TPTP Problem File: ITP206^1.p
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%------------------------------------------------------------------------------
% File : ITP206^1 : TPTP v8.2.0. Released v8.1.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer problem Imperative_HOL_Add 00014_000455
% Version : [Des22] axioms.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des22] Desharnais (2022), Email to Geoff Sutcliffe
% Source : [Des22]
% Names : 0017_Imperative_HOL_Add_00014_000455 [Des22]
% Status : Theorem
% Rating : 0.50 v8.2.0, 0.62 v8.1.0
% Syntax : Number of formulae : 6932 (1744 unt;2602 typ; 0 def)
% Number of atoms : 13690 (5570 equ; 1 cnn)
% Maximal formula atoms : 11 ( 3 avg)
% Number of connectives : 50597 (1150 ~; 79 |; 813 &;43012 @)
% ( 0 <=>;5543 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 7 avg)
% Number of types : 205 ( 204 usr)
% Number of type conns : 17027 (17027 >; 0 *; 0 +; 0 <<)
% Number of symbols : 2402 (2398 usr; 114 con; 0-5 aty)
% Number of variables : 15786 (2761 ^;12753 !; 272 ?;15786 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% from the van Emde Boas Trees session in the Archive of Formal
% proofs -
% www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
% 2022-02-17 13:39:20.752
%------------------------------------------------------------------------------
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bNF_Ca1968104039914474786nt_nat: set_Pr958786334691620121nt_int > ( int > nat ) > $o ).
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Int__Oint_001t__Rat__Orat,type,
bNF_Ca1332973979827979050nt_rat: set_Pr958786334691620121nt_int > ( int > rat ) > $o ).
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Nat__Onat_001t__Code____Numeral__Ointeger,type,
bNF_Ca30959104405961555nteger: set_Pr1261947904930325089at_nat > ( nat > code_integer ) > $o ).
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Nat__Onat_001t__Int__Oint,type,
bNF_Ca966259857504369954at_int: set_Pr1261947904930325089at_nat > ( nat > int ) > $o ).
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Nat__Onat_001t__Nat__Onat,type,
bNF_Ca968750328013420230at_nat: set_Pr1261947904930325089at_nat > ( nat > nat ) > $o ).
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Nat__Onat_001t__Rat__Orat,type,
bNF_Ca333620267926924494at_rat: set_Pr1261947904930325089at_nat > ( nat > rat ) > $o ).
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Int__Oint,type,
bNF_Ca4946600973702282345at_int: set_Pr4329608150637261639at_nat > ( set_Pr1261947904930325089at_nat > int ) > $o ).
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
bNF_Ca4949091444211332621at_nat: set_Pr4329608150637261639at_nat > ( set_Pr1261947904930325089at_nat > nat ) > $o ).
thf(sy_c_BNF__Def_OGr_001_Eo_001_Eo,type,
bNF_Gr_o_o: set_o > ( $o > $o ) > set_Product_prod_o_o ).
thf(sy_c_BNF__Def_OGr_001t__Int__Oint_001t__Int__Oint,type,
bNF_Gr_int_int: set_int > ( int > int ) > set_Pr958786334691620121nt_int ).
thf(sy_c_BNF__Def_OGr_001t__Int__Oint_001t__Nat__Onat,type,
bNF_Gr_int_nat: set_int > ( int > nat ) > set_Pr3448869479623346877nt_nat ).
thf(sy_c_BNF__Def_OGr_001t__Nat__Onat_001t__Int__Oint,type,
bNF_Gr_nat_int: set_nat > ( nat > int ) > set_Pr7995236796853374141at_int ).
thf(sy_c_BNF__Def_OGr_001t__Nat__Onat_001t__Nat__Onat,type,
bNF_Gr_nat_nat: set_nat > ( nat > nat ) > set_Pr1261947904930325089at_nat ).
thf(sy_c_BNF__Def_OGr_001t__Nat__Onat_001t__Num__Onum,type,
bNF_Gr_nat_num: set_nat > ( nat > num ) > set_Pr6200539531224447659at_num ).
thf(sy_c_BNF__Def_OGr_001t__Nat__Onat_001t__Rat__Orat,type,
bNF_Gr_nat_rat: set_nat > ( nat > rat ) > set_Pr4105333604307423337at_rat ).
thf(sy_c_BNF__Def_OGr_001t__Num__Onum_001t__Num__Onum,type,
bNF_Gr_num_num: set_num > ( num > num ) > set_Pr8218934625190621173um_num ).
thf(sy_c_BNF__Def_OGr_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
bNF_Gr1527221351080490466at_nat: set_se7855581050983116737at_nat > ( set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ) > set_Pr4329608150637261639at_nat ).
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bNF_Gr7888059597227825506at_nat: set_se5479201105039950375at_nat > ( set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat ) > set_Pr7459493094073627847at_nat ).
thf(sy_c_BNF__Def_OGrp_001_062_It__Code____Numeral__Ointeger_M_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001_062_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J,type,
bNF_Gr1453458339979195462nteger: set_Co4375521073081744395nteger > ( ( code_integer > code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer > code_integer ) > ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > $o ).
thf(sy_c_BNF__Def_OGrp_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
bNF_Gr4092539093814444494nteger: set_Co726186806682698383nteger > ( ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer ) > ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > $o ).
thf(sy_c_BNF__Def_OGrp_001_062_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Int__Oint_Mt__Nat__Onat_J,type,
bNF_Gr2243168462153404386nt_nat: set_int_int > ( ( int > int ) > int > nat ) > ( int > int ) > ( int > nat ) > $o ).
thf(sy_c_BNF__Def_OGrp_001_062_It__Int__Oint_Mt__Nat__Onat_J_001_062_It__Int__Oint_Mt__Num__Onum_J,type,
bNF_Gr6763702254253555408nt_num: set_int_nat > ( ( int > nat ) > int > num ) > ( int > nat ) > ( int > num ) > $o ).
thf(sy_c_BNF__Def_OGrp_001_062_It__Int__Oint_Mt__Num__Onum_J_001_062_It__Int__Oint_Mt__Num__Onum_J,type,
bNF_Gr318824752125099290nt_num: set_int_num > ( ( int > num ) > int > num ) > ( int > num ) > ( int > num ) > $o ).
thf(sy_c_BNF__Def_OGrp_001_062_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
bNF_Gr952524912132828525nt_int: set_in6154042108061588260nt_int > ( ( int > product_prod_int_int ) > int > int ) > ( int > product_prod_int_int ) > ( int > int ) > $o ).
thf(sy_c_BNF__Def_OGrp_001_062_It__Int__Oint_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Int__Oint_Mt__Nat__Onat_J,type,
bNF_Gr1462982377596425817nt_nat: set_in7748886932685137004at_nat > ( ( int > product_prod_nat_nat ) > int > nat ) > ( int > product_prod_nat_nat ) > ( int > nat ) > $o ).
thf(sy_c_BNF__Def_OGrp_001_062_It__Int__Oint_Mt__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J_J_001_062_It__Int__Oint_Mt__Num__Onum_J,type,
bNF_Gr5590197745039885111nt_num: set_in3169746454734514176um_num > ( ( int > product_prod_num_num ) > int > num ) > ( int > product_prod_num_num ) > ( int > num ) > $o ).
thf(sy_c_BNF__Def_OGrp_001_062_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__List__Olist_It__Nat__Onat_J_Mt__Nat__Onat_J,type,
bNF_Gr1532745923208276441at_nat: set_li2168235667650531768at_nat > ( ( list_nat > product_prod_nat_nat ) > list_nat > nat ) > ( list_nat > product_prod_nat_nat ) > ( list_nat > nat ) > $o ).
thf(sy_c_BNF__Def_OGrp_001_062_It__Nat__Onat_M_Eo_J_001_062_It__Nat__Onat_M_Eo_J,type,
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thf(sy_c_BNF__Def_OGrp_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
bNF_Gr3847987472475283150at_nat: set_nat_nat > ( ( nat > nat ) > nat > nat ) > ( nat > nat ) > ( nat > nat ) > $o ).
thf(sy_c_BNF__Def_OGrp_001_062_It__Nat__Onat_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Nat__Onat_Mt__Int__Oint_J,type,
bNF_Gr2878093617000900205at_int: set_na7251125088705586816nt_int > ( ( nat > product_prod_int_int ) > nat > int ) > ( nat > product_prod_int_int ) > ( nat > int ) > $o ).
thf(sy_c_BNF__Def_OGrp_001_062_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
bNF_Gr3388551082464497497at_nat: set_na8845969913329135560at_nat > ( ( nat > product_prod_nat_nat ) > nat > nat ) > ( nat > product_prod_nat_nat ) > ( nat > nat ) > $o ).
thf(sy_c_BNF__Def_OGrp_001_062_It__Rat__Orat_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Rat__Orat_Mt__Int__Oint_J,type,
bNF_Gr6858100456814184557at_int: set_ra518564293221149304nt_int > ( ( rat > product_prod_int_int ) > rat > int ) > ( rat > product_prod_int_int ) > ( rat > int ) > $o ).
thf(sy_c_BNF__Def_OGrp_001_062_It__Rat__Orat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Rat__Orat_Mt__Nat__Onat_J,type,
bNF_Gr7368557922277781849at_nat: set_ra2113409117844698048at_nat > ( ( rat > product_prod_nat_nat ) > rat > nat ) > ( rat > product_prod_nat_nat ) > ( rat > nat ) > $o ).
thf(sy_c_BNF__Def_OGrp_001_062_It__String__Ochar_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__String__Ochar_Mt__Int__Oint_J,type,
bNF_Gr6947267152677263533ar_int: set_ch6383161263842509759nt_int > ( ( char > product_prod_int_int ) > char > int ) > ( char > product_prod_int_int ) > ( char > int ) > $o ).
thf(sy_c_BNF__Def_OGrp_001_062_It__String__Ochar_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__String__Ochar_Mt__Nat__Onat_J,type,
bNF_Gr1779741795384719769ar_nat: set_ch7978006088466058503at_nat > ( ( char > product_prod_nat_nat ) > char > nat ) > ( char > product_prod_nat_nat ) > ( char > nat ) > $o ).
thf(sy_c_BNF__Def_OGrp_001_Eo_001_Eo,type,
bNF_Grp_o_o: set_o > ( $o > $o ) > $o > $o > $o ).
thf(sy_c_BNF__Def_OGrp_001_Eo_001t__Product____Type__Ounit,type,
bNF_Gr395642070819736771t_unit: set_o > ( $o > product_unit ) > $o > product_unit > $o ).
thf(sy_c_BNF__Def_OGrp_001_Eo_001t__Rat__Orat,type,
bNF_Grp_o_rat: set_o > ( $o > rat ) > $o > rat > $o ).
thf(sy_c_BNF__Def_OGrp_001_Eo_001t__String__Ochar,type,
bNF_Grp_o_char: set_o > ( $o > char ) > $o > char > $o ).
thf(sy_c_BNF__Def_OGrp_001t__Code____Numeral__Ointeger_001t__Int__Oint,type,
bNF_Gr6045950513751025625er_int: set_Code_integer > ( code_integer > int ) > code_integer > int > $o ).
thf(sy_c_BNF__Def_OGrp_001t__Heap__Oref_Itf__a_J_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J,type,
bNF_Gr6510375311308651680t_unit: set_ref_a > ( ref_a > heap_e7401611519738050253t_unit ) > ref_a > heap_e7401611519738050253t_unit > $o ).
thf(sy_c_BNF__Def_OGrp_001t__Int__Oint_001t__Code____Numeral__Ointeger,type,
bNF_Gr6684649993275963353nteger: set_int > ( int > code_integer ) > int > code_integer > $o ).
thf(sy_c_BNF__Def_OGrp_001t__Int__Oint_001t__Int__Oint,type,
bNF_Grp_int_int: set_int > ( int > int ) > int > int > $o ).
thf(sy_c_BNF__Def_OGrp_001t__Int__Oint_001t__Nat__Onat,type,
bNF_Grp_int_nat: set_int > ( int > nat ) > int > nat > $o ).
thf(sy_c_BNF__Def_OGrp_001t__Int__Oint_001t__Num__Onum,type,
bNF_Grp_int_num: set_int > ( int > num ) > int > num > $o ).
thf(sy_c_BNF__Def_OGrp_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
bNF_Gr6591314147588017797at_nat: set_int > ( int > product_prod_nat_nat ) > int > product_prod_nat_nat > $o ).
thf(sy_c_BNF__Def_OGrp_001t__Int__Oint_001t__Rat__Orat,type,
bNF_Grp_int_rat: set_int > ( int > rat ) > int > rat > $o ).
thf(sy_c_BNF__Def_OGrp_001t__Int__Oint_001t__String__Ochar,type,
bNF_Grp_int_char: set_int > ( int > char ) > int > char > $o ).
