TPTP Problem File: SLH0325^1.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    : Pluennecke_Ruzsa_Inequality/0003_Pluennecke_Ruzsa_Inequality/prob_00364_012993__12214554_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1694 ( 394 unt; 421 typ;   0 def)
%            Number of atoms       : 4386 (1192 equ;   0 cnn)
%            Maximal formula atoms :   10 (   3 avg)
%            Number of connectives : 15572 ( 518   ~;  84   |; 476   &;12307   @)
%                                         (   0 <=>;2187  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   19 (   8 avg)
%            Number of types       :   71 (  70 usr)
%            Number of type conns  : 3755 (3755   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  352 ( 351 usr;  21 con; 0-5 aty)
%            Number of variables   : 4742 ( 410   ^;4165   !; 167   ?;4742   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-18 16:21:09.302
%------------------------------------------------------------------------------
% Could-be-implicit typings (70)
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thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_Itf__a_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
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thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_Itf__a_Mtf__a_J_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__Set__Oset_Itf__a_J,type,
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thf(ty_n_t__Nat__Onat,type,
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thf(ty_n_tf__a,type,
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% Explicit typings (351)
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thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
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thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
    inj_on_nat_nat_nat: ( nat > nat > nat ) > set_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001_062_It__Nat__Onat_Mtf__a_J,type,
    inj_on_nat_nat_a: ( nat > nat > a ) > set_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
    inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
    inj_on8964604748314331044od_a_a: ( nat > product_prod_a_a ) > set_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001tf__a,type,
    inj_on_nat_a: ( nat > a ) > set_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Nat__Onat,type,
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thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
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thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001tf__a,type,
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thf(sy_c_Fun_Oinj__on_001tf__a_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
    inj_on_a_nat_nat: ( a > nat > nat ) > set_a > $o ).

thf(sy_c_Fun_Oinj__on_001tf__a_001_062_It__Nat__Onat_Mtf__a_J,type,
    inj_on_a_nat_a: ( a > nat > a ) > set_a > $o ).

thf(sy_c_Fun_Oinj__on_001tf__a_001_062_Itf__a_Mt__Nat__Onat_J,type,
    inj_on_a_a_nat: ( a > a > nat ) > set_a > $o ).

thf(sy_c_Fun_Oinj__on_001tf__a_001_062_Itf__a_Mtf__a_J,type,
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thf(sy_c_Fun_Oinj__on_001tf__a_001t__Nat__Onat,type,
    inj_on_a_nat: ( a > nat ) > set_a > $o ).

thf(sy_c_Fun_Oinj__on_001tf__a_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
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thf(sy_c_Fun_Oinj__on_001tf__a_001t__Set__Oset_Itf__a_J,type,
    inj_on_a_set_a: ( a > set_a ) > set_a > $o ).

thf(sy_c_Fun_Oinj__on_001tf__a_001tf__a,type,
    inj_on_a_a: ( a > a ) > set_a > $o ).

thf(sy_c_FuncSet_OPi_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
    pi_nat_nat_nat: set_nat_nat > ( ( nat > nat ) > set_nat ) > set_nat_nat_nat2 ).

thf(sy_c_FuncSet_OPi_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001tf__a,type,
    pi_nat_nat_a: set_nat_nat > ( ( nat > nat ) > set_a ) > set_nat_nat_a2 ).

thf(sy_c_FuncSet_OPi_001_062_It__Nat__Onat_Mtf__a_J_001t__Nat__Onat,type,
    pi_nat_a_nat: set_nat_a > ( ( nat > a ) > set_nat ) > set_nat_a_nat ).

thf(sy_c_FuncSet_OPi_001_062_It__Nat__Onat_Mtf__a_J_001tf__a,type,
    pi_nat_a_a: set_nat_a > ( ( nat > a ) > set_a ) > set_nat_a_a ).

thf(sy_c_FuncSet_OPi_001_062_Itf__a_Mt__Nat__Onat_J_001tf__a,type,
    pi_a_nat_a: set_a_nat > ( ( a > nat ) > set_a ) > set_a_nat_a2 ).

thf(sy_c_FuncSet_OPi_001_062_Itf__a_Mtf__a_J_001tf__a,type,
    pi_a_a_a: set_a_a > ( ( a > a ) > set_a ) > set_a_a_a2 ).

thf(sy_c_FuncSet_OPi_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_FuncSet_OPi_001t__Nat__Onat_001_062_It__Nat__Onat_Mtf__a_J,type,
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thf(sy_c_FuncSet_OPi_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_FuncSet_OPi_001t__Nat__Onat_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
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thf(sy_c_FuncSet_OPi_001t__Nat__Onat_001tf__a,type,
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thf(sy_c_FuncSet_OPi_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Nat__Onat,type,
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thf(sy_c_FuncSet_OPi_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
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thf(sy_c_FuncSet_OPi_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001tf__a,type,
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thf(sy_c_FuncSet_OPi_001tf__a_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
    pi_a_nat_nat: set_a > ( a > set_nat_nat ) > set_a_nat_nat ).

thf(sy_c_FuncSet_OPi_001tf__a_001_062_It__Nat__Onat_Mtf__a_J,type,
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thf(sy_c_FuncSet_OPi_001tf__a_001_062_Itf__a_Mt__Nat__Onat_J,type,
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thf(sy_c_FuncSet_OPi_001tf__a_001_062_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
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thf(sy_c_FuncSet_OPi_001tf__a_001_062_Itf__a_Mtf__a_J,type,
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thf(sy_c_FuncSet_OPi_001tf__a_001t__Nat__Onat,type,
    pi_a_nat: set_a > ( a > set_nat ) > set_a_nat ).

thf(sy_c_FuncSet_OPi_001tf__a_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
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thf(sy_c_FuncSet_OPi_001tf__a_001tf__a,type,
    pi_a_a: set_a > ( a > set_a ) > set_a_a ).

thf(sy_c_Group__Theory_Oabelian__group_001tf__a,type,
    group_201663378560352916roup_a: set_a > ( a > a > a ) > a > $o ).

thf(sy_c_Group__Theory_Ocommutative__monoid_001tf__a,type,
    group_4866109990395492029noid_a: set_a > ( a > a > a ) > a > $o ).

thf(sy_c_Group__Theory_Ogroup_001tf__a,type,
    group_group_a: set_a > ( a > a > a ) > a > $o ).

thf(sy_c_Group__Theory_Omonoid_OUnits_001tf__a,type,
    group_Units_a: set_a > ( a > a > a ) > a > set_a ).

thf(sy_c_Group__Theory_Omonoid_Oinverse_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
    group_5492157109111387274at_nat: set_nat_nat > ( ( nat > nat ) > ( nat > nat ) > nat > nat ) > ( nat > nat ) > ( nat > nat ) > nat > nat ).

thf(sy_c_Group__Theory_Omonoid_Oinverse_001_062_It__Nat__Onat_Mtf__a_J,type,
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thf(sy_c_Group__Theory_Omonoid_Oinverse_001_062_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
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thf(sy_c_Group__Theory_Omonoid_Oinverse_001_062_Itf__a_Mt__Nat__Onat_J,type,
    group_inverse_a_nat: set_a_nat > ( ( a > nat ) > ( a > nat ) > a > nat ) > ( a > nat ) > ( a > nat ) > a > nat ).

thf(sy_c_Group__Theory_Omonoid_Oinverse_001_062_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
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thf(sy_c_Group__Theory_Omonoid_Oinverse_001_062_Itf__a_Mtf__a_J,type,
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thf(sy_c_Group__Theory_Omonoid_Oinverse_001t__Nat__Onat,type,
    group_inverse_nat: set_nat > ( nat > nat > nat ) > nat > nat > nat ).

thf(sy_c_Group__Theory_Omonoid_Oinverse_001tf__a,type,
    group_inverse_a: set_a > ( a > a > a ) > a > a > a ).

thf(sy_c_Group__Theory_Omonoid_Oinvertible_001tf__a,type,
    group_invertible_a: set_a > ( a > a > a ) > a > a > $o ).

thf(sy_c_Group__Theory_Osubgroup_001tf__a,type,
    group_subgroup_a: set_a > set_a > ( a > a > a ) > a > $o ).

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

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
    minus_minus_set_a: set_a > set_a > set_a ).

thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
    one_one_nat: nat ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
    zero_zero_nat: nat ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_062_Itf__a_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_Itf__a_Mt__Nat__Onat_J,type,
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thf(sy_c_Set_Oimage_001tf__a_001_062_It__Nat__Onat_Mtf__a_J,type,
    image_a_nat_a2: ( a > nat > a ) > set_a > set_nat_a ).

thf(sy_c_Set_Oimage_001tf__a_001_062_Itf__a_Mt__Nat__Onat_J,type,
    image_a_a_nat: ( a > a > nat ) > set_a > set_a_nat ).

thf(sy_c_Set_Oimage_001tf__a_001_062_Itf__a_Mtf__a_J,type,
    image_a_a_a2: ( a > a > a ) > set_a > set_a_a ).

thf(sy_c_Set_Oimage_001tf__a_001t__Nat__Onat,type,
    image_a_nat: ( a > nat ) > set_a > set_nat ).

thf(sy_c_Set_Oimage_001tf__a_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
    image_7400625782589995694od_a_a: ( a > product_prod_a_a ) > set_a > set_Product_prod_a_a ).

thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_Itf__a_J,type,
    image_a_set_a: ( a > set_a ) > set_a > set_set_a ).

thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
    image_a_a: ( a > a ) > set_a > set_a ).

thf(sy_c_Set_Oinsert_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
    insert_nat_nat: ( nat > nat ) > set_nat_nat > set_nat_nat ).

thf(sy_c_Set_Oinsert_001_062_It__Nat__Onat_Mtf__a_J,type,
    insert_nat_a: ( nat > a ) > set_nat_a > set_nat_a ).

thf(sy_c_Set_Oinsert_001_062_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
    insert1583624352380834389od_a_a: ( product_prod_a_a > product_prod_a_a ) > set_Pr8826267807999420763od_a_a > set_Pr8826267807999420763od_a_a ).

thf(sy_c_Set_Oinsert_001_062_Itf__a_Mt__Nat__Onat_J,type,
    insert_a_nat: ( a > nat ) > set_a_nat > set_a_nat ).

thf(sy_c_Set_Oinsert_001_062_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
    insert5082226857754029630od_a_a: ( a > product_prod_a_a ) > set_a_6829686330177631172od_a_a > set_a_6829686330177631172od_a_a ).

thf(sy_c_Set_Oinsert_001_062_Itf__a_Mtf__a_J,type,
    insert_a_a: ( a > a ) > set_a_a > set_a_a ).

thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
    insert_nat: nat > set_nat > set_nat ).

thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
    insert4534936382041156343od_a_a: product_prod_a_a > set_Product_prod_a_a > set_Product_prod_a_a ).

thf(sy_c_Set_Oinsert_001t__Set__Oset_Itf__a_J,type,
    insert_set_a: set_a > set_set_a > set_set_a ).

thf(sy_c_Set_Oinsert_001tf__a,type,
    insert_a: a > set_a > set_a ).

thf(sy_c_member_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
    member_nat_nat_nat: ( ( nat > nat ) > nat ) > set_nat_nat_nat2 > $o ).

thf(sy_c_member_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mtf__a_J,type,
    member_nat_nat_a: ( ( nat > nat ) > a ) > set_nat_nat_a2 > $o ).

thf(sy_c_member_001_062_I_062_It__Nat__Onat_Mtf__a_J_Mt__Nat__Onat_J,type,
    member_nat_a_nat: ( ( nat > a ) > nat ) > set_nat_a_nat > $o ).

thf(sy_c_member_001_062_I_062_It__Nat__Onat_Mtf__a_J_Mtf__a_J,type,
    member_nat_a_a: ( ( nat > a ) > a ) > set_nat_a_a > $o ).

thf(sy_c_member_001_062_I_062_Itf__a_Mt__Nat__Onat_J_Mtf__a_J,type,
    member_a_nat_a: ( ( a > nat ) > a ) > set_a_nat_a2 > $o ).

thf(sy_c_member_001_062_I_062_Itf__a_Mtf__a_J_Mtf__a_J,type,
    member_a_a_a: ( ( a > a ) > a ) > set_a_a_a2 > $o ).

thf(sy_c_member_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    member_nat_nat_nat2: ( nat > nat > nat ) > set_nat_nat_nat > $o ).

thf(sy_c_member_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__a_J_J,type,
    member_nat_nat_a2: ( nat > nat > a ) > set_nat_nat_a > $o ).

thf(sy_c_member_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
    member_nat_nat: ( nat > nat ) > set_nat_nat > $o ).

thf(sy_c_member_001_062_It__Nat__Onat_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
    member909238274692171063od_a_a: ( nat > product_prod_a_a ) > set_na6140580400147104110od_a_a > $o ).

thf(sy_c_member_001_062_It__Nat__Onat_Mtf__a_J,type,
    member_nat_a: ( nat > a ) > set_nat_a > $o ).

thf(sy_c_member_001_062_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Nat__Onat_J,type,
    member4881838279615896465_a_nat: ( product_prod_a_a > nat ) > set_Pr194737476958849736_a_nat > $o ).

thf(sy_c_member_001_062_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
    member4020126937092221116od_a_a: ( product_prod_a_a > product_prod_a_a ) > set_Pr8826267807999420763od_a_a > $o ).

thf(sy_c_member_001_062_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J,type,
    member1716570166360300819_a_a_a: ( product_prod_a_a > a ) > set_Pr952751117562918450_a_a_a > $o ).

thf(sy_c_member_001_062_Itf__a_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    member_a_nat_nat: ( a > nat > nat ) > set_a_nat_nat > $o ).

thf(sy_c_member_001_062_Itf__a_M_062_It__Nat__Onat_Mtf__a_J_J,type,
    member_a_nat_a2: ( a > nat > a ) > set_a_nat_a > $o ).

thf(sy_c_member_001_062_Itf__a_M_062_Itf__a_Mt__Nat__Onat_J_J,type,
    member_a_a_nat: ( a > a > nat ) > set_a_a_nat > $o ).

thf(sy_c_member_001_062_Itf__a_M_062_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
    member2522163699605435814od_a_a: ( a > a > product_prod_a_a ) > set_a_5030874611358290525od_a_a > $o ).

thf(sy_c_member_001_062_Itf__a_M_062_Itf__a_Mtf__a_J_J,type,
    member_a_a_a2: ( a > a > a ) > set_a_a_a > $o ).

thf(sy_c_member_001_062_Itf__a_Mt__Nat__Onat_J,type,
    member_a_nat: ( a > nat ) > set_a_nat > $o ).

thf(sy_c_member_001_062_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
    member1957775702407316389od_a_a: ( a > product_prod_a_a ) > set_a_6829686330177631172od_a_a > $o ).

thf(sy_c_member_001_062_Itf__a_Mtf__a_J,type,
    member_a_a: ( a > a ) > set_a_a > $o ).

thf(sy_c_member_001t__Nat__Onat,type,
    member_nat: nat > set_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    member8336108448496249625at_nat: produc85711943791777264at_nat > set_Pr6370437063884598352at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__a_J_J,type,
    member2811451201825703969_nat_a: produc7123000486447228170_nat_a > set_Pr5513774608395838528_nat_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J,type,
    member8962352052110095674_nat_a: product_prod_nat_a > set_Pr4193341848836149977_nat_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    member5463269792551873667at_nat: produc551447040318622060at_nat > set_Pr6841066879934740386at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_M_062_It__Nat__Onat_Mtf__a_J_J,type,
    member5853765858310737207_nat_a: product_prod_a_nat_a > set_Pr7263265662379690350_nat_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_M_062_Itf__a_Mt__Nat__Onat_J_J,type,
    member6594859421832710929_a_nat: product_prod_a_a_nat > set_Pr5501298576112220744_a_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_M_062_Itf__a_Mtf__a_J_J,type,
    member681807718693926441_a_a_a: product_prod_a_a_a > set_Pr2596522554037969992_a_a_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
    member5724188588386418708_a_nat: product_prod_a_nat > set_Pr4934435412358123699_a_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
    member1426531477525435216od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).

thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
    member_set_a: set_a > set_set_a > $o ).

thf(sy_c_member_001tf__a,type,
    member_a: a > set_a > $o ).

thf(sy_v_G,type,
    g: set_a ).

thf(sy_v_U,type,
    u: set_a ).

thf(sy_v_V,type,
    v: set_a ).

thf(sy_v_W,type,
    w: set_a ).

thf(sy_v__092_060phi_062____,type,
    phi: product_prod_a_a > product_prod_a_a ).

thf(sy_v_addition,type,
    addition: a > a > a ).

thf(sy_v_v____,type,
    v2: a > a ).

thf(sy_v_w____,type,
    w2: a > a ).

thf(sy_v_zero,type,
    zero: a ).

% Relevant facts (1272)
thf(fact_0_assms_I5_J,axiom,
    finite_finite_a @ w ).

% assms(5)
thf(fact_1_assms_I3_J,axiom,
    finite_finite_a @ v ).

% assms(3)
thf(fact_2_assms_I1_J,axiom,
    finite_finite_a @ u ).

% assms(1)
thf(fact_3_card__inj,axiom,
    ! [F: product_prod_a_a > product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
      ( ( member4020126937092221116od_a_a @ F
        @ ( pi_Pro6370639526499058571od_a_a @ A
          @ ^ [Uu: product_prod_a_a] : B ) )
     => ( ( inj_on2566144670800592689od_a_a @ F @ A )
       => ( ( finite6544458595007987280od_a_a @ B )
         => ( ord_less_eq_nat @ ( finite4795055649997197647od_a_a @ A ) @ ( finite4795055649997197647od_a_a @ B ) ) ) ) ) ).

% card_inj
thf(fact_4_card__inj,axiom,
    ! [F: a > product_prod_a_a,A: set_a,B: set_Product_prod_a_a] :
      ( ( member1957775702407316389od_a_a @ F
        @ ( pi_a_P2178097759547960436od_a_a @ A
          @ ^ [Uu: a] : B ) )
     => ( ( inj_on8941660083241582106od_a_a @ F @ A )
       => ( ( finite6544458595007987280od_a_a @ B )
         => ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite4795055649997197647od_a_a @ B ) ) ) ) ) ).

% card_inj
thf(fact_5_card__inj,axiom,
    ! [F: nat > product_prod_a_a,A: set_nat,B: set_Product_prod_a_a] :
      ( ( member909238274692171063od_a_a @ F
        @ ( pi_nat2514139106461652682od_a_a @ A
          @ ^ [Uu: nat] : B ) )
     => ( ( inj_on8964604748314331044od_a_a @ F @ A )
       => ( ( finite6544458595007987280od_a_a @ B )
         => ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite4795055649997197647od_a_a @ B ) ) ) ) ) ).

% card_inj
thf(fact_6_card__inj,axiom,
    ! [F: product_prod_a_a > a,A: set_Product_prod_a_a,B: set_a] :
      ( ( member1716570166360300819_a_a_a @ F
        @ ( pi_Pro7438789266712198498_a_a_a @ A
          @ ^ [Uu: product_prod_a_a] : B ) )
     => ( ( inj_on4978979553551044360_a_a_a @ F @ A )
       => ( ( finite_finite_a @ B )
         => ( ord_less_eq_nat @ ( finite4795055649997197647od_a_a @ A ) @ ( finite_card_a @ B ) ) ) ) ) ).

% card_inj
thf(fact_7_card__inj,axiom,
    ! [F: a > a,A: set_a,B: set_a] :
      ( ( member_a_a @ F
        @ ( pi_a_a @ A
          @ ^ [Uu: a] : B ) )
     => ( ( inj_on_a_a @ F @ A )
       => ( ( finite_finite_a @ B )
         => ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ B ) ) ) ) ) ).

% card_inj
thf(fact_8_card__inj,axiom,
    ! [F: nat > a,A: set_nat,B: set_a] :
      ( ( member_nat_a @ F
        @ ( pi_nat_a @ A
          @ ^ [Uu: nat] : B ) )
     => ( ( inj_on_nat_a @ F @ A )
       => ( ( finite_finite_a @ B )
         => ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_a @ B ) ) ) ) ) ).

% card_inj
thf(fact_9_card__inj,axiom,
    ! [F: product_prod_a_a > nat,A: set_Product_prod_a_a,B: set_nat] :
      ( ( member4881838279615896465_a_nat @ F
        @ ( pi_Pro1971496080287246444_a_nat @ A
          @ ^ [Uu: product_prod_a_a] : B ) )
     => ( ( inj_on8421961722139924806_a_nat @ F @ A )
       => ( ( finite_finite_nat @ B )
         => ( ord_less_eq_nat @ ( finite4795055649997197647od_a_a @ A ) @ ( finite_card_nat @ B ) ) ) ) ) ).

% card_inj
thf(fact_10_card__inj,axiom,
    ! [F: a > nat,A: set_a,B: set_nat] :
      ( ( member_a_nat @ F
        @ ( pi_a_nat @ A
          @ ^ [Uu: a] : B ) )
     => ( ( inj_on_a_nat @ F @ A )
       => ( ( finite_finite_nat @ B )
         => ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_nat @ B ) ) ) ) ) ).

% card_inj
thf(fact_11_card__inj,axiom,
    ! [F: nat > nat,A: set_nat,B: set_nat] :
      ( ( member_nat_nat @ F
        @ ( pi_nat_nat @ A
          @ ^ [Uu: nat] : B ) )
     => ( ( inj_on_nat_nat @ F @ A )
       => ( ( finite_finite_nat @ B )
         => ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) ) ) ) ) ).

% card_inj
thf(fact_12_commutative,axiom,
    ! [X: a,Y: a] :
      ( ( member_a @ X @ g )
     => ( ( member_a @ Y @ g )
       => ( ( addition @ X @ Y )
          = ( addition @ Y @ X ) ) ) ) ).

% commutative
thf(fact_13_sumset_Ocases,axiom,
    ! [A2: a,A: set_a,B: set_a] :
      ( ( member_a @ A2 @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) )
     => ~ ! [A3: a,B2: a] :
            ( ( A2
              = ( addition @ A3 @ B2 ) )
           => ( ( member_a @ A3 @ A )
             => ( ( member_a @ A3 @ g )
               => ( ( member_a @ B2 @ B )
                 => ~ ( member_a @ B2 @ g ) ) ) ) ) ) ).

% sumset.cases
thf(fact_14_sumset_Osimps,axiom,
    ! [A2: a,A: set_a,B: set_a] :
      ( ( member_a @ A2 @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) )
      = ( ? [A4: a,B3: a] :
            ( ( A2
              = ( addition @ A4 @ B3 ) )
            & ( member_a @ A4 @ A )
            & ( member_a @ A4 @ g )
            & ( member_a @ B3 @ B )
            & ( member_a @ B3 @ g ) ) ) ) ).

% sumset.simps
thf(fact_15_sumset_OsumsetI,axiom,
    ! [A2: a,A: set_a,B4: a,B: set_a] :
      ( ( member_a @ A2 @ A )
     => ( ( member_a @ A2 @ g )
       => ( ( member_a @ B4 @ B )
         => ( ( member_a @ B4 @ g )
           => ( member_a @ ( addition @ A2 @ B4 ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) ) ) ) ) ) ).

% sumset.sumsetI
thf(fact_16_sumset__assoc,axiom,
    ! [A: set_a,B: set_a,C: set_a] :
      ( ( pluenn3038260743871226533mset_a @ g @ addition @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) @ C )
      = ( pluenn3038260743871226533mset_a @ g @ addition @ A @ ( pluenn3038260743871226533mset_a @ g @ addition @ B @ C ) ) ) ).

% sumset_assoc
thf(fact_17_sumset__commute,axiom,
    ! [A: set_a,B: set_a] :
      ( ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B )
      = ( pluenn3038260743871226533mset_a @ g @ addition @ B @ A ) ) ).

% sumset_commute
thf(fact_18_fin_I2_J,axiom,
    finite_finite_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ u @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ w ) ) ).

% fin(2)
thf(fact_19_fin_I1_J,axiom,
    finite_finite_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ u @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ v ) ) ).

% fin(1)
thf(fact_20_local_Oinverse__unique,axiom,
    ! [U: a,V: a,V2: a] :
      ( ( ( addition @ U @ V )
        = zero )
     => ( ( ( addition @ V2 @ U )
          = zero )
       => ( ( member_a @ U @ g )
         => ( ( member_a @ V2 @ g )
           => ( ( member_a @ V @ g )
             => ( V2 = V ) ) ) ) ) ) ).

% local.inverse_unique
thf(fact_21_minusset__distrib__sum,axiom,
    ! [A: set_a,B: set_a] :
      ( ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) )
      = ( pluenn3038260743871226533mset_a @ g @ addition @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A ) @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ B ) ) ) ).

% minusset_distrib_sum
thf(fact_22__092_060open_062_092_060phi_062_A_092_060in_062_AU_A_092_060times_062_Adifferenceset_AV_AW_A_092_060rightarrow_062_Adifferenceset_AU_AV_A_092_060times_062_Adifferenceset_AU_AW_092_060close_062,axiom,
    ( member4020126937092221116od_a_a @ phi
    @ ( pi_Pro6370639526499058571od_a_a
      @ ( product_Sigma_a_a @ u
        @ ^ [Uu: a] : ( pluenn3038260743871226533mset_a @ g @ addition @ v @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ w ) ) )
      @ ^ [Uu: product_prod_a_a] :
          ( product_Sigma_a_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ u @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ v ) )
          @ ^ [Uv: a] : ( pluenn3038260743871226533mset_a @ g @ addition @ u @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ w ) ) ) ) ) ).

% \<open>\<phi> \<in> U \<times> differenceset V W \<rightarrow> differenceset U V \<times> differenceset U W\<close>
thf(fact_23__092_060open_062inj__on_A_092_060phi_062_A_IU_A_092_060times_062_Adifferenceset_AV_AW_J_092_060close_062,axiom,
    ( inj_on2566144670800592689od_a_a @ phi
    @ ( product_Sigma_a_a @ u
      @ ^ [Uu: a] : ( pluenn3038260743871226533mset_a @ g @ addition @ v @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ w ) ) ) ) ).

% \<open>inj_on \<phi> (U \<times> differenceset V W)\<close>
thf(fact_24_finite__sumset,axiom,
    ! [A: set_a,B: set_a] :
      ( ( finite_finite_a @ A )
     => ( ( finite_finite_a @ B )
       => ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) ) ) ) ).

% finite_sumset
thf(fact_25_finite__minusset,axiom,
    ! [A: set_a] :
      ( ( finite_finite_a @ A )
     => ( finite_finite_a @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A ) ) ) ).

% finite_minusset
thf(fact_26_assms_I6_J,axiom,
    ord_less_eq_set_a @ w @ g ).

% assms(6)
thf(fact_27_assms_I4_J,axiom,
    ord_less_eq_set_a @ v @ g ).

% assms(4)
thf(fact_28_finite__differenceset,axiom,
    ! [A: set_a,B: set_a] :
      ( ( finite_finite_a @ A )
     => ( ( finite_finite_a @ B )
       => ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ B ) ) ) ) ) ).

% finite_differenceset
thf(fact_29_assms_I2_J,axiom,
    ord_less_eq_set_a @ u @ g ).

% assms(2)
thf(fact_30_associative,axiom,
    ! [A2: a,B4: a,C2: a] :
      ( ( member_a @ A2 @ g )
     => ( ( member_a @ B4 @ g )
       => ( ( member_a @ C2 @ g )
         => ( ( addition @ ( addition @ A2 @ B4 ) @ C2 )
            = ( addition @ A2 @ ( addition @ B4 @ C2 ) ) ) ) ) ) ).

% associative
thf(fact_31_composition__closed,axiom,
    ! [A2: a,B4: a] :
      ( ( member_a @ A2 @ g )
     => ( ( member_a @ B4 @ g )
       => ( member_a @ ( addition @ A2 @ B4 ) @ g ) ) ) ).

% composition_closed
thf(fact_32_unit__closed,axiom,
    member_a @ zero @ g ).

% unit_closed
thf(fact_33_left__unit,axiom,
    ! [A2: a] :
      ( ( member_a @ A2 @ g )
     => ( ( addition @ zero @ A2 )
        = A2 ) ) ).

% left_unit
thf(fact_34_right__unit,axiom,
    ! [A2: a] :
      ( ( member_a @ A2 @ g )
     => ( ( addition @ A2 @ zero )
        = A2 ) ) ).

% right_unit
thf(fact_35_differenceset__commute,axiom,
    ! [B: set_a,A: set_a] :
      ( ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ ( pluenn3038260743871226533mset_a @ g @ addition @ B @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A ) ) )
      = ( pluenn3038260743871226533mset_a @ g @ addition @ A @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ B ) ) ) ).

% differenceset_commute
thf(fact_36_additive__abelian__group__axioms,axiom,
    pluenn1164192988769422572roup_a @ g @ addition @ zero ).

% additive_abelian_group_axioms
thf(fact_37_commutative__monoid__axioms,axiom,
    group_4866109990395492029noid_a @ g @ addition @ zero ).

% commutative_monoid_axioms
thf(fact_38_vinG,axiom,
    ! [X: a] :
      ( ( member_a @ X @ ( pluenn3038260743871226533mset_a @ g @ addition @ v @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ w ) ) )
     => ( member_a @ ( v2 @ X ) @ g ) ) ).

% vinG
thf(fact_39_vinV,axiom,
    ! [X: a] :
      ( ( member_a @ X @ ( pluenn3038260743871226533mset_a @ g @ addition @ v @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ w ) ) )
     => ( member_a @ ( v2 @ X ) @ v ) ) ).

% vinV
thf(fact_40_winG,axiom,
    ! [X: a] :
      ( ( member_a @ X @ ( pluenn3038260743871226533mset_a @ g @ addition @ v @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ w ) ) )
     => ( member_a @ ( w2 @ X ) @ g ) ) ).

% winG
thf(fact_41_winW,axiom,
    ! [X: a] :
      ( ( member_a @ X @ ( pluenn3038260743871226533mset_a @ g @ addition @ v @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ w ) ) )
     => ( member_a @ ( w2 @ X ) @ w ) ) ).

% winW
thf(fact_42_additive__abelian__group_Osumset_Ocong,axiom,
    pluenn3038260743871226533mset_a = pluenn3038260743871226533mset_a ).

% additive_abelian_group.sumset.cong
thf(fact_43_additive__abelian__group_Ominusset_Ocong,axiom,
    pluenn2534204936789923946sset_a = pluenn2534204936789923946sset_a ).

% additive_abelian_group.minusset.cong
thf(fact_44_minusset__def,axiom,
    ( ( pluenn2534204936789923946sset_a @ g @ addition @ zero )
    = ( ^ [A5: set_a] :
          ( collect_a
          @ ( pluenn1126946703085653920setp_a @ g @ addition @ zero
            @ ^ [X2: a] : ( member_a @ X2 @ A5 ) ) ) ) ) ).

% minusset_def
thf(fact_45_minussetp__minusset__eq,axiom,
    ! [A: set_a] :
      ( ( pluenn1126946703085653920setp_a @ g @ addition @ zero
        @ ^ [X2: a] : ( member_a @ X2 @ A ) )
      = ( ^ [X2: a] : ( member_a @ X2 @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A ) ) ) ) ).

% minussetp_minusset_eq
thf(fact_46_abelian__group__axioms,axiom,
    group_201663378560352916roup_a @ g @ addition @ zero ).

% abelian_group_axioms
thf(fact_47_sumset__def,axiom,
    ( ( pluenn3038260743871226533mset_a @ g @ addition )
    = ( ^ [A5: set_a,B5: set_a] :
          ( collect_a
          @ ( pluenn895083305082786853setp_a @ g @ addition
            @ ^ [X2: a] : ( member_a @ X2 @ A5 )
            @ ^ [X2: a] : ( member_a @ X2 @ B5 ) ) ) ) ) ).

% sumset_def
thf(fact_48_sumsetp__sumset__eq,axiom,
    ! [A: set_a,B: set_a] :
      ( ( pluenn895083305082786853setp_a @ g @ addition
        @ ^ [X2: a] : ( member_a @ X2 @ A )
        @ ^ [X2: a] : ( member_a @ X2 @ B ) )
      = ( ^ [X2: a] : ( member_a @ X2 @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) ) ) ) ).

% sumsetp_sumset_eq
thf(fact_49_sumsetp_Ocases,axiom,
    ! [A: a > $o,B: a > $o,A2: a] :
      ( ( pluenn895083305082786853setp_a @ g @ addition @ A @ B @ A2 )
     => ~ ! [A3: a,B2: a] :
            ( ( A2
              = ( addition @ A3 @ B2 ) )
           => ( ( A @ A3 )
             => ( ( member_a @ A3 @ g )
               => ( ( B @ B2 )
                 => ~ ( member_a @ B2 @ g ) ) ) ) ) ) ).

% sumsetp.cases
thf(fact_50_sumsetp_Osimps,axiom,
    ! [A: a > $o,B: a > $o,A2: a] :
      ( ( pluenn895083305082786853setp_a @ g @ addition @ A @ B @ A2 )
      = ( ? [A4: a,B3: a] :
            ( ( A2
              = ( addition @ A4 @ B3 ) )
            & ( A @ A4 )
            & ( member_a @ A4 @ g )
            & ( B @ B3 )
            & ( member_a @ B3 @ g ) ) ) ) ).

% sumsetp.simps
thf(fact_51_mem__Collect__eq,axiom,
    ! [A2: product_prod_a_a > product_prod_a_a,P: ( product_prod_a_a > product_prod_a_a ) > $o] :
      ( ( member4020126937092221116od_a_a @ A2 @ ( collec8125451137695935482od_a_a @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_52_mem__Collect__eq,axiom,
    ! [A2: nat > nat,P: ( nat > nat ) > $o] :
      ( ( member_nat_nat @ A2 @ ( collect_nat_nat @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_53_mem__Collect__eq,axiom,
    ! [A2: nat > a,P: ( nat > a ) > $o] :
      ( ( member_nat_a @ A2 @ ( collect_nat_a @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_54_mem__Collect__eq,axiom,
    ! [A2: a > product_prod_a_a,P: ( a > product_prod_a_a ) > $o] :
      ( ( member1957775702407316389od_a_a @ A2 @ ( collec3127584877517960419od_a_a @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_55_mem__Collect__eq,axiom,
    ! [A2: a > nat,P: ( a > nat ) > $o] :
      ( ( member_a_nat @ A2 @ ( collect_a_nat @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_56_mem__Collect__eq,axiom,
    ! [A2: a > a,P: ( a > a ) > $o] :
      ( ( member_a_a @ A2 @ ( collect_a_a @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_57_mem__Collect__eq,axiom,
    ! [A2: a,P: a > $o] :
      ( ( member_a @ A2 @ ( collect_a @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_58_mem__Collect__eq,axiom,
    ! [A2: nat,P: nat > $o] :
      ( ( member_nat @ A2 @ ( collect_nat @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_59_Collect__mem__eq,axiom,
    ! [A: set_Pr8826267807999420763od_a_a] :
      ( ( collec8125451137695935482od_a_a
        @ ^ [X2: product_prod_a_a > product_prod_a_a] : ( member4020126937092221116od_a_a @ X2 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_60_Collect__mem__eq,axiom,
    ! [A: set_nat_nat] :
      ( ( collect_nat_nat
        @ ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_61_Collect__mem__eq,axiom,
    ! [A: set_nat_a] :
      ( ( collect_nat_a
        @ ^ [X2: nat > a] : ( member_nat_a @ X2 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_62_Collect__mem__eq,axiom,
    ! [A: set_a_6829686330177631172od_a_a] :
      ( ( collec3127584877517960419od_a_a
        @ ^ [X2: a > product_prod_a_a] : ( member1957775702407316389od_a_a @ X2 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_63_Collect__mem__eq,axiom,
    ! [A: set_a_nat] :
      ( ( collect_a_nat
        @ ^ [X2: a > nat] : ( member_a_nat @ X2 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_64_Collect__mem__eq,axiom,
    ! [A: set_a_a] :
      ( ( collect_a_a
        @ ^ [X2: a > a] : ( member_a_a @ X2 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_65_Collect__mem__eq,axiom,
    ! [A: set_a] :
      ( ( collect_a
        @ ^ [X2: a] : ( member_a @ X2 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_66_Collect__mem__eq,axiom,
    ! [A: set_nat] :
      ( ( collect_nat
        @ ^ [X2: nat] : ( member_nat @ X2 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_67_Collect__cong,axiom,
    ! [P: a > $o,Q: a > $o] :
      ( ! [X3: a] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect_a @ P )
        = ( collect_a @ Q ) ) ) ).

% Collect_cong
thf(fact_68_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_69_sumsetp_OsumsetI,axiom,
    ! [A: a > $o,A2: a,B: a > $o,B4: a] :
      ( ( A @ A2 )
     => ( ( member_a @ A2 @ g )
       => ( ( B @ B4 )
         => ( ( member_a @ B4 @ g )
           => ( pluenn895083305082786853setp_a @ g @ addition @ A @ B @ ( addition @ A2 @ B4 ) ) ) ) ) ) ).

% sumsetp.sumsetI
thf(fact_70_minusset__iterated__minusset,axiom,
    ! [A: set_a,K: nat] :
      ( ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A ) @ K )
      = ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ A @ K ) ) ) ).

% minusset_iterated_minusset
thf(fact_71__092_060open_062_092_060And_062x_O_Ax_A_092_060in_062_Adifferenceset_AV_AW_A_092_060Longrightarrow_062_A_092_060exists_062v_Aw_O_Av_A_092_060in_062_AV_A_092_060and_062_Aw_A_092_060in_062_AW_A_092_060and_062_Ax_A_061_Av_A_092_060ominus_062_Aw_092_060close_062,axiom,
    ! [X: a] :
      ( ( member_a @ X @ ( pluenn3038260743871226533mset_a @ g @ addition @ v @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ w ) ) )
     => ? [V3: a,W: a] :
          ( ( member_a @ V3 @ v )
          & ( member_a @ W @ w )
          & ( X
            = ( addition @ V3 @ ( group_inverse_a @ g @ addition @ zero @ W ) ) ) ) ) ).

% \<open>\<And>x. x \<in> differenceset V W \<Longrightarrow> \<exists>v w. v \<in> V \<and> w \<in> W \<and> x = v \<ominus> w\<close>
thf(fact_72__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062v_Aw_O_A_092_060lbrakk_062_092_060And_062x_O_Ax_A_092_060in_062_Adifferenceset_AV_AW_A_092_060Longrightarrow_062_Av_Ax_A_092_060in_062_AV_059_A_092_060And_062x_O_Ax_A_092_060in_062_Adifferenceset_AV_AW_A_092_060Longrightarrow_062_Aw_Ax_A_092_060in_062_AW_059_A_092_060And_062x_O_Ax_A_092_060in_062_Adifferenceset_AV_AW_A_092_060Longrightarrow_062_Av_Ax_A_092_060ominus_062_Aw_Ax_A_061_Ax_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [V3: a > a] :
        ( ! [X4: a] :
            ( ( member_a @ X4 @ ( pluenn3038260743871226533mset_a @ g @ addition @ v @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ w ) ) )
           => ( member_a @ ( V3 @ X4 ) @ v ) )
       => ! [W: a > a] :
            ( ! [X4: a] :
                ( ( member_a @ X4 @ ( pluenn3038260743871226533mset_a @ g @ addition @ v @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ w ) ) )
               => ( member_a @ ( W @ X4 ) @ w ) )
           => ~ ! [X4: a] :
                  ( ( member_a @ X4 @ ( pluenn3038260743871226533mset_a @ g @ addition @ v @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ w ) ) )
                 => ( ( addition @ ( V3 @ X4 ) @ ( group_inverse_a @ g @ addition @ zero @ ( W @ X4 ) ) )
                    = X4 ) ) ) ) ).

% \<open>\<And>thesis. (\<And>v w. \<lbrakk>\<And>x. x \<in> differenceset V W \<Longrightarrow> v x \<in> V; \<And>x. x \<in> differenceset V W \<Longrightarrow> w x \<in> W; \<And>x. x \<in> differenceset V W \<Longrightarrow> v x \<ominus> w x = x\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_73_Pi__I,axiom,
    ! [A: set_a,F: a > a,B: a > set_a] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ A )
         => ( member_a @ ( F @ X3 ) @ ( B @ X3 ) ) )
     => ( member_a_a @ F @ ( pi_a_a @ A @ B ) ) ) ).

% Pi_I
thf(fact_74_Pi__I,axiom,
    ! [A: set_a,F: a > nat,B: a > set_nat] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ A )
         => ( member_nat @ ( F @ X3 ) @ ( B @ X3 ) ) )
     => ( member_a_nat @ F @ ( pi_a_nat @ A @ B ) ) ) ).

% Pi_I
thf(fact_75_Pi__I,axiom,
    ! [A: set_nat,F: nat > a,B: nat > set_a] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A )
         => ( member_a @ ( F @ X3 ) @ ( B @ X3 ) ) )
     => ( member_nat_a @ F @ ( pi_nat_a @ A @ B ) ) ) ).

% Pi_I
thf(fact_76_Pi__I,axiom,
    ! [A: set_nat,F: nat > nat,B: nat > set_nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A )
         => ( member_nat @ ( F @ X3 ) @ ( B @ X3 ) ) )
     => ( member_nat_nat @ F @ ( pi_nat_nat @ A @ B ) ) ) ).

% Pi_I
thf(fact_77_Pi__I,axiom,
    ! [A: set_a,F: a > product_prod_a_a,B: a > set_Product_prod_a_a] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ A )
         => ( member1426531477525435216od_a_a @ ( F @ X3 ) @ ( B @ X3 ) ) )
     => ( member1957775702407316389od_a_a @ F @ ( pi_a_P2178097759547960436od_a_a @ A @ B ) ) ) ).

% Pi_I
thf(fact_78_Pi__I,axiom,
    ! [A: set_a,F: a > nat > nat,B: a > set_nat_nat] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ A )
         => ( member_nat_nat @ ( F @ X3 ) @ ( B @ X3 ) ) )
     => ( member_a_nat_nat @ F @ ( pi_a_nat_nat @ A @ B ) ) ) ).

% Pi_I
thf(fact_79_Pi__I,axiom,
    ! [A: set_a,F: a > nat > a,B: a > set_nat_a] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ A )
         => ( member_nat_a @ ( F @ X3 ) @ ( B @ X3 ) ) )
     => ( member_a_nat_a2 @ F @ ( pi_a_nat_a2 @ A @ B ) ) ) ).

% Pi_I
thf(fact_80_Pi__I,axiom,
    ! [A: set_a,F: a > a > nat,B: a > set_a_nat] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ A )
         => ( member_a_nat @ ( F @ X3 ) @ ( B @ X3 ) ) )
     => ( member_a_a_nat @ F @ ( pi_a_a_nat @ A @ B ) ) ) ).

% Pi_I
thf(fact_81_Pi__I,axiom,
    ! [A: set_a,F: a > a > a,B: a > set_a_a] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ A )
         => ( member_a_a @ ( F @ X3 ) @ ( B @ X3 ) ) )
     => ( member_a_a_a2 @ F @ ( pi_a_a_a2 @ A @ B ) ) ) ).

% Pi_I
thf(fact_82_Pi__I,axiom,
    ! [A: set_nat,F: nat > nat > nat,B: nat > set_nat_nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A )
         => ( member_nat_nat @ ( F @ X3 ) @ ( B @ X3 ) ) )
     => ( member_nat_nat_nat2 @ F @ ( pi_nat_nat_nat2 @ A @ B ) ) ) ).

% Pi_I
thf(fact_83_sumset__subset__carrier,axiom,
    ! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) @ g ) ).

% sumset_subset_carrier
thf(fact_84_sumset__mono,axiom,
    ! [A6: set_a,A: set_a,B6: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ A6 @ A )
     => ( ( ord_less_eq_set_a @ B6 @ B )
       => ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A6 @ B6 ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) ) ) ) ).

% sumset_mono
thf(fact_85_inverse__equality,axiom,
    ! [U: a,V2: a] :
      ( ( ( addition @ U @ V2 )
        = zero )
     => ( ( ( addition @ V2 @ U )
          = zero )
       => ( ( member_a @ U @ g )
         => ( ( member_a @ V2 @ g )
           => ( ( group_inverse_a @ g @ addition @ zero @ U )
              = V2 ) ) ) ) ) ).

% inverse_equality
thf(fact_86_inverse__closed,axiom,
    ! [X: a] :
      ( ( member_a @ X @ g )
     => ( member_a @ ( group_inverse_a @ g @ addition @ zero @ X ) @ g ) ) ).

% inverse_closed
thf(fact_87_card__differenceset__commute,axiom,
    ! [B: set_a,A: set_a] :
      ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ B @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A ) ) )
      = ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ B ) ) ) ) ).

% card_differenceset_commute
thf(fact_88_minusset_Osimps,axiom,
    ! [A2: a,A: set_a] :
      ( ( member_a @ A2 @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A ) )
      = ( ? [A4: a] :
            ( ( A2
              = ( group_inverse_a @ g @ addition @ zero @ A4 ) )
            & ( member_a @ A4 @ A )
            & ( member_a @ A4 @ g ) ) ) ) ).

% minusset.simps
thf(fact_89_minusset_OminussetI,axiom,
    ! [A2: a,A: set_a] :
      ( ( member_a @ A2 @ A )
     => ( ( member_a @ A2 @ g )
       => ( member_a @ ( group_inverse_a @ g @ addition @ zero @ A2 ) @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A ) ) ) ) ).

% minusset.minussetI
thf(fact_90_minusset_Ocases,axiom,
    ! [A2: a,A: set_a] :
      ( ( member_a @ A2 @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A ) )
     => ~ ! [A3: a] :
            ( ( A2
              = ( group_inverse_a @ g @ addition @ zero @ A3 ) )
           => ( ( member_a @ A3 @ A )
             => ~ ( member_a @ A3 @ g ) ) ) ) ).

% minusset.cases
thf(fact_91_card__minusset_H,axiom,
    ! [A: set_a] :
      ( ( ord_less_eq_set_a @ A @ g )
     => ( ( finite_card_a @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A ) )
        = ( finite_card_a @ A ) ) ) ).

% card_minusset'
thf(fact_92_minusset__subset__carrier,axiom,
    ! [A: set_a] : ( ord_less_eq_set_a @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A ) @ g ) ).

% minusset_subset_carrier
thf(fact_93_card__sumset__iterated__minusset,axiom,
    ! [A: set_a,K: nat] :
      ( ( finite_card_a @ ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A ) @ K ) )
      = ( finite_card_a @ ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ A @ K ) ) ) ).

% card_sumset_iterated_minusset
thf(fact_94_finite__sumset__iterated,axiom,
    ! [A: set_a,R: nat] :
      ( ( finite_finite_a @ A )
     => ( finite_finite_a @ ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ A @ R ) ) ) ).

% finite_sumset_iterated
thf(fact_95_sumset__iterated__subset__carrier,axiom,
    ! [A: set_a,K: nat] : ( ord_less_eq_set_a @ ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ A @ K ) @ g ) ).

% sumset_iterated_subset_carrier
thf(fact_96_minussetp_Osimps,axiom,
    ! [A: a > $o,A2: a] :
      ( ( pluenn1126946703085653920setp_a @ g @ addition @ zero @ A @ A2 )
      = ( ? [A4: a] :
            ( ( A2
              = ( group_inverse_a @ g @ addition @ zero @ A4 ) )
            & ( A @ A4 )
            & ( member_a @ A4 @ g ) ) ) ) ).

% minussetp.simps
thf(fact_97_minussetp_OminussetI,axiom,
    ! [A: a > $o,A2: a] :
      ( ( A @ A2 )
     => ( ( member_a @ A2 @ g )
       => ( pluenn1126946703085653920setp_a @ g @ addition @ zero @ A @ ( group_inverse_a @ g @ addition @ zero @ A2 ) ) ) ) ).

% minussetp.minussetI
thf(fact_98_minussetp_Ocases,axiom,
    ! [A: a > $o,A2: a] :
      ( ( pluenn1126946703085653920setp_a @ g @ addition @ zero @ A @ A2 )
     => ~ ! [A3: a] :
            ( ( A2
              = ( group_inverse_a @ g @ addition @ zero @ A3 ) )
           => ( ( A @ A3 )
             => ~ ( member_a @ A3 @ g ) ) ) ) ).

% minussetp.cases
thf(fact_99_card__le__sumset,axiom,
    ! [A: set_a,A2: a,B: set_a] :
      ( ( finite_finite_a @ A )
     => ( ( member_a @ A2 @ A )
       => ( ( member_a @ A2 @ g )
         => ( ( finite_finite_a @ B )
           => ( ( ord_less_eq_set_a @ B @ g )
             => ( ord_less_eq_nat @ ( finite_card_a @ B ) @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) ) ) ) ) ) ) ) ).

% card_le_sumset
thf(fact_100_vw__eq,axiom,
    ! [X: a] :
      ( ( member_a @ X @ ( pluenn3038260743871226533mset_a @ g @ addition @ v @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ w ) ) )
     => ( ( addition @ ( v2 @ X ) @ ( group_inverse_a @ g @ addition @ zero @ ( w2 @ X ) ) )
        = X ) ) ).

% vw_eq
thf(fact_101_inverse__unit,axiom,
    ( ( group_inverse_a @ g @ addition @ zero @ zero )
    = zero ) ).

% inverse_unit
thf(fact_102_additive__abelian__group_Ointro,axiom,
    ! [G: set_a,Addition: a > a > a,Zero: a] :
      ( ( group_201663378560352916roup_a @ G @ Addition @ Zero )
     => ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero ) ) ).

% additive_abelian_group.intro
thf(fact_103_additive__abelian__group_Oaxioms,axiom,
    ! [G: set_a,Addition: a > a > a,Zero: a] :
      ( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
     => ( group_201663378560352916roup_a @ G @ Addition @ Zero ) ) ).

% additive_abelian_group.axioms
thf(fact_104_additive__abelian__group_Osumsetp_Ocases,axiom,
    ! [G: set_Pr8826267807999420763od_a_a,Addition: ( product_prod_a_a > product_prod_a_a ) > ( product_prod_a_a > product_prod_a_a ) > product_prod_a_a > product_prod_a_a,Zero: product_prod_a_a > product_prod_a_a,A: ( product_prod_a_a > product_prod_a_a ) > $o,B: ( product_prod_a_a > product_prod_a_a ) > $o,A2: product_prod_a_a > product_prod_a_a] :
      ( ( pluenn7398882097854948023od_a_a @ G @ Addition @ Zero )
     => ( ( pluenn6245878296707294014od_a_a @ G @ Addition @ A @ B @ A2 )
       => ~ ! [A3: product_prod_a_a > product_prod_a_a,B2: product_prod_a_a > product_prod_a_a] :
              ( ( A2
                = ( Addition @ A3 @ B2 ) )
             => ( ( A @ A3 )
               => ( ( member4020126937092221116od_a_a @ A3 @ G )
                 => ( ( B @ B2 )
                   => ~ ( member4020126937092221116od_a_a @ B2 @ G ) ) ) ) ) ) ) ).

% additive_abelian_group.sumsetp.cases
thf(fact_105_additive__abelian__group_Osumsetp_Ocases,axiom,
    ! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: nat > $o,B: nat > $o,A2: nat] :
      ( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
     => ( ( pluenn5670965976768739049tp_nat @ G @ Addition @ A @ B @ A2 )
       => ~ ! [A3: nat,B2: nat] :
              ( ( A2
                = ( Addition @ A3 @ B2 ) )
             => ( ( A @ A3 )
               => ( ( member_nat @ A3 @ G )
                 => ( ( B @ B2 )
                   => ~ ( member_nat @ B2 @ G ) ) ) ) ) ) ) ).

% additive_abelian_group.sumsetp.cases
thf(fact_106_additive__abelian__group_Osumsetp_Ocases,axiom,
    ! [G: set_nat_nat,Addition: ( nat > nat ) > ( nat > nat ) > nat > nat,Zero: nat > nat,A: ( nat > nat ) > $o,B: ( nat > nat ) > $o,A2: nat > nat] :
      ( ( pluenn2593311475829585105at_nat @ G @ Addition @ Zero )
     => ( ( pluenn8622197827967232728at_nat @ G @ Addition @ A @ B @ A2 )
       => ~ ! [A3: nat > nat,B2: nat > nat] :
              ( ( A2
                = ( Addition @ A3 @ B2 ) )
             => ( ( A @ A3 )
               => ( ( member_nat_nat @ A3 @ G )
                 => ( ( B @ B2 )
                   => ~ ( member_nat_nat @ B2 @ G ) ) ) ) ) ) ) ).

% additive_abelian_group.sumsetp.cases
thf(fact_107_additive__abelian__group_Osumsetp_Ocases,axiom,
    ! [G: set_nat_a,Addition: ( nat > a ) > ( nat > a ) > nat > a,Zero: nat > a,A: ( nat > a ) > $o,B: ( nat > a ) > $o,A2: nat > a] :
      ( ( pluenn1914567342357533651_nat_a @ G @ Addition @ Zero )
     => ( ( pluenn5759993693271462284_nat_a @ G @ Addition @ A @ B @ A2 )
       => ~ ! [A3: nat > a,B2: nat > a] :
              ( ( A2
                = ( Addition @ A3 @ B2 ) )
             => ( ( A @ A3 )
               => ( ( member_nat_a @ A3 @ G )
                 => ( ( B @ B2 )
                   => ~ ( member_nat_a @ B2 @ G ) ) ) ) ) ) ) ).

% additive_abelian_group.sumsetp.cases
thf(fact_108_additive__abelian__group_Osumsetp_Ocases,axiom,
    ! [G: set_a_6829686330177631172od_a_a,Addition: ( a > product_prod_a_a ) > ( a > product_prod_a_a ) > a > product_prod_a_a,Zero: a > product_prod_a_a,A: ( a > product_prod_a_a ) > $o,B: ( a > product_prod_a_a ) > $o,A2: a > product_prod_a_a] :
      ( ( pluenn3290985442670702752od_a_a @ G @ Addition @ Zero )
     => ( ( pluenn3409616955423707175od_a_a @ G @ Addition @ A @ B @ A2 )
       => ~ ! [A3: a > product_prod_a_a,B2: a > product_prod_a_a] :
              ( ( A2
                = ( Addition @ A3 @ B2 ) )
             => ( ( A @ A3 )
               => ( ( member1957775702407316389od_a_a @ A3 @ G )
                 => ( ( B @ B2 )
                   => ~ ( member1957775702407316389od_a_a @ B2 @ G ) ) ) ) ) ) ) ).

% additive_abelian_group.sumsetp.cases
thf(fact_109_additive__abelian__group_Osumsetp_Ocases,axiom,
    ! [G: set_a_nat,Addition: ( a > nat ) > ( a > nat ) > a > nat,Zero: a > nat,A: ( a > nat ) > $o,B: ( a > nat ) > $o,A2: a > nat] :
      ( ( pluenn7899775915488632493_a_nat @ G @ Addition @ Zero )
     => ( ( pluenn2521830229547785318_a_nat @ G @ Addition @ A @ B @ A2 )
       => ~ ! [A3: a > nat,B2: a > nat] :
              ( ( A2
                = ( Addition @ A3 @ B2 ) )
             => ( ( A @ A3 )
               => ( ( member_a_nat @ A3 @ G )
                 => ( ( B @ B2 )
                   => ~ ( member_a_nat @ B2 @ G ) ) ) ) ) ) ) ).

% additive_abelian_group.sumsetp.cases
thf(fact_110_additive__abelian__group_Osumsetp_Ocases,axiom,
    ! [G: set_a_a,Addition: ( a > a ) > ( a > a ) > a > a,Zero: a > a,A: ( a > a ) > $o,B: ( a > a ) > $o,A2: a > a] :
      ( ( pluenn1959071899916691703up_a_a @ G @ Addition @ Zero )
     => ( ( pluenn676121060179339134tp_a_a @ G @ Addition @ A @ B @ A2 )
       => ~ ! [A3: a > a,B2: a > a] :
              ( ( A2
                = ( Addition @ A3 @ B2 ) )
             => ( ( A @ A3 )
               => ( ( member_a_a @ A3 @ G )
                 => ( ( B @ B2 )
                   => ~ ( member_a_a @ B2 @ G ) ) ) ) ) ) ) ).

% additive_abelian_group.sumsetp.cases
thf(fact_111_additive__abelian__group_Osumsetp_Ocases,axiom,
    ! [G: set_a,Addition: a > a > a,Zero: a,A: a > $o,B: a > $o,A2: a] :
      ( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
     => ( ( pluenn895083305082786853setp_a @ G @ Addition @ A @ B @ A2 )
       => ~ ! [A3: a,B2: a] :
              ( ( A2
                = ( Addition @ A3 @ B2 ) )
             => ( ( A @ A3 )
               => ( ( member_a @ A3 @ G )
                 => ( ( B @ B2 )
                   => ~ ( member_a @ B2 @ G ) ) ) ) ) ) ) ).

% additive_abelian_group.sumsetp.cases
thf(fact_112_additive__abelian__group_Osumsetp_Osimps,axiom,
    ! [G: set_Pr8826267807999420763od_a_a,Addition: ( product_prod_a_a > product_prod_a_a ) > ( product_prod_a_a > product_prod_a_a ) > product_prod_a_a > product_prod_a_a,Zero: product_prod_a_a > product_prod_a_a,A: ( product_prod_a_a > product_prod_a_a ) > $o,B: ( product_prod_a_a > product_prod_a_a ) > $o,A2: product_prod_a_a > product_prod_a_a] :
      ( ( pluenn7398882097854948023od_a_a @ G @ Addition @ Zero )
     => ( ( pluenn6245878296707294014od_a_a @ G @ Addition @ A @ B @ A2 )
        = ( ? [A4: product_prod_a_a > product_prod_a_a,B3: product_prod_a_a > product_prod_a_a] :
              ( ( A2
                = ( Addition @ A4 @ B3 ) )
              & ( A @ A4 )
              & ( member4020126937092221116od_a_a @ A4 @ G )
              & ( B @ B3 )
              & ( member4020126937092221116od_a_a @ B3 @ G ) ) ) ) ) ).

% additive_abelian_group.sumsetp.simps
thf(fact_113_additive__abelian__group_Osumsetp_Osimps,axiom,
    ! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: nat > $o,B: nat > $o,A2: nat] :
      ( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
     => ( ( pluenn5670965976768739049tp_nat @ G @ Addition @ A @ B @ A2 )
        = ( ? [A4: nat,B3: nat] :
              ( ( A2
                = ( Addition @ A4 @ B3 ) )
              & ( A @ A4 )
              & ( member_nat @ A4 @ G )
              & ( B @ B3 )
              & ( member_nat @ B3 @ G ) ) ) ) ) ).

% additive_abelian_group.sumsetp.simps
thf(fact_114_additive__abelian__group_Osumsetp_Osimps,axiom,
    ! [G: set_nat_nat,Addition: ( nat > nat ) > ( nat > nat ) > nat > nat,Zero: nat > nat,A: ( nat > nat ) > $o,B: ( nat > nat ) > $o,A2: nat > nat] :
      ( ( pluenn2593311475829585105at_nat @ G @ Addition @ Zero )
     => ( ( pluenn8622197827967232728at_nat @ G @ Addition @ A @ B @ A2 )
        = ( ? [A4: nat > nat,B3: nat > nat] :
              ( ( A2
                = ( Addition @ A4 @ B3 ) )
              & ( A @ A4 )
              & ( member_nat_nat @ A4 @ G )
              & ( B @ B3 )
              & ( member_nat_nat @ B3 @ G ) ) ) ) ) ).

% additive_abelian_group.sumsetp.simps
thf(fact_115_additive__abelian__group_Osumsetp_Osimps,axiom,
    ! [G: set_nat_a,Addition: ( nat > a ) > ( nat > a ) > nat > a,Zero: nat > a,A: ( nat > a ) > $o,B: ( nat > a ) > $o,A2: nat > a] :
      ( ( pluenn1914567342357533651_nat_a @ G @ Addition @ Zero )
     => ( ( pluenn5759993693271462284_nat_a @ G @ Addition @ A @ B @ A2 )
        = ( ? [A4: nat > a,B3: nat > a] :
              ( ( A2
                = ( Addition @ A4 @ B3 ) )
              & ( A @ A4 )
              & ( member_nat_a @ A4 @ G )
              & ( B @ B3 )
              & ( member_nat_a @ B3 @ G ) ) ) ) ) ).

% additive_abelian_group.sumsetp.simps
thf(fact_116_additive__abelian__group_Osumsetp_Osimps,axiom,
    ! [G: set_a_6829686330177631172od_a_a,Addition: ( a > product_prod_a_a ) > ( a > product_prod_a_a ) > a > product_prod_a_a,Zero: a > product_prod_a_a,A: ( a > product_prod_a_a ) > $o,B: ( a > product_prod_a_a ) > $o,A2: a > product_prod_a_a] :
      ( ( pluenn3290985442670702752od_a_a @ G @ Addition @ Zero )
     => ( ( pluenn3409616955423707175od_a_a @ G @ Addition @ A @ B @ A2 )
        = ( ? [A4: a > product_prod_a_a,B3: a > product_prod_a_a] :
              ( ( A2
                = ( Addition @ A4 @ B3 ) )
              & ( A @ A4 )
              & ( member1957775702407316389od_a_a @ A4 @ G )
              & ( B @ B3 )
              & ( member1957775702407316389od_a_a @ B3 @ G ) ) ) ) ) ).

% additive_abelian_group.sumsetp.simps
thf(fact_117_additive__abelian__group_Osumsetp_Osimps,axiom,
    ! [G: set_a_nat,Addition: ( a > nat ) > ( a > nat ) > a > nat,Zero: a > nat,A: ( a > nat ) > $o,B: ( a > nat ) > $o,A2: a > nat] :
      ( ( pluenn7899775915488632493_a_nat @ G @ Addition @ Zero )
     => ( ( pluenn2521830229547785318_a_nat @ G @ Addition @ A @ B @ A2 )
        = ( ? [A4: a > nat,B3: a > nat] :
              ( ( A2
                = ( Addition @ A4 @ B3 ) )
              & ( A @ A4 )
              & ( member_a_nat @ A4 @ G )
              & ( B @ B3 )
              & ( member_a_nat @ B3 @ G ) ) ) ) ) ).

% additive_abelian_group.sumsetp.simps
thf(fact_118_additive__abelian__group_Osumsetp_Osimps,axiom,
    ! [G: set_a_a,Addition: ( a > a ) > ( a > a ) > a > a,Zero: a > a,A: ( a > a ) > $o,B: ( a > a ) > $o,A2: a > a] :
      ( ( pluenn1959071899916691703up_a_a @ G @ Addition @ Zero )
     => ( ( pluenn676121060179339134tp_a_a @ G @ Addition @ A @ B @ A2 )
        = ( ? [A4: a > a,B3: a > a] :
              ( ( A2
                = ( Addition @ A4 @ B3 ) )
              & ( A @ A4 )
              & ( member_a_a @ A4 @ G )
              & ( B @ B3 )
              & ( member_a_a @ B3 @ G ) ) ) ) ) ).

% additive_abelian_group.sumsetp.simps
thf(fact_119_additive__abelian__group_Osumsetp_Osimps,axiom,
    ! [G: set_a,Addition: a > a > a,Zero: a,A: a > $o,B: a > $o,A2: a] :
      ( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
     => ( ( pluenn895083305082786853setp_a @ G @ Addition @ A @ B @ A2 )
        = ( ? [A4: a,B3: a] :
              ( ( A2
                = ( Addition @ A4 @ B3 ) )
              & ( A @ A4 )
              & ( member_a @ A4 @ G )
              & ( B @ B3 )
              & ( member_a @ B3 @ G ) ) ) ) ) ).

% additive_abelian_group.sumsetp.simps
thf(fact_120_additive__abelian__group_Oinverse__closed,axiom,
    ! [G: set_Pr8826267807999420763od_a_a,Addition: ( product_prod_a_a > product_prod_a_a ) > ( product_prod_a_a > product_prod_a_a ) > product_prod_a_a > product_prod_a_a,Zero: product_prod_a_a > product_prod_a_a,X: product_prod_a_a > product_prod_a_a] :
      ( ( pluenn7398882097854948023od_a_a @ G @ Addition @ Zero )
     => ( ( member4020126937092221116od_a_a @ X @ G )
       => ( member4020126937092221116od_a_a @ ( group_1100041695951271920od_a_a @ G @ Addition @ Zero @ X ) @ G ) ) ) ).

% additive_abelian_group.inverse_closed
thf(fact_121_additive__abelian__group_Oinverse__closed,axiom,
    ! [G: set_nat,Addition: nat > nat > nat,Zero: nat,X: nat] :
      ( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
     => ( ( member_nat @ X @ G )
       => ( member_nat @ ( group_inverse_nat @ G @ Addition @ Zero @ X ) @ G ) ) ) ).

% additive_abelian_group.inverse_closed
thf(fact_122_additive__abelian__group_Oinverse__closed,axiom,
    ! [G: set_nat_nat,Addition: ( nat > nat ) > ( nat > nat ) > nat > nat,Zero: nat > nat,X: nat > nat] :
      ( ( pluenn2593311475829585105at_nat @ G @ Addition @ Zero )
     => ( ( member_nat_nat @ X @ G )
       => ( member_nat_nat @ ( group_5492157109111387274at_nat @ G @ Addition @ Zero @ X ) @ G ) ) ) ).

% additive_abelian_group.inverse_closed
thf(fact_123_additive__abelian__group_Oinverse__closed,axiom,
    ! [G: set_nat_a,Addition: ( nat > a ) > ( nat > a ) > nat > a,Zero: nat > a,X: nat > a] :
      ( ( pluenn1914567342357533651_nat_a @ G @ Addition @ Zero )
     => ( ( member_nat_a @ X @ G )
       => ( member_nat_a @ ( group_inverse_nat_a @ G @ Addition @ Zero @ X ) @ G ) ) ) ).

% additive_abelian_group.inverse_closed
thf(fact_124_additive__abelian__group_Oinverse__closed,axiom,
    ! [G: set_a_6829686330177631172od_a_a,Addition: ( a > product_prod_a_a ) > ( a > product_prod_a_a ) > a > product_prod_a_a,Zero: a > product_prod_a_a,X: a > product_prod_a_a] :
      ( ( pluenn3290985442670702752od_a_a @ G @ Addition @ Zero )
     => ( ( member1957775702407316389od_a_a @ X @ G )
       => ( member1957775702407316389od_a_a @ ( group_2336095992253589209od_a_a @ G @ Addition @ Zero @ X ) @ G ) ) ) ).

% additive_abelian_group.inverse_closed
thf(fact_125_additive__abelian__group_Oinverse__closed,axiom,
    ! [G: set_a_nat,Addition: ( a > nat ) > ( a > nat ) > a > nat,Zero: a > nat,X: a > nat] :
      ( ( pluenn7899775915488632493_a_nat @ G @ Addition @ Zero )
     => ( ( member_a_nat @ X @ G )
       => ( member_a_nat @ ( group_inverse_a_nat @ G @ Addition @ Zero @ X ) @ G ) ) ) ).

% additive_abelian_group.inverse_closed
thf(fact_126_additive__abelian__group_Oinverse__closed,axiom,
    ! [G: set_a_a,Addition: ( a > a ) > ( a > a ) > a > a,Zero: a > a,X: a > a] :
      ( ( pluenn1959071899916691703up_a_a @ G @ Addition @ Zero )
     => ( ( member_a_a @ X @ G )
       => ( member_a_a @ ( group_inverse_a_a @ G @ Addition @ Zero @ X ) @ G ) ) ) ).

% additive_abelian_group.inverse_closed
thf(fact_127_additive__abelian__group_Oinverse__closed,axiom,
    ! [G: set_a,Addition: a > a > a,Zero: a,X: a] :
      ( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
     => ( ( member_a @ X @ G )
       => ( member_a @ ( group_inverse_a @ G @ Addition @ Zero @ X ) @ G ) ) ) ).

% additive_abelian_group.inverse_closed
thf(fact_128_additive__abelian__group_Ominussetp_Ocases,axiom,
    ! [G: set_Pr8826267807999420763od_a_a,Addition: ( product_prod_a_a > product_prod_a_a ) > ( product_prod_a_a > product_prod_a_a ) > product_prod_a_a > product_prod_a_a,Zero: product_prod_a_a > product_prod_a_a,A: ( product_prod_a_a > product_prod_a_a ) > $o,A2: product_prod_a_a > product_prod_a_a] :
      ( ( pluenn7398882097854948023od_a_a @ G @ Addition @ Zero )
     => ( ( pluenn2777150465843325699od_a_a @ G @ Addition @ Zero @ A @ A2 )
       => ~ ! [A3: product_prod_a_a > product_prod_a_a] :
              ( ( A2
                = ( group_1100041695951271920od_a_a @ G @ Addition @ Zero @ A3 ) )
             => ( ( A @ A3 )
               => ~ ( member4020126937092221116od_a_a @ A3 @ G ) ) ) ) ) ).

% additive_abelian_group.minussetp.cases
thf(fact_129_additive__abelian__group_Ominussetp_Ocases,axiom,
    ! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: nat > $o,A2: nat] :
      ( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
     => ( ( pluenn8372939692575285934tp_nat @ G @ Addition @ Zero @ A @ A2 )
       => ~ ! [A3: nat] :
              ( ( A2
                = ( group_inverse_nat @ G @ Addition @ Zero @ A3 ) )
             => ( ( A @ A3 )
               => ~ ( member_nat @ A3 @ G ) ) ) ) ) ).

% additive_abelian_group.minussetp.cases
thf(fact_130_additive__abelian__group_Ominussetp_Ocases,axiom,
    ! [G: set_nat_nat,Addition: ( nat > nat ) > ( nat > nat ) > nat > nat,Zero: nat > nat,A: ( nat > nat ) > $o,A2: nat > nat] :
      ( ( pluenn2593311475829585105at_nat @ G @ Addition @ Zero )
     => ( ( pluenn6500390917429955357at_nat @ G @ Addition @ Zero @ A @ A2 )
       => ~ ! [A3: nat > nat] :
              ( ( A2
                = ( group_5492157109111387274at_nat @ G @ Addition @ Zero @ A3 ) )
             => ( ( A @ A3 )
               => ~ ( member_nat_nat @ A3 @ G ) ) ) ) ) ).

% additive_abelian_group.minussetp.cases
thf(fact_131_additive__abelian__group_Ominussetp_Ocases,axiom,
    ! [G: set_nat_a,Addition: ( nat > a ) > ( nat > a ) > nat > a,Zero: nat > a,A: ( nat > a ) > $o,A2: nat > a] :
      ( ( pluenn1914567342357533651_nat_a @ G @ Addition @ Zero )
     => ( ( pluenn2964913278637471879_nat_a @ G @ Addition @ Zero @ A @ A2 )
       => ~ ! [A3: nat > a] :
              ( ( A2
                = ( group_inverse_nat_a @ G @ Addition @ Zero @ A3 ) )
             => ( ( A @ A3 )
               => ~ ( member_nat_a @ A3 @ G ) ) ) ) ) ).

% additive_abelian_group.minussetp.cases
thf(fact_132_additive__abelian__group_Ominussetp_Ocases,axiom,
    ! [G: set_a_6829686330177631172od_a_a,Addition: ( a > product_prod_a_a ) > ( a > product_prod_a_a ) > a > product_prod_a_a,Zero: a > product_prod_a_a,A: ( a > product_prod_a_a ) > $o,A2: a > product_prod_a_a] :
      ( ( pluenn3290985442670702752od_a_a @ G @ Addition @ Zero )
     => ( ( pluenn5763864222185776364od_a_a @ G @ Addition @ Zero @ A @ A2 )
       => ~ ! [A3: a > product_prod_a_a] :
              ( ( A2
                = ( group_2336095992253589209od_a_a @ G @ Addition @ Zero @ A3 ) )
             => ( ( A @ A3 )
               => ~ ( member1957775702407316389od_a_a @ A3 @ G ) ) ) ) ) ).

% additive_abelian_group.minussetp.cases
thf(fact_133_additive__abelian__group_Ominussetp_Ocases,axiom,
    ! [G: set_a_nat,Addition: ( a > nat ) > ( a > nat ) > a > nat,Zero: a > nat,A: ( a > nat ) > $o,A2: a > nat] :
      ( ( pluenn7899775915488632493_a_nat @ G @ Addition @ Zero )
     => ( ( pluenn8950121851768570721_a_nat @ G @ Addition @ Zero @ A @ A2 )
       => ~ ! [A3: a > nat] :
              ( ( A2
                = ( group_inverse_a_nat @ G @ Addition @ Zero @ A3 ) )
             => ( ( A @ A3 )
               => ~ ( member_a_nat @ A3 @ G ) ) ) ) ) ).

% additive_abelian_group.minussetp.cases
thf(fact_134_additive__abelian__group_Ominussetp_Ocases,axiom,
    ! [G: set_a_a,Addition: ( a > a ) > ( a > a ) > a > a,Zero: a > a,A: ( a > a ) > $o,A2: a > a] :
      ( ( pluenn1959071899916691703up_a_a @ G @ Addition @ Zero )
     => ( ( pluenn1140692617289730371tp_a_a @ G @ Addition @ Zero @ A @ A2 )
       => ~ ! [A3: a > a] :
              ( ( A2
                = ( group_inverse_a_a @ G @ Addition @ Zero @ A3 ) )
             => ( ( A @ A3 )
               => ~ ( member_a_a @ A3 @ G ) ) ) ) ) ).

