TPTP Problem File: SLH0475^1.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    : Safe_Range_RC/0021_Relational_Calculus/prob_00812_029869__16849758_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1810 ( 772 unt; 522 typ;   0 def)
%            Number of atoms       : 3939 (2199 equ;   0 cnn)
%            Maximal formula atoms :   37 (   3 avg)
%            Number of connectives : 14139 ( 752   ~; 109   |; 327   &;11384   @)
%                                         (   0 <=>;1567  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   30 (   6 avg)
%            Number of types       :   89 (  88 usr)
%            Number of type conns  : 2291 (2291   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  437 ( 434 usr;  17 con; 0-4 aty)
%            Number of variables   : 4706 ( 641   ^;3900   !; 165   ?;4706   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 14:25:37.728
%------------------------------------------------------------------------------
% Could-be-implicit typings (88)
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thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001t__Nat__Onat,type,
    condit2214826472909112428ve_nat: set_nat > $o ).

thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__below_001t__Nat__Onat,type,
    condit1738341127787009408ow_nat: set_nat > $o ).

thf(sy_c_Fun_Ofun__upd_001_Eo_001_Eo,type,
    fun_upd_o_o: ( $o > $o ) > $o > $o > $o > $o ).

thf(sy_c_Fun_Ofun__upd_001_Eo_001t__Set__Oset_I_Eo_J,type,
    fun_upd_o_set_o: ( $o > set_o ) > $o > set_o > $o > set_o ).

thf(sy_c_Fun_Ofun__upd_001t__Nat__Onat_001tf__a,type,
    fun_upd_nat_a: ( nat > a ) > nat > a > nat > a ).

thf(sy_c_Fun_Oinj__on_001_Eo_001_Eo,type,
    inj_on_o_o: ( $o > $o ) > set_o > $o ).

thf(sy_c_Fun_Oinj__on_001_Eo_001t__Set__Oset_I_Eo_J,type,
    inj_on_o_set_o: ( $o > set_o ) > set_o > $o ).

thf(sy_c_Groups_Ominus__class_Ominus_001_062_I_Eo_M_Eo_J,type,
    minus_minus_o_o: ( $o > $o ) > ( $o > $o ) > $o > $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_I_Eo_J,type,
    minus_minus_set_o: set_o > set_o > set_o ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
    minus_minus_set_nat: set_nat > set_nat > set_nat ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_I_Eo_M_Eo_J,type,
    uminus_uminus_o_o: ( $o > $o ) > $o > $o ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_I_Eo_J,type,
    uminus_uminus_set_o: set_o > set_o ).

thf(sy_c_HOL_OThe_001_Eo,type,
    the_o: ( $o > $o ) > $o ).

thf(sy_c_If_001t__Nat__Onat,type,
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thf(sy_c_If_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
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thf(sy_c_If_001t__Set__Oset_I_Eo_J,type,
    if_set_o: $o > set_o > set_o > set_o ).

thf(sy_c_If_001tf__a,type,
    if_a: $o > a > a > a ).

thf(sy_c_Lattices_Oinf__class_Oinf_001_062_I_Eo_M_Eo_J,type,
    inf_inf_o_o: ( $o > $o ) > ( $o > $o ) > $o > $o ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_Eo_J,type,
    inf_inf_set_o: set_o > set_o > set_o ).

thf(sy_c_Lattices_Osup__class_Osup_001_062_I_Eo_M_Eo_J,type,
    sup_sup_o_o: ( $o > $o ) > ( $o > $o ) > $o > $o ).

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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_I_Eo_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
    sup_sup_set_list_a: set_list_a > set_list_a > set_list_a ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__b_J,type,
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thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001_Eo,type,
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thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Set_Oimage_001_Eo_001t__Set__Oset_I_Eo_J,type,
    image_o_set_o: ( $o > set_o ) > set_o > set_set_o ).

thf(sy_c_Set_Oimage_001_Eo_001tf__b,type,
    image_o_b: ( $o > b ) > set_o > set_b ).

thf(sy_c_Set_Oimage_001t__List__Olist_Itf__a_J_001_Eo,type,
    image_list_a_o: ( list_a > $o ) > set_list_a > set_o ).

thf(sy_c_Set_Oimage_001t__List__Olist_Itf__a_J_001t__Nat__Onat,type,
    image_list_a_nat: ( list_a > nat ) > set_list_a > set_nat ).

thf(sy_c_Set_Oimage_001t__Nat__Onat_001_Eo,type,
    image_nat_o: ( nat > $o ) > set_nat > set_o ).

thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__List__Olist_Itf__a_J,type,
    image_nat_list_a: ( nat > list_a ) > set_nat > set_list_a ).

thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
    image_nat_nat: ( nat > nat ) > set_nat > set_nat ).

thf(sy_c_Set_Oimage_001t__Nat__Onat_001tf__b,type,
    image_nat_b: ( nat > b ) > set_nat > set_b ).

thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mtf__a_J_M_062_It__Product____Type__Oprod_Itf__b_Mt__Nat__Onat_J_Mt__Set__Oset_It__List__Olist_Itf__a_J_J_J_J_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_Itf__b_Mt__Nat__Onat_J_Mt__Set__Oset_It__List__Olist_Itf__a_J_J_J_M_062_It__Nat__Onat_Mtf__a_J_J,type,
    image_3727386206174543445_nat_a: ( produc166345656740828447list_a > produc5835360497134304175_nat_a ) > set_Pr4091103320399850111list_a > set_Pr6389665502131816719_nat_a ).

thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_Itf__b_Mt__Nat__Onat_J_Mt__Set__Oset_It__List__Olist_Itf__a_J_J_J_M_062_It__Nat__Onat_Mtf__a_J_J_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mtf__a_J_M_062_It__Product____Type__Oprod_Itf__b_Mt__Nat__Onat_J_Mt__Set__Oset_It__List__Olist_Itf__a_J_J_J_J,type,
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thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_I_Eo_M_Eo_J_001_Eo,type,
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thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_I_Eo_M_Eo_J_001t__Nat__Onat,type,
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thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_I_Eo_M_Eo_J_001t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
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thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_I_Eo_M_Eo_J_001tf__b,type,
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thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_Mt__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
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thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_Eo,type,
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thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
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thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    image_5168914502847457605at_nat: ( product_prod_nat_nat > product_prod_nat_nat ) > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
    image_5019456213502855771la_a_b: ( product_prod_nat_nat > relational_fmla_a_b ) > set_Pr1261947904930325089at_nat > set_Re381260168593705685la_a_b ).

thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001tf__b,type,
    image_5995000214508162371_nat_b: ( product_prod_nat_nat > b ) > set_Pr1261947904930325089at_nat > set_b ).

thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
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thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Nat__Onat_Mtf__b_J_001t__Product____Type__Oprod_Itf__b_Mt__Nat__Onat_J,type,
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thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_Itf__b_Mt__Nat__Onat_J_001_Eo,type,
    image_191264095111578067_nat_o: ( product_prod_b_nat > $o ) > set_Pr1307281990691478580_b_nat > set_o ).

thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_Itf__b_Mt__Nat__Onat_J_001t__Nat__Onat,type,
    image_3660612598617633365at_nat: ( product_prod_b_nat > nat ) > set_Pr1307281990691478580_b_nat > set_nat ).

thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_Itf__b_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mtf__b_J,type,
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thf(sy_c_Set_Oimage_001t__Product____Type__Ounit_001_Eo,type,
    image_Product_unit_o: ( product_unit > $o ) > set_Product_unit > set_o ).

thf(sy_c_Set_Oimage_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_001_Eo,type,
    image_1316678413157792882_a_b_o: ( relational_fmla_a_b > $o ) > set_Re381260168593705685la_a_b > set_o ).

thf(sy_c_Set_Oimage_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_001t__Nat__Onat,type,
    image_341122591648980342_b_nat: ( relational_fmla_a_b > nat ) > set_Re381260168593705685la_a_b > set_nat ).

thf(sy_c_Set_Oimage_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
    image_6790371041703824709la_a_b: ( relational_fmla_a_b > relational_fmla_a_b ) > set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b ).

thf(sy_c_Set_Oimage_001t__Set__Oset_I_Eo_J_001_062_I_Eo_M_Eo_J,type,
    image_set_o_o_o: ( set_o > $o > $o ) > set_set_o > set_o_o ).

thf(sy_c_Set_Oimage_001t__Set__Oset_I_Eo_J_001_Eo,type,
    image_set_o_o: ( set_o > $o ) > set_set_o > set_o ).

thf(sy_c_Set_Oimage_001t__Set__Oset_I_Eo_J_001t__Nat__Onat,type,
    image_set_o_nat: ( set_o > nat ) > set_set_o > set_nat ).

thf(sy_c_Set_Oimage_001t__Set__Oset_I_Eo_J_001t__Set__Oset_I_Eo_J,type,
    image_set_o_set_o: ( set_o > set_o ) > set_set_o > set_set_o ).

thf(sy_c_Set_Oimage_001tf__b_001_Eo,type,
    image_b_o: ( b > $o ) > set_b > set_o ).

thf(sy_c_Set_Oimage_001tf__b_001t__Nat__Onat,type,
    image_b_nat: ( b > nat ) > set_b > set_nat ).

thf(sy_c_Set_Oimage_001tf__b_001tf__b,type,
    image_b_b: ( b > b ) > set_b > set_b ).

thf(sy_c_Set_Oinsert_001_062_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_Mt__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
    insert8904949763332019597la_a_b: ( relational_fmla_a_b > relational_fmla_a_b ) > set_Re1288005135514575379la_a_b > set_Re1288005135514575379la_a_b ).

thf(sy_c_Set_Oinsert_001_Eo,type,
    insert_o: $o > set_o > set_o ).

thf(sy_c_Set_Oinsert_001t__List__Olist_Itf__a_J,type,
    insert_list_a: list_a > set_list_a > set_list_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_I_Eo_M_Eo_J,type,
    insert6201435330877294327od_o_o: product_prod_o_o > set_Product_prod_o_o > set_Product_prod_o_o ).

thf(sy_c_Set_Oinsert_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
    insert7010464514620295119la_a_b: relational_fmla_a_b > set_Re381260168593705685la_a_b > set_Re381260168593705685la_a_b ).

thf(sy_c_Set_Oinsert_001t__Set__Oset_I_Eo_J,type,
    insert_set_o: set_o > set_set_o > set_set_o ).

thf(sy_c_Set_Oinsert_001tf__b,type,
    insert_b: b > set_b > set_b ).

thf(sy_c_Set_Ois__empty_001_Eo,type,
    is_empty_o: set_o > $o ).

thf(sy_c_Set_Ois__singleton_001_Eo,type,
    is_singleton_o: set_o > $o ).

thf(sy_c_Set_Oremove_001_Eo,type,
    remove_o: $o > set_o > set_o ).

thf(sy_c_Set_Othe__elem_001_Eo,type,
    the_elem_o: set_o > $o ).

thf(sy_c_Set_Ovimage_001_Eo_001_Eo,type,
    vimage_o_o: ( $o > $o ) > set_o > set_o ).

thf(sy_c_Set_Ovimage_001_Eo_001t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
    vimage8945963521958007626od_o_o: ( $o > product_prod_o_o ) > set_Product_prod_o_o > set_o ).

thf(sy_c_Typedef_Otype__definition_001t__Product____Type__Ounit_001_Eo,type,
    type_d6188575255521822967unit_o: ( product_unit > $o ) > ( $o > product_unit ) > set_o > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Relational____Calculus__Ofmla_Itf__a_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Set__Oset_It__List__Olist_Itf__a_J_J_J_M_062_It__Nat__Onat_Mtf__a_J_J_J,type,
    accp_P3115975753873020414_nat_a: ( produc3037992005704992583_nat_a > produc3037992005704992583_nat_a > $o ) > produc3037992005704992583_nat_a > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_Mt__Product____Type__Oprod_I_062_It__Product____Type__Oprod_Itf__b_Mt__Nat__Onat_J_Mt__Set__Oset_It__List__Olist_Itf__a_J_J_J_M_062_It__Nat__Onat_Mtf__a_J_J_J,type,
    accp_P6721201822162371452_nat_a: ( produc1132964494702330949_nat_a > produc1132964494702330949_nat_a > $o ) > produc1132964494702330949_nat_a > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    accp_P2470304046166516174at_nat: ( produc8867654947514737559at_nat > produc8867654947514737559at_nat > $o ) > produc8867654947514737559at_nat > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Relational____Calculus__Oterm_Itf__a_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    accp_P7512640865500879912at_nat: ( produc6058688428250151583at_nat > produc6058688428250151583at_nat > $o ) > produc6058688428250151583at_nat > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
    accp_R989495437599811158la_a_b: ( relational_fmla_a_b > relational_fmla_a_b > $o ) > relational_fmla_a_b > $o ).

thf(sy_c_member_001_062_I_Eo_M_Eo_J,type,
    member_o_o: ( $o > $o ) > set_o_o > $o ).

thf(sy_c_member_001_062_It__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_Mt__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_J,type,
    member8433577210552456052la_a_b: ( relational_fmla_a_b > relational_fmla_a_b ) > set_Re1288005135514575379la_a_b > $o ).

thf(sy_c_member_001_Eo,type,
    member_o: $o > set_o > $o ).

thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
    member_list_a: list_a > set_list_a > $o ).

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

thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mtf__a_J_M_062_It__Product____Type__Oprod_Itf__b_Mt__Nat__Onat_J_Mt__Set__Oset_It__List__Olist_Itf__a_J_J_J_J,type,
    member3529051575741102792list_a: produc166345656740828447list_a > set_Pr4091103320399850111list_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_Itf__b_Mt__Nat__Onat_J_Mt__Set__Oset_It__List__Olist_Itf__a_J_J_J_M_062_It__Nat__Onat_Mtf__a_J_J,type,
    member9198066416134578520_nat_a: produc5835360497134304175_nat_a > set_Pr6389665502131816719_nat_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
    member7466972457876170832od_o_o: product_prod_o_o > set_Product_prod_o_o > $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_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mtf__b_J,type,
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thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
    member3348759134392003351at_nat: produc8373899037510109440at_nat > set_Pr2539167527615954998at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__b_Mt__Nat__Onat_J,type,
    member6959632917342813205_b_nat: product_prod_b_nat > set_Pr1307281990691478580_b_nat > $o ).

thf(sy_c_member_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
    member4680049679412964150la_a_b: relational_fmla_a_b > set_Re381260168593705685la_a_b > $o ).

thf(sy_c_member_001t__Set__Oset_I_Eo_J,type,
    member_set_o: set_o > set_set_o > $o ).

thf(sy_c_member_001tf__b,type,
    member_b: b > set_b > $o ).

thf(sy_v_Q,type,
    q: relational_fmla_a_b ).

thf(sy_v_xs,type,
    xs: list_nat ).

thf(sy_v_y,type,
    y: nat ).

thf(sy_v_ys,type,
    ys: list_nat ).

% Relevant facts (1276)
thf(fact_0_fmla_Oinject_I7_J,axiom,
    ! [X71: nat,X72: relational_fmla_a_b,Y71: nat,Y72: relational_fmla_a_b] :
      ( ( ( relati591517084277583526ts_a_b @ X71 @ X72 )
        = ( relati591517084277583526ts_a_b @ Y71 @ Y72 ) )
      = ( ( X71 = Y71 )
        & ( X72 = Y72 ) ) ) ).

% fmla.inject(7)
thf(fact_1_genempty_Ointros_I9_J,axiom,
    ! [Q: relational_fmla_a_b,Y: nat] :
      ( ( relati5999705594545617851ty_a_b @ Q )
     => ( relati5999705594545617851ty_a_b @ ( relati591517084277583526ts_a_b @ Y @ Q ) ) ) ).

% genempty.intros(9)
thf(fact_2_genempty__substs,axiom,
    ! [Q: relational_fmla_a_b,Xs: list_nat,Ys: list_nat] :
      ( ( relati5999705594545617851ty_a_b @ Q )
     => ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_nat @ Ys ) )
       => ( relati5999705594545617851ty_a_b
          @ ( fold_P7970104616371074773la_a_b
            @ ( produc5586541307551673003la_a_b
              @ ^ [X: nat,Y2: nat,Q2: relational_fmla_a_b] : ( relational_subst_a_b @ Q2 @ X @ Y2 ) )
            @ ( zip_nat_nat @ Xs @ Ys )
            @ Q ) ) ) ) ).

% genempty_substs
thf(fact_3_case__prod__app,axiom,
    ( produc6628518323692928499term_a
    = ( ^ [F: nat > nat > relational_term_a > relational_term_a,X: product_prod_nat_nat,Y2: relational_term_a] :
          ( produc98462517025009019term_a
          @ ^ [L: nat,R: nat] : ( F @ L @ R @ Y2 )
          @ X ) ) ) ).

% case_prod_app
thf(fact_4_case__prod__app,axiom,
    ( produc7810592499157111267at_nat
    = ( ^ [F: nat > nat > nat > produc7248412053542808358at_nat,X: product_prod_nat_nat,Y2: nat] :
          ( produc9083241971206738548at_nat
          @ ^ [L: nat,R: nat] : ( F @ L @ R @ Y2 )
          @ X ) ) ) ).

% case_prod_app
thf(fact_5_case__prod__app,axiom,
    ( produc5586541307551673003la_a_b
    = ( ^ [F: nat > nat > relational_fmla_a_b > relational_fmla_a_b,X: product_prod_nat_nat,Y2: relational_fmla_a_b] :
          ( produc3270801013941088237la_a_b
          @ ^ [L: nat,R: nat] : ( F @ L @ R @ Y2 )
          @ X ) ) ) ).

% case_prod_app
thf(fact_6_ap__substs,axiom,
    ! [Q: relational_fmla_a_b,Xs: list_nat,Ys: list_nat] :
      ( ( relational_ap_a_b @ Q )
     => ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_nat @ Ys ) )
       => ( relational_ap_a_b
          @ ( fold_P7970104616371074773la_a_b
            @ ( produc5586541307551673003la_a_b
              @ ^ [X: nat,Y2: nat,Q2: relational_fmla_a_b] : ( relational_subst_a_b @ Q2 @ X @ Y2 ) )
            @ ( zip_nat_nat @ Xs @ Ys )
            @ Q ) ) ) ) ).

% ap_substs
thf(fact_7_substs__Neg,axiom,
    ! [Xs: list_nat,Ys: list_nat,Q: relational_fmla_a_b] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( fold_P7970104616371074773la_a_b
          @ ( produc5586541307551673003la_a_b
            @ ^ [X: nat,Y2: nat,Q2: relational_fmla_a_b] : ( relational_subst_a_b @ Q2 @ X @ Y2 ) )
          @ ( zip_nat_nat @ Xs @ Ys )
          @ ( relational_Neg_a_b @ Q ) )
        = ( relational_Neg_a_b
          @ ( fold_P7970104616371074773la_a_b
            @ ( produc5586541307551673003la_a_b
              @ ^ [X: nat,Y2: nat,Q2: relational_fmla_a_b] : ( relational_subst_a_b @ Q2 @ X @ Y2 ) )
            @ ( zip_nat_nat @ Xs @ Ys )
            @ Q ) ) ) ) ).

% substs_Neg
thf(fact_8_substs__Disj,axiom,
    ! [Xs: list_nat,Ys: list_nat,Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( fold_P7970104616371074773la_a_b
          @ ( produc5586541307551673003la_a_b
            @ ^ [X: nat,Y2: nat,Q2: relational_fmla_a_b] : ( relational_subst_a_b @ Q2 @ X @ Y2 ) )
          @ ( zip_nat_nat @ Xs @ Ys )
          @ ( relational_Disj_a_b @ Q1 @ Q22 ) )
        = ( relational_Disj_a_b
          @ ( fold_P7970104616371074773la_a_b
            @ ( produc5586541307551673003la_a_b
              @ ^ [X: nat,Y2: nat,Q2: relational_fmla_a_b] : ( relational_subst_a_b @ Q2 @ X @ Y2 ) )
            @ ( zip_nat_nat @ Xs @ Ys )
            @ Q1 )
          @ ( fold_P7970104616371074773la_a_b
            @ ( produc5586541307551673003la_a_b
              @ ^ [X: nat,Y2: nat,Q2: relational_fmla_a_b] : ( relational_subst_a_b @ Q2 @ X @ Y2 ) )
            @ ( zip_nat_nat @ Xs @ Ys )
            @ Q22 ) ) ) ) ).

% substs_Disj
thf(fact_9_substs__Conj,axiom,
    ! [Xs: list_nat,Ys: list_nat,Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( fold_P7970104616371074773la_a_b
          @ ( produc5586541307551673003la_a_b
            @ ^ [X: nat,Y2: nat,Q2: relational_fmla_a_b] : ( relational_subst_a_b @ Q2 @ X @ Y2 ) )
          @ ( zip_nat_nat @ Xs @ Ys )
          @ ( relational_Conj_a_b @ Q1 @ Q22 ) )
        = ( relational_Conj_a_b
          @ ( fold_P7970104616371074773la_a_b
            @ ( produc5586541307551673003la_a_b
              @ ^ [X: nat,Y2: nat,Q2: relational_fmla_a_b] : ( relational_subst_a_b @ Q2 @ X @ Y2 ) )
            @ ( zip_nat_nat @ Xs @ Ys )
            @ Q1 )
          @ ( fold_P7970104616371074773la_a_b
            @ ( produc5586541307551673003la_a_b
              @ ^ [X: nat,Y2: nat,Q2: relational_fmla_a_b] : ( relational_subst_a_b @ Q2 @ X @ Y2 ) )
            @ ( zip_nat_nat @ Xs @ Ys )
            @ Q22 ) ) ) ) ).

% substs_Conj
thf(fact_10_substs__Bool,axiom,
    ! [Xs: list_nat,Ys: list_nat,B: $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( fold_P7970104616371074773la_a_b
          @ ( produc5586541307551673003la_a_b
            @ ^ [X: nat,Y2: nat,Q2: relational_fmla_a_b] : ( relational_subst_a_b @ Q2 @ X @ Y2 ) )
          @ ( zip_nat_nat @ Xs @ Ys )
          @ ( relational_Bool_a_b @ B ) )
        = ( relational_Bool_a_b @ B ) ) ) ).

% substs_Bool
thf(fact_11_qp__substs,axiom,
    ! [Q: relational_fmla_a_b,Xs: list_nat,Ys: list_nat] :
      ( ( relational_qp_a_b @ Q )
     => ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_nat @ Ys ) )
       => ( relational_qp_a_b
          @ ( fold_P7970104616371074773la_a_b
            @ ( produc5586541307551673003la_a_b
              @ ^ [X: nat,Y2: nat,Q2: relational_fmla_a_b] : ( relational_subst_a_b @ Q2 @ X @ Y2 ) )
            @ ( zip_nat_nat @ Xs @ Ys )
            @ Q ) ) ) ) ).

% qp_substs
thf(fact_12_prod_Ocase__distrib,axiom,
    ! [H: relational_fmla_a_b > relational_fmla_a_b,F2: nat > nat > relational_fmla_a_b,Prod: product_prod_nat_nat] :
      ( ( H @ ( produc3270801013941088237la_a_b @ F2 @ Prod ) )
      = ( produc3270801013941088237la_a_b
        @ ^ [X1: nat,X2: nat] : ( H @ ( F2 @ X1 @ X2 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_13_prod_Ocase__distrib,axiom,
    ! [H: ( relational_fmla_a_b > relational_fmla_a_b ) > relational_fmla_a_b,F2: nat > nat > relational_fmla_a_b > relational_fmla_a_b,Prod: product_prod_nat_nat] :
      ( ( H @ ( produc5586541307551673003la_a_b @ F2 @ Prod ) )
      = ( produc3270801013941088237la_a_b
        @ ^ [X1: nat,X2: nat] : ( H @ ( F2 @ X1 @ X2 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_14_prod_Ocase__distrib,axiom,
    ! [H: relational_fmla_a_b > relational_fmla_a_b > relational_fmla_a_b,F2: nat > nat > relational_fmla_a_b,Prod: product_prod_nat_nat] :
      ( ( H @ ( produc3270801013941088237la_a_b @ F2 @ Prod ) )
      = ( produc5586541307551673003la_a_b
        @ ^ [X1: nat,X2: nat] : ( H @ ( F2 @ X1 @ X2 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_15_prod_Ocase__distrib,axiom,
    ! [H: ( relational_fmla_a_b > relational_fmla_a_b ) > relational_fmla_a_b > relational_fmla_a_b,F2: nat > nat > relational_fmla_a_b > relational_fmla_a_b,Prod: product_prod_nat_nat] :
      ( ( H @ ( produc5586541307551673003la_a_b @ F2 @ Prod ) )
      = ( produc5586541307551673003la_a_b
        @ ^ [X1: nat,X2: nat] : ( H @ ( F2 @ X1 @ X2 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_16_prod_Ocase__distrib,axiom,
    ! [H: relational_fmla_a_b > product_prod_nat_nat,F2: nat > nat > relational_fmla_a_b,Prod: product_prod_nat_nat] :
      ( ( H @ ( produc3270801013941088237la_a_b @ F2 @ Prod ) )
      = ( produc2626176000494625587at_nat
        @ ^ [X1: nat,X2: nat] : ( H @ ( F2 @ X1 @ X2 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_17_prod_Ocase__distrib,axiom,
    ! [H: product_prod_nat_nat > relational_fmla_a_b,F2: nat > nat > product_prod_nat_nat,Prod: product_prod_nat_nat] :
      ( ( H @ ( produc2626176000494625587at_nat @ F2 @ Prod ) )
      = ( produc3270801013941088237la_a_b
        @ ^ [X1: nat,X2: nat] : ( H @ ( F2 @ X1 @ X2 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_18_prod_Ocase__distrib,axiom,
    ! [H: product_prod_nat_nat > product_prod_nat_nat,F2: nat > nat > product_prod_nat_nat,Prod: product_prod_nat_nat] :
      ( ( H @ ( produc2626176000494625587at_nat @ F2 @ Prod ) )
      = ( produc2626176000494625587at_nat
        @ ^ [X1: nat,X2: nat] : ( H @ ( F2 @ X1 @ X2 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_19_prod_Ocase__distrib,axiom,
    ! [H: relational_fmla_a_b > produc7248412053542808358at_nat,F2: nat > nat > relational_fmla_a_b,Prod: product_prod_nat_nat] :
      ( ( H @ ( produc3270801013941088237la_a_b @ F2 @ Prod ) )
      = ( produc9083241971206738548at_nat
        @ ^ [X1: nat,X2: nat] : ( H @ ( F2 @ X1 @ X2 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_20_prod_Ocase__distrib,axiom,
    ! [H: relational_fmla_a_b > relational_term_a > relational_term_a,F2: nat > nat > relational_fmla_a_b,Prod: product_prod_nat_nat] :
      ( ( H @ ( produc3270801013941088237la_a_b @ F2 @ Prod ) )
      = ( produc6628518323692928499term_a
        @ ^ [X1: nat,X2: nat] : ( H @ ( F2 @ X1 @ X2 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_21_prod_Ocase__distrib,axiom,
    ! [H: produc7248412053542808358at_nat > relational_fmla_a_b,F2: nat > nat > produc7248412053542808358at_nat,Prod: product_prod_nat_nat] :
      ( ( H @ ( produc9083241971206738548at_nat @ F2 @ Prod ) )
      = ( produc3270801013941088237la_a_b
        @ ^ [X1: nat,X2: nat] : ( H @ ( F2 @ X1 @ X2 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_22_substs__Exists,axiom,
    ! [Xs: list_nat,Ys: list_nat,Z: nat,Q: relational_fmla_a_b] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( fold_P7970104616371074773la_a_b
          @ ( produc5586541307551673003la_a_b
            @ ^ [X: nat,Y2: nat,Q2: relational_fmla_a_b] : ( relational_subst_a_b @ Q2 @ X @ Y2 ) )
          @ ( zip_nat_nat @ Xs @ Ys )
          @ ( relati591517084277583526ts_a_b @ Z @ Q ) )
        = ( relati591517084277583526ts_a_b @ ( relati3061153751618241490bd_a_b @ Z @ Xs @ Ys @ Q )
          @ ( fold_P7970104616371074773la_a_b
            @ ( produc5586541307551673003la_a_b
              @ ^ [X: nat,Y2: nat,Q2: relational_fmla_a_b] : ( relational_subst_a_b @ Q2 @ X @ Y2 ) )
            @ ( zip_nat_nat @ ( relati8440516942800864086rc_a_b @ Z @ Xs @ Ys @ Q ) @ ( relati4597344298476144279st_a_b @ Z @ Xs @ Ys @ Q ) )
            @ Q ) ) ) ) ).

% substs_Exists
thf(fact_23_in__fv__substs,axiom,
    ! [Xs: list_nat,Ys: list_nat,X3: nat,Q: relational_fmla_a_b] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member_nat @ X3 @ ( relational_fv_a_b @ Q ) )
       => ( member_nat @ ( relati8128731020529265620ar_nat @ Xs @ Ys @ X3 )
          @ ( relational_fv_a_b
            @ ( fold_P7970104616371074773la_a_b
              @ ( produc5586541307551673003la_a_b
                @ ^ [X: nat,Y2: nat,Q2: relational_fmla_a_b] : ( relational_subst_a_b @ Q2 @ X @ Y2 ) )
              @ ( zip_nat_nat @ Xs @ Ys )
              @ Q ) ) ) ) ) ).

% in_fv_substs
thf(fact_24_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_R6823256787227418703term_a] :
      ( ( size_s88622898042387131term_a @ Xs2 )
      = N ) ).

% Ex_list_of_length
thf(fact_25_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_nat] :
      ( ( size_size_list_nat @ Xs2 )
      = N ) ).

% Ex_list_of_length
thf(fact_26_neq__if__length__neq,axiom,
    ! [Xs: list_R6823256787227418703term_a,Ys: list_R6823256787227418703term_a] :
      ( ( ( size_s88622898042387131term_a @ Xs )
       != ( size_s88622898042387131term_a @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_27_neq__if__length__neq,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
       != ( size_size_list_nat @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_28_fmla_Oinject_I4_J,axiom,
    ! [X4: relational_fmla_a_b,Y4: relational_fmla_a_b] :
      ( ( ( relational_Neg_a_b @ X4 )
        = ( relational_Neg_a_b @ Y4 ) )
      = ( X4 = Y4 ) ) ).

% fmla.inject(4)
thf(fact_29_fmla_Oinject_I6_J,axiom,
    ! [X61: relational_fmla_a_b,X62: relational_fmla_a_b,Y61: relational_fmla_a_b,Y62: relational_fmla_a_b] :
      ( ( ( relational_Disj_a_b @ X61 @ X62 )
        = ( relational_Disj_a_b @ Y61 @ Y62 ) )
      = ( ( X61 = Y61 )
        & ( X62 = Y62 ) ) ) ).

% fmla.inject(6)
thf(fact_30_fmla_Oinject_I5_J,axiom,
    ! [X51: relational_fmla_a_b,X52: relational_fmla_a_b,Y51: relational_fmla_a_b,Y52: relational_fmla_a_b] :
      ( ( ( relational_Conj_a_b @ X51 @ X52 )
        = ( relational_Conj_a_b @ Y51 @ Y52 ) )
      = ( ( X51 = Y51 )
        & ( X52 = Y52 ) ) ) ).

% fmla.inject(5)
thf(fact_31_fmla_Oinject_I2_J,axiom,
    ! [X22: $o,Y22: $o] :
      ( ( ( relational_Bool_a_b @ X22 )
        = ( relational_Bool_a_b @ Y22 ) )
      = ( X22 = Y22 ) ) ).

% fmla.inject(2)
thf(fact_32_qp__subst__eq,axiom,
    ! [Q: relational_fmla_a_b,X3: nat,Y: nat] :
      ( ( relational_qp_a_b @ ( relational_subst_a_b @ Q @ X3 @ Y ) )
      = ( relational_qp_a_b @ Q ) ) ).

% qp_subst_eq
thf(fact_33_length__substs,axiom,
    ! [Xs: list_nat,Ys: list_nat,Z: nat,Q: relational_fmla_a_b] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( size_size_list_nat @ ( relati8440516942800864086rc_a_b @ Z @ Xs @ Ys @ Q ) )
        = ( size_size_list_nat @ ( relati4597344298476144279st_a_b @ Z @ Xs @ Ys @ Q ) ) ) ) ).

% length_substs
thf(fact_34_fv_Osimps_I4_J,axiom,
    ! [Phi: relational_fmla_a_b] :
      ( ( relational_fv_a_b @ ( relational_Neg_a_b @ Phi ) )
      = ( relational_fv_a_b @ Phi ) ) ).

% fv.simps(4)
thf(fact_35_fmla_Odistinct_I37_J,axiom,
    ! [X51: relational_fmla_a_b,X52: relational_fmla_a_b,X61: relational_fmla_a_b,X62: relational_fmla_a_b] :
      ( ( relational_Conj_a_b @ X51 @ X52 )
     != ( relational_Disj_a_b @ X61 @ X62 ) ) ).

% fmla.distinct(37)
thf(fact_36_fmla_Odistinct_I33_J,axiom,
    ! [X4: relational_fmla_a_b,X61: relational_fmla_a_b,X62: relational_fmla_a_b] :
      ( ( relational_Neg_a_b @ X4 )
     != ( relational_Disj_a_b @ X61 @ X62 ) ) ).

% fmla.distinct(33)
thf(fact_37_fmla_Odistinct_I31_J,axiom,
    ! [X4: relational_fmla_a_b,X51: relational_fmla_a_b,X52: relational_fmla_a_b] :
      ( ( relational_Neg_a_b @ X4 )
     != ( relational_Conj_a_b @ X51 @ X52 ) ) ).

% fmla.distinct(31)
thf(fact_38_fmla_Odistinct_I19_J,axiom,
    ! [X22: $o,X61: relational_fmla_a_b,X62: relational_fmla_a_b] :
      ( ( relational_Bool_a_b @ X22 )
     != ( relational_Disj_a_b @ X61 @ X62 ) ) ).

% fmla.distinct(19)
thf(fact_39_fmla_Odistinct_I17_J,axiom,
    ! [X22: $o,X51: relational_fmla_a_b,X52: relational_fmla_a_b] :
      ( ( relational_Bool_a_b @ X22 )
     != ( relational_Conj_a_b @ X51 @ X52 ) ) ).

% fmla.distinct(17)
thf(fact_40_fmla_Odistinct_I15_J,axiom,
    ! [X22: $o,X4: relational_fmla_a_b] :
      ( ( relational_Bool_a_b @ X22 )
     != ( relational_Neg_a_b @ X4 ) ) ).

% fmla.distinct(15)
thf(fact_41_genempty_Ointros_I4_J,axiom,
    ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
      ( ( relati5999705594545617851ty_a_b @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q1 ) @ ( relational_Neg_a_b @ Q22 ) ) )
     => ( relati5999705594545617851ty_a_b @ ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q1 @ Q22 ) ) ) ) ).

% genempty.intros(4)
thf(fact_42_genempty_Ointros_I3_J,axiom,
    ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
      ( ( relati5999705594545617851ty_a_b @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q1 ) @ ( relational_Neg_a_b @ Q22 ) ) )
     => ( relati5999705594545617851ty_a_b @ ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) ) ) ) ).

% genempty.intros(3)
thf(fact_43_ap,axiom,
    ! [Q: relational_fmla_a_b] :
      ( ( relational_ap_a_b @ Q )
     => ( relational_qp_a_b @ Q ) ) ).

% ap
thf(fact_44_qp__Neg,axiom,
    ! [Q: relational_fmla_a_b] :
      ~ ( relational_qp_a_b @ ( relational_Neg_a_b @ Q ) ) ).

% qp_Neg
thf(fact_45_qp__Conj,axiom,
    ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
      ~ ( relational_qp_a_b @ ( relational_Conj_a_b @ Q1 @ Q22 ) ) ).

% qp_Conj
thf(fact_46_qp__Disj,axiom,
    ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
      ~ ( relational_qp_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) ) ).

% qp_Disj
thf(fact_47_qp__ExistsE,axiom,
    ! [X3: nat,Q: relational_fmla_a_b] :
      ( ( relational_qp_a_b @ ( relati591517084277583526ts_a_b @ X3 @ Q ) )
     => ~ ( ( relational_qp_a_b @ Q )
         => ~ ( member_nat @ X3 @ ( relational_fv_a_b @ Q ) ) ) ) ).

% qp_ExistsE
thf(fact_48_qp__Exists,axiom,
    ! [Q: relational_fmla_a_b,X3: nat] :
      ( ( relational_qp_a_b @ Q )
     => ( ( member_nat @ X3 @ ( relational_fv_a_b @ Q ) )
       => ( relational_qp_a_b @ ( relati591517084277583526ts_a_b @ X3 @ Q ) ) ) ) ).

% qp_Exists
thf(fact_49_fmla_Odistinct_I21_J,axiom,
    ! [X22: $o,X71: nat,X72: relational_fmla_a_b] :
      ( ( relational_Bool_a_b @ X22 )
     != ( relati591517084277583526ts_a_b @ X71 @ X72 ) ) ).

% fmla.distinct(21)
thf(fact_50_fmla_Odistinct_I39_J,axiom,
    ! [X51: relational_fmla_a_b,X52: relational_fmla_a_b,X71: nat,X72: relational_fmla_a_b] :
      ( ( relational_Conj_a_b @ X51 @ X52 )
     != ( relati591517084277583526ts_a_b @ X71 @ X72 ) ) ).

% fmla.distinct(39)
thf(fact_51_fmla_Odistinct_I41_J,axiom,
    ! [X61: relational_fmla_a_b,X62: relational_fmla_a_b,X71: nat,X72: relational_fmla_a_b] :
      ( ( relational_Disj_a_b @ X61 @ X62 )
     != ( relati591517084277583526ts_a_b @ X71 @ X72 ) ) ).

% fmla.distinct(41)
thf(fact_52_subst_Osimps_I1_J,axiom,
    ! [T: $o,X3: nat,Y: nat] :
      ( ( relational_subst_a_b @ ( relational_Bool_a_b @ T ) @ X3 @ Y )
      = ( relational_Bool_a_b @ T ) ) ).

% subst.simps(1)
thf(fact_53_fmla_Odistinct_I35_J,axiom,
    ! [X4: relational_fmla_a_b,X71: nat,X72: relational_fmla_a_b] :
      ( ( relational_Neg_a_b @ X4 )
     != ( relati591517084277583526ts_a_b @ X71 @ X72 ) ) ).

% fmla.distinct(35)
thf(fact_54_subst_Osimps_I5_J,axiom,
    ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b,X3: nat,Y: nat] :
      ( ( relational_subst_a_b @ ( relational_Conj_a_b @ Q1 @ Q22 ) @ X3 @ Y )
      = ( relational_Conj_a_b @ ( relational_subst_a_b @ Q1 @ X3 @ Y ) @ ( relational_subst_a_b @ Q22 @ X3 @ Y ) ) ) ).

% subst.simps(5)
thf(fact_55_subst_Osimps_I6_J,axiom,
    ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b,X3: nat,Y: nat] :
      ( ( relational_subst_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) @ X3 @ Y )
      = ( relational_Disj_a_b @ ( relational_subst_a_b @ Q1 @ X3 @ Y ) @ ( relational_subst_a_b @ Q22 @ X3 @ Y ) ) ) ).

% subst.simps(6)
thf(fact_56_subst_Osimps_I4_J,axiom,
    ! [Q: relational_fmla_a_b,X3: nat,Y: nat] :
      ( ( relational_subst_a_b @ ( relational_Neg_a_b @ Q ) @ X3 @ Y )
      = ( relational_Neg_a_b @ ( relational_subst_a_b @ Q @ X3 @ Y ) ) ) ).

% subst.simps(4)
thf(fact_57_genempty_Ointros_I1_J,axiom,
    relati5999705594545617851ty_a_b @ ( relational_Bool_a_b @ $false ) ).

% genempty.intros(1)
thf(fact_58_mem__Collect__eq,axiom,
    ! [A: list_a,P: list_a > $o] :
      ( ( member_list_a @ A @ ( collect_list_a @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_59_mem__Collect__eq,axiom,
    ! [A: set_o,P: set_o > $o] :
      ( ( member_set_o @ A @ ( collect_set_o @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_60_mem__Collect__eq,axiom,
    ! [A: nat,P: nat > $o] :
      ( ( member_nat @ A @ ( collect_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_61_mem__Collect__eq,axiom,
    ! [A: relational_fmla_a_b > relational_fmla_a_b,P: ( relational_fmla_a_b > relational_fmla_a_b ) > $o] :
      ( ( member8433577210552456052la_a_b @ A @ ( collec5041345499257167282la_a_b @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_62_mem__Collect__eq,axiom,
    ! [A: b,P: b > $o] :
      ( ( member_b @ A @ ( collect_b @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_63_mem__Collect__eq,axiom,
    ! [A: $o,P: $o > $o] :
      ( ( member_o @ A @ ( collect_o @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_64_Collect__mem__eq,axiom,
    ! [A2: set_list_a] :
      ( ( collect_list_a
        @ ^ [X: list_a] : ( member_list_a @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_65_Collect__mem__eq,axiom,
    ! [A2: set_set_o] :
      ( ( collect_set_o
        @ ^ [X: set_o] : ( member_set_o @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_66_Collect__mem__eq,axiom,
    ! [A2: set_nat] :
      ( ( collect_nat
        @ ^ [X: nat] : ( member_nat @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_67_Collect__mem__eq,axiom,
    ! [A2: set_Re1288005135514575379la_a_b] :
      ( ( collec5041345499257167282la_a_b
        @ ^ [X: relational_fmla_a_b > relational_fmla_a_b] : ( member8433577210552456052la_a_b @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_68_Collect__mem__eq,axiom,
    ! [A2: set_b] :
      ( ( collect_b
        @ ^ [X: b] : ( member_b @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_69_Collect__mem__eq,axiom,
    ! [A2: set_o] :
      ( ( collect_o
        @ ^ [X: $o] : ( member_o @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_70_Collect__cong,axiom,
    ! [P: $o > $o,Q: $o > $o] :
      ( ! [X5: $o] :
          ( ( P @ X5 )
          = ( Q @ X5 ) )
     => ( ( collect_o @ P )
        = ( collect_o @ Q ) ) ) ).

% Collect_cong
thf(fact_71_genempty_Ointros_I6_J,axiom,
    ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
      ( ( ( relati5999705594545617851ty_a_b @ Q1 )
        | ( relati5999705594545617851ty_a_b @ Q22 ) )
     => ( relati5999705594545617851ty_a_b @ ( relational_Conj_a_b @ Q1 @ Q22 ) ) ) ).

% genempty.intros(6)
thf(fact_72_genempty_Ointros_I5_J,axiom,
    ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
      ( ( relati5999705594545617851ty_a_b @ Q1 )
     => ( ( relati5999705594545617851ty_a_b @ Q22 )
       => ( relati5999705594545617851ty_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) ) ) ) ).

% genempty.intros(5)
thf(fact_73_genempty_Ointros_I2_J,axiom,
    ! [Q: relational_fmla_a_b] :
      ( ( relati5999705594545617851ty_a_b @ Q )
     => ( relati5999705594545617851ty_a_b @ ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q ) ) ) ) ).

% genempty.intros(2)
thf(fact_74_qp__subst_H,axiom,
    ! [Q: relational_fmla_a_b,X3: nat,Y: nat] :
      ( ( relational_qp_a_b @ ( relational_subst_a_b @ Q @ X3 @ Y ) )
     => ( relational_qp_a_b @ Q ) ) ).

% qp_subst'
thf(fact_75_qp__subst,axiom,
    ! [Q: relational_fmla_a_b,X3: nat,Y: nat] :
      ( ( relational_qp_a_b @ Q )
     => ( relational_qp_a_b @ ( relational_subst_a_b @ Q @ X3 @ Y ) ) ) ).

% qp_subst
thf(fact_76_ap__subst_H,axiom,
    ! [Q: relational_fmla_a_b,X3: nat,Y: nat] :
      ( ( relational_ap_a_b @ ( relational_subst_a_b @ Q @ X3 @ Y ) )
     => ( relational_ap_a_b @ Q ) ) ).

% ap_subst'
thf(fact_77_qp__impl_Osimps_I3_J,axiom,
    ! [X3: nat,Q: relational_fmla_a_b] :
      ( ( relati3725921752842749053pl_a_b @ ( relati591517084277583526ts_a_b @ X3 @ Q ) )
      = ( ( member_nat @ X3 @ ( relational_fv_a_b @ Q ) )
        & ( relational_qp_a_b @ Q ) ) ) ).

% qp_impl.simps(3)
thf(fact_78_internal__case__prod__def,axiom,
    produc3133647842178331349at_nat = produc968775922737392939at_nat ).

% internal_case_prod_def
thf(fact_79_internal__case__prod__def,axiom,
    produc1511379720124852554at_nat = produc9083241971206738548at_nat ).

% internal_case_prod_def
thf(fact_80_internal__case__prod__def,axiom,
    produc7900996641596641245at_nat = produc2626176000494625587at_nat ).

% internal_case_prod_def
thf(fact_81_internal__case__prod__def,axiom,
    produc6183066805105636809term_a = produc6628518323692928499term_a ).

% internal_case_prod_def
thf(fact_82_internal__case__prod__def,axiom,
    produc414753464042270649at_nat = produc7810592499157111267at_nat ).

% internal_case_prod_def
thf(fact_83_internal__case__prod__def,axiom,
    produc7208501889184683393la_a_b = produc5586541307551673003la_a_b ).

% internal_case_prod_def
thf(fact_84_internal__case__prod__def,axiom,
    produc2167329440614046147la_a_b = produc3270801013941088237la_a_b ).

% internal_case_prod_def
thf(fact_85_curry__case__prod,axiom,
    ! [F2: nat > product_prod_nat_nat > produc7248412053542808358at_nat] :
      ( ( produc7023161500789126988at_nat @ ( produc968775922737392939at_nat @ F2 ) )
      = F2 ) ).

% curry_case_prod
thf(fact_86_curry__case__prod,axiom,
    ! [F2: nat > nat > produc7248412053542808358at_nat] :
      ( ( produc8208958989098927507at_nat @ ( produc9083241971206738548at_nat @ F2 ) )
      = F2 ) ).

% curry_case_prod
thf(fact_87_curry__case__prod,axiom,
    ! [F2: nat > nat > product_prod_nat_nat] :
      ( ( produc6629854527392350932at_nat @ ( produc2626176000494625587at_nat @ F2 ) )
      = F2 ) ).

% curry_case_prod
thf(fact_88_curry__case__prod,axiom,
    ! [F2: nat > nat > relational_term_a > relational_term_a] :
      ( ( produc6250739812019294098term_a @ ( produc6628518323692928499term_a @ F2 ) )
      = F2 ) ).

% curry_case_prod
thf(fact_89_curry__case__prod,axiom,
    ! [F2: nat > nat > nat > produc7248412053542808358at_nat] :
      ( ( produc2170021319937026690at_nat @ ( produc7810592499157111267at_nat @ F2 ) )
      = F2 ) ).

% curry_case_prod
thf(fact_90_curry__case__prod,axiom,
    ! [F2: nat > nat > relational_fmla_a_b > relational_fmla_a_b] :
      ( ( produc7541201833284165578la_a_b @ ( produc5586541307551673003la_a_b @ F2 ) )
      = F2 ) ).

% curry_case_prod
thf(fact_91_curry__case__prod,axiom,
    ! [F2: nat > nat > relational_fmla_a_b] :
      ( ( produc858456811296061068la_a_b @ ( produc3270801013941088237la_a_b @ F2 ) )
      = F2 ) ).

% curry_case_prod
thf(fact_92_case__prod__curry,axiom,
    ! [F2: produc7248412053542808358at_nat > produc7248412053542808358at_nat] :
      ( ( produc968775922737392939at_nat @ ( produc7023161500789126988at_nat @ F2 ) )
      = F2 ) ).

% case_prod_curry
thf(fact_93_case__prod__curry,axiom,
    ! [F2: product_prod_nat_nat > produc7248412053542808358at_nat] :
      ( ( produc9083241971206738548at_nat @ ( produc8208958989098927507at_nat @ F2 ) )
      = F2 ) ).

% case_prod_curry
thf(fact_94_case__prod__curry,axiom,
    ! [F2: product_prod_nat_nat > product_prod_nat_nat] :
      ( ( produc2626176000494625587at_nat @ ( produc6629854527392350932at_nat @ F2 ) )
      = F2 ) ).

% case_prod_curry
thf(fact_95_case__prod__curry,axiom,
    ! [F2: product_prod_nat_nat > relational_term_a > relational_term_a] :
      ( ( produc6628518323692928499term_a @ ( produc6250739812019294098term_a @ F2 ) )
      = F2 ) ).

% case_prod_curry
thf(fact_96_case__prod__curry,axiom,
    ! [F2: product_prod_nat_nat > nat > produc7248412053542808358at_nat] :
      ( ( produc7810592499157111267at_nat @ ( produc2170021319937026690at_nat @ F2 ) )
      = F2 ) ).

% case_prod_curry
thf(fact_97_case__prod__curry,axiom,
    ! [F2: product_prod_nat_nat > relational_fmla_a_b > relational_fmla_a_b] :
      ( ( produc5586541307551673003la_a_b @ ( produc7541201833284165578la_a_b @ F2 ) )
      = F2 ) ).

% case_prod_curry
thf(fact_98_case__prod__curry,axiom,
    ! [F2: product_prod_nat_nat > relational_fmla_a_b] :
      ( ( produc3270801013941088237la_a_b @ ( produc858456811296061068la_a_b @ F2 ) )
      = F2 ) ).

% case_prod_curry
thf(fact_99_size__neq__size__imp__neq,axiom,
    ! [X3: relational_fmla_a_b,Y: relational_fmla_a_b] :
      ( ( ( size_s453432777765377587la_a_b @ X3 )
       != ( size_s453432777765377587la_a_b @ Y ) )
     => ( X3 != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_100_size__neq__size__imp__neq,axiom,
    ! [X3: list_R6823256787227418703term_a,Y: list_R6823256787227418703term_a] :
      ( ( ( size_s88622898042387131term_a @ X3 )
       != ( size_s88622898042387131term_a @ Y ) )
     => ( X3 != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_101_size__neq__size__imp__neq,axiom,
    ! [X3: list_nat,Y: list_nat] :
      ( ( ( size_size_list_nat @ X3 )
       != ( size_size_list_nat @ Y ) )
     => ( X3 != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_102_case__swap,axiom,
    ! [F2: nat > product_prod_nat_nat > produc7248412053542808358at_nat,P2: produc7248412053542808358at_nat] :
      ( ( produc3206169289476954189at_nat
        @ ^ [Y2: product_prod_nat_nat,X: nat] : ( F2 @ X @ Y2 )
        @ ( produc4032600223772806584at_nat @ P2 ) )
      = ( produc968775922737392939at_nat @ F2 @ P2 ) ) ).

% case_swap
thf(fact_103_case__swap,axiom,
    ! [F2: product_prod_nat_nat > nat > produc7248412053542808358at_nat,P2: produc8373899037510109440at_nat] :
      ( ( produc968775922737392939at_nat
        @ ^ [Y2: nat,X: product_prod_nat_nat] : ( F2 @ X @ Y2 )
        @ ( produc672552830730091482at_nat @ P2 ) )
      = ( produc3206169289476954189at_nat @ F2 @ P2 ) ) ).

% case_swap
thf(fact_104_case__swap,axiom,
    ! [F2: nat > nat > produc7248412053542808358at_nat,P2: product_prod_nat_nat] :
      ( ( produc9083241971206738548at_nat
        @ ^ [Y2: nat,X: nat] : ( F2 @ X @ Y2 )
        @ ( product_swap_nat_nat @ P2 ) )
      = ( produc9083241971206738548at_nat @ F2 @ P2 ) ) ).

% case_swap
thf(fact_105_case__swap,axiom,
    ! [F2: nat > nat > product_prod_nat_nat,P2: product_prod_nat_nat] :
      ( ( produc2626176000494625587at_nat
        @ ^ [Y2: nat,X: nat] : ( F2 @ X @ Y2 )
        @ ( product_swap_nat_nat @ P2 ) )
      = ( produc2626176000494625587at_nat @ F2 @ P2 ) ) ).

% case_swap
thf(fact_106_case__swap,axiom,
    ! [F2: nat > nat > relational_term_a > relational_term_a,P2: product_prod_nat_nat] :
      ( ( produc6628518323692928499term_a
        @ ^ [Y2: nat,X: nat] : ( F2 @ X @ Y2 )
        @ ( product_swap_nat_nat @ P2 ) )
      = ( produc6628518323692928499term_a @ F2 @ P2 ) ) ).

% case_swap
thf(fact_107_case__swap,axiom,
    ! [F2: nat > nat > nat > produc7248412053542808358at_nat,P2: product_prod_nat_nat] :
      ( ( produc7810592499157111267at_nat
        @ ^ [Y2: nat,X: nat] : ( F2 @ X @ Y2 )
        @ ( product_swap_nat_nat @ P2 ) )
      = ( produc7810592499157111267at_nat @ F2 @ P2 ) ) ).

% case_swap
thf(fact_108_case__swap,axiom,
    ! [F2: nat > nat > relational_fmla_a_b > relational_fmla_a_b,P2: product_prod_nat_nat] :
      ( ( produc5586541307551673003la_a_b
        @ ^ [Y2: nat,X: nat] : ( F2 @ X @ Y2 )
        @ ( product_swap_nat_nat @ P2 ) )
      = ( produc5586541307551673003la_a_b @ F2 @ P2 ) ) ).

% case_swap
thf(fact_109_case__swap,axiom,
    ! [F2: nat > nat > relational_fmla_a_b,P2: product_prod_nat_nat] :
      ( ( produc3270801013941088237la_a_b
        @ ^ [Y2: nat,X: nat] : ( F2 @ X @ Y2 )
        @ ( product_swap_nat_nat @ P2 ) )
      = ( produc3270801013941088237la_a_b @ F2 @ P2 ) ) ).

% case_swap
thf(fact_110_gen_H_Ointros_I5_J,axiom,
    ! [X3: nat,Q1: relational_fmla_a_b,Q22: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
      ( ( relational_gen_a_b2 @ X3 @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q1 ) @ ( relational_Neg_a_b @ Q22 ) ) @ G )
     => ( relational_gen_a_b2 @ X3 @ ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q1 @ Q22 ) ) @ G ) ) ).

% gen'.intros(5)
thf(fact_111_gen_H_Ointros_I4_J,axiom,
    ! [X3: nat,Q1: relational_fmla_a_b,Q22: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
      ( ( relational_gen_a_b2 @ X3 @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q1 ) @ ( relational_Neg_a_b @ Q22 ) ) @ G )
     => ( relational_gen_a_b2 @ X3 @ ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) ) @ G ) ) ).

% gen'.intros(4)
thf(fact_112_substs__Eq,axiom,
    ! [Xs: list_nat,Ys: list_nat,X3: nat,T: relational_term_a] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( fold_P7970104616371074773la_a_b
          @ ( produc5586541307551673003la_a_b
            @ ^ [X: nat,Y2: nat,Q2: relational_fmla_a_b] : ( relational_subst_a_b @ Q2 @ X @ Y2 ) )
          @ ( zip_nat_nat @ Xs @ Ys )
          @ ( relational_Eq_a_b @ X3 @ T ) )
        = ( relational_Eq_a_b @ ( relati8128731020529265620ar_nat @ Xs @ Ys @ X3 )
          @ ( fold_P2653167865486626963term_a
            @ ( produc6628518323692928499term_a
              @ ^ [X: nat,Y2: nat,T2: relational_term_a] : ( relati7175845559408349773term_a @ T2 @ X @ Y2 ) )
            @ ( zip_nat_nat @ Xs @ Ys )
            @ T ) ) ) ) ).

% substs_Eq
thf(fact_113_subst_Osimps_I7_J,axiom,
    ! [X3: nat,Z: nat,Q: relational_fmla_a_b,Y: nat] :
      ( ( ( X3 = Z )
       => ( ( relational_subst_a_b @ ( relati591517084277583526ts_a_b @ Z @ Q ) @ X3 @ Y )
          = ( relati591517084277583526ts_a_b @ X3 @ Q ) ) )
      & ( ( X3 != Z )
       => ( ( ( Z = Y )
           => ( ( relational_subst_a_b @ ( relati591517084277583526ts_a_b @ Z @ Q ) @ X3 @ Y )
              = ( relati591517084277583526ts_a_b @ ( relati2677767559083392098h2_a_b @ X3 @ Y @ Q ) @ ( relational_subst_a_b @ ( relational_subst_a_b @ Q @ Z @ ( relati2677767559083392098h2_a_b @ X3 @ Y @ Q ) ) @ X3 @ Y ) ) ) )
          & ( ( Z != Y )
           => ( ( relational_subst_a_b @ ( relati591517084277583526ts_a_b @ Z @ Q ) @ X3 @ Y )
              = ( relati591517084277583526ts_a_b @ Z @ ( relational_subst_a_b @ Q @ X3 @ Y ) ) ) ) ) ) ) ).

% subst.simps(7)
thf(fact_114_gen_Ointros_I5_J,axiom,
    ! [X3: nat,Q1: relational_fmla_a_b,Q22: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
      ( ( relational_gen_a_b @ X3 @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q1 ) @ ( relational_Neg_a_b @ Q22 ) ) @ G )
     => ( relational_gen_a_b @ X3 @ ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q1 @ Q22 ) ) @ G ) ) ).

% gen.intros(5)
thf(fact_115_gen_Ointros_I4_J,axiom,
    ! [X3: nat,Q1: relational_fmla_a_b,Q22: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
      ( ( relational_gen_a_b @ X3 @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q1 ) @ ( relational_Neg_a_b @ Q22 ) ) @ G )
     => ( relational_gen_a_b @ X3 @ ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) ) @ G ) ) ).

% gen.intros(4)
thf(fact_116_fmla_Oinject_I3_J,axiom,
    ! [X31: nat,X32: relational_term_a,Y31: nat,Y32: relational_term_a] :
      ( ( ( relational_Eq_a_b @ X31 @ X32 )
        = ( relational_Eq_a_b @ Y31 @ Y32 ) )
      = ( ( X31 = Y31 )
        & ( X32 = Y32 ) ) ) ).

% fmla.inject(3)
thf(fact_117_size__subst,axiom,
    ! [Q: relational_fmla_a_b,X3: nat,Y: nat] :
      ( ( size_s453432777765377587la_a_b @ ( relational_subst_a_b @ Q @ X3 @ Y ) )
      = ( size_s453432777765377587la_a_b @ Q ) ) ).

% size_subst
thf(fact_118_size__subst__term,axiom,
    ! [T: relational_term_a,X3: nat,Y: nat] :
      ( ( size_s49661629988129973term_a @ ( relati7175845559408349773term_a @ T @ X3 @ Y ) )
      = ( size_s49661629988129973term_a @ T ) ) ).

% size_subst_term
thf(fact_119_swap__swap,axiom,
    ! [P2: product_prod_nat_nat] :
      ( ( product_swap_nat_nat @ ( product_swap_nat_nat @ P2 ) )
      = P2 ) ).

% swap_swap
thf(fact_120_gen__Bool__True,axiom,
    ! [X3: nat,G: set_Re381260168593705685la_a_b] :
      ~ ( relational_gen_a_b @ X3 @ ( relational_Bool_a_b @ $true ) @ G ) ).

% gen_Bool_True
thf(fact_121_gen__eq__gen_H,axiom,
    relational_gen_a_b = relational_gen_a_b2 ).

% gen_eq_gen'
thf(fact_122_gen__gen_H,axiom,
    ! [X3: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
      ( ( relational_gen_a_b @ X3 @ Q @ G )
     => ( relational_gen_a_b2 @ X3 @ Q @ G ) ) ).

% gen_gen'
thf(fact_123_gen_H__gen,axiom,
    ! [X3: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
      ( ( relational_gen_a_b2 @ X3 @ Q @ G )
     => ( relational_gen_a_b @ X3 @ Q @ G ) ) ).

% gen'_gen
thf(fact_124_fresh2_I1_J,axiom,
    ! [X3: nat,Y: nat,Q: relational_fmla_a_b] :
      ( X3
     != ( relati2677767559083392098h2_a_b @ X3 @ Y @ Q ) ) ).

% fresh2(1)
thf(fact_125_fresh2_I2_J,axiom,
    ! [Y: nat,X3: nat,Q: relational_fmla_a_b] :
      ( Y
     != ( relati2677767559083392098h2_a_b @ X3 @ Y @ Q ) ) ).

% fresh2(2)
thf(fact_126_subst_Osimps_I3_J,axiom,
    ! [Z: nat,T: relational_term_a,X3: nat,Y: nat] :
      ( ( relational_subst_a_b @ ( relational_Eq_a_b @ Z @ T ) @ X3 @ Y )
      = ( relational_Eq_a_b @ ( if_nat @ ( Z = X3 ) @ Y @ Z ) @ ( relati7175845559408349773term_a @ T @ X3 @ Y ) ) ) ).

% subst.simps(3)
thf(fact_127_gen_Ointros_I3_J,axiom,
    ! [X3: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
      ( ( relational_gen_a_b @ X3 @ Q @ G )
     => ( relational_gen_a_b @ X3 @ ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q ) ) @ G ) ) ).

% gen.intros(3)
thf(fact_128_gen_Ointros_I7_J,axiom,
    ! [X3: nat,Q1: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Q22: relational_fmla_a_b] :
      ( ( ( relational_gen_a_b @ X3 @ Q1 @ G )
        | ( relational_gen_a_b @ X3 @ Q22 @ G ) )
     => ( relational_gen_a_b @ X3 @ ( relational_Conj_a_b @ Q1 @ Q22 ) @ G ) ) ).

% gen.intros(7)
thf(fact_129_fmla_Odistinct_I23_J,axiom,
    ! [X31: nat,X32: relational_term_a,X4: relational_fmla_a_b] :
      ( ( relational_Eq_a_b @ X31 @ X32 )
     != ( relational_Neg_a_b @ X4 ) ) ).

% fmla.distinct(23)
thf(fact_130_fmla_Odistinct_I27_J,axiom,
    ! [X31: nat,X32: relational_term_a,X61: relational_fmla_a_b,X62: relational_fmla_a_b] :
      ( ( relational_Eq_a_b @ X31 @ X32 )
     != ( relational_Disj_a_b @ X61 @ X62 ) ) ).

% fmla.distinct(27)
thf(fact_131_fmla_Odistinct_I25_J,axiom,
    ! [X31: nat,X32: relational_term_a,X51: relational_fmla_a_b,X52: relational_fmla_a_b] :
      ( ( relational_Eq_a_b @ X31 @ X32 )
     != ( relational_Conj_a_b @ X51 @ X52 ) ) ).

% fmla.distinct(25)
thf(fact_132_fmla_Odistinct_I29_J,axiom,
    ! [X31: nat,X32: relational_term_a,X71: nat,X72: relational_fmla_a_b] :
      ( ( relational_Eq_a_b @ X31 @ X32 )
     != ( relati591517084277583526ts_a_b @ X71 @ X72 ) ) ).

% fmla.distinct(29)
thf(fact_133_fmla_Odistinct_I13_J,axiom,
    ! [X22: $o,X31: nat,X32: relational_term_a] :
      ( ( relational_Bool_a_b @ X22 )
     != ( relational_Eq_a_b @ X31 @ X32 ) ) ).

% fmla.distinct(13)
thf(fact_134_curry__K,axiom,
    ! [C: relational_fmla_a_b] :
      ( ( produc858456811296061068la_a_b
        @ ^ [X: product_prod_nat_nat] : C )
      = ( ^ [X: nat,Y2: nat] : C ) ) ).

% curry_K
thf(fact_135_curry__K,axiom,
    ! [C: relational_fmla_a_b > relational_fmla_a_b] :
      ( ( produc7541201833284165578la_a_b
        @ ^ [X: product_prod_nat_nat] : C )
      = ( ^ [X: nat,Y2: nat] : C ) ) ).

% curry_K
thf(fact_136_gen__qp,axiom,
    ! [X3: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Qqp: relational_fmla_a_b] :
      ( ( relational_gen_a_b @ X3 @ Q @ G )
     => ( ( member4680049679412964150la_a_b @ Qqp @ G )
       => ( relational_qp_a_b @ Qqp ) ) ) ).

% gen_qp
thf(fact_137_gen_H_Ointros_I3_J,axiom,
    ! [X3: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
      ( ( relational_gen_a_b2 @ X3 @ Q @ G )
     => ( relational_gen_a_b2 @ X3 @ ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q ) ) @ G ) ) ).

% gen'.intros(3)
thf(fact_138_gen_H_Ointros_I7_J,axiom,
    ! [X3: nat,Q1: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Q22: relational_fmla_a_b] :
      ( ( ( relational_gen_a_b2 @ X3 @ Q1 @ G )
        | ( relational_gen_a_b2 @ X3 @ Q22 @ G ) )
     => ( relational_gen_a_b2 @ X3 @ ( relational_Conj_a_b @ Q1 @ Q22 ) @ G ) ) ).

% gen'.intros(7)
thf(fact_139_gen_H__qp,axiom,
    ! [X3: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Qqp: relational_fmla_a_b] :
      ( ( relational_gen_a_b2 @ X3 @ Q @ G )
     => ( ( member4680049679412964150la_a_b @ Qqp @ G )
       => ( relational_qp_a_b @ Qqp ) ) ) ).

% gen'_qp
thf(fact_140_fresh2_I3_J,axiom,
    ! [X3: nat,Y: nat,Q: relational_fmla_a_b] :
      ~ ( member_nat @ ( relati2677767559083392098h2_a_b @ X3 @ Y @ Q ) @ ( relational_fv_a_b @ Q ) ) ).

% fresh2(3)
thf(fact_141_qp__impl_Osimps_I6_J,axiom,
    ! [V: relational_fmla_a_b] :
      ~ ( relati3725921752842749053pl_a_b @ ( relational_Neg_a_b @ V ) ) ).

% qp_impl.simps(6)
thf(fact_142_qp__impl_Osimps_I8_J,axiom,
    ! [V: relational_fmla_a_b,Va: relational_fmla_a_b] :
      ~ ( relati3725921752842749053pl_a_b @ ( relational_Disj_a_b @ V @ Va ) ) ).

% qp_impl.simps(8)
thf(fact_143_qp__impl_Osimps_I7_J,axiom,
    ! [V: relational_fmla_a_b,Va: relational_fmla_a_b] :
      ~ ( relati3725921752842749053pl_a_b @ ( relational_Conj_a_b @ V @ Va ) ) ).

% qp_impl.simps(7)
thf(fact_144_qp__impl_Osimps_I4_J,axiom,
    ! [V: $o] :
      ~ ( relati3725921752842749053pl_a_b @ ( relational_Bool_a_b @ V ) ) ).

% qp_impl.simps(4)
thf(fact_145_qp__impl__imp__qp,axiom,
    ! [Q: relational_fmla_a_b] :
      ( ( relati3725921752842749053pl_a_b @ Q )
     => ( relational_qp_a_b @ Q ) ) ).

% qp_impl_imp_qp
thf(fact_146_qp__imp__qp__impl,axiom,
    ! [Q: relational_fmla_a_b] :
      ( ( relational_qp_a_b @ Q )
     => ( relati3725921752842749053pl_a_b @ Q ) ) ).

% qp_imp_qp_impl
thf(fact_147_qp__code,axiom,
    relational_qp_a_b = relati3725921752842749053pl_a_b ).

% qp_code
thf(fact_148_qp__Gen,axiom,
    ! [Q: relational_fmla_a_b,X3: nat] :
      ( ( relational_qp_a_b @ Q )
     => ( ( member_nat @ X3 @ ( relational_fv_a_b @ Q ) )
       => ? [X_1: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X3 @ Q @ X_1 ) ) ) ).

% qp_Gen
thf(fact_149_substs__term__Var,axiom,
    ! [Xs: list_nat,Ys: list_nat,X3: nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( fold_P2653167865486626963term_a
          @ ( produc6628518323692928499term_a
            @ ^ [X: nat,Y2: nat,T2: relational_term_a] : ( relati7175845559408349773term_a @ T2 @ X @ Y2 ) )
          @ ( zip_nat_nat @ Xs @ Ys )
          @ ( relational_Var_a @ X3 ) )
        = ( relational_Var_a @ ( relati8128731020529265620ar_nat @ Xs @ Ys @ X3 ) ) ) ) ).

% substs_term_Var
thf(fact_150_substs__term__Const,axiom,
    ! [Xs: list_nat,Ys: list_nat,C: a] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( fold_P2653167865486626963term_a
          @ ( produc6628518323692928499term_a
            @ ^ [X: nat,Y2: nat,T2: relational_term_a] : ( relati7175845559408349773term_a @ T2 @ X @ Y2 ) )
          @ ( zip_nat_nat @ Xs @ Ys )
          @ ( relational_Const_a @ C ) )
        = ( relational_Const_a @ C ) ) ) ).

% substs_term_Const
thf(fact_151_qp__impl_Oelims_I3_J,axiom,
    ! [X3: relational_fmla_a_b] :
      ( ~ ( relati3725921752842749053pl_a_b @ X3 )
     => ( ! [X5: nat,Q3: relational_fmla_a_b] :
            ( ( X3
              = ( relati591517084277583526ts_a_b @ X5 @ Q3 ) )
           => ( ( member_nat @ X5 @ ( relational_fv_a_b @ Q3 ) )
              & ( relational_qp_a_b @ Q3 ) ) )
       => ( ! [V2: $o] :
              ( X3
             != ( relational_Bool_a_b @ V2 ) )
         => ( ! [V2: nat,Vb: nat] :
                ( X3
               != ( relational_Eq_a_b @ V2 @ ( relational_Var_a @ Vb ) ) )
           => ( ! [V2: relational_fmla_a_b] :
                  ( X3
                 != ( relational_Neg_a_b @ V2 ) )
             => ( ! [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
                    ( X3
                   != ( relational_Conj_a_b @ V2 @ Va2 ) )
               => ~ ! [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
                      ( X3
                     != ( relational_Disj_a_b @ V2 @ Va2 ) ) ) ) ) ) ) ) ).

% qp_impl.elims(3)
thf(fact_152_genempty_Osimps,axiom,
    ( relati5999705594545617851ty_a_b
    = ( ^ [A3: relational_fmla_a_b] :
          ( ( A3
            = ( relational_Bool_a_b @ $false ) )
          | ? [Q2: relational_fmla_a_b] :
              ( ( A3
                = ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q2 ) ) )
              & ( relati5999705594545617851ty_a_b @ Q2 ) )
          | ? [Q12: relational_fmla_a_b,Q23: relational_fmla_a_b] :
              ( ( A3
                = ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q12 @ Q23 ) ) )
              & ( relati5999705594545617851ty_a_b @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q12 ) @ ( relational_Neg_a_b @ Q23 ) ) ) )
          | ? [Q12: relational_fmla_a_b,Q23: relational_fmla_a_b] :
              ( ( A3
                = ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q12 @ Q23 ) ) )
              & ( relati5999705594545617851ty_a_b @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q12 ) @ ( relational_Neg_a_b @ Q23 ) ) ) )
          | ? [Q12: relational_fmla_a_b,Q23: relational_fmla_a_b] :
              ( ( A3
                = ( relational_Disj_a_b @ Q12 @ Q23 ) )
              & ( relati5999705594545617851ty_a_b @ Q12 )
              & ( relati5999705594545617851ty_a_b @ Q23 ) )
          | ? [Q12: relational_fmla_a_b,Q23: relational_fmla_a_b] :
              ( ( A3
                = ( relational_Conj_a_b @ Q12 @ Q23 ) )
              & ( ( relati5999705594545617851ty_a_b @ Q12 )
                | ( relati5999705594545617851ty_a_b @ Q23 ) ) )
          | ? [Q2: relational_fmla_a_b] :
              ( ? [X: nat,Y2: nat] :
                  ( A3
                  = ( relational_Conj_a_b @ Q2 @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y2 ) ) ) )
              & ( relati5999705594545617851ty_a_b @ Q2 ) )
          | ? [Q2: relational_fmla_a_b] :
              ( ? [Y2: nat] :
                  ( A3
                  = ( relati591517084277583526ts_a_b @ Y2 @ Q2 ) )
              & ( relati5999705594545617851ty_a_b @ Q2 ) ) ) ) ) ).

% genempty.simps
thf(fact_153_genempty_Ocases,axiom,
    ! [A: relational_fmla_a_b] :
      ( ( relati5999705594545617851ty_a_b @ A )
     => ( ( A
         != ( relational_Bool_a_b @ $false ) )
       => ( ! [Q3: relational_fmla_a_b] :
              ( ( A
                = ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q3 ) ) )
             => ~ ( relati5999705594545617851ty_a_b @ Q3 ) )
         => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                ( ( A
                  = ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q13 @ Q24 ) ) )
               => ~ ( relati5999705594545617851ty_a_b @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q13 ) @ ( relational_Neg_a_b @ Q24 ) ) ) )
           => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                  ( ( A
                    = ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q13 @ Q24 ) ) )
                 => ~ ( relati5999705594545617851ty_a_b @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q13 ) @ ( relational_Neg_a_b @ Q24 ) ) ) )
             => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                    ( ( A
                      = ( relational_Disj_a_b @ Q13 @ Q24 ) )
                   => ( ( relati5999705594545617851ty_a_b @ Q13 )
                     => ~ ( relati5999705594545617851ty_a_b @ Q24 ) ) )
               => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                      ( ( A
                        = ( relational_Conj_a_b @ Q13 @ Q24 ) )
                     => ~ ( ( relati5999705594545617851ty_a_b @ Q13 )
                          | ( relati5999705594545617851ty_a_b @ Q24 ) ) )
                 => ( ! [Q3: relational_fmla_a_b] :
                        ( ? [X5: nat,Y3: nat] :
                            ( A
                            = ( relational_Conj_a_b @ Q3 @ ( relational_Eq_a_b @ X5 @ ( relational_Var_a @ Y3 ) ) ) )
                       => ~ ( relati5999705594545617851ty_a_b @ Q3 ) )
                   => ( ! [Q3: relational_fmla_a_b] :
                          ( ? [Y3: nat,X5: nat] :
                              ( A
                              = ( relational_Conj_a_b @ Q3 @ ( relational_Eq_a_b @ Y3 @ ( relational_Var_a @ X5 ) ) ) )
                         => ~ ( relati5999705594545617851ty_a_b @ Q3 ) )
                     => ~ ! [Q3: relational_fmla_a_b] :
                            ( ? [Y3: nat] :
                                ( A
                                = ( relati591517084277583526ts_a_b @ Y3 @ Q3 ) )
                           => ~ ( relati5999705594545617851ty_a_b @ Q3 ) ) ) ) ) ) ) ) ) ) ) ).

% genempty.cases
thf(fact_154_fmla_Oexhaust,axiom,
    ! [Y: relational_fmla_a_b] :
      ( ! [X11: b,X12: list_R6823256787227418703term_a] :
          ( Y
         != ( relational_Pred_b_a @ X11 @ X12 ) )
     => ( ! [X23: $o] :
            ( Y
           != ( relational_Bool_a_b @ X23 ) )
       => ( ! [X312: nat,X322: relational_term_a] :
              ( Y
             != ( relational_Eq_a_b @ X312 @ X322 ) )
         => ( ! [X42: relational_fmla_a_b] :
                ( Y
               != ( relational_Neg_a_b @ X42 ) )
           => ( ! [X512: relational_fmla_a_b,X522: relational_fmla_a_b] :
                  ( Y
                 != ( relational_Conj_a_b @ X512 @ X522 ) )
             => ( ! [X612: relational_fmla_a_b,X622: relational_fmla_a_b] :
                    ( Y
                   != ( relational_Disj_a_b @ X612 @ X622 ) )
               => ~ ! [X712: nat,X722: relational_fmla_a_b] :
                      ( Y
                     != ( relati591517084277583526ts_a_b @ X712 @ X722 ) ) ) ) ) ) ) ) ).

% fmla.exhaust
thf(fact_155_fmla_Oinject_I1_J,axiom,
    ! [X112: b,X122: list_R6823256787227418703term_a,Y11: b,Y12: list_R6823256787227418703term_a] :
      ( ( ( relational_Pred_b_a @ X112 @ X122 )
        = ( relational_Pred_b_a @ Y11 @ Y12 ) )
      = ( ( X112 = Y11 )
        & ( X122 = Y12 ) ) ) ).

% fmla.inject(1)
thf(fact_156_term_Oinject_I2_J,axiom,
    ! [X22: nat,Y22: nat] :
      ( ( ( relational_Var_a @ X22 )
        = ( relational_Var_a @ Y22 ) )
      = ( X22 = Y22 ) ) ).

% term.inject(2)
thf(fact_157_term_Oinject_I1_J,axiom,
    ! [X13: a,Y1: a] :
      ( ( ( relational_Const_a @ X13 )
        = ( relational_Const_a @ Y1 ) )
      = ( X13 = Y1 ) ) ).

% term.inject(1)
thf(fact_158_term_Odistinct_I1_J,axiom,
    ! [X13: a,X22: nat] :
      ( ( relational_Const_a @ X13 )
     != ( relational_Var_a @ X22 ) ) ).

% term.distinct(1)
thf(fact_159_Relational__Calculus_Oterm_Oexhaust,axiom,
    ! [Y: relational_term_a] :
      ( ! [X14: a] :
          ( Y
         != ( relational_Const_a @ X14 ) )
     => ~ ! [X23: nat] :
            ( Y
           != ( relational_Var_a @ X23 ) ) ) ).

% Relational_Calculus.term.exhaust
thf(fact_160_fv__fo__term__list_Ocases,axiom,
    ! [X3: relational_term_a] :
      ( ! [N2: nat] :
          ( X3
         != ( relational_Var_a @ N2 ) )
     => ~ ! [V2: a] :
            ( X3
           != ( relational_Const_a @ V2 ) ) ) ).

% fv_fo_term_list.cases
thf(fact_161_subst__term_Oelims,axiom,
    ! [X3: relational_term_a,Xa: nat,Xb: nat,Y: relational_term_a] :
      ( ( ( relati7175845559408349773term_a @ X3 @ Xa @ Xb )
        = Y )
     => ( ! [Z2: nat] :
            ( ( X3
              = ( relational_Var_a @ Z2 ) )
           => ( Y
             != ( relational_Var_a @ ( if_nat @ ( Xa = Z2 ) @ Xb @ Z2 ) ) ) )
       => ~ ! [C2: a] :
              ( ( X3
                = ( relational_Const_a @ C2 ) )
             => ( Y
               != ( relational_Const_a @ C2 ) ) ) ) ) ).

% subst_term.elims
thf(fact_162_ap_Osimps,axiom,
    ( relational_ap_a_b
    = ( ^ [A3: relational_fmla_a_b] :
          ( ? [P3: b,Ts: list_R6823256787227418703term_a] :
              ( A3
              = ( relational_Pred_b_a @ P3 @ Ts ) )
          | ? [X: nat,C3: a] :
              ( A3
              = ( relational_Eq_a_b @ X @ ( relational_Const_a @ C3 ) ) ) ) ) ) ).