thf(sy_c_BNF__Def_OGrp_001t__List__Olist_It__Nat__Onat_J_001t__Int__Oint,type,
bNF_Grp_list_nat_int: set_list_nat > ( list_nat > int ) > list_nat > int > $o ).
thf(sy_c_BNF__Def_OGrp_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
bNF_Gr600725020315365264st_nat: set_list_nat > ( list_nat > list_nat ) > list_nat > list_nat > $o ).
thf(sy_c_BNF__Def_OGrp_001t__List__Olist_It__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
bNF_Gr4119388860389678161at_nat: set_list_nat > ( list_nat > product_prod_nat_nat ) > list_nat > product_prod_nat_nat > $o ).
thf(sy_c_BNF__Def_OGrp_001t__Nat__Onat_001_Eo,type,
bNF_Grp_nat_o: set_nat > ( nat > $o ) > nat > $o > $o ).
thf(sy_c_BNF__Def_OGrp_001t__Nat__Onat_001t__Int__Oint,type,
bNF_Grp_nat_int: set_nat > ( nat > int ) > nat > int > $o ).
thf(sy_c_BNF__Def_OGrp_001t__Nat__Onat_001t__Nat__Onat,type,
bNF_Grp_nat_nat: set_nat > ( nat > nat ) > nat > nat > $o ).
thf(sy_c_BNF__Def_OGrp_001t__Nat__Onat_001t__Num__Onum,type,
bNF_Grp_nat_num: set_nat > ( nat > num ) > nat > num > $o ).
thf(sy_c_BNF__Def_OGrp_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
bNF_Gr3553472063123884001at_nat: set_nat > ( nat > product_prod_nat_nat ) > nat > product_prod_nat_nat > $o ).
thf(sy_c_BNF__Def_OGrp_001t__Nat__Onat_001t__Product____Type__Ounit,type,
bNF_Gr1993012052113314029t_unit: set_nat > ( nat > product_unit ) > nat > product_unit > $o ).
thf(sy_c_BNF__Def_OGrp_001t__Nat__Onat_001t__Rat__Orat,type,
bNF_Grp_nat_rat: set_nat > ( nat > rat ) > nat > rat > $o ).
thf(sy_c_BNF__Def_OGrp_001t__Nat__Onat_001t__String__Ochar,type,
bNF_Grp_nat_char: set_nat > ( nat > char ) > nat > char > $o ).
thf(sy_c_BNF__Def_OGrp_001t__Num__Onum_001t__Num__Onum,type,
bNF_Grp_num_num: set_num > ( num > num ) > num > num > $o ).
thf(sy_c_BNF__Def_OGrp_001t__Product____Type__Oprod_It__Heap__Oref_Itf__a_J_Mt__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_J_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J,type,
bNF_Gr5237703264515865432t_unit: set_Pr1133966238863718811t_unit > ( produc7812382424243267045t_unit > heap_e7401611519738050253t_unit ) > produc7812382424243267045t_unit > heap_e7401611519738050253t_unit > $o ).
thf(sy_c_BNF__Def_OGrp_001t__Product____Type__Oprod_It__Heap__Oref_Itf__a_J_Mt__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_J_001t__Heap__Oref_Itf__a_J,type,
bNF_Gr5964920315653434322_ref_a: set_Pr1133966238863718811t_unit > ( produc7812382424243267045t_unit > ref_a ) > produc7812382424243267045t_unit > ref_a > $o ).
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bNF_Gr2749509334501985047nt_int: set_Pr958786334691620121nt_int > ( product_prod_int_int > int ) > product_prod_int_int > int > $o ).
thf(sy_c_BNF__Def_OGrp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Num__Onum,type,
bNF_Gr8532703628021589765nt_num: set_Pr958786334691620121nt_int > ( product_prod_int_int > num ) > product_prod_int_int > num > $o ).
thf(sy_c_BNF__Def_OGrp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_Eo,type,
bNF_Gr8859634586176591525_nat_o: set_Pr1261947904930325089at_nat > ( product_prod_nat_nat > $o ) > product_prod_nat_nat > $o > $o ).
thf(sy_c_BNF__Def_OGrp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
bNF_Gr190934199572118623at_int: set_Pr1261947904930325089at_nat > ( product_prod_nat_nat > int ) > product_prod_nat_nat > int > $o ).
thf(sy_c_BNF__Def_OGrp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
bNF_Gr193424670081168899at_nat: set_Pr1261947904930325089at_nat > ( product_prod_nat_nat > nat ) > product_prod_nat_nat > nat > $o ).
thf(sy_c_BNF__Def_OGrp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Num__Onum,type,
bNF_Gr5974128493091723341at_num: set_Pr1261947904930325089at_nat > ( product_prod_nat_nat > num ) > product_prod_nat_nat > num > $o ).
thf(sy_c_BNF__Def_OGrp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
bNF_Gr7275288873855648526at_nat: set_Pr1261947904930325089at_nat > ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat > $o ).
thf(sy_c_BNF__Def_OGrp_001t__Rat__Orat_001t__Int__Oint,type,
bNF_Grp_rat_int: set_rat > ( rat > int ) > rat > int > $o ).
thf(sy_c_BNF__Def_OGrp_001t__Rat__Orat_001t__Nat__Onat,type,
bNF_Grp_rat_nat: set_rat > ( rat > nat ) > rat > nat > $o ).
thf(sy_c_BNF__Def_OGrp_001t__Rat__Orat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
bNF_Gr6883301533820405721at_nat: set_rat > ( rat > product_prod_nat_nat ) > rat > product_prod_nat_nat > $o ).
thf(sy_c_BNF__Def_OGrp_001t__Rat__Orat_001t__Rat__Orat,type,
bNF_Grp_rat_rat: set_rat > ( rat > rat ) > rat > rat > $o ).
thf(sy_c_BNF__Def_OGrp_001t__Rat__Orat_001t__String__Ochar,type,
bNF_Grp_rat_char: set_rat > ( rat > char ) > rat > char > $o ).
thf(sy_c_BNF__Def_OGrp_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
bNF_Gr6677838525845560078at_nat: set_se7855581050983116737at_nat > ( set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ) > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_BNF__Def_OGrp_001t__String__Ochar_001t__Int__Oint,type,
bNF_Grp_char_int: set_char > ( char > int ) > char > int > $o ).
thf(sy_c_BNF__Def_OGrp_001t__String__Ochar_001t__Nat__Onat,type,
bNF_Grp_char_nat: set_char > ( char > nat ) > char > nat > $o ).
thf(sy_c_BNF__Def_OGrp_001t__String__Ochar_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
bNF_Gr8982933507515605408at_nat: set_char > ( char > product_prod_nat_nat ) > char > product_prod_nat_nat > $o ).
thf(sy_c_BNF__Def_OGrp_001t__String__Ochar_001t__Rat__Orat,type,
bNF_Grp_char_rat: set_char > ( char > rat ) > char > rat > $o ).
thf(sy_c_BNF__Def_OGrp_001t__String__Ochar_001t__String__Ochar,type,
bNF_Grp_char_char: set_char > ( char > char ) > char > char > $o ).
thf(sy_c_BNF__Def_Oconvol_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
bNF_co1764293801198077070nteger: ( ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer ) > produc101387210554329899nteger ).
thf(sy_c_BNF__Def_Oconvol_001_Eo_001_Eo_001_Eo,type,
bNF_convol_o_o_o: ( $o > $o ) > ( $o > $o ) > $o > product_prod_o_o ).
thf(sy_c_BNF__Def_Oconvol_001_Eo_001_Eo_001t__Nat__Onat,type,
bNF_convol_o_o_nat: ( $o > $o ) > ( $o > nat ) > $o > product_prod_o_nat ).
thf(sy_c_BNF__Def_Oconvol_001_Eo_001t__Nat__Onat_001_Eo,type,
bNF_convol_o_nat_o: ( $o > nat ) > ( $o > $o ) > $o > product_prod_nat_o ).
thf(sy_c_BNF__Def_Oconvol_001_Eo_001t__Nat__Onat_001t__Nat__Onat,type,
bNF_convol_o_nat_nat: ( $o > nat ) > ( $o > nat ) > $o > product_prod_nat_nat ).
thf(sy_c_BNF__Def_Oconvol_001_Eo_001t__Num__Onum_001t__Nat__Onat,type,
bNF_convol_o_num_nat: ( $o > num ) > ( $o > nat ) > $o > product_prod_num_nat ).
thf(sy_c_BNF__Def_Oconvol_001t__Code____Numeral__Ointeger_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
bNF_co6169604788974073338nteger: ( code_integer > code_integer > code_integer ) > ( code_integer > code_integer > code_integer ) > code_integer > produc723880050079548563nteger ).
thf(sy_c_BNF__Def_Oconvol_001t__Code____Numeral__Ointeger_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
bNF_co3156833790671340178nteger: ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > code_integer > produc101387210554329899nteger ).
thf(sy_c_BNF__Def_Oconvol_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
bNF_co2174166945328636015nt_int: ( int > int ) > ( int > int ) > int > product_prod_int_int ).
thf(sy_c_BNF__Def_Oconvol_001t__Int__Oint_001t__Int__Oint_001t__Nat__Onat,type,
bNF_co2176657415837686291nt_nat: ( int > int ) > ( int > nat ) > int > product_prod_int_nat ).
thf(sy_c_BNF__Def_Oconvol_001t__Int__Oint_001t__Nat__Onat_001t__Int__Oint,type,
bNF_co1174813233427581459at_int: ( int > nat ) > ( int > int ) > int > product_prod_nat_int ).
thf(sy_c_BNF__Def_Oconvol_001t__Int__Oint_001t__Nat__Onat_001t__Nat__Onat,type,
bNF_co1177303703936631735at_nat: ( int > nat ) > ( int > nat ) > int > product_prod_nat_nat ).
thf(sy_c_BNF__Def_Oconvol_001t__Int__Oint_001t__Nat__Onat_001t__Num__Onum,type,
bNF_co6958007526947186177at_num: ( int > nat ) > ( int > num ) > int > product_prod_nat_num ).
thf(sy_c_BNF__Def_Oconvol_001t__Int__Oint_001t__Num__Onum_001t__Nat__Onat,type,
bNF_co8531006144098876929um_nat: ( int > num ) > ( int > nat ) > int > product_prod_num_nat ).
thf(sy_c_BNF__Def_Oconvol_001t__Int__Oint_001t__Num__Onum_001t__Num__Onum,type,
bNF_co5088337930254655563um_num: ( int > num ) > ( int > num ) > int > product_prod_num_num ).
thf(sy_c_BNF__Def_Oconvol_001t__Nat__Onat_001_Eo_001_Eo,type,
bNF_convol_nat_o_o: ( nat > $o ) > ( nat > $o ) > nat > product_prod_o_o ).
thf(sy_c_BNF__Def_Oconvol_001t__Nat__Onat_001_Eo_001t__Nat__Onat,type,
bNF_convol_nat_o_nat: ( nat > $o ) > ( nat > nat ) > nat > product_prod_o_nat ).
thf(sy_c_BNF__Def_Oconvol_001t__Nat__Onat_001t__Nat__Onat_001_Eo,type,
bNF_convol_nat_nat_o: ( nat > nat ) > ( nat > $o ) > nat > product_prod_nat_o ).
thf(sy_c_BNF__Def_Oconvol_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
bNF_co805650143699787099at_nat: ( nat > nat ) > ( nat > nat ) > nat > product_prod_nat_nat ).
thf(sy_c_BNF__Def_Oconvol_001t__Nat__Onat_001t__Num__Onum_001_Eo,type,
bNF_convol_nat_num_o: ( nat > num ) > ( nat > $o ) > nat > product_prod_num_o ).
thf(sy_c_BNF__Def_Oconvol_001t__Nat__Onat_001t__Num__Onum_001t__Nat__Onat,type,
bNF_co8159352583862032293um_nat: ( nat > num ) > ( nat > nat ) > nat > product_prod_num_nat ).
thf(sy_c_BNF__Def_Oconvol_001t__Nat__Onat_001t__Num__Onum_001t__Num__Onum,type,
bNF_co4716684370017810927um_num: ( nat > num ) > ( nat > num ) > nat > product_prod_num_num ).
thf(sy_c_BNF__Def_Oconvol_001t__Num__Onum_001_Eo_001t__Nat__Onat,type,
bNF_convol_num_o_nat: ( num > $o ) > ( num > nat ) > num > product_prod_o_nat ).
thf(sy_c_BNF__Def_Oconvol_001t__Num__Onum_001t__Nat__Onat_001t__Nat__Onat,type,
bNF_co7969830008525221797at_nat: ( num > nat ) > ( num > nat ) > num > product_prod_nat_nat ).
thf(sy_c_BNF__Def_Oconvol_001t__Num__Onum_001t__Num__Onum_001t__Nat__Onat,type,
bNF_co6100160411832691183um_nat: ( num > num ) > ( num > nat ) > num > product_prod_num_nat ).