% additive_abelian_group.minussetp.cases
thf(fact_135_additive__abelian__group_Ominussetp_Ocases,axiom,
    ! [G: set_a,Addition: a > a > a,Zero: a,A: a > $o,A2: a] :
      ( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
     => ( ( pluenn1126946703085653920setp_a @ G @ Addition @ Zero @ A @ A2 )
       => ~ ! [A3: a] :
              ( ( A2
                = ( group_inverse_a @ G @ Addition @ Zero @ A3 ) )
             => ( ( A @ A3 )
               => ~ ( member_a @ A3 @ G ) ) ) ) ) ).

% additive_abelian_group.minussetp.cases
thf(fact_136_additive__abelian__group_Ominussetp_Osimps,axiom,
    ! [G: set_Pr8826267807999420763od_a_a,Addition: ( product_prod_a_a > product_prod_a_a ) > ( product_prod_a_a > product_prod_a_a ) > product_prod_a_a > product_prod_a_a,Zero: product_prod_a_a > product_prod_a_a,A: ( product_prod_a_a > product_prod_a_a ) > $o,A2: product_prod_a_a > product_prod_a_a] :
      ( ( pluenn7398882097854948023od_a_a @ G @ Addition @ Zero )
     => ( ( pluenn2777150465843325699od_a_a @ G @ Addition @ Zero @ A @ A2 )
        = ( ? [A4: product_prod_a_a > product_prod_a_a] :
              ( ( A2
                = ( group_1100041695951271920od_a_a @ G @ Addition @ Zero @ A4 ) )
              & ( A @ A4 )
              & ( member4020126937092221116od_a_a @ A4 @ G ) ) ) ) ) ).

% additive_abelian_group.minussetp.simps
thf(fact_137_additive__abelian__group_Ominussetp_Osimps,axiom,
    ! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: nat > $o,A2: nat] :
      ( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
     => ( ( pluenn8372939692575285934tp_nat @ G @ Addition @ Zero @ A @ A2 )
        = ( ? [A4: nat] :
              ( ( A2
                = ( group_inverse_nat @ G @ Addition @ Zero @ A4 ) )
              & ( A @ A4 )
              & ( member_nat @ A4 @ G ) ) ) ) ) ).

% additive_abelian_group.minussetp.simps
thf(fact_138_additive__abelian__group_Ominussetp_Osimps,axiom,
    ! [G: set_nat_nat,Addition: ( nat > nat ) > ( nat > nat ) > nat > nat,Zero: nat > nat,A: ( nat > nat ) > $o,A2: nat > nat] :
      ( ( pluenn2593311475829585105at_nat @ G @ Addition @ Zero )
     => ( ( pluenn6500390917429955357at_nat @ G @ Addition @ Zero @ A @ A2 )
        = ( ? [A4: nat > nat] :
              ( ( A2
                = ( group_5492157109111387274at_nat @ G @ Addition @ Zero @ A4 ) )
              & ( A @ A4 )
              & ( member_nat_nat @ A4 @ G ) ) ) ) ) ).

% additive_abelian_group.minussetp.simps
thf(fact_139_additive__abelian__group_Ominussetp_Osimps,axiom,
    ! [G: set_nat_a,Addition: ( nat > a ) > ( nat > a ) > nat > a,Zero: nat > a,A: ( nat > a ) > $o,A2: nat > a] :
      ( ( pluenn1914567342357533651_nat_a @ G @ Addition @ Zero )
     => ( ( pluenn2964913278637471879_nat_a @ G @ Addition @ Zero @ A @ A2 )
        = ( ? [A4: nat > a] :
              ( ( A2
                = ( group_inverse_nat_a @ G @ Addition @ Zero @ A4 ) )
              & ( A @ A4 )
              & ( member_nat_a @ A4 @ G ) ) ) ) ) ).

% additive_abelian_group.minussetp.simps
thf(fact_140_additive__abelian__group_Ominussetp_Osimps,axiom,
    ! [G: set_a_6829686330177631172od_a_a,Addition: ( a > product_prod_a_a ) > ( a > product_prod_a_a ) > a > product_prod_a_a,Zero: a > product_prod_a_a,A: ( a > product_prod_a_a ) > $o,A2: a > product_prod_a_a] :
      ( ( pluenn3290985442670702752od_a_a @ G @ Addition @ Zero )
     => ( ( pluenn5763864222185776364od_a_a @ G @ Addition @ Zero @ A @ A2 )
        = ( ? [A4: a > product_prod_a_a] :
              ( ( A2
                = ( group_2336095992253589209od_a_a @ G @ Addition @ Zero @ A4 ) )
              & ( A @ A4 )
              & ( member1957775702407316389od_a_a @ A4 @ G ) ) ) ) ) ).

% additive_abelian_group.minussetp.simps
thf(fact_141_additive__abelian__group_Ominussetp_Osimps,axiom,
    ! [G: set_a_nat,Addition: ( a > nat ) > ( a > nat ) > a > nat,Zero: a > nat,A: ( a > nat ) > $o,A2: a > nat] :
      ( ( pluenn7899775915488632493_a_nat @ G @ Addition @ Zero )
     => ( ( pluenn8950121851768570721_a_nat @ G @ Addition @ Zero @ A @ A2 )
        = ( ? [A4: a > nat] :
              ( ( A2
                = ( group_inverse_a_nat @ G @ Addition @ Zero @ A4 ) )
              & ( A @ A4 )
              & ( member_a_nat @ A4 @ G ) ) ) ) ) ).

% additive_abelian_group.minussetp.simps
thf(fact_142_additive__abelian__group_Ominussetp_Osimps,axiom,
    ! [G: set_a_a,Addition: ( a > a ) > ( a > a ) > a > a,Zero: a > a,A: ( a > a ) > $o,A2: a > a] :
      ( ( pluenn1959071899916691703up_a_a @ G @ Addition @ Zero )
     => ( ( pluenn1140692617289730371tp_a_a @ G @ Addition @ Zero @ A @ A2 )
        = ( ? [A4: a > a] :
              ( ( A2
                = ( group_inverse_a_a @ G @ Addition @ Zero @ A4 ) )
              & ( A @ A4 )
              & ( member_a_a @ A4 @ G ) ) ) ) ) ).

% additive_abelian_group.minussetp.simps
thf(fact_143_additive__abelian__group_Ominussetp_Osimps,axiom,
    ! [G: set_a,Addition: a > a > a,Zero: a,A: a > $o,A2: a] :
      ( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
     => ( ( pluenn1126946703085653920setp_a @ G @ Addition @ Zero @ A @ A2 )
        = ( ? [A4: a] :
              ( ( A2
                = ( group_inverse_a @ G @ Addition @ Zero @ A4 ) )
              & ( A @ A4 )
              & ( member_a @ A4 @ G ) ) ) ) ) ).

% additive_abelian_group.minussetp.simps
thf(fact_144_additive__abelian__group_Osumsetp_OsumsetI,axiom,
    ! [G: set_Pr8826267807999420763od_a_a,Addition: ( product_prod_a_a > product_prod_a_a ) > ( product_prod_a_a > product_prod_a_a ) > product_prod_a_a > product_prod_a_a,Zero: product_prod_a_a > product_prod_a_a,A: ( product_prod_a_a > product_prod_a_a ) > $o,A2: product_prod_a_a > product_prod_a_a,B: ( product_prod_a_a > product_prod_a_a ) > $o,B4: product_prod_a_a > product_prod_a_a] :
      ( ( pluenn7398882097854948023od_a_a @ G @ Addition @ Zero )
     => ( ( A @ A2 )
       => ( ( member4020126937092221116od_a_a @ A2 @ G )
         => ( ( B @ B4 )
           => ( ( member4020126937092221116od_a_a @ B4 @ G )
             => ( pluenn6245878296707294014od_a_a @ G @ Addition @ A @ B @ ( Addition @ A2 @ B4 ) ) ) ) ) ) ) ).

% additive_abelian_group.sumsetp.sumsetI
thf(fact_145_additive__abelian__group_Osumsetp_OsumsetI,axiom,
    ! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: nat > $o,A2: nat,B: nat > $o,B4: nat] :
      ( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
     => ( ( A @ A2 )
       => ( ( member_nat @ A2 @ G )
         => ( ( B @ B4 )
           => ( ( member_nat @ B4 @ G )
             => ( pluenn5670965976768739049tp_nat @ G @ Addition @ A @ B @ ( Addition @ A2 @ B4 ) ) ) ) ) ) ) ).

% additive_abelian_group.sumsetp.sumsetI
thf(fact_146_additive__abelian__group_Osumsetp_OsumsetI,axiom,
    ! [G: set_nat_nat,Addition: ( nat > nat ) > ( nat > nat ) > nat > nat,Zero: nat > nat,A: ( nat > nat ) > $o,A2: nat > nat,B: ( nat > nat ) > $o,B4: nat > nat] :
      ( ( pluenn2593311475829585105at_nat @ G @ Addition @ Zero )
     => ( ( A @ A2 )
       => ( ( member_nat_nat @ A2 @ G )
         => ( ( B @ B4 )
           => ( ( member_nat_nat @ B4 @ G )
             => ( pluenn8622197827967232728at_nat @ G @ Addition @ A @ B @ ( Addition @ A2 @ B4 ) ) ) ) ) ) ) ).

% additive_abelian_group.sumsetp.sumsetI
thf(fact_147_additive__abelian__group_Osumsetp_OsumsetI,axiom,
    ! [G: set_nat_a,Addition: ( nat > a ) > ( nat > a ) > nat > a,Zero: nat > a,A: ( nat > a ) > $o,A2: nat > a,B: ( nat > a ) > $o,B4: nat > a] :
      ( ( pluenn1914567342357533651_nat_a @ G @ Addition @ Zero )
     => ( ( A @ A2 )
       => ( ( member_nat_a @ A2 @ G )
         => ( ( B @ B4 )
           => ( ( member_nat_a @ B4 @ G )
             => ( pluenn5759993693271462284_nat_a @ G @ Addition @ A @ B @ ( Addition @ A2 @ B4 ) ) ) ) ) ) ) ).

% additive_abelian_group.sumsetp.sumsetI
thf(fact_148_additive__abelian__group_Osumsetp_OsumsetI,axiom,
    ! [G: set_a_6829686330177631172od_a_a,Addition: ( a > product_prod_a_a ) > ( a > product_prod_a_a ) > a > product_prod_a_a,Zero: a > product_prod_a_a,A: ( a > product_prod_a_a ) > $o,A2: a > product_prod_a_a,B: ( a > product_prod_a_a ) > $o,B4: a > product_prod_a_a] :
      ( ( pluenn3290985442670702752od_a_a @ G @ Addition @ Zero )
     => ( ( A @ A2 )
       => ( ( member1957775702407316389od_a_a @ A2 @ G )
         => ( ( B @ B4 )
           => ( ( member1957775702407316389od_a_a @ B4 @ G )
             => ( pluenn3409616955423707175od_a_a @ G @ Addition @ A @ B @ ( Addition @ A2 @ B4 ) ) ) ) ) ) ) ).

% additive_abelian_group.sumsetp.sumsetI
thf(fact_149_additive__abelian__group_Osumsetp_OsumsetI,axiom,
    ! [G: set_a_nat,Addition: ( a > nat ) > ( a > nat ) > a > nat,Zero: a > nat,A: ( a > nat ) > $o,A2: a > nat,B: ( a > nat ) > $o,B4: a > nat] :
      ( ( pluenn7899775915488632493_a_nat @ G @ Addition @ Zero )
     => ( ( A @ A2 )
       => ( ( member_a_nat @ A2 @ G )
         => ( ( B @ B4 )
           => ( ( member_a_nat @ B4 @ G )
             => ( pluenn2521830229547785318_a_nat @ G @ Addition @ A @ B @ ( Addition @ A2 @ B4 ) ) ) ) ) ) ) ).

% additive_abelian_group.sumsetp.sumsetI
thf(fact_150_additive__abelian__group_Osumsetp_OsumsetI,axiom,
    ! [G: set_a_a,Addition: ( a > a ) > ( a > a ) > a > a,Zero: a > a,A: ( a > a ) > $o,A2: a > a,B: ( a > a ) > $o,B4: a > a] :
      ( ( pluenn1959071899916691703up_a_a @ G @ Addition @ Zero )
     => ( ( A @ A2 )
       => ( ( member_a_a @ A2 @ G )
         => ( ( B @ B4 )
           => ( ( member_a_a @ B4 @ G )
             => ( pluenn676121060179339134tp_a_a @ G @ Addition @ A @ B @ ( Addition @ A2 @ B4 ) ) ) ) ) ) ) ).

% additive_abelian_group.sumsetp.sumsetI
thf(fact_151_additive__abelian__group_Osumsetp_OsumsetI,axiom,
    ! [G: set_a,Addition: a > a > a,Zero: a,A: a > $o,A2: a,B: a > $o,B4: a] :
      ( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
     => ( ( A @ A2 )
       => ( ( member_a @ A2 @ G )
         => ( ( B @ B4 )
           => ( ( member_a @ B4 @ G )
             => ( pluenn895083305082786853setp_a @ G @ Addition @ A @ B @ ( Addition @ A2 @ B4 ) ) ) ) ) ) ) ).

% additive_abelian_group.sumsetp.sumsetI
thf(fact_152_additive__abelian__group_Ominussetp_OminussetI,axiom,
    ! [G: set_Pr8826267807999420763od_a_a,Addition: ( product_prod_a_a > product_prod_a_a ) > ( product_prod_a_a > product_prod_a_a ) > product_prod_a_a > product_prod_a_a,Zero: product_prod_a_a > product_prod_a_a,A: ( product_prod_a_a > product_prod_a_a ) > $o,A2: product_prod_a_a > product_prod_a_a] :
      ( ( pluenn7398882097854948023od_a_a @ G @ Addition @ Zero )
     => ( ( A @ A2 )
       => ( ( member4020126937092221116od_a_a @ A2 @ G )
         => ( pluenn2777150465843325699od_a_a @ G @ Addition @ Zero @ A @ ( group_1100041695951271920od_a_a @ G @ Addition @ Zero @ A2 ) ) ) ) ) ).

% additive_abelian_group.minussetp.minussetI
thf(fact_153_additive__abelian__group_Ominussetp_OminussetI,axiom,
    ! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: nat > $o,A2: nat] :
      ( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
     => ( ( A @ A2 )
       => ( ( member_nat @ A2 @ G )
         => ( pluenn8372939692575285934tp_nat @ G @ Addition @ Zero @ A @ ( group_inverse_nat @ G @ Addition @ Zero @ A2 ) ) ) ) ) ).

% additive_abelian_group.minussetp.minussetI
thf(fact_154_additive__abelian__group_Ominussetp_OminussetI,axiom,
    ! [G: set_nat_nat,Addition: ( nat > nat ) > ( nat > nat ) > nat > nat,Zero: nat > nat,A: ( nat > nat ) > $o,A2: nat > nat] :
      ( ( pluenn2593311475829585105at_nat @ G @ Addition @ Zero )
     => ( ( A @ A2 )
       => ( ( member_nat_nat @ A2 @ G )
         => ( pluenn6500390917429955357at_nat @ G @ Addition @ Zero @ A @ ( group_5492157109111387274at_nat @ G @ Addition @ Zero @ A2 ) ) ) ) ) ).

% additive_abelian_group.minussetp.minussetI
thf(fact_155_additive__abelian__group_Ominussetp_OminussetI,axiom,
    ! [G: set_nat_a,Addition: ( nat > a ) > ( nat > a ) > nat > a,Zero: nat > a,A: ( nat > a ) > $o,A2: nat > a] :
      ( ( pluenn1914567342357533651_nat_a @ G @ Addition @ Zero )
     => ( ( A @ A2 )
       => ( ( member_nat_a @ A2 @ G )
         => ( pluenn2964913278637471879_nat_a @ G @ Addition @ Zero @ A @ ( group_inverse_nat_a @ G @ Addition @ Zero @ A2 ) ) ) ) ) ).

% additive_abelian_group.minussetp.minussetI
thf(fact_156_additive__abelian__group_Ominussetp_OminussetI,axiom,
    ! [G: set_a_6829686330177631172od_a_a,Addition: ( a > product_prod_a_a ) > ( a > product_prod_a_a ) > a > product_prod_a_a,Zero: a > product_prod_a_a,A: ( a > product_prod_a_a ) > $o,A2: a > product_prod_a_a] :
      ( ( pluenn3290985442670702752od_a_a @ G @ Addition @ Zero )
     => ( ( A @ A2 )
       => ( ( member1957775702407316389od_a_a @ A2 @ G )
         => ( pluenn5763864222185776364od_a_a @ G @ Addition @ Zero @ A @ ( group_2336095992253589209od_a_a @ G @ Addition @ Zero @ A2 ) ) ) ) ) ).

% additive_abelian_group.minussetp.minussetI
thf(fact_157_additive__abelian__group_Ominussetp_OminussetI,axiom,
    ! [G: set_a_nat,Addition: ( a > nat ) > ( a > nat ) > a > nat,Zero: a > nat,A: ( a > nat ) > $o,A2: a > nat] :
      ( ( pluenn7899775915488632493_a_nat @ G @ Addition @ Zero )
     => ( ( A @ A2 )
       => ( ( member_a_nat @ A2 @ G )
         => ( pluenn8950121851768570721_a_nat @ G @ Addition @ Zero @ A @ ( group_inverse_a_nat @ G @ Addition @ Zero @ A2 ) ) ) ) ) ).

% additive_abelian_group.minussetp.minussetI
thf(fact_158_additive__abelian__group_Ominussetp_OminussetI,axiom,
    ! [G: set_a_a,Addition: ( a > a ) > ( a > a ) > a > a,Zero: a > a,A: ( a > a ) > $o,A2: a > a] :
      ( ( pluenn1959071899916691703up_a_a @ G @ Addition @ Zero )
     => ( ( A @ A2 )
       => ( ( member_a_a @ A2 @ G )
         => ( pluenn1140692617289730371tp_a_a @ G @ Addition @ Zero @ A @ ( group_inverse_a_a @ G @ Addition @ Zero @ A2 ) ) ) ) ) ).

% additive_abelian_group.minussetp.minussetI
thf(fact_159_additive__abelian__group_Ominussetp_OminussetI,axiom,
    ! [G: set_a,Addition: a > a > a,Zero: a,A: a > $o,A2: a] :
      ( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
     => ( ( A @ A2 )
       => ( ( member_a @ A2 @ G )
         => ( pluenn1126946703085653920setp_a @ G @ Addition @ Zero @ A @ ( group_inverse_a @ G @ Addition @ Zero @ A2 ) ) ) ) ) ).

% additive_abelian_group.minussetp.minussetI
thf(fact_160_additive__abelian__group_Osumset__iterated_Ocong,axiom,
    pluenn1960970773371692859ated_a = pluenn1960970773371692859ated_a ).

% additive_abelian_group.sumset_iterated.cong
thf(fact_161_additive__abelian__group_Ofinite__sumset__iterated,axiom,
    ! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: set_nat,R: nat] :
      ( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
     => ( ( finite_finite_nat @ A )
       => ( finite_finite_nat @ ( pluenn7055013279391836755ed_nat @ G @ Addition @ Zero @ A @ R ) ) ) ) ).

% additive_abelian_group.finite_sumset_iterated
thf(fact_162_additive__abelian__group_Ofinite__sumset__iterated,axiom,
    ! [G: set_Product_prod_a_a,Addition: product_prod_a_a > product_prod_a_a > product_prod_a_a,Zero: product_prod_a_a,A: set_Product_prod_a_a,R: nat] :
      ( ( pluenn6837661381354551573od_a_a @ G @ Addition @ Zero )
     => ( ( finite6544458595007987280od_a_a @ A )
       => ( finite6544458595007987280od_a_a @ ( pluenn2164957731648007268od_a_a @ G @ Addition @ Zero @ A @ R ) ) ) ) ).

% additive_abelian_group.finite_sumset_iterated
thf(fact_163_additive__abelian__group_Ofinite__sumset__iterated,axiom,
    ! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,R: nat] :
      ( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
     => ( ( finite_finite_a @ A )
       => ( finite_finite_a @ ( pluenn1960970773371692859ated_a @ G @ Addition @ Zero @ A @ R ) ) ) ) ).

% additive_abelian_group.finite_sumset_iterated
thf(fact_164_additive__abelian__group_Osumset__iterated__subset__carrier,axiom,
    ! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,K: nat] :
      ( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
     => ( ord_less_eq_set_a @ ( pluenn1960970773371692859ated_a @ G @ Addition @ Zero @ A @ K ) @ G ) ) ).

% additive_abelian_group.sumset_iterated_subset_carrier
thf(fact_165_PiE,axiom,
    ! [F: nat > nat,A: set_nat,B: nat > set_nat,X: nat] :
      ( ( member_nat_nat @ F @ ( pi_nat_nat @ A @ B ) )
     => ( ~ ( member_nat @ ( F @ X ) @ ( B @ X ) )
       => ~ ( member_nat @ X @ A ) ) ) ).

% PiE
thf(fact_166_PiE,axiom,
    ! [F: nat > a,A: set_nat,B: nat > set_a,X: nat] :
      ( ( member_nat_a @ F @ ( pi_nat_a @ A @ B ) )
     => ( ~ ( member_a @ ( F @ X ) @ ( B @ X ) )
       => ~ ( member_nat @ X @ A ) ) ) ).

% PiE
thf(fact_167_PiE,axiom,
    ! [F: a > nat,A: set_a,B: a > set_nat,X: a] :
      ( ( member_a_nat @ F @ ( pi_a_nat @ A @ B ) )
     => ( ~ ( member_nat @ ( F @ X ) @ ( B @ X ) )
       => ~ ( member_a @ X @ A ) ) ) ).

% PiE
thf(fact_168_PiE,axiom,
    ! [F: a > a,A: set_a,B: a > set_a,X: a] :
      ( ( member_a_a @ F @ ( pi_a_a @ A @ B ) )
     => ( ~ ( member_a @ ( F @ X ) @ ( B @ X ) )
       => ~ ( member_a @ X @ A ) ) ) ).

% PiE
thf(fact_169_PiE,axiom,
    ! [F: ( nat > nat ) > a,A: set_nat_nat,B: ( nat > nat ) > set_a,X: nat > nat] :
      ( ( member_nat_nat_a @ F @ ( pi_nat_nat_a @ A @ B ) )
     => ( ~ ( member_a @ ( F @ X ) @ ( B @ X ) )
       => ~ ( member_nat_nat @ X @ A ) ) ) ).

% PiE
thf(fact_170_PiE,axiom,
    ! [F: ( nat > a ) > a,A: set_nat_a,B: ( nat > a ) > set_a,X: nat > a] :
      ( ( member_nat_a_a @ F @ ( pi_nat_a_a @ A @ B ) )
     => ( ~ ( member_a @ ( F @ X ) @ ( B @ X ) )
       => ~ ( member_nat_a @ X @ A ) ) ) ).

% PiE
thf(fact_171_PiE,axiom,
    ! [F: ( a > nat ) > a,A: set_a_nat,B: ( a > nat ) > set_a,X: a > nat] :
      ( ( member_a_nat_a @ F @ ( pi_a_nat_a @ A @ B ) )
     => ( ~ ( member_a @ ( F @ X ) @ ( B @ X ) )
       => ~ ( member_a_nat @ X @ A ) ) ) ).

% PiE
thf(fact_172_PiE,axiom,
    ! [F: ( a > a ) > a,A: set_a_a,B: ( a > a ) > set_a,X: a > a] :
      ( ( member_a_a_a @ F @ ( pi_a_a_a @ A @ B ) )
     => ( ~ ( member_a @ ( F @ X ) @ ( B @ X ) )
       => ~ ( member_a_a @ X @ A ) ) ) ).

% PiE
thf(fact_173_PiE,axiom,
    ! [F: ( nat > nat ) > nat,A: set_nat_nat,B: ( nat > nat ) > set_nat,X: nat > nat] :
      ( ( member_nat_nat_nat @ F @ ( pi_nat_nat_nat @ A @ B ) )
     => ( ~ ( member_nat @ ( F @ X ) @ ( B @ X ) )
       => ~ ( member_nat_nat @ X @ A ) ) ) ).

% PiE
thf(fact_174_PiE,axiom,
    ! [F: ( nat > a ) > nat,A: set_nat_a,B: ( nat > a ) > set_nat,X: nat > a] :
      ( ( member_nat_a_nat @ F @ ( pi_nat_a_nat @ A @ B ) )
     => ( ~ ( member_nat @ ( F @ X ) @ ( B @ X ) )
       => ~ ( member_nat_a @ X @ A ) ) ) ).

% PiE
thf(fact_175_Pi__I_H,axiom,
    ! [A: set_a,F: a > a,B: a > set_a] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ A )
         => ( member_a @ ( F @ X3 ) @ ( B @ X3 ) ) )
     => ( member_a_a @ F @ ( pi_a_a @ A @ B ) ) ) ).

% Pi_I'
thf(fact_176_Pi__I_H,axiom,
    ! [A: set_a,F: a > nat,B: a > set_nat] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ A )
         => ( member_nat @ ( F @ X3 ) @ ( B @ X3 ) ) )
     => ( member_a_nat @ F @ ( pi_a_nat @ A @ B ) ) ) ).

% Pi_I'
thf(fact_177_Pi__I_H,axiom,
    ! [A: set_nat,F: nat > a,B: nat > set_a] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A )
         => ( member_a @ ( F @ X3 ) @ ( B @ X3 ) ) )
     => ( member_nat_a @ F @ ( pi_nat_a @ A @ B ) ) ) ).

% Pi_I'
thf(fact_178_Pi__I_H,axiom,
    ! [A: set_nat,F: nat > nat,B: nat > set_nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A )
         => ( member_nat @ ( F @ X3 ) @ ( B @ X3 ) ) )
     => ( member_nat_nat @ F @ ( pi_nat_nat @ A @ B ) ) ) ).

% Pi_I'
thf(fact_179_Pi__I_H,axiom,
    ! [A: set_a,F: a > product_prod_a_a,B: a > set_Product_prod_a_a] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ A )
         => ( member1426531477525435216od_a_a @ ( F @ X3 ) @ ( B @ X3 ) ) )
     => ( member1957775702407316389od_a_a @ F @ ( pi_a_P2178097759547960436od_a_a @ A @ B ) ) ) ).

% Pi_I'
thf(fact_180_Pi__I_H,axiom,
    ! [A: set_a,F: a > nat > nat,B: a > set_nat_nat] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ A )
         => ( member_nat_nat @ ( F @ X3 ) @ ( B @ X3 ) ) )
     => ( member_a_nat_nat @ F @ ( pi_a_nat_nat @ A @ B ) ) ) ).

% Pi_I'
thf(fact_181_Pi__I_H,axiom,
    ! [A: set_a,F: a > nat > a,B: a > set_nat_a] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ A )
         => ( member_nat_a @ ( F @ X3 ) @ ( B @ X3 ) ) )
     => ( member_a_nat_a2 @ F @ ( pi_a_nat_a2 @ A @ B ) ) ) ).

% Pi_I'
thf(fact_182_Pi__I_H,axiom,
    ! [A: set_a,F: a > a > nat,B: a > set_a_nat] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ A )
         => ( member_a_nat @ ( F @ X3 ) @ ( B @ X3 ) ) )
     => ( member_a_a_nat @ F @ ( pi_a_a_nat @ A @ B ) ) ) ).

% Pi_I'
thf(fact_183_Pi__I_H,axiom,
    ! [A: set_a,F: a > a > a,B: a > set_a_a] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ A )
         => ( member_a_a @ ( F @ X3 ) @ ( B @ X3 ) ) )
     => ( member_a_a_a2 @ F @ ( pi_a_a_a2 @ A @ B ) ) ) ).

% Pi_I'
thf(fact_184_Pi__I_H,axiom,
    ! [A: set_nat,F: nat > nat > nat,B: nat > set_nat_nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A )
         => ( member_nat_nat @ ( F @ X3 ) @ ( B @ X3 ) ) )
     => ( member_nat_nat_nat2 @ F @ ( pi_nat_nat_nat2 @ A @ B ) ) ) ).

% Pi_I'
thf(fact_185_Pi__iff,axiom,
    ! [F: product_prod_a_a > product_prod_a_a,I: set_Product_prod_a_a,X5: product_prod_a_a > set_Product_prod_a_a] :
      ( ( member4020126937092221116od_a_a @ F @ ( pi_Pro6370639526499058571od_a_a @ I @ X5 ) )
      = ( ! [X2: product_prod_a_a] :
            ( ( member1426531477525435216od_a_a @ X2 @ I )
           => ( member1426531477525435216od_a_a @ ( F @ X2 ) @ ( X5 @ X2 ) ) ) ) ) ).

% Pi_iff
thf(fact_186_Pi__iff,axiom,
    ! [F: nat > nat,I: set_nat,X5: nat > set_nat] :
      ( ( member_nat_nat @ F @ ( pi_nat_nat @ I @ X5 ) )
      = ( ! [X2: nat] :
            ( ( member_nat @ X2 @ I )
           => ( member_nat @ ( F @ X2 ) @ ( X5 @ X2 ) ) ) ) ) ).

% Pi_iff
thf(fact_187_Pi__iff,axiom,
    ! [F: nat > a,I: set_nat,X5: nat > set_a] :
      ( ( member_nat_a @ F @ ( pi_nat_a @ I @ X5 ) )
      = ( ! [X2: nat] :
            ( ( member_nat @ X2 @ I )
           => ( member_a @ ( F @ X2 ) @ ( X5 @ X2 ) ) ) ) ) ).

% Pi_iff
thf(fact_188_Pi__iff,axiom,
    ! [F: a > product_prod_a_a,I: set_a,X5: a > set_Product_prod_a_a] :
      ( ( member1957775702407316389od_a_a @ F @ ( pi_a_P2178097759547960436od_a_a @ I @ X5 ) )
      = ( ! [X2: a] :
            ( ( member_a @ X2 @ I )
           => ( member1426531477525435216od_a_a @ ( F @ X2 ) @ ( X5 @ X2 ) ) ) ) ) ).

% Pi_iff
thf(fact_189_Pi__iff,axiom,
    ! [F: a > nat,I: set_a,X5: a > set_nat] :
      ( ( member_a_nat @ F @ ( pi_a_nat @ I @ X5 ) )
      = ( ! [X2: a] :
            ( ( member_a @ X2 @ I )
           => ( member_nat @ ( F @ X2 ) @ ( X5 @ X2 ) ) ) ) ) ).

% Pi_iff
thf(fact_190_Pi__iff,axiom,
    ! [F: a > a,I: set_a,X5: a > set_a] :
      ( ( member_a_a @ F @ ( pi_a_a @ I @ X5 ) )
      = ( ! [X2: a] :
            ( ( member_a @ X2 @ I )
           => ( member_a @ ( F @ X2 ) @ ( X5 @ X2 ) ) ) ) ) ).

% Pi_iff
thf(fact_191_Pi__mem,axiom,
    ! [F: nat > nat,A: set_nat,B: nat > set_nat,X: nat] :
      ( ( member_nat_nat @ F @ ( pi_nat_nat @ A @ B ) )
     => ( ( member_nat @ X @ A )
       => ( member_nat @ ( F @ X ) @ ( B @ X ) ) ) ) ).

% Pi_mem
thf(fact_192_Pi__mem,axiom,
    ! [F: nat > a,A: set_nat,B: nat > set_a,X: nat] :
      ( ( member_nat_a @ F @ ( pi_nat_a @ A @ B ) )
     => ( ( member_nat @ X @ A )
       => ( member_a @ ( F @ X ) @ ( B @ X ) ) ) ) ).

% Pi_mem
thf(fact_193_Pi__mem,axiom,
    ! [F: a > nat,A: set_a,B: a > set_nat,X: a] :
      ( ( member_a_nat @ F @ ( pi_a_nat @ A @ B ) )
     => ( ( member_a @ X @ A )
       => ( member_nat @ ( F @ X ) @ ( B @ X ) ) ) ) ).

% Pi_mem
thf(fact_194_Pi__mem,axiom,
    ! [F: a > a,A: set_a,B: a > set_a,X: a] :
      ( ( member_a_a @ F @ ( pi_a_a @ A @ B ) )
     => ( ( member_a @ X @ A )
       => ( member_a @ ( F @ X ) @ ( B @ X ) ) ) ) ).

% Pi_mem
thf(fact_195_Pi__mem,axiom,
    ! [F: a > nat > nat,A: set_a,B: a > set_nat_nat,X: a] :
      ( ( member_a_nat_nat @ F @ ( pi_a_nat_nat @ A @ B ) )
     => ( ( member_a @ X @ A )
       => ( member_nat_nat @ ( F @ X ) @ ( B @ X ) ) ) ) ).

% Pi_mem
thf(fact_196_Pi__mem,axiom,
    ! [F: a > nat > a,A: set_a,B: a > set_nat_a,X: a] :
      ( ( member_a_nat_a2 @ F @ ( pi_a_nat_a2 @ A @ B ) )
     => ( ( member_a @ X @ A )
       => ( member_nat_a @ ( F @ X ) @ ( B @ X ) ) ) ) ).

% Pi_mem
thf(fact_197_Pi__mem,axiom,
    ! [F: a > a > nat,A: set_a,B: a > set_a_nat,X: a] :
      ( ( member_a_a_nat @ F @ ( pi_a_a_nat @ A @ B ) )
     => ( ( member_a @ X @ A )
       => ( member_a_nat @ ( F @ X ) @ ( B @ X ) ) ) ) ).

% Pi_mem
thf(fact_198_Pi__mem,axiom,
    ! [F: a > a > a,A: set_a,B: a > set_a_a,X: a] :
      ( ( member_a_a_a2 @ F @ ( pi_a_a_a2 @ A @ B ) )
     => ( ( member_a @ X @ A )
       => ( member_a_a @ ( F @ X ) @ ( B @ X ) ) ) ) ).

% Pi_mem
thf(fact_199_Pi__mem,axiom,
    ! [F: nat > nat > nat,A: set_nat,B: nat > set_nat_nat,X: nat] :
      ( ( member_nat_nat_nat2 @ F @ ( pi_nat_nat_nat2 @ A @ B ) )
     => ( ( member_nat @ X @ A )
       => ( member_nat_nat @ ( F @ X ) @ ( B @ X ) ) ) ) ).

% Pi_mem
thf(fact_200_Pi__mem,axiom,
    ! [F: nat > nat > a,A: set_nat,B: nat > set_nat_a,X: nat] :
      ( ( member_nat_nat_a2 @ F @ ( pi_nat_nat_a2 @ A @ B ) )
     => ( ( member_nat @ X @ A )
       => ( member_nat_a @ ( F @ X ) @ ( B @ X ) ) ) ) ).

% Pi_mem
thf(fact_201_Pi__cong,axiom,
    ! [A: set_Product_prod_a_a,F: product_prod_a_a > product_prod_a_a,G2: product_prod_a_a > product_prod_a_a,B: product_prod_a_a > set_Product_prod_a_a] :
      ( ! [W: product_prod_a_a] :
          ( ( member1426531477525435216od_a_a @ W @ A )
         => ( ( F @ W )
            = ( G2 @ W ) ) )
     => ( ( member4020126937092221116od_a_a @ F @ ( pi_Pro6370639526499058571od_a_a @ A @ B ) )
        = ( member4020126937092221116od_a_a @ G2 @ ( pi_Pro6370639526499058571od_a_a @ A @ B ) ) ) ) ).

% Pi_cong
thf(fact_202_Pi__cong,axiom,
    ! [A: set_a,F: a > product_prod_a_a,G2: a > product_prod_a_a,B: a > set_Product_prod_a_a] :
      ( ! [W: a] :
          ( ( member_a @ W @ A )
         => ( ( F @ W )
            = ( G2 @ W ) ) )
     => ( ( member1957775702407316389od_a_a @ F @ ( pi_a_P2178097759547960436od_a_a @ A @ B ) )
        = ( member1957775702407316389od_a_a @ G2 @ ( pi_a_P2178097759547960436od_a_a @ A @ B ) ) ) ) ).

% Pi_cong
thf(fact_203_Pi__cong,axiom,
    ! [A: set_a,F: a > nat,G2: a > nat,B: a > set_nat] :
      ( ! [W: a] :
          ( ( member_a @ W @ A )
         => ( ( F @ W )
            = ( G2 @ W ) ) )
     => ( ( member_a_nat @ F @ ( pi_a_nat @ A @ B ) )
        = ( member_a_nat @ G2 @ ( pi_a_nat @ A @ B ) ) ) ) ).

% Pi_cong
thf(fact_204_Pi__cong,axiom,
    ! [A: set_a,F: a > a,G2: a > a,B: a > set_a] :
      ( ! [W: a] :
          ( ( member_a @ W @ A )
         => ( ( F @ W )
            = ( G2 @ W ) ) )
     => ( ( member_a_a @ F @ ( pi_a_a @ A @ B ) )
        = ( member_a_a @ G2 @ ( pi_a_a @ A @ B ) ) ) ) ).

% Pi_cong
thf(fact_205_Pi__cong,axiom,
    ! [A: set_nat,F: nat > nat,G2: nat > nat,B: nat > set_nat] :
      ( ! [W: nat] :
          ( ( member_nat @ W @ A )
         => ( ( F @ W )
            = ( G2 @ W ) ) )
     => ( ( member_nat_nat @ F @ ( pi_nat_nat @ A @ B ) )
        = ( member_nat_nat @ G2 @ ( pi_nat_nat @ A @ B ) ) ) ) ).

% Pi_cong
thf(fact_206_Pi__cong,axiom,
    ! [A: set_nat,F: nat > a,G2: nat > a,B: nat > set_a] :
      ( ! [W: nat] :
          ( ( member_nat @ W @ A )
         => ( ( F @ W )
            = ( G2 @ W ) ) )
     => ( ( member_nat_a @ F @ ( pi_nat_a @ A @ B ) )
        = ( member_nat_a @ G2 @ ( pi_nat_a @ A @ B ) ) ) ) ).

% Pi_cong
thf(fact_207_Pi__mono,axiom,
    ! [A: set_a,B: a > set_nat,C: a > set_nat] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ A )
         => ( ord_less_eq_set_nat @ ( B @ X3 ) @ ( C @ X3 ) ) )
     => ( ord_le1612561287239139007_a_nat @ ( pi_a_nat @ A @ B ) @ ( pi_a_nat @ A @ C ) ) ) ).

% Pi_mono
thf(fact_208_Pi__mono,axiom,
    ! [A: set_nat,B: nat > set_nat,C: nat > set_nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A )
         => ( ord_less_eq_set_nat @ ( B @ X3 ) @ ( C @ X3 ) ) )
     => ( ord_le9059583361652607317at_nat @ ( pi_nat_nat @ A @ B ) @ ( pi_nat_nat @ A @ C ) ) ) ).

% Pi_mono
thf(fact_209_Pi__mono,axiom,
    ! [A: set_a,B: a > set_a,C: a > set_a] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ A )
         => ( ord_less_eq_set_a @ ( B @ X3 ) @ ( C @ X3 ) ) )
     => ( ord_less_eq_set_a_a @ ( pi_a_a @ A @ B ) @ ( pi_a_a @ A @ C ) ) ) ).

% Pi_mono
thf(fact_210_Pi__mono,axiom,
    ! [A: set_nat,B: nat > set_a,C: nat > set_a] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A )
         => ( ord_less_eq_set_a @ ( B @ X3 ) @ ( C @ X3 ) ) )
     => ( ord_le871467723717165285_nat_a @ ( pi_nat_a @ A @ B ) @ ( pi_nat_a @ A @ C ) ) ) ).

% Pi_mono
thf(fact_211_Pi__mono,axiom,
    ! [A: set_a,B: a > set_Product_prod_a_a,C: a > set_Product_prod_a_a] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ A )
         => ( ord_le746702958409616551od_a_a @ ( B @ X3 ) @ ( C @ X3 ) ) )
     => ( ord_le2371270124811097124od_a_a @ ( pi_a_P2178097759547960436od_a_a @ A @ B ) @ ( pi_a_P2178097759547960436od_a_a @ A @ C ) ) ) ).

% Pi_mono
thf(fact_212_Pi__mono,axiom,
    ! [A: set_nat_nat,B: ( nat > nat ) > set_a,C: ( nat > nat ) > set_a] :
      ( ! [X3: nat > nat] :
          ( ( member_nat_nat @ X3 @ A )
         => ( ord_less_eq_set_a @ ( B @ X3 ) @ ( C @ X3 ) ) )
     => ( ord_le1083773273886207926_nat_a @ ( pi_nat_nat_a @ A @ B ) @ ( pi_nat_nat_a @ A @ C ) ) ) ).

% Pi_mono
thf(fact_213_Pi__mono,axiom,
    ! [A: set_nat_a,B: ( nat > a ) > set_a,C: ( nat > a ) > set_a] :
      ( ! [X3: nat > a] :
          ( ( member_nat_a @ X3 @ A )
         => ( ord_less_eq_set_a @ ( B @ X3 ) @ ( C @ X3 ) ) )
     => ( ord_le3509452538356653652at_a_a @ ( pi_nat_a_a @ A @ B ) @ ( pi_nat_a_a @ A @ C ) ) ) ).

% Pi_mono
thf(fact_214_Pi__mono,axiom,
    ! [A: set_a_nat,B: ( a > nat ) > set_a,C: ( a > nat ) > set_a] :
      ( ! [X3: a > nat] :
          ( ( member_a_nat @ X3 @ A )
         => ( ord_less_eq_set_a @ ( B @ X3 ) @ ( C @ X3 ) ) )
     => ( ord_le6383172514208795386_nat_a @ ( pi_a_nat_a @ A @ B ) @ ( pi_a_nat_a @ A @ C ) ) ) ).

% Pi_mono
thf(fact_215_Pi__mono,axiom,
    ! [A: set_a_a,B: ( a > a ) > set_a,C: ( a > a ) > set_a] :
      ( ! [X3: a > a] :
          ( ( member_a_a @ X3 @ A )
         => ( ord_less_eq_set_a @ ( B @ X3 ) @ ( C @ X3 ) ) )
     => ( ord_le7181591058469194768_a_a_a @ ( pi_a_a_a @ A @ B ) @ ( pi_a_a_a @ A @ C ) ) ) ).

% Pi_mono
thf(fact_216_Pi__mono,axiom,
    ! [A: set_Product_prod_a_a,B: product_prod_a_a > set_Product_prod_a_a,C: product_prod_a_a > set_Product_prod_a_a] :
      ( ! [X3: product_prod_a_a] :
          ( ( member1426531477525435216od_a_a @ X3 @ A )
         => ( ord_le746702958409616551od_a_a @ ( B @ X3 ) @ ( C @ X3 ) ) )
     => ( ord_le741722312431004091od_a_a @ ( pi_Pro6370639526499058571od_a_a @ A @ B ) @ ( pi_Pro6370639526499058571od_a_a @ A @ C ) ) ) ).

% Pi_mono
thf(fact_217_Pi__anti__mono,axiom,
    ! [A6: set_Product_prod_a_a,A: set_Product_prod_a_a,B: product_prod_a_a > set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ A6 @ A )
     => ( ord_le741722312431004091od_a_a @ ( pi_Pro6370639526499058571od_a_a @ A @ B ) @ ( pi_Pro6370639526499058571od_a_a @ A6 @ B ) ) ) ).

% Pi_anti_mono
thf(fact_218_Pi__anti__mono,axiom,
    ! [A6: set_nat,A: set_nat,B: nat > set_nat] :
      ( ( ord_less_eq_set_nat @ A6 @ A )
     => ( ord_le9059583361652607317at_nat @ ( pi_nat_nat @ A @ B ) @ ( pi_nat_nat @ A6 @ B ) ) ) ).

% Pi_anti_mono
thf(fact_219_Pi__anti__mono,axiom,
    ! [A6: set_nat,A: set_nat,B: nat > set_a] :
      ( ( ord_less_eq_set_nat @ A6 @ A )
     => ( ord_le871467723717165285_nat_a @ ( pi_nat_a @ A @ B ) @ ( pi_nat_a @ A6 @ B ) ) ) ).

% Pi_anti_mono
thf(fact_220_Pi__anti__mono,axiom,
    ! [A6: set_a,A: set_a,B: a > set_Product_prod_a_a] :
      ( ( ord_less_eq_set_a @ A6 @ A )
     => ( ord_le2371270124811097124od_a_a @ ( pi_a_P2178097759547960436od_a_a @ A @ B ) @ ( pi_a_P2178097759547960436od_a_a @ A6 @ B ) ) ) ).

% Pi_anti_mono
thf(fact_221_Pi__anti__mono,axiom,
    ! [A6: set_a,A: set_a,B: a > set_nat] :
      ( ( ord_less_eq_set_a @ A6 @ A )
     => ( ord_le1612561287239139007_a_nat @ ( pi_a_nat @ A @ B ) @ ( pi_a_nat @ A6 @ B ) ) ) ).

% Pi_anti_mono
thf(fact_222_Pi__anti__mono,axiom,
    ! [A6: set_a,A: set_a,B: a > set_a] :
      ( ( ord_less_eq_set_a @ A6 @ A )
     => ( ord_less_eq_set_a_a @ ( pi_a_a @ A @ B ) @ ( pi_a_a @ A6 @ B ) ) ) ).

% Pi_anti_mono
thf(fact_223_additive__abelian__group__def,axiom,
    pluenn1164192988769422572roup_a = group_201663378560352916roup_a ).

% additive_abelian_group_def
thf(fact_224_additive__abelian__group_Ominussetp_Ocong,axiom,
    pluenn1126946703085653920setp_a = pluenn1126946703085653920setp_a ).

% additive_abelian_group.minussetp.cong
thf(fact_225_additive__abelian__group_Osumsetp_Ocong,axiom,
    pluenn895083305082786853setp_a = pluenn895083305082786853setp_a ).

% additive_abelian_group.sumsetp.cong
thf(fact_226_additive__abelian__group_Osumset__subset__carrier,axiom,
    ! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,B: set_a] :
      ( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
     => ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) @ G ) ) ).

% additive_abelian_group.sumset_subset_carrier
thf(fact_227_additive__abelian__group_Osumset__mono,axiom,
    ! [G: set_a,Addition: a > a > a,Zero: a,A6: set_a,A: set_a,B6: set_a,B: set_a] :
      ( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
     => ( ( ord_less_eq_set_a @ A6 @ A )
       => ( ( ord_less_eq_set_a @ B6 @ B )
         => ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A6 @ B6 ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) ) ) ) ) ).

% additive_abelian_group.sumset_mono
thf(fact_228_additive__abelian__group_Ominusset_OminussetI,axiom,
    ! [G: set_Pr8826267807999420763od_a_a,Addition: ( product_prod_a_a > product_prod_a_a ) > ( product_prod_a_a > product_prod_a_a ) > product_prod_a_a > product_prod_a_a,Zero: product_prod_a_a > product_prod_a_a,A2: product_prod_a_a > product_prod_a_a,A: set_Pr8826267807999420763od_a_a] :
      ( ( pluenn7398882097854948023od_a_a @ G @ Addition @ Zero )
     => ( ( member4020126937092221116od_a_a @ A2 @ A )
       => ( ( member4020126937092221116od_a_a @ A2 @ G )
         => ( member4020126937092221116od_a_a @ ( group_1100041695951271920od_a_a @ G @ Addition @ Zero @ A2 ) @ ( pluenn3241789754909797305od_a_a @ G @ Addition @ Zero @ A ) ) ) ) ) ).

% additive_abelian_group.minusset.minussetI
thf(fact_229_additive__abelian__group_Ominusset_OminussetI,axiom,
    ! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A2: nat,A: set_nat] :
      ( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
     => ( ( member_nat @ A2 @ A )
       => ( ( member_nat @ A2 @ G )
         => ( member_nat @ ( group_inverse_nat @ G @ Addition @ Zero @ A2 ) @ ( pluenn7323955030898006884et_nat @ G @ Addition @ Zero @ A ) ) ) ) ) ).

% additive_abelian_group.minusset.minussetI
thf(fact_230_additive__abelian__group_Ominusset_OminussetI,axiom,
    ! [G: set_nat_nat,Addition: ( nat > nat ) > ( nat > nat ) > nat > nat,Zero: nat > nat,A2: nat > nat,A: set_nat_nat] :
      ( ( pluenn2593311475829585105at_nat @ G @ Addition @ Zero )
     => ( ( member_nat_nat @ A2 @ A )
       => ( ( member_nat_nat @ A2 @ G )
         => ( member_nat_nat @ ( group_5492157109111387274at_nat @ G @ Addition @ Zero @ A2 ) @ ( pluenn9018335828495219411at_nat @ G @ Addition @ Zero @ A ) ) ) ) ) ).

% additive_abelian_group.minusset.minussetI
thf(fact_231_additive__abelian__group_Ominusset_OminussetI,axiom,
    ! [G: set_nat_a,Addition: ( nat > a ) > ( nat > a ) > nat > a,Zero: nat > a,A2: nat > a,A: set_nat_a] :
      ( ( pluenn1914567342357533651_nat_a @ G @ Addition @ Zero )
     => ( ( member_nat_a @ A2 @ A )
       => ( ( member_nat_a @ A2 @ G )
         => ( member_nat_a @ ( group_inverse_nat_a @ G @ Addition @ Zero @ A2 ) @ ( pluenn1683855782372592209_nat_a @ G @ Addition @ Zero @ A ) ) ) ) ) ).

% additive_abelian_group.minusset.minussetI
thf(fact_232_additive__abelian__group_Ominusset_OminussetI,axiom,
    ! [G: set_a_6829686330177631172od_a_a,Addition: ( a > product_prod_a_a ) > ( a > product_prod_a_a ) > a > product_prod_a_a,Zero: a > product_prod_a_a,A2: a > product_prod_a_a,A: set_a_6829686330177631172od_a_a] :
      ( ( pluenn3290985442670702752od_a_a @ G @ Addition @ Zero )
     => ( ( member1957775702407316389od_a_a @ A2 @ A )
       => ( ( member1957775702407316389od_a_a @ A2 @ G )
         => ( member1957775702407316389od_a_a @ ( group_2336095992253589209od_a_a @ G @ Addition @ Zero @ A2 ) @ ( pluenn5371217042752719778od_a_a @ G @ Addition @ Zero @ A ) ) ) ) ) ).

% additive_abelian_group.minusset.minussetI
thf(fact_233_additive__abelian__group_Ominusset_OminussetI,axiom,
    ! [G: set_a_nat,Addition: ( a > nat ) > ( a > nat ) > a > nat,Zero: a > nat,A2: a > nat,A: set_a_nat] :
      ( ( pluenn7899775915488632493_a_nat @ G @ Addition @ Zero )
     => ( ( member_a_nat @ A2 @ A )
       => ( ( member_a_nat @ A2 @ G )
         => ( member_a_nat @ ( group_inverse_a_nat @ G @ Addition @ Zero @ A2 ) @ ( pluenn7669064355503691051_a_nat @ G @ Addition @ Zero @ A ) ) ) ) ) ).

% additive_abelian_group.minusset.minussetI
thf(fact_234_additive__abelian__group_Ominusset_OminussetI,axiom,
    ! [G: set_a_a,Addition: ( a > a ) > ( a > a ) > a > a,Zero: a > a,A2: a > a,A: set_a_a] :
      ( ( pluenn1959071899916691703up_a_a @ G @ Addition @ Zero )
     => ( ( member_a_a @ A2 @ A )
       => ( ( member_a_a @ A2 @ G )
         => ( member_a_a @ ( group_inverse_a_a @ G @ Addition @ Zero @ A2 ) @ ( pluenn2066645275496715769et_a_a @ G @ Addition @ Zero @ A ) ) ) ) ) ).

% additive_abelian_group.minusset.minussetI
thf(fact_235_additive__abelian__group_Ominusset_OminussetI,axiom,
    ! [G: set_a,Addition: a > a > a,Zero: a,A2: a,A: set_a] :
      ( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
     => ( ( member_a @ A2 @ A )
       => ( ( member_a @ A2 @ G )
         => ( member_a @ ( group_inverse_a @ G @ Addition @ Zero @ A2 ) @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A ) ) ) ) ) ).

% additive_abelian_group.minusset.minussetI
thf(fact_236_additive__abelian__group_Ominusset_Osimps,axiom,
    ! [G: set_Pr8826267807999420763od_a_a,Addition: ( product_prod_a_a > product_prod_a_a ) > ( product_prod_a_a > product_prod_a_a ) > product_prod_a_a > product_prod_a_a,Zero: product_prod_a_a > product_prod_a_a,A2: product_prod_a_a > product_prod_a_a,A: set_Pr8826267807999420763od_a_a] :
      ( ( pluenn7398882097854948023od_a_a @ G @ Addition @ Zero )
     => ( ( member4020126937092221116od_a_a @ A2 @ ( pluenn3241789754909797305od_a_a @ G @ Addition @ Zero @ A ) )
        = ( ? [A4: product_prod_a_a > product_prod_a_a] :
              ( ( A2
                = ( group_1100041695951271920od_a_a @ G @ Addition @ Zero @ A4 ) )
              & ( member4020126937092221116od_a_a @ A4 @ A )
              & ( member4020126937092221116od_a_a @ A4 @ G ) ) ) ) ) ).

% additive_abelian_group.minusset.simps
thf(fact_237_additive__abelian__group_Ominusset_Osimps,axiom,
    ! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A2: nat,A: set_nat] :
      ( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
     => ( ( member_nat @ A2 @ ( pluenn7323955030898006884et_nat @ G @ Addition @ Zero @ A ) )
        = ( ? [A4: nat] :
              ( ( A2
                = ( group_inverse_nat @ G @ Addition @ Zero @ A4 ) )
              & ( member_nat @ A4 @ A )
              & ( member_nat @ A4 @ G ) ) ) ) ) ).

% additive_abelian_group.minusset.simps
thf(fact_238_additive__abelian__group_Ominusset_Osimps,axiom,
    ! [G: set_nat_nat,Addition: ( nat > nat ) > ( nat > nat ) > nat > nat,Zero: nat > nat,A2: nat > nat,A: set_nat_nat] :
      ( ( pluenn2593311475829585105at_nat @ G @ Addition @ Zero )
     => ( ( member_nat_nat @ A2 @ ( pluenn9018335828495219411at_nat @ G @ Addition @ Zero @ A ) )
        = ( ? [A4: nat > nat] :
              ( ( A2
                = ( group_5492157109111387274at_nat @ G @ Addition @ Zero @ A4 ) )
              & ( member_nat_nat @ A4 @ A )
              & ( member_nat_nat @ A4 @ G ) ) ) ) ) ).

% additive_abelian_group.minusset.simps
thf(fact_239_additive__abelian__group_Ominusset_Osimps,axiom,
    ! [G: set_nat_a,Addition: ( nat > a ) > ( nat > a ) > nat > a,Zero: nat > a,A2: nat > a,A: set_nat_a] :
      ( ( pluenn1914567342357533651_nat_a @ G @ Addition @ Zero )
     => ( ( member_nat_a @ A2 @ ( pluenn1683855782372592209_nat_a @ G @ Addition @ Zero @ A ) )
        = ( ? [A4: nat > a] :
              ( ( A2
                = ( group_inverse_nat_a @ G @ Addition @ Zero @ A4 ) )
              & ( member_nat_a @ A4 @ A )
              & ( member_nat_a @ A4 @ G ) ) ) ) ) ).

% additive_abelian_group.minusset.simps
thf(fact_240_additive__abelian__group_Ominusset_Osimps,axiom,
    ! [G: set_a_6829686330177631172od_a_a,Addition: ( a > product_prod_a_a ) > ( a > product_prod_a_a ) > a > product_prod_a_a,Zero: a > product_prod_a_a,A2: a > product_prod_a_a,A: set_a_6829686330177631172od_a_a] :
      ( ( pluenn3290985442670702752od_a_a @ G @ Addition @ Zero )
     => ( ( member1957775702407316389od_a_a @ A2 @ ( pluenn5371217042752719778od_a_a @ G @ Addition @ Zero @ A ) )
        = ( ? [A4: a > product_prod_a_a] :
              ( ( A2
                = ( group_2336095992253589209od_a_a @ G @ Addition @ Zero @ A4 ) )
              & ( member1957775702407316389od_a_a @ A4 @ A )
              & ( member1957775702407316389od_a_a @ A4 @ G ) ) ) ) ) ).

% additive_abelian_group.minusset.simps
thf(fact_241_additive__abelian__group_Ominusset_Osimps,axiom,
    ! [G: set_a_nat,Addition: ( a > nat ) > ( a > nat ) > a > nat,Zero: a > nat,A2: a > nat,A: set_a_nat] :
      ( ( pluenn7899775915488632493_a_nat @ G @ Addition @ Zero )
     => ( ( member_a_nat @ A2 @ ( pluenn7669064355503691051_a_nat @ G @ Addition @ Zero @ A ) )
        = ( ? [A4: a > nat] :
              ( ( A2
                = ( group_inverse_a_nat @ G @ Addition @ Zero @ A4 ) )
              & ( member_a_nat @ A4 @ A )
              & ( member_a_nat @ A4 @ G ) ) ) ) ) ).

% additive_abelian_group.minusset.simps
thf(fact_242_additive__abelian__group_Ominusset_Osimps,axiom,
    ! [G: set_a_a,Addition: ( a > a ) > ( a > a ) > a > a,Zero: a > a,A2: a > a,A: set_a_a] :
      ( ( pluenn1959071899916691703up_a_a @ G @ Addition @ Zero )
     => ( ( member_a_a @ A2 @ ( pluenn2066645275496715769et_a_a @ G @ Addition @ Zero @ A ) )
        = ( ? [A4: a > a] :
              ( ( A2
                = ( group_inverse_a_a @ G @ Addition @ Zero @ A4 ) )
              & ( member_a_a @ A4 @ A )
              & ( member_a_a @ A4 @ G ) ) ) ) ) ).

% additive_abelian_group.minusset.simps
thf(fact_243_additive__abelian__group_Ominusset_Osimps,axiom,
    ! [G: set_a,Addition: a > a > a,Zero: a,A2: a,A: set_a] :
      ( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
     => ( ( member_a @ A2 @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A ) )
        = ( ? [A4: a] :
              ( ( A2
                = ( group_inverse_a @ G @ Addition @ Zero @ A4 ) )
              & ( member_a @ A4 @ A )
              & ( member_a @ A4 @ G ) ) ) ) ) ).

% additive_abelian_group.minusset.simps
thf(fact_244_additive__abelian__group_Ominusset_Ocases,axiom,
    ! [G: set_Pr8826267807999420763od_a_a,Addition: ( product_prod_a_a > product_prod_a_a ) > ( product_prod_a_a > product_prod_a_a ) > product_prod_a_a > product_prod_a_a,Zero: product_prod_a_a > product_prod_a_a,A2: product_prod_a_a > product_prod_a_a,A: set_Pr8826267807999420763od_a_a] :
      ( ( pluenn7398882097854948023od_a_a @ G @ Addition @ Zero )
     => ( ( member4020126937092221116od_a_a @ A2 @ ( pluenn3241789754909797305od_a_a @ G @ Addition @ Zero @ A ) )
       => ~ ! [A3: product_prod_a_a > product_prod_a_a] :
              ( ( A2
                = ( group_1100041695951271920od_a_a @ G @ Addition @ Zero @ A3 ) )
             => ( ( member4020126937092221116od_a_a @ A3 @ A )
               => ~ ( member4020126937092221116od_a_a @ A3 @ G ) ) ) ) ) ).

% additive_abelian_group.minusset.cases
thf(fact_245_additive__abelian__group_Ominusset_Ocases,axiom,
    ! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A2: nat,A: set_nat] :
      ( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
     => ( ( member_nat @ A2 @ ( pluenn7323955030898006884et_nat @ G @ Addition @ Zero @ A ) )
       => ~ ! [A3: nat] :
              ( ( A2
                = ( group_inverse_nat @ G @ Addition @ Zero @ A3 ) )
             => ( ( member_nat @ A3 @ A )
               => ~ ( member_nat @ A3 @ G ) ) ) ) ) ).

% additive_abelian_group.minusset.cases
thf(fact_246_additive__abelian__group_Ominusset_Ocases,axiom,
    ! [G: set_nat_nat,Addition: ( nat > nat ) > ( nat > nat ) > nat > nat,Zero: nat > nat,A2: nat > nat,A: set_nat_nat] :
      ( ( pluenn2593311475829585105at_nat @ G @ Addition @ Zero )
     => ( ( member_nat_nat @ A2 @ ( pluenn9018335828495219411at_nat @ G @ Addition @ Zero @ A ) )
       => ~ ! [A3: nat > nat] :
              ( ( A2
                = ( group_5492157109111387274at_nat @ G @ Addition @ Zero @ A3 ) )
             => ( ( member_nat_nat @ A3 @ A )
               => ~ ( member_nat_nat @ A3 @ G ) ) ) ) ) ).

% additive_abelian_group.minusset.cases
thf(fact_247_additive__abelian__group_Ominusset_Ocases,axiom,
    ! [G: set_nat_a,Addition: ( nat > a ) > ( nat > a ) > nat > a,Zero: nat > a,A2: nat > a,A: set_nat_a] :
      ( ( pluenn1914567342357533651_nat_a @ G @ Addition @ Zero )
     => ( ( member_nat_a @ A2 @ ( pluenn1683855782372592209_nat_a @ G @ Addition @ Zero @ A ) )
       => ~ ! [A3: nat > a] :
              ( ( A2
                = ( group_inverse_nat_a @ G @ Addition @ Zero @ A3 ) )
             => ( ( member_nat_a @ A3 @ A )
               => ~ ( member_nat_a @ A3 @ G ) ) ) ) ) ).

% additive_abelian_group.minusset.cases
thf(fact_248_additive__abelian__group_Ominusset_Ocases,axiom,
    ! [G: set_a_6829686330177631172od_a_a,Addition: ( a > product_prod_a_a ) > ( a > product_prod_a_a ) > a > product_prod_a_a,Zero: a > product_prod_a_a,A2: a > product_prod_a_a,A: set_a_6829686330177631172od_a_a] :
      ( ( pluenn3290985442670702752od_a_a @ G @ Addition @ Zero )
     => ( ( member1957775702407316389od_a_a @ A2 @ ( pluenn5371217042752719778od_a_a @ G @ Addition @ Zero @ A ) )
       => ~ ! [A3: a > product_prod_a_a] :
              ( ( A2
                = ( group_2336095992253589209od_a_a @ G @ Addition @ Zero @ A3 ) )
             => ( ( member1957775702407316389od_a_a @ A3 @ A )
               => ~ ( member1957775702407316389od_a_a @ A3 @ G ) ) ) ) ) ).

% additive_abelian_group.minusset.cases
thf(fact_249_additive__abelian__group_Ominusset_Ocases,axiom,
    ! [G: set_a_nat,Addition: ( a > nat ) > ( a > nat ) > a > nat,Zero: a > nat,A2: a > nat,A: set_a_nat] :
      ( ( pluenn7899775915488632493_a_nat @ G @ Addition @ Zero )
     => ( ( member_a_nat @ A2 @ ( pluenn7669064355503691051_a_nat @ G @ Addition @ Zero @ A ) )
       => ~ ! [A3: a > nat] :
              ( ( A2
                = ( group_inverse_a_nat @ G @ Addition @ Zero @ A3 ) )
             => ( ( member_a_nat @ A3 @ A )
               => ~ ( member_a_nat @ A3 @ G ) ) ) ) ) ).

% additive_abelian_group.minusset.cases
thf(fact_250_additive__abelian__group_Ominusset_Ocases,axiom,
    ! [G: set_a_a,Addition: ( a > a ) > ( a > a ) > a > a,Zero: a > a,A2: a > a,A: set_a_a] :
      ( ( pluenn1959071899916691703up_a_a @ G @ Addition @ Zero )
     => ( ( member_a_a @ A2 @ ( pluenn2066645275496715769et_a_a @ G @ Addition @ Zero @ A ) )
       => ~ ! [A3: a > a] :
              ( ( A2
                = ( group_inverse_a_a @ G @ Addition @ Zero @ A3 ) )
             => ( ( member_a_a @ A3 @ A )
               => ~ ( member_a_a @ A3 @ G ) ) ) ) ) ).

% additive_abelian_group.minusset.cases
thf(fact_251_additive__abelian__group_Ominusset_Ocases,axiom,
    ! [G: set_a,Addition: a > a > a,Zero: a,A2: a,A: set_a] :
      ( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
     => ( ( member_a @ A2 @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A ) )
       => ~ ! [A3: a] :
              ( ( A2
                = ( group_inverse_a @ G @ Addition @ Zero @ A3 ) )
             => ( ( member_a @ A3 @ A )
               => ~ ( member_a @ A3 @ G ) ) ) ) ) ).

% additive_abelian_group.minusset.cases
thf(fact_252_funcset__mem,axiom,
    ! [F: nat > nat,A: set_nat,B: set_nat,X: nat] :
      ( ( member_nat_nat @ F
        @ ( pi_nat_nat @ A
          @ ^ [Uu: nat] : B ) )
     => ( ( member_nat @ X @ A )
       => ( member_nat @ ( F @ X ) @ B ) ) ) ).

% funcset_mem
thf(fact_253_funcset__mem,axiom,
    ! [F: nat > a,A: set_nat,B: set_a,X: nat] :
      ( ( member_nat_a @ F
        @ ( pi_nat_a @ A
          @ ^ [Uu: nat] : B ) )
     => ( ( member_nat @ X @ A )
       => ( member_a @ ( F @ X ) @ B ) ) ) ).

% funcset_mem
thf(fact_254_funcset__mem,axiom,
    ! [F: a > nat,A: set_a,B: set_nat,X: a] :
      ( ( member_a_nat @ F
        @ ( pi_a_nat @ A
          @ ^ [Uu: a] : B ) )
     => ( ( member_a @ X @ A )
       => ( member_nat @ ( F @ X ) @ B ) ) ) ).

% funcset_mem
thf(fact_255_funcset__mem,axiom,
    ! [F: a > a,A: set_a,B: set_a,X: a] :
      ( ( member_a_a @ F
        @ ( pi_a_a @ A
          @ ^ [Uu: a] : B ) )
     => ( ( member_a @ X @ A )
       => ( member_a @ ( F @ X ) @ B ) ) ) ).

% funcset_mem
thf(fact_256_funcset__mem,axiom,
    ! [F: a > nat > nat,A: set_a,B: set_nat_nat,X: a] :
      ( ( member_a_nat_nat @ F
        @ ( pi_a_nat_nat @ A
          @ ^ [Uu: a] : B ) )
     => ( ( member_a @ X @ A )
       => ( member_nat_nat @ ( F @ X ) @ B ) ) ) ).

% funcset_mem
thf(fact_257_funcset__mem,axiom,
    ! [F: a > nat > a,A: set_a,B: set_nat_a,X: a] :
      ( ( member_a_nat_a2 @ F
        @ ( pi_a_nat_a2 @ A
          @ ^ [Uu: a] : B ) )
     => ( ( member_a @ X @ A )
       => ( member_nat_a @ ( F @ X ) @ B ) ) ) ).

% funcset_mem
thf(fact_258_funcset__mem,axiom,
    ! [F: a > a > nat,A: set_a,B: set_a_nat,X: a] :
      ( ( member_a_a_nat @ F
        @ ( pi_a_a_nat @ A
          @ ^ [Uu: a] : B ) )
     => ( ( member_a @ X @ A )
       => ( member_a_nat @ ( F @ X ) @ B ) ) ) ).

% funcset_mem
thf(fact_259_funcset__mem,axiom,
    ! [F: a > a > a,A: set_a,B: set_a_a,X: a] :
      ( ( member_a_a_a2 @ F
        @ ( pi_a_a_a2 @ A
          @ ^ [Uu: a] : B ) )
     => ( ( member_a @ X @ A )
       => ( member_a_a @ ( F @ X ) @ B ) ) ) ).

% funcset_mem
thf(fact_260_funcset__mem,axiom,
    ! [F: nat > nat > nat,A: set_nat,B: set_nat_nat,X: nat] :
      ( ( member_nat_nat_nat2 @ F
        @ ( pi_nat_nat_nat2 @ A
          @ ^ [Uu: nat] : B ) )
     => ( ( member_nat @ X @ A )
       => ( member_nat_nat @ ( F @ X ) @ B ) ) ) ).

% funcset_mem
thf(fact_261_funcset__mem,axiom,
    ! [F: nat > nat > a,A: set_nat,B: set_nat_a,X: nat] :
      ( ( member_nat_nat_a2 @ F
        @ ( pi_nat_nat_a2 @ A
          @ ^ [Uu: nat] : B ) )
     => ( ( member_nat @ X @ A )
       => ( member_nat_a @ ( F @ X ) @ B ) ) ) ).

% funcset_mem
thf(fact_262_funcset__id,axiom,
    ! [A: set_Product_prod_a_a] :
      ( member4020126937092221116od_a_a
      @ ^ [X2: product_prod_a_a] : X2
      @ ( pi_Pro6370639526499058571od_a_a @ A
        @ ^ [Uu: product_prod_a_a] : A ) ) ).

% funcset_id
thf(fact_263_funcset__id,axiom,
    ! [A: set_nat] :
      ( member_nat_nat
      @ ^ [X2: nat] : X2
      @ ( pi_nat_nat @ A
        @ ^ [Uu: nat] : A ) ) ).

% funcset_id
thf(fact_264_funcset__id,axiom,
    ! [A: set_a] :
      ( member_a_a
      @ ^ [X2: a] : X2
      @ ( pi_a_a @ A
        @ ^ [Uu: a] : A ) ) ).

% funcset_id
thf(fact_265_funcsetI,axiom,
    ! [A: set_a,F: a > a,B: set_a] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ A )
         => ( member_a @ ( F @ X3 ) @ B ) )
     => ( member_a_a @ F
        @ ( pi_a_a @ A
          @ ^ [Uu: a] : B ) ) ) ).

% funcsetI
thf(fact_266_funcsetI,axiom,
    ! [A: set_a,F: a > nat,B: set_nat] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ A )
         => ( member_nat @ ( F @ X3 ) @ B ) )
     => ( member_a_nat @ F
        @ ( pi_a_nat @ A
          @ ^ [Uu: a] : B ) ) ) ).

% funcsetI
thf(fact_267_funcsetI,axiom,
    ! [A: set_nat,F: nat > a,B: set_a] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A )
         => ( member_a @ ( F @ X3 ) @ B ) )
     => ( member_nat_a @ F
        @ ( pi_nat_a @ A
          @ ^ [Uu: nat] : B ) ) ) ).

% funcsetI
thf(fact_268_funcsetI,axiom,
    ! [A: set_nat,F: nat > nat,B: set_nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A )
         => ( member_nat @ ( F @ X3 ) @ B ) )
     => ( member_nat_nat @ F
        @ ( pi_nat_nat @ A
          @ ^ [Uu: nat] : B ) ) ) ).

% funcsetI
thf(fact_269_funcsetI,axiom,
    ! [A: set_a,F: a > product_prod_a_a,B: set_Product_prod_a_a] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ A )
         => ( member1426531477525435216od_a_a @ ( F @ X3 ) @ B ) )
     => ( member1957775702407316389od_a_a @ F
        @ ( pi_a_P2178097759547960436od_a_a @ A
          @ ^ [Uu: a] : B ) ) ) ).

% funcsetI
thf(fact_270_funcsetI,axiom,
    ! [A: set_a,F: a > nat > nat,B: set_nat_nat] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ A )
         => ( member_nat_nat @ ( F @ X3 ) @ B ) )
     => ( member_a_nat_nat @ F
        @ ( pi_a_nat_nat @ A
          @ ^ [Uu: a] : B ) ) ) ).

% funcsetI
thf(fact_271_funcsetI,axiom,
    ! [A: set_a,F: a > nat > a,B: set_nat_a] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ A )
         => ( member_nat_a @ ( F @ X3 ) @ B ) )
     => ( member_a_nat_a2 @ F
        @ ( pi_a_nat_a2 @ A
          @ ^ [Uu: a] : B ) ) ) ).

% funcsetI
thf(fact_272_funcsetI,axiom,
    ! [A: set_a,F: a > a > nat,B: set_a_nat] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ A )
         => ( member_a_nat @ ( F @ X3 ) @ B ) )
     => ( member_a_a_nat @ F
        @ ( pi_a_a_nat @ A
          @ ^ [Uu: a] : B ) ) ) ).

% funcsetI
thf(fact_273_funcsetI,axiom,
    ! [A: set_a,F: a > a > a,B: set_a_a] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ A )
         => ( member_a_a @ ( F @ X3 ) @ B ) )
     => ( member_a_a_a2 @ F
        @ ( pi_a_a_a2 @ A
          @ ^ [Uu: a] : B ) ) ) ).

% funcsetI
thf(fact_274_funcsetI,axiom,
    ! [A: set_nat,F: nat > nat > nat,B: set_nat_nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A )
         => ( member_nat_nat @ ( F @ X3 ) @ B ) )
     => ( member_nat_nat_nat2 @ F
        @ ( pi_nat_nat_nat2 @ A
          @ ^ [Uu: nat] : B ) ) ) ).

% funcsetI
thf(fact_275_additive__abelian__group_Ominusset__subset__carrier,axiom,
    ! [G: set_a,Addition: a > a > a,Zero: a,A: set_a] :
      ( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
     => ( ord_less_eq_set_a @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A ) @ G ) ) ).