% ap.simps
thf(fact_163_ap_Ocases,axiom,
    ! [A: relational_fmla_a_b] :
      ( ( relational_ap_a_b @ A )
     => ( ! [P4: b,Ts2: list_R6823256787227418703term_a] :
            ( A
           != ( relational_Pred_b_a @ P4 @ Ts2 ) )
       => ~ ! [X5: nat,C2: a] :
              ( A
             != ( relational_Eq_a_b @ X5 @ ( relational_Const_a @ C2 ) ) ) ) ) ).

% ap.cases
thf(fact_164_fmla_Odistinct_I5_J,axiom,
    ! [X112: b,X122: list_R6823256787227418703term_a,X4: relational_fmla_a_b] :
      ( ( relational_Pred_b_a @ X112 @ X122 )
     != ( relational_Neg_a_b @ X4 ) ) ).

% fmla.distinct(5)
thf(fact_165_fmla_Odistinct_I9_J,axiom,
    ! [X112: b,X122: list_R6823256787227418703term_a,X61: relational_fmla_a_b,X62: relational_fmla_a_b] :
      ( ( relational_Pred_b_a @ X112 @ X122 )
     != ( relational_Disj_a_b @ X61 @ X62 ) ) ).

% fmla.distinct(9)
thf(fact_166_fmla_Odistinct_I7_J,axiom,
    ! [X112: b,X122: list_R6823256787227418703term_a,X51: relational_fmla_a_b,X52: relational_fmla_a_b] :
      ( ( relational_Pred_b_a @ X112 @ X122 )
     != ( relational_Conj_a_b @ X51 @ X52 ) ) ).

% fmla.distinct(7)
thf(fact_167_fmla_Odistinct_I11_J,axiom,
    ! [X112: b,X122: list_R6823256787227418703term_a,X71: nat,X72: relational_fmla_a_b] :
      ( ( relational_Pred_b_a @ X112 @ X122 )
     != ( relati591517084277583526ts_a_b @ X71 @ X72 ) ) ).

% fmla.distinct(11)
thf(fact_168_fmla_Odistinct_I3_J,axiom,
    ! [X112: b,X122: list_R6823256787227418703term_a,X31: nat,X32: relational_term_a] :
      ( ( relational_Pred_b_a @ X112 @ X122 )
     != ( relational_Eq_a_b @ X31 @ X32 ) ) ).

% fmla.distinct(3)
thf(fact_169_fmla_Odistinct_I1_J,axiom,
    ! [X112: b,X122: list_R6823256787227418703term_a,X22: $o] :
      ( ( relational_Pred_b_a @ X112 @ X122 )
     != ( relational_Bool_a_b @ X22 ) ) ).

% fmla.distinct(1)
thf(fact_170_Pred,axiom,
    ! [P2: b,Ts3: list_R6823256787227418703term_a] : ( relational_ap_a_b @ ( relational_Pred_b_a @ P2 @ Ts3 ) ) ).

% Pred
thf(fact_171_subst__term_Osimps_I1_J,axiom,
    ! [Z: nat,X3: nat,Y: nat] :
      ( ( relati7175845559408349773term_a @ ( relational_Var_a @ Z ) @ X3 @ Y )
      = ( relational_Var_a @ ( if_nat @ ( X3 = Z ) @ Y @ Z ) ) ) ).

% subst_term.simps(1)
thf(fact_172_qp__impl_Osimps_I2_J,axiom,
    ! [X3: b,Ts3: list_R6823256787227418703term_a] : ( relati3725921752842749053pl_a_b @ ( relational_Pred_b_a @ X3 @ Ts3 ) ) ).

% qp_impl.simps(2)
thf(fact_173_subst__term_Osimps_I2_J,axiom,
    ! [C: a,X3: nat,Y: nat] :
      ( ( relati7175845559408349773term_a @ ( relational_Const_a @ C ) @ X3 @ Y )
      = ( relational_Const_a @ C ) ) ).

% subst_term.simps(2)
thf(fact_174_qp__impl_Ocases,axiom,
    ! [X3: relational_fmla_a_b] :
      ( ! [X5: nat,C2: a] :
          ( X3
         != ( relational_Eq_a_b @ X5 @ ( relational_Const_a @ C2 ) ) )
     => ( ! [X5: b,Ts2: list_R6823256787227418703term_a] :
            ( X3
           != ( relational_Pred_b_a @ X5 @ Ts2 ) )
       => ( ! [X5: nat,Q3: relational_fmla_a_b] :
              ( X3
             != ( relati591517084277583526ts_a_b @ X5 @ Q3 ) )
         => ( ! [V2: $o] :
                ( X3
               != ( relational_Bool_a_b @ V2 ) )
           => ( ! [V2: nat,Vb: nat] :
                  ( X3
                 != ( relational_Eq_a_b @ V2 @ ( relational_Var_a @ Vb ) ) )
             => ( ! [V2: relational_fmla_a_b] :
                    ( X3
                   != ( relational_Neg_a_b @ V2 ) )
               => ( ! [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
                      ( X3
                     != ( relational_Conj_a_b @ V2 @ Va2 ) )
                 => ~ ! [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
                        ( X3
                       != ( relational_Disj_a_b @ V2 @ Va2 ) ) ) ) ) ) ) ) ) ).

% qp_impl.cases
thf(fact_175_qp__eq,axiom,
    ! [X3: nat,Y: nat] :
      ~ ( relational_qp_a_b @ ( relational_Eq_a_b @ X3 @ ( relational_Var_a @ Y ) ) ) ).

% qp_eq
thf(fact_176_qp__impl_Osimps_I5_J,axiom,
    ! [V: nat,Vb2: nat] :
      ~ ( relati3725921752842749053pl_a_b @ ( relational_Eq_a_b @ V @ ( relational_Var_a @ Vb2 ) ) ) ).

% qp_impl.simps(5)
thf(fact_177_Eqc,axiom,
    ! [X3: nat,C: a] : ( relational_ap_a_b @ ( relational_Eq_a_b @ X3 @ ( relational_Const_a @ C ) ) ) ).

% Eqc
thf(fact_178_qp__impl_Osimps_I1_J,axiom,
    ! [X3: nat,C: a] : ( relati3725921752842749053pl_a_b @ ( relational_Eq_a_b @ X3 @ ( relational_Const_a @ C ) ) ) ).

% qp_impl.simps(1)
thf(fact_179_qp__impl_Oelims_I2_J,axiom,
    ! [X3: relational_fmla_a_b] :
      ( ( relati3725921752842749053pl_a_b @ X3 )
     => ( ! [X5: nat,C2: a] :
            ( X3
           != ( relational_Eq_a_b @ X5 @ ( relational_Const_a @ C2 ) ) )
       => ( ! [X5: b,Ts2: list_R6823256787227418703term_a] :
              ( X3
             != ( relational_Pred_b_a @ X5 @ Ts2 ) )
         => ~ ! [X5: nat,Q3: relational_fmla_a_b] :
                ( ( X3
                  = ( relati591517084277583526ts_a_b @ X5 @ Q3 ) )
               => ~ ( ( member_nat @ X5 @ ( relational_fv_a_b @ Q3 ) )
                    & ( relational_qp_a_b @ Q3 ) ) ) ) ) ) ).

% qp_impl.elims(2)
thf(fact_180_qp__impl_Oelims_I1_J,axiom,
    ! [X3: relational_fmla_a_b,Y: $o] :
      ( ( ( relati3725921752842749053pl_a_b @ X3 )
        = Y )
     => ( ( ? [X5: nat,C2: a] :
              ( X3
              = ( relational_Eq_a_b @ X5 @ ( relational_Const_a @ C2 ) ) )
         => ~ Y )
       => ( ( ? [X5: b,Ts2: list_R6823256787227418703term_a] :
                ( X3
                = ( relational_Pred_b_a @ X5 @ Ts2 ) )
           => ~ Y )
         => ( ! [X5: nat,Q3: relational_fmla_a_b] :
                ( ( X3
                  = ( relati591517084277583526ts_a_b @ X5 @ Q3 ) )
               => ( Y
                  = ( ~ ( ( member_nat @ X5 @ ( relational_fv_a_b @ Q3 ) )
                        & ( relational_qp_a_b @ Q3 ) ) ) ) )
           => ( ( ? [V2: $o] :
                    ( X3
                    = ( relational_Bool_a_b @ V2 ) )
               => Y )
             => ( ( ? [V2: nat,Vb: nat] :
                      ( X3
                      = ( relational_Eq_a_b @ V2 @ ( relational_Var_a @ Vb ) ) )
                 => Y )
               => ( ( ? [V2: relational_fmla_a_b] :
                        ( X3
                        = ( relational_Neg_a_b @ V2 ) )
                   => Y )
                 => ( ( ? [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
                          ( X3
                          = ( relational_Conj_a_b @ V2 @ Va2 ) )
                     => Y )
                   => ~ ( ? [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
                            ( X3
                            = ( relational_Disj_a_b @ V2 @ Va2 ) )
                       => Y ) ) ) ) ) ) ) ) ) ).

% qp_impl.elims(1)
thf(fact_181_genempty_Ointros_I8_J,axiom,
    ! [Q: relational_fmla_a_b,Y: nat,X3: nat] :
      ( ( relati5999705594545617851ty_a_b @ Q )
     => ( relati5999705594545617851ty_a_b @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ Y @ ( relational_Var_a @ X3 ) ) ) ) ) ).

% genempty.intros(8)
thf(fact_182_genempty_Ointros_I7_J,axiom,
    ! [Q: relational_fmla_a_b,X3: nat,Y: nat] :
      ( ( relati5999705594545617851ty_a_b @ Q )
     => ( relati5999705594545617851ty_a_b @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ X3 @ ( relational_Var_a @ Y ) ) ) ) ) ).

% genempty.intros(7)
thf(fact_183_cp_Ocases,axiom,
    ! [X3: relational_fmla_a_b] :
      ( ! [X5: nat,T3: relational_term_a] :
          ( X3
         != ( relational_Eq_a_b @ X5 @ T3 ) )
     => ( ! [Q3: relational_fmla_a_b] :
            ( X3
           != ( relational_Neg_a_b @ Q3 ) )
       => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
              ( X3
             != ( relational_Conj_a_b @ Q13 @ Q24 ) )
         => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                ( X3
               != ( relational_Disj_a_b @ Q13 @ Q24 ) )
           => ( ! [X5: nat,Q3: relational_fmla_a_b] :
                  ( X3
                 != ( relati591517084277583526ts_a_b @ X5 @ Q3 ) )
             => ( ! [V2: b,Va2: list_R6823256787227418703term_a] :
                    ( X3
                   != ( relational_Pred_b_a @ V2 @ Va2 ) )
               => ~ ! [V2: $o] :
                      ( X3
                     != ( relational_Bool_a_b @ V2 ) ) ) ) ) ) ) ) ).

% cp.cases
thf(fact_184_fv_Ocases,axiom,
    ! [X3: relational_fmla_a_b] :
      ( ! [Uu: b,Ts2: list_R6823256787227418703term_a] :
          ( X3
         != ( relational_Pred_b_a @ Uu @ Ts2 ) )
     => ( ! [B2: $o] :
            ( X3
           != ( relational_Bool_a_b @ B2 ) )
       => ( ! [X5: nat,T4: relational_term_a] :
              ( X3
             != ( relational_Eq_a_b @ X5 @ T4 ) )
         => ( ! [Phi2: relational_fmla_a_b] :
                ( X3
               != ( relational_Neg_a_b @ Phi2 ) )
           => ( ! [Phi2: relational_fmla_a_b,Psi: relational_fmla_a_b] :
                  ( X3
                 != ( relational_Conj_a_b @ Phi2 @ Psi ) )
             => ( ! [Phi2: relational_fmla_a_b,Psi: relational_fmla_a_b] :
                    ( X3
                   != ( relational_Disj_a_b @ Phi2 @ Psi ) )
               => ~ ! [Z2: nat,Phi2: relational_fmla_a_b] :
                      ( X3
                     != ( relati591517084277583526ts_a_b @ Z2 @ Phi2 ) ) ) ) ) ) ) ) ).

% fv.cases
thf(fact_185_nocp_Ocases,axiom,
    ! [X3: relational_fmla_a_b] :
      ( ! [B2: $o] :
          ( X3
         != ( relational_Bool_a_b @ B2 ) )
     => ( ! [P4: b,Ts2: list_R6823256787227418703term_a] :
            ( X3
           != ( relational_Pred_b_a @ P4 @ Ts2 ) )
       => ( ! [X5: nat,T3: relational_term_a] :
              ( X3
             != ( relational_Eq_a_b @ X5 @ T3 ) )
         => ( ! [Q3: relational_fmla_a_b] :
                ( X3
               != ( relational_Neg_a_b @ Q3 ) )
           => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                  ( X3
                 != ( relational_Conj_a_b @ Q13 @ Q24 ) )
             => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                    ( X3
                   != ( relational_Disj_a_b @ Q13 @ Q24 ) )
               => ~ ! [X5: nat,Q3: relational_fmla_a_b] :
                      ( X3
                     != ( relati591517084277583526ts_a_b @ X5 @ Q3 ) ) ) ) ) ) ) ) ).

% nocp.cases
thf(fact_186_subst_Oelims,axiom,
    ! [X3: relational_fmla_a_b,Xa: nat,Xb: nat,Y: relational_fmla_a_b] :
      ( ( ( relational_subst_a_b @ X3 @ Xa @ Xb )
        = Y )
     => ( ! [T3: $o] :
            ( ( X3
              = ( relational_Bool_a_b @ T3 ) )
           => ( Y
             != ( relational_Bool_a_b @ T3 ) ) )
       => ( ! [P4: b,Ts2: list_R6823256787227418703term_a] :
              ( ( X3
                = ( relational_Pred_b_a @ P4 @ Ts2 ) )
             => ( Y
               != ( relational_Pred_b_a @ P4
                  @ ( map_Re5736185711816362116term_a
                    @ ^ [T2: relational_term_a] : ( relati7175845559408349773term_a @ T2 @ Xa @ Xb )
                    @ Ts2 ) ) ) )
         => ( ! [Z2: nat,T3: relational_term_a] :
                ( ( X3
                  = ( relational_Eq_a_b @ Z2 @ T3 ) )
               => ( Y
                 != ( relational_Eq_a_b @ ( if_nat @ ( Z2 = Xa ) @ Xb @ Z2 ) @ ( relati7175845559408349773term_a @ T3 @ Xa @ Xb ) ) ) )
           => ( ! [Q3: relational_fmla_a_b] :
                  ( ( X3
                    = ( relational_Neg_a_b @ Q3 ) )
                 => ( Y
                   != ( relational_Neg_a_b @ ( relational_subst_a_b @ Q3 @ Xa @ Xb ) ) ) )
             => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                    ( ( X3
                      = ( relational_Conj_a_b @ Q13 @ Q24 ) )
                   => ( Y
                     != ( relational_Conj_a_b @ ( relational_subst_a_b @ Q13 @ Xa @ Xb ) @ ( relational_subst_a_b @ Q24 @ Xa @ Xb ) ) ) )
               => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                      ( ( X3
                        = ( relational_Disj_a_b @ Q13 @ Q24 ) )
                     => ( Y
                       != ( relational_Disj_a_b @ ( relational_subst_a_b @ Q13 @ Xa @ Xb ) @ ( relational_subst_a_b @ Q24 @ Xa @ Xb ) ) ) )
                 => ~ ! [Z2: nat,Q3: relational_fmla_a_b] :
                        ( ( X3
                          = ( relati591517084277583526ts_a_b @ Z2 @ Q3 ) )
                       => ~ ( ( ( Xa = Z2 )
                             => ( Y
                                = ( relati591517084277583526ts_a_b @ Xa @ Q3 ) ) )
                            & ( ( Xa != Z2 )
                             => ( ( ( Z2 = Xb )
                                 => ( Y
                                    = ( relati591517084277583526ts_a_b @ ( relati2677767559083392098h2_a_b @ Xa @ Xb @ Q3 ) @ ( relational_subst_a_b @ ( relational_subst_a_b @ Q3 @ Z2 @ ( relati2677767559083392098h2_a_b @ Xa @ Xb @ Q3 ) ) @ Xa @ Xb ) ) ) )
                                & ( ( Z2 != Xb )
                                 => ( Y
                                    = ( relati591517084277583526ts_a_b @ Z2 @ ( relational_subst_a_b @ Q3 @ Xa @ Xb ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% subst.elims
thf(fact_187_nocp_Oelims_I1_J,axiom,
    ! [X3: relational_fmla_a_b,Y: $o] :
      ( ( ( relational_nocp_a_b @ X3 )
        = Y )
     => ( ( ? [B2: $o] :
              ( X3
              = ( relational_Bool_a_b @ B2 ) )
         => Y )
       => ( ( ? [P4: b,Ts2: list_R6823256787227418703term_a] :
                ( X3
                = ( relational_Pred_b_a @ P4 @ Ts2 ) )
           => ~ Y )
         => ( ! [X5: nat,T3: relational_term_a] :
                ( ( X3
                  = ( relational_Eq_a_b @ X5 @ T3 ) )
               => ( Y
                  = ( T3
                    = ( relational_Var_a @ X5 ) ) ) )
           => ( ! [Q3: relational_fmla_a_b] :
                  ( ( X3
                    = ( relational_Neg_a_b @ Q3 ) )
                 => ( Y
                    = ( ~ ( relational_nocp_a_b @ Q3 ) ) ) )
             => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                    ( ( X3
                      = ( relational_Conj_a_b @ Q13 @ Q24 ) )
                   => ( Y
                      = ( ~ ( ( relational_nocp_a_b @ Q13 )
                            & ( relational_nocp_a_b @ Q24 ) ) ) ) )
               => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                      ( ( X3
                        = ( relational_Disj_a_b @ Q13 @ Q24 ) )
                     => ( Y
                        = ( ~ ( ( relational_nocp_a_b @ Q13 )
                              & ( relational_nocp_a_b @ Q24 ) ) ) ) )
                 => ~ ! [X5: nat,Q3: relational_fmla_a_b] :
                        ( ( X3
                          = ( relati591517084277583526ts_a_b @ X5 @ Q3 ) )
                       => ( Y
                          = ( ~ ( ( member_nat @ X5 @ ( relational_fv_a_b @ Q3 ) )
                                & ( relational_nocp_a_b @ Q3 ) ) ) ) ) ) ) ) ) ) ) ) ).

% nocp.elims(1)
thf(fact_188_nocp_Oelims_I2_J,axiom,
    ! [X3: relational_fmla_a_b] :
      ( ( relational_nocp_a_b @ X3 )
     => ( ! [P4: b,Ts2: list_R6823256787227418703term_a] :
            ( X3
           != ( relational_Pred_b_a @ P4 @ Ts2 ) )
       => ( ! [X5: nat,T3: relational_term_a] :
              ( ( X3
                = ( relational_Eq_a_b @ X5 @ T3 ) )
             => ( T3
                = ( relational_Var_a @ X5 ) ) )
         => ( ! [Q3: relational_fmla_a_b] :
                ( ( X3
                  = ( relational_Neg_a_b @ Q3 ) )
               => ~ ( relational_nocp_a_b @ Q3 ) )
           => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                  ( ( X3
                    = ( relational_Conj_a_b @ Q13 @ Q24 ) )
                 => ~ ( ( relational_nocp_a_b @ Q13 )
                      & ( relational_nocp_a_b @ Q24 ) ) )
             => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                    ( ( X3
                      = ( relational_Disj_a_b @ Q13 @ Q24 ) )
                   => ~ ( ( relational_nocp_a_b @ Q13 )
                        & ( relational_nocp_a_b @ Q24 ) ) )
               => ~ ! [X5: nat,Q3: relational_fmla_a_b] :
                      ( ( X3
                        = ( relati591517084277583526ts_a_b @ X5 @ Q3 ) )
                     => ~ ( ( member_nat @ X5 @ ( relational_fv_a_b @ Q3 ) )
                          & ( relational_nocp_a_b @ Q3 ) ) ) ) ) ) ) ) ) ).

% nocp.elims(2)
thf(fact_189_nocp_Oelims_I3_J,axiom,
    ! [X3: relational_fmla_a_b] :
      ( ~ ( relational_nocp_a_b @ X3 )
     => ( ! [B2: $o] :
            ( X3
           != ( relational_Bool_a_b @ B2 ) )
       => ( ! [X5: nat,T3: relational_term_a] :
              ( ( X3
                = ( relational_Eq_a_b @ X5 @ T3 ) )
             => ( T3
               != ( relational_Var_a @ X5 ) ) )
         => ( ! [Q3: relational_fmla_a_b] :
                ( ( X3
                  = ( relational_Neg_a_b @ Q3 ) )
               => ( relational_nocp_a_b @ Q3 ) )
           => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                  ( ( X3
                    = ( relational_Conj_a_b @ Q13 @ Q24 ) )
                 => ( ( relational_nocp_a_b @ Q13 )
                    & ( relational_nocp_a_b @ Q24 ) ) )
             => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                    ( ( X3
                      = ( relational_Disj_a_b @ Q13 @ Q24 ) )
                   => ( ( relational_nocp_a_b @ Q13 )
                      & ( relational_nocp_a_b @ Q24 ) ) )
               => ~ ! [X5: nat,Q3: relational_fmla_a_b] :
                      ( ( X3
                        = ( relati591517084277583526ts_a_b @ X5 @ Q3 ) )
                     => ( ( member_nat @ X5 @ ( relational_fv_a_b @ Q3 ) )
                        & ( relational_nocp_a_b @ Q3 ) ) ) ) ) ) ) ) ) ).

% nocp.elims(3)
thf(fact_190_qp__impl_Opelims_I1_J,axiom,
    ! [X3: relational_fmla_a_b,Y: $o] :
      ( ( ( relati3725921752842749053pl_a_b @ X3 )
        = Y )
     => ( ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ X3 )
       => ( ! [X5: nat,C2: a] :
              ( ( X3
                = ( relational_Eq_a_b @ X5 @ ( relational_Const_a @ C2 ) ) )
             => ( Y
               => ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Eq_a_b @ X5 @ ( relational_Const_a @ C2 ) ) ) ) )
         => ( ! [X5: b,Ts2: list_R6823256787227418703term_a] :
                ( ( X3
                  = ( relational_Pred_b_a @ X5 @ Ts2 ) )
               => ( Y
                 => ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Pred_b_a @ X5 @ Ts2 ) ) ) )
           => ( ! [X5: nat,Q3: relational_fmla_a_b] :
                  ( ( X3
                    = ( relati591517084277583526ts_a_b @ X5 @ Q3 ) )
                 => ( ( Y
                      = ( ( member_nat @ X5 @ ( relational_fv_a_b @ Q3 ) )
                        & ( relational_qp_a_b @ Q3 ) ) )
                   => ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relati591517084277583526ts_a_b @ X5 @ Q3 ) ) ) )
             => ( ! [V2: $o] :
                    ( ( X3
                      = ( relational_Bool_a_b @ V2 ) )
                   => ( ~ Y
                     => ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Bool_a_b @ V2 ) ) ) )
               => ( ! [V2: nat,Vb: nat] :
                      ( ( X3
                        = ( relational_Eq_a_b @ V2 @ ( relational_Var_a @ Vb ) ) )
                     => ( ~ Y
                       => ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Eq_a_b @ V2 @ ( relational_Var_a @ Vb ) ) ) ) )
                 => ( ! [V2: relational_fmla_a_b] :
                        ( ( X3
                          = ( relational_Neg_a_b @ V2 ) )
                       => ( ~ Y
                         => ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Neg_a_b @ V2 ) ) ) )
                   => ( ! [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
                          ( ( X3
                            = ( relational_Conj_a_b @ V2 @ Va2 ) )
                         => ( ~ Y
                           => ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Conj_a_b @ V2 @ Va2 ) ) ) )
                     => ~ ! [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
                            ( ( X3
                              = ( relational_Disj_a_b @ V2 @ Va2 ) )
                           => ( ~ Y
                             => ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Disj_a_b @ V2 @ Va2 ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% qp_impl.pelims(1)
thf(fact_191_qp__impl_Opelims_I3_J,axiom,
    ! [X3: relational_fmla_a_b] :
      ( ~ ( relati3725921752842749053pl_a_b @ X3 )
     => ( ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ X3 )
       => ( ! [X5: nat,Q3: relational_fmla_a_b] :
              ( ( X3
                = ( relati591517084277583526ts_a_b @ X5 @ Q3 ) )
             => ( ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relati591517084277583526ts_a_b @ X5 @ Q3 ) )
               => ( ( member_nat @ X5 @ ( relational_fv_a_b @ Q3 ) )
                  & ( relational_qp_a_b @ Q3 ) ) ) )
         => ( ! [V2: $o] :
                ( ( X3
                  = ( relational_Bool_a_b @ V2 ) )
               => ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Bool_a_b @ V2 ) ) )
           => ( ! [V2: nat,Vb: nat] :
                  ( ( X3
                    = ( relational_Eq_a_b @ V2 @ ( relational_Var_a @ Vb ) ) )
                 => ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Eq_a_b @ V2 @ ( relational_Var_a @ Vb ) ) ) )
             => ( ! [V2: relational_fmla_a_b] :
                    ( ( X3
                      = ( relational_Neg_a_b @ V2 ) )
                   => ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Neg_a_b @ V2 ) ) )
               => ( ! [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
                      ( ( X3
                        = ( relational_Conj_a_b @ V2 @ Va2 ) )
                     => ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Conj_a_b @ V2 @ Va2 ) ) )
                 => ~ ! [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
                        ( ( X3
                          = ( relational_Disj_a_b @ V2 @ Va2 ) )
                       => ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Disj_a_b @ V2 @ Va2 ) ) ) ) ) ) ) ) ) ) ).

% qp_impl.pelims(3)
thf(fact_192_gen_H_Ointros_I9_J,axiom,
    ! [Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,X3: nat] :
      ( ( relational_gen_a_b2 @ Y @ Q @ G )
     => ( relational_gen_a_b2 @ X3 @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ Y @ ( relational_Var_a @ X3 ) ) )
        @ ( image_6790371041703824709la_a_b
          @ ^ [Q2: relational_fmla_a_b] : ( relational_subst_a_b @ Q2 @ Y @ X3 )
          @ G ) ) ) ).

% gen'.intros(9)
thf(fact_193_cp__idem,axiom,
    ! [Q: relational_fmla_a_b] :
      ( ( relational_cp_a_b @ ( relational_cp_a_b @ Q ) )
      = ( relational_cp_a_b @ Q ) ) ).

% cp_idem
thf(fact_194_map__ident,axiom,
    ( ( map_Pr8058819605623181956at_nat
      @ ^ [X: product_prod_nat_nat] : X )
    = ( ^ [Xs3: list_P6011104703257516679at_nat] : Xs3 ) ) ).

% map_ident
thf(fact_195_map__ident,axiom,
    ( ( map_nat_nat
      @ ^ [X: nat] : X )
    = ( ^ [Xs3: list_nat] : Xs3 ) ) ).

% map_ident
thf(fact_196_length__map,axiom,
    ! [F2: product_prod_nat_nat > relational_fmla_a_b,Xs: list_P6011104703257516679at_nat] :
      ( ( size_s5655194420294726339la_a_b @ ( map_Pr2810398200501793500la_a_b @ F2 @ Xs ) )
      = ( size_s5460976970255530739at_nat @ Xs ) ) ).

% length_map
thf(fact_197_length__map,axiom,
    ! [F2: product_prod_nat_nat > product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( size_s5460976970255530739at_nat @ ( map_Pr8058819605623181956at_nat @ F2 @ Xs ) )
      = ( size_s5460976970255530739at_nat @ Xs ) ) ).

% length_map
thf(fact_198_length__map,axiom,
    ! [F2: product_prod_nat_nat > relational_fmla_a_b > relational_fmla_a_b,Xs: list_P6011104703257516679at_nat] :
      ( ( size_s1780709169027788545la_a_b @ ( map_Pr591601166967198746la_a_b @ F2 @ Xs ) )
      = ( size_s5460976970255530739at_nat @ Xs ) ) ).

% length_map
thf(fact_199_length__map,axiom,
    ! [F2: nat > product_prod_nat_nat,Xs: list_nat] :
      ( ( size_s5460976970255530739at_nat @ ( map_na7298421622053143531at_nat @ F2 @ Xs ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_map
thf(fact_200_length__map,axiom,
    ! [F2: relational_term_a > nat,Xs: list_R6823256787227418703term_a] :
      ( ( size_size_list_nat @ ( map_Re700991392905337813_a_nat @ F2 @ Xs ) )
      = ( size_s88622898042387131term_a @ Xs ) ) ).

% length_map
thf(fact_201_length__map,axiom,
    ! [F2: nat > relational_term_a,Xs: list_nat] :
      ( ( size_s88622898042387131term_a @ ( map_na4360460618005746227term_a @ F2 @ Xs ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_map
thf(fact_202_length__map,axiom,
    ! [F2: relational_term_a > relational_term_a,Xs: list_R6823256787227418703term_a] :
      ( ( size_s88622898042387131term_a @ ( map_Re5736185711816362116term_a @ F2 @ Xs ) )
      = ( size_s88622898042387131term_a @ Xs ) ) ).

% length_map
thf(fact_203_length__map,axiom,
    ! [F2: nat > nat,Xs: list_nat] :
      ( ( size_size_list_nat @ ( map_nat_nat @ F2 @ Xs ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_map
thf(fact_204_list_Omap__ident,axiom,
    ! [T: list_P6011104703257516679at_nat] :
      ( ( map_Pr8058819605623181956at_nat
        @ ^ [X: product_prod_nat_nat] : X
        @ T )
      = T ) ).

% list.map_ident
thf(fact_205_list_Omap__ident,axiom,
    ! [T: list_nat] :
      ( ( map_nat_nat
        @ ^ [X: nat] : X
        @ T )
      = T ) ).

% list.map_ident
thf(fact_206_nocp__cp__triv,axiom,
    ! [Q: relational_fmla_a_b] :
      ( ( relational_nocp_a_b @ Q )
     => ( ( relational_cp_a_b @ Q )
        = Q ) ) ).

% nocp_cp_triv
thf(fact_207_map__eq__imp__length__eq,axiom,
    ! [F2: nat > nat,Xs: list_nat,G2: nat > nat,Ys: list_nat] :
      ( ( ( map_nat_nat @ F2 @ Xs )
        = ( map_nat_nat @ G2 @ Ys ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_208_map__eq__imp__length__eq,axiom,
    ! [F2: nat > nat,Xs: list_nat,G2: relational_term_a > nat,Ys: list_R6823256787227418703term_a] :
      ( ( ( map_nat_nat @ F2 @ Xs )
        = ( map_Re700991392905337813_a_nat @ G2 @ Ys ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_s88622898042387131term_a @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_209_map__eq__imp__length__eq,axiom,
    ! [F2: relational_term_a > nat,Xs: list_R6823256787227418703term_a,G2: nat > nat,Ys: list_nat] :
      ( ( ( map_Re700991392905337813_a_nat @ F2 @ Xs )
        = ( map_nat_nat @ G2 @ Ys ) )
     => ( ( size_s88622898042387131term_a @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_210_map__eq__imp__length__eq,axiom,
    ! [F2: nat > product_prod_nat_nat,Xs: list_nat,G2: nat > product_prod_nat_nat,Ys: list_nat] :
      ( ( ( map_na7298421622053143531at_nat @ F2 @ Xs )
        = ( map_na7298421622053143531at_nat @ G2 @ Ys ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_211_map__eq__imp__length__eq,axiom,
    ! [F2: nat > product_prod_nat_nat,Xs: list_nat,G2: relational_term_a > product_prod_nat_nat,Ys: list_R6823256787227418703term_a] :
      ( ( ( map_na7298421622053143531at_nat @ F2 @ Xs )
        = ( map_Re7128850613395181244at_nat @ G2 @ Ys ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_s88622898042387131term_a @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_212_map__eq__imp__length__eq,axiom,
    ! [F2: relational_term_a > product_prod_nat_nat,Xs: list_R6823256787227418703term_a,G2: nat > product_prod_nat_nat,Ys: list_nat] :
      ( ( ( map_Re7128850613395181244at_nat @ F2 @ Xs )
        = ( map_na7298421622053143531at_nat @ G2 @ Ys ) )
     => ( ( size_s88622898042387131term_a @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_213_map__eq__imp__length__eq,axiom,
    ! [F2: product_prod_nat_nat > relational_fmla_a_b,Xs: list_P6011104703257516679at_nat,G2: nat > relational_fmla_a_b,Ys: list_nat] :
      ( ( ( map_Pr2810398200501793500la_a_b @ F2 @ Xs )
        = ( map_na3635843195271662901la_a_b @ G2 @ Ys ) )
     => ( ( size_s5460976970255530739at_nat @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_214_map__eq__imp__length__eq,axiom,
    ! [F2: product_prod_nat_nat > product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,G2: nat > product_prod_nat_nat,Ys: list_nat] :
      ( ( ( map_Pr8058819605623181956at_nat @ F2 @ Xs )
        = ( map_na7298421622053143531at_nat @ G2 @ Ys ) )
     => ( ( size_s5460976970255530739at_nat @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_215_map__eq__imp__length__eq,axiom,
    ! [F2: nat > relational_fmla_a_b,Xs: list_nat,G2: product_prod_nat_nat > relational_fmla_a_b,Ys: list_P6011104703257516679at_nat] :
      ( ( ( map_na3635843195271662901la_a_b @ F2 @ Xs )
        = ( map_Pr2810398200501793500la_a_b @ G2 @ Ys ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_s5460976970255530739at_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_216_map__eq__imp__length__eq,axiom,
    ! [F2: nat > product_prod_nat_nat,Xs: list_nat,G2: product_prod_nat_nat > product_prod_nat_nat,Ys: list_P6011104703257516679at_nat] :
      ( ( ( map_na7298421622053143531at_nat @ F2 @ Xs )
        = ( map_Pr8058819605623181956at_nat @ G2 @ Ys ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_s5460976970255530739at_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_217_map2__map__map,axiom,
    ! [H: nat > nat > nat,F2: nat > nat,Xs: list_nat,G2: nat > nat] :
      ( ( map_Pr3938374229010428429at_nat @ ( produc6842872674320459806at_nat @ H ) @ ( zip_nat_nat @ ( map_nat_nat @ F2 @ Xs ) @ ( map_nat_nat @ G2 @ Xs ) ) )
      = ( map_nat_nat
        @ ^ [X: nat] : ( H @ ( F2 @ X ) @ ( G2 @ X ) )
        @ Xs ) ) ).

% map2_map_map
thf(fact_218_map2__map__map,axiom,
    ! [H: product_prod_nat_nat > nat > nat,F2: nat > product_prod_nat_nat,Xs: list_nat,G2: nat > nat] :
      ( ( map_Pr4661564954218501910at_nat @ ( produc1870545637363562359at_nat @ H ) @ ( zip_Pr6869450617852699226at_nat @ ( map_na7298421622053143531at_nat @ F2 @ Xs ) @ ( map_nat_nat @ G2 @ Xs ) ) )
      = ( map_nat_nat
        @ ^ [X: nat] : ( H @ ( F2 @ X ) @ ( G2 @ X ) )
        @ Xs ) ) ).

% map2_map_map
thf(fact_219_map2__map__map,axiom,
    ! [H: nat > product_prod_nat_nat > nat,F2: nat > nat,Xs: list_nat,G2: nat > product_prod_nat_nat] :
      ( ( map_Pr5850356042547634492at_nat @ ( produc8489500644306686293at_nat @ H ) @ ( zip_na1006125974040638520at_nat @ ( map_nat_nat @ F2 @ Xs ) @ ( map_na7298421622053143531at_nat @ G2 @ Xs ) ) )
      = ( map_nat_nat
        @ ^ [X: nat] : ( H @ ( F2 @ X ) @ ( G2 @ X ) )
        @ Xs ) ) ).

% map2_map_map
thf(fact_220_map2__map__map,axiom,
    ! [H: nat > nat > relational_fmla_a_b,F2: nat > nat,Xs: list_nat,G2: nat > nat] :
      ( ( map_Pr2810398200501793500la_a_b @ ( produc3270801013941088237la_a_b @ H ) @ ( zip_nat_nat @ ( map_nat_nat @ F2 @ Xs ) @ ( map_nat_nat @ G2 @ Xs ) ) )
      = ( map_na3635843195271662901la_a_b
        @ ^ [X: nat] : ( H @ ( F2 @ X ) @ ( G2 @ X ) )
        @ Xs ) ) ).

% map2_map_map
thf(fact_221_map2__map__map,axiom,
    ! [H: nat > nat > product_prod_nat_nat,F2: nat > nat,Xs: list_nat,G2: nat > nat] :
      ( ( map_Pr8058819605623181956at_nat @ ( produc2626176000494625587at_nat @ H ) @ ( zip_nat_nat @ ( map_nat_nat @ F2 @ Xs ) @ ( map_nat_nat @ G2 @ Xs ) ) )
      = ( map_na7298421622053143531at_nat
        @ ^ [X: nat] : ( H @ ( F2 @ X ) @ ( G2 @ X ) )
        @ Xs ) ) ).

% map2_map_map
thf(fact_222_map2__map__map,axiom,
    ! [H: product_prod_nat_nat > product_prod_nat_nat > nat,F2: nat > product_prod_nat_nat,Xs: list_nat,G2: nat > product_prod_nat_nat] :
      ( ( map_Pr2610988902043053299at_nat @ ( produc6237124055692578492at_nat @ H ) @ ( zip_Pr4664179122662387191at_nat @ ( map_na7298421622053143531at_nat @ F2 @ Xs ) @ ( map_na7298421622053143531at_nat @ G2 @ Xs ) ) )
      = ( map_nat_nat
        @ ^ [X: nat] : ( H @ ( F2 @ X ) @ ( G2 @ X ) )
        @ Xs ) ) ).

% map2_map_map
thf(fact_223_map2__map__map,axiom,
    ! [H: product_prod_nat_nat > nat > product_prod_nat_nat,F2: nat > product_prod_nat_nat,Xs: list_nat,G2: nat > nat] :
      ( ( map_Pr4819452465118600763at_nat @ ( produc373799411880517786at_nat @ H ) @ ( zip_Pr6869450617852699226at_nat @ ( map_na7298421622053143531at_nat @ F2 @ Xs ) @ ( map_nat_nat @ G2 @ Xs ) ) )
      = ( map_na7298421622053143531at_nat
        @ ^ [X: nat] : ( H @ ( F2 @ X ) @ ( G2 @ X ) )
        @ Xs ) ) ).

% map2_map_map
thf(fact_224_map2__map__map,axiom,
    ! [H: nat > product_prod_nat_nat > product_prod_nat_nat,F2: nat > nat,Xs: list_nat,G2: nat > product_prod_nat_nat] :
      ( ( map_Pr2617240807308709013at_nat @ ( produc8859641928216934716at_nat @ H ) @ ( zip_na1006125974040638520at_nat @ ( map_nat_nat @ F2 @ Xs ) @ ( map_na7298421622053143531at_nat @ G2 @ Xs ) ) )
      = ( map_na7298421622053143531at_nat
        @ ^ [X: nat] : ( H @ ( F2 @ X ) @ ( G2 @ X ) )
        @ Xs ) ) ).

% map2_map_map
thf(fact_225_map2__map__map,axiom,
    ! [H: nat > nat > relational_fmla_a_b,F2: product_prod_nat_nat > nat,Xs: list_P6011104703257516679at_nat,G2: product_prod_nat_nat > nat] :
      ( ( map_Pr2810398200501793500la_a_b @ ( produc3270801013941088237la_a_b @ H ) @ ( zip_nat_nat @ ( map_Pr3938374229010428429at_nat @ F2 @ Xs ) @ ( map_Pr3938374229010428429at_nat @ G2 @ Xs ) ) )
      = ( map_Pr2810398200501793500la_a_b
        @ ^ [X: product_prod_nat_nat] : ( H @ ( F2 @ X ) @ ( G2 @ X ) )
        @ Xs ) ) ).

% map2_map_map
thf(fact_226_map2__map__map,axiom,
    ! [H: nat > nat > produc7248412053542808358at_nat,F2: nat > nat,Xs: list_nat,G2: nat > nat] :
      ( ( map_Pr7536285556892129763at_nat @ ( produc9083241971206738548at_nat @ H ) @ ( zip_nat_nat @ ( map_nat_nat @ F2 @ Xs ) @ ( map_nat_nat @ G2 @ Xs ) ) )
      = ( map_na8282419648612985788at_nat
        @ ^ [X: nat] : ( H @ ( F2 @ X ) @ ( G2 @ X ) )
        @ Xs ) ) ).

% map2_map_map
thf(fact_227_cp_Osimps_I7_J,axiom,
    ! [V: $o] :
      ( ( relational_cp_a_b @ ( relational_Bool_a_b @ V ) )
      = ( relational_Bool_a_b @ V ) ) ).

% cp.simps(7)
thf(fact_228_cp_Osimps_I6_J,axiom,
    ! [V: b,Va: list_R6823256787227418703term_a] :
      ( ( relational_cp_a_b @ ( relational_Pred_b_a @ V @ Va ) )
      = ( relational_Pred_b_a @ V @ Va ) ) ).

% cp.simps(6)
thf(fact_229_nocp_Osimps_I4_J,axiom,
    ! [Q: relational_fmla_a_b] :
      ( ( relational_nocp_a_b @ ( relational_Neg_a_b @ Q ) )
      = ( relational_nocp_a_b @ Q ) ) ).

% nocp.simps(4)
thf(fact_230_gen__Gen__cp,axiom,
    ! [X3: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
      ( ( relational_gen_a_b @ X3 @ Q @ G )
     => ? [X_1: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X3 @ ( relational_cp_a_b @ Q ) @ X_1 ) ) ).

% gen_Gen_cp
thf(fact_231_Gen__cp,axiom,
    ! [X3: nat,Q: relational_fmla_a_b] :
      ( ? [X_12: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X3 @ Q @ X_12 )
     => ? [X_1: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X3 @ ( relational_cp_a_b @ Q ) @ X_1 ) ) ).

% Gen_cp
thf(fact_232_nocp_Osimps_I6_J,axiom,
    ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
      ( ( relational_nocp_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) )
      = ( ( relational_nocp_a_b @ Q1 )
        & ( relational_nocp_a_b @ Q22 ) ) ) ).

% nocp.simps(6)
thf(fact_233_nocp_Osimps_I5_J,axiom,
    ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
      ( ( relational_nocp_a_b @ ( relational_Conj_a_b @ Q1 @ Q22 ) )
      = ( ( relational_nocp_a_b @ Q1 )
        & ( relational_nocp_a_b @ Q22 ) ) ) ).

% nocp.simps(5)
thf(fact_234_nocp_Osimps_I1_J,axiom,
    ! [B: $o] :
      ~ ( relational_nocp_a_b @ ( relational_Bool_a_b @ B ) ) ).

% nocp.simps(1)
thf(fact_235_nocp_Osimps_I2_J,axiom,
    ! [P2: b,Ts3: list_R6823256787227418703term_a] : ( relational_nocp_a_b @ ( relational_Pred_b_a @ P2 @ Ts3 ) ) ).

% nocp.simps(2)
thf(fact_236_qp__cp__triv,axiom,
    ! [Q: relational_fmla_a_b] :
      ( ( relational_qp_a_b @ Q )
     => ( ( relational_cp_a_b @ Q )
        = Q ) ) ).

% qp_cp_triv
thf(fact_237_qp__cp,axiom,
    ! [Q: relational_fmla_a_b] :
      ( ( relational_qp_a_b @ Q )
     => ( relational_qp_a_b @ ( relational_cp_a_b @ Q ) ) ) ).

% qp_cp
thf(fact_238_ap__cp__triv,axiom,
    ! [Q: relational_fmla_a_b] :
      ( ( relational_ap_a_b @ Q )
     => ( ( relational_cp_a_b @ Q )
        = Q ) ) ).

% ap_cp_triv
thf(fact_239_ap__cp,axiom,
    ! [Q: relational_fmla_a_b] :
      ( ( relational_ap_a_b @ Q )
     => ( relational_ap_a_b @ ( relational_cp_a_b @ Q ) ) ) ).

% ap_cp
thf(fact_240_qp__cp__subst__triv,axiom,
    ! [Q: relational_fmla_a_b,X3: nat,Y: nat] :
      ( ( relational_qp_a_b @ Q )
     => ( ( relational_cp_a_b @ ( relational_subst_a_b @ Q @ X3 @ Y ) )
        = ( relational_subst_a_b @ Q @ X3 @ Y ) ) ) ).

% qp_cp_subst_triv
thf(fact_241_nocp_Osimps_I3_J,axiom,
    ! [X3: nat,T: relational_term_a] :
      ( ( relational_nocp_a_b @ ( relational_Eq_a_b @ X3 @ T ) )
      = ( T
       != ( relational_Var_a @ X3 ) ) ) ).

% nocp.simps(3)
thf(fact_242_nocp_Osimps_I7_J,axiom,
    ! [X3: nat,Q: relational_fmla_a_b] :
      ( ( relational_nocp_a_b @ ( relati591517084277583526ts_a_b @ X3 @ Q ) )
      = ( ( member_nat @ X3 @ ( relational_fv_a_b @ Q ) )
        & ( relational_nocp_a_b @ Q ) ) ) ).

% nocp.simps(7)
thf(fact_243_ap__cp__subst__triv,axiom,
    ! [Q: relational_fmla_a_b,X3: nat,Y: nat] :
      ( ( relational_ap_a_b @ Q )
     => ( ( relational_cp_a_b @ ( relational_subst_a_b @ Q @ X3 @ Y ) )
        = ( relational_subst_a_b @ Q @ X3 @ Y ) ) ) ).

% ap_cp_subst_triv
thf(fact_244_gen_Ointros_I9_J,axiom,
    ! [Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,X3: nat] :
      ( ( relational_gen_a_b @ Y @ Q @ G )
     => ( relational_gen_a_b @ X3 @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ Y @ ( relational_Var_a @ X3 ) ) )
        @ ( image_6790371041703824709la_a_b
          @ ^ [Q2: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Q2 @ Y @ X3 ) )
          @ G ) ) ) ).

% gen.intros(9)
thf(fact_245_gen_Ointros_I8_J,axiom,
    ! [Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,X3: nat] :
      ( ( relational_gen_a_b @ Y @ Q @ G )
     => ( relational_gen_a_b @ X3 @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ X3 @ ( relational_Var_a @ Y ) ) )
        @ ( image_6790371041703824709la_a_b
          @ ^ [Q2: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Q2 @ Y @ X3 ) )
          @ G ) ) ) ).

% gen.intros(8)
thf(fact_246_gen_H__cp__intros_I2_J,axiom,
    ! [Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,X3: nat] :
      ( ( relational_gen_a_b2 @ Y @ Q @ G )
     => ( relational_gen_a_b2 @ X3 @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ Y @ ( relational_Var_a @ X3 ) ) )
        @ ( image_6790371041703824709la_a_b
          @ ^ [Q2: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Q2 @ Y @ X3 ) )
          @ G ) ) ) ).

% gen'_cp_intros(2)
thf(fact_247_gen_H__cp__intros_I1_J,axiom,
    ! [Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,X3: nat] :
      ( ( relational_gen_a_b2 @ Y @ Q @ G )
     => ( relational_gen_a_b2 @ X3 @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ X3 @ ( relational_Var_a @ Y ) ) )
        @ ( image_6790371041703824709la_a_b
          @ ^ [Q2: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Q2 @ Y @ X3 ) )
          @ G ) ) ) ).

% gen'_cp_intros(1)
thf(fact_248_subst_Osimps_I2_J,axiom,
    ! [P2: b,Ts3: list_R6823256787227418703term_a,X3: nat,Y: nat] :
      ( ( relational_subst_a_b @ ( relational_Pred_b_a @ P2 @ Ts3 ) @ X3 @ Y )
      = ( relational_Pred_b_a @ P2
        @ ( map_Re5736185711816362116term_a
          @ ^ [T2: relational_term_a] : ( relati7175845559408349773term_a @ T2 @ X3 @ Y )
          @ Ts3 ) ) ) ).

% subst.simps(2)
thf(fact_249_qp__impl_Opelims_I2_J,axiom,
    ! [X3: relational_fmla_a_b] :
      ( ( relati3725921752842749053pl_a_b @ X3 )
     => ( ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ X3 )
       => ( ! [X5: nat,C2: a] :
              ( ( X3
                = ( relational_Eq_a_b @ X5 @ ( relational_Const_a @ C2 ) ) )
             => ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Eq_a_b @ X5 @ ( relational_Const_a @ C2 ) ) ) )
         => ( ! [X5: b,Ts2: list_R6823256787227418703term_a] :
                ( ( X3
                  = ( relational_Pred_b_a @ X5 @ Ts2 ) )
               => ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Pred_b_a @ X5 @ Ts2 ) ) )
           => ~ ! [X5: nat,Q3: relational_fmla_a_b] :
                  ( ( X3
                    = ( relati591517084277583526ts_a_b @ X5 @ Q3 ) )
                 => ( ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relati591517084277583526ts_a_b @ X5 @ Q3 ) )
                   => ~ ( ( member_nat @ X5 @ ( relational_fv_a_b @ Q3 ) )
                        & ( relational_qp_a_b @ Q3 ) ) ) ) ) ) ) ) ).

% qp_impl.pelims(2)
thf(fact_250_gen__nocp__intros_I1_J,axiom,
    ! [Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,X3: nat] :
      ( ( relational_gen_a_b @ Y @ Q @ G )
     => ( relational_gen_a_b @ X3 @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ X3 @ ( relational_Var_a @ Y ) ) )
        @ ( image_6790371041703824709la_a_b
          @ ^ [Q2: relational_fmla_a_b] : ( relational_subst_a_b @ Q2 @ Y @ X3 )
          @ G ) ) ) ).

% gen_nocp_intros(1)
thf(fact_251_gen__nocp__intros_I2_J,axiom,
    ! [Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,X3: nat] :
      ( ( relational_gen_a_b @ Y @ Q @ G )
     => ( relational_gen_a_b @ X3 @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ Y @ ( relational_Var_a @ X3 ) ) )
        @ ( image_6790371041703824709la_a_b
          @ ^ [Q2: relational_fmla_a_b] : ( relational_subst_a_b @ Q2 @ Y @ X3 )
          @ G ) ) ) ).

% gen_nocp_intros(2)
thf(fact_252_gen_H_Ointros_I8_J,axiom,
    ! [Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,X3: nat] :
      ( ( relational_gen_a_b2 @ Y @ Q @ G )
     => ( relational_gen_a_b2 @ X3 @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ X3 @ ( relational_Var_a @ Y ) ) )
        @ ( image_6790371041703824709la_a_b
          @ ^ [Q2: relational_fmla_a_b] : ( relational_subst_a_b @ Q2 @ Y @ X3 )
          @ G ) ) ) ).

% gen'.intros(8)
thf(fact_253_nocp_Opelims_I1_J,axiom,
    ! [X3: relational_fmla_a_b,Y: $o] :
      ( ( ( relational_nocp_a_b @ X3 )
        = Y )
     => ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ X3 )
       => ( ! [B2: $o] :
              ( ( X3
                = ( relational_Bool_a_b @ B2 ) )
             => ( ~ Y
               => ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Bool_a_b @ B2 ) ) ) )
         => ( ! [P4: b,Ts2: list_R6823256787227418703term_a] :
                ( ( X3
                  = ( relational_Pred_b_a @ P4 @ Ts2 ) )
               => ( Y
                 => ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Pred_b_a @ P4 @ Ts2 ) ) ) )
           => ( ! [X5: nat,T3: relational_term_a] :
                  ( ( X3
                    = ( relational_Eq_a_b @ X5 @ T3 ) )
                 => ( ( Y
                      = ( T3
                       != ( relational_Var_a @ X5 ) ) )
                   => ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Eq_a_b @ X5 @ T3 ) ) ) )
             => ( ! [Q3: relational_fmla_a_b] :
                    ( ( X3
                      = ( relational_Neg_a_b @ Q3 ) )
                   => ( ( Y
                        = ( relational_nocp_a_b @ Q3 ) )
                     => ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Neg_a_b @ Q3 ) ) ) )
               => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                      ( ( X3
                        = ( relational_Conj_a_b @ Q13 @ Q24 ) )
                     => ( ( Y
                          = ( ( relational_nocp_a_b @ Q13 )
                            & ( relational_nocp_a_b @ Q24 ) ) )
                       => ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Conj_a_b @ Q13 @ Q24 ) ) ) )
                 => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                        ( ( X3
                          = ( relational_Disj_a_b @ Q13 @ Q24 ) )
                       => ( ( Y
                            = ( ( relational_nocp_a_b @ Q13 )
                              & ( relational_nocp_a_b @ Q24 ) ) )
                         => ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Disj_a_b @ Q13 @ Q24 ) ) ) )
                   => ~ ! [X5: nat,Q3: relational_fmla_a_b] :
                          ( ( X3
                            = ( relati591517084277583526ts_a_b @ X5 @ Q3 ) )
                         => ( ( Y
                              = ( ( member_nat @ X5 @ ( relational_fv_a_b @ Q3 ) )
                                & ( relational_nocp_a_b @ Q3 ) ) )
                           => ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relati591517084277583526ts_a_b @ X5 @ Q3 ) ) ) ) ) ) ) ) ) ) ) ) ).

% nocp.pelims(1)
thf(fact_254_nocp_Opelims_I2_J,axiom,
    ! [X3: relational_fmla_a_b] :
      ( ( relational_nocp_a_b @ X3 )
     => ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ X3 )
       => ( ! [P4: b,Ts2: list_R6823256787227418703term_a] :
              ( ( X3
                = ( relational_Pred_b_a @ P4 @ Ts2 ) )
             => ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Pred_b_a @ P4 @ Ts2 ) ) )
         => ( ! [X5: nat,T3: relational_term_a] :
                ( ( X3
                  = ( relational_Eq_a_b @ X5 @ T3 ) )
               => ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Eq_a_b @ X5 @ T3 ) )
                 => ( T3
                    = ( relational_Var_a @ X5 ) ) ) )
           => ( ! [Q3: relational_fmla_a_b] :
                  ( ( X3
                    = ( relational_Neg_a_b @ Q3 ) )
                 => ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Neg_a_b @ Q3 ) )
                   => ~ ( relational_nocp_a_b @ Q3 ) ) )
             => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                    ( ( X3
                      = ( relational_Conj_a_b @ Q13 @ Q24 ) )
                   => ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Conj_a_b @ Q13 @ Q24 ) )
                     => ~ ( ( relational_nocp_a_b @ Q13 )
                          & ( relational_nocp_a_b @ Q24 ) ) ) )
               => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                      ( ( X3
                        = ( relational_Disj_a_b @ Q13 @ Q24 ) )
                     => ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Disj_a_b @ Q13 @ Q24 ) )
                       => ~ ( ( relational_nocp_a_b @ Q13 )
                            & ( relational_nocp_a_b @ Q24 ) ) ) )
                 => ~ ! [X5: nat,Q3: relational_fmla_a_b] :
                        ( ( X3
                          = ( relati591517084277583526ts_a_b @ X5 @ Q3 ) )
                       => ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relati591517084277583526ts_a_b @ X5 @ Q3 ) )
                         => ~ ( ( member_nat @ X5 @ ( relational_fv_a_b @ Q3 ) )
                              & ( relational_nocp_a_b @ Q3 ) ) ) ) ) ) ) ) ) ) ) ).

% nocp.pelims(2)
thf(fact_255_nocp_Opelims_I3_J,axiom,
    ! [X3: relational_fmla_a_b] :
      ( ~ ( relational_nocp_a_b @ X3 )
     => ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ X3 )
       => ( ! [B2: $o] :
              ( ( X3
                = ( relational_Bool_a_b @ B2 ) )
             => ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Bool_a_b @ B2 ) ) )
         => ( ! [X5: nat,T3: relational_term_a] :
                ( ( X3
                  = ( relational_Eq_a_b @ X5 @ T3 ) )
               => ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Eq_a_b @ X5 @ T3 ) )
                 => ( T3
                   != ( relational_Var_a @ X5 ) ) ) )
           => ( ! [Q3: relational_fmla_a_b] :
                  ( ( X3
                    = ( relational_Neg_a_b @ Q3 ) )
                 => ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Neg_a_b @ Q3 ) )
                   => ( relational_nocp_a_b @ Q3 ) ) )
             => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                    ( ( X3
                      = ( relational_Conj_a_b @ Q13 @ Q24 ) )
                   => ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Conj_a_b @ Q13 @ Q24 ) )
                     => ( ( relational_nocp_a_b @ Q13 )
                        & ( relational_nocp_a_b @ Q24 ) ) ) )
               => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                      ( ( X3
                        = ( relational_Disj_a_b @ Q13 @ Q24 ) )
                     => ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Disj_a_b @ Q13 @ Q24 ) )
                       => ( ( relational_nocp_a_b @ Q13 )
                          & ( relational_nocp_a_b @ Q24 ) ) ) )
                 => ~ ! [X5: nat,Q3: relational_fmla_a_b] :
                        ( ( X3
                          = ( relati591517084277583526ts_a_b @ X5 @ Q3 ) )
                       => ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relati591517084277583526ts_a_b @ X5 @ Q3 ) )
                         => ( ( member_nat @ X5 @ ( relational_fv_a_b @ Q3 ) )
                            & ( relational_nocp_a_b @ Q3 ) ) ) ) ) ) ) ) ) ) ) ).

% nocp.pelims(3)
thf(fact_256_image__ident,axiom,
    ! [Y5: set_set_o] :
      ( ( image_set_o_set_o
        @ ^ [X: set_o] : X
        @ Y5 )
      = Y5 ) ).

% image_ident
thf(fact_257_fun__upd__apply,axiom,
    ( fun_upd_nat_a
    = ( ^ [F: nat > a,X: nat,Y2: a,Z3: nat] : ( if_a @ ( Z3 = X ) @ Y2 @ ( F @ Z3 ) ) ) ) ).

% fun_upd_apply
thf(fact_258_fun__upd__apply,axiom,
    ( fun_upd_o_o
    = ( ^ [F: $o > $o,X: $o,Y2: $o,Z3: $o] :
          ( ( ( Z3 = X )
           => Y2 )
          & ( ( Z3 = (~ X) )
           => ( F @ Z3 ) ) ) ) ) ).

% fun_upd_apply
thf(fact_259_fun__upd__apply,axiom,
    ( fun_upd_o_set_o
    = ( ^ [F: $o > set_o,X: $o,Y2: set_o,Z3: $o] : ( if_set_o @ ( Z3 = X ) @ Y2 @ ( F @ Z3 ) ) ) ) ).

% fun_upd_apply
thf(fact_260_fun__upd__triv,axiom,
    ! [F2: nat > a,X3: nat] :
      ( ( fun_upd_nat_a @ F2 @ X3 @ ( F2 @ X3 ) )
      = F2 ) ).

% fun_upd_triv
thf(fact_261_fun__upd__triv,axiom,
    ! [F2: $o > $o,X3: $o] :
      ( ( fun_upd_o_o @ F2 @ X3 @ ( F2 @ X3 ) )
      = F2 ) ).

% fun_upd_triv
thf(fact_262_fun__upd__triv,axiom,
    ! [F2: $o > set_o,X3: $o] :
      ( ( fun_upd_o_set_o @ F2 @ X3 @ ( F2 @ X3 ) )
      = F2 ) ).

% fun_upd_triv
thf(fact_263_fun__upd__upd,axiom,
    ! [F2: nat > a,X3: nat,Y: a,Z: a] :
      ( ( fun_upd_nat_a @ ( fun_upd_nat_a @ F2 @ X3 @ Y ) @ X3 @ Z )
      = ( fun_upd_nat_a @ F2 @ X3 @ Z ) ) ).

% fun_upd_upd
thf(fact_264_fun__upd__upd,axiom,
    ! [F2: $o > $o,X3: $o,Y: $o,Z: $o] :
      ( ( fun_upd_o_o @ ( fun_upd_o_o @ F2 @ X3 @ Y ) @ X3 @ Z )
      = ( fun_upd_o_o @ F2 @ X3 @ Z ) ) ).

% fun_upd_upd
thf(fact_265_fun__upd__upd,axiom,
    ! [F2: $o > set_o,X3: $o,Y: set_o,Z: set_o] :
      ( ( fun_upd_o_set_o @ ( fun_upd_o_set_o @ F2 @ X3 @ Y ) @ X3 @ Z )
      = ( fun_upd_o_set_o @ F2 @ X3 @ Z ) ) ).

% fun_upd_upd
thf(fact_266_image__eqI,axiom,
    ! [B: $o,F2: $o > $o,X3: $o,A2: set_o] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( ( member_o @ X3 @ A2 )
       => ( member_o @ B @ ( image_o_o @ F2 @ A2 ) ) ) ) ).

% image_eqI
thf(fact_267_image__eqI,axiom,
    ! [B: nat,F2: $o > nat,X3: $o,A2: set_o] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( ( member_o @ X3 @ A2 )
       => ( member_nat @ B @ ( image_o_nat @ F2 @ A2 ) ) ) ) ).

% image_eqI
thf(fact_268_image__eqI,axiom,
    ! [B: b,F2: $o > b,X3: $o,A2: set_o] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( ( member_o @ X3 @ A2 )
       => ( member_b @ B @ ( image_o_b @ F2 @ A2 ) ) ) ) ).

% image_eqI
thf(fact_269_image__eqI,axiom,
    ! [B: $o,F2: nat > $o,X3: nat,A2: set_nat] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( ( member_nat @ X3 @ A2 )
       => ( member_o @ B @ ( image_nat_o @ F2 @ A2 ) ) ) ) ).

% image_eqI
thf(fact_270_image__eqI,axiom,
    ! [B: nat,F2: nat > nat,X3: nat,A2: set_nat] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( ( member_nat @ X3 @ A2 )
       => ( member_nat @ B @ ( image_nat_nat @ F2 @ A2 ) ) ) ) ).

% image_eqI
thf(fact_271_image__eqI,axiom,
    ! [B: b,F2: nat > b,X3: nat,A2: set_nat] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( ( member_nat @ X3 @ A2 )
       => ( member_b @ B @ ( image_nat_b @ F2 @ A2 ) ) ) ) ).

% image_eqI
thf(fact_272_image__eqI,axiom,
    ! [B: $o,F2: b > $o,X3: b,A2: set_b] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( ( member_b @ X3 @ A2 )
       => ( member_o @ B @ ( image_b_o @ F2 @ A2 ) ) ) ) ).

% image_eqI
thf(fact_273_image__eqI,axiom,
    ! [B: nat,F2: b > nat,X3: b,A2: set_b] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( ( member_b @ X3 @ A2 )
       => ( member_nat @ B @ ( image_b_nat @ F2 @ A2 ) ) ) ) ).

% image_eqI
thf(fact_274_image__eqI,axiom,
    ! [B: b,F2: b > b,X3: b,A2: set_b] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( ( member_b @ X3 @ A2 )
       => ( member_b @ B @ ( image_b_b @ F2 @ A2 ) ) ) ) ).

% image_eqI
thf(fact_275_image__eqI,axiom,
    ! [B: list_a,F2: $o > list_a,X3: $o,A2: set_o] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( ( member_o @ X3 @ A2 )
       => ( member_list_a @ B @ ( image_o_list_a @ F2 @ A2 ) ) ) ) ).

% image_eqI
thf(fact_276_cp_Osimps_I1_J,axiom,
    ! [X3: nat,T: relational_term_a] :
      ( ( relational_cp_a_b @ ( relational_Eq_a_b @ X3 @ T ) )
      = ( relati582353067970734056la_a_b
        @ ^ [A3: a] : ( relational_Eq_a_b @ X3 @ T )
        @ ^ [Y2: nat] : ( if_Rel1279876242545935705la_a_b @ ( X3 = Y2 ) @ ( relational_Bool_a_b @ $true ) @ ( relational_Eq_a_b @ X3 @ ( relational_Var_a @ Y2 ) ) )
        @ T ) ) ).

% cp.simps(1)
thf(fact_277_Relational__Calculus_Oterm_Ocase__distrib,axiom,
    ! [H: relational_fmla_a_b > relational_fmla_a_b,F1: a > relational_fmla_a_b,F22: nat > relational_fmla_a_b,Term: relational_term_a] :
      ( ( H @ ( relati582353067970734056la_a_b @ F1 @ F22 @ Term ) )
      = ( relati582353067970734056la_a_b
        @ ^ [X: a] : ( H @ ( F1 @ X ) )
        @ ^ [X: nat] : ( H @ ( F22 @ X ) )
        @ Term ) ) ).

% Relational_Calculus.term.case_distrib
thf(fact_278_Relational__Calculus_Oterm_Osimps_I6_J,axiom,
    ! [F1: a > relational_fmla_a_b,F22: nat > relational_fmla_a_b,X22: nat] :
      ( ( relati582353067970734056la_a_b @ F1 @ F22 @ ( relational_Var_a @ X22 ) )
      = ( F22 @ X22 ) ) ).

% Relational_Calculus.term.simps(6)
thf(fact_279_Relational__Calculus_Oterm_Osimps_I5_J,axiom,
    ! [F1: a > relational_fmla_a_b,F22: nat > relational_fmla_a_b,X13: a] :
      ( ( relati582353067970734056la_a_b @ F1 @ F22 @ ( relational_Const_a @ X13 ) )
      = ( F1 @ X13 ) ) ).

% Relational_Calculus.term.simps(5)
thf(fact_280_imageI,axiom,
    ! [X3: $o,A2: set_o,F2: $o > $o] :
      ( ( member_o @ X3 @ A2 )
     => ( member_o @ ( F2 @ X3 ) @ ( image_o_o @ F2 @ A2 ) ) ) ).

% imageI
thf(fact_281_imageI,axiom,
    ! [X3: $o,A2: set_o,F2: $o > nat] :
      ( ( member_o @ X3 @ A2 )
     => ( member_nat @ ( F2 @ X3 ) @ ( image_o_nat @ F2 @ A2 ) ) ) ).

% imageI
thf(fact_282_imageI,axiom,
    ! [X3: $o,A2: set_o,F2: $o > b] :
      ( ( member_o @ X3 @ A2 )
     => ( member_b @ ( F2 @ X3 ) @ ( image_o_b @ F2 @ A2 ) ) ) ).

% imageI
thf(fact_283_imageI,axiom,
    ! [X3: nat,A2: set_nat,F2: nat > $o] :
      ( ( member_nat @ X3 @ A2 )
     => ( member_o @ ( F2 @ X3 ) @ ( image_nat_o @ F2 @ A2 ) ) ) ).

% imageI
thf(fact_284_imageI,axiom,
    ! [X3: nat,A2: set_nat,F2: nat > nat] :
      ( ( member_nat @ X3 @ A2 )
     => ( member_nat @ ( F2 @ X3 ) @ ( image_nat_nat @ F2 @ A2 ) ) ) ).

% imageI
thf(fact_285_imageI,axiom,
    ! [X3: nat,A2: set_nat,F2: nat > b] :
      ( ( member_nat @ X3 @ A2 )
     => ( member_b @ ( F2 @ X3 ) @ ( image_nat_b @ F2 @ A2 ) ) ) ).

% imageI
thf(fact_286_imageI,axiom,
    ! [X3: b,A2: set_b,F2: b > $o] :
      ( ( member_b @ X3 @ A2 )
     => ( member_o @ ( F2 @ X3 ) @ ( image_b_o @ F2 @ A2 ) ) ) ).

% imageI
thf(fact_287_imageI,axiom,
    ! [X3: b,A2: set_b,F2: b > nat] :
      ( ( member_b @ X3 @ A2 )
     => ( member_nat @ ( F2 @ X3 ) @ ( image_b_nat @ F2 @ A2 ) ) ) ).

% imageI
thf(fact_288_imageI,axiom,
    ! [X3: b,A2: set_b,F2: b > b] :
      ( ( member_b @ X3 @ A2 )
     => ( member_b @ ( F2 @ X3 ) @ ( image_b_b @ F2 @ A2 ) ) ) ).

% imageI
thf(fact_289_imageI,axiom,
    ! [X3: $o,A2: set_o,F2: $o > list_a] :
      ( ( member_o @ X3 @ A2 )
     => ( member_list_a @ ( F2 @ X3 ) @ ( image_o_list_a @ F2 @ A2 ) ) ) ).

% imageI
thf(fact_290_image__iff,axiom,
    ! [Z: $o > $o,F2: set_o > $o > $o,A2: set_set_o] :
      ( ( member_o_o @ Z @ ( image_set_o_o_o @ F2 @ A2 ) )
      = ( ? [X: set_o] :
            ( ( member_set_o @ X @ A2 )
            & ( Z
              = ( F2 @ X ) ) ) ) ) ).

% image_iff
thf(fact_291_image__iff,axiom,
    ! [Z: $o,F2: set_o > $o,A2: set_set_o] :
      ( ( member_o @ Z @ ( image_set_o_o @ F2 @ A2 ) )
      = ( ? [X: set_o] :
            ( ( member_set_o @ X @ A2 )
            & ( Z
              = ( F2 @ X ) ) ) ) ) ).

% image_iff
thf(fact_292_image__iff,axiom,
    ! [Z: set_o,F2: set_o > set_o,A2: set_set_o] :
      ( ( member_set_o @ Z @ ( image_set_o_set_o @ F2 @ A2 ) )
      = ( ? [X: set_o] :
            ( ( member_set_o @ X @ A2 )
            & ( Z
              = ( F2 @ X ) ) ) ) ) ).

% image_iff
thf(fact_293_image__iff,axiom,
    ! [Z: set_o,F2: ( $o > $o ) > set_o,A2: set_o_o] :
      ( ( member_set_o @ Z @ ( image_o_o_set_o @ F2 @ A2 ) )
      = ( ? [X: $o > $o] :
            ( ( member_o_o @ X @ A2 )
            & ( Z
              = ( F2 @ X ) ) ) ) ) ).

% image_iff
thf(fact_294_image__iff,axiom,
    ! [Z: nat,F2: $o > nat,A2: set_o] :
      ( ( member_nat @ Z @ ( image_o_nat @ F2 @ A2 ) )
      = ( ? [X: $o] :
            ( ( member_o @ X @ A2 )
            & ( Z
              = ( F2 @ X ) ) ) ) ) ).

% image_iff
thf(fact_295_bex__imageD,axiom,
    ! [F2: $o > nat,A2: set_o,P: nat > $o] :
      ( ? [X6: nat] :
          ( ( member_nat @ X6 @ ( image_o_nat @ F2 @ A2 ) )
          & ( P @ X6 ) )
     => ? [X5: $o] :
          ( ( member_o @ X5 @ A2 )
          & ( P @ ( F2 @ X5 ) ) ) ) ).

% bex_imageD
thf(fact_296_bex__imageD,axiom,
    ! [F2: set_o > $o,A2: set_set_o,P: $o > $o] :
      ( ? [X6: $o] :
          ( ( member_o @ X6 @ ( image_set_o_o @ F2 @ A2 ) )
          & ( P @ X6 ) )
     => ? [X5: set_o] :
          ( ( member_set_o @ X5 @ A2 )
          & ( P @ ( F2 @ X5 ) ) ) ) ).

% bex_imageD
thf(fact_297_bex__imageD,axiom,
    ! [F2: set_o > set_o,A2: set_set_o,P: set_o > $o] :
      ( ? [X6: set_o] :
          ( ( member_set_o @ X6 @ ( image_set_o_set_o @ F2 @ A2 ) )
          & ( P @ X6 ) )
     => ? [X5: set_o] :
          ( ( member_set_o @ X5 @ A2 )
          & ( P @ ( F2 @ X5 ) ) ) ) ).

% bex_imageD
thf(fact_298_bex__imageD,axiom,
    ! [F2: set_o > $o > $o,A2: set_set_o,P: ( $o > $o ) > $o] :
      ( ? [X6: $o > $o] :
          ( ( member_o_o @ X6 @ ( image_set_o_o_o @ F2 @ A2 ) )
          & ( P @ X6 ) )
     => ? [X5: set_o] :
          ( ( member_set_o @ X5 @ A2 )
          & ( P @ ( F2 @ X5 ) ) ) ) ).

% bex_imageD
thf(fact_299_bex__imageD,axiom,
    ! [F2: ( $o > $o ) > set_o,A2: set_o_o,P: set_o > $o] :
      ( ? [X6: set_o] :
          ( ( member_set_o @ X6 @ ( image_o_o_set_o @ F2 @ A2 ) )
          & ( P @ X6 ) )
     => ? [X5: $o > $o] :
          ( ( member_o_o @ X5 @ A2 )
          & ( P @ ( F2 @ X5 ) ) ) ) ).

% bex_imageD
thf(fact_300_image__cong,axiom,
    ! [M: set_o_o,N3: set_o_o,F2: ( $o > $o ) > set_o,G2: ( $o > $o ) > set_o] :
      ( ( M = N3 )
     => ( ! [X5: $o > $o] :
            ( ( member_o_o @ X5 @ N3 )
           => ( ( F2 @ X5 )
              = ( G2 @ X5 ) ) )
       => ( ( image_o_o_set_o @ F2 @ M )
          = ( image_o_o_set_o @ G2 @ N3 ) ) ) ) ).

% image_cong
thf(fact_301_image__cong,axiom,
    ! [M: set_o,N3: set_o,F2: $o > nat,G2: $o > nat] :
      ( ( M = N3 )
     => ( ! [X5: $o] :
            ( ( member_o @ X5 @ N3 )
           => ( ( F2 @ X5 )
              = ( G2 @ X5 ) ) )
       => ( ( image_o_nat @ F2 @ M )
          = ( image_o_nat @ G2 @ N3 ) ) ) ) ).

% image_cong
thf(fact_302_image__cong,axiom,
    ! [M: set_set_o,N3: set_set_o,F2: set_o > $o,G2: set_o > $o] :
      ( ( M = N3 )
     => ( ! [X5: set_o] :
            ( ( member_set_o @ X5 @ N3 )
           => ( ( F2 @ X5 )
              = ( G2 @ X5 ) ) )
       => ( ( image_set_o_o @ F2 @ M )
          = ( image_set_o_o @ G2 @ N3 ) ) ) ) ).

% image_cong
thf(fact_303_image__cong,axiom,
    ! [M: set_set_o,N3: set_set_o,F2: set_o > set_o,G2: set_o > set_o] :
      ( ( M = N3 )
     => ( ! [X5: set_o] :
            ( ( member_set_o @ X5 @ N3 )
           => ( ( F2 @ X5 )
              = ( G2 @ X5 ) ) )
       => ( ( image_set_o_set_o @ F2 @ M )
          = ( image_set_o_set_o @ G2 @ N3 ) ) ) ) ).

% image_cong
thf(fact_304_image__cong,axiom,
    ! [M: set_set_o,N3: set_set_o,F2: set_o > $o > $o,G2: set_o > $o > $o] :
      ( ( M = N3 )
     => ( ! [X5: set_o] :
            ( ( member_set_o @ X5 @ N3 )
           => ( ( F2 @ X5 )
              = ( G2 @ X5 ) ) )
       => ( ( image_set_o_o_o @ F2 @ M )
          = ( image_set_o_o_o @ G2 @ N3 ) ) ) ) ).

% image_cong
thf(fact_305_ball__imageD,axiom,
    ! [F2: $o > nat,A2: set_o,P: nat > $o] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ ( image_o_nat @ F2 @ A2 ) )
         => ( P @ X5 ) )
     => ! [X6: $o] :
          ( ( member_o @ X6 @ A2 )
         => ( P @ ( F2 @ X6 ) ) ) ) ).

% ball_imageD
thf(fact_306_ball__imageD,axiom,
    ! [F2: set_o > $o,A2: set_set_o,P: $o > $o] :
      ( ! [X5: $o] :
          ( ( member_o @ X5 @ ( image_set_o_o @ F2 @ A2 ) )
         => ( P @ X5 ) )
     => ! [X6: set_o] :
          ( ( member_set_o @ X6 @ A2 )
         => ( P @ ( F2 @ X6 ) ) ) ) ).

% ball_imageD
thf(fact_307_ball__imageD,axiom,
    ! [F2: set_o > set_o,A2: set_set_o,P: set_o > $o] :
      ( ! [X5: set_o] :
          ( ( member_set_o @ X5 @ ( image_set_o_set_o @ F2 @ A2 ) )
         => ( P @ X5 ) )
     => ! [X6: set_o] :
          ( ( member_set_o @ X6 @ A2 )
         => ( P @ ( F2 @ X6 ) ) ) ) ).

% ball_imageD
thf(fact_308_ball__imageD,axiom,
    ! [F2: set_o > $o > $o,A2: set_set_o,P: ( $o > $o ) > $o] :
      ( ! [X5: $o > $o] :
          ( ( member_o_o @ X5 @ ( image_set_o_o_o @ F2 @ A2 ) )
         => ( P @ X5 ) )
     => ! [X6: set_o] :
          ( ( member_set_o @ X6 @ A2 )
         => ( P @ ( F2 @ X6 ) ) ) ) ).

% ball_imageD
thf(fact_309_ball__imageD,axiom,
    ! [F2: ( $o > $o ) > set_o,A2: set_o_o,P: set_o > $o] :
      ( ! [X5: set_o] :
          ( ( member_set_o @ X5 @ ( image_o_o_set_o @ F2 @ A2 ) )
         => ( P @ X5 ) )
     => ! [X6: $o > $o] :
          ( ( member_o_o @ X6 @ A2 )
         => ( P @ ( F2 @ X6 ) ) ) ) ).

% ball_imageD
thf(fact_310_rev__image__eqI,axiom,
    ! [X3: $o,A2: set_o,B: $o,F2: $o > $o] :
      ( ( member_o @ X3 @ A2 )
     => ( ( B
          = ( F2 @ X3 ) )
       => ( member_o @ B @ ( image_o_o @ F2 @ A2 ) ) ) ) ).

% rev_image_eqI
thf(fact_311_rev__image__eqI,axiom,
    ! [X3: $o,A2: set_o,B: nat,F2: $o > nat] :
      ( ( member_o @ X3 @ A2 )
     => ( ( B
          = ( F2 @ X3 ) )
       => ( member_nat @ B @ ( image_o_nat @ F2 @ A2 ) ) ) ) ).

% rev_image_eqI
thf(fact_312_rev__image__eqI,axiom,
    ! [X3: $o,A2: set_o,B: b,F2: $o > b] :
      ( ( member_o @ X3 @ A2 )
     => ( ( B
          = ( F2 @ X3 ) )
       => ( member_b @ B @ ( image_o_b @ F2 @ A2 ) ) ) ) ).

% rev_image_eqI
thf(fact_313_rev__image__eqI,axiom,
    ! [X3: nat,A2: set_nat,B: $o,F2: nat > $o] :
      ( ( member_nat @ X3 @ A2 )
     => ( ( B
          = ( F2 @ X3 ) )
       => ( member_o @ B @ ( image_nat_o @ F2 @ A2 ) ) ) ) ).