thf(sy_c_BNF__Def_Oconvol_001t__Product____Type__Oprod_It__Heap__Oref_Itf__a_J_Mt__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_J_001t__Int__Oint_001t__Int__Oint,type,
bNF_co6607689160745551784nt_int: ( produc7812382424243267045t_unit > int ) > ( produc7812382424243267045t_unit > int ) > produc7812382424243267045t_unit > product_prod_int_int ).
thf(sy_c_BNF__Def_Oconvol_001t__Product____Type__Oprod_It__Heap__Oref_Itf__a_J_Mt__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_J_001t__Nat__Onat_001t__Nat__Onat,type,
bNF_co5610825919353547504at_nat: ( produc7812382424243267045t_unit > nat ) > ( produc7812382424243267045t_unit > nat ) > produc7812382424243267045t_unit > product_prod_nat_nat ).
thf(sy_c_BNF__Def_Oconvol_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Int__Oint_001t__Int__Oint,type,
bNF_co3748779267718262122nt_int: ( product_prod_int_int > int ) > ( product_prod_int_int > int ) > product_prod_int_int > product_prod_int_int ).
thf(sy_c_BNF__Def_Oconvol_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Nat__Onat_001t__Nat__Onat,type,
bNF_co2751916026326257842at_nat: ( product_prod_int_int > nat ) > ( product_prod_int_int > nat ) > product_prod_int_int > product_prod_nat_nat ).
thf(sy_c_BNF__Def_Oconvol_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001t__Int__Oint,type,
bNF_co49787548815638194nt_int: ( product_prod_nat_nat > int ) > ( product_prod_nat_nat > int ) > product_prod_nat_nat > product_prod_int_int ).
thf(sy_c_BNF__Def_Oconvol_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat_001t__Nat__Onat,type,
bNF_co8276296344278409722at_nat: ( product_prod_nat_nat > nat ) > ( product_prod_nat_nat > nat ) > product_prod_nat_nat > product_prod_nat_nat ).
thf(sy_c_BNF__Def_Oconvol_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
bNF_co147202357524637304at_nat: ( set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ) > ( set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ) > set_Pr1261947904930325089at_nat > produc3843707927480180839at_nat ).
thf(sy_c_BNF__Def_Ocsquare_001t__Product____Type__Oprod_It__Int__Oint_Mt__Code____Numeral__Ointeger_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Code____Numeral__Ointeger_J,type,
bNF_cs4943907971211754724nteger: set_Pr8415741480598773322nteger > ( product_prod_int_int > int ) > ( produc2963269191615333738nteger > int ) > ( produc2963269191615333738nteger > product_prod_int_int ) > ( produc2963269191615333738nteger > produc2963269191615333738nteger ) > $o ).
thf(sy_c_BNF__Def_Ocsquare_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Int__Oint_J,type,
bNF_cs7194553038297458973at_int: set_Pr49084382684228242at_int > ( produc859450856879609959at_nat > product_prod_nat_nat ) > ( produc4196048018000912732at_int > product_prod_nat_nat ) > ( produc4196048018000912732at_int > produc859450856879609959at_nat ) > ( produc4196048018000912732at_int > produc4196048018000912732at_int ) > $o ).
thf(sy_c_BNF__Def_OfstOp_001t__Heap__Oref_Itf__a_J_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J,type,
bNF_fs409234558581511823t_unit: ( ref_a > heap_e7401611519738050253t_unit > $o ) > ( heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit > $o ) > produc7812382424243267045t_unit > produc7812382424243267045t_unit ).
thf(sy_c_BNF__Def_OfstOp_001t__Int__Oint_001t__Int__Oint_001t__Code____Numeral__Ointeger,type,
bNF_fs4543836202474805097nteger: ( int > int > $o ) > ( int > code_integer > $o ) > produc2963269191615333738nteger > product_prod_int_int ).
thf(sy_c_BNF__Def_OfstOp_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
bNF_fs8167890477030535480nt_int: ( int > int > $o ) > ( int > int > $o ) > product_prod_int_int > product_prod_int_int ).
thf(sy_c_BNF__Def_OfstOp_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
bNF_fs6799373675401686564at_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > product_prod_nat_nat > product_prod_nat_nat ).
thf(sy_c_BNF__Def_OfstOp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
bNF_fs6342258632037972510at_int: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > int > $o ) > produc4196048018000912732at_int > produc859450856879609959at_nat ).
thf(sy_c_BNF__Def_OfstOp_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
bNF_fs1946732770515719151at_nat: ( set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ) > ( set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ) > produc3843707927480180839at_nat > produc3843707927480180839at_nat ).
thf(sy_c_BNF__Def_Opick__middlep_001t__Int__Oint_001t__Int__Oint_001t__Code____Numeral__Ointeger,type,
bNF_pi4515785505327359899nteger: ( int > int > $o ) > ( int > code_integer > $o ) > int > code_integer > int ).
thf(sy_c_BNF__Def_Opick__middlep_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
bNF_pi770922605693484880at_int: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > int > $o ) > product_prod_nat_nat > int > product_prod_nat_nat ).
thf(sy_c_BNF__Def_Orel__fun_001_062_I_Eo_M_Eo_J_001_062_I_Eo_M_Eo_J_001_062_I_062_It__Nat__Onat_M_Eo_J_M_062_It__Nat__Onat_M_Eo_J_J_001_062_I_062_It__Nat__Onat_M_Eo_J_M_062_It__Nat__Onat_M_Eo_J_J,type,
bNF_re6580757339198446453_nat_o: ( ( $o > $o ) > ( $o > $o ) > $o ) > ( ( ( nat > $o ) > nat > $o ) > ( ( nat > $o ) > nat > $o ) > $o ) > ( ( $o > $o ) > ( nat > $o ) > nat > $o ) > ( ( $o > $o ) > ( nat > $o ) > nat > $o ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_062_I_Eo_M_Eo_J_001_062_It__Int__Oint_Mt__Nat__Onat_J_001_062_I_062_It__Nat__Onat_M_Eo_J_M_062_It__Nat__Onat_M_Eo_J_J_001_062_I_062_It__Int__Oint_Mt__Int__Oint_J_M_062_It__Int__Oint_Mt__Nat__Onat_J_J,type,
bNF_re7413876800493218871nt_nat: ( ( $o > $o ) > ( int > nat ) > $o ) > ( ( ( nat > $o ) > nat > $o ) > ( ( int > int ) > int > nat ) > $o ) > ( ( $o > $o ) > ( nat > $o ) > nat > $o ) > ( ( int > nat ) > ( int > int ) > int > nat ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_062_I_Eo_M_Eo_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_I_062_It__Nat__Onat_M_Eo_J_M_062_It__Nat__Onat_M_Eo_J_J_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
bNF_re1515921786380824775at_nat: ( ( $o > $o ) > ( nat > nat ) > $o ) > ( ( ( nat > $o ) > nat > $o ) > ( ( nat > nat ) > nat > nat ) > $o ) > ( ( $o > $o ) > ( nat > $o ) > nat > $o ) > ( ( nat > nat ) > ( nat > nat ) > nat > nat ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_062_I_Eo_M_Eo_J_001_062_It__Nat__Onat_Mt__Num__Onum_J_001_062_I_062_It__Nat__Onat_M_Eo_J_M_062_It__Nat__Onat_M_Eo_J_J_001_062_I_062_It__Int__Oint_Mt__Nat__Onat_J_M_062_It__Int__Oint_Mt__Num__Onum_J_J,type,
bNF_re7387973359137000979nt_num: ( ( $o > $o ) > ( nat > num ) > $o ) > ( ( ( nat > $o ) > nat > $o ) > ( ( int > nat ) > int > num ) > $o ) > ( ( $o > $o ) > ( nat > $o ) > nat > $o ) > ( ( nat > num ) > ( int > nat ) > int > num ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_062_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Int__Oint_Mt__Code____Numeral__Ointeger_J_001_062_It__Int__Oint_Mt__Num__Onum_J_001_062_It__Int__Oint_Mt__Num__Onum_J,type,
bNF_re2961086658867094474nt_num: ( ( int > int ) > ( int > code_integer ) > $o ) > ( ( int > num ) > ( int > num ) > $o ) > ( ( int > int ) > int > num ) > ( ( int > code_integer ) > int > num ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_062_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Int__Oint_Mt__Nat__Onat_J_001_062_It__Int__Oint_Mt__Nat__Onat_J,type,
bNF_re2034674868506727813nt_nat: ( ( int > int ) > ( int > int ) > $o ) > ( ( int > nat ) > ( int > nat ) > $o ) > ( ( int > int ) > int > nat ) > ( ( int > int ) > int > nat ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_062_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Int__Oint_Mt__Nat__Onat_J_001_062_It__Int__Oint_Mt__Nat__Onat_J_001_062_It__Int__Oint_Mt__Num__Onum_J,type,
bNF_re7492190849051111027nt_num: ( ( int > int ) > ( int > nat ) > $o ) > ( ( int > nat ) > ( int > num ) > $o ) > ( ( int > int ) > int > nat ) > ( ( int > nat ) > int > num ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_062_It__Int__Oint_Mt__Nat__Onat_J_001_062_It__Code____Numeral__Ointeger_Mt__Nat__Onat_J_001_062_It__Int__Oint_Mt__Num__Onum_J_001_062_It__Code____Numeral__Ointeger_Mt__Num__Onum_J,type,
bNF_re5502197115626545091er_num: ( ( int > nat ) > ( code_integer > nat ) > $o ) > ( ( int > num ) > ( code_integer > num ) > $o ) > ( ( int > nat ) > int > num ) > ( ( code_integer > nat ) > code_integer > num ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_062_It__Int__Oint_Mt__Nat__Onat_J_001_062_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Int__Oint_Mt__Num__Onum_J_001_062_It__Int__Oint_Mt__Nat__Onat_J,type,
bNF_re3642433447925381491nt_nat: ( ( int > nat ) > ( int > int ) > $o ) > ( ( int > num ) > ( int > nat ) > $o ) > ( ( int > nat ) > int > num ) > ( ( int > int ) > int > nat ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_062_It__Int__Oint_Mt__Nat__Onat_J_001_062_It__Int__Oint_Mt__Nat__Onat_J_001_062_I_062_It__Int__Oint_Mt__Int__Oint_J_M_062_It__Int__Oint_Mt__Nat__Onat_J_J_001_062_I_062_It__Int__Oint_Mt__Int__Oint_J_M_062_It__Int__Oint_Mt__Nat__Onat_J_J,type,
bNF_re6307232852808291833nt_nat: ( ( int > nat ) > ( int > nat ) > $o ) > ( ( ( int > int ) > int > nat ) > ( ( int > int ) > int > nat ) > $o ) > ( ( int > nat ) > ( int > int ) > int > nat ) > ( ( int > nat ) > ( int > int ) > int > nat ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_062_It__Int__Oint_Mt__Nat__Onat_J_001_062_It__Int__Oint_Mt__Nat__Onat_J_001_062_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Int__Oint_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J_001_062_I_062_It__Int__Oint_Mt__Int__Oint_J_M_062_It__Int__Oint_Mt__Nat__Onat_J_J,type,
bNF_re4099335476084980305nt_nat: ( ( int > nat ) > ( int > nat ) > $o ) > ( ( ( product_prod_nat_nat > int ) > product_prod_nat_nat > nat ) > ( ( int > int ) > int > nat ) > $o ) > ( ( int > nat ) > ( product_prod_nat_nat > int ) > product_prod_nat_nat > nat ) > ( ( int > nat ) > ( int > int ) > int > nat ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_062_It__Int__Oint_Mt__Nat__Onat_J_001_062_It__Int__Oint_Mt__Nat__Onat_J_001_062_It__Int__Oint_Mt__Num__Onum_J_001_062_It__Int__Oint_Mt__Num__Onum_J,type,
bNF_re9099949428469764705nt_num: ( ( int > nat ) > ( int > nat ) > $o ) > ( ( int > num ) > ( int > num ) > $o ) > ( ( int > nat ) > int > num ) > ( ( int > nat ) > int > num ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_062_It__Int__Oint_Mt__Nat__Onat_J_001_062_It__Nat__Onat_M_Eo_J_001_062_It__Int__Oint_Mt__Num__Onum_J_001_062_It__Nat__Onat_M_Eo_J,type,
bNF_re7435575058205011731_nat_o: ( ( int > nat ) > ( nat > $o ) > $o ) > ( ( int > num ) > ( nat > $o ) > $o ) > ( ( int > nat ) > int > num ) > ( ( nat > $o ) > nat > $o ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_062_It__Int__Oint_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Int__Oint_Mt__Num__Onum_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
bNF_re717418264989850719at_nat: ( ( int > nat ) > ( nat > nat ) > $o ) > ( ( int > num ) > ( nat > nat ) > $o ) > ( ( int > nat ) > int > num ) > ( ( nat > nat ) > nat > nat ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_062_It__Int__Oint_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Num__Onum_J_001_062_I_062_It__Int__Oint_Mt__Int__Oint_J_M_062_It__Int__Oint_Mt__Nat__Onat_J_J_001_062_I_062_It__Int__Oint_Mt__Nat__Onat_J_M_062_It__Int__Oint_Mt__Num__Onum_J_J,type,
bNF_re4310571189260822741nt_num: ( ( int > nat ) > ( nat > num ) > $o ) > ( ( ( int > int ) > int > nat ) > ( ( int > nat ) > int > num ) > $o ) > ( ( int > nat ) > ( int > int ) > int > nat ) > ( ( nat > num ) > ( int > nat ) > int > num ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_062_It__Int__Oint_Mt__Num__Onum_J_001_062_It__Code____Numeral__Ointeger_Mt__Num__Onum_J_001_062_I_062_It__Int__Oint_Mt__Int__Oint_J_M_062_It__Int__Oint_Mt__Num__Onum_J_J_001_062_I_062_It__Int__Oint_Mt__Code____Numeral__Ointeger_J_M_062_It__Int__Oint_Mt__Num__Onum_J_J,type,
