904930325089at_nat,X5: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
      ( ( member8440522571783428010at_nat @ X @ Z4 )
     => ( ( ord_le3146513528884898305at_nat @ Z4
          @ ( collec3392354462482085612at_nat
            @ ^ [X4: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X4 @ X5 )
                & ( P @ X4 ) ) ) )
       => ( P @ X ) ) ) ).

% prop_restrict
thf(fact_822_Collect__restrict,axiom,
    ! [X5: set_Pr8693737435421807431at_nat,P: produc859450856879609959at_nat > $o] :
      ( ord_le3000389064537975527at_nat
      @ ( collec7088162979684241874at_nat
        @ ^ [X4: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X4 @ X5 )
            & ( P @ X4 ) ) )
      @ X5 ) ).

% Collect_restrict
thf(fact_823_Collect__restrict,axiom,
    ! [X5: set_Pr4329608150637261639at_nat,P: produc3843707927480180839at_nat > $o] :
      ( ord_le1268244103169919719at_nat
      @ ( collec6321179662152712658at_nat
        @ ^ [X4: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X4 @ X5 )
            & ( P @ X4 ) ) )
      @ X5 ) ).

% Collect_restrict
thf(fact_824_Collect__restrict,axiom,
    ! [X5: set_nat,P: nat > $o] :
      ( ord_less_eq_set_nat
      @ ( collect_nat
        @ ^ [X4: nat] :
            ( ( member_nat @ X4 @ X5 )
            & ( P @ X4 ) ) )
      @ X5 ) ).

% Collect_restrict
thf(fact_825_Collect__restrict,axiom,
    ! [X5: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
      ( ord_le3146513528884898305at_nat
      @ ( collec3392354462482085612at_nat
        @ ^ [X4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X4 @ X5 )
            & ( P @ X4 ) ) )
      @ X5 ) ).

% Collect_restrict
thf(fact_826_min__rpair__set,axiom,
    fun_re2478310338295953701at_nat @ ( produc9060074326276436823at_nat @ fun_min_strict @ fun_min_weak ) ).

% min_rpair_set
thf(fact_827_max__rpair__set,axiom,
    fun_re2478310338295953701at_nat @ ( produc9060074326276436823at_nat @ fun_max_strict @ fun_max_weak ) ).

% max_rpair_set
thf(fact_828_finite__indexed__bound,axiom,
    ! [A4: set_Pr8693737435421807431at_nat,P: produc859450856879609959at_nat > nat > $o] :
      ( ( finite4392333629123659920at_nat @ A4 )
     => ( ! [X3: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X3 @ A4 )
           => ? [X_12: nat] : ( P @ X3 @ X_12 ) )
       => ? [M2: nat] :
          ! [X6: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X6 @ A4 )
           => ? [K3: nat] :
                ( ( ord_less_eq_nat @ K3 @ M2 )
                & ( P @ X6 @ K3 ) ) ) ) ) ).

% finite_indexed_bound
thf(fact_829_finite__indexed__bound,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,P: produc3843707927480180839at_nat > nat > $o] :
      ( ( finite4343798906461161616at_nat @ A4 )
     => ( ! [X3: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X3 @ A4 )
           => ? [X_12: nat] : ( P @ X3 @ X_12 ) )
       => ? [M2: nat] :
          ! [X6: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X6 @ A4 )
           => ? [K3: nat] :
                ( ( ord_less_eq_nat @ K3 @ M2 )
                & ( P @ X6 @ K3 ) ) ) ) ) ).

% finite_indexed_bound
thf(fact_830_finite__indexed__bound,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > nat > $o] :
      ( ( finite6177210948735845034at_nat @ A4 )
     => ( ! [X3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ A4 )
           => ? [X_12: nat] : ( P @ X3 @ X_12 ) )
       => ? [M2: nat] :
          ! [X6: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X6 @ A4 )
           => ? [K3: nat] :
                ( ( ord_less_eq_nat @ K3 @ M2 )
                & ( P @ X6 @ K3 ) ) ) ) ) ).

% finite_indexed_bound
thf(fact_831_finite__indexed__bound,axiom,
    ! [A4: set_nat,P: nat > nat > $o] :
      ( ( finite_finite_nat @ A4 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ A4 )
           => ? [X_12: nat] : ( P @ X3 @ X_12 ) )
       => ? [M2: nat] :
          ! [X6: nat] :
            ( ( member_nat @ X6 @ A4 )
           => ? [K3: nat] :
                ( ( ord_less_eq_nat @ K3 @ M2 )
                & ( P @ X6 @ K3 ) ) ) ) ) ).

% finite_indexed_bound
thf(fact_832_min__strict__def,axiom,
    ( fun_min_strict
    = ( min_ex6901939911449802026at_nat @ fun_pair_less ) ) ).

% min_strict_def
thf(fact_833_rp__inv__image__rp,axiom,
    ! [P: produc1319942482725812455at_nat,F2: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat] :
      ( ( fun_re2478310338295953701at_nat @ P )
     => ( fun_re2478310338295953701at_nat @ ( fun_rp8445993449185644698at_nat @ P @ F2 ) ) ) ).

% rp_inv_image_rp
thf(fact_834_max__ext_Omax__extI,axiom,
    ! [X5: set_Pr8693737435421807431at_nat,Y7: set_Pr8693737435421807431at_nat,R: set_Pr553994874890374343at_nat] :
      ( ( finite4392333629123659920at_nat @ X5 )
     => ( ( finite4392333629123659920at_nat @ Y7 )
       => ( ( Y7 != bot_bo5327735625951526323at_nat )
         => ( ! [X3: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ X3 @ X5 )
               => ? [Xa2: produc859450856879609959at_nat] :
                    ( ( member8206827879206165904at_nat @ Xa2 @ Y7 )
                    & ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X3 @ Xa2 ) @ R ) ) )
           => ( member5855424355840516880at_nat @ ( produc3236233026405413719at_nat @ X5 @ Y7 ) @ ( max_ex4511810952740877858at_nat @ R ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_835_max__ext_Omax__extI,axiom,
    ! [X5: set_nat,Y7: set_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( finite_finite_nat @ X5 )
     => ( ( finite_finite_nat @ Y7 )
       => ( ( Y7 != bot_bot_set_nat )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ X5 )
               => ? [Xa2: nat] :
                    ( ( member_nat @ Xa2 @ Y7 )
                    & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Xa2 ) @ R ) ) )
           => ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X5 @ Y7 ) @ ( max_ext_nat @ R ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_836_max__ext_Omax__extI,axiom,
    ! [X5: set_se7855581050983116737at_nat,Y7: set_se7855581050983116737at_nat,R: set_Pr4329608150637261639at_nat] :
      ( ( finite9047747110432174090at_nat @ X5 )
     => ( ( finite9047747110432174090at_nat @ Y7 )
       => ( ( Y7 != bot_bo3083307316010499117at_nat )
         => ( ! [X3: set_Pr1261947904930325089at_nat] :
                ( ( member2643936169264416010at_nat @ X3 @ X5 )
               => ? [Xa2: set_Pr1261947904930325089at_nat] :
                    ( ( member2643936169264416010at_nat @ Xa2 @ Y7 )
                    & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Xa2 ) @ R ) ) )
           => ( member4300370950623927696at_nat @ ( produc7462898802507495127at_nat @ X5 @ Y7 ) @ ( max_ex4535816423174879388at_nat @ R ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_837_max__ext_Omax__extI,axiom,
    ! [X5: set_se5479201105039950375at_nat,Y7: set_se5479201105039950375at_nat,R: set_Pr7459493094073627847at_nat] :
      ( ( finite4768004253697144688at_nat @ X5 )
     => ( ( finite4768004253697144688at_nat @ Y7 )
       => ( ( Y7 != bot_bo1534825418459847827at_nat )
         => ( ! [X3: set_Pr4329608150637261639at_nat] :
                ( ( member6273874620674182256at_nat @ X3 @ X5 )
               => ? [Xa2: set_Pr4329608150637261639at_nat] :
                    ( ( member6273874620674182256at_nat @ Xa2 @ Y7 )
                    & ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ Xa2 ) @ R ) ) )
           => ( member6436561159828024080at_nat @ ( produc2492784117751810903at_nat @ X5 @ Y7 ) @ ( max_ex4000160833828605186at_nat @ R ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_838_max__ext_Omax__extI,axiom,
    ! [X5: set_Pr1261947904930325089at_nat,Y7: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat] :
      ( ( finite6177210948735845034at_nat @ X5 )
     => ( ( finite6177210948735845034at_nat @ Y7 )
       => ( ( Y7 != bot_bo2099793752762293965at_nat )
         => ( ! [X3: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X3 @ X5 )
               => ? [Xa2: product_prod_nat_nat] :
                    ( ( member8440522571783428010at_nat @ Xa2 @ Y7 )
                    & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Xa2 ) @ R ) ) )
           => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X5 @ Y7 ) @ ( max_ex8135407076693332796at_nat @ R ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_839_max__ext_Omax__extI,axiom,
    ! [X5: set_Pr4329608150637261639at_nat,Y7: set_Pr4329608150637261639at_nat,R: set_Pr5564308138774400199at_nat] :
      ( ( finite4343798906461161616at_nat @ X5 )
     => ( ( finite4343798906461161616at_nat @ Y7 )
       => ( ( Y7 != bot_bo228742789529271731at_nat )
         => ( ! [X3: produc3843707927480180839at_nat] :
                ( ( member8757157785044589968at_nat @ X3 @ X5 )
               => ? [Xa2: produc3843707927480180839at_nat] :
                    ( ( member8757157785044589968at_nat @ Xa2 @ Y7 )
                    & ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X3 @ Xa2 ) @ R ) ) )
           => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X5 @ Y7 ) @ ( max_ex8833738776070432290at_nat @ R ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_840_max__strict__def,axiom,
    ( fun_max_strict
    = ( max_ex8135407076693332796at_nat @ fun_pair_less ) ) ).

% max_strict_def
thf(fact_841_max__ext_Ocases,axiom,
    ! [A1: set_Pr8693737435421807431at_nat,A22: set_Pr8693737435421807431at_nat,R: set_Pr553994874890374343at_nat] :
      ( ( member5855424355840516880at_nat @ ( produc3236233026405413719at_nat @ A1 @ A22 ) @ ( max_ex4511810952740877858at_nat @ R ) )
     => ~ ( ( finite4392333629123659920at_nat @ A1 )
         => ( ( finite4392333629123659920at_nat @ A22 )
           => ( ( A22 != bot_bo5327735625951526323at_nat )
             => ~ ! [X6: produc859450856879609959at_nat] :
                    ( ( member8206827879206165904at_nat @ X6 @ A1 )
                   => ? [Xa3: produc859450856879609959at_nat] :
                        ( ( member8206827879206165904at_nat @ Xa3 @ A22 )
                        & ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X6 @ Xa3 ) @ R ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_842_max__ext_Ocases,axiom,
    ! [A1: set_nat,A22: set_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A1 @ A22 ) @ ( max_ext_nat @ R ) )
     => ~ ( ( finite_finite_nat @ A1 )
         => ( ( finite_finite_nat @ A22 )
           => ( ( A22 != bot_bot_set_nat )
             => ~ ! [X6: nat] :
                    ( ( member_nat @ X6 @ A1 )
                   => ? [Xa3: nat] :
                        ( ( member_nat @ Xa3 @ A22 )
                        & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X6 @ Xa3 ) @ R ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_843_max__ext_Ocases,axiom,
    ! [A1: set_se7855581050983116737at_nat,A22: set_se7855581050983116737at_nat,R: set_Pr4329608150637261639at_nat] :
      ( ( member4300370950623927696at_nat @ ( produc7462898802507495127at_nat @ A1 @ A22 ) @ ( max_ex4535816423174879388at_nat @ R ) )
     => ~ ( ( finite9047747110432174090at_nat @ A1 )
         => ( ( finite9047747110432174090at_nat @ A22 )
           => ( ( A22 != bot_bo3083307316010499117at_nat )
             => ~ ! [X6: set_Pr1261947904930325089at_nat] :
                    ( ( member2643936169264416010at_nat @ X6 @ A1 )
                   => ? [Xa3: set_Pr1261947904930325089at_nat] :
                        ( ( member2643936169264416010at_nat @ Xa3 @ A22 )
                        & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X6 @ Xa3 ) @ R ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_844_max__ext_Ocases,axiom,
    ! [A1: set_se5479201105039950375at_nat,A22: set_se5479201105039950375at_nat,R: set_Pr7459493094073627847at_nat] :
      ( ( member6436561159828024080at_nat @ ( produc2492784117751810903at_nat @ A1 @ A22 ) @ ( max_ex4000160833828605186at_nat @ R ) )
     => ~ ( ( finite4768004253697144688at_nat @ A1 )
         => ( ( finite4768004253697144688at_nat @ A22 )
           => ( ( A22 != bot_bo1534825418459847827at_nat )
             => ~ ! [X6: set_Pr4329608150637261639at_nat] :
                    ( ( member6273874620674182256at_nat @ X6 @ A1 )
                   => ? [Xa3: set_Pr4329608150637261639at_nat] :
                        ( ( member6273874620674182256at_nat @ Xa3 @ A22 )
                        & ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X6 @ Xa3 ) @ R ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_845_max__ext_Ocases,axiom,
    ! [A1: set_Pr1261947904930325089at_nat,A22: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A1 @ A22 ) @ ( max_ex8135407076693332796at_nat @ R ) )
     => ~ ( ( finite6177210948735845034at_nat @ A1 )
         => ( ( finite6177210948735845034at_nat @ A22 )
           => ( ( A22 != bot_bo2099793752762293965at_nat )
             => ~ ! [X6: product_prod_nat_nat] :
                    ( ( member8440522571783428010at_nat @ X6 @ A1 )
                   => ? [Xa3: product_prod_nat_nat] :
                        ( ( member8440522571783428010at_nat @ Xa3 @ A22 )
                        & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X6 @ Xa3 ) @ R ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_846_max__ext_Ocases,axiom,
    ! [A1: set_Pr4329608150637261639at_nat,A22: set_Pr4329608150637261639at_nat,R: set_Pr5564308138774400199at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A1 @ A22 ) @ ( max_ex8833738776070432290at_nat @ R ) )
     => ~ ( ( finite4343798906461161616at_nat @ A1 )
         => ( ( finite4343798906461161616at_nat @ A22 )
           => ( ( A22 != bot_bo228742789529271731at_nat )
             => ~ ! [X6: produc3843707927480180839at_nat] :
                    ( ( member8757157785044589968at_nat @ X6 @ A1 )
                   => ? [Xa3: produc3843707927480180839at_nat] :
                        ( ( member8757157785044589968at_nat @ Xa3 @ A22 )
                        & ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X6 @ Xa3 ) @ R ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_847_max__ext_Osimps,axiom,
    ! [A1: set_Pr8693737435421807431at_nat,A22: set_Pr8693737435421807431at_nat,R: set_Pr553994874890374343at_nat] :
      ( ( member5855424355840516880at_nat @ ( produc3236233026405413719at_nat @ A1 @ A22 ) @ ( max_ex4511810952740877858at_nat @ R ) )
      = ( ( finite4392333629123659920at_nat @ A1 )
        & ( finite4392333629123659920at_nat @ A22 )
        & ( A22 != bot_bo5327735625951526323at_nat )
        & ! [X4: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X4 @ A1 )
           => ? [Y4: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ Y4 @ A22 )
                & ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X4 @ Y4 ) @ R ) ) ) ) ) ).