% additive_abelian_group.minusset_subset_carrier
thf(fact_276_additive__abelian__group_Ominusset__iterated__minusset,axiom,
    ! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,K: nat] :
      ( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
     => ( ( pluenn1960970773371692859ated_a @ G @ Addition @ Zero @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A ) @ K )
        = ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ ( pluenn1960970773371692859ated_a @ G @ Addition @ Zero @ A @ K ) ) ) ) ).

% additive_abelian_group.minusset_iterated_minusset
thf(fact_277_additive__abelian__group_Ofinite__sumset,axiom,
    ! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: set_nat,B: set_nat] :
      ( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
     => ( ( finite_finite_nat @ A )
       => ( ( finite_finite_nat @ B )
         => ( finite_finite_nat @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A @ B ) ) ) ) ) ).

% additive_abelian_group.finite_sumset
thf(fact_278_additive__abelian__group_Ofinite__sumset,axiom,
    ! [G: set_Product_prod_a_a,Addition: product_prod_a_a > product_prod_a_a > product_prod_a_a,Zero: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
      ( ( pluenn6837661381354551573od_a_a @ G @ Addition @ Zero )
     => ( ( finite6544458595007987280od_a_a @ A )
       => ( ( finite6544458595007987280od_a_a @ B )
         => ( finite6544458595007987280od_a_a @ ( pluenn659090379318209998od_a_a @ G @ Addition @ A @ B ) ) ) ) ) ).

% additive_abelian_group.finite_sumset
thf(fact_279_additive__abelian__group_Ofinite__sumset,axiom,
    ! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,B: set_a] :
      ( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
     => ( ( finite_finite_a @ A )
       => ( ( finite_finite_a @ B )
         => ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) ) ) ) ) ).

% additive_abelian_group.finite_sumset
thf(fact_280_additive__abelian__group_Osumsetp__sumset__eq,axiom,
    ! [G: set_Pr8826267807999420763od_a_a,Addition: ( product_prod_a_a > product_prod_a_a ) > ( product_prod_a_a > product_prod_a_a ) > product_prod_a_a > product_prod_a_a,Zero: product_prod_a_a > product_prod_a_a,A: set_Pr8826267807999420763od_a_a,B: set_Pr8826267807999420763od_a_a] :
      ( ( pluenn7398882097854948023od_a_a @ G @ Addition @ Zero )
     => ( ( pluenn6245878296707294014od_a_a @ G @ Addition
          @ ^ [X2: product_prod_a_a > product_prod_a_a] : ( member4020126937092221116od_a_a @ X2 @ A )
          @ ^ [X2: product_prod_a_a > product_prod_a_a] : ( member4020126937092221116od_a_a @ X2 @ B ) )
        = ( ^ [X2: product_prod_a_a > product_prod_a_a] : ( member4020126937092221116od_a_a @ X2 @ ( pluenn6825655340897727678od_a_a @ G @ Addition @ A @ B ) ) ) ) ) ).

% additive_abelian_group.sumsetp_sumset_eq
thf(fact_281_additive__abelian__group_Osumsetp__sumset__eq,axiom,
    ! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: set_nat,B: set_nat] :
      ( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
     => ( ( pluenn5670965976768739049tp_nat @ G @ Addition
          @ ^ [X2: nat] : ( member_nat @ X2 @ A )
          @ ^ [X2: nat] : ( member_nat @ X2 @ B ) )
        = ( ^ [X2: nat] : ( member_nat @ X2 @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A @ B ) ) ) ) ) ).

% additive_abelian_group.sumsetp_sumset_eq
thf(fact_282_additive__abelian__group_Osumsetp__sumset__eq,axiom,
    ! [G: set_nat_nat,Addition: ( nat > nat ) > ( nat > nat ) > nat > nat,Zero: nat > nat,A: set_nat_nat,B: set_nat_nat] :
      ( ( pluenn2593311475829585105at_nat @ G @ Addition @ Zero )
     => ( ( pluenn8622197827967232728at_nat @ G @ Addition
          @ ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ A )
          @ ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ B ) )
        = ( ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ ( pluenn8091236430575893592at_nat @ G @ Addition @ A @ B ) ) ) ) ) ).

% additive_abelian_group.sumsetp_sumset_eq
thf(fact_283_additive__abelian__group_Osumsetp__sumset__eq,axiom,
    ! [G: set_nat_a,Addition: ( nat > a ) > ( nat > a ) > nat > a,Zero: nat > a,A: set_nat_a,B: set_nat_a] :
      ( ( pluenn1914567342357533651_nat_a @ G @ Addition @ Zero )
     => ( ( pluenn5759993693271462284_nat_a @ G @ Addition
          @ ^ [X2: nat > a] : ( member_nat_a @ X2 @ A )
          @ ^ [X2: nat > a] : ( member_nat_a @ X2 @ B ) )
        = ( ^ [X2: nat > a] : ( member_nat_a @ X2 @ ( pluenn551768455426449420_nat_a @ G @ Addition @ A @ B ) ) ) ) ) ).

% additive_abelian_group.sumsetp_sumset_eq
thf(fact_284_additive__abelian__group_Osumsetp__sumset__eq,axiom,
    ! [G: set_a_6829686330177631172od_a_a,Addition: ( a > product_prod_a_a ) > ( a > product_prod_a_a ) > a > product_prod_a_a,Zero: a > product_prod_a_a,A: set_a_6829686330177631172od_a_a,B: set_a_6829686330177631172od_a_a] :
      ( ( pluenn3290985442670702752od_a_a @ G @ Addition @ Zero )
     => ( ( pluenn3409616955423707175od_a_a @ G @ Addition
          @ ^ [X2: a > product_prod_a_a] : ( member1957775702407316389od_a_a @ X2 @ A )
          @ ^ [X2: a > product_prod_a_a] : ( member1957775702407316389od_a_a @ X2 @ B ) )
        = ( ^ [X2: a > product_prod_a_a] : ( member1957775702407316389od_a_a @ X2 @ ( pluenn6474628215148175783od_a_a @ G @ Addition @ A @ B ) ) ) ) ) ).

% additive_abelian_group.sumsetp_sumset_eq
thf(fact_285_additive__abelian__group_Osumsetp__sumset__eq,axiom,
    ! [G: set_a_nat,Addition: ( a > nat ) > ( a > nat ) > a > nat,Zero: a > nat,A: set_a_nat,B: set_a_nat] :
      ( ( pluenn7899775915488632493_a_nat @ G @ Addition @ Zero )
     => ( ( pluenn2521830229547785318_a_nat @ G @ Addition
          @ ^ [X2: a > nat] : ( member_a_nat @ X2 @ A )
          @ ^ [X2: a > nat] : ( member_a_nat @ X2 @ B ) )
        = ( ^ [X2: a > nat] : ( member_a_nat @ X2 @ ( pluenn6536977028557548262_a_nat @ G @ Addition @ A @ B ) ) ) ) ) ).

% additive_abelian_group.sumsetp_sumset_eq
thf(fact_286_additive__abelian__group_Osumsetp__sumset__eq,axiom,
    ! [G: set_a_a,Addition: ( a > a ) > ( a > a ) > a > a,Zero: a > a,A: set_a_a,B: set_a_a] :
      ( ( pluenn1959071899916691703up_a_a @ G @ Addition @ Zero )
     => ( ( pluenn676121060179339134tp_a_a @ G @ Addition
          @ ^ [X2: a > a] : ( member_a_a @ X2 @ A )
          @ ^ [X2: a > a] : ( member_a_a @ X2 @ B ) )
        = ( ^ [X2: a > a] : ( member_a_a @ X2 @ ( pluenn7679623682358442238et_a_a @ G @ Addition @ A @ B ) ) ) ) ) ).

% additive_abelian_group.sumsetp_sumset_eq
thf(fact_287_additive__abelian__group_Osumsetp__sumset__eq,axiom,
    ! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,B: set_a] :
      ( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
     => ( ( pluenn895083305082786853setp_a @ G @ Addition
          @ ^ [X2: a] : ( member_a @ X2 @ A )
          @ ^ [X2: a] : ( member_a @ X2 @ B ) )
        = ( ^ [X2: a] : ( member_a @ X2 @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) ) ) ) ) ).

% additive_abelian_group.sumsetp_sumset_eq
thf(fact_288_additive__abelian__group_Osumset__def,axiom,
    ! [G: set_Pr8826267807999420763od_a_a,Addition: ( product_prod_a_a > product_prod_a_a ) > ( product_prod_a_a > product_prod_a_a ) > product_prod_a_a > product_prod_a_a,Zero: product_prod_a_a > product_prod_a_a] :
      ( ( pluenn7398882097854948023od_a_a @ G @ Addition @ Zero )
     => ( ( pluenn6825655340897727678od_a_a @ G @ Addition )
        = ( ^ [A5: set_Pr8826267807999420763od_a_a,B5: set_Pr8826267807999420763od_a_a] :
              ( collec8125451137695935482od_a_a
              @ ( pluenn6245878296707294014od_a_a @ G @ Addition
                @ ^ [X2: product_prod_a_a > product_prod_a_a] : ( member4020126937092221116od_a_a @ X2 @ A5 )
                @ ^ [X2: product_prod_a_a > product_prod_a_a] : ( member4020126937092221116od_a_a @ X2 @ B5 ) ) ) ) ) ) ).

% additive_abelian_group.sumset_def
thf(fact_289_additive__abelian__group_Osumset__def,axiom,
    ! [G: set_nat_nat,Addition: ( nat > nat ) > ( nat > nat ) > nat > nat,Zero: nat > nat] :
      ( ( pluenn2593311475829585105at_nat @ G @ Addition @ Zero )
     => ( ( pluenn8091236430575893592at_nat @ G @ Addition )
        = ( ^ [A5: set_nat_nat,B5: set_nat_nat] :
              ( collect_nat_nat
              @ ( pluenn8622197827967232728at_nat @ G @ Addition
                @ ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ A5 )
                @ ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ B5 ) ) ) ) ) ) ).

% additive_abelian_group.sumset_def
thf(fact_290_additive__abelian__group_Osumset__def,axiom,
    ! [G: set_nat_a,Addition: ( nat > a ) > ( nat > a ) > nat > a,Zero: nat > a] :
      ( ( pluenn1914567342357533651_nat_a @ G @ Addition @ Zero )
     => ( ( pluenn551768455426449420_nat_a @ G @ Addition )
        = ( ^ [A5: set_nat_a,B5: set_nat_a] :
              ( collect_nat_a
              @ ( pluenn5759993693271462284_nat_a @ G @ Addition
                @ ^ [X2: nat > a] : ( member_nat_a @ X2 @ A5 )
                @ ^ [X2: nat > a] : ( member_nat_a @ X2 @ B5 ) ) ) ) ) ) ).

% additive_abelian_group.sumset_def
thf(fact_291_additive__abelian__group_Osumset__def,axiom,
    ! [G: set_a_6829686330177631172od_a_a,Addition: ( a > product_prod_a_a ) > ( a > product_prod_a_a ) > a > product_prod_a_a,Zero: a > product_prod_a_a] :
      ( ( pluenn3290985442670702752od_a_a @ G @ Addition @ Zero )
     => ( ( pluenn6474628215148175783od_a_a @ G @ Addition )
        = ( ^ [A5: set_a_6829686330177631172od_a_a,B5: set_a_6829686330177631172od_a_a] :
              ( collec3127584877517960419od_a_a
              @ ( pluenn3409616955423707175od_a_a @ G @ Addition
                @ ^ [X2: a > product_prod_a_a] : ( member1957775702407316389od_a_a @ X2 @ A5 )
                @ ^ [X2: a > product_prod_a_a] : ( member1957775702407316389od_a_a @ X2 @ B5 ) ) ) ) ) ) ).

% additive_abelian_group.sumset_def
thf(fact_292_additive__abelian__group_Osumset__def,axiom,
    ! [G: set_a_nat,Addition: ( a > nat ) > ( a > nat ) > a > nat,Zero: a > nat] :
      ( ( pluenn7899775915488632493_a_nat @ G @ Addition @ Zero )
     => ( ( pluenn6536977028557548262_a_nat @ G @ Addition )
        = ( ^ [A5: set_a_nat,B5: set_a_nat] :
              ( collect_a_nat
              @ ( pluenn2521830229547785318_a_nat @ G @ Addition
                @ ^ [X2: a > nat] : ( member_a_nat @ X2 @ A5 )
                @ ^ [X2: a > nat] : ( member_a_nat @ X2 @ B5 ) ) ) ) ) ) ).

% additive_abelian_group.sumset_def
thf(fact_293_additive__abelian__group_Osumset__def,axiom,
    ! [G: set_a_a,Addition: ( a > a ) > ( a > a ) > a > a,Zero: a > a] :
      ( ( pluenn1959071899916691703up_a_a @ G @ Addition @ Zero )
     => ( ( pluenn7679623682358442238et_a_a @ G @ Addition )
        = ( ^ [A5: set_a_a,B5: set_a_a] :
              ( collect_a_a
              @ ( pluenn676121060179339134tp_a_a @ G @ Addition
                @ ^ [X2: a > a] : ( member_a_a @ X2 @ A5 )
                @ ^ [X2: a > a] : ( member_a_a @ X2 @ B5 ) ) ) ) ) ) ).

% additive_abelian_group.sumset_def
thf(fact_294_additive__abelian__group_Osumset__def,axiom,
    ! [G: set_nat,Addition: nat > nat > nat,Zero: nat] :
      ( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
     => ( ( pluenn3669378163024332905et_nat @ G @ Addition )
        = ( ^ [A5: set_nat,B5: set_nat] :
              ( collect_nat
              @ ( pluenn5670965976768739049tp_nat @ G @ Addition
                @ ^ [X2: nat] : ( member_nat @ X2 @ A5 )
                @ ^ [X2: nat] : ( member_nat @ X2 @ B5 ) ) ) ) ) ) ).

% additive_abelian_group.sumset_def
thf(fact_295_additive__abelian__group_Osumset__def,axiom,
    ! [G: set_a,Addition: a > a > a,Zero: a] :
      ( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
     => ( ( pluenn3038260743871226533mset_a @ G @ Addition )
        = ( ^ [A5: set_a,B5: set_a] :
              ( collect_a
              @ ( pluenn895083305082786853setp_a @ G @ Addition
                @ ^ [X2: a] : ( member_a @ X2 @ A5 )
                @ ^ [X2: a] : ( member_a @ X2 @ B5 ) ) ) ) ) ) ).

% additive_abelian_group.sumset_def
thf(fact_296_additive__abelian__group_Ofinite__minusset,axiom,
    ! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: set_nat] :
      ( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
     => ( ( finite_finite_nat @ A )
       => ( finite_finite_nat @ ( pluenn7323955030898006884et_nat @ G @ Addition @ Zero @ A ) ) ) ) ).

% additive_abelian_group.finite_minusset
thf(fact_297_additive__abelian__group_Ofinite__minusset,axiom,
    ! [G: set_Product_prod_a_a,Addition: product_prod_a_a > product_prod_a_a > product_prod_a_a,Zero: product_prod_a_a,A: set_Product_prod_a_a] :
      ( ( pluenn6837661381354551573od_a_a @ G @ Addition @ Zero )
     => ( ( finite6544458595007987280od_a_a @ A )
       => ( finite6544458595007987280od_a_a @ ( pluenn2765658404358379155od_a_a @ G @ Addition @ Zero @ A ) ) ) ) ).

% additive_abelian_group.finite_minusset
thf(fact_298_additive__abelian__group_Ofinite__minusset,axiom,
    ! [G: set_a,Addition: a > a > a,Zero: a,A: set_a] :
      ( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
     => ( ( finite_finite_a @ A )
       => ( finite_finite_a @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A ) ) ) ) ).

% additive_abelian_group.finite_minusset
thf(fact_299_additive__abelian__group_Ominussetp__minusset__eq,axiom,
    ! [G: set_Pr8826267807999420763od_a_a,Addition: ( product_prod_a_a > product_prod_a_a ) > ( product_prod_a_a > product_prod_a_a ) > product_prod_a_a > product_prod_a_a,Zero: product_prod_a_a > product_prod_a_a,A: set_Pr8826267807999420763od_a_a] :
      ( ( pluenn7398882097854948023od_a_a @ G @ Addition @ Zero )
     => ( ( pluenn2777150465843325699od_a_a @ G @ Addition @ Zero
          @ ^ [X2: product_prod_a_a > product_prod_a_a] : ( member4020126937092221116od_a_a @ X2 @ A ) )
        = ( ^ [X2: product_prod_a_a > product_prod_a_a] : ( member4020126937092221116od_a_a @ X2 @ ( pluenn3241789754909797305od_a_a @ G @ Addition @ Zero @ A ) ) ) ) ) ).

% additive_abelian_group.minussetp_minusset_eq
thf(fact_300_additive__abelian__group_Ominussetp__minusset__eq,axiom,
    ! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: set_nat] :
      ( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
     => ( ( pluenn8372939692575285934tp_nat @ G @ Addition @ Zero
          @ ^ [X2: nat] : ( member_nat @ X2 @ A ) )
        = ( ^ [X2: nat] : ( member_nat @ X2 @ ( pluenn7323955030898006884et_nat @ G @ Addition @ Zero @ A ) ) ) ) ) ).

% additive_abelian_group.minussetp_minusset_eq
thf(fact_301_additive__abelian__group_Ominussetp__minusset__eq,axiom,
    ! [G: set_nat_nat,Addition: ( nat > nat ) > ( nat > nat ) > nat > nat,Zero: nat > nat,A: set_nat_nat] :
      ( ( pluenn2593311475829585105at_nat @ G @ Addition @ Zero )
     => ( ( pluenn6500390917429955357at_nat @ G @ Addition @ Zero
          @ ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ A ) )
        = ( ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ ( pluenn9018335828495219411at_nat @ G @ Addition @ Zero @ A ) ) ) ) ) ).

% additive_abelian_group.minussetp_minusset_eq
thf(fact_302_additive__abelian__group_Ominussetp__minusset__eq,axiom,
    ! [G: set_nat_a,Addition: ( nat > a ) > ( nat > a ) > nat > a,Zero: nat > a,A: set_nat_a] :
      ( ( pluenn1914567342357533651_nat_a @ G @ Addition @ Zero )
     => ( ( pluenn2964913278637471879_nat_a @ G @ Addition @ Zero
          @ ^ [X2: nat > a] : ( member_nat_a @ X2 @ A ) )
        = ( ^ [X2: nat > a] : ( member_nat_a @ X2 @ ( pluenn1683855782372592209_nat_a @ G @ Addition @ Zero @ A ) ) ) ) ) ).

% additive_abelian_group.minussetp_minusset_eq
thf(fact_303_additive__abelian__group_Ominussetp__minusset__eq,axiom,
    ! [G: set_a_6829686330177631172od_a_a,Addition: ( a > product_prod_a_a ) > ( a > product_prod_a_a ) > a > product_prod_a_a,Zero: a > product_prod_a_a,A: set_a_6829686330177631172od_a_a] :
      ( ( pluenn3290985442670702752od_a_a @ G @ Addition @ Zero )
     => ( ( pluenn5763864222185776364od_a_a @ G @ Addition @ Zero
          @ ^ [X2: a > product_prod_a_a] : ( member1957775702407316389od_a_a @ X2 @ A ) )
        = ( ^ [X2: a > product_prod_a_a] : ( member1957775702407316389od_a_a @ X2 @ ( pluenn5371217042752719778od_a_a @ G @ Addition @ Zero @ A ) ) ) ) ) ).

% additive_abelian_group.minussetp_minusset_eq
thf(fact_304_additive__abelian__group_Ominussetp__minusset__eq,axiom,
    ! [G: set_a_nat,Addition: ( a > nat ) > ( a > nat ) > a > nat,Zero: a > nat,A: set_a_nat] :
      ( ( pluenn7899775915488632493_a_nat @ G @ Addition @ Zero )
     => ( ( pluenn8950121851768570721_a_nat @ G @ Addition @ Zero
          @ ^ [X2: a > nat] : ( member_a_nat @ X2 @ A ) )
        = ( ^ [X2: a > nat] : ( member_a_nat @ X2 @ ( pluenn7669064355503691051_a_nat @ G @ Addition @ Zero @ A ) ) ) ) ) ).

% additive_abelian_group.minussetp_minusset_eq
thf(fact_305_additive__abelian__group_Ominussetp__minusset__eq,axiom,
    ! [G: set_a_a,Addition: ( a > a ) > ( a > a ) > a > a,Zero: a > a,A: set_a_a] :
      ( ( pluenn1959071899916691703up_a_a @ G @ Addition @ Zero )
     => ( ( pluenn1140692617289730371tp_a_a @ G @ Addition @ Zero
          @ ^ [X2: a > a] : ( member_a_a @ X2 @ A ) )
        = ( ^ [X2: a > a] : ( member_a_a @ X2 @ ( pluenn2066645275496715769et_a_a @ G @ Addition @ Zero @ A ) ) ) ) ) ).

% additive_abelian_group.minussetp_minusset_eq
thf(fact_306_additive__abelian__group_Ominussetp__minusset__eq,axiom,
    ! [G: set_a,Addition: a > a > a,Zero: a,A: set_a] :
      ( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
     => ( ( pluenn1126946703085653920setp_a @ G @ Addition @ Zero
          @ ^ [X2: a] : ( member_a @ X2 @ A ) )
        = ( ^ [X2: a] : ( member_a @ X2 @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A ) ) ) ) ) ).

% additive_abelian_group.minussetp_minusset_eq
thf(fact_307_additive__abelian__group_Ominusset__def,axiom,
    ! [G: set_Pr8826267807999420763od_a_a,Addition: ( product_prod_a_a > product_prod_a_a ) > ( product_prod_a_a > product_prod_a_a ) > product_prod_a_a > product_prod_a_a,Zero: product_prod_a_a > product_prod_a_a] :
      ( ( pluenn7398882097854948023od_a_a @ G @ Addition @ Zero )
     => ( ( pluenn3241789754909797305od_a_a @ G @ Addition @ Zero )
        = ( ^ [A5: set_Pr8826267807999420763od_a_a] :
              ( collec8125451137695935482od_a_a
              @ ( pluenn2777150465843325699od_a_a @ G @ Addition @ Zero
                @ ^ [X2: product_prod_a_a > product_prod_a_a] : ( member4020126937092221116od_a_a @ X2 @ A5 ) ) ) ) ) ) ).

% additive_abelian_group.minusset_def
thf(fact_308_additive__abelian__group_Ominusset__def,axiom,
    ! [G: set_nat_nat,Addition: ( nat > nat ) > ( nat > nat ) > nat > nat,Zero: nat > nat] :
      ( ( pluenn2593311475829585105at_nat @ G @ Addition @ Zero )
     => ( ( pluenn9018335828495219411at_nat @ G @ Addition @ Zero )
        = ( ^ [A5: set_nat_nat] :
              ( collect_nat_nat
              @ ( pluenn6500390917429955357at_nat @ G @ Addition @ Zero
                @ ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ A5 ) ) ) ) ) ) ).

% additive_abelian_group.minusset_def
thf(fact_309_additive__abelian__group_Ominusset__def,axiom,
    ! [G: set_nat_a,Addition: ( nat > a ) > ( nat > a ) > nat > a,Zero: nat > a] :
      ( ( pluenn1914567342357533651_nat_a @ G @ Addition @ Zero )
     => ( ( pluenn1683855782372592209_nat_a @ G @ Addition @ Zero )
        = ( ^ [A5: set_nat_a] :
              ( collect_nat_a
              @ ( pluenn2964913278637471879_nat_a @ G @ Addition @ Zero
                @ ^ [X2: nat > a] : ( member_nat_a @ X2 @ A5 ) ) ) ) ) ) ).

% additive_abelian_group.minusset_def
thf(fact_310_additive__abelian__group_Ominusset__def,axiom,
    ! [G: set_a_6829686330177631172od_a_a,Addition: ( a > product_prod_a_a ) > ( a > product_prod_a_a ) > a > product_prod_a_a,Zero: a > product_prod_a_a] :
      ( ( pluenn3290985442670702752od_a_a @ G @ Addition @ Zero )
     => ( ( pluenn5371217042752719778od_a_a @ G @ Addition @ Zero )
        = ( ^ [A5: set_a_6829686330177631172od_a_a] :
              ( collec3127584877517960419od_a_a
              @ ( pluenn5763864222185776364od_a_a @ G @ Addition @ Zero
                @ ^ [X2: a > product_prod_a_a] : ( member1957775702407316389od_a_a @ X2 @ A5 ) ) ) ) ) ) ).

% additive_abelian_group.minusset_def
thf(fact_311_additive__abelian__group_Ominusset__def,axiom,
    ! [G: set_a_nat,Addition: ( a > nat ) > ( a > nat ) > a > nat,Zero: a > nat] :
      ( ( pluenn7899775915488632493_a_nat @ G @ Addition @ Zero )
     => ( ( pluenn7669064355503691051_a_nat @ G @ Addition @ Zero )
        = ( ^ [A5: set_a_nat] :
              ( collect_a_nat
              @ ( pluenn8950121851768570721_a_nat @ G @ Addition @ Zero
                @ ^ [X2: a > nat] : ( member_a_nat @ X2 @ A5 ) ) ) ) ) ) ).

% additive_abelian_group.minusset_def
thf(fact_312_additive__abelian__group_Ominusset__def,axiom,
    ! [G: set_a_a,Addition: ( a > a ) > ( a > a ) > a > a,Zero: a > a] :
      ( ( pluenn1959071899916691703up_a_a @ G @ Addition @ Zero )
     => ( ( pluenn2066645275496715769et_a_a @ G @ Addition @ Zero )
        = ( ^ [A5: set_a_a] :
              ( collect_a_a
              @ ( pluenn1140692617289730371tp_a_a @ G @ Addition @ Zero
                @ ^ [X2: a > a] : ( member_a_a @ X2 @ A5 ) ) ) ) ) ) ).

% additive_abelian_group.minusset_def
thf(fact_313_additive__abelian__group_Ominusset__def,axiom,
    ! [G: set_nat,Addition: nat > nat > nat,Zero: nat] :
      ( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
     => ( ( pluenn7323955030898006884et_nat @ G @ Addition @ Zero )
        = ( ^ [A5: set_nat] :
              ( collect_nat
              @ ( pluenn8372939692575285934tp_nat @ G @ Addition @ Zero
                @ ^ [X2: nat] : ( member_nat @ X2 @ A5 ) ) ) ) ) ) ).

% additive_abelian_group.minusset_def
thf(fact_314_additive__abelian__group_Ominusset__def,axiom,
    ! [G: set_a,Addition: a > a > a,Zero: a] :
      ( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
     => ( ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero )
        = ( ^ [A5: set_a] :
              ( collect_a
              @ ( pluenn1126946703085653920setp_a @ G @ Addition @ Zero
                @ ^ [X2: a] : ( member_a @ X2 @ A5 ) ) ) ) ) ) ).

% additive_abelian_group.minusset_def
thf(fact_315_additive__abelian__group_Ocard__minusset_H,axiom,
    ! [G: set_Product_prod_a_a,Addition: product_prod_a_a > product_prod_a_a > product_prod_a_a,Zero: product_prod_a_a,A: set_Product_prod_a_a] :
      ( ( pluenn6837661381354551573od_a_a @ G @ Addition @ Zero )
     => ( ( ord_le746702958409616551od_a_a @ A @ G )
       => ( ( finite4795055649997197647od_a_a @ ( pluenn2765658404358379155od_a_a @ G @ Addition @ Zero @ A ) )
          = ( finite4795055649997197647od_a_a @ A ) ) ) ) ).

% additive_abelian_group.card_minusset'
thf(fact_316_additive__abelian__group_Ocard__minusset_H,axiom,
    ! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: set_nat] :
      ( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
     => ( ( ord_less_eq_set_nat @ A @ G )
       => ( ( finite_card_nat @ ( pluenn7323955030898006884et_nat @ G @ Addition @ Zero @ A ) )
          = ( finite_card_nat @ A ) ) ) ) ).

% additive_abelian_group.card_minusset'
thf(fact_317_additive__abelian__group_Ocard__minusset_H,axiom,
    ! [G: set_a,Addition: a > a > a,Zero: a,A: set_a] :
      ( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
     => ( ( ord_less_eq_set_a @ A @ G )
       => ( ( finite_card_a @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A ) )
          = ( finite_card_a @ A ) ) ) ) ).

% additive_abelian_group.card_minusset'
thf(fact_318_additive__abelian__group_Osumset__commute,axiom,
    ! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,B: set_a] :
      ( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
     => ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B )
        = ( pluenn3038260743871226533mset_a @ G @ Addition @ B @ A ) ) ) ).

% additive_abelian_group.sumset_commute
thf(fact_319_additive__abelian__group_Osumset_OsumsetI,axiom,
    ! [G: set_Pr8826267807999420763od_a_a,Addition: ( product_prod_a_a > product_prod_a_a ) > ( product_prod_a_a > product_prod_a_a ) > product_prod_a_a > product_prod_a_a,Zero: product_prod_a_a > product_prod_a_a,A2: product_prod_a_a > product_prod_a_a,A: set_Pr8826267807999420763od_a_a,B4: product_prod_a_a > product_prod_a_a,B: set_Pr8826267807999420763od_a_a] :
      ( ( pluenn7398882097854948023od_a_a @ G @ Addition @ Zero )
     => ( ( member4020126937092221116od_a_a @ A2 @ A )
       => ( ( member4020126937092221116od_a_a @ A2 @ G )
         => ( ( member4020126937092221116od_a_a @ B4 @ B )
           => ( ( member4020126937092221116od_a_a @ B4 @ G )
             => ( member4020126937092221116od_a_a @ ( Addition @ A2 @ B4 ) @ ( pluenn6825655340897727678od_a_a @ G @ Addition @ A @ B ) ) ) ) ) ) ) ).

% additive_abelian_group.sumset.sumsetI
thf(fact_320_additive__abelian__group_Osumset_OsumsetI,axiom,
    ! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A2: nat,A: set_nat,B4: nat,B: set_nat] :
      ( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
     => ( ( member_nat @ A2 @ A )
       => ( ( member_nat @ A2 @ G )
         => ( ( member_nat @ B4 @ B )
           => ( ( member_nat @ B4 @ G )
             => ( member_nat @ ( Addition @ A2 @ B4 ) @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A @ B ) ) ) ) ) ) ) ).

% additive_abelian_group.sumset.sumsetI
thf(fact_321_additive__abelian__group_Osumset_OsumsetI,axiom,
    ! [G: set_nat_nat,Addition: ( nat > nat ) > ( nat > nat ) > nat > nat,Zero: nat > nat,A2: nat > nat,A: set_nat_nat,B4: nat > nat,B: set_nat_nat] :
      ( ( pluenn2593311475829585105at_nat @ G @ Addition @ Zero )
     => ( ( member_nat_nat @ A2 @ A )
       => ( ( member_nat_nat @ A2 @ G )
         => ( ( member_nat_nat @ B4 @ B )
           => ( ( member_nat_nat @ B4 @ G )
             => ( member_nat_nat @ ( Addition @ A2 @ B4 ) @ ( pluenn8091236430575893592at_nat @ G @ Addition @ A @ B ) ) ) ) ) ) ) ).

% additive_abelian_group.sumset.sumsetI
thf(fact_322_additive__abelian__group_Osumset_OsumsetI,axiom,
    ! [G: set_nat_a,Addition: ( nat > a ) > ( nat > a ) > nat > a,Zero: nat > a,A2: nat > a,A: set_nat_a,B4: nat > a,B: set_nat_a] :
      ( ( pluenn1914567342357533651_nat_a @ G @ Addition @ Zero )
     => ( ( member_nat_a @ A2 @ A )
       => ( ( member_nat_a @ A2 @ G )
         => ( ( member_nat_a @ B4 @ B )
           => ( ( member_nat_a @ B4 @ G )
             => ( member_nat_a @ ( Addition @ A2 @ B4 ) @ ( pluenn551768455426449420_nat_a @ G @ Addition @ A @ B ) ) ) ) ) ) ) ).

% additive_abelian_group.sumset.sumsetI
thf(fact_323_additive__abelian__group_Osumset_OsumsetI,axiom,
    ! [G: set_a_6829686330177631172od_a_a,Addition: ( a > product_prod_a_a ) > ( a > product_prod_a_a ) > a > product_prod_a_a,Zero: a > product_prod_a_a,A2: a > product_prod_a_a,A: set_a_6829686330177631172od_a_a,B4: a > product_prod_a_a,B: set_a_6829686330177631172od_a_a] :
      ( ( pluenn3290985442670702752od_a_a @ G @ Addition @ Zero )
     => ( ( member1957775702407316389od_a_a @ A2 @ A )
       => ( ( member1957775702407316389od_a_a @ A2 @ G )
         => ( ( member1957775702407316389od_a_a @ B4 @ B )
           => ( ( member1957775702407316389od_a_a @ B4 @ G )
             => ( member1957775702407316389od_a_a @ ( Addition @ A2 @ B4 ) @ ( pluenn6474628215148175783od_a_a @ G @ Addition @ A @ B ) ) ) ) ) ) ) ).

% additive_abelian_group.sumset.sumsetI
thf(fact_324_additive__abelian__group_Osumset_OsumsetI,axiom,
    ! [G: set_a_nat,Addition: ( a > nat ) > ( a > nat ) > a > nat,Zero: a > nat,A2: a > nat,A: set_a_nat,B4: a > nat,B: set_a_nat] :
      ( ( pluenn7899775915488632493_a_nat @ G @ Addition @ Zero )
     => ( ( member_a_nat @ A2 @ A )
       => ( ( member_a_nat @ A2 @ G )
         => ( ( member_a_nat @ B4 @ B )
           => ( ( member_a_nat @ B4 @ G )
             => ( member_a_nat @ ( Addition @ A2 @ B4 ) @ ( pluenn6536977028557548262_a_nat @ G @ Addition @ A @ B ) ) ) ) ) ) ) ).

% additive_abelian_group.sumset.sumsetI
thf(fact_325_additive__abelian__group_Osumset_OsumsetI,axiom,
    ! [G: set_a_a,Addition: ( a > a ) > ( a > a ) > a > a,Zero: a > a,A2: a > a,A: set_a_a,B4: a > a,B: set_a_a] :
      ( ( pluenn1959071899916691703up_a_a @ G @ Addition @ Zero )
     => ( ( member_a_a @ A2 @ A )
       => ( ( member_a_a @ A2 @ G )
         => ( ( member_a_a @ B4 @ B )
           => ( ( member_a_a @ B4 @ G )
             => ( member_a_a @ ( Addition @ A2 @ B4 ) @ ( pluenn7679623682358442238et_a_a @ G @ Addition @ A @ B ) ) ) ) ) ) ) ).

% additive_abelian_group.sumset.sumsetI
thf(fact_326_additive__abelian__group_Osumset_OsumsetI,axiom,
    ! [G: set_a,Addition: a > a > a,Zero: a,A2: a,A: set_a,B4: a,B: set_a] :
      ( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
     => ( ( member_a @ A2 @ A )
       => ( ( member_a @ A2 @ G )
         => ( ( member_a @ B4 @ B )
           => ( ( member_a @ B4 @ G )
             => ( member_a @ ( Addition @ A2 @ B4 ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) ) ) ) ) ) ) ).

% additive_abelian_group.sumset.sumsetI
thf(fact_327_additive__abelian__group_Osumset__assoc,axiom,
    ! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,B: set_a,C: set_a] :
      ( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
     => ( ( pluenn3038260743871226533mset_a @ G @ Addition @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) @ C )
        = ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ ( pluenn3038260743871226533mset_a @ G @ Addition @ B @ C ) ) ) ) ).

% additive_abelian_group.sumset_assoc
thf(fact_328_additive__abelian__group_Osumset_Osimps,axiom,
    ! [G: set_Pr8826267807999420763od_a_a,Addition: ( product_prod_a_a > product_prod_a_a ) > ( product_prod_a_a > product_prod_a_a ) > product_prod_a_a > product_prod_a_a,Zero: product_prod_a_a > product_prod_a_a,A2: product_prod_a_a > product_prod_a_a,A: set_Pr8826267807999420763od_a_a,B: set_Pr8826267807999420763od_a_a] :
      ( ( pluenn7398882097854948023od_a_a @ G @ Addition @ Zero )
     => ( ( member4020126937092221116od_a_a @ A2 @ ( pluenn6825655340897727678od_a_a @ G @ Addition @ A @ B ) )
        = ( ? [A4: product_prod_a_a > product_prod_a_a,B3: product_prod_a_a > product_prod_a_a] :
              ( ( A2
                = ( Addition @ A4 @ B3 ) )
              & ( member4020126937092221116od_a_a @ A4 @ A )
              & ( member4020126937092221116od_a_a @ A4 @ G )
              & ( member4020126937092221116od_a_a @ B3 @ B )
              & ( member4020126937092221116od_a_a @ B3 @ G ) ) ) ) ) ).

% additive_abelian_group.sumset.simps
thf(fact_329_additive__abelian__group_Osumset_Osimps,axiom,
    ! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A2: nat,A: set_nat,B: set_nat] :
      ( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
     => ( ( member_nat @ A2 @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A @ B ) )
        = ( ? [A4: nat,B3: nat] :
              ( ( A2
                = ( Addition @ A4 @ B3 ) )
              & ( member_nat @ A4 @ A )
              & ( member_nat @ A4 @ G )
              & ( member_nat @ B3 @ B )
              & ( member_nat @ B3 @ G ) ) ) ) ) ).

% additive_abelian_group.sumset.simps
thf(fact_330_additive__abelian__group_Osumset_Osimps,axiom,
    ! [G: set_nat_nat,Addition: ( nat > nat ) > ( nat > nat ) > nat > nat,Zero: nat > nat,A2: nat > nat,A: set_nat_nat,B: set_nat_nat] :
      ( ( pluenn2593311475829585105at_nat @ G @ Addition @ Zero )
     => ( ( member_nat_nat @ A2 @ ( pluenn8091236430575893592at_nat @ G @ Addition @ A @ B ) )
        = ( ? [A4: nat > nat,B3: nat > nat] :
              ( ( A2
                = ( Addition @ A4 @ B3 ) )
              & ( member_nat_nat @ A4 @ A )
              & ( member_nat_nat @ A4 @ G )
              & ( member_nat_nat @ B3 @ B )
              & ( member_nat_nat @ B3 @ G ) ) ) ) ) ).

% additive_abelian_group.sumset.simps
thf(fact_331_additive__abelian__group_Osumset_Osimps,axiom,
    ! [G: set_nat_a,Addition: ( nat > a ) > ( nat > a ) > nat > a,Zero: nat > a,A2: nat > a,A: set_nat_a,B: set_nat_a] :
      ( ( pluenn1914567342357533651_nat_a @ G @ Addition @ Zero )
     => ( ( member_nat_a @ A2 @ ( pluenn551768455426449420_nat_a @ G @ Addition @ A @ B ) )
        = ( ? [A4: nat > a,B3: nat > a] :
              ( ( A2
                = ( Addition @ A4 @ B3 ) )
              & ( member_nat_a @ A4 @ A )
              & ( member_nat_a @ A4 @ G )
              & ( member_nat_a @ B3 @ B )
              & ( member_nat_a @ B3 @ G ) ) ) ) ) ).

% additive_abelian_group.sumset.simps
thf(fact_332_additive__abelian__group_Osumset_Osimps,axiom,
    ! [G: set_a_6829686330177631172od_a_a,Addition: ( a > product_prod_a_a ) > ( a > product_prod_a_a ) > a > product_prod_a_a,Zero: a > product_prod_a_a,A2: a > product_prod_a_a,A: set_a_6829686330177631172od_a_a,B: set_a_6829686330177631172od_a_a] :
      ( ( pluenn3290985442670702752od_a_a @ G @ Addition @ Zero )
     => ( ( member1957775702407316389od_a_a @ A2 @ ( pluenn6474628215148175783od_a_a @ G @ Addition @ A @ B ) )
        = ( ? [A4: a > product_prod_a_a,B3: a > product_prod_a_a] :
              ( ( A2
                = ( Addition @ A4 @ B3 ) )
              & ( member1957775702407316389od_a_a @ A4 @ A )
              & ( member1957775702407316389od_a_a @ A4 @ G )
              & ( member1957775702407316389od_a_a @ B3 @ B )
              & ( member1957775702407316389od_a_a @ B3 @ G ) ) ) ) ) ).

% additive_abelian_group.sumset.simps
thf(fact_333_additive__abelian__group_Osumset_Osimps,axiom,
    ! [G: set_a_nat,Addition: ( a > nat ) > ( a > nat ) > a > nat,Zero: a > nat,A2: a > nat,A: set_a_nat,B: set_a_nat] :
      ( ( pluenn7899775915488632493_a_nat @ G @ Addition @ Zero )
     => ( ( member_a_nat @ A2 @ ( pluenn6536977028557548262_a_nat @ G @ Addition @ A @ B ) )
        = ( ? [A4: a > nat,B3: a > nat] :
              ( ( A2
                = ( Addition @ A4 @ B3 ) )
              & ( member_a_nat @ A4 @ A )
              & ( member_a_nat @ A4 @ G )
              & ( member_a_nat @ B3 @ B )
              & ( member_a_nat @ B3 @ G ) ) ) ) ) ).

% additive_abelian_group.sumset.simps
thf(fact_334_additive__abelian__group_Osumset_Osimps,axiom,
    ! [G: set_a_a,Addition: ( a > a ) > ( a > a ) > a > a,Zero: a > a,A2: a > a,A: set_a_a,B: set_a_a] :
      ( ( pluenn1959071899916691703up_a_a @ G @ Addition @ Zero )
     => ( ( member_a_a @ A2 @ ( pluenn7679623682358442238et_a_a @ G @ Addition @ A @ B ) )
        = ( ? [A4: a > a,B3: a > a] :
              ( ( A2
                = ( Addition @ A4 @ B3 ) )
              & ( member_a_a @ A4 @ A )
              & ( member_a_a @ A4 @ G )
              & ( member_a_a @ B3 @ B )
              & ( member_a_a @ B3 @ G ) ) ) ) ) ).

% additive_abelian_group.sumset.simps
thf(fact_335_additive__abelian__group_Osumset_Osimps,axiom,
    ! [G: set_a,Addition: a > a > a,Zero: a,A2: a,A: set_a,B: set_a] :
      ( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
     => ( ( member_a @ A2 @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) )
        = ( ? [A4: a,B3: a] :
              ( ( A2
                = ( Addition @ A4 @ B3 ) )
              & ( member_a @ A4 @ A )
              & ( member_a @ A4 @ G )
              & ( member_a @ B3 @ B )
              & ( member_a @ B3 @ G ) ) ) ) ) ).

% additive_abelian_group.sumset.simps
thf(fact_336_additive__abelian__group_Osumset_Ocases,axiom,
    ! [G: set_Pr8826267807999420763od_a_a,Addition: ( product_prod_a_a > product_prod_a_a ) > ( product_prod_a_a > product_prod_a_a ) > product_prod_a_a > product_prod_a_a,Zero: product_prod_a_a > product_prod_a_a,A2: product_prod_a_a > product_prod_a_a,A: set_Pr8826267807999420763od_a_a,B: set_Pr8826267807999420763od_a_a] :
      ( ( pluenn7398882097854948023od_a_a @ G @ Addition @ Zero )
     => ( ( member4020126937092221116od_a_a @ A2 @ ( pluenn6825655340897727678od_a_a @ G @ Addition @ A @ B ) )
       => ~ ! [A3: product_prod_a_a > product_prod_a_a,B2: product_prod_a_a > product_prod_a_a] :
              ( ( A2
                = ( Addition @ A3 @ B2 ) )
             => ( ( member4020126937092221116od_a_a @ A3 @ A )
               => ( ( member4020126937092221116od_a_a @ A3 @ G )
                 => ( ( member4020126937092221116od_a_a @ B2 @ B )
                   => ~ ( member4020126937092221116od_a_a @ B2 @ G ) ) ) ) ) ) ) ).

% additive_abelian_group.sumset.cases
thf(fact_337_additive__abelian__group_Osumset_Ocases,axiom,
    ! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A2: nat,A: set_nat,B: set_nat] :
      ( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
     => ( ( member_nat @ A2 @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A @ B ) )
       => ~ ! [A3: nat,B2: nat] :
              ( ( A2
                = ( Addition @ A3 @ B2 ) )
             => ( ( member_nat @ A3 @ A )
               => ( ( member_nat @ A3 @ G )
                 => ( ( member_nat @ B2 @ B )
                   => ~ ( member_nat @ B2 @ G ) ) ) ) ) ) ) ).

% additive_abelian_group.sumset.cases
thf(fact_338_additive__abelian__group_Osumset_Ocases,axiom,
    ! [G: set_nat_nat,Addition: ( nat > nat ) > ( nat > nat ) > nat > nat,Zero: nat > nat,A2: nat > nat,A: set_nat_nat,B: set_nat_nat] :
      ( ( pluenn2593311475829585105at_nat @ G @ Addition @ Zero )
     => ( ( member_nat_nat @ A2 @ ( pluenn8091236430575893592at_nat @ G @ Addition @ A @ B ) )
       => ~ ! [A3: nat > nat,B2: nat > nat] :
              ( ( A2
                = ( Addition @ A3 @ B2 ) )
             => ( ( member_nat_nat @ A3 @ A )
               => ( ( member_nat_nat @ A3 @ G )
                 => ( ( member_nat_nat @ B2 @ B )
                   => ~ ( member_nat_nat @ B2 @ G ) ) ) ) ) ) ) ).

% additive_abelian_group.sumset.cases
thf(fact_339_additive__abelian__group_Osumset_Ocases,axiom,
    ! [G: set_nat_a,Addition: ( nat > a ) > ( nat > a ) > nat > a,Zero: nat > a,A2: nat > a,A: set_nat_a,B: set_nat_a] :
      ( ( pluenn1914567342357533651_nat_a @ G @ Addition @ Zero )
     => ( ( member_nat_a @ A2 @ ( pluenn551768455426449420_nat_a @ G @ Addition @ A @ B ) )
       => ~ ! [A3: nat > a,B2: nat > a] :
              ( ( A2
                = ( Addition @ A3 @ B2 ) )
             => ( ( member_nat_a @ A3 @ A )
               => ( ( member_nat_a @ A3 @ G )
                 => ( ( member_nat_a @ B2 @ B )
                   => ~ ( member_nat_a @ B2 @ G ) ) ) ) ) ) ) ).

% additive_abelian_group.sumset.cases
thf(fact_340_additive__abelian__group_Osumset_Ocases,axiom,
    ! [G: set_a_6829686330177631172od_a_a,Addition: ( a > product_prod_a_a ) > ( a > product_prod_a_a ) > a > product_prod_a_a,Zero: a > product_prod_a_a,A2: a > product_prod_a_a,A: set_a_6829686330177631172od_a_a,B: set_a_6829686330177631172od_a_a] :
      ( ( pluenn3290985442670702752od_a_a @ G @ Addition @ Zero )
     => ( ( member1957775702407316389od_a_a @ A2 @ ( pluenn6474628215148175783od_a_a @ G @ Addition @ A @ B ) )
       => ~ ! [A3: a > product_prod_a_a,B2: a > product_prod_a_a] :
              ( ( A2
                = ( Addition @ A3 @ B2 ) )
             => ( ( member1957775702407316389od_a_a @ A3 @ A )
               => ( ( member1957775702407316389od_a_a @ A3 @ G )
                 => ( ( member1957775702407316389od_a_a @ B2 @ B )
                   => ~ ( member1957775702407316389od_a_a @ B2 @ G ) ) ) ) ) ) ) ).

% additive_abelian_group.sumset.cases
thf(fact_341_additive__abelian__group_Osumset_Ocases,axiom,
    ! [G: set_a_nat,Addition: ( a > nat ) > ( a > nat ) > a > nat,Zero: a > nat,A2: a > nat,A: set_a_nat,B: set_a_nat] :
      ( ( pluenn7899775915488632493_a_nat @ G @ Addition @ Zero )
     => ( ( member_a_nat @ A2 @ ( pluenn6536977028557548262_a_nat @ G @ Addition @ A @ B ) )
       => ~ ! [A3: a > nat,B2: a > nat] :
              ( ( A2
                = ( Addition @ A3 @ B2 ) )
             => ( ( member_a_nat @ A3 @ A )
               => ( ( member_a_nat @ A3 @ G )
                 => ( ( member_a_nat @ B2 @ B )
                   => ~ ( member_a_nat @ B2 @ G ) ) ) ) ) ) ) ).

% additive_abelian_group.sumset.cases
thf(fact_342_additive__abelian__group_Osumset_Ocases,axiom,
    ! [G: set_a_a,Addition: ( a > a ) > ( a > a ) > a > a,Zero: a > a,A2: a > a,A: set_a_a,B: set_a_a] :
      ( ( pluenn1959071899916691703up_a_a @ G @ Addition @ Zero )
     => ( ( member_a_a @ A2 @ ( pluenn7679623682358442238et_a_a @ G @ Addition @ A @ B ) )
       => ~ ! [A3: a > a,B2: a > a] :
              ( ( A2
                = ( Addition @ A3 @ B2 ) )
             => ( ( member_a_a @ A3 @ A )
               => ( ( member_a_a @ A3 @ G )
                 => ( ( member_a_a @ B2 @ B )
                   => ~ ( member_a_a @ B2 @ G ) ) ) ) ) ) ) ).

% additive_abelian_group.sumset.cases
thf(fact_343_additive__abelian__group_Osumset_Ocases,axiom,
    ! [G: set_a,Addition: a > a > a,Zero: a,A2: a,A: set_a,B: set_a] :
      ( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
     => ( ( member_a @ A2 @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) )
       => ~ ! [A3: a,B2: a] :
              ( ( A2
                = ( Addition @ A3 @ B2 ) )
             => ( ( member_a @ A3 @ A )
               => ( ( member_a @ A3 @ G )
                 => ( ( member_a @ B2 @ B )
                   => ~ ( member_a @ B2 @ G ) ) ) ) ) ) ) ).

% additive_abelian_group.sumset.cases
thf(fact_344_card__bij,axiom,
    ! [F: a > a,A: set_a,B: set_a,G2: a > a] :
      ( ( member_a_a @ F
        @ ( pi_a_a @ A
          @ ^ [Uu: a] : B ) )
     => ( ( inj_on_a_a @ F @ A )
       => ( ( member_a_a @ G2
            @ ( pi_a_a @ B
              @ ^ [Uu: a] : A ) )
         => ( ( inj_on_a_a @ G2 @ B )
           => ( ( finite_finite_a @ A )
             => ( ( finite_finite_a @ B )
               => ( ( finite_card_a @ A )
                  = ( finite_card_a @ B ) ) ) ) ) ) ) ) ).

% card_bij
thf(fact_345_card__bij,axiom,
    ! [F: a > nat,A: set_a,B: set_nat,G2: nat > a] :
      ( ( member_a_nat @ F
        @ ( pi_a_nat @ A
          @ ^ [Uu: a] : B ) )
     => ( ( inj_on_a_nat @ F @ A )
       => ( ( member_nat_a @ G2
            @ ( pi_nat_a @ B
              @ ^ [Uu: nat] : A ) )
         => ( ( inj_on_nat_a @ G2 @ B )
           => ( ( finite_finite_a @ A )
             => ( ( finite_finite_nat @ B )
               => ( ( finite_card_a @ A )
                  = ( finite_card_nat @ B ) ) ) ) ) ) ) ) ).

% card_bij
thf(fact_346_card__bij,axiom,
    ! [F: a > product_prod_a_a,A: set_a,B: set_Product_prod_a_a,G2: product_prod_a_a > a] :
      ( ( member1957775702407316389od_a_a @ F
        @ ( pi_a_P2178097759547960436od_a_a @ A
          @ ^ [Uu: a] : B ) )
     => ( ( inj_on8941660083241582106od_a_a @ F @ A )
       => ( ( member1716570166360300819_a_a_a @ G2
            @ ( pi_Pro7438789266712198498_a_a_a @ B
              @ ^ [Uu: product_prod_a_a] : A ) )
         => ( ( inj_on4978979553551044360_a_a_a @ G2 @ B )
           => ( ( finite_finite_a @ A )
             => ( ( finite6544458595007987280od_a_a @ B )
               => ( ( finite_card_a @ A )
                  = ( finite4795055649997197647od_a_a @ B ) ) ) ) ) ) ) ) ).

% card_bij
thf(fact_347_card__bij,axiom,
    ! [F: nat > a,A: set_nat,B: set_a,G2: a > nat] :
      ( ( member_nat_a @ F
        @ ( pi_nat_a @ A
          @ ^ [Uu: nat] : B ) )
     => ( ( inj_on_nat_a @ F @ A )
       => ( ( member_a_nat @ G2
            @ ( pi_a_nat @ B
              @ ^ [Uu: a] : A ) )
         => ( ( inj_on_a_nat @ G2 @ B )
           => ( ( finite_finite_nat @ A )
             => ( ( finite_finite_a @ B )
               => ( ( finite_card_nat @ A )
                  = ( finite_card_a @ B ) ) ) ) ) ) ) ) ).

% card_bij
thf(fact_348_card__bij,axiom,
    ! [F: nat > nat,A: set_nat,B: set_nat,G2: nat > nat] :
      ( ( member_nat_nat @ F
        @ ( pi_nat_nat @ A
          @ ^ [Uu: nat] : B ) )
     => ( ( inj_on_nat_nat @ F @ A )
       => ( ( member_nat_nat @ G2
            @ ( pi_nat_nat @ B
              @ ^ [Uu: nat] : A ) )
         => ( ( inj_on_nat_nat @ G2 @ B )
           => ( ( finite_finite_nat @ A )
             => ( ( finite_finite_nat @ B )
               => ( ( finite_card_nat @ A )
                  = ( finite_card_nat @ B ) ) ) ) ) ) ) ) ).

% card_bij
thf(fact_349_card__bij,axiom,
    ! [F: nat > product_prod_a_a,A: set_nat,B: set_Product_prod_a_a,G2: product_prod_a_a > nat] :
      ( ( member909238274692171063od_a_a @ F
        @ ( pi_nat2514139106461652682od_a_a @ A
          @ ^ [Uu: nat] : B ) )
     => ( ( inj_on8964604748314331044od_a_a @ F @ A )
       => ( ( member4881838279615896465_a_nat @ G2
            @ ( pi_Pro1971496080287246444_a_nat @ B
              @ ^ [Uu: product_prod_a_a] : A ) )
         => ( ( inj_on8421961722139924806_a_nat @ G2 @ B )
           => ( ( finite_finite_nat @ A )
             => ( ( finite6544458595007987280od_a_a @ B )
               => ( ( finite_card_nat @ A )
                  = ( finite4795055649997197647od_a_a @ B ) ) ) ) ) ) ) ) ).

% card_bij
thf(fact_350_card__bij,axiom,
    ! [F: product_prod_a_a > a,A: set_Product_prod_a_a,B: set_a,G2: a > product_prod_a_a] :
      ( ( member1716570166360300819_a_a_a @ F
        @ ( pi_Pro7438789266712198498_a_a_a @ A
          @ ^ [Uu: product_prod_a_a] : B ) )
     => ( ( inj_on4978979553551044360_a_a_a @ F @ A )
       => ( ( member1957775702407316389od_a_a @ G2
            @ ( pi_a_P2178097759547960436od_a_a @ B
              @ ^ [Uu: a] : A ) )
         => ( ( inj_on8941660083241582106od_a_a @ G2 @ B )
           => ( ( finite6544458595007987280od_a_a @ A )
             => ( ( finite_finite_a @ B )
               => ( ( finite4795055649997197647od_a_a @ A )
                  = ( finite_card_a @ B ) ) ) ) ) ) ) ) ).

% card_bij
thf(fact_351_card__bij,axiom,
    ! [F: product_prod_a_a > nat,A: set_Product_prod_a_a,B: set_nat,G2: nat > product_prod_a_a] :
      ( ( member4881838279615896465_a_nat @ F
        @ ( pi_Pro1971496080287246444_a_nat @ A
          @ ^ [Uu: product_prod_a_a] : B ) )
     => ( ( inj_on8421961722139924806_a_nat @ F @ A )
       => ( ( member909238274692171063od_a_a @ G2
            @ ( pi_nat2514139106461652682od_a_a @ B
              @ ^ [Uu: nat] : A ) )
         => ( ( inj_on8964604748314331044od_a_a @ G2 @ B )
           => ( ( finite6544458595007987280od_a_a @ A )
             => ( ( finite_finite_nat @ B )
               => ( ( finite4795055649997197647od_a_a @ A )
                  = ( finite_card_nat @ B ) ) ) ) ) ) ) ) ).

% card_bij
thf(fact_352_card__bij,axiom,
    ! [F: product_prod_a_a > product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a,G2: product_prod_a_a > product_prod_a_a] :
      ( ( member4020126937092221116od_a_a @ F
        @ ( pi_Pro6370639526499058571od_a_a @ A
          @ ^ [Uu: product_prod_a_a] : B ) )
     => ( ( inj_on2566144670800592689od_a_a @ F @ A )
       => ( ( member4020126937092221116od_a_a @ G2
            @ ( pi_Pro6370639526499058571od_a_a @ B
              @ ^ [Uu: product_prod_a_a] : A ) )
         => ( ( inj_on2566144670800592689od_a_a @ G2 @ B )
           => ( ( finite6544458595007987280od_a_a @ A )
             => ( ( finite6544458595007987280od_a_a @ B )
               => ( ( finite4795055649997197647od_a_a @ A )
                  = ( finite4795055649997197647od_a_a @ B ) ) ) ) ) ) ) ) ).

% card_bij
thf(fact_353_additive__abelian__group_Ocard__sumset__iterated__minusset,axiom,
    ! [G: set_Product_prod_a_a,Addition: product_prod_a_a > product_prod_a_a > product_prod_a_a,Zero: product_prod_a_a,A: set_Product_prod_a_a,K: nat] :
      ( ( pluenn6837661381354551573od_a_a @ G @ Addition @ Zero )
     => ( ( finite4795055649997197647od_a_a @ ( pluenn2164957731648007268od_a_a @ G @ Addition @ Zero @ ( pluenn2765658404358379155od_a_a @ G @ Addition @ Zero @ A ) @ K ) )
        = ( finite4795055649997197647od_a_a @ ( pluenn2164957731648007268od_a_a @ G @ Addition @ Zero @ A @ K ) ) ) ) ).

% additive_abelian_group.card_sumset_iterated_minusset
thf(fact_354_additive__abelian__group_Ocard__sumset__iterated__minusset,axiom,
    ! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: set_nat,K: nat] :
      ( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
     => ( ( finite_card_nat @ ( pluenn7055013279391836755ed_nat @ G @ Addition @ Zero @ ( pluenn7323955030898006884et_nat @ G @ Addition @ Zero @ A ) @ K ) )
        = ( finite_card_nat @ ( pluenn7055013279391836755ed_nat @ G @ Addition @ Zero @ A @ K ) ) ) ) ).

% additive_abelian_group.card_sumset_iterated_minusset
thf(fact_355_additive__abelian__group_Ocard__sumset__iterated__minusset,axiom,
    ! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,K: nat] :
      ( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
     => ( ( finite_card_a @ ( pluenn1960970773371692859ated_a @ G @ Addition @ Zero @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A ) @ K ) )
        = ( finite_card_a @ ( pluenn1960970773371692859ated_a @ G @ Addition @ Zero @ A @ K ) ) ) ) ).

% additive_abelian_group.card_sumset_iterated_minusset
thf(fact_356_additive__abelian__group_Ocard__le__sumset,axiom,
    ! [G: set_Pr8826267807999420763od_a_a,Addition: ( product_prod_a_a > product_prod_a_a ) > ( product_prod_a_a > product_prod_a_a ) > product_prod_a_a > product_prod_a_a,Zero: product_prod_a_a > product_prod_a_a,A: set_Pr8826267807999420763od_a_a,A2: product_prod_a_a > product_prod_a_a,B: set_Pr8826267807999420763od_a_a] :
      ( ( pluenn7398882097854948023od_a_a @ G @ Addition @ Zero )
     => ( ( finite2714571839071174076od_a_a @ A )
       => ( ( member4020126937092221116od_a_a @ A2 @ A )
         => ( ( member4020126937092221116od_a_a @ A2 @ G )
           => ( ( finite2714571839071174076od_a_a @ B )
             => ( ( ord_le741722312431004091od_a_a @ B @ G )
               => ( ord_less_eq_nat @ ( finite3311474722152943101od_a_a @ B ) @ ( finite3311474722152943101od_a_a @ ( pluenn6825655340897727678od_a_a @ G @ Addition @ A @ B ) ) ) ) ) ) ) ) ) ).

% additive_abelian_group.card_le_sumset
thf(fact_357_additive__abelian__group_Ocard__le__sumset,axiom,
    ! [G: set_nat_nat,Addition: ( nat > nat ) > ( nat > nat ) > nat > nat,Zero: nat > nat,A: set_nat_nat,A2: nat > nat,B: set_nat_nat] :
      ( ( pluenn2593311475829585105at_nat @ G @ Addition @ Zero )
     => ( ( finite2115694454571419734at_nat @ A )
       => ( ( member_nat_nat @ A2 @ A )
         => ( ( member_nat_nat @ A2 @ G )
           => ( ( finite2115694454571419734at_nat @ B )
             => ( ( ord_le9059583361652607317at_nat @ B @ G )
               => ( ord_less_eq_nat @ ( finite_card_nat_nat @ B ) @ ( finite_card_nat_nat @ ( pluenn8091236430575893592at_nat @ G @ Addition @ A @ B ) ) ) ) ) ) ) ) ) ).

% additive_abelian_group.card_le_sumset
thf(fact_358_additive__abelian__group_Ocard__le__sumset,axiom,
    ! [G: set_nat_a,Addition: ( nat > a ) > ( nat > a ) > nat > a,Zero: nat > a,A: set_nat_a,A2: nat > a,B: set_nat_a] :
      ( ( pluenn1914567342357533651_nat_a @ G @ Addition @ Zero )
     => ( ( finite_finite_nat_a @ A )
       => ( ( member_nat_a @ A2 @ A )
         => ( ( member_nat_a @ A2 @ G )
           => ( ( finite_finite_nat_a @ B )
             => ( ( ord_le871467723717165285_nat_a @ B @ G )
               => ( ord_less_eq_nat @ ( finite_card_nat_a @ B ) @ ( finite_card_nat_a @ ( pluenn551768455426449420_nat_a @ G @ Addition @ A @ B ) ) ) ) ) ) ) ) ) ).

% additive_abelian_group.card_le_sumset
thf(fact_359_additive__abelian__group_Ocard__le__sumset,axiom,
    ! [G: set_a_6829686330177631172od_a_a,Addition: ( a > product_prod_a_a ) > ( a > product_prod_a_a ) > a > product_prod_a_a,Zero: a > product_prod_a_a,A: set_a_6829686330177631172od_a_a,A2: a > product_prod_a_a,B: set_a_6829686330177631172od_a_a] :
      ( ( pluenn3290985442670702752od_a_a @ G @ Addition @ Zero )
     => ( ( finite8821664692192630949od_a_a @ A )
       => ( ( member1957775702407316389od_a_a @ A2 @ A )
         => ( ( member1957775702407316389od_a_a @ A2 @ G )
           => ( ( finite8821664692192630949od_a_a @ B )
             => ( ( ord_le2371270124811097124od_a_a @ B @ G )
               => ( ord_less_eq_nat @ ( finite6862662396756234726od_a_a @ B ) @ ( finite6862662396756234726od_a_a @ ( pluenn6474628215148175783od_a_a @ G @ Addition @ A @ B ) ) ) ) ) ) ) ) ) ).

% additive_abelian_group.card_le_sumset
thf(fact_360_additive__abelian__group_Ocard__le__sumset,axiom,
    ! [G: set_a_nat,Addition: ( a > nat ) > ( a > nat ) > a > nat,Zero: a > nat,A: set_a_nat,A2: a > nat,B: set_a_nat] :
      ( ( pluenn7899775915488632493_a_nat @ G @ Addition @ Zero )
     => ( ( finite_finite_a_nat @ A )
       => ( ( member_a_nat @ A2 @ A )
         => ( ( member_a_nat @ A2 @ G )
           => ( ( finite_finite_a_nat @ B )
             => ( ( ord_le1612561287239139007_a_nat @ B @ G )
               => ( ord_less_eq_nat @ ( finite_card_a_nat @ B ) @ ( finite_card_a_nat @ ( pluenn6536977028557548262_a_nat @ G @ Addition @ A @ B ) ) ) ) ) ) ) ) ) ).

% additive_abelian_group.card_le_sumset
thf(fact_361_additive__abelian__group_Ocard__le__sumset,axiom,
    ! [G: set_a_a,Addition: ( a > a ) > ( a > a ) > a > a,Zero: a > a,A: set_a_a,A2: a > a,B: set_a_a] :
      ( ( pluenn1959071899916691703up_a_a @ G @ Addition @ Zero )
     => ( ( finite_finite_a_a @ A )
       => ( ( member_a_a @ A2 @ A )
         => ( ( member_a_a @ A2 @ G )
           => ( ( finite_finite_a_a @ B )
             => ( ( ord_less_eq_set_a_a @ B @ G )
               => ( ord_less_eq_nat @ ( finite_card_a_a @ B ) @ ( finite_card_a_a @ ( pluenn7679623682358442238et_a_a @ G @ Addition @ A @ B ) ) ) ) ) ) ) ) ) ).

% additive_abelian_group.card_le_sumset
thf(fact_362_additive__abelian__group_Ocard__le__sumset,axiom,
    ! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: set_nat,A2: nat,B: set_nat] :
      ( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
     => ( ( finite_finite_nat @ A )
       => ( ( member_nat @ A2 @ A )
         => ( ( member_nat @ A2 @ G )
           => ( ( finite_finite_nat @ B )
             => ( ( ord_less_eq_set_nat @ B @ G )
               => ( ord_less_eq_nat @ ( finite_card_nat @ B ) @ ( finite_card_nat @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A @ B ) ) ) ) ) ) ) ) ) ).

% additive_abelian_group.card_le_sumset
thf(fact_363_additive__abelian__group_Ocard__le__sumset,axiom,
    ! [G: set_Product_prod_a_a,Addition: product_prod_a_a > product_prod_a_a > product_prod_a_a,Zero: product_prod_a_a,A: set_Product_prod_a_a,A2: product_prod_a_a,B: set_Product_prod_a_a] :
      ( ( pluenn6837661381354551573od_a_a @ G @ Addition @ Zero )
     => ( ( finite6544458595007987280od_a_a @ A )
       => ( ( member1426531477525435216od_a_a @ A2 @ A )
         => ( ( member1426531477525435216od_a_a @ A2 @ G )
           => ( ( finite6544458595007987280od_a_a @ B )
             => ( ( ord_le746702958409616551od_a_a @ B @ G )
               => ( ord_less_eq_nat @ ( finite4795055649997197647od_a_a @ B ) @ ( finite4795055649997197647od_a_a @ ( pluenn659090379318209998od_a_a @ G @ Addition @ A @ B ) ) ) ) ) ) ) ) ) ).

% additive_abelian_group.card_le_sumset
thf(fact_364_additive__abelian__group_Ocard__le__sumset,axiom,
    ! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,A2: a,B: set_a] :
      ( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
     => ( ( finite_finite_a @ A )
       => ( ( member_a @ A2 @ A )
         => ( ( member_a @ A2 @ G )
           => ( ( finite_finite_a @ B )
             => ( ( ord_less_eq_set_a @ B @ G )
               => ( ord_less_eq_nat @ ( finite_card_a @ B ) @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) ) ) ) ) ) ) ) ) ).

% additive_abelian_group.card_le_sumset
thf(fact_365_additive__abelian__group_Ofinite__differenceset,axiom,
    ! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: set_nat,B: set_nat] :
      ( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
     => ( ( finite_finite_nat @ A )
       => ( ( finite_finite_nat @ B )
         => ( finite_finite_nat @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A @ ( pluenn7323955030898006884et_nat @ G @ Addition @ Zero @ B ) ) ) ) ) ) ).

% additive_abelian_group.finite_differenceset
thf(fact_366_additive__abelian__group_Ofinite__differenceset,axiom,
    ! [G: set_Product_prod_a_a,Addition: product_prod_a_a > product_prod_a_a > product_prod_a_a,Zero: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
      ( ( pluenn6837661381354551573od_a_a @ G @ Addition @ Zero )
     => ( ( finite6544458595007987280od_a_a @ A )
       => ( ( finite6544458595007987280od_a_a @ B )
         => ( finite6544458595007987280od_a_a @ ( pluenn659090379318209998od_a_a @ G @ Addition @ A @ ( pluenn2765658404358379155od_a_a @ G @ Addition @ Zero @ B ) ) ) ) ) ) ).

% additive_abelian_group.finite_differenceset
thf(fact_367_additive__abelian__group_Ofinite__differenceset,axiom,
    ! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,B: set_a] :
      ( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
     => ( ( finite_finite_a @ A )
       => ( ( finite_finite_a @ B )
         => ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ B ) ) ) ) ) ) ).

% additive_abelian_group.finite_differenceset
thf(fact_368_additive__abelian__group_Odifferenceset__commute,axiom,
    ! [G: set_a,Addition: a > a > a,Zero: a,B: set_a,A: set_a] :
      ( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
     => ( ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ ( pluenn3038260743871226533mset_a @ G @ Addition @ B @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A ) ) )
        = ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ B ) ) ) ) ).

% additive_abelian_group.differenceset_commute
thf(fact_369_additive__abelian__group_Ominusset__distrib__sum,axiom,
    ! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,B: set_a] :
      ( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
     => ( ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) )
        = ( pluenn3038260743871226533mset_a @ G @ Addition @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A ) @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ B ) ) ) ) ).

% additive_abelian_group.minusset_distrib_sum
thf(fact_370_additive__abelian__group_Odiff__minus__set,axiom,
    ! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,B: set_a] :
      ( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
     => ( ( pluenn3038260743871226533mset_a @ G @ Addition @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A ) @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ B ) )
        = ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) ) ) ) ).

% additive_abelian_group.diff_minus_set
thf(fact_371_additive__abelian__group_Ocard__differenceset__commute,axiom,
    ! [G: set_Product_prod_a_a,Addition: product_prod_a_a > product_prod_a_a > product_prod_a_a,Zero: product_prod_a_a,B: set_Product_prod_a_a,A: set_Product_prod_a_a] :
      ( ( pluenn6837661381354551573od_a_a @ G @ Addition @ Zero )
     => ( ( finite4795055649997197647od_a_a @ ( pluenn659090379318209998od_a_a @ G @ Addition @ B @ ( pluenn2765658404358379155od_a_a @ G @ Addition @ Zero @ A ) ) )
        = ( finite4795055649997197647od_a_a @ ( pluenn659090379318209998od_a_a @ G @ Addition @ A @ ( pluenn2765658404358379155od_a_a @ G @ Addition @ Zero @ B ) ) ) ) ) ).

% additive_abelian_group.card_differenceset_commute
thf(fact_372_additive__abelian__group_Ocard__differenceset__commute,axiom,
    ! [G: set_nat,Addition: nat > nat > nat,Zero: nat,B: set_nat,A: set_nat] :
      ( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
     => ( ( finite_card_nat @ ( pluenn3669378163024332905et_nat @ G @ Addition @ B @ ( pluenn7323955030898006884et_nat @ G @ Addition @ Zero @ A ) ) )
        = ( finite_card_nat @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A @ ( pluenn7323955030898006884et_nat @ G @ Addition @ Zero @ B ) ) ) ) ) ).

% additive_abelian_group.card_differenceset_commute
thf(fact_373_additive__abelian__group_Ocard__differenceset__commute,axiom,
    ! [G: set_a,Addition: a > a > a,Zero: a,B: set_a,A: set_a] :
      ( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
     => ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ B @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A ) ) )
        = ( finite_card_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ B ) ) ) ) ) ).

% additive_abelian_group.card_differenceset_commute
thf(fact_374_finite__SigmaI,axiom,
    ! [A: set_a,B: a > set_nat] :
      ( ( finite_finite_a @ A )
     => ( ! [A3: a] :
            ( ( member_a @ A3 @ A )
           => ( finite_finite_nat @ ( B @ A3 ) ) )
       => ( finite6644898363146130708_a_nat @ ( product_Sigma_a_nat @ A @ B ) ) ) ) ).

% finite_SigmaI
thf(fact_375_finite__SigmaI,axiom,
    ! [A: set_nat,B: nat > set_a] :
      ( ( finite_finite_nat @ A )
     => ( ! [A3: nat] :
            ( ( member_nat @ A3 @ A )
           => ( finite_finite_a @ ( B @ A3 ) ) )
       => ( finite659689790015031866_nat_a @ ( product_Sigma_nat_a @ A @ B ) ) ) ) ).

% finite_SigmaI
thf(fact_376_finite__SigmaI,axiom,
    ! [A: set_nat,B: nat > set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ! [A3: nat] :
            ( ( member_nat @ A3 @ A )
           => ( finite_finite_nat @ ( B @ A3 ) ) )
       => ( finite6177210948735845034at_nat @ ( produc457027306803732586at_nat @ A @ B ) ) ) ) ).

% finite_SigmaI
thf(fact_377_finite__SigmaI,axiom,
    ! [A: set_a,B: a > set_a] :
      ( ( finite_finite_a @ A )
     => ( ! [A3: a] :
            ( ( member_a @ A3 @ A )
           => ( finite_finite_a @ ( B @ A3 ) ) )
       => ( finite6544458595007987280od_a_a @ ( product_Sigma_a_a @ A @ B ) ) ) ) ).