bNF_re7839584876883444355nt_num: ( ( int > num ) > ( code_integer > num ) > $o ) > ( ( ( int > int ) > int > num ) > ( ( int > code_integer ) > int > num ) > $o ) > ( ( int > num ) > ( int > int ) > int > num ) > ( ( code_integer > num ) > ( int > code_integer ) > int > num ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_062_It__Int__Oint_Mt__Num__Onum_J_001_062_It__Code____Numeral__Ointeger_Mt__Num__Onum_J_001_062_It__Int__Oint_Mt__Num__Onum_J_001_062_It__Code____Numeral__Ointeger_Mt__Num__Onum_J,type,
bNF_re6292805115969292119er_num: ( ( int > num ) > ( code_integer > num ) > $o ) > ( ( int > num ) > ( code_integer > num ) > $o ) > ( ( int > num ) > int > num ) > ( ( code_integer > num ) > code_integer > num ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_062_It__Int__Oint_Mt__Num__Onum_J_001_062_It__Int__Oint_Mt__Num__Onum_J_001_062_It__Int__Oint_Mt__Num__Onum_J_001_062_It__Int__Oint_Mt__Num__Onum_J,type,
bNF_re4768042542643474933nt_num: ( ( int > num ) > ( int > num ) > $o ) > ( ( int > num ) > ( int > num ) > $o ) > ( ( int > num ) > int > num ) > ( ( int > num ) > int > num ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_M_Eo_J_001_062_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Nat__Onat_M_Eo_J_001_062_It__Int__Oint_Mt__Nat__Onat_J,type,
bNF_re1982657550219213441nt_nat: ( ( nat > $o ) > ( int > int ) > $o ) > ( ( nat > $o ) > ( int > nat ) > $o ) > ( ( nat > $o ) > nat > $o ) > ( ( int > int ) > int > nat ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_M_Eo_J_001_062_It__Int__Oint_Mt__Nat__Onat_J_001_062_It__Nat__Onat_M_Eo_J_001_062_It__Int__Oint_Mt__Num__Onum_J,type,
bNF_re4102725726183926567nt_num: ( ( nat > $o ) > ( int > nat ) > $o ) > ( ( nat > $o ) > ( int > num ) > $o ) > ( ( nat > $o ) > nat > $o ) > ( ( int > nat ) > int > num ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_M_Eo_J_001_062_It__Nat__Onat_M_Eo_J_001_062_It__Nat__Onat_M_Eo_J_001_062_It__Nat__Onat_M_Eo_J,type,
bNF_re2697311982885662685_nat_o: ( ( nat > $o ) > ( nat > $o ) > $o ) > ( ( nat > $o ) > ( nat > $o ) > $o ) > ( ( nat > $o ) > nat > $o ) > ( ( nat > $o ) > nat > $o ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_M_Eo_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_M_Eo_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
bNF_re2144779315220482269at_nat: ( ( nat > $o ) > ( nat > nat ) > $o ) > ( ( nat > $o ) > ( nat > nat ) > $o ) > ( ( nat > $o ) > nat > $o ) > ( ( nat > nat ) > nat > nat ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_I_Eo_M_Eo_J_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_I_062_It__Nat__Onat_M_Eo_J_M_062_It__Nat__Onat_M_Eo_J_J,type,
bNF_re3930744952143938759_nat_o: ( ( nat > nat ) > ( $o > $o ) > $o ) > ( ( ( nat > nat ) > nat > nat ) > ( ( nat > $o ) > nat > $o ) > $o ) > ( ( nat > nat ) > ( nat > nat ) > nat > nat ) > ( ( $o > $o ) > ( nat > $o ) > nat > $o ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Int__Oint_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Int__Oint_Mt__Num__Onum_J,type,
bNF_re8622501330148003679nt_num: ( ( nat > nat ) > ( int > nat ) > $o ) > ( ( nat > nat ) > ( int > num ) > $o ) > ( ( nat > nat ) > nat > nat ) > ( ( int > nat ) > int > num ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_M_Eo_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_M_Eo_J,type,
bNF_re7744223787735169245_nat_o: ( ( nat > nat ) > ( nat > $o ) > $o ) > ( ( nat > nat ) > ( nat > $o ) > $o ) > ( ( nat > nat ) > nat > nat ) > ( ( nat > $o ) > nat > $o ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
bNF_re3262823321055862553at_nat: ( ( nat > nat ) > ( nat > nat ) > $o ) > ( ( ( nat > nat ) > nat > nat ) > ( ( nat > nat ) > nat > nat ) > $o ) > ( ( nat > nat ) > ( nat > nat ) > nat > nat ) > ( ( nat > nat ) > ( nat > nat ) > nat > nat ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J_001_062_I_062_It__Int__Oint_Mt__Nat__Onat_J_M_062_It__Int__Oint_Mt__Nat__Onat_J_J,type,
bNF_re1160762158786734233nt_nat: ( ( nat > nat ) > ( nat > nat ) > $o ) > ( ( ( product_prod_nat_nat > nat ) > product_prod_nat_nat > nat ) > ( ( int > nat ) > int > nat ) > $o ) > ( ( nat > nat ) > ( product_prod_nat_nat > nat ) > product_prod_nat_nat > nat ) > ( ( nat > nat ) > ( int > nat ) > int > nat ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
bNF_re239970166668089693at_nat: ( ( nat > nat ) > ( nat > nat ) > $o ) > ( ( nat > nat ) > ( nat > nat ) > $o ) > ( ( nat > nat ) > nat > nat ) > ( ( nat > nat ) > nat > nat ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Num__Onum_J_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_I_062_It__Int__Oint_Mt__Nat__Onat_J_M_062_It__Int__Oint_Mt__Num__Onum_J_J,type,
bNF_re584878643992547173nt_num: ( ( nat > nat ) > ( nat > num ) > $o ) > ( ( ( nat > nat ) > nat > nat ) > ( ( int > nat ) > int > num ) > $o ) > ( ( nat > nat ) > ( nat > nat ) > nat > nat ) > ( ( nat > num ) > ( int > nat ) > int > num ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Num__Onum_J_001_062_I_Eo_M_Eo_J_001_062_I_062_It__Int__Oint_Mt__Nat__Onat_J_M_062_It__Int__Oint_Mt__Num__Onum_J_J_001_062_I_062_It__Nat__Onat_M_Eo_J_M_062_It__Nat__Onat_M_Eo_J_J,type,
bNF_re3931674169038344211_nat_o: ( ( nat > num ) > ( $o > $o ) > $o ) > ( ( ( int > nat ) > int > num ) > ( ( nat > $o ) > nat > $o ) > $o ) > ( ( nat > num ) > ( int > nat ) > int > num ) > ( ( $o > $o ) > ( nat > $o ) > nat > $o ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Num__Onum_J_001_062_It__Int__Oint_Mt__Nat__Onat_J_001_062_I_062_It__Int__Oint_Mt__Nat__Onat_J_M_062_It__Int__Oint_Mt__Num__Onum_J_J_001_062_I_062_It__Int__Oint_Mt__Int__Oint_J_M_062_It__Int__Oint_Mt__Nat__Onat_J_J,type,
bNF_re5843449549409830101nt_nat: ( ( nat > num ) > ( int > nat ) > $o ) > ( ( ( int > nat ) > int > num ) > ( ( int > int ) > int > nat ) > $o ) > ( ( nat > num ) > ( int > nat ) > int > num ) > ( ( int > nat ) > ( int > int ) > int > nat ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Num__Onum_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_I_062_It__Int__Oint_Mt__Nat__Onat_J_M_062_It__Int__Oint_Mt__Num__Onum_J_J_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
bNF_re6524732562925676389at_nat: ( ( nat > num ) > ( nat > nat ) > $o ) > ( ( ( int > nat ) > int > num ) > ( ( nat > nat ) > nat > nat ) > $o ) > ( ( nat > num ) > ( int > nat ) > int > num ) > ( ( nat > nat ) > ( nat > nat ) > nat > nat ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Num__Onum_J_001_062_It__Nat__Onat_Mt__Num__Onum_J_001_062_I_062_It__Int__Oint_Mt__Nat__Onat_J_M_062_It__Int__Oint_Mt__Num__Onum_J_J_001_062_I_062_It__Code____Numeral__Ointeger_Mt__Nat__Onat_J_M_062_It__Code____Numeral__Ointeger_Mt__Num__Onum_J_J,type,
bNF_re2655444144887082515er_num: ( ( nat > num ) > ( nat > num ) > $o ) > ( ( ( int > nat ) > int > num ) > ( ( code_integer > nat ) > code_integer > num ) > $o ) > ( ( nat > num ) > ( int > nat ) > int > num ) > ( ( nat > num ) > ( code_integer > nat ) > code_integer > num ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Num__Onum_J_001_062_It__Nat__Onat_Mt__Num__Onum_J_001_062_I_062_It__Int__Oint_Mt__Nat__Onat_J_M_062_It__Int__Oint_Mt__Num__Onum_J_J_001_062_I_062_It__Int__Oint_Mt__Nat__Onat_J_M_062_It__Int__Oint_Mt__Num__Onum_J_J,type,
bNF_re3846787885862361009nt_num: ( ( nat > num ) > ( nat > num ) > $o ) > ( ( ( int > nat ) > int > num ) > ( ( int > nat ) > int > num ) > $o ) > ( ( nat > num ) > ( int > nat ) > int > num ) > ( ( nat > num ) > ( int > nat ) > int > num ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Num__Onum_J_001_062_It__Nat__Onat_Mt__Num__Onum_J_001_062_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Num__Onum_J_J_001_062_I_062_It__Int__Oint_Mt__Nat__Onat_J_M_062_It__Int__Oint_Mt__Num__Onum_J_J,type,
bNF_re2742285375668209089nt_num: ( ( nat > num ) > ( nat > num ) > $o ) > ( ( ( product_prod_nat_nat > nat ) > product_prod_nat_nat > num ) > ( ( int > nat ) > int > num ) > $o ) > ( ( nat > num ) > ( product_prod_nat_nat > nat ) > product_prod_nat_nat > num ) > ( ( nat > num ) > ( int > nat ) > int > num ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_062_It__Num__Onum_Mt__Num__Onum_J_001_062_It__Num__Onum_Mt__Num__Onum_J_001_062_I_062_It__Int__Oint_Mt__Num__Onum_J_M_062_It__Int__Oint_Mt__Num__Onum_J_J_001_062_I_062_It__Code____Numeral__Ointeger_Mt__Num__Onum_J_M_062_It__Code____Numeral__Ointeger_Mt__Num__Onum_J_J,type,
bNF_re4105200416192800571er_num: ( ( num > num ) > ( num > num ) > $o ) > ( ( ( int > num ) > int > num ) > ( ( code_integer > num ) > code_integer > num ) > $o ) > ( ( num > num ) > ( int > num ) > int > num ) > ( ( num > num ) > ( code_integer > num ) > code_integer > num ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_062_It__Num__Onum_Mt__Num__Onum_J_001_062_It__Num__Onum_Mt__Num__Onum_J_001_062_I_062_It__Int__Oint_Mt__Num__Onum_J_M_062_It__Int__Oint_Mt__Num__Onum_J_J_001_062_I_062_It__Int__Oint_Mt__Num__Onum_J_M_062_It__Int__Oint_Mt__Num__Onum_J_J,type,
bNF_re5446066262214944985nt_num: ( ( num > num ) > ( num > num ) > $o ) > ( ( ( int > num ) > int > num ) > ( ( int > num ) > int > num ) > $o ) > ( ( num > num ) > ( int > num ) > int > num ) > ( ( num > num ) > ( int > num ) > int > num ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Int__Oint_J_001_062_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_001_062_It__Int__Oint_Mt__Nat__Onat_J,type,
bNF_re3284331099615978085nt_nat: ( ( product_prod_nat_nat > int ) > ( int > int ) > $o ) > ( ( product_prod_nat_nat > nat ) > ( int > nat ) > $o ) > ( ( product_prod_nat_nat > int ) > product_prod_nat_nat > nat ) > ( ( int > int ) > int > nat ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_001_062_It__Int__Oint_Mt__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_001_062_It__Int__Oint_Mt__Nat__Onat_J,type,
bNF_re8419181437172746013nt_nat: ( ( product_prod_nat_nat > nat ) > ( int > nat ) > $o ) > ( ( product_prod_nat_nat > nat ) > ( int > nat ) > $o ) > ( ( product_prod_nat_nat > nat ) > product_prod_nat_nat > nat ) > ( ( int > nat ) > int > nat ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_001_062_It__Int__Oint_Mt__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Num__Onum_J_001_062_It__Int__Oint_Mt__Num__Onum_J,type,
bNF_re2682547871993127601nt_num: ( ( product_prod_nat_nat > nat ) > ( int > nat ) > $o ) > ( ( product_prod_nat_nat > num ) > ( int > num ) > $o ) > ( ( product_prod_nat_nat > nat ) > product_prod_nat_nat > num ) > ( ( int > nat ) > int > num ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_Eo_001_Eo_001_Eo_001_Eo,type,
bNF_rel_fun_o_o_o_o: ( $o > $o > $o ) > ( $o > $o > $o ) > ( $o > $o ) > ( $o > $o ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_Eo_001t__Int__Oint_001_Eo_001t__Nat__Onat,type,
bNF_re4895681925607069953_o_nat: ( $o > int > $o ) > ( $o > nat > $o ) > ( $o > $o ) > ( int > nat ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_Eo_001t__Nat__Onat_001_Eo_001t__Nat__Onat,type,
bNF_re4209105892566519389_o_nat: ( $o > nat > $o ) > ( $o > nat > $o ) > ( $o > $o ) > ( nat > nat ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001_Eo_001t__Nat__Onat_001_Eo_001t__Num__Onum,type,
bNF_re766437678722298023_o_num: ( $o > nat > $o ) > ( $o > num > $o ) > ( $o > $o ) > ( nat > num ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
bNF_re1181178334029315097nteger: ( code_integer > code_integer > $o ) > ( ( code_integer > code_integer ) > ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > $o ) > ( code_integer > code_integer > code_integer ) > ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Num__Onum_001t__Num__Onum,type,
bNF_re40212662316476751um_num: ( code_integer > code_integer > $o ) > ( num > num > $o ) > ( code_integer > num ) > ( code_integer > num ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Code____Numeral__Ointeger_001t__Int__Oint_001t__Num__Onum_001t__Num__Onum,type,
bNF_re8313794964846168862um_num: ( code_integer > int > $o ) > ( num > num > $o ) > ( code_integer > num ) > ( int > num ) > $o ).