% rev_image_eqI
thf(fact_314_rev__image__eqI,axiom,
    ! [X3: nat,A2: set_nat,B: nat,F2: nat > nat] :
      ( ( member_nat @ X3 @ A2 )
     => ( ( B
          = ( F2 @ X3 ) )
       => ( member_nat @ B @ ( image_nat_nat @ F2 @ A2 ) ) ) ) ).

% rev_image_eqI
thf(fact_315_rev__image__eqI,axiom,
    ! [X3: nat,A2: set_nat,B: b,F2: nat > b] :
      ( ( member_nat @ X3 @ A2 )
     => ( ( B
          = ( F2 @ X3 ) )
       => ( member_b @ B @ ( image_nat_b @ F2 @ A2 ) ) ) ) ).

% rev_image_eqI
thf(fact_316_rev__image__eqI,axiom,
    ! [X3: b,A2: set_b,B: $o,F2: b > $o] :
      ( ( member_b @ X3 @ A2 )
     => ( ( B
          = ( F2 @ X3 ) )
       => ( member_o @ B @ ( image_b_o @ F2 @ A2 ) ) ) ) ).

% rev_image_eqI
thf(fact_317_rev__image__eqI,axiom,
    ! [X3: b,A2: set_b,B: nat,F2: b > nat] :
      ( ( member_b @ X3 @ A2 )
     => ( ( B
          = ( F2 @ X3 ) )
       => ( member_nat @ B @ ( image_b_nat @ F2 @ A2 ) ) ) ) ).

% rev_image_eqI
thf(fact_318_rev__image__eqI,axiom,
    ! [X3: b,A2: set_b,B: b,F2: b > b] :
      ( ( member_b @ X3 @ A2 )
     => ( ( B
          = ( F2 @ X3 ) )
       => ( member_b @ B @ ( image_b_b @ F2 @ A2 ) ) ) ) ).

% rev_image_eqI
thf(fact_319_rev__image__eqI,axiom,
    ! [X3: $o,A2: set_o,B: list_a,F2: $o > list_a] :
      ( ( member_o @ X3 @ A2 )
     => ( ( B
          = ( F2 @ X3 ) )
       => ( member_list_a @ B @ ( image_o_list_a @ F2 @ A2 ) ) ) ) ).

% rev_image_eqI
thf(fact_320_fun__upd__def,axiom,
    ( fun_upd_nat_a
    = ( ^ [F: nat > a,A3: nat,B3: a,X: nat] : ( if_a @ ( X = A3 ) @ B3 @ ( F @ X ) ) ) ) ).

% fun_upd_def
thf(fact_321_fun__upd__def,axiom,
    ( fun_upd_o_o
    = ( ^ [F: $o > $o,A3: $o,B3: $o,X: $o] :
          ( ( ( X = A3 )
           => B3 )
          & ( ( X = (~ A3) )
           => ( F @ X ) ) ) ) ) ).

% fun_upd_def
thf(fact_322_fun__upd__def,axiom,
    ( fun_upd_o_set_o
    = ( ^ [F: $o > set_o,A3: $o,B3: set_o,X: $o] : ( if_set_o @ ( X = A3 ) @ B3 @ ( F @ X ) ) ) ) ).

% fun_upd_def
thf(fact_323_fun__upd__eqD,axiom,
    ! [F2: nat > a,X3: nat,Y: a,G2: nat > a,Z: a] :
      ( ( ( fun_upd_nat_a @ F2 @ X3 @ Y )
        = ( fun_upd_nat_a @ G2 @ X3 @ Z ) )
     => ( Y = Z ) ) ).

% fun_upd_eqD
thf(fact_324_fun__upd__eqD,axiom,
    ! [F2: $o > $o,X3: $o,Y: $o,G2: $o > $o,Z: $o] :
      ( ( ( fun_upd_o_o @ F2 @ X3 @ Y )
        = ( fun_upd_o_o @ G2 @ X3 @ Z ) )
     => ( Y = Z ) ) ).

% fun_upd_eqD
thf(fact_325_fun__upd__eqD,axiom,
    ! [F2: $o > set_o,X3: $o,Y: set_o,G2: $o > set_o,Z: set_o] :
      ( ( ( fun_upd_o_set_o @ F2 @ X3 @ Y )
        = ( fun_upd_o_set_o @ G2 @ X3 @ Z ) )
     => ( Y = Z ) ) ).

% fun_upd_eqD
thf(fact_326_fun__upd__idem,axiom,
    ! [F2: nat > a,X3: nat,Y: a] :
      ( ( ( F2 @ X3 )
        = Y )
     => ( ( fun_upd_nat_a @ F2 @ X3 @ Y )
        = F2 ) ) ).

% fun_upd_idem
thf(fact_327_fun__upd__idem,axiom,
    ! [F2: $o > $o,X3: $o,Y: $o] :
      ( ( ( F2 @ X3 )
        = Y )
     => ( ( fun_upd_o_o @ F2 @ X3 @ Y )
        = F2 ) ) ).

% fun_upd_idem
thf(fact_328_fun__upd__idem,axiom,
    ! [F2: $o > set_o,X3: $o,Y: set_o] :
      ( ( ( F2 @ X3 )
        = Y )
     => ( ( fun_upd_o_set_o @ F2 @ X3 @ Y )
        = F2 ) ) ).

% fun_upd_idem
thf(fact_329_fun__upd__same,axiom,
    ! [F2: nat > a,X3: nat,Y: a] :
      ( ( fun_upd_nat_a @ F2 @ X3 @ Y @ X3 )
      = Y ) ).

% fun_upd_same
thf(fact_330_fun__upd__same,axiom,
    ! [F2: $o > $o,X3: $o,Y: $o] :
      ( ( fun_upd_o_o @ F2 @ X3 @ Y @ X3 )
      = Y ) ).

% fun_upd_same
thf(fact_331_fun__upd__same,axiom,
    ! [F2: $o > set_o,X3: $o,Y: set_o] :
      ( ( fun_upd_o_set_o @ F2 @ X3 @ Y @ X3 )
      = Y ) ).

% fun_upd_same
thf(fact_332_fun__upd__other,axiom,
    ! [Z: nat,X3: nat,F2: nat > a,Y: a] :
      ( ( Z != X3 )
     => ( ( fun_upd_nat_a @ F2 @ X3 @ Y @ Z )
        = ( F2 @ Z ) ) ) ).

% fun_upd_other
thf(fact_333_fun__upd__other,axiom,
    ! [Z: $o,X3: $o,F2: $o > $o,Y: $o] :
      ( ( Z != X3 )
     => ( ( fun_upd_o_o @ F2 @ X3 @ Y @ Z )
        = ( F2 @ Z ) ) ) ).

% fun_upd_other
thf(fact_334_fun__upd__other,axiom,
    ! [Z: $o,X3: $o,F2: $o > set_o,Y: set_o] :
      ( ( Z != X3 )
     => ( ( fun_upd_o_set_o @ F2 @ X3 @ Y @ Z )
        = ( F2 @ Z ) ) ) ).

% fun_upd_other
thf(fact_335_fun__upd__twist,axiom,
    ! [A: nat,C: nat,M2: nat > a,B: a,D: a] :
      ( ( A != C )
     => ( ( fun_upd_nat_a @ ( fun_upd_nat_a @ M2 @ A @ B ) @ C @ D )
        = ( fun_upd_nat_a @ ( fun_upd_nat_a @ M2 @ C @ D ) @ A @ B ) ) ) ).

% fun_upd_twist
thf(fact_336_fun__upd__twist,axiom,
    ! [A: $o,C: $o,M2: $o > $o,B: $o,D: $o] :
      ( ( A != C )
     => ( ( fun_upd_o_o @ ( fun_upd_o_o @ M2 @ A @ B ) @ C @ D )
        = ( fun_upd_o_o @ ( fun_upd_o_o @ M2 @ C @ D ) @ A @ B ) ) ) ).

% fun_upd_twist
thf(fact_337_fun__upd__twist,axiom,
    ! [A: $o,C: $o,M2: $o > set_o,B: set_o,D: set_o] :
      ( ( A != C )
     => ( ( fun_upd_o_set_o @ ( fun_upd_o_set_o @ M2 @ A @ B ) @ C @ D )
        = ( fun_upd_o_set_o @ ( fun_upd_o_set_o @ M2 @ C @ D ) @ A @ B ) ) ) ).

% fun_upd_twist
thf(fact_338_fun__upd__idem__iff,axiom,
    ! [F2: nat > a,X3: nat,Y: a] :
      ( ( ( fun_upd_nat_a @ F2 @ X3 @ Y )
        = F2 )
      = ( ( F2 @ X3 )
        = Y ) ) ).

% fun_upd_idem_iff
thf(fact_339_fun__upd__idem__iff,axiom,
    ! [F2: $o > $o,X3: $o,Y: $o] :
      ( ( ( fun_upd_o_o @ F2 @ X3 @ Y )
        = F2 )
      = ( ( F2 @ X3 )
        = Y ) ) ).

% fun_upd_idem_iff
thf(fact_340_fun__upd__idem__iff,axiom,
    ! [F2: $o > set_o,X3: $o,Y: set_o] :
      ( ( ( fun_upd_o_set_o @ F2 @ X3 @ Y )
        = F2 )
      = ( ( F2 @ X3 )
        = Y ) ) ).

% fun_upd_idem_iff
thf(fact_341_imageE,axiom,
    ! [B: $o,F2: $o > $o,A2: set_o] :
      ( ( member_o @ B @ ( image_o_o @ F2 @ A2 ) )
     => ~ ! [X5: $o] :
            ( ( B
              = ( F2 @ X5 ) )
           => ~ ( member_o @ X5 @ A2 ) ) ) ).

% imageE
thf(fact_342_imageE,axiom,
    ! [B: $o,F2: nat > $o,A2: set_nat] :
      ( ( member_o @ B @ ( image_nat_o @ F2 @ A2 ) )
     => ~ ! [X5: nat] :
            ( ( B
              = ( F2 @ X5 ) )
           => ~ ( member_nat @ X5 @ A2 ) ) ) ).

% imageE
thf(fact_343_imageE,axiom,
    ! [B: $o,F2: b > $o,A2: set_b] :
      ( ( member_o @ B @ ( image_b_o @ F2 @ A2 ) )
     => ~ ! [X5: b] :
            ( ( B
              = ( F2 @ X5 ) )
           => ~ ( member_b @ X5 @ A2 ) ) ) ).

% imageE
thf(fact_344_imageE,axiom,
    ! [B: nat,F2: $o > nat,A2: set_o] :
      ( ( member_nat @ B @ ( image_o_nat @ F2 @ A2 ) )
     => ~ ! [X5: $o] :
            ( ( B
              = ( F2 @ X5 ) )
           => ~ ( member_o @ X5 @ A2 ) ) ) ).

% imageE
thf(fact_345_imageE,axiom,
    ! [B: nat,F2: nat > nat,A2: set_nat] :
      ( ( member_nat @ B @ ( image_nat_nat @ F2 @ A2 ) )
     => ~ ! [X5: nat] :
            ( ( B
              = ( F2 @ X5 ) )
           => ~ ( member_nat @ X5 @ A2 ) ) ) ).

% imageE
thf(fact_346_imageE,axiom,
    ! [B: nat,F2: b > nat,A2: set_b] :
      ( ( member_nat @ B @ ( image_b_nat @ F2 @ A2 ) )
     => ~ ! [X5: b] :
            ( ( B
              = ( F2 @ X5 ) )
           => ~ ( member_b @ X5 @ A2 ) ) ) ).

% imageE
thf(fact_347_imageE,axiom,
    ! [B: b,F2: $o > b,A2: set_o] :
      ( ( member_b @ B @ ( image_o_b @ F2 @ A2 ) )
     => ~ ! [X5: $o] :
            ( ( B
              = ( F2 @ X5 ) )
           => ~ ( member_o @ X5 @ A2 ) ) ) ).

% imageE
thf(fact_348_imageE,axiom,
    ! [B: b,F2: nat > b,A2: set_nat] :
      ( ( member_b @ B @ ( image_nat_b @ F2 @ A2 ) )
     => ~ ! [X5: nat] :
            ( ( B
              = ( F2 @ X5 ) )
           => ~ ( member_nat @ X5 @ A2 ) ) ) ).

% imageE
thf(fact_349_imageE,axiom,
    ! [B: b,F2: b > b,A2: set_b] :
      ( ( member_b @ B @ ( image_b_b @ F2 @ A2 ) )
     => ~ ! [X5: b] :
            ( ( B
              = ( F2 @ X5 ) )
           => ~ ( member_b @ X5 @ A2 ) ) ) ).

% imageE
thf(fact_350_imageE,axiom,
    ! [B: $o,F2: list_a > $o,A2: set_list_a] :
      ( ( member_o @ B @ ( image_list_a_o @ F2 @ A2 ) )
     => ~ ! [X5: list_a] :
            ( ( B
              = ( F2 @ X5 ) )
           => ~ ( member_list_a @ X5 @ A2 ) ) ) ).

% imageE
thf(fact_351_image__image,axiom,
    ! [F2: nat > nat,G2: $o > nat,A2: set_o] :
      ( ( image_nat_nat @ F2 @ ( image_o_nat @ G2 @ A2 ) )
      = ( image_o_nat
        @ ^ [X: $o] : ( F2 @ ( G2 @ X ) )
        @ A2 ) ) ).

% image_image
thf(fact_352_image__image,axiom,
    ! [F2: $o > nat,G2: $o > $o,A2: set_o] :
      ( ( image_o_nat @ F2 @ ( image_o_o @ G2 @ A2 ) )
      = ( image_o_nat
        @ ^ [X: $o] : ( F2 @ ( G2 @ X ) )
        @ A2 ) ) ).

% image_image
thf(fact_353_image__image,axiom,
    ! [F2: $o > $o,G2: set_o > $o,A2: set_set_o] :
      ( ( image_o_o @ F2 @ ( image_set_o_o @ G2 @ A2 ) )
      = ( image_set_o_o
        @ ^ [X: set_o] : ( F2 @ ( G2 @ X ) )
        @ A2 ) ) ).

% image_image
thf(fact_354_image__image,axiom,
    ! [F2: $o > nat,G2: set_o > $o,A2: set_set_o] :
      ( ( image_o_nat @ F2 @ ( image_set_o_o @ G2 @ A2 ) )
      = ( image_set_o_nat
        @ ^ [X: set_o] : ( F2 @ ( G2 @ X ) )
        @ A2 ) ) ).

% image_image
thf(fact_355_image__image,axiom,
    ! [F2: $o > set_o,G2: set_o > $o,A2: set_set_o] :
      ( ( image_o_set_o @ F2 @ ( image_set_o_o @ G2 @ A2 ) )
      = ( image_set_o_set_o
        @ ^ [X: set_o] : ( F2 @ ( G2 @ X ) )
        @ A2 ) ) ).

% image_image
thf(fact_356_image__image,axiom,
    ! [F2: set_o > $o,G2: set_o > set_o,A2: set_set_o] :
      ( ( image_set_o_o @ F2 @ ( image_set_o_set_o @ G2 @ A2 ) )
      = ( image_set_o_o
        @ ^ [X: set_o] : ( F2 @ ( G2 @ X ) )
        @ A2 ) ) ).

% image_image
thf(fact_357_image__image,axiom,
    ! [F2: $o > $o > $o,G2: set_o > $o,A2: set_set_o] :
      ( ( image_o_o_o2 @ F2 @ ( image_set_o_o @ G2 @ A2 ) )
      = ( image_set_o_o_o
        @ ^ [X: set_o] : ( F2 @ ( G2 @ X ) )
        @ A2 ) ) ).

% image_image
thf(fact_358_image__image,axiom,
    ! [F2: ( $o > $o ) > $o,G2: set_o > $o > $o,A2: set_set_o] :
      ( ( image_o_o_o @ F2 @ ( image_set_o_o_o @ G2 @ A2 ) )
      = ( image_set_o_o
        @ ^ [X: set_o] : ( F2 @ ( G2 @ X ) )
        @ A2 ) ) ).

% image_image
thf(fact_359_image__image,axiom,
    ! [F2: set_o > $o,G2: ( $o > $o ) > set_o,A2: set_o_o] :
      ( ( image_set_o_o @ F2 @ ( image_o_o_set_o @ G2 @ A2 ) )
      = ( image_o_o_o
        @ ^ [X: $o > $o] : ( F2 @ ( G2 @ X ) )
        @ A2 ) ) ).

% image_image
thf(fact_360_image__image,axiom,
    ! [F2: set_o > set_o,G2: set_o > set_o,A2: set_set_o] :
      ( ( image_set_o_set_o @ F2 @ ( image_set_o_set_o @ G2 @ A2 ) )
      = ( image_set_o_set_o
        @ ^ [X: set_o] : ( F2 @ ( G2 @ X ) )
        @ A2 ) ) ).

% image_image
thf(fact_361_Compr__image__eq,axiom,
    ! [F2: $o > $o,A2: set_o,P: $o > $o] :
      ( ( collect_o
        @ ^ [X: $o] :
            ( ( member_o @ X @ ( image_o_o @ F2 @ A2 ) )
            & ( P @ X ) ) )
      = ( image_o_o @ F2
        @ ( collect_o
          @ ^ [X: $o] :
              ( ( member_o @ X @ A2 )
              & ( P @ ( F2 @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_362_Compr__image__eq,axiom,
    ! [F2: nat > nat,A2: set_nat,P: nat > $o] :
      ( ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ ( image_nat_nat @ F2 @ A2 ) )
            & ( P @ X ) ) )
      = ( image_nat_nat @ F2
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A2 )
              & ( P @ ( F2 @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_363_Compr__image__eq,axiom,
    ! [F2: b > nat,A2: set_b,P: nat > $o] :
      ( ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ ( image_b_nat @ F2 @ A2 ) )
            & ( P @ X ) ) )
      = ( image_b_nat @ F2
        @ ( collect_b
          @ ^ [X: b] :
              ( ( member_b @ X @ A2 )
              & ( P @ ( F2 @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_364_Compr__image__eq,axiom,
    ! [F2: nat > b,A2: set_nat,P: b > $o] :
      ( ( collect_b
        @ ^ [X: b] :
            ( ( member_b @ X @ ( image_nat_b @ F2 @ A2 ) )
            & ( P @ X ) ) )
      = ( image_nat_b @ F2
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A2 )
              & ( P @ ( F2 @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_365_Compr__image__eq,axiom,
    ! [F2: b > b,A2: set_b,P: b > $o] :
      ( ( collect_b
        @ ^ [X: b] :
            ( ( member_b @ X @ ( image_b_b @ F2 @ A2 ) )
            & ( P @ X ) ) )
      = ( image_b_b @ F2
        @ ( collect_b
          @ ^ [X: b] :
              ( ( member_b @ X @ A2 )
              & ( P @ ( F2 @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_366_Compr__image__eq,axiom,
    ! [F2: $o > nat,A2: set_o,P: nat > $o] :
      ( ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ ( image_o_nat @ F2 @ A2 ) )
            & ( P @ X ) ) )
      = ( image_o_nat @ F2
        @ ( collect_o
          @ ^ [X: $o] :
              ( ( member_o @ X @ A2 )
              & ( P @ ( F2 @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_367_Compr__image__eq,axiom,
    ! [F2: $o > b,A2: set_o,P: b > $o] :
      ( ( collect_b
        @ ^ [X: b] :
            ( ( member_b @ X @ ( image_o_b @ F2 @ A2 ) )
            & ( P @ X ) ) )
      = ( image_o_b @ F2
        @ ( collect_o
          @ ^ [X: $o] :
              ( ( member_o @ X @ A2 )
              & ( P @ ( F2 @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_368_Compr__image__eq,axiom,
    ! [F2: nat > $o,A2: set_nat,P: $o > $o] :
      ( ( collect_o
        @ ^ [X: $o] :
            ( ( member_o @ X @ ( image_nat_o @ F2 @ A2 ) )
            & ( P @ X ) ) )
      = ( image_nat_o @ F2
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A2 )
              & ( P @ ( F2 @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_369_Compr__image__eq,axiom,
    ! [F2: b > $o,A2: set_b,P: $o > $o] :
      ( ( collect_o
        @ ^ [X: $o] :
            ( ( member_o @ X @ ( image_b_o @ F2 @ A2 ) )
            & ( P @ X ) ) )
      = ( image_b_o @ F2
        @ ( collect_b
          @ ^ [X: b] :
              ( ( member_b @ X @ A2 )
              & ( P @ ( F2 @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_370_Compr__image__eq,axiom,
    ! [F2: nat > list_a,A2: set_nat,P: list_a > $o] :
      ( ( collect_list_a
        @ ^ [X: list_a] :
            ( ( member_list_a @ X @ ( image_nat_list_a @ F2 @ A2 ) )
            & ( P @ X ) ) )
      = ( image_nat_list_a @ F2
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A2 )
              & ( P @ ( F2 @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_371_cpropagated__simps_I7_J,axiom,
    ! [X3: nat,Q: relational_fmla_a_b] :
      ( ( relati1591879772219623554ed_a_b @ ( relati591517084277583526ts_a_b @ X3 @ Q ) )
      = ( ( member_nat @ X3 @ ( relational_fv_a_b @ Q ) )
        & ( relational_nocp_a_b @ Q ) ) ) ).

% cpropagated_simps(7)
thf(fact_372_fmla_Oset__cases_I2_J,axiom,
    ! [E: b,A: relational_fmla_a_b] :
      ( ( member_b @ E @ ( relati8924981150291758614la_a_b @ A ) )
     => ( ! [Z22: list_R6823256787227418703term_a] :
            ( A
           != ( relational_Pred_b_a @ E @ Z22 ) )
       => ( ! [Z2: relational_fmla_a_b] :
              ( ( A
                = ( relational_Neg_a_b @ Z2 ) )
             => ~ ( member_b @ E @ ( relati8924981150291758614la_a_b @ Z2 ) ) )
         => ( ! [Z1: relational_fmla_a_b] :
                ( ? [Z22: relational_fmla_a_b] :
                    ( A
                    = ( relational_Conj_a_b @ Z1 @ Z22 ) )
               => ~ ( member_b @ E @ ( relati8924981150291758614la_a_b @ Z1 ) ) )
           => ( ! [Z1: relational_fmla_a_b,Z22: relational_fmla_a_b] :
                  ( ( A
                    = ( relational_Conj_a_b @ Z1 @ Z22 ) )
                 => ~ ( member_b @ E @ ( relati8924981150291758614la_a_b @ Z22 ) ) )
             => ( ! [Z1: relational_fmla_a_b] :
                    ( ? [Z22: relational_fmla_a_b] :
                        ( A
                        = ( relational_Disj_a_b @ Z1 @ Z22 ) )
                   => ~ ( member_b @ E @ ( relati8924981150291758614la_a_b @ Z1 ) ) )
               => ( ! [Z1: relational_fmla_a_b,Z22: relational_fmla_a_b] :
                      ( ( A
                        = ( relational_Disj_a_b @ Z1 @ Z22 ) )
                     => ~ ( member_b @ E @ ( relati8924981150291758614la_a_b @ Z22 ) ) )
                 => ~ ! [Z1: nat,Z22: relational_fmla_a_b] :
                        ( ( A
                          = ( relati591517084277583526ts_a_b @ Z1 @ Z22 ) )
                       => ~ ( member_b @ E @ ( relati8924981150291758614la_a_b @ Z22 ) ) ) ) ) ) ) ) ) ) ).

% fmla.set_cases(2)
thf(fact_373_eval__subst,axiom,
    ! [Sigma: nat > a,T: relational_term_a,X3: nat,Y: nat] :
      ( ( relati1177013128715261720term_a @ Sigma @ ( relati7175845559408349773term_a @ T @ X3 @ Y ) )
      = ( relati1177013128715261720term_a @ ( fun_upd_nat_a @ Sigma @ X3 @ ( Sigma @ Y ) ) @ T ) ) ).

% eval_subst
thf(fact_374_map__eq__map__tailrec,axiom,
    map_Pr2810398200501793500la_a_b = map_ta546366608199281938la_a_b ).

% map_eq_map_tailrec
thf(fact_375_map__eq__map__tailrec,axiom,
    map_Pr8058819605623181956at_nat = map_ta258029862405838542at_nat ).

% map_eq_map_tailrec
thf(fact_376_map__eq__map__tailrec,axiom,
    map_Pr591601166967198746la_a_b = map_ta5757639970438782800la_a_b ).

% map_eq_map_tailrec
thf(fact_377_map__eq__map__tailrec,axiom,
    map_na7298421622053143531at_nat = map_ta2717133102674191393at_nat ).

% map_eq_map_tailrec
thf(fact_378_map__eq__map__tailrec,axiom,
    map_nat_nat = map_tailrec_nat_nat ).

% map_eq_map_tailrec
thf(fact_379_map__zip__map2,axiom,
    ! [F2: product_prod_nat_nat > relational_fmla_a_b,Xs: list_nat,G2: nat > nat,Ys: list_nat] :
      ( ( map_Pr2810398200501793500la_a_b @ F2 @ ( zip_nat_nat @ Xs @ ( map_nat_nat @ G2 @ Ys ) ) )
      = ( map_Pr2810398200501793500la_a_b
        @ ( produc3270801013941088237la_a_b
          @ ^ [X: nat,Y2: nat] : ( F2 @ ( product_Pair_nat_nat @ X @ ( G2 @ Y2 ) ) ) )
        @ ( zip_nat_nat @ Xs @ Ys ) ) ) ).

% map_zip_map2
thf(fact_380_map__zip__map2,axiom,
    ! [F2: product_prod_nat_nat > relational_fmla_a_b > relational_fmla_a_b,Xs: list_nat,G2: nat > nat,Ys: list_nat] :
      ( ( map_Pr591601166967198746la_a_b @ F2 @ ( zip_nat_nat @ Xs @ ( map_nat_nat @ G2 @ Ys ) ) )
      = ( map_Pr591601166967198746la_a_b
        @ ( produc5586541307551673003la_a_b
          @ ^ [X: nat,Y2: nat] : ( F2 @ ( product_Pair_nat_nat @ X @ ( G2 @ Y2 ) ) ) )
        @ ( zip_nat_nat @ Xs @ Ys ) ) ) ).

% map_zip_map2
thf(fact_381_map__zip__map2,axiom,
    ! [F2: product_prod_nat_nat > product_prod_nat_nat,Xs: list_nat,G2: nat > nat,Ys: list_nat] :
      ( ( map_Pr8058819605623181956at_nat @ F2 @ ( zip_nat_nat @ Xs @ ( map_nat_nat @ G2 @ Ys ) ) )
      = ( map_Pr8058819605623181956at_nat
        @ ( produc2626176000494625587at_nat
          @ ^ [X: nat,Y2: nat] : ( F2 @ ( product_Pair_nat_nat @ X @ ( G2 @ Y2 ) ) ) )
        @ ( zip_nat_nat @ Xs @ Ys ) ) ) ).

% map_zip_map2
thf(fact_382_map__zip__map2,axiom,
    ! [F2: product_prod_nat_nat > relational_fmla_a_b,Xs: list_nat,G2: product_prod_nat_nat > nat,Ys: list_P6011104703257516679at_nat] :
      ( ( map_Pr2810398200501793500la_a_b @ F2 @ ( zip_nat_nat @ Xs @ ( map_Pr3938374229010428429at_nat @ G2 @ Ys ) ) )
      = ( map_Pr6825106633284705035la_a_b
        @ ( produc4829065610495871524la_a_b
          @ ^ [X: nat,Y2: product_prod_nat_nat] : ( F2 @ ( product_Pair_nat_nat @ X @ ( G2 @ Y2 ) ) ) )
        @ ( zip_na1006125974040638520at_nat @ Xs @ Ys ) ) ) ).

% map_zip_map2
thf(fact_383_map__zip__map2,axiom,
    ! [F2: product_prod_nat_nat > product_prod_nat_nat,Xs: list_nat,G2: product_prod_nat_nat > nat,Ys: list_P6011104703257516679at_nat] :
      ( ( map_Pr8058819605623181956at_nat @ F2 @ ( zip_nat_nat @ Xs @ ( map_Pr3938374229010428429at_nat @ G2 @ Ys ) ) )
      = ( map_Pr2617240807308709013at_nat
        @ ( produc8859641928216934716at_nat
          @ ^ [X: nat,Y2: product_prod_nat_nat] : ( F2 @ ( product_Pair_nat_nat @ X @ ( G2 @ Y2 ) ) ) )
        @ ( zip_na1006125974040638520at_nat @ Xs @ Ys ) ) ) ).

% map_zip_map2
thf(fact_384_map__zip__map2,axiom,
    ! [F2: produc7248412053542808358at_nat > relational_fmla_a_b,Xs: list_nat,G2: nat > product_prod_nat_nat,Ys: list_nat] :
      ( ( map_Pr6825106633284705035la_a_b @ F2 @ ( zip_na1006125974040638520at_nat @ Xs @ ( map_na7298421622053143531at_nat @ G2 @ Ys ) ) )
      = ( map_Pr2810398200501793500la_a_b
        @ ( produc3270801013941088237la_a_b
          @ ^ [X: nat,Y2: nat] : ( F2 @ ( produc487386426758144856at_nat @ X @ ( G2 @ Y2 ) ) ) )
        @ ( zip_nat_nat @ Xs @ Ys ) ) ) ).

% map_zip_map2
thf(fact_385_map__zip__map2,axiom,
    ! [F2: product_prod_nat_nat > produc7248412053542808358at_nat,Xs: list_nat,G2: nat > nat,Ys: list_nat] :
      ( ( map_Pr7536285556892129763at_nat @ F2 @ ( zip_nat_nat @ Xs @ ( map_nat_nat @ G2 @ Ys ) ) )
      = ( map_Pr7536285556892129763at_nat
        @ ( produc9083241971206738548at_nat
          @ ^ [X: nat,Y2: nat] : ( F2 @ ( product_Pair_nat_nat @ X @ ( G2 @ Y2 ) ) ) )
        @ ( zip_nat_nat @ Xs @ Ys ) ) ) ).

% map_zip_map2
thf(fact_386_map__zip__map2,axiom,
    ! [F2: produc7248412053542808358at_nat > product_prod_nat_nat,Xs: list_nat,G2: nat > product_prod_nat_nat,Ys: list_nat] :
      ( ( map_Pr2617240807308709013at_nat @ F2 @ ( zip_na1006125974040638520at_nat @ Xs @ ( map_na7298421622053143531at_nat @ G2 @ Ys ) ) )
      = ( map_Pr8058819605623181956at_nat
        @ ( produc2626176000494625587at_nat
          @ ^ [X: nat,Y2: nat] : ( F2 @ ( produc487386426758144856at_nat @ X @ ( G2 @ Y2 ) ) ) )
        @ ( zip_nat_nat @ Xs @ Ys ) ) ) ).

% map_zip_map2
thf(fact_387_map__zip__map2,axiom,
    ! [F2: product_prod_nat_nat > relational_term_a > relational_term_a,Xs: list_nat,G2: nat > nat,Ys: list_nat] :
      ( ( map_Pr244373919324003298term_a @ F2 @ ( zip_nat_nat @ Xs @ ( map_nat_nat @ G2 @ Ys ) ) )
      = ( map_Pr244373919324003298term_a
        @ ( produc6628518323692928499term_a
          @ ^ [X: nat,Y2: nat] : ( F2 @ ( product_Pair_nat_nat @ X @ ( G2 @ Y2 ) ) ) )
        @ ( zip_nat_nat @ Xs @ Ys ) ) ) ).

% map_zip_map2
thf(fact_388_map__zip__map2,axiom,
    ! [F2: product_prod_nat_nat > produc7248412053542808358at_nat,Xs: list_nat,G2: product_prod_nat_nat > nat,Ys: list_P6011104703257516679at_nat] :
      ( ( map_Pr7536285556892129763at_nat @ F2 @ ( zip_nat_nat @ Xs @ ( map_Pr3938374229010428429at_nat @ G2 @ Ys ) ) )
      = ( map_Pr6261813372141627026at_nat
        @ ( produc968775922737392939at_nat
          @ ^ [X: nat,Y2: product_prod_nat_nat] : ( F2 @ ( product_Pair_nat_nat @ X @ ( G2 @ Y2 ) ) ) )
        @ ( zip_na1006125974040638520at_nat @ Xs @ Ys ) ) ) ).

% map_zip_map2
thf(fact_389_map__zip__map,axiom,
    ! [F2: product_prod_nat_nat > relational_fmla_a_b,G2: nat > nat,Xs: list_nat,Ys: list_nat] :
      ( ( map_Pr2810398200501793500la_a_b @ F2 @ ( zip_nat_nat @ ( map_nat_nat @ G2 @ Xs ) @ Ys ) )
      = ( map_Pr2810398200501793500la_a_b
        @ ( produc3270801013941088237la_a_b
          @ ^ [X: nat,Y2: nat] : ( F2 @ ( product_Pair_nat_nat @ ( G2 @ X ) @ Y2 ) ) )
        @ ( zip_nat_nat @ Xs @ Ys ) ) ) ).

% map_zip_map
thf(fact_390_map__zip__map,axiom,
    ! [F2: product_prod_nat_nat > relational_fmla_a_b > relational_fmla_a_b,G2: nat > nat,Xs: list_nat,Ys: list_nat] :
      ( ( map_Pr591601166967198746la_a_b @ F2 @ ( zip_nat_nat @ ( map_nat_nat @ G2 @ Xs ) @ Ys ) )
      = ( map_Pr591601166967198746la_a_b
        @ ( produc5586541307551673003la_a_b
          @ ^ [X: nat,Y2: nat] : ( F2 @ ( product_Pair_nat_nat @ ( G2 @ X ) @ Y2 ) ) )
        @ ( zip_nat_nat @ Xs @ Ys ) ) ) ).

% map_zip_map
thf(fact_391_map__zip__map,axiom,
    ! [F2: product_prod_b_nat > relational_fmla_a_b,G2: nat > b,Xs: list_nat,Ys: list_nat] :
      ( ( map_Pr5462649024688169187la_a_b @ F2 @ ( zip_b_nat @ ( map_nat_b @ G2 @ Xs ) @ Ys ) )
      = ( map_Pr2810398200501793500la_a_b
        @ ( produc3270801013941088237la_a_b
          @ ^ [X: nat,Y2: nat] : ( F2 @ ( product_Pair_b_nat @ ( G2 @ X ) @ Y2 ) ) )
        @ ( zip_nat_nat @ Xs @ Ys ) ) ) ).

% map_zip_map
thf(fact_392_map__zip__map,axiom,
    ! [F2: product_prod_nat_nat > product_prod_nat_nat,G2: nat > nat,Xs: list_nat,Ys: list_nat] :
      ( ( map_Pr8058819605623181956at_nat @ F2 @ ( zip_nat_nat @ ( map_nat_nat @ G2 @ Xs ) @ Ys ) )
      = ( map_Pr8058819605623181956at_nat
        @ ( produc2626176000494625587at_nat
          @ ^ [X: nat,Y2: nat] : ( F2 @ ( product_Pair_nat_nat @ ( G2 @ X ) @ Y2 ) ) )
        @ ( zip_nat_nat @ Xs @ Ys ) ) ) ).

% map_zip_map
thf(fact_393_map__zip__map,axiom,
    ! [F2: product_prod_b_nat > product_prod_nat_nat,G2: nat > b,Xs: list_nat,Ys: list_nat] :
      ( ( map_Pr3466128418965176509at_nat @ F2 @ ( zip_b_nat @ ( map_nat_b @ G2 @ Xs ) @ Ys ) )
      = ( map_Pr8058819605623181956at_nat
        @ ( produc2626176000494625587at_nat
          @ ^ [X: nat,Y2: nat] : ( F2 @ ( product_Pair_b_nat @ ( G2 @ X ) @ Y2 ) ) )
        @ ( zip_nat_nat @ Xs @ Ys ) ) ) ).

% map_zip_map
thf(fact_394_map__zip__map,axiom,
    ! [F2: product_prod_nat_nat > relational_fmla_a_b,G2: product_prod_nat_nat > nat,Xs: list_P6011104703257516679at_nat,Ys: list_nat] :
      ( ( map_Pr2810398200501793500la_a_b @ F2 @ ( zip_nat_nat @ ( map_Pr3938374229010428429at_nat @ G2 @ Xs ) @ Ys ) )
      = ( map_Pr5367276110915692261la_a_b
        @ ( produc7360703905887518278la_a_b
          @ ^ [X: product_prod_nat_nat,Y2: nat] : ( F2 @ ( product_Pair_nat_nat @ ( G2 @ X ) @ Y2 ) ) )
        @ ( zip_Pr6869450617852699226at_nat @ Xs @ Ys ) ) ) ).

% map_zip_map
thf(fact_395_map__zip__map,axiom,
    ! [F2: product_prod_nat_nat > product_prod_nat_nat,G2: product_prod_nat_nat > nat,Xs: list_P6011104703257516679at_nat,Ys: list_nat] :
      ( ( map_Pr8058819605623181956at_nat @ F2 @ ( zip_nat_nat @ ( map_Pr3938374229010428429at_nat @ G2 @ Xs ) @ Ys ) )
      = ( map_Pr4819452465118600763at_nat
        @ ( produc373799411880517786at_nat
          @ ^ [X: product_prod_nat_nat,Y2: nat] : ( F2 @ ( product_Pair_nat_nat @ ( G2 @ X ) @ Y2 ) ) )
        @ ( zip_Pr6869450617852699226at_nat @ Xs @ Ys ) ) ) ).

% map_zip_map
thf(fact_396_map__zip__map,axiom,
    ! [F2: produc8373899037510109440at_nat > relational_fmla_a_b,G2: nat > product_prod_nat_nat,Xs: list_nat,Ys: list_nat] :
      ( ( map_Pr5367276110915692261la_a_b @ F2 @ ( zip_Pr6869450617852699226at_nat @ ( map_na7298421622053143531at_nat @ G2 @ Xs ) @ Ys ) )
      = ( map_Pr2810398200501793500la_a_b
        @ ( produc3270801013941088237la_a_b
          @ ^ [X: nat,Y2: nat] : ( F2 @ ( produc6350711070570205562at_nat @ ( G2 @ X ) @ Y2 ) ) )
        @ ( zip_nat_nat @ Xs @ Ys ) ) ) ).

% map_zip_map
thf(fact_397_map__zip__map,axiom,
    ! [F2: product_prod_nat_nat > produc7248412053542808358at_nat,G2: nat > nat,Xs: list_nat,Ys: list_nat] :
      ( ( map_Pr7536285556892129763at_nat @ F2 @ ( zip_nat_nat @ ( map_nat_nat @ G2 @ Xs ) @ Ys ) )
      = ( map_Pr7536285556892129763at_nat
        @ ( produc9083241971206738548at_nat
          @ ^ [X: nat,Y2: nat] : ( F2 @ ( product_Pair_nat_nat @ ( G2 @ X ) @ Y2 ) ) )
        @ ( zip_nat_nat @ Xs @ Ys ) ) ) ).

% map_zip_map
thf(fact_398_map__zip__map,axiom,
    ! [F2: product_prod_b_nat > produc7248412053542808358at_nat,G2: nat > b,Xs: list_nat,Ys: list_nat] :
      ( ( map_Pr7862426794231505002at_nat @ F2 @ ( zip_b_nat @ ( map_nat_b @ G2 @ Xs ) @ Ys ) )
      = ( map_Pr7536285556892129763at_nat
        @ ( produc9083241971206738548at_nat
          @ ^ [X: nat,Y2: nat] : ( F2 @ ( product_Pair_b_nat @ ( G2 @ X ) @ Y2 ) ) )
        @ ( zip_nat_nat @ Xs @ Ys ) ) ) ).

% map_zip_map
thf(fact_399_Inf_OINF__identity__eq,axiom,
    ! [Inf: set_set_o > set_o,A2: set_set_o] :
      ( ( Inf
        @ ( image_set_o_set_o
          @ ^ [X: set_o] : X
          @ A2 ) )
      = ( Inf @ A2 ) ) ).

% Inf.INF_identity_eq
thf(fact_400_Sup_OSUP__identity__eq,axiom,
    ! [Sup: set_set_o > set_o,A2: set_set_o] :
      ( ( Sup
        @ ( image_set_o_set_o
          @ ^ [X: set_o] : X
          @ A2 ) )
      = ( Sup @ A2 ) ) ).

% Sup.SUP_identity_eq
thf(fact_401_prod_Oinject,axiom,
    ! [X13: $o,X22: $o,Y1: $o,Y22: $o] :
      ( ( ( product_Pair_o_o @ X13 @ X22 )
        = ( product_Pair_o_o @ Y1 @ Y22 ) )
      = ( ( X13 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_402_prod_Oinject,axiom,
    ! [X13: nat,X22: product_prod_nat_nat,Y1: nat,Y22: product_prod_nat_nat] :
      ( ( ( produc487386426758144856at_nat @ X13 @ X22 )
        = ( produc487386426758144856at_nat @ Y1 @ Y22 ) )
      = ( ( X13 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_403_prod_Oinject,axiom,
    ! [X13: nat,X22: nat,Y1: nat,Y22: nat] :
      ( ( ( product_Pair_nat_nat @ X13 @ X22 )
        = ( product_Pair_nat_nat @ Y1 @ Y22 ) )
      = ( ( X13 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_404_prod_Oinject,axiom,
    ! [X13: product_prod_b_nat > set_list_a,X22: nat > a,Y1: product_prod_b_nat > set_list_a,Y22: nat > a] :
      ( ( ( produc2895298938842563487_nat_a @ X13 @ X22 )
        = ( produc2895298938842563487_nat_a @ Y1 @ Y22 ) )
      = ( ( X13 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_405_prod_Oinject,axiom,
    ! [X13: b,X22: nat,Y1: b,Y22: nat] :
      ( ( ( product_Pair_b_nat @ X13 @ X22 )
        = ( product_Pair_b_nat @ Y1 @ Y22 ) )
      = ( ( X13 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_406_old_Oprod_Oinject,axiom,
    ! [A: $o,B: $o,A4: $o,B4: $o] :
      ( ( ( product_Pair_o_o @ A @ B )
        = ( product_Pair_o_o @ A4 @ B4 ) )
      = ( ( A = A4 )
        & ( B = B4 ) ) ) ).

% old.prod.inject
thf(fact_407_old_Oprod_Oinject,axiom,
    ! [A: nat,B: product_prod_nat_nat,A4: nat,B4: product_prod_nat_nat] :
      ( ( ( produc487386426758144856at_nat @ A @ B )
        = ( produc487386426758144856at_nat @ A4 @ B4 ) )
      = ( ( A = A4 )
        & ( B = B4 ) ) ) ).

% old.prod.inject
thf(fact_408_old_Oprod_Oinject,axiom,
    ! [A: nat,B: nat,A4: nat,B4: nat] :
      ( ( ( product_Pair_nat_nat @ A @ B )
        = ( product_Pair_nat_nat @ A4 @ B4 ) )
      = ( ( A = A4 )
        & ( B = B4 ) ) ) ).

% old.prod.inject
thf(fact_409_old_Oprod_Oinject,axiom,
    ! [A: product_prod_b_nat > set_list_a,B: nat > a,A4: product_prod_b_nat > set_list_a,B4: nat > a] :
      ( ( ( produc2895298938842563487_nat_a @ A @ B )
        = ( produc2895298938842563487_nat_a @ A4 @ B4 ) )
      = ( ( A = A4 )
        & ( B = B4 ) ) ) ).

% old.prod.inject
thf(fact_410_old_Oprod_Oinject,axiom,
    ! [A: b,B: nat,A4: b,B4: nat] :
      ( ( ( product_Pair_b_nat @ A @ B )
        = ( product_Pair_b_nat @ A4 @ B4 ) )
      = ( ( A = A4 )
        & ( B = B4 ) ) ) ).

% old.prod.inject
thf(fact_411_curryI,axiom,
    ! [F2: product_prod_o_o > $o,A: $o,B: $o] :
      ( ( F2 @ ( product_Pair_o_o @ A @ B ) )
     => ( product_curry_o_o_o @ F2 @ A @ B ) ) ).

% curryI
thf(fact_412_curryI,axiom,
    ! [F2: produc7248412053542808358at_nat > $o,A: nat,B: product_prod_nat_nat] :
      ( ( F2 @ ( produc487386426758144856at_nat @ A @ B ) )
     => ( produc7331361819228145906_nat_o @ F2 @ A @ B ) ) ).

% curryI
thf(fact_413_curryI,axiom,
    ! [F2: product_prod_nat_nat > $o,A: nat,B: nat] :
      ( ( F2 @ ( product_Pair_nat_nat @ A @ B ) )
     => ( produc1310100445399344235_nat_o @ F2 @ A @ B ) ) ).

% curryI
thf(fact_414_curryI,axiom,
    ! [F2: produc5835360497134304175_nat_a > $o,A: product_prod_b_nat > set_list_a,B: nat > a] :
      ( ( F2 @ ( produc2895298938842563487_nat_a @ A @ B ) )
     => ( produc5059092166261958789at_a_o @ F2 @ A @ B ) ) ).

% curryI
thf(fact_415_curryI,axiom,
    ! [F2: product_prod_b_nat > $o,A: b,B: nat] :
      ( ( F2 @ ( product_Pair_b_nat @ A @ B ) )
     => ( produc2461434047082304082_nat_o @ F2 @ A @ B ) ) ).

% curryI
thf(fact_416_mem__case__prodI2,axiom,
    ! [P2: product_prod_o_o,Z: $o,C: $o > $o > set_o] :
      ( ! [A5: $o,B2: $o] :
          ( ( P2
            = ( product_Pair_o_o @ A5 @ B2 ) )
         => ( member_o @ Z @ ( C @ A5 @ B2 ) ) )
     => ( member_o @ Z @ ( produc1238384690215476812_set_o @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_417_mem__case__prodI2,axiom,
    ! [P2: product_prod_o_o,Z: nat,C: $o > $o > set_nat] :
      ( ! [A5: $o,B2: $o] :
          ( ( P2
            = ( product_Pair_o_o @ A5 @ B2 ) )
         => ( member_nat @ Z @ ( C @ A5 @ B2 ) ) )
     => ( member_nat @ Z @ ( produc6723186405834743986et_nat @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_418_mem__case__prodI2,axiom,
    ! [P2: product_prod_o_o,Z: b,C: $o > $o > set_b] :
      ( ! [A5: $o,B2: $o] :
          ( ( P2
            = ( product_Pair_o_o @ A5 @ B2 ) )
         => ( member_b @ Z @ ( C @ A5 @ B2 ) ) )
     => ( member_b @ Z @ ( produc8162358544190516659_set_b @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_419_mem__case__prodI2,axiom,
    ! [P2: product_prod_nat_nat,Z: $o,C: nat > nat > set_o] :
      ( ! [A5: nat,B2: nat] :
          ( ( P2
            = ( product_Pair_nat_nat @ A5 @ B2 ) )
         => ( member_o @ Z @ ( C @ A5 @ B2 ) ) )
     => ( member_o @ Z @ ( produc59986286002894506_set_o @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_420_mem__case__prodI2,axiom,
    ! [P2: product_prod_nat_nat,Z: nat,C: nat > nat > set_nat] :
      ( ! [A5: nat,B2: nat] :
          ( ( P2
            = ( product_Pair_nat_nat @ A5 @ B2 ) )
         => ( member_nat @ Z @ ( C @ A5 @ B2 ) ) )
     => ( member_nat @ Z @ ( produc6189476227299908564et_nat @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_421_mem__case__prodI2,axiom,
    ! [P2: product_prod_nat_nat,Z: b,C: nat > nat > set_b] :
      ( ! [A5: nat,B2: nat] :
          ( ( P2
            = ( product_Pair_nat_nat @ A5 @ B2 ) )
         => ( member_b @ Z @ ( C @ A5 @ B2 ) ) )
     => ( member_b @ Z @ ( produc8052394788132812561_set_b @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_422_mem__case__prodI2,axiom,
    ! [P2: product_prod_b_nat,Z: $o,C: b > nat > set_o] :
      ( ! [A5: b,B2: nat] :
          ( ( P2
            = ( product_Pair_b_nat @ A5 @ B2 ) )
         => ( member_o @ Z @ ( C @ A5 @ B2 ) ) )
     => ( member_o @ Z @ ( produc849617430097769363_set_o @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_423_mem__case__prodI2,axiom,
    ! [P2: product_prod_b_nat,Z: nat,C: b > nat > set_nat] :
      ( ! [A5: b,B2: nat] :
          ( ( P2
            = ( product_Pair_b_nat @ A5 @ B2 ) )
         => ( member_nat @ Z @ ( C @ A5 @ B2 ) ) )
     => ( member_nat @ Z @ ( produc7337630463249427243et_nat @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_424_mem__case__prodI2,axiom,
    ! [P2: product_prod_b_nat,Z: b,C: b > nat > set_b] :
      ( ! [A5: b,B2: nat] :
          ( ( P2
            = ( product_Pair_b_nat @ A5 @ B2 ) )
         => ( member_b @ Z @ ( C @ A5 @ B2 ) ) )
     => ( member_b @ Z @ ( produc6760141351533629306_set_b @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_425_mem__case__prodI2,axiom,
    ! [P2: product_prod_o_o,Z: list_a,C: $o > $o > set_list_a] :
      ( ! [A5: $o,B2: $o] :
          ( ( P2
            = ( product_Pair_o_o @ A5 @ B2 ) )
         => ( member_list_a @ Z @ ( C @ A5 @ B2 ) ) )
     => ( member_list_a @ Z @ ( produc1636764921201983288list_a @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_426_mem__case__prodI,axiom,
    ! [Z: $o,C: $o > $o > set_o,A: $o,B: $o] :
      ( ( member_o @ Z @ ( C @ A @ B ) )
     => ( member_o @ Z @ ( produc1238384690215476812_set_o @ C @ ( product_Pair_o_o @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_427_mem__case__prodI,axiom,
    ! [Z: nat,C: $o > $o > set_nat,A: $o,B: $o] :
      ( ( member_nat @ Z @ ( C @ A @ B ) )
     => ( member_nat @ Z @ ( produc6723186405834743986et_nat @ C @ ( product_Pair_o_o @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_428_mem__case__prodI,axiom,
    ! [Z: b,C: $o > $o > set_b,A: $o,B: $o] :
      ( ( member_b @ Z @ ( C @ A @ B ) )
     => ( member_b @ Z @ ( produc8162358544190516659_set_b @ C @ ( product_Pair_o_o @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_429_mem__case__prodI,axiom,
    ! [Z: $o,C: nat > nat > set_o,A: nat,B: nat] :
      ( ( member_o @ Z @ ( C @ A @ B ) )
     => ( member_o @ Z @ ( produc59986286002894506_set_o @ C @ ( product_Pair_nat_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_430_mem__case__prodI,axiom,
    ! [Z: nat,C: nat > nat > set_nat,A: nat,B: nat] :
      ( ( member_nat @ Z @ ( C @ A @ B ) )
     => ( member_nat @ Z @ ( produc6189476227299908564et_nat @ C @ ( product_Pair_nat_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_431_mem__case__prodI,axiom,
    ! [Z: b,C: nat > nat > set_b,A: nat,B: nat] :
      ( ( member_b @ Z @ ( C @ A @ B ) )
     => ( member_b @ Z @ ( produc8052394788132812561_set_b @ C @ ( product_Pair_nat_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_432_mem__case__prodI,axiom,
    ! [Z: $o,C: b > nat > set_o,A: b,B: nat] :
      ( ( member_o @ Z @ ( C @ A @ B ) )
     => ( member_o @ Z @ ( produc849617430097769363_set_o @ C @ ( product_Pair_b_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_433_mem__case__prodI,axiom,
    ! [Z: nat,C: b > nat > set_nat,A: b,B: nat] :
      ( ( member_nat @ Z @ ( C @ A @ B ) )
     => ( member_nat @ Z @ ( produc7337630463249427243et_nat @ C @ ( product_Pair_b_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_434_mem__case__prodI,axiom,
    ! [Z: b,C: b > nat > set_b,A: b,B: nat] :
      ( ( member_b @ Z @ ( C @ A @ B ) )
     => ( member_b @ Z @ ( produc6760141351533629306_set_b @ C @ ( product_Pair_b_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_435_mem__case__prodI,axiom,
    ! [Z: list_a,C: $o > $o > set_list_a,A: $o,B: $o] :
      ( ( member_list_a @ Z @ ( C @ A @ B ) )
     => ( member_list_a @ Z @ ( produc1636764921201983288list_a @ C @ ( product_Pair_o_o @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_436_case__prodI2,axiom,
    ! [P2: product_prod_o_o,C: $o > $o > $o] :
      ( ! [A5: $o,B2: $o] :
          ( ( P2
            = ( product_Pair_o_o @ A5 @ B2 ) )
         => ( C @ A5 @ B2 ) )
     => ( produc6197397395684419436_o_o_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_437_case__prodI2,axiom,
    ! [P2: produc7248412053542808358at_nat,C: nat > product_prod_nat_nat > $o] :
      ( ! [A5: nat,B2: product_prod_nat_nat] :
          ( ( P2
            = ( produc487386426758144856at_nat @ A5 @ B2 ) )
         => ( C @ A5 @ B2 ) )
     => ( produc5864757623865647827_nat_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_438_case__prodI2,axiom,
    ! [P2: product_prod_nat_nat,C: nat > nat > $o] :
      ( ! [A5: nat,B2: nat] :
          ( ( P2
            = ( product_Pair_nat_nat @ A5 @ B2 ) )
         => ( C @ A5 @ B2 ) )
     => ( produc6081775807080527818_nat_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_439_case__prodI2,axiom,
    ! [P2: produc5835360497134304175_nat_a,C: ( product_prod_b_nat > set_list_a ) > ( nat > a ) > $o] :
      ( ! [A5: product_prod_b_nat > set_list_a,B2: nat > a] :
          ( ( P2
            = ( produc2895298938842563487_nat_a @ A5 @ B2 ) )
         => ( C @ A5 @ B2 ) )
     => ( produc7664446012336723172at_a_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_440_case__prodI2,axiom,
    ! [P2: product_prod_b_nat,C: b > nat > $o] :
      ( ! [A5: b,B2: nat] :
          ( ( P2
            = ( product_Pair_b_nat @ A5 @ B2 ) )
         => ( C @ A5 @ B2 ) )
     => ( produc795641402153621683_nat_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_441_case__prodI,axiom,
    ! [F2: $o > $o > $o,A: $o,B: $o] :
      ( ( F2 @ A @ B )
     => ( produc6197397395684419436_o_o_o @ F2 @ ( product_Pair_o_o @ A @ B ) ) ) ).

% case_prodI
thf(fact_442_case__prodI,axiom,
    ! [F2: nat > product_prod_nat_nat > $o,A: nat,B: product_prod_nat_nat] :
      ( ( F2 @ A @ B )
     => ( produc5864757623865647827_nat_o @ F2 @ ( produc487386426758144856at_nat @ A @ B ) ) ) ).

% case_prodI
thf(fact_443_case__prodI,axiom,
    ! [F2: nat > nat > $o,A: nat,B: nat] :
      ( ( F2 @ A @ B )
     => ( produc6081775807080527818_nat_o @ F2 @ ( product_Pair_nat_nat @ A @ B ) ) ) ).

% case_prodI
thf(fact_444_case__prodI,axiom,
    ! [F2: ( product_prod_b_nat > set_list_a ) > ( nat > a ) > $o,A: product_prod_b_nat > set_list_a,B: nat > a] :
      ( ( F2 @ A @ B )
     => ( produc7664446012336723172at_a_o @ F2 @ ( produc2895298938842563487_nat_a @ A @ B ) ) ) ).

% case_prodI
thf(fact_445_case__prodI,axiom,
    ! [F2: b > nat > $o,A: b,B: nat] :
      ( ( F2 @ A @ B )
     => ( produc795641402153621683_nat_o @ F2 @ ( product_Pair_b_nat @ A @ B ) ) ) ).

% case_prodI
thf(fact_446_swap__simp,axiom,
    ! [X3: product_prod_nat_nat,Y: nat] :
      ( ( produc672552830730091482at_nat @ ( produc6350711070570205562at_nat @ X3 @ Y ) )
      = ( produc487386426758144856at_nat @ Y @ X3 ) ) ).

% swap_simp
thf(fact_447_swap__simp,axiom,
    ! [X3: nat > a,Y: product_prod_b_nat > set_list_a] :
      ( ( produc7690703768137596079list_a @ ( produc1469764172246224655list_a @ X3 @ Y ) )
      = ( produc2895298938842563487_nat_a @ Y @ X3 ) ) ).

% swap_simp
thf(fact_448_swap__simp,axiom,
    ! [X3: nat,Y: b] :
      ( ( product_swap_nat_b @ ( product_Pair_nat_b @ X3 @ Y ) )
      = ( product_Pair_b_nat @ Y @ X3 ) ) ).

% swap_simp
thf(fact_449_swap__simp,axiom,
    ! [X3: $o,Y: $o] :
      ( ( product_swap_o_o @ ( product_Pair_o_o @ X3 @ Y ) )
      = ( product_Pair_o_o @ Y @ X3 ) ) ).

% swap_simp
thf(fact_450_swap__simp,axiom,
    ! [X3: nat,Y: product_prod_nat_nat] :
      ( ( produc4032600223772806584at_nat @ ( produc487386426758144856at_nat @ X3 @ Y ) )
      = ( produc6350711070570205562at_nat @ Y @ X3 ) ) ).

% swap_simp
thf(fact_451_swap__simp,axiom,
    ! [X3: product_prod_b_nat > set_list_a,Y: nat > a] :
      ( ( produc9116238534733934911_nat_a @ ( produc2895298938842563487_nat_a @ X3 @ Y ) )
      = ( produc1469764172246224655list_a @ Y @ X3 ) ) ).

% swap_simp
thf(fact_452_swap__simp,axiom,
    ! [X3: b,Y: nat] :
      ( ( product_swap_b_nat @ ( product_Pair_b_nat @ X3 @ Y ) )
      = ( product_Pair_nat_b @ Y @ X3 ) ) ).

% swap_simp
thf(fact_453_swap__simp,axiom,
    ! [X3: nat,Y: nat] :
      ( ( product_swap_nat_nat @ ( product_Pair_nat_nat @ X3 @ Y ) )
      = ( product_Pair_nat_nat @ Y @ X3 ) ) ).

% swap_simp
thf(fact_454_pair__in__swap__image,axiom,
    ! [Y: product_prod_nat_nat,X3: nat,A2: set_Pr7717912310451564380at_nat] :
      ( ( member3348759134392003351at_nat @ ( produc6350711070570205562at_nat @ Y @ X3 ) @ ( image_8624973904636368301at_nat @ produc4032600223772806584at_nat @ A2 ) )
      = ( member2223272150424702269at_nat @ ( produc487386426758144856at_nat @ X3 @ Y ) @ A2 ) ) ).

% pair_in_swap_image
thf(fact_455_pair__in__swap__image,axiom,
    ! [Y: nat > a,X3: product_prod_b_nat > set_list_a,A2: set_Pr6389665502131816719_nat_a] :
      ( ( member3529051575741102792list_a @ ( produc1469764172246224655list_a @ Y @ X3 ) @ ( image_5395764133486988597list_a @ produc9116238534733934911_nat_a @ A2 ) )
      = ( member9198066416134578520_nat_a @ ( produc2895298938842563487_nat_a @ X3 @ Y ) @ A2 ) ) ).

% pair_in_swap_image
thf(fact_456_pair__in__swap__image,axiom,
    ! [Y: nat,X3: b,A2: set_Pr1307281990691478580_b_nat] :
      ( ( member8962352056413324475_nat_b @ ( product_Pair_nat_b @ Y @ X3 ) @ ( image_2254923467274978473_nat_b @ product_swap_b_nat @ A2 ) )
      = ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ X3 @ Y ) @ A2 ) ) ).

% pair_in_swap_image
thf(fact_457_pair__in__swap__image,axiom,
    ! [Y: $o,X3: $o,A2: set_Product_prod_o_o] :
      ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y @ X3 ) @ ( image_9131363867636255685od_o_o @ product_swap_o_o @ A2 ) )
      = ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X3 @ Y ) @ A2 ) ) ).

% pair_in_swap_image
thf(fact_458_pair__in__swap__image,axiom,
    ! [Y: nat,X3: product_prod_nat_nat,A2: set_Pr2539167527615954998at_nat] :
      ( ( member2223272150424702269at_nat @ ( produc487386426758144856at_nat @ Y @ X3 ) @ ( image_2402546415023586989at_nat @ produc672552830730091482at_nat @ A2 ) )
      = ( member3348759134392003351at_nat @ ( produc6350711070570205562at_nat @ X3 @ Y ) @ A2 ) ) ).

% pair_in_swap_image
thf(fact_459_pair__in__swap__image,axiom,
    ! [Y: product_prod_b_nat > set_list_a,X3: nat > a,A2: set_Pr4091103320399850111list_a] :
      ( ( member9198066416134578520_nat_a @ ( produc2895298938842563487_nat_a @ Y @ X3 ) @ ( image_3727386206174543445_nat_a @ produc7690703768137596079list_a @ A2 ) )
      = ( member3529051575741102792list_a @ ( produc1469764172246224655list_a @ X3 @ Y ) @ A2 ) ) ).

% pair_in_swap_image
thf(fact_460_pair__in__swap__image,axiom,
    ! [Y: b,X3: nat,A2: set_Pr4264375888882495962_nat_b] :
      ( ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ Y @ X3 ) @ ( image_1100448597386909353_b_nat @ product_swap_nat_b @ A2 ) )
      = ( member8962352056413324475_nat_b @ ( product_Pair_nat_b @ X3 @ Y ) @ A2 ) ) ).

% pair_in_swap_image
thf(fact_461_pair__in__swap__image,axiom,
    ! [Y: nat,X3: nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ X3 ) @ ( image_5168914502847457605at_nat @ product_swap_nat_nat @ A2 ) )
      = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ A2 ) ) ).

% pair_in_swap_image
thf(fact_462_curry__conv,axiom,
    ( produc858456811296061068la_a_b
    = ( ^ [F: product_prod_nat_nat > relational_fmla_a_b,A3: nat,B3: nat] : ( F @ ( product_Pair_nat_nat @ A3 @ B3 ) ) ) ) ).

% curry_conv
thf(fact_463_curry__conv,axiom,
    ( produc7541201833284165578la_a_b
    = ( ^ [F: product_prod_nat_nat > relational_fmla_a_b > relational_fmla_a_b,A3: nat,B3: nat] : ( F @ ( product_Pair_nat_nat @ A3 @ B3 ) ) ) ) ).

% curry_conv
thf(fact_464_cpropagated__simps_I1_J,axiom,
    ! [B: $o] : ( relati1591879772219623554ed_a_b @ ( relational_Bool_a_b @ B ) ) ).

% cpropagated_simps(1)
thf(fact_465_case__prod__conv,axiom,
    ! [F2: nat > product_prod_nat_nat > produc7248412053542808358at_nat,A: nat,B: product_prod_nat_nat] :
      ( ( produc968775922737392939at_nat @ F2 @ ( produc487386426758144856at_nat @ A @ B ) )
      = ( F2 @ A @ B ) ) ).

% case_prod_conv
thf(fact_466_case__prod__conv,axiom,
    ! [F2: nat > nat > produc7248412053542808358at_nat,A: nat,B: nat] :
      ( ( produc9083241971206738548at_nat @ F2 @ ( product_Pair_nat_nat @ A @ B ) )
      = ( F2 @ A @ B ) ) ).

% case_prod_conv
thf(fact_467_case__prod__conv,axiom,
    ! [F2: nat > nat > product_prod_nat_nat,A: nat,B: nat] :
      ( ( produc2626176000494625587at_nat @ F2 @ ( product_Pair_nat_nat @ A @ B ) )
      = ( F2 @ A @ B ) ) ).

% case_prod_conv
thf(fact_468_case__prod__conv,axiom,
    ! [F2: nat > nat > relational_term_a > relational_term_a,A: nat,B: nat] :
      ( ( produc6628518323692928499term_a @ F2 @ ( product_Pair_nat_nat @ A @ B ) )
      = ( F2 @ A @ B ) ) ).

% case_prod_conv
thf(fact_469_case__prod__conv,axiom,
    ! [F2: nat > nat > nat > produc7248412053542808358at_nat,A: nat,B: nat] :
      ( ( produc7810592499157111267at_nat @ F2 @ ( product_Pair_nat_nat @ A @ B ) )
      = ( F2 @ A @ B ) ) ).

% case_prod_conv
thf(fact_470_case__prod__conv,axiom,
    ! [F2: nat > nat > relational_fmla_a_b > relational_fmla_a_b,A: nat,B: nat] :
      ( ( produc5586541307551673003la_a_b @ F2 @ ( product_Pair_nat_nat @ A @ B ) )
      = ( F2 @ A @ B ) ) ).

% case_prod_conv
thf(fact_471_case__prod__conv,axiom,
    ! [F2: nat > nat > relational_fmla_a_b,A: nat,B: nat] :
      ( ( produc3270801013941088237la_a_b @ F2 @ ( product_Pair_nat_nat @ A @ B ) )
      = ( F2 @ A @ B ) ) ).

% case_prod_conv
thf(fact_472_pair__imageI,axiom,
    ! [A: nat,B: nat,A2: set_Pr1261947904930325089at_nat,F2: nat > nat > relational_fmla_a_b] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ A2 )
     => ( member4680049679412964150la_a_b @ ( F2 @ A @ B ) @ ( image_5019456213502855771la_a_b @ ( produc3270801013941088237la_a_b @ F2 ) @ A2 ) ) ) ).

% pair_imageI
thf(fact_473_pair__imageI,axiom,
    ! [A: nat,B: nat,A2: set_Pr1261947904930325089at_nat,F2: nat > nat > relational_fmla_a_b > relational_fmla_a_b] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ A2 )
     => ( member8433577210552456052la_a_b @ ( F2 @ A @ B ) @ ( image_4309136164212173337la_a_b @ ( produc5586541307551673003la_a_b @ F2 ) @ A2 ) ) ) ).

% pair_imageI
thf(fact_474_pair__imageI,axiom,
    ! [A: $o,B: $o,A2: set_Product_prod_o_o,F2: $o > $o > $o] :
      ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ A2 )
     => ( member_o @ ( F2 @ A @ B ) @ ( image_7896445794123959606_o_o_o @ ( produc6197397395684419436_o_o_o @ F2 ) @ A2 ) ) ) ).

% pair_imageI
thf(fact_475_pair__imageI,axiom,
    ! [A: $o,B: $o,A2: set_Product_prod_o_o,F2: $o > $o > nat] :
      ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ A2 )
     => ( member_nat @ ( F2 @ A @ B ) @ ( image_3818509671500154290_o_nat @ ( produc5300922066696922364_o_nat @ F2 ) @ A2 ) ) ) ).

% pair_imageI
thf(fact_476_pair__imageI,axiom,
    ! [A: $o,B: $o,A2: set_Product_prod_o_o,F2: $o > $o > b] :
      ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ A2 )
     => ( member_b @ ( F2 @ A @ B ) @ ( image_1625555692148909469_o_o_b @ ( produc368963595484087507_o_o_b @ F2 ) @ A2 ) ) ) ).

% pair_imageI
thf(fact_477_pair__imageI,axiom,
    ! [A: nat,B: nat,A2: set_Pr1261947904930325089at_nat,F2: nat > nat > $o] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ A2 )
     => ( member_o @ ( F2 @ A @ B ) @ ( image_3693632289388996572_nat_o @ ( produc6081775807080527818_nat_o @ F2 ) @ A2 ) ) ) ).

% pair_imageI
thf(fact_478_pair__imageI,axiom,
    ! [A: nat,B: nat,A2: set_Pr1261947904930325089at_nat,F2: nat > nat > nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ A2 )
     => ( member_nat @ ( F2 @ A @ B ) @ ( image_2486076414777270412at_nat @ ( produc6842872674320459806at_nat @ F2 ) @ A2 ) ) ) ).

% pair_imageI
thf(fact_479_pair__imageI,axiom,
    ! [A: nat,B: nat,A2: set_Pr1261947904930325089at_nat,F2: nat > nat > b] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ A2 )
     => ( member_b @ ( F2 @ A @ B ) @ ( image_5995000214508162371_nat_b @ ( produc3276484115406849585_nat_b @ F2 ) @ A2 ) ) ) ).

% pair_imageI
thf(fact_480_pair__imageI,axiom,
    ! [A: b,B: nat,A2: set_Pr1307281990691478580_b_nat,F2: b > nat > $o] :
      ( ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ A @ B ) @ A2 )
     => ( member_o @ ( F2 @ A @ B ) @ ( image_191264095111578067_nat_o @ ( produc795641402153621683_nat_o @ F2 ) @ A2 ) ) ) ).

% pair_imageI
thf(fact_481_pair__imageI,axiom,
    ! [A: b,B: nat,A2: set_Pr1307281990691478580_b_nat,F2: b > nat > nat] :
      ( ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ A @ B ) @ A2 )
     => ( member_nat @ ( F2 @ A @ B ) @ ( image_3660612598617633365at_nat @ ( produc566631499068838773at_nat @ F2 ) @ A2 ) ) ) ).

% pair_imageI
thf(fact_482_cpropagated__simps_I2_J,axiom,
    ! [P2: b,Ts3: list_R6823256787227418703term_a] : ( relati1591879772219623554ed_a_b @ ( relational_Pred_b_a @ P2 @ Ts3 ) ) ).

% cpropagated_simps(2)
thf(fact_483_cpropagated__simps_I3_J,axiom,
    ! [X3: nat,T: relational_term_a] :
      ( ( relati1591879772219623554ed_a_b @ ( relational_Eq_a_b @ X3 @ T ) )
      = ( T
       != ( relational_Var_a @ X3 ) ) ) ).

% cpropagated_simps(3)
thf(fact_484_cpropagated__simps_I4_J,axiom,
    ! [Q: relational_fmla_a_b] :
      ( ( relati1591879772219623554ed_a_b @ ( relational_Neg_a_b @ Q ) )
      = ( relational_nocp_a_b @ Q ) ) ).

% cpropagated_simps(4)
thf(fact_485_cpropagated__simps_I6_J,axiom,
    ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
      ( ( relati1591879772219623554ed_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) )
      = ( ( relational_nocp_a_b @ Q1 )
        & ( relational_nocp_a_b @ Q22 ) ) ) ).

% cpropagated_simps(6)
thf(fact_486_cpropagated__simps_I5_J,axiom,
    ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
      ( ( relati1591879772219623554ed_a_b @ ( relational_Conj_a_b @ Q1 @ Q22 ) )
      = ( ( relational_nocp_a_b @ Q1 )
        & ( relational_nocp_a_b @ Q22 ) ) ) ).

% cpropagated_simps(5)
thf(fact_487_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_o_o] :
      ~ ! [A5: $o,B2: $o] :
          ( Y
         != ( product_Pair_o_o @ A5 @ B2 ) ) ).

% old.prod.exhaust
thf(fact_488_old_Oprod_Oexhaust,axiom,
    ! [Y: produc7248412053542808358at_nat] :
      ~ ! [A5: nat,B2: product_prod_nat_nat] :
          ( Y
         != ( produc487386426758144856at_nat @ A5 @ B2 ) ) ).

% old.prod.exhaust
thf(fact_489_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_nat_nat] :
      ~ ! [A5: nat,B2: nat] :
          ( Y
         != ( product_Pair_nat_nat @ A5 @ B2 ) ) ).

% old.prod.exhaust
thf(fact_490_old_Oprod_Oexhaust,axiom,
    ! [Y: produc5835360497134304175_nat_a] :
      ~ ! [A5: product_prod_b_nat > set_list_a,B2: nat > a] :
          ( Y
         != ( produc2895298938842563487_nat_a @ A5 @ B2 ) ) ).

% old.prod.exhaust
thf(fact_491_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_b_nat] :
      ~ ! [A5: b,B2: nat] :
          ( Y
         != ( product_Pair_b_nat @ A5 @ B2 ) ) ).

% old.prod.exhaust
thf(fact_492_surj__pair,axiom,
    ! [P2: product_prod_o_o] :
    ? [X5: $o,Y3: $o] :
      ( P2
      = ( product_Pair_o_o @ X5 @ Y3 ) ) ).

% surj_pair
thf(fact_493_surj__pair,axiom,
    ! [P2: produc7248412053542808358at_nat] :
    ? [X5: nat,Y3: product_prod_nat_nat] :
      ( P2
      = ( produc487386426758144856at_nat @ X5 @ Y3 ) ) ).

% surj_pair
thf(fact_494_surj__pair,axiom,
    ! [P2: product_prod_nat_nat] :
    ? [X5: nat,Y3: nat] :
      ( P2
      = ( product_Pair_nat_nat @ X5 @ Y3 ) ) ).

% surj_pair
thf(fact_495_surj__pair,axiom,
    ! [P2: produc5835360497134304175_nat_a] :
    ? [X5: product_prod_b_nat > set_list_a,Y3: nat > a] :
      ( P2
      = ( produc2895298938842563487_nat_a @ X5 @ Y3 ) ) ).

% surj_pair
thf(fact_496_surj__pair,axiom,
    ! [P2: product_prod_b_nat] :
    ? [X5: b,Y3: nat] :
      ( P2
      = ( product_Pair_b_nat @ X5 @ Y3 ) ) ).

% surj_pair
thf(fact_497_prod__cases,axiom,
    ! [P: product_prod_o_o > $o,P2: product_prod_o_o] :
      ( ! [A5: $o,B2: $o] : ( P @ ( product_Pair_o_o @ A5 @ B2 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_498_prod__cases,axiom,
    ! [P: produc7248412053542808358at_nat > $o,P2: produc7248412053542808358at_nat] :
      ( ! [A5: nat,B2: product_prod_nat_nat] : ( P @ ( produc487386426758144856at_nat @ A5 @ B2 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_499_prod__cases,axiom,
    ! [P: product_prod_nat_nat > $o,P2: product_prod_nat_nat] :
      ( ! [A5: nat,B2: nat] : ( P @ ( product_Pair_nat_nat @ A5 @ B2 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_500_prod__cases,axiom,
    ! [P: produc5835360497134304175_nat_a > $o,P2: produc5835360497134304175_nat_a] :
      ( ! [A5: product_prod_b_nat > set_list_a,B2: nat > a] : ( P @ ( produc2895298938842563487_nat_a @ A5 @ B2 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_501_prod__cases,axiom,
    ! [P: product_prod_b_nat > $o,P2: product_prod_b_nat] :
      ( ! [A5: b,B2: nat] : ( P @ ( product_Pair_b_nat @ A5 @ B2 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_502_Pair__inject,axiom,
    ! [A: $o,B: $o,A4: $o,B4: $o] :
      ( ( ( product_Pair_o_o @ A @ B )
        = ( product_Pair_o_o @ A4 @ B4 ) )
     => ~ ( ( A = A4 )
         => ( B = (~ B4) ) ) ) ).

% Pair_inject
thf(fact_503_Pair__inject,axiom,
    ! [A: nat,B: product_prod_nat_nat,A4: nat,B4: product_prod_nat_nat] :
      ( ( ( produc487386426758144856at_nat @ A @ B )
        = ( produc487386426758144856at_nat @ A4 @ B4 ) )
     => ~ ( ( A = A4 )
         => ( B != B4 ) ) ) ).

% Pair_inject
thf(fact_504_Pair__inject,axiom,
    ! [A: nat,B: nat,A4: nat,B4: nat] :
      ( ( ( product_Pair_nat_nat @ A @ B )
        = ( product_Pair_nat_nat @ A4 @ B4 ) )
     => ~ ( ( A = A4 )
         => ( B != B4 ) ) ) ).

% Pair_inject
thf(fact_505_Pair__inject,axiom,
    ! [A: product_prod_b_nat > set_list_a,B: nat > a,A4: product_prod_b_nat > set_list_a,B4: nat > a] :
      ( ( ( produc2895298938842563487_nat_a @ A @ B )
        = ( produc2895298938842563487_nat_a @ A4 @ B4 ) )
     => ~ ( ( A = A4 )
         => ( B != B4 ) ) ) ).

% Pair_inject
thf(fact_506_Pair__inject,axiom,
    ! [A: b,B: nat,A4: b,B4: nat] :
      ( ( ( product_Pair_b_nat @ A @ B )
        = ( product_Pair_b_nat @ A4 @ B4 ) )
     => ~ ( ( A = A4 )
         => ( B != B4 ) ) ) ).

% Pair_inject
thf(fact_507_prod__cases3,axiom,
    ! [Y: produc7248412053542808358at_nat] :
      ~ ! [A5: nat,B2: nat,C2: nat] :
          ( Y
         != ( produc487386426758144856at_nat @ A5 @ ( product_Pair_nat_nat @ B2 @ C2 ) ) ) ).

% prod_cases3
thf(fact_508_prod__induct3,axiom,
    ! [P: produc7248412053542808358at_nat > $o,X3: produc7248412053542808358at_nat] :
      ( ! [A5: nat,B2: nat,C2: nat] : ( P @ ( produc487386426758144856at_nat @ A5 @ ( product_Pair_nat_nat @ B2 @ C2 ) ) )
     => ( P @ X3 ) ) ).