% finite_SigmaI
thf(fact_378_finite__SigmaI,axiom,
    ! [A: set_nat_nat,B: ( nat > nat ) > set_a] :
      ( ( finite2115694454571419734at_nat @ A )
     => ( ! [A3: nat > nat] :
            ( ( member_nat_nat @ A3 @ A )
           => ( finite_finite_a @ ( B @ A3 ) ) )
       => ( finite6410384693344227723_nat_a @ ( produc2309334610237048501_nat_a @ A @ B ) ) ) ) ).

% finite_SigmaI
thf(fact_379_finite__SigmaI,axiom,
    ! [A: set_nat_a,B: ( nat > a ) > set_a] :
      ( ( finite_finite_nat_a @ A )
     => ( ! [A3: nat > a] :
            ( ( member_nat_a @ A3 @ A )
           => ( finite_finite_a @ ( B @ A3 ) ) )
       => ( finite1819187048435525417at_a_a @ ( produc6085074725259719457at_a_a @ A @ B ) ) ) ) ).

% finite_SigmaI
thf(fact_380_finite__SigmaI,axiom,
    ! [A: set_a_nat,B: ( a > nat ) > set_a] :
      ( ( finite_finite_a_nat @ A )
     => ( ! [A3: a > nat] :
            ( ( member_a_nat @ A3 @ A )
           => ( finite_finite_a @ ( B @ A3 ) ) )
       => ( finite4771698143452169935_nat_a @ ( produc9037585820276363975_nat_a @ A @ B ) ) ) ) ).

% finite_SigmaI
thf(fact_381_finite__SigmaI,axiom,
    ! [A: set_a_a,B: ( a > a ) > set_a] :
      ( ( finite_finite_a_a @ A )
     => ( ! [A3: a > a] :
            ( ( member_a_a @ A3 @ A )
           => ( finite_finite_a @ ( B @ A3 ) ) )
       => ( finite4229576983758256869_a_a_a @ ( product_Sigma_a_a_a @ A @ B ) ) ) ) ).

% finite_SigmaI
thf(fact_382_finite__SigmaI,axiom,
    ! [A: set_nat_nat,B: ( nat > nat ) > set_nat] :
      ( ( finite2115694454571419734at_nat @ A )
     => ( ! [A3: nat > nat] :
            ( ( member_nat_nat @ A3 @ A )
           => ( finite_finite_nat @ ( B @ A3 ) ) )
       => ( finite781204624771987737at_nat @ ( produc5982696620300550233at_nat @ A @ B ) ) ) ) ).

% finite_SigmaI
thf(fact_383_finite__SigmaI,axiom,
    ! [A: set_nat_a,B: ( nat > a ) > set_nat] :
      ( ( finite_finite_nat_a @ A )
     => ( ! [A3: nat > a] :
            ( ( member_nat_a @ A3 @ A )
           => ( finite_finite_nat @ ( B @ A3 ) ) )
       => ( finite375242429164574715_a_nat @ ( produc4703796345457561837_a_nat @ A @ B ) ) ) ) ).

% finite_SigmaI
thf(fact_384_finite__Collect__subsets,axiom,
    ! [A: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [B5: set_nat] : ( ord_less_eq_set_nat @ B5 @ A ) ) ) ) ).

% finite_Collect_subsets
thf(fact_385_finite__Collect__subsets,axiom,
    ! [A: set_Product_prod_a_a] :
      ( ( finite6544458595007987280od_a_a @ A )
     => ( finite8717734299975451184od_a_a
        @ ( collec1673347964119250290od_a_a
          @ ^ [B5: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ B5 @ A ) ) ) ) ).

% finite_Collect_subsets
thf(fact_386_finite__Collect__subsets,axiom,
    ! [A: set_a] :
      ( ( finite_finite_a @ A )
     => ( finite_finite_set_a
        @ ( collect_set_a
          @ ^ [B5: set_a] : ( ord_less_eq_set_a @ B5 @ A ) ) ) ) ).

% finite_Collect_subsets
thf(fact_387_card__sumset__0__iff,axiom,
    ! [A: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ A @ g )
     => ( ( ord_less_eq_set_a @ B @ g )
       => ( ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) )
            = zero_zero_nat )
          = ( ( ( finite_card_a @ A )
              = zero_zero_nat )
            | ( ( finite_card_a @ B )
              = zero_zero_nat ) ) ) ) ) ).

% card_sumset_0_iff
thf(fact_388_group__axioms,axiom,
    group_group_a @ g @ addition @ zero ).

% group_axioms
thf(fact_389__092_060phi_062__def,axiom,
    ( phi
    = ( produc408267641121961211od_a_a
      @ ^ [U2: a,X2: a] : ( product_Pair_a_a @ ( addition @ U2 @ ( group_inverse_a @ g @ addition @ zero @ ( v2 @ X2 ) ) ) @ ( addition @ U2 @ ( group_inverse_a @ g @ addition @ zero @ ( w2 @ X2 ) ) ) ) ) ) ).

% \<phi>_def
thf(fact_390_sumset__subset__insert_I1_J,axiom,
    ! [A: set_a,B: set_a,X: a] : ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ ( insert_a @ X @ B ) ) ) ).

% sumset_subset_insert(1)
thf(fact_391_sumset__subset__insert_I2_J,axiom,
    ! [A: set_a,B: set_a,X: a] : ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ ( insert_a @ X @ A ) @ B ) ) ).

% sumset_subset_insert(2)
thf(fact_392_finite__sumset_H,axiom,
    ! [A: set_a,B: set_a] :
      ( ( finite_finite_a @ ( inf_inf_set_a @ A @ g ) )
     => ( ( finite_finite_a @ ( inf_inf_set_a @ B @ g ) )
       => ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) ) ) ) ).

% finite_sumset'
thf(fact_393_invertible__right__inverse2,axiom,
    ! [U: a,V2: a] :
      ( ( group_invertible_a @ g @ addition @ zero @ U )
     => ( ( member_a @ U @ g )
       => ( ( member_a @ V2 @ g )
         => ( ( addition @ U @ ( addition @ ( group_inverse_a @ g @ addition @ zero @ U ) @ V2 ) )
            = V2 ) ) ) ) ).

% invertible_right_inverse2
thf(fact_394_invertible__left__inverse2,axiom,
    ! [U: a,V2: a] :
      ( ( group_invertible_a @ g @ addition @ zero @ U )
     => ( ( member_a @ U @ g )
       => ( ( member_a @ V2 @ g )
         => ( ( addition @ ( group_inverse_a @ g @ addition @ zero @ U ) @ ( addition @ U @ V2 ) )
            = V2 ) ) ) ) ).

% invertible_left_inverse2
thf(fact_395_inverse__composition__commute,axiom,
    ! [X: a,Y: a] :
      ( ( group_invertible_a @ g @ addition @ zero @ X )
     => ( ( group_invertible_a @ g @ addition @ zero @ Y )
       => ( ( member_a @ X @ g )
         => ( ( member_a @ Y @ g )
           => ( ( group_inverse_a @ g @ addition @ zero @ ( addition @ X @ Y ) )
              = ( addition @ ( group_inverse_a @ g @ addition @ zero @ Y ) @ ( group_inverse_a @ g @ addition @ zero @ X ) ) ) ) ) ) ) ).

% inverse_composition_commute
thf(fact_396_invertibleE,axiom,
    ! [U: a] :
      ( ( group_invertible_a @ g @ addition @ zero @ U )
     => ( ! [V3: a] :
            ( ( ( ( addition @ U @ V3 )
                = zero )
              & ( ( addition @ V3 @ U )
                = zero ) )
           => ~ ( member_a @ V3 @ g ) )
       => ~ ( member_a @ U @ g ) ) ) ).

% invertibleE
thf(fact_397_invertible__def,axiom,
    ! [U: a] :
      ( ( member_a @ U @ g )
     => ( ( group_invertible_a @ g @ addition @ zero @ U )
        = ( ? [X2: a] :
              ( ( member_a @ X2 @ g )
              & ( ( addition @ U @ X2 )
                = zero )
              & ( ( addition @ X2 @ U )
                = zero ) ) ) ) ) ).

% invertible_def
thf(fact_398_unit__invertible,axiom,
    group_invertible_a @ g @ addition @ zero @ zero ).

% unit_invertible
thf(fact_399_finite__Collect__conjI,axiom,
    ! [P: a > $o,Q: a > $o] :
      ( ( ( finite_finite_a @ ( collect_a @ P ) )
        | ( finite_finite_a @ ( collect_a @ Q ) ) )
     => ( finite_finite_a
        @ ( collect_a
          @ ^ [X2: a] :
              ( ( P @ X2 )
              & ( Q @ X2 ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_400_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
          @ ^ [X2: nat] :
              ( ( P @ X2 )
              & ( Q @ X2 ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_401_finite__Collect__conjI,axiom,
    ! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
      ( ( ( finite6544458595007987280od_a_a @ ( collec3336397797384452498od_a_a @ P ) )
        | ( finite6544458595007987280od_a_a @ ( collec3336397797384452498od_a_a @ Q ) ) )
     => ( finite6544458595007987280od_a_a
        @ ( collec3336397797384452498od_a_a
          @ ^ [X2: product_prod_a_a] :
              ( ( P @ X2 )
              & ( Q @ X2 ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_402_finite__Collect__disjI,axiom,
    ! [P: a > $o,Q: a > $o] :
      ( ( finite_finite_a
        @ ( collect_a
          @ ^ [X2: a] :
              ( ( P @ X2 )
              | ( Q @ X2 ) ) ) )
      = ( ( finite_finite_a @ ( collect_a @ P ) )
        & ( finite_finite_a @ ( collect_a @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_403_finite__Collect__disjI,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X2: nat] :
              ( ( P @ X2 )
              | ( Q @ X2 ) ) ) )
      = ( ( finite_finite_nat @ ( collect_nat @ P ) )
        & ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_404_finite__Collect__disjI,axiom,
    ! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
      ( ( finite6544458595007987280od_a_a
        @ ( collec3336397797384452498od_a_a
          @ ^ [X2: product_prod_a_a] :
              ( ( P @ X2 )
              | ( Q @ X2 ) ) ) )
      = ( ( finite6544458595007987280od_a_a @ ( collec3336397797384452498od_a_a @ P ) )
        & ( finite6544458595007987280od_a_a @ ( collec3336397797384452498od_a_a @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_405_finite__Collect__le__nat,axiom,
    ! [K: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N: nat] : ( ord_less_eq_nat @ N @ K ) ) ) ).

% finite_Collect_le_nat
thf(fact_406_card__sumset__0__iff_H,axiom,
    ! [A: set_a,B: set_a] :
      ( ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) )
        = zero_zero_nat )
      = ( ( ( finite_card_a @ ( inf_inf_set_a @ A @ g ) )
          = zero_zero_nat )
        | ( ( finite_card_a @ ( inf_inf_set_a @ B @ g ) )
          = zero_zero_nat ) ) ) ).

% card_sumset_0_iff'
thf(fact_407_finite__insert,axiom,
    ! [A2: a,A: set_a] :
      ( ( finite_finite_a @ ( insert_a @ A2 @ A ) )
      = ( finite_finite_a @ A ) ) ).

% finite_insert
thf(fact_408_finite__insert,axiom,
    ! [A2: nat,A: set_nat] :
      ( ( finite_finite_nat @ ( insert_nat @ A2 @ A ) )
      = ( finite_finite_nat @ A ) ) ).

% finite_insert
thf(fact_409_finite__insert,axiom,
    ! [A2: product_prod_a_a,A: set_Product_prod_a_a] :
      ( ( finite6544458595007987280od_a_a @ ( insert4534936382041156343od_a_a @ A2 @ A ) )
      = ( finite6544458595007987280od_a_a @ A ) ) ).

% finite_insert
thf(fact_410_finite__Int,axiom,
    ! [F2: set_nat,G: set_nat] :
      ( ( ( finite_finite_nat @ F2 )
        | ( finite_finite_nat @ G ) )
     => ( finite_finite_nat @ ( inf_inf_set_nat @ F2 @ G ) ) ) ).

% finite_Int
thf(fact_411_finite__Int,axiom,
    ! [F2: set_Product_prod_a_a,G: set_Product_prod_a_a] :
      ( ( ( finite6544458595007987280od_a_a @ F2 )
        | ( finite6544458595007987280od_a_a @ G ) )
     => ( finite6544458595007987280od_a_a @ ( inf_in8905007599844390133od_a_a @ F2 @ G ) ) ) ).

% finite_Int
thf(fact_412_finite__Int,axiom,
    ! [F2: set_a,G: set_a] :
      ( ( ( finite_finite_a @ F2 )
        | ( finite_finite_a @ G ) )
     => ( finite_finite_a @ ( inf_inf_set_a @ F2 @ G ) ) ) ).

% finite_Int
thf(fact_413_Pi__split__insert__domain,axiom,
    ! [X: nat > nat,I2: nat,I: set_nat,X5: nat > set_nat] :
      ( ( member_nat_nat @ X @ ( pi_nat_nat @ ( insert_nat @ I2 @ I ) @ X5 ) )
      = ( ( member_nat_nat @ X @ ( pi_nat_nat @ I @ X5 ) )
        & ( member_nat @ ( X @ I2 ) @ ( X5 @ I2 ) ) ) ) ).

% Pi_split_insert_domain
thf(fact_414_Pi__split__insert__domain,axiom,
    ! [X: nat > a,I2: nat,I: set_nat,X5: nat > set_a] :
      ( ( member_nat_a @ X @ ( pi_nat_a @ ( insert_nat @ I2 @ I ) @ X5 ) )
      = ( ( member_nat_a @ X @ ( pi_nat_a @ I @ X5 ) )
        & ( member_a @ ( X @ I2 ) @ ( X5 @ I2 ) ) ) ) ).

% Pi_split_insert_domain
thf(fact_415_Pi__split__insert__domain,axiom,
    ! [X: a > nat,I2: a,I: set_a,X5: a > set_nat] :
      ( ( member_a_nat @ X @ ( pi_a_nat @ ( insert_a @ I2 @ I ) @ X5 ) )
      = ( ( member_a_nat @ X @ ( pi_a_nat @ I @ X5 ) )
        & ( member_nat @ ( X @ I2 ) @ ( X5 @ I2 ) ) ) ) ).

% Pi_split_insert_domain
thf(fact_416_Pi__split__insert__domain,axiom,
    ! [X: a > a,I2: a,I: set_a,X5: a > set_a] :
      ( ( member_a_a @ X @ ( pi_a_a @ ( insert_a @ I2 @ I ) @ X5 ) )
      = ( ( member_a_a @ X @ ( pi_a_a @ I @ X5 ) )
        & ( member_a @ ( X @ I2 ) @ ( X5 @ I2 ) ) ) ) ).

% Pi_split_insert_domain
thf(fact_417_Pi__split__insert__domain,axiom,
    ! [X: a > nat > nat,I2: a,I: set_a,X5: a > set_nat_nat] :
      ( ( member_a_nat_nat @ X @ ( pi_a_nat_nat @ ( insert_a @ I2 @ I ) @ X5 ) )
      = ( ( member_a_nat_nat @ X @ ( pi_a_nat_nat @ I @ X5 ) )
        & ( member_nat_nat @ ( X @ I2 ) @ ( X5 @ I2 ) ) ) ) ).

% Pi_split_insert_domain
thf(fact_418_Pi__split__insert__domain,axiom,
    ! [X: a > nat > a,I2: a,I: set_a,X5: a > set_nat_a] :
      ( ( member_a_nat_a2 @ X @ ( pi_a_nat_a2 @ ( insert_a @ I2 @ I ) @ X5 ) )
      = ( ( member_a_nat_a2 @ X @ ( pi_a_nat_a2 @ I @ X5 ) )
        & ( member_nat_a @ ( X @ I2 ) @ ( X5 @ I2 ) ) ) ) ).

% Pi_split_insert_domain
thf(fact_419_Pi__split__insert__domain,axiom,
    ! [X: a > a > nat,I2: a,I: set_a,X5: a > set_a_nat] :
      ( ( member_a_a_nat @ X @ ( pi_a_a_nat @ ( insert_a @ I2 @ I ) @ X5 ) )
      = ( ( member_a_a_nat @ X @ ( pi_a_a_nat @ I @ X5 ) )
        & ( member_a_nat @ ( X @ I2 ) @ ( X5 @ I2 ) ) ) ) ).

% Pi_split_insert_domain
thf(fact_420_Pi__split__insert__domain,axiom,
    ! [X: a > a > a,I2: a,I: set_a,X5: a > set_a_a] :
      ( ( member_a_a_a2 @ X @ ( pi_a_a_a2 @ ( insert_a @ I2 @ I ) @ X5 ) )
      = ( ( member_a_a_a2 @ X @ ( pi_a_a_a2 @ I @ X5 ) )
        & ( member_a_a @ ( X @ I2 ) @ ( X5 @ I2 ) ) ) ) ).

% Pi_split_insert_domain
thf(fact_421_Pi__split__insert__domain,axiom,
    ! [X: a > product_prod_a_a,I2: a,I: set_a,X5: a > set_Product_prod_a_a] :
      ( ( member1957775702407316389od_a_a @ X @ ( pi_a_P2178097759547960436od_a_a @ ( insert_a @ I2 @ I ) @ X5 ) )
      = ( ( member1957775702407316389od_a_a @ X @ ( pi_a_P2178097759547960436od_a_a @ I @ X5 ) )
        & ( member1426531477525435216od_a_a @ ( X @ I2 ) @ ( X5 @ I2 ) ) ) ) ).

% Pi_split_insert_domain
thf(fact_422_Pi__split__insert__domain,axiom,
    ! [X: a > a > product_prod_a_a,I2: a,I: set_a,X5: a > set_a_6829686330177631172od_a_a] :
      ( ( member2522163699605435814od_a_a @ X @ ( pi_a_a3761179661865972097od_a_a @ ( insert_a @ I2 @ I ) @ X5 ) )
      = ( ( member2522163699605435814od_a_a @ X @ ( pi_a_a3761179661865972097od_a_a @ I @ X5 ) )
        & ( member1957775702407316389od_a_a @ ( X @ I2 ) @ ( X5 @ I2 ) ) ) ) ).

% Pi_split_insert_domain
thf(fact_423_card_Oinfinite,axiom,
    ! [A: set_a] :
      ( ~ ( finite_finite_a @ A )
     => ( ( finite_card_a @ A )
        = zero_zero_nat ) ) ).

% card.infinite
thf(fact_424_card_Oinfinite,axiom,
    ! [A: set_nat] :
      ( ~ ( finite_finite_nat @ A )
     => ( ( finite_card_nat @ A )
        = zero_zero_nat ) ) ).

% card.infinite
thf(fact_425_card_Oinfinite,axiom,
    ! [A: set_Product_prod_a_a] :
      ( ~ ( finite6544458595007987280od_a_a @ A )
     => ( ( finite4795055649997197647od_a_a @ A )
        = zero_zero_nat ) ) ).

% card.infinite
thf(fact_426_sumset__Int__carrier__eq_I2_J,axiom,
    ! [A: set_a,B: set_a] :
      ( ( pluenn3038260743871226533mset_a @ g @ addition @ ( inf_inf_set_a @ A @ g ) @ B )
      = ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) ) ).

% sumset_Int_carrier_eq(2)
thf(fact_427_sumset__Int__carrier__eq_I1_J,axiom,
    ! [A: set_a,B: set_a] :
      ( ( pluenn3038260743871226533mset_a @ g @ addition @ A @ ( inf_inf_set_a @ B @ g ) )
      = ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) ) ).

% sumset_Int_carrier_eq(1)
thf(fact_428_sumset__Int__carrier,axiom,
    ! [A: set_a,B: set_a] :
      ( ( inf_inf_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) @ g )
      = ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) ) ).

% sumset_Int_carrier
thf(fact_429_composition__invertible,axiom,
    ! [X: a,Y: a] :
      ( ( group_invertible_a @ g @ addition @ zero @ X )
     => ( ( group_invertible_a @ g @ addition @ zero @ Y )
       => ( ( member_a @ X @ g )
         => ( ( member_a @ Y @ g )
           => ( group_invertible_a @ g @ addition @ zero @ ( addition @ X @ Y ) ) ) ) ) ) ).

% composition_invertible
thf(fact_430_invertible,axiom,
    ! [U: a] :
      ( ( member_a @ U @ g )
     => ( group_invertible_a @ g @ addition @ zero @ U ) ) ).

% invertible
thf(fact_431_invertibleI,axiom,
    ! [U: a,V2: a] :
      ( ( ( addition @ U @ V2 )
        = zero )
     => ( ( ( addition @ V2 @ U )
          = zero )
       => ( ( member_a @ U @ g )
         => ( ( member_a @ V2 @ g )
           => ( group_invertible_a @ g @ addition @ zero @ U ) ) ) ) ) ).

% invertibleI
thf(fact_432_invertible__left__cancel,axiom,
    ! [X: a,Y: a,Z: a] :
      ( ( group_invertible_a @ g @ addition @ zero @ X )
     => ( ( member_a @ X @ g )
       => ( ( member_a @ Y @ g )
         => ( ( member_a @ Z @ g )
           => ( ( ( addition @ X @ Y )
                = ( addition @ X @ Z ) )
              = ( Y = Z ) ) ) ) ) ) ).

% invertible_left_cancel
thf(fact_433_invertible__right__cancel,axiom,
    ! [X: a,Y: a,Z: a] :
      ( ( group_invertible_a @ g @ addition @ zero @ X )
     => ( ( member_a @ X @ g )
       => ( ( member_a @ Y @ g )
         => ( ( member_a @ Z @ g )
           => ( ( ( addition @ Y @ X )
                = ( addition @ Z @ X ) )
              = ( Y = Z ) ) ) ) ) ) ).

% invertible_right_cancel
thf(fact_434_minus__minusset,axiom,
    ! [A: set_a] :
      ( ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A ) )
      = ( inf_inf_set_a @ A @ g ) ) ).

% minus_minusset
thf(fact_435_invertible__inverse__closed,axiom,
    ! [U: a] :
      ( ( group_invertible_a @ g @ addition @ zero @ U )
     => ( ( member_a @ U @ g )
       => ( member_a @ ( group_inverse_a @ g @ addition @ zero @ U ) @ g ) ) ) ).

% invertible_inverse_closed
thf(fact_436_invertible__inverse__inverse,axiom,
    ! [U: a] :
      ( ( group_invertible_a @ g @ addition @ zero @ U )
     => ( ( member_a @ U @ g )
       => ( ( group_inverse_a @ g @ addition @ zero @ ( group_inverse_a @ g @ addition @ zero @ U ) )
          = U ) ) ) ).

% invertible_inverse_inverse
thf(fact_437_invertible__inverse__invertible,axiom,
    ! [U: a] :
      ( ( group_invertible_a @ g @ addition @ zero @ U )
     => ( ( member_a @ U @ g )
       => ( group_invertible_a @ g @ addition @ zero @ ( group_inverse_a @ g @ addition @ zero @ U ) ) ) ) ).

% invertible_inverse_invertible
thf(fact_438_invertible__left__inverse,axiom,
    ! [U: a] :
      ( ( group_invertible_a @ g @ addition @ zero @ U )
     => ( ( member_a @ U @ g )
       => ( ( addition @ ( group_inverse_a @ g @ addition @ zero @ U ) @ U )
          = zero ) ) ) ).

% invertible_left_inverse
thf(fact_439_invertible__right__inverse,axiom,
    ! [U: a] :
      ( ( group_invertible_a @ g @ addition @ zero @ U )
     => ( ( member_a @ U @ g )
       => ( ( addition @ U @ ( group_inverse_a @ g @ addition @ zero @ U ) )
          = zero ) ) ) ).

% invertible_right_inverse
thf(fact_440_card__minusset,axiom,
    ! [A: set_a] :
      ( ( finite_card_a @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A ) )
      = ( finite_card_a @ ( inf_inf_set_a @ A @ g ) ) ) ).

% card_minusset
thf(fact_441_finite_OinsertI,axiom,
    ! [A: set_a,A2: a] :
      ( ( finite_finite_a @ A )
     => ( finite_finite_a @ ( insert_a @ A2 @ A ) ) ) ).

% finite.insertI
thf(fact_442_finite_OinsertI,axiom,
    ! [A: set_nat,A2: nat] :
      ( ( finite_finite_nat @ A )
     => ( finite_finite_nat @ ( insert_nat @ A2 @ A ) ) ) ).

% finite.insertI
thf(fact_443_finite_OinsertI,axiom,
    ! [A: set_Product_prod_a_a,A2: product_prod_a_a] :
      ( ( finite6544458595007987280od_a_a @ A )
     => ( finite6544458595007987280od_a_a @ ( insert4534936382041156343od_a_a @ A2 @ A ) ) ) ).

% finite.insertI
thf(fact_444_Pi__Int,axiom,
    ! [I: set_Product_prod_a_a,E: product_prod_a_a > set_Product_prod_a_a,F2: product_prod_a_a > set_Product_prod_a_a] :
      ( ( inf_in3507259597588448237od_a_a @ ( pi_Pro6370639526499058571od_a_a @ I @ E ) @ ( pi_Pro6370639526499058571od_a_a @ I @ F2 ) )
      = ( pi_Pro6370639526499058571od_a_a @ I
        @ ^ [I3: product_prod_a_a] : ( inf_in8905007599844390133od_a_a @ ( E @ I3 ) @ ( F2 @ I3 ) ) ) ) ).

% Pi_Int
thf(fact_445_Pi__Int,axiom,
    ! [I: set_nat,E: nat > set_nat,F2: nat > set_nat] :
      ( ( inf_inf_set_nat_nat @ ( pi_nat_nat @ I @ E ) @ ( pi_nat_nat @ I @ F2 ) )
      = ( pi_nat_nat @ I
        @ ^ [I3: nat] : ( inf_inf_set_nat @ ( E @ I3 ) @ ( F2 @ I3 ) ) ) ) ).

% Pi_Int
thf(fact_446_Pi__Int,axiom,
    ! [I: set_a,E: a > set_Product_prod_a_a,F2: a > set_Product_prod_a_a] :
      ( ( inf_in4434067354547777622od_a_a @ ( pi_a_P2178097759547960436od_a_a @ I @ E ) @ ( pi_a_P2178097759547960436od_a_a @ I @ F2 ) )
      = ( pi_a_P2178097759547960436od_a_a @ I
        @ ^ [I3: a] : ( inf_in8905007599844390133od_a_a @ ( E @ I3 ) @ ( F2 @ I3 ) ) ) ) ).

% Pi_Int
thf(fact_447_Pi__Int,axiom,
    ! [I: set_a,E: a > set_nat,F2: a > set_nat] :
      ( ( inf_inf_set_a_nat @ ( pi_a_nat @ I @ E ) @ ( pi_a_nat @ I @ F2 ) )
      = ( pi_a_nat @ I
        @ ^ [I3: a] : ( inf_inf_set_nat @ ( E @ I3 ) @ ( F2 @ I3 ) ) ) ) ).

% Pi_Int
thf(fact_448_Pi__Int,axiom,
    ! [I: set_nat,E: nat > set_a,F2: nat > set_a] :
      ( ( inf_inf_set_nat_a @ ( pi_nat_a @ I @ E ) @ ( pi_nat_a @ I @ F2 ) )
      = ( pi_nat_a @ I
        @ ^ [I3: nat] : ( inf_inf_set_a @ ( E @ I3 ) @ ( F2 @ I3 ) ) ) ) ).

% Pi_Int
thf(fact_449_Pi__Int,axiom,
    ! [I: set_a,E: a > set_a,F2: a > set_a] :
      ( ( inf_inf_set_a_a @ ( pi_a_a @ I @ E ) @ ( pi_a_a @ I @ F2 ) )
      = ( pi_a_a @ I
        @ ^ [I3: a] : ( inf_inf_set_a @ ( E @ I3 ) @ ( F2 @ I3 ) ) ) ) ).

% Pi_Int
thf(fact_450_card__insert__le,axiom,
    ! [A: set_Product_prod_a_a,X: product_prod_a_a] : ( ord_less_eq_nat @ ( finite4795055649997197647od_a_a @ A ) @ ( finite4795055649997197647od_a_a @ ( insert4534936382041156343od_a_a @ X @ A ) ) ) ).

% card_insert_le
thf(fact_451_card__insert__le,axiom,
    ! [A: set_a,X: a] : ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ ( insert_a @ X @ A ) ) ) ).

% card_insert_le
thf(fact_452_card__insert__le,axiom,
    ! [A: set_nat,X: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ ( insert_nat @ X @ A ) ) ) ).

% card_insert_le
thf(fact_453_additive__abelian__group_Ocard__sumset__0__iff_H,axiom,
    ! [G: set_Product_prod_a_a,Addition: product_prod_a_a > product_prod_a_a > product_prod_a_a,Zero: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
      ( ( pluenn6837661381354551573od_a_a @ G @ Addition @ Zero )
     => ( ( ( finite4795055649997197647od_a_a @ ( pluenn659090379318209998od_a_a @ G @ Addition @ A @ B ) )
          = zero_zero_nat )
        = ( ( ( finite4795055649997197647od_a_a @ ( inf_in8905007599844390133od_a_a @ A @ G ) )
            = zero_zero_nat )
          | ( ( finite4795055649997197647od_a_a @ ( inf_in8905007599844390133od_a_a @ B @ G ) )
            = zero_zero_nat ) ) ) ) ).

% additive_abelian_group.card_sumset_0_iff'
thf(fact_454_additive__abelian__group_Ocard__sumset__0__iff_H,axiom,
    ! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: set_nat,B: set_nat] :
      ( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
     => ( ( ( finite_card_nat @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A @ B ) )
          = zero_zero_nat )
        = ( ( ( finite_card_nat @ ( inf_inf_set_nat @ A @ G ) )
            = zero_zero_nat )
          | ( ( finite_card_nat @ ( inf_inf_set_nat @ B @ G ) )
            = zero_zero_nat ) ) ) ) ).

% additive_abelian_group.card_sumset_0_iff'
thf(fact_455_additive__abelian__group_Ocard__sumset__0__iff_H,axiom,
    ! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,B: set_a] :
      ( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
     => ( ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) )
          = zero_zero_nat )
        = ( ( ( finite_card_a @ ( inf_inf_set_a @ A @ G ) )
            = zero_zero_nat )
          | ( ( finite_card_a @ ( inf_inf_set_a @ B @ G ) )
            = zero_zero_nat ) ) ) ) ).

% additive_abelian_group.card_sumset_0_iff'
thf(fact_456_additive__abelian__group_Osumset__Int__carrier__eq_I2_J,axiom,
    ! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,B: set_a] :
      ( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
     => ( ( pluenn3038260743871226533mset_a @ G @ Addition @ ( inf_inf_set_a @ A @ G ) @ B )
        = ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) ) ) ).

% additive_abelian_group.sumset_Int_carrier_eq(2)
thf(fact_457_additive__abelian__group_Osumset__Int__carrier__eq_I1_J,axiom,
    ! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,B: set_a] :
      ( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
     => ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ ( inf_inf_set_a @ B @ G ) )
        = ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) ) ) ).

% additive_abelian_group.sumset_Int_carrier_eq(1)
thf(fact_458_additive__abelian__group_Osumset__Int__carrier,axiom,
    ! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,B: set_a] :
      ( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
     => ( ( inf_inf_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) @ G )
        = ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) ) ) ).

% additive_abelian_group.sumset_Int_carrier
thf(fact_459_additive__abelian__group_Ominus__minusset,axiom,
    ! [G: set_a,Addition: a > a > a,Zero: a,A: set_a] :
      ( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
     => ( ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A ) )
        = ( inf_inf_set_a @ A @ G ) ) ) ).

% additive_abelian_group.minus_minusset
thf(fact_460_additive__abelian__group_Ofinite__sumset_H,axiom,
    ! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: set_nat,B: set_nat] :
      ( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
     => ( ( finite_finite_nat @ ( inf_inf_set_nat @ A @ G ) )
       => ( ( finite_finite_nat @ ( inf_inf_set_nat @ B @ G ) )
         => ( finite_finite_nat @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A @ B ) ) ) ) ) ).

% additive_abelian_group.finite_sumset'
thf(fact_461_additive__abelian__group_Ofinite__sumset_H,axiom,
    ! [G: set_Product_prod_a_a,Addition: product_prod_a_a > product_prod_a_a > product_prod_a_a,Zero: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
      ( ( pluenn6837661381354551573od_a_a @ G @ Addition @ Zero )
     => ( ( finite6544458595007987280od_a_a @ ( inf_in8905007599844390133od_a_a @ A @ G ) )
       => ( ( finite6544458595007987280od_a_a @ ( inf_in8905007599844390133od_a_a @ B @ G ) )
         => ( finite6544458595007987280od_a_a @ ( pluenn659090379318209998od_a_a @ G @ Addition @ A @ B ) ) ) ) ) ).

% additive_abelian_group.finite_sumset'
thf(fact_462_additive__abelian__group_Ofinite__sumset_H,axiom,
    ! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,B: set_a] :
      ( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
     => ( ( finite_finite_a @ ( inf_inf_set_a @ A @ G ) )
       => ( ( finite_finite_a @ ( inf_inf_set_a @ B @ G ) )
         => ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) ) ) ) ) ).

% additive_abelian_group.finite_sumset'
thf(fact_463_additive__abelian__group_Osumset__subset__insert_I1_J,axiom,
    ! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,B: set_a,X: a] :
      ( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
     => ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ ( insert_a @ X @ B ) ) ) ) ).

% additive_abelian_group.sumset_subset_insert(1)
thf(fact_464_additive__abelian__group_Osumset__subset__insert_I2_J,axiom,
    ! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,B: set_a,X: a] :
      ( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
     => ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) @ ( pluenn3038260743871226533mset_a @ G @ Addition @ ( insert_a @ X @ A ) @ B ) ) ) ).

% additive_abelian_group.sumset_subset_insert(2)
thf(fact_465_additive__abelian__group_Ocard__minusset,axiom,
    ! [G: set_Product_prod_a_a,Addition: product_prod_a_a > product_prod_a_a > product_prod_a_a,Zero: product_prod_a_a,A: set_Product_prod_a_a] :
      ( ( pluenn6837661381354551573od_a_a @ G @ Addition @ Zero )
     => ( ( finite4795055649997197647od_a_a @ ( pluenn2765658404358379155od_a_a @ G @ Addition @ Zero @ A ) )
        = ( finite4795055649997197647od_a_a @ ( inf_in8905007599844390133od_a_a @ A @ G ) ) ) ) ).

% additive_abelian_group.card_minusset
thf(fact_466_additive__abelian__group_Ocard__minusset,axiom,
    ! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: set_nat] :
      ( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
     => ( ( finite_card_nat @ ( pluenn7323955030898006884et_nat @ G @ Addition @ Zero @ A ) )
        = ( finite_card_nat @ ( inf_inf_set_nat @ A @ G ) ) ) ) ).

% additive_abelian_group.card_minusset
thf(fact_467_additive__abelian__group_Ocard__minusset,axiom,
    ! [G: set_a,Addition: a > a > a,Zero: a,A: set_a] :
      ( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
     => ( ( finite_card_a @ ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A ) )
        = ( finite_card_a @ ( inf_inf_set_a @ A @ G ) ) ) ) ).

% additive_abelian_group.card_minusset
thf(fact_468_not__finite__existsD,axiom,
    ! [P: a > $o] :
      ( ~ ( finite_finite_a @ ( collect_a @ P ) )
     => ? [X_1: a] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_469_not__finite__existsD,axiom,
    ! [P: nat > $o] :
      ( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
     => ? [X_1: nat] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_470_not__finite__existsD,axiom,
    ! [P: product_prod_a_a > $o] :
      ( ~ ( finite6544458595007987280od_a_a @ ( collec3336397797384452498od_a_a @ P ) )
     => ? [X_1: product_prod_a_a] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_471_pigeonhole__infinite__rel,axiom,
    ! [A: set_a,B: set_a,R2: a > a > $o] :
      ( ~ ( finite_finite_a @ A )
     => ( ( finite_finite_a @ B )
       => ( ! [X3: a] :
              ( ( member_a @ X3 @ A )
             => ? [Xa: a] :
                  ( ( member_a @ Xa @ B )
                  & ( R2 @ X3 @ Xa ) ) )
         => ? [X3: a] :
              ( ( member_a @ X3 @ B )
              & ~ ( finite_finite_a
                  @ ( collect_a
                    @ ^ [A4: a] :
                        ( ( member_a @ A4 @ A )
                        & ( R2 @ A4 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_472_pigeonhole__infinite__rel,axiom,
    ! [A: set_a,B: set_nat,R2: a > nat > $o] :
      ( ~ ( finite_finite_a @ A )
     => ( ( finite_finite_nat @ B )
       => ( ! [X3: a] :
              ( ( member_a @ X3 @ A )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B )
                  & ( R2 @ X3 @ Xa ) ) )
         => ? [X3: nat] :
              ( ( member_nat @ X3 @ B )
              & ~ ( finite_finite_a
                  @ ( collect_a
                    @ ^ [A4: a] :
                        ( ( member_a @ A4 @ A )
                        & ( R2 @ A4 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_473_pigeonhole__infinite__rel,axiom,
    ! [A: set_nat,B: set_a,R2: nat > a > $o] :
      ( ~ ( finite_finite_nat @ A )
     => ( ( finite_finite_a @ B )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ A )
             => ? [Xa: a] :
                  ( ( member_a @ Xa @ B )
                  & ( R2 @ X3 @ Xa ) ) )
         => ? [X3: a] :
              ( ( member_a @ X3 @ B )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A4: nat] :
                        ( ( member_nat @ A4 @ A )
                        & ( R2 @ A4 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_474_pigeonhole__infinite__rel,axiom,
    ! [A: set_nat,B: set_nat,R2: nat > nat > $o] :
      ( ~ ( finite_finite_nat @ A )
     => ( ( finite_finite_nat @ B )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ A )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B )
                  & ( R2 @ X3 @ Xa ) ) )
         => ? [X3: nat] :
              ( ( member_nat @ X3 @ B )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A4: nat] :
                        ( ( member_nat @ A4 @ A )
                        & ( R2 @ A4 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_475_pigeonhole__infinite__rel,axiom,
    ! [A: set_nat_nat,B: set_a,R2: ( nat > nat ) > a > $o] :
      ( ~ ( finite2115694454571419734at_nat @ A )
     => ( ( finite_finite_a @ B )
       => ( ! [X3: nat > nat] :
              ( ( member_nat_nat @ X3 @ A )
             => ? [Xa: a] :
                  ( ( member_a @ Xa @ B )
                  & ( R2 @ X3 @ Xa ) ) )
         => ? [X3: a] :
              ( ( member_a @ X3 @ B )
              & ~ ( finite2115694454571419734at_nat
                  @ ( collect_nat_nat
                    @ ^ [A4: nat > nat] :
                        ( ( member_nat_nat @ A4 @ A )
                        & ( R2 @ A4 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_476_pigeonhole__infinite__rel,axiom,
    ! [A: set_nat_a,B: set_a,R2: ( nat > a ) > a > $o] :
      ( ~ ( finite_finite_nat_a @ A )
     => ( ( finite_finite_a @ B )
       => ( ! [X3: nat > a] :
              ( ( member_nat_a @ X3 @ A )
             => ? [Xa: a] :
                  ( ( member_a @ Xa @ B )
                  & ( R2 @ X3 @ Xa ) ) )
         => ? [X3: a] :
              ( ( member_a @ X3 @ B )
              & ~ ( finite_finite_nat_a
                  @ ( collect_nat_a
                    @ ^ [A4: nat > a] :
                        ( ( member_nat_a @ A4 @ A )
                        & ( R2 @ A4 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_477_pigeonhole__infinite__rel,axiom,
    ! [A: set_a_nat,B: set_a,R2: ( a > nat ) > a > $o] :
      ( ~ ( finite_finite_a_nat @ A )
     => ( ( finite_finite_a @ B )
       => ( ! [X3: a > nat] :
              ( ( member_a_nat @ X3 @ A )
             => ? [Xa: a] :
                  ( ( member_a @ Xa @ B )
                  & ( R2 @ X3 @ Xa ) ) )
         => ? [X3: a] :
              ( ( member_a @ X3 @ B )
              & ~ ( finite_finite_a_nat
                  @ ( collect_a_nat
                    @ ^ [A4: a > nat] :
                        ( ( member_a_nat @ A4 @ A )
                        & ( R2 @ A4 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_478_pigeonhole__infinite__rel,axiom,
    ! [A: set_a_a,B: set_a,R2: ( a > a ) > a > $o] :
      ( ~ ( finite_finite_a_a @ A )
     => ( ( finite_finite_a @ B )
       => ( ! [X3: a > a] :
              ( ( member_a_a @ X3 @ A )
             => ? [Xa: a] :
                  ( ( member_a @ Xa @ B )
                  & ( R2 @ X3 @ Xa ) ) )
         => ? [X3: a] :
              ( ( member_a @ X3 @ B )
              & ~ ( finite_finite_a_a
                  @ ( collect_a_a
                    @ ^ [A4: a > a] :
                        ( ( member_a_a @ A4 @ A )
                        & ( R2 @ A4 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_479_pigeonhole__infinite__rel,axiom,
    ! [A: set_nat_nat,B: set_nat,R2: ( nat > nat ) > nat > $o] :
      ( ~ ( finite2115694454571419734at_nat @ A )
     => ( ( finite_finite_nat @ B )
       => ( ! [X3: nat > nat] :
              ( ( member_nat_nat @ X3 @ A )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B )
                  & ( R2 @ X3 @ Xa ) ) )
         => ? [X3: nat] :
              ( ( member_nat @ X3 @ B )
              & ~ ( finite2115694454571419734at_nat
                  @ ( collect_nat_nat
                    @ ^ [A4: nat > nat] :
                        ( ( member_nat_nat @ A4 @ A )
                        & ( R2 @ A4 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_480_pigeonhole__infinite__rel,axiom,
    ! [A: set_nat_a,B: set_nat,R2: ( nat > a ) > nat > $o] :
      ( ~ ( finite_finite_nat_a @ A )
     => ( ( finite_finite_nat @ B )
       => ( ! [X3: nat > a] :
              ( ( member_nat_a @ X3 @ A )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B )
                  & ( R2 @ X3 @ Xa ) ) )
         => ? [X3: nat] :
              ( ( member_nat @ X3 @ B )
              & ~ ( finite_finite_nat_a
                  @ ( collect_nat_a
                    @ ^ [A4: nat > a] :
                        ( ( member_nat_a @ A4 @ A )
                        & ( R2 @ A4 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_481_additive__abelian__group_Ocard__sumset__0__iff,axiom,
    ! [G: set_Product_prod_a_a,Addition: product_prod_a_a > product_prod_a_a > product_prod_a_a,Zero: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
      ( ( pluenn6837661381354551573od_a_a @ G @ Addition @ Zero )
     => ( ( ord_le746702958409616551od_a_a @ A @ G )
       => ( ( ord_le746702958409616551od_a_a @ B @ G )
         => ( ( ( finite4795055649997197647od_a_a @ ( pluenn659090379318209998od_a_a @ G @ Addition @ A @ B ) )
              = zero_zero_nat )
            = ( ( ( finite4795055649997197647od_a_a @ A )
                = zero_zero_nat )
              | ( ( finite4795055649997197647od_a_a @ B )
                = zero_zero_nat ) ) ) ) ) ) ).

% additive_abelian_group.card_sumset_0_iff
thf(fact_482_additive__abelian__group_Ocard__sumset__0__iff,axiom,
    ! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: set_nat,B: set_nat] :
      ( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
     => ( ( ord_less_eq_set_nat @ A @ G )
       => ( ( ord_less_eq_set_nat @ B @ G )
         => ( ( ( finite_card_nat @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A @ B ) )
              = zero_zero_nat )
            = ( ( ( finite_card_nat @ A )
                = zero_zero_nat )
              | ( ( finite_card_nat @ B )
                = zero_zero_nat ) ) ) ) ) ) ).

% additive_abelian_group.card_sumset_0_iff
thf(fact_483_additive__abelian__group_Ocard__sumset__0__iff,axiom,
    ! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,B: set_a] :
      ( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
     => ( ( ord_less_eq_set_a @ A @ G )
       => ( ( ord_less_eq_set_a @ B @ G )
         => ( ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) )
              = zero_zero_nat )
            = ( ( ( finite_card_a @ A )
                = zero_zero_nat )
              | ( ( finite_card_a @ B )
                = zero_zero_nat ) ) ) ) ) ) ).

% additive_abelian_group.card_sumset_0_iff
thf(fact_484_finite__has__maximal2,axiom,
    ! [A: set_nat_nat,A2: nat > nat] :
      ( ( finite2115694454571419734at_nat @ A )
     => ( ( member_nat_nat @ A2 @ A )
       => ? [X3: nat > nat] :
            ( ( member_nat_nat @ X3 @ A )
            & ( ord_less_eq_nat_nat @ A2 @ X3 )
            & ! [Xa: nat > nat] :
                ( ( member_nat_nat @ Xa @ A )
               => ( ( ord_less_eq_nat_nat @ X3 @ Xa )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_485_finite__has__maximal2,axiom,
    ! [A: set_a_nat,A2: a > nat] :
      ( ( finite_finite_a_nat @ A )
     => ( ( member_a_nat @ A2 @ A )
       => ? [X3: a > nat] :
            ( ( member_a_nat @ X3 @ A )
            & ( ord_less_eq_a_nat @ A2 @ X3 )
            & ! [Xa: a > nat] :
                ( ( member_a_nat @ Xa @ A )
               => ( ( ord_less_eq_a_nat @ X3 @ Xa )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_486_finite__has__maximal2,axiom,
    ! [A: set_nat,A2: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( member_nat @ A2 @ A )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A )
            & ( ord_less_eq_nat @ A2 @ X3 )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A )
               => ( ( ord_less_eq_nat @ X3 @ Xa )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_487_finite__has__maximal2,axiom,
    ! [A: set_set_a,A2: set_a] :
      ( ( finite_finite_set_a @ A )
     => ( ( member_set_a @ A2 @ A )
       => ? [X3: set_a] :
            ( ( member_set_a @ X3 @ A )
            & ( ord_less_eq_set_a @ A2 @ X3 )
            & ! [Xa: set_a] :
                ( ( member_set_a @ Xa @ A )
               => ( ( ord_less_eq_set_a @ X3 @ Xa )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_488_finite__has__minimal2,axiom,
    ! [A: set_nat_nat,A2: nat > nat] :
      ( ( finite2115694454571419734at_nat @ A )
     => ( ( member_nat_nat @ A2 @ A )
       => ? [X3: nat > nat] :
            ( ( member_nat_nat @ X3 @ A )
            & ( ord_less_eq_nat_nat @ X3 @ A2 )
            & ! [Xa: nat > nat] :
                ( ( member_nat_nat @ Xa @ A )
               => ( ( ord_less_eq_nat_nat @ Xa @ X3 )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_489_finite__has__minimal2,axiom,
    ! [A: set_a_nat,A2: a > nat] :
      ( ( finite_finite_a_nat @ A )
     => ( ( member_a_nat @ A2 @ A )
       => ? [X3: a > nat] :
            ( ( member_a_nat @ X3 @ A )
            & ( ord_less_eq_a_nat @ X3 @ A2 )
            & ! [Xa: a > nat] :
                ( ( member_a_nat @ Xa @ A )
               => ( ( ord_less_eq_a_nat @ Xa @ X3 )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_490_finite__has__minimal2,axiom,
    ! [A: set_nat,A2: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( member_nat @ A2 @ A )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A )
            & ( ord_less_eq_nat @ X3 @ A2 )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A )
               => ( ( ord_less_eq_nat @ Xa @ X3 )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_491_finite__has__minimal2,axiom,
    ! [A: set_set_a,A2: set_a] :
      ( ( finite_finite_set_a @ A )
     => ( ( member_set_a @ A2 @ A )
       => ? [X3: set_a] :
            ( ( member_set_a @ X3 @ A )
            & ( ord_less_eq_set_a @ X3 @ A2 )
            & ! [Xa: set_a] :
                ( ( member_set_a @ Xa @ A )
               => ( ( ord_less_eq_set_a @ Xa @ X3 )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_492_rev__finite__subset,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( finite_finite_nat @ B )
     => ( ( ord_less_eq_set_nat @ A @ B )
       => ( finite_finite_nat @ A ) ) ) ).

% rev_finite_subset
thf(fact_493_rev__finite__subset,axiom,
    ! [B: set_Product_prod_a_a,A: set_Product_prod_a_a] :
      ( ( finite6544458595007987280od_a_a @ B )
     => ( ( ord_le746702958409616551od_a_a @ A @ B )
       => ( finite6544458595007987280od_a_a @ A ) ) ) ).

% rev_finite_subset
thf(fact_494_rev__finite__subset,axiom,
    ! [B: set_a,A: set_a] :
      ( ( finite_finite_a @ B )
     => ( ( ord_less_eq_set_a @ A @ B )
       => ( finite_finite_a @ A ) ) ) ).

% rev_finite_subset
thf(fact_495_infinite__super,axiom,
    ! [S: set_nat,T: set_nat] :
      ( ( ord_less_eq_set_nat @ S @ T )
     => ( ~ ( finite_finite_nat @ S )
       => ~ ( finite_finite_nat @ T ) ) ) ).

% infinite_super
thf(fact_496_infinite__super,axiom,
    ! [S: set_Product_prod_a_a,T: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ S @ T )
     => ( ~ ( finite6544458595007987280od_a_a @ S )
       => ~ ( finite6544458595007987280od_a_a @ T ) ) ) ).

% infinite_super
thf(fact_497_infinite__super,axiom,
    ! [S: set_a,T: set_a] :
      ( ( ord_less_eq_set_a @ S @ T )
     => ( ~ ( finite_finite_a @ S )
       => ~ ( finite_finite_a @ T ) ) ) ).

% infinite_super
thf(fact_498_finite__subset,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( finite_finite_nat @ B )
       => ( finite_finite_nat @ A ) ) ) ).

% finite_subset
thf(fact_499_finite__subset,axiom,
    ! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ A @ B )
     => ( ( finite6544458595007987280od_a_a @ B )
       => ( finite6544458595007987280od_a_a @ A ) ) ) ).

% finite_subset
thf(fact_500_finite__subset,axiom,
    ! [A: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ( finite_finite_a @ B )
       => ( finite_finite_a @ A ) ) ) ).

% finite_subset
thf(fact_501_finite__inverse__image__gen,axiom,
    ! [A: set_a,F: a > a,D: set_a] :
      ( ( finite_finite_a @ A )
     => ( ( inj_on_a_a @ F @ D )
       => ( finite_finite_a
          @ ( collect_a
            @ ^ [J: a] :
                ( ( member_a @ J @ D )
                & ( member_a @ ( F @ J ) @ A ) ) ) ) ) ) ).

% finite_inverse_image_gen
thf(fact_502_finite__inverse__image__gen,axiom,
    ! [A: set_a,F: nat > a,D: set_nat] :
      ( ( finite_finite_a @ A )
     => ( ( inj_on_nat_a @ F @ D )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [J: nat] :
                ( ( member_nat @ J @ D )
                & ( member_a @ ( F @ J ) @ A ) ) ) ) ) ) ).

% finite_inverse_image_gen
thf(fact_503_finite__inverse__image__gen,axiom,
    ! [A: set_nat,F: a > nat,D: set_a] :
      ( ( finite_finite_nat @ A )
     => ( ( inj_on_a_nat @ F @ D )
       => ( finite_finite_a
          @ ( collect_a
            @ ^ [J: a] :
                ( ( member_a @ J @ D )
                & ( member_nat @ ( F @ J ) @ A ) ) ) ) ) ) ).

% finite_inverse_image_gen
thf(fact_504_finite__inverse__image__gen,axiom,
    ! [A: set_nat,F: nat > nat,D: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( inj_on_nat_nat @ F @ D )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [J: nat] :
                ( ( member_nat @ J @ D )
                & ( member_nat @ ( F @ J ) @ A ) ) ) ) ) ) ).

% finite_inverse_image_gen
thf(fact_505_finite__inverse__image__gen,axiom,
    ! [A: set_nat_nat,F: a > nat > nat,D: set_a] :
      ( ( finite2115694454571419734at_nat @ A )
     => ( ( inj_on_a_nat_nat @ F @ D )
       => ( finite_finite_a
          @ ( collect_a
            @ ^ [J: a] :
                ( ( member_a @ J @ D )
                & ( member_nat_nat @ ( F @ J ) @ A ) ) ) ) ) ) ).

% finite_inverse_image_gen
thf(fact_506_finite__inverse__image__gen,axiom,
    ! [A: set_nat_a,F: a > nat > a,D: set_a] :
      ( ( finite_finite_nat_a @ A )
     => ( ( inj_on_a_nat_a @ F @ D )
       => ( finite_finite_a
          @ ( collect_a
            @ ^ [J: a] :
                ( ( member_a @ J @ D )
                & ( member_nat_a @ ( F @ J ) @ A ) ) ) ) ) ) ).

% finite_inverse_image_gen
thf(fact_507_finite__inverse__image__gen,axiom,
    ! [A: set_a_nat,F: a > a > nat,D: set_a] :
      ( ( finite_finite_a_nat @ A )
     => ( ( inj_on_a_a_nat @ F @ D )
       => ( finite_finite_a
          @ ( collect_a
            @ ^ [J: a] :
                ( ( member_a @ J @ D )
                & ( member_a_nat @ ( F @ J ) @ A ) ) ) ) ) ) ).

% finite_inverse_image_gen
thf(fact_508_finite__inverse__image__gen,axiom,
    ! [A: set_a_a,F: a > a > a,D: set_a] :
      ( ( finite_finite_a_a @ A )
     => ( ( inj_on_a_a_a @ F @ D )
       => ( finite_finite_a
          @ ( collect_a
            @ ^ [J: a] :
                ( ( member_a @ J @ D )
                & ( member_a_a @ ( F @ J ) @ A ) ) ) ) ) ) ).

% finite_inverse_image_gen
thf(fact_509_finite__inverse__image__gen,axiom,
    ! [A: set_nat_nat,F: nat > nat > nat,D: set_nat] :
      ( ( finite2115694454571419734at_nat @ A )
     => ( ( inj_on_nat_nat_nat @ F @ D )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [J: nat] :
                ( ( member_nat @ J @ D )
                & ( member_nat_nat @ ( F @ J ) @ A ) ) ) ) ) ) ).

% finite_inverse_image_gen
thf(fact_510_finite__inverse__image__gen,axiom,
    ! [A: set_nat_a,F: nat > nat > a,D: set_nat] :
      ( ( finite_finite_nat_a @ A )
     => ( ( inj_on_nat_nat_a @ F @ D )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [J: nat] :
                ( ( member_nat @ J @ D )
                & ( member_nat_a @ ( F @ J ) @ A ) ) ) ) ) ) ).

% finite_inverse_image_gen
thf(fact_511_finite__cartesian__product,axiom,
    ! [A: set_a,B: set_nat] :
      ( ( finite_finite_a @ A )
     => ( ( finite_finite_nat @ B )
       => ( finite6644898363146130708_a_nat
          @ ( product_Sigma_a_nat @ A
            @ ^ [Uu: a] : B ) ) ) ) ).

% finite_cartesian_product
thf(fact_512_finite__cartesian__product,axiom,
    ! [A: set_a,B: set_Product_prod_a_a] :
      ( ( finite_finite_a @ A )
     => ( ( finite6544458595007987280od_a_a @ B )
       => ( finite5848958031409366265od_a_a
          @ ( produc6342321021181284593od_a_a @ A
            @ ^ [Uu: a] : B ) ) ) ) ).

% finite_cartesian_product
thf(fact_513_finite__cartesian__product,axiom,
    ! [A: set_nat,B: set_a] :
      ( ( finite_finite_nat @ A )
     => ( ( finite_finite_a @ B )
       => ( finite659689790015031866_nat_a
          @ ( product_Sigma_nat_a @ A
            @ ^ [Uu: nat] : B ) ) ) ) ).

% finite_cartesian_product
thf(fact_514_finite__cartesian__product,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( finite_finite_nat @ B )
       => ( finite6177210948735845034at_nat
          @ ( produc457027306803732586at_nat @ A
            @ ^ [Uu: nat] : B ) ) ) ) ).

% finite_cartesian_product
thf(fact_515_finite__cartesian__product,axiom,
    ! [A: set_nat,B: set_Product_prod_a_a] :
      ( ( finite_finite_nat @ A )
     => ( ( finite6544458595007987280od_a_a @ B )
       => ( finite8333580178073949923od_a_a
          @ ( produc1591714161673420045od_a_a @ A
            @ ^ [Uu: nat] : B ) ) ) ) ).

% finite_cartesian_product
thf(fact_516_finite__cartesian__product,axiom,
    ! [A: set_Product_prod_a_a,B: set_a] :
      ( ( finite6544458595007987280od_a_a @ A )
     => ( ( finite_finite_a @ B )
       => ( finite5607752495362350695_a_a_a
          @ ( produc2379640491490746847_a_a_a @ A
            @ ^ [Uu: product_prod_a_a] : B ) ) ) ) ).

% finite_cartesian_product
thf(fact_517_finite__cartesian__product,axiom,
    ! [A: set_Product_prod_a_a,B: set_nat] :
      ( ( finite6544458595007987280od_a_a @ A )
     => ( ( finite_finite_nat @ B )
       => ( finite3082808146142899517_a_nat
          @ ( produc1049071135499013807_a_nat @ A
            @ ^ [Uu: product_prod_a_a] : B ) ) ) ) ).

% finite_cartesian_product
thf(fact_518_finite__cartesian__product,axiom,
    ! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
      ( ( finite6544458595007987280od_a_a @ A )
     => ( ( finite6544458595007987280od_a_a @ B )
       => ( finite20598220287489552od_a_a
          @ ( produc5899993699339346696od_a_a @ A
            @ ^ [Uu: product_prod_a_a] : B ) ) ) ) ).

% finite_cartesian_product
thf(fact_519_finite__cartesian__product,axiom,
    ! [A: set_a,B: set_a] :
      ( ( finite_finite_a @ A )
     => ( ( finite_finite_a @ B )
       => ( finite6544458595007987280od_a_a
          @ ( product_Sigma_a_a @ A
            @ ^ [Uu: a] : B ) ) ) ) ).

% finite_cartesian_product
thf(fact_520_card__subset__eq,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( finite_finite_nat @ B )
     => ( ( ord_less_eq_set_nat @ A @ B )
       => ( ( ( finite_card_nat @ A )
            = ( finite_card_nat @ B ) )
         => ( A = B ) ) ) ) ).

% card_subset_eq
thf(fact_521_card__subset__eq,axiom,
    ! [B: set_Product_prod_a_a,A: set_Product_prod_a_a] :
      ( ( finite6544458595007987280od_a_a @ B )
     => ( ( ord_le746702958409616551od_a_a @ A @ B )
       => ( ( ( finite4795055649997197647od_a_a @ A )
            = ( finite4795055649997197647od_a_a @ B ) )
         => ( A = B ) ) ) ) ).

% card_subset_eq
thf(fact_522_card__subset__eq,axiom,
    ! [B: set_a,A: set_a] :
      ( ( finite_finite_a @ B )
     => ( ( ord_less_eq_set_a @ A @ B )
       => ( ( ( finite_card_a @ A )
            = ( finite_card_a @ B ) )
         => ( A = B ) ) ) ) ).

% card_subset_eq
thf(fact_523_infinite__arbitrarily__large,axiom,
    ! [A: set_nat,N2: nat] :
      ( ~ ( finite_finite_nat @ A )
     => ? [B7: set_nat] :
          ( ( finite_finite_nat @ B7 )
          & ( ( finite_card_nat @ B7 )
            = N2 )
          & ( ord_less_eq_set_nat @ B7 @ A ) ) ) ).

% infinite_arbitrarily_large
thf(fact_524_infinite__arbitrarily__large,axiom,
    ! [A: set_Product_prod_a_a,N2: nat] :
      ( ~ ( finite6544458595007987280od_a_a @ A )
     => ? [B7: set_Product_prod_a_a] :
          ( ( finite6544458595007987280od_a_a @ B7 )
          & ( ( finite4795055649997197647od_a_a @ B7 )
            = N2 )
          & ( ord_le746702958409616551od_a_a @ B7 @ A ) ) ) ).

% infinite_arbitrarily_large
thf(fact_525_infinite__arbitrarily__large,axiom,
    ! [A: set_a,N2: nat] :
      ( ~ ( finite_finite_a @ A )
     => ? [B7: set_a] :
          ( ( finite_finite_a @ B7 )
          & ( ( finite_card_a @ B7 )
            = N2 )
          & ( ord_less_eq_set_a @ B7 @ A ) ) ) ).

% infinite_arbitrarily_large
thf(fact_526_card__mono,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( finite_finite_nat @ B )
     => ( ( ord_less_eq_set_nat @ A @ B )
       => ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) ) ) ) ).

% card_mono
thf(fact_527_card__mono,axiom,
    ! [B: set_Product_prod_a_a,A: set_Product_prod_a_a] :
      ( ( finite6544458595007987280od_a_a @ B )
     => ( ( ord_le746702958409616551od_a_a @ A @ B )
       => ( ord_less_eq_nat @ ( finite4795055649997197647od_a_a @ A ) @ ( finite4795055649997197647od_a_a @ B ) ) ) ) ).

% card_mono
thf(fact_528_card__mono,axiom,
    ! [B: set_a,A: set_a] :
      ( ( finite_finite_a @ B )
     => ( ( ord_less_eq_set_a @ A @ B )
       => ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ B ) ) ) ) ).

% card_mono
thf(fact_529_card__seteq,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( finite_finite_nat @ B )
     => ( ( ord_less_eq_set_nat @ A @ B )
       => ( ( ord_less_eq_nat @ ( finite_card_nat @ B ) @ ( finite_card_nat @ A ) )
         => ( A = B ) ) ) ) ).

% card_seteq
thf(fact_530_card__seteq,axiom,
    ! [B: set_Product_prod_a_a,A: set_Product_prod_a_a] :
      ( ( finite6544458595007987280od_a_a @ B )
     => ( ( ord_le746702958409616551od_a_a @ A @ B )
       => ( ( ord_less_eq_nat @ ( finite4795055649997197647od_a_a @ B ) @ ( finite4795055649997197647od_a_a @ A ) )
         => ( A = B ) ) ) ) ).

% card_seteq
thf(fact_531_card__seteq,axiom,
    ! [B: set_a,A: set_a] :
      ( ( finite_finite_a @ B )
     => ( ( ord_less_eq_set_a @ A @ B )
       => ( ( ord_less_eq_nat @ ( finite_card_a @ B ) @ ( finite_card_a @ A ) )
         => ( A = B ) ) ) ) ).

% card_seteq
thf(fact_532_exists__subset__between,axiom,
    ! [A: set_nat,N2: nat,C: set_nat] :
      ( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ ( finite_card_nat @ C ) )
       => ( ( ord_less_eq_set_nat @ A @ C )
         => ( ( finite_finite_nat @ C )
           => ? [B7: set_nat] :
                ( ( ord_less_eq_set_nat @ A @ B7 )
                & ( ord_less_eq_set_nat @ B7 @ C )
                & ( ( finite_card_nat @ B7 )
                  = N2 ) ) ) ) ) ) ).

% exists_subset_between
thf(fact_533_exists__subset__between,axiom,
    ! [A: set_Product_prod_a_a,N2: nat,C: set_Product_prod_a_a] :
      ( ( ord_less_eq_nat @ ( finite4795055649997197647od_a_a @ A ) @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ ( finite4795055649997197647od_a_a @ C ) )
       => ( ( ord_le746702958409616551od_a_a @ A @ C )
         => ( ( finite6544458595007987280od_a_a @ C )
           => ? [B7: set_Product_prod_a_a] :
                ( ( ord_le746702958409616551od_a_a @ A @ B7 )
                & ( ord_le746702958409616551od_a_a @ B7 @ C )
                & ( ( finite4795055649997197647od_a_a @ B7 )
                  = N2 ) ) ) ) ) ) ).

% exists_subset_between
thf(fact_534_exists__subset__between,axiom,
    ! [A: set_a,N2: nat,C: set_a] :
      ( ( ord_less_eq_nat @ ( finite_card_a @ A ) @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ ( finite_card_a @ C ) )
       => ( ( ord_less_eq_set_a @ A @ C )
         => ( ( finite_finite_a @ C )
           => ? [B7: set_a] :
                ( ( ord_less_eq_set_a @ A @ B7 )
                & ( ord_less_eq_set_a @ B7 @ C )
                & ( ( finite_card_a @ B7 )
                  = N2 ) ) ) ) ) ) ).

% exists_subset_between
thf(fact_535_obtain__subset__with__card__n,axiom,
    ! [N2: nat,S: set_nat] :
      ( ( ord_less_eq_nat @ N2 @ ( finite_card_nat @ S ) )
     => ~ ! [T2: set_nat] :
            ( ( ord_less_eq_set_nat @ T2 @ S )
           => ( ( ( finite_card_nat @ T2 )
                = N2 )
             => ~ ( finite_finite_nat @ T2 ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_536_obtain__subset__with__card__n,axiom,
    ! [N2: nat,S: set_Product_prod_a_a] :
      ( ( ord_less_eq_nat @ N2 @ ( finite4795055649997197647od_a_a @ S ) )
     => ~ ! [T2: set_Product_prod_a_a] :
            ( ( ord_le746702958409616551od_a_a @ T2 @ S )
           => ( ( ( finite4795055649997197647od_a_a @ T2 )
                = N2 )
             => ~ ( finite6544458595007987280od_a_a @ T2 ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_537_obtain__subset__with__card__n,axiom,
    ! [N2: nat,S: set_a] :
      ( ( ord_less_eq_nat @ N2 @ ( finite_card_a @ S ) )
     => ~ ! [T2: set_a] :
            ( ( ord_less_eq_set_a @ T2 @ S )
           => ( ( ( finite_card_a @ T2 )
                = N2 )
             => ~ ( finite_finite_a @ T2 ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_538_finite__if__finite__subsets__card__bdd,axiom,
    ! [F2: set_nat,C: nat] :
      ( ! [G3: set_nat] :
          ( ( ord_less_eq_set_nat @ G3 @ F2 )
         => ( ( finite_finite_nat @ G3 )
           => ( ord_less_eq_nat @ ( finite_card_nat @ G3 ) @ C ) ) )
     => ( ( finite_finite_nat @ F2 )
        & ( ord_less_eq_nat @ ( finite_card_nat @ F2 ) @ C ) ) ) ).

% finite_if_finite_subsets_card_bdd
thf(fact_539_finite__if__finite__subsets__card__bdd,axiom,
    ! [F2: set_Product_prod_a_a,C: nat] :
      ( ! [G3: set_Product_prod_a_a] :
          ( ( ord_le746702958409616551od_a_a @ G3 @ F2 )
         => ( ( finite6544458595007987280od_a_a @ G3 )
           => ( ord_less_eq_nat @ ( finite4795055649997197647od_a_a @ G3 ) @ C ) ) )
     => ( ( finite6544458595007987280od_a_a @ F2 )
        & ( ord_less_eq_nat @ ( finite4795055649997197647od_a_a @ F2 ) @ C ) ) ) ).

% finite_if_finite_subsets_card_bdd
thf(fact_540_finite__if__finite__subsets__card__bdd,axiom,
    ! [F2: set_a,C: nat] :
      ( ! [G3: set_a] :
          ( ( ord_less_eq_set_a @ G3 @ F2 )
         => ( ( finite_finite_a @ G3 )
           => ( ord_less_eq_nat @ ( finite_card_a @ G3 ) @ C ) ) )
     => ( ( finite_finite_a @ F2 )
        & ( ord_less_eq_nat @ ( finite_card_a @ F2 ) @ C ) ) ) ).

% finite_if_finite_subsets_card_bdd
thf(fact_541_Units__def,axiom,
    ( ( group_Units_a @ g @ addition @ zero )
    = ( collect_a
      @ ^ [U2: a] :
          ( ( member_a @ U2 @ g )
          & ( group_invertible_a @ g @ addition @ zero @ U2 ) ) ) ) ).

% Units_def
thf(fact_542_mem__UnitsD,axiom,
    ! [U: a] :
      ( ( member_a @ U @ ( group_Units_a @ g @ addition @ zero ) )
     => ( ( group_invertible_a @ g @ addition @ zero @ U )
        & ( member_a @ U @ g ) ) ) ).

% mem_UnitsD
thf(fact_543_mem__UnitsI,axiom,
    ! [U: a] :
      ( ( group_invertible_a @ g @ addition @ zero @ U )
     => ( ( member_a @ U @ g )
       => ( member_a @ U @ ( group_Units_a @ g @ addition @ zero ) ) ) ) ).

% mem_UnitsI
thf(fact_544_subgroupI,axiom,
    ! [G: set_a] :
      ( ( ord_less_eq_set_a @ G @ g )
     => ( ( member_a @ zero @ G )
       => ( ! [G4: a,H: a] :
              ( ( member_a @ G4 @ G )
             => ( ( member_a @ H @ G )
               => ( member_a @ ( addition @ G4 @ H ) @ G ) ) )
         => ( ! [G4: a] :
                ( ( member_a @ G4 @ G )
               => ( group_invertible_a @ g @ addition @ zero @ G4 ) )
           => ( ! [G4: a] :
                  ( ( member_a @ G4 @ G )
                 => ( member_a @ ( group_inverse_a @ g @ addition @ zero @ G4 ) @ G ) )
             => ( group_subgroup_a @ G @ g @ addition @ zero ) ) ) ) ) ) ).

% subgroupI
thf(fact_545_card__sumset__singleton__eq,axiom,
    ! [A: set_a,A2: a] :
      ( ( finite_finite_a @ A )
     => ( ( ( member_a @ A2 @ g )
         => ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ ( insert_a @ A2 @ bot_bot_set_a ) ) )
            = ( finite_card_a @ ( inf_inf_set_a @ A @ g ) ) ) )
        & ( ~ ( member_a @ A2 @ g )
         => ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ ( insert_a @ A2 @ bot_bot_set_a ) ) )
            = zero_zero_nat ) ) ) ) ).

% card_sumset_singleton_eq
thf(fact_546_card__sumset__le,axiom,
    ! [A: set_a,A2: a] :
      ( ( finite_finite_a @ A )
     => ( ord_less_eq_nat @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ ( insert_a @ A2 @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A ) ) ) ).

% card_sumset_le
thf(fact_547_case__prod__conv,axiom,
    ! [F: a > a > product_prod_a_a,A2: a,B4: a] :
      ( ( produc408267641121961211od_a_a @ F @ ( product_Pair_a_a @ A2 @ B4 ) )
      = ( F @ A2 @ B4 ) ) ).

% case_prod_conv
thf(fact_548_infinite__sumset__aux,axiom,
    ! [A: set_a,B: set_a] :
      ( ~ ( finite_finite_a @ ( inf_inf_set_a @ A @ g ) )
     => ( ( ~ ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) ) )
        = ( ( inf_inf_set_a @ B @ g )
         != bot_bot_set_a ) ) ) ).

% infinite_sumset_aux
thf(fact_549_infinite__sumset__iff,axiom,
    ! [A: set_a,B: set_a] :
      ( ( ~ ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) ) )
      = ( ( ~ ( finite_finite_a @ ( inf_inf_set_a @ A @ g ) )
          & ( ( inf_inf_set_a @ B @ g )
           != bot_bot_set_a ) )
        | ( ( ( inf_inf_set_a @ A @ g )
           != bot_bot_set_a )
          & ~ ( finite_finite_a @ ( inf_inf_set_a @ B @ g ) ) ) ) ) ).

% infinite_sumset_iff
thf(fact_550_prod_Oinject,axiom,
    ! [X1: a,X22: a,Y1: a,Y2: a] :
      ( ( ( product_Pair_a_a @ X1 @ X22 )
        = ( product_Pair_a_a @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_551_old_Oprod_Oinject,axiom,
    ! [A2: a,B4: a,A7: a,B8: a] :
      ( ( ( product_Pair_a_a @ A2 @ B4 )
        = ( product_Pair_a_a @ A7 @ B8 ) )
      = ( ( A2 = A7 )
        & ( B4 = B8 ) ) ) ).

% old.prod.inject
thf(fact_552_case__prodI,axiom,
    ! [F: a > a > $o,A2: a,B4: a] :
      ( ( F @ A2 @ B4 )
     => ( produc6436628058953941356_a_a_o @ F @ ( product_Pair_a_a @ A2 @ B4 ) ) ) ).

% case_prodI
thf(fact_553_case__prodI2,axiom,
    ! [P2: product_prod_a_a,C2: a > a > $o] :
      ( ! [A3: a,B2: a] :
          ( ( P2
            = ( product_Pair_a_a @ A3 @ B2 ) )
         => ( C2 @ A3 @ B2 ) )
     => ( produc6436628058953941356_a_a_o @ C2 @ P2 ) ) ).

% case_prodI2
thf(fact_554_mem__case__prodI,axiom,
    ! [Z: a,C2: a > a > set_a,A2: a,B4: a] :
      ( ( member_a @ Z @ ( C2 @ A2 @ B4 ) )
     => ( member_a @ Z @ ( produc9217457822752978994_set_a @ C2 @ ( product_Pair_a_a @ A2 @ B4 ) ) ) ) ).