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thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Int__Oint_M_Eo_J_J,type,
sup_su5231854730092680051_int_o: ( product_prod_nat_nat > int > $o ) > ( product_prod_nat_nat > int > $o ) > product_prod_nat_nat > int > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J,type,
sup_su362511073950362882_nat_o: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > product_prod_nat_nat > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_M_062_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J_J,type,
sup_su7519161239522478338_nat_o: ( set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ) > ( set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ) > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_M_062_It__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_M_Eo_J_J,type,
sup_su6164525874564267778_nat_o: ( set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o ) > ( set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o ) > set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Int__Oint,type,
sup_sup_int: int > int > int ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
sup_sup_nat: nat > nat > nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Product____Type__Ounit,type,
sup_sup_Product_unit: product_unit > product_unit > product_unit ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Rat__Orat,type,
sup_sup_rat: rat > rat > rat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_I_Eo_J,type,
sup_sup_set_o: set_o > set_o > set_o ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Int__Oint_J,type,
sup_sup_set_int: set_int > set_int > set_int ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
sup_sup_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
sup_su5769328420594410459od_o_o: set_Product_prod_o_o > set_Product_prod_o_o > set_Product_prod_o_o ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J_J,type,
sup_su8679140417860611193_o_nat: set_Pr2101469702781467981_o_nat > set_Pr2101469702781467981_o_nat > set_Pr2101469702781467981_o_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_I_Eo_Mt__Product____Type__Ounit_J_J,type,
sup_su4058753791541417424t_unit: set_Pr6671815714118170108t_unit > set_Pr6671815714118170108t_unit > set_Pr6671815714118170108t_unit ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
sup_su6024340866399070445nt_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_M_Eo_J_J,type,
sup_su503371503184139039_nat_o: set_Pr3149072824959771635_nat_o > set_Pr3149072824959771635_nat_o > set_Pr3149072824959771635_nat_o ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
sup_su6327502436637775413at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J_J,type,
sup_su2042722026077122175at_num: set_Pr6200539531224447659at_num > set_Pr6200539531224447659at_num > set_Pr6200539531224447659at_num ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Ounit_J_J,type,
sup_su4209953369189666580t_unit: set_Pr4334478416066269672t_unit > set_Pr4334478416066269672t_unit > set_Pr4334478416066269672t_unit ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
sup_su718114333110466843at_nat: set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Ounit_M_Eo_J_J,type,
sup_su2851927382625014182unit_o: set_Pr5464989305201766866unit_o > set_Pr5464989305201766866unit_o > set_Pr5464989305201766866unit_o ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Nat__Onat_J_J,type,
sup_su1639320892072265582it_nat: set_Pr1763845938948868674it_nat > set_Pr1763845938948868674it_nat > set_Pr1763845938948868674it_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Ounit_J_J,type,
sup_su6481247232241170203t_unit: set_Pr5094982260447487303t_unit > set_Pr5094982260447487303t_unit > set_Pr5094982260447487303t_unit ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
sup_su5525570899277871387at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_J_J,type,
sup_su5395584912770621083at_nat: set_Pr7459493094073627847at_nat > set_Pr7459493094073627847at_nat > set_Pr7459493094073627847at_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Ounit_J,type,
sup_su793286257634532545t_unit: set_Product_unit > set_Product_unit > set_Product_unit ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Rat__Orat_J,type,
sup_sup_set_rat: set_rat > set_rat > set_rat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
sup_sup_set_set_o: set_set_o > set_set_o > set_set_o ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
sup_sup_set_set_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
sup_su3642409539654194069at_nat: set_se7855581050983116737at_nat > set_se7855581050983116737at_nat > set_se7855581050983116737at_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_J,type,
sup_su474284620922681339at_nat: set_se5479201105039950375at_nat > set_se5479201105039950375at_nat > set_se5479201105039950375at_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Product____Type__Ounit_J_J,type,
sup_su5975557051335651745t_unit: set_set_Product_unit > set_set_Product_unit > set_set_Product_unit ).
thf(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Nat__Onat,type,
lattic8265883725875713057ax_nat: set_nat > nat ).
thf(sy_c_Lifting_OQuotient_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Code____Numeral__Ointeger,type,
quotie9062115205763428814nteger: ( ( code_integer > code_integer ) > ( code_integer > code_integer ) > $o ) > ( ( code_integer > code_integer ) > code_integer ) > ( code_integer > code_integer > code_integer ) > ( ( code_integer > code_integer ) > code_integer > $o ) > $o ).
thf(sy_c_Lifting_OQuotient_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
quotie8207154290130328982nteger: ( ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > $o ) > ( ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > code_integer > code_integer ) > ( ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer ) > $o ) > $o ).
thf(sy_c_Lifting_OQuotient_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Code____Numeral__Ointeger,type,
quotie8803226531959296410nteger: ( ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > $o ) > ( ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > code_integer ) > ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > code_integer > $o ) > $o ).
thf(sy_c_Lifting_OQuotient_001_Eo_001_Eo,type,
quotient_o_o: ( $o > $o > $o ) > ( $o > $o ) > ( $o > $o ) > ( $o > $o > $o ) > $o ).
thf(sy_c_Lifting_OQuotient_001_Eo_001t__Nat__Onat,type,
quotient_o_nat: ( $o > $o > $o ) > ( $o > nat ) > ( nat > $o ) > ( $o > nat > $o ) > $o ).
thf(sy_c_Lifting_OQuotient_001t__Int__Oint_001t__Code____Numeral__Ointeger,type,
quotie2291698863146621601nteger: ( int > int > $o ) > ( int > code_integer ) > ( code_integer > int ) > ( int > code_integer > $o ) > $o ).
thf(sy_c_Lifting_OQuotient_001t__Int__Oint_001t__Int__Oint,type,
quotient_int_int: ( int > int > $o ) > ( int > int ) > ( int > int ) > ( int > int > $o ) > $o ).
thf(sy_c_Lifting_OQuotient_001t__Int__Oint_001t__Nat__Onat,type,
quotient_int_nat: ( int > int > $o ) > ( int > nat ) > ( nat > int ) > ( int > nat > $o ) > $o ).
thf(sy_c_Lifting_OQuotient_001t__Int__Oint_001t__Num__Onum,type,
quotient_int_num: ( int > int > $o ) > ( int > num ) > ( num > int ) > ( int > num > $o ) > $o ).
thf(sy_c_Lifting_OQuotient_001t__Nat__Onat_001_Eo,type,
quotient_nat_o: ( nat > nat > $o ) > ( nat > $o ) > ( $o > nat ) > ( nat > $o > $o ) > $o ).
thf(sy_c_Lifting_OQuotient_001t__Nat__Onat_001t__Int__Oint,type,
quotient_nat_int: ( nat > nat > $o ) > ( nat > int ) > ( int > nat ) > ( nat > int > $o ) > $o ).
thf(sy_c_Lifting_OQuotient_001t__Nat__Onat_001t__Nat__Onat,type,
quotient_nat_nat: ( nat > nat > $o ) > ( nat > nat ) > ( nat > nat ) > ( nat > nat > $o ) > $o ).
thf(sy_c_Lifting_OQuotient_001t__Nat__Onat_001t__Num__Onum,type,
quotient_nat_num: ( nat > nat > $o ) > ( nat > num ) > ( num > nat ) > ( nat > num > $o ) > $o ).
thf(sy_c_Lifting_OQuotient_001t__Nat__Onat_001t__Rat__Orat,type,
quotient_nat_rat: ( nat > nat > $o ) > ( nat > rat ) > ( rat > nat ) > ( nat > rat > $o ) > $o ).
thf(sy_c_Lifting_OQuotient_001t__Nat__Onat_001t__String__Ochar,type,
quotient_nat_char: ( nat > nat > $o ) > ( nat > char ) > ( char > nat ) > ( nat > char > $o ) > $o ).
thf(sy_c_Lifting_OQuotient_001t__Num__Onum_001t__Int__Oint,type,
quotient_num_int: ( num > num > $o ) > ( num > int ) > ( int > num ) > ( num > int > $o ) > $o ).
thf(sy_c_Lifting_OQuotient_001t__Num__Onum_001t__Nat__Onat,type,
quotient_num_nat: ( num > num > $o ) > ( num > nat ) > ( nat > num ) > ( num > nat > $o ) > $o ).
thf(sy_c_Lifting_OQuotient_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
quotie1194848508323700631at_int: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > int ) > ( int > product_prod_nat_nat ) > ( product_prod_nat_nat > int > $o ) > $o ).
thf(sy_c_Lifting_OQuotient_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
quotie4166942682936818134at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > $o ).
thf(sy_c_Lifting_OQuotient_001t__Rat__Orat_001t__Int__Oint,type,
quotient_rat_int: ( rat > rat > $o ) > ( rat > int ) > ( int > rat ) > ( rat > int > $o ) > $o ).
thf(sy_c_Lifting_OQuotient_001t__Rat__Orat_001t__Nat__Onat,type,
quotient_rat_nat: ( rat > rat > $o ) > ( rat > nat ) > ( nat > rat ) > ( rat > nat > $o ) > $o ).
thf(sy_c_Lifting_OQuotient_001t__Rat__Orat_001t__Rat__Orat,type,
quotient_rat_rat: ( rat > rat > $o ) > ( rat > rat ) > ( rat > rat ) > ( rat > rat > $o ) > $o ).
thf(sy_c_Lifting_OQuotient_001t__Rat__Orat_001t__String__Ochar,type,
quotient_rat_char: ( rat > rat > $o ) > ( rat > char ) > ( char > rat ) > ( rat > char > $o ) > $o ).
thf(sy_c_Lifting_OQuotient_001t__String__Ochar_001t__Int__Oint,type,
quotient_char_int: ( char > char > $o ) > ( char > int ) > ( int > char ) > ( char > int > $o ) > $o ).