% max_ext.simps
thf(fact_848_max__ext_Osimps,axiom,
    ! [A1: set_nat,A22: set_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A1 @ A22 ) @ ( max_ext_nat @ R ) )
      = ( ( finite_finite_nat @ A1 )
        & ( finite_finite_nat @ A22 )
        & ( A22 != bot_bot_set_nat )
        & ! [X4: nat] :
            ( ( member_nat @ X4 @ A1 )
           => ? [Y4: nat] :
                ( ( member_nat @ Y4 @ A22 )
                & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ R ) ) ) ) ) ).

% max_ext.simps
thf(fact_849_max__ext_Osimps,axiom,
    ! [A1: set_se7855581050983116737at_nat,A22: set_se7855581050983116737at_nat,R: set_Pr4329608150637261639at_nat] :
      ( ( member4300370950623927696at_nat @ ( produc7462898802507495127at_nat @ A1 @ A22 ) @ ( max_ex4535816423174879388at_nat @ R ) )
      = ( ( finite9047747110432174090at_nat @ A1 )
        & ( finite9047747110432174090at_nat @ A22 )
        & ( A22 != bot_bo3083307316010499117at_nat )
        & ! [X4: set_Pr1261947904930325089at_nat] :
            ( ( member2643936169264416010at_nat @ X4 @ A1 )
           => ? [Y4: set_Pr1261947904930325089at_nat] :
                ( ( member2643936169264416010at_nat @ Y4 @ A22 )
                & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y4 ) @ R ) ) ) ) ) ).

% max_ext.simps
thf(fact_850_max__ext_Osimps,axiom,
    ! [A1: set_se5479201105039950375at_nat,A22: set_se5479201105039950375at_nat,R: set_Pr7459493094073627847at_nat] :
      ( ( member6436561159828024080at_nat @ ( produc2492784117751810903at_nat @ A1 @ A22 ) @ ( max_ex4000160833828605186at_nat @ R ) )
      = ( ( finite4768004253697144688at_nat @ A1 )
        & ( finite4768004253697144688at_nat @ A22 )
        & ( A22 != bot_bo1534825418459847827at_nat )
        & ! [X4: set_Pr4329608150637261639at_nat] :
            ( ( member6273874620674182256at_nat @ X4 @ A1 )
           => ? [Y4: set_Pr4329608150637261639at_nat] :
                ( ( member6273874620674182256at_nat @ Y4 @ A22 )
                & ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X4 @ Y4 ) @ R ) ) ) ) ) ).

% max_ext.simps
thf(fact_851_max__ext_Osimps,axiom,
    ! [A1: set_Pr1261947904930325089at_nat,A22: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A1 @ A22 ) @ ( max_ex8135407076693332796at_nat @ R ) )
      = ( ( finite6177210948735845034at_nat @ A1 )
        & ( finite6177210948735845034at_nat @ A22 )
        & ( A22 != bot_bo2099793752762293965at_nat )
        & ! [X4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X4 @ A1 )
           => ? [Y4: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ Y4 @ A22 )
                & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y4 ) @ R ) ) ) ) ) ).

% max_ext.simps
thf(fact_852_max__ext_Osimps,axiom,
    ! [A1: set_Pr4329608150637261639at_nat,A22: set_Pr4329608150637261639at_nat,R: set_Pr5564308138774400199at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A1 @ A22 ) @ ( max_ex8833738776070432290at_nat @ R ) )
      = ( ( finite4343798906461161616at_nat @ A1 )
        & ( finite4343798906461161616at_nat @ A22 )
        & ( A22 != bot_bo228742789529271731at_nat )
        & ! [X4: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X4 @ A1 )
           => ? [Y4: produc3843707927480180839at_nat] :
                ( ( member8757157785044589968at_nat @ Y4 @ A22 )
                & ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X4 @ Y4 ) @ R ) ) ) ) ) ).

% max_ext.simps
thf(fact_853_max__ext__def,axiom,
    ( max_ext_nat
    = ( ^ [R3: set_Pr1261947904930325089at_nat] :
          ( collec6662362479098859352et_nat
          @ ( produc6247414631856714078_nat_o
            @ ( max_extp_nat
              @ ^ [X4: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ R3 ) ) ) ) ) ) ).

% max_ext_def
thf(fact_854_max__ext__def,axiom,
    ( max_ex4535816423174879388at_nat
    = ( ^ [R3: set_Pr4329608150637261639at_nat] :
          ( collec2601550354786890194at_nat
          @ ( produc1491100424724016556_nat_o
            @ ( max_ex4845989503969680116at_nat
              @ ^ [X4: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y4 ) @ R3 ) ) ) ) ) ) ).

% max_ext_def
thf(fact_855_max__ext__def,axiom,
    ( max_ex4000160833828605186at_nat
    = ( ^ [R3: set_Pr7459493094073627847at_nat] :
          ( collec1646449821761096018at_nat
          @ ( produc3723810296477300012_nat_o
            @ ( max_ex8849487260381072218at_nat
              @ ^ [X4: set_Pr4329608150637261639at_nat,Y4: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X4 @ Y4 ) @ R3 ) ) ) ) ) ) ).

% max_ext_def
thf(fact_856_max__ext__def,axiom,
    ( max_ex8135407076693332796at_nat
    = ( ^ [R3: set_Pr8693737435421807431at_nat] :
          ( collec6321179662152712658at_nat
          @ ( produc410239310623530412_nat_o
            @ ( max_ex4864111882549613972at_nat
              @ ^ [X4: product_prod_nat_nat,Y4: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y4 ) @ R3 ) ) ) ) ) ) ).

% max_ext_def
thf(fact_857_max__extp__max__ext__eq,axiom,
    ! [R: set_Pr5564308138774400199at_nat] :
      ( ( max_ex6719288698447201402at_nat
        @ ^ [X4: produc3843707927480180839at_nat,Y4: produc3843707927480180839at_nat] : ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X4 @ Y4 ) @ R ) )
      = ( ^ [X4: set_Pr4329608150637261639at_nat,Y4: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X4 @ Y4 ) @ ( max_ex8833738776070432290at_nat @ R ) ) ) ) ).

% max_extp_max_ext_eq
thf(fact_858_max__extp__max__ext__eq,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( max_extp_nat
        @ ^ [X4: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ R ) )
      = ( ^ [X4: set_nat,Y4: set_nat] : ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X4 @ Y4 ) @ ( max_ext_nat @ R ) ) ) ) ).

% max_extp_max_ext_eq
thf(fact_859_max__extp__max__ext__eq,axiom,
    ! [R: set_Pr4329608150637261639at_nat] :
      ( ( max_ex4845989503969680116at_nat
        @ ^ [X4: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y4 ) @ R ) )
      = ( ^ [X4: set_se7855581050983116737at_nat,Y4: set_se7855581050983116737at_nat] : ( member4300370950623927696at_nat @ ( produc7462898802507495127at_nat @ X4 @ Y4 ) @ ( max_ex4535816423174879388at_nat @ R ) ) ) ) ).

% max_extp_max_ext_eq
thf(fact_860_max__extp__max__ext__eq,axiom,
    ! [R: set_Pr7459493094073627847at_nat] :
      ( ( max_ex8849487260381072218at_nat
        @ ^ [X4: set_Pr4329608150637261639at_nat,Y4: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X4 @ Y4 ) @ R ) )
      = ( ^ [X4: set_se5479201105039950375at_nat,Y4: set_se5479201105039950375at_nat] : ( member6436561159828024080at_nat @ ( produc2492784117751810903at_nat @ X4 @ Y4 ) @ ( max_ex4000160833828605186at_nat @ R ) ) ) ) ).

% max_extp_max_ext_eq
thf(fact_861_max__extp__max__ext__eq,axiom,
    ! [R: set_Pr8693737435421807431at_nat] :
      ( ( max_ex4864111882549613972at_nat
        @ ^ [X4: product_prod_nat_nat,Y4: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y4 ) @ R ) )
      = ( ^ [X4: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y4 ) @ ( max_ex8135407076693332796at_nat @ R ) ) ) ) ).

% max_extp_max_ext_eq
thf(fact_862_in__inv__image,axiom,
    ! [X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat,F2: nat > nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( inv_image_nat_nat @ R2 @ F2 ) )
      = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R2 ) ) ).

% in_inv_image
thf(fact_863_in__inv__image,axiom,
    ! [X: nat,Y: nat,R2: set_Pr8693737435421807431at_nat,F2: nat > product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( inv_im8721129524585065343at_nat @ R2 @ F2 ) )
      = ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R2 ) ) ).

% in_inv_image
thf(fact_864_in__inv__image,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,R2: set_Pr1261947904930325089at_nat,F2: product_prod_nat_nat > nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( inv_im2857804880773004637at_nat @ R2 @ F2 ) )
      = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R2 ) ) ).

% in_inv_image
thf(fact_865_in__inv__image,axiom,
    ! [X: nat,Y: nat,R2: set_Pr4329608150637261639at_nat,F2: nat > set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( inv_im4437241714082187487at_nat @ R2 @ F2 ) )
      = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R2 ) ) ).

% in_inv_image
thf(fact_866_in__inv__image,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,R2: set_Pr1261947904930325089at_nat,F2: set_Pr1261947904930325089at_nat > nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ ( inv_im6750873321330482877at_nat @ R2 @ F2 ) )
      = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R2 ) ) ).

% in_inv_image
thf(fact_867_in__inv__image,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat,F2: product_prod_nat_nat > product_prod_nat_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( inv_im5896302226278515346at_nat @ R2 @ F2 ) )
      = ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R2 ) ) ).

% in_inv_image
thf(fact_868_in__inv__image,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,R2: set_Pr4329608150637261639at_nat,F2: product_prod_nat_nat > set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( inv_im8562125120543730994at_nat @ R2 @ F2 ) )
      = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R2 ) ) ).

% in_inv_image
thf(fact_869_in__inv__image,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,R2: set_Pr8693737435421807431at_nat,F2: set_Pr1261947904930325089at_nat > product_prod_nat_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ ( inv_im984280340187844594at_nat @ R2 @ F2 ) )
      = ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R2 ) ) ).

% in_inv_image
thf(fact_870_in__inv__image,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat,F2: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ ( inv_im7023619800666061458at_nat @ R2 @ F2 ) )
      = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R2 ) ) ).

% in_inv_image
thf(fact_871_in__inv__image,axiom,
    ! [X: nat,Y: nat,R2: set_Pr7459493094073627847at_nat,F2: nat > set_Pr4329608150637261639at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( inv_im2903009620893676101at_nat @ R2 @ F2 ) )
      = ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R2 ) ) ).

% in_inv_image
thf(fact_872_inv__image__def,axiom,
    ( inv_image_nat_nat
    = ( ^ [R4: set_Pr1261947904930325089at_nat,F3: nat > nat] :
          ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [X4: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F3 @ X4 ) @ ( F3 @ Y4 ) ) @ R4 ) ) ) ) ) ).

% inv_image_def
thf(fact_873_inv__image__def,axiom,
    ( inv_im8721129524585065343at_nat
    = ( ^ [R4: set_Pr8693737435421807431at_nat,F3: nat > product_prod_nat_nat] :
          ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [X4: nat,Y4: nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( F3 @ X4 ) @ ( F3 @ Y4 ) ) @ R4 ) ) ) ) ) ).

% inv_image_def
thf(fact_874_inv__image__def,axiom,
    ( inv_im4437241714082187487at_nat
    = ( ^ [R4: set_Pr4329608150637261639at_nat,F3: nat > set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [X4: nat,Y4: nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( F3 @ X4 ) @ ( F3 @ Y4 ) ) @ R4 ) ) ) ) ) ).

% inv_image_def
thf(fact_875_inv__image__def,axiom,
    ( inv_im2903009620893676101at_nat
    = ( ^ [R4: set_Pr7459493094073627847at_nat,F3: nat > set_Pr4329608150637261639at_nat] :
          ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [X4: nat,Y4: nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ ( F3 @ X4 ) @ ( F3 @ Y4 ) ) @ R4 ) ) ) ) ) ).

% inv_image_def
thf(fact_876_max__extp__eq,axiom,
    ( max_ex6719288698447201402at_nat
    = ( ^ [R4: produc3843707927480180839at_nat > produc3843707927480180839at_nat > $o,X4: set_Pr4329608150637261639at_nat,Y4: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X4 @ Y4 ) @ ( max_ex8833738776070432290at_nat @ ( collec223960383693904210at_nat @ ( produc1561160019221259564_nat_o @ R4 ) ) ) ) ) ) ).

% max_extp_eq
thf(fact_877_max__extp__eq,axiom,
    ( max_ex4864111882549613972at_nat
    = ( ^ [R4: product_prod_nat_nat > product_prod_nat_nat > $o,X4: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y4 ) @ ( max_ex8135407076693332796at_nat @ ( collec7088162979684241874at_nat @ ( produc6590410687421337004_nat_o @ R4 ) ) ) ) ) ) ).