% mem_case_prodI
thf(fact_555_mem__case__prodI,axiom,
    ! [Z: product_prod_a_a > product_prod_a_a,C2: a > a > set_Pr8826267807999420763od_a_a,A2: a,B4: a] :
      ( ( member4020126937092221116od_a_a @ Z @ ( C2 @ A2 @ B4 ) )
     => ( member4020126937092221116od_a_a @ Z @ ( produc797403868875161735od_a_a @ C2 @ ( product_Pair_a_a @ A2 @ B4 ) ) ) ) ).

% mem_case_prodI
thf(fact_556_mem__case__prodI,axiom,
    ! [Z: nat,C2: a > a > set_nat,A2: a,B4: a] :
      ( ( member_nat @ Z @ ( C2 @ A2 @ B4 ) )
     => ( member_nat @ Z @ ( produc153843693180602034et_nat @ C2 @ ( product_Pair_a_a @ A2 @ B4 ) ) ) ) ).

% mem_case_prodI
thf(fact_557_mem__case__prodI,axiom,
    ! [Z: nat > nat,C2: a > a > set_nat_nat,A2: a,B4: a] :
      ( ( member_nat_nat @ Z @ ( C2 @ A2 @ B4 ) )
     => ( member_nat_nat @ Z @ ( produc1670433337670807073at_nat @ C2 @ ( product_Pair_a_a @ A2 @ B4 ) ) ) ) ).

% mem_case_prodI
thf(fact_558_mem__case__prodI,axiom,
    ! [Z: nat > a,C2: a > a > set_nat_a,A2: a,B4: a] :
      ( ( member_nat_a @ Z @ ( C2 @ A2 @ B4 ) )
     => ( member_nat_a @ Z @ ( produc2694066087470364441_nat_a @ C2 @ ( product_Pair_a_a @ A2 @ B4 ) ) ) ) ).

% mem_case_prodI
thf(fact_559_mem__case__prodI,axiom,
    ! [Z: a > product_prod_a_a,C2: a > a > set_a_6829686330177631172od_a_a,A2: a,B4: a] :
      ( ( member1957775702407316389od_a_a @ Z @ ( C2 @ A2 @ B4 ) )
     => ( member1957775702407316389od_a_a @ Z @ ( produc2933953731230324464od_a_a @ C2 @ ( product_Pair_a_a @ A2 @ B4 ) ) ) ) ).

% mem_case_prodI
thf(fact_560_mem__case__prodI,axiom,
    ! [Z: a > nat,C2: a > a > set_a_nat,A2: a,B4: a] :
      ( ( member_a_nat @ Z @ ( C2 @ A2 @ B4 ) )
     => ( member_a_nat @ Z @ ( produc3435159650992338163_a_nat @ C2 @ ( product_Pair_a_a @ A2 @ B4 ) ) ) ) ).

% mem_case_prodI
thf(fact_561_mem__case__prodI,axiom,
    ! [Z: a > a,C2: a > a > set_a_a,A2: a,B4: a] :
      ( ( member_a_a @ Z @ ( C2 @ A2 @ B4 ) )
     => ( member_a_a @ Z @ ( produc1550278517953855687et_a_a @ C2 @ ( product_Pair_a_a @ A2 @ B4 ) ) ) ) ).

% mem_case_prodI
thf(fact_562_mem__case__prodI2,axiom,
    ! [P2: product_prod_a_a,Z: a,C2: a > a > set_a] :
      ( ! [A3: a,B2: a] :
          ( ( P2
            = ( product_Pair_a_a @ A3 @ B2 ) )
         => ( member_a @ Z @ ( C2 @ A3 @ B2 ) ) )
     => ( member_a @ Z @ ( produc9217457822752978994_set_a @ C2 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_563_mem__case__prodI2,axiom,
    ! [P2: product_prod_a_a,Z: product_prod_a_a > product_prod_a_a,C2: a > a > set_Pr8826267807999420763od_a_a] :
      ( ! [A3: a,B2: a] :
          ( ( P2
            = ( product_Pair_a_a @ A3 @ B2 ) )
         => ( member4020126937092221116od_a_a @ Z @ ( C2 @ A3 @ B2 ) ) )
     => ( member4020126937092221116od_a_a @ Z @ ( produc797403868875161735od_a_a @ C2 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_564_mem__case__prodI2,axiom,
    ! [P2: product_prod_a_a,Z: nat,C2: a > a > set_nat] :
      ( ! [A3: a,B2: a] :
          ( ( P2
            = ( product_Pair_a_a @ A3 @ B2 ) )
         => ( member_nat @ Z @ ( C2 @ A3 @ B2 ) ) )
     => ( member_nat @ Z @ ( produc153843693180602034et_nat @ C2 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_565_mem__case__prodI2,axiom,
    ! [P2: product_prod_a_a,Z: nat > nat,C2: a > a > set_nat_nat] :
      ( ! [A3: a,B2: a] :
          ( ( P2
            = ( product_Pair_a_a @ A3 @ B2 ) )
         => ( member_nat_nat @ Z @ ( C2 @ A3 @ B2 ) ) )
     => ( member_nat_nat @ Z @ ( produc1670433337670807073at_nat @ C2 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_566_mem__case__prodI2,axiom,
    ! [P2: product_prod_a_a,Z: nat > a,C2: a > a > set_nat_a] :
      ( ! [A3: a,B2: a] :
          ( ( P2
            = ( product_Pair_a_a @ A3 @ B2 ) )
         => ( member_nat_a @ Z @ ( C2 @ A3 @ B2 ) ) )
     => ( member_nat_a @ Z @ ( produc2694066087470364441_nat_a @ C2 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_567_mem__case__prodI2,axiom,
    ! [P2: product_prod_a_a,Z: a > product_prod_a_a,C2: a > a > set_a_6829686330177631172od_a_a] :
      ( ! [A3: a,B2: a] :
          ( ( P2
            = ( product_Pair_a_a @ A3 @ B2 ) )
         => ( member1957775702407316389od_a_a @ Z @ ( C2 @ A3 @ B2 ) ) )
     => ( member1957775702407316389od_a_a @ Z @ ( produc2933953731230324464od_a_a @ C2 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_568_mem__case__prodI2,axiom,
    ! [P2: product_prod_a_a,Z: a > nat,C2: a > a > set_a_nat] :
      ( ! [A3: a,B2: a] :
          ( ( P2
            = ( product_Pair_a_a @ A3 @ B2 ) )
         => ( member_a_nat @ Z @ ( C2 @ A3 @ B2 ) ) )
     => ( member_a_nat @ Z @ ( produc3435159650992338163_a_nat @ C2 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_569_mem__case__prodI2,axiom,
    ! [P2: product_prod_a_a,Z: a > a,C2: a > a > set_a_a] :
      ( ! [A3: a,B2: a] :
          ( ( P2
            = ( product_Pair_a_a @ A3 @ B2 ) )
         => ( member_a_a @ Z @ ( C2 @ A3 @ B2 ) ) )
     => ( member_a_a @ Z @ ( produc1550278517953855687et_a_a @ C2 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_570_Collect__case__prod,axiom,
    ! [P: a > $o,Q: nat > $o] :
      ( ( collec4464134535221767506_a_nat
        @ ( produc3680711911437148916_nat_o
          @ ^ [A4: a,B3: nat] :
              ( ( P @ A4 )
              & ( Q @ B3 ) ) ) )
      = ( product_Sigma_a_nat @ ( collect_a @ P )
        @ ^ [Uu: a] : ( collect_nat @ Q ) ) ) ).

% Collect_case_prod
thf(fact_571_Collect__case__prod,axiom,
    ! [P: nat > $o,Q: a > $o] :
      ( ( collec7702297998945444472_nat_a
        @ ( produc2746933349376800278at_a_o
          @ ^ [A4: nat,B3: a] :
              ( ( P @ A4 )
              & ( Q @ B3 ) ) ) )
      = ( product_Sigma_nat_a @ ( collect_nat @ P )
        @ ^ [Uu: nat] : ( collect_a @ Q ) ) ) ).

% Collect_case_prod
thf(fact_572_Collect__case__prod,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [A4: nat,B3: nat] :
              ( ( P @ A4 )
              & ( Q @ B3 ) ) ) )
      = ( produc457027306803732586at_nat @ ( collect_nat @ P )
        @ ^ [Uu: nat] : ( collect_nat @ Q ) ) ) ).

% Collect_case_prod
thf(fact_573_Collect__case__prod,axiom,
    ! [P: a > $o,Q: a > $o] :
      ( ( collec3336397797384452498od_a_a
        @ ( produc6436628058953941356_a_a_o
          @ ^ [A4: a,B3: a] :
              ( ( P @ A4 )
              & ( Q @ B3 ) ) ) )
      = ( product_Sigma_a_a @ ( collect_a @ P )
        @ ^ [Uu: a] : ( collect_a @ Q ) ) ) ).

% Collect_case_prod
thf(fact_574_sumset__empty_H_I1_J,axiom,
    ! [A: set_a,B: set_a] :
      ( ( ( inf_inf_set_a @ A @ g )
        = bot_bot_set_a )
     => ( ( pluenn3038260743871226533mset_a @ g @ addition @ B @ A )
        = bot_bot_set_a ) ) ).

% sumset_empty'(1)
thf(fact_575_sumset__empty_H_I2_J,axiom,
    ! [A: set_a,B: set_a] :
      ( ( ( inf_inf_set_a @ A @ g )
        = bot_bot_set_a )
     => ( ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B )
        = bot_bot_set_a ) ) ).

% sumset_empty'(2)
thf(fact_576_mem__Sigma__iff,axiom,
    ! [A2: a,B4: nat,A: set_a,B: a > set_nat] :
      ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A2 @ B4 ) @ ( product_Sigma_a_nat @ A @ B ) )
      = ( ( member_a @ A2 @ A )
        & ( member_nat @ B4 @ ( B @ A2 ) ) ) ) ).

% mem_Sigma_iff
thf(fact_577_mem__Sigma__iff,axiom,
    ! [A2: nat,B4: a,A: set_nat,B: nat > set_a] :
      ( ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ A2 @ B4 ) @ ( product_Sigma_nat_a @ A @ B ) )
      = ( ( member_nat @ A2 @ A )
        & ( member_a @ B4 @ ( B @ A2 ) ) ) ) ).

% mem_Sigma_iff
thf(fact_578_mem__Sigma__iff,axiom,
    ! [A2: nat,B4: nat,A: set_nat,B: nat > set_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B4 ) @ ( produc457027306803732586at_nat @ A @ B ) )
      = ( ( member_nat @ A2 @ A )
        & ( member_nat @ B4 @ ( B @ A2 ) ) ) ) ).

% mem_Sigma_iff
thf(fact_579_mem__Sigma__iff,axiom,
    ! [A2: a,B4: a,A: set_a,B: a > set_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B4 ) @ ( product_Sigma_a_a @ A @ B ) )
      = ( ( member_a @ A2 @ A )
        & ( member_a @ B4 @ ( B @ A2 ) ) ) ) ).

% mem_Sigma_iff
thf(fact_580_mem__Sigma__iff,axiom,
    ! [A2: a,B4: nat > nat,A: set_a,B: a > set_nat_nat] :
      ( ( member5463269792551873667at_nat @ ( produc7514922136058452582at_nat @ A2 @ B4 ) @ ( produc5882319205326396405at_nat @ A @ B ) )
      = ( ( member_a @ A2 @ A )
        & ( member_nat_nat @ B4 @ ( B @ A2 ) ) ) ) ).

% mem_Sigma_iff
thf(fact_581_mem__Sigma__iff,axiom,
    ! [A2: a,B4: nat > a,A: set_a,B: a > set_nat_a] :
      ( ( member5853765858310737207_nat_a @ ( product_Pair_a_nat_a @ A2 @ B4 ) @ ( produc830630580412313007_nat_a @ A @ B ) )
      = ( ( member_a @ A2 @ A )
        & ( member_nat_a @ B4 @ ( B @ A2 ) ) ) ) ).

% mem_Sigma_iff
thf(fact_582_mem__Sigma__iff,axiom,
    ! [A2: a,B4: a > nat,A: set_a,B: a > set_a_nat] :
      ( ( member6594859421832710929_a_nat @ ( product_Pair_a_a_nat @ A2 @ B4 ) @ ( produc6815839153543411849_a_nat @ A @ B ) )
      = ( ( member_a @ A2 @ A )
        & ( member_a_nat @ B4 @ ( B @ A2 ) ) ) ) ).

% mem_Sigma_iff
thf(fact_583_mem__Sigma__iff,axiom,
    ! [A2: a,B4: a > a,A: set_a,B: a > set_a_a] :
      ( ( member681807718693926441_a_a_a @ ( product_Pair_a_a_a @ A2 @ B4 ) @ ( product_Sigma_a_a_a2 @ A @ B ) )
      = ( ( member_a @ A2 @ A )
        & ( member_a_a @ B4 @ ( B @ A2 ) ) ) ) ).

% mem_Sigma_iff
thf(fact_584_mem__Sigma__iff,axiom,
    ! [A2: nat,B4: nat > nat,A: set_nat,B: nat > set_nat_nat] :
      ( ( member8336108448496249625at_nat @ ( produc7839516862119294504at_nat @ A2 @ B4 ) @ ( produc4526620505022529241at_nat @ A @ B ) )
      = ( ( member_nat @ A2 @ A )
        & ( member_nat_nat @ B4 @ ( B @ A2 ) ) ) ) ).

% mem_Sigma_iff
thf(fact_585_mem__Sigma__iff,axiom,
    ! [A2: nat,B4: nat > a,A: set_nat,B: nat > set_nat_a] :
      ( ( member2811451201825703969_nat_a @ ( produc5292568359338195516_nat_a @ A2 @ B4 ) @ ( produc3659965428606139339_nat_a @ A @ B ) )
      = ( ( member_nat @ A2 @ A )
        & ( member_nat_a @ B4 @ ( B @ A2 ) ) ) ) ).

% mem_Sigma_iff
thf(fact_586_SigmaI,axiom,
    ! [A2: a,A: set_a,B4: nat,B: a > set_nat] :
      ( ( member_a @ A2 @ A )
     => ( ( member_nat @ B4 @ ( B @ A2 ) )
       => ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A2 @ B4 ) @ ( product_Sigma_a_nat @ A @ B ) ) ) ) ).

% SigmaI
thf(fact_587_SigmaI,axiom,
    ! [A2: nat,A: set_nat,B4: a,B: nat > set_a] :
      ( ( member_nat @ A2 @ A )
     => ( ( member_a @ B4 @ ( B @ A2 ) )
       => ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ A2 @ B4 ) @ ( product_Sigma_nat_a @ A @ B ) ) ) ) ).

% SigmaI
thf(fact_588_SigmaI,axiom,
    ! [A2: nat,A: set_nat,B4: nat,B: nat > set_nat] :
      ( ( member_nat @ A2 @ A )
     => ( ( member_nat @ B4 @ ( B @ A2 ) )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B4 ) @ ( produc457027306803732586at_nat @ A @ B ) ) ) ) ).

% SigmaI
thf(fact_589_SigmaI,axiom,
    ! [A2: a,A: set_a,B4: a,B: a > set_a] :
      ( ( member_a @ A2 @ A )
     => ( ( member_a @ B4 @ ( B @ A2 ) )
       => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B4 ) @ ( product_Sigma_a_a @ A @ B ) ) ) ) ).

% SigmaI
thf(fact_590_SigmaI,axiom,
    ! [A2: a,A: set_a,B4: nat > nat,B: a > set_nat_nat] :
      ( ( member_a @ A2 @ A )
     => ( ( member_nat_nat @ B4 @ ( B @ A2 ) )
       => ( member5463269792551873667at_nat @ ( produc7514922136058452582at_nat @ A2 @ B4 ) @ ( produc5882319205326396405at_nat @ A @ B ) ) ) ) ).

% SigmaI
thf(fact_591_SigmaI,axiom,
    ! [A2: a,A: set_a,B4: nat > a,B: a > set_nat_a] :
      ( ( member_a @ A2 @ A )
     => ( ( member_nat_a @ B4 @ ( B @ A2 ) )
       => ( member5853765858310737207_nat_a @ ( product_Pair_a_nat_a @ A2 @ B4 ) @ ( produc830630580412313007_nat_a @ A @ B ) ) ) ) ).

% SigmaI
thf(fact_592_SigmaI,axiom,
    ! [A2: a,A: set_a,B4: a > nat,B: a > set_a_nat] :
      ( ( member_a @ A2 @ A )
     => ( ( member_a_nat @ B4 @ ( B @ A2 ) )
       => ( member6594859421832710929_a_nat @ ( product_Pair_a_a_nat @ A2 @ B4 ) @ ( produc6815839153543411849_a_nat @ A @ B ) ) ) ) ).

% SigmaI
thf(fact_593_SigmaI,axiom,
    ! [A2: a,A: set_a,B4: a > a,B: a > set_a_a] :
      ( ( member_a @ A2 @ A )
     => ( ( member_a_a @ B4 @ ( B @ A2 ) )
       => ( member681807718693926441_a_a_a @ ( product_Pair_a_a_a @ A2 @ B4 ) @ ( product_Sigma_a_a_a2 @ A @ B ) ) ) ) ).

% SigmaI
thf(fact_594_SigmaI,axiom,
    ! [A2: nat,A: set_nat,B4: nat > nat,B: nat > set_nat_nat] :
      ( ( member_nat @ A2 @ A )
     => ( ( member_nat_nat @ B4 @ ( B @ A2 ) )
       => ( member8336108448496249625at_nat @ ( produc7839516862119294504at_nat @ A2 @ B4 ) @ ( produc4526620505022529241at_nat @ A @ B ) ) ) ) ).

% SigmaI
thf(fact_595_SigmaI,axiom,
    ! [A2: nat,A: set_nat,B4: nat > a,B: nat > set_nat_a] :
      ( ( member_nat @ A2 @ A )
     => ( ( member_nat_a @ B4 @ ( B @ A2 ) )
       => ( member2811451201825703969_nat_a @ ( produc5292568359338195516_nat_a @ A2 @ B4 ) @ ( produc3659965428606139339_nat_a @ A @ B ) ) ) ) ).

% SigmaI
thf(fact_596_Sigma__empty1,axiom,
    ! [B: a > set_a] :
      ( ( product_Sigma_a_a @ bot_bot_set_a @ B )
      = bot_bo3357376287454694259od_a_a ) ).

% Sigma_empty1
thf(fact_597_Times__empty,axiom,
    ! [A: set_a,B: set_a] :
      ( ( ( product_Sigma_a_a @ A
          @ ^ [Uu: a] : B )
        = bot_bo3357376287454694259od_a_a )
      = ( ( A = bot_bot_set_a )
        | ( B = bot_bot_set_a ) ) ) ).

% Times_empty
thf(fact_598_Times__empty,axiom,
    ! [A: set_a,B: set_nat] :
      ( ( ( product_Sigma_a_nat @ A
          @ ^ [Uu: a] : B )
        = bot_bo9049108969261143879_a_nat )
      = ( ( A = bot_bot_set_a )
        | ( B = bot_bot_set_nat ) ) ) ).

% Times_empty
thf(fact_599_Times__empty,axiom,
    ! [A: set_nat,B: set_a] :
      ( ( ( product_Sigma_nat_a @ A
          @ ^ [Uu: nat] : B )
        = bot_bo8308015405739170157_nat_a )
      = ( ( A = bot_bot_set_nat )
        | ( B = bot_bot_set_a ) ) ) ).

% Times_empty
thf(fact_600_Times__empty,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ( produc457027306803732586at_nat @ A
          @ ^ [Uu: nat] : B )
        = bot_bo2099793752762293965at_nat )
      = ( ( A = bot_bot_set_nat )
        | ( B = bot_bot_set_nat ) ) ) ).

% Times_empty
thf(fact_601_Sigma__empty2,axiom,
    ! [A: set_a] :
      ( ( product_Sigma_a_a @ A
        @ ^ [Uu: a] : bot_bot_set_a )
      = bot_bo3357376287454694259od_a_a ) ).

% Sigma_empty2
thf(fact_602_Pi__eq__empty,axiom,
    ! [A: set_Product_prod_a_a,B: product_prod_a_a > set_Product_prod_a_a] :
      ( ( ( pi_Pro6370639526499058571od_a_a @ A @ B )
        = bot_bo2841618473486996463od_a_a )
      = ( ? [X2: product_prod_a_a] :
            ( ( member1426531477525435216od_a_a @ X2 @ A )
            & ( ( B @ X2 )
              = bot_bo3357376287454694259od_a_a ) ) ) ) ).

% Pi_eq_empty
thf(fact_603_Pi__eq__empty,axiom,
    ! [A: set_a,B: a > set_Product_prod_a_a] :
      ( ( ( pi_a_P2178097759547960436od_a_a @ A @ B )
        = bot_bo6605490641894888024od_a_a )
      = ( ? [X2: a] :
            ( ( member_a @ X2 @ A )
            & ( ( B @ X2 )
              = bot_bo3357376287454694259od_a_a ) ) ) ) ).

% Pi_eq_empty
thf(fact_604_Pi__eq__empty,axiom,
    ! [A: set_nat,B: nat > set_a] :
      ( ( ( pi_nat_a @ A @ B )
        = bot_bot_set_nat_a )
      = ( ? [X2: nat] :
            ( ( member_nat @ X2 @ A )
            & ( ( B @ X2 )
              = bot_bot_set_a ) ) ) ) ).

% Pi_eq_empty
thf(fact_605_Pi__eq__empty,axiom,
    ! [A: set_a,B: a > set_a] :
      ( ( ( pi_a_a @ A @ B )
        = bot_bot_set_a_a )
      = ( ? [X2: a] :
            ( ( member_a @ X2 @ A )
            & ( ( B @ X2 )
              = bot_bot_set_a ) ) ) ) ).

% Pi_eq_empty
thf(fact_606_Pi__eq__empty,axiom,
    ! [A: set_nat,B: nat > set_nat] :
      ( ( ( pi_nat_nat @ A @ B )
        = bot_bot_set_nat_nat )
      = ( ? [X2: nat] :
            ( ( member_nat @ X2 @ A )
            & ( ( B @ X2 )
              = bot_bot_set_nat ) ) ) ) ).

% Pi_eq_empty
thf(fact_607_Pi__eq__empty,axiom,
    ! [A: set_a,B: a > set_nat] :
      ( ( ( pi_a_nat @ A @ B )
        = bot_bot_set_a_nat )
      = ( ? [X2: a] :
            ( ( member_a @ X2 @ A )
            & ( ( B @ X2 )
              = bot_bot_set_nat ) ) ) ) ).

% Pi_eq_empty
thf(fact_608_card_Oempty,axiom,
    ( ( finite4795055649997197647od_a_a @ bot_bo3357376287454694259od_a_a )
    = zero_zero_nat ) ).

% card.empty
thf(fact_609_card_Oempty,axiom,
    ( ( finite_card_a @ bot_bot_set_a )
    = zero_zero_nat ) ).

% card.empty
thf(fact_610_card_Oempty,axiom,
    ( ( finite_card_nat @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% card.empty
thf(fact_611_card__0__eq,axiom,
    ! [A: set_Product_prod_a_a] :
      ( ( finite6544458595007987280od_a_a @ A )
     => ( ( ( finite4795055649997197647od_a_a @ A )
          = zero_zero_nat )
        = ( A = bot_bo3357376287454694259od_a_a ) ) ) ).

% card_0_eq
thf(fact_612_card__0__eq,axiom,
    ! [A: set_a] :
      ( ( finite_finite_a @ A )
     => ( ( ( finite_card_a @ A )
          = zero_zero_nat )
        = ( A = bot_bot_set_a ) ) ) ).

% card_0_eq
thf(fact_613_card__0__eq,axiom,
    ! [A: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( ( finite_card_nat @ A )
          = zero_zero_nat )
        = ( A = bot_bot_set_nat ) ) ) ).

% card_0_eq
thf(fact_614_sumset__empty_I1_J,axiom,
    ! [A: set_a] :
      ( ( pluenn3038260743871226533mset_a @ g @ addition @ A @ bot_bot_set_a )
      = bot_bot_set_a ) ).

% sumset_empty(1)
thf(fact_615_sumset__empty_I2_J,axiom,
    ! [A: set_a] :
      ( ( pluenn3038260743871226533mset_a @ g @ addition @ bot_bot_set_a @ A )
      = bot_bot_set_a ) ).

% sumset_empty(2)
thf(fact_616_sumset__is__empty__iff,axiom,
    ! [A: set_a,B: set_a] :
      ( ( ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B )
        = bot_bot_set_a )
      = ( ( ( inf_inf_set_a @ A @ g )
          = bot_bot_set_a )
        | ( ( inf_inf_set_a @ B @ g )
          = bot_bot_set_a ) ) ) ).

% sumset_is_empty_iff
thf(fact_617_minusset__is__empty__iff,axiom,
    ! [A: set_a] :
      ( ( ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A )
        = bot_bot_set_a )
      = ( ( inf_inf_set_a @ A @ g )
        = bot_bot_set_a ) ) ).

% minusset_is_empty_iff
thf(fact_618_minusset__triv,axiom,
    ( ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ ( insert_a @ zero @ bot_bot_set_a ) )
    = ( insert_a @ zero @ bot_bot_set_a ) ) ).

% minusset_triv
thf(fact_619_group__of__Units,axiom,
    group_group_a @ ( group_Units_a @ g @ addition @ zero ) @ addition @ zero ).

% group_of_Units
thf(fact_620_sumset__iterated__0,axiom,
    ! [A: set_a] :
      ( ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ A @ zero_zero_nat )
      = ( insert_a @ zero @ bot_bot_set_a ) ) ).

% sumset_iterated_0
thf(fact_621_sumset__D_I1_J,axiom,
    ! [A: set_a] :
      ( ( pluenn3038260743871226533mset_a @ g @ addition @ A @ ( insert_a @ zero @ bot_bot_set_a ) )
      = ( inf_inf_set_a @ A @ g ) ) ).

% sumset_D(1)
thf(fact_622_sumset__D_I2_J,axiom,
    ! [A: set_a] :
      ( ( pluenn3038260743871226533mset_a @ g @ addition @ ( insert_a @ zero @ bot_bot_set_a ) @ A )
      = ( inf_inf_set_a @ A @ g ) ) ).

% sumset_D(2)
thf(fact_623_mem__case__prodE,axiom,
    ! [Z: a,C2: a > a > set_a,P2: product_prod_a_a] :
      ( ( member_a @ Z @ ( produc9217457822752978994_set_a @ C2 @ P2 ) )
     => ~ ! [X3: a,Y3: a] :
            ( ( P2
              = ( product_Pair_a_a @ X3 @ Y3 ) )
           => ~ ( member_a @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_624_mem__case__prodE,axiom,
    ! [Z: product_prod_a_a > product_prod_a_a,C2: a > a > set_Pr8826267807999420763od_a_a,P2: product_prod_a_a] :
      ( ( member4020126937092221116od_a_a @ Z @ ( produc797403868875161735od_a_a @ C2 @ P2 ) )
     => ~ ! [X3: a,Y3: a] :
            ( ( P2
              = ( product_Pair_a_a @ X3 @ Y3 ) )
           => ~ ( member4020126937092221116od_a_a @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_625_mem__case__prodE,axiom,
    ! [Z: nat,C2: a > a > set_nat,P2: product_prod_a_a] :
      ( ( member_nat @ Z @ ( produc153843693180602034et_nat @ C2 @ P2 ) )
     => ~ ! [X3: a,Y3: a] :
            ( ( P2
              = ( product_Pair_a_a @ X3 @ Y3 ) )
           => ~ ( member_nat @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_626_mem__case__prodE,axiom,
    ! [Z: nat > nat,C2: a > a > set_nat_nat,P2: product_prod_a_a] :
      ( ( member_nat_nat @ Z @ ( produc1670433337670807073at_nat @ C2 @ P2 ) )
     => ~ ! [X3: a,Y3: a] :
            ( ( P2
              = ( product_Pair_a_a @ X3 @ Y3 ) )
           => ~ ( member_nat_nat @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_627_mem__case__prodE,axiom,
    ! [Z: nat > a,C2: a > a > set_nat_a,P2: product_prod_a_a] :
      ( ( member_nat_a @ Z @ ( produc2694066087470364441_nat_a @ C2 @ P2 ) )
     => ~ ! [X3: a,Y3: a] :
            ( ( P2
              = ( product_Pair_a_a @ X3 @ Y3 ) )
           => ~ ( member_nat_a @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_628_mem__case__prodE,axiom,
    ! [Z: a > product_prod_a_a,C2: a > a > set_a_6829686330177631172od_a_a,P2: product_prod_a_a] :
      ( ( member1957775702407316389od_a_a @ Z @ ( produc2933953731230324464od_a_a @ C2 @ P2 ) )
     => ~ ! [X3: a,Y3: a] :
            ( ( P2
              = ( product_Pair_a_a @ X3 @ Y3 ) )
           => ~ ( member1957775702407316389od_a_a @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_629_mem__case__prodE,axiom,
    ! [Z: a > nat,C2: a > a > set_a_nat,P2: product_prod_a_a] :
      ( ( member_a_nat @ Z @ ( produc3435159650992338163_a_nat @ C2 @ P2 ) )
     => ~ ! [X3: a,Y3: a] :
            ( ( P2
              = ( product_Pair_a_a @ X3 @ Y3 ) )
           => ~ ( member_a_nat @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_630_mem__case__prodE,axiom,
    ! [Z: a > a,C2: a > a > set_a_a,P2: product_prod_a_a] :
      ( ( member_a_a @ Z @ ( produc1550278517953855687et_a_a @ C2 @ P2 ) )
     => ~ ! [X3: a,Y3: a] :
            ( ( P2
              = ( product_Pair_a_a @ X3 @ Y3 ) )
           => ~ ( member_a_a @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_631_Sigma__empty__iff,axiom,
    ! [I: set_a,X5: a > set_a] :
      ( ( ( product_Sigma_a_a @ I @ X5 )
        = bot_bo3357376287454694259od_a_a )
      = ( ! [X2: a] :
            ( ( member_a @ X2 @ I )
           => ( ( X5 @ X2 )
              = bot_bot_set_a ) ) ) ) ).

% Sigma_empty_iff
thf(fact_632_case__prodD,axiom,
    ! [F: a > a > $o,A2: a,B4: a] :
      ( ( produc6436628058953941356_a_a_o @ F @ ( product_Pair_a_a @ A2 @ B4 ) )
     => ( F @ A2 @ B4 ) ) ).

% case_prodD
thf(fact_633_case__prodE,axiom,
    ! [C2: a > a > $o,P2: product_prod_a_a] :
      ( ( produc6436628058953941356_a_a_o @ C2 @ P2 )
     => ~ ! [X3: a,Y3: a] :
            ( ( P2
              = ( product_Pair_a_a @ X3 @ Y3 ) )
           => ~ ( C2 @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_634_Collect__case__prod__Sigma,axiom,
    ! [P: a > $o,Q: a > nat > $o] :
      ( ( collec4464134535221767506_a_nat
        @ ( produc3680711911437148916_nat_o
          @ ^ [X2: a,Y4: nat] :
              ( ( P @ X2 )
              & ( Q @ X2 @ Y4 ) ) ) )
      = ( product_Sigma_a_nat @ ( collect_a @ P )
        @ ^ [X2: a] : ( collect_nat @ ( Q @ X2 ) ) ) ) ).

% Collect_case_prod_Sigma
thf(fact_635_Collect__case__prod__Sigma,axiom,
    ! [P: nat > $o,Q: nat > a > $o] :
      ( ( collec7702297998945444472_nat_a
        @ ( produc2746933349376800278at_a_o
          @ ^ [X2: nat,Y4: a] :
              ( ( P @ X2 )
              & ( Q @ X2 @ Y4 ) ) ) )
      = ( product_Sigma_nat_a @ ( collect_nat @ P )
        @ ^ [X2: nat] : ( collect_a @ ( Q @ X2 ) ) ) ) ).

% Collect_case_prod_Sigma
thf(fact_636_Collect__case__prod__Sigma,axiom,
    ! [P: nat > $o,Q: nat > nat > $o] :
      ( ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X2: nat,Y4: nat] :
              ( ( P @ X2 )
              & ( Q @ X2 @ Y4 ) ) ) )
      = ( produc457027306803732586at_nat @ ( collect_nat @ P )
        @ ^ [X2: nat] : ( collect_nat @ ( Q @ X2 ) ) ) ) ).

% Collect_case_prod_Sigma
thf(fact_637_Collect__case__prod__Sigma,axiom,
    ! [P: a > $o,Q: a > a > $o] :
      ( ( collec3336397797384452498od_a_a
        @ ( produc6436628058953941356_a_a_o
          @ ^ [X2: a,Y4: a] :
              ( ( P @ X2 )
              & ( Q @ X2 @ Y4 ) ) ) )
      = ( product_Sigma_a_a @ ( collect_a @ P )
        @ ^ [X2: a] : ( collect_a @ ( Q @ X2 ) ) ) ) ).

% Collect_case_prod_Sigma
thf(fact_638_times__eq__iff,axiom,
    ! [A: set_a,B: set_a,C: set_a,D: set_a] :
      ( ( ( product_Sigma_a_a @ A
          @ ^ [Uu: a] : B )
        = ( product_Sigma_a_a @ C
          @ ^ [Uu: a] : D ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( ( A = bot_bot_set_a )
            | ( B = bot_bot_set_a ) )
          & ( ( C = bot_bot_set_a )
            | ( D = bot_bot_set_a ) ) ) ) ) ).

% times_eq_iff
thf(fact_639_times__eq__iff,axiom,
    ! [A: set_a,B: set_nat,C: set_a,D: set_nat] :
      ( ( ( product_Sigma_a_nat @ A
          @ ^ [Uu: a] : B )
        = ( product_Sigma_a_nat @ C
          @ ^ [Uu: a] : D ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( ( A = bot_bot_set_a )
            | ( B = bot_bot_set_nat ) )
          & ( ( C = bot_bot_set_a )
            | ( D = bot_bot_set_nat ) ) ) ) ) ).

% times_eq_iff
thf(fact_640_times__eq__iff,axiom,
    ! [A: set_nat,B: set_a,C: set_nat,D: set_a] :
      ( ( ( product_Sigma_nat_a @ A
          @ ^ [Uu: nat] : B )
        = ( product_Sigma_nat_a @ C
          @ ^ [Uu: nat] : D ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( ( A = bot_bot_set_nat )
            | ( B = bot_bot_set_a ) )
          & ( ( C = bot_bot_set_nat )
            | ( D = bot_bot_set_a ) ) ) ) ) ).

% times_eq_iff
thf(fact_641_times__eq__iff,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat,D: set_nat] :
      ( ( ( produc457027306803732586at_nat @ A
          @ ^ [Uu: nat] : B )
        = ( produc457027306803732586at_nat @ C
          @ ^ [Uu: nat] : D ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( ( A = bot_bot_set_nat )
            | ( B = bot_bot_set_nat ) )
          & ( ( C = bot_bot_set_nat )
            | ( D = bot_bot_set_nat ) ) ) ) ) ).

% times_eq_iff
thf(fact_642_finite_OemptyI,axiom,
    finite6544458595007987280od_a_a @ bot_bo3357376287454694259od_a_a ).

% finite.emptyI
thf(fact_643_finite_OemptyI,axiom,
    finite_finite_a @ bot_bot_set_a ).

% finite.emptyI
thf(fact_644_finite_OemptyI,axiom,
    finite_finite_nat @ bot_bot_set_nat ).

% finite.emptyI
thf(fact_645_infinite__imp__nonempty,axiom,
    ! [S: set_Product_prod_a_a] :
      ( ~ ( finite6544458595007987280od_a_a @ S )
     => ( S != bot_bo3357376287454694259od_a_a ) ) ).

% infinite_imp_nonempty
thf(fact_646_infinite__imp__nonempty,axiom,
    ! [S: set_a] :
      ( ~ ( finite_finite_a @ S )
     => ( S != bot_bot_set_a ) ) ).

% infinite_imp_nonempty
thf(fact_647_infinite__imp__nonempty,axiom,
    ! [S: set_nat] :
      ( ~ ( finite_finite_nat @ S )
     => ( S != bot_bot_set_nat ) ) ).

% infinite_imp_nonempty
thf(fact_648_times__subset__iff,axiom,
    ! [A: set_nat,C: set_nat,B: set_nat,D: set_nat] :
      ( ( ord_le3146513528884898305at_nat
        @ ( produc457027306803732586at_nat @ A
          @ ^ [Uu: nat] : C )
        @ ( produc457027306803732586at_nat @ B
          @ ^ [Uu: nat] : D ) )
      = ( ( A = bot_bot_set_nat )
        | ( C = bot_bot_set_nat )
        | ( ( ord_less_eq_set_nat @ A @ B )
          & ( ord_less_eq_set_nat @ C @ D ) ) ) ) ).

% times_subset_iff
thf(fact_649_times__subset__iff,axiom,
    ! [A: set_nat,C: set_a,B: set_nat,D: set_a] :
      ( ( ord_le7924913712489149241_nat_a
        @ ( product_Sigma_nat_a @ A
          @ ^ [Uu: nat] : C )
        @ ( product_Sigma_nat_a @ B
          @ ^ [Uu: nat] : D ) )
      = ( ( A = bot_bot_set_nat )
        | ( C = bot_bot_set_a )
        | ( ( ord_less_eq_set_nat @ A @ B )
          & ( ord_less_eq_set_a @ C @ D ) ) ) ) ).

% times_subset_iff
thf(fact_650_times__subset__iff,axiom,
    ! [A: set_a,C: set_nat,B: set_a,D: set_nat] :
      ( ( ord_le8666007276011122963_a_nat
        @ ( product_Sigma_a_nat @ A
          @ ^ [Uu: a] : C )
        @ ( product_Sigma_a_nat @ B
          @ ^ [Uu: a] : D ) )
      = ( ( A = bot_bot_set_a )
        | ( C = bot_bot_set_nat )
        | ( ( ord_less_eq_set_a @ A @ B )
          & ( ord_less_eq_set_nat @ C @ D ) ) ) ) ).

% times_subset_iff
thf(fact_651_times__subset__iff,axiom,
    ! [A: set_a,C: set_a,B: set_a,D: set_a] :
      ( ( ord_le746702958409616551od_a_a
        @ ( product_Sigma_a_a @ A
          @ ^ [Uu: a] : C )
        @ ( product_Sigma_a_a @ B
          @ ^ [Uu: a] : D ) )
      = ( ( A = bot_bot_set_a )
        | ( C = bot_bot_set_a )
        | ( ( ord_less_eq_set_a @ A @ B )
          & ( ord_less_eq_set_a @ C @ D ) ) ) ) ).

% times_subset_iff
thf(fact_652_finite__has__minimal,axiom,
    ! [A: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A )
               => ( ( ord_less_eq_nat @ Xa @ X3 )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_653_finite__has__minimal,axiom,
    ! [A: set_set_a] :
      ( ( finite_finite_set_a @ A )
     => ( ( A != bot_bot_set_set_a )
       => ? [X3: set_a] :
            ( ( member_set_a @ X3 @ A )
            & ! [Xa: set_a] :
                ( ( member_set_a @ Xa @ A )
               => ( ( ord_less_eq_set_a @ Xa @ X3 )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_654_finite__has__maximal,axiom,
    ! [A: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A )
               => ( ( ord_less_eq_nat @ X3 @ Xa )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_655_finite__has__maximal,axiom,
    ! [A: set_set_a] :
      ( ( finite_finite_set_a @ A )
     => ( ( A != bot_bot_set_set_a )
       => ? [X3: set_a] :
            ( ( member_set_a @ X3 @ A )
            & ! [Xa: set_a] :
                ( ( member_set_a @ Xa @ A )
               => ( ( ord_less_eq_set_a @ X3 @ Xa )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_656_infinite__finite__induct,axiom,
    ! [P: set_Pr8826267807999420763od_a_a > $o,A: set_Pr8826267807999420763od_a_a] :
      ( ! [A8: set_Pr8826267807999420763od_a_a] :
          ( ~ ( finite2714571839071174076od_a_a @ A8 )
         => ( P @ A8 ) )
     => ( ( P @ bot_bo2841618473486996463od_a_a )
       => ( ! [X3: product_prod_a_a > product_prod_a_a,F3: set_Pr8826267807999420763od_a_a] :
              ( ( finite2714571839071174076od_a_a @ F3 )
             => ( ~ ( member4020126937092221116od_a_a @ X3 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert1583624352380834389od_a_a @ X3 @ F3 ) ) ) ) )
         => ( P @ A ) ) ) ) ).

% infinite_finite_induct
thf(fact_657_infinite__finite__induct,axiom,
    ! [P: set_nat_nat > $o,A: set_nat_nat] :
      ( ! [A8: set_nat_nat] :
          ( ~ ( finite2115694454571419734at_nat @ A8 )
         => ( P @ A8 ) )
     => ( ( P @ bot_bot_set_nat_nat )
       => ( ! [X3: nat > nat,F3: set_nat_nat] :
              ( ( finite2115694454571419734at_nat @ F3 )
             => ( ~ ( member_nat_nat @ X3 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert_nat_nat @ X3 @ F3 ) ) ) ) )
         => ( P @ A ) ) ) ) ).

% infinite_finite_induct
thf(fact_658_infinite__finite__induct,axiom,
    ! [P: set_nat_a > $o,A: set_nat_a] :
      ( ! [A8: set_nat_a] :
          ( ~ ( finite_finite_nat_a @ A8 )
         => ( P @ A8 ) )
     => ( ( P @ bot_bot_set_nat_a )
       => ( ! [X3: nat > a,F3: set_nat_a] :
              ( ( finite_finite_nat_a @ F3 )
             => ( ~ ( member_nat_a @ X3 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert_nat_a @ X3 @ F3 ) ) ) ) )
         => ( P @ A ) ) ) ) ).

% infinite_finite_induct
thf(fact_659_infinite__finite__induct,axiom,
    ! [P: set_a_6829686330177631172od_a_a > $o,A: set_a_6829686330177631172od_a_a] :
      ( ! [A8: set_a_6829686330177631172od_a_a] :
          ( ~ ( finite8821664692192630949od_a_a @ A8 )
         => ( P @ A8 ) )
     => ( ( P @ bot_bo6605490641894888024od_a_a )
       => ( ! [X3: a > product_prod_a_a,F3: set_a_6829686330177631172od_a_a] :
              ( ( finite8821664692192630949od_a_a @ F3 )
             => ( ~ ( member1957775702407316389od_a_a @ X3 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert5082226857754029630od_a_a @ X3 @ F3 ) ) ) ) )
         => ( P @ A ) ) ) ) ).

% infinite_finite_induct
thf(fact_660_infinite__finite__induct,axiom,
    ! [P: set_a_nat > $o,A: set_a_nat] :
      ( ! [A8: set_a_nat] :
          ( ~ ( finite_finite_a_nat @ A8 )
         => ( P @ A8 ) )
     => ( ( P @ bot_bot_set_a_nat )
       => ( ! [X3: a > nat,F3: set_a_nat] :
              ( ( finite_finite_a_nat @ F3 )
             => ( ~ ( member_a_nat @ X3 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert_a_nat @ X3 @ F3 ) ) ) ) )
         => ( P @ A ) ) ) ) ).

% infinite_finite_induct
thf(fact_661_infinite__finite__induct,axiom,
    ! [P: set_a_a > $o,A: set_a_a] :
      ( ! [A8: set_a_a] :
          ( ~ ( finite_finite_a_a @ A8 )
         => ( P @ A8 ) )
     => ( ( P @ bot_bot_set_a_a )
       => ( ! [X3: a > a,F3: set_a_a] :
              ( ( finite_finite_a_a @ F3 )
             => ( ~ ( member_a_a @ X3 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert_a_a @ X3 @ F3 ) ) ) ) )
         => ( P @ A ) ) ) ) ).

% infinite_finite_induct
thf(fact_662_infinite__finite__induct,axiom,
    ! [P: set_Product_prod_a_a > $o,A: set_Product_prod_a_a] :
      ( ! [A8: set_Product_prod_a_a] :
          ( ~ ( finite6544458595007987280od_a_a @ A8 )
         => ( P @ A8 ) )
     => ( ( P @ bot_bo3357376287454694259od_a_a )
       => ( ! [X3: product_prod_a_a,F3: set_Product_prod_a_a] :
              ( ( finite6544458595007987280od_a_a @ F3 )
             => ( ~ ( member1426531477525435216od_a_a @ X3 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert4534936382041156343od_a_a @ X3 @ F3 ) ) ) ) )
         => ( P @ A ) ) ) ) ).

% infinite_finite_induct
thf(fact_663_infinite__finite__induct,axiom,
    ! [P: set_a > $o,A: set_a] :
      ( ! [A8: set_a] :
          ( ~ ( finite_finite_a @ A8 )
         => ( P @ A8 ) )
     => ( ( P @ bot_bot_set_a )
       => ( ! [X3: a,F3: set_a] :
              ( ( finite_finite_a @ F3 )
             => ( ~ ( member_a @ X3 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert_a @ X3 @ F3 ) ) ) ) )
         => ( P @ A ) ) ) ) ).

% infinite_finite_induct
thf(fact_664_infinite__finite__induct,axiom,
    ! [P: set_nat > $o,A: set_nat] :
      ( ! [A8: set_nat] :
          ( ~ ( finite_finite_nat @ A8 )
         => ( P @ A8 ) )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X3: nat,F3: set_nat] :
              ( ( finite_finite_nat @ F3 )
             => ( ~ ( member_nat @ X3 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert_nat @ X3 @ F3 ) ) ) ) )
         => ( P @ A ) ) ) ) ).

% infinite_finite_induct
thf(fact_665_finite__ne__induct,axiom,
    ! [F2: set_Pr8826267807999420763od_a_a,P: set_Pr8826267807999420763od_a_a > $o] :
      ( ( finite2714571839071174076od_a_a @ F2 )
     => ( ( F2 != bot_bo2841618473486996463od_a_a )
       => ( ! [X3: product_prod_a_a > product_prod_a_a] : ( P @ ( insert1583624352380834389od_a_a @ X3 @ bot_bo2841618473486996463od_a_a ) )
         => ( ! [X3: product_prod_a_a > product_prod_a_a,F3: set_Pr8826267807999420763od_a_a] :
                ( ( finite2714571839071174076od_a_a @ F3 )
               => ( ( F3 != bot_bo2841618473486996463od_a_a )
                 => ( ~ ( member4020126937092221116od_a_a @ X3 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert1583624352380834389od_a_a @ X3 @ F3 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_666_finite__ne__induct,axiom,
    ! [F2: set_nat_nat,P: set_nat_nat > $o] :
      ( ( finite2115694454571419734at_nat @ F2 )
     => ( ( F2 != bot_bot_set_nat_nat )
       => ( ! [X3: nat > nat] : ( P @ ( insert_nat_nat @ X3 @ bot_bot_set_nat_nat ) )
         => ( ! [X3: nat > nat,F3: set_nat_nat] :
                ( ( finite2115694454571419734at_nat @ F3 )
               => ( ( F3 != bot_bot_set_nat_nat )
                 => ( ~ ( member_nat_nat @ X3 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert_nat_nat @ X3 @ F3 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_667_finite__ne__induct,axiom,
    ! [F2: set_nat_a,P: set_nat_a > $o] :
      ( ( finite_finite_nat_a @ F2 )
     => ( ( F2 != bot_bot_set_nat_a )
       => ( ! [X3: nat > a] : ( P @ ( insert_nat_a @ X3 @ bot_bot_set_nat_a ) )
         => ( ! [X3: nat > a,F3: set_nat_a] :
                ( ( finite_finite_nat_a @ F3 )
               => ( ( F3 != bot_bot_set_nat_a )
                 => ( ~ ( member_nat_a @ X3 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert_nat_a @ X3 @ F3 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_668_finite__ne__induct,axiom,
    ! [F2: set_a_6829686330177631172od_a_a,P: set_a_6829686330177631172od_a_a > $o] :
      ( ( finite8821664692192630949od_a_a @ F2 )
     => ( ( F2 != bot_bo6605490641894888024od_a_a )
       => ( ! [X3: a > product_prod_a_a] : ( P @ ( insert5082226857754029630od_a_a @ X3 @ bot_bo6605490641894888024od_a_a ) )
         => ( ! [X3: a > product_prod_a_a,F3: set_a_6829686330177631172od_a_a] :
                ( ( finite8821664692192630949od_a_a @ F3 )
               => ( ( F3 != bot_bo6605490641894888024od_a_a )
                 => ( ~ ( member1957775702407316389od_a_a @ X3 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert5082226857754029630od_a_a @ X3 @ F3 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_669_finite__ne__induct,axiom,
    ! [F2: set_a_nat,P: set_a_nat > $o] :
      ( ( finite_finite_a_nat @ F2 )
     => ( ( F2 != bot_bot_set_a_nat )
       => ( ! [X3: a > nat] : ( P @ ( insert_a_nat @ X3 @ bot_bot_set_a_nat ) )
         => ( ! [X3: a > nat,F3: set_a_nat] :
                ( ( finite_finite_a_nat @ F3 )
               => ( ( F3 != bot_bot_set_a_nat )
                 => ( ~ ( member_a_nat @ X3 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert_a_nat @ X3 @ F3 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_670_finite__ne__induct,axiom,
    ! [F2: set_a_a,P: set_a_a > $o] :
      ( ( finite_finite_a_a @ F2 )
     => ( ( F2 != bot_bot_set_a_a )
       => ( ! [X3: a > a] : ( P @ ( insert_a_a @ X3 @ bot_bot_set_a_a ) )
         => ( ! [X3: a > a,F3: set_a_a] :
                ( ( finite_finite_a_a @ F3 )
               => ( ( F3 != bot_bot_set_a_a )
                 => ( ~ ( member_a_a @ X3 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert_a_a @ X3 @ F3 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_671_finite__ne__induct,axiom,
    ! [F2: set_Product_prod_a_a,P: set_Product_prod_a_a > $o] :
      ( ( finite6544458595007987280od_a_a @ F2 )
     => ( ( F2 != bot_bo3357376287454694259od_a_a )
       => ( ! [X3: product_prod_a_a] : ( P @ ( insert4534936382041156343od_a_a @ X3 @ bot_bo3357376287454694259od_a_a ) )
         => ( ! [X3: product_prod_a_a,F3: set_Product_prod_a_a] :
                ( ( finite6544458595007987280od_a_a @ F3 )
               => ( ( F3 != bot_bo3357376287454694259od_a_a )
                 => ( ~ ( member1426531477525435216od_a_a @ X3 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert4534936382041156343od_a_a @ X3 @ F3 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_672_finite__ne__induct,axiom,
    ! [F2: set_a,P: set_a > $o] :
      ( ( finite_finite_a @ F2 )
     => ( ( F2 != bot_bot_set_a )
       => ( ! [X3: a] : ( P @ ( insert_a @ X3 @ bot_bot_set_a ) )
         => ( ! [X3: a,F3: set_a] :
                ( ( finite_finite_a @ F3 )
               => ( ( F3 != bot_bot_set_a )
                 => ( ~ ( member_a @ X3 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert_a @ X3 @ F3 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_673_finite__ne__induct,axiom,
    ! [F2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F2 )
     => ( ( F2 != bot_bot_set_nat )
       => ( ! [X3: nat] : ( P @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
         => ( ! [X3: nat,F3: set_nat] :
                ( ( finite_finite_nat @ F3 )
               => ( ( F3 != bot_bot_set_nat )
                 => ( ~ ( member_nat @ X3 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert_nat @ X3 @ F3 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_674_finite__induct,axiom,
    ! [F2: set_Pr8826267807999420763od_a_a,P: set_Pr8826267807999420763od_a_a > $o] :
      ( ( finite2714571839071174076od_a_a @ F2 )
     => ( ( P @ bot_bo2841618473486996463od_a_a )
       => ( ! [X3: product_prod_a_a > product_prod_a_a,F3: set_Pr8826267807999420763od_a_a] :
              ( ( finite2714571839071174076od_a_a @ F3 )
             => ( ~ ( member4020126937092221116od_a_a @ X3 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert1583624352380834389od_a_a @ X3 @ F3 ) ) ) ) )
         => ( P @ F2 ) ) ) ) ).

% finite_induct
thf(fact_675_finite__induct,axiom,
    ! [F2: set_nat_nat,P: set_nat_nat > $o] :
      ( ( finite2115694454571419734at_nat @ F2 )
     => ( ( P @ bot_bot_set_nat_nat )
       => ( ! [X3: nat > nat,F3: set_nat_nat] :
              ( ( finite2115694454571419734at_nat @ F3 )
             => ( ~ ( member_nat_nat @ X3 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert_nat_nat @ X3 @ F3 ) ) ) ) )
         => ( P @ F2 ) ) ) ) ).

% finite_induct
thf(fact_676_finite__induct,axiom,
    ! [F2: set_nat_a,P: set_nat_a > $o] :
      ( ( finite_finite_nat_a @ F2 )
     => ( ( P @ bot_bot_set_nat_a )
       => ( ! [X3: nat > a,F3: set_nat_a] :
              ( ( finite_finite_nat_a @ F3 )
             => ( ~ ( member_nat_a @ X3 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert_nat_a @ X3 @ F3 ) ) ) ) )
         => ( P @ F2 ) ) ) ) ).

% finite_induct
thf(fact_677_finite__induct,axiom,
    ! [F2: set_a_6829686330177631172od_a_a,P: set_a_6829686330177631172od_a_a > $o] :
      ( ( finite8821664692192630949od_a_a @ F2 )
     => ( ( P @ bot_bo6605490641894888024od_a_a )
       => ( ! [X3: a > product_prod_a_a,F3: set_a_6829686330177631172od_a_a] :
              ( ( finite8821664692192630949od_a_a @ F3 )
             => ( ~ ( member1957775702407316389od_a_a @ X3 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert5082226857754029630od_a_a @ X3 @ F3 ) ) ) ) )
         => ( P @ F2 ) ) ) ) ).

% finite_induct
thf(fact_678_finite__induct,axiom,
    ! [F2: set_a_nat,P: set_a_nat > $o] :
      ( ( finite_finite_a_nat @ F2 )
     => ( ( P @ bot_bot_set_a_nat )
       => ( ! [X3: a > nat,F3: set_a_nat] :
              ( ( finite_finite_a_nat @ F3 )
             => ( ~ ( member_a_nat @ X3 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert_a_nat @ X3 @ F3 ) ) ) ) )
         => ( P @ F2 ) ) ) ) ).

% finite_induct
thf(fact_679_finite__induct,axiom,
    ! [F2: set_a_a,P: set_a_a > $o] :
      ( ( finite_finite_a_a @ F2 )
     => ( ( P @ bot_bot_set_a_a )
       => ( ! [X3: a > a,F3: set_a_a] :
              ( ( finite_finite_a_a @ F3 )
             => ( ~ ( member_a_a @ X3 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert_a_a @ X3 @ F3 ) ) ) ) )
         => ( P @ F2 ) ) ) ) ).

% finite_induct
thf(fact_680_finite__induct,axiom,
    ! [F2: set_Product_prod_a_a,P: set_Product_prod_a_a > $o] :
      ( ( finite6544458595007987280od_a_a @ F2 )
     => ( ( P @ bot_bo3357376287454694259od_a_a )
       => ( ! [X3: product_prod_a_a,F3: set_Product_prod_a_a] :
              ( ( finite6544458595007987280od_a_a @ F3 )
             => ( ~ ( member1426531477525435216od_a_a @ X3 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert4534936382041156343od_a_a @ X3 @ F3 ) ) ) ) )
         => ( P @ F2 ) ) ) ) ).

% finite_induct
thf(fact_681_finite__induct,axiom,
    ! [F2: set_a,P: set_a > $o] :
      ( ( finite_finite_a @ F2 )
     => ( ( P @ bot_bot_set_a )
       => ( ! [X3: a,F3: set_a] :
              ( ( finite_finite_a @ F3 )
             => ( ~ ( member_a @ X3 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert_a @ X3 @ F3 ) ) ) ) )
         => ( P @ F2 ) ) ) ) ).

% finite_induct
thf(fact_682_finite__induct,axiom,
    ! [F2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F2 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X3: nat,F3: set_nat] :
              ( ( finite_finite_nat @ F3 )
             => ( ~ ( member_nat @ X3 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert_nat @ X3 @ F3 ) ) ) ) )
         => ( P @ F2 ) ) ) ) ).

% finite_induct
thf(fact_683_finite_Osimps,axiom,
    ( finite6544458595007987280od_a_a
    = ( ^ [A4: set_Product_prod_a_a] :
          ( ( A4 = bot_bo3357376287454694259od_a_a )
          | ? [A5: set_Product_prod_a_a,B3: product_prod_a_a] :
              ( ( A4
                = ( insert4534936382041156343od_a_a @ B3 @ A5 ) )
              & ( finite6544458595007987280od_a_a @ A5 ) ) ) ) ) ).

% finite.simps
thf(fact_684_finite_Osimps,axiom,
    ( finite_finite_a
    = ( ^ [A4: set_a] :
          ( ( A4 = bot_bot_set_a )
          | ? [A5: set_a,B3: a] :
              ( ( A4
                = ( insert_a @ B3 @ A5 ) )
              & ( finite_finite_a @ A5 ) ) ) ) ) ).

% finite.simps
thf(fact_685_finite_Osimps,axiom,
    ( finite_finite_nat
    = ( ^ [A4: set_nat] :
          ( ( A4 = bot_bot_set_nat )
          | ? [A5: set_nat,B3: nat] :
              ( ( A4
                = ( insert_nat @ B3 @ A5 ) )
              & ( finite_finite_nat @ A5 ) ) ) ) ) ).

% finite.simps
thf(fact_686_finite_Ocases,axiom,
    ! [A2: set_Product_prod_a_a] :
      ( ( finite6544458595007987280od_a_a @ A2 )
     => ( ( A2 != bot_bo3357376287454694259od_a_a )
       => ~ ! [A8: set_Product_prod_a_a] :
              ( ? [A3: product_prod_a_a] :
                  ( A2
                  = ( insert4534936382041156343od_a_a @ A3 @ A8 ) )
             => ~ ( finite6544458595007987280od_a_a @ A8 ) ) ) ) ).

% finite.cases
thf(fact_687_finite_Ocases,axiom,
    ! [A2: set_a] :
      ( ( finite_finite_a @ A2 )
     => ( ( A2 != bot_bot_set_a )
       => ~ ! [A8: set_a] :
              ( ? [A3: a] :
                  ( A2
                  = ( insert_a @ A3 @ A8 ) )
             => ~ ( finite_finite_a @ A8 ) ) ) ) ).

% finite.cases
thf(fact_688_finite_Ocases,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( A2 != bot_bot_set_nat )
       => ~ ! [A8: set_nat] :
              ( ? [A3: nat] :
                  ( A2
                  = ( insert_nat @ A3 @ A8 ) )
             => ~ ( finite_finite_nat @ A8 ) ) ) ) ).

% finite.cases
thf(fact_689_additive__abelian__group_Osumset__empty_I1_J,axiom,
    ! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: set_nat] :
      ( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
     => ( ( pluenn3669378163024332905et_nat @ G @ Addition @ A @ bot_bot_set_nat )
        = bot_bot_set_nat ) ) ).

% additive_abelian_group.sumset_empty(1)
thf(fact_690_additive__abelian__group_Osumset__empty_I1_J,axiom,
    ! [G: set_a,Addition: a > a > a,Zero: a,A: set_a] :
      ( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
     => ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ bot_bot_set_a )
        = bot_bot_set_a ) ) ).

% additive_abelian_group.sumset_empty(1)
thf(fact_691_additive__abelian__group_Osumset__empty_I2_J,axiom,
    ! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: set_nat] :
      ( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
     => ( ( pluenn3669378163024332905et_nat @ G @ Addition @ bot_bot_set_nat @ A )
        = bot_bot_set_nat ) ) ).

% additive_abelian_group.sumset_empty(2)
thf(fact_692_additive__abelian__group_Osumset__empty_I2_J,axiom,
    ! [G: set_a,Addition: a > a > a,Zero: a,A: set_a] :
      ( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
     => ( ( pluenn3038260743871226533mset_a @ G @ Addition @ bot_bot_set_a @ A )
        = bot_bot_set_a ) ) ).

% additive_abelian_group.sumset_empty(2)
thf(fact_693_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_a_a] :
      ~ ! [A3: a,B2: a] :
          ( Y
         != ( product_Pair_a_a @ A3 @ B2 ) ) ).

% old.prod.exhaust
thf(fact_694_surj__pair,axiom,
    ! [P2: product_prod_a_a] :
    ? [X3: a,Y3: a] :
      ( P2
      = ( product_Pair_a_a @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_695_prod__cases,axiom,
    ! [P: product_prod_a_a > $o,P2: product_prod_a_a] :
      ( ! [A3: a,B2: a] : ( P @ ( product_Pair_a_a @ A3 @ B2 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_696_Pair__inject,axiom,
    ! [A2: a,B4: a,A7: a,B8: a] :
      ( ( ( product_Pair_a_a @ A2 @ B4 )
        = ( product_Pair_a_a @ A7 @ B8 ) )
     => ~ ( ( A2 = A7 )
         => ( B4 != B8 ) ) ) ).

% Pair_inject
thf(fact_697_finite__subset__induct,axiom,
    ! [F2: set_Pr8826267807999420763od_a_a,A: set_Pr8826267807999420763od_a_a,P: set_Pr8826267807999420763od_a_a > $o] :
      ( ( finite2714571839071174076od_a_a @ F2 )
     => ( ( ord_le741722312431004091od_a_a @ F2 @ A )
       => ( ( P @ bot_bo2841618473486996463od_a_a )
         => ( ! [A3: product_prod_a_a > product_prod_a_a,F3: set_Pr8826267807999420763od_a_a] :
                ( ( finite2714571839071174076od_a_a @ F3 )
               => ( ( member4020126937092221116od_a_a @ A3 @ A )
                 => ( ~ ( member4020126937092221116od_a_a @ A3 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert1583624352380834389od_a_a @ A3 @ F3 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_698_finite__subset__induct,axiom,
    ! [F2: set_nat_nat,A: set_nat_nat,P: set_nat_nat > $o] :
      ( ( finite2115694454571419734at_nat @ F2 )
     => ( ( ord_le9059583361652607317at_nat @ F2 @ A )
       => ( ( P @ bot_bot_set_nat_nat )
         => ( ! [A3: nat > nat,F3: set_nat_nat] :
                ( ( finite2115694454571419734at_nat @ F3 )
               => ( ( member_nat_nat @ A3 @ A )
                 => ( ~ ( member_nat_nat @ A3 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert_nat_nat @ A3 @ F3 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_699_finite__subset__induct,axiom,
    ! [F2: set_nat_a,A: set_nat_a,P: set_nat_a > $o] :
      ( ( finite_finite_nat_a @ F2 )
     => ( ( ord_le871467723717165285_nat_a @ F2 @ A )
       => ( ( P @ bot_bot_set_nat_a )
         => ( ! [A3: nat > a,F3: set_nat_a] :
                ( ( finite_finite_nat_a @ F3 )
               => ( ( member_nat_a @ A3 @ A )
                 => ( ~ ( member_nat_a @ A3 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert_nat_a @ A3 @ F3 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_700_finite__subset__induct,axiom,
    ! [F2: set_a_6829686330177631172od_a_a,A: set_a_6829686330177631172od_a_a,P: set_a_6829686330177631172od_a_a > $o] :
      ( ( finite8821664692192630949od_a_a @ F2 )
     => ( ( ord_le2371270124811097124od_a_a @ F2 @ A )
       => ( ( P @ bot_bo6605490641894888024od_a_a )
         => ( ! [A3: a > product_prod_a_a,F3: set_a_6829686330177631172od_a_a] :
                ( ( finite8821664692192630949od_a_a @ F3 )
               => ( ( member1957775702407316389od_a_a @ A3 @ A )
                 => ( ~ ( member1957775702407316389od_a_a @ A3 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert5082226857754029630od_a_a @ A3 @ F3 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_701_finite__subset__induct,axiom,
    ! [F2: set_a_nat,A: set_a_nat,P: set_a_nat > $o] :
      ( ( finite_finite_a_nat @ F2 )
     => ( ( ord_le1612561287239139007_a_nat @ F2 @ A )
       => ( ( P @ bot_bot_set_a_nat )
         => ( ! [A3: a > nat,F3: set_a_nat] :
                ( ( finite_finite_a_nat @ F3 )
               => ( ( member_a_nat @ A3 @ A )
                 => ( ~ ( member_a_nat @ A3 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert_a_nat @ A3 @ F3 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_702_finite__subset__induct,axiom,
    ! [F2: set_a_a,A: set_a_a,P: set_a_a > $o] :
      ( ( finite_finite_a_a @ F2 )
     => ( ( ord_less_eq_set_a_a @ F2 @ A )
       => ( ( P @ bot_bot_set_a_a )
         => ( ! [A3: a > a,F3: set_a_a] :
                ( ( finite_finite_a_a @ F3 )
               => ( ( member_a_a @ A3 @ A )
                 => ( ~ ( member_a_a @ A3 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert_a_a @ A3 @ F3 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_703_finite__subset__induct,axiom,
    ! [F2: set_Product_prod_a_a,A: set_Product_prod_a_a,P: set_Product_prod_a_a > $o] :
      ( ( finite6544458595007987280od_a_a @ F2 )
     => ( ( ord_le746702958409616551od_a_a @ F2 @ A )
       => ( ( P @ bot_bo3357376287454694259od_a_a )
         => ( ! [A3: product_prod_a_a,F3: set_Product_prod_a_a] :
                ( ( finite6544458595007987280od_a_a @ F3 )
               => ( ( member1426531477525435216od_a_a @ A3 @ A )
                 => ( ~ ( member1426531477525435216od_a_a @ A3 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert4534936382041156343od_a_a @ A3 @ F3 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_704_finite__subset__induct,axiom,
    ! [F2: set_nat,A: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F2 )
     => ( ( ord_less_eq_set_nat @ F2 @ A )
       => ( ( P @ bot_bot_set_nat )
         => ( ! [A3: nat,F3: set_nat] :
                ( ( finite_finite_nat @ F3 )
               => ( ( member_nat @ A3 @ A )
                 => ( ~ ( member_nat @ A3 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert_nat @ A3 @ F3 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_705_finite__subset__induct,axiom,
    ! [F2: set_a,A: set_a,P: set_a > $o] :
      ( ( finite_finite_a @ F2 )
     => ( ( ord_less_eq_set_a @ F2 @ A )
       => ( ( P @ bot_bot_set_a )
         => ( ! [A3: a,F3: set_a] :
                ( ( finite_finite_a @ F3 )
               => ( ( member_a @ A3 @ A )
                 => ( ~ ( member_a @ A3 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert_a @ A3 @ F3 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_706_finite__subset__induct_H,axiom,
    ! [F2: set_Pr8826267807999420763od_a_a,A: set_Pr8826267807999420763od_a_a,P: set_Pr8826267807999420763od_a_a > $o] :
      ( ( finite2714571839071174076od_a_a @ F2 )
     => ( ( ord_le741722312431004091od_a_a @ F2 @ A )
       => ( ( P @ bot_bo2841618473486996463od_a_a )
         => ( ! [A3: product_prod_a_a > product_prod_a_a,F3: set_Pr8826267807999420763od_a_a] :
                ( ( finite2714571839071174076od_a_a @ F3 )
               => ( ( member4020126937092221116od_a_a @ A3 @ A )
                 => ( ( ord_le741722312431004091od_a_a @ F3 @ A )
                   => ( ~ ( member4020126937092221116od_a_a @ A3 @ F3 )
                     => ( ( P @ F3 )
                       => ( P @ ( insert1583624352380834389od_a_a @ A3 @ F3 ) ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_707_finite__subset__induct_H,axiom,
    ! [F2: set_nat_nat,A: set_nat_nat,P: set_nat_nat > $o] :
      ( ( finite2115694454571419734at_nat @ F2 )
     => ( ( ord_le9059583361652607317at_nat @ F2 @ A )
       => ( ( P @ bot_bot_set_nat_nat )
         => ( ! [A3: nat > nat,F3: set_nat_nat] :
                ( ( finite2115694454571419734at_nat @ F3 )
               => ( ( member_nat_nat @ A3 @ A )
                 => ( ( ord_le9059583361652607317at_nat @ F3 @ A )
                   => ( ~ ( member_nat_nat @ A3 @ F3 )
                     => ( ( P @ F3 )
                       => ( P @ ( insert_nat_nat @ A3 @ F3 ) ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_708_finite__subset__induct_H,axiom,
    ! [F2: set_nat_a,A: set_nat_a,P: set_nat_a > $o] :
      ( ( finite_finite_nat_a @ F2 )
     => ( ( ord_le871467723717165285_nat_a @ F2 @ A )
       => ( ( P @ bot_bot_set_nat_a )
         => ( ! [A3: nat > a,F3: set_nat_a] :
                ( ( finite_finite_nat_a @ F3 )
               => ( ( member_nat_a @ A3 @ A )
                 => ( ( ord_le871467723717165285_nat_a @ F3 @ A )
                   => ( ~ ( member_nat_a @ A3 @ F3 )
                     => ( ( P @ F3 )
                       => ( P @ ( insert_nat_a @ A3 @ F3 ) ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_709_finite__subset__induct_H,axiom,
    ! [F2: set_a_6829686330177631172od_a_a,A: set_a_6829686330177631172od_a_a,P: set_a_6829686330177631172od_a_a > $o] :
      ( ( finite8821664692192630949od_a_a @ F2 )
     => ( ( ord_le2371270124811097124od_a_a @ F2 @ A )
       => ( ( P @ bot_bo6605490641894888024od_a_a )
         => ( ! [A3: a > product_prod_a_a,F3: set_a_6829686330177631172od_a_a] :
                ( ( finite8821664692192630949od_a_a @ F3 )
               => ( ( member1957775702407316389od_a_a @ A3 @ A )
                 => ( ( ord_le2371270124811097124od_a_a @ F3 @ A )
                   => ( ~ ( member1957775702407316389od_a_a @ A3 @ F3 )
                     => ( ( P @ F3 )
                       => ( P @ ( insert5082226857754029630od_a_a @ A3 @ F3 ) ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_710_finite__subset__induct_H,axiom,
    ! [F2: set_a_nat,A: set_a_nat,P: set_a_nat > $o] :
      ( ( finite_finite_a_nat @ F2 )
     => ( ( ord_le1612561287239139007_a_nat @ F2 @ A )
       => ( ( P @ bot_bot_set_a_nat )
         => ( ! [A3: a > nat,F3: set_a_nat] :
                ( ( finite_finite_a_nat @ F3 )
               => ( ( member_a_nat @ A3 @ A )
                 => ( ( ord_le1612561287239139007_a_nat @ F3 @ A )
                   => ( ~ ( member_a_nat @ A3 @ F3 )
                     => ( ( P @ F3 )
                       => ( P @ ( insert_a_nat @ A3 @ F3 ) ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_711_finite__subset__induct_H,axiom,
    ! [F2: set_a_a,A: set_a_a,P: set_a_a > $o] :
      ( ( finite_finite_a_a @ F2 )
     => ( ( ord_less_eq_set_a_a @ F2 @ A )
       => ( ( P @ bot_bot_set_a_a )
         => ( ! [A3: a > a,F3: set_a_a] :
                ( ( finite_finite_a_a @ F3 )
               => ( ( member_a_a @ A3 @ A )
                 => ( ( ord_less_eq_set_a_a @ F3 @ A )
                   => ( ~ ( member_a_a @ A3 @ F3 )
                     => ( ( P @ F3 )
                       => ( P @ ( insert_a_a @ A3 @ F3 ) ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_712_finite__subset__induct_H,axiom,
    ! [F2: set_Product_prod_a_a,A: set_Product_prod_a_a,P: set_Product_prod_a_a > $o] :
      ( ( finite6544458595007987280od_a_a @ F2 )
     => ( ( ord_le746702958409616551od_a_a @ F2 @ A )
       => ( ( P @ bot_bo3357376287454694259od_a_a )
         => ( ! [A3: product_prod_a_a,F3: set_Product_prod_a_a] :
                ( ( finite6544458595007987280od_a_a @ F3 )
               => ( ( member1426531477525435216od_a_a @ A3 @ A )
                 => ( ( ord_le746702958409616551od_a_a @ F3 @ A )
                   => ( ~ ( member1426531477525435216od_a_a @ A3 @ F3 )
                     => ( ( P @ F3 )
                       => ( P @ ( insert4534936382041156343od_a_a @ A3 @ F3 ) ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_713_finite__subset__induct_H,axiom,
    ! [F2: set_nat,A: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F2 )
     => ( ( ord_less_eq_set_nat @ F2 @ A )
       => ( ( P @ bot_bot_set_nat )
         => ( ! [A3: nat,F3: set_nat] :
                ( ( finite_finite_nat @ F3 )
               => ( ( member_nat @ A3 @ A )
                 => ( ( ord_less_eq_set_nat @ F3 @ A )
                   => ( ~ ( member_nat @ A3 @ F3 )
                     => ( ( P @ F3 )
                       => ( P @ ( insert_nat @ A3 @ F3 ) ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_714_finite__subset__induct_H,axiom,
    ! [F2: set_a,A: set_a,P: set_a > $o] :
      ( ( finite_finite_a @ F2 )
     => ( ( ord_less_eq_set_a @ F2 @ A )
       => ( ( P @ bot_bot_set_a )
         => ( ! [A3: a,F3: set_a] :
                ( ( finite_finite_a @ F3 )
               => ( ( member_a @ A3 @ A )
                 => ( ( ord_less_eq_set_a @ F3 @ A )
                   => ( ~ ( member_a @ A3 @ F3 )
                     => ( ( P @ F3 )
                       => ( P @ ( insert_a @ A3 @ F3 ) ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_715_card__eq__0__iff,axiom,
    ! [A: set_Product_prod_a_a] :
      ( ( ( finite4795055649997197647od_a_a @ A )
        = zero_zero_nat )
      = ( ( A = bot_bo3357376287454694259od_a_a )
        | ~ ( finite6544458595007987280od_a_a @ A ) ) ) ).