thf(sy_c_Lifting_OQuotient_001t__String__Ochar_001t__Nat__Onat,type,
quotient_char_nat: ( char > char > $o ) > ( char > nat ) > ( nat > char ) > ( char > nat > $o ) > $o ).
thf(sy_c_Lifting_OQuotient_001t__String__Ochar_001t__Rat__Orat,type,
quotient_char_rat: ( char > char > $o ) > ( char > rat ) > ( rat > char ) > ( char > rat > $o ) > $o ).
thf(sy_c_Lifting_OQuotient_001t__String__Ochar_001t__String__Ochar,type,
quotient_char_char: ( char > char > $o ) > ( char > char ) > ( char > char ) > ( char > char > $o ) > $o ).
thf(sy_c_List_Oall__interval__nat,type,
all_interval_nat: ( nat > $o ) > nat > nat > $o ).
thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
cons_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
nil_nat: list_nat ).
thf(sy_c_List_Olist__ex_001t__Nat__Onat,type,
list_ex_nat: ( nat > $o ) > list_nat > $o ).
thf(sy_c_List_Oupt,type,
upt: nat > nat > list_nat ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Onat_Ocase__nat_001t__Option__Ooption_It__Num__Onum_J,type,
case_nat_option_num: option_num > ( nat > option_num ) > nat > option_num ).
thf(sy_c_Nat_Osemiring__1__class_ONats_001t__Int__Oint,type,
semiring_1_Nats_int: set_int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat__Bijection_Oint__decode,type,
nat_int_decode: nat > int ).
thf(sy_c_Nat__Bijection_Oint__encode,type,
nat_int_encode: int > nat ).
thf(sy_c_Nat__Bijection_Olist__decode,type,
nat_list_decode: nat > list_nat ).
thf(sy_c_Nat__Bijection_Olist__decode__rel,type,
nat_list_decode_rel: nat > nat > $o ).
thf(sy_c_Nat__Bijection_Olist__encode,type,
nat_list_encode: list_nat > nat ).
thf(sy_c_Nat__Bijection_Oprod__decode,type,
nat_prod_decode: nat > product_prod_nat_nat ).
thf(sy_c_Nat__Bijection_Oprod__encode,type,
nat_prod_encode: product_prod_nat_nat > nat ).
thf(sy_c_Nat__Bijection_Osum__decode,type,
nat_sum_decode: nat > sum_sum_nat_nat ).
thf(sy_c_Nat__Bijection_Osum__encode,type,
nat_sum_encode: sum_sum_nat_nat > nat ).
thf(sy_c_Nat__Bijection_Otriangle,type,
nat_triangle: nat > nat ).
thf(sy_c_Num_Onum_OBit0,type,
bit0: num > num ).
thf(sy_c_Num_Onum_OBit1,type,
bit1: num > num ).
thf(sy_c_Num_Onum_OOne,type,
one: num ).
thf(sy_c_Num_Onum_Ocase__num_001t__Option__Ooption_It__Num__Onum_J,type,
case_num_option_num: option_num > ( num > option_num ) > ( num > option_num ) > num > option_num ).
thf(sy_c_Num_Onum__of__nat,type,
num_of_nat: nat > num ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Code____Numeral__Ointeger,type,
numera6620942414471956472nteger: num > code_integer ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
numeral_numeral_int: num > int ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
numeral_numeral_nat: num > nat ).
thf(sy_c_Option_Ooption_ONone_001t__Num__Onum,type,
none_num: option_num ).
thf(sy_c_Option_Ooption_OSome_001t__Num__Onum,type,
some_num: num > option_num ).
thf(sy_c_Option_Ooption_Ocase__option_001t__Num__Onum_001t__Num__Onum,type,
case_option_num_num: num > ( num > num ) > option_num > num ).
thf(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_It__Num__Onum_J_001t__Num__Onum,type,
case_o6005452278849405969um_num: option_num > ( num > option_num ) > option_num > option_num ).
thf(sy_c_Order__Relation_OaboveS_001t__Int__Oint,type,
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product_swap_nat_nat: product_prod_nat_nat > product_prod_nat_nat ).
thf(sy_c_Product__Type_Oprod_Oswap_001t__Nat__Onat_001t__Num__Onum,type,
product_swap_nat_num: product_prod_nat_num > product_prod_num_nat ).
thf(sy_c_Product__Type_Oprod_Oswap_001t__Num__Onum_001t__Nat__Onat,type,
product_swap_num_nat: product_prod_num_nat > product_prod_nat_num ).
thf(sy_c_Product__Type_Oprod_Oswap_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
produc3946833401584108663at_nat: produc3843707927480180839at_nat > produc3843707927480180839at_nat ).
thf(sy_c_Product__Type_Oprod_Oswap_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
produc5396952528824381687at_nat: produc1319942482725812455at_nat > produc1319942482725812455at_nat ).
thf(sy_c_Product__Type_Oproduct_001t__Nat__Onat_001t__Nat__Onat,type,
produc929334515565554804at_nat: set_nat > set_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_Product__Type_Oproduct_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
produc3241716407206042770at_nat: set_se7855581050983116737at_nat > set_se7855581050983116737at_nat > set_Pr4329608150637261639at_nat ).
thf(sy_c_Product__Type_Ounit_OAbs__unit,type,
product_Abs_unit: $o > product_unit ).
thf(sy_c_Product__Type_Ounit_ORep__unit,type,
product_Rep_unit: product_unit > $o ).
thf(sy_c_Random__Pred_Onot__randompred,type,
random6974930770145893639ompred: ( produc7822875418678951345atural > produc4675096598859438275atural ) > produc7822875418678951345atural > produc4675096598859438275atural ).
thf(sy_c_Rat_OFract,type,
fract: int > int > rat ).
thf(sy_c_Rat_Ofield__char__0__class_ORats_001t__Rat__Orat,type,
field_6020823756834552118ts_rat: set_rat ).
thf(sy_c_Rat_Onormalize,type,
normalize: product_prod_int_int > product_prod_int_int ).
thf(sy_c_Rat_Oquotient__of,type,
quotient_of: rat > product_prod_int_int ).
thf(sy_c_Ref_Oalloc_001tf__a,type,
alloc_a: a > heap_e7401611519738050253t_unit > produc7812382424243267045t_unit ).
thf(sy_c_Ref_Ochange_001t__Heap__Oref_Itf__a_J,type,
change_ref_a: ( ref_a > ref_a ) > ref_ref_a > heap_Heap_ref_a ).
thf(sy_c_Ref_Ochange_001tf__a,type,
change_a: ( a > a ) > ref_a > heap_Heap_a ).
thf(sy_c_Ref_Oget_001t__Heap__Oref_Itf__a_J,type,
get_ref_a: heap_e7401611519738050253t_unit > ref_ref_a > ref_a ).
thf(sy_c_Ref_Oget_001tf__a,type,
get_a: heap_e7401611519738050253t_unit > ref_a > a ).
thf(sy_c_Ref_Olookup_001t__Heap__Oref_Itf__a_J,type,
lookup_ref_a: ref_ref_a > heap_Heap_ref_a ).
thf(sy_c_Ref_Olookup_001tf__a,type,
lookup_a: ref_a > heap_Heap_a ).
thf(sy_c_Ref_Onoteq_001tf__a_001tf__a,type,
noteq_a_a: ref_a > ref_a > $o ).
thf(sy_c_Ref_Opresent_001tf__a,type,
present_a: heap_e7401611519738050253t_unit > ref_a > $o ).
thf(sy_c_Ref_Oref_001tf__a,type,
ref_a2: a > heap_Heap_ref_a ).
thf(sy_c_Ref_Oref_H_001tf__a,type,
ref_a3: a > heap_Heap_ref_a ).
thf(sy_c_Ref_Oset_001t__Heap__Oref_Itf__a_J,type,
set_ref_a2: ref_ref_a > ref_a > heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit ).
thf(sy_c_Ref_Oset_001tf__a,type,
set_a: ref_a > a > heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit ).
thf(sy_c_Ref_Oupdate_001tf__a,type,
update_a: ref_a > a > heap_H6156071679106078628t_unit ).
thf(sy_c_Relation_OField_001_Eo,type,
field_o: set_Product_prod_o_o > set_o ).
thf(sy_c_Relation_OField_001t__Int__Oint,type,
field_int: set_Pr958786334691620121nt_int > set_int ).
thf(sy_c_Relation_OField_001t__Nat__Onat,type,
field_nat: set_Pr1261947904930325089at_nat > set_nat ).
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field_3044959742847878580at_nat: set_Pr553994874890374343at_nat > set_Pr8693737435421807431at_nat ).
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field_2862844103579723188at_nat: set_Pr5564308138774400199at_nat > set_Pr4329608150637261639at_nat ).
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field_3145993188331922222at_nat: set_Pr4329608150637261639at_nat > set_se7855581050983116737at_nat ).
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field_1428494348145443988at_nat: set_Pr7459493094073627847at_nat > set_se5479201105039950375at_nat ).
thf(sy_c_Relation_OId_001_Eo,type,
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thf(sy_c_Relation_OId_001t__Int__Oint,type,
id_int2: set_Pr958786334691620121nt_int ).
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id_set4888139568991089827at_nat: set_Pr7459493094073627847at_nat ).
thf(sy_c_Relation_OId__on_001_Eo,type,
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id_on_int: set_int > set_Pr958786334691620121nt_int ).
thf(sy_c_Relation_OId__on_001t__Nat__Onat,type,
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thf(sy_c_Relation_OId__on_001t__Num__Onum,type,
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thf(sy_c_Relation_Oconversep_001_Eo_001_Eo,type,
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thf(sy_c_Relation_Oconversep_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_Relation_Oconversep_001t__Nat__Onat_001_Eo,type,
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thf(sy_c_Relation_Oconversep_001t__Nat__Onat_001t__Int__Oint,type,
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thf(sy_c_member_001_062_It__String__Ochar_Mt__Nat__Onat_J,type,
member_char_nat: ( char > nat ) > set_char_nat > $o ).
thf(sy_c_member_001_062_It__String__Ochar_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
member2992454361780672008nt_int: ( char > product_prod_int_int ) > set_ch6383161263842509759nt_int > $o ).
thf(sy_c_member_001_062_It__String__Ochar_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member3295615932019376976at_nat: ( char > product_prod_nat_nat ) > set_ch7978006088466058503at_nat > $o ).
thf(sy_c_member_001_062_It__String__Ochar_Mt__Product____Type__Ounit_J,type,
member379592906918970460t_unit: ( char > product_unit ) > set_ch7814610555394875923t_unit > $o ).
thf(sy_c_member_001_Eo,type,
member_o: $o > set_o > $o ).
thf(sy_c_member_001t__Code____Numeral__Ointeger,type,
member_Code_integer: code_integer > set_Code_integer > $o ).
thf(sy_c_member_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J,type,
member242389114213480694t_unit: heap_e7401611519738050253t_unit > set_he8644174446132254637t_unit > $o ).
thf(sy_c_member_001t__Heap__Oref_Itf__a_J,type,
member_ref_a: ref_a > set_ref_a > $o ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Num__Onum,type,
member_num: num > set_num > $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_I_Eo_Mt__Int__Oint_J,type,
member7847949116333733898_o_int: product_prod_o_int > set_Pr8834758594704517033_o_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J,type,
member2802428098988154798_o_nat: product_prod_o_nat > set_Pr2101469702781467981_o_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Heap__Oref_Itf__a_J_Mt__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_J,type,
member7849944260330398460t_unit: produc7812382424243267045t_unit > set_Pr1133966238863718811t_unit > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_M_Eo_J,type,
member4489920277610959864_int_o: product_prod_int_o > set_Pr903927857289325719_int_o > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Code____Numeral__Ointeger_J,type,
member4345257455949424659nteger: produc2963269191615333738nteger > set_Pr8415741480598773322nteger > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
member5262025264175285858nt_int: product_prod_int_int > set_Pr958786334691620121nt_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Nat__Onat_J,type,
member216504246829706758nt_nat: product_prod_int_nat > set_Pr3448869479623346877nt_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_M_Eo_J,type,
member6310962623043647828_nat_o: product_prod_nat_o > set_Pr3149072824959771635_nat_o > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J,type,
member4262671552274231302at_int: product_prod_nat_int > set_Pr7995236796853374141at_int > $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__Num__Onum_J,type,
member9148766508732265716at_num: product_prod_nat_num > set_Pr6200539531224447659at_num > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Num__Onum_Mt__Nat__Onat_J,type,
member6570852975090897396um_nat: product_prod_num_nat > set_Pr3280342998896498603um_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
member8566619992076573584nt_int: produc1219242969750017639nt_int > set_Pr2560585780119916871nt_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
member2200362271989506215_nat_o: produc652263959640188542_nat_o > set_Pr6558897299634722526_nat_o > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Int__Oint_J,type,
member8394280151737582451at_int: produc4196048018000912732at_int > set_Pr49084382684228242at_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member8206827879206165904at_nat: produc859450856879609959at_nat > set_Pr8693737435421807431at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
member6009668390874684176at_nat: produc1906917888121415911at_nat > set_Pr5564308138774400199at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Ounit_J,type,
member7821138191923116944t_unit: produc8459935480633519975t_unit > set_Pr5094982260447487303t_unit > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
member8757157785044589968at_nat: produc3843707927480180839at_nat > set_Pr4329608150637261639at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_J,type,
member1466754251312161552at_nat: produc1319942482725812455at_nat > set_Pr7459493094073627847at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Ounit,type,
member_Product_unit: product_unit > set_Product_unit > $o ).
thf(sy_c_member_001t__Rat__Orat,type,
member_rat: rat > set_rat > $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_001t__Set__Oset_It__Int__Oint_J,type,
member_set_int: set_int > set_set_int > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member2643936169264416010at_nat: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
member6273874620674182256at_nat: set_Pr4329608150637261639at_nat > set_se5479201105039950375at_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Product____Type__Ounit_J,type,
member5877623283571906838t_unit: set_Product_unit > set_set_Product_unit > $o ).
thf(sy_c_member_001t__String__Ochar,type,
member_char: char > set_char > $o ).
thf(sy_v_h,type,
h: heap_e7401611519738050253t_unit ).
thf(sy_v_h_H,type,
h2: heap_e7401611519738050253t_unit ).
thf(sy_v_r,type,
r: ref_a ).
thf(sy_v_thesis,type,
thesis: $o ).
thf(sy_v_x,type,
x: a ).