% max_extp_eq
thf(fact_878_max__extp__eq,axiom,
    ( max_extp_nat
    = ( ^ [R4: nat > nat > $o,X4: set_nat,Y4: set_nat] : ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X4 @ Y4 ) @ ( max_ext_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R4 ) ) ) ) ) ) ).

% max_extp_eq
thf(fact_879_mlex__prod__def,axiom,
    ( mlex_prod_nat
    = ( ^ [F3: nat > nat,R3: set_Pr1261947904930325089at_nat] :
          ( inv_im8721129524585065343at_nat @ ( lex_prod_nat_nat @ less_than @ R3 )
          @ ^ [X4: nat] : ( product_Pair_nat_nat @ ( F3 @ X4 ) @ X4 ) ) ) ) ).

% mlex_prod_def
thf(fact_880_smin__insertI,axiom,
    ! [X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ X @ XS )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ fun_pair_less )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_min_strict )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ ( insert8211810215607154385at_nat @ Y @ YS ) ) @ fun_min_strict ) ) ) ) ).

% smin_insertI
thf(fact_881_insert__absorb2,axiom,
    ! [X: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( insert8211810215607154385at_nat @ X @ ( insert8211810215607154385at_nat @ X @ A4 ) )
      = ( insert8211810215607154385at_nat @ X @ A4 ) ) ).

% insert_absorb2
thf(fact_882_insert__absorb2,axiom,
    ! [X: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ( insert9069300056098147895at_nat @ X @ ( insert9069300056098147895at_nat @ X @ A4 ) )
      = ( insert9069300056098147895at_nat @ X @ A4 ) ) ).

% insert_absorb2
thf(fact_883_insert__iff,axiom,
    ! [A: produc859450856879609959at_nat,B: produc859450856879609959at_nat,A4: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ A @ ( insert5050368324300391991at_nat @ B @ A4 ) )
      = ( ( A = B )
        | ( member8206827879206165904at_nat @ A @ A4 ) ) ) ).

% insert_iff
thf(fact_884_insert__iff,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ B @ A4 ) )
      = ( ( A = B )
        | ( member8440522571783428010at_nat @ A @ A4 ) ) ) ).

% insert_iff
thf(fact_885_insert__iff,axiom,
    ! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ A @ ( insert9069300056098147895at_nat @ B @ A4 ) )
      = ( ( A = B )
        | ( member8757157785044589968at_nat @ A @ A4 ) ) ) ).

% insert_iff
thf(fact_886_insert__iff,axiom,
    ! [A: nat,B: nat,A4: set_nat] :
      ( ( member_nat @ A @ ( insert_nat @ B @ A4 ) )
      = ( ( A = B )
        | ( member_nat @ A @ A4 ) ) ) ).

% insert_iff
thf(fact_887_insertCI,axiom,
    ! [A: produc859450856879609959at_nat,B4: set_Pr8693737435421807431at_nat,B: produc859450856879609959at_nat] :
      ( ( ~ ( member8206827879206165904at_nat @ A @ B4 )
       => ( A = B ) )
     => ( member8206827879206165904at_nat @ A @ ( insert5050368324300391991at_nat @ B @ B4 ) ) ) ).

% insertCI
thf(fact_888_insertCI,axiom,
    ! [A: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
      ( ( ~ ( member8440522571783428010at_nat @ A @ B4 )
       => ( A = B ) )
     => ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ B @ B4 ) ) ) ).

% insertCI
thf(fact_889_insertCI,axiom,
    ! [A: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
      ( ( ~ ( member8757157785044589968at_nat @ A @ B4 )
       => ( A = B ) )
     => ( member8757157785044589968at_nat @ A @ ( insert9069300056098147895at_nat @ B @ B4 ) ) ) ).

% insertCI
thf(fact_890_insertCI,axiom,
    ! [A: nat,B4: set_nat,B: nat] :
      ( ( ~ ( member_nat @ A @ B4 )
       => ( A = B ) )
     => ( member_nat @ A @ ( insert_nat @ B @ B4 ) ) ) ).

% insertCI
thf(fact_891_insert__subset,axiom,
    ! [X: produc859450856879609959at_nat,A4: set_Pr8693737435421807431at_nat,B4: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ ( insert5050368324300391991at_nat @ X @ A4 ) @ B4 )
      = ( ( member8206827879206165904at_nat @ X @ B4 )
        & ( ord_le3000389064537975527at_nat @ A4 @ B4 ) ) ) ).

% insert_subset
thf(fact_892_insert__subset,axiom,
    ! [X: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ ( insert8211810215607154385at_nat @ X @ A4 ) @ B4 )
      = ( ( member8440522571783428010at_nat @ X @ B4 )
        & ( ord_le3146513528884898305at_nat @ A4 @ B4 ) ) ) ).

% insert_subset
thf(fact_893_insert__subset,axiom,
    ! [X: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ ( insert9069300056098147895at_nat @ X @ A4 ) @ B4 )
      = ( ( member8757157785044589968at_nat @ X @ B4 )
        & ( ord_le1268244103169919719at_nat @ A4 @ B4 ) ) ) ).

% insert_subset
thf(fact_894_insert__subset,axiom,
    ! [X: nat,A4: set_nat,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ ( insert_nat @ X @ A4 ) @ B4 )
      = ( ( member_nat @ X @ B4 )
        & ( ord_less_eq_set_nat @ A4 @ B4 ) ) ) ).

% insert_subset
thf(fact_895_singletonI,axiom,
    ! [A: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ A @ ( insert5050368324300391991at_nat @ A @ bot_bo5327735625951526323at_nat ) ) ).

% singletonI
thf(fact_896_singletonI,axiom,
    ! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% singletonI
thf(fact_897_singletonI,axiom,
    ! [A: product_prod_nat_nat] : ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ).

% singletonI
thf(fact_898_singletonI,axiom,
    ! [A: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ A @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ).

% singletonI
thf(fact_899_singleton__conv,axiom,
    ! [A: nat] :
      ( ( collect_nat
        @ ^ [X4: nat] : ( X4 = A ) )
      = ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% singleton_conv
thf(fact_900_singleton__conv,axiom,
    ! [A: product_prod_nat_nat] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X4: product_prod_nat_nat] : ( X4 = A ) )
      = ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ).

% singleton_conv
thf(fact_901_singleton__conv,axiom,
    ! [A: produc3843707927480180839at_nat] :
      ( ( collec6321179662152712658at_nat
        @ ^ [X4: produc3843707927480180839at_nat] : ( X4 = A ) )
      = ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ).

% singleton_conv
thf(fact_902_singleton__conv2,axiom,
    ! [A: nat] :
      ( ( collect_nat
        @ ( ^ [Y2: nat,Z2: nat] : ( Y2 = Z2 )
          @ A ) )
      = ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% singleton_conv2
thf(fact_903_singleton__conv2,axiom,
    ! [A: product_prod_nat_nat] :
      ( ( collec3392354462482085612at_nat
        @ ( ^ [Y2: product_prod_nat_nat,Z2: product_prod_nat_nat] : ( Y2 = Z2 )
          @ A ) )
      = ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ).

% singleton_conv2
thf(fact_904_singleton__conv2,axiom,
    ! [A: produc3843707927480180839at_nat] :
      ( ( collec6321179662152712658at_nat
        @ ( ^ [Y2: produc3843707927480180839at_nat,Z2: produc3843707927480180839at_nat] : ( Y2 = Z2 )
          @ A ) )
      = ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ).

% singleton_conv2
thf(fact_905_singleton__insert__inj__eq_H,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
      ( ( ( insert8211810215607154385at_nat @ A @ A4 )
        = ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) )
      = ( ( A = B )
        & ( ord_le3146513528884898305at_nat @ A4 @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_906_singleton__insert__inj__eq_H,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
      ( ( ( insert9069300056098147895at_nat @ A @ A4 )
        = ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) )
      = ( ( A = B )
        & ( ord_le1268244103169919719at_nat @ A4 @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_907_singleton__insert__inj__eq,axiom,
    ! [B: product_prod_nat_nat,A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat )
        = ( insert8211810215607154385at_nat @ A @ A4 ) )
      = ( ( A = B )
        & ( ord_le3146513528884898305at_nat @ A4 @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_908_singleton__insert__inj__eq,axiom,
    ! [B: produc3843707927480180839at_nat,A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ( ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat )
        = ( insert9069300056098147895at_nat @ A @ A4 ) )
      = ( ( A = B )
        & ( ord_le1268244103169919719at_nat @ A4 @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_909_Collect__conv__if,axiom,
    ! [P: nat > $o,A: nat] :
      ( ( ( P @ A )
       => ( ( collect_nat
            @ ^ [X4: nat] :
                ( ( X4 = A )
                & ( P @ X4 ) ) )
          = ( insert_nat @ A @ bot_bot_set_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_nat
            @ ^ [X4: nat] :
                ( ( X4 = A )
                & ( P @ X4 ) ) )
          = bot_bot_set_nat ) ) ) ).

% Collect_conv_if
thf(fact_910_Collect__conv__if,axiom,
    ! [P: product_prod_nat_nat > $o,A: product_prod_nat_nat] :
      ( ( ( P @ A )
       => ( ( collec3392354462482085612at_nat
            @ ^ [X4: product_prod_nat_nat] :
                ( ( X4 = A )
                & ( P @ X4 ) ) )
          = ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collec3392354462482085612at_nat
            @ ^ [X4: product_prod_nat_nat] :
                ( ( X4 = A )
                & ( P @ X4 ) ) )
          = bot_bo2099793752762293965at_nat ) ) ) ).

% Collect_conv_if
thf(fact_911_Collect__conv__if,axiom,
    ! [P: produc3843707927480180839at_nat > $o,A: produc3843707927480180839at_nat] :
      ( ( ( P @ A )
       => ( ( collec6321179662152712658at_nat
            @ ^ [X4: produc3843707927480180839at_nat] :
                ( ( X4 = A )
                & ( P @ X4 ) ) )
          = ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collec6321179662152712658at_nat
            @ ^ [X4: produc3843707927480180839at_nat] :
                ( ( X4 = A )
                & ( P @ X4 ) ) )
          = bot_bo228742789529271731at_nat ) ) ) ).

% Collect_conv_if
thf(fact_912_Collect__conv__if2,axiom,
    ! [P: nat > $o,A: nat] :
      ( ( ( P @ A )
       => ( ( collect_nat
            @ ^ [X4: nat] :
                ( ( A = X4 )
                & ( P @ X4 ) ) )
          = ( insert_nat @ A @ bot_bot_set_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_nat
            @ ^ [X4: nat] :
                ( ( A = X4 )
                & ( P @ X4 ) ) )
          = bot_bot_set_nat ) ) ) ).

% Collect_conv_if2
thf(fact_913_Collect__conv__if2,axiom,
    ! [P: product_prod_nat_nat > $o,A: product_prod_nat_nat] :
      ( ( ( P @ A )
       => ( ( collec3392354462482085612at_nat
            @ ^ [X4: product_prod_nat_nat] :
                ( ( A = X4 )
                & ( P @ X4 ) ) )
          = ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collec3392354462482085612at_nat
            @ ^ [X4: product_prod_nat_nat] :
                ( ( A = X4 )
                & ( P @ X4 ) ) )
          = bot_bo2099793752762293965at_nat ) ) ) ).

% Collect_conv_if2
thf(fact_914_Collect__conv__if2,axiom,
    ! [P: produc3843707927480180839at_nat > $o,A: produc3843707927480180839at_nat] :
      ( ( ( P @ A )
       => ( ( collec6321179662152712658at_nat
            @ ^ [X4: produc3843707927480180839at_nat] :
                ( ( A = X4 )
                & ( P @ X4 ) ) )
          = ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collec6321179662152712658at_nat
            @ ^ [X4: produc3843707927480180839at_nat] :
                ( ( A = X4 )
                & ( P @ X4 ) ) )
          = bot_bo228742789529271731at_nat ) ) ) ).

% Collect_conv_if2
thf(fact_915_inj__singleton,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( inj_on8528517097230073361at_nat
      @ ^ [X4: product_prod_nat_nat] : ( insert8211810215607154385at_nat @ X4 @ bot_bo2099793752762293965at_nat )
      @ A4 ) ).

% inj_singleton
thf(fact_916_inj__singleton,axiom,
    ! [A4: set_Pr4329608150637261639at_nat] :
      ( inj_on104525631246638609at_nat
      @ ^ [X4: produc3843707927480180839at_nat] : ( insert9069300056098147895at_nat @ X4 @ bot_bo228742789529271731at_nat )
      @ A4 ) ).

% inj_singleton
thf(fact_917_insert__compr,axiom,
    ( insert5050368324300391991at_nat
    = ( ^ [A3: produc859450856879609959at_nat,B6: set_Pr8693737435421807431at_nat] :
          ( collec7088162979684241874at_nat
          @ ^ [X4: produc859450856879609959at_nat] :
              ( ( X4 = A3 )
              | ( member8206827879206165904at_nat @ X4 @ B6 ) ) ) ) ) ).

% insert_compr
thf(fact_918_insert__compr,axiom,
    ( insert9069300056098147895at_nat
    = ( ^ [A3: produc3843707927480180839at_nat,B6: set_Pr4329608150637261639at_nat] :
          ( collec6321179662152712658at_nat
          @ ^ [X4: produc3843707927480180839at_nat] :
              ( ( X4 = A3 )
              | ( member8757157785044589968at_nat @ X4 @ B6 ) ) ) ) ) ).

% insert_compr
thf(fact_919_insert__compr,axiom,
    ( insert_nat
    = ( ^ [A3: nat,B6: set_nat] :
          ( collect_nat
          @ ^ [X4: nat] :
              ( ( X4 = A3 )
              | ( member_nat @ X4 @ B6 ) ) ) ) ) ).

% insert_compr
thf(fact_920_insert__compr,axiom,
    ( insert8211810215607154385at_nat
    = ( ^ [A3: product_prod_nat_nat,B6: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [X4: product_prod_nat_nat] :
              ( ( X4 = A3 )
              | ( member8440522571783428010at_nat @ X4 @ B6 ) ) ) ) ) ).

% insert_compr
thf(fact_921_insert__Collect,axiom,
    ! [A: produc3843707927480180839at_nat,P: produc3843707927480180839at_nat > $o] :
      ( ( insert9069300056098147895at_nat @ A @ ( collec6321179662152712658at_nat @ P ) )
      = ( collec6321179662152712658at_nat
        @ ^ [U: produc3843707927480180839at_nat] :
            ( ( U != A )
           => ( P @ U ) ) ) ) ).