% card_eq_0_iff
thf(fact_716_card__eq__0__iff,axiom,
    ! [A: set_a] :
      ( ( ( finite_card_a @ A )
        = zero_zero_nat )
      = ( ( A = bot_bot_set_a )
        | ~ ( finite_finite_a @ A ) ) ) ).

% card_eq_0_iff
thf(fact_717_card__eq__0__iff,axiom,
    ! [A: set_nat] :
      ( ( ( finite_card_nat @ A )
        = zero_zero_nat )
      = ( ( A = bot_bot_set_nat )
        | ~ ( finite_finite_nat @ A ) ) ) ).

% card_eq_0_iff
thf(fact_718_additive__abelian__group_Osumset__empty_H_I2_J,axiom,
    ! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: set_nat,B: set_nat] :
      ( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
     => ( ( ( inf_inf_set_nat @ A @ G )
          = bot_bot_set_nat )
       => ( ( pluenn3669378163024332905et_nat @ G @ Addition @ A @ B )
          = bot_bot_set_nat ) ) ) ).

% additive_abelian_group.sumset_empty'(2)
thf(fact_719_additive__abelian__group_Osumset__empty_H_I2_J,axiom,
    ! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,B: set_a] :
      ( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
     => ( ( ( inf_inf_set_a @ A @ G )
          = bot_bot_set_a )
       => ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B )
          = bot_bot_set_a ) ) ) ).

% additive_abelian_group.sumset_empty'(2)
thf(fact_720_additive__abelian__group_Osumset__empty_H_I1_J,axiom,
    ! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: set_nat,B: set_nat] :
      ( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
     => ( ( ( inf_inf_set_nat @ A @ G )
          = bot_bot_set_nat )
       => ( ( pluenn3669378163024332905et_nat @ G @ Addition @ B @ A )
          = bot_bot_set_nat ) ) ) ).

% additive_abelian_group.sumset_empty'(1)
thf(fact_721_additive__abelian__group_Osumset__empty_H_I1_J,axiom,
    ! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,B: set_a] :
      ( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
     => ( ( ( inf_inf_set_a @ A @ G )
          = bot_bot_set_a )
       => ( ( pluenn3038260743871226533mset_a @ G @ Addition @ B @ A )
          = bot_bot_set_a ) ) ) ).

% additive_abelian_group.sumset_empty'(1)
thf(fact_722_additive__abelian__group_Osumset__is__empty__iff,axiom,
    ! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: set_nat,B: set_nat] :
      ( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
     => ( ( ( pluenn3669378163024332905et_nat @ G @ Addition @ A @ B )
          = bot_bot_set_nat )
        = ( ( ( inf_inf_set_nat @ A @ G )
            = bot_bot_set_nat )
          | ( ( inf_inf_set_nat @ B @ G )
            = bot_bot_set_nat ) ) ) ) ).

% additive_abelian_group.sumset_is_empty_iff
thf(fact_723_additive__abelian__group_Osumset__is__empty__iff,axiom,
    ! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,B: set_a] :
      ( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
     => ( ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B )
          = bot_bot_set_a )
        = ( ( ( inf_inf_set_a @ A @ G )
            = bot_bot_set_a )
          | ( ( inf_inf_set_a @ B @ G )
            = bot_bot_set_a ) ) ) ) ).

% additive_abelian_group.sumset_is_empty_iff
thf(fact_724_additive__abelian__group_Ominusset__triv,axiom,
    ! [G: set_nat,Addition: nat > nat > nat,Zero: nat] :
      ( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
     => ( ( pluenn7323955030898006884et_nat @ G @ Addition @ Zero @ ( insert_nat @ Zero @ bot_bot_set_nat ) )
        = ( insert_nat @ Zero @ bot_bot_set_nat ) ) ) ).

% additive_abelian_group.minusset_triv
thf(fact_725_additive__abelian__group_Ominusset__triv,axiom,
    ! [G: set_a,Addition: a > a > a,Zero: a] :
      ( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
     => ( ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ ( insert_a @ Zero @ bot_bot_set_a ) )
        = ( insert_a @ Zero @ bot_bot_set_a ) ) ) ).

% additive_abelian_group.minusset_triv
thf(fact_726_additive__abelian__group_Ominusset__is__empty__iff,axiom,
    ! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: set_nat] :
      ( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
     => ( ( ( pluenn7323955030898006884et_nat @ G @ Addition @ Zero @ A )
          = bot_bot_set_nat )
        = ( ( inf_inf_set_nat @ A @ G )
          = bot_bot_set_nat ) ) ) ).

% additive_abelian_group.minusset_is_empty_iff
thf(fact_727_additive__abelian__group_Ominusset__is__empty__iff,axiom,
    ! [G: set_a,Addition: a > a > a,Zero: a,A: set_a] :
      ( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
     => ( ( ( pluenn2534204936789923946sset_a @ G @ Addition @ Zero @ A )
          = bot_bot_set_a )
        = ( ( inf_inf_set_a @ A @ G )
          = bot_bot_set_a ) ) ) ).

% additive_abelian_group.minusset_is_empty_iff
thf(fact_728_finite__cartesian__product__iff,axiom,
    ! [A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
      ( ( finite20598220287489552od_a_a
        @ ( produc5899993699339346696od_a_a @ A
          @ ^ [Uu: product_prod_a_a] : B ) )
      = ( ( A = bot_bo3357376287454694259od_a_a )
        | ( B = bot_bo3357376287454694259od_a_a )
        | ( ( finite6544458595007987280od_a_a @ A )
          & ( finite6544458595007987280od_a_a @ B ) ) ) ) ).

% finite_cartesian_product_iff
thf(fact_729_finite__cartesian__product__iff,axiom,
    ! [A: set_Product_prod_a_a,B: set_a] :
      ( ( finite5607752495362350695_a_a_a
        @ ( produc2379640491490746847_a_a_a @ A
          @ ^ [Uu: product_prod_a_a] : B ) )
      = ( ( A = bot_bo3357376287454694259od_a_a )
        | ( B = bot_bot_set_a )
        | ( ( finite6544458595007987280od_a_a @ A )
          & ( finite_finite_a @ B ) ) ) ) ).

% finite_cartesian_product_iff
thf(fact_730_finite__cartesian__product__iff,axiom,
    ! [A: set_Product_prod_a_a,B: set_nat] :
      ( ( finite3082808146142899517_a_nat
        @ ( produc1049071135499013807_a_nat @ A
          @ ^ [Uu: product_prod_a_a] : B ) )
      = ( ( A = bot_bo3357376287454694259od_a_a )
        | ( B = bot_bot_set_nat )
        | ( ( finite6544458595007987280od_a_a @ A )
          & ( finite_finite_nat @ B ) ) ) ) ).

% finite_cartesian_product_iff
thf(fact_731_finite__cartesian__product__iff,axiom,
    ! [A: set_a,B: set_Product_prod_a_a] :
      ( ( finite5848958031409366265od_a_a
        @ ( produc6342321021181284593od_a_a @ A
          @ ^ [Uu: a] : B ) )
      = ( ( A = bot_bot_set_a )
        | ( B = bot_bo3357376287454694259od_a_a )
        | ( ( finite_finite_a @ A )
          & ( finite6544458595007987280od_a_a @ B ) ) ) ) ).

% finite_cartesian_product_iff
thf(fact_732_finite__cartesian__product__iff,axiom,
    ! [A: set_a,B: set_a] :
      ( ( finite6544458595007987280od_a_a
        @ ( product_Sigma_a_a @ A
          @ ^ [Uu: a] : B ) )
      = ( ( A = bot_bot_set_a )
        | ( B = bot_bot_set_a )
        | ( ( finite_finite_a @ A )
          & ( finite_finite_a @ B ) ) ) ) ).

% finite_cartesian_product_iff
thf(fact_733_finite__cartesian__product__iff,axiom,
    ! [A: set_a,B: set_nat] :
      ( ( finite6644898363146130708_a_nat
        @ ( product_Sigma_a_nat @ A
          @ ^ [Uu: a] : B ) )
      = ( ( A = bot_bot_set_a )
        | ( B = bot_bot_set_nat )
        | ( ( finite_finite_a @ A )
          & ( finite_finite_nat @ B ) ) ) ) ).

% finite_cartesian_product_iff
thf(fact_734_finite__cartesian__product__iff,axiom,
    ! [A: set_nat,B: set_Product_prod_a_a] :
      ( ( finite8333580178073949923od_a_a
        @ ( produc1591714161673420045od_a_a @ A
          @ ^ [Uu: nat] : B ) )
      = ( ( A = bot_bot_set_nat )
        | ( B = bot_bo3357376287454694259od_a_a )
        | ( ( finite_finite_nat @ A )
          & ( finite6544458595007987280od_a_a @ B ) ) ) ) ).

% finite_cartesian_product_iff
thf(fact_735_finite__cartesian__product__iff,axiom,
    ! [A: set_nat,B: set_a] :
      ( ( finite659689790015031866_nat_a
        @ ( product_Sigma_nat_a @ A
          @ ^ [Uu: nat] : B ) )
      = ( ( A = bot_bot_set_nat )
        | ( B = bot_bot_set_a )
        | ( ( finite_finite_nat @ A )
          & ( finite_finite_a @ B ) ) ) ) ).

% finite_cartesian_product_iff
thf(fact_736_finite__cartesian__product__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( finite6177210948735845034at_nat
        @ ( produc457027306803732586at_nat @ A
          @ ^ [Uu: nat] : B ) )
      = ( ( A = bot_bot_set_nat )
        | ( B = bot_bot_set_nat )
        | ( ( finite_finite_nat @ A )
          & ( finite_finite_nat @ B ) ) ) ) ).

% finite_cartesian_product_iff
thf(fact_737_finite__cartesian__productD2,axiom,
    ! [A: set_a,B: set_nat] :
      ( ( finite6644898363146130708_a_nat
        @ ( product_Sigma_a_nat @ A
          @ ^ [Uu: a] : B ) )
     => ( ( A != bot_bot_set_a )
       => ( finite_finite_nat @ B ) ) ) ).

% finite_cartesian_productD2
thf(fact_738_finite__cartesian__productD2,axiom,
    ! [A: set_a,B: set_Product_prod_a_a] :
      ( ( finite5848958031409366265od_a_a
        @ ( produc6342321021181284593od_a_a @ A
          @ ^ [Uu: a] : B ) )
     => ( ( A != bot_bot_set_a )
       => ( finite6544458595007987280od_a_a @ B ) ) ) ).

% finite_cartesian_productD2
thf(fact_739_finite__cartesian__productD2,axiom,
    ! [A: set_a,B: set_a] :
      ( ( finite6544458595007987280od_a_a
        @ ( product_Sigma_a_a @ A
          @ ^ [Uu: a] : B ) )
     => ( ( A != bot_bot_set_a )
       => ( finite_finite_a @ B ) ) ) ).

% finite_cartesian_productD2
thf(fact_740_finite__cartesian__productD2,axiom,
    ! [A: set_nat,B: set_a] :
      ( ( finite659689790015031866_nat_a
        @ ( product_Sigma_nat_a @ A
          @ ^ [Uu: nat] : B ) )
     => ( ( A != bot_bot_set_nat )
       => ( finite_finite_a @ B ) ) ) ).

% finite_cartesian_productD2
thf(fact_741_finite__cartesian__productD2,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( finite6177210948735845034at_nat
        @ ( produc457027306803732586at_nat @ A
          @ ^ [Uu: nat] : B ) )
     => ( ( A != bot_bot_set_nat )
       => ( finite_finite_nat @ B ) ) ) ).

% finite_cartesian_productD2
thf(fact_742_finite__cartesian__productD2,axiom,
    ! [A: set_nat,B: set_Product_prod_a_a] :
      ( ( finite8333580178073949923od_a_a
        @ ( produc1591714161673420045od_a_a @ A
          @ ^ [Uu: nat] : B ) )
     => ( ( A != bot_bot_set_nat )
       => ( finite6544458595007987280od_a_a @ B ) ) ) ).

% finite_cartesian_productD2
thf(fact_743_finite__cartesian__productD1,axiom,
    ! [A: set_nat,B: set_a] :
      ( ( finite659689790015031866_nat_a
        @ ( product_Sigma_nat_a @ A
          @ ^ [Uu: nat] : B ) )
     => ( ( B != bot_bot_set_a )
       => ( finite_finite_nat @ A ) ) ) ).

% finite_cartesian_productD1
thf(fact_744_finite__cartesian__productD1,axiom,
    ! [A: set_Product_prod_a_a,B: set_a] :
      ( ( finite5607752495362350695_a_a_a
        @ ( produc2379640491490746847_a_a_a @ A
          @ ^ [Uu: product_prod_a_a] : B ) )
     => ( ( B != bot_bot_set_a )
       => ( finite6544458595007987280od_a_a @ A ) ) ) ).

% finite_cartesian_productD1
thf(fact_745_finite__cartesian__productD1,axiom,
    ! [A: set_a,B: set_a] :
      ( ( finite6544458595007987280od_a_a
        @ ( product_Sigma_a_a @ A
          @ ^ [Uu: a] : B ) )
     => ( ( B != bot_bot_set_a )
       => ( finite_finite_a @ A ) ) ) ).

% finite_cartesian_productD1
thf(fact_746_finite__cartesian__productD1,axiom,
    ! [A: set_a,B: set_nat] :
      ( ( finite6644898363146130708_a_nat
        @ ( product_Sigma_a_nat @ A
          @ ^ [Uu: a] : B ) )
     => ( ( B != bot_bot_set_nat )
       => ( finite_finite_a @ A ) ) ) ).

% finite_cartesian_productD1
thf(fact_747_finite__cartesian__productD1,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( finite6177210948735845034at_nat
        @ ( produc457027306803732586at_nat @ A
          @ ^ [Uu: nat] : B ) )
     => ( ( B != bot_bot_set_nat )
       => ( finite_finite_nat @ A ) ) ) ).

% finite_cartesian_productD1
thf(fact_748_finite__cartesian__productD1,axiom,
    ! [A: set_Product_prod_a_a,B: set_nat] :
      ( ( finite3082808146142899517_a_nat
        @ ( produc1049071135499013807_a_nat @ A
          @ ^ [Uu: product_prod_a_a] : B ) )
     => ( ( B != bot_bot_set_nat )
       => ( finite6544458595007987280od_a_a @ A ) ) ) ).

% finite_cartesian_productD1
thf(fact_749_finite__SigmaI2,axiom,
    ! [A: set_nat,B: nat > set_a] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X2: nat] :
              ( ( member_nat @ X2 @ A )
              & ( ( B @ X2 )
               != bot_bot_set_a ) ) ) )
     => ( ! [A3: nat] :
            ( ( member_nat @ A3 @ A )
           => ( finite_finite_a @ ( B @ A3 ) ) )
       => ( finite659689790015031866_nat_a @ ( product_Sigma_nat_a @ A @ B ) ) ) ) ).

% finite_SigmaI2
thf(fact_750_finite__SigmaI2,axiom,
    ! [A: set_a,B: a > set_a] :
      ( ( finite_finite_a
        @ ( collect_a
          @ ^ [X2: a] :
              ( ( member_a @ X2 @ A )
              & ( ( B @ X2 )
               != bot_bot_set_a ) ) ) )
     => ( ! [A3: a] :
            ( ( member_a @ A3 @ A )
           => ( finite_finite_a @ ( B @ A3 ) ) )
       => ( finite6544458595007987280od_a_a @ ( product_Sigma_a_a @ A @ B ) ) ) ) ).

% finite_SigmaI2
thf(fact_751_finite__SigmaI2,axiom,
    ! [A: set_a,B: a > set_nat] :
      ( ( finite_finite_a
        @ ( collect_a
          @ ^ [X2: a] :
              ( ( member_a @ X2 @ A )
              & ( ( B @ X2 )
               != bot_bot_set_nat ) ) ) )
     => ( ! [A3: a] :
            ( ( member_a @ A3 @ A )
           => ( finite_finite_nat @ ( B @ A3 ) ) )
       => ( finite6644898363146130708_a_nat @ ( product_Sigma_a_nat @ A @ B ) ) ) ) ).

% finite_SigmaI2
thf(fact_752_finite__SigmaI2,axiom,
    ! [A: set_nat,B: nat > set_nat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X2: nat] :
              ( ( member_nat @ X2 @ A )
              & ( ( B @ X2 )
               != bot_bot_set_nat ) ) ) )
     => ( ! [A3: nat] :
            ( ( member_nat @ A3 @ A )
           => ( finite_finite_nat @ ( B @ A3 ) ) )
       => ( finite6177210948735845034at_nat @ ( produc457027306803732586at_nat @ A @ B ) ) ) ) ).

% finite_SigmaI2
thf(fact_753_finite__SigmaI2,axiom,
    ! [A: set_a,B: a > set_Product_prod_a_a] :
      ( ( finite_finite_a
        @ ( collect_a
          @ ^ [X2: a] :
              ( ( member_a @ X2 @ A )
              & ( ( B @ X2 )
               != bot_bo3357376287454694259od_a_a ) ) ) )
     => ( ! [A3: a] :
            ( ( member_a @ A3 @ A )
           => ( finite6544458595007987280od_a_a @ ( B @ A3 ) ) )
       => ( finite5848958031409366265od_a_a @ ( produc6342321021181284593od_a_a @ A @ B ) ) ) ) ).

% finite_SigmaI2
thf(fact_754_finite__SigmaI2,axiom,
    ! [A: set_nat,B: nat > set_Product_prod_a_a] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X2: nat] :
              ( ( member_nat @ X2 @ A )
              & ( ( B @ X2 )
               != bot_bo3357376287454694259od_a_a ) ) ) )
     => ( ! [A3: nat] :
            ( ( member_nat @ A3 @ A )
           => ( finite6544458595007987280od_a_a @ ( B @ A3 ) ) )
       => ( finite8333580178073949923od_a_a @ ( produc1591714161673420045od_a_a @ A @ B ) ) ) ) ).

% finite_SigmaI2
thf(fact_755_finite__SigmaI2,axiom,
    ! [A: set_nat_nat,B: ( nat > nat ) > set_a] :
      ( ( finite2115694454571419734at_nat
        @ ( collect_nat_nat
          @ ^ [X2: nat > nat] :
              ( ( member_nat_nat @ X2 @ A )
              & ( ( B @ X2 )
               != bot_bot_set_a ) ) ) )
     => ( ! [A3: nat > nat] :
            ( ( member_nat_nat @ A3 @ A )
           => ( finite_finite_a @ ( B @ A3 ) ) )
       => ( finite6410384693344227723_nat_a @ ( produc2309334610237048501_nat_a @ A @ B ) ) ) ) ).

% finite_SigmaI2
thf(fact_756_finite__SigmaI2,axiom,
    ! [A: set_nat_a,B: ( nat > a ) > set_a] :
      ( ( finite_finite_nat_a
        @ ( collect_nat_a
          @ ^ [X2: nat > a] :
              ( ( member_nat_a @ X2 @ A )
              & ( ( B @ X2 )
               != bot_bot_set_a ) ) ) )
     => ( ! [A3: nat > a] :
            ( ( member_nat_a @ A3 @ A )
           => ( finite_finite_a @ ( B @ A3 ) ) )
       => ( finite1819187048435525417at_a_a @ ( produc6085074725259719457at_a_a @ A @ B ) ) ) ) ).

% finite_SigmaI2
thf(fact_757_finite__SigmaI2,axiom,
    ! [A: set_a_nat,B: ( a > nat ) > set_a] :
      ( ( finite_finite_a_nat
        @ ( collect_a_nat
          @ ^ [X2: a > nat] :
              ( ( member_a_nat @ X2 @ A )
              & ( ( B @ X2 )
               != bot_bot_set_a ) ) ) )
     => ( ! [A3: a > nat] :
            ( ( member_a_nat @ A3 @ A )
           => ( finite_finite_a @ ( B @ A3 ) ) )
       => ( finite4771698143452169935_nat_a @ ( produc9037585820276363975_nat_a @ A @ B ) ) ) ) ).

% finite_SigmaI2
thf(fact_758_finite__SigmaI2,axiom,
    ! [A: set_a_a,B: ( a > a ) > set_a] :
      ( ( finite_finite_a_a
        @ ( collect_a_a
          @ ^ [X2: a > a] :
              ( ( member_a_a @ X2 @ A )
              & ( ( B @ X2 )
               != bot_bot_set_a ) ) ) )
     => ( ! [A3: a > a] :
            ( ( member_a_a @ A3 @ A )
           => ( finite_finite_a @ ( B @ A3 ) ) )
       => ( finite4229576983758256869_a_a_a @ ( product_Sigma_a_a_a @ A @ B ) ) ) ) ).

% finite_SigmaI2
thf(fact_759_additive__abelian__group_Oinfinite__sumset__aux,axiom,
    ! [G: set_Product_prod_a_a,Addition: product_prod_a_a > product_prod_a_a > product_prod_a_a,Zero: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
      ( ( pluenn6837661381354551573od_a_a @ G @ Addition @ Zero )
     => ( ~ ( finite6544458595007987280od_a_a @ ( inf_in8905007599844390133od_a_a @ A @ G ) )
       => ( ( ~ ( finite6544458595007987280od_a_a @ ( pluenn659090379318209998od_a_a @ G @ Addition @ A @ B ) ) )
          = ( ( inf_in8905007599844390133od_a_a @ B @ G )
           != bot_bo3357376287454694259od_a_a ) ) ) ) ).

% additive_abelian_group.infinite_sumset_aux
thf(fact_760_additive__abelian__group_Oinfinite__sumset__aux,axiom,
    ! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: set_nat,B: set_nat] :
      ( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
     => ( ~ ( finite_finite_nat @ ( inf_inf_set_nat @ A @ G ) )
       => ( ( ~ ( finite_finite_nat @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A @ B ) ) )
          = ( ( inf_inf_set_nat @ B @ G )
           != bot_bot_set_nat ) ) ) ) ).

% additive_abelian_group.infinite_sumset_aux
thf(fact_761_additive__abelian__group_Oinfinite__sumset__aux,axiom,
    ! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,B: set_a] :
      ( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
     => ( ~ ( finite_finite_a @ ( inf_inf_set_a @ A @ G ) )
       => ( ( ~ ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) ) )
          = ( ( inf_inf_set_a @ B @ G )
           != bot_bot_set_a ) ) ) ) ).

% additive_abelian_group.infinite_sumset_aux
thf(fact_762_additive__abelian__group_Oinfinite__sumset__iff,axiom,
    ! [G: set_Product_prod_a_a,Addition: product_prod_a_a > product_prod_a_a > product_prod_a_a,Zero: product_prod_a_a,A: set_Product_prod_a_a,B: set_Product_prod_a_a] :
      ( ( pluenn6837661381354551573od_a_a @ G @ Addition @ Zero )
     => ( ( ~ ( finite6544458595007987280od_a_a @ ( pluenn659090379318209998od_a_a @ G @ Addition @ A @ B ) ) )
        = ( ( ~ ( finite6544458595007987280od_a_a @ ( inf_in8905007599844390133od_a_a @ A @ G ) )
            & ( ( inf_in8905007599844390133od_a_a @ B @ G )
             != bot_bo3357376287454694259od_a_a ) )
          | ( ( ( inf_in8905007599844390133od_a_a @ A @ G )
             != bot_bo3357376287454694259od_a_a )
            & ~ ( finite6544458595007987280od_a_a @ ( inf_in8905007599844390133od_a_a @ B @ G ) ) ) ) ) ) ).

% additive_abelian_group.infinite_sumset_iff
thf(fact_763_additive__abelian__group_Oinfinite__sumset__iff,axiom,
    ! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: set_nat,B: set_nat] :
      ( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
     => ( ( ~ ( finite_finite_nat @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A @ B ) ) )
        = ( ( ~ ( finite_finite_nat @ ( inf_inf_set_nat @ A @ G ) )
            & ( ( inf_inf_set_nat @ B @ G )
             != bot_bot_set_nat ) )
          | ( ( ( inf_inf_set_nat @ A @ G )
             != bot_bot_set_nat )
            & ~ ( finite_finite_nat @ ( inf_inf_set_nat @ B @ G ) ) ) ) ) ) ).

% additive_abelian_group.infinite_sumset_iff
thf(fact_764_additive__abelian__group_Oinfinite__sumset__iff,axiom,
    ! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,B: set_a] :
      ( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
     => ( ( ~ ( finite_finite_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ B ) ) )
        = ( ( ~ ( finite_finite_a @ ( inf_inf_set_a @ A @ G ) )
            & ( ( inf_inf_set_a @ B @ G )
             != bot_bot_set_a ) )
          | ( ( ( inf_inf_set_a @ A @ G )
             != bot_bot_set_a )
            & ~ ( finite_finite_a @ ( inf_inf_set_a @ B @ G ) ) ) ) ) ) ).

% additive_abelian_group.infinite_sumset_iff
thf(fact_765_additive__abelian__group_Osumset__D_I2_J,axiom,
    ! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: set_nat] :
      ( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
     => ( ( pluenn3669378163024332905et_nat @ G @ Addition @ ( insert_nat @ Zero @ bot_bot_set_nat ) @ A )
        = ( inf_inf_set_nat @ A @ G ) ) ) ).

% additive_abelian_group.sumset_D(2)
thf(fact_766_additive__abelian__group_Osumset__D_I2_J,axiom,
    ! [G: set_a,Addition: a > a > a,Zero: a,A: set_a] :
      ( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
     => ( ( pluenn3038260743871226533mset_a @ G @ Addition @ ( insert_a @ Zero @ bot_bot_set_a ) @ A )
        = ( inf_inf_set_a @ A @ G ) ) ) ).

% additive_abelian_group.sumset_D(2)
thf(fact_767_additive__abelian__group_Osumset__D_I1_J,axiom,
    ! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: set_nat] :
      ( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
     => ( ( pluenn3669378163024332905et_nat @ G @ Addition @ A @ ( insert_nat @ Zero @ bot_bot_set_nat ) )
        = ( inf_inf_set_nat @ A @ G ) ) ) ).

% additive_abelian_group.sumset_D(1)
thf(fact_768_additive__abelian__group_Osumset__D_I1_J,axiom,
    ! [G: set_a,Addition: a > a > a,Zero: a,A: set_a] :
      ( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
     => ( ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ ( insert_a @ Zero @ bot_bot_set_a ) )
        = ( inf_inf_set_a @ A @ G ) ) ) ).

% additive_abelian_group.sumset_D(1)
thf(fact_769_additive__abelian__group_Osumset__iterated__0,axiom,
    ! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: set_nat] :
      ( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
     => ( ( pluenn7055013279391836755ed_nat @ G @ Addition @ Zero @ A @ zero_zero_nat )
        = ( insert_nat @ Zero @ bot_bot_set_nat ) ) ) ).

% additive_abelian_group.sumset_iterated_0
thf(fact_770_additive__abelian__group_Osumset__iterated__0,axiom,
    ! [G: set_a,Addition: a > a > a,Zero: a,A: set_a] :
      ( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
     => ( ( pluenn1960970773371692859ated_a @ G @ Addition @ Zero @ A @ zero_zero_nat )
        = ( insert_a @ Zero @ bot_bot_set_a ) ) ) ).

% additive_abelian_group.sumset_iterated_0
thf(fact_771_prod_Ocase__distrib,axiom,
    ! [H2: product_prod_a_a > product_prod_a_a,F: a > a > product_prod_a_a,Prod: product_prod_a_a] :
      ( ( H2 @ ( produc408267641121961211od_a_a @ F @ Prod ) )
      = ( produc408267641121961211od_a_a
        @ ^ [X12: a,X23: a] : ( H2 @ ( F @ X12 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_772_Times__eq__cancel2,axiom,
    ! [X: a,C: set_a,A: set_a,B: set_a] :
      ( ( member_a @ X @ C )
     => ( ( ( product_Sigma_a_a @ A
            @ ^ [Uu: a] : C )
          = ( product_Sigma_a_a @ B
            @ ^ [Uu: a] : C ) )
        = ( A = B ) ) ) ).

% Times_eq_cancel2
thf(fact_773_Sigma__cong,axiom,
    ! [A: set_a,B: set_a,C: a > set_a,D: a > set_a] :
      ( ( A = B )
     => ( ! [X3: a] :
            ( ( member_a @ X3 @ B )
           => ( ( C @ X3 )
              = ( D @ X3 ) ) )
       => ( ( product_Sigma_a_a @ A @ C )
          = ( product_Sigma_a_a @ B @ D ) ) ) ) ).

% Sigma_cong
thf(fact_774_additive__abelian__group_Ocard__sumset__le,axiom,
    ! [G: set_Product_prod_a_a,Addition: product_prod_a_a > product_prod_a_a > product_prod_a_a,Zero: product_prod_a_a,A: set_Product_prod_a_a,A2: product_prod_a_a] :
      ( ( pluenn6837661381354551573od_a_a @ G @ Addition @ Zero )
     => ( ( finite6544458595007987280od_a_a @ A )
       => ( ord_less_eq_nat @ ( finite4795055649997197647od_a_a @ ( pluenn659090379318209998od_a_a @ G @ Addition @ A @ ( insert4534936382041156343od_a_a @ A2 @ bot_bo3357376287454694259od_a_a ) ) ) @ ( finite4795055649997197647od_a_a @ A ) ) ) ) ).

% additive_abelian_group.card_sumset_le
thf(fact_775_additive__abelian__group_Ocard__sumset__le,axiom,
    ! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: set_nat,A2: nat] :
      ( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
     => ( ( finite_finite_nat @ A )
       => ( ord_less_eq_nat @ ( finite_card_nat @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A ) ) ) ) ).

% additive_abelian_group.card_sumset_le
thf(fact_776_additive__abelian__group_Ocard__sumset__le,axiom,
    ! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,A2: a] :
      ( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
     => ( ( finite_finite_a @ A )
       => ( ord_less_eq_nat @ ( finite_card_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ ( insert_a @ A2 @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A ) ) ) ) ).

% additive_abelian_group.card_sumset_le
thf(fact_777_old_Oprod_Ocase,axiom,
    ! [F: a > a > product_prod_a_a,X1: a,X22: a] :
      ( ( produc408267641121961211od_a_a @ F @ ( product_Pair_a_a @ X1 @ X22 ) )
      = ( F @ X1 @ X22 ) ) ).

% old.prod.case
thf(fact_778_SigmaE2,axiom,
    ! [A2: a,B4: nat,A: set_a,B: a > set_nat] :
      ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A2 @ B4 ) @ ( product_Sigma_a_nat @ A @ B ) )
     => ~ ( ( member_a @ A2 @ A )
         => ~ ( member_nat @ B4 @ ( B @ A2 ) ) ) ) ).

% SigmaE2
thf(fact_779_SigmaE2,axiom,
    ! [A2: nat,B4: a,A: set_nat,B: nat > set_a] :
      ( ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ A2 @ B4 ) @ ( product_Sigma_nat_a @ A @ B ) )
     => ~ ( ( member_nat @ A2 @ A )
         => ~ ( member_a @ B4 @ ( B @ A2 ) ) ) ) ).

% SigmaE2
thf(fact_780_SigmaE2,axiom,
    ! [A2: nat,B4: nat,A: set_nat,B: nat > set_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B4 ) @ ( produc457027306803732586at_nat @ A @ B ) )
     => ~ ( ( member_nat @ A2 @ A )
         => ~ ( member_nat @ B4 @ ( B @ A2 ) ) ) ) ).

% SigmaE2
thf(fact_781_SigmaE2,axiom,
    ! [A2: a,B4: a,A: set_a,B: a > set_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B4 ) @ ( product_Sigma_a_a @ A @ B ) )
     => ~ ( ( member_a @ A2 @ A )
         => ~ ( member_a @ B4 @ ( B @ A2 ) ) ) ) ).

% SigmaE2
thf(fact_782_SigmaE2,axiom,
    ! [A2: a,B4: nat > nat,A: set_a,B: a > set_nat_nat] :
      ( ( member5463269792551873667at_nat @ ( produc7514922136058452582at_nat @ A2 @ B4 ) @ ( produc5882319205326396405at_nat @ A @ B ) )
     => ~ ( ( member_a @ A2 @ A )
         => ~ ( member_nat_nat @ B4 @ ( B @ A2 ) ) ) ) ).

% SigmaE2
thf(fact_783_SigmaE2,axiom,
    ! [A2: a,B4: nat > a,A: set_a,B: a > set_nat_a] :
      ( ( member5853765858310737207_nat_a @ ( product_Pair_a_nat_a @ A2 @ B4 ) @ ( produc830630580412313007_nat_a @ A @ B ) )
     => ~ ( ( member_a @ A2 @ A )
         => ~ ( member_nat_a @ B4 @ ( B @ A2 ) ) ) ) ).

% SigmaE2
thf(fact_784_SigmaE2,axiom,
    ! [A2: a,B4: a > nat,A: set_a,B: a > set_a_nat] :
      ( ( member6594859421832710929_a_nat @ ( product_Pair_a_a_nat @ A2 @ B4 ) @ ( produc6815839153543411849_a_nat @ A @ B ) )
     => ~ ( ( member_a @ A2 @ A )
         => ~ ( member_a_nat @ B4 @ ( B @ A2 ) ) ) ) ).

% SigmaE2
thf(fact_785_SigmaE2,axiom,
    ! [A2: a,B4: a > a,A: set_a,B: a > set_a_a] :
      ( ( member681807718693926441_a_a_a @ ( product_Pair_a_a_a @ A2 @ B4 ) @ ( product_Sigma_a_a_a2 @ A @ B ) )
     => ~ ( ( member_a @ A2 @ A )
         => ~ ( member_a_a @ B4 @ ( B @ A2 ) ) ) ) ).

% SigmaE2
thf(fact_786_SigmaE2,axiom,
    ! [A2: nat,B4: nat > nat,A: set_nat,B: nat > set_nat_nat] :
      ( ( member8336108448496249625at_nat @ ( produc7839516862119294504at_nat @ A2 @ B4 ) @ ( produc4526620505022529241at_nat @ A @ B ) )
     => ~ ( ( member_nat @ A2 @ A )
         => ~ ( member_nat_nat @ B4 @ ( B @ A2 ) ) ) ) ).

% SigmaE2
thf(fact_787_SigmaE2,axiom,
    ! [A2: nat,B4: nat > a,A: set_nat,B: nat > set_nat_a] :
      ( ( member2811451201825703969_nat_a @ ( produc5292568359338195516_nat_a @ A2 @ B4 ) @ ( produc3659965428606139339_nat_a @ A @ B ) )
     => ~ ( ( member_nat @ A2 @ A )
         => ~ ( member_nat_a @ B4 @ ( B @ A2 ) ) ) ) ).

% SigmaE2
thf(fact_788_SigmaD2,axiom,
    ! [A2: a,B4: a,A: set_a,B: a > set_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B4 ) @ ( product_Sigma_a_a @ A @ B ) )
     => ( member_a @ B4 @ ( B @ A2 ) ) ) ).

% SigmaD2
thf(fact_789_SigmaD1,axiom,
    ! [A2: a,B4: a,A: set_a,B: a > set_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B4 ) @ ( product_Sigma_a_a @ A @ B ) )
     => ( member_a @ A2 @ A ) ) ).

% SigmaD1
thf(fact_790_SigmaE,axiom,
    ! [C2: product_prod_a_nat,A: set_a,B: a > set_nat] :
      ( ( member5724188588386418708_a_nat @ C2 @ ( product_Sigma_a_nat @ A @ B ) )
     => ~ ! [X3: a] :
            ( ( member_a @ X3 @ A )
           => ! [Y3: nat] :
                ( ( member_nat @ Y3 @ ( B @ X3 ) )
               => ( C2
                 != ( product_Pair_a_nat @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_791_SigmaE,axiom,
    ! [C2: product_prod_nat_a,A: set_nat,B: nat > set_a] :
      ( ( member8962352052110095674_nat_a @ C2 @ ( product_Sigma_nat_a @ A @ B ) )
     => ~ ! [X3: nat] :
            ( ( member_nat @ X3 @ A )
           => ! [Y3: a] :
                ( ( member_a @ Y3 @ ( B @ X3 ) )
               => ( C2
                 != ( product_Pair_nat_a @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_792_SigmaE,axiom,
    ! [C2: product_prod_nat_nat,A: set_nat,B: nat > set_nat] :
      ( ( member8440522571783428010at_nat @ C2 @ ( produc457027306803732586at_nat @ A @ B ) )
     => ~ ! [X3: nat] :
            ( ( member_nat @ X3 @ A )
           => ! [Y3: nat] :
                ( ( member_nat @ Y3 @ ( B @ X3 ) )
               => ( C2
                 != ( product_Pair_nat_nat @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_793_SigmaE,axiom,
    ! [C2: product_prod_a_a,A: set_a,B: a > set_a] :
      ( ( member1426531477525435216od_a_a @ C2 @ ( product_Sigma_a_a @ A @ B ) )
     => ~ ! [X3: a] :
            ( ( member_a @ X3 @ A )
           => ! [Y3: a] :
                ( ( member_a @ Y3 @ ( B @ X3 ) )
               => ( C2
                 != ( product_Pair_a_a @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_794_SigmaE,axiom,
    ! [C2: produc551447040318622060at_nat,A: set_a,B: a > set_nat_nat] :
      ( ( member5463269792551873667at_nat @ C2 @ ( produc5882319205326396405at_nat @ A @ B ) )
     => ~ ! [X3: a] :
            ( ( member_a @ X3 @ A )
           => ! [Y3: nat > nat] :
                ( ( member_nat_nat @ Y3 @ ( B @ X3 ) )
               => ( C2
                 != ( produc7514922136058452582at_nat @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_795_SigmaE,axiom,
    ! [C2: product_prod_a_nat_a,A: set_a,B: a > set_nat_a] :
      ( ( member5853765858310737207_nat_a @ C2 @ ( produc830630580412313007_nat_a @ A @ B ) )
     => ~ ! [X3: a] :
            ( ( member_a @ X3 @ A )
           => ! [Y3: nat > a] :
                ( ( member_nat_a @ Y3 @ ( B @ X3 ) )
               => ( C2
                 != ( product_Pair_a_nat_a @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_796_SigmaE,axiom,
    ! [C2: product_prod_a_a_nat,A: set_a,B: a > set_a_nat] :
      ( ( member6594859421832710929_a_nat @ C2 @ ( produc6815839153543411849_a_nat @ A @ B ) )
     => ~ ! [X3: a] :
            ( ( member_a @ X3 @ A )
           => ! [Y3: a > nat] :
                ( ( member_a_nat @ Y3 @ ( B @ X3 ) )
               => ( C2
                 != ( product_Pair_a_a_nat @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_797_SigmaE,axiom,
    ! [C2: product_prod_a_a_a,A: set_a,B: a > set_a_a] :
      ( ( member681807718693926441_a_a_a @ C2 @ ( product_Sigma_a_a_a2 @ A @ B ) )
     => ~ ! [X3: a] :
            ( ( member_a @ X3 @ A )
           => ! [Y3: a > a] :
                ( ( member_a_a @ Y3 @ ( B @ X3 ) )
               => ( C2
                 != ( product_Pair_a_a_a @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_798_SigmaE,axiom,
    ! [C2: produc85711943791777264at_nat,A: set_nat,B: nat > set_nat_nat] :
      ( ( member8336108448496249625at_nat @ C2 @ ( produc4526620505022529241at_nat @ A @ B ) )
     => ~ ! [X3: nat] :
            ( ( member_nat @ X3 @ A )
           => ! [Y3: nat > nat] :
                ( ( member_nat_nat @ Y3 @ ( B @ X3 ) )
               => ( C2
                 != ( produc7839516862119294504at_nat @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_799_SigmaE,axiom,
    ! [C2: produc7123000486447228170_nat_a,A: set_nat,B: nat > set_nat_a] :
      ( ( member2811451201825703969_nat_a @ C2 @ ( produc3659965428606139339_nat_a @ A @ B ) )
     => ~ ! [X3: nat] :
            ( ( member_nat @ X3 @ A )
           => ! [Y3: nat > a] :
                ( ( member_nat_a @ Y3 @ ( B @ X3 ) )
               => ( C2
                 != ( produc5292568359338195516_nat_a @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_800_Sigma__mono,axiom,
    ! [A: set_Pr8826267807999420763od_a_a,C: set_Pr8826267807999420763od_a_a,B: ( product_prod_a_a > product_prod_a_a ) > set_a,D: ( product_prod_a_a > product_prod_a_a ) > set_a] :
      ( ( ord_le741722312431004091od_a_a @ A @ C )
     => ( ! [X3: product_prod_a_a > product_prod_a_a] :
            ( ( member4020126937092221116od_a_a @ X3 @ A )
           => ( ord_less_eq_set_a @ ( B @ X3 ) @ ( D @ X3 ) ) )
       => ( ord_le8233391959629702308_a_a_a @ ( produc2952938353304394703_a_a_a @ A @ B ) @ ( produc2952938353304394703_a_a_a @ C @ D ) ) ) ) ).

% Sigma_mono
thf(fact_801_Sigma__mono,axiom,
    ! [A: set_nat,C: set_nat,B: nat > set_a,D: nat > set_a] :
      ( ( ord_less_eq_set_nat @ A @ C )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ A )
           => ( ord_less_eq_set_a @ ( B @ X3 ) @ ( D @ X3 ) ) )
       => ( ord_le7924913712489149241_nat_a @ ( product_Sigma_nat_a @ A @ B ) @ ( product_Sigma_nat_a @ C @ D ) ) ) ) ).

% Sigma_mono
thf(fact_802_Sigma__mono,axiom,
    ! [A: set_nat_nat,C: set_nat_nat,B: ( nat > nat ) > set_a,D: ( nat > nat ) > set_a] :
      ( ( ord_le9059583361652607317at_nat @ A @ C )
     => ( ! [X3: nat > nat] :
            ( ( member_nat_nat @ X3 @ A )
           => ( ord_less_eq_set_a @ ( B @ X3 ) @ ( D @ X3 ) ) )
       => ( ord_le7986397757451867146_nat_a @ ( produc2309334610237048501_nat_a @ A @ B ) @ ( produc2309334610237048501_nat_a @ C @ D ) ) ) ) ).

% Sigma_mono
thf(fact_803_Sigma__mono,axiom,
    ! [A: set_nat_a,C: set_nat_a,B: ( nat > a ) > set_a,D: ( nat > a ) > set_a] :
      ( ( ord_le871467723717165285_nat_a @ A @ C )
     => ( ! [X3: nat > a] :
            ( ( member_nat_a @ X3 @ A )
           => ( ord_less_eq_set_a @ ( B @ X3 ) @ ( D @ X3 ) ) )
       => ( ord_le6236358127490459392at_a_a @ ( produc6085074725259719457at_a_a @ A @ B ) @ ( produc6085074725259719457at_a_a @ C @ D ) ) ) ) ).

% Sigma_mono
thf(fact_804_Sigma__mono,axiom,
    ! [A: set_a_6829686330177631172od_a_a,C: set_a_6829686330177631172od_a_a,B: ( a > product_prod_a_a ) > set_a,D: ( a > product_prod_a_a ) > set_a] :
      ( ( ord_le2371270124811097124od_a_a @ A @ C )
     => ( ! [X3: a > product_prod_a_a] :
            ( ( member1957775702407316389od_a_a @ X3 @ A )
           => ( ord_less_eq_set_a @ ( B @ X3 ) @ ( D @ X3 ) ) )
       => ( ord_le1927655382401944443_a_a_a @ ( produc3311275962944301862_a_a_a @ A @ B ) @ ( produc3311275962944301862_a_a_a @ C @ D ) ) ) ) ).

% Sigma_mono
thf(fact_805_Sigma__mono,axiom,
    ! [A: set_a_nat,C: set_a_nat,B: ( a > nat ) > set_a,D: ( a > nat ) > set_a] :
      ( ( ord_le1612561287239139007_a_nat @ A @ C )
     => ( ! [X3: a > nat] :
            ( ( member_a_nat @ X3 @ A )
           => ( ord_less_eq_set_a @ ( B @ X3 ) @ ( D @ X3 ) ) )
       => ( ord_le9110078103342601126_nat_a @ ( produc9037585820276363975_nat_a @ A @ B ) @ ( produc9037585820276363975_nat_a @ C @ D ) ) ) ) ).

% Sigma_mono
thf(fact_806_Sigma__mono,axiom,
    ! [A: set_a_a,C: set_a_a,B: ( a > a ) > set_a,D: ( a > a ) > set_a] :
      ( ( ord_less_eq_set_a_a @ A @ C )
     => ( ! [X3: a > a] :
            ( ( member_a_a @ X3 @ A )
           => ( ord_less_eq_set_a @ ( B @ X3 ) @ ( D @ X3 ) ) )
       => ( ord_le4624641257122322020_a_a_a @ ( product_Sigma_a_a_a @ A @ B ) @ ( product_Sigma_a_a_a @ C @ D ) ) ) ) ).

% Sigma_mono
thf(fact_807_Sigma__mono,axiom,
    ! [A: set_a,C: set_a,B: a > set_a,D: a > set_a] :
      ( ( ord_less_eq_set_a @ A @ C )
     => ( ! [X3: a] :
            ( ( member_a @ X3 @ A )
           => ( ord_less_eq_set_a @ ( B @ X3 ) @ ( D @ X3 ) ) )
       => ( ord_le746702958409616551od_a_a @ ( product_Sigma_a_a @ A @ B ) @ ( product_Sigma_a_a @ C @ D ) ) ) ) ).

% Sigma_mono
thf(fact_808_Sigma__Int__distrib1,axiom,
    ! [I: set_a,J2: set_a,C: a > set_a] :
      ( ( product_Sigma_a_a @ ( inf_inf_set_a @ I @ J2 ) @ C )
      = ( inf_in8905007599844390133od_a_a @ ( product_Sigma_a_a @ I @ C ) @ ( product_Sigma_a_a @ J2 @ C ) ) ) ).

% Sigma_Int_distrib1
thf(fact_809_additive__abelian__group_Ocard__sumset__singleton__eq,axiom,
    ! [G: set_Pr8826267807999420763od_a_a,Addition: ( product_prod_a_a > product_prod_a_a ) > ( product_prod_a_a > product_prod_a_a ) > product_prod_a_a > product_prod_a_a,Zero: product_prod_a_a > product_prod_a_a,A: set_Pr8826267807999420763od_a_a,A2: product_prod_a_a > product_prod_a_a] :
      ( ( pluenn7398882097854948023od_a_a @ G @ Addition @ Zero )
     => ( ( finite2714571839071174076od_a_a @ A )
       => ( ( ( member4020126937092221116od_a_a @ A2 @ G )
           => ( ( finite3311474722152943101od_a_a @ ( pluenn6825655340897727678od_a_a @ G @ Addition @ A @ ( insert1583624352380834389od_a_a @ A2 @ bot_bo2841618473486996463od_a_a ) ) )
              = ( finite3311474722152943101od_a_a @ ( inf_in3507259597588448237od_a_a @ A @ G ) ) ) )
          & ( ~ ( member4020126937092221116od_a_a @ A2 @ G )
           => ( ( finite3311474722152943101od_a_a @ ( pluenn6825655340897727678od_a_a @ G @ Addition @ A @ ( insert1583624352380834389od_a_a @ A2 @ bot_bo2841618473486996463od_a_a ) ) )
              = zero_zero_nat ) ) ) ) ) ).

% additive_abelian_group.card_sumset_singleton_eq
thf(fact_810_additive__abelian__group_Ocard__sumset__singleton__eq,axiom,
    ! [G: set_nat_nat,Addition: ( nat > nat ) > ( nat > nat ) > nat > nat,Zero: nat > nat,A: set_nat_nat,A2: nat > nat] :
      ( ( pluenn2593311475829585105at_nat @ G @ Addition @ Zero )
     => ( ( finite2115694454571419734at_nat @ A )
       => ( ( ( member_nat_nat @ A2 @ G )
           => ( ( finite_card_nat_nat @ ( pluenn8091236430575893592at_nat @ G @ Addition @ A @ ( insert_nat_nat @ A2 @ bot_bot_set_nat_nat ) ) )
              = ( finite_card_nat_nat @ ( inf_inf_set_nat_nat @ A @ G ) ) ) )
          & ( ~ ( member_nat_nat @ A2 @ G )
           => ( ( finite_card_nat_nat @ ( pluenn8091236430575893592at_nat @ G @ Addition @ A @ ( insert_nat_nat @ A2 @ bot_bot_set_nat_nat ) ) )
              = zero_zero_nat ) ) ) ) ) ).

% additive_abelian_group.card_sumset_singleton_eq
thf(fact_811_additive__abelian__group_Ocard__sumset__singleton__eq,axiom,
    ! [G: set_nat_a,Addition: ( nat > a ) > ( nat > a ) > nat > a,Zero: nat > a,A: set_nat_a,A2: nat > a] :
      ( ( pluenn1914567342357533651_nat_a @ G @ Addition @ Zero )
     => ( ( finite_finite_nat_a @ A )
       => ( ( ( member_nat_a @ A2 @ G )
           => ( ( finite_card_nat_a @ ( pluenn551768455426449420_nat_a @ G @ Addition @ A @ ( insert_nat_a @ A2 @ bot_bot_set_nat_a ) ) )
              = ( finite_card_nat_a @ ( inf_inf_set_nat_a @ A @ G ) ) ) )
          & ( ~ ( member_nat_a @ A2 @ G )
           => ( ( finite_card_nat_a @ ( pluenn551768455426449420_nat_a @ G @ Addition @ A @ ( insert_nat_a @ A2 @ bot_bot_set_nat_a ) ) )
              = zero_zero_nat ) ) ) ) ) ).

% additive_abelian_group.card_sumset_singleton_eq
thf(fact_812_additive__abelian__group_Ocard__sumset__singleton__eq,axiom,
    ! [G: set_a_6829686330177631172od_a_a,Addition: ( a > product_prod_a_a ) > ( a > product_prod_a_a ) > a > product_prod_a_a,Zero: a > product_prod_a_a,A: set_a_6829686330177631172od_a_a,A2: a > product_prod_a_a] :
      ( ( pluenn3290985442670702752od_a_a @ G @ Addition @ Zero )
     => ( ( finite8821664692192630949od_a_a @ A )
       => ( ( ( member1957775702407316389od_a_a @ A2 @ G )
           => ( ( finite6862662396756234726od_a_a @ ( pluenn6474628215148175783od_a_a @ G @ Addition @ A @ ( insert5082226857754029630od_a_a @ A2 @ bot_bo6605490641894888024od_a_a ) ) )
              = ( finite6862662396756234726od_a_a @ ( inf_in4434067354547777622od_a_a @ A @ G ) ) ) )
          & ( ~ ( member1957775702407316389od_a_a @ A2 @ G )
           => ( ( finite6862662396756234726od_a_a @ ( pluenn6474628215148175783od_a_a @ G @ Addition @ A @ ( insert5082226857754029630od_a_a @ A2 @ bot_bo6605490641894888024od_a_a ) ) )
              = zero_zero_nat ) ) ) ) ) ).

% additive_abelian_group.card_sumset_singleton_eq
thf(fact_813_additive__abelian__group_Ocard__sumset__singleton__eq,axiom,
    ! [G: set_a_nat,Addition: ( a > nat ) > ( a > nat ) > a > nat,Zero: a > nat,A: set_a_nat,A2: a > nat] :
      ( ( pluenn7899775915488632493_a_nat @ G @ Addition @ Zero )
     => ( ( finite_finite_a_nat @ A )
       => ( ( ( member_a_nat @ A2 @ G )
           => ( ( finite_card_a_nat @ ( pluenn6536977028557548262_a_nat @ G @ Addition @ A @ ( insert_a_nat @ A2 @ bot_bot_set_a_nat ) ) )
              = ( finite_card_a_nat @ ( inf_inf_set_a_nat @ A @ G ) ) ) )
          & ( ~ ( member_a_nat @ A2 @ G )
           => ( ( finite_card_a_nat @ ( pluenn6536977028557548262_a_nat @ G @ Addition @ A @ ( insert_a_nat @ A2 @ bot_bot_set_a_nat ) ) )
              = zero_zero_nat ) ) ) ) ) ).

% additive_abelian_group.card_sumset_singleton_eq
thf(fact_814_additive__abelian__group_Ocard__sumset__singleton__eq,axiom,
    ! [G: set_a_a,Addition: ( a > a ) > ( a > a ) > a > a,Zero: a > a,A: set_a_a,A2: a > a] :
      ( ( pluenn1959071899916691703up_a_a @ G @ Addition @ Zero )
     => ( ( finite_finite_a_a @ A )
       => ( ( ( member_a_a @ A2 @ G )
           => ( ( finite_card_a_a @ ( pluenn7679623682358442238et_a_a @ G @ Addition @ A @ ( insert_a_a @ A2 @ bot_bot_set_a_a ) ) )
              = ( finite_card_a_a @ ( inf_inf_set_a_a @ A @ G ) ) ) )
          & ( ~ ( member_a_a @ A2 @ G )
           => ( ( finite_card_a_a @ ( pluenn7679623682358442238et_a_a @ G @ Addition @ A @ ( insert_a_a @ A2 @ bot_bot_set_a_a ) ) )
              = zero_zero_nat ) ) ) ) ) ).

% additive_abelian_group.card_sumset_singleton_eq
thf(fact_815_additive__abelian__group_Ocard__sumset__singleton__eq,axiom,
    ! [G: set_Product_prod_a_a,Addition: product_prod_a_a > product_prod_a_a > product_prod_a_a,Zero: product_prod_a_a,A: set_Product_prod_a_a,A2: product_prod_a_a] :
      ( ( pluenn6837661381354551573od_a_a @ G @ Addition @ Zero )
     => ( ( finite6544458595007987280od_a_a @ A )
       => ( ( ( member1426531477525435216od_a_a @ A2 @ G )
           => ( ( finite4795055649997197647od_a_a @ ( pluenn659090379318209998od_a_a @ G @ Addition @ A @ ( insert4534936382041156343od_a_a @ A2 @ bot_bo3357376287454694259od_a_a ) ) )
              = ( finite4795055649997197647od_a_a @ ( inf_in8905007599844390133od_a_a @ A @ G ) ) ) )
          & ( ~ ( member1426531477525435216od_a_a @ A2 @ G )
           => ( ( finite4795055649997197647od_a_a @ ( pluenn659090379318209998od_a_a @ G @ Addition @ A @ ( insert4534936382041156343od_a_a @ A2 @ bot_bo3357376287454694259od_a_a ) ) )
              = zero_zero_nat ) ) ) ) ) ).

% additive_abelian_group.card_sumset_singleton_eq
thf(fact_816_additive__abelian__group_Ocard__sumset__singleton__eq,axiom,
    ! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: set_nat,A2: nat] :
      ( ( pluenn2073725187428264546up_nat @ G @ Addition @ Zero )
     => ( ( finite_finite_nat @ A )
       => ( ( ( member_nat @ A2 @ G )
           => ( ( finite_card_nat @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
              = ( finite_card_nat @ ( inf_inf_set_nat @ A @ G ) ) ) )
          & ( ~ ( member_nat @ A2 @ G )
           => ( ( finite_card_nat @ ( pluenn3669378163024332905et_nat @ G @ Addition @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
              = zero_zero_nat ) ) ) ) ) ).

% additive_abelian_group.card_sumset_singleton_eq
thf(fact_817_additive__abelian__group_Ocard__sumset__singleton__eq,axiom,
    ! [G: set_a,Addition: a > a > a,Zero: a,A: set_a,A2: a] :
      ( ( pluenn1164192988769422572roup_a @ G @ Addition @ Zero )
     => ( ( finite_finite_a @ A )
       => ( ( ( member_a @ A2 @ G )
           => ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ ( insert_a @ A2 @ bot_bot_set_a ) ) )
              = ( finite_card_a @ ( inf_inf_set_a @ A @ G ) ) ) )
          & ( ~ ( member_a @ A2 @ G )
           => ( ( finite_card_a @ ( pluenn3038260743871226533mset_a @ G @ Addition @ A @ ( insert_a @ A2 @ bot_bot_set_a ) ) )
              = zero_zero_nat ) ) ) ) ) ).

% additive_abelian_group.card_sumset_singleton_eq
thf(fact_818_case__prodE2,axiom,
    ! [Q: product_prod_a_a > $o,P: a > a > product_prod_a_a,Z: product_prod_a_a] :
      ( ( Q @ ( produc408267641121961211od_a_a @ P @ Z ) )
     => ~ ! [X3: a,Y3: a] :
            ( ( Z
              = ( product_Pair_a_a @ X3 @ Y3 ) )
           => ~ ( Q @ ( P @ X3 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_819_case__prod__eta,axiom,
    ! [F: product_prod_a_a > product_prod_a_a] :
      ( ( produc408267641121961211od_a_a
        @ ^ [X2: a,Y4: a] : ( F @ ( product_Pair_a_a @ X2 @ Y4 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_820_cond__case__prod__eta,axiom,
    ! [F: a > a > product_prod_a_a,G2: product_prod_a_a > product_prod_a_a] :
      ( ! [X3: a,Y3: a] :
          ( ( F @ X3 @ Y3 )
          = ( G2 @ ( product_Pair_a_a @ X3 @ Y3 ) ) )
     => ( ( produc408267641121961211od_a_a @ F )
        = G2 ) ) ).

% cond_case_prod_eta
thf(fact_821_Times__subset__cancel2,axiom,
    ! [X: product_prod_a_a > product_prod_a_a,C: set_Pr8826267807999420763od_a_a,A: set_a,B: set_a] :
      ( ( member4020126937092221116od_a_a @ X @ C )
     => ( ( ord_le323047320351004776od_a_a
          @ ( produc5709663546972291035od_a_a @ A
            @ ^ [Uu: a] : C )
          @ ( produc5709663546972291035od_a_a @ B
            @ ^ [Uu: a] : C ) )
        = ( ord_less_eq_set_a @ A @ B ) ) ) ).

% Times_subset_cancel2
thf(fact_822_Times__subset__cancel2,axiom,
    ! [X: nat,C: set_nat,A: set_a,B: set_a] :
      ( ( member_nat @ X @ C )
     => ( ( ord_le8666007276011122963_a_nat
          @ ( product_Sigma_a_nat @ A
            @ ^ [Uu: a] : C )
          @ ( product_Sigma_a_nat @ B
            @ ^ [Uu: a] : C ) )
        = ( ord_less_eq_set_a @ A @ B ) ) ) ).

% Times_subset_cancel2
thf(fact_823_Times__subset__cancel2,axiom,
    ! [X: nat > nat,C: set_nat_nat,A: set_a,B: set_a] :
      ( ( member_nat_nat @ X @ C )
     => ( ( ord_le2349859280813574146at_nat
          @ ( produc5882319205326396405at_nat @ A
            @ ^ [Uu: a] : C )
          @ ( produc5882319205326396405at_nat @ B
            @ ^ [Uu: a] : C ) )
        = ( ord_less_eq_set_a @ A @ B ) ) ) ).

% Times_subset_cancel2
thf(fact_824_Times__subset__cancel2,axiom,
    ! [X: nat > a,C: set_nat_a,A: set_a,B: set_a] :
      ( ( member_nat_a @ X @ C )
     => ( ( ord_le5235418285678350606_nat_a
          @ ( produc830630580412313007_nat_a @ A
            @ ^ [Uu: a] : C )
          @ ( produc830630580412313007_nat_a @ B
            @ ^ [Uu: a] : C ) )
        = ( ord_less_eq_set_a @ A @ B ) ) ) ).

% Times_subset_cancel2
thf(fact_825_Times__subset__cancel2,axiom,
    ! [X: a > product_prod_a_a,C: set_a_6829686330177631172od_a_a,A: set_a,B: set_a] :
      ( ( member1957775702407316389od_a_a @ X @ C )
     => ( ( ord_le4951836290565484625od_a_a
          @ ( produc6335909599342940612od_a_a @ A
            @ ^ [Uu: a] : C )
          @ ( produc6335909599342940612od_a_a @ B
            @ ^ [Uu: a] : C ) )
        = ( ord_less_eq_set_a @ A @ B ) ) ) ).

% Times_subset_cancel2
thf(fact_826_Times__subset__cancel2,axiom,
    ! [X: a > nat,C: set_a_nat,A: set_a,B: set_a] :
      ( ( member_a_nat @ X @ C )
     => ( ( ord_le3473451199410881000_a_nat
          @ ( produc6815839153543411849_a_nat @ A
            @ ^ [Uu: a] : C )
          @ ( produc6815839153543411849_a_nat @ B
            @ ^ [Uu: a] : C ) )
        = ( ord_less_eq_set_a @ A @ B ) ) ) ).

% Times_subset_cancel2
thf(fact_827_Times__subset__cancel2,axiom,
    ! [X: a > a,C: set_a_a,A: set_a,B: set_a] :
      ( ( member_a_a @ X @ C )
     => ( ( ord_le7361478385526211752_a_a_a
          @ ( product_Sigma_a_a_a2 @ A
            @ ^ [Uu: a] : C )
          @ ( product_Sigma_a_a_a2 @ B
            @ ^ [Uu: a] : C ) )
        = ( ord_less_eq_set_a @ A @ B ) ) ) ).

% Times_subset_cancel2
thf(fact_828_Times__subset__cancel2,axiom,
    ! [X: a,C: set_a,A: set_a,B: set_a] :
      ( ( member_a @ X @ C )
     => ( ( ord_le746702958409616551od_a_a
          @ ( product_Sigma_a_a @ A
            @ ^ [Uu: a] : C )
          @ ( product_Sigma_a_a @ B
            @ ^ [Uu: a] : C ) )
        = ( ord_less_eq_set_a @ A @ B ) ) ) ).

% Times_subset_cancel2
thf(fact_829_Times__Int__Times,axiom,
    ! [A: set_a,B: set_a,C: set_a,D: set_a] :
      ( ( inf_in8905007599844390133od_a_a
        @ ( product_Sigma_a_a @ A
          @ ^ [Uu: a] : B )
        @ ( product_Sigma_a_a @ C
          @ ^ [Uu: a] : D ) )
      = ( product_Sigma_a_a @ ( inf_inf_set_a @ A @ C )
        @ ^ [Uu: a] : ( inf_inf_set_a @ B @ D ) ) ) ).

% Times_Int_Times
thf(fact_830_Sigma__Int__distrib2,axiom,
    ! [I: set_a,A: a > set_a,B: a > set_a] :
      ( ( product_Sigma_a_a @ I
        @ ^ [I3: a] : ( inf_inf_set_a @ ( A @ I3 ) @ ( B @ I3 ) ) )
      = ( inf_in8905007599844390133od_a_a @ ( product_Sigma_a_a @ I @ A ) @ ( product_Sigma_a_a @ I @ B ) ) ) ).

% Sigma_Int_distrib2
thf(fact_831_Times__Int__distrib1,axiom,
    ! [A: set_a,B: set_a,C: set_a] :
      ( ( product_Sigma_a_a @ ( inf_inf_set_a @ A @ B )
        @ ^ [Uu: a] : C )
      = ( inf_in8905007599844390133od_a_a
        @ ( product_Sigma_a_a @ A
          @ ^ [Uu: a] : C )
        @ ( product_Sigma_a_a @ B
          @ ^ [Uu: a] : C ) ) ) ).

% Times_Int_distrib1
thf(fact_832_swap__inj__on,axiom,
    ! [A: set_Product_prod_a_a] :
      ( inj_on2566144670800592689od_a_a
      @ ( produc408267641121961211od_a_a
        @ ^ [I3: a,J: a] : ( product_Pair_a_a @ J @ I3 ) )
      @ A ) ).

% swap_inj_on
thf(fact_833_disjoint__insert_I2_J,axiom,
    ! [A: set_Pr8826267807999420763od_a_a,B4: product_prod_a_a > product_prod_a_a,B: set_Pr8826267807999420763od_a_a] :
      ( ( bot_bo2841618473486996463od_a_a
        = ( inf_in3507259597588448237od_a_a @ A @ ( insert1583624352380834389od_a_a @ B4 @ B ) ) )
      = ( ~ ( member4020126937092221116od_a_a @ B4 @ A )
        & ( bot_bo2841618473486996463od_a_a
          = ( inf_in3507259597588448237od_a_a @ A @ B ) ) ) ) ).

% disjoint_insert(2)
thf(fact_834_disjoint__insert_I2_J,axiom,
    ! [A: set_nat_nat,B4: nat > nat,B: set_nat_nat] :
      ( ( bot_bot_set_nat_nat
        = ( inf_inf_set_nat_nat @ A @ ( insert_nat_nat @ B4 @ B ) ) )
      = ( ~ ( member_nat_nat @ B4 @ A )
        & ( bot_bot_set_nat_nat
          = ( inf_inf_set_nat_nat @ A @ B ) ) ) ) ).

% disjoint_insert(2)
thf(fact_835_disjoint__insert_I2_J,axiom,
    ! [A: set_nat_a,B4: nat > a,B: set_nat_a] :
      ( ( bot_bot_set_nat_a
        = ( inf_inf_set_nat_a @ A @ ( insert_nat_a @ B4 @ B ) ) )
      = ( ~ ( member_nat_a @ B4 @ A )
        & ( bot_bot_set_nat_a
          = ( inf_inf_set_nat_a @ A @ B ) ) ) ) ).

% disjoint_insert(2)
thf(fact_836_disjoint__insert_I2_J,axiom,
    ! [A: set_a_6829686330177631172od_a_a,B4: a > product_prod_a_a,B: set_a_6829686330177631172od_a_a] :
      ( ( bot_bo6605490641894888024od_a_a
        = ( inf_in4434067354547777622od_a_a @ A @ ( insert5082226857754029630od_a_a @ B4 @ B ) ) )
      = ( ~ ( member1957775702407316389od_a_a @ B4 @ A )
        & ( bot_bo6605490641894888024od_a_a
          = ( inf_in4434067354547777622od_a_a @ A @ B ) ) ) ) ).

% disjoint_insert(2)
thf(fact_837_disjoint__insert_I2_J,axiom,
    ! [A: set_a_nat,B4: a > nat,B: set_a_nat] :
      ( ( bot_bot_set_a_nat
        = ( inf_inf_set_a_nat @ A @ ( insert_a_nat @ B4 @ B ) ) )
      = ( ~ ( member_a_nat @ B4 @ A )
        & ( bot_bot_set_a_nat
          = ( inf_inf_set_a_nat @ A @ B ) ) ) ) ).

% disjoint_insert(2)
thf(fact_838_disjoint__insert_I2_J,axiom,
    ! [A: set_a_a,B4: a > a,B: set_a_a] :
      ( ( bot_bot_set_a_a
        = ( inf_inf_set_a_a @ A @ ( insert_a_a @ B4 @ B ) ) )
      = ( ~ ( member_a_a @ B4 @ A )
        & ( bot_bot_set_a_a
          = ( inf_inf_set_a_a @ A @ B ) ) ) ) ).

% disjoint_insert(2)
thf(fact_839_disjoint__insert_I2_J,axiom,
    ! [A: set_a,B4: a,B: set_a] :
      ( ( bot_bot_set_a
        = ( inf_inf_set_a @ A @ ( insert_a @ B4 @ B ) ) )
      = ( ~ ( member_a @ B4 @ A )
        & ( bot_bot_set_a
          = ( inf_inf_set_a @ A @ B ) ) ) ) ).

% disjoint_insert(2)
thf(fact_840_disjoint__insert_I2_J,axiom,
    ! [A: set_nat,B4: nat,B: set_nat] :
      ( ( bot_bot_set_nat
        = ( inf_inf_set_nat @ A @ ( insert_nat @ B4 @ B ) ) )
      = ( ~ ( member_nat @ B4 @ A )
        & ( bot_bot_set_nat
          = ( inf_inf_set_nat @ A @ B ) ) ) ) ).

% disjoint_insert(2)
thf(fact_841_disjoint__insert_I1_J,axiom,
    ! [B: set_Pr8826267807999420763od_a_a,A2: product_prod_a_a > product_prod_a_a,A: set_Pr8826267807999420763od_a_a] :
      ( ( ( inf_in3507259597588448237od_a_a @ B @ ( insert1583624352380834389od_a_a @ A2 @ A ) )
        = bot_bo2841618473486996463od_a_a )
      = ( ~ ( member4020126937092221116od_a_a @ A2 @ B )
        & ( ( inf_in3507259597588448237od_a_a @ B @ A )
          = bot_bo2841618473486996463od_a_a ) ) ) ).

% disjoint_insert(1)
thf(fact_842_disjoint__insert_I1_J,axiom,
    ! [B: set_nat_nat,A2: nat > nat,A: set_nat_nat] :
      ( ( ( inf_inf_set_nat_nat @ B @ ( insert_nat_nat @ A2 @ A ) )
        = bot_bot_set_nat_nat )
      = ( ~ ( member_nat_nat @ A2 @ B )
        & ( ( inf_inf_set_nat_nat @ B @ A )
          = bot_bot_set_nat_nat ) ) ) ).

% disjoint_insert(1)
thf(fact_843_disjoint__insert_I1_J,axiom,
    ! [B: set_nat_a,A2: nat > a,A: set_nat_a] :
      ( ( ( inf_inf_set_nat_a @ B @ ( insert_nat_a @ A2 @ A ) )
        = bot_bot_set_nat_a )
      = ( ~ ( member_nat_a @ A2 @ B )
        & ( ( inf_inf_set_nat_a @ B @ A )
          = bot_bot_set_nat_a ) ) ) ).

% disjoint_insert(1)
thf(fact_844_disjoint__insert_I1_J,axiom,
    ! [B: set_a_6829686330177631172od_a_a,A2: a > product_prod_a_a,A: set_a_6829686330177631172od_a_a] :
      ( ( ( inf_in4434067354547777622od_a_a @ B @ ( insert5082226857754029630od_a_a @ A2 @ A ) )
        = bot_bo6605490641894888024od_a_a )
      = ( ~ ( member1957775702407316389od_a_a @ A2 @ B )
        & ( ( inf_in4434067354547777622od_a_a @ B @ A )
          = bot_bo6605490641894888024od_a_a ) ) ) ).

% disjoint_insert(1)
thf(fact_845_disjoint__insert_I1_J,axiom,
    ! [B: set_a_nat,A2: a > nat,A: set_a_nat] :
      ( ( ( inf_inf_set_a_nat @ B @ ( insert_a_nat @ A2 @ A ) )
        = bot_bot_set_a_nat )
      = ( ~ ( member_a_nat @ A2 @ B )
        & ( ( inf_inf_set_a_nat @ B @ A )
          = bot_bot_set_a_nat ) ) ) ).

% disjoint_insert(1)
thf(fact_846_disjoint__insert_I1_J,axiom,
    ! [B: set_a_a,A2: a > a,A: set_a_a] :
      ( ( ( inf_inf_set_a_a @ B @ ( insert_a_a @ A2 @ A ) )
        = bot_bot_set_a_a )
      = ( ~ ( member_a_a @ A2 @ B )
        & ( ( inf_inf_set_a_a @ B @ A )
          = bot_bot_set_a_a ) ) ) ).

% disjoint_insert(1)
thf(fact_847_disjoint__insert_I1_J,axiom,
    ! [B: set_a,A2: a,A: set_a] :
      ( ( ( inf_inf_set_a @ B @ ( insert_a @ A2 @ A ) )
        = bot_bot_set_a )
      = ( ~ ( member_a @ A2 @ B )
        & ( ( inf_inf_set_a @ B @ A )
          = bot_bot_set_a ) ) ) ).

% disjoint_insert(1)
thf(fact_848_disjoint__insert_I1_J,axiom,
    ! [B: set_nat,A2: nat,A: set_nat] :
      ( ( ( inf_inf_set_nat @ B @ ( insert_nat @ A2 @ A ) )
        = bot_bot_set_nat )
      = ( ~ ( member_nat @ A2 @ B )
        & ( ( inf_inf_set_nat @ B @ A )
          = bot_bot_set_nat ) ) ) ).

% disjoint_insert(1)
thf(fact_849_insert__disjoint_I2_J,axiom,
    ! [A2: product_prod_a_a > product_prod_a_a,A: set_Pr8826267807999420763od_a_a,B: set_Pr8826267807999420763od_a_a] :
      ( ( bot_bo2841618473486996463od_a_a
        = ( inf_in3507259597588448237od_a_a @ ( insert1583624352380834389od_a_a @ A2 @ A ) @ B ) )
      = ( ~ ( member4020126937092221116od_a_a @ A2 @ B )
        & ( bot_bo2841618473486996463od_a_a
          = ( inf_in3507259597588448237od_a_a @ A @ B ) ) ) ) ).

% insert_disjoint(2)
thf(fact_850_insert__disjoint_I2_J,axiom,
    ! [A2: nat > nat,A: set_nat_nat,B: set_nat_nat] :
      ( ( bot_bot_set_nat_nat
        = ( inf_inf_set_nat_nat @ ( insert_nat_nat @ A2 @ A ) @ B ) )
      = ( ~ ( member_nat_nat @ A2 @ B )
        & ( bot_bot_set_nat_nat
          = ( inf_inf_set_nat_nat @ A @ B ) ) ) ) ).