% Relevant facts (4289)
thf(fact_0_that,axiom,
( ( r
= ( produc473321116828791851t_unit @ ( alloc_a @ x @ h ) ) )
=> ( ( h2
= ( produc1126150082675451753t_unit @ ( alloc_a @ x @ h ) ) )
=> thesis ) ) ).
% that
thf(fact_1_assms,axiom,
heap_effect_ref_a @ ( ref_a2 @ x ) @ h @ h2 @ r ).
% assms
thf(fact_2_prod_Oexpand,axiom,
! [Prod: produc7812382424243267045t_unit,Prod2: produc7812382424243267045t_unit] :
( ( ( ( produc473321116828791851t_unit @ Prod )
= ( produc473321116828791851t_unit @ Prod2 ) )
& ( ( produc1126150082675451753t_unit @ Prod )
= ( produc1126150082675451753t_unit @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_3_prod__eqI,axiom,
! [P: produc7812382424243267045t_unit,Q: produc7812382424243267045t_unit] :
( ( ( produc473321116828791851t_unit @ P )
= ( produc473321116828791851t_unit @ Q ) )
=> ( ( ( produc1126150082675451753t_unit @ P )
= ( produc1126150082675451753t_unit @ Q ) )
=> ( P = Q ) ) ) ).
% prod_eqI
thf(fact_4_exE__realizer_H,axiom,
! [P2: heap_e7401611519738050253t_unit > ref_a > $o,P: produc7812382424243267045t_unit] :
( ( P2 @ ( produc1126150082675451753t_unit @ P ) @ ( produc473321116828791851t_unit @ P ) )
=> ~ ! [X: ref_a,Y: heap_e7401611519738050253t_unit] :
~ ( P2 @ Y @ X ) ) ).
% exE_realizer'
thf(fact_5_prod__eq__iff,axiom,
( ( ^ [Y2: produc7812382424243267045t_unit,Z: produc7812382424243267045t_unit] : Y2 = Z )
= ( ^ [S: produc7812382424243267045t_unit,T: produc7812382424243267045t_unit] :
( ( ( produc473321116828791851t_unit @ S )
= ( produc473321116828791851t_unit @ T ) )
& ( ( produc1126150082675451753t_unit @ S )
= ( produc1126150082675451753t_unit @ T ) ) ) ) ) ).
% prod_eq_iff
thf(fact_6_Ref_Oget__alloc,axiom,
! [X2: a,H: heap_e7401611519738050253t_unit,X3: a] :
( ( get_a @ ( produc1126150082675451753t_unit @ ( alloc_a @ X2 @ H ) ) @ ( produc473321116828791851t_unit @ ( alloc_a @ X3 @ H ) ) )
= X2 ) ).
% Ref.get_alloc
thf(fact_7_set__alloc,axiom,
! [V: a,H: heap_e7401611519738050253t_unit,V2: a] :
( ( set_a @ ( produc473321116828791851t_unit @ ( alloc_a @ V @ H ) ) @ V2 @ ( produc1126150082675451753t_unit @ ( alloc_a @ V @ H ) ) )
= ( produc1126150082675451753t_unit @ ( alloc_a @ V2 @ H ) ) ) ).
% set_alloc
thf(fact_8_Ref_Opresent__alloc,axiom,
! [H: heap_e7401611519738050253t_unit,R: ref_a,V: a] :
( ( present_a @ H @ R )
=> ( present_a @ ( produc1126150082675451753t_unit @ ( alloc_a @ V @ H ) ) @ R ) ) ).
% Ref.present_alloc
thf(fact_9_Ref_Onot__present__alloc,axiom,
! [H: heap_e7401611519738050253t_unit,V: a] :
~ ( present_a @ H @ ( produc473321116828791851t_unit @ ( alloc_a @ V @ H ) ) ) ).
% Ref.not_present_alloc
thf(fact_10_Ref_Oset__same,axiom,
! [R: ref_a,X2: a,Y3: a,H: heap_e7401611519738050253t_unit] :
( ( set_a @ R @ X2 @ ( set_a @ R @ Y3 @ H ) )
= ( set_a @ R @ X2 @ H ) ) ).
% Ref.set_same
thf(fact_11_Ref_Oget__set__eq,axiom,
! [R: ref_a,X2: a,H: heap_e7401611519738050253t_unit] :
( ( get_a @ ( set_a @ R @ X2 @ H ) @ R )
= X2 ) ).
% Ref.get_set_eq
thf(fact_12_present__set,axiom,
! [R: ref_a,V: a,H: heap_e7401611519738050253t_unit] :
( ( present_a @ ( set_a @ R @ V @ H ) )
= ( present_a @ H ) ) ).
% present_set
thf(fact_13_effect__refE,axiom,
! [V: a,H: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,R: ref_a] :
( ( heap_effect_ref_a @ ( ref_a2 @ V ) @ H @ H2 @ R )
=> ~ ( ( ( get_a @ H2 @ R )
= V )
=> ( ( present_a @ H2 @ R )
=> ( present_a @ H @ R ) ) ) ) ).
% effect_refE
thf(fact_14_alloc__set,axiom,
! [X2: a,R: ref_a,X3: a,H: heap_e7401611519738050253t_unit] :
( ( produc473321116828791851t_unit @ ( alloc_a @ X2 @ ( set_a @ R @ X3 @ H ) ) )
= ( produc473321116828791851t_unit @ ( alloc_a @ X2 @ H ) ) ) ).
% alloc_set
thf(fact_15_ref_H__def,axiom,
ref_a3 = ref_a2 ).
% ref'_def
thf(fact_16_get__alloc__neq,axiom,
! [R: ref_a,V: a,H: heap_e7401611519738050253t_unit] :
( ( noteq_a_a @ R @ ( produc473321116828791851t_unit @ ( alloc_a @ V @ H ) ) )
=> ( ( get_a @ ( produc1126150082675451753t_unit @ ( alloc_a @ V @ H ) ) @ R )
= ( get_a @ H @ R ) ) ) ).
% get_alloc_neq
thf(fact_17_effect__changeE,axiom,
! [F: a > a,R2: ref_a,H: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,R: a] :
( ( heap_effect_a @ ( change_a @ F @ R2 ) @ H @ H2 @ R )
=> ~ ( ( H2
= ( set_a @ R2 @ ( F @ ( get_a @ H @ R2 ) ) @ H ) )
=> ( R
!= ( F @ ( get_a @ H @ R2 ) ) ) ) ) ).
% effect_changeE
thf(fact_18_effect__changeE,axiom,
! [F: ref_a > ref_a,R2: ref_ref_a,H: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,R: ref_a] :
( ( heap_effect_ref_a @ ( change_ref_a @ F @ R2 ) @ H @ H2 @ R )
=> ~ ( ( H2
= ( set_ref_a2 @ R2 @ ( F @ ( get_ref_a @ H @ R2 ) ) @ H ) )
=> ( R
!= ( F @ ( get_ref_a @ H @ R2 ) ) ) ) ) ).
% effect_changeE
thf(fact_19_effect__changeI,axiom,
! [H2: heap_e7401611519738050253t_unit,R: ref_a,F: a > a,H: heap_e7401611519738050253t_unit,X2: a] :
( ( H2
= ( set_a @ R @ ( F @ ( get_a @ H @ R ) ) @ H ) )
=> ( ( X2
= ( F @ ( get_a @ H @ R ) ) )
=> ( heap_effect_a @ ( change_a @ F @ R ) @ H @ H2 @ X2 ) ) ) ).
% effect_changeI
thf(fact_20_effect__changeI,axiom,
! [H2: heap_e7401611519738050253t_unit,R: ref_ref_a,F: ref_a > ref_a,H: heap_e7401611519738050253t_unit,X2: ref_a] :
( ( H2
= ( set_ref_a2 @ R @ ( F @ ( get_ref_a @ H @ R ) ) @ H ) )
=> ( ( X2
= ( F @ ( get_ref_a @ H @ R ) ) )
=> ( heap_effect_ref_a @ ( change_ref_a @ F @ R ) @ H @ H2 @ X2 ) ) ) ).
% effect_changeI
thf(fact_21_present__alloc__neq,axiom,
! [H: heap_e7401611519738050253t_unit,R: ref_a,V: a] :
( ( present_a @ H @ R )
=> ( noteq_a_a @ R @ ( produc473321116828791851t_unit @ ( alloc_a @ V @ H ) ) ) ) ).
% present_alloc_neq
thf(fact_22_Ref_Oget__set__neq,axiom,
! [R: ref_a,S2: ref_a,X2: a,H: heap_e7401611519738050253t_unit] :
( ( noteq_a_a @ R @ S2 )
=> ( ( get_a @ ( set_a @ S2 @ X2 @ H ) @ R )
= ( get_a @ H @ R ) ) ) ).
% Ref.get_set_neq
thf(fact_23_ref__def,axiom,
( ref_a2
= ( ^ [V3: a] : ( heap_heap_ref_a @ ( alloc_a @ V3 ) ) ) ) ).
% ref_def
thf(fact_24_effect__lookupE,axiom,
! [R: ref_a,H: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,X2: a] :
( ( heap_effect_a @ ( lookup_a @ R ) @ H @ H2 @ X2 )
=> ~ ( ( H2 = H )
=> ( X2
!= ( get_a @ H @ R ) ) ) ) ).
% effect_lookupE
thf(fact_25_effect__lookupE,axiom,
! [R: ref_ref_a,H: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,X2: ref_a] :
( ( heap_effect_ref_a @ ( lookup_ref_a @ R ) @ H @ H2 @ X2 )
=> ~ ( ( H2 = H )
=> ( X2
!= ( get_ref_a @ H @ R ) ) ) ) ).
% effect_lookupE
thf(fact_26_effect__lookupI,axiom,
! [H2: heap_e7401611519738050253t_unit,H: heap_e7401611519738050253t_unit,X2: a,R: ref_a] :
( ( H2 = H )
=> ( ( X2
= ( get_a @ H @ R ) )
=> ( heap_effect_a @ ( lookup_a @ R ) @ H @ H2 @ X2 ) ) ) ).
% effect_lookupI
thf(fact_27_effect__lookupI,axiom,
! [H2: heap_e7401611519738050253t_unit,H: heap_e7401611519738050253t_unit,X2: ref_a,R: ref_ref_a] :
( ( H2 = H )
=> ( ( X2
= ( get_ref_a @ H @ R ) )
=> ( heap_effect_ref_a @ ( lookup_ref_a @ R ) @ H @ H2 @ X2 ) ) ) ).
% effect_lookupI
thf(fact_28_effect__refI,axiom,
! [R: ref_a,H2: heap_e7401611519738050253t_unit,V: a,H: heap_e7401611519738050253t_unit] :
( ( ( produc6544324848051098391t_unit @ R @ H2 )
= ( alloc_a @ V @ H ) )
=> ( heap_effect_ref_a @ ( ref_a2 @ V ) @ H @ H2 @ R ) ) ).
% effect_refI
thf(fact_29_snd__swap,axiom,
! [X2: produc7812382424243267045t_unit] :
( ( produc6369869760173755749_ref_a @ ( produc970745769005243767t_unit @ X2 ) )
= ( produc473321116828791851t_unit @ X2 ) ) ).