% insert_Collect
thf(fact_922_insert__Collect,axiom,
    ! [A: nat,P: nat > $o] :
      ( ( insert_nat @ A @ ( collect_nat @ P ) )
      = ( collect_nat
        @ ^ [U: nat] :
            ( ( U != A )
           => ( P @ U ) ) ) ) ).

% insert_Collect
thf(fact_923_insert__Collect,axiom,
    ! [A: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
      ( ( insert8211810215607154385at_nat @ A @ ( collec3392354462482085612at_nat @ P ) )
      = ( collec3392354462482085612at_nat
        @ ^ [U: product_prod_nat_nat] :
            ( ( U != A )
           => ( P @ U ) ) ) ) ).

% insert_Collect
thf(fact_924_card__insert__le,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat] : ( ord_less_eq_nat @ ( finite3771342082235030671at_nat @ A4 ) @ ( finite3771342082235030671at_nat @ ( insert9069300056098147895at_nat @ X @ A4 ) ) ) ).

% card_insert_le
thf(fact_925_card__insert__le,axiom,
    ! [A4: set_nat,X: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ ( insert_nat @ X @ A4 ) ) ) ).

% card_insert_le
thf(fact_926_card__insert__le,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] : ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ A4 ) @ ( finite711546835091564841at_nat @ ( insert8211810215607154385at_nat @ X @ A4 ) ) ) ).

% card_insert_le
thf(fact_927_Set_Oinsert__mono,axiom,
    ! [C2: set_Pr1261947904930325089at_nat,D2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( ( ord_le3146513528884898305at_nat @ C2 @ D2 )
     => ( ord_le3146513528884898305at_nat @ ( insert8211810215607154385at_nat @ A @ C2 ) @ ( insert8211810215607154385at_nat @ A @ D2 ) ) ) ).

% Set.insert_mono
thf(fact_928_Set_Oinsert__mono,axiom,
    ! [C2: set_Pr4329608150637261639at_nat,D2: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
      ( ( ord_le1268244103169919719at_nat @ C2 @ D2 )
     => ( ord_le1268244103169919719at_nat @ ( insert9069300056098147895at_nat @ A @ C2 ) @ ( insert9069300056098147895at_nat @ A @ D2 ) ) ) ).

% Set.insert_mono
thf(fact_929_subset__insert,axiom,
    ! [X: produc859450856879609959at_nat,A4: set_Pr8693737435421807431at_nat,B4: set_Pr8693737435421807431at_nat] :
      ( ~ ( member8206827879206165904at_nat @ X @ A4 )
     => ( ( ord_le3000389064537975527at_nat @ A4 @ ( insert5050368324300391991at_nat @ X @ B4 ) )
        = ( ord_le3000389064537975527at_nat @ A4 @ B4 ) ) ) ).

% subset_insert
thf(fact_930_subset__insert,axiom,
    ! [X: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ~ ( member8440522571783428010at_nat @ X @ A4 )
     => ( ( ord_le3146513528884898305at_nat @ A4 @ ( insert8211810215607154385at_nat @ X @ B4 ) )
        = ( ord_le3146513528884898305at_nat @ A4 @ B4 ) ) ) ).

% subset_insert
thf(fact_931_subset__insert,axiom,
    ! [X: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ~ ( member8757157785044589968at_nat @ X @ A4 )
     => ( ( ord_le1268244103169919719at_nat @ A4 @ ( insert9069300056098147895at_nat @ X @ B4 ) )
        = ( ord_le1268244103169919719at_nat @ A4 @ B4 ) ) ) ).

% subset_insert
thf(fact_932_subset__insert,axiom,
    ! [X: nat,A4: set_nat,B4: set_nat] :
      ( ~ ( member_nat @ X @ A4 )
     => ( ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ X @ B4 ) )
        = ( ord_less_eq_set_nat @ A4 @ B4 ) ) ) ).

% subset_insert
thf(fact_933_subset__insertI,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] : ( ord_le3146513528884898305at_nat @ B4 @ ( insert8211810215607154385at_nat @ A @ B4 ) ) ).

% subset_insertI
thf(fact_934_subset__insertI,axiom,
    ! [B4: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] : ( ord_le1268244103169919719at_nat @ B4 @ ( insert9069300056098147895at_nat @ A @ B4 ) ) ).

% subset_insertI
thf(fact_935_subset__insertI2,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
      ( ( ord_le3146513528884898305at_nat @ A4 @ B4 )
     => ( ord_le3146513528884898305at_nat @ A4 @ ( insert8211810215607154385at_nat @ B @ B4 ) ) ) ).

% subset_insertI2
thf(fact_936_subset__insertI2,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A4 @ B4 )
     => ( ord_le1268244103169919719at_nat @ A4 @ ( insert9069300056098147895at_nat @ B @ B4 ) ) ) ).

% subset_insertI2
thf(fact_937_mk__disjoint__insert,axiom,
    ! [A: produc859450856879609959at_nat,A4: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ A @ A4 )
     => ? [B8: set_Pr8693737435421807431at_nat] :
          ( ( A4
            = ( insert5050368324300391991at_nat @ A @ B8 ) )
          & ~ ( member8206827879206165904at_nat @ A @ B8 ) ) ) ).

% mk_disjoint_insert
thf(fact_938_mk__disjoint__insert,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ A @ A4 )
     => ? [B8: set_Pr1261947904930325089at_nat] :
          ( ( A4
            = ( insert8211810215607154385at_nat @ A @ B8 ) )
          & ~ ( member8440522571783428010at_nat @ A @ B8 ) ) ) ).

% mk_disjoint_insert
thf(fact_939_mk__disjoint__insert,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ A @ A4 )
     => ? [B8: set_Pr4329608150637261639at_nat] :
          ( ( A4
            = ( insert9069300056098147895at_nat @ A @ B8 ) )
          & ~ ( member8757157785044589968at_nat @ A @ B8 ) ) ) ).

% mk_disjoint_insert
thf(fact_940_mk__disjoint__insert,axiom,
    ! [A: nat,A4: set_nat] :
      ( ( member_nat @ A @ A4 )
     => ? [B8: set_nat] :
          ( ( A4
            = ( insert_nat @ A @ B8 ) )
          & ~ ( member_nat @ A @ B8 ) ) ) ).

% mk_disjoint_insert
thf(fact_941_insert__commute,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( insert8211810215607154385at_nat @ X @ ( insert8211810215607154385at_nat @ Y @ A4 ) )
      = ( insert8211810215607154385at_nat @ Y @ ( insert8211810215607154385at_nat @ X @ A4 ) ) ) ).

% insert_commute
thf(fact_942_insert__commute,axiom,
    ! [X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ( insert9069300056098147895at_nat @ X @ ( insert9069300056098147895at_nat @ Y @ A4 ) )
      = ( insert9069300056098147895at_nat @ Y @ ( insert9069300056098147895at_nat @ X @ A4 ) ) ) ).

% insert_commute
thf(fact_943_insert__eq__iff,axiom,
    ! [A: produc859450856879609959at_nat,A4: set_Pr8693737435421807431at_nat,B: produc859450856879609959at_nat,B4: set_Pr8693737435421807431at_nat] :
      ( ~ ( member8206827879206165904at_nat @ A @ A4 )
     => ( ~ ( member8206827879206165904at_nat @ B @ B4 )
       => ( ( ( insert5050368324300391991at_nat @ A @ A4 )
            = ( insert5050368324300391991at_nat @ B @ B4 ) )
          = ( ( ( A = B )
             => ( A4 = B4 ) )
            & ( ( A != B )
             => ? [C5: set_Pr8693737435421807431at_nat] :
                  ( ( A4
                    = ( insert5050368324300391991at_nat @ B @ C5 ) )
                  & ~ ( member8206827879206165904at_nat @ B @ C5 )
                  & ( B4
                    = ( insert5050368324300391991at_nat @ A @ C5 ) )
                  & ~ ( member8206827879206165904at_nat @ A @ C5 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_944_insert__eq__iff,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ~ ( member8440522571783428010at_nat @ A @ A4 )
     => ( ~ ( member8440522571783428010at_nat @ B @ B4 )
       => ( ( ( insert8211810215607154385at_nat @ A @ A4 )
            = ( insert8211810215607154385at_nat @ B @ B4 ) )
          = ( ( ( A = B )
             => ( A4 = B4 ) )
            & ( ( A != B )
             => ? [C5: set_Pr1261947904930325089at_nat] :
                  ( ( A4
                    = ( insert8211810215607154385at_nat @ B @ C5 ) )
                  & ~ ( member8440522571783428010at_nat @ B @ C5 )
                  & ( B4
                    = ( insert8211810215607154385at_nat @ A @ C5 ) )
                  & ~ ( member8440522571783428010at_nat @ A @ C5 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_945_insert__eq__iff,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ~ ( member8757157785044589968at_nat @ A @ A4 )
     => ( ~ ( member8757157785044589968at_nat @ B @ B4 )
       => ( ( ( insert9069300056098147895at_nat @ A @ A4 )
            = ( insert9069300056098147895at_nat @ B @ B4 ) )
          = ( ( ( A = B )
             => ( A4 = B4 ) )
            & ( ( A != B )
             => ? [C5: set_Pr4329608150637261639at_nat] :
                  ( ( A4
                    = ( insert9069300056098147895at_nat @ B @ C5 ) )
                  & ~ ( member8757157785044589968at_nat @ B @ C5 )
                  & ( B4
                    = ( insert9069300056098147895at_nat @ A @ C5 ) )
                  & ~ ( member8757157785044589968at_nat @ A @ C5 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_946_insert__eq__iff,axiom,
    ! [A: nat,A4: set_nat,B: nat,B4: set_nat] :
      ( ~ ( member_nat @ A @ A4 )
     => ( ~ ( member_nat @ B @ B4 )
       => ( ( ( insert_nat @ A @ A4 )
            = ( insert_nat @ B @ B4 ) )
          = ( ( ( A = B )
             => ( A4 = B4 ) )
            & ( ( A != B )
             => ? [C5: set_nat] :
                  ( ( A4
                    = ( insert_nat @ B @ C5 ) )
                  & ~ ( member_nat @ B @ C5 )
                  & ( B4
                    = ( insert_nat @ A @ C5 ) )
                  & ~ ( member_nat @ A @ C5 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_947_insert__absorb,axiom,
    ! [A: produc859450856879609959at_nat,A4: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ A @ A4 )
     => ( ( insert5050368324300391991at_nat @ A @ A4 )
        = A4 ) ) ).

% insert_absorb
thf(fact_948_insert__absorb,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ A @ A4 )
     => ( ( insert8211810215607154385at_nat @ A @ A4 )
        = A4 ) ) ).

% insert_absorb
thf(fact_949_insert__absorb,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ A @ A4 )
     => ( ( insert9069300056098147895at_nat @ A @ A4 )
        = A4 ) ) ).

% insert_absorb
thf(fact_950_insert__absorb,axiom,
    ! [A: nat,A4: set_nat] :
      ( ( member_nat @ A @ A4 )
     => ( ( insert_nat @ A @ A4 )
        = A4 ) ) ).

% insert_absorb
thf(fact_951_insert__ident,axiom,
    ! [X: produc859450856879609959at_nat,A4: set_Pr8693737435421807431at_nat,B4: set_Pr8693737435421807431at_nat] :
      ( ~ ( member8206827879206165904at_nat @ X @ A4 )
     => ( ~ ( member8206827879206165904at_nat @ X @ B4 )
       => ( ( ( insert5050368324300391991at_nat @ X @ A4 )
            = ( insert5050368324300391991at_nat @ X @ B4 ) )
          = ( A4 = B4 ) ) ) ) ).

% insert_ident
thf(fact_952_insert__ident,axiom,
    ! [X: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ~ ( member8440522571783428010at_nat @ X @ A4 )
     => ( ~ ( member8440522571783428010at_nat @ X @ B4 )
       => ( ( ( insert8211810215607154385at_nat @ X @ A4 )
            = ( insert8211810215607154385at_nat @ X @ B4 ) )
          = ( A4 = B4 ) ) ) ) ).

% insert_ident
thf(fact_953_insert__ident,axiom,
    ! [X: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ~ ( member8757157785044589968at_nat @ X @ A4 )
     => ( ~ ( member8757157785044589968at_nat @ X @ B4 )
       => ( ( ( insert9069300056098147895at_nat @ X @ A4 )
            = ( insert9069300056098147895at_nat @ X @ B4 ) )
          = ( A4 = B4 ) ) ) ) ).

% insert_ident
thf(fact_954_insert__ident,axiom,
    ! [X: nat,A4: set_nat,B4: set_nat] :
      ( ~ ( member_nat @ X @ A4 )
     => ( ~ ( member_nat @ X @ B4 )
       => ( ( ( insert_nat @ X @ A4 )
            = ( insert_nat @ X @ B4 ) )
          = ( A4 = B4 ) ) ) ) ).

% insert_ident
thf(fact_955_Set_Oset__insert,axiom,
    ! [X: produc859450856879609959at_nat,A4: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ X @ A4 )
     => ~ ! [B8: set_Pr8693737435421807431at_nat] :
            ( ( A4
              = ( insert5050368324300391991at_nat @ X @ B8 ) )
           => ( member8206827879206165904at_nat @ X @ B8 ) ) ) ).

% Set.set_insert
thf(fact_956_Set_Oset__insert,axiom,
    ! [X: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ X @ A4 )
     => ~ ! [B8: set_Pr1261947904930325089at_nat] :
            ( ( A4
              = ( insert8211810215607154385at_nat @ X @ B8 ) )
           => ( member8440522571783428010at_nat @ X @ B8 ) ) ) ).

% Set.set_insert
thf(fact_957_Set_Oset__insert,axiom,
    ! [X: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ X @ A4 )
     => ~ ! [B8: set_Pr4329608150637261639at_nat] :
            ( ( A4
              = ( insert9069300056098147895at_nat @ X @ B8 ) )
           => ( member8757157785044589968at_nat @ X @ B8 ) ) ) ).