% insert_disjoint(2)
thf(fact_851_insert__disjoint_I2_J,axiom,
    ! [A2: nat > a,A: set_nat_a,B: set_nat_a] :
      ( ( bot_bot_set_nat_a
        = ( inf_inf_set_nat_a @ ( insert_nat_a @ A2 @ A ) @ B ) )
      = ( ~ ( member_nat_a @ A2 @ B )
        & ( bot_bot_set_nat_a
          = ( inf_inf_set_nat_a @ A @ B ) ) ) ) ).

% insert_disjoint(2)
thf(fact_852_insert__disjoint_I2_J,axiom,
    ! [A2: a > product_prod_a_a,A: set_a_6829686330177631172od_a_a,B: set_a_6829686330177631172od_a_a] :
      ( ( bot_bo6605490641894888024od_a_a
        = ( inf_in4434067354547777622od_a_a @ ( insert5082226857754029630od_a_a @ A2 @ A ) @ B ) )
      = ( ~ ( member1957775702407316389od_a_a @ A2 @ B )
        & ( bot_bo6605490641894888024od_a_a
          = ( inf_in4434067354547777622od_a_a @ A @ B ) ) ) ) ).

% insert_disjoint(2)
thf(fact_853_insert__disjoint_I2_J,axiom,
    ! [A2: a > nat,A: set_a_nat,B: set_a_nat] :
      ( ( bot_bot_set_a_nat
        = ( inf_inf_set_a_nat @ ( insert_a_nat @ A2 @ A ) @ B ) )
      = ( ~ ( member_a_nat @ A2 @ B )
        & ( bot_bot_set_a_nat
          = ( inf_inf_set_a_nat @ A @ B ) ) ) ) ).

% insert_disjoint(2)
thf(fact_854_insert__disjoint_I2_J,axiom,
    ! [A2: a > a,A: set_a_a,B: set_a_a] :
      ( ( bot_bot_set_a_a
        = ( inf_inf_set_a_a @ ( insert_a_a @ A2 @ A ) @ B ) )
      = ( ~ ( member_a_a @ A2 @ B )
        & ( bot_bot_set_a_a
          = ( inf_inf_set_a_a @ A @ B ) ) ) ) ).

% insert_disjoint(2)
thf(fact_855_insert__disjoint_I2_J,axiom,
    ! [A2: a,A: set_a,B: set_a] :
      ( ( bot_bot_set_a
        = ( inf_inf_set_a @ ( insert_a @ A2 @ A ) @ B ) )
      = ( ~ ( member_a @ A2 @ B )
        & ( bot_bot_set_a
          = ( inf_inf_set_a @ A @ B ) ) ) ) ).

% insert_disjoint(2)
thf(fact_856_insert__disjoint_I2_J,axiom,
    ! [A2: nat,A: set_nat,B: set_nat] :
      ( ( bot_bot_set_nat
        = ( inf_inf_set_nat @ ( insert_nat @ A2 @ A ) @ B ) )
      = ( ~ ( member_nat @ A2 @ B )
        & ( bot_bot_set_nat
          = ( inf_inf_set_nat @ A @ B ) ) ) ) ).

% insert_disjoint(2)
thf(fact_857_insert__disjoint_I1_J,axiom,
    ! [A2: product_prod_a_a > product_prod_a_a,A: set_Pr8826267807999420763od_a_a,B: set_Pr8826267807999420763od_a_a] :
      ( ( ( inf_in3507259597588448237od_a_a @ ( insert1583624352380834389od_a_a @ A2 @ A ) @ B )
        = bot_bo2841618473486996463od_a_a )
      = ( ~ ( member4020126937092221116od_a_a @ A2 @ B )
        & ( ( inf_in3507259597588448237od_a_a @ A @ B )
          = bot_bo2841618473486996463od_a_a ) ) ) ).

% insert_disjoint(1)
thf(fact_858_insert__disjoint_I1_J,axiom,
    ! [A2: nat > nat,A: set_nat_nat,B: set_nat_nat] :
      ( ( ( inf_inf_set_nat_nat @ ( insert_nat_nat @ A2 @ A ) @ B )
        = bot_bot_set_nat_nat )
      = ( ~ ( member_nat_nat @ A2 @ B )
        & ( ( inf_inf_set_nat_nat @ A @ B )
          = bot_bot_set_nat_nat ) ) ) ).

% insert_disjoint(1)
thf(fact_859_insert__disjoint_I1_J,axiom,
    ! [A2: nat > a,A: set_nat_a,B: set_nat_a] :
      ( ( ( inf_inf_set_nat_a @ ( insert_nat_a @ A2 @ A ) @ B )
        = bot_bot_set_nat_a )
      = ( ~ ( member_nat_a @ A2 @ B )
        & ( ( inf_inf_set_nat_a @ A @ B )
          = bot_bot_set_nat_a ) ) ) ).

% insert_disjoint(1)
thf(fact_860_insert__disjoint_I1_J,axiom,
    ! [A2: a > product_prod_a_a,A: set_a_6829686330177631172od_a_a,B: set_a_6829686330177631172od_a_a] :
      ( ( ( inf_in4434067354547777622od_a_a @ ( insert5082226857754029630od_a_a @ A2 @ A ) @ B )
        = bot_bo6605490641894888024od_a_a )
      = ( ~ ( member1957775702407316389od_a_a @ A2 @ B )
        & ( ( inf_in4434067354547777622od_a_a @ A @ B )
          = bot_bo6605490641894888024od_a_a ) ) ) ).

% insert_disjoint(1)
thf(fact_861_insert__disjoint_I1_J,axiom,
    ! [A2: a > nat,A: set_a_nat,B: set_a_nat] :
      ( ( ( inf_inf_set_a_nat @ ( insert_a_nat @ A2 @ A ) @ B )
        = bot_bot_set_a_nat )
      = ( ~ ( member_a_nat @ A2 @ B )
        & ( ( inf_inf_set_a_nat @ A @ B )
          = bot_bot_set_a_nat ) ) ) ).

% insert_disjoint(1)
thf(fact_862_insert__disjoint_I1_J,axiom,
    ! [A2: a > a,A: set_a_a,B: set_a_a] :
      ( ( ( inf_inf_set_a_a @ ( insert_a_a @ A2 @ A ) @ B )
        = bot_bot_set_a_a )
      = ( ~ ( member_a_a @ A2 @ B )
        & ( ( inf_inf_set_a_a @ A @ B )
          = bot_bot_set_a_a ) ) ) ).

% insert_disjoint(1)
thf(fact_863_insert__disjoint_I1_J,axiom,
    ! [A2: a,A: set_a,B: set_a] :
      ( ( ( inf_inf_set_a @ ( insert_a @ A2 @ A ) @ B )
        = bot_bot_set_a )
      = ( ~ ( member_a @ A2 @ B )
        & ( ( inf_inf_set_a @ A @ B )
          = bot_bot_set_a ) ) ) ).

% insert_disjoint(1)
thf(fact_864_insert__disjoint_I1_J,axiom,
    ! [A2: nat,A: set_nat,B: set_nat] :
      ( ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ A ) @ B )
        = bot_bot_set_nat )
      = ( ~ ( member_nat @ A2 @ B )
        & ( ( inf_inf_set_nat @ A @ B )
          = bot_bot_set_nat ) ) ) ).

% insert_disjoint(1)
thf(fact_865_singleton__insert__inj__eq_H,axiom,
    ! [A2: nat,A: set_nat,B4: nat] :
      ( ( ( insert_nat @ A2 @ A )
        = ( insert_nat @ B4 @ bot_bot_set_nat ) )
      = ( ( A2 = B4 )
        & ( ord_less_eq_set_nat @ A @ ( insert_nat @ B4 @ bot_bot_set_nat ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_866_singleton__insert__inj__eq_H,axiom,
    ! [A2: a,A: set_a,B4: a] :
      ( ( ( insert_a @ A2 @ A )
        = ( insert_a @ B4 @ bot_bot_set_a ) )
      = ( ( A2 = B4 )
        & ( ord_less_eq_set_a @ A @ ( insert_a @ B4 @ bot_bot_set_a ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_867_singleton__insert__inj__eq,axiom,
    ! [B4: nat,A2: nat,A: set_nat] :
      ( ( ( insert_nat @ B4 @ bot_bot_set_nat )
        = ( insert_nat @ A2 @ A ) )
      = ( ( A2 = B4 )
        & ( ord_less_eq_set_nat @ A @ ( insert_nat @ B4 @ bot_bot_set_nat ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_868_singleton__insert__inj__eq,axiom,
    ! [B4: a,A2: a,A: set_a] :
      ( ( ( insert_a @ B4 @ bot_bot_set_a )
        = ( insert_a @ A2 @ A ) )
      = ( ( A2 = B4 )
        & ( ord_less_eq_set_a @ A @ ( insert_a @ B4 @ bot_bot_set_a ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_869_inverse__subgroupD,axiom,
    ! [H3: set_a] :
      ( ( group_subgroup_a @ ( image_a_a @ ( group_inverse_a @ g @ addition @ zero ) @ H3 ) @ g @ addition @ zero )
     => ( ( ord_less_eq_set_a @ H3 @ ( group_Units_a @ g @ addition @ zero ) )
       => ( group_subgroup_a @ H3 @ g @ addition @ zero ) ) ) ).

% inverse_subgroupD
thf(fact_870_sumset__iterated__empty,axiom,
    ! [R: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ R )
     => ( ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ bot_bot_set_a @ R )
        = bot_bot_set_a ) ) ).

% sumset_iterated_empty
thf(fact_871_image__eqI,axiom,
    ! [B4: a,F: a > a,X: a,A: set_a] :
      ( ( B4
        = ( F @ X ) )
     => ( ( member_a @ X @ A )
       => ( member_a @ B4 @ ( image_a_a @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_872_image__eqI,axiom,
    ! [B4: nat,F: a > nat,X: a,A: set_a] :
      ( ( B4
        = ( F @ X ) )
     => ( ( member_a @ X @ A )
       => ( member_nat @ B4 @ ( image_a_nat @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_873_image__eqI,axiom,
    ! [B4: a,F: nat > a,X: nat,A: set_nat] :
      ( ( B4
        = ( F @ X ) )
     => ( ( member_nat @ X @ A )
       => ( member_a @ B4 @ ( image_nat_a @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_874_image__eqI,axiom,
    ! [B4: nat,F: nat > nat,X: nat,A: set_nat] :
      ( ( B4
        = ( F @ X ) )
     => ( ( member_nat @ X @ A )
       => ( member_nat @ B4 @ ( image_nat_nat @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_875_image__eqI,axiom,
    ! [B4: set_a,F: a > set_a,X: a,A: set_a] :
      ( ( B4
        = ( F @ X ) )
     => ( ( member_a @ X @ A )
       => ( member_set_a @ B4 @ ( image_a_set_a @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_876_image__eqI,axiom,
    ! [B4: nat > nat,F: a > nat > nat,X: a,A: set_a] :
      ( ( B4
        = ( F @ X ) )
     => ( ( member_a @ X @ A )
       => ( member_nat_nat @ B4 @ ( image_a_nat_nat @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_877_image__eqI,axiom,
    ! [B4: nat > a,F: a > nat > a,X: a,A: set_a] :
      ( ( B4
        = ( F @ X ) )
     => ( ( member_a @ X @ A )
       => ( member_nat_a @ B4 @ ( image_a_nat_a2 @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_878_image__eqI,axiom,
    ! [B4: a > nat,F: a > a > nat,X: a,A: set_a] :
      ( ( B4
        = ( F @ X ) )
     => ( ( member_a @ X @ A )
       => ( member_a_nat @ B4 @ ( image_a_a_nat @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_879_image__eqI,axiom,
    ! [B4: a > a,F: a > a > a,X: a,A: set_a] :
      ( ( B4
        = ( F @ X ) )
     => ( ( member_a @ X @ A )
       => ( member_a_a @ B4 @ ( image_a_a_a2 @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_880_image__eqI,axiom,
    ! [B4: nat > nat,F: nat > nat > nat,X: nat,A: set_nat] :
      ( ( B4
        = ( F @ X ) )
     => ( ( member_nat @ X @ A )
       => ( member_nat_nat @ B4 @ ( image_nat_nat_nat2 @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_881_empty__Collect__eq,axiom,
    ! [P: a > $o] :
      ( ( bot_bot_set_a
        = ( collect_a @ P ) )
      = ( ! [X2: a] :
            ~ ( P @ X2 ) ) ) ).

% empty_Collect_eq
thf(fact_882_empty__Collect__eq,axiom,
    ! [P: nat > $o] :
      ( ( bot_bot_set_nat
        = ( collect_nat @ P ) )
      = ( ! [X2: nat] :
            ~ ( P @ X2 ) ) ) ).

% empty_Collect_eq
thf(fact_883_Collect__empty__eq,axiom,
    ! [P: a > $o] :
      ( ( ( collect_a @ P )
        = bot_bot_set_a )
      = ( ! [X2: a] :
            ~ ( P @ X2 ) ) ) ).

% Collect_empty_eq
thf(fact_884_Collect__empty__eq,axiom,
    ! [P: nat > $o] :
      ( ( ( collect_nat @ P )
        = bot_bot_set_nat )
      = ( ! [X2: nat] :
            ~ ( P @ X2 ) ) ) ).

% Collect_empty_eq
thf(fact_885_all__not__in__conv,axiom,
    ! [A: set_Pr8826267807999420763od_a_a] :
      ( ( ! [X2: product_prod_a_a > product_prod_a_a] :
            ~ ( member4020126937092221116od_a_a @ X2 @ A ) )
      = ( A = bot_bo2841618473486996463od_a_a ) ) ).

% all_not_in_conv
thf(fact_886_all__not__in__conv,axiom,
    ! [A: set_nat_nat] :
      ( ( ! [X2: nat > nat] :
            ~ ( member_nat_nat @ X2 @ A ) )
      = ( A = bot_bot_set_nat_nat ) ) ).

% all_not_in_conv
thf(fact_887_all__not__in__conv,axiom,
    ! [A: set_nat_a] :
      ( ( ! [X2: nat > a] :
            ~ ( member_nat_a @ X2 @ A ) )
      = ( A = bot_bot_set_nat_a ) ) ).

% all_not_in_conv
thf(fact_888_all__not__in__conv,axiom,
    ! [A: set_a_6829686330177631172od_a_a] :
      ( ( ! [X2: a > product_prod_a_a] :
            ~ ( member1957775702407316389od_a_a @ X2 @ A ) )
      = ( A = bot_bo6605490641894888024od_a_a ) ) ).

% all_not_in_conv
thf(fact_889_all__not__in__conv,axiom,
    ! [A: set_a_nat] :
      ( ( ! [X2: a > nat] :
            ~ ( member_a_nat @ X2 @ A ) )
      = ( A = bot_bot_set_a_nat ) ) ).

% all_not_in_conv
thf(fact_890_all__not__in__conv,axiom,
    ! [A: set_a_a] :
      ( ( ! [X2: a > a] :
            ~ ( member_a_a @ X2 @ A ) )
      = ( A = bot_bot_set_a_a ) ) ).

% all_not_in_conv
thf(fact_891_all__not__in__conv,axiom,
    ! [A: set_a] :
      ( ( ! [X2: a] :
            ~ ( member_a @ X2 @ A ) )
      = ( A = bot_bot_set_a ) ) ).

% all_not_in_conv
thf(fact_892_all__not__in__conv,axiom,
    ! [A: set_nat] :
      ( ( ! [X2: nat] :
            ~ ( member_nat @ X2 @ A ) )
      = ( A = bot_bot_set_nat ) ) ).

% all_not_in_conv
thf(fact_893_empty__iff,axiom,
    ! [C2: product_prod_a_a > product_prod_a_a] :
      ~ ( member4020126937092221116od_a_a @ C2 @ bot_bo2841618473486996463od_a_a ) ).

% empty_iff
thf(fact_894_empty__iff,axiom,
    ! [C2: nat > nat] :
      ~ ( member_nat_nat @ C2 @ bot_bot_set_nat_nat ) ).

% empty_iff
thf(fact_895_empty__iff,axiom,
    ! [C2: nat > a] :
      ~ ( member_nat_a @ C2 @ bot_bot_set_nat_a ) ).

% empty_iff
thf(fact_896_empty__iff,axiom,
    ! [C2: a > product_prod_a_a] :
      ~ ( member1957775702407316389od_a_a @ C2 @ bot_bo6605490641894888024od_a_a ) ).

% empty_iff
thf(fact_897_empty__iff,axiom,
    ! [C2: a > nat] :
      ~ ( member_a_nat @ C2 @ bot_bot_set_a_nat ) ).

% empty_iff
thf(fact_898_empty__iff,axiom,
    ! [C2: a > a] :
      ~ ( member_a_a @ C2 @ bot_bot_set_a_a ) ).

% empty_iff
thf(fact_899_empty__iff,axiom,
    ! [C2: a] :
      ~ ( member_a @ C2 @ bot_bot_set_a ) ).

% empty_iff
thf(fact_900_empty__iff,axiom,
    ! [C2: nat] :
      ~ ( member_nat @ C2 @ bot_bot_set_nat ) ).

% empty_iff
thf(fact_901_subset__antisym,axiom,
    ! [A: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ( ord_less_eq_set_a @ B @ A )
       => ( A = B ) ) ) ).

% subset_antisym
thf(fact_902_subsetI,axiom,
    ! [A: set_Pr8826267807999420763od_a_a,B: set_Pr8826267807999420763od_a_a] :
      ( ! [X3: product_prod_a_a > product_prod_a_a] :
          ( ( member4020126937092221116od_a_a @ X3 @ A )
         => ( member4020126937092221116od_a_a @ X3 @ B ) )
     => ( ord_le741722312431004091od_a_a @ A @ B ) ) ).

% subsetI
thf(fact_903_subsetI,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A )
         => ( member_nat @ X3 @ B ) )
     => ( ord_less_eq_set_nat @ A @ B ) ) ).

% subsetI
thf(fact_904_subsetI,axiom,
    ! [A: set_nat_nat,B: set_nat_nat] :
      ( ! [X3: nat > nat] :
          ( ( member_nat_nat @ X3 @ A )
         => ( member_nat_nat @ X3 @ B ) )
     => ( ord_le9059583361652607317at_nat @ A @ B ) ) ).

% subsetI
thf(fact_905_subsetI,axiom,
    ! [A: set_nat_a,B: set_nat_a] :
      ( ! [X3: nat > a] :
          ( ( member_nat_a @ X3 @ A )
         => ( member_nat_a @ X3 @ B ) )
     => ( ord_le871467723717165285_nat_a @ A @ B ) ) ).

% subsetI
thf(fact_906_subsetI,axiom,
    ! [A: set_a_6829686330177631172od_a_a,B: set_a_6829686330177631172od_a_a] :
      ( ! [X3: a > product_prod_a_a] :
          ( ( member1957775702407316389od_a_a @ X3 @ A )
         => ( member1957775702407316389od_a_a @ X3 @ B ) )
     => ( ord_le2371270124811097124od_a_a @ A @ B ) ) ).

% subsetI
thf(fact_907_subsetI,axiom,
    ! [A: set_a_nat,B: set_a_nat] :
      ( ! [X3: a > nat] :
          ( ( member_a_nat @ X3 @ A )
         => ( member_a_nat @ X3 @ B ) )
     => ( ord_le1612561287239139007_a_nat @ A @ B ) ) ).

% subsetI
thf(fact_908_subsetI,axiom,
    ! [A: set_a_a,B: set_a_a] :
      ( ! [X3: a > a] :
          ( ( member_a_a @ X3 @ A )
         => ( member_a_a @ X3 @ B ) )
     => ( ord_less_eq_set_a_a @ A @ B ) ) ).

% subsetI
thf(fact_909_subsetI,axiom,
    ! [A: set_a,B: set_a] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ A )
         => ( member_a @ X3 @ B ) )
     => ( ord_less_eq_set_a @ A @ B ) ) ).

% subsetI
thf(fact_910_insert__absorb2,axiom,
    ! [X: a,A: set_a] :
      ( ( insert_a @ X @ ( insert_a @ X @ A ) )
      = ( insert_a @ X @ A ) ) ).

% insert_absorb2
thf(fact_911_insert__iff,axiom,
    ! [A2: a,B4: a,A: set_a] :
      ( ( member_a @ A2 @ ( insert_a @ B4 @ A ) )
      = ( ( A2 = B4 )
        | ( member_a @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_912_insert__iff,axiom,
    ! [A2: product_prod_a_a > product_prod_a_a,B4: product_prod_a_a > product_prod_a_a,A: set_Pr8826267807999420763od_a_a] :
      ( ( member4020126937092221116od_a_a @ A2 @ ( insert1583624352380834389od_a_a @ B4 @ A ) )
      = ( ( A2 = B4 )
        | ( member4020126937092221116od_a_a @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_913_insert__iff,axiom,
    ! [A2: nat,B4: nat,A: set_nat] :
      ( ( member_nat @ A2 @ ( insert_nat @ B4 @ A ) )
      = ( ( A2 = B4 )
        | ( member_nat @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_914_insert__iff,axiom,
    ! [A2: nat > nat,B4: nat > nat,A: set_nat_nat] :
      ( ( member_nat_nat @ A2 @ ( insert_nat_nat @ B4 @ A ) )
      = ( ( A2 = B4 )
        | ( member_nat_nat @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_915_insert__iff,axiom,
    ! [A2: nat > a,B4: nat > a,A: set_nat_a] :
      ( ( member_nat_a @ A2 @ ( insert_nat_a @ B4 @ A ) )
      = ( ( A2 = B4 )
        | ( member_nat_a @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_916_insert__iff,axiom,
    ! [A2: a > product_prod_a_a,B4: a > product_prod_a_a,A: set_a_6829686330177631172od_a_a] :
      ( ( member1957775702407316389od_a_a @ A2 @ ( insert5082226857754029630od_a_a @ B4 @ A ) )
      = ( ( A2 = B4 )
        | ( member1957775702407316389od_a_a @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_917_insert__iff,axiom,
    ! [A2: a > nat,B4: a > nat,A: set_a_nat] :
      ( ( member_a_nat @ A2 @ ( insert_a_nat @ B4 @ A ) )
      = ( ( A2 = B4 )
        | ( member_a_nat @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_918_insert__iff,axiom,
    ! [A2: a > a,B4: a > a,A: set_a_a] :
      ( ( member_a_a @ A2 @ ( insert_a_a @ B4 @ A ) )
      = ( ( A2 = B4 )
        | ( member_a_a @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_919_insertCI,axiom,
    ! [A2: a,B: set_a,B4: a] :
      ( ( ~ ( member_a @ A2 @ B )
       => ( A2 = B4 ) )
     => ( member_a @ A2 @ ( insert_a @ B4 @ B ) ) ) ).

% insertCI
thf(fact_920_insertCI,axiom,
    ! [A2: product_prod_a_a > product_prod_a_a,B: set_Pr8826267807999420763od_a_a,B4: product_prod_a_a > product_prod_a_a] :
      ( ( ~ ( member4020126937092221116od_a_a @ A2 @ B )
       => ( A2 = B4 ) )
     => ( member4020126937092221116od_a_a @ A2 @ ( insert1583624352380834389od_a_a @ B4 @ B ) ) ) ).

% insertCI
thf(fact_921_insertCI,axiom,
    ! [A2: nat,B: set_nat,B4: nat] :
      ( ( ~ ( member_nat @ A2 @ B )
       => ( A2 = B4 ) )
     => ( member_nat @ A2 @ ( insert_nat @ B4 @ B ) ) ) ).

% insertCI
thf(fact_922_insertCI,axiom,
    ! [A2: nat > nat,B: set_nat_nat,B4: nat > nat] :
      ( ( ~ ( member_nat_nat @ A2 @ B )
       => ( A2 = B4 ) )
     => ( member_nat_nat @ A2 @ ( insert_nat_nat @ B4 @ B ) ) ) ).

% insertCI
thf(fact_923_insertCI,axiom,
    ! [A2: nat > a,B: set_nat_a,B4: nat > a] :
      ( ( ~ ( member_nat_a @ A2 @ B )
       => ( A2 = B4 ) )
     => ( member_nat_a @ A2 @ ( insert_nat_a @ B4 @ B ) ) ) ).

% insertCI
thf(fact_924_insertCI,axiom,
    ! [A2: a > product_prod_a_a,B: set_a_6829686330177631172od_a_a,B4: a > product_prod_a_a] :
      ( ( ~ ( member1957775702407316389od_a_a @ A2 @ B )
       => ( A2 = B4 ) )
     => ( member1957775702407316389od_a_a @ A2 @ ( insert5082226857754029630od_a_a @ B4 @ B ) ) ) ).

% insertCI
thf(fact_925_insertCI,axiom,
    ! [A2: a > nat,B: set_a_nat,B4: a > nat] :
      ( ( ~ ( member_a_nat @ A2 @ B )
       => ( A2 = B4 ) )
     => ( member_a_nat @ A2 @ ( insert_a_nat @ B4 @ B ) ) ) ).

% insertCI
thf(fact_926_insertCI,axiom,
    ! [A2: a > a,B: set_a_a,B4: a > a] :
      ( ( ~ ( member_a_a @ A2 @ B )
       => ( A2 = B4 ) )
     => ( member_a_a @ A2 @ ( insert_a_a @ B4 @ B ) ) ) ).

% insertCI
thf(fact_927_Int__iff,axiom,
    ! [C2: product_prod_a_a > product_prod_a_a,A: set_Pr8826267807999420763od_a_a,B: set_Pr8826267807999420763od_a_a] :
      ( ( member4020126937092221116od_a_a @ C2 @ ( inf_in3507259597588448237od_a_a @ A @ B ) )
      = ( ( member4020126937092221116od_a_a @ C2 @ A )
        & ( member4020126937092221116od_a_a @ C2 @ B ) ) ) ).

% Int_iff
thf(fact_928_Int__iff,axiom,
    ! [C2: nat,A: set_nat,B: set_nat] :
      ( ( member_nat @ C2 @ ( inf_inf_set_nat @ A @ B ) )
      = ( ( member_nat @ C2 @ A )
        & ( member_nat @ C2 @ B ) ) ) ).

% Int_iff
thf(fact_929_Int__iff,axiom,
    ! [C2: nat > nat,A: set_nat_nat,B: set_nat_nat] :
      ( ( member_nat_nat @ C2 @ ( inf_inf_set_nat_nat @ A @ B ) )
      = ( ( member_nat_nat @ C2 @ A )
        & ( member_nat_nat @ C2 @ B ) ) ) ).

% Int_iff
thf(fact_930_Int__iff,axiom,
    ! [C2: nat > a,A: set_nat_a,B: set_nat_a] :
      ( ( member_nat_a @ C2 @ ( inf_inf_set_nat_a @ A @ B ) )
      = ( ( member_nat_a @ C2 @ A )
        & ( member_nat_a @ C2 @ B ) ) ) ).

% Int_iff
thf(fact_931_Int__iff,axiom,
    ! [C2: a > product_prod_a_a,A: set_a_6829686330177631172od_a_a,B: set_a_6829686330177631172od_a_a] :
      ( ( member1957775702407316389od_a_a @ C2 @ ( inf_in4434067354547777622od_a_a @ A @ B ) )
      = ( ( member1957775702407316389od_a_a @ C2 @ A )
        & ( member1957775702407316389od_a_a @ C2 @ B ) ) ) ).

% Int_iff
thf(fact_932_Int__iff,axiom,
    ! [C2: a > nat,A: set_a_nat,B: set_a_nat] :
      ( ( member_a_nat @ C2 @ ( inf_inf_set_a_nat @ A @ B ) )
      = ( ( member_a_nat @ C2 @ A )
        & ( member_a_nat @ C2 @ B ) ) ) ).

% Int_iff
thf(fact_933_Int__iff,axiom,
    ! [C2: a > a,A: set_a_a,B: set_a_a] :
      ( ( member_a_a @ C2 @ ( inf_inf_set_a_a @ A @ B ) )
      = ( ( member_a_a @ C2 @ A )
        & ( member_a_a @ C2 @ B ) ) ) ).

% Int_iff
thf(fact_934_Int__iff,axiom,
    ! [C2: a,A: set_a,B: set_a] :
      ( ( member_a @ C2 @ ( inf_inf_set_a @ A @ B ) )
      = ( ( member_a @ C2 @ A )
        & ( member_a @ C2 @ B ) ) ) ).

% Int_iff
thf(fact_935_IntI,axiom,
    ! [C2: product_prod_a_a > product_prod_a_a,A: set_Pr8826267807999420763od_a_a,B: set_Pr8826267807999420763od_a_a] :
      ( ( member4020126937092221116od_a_a @ C2 @ A )
     => ( ( member4020126937092221116od_a_a @ C2 @ B )
       => ( member4020126937092221116od_a_a @ C2 @ ( inf_in3507259597588448237od_a_a @ A @ B ) ) ) ) ).

% IntI
thf(fact_936_IntI,axiom,
    ! [C2: nat,A: set_nat,B: set_nat] :
      ( ( member_nat @ C2 @ A )
     => ( ( member_nat @ C2 @ B )
       => ( member_nat @ C2 @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).

% IntI
thf(fact_937_IntI,axiom,
    ! [C2: nat > nat,A: set_nat_nat,B: set_nat_nat] :
      ( ( member_nat_nat @ C2 @ A )
     => ( ( member_nat_nat @ C2 @ B )
       => ( member_nat_nat @ C2 @ ( inf_inf_set_nat_nat @ A @ B ) ) ) ) ).

% IntI
thf(fact_938_IntI,axiom,
    ! [C2: nat > a,A: set_nat_a,B: set_nat_a] :
      ( ( member_nat_a @ C2 @ A )
     => ( ( member_nat_a @ C2 @ B )
       => ( member_nat_a @ C2 @ ( inf_inf_set_nat_a @ A @ B ) ) ) ) ).

% IntI
thf(fact_939_IntI,axiom,
    ! [C2: a > product_prod_a_a,A: set_a_6829686330177631172od_a_a,B: set_a_6829686330177631172od_a_a] :
      ( ( member1957775702407316389od_a_a @ C2 @ A )
     => ( ( member1957775702407316389od_a_a @ C2 @ B )
       => ( member1957775702407316389od_a_a @ C2 @ ( inf_in4434067354547777622od_a_a @ A @ B ) ) ) ) ).

% IntI
thf(fact_940_IntI,axiom,
    ! [C2: a > nat,A: set_a_nat,B: set_a_nat] :
      ( ( member_a_nat @ C2 @ A )
     => ( ( member_a_nat @ C2 @ B )
       => ( member_a_nat @ C2 @ ( inf_inf_set_a_nat @ A @ B ) ) ) ) ).

% IntI
thf(fact_941_IntI,axiom,
    ! [C2: a > a,A: set_a_a,B: set_a_a] :
      ( ( member_a_a @ C2 @ A )
     => ( ( member_a_a @ C2 @ B )
       => ( member_a_a @ C2 @ ( inf_inf_set_a_a @ A @ B ) ) ) ) ).

% IntI
thf(fact_942_IntI,axiom,
    ! [C2: a,A: set_a,B: set_a] :
      ( ( member_a @ C2 @ A )
     => ( ( member_a @ C2 @ B )
       => ( member_a @ C2 @ ( inf_inf_set_a @ A @ B ) ) ) ) ).

% IntI
thf(fact_943_image__ident,axiom,
    ! [Y5: set_a] :
      ( ( image_a_a
        @ ^ [X2: a] : X2
        @ Y5 )
      = Y5 ) ).

% image_ident
thf(fact_944_card__Collect__less__nat,axiom,
    ! [N2: nat] :
      ( ( finite_card_nat
        @ ( collect_nat
          @ ^ [I3: nat] : ( ord_less_nat @ I3 @ N2 ) ) )
      = N2 ) ).

% card_Collect_less_nat
thf(fact_945_image__empty,axiom,
    ! [F: a > set_a] :
      ( ( image_a_set_a @ F @ bot_bot_set_a )
      = bot_bot_set_set_a ) ).

% image_empty
thf(fact_946_image__empty,axiom,
    ! [F: a > a] :
      ( ( image_a_a @ F @ bot_bot_set_a )
      = bot_bot_set_a ) ).

% image_empty
thf(fact_947_image__empty,axiom,
    ! [F: a > nat] :
      ( ( image_a_nat @ F @ bot_bot_set_a )
      = bot_bot_set_nat ) ).

% image_empty
thf(fact_948_image__empty,axiom,
    ! [F: nat > a] :
      ( ( image_nat_a @ F @ bot_bot_set_nat )
      = bot_bot_set_a ) ).

% image_empty
thf(fact_949_image__empty,axiom,
    ! [F: nat > nat] :
      ( ( image_nat_nat @ F @ bot_bot_set_nat )
      = bot_bot_set_nat ) ).

% image_empty
thf(fact_950_empty__is__image,axiom,
    ! [F: a > set_a,A: set_a] :
      ( ( bot_bot_set_set_a
        = ( image_a_set_a @ F @ A ) )
      = ( A = bot_bot_set_a ) ) ).

% empty_is_image
thf(fact_951_empty__is__image,axiom,
    ! [F: a > a,A: set_a] :
      ( ( bot_bot_set_a
        = ( image_a_a @ F @ A ) )
      = ( A = bot_bot_set_a ) ) ).

% empty_is_image
thf(fact_952_empty__is__image,axiom,
    ! [F: nat > a,A: set_nat] :
      ( ( bot_bot_set_a
        = ( image_nat_a @ F @ A ) )
      = ( A = bot_bot_set_nat ) ) ).

% empty_is_image
thf(fact_953_empty__is__image,axiom,
    ! [F: a > nat,A: set_a] :
      ( ( bot_bot_set_nat
        = ( image_a_nat @ F @ A ) )
      = ( A = bot_bot_set_a ) ) ).

% empty_is_image
thf(fact_954_empty__is__image,axiom,
    ! [F: nat > nat,A: set_nat] :
      ( ( bot_bot_set_nat
        = ( image_nat_nat @ F @ A ) )
      = ( A = bot_bot_set_nat ) ) ).

% empty_is_image
thf(fact_955_image__is__empty,axiom,
    ! [F: a > set_a,A: set_a] :
      ( ( ( image_a_set_a @ F @ A )
        = bot_bot_set_set_a )
      = ( A = bot_bot_set_a ) ) ).

% image_is_empty
thf(fact_956_image__is__empty,axiom,
    ! [F: a > a,A: set_a] :
      ( ( ( image_a_a @ F @ A )
        = bot_bot_set_a )
      = ( A = bot_bot_set_a ) ) ).

% image_is_empty
thf(fact_957_image__is__empty,axiom,
    ! [F: nat > a,A: set_nat] :
      ( ( ( image_nat_a @ F @ A )
        = bot_bot_set_a )
      = ( A = bot_bot_set_nat ) ) ).

% image_is_empty
thf(fact_958_image__is__empty,axiom,
    ! [F: a > nat,A: set_a] :
      ( ( ( image_a_nat @ F @ A )
        = bot_bot_set_nat )
      = ( A = bot_bot_set_a ) ) ).

% image_is_empty
thf(fact_959_image__is__empty,axiom,
    ! [F: nat > nat,A: set_nat] :
      ( ( ( image_nat_nat @ F @ A )
        = bot_bot_set_nat )
      = ( A = bot_bot_set_nat ) ) ).

% image_is_empty
thf(fact_960_finite__imageI,axiom,
    ! [F2: set_a,H2: a > set_a] :
      ( ( finite_finite_a @ F2 )
     => ( finite_finite_set_a @ ( image_a_set_a @ H2 @ F2 ) ) ) ).

% finite_imageI
thf(fact_961_finite__imageI,axiom,
    ! [F2: set_a,H2: a > a] :
      ( ( finite_finite_a @ F2 )
     => ( finite_finite_a @ ( image_a_a @ H2 @ F2 ) ) ) ).

% finite_imageI
thf(fact_962_finite__imageI,axiom,
    ! [F2: set_a,H2: a > nat] :
      ( ( finite_finite_a @ F2 )
     => ( finite_finite_nat @ ( image_a_nat @ H2 @ F2 ) ) ) ).

% finite_imageI
thf(fact_963_finite__imageI,axiom,
    ! [F2: set_a,H2: a > product_prod_a_a] :
      ( ( finite_finite_a @ F2 )
     => ( finite6544458595007987280od_a_a @ ( image_7400625782589995694od_a_a @ H2 @ F2 ) ) ) ).

% finite_imageI
thf(fact_964_finite__imageI,axiom,
    ! [F2: set_nat,H2: nat > a] :
      ( ( finite_finite_nat @ F2 )
     => ( finite_finite_a @ ( image_nat_a @ H2 @ F2 ) ) ) ).

% finite_imageI
thf(fact_965_finite__imageI,axiom,
    ! [F2: set_nat,H2: nat > nat] :
      ( ( finite_finite_nat @ F2 )
     => ( finite_finite_nat @ ( image_nat_nat @ H2 @ F2 ) ) ) ).

% finite_imageI
thf(fact_966_finite__imageI,axiom,
    ! [F2: set_nat,H2: nat > product_prod_a_a] :
      ( ( finite_finite_nat @ F2 )
     => ( finite6544458595007987280od_a_a @ ( image_372941888232738320od_a_a @ H2 @ F2 ) ) ) ).

% finite_imageI
thf(fact_967_finite__imageI,axiom,
    ! [F2: set_Product_prod_a_a,H2: product_prod_a_a > a] :
      ( ( finite6544458595007987280od_a_a @ F2 )
     => ( finite_finite_a @ ( image_3437945252899457948_a_a_a @ H2 @ F2 ) ) ) ).

% finite_imageI
thf(fact_968_finite__imageI,axiom,
    ! [F2: set_Product_prod_a_a,H2: product_prod_a_a > nat] :
      ( ( finite6544458595007987280od_a_a @ F2 )
     => ( finite_finite_nat @ ( image_9053670898913107890_a_nat @ H2 @ F2 ) ) ) ).

% finite_imageI
thf(fact_969_finite__imageI,axiom,
    ! [F2: set_Product_prod_a_a,H2: product_prod_a_a > product_prod_a_a] :
      ( ( finite6544458595007987280od_a_a @ F2 )
     => ( finite6544458595007987280od_a_a @ ( image_4636654165204879301od_a_a @ H2 @ F2 ) ) ) ).

% finite_imageI
thf(fact_970_empty__subsetI,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).

% empty_subsetI
thf(fact_971_empty__subsetI,axiom,
    ! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).

% empty_subsetI
thf(fact_972_subset__empty,axiom,
    ! [A: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
      = ( A = bot_bot_set_nat ) ) ).

% subset_empty
thf(fact_973_subset__empty,axiom,
    ! [A: set_a] :
      ( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
      = ( A = bot_bot_set_a ) ) ).

% subset_empty
thf(fact_974_image__insert,axiom,
    ! [F: a > set_a,A2: a,B: set_a] :
      ( ( image_a_set_a @ F @ ( insert_a @ A2 @ B ) )
      = ( insert_set_a @ ( F @ A2 ) @ ( image_a_set_a @ F @ B ) ) ) ).

% image_insert
thf(fact_975_image__insert,axiom,
    ! [F: a > a,A2: a,B: set_a] :
      ( ( image_a_a @ F @ ( insert_a @ A2 @ B ) )
      = ( insert_a @ ( F @ A2 ) @ ( image_a_a @ F @ B ) ) ) ).

% image_insert
thf(fact_976_insert__image,axiom,
    ! [X: a,A: set_a,F: a > set_a] :
      ( ( member_a @ X @ A )
     => ( ( insert_set_a @ ( F @ X ) @ ( image_a_set_a @ F @ A ) )
        = ( image_a_set_a @ F @ A ) ) ) ).

% insert_image
thf(fact_977_insert__image,axiom,
    ! [X: a,A: set_a,F: a > a] :
      ( ( member_a @ X @ A )
     => ( ( insert_a @ ( F @ X ) @ ( image_a_a @ F @ A ) )
        = ( image_a_a @ F @ A ) ) ) ).

% insert_image
thf(fact_978_insert__image,axiom,
    ! [X: product_prod_a_a > product_prod_a_a,A: set_Pr8826267807999420763od_a_a,F: ( product_prod_a_a > product_prod_a_a ) > a] :
      ( ( member4020126937092221116od_a_a @ X @ A )
     => ( ( insert_a @ ( F @ X ) @ ( image_6673660663317998546_a_a_a @ F @ A ) )
        = ( image_6673660663317998546_a_a_a @ F @ A ) ) ) ).

% insert_image
thf(fact_979_insert__image,axiom,
    ! [X: nat,A: set_nat,F: nat > a] :
      ( ( member_nat @ X @ A )
     => ( ( insert_a @ ( F @ X ) @ ( image_nat_a @ F @ A ) )
        = ( image_nat_a @ F @ A ) ) ) ).

% insert_image
thf(fact_980_insert__image,axiom,
    ! [X: nat > nat,A: set_nat_nat,F: ( nat > nat ) > a] :
      ( ( member_nat_nat @ X @ A )
     => ( ( insert_a @ ( F @ X ) @ ( image_nat_nat_a @ F @ A ) )
        = ( image_nat_nat_a @ F @ A ) ) ) ).

% insert_image
thf(fact_981_insert__image,axiom,
    ! [X: nat > a,A: set_nat_a,F: ( nat > a ) > a] :
      ( ( member_nat_a @ X @ A )
     => ( ( insert_a @ ( F @ X ) @ ( image_nat_a_a @ F @ A ) )
        = ( image_nat_a_a @ F @ A ) ) ) ).

% insert_image
thf(fact_982_insert__image,axiom,
    ! [X: a > product_prod_a_a,A: set_a_6829686330177631172od_a_a,F: ( a > product_prod_a_a ) > a] :
      ( ( member1957775702407316389od_a_a @ X @ A )
     => ( ( insert_a @ ( F @ X ) @ ( image_744702154072782377_a_a_a @ F @ A ) )
        = ( image_744702154072782377_a_a_a @ F @ A ) ) ) ).

% insert_image
thf(fact_983_insert__image,axiom,
    ! [X: a > nat,A: set_a_nat,F: ( a > nat ) > a] :
      ( ( member_a_nat @ X @ A )
     => ( ( insert_a @ ( F @ X ) @ ( image_a_nat_a @ F @ A ) )
        = ( image_a_nat_a @ F @ A ) ) ) ).

% insert_image
thf(fact_984_insert__image,axiom,
    ! [X: a > a,A: set_a_a,F: ( a > a ) > a] :
      ( ( member_a_a @ X @ A )
     => ( ( insert_a @ ( F @ X ) @ ( image_a_a_a @ F @ A ) )
        = ( image_a_a_a @ F @ A ) ) ) ).

% insert_image
thf(fact_985_singletonI,axiom,
    ! [A2: product_prod_a_a > product_prod_a_a] : ( member4020126937092221116od_a_a @ A2 @ ( insert1583624352380834389od_a_a @ A2 @ bot_bo2841618473486996463od_a_a ) ) ).

% singletonI
thf(fact_986_singletonI,axiom,
    ! [A2: nat > nat] : ( member_nat_nat @ A2 @ ( insert_nat_nat @ A2 @ bot_bot_set_nat_nat ) ) ).

% singletonI
thf(fact_987_singletonI,axiom,
    ! [A2: nat > a] : ( member_nat_a @ A2 @ ( insert_nat_a @ A2 @ bot_bot_set_nat_a ) ) ).

% singletonI
thf(fact_988_singletonI,axiom,
    ! [A2: a > product_prod_a_a] : ( member1957775702407316389od_a_a @ A2 @ ( insert5082226857754029630od_a_a @ A2 @ bot_bo6605490641894888024od_a_a ) ) ).

% singletonI
thf(fact_989_singletonI,axiom,
    ! [A2: a > nat] : ( member_a_nat @ A2 @ ( insert_a_nat @ A2 @ bot_bot_set_a_nat ) ) ).

% singletonI
thf(fact_990_singletonI,axiom,
    ! [A2: a > a] : ( member_a_a @ A2 @ ( insert_a_a @ A2 @ bot_bot_set_a_a ) ) ).

% singletonI
thf(fact_991_singletonI,axiom,
    ! [A2: a] : ( member_a @ A2 @ ( insert_a @ A2 @ bot_bot_set_a ) ) ).

% singletonI
thf(fact_992_singletonI,axiom,
    ! [A2: nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).

% singletonI
thf(fact_993_insert__subset,axiom,
    ! [X: product_prod_a_a > product_prod_a_a,A: set_Pr8826267807999420763od_a_a,B: set_Pr8826267807999420763od_a_a] :
      ( ( ord_le741722312431004091od_a_a @ ( insert1583624352380834389od_a_a @ X @ A ) @ B )
      = ( ( member4020126937092221116od_a_a @ X @ B )
        & ( ord_le741722312431004091od_a_a @ A @ B ) ) ) ).

% insert_subset
thf(fact_994_insert__subset,axiom,
    ! [X: nat,A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ ( insert_nat @ X @ A ) @ B )
      = ( ( member_nat @ X @ B )
        & ( ord_less_eq_set_nat @ A @ B ) ) ) ).

% insert_subset
thf(fact_995_insert__subset,axiom,
    ! [X: nat > nat,A: set_nat_nat,B: set_nat_nat] :
      ( ( ord_le9059583361652607317at_nat @ ( insert_nat_nat @ X @ A ) @ B )
      = ( ( member_nat_nat @ X @ B )
        & ( ord_le9059583361652607317at_nat @ A @ B ) ) ) ).

% insert_subset
thf(fact_996_insert__subset,axiom,
    ! [X: nat > a,A: set_nat_a,B: set_nat_a] :
      ( ( ord_le871467723717165285_nat_a @ ( insert_nat_a @ X @ A ) @ B )
      = ( ( member_nat_a @ X @ B )
        & ( ord_le871467723717165285_nat_a @ A @ B ) ) ) ).

% insert_subset
thf(fact_997_insert__subset,axiom,
    ! [X: a > product_prod_a_a,A: set_a_6829686330177631172od_a_a,B: set_a_6829686330177631172od_a_a] :
      ( ( ord_le2371270124811097124od_a_a @ ( insert5082226857754029630od_a_a @ X @ A ) @ B )
      = ( ( member1957775702407316389od_a_a @ X @ B )
        & ( ord_le2371270124811097124od_a_a @ A @ B ) ) ) ).

% insert_subset
thf(fact_998_insert__subset,axiom,
    ! [X: a > nat,A: set_a_nat,B: set_a_nat] :
      ( ( ord_le1612561287239139007_a_nat @ ( insert_a_nat @ X @ A ) @ B )
      = ( ( member_a_nat @ X @ B )
        & ( ord_le1612561287239139007_a_nat @ A @ B ) ) ) ).

% insert_subset
thf(fact_999_insert__subset,axiom,
    ! [X: a > a,A: set_a_a,B: set_a_a] :
      ( ( ord_less_eq_set_a_a @ ( insert_a_a @ X @ A ) @ B )
      = ( ( member_a_a @ X @ B )
        & ( ord_less_eq_set_a_a @ A @ B ) ) ) ).

% insert_subset
thf(fact_1000_insert__subset,axiom,
    ! [X: a,A: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ ( insert_a @ X @ A ) @ B )
      = ( ( member_a @ X @ B )
        & ( ord_less_eq_set_a @ A @ B ) ) ) ).

% insert_subset
thf(fact_1001_Int__subset__iff,axiom,
    ! [C: set_a,A: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ C @ ( inf_inf_set_a @ A @ B ) )
      = ( ( ord_less_eq_set_a @ C @ A )
        & ( ord_less_eq_set_a @ C @ B ) ) ) ).

% Int_subset_iff
thf(fact_1002_Int__insert__right__if1,axiom,
    ! [A2: product_prod_a_a > product_prod_a_a,A: set_Pr8826267807999420763od_a_a,B: set_Pr8826267807999420763od_a_a] :
      ( ( member4020126937092221116od_a_a @ A2 @ A )
     => ( ( inf_in3507259597588448237od_a_a @ A @ ( insert1583624352380834389od_a_a @ A2 @ B ) )
        = ( insert1583624352380834389od_a_a @ A2 @ ( inf_in3507259597588448237od_a_a @ A @ B ) ) ) ) ).

% Int_insert_right_if1
thf(fact_1003_Int__insert__right__if1,axiom,
    ! [A2: nat,A: set_nat,B: set_nat] :
      ( ( member_nat @ A2 @ A )
     => ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B ) )
        = ( insert_nat @ A2 @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).

% Int_insert_right_if1
thf(fact_1004_Int__insert__right__if1,axiom,
    ! [A2: nat > nat,A: set_nat_nat,B: set_nat_nat] :
      ( ( member_nat_nat @ A2 @ A )
     => ( ( inf_inf_set_nat_nat @ A @ ( insert_nat_nat @ A2 @ B ) )
        = ( insert_nat_nat @ A2 @ ( inf_inf_set_nat_nat @ A @ B ) ) ) ) ).

% Int_insert_right_if1
thf(fact_1005_Int__insert__right__if1,axiom,
    ! [A2: nat > a,A: set_nat_a,B: set_nat_a] :
      ( ( member_nat_a @ A2 @ A )
     => ( ( inf_inf_set_nat_a @ A @ ( insert_nat_a @ A2 @ B ) )
        = ( insert_nat_a @ A2 @ ( inf_inf_set_nat_a @ A @ B ) ) ) ) ).

% Int_insert_right_if1
thf(fact_1006_Int__insert__right__if1,axiom,
    ! [A2: a > product_prod_a_a,A: set_a_6829686330177631172od_a_a,B: set_a_6829686330177631172od_a_a] :
      ( ( member1957775702407316389od_a_a @ A2 @ A )
     => ( ( inf_in4434067354547777622od_a_a @ A @ ( insert5082226857754029630od_a_a @ A2 @ B ) )
        = ( insert5082226857754029630od_a_a @ A2 @ ( inf_in4434067354547777622od_a_a @ A @ B ) ) ) ) ).

% Int_insert_right_if1
thf(fact_1007_Int__insert__right__if1,axiom,
    ! [A2: a > nat,A: set_a_nat,B: set_a_nat] :
      ( ( member_a_nat @ A2 @ A )
     => ( ( inf_inf_set_a_nat @ A @ ( insert_a_nat @ A2 @ B ) )
        = ( insert_a_nat @ A2 @ ( inf_inf_set_a_nat @ A @ B ) ) ) ) ).

% Int_insert_right_if1
thf(fact_1008_Int__insert__right__if1,axiom,
    ! [A2: a > a,A: set_a_a,B: set_a_a] :
      ( ( member_a_a @ A2 @ A )
     => ( ( inf_inf_set_a_a @ A @ ( insert_a_a @ A2 @ B ) )
        = ( insert_a_a @ A2 @ ( inf_inf_set_a_a @ A @ B ) ) ) ) ).

% Int_insert_right_if1
thf(fact_1009_Int__insert__right__if1,axiom,
    ! [A2: a,A: set_a,B: set_a] :
      ( ( member_a @ A2 @ A )
     => ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
        = ( insert_a @ A2 @ ( inf_inf_set_a @ A @ B ) ) ) ) ).

% Int_insert_right_if1
thf(fact_1010_Int__insert__right__if0,axiom,
    ! [A2: product_prod_a_a > product_prod_a_a,A: set_Pr8826267807999420763od_a_a,B: set_Pr8826267807999420763od_a_a] :
      ( ~ ( member4020126937092221116od_a_a @ A2 @ A )
     => ( ( inf_in3507259597588448237od_a_a @ A @ ( insert1583624352380834389od_a_a @ A2 @ B ) )
        = ( inf_in3507259597588448237od_a_a @ A @ B ) ) ) ).

% Int_insert_right_if0
thf(fact_1011_Int__insert__right__if0,axiom,
    ! [A2: nat,A: set_nat,B: set_nat] :
      ( ~ ( member_nat @ A2 @ A )
     => ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B ) )
        = ( inf_inf_set_nat @ A @ B ) ) ) ).

% Int_insert_right_if0
thf(fact_1012_Int__insert__right__if0,axiom,
    ! [A2: nat > nat,A: set_nat_nat,B: set_nat_nat] :
      ( ~ ( member_nat_nat @ A2 @ A )
     => ( ( inf_inf_set_nat_nat @ A @ ( insert_nat_nat @ A2 @ B ) )
        = ( inf_inf_set_nat_nat @ A @ B ) ) ) ).

% Int_insert_right_if0
thf(fact_1013_Int__insert__right__if0,axiom,
    ! [A2: nat > a,A: set_nat_a,B: set_nat_a] :
      ( ~ ( member_nat_a @ A2 @ A )
     => ( ( inf_inf_set_nat_a @ A @ ( insert_nat_a @ A2 @ B ) )
        = ( inf_inf_set_nat_a @ A @ B ) ) ) ).

% Int_insert_right_if0
thf(fact_1014_Int__insert__right__if0,axiom,
    ! [A2: a > product_prod_a_a,A: set_a_6829686330177631172od_a_a,B: set_a_6829686330177631172od_a_a] :
      ( ~ ( member1957775702407316389od_a_a @ A2 @ A )
     => ( ( inf_in4434067354547777622od_a_a @ A @ ( insert5082226857754029630od_a_a @ A2 @ B ) )
        = ( inf_in4434067354547777622od_a_a @ A @ B ) ) ) ).

% Int_insert_right_if0
thf(fact_1015_Int__insert__right__if0,axiom,
    ! [A2: a > nat,A: set_a_nat,B: set_a_nat] :
      ( ~ ( member_a_nat @ A2 @ A )
     => ( ( inf_inf_set_a_nat @ A @ ( insert_a_nat @ A2 @ B ) )
        = ( inf_inf_set_a_nat @ A @ B ) ) ) ).

% Int_insert_right_if0
thf(fact_1016_Int__insert__right__if0,axiom,
    ! [A2: a > a,A: set_a_a,B: set_a_a] :
      ( ~ ( member_a_a @ A2 @ A )
     => ( ( inf_inf_set_a_a @ A @ ( insert_a_a @ A2 @ B ) )
        = ( inf_inf_set_a_a @ A @ B ) ) ) ).

% Int_insert_right_if0
thf(fact_1017_Int__insert__right__if0,axiom,
    ! [A2: a,A: set_a,B: set_a] :
      ( ~ ( member_a @ A2 @ A )
     => ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
        = ( inf_inf_set_a @ A @ B ) ) ) ).

% Int_insert_right_if0
thf(fact_1018_insert__inter__insert,axiom,
    ! [A2: a,A: set_a,B: set_a] :
      ( ( inf_inf_set_a @ ( insert_a @ A2 @ A ) @ ( insert_a @ A2 @ B ) )
      = ( insert_a @ A2 @ ( inf_inf_set_a @ A @ B ) ) ) ).

% insert_inter_insert
thf(fact_1019_Int__insert__left__if1,axiom,
    ! [A2: product_prod_a_a > product_prod_a_a,C: set_Pr8826267807999420763od_a_a,B: set_Pr8826267807999420763od_a_a] :
      ( ( member4020126937092221116od_a_a @ A2 @ C )
     => ( ( inf_in3507259597588448237od_a_a @ ( insert1583624352380834389od_a_a @ A2 @ B ) @ C )
        = ( insert1583624352380834389od_a_a @ A2 @ ( inf_in3507259597588448237od_a_a @ B @ C ) ) ) ) ).

% Int_insert_left_if1
thf(fact_1020_Int__insert__left__if1,axiom,
    ! [A2: nat,C: set_nat,B: set_nat] :
      ( ( member_nat @ A2 @ C )
     => ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B ) @ C )
        = ( insert_nat @ A2 @ ( inf_inf_set_nat @ B @ C ) ) ) ) ).

% Int_insert_left_if1
thf(fact_1021_Int__insert__left__if1,axiom,
    ! [A2: nat > nat,C: set_nat_nat,B: set_nat_nat] :
      ( ( member_nat_nat @ A2 @ C )
     => ( ( inf_inf_set_nat_nat @ ( insert_nat_nat @ A2 @ B ) @ C )
        = ( insert_nat_nat @ A2 @ ( inf_inf_set_nat_nat @ B @ C ) ) ) ) ).

% Int_insert_left_if1
thf(fact_1022_Int__insert__left__if1,axiom,
    ! [A2: nat > a,C: set_nat_a,B: set_nat_a] :
      ( ( member_nat_a @ A2 @ C )
     => ( ( inf_inf_set_nat_a @ ( insert_nat_a @ A2 @ B ) @ C )
        = ( insert_nat_a @ A2 @ ( inf_inf_set_nat_a @ B @ C ) ) ) ) ).

% Int_insert_left_if1
thf(fact_1023_Int__insert__left__if1,axiom,
    ! [A2: a > product_prod_a_a,C: set_a_6829686330177631172od_a_a,B: set_a_6829686330177631172od_a_a] :
      ( ( member1957775702407316389od_a_a @ A2 @ C )
     => ( ( inf_in4434067354547777622od_a_a @ ( insert5082226857754029630od_a_a @ A2 @ B ) @ C )
        = ( insert5082226857754029630od_a_a @ A2 @ ( inf_in4434067354547777622od_a_a @ B @ C ) ) ) ) ).

% Int_insert_left_if1
thf(fact_1024_Int__insert__left__if1,axiom,
    ! [A2: a > nat,C: set_a_nat,B: set_a_nat] :
      ( ( member_a_nat @ A2 @ C )
     => ( ( inf_inf_set_a_nat @ ( insert_a_nat @ A2 @ B ) @ C )
        = ( insert_a_nat @ A2 @ ( inf_inf_set_a_nat @ B @ C ) ) ) ) ).

% Int_insert_left_if1
thf(fact_1025_Int__insert__left__if1,axiom,
    ! [A2: a > a,C: set_a_a,B: set_a_a] :
      ( ( member_a_a @ A2 @ C )
     => ( ( inf_inf_set_a_a @ ( insert_a_a @ A2 @ B ) @ C )
        = ( insert_a_a @ A2 @ ( inf_inf_set_a_a @ B @ C ) ) ) ) ).

% Int_insert_left_if1
thf(fact_1026_Int__insert__left__if1,axiom,
    ! [A2: a,C: set_a,B: set_a] :
      ( ( member_a @ A2 @ C )
     => ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C )
        = ( insert_a @ A2 @ ( inf_inf_set_a @ B @ C ) ) ) ) ).

% Int_insert_left_if1
thf(fact_1027_Int__insert__left__if0,axiom,
    ! [A2: product_prod_a_a > product_prod_a_a,C: set_Pr8826267807999420763od_a_a,B: set_Pr8826267807999420763od_a_a] :
      ( ~ ( member4020126937092221116od_a_a @ A2 @ C )
     => ( ( inf_in3507259597588448237od_a_a @ ( insert1583624352380834389od_a_a @ A2 @ B ) @ C )
        = ( inf_in3507259597588448237od_a_a @ B @ C ) ) ) ).

% Int_insert_left_if0
thf(fact_1028_Int__insert__left__if0,axiom,
    ! [A2: nat,C: set_nat,B: set_nat] :
      ( ~ ( member_nat @ A2 @ C )
     => ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B ) @ C )
        = ( inf_inf_set_nat @ B @ C ) ) ) ).

% Int_insert_left_if0
thf(fact_1029_Int__insert__left__if0,axiom,
    ! [A2: nat > nat,C: set_nat_nat,B: set_nat_nat] :
      ( ~ ( member_nat_nat @ A2 @ C )
     => ( ( inf_inf_set_nat_nat @ ( insert_nat_nat @ A2 @ B ) @ C )
        = ( inf_inf_set_nat_nat @ B @ C ) ) ) ).

% Int_insert_left_if0
thf(fact_1030_Int__insert__left__if0,axiom,
    ! [A2: nat > a,C: set_nat_a,B: set_nat_a] :
      ( ~ ( member_nat_a @ A2 @ C )
     => ( ( inf_inf_set_nat_a @ ( insert_nat_a @ A2 @ B ) @ C )
        = ( inf_inf_set_nat_a @ B @ C ) ) ) ).

% Int_insert_left_if0
thf(fact_1031_Int__insert__left__if0,axiom,
    ! [A2: a > product_prod_a_a,C: set_a_6829686330177631172od_a_a,B: set_a_6829686330177631172od_a_a] :
      ( ~ ( member1957775702407316389od_a_a @ A2 @ C )
     => ( ( inf_in4434067354547777622od_a_a @ ( insert5082226857754029630od_a_a @ A2 @ B ) @ C )
        = ( inf_in4434067354547777622od_a_a @ B @ C ) ) ) ).

% Int_insert_left_if0
thf(fact_1032_Int__insert__left__if0,axiom,
    ! [A2: a > nat,C: set_a_nat,B: set_a_nat] :
      ( ~ ( member_a_nat @ A2 @ C )
     => ( ( inf_inf_set_a_nat @ ( insert_a_nat @ A2 @ B ) @ C )
        = ( inf_inf_set_a_nat @ B @ C ) ) ) ).

% Int_insert_left_if0
thf(fact_1033_Int__insert__left__if0,axiom,
    ! [A2: a > a,C: set_a_a,B: set_a_a] :
      ( ~ ( member_a_a @ A2 @ C )
     => ( ( inf_inf_set_a_a @ ( insert_a_a @ A2 @ B ) @ C )
        = ( inf_inf_set_a_a @ B @ C ) ) ) ).

% Int_insert_left_if0
thf(fact_1034_Int__insert__left__if0,axiom,
    ! [A2: a,C: set_a,B: set_a] :
      ( ~ ( member_a @ A2 @ C )
     => ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C )
        = ( inf_inf_set_a @ B @ C ) ) ) ).

% Int_insert_left_if0
thf(fact_1035_inverse__subgroupI,axiom,
    ! [H3: set_a] :
      ( ( group_subgroup_a @ H3 @ g @ addition @ zero )
     => ( group_subgroup_a @ ( image_a_a @ ( group_inverse_a @ g @ addition @ zero ) @ H3 ) @ g @ addition @ zero ) ) ).

% inverse_subgroupI
thf(fact_1036_singleton__conv2,axiom,
    ! [A2: a] :
      ( ( collect_a
        @ ( ^ [Y6: a,Z2: a] : ( Y6 = Z2 )
          @ A2 ) )
      = ( insert_a @ A2 @ bot_bot_set_a ) ) ).

% singleton_conv2
thf(fact_1037_singleton__conv2,axiom,
    ! [A2: nat] :
      ( ( collect_nat
        @ ( ^ [Y6: nat,Z2: nat] : ( Y6 = Z2 )
          @ A2 ) )
      = ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).

% singleton_conv2
thf(fact_1038_singleton__conv,axiom,
    ! [A2: a] :
      ( ( collect_a
        @ ^ [X2: a] : ( X2 = A2 ) )
      = ( insert_a @ A2 @ bot_bot_set_a ) ) ).

% singleton_conv
thf(fact_1039_singleton__conv,axiom,
    ! [A2: nat] :
      ( ( collect_nat
        @ ^ [X2: nat] : ( X2 = A2 ) )
      = ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).

% singleton_conv
thf(fact_1040_minusset__eq,axiom,
    ! [A: set_a] :
      ( ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ A )
      = ( image_a_a @ ( group_inverse_a @ g @ addition @ zero ) @ ( inf_inf_set_a @ A @ g ) ) ) ).

% minusset_eq
thf(fact_1041_finite__Collect__less__nat,axiom,
    ! [K: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N: nat] : ( ord_less_nat @ N @ K ) ) ) ).

% finite_Collect_less_nat
thf(fact_1042_subset__image__iff,axiom,
    ! [B: set_set_a,F: a > set_a,A: set_a] :
      ( ( ord_le3724670747650509150_set_a @ B @ ( image_a_set_a @ F @ A ) )
      = ( ? [AA: set_a] :
            ( ( ord_less_eq_set_a @ AA @ A )
            & ( B
              = ( image_a_set_a @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_1043_subset__image__iff,axiom,
    ! [B: set_a,F: a > a,A: set_a] :
      ( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A ) )
      = ( ? [AA: set_a] :
            ( ( ord_less_eq_set_a @ AA @ A )
            & ( B
              = ( image_a_a @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_1044_image__subset__iff,axiom,
    ! [F: a > set_a,A: set_a,B: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ A ) @ B )
      = ( ! [X2: a] :
            ( ( member_a @ X2 @ A )
           => ( member_set_a @ ( F @ X2 ) @ B ) ) ) ) ).

% image_subset_iff
thf(fact_1045_image__subset__iff,axiom,
    ! [F: a > a,A: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ B )
      = ( ! [X2: a] :
            ( ( member_a @ X2 @ A )
           => ( member_a @ ( F @ X2 ) @ B ) ) ) ) ).

% image_subset_iff
thf(fact_1046_subset__imageE,axiom,
    ! [B: set_set_a,F: a > set_a,A: set_a] :
      ( ( ord_le3724670747650509150_set_a @ B @ ( image_a_set_a @ F @ A ) )
     => ~ ! [C3: set_a] :
            ( ( ord_less_eq_set_a @ C3 @ A )
           => ( B
             != ( image_a_set_a @ F @ C3 ) ) ) ) ).

% subset_imageE
thf(fact_1047_subset__imageE,axiom,
    ! [B: set_a,F: a > a,A: set_a] :
      ( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A ) )
     => ~ ! [C3: set_a] :
            ( ( ord_less_eq_set_a @ C3 @ A )
           => ( B
             != ( image_a_a @ F @ C3 ) ) ) ) ).

% subset_imageE
thf(fact_1048_image__subsetI,axiom,
    ! [A: set_a,F: a > nat,B: set_nat] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ A )
         => ( member_nat @ ( F @ X3 ) @ B ) )
     => ( ord_less_eq_set_nat @ ( image_a_nat @ F @ A ) @ B ) ) ).

% image_subsetI
thf(fact_1049_image__subsetI,axiom,
    ! [A: set_nat,F: nat > nat,B: set_nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A )
         => ( member_nat @ ( F @ X3 ) @ B ) )
     => ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B ) ) ).

% image_subsetI
thf(fact_1050_image__subsetI,axiom,
    ! [A: set_a,F: a > a,B: set_a] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ A )
         => ( member_a @ ( F @ X3 ) @ B ) )
     => ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ B ) ) ).

% image_subsetI
thf(fact_1051_image__subsetI,axiom,
    ! [A: set_nat,F: nat > a,B: set_a] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A )
         => ( member_a @ ( F @ X3 ) @ B ) )
     => ( ord_less_eq_set_a @ ( image_nat_a @ F @ A ) @ B ) ) ).

% image_subsetI
thf(fact_1052_image__subsetI,axiom,
    ! [A: set_a,F: a > set_a,B: set_set_a] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ A )
         => ( member_set_a @ ( F @ X3 ) @ B ) )
     => ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ A ) @ B ) ) ).

% image_subsetI
thf(fact_1053_image__subsetI,axiom,
    ! [A: set_a,F: a > nat > nat,B: set_nat_nat] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ A )
         => ( member_nat_nat @ ( F @ X3 ) @ B ) )
     => ( ord_le9059583361652607317at_nat @ ( image_a_nat_nat @ F @ A ) @ B ) ) ).

% image_subsetI
thf(fact_1054_image__subsetI,axiom,
    ! [A: set_a,F: a > nat > a,B: set_nat_a] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ A )
         => ( member_nat_a @ ( F @ X3 ) @ B ) )
     => ( ord_le871467723717165285_nat_a @ ( image_a_nat_a2 @ F @ A ) @ B ) ) ).

% image_subsetI
thf(fact_1055_image__subsetI,axiom,
    ! [A: set_a,F: a > a > nat,B: set_a_nat] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ A )
         => ( member_a_nat @ ( F @ X3 ) @ B ) )
     => ( ord_le1612561287239139007_a_nat @ ( image_a_a_nat @ F @ A ) @ B ) ) ).

% image_subsetI
thf(fact_1056_image__subsetI,axiom,
    ! [A: set_a,F: a > a > a,B: set_a_a] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ A )
         => ( member_a_a @ ( F @ X3 ) @ B ) )
     => ( ord_less_eq_set_a_a @ ( image_a_a_a2 @ F @ A ) @ B ) ) ).

% image_subsetI
thf(fact_1057_image__subsetI,axiom,
    ! [A: set_nat,F: nat > nat > nat,B: set_nat_nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A )
         => ( member_nat_nat @ ( F @ X3 ) @ B ) )
     => ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F @ A ) @ B ) ) ).

% image_subsetI
thf(fact_1058_image__mono,axiom,
    ! [A: set_a,B: set_a,F: a > set_a] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ A ) @ ( image_a_set_a @ F @ B ) ) ) ).

% image_mono
thf(fact_1059_image__mono,axiom,
    ! [A: set_a,B: set_a,F: a > a] :
      ( ( ord_less_eq_set_a @ A @ B )
     => ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ ( image_a_a @ F @ B ) ) ) ).

% image_mono
thf(fact_1060_Compr__image__eq,axiom,
    ! [F: a > a,A: set_a,P: a > $o] :
      ( ( collect_a
        @ ^ [X2: a] :
            ( ( member_a @ X2 @ ( image_a_a @ F @ A ) )
            & ( P @ X2 ) ) )
      = ( image_a_a @ F
        @ ( collect_a
          @ ^ [X2: a] :
              ( ( member_a @ X2 @ A )
              & ( P @ ( F @ X2 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_1061_Compr__image__eq,axiom,
    ! [F: nat > a,A: set_nat,P: a > $o] :
      ( ( collect_a
        @ ^ [X2: a] :
            ( ( member_a @ X2 @ ( image_nat_a @ F @ A ) )
            & ( P @ X2 ) ) )
      = ( image_nat_a @ F
        @ ( collect_nat
          @ ^ [X2: nat] :
              ( ( member_nat @ X2 @ A )
              & ( P @ ( F @ X2 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_1062_Compr__image__eq,axiom,
    ! [F: a > nat,A: set_a,P: nat > $o] :
      ( ( collect_nat
        @ ^ [X2: nat] :
            ( ( member_nat @ X2 @ ( image_a_nat @ F @ A ) )
            & ( P @ X2 ) ) )
      = ( image_a_nat @ F
        @ ( collect_a
          @ ^ [X2: a] :
              ( ( member_a @ X2 @ A )
              & ( P @ ( F @ X2 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_1063_Compr__image__eq,axiom,
    ! [F: nat > nat,A: set_nat,P: nat > $o] :
      ( ( collect_nat
        @ ^ [X2: nat] :
            ( ( member_nat @ X2 @ ( image_nat_nat @ F @ A ) )
            & ( P @ X2 ) ) )
      = ( image_nat_nat @ F
        @ ( collect_nat
          @ ^ [X2: nat] :
              ( ( member_nat @ X2 @ A )
              & ( P @ ( F @ X2 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_1064_Compr__image__eq,axiom,
    ! [F: a > set_a,A: set_a,P: set_a > $o] :
      ( ( collect_set_a
        @ ^ [X2: set_a] :
            ( ( member_set_a @ X2 @ ( image_a_set_a @ F @ A ) )
            & ( P @ X2 ) ) )
      = ( image_a_set_a @ F
        @ ( collect_a
          @ ^ [X2: a] :
              ( ( member_a @ X2 @ A )
              & ( P @ ( F @ X2 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_1065_Compr__image__eq,axiom,
    ! [F: a > nat > nat,A: set_a,P: ( nat > nat ) > $o] :
      ( ( collect_nat_nat
        @ ^ [X2: nat > nat] :
            ( ( member_nat_nat @ X2 @ ( image_a_nat_nat @ F @ A ) )
            & ( P @ X2 ) ) )
      = ( image_a_nat_nat @ F
        @ ( collect_a
          @ ^ [X2: a] :
              ( ( member_a @ X2 @ A )
              & ( P @ ( F @ X2 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_1066_Compr__image__eq,axiom,
    ! [F: a > nat > a,A: set_a,P: ( nat > a ) > $o] :
      ( ( collect_nat_a
        @ ^ [X2: nat > a] :
            ( ( member_nat_a @ X2 @ ( image_a_nat_a2 @ F @ A ) )
            & ( P @ X2 ) ) )
      = ( image_a_nat_a2 @ F
        @ ( collect_a
          @ ^ [X2: a] :
              ( ( member_a @ X2 @ A )
              & ( P @ ( F @ X2 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_1067_Compr__image__eq,axiom,
    ! [F: a > a > nat,A: set_a,P: ( a > nat ) > $o] :
      ( ( collect_a_nat
        @ ^ [X2: a > nat] :
            ( ( member_a_nat @ X2 @ ( image_a_a_nat @ F @ A ) )
            & ( P @ X2 ) ) )
      = ( image_a_a_nat @ F
        @ ( collect_a
          @ ^ [X2: a] :
              ( ( member_a @ X2 @ A )
              & ( P @ ( F @ X2 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_1068_Compr__image__eq,axiom,
    ! [F: a > a > a,A: set_a,P: ( a > a ) > $o] :
      ( ( collect_a_a
        @ ^ [X2: a > a] :
            ( ( member_a_a @ X2 @ ( image_a_a_a2 @ F @ A ) )
            & ( P @ X2 ) ) )
      = ( image_a_a_a2 @ F
        @ ( collect_a
          @ ^ [X2: a] :
              ( ( member_a @ X2 @ A )
              & ( P @ ( F @ X2 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_1069_Compr__image__eq,axiom,
    ! [F: nat > nat > nat,A: set_nat,P: ( nat > nat ) > $o] :
      ( ( collect_nat_nat
        @ ^ [X2: nat > nat] :
            ( ( member_nat_nat @ X2 @ ( image_nat_nat_nat2 @ F @ A ) )
            & ( P @ X2 ) ) )
      = ( image_nat_nat_nat2 @ F
        @ ( collect_nat
          @ ^ [X2: nat] :
              ( ( member_nat @ X2 @ A )
              & ( P @ ( F @ X2 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_1070_rev__image__eqI,axiom,
    ! [X: a,A: set_a,B4: a,F: a > a] :
      ( ( member_a @ X @ A )
     => ( ( B4
          = ( F @ X ) )
       => ( member_a @ B4 @ ( image_a_a @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_1071_rev__image__eqI,axiom,
    ! [X: a,A: set_a,B4: nat,F: a > nat] :
      ( ( member_a @ X @ A )
     => ( ( B4
          = ( F @ X ) )
       => ( member_nat @ B4 @ ( image_a_nat @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_1072_rev__image__eqI,axiom,
    ! [X: nat,A: set_nat,B4: a,F: nat > a] :
      ( ( member_nat @ X @ A )
     => ( ( B4
          = ( F @ X ) )
       => ( member_a @ B4 @ ( image_nat_a @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_1073_rev__image__eqI,axiom,
    ! [X: nat,A: set_nat,B4: nat,F: nat > nat] :
      ( ( member_nat @ X @ A )
     => ( ( B4
          = ( F @ X ) )
       => ( member_nat @ B4 @ ( image_nat_nat @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_1074_rev__image__eqI,axiom,
    ! [X: a,A: set_a,B4: set_a,F: a > set_a] :
      ( ( member_a @ X @ A )
     => ( ( B4
          = ( F @ X ) )
       => ( member_set_a @ B4 @ ( image_a_set_a @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_1075_rev__image__eqI,axiom,
    ! [X: a,A: set_a,B4: nat > nat,F: a > nat > nat] :
      ( ( member_a @ X @ A )
     => ( ( B4
          = ( F @ X ) )
       => ( member_nat_nat @ B4 @ ( image_a_nat_nat @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_1076_rev__image__eqI,axiom,
    ! [X: a,A: set_a,B4: nat > a,F: a > nat > a] :
      ( ( member_a @ X @ A )
     => ( ( B4
          = ( F @ X ) )
       => ( member_nat_a @ B4 @ ( image_a_nat_a2 @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_1077_rev__image__eqI,axiom,
    ! [X: a,A: set_a,B4: a > nat,F: a > a > nat] :
      ( ( member_a @ X @ A )
     => ( ( B4
          = ( F @ X ) )
       => ( member_a_nat @ B4 @ ( image_a_a_nat @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_1078_rev__image__eqI,axiom,
    ! [X: a,A: set_a,B4: a > a,F: a > a > a] :
      ( ( member_a @ X @ A )
     => ( ( B4
          = ( F @ X ) )
       => ( member_a_a @ B4 @ ( image_a_a_a2 @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_1079_rev__image__eqI,axiom,
    ! [X: nat,A: set_nat,B4: nat > nat,F: nat > nat > nat] :
      ( ( member_nat @ X @ A )
     => ( ( B4
          = ( F @ X ) )
       => ( member_nat_nat @ B4 @ ( image_nat_nat_nat2 @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_1080_image__image,axiom,
    ! [F: set_a > a,G2: a > set_a,A: set_a] :
      ( ( image_set_a_a @ F @ ( image_a_set_a @ G2 @ A ) )
      = ( image_a_a
        @ ^ [X2: a] : ( F @ ( G2 @ X2 ) )
        @ A ) ) ).