% prod_induct3
thf(fact_509_mem__case__prodE,axiom,
    ! [Z: $o,C: $o > $o > set_o,P2: product_prod_o_o] :
      ( ( member_o @ Z @ ( produc1238384690215476812_set_o @ C @ P2 ) )
     => ~ ! [X5: $o,Y3: $o] :
            ( ( P2
              = ( product_Pair_o_o @ X5 @ Y3 ) )
           => ~ ( member_o @ Z @ ( C @ X5 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_510_mem__case__prodE,axiom,
    ! [Z: nat,C: $o > $o > set_nat,P2: product_prod_o_o] :
      ( ( member_nat @ Z @ ( produc6723186405834743986et_nat @ C @ P2 ) )
     => ~ ! [X5: $o,Y3: $o] :
            ( ( P2
              = ( product_Pair_o_o @ X5 @ Y3 ) )
           => ~ ( member_nat @ Z @ ( C @ X5 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_511_mem__case__prodE,axiom,
    ! [Z: b,C: $o > $o > set_b,P2: product_prod_o_o] :
      ( ( member_b @ Z @ ( produc8162358544190516659_set_b @ C @ P2 ) )
     => ~ ! [X5: $o,Y3: $o] :
            ( ( P2
              = ( product_Pair_o_o @ X5 @ Y3 ) )
           => ~ ( member_b @ Z @ ( C @ X5 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_512_mem__case__prodE,axiom,
    ! [Z: $o,C: nat > nat > set_o,P2: product_prod_nat_nat] :
      ( ( member_o @ Z @ ( produc59986286002894506_set_o @ C @ P2 ) )
     => ~ ! [X5: nat,Y3: nat] :
            ( ( P2
              = ( product_Pair_nat_nat @ X5 @ Y3 ) )
           => ~ ( member_o @ Z @ ( C @ X5 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_513_mem__case__prodE,axiom,
    ! [Z: nat,C: nat > nat > set_nat,P2: product_prod_nat_nat] :
      ( ( member_nat @ Z @ ( produc6189476227299908564et_nat @ C @ P2 ) )
     => ~ ! [X5: nat,Y3: nat] :
            ( ( P2
              = ( product_Pair_nat_nat @ X5 @ Y3 ) )
           => ~ ( member_nat @ Z @ ( C @ X5 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_514_mem__case__prodE,axiom,
    ! [Z: b,C: nat > nat > set_b,P2: product_prod_nat_nat] :
      ( ( member_b @ Z @ ( produc8052394788132812561_set_b @ C @ P2 ) )
     => ~ ! [X5: nat,Y3: nat] :
            ( ( P2
              = ( product_Pair_nat_nat @ X5 @ Y3 ) )
           => ~ ( member_b @ Z @ ( C @ X5 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_515_mem__case__prodE,axiom,
    ! [Z: $o,C: b > nat > set_o,P2: product_prod_b_nat] :
      ( ( member_o @ Z @ ( produc849617430097769363_set_o @ C @ P2 ) )
     => ~ ! [X5: b,Y3: nat] :
            ( ( P2
              = ( product_Pair_b_nat @ X5 @ Y3 ) )
           => ~ ( member_o @ Z @ ( C @ X5 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_516_mem__case__prodE,axiom,
    ! [Z: nat,C: b > nat > set_nat,P2: product_prod_b_nat] :
      ( ( member_nat @ Z @ ( produc7337630463249427243et_nat @ C @ P2 ) )
     => ~ ! [X5: b,Y3: nat] :
            ( ( P2
              = ( product_Pair_b_nat @ X5 @ Y3 ) )
           => ~ ( member_nat @ Z @ ( C @ X5 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_517_mem__case__prodE,axiom,
    ! [Z: b,C: b > nat > set_b,P2: product_prod_b_nat] :
      ( ( member_b @ Z @ ( produc6760141351533629306_set_b @ C @ P2 ) )
     => ~ ! [X5: b,Y3: nat] :
            ( ( P2
              = ( product_Pair_b_nat @ X5 @ Y3 ) )
           => ~ ( member_b @ Z @ ( C @ X5 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_518_mem__case__prodE,axiom,
    ! [Z: list_a,C: $o > $o > set_list_a,P2: product_prod_o_o] :
      ( ( member_list_a @ Z @ ( produc1636764921201983288list_a @ C @ P2 ) )
     => ~ ! [X5: $o,Y3: $o] :
            ( ( P2
              = ( product_Pair_o_o @ X5 @ Y3 ) )
           => ~ ( member_list_a @ Z @ ( C @ X5 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_519_curryE,axiom,
    ! [F2: product_prod_o_o > $o,A: $o,B: $o] :
      ( ( product_curry_o_o_o @ F2 @ A @ B )
     => ( F2 @ ( product_Pair_o_o @ A @ B ) ) ) ).

% curryE
thf(fact_520_curryE,axiom,
    ! [F2: produc7248412053542808358at_nat > $o,A: nat,B: product_prod_nat_nat] :
      ( ( produc7331361819228145906_nat_o @ F2 @ A @ B )
     => ( F2 @ ( produc487386426758144856at_nat @ A @ B ) ) ) ).

% curryE
thf(fact_521_curryE,axiom,
    ! [F2: product_prod_nat_nat > $o,A: nat,B: nat] :
      ( ( produc1310100445399344235_nat_o @ F2 @ A @ B )
     => ( F2 @ ( product_Pair_nat_nat @ A @ B ) ) ) ).

% curryE
thf(fact_522_curryE,axiom,
    ! [F2: produc5835360497134304175_nat_a > $o,A: product_prod_b_nat > set_list_a,B: nat > a] :
      ( ( produc5059092166261958789at_a_o @ F2 @ A @ B )
     => ( F2 @ ( produc2895298938842563487_nat_a @ A @ B ) ) ) ).

% curryE
thf(fact_523_curryE,axiom,
    ! [F2: product_prod_b_nat > $o,A: b,B: nat] :
      ( ( produc2461434047082304082_nat_o @ F2 @ A @ B )
     => ( F2 @ ( product_Pair_b_nat @ A @ B ) ) ) ).

% curryE
thf(fact_524_curryD,axiom,
    ! [F2: product_prod_o_o > $o,A: $o,B: $o] :
      ( ( product_curry_o_o_o @ F2 @ A @ B )
     => ( F2 @ ( product_Pair_o_o @ A @ B ) ) ) ).

% curryD
thf(fact_525_curryD,axiom,
    ! [F2: produc7248412053542808358at_nat > $o,A: nat,B: product_prod_nat_nat] :
      ( ( produc7331361819228145906_nat_o @ F2 @ A @ B )
     => ( F2 @ ( produc487386426758144856at_nat @ A @ B ) ) ) ).

% curryD
thf(fact_526_curryD,axiom,
    ! [F2: product_prod_nat_nat > $o,A: nat,B: nat] :
      ( ( produc1310100445399344235_nat_o @ F2 @ A @ B )
     => ( F2 @ ( product_Pair_nat_nat @ A @ B ) ) ) ).

% curryD
thf(fact_527_curryD,axiom,
    ! [F2: produc5835360497134304175_nat_a > $o,A: product_prod_b_nat > set_list_a,B: nat > a] :
      ( ( produc5059092166261958789at_a_o @ F2 @ A @ B )
     => ( F2 @ ( produc2895298938842563487_nat_a @ A @ B ) ) ) ).

% curryD
thf(fact_528_curryD,axiom,
    ! [F2: product_prod_b_nat > $o,A: b,B: nat] :
      ( ( produc2461434047082304082_nat_o @ F2 @ A @ B )
     => ( F2 @ ( product_Pair_b_nat @ A @ B ) ) ) ).

% curryD
thf(fact_529_case__prodD,axiom,
    ! [F2: $o > $o > $o,A: $o,B: $o] :
      ( ( produc6197397395684419436_o_o_o @ F2 @ ( product_Pair_o_o @ A @ B ) )
     => ( F2 @ A @ B ) ) ).

% case_prodD
thf(fact_530_case__prodD,axiom,
    ! [F2: nat > product_prod_nat_nat > $o,A: nat,B: product_prod_nat_nat] :
      ( ( produc5864757623865647827_nat_o @ F2 @ ( produc487386426758144856at_nat @ A @ B ) )
     => ( F2 @ A @ B ) ) ).

% case_prodD
thf(fact_531_case__prodD,axiom,
    ! [F2: nat > nat > $o,A: nat,B: nat] :
      ( ( produc6081775807080527818_nat_o @ F2 @ ( product_Pair_nat_nat @ A @ B ) )
     => ( F2 @ A @ B ) ) ).

% case_prodD
thf(fact_532_case__prodD,axiom,
    ! [F2: ( product_prod_b_nat > set_list_a ) > ( nat > a ) > $o,A: product_prod_b_nat > set_list_a,B: nat > a] :
      ( ( produc7664446012336723172at_a_o @ F2 @ ( produc2895298938842563487_nat_a @ A @ B ) )
     => ( F2 @ A @ B ) ) ).

% case_prodD
thf(fact_533_case__prodD,axiom,
    ! [F2: b > nat > $o,A: b,B: nat] :
      ( ( produc795641402153621683_nat_o @ F2 @ ( product_Pair_b_nat @ A @ B ) )
     => ( F2 @ A @ B ) ) ).

% case_prodD
thf(fact_534_case__prodE,axiom,
    ! [C: $o > $o > $o,P2: product_prod_o_o] :
      ( ( produc6197397395684419436_o_o_o @ C @ P2 )
     => ~ ! [X5: $o,Y3: $o] :
            ( ( P2
              = ( product_Pair_o_o @ X5 @ Y3 ) )
           => ~ ( C @ X5 @ Y3 ) ) ) ).

% case_prodE
thf(fact_535_case__prodE,axiom,
    ! [C: nat > product_prod_nat_nat > $o,P2: produc7248412053542808358at_nat] :
      ( ( produc5864757623865647827_nat_o @ C @ P2 )
     => ~ ! [X5: nat,Y3: product_prod_nat_nat] :
            ( ( P2
              = ( produc487386426758144856at_nat @ X5 @ Y3 ) )
           => ~ ( C @ X5 @ Y3 ) ) ) ).

% case_prodE
thf(fact_536_case__prodE,axiom,
    ! [C: nat > nat > $o,P2: product_prod_nat_nat] :
      ( ( produc6081775807080527818_nat_o @ C @ P2 )
     => ~ ! [X5: nat,Y3: nat] :
            ( ( P2
              = ( product_Pair_nat_nat @ X5 @ Y3 ) )
           => ~ ( C @ X5 @ Y3 ) ) ) ).

% case_prodE
thf(fact_537_case__prodE,axiom,
    ! [C: ( product_prod_b_nat > set_list_a ) > ( nat > a ) > $o,P2: produc5835360497134304175_nat_a] :
      ( ( produc7664446012336723172at_a_o @ C @ P2 )
     => ~ ! [X5: product_prod_b_nat > set_list_a,Y3: nat > a] :
            ( ( P2
              = ( produc2895298938842563487_nat_a @ X5 @ Y3 ) )
           => ~ ( C @ X5 @ Y3 ) ) ) ).

% case_prodE
thf(fact_538_case__prodE,axiom,
    ! [C: b > nat > $o,P2: product_prod_b_nat] :
      ( ( produc795641402153621683_nat_o @ C @ P2 )
     => ~ ! [X5: b,Y3: nat] :
            ( ( P2
              = ( product_Pair_b_nat @ X5 @ Y3 ) )
           => ~ ( C @ X5 @ Y3 ) ) ) ).

% case_prodE
thf(fact_539_case__prod__Pair__iden,axiom,
    ! [P2: product_prod_o_o] :
      ( ( produc7436348682273225467od_o_o @ product_Pair_o_o @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_540_case__prod__Pair__iden,axiom,
    ! [P2: produc5835360497134304175_nat_a] :
      ( ( produc5978674056842407163_nat_a @ produc2895298938842563487_nat_a @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_541_case__prod__Pair__iden,axiom,
    ! [P2: product_prod_b_nat] :
      ( ( produc282185899741183267_b_nat @ product_Pair_b_nat @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_542_case__prod__Pair__iden,axiom,
    ! [P2: produc7248412053542808358at_nat] :
      ( ( produc968775922737392939at_nat @ produc487386426758144856at_nat @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_543_case__prod__Pair__iden,axiom,
    ! [P2: product_prod_nat_nat] :
      ( ( produc2626176000494625587at_nat @ product_Pair_nat_nat @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_544_zip__left__commute,axiom,
    ! [Xs: list_nat,Ys: list_P6011104703257516679at_nat,Zs: list_nat] :
      ( ( zip_na6013266149619492105at_nat @ Xs @ ( zip_Pr6869450617852699226at_nat @ Ys @ Zs ) )
      = ( map_Pr487744158910307534at_nat
        @ ( produc125514667395075397at_nat
          @ ^ [Y2: product_prod_nat_nat] :
              ( produc4987048382248537187at_nat
              @ ^ [X: nat,Z3: nat] : ( produc7510937029849927145at_nat @ X @ ( produc6350711070570205562at_nat @ Y2 @ Z3 ) ) ) )
        @ ( zip_Pr4664179122662387191at_nat @ Ys @ ( zip_nat_nat @ Xs @ Zs ) ) ) ) ).

% zip_left_commute
thf(fact_545_zip__left__commute,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,Zs: list_nat] :
      ( ( zip_Pr1012031569167933578at_nat @ Xs @ ( zip_Pr6869450617852699226at_nat @ Ys @ Zs ) )
      = ( map_Pr6237365107028939302at_nat
        @ ( produc8149036724686593415at_nat
          @ ^ [Y2: product_prod_nat_nat] :
              ( produc1019913499849056599at_nat
              @ ^ [X: product_prod_nat_nat,Z3: nat] : ( produc7194719449938452394at_nat @ X @ ( produc6350711070570205562at_nat @ Y2 @ Z3 ) ) ) )
        @ ( zip_Pr1012031569167933578at_nat @ Ys @ ( zip_Pr6869450617852699226at_nat @ Xs @ Zs ) ) ) ) ).

% zip_left_commute
thf(fact_546_zip__left__commute,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_P6011104703257516679at_nat] :
      ( ( zip_na4887779165652191023at_nat @ Xs @ ( zip_na1006125974040638520at_nat @ Ys @ Zs ) )
      = ( map_Pr1222948676352107524at_nat
        @ ( produc6670672579486093619at_nat
          @ ^ [Y2: nat] :
              ( produc3956061956167368722at_nat
              @ ^ [X: nat,Z3: product_prod_nat_nat] : ( produc6385450045882626063at_nat @ X @ ( produc487386426758144856at_nat @ Y2 @ Z3 ) ) ) )
        @ ( zip_na4887779165652191023at_nat @ Ys @ ( zip_na1006125974040638520at_nat @ Xs @ Zs ) ) ) ) ).

% zip_left_commute
thf(fact_547_zip__left__commute,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys: list_nat,Zs: list_nat] :
      ( ( zip_Pr4664179122662387191at_nat @ Xs @ ( zip_nat_nat @ Ys @ Zs ) )
      = ( map_Pr5249061301477117370at_nat
        @ ( produc6955795805150227945at_nat
          @ ^ [Y2: nat] :
              ( produc4883074260277788288at_nat
              @ ^ [X: product_prod_nat_nat,Z3: nat] : ( produc6161850002892822231at_nat @ X @ ( product_Pair_nat_nat @ Y2 @ Z3 ) ) ) )
        @ ( zip_na6013266149619492105at_nat @ Ys @ ( zip_Pr6869450617852699226at_nat @ Xs @ Zs ) ) ) ) ).

% zip_left_commute
thf(fact_548_zip__left__commute,axiom,
    ! [Xs: list_nat,Ys: list_b,Zs: list_nat] :
      ( ( zip_na5846944185084405511_b_nat @ Xs @ ( zip_b_nat @ Ys @ Zs ) )
      = ( map_Pr6582810724067715530_b_nat
        @ ( produc1256931253707451841_b_nat
          @ ^ [Y2: b] :
              ( produc1217974579069012705_b_nat
              @ ^ [X: nat,Z3: nat] : ( produc1383517840785260519_b_nat @ X @ ( product_Pair_b_nat @ Y2 @ Z3 ) ) ) )
        @ ( zip_b_4974252998908922489at_nat @ Ys @ ( zip_nat_nat @ Xs @ Zs ) ) ) ) ).

% zip_left_commute
thf(fact_549_zip__left__commute,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys: list_b,Zs: list_nat] :
      ( ( zip_Pr8182399961179643656_b_nat @ Xs @ ( zip_b_nat @ Ys @ Zs ) )
      = ( map_Pr2834422924150617634_b_nat
        @ ( produc1052448447240954371_b_nat
          @ ^ [Y2: b] :
              ( produc2053326735986100693_b_nat
              @ ^ [X: product_prod_nat_nat,Z3: nat] : ( produc8359732309707189800_b_nat @ X @ ( product_Pair_b_nat @ Y2 @ Z3 ) ) ) )
        @ ( zip_b_5947110922674133384at_nat @ Ys @ ( zip_Pr6869450617852699226at_nat @ Xs @ Zs ) ) ) ) ).

% zip_left_commute
thf(fact_550_zip__left__commute,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
      ( ( zip_na1006125974040638520at_nat @ Xs @ ( zip_nat_nat @ Ys @ Zs ) )
      = ( map_Pr6261813372141627026at_nat
        @ ( produc968775922737392939at_nat
          @ ^ [Y2: nat] :
              ( produc9083241971206738548at_nat
              @ ^ [X: nat,Z3: nat] : ( produc487386426758144856at_nat @ X @ ( product_Pair_nat_nat @ Y2 @ Z3 ) ) ) )
        @ ( zip_na1006125974040638520at_nat @ Ys @ ( zip_nat_nat @ Xs @ Zs ) ) ) ) ).

% zip_left_commute
thf(fact_551_zip__assoc,axiom,
    ! [Xs: list_nat,Ys: list_P6011104703257516679at_nat,Zs: list_nat] :
      ( ( zip_na6013266149619492105at_nat @ Xs @ ( zip_Pr6869450617852699226at_nat @ Ys @ Zs ) )
      = ( map_Pr1042041867848710750at_nat
        @ ( produc1321701454779763853at_nat
          @ ( produc2343800788440956627at_nat
            @ ^ [X: nat,Y2: product_prod_nat_nat,Z3: nat] : ( produc7510937029849927145at_nat @ X @ ( produc6350711070570205562at_nat @ Y2 @ Z3 ) ) ) )
        @ ( zip_Pr2455715559586839727at_nat @ ( zip_na1006125974040638520at_nat @ Xs @ Ys ) @ Zs ) ) ) ).

% zip_assoc
thf(fact_552_zip__assoc,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_P6011104703257516679at_nat] :
      ( ( zip_na4887779165652191023at_nat @ Xs @ ( zip_na1006125974040638520at_nat @ Ys @ Zs ) )
      = ( map_Pr5666488941745916916at_nat
        @ ( produc5304259450230684779at_nat
          @ ( produc8968847002844571593at_nat
            @ ^ [X: nat,Y2: nat,Z3: product_prod_nat_nat] : ( produc6385450045882626063at_nat @ X @ ( produc487386426758144856at_nat @ Y2 @ Z3 ) ) ) )
        @ ( zip_Pr4664179122662387191at_nat @ ( zip_nat_nat @ Xs @ Ys ) @ Zs ) ) ) ).

% zip_assoc
thf(fact_553_zip__assoc,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys: list_nat,Zs: list_P6011104703257516679at_nat] :
      ( ( zip_Pr9109916622055408304at_nat @ Xs @ ( zip_na1006125974040638520at_nat @ Ys @ Zs ) )
      = ( map_Pr7199885541783162898at_nat
        @ ( produc2557693695688233259at_nat
          @ ( produc6561792249752071265at_nat
            @ ^ [X: product_prod_nat_nat,Y2: nat,Z3: product_prod_nat_nat] : ( produc6069232465971151312at_nat @ X @ ( produc487386426758144856at_nat @ Y2 @ Z3 ) ) ) )
        @ ( zip_Pr6393083530281879304at_nat @ ( zip_Pr6869450617852699226at_nat @ Xs @ Ys ) @ Zs ) ) ) ).

% zip_assoc
thf(fact_554_zip__assoc,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys: list_nat,Zs: list_nat] :
      ( ( zip_Pr4664179122662387191at_nat @ Xs @ ( zip_nat_nat @ Ys @ Zs ) )
      = ( map_Pr2411667379968732730at_nat
        @ ( produc2691326966899785833at_nat
          @ ( produc923811603121015911at_nat
            @ ^ [X: product_prod_nat_nat,Y2: nat,Z3: nat] : ( produc6161850002892822231at_nat @ X @ ( product_Pair_nat_nat @ Y2 @ Z3 ) ) ) )
        @ ( zip_Pr1266924471257707145at_nat @ ( zip_Pr6869450617852699226at_nat @ Xs @ Ys ) @ Zs ) ) ) ).

% zip_assoc
thf(fact_555_zip__assoc,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
      ( ( zip_na1006125974040638520at_nat @ Xs @ ( zip_nat_nat @ Ys @ Zs ) )
      = ( map_Pr1164872866496146796at_nat
        @ ( produc3206169289476954189at_nat
          @ ( produc7810592499157111267at_nat
            @ ^ [X: nat,Y2: nat,Z3: nat] : ( produc487386426758144856at_nat @ X @ ( product_Pair_nat_nat @ Y2 @ Z3 ) ) ) )
        @ ( zip_Pr6869450617852699226at_nat @ ( zip_nat_nat @ Xs @ Ys ) @ Zs ) ) ) ).

% zip_assoc
thf(fact_556_subst__term_Ocases,axiom,
    ! [X3: produc6058688428250151583at_nat] :
      ( ! [Z2: nat,X5: nat,Y3: nat] :
          ( X3
         != ( produc2180204704594896271at_nat @ ( relational_Var_a @ Z2 ) @ ( product_Pair_nat_nat @ X5 @ Y3 ) ) )
     => ~ ! [C2: a,X5: nat,Y3: nat] :
            ( X3
           != ( produc2180204704594896271at_nat @ ( relational_Const_a @ C2 ) @ ( product_Pair_nat_nat @ X5 @ Y3 ) ) ) ) ).

% subst_term.cases
thf(fact_557_old_Oprod_Ocase,axiom,
    ! [F2: nat > product_prod_nat_nat > produc7248412053542808358at_nat,X13: nat,X22: product_prod_nat_nat] :
      ( ( produc968775922737392939at_nat @ F2 @ ( produc487386426758144856at_nat @ X13 @ X22 ) )
      = ( F2 @ X13 @ X22 ) ) ).

% old.prod.case
thf(fact_558_old_Oprod_Ocase,axiom,
    ! [F2: nat > nat > produc7248412053542808358at_nat,X13: nat,X22: nat] :
      ( ( produc9083241971206738548at_nat @ F2 @ ( product_Pair_nat_nat @ X13 @ X22 ) )
      = ( F2 @ X13 @ X22 ) ) ).

% old.prod.case
thf(fact_559_old_Oprod_Ocase,axiom,
    ! [F2: nat > nat > product_prod_nat_nat,X13: nat,X22: nat] :
      ( ( produc2626176000494625587at_nat @ F2 @ ( product_Pair_nat_nat @ X13 @ X22 ) )
      = ( F2 @ X13 @ X22 ) ) ).

% old.prod.case
thf(fact_560_old_Oprod_Ocase,axiom,
    ! [F2: nat > nat > relational_term_a > relational_term_a,X13: nat,X22: nat] :
      ( ( produc6628518323692928499term_a @ F2 @ ( product_Pair_nat_nat @ X13 @ X22 ) )
      = ( F2 @ X13 @ X22 ) ) ).

% old.prod.case
thf(fact_561_old_Oprod_Ocase,axiom,
    ! [F2: nat > nat > nat > produc7248412053542808358at_nat,X13: nat,X22: nat] :
      ( ( produc7810592499157111267at_nat @ F2 @ ( product_Pair_nat_nat @ X13 @ X22 ) )
      = ( F2 @ X13 @ X22 ) ) ).

% old.prod.case
thf(fact_562_old_Oprod_Ocase,axiom,
    ! [F2: nat > nat > relational_fmla_a_b > relational_fmla_a_b,X13: nat,X22: nat] :
      ( ( produc5586541307551673003la_a_b @ F2 @ ( product_Pair_nat_nat @ X13 @ X22 ) )
      = ( F2 @ X13 @ X22 ) ) ).

% old.prod.case
thf(fact_563_old_Oprod_Ocase,axiom,
    ! [F2: nat > nat > relational_fmla_a_b,X13: nat,X22: nat] :
      ( ( produc3270801013941088237la_a_b @ F2 @ ( product_Pair_nat_nat @ X13 @ X22 ) )
      = ( F2 @ X13 @ X22 ) ) ).

% old.prod.case
thf(fact_564_fmla_Oset__intros_I10_J,axiom,
    ! [Ys: b,X4: relational_fmla_a_b] :
      ( ( member_b @ Ys @ ( relati8924981150291758614la_a_b @ X4 ) )
     => ( member_b @ Ys @ ( relati8924981150291758614la_a_b @ ( relational_Neg_a_b @ X4 ) ) ) ) ).

% fmla.set_intros(10)
thf(fact_565_fmla_Osimps_I130_J,axiom,
    ! [X4: relational_fmla_a_b] :
      ( ( relati8924981150291758614la_a_b @ ( relational_Neg_a_b @ X4 ) )
      = ( relati8924981150291758614la_a_b @ X4 ) ) ).

% fmla.simps(130)
thf(fact_566_fmla_Oset__intros_I13_J,axiom,
    ! [Yy: b,X61: relational_fmla_a_b,X62: relational_fmla_a_b] :
      ( ( member_b @ Yy @ ( relati8924981150291758614la_a_b @ X61 ) )
     => ( member_b @ Yy @ ( relati8924981150291758614la_a_b @ ( relational_Disj_a_b @ X61 @ X62 ) ) ) ) ).

% fmla.set_intros(13)
thf(fact_567_fmla_Oset__intros_I14_J,axiom,
    ! [Za: b,X62: relational_fmla_a_b,X61: relational_fmla_a_b] :
      ( ( member_b @ Za @ ( relati8924981150291758614la_a_b @ X62 ) )
     => ( member_b @ Za @ ( relati8924981150291758614la_a_b @ ( relational_Disj_a_b @ X61 @ X62 ) ) ) ) ).

% fmla.set_intros(14)
thf(fact_568_fmla_Oset__intros_I11_J,axiom,
    ! [Yu: b,X51: relational_fmla_a_b,X52: relational_fmla_a_b] :
      ( ( member_b @ Yu @ ( relati8924981150291758614la_a_b @ X51 ) )
     => ( member_b @ Yu @ ( relati8924981150291758614la_a_b @ ( relational_Conj_a_b @ X51 @ X52 ) ) ) ) ).

% fmla.set_intros(11)
thf(fact_569_fmla_Oset__intros_I12_J,axiom,
    ! [Yw: b,X52: relational_fmla_a_b,X51: relational_fmla_a_b] :
      ( ( member_b @ Yw @ ( relati8924981150291758614la_a_b @ X52 ) )
     => ( member_b @ Yw @ ( relati8924981150291758614la_a_b @ ( relational_Conj_a_b @ X51 @ X52 ) ) ) ) ).

% fmla.set_intros(12)
thf(fact_570_fmla_Osimps_I133_J,axiom,
    ! [X71: nat,X72: relational_fmla_a_b] :
      ( ( relati8924981150291758614la_a_b @ ( relati591517084277583526ts_a_b @ X71 @ X72 ) )
      = ( relati8924981150291758614la_a_b @ X72 ) ) ).

% fmla.simps(133)
thf(fact_571_fmla_Oset__intros_I15_J,axiom,
    ! [Zc: b,X72: relational_fmla_a_b,X71: nat] :
      ( ( member_b @ Zc @ ( relati8924981150291758614la_a_b @ X72 ) )
     => ( member_b @ Zc @ ( relati8924981150291758614la_a_b @ ( relati591517084277583526ts_a_b @ X71 @ X72 ) ) ) ) ).

% fmla.set_intros(15)
thf(fact_572_fmla_Oset__intros_I9_J,axiom,
    ! [X112: b,X122: list_R6823256787227418703term_a] : ( member_b @ X112 @ ( relati8924981150291758614la_a_b @ ( relational_Pred_b_a @ X112 @ X122 ) ) ) ).

% fmla.set_intros(9)
thf(fact_573_case__prodE2,axiom,
    ! [Q: produc7248412053542808358at_nat > $o,P: nat > product_prod_nat_nat > produc7248412053542808358at_nat,Z: produc7248412053542808358at_nat] :
      ( ( Q @ ( produc968775922737392939at_nat @ P @ Z ) )
     => ~ ! [X5: nat,Y3: product_prod_nat_nat] :
            ( ( Z
              = ( produc487386426758144856at_nat @ X5 @ Y3 ) )
           => ~ ( Q @ ( P @ X5 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_574_case__prodE2,axiom,
    ! [Q: produc7248412053542808358at_nat > $o,P: nat > nat > produc7248412053542808358at_nat,Z: product_prod_nat_nat] :
      ( ( Q @ ( produc9083241971206738548at_nat @ P @ Z ) )
     => ~ ! [X5: nat,Y3: nat] :
            ( ( Z
              = ( product_Pair_nat_nat @ X5 @ Y3 ) )
           => ~ ( Q @ ( P @ X5 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_575_case__prodE2,axiom,
    ! [Q: product_prod_nat_nat > $o,P: nat > nat > product_prod_nat_nat,Z: product_prod_nat_nat] :
      ( ( Q @ ( produc2626176000494625587at_nat @ P @ Z ) )
     => ~ ! [X5: nat,Y3: nat] :
            ( ( Z
              = ( product_Pair_nat_nat @ X5 @ Y3 ) )
           => ~ ( Q @ ( P @ X5 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_576_case__prodE2,axiom,
    ! [Q: ( relational_term_a > relational_term_a ) > $o,P: nat > nat > relational_term_a > relational_term_a,Z: product_prod_nat_nat] :
      ( ( Q @ ( produc6628518323692928499term_a @ P @ Z ) )
     => ~ ! [X5: nat,Y3: nat] :
            ( ( Z
              = ( product_Pair_nat_nat @ X5 @ Y3 ) )
           => ~ ( Q @ ( P @ X5 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_577_case__prodE2,axiom,
    ! [Q: ( nat > produc7248412053542808358at_nat ) > $o,P: nat > nat > nat > produc7248412053542808358at_nat,Z: product_prod_nat_nat] :
      ( ( Q @ ( produc7810592499157111267at_nat @ P @ Z ) )
     => ~ ! [X5: nat,Y3: nat] :
            ( ( Z
              = ( product_Pair_nat_nat @ X5 @ Y3 ) )
           => ~ ( Q @ ( P @ X5 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_578_case__prodE2,axiom,
    ! [Q: ( relational_fmla_a_b > relational_fmla_a_b ) > $o,P: nat > nat > relational_fmla_a_b > relational_fmla_a_b,Z: product_prod_nat_nat] :
      ( ( Q @ ( produc5586541307551673003la_a_b @ P @ Z ) )
     => ~ ! [X5: nat,Y3: nat] :
            ( ( Z
              = ( product_Pair_nat_nat @ X5 @ Y3 ) )
           => ~ ( Q @ ( P @ X5 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_579_case__prodE2,axiom,
    ! [Q: relational_fmla_a_b > $o,P: nat > nat > relational_fmla_a_b,Z: product_prod_nat_nat] :
      ( ( Q @ ( produc3270801013941088237la_a_b @ P @ Z ) )
     => ~ ! [X5: nat,Y3: nat] :
            ( ( Z
              = ( product_Pair_nat_nat @ X5 @ Y3 ) )
           => ~ ( Q @ ( P @ X5 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_580_case__prod__eta,axiom,
    ! [F2: produc7248412053542808358at_nat > produc7248412053542808358at_nat] :
      ( ( produc968775922737392939at_nat
        @ ^ [X: nat,Y2: product_prod_nat_nat] : ( F2 @ ( produc487386426758144856at_nat @ X @ Y2 ) ) )
      = F2 ) ).

% case_prod_eta
thf(fact_581_case__prod__eta,axiom,
    ! [F2: product_prod_nat_nat > produc7248412053542808358at_nat] :
      ( ( produc9083241971206738548at_nat
        @ ^ [X: nat,Y2: nat] : ( F2 @ ( product_Pair_nat_nat @ X @ Y2 ) ) )
      = F2 ) ).

% case_prod_eta
thf(fact_582_case__prod__eta,axiom,
    ! [F2: product_prod_nat_nat > product_prod_nat_nat] :
      ( ( produc2626176000494625587at_nat
        @ ^ [X: nat,Y2: nat] : ( F2 @ ( product_Pair_nat_nat @ X @ Y2 ) ) )
      = F2 ) ).

% case_prod_eta
thf(fact_583_case__prod__eta,axiom,
    ! [F2: product_prod_nat_nat > relational_term_a > relational_term_a] :
      ( ( produc6628518323692928499term_a
        @ ^ [X: nat,Y2: nat] : ( F2 @ ( product_Pair_nat_nat @ X @ Y2 ) ) )
      = F2 ) ).

% case_prod_eta
thf(fact_584_case__prod__eta,axiom,
    ! [F2: product_prod_nat_nat > nat > produc7248412053542808358at_nat] :
      ( ( produc7810592499157111267at_nat
        @ ^ [X: nat,Y2: nat] : ( F2 @ ( product_Pair_nat_nat @ X @ Y2 ) ) )
      = F2 ) ).

% case_prod_eta
thf(fact_585_case__prod__eta,axiom,
    ! [F2: product_prod_nat_nat > relational_fmla_a_b > relational_fmla_a_b] :
      ( ( produc5586541307551673003la_a_b
        @ ^ [X: nat,Y2: nat] : ( F2 @ ( product_Pair_nat_nat @ X @ Y2 ) ) )
      = F2 ) ).

% case_prod_eta
thf(fact_586_case__prod__eta,axiom,
    ! [F2: product_prod_nat_nat > relational_fmla_a_b] :
      ( ( produc3270801013941088237la_a_b
        @ ^ [X: nat,Y2: nat] : ( F2 @ ( product_Pair_nat_nat @ X @ Y2 ) ) )
      = F2 ) ).

% case_prod_eta
thf(fact_587_cond__case__prod__eta,axiom,
    ! [F2: nat > product_prod_nat_nat > produc7248412053542808358at_nat,G2: produc7248412053542808358at_nat > produc7248412053542808358at_nat] :
      ( ! [X5: nat,Y3: product_prod_nat_nat] :
          ( ( F2 @ X5 @ Y3 )
          = ( G2 @ ( produc487386426758144856at_nat @ X5 @ Y3 ) ) )
     => ( ( produc968775922737392939at_nat @ F2 )
        = G2 ) ) ).

% cond_case_prod_eta
thf(fact_588_cond__case__prod__eta,axiom,
    ! [F2: nat > nat > produc7248412053542808358at_nat,G2: product_prod_nat_nat > produc7248412053542808358at_nat] :
      ( ! [X5: nat,Y3: nat] :
          ( ( F2 @ X5 @ Y3 )
          = ( G2 @ ( product_Pair_nat_nat @ X5 @ Y3 ) ) )
     => ( ( produc9083241971206738548at_nat @ F2 )
        = G2 ) ) ).

% cond_case_prod_eta
thf(fact_589_cond__case__prod__eta,axiom,
    ! [F2: nat > nat > product_prod_nat_nat,G2: product_prod_nat_nat > product_prod_nat_nat] :
      ( ! [X5: nat,Y3: nat] :
          ( ( F2 @ X5 @ Y3 )
          = ( G2 @ ( product_Pair_nat_nat @ X5 @ Y3 ) ) )
     => ( ( produc2626176000494625587at_nat @ F2 )
        = G2 ) ) ).

% cond_case_prod_eta
thf(fact_590_cond__case__prod__eta,axiom,
    ! [F2: nat > nat > relational_term_a > relational_term_a,G2: product_prod_nat_nat > relational_term_a > relational_term_a] :
      ( ! [X5: nat,Y3: nat] :
          ( ( F2 @ X5 @ Y3 )
          = ( G2 @ ( product_Pair_nat_nat @ X5 @ Y3 ) ) )
     => ( ( produc6628518323692928499term_a @ F2 )
        = G2 ) ) ).

% cond_case_prod_eta
thf(fact_591_cond__case__prod__eta,axiom,
    ! [F2: nat > nat > nat > produc7248412053542808358at_nat,G2: product_prod_nat_nat > nat > produc7248412053542808358at_nat] :
      ( ! [X5: nat,Y3: nat] :
          ( ( F2 @ X5 @ Y3 )
          = ( G2 @ ( product_Pair_nat_nat @ X5 @ Y3 ) ) )
     => ( ( produc7810592499157111267at_nat @ F2 )
        = G2 ) ) ).

% cond_case_prod_eta
thf(fact_592_cond__case__prod__eta,axiom,
    ! [F2: nat > nat > relational_fmla_a_b > relational_fmla_a_b,G2: product_prod_nat_nat > relational_fmla_a_b > relational_fmla_a_b] :
      ( ! [X5: nat,Y3: nat] :
          ( ( F2 @ X5 @ Y3 )
          = ( G2 @ ( product_Pair_nat_nat @ X5 @ Y3 ) ) )
     => ( ( produc5586541307551673003la_a_b @ F2 )
        = G2 ) ) ).

% cond_case_prod_eta
thf(fact_593_cond__case__prod__eta,axiom,
    ! [F2: nat > nat > relational_fmla_a_b,G2: product_prod_nat_nat > relational_fmla_a_b] :
      ( ! [X5: nat,Y3: nat] :
          ( ( F2 @ X5 @ Y3 )
          = ( G2 @ ( product_Pair_nat_nat @ X5 @ Y3 ) ) )
     => ( ( produc3270801013941088237la_a_b @ F2 )
        = G2 ) ) ).

% cond_case_prod_eta
thf(fact_594_zip__commute,axiom,
    ( zip_Pr6869450617852699226at_nat
    = ( ^ [Xs3: list_P6011104703257516679at_nat,Ys2: list_nat] :
          ( map_Pr7387300356108928108at_nat
          @ ( produc2094262906704694021at_nat
            @ ^ [X: nat,Y2: product_prod_nat_nat] : ( produc6350711070570205562at_nat @ Y2 @ X ) )
          @ ( zip_na1006125974040638520at_nat @ Ys2 @ Xs3 ) ) ) ) ).

% zip_commute
thf(fact_595_zip__commute,axiom,
    ( zip_o_o
    = ( ^ [Xs3: list_o,Ys2: list_o] :
          ( map_Pr8243895103994891268od_o_o
          @ ( produc7436348682273225467od_o_o
            @ ^ [X: $o,Y2: $o] : ( product_Pair_o_o @ Y2 @ X ) )
          @ ( zip_o_o @ Ys2 @ Xs3 ) ) ) ) ).

% zip_commute
thf(fact_596_zip__commute,axiom,
    ( zip_na1006125974040638520at_nat
    = ( ^ [Xs3: list_nat,Ys2: list_P6011104703257516679at_nat] :
          ( map_Pr1164872866496146796at_nat
          @ ( produc3206169289476954189at_nat
            @ ^ [X: product_prod_nat_nat,Y2: nat] : ( produc487386426758144856at_nat @ Y2 @ X ) )
          @ ( zip_Pr6869450617852699226at_nat @ Ys2 @ Xs3 ) ) ) ) ).

% zip_commute
thf(fact_597_zip__commute,axiom,
    ( zip_Pr5387723086728981183_nat_a
    = ( ^ [Xs3: list_P6049048235159712035list_a,Ys2: list_nat_a] :
          ( map_Pr8478378587642173844_nat_a
          @ ( produc4690939927037265291_nat_a
            @ ^ [X: nat > a,Y2: product_prod_b_nat > set_list_a] : ( produc2895298938842563487_nat_a @ Y2 @ X ) )
          @ ( zip_na3962188320132642351list_a @ Ys2 @ Xs3 ) ) ) ) ).

% zip_commute
thf(fact_598_zip__commute,axiom,
    ( zip_b_nat
    = ( ^ [Xs3: list_b,Ys2: list_nat] :
          ( map_Pr8590476531800460392_b_nat
          @ ( produc9075294634372754561_b_nat
            @ ^ [X: nat,Y2: b] : ( product_Pair_b_nat @ Y2 @ X ) )
          @ ( zip_nat_b @ Ys2 @ Xs3 ) ) ) ) ).

% zip_commute
thf(fact_599_zip__commute,axiom,
    ( zip_nat_nat
    = ( ^ [Xs3: list_nat,Ys2: list_nat] :
          ( map_Pr8058819605623181956at_nat
          @ ( produc2626176000494625587at_nat
            @ ^ [X: nat,Y2: nat] : ( product_Pair_nat_nat @ Y2 @ X ) )
          @ ( zip_nat_nat @ Ys2 @ Xs3 ) ) ) ) ).

% zip_commute
thf(fact_600_zip__same__conv__map,axiom,
    ! [Xs: list_o] :
      ( ( zip_o_o @ Xs @ Xs )
      = ( map_o_3702434973371374163od_o_o
        @ ^ [X: $o] : ( product_Pair_o_o @ X @ X )
        @ Xs ) ) ).

% zip_same_conv_map
thf(fact_601_zip__same__conv__map,axiom,
    ! [Xs: list_nat] :
      ( ( zip_nat_nat @ Xs @ Xs )
      = ( map_na7298421622053143531at_nat
        @ ^ [X: nat] : ( product_Pair_nat_nat @ X @ X )
        @ Xs ) ) ).

% zip_same_conv_map
thf(fact_602_eval__term_Osimps_I2_J,axiom,
    ! [Sigma: nat > a,N: nat] :
      ( ( relati1177013128715261720term_a @ Sigma @ ( relational_Var_a @ N ) )
      = ( Sigma @ N ) ) ).

% eval_term.simps(2)
thf(fact_603_eval__term_Osimps_I1_J,axiom,
    ! [Sigma: nat > a,C: a] :
      ( ( relati1177013128715261720term_a @ Sigma @ ( relational_Const_a @ C ) )
      = C ) ).

% eval_term.simps(1)
thf(fact_604_curry__def,axiom,
    ( produc858456811296061068la_a_b
    = ( ^ [C3: product_prod_nat_nat > relational_fmla_a_b,X: nat,Y2: nat] : ( C3 @ ( product_Pair_nat_nat @ X @ Y2 ) ) ) ) ).

% curry_def
thf(fact_605_curry__def,axiom,
    ( produc7541201833284165578la_a_b
    = ( ^ [C3: product_prod_nat_nat > relational_fmla_a_b > relational_fmla_a_b,X: nat,Y2: nat] : ( C3 @ ( product_Pair_nat_nat @ X @ Y2 ) ) ) ) ).

% curry_def
thf(fact_606_cpropagated__cp__triv,axiom,
    ! [Q: relational_fmla_a_b] :
      ( ( relati1591879772219623554ed_a_b @ Q )
     => ( ( relational_cp_a_b @ Q )
        = Q ) ) ).

% cpropagated_cp_triv
thf(fact_607_cpropagated__cp,axiom,
    ! [Q: relational_fmla_a_b] : ( relati1591879772219623554ed_a_b @ ( relational_cp_a_b @ Q ) ) ).

% cpropagated_cp
thf(fact_608_nocp__cpropagated,axiom,
    ! [Q: relational_fmla_a_b] :
      ( ( relational_nocp_a_b @ Q )
     => ( relati1591879772219623554ed_a_b @ Q ) ) ).

% nocp_cpropagated
thf(fact_609_eval__term_Ocases,axiom,
    ! [X3: produc8608687409264118859term_a] :
      ( ! [Sigma2: nat > a,C2: a] :
          ( X3
         != ( produc8917778089171359291term_a @ Sigma2 @ ( relational_Const_a @ C2 ) ) )
     => ~ ! [Sigma2: nat > a,N2: nat] :
            ( X3
           != ( produc8917778089171359291term_a @ Sigma2 @ ( relational_Var_a @ N2 ) ) ) ) ).

% eval_term.cases
thf(fact_610_internal__case__prod__conv,axiom,
    ! [C: nat > nat > relational_fmla_a_b,A: nat,B: nat] :
      ( ( produc2167329440614046147la_a_b @ C @ ( product_Pair_nat_nat @ A @ B ) )
      = ( C @ A @ B ) ) ).

% internal_case_prod_conv
thf(fact_611_internal__case__prod__conv,axiom,
    ! [C: nat > nat > relational_fmla_a_b > relational_fmla_a_b,A: nat,B: nat] :
      ( ( produc7208501889184683393la_a_b @ C @ ( product_Pair_nat_nat @ A @ B ) )
      = ( C @ A @ B ) ) ).

% internal_case_prod_conv
thf(fact_612_zip__map__map,axiom,
    ! [F2: nat > nat,Xs: list_nat,G2: nat > nat,Ys: list_nat] :
      ( ( zip_nat_nat @ ( map_nat_nat @ F2 @ Xs ) @ ( map_nat_nat @ G2 @ Ys ) )
      = ( map_Pr8058819605623181956at_nat
        @ ( produc2626176000494625587at_nat
          @ ^ [X: nat,Y2: nat] : ( product_Pair_nat_nat @ ( F2 @ X ) @ ( G2 @ Y2 ) ) )
        @ ( zip_nat_nat @ Xs @ Ys ) ) ) ).

% zip_map_map
thf(fact_613_zip__map__map,axiom,
    ! [F2: nat > $o,Xs: list_nat,G2: nat > $o,Ys: list_nat] :
      ( ( zip_o_o @ ( map_nat_o @ F2 @ Xs ) @ ( map_nat_o @ G2 @ Ys ) )
      = ( map_Pr1728080061549680682od_o_o
        @ ( produc2065404711520291929od_o_o
          @ ^ [X: nat,Y2: nat] : ( product_Pair_o_o @ ( F2 @ X ) @ ( G2 @ Y2 ) ) )
        @ ( zip_nat_nat @ Xs @ Ys ) ) ) ).

% zip_map_map
thf(fact_614_zip__map__map,axiom,
    ! [F2: nat > b,Xs: list_nat,G2: nat > nat,Ys: list_nat] :
      ( ( zip_b_nat @ ( map_nat_b @ F2 @ Xs ) @ ( map_nat_nat @ G2 @ Ys ) )
      = ( map_Pr5089981438431642555_b_nat
        @ ( produc5550384251870937292_b_nat
          @ ^ [X: nat,Y2: nat] : ( product_Pair_b_nat @ ( F2 @ X ) @ ( G2 @ Y2 ) ) )
        @ ( zip_nat_nat @ Xs @ Ys ) ) ) ).

% zip_map_map
thf(fact_615_zip__map__map,axiom,
    ! [F2: nat > product_prod_nat_nat,Xs: list_nat,G2: nat > nat,Ys: list_nat] :
      ( ( zip_Pr6869450617852699226at_nat @ ( map_na7298421622053143531at_nat @ F2 @ Xs ) @ ( map_nat_nat @ G2 @ Ys ) )
      = ( map_Pr8661772540859430845at_nat
        @ ( produc985356918319263822at_nat
          @ ^ [X: nat,Y2: nat] : ( produc6350711070570205562at_nat @ ( F2 @ X ) @ ( G2 @ Y2 ) ) )
        @ ( zip_nat_nat @ Xs @ Ys ) ) ) ).

% zip_map_map
thf(fact_616_zip__map__map,axiom,
    ! [F2: nat > $o,Xs: list_nat,G2: product_prod_nat_nat > $o,Ys: list_P6011104703257516679at_nat] :
      ( ( zip_o_o @ ( map_nat_o @ F2 @ Xs ) @ ( map_Pr9219130883924091931_nat_o @ G2 @ Ys ) )
      = ( map_Pr5892519504250064571od_o_o
        @ ( produc471732273315494626od_o_o
          @ ^ [X: nat,Y2: product_prod_nat_nat] : ( product_Pair_o_o @ ( F2 @ X ) @ ( G2 @ Y2 ) ) )
        @ ( zip_na1006125974040638520at_nat @ Xs @ Ys ) ) ) ).

% zip_map_map
thf(fact_617_zip__map__map,axiom,
    ! [F2: product_prod_nat_nat > $o,Xs: list_P6011104703257516679at_nat,G2: nat > $o,Ys: list_nat] :
      ( ( zip_o_o @ ( map_Pr9219130883924091931_nat_o @ F2 @ Xs ) @ ( map_nat_o @ G2 @ Ys ) )
      = ( map_Pr1711353591275973473od_o_o
        @ ( produc2653038494237387072od_o_o
          @ ^ [X: product_prod_nat_nat,Y2: nat] : ( product_Pair_o_o @ ( F2 @ X ) @ ( G2 @ Y2 ) ) )
        @ ( zip_Pr6869450617852699226at_nat @ Xs @ Ys ) ) ) ).

% zip_map_map
thf(fact_618_zip__map__map,axiom,
    ! [F2: product_prod_nat_nat > nat,Xs: list_P6011104703257516679at_nat,G2: nat > nat,Ys: list_nat] :
      ( ( zip_nat_nat @ ( map_Pr3938374229010428429at_nat @ F2 @ Xs ) @ ( map_nat_nat @ G2 @ Ys ) )
      = ( map_Pr4819452465118600763at_nat
        @ ( produc373799411880517786at_nat
          @ ^ [X: product_prod_nat_nat,Y2: nat] : ( product_Pair_nat_nat @ ( F2 @ X ) @ ( G2 @ Y2 ) ) )
        @ ( zip_Pr6869450617852699226at_nat @ Xs @ Ys ) ) ) ).

% zip_map_map
thf(fact_619_zip__map__map,axiom,
    ! [F2: nat > nat,Xs: list_nat,G2: product_prod_nat_nat > nat,Ys: list_P6011104703257516679at_nat] :
      ( ( zip_nat_nat @ ( map_nat_nat @ F2 @ Xs ) @ ( map_Pr3938374229010428429at_nat @ G2 @ Ys ) )
      = ( map_Pr2617240807308709013at_nat
        @ ( produc8859641928216934716at_nat
          @ ^ [X: nat,Y2: product_prod_nat_nat] : ( product_Pair_nat_nat @ ( F2 @ X ) @ ( G2 @ Y2 ) ) )
        @ ( zip_na1006125974040638520at_nat @ Xs @ Ys ) ) ) ).

% zip_map_map
thf(fact_620_zip__map__map,axiom,
    ! [F2: nat > b,Xs: list_nat,G2: product_prod_nat_nat > nat,Ys: list_P6011104703257516679at_nat] :
      ( ( zip_b_nat @ ( map_nat_b @ F2 @ Xs ) @ ( map_Pr3938374229010428429at_nat @ G2 @ Ys ) )
      = ( map_Pr9104689871214554090_b_nat
        @ ( produc7108648848425720579_b_nat
          @ ^ [X: nat,Y2: product_prod_nat_nat] : ( product_Pair_b_nat @ ( F2 @ X ) @ ( G2 @ Y2 ) ) )
        @ ( zip_na1006125974040638520at_nat @ Xs @ Ys ) ) ) ).

% zip_map_map
thf(fact_621_zip__map__map,axiom,
    ! [F2: product_prod_nat_nat > b,Xs: list_P6011104703257516679at_nat,G2: nat > nat,Ys: list_nat] :
      ( ( zip_b_nat @ ( map_Pr5244471862779271682_nat_b @ F2 @ Xs ) @ ( map_nat_nat @ G2 @ Ys ) )
      = ( map_Pr7646859348845541316_b_nat
        @ ( produc416915106962591525_b_nat
          @ ^ [X: product_prod_nat_nat,Y2: nat] : ( product_Pair_b_nat @ ( F2 @ X ) @ ( G2 @ Y2 ) ) )
        @ ( zip_Pr6869450617852699226at_nat @ Xs @ Ys ) ) ) ).

% zip_map_map
thf(fact_622_zip__map2,axiom,
    ! [Xs: list_nat,F2: product_prod_nat_nat > relational_fmla_a_b,Ys: list_P6011104703257516679at_nat] :
      ( ( zip_na3567360947154556456la_a_b @ Xs @ ( map_Pr2810398200501793500la_a_b @ F2 @ Ys ) )
      = ( map_Pr1524493114334851300la_a_b
        @ ( produc3281652006734995083la_a_b
          @ ^ [X: nat,Y2: product_prod_nat_nat] : ( produc8327306639710187272la_a_b @ X @ ( F2 @ Y2 ) ) )
        @ ( zip_na1006125974040638520at_nat @ Xs @ Ys ) ) ) ).

% zip_map2
thf(fact_623_zip__map2,axiom,
    ! [Xs: list_nat,F2: product_prod_nat_nat > relational_fmla_a_b > relational_fmla_a_b,Ys: list_P6011104703257516679at_nat] :
      ( ( zip_na1289494917740915046la_a_b @ Xs @ ( map_Pr591601166967198746la_a_b @ F2 @ Ys ) )
      = ( map_Pr2449914881779310242la_a_b
        @ ( produc433188378764986057la_a_b
          @ ^ [X: nat,Y2: product_prod_nat_nat] : ( produc5257537519060881990la_a_b @ X @ ( F2 @ Y2 ) ) )
        @ ( zip_na1006125974040638520at_nat @ Xs @ Ys ) ) ) ).

% zip_map2
thf(fact_624_zip__map2,axiom,
    ! [Xs: list_P6011104703257516679at_nat,F2: nat > product_prod_nat_nat,Ys: list_nat] :
      ( ( zip_Pr4664179122662387191at_nat @ Xs @ ( map_na7298421622053143531at_nat @ F2 @ Ys ) )
      = ( map_Pr1933158075150132897at_nat
        @ ( produc4883074260277788288at_nat
          @ ^ [X: product_prod_nat_nat,Y2: nat] : ( produc6161850002892822231at_nat @ X @ ( F2 @ Y2 ) ) )
        @ ( zip_Pr6869450617852699226at_nat @ Xs @ Ys ) ) ) ).

% zip_map2
thf(fact_625_zip__map2,axiom,
    ! [Xs: list_P6011104703257516679at_nat,F2: nat > nat,Ys: list_nat] :
      ( ( zip_Pr6869450617852699226at_nat @ Xs @ ( map_nat_nat @ F2 @ Ys ) )
      = ( map_Pr2290359850463447878at_nat
        @ ( produc4331656273444255271at_nat
          @ ^ [X: product_prod_nat_nat,Y2: nat] : ( produc6350711070570205562at_nat @ X @ ( F2 @ Y2 ) ) )
        @ ( zip_Pr6869450617852699226at_nat @ Xs @ Ys ) ) ) ).

% zip_map2
thf(fact_626_zip__map2,axiom,
    ! [Xs: list_nat,F2: product_prod_nat_nat > nat,Ys: list_P6011104703257516679at_nat] :
      ( ( zip_nat_nat @ Xs @ ( map_Pr3938374229010428429at_nat @ F2 @ Ys ) )
      = ( map_Pr2617240807308709013at_nat
        @ ( produc8859641928216934716at_nat
          @ ^ [X: nat,Y2: product_prod_nat_nat] : ( product_Pair_nat_nat @ X @ ( F2 @ Y2 ) ) )
        @ ( zip_na1006125974040638520at_nat @ Xs @ Ys ) ) ) ).

% zip_map2
thf(fact_627_zip__map2,axiom,
    ! [Xs: list_b,F2: nat > nat,Ys: list_nat] :
      ( ( zip_b_nat @ Xs @ ( map_nat_nat @ F2 @ Ys ) )
      = ( map_Pr7742232262618018242_b_nat
        @ ( produc282185899741183267_b_nat
          @ ^ [X: b,Y2: nat] : ( product_Pair_b_nat @ X @ ( F2 @ Y2 ) ) )
        @ ( zip_b_nat @ Xs @ Ys ) ) ) ).

% zip_map2
thf(fact_628_zip__map2,axiom,
    ! [Xs: list_nat,F2: product_prod_nat_nat > product_prod_nat_nat,Ys: list_P6011104703257516679at_nat] :
      ( ( zip_na1006125974040638520at_nat @ Xs @ ( map_Pr8058819605623181956at_nat @ F2 @ Ys ) )
      = ( map_Pr6261813372141627026at_nat
        @ ( produc968775922737392939at_nat
          @ ^ [X: nat,Y2: product_prod_nat_nat] : ( produc487386426758144856at_nat @ X @ ( F2 @ Y2 ) ) )
        @ ( zip_na1006125974040638520at_nat @ Xs @ Ys ) ) ) ).

% zip_map2
thf(fact_629_zip__map2,axiom,
    ! [Xs: list_nat,F2: nat > product_prod_nat_nat,Ys: list_nat] :
      ( ( zip_na1006125974040638520at_nat @ Xs @ ( map_na7298421622053143531at_nat @ F2 @ Ys ) )
      = ( map_Pr7536285556892129763at_nat
        @ ( produc9083241971206738548at_nat
          @ ^ [X: nat,Y2: nat] : ( produc487386426758144856at_nat @ X @ ( F2 @ Y2 ) ) )
        @ ( zip_nat_nat @ Xs @ Ys ) ) ) ).

% zip_map2
thf(fact_630_zip__map2,axiom,
    ! [Xs: list_nat,F2: nat > nat,Ys: list_nat] :
      ( ( zip_nat_nat @ Xs @ ( map_nat_nat @ F2 @ Ys ) )
      = ( map_Pr8058819605623181956at_nat
        @ ( produc2626176000494625587at_nat
          @ ^ [X: nat,Y2: nat] : ( product_Pair_nat_nat @ X @ ( F2 @ Y2 ) ) )
        @ ( zip_nat_nat @ Xs @ Ys ) ) ) ).

% zip_map2
thf(fact_631_zip__map1,axiom,
    ! [F2: product_prod_nat_nat > relational_fmla_a_b,Xs: list_P6011104703257516679at_nat,Ys: list_nat] :
      ( ( zip_Re8745484028657759016_b_nat @ ( map_Pr2810398200501793500la_a_b @ F2 @ Xs ) @ Ys )
      = ( map_Pr987686398805048586_b_nat
        @ ( produc8806289362176886889_b_nat
          @ ^ [X: product_prod_nat_nat] : ( produc4282057684358614024_b_nat @ ( F2 @ X ) ) )
        @ ( zip_Pr6869450617852699226at_nat @ Xs @ Ys ) ) ) ).

% zip_map1
thf(fact_632_zip__map1,axiom,
    ! [F2: product_prod_nat_nat > product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Ys: list_nat] :
      ( ( zip_Pr6869450617852699226at_nat @ ( map_Pr8058819605623181956at_nat @ F2 @ Xs ) @ Ys )
      = ( map_Pr2290359850463447878at_nat
        @ ( produc4331656273444255271at_nat
          @ ^ [X: product_prod_nat_nat] : ( produc6350711070570205562at_nat @ ( F2 @ X ) ) )
        @ ( zip_Pr6869450617852699226at_nat @ Xs @ Ys ) ) ) ).

% zip_map1
thf(fact_633_zip__map1,axiom,
    ! [F2: product_prod_nat_nat > relational_fmla_a_b > relational_fmla_a_b,Xs: list_P6011104703257516679at_nat,Ys: list_nat] :
      ( ( zip_Re412449016089142502_b_nat @ ( map_Pr591601166967198746la_a_b @ F2 @ Xs ) @ Ys )
      = ( map_Pr5610943684965357384_b_nat
        @ ( produc542637198479520039_b_nat
          @ ^ [X: product_prod_nat_nat] : ( produc4380491617409109446_b_nat @ ( F2 @ X ) ) )
        @ ( zip_Pr6869450617852699226at_nat @ Xs @ Ys ) ) ) ).

% zip_map1
thf(fact_634_zip__map1,axiom,
    ! [F2: nat > product_prod_nat_nat,Xs: list_nat,Ys: list_P6011104703257516679at_nat] :
      ( ( zip_Pr4664179122662387191at_nat @ ( map_na7298421622053143531at_nat @ F2 @ Xs ) @ Ys )
      = ( map_Pr3088088100884424187at_nat
        @ ( produc5396115207565794338at_nat
          @ ^ [X: nat] : ( produc6161850002892822231at_nat @ ( F2 @ X ) ) )
        @ ( zip_na1006125974040638520at_nat @ Xs @ Ys ) ) ) ).

% zip_map1
thf(fact_635_zip__map1,axiom,
    ! [F2: nat > product_prod_nat_nat,Xs: list_nat,Ys: list_nat] :
      ( ( zip_Pr6869450617852699226at_nat @ ( map_na7298421622053143531at_nat @ F2 @ Xs ) @ Ys )
      = ( map_Pr8661772540859430845at_nat
        @ ( produc985356918319263822at_nat
          @ ^ [X: nat] : ( produc6350711070570205562at_nat @ ( F2 @ X ) ) )
        @ ( zip_nat_nat @ Xs @ Ys ) ) ) ).

% zip_map1
thf(fact_636_zip__map1,axiom,
    ! [F2: product_prod_nat_nat > nat,Xs: list_P6011104703257516679at_nat,Ys: list_nat] :
      ( ( zip_nat_nat @ ( map_Pr3938374229010428429at_nat @ F2 @ Xs ) @ Ys )
      = ( map_Pr4819452465118600763at_nat
        @ ( produc373799411880517786at_nat
          @ ^ [X: product_prod_nat_nat] : ( product_Pair_nat_nat @ ( F2 @ X ) ) )
        @ ( zip_Pr6869450617852699226at_nat @ Xs @ Ys ) ) ) ).

% zip_map1
thf(fact_637_zip__map1,axiom,
    ! [F2: nat > b,Xs: list_nat,Ys: list_nat] :
      ( ( zip_b_nat @ ( map_nat_b @ F2 @ Xs ) @ Ys )
      = ( map_Pr5089981438431642555_b_nat
        @ ( produc5550384251870937292_b_nat
          @ ^ [X: nat] : ( product_Pair_b_nat @ ( F2 @ X ) ) )
        @ ( zip_nat_nat @ Xs @ Ys ) ) ) ).

% zip_map1
thf(fact_638_zip__map1,axiom,
    ! [F2: product_prod_nat_nat > b,Xs: list_P6011104703257516679at_nat,Ys: list_nat] :
      ( ( zip_b_nat @ ( map_Pr5244471862779271682_nat_b @ F2 @ Xs ) @ Ys )
      = ( map_Pr7646859348845541316_b_nat
        @ ( produc416915106962591525_b_nat
          @ ^ [X: product_prod_nat_nat] : ( product_Pair_b_nat @ ( F2 @ X ) ) )
        @ ( zip_Pr6869450617852699226at_nat @ Xs @ Ys ) ) ) ).

% zip_map1
thf(fact_639_zip__map1,axiom,
    ! [F2: nat > nat,Xs: list_nat,Ys: list_P6011104703257516679at_nat] :
      ( ( zip_na1006125974040638520at_nat @ ( map_nat_nat @ F2 @ Xs ) @ Ys )
      = ( map_Pr6261813372141627026at_nat
        @ ( produc968775922737392939at_nat
          @ ^ [X: nat] : ( produc487386426758144856at_nat @ ( F2 @ X ) ) )
        @ ( zip_na1006125974040638520at_nat @ Xs @ Ys ) ) ) ).

% zip_map1
thf(fact_640_zip__map1,axiom,
    ! [F2: nat > nat,Xs: list_nat,Ys: list_nat] :
      ( ( zip_nat_nat @ ( map_nat_nat @ F2 @ Xs ) @ Ys )
      = ( map_Pr8058819605623181956at_nat
        @ ( produc2626176000494625587at_nat
          @ ^ [X: nat] : ( product_Pair_nat_nat @ ( F2 @ X ) ) )
        @ ( zip_nat_nat @ Xs @ Ys ) ) ) ).

% zip_map1
thf(fact_641_Sup_OSUP__cong,axiom,
    ! [A2: set_o_o,B5: set_o_o,C4: ( $o > $o ) > set_o,D2: ( $o > $o ) > set_o,Sup: set_set_o > set_o] :
      ( ( A2 = B5 )
     => ( ! [X5: $o > $o] :
            ( ( member_o_o @ X5 @ B5 )
           => ( ( C4 @ X5 )
              = ( D2 @ X5 ) ) )
       => ( ( Sup @ ( image_o_o_set_o @ C4 @ A2 ) )
          = ( Sup @ ( image_o_o_set_o @ D2 @ B5 ) ) ) ) ) ).

% Sup.SUP_cong
thf(fact_642_Sup_OSUP__cong,axiom,
    ! [A2: set_o,B5: set_o,C4: $o > nat,D2: $o > nat,Sup: set_nat > nat] :
      ( ( A2 = B5 )
     => ( ! [X5: $o] :
            ( ( member_o @ X5 @ B5 )
           => ( ( C4 @ X5 )
              = ( D2 @ X5 ) ) )
       => ( ( Sup @ ( image_o_nat @ C4 @ A2 ) )
          = ( Sup @ ( image_o_nat @ D2 @ B5 ) ) ) ) ) ).

% Sup.SUP_cong
thf(fact_643_Sup_OSUP__cong,axiom,
    ! [A2: set_set_o,B5: set_set_o,C4: set_o > $o,D2: set_o > $o,Sup: set_o > $o] :
      ( ( A2 = B5 )
     => ( ! [X5: set_o] :
            ( ( member_set_o @ X5 @ B5 )
           => ( ( C4 @ X5 )
              = ( D2 @ X5 ) ) )
       => ( ( Sup @ ( image_set_o_o @ C4 @ A2 ) )
          = ( Sup @ ( image_set_o_o @ D2 @ B5 ) ) ) ) ) ).

% Sup.SUP_cong
thf(fact_644_Sup_OSUP__cong,axiom,
    ! [A2: set_set_o,B5: set_set_o,C4: set_o > set_o,D2: set_o > set_o,Sup: set_set_o > set_o] :
      ( ( A2 = B5 )
     => ( ! [X5: set_o] :
            ( ( member_set_o @ X5 @ B5 )
           => ( ( C4 @ X5 )
              = ( D2 @ X5 ) ) )
       => ( ( Sup @ ( image_set_o_set_o @ C4 @ A2 ) )
          = ( Sup @ ( image_set_o_set_o @ D2 @ B5 ) ) ) ) ) ).

% Sup.SUP_cong
thf(fact_645_Sup_OSUP__cong,axiom,
    ! [A2: set_set_o,B5: set_set_o,C4: set_o > $o > $o,D2: set_o > $o > $o,Sup: set_o_o > $o > $o] :
      ( ( A2 = B5 )
     => ( ! [X5: set_o] :
            ( ( member_set_o @ X5 @ B5 )
           => ( ( C4 @ X5 )
              = ( D2 @ X5 ) ) )
       => ( ( Sup @ ( image_set_o_o_o @ C4 @ A2 ) )
          = ( Sup @ ( image_set_o_o_o @ D2 @ B5 ) ) ) ) ) ).

% Sup.SUP_cong
thf(fact_646_Inf_OINF__cong,axiom,
    ! [A2: set_o_o,B5: set_o_o,C4: ( $o > $o ) > set_o,D2: ( $o > $o ) > set_o,Inf: set_set_o > set_o] :
      ( ( A2 = B5 )
     => ( ! [X5: $o > $o] :
            ( ( member_o_o @ X5 @ B5 )
           => ( ( C4 @ X5 )
              = ( D2 @ X5 ) ) )
       => ( ( Inf @ ( image_o_o_set_o @ C4 @ A2 ) )
          = ( Inf @ ( image_o_o_set_o @ D2 @ B5 ) ) ) ) ) ).

% Inf.INF_cong
thf(fact_647_Inf_OINF__cong,axiom,
    ! [A2: set_o,B5: set_o,C4: $o > nat,D2: $o > nat,Inf: set_nat > nat] :
      ( ( A2 = B5 )
     => ( ! [X5: $o] :
            ( ( member_o @ X5 @ B5 )
           => ( ( C4 @ X5 )
              = ( D2 @ X5 ) ) )
       => ( ( Inf @ ( image_o_nat @ C4 @ A2 ) )
          = ( Inf @ ( image_o_nat @ D2 @ B5 ) ) ) ) ) ).

% Inf.INF_cong
thf(fact_648_Inf_OINF__cong,axiom,
    ! [A2: set_set_o,B5: set_set_o,C4: set_o > $o,D2: set_o > $o,Inf: set_o > $o] :
      ( ( A2 = B5 )
     => ( ! [X5: set_o] :
            ( ( member_set_o @ X5 @ B5 )
           => ( ( C4 @ X5 )
              = ( D2 @ X5 ) ) )
       => ( ( Inf @ ( image_set_o_o @ C4 @ A2 ) )
          = ( Inf @ ( image_set_o_o @ D2 @ B5 ) ) ) ) ) ).

% Inf.INF_cong
thf(fact_649_Inf_OINF__cong,axiom,
    ! [A2: set_set_o,B5: set_set_o,C4: set_o > set_o,D2: set_o > set_o,Inf: set_set_o > set_o] :
      ( ( A2 = B5 )
     => ( ! [X5: set_o] :
            ( ( member_set_o @ X5 @ B5 )
           => ( ( C4 @ X5 )
              = ( D2 @ X5 ) ) )
       => ( ( Inf @ ( image_set_o_set_o @ C4 @ A2 ) )
          = ( Inf @ ( image_set_o_set_o @ D2 @ B5 ) ) ) ) ) ).

% Inf.INF_cong
thf(fact_650_Inf_OINF__cong,axiom,
    ! [A2: set_set_o,B5: set_set_o,C4: set_o > $o > $o,D2: set_o > $o > $o,Inf: set_o_o > $o > $o] :
      ( ( A2 = B5 )
     => ( ! [X5: set_o] :
            ( ( member_set_o @ X5 @ B5 )
           => ( ( C4 @ X5 )
              = ( D2 @ X5 ) ) )
       => ( ( Inf @ ( image_set_o_o_o @ C4 @ A2 ) )
          = ( Inf @ ( image_set_o_o_o @ D2 @ B5 ) ) ) ) ) ).

% Inf.INF_cong
thf(fact_651_eval__term_Oelims,axiom,
    ! [X3: nat > a,Xa: relational_term_a,Y: a] :
      ( ( ( relati1177013128715261720term_a @ X3 @ Xa )
        = Y )
     => ( ! [C2: a] :
            ( ( Xa
              = ( relational_Const_a @ C2 ) )
           => ( Y != C2 ) )
       => ~ ! [N2: nat] :
              ( ( Xa
                = ( relational_Var_a @ N2 ) )
             => ( Y
               != ( X3 @ N2 ) ) ) ) ) ).

% eval_term.elims
thf(fact_652_sat_Ocases,axiom,
    ! [X3: produc1132964494702330949_nat_a] :
      ( ! [R2: b,Ts2: list_R6823256787227418703term_a,I: product_prod_b_nat > set_list_a,Sigma2: nat > a] :
          ( X3
         != ( produc6598558901832717687_nat_a @ ( relational_Pred_b_a @ R2 @ Ts2 ) @ ( produc2895298938842563487_nat_a @ I @ Sigma2 ) ) )
     => ( ! [B2: $o,I: product_prod_b_nat > set_list_a,Sigma2: nat > a] :
            ( X3
           != ( produc6598558901832717687_nat_a @ ( relational_Bool_a_b @ B2 ) @ ( produc2895298938842563487_nat_a @ I @ Sigma2 ) ) )
       => ( ! [X5: nat,T4: relational_term_a,I: product_prod_b_nat > set_list_a,Sigma2: nat > a] :
              ( X3
             != ( produc6598558901832717687_nat_a @ ( relational_Eq_a_b @ X5 @ T4 ) @ ( produc2895298938842563487_nat_a @ I @ Sigma2 ) ) )
         => ( ! [Phi2: relational_fmla_a_b,I: product_prod_b_nat > set_list_a,Sigma2: nat > a] :
                ( X3
               != ( produc6598558901832717687_nat_a @ ( relational_Neg_a_b @ Phi2 ) @ ( produc2895298938842563487_nat_a @ I @ Sigma2 ) ) )
           => ( ! [Phi2: relational_fmla_a_b,Psi: relational_fmla_a_b,I: product_prod_b_nat > set_list_a,Sigma2: nat > a] :
                  ( X3
                 != ( produc6598558901832717687_nat_a @ ( relational_Conj_a_b @ Phi2 @ Psi ) @ ( produc2895298938842563487_nat_a @ I @ Sigma2 ) ) )
             => ( ! [Phi2: relational_fmla_a_b,Psi: relational_fmla_a_b,I: product_prod_b_nat > set_list_a,Sigma2: nat > a] :
                    ( X3
                   != ( produc6598558901832717687_nat_a @ ( relational_Disj_a_b @ Phi2 @ Psi ) @ ( produc2895298938842563487_nat_a @ I @ Sigma2 ) ) )
               => ~ ! [Z2: nat,Phi2: relational_fmla_a_b,I: product_prod_b_nat > set_list_a,Sigma2: nat > a] :
                      ( X3
                     != ( produc6598558901832717687_nat_a @ ( relati591517084277583526ts_a_b @ Z2 @ Phi2 ) @ ( produc2895298938842563487_nat_a @ I @ Sigma2 ) ) ) ) ) ) ) ) ) ).

% sat.cases
thf(fact_653_cpropagated__nocp,axiom,
    ! [Q: relational_fmla_a_b,X3: nat] :
      ( ( relati1591879772219623554ed_a_b @ Q )
     => ( ( member_nat @ X3 @ ( relational_fv_a_b @ Q ) )
       => ( relational_nocp_a_b @ Q ) ) ) ).

% cpropagated_nocp
thf(fact_654_subst_Ocases,axiom,
    ! [X3: produc8867654947514737559at_nat] :
      ( ! [T3: $o,X5: nat,Y3: nat] :
          ( X3
         != ( produc6913411929637712585at_nat @ ( relational_Bool_a_b @ T3 ) @ ( product_Pair_nat_nat @ X5 @ Y3 ) ) )
     => ( ! [P4: b,Ts2: list_R6823256787227418703term_a,X5: nat,Y3: nat] :
            ( X3
           != ( produc6913411929637712585at_nat @ ( relational_Pred_b_a @ P4 @ Ts2 ) @ ( product_Pair_nat_nat @ X5 @ Y3 ) ) )
       => ( ! [Z2: nat,T3: relational_term_a,X5: nat,Y3: nat] :
              ( X3
             != ( produc6913411929637712585at_nat @ ( relational_Eq_a_b @ Z2 @ T3 ) @ ( product_Pair_nat_nat @ X5 @ Y3 ) ) )
         => ( ! [Q3: relational_fmla_a_b,X5: nat,Y3: nat] :
                ( X3
               != ( produc6913411929637712585at_nat @ ( relational_Neg_a_b @ Q3 ) @ ( product_Pair_nat_nat @ X5 @ Y3 ) ) )
           => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b,X5: nat,Y3: nat] :
                  ( X3
                 != ( produc6913411929637712585at_nat @ ( relational_Conj_a_b @ Q13 @ Q24 ) @ ( product_Pair_nat_nat @ X5 @ Y3 ) ) )
             => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b,X5: nat,Y3: nat] :
                    ( X3
                   != ( produc6913411929637712585at_nat @ ( relational_Disj_a_b @ Q13 @ Q24 ) @ ( product_Pair_nat_nat @ X5 @ Y3 ) ) )
               => ~ ! [Z2: nat,Q3: relational_fmla_a_b,X5: nat,Y3: nat] :
                      ( X3
                     != ( produc6913411929637712585at_nat @ ( relati591517084277583526ts_a_b @ Z2 @ Q3 ) @ ( product_Pair_nat_nat @ X5 @ Y3 ) ) ) ) ) ) ) ) ) ).

% subst.cases
thf(fact_655_subst_Opelims,axiom,
    ! [X3: relational_fmla_a_b,Xa: nat,Xb: nat,Y: relational_fmla_a_b] :
      ( ( ( relational_subst_a_b @ X3 @ Xa @ Xb )
        = Y )
     => ( ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ X3 @ ( product_Pair_nat_nat @ Xa @ Xb ) ) )
       => ( ! [T3: $o] :
              ( ( X3
                = ( relational_Bool_a_b @ T3 ) )
             => ( ( Y
                  = ( relational_Bool_a_b @ T3 ) )
               => ~ ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relational_Bool_a_b @ T3 ) @ ( product_Pair_nat_nat @ Xa @ Xb ) ) ) ) )
         => ( ! [P4: b,Ts2: list_R6823256787227418703term_a] :
                ( ( X3
                  = ( relational_Pred_b_a @ P4 @ Ts2 ) )
               => ( ( Y
                    = ( relational_Pred_b_a @ P4
                      @ ( map_Re5736185711816362116term_a
                        @ ^ [T2: relational_term_a] : ( relati7175845559408349773term_a @ T2 @ Xa @ Xb )
                        @ Ts2 ) ) )
                 => ~ ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relational_Pred_b_a @ P4 @ Ts2 ) @ ( product_Pair_nat_nat @ Xa @ Xb ) ) ) ) )
           => ( ! [Z2: nat,T3: relational_term_a] :
                  ( ( X3
                    = ( relational_Eq_a_b @ Z2 @ T3 ) )
                 => ( ( Y
                      = ( relational_Eq_a_b @ ( if_nat @ ( Z2 = Xa ) @ Xb @ Z2 ) @ ( relati7175845559408349773term_a @ T3 @ Xa @ Xb ) ) )
                   => ~ ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relational_Eq_a_b @ Z2 @ T3 ) @ ( product_Pair_nat_nat @ Xa @ Xb ) ) ) ) )
             => ( ! [Q3: relational_fmla_a_b] :
                    ( ( X3
                      = ( relational_Neg_a_b @ Q3 ) )
                   => ( ( Y
                        = ( relational_Neg_a_b @ ( relational_subst_a_b @ Q3 @ Xa @ Xb ) ) )
                     => ~ ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relational_Neg_a_b @ Q3 ) @ ( product_Pair_nat_nat @ Xa @ Xb ) ) ) ) )
               => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                      ( ( X3
                        = ( relational_Conj_a_b @ Q13 @ Q24 ) )
                     => ( ( Y
                          = ( relational_Conj_a_b @ ( relational_subst_a_b @ Q13 @ Xa @ Xb ) @ ( relational_subst_a_b @ Q24 @ Xa @ Xb ) ) )
                       => ~ ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relational_Conj_a_b @ Q13 @ Q24 ) @ ( product_Pair_nat_nat @ Xa @ Xb ) ) ) ) )
                 => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                        ( ( X3
                          = ( relational_Disj_a_b @ Q13 @ Q24 ) )
                       => ( ( Y
                            = ( relational_Disj_a_b @ ( relational_subst_a_b @ Q13 @ Xa @ Xb ) @ ( relational_subst_a_b @ Q24 @ Xa @ Xb ) ) )
                         => ~ ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relational_Disj_a_b @ Q13 @ Q24 ) @ ( product_Pair_nat_nat @ Xa @ Xb ) ) ) ) )
                   => ~ ! [Z2: nat,Q3: relational_fmla_a_b] :
                          ( ( X3
                            = ( relati591517084277583526ts_a_b @ Z2 @ Q3 ) )
                         => ( ( ( ( Xa = Z2 )
                               => ( Y
                                  = ( relati591517084277583526ts_a_b @ Xa @ Q3 ) ) )
                              & ( ( Xa != Z2 )
                               => ( ( ( Z2 = Xb )
                                   => ( Y
                                      = ( relati591517084277583526ts_a_b @ ( relati2677767559083392098h2_a_b @ Xa @ Xb @ Q3 ) @ ( relational_subst_a_b @ ( relational_subst_a_b @ Q3 @ Z2 @ ( relati2677767559083392098h2_a_b @ Xa @ Xb @ Q3 ) ) @ Xa @ Xb ) ) ) )
                                  & ( ( Z2 != Xb )
                                   => ( Y
                                      = ( relati591517084277583526ts_a_b @ Z2 @ ( relational_subst_a_b @ Q3 @ Xa @ Xb ) ) ) ) ) ) )
                           => ~ ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relati591517084277583526ts_a_b @ Z2 @ Q3 ) @ ( product_Pair_nat_nat @ Xa @ Xb ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% subst.pelims
thf(fact_656_split__cong,axiom,
    ! [Q4: produc7248412053542808358at_nat,F2: nat > product_prod_nat_nat > produc7248412053542808358at_nat,G2: nat > product_prod_nat_nat > produc7248412053542808358at_nat,P2: produc7248412053542808358at_nat] :
      ( ! [X5: nat,Y3: product_prod_nat_nat] :
          ( ( ( produc487386426758144856at_nat @ X5 @ Y3 )
            = Q4 )
         => ( ( F2 @ X5 @ Y3 )
            = ( G2 @ X5 @ Y3 ) ) )
     => ( ( P2 = Q4 )
       => ( ( produc968775922737392939at_nat @ F2 @ P2 )
          = ( produc968775922737392939at_nat @ G2 @ Q4 ) ) ) ) ).