% Set.set_insert
thf(fact_958_Set_Oset__insert,axiom,
    ! [X: nat,A4: set_nat] :
      ( ( member_nat @ X @ A4 )
     => ~ ! [B8: set_nat] :
            ( ( A4
              = ( insert_nat @ X @ B8 ) )
           => ( member_nat @ X @ B8 ) ) ) ).

% Set.set_insert
thf(fact_959_insertI2,axiom,
    ! [A: produc859450856879609959at_nat,B4: set_Pr8693737435421807431at_nat,B: produc859450856879609959at_nat] :
      ( ( member8206827879206165904at_nat @ A @ B4 )
     => ( member8206827879206165904at_nat @ A @ ( insert5050368324300391991at_nat @ B @ B4 ) ) ) ).

% insertI2
thf(fact_960_insertI2,axiom,
    ! [A: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ A @ B4 )
     => ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ B @ B4 ) ) ) ).

% insertI2
thf(fact_961_insertI2,axiom,
    ! [A: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
      ( ( member8757157785044589968at_nat @ A @ B4 )
     => ( member8757157785044589968at_nat @ A @ ( insert9069300056098147895at_nat @ B @ B4 ) ) ) ).

% insertI2
thf(fact_962_insertI2,axiom,
    ! [A: nat,B4: set_nat,B: nat] :
      ( ( member_nat @ A @ B4 )
     => ( member_nat @ A @ ( insert_nat @ B @ B4 ) ) ) ).

% insertI2
thf(fact_963_insertI1,axiom,
    ! [A: produc859450856879609959at_nat,B4: set_Pr8693737435421807431at_nat] : ( member8206827879206165904at_nat @ A @ ( insert5050368324300391991at_nat @ A @ B4 ) ) ).

% insertI1
thf(fact_964_insertI1,axiom,
    ! [A: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] : ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ A @ B4 ) ) ).

% insertI1
thf(fact_965_insertI1,axiom,
    ! [A: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat] : ( member8757157785044589968at_nat @ A @ ( insert9069300056098147895at_nat @ A @ B4 ) ) ).

% insertI1
thf(fact_966_insertI1,axiom,
    ! [A: nat,B4: set_nat] : ( member_nat @ A @ ( insert_nat @ A @ B4 ) ) ).

% insertI1
thf(fact_967_insertE,axiom,
    ! [A: produc859450856879609959at_nat,B: produc859450856879609959at_nat,A4: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ A @ ( insert5050368324300391991at_nat @ B @ A4 ) )
     => ( ( A != B )
       => ( member8206827879206165904at_nat @ A @ A4 ) ) ) ).

% insertE
thf(fact_968_insertE,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ B @ A4 ) )
     => ( ( A != B )
       => ( member8440522571783428010at_nat @ A @ A4 ) ) ) ).

% insertE
thf(fact_969_insertE,axiom,
    ! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ A @ ( insert9069300056098147895at_nat @ B @ A4 ) )
     => ( ( A != B )
       => ( member8757157785044589968at_nat @ A @ A4 ) ) ) ).

% insertE
thf(fact_970_insertE,axiom,
    ! [A: nat,B: nat,A4: set_nat] :
      ( ( member_nat @ A @ ( insert_nat @ B @ A4 ) )
     => ( ( A != B )
       => ( member_nat @ A @ A4 ) ) ) ).

% insertE
thf(fact_971_subset__singletonD,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( ( ord_le3146513528884898305at_nat @ A4 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
     => ( ( A4 = bot_bo2099793752762293965at_nat )
        | ( A4
          = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% subset_singletonD
thf(fact_972_subset__singletonD,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A4 @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) )
     => ( ( A4 = bot_bo228742789529271731at_nat )
        | ( A4
          = ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ) ) ).

% subset_singletonD
thf(fact_973_subset__singleton__iff,axiom,
    ! [X5: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( ( ord_le3146513528884898305at_nat @ X5 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) )
      = ( ( X5 = bot_bo2099793752762293965at_nat )
        | ( X5
          = ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% subset_singleton_iff
thf(fact_974_subset__singleton__iff,axiom,
    ! [X5: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
      ( ( ord_le1268244103169919719at_nat @ X5 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) )
      = ( ( X5 = bot_bo228742789529271731at_nat )
        | ( X5
          = ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ) ) ).

% subset_singleton_iff
thf(fact_975_set__one,axiom,
    ( one_one_set_nat
    = ( insert_nat @ one_one_nat @ bot_bot_set_nat ) ) ).

% set_one
thf(fact_976_singletonD,axiom,
    ! [B: produc859450856879609959at_nat,A: produc859450856879609959at_nat] :
      ( ( member8206827879206165904at_nat @ B @ ( insert5050368324300391991at_nat @ A @ bot_bo5327735625951526323at_nat ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_977_singletonD,axiom,
    ! [B: nat,A: nat] :
      ( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_978_singletonD,axiom,
    ! [B: product_prod_nat_nat,A: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ B @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_979_singletonD,axiom,
    ! [B: produc3843707927480180839at_nat,A: produc3843707927480180839at_nat] :
      ( ( member8757157785044589968at_nat @ B @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_980_singleton__iff,axiom,
    ! [B: produc859450856879609959at_nat,A: produc859450856879609959at_nat] :
      ( ( member8206827879206165904at_nat @ B @ ( insert5050368324300391991at_nat @ A @ bot_bo5327735625951526323at_nat ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_981_singleton__iff,axiom,
    ! [B: nat,A: nat] :
      ( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_982_singleton__iff,axiom,
    ! [B: product_prod_nat_nat,A: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ B @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_983_singleton__iff,axiom,
    ! [B: produc3843707927480180839at_nat,A: produc3843707927480180839at_nat] :
      ( ( member8757157785044589968at_nat @ B @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_984_doubleton__eq__iff,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,C: product_prod_nat_nat,D: product_prod_nat_nat] :
      ( ( ( insert8211810215607154385at_nat @ A @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) )
        = ( insert8211810215607154385at_nat @ C @ ( insert8211810215607154385at_nat @ D @ bot_bo2099793752762293965at_nat ) ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_985_doubleton__eq__iff,axiom,
    ! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat,C: produc3843707927480180839at_nat,D: produc3843707927480180839at_nat] :
      ( ( ( insert9069300056098147895at_nat @ A @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) )
        = ( insert9069300056098147895at_nat @ C @ ( insert9069300056098147895at_nat @ D @ bot_bo228742789529271731at_nat ) ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_986_empty__not__insert,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( bot_bo2099793752762293965at_nat
     != ( insert8211810215607154385at_nat @ A @ A4 ) ) ).

% empty_not_insert
thf(fact_987_empty__not__insert,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( bot_bo228742789529271731at_nat
     != ( insert9069300056098147895at_nat @ A @ A4 ) ) ).

% empty_not_insert
thf(fact_988_singleton__inject,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( ( ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat )
        = ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) )
     => ( A = B ) ) ).

% singleton_inject
thf(fact_989_singleton__inject,axiom,
    ! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
      ( ( ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat )
        = ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) )
     => ( A = B ) ) ).

% singleton_inject
thf(fact_990_insert__subsetI,axiom,
    ! [X: produc859450856879609959at_nat,A4: set_Pr8693737435421807431at_nat,X5: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ X @ A4 )
     => ( ( ord_le3000389064537975527at_nat @ X5 @ A4 )
       => ( ord_le3000389064537975527at_nat @ ( insert5050368324300391991at_nat @ X @ X5 ) @ A4 ) ) ) ).

% insert_subsetI
thf(fact_991_insert__subsetI,axiom,
    ! [X: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,X5: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ X @ A4 )
     => ( ( ord_le3146513528884898305at_nat @ X5 @ A4 )
       => ( ord_le3146513528884898305at_nat @ ( insert8211810215607154385at_nat @ X @ X5 ) @ A4 ) ) ) ).

% insert_subsetI
thf(fact_992_insert__subsetI,axiom,
    ! [X: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,X5: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ X @ A4 )
     => ( ( ord_le1268244103169919719at_nat @ X5 @ A4 )
       => ( ord_le1268244103169919719at_nat @ ( insert9069300056098147895at_nat @ X @ X5 ) @ A4 ) ) ) ).

% insert_subsetI
thf(fact_993_insert__subsetI,axiom,
    ! [X: nat,A4: set_nat,X5: set_nat] :
      ( ( member_nat @ X @ A4 )
     => ( ( ord_less_eq_set_nat @ X5 @ A4 )
       => ( ord_less_eq_set_nat @ ( insert_nat @ X @ X5 ) @ A4 ) ) ) ).

% insert_subsetI
thf(fact_994_finite__ranking__induct,axiom,
    ! [S2: set_Pr8693737435421807431at_nat,P: set_Pr8693737435421807431at_nat > $o,F2: produc859450856879609959at_nat > nat] :
      ( ( finite4392333629123659920at_nat @ S2 )
     => ( ( P @ bot_bo5327735625951526323at_nat )
       => ( ! [X3: produc859450856879609959at_nat,S3: set_Pr8693737435421807431at_nat] :
              ( ( finite4392333629123659920at_nat @ S3 )
             => ( ! [Y5: produc859450856879609959at_nat] :
                    ( ( member8206827879206165904at_nat @ Y5 @ S3 )
                   => ( ord_less_eq_nat @ ( F2 @ Y5 ) @ ( F2 @ X3 ) ) )
               => ( ( P @ S3 )
                 => ( P @ ( insert5050368324300391991at_nat @ X3 @ S3 ) ) ) ) )
         => ( P @ S2 ) ) ) ) ).

% finite_ranking_induct
thf(fact_995_finite__ranking__induct,axiom,
    ! [S2: set_nat,P: set_nat > $o,F2: nat > nat] :
      ( ( finite_finite_nat @ S2 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X3: nat,S3: set_nat] :
              ( ( finite_finite_nat @ S3 )
             => ( ! [Y5: nat] :
                    ( ( member_nat @ Y5 @ S3 )
                   => ( ord_less_eq_nat @ ( F2 @ Y5 ) @ ( F2 @ X3 ) ) )
               => ( ( P @ S3 )
                 => ( P @ ( insert_nat @ X3 @ S3 ) ) ) ) )
         => ( P @ S2 ) ) ) ) ).

% finite_ranking_induct
thf(fact_996_finite__ranking__induct,axiom,
    ! [S2: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o,F2: product_prod_nat_nat > nat] :
      ( ( finite6177210948735845034at_nat @ S2 )
     => ( ( P @ bot_bo2099793752762293965at_nat )
       => ( ! [X3: product_prod_nat_nat,S3: set_Pr1261947904930325089at_nat] :
              ( ( finite6177210948735845034at_nat @ S3 )
             => ( ! [Y5: product_prod_nat_nat] :
                    ( ( member8440522571783428010at_nat @ Y5 @ S3 )
                   => ( ord_less_eq_nat @ ( F2 @ Y5 ) @ ( F2 @ X3 ) ) )
               => ( ( P @ S3 )
                 => ( P @ ( insert8211810215607154385at_nat @ X3 @ S3 ) ) ) ) )
         => ( P @ S2 ) ) ) ) ).

% finite_ranking_induct
thf(fact_997_finite__ranking__induct,axiom,
    ! [S2: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o,F2: produc3843707927480180839at_nat > nat] :
      ( ( finite4343798906461161616at_nat @ S2 )
     => ( ( P @ bot_bo228742789529271731at_nat )
       => ( ! [X3: produc3843707927480180839at_nat,S3: set_Pr4329608150637261639at_nat] :
              ( ( finite4343798906461161616at_nat @ S3 )
             => ( ! [Y5: produc3843707927480180839at_nat] :
                    ( ( member8757157785044589968at_nat @ Y5 @ S3 )
                   => ( ord_less_eq_nat @ ( F2 @ Y5 ) @ ( F2 @ X3 ) ) )
               => ( ( P @ S3 )
                 => ( P @ ( insert9069300056098147895at_nat @ X3 @ S3 ) ) ) ) )
         => ( P @ S2 ) ) ) ) ).