% image_image
thf(fact_1081_image__image,axiom,
    ! [F: set_a > set_a,G2: a > set_a,A: set_a] :
      ( ( image_set_a_set_a @ F @ ( image_a_set_a @ G2 @ A ) )
      = ( image_a_set_a
        @ ^ [X2: a] : ( F @ ( G2 @ X2 ) )
        @ A ) ) ).

% image_image
thf(fact_1082_image__image,axiom,
    ! [F: a > a,G2: a > a,A: set_a] :
      ( ( image_a_a @ F @ ( image_a_a @ G2 @ A ) )
      = ( image_a_a
        @ ^ [X2: a] : ( F @ ( G2 @ X2 ) )
        @ A ) ) ).

% image_image
thf(fact_1083_image__image,axiom,
    ! [F: a > set_a,G2: a > a,A: set_a] :
      ( ( image_a_set_a @ F @ ( image_a_a @ G2 @ A ) )
      = ( image_a_set_a
        @ ^ [X2: a] : ( F @ ( G2 @ X2 ) )
        @ A ) ) ).

% image_image
thf(fact_1084_ball__imageD,axiom,
    ! [F: a > a,A: set_a,P: a > $o] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ ( image_a_a @ F @ A ) )
         => ( P @ X3 ) )
     => ! [X4: a] :
          ( ( member_a @ X4 @ A )
         => ( P @ ( F @ X4 ) ) ) ) ).

% ball_imageD
thf(fact_1085_ball__imageD,axiom,
    ! [F: a > set_a,A: set_a,P: set_a > $o] :
      ( ! [X3: set_a] :
          ( ( member_set_a @ X3 @ ( image_a_set_a @ F @ A ) )
         => ( P @ X3 ) )
     => ! [X4: a] :
          ( ( member_a @ X4 @ A )
         => ( P @ ( F @ X4 ) ) ) ) ).

% ball_imageD
thf(fact_1086_image__cong,axiom,
    ! [M: set_a,N3: set_a,F: a > a,G2: a > a] :
      ( ( M = N3 )
     => ( ! [X3: a] :
            ( ( member_a @ X3 @ N3 )
           => ( ( F @ X3 )
              = ( G2 @ X3 ) ) )
       => ( ( image_a_a @ F @ M )
          = ( image_a_a @ G2 @ N3 ) ) ) ) ).

% image_cong
thf(fact_1087_image__cong,axiom,
    ! [M: set_a,N3: set_a,F: a > set_a,G2: a > set_a] :
      ( ( M = N3 )
     => ( ! [X3: a] :
            ( ( member_a @ X3 @ N3 )
           => ( ( F @ X3 )
              = ( G2 @ X3 ) ) )
       => ( ( image_a_set_a @ F @ M )
          = ( image_a_set_a @ G2 @ N3 ) ) ) ) ).

% image_cong
thf(fact_1088_bex__imageD,axiom,
    ! [F: a > a,A: set_a,P: a > $o] :
      ( ? [X4: a] :
          ( ( member_a @ X4 @ ( image_a_a @ F @ A ) )
          & ( P @ X4 ) )
     => ? [X3: a] :
          ( ( member_a @ X3 @ A )
          & ( P @ ( F @ X3 ) ) ) ) ).

% bex_imageD
thf(fact_1089_bex__imageD,axiom,
    ! [F: a > set_a,A: set_a,P: set_a > $o] :
      ( ? [X4: set_a] :
          ( ( member_set_a @ X4 @ ( image_a_set_a @ F @ A ) )
          & ( P @ X4 ) )
     => ? [X3: a] :
          ( ( member_a @ X3 @ A )
          & ( P @ ( F @ X3 ) ) ) ) ).

% bex_imageD
thf(fact_1090_image__iff,axiom,
    ! [Z: set_a,F: a > set_a,A: set_a] :
      ( ( member_set_a @ Z @ ( image_a_set_a @ F @ A ) )
      = ( ? [X2: a] :
            ( ( member_a @ X2 @ A )
            & ( Z
              = ( F @ X2 ) ) ) ) ) ).

% image_iff
thf(fact_1091_image__iff,axiom,
    ! [Z: a,F: a > a,A: set_a] :
      ( ( member_a @ Z @ ( image_a_a @ F @ A ) )
      = ( ? [X2: a] :
            ( ( member_a @ X2 @ A )
            & ( Z
              = ( F @ X2 ) ) ) ) ) ).

% image_iff
thf(fact_1092_imageI,axiom,
    ! [X: a,A: set_a,F: a > a] :
      ( ( member_a @ X @ A )
     => ( member_a @ ( F @ X ) @ ( image_a_a @ F @ A ) ) ) ).

% imageI
thf(fact_1093_imageI,axiom,
    ! [X: a,A: set_a,F: a > nat] :
      ( ( member_a @ X @ A )
     => ( member_nat @ ( F @ X ) @ ( image_a_nat @ F @ A ) ) ) ).

% imageI
thf(fact_1094_imageI,axiom,
    ! [X: nat,A: set_nat,F: nat > a] :
      ( ( member_nat @ X @ A )
     => ( member_a @ ( F @ X ) @ ( image_nat_a @ F @ A ) ) ) ).

% imageI
thf(fact_1095_imageI,axiom,
    ! [X: nat,A: set_nat,F: nat > nat] :
      ( ( member_nat @ X @ A )
     => ( member_nat @ ( F @ X ) @ ( image_nat_nat @ F @ A ) ) ) ).

% imageI
thf(fact_1096_imageI,axiom,
    ! [X: a,A: set_a,F: a > set_a] :
      ( ( member_a @ X @ A )
     => ( member_set_a @ ( F @ X ) @ ( image_a_set_a @ F @ A ) ) ) ).

% imageI
thf(fact_1097_imageI,axiom,
    ! [X: a,A: set_a,F: a > nat > nat] :
      ( ( member_a @ X @ A )
     => ( member_nat_nat @ ( F @ X ) @ ( image_a_nat_nat @ F @ A ) ) ) ).

% imageI
thf(fact_1098_imageI,axiom,
    ! [X: a,A: set_a,F: a > nat > a] :
      ( ( member_a @ X @ A )
     => ( member_nat_a @ ( F @ X ) @ ( image_a_nat_a2 @ F @ A ) ) ) ).

% imageI
thf(fact_1099_imageI,axiom,
    ! [X: a,A: set_a,F: a > a > nat] :
      ( ( member_a @ X @ A )
     => ( member_a_nat @ ( F @ X ) @ ( image_a_a_nat @ F @ A ) ) ) ).

% imageI
thf(fact_1100_imageI,axiom,
    ! [X: a,A: set_a,F: a > a > a] :
      ( ( member_a @ X @ A )
     => ( member_a_a @ ( F @ X ) @ ( image_a_a_a2 @ F @ A ) ) ) ).

% imageI
thf(fact_1101_imageI,axiom,
    ! [X: nat,A: set_nat,F: nat > nat > nat] :
      ( ( member_nat @ X @ A )
     => ( member_nat_nat @ ( F @ X ) @ ( image_nat_nat_nat2 @ F @ A ) ) ) ).

% imageI
thf(fact_1102_imageE,axiom,
    ! [B4: a,F: a > a,A: set_a] :
      ( ( member_a @ B4 @ ( image_a_a @ F @ A ) )
     => ~ ! [X3: a] :
            ( ( B4
              = ( F @ X3 ) )
           => ~ ( member_a @ X3 @ A ) ) ) ).

% imageE
thf(fact_1103_imageE,axiom,
    ! [B4: a,F: nat > a,A: set_nat] :
      ( ( member_a @ B4 @ ( image_nat_a @ F @ A ) )
     => ~ ! [X3: nat] :
            ( ( B4
              = ( F @ X3 ) )
           => ~ ( member_nat @ X3 @ A ) ) ) ).

% imageE
thf(fact_1104_imageE,axiom,
    ! [B4: nat,F: a > nat,A: set_a] :
      ( ( member_nat @ B4 @ ( image_a_nat @ F @ A ) )
     => ~ ! [X3: a] :
            ( ( B4
              = ( F @ X3 ) )
           => ~ ( member_a @ X3 @ A ) ) ) ).

% imageE
thf(fact_1105_imageE,axiom,
    ! [B4: nat,F: nat > nat,A: set_nat] :
      ( ( member_nat @ B4 @ ( image_nat_nat @ F @ A ) )
     => ~ ! [X3: nat] :
            ( ( B4
              = ( F @ X3 ) )
           => ~ ( member_nat @ X3 @ A ) ) ) ).

% imageE
thf(fact_1106_imageE,axiom,
    ! [B4: set_a,F: a > set_a,A: set_a] :
      ( ( member_set_a @ B4 @ ( image_a_set_a @ F @ A ) )
     => ~ ! [X3: a] :
            ( ( B4
              = ( F @ X3 ) )
           => ~ ( member_a @ X3 @ A ) ) ) ).

% imageE
thf(fact_1107_imageE,axiom,
    ! [B4: a,F: ( nat > nat ) > a,A: set_nat_nat] :
      ( ( member_a @ B4 @ ( image_nat_nat_a @ F @ A ) )
     => ~ ! [X3: nat > nat] :
            ( ( B4
              = ( F @ X3 ) )
           => ~ ( member_nat_nat @ X3 @ A ) ) ) ).

% imageE
thf(fact_1108_imageE,axiom,
    ! [B4: a,F: ( nat > a ) > a,A: set_nat_a] :
      ( ( member_a @ B4 @ ( image_nat_a_a @ F @ A ) )
     => ~ ! [X3: nat > a] :
            ( ( B4
              = ( F @ X3 ) )
           => ~ ( member_nat_a @ X3 @ A ) ) ) ).

% imageE
thf(fact_1109_imageE,axiom,
    ! [B4: a,F: ( a > nat ) > a,A: set_a_nat] :
      ( ( member_a @ B4 @ ( image_a_nat_a @ F @ A ) )
     => ~ ! [X3: a > nat] :
            ( ( B4
              = ( F @ X3 ) )
           => ~ ( member_a_nat @ X3 @ A ) ) ) ).

% imageE
thf(fact_1110_imageE,axiom,
    ! [B4: a,F: ( a > a ) > a,A: set_a_a] :
      ( ( member_a @ B4 @ ( image_a_a_a @ F @ A ) )
     => ~ ! [X3: a > a] :
            ( ( B4
              = ( F @ X3 ) )
           => ~ ( member_a_a @ X3 @ A ) ) ) ).

% imageE
thf(fact_1111_imageE,axiom,
    ! [B4: nat,F: ( nat > nat ) > nat,A: set_nat_nat] :
      ( ( member_nat @ B4 @ ( image_nat_nat_nat @ F @ A ) )
     => ~ ! [X3: nat > nat] :
            ( ( B4
              = ( F @ X3 ) )
           => ~ ( member_nat_nat @ X3 @ A ) ) ) ).

% imageE
thf(fact_1112_pigeonhole,axiom,
    ! [F: a > set_a,A: set_a] :
      ( ( ord_less_nat @ ( finite_card_set_a @ ( image_a_set_a @ F @ A ) ) @ ( finite_card_a @ A ) )
     => ~ ( inj_on_a_set_a @ F @ A ) ) ).

% pigeonhole
thf(fact_1113_pigeonhole,axiom,
    ! [F: product_prod_a_a > product_prod_a_a,A: set_Product_prod_a_a] :
      ( ( ord_less_nat @ ( finite4795055649997197647od_a_a @ ( image_4636654165204879301od_a_a @ F @ A ) ) @ ( finite4795055649997197647od_a_a @ A ) )
     => ~ ( inj_on2566144670800592689od_a_a @ F @ A ) ) ).

% pigeonhole
thf(fact_1114_pigeonhole,axiom,
    ! [F: a > product_prod_a_a,A: set_a] :
      ( ( ord_less_nat @ ( finite4795055649997197647od_a_a @ ( image_7400625782589995694od_a_a @ F @ A ) ) @ ( finite_card_a @ A ) )
     => ~ ( inj_on8941660083241582106od_a_a @ F @ A ) ) ).

% pigeonhole
thf(fact_1115_pigeonhole,axiom,
    ! [F: nat > product_prod_a_a,A: set_nat] :
      ( ( ord_less_nat @ ( finite4795055649997197647od_a_a @ ( image_372941888232738320od_a_a @ F @ A ) ) @ ( finite_card_nat @ A ) )
     => ~ ( inj_on8964604748314331044od_a_a @ F @ A ) ) ).

% pigeonhole
thf(fact_1116_pigeonhole,axiom,
    ! [F: product_prod_a_a > a,A: set_Product_prod_a_a] :
      ( ( ord_less_nat @ ( finite_card_a @ ( image_3437945252899457948_a_a_a @ F @ A ) ) @ ( finite4795055649997197647od_a_a @ A ) )
     => ~ ( inj_on4978979553551044360_a_a_a @ F @ A ) ) ).

% pigeonhole
thf(fact_1117_pigeonhole,axiom,
    ! [F: a > a,A: set_a] :
      ( ( ord_less_nat @ ( finite_card_a @ ( image_a_a @ F @ A ) ) @ ( finite_card_a @ A ) )
     => ~ ( inj_on_a_a @ F @ A ) ) ).

% pigeonhole
thf(fact_1118_pigeonhole,axiom,
    ! [F: nat > a,A: set_nat] :
      ( ( ord_less_nat @ ( finite_card_a @ ( image_nat_a @ F @ A ) ) @ ( finite_card_nat @ A ) )
     => ~ ( inj_on_nat_a @ F @ A ) ) ).

% pigeonhole
thf(fact_1119_pigeonhole,axiom,
    ! [F: product_prod_a_a > nat,A: set_Product_prod_a_a] :
      ( ( ord_less_nat @ ( finite_card_nat @ ( image_9053670898913107890_a_nat @ F @ A ) ) @ ( finite4795055649997197647od_a_a @ A ) )
     => ~ ( inj_on8421961722139924806_a_nat @ F @ A ) ) ).

% pigeonhole
thf(fact_1120_pigeonhole,axiom,
    ! [F: a > nat,A: set_a] :
      ( ( ord_less_nat @ ( finite_card_nat @ ( image_a_nat @ F @ A ) ) @ ( finite_card_a @ A ) )
     => ~ ( inj_on_a_nat @ F @ A ) ) ).

% pigeonhole
thf(fact_1121_pigeonhole,axiom,
    ! [F: nat > nat,A: set_nat] :
      ( ( ord_less_nat @ ( finite_card_nat @ ( image_nat_nat @ F @ A ) ) @ ( finite_card_nat @ A ) )
     => ~ ( inj_on_nat_nat @ F @ A ) ) ).

% pigeonhole
thf(fact_1122_all__subset__image,axiom,
    ! [F: a > set_a,A: set_a,P: set_set_a > $o] :
      ( ( ! [B5: set_set_a] :
            ( ( ord_le3724670747650509150_set_a @ B5 @ ( image_a_set_a @ F @ A ) )
           => ( P @ B5 ) ) )
      = ( ! [B5: set_a] :
            ( ( ord_less_eq_set_a @ B5 @ A )
           => ( P @ ( image_a_set_a @ F @ B5 ) ) ) ) ) ).

% all_subset_image
thf(fact_1123_all__subset__image,axiom,
    ! [F: a > a,A: set_a,P: set_a > $o] :
      ( ( ! [B5: set_a] :
            ( ( ord_less_eq_set_a @ B5 @ ( image_a_a @ F @ A ) )
           => ( P @ B5 ) ) )
      = ( ! [B5: set_a] :
            ( ( ord_less_eq_set_a @ B5 @ A )
           => ( P @ ( image_a_a @ F @ B5 ) ) ) ) ) ).

% all_subset_image
thf(fact_1124_image__Int__subset,axiom,
    ! [F: a > set_a,A: set_a,B: set_a] : ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ ( inf_inf_set_a @ A @ B ) ) @ ( inf_inf_set_set_a @ ( image_a_set_a @ F @ A ) @ ( image_a_set_a @ F @ B ) ) ) ).

% image_Int_subset
thf(fact_1125_image__Int__subset,axiom,
    ! [F: a > a,A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( image_a_a @ F @ ( inf_inf_set_a @ A @ B ) ) @ ( inf_inf_set_a @ ( image_a_a @ F @ A ) @ ( image_a_a @ F @ B ) ) ) ).

% image_Int_subset
thf(fact_1126_pigeonhole__infinite,axiom,
    ! [A: set_a,F: a > a] :
      ( ~ ( finite_finite_a @ A )
     => ( ( finite_finite_a @ ( image_a_a @ F @ A ) )
       => ? [X3: a] :
            ( ( member_a @ X3 @ A )
            & ~ ( finite_finite_a
                @ ( collect_a
                  @ ^ [A4: a] :
                      ( ( member_a @ A4 @ A )
                      & ( ( F @ A4 )
                        = ( F @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_1127_pigeonhole__infinite,axiom,
    ! [A: set_a,F: a > nat] :
      ( ~ ( finite_finite_a @ A )
     => ( ( finite_finite_nat @ ( image_a_nat @ F @ A ) )
       => ? [X3: a] :
            ( ( member_a @ X3 @ A )
            & ~ ( finite_finite_a
                @ ( collect_a
                  @ ^ [A4: a] :
                      ( ( member_a @ A4 @ A )
                      & ( ( F @ A4 )
                        = ( F @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_1128_pigeonhole__infinite,axiom,
    ! [A: set_nat,F: nat > a] :
      ( ~ ( finite_finite_nat @ A )
     => ( ( finite_finite_a @ ( image_nat_a @ F @ A ) )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A )
            & ~ ( finite_finite_nat
                @ ( collect_nat
                  @ ^ [A4: nat] :
                      ( ( member_nat @ A4 @ A )
                      & ( ( F @ A4 )
                        = ( F @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_1129_pigeonhole__infinite,axiom,
    ! [A: set_nat,F: nat > nat] :
      ( ~ ( finite_finite_nat @ A )
     => ( ( finite_finite_nat @ ( image_nat_nat @ F @ A ) )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A )
            & ~ ( finite_finite_nat
                @ ( collect_nat
                  @ ^ [A4: nat] :
                      ( ( member_nat @ A4 @ A )
                      & ( ( F @ A4 )
                        = ( F @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_1130_pigeonhole__infinite,axiom,
    ! [A: set_a,F: a > set_a] :
      ( ~ ( finite_finite_a @ A )
     => ( ( finite_finite_set_a @ ( image_a_set_a @ F @ A ) )
       => ? [X3: a] :
            ( ( member_a @ X3 @ A )
            & ~ ( finite_finite_a
                @ ( collect_a
                  @ ^ [A4: a] :
                      ( ( member_a @ A4 @ A )
                      & ( ( F @ A4 )
                        = ( F @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_1131_pigeonhole__infinite,axiom,
    ! [A: set_nat_nat,F: ( nat > nat ) > a] :
      ( ~ ( finite2115694454571419734at_nat @ A )
     => ( ( finite_finite_a @ ( image_nat_nat_a @ F @ A ) )
       => ? [X3: nat > nat] :
            ( ( member_nat_nat @ X3 @ A )
            & ~ ( finite2115694454571419734at_nat
                @ ( collect_nat_nat
                  @ ^ [A4: nat > nat] :
                      ( ( member_nat_nat @ A4 @ A )
                      & ( ( F @ A4 )
                        = ( F @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_1132_pigeonhole__infinite,axiom,
    ! [A: set_nat_a,F: ( nat > a ) > a] :
      ( ~ ( finite_finite_nat_a @ A )
     => ( ( finite_finite_a @ ( image_nat_a_a @ F @ A ) )
       => ? [X3: nat > a] :
            ( ( member_nat_a @ X3 @ A )
            & ~ ( finite_finite_nat_a
                @ ( collect_nat_a
                  @ ^ [A4: nat > a] :
                      ( ( member_nat_a @ A4 @ A )
                      & ( ( F @ A4 )
                        = ( F @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_1133_pigeonhole__infinite,axiom,
    ! [A: set_a_nat,F: ( a > nat ) > a] :
      ( ~ ( finite_finite_a_nat @ A )
     => ( ( finite_finite_a @ ( image_a_nat_a @ F @ A ) )
       => ? [X3: a > nat] :
            ( ( member_a_nat @ X3 @ A )
            & ~ ( finite_finite_a_nat
                @ ( collect_a_nat
                  @ ^ [A4: a > nat] :
                      ( ( member_a_nat @ A4 @ A )
                      & ( ( F @ A4 )
                        = ( F @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_1134_pigeonhole__infinite,axiom,
    ! [A: set_a_a,F: ( a > a ) > a] :
      ( ~ ( finite_finite_a_a @ A )
     => ( ( finite_finite_a @ ( image_a_a_a @ F @ A ) )
       => ? [X3: a > a] :
            ( ( member_a_a @ X3 @ A )
            & ~ ( finite_finite_a_a
                @ ( collect_a_a
                  @ ^ [A4: a > a] :
                      ( ( member_a_a @ A4 @ A )
                      & ( ( F @ A4 )
                        = ( F @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_1135_pigeonhole__infinite,axiom,
    ! [A: set_nat_nat,F: ( nat > nat ) > nat] :
      ( ~ ( finite2115694454571419734at_nat @ A )
     => ( ( finite_finite_nat @ ( image_nat_nat_nat @ F @ A ) )
       => ? [X3: nat > nat] :
            ( ( member_nat_nat @ X3 @ A )
            & ~ ( finite2115694454571419734at_nat
                @ ( collect_nat_nat
                  @ ^ [A4: nat > nat] :
                      ( ( member_nat_nat @ A4 @ A )
                      & ( ( F @ A4 )
                        = ( F @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_1136_image__constant,axiom,
    ! [X: a,A: set_a,C2: a] :
      ( ( member_a @ X @ A )
     => ( ( image_a_a
          @ ^ [X2: a] : C2
          @ A )
        = ( insert_a @ C2 @ bot_bot_set_a ) ) ) ).

% image_constant
thf(fact_1137_image__constant,axiom,
    ! [X: nat,A: set_nat,C2: a] :
      ( ( member_nat @ X @ A )
     => ( ( image_nat_a
          @ ^ [X2: nat] : C2
          @ A )
        = ( insert_a @ C2 @ bot_bot_set_a ) ) ) ).

% image_constant
thf(fact_1138_image__constant,axiom,
    ! [X: a,A: set_a,C2: nat] :
      ( ( member_a @ X @ A )
     => ( ( image_a_nat
          @ ^ [X2: a] : C2
          @ A )
        = ( insert_nat @ C2 @ bot_bot_set_nat ) ) ) ).

% image_constant
thf(fact_1139_image__constant,axiom,
    ! [X: nat,A: set_nat,C2: nat] :
      ( ( member_nat @ X @ A )
     => ( ( image_nat_nat
          @ ^ [X2: nat] : C2
          @ A )
        = ( insert_nat @ C2 @ bot_bot_set_nat ) ) ) ).

% image_constant
thf(fact_1140_image__constant,axiom,
    ! [X: a,A: set_a,C2: set_a] :
      ( ( member_a @ X @ A )
     => ( ( image_a_set_a
          @ ^ [X2: a] : C2
          @ A )
        = ( insert_set_a @ C2 @ bot_bot_set_set_a ) ) ) ).

% image_constant
thf(fact_1141_image__constant,axiom,
    ! [X: nat > nat,A: set_nat_nat,C2: a] :
      ( ( member_nat_nat @ X @ A )
     => ( ( image_nat_nat_a
          @ ^ [X2: nat > nat] : C2
          @ A )
        = ( insert_a @ C2 @ bot_bot_set_a ) ) ) ).

% image_constant
thf(fact_1142_image__constant,axiom,
    ! [X: nat > a,A: set_nat_a,C2: a] :
      ( ( member_nat_a @ X @ A )
     => ( ( image_nat_a_a
          @ ^ [X2: nat > a] : C2
          @ A )
        = ( insert_a @ C2 @ bot_bot_set_a ) ) ) ).

% image_constant
thf(fact_1143_image__constant,axiom,
    ! [X: a > nat,A: set_a_nat,C2: a] :
      ( ( member_a_nat @ X @ A )
     => ( ( image_a_nat_a
          @ ^ [X2: a > nat] : C2
          @ A )
        = ( insert_a @ C2 @ bot_bot_set_a ) ) ) ).

% image_constant
thf(fact_1144_image__constant,axiom,
    ! [X: a > a,A: set_a_a,C2: a] :
      ( ( member_a_a @ X @ A )
     => ( ( image_a_a_a
          @ ^ [X2: a > a] : C2
          @ A )
        = ( insert_a @ C2 @ bot_bot_set_a ) ) ) ).

% image_constant
thf(fact_1145_image__constant,axiom,
    ! [X: nat > nat,A: set_nat_nat,C2: nat] :
      ( ( member_nat_nat @ X @ A )
     => ( ( image_nat_nat_nat
          @ ^ [X2: nat > nat] : C2
          @ A )
        = ( insert_nat @ C2 @ bot_bot_set_nat ) ) ) ).

% image_constant
thf(fact_1146_image__constant__conv,axiom,
    ! [A: set_a,C2: set_a] :
      ( ( ( A = bot_bot_set_a )
       => ( ( image_a_set_a
            @ ^ [X2: a] : C2
            @ A )
          = bot_bot_set_set_a ) )
      & ( ( A != bot_bot_set_a )
       => ( ( image_a_set_a
            @ ^ [X2: a] : C2
            @ A )
          = ( insert_set_a @ C2 @ bot_bot_set_set_a ) ) ) ) ).

% image_constant_conv
thf(fact_1147_image__constant__conv,axiom,
    ! [A: set_a,C2: a] :
      ( ( ( A = bot_bot_set_a )
       => ( ( image_a_a
            @ ^ [X2: a] : C2
            @ A )
          = bot_bot_set_a ) )
      & ( ( A != bot_bot_set_a )
       => ( ( image_a_a
            @ ^ [X2: a] : C2
            @ A )
          = ( insert_a @ C2 @ bot_bot_set_a ) ) ) ) ).

% image_constant_conv
thf(fact_1148_image__constant__conv,axiom,
    ! [A: set_a,C2: nat] :
      ( ( ( A = bot_bot_set_a )
       => ( ( image_a_nat
            @ ^ [X2: a] : C2
            @ A )
          = bot_bot_set_nat ) )
      & ( ( A != bot_bot_set_a )
       => ( ( image_a_nat
            @ ^ [X2: a] : C2
            @ A )
          = ( insert_nat @ C2 @ bot_bot_set_nat ) ) ) ) ).

% image_constant_conv
thf(fact_1149_image__constant__conv,axiom,
    ! [A: set_nat,C2: a] :
      ( ( ( A = bot_bot_set_nat )
       => ( ( image_nat_a
            @ ^ [X2: nat] : C2
            @ A )
          = bot_bot_set_a ) )
      & ( ( A != bot_bot_set_nat )
       => ( ( image_nat_a
            @ ^ [X2: nat] : C2
            @ A )
          = ( insert_a @ C2 @ bot_bot_set_a ) ) ) ) ).

% image_constant_conv
thf(fact_1150_image__constant__conv,axiom,
    ! [A: set_nat,C2: nat] :
      ( ( ( A = bot_bot_set_nat )
       => ( ( image_nat_nat
            @ ^ [X2: nat] : C2
            @ A )
          = bot_bot_set_nat ) )
      & ( ( A != bot_bot_set_nat )
       => ( ( image_nat_nat
            @ ^ [X2: nat] : C2
            @ A )
          = ( insert_nat @ C2 @ bot_bot_set_nat ) ) ) ) ).

% image_constant_conv
thf(fact_1151_image__paired__Times,axiom,
    ! [F: a > a,G2: a > set_a,A: set_a,B: set_a] :
      ( ( image_422237461394870565_set_a
        @ ( produc8721876326212642395_set_a
          @ ^ [X2: a,Y4: a] : ( product_Pair_a_set_a @ ( F @ X2 ) @ ( G2 @ Y4 ) ) )
        @ ( product_Sigma_a_a @ A
          @ ^ [Uu: a] : B ) )
      = ( produc7797748338049884712_set_a @ ( image_a_a @ F @ A )
        @ ^ [Uu: a] : ( image_a_set_a @ G2 @ B ) ) ) ).

% image_paired_Times
thf(fact_1152_image__paired__Times,axiom,
    ! [F: a > set_a,G2: a > a,A: set_a,B: set_a] :
      ( ( image_7471988017431894629et_a_a
        @ ( produc6548254845394890651et_a_a
          @ ^ [X2: a,Y4: a] : ( product_Pair_set_a_a @ ( F @ X2 ) @ ( G2 @ Y4 ) ) )
        @ ( product_Sigma_a_a @ A
          @ ^ [Uu: a] : B ) )
      = ( produc8014260396509227880et_a_a @ ( image_a_set_a @ F @ A )
        @ ^ [Uu: set_a] : ( image_a_a @ G2 @ B ) ) ) ).

% image_paired_Times
thf(fact_1153_image__paired__Times,axiom,
    ! [F: a > set_a,G2: a > set_a,A: set_a,B: set_a] :
      ( ( image_3365282634368607173_set_a
        @ ( produc6733207012205814011_set_a
          @ ^ [X2: a,Y4: a] : ( produc9088192753505129239_set_a @ ( F @ X2 ) @ ( G2 @ Y4 ) ) )
        @ ( product_Sigma_a_a @ A
          @ ^ [Uu: a] : B ) )
      = ( produc6033315442965015752_set_a @ ( image_a_set_a @ F @ A )
        @ ^ [Uu: set_a] : ( image_a_set_a @ G2 @ B ) ) ) ).

% image_paired_Times
thf(fact_1154_image__paired__Times,axiom,
    ! [F: a > a,G2: a > a,A: set_a,B: set_a] :
      ( ( image_4636654165204879301od_a_a
        @ ( produc408267641121961211od_a_a
          @ ^ [X2: a,Y4: a] : ( product_Pair_a_a @ ( F @ X2 ) @ ( G2 @ Y4 ) ) )
        @ ( product_Sigma_a_a @ A
          @ ^ [Uu: a] : B ) )
      = ( product_Sigma_a_a @ ( image_a_a @ F @ A )
        @ ^ [Uu: a] : ( image_a_a @ G2 @ B ) ) ) ).

% image_paired_Times
thf(fact_1155_all__finite__subset__image,axiom,
    ! [F: nat > nat,A: set_nat,P: set_nat > $o] :
      ( ( ! [B5: set_nat] :
            ( ( ( finite_finite_nat @ B5 )
              & ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat @ F @ A ) ) )
           => ( P @ B5 ) ) )
      = ( ! [B5: set_nat] :
            ( ( ( finite_finite_nat @ B5 )
              & ( ord_less_eq_set_nat @ B5 @ A ) )
           => ( P @ ( image_nat_nat @ F @ B5 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_1156_all__finite__subset__image,axiom,
    ! [F: product_prod_a_a > nat,A: set_Product_prod_a_a,P: set_nat > $o] :
      ( ( ! [B5: set_nat] :
            ( ( ( finite_finite_nat @ B5 )
              & ( ord_less_eq_set_nat @ B5 @ ( image_9053670898913107890_a_nat @ F @ A ) ) )
           => ( P @ B5 ) ) )
      = ( ! [B5: set_Product_prod_a_a] :
            ( ( ( finite6544458595007987280od_a_a @ B5 )
              & ( ord_le746702958409616551od_a_a @ B5 @ A ) )
           => ( P @ ( image_9053670898913107890_a_nat @ F @ B5 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_1157_all__finite__subset__image,axiom,
    ! [F: nat > product_prod_a_a,A: set_nat,P: set_Product_prod_a_a > $o] :
      ( ( ! [B5: set_Product_prod_a_a] :
            ( ( ( finite6544458595007987280od_a_a @ B5 )
              & ( ord_le746702958409616551od_a_a @ B5 @ ( image_372941888232738320od_a_a @ F @ A ) ) )
           => ( P @ B5 ) ) )
      = ( ! [B5: set_nat] :
            ( ( ( finite_finite_nat @ B5 )
              & ( ord_less_eq_set_nat @ B5 @ A ) )
           => ( P @ ( image_372941888232738320od_a_a @ F @ B5 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_1158_all__finite__subset__image,axiom,
    ! [F: product_prod_a_a > product_prod_a_a,A: set_Product_prod_a_a,P: set_Product_prod_a_a > $o] :
      ( ( ! [B5: set_Product_prod_a_a] :
            ( ( ( finite6544458595007987280od_a_a @ B5 )
              & ( ord_le746702958409616551od_a_a @ B5 @ ( image_4636654165204879301od_a_a @ F @ A ) ) )
           => ( P @ B5 ) ) )
      = ( ! [B5: set_Product_prod_a_a] :
            ( ( ( finite6544458595007987280od_a_a @ B5 )
              & ( ord_le746702958409616551od_a_a @ B5 @ A ) )
           => ( P @ ( image_4636654165204879301od_a_a @ F @ B5 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_1159_all__finite__subset__image,axiom,
    ! [F: a > set_a,A: set_a,P: set_set_a > $o] :
      ( ( ! [B5: set_set_a] :
            ( ( ( finite_finite_set_a @ B5 )
              & ( ord_le3724670747650509150_set_a @ B5 @ ( image_a_set_a @ F @ A ) ) )
           => ( P @ B5 ) ) )
      = ( ! [B5: set_a] :
            ( ( ( finite_finite_a @ B5 )
              & ( ord_less_eq_set_a @ B5 @ A ) )
           => ( P @ ( image_a_set_a @ F @ B5 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_1160_all__finite__subset__image,axiom,
    ! [F: a > nat,A: set_a,P: set_nat > $o] :
      ( ( ! [B5: set_nat] :
            ( ( ( finite_finite_nat @ B5 )
              & ( ord_less_eq_set_nat @ B5 @ ( image_a_nat @ F @ A ) ) )
           => ( P @ B5 ) ) )
      = ( ! [B5: set_a] :
            ( ( ( finite_finite_a @ B5 )
              & ( ord_less_eq_set_a @ B5 @ A ) )
           => ( P @ ( image_a_nat @ F @ B5 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_1161_all__finite__subset__image,axiom,
    ! [F: a > product_prod_a_a,A: set_a,P: set_Product_prod_a_a > $o] :
      ( ( ! [B5: set_Product_prod_a_a] :
            ( ( ( finite6544458595007987280od_a_a @ B5 )
              & ( ord_le746702958409616551od_a_a @ B5 @ ( image_7400625782589995694od_a_a @ F @ A ) ) )
           => ( P @ B5 ) ) )
      = ( ! [B5: set_a] :
            ( ( ( finite_finite_a @ B5 )
              & ( ord_less_eq_set_a @ B5 @ A ) )
           => ( P @ ( image_7400625782589995694od_a_a @ F @ B5 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_1162_all__finite__subset__image,axiom,
    ! [F: nat > a,A: set_nat,P: set_a > $o] :
      ( ( ! [B5: set_a] :
            ( ( ( finite_finite_a @ B5 )
              & ( ord_less_eq_set_a @ B5 @ ( image_nat_a @ F @ A ) ) )
           => ( P @ B5 ) ) )
      = ( ! [B5: set_nat] :
            ( ( ( finite_finite_nat @ B5 )
              & ( ord_less_eq_set_nat @ B5 @ A ) )
           => ( P @ ( image_nat_a @ F @ B5 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_1163_all__finite__subset__image,axiom,
    ! [F: product_prod_a_a > a,A: set_Product_prod_a_a,P: set_a > $o] :
      ( ( ! [B5: set_a] :
            ( ( ( finite_finite_a @ B5 )
              & ( ord_less_eq_set_a @ B5 @ ( image_3437945252899457948_a_a_a @ F @ A ) ) )
           => ( P @ B5 ) ) )
      = ( ! [B5: set_Product_prod_a_a] :
            ( ( ( finite6544458595007987280od_a_a @ B5 )
              & ( ord_le746702958409616551od_a_a @ B5 @ A ) )
           => ( P @ ( image_3437945252899457948_a_a_a @ F @ B5 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_1164_all__finite__subset__image,axiom,
    ! [F: a > a,A: set_a,P: set_a > $o] :
      ( ( ! [B5: set_a] :
            ( ( ( finite_finite_a @ B5 )
              & ( ord_less_eq_set_a @ B5 @ ( image_a_a @ F @ A ) ) )
           => ( P @ B5 ) ) )
      = ( ! [B5: set_a] :
            ( ( ( finite_finite_a @ B5 )
              & ( ord_less_eq_set_a @ B5 @ A ) )
           => ( P @ ( image_a_a @ F @ B5 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_1165_ex__finite__subset__image,axiom,
    ! [F: nat > nat,A: set_nat,P: set_nat > $o] :
      ( ( ? [B5: set_nat] :
            ( ( finite_finite_nat @ B5 )
            & ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat @ F @ A ) )
            & ( P @ B5 ) ) )
      = ( ? [B5: set_nat] :
            ( ( finite_finite_nat @ B5 )
            & ( ord_less_eq_set_nat @ B5 @ A )
            & ( P @ ( image_nat_nat @ F @ B5 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_1166_ex__finite__subset__image,axiom,
    ! [F: product_prod_a_a > nat,A: set_Product_prod_a_a,P: set_nat > $o] :
      ( ( ? [B5: set_nat] :
            ( ( finite_finite_nat @ B5 )
            & ( ord_less_eq_set_nat @ B5 @ ( image_9053670898913107890_a_nat @ F @ A ) )
            & ( P @ B5 ) ) )
      = ( ? [B5: set_Product_prod_a_a] :
            ( ( finite6544458595007987280od_a_a @ B5 )
            & ( ord_le746702958409616551od_a_a @ B5 @ A )
            & ( P @ ( image_9053670898913107890_a_nat @ F @ B5 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_1167_ex__finite__subset__image,axiom,
    ! [F: nat > product_prod_a_a,A: set_nat,P: set_Product_prod_a_a > $o] :
      ( ( ? [B5: set_Product_prod_a_a] :
            ( ( finite6544458595007987280od_a_a @ B5 )
            & ( ord_le746702958409616551od_a_a @ B5 @ ( image_372941888232738320od_a_a @ F @ A ) )
            & ( P @ B5 ) ) )
      = ( ? [B5: set_nat] :
            ( ( finite_finite_nat @ B5 )
            & ( ord_less_eq_set_nat @ B5 @ A )
            & ( P @ ( image_372941888232738320od_a_a @ F @ B5 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_1168_ex__finite__subset__image,axiom,
    ! [F: product_prod_a_a > product_prod_a_a,A: set_Product_prod_a_a,P: set_Product_prod_a_a > $o] :
      ( ( ? [B5: set_Product_prod_a_a] :
            ( ( finite6544458595007987280od_a_a @ B5 )
            & ( ord_le746702958409616551od_a_a @ B5 @ ( image_4636654165204879301od_a_a @ F @ A ) )
            & ( P @ B5 ) ) )
      = ( ? [B5: set_Product_prod_a_a] :
            ( ( finite6544458595007987280od_a_a @ B5 )
            & ( ord_le746702958409616551od_a_a @ B5 @ A )
            & ( P @ ( image_4636654165204879301od_a_a @ F @ B5 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_1169_ex__finite__subset__image,axiom,
    ! [F: a > set_a,A: set_a,P: set_set_a > $o] :
      ( ( ? [B5: set_set_a] :
            ( ( finite_finite_set_a @ B5 )
            & ( ord_le3724670747650509150_set_a @ B5 @ ( image_a_set_a @ F @ A ) )
            & ( P @ B5 ) ) )
      = ( ? [B5: set_a] :
            ( ( finite_finite_a @ B5 )
            & ( ord_less_eq_set_a @ B5 @ A )
            & ( P @ ( image_a_set_a @ F @ B5 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_1170_ex__finite__subset__image,axiom,
    ! [F: a > nat,A: set_a,P: set_nat > $o] :
      ( ( ? [B5: set_nat] :
            ( ( finite_finite_nat @ B5 )
            & ( ord_less_eq_set_nat @ B5 @ ( image_a_nat @ F @ A ) )
            & ( P @ B5 ) ) )
      = ( ? [B5: set_a] :
            ( ( finite_finite_a @ B5 )
            & ( ord_less_eq_set_a @ B5 @ A )
            & ( P @ ( image_a_nat @ F @ B5 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_1171_ex__finite__subset__image,axiom,
    ! [F: a > product_prod_a_a,A: set_a,P: set_Product_prod_a_a > $o] :
      ( ( ? [B5: set_Product_prod_a_a] :
            ( ( finite6544458595007987280od_a_a @ B5 )
            & ( ord_le746702958409616551od_a_a @ B5 @ ( image_7400625782589995694od_a_a @ F @ A ) )
            & ( P @ B5 ) ) )
      = ( ? [B5: set_a] :
            ( ( finite_finite_a @ B5 )
            & ( ord_less_eq_set_a @ B5 @ A )
            & ( P @ ( image_7400625782589995694od_a_a @ F @ B5 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_1172_ex__finite__subset__image,axiom,
    ! [F: nat > a,A: set_nat,P: set_a > $o] :
      ( ( ? [B5: set_a] :
            ( ( finite_finite_a @ B5 )
            & ( ord_less_eq_set_a @ B5 @ ( image_nat_a @ F @ A ) )
            & ( P @ B5 ) ) )
      = ( ? [B5: set_nat] :
            ( ( finite_finite_nat @ B5 )
            & ( ord_less_eq_set_nat @ B5 @ A )
            & ( P @ ( image_nat_a @ F @ B5 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_1173_ex__finite__subset__image,axiom,
    ! [F: product_prod_a_a > a,A: set_Product_prod_a_a,P: set_a > $o] :
      ( ( ? [B5: set_a] :
            ( ( finite_finite_a @ B5 )
            & ( ord_less_eq_set_a @ B5 @ ( image_3437945252899457948_a_a_a @ F @ A ) )
            & ( P @ B5 ) ) )
      = ( ? [B5: set_Product_prod_a_a] :
            ( ( finite6544458595007987280od_a_a @ B5 )
            & ( ord_le746702958409616551od_a_a @ B5 @ A )
            & ( P @ ( image_3437945252899457948_a_a_a @ F @ B5 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_1174_ex__finite__subset__image,axiom,
    ! [F: a > a,A: set_a,P: set_a > $o] :
      ( ( ? [B5: set_a] :
            ( ( finite_finite_a @ B5 )
            & ( ord_less_eq_set_a @ B5 @ ( image_a_a @ F @ A ) )
            & ( P @ B5 ) ) )
      = ( ? [B5: set_a] :
            ( ( finite_finite_a @ B5 )
            & ( ord_less_eq_set_a @ B5 @ A )
            & ( P @ ( image_a_a @ F @ B5 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_1175_finite__subset__image,axiom,
    ! [B: set_nat,F: nat > nat,A: set_nat] :
      ( ( finite_finite_nat @ B )
     => ( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
       => ? [C3: set_nat] :
            ( ( ord_less_eq_set_nat @ C3 @ A )
            & ( finite_finite_nat @ C3 )
            & ( B
              = ( image_nat_nat @ F @ C3 ) ) ) ) ) ).

% finite_subset_image
thf(fact_1176_finite__subset__image,axiom,
    ! [B: set_nat,F: product_prod_a_a > nat,A: set_Product_prod_a_a] :
      ( ( finite_finite_nat @ B )
     => ( ( ord_less_eq_set_nat @ B @ ( image_9053670898913107890_a_nat @ F @ A ) )
       => ? [C3: set_Product_prod_a_a] :
            ( ( ord_le746702958409616551od_a_a @ C3 @ A )
            & ( finite6544458595007987280od_a_a @ C3 )
            & ( B
              = ( image_9053670898913107890_a_nat @ F @ C3 ) ) ) ) ) ).

% finite_subset_image
thf(fact_1177_finite__subset__image,axiom,
    ! [B: set_Product_prod_a_a,F: nat > product_prod_a_a,A: set_nat] :
      ( ( finite6544458595007987280od_a_a @ B )
     => ( ( ord_le746702958409616551od_a_a @ B @ ( image_372941888232738320od_a_a @ F @ A ) )
       => ? [C3: set_nat] :
            ( ( ord_less_eq_set_nat @ C3 @ A )
            & ( finite_finite_nat @ C3 )
            & ( B
              = ( image_372941888232738320od_a_a @ F @ C3 ) ) ) ) ) ).

% finite_subset_image
thf(fact_1178_finite__subset__image,axiom,
    ! [B: set_Product_prod_a_a,F: product_prod_a_a > product_prod_a_a,A: set_Product_prod_a_a] :
      ( ( finite6544458595007987280od_a_a @ B )
     => ( ( ord_le746702958409616551od_a_a @ B @ ( image_4636654165204879301od_a_a @ F @ A ) )
       => ? [C3: set_Product_prod_a_a] :
            ( ( ord_le746702958409616551od_a_a @ C3 @ A )
            & ( finite6544458595007987280od_a_a @ C3 )
            & ( B
              = ( image_4636654165204879301od_a_a @ F @ C3 ) ) ) ) ) ).

% finite_subset_image
thf(fact_1179_finite__subset__image,axiom,
    ! [B: set_set_a,F: a > set_a,A: set_a] :
      ( ( finite_finite_set_a @ B )
     => ( ( ord_le3724670747650509150_set_a @ B @ ( image_a_set_a @ F @ A ) )
       => ? [C3: set_a] :
            ( ( ord_less_eq_set_a @ C3 @ A )
            & ( finite_finite_a @ C3 )
            & ( B
              = ( image_a_set_a @ F @ C3 ) ) ) ) ) ).

% finite_subset_image
thf(fact_1180_finite__subset__image,axiom,
    ! [B: set_nat,F: a > nat,A: set_a] :
      ( ( finite_finite_nat @ B )
     => ( ( ord_less_eq_set_nat @ B @ ( image_a_nat @ F @ A ) )
       => ? [C3: set_a] :
            ( ( ord_less_eq_set_a @ C3 @ A )
            & ( finite_finite_a @ C3 )
            & ( B
              = ( image_a_nat @ F @ C3 ) ) ) ) ) ).

% finite_subset_image
thf(fact_1181_finite__subset__image,axiom,
    ! [B: set_Product_prod_a_a,F: a > product_prod_a_a,A: set_a] :
      ( ( finite6544458595007987280od_a_a @ B )
     => ( ( ord_le746702958409616551od_a_a @ B @ ( image_7400625782589995694od_a_a @ F @ A ) )
       => ? [C3: set_a] :
            ( ( ord_less_eq_set_a @ C3 @ A )
            & ( finite_finite_a @ C3 )
            & ( B
              = ( image_7400625782589995694od_a_a @ F @ C3 ) ) ) ) ) ).

% finite_subset_image
thf(fact_1182_finite__subset__image,axiom,
    ! [B: set_a,F: nat > a,A: set_nat] :
      ( ( finite_finite_a @ B )
     => ( ( ord_less_eq_set_a @ B @ ( image_nat_a @ F @ A ) )
       => ? [C3: set_nat] :
            ( ( ord_less_eq_set_nat @ C3 @ A )
            & ( finite_finite_nat @ C3 )
            & ( B
              = ( image_nat_a @ F @ C3 ) ) ) ) ) ).

% finite_subset_image
thf(fact_1183_finite__subset__image,axiom,
    ! [B: set_a,F: product_prod_a_a > a,A: set_Product_prod_a_a] :
      ( ( finite_finite_a @ B )
     => ( ( ord_less_eq_set_a @ B @ ( image_3437945252899457948_a_a_a @ F @ A ) )
       => ? [C3: set_Product_prod_a_a] :
            ( ( ord_le746702958409616551od_a_a @ C3 @ A )
            & ( finite6544458595007987280od_a_a @ C3 )
            & ( B
              = ( image_3437945252899457948_a_a_a @ F @ C3 ) ) ) ) ) ).

% finite_subset_image
thf(fact_1184_finite__subset__image,axiom,
    ! [B: set_a,F: a > a,A: set_a] :
      ( ( finite_finite_a @ B )
     => ( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A ) )
       => ? [C3: set_a] :
            ( ( ord_less_eq_set_a @ C3 @ A )
            & ( finite_finite_a @ C3 )
            & ( B
              = ( image_a_a @ F @ C3 ) ) ) ) ) ).

% finite_subset_image
thf(fact_1185_finite__surj,axiom,
    ! [A: set_a,B: set_set_a,F: a > set_a] :
      ( ( finite_finite_a @ A )
     => ( ( ord_le3724670747650509150_set_a @ B @ ( image_a_set_a @ F @ A ) )
       => ( finite_finite_set_a @ B ) ) ) ).

% finite_surj
thf(fact_1186_finite__surj,axiom,
    ! [A: set_a,B: set_nat,F: a > nat] :
      ( ( finite_finite_a @ A )
     => ( ( ord_less_eq_set_nat @ B @ ( image_a_nat @ F @ A ) )
       => ( finite_finite_nat @ B ) ) ) ).

% finite_surj
thf(fact_1187_finite__surj,axiom,
    ! [A: set_a,B: set_Product_prod_a_a,F: a > product_prod_a_a] :
      ( ( finite_finite_a @ A )
     => ( ( ord_le746702958409616551od_a_a @ B @ ( image_7400625782589995694od_a_a @ F @ A ) )
       => ( finite6544458595007987280od_a_a @ B ) ) ) ).

% finite_surj
thf(fact_1188_finite__surj,axiom,
    ! [A: set_nat,B: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A )
     => ( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
       => ( finite_finite_nat @ B ) ) ) ).

% finite_surj
thf(fact_1189_finite__surj,axiom,
    ! [A: set_nat,B: set_Product_prod_a_a,F: nat > product_prod_a_a] :
      ( ( finite_finite_nat @ A )
     => ( ( ord_le746702958409616551od_a_a @ B @ ( image_372941888232738320od_a_a @ F @ A ) )
       => ( finite6544458595007987280od_a_a @ B ) ) ) ).

% finite_surj
thf(fact_1190_finite__surj,axiom,
    ! [A: set_Product_prod_a_a,B: set_nat,F: product_prod_a_a > nat] :
      ( ( finite6544458595007987280od_a_a @ A )
     => ( ( ord_less_eq_set_nat @ B @ ( image_9053670898913107890_a_nat @ F @ A ) )
       => ( finite_finite_nat @ B ) ) ) ).

% finite_surj
thf(fact_1191_finite__surj,axiom,
    ! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,F: product_prod_a_a > product_prod_a_a] :
      ( ( finite6544458595007987280od_a_a @ A )
     => ( ( ord_le746702958409616551od_a_a @ B @ ( image_4636654165204879301od_a_a @ F @ A ) )
       => ( finite6544458595007987280od_a_a @ B ) ) ) ).

% finite_surj
thf(fact_1192_finite__surj,axiom,
    ! [A: set_a,B: set_a,F: a > a] :
      ( ( finite_finite_a @ A )
     => ( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A ) )
       => ( finite_finite_a @ B ) ) ) ).

% finite_surj
thf(fact_1193_finite__surj,axiom,
    ! [A: set_nat,B: set_a,F: nat > a] :
      ( ( finite_finite_nat @ A )
     => ( ( ord_less_eq_set_a @ B @ ( image_nat_a @ F @ A ) )
       => ( finite_finite_a @ B ) ) ) ).

% finite_surj
thf(fact_1194_finite__surj,axiom,
    ! [A: set_Product_prod_a_a,B: set_a,F: product_prod_a_a > a] :
      ( ( finite6544458595007987280od_a_a @ A )
     => ( ( ord_less_eq_set_a @ B @ ( image_3437945252899457948_a_a_a @ F @ A ) )
       => ( finite_finite_a @ B ) ) ) ).

% finite_surj
thf(fact_1195_finite__image__iff,axiom,
    ! [F: nat > a,A: set_nat] :
      ( ( inj_on_nat_a @ F @ A )
     => ( ( finite_finite_a @ ( image_nat_a @ F @ A ) )
        = ( finite_finite_nat @ A ) ) ) ).

% finite_image_iff
thf(fact_1196_finite__image__iff,axiom,
    ! [F: product_prod_a_a > a,A: set_Product_prod_a_a] :
      ( ( inj_on4978979553551044360_a_a_a @ F @ A )
     => ( ( finite_finite_a @ ( image_3437945252899457948_a_a_a @ F @ A ) )
        = ( finite6544458595007987280od_a_a @ A ) ) ) ).

% finite_image_iff
thf(fact_1197_finite__image__iff,axiom,
    ! [F: a > nat,A: set_a] :
      ( ( inj_on_a_nat @ F @ A )
     => ( ( finite_finite_nat @ ( image_a_nat @ F @ A ) )
        = ( finite_finite_a @ A ) ) ) ).

% finite_image_iff
thf(fact_1198_finite__image__iff,axiom,
    ! [F: nat > nat,A: set_nat] :
      ( ( inj_on_nat_nat @ F @ A )
     => ( ( finite_finite_nat @ ( image_nat_nat @ F @ A ) )
        = ( finite_finite_nat @ A ) ) ) ).

% finite_image_iff
thf(fact_1199_finite__image__iff,axiom,
    ! [F: product_prod_a_a > nat,A: set_Product_prod_a_a] :
      ( ( inj_on8421961722139924806_a_nat @ F @ A )
     => ( ( finite_finite_nat @ ( image_9053670898913107890_a_nat @ F @ A ) )
        = ( finite6544458595007987280od_a_a @ A ) ) ) ).

% finite_image_iff
thf(fact_1200_finite__image__iff,axiom,
    ! [F: a > product_prod_a_a,A: set_a] :
      ( ( inj_on8941660083241582106od_a_a @ F @ A )
     => ( ( finite6544458595007987280od_a_a @ ( image_7400625782589995694od_a_a @ F @ A ) )
        = ( finite_finite_a @ A ) ) ) ).

% finite_image_iff
thf(fact_1201_finite__image__iff,axiom,
    ! [F: nat > product_prod_a_a,A: set_nat] :
      ( ( inj_on8964604748314331044od_a_a @ F @ A )
     => ( ( finite6544458595007987280od_a_a @ ( image_372941888232738320od_a_a @ F @ A ) )
        = ( finite_finite_nat @ A ) ) ) ).

% finite_image_iff
thf(fact_1202_finite__image__iff,axiom,
    ! [F: product_prod_a_a > product_prod_a_a,A: set_Product_prod_a_a] :
      ( ( inj_on2566144670800592689od_a_a @ F @ A )
     => ( ( finite6544458595007987280od_a_a @ ( image_4636654165204879301od_a_a @ F @ A ) )
        = ( finite6544458595007987280od_a_a @ A ) ) ) ).

% finite_image_iff
thf(fact_1203_sumsetdiff__sing,axiom,
    ! [A: set_a,B: set_a,X: a] :
      ( ( pluenn3038260743871226533mset_a @ g @ addition @ ( minus_minus_set_a @ A @ B ) @ ( insert_a @ X @ bot_bot_set_a ) )
      = ( minus_minus_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ ( insert_a @ X @ bot_bot_set_a ) ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ B @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).

% sumsetdiff_sing
thf(fact_1204_bot__nat__0_Oextremum,axiom,
    ! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).

% bot_nat_0.extremum
thf(fact_1205_le0,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).

% le0
thf(fact_1206_less__nat__zero__code,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% less_nat_zero_code
thf(fact_1207_neq0__conv,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% neq0_conv
thf(fact_1208_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A2: nat] :
      ( ( A2 != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ A2 ) ) ).

% bot_nat_0.not_eq_extremum
thf(fact_1209_nat__neq__iff,axiom,
    ! [M2: nat,N2: nat] :
      ( ( M2 != N2 )
      = ( ( ord_less_nat @ M2 @ N2 )
        | ( ord_less_nat @ N2 @ M2 ) ) ) ).

% nat_neq_iff
thf(fact_1210_less__not__refl,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ N2 ) ).

% less_not_refl
thf(fact_1211_less__not__refl2,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ord_less_nat @ N2 @ M2 )
     => ( M2 != N2 ) ) ).

% less_not_refl2
thf(fact_1212_less__not__refl3,axiom,
    ! [S2: nat,T3: nat] :
      ( ( ord_less_nat @ S2 @ T3 )
     => ( S2 != T3 ) ) ).

% less_not_refl3
thf(fact_1213_less__irrefl__nat,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ N2 ) ).

% less_irrefl_nat
thf(fact_1214_nat__less__induct,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ! [N4: nat] :
          ( ! [M3: nat] :
              ( ( ord_less_nat @ M3 @ N4 )
             => ( P @ M3 ) )
         => ( P @ N4 ) )
     => ( P @ N2 ) ) ).

% nat_less_induct
thf(fact_1215_infinite__descent,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ! [N4: nat] :
          ( ~ ( P @ N4 )
         => ? [M3: nat] :
              ( ( ord_less_nat @ M3 @ N4 )
              & ~ ( P @ M3 ) ) )
     => ( P @ N2 ) ) ).

% infinite_descent
thf(fact_1216_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_1217_Nat_Oex__has__greatest__nat,axiom,
    ! [P: nat > $o,K: nat,B4: nat] :
      ( ( P @ K )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B4 ) )
       => ? [X3: nat] :
            ( ( P @ X3 )
            & ! [Y7: nat] :
                ( ( P @ Y7 )
               => ( ord_less_eq_nat @ Y7 @ X3 ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_1218_nat__le__linear,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
      | ( ord_less_eq_nat @ N2 @ M2 ) ) ).

% nat_le_linear
thf(fact_1219_le__antisym,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ M2 )
       => ( M2 = N2 ) ) ) ).

% le_antisym
thf(fact_1220_eq__imp__le,axiom,
    ! [M2: nat,N2: nat] :
      ( ( M2 = N2 )
     => ( ord_less_eq_nat @ M2 @ N2 ) ) ).

% eq_imp_le
thf(fact_1221_le__trans,axiom,
    ! [I2: nat,J3: nat,K: nat] :
      ( ( ord_less_eq_nat @ I2 @ J3 )
     => ( ( ord_less_eq_nat @ J3 @ K )
       => ( ord_less_eq_nat @ I2 @ K ) ) ) ).

% le_trans
thf(fact_1222_le__refl,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).

% le_refl
thf(fact_1223_infinite__descent0,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N4: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N4 )
           => ( ~ ( P @ N4 )
             => ? [M3: nat] :
                  ( ( ord_less_nat @ M3 @ N4 )
                  & ~ ( P @ M3 ) ) ) )
       => ( P @ N2 ) ) ) ).

% infinite_descent0
thf(fact_1224_gr__implies__not0,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ M2 @ N2 )
     => ( N2 != zero_zero_nat ) ) ).

% gr_implies_not0
thf(fact_1225_less__zeroE,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% less_zeroE
thf(fact_1226_not__less0,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% not_less0
thf(fact_1227_not__gr0,axiom,
    ! [N2: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
      = ( N2 = zero_zero_nat ) ) ).

% not_gr0
thf(fact_1228_gr0I,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% gr0I
thf(fact_1229_bot__nat__0_Oextremum__strict,axiom,
    ! [A2: nat] :
      ~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).

% bot_nat_0.extremum_strict
thf(fact_1230_le__0__eq,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
      = ( N2 = zero_zero_nat ) ) ).

% le_0_eq
thf(fact_1231_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A2: nat] :
      ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
     => ( A2 = zero_zero_nat ) ) ).

% bot_nat_0.extremum_uniqueI
thf(fact_1232_bot__nat__0_Oextremum__unique,axiom,
    ! [A2: nat] :
      ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
      = ( A2 = zero_zero_nat ) ) ).

% bot_nat_0.extremum_unique
thf(fact_1233_less__eq__nat_Osimps_I1_J,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).

% less_eq_nat.simps(1)
thf(fact_1234_less__mono__imp__le__mono,axiom,
    ! [F: nat > nat,I2: nat,J3: nat] :
      ( ! [I4: nat,J4: nat] :
          ( ( ord_less_nat @ I4 @ J4 )
         => ( ord_less_nat @ ( F @ I4 ) @ ( F @ J4 ) ) )
     => ( ( ord_less_eq_nat @ I2 @ J3 )
       => ( ord_less_eq_nat @ ( F @ I2 ) @ ( F @ J3 ) ) ) ) ).

% less_mono_imp_le_mono
thf(fact_1235_le__neq__implies__less,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( M2 != N2 )
       => ( ord_less_nat @ M2 @ N2 ) ) ) ).

% le_neq_implies_less
thf(fact_1236_less__or__eq__imp__le,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ( ord_less_nat @ M2 @ N2 )
        | ( M2 = N2 ) )
     => ( ord_less_eq_nat @ M2 @ N2 ) ) ).

% less_or_eq_imp_le
thf(fact_1237_le__eq__less__or__eq,axiom,
    ( ord_less_eq_nat
    = ( ^ [M4: nat,N: nat] :
          ( ( ord_less_nat @ M4 @ N )
          | ( M4 = N ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_1238_less__imp__le__nat,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ M2 @ N2 )
     => ( ord_less_eq_nat @ M2 @ N2 ) ) ).

% less_imp_le_nat
thf(fact_1239_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M4: nat,N: nat] :
          ( ( ord_less_eq_nat @ M4 @ N )
          & ( M4 != N ) ) ) ) ).

% nat_less_le
thf(fact_1240_ex__least__nat__le,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ N2 )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K2: nat] :
            ( ( ord_less_eq_nat @ K2 @ N2 )
            & ! [I5: nat] :
                ( ( ord_less_nat @ I5 @ K2 )
               => ~ ( P @ I5 ) )
            & ( P @ K2 ) ) ) ) ).

% ex_least_nat_le
thf(fact_1241_sumset,axiom,
    ! [A: set_a,B: set_a] :
      ( ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B )
      = ( comple2307003609928055243_set_a
        @ ( image_a_set_a
          @ ^ [A4: a] :
              ( comple2307003609928055243_set_a
              @ ( image_a_set_a
                @ ^ [B3: a] : ( insert_a @ ( addition @ A4 @ B3 ) @ bot_bot_set_a )
                @ ( inf_inf_set_a @ B @ g ) ) )
          @ ( inf_inf_set_a @ A @ g ) ) ) ) ).

% sumset
thf(fact_1242_diff__0__eq__0,axiom,
    ! [N2: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ N2 )
      = zero_zero_nat ) ).

% diff_0_eq_0
thf(fact_1243_diff__self__eq__0,axiom,
    ! [M2: nat] :
      ( ( minus_minus_nat @ M2 @ M2 )
      = zero_zero_nat ) ).

% diff_self_eq_0
thf(fact_1244_diff__diff__cancel,axiom,
    ! [I2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ I2 @ N2 )
     => ( ( minus_minus_nat @ N2 @ ( minus_minus_nat @ N2 @ I2 ) )
        = I2 ) ) ).

% diff_diff_cancel
thf(fact_1245_zero__less__diff,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M2 ) )
      = ( ord_less_nat @ M2 @ N2 ) ) ).

% zero_less_diff
thf(fact_1246_diff__is__0__eq,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ( minus_minus_nat @ M2 @ N2 )
        = zero_zero_nat )
      = ( ord_less_eq_nat @ M2 @ N2 ) ) ).

% diff_is_0_eq
thf(fact_1247_diff__is__0__eq_H,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( minus_minus_nat @ M2 @ N2 )
        = zero_zero_nat ) ) ).

% diff_is_0_eq'
thf(fact_1248_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
thf(fact_1249_eq__diff__iff,axiom,
    ! [K: nat,M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ M2 )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( ( ( minus_minus_nat @ M2 @ K )
            = ( minus_minus_nat @ N2 @ K ) )
          = ( M2 = N2 ) ) ) ) ).

% eq_diff_iff
thf(fact_1250_le__diff__iff,axiom,
    ! [K: nat,M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ M2 )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
          = ( ord_less_eq_nat @ M2 @ N2 ) ) ) ) ).

% le_diff_iff
thf(fact_1251_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ M2 )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
          = ( minus_minus_nat @ M2 @ N2 ) ) ) ) ).

% Nat.diff_diff_eq
thf(fact_1252_diff__le__mono,axiom,
    ! [M2: nat,N2: nat,L: nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N2 @ L ) ) ) ).

% diff_le_mono
thf(fact_1253_diff__le__self,axiom,
    ! [M2: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N2 ) @ M2 ) ).

% diff_le_self
thf(fact_1254_le__diff__iff_H,axiom,
    ! [A2: nat,C2: nat,B4: nat] :
      ( ( ord_less_eq_nat @ A2 @ C2 )
     => ( ( ord_less_eq_nat @ B4 @ C2 )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A2 ) @ ( minus_minus_nat @ C2 @ B4 ) )
          = ( ord_less_eq_nat @ B4 @ A2 ) ) ) ) ).

% le_diff_iff'
thf(fact_1255_diff__le__mono2,axiom,
    ! [M2: nat,N2: nat,L: nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).

% diff_le_mono2
thf(fact_1256_less__imp__diff__less,axiom,
    ! [J3: nat,K: nat,N2: nat] :
      ( ( ord_less_nat @ J3 @ K )
     => ( ord_less_nat @ ( minus_minus_nat @ J3 @ N2 ) @ K ) ) ).

% less_imp_diff_less
thf(fact_1257_diff__less__mono2,axiom,
    ! [M2: nat,N2: nat,L: nat] :
      ( ( ord_less_nat @ M2 @ N2 )
     => ( ( ord_less_nat @ M2 @ L )
       => ( ord_less_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M2 ) ) ) ) ).

% diff_less_mono2
thf(fact_1258_diffs0__imp__equal,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ( minus_minus_nat @ M2 @ N2 )
        = zero_zero_nat )
     => ( ( ( minus_minus_nat @ N2 @ M2 )
          = zero_zero_nat )
       => ( M2 = N2 ) ) ) ).

% diffs0_imp_equal
thf(fact_1259_minus__nat_Odiff__0,axiom,
    ! [M2: nat] :
      ( ( minus_minus_nat @ M2 @ zero_zero_nat )
      = M2 ) ).

% minus_nat.diff_0
thf(fact_1260_inj__on__diff__nat,axiom,
    ! [N3: set_nat,K: nat] :
      ( ! [N4: nat] :
          ( ( member_nat @ N4 @ N3 )
         => ( ord_less_eq_nat @ K @ N4 ) )
     => ( inj_on_nat_nat
        @ ^ [N: nat] : ( minus_minus_nat @ N @ K )
        @ N3 ) ) ).

% inj_on_diff_nat
thf(fact_1261_diff__less,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_nat @ zero_zero_nat @ M2 )
       => ( ord_less_nat @ ( minus_minus_nat @ M2 @ N2 ) @ M2 ) ) ) ).

% diff_less
thf(fact_1262_less__diff__iff,axiom,
    ! [K: nat,M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ M2 )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
          = ( ord_less_nat @ M2 @ N2 ) ) ) ) ).

% less_diff_iff
thf(fact_1263_diff__less__mono,axiom,
    ! [A2: nat,B4: nat,C2: nat] :
      ( ( ord_less_nat @ A2 @ B4 )
     => ( ( ord_less_eq_nat @ C2 @ A2 )
       => ( ord_less_nat @ ( minus_minus_nat @ A2 @ C2 ) @ ( minus_minus_nat @ B4 @ C2 ) ) ) ) ).

% diff_less_mono
thf(fact_1264_diff__commute,axiom,
    ! [I2: nat,J3: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J3 ) @ K )
      = ( minus_minus_nat @ ( minus_minus_nat @ I2 @ K ) @ J3 ) ) ).

% diff_commute
thf(fact_1265_sumset__iterated__r,axiom,
    ! [R: nat,A: set_a] :
      ( ( ord_less_nat @ zero_zero_nat @ R )
     => ( ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ A @ R )
        = ( pluenn3038260743871226533mset_a @ g @ addition @ A @ ( pluenn1960970773371692859ated_a @ g @ addition @ zero @ A @ ( minus_minus_nat @ R @ one_one_nat ) ) ) ) ) ).

% sumset_iterated_r
thf(fact_1266_Sup__nat__empty,axiom,
    ( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% Sup_nat_empty
thf(fact_1267_less__one,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ N2 @ one_one_nat )
      = ( N2 = zero_zero_nat ) ) ).

% less_one
thf(fact_1268_sumset__subset__Un_I1_J,axiom,
    ! [A: set_a,B: set_a,C: set_a] : ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ ( sup_sup_set_a @ B @ C ) ) ) ).

% sumset_subset_Un(1)
thf(fact_1269_sumset__subset__Un1,axiom,
    ! [A: set_a,A6: set_a,B: set_a] :
      ( ( pluenn3038260743871226533mset_a @ g @ addition @ ( sup_sup_set_a @ A @ A6 ) @ B )
      = ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A6 @ B ) ) ) ).

% sumset_subset_Un1
thf(fact_1270_sumset__subset__Un2,axiom,
    ! [A: set_a,B: set_a,B6: set_a] :
      ( ( pluenn3038260743871226533mset_a @ g @ addition @ A @ ( sup_sup_set_a @ B @ B6 ) )
      = ( sup_sup_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B6 ) ) ) ).

% sumset_subset_Un2
thf(fact_1271_sumset__subset__Un_I2_J,axiom,
    ! [A: set_a,B: set_a,C: set_a] : ( ord_less_eq_set_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ A @ B ) @ ( pluenn3038260743871226533mset_a @ g @ addition @ ( sup_sup_set_a @ A @ C ) @ B ) ) ).

% sumset_subset_Un(2)

% Conjectures (1)
thf(conj_0,conjecture,
    ( ord_less_eq_nat
    @ ( finite4795055649997197647od_a_a
      @ ( product_Sigma_a_a @ u
        @ ^ [Uu: a] : ( pluenn3038260743871226533mset_a @ g @ addition @ v @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ w ) ) ) )
    @ ( finite4795055649997197647od_a_a
      @ ( product_Sigma_a_a @ ( pluenn3038260743871226533mset_a @ g @ addition @ u @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ v ) )
        @ ^ [Uu: a] : ( pluenn3038260743871226533mset_a @ g @ addition @ u @ ( pluenn2534204936789923946sset_a @ g @ addition @ zero @ w ) ) ) ) ) ).

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