% split_cong
thf(fact_657_split__cong,axiom,
    ! [Q4: product_prod_nat_nat,F2: nat > nat > produc7248412053542808358at_nat,G2: nat > nat > produc7248412053542808358at_nat,P2: product_prod_nat_nat] :
      ( ! [X5: nat,Y3: nat] :
          ( ( ( product_Pair_nat_nat @ X5 @ Y3 )
            = Q4 )
         => ( ( F2 @ X5 @ Y3 )
            = ( G2 @ X5 @ Y3 ) ) )
     => ( ( P2 = Q4 )
       => ( ( produc9083241971206738548at_nat @ F2 @ P2 )
          = ( produc9083241971206738548at_nat @ G2 @ Q4 ) ) ) ) ).

% split_cong
thf(fact_658_split__cong,axiom,
    ! [Q4: product_prod_nat_nat,F2: nat > nat > product_prod_nat_nat,G2: nat > nat > product_prod_nat_nat,P2: product_prod_nat_nat] :
      ( ! [X5: nat,Y3: nat] :
          ( ( ( product_Pair_nat_nat @ X5 @ Y3 )
            = Q4 )
         => ( ( F2 @ X5 @ Y3 )
            = ( G2 @ X5 @ Y3 ) ) )
     => ( ( P2 = Q4 )
       => ( ( produc2626176000494625587at_nat @ F2 @ P2 )
          = ( produc2626176000494625587at_nat @ G2 @ Q4 ) ) ) ) ).

% split_cong
thf(fact_659_split__cong,axiom,
    ! [Q4: product_prod_nat_nat,F2: nat > nat > relational_term_a > relational_term_a,G2: nat > nat > relational_term_a > relational_term_a,P2: product_prod_nat_nat] :
      ( ! [X5: nat,Y3: nat] :
          ( ( ( product_Pair_nat_nat @ X5 @ Y3 )
            = Q4 )
         => ( ( F2 @ X5 @ Y3 )
            = ( G2 @ X5 @ Y3 ) ) )
     => ( ( P2 = Q4 )
       => ( ( produc6628518323692928499term_a @ F2 @ P2 )
          = ( produc6628518323692928499term_a @ G2 @ Q4 ) ) ) ) ).

% split_cong
thf(fact_660_split__cong,axiom,
    ! [Q4: product_prod_nat_nat,F2: nat > nat > nat > produc7248412053542808358at_nat,G2: nat > nat > nat > produc7248412053542808358at_nat,P2: product_prod_nat_nat] :
      ( ! [X5: nat,Y3: nat] :
          ( ( ( product_Pair_nat_nat @ X5 @ Y3 )
            = Q4 )
         => ( ( F2 @ X5 @ Y3 )
            = ( G2 @ X5 @ Y3 ) ) )
     => ( ( P2 = Q4 )
       => ( ( produc7810592499157111267at_nat @ F2 @ P2 )
          = ( produc7810592499157111267at_nat @ G2 @ Q4 ) ) ) ) ).

% split_cong
thf(fact_661_split__cong,axiom,
    ! [Q4: product_prod_nat_nat,F2: nat > nat > relational_fmla_a_b > relational_fmla_a_b,G2: nat > nat > relational_fmla_a_b > relational_fmla_a_b,P2: product_prod_nat_nat] :
      ( ! [X5: nat,Y3: nat] :
          ( ( ( product_Pair_nat_nat @ X5 @ Y3 )
            = Q4 )
         => ( ( F2 @ X5 @ Y3 )
            = ( G2 @ X5 @ Y3 ) ) )
     => ( ( P2 = Q4 )
       => ( ( produc5586541307551673003la_a_b @ F2 @ P2 )
          = ( produc5586541307551673003la_a_b @ G2 @ Q4 ) ) ) ) ).

% split_cong
thf(fact_662_split__cong,axiom,
    ! [Q4: product_prod_nat_nat,F2: nat > nat > relational_fmla_a_b,G2: nat > nat > relational_fmla_a_b,P2: product_prod_nat_nat] :
      ( ! [X5: nat,Y3: nat] :
          ( ( ( product_Pair_nat_nat @ X5 @ Y3 )
            = Q4 )
         => ( ( F2 @ X5 @ Y3 )
            = ( G2 @ X5 @ Y3 ) ) )
     => ( ( P2 = Q4 )
       => ( ( produc3270801013941088237la_a_b @ F2 @ P2 )
          = ( produc3270801013941088237la_a_b @ G2 @ Q4 ) ) ) ) ).

% split_cong
thf(fact_663_sat_Oelims_I1_J,axiom,
    ! [X3: relati9047081815478866374_a_nat,Xa: product_prod_nat_nat > set_list_a,Xb: nat > a,Y: $o] :
      ( ( ( relational_sat_a_nat @ X3 @ Xa @ Xb )
        = Y )
     => ( ! [R2: nat,Ts2: list_R6823256787227418703term_a] :
            ( ( X3
              = ( relati6362048942677509346_nat_a @ R2 @ Ts2 ) )
           => ( Y
              = ( ~ ( member_list_a @ ( relati4772805863405912879erms_a @ Xb @ Ts2 ) @ ( Xa @ ( product_Pair_nat_nat @ R2 @ ( size_s88622898042387131term_a @ Ts2 ) ) ) ) ) ) )
       => ( ! [B2: $o] :
              ( ( X3
                = ( relati9034565498597818939_a_nat @ B2 ) )
             => ( Y = (~ B2) ) )
         => ( ! [X5: nat,T4: relational_term_a] :
                ( ( X3
                  = ( relational_Eq_a_nat @ X5 @ T4 ) )
               => ( Y
                  = ( ( Xb @ X5 )
                   != ( relati1177013128715261720term_a @ Xb @ T4 ) ) ) )
           => ( ! [Phi2: relati9047081815478866374_a_nat] :
                  ( ( X3
                    = ( relational_Neg_a_nat @ Phi2 ) )
                 => ( Y
                    = ( relational_sat_a_nat @ Phi2 @ Xa @ Xb ) ) )
             => ( ! [Phi2: relati9047081815478866374_a_nat,Psi: relati9047081815478866374_a_nat] :
                    ( ( X3
                      = ( relati2542520632142267709_a_nat @ Phi2 @ Psi ) )
                   => ( Y
                      = ( ~ ( ( relational_sat_a_nat @ Phi2 @ Xa @ Xb )
                            & ( relational_sat_a_nat @ Psi @ Xa @ Xb ) ) ) ) )
               => ( ! [Phi2: relati9047081815478866374_a_nat,Psi: relati9047081815478866374_a_nat] :
                      ( ( X3
                        = ( relati9106205213788308809_a_nat @ Phi2 @ Psi ) )
                     => ( Y
                        = ( ~ ( ( relational_sat_a_nat @ Phi2 @ Xa @ Xb )
                              | ( relational_sat_a_nat @ Psi @ Xa @ Xb ) ) ) ) )
                 => ~ ! [Z2: nat,Phi2: relati9047081815478866374_a_nat] :
                        ( ( X3
                          = ( relati6314223733442460777_a_nat @ Z2 @ Phi2 ) )
                       => ( Y
                          = ( ~ ? [X: a] : ( relational_sat_a_nat @ Phi2 @ Xa @ ( fun_upd_nat_a @ Xb @ Z2 @ X ) ) ) ) ) ) ) ) ) ) ) ) ).

% sat.elims(1)
thf(fact_664_sat_Oelims_I1_J,axiom,
    ! [X3: relational_fmla_a_b,Xa: product_prod_b_nat > set_list_a,Xb: nat > a,Y: $o] :
      ( ( ( relational_sat_a_b @ X3 @ Xa @ Xb )
        = Y )
     => ( ! [R2: b,Ts2: list_R6823256787227418703term_a] :
            ( ( X3
              = ( relational_Pred_b_a @ R2 @ Ts2 ) )
           => ( Y
              = ( ~ ( member_list_a @ ( relati4772805863405912879erms_a @ Xb @ Ts2 ) @ ( Xa @ ( product_Pair_b_nat @ R2 @ ( size_s88622898042387131term_a @ Ts2 ) ) ) ) ) ) )
       => ( ! [B2: $o] :
              ( ( X3
                = ( relational_Bool_a_b @ B2 ) )
             => ( Y = (~ B2) ) )
         => ( ! [X5: nat,T4: relational_term_a] :
                ( ( X3
                  = ( relational_Eq_a_b @ X5 @ T4 ) )
               => ( Y
                  = ( ( Xb @ X5 )
                   != ( relati1177013128715261720term_a @ Xb @ T4 ) ) ) )
           => ( ! [Phi2: relational_fmla_a_b] :
                  ( ( X3
                    = ( relational_Neg_a_b @ Phi2 ) )
                 => ( Y
                    = ( relational_sat_a_b @ Phi2 @ Xa @ Xb ) ) )
             => ( ! [Phi2: relational_fmla_a_b,Psi: relational_fmla_a_b] :
                    ( ( X3
                      = ( relational_Conj_a_b @ Phi2 @ Psi ) )
                   => ( Y
                      = ( ~ ( ( relational_sat_a_b @ Phi2 @ Xa @ Xb )
                            & ( relational_sat_a_b @ Psi @ Xa @ Xb ) ) ) ) )
               => ( ! [Phi2: relational_fmla_a_b,Psi: relational_fmla_a_b] :
                      ( ( X3
                        = ( relational_Disj_a_b @ Phi2 @ Psi ) )
                     => ( Y
                        = ( ~ ( ( relational_sat_a_b @ Phi2 @ Xa @ Xb )
                              | ( relational_sat_a_b @ Psi @ Xa @ Xb ) ) ) ) )
                 => ~ ! [Z2: nat,Phi2: relational_fmla_a_b] :
                        ( ( X3
                          = ( relati591517084277583526ts_a_b @ Z2 @ Phi2 ) )
                       => ( Y
                          = ( ~ ? [X: a] : ( relational_sat_a_b @ Phi2 @ Xa @ ( fun_upd_nat_a @ Xb @ Z2 @ X ) ) ) ) ) ) ) ) ) ) ) ) ).

% sat.elims(1)
thf(fact_665_sat_Oelims_I2_J,axiom,
    ! [X3: relati9047081815478866374_a_nat,Xa: product_prod_nat_nat > set_list_a,Xb: nat > a] :
      ( ( relational_sat_a_nat @ X3 @ Xa @ Xb )
     => ( ! [R2: nat,Ts2: list_R6823256787227418703term_a] :
            ( ( X3
              = ( relati6362048942677509346_nat_a @ R2 @ Ts2 ) )
           => ~ ( member_list_a @ ( relati4772805863405912879erms_a @ Xb @ Ts2 ) @ ( Xa @ ( product_Pair_nat_nat @ R2 @ ( size_s88622898042387131term_a @ Ts2 ) ) ) ) )
       => ( ! [B2: $o] :
              ( ( X3
                = ( relati9034565498597818939_a_nat @ B2 ) )
             => ~ B2 )
         => ( ! [X5: nat,T4: relational_term_a] :
                ( ( X3
                  = ( relational_Eq_a_nat @ X5 @ T4 ) )
               => ( ( Xb @ X5 )
                 != ( relati1177013128715261720term_a @ Xb @ T4 ) ) )
           => ( ! [Phi2: relati9047081815478866374_a_nat] :
                  ( ( X3
                    = ( relational_Neg_a_nat @ Phi2 ) )
                 => ( relational_sat_a_nat @ Phi2 @ Xa @ Xb ) )
             => ( ! [Phi2: relati9047081815478866374_a_nat,Psi: relati9047081815478866374_a_nat] :
                    ( ( X3
                      = ( relati2542520632142267709_a_nat @ Phi2 @ Psi ) )
                   => ~ ( ( relational_sat_a_nat @ Phi2 @ Xa @ Xb )
                        & ( relational_sat_a_nat @ Psi @ Xa @ Xb ) ) )
               => ( ! [Phi2: relati9047081815478866374_a_nat,Psi: relati9047081815478866374_a_nat] :
                      ( ( X3
                        = ( relati9106205213788308809_a_nat @ Phi2 @ Psi ) )
                     => ~ ( ( relational_sat_a_nat @ Phi2 @ Xa @ Xb )
                          | ( relational_sat_a_nat @ Psi @ Xa @ Xb ) ) )
                 => ~ ! [Z2: nat,Phi2: relati9047081815478866374_a_nat] :
                        ( ( X3
                          = ( relati6314223733442460777_a_nat @ Z2 @ Phi2 ) )
                       => ~ ? [X5: a] : ( relational_sat_a_nat @ Phi2 @ Xa @ ( fun_upd_nat_a @ Xb @ Z2 @ X5 ) ) ) ) ) ) ) ) ) ) ).

% sat.elims(2)
thf(fact_666_sat_Oelims_I2_J,axiom,
    ! [X3: relational_fmla_a_b,Xa: product_prod_b_nat > set_list_a,Xb: nat > a] :
      ( ( relational_sat_a_b @ X3 @ Xa @ Xb )
     => ( ! [R2: b,Ts2: list_R6823256787227418703term_a] :
            ( ( X3
              = ( relational_Pred_b_a @ R2 @ Ts2 ) )
           => ~ ( member_list_a @ ( relati4772805863405912879erms_a @ Xb @ Ts2 ) @ ( Xa @ ( product_Pair_b_nat @ R2 @ ( size_s88622898042387131term_a @ Ts2 ) ) ) ) )
       => ( ! [B2: $o] :
              ( ( X3
                = ( relational_Bool_a_b @ B2 ) )
             => ~ B2 )
         => ( ! [X5: nat,T4: relational_term_a] :
                ( ( X3
                  = ( relational_Eq_a_b @ X5 @ T4 ) )
               => ( ( Xb @ X5 )
                 != ( relati1177013128715261720term_a @ Xb @ T4 ) ) )
           => ( ! [Phi2: relational_fmla_a_b] :
                  ( ( X3
                    = ( relational_Neg_a_b @ Phi2 ) )
                 => ( relational_sat_a_b @ Phi2 @ Xa @ Xb ) )
             => ( ! [Phi2: relational_fmla_a_b,Psi: relational_fmla_a_b] :
                    ( ( X3
                      = ( relational_Conj_a_b @ Phi2 @ Psi ) )
                   => ~ ( ( relational_sat_a_b @ Phi2 @ Xa @ Xb )
                        & ( relational_sat_a_b @ Psi @ Xa @ Xb ) ) )
               => ( ! [Phi2: relational_fmla_a_b,Psi: relational_fmla_a_b] :
                      ( ( X3
                        = ( relational_Disj_a_b @ Phi2 @ Psi ) )
                     => ~ ( ( relational_sat_a_b @ Phi2 @ Xa @ Xb )
                          | ( relational_sat_a_b @ Psi @ Xa @ Xb ) ) )
                 => ~ ! [Z2: nat,Phi2: relational_fmla_a_b] :
                        ( ( X3
                          = ( relati591517084277583526ts_a_b @ Z2 @ Phi2 ) )
                       => ~ ? [X5: a] : ( relational_sat_a_b @ Phi2 @ Xa @ ( fun_upd_nat_a @ Xb @ Z2 @ X5 ) ) ) ) ) ) ) ) ) ) ).

% sat.elims(2)
thf(fact_667_sat_Oelims_I3_J,axiom,
    ! [X3: relati9047081815478866374_a_nat,Xa: product_prod_nat_nat > set_list_a,Xb: nat > a] :
      ( ~ ( relational_sat_a_nat @ X3 @ Xa @ Xb )
     => ( ! [R2: nat,Ts2: list_R6823256787227418703term_a] :
            ( ( X3
              = ( relati6362048942677509346_nat_a @ R2 @ Ts2 ) )
           => ( member_list_a @ ( relati4772805863405912879erms_a @ Xb @ Ts2 ) @ ( Xa @ ( product_Pair_nat_nat @ R2 @ ( size_s88622898042387131term_a @ Ts2 ) ) ) ) )
       => ( ! [B2: $o] :
              ( ( X3
                = ( relati9034565498597818939_a_nat @ B2 ) )
             => B2 )
         => ( ! [X5: nat,T4: relational_term_a] :
                ( ( X3
                  = ( relational_Eq_a_nat @ X5 @ T4 ) )
               => ( ( Xb @ X5 )
                  = ( relati1177013128715261720term_a @ Xb @ T4 ) ) )
           => ( ! [Phi2: relati9047081815478866374_a_nat] :
                  ( ( X3
                    = ( relational_Neg_a_nat @ Phi2 ) )
                 => ~ ( relational_sat_a_nat @ Phi2 @ Xa @ Xb ) )
             => ( ! [Phi2: relati9047081815478866374_a_nat,Psi: relati9047081815478866374_a_nat] :
                    ( ( X3
                      = ( relati2542520632142267709_a_nat @ Phi2 @ Psi ) )
                   => ( ( relational_sat_a_nat @ Phi2 @ Xa @ Xb )
                      & ( relational_sat_a_nat @ Psi @ Xa @ Xb ) ) )
               => ( ! [Phi2: relati9047081815478866374_a_nat,Psi: relati9047081815478866374_a_nat] :
                      ( ( X3
                        = ( relati9106205213788308809_a_nat @ Phi2 @ Psi ) )
                     => ( ( relational_sat_a_nat @ Phi2 @ Xa @ Xb )
                        | ( relational_sat_a_nat @ Psi @ Xa @ Xb ) ) )
                 => ~ ! [Z2: nat,Phi2: relati9047081815478866374_a_nat] :
                        ( ( X3
                          = ( relati6314223733442460777_a_nat @ Z2 @ Phi2 ) )
                       => ? [X6: a] : ( relational_sat_a_nat @ Phi2 @ Xa @ ( fun_upd_nat_a @ Xb @ Z2 @ X6 ) ) ) ) ) ) ) ) ) ) ).

% sat.elims(3)
thf(fact_668_sat_Oelims_I3_J,axiom,
    ! [X3: relational_fmla_a_b,Xa: product_prod_b_nat > set_list_a,Xb: nat > a] :
      ( ~ ( relational_sat_a_b @ X3 @ Xa @ Xb )
     => ( ! [R2: b,Ts2: list_R6823256787227418703term_a] :
            ( ( X3
              = ( relational_Pred_b_a @ R2 @ Ts2 ) )
           => ( member_list_a @ ( relati4772805863405912879erms_a @ Xb @ Ts2 ) @ ( Xa @ ( product_Pair_b_nat @ R2 @ ( size_s88622898042387131term_a @ Ts2 ) ) ) ) )
       => ( ! [B2: $o] :
              ( ( X3
                = ( relational_Bool_a_b @ B2 ) )
             => B2 )
         => ( ! [X5: nat,T4: relational_term_a] :
                ( ( X3
                  = ( relational_Eq_a_b @ X5 @ T4 ) )
               => ( ( Xb @ X5 )
                  = ( relati1177013128715261720term_a @ Xb @ T4 ) ) )
           => ( ! [Phi2: relational_fmla_a_b] :
                  ( ( X3
                    = ( relational_Neg_a_b @ Phi2 ) )
                 => ~ ( relational_sat_a_b @ Phi2 @ Xa @ Xb ) )
             => ( ! [Phi2: relational_fmla_a_b,Psi: relational_fmla_a_b] :
                    ( ( X3
                      = ( relational_Conj_a_b @ Phi2 @ Psi ) )
                   => ( ( relational_sat_a_b @ Phi2 @ Xa @ Xb )
                      & ( relational_sat_a_b @ Psi @ Xa @ Xb ) ) )
               => ( ! [Phi2: relational_fmla_a_b,Psi: relational_fmla_a_b] :
                      ( ( X3
                        = ( relational_Disj_a_b @ Phi2 @ Psi ) )
                     => ( ( relational_sat_a_b @ Phi2 @ Xa @ Xb )
                        | ( relational_sat_a_b @ Psi @ Xa @ Xb ) ) )
                 => ~ ! [Z2: nat,Phi2: relational_fmla_a_b] :
                        ( ( X3
                          = ( relati591517084277583526ts_a_b @ Z2 @ Phi2 ) )
                       => ? [X6: a] : ( relational_sat_a_b @ Phi2 @ Xa @ ( fun_upd_nat_a @ Xb @ Z2 @ X6 ) ) ) ) ) ) ) ) ) ) ).

% sat.elims(3)
thf(fact_669_subst_Opinduct,axiom,
    ! [A0: relational_fmla_a_b,A1: nat,A22: nat,P: relational_fmla_a_b > nat > nat > $o] :
      ( ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ A0 @ ( product_Pair_nat_nat @ A1 @ A22 ) ) )
     => ( ! [T3: $o,X5: nat,Y3: nat] :
            ( ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relational_Bool_a_b @ T3 ) @ ( product_Pair_nat_nat @ X5 @ Y3 ) ) )
           => ( P @ ( relational_Bool_a_b @ T3 ) @ X5 @ Y3 ) )
       => ( ! [P4: b,Ts2: list_R6823256787227418703term_a,X5: nat,Y3: nat] :
              ( ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relational_Pred_b_a @ P4 @ Ts2 ) @ ( product_Pair_nat_nat @ X5 @ Y3 ) ) )
             => ( P @ ( relational_Pred_b_a @ P4 @ Ts2 ) @ X5 @ Y3 ) )
         => ( ! [Z2: nat,T3: relational_term_a,X5: nat,Y3: nat] :
                ( ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relational_Eq_a_b @ Z2 @ T3 ) @ ( product_Pair_nat_nat @ X5 @ Y3 ) ) )
               => ( P @ ( relational_Eq_a_b @ Z2 @ T3 ) @ X5 @ Y3 ) )
           => ( ! [Q3: relational_fmla_a_b,X5: nat,Y3: nat] :
                  ( ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relational_Neg_a_b @ Q3 ) @ ( product_Pair_nat_nat @ X5 @ Y3 ) ) )
                 => ( ( P @ Q3 @ X5 @ Y3 )
                   => ( P @ ( relational_Neg_a_b @ Q3 ) @ X5 @ Y3 ) ) )
             => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b,X5: nat,Y3: nat] :
                    ( ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relational_Conj_a_b @ Q13 @ Q24 ) @ ( product_Pair_nat_nat @ X5 @ Y3 ) ) )
                   => ( ( P @ Q13 @ X5 @ Y3 )
                     => ( ( P @ Q24 @ X5 @ Y3 )
                       => ( P @ ( relational_Conj_a_b @ Q13 @ Q24 ) @ X5 @ Y3 ) ) ) )
               => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b,X5: nat,Y3: nat] :
                      ( ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relational_Disj_a_b @ Q13 @ Q24 ) @ ( product_Pair_nat_nat @ X5 @ Y3 ) ) )
                     => ( ( P @ Q13 @ X5 @ Y3 )
                       => ( ( P @ Q24 @ X5 @ Y3 )
                         => ( P @ ( relational_Disj_a_b @ Q13 @ Q24 ) @ X5 @ Y3 ) ) ) )
                 => ( ! [Z2: nat,Q3: relational_fmla_a_b,X5: nat,Y3: nat] :
                        ( ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relati591517084277583526ts_a_b @ Z2 @ Q3 ) @ ( product_Pair_nat_nat @ X5 @ Y3 ) ) )
                       => ( ! [Xa2: nat] :
                              ( ( X5 != Z2 )
                             => ( ( Z2 = Y3 )
                               => ( ( Xa2
                                    = ( relati2677767559083392098h2_a_b @ X5 @ Y3 @ Q3 ) )
                                 => ( P @ Q3 @ Z2 @ Xa2 ) ) ) )
                         => ( ! [Xa2: nat] :
                                ( ( X5 != Z2 )
                               => ( ( Z2 = Y3 )
                                 => ( ( Xa2
                                      = ( relati2677767559083392098h2_a_b @ X5 @ Y3 @ Q3 ) )
                                   => ( P @ ( relational_subst_a_b @ Q3 @ Z2 @ Xa2 ) @ X5 @ Y3 ) ) ) )
                           => ( ( ( X5 != Z2 )
                               => ( ( Z2 != Y3 )
                                 => ( P @ Q3 @ X5 @ Y3 ) ) )
                             => ( P @ ( relati591517084277583526ts_a_b @ Z2 @ Q3 ) @ X5 @ Y3 ) ) ) ) )
                   => ( P @ A0 @ A1 @ A22 ) ) ) ) ) ) ) ) ) ).

% subst.pinduct
thf(fact_670_sat__cp,axiom,
    ! [Q: relational_fmla_a_b,I2: product_prod_b_nat > set_list_a,Sigma: nat > a] :
      ( ( relational_sat_a_b @ ( relational_cp_a_b @ Q ) @ I2 @ Sigma )
      = ( relational_sat_a_b @ Q @ I2 @ Sigma ) ) ).

% sat_cp
thf(fact_671_sat__subst,axiom,
    ! [Q: relational_fmla_a_b,X3: nat,Y: nat,I2: product_prod_b_nat > set_list_a,Sigma: nat > a] :
      ( ( relational_sat_a_b @ ( relational_subst_a_b @ Q @ X3 @ Y ) @ I2 @ Sigma )
      = ( relational_sat_a_b @ Q @ I2 @ ( fun_upd_nat_a @ Sigma @ X3 @ ( Sigma @ Y ) ) ) ) ).

% sat_subst
thf(fact_672_size__subst__p,axiom,
    ! [Q: relational_fmla_a_b,X3: nat,Y: nat] :
      ( ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ Q @ ( product_Pair_nat_nat @ X3 @ Y ) ) )
     => ( ( size_s453432777765377587la_a_b @ ( relational_subst_a_b @ Q @ X3 @ Y ) )
        = ( size_s453432777765377587la_a_b @ Q ) ) ) ).

% size_subst_p
thf(fact_673_sat_Osimps_I1_J,axiom,
    ! [R3: nat,Ts3: list_R6823256787227418703term_a,I2: product_prod_nat_nat > set_list_a,Sigma: nat > a] :
      ( ( relational_sat_a_nat @ ( relati6362048942677509346_nat_a @ R3 @ Ts3 ) @ I2 @ Sigma )
      = ( member_list_a @ ( relati4772805863405912879erms_a @ Sigma @ Ts3 ) @ ( I2 @ ( product_Pair_nat_nat @ R3 @ ( size_s88622898042387131term_a @ Ts3 ) ) ) ) ) ).

% sat.simps(1)
thf(fact_674_sat_Osimps_I1_J,axiom,
    ! [R3: b,Ts3: list_R6823256787227418703term_a,I2: product_prod_b_nat > set_list_a,Sigma: nat > a] :
      ( ( relational_sat_a_b @ ( relational_Pred_b_a @ R3 @ Ts3 ) @ I2 @ Sigma )
      = ( member_list_a @ ( relati4772805863405912879erms_a @ Sigma @ Ts3 ) @ ( I2 @ ( product_Pair_b_nat @ R3 @ ( size_s88622898042387131term_a @ Ts3 ) ) ) ) ) ).

% sat.simps(1)
thf(fact_675_sat_Osimps_I4_J,axiom,
    ! [Phi: relational_fmla_a_b,I2: product_prod_b_nat > set_list_a,Sigma: nat > a] :
      ( ( relational_sat_a_b @ ( relational_Neg_a_b @ Phi ) @ I2 @ Sigma )
      = ( ~ ( relational_sat_a_b @ Phi @ I2 @ Sigma ) ) ) ).

% sat.simps(4)
thf(fact_676_sat_Osimps_I6_J,axiom,
    ! [Phi: relational_fmla_a_b,Psi2: relational_fmla_a_b,I2: product_prod_b_nat > set_list_a,Sigma: nat > a] :
      ( ( relational_sat_a_b @ ( relational_Disj_a_b @ Phi @ Psi2 ) @ I2 @ Sigma )
      = ( ( relational_sat_a_b @ Phi @ I2 @ Sigma )
        | ( relational_sat_a_b @ Psi2 @ I2 @ Sigma ) ) ) ).

% sat.simps(6)
thf(fact_677_sat_Osimps_I5_J,axiom,
    ! [Phi: relational_fmla_a_b,Psi2: relational_fmla_a_b,I2: product_prod_b_nat > set_list_a,Sigma: nat > a] :
      ( ( relational_sat_a_b @ ( relational_Conj_a_b @ Phi @ Psi2 ) @ I2 @ Sigma )
      = ( ( relational_sat_a_b @ Phi @ I2 @ Sigma )
        & ( relational_sat_a_b @ Psi2 @ I2 @ Sigma ) ) ) ).

% sat.simps(5)
thf(fact_678_sat_Osimps_I2_J,axiom,
    ! [B: $o,I2: product_prod_b_nat > set_list_a,Sigma: nat > a] :
      ( ( relational_sat_a_b @ ( relational_Bool_a_b @ B ) @ I2 @ Sigma )
      = B ) ).

% sat.simps(2)
thf(fact_679_sat__fv__cong,axiom,
    ! [Phi: relational_fmla_a_b,Sigma: nat > a,Sigma3: nat > a,I2: product_prod_b_nat > set_list_a] :
      ( ! [N2: nat] :
          ( ( member_nat @ N2 @ ( relational_fv_a_b @ Phi ) )
         => ( ( Sigma @ N2 )
            = ( Sigma3 @ N2 ) ) )
     => ( ( relational_sat_a_b @ Phi @ I2 @ Sigma )
        = ( relational_sat_a_b @ Phi @ I2 @ Sigma3 ) ) ) ).

% sat_fv_cong
thf(fact_680_subst_Opsimps_I4_J,axiom,
    ! [Q: relational_fmla_a_b,X3: nat,Y: nat] :
      ( ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relational_Neg_a_b @ Q ) @ ( product_Pair_nat_nat @ X3 @ Y ) ) )
     => ( ( relational_subst_a_b @ ( relational_Neg_a_b @ Q ) @ X3 @ Y )
        = ( relational_Neg_a_b @ ( relational_subst_a_b @ Q @ X3 @ Y ) ) ) ) ).

% subst.psimps(4)
thf(fact_681_subst_Opsimps_I6_J,axiom,
    ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b,X3: nat,Y: nat] :
      ( ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relational_Disj_a_b @ Q1 @ Q22 ) @ ( product_Pair_nat_nat @ X3 @ Y ) ) )
     => ( ( relational_subst_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) @ X3 @ Y )
        = ( relational_Disj_a_b @ ( relational_subst_a_b @ Q1 @ X3 @ Y ) @ ( relational_subst_a_b @ Q22 @ X3 @ Y ) ) ) ) ).

% subst.psimps(6)
thf(fact_682_subst_Opsimps_I5_J,axiom,
    ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b,X3: nat,Y: nat] :
      ( ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relational_Conj_a_b @ Q1 @ Q22 ) @ ( product_Pair_nat_nat @ X3 @ Y ) ) )
     => ( ( relational_subst_a_b @ ( relational_Conj_a_b @ Q1 @ Q22 ) @ X3 @ Y )
        = ( relational_Conj_a_b @ ( relational_subst_a_b @ Q1 @ X3 @ Y ) @ ( relational_subst_a_b @ Q22 @ X3 @ Y ) ) ) ) ).

% subst.psimps(5)
thf(fact_683_subst_Opsimps_I1_J,axiom,
    ! [T: $o,X3: nat,Y: nat] :
      ( ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relational_Bool_a_b @ T ) @ ( product_Pair_nat_nat @ X3 @ Y ) ) )
     => ( ( relational_subst_a_b @ ( relational_Bool_a_b @ T ) @ X3 @ Y )
        = ( relational_Bool_a_b @ T ) ) ) ).

% subst.psimps(1)
thf(fact_684_subst_Opsimps_I3_J,axiom,
    ! [Z: nat,T: relational_term_a,X3: nat,Y: nat] :
      ( ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relational_Eq_a_b @ Z @ T ) @ ( product_Pair_nat_nat @ X3 @ Y ) ) )
     => ( ( relational_subst_a_b @ ( relational_Eq_a_b @ Z @ T ) @ X3 @ Y )
        = ( relational_Eq_a_b @ ( if_nat @ ( Z = X3 ) @ Y @ Z ) @ ( relati7175845559408349773term_a @ T @ X3 @ Y ) ) ) ) ).

% subst.psimps(3)
thf(fact_685_subst_Opsimps_I7_J,axiom,
    ! [Z: nat,Q: relational_fmla_a_b,X3: nat,Y: nat] :
      ( ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relati591517084277583526ts_a_b @ Z @ Q ) @ ( product_Pair_nat_nat @ X3 @ Y ) ) )
     => ( ( ( X3 = Z )
         => ( ( relational_subst_a_b @ ( relati591517084277583526ts_a_b @ Z @ Q ) @ X3 @ Y )
            = ( relati591517084277583526ts_a_b @ X3 @ Q ) ) )
        & ( ( X3 != Z )
         => ( ( ( Z = Y )
             => ( ( relational_subst_a_b @ ( relati591517084277583526ts_a_b @ Z @ Q ) @ X3 @ Y )
                = ( relati591517084277583526ts_a_b @ ( relati2677767559083392098h2_a_b @ X3 @ Y @ Q ) @ ( relational_subst_a_b @ ( relational_subst_a_b @ Q @ Z @ ( relati2677767559083392098h2_a_b @ X3 @ Y @ Q ) ) @ X3 @ Y ) ) ) )
            & ( ( Z != Y )
             => ( ( relational_subst_a_b @ ( relati591517084277583526ts_a_b @ Z @ Q ) @ X3 @ Y )
                = ( relati591517084277583526ts_a_b @ Z @ ( relational_subst_a_b @ Q @ X3 @ Y ) ) ) ) ) ) ) ) ).

% subst.psimps(7)
thf(fact_686_sat_Osimps_I3_J,axiom,
    ! [X3: nat,T5: relational_term_a,I2: product_prod_b_nat > set_list_a,Sigma: nat > a] :
      ( ( relational_sat_a_b @ ( relational_Eq_a_b @ X3 @ T5 ) @ I2 @ Sigma )
      = ( ( Sigma @ X3 )
        = ( relati1177013128715261720term_a @ Sigma @ T5 ) ) ) ).

% sat.simps(3)
thf(fact_687_sat_Osimps_I7_J,axiom,
    ! [Z: nat,Phi: relational_fmla_a_b,I2: product_prod_b_nat > set_list_a,Sigma: nat > a] :
      ( ( relational_sat_a_b @ ( relati591517084277583526ts_a_b @ Z @ Phi ) @ I2 @ Sigma )
      = ( ? [X: a] : ( relational_sat_a_b @ Phi @ I2 @ ( fun_upd_nat_a @ Sigma @ Z @ X ) ) ) ) ).

% sat.simps(7)
thf(fact_688_sat__fun__upd,axiom,
    ! [N: nat,Q: relational_fmla_a_b,I2: product_prod_b_nat > set_list_a,Sigma: nat > a,Z: a] :
      ( ~ ( member_nat @ N @ ( relational_fv_a_b @ Q ) )
     => ( ( relational_sat_a_b @ Q @ I2 @ ( fun_upd_nat_a @ Sigma @ N @ Z ) )
        = ( relational_sat_a_b @ Q @ I2 @ Sigma ) ) ) ).

% sat_fun_upd
thf(fact_689_subst_Opsimps_I2_J,axiom,
    ! [P2: b,Ts3: list_R6823256787227418703term_a,X3: nat,Y: nat] :
      ( ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relational_Pred_b_a @ P2 @ Ts3 ) @ ( product_Pair_nat_nat @ X3 @ Y ) ) )
     => ( ( relational_subst_a_b @ ( relational_Pred_b_a @ P2 @ Ts3 ) @ X3 @ Y )
        = ( relational_Pred_b_a @ P2
          @ ( map_Re5736185711816362116term_a
            @ ^ [T2: relational_term_a] : ( relati7175845559408349773term_a @ T2 @ X3 @ Y )
            @ Ts3 ) ) ) ) ).

% subst.psimps(2)
thf(fact_690_eval__terms__def,axiom,
    ( relati4772805863405912879erms_a
    = ( ^ [Sigma4: nat > a] : ( map_Re419313091343012409rm_a_a @ ( relati1177013128715261720term_a @ Sigma4 ) ) ) ) ).

% eval_terms_def
thf(fact_691_sat_Opelims_I1_J,axiom,
    ! [X3: relati9047081815478866374_a_nat,Xa: product_prod_nat_nat > set_list_a,Xb: nat > a,Y: $o] :
      ( ( ( relational_sat_a_nat @ X3 @ Xa @ Xb )
        = Y )
     => ( ( accp_P3115975753873020414_nat_a @ relati9040413291598363742_a_nat @ ( produc2574091134533170497_nat_a @ X3 @ ( produc4953239424914856836_nat_a @ Xa @ Xb ) ) )
       => ( ! [R2: nat,Ts2: list_R6823256787227418703term_a] :
              ( ( X3
                = ( relati6362048942677509346_nat_a @ R2 @ Ts2 ) )
             => ( ( Y
                  = ( member_list_a @ ( relati4772805863405912879erms_a @ Xb @ Ts2 ) @ ( Xa @ ( product_Pair_nat_nat @ R2 @ ( size_s88622898042387131term_a @ Ts2 ) ) ) ) )
               => ~ ( accp_P3115975753873020414_nat_a @ relati9040413291598363742_a_nat @ ( produc2574091134533170497_nat_a @ ( relati6362048942677509346_nat_a @ R2 @ Ts2 ) @ ( produc4953239424914856836_nat_a @ Xa @ Xb ) ) ) ) )
         => ( ! [B2: $o] :
                ( ( X3
                  = ( relati9034565498597818939_a_nat @ B2 ) )
               => ( ( Y = B2 )
                 => ~ ( accp_P3115975753873020414_nat_a @ relati9040413291598363742_a_nat @ ( produc2574091134533170497_nat_a @ ( relati9034565498597818939_a_nat @ B2 ) @ ( produc4953239424914856836_nat_a @ Xa @ Xb ) ) ) ) )
           => ( ! [X5: nat,T4: relational_term_a] :
                  ( ( X3
                    = ( relational_Eq_a_nat @ X5 @ T4 ) )
                 => ( ( Y
                      = ( ( Xb @ X5 )
                        = ( relati1177013128715261720term_a @ Xb @ T4 ) ) )
                   => ~ ( accp_P3115975753873020414_nat_a @ relati9040413291598363742_a_nat @ ( produc2574091134533170497_nat_a @ ( relational_Eq_a_nat @ X5 @ T4 ) @ ( produc4953239424914856836_nat_a @ Xa @ Xb ) ) ) ) )
             => ( ! [Phi2: relati9047081815478866374_a_nat] :
                    ( ( X3
                      = ( relational_Neg_a_nat @ Phi2 ) )
                   => ( ( Y
                        = ( ~ ( relational_sat_a_nat @ Phi2 @ Xa @ Xb ) ) )
                     => ~ ( accp_P3115975753873020414_nat_a @ relati9040413291598363742_a_nat @ ( produc2574091134533170497_nat_a @ ( relational_Neg_a_nat @ Phi2 ) @ ( produc4953239424914856836_nat_a @ Xa @ Xb ) ) ) ) )
               => ( ! [Phi2: relati9047081815478866374_a_nat,Psi: relati9047081815478866374_a_nat] :
                      ( ( X3
                        = ( relati2542520632142267709_a_nat @ Phi2 @ Psi ) )
                     => ( ( Y
                          = ( ( relational_sat_a_nat @ Phi2 @ Xa @ Xb )
                            & ( relational_sat_a_nat @ Psi @ Xa @ Xb ) ) )
                       => ~ ( accp_P3115975753873020414_nat_a @ relati9040413291598363742_a_nat @ ( produc2574091134533170497_nat_a @ ( relati2542520632142267709_a_nat @ Phi2 @ Psi ) @ ( produc4953239424914856836_nat_a @ Xa @ Xb ) ) ) ) )
                 => ( ! [Phi2: relati9047081815478866374_a_nat,Psi: relati9047081815478866374_a_nat] :
                        ( ( X3
                          = ( relati9106205213788308809_a_nat @ Phi2 @ Psi ) )
                       => ( ( Y
                            = ( ( relational_sat_a_nat @ Phi2 @ Xa @ Xb )
                              | ( relational_sat_a_nat @ Psi @ Xa @ Xb ) ) )
                         => ~ ( accp_P3115975753873020414_nat_a @ relati9040413291598363742_a_nat @ ( produc2574091134533170497_nat_a @ ( relati9106205213788308809_a_nat @ Phi2 @ Psi ) @ ( produc4953239424914856836_nat_a @ Xa @ Xb ) ) ) ) )
                   => ~ ! [Z2: nat,Phi2: relati9047081815478866374_a_nat] :
                          ( ( X3
                            = ( relati6314223733442460777_a_nat @ Z2 @ Phi2 ) )
                         => ( ( Y
                              = ( ? [X: a] : ( relational_sat_a_nat @ Phi2 @ Xa @ ( fun_upd_nat_a @ Xb @ Z2 @ X ) ) ) )
                           => ~ ( accp_P3115975753873020414_nat_a @ relati9040413291598363742_a_nat @ ( produc2574091134533170497_nat_a @ ( relati6314223733442460777_a_nat @ Z2 @ Phi2 ) @ ( produc4953239424914856836_nat_a @ Xa @ Xb ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% sat.pelims(1)
thf(fact_692_sat_Opelims_I1_J,axiom,
    ! [X3: relational_fmla_a_b,Xa: product_prod_b_nat > set_list_a,Xb: nat > a,Y: $o] :
      ( ( ( relational_sat_a_b @ X3 @ Xa @ Xb )
        = Y )
     => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ X3 @ ( produc2895298938842563487_nat_a @ Xa @ Xb ) ) )
       => ( ! [R2: b,Ts2: list_R6823256787227418703term_a] :
              ( ( X3
                = ( relational_Pred_b_a @ R2 @ Ts2 ) )
             => ( ( Y
                  = ( member_list_a @ ( relati4772805863405912879erms_a @ Xb @ Ts2 ) @ ( Xa @ ( product_Pair_b_nat @ R2 @ ( size_s88622898042387131term_a @ Ts2 ) ) ) ) )
               => ~ ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Pred_b_a @ R2 @ Ts2 ) @ ( produc2895298938842563487_nat_a @ Xa @ Xb ) ) ) ) )
         => ( ! [B2: $o] :
                ( ( X3
                  = ( relational_Bool_a_b @ B2 ) )
               => ( ( Y = B2 )
                 => ~ ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Bool_a_b @ B2 ) @ ( produc2895298938842563487_nat_a @ Xa @ Xb ) ) ) ) )
           => ( ! [X5: nat,T4: relational_term_a] :
                  ( ( X3
                    = ( relational_Eq_a_b @ X5 @ T4 ) )
                 => ( ( Y
                      = ( ( Xb @ X5 )
                        = ( relati1177013128715261720term_a @ Xb @ T4 ) ) )
                   => ~ ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Eq_a_b @ X5 @ T4 ) @ ( produc2895298938842563487_nat_a @ Xa @ Xb ) ) ) ) )
             => ( ! [Phi2: relational_fmla_a_b] :
                    ( ( X3
                      = ( relational_Neg_a_b @ Phi2 ) )
                   => ( ( Y
                        = ( ~ ( relational_sat_a_b @ Phi2 @ Xa @ Xb ) ) )
                     => ~ ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Neg_a_b @ Phi2 ) @ ( produc2895298938842563487_nat_a @ Xa @ Xb ) ) ) ) )
               => ( ! [Phi2: relational_fmla_a_b,Psi: relational_fmla_a_b] :
                      ( ( X3
                        = ( relational_Conj_a_b @ Phi2 @ Psi ) )
                     => ( ( Y
                          = ( ( relational_sat_a_b @ Phi2 @ Xa @ Xb )
                            & ( relational_sat_a_b @ Psi @ Xa @ Xb ) ) )
                       => ~ ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Conj_a_b @ Phi2 @ Psi ) @ ( produc2895298938842563487_nat_a @ Xa @ Xb ) ) ) ) )
                 => ( ! [Phi2: relational_fmla_a_b,Psi: relational_fmla_a_b] :
                        ( ( X3
                          = ( relational_Disj_a_b @ Phi2 @ Psi ) )
                       => ( ( Y
                            = ( ( relational_sat_a_b @ Phi2 @ Xa @ Xb )
                              | ( relational_sat_a_b @ Psi @ Xa @ Xb ) ) )
                         => ~ ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Disj_a_b @ Phi2 @ Psi ) @ ( produc2895298938842563487_nat_a @ Xa @ Xb ) ) ) ) )
                   => ~ ! [Z2: nat,Phi2: relational_fmla_a_b] :
                          ( ( X3
                            = ( relati591517084277583526ts_a_b @ Z2 @ Phi2 ) )
                         => ( ( Y
                              = ( ? [X: a] : ( relational_sat_a_b @ Phi2 @ Xa @ ( fun_upd_nat_a @ Xb @ Z2 @ X ) ) ) )
                           => ~ ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relati591517084277583526ts_a_b @ Z2 @ Phi2 ) @ ( produc2895298938842563487_nat_a @ Xa @ Xb ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% sat.pelims(1)
thf(fact_693_sat_Opelims_I2_J,axiom,
    ! [X3: relati9047081815478866374_a_nat,Xa: product_prod_nat_nat > set_list_a,Xb: nat > a] :
      ( ( relational_sat_a_nat @ X3 @ Xa @ Xb )
     => ( ( accp_P3115975753873020414_nat_a @ relati9040413291598363742_a_nat @ ( produc2574091134533170497_nat_a @ X3 @ ( produc4953239424914856836_nat_a @ Xa @ Xb ) ) )
       => ( ! [R2: nat,Ts2: list_R6823256787227418703term_a] :
              ( ( X3
                = ( relati6362048942677509346_nat_a @ R2 @ Ts2 ) )
             => ( ( accp_P3115975753873020414_nat_a @ relati9040413291598363742_a_nat @ ( produc2574091134533170497_nat_a @ ( relati6362048942677509346_nat_a @ R2 @ Ts2 ) @ ( produc4953239424914856836_nat_a @ Xa @ Xb ) ) )
               => ~ ( member_list_a @ ( relati4772805863405912879erms_a @ Xb @ Ts2 ) @ ( Xa @ ( product_Pair_nat_nat @ R2 @ ( size_s88622898042387131term_a @ Ts2 ) ) ) ) ) )
         => ( ! [B2: $o] :
                ( ( X3
                  = ( relati9034565498597818939_a_nat @ B2 ) )
               => ( ( accp_P3115975753873020414_nat_a @ relati9040413291598363742_a_nat @ ( produc2574091134533170497_nat_a @ ( relati9034565498597818939_a_nat @ B2 ) @ ( produc4953239424914856836_nat_a @ Xa @ Xb ) ) )
                 => ~ B2 ) )
           => ( ! [X5: nat,T4: relational_term_a] :
                  ( ( X3
                    = ( relational_Eq_a_nat @ X5 @ T4 ) )
                 => ( ( accp_P3115975753873020414_nat_a @ relati9040413291598363742_a_nat @ ( produc2574091134533170497_nat_a @ ( relational_Eq_a_nat @ X5 @ T4 ) @ ( produc4953239424914856836_nat_a @ Xa @ Xb ) ) )
                   => ( ( Xb @ X5 )
                     != ( relati1177013128715261720term_a @ Xb @ T4 ) ) ) )
             => ( ! [Phi2: relati9047081815478866374_a_nat] :
                    ( ( X3
                      = ( relational_Neg_a_nat @ Phi2 ) )
                   => ( ( accp_P3115975753873020414_nat_a @ relati9040413291598363742_a_nat @ ( produc2574091134533170497_nat_a @ ( relational_Neg_a_nat @ Phi2 ) @ ( produc4953239424914856836_nat_a @ Xa @ Xb ) ) )
                     => ( relational_sat_a_nat @ Phi2 @ Xa @ Xb ) ) )
               => ( ! [Phi2: relati9047081815478866374_a_nat,Psi: relati9047081815478866374_a_nat] :
                      ( ( X3
                        = ( relati2542520632142267709_a_nat @ Phi2 @ Psi ) )
                     => ( ( accp_P3115975753873020414_nat_a @ relati9040413291598363742_a_nat @ ( produc2574091134533170497_nat_a @ ( relati2542520632142267709_a_nat @ Phi2 @ Psi ) @ ( produc4953239424914856836_nat_a @ Xa @ Xb ) ) )
                       => ~ ( ( relational_sat_a_nat @ Phi2 @ Xa @ Xb )
                            & ( relational_sat_a_nat @ Psi @ Xa @ Xb ) ) ) )
                 => ( ! [Phi2: relati9047081815478866374_a_nat,Psi: relati9047081815478866374_a_nat] :
                        ( ( X3
                          = ( relati9106205213788308809_a_nat @ Phi2 @ Psi ) )
                       => ( ( accp_P3115975753873020414_nat_a @ relati9040413291598363742_a_nat @ ( produc2574091134533170497_nat_a @ ( relati9106205213788308809_a_nat @ Phi2 @ Psi ) @ ( produc4953239424914856836_nat_a @ Xa @ Xb ) ) )
                         => ~ ( ( relational_sat_a_nat @ Phi2 @ Xa @ Xb )
                              | ( relational_sat_a_nat @ Psi @ Xa @ Xb ) ) ) )
                   => ~ ! [Z2: nat,Phi2: relati9047081815478866374_a_nat] :
                          ( ( X3
                            = ( relati6314223733442460777_a_nat @ Z2 @ Phi2 ) )
                         => ( ( accp_P3115975753873020414_nat_a @ relati9040413291598363742_a_nat @ ( produc2574091134533170497_nat_a @ ( relati6314223733442460777_a_nat @ Z2 @ Phi2 ) @ ( produc4953239424914856836_nat_a @ Xa @ Xb ) ) )
                           => ~ ? [X5: a] : ( relational_sat_a_nat @ Phi2 @ Xa @ ( fun_upd_nat_a @ Xb @ Z2 @ X5 ) ) ) ) ) ) ) ) ) ) ) ) ).

% sat.pelims(2)
thf(fact_694_sat_Opelims_I2_J,axiom,
    ! [X3: relational_fmla_a_b,Xa: product_prod_b_nat > set_list_a,Xb: nat > a] :
      ( ( relational_sat_a_b @ X3 @ Xa @ Xb )
     => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ X3 @ ( produc2895298938842563487_nat_a @ Xa @ Xb ) ) )
       => ( ! [R2: b,Ts2: list_R6823256787227418703term_a] :
              ( ( X3
                = ( relational_Pred_b_a @ R2 @ Ts2 ) )
             => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Pred_b_a @ R2 @ Ts2 ) @ ( produc2895298938842563487_nat_a @ Xa @ Xb ) ) )
               => ~ ( member_list_a @ ( relati4772805863405912879erms_a @ Xb @ Ts2 ) @ ( Xa @ ( product_Pair_b_nat @ R2 @ ( size_s88622898042387131term_a @ Ts2 ) ) ) ) ) )
         => ( ! [B2: $o] :
                ( ( X3
                  = ( relational_Bool_a_b @ B2 ) )
               => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Bool_a_b @ B2 ) @ ( produc2895298938842563487_nat_a @ Xa @ Xb ) ) )
                 => ~ B2 ) )
           => ( ! [X5: nat,T4: relational_term_a] :
                  ( ( X3
                    = ( relational_Eq_a_b @ X5 @ T4 ) )
                 => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Eq_a_b @ X5 @ T4 ) @ ( produc2895298938842563487_nat_a @ Xa @ Xb ) ) )
                   => ( ( Xb @ X5 )
                     != ( relati1177013128715261720term_a @ Xb @ T4 ) ) ) )
             => ( ! [Phi2: relational_fmla_a_b] :
                    ( ( X3
                      = ( relational_Neg_a_b @ Phi2 ) )
                   => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Neg_a_b @ Phi2 ) @ ( produc2895298938842563487_nat_a @ Xa @ Xb ) ) )
                     => ( relational_sat_a_b @ Phi2 @ Xa @ Xb ) ) )
               => ( ! [Phi2: relational_fmla_a_b,Psi: relational_fmla_a_b] :
                      ( ( X3
                        = ( relational_Conj_a_b @ Phi2 @ Psi ) )
                     => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Conj_a_b @ Phi2 @ Psi ) @ ( produc2895298938842563487_nat_a @ Xa @ Xb ) ) )
                       => ~ ( ( relational_sat_a_b @ Phi2 @ Xa @ Xb )
                            & ( relational_sat_a_b @ Psi @ Xa @ Xb ) ) ) )
                 => ( ! [Phi2: relational_fmla_a_b,Psi: relational_fmla_a_b] :
                        ( ( X3
                          = ( relational_Disj_a_b @ Phi2 @ Psi ) )
                       => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Disj_a_b @ Phi2 @ Psi ) @ ( produc2895298938842563487_nat_a @ Xa @ Xb ) ) )
                         => ~ ( ( relational_sat_a_b @ Phi2 @ Xa @ Xb )
                              | ( relational_sat_a_b @ Psi @ Xa @ Xb ) ) ) )
                   => ~ ! [Z2: nat,Phi2: relational_fmla_a_b] :
                          ( ( X3
                            = ( relati591517084277583526ts_a_b @ Z2 @ Phi2 ) )
                         => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relati591517084277583526ts_a_b @ Z2 @ Phi2 ) @ ( produc2895298938842563487_nat_a @ Xa @ Xb ) ) )
                           => ~ ? [X5: a] : ( relational_sat_a_b @ Phi2 @ Xa @ ( fun_upd_nat_a @ Xb @ Z2 @ X5 ) ) ) ) ) ) ) ) ) ) ) ) ).

% sat.pelims(2)
thf(fact_695_sat_Opelims_I3_J,axiom,
    ! [X3: relati9047081815478866374_a_nat,Xa: product_prod_nat_nat > set_list_a,Xb: nat > a] :
      ( ~ ( relational_sat_a_nat @ X3 @ Xa @ Xb )
     => ( ( accp_P3115975753873020414_nat_a @ relati9040413291598363742_a_nat @ ( produc2574091134533170497_nat_a @ X3 @ ( produc4953239424914856836_nat_a @ Xa @ Xb ) ) )
       => ( ! [R2: nat,Ts2: list_R6823256787227418703term_a] :
              ( ( X3
                = ( relati6362048942677509346_nat_a @ R2 @ Ts2 ) )
             => ( ( accp_P3115975753873020414_nat_a @ relati9040413291598363742_a_nat @ ( produc2574091134533170497_nat_a @ ( relati6362048942677509346_nat_a @ R2 @ Ts2 ) @ ( produc4953239424914856836_nat_a @ Xa @ Xb ) ) )
               => ( member_list_a @ ( relati4772805863405912879erms_a @ Xb @ Ts2 ) @ ( Xa @ ( product_Pair_nat_nat @ R2 @ ( size_s88622898042387131term_a @ Ts2 ) ) ) ) ) )
         => ( ! [B2: $o] :
                ( ( X3
                  = ( relati9034565498597818939_a_nat @ B2 ) )
               => ( ( accp_P3115975753873020414_nat_a @ relati9040413291598363742_a_nat @ ( produc2574091134533170497_nat_a @ ( relati9034565498597818939_a_nat @ B2 ) @ ( produc4953239424914856836_nat_a @ Xa @ Xb ) ) )
                 => B2 ) )
           => ( ! [X5: nat,T4: relational_term_a] :
                  ( ( X3
                    = ( relational_Eq_a_nat @ X5 @ T4 ) )
                 => ( ( accp_P3115975753873020414_nat_a @ relati9040413291598363742_a_nat @ ( produc2574091134533170497_nat_a @ ( relational_Eq_a_nat @ X5 @ T4 ) @ ( produc4953239424914856836_nat_a @ Xa @ Xb ) ) )
                   => ( ( Xb @ X5 )
                      = ( relati1177013128715261720term_a @ Xb @ T4 ) ) ) )
             => ( ! [Phi2: relati9047081815478866374_a_nat] :
                    ( ( X3
                      = ( relational_Neg_a_nat @ Phi2 ) )
                   => ( ( accp_P3115975753873020414_nat_a @ relati9040413291598363742_a_nat @ ( produc2574091134533170497_nat_a @ ( relational_Neg_a_nat @ Phi2 ) @ ( produc4953239424914856836_nat_a @ Xa @ Xb ) ) )
                     => ~ ( relational_sat_a_nat @ Phi2 @ Xa @ Xb ) ) )
               => ( ! [Phi2: relati9047081815478866374_a_nat,Psi: relati9047081815478866374_a_nat] :
                      ( ( X3
                        = ( relati2542520632142267709_a_nat @ Phi2 @ Psi ) )
                     => ( ( accp_P3115975753873020414_nat_a @ relati9040413291598363742_a_nat @ ( produc2574091134533170497_nat_a @ ( relati2542520632142267709_a_nat @ Phi2 @ Psi ) @ ( produc4953239424914856836_nat_a @ Xa @ Xb ) ) )
                       => ( ( relational_sat_a_nat @ Phi2 @ Xa @ Xb )
                          & ( relational_sat_a_nat @ Psi @ Xa @ Xb ) ) ) )
                 => ( ! [Phi2: relati9047081815478866374_a_nat,Psi: relati9047081815478866374_a_nat] :
                        ( ( X3
                          = ( relati9106205213788308809_a_nat @ Phi2 @ Psi ) )
                       => ( ( accp_P3115975753873020414_nat_a @ relati9040413291598363742_a_nat @ ( produc2574091134533170497_nat_a @ ( relati9106205213788308809_a_nat @ Phi2 @ Psi ) @ ( produc4953239424914856836_nat_a @ Xa @ Xb ) ) )
                         => ( ( relational_sat_a_nat @ Phi2 @ Xa @ Xb )
                            | ( relational_sat_a_nat @ Psi @ Xa @ Xb ) ) ) )
                   => ~ ! [Z2: nat,Phi2: relati9047081815478866374_a_nat] :
                          ( ( X3
                            = ( relati6314223733442460777_a_nat @ Z2 @ Phi2 ) )
                         => ( ( accp_P3115975753873020414_nat_a @ relati9040413291598363742_a_nat @ ( produc2574091134533170497_nat_a @ ( relati6314223733442460777_a_nat @ Z2 @ Phi2 ) @ ( produc4953239424914856836_nat_a @ Xa @ Xb ) ) )
                           => ? [X6: a] : ( relational_sat_a_nat @ Phi2 @ Xa @ ( fun_upd_nat_a @ Xb @ Z2 @ X6 ) ) ) ) ) ) ) ) ) ) ) ) ).

% sat.pelims(3)
thf(fact_696_sat_Opelims_I3_J,axiom,
    ! [X3: relational_fmla_a_b,Xa: product_prod_b_nat > set_list_a,Xb: nat > a] :
      ( ~ ( relational_sat_a_b @ X3 @ Xa @ Xb )
     => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ X3 @ ( produc2895298938842563487_nat_a @ Xa @ Xb ) ) )
       => ( ! [R2: b,Ts2: list_R6823256787227418703term_a] :
              ( ( X3
                = ( relational_Pred_b_a @ R2 @ Ts2 ) )
             => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Pred_b_a @ R2 @ Ts2 ) @ ( produc2895298938842563487_nat_a @ Xa @ Xb ) ) )
               => ( member_list_a @ ( relati4772805863405912879erms_a @ Xb @ Ts2 ) @ ( Xa @ ( product_Pair_b_nat @ R2 @ ( size_s88622898042387131term_a @ Ts2 ) ) ) ) ) )
         => ( ! [B2: $o] :
                ( ( X3
                  = ( relational_Bool_a_b @ B2 ) )
               => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Bool_a_b @ B2 ) @ ( produc2895298938842563487_nat_a @ Xa @ Xb ) ) )
                 => B2 ) )
           => ( ! [X5: nat,T4: relational_term_a] :
                  ( ( X3
                    = ( relational_Eq_a_b @ X5 @ T4 ) )
                 => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Eq_a_b @ X5 @ T4 ) @ ( produc2895298938842563487_nat_a @ Xa @ Xb ) ) )
                   => ( ( Xb @ X5 )
                      = ( relati1177013128715261720term_a @ Xb @ T4 ) ) ) )
             => ( ! [Phi2: relational_fmla_a_b] :
                    ( ( X3
                      = ( relational_Neg_a_b @ Phi2 ) )
                   => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Neg_a_b @ Phi2 ) @ ( produc2895298938842563487_nat_a @ Xa @ Xb ) ) )
                     => ~ ( relational_sat_a_b @ Phi2 @ Xa @ Xb ) ) )
               => ( ! [Phi2: relational_fmla_a_b,Psi: relational_fmla_a_b] :
                      ( ( X3
                        = ( relational_Conj_a_b @ Phi2 @ Psi ) )
                     => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Conj_a_b @ Phi2 @ Psi ) @ ( produc2895298938842563487_nat_a @ Xa @ Xb ) ) )
                       => ( ( relational_sat_a_b @ Phi2 @ Xa @ Xb )
                          & ( relational_sat_a_b @ Psi @ Xa @ Xb ) ) ) )
                 => ( ! [Phi2: relational_fmla_a_b,Psi: relational_fmla_a_b] :
                        ( ( X3
                          = ( relational_Disj_a_b @ Phi2 @ Psi ) )
                       => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Disj_a_b @ Phi2 @ Psi ) @ ( produc2895298938842563487_nat_a @ Xa @ Xb ) ) )
                         => ( ( relational_sat_a_b @ Phi2 @ Xa @ Xb )
                            | ( relational_sat_a_b @ Psi @ Xa @ Xb ) ) ) )
                   => ~ ! [Z2: nat,Phi2: relational_fmla_a_b] :
                          ( ( X3
                            = ( relati591517084277583526ts_a_b @ Z2 @ Phi2 ) )
                         => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relati591517084277583526ts_a_b @ Z2 @ Phi2 ) @ ( produc2895298938842563487_nat_a @ Xa @ Xb ) ) )
                           => ? [X6: a] : ( relational_sat_a_b @ Phi2 @ Xa @ ( fun_upd_nat_a @ Xb @ Z2 @ X6 ) ) ) ) ) ) ) ) ) ) ) ) ).

% sat.pelims(3)
thf(fact_697_subst__term_Opelims,axiom,
    ! [X3: relational_term_a,Xa: nat,Xb: nat,Y: relational_term_a] :
      ( ( ( relati7175845559408349773term_a @ X3 @ Xa @ Xb )
        = Y )
     => ( ( accp_P7512640865500879912at_nat @ relati1089594453538404916_rel_a @ ( produc2180204704594896271at_nat @ X3 @ ( product_Pair_nat_nat @ Xa @ Xb ) ) )
       => ( ! [Z2: nat] :
              ( ( X3
                = ( relational_Var_a @ Z2 ) )
             => ( ( Y
                  = ( relational_Var_a @ ( if_nat @ ( Xa = Z2 ) @ Xb @ Z2 ) ) )
               => ~ ( accp_P7512640865500879912at_nat @ relati1089594453538404916_rel_a @ ( produc2180204704594896271at_nat @ ( relational_Var_a @ Z2 ) @ ( product_Pair_nat_nat @ Xa @ Xb ) ) ) ) )
         => ~ ! [C2: a] :
                ( ( X3
                  = ( relational_Const_a @ C2 ) )
               => ( ( Y
                    = ( relational_Const_a @ C2 ) )
                 => ~ ( accp_P7512640865500879912at_nat @ relati1089594453538404916_rel_a @ ( produc2180204704594896271at_nat @ ( relational_Const_a @ C2 ) @ ( product_Pair_nat_nat @ Xa @ Xb ) ) ) ) ) ) ) ) ).

% subst_term.pelims
thf(fact_698_pred__equals__eq2,axiom,
    ! [R4: set_Product_prod_o_o,S: set_Product_prod_o_o] :
      ( ( ( ^ [X: $o,Y2: $o] : ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y2 ) @ R4 ) )
        = ( ^ [X: $o,Y2: $o] : ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y2 ) @ S ) ) )
      = ( R4 = S ) ) ).

% pred_equals_eq2
thf(fact_699_pred__equals__eq2,axiom,
    ! [R4: set_Pr7717912310451564380at_nat,S: set_Pr7717912310451564380at_nat] :
      ( ( ( ^ [X: nat,Y2: product_prod_nat_nat] : ( member2223272150424702269at_nat @ ( produc487386426758144856at_nat @ X @ Y2 ) @ R4 ) )
        = ( ^ [X: nat,Y2: product_prod_nat_nat] : ( member2223272150424702269at_nat @ ( produc487386426758144856at_nat @ X @ Y2 ) @ S ) ) )
      = ( R4 = S ) ) ).

% pred_equals_eq2
thf(fact_700_pred__equals__eq2,axiom,
    ! [R4: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( ( ^ [X: nat,Y2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y2 ) @ R4 ) )
        = ( ^ [X: nat,Y2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y2 ) @ S ) ) )
      = ( R4 = S ) ) ).

% pred_equals_eq2
thf(fact_701_pred__equals__eq2,axiom,
    ! [R4: set_Pr6389665502131816719_nat_a,S: set_Pr6389665502131816719_nat_a] :
      ( ( ( ^ [X: product_prod_b_nat > set_list_a,Y2: nat > a] : ( member9198066416134578520_nat_a @ ( produc2895298938842563487_nat_a @ X @ Y2 ) @ R4 ) )
        = ( ^ [X: product_prod_b_nat > set_list_a,Y2: nat > a] : ( member9198066416134578520_nat_a @ ( produc2895298938842563487_nat_a @ X @ Y2 ) @ S ) ) )
      = ( R4 = S ) ) ).

% pred_equals_eq2
thf(fact_702_pred__equals__eq2,axiom,
    ! [R4: set_Pr1307281990691478580_b_nat,S: set_Pr1307281990691478580_b_nat] :
      ( ( ( ^ [X: b,Y2: nat] : ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ X @ Y2 ) @ R4 ) )
        = ( ^ [X: b,Y2: nat] : ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ X @ Y2 ) @ S ) ) )
      = ( R4 = S ) ) ).

% pred_equals_eq2
thf(fact_703_subst__exists,axiom,
    ! [Z: nat,Q: relational_fmla_a_b,X3: nat,Y: nat] :
      ( ( ( member_nat @ Z @ ( relational_fv_a_b @ Q ) )
       => ( ( ( X3 = Z )
           => ( ( relational_subst_a_b @ ( relati3989891337220013914ts_a_b @ Z @ Q ) @ X3 @ Y )
              = ( relati3989891337220013914ts_a_b @ X3 @ Q ) ) )
          & ( ( X3 != Z )
           => ( ( ( Z = Y )
               => ( ( relational_subst_a_b @ ( relati3989891337220013914ts_a_b @ Z @ Q ) @ X3 @ Y )
                  = ( relati3989891337220013914ts_a_b @ ( relati2677767559083392098h2_a_b @ X3 @ Y @ Q ) @ ( relational_subst_a_b @ ( relational_subst_a_b @ Q @ Z @ ( relati2677767559083392098h2_a_b @ X3 @ Y @ Q ) ) @ X3 @ Y ) ) ) )
              & ( ( Z != Y )
               => ( ( relational_subst_a_b @ ( relati3989891337220013914ts_a_b @ Z @ Q ) @ X3 @ Y )
                  = ( relati3989891337220013914ts_a_b @ Z @ ( relational_subst_a_b @ Q @ X3 @ Y ) ) ) ) ) ) ) )
      & ( ~ ( member_nat @ Z @ ( relational_fv_a_b @ Q ) )
       => ( ( relational_subst_a_b @ ( relati3989891337220013914ts_a_b @ Z @ Q ) @ X3 @ Y )
          = ( relational_subst_a_b @ Q @ X3 @ Y ) ) ) ) ).

% subst_exists
thf(fact_704_cp__exists,axiom,
    ! [X3: nat,Q: relational_fmla_a_b] :
      ( ( relational_cp_a_b @ ( relati3989891337220013914ts_a_b @ X3 @ Q ) )
      = ( relati3989891337220013914ts_a_b @ X3 @ ( relational_cp_a_b @ Q ) ) ) ).

% cp_exists
thf(fact_705_nocp__exists,axiom,
    ! [X3: nat,Q: relational_fmla_a_b] :
      ( ( relational_nocp_a_b @ ( relati3989891337220013914ts_a_b @ X3 @ Q ) )
      = ( relational_nocp_a_b @ Q ) ) ).

% nocp_exists
thf(fact_706_sat__exists,axiom,
    ! [N: nat,Q: relational_fmla_a_b,I2: product_prod_b_nat > set_list_a,Sigma: nat > a] :
      ( ( relational_sat_a_b @ ( relati3989891337220013914ts_a_b @ N @ Q ) @ I2 @ Sigma )
      = ( ? [X: a] : ( relational_sat_a_b @ Q @ I2 @ ( fun_upd_nat_a @ Sigma @ N @ X ) ) ) ) ).

% sat_exists
thf(fact_707_exists,axiom,
    ! [Q: relational_fmla_a_b,X3: nat] :
      ( ( relational_qp_a_b @ Q )
     => ( relational_qp_a_b @ ( relati3989891337220013914ts_a_b @ X3 @ Q ) ) ) ).

% exists
thf(fact_708_exists__def,axiom,
    ( relati3989891337220013914ts_a_b
    = ( ^ [X: nat,Q2: relational_fmla_a_b] : ( if_Rel1279876242545935705la_a_b @ ( member_nat @ X @ ( relational_fv_a_b @ Q2 ) ) @ ( relati591517084277583526ts_a_b @ X @ Q2 ) @ Q2 ) ) ) ).

% exists_def
thf(fact_709_exists__Exists,axiom,
    ! [X3: nat,Q: relational_fmla_a_b] :
      ( ( member_nat @ X3 @ ( relational_fv_a_b @ Q ) )
     => ( ( relati3989891337220013914ts_a_b @ X3 @ Q )
        = ( relati591517084277583526ts_a_b @ X3 @ Q ) ) ) ).

% exists_Exists
thf(fact_710_cp_Osimps_I5_J,axiom,
    ! [X3: nat,Q: relational_fmla_a_b] :
      ( ( relational_cp_a_b @ ( relati591517084277583526ts_a_b @ X3 @ Q ) )
      = ( relati3989891337220013914ts_a_b @ X3 @ ( relational_cp_a_b @ Q ) ) ) ).

% cp.simps(5)
thf(fact_711_exists__cp__subst,axiom,
    ! [X3: nat,Y: nat,Q: relational_fmla_a_b] :
      ( ( X3 != Y )
     => ( ( relati3989891337220013914ts_a_b @ X3 @ ( relational_cp_a_b @ ( relational_subst_a_b @ Q @ X3 @ Y ) ) )
        = ( relational_cp_a_b @ ( relational_subst_a_b @ Q @ X3 @ Y ) ) ) ) ).

% exists_cp_subst
thf(fact_712_qp_Osimps,axiom,
    ( relational_qp_a_b
    = ( ^ [A3: relational_fmla_a_b] :
          ( ? [Q2: relational_fmla_a_b] :
              ( ( A3 = Q2 )
              & ( relational_ap_a_b @ Q2 ) )
          | ? [Q2: relational_fmla_a_b,X: nat] :
              ( ( A3
                = ( relati3989891337220013914ts_a_b @ X @ Q2 ) )
              & ( relational_qp_a_b @ Q2 ) ) ) ) ) ).

% qp.simps
thf(fact_713_qp_Ocases,axiom,
    ! [A: relational_fmla_a_b] :
      ( ( relational_qp_a_b @ A )
     => ( ~ ( relational_ap_a_b @ A )
       => ~ ! [Q3: relational_fmla_a_b] :
              ( ? [X5: nat] :
                  ( A
                  = ( relati3989891337220013914ts_a_b @ X5 @ Q3 ) )
             => ~ ( relational_qp_a_b @ Q3 ) ) ) ) ).

% qp.cases
thf(fact_714_gen_Ointros_I10_J,axiom,
    ! [X3: nat,Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
      ( ( X3 != Y )
     => ( ( relational_gen_a_b @ X3 @ Q @ G )
       => ( relational_gen_a_b @ X3 @ ( relati591517084277583526ts_a_b @ Y @ Q ) @ ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y ) @ G ) ) ) ) ).

% gen.intros(10)
thf(fact_715_gen_H_Ointros_I10_J,axiom,
    ! [X3: nat,Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
      ( ( X3 != Y )
     => ( ( relational_gen_a_b2 @ X3 @ Q @ G )
       => ( relational_gen_a_b2 @ X3 @ ( relati591517084277583526ts_a_b @ Y @ Q ) @ ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y ) @ G ) ) ) ) ).

% gen'.intros(10)
thf(fact_716_cp_Oelims,axiom,
    ! [X3: relational_fmla_a_b,Y: relational_fmla_a_b] :
      ( ( ( relational_cp_a_b @ X3 )
        = Y )
     => ( ! [X5: nat,T3: relational_term_a] :
            ( ( X3
              = ( relational_Eq_a_b @ X5 @ T3 ) )
           => ( Y
             != ( relati582353067970734056la_a_b
                @ ^ [A3: a] : ( relational_Eq_a_b @ X5 @ T3 )
                @ ^ [Y2: nat] : ( if_Rel1279876242545935705la_a_b @ ( X5 = Y2 ) @ ( relational_Bool_a_b @ $true ) @ ( relational_Eq_a_b @ X5 @ ( relational_Var_a @ Y2 ) ) )
                @ T3 ) ) )
       => ( ! [Q3: relational_fmla_a_b] :
              ( ( X3
                = ( relational_Neg_a_b @ Q3 ) )
             => ( Y
               != ( if_Rel1279876242545935705la_a_b @ ( relati6551038146797045342ol_a_b @ ( relational_cp_a_b @ Q3 ) )
                  @ ( relational_Bool_a_b
                    @ ~ ( relati2638701775882563405ol_a_b @ ( relational_cp_a_b @ Q3 ) ) )
                  @ ( relational_Neg_a_b @ ( relational_cp_a_b @ Q3 ) ) ) ) )
         => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                ( ( X3
                  = ( relational_Conj_a_b @ Q13 @ Q24 ) )
               => ( Y
                 != ( if_Rel1279876242545935705la_a_b @ ( relati6551038146797045342ol_a_b @ ( relational_cp_a_b @ Q13 ) ) @ ( if_Rel1279876242545935705la_a_b @ ( relati2638701775882563405ol_a_b @ ( relational_cp_a_b @ Q13 ) ) @ ( relational_cp_a_b @ Q24 ) @ ( relational_Bool_a_b @ $false ) ) @ ( if_Rel1279876242545935705la_a_b @ ( relati6551038146797045342ol_a_b @ ( relational_cp_a_b @ Q24 ) ) @ ( if_Rel1279876242545935705la_a_b @ ( relati2638701775882563405ol_a_b @ ( relational_cp_a_b @ Q24 ) ) @ ( relational_cp_a_b @ Q13 ) @ ( relational_Bool_a_b @ $false ) ) @ ( relational_Conj_a_b @ ( relational_cp_a_b @ Q13 ) @ ( relational_cp_a_b @ Q24 ) ) ) ) ) )
           => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                  ( ( X3
                    = ( relational_Disj_a_b @ Q13 @ Q24 ) )
                 => ( Y
                   != ( if_Rel1279876242545935705la_a_b @ ( relati6551038146797045342ol_a_b @ ( relational_cp_a_b @ Q13 ) ) @ ( if_Rel1279876242545935705la_a_b @ ( relati2638701775882563405ol_a_b @ ( relational_cp_a_b @ Q13 ) ) @ ( relational_Bool_a_b @ $true ) @ ( relational_cp_a_b @ Q24 ) ) @ ( if_Rel1279876242545935705la_a_b @ ( relati6551038146797045342ol_a_b @ ( relational_cp_a_b @ Q24 ) ) @ ( if_Rel1279876242545935705la_a_b @ ( relati2638701775882563405ol_a_b @ ( relational_cp_a_b @ Q24 ) ) @ ( relational_Bool_a_b @ $true ) @ ( relational_cp_a_b @ Q13 ) ) @ ( relational_Disj_a_b @ ( relational_cp_a_b @ Q13 ) @ ( relational_cp_a_b @ Q24 ) ) ) ) ) )
             => ( ! [X5: nat,Q3: relational_fmla_a_b] :
                    ( ( X3
                      = ( relati591517084277583526ts_a_b @ X5 @ Q3 ) )
                   => ( Y
                     != ( relati3989891337220013914ts_a_b @ X5 @ ( relational_cp_a_b @ Q3 ) ) ) )
               => ( ! [V2: b,Va2: list_R6823256787227418703term_a] :
                      ( ( X3
                        = ( relational_Pred_b_a @ V2 @ Va2 ) )
                     => ( Y
                       != ( relational_Pred_b_a @ V2 @ Va2 ) ) )
                 => ~ ! [V2: $o] :
                        ( ( X3
                          = ( relational_Bool_a_b @ V2 ) )
                       => ( Y
                         != ( relational_Bool_a_b @ V2 ) ) ) ) ) ) ) ) ) ) ).