% finite_ranking_induct
thf(fact_998_finite__linorder__min__induct,axiom,
    ! [A4: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A4 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [B2: nat,A7: set_nat] :
              ( ( finite_finite_nat @ A7 )
             => ( ! [X6: nat] :
                    ( ( member_nat @ X6 @ A7 )
                   => ( ord_less_nat @ B2 @ X6 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_nat @ B2 @ A7 ) ) ) ) )
         => ( P @ A4 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_999_finite__linorder__max__induct,axiom,
    ! [A4: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A4 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [B2: nat,A7: set_nat] :
              ( ( finite_finite_nat @ A7 )
             => ( ! [X6: nat] :
                    ( ( member_nat @ X6 @ A7 )
                   => ( ord_less_nat @ X6 @ B2 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_nat @ B2 @ A7 ) ) ) ) )
         => ( P @ A4 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_1000_finite__subset__induct,axiom,
    ! [F4: set_Pr8693737435421807431at_nat,A4: set_Pr8693737435421807431at_nat,P: set_Pr8693737435421807431at_nat > $o] :
      ( ( finite4392333629123659920at_nat @ F4 )
     => ( ( ord_le3000389064537975527at_nat @ F4 @ A4 )
       => ( ( P @ bot_bo5327735625951526323at_nat )
         => ( ! [A2: produc859450856879609959at_nat,F5: set_Pr8693737435421807431at_nat] :
                ( ( finite4392333629123659920at_nat @ F5 )
               => ( ( member8206827879206165904at_nat @ A2 @ A4 )
                 => ( ~ ( member8206827879206165904at_nat @ A2 @ F5 )
                   => ( ( P @ F5 )
                     => ( P @ ( insert5050368324300391991at_nat @ A2 @ F5 ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_1001_finite__subset__induct,axiom,
    ! [F4: set_nat,A4: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F4 )
     => ( ( ord_less_eq_set_nat @ F4 @ A4 )
       => ( ( P @ bot_bot_set_nat )
         => ( ! [A2: nat,F5: set_nat] :
                ( ( finite_finite_nat @ F5 )
               => ( ( member_nat @ A2 @ A4 )
                 => ( ~ ( member_nat @ A2 @ F5 )
                   => ( ( P @ F5 )
                     => ( P @ ( insert_nat @ A2 @ F5 ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_1002_finite__subset__induct,axiom,
    ! [F4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
      ( ( finite6177210948735845034at_nat @ F4 )
     => ( ( ord_le3146513528884898305at_nat @ F4 @ A4 )
       => ( ( P @ bot_bo2099793752762293965at_nat )
         => ( ! [A2: product_prod_nat_nat,F5: set_Pr1261947904930325089at_nat] :
                ( ( finite6177210948735845034at_nat @ F5 )
               => ( ( member8440522571783428010at_nat @ A2 @ A4 )
                 => ( ~ ( member8440522571783428010at_nat @ A2 @ F5 )
                   => ( ( P @ F5 )
                     => ( P @ ( insert8211810215607154385at_nat @ A2 @ F5 ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_1003_finite__subset__induct,axiom,
    ! [F4: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o] :
      ( ( finite4343798906461161616at_nat @ F4 )
     => ( ( ord_le1268244103169919719at_nat @ F4 @ A4 )
       => ( ( P @ bot_bo228742789529271731at_nat )
         => ( ! [A2: produc3843707927480180839at_nat,F5: set_Pr4329608150637261639at_nat] :
                ( ( finite4343798906461161616at_nat @ F5 )
               => ( ( member8757157785044589968at_nat @ A2 @ A4 )
                 => ( ~ ( member8757157785044589968at_nat @ A2 @ F5 )
                   => ( ( P @ F5 )
                     => ( P @ ( insert9069300056098147895at_nat @ A2 @ F5 ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_1004_finite__subset__induct_H,axiom,
    ! [F4: set_Pr8693737435421807431at_nat,A4: set_Pr8693737435421807431at_nat,P: set_Pr8693737435421807431at_nat > $o] :
      ( ( finite4392333629123659920at_nat @ F4 )
     => ( ( ord_le3000389064537975527at_nat @ F4 @ A4 )
       => ( ( P @ bot_bo5327735625951526323at_nat )
         => ( ! [A2: produc859450856879609959at_nat,F5: set_Pr8693737435421807431at_nat] :
                ( ( finite4392333629123659920at_nat @ F5 )
               => ( ( member8206827879206165904at_nat @ A2 @ A4 )
                 => ( ( ord_le3000389064537975527at_nat @ F5 @ A4 )
                   => ( ~ ( member8206827879206165904at_nat @ A2 @ F5 )
                     => ( ( P @ F5 )
                       => ( P @ ( insert5050368324300391991at_nat @ A2 @ F5 ) ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_1005_finite__subset__induct_H,axiom,
    ! [F4: set_nat,A4: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F4 )
     => ( ( ord_less_eq_set_nat @ F4 @ A4 )
       => ( ( P @ bot_bot_set_nat )
         => ( ! [A2: nat,F5: set_nat] :
                ( ( finite_finite_nat @ F5 )
               => ( ( member_nat @ A2 @ A4 )
                 => ( ( ord_less_eq_set_nat @ F5 @ A4 )
                   => ( ~ ( member_nat @ A2 @ F5 )
                     => ( ( P @ F5 )
                       => ( P @ ( insert_nat @ A2 @ F5 ) ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_1006_finite__subset__induct_H,axiom,
    ! [F4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
      ( ( finite6177210948735845034at_nat @ F4 )
     => ( ( ord_le3146513528884898305at_nat @ F4 @ A4 )
       => ( ( P @ bot_bo2099793752762293965at_nat )
         => ( ! [A2: product_prod_nat_nat,F5: set_Pr1261947904930325089at_nat] :
                ( ( finite6177210948735845034at_nat @ F5 )
               => ( ( member8440522571783428010at_nat @ A2 @ A4 )
                 => ( ( ord_le3146513528884898305at_nat @ F5 @ A4 )
                   => ( ~ ( member8440522571783428010at_nat @ A2 @ F5 )
                     => ( ( P @ F5 )
                       => ( P @ ( insert8211810215607154385at_nat @ A2 @ F5 ) ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_1007_finite__subset__induct_H,axiom,
    ! [F4: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o] :
      ( ( finite4343798906461161616at_nat @ F4 )
     => ( ( ord_le1268244103169919719at_nat @ F4 @ A4 )
       => ( ( P @ bot_bo228742789529271731at_nat )
         => ( ! [A2: produc3843707927480180839at_nat,F5: set_Pr4329608150637261639at_nat] :
                ( ( finite4343798906461161616at_nat @ F5 )
               => ( ( member8757157785044589968at_nat @ A2 @ A4 )
                 => ( ( ord_le1268244103169919719at_nat @ F5 @ A4 )
                   => ( ~ ( member8757157785044589968at_nat @ A2 @ F5 )
                     => ( ( P @ F5 )
                       => ( P @ ( insert9069300056098147895at_nat @ A2 @ F5 ) ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_1008_card__1__singletonE,axiom,
    ! [A4: set_nat] :
      ( ( ( finite_card_nat @ A4 )
        = one_one_nat )
     => ~ ! [X3: nat] :
            ( A4
           != ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ).

% card_1_singletonE
thf(fact_1009_card__1__singletonE,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ( ( finite711546835091564841at_nat @ A4 )
        = one_one_nat )
     => ~ ! [X3: product_prod_nat_nat] :
            ( A4
           != ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) ) ) ).

% card_1_singletonE
thf(fact_1010_card__1__singletonE,axiom,
    ! [A4: set_Pr4329608150637261639at_nat] :
      ( ( ( finite3771342082235030671at_nat @ A4 )
        = one_one_nat )
     => ~ ! [X3: produc3843707927480180839at_nat] :
            ( A4
           != ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) ) ) ).

% card_1_singletonE
thf(fact_1011_card__le__Suc__iff,axiom,
    ! [N: nat,A4: set_Pr8693737435421807431at_nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite1207074278014112911at_nat @ A4 ) )
      = ( ? [A3: produc859450856879609959at_nat,B6: set_Pr8693737435421807431at_nat] :
            ( ( A4
              = ( insert5050368324300391991at_nat @ A3 @ B6 ) )
            & ~ ( member8206827879206165904at_nat @ A3 @ B6 )
            & ( ord_less_eq_nat @ N @ ( finite1207074278014112911at_nat @ B6 ) )
            & ( finite4392333629123659920at_nat @ B6 ) ) ) ) ).

% card_le_Suc_iff
thf(fact_1012_card__le__Suc__iff,axiom,
    ! [N: nat,A4: set_Pr4329608150637261639at_nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite3771342082235030671at_nat @ A4 ) )
      = ( ? [A3: produc3843707927480180839at_nat,B6: set_Pr4329608150637261639at_nat] :
            ( ( A4
              = ( insert9069300056098147895at_nat @ A3 @ B6 ) )
            & ~ ( member8757157785044589968at_nat @ A3 @ B6 )
            & ( ord_less_eq_nat @ N @ ( finite3771342082235030671at_nat @ B6 ) )
            & ( finite4343798906461161616at_nat @ B6 ) ) ) ) ).

% card_le_Suc_iff
thf(fact_1013_card__le__Suc__iff,axiom,
    ! [N: nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite711546835091564841at_nat @ A4 ) )
      = ( ? [A3: product_prod_nat_nat,B6: set_Pr1261947904930325089at_nat] :
            ( ( A4
              = ( insert8211810215607154385at_nat @ A3 @ B6 ) )
            & ~ ( member8440522571783428010at_nat @ A3 @ B6 )
            & ( ord_less_eq_nat @ N @ ( finite711546835091564841at_nat @ B6 ) )
            & ( finite6177210948735845034at_nat @ B6 ) ) ) ) ).

% card_le_Suc_iff
thf(fact_1014_card__le__Suc__iff,axiom,
    ! [N: nat,A4: set_nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_nat @ A4 ) )
      = ( ? [A3: nat,B6: set_nat] :
            ( ( A4
              = ( insert_nat @ A3 @ B6 ) )
            & ~ ( member_nat @ A3 @ B6 )
            & ( ord_less_eq_nat @ N @ ( finite_card_nat @ B6 ) )
            & ( finite_finite_nat @ B6 ) ) ) ) ).

% card_le_Suc_iff
thf(fact_1015_mlex__less,axiom,
    ! [F2: nat > nat,X: nat,Y: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y ) )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( mlex_prod_nat @ F2 @ R ) ) ) ).

% mlex_less
thf(fact_1016_mlex__less,axiom,
    ! [F2: product_prod_nat_nat > nat,X: product_prod_nat_nat,Y: product_prod_nat_nat,R: set_Pr8693737435421807431at_nat] :
      ( ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y ) )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( mlex_p6366001652026297872at_nat @ F2 @ R ) ) ) ).

% mlex_less
thf(fact_1017_mlex__less,axiom,
    ! [F2: set_Pr1261947904930325089at_nat > nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat] :
      ( ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y ) )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ ( mlex_p3502713205431853936at_nat @ F2 @ R ) ) ) ).

% mlex_less
thf(fact_1018_mlex__less,axiom,
    ! [F2: set_Pr4329608150637261639at_nat > nat,X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,R: set_Pr7459493094073627847at_nat] :
      ( ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y ) )
     => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ ( mlex_p4791090246024907222at_nat @ F2 @ R ) ) ) ).

% mlex_less
thf(fact_1019_mlex__iff,axiom,
    ! [X: nat,Y: nat,F2: nat > nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( mlex_prod_nat @ F2 @ R ) )
      = ( ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y ) )
        | ( ( ( F2 @ X )
            = ( F2 @ Y ) )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R ) ) ) ) ).

% mlex_iff
thf(fact_1020_mlex__iff,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,F2: product_prod_nat_nat > nat,R: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( mlex_p6366001652026297872at_nat @ F2 @ R ) )
      = ( ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y ) )
        | ( ( ( F2 @ X )
            = ( F2 @ Y ) )
          & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R ) ) ) ) ).

% mlex_iff
thf(fact_1021_mlex__iff,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,F2: set_Pr1261947904930325089at_nat > nat,R: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ ( mlex_p3502713205431853936at_nat @ F2 @ R ) )
      = ( ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y ) )
        | ( ( ( F2 @ X )
            = ( F2 @ Y ) )
          & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R ) ) ) ) ).

% mlex_iff
thf(fact_1022_mlex__iff,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,F2: set_Pr4329608150637261639at_nat > nat,R: set_Pr7459493094073627847at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ ( mlex_p4791090246024907222at_nat @ F2 @ R ) )
      = ( ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y ) )
        | ( ( ( F2 @ X )
            = ( F2 @ Y ) )
          & ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ R ) ) ) ) ).

% mlex_iff
thf(fact_1023_mlex__leq,axiom,
    ! [F2: nat > nat,X: nat,Y: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y ) )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( mlex_prod_nat @ F2 @ R ) ) ) ) ).

% mlex_leq
thf(fact_1024_mlex__leq,axiom,
    ! [F2: product_prod_nat_nat > nat,X: product_prod_nat_nat,Y: product_prod_nat_nat,R: set_Pr8693737435421807431at_nat] :
      ( ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y ) )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( mlex_p6366001652026297872at_nat @ F2 @ R ) ) ) ) ).

% mlex_leq
thf(fact_1025_mlex__leq,axiom,
    ! [F2: set_Pr1261947904930325089at_nat > nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat] :
      ( ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y ) )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ ( mlex_p3502713205431853936at_nat @ F2 @ R ) ) ) ) ).

% mlex_leq
thf(fact_1026_mlex__leq,axiom,
    ! [F2: set_Pr4329608150637261639at_nat > nat,X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,R: set_Pr7459493094073627847at_nat] :
      ( ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y ) )
     => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ R )
       => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ ( mlex_p4791090246024907222at_nat @ F2 @ R ) ) ) ) ).

% mlex_leq
thf(fact_1027_mlex__eq,axiom,
    ( mlex_p6366001652026297872at_nat
    = ( ^ [F3: product_prod_nat_nat > nat,R3: set_Pr8693737435421807431at_nat] :
          ( collec7088162979684241874at_nat
          @ ( produc6590410687421337004_nat_o
            @ ^ [X4: product_prod_nat_nat,Y4: product_prod_nat_nat] :
                ( ( ord_less_nat @ ( F3 @ X4 ) @ ( F3 @ Y4 ) )
                | ( ( ord_less_eq_nat @ ( F3 @ X4 ) @ ( F3 @ Y4 ) )
                  & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y4 ) @ R3 ) ) ) ) ) ) ) ).

% mlex_eq
thf(fact_1028_mlex__eq,axiom,
    ( mlex_p3502713205431853936at_nat
    = ( ^ [F3: set_Pr1261947904930325089at_nat > nat,R3: set_Pr4329608150637261639at_nat] :
          ( collec6321179662152712658at_nat
          @ ( produc410239310623530412_nat_o
            @ ^ [X4: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] :
                ( ( ord_less_nat @ ( F3 @ X4 ) @ ( F3 @ Y4 ) )
                | ( ( ord_less_eq_nat @ ( F3 @ X4 ) @ ( F3 @ Y4 ) )
                  & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y4 ) @ R3 ) ) ) ) ) ) ) ).

% mlex_eq
thf(fact_1029_mlex__eq,axiom,
    ( mlex_p4791090246024907222at_nat
    = ( ^ [F3: set_Pr4329608150637261639at_nat > nat,R3: set_Pr7459493094073627847at_nat] :
          ( collec7872792499061297490at_nat
          @ ( produc9020218426428501292_nat_o
            @ ^ [X4: set_Pr4329608150637261639at_nat,Y4: set_Pr4329608150637261639at_nat] :
                ( ( ord_less_nat @ ( F3 @ X4 ) @ ( F3 @ Y4 ) )
                | ( ( ord_less_eq_nat @ ( F3 @ X4 ) @ ( F3 @ Y4 ) )
                  & ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X4 @ Y4 ) @ R3 ) ) ) ) ) ) ) ).

% mlex_eq
thf(fact_1030_mlex__eq,axiom,
    ( mlex_prod_nat
    = ( ^ [F3: nat > nat,R3: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [X4: nat,Y4: nat] :
                ( ( ord_less_nat @ ( F3 @ X4 ) @ ( F3 @ Y4 ) )
                | ( ( ord_less_eq_nat @ ( F3 @ X4 ) @ ( F3 @ Y4 ) )
                  & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ R3 ) ) ) ) ) ) ) ).

% mlex_eq
thf(fact_1031_smax__insertI,axiom,
    ! [Y: product_prod_nat_nat,Y7: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,X5: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ Y @ Y7 )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ fun_pair_less )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X5 @ Y7 ) @ fun_max_strict )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( insert8211810215607154385at_nat @ X @ X5 ) @ Y7 ) @ fun_max_strict ) ) ) ) ).