% cp.elims
thf(fact_717_cp_Opelims,axiom,
    ! [X3: relational_fmla_a_b,Y: relational_fmla_a_b] :
      ( ( ( relational_cp_a_b @ X3 )
        = Y )
     => ( ( accp_R989495437599811158la_a_b @ relati5452380258257131376el_a_b @ X3 )
       => ( ! [X5: nat,T3: relational_term_a] :
              ( ( X3
                = ( relational_Eq_a_b @ X5 @ T3 ) )
             => ( ( Y
                  = ( relati582353067970734056la_a_b
                    @ ^ [A3: a] : ( relational_Eq_a_b @ X5 @ T3 )
                    @ ^ [Y2: nat] : ( if_Rel1279876242545935705la_a_b @ ( X5 = Y2 ) @ ( relational_Bool_a_b @ $true ) @ ( relational_Eq_a_b @ X5 @ ( relational_Var_a @ Y2 ) ) )
                    @ T3 ) )
               => ~ ( accp_R989495437599811158la_a_b @ relati5452380258257131376el_a_b @ ( relational_Eq_a_b @ X5 @ T3 ) ) ) )
         => ( ! [Q3: relational_fmla_a_b] :
                ( ( X3
                  = ( relational_Neg_a_b @ Q3 ) )
               => ( ( Y
                    = ( if_Rel1279876242545935705la_a_b @ ( relati6551038146797045342ol_a_b @ ( relational_cp_a_b @ Q3 ) )
                      @ ( relational_Bool_a_b
                        @ ~ ( relati2638701775882563405ol_a_b @ ( relational_cp_a_b @ Q3 ) ) )
                      @ ( relational_Neg_a_b @ ( relational_cp_a_b @ Q3 ) ) ) )
                 => ~ ( accp_R989495437599811158la_a_b @ relati5452380258257131376el_a_b @ ( relational_Neg_a_b @ Q3 ) ) ) )
           => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                  ( ( X3
                    = ( relational_Conj_a_b @ Q13 @ Q24 ) )
                 => ( ( Y
                      = ( if_Rel1279876242545935705la_a_b @ ( relati6551038146797045342ol_a_b @ ( relational_cp_a_b @ Q13 ) ) @ ( if_Rel1279876242545935705la_a_b @ ( relati2638701775882563405ol_a_b @ ( relational_cp_a_b @ Q13 ) ) @ ( relational_cp_a_b @ Q24 ) @ ( relational_Bool_a_b @ $false ) ) @ ( if_Rel1279876242545935705la_a_b @ ( relati6551038146797045342ol_a_b @ ( relational_cp_a_b @ Q24 ) ) @ ( if_Rel1279876242545935705la_a_b @ ( relati2638701775882563405ol_a_b @ ( relational_cp_a_b @ Q24 ) ) @ ( relational_cp_a_b @ Q13 ) @ ( relational_Bool_a_b @ $false ) ) @ ( relational_Conj_a_b @ ( relational_cp_a_b @ Q13 ) @ ( relational_cp_a_b @ Q24 ) ) ) ) )
                   => ~ ( accp_R989495437599811158la_a_b @ relati5452380258257131376el_a_b @ ( relational_Conj_a_b @ Q13 @ Q24 ) ) ) )
             => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                    ( ( X3
                      = ( relational_Disj_a_b @ Q13 @ Q24 ) )
                   => ( ( Y
                        = ( if_Rel1279876242545935705la_a_b @ ( relati6551038146797045342ol_a_b @ ( relational_cp_a_b @ Q13 ) ) @ ( if_Rel1279876242545935705la_a_b @ ( relati2638701775882563405ol_a_b @ ( relational_cp_a_b @ Q13 ) ) @ ( relational_Bool_a_b @ $true ) @ ( relational_cp_a_b @ Q24 ) ) @ ( if_Rel1279876242545935705la_a_b @ ( relati6551038146797045342ol_a_b @ ( relational_cp_a_b @ Q24 ) ) @ ( if_Rel1279876242545935705la_a_b @ ( relati2638701775882563405ol_a_b @ ( relational_cp_a_b @ Q24 ) ) @ ( relational_Bool_a_b @ $true ) @ ( relational_cp_a_b @ Q13 ) ) @ ( relational_Disj_a_b @ ( relational_cp_a_b @ Q13 ) @ ( relational_cp_a_b @ Q24 ) ) ) ) )
                     => ~ ( accp_R989495437599811158la_a_b @ relati5452380258257131376el_a_b @ ( relational_Disj_a_b @ Q13 @ Q24 ) ) ) )
               => ( ! [X5: nat,Q3: relational_fmla_a_b] :
                      ( ( X3
                        = ( relati591517084277583526ts_a_b @ X5 @ Q3 ) )
                     => ( ( Y
                          = ( relati3989891337220013914ts_a_b @ X5 @ ( relational_cp_a_b @ Q3 ) ) )
                       => ~ ( accp_R989495437599811158la_a_b @ relati5452380258257131376el_a_b @ ( relati591517084277583526ts_a_b @ X5 @ Q3 ) ) ) )
                 => ( ! [V2: b,Va2: list_R6823256787227418703term_a] :
                        ( ( X3
                          = ( relational_Pred_b_a @ V2 @ Va2 ) )
                       => ( ( Y
                            = ( relational_Pred_b_a @ V2 @ Va2 ) )
                         => ~ ( accp_R989495437599811158la_a_b @ relati5452380258257131376el_a_b @ ( relational_Pred_b_a @ V2 @ Va2 ) ) ) )
                   => ~ ! [V2: $o] :
                          ( ( X3
                            = ( relational_Bool_a_b @ V2 ) )
                         => ( ( Y
                              = ( relational_Bool_a_b @ V2 ) )
                           => ~ ( accp_R989495437599811158la_a_b @ relati5452380258257131376el_a_b @ ( relational_Bool_a_b @ V2 ) ) ) ) ) ) ) ) ) ) ) ) ).

% cp.pelims
thf(fact_718_gen_Osimps,axiom,
    ( relational_gen_a_b
    = ( ^ [A12: nat,A23: relational_fmla_a_b,A32: set_Re381260168593705685la_a_b] :
          ( ( ( A23
              = ( relational_Bool_a_b @ $false ) )
            & ( A32 = bot_bo4495933725496725865la_a_b ) )
          | ( ( A32
              = ( insert7010464514620295119la_a_b @ A23 @ bot_bo4495933725496725865la_a_b ) )
            & ( relational_ap_a_b @ A23 )
            & ( member_nat @ A12 @ ( relational_fv_a_b @ A23 ) ) )
          | ? [Q2: relational_fmla_a_b] :
              ( ( A23
                = ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q2 ) ) )
              & ( relational_gen_a_b @ A12 @ Q2 @ A32 ) )
          | ? [Q12: relational_fmla_a_b,Q23: relational_fmla_a_b] :
              ( ( A23
                = ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q12 @ Q23 ) ) )
              & ( relational_gen_a_b @ A12 @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q12 ) @ ( relational_Neg_a_b @ Q23 ) ) @ A32 ) )
          | ? [Q12: relational_fmla_a_b,Q23: relational_fmla_a_b] :
              ( ( A23
                = ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q12 @ Q23 ) ) )
              & ( relational_gen_a_b @ A12 @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q12 ) @ ( relational_Neg_a_b @ Q23 ) ) @ A32 ) )
          | ? [Q12: relational_fmla_a_b,G1: set_Re381260168593705685la_a_b,Q23: relational_fmla_a_b] :
              ( ( A23
                = ( relational_Disj_a_b @ Q12 @ Q23 ) )
              & ? [G22: set_Re381260168593705685la_a_b] :
                  ( ( A32
                    = ( sup_su5130108678486352897la_a_b @ G1 @ G22 ) )
                  & ( relational_gen_a_b @ A12 @ Q12 @ G1 )
                  & ( relational_gen_a_b @ A12 @ Q23 @ G22 ) ) )
          | ? [Q12: relational_fmla_a_b,Q23: relational_fmla_a_b] :
              ( ( A23
                = ( relational_Conj_a_b @ Q12 @ Q23 ) )
              & ( ( relational_gen_a_b @ A12 @ Q12 @ A32 )
                | ( relational_gen_a_b @ A12 @ Q23 @ A32 ) ) )
          | ? [Y2: nat,Q2: relational_fmla_a_b] :
              ( ( A23
                = ( relational_Conj_a_b @ Q2 @ ( relational_Eq_a_b @ A12 @ ( relational_Var_a @ Y2 ) ) ) )
              & ? [G3: set_Re381260168593705685la_a_b] :
                  ( ( A32
                    = ( image_6790371041703824709la_a_b
                      @ ^ [Qa: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Qa @ Y2 @ A12 ) )
                      @ G3 ) )
                  & ( relational_gen_a_b @ Y2 @ Q2 @ G3 ) ) )
          | ? [Y2: nat,Q2: relational_fmla_a_b] :
              ( ( A23
                = ( relational_Conj_a_b @ Q2 @ ( relational_Eq_a_b @ Y2 @ ( relational_Var_a @ A12 ) ) ) )
              & ? [G3: set_Re381260168593705685la_a_b] :
                  ( ( A32
                    = ( image_6790371041703824709la_a_b
                      @ ^ [Qa: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Qa @ Y2 @ A12 ) )
                      @ G3 ) )
                  & ( relational_gen_a_b @ Y2 @ Q2 @ G3 ) ) )
          | ? [Y2: nat,Q2: relational_fmla_a_b] :
              ( ( A23
                = ( relati591517084277583526ts_a_b @ Y2 @ Q2 ) )
              & ? [G3: set_Re381260168593705685la_a_b] :
                  ( ( A32
                    = ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y2 ) @ G3 ) )
                  & ( A12 != Y2 )
                  & ( relational_gen_a_b @ A12 @ Q2 @ G3 ) ) ) ) ) ) ).

% gen.simps
thf(fact_719_gen_Ocases,axiom,
    ! [A1: nat,A22: relational_fmla_a_b,A33: set_Re381260168593705685la_a_b] :
      ( ( relational_gen_a_b @ A1 @ A22 @ A33 )
     => ( ( ( A22
            = ( relational_Bool_a_b @ $false ) )
         => ( A33 != bot_bo4495933725496725865la_a_b ) )
       => ( ( ( A33
              = ( insert7010464514620295119la_a_b @ A22 @ bot_bo4495933725496725865la_a_b ) )
           => ( ( relational_ap_a_b @ A22 )
             => ~ ( member_nat @ A1 @ ( relational_fv_a_b @ A22 ) ) ) )
         => ( ! [Q3: relational_fmla_a_b] :
                ( ( A22
                  = ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q3 ) ) )
               => ~ ( relational_gen_a_b @ A1 @ Q3 @ A33 ) )
           => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                  ( ( A22
                    = ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q13 @ Q24 ) ) )
                 => ~ ( relational_gen_a_b @ A1 @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q13 ) @ ( relational_Neg_a_b @ Q24 ) ) @ A33 ) )
             => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                    ( ( A22
                      = ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q13 @ Q24 ) ) )
                   => ~ ( relational_gen_a_b @ A1 @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q13 ) @ ( relational_Neg_a_b @ Q24 ) ) @ A33 ) )
               => ( ! [Q13: relational_fmla_a_b,G12: set_Re381260168593705685la_a_b,Q24: relational_fmla_a_b] :
                      ( ( A22
                        = ( relational_Disj_a_b @ Q13 @ Q24 ) )
                     => ! [G23: set_Re381260168593705685la_a_b] :
                          ( ( A33
                            = ( sup_su5130108678486352897la_a_b @ G12 @ G23 ) )
                         => ( ( relational_gen_a_b @ A1 @ Q13 @ G12 )
                           => ~ ( relational_gen_a_b @ A1 @ Q24 @ G23 ) ) ) )
                 => ( ! [Q13: relational_fmla_a_b,G4: set_Re381260168593705685la_a_b,Q24: relational_fmla_a_b] :
                        ( ( A22
                          = ( relational_Conj_a_b @ Q13 @ Q24 ) )
                       => ( ( A33 = G4 )
                         => ~ ( ( relational_gen_a_b @ A1 @ Q13 @ G4 )
                              | ( relational_gen_a_b @ A1 @ Q24 @ G4 ) ) ) )
                   => ( ! [Y3: nat,Q3: relational_fmla_a_b] :
                          ( ( A22
                            = ( relational_Conj_a_b @ Q3 @ ( relational_Eq_a_b @ A1 @ ( relational_Var_a @ Y3 ) ) ) )
                         => ! [G4: set_Re381260168593705685la_a_b] :
                              ( ( A33
                                = ( image_6790371041703824709la_a_b
                                  @ ^ [Qa: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Qa @ Y3 @ A1 ) )
                                  @ G4 ) )
                             => ~ ( relational_gen_a_b @ Y3 @ Q3 @ G4 ) ) )
                     => ( ! [Y3: nat,Q3: relational_fmla_a_b] :
                            ( ( A22
                              = ( relational_Conj_a_b @ Q3 @ ( relational_Eq_a_b @ Y3 @ ( relational_Var_a @ A1 ) ) ) )
                           => ! [G4: set_Re381260168593705685la_a_b] :
                                ( ( A33
                                  = ( image_6790371041703824709la_a_b
                                    @ ^ [Qa: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Qa @ Y3 @ A1 ) )
                                    @ G4 ) )
                               => ~ ( relational_gen_a_b @ Y3 @ Q3 @ G4 ) ) )
                       => ~ ! [Y3: nat,Q3: relational_fmla_a_b] :
                              ( ( A22
                                = ( relati591517084277583526ts_a_b @ Y3 @ Q3 ) )
                             => ! [G4: set_Re381260168593705685la_a_b] :
                                  ( ( A33
                                    = ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y3 ) @ G4 ) )
                                 => ( ( A1 != Y3 )
                                   => ~ ( relational_gen_a_b @ A1 @ Q3 @ G4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% gen.cases
thf(fact_720_empty__Collect__eq,axiom,
    ! [P: relational_fmla_a_b > $o] :
      ( ( bot_bo4495933725496725865la_a_b
        = ( collec3419995626248312948la_a_b @ P ) )
      = ( ! [X: relational_fmla_a_b] :
            ~ ( P @ X ) ) ) ).

% empty_Collect_eq
thf(fact_721_empty__Collect__eq,axiom,
    ! [P: b > $o] :
      ( ( bot_bot_set_b
        = ( collect_b @ P ) )
      = ( ! [X: b] :
            ~ ( P @ X ) ) ) ).

% empty_Collect_eq
thf(fact_722_empty__Collect__eq,axiom,
    ! [P: nat > $o] :
      ( ( bot_bot_set_nat
        = ( collect_nat @ P ) )
      = ( ! [X: nat] :
            ~ ( P @ X ) ) ) ).

% empty_Collect_eq
thf(fact_723_empty__Collect__eq,axiom,
    ! [P: product_prod_o_o > $o] :
      ( ( bot_bo7073875226086086771od_o_o
        = ( collec3167064739498627218od_o_o @ P ) )
      = ( ! [X: product_prod_o_o] :
            ~ ( P @ X ) ) ) ).

% empty_Collect_eq
thf(fact_724_empty__Collect__eq,axiom,
    ! [P: $o > $o] :
      ( ( bot_bot_set_o
        = ( collect_o @ P ) )
      = ( ! [X: $o] :
            ~ ( P @ X ) ) ) ).

% empty_Collect_eq
thf(fact_725_Collect__empty__eq,axiom,
    ! [P: relational_fmla_a_b > $o] :
      ( ( ( collec3419995626248312948la_a_b @ P )
        = bot_bo4495933725496725865la_a_b )
      = ( ! [X: relational_fmla_a_b] :
            ~ ( P @ X ) ) ) ).

% Collect_empty_eq
thf(fact_726_Collect__empty__eq,axiom,
    ! [P: b > $o] :
      ( ( ( collect_b @ P )
        = bot_bot_set_b )
      = ( ! [X: b] :
            ~ ( P @ X ) ) ) ).

% Collect_empty_eq
thf(fact_727_Collect__empty__eq,axiom,
    ! [P: nat > $o] :
      ( ( ( collect_nat @ P )
        = bot_bot_set_nat )
      = ( ! [X: nat] :
            ~ ( P @ X ) ) ) ).

% Collect_empty_eq
thf(fact_728_Collect__empty__eq,axiom,
    ! [P: product_prod_o_o > $o] :
      ( ( ( collec3167064739498627218od_o_o @ P )
        = bot_bo7073875226086086771od_o_o )
      = ( ! [X: product_prod_o_o] :
            ~ ( P @ X ) ) ) ).

% Collect_empty_eq
thf(fact_729_Collect__empty__eq,axiom,
    ! [P: $o > $o] :
      ( ( ( collect_o @ P )
        = bot_bot_set_o )
      = ( ! [X: $o] :
            ~ ( P @ X ) ) ) ).

% Collect_empty_eq
thf(fact_730_all__not__in__conv,axiom,
    ! [A2: set_list_a] :
      ( ( ! [X: list_a] :
            ~ ( member_list_a @ X @ A2 ) )
      = ( A2 = bot_bot_set_list_a ) ) ).

% all_not_in_conv
thf(fact_731_all__not__in__conv,axiom,
    ! [A2: set_set_o] :
      ( ( ! [X: set_o] :
            ~ ( member_set_o @ X @ A2 ) )
      = ( A2 = bot_bot_set_set_o ) ) ).

% all_not_in_conv
thf(fact_732_all__not__in__conv,axiom,
    ! [A2: set_Re1288005135514575379la_a_b] :
      ( ( ! [X: relational_fmla_a_b > relational_fmla_a_b] :
            ~ ( member8433577210552456052la_a_b @ X @ A2 ) )
      = ( A2 = bot_bo9179849999556691623la_a_b ) ) ).

% all_not_in_conv
thf(fact_733_all__not__in__conv,axiom,
    ! [A2: set_Re381260168593705685la_a_b] :
      ( ( ! [X: relational_fmla_a_b] :
            ~ ( member4680049679412964150la_a_b @ X @ A2 ) )
      = ( A2 = bot_bo4495933725496725865la_a_b ) ) ).

% all_not_in_conv
thf(fact_734_all__not__in__conv,axiom,
    ! [A2: set_b] :
      ( ( ! [X: b] :
            ~ ( member_b @ X @ A2 ) )
      = ( A2 = bot_bot_set_b ) ) ).

% all_not_in_conv
thf(fact_735_all__not__in__conv,axiom,
    ! [A2: set_nat] :
      ( ( ! [X: nat] :
            ~ ( member_nat @ X @ A2 ) )
      = ( A2 = bot_bot_set_nat ) ) ).

% all_not_in_conv
thf(fact_736_all__not__in__conv,axiom,
    ! [A2: set_Product_prod_o_o] :
      ( ( ! [X: product_prod_o_o] :
            ~ ( member7466972457876170832od_o_o @ X @ A2 ) )
      = ( A2 = bot_bo7073875226086086771od_o_o ) ) ).

% all_not_in_conv
thf(fact_737_all__not__in__conv,axiom,
    ! [A2: set_o] :
      ( ( ! [X: $o] :
            ~ ( member_o @ X @ A2 ) )
      = ( A2 = bot_bot_set_o ) ) ).

% all_not_in_conv
thf(fact_738_empty__iff,axiom,
    ! [C: list_a] :
      ~ ( member_list_a @ C @ bot_bot_set_list_a ) ).

% empty_iff
thf(fact_739_empty__iff,axiom,
    ! [C: set_o] :
      ~ ( member_set_o @ C @ bot_bot_set_set_o ) ).

% empty_iff
thf(fact_740_empty__iff,axiom,
    ! [C: relational_fmla_a_b > relational_fmla_a_b] :
      ~ ( member8433577210552456052la_a_b @ C @ bot_bo9179849999556691623la_a_b ) ).

% empty_iff
thf(fact_741_empty__iff,axiom,
    ! [C: relational_fmla_a_b] :
      ~ ( member4680049679412964150la_a_b @ C @ bot_bo4495933725496725865la_a_b ) ).

% empty_iff
thf(fact_742_empty__iff,axiom,
    ! [C: b] :
      ~ ( member_b @ C @ bot_bot_set_b ) ).

% empty_iff
thf(fact_743_empty__iff,axiom,
    ! [C: nat] :
      ~ ( member_nat @ C @ bot_bot_set_nat ) ).

% empty_iff
thf(fact_744_empty__iff,axiom,
    ! [C: product_prod_o_o] :
      ~ ( member7466972457876170832od_o_o @ C @ bot_bo7073875226086086771od_o_o ) ).

% empty_iff
thf(fact_745_empty__iff,axiom,
    ! [C: $o] :
      ~ ( member_o @ C @ bot_bot_set_o ) ).

% empty_iff
thf(fact_746_insert__absorb2,axiom,
    ! [X3: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
      ( ( insert7010464514620295119la_a_b @ X3 @ ( insert7010464514620295119la_a_b @ X3 @ A2 ) )
      = ( insert7010464514620295119la_a_b @ X3 @ A2 ) ) ).

% insert_absorb2
thf(fact_747_insert__absorb2,axiom,
    ! [X3: nat,A2: set_nat] :
      ( ( insert_nat @ X3 @ ( insert_nat @ X3 @ A2 ) )
      = ( insert_nat @ X3 @ A2 ) ) ).

% insert_absorb2
thf(fact_748_insert__absorb2,axiom,
    ! [X3: product_prod_o_o,A2: set_Product_prod_o_o] :
      ( ( insert6201435330877294327od_o_o @ X3 @ ( insert6201435330877294327od_o_o @ X3 @ A2 ) )
      = ( insert6201435330877294327od_o_o @ X3 @ A2 ) ) ).

% insert_absorb2
thf(fact_749_insert__absorb2,axiom,
    ! [X3: $o,A2: set_o] :
      ( ( insert_o @ X3 @ ( insert_o @ X3 @ A2 ) )
      = ( insert_o @ X3 @ A2 ) ) ).

% insert_absorb2
thf(fact_750_insert__iff,axiom,
    ! [A: relational_fmla_a_b,B: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
      ( ( member4680049679412964150la_a_b @ A @ ( insert7010464514620295119la_a_b @ B @ A2 ) )
      = ( ( A = B )
        | ( member4680049679412964150la_a_b @ A @ A2 ) ) ) ).

% insert_iff
thf(fact_751_insert__iff,axiom,
    ! [A: product_prod_o_o,B: product_prod_o_o,A2: set_Product_prod_o_o] :
      ( ( member7466972457876170832od_o_o @ A @ ( insert6201435330877294327od_o_o @ B @ A2 ) )
      = ( ( A = B )
        | ( member7466972457876170832od_o_o @ A @ A2 ) ) ) ).

% insert_iff
thf(fact_752_insert__iff,axiom,
    ! [A: list_a,B: list_a,A2: set_list_a] :
      ( ( member_list_a @ A @ ( insert_list_a @ B @ A2 ) )
      = ( ( A = B )
        | ( member_list_a @ A @ A2 ) ) ) ).

% insert_iff
thf(fact_753_insert__iff,axiom,
    ! [A: set_o,B: set_o,A2: set_set_o] :
      ( ( member_set_o @ A @ ( insert_set_o @ B @ A2 ) )
      = ( ( A = B )
        | ( member_set_o @ A @ A2 ) ) ) ).

% insert_iff
thf(fact_754_insert__iff,axiom,
    ! [A: nat,B: nat,A2: set_nat] :
      ( ( member_nat @ A @ ( insert_nat @ B @ A2 ) )
      = ( ( A = B )
        | ( member_nat @ A @ A2 ) ) ) ).

% insert_iff
thf(fact_755_insert__iff,axiom,
    ! [A: relational_fmla_a_b > relational_fmla_a_b,B: relational_fmla_a_b > relational_fmla_a_b,A2: set_Re1288005135514575379la_a_b] :
      ( ( member8433577210552456052la_a_b @ A @ ( insert8904949763332019597la_a_b @ B @ A2 ) )
      = ( ( A = B )
        | ( member8433577210552456052la_a_b @ A @ A2 ) ) ) ).

% insert_iff
thf(fact_756_insert__iff,axiom,
    ! [A: b,B: b,A2: set_b] :
      ( ( member_b @ A @ ( insert_b @ B @ A2 ) )
      = ( ( A = B )
        | ( member_b @ A @ A2 ) ) ) ).

% insert_iff
thf(fact_757_insert__iff,axiom,
    ! [A: $o,B: $o,A2: set_o] :
      ( ( member_o @ A @ ( insert_o @ B @ A2 ) )
      = ( ( A = B )
        | ( member_o @ A @ A2 ) ) ) ).

% insert_iff
thf(fact_758_insertCI,axiom,
    ! [A: relational_fmla_a_b,B5: set_Re381260168593705685la_a_b,B: relational_fmla_a_b] :
      ( ( ~ ( member4680049679412964150la_a_b @ A @ B5 )
       => ( A = B ) )
     => ( member4680049679412964150la_a_b @ A @ ( insert7010464514620295119la_a_b @ B @ B5 ) ) ) ).

% insertCI
thf(fact_759_insertCI,axiom,
    ! [A: product_prod_o_o,B5: set_Product_prod_o_o,B: product_prod_o_o] :
      ( ( ~ ( member7466972457876170832od_o_o @ A @ B5 )
       => ( A = B ) )
     => ( member7466972457876170832od_o_o @ A @ ( insert6201435330877294327od_o_o @ B @ B5 ) ) ) ).

% insertCI
thf(fact_760_insertCI,axiom,
    ! [A: list_a,B5: set_list_a,B: list_a] :
      ( ( ~ ( member_list_a @ A @ B5 )
       => ( A = B ) )
     => ( member_list_a @ A @ ( insert_list_a @ B @ B5 ) ) ) ).

% insertCI
thf(fact_761_insertCI,axiom,
    ! [A: set_o,B5: set_set_o,B: set_o] :
      ( ( ~ ( member_set_o @ A @ B5 )
       => ( A = B ) )
     => ( member_set_o @ A @ ( insert_set_o @ B @ B5 ) ) ) ).

% insertCI
thf(fact_762_insertCI,axiom,
    ! [A: nat,B5: set_nat,B: nat] :
      ( ( ~ ( member_nat @ A @ B5 )
       => ( A = B ) )
     => ( member_nat @ A @ ( insert_nat @ B @ B5 ) ) ) ).

% insertCI
thf(fact_763_insertCI,axiom,
    ! [A: relational_fmla_a_b > relational_fmla_a_b,B5: set_Re1288005135514575379la_a_b,B: relational_fmla_a_b > relational_fmla_a_b] :
      ( ( ~ ( member8433577210552456052la_a_b @ A @ B5 )
       => ( A = B ) )
     => ( member8433577210552456052la_a_b @ A @ ( insert8904949763332019597la_a_b @ B @ B5 ) ) ) ).

% insertCI
thf(fact_764_insertCI,axiom,
    ! [A: b,B5: set_b,B: b] :
      ( ( ~ ( member_b @ A @ B5 )
       => ( A = B ) )
     => ( member_b @ A @ ( insert_b @ B @ B5 ) ) ) ).

% insertCI
thf(fact_765_insertCI,axiom,
    ! [A: $o,B5: set_o,B: $o] :
      ( ( ~ ( member_o @ A @ B5 )
       => ( A = B ) )
     => ( member_o @ A @ ( insert_o @ B @ B5 ) ) ) ).

% insertCI
thf(fact_766_Un__iff,axiom,
    ! [C: list_a,A2: set_list_a,B5: set_list_a] :
      ( ( member_list_a @ C @ ( sup_sup_set_list_a @ A2 @ B5 ) )
      = ( ( member_list_a @ C @ A2 )
        | ( member_list_a @ C @ B5 ) ) ) ).

% Un_iff
thf(fact_767_Un__iff,axiom,
    ! [C: set_o,A2: set_set_o,B5: set_set_o] :
      ( ( member_set_o @ C @ ( sup_sup_set_set_o @ A2 @ B5 ) )
      = ( ( member_set_o @ C @ A2 )
        | ( member_set_o @ C @ B5 ) ) ) ).

% Un_iff
thf(fact_768_Un__iff,axiom,
    ! [C: relational_fmla_a_b > relational_fmla_a_b,A2: set_Re1288005135514575379la_a_b,B5: set_Re1288005135514575379la_a_b] :
      ( ( member8433577210552456052la_a_b @ C @ ( sup_su3223031376161914175la_a_b @ A2 @ B5 ) )
      = ( ( member8433577210552456052la_a_b @ C @ A2 )
        | ( member8433577210552456052la_a_b @ C @ B5 ) ) ) ).

% Un_iff
thf(fact_769_Un__iff,axiom,
    ! [C: product_prod_o_o,A2: set_Product_prod_o_o,B5: set_Product_prod_o_o] :
      ( ( member7466972457876170832od_o_o @ C @ ( sup_su5769328420594410459od_o_o @ A2 @ B5 ) )
      = ( ( member7466972457876170832od_o_o @ C @ A2 )
        | ( member7466972457876170832od_o_o @ C @ B5 ) ) ) ).

% Un_iff
thf(fact_770_Un__iff,axiom,
    ! [C: nat,A2: set_nat,B5: set_nat] :
      ( ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B5 ) )
      = ( ( member_nat @ C @ A2 )
        | ( member_nat @ C @ B5 ) ) ) ).

% Un_iff
thf(fact_771_Un__iff,axiom,
    ! [C: b,A2: set_b,B5: set_b] :
      ( ( member_b @ C @ ( sup_sup_set_b @ A2 @ B5 ) )
      = ( ( member_b @ C @ A2 )
        | ( member_b @ C @ B5 ) ) ) ).

% Un_iff
thf(fact_772_Un__iff,axiom,
    ! [C: $o,A2: set_o,B5: set_o] :
      ( ( member_o @ C @ ( sup_sup_set_o @ A2 @ B5 ) )
      = ( ( member_o @ C @ A2 )
        | ( member_o @ C @ B5 ) ) ) ).

% Un_iff
thf(fact_773_UnCI,axiom,
    ! [C: list_a,B5: set_list_a,A2: set_list_a] :
      ( ( ~ ( member_list_a @ C @ B5 )
       => ( member_list_a @ C @ A2 ) )
     => ( member_list_a @ C @ ( sup_sup_set_list_a @ A2 @ B5 ) ) ) ).

% UnCI
thf(fact_774_UnCI,axiom,
    ! [C: set_o,B5: set_set_o,A2: set_set_o] :
      ( ( ~ ( member_set_o @ C @ B5 )
       => ( member_set_o @ C @ A2 ) )
     => ( member_set_o @ C @ ( sup_sup_set_set_o @ A2 @ B5 ) ) ) ).

% UnCI
thf(fact_775_UnCI,axiom,
    ! [C: relational_fmla_a_b > relational_fmla_a_b,B5: set_Re1288005135514575379la_a_b,A2: set_Re1288005135514575379la_a_b] :
      ( ( ~ ( member8433577210552456052la_a_b @ C @ B5 )
       => ( member8433577210552456052la_a_b @ C @ A2 ) )
     => ( member8433577210552456052la_a_b @ C @ ( sup_su3223031376161914175la_a_b @ A2 @ B5 ) ) ) ).

% UnCI
thf(fact_776_UnCI,axiom,
    ! [C: product_prod_o_o,B5: set_Product_prod_o_o,A2: set_Product_prod_o_o] :
      ( ( ~ ( member7466972457876170832od_o_o @ C @ B5 )
       => ( member7466972457876170832od_o_o @ C @ A2 ) )
     => ( member7466972457876170832od_o_o @ C @ ( sup_su5769328420594410459od_o_o @ A2 @ B5 ) ) ) ).

% UnCI
thf(fact_777_UnCI,axiom,
    ! [C: nat,B5: set_nat,A2: set_nat] :
      ( ( ~ ( member_nat @ C @ B5 )
       => ( member_nat @ C @ A2 ) )
     => ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B5 ) ) ) ).

% UnCI
thf(fact_778_UnCI,axiom,
    ! [C: b,B5: set_b,A2: set_b] :
      ( ( ~ ( member_b @ C @ B5 )
       => ( member_b @ C @ A2 ) )
     => ( member_b @ C @ ( sup_sup_set_b @ A2 @ B5 ) ) ) ).

% UnCI
thf(fact_779_UnCI,axiom,
    ! [C: $o,B5: set_o,A2: set_o] :
      ( ( ~ ( member_o @ C @ B5 )
       => ( member_o @ C @ A2 ) )
     => ( member_o @ C @ ( sup_sup_set_o @ A2 @ B5 ) ) ) ).

% UnCI
thf(fact_780_image__empty,axiom,
    ! [F2: $o > $o] :
      ( ( image_o_o @ F2 @ bot_bot_set_o )
      = bot_bot_set_o ) ).

% image_empty
thf(fact_781_image__empty,axiom,
    ! [F2: $o > b] :
      ( ( image_o_b @ F2 @ bot_bot_set_o )
      = bot_bot_set_b ) ).

% image_empty
thf(fact_782_image__empty,axiom,
    ! [F2: $o > nat] :
      ( ( image_o_nat @ F2 @ bot_bot_set_o )
      = bot_bot_set_nat ) ).

% image_empty
thf(fact_783_image__empty,axiom,
    ! [F2: b > $o] :
      ( ( image_b_o @ F2 @ bot_bot_set_b )
      = bot_bot_set_o ) ).

% image_empty
thf(fact_784_image__empty,axiom,
    ! [F2: b > b] :
      ( ( image_b_b @ F2 @ bot_bot_set_b )
      = bot_bot_set_b ) ).

% image_empty
thf(fact_785_image__empty,axiom,
    ! [F2: b > nat] :
      ( ( image_b_nat @ F2 @ bot_bot_set_b )
      = bot_bot_set_nat ) ).

% image_empty
thf(fact_786_image__empty,axiom,
    ! [F2: nat > $o] :
      ( ( image_nat_o @ F2 @ bot_bot_set_nat )
      = bot_bot_set_o ) ).

% image_empty
thf(fact_787_image__empty,axiom,
    ! [F2: nat > b] :
      ( ( image_nat_b @ F2 @ bot_bot_set_nat )
      = bot_bot_set_b ) ).

% image_empty
thf(fact_788_image__empty,axiom,
    ! [F2: nat > nat] :
      ( ( image_nat_nat @ F2 @ bot_bot_set_nat )
      = bot_bot_set_nat ) ).

% image_empty
thf(fact_789_image__empty,axiom,
    ! [F2: set_o > $o] :
      ( ( image_set_o_o @ F2 @ bot_bot_set_set_o )
      = bot_bot_set_o ) ).

% image_empty
thf(fact_790_empty__is__image,axiom,
    ! [F2: $o > $o,A2: set_o] :
      ( ( bot_bot_set_o
        = ( image_o_o @ F2 @ A2 ) )
      = ( A2 = bot_bot_set_o ) ) ).

% empty_is_image
thf(fact_791_empty__is__image,axiom,
    ! [F2: b > $o,A2: set_b] :
      ( ( bot_bot_set_o
        = ( image_b_o @ F2 @ A2 ) )
      = ( A2 = bot_bot_set_b ) ) ).

% empty_is_image
thf(fact_792_empty__is__image,axiom,
    ! [F2: nat > $o,A2: set_nat] :
      ( ( bot_bot_set_o
        = ( image_nat_o @ F2 @ A2 ) )
      = ( A2 = bot_bot_set_nat ) ) ).

% empty_is_image
thf(fact_793_empty__is__image,axiom,
    ! [F2: $o > b,A2: set_o] :
      ( ( bot_bot_set_b
        = ( image_o_b @ F2 @ A2 ) )
      = ( A2 = bot_bot_set_o ) ) ).

% empty_is_image
thf(fact_794_empty__is__image,axiom,
    ! [F2: b > b,A2: set_b] :
      ( ( bot_bot_set_b
        = ( image_b_b @ F2 @ A2 ) )
      = ( A2 = bot_bot_set_b ) ) ).

% empty_is_image
thf(fact_795_empty__is__image,axiom,
    ! [F2: nat > b,A2: set_nat] :
      ( ( bot_bot_set_b
        = ( image_nat_b @ F2 @ A2 ) )
      = ( A2 = bot_bot_set_nat ) ) ).

% empty_is_image
thf(fact_796_empty__is__image,axiom,
    ! [F2: $o > nat,A2: set_o] :
      ( ( bot_bot_set_nat
        = ( image_o_nat @ F2 @ A2 ) )
      = ( A2 = bot_bot_set_o ) ) ).

% empty_is_image
thf(fact_797_empty__is__image,axiom,
    ! [F2: b > nat,A2: set_b] :
      ( ( bot_bot_set_nat
        = ( image_b_nat @ F2 @ A2 ) )
      = ( A2 = bot_bot_set_b ) ) ).

% empty_is_image
thf(fact_798_empty__is__image,axiom,
    ! [F2: nat > nat,A2: set_nat] :
      ( ( bot_bot_set_nat
        = ( image_nat_nat @ F2 @ A2 ) )
      = ( A2 = bot_bot_set_nat ) ) ).

% empty_is_image
thf(fact_799_empty__is__image,axiom,
    ! [F2: set_o > $o,A2: set_set_o] :
      ( ( bot_bot_set_o
        = ( image_set_o_o @ F2 @ A2 ) )
      = ( A2 = bot_bot_set_set_o ) ) ).

% empty_is_image
thf(fact_800_image__is__empty,axiom,
    ! [F2: $o > $o,A2: set_o] :
      ( ( ( image_o_o @ F2 @ A2 )
        = bot_bot_set_o )
      = ( A2 = bot_bot_set_o ) ) ).

% image_is_empty
thf(fact_801_image__is__empty,axiom,
    ! [F2: b > $o,A2: set_b] :
      ( ( ( image_b_o @ F2 @ A2 )
        = bot_bot_set_o )
      = ( A2 = bot_bot_set_b ) ) ).

% image_is_empty
thf(fact_802_image__is__empty,axiom,
    ! [F2: nat > $o,A2: set_nat] :
      ( ( ( image_nat_o @ F2 @ A2 )
        = bot_bot_set_o )
      = ( A2 = bot_bot_set_nat ) ) ).

% image_is_empty
thf(fact_803_image__is__empty,axiom,
    ! [F2: $o > b,A2: set_o] :
      ( ( ( image_o_b @ F2 @ A2 )
        = bot_bot_set_b )
      = ( A2 = bot_bot_set_o ) ) ).

% image_is_empty
thf(fact_804_image__is__empty,axiom,
    ! [F2: b > b,A2: set_b] :
      ( ( ( image_b_b @ F2 @ A2 )
        = bot_bot_set_b )
      = ( A2 = bot_bot_set_b ) ) ).

% image_is_empty
thf(fact_805_image__is__empty,axiom,
    ! [F2: nat > b,A2: set_nat] :
      ( ( ( image_nat_b @ F2 @ A2 )
        = bot_bot_set_b )
      = ( A2 = bot_bot_set_nat ) ) ).

% image_is_empty
thf(fact_806_image__is__empty,axiom,
    ! [F2: $o > nat,A2: set_o] :
      ( ( ( image_o_nat @ F2 @ A2 )
        = bot_bot_set_nat )
      = ( A2 = bot_bot_set_o ) ) ).

% image_is_empty
thf(fact_807_image__is__empty,axiom,
    ! [F2: b > nat,A2: set_b] :
      ( ( ( image_b_nat @ F2 @ A2 )
        = bot_bot_set_nat )
      = ( A2 = bot_bot_set_b ) ) ).

% image_is_empty
thf(fact_808_image__is__empty,axiom,
    ! [F2: nat > nat,A2: set_nat] :
      ( ( ( image_nat_nat @ F2 @ A2 )
        = bot_bot_set_nat )
      = ( A2 = bot_bot_set_nat ) ) ).

% image_is_empty
thf(fact_809_image__is__empty,axiom,
    ! [F2: set_o > $o,A2: set_set_o] :
      ( ( ( image_set_o_o @ F2 @ A2 )
        = bot_bot_set_o )
      = ( A2 = bot_bot_set_set_o ) ) ).

% image_is_empty
thf(fact_810_image__insert,axiom,
    ! [F2: $o > $o,A: $o,B5: set_o] :
      ( ( image_o_o @ F2 @ ( insert_o @ A @ B5 ) )
      = ( insert_o @ ( F2 @ A ) @ ( image_o_o @ F2 @ B5 ) ) ) ).

% image_insert
thf(fact_811_image__insert,axiom,
    ! [F2: $o > nat,A: $o,B5: set_o] :
      ( ( image_o_nat @ F2 @ ( insert_o @ A @ B5 ) )
      = ( insert_nat @ ( F2 @ A ) @ ( image_o_nat @ F2 @ B5 ) ) ) ).

% image_insert
thf(fact_812_image__insert,axiom,
    ! [F2: nat > $o,A: nat,B5: set_nat] :
      ( ( image_nat_o @ F2 @ ( insert_nat @ A @ B5 ) )
      = ( insert_o @ ( F2 @ A ) @ ( image_nat_o @ F2 @ B5 ) ) ) ).

% image_insert
thf(fact_813_image__insert,axiom,
    ! [F2: nat > nat,A: nat,B5: set_nat] :
      ( ( image_nat_nat @ F2 @ ( insert_nat @ A @ B5 ) )
      = ( insert_nat @ ( F2 @ A ) @ ( image_nat_nat @ F2 @ B5 ) ) ) ).

% image_insert
thf(fact_814_image__insert,axiom,
    ! [F2: set_o > $o,A: set_o,B5: set_set_o] :
      ( ( image_set_o_o @ F2 @ ( insert_set_o @ A @ B5 ) )
      = ( insert_o @ ( F2 @ A ) @ ( image_set_o_o @ F2 @ B5 ) ) ) ).

% image_insert
thf(fact_815_image__insert,axiom,
    ! [F2: set_o > set_o,A: set_o,B5: set_set_o] :
      ( ( image_set_o_set_o @ F2 @ ( insert_set_o @ A @ B5 ) )
      = ( insert_set_o @ ( F2 @ A ) @ ( image_set_o_set_o @ F2 @ B5 ) ) ) ).

% image_insert
thf(fact_816_image__insert,axiom,
    ! [F2: $o > relational_fmla_a_b,A: $o,B5: set_o] :
      ( ( image_6295478110982790130la_a_b @ F2 @ ( insert_o @ A @ B5 ) )
      = ( insert7010464514620295119la_a_b @ ( F2 @ A ) @ ( image_6295478110982790130la_a_b @ F2 @ B5 ) ) ) ).

% image_insert
thf(fact_817_image__insert,axiom,
    ! [F2: $o > product_prod_o_o,A: $o,B5: set_o] :
      ( ( image_4057150146340385428od_o_o @ F2 @ ( insert_o @ A @ B5 ) )
      = ( insert6201435330877294327od_o_o @ ( F2 @ A ) @ ( image_4057150146340385428od_o_o @ F2 @ B5 ) ) ) ).

% image_insert
thf(fact_818_image__insert,axiom,
    ! [F2: relational_fmla_a_b > $o,A: relational_fmla_a_b,B5: set_Re381260168593705685la_a_b] :
      ( ( image_1316678413157792882_a_b_o @ F2 @ ( insert7010464514620295119la_a_b @ A @ B5 ) )
      = ( insert_o @ ( F2 @ A ) @ ( image_1316678413157792882_a_b_o @ F2 @ B5 ) ) ) ).

% image_insert
thf(fact_819_image__insert,axiom,
    ! [F2: relational_fmla_a_b > nat,A: relational_fmla_a_b,B5: set_Re381260168593705685la_a_b] :
      ( ( image_341122591648980342_b_nat @ F2 @ ( insert7010464514620295119la_a_b @ A @ B5 ) )
      = ( insert_nat @ ( F2 @ A ) @ ( image_341122591648980342_b_nat @ F2 @ B5 ) ) ) ).

% image_insert
thf(fact_820_insert__image,axiom,
    ! [X3: $o,A2: set_o,F2: $o > $o] :
      ( ( member_o @ X3 @ A2 )
     => ( ( insert_o @ ( F2 @ X3 ) @ ( image_o_o @ F2 @ A2 ) )
        = ( image_o_o @ F2 @ A2 ) ) ) ).

% insert_image
thf(fact_821_insert__image,axiom,
    ! [X3: $o,A2: set_o,F2: $o > nat] :
      ( ( member_o @ X3 @ A2 )
     => ( ( insert_nat @ ( F2 @ X3 ) @ ( image_o_nat @ F2 @ A2 ) )
        = ( image_o_nat @ F2 @ A2 ) ) ) ).

% insert_image
thf(fact_822_insert__image,axiom,
    ! [X3: nat,A2: set_nat,F2: nat > $o] :
      ( ( member_nat @ X3 @ A2 )
     => ( ( insert_o @ ( F2 @ X3 ) @ ( image_nat_o @ F2 @ A2 ) )
        = ( image_nat_o @ F2 @ A2 ) ) ) ).

% insert_image
thf(fact_823_insert__image,axiom,
    ! [X3: nat,A2: set_nat,F2: nat > nat] :
      ( ( member_nat @ X3 @ A2 )
     => ( ( insert_nat @ ( F2 @ X3 ) @ ( image_nat_nat @ F2 @ A2 ) )
        = ( image_nat_nat @ F2 @ A2 ) ) ) ).

% insert_image
thf(fact_824_insert__image,axiom,
    ! [X3: b,A2: set_b,F2: b > $o] :
      ( ( member_b @ X3 @ A2 )
     => ( ( insert_o @ ( F2 @ X3 ) @ ( image_b_o @ F2 @ A2 ) )
        = ( image_b_o @ F2 @ A2 ) ) ) ).

% insert_image
thf(fact_825_insert__image,axiom,
    ! [X3: b,A2: set_b,F2: b > nat] :
      ( ( member_b @ X3 @ A2 )
     => ( ( insert_nat @ ( F2 @ X3 ) @ ( image_b_nat @ F2 @ A2 ) )
        = ( image_b_nat @ F2 @ A2 ) ) ) ).

% insert_image
thf(fact_826_insert__image,axiom,
    ! [X3: list_a,A2: set_list_a,F2: list_a > $o] :
      ( ( member_list_a @ X3 @ A2 )
     => ( ( insert_o @ ( F2 @ X3 ) @ ( image_list_a_o @ F2 @ A2 ) )
        = ( image_list_a_o @ F2 @ A2 ) ) ) ).

% insert_image
thf(fact_827_insert__image,axiom,
    ! [X3: list_a,A2: set_list_a,F2: list_a > nat] :
      ( ( member_list_a @ X3 @ A2 )
     => ( ( insert_nat @ ( F2 @ X3 ) @ ( image_list_a_nat @ F2 @ A2 ) )
        = ( image_list_a_nat @ F2 @ A2 ) ) ) ).

% insert_image
thf(fact_828_insert__image,axiom,
    ! [X3: set_o,A2: set_set_o,F2: set_o > $o] :
      ( ( member_set_o @ X3 @ A2 )
     => ( ( insert_o @ ( F2 @ X3 ) @ ( image_set_o_o @ F2 @ A2 ) )
        = ( image_set_o_o @ F2 @ A2 ) ) ) ).

% insert_image
thf(fact_829_insert__image,axiom,
    ! [X3: set_o,A2: set_set_o,F2: set_o > nat] :
      ( ( member_set_o @ X3 @ A2 )
     => ( ( insert_nat @ ( F2 @ X3 ) @ ( image_set_o_nat @ F2 @ A2 ) )
        = ( image_set_o_nat @ F2 @ A2 ) ) ) ).

% insert_image
thf(fact_830_singletonI,axiom,
    ! [A: list_a] : ( member_list_a @ A @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) ).

% singletonI
thf(fact_831_singletonI,axiom,
    ! [A: set_o] : ( member_set_o @ A @ ( insert_set_o @ A @ bot_bot_set_set_o ) ) ).

% singletonI
thf(fact_832_singletonI,axiom,
    ! [A: relational_fmla_a_b > relational_fmla_a_b] : ( member8433577210552456052la_a_b @ A @ ( insert8904949763332019597la_a_b @ A @ bot_bo9179849999556691623la_a_b ) ) ).

% singletonI
thf(fact_833_singletonI,axiom,
    ! [A: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ A @ ( insert7010464514620295119la_a_b @ A @ bot_bo4495933725496725865la_a_b ) ) ).

% singletonI
thf(fact_834_singletonI,axiom,
    ! [A: b] : ( member_b @ A @ ( insert_b @ A @ bot_bot_set_b ) ) ).

% singletonI
thf(fact_835_singletonI,axiom,
    ! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% singletonI
thf(fact_836_singletonI,axiom,
    ! [A: product_prod_o_o] : ( member7466972457876170832od_o_o @ A @ ( insert6201435330877294327od_o_o @ A @ bot_bo7073875226086086771od_o_o ) ) ).

% singletonI
thf(fact_837_singletonI,axiom,
    ! [A: $o] : ( member_o @ A @ ( insert_o @ A @ bot_bot_set_o ) ) ).

% singletonI
thf(fact_838_Un__empty,axiom,
    ! [A2: set_Re381260168593705685la_a_b,B5: set_Re381260168593705685la_a_b] :
      ( ( ( sup_su5130108678486352897la_a_b @ A2 @ B5 )
        = bot_bo4495933725496725865la_a_b )
      = ( ( A2 = bot_bo4495933725496725865la_a_b )
        & ( B5 = bot_bo4495933725496725865la_a_b ) ) ) ).

% Un_empty
thf(fact_839_Un__empty,axiom,
    ! [A2: set_b,B5: set_b] :
      ( ( ( sup_sup_set_b @ A2 @ B5 )
        = bot_bot_set_b )
      = ( ( A2 = bot_bot_set_b )
        & ( B5 = bot_bot_set_b ) ) ) ).

% Un_empty
thf(fact_840_Un__empty,axiom,
    ! [A2: set_nat,B5: set_nat] :
      ( ( ( sup_sup_set_nat @ A2 @ B5 )
        = bot_bot_set_nat )
      = ( ( A2 = bot_bot_set_nat )
        & ( B5 = bot_bot_set_nat ) ) ) ).

% Un_empty
thf(fact_841_Un__empty,axiom,
    ! [A2: set_Product_prod_o_o,B5: set_Product_prod_o_o] :
      ( ( ( sup_su5769328420594410459od_o_o @ A2 @ B5 )
        = bot_bo7073875226086086771od_o_o )
      = ( ( A2 = bot_bo7073875226086086771od_o_o )
        & ( B5 = bot_bo7073875226086086771od_o_o ) ) ) ).

% Un_empty
thf(fact_842_Un__empty,axiom,
    ! [A2: set_o,B5: set_o] :
      ( ( ( sup_sup_set_o @ A2 @ B5 )
        = bot_bot_set_o )
      = ( ( A2 = bot_bot_set_o )
        & ( B5 = bot_bot_set_o ) ) ) ).

% Un_empty
thf(fact_843_Un__insert__right,axiom,
    ! [A2: set_Re381260168593705685la_a_b,A: relational_fmla_a_b,B5: set_Re381260168593705685la_a_b] :
      ( ( sup_su5130108678486352897la_a_b @ A2 @ ( insert7010464514620295119la_a_b @ A @ B5 ) )
      = ( insert7010464514620295119la_a_b @ A @ ( sup_su5130108678486352897la_a_b @ A2 @ B5 ) ) ) ).

% Un_insert_right
thf(fact_844_Un__insert__right,axiom,
    ! [A2: set_Product_prod_o_o,A: product_prod_o_o,B5: set_Product_prod_o_o] :
      ( ( sup_su5769328420594410459od_o_o @ A2 @ ( insert6201435330877294327od_o_o @ A @ B5 ) )
      = ( insert6201435330877294327od_o_o @ A @ ( sup_su5769328420594410459od_o_o @ A2 @ B5 ) ) ) ).

% Un_insert_right
thf(fact_845_Un__insert__right,axiom,
    ! [A2: set_nat,A: nat,B5: set_nat] :
      ( ( sup_sup_set_nat @ A2 @ ( insert_nat @ A @ B5 ) )
      = ( insert_nat @ A @ ( sup_sup_set_nat @ A2 @ B5 ) ) ) ).

% Un_insert_right
thf(fact_846_Un__insert__right,axiom,
    ! [A2: set_b,A: b,B5: set_b] :
      ( ( sup_sup_set_b @ A2 @ ( insert_b @ A @ B5 ) )
      = ( insert_b @ A @ ( sup_sup_set_b @ A2 @ B5 ) ) ) ).

% Un_insert_right
thf(fact_847_Un__insert__right,axiom,
    ! [A2: set_o,A: $o,B5: set_o] :
      ( ( sup_sup_set_o @ A2 @ ( insert_o @ A @ B5 ) )
      = ( insert_o @ A @ ( sup_sup_set_o @ A2 @ B5 ) ) ) ).

% Un_insert_right
thf(fact_848_Un__insert__left,axiom,
    ! [A: relational_fmla_a_b,B5: set_Re381260168593705685la_a_b,C4: set_Re381260168593705685la_a_b] :
      ( ( sup_su5130108678486352897la_a_b @ ( insert7010464514620295119la_a_b @ A @ B5 ) @ C4 )
      = ( insert7010464514620295119la_a_b @ A @ ( sup_su5130108678486352897la_a_b @ B5 @ C4 ) ) ) ).

% Un_insert_left
thf(fact_849_Un__insert__left,axiom,
    ! [A: product_prod_o_o,B5: set_Product_prod_o_o,C4: set_Product_prod_o_o] :
      ( ( sup_su5769328420594410459od_o_o @ ( insert6201435330877294327od_o_o @ A @ B5 ) @ C4 )
      = ( insert6201435330877294327od_o_o @ A @ ( sup_su5769328420594410459od_o_o @ B5 @ C4 ) ) ) ).

% Un_insert_left
thf(fact_850_Un__insert__left,axiom,
    ! [A: nat,B5: set_nat,C4: set_nat] :
      ( ( sup_sup_set_nat @ ( insert_nat @ A @ B5 ) @ C4 )
      = ( insert_nat @ A @ ( sup_sup_set_nat @ B5 @ C4 ) ) ) ).

% Un_insert_left
thf(fact_851_Un__insert__left,axiom,
    ! [A: b,B5: set_b,C4: set_b] :
      ( ( sup_sup_set_b @ ( insert_b @ A @ B5 ) @ C4 )
      = ( insert_b @ A @ ( sup_sup_set_b @ B5 @ C4 ) ) ) ).

% Un_insert_left
thf(fact_852_Un__insert__left,axiom,
    ! [A: $o,B5: set_o,C4: set_o] :
      ( ( sup_sup_set_o @ ( insert_o @ A @ B5 ) @ C4 )
      = ( insert_o @ A @ ( sup_sup_set_o @ B5 @ C4 ) ) ) ).

% Un_insert_left
thf(fact_853_is__Bool__exists,axiom,
    ! [X3: nat,Q: relational_fmla_a_b] :
      ( ( relati6551038146797045342ol_a_b @ ( relati3989891337220013914ts_a_b @ X3 @ Q ) )
      = ( relati6551038146797045342ol_a_b @ Q ) ) ).

% is_Bool_exists
thf(fact_854_singleton__conv2,axiom,
    ! [A: relational_fmla_a_b] :
      ( ( collec3419995626248312948la_a_b
        @ ( ^ [Y6: relational_fmla_a_b,Z4: relational_fmla_a_b] : ( Y6 = Z4 )
          @ A ) )
      = ( insert7010464514620295119la_a_b @ A @ bot_bo4495933725496725865la_a_b ) ) ).

% singleton_conv2
thf(fact_855_singleton__conv2,axiom,
    ! [A: b] :
      ( ( collect_b
        @ ( ^ [Y6: b,Z4: b] : ( Y6 = Z4 )
          @ A ) )
      = ( insert_b @ A @ bot_bot_set_b ) ) ).

% singleton_conv2
thf(fact_856_singleton__conv2,axiom,
    ! [A: nat] :
      ( ( collect_nat
        @ ( ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
          @ A ) )
      = ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% singleton_conv2
thf(fact_857_singleton__conv2,axiom,
    ! [A: product_prod_o_o] :
      ( ( collec3167064739498627218od_o_o
        @ ( ^ [Y6: product_prod_o_o,Z4: product_prod_o_o] : ( Y6 = Z4 )
          @ A ) )
      = ( insert6201435330877294327od_o_o @ A @ bot_bo7073875226086086771od_o_o ) ) ).

% singleton_conv2
thf(fact_858_singleton__conv2,axiom,
    ! [A: $o] :
      ( ( collect_o
        @ ( ^ [Y6: $o,Z4: $o] : ( Y6 = Z4 )
          @ A ) )
      = ( insert_o @ A @ bot_bot_set_o ) ) ).

% singleton_conv2
thf(fact_859_singleton__conv,axiom,
    ! [A: relational_fmla_a_b] :
      ( ( collec3419995626248312948la_a_b
        @ ^ [X: relational_fmla_a_b] : ( X = A ) )
      = ( insert7010464514620295119la_a_b @ A @ bot_bo4495933725496725865la_a_b ) ) ).

% singleton_conv
thf(fact_860_singleton__conv,axiom,
    ! [A: b] :
      ( ( collect_b
        @ ^ [X: b] : ( X = A ) )
      = ( insert_b @ A @ bot_bot_set_b ) ) ).

% singleton_conv
thf(fact_861_singleton__conv,axiom,
    ! [A: nat] :
      ( ( collect_nat
        @ ^ [X: nat] : ( X = A ) )
      = ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% singleton_conv
thf(fact_862_singleton__conv,axiom,
    ! [A: product_prod_o_o] :
      ( ( collec3167064739498627218od_o_o
        @ ^ [X: product_prod_o_o] : ( X = A ) )
      = ( insert6201435330877294327od_o_o @ A @ bot_bo7073875226086086771od_o_o ) ) ).

% singleton_conv
thf(fact_863_singleton__conv,axiom,
    ! [A: $o] :
      ( ( collect_o
        @ ^ [X: $o] : ( X = A ) )
      = ( insert_o @ A @ bot_bot_set_o ) ) ).

% singleton_conv
thf(fact_864_gen__Bool__False,axiom,
    ! [X3: nat,G: set_Re381260168593705685la_a_b] :
      ( ( relational_gen_a_b @ X3 @ ( relational_Bool_a_b @ $false ) @ G )
      = ( G = bot_bo4495933725496725865la_a_b ) ) ).

% gen_Bool_False
thf(fact_865_fmla_Ocollapse_I2_J,axiom,
    ! [Fmla: relational_fmla_a_b] :
      ( ( relati6551038146797045342ol_a_b @ Fmla )
     => ( ( relational_Bool_a_b @ ( relati2638701775882563405ol_a_b @ Fmla ) )
        = Fmla ) ) ).

% fmla.collapse(2)
thf(fact_866_gen__Eq,axiom,
    ! [Z: nat,A: nat,T: relational_term_a,G: set_Re381260168593705685la_a_b] :
      ( ( relational_gen_a_b @ Z @ ( relational_Eq_a_b @ A @ T ) @ G )
      = ( ( Z = A )
        & ? [C3: a] :
            ( ( T
              = ( relational_Const_a @ C3 ) )
            & ( G
              = ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ A @ T ) @ bot_bo4495933725496725865la_a_b ) ) ) ) ) ).

% gen_Eq
thf(fact_867_mk__disjoint__insert,axiom,
    ! [A: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
      ( ( member4680049679412964150la_a_b @ A @ A2 )
     => ? [B6: set_Re381260168593705685la_a_b] :
          ( ( A2
            = ( insert7010464514620295119la_a_b @ A @ B6 ) )
          & ~ ( member4680049679412964150la_a_b @ A @ B6 ) ) ) ).

% mk_disjoint_insert
thf(fact_868_mk__disjoint__insert,axiom,
    ! [A: product_prod_o_o,A2: set_Product_prod_o_o] :
      ( ( member7466972457876170832od_o_o @ A @ A2 )
     => ? [B6: set_Product_prod_o_o] :
          ( ( A2
            = ( insert6201435330877294327od_o_o @ A @ B6 ) )
          & ~ ( member7466972457876170832od_o_o @ A @ B6 ) ) ) ).

% mk_disjoint_insert
thf(fact_869_mk__disjoint__insert,axiom,
    ! [A: list_a,A2: set_list_a] :
      ( ( member_list_a @ A @ A2 )
     => ? [B6: set_list_a] :
          ( ( A2
            = ( insert_list_a @ A @ B6 ) )
          & ~ ( member_list_a @ A @ B6 ) ) ) ).

% mk_disjoint_insert
thf(fact_870_mk__disjoint__insert,axiom,
    ! [A: set_o,A2: set_set_o] :
      ( ( member_set_o @ A @ A2 )
     => ? [B6: set_set_o] :
          ( ( A2
            = ( insert_set_o @ A @ B6 ) )
          & ~ ( member_set_o @ A @ B6 ) ) ) ).

% mk_disjoint_insert
thf(fact_871_mk__disjoint__insert,axiom,
    ! [A: nat,A2: set_nat] :
      ( ( member_nat @ A @ A2 )
     => ? [B6: set_nat] :
          ( ( A2
            = ( insert_nat @ A @ B6 ) )
          & ~ ( member_nat @ A @ B6 ) ) ) ).

% mk_disjoint_insert
thf(fact_872_mk__disjoint__insert,axiom,
    ! [A: relational_fmla_a_b > relational_fmla_a_b,A2: set_Re1288005135514575379la_a_b] :
      ( ( member8433577210552456052la_a_b @ A @ A2 )
     => ? [B6: set_Re1288005135514575379la_a_b] :
          ( ( A2
            = ( insert8904949763332019597la_a_b @ A @ B6 ) )
          & ~ ( member8433577210552456052la_a_b @ A @ B6 ) ) ) ).

% mk_disjoint_insert
thf(fact_873_mk__disjoint__insert,axiom,
    ! [A: b,A2: set_b] :
      ( ( member_b @ A @ A2 )
     => ? [B6: set_b] :
          ( ( A2
            = ( insert_b @ A @ B6 ) )
          & ~ ( member_b @ A @ B6 ) ) ) ).

% mk_disjoint_insert
thf(fact_874_mk__disjoint__insert,axiom,
    ! [A: $o,A2: set_o] :
      ( ( member_o @ A @ A2 )
     => ? [B6: set_o] :
          ( ( A2
            = ( insert_o @ A @ B6 ) )
          & ~ ( member_o @ A @ B6 ) ) ) ).

% mk_disjoint_insert
thf(fact_875_singleton__inject,axiom,
    ! [A: relational_fmla_a_b,B: relational_fmla_a_b] :
      ( ( ( insert7010464514620295119la_a_b @ A @ bot_bo4495933725496725865la_a_b )
        = ( insert7010464514620295119la_a_b @ B @ bot_bo4495933725496725865la_a_b ) )
     => ( A = B ) ) ).

% singleton_inject
thf(fact_876_singleton__inject,axiom,
    ! [A: b,B: b] :
      ( ( ( insert_b @ A @ bot_bot_set_b )
        = ( insert_b @ B @ bot_bot_set_b ) )
     => ( A = B ) ) ).

% singleton_inject
thf(fact_877_singleton__inject,axiom,
    ! [A: nat,B: nat] :
      ( ( ( insert_nat @ A @ bot_bot_set_nat )
        = ( insert_nat @ B @ bot_bot_set_nat ) )
     => ( A = B ) ) ).

% singleton_inject
thf(fact_878_singleton__inject,axiom,
    ! [A: product_prod_o_o,B: product_prod_o_o] :
      ( ( ( insert6201435330877294327od_o_o @ A @ bot_bo7073875226086086771od_o_o )
        = ( insert6201435330877294327od_o_o @ B @ bot_bo7073875226086086771od_o_o ) )
     => ( A = B ) ) ).

% singleton_inject
thf(fact_879_singleton__inject,axiom,
    ! [A: $o,B: $o] :
      ( ( ( insert_o @ A @ bot_bot_set_o )
        = ( insert_o @ B @ bot_bot_set_o ) )
     => ( A = B ) ) ).

% singleton_inject
thf(fact_880_singleton__Un__iff,axiom,
    ! [X3: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B5: set_Re381260168593705685la_a_b] :
      ( ( ( insert7010464514620295119la_a_b @ X3 @ bot_bo4495933725496725865la_a_b )
        = ( sup_su5130108678486352897la_a_b @ A2 @ B5 ) )
      = ( ( ( A2 = bot_bo4495933725496725865la_a_b )
          & ( B5
            = ( insert7010464514620295119la_a_b @ X3 @ bot_bo4495933725496725865la_a_b ) ) )
        | ( ( A2
            = ( insert7010464514620295119la_a_b @ X3 @ bot_bo4495933725496725865la_a_b ) )
          & ( B5 = bot_bo4495933725496725865la_a_b ) )
        | ( ( A2
            = ( insert7010464514620295119la_a_b @ X3 @ bot_bo4495933725496725865la_a_b ) )
          & ( B5
            = ( insert7010464514620295119la_a_b @ X3 @ bot_bo4495933725496725865la_a_b ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_881_singleton__Un__iff,axiom,
    ! [X3: b,A2: set_b,B5: set_b] :
      ( ( ( insert_b @ X3 @ bot_bot_set_b )
        = ( sup_sup_set_b @ A2 @ B5 ) )
      = ( ( ( A2 = bot_bot_set_b )
          & ( B5
            = ( insert_b @ X3 @ bot_bot_set_b ) ) )
        | ( ( A2
            = ( insert_b @ X3 @ bot_bot_set_b ) )
          & ( B5 = bot_bot_set_b ) )
        | ( ( A2
            = ( insert_b @ X3 @ bot_bot_set_b ) )
          & ( B5
            = ( insert_b @ X3 @ bot_bot_set_b ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_882_singleton__Un__iff,axiom,
    ! [X3: nat,A2: set_nat,B5: set_nat] :
      ( ( ( insert_nat @ X3 @ bot_bot_set_nat )
        = ( sup_sup_set_nat @ A2 @ B5 ) )
      = ( ( ( A2 = bot_bot_set_nat )
          & ( B5
            = ( insert_nat @ X3 @ bot_bot_set_nat ) ) )
        | ( ( A2
            = ( insert_nat @ X3 @ bot_bot_set_nat ) )
          & ( B5 = bot_bot_set_nat ) )
        | ( ( A2
            = ( insert_nat @ X3 @ bot_bot_set_nat ) )
          & ( B5
            = ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_883_singleton__Un__iff,axiom,
    ! [X3: product_prod_o_o,A2: set_Product_prod_o_o,B5: set_Product_prod_o_o] :
      ( ( ( insert6201435330877294327od_o_o @ X3 @ bot_bo7073875226086086771od_o_o )
        = ( sup_su5769328420594410459od_o_o @ A2 @ B5 ) )
      = ( ( ( A2 = bot_bo7073875226086086771od_o_o )
          & ( B5
            = ( insert6201435330877294327od_o_o @ X3 @ bot_bo7073875226086086771od_o_o ) ) )
        | ( ( A2
            = ( insert6201435330877294327od_o_o @ X3 @ bot_bo7073875226086086771od_o_o ) )
          & ( B5 = bot_bo7073875226086086771od_o_o ) )
        | ( ( A2
            = ( insert6201435330877294327od_o_o @ X3 @ bot_bo7073875226086086771od_o_o ) )
          & ( B5
            = ( insert6201435330877294327od_o_o @ X3 @ bot_bo7073875226086086771od_o_o ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_884_singleton__Un__iff,axiom,
    ! [X3: $o,A2: set_o,B5: set_o] :
      ( ( ( insert_o @ X3 @ bot_bot_set_o )
        = ( sup_sup_set_o @ A2 @ B5 ) )
      = ( ( ( A2 = bot_bot_set_o )
          & ( B5
            = ( insert_o @ X3 @ bot_bot_set_o ) ) )
        | ( ( A2
            = ( insert_o @ X3 @ bot_bot_set_o ) )
          & ( B5 = bot_bot_set_o ) )
        | ( ( A2
            = ( insert_o @ X3 @ bot_bot_set_o ) )
          & ( B5
            = ( insert_o @ X3 @ bot_bot_set_o ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_885_insert__not__empty,axiom,
    ! [A: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
      ( ( insert7010464514620295119la_a_b @ A @ A2 )
     != bot_bo4495933725496725865la_a_b ) ).

% insert_not_empty
thf(fact_886_insert__not__empty,axiom,
    ! [A: b,A2: set_b] :
      ( ( insert_b @ A @ A2 )
     != bot_bot_set_b ) ).

% insert_not_empty
thf(fact_887_insert__not__empty,axiom,
    ! [A: nat,A2: set_nat] :
      ( ( insert_nat @ A @ A2 )
     != bot_bot_set_nat ) ).

% insert_not_empty
thf(fact_888_insert__not__empty,axiom,
    ! [A: product_prod_o_o,A2: set_Product_prod_o_o] :
      ( ( insert6201435330877294327od_o_o @ A @ A2 )
     != bot_bo7073875226086086771od_o_o ) ).

% insert_not_empty
thf(fact_889_insert__not__empty,axiom,
    ! [A: $o,A2: set_o] :
      ( ( insert_o @ A @ A2 )
     != bot_bot_set_o ) ).

% insert_not_empty
thf(fact_890_doubleton__eq__iff,axiom,
    ! [A: relational_fmla_a_b,B: relational_fmla_a_b,C: relational_fmla_a_b,D: relational_fmla_a_b] :
      ( ( ( insert7010464514620295119la_a_b @ A @ ( insert7010464514620295119la_a_b @ B @ bot_bo4495933725496725865la_a_b ) )
        = ( insert7010464514620295119la_a_b @ C @ ( insert7010464514620295119la_a_b @ D @ bot_bo4495933725496725865la_a_b ) ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_891_doubleton__eq__iff,axiom,
    ! [A: b,B: b,C: b,D: b] :
      ( ( ( insert_b @ A @ ( insert_b @ B @ bot_bot_set_b ) )
        = ( insert_b @ C @ ( insert_b @ D @ bot_bot_set_b ) ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_892_doubleton__eq__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) )
        = ( insert_nat @ C @ ( insert_nat @ D @ bot_bot_set_nat ) ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_893_doubleton__eq__iff,axiom,
    ! [A: product_prod_o_o,B: product_prod_o_o,C: product_prod_o_o,D: product_prod_o_o] :
      ( ( ( insert6201435330877294327od_o_o @ A @ ( insert6201435330877294327od_o_o @ B @ bot_bo7073875226086086771od_o_o ) )
        = ( insert6201435330877294327od_o_o @ C @ ( insert6201435330877294327od_o_o @ D @ bot_bo7073875226086086771od_o_o ) ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_894_doubleton__eq__iff,axiom,
    ! [A: $o,B: $o,C: $o,D: $o] :
      ( ( ( insert_o @ A @ ( insert_o @ B @ bot_bot_set_o ) )
        = ( insert_o @ C @ ( insert_o @ D @ bot_bot_set_o ) ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_895_Un__singleton__iff,axiom,
    ! [A2: set_Re381260168593705685la_a_b,B5: set_Re381260168593705685la_a_b,X3: relational_fmla_a_b] :
      ( ( ( sup_su5130108678486352897la_a_b @ A2 @ B5 )
        = ( insert7010464514620295119la_a_b @ X3 @ bot_bo4495933725496725865la_a_b ) )
      = ( ( ( A2 = bot_bo4495933725496725865la_a_b )
          & ( B5
            = ( insert7010464514620295119la_a_b @ X3 @ bot_bo4495933725496725865la_a_b ) ) )
        | ( ( A2
            = ( insert7010464514620295119la_a_b @ X3 @ bot_bo4495933725496725865la_a_b ) )
          & ( B5 = bot_bo4495933725496725865la_a_b ) )
        | ( ( A2
            = ( insert7010464514620295119la_a_b @ X3 @ bot_bo4495933725496725865la_a_b ) )
          & ( B5
            = ( insert7010464514620295119la_a_b @ X3 @ bot_bo4495933725496725865la_a_b ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_896_Un__singleton__iff,axiom,
    ! [A2: set_b,B5: set_b,X3: b] :
      ( ( ( sup_sup_set_b @ A2 @ B5 )
        = ( insert_b @ X3 @ bot_bot_set_b ) )
      = ( ( ( A2 = bot_bot_set_b )
          & ( B5
            = ( insert_b @ X3 @ bot_bot_set_b ) ) )
        | ( ( A2
            = ( insert_b @ X3 @ bot_bot_set_b ) )
          & ( B5 = bot_bot_set_b ) )
        | ( ( A2
            = ( insert_b @ X3 @ bot_bot_set_b ) )
          & ( B5
            = ( insert_b @ X3 @ bot_bot_set_b ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_897_Un__singleton__iff,axiom,
    ! [A2: set_nat,B5: set_nat,X3: nat] :
      ( ( ( sup_sup_set_nat @ A2 @ B5 )
        = ( insert_nat @ X3 @ bot_bot_set_nat ) )
      = ( ( ( A2 = bot_bot_set_nat )
          & ( B5
            = ( insert_nat @ X3 @ bot_bot_set_nat ) ) )
        | ( ( A2
            = ( insert_nat @ X3 @ bot_bot_set_nat ) )
          & ( B5 = bot_bot_set_nat ) )
        | ( ( A2
            = ( insert_nat @ X3 @ bot_bot_set_nat ) )
          & ( B5
            = ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_898_Un__singleton__iff,axiom,
    ! [A2: set_Product_prod_o_o,B5: set_Product_prod_o_o,X3: product_prod_o_o] :
      ( ( ( sup_su5769328420594410459od_o_o @ A2 @ B5 )
        = ( insert6201435330877294327od_o_o @ X3 @ bot_bo7073875226086086771od_o_o ) )
      = ( ( ( A2 = bot_bo7073875226086086771od_o_o )
          & ( B5
            = ( insert6201435330877294327od_o_o @ X3 @ bot_bo7073875226086086771od_o_o ) ) )
        | ( ( A2
            = ( insert6201435330877294327od_o_o @ X3 @ bot_bo7073875226086086771od_o_o ) )
          & ( B5 = bot_bo7073875226086086771od_o_o ) )
        | ( ( A2
            = ( insert6201435330877294327od_o_o @ X3 @ bot_bo7073875226086086771od_o_o ) )
          & ( B5
            = ( insert6201435330877294327od_o_o @ X3 @ bot_bo7073875226086086771od_o_o ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_899_Un__singleton__iff,axiom,
    ! [A2: set_o,B5: set_o,X3: $o] :
      ( ( ( sup_sup_set_o @ A2 @ B5 )
        = ( insert_o @ X3 @ bot_bot_set_o ) )
      = ( ( ( A2 = bot_bot_set_o )
          & ( B5
            = ( insert_o @ X3 @ bot_bot_set_o ) ) )
        | ( ( A2
            = ( insert_o @ X3 @ bot_bot_set_o ) )
          & ( B5 = bot_bot_set_o ) )
        | ( ( A2
            = ( insert_o @ X3 @ bot_bot_set_o ) )
          & ( B5
            = ( insert_o @ X3 @ bot_bot_set_o ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_900_Collect__conv__if2,axiom,
    ! [P: relational_fmla_a_b > $o,A: relational_fmla_a_b] :
      ( ( ( P @ A )
       => ( ( collec3419995626248312948la_a_b
            @ ^ [X: relational_fmla_a_b] :
                ( ( A = X )
                & ( P @ X ) ) )
          = ( insert7010464514620295119la_a_b @ A @ bot_bo4495933725496725865la_a_b ) ) )
      & ( ~ ( P @ A )
       => ( ( collec3419995626248312948la_a_b
            @ ^ [X: relational_fmla_a_b] :
                ( ( A = X )
                & ( P @ X ) ) )
          = bot_bo4495933725496725865la_a_b ) ) ) ).

% Collect_conv_if2
thf(fact_901_Collect__conv__if2,axiom,
    ! [P: b > $o,A: b] :
      ( ( ( P @ A )
       => ( ( collect_b
            @ ^ [X: b] :
                ( ( A = X )
                & ( P @ X ) ) )
          = ( insert_b @ A @ bot_bot_set_b ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_b
            @ ^ [X: b] :
                ( ( A = X )
                & ( P @ X ) ) )
          = bot_bot_set_b ) ) ) ).

% Collect_conv_if2
thf(fact_902_Collect__conv__if2,axiom,
    ! [P: nat > $o,A: nat] :
      ( ( ( P @ A )
       => ( ( collect_nat
            @ ^ [X: nat] :
                ( ( A = X )
                & ( P @ X ) ) )
          = ( insert_nat @ A @ bot_bot_set_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_nat
            @ ^ [X: nat] :
                ( ( A = X )
                & ( P @ X ) ) )
          = bot_bot_set_nat ) ) ) ).

% Collect_conv_if2
thf(fact_903_Collect__conv__if2,axiom,
    ! [P: product_prod_o_o > $o,A: product_prod_o_o] :
      ( ( ( P @ A )
       => ( ( collec3167064739498627218od_o_o
            @ ^ [X: product_prod_o_o] :
                ( ( A = X )
                & ( P @ X ) ) )
          = ( insert6201435330877294327od_o_o @ A @ bot_bo7073875226086086771od_o_o ) ) )
      & ( ~ ( P @ A )
       => ( ( collec3167064739498627218od_o_o
            @ ^ [X: product_prod_o_o] :
                ( ( A = X )
                & ( P @ X ) ) )
          = bot_bo7073875226086086771od_o_o ) ) ) ).

% Collect_conv_if2
thf(fact_904_Collect__conv__if2,axiom,
    ! [P: $o > $o,A: $o] :
      ( ( ( P @ A )
       => ( ( collect_o
            @ ^ [X: $o] :
                ( ( A = X )
                & ( P @ X ) ) )
          = ( insert_o @ A @ bot_bot_set_o ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_o
            @ ^ [X: $o] :
                ( ( A = X )
                & ( P @ X ) ) )
          = bot_bot_set_o ) ) ) ).

% Collect_conv_if2
thf(fact_905_Un__left__commute,axiom,
    ! [A2: set_Product_prod_o_o,B5: set_Product_prod_o_o,C4: set_Product_prod_o_o] :
      ( ( sup_su5769328420594410459od_o_o @ A2 @ ( sup_su5769328420594410459od_o_o @ B5 @ C4 ) )
      = ( sup_su5769328420594410459od_o_o @ B5 @ ( sup_su5769328420594410459od_o_o @ A2 @ C4 ) ) ) ).

% Un_left_commute
thf(fact_906_Un__left__commute,axiom,
    ! [A2: set_o,B5: set_o,C4: set_o] :
      ( ( sup_sup_set_o @ A2 @ ( sup_sup_set_o @ B5 @ C4 ) )
      = ( sup_sup_set_o @ B5 @ ( sup_sup_set_o @ A2 @ C4 ) ) ) ).

% Un_left_commute
thf(fact_907_Un__left__commute,axiom,
    ! [A2: set_nat,B5: set_nat,C4: set_nat] :
      ( ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B5 @ C4 ) )
      = ( sup_sup_set_nat @ B5 @ ( sup_sup_set_nat @ A2 @ C4 ) ) ) ).

% Un_left_commute
thf(fact_908_Un__left__commute,axiom,
    ! [A2: set_b,B5: set_b,C4: set_b] :
      ( ( sup_sup_set_b @ A2 @ ( sup_sup_set_b @ B5 @ C4 ) )
      = ( sup_sup_set_b @ B5 @ ( sup_sup_set_b @ A2 @ C4 ) ) ) ).

% Un_left_commute
thf(fact_909_Collect__disj__eq,axiom,
    ! [P: product_prod_o_o > $o,Q: product_prod_o_o > $o] :
      ( ( collec3167064739498627218od_o_o
        @ ^ [X: product_prod_o_o] :
            ( ( P @ X )
            | ( Q @ X ) ) )
      = ( sup_su5769328420594410459od_o_o @ ( collec3167064739498627218od_o_o @ P ) @ ( collec3167064739498627218od_o_o @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_910_Collect__disj__eq,axiom,
    ! [P: $o > $o,Q: $o > $o] :
      ( ( collect_o
        @ ^ [X: $o] :
            ( ( P @ X )
            | ( Q @ X ) ) )
      = ( sup_sup_set_o @ ( collect_o @ P ) @ ( collect_o @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_911_Collect__disj__eq,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( collect_nat
        @ ^ [X: nat] :
            ( ( P @ X )
            | ( Q @ X ) ) )
      = ( sup_sup_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_912_Collect__disj__eq,axiom,
    ! [P: b > $o,Q: b > $o] :
      ( ( collect_b
        @ ^ [X: b] :
            ( ( P @ X )
            | ( Q @ X ) ) )
      = ( sup_sup_set_b @ ( collect_b @ P ) @ ( collect_b @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_913_Collect__conv__if,axiom,
    ! [P: relational_fmla_a_b > $o,A: relational_fmla_a_b] :
      ( ( ( P @ A )
       => ( ( collec3419995626248312948la_a_b
            @ ^ [X: relational_fmla_a_b] :
                ( ( X = A )
                & ( P @ X ) ) )
          = ( insert7010464514620295119la_a_b @ A @ bot_bo4495933725496725865la_a_b ) ) )
      & ( ~ ( P @ A )
       => ( ( collec3419995626248312948la_a_b
            @ ^ [X: relational_fmla_a_b] :
                ( ( X = A )
                & ( P @ X ) ) )
          = bot_bo4495933725496725865la_a_b ) ) ) ).