% smax_insertI
thf(fact_1032_wmin__insertI,axiom,
    ! [X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ X @ XS )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ fun_pair_leq )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_min_weak )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ ( insert8211810215607154385at_nat @ Y @ YS ) ) @ fun_min_weak ) ) ) ) ).

% wmin_insertI
thf(fact_1033_wmax__insertI,axiom,
    ! [Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ Y @ YS )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ fun_pair_leq )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_max_weak )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( insert8211810215607154385at_nat @ X @ XS ) @ YS ) @ fun_max_weak ) ) ) ) ).

% wmax_insertI
thf(fact_1034_is__singletonI,axiom,
    ! [X: product_prod_nat_nat] : ( is_sin2850979758926227957at_nat @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ).

% is_singletonI
thf(fact_1035_is__singletonI,axiom,
    ! [X: produc3843707927480180839at_nat] : ( is_sin2937591304547752795at_nat @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ).

% is_singletonI
thf(fact_1036_is__singletonI_H,axiom,
    ! [A4: set_Pr8693737435421807431at_nat] :
      ( ( A4 != bot_bo5327735625951526323at_nat )
     => ( ! [X3: produc859450856879609959at_nat,Y3: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X3 @ A4 )
           => ( ( member8206827879206165904at_nat @ Y3 @ A4 )
             => ( X3 = Y3 ) ) )
       => ( is_sin3461952106595595099at_nat @ A4 ) ) ) ).

% is_singletonI'
thf(fact_1037_is__singletonI_H,axiom,
    ! [A4: set_nat] :
      ( ( A4 != bot_bot_set_nat )
     => ( ! [X3: nat,Y3: nat] :
            ( ( member_nat @ X3 @ A4 )
           => ( ( member_nat @ Y3 @ A4 )
             => ( X3 = Y3 ) ) )
       => ( is_singleton_nat @ A4 ) ) ) ).

% is_singletonI'
thf(fact_1038_is__singletonI_H,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ( A4 != bot_bo2099793752762293965at_nat )
     => ( ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ A4 )
           => ( ( member8440522571783428010at_nat @ Y3 @ A4 )
             => ( X3 = Y3 ) ) )
       => ( is_sin2850979758926227957at_nat @ A4 ) ) ) ).

% is_singletonI'
thf(fact_1039_is__singletonI_H,axiom,
    ! [A4: set_Pr4329608150637261639at_nat] :
      ( ( A4 != bot_bo228742789529271731at_nat )
     => ( ! [X3: produc3843707927480180839at_nat,Y3: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X3 @ A4 )
           => ( ( member8757157785044589968at_nat @ Y3 @ A4 )
             => ( X3 = Y3 ) ) )
       => ( is_sin2937591304547752795at_nat @ A4 ) ) ) ).

% is_singletonI'
thf(fact_1040_is__singleton__def,axiom,
    ( is_sin2850979758926227957at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat] :
        ? [X4: product_prod_nat_nat] :
          ( A6
          = ( insert8211810215607154385at_nat @ X4 @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% is_singleton_def
thf(fact_1041_is__singleton__def,axiom,
    ( is_sin2937591304547752795at_nat
    = ( ^ [A6: set_Pr4329608150637261639at_nat] :
        ? [X4: produc3843707927480180839at_nat] :
          ( A6
          = ( insert9069300056098147895at_nat @ X4 @ bot_bo228742789529271731at_nat ) ) ) ) ).

% is_singleton_def
thf(fact_1042_is__singletonE,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ( is_sin2850979758926227957at_nat @ A4 )
     => ~ ! [X3: product_prod_nat_nat] :
            ( A4
           != ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) ) ) ).

% is_singletonE
thf(fact_1043_is__singletonE,axiom,
    ! [A4: set_Pr4329608150637261639at_nat] :
      ( ( is_sin2937591304547752795at_nat @ A4 )
     => ~ ! [X3: produc3843707927480180839at_nat] :
            ( A4
           != ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) ) ) ).

% is_singletonE
thf(fact_1044_is__singleton__altdef,axiom,
    ( is_singleton_nat
    = ( ^ [A6: set_nat] :
          ( ( finite_card_nat @ A6 )
          = one_one_nat ) ) ) ).

% is_singleton_altdef
thf(fact_1045_is__singleton__altdef,axiom,
    ( is_sin2850979758926227957at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat] :
          ( ( finite711546835091564841at_nat @ A6 )
          = one_one_nat ) ) ) ).

% is_singleton_altdef
thf(fact_1046_max__weak__def,axiom,
    ( fun_max_weak
    = ( sup_su5525570899277871387at_nat @ ( max_ex8135407076693332796at_nat @ fun_pair_leq ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ bot_bo2099793752762293965at_nat ) @ bot_bo228742789529271731at_nat ) ) ) ).

% max_weak_def
thf(fact_1047_min__weak__def,axiom,
    ( fun_min_weak
    = ( sup_su5525570899277871387at_nat @ ( min_ex6901939911449802026at_nat @ fun_pair_leq ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ bot_bo2099793752762293965at_nat ) @ bot_bo228742789529271731at_nat ) ) ) ).

% min_weak_def
thf(fact_1048_Un__iff,axiom,
    ! [C: produc859450856879609959at_nat,A4: set_Pr8693737435421807431at_nat,B4: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ C @ ( sup_su718114333110466843at_nat @ A4 @ B4 ) )
      = ( ( member8206827879206165904at_nat @ C @ A4 )
        | ( member8206827879206165904at_nat @ C @ B4 ) ) ) ).

% Un_iff
thf(fact_1049_Un__iff,axiom,
    ! [C: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C @ ( sup_su6327502436637775413at_nat @ A4 @ B4 ) )
      = ( ( member8440522571783428010at_nat @ C @ A4 )
        | ( member8440522571783428010at_nat @ C @ B4 ) ) ) ).

% Un_iff
thf(fact_1050_Un__iff,axiom,
    ! [C: nat,A4: set_nat,B4: set_nat] :
      ( ( member_nat @ C @ ( sup_sup_set_nat @ A4 @ B4 ) )
      = ( ( member_nat @ C @ A4 )
        | ( member_nat @ C @ B4 ) ) ) ).

% Un_iff
thf(fact_1051_Un__iff,axiom,
    ! [C: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ C @ ( sup_su5525570899277871387at_nat @ A4 @ B4 ) )
      = ( ( member8757157785044589968at_nat @ C @ A4 )
        | ( member8757157785044589968at_nat @ C @ B4 ) ) ) ).

% Un_iff
thf(fact_1052_UnCI,axiom,
    ! [C: produc859450856879609959at_nat,B4: set_Pr8693737435421807431at_nat,A4: set_Pr8693737435421807431at_nat] :
      ( ( ~ ( member8206827879206165904at_nat @ C @ B4 )
       => ( member8206827879206165904at_nat @ C @ A4 ) )
     => ( member8206827879206165904at_nat @ C @ ( sup_su718114333110466843at_nat @ A4 @ B4 ) ) ) ).

% UnCI
thf(fact_1053_UnCI,axiom,
    ! [C: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( ~ ( member8440522571783428010at_nat @ C @ B4 )
       => ( member8440522571783428010at_nat @ C @ A4 ) )
     => ( member8440522571783428010at_nat @ C @ ( sup_su6327502436637775413at_nat @ A4 @ B4 ) ) ) ).

% UnCI
thf(fact_1054_UnCI,axiom,
    ! [C: nat,B4: set_nat,A4: set_nat] :
      ( ( ~ ( member_nat @ C @ B4 )
       => ( member_nat @ C @ A4 ) )
     => ( member_nat @ C @ ( sup_sup_set_nat @ A4 @ B4 ) ) ) ).

% UnCI
thf(fact_1055_UnCI,axiom,
    ! [C: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ( ~ ( member8757157785044589968at_nat @ C @ B4 )
       => ( member8757157785044589968at_nat @ C @ A4 ) )
     => ( member8757157785044589968at_nat @ C @ ( sup_su5525570899277871387at_nat @ A4 @ B4 ) ) ) ).

% UnCI
thf(fact_1056_sup__Pair__Pair,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( sup_su4120719815643632853at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( product_Pair_nat_nat @ C @ D ) )
      = ( product_Pair_nat_nat @ ( sup_sup_nat @ A @ C ) @ ( sup_sup_nat @ B @ D ) ) ) ).

% sup_Pair_Pair
thf(fact_1057_sup__Pair__Pair,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,C: product_prod_nat_nat,D: product_prod_nat_nat] :
      ( ( sup_su8663239211571056187at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( produc6161850002892822231at_nat @ C @ D ) )
      = ( produc6161850002892822231at_nat @ ( sup_su4120719815643632853at_nat @ A @ C ) @ ( sup_su4120719815643632853at_nat @ B @ D ) ) ) ).

% sup_Pair_Pair
thf(fact_1058_sup__Pair__Pair,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,D: set_Pr1261947904930325089at_nat] :
      ( ( sup_su4419393189987956283at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ ( produc2922128104949294807at_nat @ C @ D ) )
      = ( produc2922128104949294807at_nat @ ( sup_su6327502436637775413at_nat @ A @ C ) @ ( sup_su6327502436637775413at_nat @ B @ D ) ) ) ).

% sup_Pair_Pair
thf(fact_1059_sup__Pair__Pair,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat,D: set_Pr4329608150637261639at_nat] :
      ( ( sup_su3216838554834203323at_nat @ ( produc9060074326276436823at_nat @ A @ B ) @ ( produc9060074326276436823at_nat @ C @ D ) )
      = ( produc9060074326276436823at_nat @ ( sup_su5525570899277871387at_nat @ A @ C ) @ ( sup_su5525570899277871387at_nat @ B @ D ) ) ) ).

% sup_Pair_Pair
thf(fact_1060_Un__subset__iff,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ A4 @ B4 ) @ C2 )
      = ( ( ord_le1268244103169919719at_nat @ A4 @ C2 )
        & ( ord_le1268244103169919719at_nat @ B4 @ C2 ) ) ) ).

% Un_subset_iff
thf(fact_1061_Un__empty,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( ( sup_su6327502436637775413at_nat @ A4 @ B4 )
        = bot_bo2099793752762293965at_nat )
      = ( ( A4 = bot_bo2099793752762293965at_nat )
        & ( B4 = bot_bo2099793752762293965at_nat ) ) ) ).

% Un_empty
thf(fact_1062_Un__empty,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( ( sup_su5525570899277871387at_nat @ A4 @ B4 )
        = bot_bo228742789529271731at_nat )
      = ( ( A4 = bot_bo228742789529271731at_nat )
        & ( B4 = bot_bo228742789529271731at_nat ) ) ) ).

% Un_empty
thf(fact_1063_Un__insert__right,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ A4 @ ( insert8211810215607154385at_nat @ A @ B4 ) )
      = ( insert8211810215607154385at_nat @ A @ ( sup_su6327502436637775413at_nat @ A4 @ B4 ) ) ) ).

% Un_insert_right
thf(fact_1064_Un__insert__right,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ A4 @ ( insert9069300056098147895at_nat @ A @ B4 ) )
      = ( insert9069300056098147895at_nat @ A @ ( sup_su5525570899277871387at_nat @ A4 @ B4 ) ) ) ).

% Un_insert_right
thf(fact_1065_Un__insert__left,axiom,
    ! [A: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ ( insert8211810215607154385at_nat @ A @ B4 ) @ C2 )
      = ( insert8211810215607154385at_nat @ A @ ( sup_su6327502436637775413at_nat @ B4 @ C2 ) ) ) ).

% Un_insert_left
thf(fact_1066_Un__insert__left,axiom,
    ! [A: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( insert9069300056098147895at_nat @ A @ B4 ) @ C2 )
      = ( insert9069300056098147895at_nat @ A @ ( sup_su5525570899277871387at_nat @ B4 @ C2 ) ) ) ).

% Un_insert_left
thf(fact_1067_insert__def,axiom,
    ( insert_nat
    = ( ^ [A3: nat] :
          ( sup_sup_set_nat
          @ ( collect_nat
            @ ^ [X4: nat] : ( X4 = A3 ) ) ) ) ) ).

% insert_def
thf(fact_1068_insert__def,axiom,
    ( insert8211810215607154385at_nat
    = ( ^ [A3: product_prod_nat_nat] :
          ( sup_su6327502436637775413at_nat
          @ ( collec3392354462482085612at_nat
            @ ^ [X4: product_prod_nat_nat] : ( X4 = A3 ) ) ) ) ) ).

% insert_def
thf(fact_1069_insert__def,axiom,
    ( insert9069300056098147895at_nat
    = ( ^ [A3: produc3843707927480180839at_nat] :
          ( sup_su5525570899277871387at_nat
          @ ( collec6321179662152712658at_nat
            @ ^ [X4: produc3843707927480180839at_nat] : ( X4 = A3 ) ) ) ) ) ).

% insert_def
thf(fact_1070_insert__is__Un,axiom,
    ( insert8211810215607154385at_nat
    = ( ^ [A3: product_prod_nat_nat] : ( sup_su6327502436637775413at_nat @ ( insert8211810215607154385at_nat @ A3 @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% insert_is_Un
thf(fact_1071_insert__is__Un,axiom,
    ( insert9069300056098147895at_nat
    = ( ^ [A3: produc3843707927480180839at_nat] : ( sup_su5525570899277871387at_nat @ ( insert9069300056098147895at_nat @ A3 @ bot_bo228742789529271731at_nat ) ) ) ) ).