% Collect_conv_if
thf(fact_914_Collect__conv__if,axiom,
    ! [P: b > $o,A: b] :
      ( ( ( P @ A )
       => ( ( collect_b
            @ ^ [X: b] :
                ( ( X = A )
                & ( P @ X ) ) )
          = ( insert_b @ A @ bot_bot_set_b ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_b
            @ ^ [X: b] :
                ( ( X = A )
                & ( P @ X ) ) )
          = bot_bot_set_b ) ) ) ).

% Collect_conv_if
thf(fact_915_Collect__conv__if,axiom,
    ! [P: nat > $o,A: nat] :
      ( ( ( P @ A )
       => ( ( collect_nat
            @ ^ [X: nat] :
                ( ( X = A )
                & ( P @ X ) ) )
          = ( insert_nat @ A @ bot_bot_set_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_nat
            @ ^ [X: nat] :
                ( ( X = A )
                & ( P @ X ) ) )
          = bot_bot_set_nat ) ) ) ).

% Collect_conv_if
thf(fact_916_Collect__conv__if,axiom,
    ! [P: product_prod_o_o > $o,A: product_prod_o_o] :
      ( ( ( P @ A )
       => ( ( collec3167064739498627218od_o_o
            @ ^ [X: product_prod_o_o] :
                ( ( X = A )
                & ( P @ X ) ) )
          = ( insert6201435330877294327od_o_o @ A @ bot_bo7073875226086086771od_o_o ) ) )
      & ( ~ ( P @ A )
       => ( ( collec3167064739498627218od_o_o
            @ ^ [X: product_prod_o_o] :
                ( ( X = A )
                & ( P @ X ) ) )
          = bot_bo7073875226086086771od_o_o ) ) ) ).

% Collect_conv_if
thf(fact_917_Collect__conv__if,axiom,
    ! [P: $o > $o,A: $o] :
      ( ( ( P @ A )
       => ( ( collect_o
            @ ^ [X: $o] :
                ( ( X = A )
                & ( P @ X ) ) )
          = ( insert_o @ A @ bot_bot_set_o ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_o
            @ ^ [X: $o] :
                ( ( X = A )
                & ( P @ X ) ) )
          = bot_bot_set_o ) ) ) ).

% Collect_conv_if
thf(fact_918_insert__commute,axiom,
    ! [X3: relational_fmla_a_b,Y: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
      ( ( insert7010464514620295119la_a_b @ X3 @ ( insert7010464514620295119la_a_b @ Y @ A2 ) )
      = ( insert7010464514620295119la_a_b @ Y @ ( insert7010464514620295119la_a_b @ X3 @ A2 ) ) ) ).

% insert_commute
thf(fact_919_insert__commute,axiom,
    ! [X3: nat,Y: nat,A2: set_nat] :
      ( ( insert_nat @ X3 @ ( insert_nat @ Y @ A2 ) )
      = ( insert_nat @ Y @ ( insert_nat @ X3 @ A2 ) ) ) ).

% insert_commute
thf(fact_920_insert__commute,axiom,
    ! [X3: product_prod_o_o,Y: product_prod_o_o,A2: set_Product_prod_o_o] :
      ( ( insert6201435330877294327od_o_o @ X3 @ ( insert6201435330877294327od_o_o @ Y @ A2 ) )
      = ( insert6201435330877294327od_o_o @ Y @ ( insert6201435330877294327od_o_o @ X3 @ A2 ) ) ) ).

% insert_commute
thf(fact_921_insert__commute,axiom,
    ! [X3: $o,Y: $o,A2: set_o] :
      ( ( insert_o @ X3 @ ( insert_o @ Y @ A2 ) )
      = ( insert_o @ Y @ ( insert_o @ X3 @ A2 ) ) ) ).

% insert_commute
thf(fact_922_insert__Collect,axiom,
    ! [A: relational_fmla_a_b,P: relational_fmla_a_b > $o] :
      ( ( insert7010464514620295119la_a_b @ A @ ( collec3419995626248312948la_a_b @ P ) )
      = ( collec3419995626248312948la_a_b
        @ ^ [U: relational_fmla_a_b] :
            ( ( U != A )
           => ( P @ U ) ) ) ) ).

% insert_Collect
thf(fact_923_insert__Collect,axiom,
    ! [A: nat,P: nat > $o] :
      ( ( insert_nat @ A @ ( collect_nat @ P ) )
      = ( collect_nat
        @ ^ [U: nat] :
            ( ( U != A )
           => ( P @ U ) ) ) ) ).

% insert_Collect
thf(fact_924_insert__Collect,axiom,
    ! [A: product_prod_o_o,P: product_prod_o_o > $o] :
      ( ( insert6201435330877294327od_o_o @ A @ ( collec3167064739498627218od_o_o @ P ) )
      = ( collec3167064739498627218od_o_o
        @ ^ [U: product_prod_o_o] :
            ( ( U != A )
           => ( P @ U ) ) ) ) ).

% insert_Collect
thf(fact_925_insert__Collect,axiom,
    ! [A: $o,P: $o > $o] :
      ( ( insert_o @ A @ ( collect_o @ P ) )
      = ( collect_o
        @ ^ [U: $o] :
            ( ( U != A )
           => ( P @ U ) ) ) ) ).

% insert_Collect
thf(fact_926_Un__left__absorb,axiom,
    ! [A2: set_Product_prod_o_o,B5: set_Product_prod_o_o] :
      ( ( sup_su5769328420594410459od_o_o @ A2 @ ( sup_su5769328420594410459od_o_o @ A2 @ B5 ) )
      = ( sup_su5769328420594410459od_o_o @ A2 @ B5 ) ) ).

% Un_left_absorb
thf(fact_927_Un__left__absorb,axiom,
    ! [A2: set_o,B5: set_o] :
      ( ( sup_sup_set_o @ A2 @ ( sup_sup_set_o @ A2 @ B5 ) )
      = ( sup_sup_set_o @ A2 @ B5 ) ) ).

% Un_left_absorb
thf(fact_928_Un__left__absorb,axiom,
    ! [A2: set_nat,B5: set_nat] :
      ( ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B5 ) )
      = ( sup_sup_set_nat @ A2 @ B5 ) ) ).

% Un_left_absorb
thf(fact_929_Un__left__absorb,axiom,
    ! [A2: set_b,B5: set_b] :
      ( ( sup_sup_set_b @ A2 @ ( sup_sup_set_b @ A2 @ B5 ) )
      = ( sup_sup_set_b @ A2 @ B5 ) ) ).

% Un_left_absorb
thf(fact_930_Un__empty__right,axiom,
    ! [A2: set_Re381260168593705685la_a_b] :
      ( ( sup_su5130108678486352897la_a_b @ A2 @ bot_bo4495933725496725865la_a_b )
      = A2 ) ).

% Un_empty_right
thf(fact_931_Un__empty__right,axiom,
    ! [A2: set_b] :
      ( ( sup_sup_set_b @ A2 @ bot_bot_set_b )
      = A2 ) ).

% Un_empty_right
thf(fact_932_Un__empty__right,axiom,
    ! [A2: set_nat] :
      ( ( sup_sup_set_nat @ A2 @ bot_bot_set_nat )
      = A2 ) ).

% Un_empty_right
thf(fact_933_Un__empty__right,axiom,
    ! [A2: set_Product_prod_o_o] :
      ( ( sup_su5769328420594410459od_o_o @ A2 @ bot_bo7073875226086086771od_o_o )
      = A2 ) ).

% Un_empty_right
thf(fact_934_Un__empty__right,axiom,
    ! [A2: set_o] :
      ( ( sup_sup_set_o @ A2 @ bot_bot_set_o )
      = A2 ) ).

% Un_empty_right
thf(fact_935_singleton__iff,axiom,
    ! [B: list_a,A: list_a] :
      ( ( member_list_a @ B @ ( insert_list_a @ A @ bot_bot_set_list_a ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_936_singleton__iff,axiom,
    ! [B: set_o,A: set_o] :
      ( ( member_set_o @ B @ ( insert_set_o @ A @ bot_bot_set_set_o ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_937_singleton__iff,axiom,
    ! [B: relational_fmla_a_b > relational_fmla_a_b,A: relational_fmla_a_b > relational_fmla_a_b] :
      ( ( member8433577210552456052la_a_b @ B @ ( insert8904949763332019597la_a_b @ A @ bot_bo9179849999556691623la_a_b ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_938_singleton__iff,axiom,
    ! [B: relational_fmla_a_b,A: relational_fmla_a_b] :
      ( ( member4680049679412964150la_a_b @ B @ ( insert7010464514620295119la_a_b @ A @ bot_bo4495933725496725865la_a_b ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_939_singleton__iff,axiom,
    ! [B: b,A: b] :
      ( ( member_b @ B @ ( insert_b @ A @ bot_bot_set_b ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_940_singleton__iff,axiom,
    ! [B: nat,A: nat] :
      ( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_941_singleton__iff,axiom,
    ! [B: product_prod_o_o,A: product_prod_o_o] :
      ( ( member7466972457876170832od_o_o @ B @ ( insert6201435330877294327od_o_o @ A @ bot_bo7073875226086086771od_o_o ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_942_singleton__iff,axiom,
    ! [B: $o,A: $o] :
      ( ( member_o @ B @ ( insert_o @ A @ bot_bot_set_o ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_943_insert__eq__iff,axiom,
    ! [A: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B: relational_fmla_a_b,B5: set_Re381260168593705685la_a_b] :
      ( ~ ( member4680049679412964150la_a_b @ A @ A2 )
     => ( ~ ( member4680049679412964150la_a_b @ B @ B5 )
       => ( ( ( insert7010464514620295119la_a_b @ A @ A2 )
            = ( insert7010464514620295119la_a_b @ B @ B5 ) )
          = ( ( ( A = B )
             => ( A2 = B5 ) )
            & ( ( A != B )
             => ? [C5: set_Re381260168593705685la_a_b] :
                  ( ( A2
                    = ( insert7010464514620295119la_a_b @ B @ C5 ) )
                  & ~ ( member4680049679412964150la_a_b @ B @ C5 )
                  & ( B5
                    = ( insert7010464514620295119la_a_b @ A @ C5 ) )
                  & ~ ( member4680049679412964150la_a_b @ A @ C5 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_944_insert__eq__iff,axiom,
    ! [A: product_prod_o_o,A2: set_Product_prod_o_o,B: product_prod_o_o,B5: set_Product_prod_o_o] :
      ( ~ ( member7466972457876170832od_o_o @ A @ A2 )
     => ( ~ ( member7466972457876170832od_o_o @ B @ B5 )
       => ( ( ( insert6201435330877294327od_o_o @ A @ A2 )
            = ( insert6201435330877294327od_o_o @ B @ B5 ) )
          = ( ( ( A = B )
             => ( A2 = B5 ) )
            & ( ( A != B )
             => ? [C5: set_Product_prod_o_o] :
                  ( ( A2
                    = ( insert6201435330877294327od_o_o @ B @ C5 ) )
                  & ~ ( member7466972457876170832od_o_o @ B @ C5 )
                  & ( B5
                    = ( insert6201435330877294327od_o_o @ A @ C5 ) )
                  & ~ ( member7466972457876170832od_o_o @ A @ C5 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_945_insert__eq__iff,axiom,
    ! [A: list_a,A2: set_list_a,B: list_a,B5: set_list_a] :
      ( ~ ( member_list_a @ A @ A2 )
     => ( ~ ( member_list_a @ B @ B5 )
       => ( ( ( insert_list_a @ A @ A2 )
            = ( insert_list_a @ B @ B5 ) )
          = ( ( ( A = B )
             => ( A2 = B5 ) )
            & ( ( A != B )
             => ? [C5: set_list_a] :
                  ( ( A2
                    = ( insert_list_a @ B @ C5 ) )
                  & ~ ( member_list_a @ B @ C5 )
                  & ( B5
                    = ( insert_list_a @ A @ C5 ) )
                  & ~ ( member_list_a @ A @ C5 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_946_insert__eq__iff,axiom,
    ! [A: set_o,A2: set_set_o,B: set_o,B5: set_set_o] :
      ( ~ ( member_set_o @ A @ A2 )
     => ( ~ ( member_set_o @ B @ B5 )
       => ( ( ( insert_set_o @ A @ A2 )
            = ( insert_set_o @ B @ B5 ) )
          = ( ( ( A = B )
             => ( A2 = B5 ) )
            & ( ( A != B )
             => ? [C5: set_set_o] :
                  ( ( A2
                    = ( insert_set_o @ B @ C5 ) )
                  & ~ ( member_set_o @ B @ C5 )
                  & ( B5
                    = ( insert_set_o @ A @ C5 ) )
                  & ~ ( member_set_o @ A @ C5 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_947_insert__eq__iff,axiom,
    ! [A: nat,A2: set_nat,B: nat,B5: set_nat] :
      ( ~ ( member_nat @ A @ A2 )
     => ( ~ ( member_nat @ B @ B5 )
       => ( ( ( insert_nat @ A @ A2 )
            = ( insert_nat @ B @ B5 ) )
          = ( ( ( A = B )
             => ( A2 = B5 ) )
            & ( ( A != B )
             => ? [C5: set_nat] :
                  ( ( A2
                    = ( insert_nat @ B @ C5 ) )
                  & ~ ( member_nat @ B @ C5 )
                  & ( B5
                    = ( insert_nat @ A @ C5 ) )
                  & ~ ( member_nat @ A @ C5 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_948_insert__eq__iff,axiom,
    ! [A: relational_fmla_a_b > relational_fmla_a_b,A2: set_Re1288005135514575379la_a_b,B: relational_fmla_a_b > relational_fmla_a_b,B5: set_Re1288005135514575379la_a_b] :
      ( ~ ( member8433577210552456052la_a_b @ A @ A2 )
     => ( ~ ( member8433577210552456052la_a_b @ B @ B5 )
       => ( ( ( insert8904949763332019597la_a_b @ A @ A2 )
            = ( insert8904949763332019597la_a_b @ B @ B5 ) )
          = ( ( ( A = B )
             => ( A2 = B5 ) )
            & ( ( A != B )
             => ? [C5: set_Re1288005135514575379la_a_b] :
                  ( ( A2
                    = ( insert8904949763332019597la_a_b @ B @ C5 ) )
                  & ~ ( member8433577210552456052la_a_b @ B @ C5 )
                  & ( B5
                    = ( insert8904949763332019597la_a_b @ A @ C5 ) )
                  & ~ ( member8433577210552456052la_a_b @ A @ C5 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_949_insert__eq__iff,axiom,
    ! [A: b,A2: set_b,B: b,B5: set_b] :
      ( ~ ( member_b @ A @ A2 )
     => ( ~ ( member_b @ B @ B5 )
       => ( ( ( insert_b @ A @ A2 )
            = ( insert_b @ B @ B5 ) )
          = ( ( ( A = B )
             => ( A2 = B5 ) )
            & ( ( A != B )
             => ? [C5: set_b] :
                  ( ( A2
                    = ( insert_b @ B @ C5 ) )
                  & ~ ( member_b @ B @ C5 )
                  & ( B5
                    = ( insert_b @ A @ C5 ) )
                  & ~ ( member_b @ A @ C5 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_950_insert__eq__iff,axiom,
    ! [A: $o,A2: set_o,B: $o,B5: set_o] :
      ( ~ ( member_o @ A @ A2 )
     => ( ~ ( member_o @ B @ B5 )
       => ( ( ( insert_o @ A @ A2 )
            = ( insert_o @ B @ B5 ) )
          = ( ( ( A = B )
             => ( A2 = B5 ) )
            & ( ( A = (~ B) )
             => ? [C5: set_o] :
                  ( ( A2
                    = ( insert_o @ B @ C5 ) )
                  & ~ ( member_o @ B @ C5 )
                  & ( B5
                    = ( insert_o @ A @ C5 ) )
                  & ~ ( member_o @ A @ C5 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_951_insert__absorb,axiom,
    ! [A: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
      ( ( member4680049679412964150la_a_b @ A @ A2 )
     => ( ( insert7010464514620295119la_a_b @ A @ A2 )
        = A2 ) ) ).

% insert_absorb
thf(fact_952_insert__absorb,axiom,
    ! [A: product_prod_o_o,A2: set_Product_prod_o_o] :
      ( ( member7466972457876170832od_o_o @ A @ A2 )
     => ( ( insert6201435330877294327od_o_o @ A @ A2 )
        = A2 ) ) ).

% insert_absorb
thf(fact_953_insert__absorb,axiom,
    ! [A: list_a,A2: set_list_a] :
      ( ( member_list_a @ A @ A2 )
     => ( ( insert_list_a @ A @ A2 )
        = A2 ) ) ).

% insert_absorb
thf(fact_954_insert__absorb,axiom,
    ! [A: set_o,A2: set_set_o] :
      ( ( member_set_o @ A @ A2 )
     => ( ( insert_set_o @ A @ A2 )
        = A2 ) ) ).

% insert_absorb
thf(fact_955_insert__absorb,axiom,
    ! [A: nat,A2: set_nat] :
      ( ( member_nat @ A @ A2 )
     => ( ( insert_nat @ A @ A2 )
        = A2 ) ) ).

% insert_absorb
thf(fact_956_insert__absorb,axiom,
    ! [A: relational_fmla_a_b > relational_fmla_a_b,A2: set_Re1288005135514575379la_a_b] :
      ( ( member8433577210552456052la_a_b @ A @ A2 )
     => ( ( insert8904949763332019597la_a_b @ A @ A2 )
        = A2 ) ) ).

% insert_absorb
thf(fact_957_insert__absorb,axiom,
    ! [A: b,A2: set_b] :
      ( ( member_b @ A @ A2 )
     => ( ( insert_b @ A @ A2 )
        = A2 ) ) ).

% insert_absorb
thf(fact_958_insert__absorb,axiom,
    ! [A: $o,A2: set_o] :
      ( ( member_o @ A @ A2 )
     => ( ( insert_o @ A @ A2 )
        = A2 ) ) ).

% insert_absorb
thf(fact_959_Un__empty__left,axiom,
    ! [B5: set_Re381260168593705685la_a_b] :
      ( ( sup_su5130108678486352897la_a_b @ bot_bo4495933725496725865la_a_b @ B5 )
      = B5 ) ).

% Un_empty_left
thf(fact_960_Un__empty__left,axiom,
    ! [B5: set_b] :
      ( ( sup_sup_set_b @ bot_bot_set_b @ B5 )
      = B5 ) ).

% Un_empty_left
thf(fact_961_Un__empty__left,axiom,
    ! [B5: set_nat] :
      ( ( sup_sup_set_nat @ bot_bot_set_nat @ B5 )
      = B5 ) ).

% Un_empty_left
thf(fact_962_Un__empty__left,axiom,
    ! [B5: set_Product_prod_o_o] :
      ( ( sup_su5769328420594410459od_o_o @ bot_bo7073875226086086771od_o_o @ B5 )
      = B5 ) ).

% Un_empty_left
thf(fact_963_Un__empty__left,axiom,
    ! [B5: set_o] :
      ( ( sup_sup_set_o @ bot_bot_set_o @ B5 )
      = B5 ) ).

% Un_empty_left
thf(fact_964_insert__is__Un,axiom,
    ( insert7010464514620295119la_a_b
    = ( ^ [A3: relational_fmla_a_b] : ( sup_su5130108678486352897la_a_b @ ( insert7010464514620295119la_a_b @ A3 @ bot_bo4495933725496725865la_a_b ) ) ) ) ).

% insert_is_Un
thf(fact_965_insert__is__Un,axiom,
    ( insert_b
    = ( ^ [A3: b] : ( sup_sup_set_b @ ( insert_b @ A3 @ bot_bot_set_b ) ) ) ) ).

% insert_is_Un
thf(fact_966_insert__is__Un,axiom,
    ( insert_nat
    = ( ^ [A3: nat] : ( sup_sup_set_nat @ ( insert_nat @ A3 @ bot_bot_set_nat ) ) ) ) ).

% insert_is_Un
thf(fact_967_insert__is__Un,axiom,
    ( insert6201435330877294327od_o_o
    = ( ^ [A3: product_prod_o_o] : ( sup_su5769328420594410459od_o_o @ ( insert6201435330877294327od_o_o @ A3 @ bot_bo7073875226086086771od_o_o ) ) ) ) ).

% insert_is_Un
thf(fact_968_insert__is__Un,axiom,
    ( insert_o
    = ( ^ [A3: $o] : ( sup_sup_set_o @ ( insert_o @ A3 @ bot_bot_set_o ) ) ) ) ).

% insert_is_Un
thf(fact_969_insert__ident,axiom,
    ! [X3: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b,B5: set_Re381260168593705685la_a_b] :
      ( ~ ( member4680049679412964150la_a_b @ X3 @ A2 )
     => ( ~ ( member4680049679412964150la_a_b @ X3 @ B5 )
       => ( ( ( insert7010464514620295119la_a_b @ X3 @ A2 )
            = ( insert7010464514620295119la_a_b @ X3 @ B5 ) )
          = ( A2 = B5 ) ) ) ) ).

% insert_ident
thf(fact_970_insert__ident,axiom,
    ! [X3: product_prod_o_o,A2: set_Product_prod_o_o,B5: set_Product_prod_o_o] :
      ( ~ ( member7466972457876170832od_o_o @ X3 @ A2 )
     => ( ~ ( member7466972457876170832od_o_o @ X3 @ B5 )
       => ( ( ( insert6201435330877294327od_o_o @ X3 @ A2 )
            = ( insert6201435330877294327od_o_o @ X3 @ B5 ) )
          = ( A2 = B5 ) ) ) ) ).

% insert_ident
thf(fact_971_insert__ident,axiom,
    ! [X3: list_a,A2: set_list_a,B5: set_list_a] :
      ( ~ ( member_list_a @ X3 @ A2 )
     => ( ~ ( member_list_a @ X3 @ B5 )
       => ( ( ( insert_list_a @ X3 @ A2 )
            = ( insert_list_a @ X3 @ B5 ) )
          = ( A2 = B5 ) ) ) ) ).

% insert_ident
thf(fact_972_insert__ident,axiom,
    ! [X3: set_o,A2: set_set_o,B5: set_set_o] :
      ( ~ ( member_set_o @ X3 @ A2 )
     => ( ~ ( member_set_o @ X3 @ B5 )
       => ( ( ( insert_set_o @ X3 @ A2 )
            = ( insert_set_o @ X3 @ B5 ) )
          = ( A2 = B5 ) ) ) ) ).

% insert_ident
thf(fact_973_insert__ident,axiom,
    ! [X3: nat,A2: set_nat,B5: set_nat] :
      ( ~ ( member_nat @ X3 @ A2 )
     => ( ~ ( member_nat @ X3 @ B5 )
       => ( ( ( insert_nat @ X3 @ A2 )
            = ( insert_nat @ X3 @ B5 ) )
          = ( A2 = B5 ) ) ) ) ).

% insert_ident
thf(fact_974_insert__ident,axiom,
    ! [X3: relational_fmla_a_b > relational_fmla_a_b,A2: set_Re1288005135514575379la_a_b,B5: set_Re1288005135514575379la_a_b] :
      ( ~ ( member8433577210552456052la_a_b @ X3 @ A2 )
     => ( ~ ( member8433577210552456052la_a_b @ X3 @ B5 )
       => ( ( ( insert8904949763332019597la_a_b @ X3 @ A2 )
            = ( insert8904949763332019597la_a_b @ X3 @ B5 ) )
          = ( A2 = B5 ) ) ) ) ).

% insert_ident
thf(fact_975_insert__ident,axiom,
    ! [X3: b,A2: set_b,B5: set_b] :
      ( ~ ( member_b @ X3 @ A2 )
     => ( ~ ( member_b @ X3 @ B5 )
       => ( ( ( insert_b @ X3 @ A2 )
            = ( insert_b @ X3 @ B5 ) )
          = ( A2 = B5 ) ) ) ) ).

% insert_ident
thf(fact_976_insert__ident,axiom,
    ! [X3: $o,A2: set_o,B5: set_o] :
      ( ~ ( member_o @ X3 @ A2 )
     => ( ~ ( member_o @ X3 @ B5 )
       => ( ( ( insert_o @ X3 @ A2 )
            = ( insert_o @ X3 @ B5 ) )
          = ( A2 = B5 ) ) ) ) ).

% insert_ident
thf(fact_977_insert__compr,axiom,
    ( insert7010464514620295119la_a_b
    = ( ^ [A3: relational_fmla_a_b,B7: set_Re381260168593705685la_a_b] :
          ( collec3419995626248312948la_a_b
          @ ^ [X: relational_fmla_a_b] :
              ( ( X = A3 )
              | ( member4680049679412964150la_a_b @ X @ B7 ) ) ) ) ) ).

% insert_compr
thf(fact_978_insert__compr,axiom,
    ( insert6201435330877294327od_o_o
    = ( ^ [A3: product_prod_o_o,B7: set_Product_prod_o_o] :
          ( collec3167064739498627218od_o_o
          @ ^ [X: product_prod_o_o] :
              ( ( X = A3 )
              | ( member7466972457876170832od_o_o @ X @ B7 ) ) ) ) ) ).

% insert_compr
thf(fact_979_insert__compr,axiom,
    ( insert_list_a
    = ( ^ [A3: list_a,B7: set_list_a] :
          ( collect_list_a
          @ ^ [X: list_a] :
              ( ( X = A3 )
              | ( member_list_a @ X @ B7 ) ) ) ) ) ).

% insert_compr
thf(fact_980_insert__compr,axiom,
    ( insert_set_o
    = ( ^ [A3: set_o,B7: set_set_o] :
          ( collect_set_o
          @ ^ [X: set_o] :
              ( ( X = A3 )
              | ( member_set_o @ X @ B7 ) ) ) ) ) ).

% insert_compr
thf(fact_981_insert__compr,axiom,
    ( insert_nat
    = ( ^ [A3: nat,B7: set_nat] :
          ( collect_nat
          @ ^ [X: nat] :
              ( ( X = A3 )
              | ( member_nat @ X @ B7 ) ) ) ) ) ).

% insert_compr
thf(fact_982_insert__compr,axiom,
    ( insert8904949763332019597la_a_b
    = ( ^ [A3: relational_fmla_a_b > relational_fmla_a_b,B7: set_Re1288005135514575379la_a_b] :
          ( collec5041345499257167282la_a_b
          @ ^ [X: relational_fmla_a_b > relational_fmla_a_b] :
              ( ( X = A3 )
              | ( member8433577210552456052la_a_b @ X @ B7 ) ) ) ) ) ).

% insert_compr
thf(fact_983_insert__compr,axiom,
    ( insert_b
    = ( ^ [A3: b,B7: set_b] :
          ( collect_b
          @ ^ [X: b] :
              ( ( X = A3 )
              | ( member_b @ X @ B7 ) ) ) ) ) ).

% insert_compr
thf(fact_984_insert__compr,axiom,
    ( insert_o
    = ( ^ [A3: $o,B7: set_o] :
          ( collect_o
          @ ^ [X: $o] :
              ( ( X = A3 )
              | ( member_o @ X @ B7 ) ) ) ) ) ).

% insert_compr
thf(fact_985_singletonD,axiom,
    ! [B: list_a,A: list_a] :
      ( ( member_list_a @ B @ ( insert_list_a @ A @ bot_bot_set_list_a ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_986_singletonD,axiom,
    ! [B: set_o,A: set_o] :
      ( ( member_set_o @ B @ ( insert_set_o @ A @ bot_bot_set_set_o ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_987_singletonD,axiom,
    ! [B: relational_fmla_a_b > relational_fmla_a_b,A: relational_fmla_a_b > relational_fmla_a_b] :
      ( ( member8433577210552456052la_a_b @ B @ ( insert8904949763332019597la_a_b @ A @ bot_bo9179849999556691623la_a_b ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_988_singletonD,axiom,
    ! [B: relational_fmla_a_b,A: relational_fmla_a_b] :
      ( ( member4680049679412964150la_a_b @ B @ ( insert7010464514620295119la_a_b @ A @ bot_bo4495933725496725865la_a_b ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_989_singletonD,axiom,
    ! [B: b,A: b] :
      ( ( member_b @ B @ ( insert_b @ A @ bot_bot_set_b ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_990_singletonD,axiom,
    ! [B: nat,A: nat] :
      ( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_991_singletonD,axiom,
    ! [B: product_prod_o_o,A: product_prod_o_o] :
      ( ( member7466972457876170832od_o_o @ B @ ( insert6201435330877294327od_o_o @ A @ bot_bo7073875226086086771od_o_o ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_992_singletonD,axiom,
    ! [B: $o,A: $o] :
      ( ( member_o @ B @ ( insert_o @ A @ bot_bot_set_o ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_993_Set_Oset__insert,axiom,
    ! [X3: relational_fmla_a_b,A2: set_Re381260168593705685la_a_b] :
      ( ( member4680049679412964150la_a_b @ X3 @ A2 )
     => ~ ! [B6: set_Re381260168593705685la_a_b] :
            ( ( A2
              = ( insert7010464514620295119la_a_b @ X3 @ B6 ) )
           => ( member4680049679412964150la_a_b @ X3 @ B6 ) ) ) ).

% Set.set_insert
thf(fact_994_Set_Oset__insert,axiom,
    ! [X3: product_prod_o_o,A2: set_Product_prod_o_o] :
      ( ( member7466972457876170832od_o_o @ X3 @ A2 )
     => ~ ! [B6: set_Product_prod_o_o] :
            ( ( A2
              = ( insert6201435330877294327od_o_o @ X3 @ B6 ) )
           => ( member7466972457876170832od_o_o @ X3 @ B6 ) ) ) ).

% Set.set_insert
thf(fact_995_Set_Oset__insert,axiom,
    ! [X3: list_a,A2: set_list_a] :
      ( ( member_list_a @ X3 @ A2 )
     => ~ ! [B6: set_list_a] :
            ( ( A2
              = ( insert_list_a @ X3 @ B6 ) )
           => ( member_list_a @ X3 @ B6 ) ) ) ).

% Set.set_insert
thf(fact_996_Set_Oset__insert,axiom,
    ! [X3: set_o,A2: set_set_o] :
      ( ( member_set_o @ X3 @ A2 )
     => ~ ! [B6: set_set_o] :
            ( ( A2
              = ( insert_set_o @ X3 @ B6 ) )
           => ( member_set_o @ X3 @ B6 ) ) ) ).

% Set.set_insert
thf(fact_997_Set_Oset__insert,axiom,
    ! [X3: nat,A2: set_nat] :
      ( ( member_nat @ X3 @ A2 )
     => ~ ! [B6: set_nat] :
            ( ( A2
              = ( insert_nat @ X3 @ B6 ) )
           => ( member_nat @ X3 @ B6 ) ) ) ).

% Set.set_insert
thf(fact_998_Set_Oset__insert,axiom,
    ! [X3: relational_fmla_a_b > relational_fmla_a_b,A2: set_Re1288005135514575379la_a_b] :
      ( ( member8433577210552456052la_a_b @ X3 @ A2 )
     => ~ ! [B6: set_Re1288005135514575379la_a_b] :
            ( ( A2
              = ( insert8904949763332019597la_a_b @ X3 @ B6 ) )
           => ( member8433577210552456052la_a_b @ X3 @ B6 ) ) ) ).

% Set.set_insert
thf(fact_999_Set_Oset__insert,axiom,
    ! [X3: b,A2: set_b] :
      ( ( member_b @ X3 @ A2 )
     => ~ ! [B6: set_b] :
            ( ( A2
              = ( insert_b @ X3 @ B6 ) )
           => ( member_b @ X3 @ B6 ) ) ) ).

% Set.set_insert
thf(fact_1000_Set_Oset__insert,axiom,
    ! [X3: $o,A2: set_o] :
      ( ( member_o @ X3 @ A2 )
     => ~ ! [B6: set_o] :
            ( ( A2
              = ( insert_o @ X3 @ B6 ) )
           => ( member_o @ X3 @ B6 ) ) ) ).

% Set.set_insert
thf(fact_1001_insert__def,axiom,
    ( insert7010464514620295119la_a_b
    = ( ^ [A3: relational_fmla_a_b] :
          ( sup_su5130108678486352897la_a_b
          @ ( collec3419995626248312948la_a_b
            @ ^ [X: relational_fmla_a_b] : ( X = A3 ) ) ) ) ) ).

% insert_def
thf(fact_1002_insert__def,axiom,
    ( insert6201435330877294327od_o_o
    = ( ^ [A3: product_prod_o_o] :
          ( sup_su5769328420594410459od_o_o
          @ ( collec3167064739498627218od_o_o
            @ ^ [X: product_prod_o_o] : ( X = A3 ) ) ) ) ) ).

% insert_def
thf(fact_1003_insert__def,axiom,
    ( insert_nat
    = ( ^ [A3: nat] :
          ( sup_sup_set_nat
          @ ( collect_nat
            @ ^ [X: nat] : ( X = A3 ) ) ) ) ) ).

% insert_def
thf(fact_1004_insert__def,axiom,
    ( insert_b
    = ( ^ [A3: b] :
          ( sup_sup_set_b
          @ ( collect_b
            @ ^ [X: b] : ( X = A3 ) ) ) ) ) ).

% insert_def
thf(fact_1005_insert__def,axiom,
    ( insert_o
    = ( ^ [A3: $o] :
          ( sup_sup_set_o
          @ ( collect_o
            @ ^ [X: $o] : ( X = A3 ) ) ) ) ) ).

% insert_def
thf(fact_1006_ex__in__conv,axiom,
    ! [A2: set_list_a] :
      ( ( ? [X: list_a] : ( member_list_a @ X @ A2 ) )
      = ( A2 != bot_bot_set_list_a ) ) ).

% ex_in_conv
thf(fact_1007_ex__in__conv,axiom,
    ! [A2: set_set_o] :
      ( ( ? [X: set_o] : ( member_set_o @ X @ A2 ) )
      = ( A2 != bot_bot_set_set_o ) ) ).

% ex_in_conv
thf(fact_1008_ex__in__conv,axiom,
    ! [A2: set_Re1288005135514575379la_a_b] :
      ( ( ? [X: relational_fmla_a_b > relational_fmla_a_b] : ( member8433577210552456052la_a_b @ X @ A2 ) )
      = ( A2 != bot_bo9179849999556691623la_a_b ) ) ).

% ex_in_conv
thf(fact_1009_ex__in__conv,axiom,
    ! [A2: set_Re381260168593705685la_a_b] :
      ( ( ? [X: relational_fmla_a_b] : ( member4680049679412964150la_a_b @ X @ A2 ) )
      = ( A2 != bot_bo4495933725496725865la_a_b ) ) ).

% ex_in_conv
thf(fact_1010_ex__in__conv,axiom,
    ! [A2: set_b] :
      ( ( ? [X: b] : ( member_b @ X @ A2 ) )
      = ( A2 != bot_bot_set_b ) ) ).

% ex_in_conv
thf(fact_1011_ex__in__conv,axiom,
    ! [A2: set_nat] :
      ( ( ? [X: nat] : ( member_nat @ X @ A2 ) )
      = ( A2 != bot_bot_set_nat ) ) ).

% ex_in_conv
thf(fact_1012_ex__in__conv,axiom,
    ! [A2: set_Product_prod_o_o] :
      ( ( ? [X: product_prod_o_o] : ( member7466972457876170832od_o_o @ X @ A2 ) )
      = ( A2 != bot_bo7073875226086086771od_o_o ) ) ).

% ex_in_conv
thf(fact_1013_ex__in__conv,axiom,
    ! [A2: set_o] :
      ( ( ? [X: $o] : ( member_o @ X @ A2 ) )
      = ( A2 != bot_bot_set_o ) ) ).

% ex_in_conv
thf(fact_1014_Un__commute,axiom,
    ( sup_sup_set_o
    = ( ^ [A6: set_o,B7: set_o] : ( sup_sup_set_o @ B7 @ A6 ) ) ) ).

% Un_commute
thf(fact_1015_Un__commute,axiom,
    ( sup_sup_set_nat
    = ( ^ [A6: set_nat,B7: set_nat] : ( sup_sup_set_nat @ B7 @ A6 ) ) ) ).

% Un_commute
thf(fact_1016_Un__commute,axiom,
    ( sup_sup_set_b
    = ( ^ [A6: set_b,B7: set_b] : ( sup_sup_set_b @ B7 @ A6 ) ) ) ).

% Un_commute
thf(fact_1017_empty__def,axiom,
    ( bot_bot_set_o
    = ( collect_o
      @ ^ [X: $o] : $false ) ) ).

% empty_def
thf(fact_1018_insertI2,axiom,
    ! [A: $o,B5: set_o,B: $o] :
      ( ( member_o @ A @ B5 )
     => ( member_o @ A @ ( insert_o @ B @ B5 ) ) ) ).

% insertI2
thf(fact_1019_insertI1,axiom,
    ! [A: $o,B5: set_o] : ( member_o @ A @ ( insert_o @ A @ B5 ) ) ).

% insertI1
thf(fact_1020_equals0I,axiom,
    ! [A2: set_o] :
      ( ! [Y3: $o] :
          ~ ( member_o @ Y3 @ A2 )
     => ( A2 = bot_bot_set_o ) ) ).

% equals0I
thf(fact_1021_equals0D,axiom,
    ! [A2: set_o,A: $o] :
      ( ( A2 = bot_bot_set_o )
     => ~ ( member_o @ A @ A2 ) ) ).

% equals0D
thf(fact_1022_insertE,axiom,
    ! [A: $o,B: $o,A2: set_o] :
      ( ( member_o @ A @ ( insert_o @ B @ A2 ) )
     => ( ( A = (~ B) )
       => ( member_o @ A @ A2 ) ) ) ).

% insertE
thf(fact_1023_emptyE,axiom,
    ! [A: $o] :
      ~ ( member_o @ A @ bot_bot_set_o ) ).

% emptyE
thf(fact_1024_Un__def,axiom,
    ( sup_sup_set_o
    = ( ^ [A6: set_o,B7: set_o] :
          ( collect_o
          @ ^ [X: $o] :
              ( ( member_o @ X @ A6 )
              | ( member_o @ X @ B7 ) ) ) ) ) ).

% Un_def
thf(fact_1025_UnI2,axiom,
    ! [C: $o,B5: set_o,A2: set_o] :
      ( ( member_o @ C @ B5 )
     => ( member_o @ C @ ( sup_sup_set_o @ A2 @ B5 ) ) ) ).

% UnI2
thf(fact_1026_UnI1,axiom,
    ! [C: $o,A2: set_o,B5: set_o] :
      ( ( member_o @ C @ A2 )
     => ( member_o @ C @ ( sup_sup_set_o @ A2 @ B5 ) ) ) ).

% UnI1
thf(fact_1027_UnE,axiom,
    ! [C: $o,A2: set_o,B5: set_o] :
      ( ( member_o @ C @ ( sup_sup_set_o @ A2 @ B5 ) )
     => ( ~ ( member_o @ C @ A2 )
       => ( member_o @ C @ B5 ) ) ) ).

% UnE
thf(fact_1028_image__constant__conv,axiom,
    ! [A2: set_o,C: $o] :
      ( ( ( A2 = bot_bot_set_o )
       => ( ( image_o_o
            @ ^ [X: $o] : C
            @ A2 )
          = bot_bot_set_o ) )
      & ( ( A2 != bot_bot_set_o )
       => ( ( image_o_o
            @ ^ [X: $o] : C
            @ A2 )
          = ( insert_o @ C @ bot_bot_set_o ) ) ) ) ).

% image_constant_conv
thf(fact_1029_image__constant,axiom,
    ! [X3: $o,A2: set_o,C: $o] :
      ( ( member_o @ X3 @ A2 )
     => ( ( image_o_o
          @ ^ [X: $o] : C
          @ A2 )
        = ( insert_o @ C @ bot_bot_set_o ) ) ) ).

% image_constant
thf(fact_1030_fmla_Odisc_I11_J,axiom,
    ! [X4: relational_fmla_a_b] :
      ~ ( relati6551038146797045342ol_a_b @ ( relational_Neg_a_b @ X4 ) ) ).

% fmla.disc(11)
thf(fact_1031_fmla_Odisc_I13_J,axiom,
    ! [X61: relational_fmla_a_b,X62: relational_fmla_a_b] :
      ~ ( relati6551038146797045342ol_a_b @ ( relational_Disj_a_b @ X61 @ X62 ) ) ).

% fmla.disc(13)
thf(fact_1032_fmla_Odisc_I12_J,axiom,
    ! [X51: relational_fmla_a_b,X52: relational_fmla_a_b] :
      ~ ( relati6551038146797045342ol_a_b @ ( relational_Conj_a_b @ X51 @ X52 ) ) ).

% fmla.disc(12)
thf(fact_1033_fmla_Odisc_I14_J,axiom,
    ! [X71: nat,X72: relational_fmla_a_b] :
      ~ ( relati6551038146797045342ol_a_b @ ( relati591517084277583526ts_a_b @ X71 @ X72 ) ) ).

% fmla.disc(14)
thf(fact_1034_fmla_Odisc_I9_J,axiom,
    ! [X22: $o] : ( relati6551038146797045342ol_a_b @ ( relational_Bool_a_b @ X22 ) ) ).

% fmla.disc(9)
thf(fact_1035_fmla_OdiscI_I2_J,axiom,
    ! [Fmla: relational_fmla_a_b,X22: $o] :
      ( ( Fmla
        = ( relational_Bool_a_b @ X22 ) )
     => ( relati6551038146797045342ol_a_b @ Fmla ) ) ).

% fmla.discI(2)
thf(fact_1036_is__Bool__def,axiom,
    ( relati6551038146797045342ol_a_b
    = ( ^ [Fmla2: relational_fmla_a_b] :
        ? [X2: $o] :
          ( Fmla2
          = ( relational_Bool_a_b @ X2 ) ) ) ) ).

% is_Bool_def
thf(fact_1037_fmla_Osel_I3_J,axiom,
    ! [X22: $o] :
      ( ( relati2638701775882563405ol_a_b @ ( relational_Bool_a_b @ X22 ) )
      = X22 ) ).

% fmla.sel(3)
thf(fact_1038_qp__gen,axiom,
    ! [Q: relational_fmla_a_b,X3: nat] :
      ( ( relational_qp_a_b @ Q )
     => ( ( member_nat @ X3 @ ( relational_fv_a_b @ Q ) )
       => ( relational_gen_a_b @ X3 @ Q @ ( insert7010464514620295119la_a_b @ Q @ bot_bo4495933725496725865la_a_b ) ) ) ) ).

% qp_gen
thf(fact_1039_gen_Ointros_I2_J,axiom,
    ! [Q: relational_fmla_a_b,X3: nat] :
      ( ( relational_ap_a_b @ Q )
     => ( ( member_nat @ X3 @ ( relational_fv_a_b @ Q ) )
       => ( relational_gen_a_b @ X3 @ Q @ ( insert7010464514620295119la_a_b @ Q @ bot_bo4495933725496725865la_a_b ) ) ) ) ).

% gen.intros(2)
thf(fact_1040_gen_H_Ointros_I2_J,axiom,
    ! [Q: relational_fmla_a_b,X3: nat] :
      ( ( relational_ap_a_b @ Q )
     => ( ( member_nat @ X3 @ ( relational_fv_a_b @ Q ) )
       => ( relational_gen_a_b2 @ X3 @ Q @ ( insert7010464514620295119la_a_b @ Q @ bot_bo4495933725496725865la_a_b ) ) ) ) ).

% gen'.intros(2)
thf(fact_1041_cp_Osimps_I2_J,axiom,
    ! [Q: relational_fmla_a_b] :
      ( ( relational_cp_a_b @ ( relational_Neg_a_b @ Q ) )
      = ( if_Rel1279876242545935705la_a_b @ ( relati6551038146797045342ol_a_b @ ( relational_cp_a_b @ Q ) )
        @ ( relational_Bool_a_b
          @ ~ ( relati2638701775882563405ol_a_b @ ( relational_cp_a_b @ Q ) ) )
        @ ( relational_Neg_a_b @ ( relational_cp_a_b @ Q ) ) ) ) ).

% cp.simps(2)
thf(fact_1042_cp_Osimps_I4_J,axiom,
    ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
      ( ( relational_cp_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) )
      = ( if_Rel1279876242545935705la_a_b @ ( relati6551038146797045342ol_a_b @ ( relational_cp_a_b @ Q1 ) ) @ ( if_Rel1279876242545935705la_a_b @ ( relati2638701775882563405ol_a_b @ ( relational_cp_a_b @ Q1 ) ) @ ( relational_Bool_a_b @ $true ) @ ( relational_cp_a_b @ Q22 ) ) @ ( if_Rel1279876242545935705la_a_b @ ( relati6551038146797045342ol_a_b @ ( relational_cp_a_b @ Q22 ) ) @ ( if_Rel1279876242545935705la_a_b @ ( relati2638701775882563405ol_a_b @ ( relational_cp_a_b @ Q22 ) ) @ ( relational_Bool_a_b @ $true ) @ ( relational_cp_a_b @ Q1 ) ) @ ( relational_Disj_a_b @ ( relational_cp_a_b @ Q1 ) @ ( relational_cp_a_b @ Q22 ) ) ) ) ) ).

% cp.simps(4)
thf(fact_1043_cp_Osimps_I3_J,axiom,
    ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
      ( ( relational_cp_a_b @ ( relational_Conj_a_b @ Q1 @ Q22 ) )
      = ( if_Rel1279876242545935705la_a_b @ ( relati6551038146797045342ol_a_b @ ( relational_cp_a_b @ Q1 ) ) @ ( if_Rel1279876242545935705la_a_b @ ( relati2638701775882563405ol_a_b @ ( relational_cp_a_b @ Q1 ) ) @ ( relational_cp_a_b @ Q22 ) @ ( relational_Bool_a_b @ $false ) ) @ ( if_Rel1279876242545935705la_a_b @ ( relati6551038146797045342ol_a_b @ ( relational_cp_a_b @ Q22 ) ) @ ( if_Rel1279876242545935705la_a_b @ ( relati2638701775882563405ol_a_b @ ( relational_cp_a_b @ Q22 ) ) @ ( relational_cp_a_b @ Q1 ) @ ( relational_Bool_a_b @ $false ) ) @ ( relational_Conj_a_b @ ( relational_cp_a_b @ Q1 ) @ ( relational_cp_a_b @ Q22 ) ) ) ) ) ).

% cp.simps(3)
thf(fact_1044_gen_Ointros_I6_J,axiom,
    ! [X3: nat,Q1: relational_fmla_a_b,G13: set_Re381260168593705685la_a_b,Q22: relational_fmla_a_b,G24: set_Re381260168593705685la_a_b] :
      ( ( relational_gen_a_b @ X3 @ Q1 @ G13 )
     => ( ( relational_gen_a_b @ X3 @ Q22 @ G24 )
       => ( relational_gen_a_b @ X3 @ ( relational_Disj_a_b @ Q1 @ Q22 ) @ ( sup_su5130108678486352897la_a_b @ G13 @ G24 ) ) ) ) ).

% gen.intros(6)
thf(fact_1045_fmla_Osimps_I132_J,axiom,
    ! [X61: relational_fmla_a_b,X62: relational_fmla_a_b] :
      ( ( relati8924981150291758614la_a_b @ ( relational_Disj_a_b @ X61 @ X62 ) )
      = ( sup_sup_set_b @ ( relati8924981150291758614la_a_b @ X61 ) @ ( relati8924981150291758614la_a_b @ X62 ) ) ) ).

% fmla.simps(132)
thf(fact_1046_fmla_Osimps_I131_J,axiom,
    ! [X51: relational_fmla_a_b,X52: relational_fmla_a_b] :
      ( ( relati8924981150291758614la_a_b @ ( relational_Conj_a_b @ X51 @ X52 ) )
      = ( sup_sup_set_b @ ( relati8924981150291758614la_a_b @ X51 ) @ ( relati8924981150291758614la_a_b @ X52 ) ) ) ).

% fmla.simps(131)
thf(fact_1047_gen_H_Ointros_I6_J,axiom,
    ! [X3: nat,Q1: relational_fmla_a_b,G13: set_Re381260168593705685la_a_b,Q22: relational_fmla_a_b,G24: set_Re381260168593705685la_a_b] :
      ( ( relational_gen_a_b2 @ X3 @ Q1 @ G13 )
     => ( ( relational_gen_a_b2 @ X3 @ Q22 @ G24 )
       => ( relational_gen_a_b2 @ X3 @ ( relational_Disj_a_b @ Q1 @ Q22 ) @ ( sup_su5130108678486352897la_a_b @ G13 @ G24 ) ) ) ) ).

% gen'.intros(6)
thf(fact_1048_gen_Ointros_I1_J,axiom,
    ! [X3: nat] : ( relational_gen_a_b @ X3 @ ( relational_Bool_a_b @ $false ) @ bot_bo4495933725496725865la_a_b ) ).

% gen.intros(1)
thf(fact_1049_fmla_Osimps_I128_J,axiom,
    ! [X22: $o] :
      ( ( relati8924981150291758614la_a_b @ ( relational_Bool_a_b @ X22 ) )
      = bot_bot_set_b ) ).

% fmla.simps(128)
thf(fact_1050_gen_H_Ointros_I1_J,axiom,
    ! [X3: nat] : ( relational_gen_a_b2 @ X3 @ ( relational_Bool_a_b @ $false ) @ bot_bo4495933725496725865la_a_b ) ).

% gen'.intros(1)
thf(fact_1051_gen__genempty,axiom,
    ! [Z: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
      ( ( relational_gen_a_b @ Z @ Q @ G )
     => ( ( G = bot_bo4495933725496725865la_a_b )
       => ( relati5999705594545617851ty_a_b @ Q ) ) ) ).

% gen_genempty
thf(fact_1052_gen__empty__cp,axiom,
    ! [Z: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
      ( ( relational_gen_a_b @ Z @ Q @ G )
     => ( ( G = bot_bo4495933725496725865la_a_b )
       => ( ( relational_cp_a_b @ Q )
          = ( relational_Bool_a_b @ $false ) ) ) ) ).

% gen_empty_cp
thf(fact_1053_gen__induct,axiom,
    ! [X13: nat,X22: relational_fmla_a_b,X33: set_Re381260168593705685la_a_b,P: nat > relational_fmla_a_b > set_Re381260168593705685la_a_b > $o] :
      ( ( relational_gen_a_b @ X13 @ X22 @ X33 )
     => ( ! [X5: nat] : ( P @ X5 @ ( relational_Bool_a_b @ $false ) @ bot_bo4495933725496725865la_a_b )
       => ( ! [Q3: relational_fmla_a_b] :
              ( ( relational_ap_a_b @ Q3 )
             => ! [X5: nat] :
                  ( ( member_nat @ X5 @ ( relational_fv_a_b @ Q3 ) )
                 => ( P @ X5 @ Q3 @ ( insert7010464514620295119la_a_b @ Q3 @ bot_bo4495933725496725865la_a_b ) ) ) )
         => ( ! [X5: nat,Q3: relational_fmla_a_b,G4: set_Re381260168593705685la_a_b] :
                ( ( relational_gen_a_b @ X5 @ Q3 @ G4 )
               => ( ( P @ X5 @ Q3 @ G4 )
                 => ( P @ X5 @ ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q3 ) ) @ G4 ) ) )
           => ( ! [X5: nat,Q13: relational_fmla_a_b,Q24: relational_fmla_a_b,G4: set_Re381260168593705685la_a_b] :
                  ( ( relational_gen_a_b @ X5 @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q13 ) @ ( relational_Neg_a_b @ Q24 ) ) @ G4 )
                 => ( ( P @ X5 @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q13 ) @ ( relational_Neg_a_b @ Q24 ) ) @ G4 )
                   => ( P @ X5 @ ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q13 @ Q24 ) ) @ G4 ) ) )
             => ( ! [X5: nat,Q13: relational_fmla_a_b,Q24: relational_fmla_a_b,G4: set_Re381260168593705685la_a_b] :
                    ( ( relational_gen_a_b @ X5 @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q13 ) @ ( relational_Neg_a_b @ Q24 ) ) @ G4 )
                   => ( ( P @ X5 @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q13 ) @ ( relational_Neg_a_b @ Q24 ) ) @ G4 )
                     => ( P @ X5 @ ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q13 @ Q24 ) ) @ G4 ) ) )
               => ( ! [X5: nat,Q13: relational_fmla_a_b,G12: set_Re381260168593705685la_a_b] :
                      ( ( relational_gen_a_b @ X5 @ Q13 @ G12 )
                     => ( ( P @ X5 @ Q13 @ G12 )
                       => ! [Q24: relational_fmla_a_b,G23: set_Re381260168593705685la_a_b] :
                            ( ( relational_gen_a_b @ X5 @ Q24 @ G23 )
                           => ( ( P @ X5 @ Q24 @ G23 )
                             => ( P @ X5 @ ( relational_Disj_a_b @ Q13 @ Q24 ) @ ( sup_su5130108678486352897la_a_b @ G12 @ G23 ) ) ) ) ) )
                 => ( ! [X5: nat,Q13: relational_fmla_a_b,G4: set_Re381260168593705685la_a_b,Q24: relational_fmla_a_b] :
                        ( ( ( ( relational_gen_a_b @ X5 @ Q13 @ G4 )
                            & ( P @ X5 @ Q13 @ G4 ) )
                          | ( ( relational_gen_a_b @ X5 @ Q24 @ G4 )
                            & ( P @ X5 @ Q24 @ G4 ) ) )
                       => ( P @ X5 @ ( relational_Conj_a_b @ Q13 @ Q24 ) @ G4 ) )
                   => ( ! [Y3: nat,Q3: relational_fmla_a_b,G4: set_Re381260168593705685la_a_b] :
                          ( ( relational_gen_a_b @ Y3 @ Q3 @ G4 )
                         => ( ( P @ Y3 @ Q3 @ G4 )
                           => ! [X5: nat] :
                                ( P @ X5 @ ( relational_Conj_a_b @ Q3 @ ( relational_Eq_a_b @ X5 @ ( relational_Var_a @ Y3 ) ) )
                                @ ( image_6790371041703824709la_a_b
                                  @ ^ [Qa: relational_fmla_a_b] : ( relational_subst_a_b @ Qa @ Y3 @ X5 )
                                  @ G4 ) ) ) )
                     => ( ! [Y3: nat,Q3: relational_fmla_a_b,G4: set_Re381260168593705685la_a_b] :
                            ( ( relational_gen_a_b @ Y3 @ Q3 @ G4 )
                           => ( ( P @ Y3 @ Q3 @ G4 )
                             => ! [X5: nat] :
                                  ( P @ X5 @ ( relational_Conj_a_b @ Q3 @ ( relational_Eq_a_b @ Y3 @ ( relational_Var_a @ X5 ) ) )
                                  @ ( image_6790371041703824709la_a_b
                                    @ ^ [Qa: relational_fmla_a_b] : ( relational_subst_a_b @ Qa @ Y3 @ X5 )
                                    @ G4 ) ) ) )
                       => ( ! [X5: nat,Y3: nat] :
                              ( ( X5 != Y3 )
                             => ! [Q3: relational_fmla_a_b,G4: set_Re381260168593705685la_a_b] :
                                  ( ( relational_gen_a_b @ X5 @ Q3 @ G4 )
                                 => ( ( P @ X5 @ Q3 @ G4 )
                                   => ( P @ X5 @ ( relati591517084277583526ts_a_b @ Y3 @ Q3 ) @ ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y3 ) @ G4 ) ) ) ) )
                         => ( P @ X13 @ X22 @ X33 ) ) ) ) ) ) ) ) ) ) ) ) ).

% gen_induct
thf(fact_1054_gen_H_Ocases,axiom,
    ! [A1: nat,A22: relational_fmla_a_b,A33: set_Re381260168593705685la_a_b] :
      ( ( relational_gen_a_b2 @ A1 @ A22 @ A33 )
     => ( ( ( A22
            = ( relational_Bool_a_b @ $false ) )
         => ( A33 != bot_bo4495933725496725865la_a_b ) )
       => ( ( ( A33
              = ( insert7010464514620295119la_a_b @ A22 @ bot_bo4495933725496725865la_a_b ) )
           => ( ( relational_ap_a_b @ A22 )
             => ~ ( member_nat @ A1 @ ( relational_fv_a_b @ A22 ) ) ) )
         => ( ! [Q3: relational_fmla_a_b] :
                ( ( A22
                  = ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q3 ) ) )
               => ~ ( relational_gen_a_b2 @ A1 @ Q3 @ A33 ) )
           => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                  ( ( A22
                    = ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q13 @ Q24 ) ) )
                 => ~ ( relational_gen_a_b2 @ A1 @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q13 ) @ ( relational_Neg_a_b @ Q24 ) ) @ A33 ) )
             => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                    ( ( A22
                      = ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q13 @ Q24 ) ) )
                   => ~ ( relational_gen_a_b2 @ A1 @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q13 ) @ ( relational_Neg_a_b @ Q24 ) ) @ A33 ) )
               => ( ! [Q13: relational_fmla_a_b,G12: set_Re381260168593705685la_a_b,Q24: relational_fmla_a_b] :
                      ( ( A22
                        = ( relational_Disj_a_b @ Q13 @ Q24 ) )
                     => ! [G23: set_Re381260168593705685la_a_b] :
                          ( ( A33
                            = ( sup_su5130108678486352897la_a_b @ G12 @ G23 ) )
                         => ( ( relational_gen_a_b2 @ A1 @ Q13 @ G12 )
                           => ~ ( relational_gen_a_b2 @ A1 @ Q24 @ G23 ) ) ) )
                 => ( ! [Q13: relational_fmla_a_b,G4: set_Re381260168593705685la_a_b,Q24: relational_fmla_a_b] :
                        ( ( A22
                          = ( relational_Conj_a_b @ Q13 @ Q24 ) )
                       => ( ( A33 = G4 )
                         => ~ ( ( relational_gen_a_b2 @ A1 @ Q13 @ G4 )
                              | ( relational_gen_a_b2 @ A1 @ Q24 @ G4 ) ) ) )
                   => ( ! [Y3: nat,Q3: relational_fmla_a_b] :
                          ( ( A22
                            = ( relational_Conj_a_b @ Q3 @ ( relational_Eq_a_b @ A1 @ ( relational_Var_a @ Y3 ) ) ) )
                         => ! [G4: set_Re381260168593705685la_a_b] :
                              ( ( A33
                                = ( image_6790371041703824709la_a_b
                                  @ ^ [Qa: relational_fmla_a_b] : ( relational_subst_a_b @ Qa @ Y3 @ A1 )
                                  @ G4 ) )
                             => ~ ( relational_gen_a_b2 @ Y3 @ Q3 @ G4 ) ) )
                     => ( ! [Y3: nat,Q3: relational_fmla_a_b] :
                            ( ( A22
                              = ( relational_Conj_a_b @ Q3 @ ( relational_Eq_a_b @ Y3 @ ( relational_Var_a @ A1 ) ) ) )
                           => ! [G4: set_Re381260168593705685la_a_b] :
                                ( ( A33
                                  = ( image_6790371041703824709la_a_b
                                    @ ^ [Qa: relational_fmla_a_b] : ( relational_subst_a_b @ Qa @ Y3 @ A1 )
                                    @ G4 ) )
                               => ~ ( relational_gen_a_b2 @ Y3 @ Q3 @ G4 ) ) )
                       => ~ ! [Y3: nat,Q3: relational_fmla_a_b] :
                              ( ( A22
                                = ( relati591517084277583526ts_a_b @ Y3 @ Q3 ) )
                             => ! [G4: set_Re381260168593705685la_a_b] :
                                  ( ( A33
                                    = ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y3 ) @ G4 ) )
                                 => ( ( A1 != Y3 )
                                   => ~ ( relational_gen_a_b2 @ A1 @ Q3 @ G4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% gen'.cases
thf(fact_1055_gen_H_Osimps,axiom,
    ( relational_gen_a_b2
    = ( ^ [A12: nat,A23: relational_fmla_a_b,A32: set_Re381260168593705685la_a_b] :
          ( ( ( A23
              = ( relational_Bool_a_b @ $false ) )
            & ( A32 = bot_bo4495933725496725865la_a_b ) )
          | ( ( A32
              = ( insert7010464514620295119la_a_b @ A23 @ bot_bo4495933725496725865la_a_b ) )
            & ( relational_ap_a_b @ A23 )
            & ( member_nat @ A12 @ ( relational_fv_a_b @ A23 ) ) )
          | ? [Q2: relational_fmla_a_b] :
              ( ( A23
                = ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q2 ) ) )
              & ( relational_gen_a_b2 @ A12 @ Q2 @ A32 ) )
          | ? [Q12: relational_fmla_a_b,Q23: relational_fmla_a_b] :
              ( ( A23
                = ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q12 @ Q23 ) ) )
              & ( relational_gen_a_b2 @ A12 @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q12 ) @ ( relational_Neg_a_b @ Q23 ) ) @ A32 ) )
          | ? [Q12: relational_fmla_a_b,Q23: relational_fmla_a_b] :
              ( ( A23
                = ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q12 @ Q23 ) ) )
              & ( relational_gen_a_b2 @ A12 @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q12 ) @ ( relational_Neg_a_b @ Q23 ) ) @ A32 ) )
          | ? [Q12: relational_fmla_a_b,G1: set_Re381260168593705685la_a_b,Q23: relational_fmla_a_b] :
              ( ( A23
                = ( relational_Disj_a_b @ Q12 @ Q23 ) )
              & ? [G22: set_Re381260168593705685la_a_b] :
                  ( ( A32
                    = ( sup_su5130108678486352897la_a_b @ G1 @ G22 ) )
                  & ( relational_gen_a_b2 @ A12 @ Q12 @ G1 )
                  & ( relational_gen_a_b2 @ A12 @ Q23 @ G22 ) ) )
          | ? [Q12: relational_fmla_a_b,Q23: relational_fmla_a_b] :
              ( ( A23
                = ( relational_Conj_a_b @ Q12 @ Q23 ) )
              & ( ( relational_gen_a_b2 @ A12 @ Q12 @ A32 )
                | ( relational_gen_a_b2 @ A12 @ Q23 @ A32 ) ) )
          | ? [Y2: nat,Q2: relational_fmla_a_b] :
              ( ( A23
                = ( relational_Conj_a_b @ Q2 @ ( relational_Eq_a_b @ A12 @ ( relational_Var_a @ Y2 ) ) ) )
              & ? [G3: set_Re381260168593705685la_a_b] :
                  ( ( A32
                    = ( image_6790371041703824709la_a_b
                      @ ^ [Qa: relational_fmla_a_b] : ( relational_subst_a_b @ Qa @ Y2 @ A12 )
                      @ G3 ) )
                  & ( relational_gen_a_b2 @ Y2 @ Q2 @ G3 ) ) )
          | ? [Y2: nat,Q2: relational_fmla_a_b] :
              ( ( A23
                = ( relational_Conj_a_b @ Q2 @ ( relational_Eq_a_b @ Y2 @ ( relational_Var_a @ A12 ) ) ) )
              & ? [G3: set_Re381260168593705685la_a_b] :
                  ( ( A32
                    = ( image_6790371041703824709la_a_b
                      @ ^ [Qa: relational_fmla_a_b] : ( relational_subst_a_b @ Qa @ Y2 @ A12 )
                      @ G3 ) )
                  & ( relational_gen_a_b2 @ Y2 @ Q2 @ G3 ) ) )
          | ? [Y2: nat,Q2: relational_fmla_a_b] :
              ( ( A23
                = ( relati591517084277583526ts_a_b @ Y2 @ Q2 ) )
              & ? [G3: set_Re381260168593705685la_a_b] :
                  ( ( A32
                    = ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y2 ) @ G3 ) )
                  & ( A12 != Y2 )
                  & ( relational_gen_a_b2 @ A12 @ Q2 @ G3 ) ) ) ) ) ) ).

% gen'.simps
thf(fact_1056_sub_Oelims,axiom,
    ! [X3: relational_fmla_a_b,Y: set_Re381260168593705685la_a_b] :
      ( ( ( relational_sub_a_b @ X3 )
        = Y )
     => ( ! [T3: $o] :
            ( ( X3
              = ( relational_Bool_a_b @ T3 ) )
           => ( Y
             != ( insert7010464514620295119la_a_b @ ( relational_Bool_a_b @ T3 ) @ bot_bo4495933725496725865la_a_b ) ) )
       => ( ! [P4: b,Ts2: list_R6823256787227418703term_a] :
              ( ( X3
                = ( relational_Pred_b_a @ P4 @ Ts2 ) )
             => ( Y
               != ( insert7010464514620295119la_a_b @ ( relational_Pred_b_a @ P4 @ Ts2 ) @ bot_bo4495933725496725865la_a_b ) ) )
         => ( ! [X5: nat,T3: relational_term_a] :
                ( ( X3
                  = ( relational_Eq_a_b @ X5 @ T3 ) )
               => ( Y
                 != ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X5 @ T3 ) @ bot_bo4495933725496725865la_a_b ) ) )
           => ( ! [Q3: relational_fmla_a_b] :
                  ( ( X3
                    = ( relational_Neg_a_b @ Q3 ) )
                 => ( Y
                   != ( insert7010464514620295119la_a_b @ ( relational_Neg_a_b @ Q3 ) @ ( relational_sub_a_b @ Q3 ) ) ) )
             => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                    ( ( X3
                      = ( relational_Conj_a_b @ Q13 @ Q24 ) )
                   => ( Y
                     != ( insert7010464514620295119la_a_b @ ( relational_Conj_a_b @ Q13 @ Q24 ) @ ( sup_su5130108678486352897la_a_b @ ( relational_sub_a_b @ Q13 ) @ ( relational_sub_a_b @ Q24 ) ) ) ) )
               => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                      ( ( X3
                        = ( relational_Disj_a_b @ Q13 @ Q24 ) )
                     => ( Y
                       != ( insert7010464514620295119la_a_b @ ( relational_Disj_a_b @ Q13 @ Q24 ) @ ( sup_su5130108678486352897la_a_b @ ( relational_sub_a_b @ Q13 ) @ ( relational_sub_a_b @ Q24 ) ) ) ) )
                 => ~ ! [Z2: nat,Q3: relational_fmla_a_b] :
                        ( ( X3
                          = ( relati591517084277583526ts_a_b @ Z2 @ Q3 ) )
                       => ( Y
                         != ( insert7010464514620295119la_a_b @ ( relati591517084277583526ts_a_b @ Z2 @ Q3 ) @ ( relational_sub_a_b @ Q3 ) ) ) ) ) ) ) ) ) ) ) ).

% sub.elims
thf(fact_1057_the__elem__eq,axiom,
    ! [X3: $o] :
      ( ( the_elem_o @ ( insert_o @ X3 @ bot_bot_set_o ) )
      = X3 ) ).

% the_elem_eq
thf(fact_1058_sub_Opelims,axiom,
    ! [X3: relational_fmla_a_b,Y: set_Re381260168593705685la_a_b] :
      ( ( ( relational_sub_a_b @ X3 )
        = Y )
     => ( ( accp_R989495437599811158la_a_b @ relati7309537865537208983el_a_b @ X3 )
       => ( ! [T3: $o] :
              ( ( X3
                = ( relational_Bool_a_b @ T3 ) )
             => ( ( Y
                  = ( insert7010464514620295119la_a_b @ ( relational_Bool_a_b @ T3 ) @ bot_bo4495933725496725865la_a_b ) )
               => ~ ( accp_R989495437599811158la_a_b @ relati7309537865537208983el_a_b @ ( relational_Bool_a_b @ T3 ) ) ) )
         => ( ! [P4: b,Ts2: list_R6823256787227418703term_a] :
                ( ( X3
                  = ( relational_Pred_b_a @ P4 @ Ts2 ) )
               => ( ( Y
                    = ( insert7010464514620295119la_a_b @ ( relational_Pred_b_a @ P4 @ Ts2 ) @ bot_bo4495933725496725865la_a_b ) )
                 => ~ ( accp_R989495437599811158la_a_b @ relati7309537865537208983el_a_b @ ( relational_Pred_b_a @ P4 @ Ts2 ) ) ) )
           => ( ! [X5: nat,T3: relational_term_a] :
                  ( ( X3
                    = ( relational_Eq_a_b @ X5 @ T3 ) )
                 => ( ( Y
                      = ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X5 @ T3 ) @ bot_bo4495933725496725865la_a_b ) )
                   => ~ ( accp_R989495437599811158la_a_b @ relati7309537865537208983el_a_b @ ( relational_Eq_a_b @ X5 @ T3 ) ) ) )
             => ( ! [Q3: relational_fmla_a_b] :
                    ( ( X3
                      = ( relational_Neg_a_b @ Q3 ) )
                   => ( ( Y
                        = ( insert7010464514620295119la_a_b @ ( relational_Neg_a_b @ Q3 ) @ ( relational_sub_a_b @ Q3 ) ) )
                     => ~ ( accp_R989495437599811158la_a_b @ relati7309537865537208983el_a_b @ ( relational_Neg_a_b @ Q3 ) ) ) )
               => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                      ( ( X3
                        = ( relational_Conj_a_b @ Q13 @ Q24 ) )
                     => ( ( Y
                          = ( insert7010464514620295119la_a_b @ ( relational_Conj_a_b @ Q13 @ Q24 ) @ ( sup_su5130108678486352897la_a_b @ ( relational_sub_a_b @ Q13 ) @ ( relational_sub_a_b @ Q24 ) ) ) )
                       => ~ ( accp_R989495437599811158la_a_b @ relati7309537865537208983el_a_b @ ( relational_Conj_a_b @ Q13 @ Q24 ) ) ) )
                 => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                        ( ( X3
                          = ( relational_Disj_a_b @ Q13 @ Q24 ) )
                       => ( ( Y
                            = ( insert7010464514620295119la_a_b @ ( relational_Disj_a_b @ Q13 @ Q24 ) @ ( sup_su5130108678486352897la_a_b @ ( relational_sub_a_b @ Q13 ) @ ( relational_sub_a_b @ Q24 ) ) ) )
                         => ~ ( accp_R989495437599811158la_a_b @ relati7309537865537208983el_a_b @ ( relational_Disj_a_b @ Q13 @ Q24 ) ) ) )
                   => ~ ! [Z2: nat,Q3: relational_fmla_a_b] :
                          ( ( X3
                            = ( relati591517084277583526ts_a_b @ Z2 @ Q3 ) )
                         => ( ( Y
                              = ( insert7010464514620295119la_a_b @ ( relati591517084277583526ts_a_b @ Z2 @ Q3 ) @ ( relational_sub_a_b @ Q3 ) ) )
                           => ~ ( accp_R989495437599811158la_a_b @ relati7309537865537208983el_a_b @ ( relati591517084277583526ts_a_b @ Z2 @ Q3 ) ) ) ) ) ) ) ) ) ) ) ) ).

% sub.pelims
thf(fact_1059_is__singletonI,axiom,
    ! [X3: $o] : ( is_singleton_o @ ( insert_o @ X3 @ bot_bot_set_o ) ) ).

% is_singletonI
thf(fact_1060_fv_Osimps_I6_J,axiom,
    ! [Phi: relational_fmla_a_b,Psi2: relational_fmla_a_b] :
      ( ( relational_fv_a_b @ ( relational_Disj_a_b @ Phi @ Psi2 ) )
      = ( sup_sup_set_nat @ ( relational_fv_a_b @ Phi ) @ ( relational_fv_a_b @ Psi2 ) ) ) ).

% fv.simps(6)
thf(fact_1061_fv_Osimps_I5_J,axiom,
    ! [Phi: relational_fmla_a_b,Psi2: relational_fmla_a_b] :
      ( ( relational_fv_a_b @ ( relational_Conj_a_b @ Phi @ Psi2 ) )
      = ( sup_sup_set_nat @ ( relational_fv_a_b @ Phi ) @ ( relational_fv_a_b @ Psi2 ) ) ) ).

% fv.simps(5)
thf(fact_1062_is__singleton__the__elem,axiom,
    ( is_singleton_o
    = ( ^ [A6: set_o] :
          ( A6
          = ( insert_o @ ( the_elem_o @ A6 ) @ bot_bot_set_o ) ) ) ) ).

% is_singleton_the_elem
thf(fact_1063_fv_Osimps_I2_J,axiom,
    ! [B: $o] :
      ( ( relational_fv_a_b @ ( relational_Bool_a_b @ B ) )
      = bot_bot_set_nat ) ).

% fv.simps(2)
thf(fact_1064_sub_Osimps_I4_J,axiom,
    ! [Q: relational_fmla_a_b] :
      ( ( relational_sub_a_b @ ( relational_Neg_a_b @ Q ) )
      = ( insert7010464514620295119la_a_b @ ( relational_Neg_a_b @ Q ) @ ( relational_sub_a_b @ Q ) ) ) ).

% sub.simps(4)
thf(fact_1065_sub_Osimps_I7_J,axiom,
    ! [Z: nat,Q: relational_fmla_a_b] :
      ( ( relational_sub_a_b @ ( relati591517084277583526ts_a_b @ Z @ Q ) )
      = ( insert7010464514620295119la_a_b @ ( relati591517084277583526ts_a_b @ Z @ Q ) @ ( relational_sub_a_b @ Q ) ) ) ).

% sub.simps(7)
thf(fact_1066_is__singletonI_H,axiom,
    ! [A2: set_o] :
      ( ( A2 != bot_bot_set_o )
     => ( ! [X5: $o,Y3: $o] :
            ( ( member_o @ X5 @ A2 )
           => ( ( member_o @ Y3 @ A2 )
             => ( X5 = Y3 ) ) )
       => ( is_singleton_o @ A2 ) ) ) ).

% is_singletonI'
thf(fact_1067_sub_Osimps_I1_J,axiom,
    ! [T: $o] :
      ( ( relational_sub_a_b @ ( relational_Bool_a_b @ T ) )
      = ( insert7010464514620295119la_a_b @ ( relational_Bool_a_b @ T ) @ bot_bo4495933725496725865la_a_b ) ) ).

% sub.simps(1)
thf(fact_1068_sub_Osimps_I6_J,axiom,
    ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
      ( ( relational_sub_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) )
      = ( insert7010464514620295119la_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) @ ( sup_su5130108678486352897la_a_b @ ( relational_sub_a_b @ Q1 ) @ ( relational_sub_a_b @ Q22 ) ) ) ) ).

% sub.simps(6)
thf(fact_1069_sub_Osimps_I5_J,axiom,
    ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
      ( ( relational_sub_a_b @ ( relational_Conj_a_b @ Q1 @ Q22 ) )
      = ( insert7010464514620295119la_a_b @ ( relational_Conj_a_b @ Q1 @ Q22 ) @ ( sup_su5130108678486352897la_a_b @ ( relational_sub_a_b @ Q1 ) @ ( relational_sub_a_b @ Q22 ) ) ) ) ).

% sub.simps(5)
thf(fact_1070_is__singletonE,axiom,
    ! [A2: set_o] :
      ( ( is_singleton_o @ A2 )
     => ~ ! [X5: $o] :
            ( A2
           != ( insert_o @ X5 @ bot_bot_set_o ) ) ) ).

% is_singletonE
thf(fact_1071_is__singleton__def,axiom,
    ( is_singleton_o
    = ( ^ [A6: set_o] :
        ? [X: $o] :
          ( A6
          = ( insert_o @ X @ bot_bot_set_o ) ) ) ) ).

% is_singleton_def
thf(fact_1072_fv_Oelims,axiom,
    ! [X3: relational_fmla_a_b,Y: set_nat] :
      ( ( ( relational_fv_a_b @ X3 )
        = Y )
     => ( ! [Uu: b,Ts2: list_R6823256787227418703term_a] :
            ( ( X3
              = ( relational_Pred_b_a @ Uu @ Ts2 ) )
           => ( Y
             != ( relati4569515538964159125_set_a @ Ts2 ) ) )
       => ( ( ? [B2: $o] :
                ( X3
                = ( relational_Bool_a_b @ B2 ) )
           => ( Y != bot_bot_set_nat ) )
         => ( ! [X5: nat,T4: relational_term_a] :
                ( ( X3
                  = ( relational_Eq_a_b @ X5 @ T4 ) )
               => ( Y
                 != ( sup_sup_set_nat @ ( insert_nat @ X5 @ bot_bot_set_nat ) @ ( relati6004689760767320788_set_a @ T4 ) ) ) )
           => ( ! [Phi2: relational_fmla_a_b] :
                  ( ( X3
                    = ( relational_Neg_a_b @ Phi2 ) )
                 => ( Y
                   != ( relational_fv_a_b @ Phi2 ) ) )
             => ( ! [Phi2: relational_fmla_a_b,Psi: relational_fmla_a_b] :
                    ( ( X3
                      = ( relational_Conj_a_b @ Phi2 @ Psi ) )
                   => ( Y
                     != ( sup_sup_set_nat @ ( relational_fv_a_b @ Phi2 ) @ ( relational_fv_a_b @ Psi ) ) ) )
               => ( ! [Phi2: relational_fmla_a_b,Psi: relational_fmla_a_b] :
                      ( ( X3
                        = ( relational_Disj_a_b @ Phi2 @ Psi ) )
                     => ( Y
                       != ( sup_sup_set_nat @ ( relational_fv_a_b @ Phi2 ) @ ( relational_fv_a_b @ Psi ) ) ) )
                 => ~ ! [Z2: nat,Phi2: relational_fmla_a_b] :
                        ( ( X3
                          = ( relati591517084277583526ts_a_b @ Z2 @ Phi2 ) )
                       => ( Y
                         != ( minus_minus_set_nat @ ( relational_fv_a_b @ Phi2 ) @ ( insert_nat @ Z2 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ) ) ).

% fv.elims
thf(fact_1073_Set_Ois__empty__def,axiom,
    ( is_empty_o
    = ( ^ [A6: set_o] : ( A6 = bot_bot_set_o ) ) ) ).