% insert_is_Un
thf(fact_1072_Un__singleton__iff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( ( ( sup_su6327502436637775413at_nat @ A4 @ B4 )
        = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
      = ( ( ( A4 = bot_bo2099793752762293965at_nat )
          & ( B4
            = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) )
        | ( ( A4
            = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
          & ( B4 = bot_bo2099793752762293965at_nat ) )
        | ( ( A4
            = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
          & ( B4
            = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_1073_Un__singleton__iff,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat] :
      ( ( ( sup_su5525570899277871387at_nat @ A4 @ B4 )
        = ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) )
      = ( ( ( A4 = bot_bo228742789529271731at_nat )
          & ( B4
            = ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) )
        | ( ( A4
            = ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) )
          & ( B4 = bot_bo228742789529271731at_nat ) )
        | ( ( A4
            = ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) )
          & ( B4
            = ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_1074_singleton__Un__iff,axiom,
    ! [X: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat )
        = ( sup_su6327502436637775413at_nat @ A4 @ B4 ) )
      = ( ( ( A4 = bot_bo2099793752762293965at_nat )
          & ( B4
            = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) )
        | ( ( A4
            = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
          & ( B4 = bot_bo2099793752762293965at_nat ) )
        | ( ( A4
            = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
          & ( B4
            = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_1075_singleton__Un__iff,axiom,
    ! [X: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat )
        = ( sup_su5525570899277871387at_nat @ A4 @ B4 ) )
      = ( ( ( A4 = bot_bo228742789529271731at_nat )
          & ( B4
            = ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) )
        | ( ( A4
            = ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) )
          & ( B4 = bot_bo228742789529271731at_nat ) )
        | ( ( A4
            = ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) )
          & ( B4
            = ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_1076_Un__def,axiom,
    ( sup_su718114333110466843at_nat
    = ( ^ [A6: set_Pr8693737435421807431at_nat,B6: set_Pr8693737435421807431at_nat] :
          ( collec7088162979684241874at_nat
          @ ^ [X4: produc859450856879609959at_nat] :
              ( ( member8206827879206165904at_nat @ X4 @ A6 )
              | ( member8206827879206165904at_nat @ X4 @ B6 ) ) ) ) ) ).

% Un_def
thf(fact_1077_Un__def,axiom,
    ( sup_sup_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
          ( collect_nat
          @ ^ [X4: nat] :
              ( ( member_nat @ X4 @ A6 )
              | ( member_nat @ X4 @ B6 ) ) ) ) ) ).

% Un_def
thf(fact_1078_Un__def,axiom,
    ( sup_su6327502436637775413at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [X4: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X4 @ A6 )
              | ( member8440522571783428010at_nat @ X4 @ B6 ) ) ) ) ) ).

% Un_def
thf(fact_1079_Un__def,axiom,
    ( sup_su5525570899277871387at_nat
    = ( ^ [A6: set_Pr4329608150637261639at_nat,B6: set_Pr4329608150637261639at_nat] :
          ( collec6321179662152712658at_nat
          @ ^ [X4: produc3843707927480180839at_nat] :
              ( ( member8757157785044589968at_nat @ X4 @ A6 )
              | ( member8757157785044589968at_nat @ X4 @ B6 ) ) ) ) ) ).

% Un_def
thf(fact_1080_Collect__disj__eq,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( collect_nat
        @ ^ [X4: nat] :
            ( ( P @ X4 )
            | ( Q @ X4 ) ) )
      = ( sup_sup_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_1081_Collect__disj__eq,axiom,
    ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X4: product_prod_nat_nat] :
            ( ( P @ X4 )
            | ( Q @ X4 ) ) )
      = ( sup_su6327502436637775413at_nat @ ( collec3392354462482085612at_nat @ P ) @ ( collec3392354462482085612at_nat @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_1082_Collect__disj__eq,axiom,
    ! [P: produc3843707927480180839at_nat > $o,Q: produc3843707927480180839at_nat > $o] :
      ( ( collec6321179662152712658at_nat
        @ ^ [X4: produc3843707927480180839at_nat] :
            ( ( P @ X4 )
            | ( Q @ X4 ) ) )
      = ( sup_su5525570899277871387at_nat @ ( collec6321179662152712658at_nat @ P ) @ ( collec6321179662152712658at_nat @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_1083_subset__Un__eq,axiom,
    ( ord_le1268244103169919719at_nat
    = ( ^ [A6: set_Pr4329608150637261639at_nat,B6: set_Pr4329608150637261639at_nat] :
          ( ( sup_su5525570899277871387at_nat @ A6 @ B6 )
          = B6 ) ) ) ).

% subset_Un_eq
thf(fact_1084_subset__UnE,axiom,
    ! [C2: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ C2 @ ( sup_su5525570899277871387at_nat @ A4 @ B4 ) )
     => ~ ! [A9: set_Pr4329608150637261639at_nat] :
            ( ( ord_le1268244103169919719at_nat @ A9 @ A4 )
           => ! [B11: set_Pr4329608150637261639at_nat] :
                ( ( ord_le1268244103169919719at_nat @ B11 @ B4 )
               => ( C2
                 != ( sup_su5525570899277871387at_nat @ A9 @ B11 ) ) ) ) ) ).

% subset_UnE
thf(fact_1085_Un__absorb2,axiom,
    ! [B4: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ B4 @ A4 )
     => ( ( sup_su5525570899277871387at_nat @ A4 @ B4 )
        = A4 ) ) ).

% Un_absorb2
thf(fact_1086_Un__absorb1,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A4 @ B4 )
     => ( ( sup_su5525570899277871387at_nat @ A4 @ B4 )
        = B4 ) ) ).

% Un_absorb1
thf(fact_1087_Un__upper2,axiom,
    ! [B4: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ B4 @ ( sup_su5525570899277871387at_nat @ A4 @ B4 ) ) ).

% Un_upper2
thf(fact_1088_Un__upper1,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ A4 @ ( sup_su5525570899277871387at_nat @ A4 @ B4 ) ) ).

% Un_upper1
thf(fact_1089_Un__least,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A4 @ C2 )
     => ( ( ord_le1268244103169919719at_nat @ B4 @ C2 )
       => ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ A4 @ B4 ) @ C2 ) ) ) ).

% Un_least
thf(fact_1090_Un__mono,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,D2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A4 @ C2 )
     => ( ( ord_le1268244103169919719at_nat @ B4 @ D2 )
       => ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ A4 @ B4 ) @ ( sup_su5525570899277871387at_nat @ C2 @ D2 ) ) ) ) ).

% Un_mono
thf(fact_1091_Un__left__commute,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ A4 @ ( sup_su5525570899277871387at_nat @ B4 @ C2 ) )
      = ( sup_su5525570899277871387at_nat @ B4 @ ( sup_su5525570899277871387at_nat @ A4 @ C2 ) ) ) ).

% Un_left_commute
thf(fact_1092_Un__left__absorb,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ A4 @ ( sup_su5525570899277871387at_nat @ A4 @ B4 ) )
      = ( sup_su5525570899277871387at_nat @ A4 @ B4 ) ) ).

% Un_left_absorb
thf(fact_1093_Un__commute,axiom,
    ( sup_su5525570899277871387at_nat
    = ( ^ [A6: set_Pr4329608150637261639at_nat,B6: set_Pr4329608150637261639at_nat] : ( sup_su5525570899277871387at_nat @ B6 @ A6 ) ) ) ).

% Un_commute
thf(fact_1094_diff__Suc__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
      = ( minus_minus_nat @ M @ N ) ) ).

% diff_Suc_Suc
thf(fact_1095_Suc__diff__diff,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).

% Suc_diff_diff
thf(fact_1096_diff__diff__cancel,axiom,
    ! [I3: nat,N: nat] :
      ( ( ord_less_eq_nat @ I3 @ N )
     => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I3 ) )
        = I3 ) ) ).

% diff_diff_cancel
thf(fact_1097_diff__diff__left,axiom,
    ! [I3: nat,J: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J ) @ K )
      = ( minus_minus_nat @ I3 @ ( plus_plus_nat @ J @ K ) ) ) ).

% diff_diff_left
thf(fact_1098_diff__Suc__1,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
      = N ) ).

% diff_Suc_1
thf(fact_1099_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J: nat,I3: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ I3 @ ( minus_minus_nat @ J @ K ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I3 @ K ) @ J ) ) ) ).

% Nat.diff_diff_right
thf(fact_1100_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J: nat,I3: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I3 )
        = ( minus_minus_nat @ ( plus_plus_nat @ J @ I3 ) @ K ) ) ) ).

% Nat.add_diff_assoc2
thf(fact_1101_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J: nat,I3: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( plus_plus_nat @ I3 @ ( minus_minus_nat @ J @ K ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I3 @ J ) @ K ) ) ) ).

% Nat.add_diff_assoc
thf(fact_1102_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J: nat,I3: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ I3 @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I3 @ K ) @ ( suc @ J ) ) ) ) ).

% diff_Suc_diff_eq1
thf(fact_1103_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J: nat,I3: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I3 )
        = ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I3 ) ) ) ) ).

% diff_Suc_diff_eq2
thf(fact_1104_diff__Suc__eq__diff__pred,axiom,
    ! [M: nat,N: nat] :
      ( ( minus_minus_nat @ M @ ( suc @ N ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).

% diff_Suc_eq_diff_pred
thf(fact_1105_Suc__diff__Suc,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ N @ M )
     => ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
        = ( minus_minus_nat @ M @ N ) ) ) ).

% Suc_diff_Suc
thf(fact_1106_diff__less__Suc,axiom,
    ! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).

% diff_less_Suc
thf(fact_1107_Suc__diff__le,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( minus_minus_nat @ ( suc @ M ) @ N )
        = ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).

% Suc_diff_le
thf(fact_1108_less__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( ord_less_nat @ M @ N ) ) ) ) ).

% less_diff_iff
thf(fact_1109_diff__less__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ A )
       => ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).

% diff_less_mono
thf(fact_1110_diff__le__mono2,axiom,
    ! [M: nat,N: nat,L2: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ L2 @ N ) @ ( minus_minus_nat @ L2 @ M ) ) ) ).

% diff_le_mono2
thf(fact_1111_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_1112_diff__le__self,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).

% diff_le_self
thf(fact_1113_diff__le__mono,axiom,
    ! [M: nat,N: nat,L2: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L2 ) @ ( minus_minus_nat @ N @ L2 ) ) ) ).

% diff_le_mono
thf(fact_1114_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_1115_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_1116_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_1117_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_1118_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_1119_diff__add__inverse,axiom,
    ! [N: nat,M: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
      = M ) ).

% diff_add_inverse
thf(fact_1120_diff__add__inverse2,axiom,
    ! [M: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
      = M ) ).

% diff_add_inverse2
thf(fact_1121_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,N: nat] :
      ( ( ord_less_nat @ J @ K )
     => ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).

% less_imp_diff_less
thf(fact_1122_diff__less__mono2,axiom,
    ! [M: nat,N: nat,L2: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( ord_less_nat @ M @ L2 )
       => ( ord_less_nat @ ( minus_minus_nat @ L2 @ N ) @ ( minus_minus_nat @ L2 @ M ) ) ) ) ).

% diff_less_mono2
thf(fact_1123_zero__induct__lemma,axiom,
    ! [P: nat > $o,K: nat,I3: nat] :
      ( ( P @ K )
     => ( ! [N2: nat] :
            ( ( P @ ( suc @ N2 ) )
           => ( P @ N2 ) )
       => ( P @ ( minus_minus_nat @ K @ I3 ) ) ) ) ).

% zero_induct_lemma
thf(fact_1124_Nat_Ole__imp__diff__is__add,axiom,
    ! [I3: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I3 @ J )
     => ( ( ( minus_minus_nat @ J @ I3 )
          = K )
        = ( J
          = ( plus_plus_nat @ K @ I3 ) ) ) ) ).

% Nat.le_imp_diff_is_add
thf(fact_1125_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J: nat,I3: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I3 ) @ K )
        = ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I3 ) ) ) ).

% Nat.diff_add_assoc2
thf(fact_1126_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J: nat,I3: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ I3 @ J ) @ K )
        = ( plus_plus_nat @ I3 @ ( minus_minus_nat @ J @ K ) ) ) ) ).

% Nat.diff_add_assoc
thf(fact_1127_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J: nat,I3: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( ord_less_eq_nat @ I3 @ ( minus_minus_nat @ J @ K ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ J ) ) ) ).

% Nat.le_diff_conv2
thf(fact_1128_le__diff__conv,axiom,
    ! [J: nat,K: nat,I3: nat] :
      ( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I3 )
      = ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I3 @ K ) ) ) ).

% le_diff_conv
thf(fact_1129_less__diff__conv,axiom,
    ! [I3: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I3 @ ( minus_minus_nat @ J @ K ) )
      = ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ J ) ) ).

% less_diff_conv
thf(fact_1130_add__diff__inverse__nat,axiom,
    ! [M: nat,N: nat] :
      ( ~ ( ord_less_nat @ M @ N )
     => ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
        = M ) ) ).

% add_diff_inverse_nat
thf(fact_1131_le__mod__geq,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( modulo_modulo_nat @ M @ N )
        = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ).

% le_mod_geq
thf(fact_1132_mod__if,axiom,
    ( modulo_modulo_nat
    = ( ^ [M5: nat,N3: nat] : ( if_nat @ ( ord_less_nat @ M5 @ N3 ) @ M5 @ ( modulo_modulo_nat @ ( minus_minus_nat @ M5 @ N3 ) @ N3 ) ) ) ) ).

% mod_if
thf(fact_1133_inj__on__diff__nat,axiom,
    ! [N4: set_nat,K: nat] :
      ( ! [N2: nat] :
          ( ( member_nat @ N2 @ N4 )
         => ( ord_less_eq_nat @ K @ N2 ) )
     => ( inj_on_nat_nat
        @ ^ [N3: nat] : ( minus_minus_nat @ N3 @ K )
        @ N4 ) ) ).

% inj_on_diff_nat
thf(fact_1134_less__diff__conv2,axiom,
    ! [K: nat,J: nat,I3: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I3 )
        = ( ord_less_nat @ J @ ( plus_plus_nat @ I3 @ K ) ) ) ) ).

% less_diff_conv2
thf(fact_1135_diff__commute,axiom,
    ! [I3: nat,J: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J ) @ K )
      = ( minus_minus_nat @ ( minus_minus_nat @ I3 @ K ) @ J ) ) ).

% diff_commute

% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [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 ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ( inj_on5538052773655684606at_nat
    @ ^ [I: nat] : ( product_Pair_nat_nat @ ( nth_nat @ as @ I ) @ ( count_list_nat @ ( drop_nat @ ( plus_plus_nat @ I @ one_one_nat ) @ as ) @ ( nth_nat @ as @ I ) ) )
    @ ( collect_nat
      @ ^ [K2: nat] : ( ord_less_nat @ K2 @ ( size_size_list_nat @ as ) ) ) ) ).

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