% Set.is_empty_def
thf(fact_1074_fv_Opelims,axiom,
    ! [X3: relational_fmla_a_b,Y: set_nat] :
      ( ( ( relational_fv_a_b @ X3 )
        = Y )
     => ( ( accp_R989495437599811158la_a_b @ relati5703530512245835757el_a_b @ X3 )
       => ( ! [Uu: b,Ts2: list_R6823256787227418703term_a] :
              ( ( X3
                = ( relational_Pred_b_a @ Uu @ Ts2 ) )
             => ( ( Y
                  = ( relati4569515538964159125_set_a @ Ts2 ) )
               => ~ ( accp_R989495437599811158la_a_b @ relati5703530512245835757el_a_b @ ( relational_Pred_b_a @ Uu @ Ts2 ) ) ) )
         => ( ! [B2: $o] :
                ( ( X3
                  = ( relational_Bool_a_b @ B2 ) )
               => ( ( Y = bot_bot_set_nat )
                 => ~ ( accp_R989495437599811158la_a_b @ relati5703530512245835757el_a_b @ ( relational_Bool_a_b @ B2 ) ) ) )
           => ( ! [X5: nat,T4: relational_term_a] :
                  ( ( X3
                    = ( relational_Eq_a_b @ X5 @ T4 ) )
                 => ( ( Y
                      = ( sup_sup_set_nat @ ( insert_nat @ X5 @ bot_bot_set_nat ) @ ( relati6004689760767320788_set_a @ T4 ) ) )
                   => ~ ( accp_R989495437599811158la_a_b @ relati5703530512245835757el_a_b @ ( relational_Eq_a_b @ X5 @ T4 ) ) ) )
             => ( ! [Phi2: relational_fmla_a_b] :
                    ( ( X3
                      = ( relational_Neg_a_b @ Phi2 ) )
                   => ( ( Y
                        = ( relational_fv_a_b @ Phi2 ) )
                     => ~ ( accp_R989495437599811158la_a_b @ relati5703530512245835757el_a_b @ ( relational_Neg_a_b @ Phi2 ) ) ) )
               => ( ! [Phi2: relational_fmla_a_b,Psi: relational_fmla_a_b] :
                      ( ( X3
                        = ( relational_Conj_a_b @ Phi2 @ Psi ) )
                     => ( ( Y
                          = ( sup_sup_set_nat @ ( relational_fv_a_b @ Phi2 ) @ ( relational_fv_a_b @ Psi ) ) )
                       => ~ ( accp_R989495437599811158la_a_b @ relati5703530512245835757el_a_b @ ( relational_Conj_a_b @ Phi2 @ Psi ) ) ) )
                 => ( ! [Phi2: relational_fmla_a_b,Psi: relational_fmla_a_b] :
                        ( ( X3
                          = ( relational_Disj_a_b @ Phi2 @ Psi ) )
                       => ( ( Y
                            = ( sup_sup_set_nat @ ( relational_fv_a_b @ Phi2 ) @ ( relational_fv_a_b @ Psi ) ) )
                         => ~ ( accp_R989495437599811158la_a_b @ relati5703530512245835757el_a_b @ ( relational_Disj_a_b @ Phi2 @ Psi ) ) ) )
                   => ~ ! [Z2: nat,Phi2: relational_fmla_a_b] :
                          ( ( X3
                            = ( relati591517084277583526ts_a_b @ Z2 @ Phi2 ) )
                         => ( ( Y
                              = ( minus_minus_set_nat @ ( relational_fv_a_b @ Phi2 ) @ ( insert_nat @ Z2 @ bot_bot_set_nat ) ) )
                           => ~ ( accp_R989495437599811158la_a_b @ relati5703530512245835757el_a_b @ ( relati591517084277583526ts_a_b @ Z2 @ Phi2 ) ) ) ) ) ) ) ) ) ) ) ) ).

% fv.pelims
thf(fact_1075_Diff__iff,axiom,
    ! [C: $o,A2: set_o,B5: set_o] :
      ( ( member_o @ C @ ( minus_minus_set_o @ A2 @ B5 ) )
      = ( ( member_o @ C @ A2 )
        & ~ ( member_o @ C @ B5 ) ) ) ).

% Diff_iff
thf(fact_1076_DiffI,axiom,
    ! [C: $o,A2: set_o,B5: set_o] :
      ( ( member_o @ C @ A2 )
     => ( ~ ( member_o @ C @ B5 )
       => ( member_o @ C @ ( minus_minus_set_o @ A2 @ B5 ) ) ) ) ).

% DiffI
thf(fact_1077_Diff__cancel,axiom,
    ! [A2: set_o] :
      ( ( minus_minus_set_o @ A2 @ A2 )
      = bot_bot_set_o ) ).

% Diff_cancel
thf(fact_1078_empty__Diff,axiom,
    ! [A2: set_o] :
      ( ( minus_minus_set_o @ bot_bot_set_o @ A2 )
      = bot_bot_set_o ) ).

% empty_Diff
thf(fact_1079_Diff__empty,axiom,
    ! [A2: set_o] :
      ( ( minus_minus_set_o @ A2 @ bot_bot_set_o )
      = A2 ) ).

% Diff_empty
thf(fact_1080_insert__Diff1,axiom,
    ! [X3: $o,B5: set_o,A2: set_o] :
      ( ( member_o @ X3 @ B5 )
     => ( ( minus_minus_set_o @ ( insert_o @ X3 @ A2 ) @ B5 )
        = ( minus_minus_set_o @ A2 @ B5 ) ) ) ).

% insert_Diff1
thf(fact_1081_Diff__insert0,axiom,
    ! [X3: $o,A2: set_o,B5: set_o] :
      ( ~ ( member_o @ X3 @ A2 )
     => ( ( minus_minus_set_o @ A2 @ ( insert_o @ X3 @ B5 ) )
        = ( minus_minus_set_o @ A2 @ B5 ) ) ) ).

% Diff_insert0
thf(fact_1082_insert__Diff__single,axiom,
    ! [A: $o,A2: set_o] :
      ( ( insert_o @ A @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ bot_bot_set_o ) ) )
      = ( insert_o @ A @ A2 ) ) ).

% insert_Diff_single
thf(fact_1083_insert__Diff__if,axiom,
    ! [X3: $o,B5: set_o,A2: set_o] :
      ( ( ( member_o @ X3 @ B5 )
       => ( ( minus_minus_set_o @ ( insert_o @ X3 @ A2 ) @ B5 )
          = ( minus_minus_set_o @ A2 @ B5 ) ) )
      & ( ~ ( member_o @ X3 @ B5 )
       => ( ( minus_minus_set_o @ ( insert_o @ X3 @ A2 ) @ B5 )
          = ( insert_o @ X3 @ ( minus_minus_set_o @ A2 @ B5 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_1084_set__diff__eq,axiom,
    ( minus_minus_set_o
    = ( ^ [A6: set_o,B7: set_o] :
          ( collect_o
          @ ^ [X: $o] :
              ( ( member_o @ X @ A6 )
              & ~ ( member_o @ X @ B7 ) ) ) ) ) ).

% set_diff_eq
thf(fact_1085_DiffD2,axiom,
    ! [C: $o,A2: set_o,B5: set_o] :
      ( ( member_o @ C @ ( minus_minus_set_o @ A2 @ B5 ) )
     => ~ ( member_o @ C @ B5 ) ) ).

% DiffD2
thf(fact_1086_DiffD1,axiom,
    ! [C: $o,A2: set_o,B5: set_o] :
      ( ( member_o @ C @ ( minus_minus_set_o @ A2 @ B5 ) )
     => ( member_o @ C @ A2 ) ) ).

% DiffD1
thf(fact_1087_DiffE,axiom,
    ! [C: $o,A2: set_o,B5: set_o] :
      ( ( member_o @ C @ ( minus_minus_set_o @ A2 @ B5 ) )
     => ~ ( ( member_o @ C @ A2 )
         => ( member_o @ C @ B5 ) ) ) ).

% DiffE
thf(fact_1088_Diff__insert__absorb,axiom,
    ! [X3: $o,A2: set_o] :
      ( ~ ( member_o @ X3 @ A2 )
     => ( ( minus_minus_set_o @ ( insert_o @ X3 @ A2 ) @ ( insert_o @ X3 @ bot_bot_set_o ) )
        = A2 ) ) ).

% Diff_insert_absorb
thf(fact_1089_Diff__insert2,axiom,
    ! [A2: set_o,A: $o,B5: set_o] :
      ( ( minus_minus_set_o @ A2 @ ( insert_o @ A @ B5 ) )
      = ( minus_minus_set_o @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ bot_bot_set_o ) ) @ B5 ) ) ).

% Diff_insert2
thf(fact_1090_insert__Diff,axiom,
    ! [A: $o,A2: set_o] :
      ( ( member_o @ A @ A2 )
     => ( ( insert_o @ A @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ bot_bot_set_o ) ) )
        = A2 ) ) ).

% insert_Diff
thf(fact_1091_Diff__insert,axiom,
    ! [A2: set_o,A: $o,B5: set_o] :
      ( ( minus_minus_set_o @ A2 @ ( insert_o @ A @ B5 ) )
      = ( minus_minus_set_o @ ( minus_minus_set_o @ A2 @ B5 ) @ ( insert_o @ A @ bot_bot_set_o ) ) ) ).

% Diff_insert
thf(fact_1092_in__image__insert__iff,axiom,
    ! [B5: set_set_o,X3: $o,A2: set_o] :
      ( ! [C6: set_o] :
          ( ( member_set_o @ C6 @ B5 )
         => ~ ( member_o @ X3 @ C6 ) )
     => ( ( member_set_o @ A2 @ ( image_set_o_set_o @ ( insert_o @ X3 ) @ B5 ) )
        = ( ( member_o @ X3 @ A2 )
          & ( member_set_o @ ( minus_minus_set_o @ A2 @ ( insert_o @ X3 @ bot_bot_set_o ) ) @ B5 ) ) ) ) ).

% in_image_insert_iff
thf(fact_1093_fv_Osimps_I7_J,axiom,
    ! [Z: nat,Phi: relational_fmla_a_b] :
      ( ( relational_fv_a_b @ ( relati591517084277583526ts_a_b @ Z @ Phi ) )
      = ( minus_minus_set_nat @ ( relational_fv_a_b @ Phi ) @ ( insert_nat @ Z @ bot_bot_set_nat ) ) ) ).

% fv.simps(7)
thf(fact_1094_fv__subst,axiom,
    ! [X3: nat,Q: relational_fmla_a_b,Y: nat] :
      ( ( ( member_nat @ X3 @ ( relational_fv_a_b @ Q ) )
       => ( ( relational_fv_a_b @ ( relational_subst_a_b @ Q @ X3 @ Y ) )
          = ( insert_nat @ Y @ ( minus_minus_set_nat @ ( relational_fv_a_b @ Q ) @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) )
      & ( ~ ( member_nat @ X3 @ ( relational_fv_a_b @ Q ) )
       => ( ( relational_fv_a_b @ ( relational_subst_a_b @ Q @ X3 @ Y ) )
          = ( relational_fv_a_b @ Q ) ) ) ) ).

% fv_subst
thf(fact_1095_fun__upd__image,axiom,
    ! [X3: $o,A2: set_o,F2: $o > $o,Y: $o] :
      ( ( ( member_o @ X3 @ A2 )
       => ( ( image_o_o @ ( fun_upd_o_o @ F2 @ X3 @ Y ) @ A2 )
          = ( insert_o @ Y @ ( image_o_o @ F2 @ ( minus_minus_set_o @ A2 @ ( insert_o @ X3 @ bot_bot_set_o ) ) ) ) ) )
      & ( ~ ( member_o @ X3 @ A2 )
       => ( ( image_o_o @ ( fun_upd_o_o @ F2 @ X3 @ Y ) @ A2 )
          = ( image_o_o @ F2 @ A2 ) ) ) ) ).

% fun_upd_image
thf(fact_1096_remove__def,axiom,
    ( remove_o
    = ( ^ [X: $o,A6: set_o] : ( minus_minus_set_o @ A6 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ).

% remove_def
thf(fact_1097_member__remove,axiom,
    ! [X3: $o,Y: $o,A2: set_o] :
      ( ( member_o @ X3 @ ( remove_o @ Y @ A2 ) )
      = ( ( member_o @ X3 @ A2 )
        & ( X3 != Y ) ) ) ).

% member_remove
thf(fact_1098_empty__bind,axiom,
    ! [F2: $o > set_o] :
      ( ( bind_o_o @ bot_bot_set_o @ F2 )
      = bot_bot_set_o ) ).

% empty_bind
thf(fact_1099_minus__set__def,axiom,
    ( minus_minus_set_o
    = ( ^ [A6: set_o,B7: set_o] :
          ( collect_o
          @ ( minus_minus_o_o
            @ ^ [X: $o] : ( member_o @ X @ A6 )
            @ ^ [X: $o] : ( member_o @ X @ B7 ) ) ) ) ) ).

% minus_set_def
thf(fact_1100_bind__const,axiom,
    ! [A2: set_o,B5: set_o] :
      ( ( ( A2 = bot_bot_set_o )
       => ( ( bind_o_o @ A2
            @ ^ [Uu2: $o] : B5 )
          = bot_bot_set_o ) )
      & ( ( A2 != bot_bot_set_o )
       => ( ( bind_o_o @ A2
            @ ^ [Uu2: $o] : B5 )
          = B5 ) ) ) ).

% bind_const
thf(fact_1101_UNION__fun__upd,axiom,
    ! [A2: $o > set_o,I3: $o,B5: set_o,J: set_o] :
      ( ( comple90263536869209701_set_o @ ( image_o_set_o @ ( fun_upd_o_set_o @ A2 @ I3 @ B5 ) @ J ) )
      = ( sup_sup_set_o @ ( comple90263536869209701_set_o @ ( image_o_set_o @ A2 @ ( minus_minus_set_o @ J @ ( insert_o @ I3 @ bot_bot_set_o ) ) ) ) @ ( if_set_o @ ( member_o @ I3 @ J ) @ B5 @ bot_bot_set_o ) ) ) ).

% UNION_fun_upd
thf(fact_1102_the__elem__def,axiom,
    ( the_elem_o
    = ( ^ [X7: set_o] :
          ( the_o
          @ ^ [X: $o] :
              ( X7
              = ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ).

% the_elem_def
thf(fact_1103_range__constant,axiom,
    ! [X3: $o] :
      ( ( image_o_o
        @ ^ [Uu2: $o] : X3
        @ top_top_set_o )
      = ( insert_o @ X3 @ bot_bot_set_o ) ) ).

% range_constant
thf(fact_1104_UNIV__I,axiom,
    ! [X3: $o] : ( member_o @ X3 @ top_top_set_o ) ).

% UNIV_I
thf(fact_1105_SigmaI,axiom,
    ! [A: $o,A2: set_o,B: $o,B5: $o > set_o] :
      ( ( member_o @ A @ A2 )
     => ( ( member_o @ B @ ( B5 @ A ) )
       => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ ( product_Sigma_o_o @ A2 @ B5 ) ) ) ) ).

% SigmaI
thf(fact_1106_mem__Sigma__iff,axiom,
    ! [A: $o,B: $o,A2: set_o,B5: $o > set_o] :
      ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ ( product_Sigma_o_o @ A2 @ B5 ) )
      = ( ( member_o @ A @ A2 )
        & ( member_o @ B @ ( B5 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_1107_Collect__const,axiom,
    ! [P: $o] :
      ( ( P
       => ( ( collect_o
            @ ^ [S2: $o] : P )
          = top_top_set_o ) )
      & ( ~ P
       => ( ( collect_o
            @ ^ [S2: $o] : P )
          = bot_bot_set_o ) ) ) ).

% Collect_const
thf(fact_1108_UNIV__Times__UNIV,axiom,
    ( ( product_Sigma_o_o @ top_top_set_o
      @ ^ [Uu2: $o] : top_top_set_o )
    = top_to7721136755696657239od_o_o ) ).

% UNIV_Times_UNIV
thf(fact_1109_Times__empty,axiom,
    ! [A2: set_o,B5: set_o] :
      ( ( ( product_Sigma_o_o @ A2
          @ ^ [Uu2: $o] : B5 )
        = bot_bo7073875226086086771od_o_o )
      = ( ( A2 = bot_bot_set_o )
        | ( B5 = bot_bot_set_o ) ) ) ).

% Times_empty
thf(fact_1110_UN__I,axiom,
    ! [A: $o,A2: set_o,B: $o,B5: $o > set_o] :
      ( ( member_o @ A @ A2 )
     => ( ( member_o @ B @ ( B5 @ A ) )
       => ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B5 @ A2 ) ) ) ) ) ).

% UN_I
thf(fact_1111_Diff__UNIV,axiom,
    ! [A2: set_o] :
      ( ( minus_minus_set_o @ A2 @ top_top_set_o )
      = bot_bot_set_o ) ).

% Diff_UNIV
thf(fact_1112_insert__Times__insert,axiom,
    ! [A: $o,A2: set_o,B: $o,B5: set_o] :
      ( ( product_Sigma_o_o @ ( insert_o @ A @ A2 )
        @ ^ [Uu2: $o] : ( insert_o @ B @ B5 ) )
      = ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ A @ B )
        @ ( sup_su5769328420594410459od_o_o
          @ ( product_Sigma_o_o @ A2
            @ ^ [Uu2: $o] : ( insert_o @ B @ B5 ) )
          @ ( product_Sigma_o_o @ ( insert_o @ A @ A2 )
            @ ^ [Uu2: $o] : B5 ) ) ) ) ).

% insert_Times_insert
thf(fact_1113_UN__constant,axiom,
    ! [A2: set_o,C: set_o] :
      ( ( ( A2 = bot_bot_set_o )
       => ( ( comple90263536869209701_set_o
            @ ( image_o_set_o
              @ ^ [Y2: $o] : C
              @ A2 ) )
          = bot_bot_set_o ) )
      & ( ( A2 != bot_bot_set_o )
       => ( ( comple90263536869209701_set_o
            @ ( image_o_set_o
              @ ^ [Y2: $o] : C
              @ A2 ) )
          = C ) ) ) ).

% UN_constant
thf(fact_1114_UN__simps_I1_J,axiom,
    ! [C4: set_o,A: $o,B5: $o > set_o] :
      ( ( ( C4 = bot_bot_set_o )
       => ( ( comple90263536869209701_set_o
            @ ( image_o_set_o
              @ ^ [X: $o] : ( insert_o @ A @ ( B5 @ X ) )
              @ C4 ) )
          = bot_bot_set_o ) )
      & ( ( C4 != bot_bot_set_o )
       => ( ( comple90263536869209701_set_o
            @ ( image_o_set_o
              @ ^ [X: $o] : ( insert_o @ A @ ( B5 @ X ) )
              @ C4 ) )
          = ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B5 @ C4 ) ) ) ) ) ) ).

% UN_simps(1)
thf(fact_1115_UN__singleton,axiom,
    ! [A2: set_o] :
      ( ( comple90263536869209701_set_o
        @ ( image_o_set_o
          @ ^ [X: $o] : ( insert_o @ X @ bot_bot_set_o )
          @ A2 ) )
      = A2 ) ).

% UN_singleton
thf(fact_1116_UN__simps_I3_J,axiom,
    ! [C4: set_o,A2: set_o,B5: $o > set_o] :
      ( ( ( C4 = bot_bot_set_o )
       => ( ( comple90263536869209701_set_o
            @ ( image_o_set_o
              @ ^ [X: $o] : ( sup_sup_set_o @ A2 @ ( B5 @ X ) )
              @ C4 ) )
          = bot_bot_set_o ) )
      & ( ( C4 != bot_bot_set_o )
       => ( ( comple90263536869209701_set_o
            @ ( image_o_set_o
              @ ^ [X: $o] : ( sup_sup_set_o @ A2 @ ( B5 @ X ) )
              @ C4 ) )
          = ( sup_sup_set_o @ A2 @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B5 @ C4 ) ) ) ) ) ) ).

% UN_simps(3)
thf(fact_1117_UN__simps_I2_J,axiom,
    ! [C4: set_o,A2: $o > set_o,B5: set_o] :
      ( ( ( C4 = bot_bot_set_o )
       => ( ( comple90263536869209701_set_o
            @ ( image_o_set_o
              @ ^ [X: $o] : ( sup_sup_set_o @ ( A2 @ X ) @ B5 )
              @ C4 ) )
          = bot_bot_set_o ) )
      & ( ( C4 != bot_bot_set_o )
       => ( ( comple90263536869209701_set_o
            @ ( image_o_set_o
              @ ^ [X: $o] : ( sup_sup_set_o @ ( A2 @ X ) @ B5 )
              @ C4 ) )
          = ( sup_sup_set_o @ ( comple90263536869209701_set_o @ ( image_o_set_o @ A2 @ C4 ) ) @ B5 ) ) ) ) ).

% UN_simps(2)
thf(fact_1118_range__eqI,axiom,
    ! [B: $o,F2: $o > $o,X3: $o] :
      ( ( B
        = ( F2 @ X3 ) )
     => ( member_o @ B @ ( image_o_o @ F2 @ top_top_set_o ) ) ) ).

% range_eqI
thf(fact_1119_surj__def,axiom,
    ! [F2: $o > $o] :
      ( ( ( image_o_o @ F2 @ top_top_set_o )
        = top_top_set_o )
      = ( ! [Y2: $o] :
          ? [X: $o] :
            ( Y2
            = ( F2 @ X ) ) ) ) ).

% surj_def
thf(fact_1120_rangeI,axiom,
    ! [F2: $o > $o,X3: $o] : ( member_o @ ( F2 @ X3 ) @ ( image_o_o @ F2 @ top_top_set_o ) ) ).

% rangeI
thf(fact_1121_surjI,axiom,
    ! [G2: $o > $o,F2: $o > $o] :
      ( ! [X5: $o] :
          ( ( G2 @ ( F2 @ X5 ) )
          = X5 )
     => ( ( image_o_o @ G2 @ top_top_set_o )
        = top_top_set_o ) ) ).

% surjI
thf(fact_1122_surjE,axiom,
    ! [F2: $o > $o,Y: $o] :
      ( ( ( image_o_o @ F2 @ top_top_set_o )
        = top_top_set_o )
     => ~ ! [X5: $o] :
            ( Y
            = ( ~ ( F2 @ X5 ) ) ) ) ).

% surjE
thf(fact_1123_surjD,axiom,
    ! [F2: $o > $o,Y: $o] :
      ( ( ( image_o_o @ F2 @ top_top_set_o )
        = top_top_set_o )
     => ? [X5: $o] :
          ( Y
          = ( F2 @ X5 ) ) ) ).

% surjD
thf(fact_1124_UNIV__witness,axiom,
    ? [X5: $o] : ( member_o @ X5 @ top_top_set_o ) ).

% UNIV_witness
thf(fact_1125_UNIV__eq__I,axiom,
    ! [A2: set_o] :
      ( ! [X5: $o] : ( member_o @ X5 @ A2 )
     => ( top_top_set_o = A2 ) ) ).

% UNIV_eq_I
thf(fact_1126_UNIV__def,axiom,
    ( top_top_set_o
    = ( collect_o
      @ ^ [X: $o] : $true ) ) ).

% UNIV_def
thf(fact_1127_SigmaE,axiom,
    ! [C: product_prod_o_o,A2: set_o,B5: $o > set_o] :
      ( ( member7466972457876170832od_o_o @ C @ ( product_Sigma_o_o @ A2 @ B5 ) )
     => ~ ! [X5: $o] :
            ( ( member_o @ X5 @ A2 )
           => ! [Y3: $o] :
                ( ( member_o @ Y3 @ ( B5 @ X5 ) )
               => ( C
                 != ( product_Pair_o_o @ X5 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_1128_SigmaE2,axiom,
    ! [A: $o,B: $o,A2: set_o,B5: $o > set_o] :
      ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ ( product_Sigma_o_o @ A2 @ B5 ) )
     => ~ ( ( member_o @ A @ A2 )
         => ~ ( member_o @ B @ ( B5 @ A ) ) ) ) ).

% SigmaE2
thf(fact_1129_times__eq__iff,axiom,
    ! [A2: set_o,B5: set_o,C4: set_o,D2: set_o] :
      ( ( ( product_Sigma_o_o @ A2
          @ ^ [Uu2: $o] : B5 )
        = ( product_Sigma_o_o @ C4
          @ ^ [Uu2: $o] : D2 ) )
      = ( ( ( A2 = C4 )
          & ( B5 = D2 ) )
        | ( ( ( A2 = bot_bot_set_o )
            | ( B5 = bot_bot_set_o ) )
          & ( ( C4 = bot_bot_set_o )
            | ( D2 = bot_bot_set_o ) ) ) ) ) ).

% times_eq_iff
thf(fact_1130_Un__UNIV__right,axiom,
    ! [A2: set_o] :
      ( ( sup_sup_set_o @ A2 @ top_top_set_o )
      = top_top_set_o ) ).

% Un_UNIV_right
thf(fact_1131_Un__UNIV__left,axiom,
    ! [B5: set_o] :
      ( ( sup_sup_set_o @ top_top_set_o @ B5 )
      = top_top_set_o ) ).

% Un_UNIV_left
thf(fact_1132_insert__UNIV,axiom,
    ! [X3: $o] :
      ( ( insert_o @ X3 @ top_top_set_o )
      = top_top_set_o ) ).

% insert_UNIV
thf(fact_1133_empty__not__UNIV,axiom,
    bot_bot_set_o != top_top_set_o ).

% empty_not_UNIV
thf(fact_1134_UN__E,axiom,
    ! [B: $o,B5: $o > set_o,A2: set_o] :
      ( ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B5 @ A2 ) ) )
     => ~ ! [X5: $o] :
            ( ( member_o @ X5 @ A2 )
           => ~ ( member_o @ B @ ( B5 @ X5 ) ) ) ) ).

% UN_E
thf(fact_1135_rangeE,axiom,
    ! [B: $o,F2: $o > $o] :
      ( ( member_o @ B @ ( image_o_o @ F2 @ top_top_set_o ) )
     => ~ ! [X5: $o] :
            ( B
            = ( ~ ( F2 @ X5 ) ) ) ) ).

% rangeE
thf(fact_1136_UN__empty,axiom,
    ! [B5: $o > set_o] :
      ( ( comple90263536869209701_set_o @ ( image_o_set_o @ B5 @ bot_bot_set_o ) )
      = bot_bot_set_o ) ).

% UN_empty
thf(fact_1137_UN__insert__distrib,axiom,
    ! [U2: $o,A2: set_o,A: $o,B5: $o > set_o] :
      ( ( member_o @ U2 @ A2 )
     => ( ( comple90263536869209701_set_o
          @ ( image_o_set_o
            @ ^ [X: $o] : ( insert_o @ A @ ( B5 @ X ) )
            @ A2 ) )
        = ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B5 @ A2 ) ) ) ) ) ).

% UN_insert_distrib
thf(fact_1138_SUP__empty,axiom,
    ! [F2: $o > set_o] :
      ( ( comple90263536869209701_set_o @ ( image_o_set_o @ F2 @ bot_bot_set_o ) )
      = bot_bot_set_o ) ).

% SUP_empty
thf(fact_1139_SUP__constant,axiom,
    ! [A2: set_o,C: set_o] :
      ( ( ( A2 = bot_bot_set_o )
       => ( ( comple90263536869209701_set_o
            @ ( image_o_set_o
              @ ^ [Y2: $o] : C
              @ A2 ) )
          = bot_bot_set_o ) )
      & ( ( A2 != bot_bot_set_o )
       => ( ( comple90263536869209701_set_o
            @ ( image_o_set_o
              @ ^ [Y2: $o] : C
              @ A2 ) )
          = C ) ) ) ).

% SUP_constant
thf(fact_1140_range__eq__singletonD,axiom,
    ! [F2: $o > $o,A: $o,X3: $o] :
      ( ( ( image_o_o @ F2 @ top_top_set_o )
        = ( insert_o @ A @ bot_bot_set_o ) )
     => ( ( F2 @ X3 )
        = A ) ) ).

% range_eq_singletonD
thf(fact_1141_UN__extend__simps_I1_J,axiom,
    ! [C4: set_o,A: $o,B5: $o > set_o] :
      ( ( ( C4 = bot_bot_set_o )
       => ( ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B5 @ C4 ) ) )
          = ( insert_o @ A @ bot_bot_set_o ) ) )
      & ( ( C4 != bot_bot_set_o )
       => ( ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B5 @ C4 ) ) )
          = ( comple90263536869209701_set_o
            @ ( image_o_set_o
              @ ^ [X: $o] : ( insert_o @ A @ ( B5 @ X ) )
              @ C4 ) ) ) ) ) ).

% UN_extend_simps(1)
thf(fact_1142_Sup__set__def,axiom,
    ( comple90263536869209701_set_o
    = ( ^ [A6: set_set_o] :
          ( collect_o
          @ ^ [X: $o] : ( complete_Sup_Sup_o @ ( image_set_o_o @ ( member_o @ X ) @ A6 ) ) ) ) ) ).

% Sup_set_def
thf(fact_1143_Sup__SUP__eq,axiom,
    ( complete_Sup_Sup_o_o
    = ( ^ [S3: set_o_o,X: $o] : ( member_o @ X @ ( comple90263536869209701_set_o @ ( image_o_o_set_o @ collect_o @ S3 ) ) ) ) ) ).

% Sup_SUP_eq
thf(fact_1144_SUP__Sup__eq,axiom,
    ! [S: set_set_o] :
      ( ( complete_Sup_Sup_o_o
        @ ( image_set_o_o_o
          @ ^ [I4: set_o,X: $o] : ( member_o @ X @ I4 )
          @ S ) )
      = ( ^ [X: $o] : ( member_o @ X @ ( comple90263536869209701_set_o @ S ) ) ) ) ).

% SUP_Sup_eq
thf(fact_1145_UNIV__bool,axiom,
    ( top_top_set_o
    = ( insert_o @ $false @ ( insert_o @ $true @ bot_bot_set_o ) ) ) ).

% UNIV_bool
thf(fact_1146_bot__set__def,axiom,
    ( bot_bot_set_o
    = ( collect_o @ bot_bot_o_o ) ) ).

% bot_set_def
thf(fact_1147_sup__Un__eq,axiom,
    ! [R4: set_o,S: set_o] :
      ( ( sup_sup_o_o
        @ ^ [X: $o] : ( member_o @ X @ R4 )
        @ ^ [X: $o] : ( member_o @ X @ S ) )
      = ( ^ [X: $o] : ( member_o @ X @ ( sup_sup_set_o @ R4 @ S ) ) ) ) ).

% sup_Un_eq
thf(fact_1148_sup__set__def,axiom,
    ( sup_sup_set_o
    = ( ^ [A6: set_o,B7: set_o] :
          ( collect_o
          @ ( sup_sup_o_o
            @ ^ [X: $o] : ( member_o @ X @ A6 )
            @ ^ [X: $o] : ( member_o @ X @ B7 ) ) ) ) ) ).

% sup_set_def
thf(fact_1149_INT__simps_I4_J,axiom,
    ! [C4: set_o,A2: set_o,B5: $o > set_o] :
      ( ( ( C4 = bot_bot_set_o )
       => ( ( comple3063163877087187839_set_o
            @ ( image_o_set_o
              @ ^ [X: $o] : ( minus_minus_set_o @ A2 @ ( B5 @ X ) )
              @ C4 ) )
          = top_top_set_o ) )
      & ( ( C4 != bot_bot_set_o )
       => ( ( comple3063163877087187839_set_o
            @ ( image_o_set_o
              @ ^ [X: $o] : ( minus_minus_set_o @ A2 @ ( B5 @ X ) )
              @ C4 ) )
          = ( minus_minus_set_o @ A2 @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B5 @ C4 ) ) ) ) ) ) ).

% INT_simps(4)
thf(fact_1150_Rep__unit__induct,axiom,
    ! [Y: $o,P: $o > $o] :
      ( ( member_o @ Y @ ( insert_o @ $true @ bot_bot_set_o ) )
     => ( ! [X5: product_unit] : ( P @ ( product_Rep_unit @ X5 ) )
       => ( P @ Y ) ) ) ).

% Rep_unit_induct
thf(fact_1151_INT__I,axiom,
    ! [A2: set_o,B: $o,B5: $o > set_o] :
      ( ! [X5: $o] :
          ( ( member_o @ X5 @ A2 )
         => ( member_o @ B @ ( B5 @ X5 ) ) )
     => ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ B5 @ A2 ) ) ) ) ).

% INT_I
thf(fact_1152_INT__constant,axiom,
    ! [A2: set_o,C: set_o] :
      ( ( ( A2 = bot_bot_set_o )
       => ( ( comple3063163877087187839_set_o
            @ ( image_o_set_o
              @ ^ [Y2: $o] : C
              @ A2 ) )
          = top_top_set_o ) )
      & ( ( A2 != bot_bot_set_o )
       => ( ( comple3063163877087187839_set_o
            @ ( image_o_set_o
              @ ^ [Y2: $o] : C
              @ A2 ) )
          = C ) ) ) ).

% INT_constant
thf(fact_1153_INT__simps_I3_J,axiom,
    ! [C4: set_o,A2: $o > set_o,B5: set_o] :
      ( ( ( C4 = bot_bot_set_o )
       => ( ( comple3063163877087187839_set_o
            @ ( image_o_set_o
              @ ^ [X: $o] : ( minus_minus_set_o @ ( A2 @ X ) @ B5 )
              @ C4 ) )
          = top_top_set_o ) )
      & ( ( C4 != bot_bot_set_o )
       => ( ( comple3063163877087187839_set_o
            @ ( image_o_set_o
              @ ^ [X: $o] : ( minus_minus_set_o @ ( A2 @ X ) @ B5 )
              @ C4 ) )
          = ( minus_minus_set_o @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ A2 @ C4 ) ) @ B5 ) ) ) ) ).

% INT_simps(3)
thf(fact_1154_INT__D,axiom,
    ! [B: $o,B5: $o > set_o,A2: set_o,A: $o] :
      ( ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ B5 @ A2 ) ) )
     => ( ( member_o @ A @ A2 )
       => ( member_o @ B @ ( B5 @ A ) ) ) ) ).

% INT_D
thf(fact_1155_INT__E,axiom,
    ! [B: $o,B5: $o > set_o,A2: set_o,A: $o] :
      ( ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ B5 @ A2 ) ) )
     => ( ~ ( member_o @ B @ ( B5 @ A ) )
       => ~ ( member_o @ A @ A2 ) ) ) ).

% INT_E
thf(fact_1156_Rep__unit__inject,axiom,
    ! [X3: product_unit,Y: product_unit] :
      ( ( ( product_Rep_unit @ X3 )
        = ( product_Rep_unit @ Y ) )
      = ( X3 = Y ) ) ).

% Rep_unit_inject
thf(fact_1157_INT__insert__distrib,axiom,
    ! [U2: $o,A2: set_o,A: $o,B5: $o > set_o] :
      ( ( member_o @ U2 @ A2 )
     => ( ( comple3063163877087187839_set_o
          @ ( image_o_set_o
            @ ^ [X: $o] : ( insert_o @ A @ ( B5 @ X ) )
            @ A2 ) )
        = ( insert_o @ A @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ B5 @ A2 ) ) ) ) ) ).

% INT_insert_distrib
thf(fact_1158_INF__empty,axiom,
    ! [F2: $o > set_o] :
      ( ( comple3063163877087187839_set_o @ ( image_o_set_o @ F2 @ bot_bot_set_o ) )
      = top_top_set_o ) ).

% INF_empty
thf(fact_1159_INF__constant,axiom,
    ! [A2: set_o,C: set_o] :
      ( ( ( A2 = bot_bot_set_o )
       => ( ( comple3063163877087187839_set_o
            @ ( image_o_set_o
              @ ^ [Y2: $o] : C
              @ A2 ) )
          = top_top_set_o ) )
      & ( ( A2 != bot_bot_set_o )
       => ( ( comple3063163877087187839_set_o
            @ ( image_o_set_o
              @ ^ [Y2: $o] : C
              @ A2 ) )
          = C ) ) ) ).

% INF_constant
thf(fact_1160_INT__empty,axiom,
    ! [B5: $o > set_o] :
      ( ( comple3063163877087187839_set_o @ ( image_o_set_o @ B5 @ bot_bot_set_o ) )
      = top_top_set_o ) ).

% INT_empty
thf(fact_1161_INT__extend__simps_I3_J,axiom,
    ! [C4: set_o,A2: $o > set_o,B5: set_o] :
      ( ( ( C4 = bot_bot_set_o )
       => ( ( minus_minus_set_o @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ A2 @ C4 ) ) @ B5 )
          = ( minus_minus_set_o @ top_top_set_o @ B5 ) ) )
      & ( ( C4 != bot_bot_set_o )
       => ( ( minus_minus_set_o @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ A2 @ C4 ) ) @ B5 )
          = ( comple3063163877087187839_set_o
            @ ( image_o_set_o
              @ ^ [X: $o] : ( minus_minus_set_o @ ( A2 @ X ) @ B5 )
              @ C4 ) ) ) ) ) ).

% INT_extend_simps(3)
thf(fact_1162_Rep__unit,axiom,
    ! [X3: product_unit] : ( member_o @ ( product_Rep_unit @ X3 ) @ ( insert_o @ $true @ bot_bot_set_o ) ) ).

% Rep_unit
thf(fact_1163_Rep__unit__cases,axiom,
    ! [Y: $o] :
      ( ( member_o @ Y @ ( insert_o @ $true @ bot_bot_set_o ) )
     => ~ ! [X5: product_unit] :
            ( Y
            = ( ~ ( product_Rep_unit @ X5 ) ) ) ) ).

% Rep_unit_cases
thf(fact_1164_top__set__def,axiom,
    ( top_top_set_o
    = ( collect_o @ top_top_o_o ) ) ).

% top_set_def
thf(fact_1165_Inf__INT__eq,axiom,
    ( complete_Inf_Inf_o_o
    = ( ^ [S3: set_o_o,X: $o] : ( member_o @ X @ ( comple3063163877087187839_set_o @ ( image_o_o_set_o @ collect_o @ S3 ) ) ) ) ) ).

% Inf_INT_eq
thf(fact_1166_INF__Int__eq,axiom,
    ! [S: set_set_o] :
      ( ( complete_Inf_Inf_o_o
        @ ( image_set_o_o_o
          @ ^ [I4: set_o,X: $o] : ( member_o @ X @ I4 )
          @ S ) )
      = ( ^ [X: $o] : ( member_o @ X @ ( comple3063163877087187839_set_o @ S ) ) ) ) ).

% INF_Int_eq
thf(fact_1167_Inf__set__def,axiom,
    ( comple3063163877087187839_set_o
    = ( ^ [A6: set_set_o] :
          ( collect_o
          @ ^ [X: $o] : ( complete_Inf_Inf_o @ ( image_set_o_o @ ( member_o @ X ) @ A6 ) ) ) ) ) ).

% Inf_set_def
thf(fact_1168_Abs__unit__inverse,axiom,
    ! [Y: $o] :
      ( ( member_o @ Y @ ( insert_o @ $true @ bot_bot_set_o ) )
     => ( ( product_Rep_unit @ ( product_Abs_unit @ Y ) )
        = Y ) ) ).

% Abs_unit_inverse
thf(fact_1169_Abs__unit__inject,axiom,
    ! [X3: $o,Y: $o] :
      ( ( member_o @ X3 @ ( insert_o @ $true @ bot_bot_set_o ) )
     => ( ( member_o @ Y @ ( insert_o @ $true @ bot_bot_set_o ) )
       => ( ( ( product_Abs_unit @ X3 )
            = ( product_Abs_unit @ Y ) )
          = ( X3 = Y ) ) ) ) ).

% Abs_unit_inject
thf(fact_1170_Abs__unit__induct,axiom,
    ! [P: product_unit > $o,X3: product_unit] :
      ( ! [Y3: $o] :
          ( ( member_o @ Y3 @ ( insert_o @ $true @ bot_bot_set_o ) )
         => ( P @ ( product_Abs_unit @ Y3 ) ) )
     => ( P @ X3 ) ) ).

% Abs_unit_induct
thf(fact_1171_Rep__unit__inverse,axiom,
    ! [X3: product_unit] :
      ( ( product_Abs_unit @ ( product_Rep_unit @ X3 ) )
      = X3 ) ).

% Rep_unit_inverse
thf(fact_1172_Abs__unit__cases,axiom,
    ! [X3: product_unit] :
      ~ ! [Y3: $o] :
          ( ( X3
            = ( product_Abs_unit @ Y3 ) )
         => ~ ( member_o @ Y3 @ ( insert_o @ $true @ bot_bot_set_o ) ) ) ).

% Abs_unit_cases
thf(fact_1173_type__definition__unit,axiom,
    type_d6188575255521822967unit_o @ product_Rep_unit @ product_Abs_unit @ ( insert_o @ $true @ bot_bot_set_o ) ).

% type_definition_unit
thf(fact_1174_INT__simps_I1_J,axiom,
    ! [C4: set_o,A2: $o > set_o,B5: set_o] :
      ( ( ( C4 = bot_bot_set_o )
       => ( ( comple3063163877087187839_set_o
            @ ( image_o_set_o
              @ ^ [X: $o] : ( inf_inf_set_o @ ( A2 @ X ) @ B5 )
              @ C4 ) )
          = top_top_set_o ) )
      & ( ( C4 != bot_bot_set_o )
       => ( ( comple3063163877087187839_set_o
            @ ( image_o_set_o
              @ ^ [X: $o] : ( inf_inf_set_o @ ( A2 @ X ) @ B5 )
              @ C4 ) )
          = ( inf_inf_set_o @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ A2 @ C4 ) ) @ B5 ) ) ) ) ).

% INT_simps(1)
thf(fact_1175_INT__simps_I2_J,axiom,
    ! [C4: set_o,A2: set_o,B5: $o > set_o] :
      ( ( ( C4 = bot_bot_set_o )
       => ( ( comple3063163877087187839_set_o
            @ ( image_o_set_o
              @ ^ [X: $o] : ( inf_inf_set_o @ A2 @ ( B5 @ X ) )
              @ C4 ) )
          = top_top_set_o ) )
      & ( ( C4 != bot_bot_set_o )
       => ( ( comple3063163877087187839_set_o
            @ ( image_o_set_o
              @ ^ [X: $o] : ( inf_inf_set_o @ A2 @ ( B5 @ X ) )
              @ C4 ) )
          = ( inf_inf_set_o @ A2 @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ B5 @ C4 ) ) ) ) ) ) ).

% INT_simps(2)
thf(fact_1176_Int__iff,axiom,
    ! [C: $o,A2: set_o,B5: set_o] :
      ( ( member_o @ C @ ( inf_inf_set_o @ A2 @ B5 ) )
      = ( ( member_o @ C @ A2 )
        & ( member_o @ C @ B5 ) ) ) ).

% Int_iff
thf(fact_1177_IntI,axiom,
    ! [C: $o,A2: set_o,B5: set_o] :
      ( ( member_o @ C @ A2 )
     => ( ( member_o @ C @ B5 )
       => ( member_o @ C @ ( inf_inf_set_o @ A2 @ B5 ) ) ) ) ).

% IntI
thf(fact_1178_Int__UNIV,axiom,
    ! [A2: set_o,B5: set_o] :
      ( ( ( inf_inf_set_o @ A2 @ B5 )
        = top_top_set_o )
      = ( ( A2 = top_top_set_o )
        & ( B5 = top_top_set_o ) ) ) ).

% Int_UNIV
thf(fact_1179_Int__insert__left__if0,axiom,
    ! [A: $o,C4: set_o,B5: set_o] :
      ( ~ ( member_o @ A @ C4 )
     => ( ( inf_inf_set_o @ ( insert_o @ A @ B5 ) @ C4 )
        = ( inf_inf_set_o @ B5 @ C4 ) ) ) ).

% Int_insert_left_if0
thf(fact_1180_Int__insert__left__if1,axiom,
    ! [A: $o,C4: set_o,B5: set_o] :
      ( ( member_o @ A @ C4 )
     => ( ( inf_inf_set_o @ ( insert_o @ A @ B5 ) @ C4 )
        = ( insert_o @ A @ ( inf_inf_set_o @ B5 @ C4 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_1181_insert__inter__insert,axiom,
    ! [A: $o,A2: set_o,B5: set_o] :
      ( ( inf_inf_set_o @ ( insert_o @ A @ A2 ) @ ( insert_o @ A @ B5 ) )
      = ( insert_o @ A @ ( inf_inf_set_o @ A2 @ B5 ) ) ) ).

% insert_inter_insert
thf(fact_1182_Int__insert__right__if0,axiom,
    ! [A: $o,A2: set_o,B5: set_o] :
      ( ~ ( member_o @ A @ A2 )
     => ( ( inf_inf_set_o @ A2 @ ( insert_o @ A @ B5 ) )
        = ( inf_inf_set_o @ A2 @ B5 ) ) ) ).

% Int_insert_right_if0
thf(fact_1183_Int__insert__right__if1,axiom,
    ! [A: $o,A2: set_o,B5: set_o] :
      ( ( member_o @ A @ A2 )
     => ( ( inf_inf_set_o @ A2 @ ( insert_o @ A @ B5 ) )
        = ( insert_o @ A @ ( inf_inf_set_o @ A2 @ B5 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_1184_insert__disjoint_I1_J,axiom,
    ! [A: $o,A2: set_o,B5: set_o] :
      ( ( ( inf_inf_set_o @ ( insert_o @ A @ A2 ) @ B5 )
        = bot_bot_set_o )
      = ( ~ ( member_o @ A @ B5 )
        & ( ( inf_inf_set_o @ A2 @ B5 )
          = bot_bot_set_o ) ) ) ).

% insert_disjoint(1)
thf(fact_1185_insert__disjoint_I2_J,axiom,
    ! [A: $o,A2: set_o,B5: set_o] :
      ( ( bot_bot_set_o
        = ( inf_inf_set_o @ ( insert_o @ A @ A2 ) @ B5 ) )
      = ( ~ ( member_o @ A @ B5 )
        & ( bot_bot_set_o
          = ( inf_inf_set_o @ A2 @ B5 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_1186_disjoint__insert_I1_J,axiom,
    ! [B5: set_o,A: $o,A2: set_o] :
      ( ( ( inf_inf_set_o @ B5 @ ( insert_o @ A @ A2 ) )
        = bot_bot_set_o )
      = ( ~ ( member_o @ A @ B5 )
        & ( ( inf_inf_set_o @ B5 @ A2 )
          = bot_bot_set_o ) ) ) ).

% disjoint_insert(1)
thf(fact_1187_disjoint__insert_I2_J,axiom,
    ! [A2: set_o,B: $o,B5: set_o] :
      ( ( bot_bot_set_o
        = ( inf_inf_set_o @ A2 @ ( insert_o @ B @ B5 ) ) )
      = ( ~ ( member_o @ B @ A2 )
        & ( bot_bot_set_o
          = ( inf_inf_set_o @ A2 @ B5 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_1188_Diff__disjoint,axiom,
    ! [A2: set_o,B5: set_o] :
      ( ( inf_inf_set_o @ A2 @ ( minus_minus_set_o @ B5 @ A2 ) )
      = bot_bot_set_o ) ).

% Diff_disjoint
thf(fact_1189_type__copy__obj__one__point__absE,axiom,
    ! [Rep: product_unit > $o,Abs: $o > product_unit,S4: product_unit] :
      ( ( type_d6188575255521822967unit_o @ Rep @ Abs @ top_top_set_o )
     => ~ ! [X5: $o] :
            ( S4
           != ( Abs @ X5 ) ) ) ).

% type_copy_obj_one_point_absE
thf(fact_1190_Int__UNIV__right,axiom,
    ! [A2: set_o] :
      ( ( inf_inf_set_o @ A2 @ top_top_set_o )
      = A2 ) ).

% Int_UNIV_right
thf(fact_1191_Int__UNIV__left,axiom,
    ! [B5: set_o] :
      ( ( inf_inf_set_o @ top_top_set_o @ B5 )
      = B5 ) ).

% Int_UNIV_left
thf(fact_1192_Int__def,axiom,
    ( inf_inf_set_o
    = ( ^ [A6: set_o,B7: set_o] :
          ( collect_o
          @ ^ [X: $o] :
              ( ( member_o @ X @ A6 )
              & ( member_o @ X @ B7 ) ) ) ) ) ).

% Int_def
thf(fact_1193_Int__Collect,axiom,
    ! [X3: $o,A2: set_o,P: $o > $o] :
      ( ( member_o @ X3 @ ( inf_inf_set_o @ A2 @ ( collect_o @ P ) ) )
      = ( ( member_o @ X3 @ A2 )
        & ( P @ X3 ) ) ) ).

% Int_Collect
thf(fact_1194_IntD2,axiom,
    ! [C: $o,A2: set_o,B5: set_o] :
      ( ( member_o @ C @ ( inf_inf_set_o @ A2 @ B5 ) )
     => ( member_o @ C @ B5 ) ) ).

% IntD2
thf(fact_1195_IntD1,axiom,
    ! [C: $o,A2: set_o,B5: set_o] :
      ( ( member_o @ C @ ( inf_inf_set_o @ A2 @ B5 ) )
     => ( member_o @ C @ A2 ) ) ).

% IntD1
thf(fact_1196_IntE,axiom,
    ! [C: $o,A2: set_o,B5: set_o] :
      ( ( member_o @ C @ ( inf_inf_set_o @ A2 @ B5 ) )
     => ~ ( ( member_o @ C @ A2 )
         => ~ ( member_o @ C @ B5 ) ) ) ).

% IntE
thf(fact_1197_Int__insert__right,axiom,
    ! [A: $o,A2: set_o,B5: set_o] :
      ( ( ( member_o @ A @ A2 )
       => ( ( inf_inf_set_o @ A2 @ ( insert_o @ A @ B5 ) )
          = ( insert_o @ A @ ( inf_inf_set_o @ A2 @ B5 ) ) ) )
      & ( ~ ( member_o @ A @ A2 )
       => ( ( inf_inf_set_o @ A2 @ ( insert_o @ A @ B5 ) )
          = ( inf_inf_set_o @ A2 @ B5 ) ) ) ) ).

% Int_insert_right
thf(fact_1198_Int__insert__left,axiom,
    ! [A: $o,C4: set_o,B5: set_o] :
      ( ( ( member_o @ A @ C4 )
       => ( ( inf_inf_set_o @ ( insert_o @ A @ B5 ) @ C4 )
          = ( insert_o @ A @ ( inf_inf_set_o @ B5 @ C4 ) ) ) )
      & ( ~ ( member_o @ A @ C4 )
       => ( ( inf_inf_set_o @ ( insert_o @ A @ B5 ) @ C4 )
          = ( inf_inf_set_o @ B5 @ C4 ) ) ) ) ).

% Int_insert_left
thf(fact_1199_disjoint__iff__not__equal,axiom,
    ! [A2: set_o,B5: set_o] :
      ( ( ( inf_inf_set_o @ A2 @ B5 )
        = bot_bot_set_o )
      = ( ! [X: $o] :
            ( ( member_o @ X @ A2 )
           => ! [Y2: $o] :
                ( ( member_o @ Y2 @ B5 )
               => ( X = (~ Y2) ) ) ) ) ) ).

% disjoint_iff_not_equal
thf(fact_1200_Int__empty__right,axiom,
    ! [A2: set_o] :
      ( ( inf_inf_set_o @ A2 @ bot_bot_set_o )
      = bot_bot_set_o ) ).

% Int_empty_right
thf(fact_1201_Int__empty__left,axiom,
    ! [B5: set_o] :
      ( ( inf_inf_set_o @ bot_bot_set_o @ B5 )
      = bot_bot_set_o ) ).

% Int_empty_left
thf(fact_1202_disjoint__iff,axiom,
    ! [A2: set_o,B5: set_o] :
      ( ( ( inf_inf_set_o @ A2 @ B5 )
        = bot_bot_set_o )
      = ( ! [X: $o] :
            ( ( member_o @ X @ A2 )
           => ~ ( member_o @ X @ B5 ) ) ) ) ).

% disjoint_iff
thf(fact_1203_Int__emptyI,axiom,
    ! [A2: set_o,B5: set_o] :
      ( ! [X5: $o] :
          ( ( member_o @ X5 @ A2 )
         => ~ ( member_o @ X5 @ B5 ) )
     => ( ( inf_inf_set_o @ A2 @ B5 )
        = bot_bot_set_o ) ) ).

% Int_emptyI
thf(fact_1204_Diff__triv,axiom,
    ! [A2: set_o,B5: set_o] :
      ( ( ( inf_inf_set_o @ A2 @ B5 )
        = bot_bot_set_o )
     => ( ( minus_minus_set_o @ A2 @ B5 )
        = A2 ) ) ).

% Diff_triv
thf(fact_1205_Int__Diff__disjoint,axiom,
    ! [A2: set_o,B5: set_o] :
      ( ( inf_inf_set_o @ ( inf_inf_set_o @ A2 @ B5 ) @ ( minus_minus_set_o @ A2 @ B5 ) )
      = bot_bot_set_o ) ).

% Int_Diff_disjoint
thf(fact_1206_type__definition_ORep__range,axiom,
    ! [Rep: product_unit > $o,Abs: $o > product_unit,A2: set_o] :
      ( ( type_d6188575255521822967unit_o @ Rep @ Abs @ A2 )
     => ( ( image_Product_unit_o @ Rep @ top_to1996260823553986621t_unit )
        = A2 ) ) ).

% type_definition.Rep_range
thf(fact_1207_type__definition_OAbs__image,axiom,
    ! [Rep: product_unit > $o,Abs: $o > product_unit,A2: set_o] :
      ( ( type_d6188575255521822967unit_o @ Rep @ Abs @ A2 )
     => ( ( image_o_Product_unit @ Abs @ A2 )
        = top_to1996260823553986621t_unit ) ) ).

% type_definition.Abs_image
thf(fact_1208_type__definition_Ouniv,axiom,
    ! [Rep: product_unit > $o,Abs: $o > product_unit,A2: set_o] :
      ( ( type_d6188575255521822967unit_o @ Rep @ Abs @ A2 )
     => ( top_to1996260823553986621t_unit
        = ( image_o_Product_unit @ Abs @ A2 ) ) ) ).

% type_definition.univ
thf(fact_1209_inf__set__def,axiom,
    ( inf_inf_set_o
    = ( ^ [A6: set_o,B7: set_o] :
          ( collect_o
          @ ( inf_inf_o_o
            @ ^ [X: $o] : ( member_o @ X @ A6 )
            @ ^ [X: $o] : ( member_o @ X @ B7 ) ) ) ) ) ).

% inf_set_def
thf(fact_1210_inf__Int__eq,axiom,
    ! [R4: set_o,S: set_o] :
      ( ( inf_inf_o_o
        @ ^ [X: $o] : ( member_o @ X @ R4 )
        @ ^ [X: $o] : ( member_o @ X @ S ) )
      = ( ^ [X: $o] : ( member_o @ X @ ( inf_inf_set_o @ R4 @ S ) ) ) ) ).

% inf_Int_eq
thf(fact_1211_vimage__eq,axiom,
    ! [A: $o,F2: $o > $o,B5: set_o] :
      ( ( member_o @ A @ ( vimage_o_o @ F2 @ B5 ) )
      = ( member_o @ ( F2 @ A ) @ B5 ) ) ).

% vimage_eq
thf(fact_1212_vimageI,axiom,
    ! [F2: $o > $o,A: $o,B: $o,B5: set_o] :
      ( ( ( F2 @ A )
        = B )
     => ( ( member_o @ B @ B5 )
       => ( member_o @ A @ ( vimage_o_o @ F2 @ B5 ) ) ) ) ).

% vimageI
thf(fact_1213_vimage__UNIV,axiom,
    ! [F2: $o > $o] :
      ( ( vimage_o_o @ F2 @ top_top_set_o )
      = top_top_set_o ) ).

% vimage_UNIV
thf(fact_1214_vimage__empty,axiom,
    ! [F2: $o > $o] :
      ( ( vimage_o_o @ F2 @ bot_bot_set_o )
      = bot_bot_set_o ) ).

% vimage_empty
thf(fact_1215_vimage__const,axiom,
    ! [C: $o,A2: set_o] :
      ( ( ( member_o @ C @ A2 )
       => ( ( vimage_o_o
            @ ^ [X: $o] : C
            @ A2 )
          = top_top_set_o ) )
      & ( ~ ( member_o @ C @ A2 )
       => ( ( vimage_o_o
            @ ^ [X: $o] : C
            @ A2 )
          = bot_bot_set_o ) ) ) ).

% vimage_const
thf(fact_1216_vimage__singleton__eq,axiom,
    ! [A: $o,F2: $o > $o,B: $o] :
      ( ( member_o @ A @ ( vimage_o_o @ F2 @ ( insert_o @ B @ bot_bot_set_o ) ) )
      = ( ( F2 @ A )
        = B ) ) ).

% vimage_singleton_eq
thf(fact_1217_surj__image__vimage__eq,axiom,
    ! [F2: $o > $o,A2: set_o] :
      ( ( ( image_o_o @ F2 @ top_top_set_o )
        = top_top_set_o )
     => ( ( image_o_o @ F2 @ ( vimage_o_o @ F2 @ A2 ) )
        = A2 ) ) ).

% surj_image_vimage_eq
thf(fact_1218_vimageI2,axiom,
    ! [F2: $o > $o,A: $o,A2: set_o] :
      ( ( member_o @ ( F2 @ A ) @ A2 )
     => ( member_o @ A @ ( vimage_o_o @ F2 @ A2 ) ) ) ).

% vimageI2
thf(fact_1219_vimageE,axiom,
    ! [A: $o,F2: $o > $o,B5: set_o] :
      ( ( member_o @ A @ ( vimage_o_o @ F2 @ B5 ) )
     => ( member_o @ ( F2 @ A ) @ B5 ) ) ).

% vimageE
thf(fact_1220_vimageD,axiom,
    ! [A: $o,F2: $o > $o,A2: set_o] :
      ( ( member_o @ A @ ( vimage_o_o @ F2 @ A2 ) )
     => ( member_o @ ( F2 @ A ) @ A2 ) ) ).

% vimageD
thf(fact_1221_surj__vimage__empty,axiom,
    ! [F2: $o > $o,A2: set_o] :
      ( ( ( image_o_o @ F2 @ top_top_set_o )
        = top_top_set_o )
     => ( ( ( vimage_o_o @ F2 @ A2 )
          = bot_bot_set_o )
        = ( A2 = bot_bot_set_o ) ) ) ).

% surj_vimage_empty
thf(fact_1222_Pair__vimage__Sigma,axiom,
    ! [X3: $o,A2: set_o,F2: $o > set_o] :
      ( ( ( member_o @ X3 @ A2 )
       => ( ( vimage8945963521958007626od_o_o @ ( product_Pair_o_o @ X3 ) @ ( product_Sigma_o_o @ A2 @ F2 ) )
          = ( F2 @ X3 ) ) )
      & ( ~ ( member_o @ X3 @ A2 )
       => ( ( vimage8945963521958007626od_o_o @ ( product_Pair_o_o @ X3 ) @ ( product_Sigma_o_o @ A2 @ F2 ) )
          = bot_bot_set_o ) ) ) ).

% Pair_vimage_Sigma
thf(fact_1223_Compl__disjoint2,axiom,
    ! [A2: set_o] :
      ( ( inf_inf_set_o @ ( uminus_uminus_set_o @ A2 ) @ A2 )
      = bot_bot_set_o ) ).

% Compl_disjoint2
thf(fact_1224_Compl__disjoint,axiom,
    ! [A2: set_o] :
      ( ( inf_inf_set_o @ A2 @ ( uminus_uminus_set_o @ A2 ) )
      = bot_bot_set_o ) ).

% Compl_disjoint
thf(fact_1225_vimage__if,axiom,
    ! [C: $o,A2: set_o,D: $o,B5: set_o] :
      ( ( ( member_o @ C @ A2 )
       => ( ( ( member_o @ D @ A2 )
           => ( ( vimage_o_o
                @ ^ [X: $o] :
                    ( ( ( member_o @ X @ B5 )
                     => C )
                    & ( ~ ( member_o @ X @ B5 )
                     => D ) )
                @ A2 )
              = top_top_set_o ) )
          & ( ~ ( member_o @ D @ A2 )
           => ( ( vimage_o_o
                @ ^ [X: $o] :
                    ( ( ( member_o @ X @ B5 )
                     => C )
                    & ( ~ ( member_o @ X @ B5 )
                     => D ) )
                @ A2 )
              = B5 ) ) ) )
      & ( ~ ( member_o @ C @ A2 )
       => ( ( ( member_o @ D @ A2 )
           => ( ( vimage_o_o
                @ ^ [X: $o] :
                    ( ( ( member_o @ X @ B5 )
                     => C )
                    & ( ~ ( member_o @ X @ B5 )
                     => D ) )
                @ A2 )
              = ( uminus_uminus_set_o @ B5 ) ) )
          & ( ~ ( member_o @ D @ A2 )
           => ( ( vimage_o_o
                @ ^ [X: $o] :
                    ( ( ( member_o @ X @ B5 )
                     => C )
                    & ( ~ ( member_o @ X @ B5 )
                     => D ) )
                @ A2 )
              = bot_bot_set_o ) ) ) ) ) ).

% vimage_if
thf(fact_1226_Compl__eq__Diff__UNIV,axiom,
    ( uminus_uminus_set_o
    = ( minus_minus_set_o @ top_top_set_o ) ) ).

% Compl_eq_Diff_UNIV
thf(fact_1227_Compl__partition,axiom,
    ! [A2: set_o] :
      ( ( sup_sup_set_o @ A2 @ ( uminus_uminus_set_o @ A2 ) )
      = top_top_set_o ) ).

% Compl_partition
thf(fact_1228_Compl__partition2,axiom,
    ! [A2: set_o] :
      ( ( sup_sup_set_o @ ( uminus_uminus_set_o @ A2 ) @ A2 )
      = top_top_set_o ) ).

% Compl_partition2
thf(fact_1229_Compl__UNIV__eq,axiom,
    ( ( uminus_uminus_set_o @ top_top_set_o )
    = bot_bot_set_o ) ).

% Compl_UNIV_eq
thf(fact_1230_Compl__empty__eq,axiom,
    ( ( uminus_uminus_set_o @ bot_bot_set_o )
    = top_top_set_o ) ).

% Compl_empty_eq
thf(fact_1231_Compl__insert,axiom,
    ! [X3: $o,A2: set_o] :
      ( ( uminus_uminus_set_o @ ( insert_o @ X3 @ A2 ) )
      = ( minus_minus_set_o @ ( uminus_uminus_set_o @ A2 ) @ ( insert_o @ X3 @ bot_bot_set_o ) ) ) ).

% Compl_insert
thf(fact_1232_ComplI,axiom,
    ! [C: $o,A2: set_o] :
      ( ~ ( member_o @ C @ A2 )
     => ( member_o @ C @ ( uminus_uminus_set_o @ A2 ) ) ) ).

% ComplI
thf(fact_1233_Compl__iff,axiom,
    ! [C: $o,A2: set_o] :
      ( ( member_o @ C @ ( uminus_uminus_set_o @ A2 ) )
      = ( ~ ( member_o @ C @ A2 ) ) ) ).

% Compl_iff
thf(fact_1234_ComplD,axiom,
    ! [C: $o,A2: set_o] :
      ( ( member_o @ C @ ( uminus_uminus_set_o @ A2 ) )
     => ~ ( member_o @ C @ A2 ) ) ).

% ComplD
thf(fact_1235_uminus__set__def,axiom,
    ( uminus_uminus_set_o
    = ( ^ [A6: set_o] :
          ( collect_o
          @ ( uminus_uminus_o_o
            @ ^ [X: $o] : ( member_o @ X @ A6 ) ) ) ) ) ).

% uminus_set_def
thf(fact_1236_Compl__eq,axiom,
    ( uminus_uminus_set_o
    = ( ^ [A6: set_o] :
          ( collect_o
          @ ^ [X: $o] :
              ~ ( member_o @ X @ A6 ) ) ) ) ).

% Compl_eq
thf(fact_1237_cINF__less__iff,axiom,
    ! [A2: set_o,F2: $o > nat,A: nat] :
      ( ( A2 != bot_bot_set_o )
     => ( ( condit1738341127787009408ow_nat @ ( image_o_nat @ F2 @ A2 ) )
       => ( ( ord_less_nat @ ( complete_Inf_Inf_nat @ ( image_o_nat @ F2 @ A2 ) ) @ A )
          = ( ? [X: $o] :
                ( ( member_o @ X @ A2 )
                & ( ord_less_nat @ ( F2 @ X ) @ A ) ) ) ) ) ) ).

% cINF_less_iff
thf(fact_1238_less__cSUP__iff,axiom,
    ! [A2: set_o,F2: $o > nat,A: nat] :
      ( ( A2 != bot_bot_set_o )
     => ( ( condit2214826472909112428ve_nat @ ( image_o_nat @ F2 @ A2 ) )
       => ( ( ord_less_nat @ A @ ( complete_Sup_Sup_nat @ ( image_o_nat @ F2 @ A2 ) ) )
          = ( ? [X: $o] :
                ( ( member_o @ X @ A2 )
                & ( ord_less_nat @ A @ ( F2 @ X ) ) ) ) ) ) ) ).

% less_cSUP_iff
thf(fact_1239_not__psubset__empty,axiom,
    ! [A2: set_o] :
      ~ ( ord_less_set_o @ A2 @ bot_bot_set_o ) ).

% not_psubset_empty
thf(fact_1240_psubset__imp__ex__mem,axiom,
    ! [A2: set_o,B5: set_o] :
      ( ( ord_less_set_o @ A2 @ B5 )
     => ? [B2: $o] : ( member_o @ B2 @ ( minus_minus_set_o @ B5 @ A2 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_1241_length__induct,axiom,
    ! [P: list_nat > $o,Xs: list_nat] :
      ( ! [Xs2: list_nat] :
          ( ! [Ys3: list_nat] :
              ( ( ord_less_nat @ ( size_size_list_nat @ Ys3 ) @ ( size_size_list_nat @ Xs2 ) )
             => ( P @ Ys3 ) )
         => ( P @ Xs2 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_1242_cSUP__lessD,axiom,
    ! [F2: $o > nat,A2: set_o,Y: nat,I3: $o] :
      ( ( condit2214826472909112428ve_nat @ ( image_o_nat @ F2 @ A2 ) )
     => ( ( ord_less_nat @ ( complete_Sup_Sup_nat @ ( image_o_nat @ F2 @ A2 ) ) @ Y )
       => ( ( member_o @ I3 @ A2 )
         => ( ord_less_nat @ ( F2 @ I3 ) @ Y ) ) ) ) ).

% cSUP_lessD
thf(fact_1243_less__cINF__D,axiom,
    ! [F2: $o > nat,A2: set_o,Y: nat,I3: $o] :
      ( ( condit1738341127787009408ow_nat @ ( image_o_nat @ F2 @ A2 ) )
     => ( ( ord_less_nat @ Y @ ( complete_Inf_Inf_nat @ ( image_o_nat @ F2 @ A2 ) ) )
       => ( ( member_o @ I3 @ A2 )
         => ( ord_less_nat @ Y @ ( F2 @ I3 ) ) ) ) ) ).

% less_cINF_D
thf(fact_1244_ball__empty,axiom,
    ! [P: $o > $o,X6: $o] :
      ( ( member_o @ X6 @ bot_bot_set_o )
     => ( P @ X6 ) ) ).

% ball_empty
thf(fact_1245_inj__on__insert,axiom,
    ! [F2: $o > $o,A: $o,A2: set_o] :
      ( ( inj_on_o_o @ F2 @ ( insert_o @ A @ A2 ) )
      = ( ( inj_on_o_o @ F2 @ A2 )
        & ~ ( member_o @ ( F2 @ A ) @ ( image_o_o @ F2 @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ) ) ).

% inj_on_insert
thf(fact_1246_less__imp__diff__less,axiom,
    ! [J2: nat,K: nat,N: nat] :
      ( ( ord_less_nat @ J2 @ K )
     => ( ord_less_nat @ ( minus_minus_nat @ J2 @ N ) @ K ) ) ).

% less_imp_diff_less
thf(fact_1247_diff__less__mono2,axiom,
    ! [M2: nat,N: nat,L2: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ( ord_less_nat @ M2 @ L2 )
       => ( ord_less_nat @ ( minus_minus_nat @ L2 @ N ) @ ( minus_minus_nat @ L2 @ M2 ) ) ) ) ).

% diff_less_mono2
thf(fact_1248_less__set__def,axiom,
    ( ord_less_set_o
    = ( ^ [A6: set_o,B7: set_o] :
          ( ord_less_o_o
          @ ^ [X: $o] : ( member_o @ X @ A6 )
          @ ^ [X: $o] : ( member_o @ X @ B7 ) ) ) ) ).

% less_set_def
thf(fact_1249_linorder__neqE__nat,axiom,
    ! [X3: nat,Y: nat] :
      ( ( X3 != Y )
     => ( ~ ( ord_less_nat @ X3 @ Y )
       => ( ord_less_nat @ Y @ X3 ) ) ) ).

% linorder_neqE_nat
thf(fact_1250_infinite__descent,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N2: nat] :
          ( ~ ( P @ N2 )
         => ? [M3: nat] :
              ( ( ord_less_nat @ M3 @ N2 )
              & ~ ( P @ M3 ) ) )
     => ( P @ N ) ) ).

% infinite_descent
thf(fact_1251_nat__less__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N2: nat] :
          ( ! [M3: nat] :
              ( ( ord_less_nat @ M3 @ N2 )
             => ( P @ M3 ) )
         => ( P @ N2 ) )
     => ( P @ N ) ) ).

% nat_less_induct
thf(fact_1252_less__irrefl__nat,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_irrefl_nat
thf(fact_1253_less__not__refl3,axiom,
    ! [S4: nat,T: nat] :
      ( ( ord_less_nat @ S4 @ T )
     => ( S4 != T ) ) ).

% less_not_refl3
thf(fact_1254_less__not__refl2,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ N @ M2 )
     => ( M2 != N ) ) ).

% less_not_refl2
thf(fact_1255_less__not__refl,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_not_refl
thf(fact_1256_nat__neq__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( M2 != N )
      = ( ( ord_less_nat @ M2 @ N )
        | ( ord_less_nat @ N @ M2 ) ) ) ).

% nat_neq_iff
thf(fact_1257_psubsetD,axiom,
    ! [A2: set_o,B5: set_o,C: $o] :
      ( ( ord_less_set_o @ A2 @ B5 )
     => ( ( member_o @ C @ A2 )
       => ( member_o @ C @ B5 ) ) ) ).

% psubsetD
thf(fact_1258_inj__image__mem__iff,axiom,
    ! [F2: $o > $o,A: $o,A2: set_o] :
      ( ( inj_on_o_o @ F2 @ top_top_set_o )
     => ( ( member_o @ ( F2 @ A ) @ ( image_o_o @ F2 @ A2 ) )
        = ( member_o @ A @ A2 ) ) ) ).

% inj_image_mem_iff
thf(fact_1259_range__ex1__eq,axiom,
    ! [F2: $o > $o,B: $o] :
      ( ( inj_on_o_o @ F2 @ top_top_set_o )
     => ( ( member_o @ B @ ( image_o_o @ F2 @ top_top_set_o ) )
        = ( ? [X: $o] :
              ( ( B
                = ( F2 @ X ) )
              & ! [Y2: $o] :
                  ( ( B
                    = ( F2 @ Y2 ) )
                 => ( Y2 = X ) ) ) ) ) ) ).

% range_ex1_eq
thf(fact_1260_inj__img__insertE,axiom,
    ! [F2: $o > $o,A2: set_o,X3: $o,B5: set_o] :
      ( ( inj_on_o_o @ F2 @ A2 )
     => ( ~ ( member_o @ X3 @ B5 )
       => ( ( ( insert_o @ X3 @ B5 )
            = ( image_o_o @ F2 @ A2 ) )
         => ~ ! [X8: $o,A7: set_o] :
                ( ~ ( member_o @ X8 @ A7 )
               => ( ( A2
                    = ( insert_o @ X8 @ A7 ) )
                 => ( ( X3
                      = ( F2 @ X8 ) )
                   => ( B5
                     != ( image_o_o @ F2 @ A7 ) ) ) ) ) ) ) ) ).

% inj_img_insertE
thf(fact_1261_Ball__def,axiom,
    ( ball_o
    = ( ^ [A6: set_o,P5: $o > $o] :
        ! [X: $o] :
          ( ( member_o @ X @ A6 )
         => ( P5 @ X ) ) ) ) ).

% Ball_def
thf(fact_1262_sorted__list__of__set_Oinj__on,axiom,
    ( inj_on_o_o
    @ ^ [X: $o] : X
    @ top_top_set_o ) ).

% sorted_list_of_set.inj_on
thf(fact_1263_Inter__eq,axiom,
    ( comple3063163877087187839_set_o
    = ( ^ [A6: set_set_o] :
          ( collect_o
          @ ^ [X: $o] :
            ! [Y2: set_o] :
              ( ( member_set_o @ Y2 @ A6 )
             => ( member_o @ X @ Y2 ) ) ) ) ) ).

% Inter_eq
thf(fact_1264_inj__singleton,axiom,
    ! [A2: set_o] :
      ( inj_on_o_set_o
      @ ^ [X: $o] : ( insert_o @ X @ bot_bot_set_o )
      @ A2 ) ).

% inj_singleton
thf(fact_1265_refl__on__def_H,axiom,
    ( refl_on_o
    = ( ^ [A6: set_o,R: set_Product_prod_o_o] :
          ( ! [X: product_prod_o_o] :
              ( ( member7466972457876170832od_o_o @ X @ R )
             => ( produc6197397395684419436_o_o_o
                @ ^ [Y2: $o,Z3: $o] :
                    ( ( member_o @ Y2 @ A6 )
                    & ( member_o @ Z3 @ A6 ) )
                @ X ) )
          & ! [X: $o] :
              ( ( member_o @ X @ A6 )
             => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ X ) @ R ) ) ) ) ) ).

% refl_on_def'
thf(fact_1266_inj__on__disjoint__Un,axiom,
    ! [F2: $o > $o,A2: set_o,G2: $o > $o,B5: set_o] :
      ( ( inj_on_o_o @ F2 @ A2 )
     => ( ( inj_on_o_o @ G2 @ B5 )
       => ( ( ( inf_inf_set_o @ ( image_o_o @ F2 @ A2 ) @ ( image_o_o @ G2 @ B5 ) )
            = bot_bot_set_o )
         => ( inj_on_o_o
            @ ^ [X: $o] :
                ( ( ( member_o @ X @ A2 )
                 => ( F2 @ X ) )
                & ( ~ ( member_o @ X @ A2 )
                 => ( G2 @ X ) ) )
            @ ( sup_sup_set_o @ A2 @ B5 ) ) ) ) ) ).

% inj_on_disjoint_Un
thf(fact_1267_inj__on__vimage__singleton,axiom,
    ! [F2: $o > $o,A2: set_o,A: $o] :
      ( ( inj_on_o_o @ F2 @ A2 )
     => ( ord_less_eq_set_o @ ( inf_inf_set_o @ ( vimage_o_o @ F2 @ ( insert_o @ A @ bot_bot_set_o ) ) @ A2 )
        @ ( insert_o
          @ ( the_o
            @ ^ [X: $o] :
                ( ( member_o @ X @ A2 )
                & ( ( F2 @ X )
                  = A ) ) )
          @ bot_bot_set_o ) ) ) ).

% inj_on_vimage_singleton
thf(fact_1268_subsetI,axiom,
    ! [A2: set_o,B5: set_o] :
      ( ! [X5: $o] :
          ( ( member_o @ X5 @ A2 )
         => ( member_o @ X5 @ B5 ) )
     => ( ord_less_eq_set_o @ A2 @ B5 ) ) ).

% subsetI
thf(fact_1269_empty__subsetI,axiom,
    ! [A2: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A2 ) ).

% empty_subsetI
thf(fact_1270_subset__empty,axiom,
    ! [A2: set_o] :
      ( ( ord_less_eq_set_o @ A2 @ bot_bot_set_o )
      = ( A2 = bot_bot_set_o ) ) ).

% subset_empty
thf(fact_1271_insert__subset,axiom,
    ! [X3: $o,A2: set_o,B5: set_o] :
      ( ( ord_less_eq_set_o @ ( insert_o @ X3 @ A2 ) @ B5 )
      = ( ( member_o @ X3 @ B5 )
        & ( ord_less_eq_set_o @ A2 @ B5 ) ) ) ).

% insert_subset
thf(fact_1272_singleton__insert__inj__eq,axiom,
    ! [B: $o,A: $o,A2: set_o] :
      ( ( ( insert_o @ B @ bot_bot_set_o )
        = ( insert_o @ A @ A2 ) )
      = ( ( A = B )
        & ( ord_less_eq_set_o @ A2 @ ( insert_o @ B @ bot_bot_set_o ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_1273_singleton__insert__inj__eq_H,axiom,
    ! [A: $o,A2: set_o,B: $o] :
      ( ( ( insert_o @ A @ A2 )
        = ( insert_o @ B @ bot_bot_set_o ) )
      = ( ( A = B )
        & ( ord_less_eq_set_o @ A2 @ ( insert_o @ B @ bot_bot_set_o ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_1274_Diff__eq__empty__iff,axiom,
    ! [A2: set_o,B5: set_o] :
      ( ( ( minus_minus_set_o @ A2 @ B5 )
        = bot_bot_set_o )
      = ( ord_less_eq_set_o @ A2 @ B5 ) ) ).

% Diff_eq_empty_iff
thf(fact_1275_subset__Compl__singleton,axiom,
    ! [A2: set_o,B: $o] :
      ( ( ord_less_eq_set_o @ A2 @ ( uminus_uminus_set_o @ ( insert_o @ B @ bot_bot_set_o ) ) )
      = ( ~ ( member_o @ B @ A2 ) ) ) ).

% subset_Compl_singleton

% Helper facts (9)
thf(help_If_2_1_If_001tf__a_T,axiom,
    ! [X3: a,Y: a] :
      ( ( if_a @ $false @ X3 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001tf__a_T,axiom,
    ! [X3: a,Y: a] :
      ( ( if_a @ $true @ X3 @ Y )
      = X3 ) ).

thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X3: nat,Y: nat] :
      ( ( if_nat @ $false @ X3 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X3: nat,Y: nat] :
      ( ( if_nat @ $true @ X3 @ Y )
      = X3 ) ).

thf(help_If_2_1_If_001t__Set__Oset_I_Eo_J_T,axiom,
    ! [X3: set_o,Y: set_o] :
      ( ( if_set_o @ $false @ X3 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Set__Oset_I_Eo_J_T,axiom,
    ! [X3: set_o,Y: set_o] :
      ( ( if_set_o @ $true @ X3 @ Y )
      = X3 ) ).

thf(help_If_3_1_If_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_T,axiom,
    ! [X3: relational_fmla_a_b,Y: relational_fmla_a_b] :
      ( ( if_Rel1279876242545935705la_a_b @ $false @ X3 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_T,axiom,
    ! [X3: relational_fmla_a_b,Y: relational_fmla_a_b] :
      ( ( if_Rel1279876242545935705la_a_b @ $true @ X3 @ Y )
      = X3 ) ).

% Conjectures (3)
thf(conj_0,hypothesis,
    relati5999705594545617851ty_a_b @ q ).

thf(conj_1,hypothesis,
    ( ( size_size_list_nat @ xs )
    = ( size_size_list_nat @ ys ) ) ).

thf(conj_2,conjecture,
    ( relati5999705594545617851ty_a_b
    @ ( fold_P7970104616371074773la_a_b
      @ ( produc5586541307551673003la_a_b
        @ ^ [X: nat,Y2: nat,Q2: relational_fmla_a_b] : ( relational_subst_a_b @ Q2 @ X @ Y2 ) )
      @ ( zip_nat_nat @ xs @ ys )
      @ ( relati591517084277583526ts_a_b @ y @ q ) ) ) ).

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