% snd_swap
thf(fact_30_snd__swap,axiom,
! [X2: produc1799199577659733785_ref_a] :
( ( produc1126150082675451753t_unit @ ( produc6214465446503547763_ref_a @ X2 ) )
= ( produc5717040794327095847_ref_a @ X2 ) ) ).
% snd_swap
thf(fact_31_fst__swap,axiom,
! [X2: produc1799199577659733785_ref_a] :
( ( produc473321116828791851t_unit @ ( produc6214465446503547763_ref_a @ X2 ) )
= ( produc6369869760173755749_ref_a @ X2 ) ) ).
% fst_swap
thf(fact_32_fst__swap,axiom,
! [X2: produc7812382424243267045t_unit] :
( ( produc5717040794327095847_ref_a @ ( produc970745769005243767t_unit @ X2 ) )
= ( produc1126150082675451753t_unit @ X2 ) ) ).
% fst_swap
thf(fact_33_prod_Oinject,axiom,
! [X1: set_Pr4329608150637261639at_nat,X22: set_Pr4329608150637261639at_nat,Y1: set_Pr4329608150637261639at_nat,Y22: set_Pr4329608150637261639at_nat] :
( ( ( produc9060074326276436823at_nat @ X1 @ X22 )
= ( produc9060074326276436823at_nat @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_34_prod_Oinject,axiom,
! [X1: set_Pr1261947904930325089at_nat,X22: set_Pr1261947904930325089at_nat,Y1: set_Pr1261947904930325089at_nat,Y22: set_Pr1261947904930325089at_nat] :
( ( ( produc2922128104949294807at_nat @ X1 @ X22 )
= ( produc2922128104949294807at_nat @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_35_prod_Oinject,axiom,
! [X1: nat,X22: num,Y1: nat,Y22: num] :
( ( ( product_Pair_nat_num @ X1 @ X22 )
= ( product_Pair_nat_num @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_36_prod_Oinject,axiom,
! [X1: nat,X22: nat,Y1: nat,Y22: nat] :
( ( ( product_Pair_nat_nat @ X1 @ X22 )
= ( product_Pair_nat_nat @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_37_prod_Oinject,axiom,
! [X1: int,X22: int,Y1: int,Y22: int] :
( ( ( product_Pair_int_int @ X1 @ X22 )
= ( product_Pair_int_int @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_38_old_Oprod_Oinject,axiom,
! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
( ( ( produc9060074326276436823at_nat @ A @ B )
= ( produc9060074326276436823at_nat @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_39_old_Oprod_Oinject,axiom,
! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( ( ( produc2922128104949294807at_nat @ A @ B )
= ( produc2922128104949294807at_nat @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_40_old_Oprod_Oinject,axiom,
! [A: nat,B: num,A2: nat,B2: num] :
( ( ( product_Pair_nat_num @ A @ B )
= ( product_Pair_nat_num @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_41_old_Oprod_Oinject,axiom,
! [A: nat,B: nat,A2: nat,B2: nat] :
( ( ( product_Pair_nat_nat @ A @ B )
= ( product_Pair_nat_nat @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_42_old_Oprod_Oinject,axiom,
! [A: int,B: int,A2: int,B2: int] :
( ( ( product_Pair_int_int @ A @ B )
= ( product_Pair_int_int @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_43_swap__simp,axiom,
! [X2: num,Y3: nat] :
( ( product_swap_num_nat @ ( product_Pair_num_nat @ X2 @ Y3 ) )
= ( product_Pair_nat_num @ Y3 @ X2 ) ) ).
% swap_simp
thf(fact_44_swap__simp,axiom,
! [X2: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] :
( ( produc5396952528824381687at_nat @ ( produc9060074326276436823at_nat @ X2 @ Y3 ) )
= ( produc9060074326276436823at_nat @ Y3 @ X2 ) ) ).
% swap_simp
thf(fact_45_swap__simp,axiom,
! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
( ( produc3946833401584108663at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y3 ) )
= ( produc2922128104949294807at_nat @ Y3 @ X2 ) ) ).
% swap_simp
thf(fact_46_swap__simp,axiom,
! [X2: nat,Y3: num] :
( ( product_swap_nat_num @ ( product_Pair_nat_num @ X2 @ Y3 ) )
= ( product_Pair_num_nat @ Y3 @ X2 ) ) ).
% swap_simp
thf(fact_47_swap__simp,axiom,
! [X2: nat,Y3: nat] :
( ( product_swap_nat_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) )
= ( product_Pair_nat_nat @ Y3 @ X2 ) ) ).
% swap_simp
thf(fact_48_swap__simp,axiom,
! [X2: int,Y3: int] :
( ( product_swap_int_int @ ( product_Pair_int_int @ X2 @ Y3 ) )
= ( product_Pair_int_int @ Y3 @ X2 ) ) ).
% swap_simp
thf(fact_49_prod_Ocollapse,axiom,
! [Prod: produc1319942482725812455at_nat] :
( ( produc9060074326276436823at_nat @ ( produc4486455499028916547at_nat @ Prod ) @ ( produc826222791318859909at_nat @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_50_prod_Ocollapse,axiom,
! [Prod: produc3843707927480180839at_nat] :
( ( produc2922128104949294807at_nat @ ( produc6011863565407846595at_nat @ Prod ) @ ( produc2937545259423399429at_nat @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_51_prod_Ocollapse,axiom,
! [Prod: product_prod_nat_num] :
( ( product_Pair_nat_num @ ( product_fst_nat_num @ Prod ) @ ( product_snd_nat_num @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_52_prod_Ocollapse,axiom,
! [Prod: product_prod_nat_nat] :
( ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_53_prod_Ocollapse,axiom,
! [Prod: product_prod_int_int] :
( ( product_Pair_int_int @ ( product_fst_int_int @ Prod ) @ ( product_snd_int_int @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_54_prod_Ocollapse,axiom,
! [Prod: produc7812382424243267045t_unit] :
( ( produc6544324848051098391t_unit @ ( produc473321116828791851t_unit @ Prod ) @ ( produc1126150082675451753t_unit @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_55_old_Oprod_Oexhaust,axiom,
! [Y3: produc1319942482725812455at_nat] :
~ ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
( Y3
!= ( produc9060074326276436823at_nat @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_56_old_Oprod_Oexhaust,axiom,
! [Y3: produc3843707927480180839at_nat] :
~ ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
( Y3
!= ( produc2922128104949294807at_nat @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_57_old_Oprod_Oexhaust,axiom,
! [Y3: product_prod_nat_num] :
~ ! [A3: nat,B3: num] :
( Y3
!= ( product_Pair_nat_num @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_58_old_Oprod_Oexhaust,axiom,
! [Y3: product_prod_nat_nat] :
~ ! [A3: nat,B3: nat] :
( Y3
!= ( product_Pair_nat_nat @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_59_old_Oprod_Oexhaust,axiom,
! [Y3: product_prod_int_int] :
~ ! [A3: int,B3: int] :
( Y3
!= ( product_Pair_int_int @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_60_surj__pair,axiom,
! [P: produc1319942482725812455at_nat] :
? [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
( P
= ( produc9060074326276436823at_nat @ X @ Y ) ) ).
% surj_pair
thf(fact_61_surj__pair,axiom,
! [P: produc3843707927480180839at_nat] :
? [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
( P
= ( produc2922128104949294807at_nat @ X @ Y ) ) ).
% surj_pair
thf(fact_62_surj__pair,axiom,
! [P: product_prod_nat_num] :
? [X: nat,Y: num] :
( P
= ( product_Pair_nat_num @ X @ Y ) ) ).
% surj_pair
thf(fact_63_surj__pair,axiom,
! [P: product_prod_nat_nat] :
? [X: nat,Y: nat] :
( P
= ( product_Pair_nat_nat @ X @ Y ) ) ).
% surj_pair
thf(fact_64_surj__pair,axiom,
! [P: product_prod_int_int] :
? [X: int,Y: int] :
( P
= ( product_Pair_int_int @ X @ Y ) ) ).
% surj_pair
thf(fact_65_prod__cases,axiom,
! [P2: produc1319942482725812455at_nat > $o,P: produc1319942482725812455at_nat] :
( ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] : ( P2 @ ( produc9060074326276436823at_nat @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_66_prod__cases,axiom,
! [P2: produc3843707927480180839at_nat > $o,P: produc3843707927480180839at_nat] :
( ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] : ( P2 @ ( produc2922128104949294807at_nat @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_67_prod__cases,axiom,
! [P2: product_prod_nat_num > $o,P: product_prod_nat_num] :
( ! [A3: nat,B3: num] : ( P2 @ ( product_Pair_nat_num @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_68_prod__cases,axiom,
! [P2: product_prod_nat_nat > $o,P: product_prod_nat_nat] :
( ! [A3: nat,B3: nat] : ( P2 @ ( product_Pair_nat_nat @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_69_prod__cases,axiom,
! [P2: product_prod_int_int > $o,P: product_prod_int_int] :
( ! [A3: int,B3: int] : ( P2 @ ( product_Pair_int_int @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_70_Pair__inject,axiom,
! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
( ( ( produc9060074326276436823at_nat @ A @ B )
= ( produc9060074326276436823at_nat @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_71_Pair__inject,axiom,
! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( ( ( produc2922128104949294807at_nat @ A @ B )
= ( produc2922128104949294807at_nat @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_72_Pair__inject,axiom,
! [A: nat,B: num,A2: nat,B2: num] :
( ( ( product_Pair_nat_num @ A @ B )
= ( product_Pair_nat_num @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_73_Pair__inject,axiom,
! [A: nat,B: nat,A2: nat,B2: nat] :
( ( ( product_Pair_nat_nat @ A @ B )
= ( product_Pair_nat_nat @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_74_Pair__inject,axiom,
! [A: int,B: int,A2: int,B2: int] :
( ( ( product_Pair_int_int @ A @ B )
= ( product_Pair_int_int @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_75_mem__Collect__eq,axiom,
! [A: produc3843707927480180839at_nat,P2: produc3843707927480180839at_nat > $o] :
( ( member8757157785044589968at_nat @ A @ ( collec6321179662152712658at_nat @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_76_mem__Collect__eq,axiom,
! [A: $o,P2: $o > $o] :
( ( member_o @ A @ ( collect_o @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_77_mem__Collect__eq,axiom,
! [A: int,P2: int > $o] :
( ( member_int @ A @ ( collect_int @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_78_mem__Collect__eq,axiom,
! [A: product_prod_nat_nat,P2: product_prod_nat_nat > $o] :
( ( member8440522571783428010at_nat @ A @ ( collec3392354462482085612at_nat @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_79_mem__Collect__eq,axiom,
! [A: nat,P2: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_80_mem__Collect__eq,axiom,
! [A: product_prod_int_int,P2: product_prod_int_int > $o] :
( ( member5262025264175285858nt_int @ A @ ( collec213857154873943460nt_int @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_81_Collect__mem__eq,axiom,
! [A4: set_Pr4329608150637261639at_nat] :
( ( collec6321179662152712658at_nat
@ ^ [X4: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_82_Collect__mem__eq,axiom,
! [A4: set_o] :
( ( collect_o
@ ^ [X4: $o] : ( member_o @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_83_Collect__mem__eq,axiom,
! [A4: set_int] :
( ( collect_int
@ ^ [X4: int] : ( member_int @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_84_Collect__mem__eq,axiom,
! [A4: set_Pr1261947904930325089at_nat] :
( ( collec3392354462482085612at_nat
@ ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_85_Collect__mem__eq,axiom,
! [A4: set_nat] :
( ( collect_nat
@ ^ [X4: nat] : ( member_nat @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_86_Collect__mem__eq,axiom,
! [A4: set_Pr958786334691620121nt_int] :
( ( collec213857154873943460nt_int
@ ^ [X4: product_prod_int_int] : ( member5262025264175285858nt_int @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_87_Collect__cong,axiom,
! [P2: product_prod_nat_nat > $o,Q2: product_prod_nat_nat > $o] :
( ! [X: product_prod_nat_nat] :
( ( P2 @ X )
= ( Q2 @ X ) )
=> ( ( collec3392354462482085612at_nat @ P2 )
= ( collec3392354462482085612at_nat @ Q2 ) ) ) ).
% Collect_cong
thf(fact_88_Collect__cong,axiom,
! [P2: nat > $o,Q2: nat > $o] :
( ! [X: nat] :
( ( P2 @ X )
= ( Q2 @ X ) )
=> ( ( collect_nat @ P2 )
= ( collect_nat @ Q2 ) ) ) ).
% Collect_cong
thf(fact_89_Collect__cong,axiom,
! [P2: product_prod_int_int > $o,Q2: product_prod_int_int > $o] :
( ! [X: product_prod_int_int] :
( ( P2 @ X )
= ( Q2 @ X ) )
=> ( ( collec213857154873943460nt_int @ P2 )
= ( collec213857154873943460nt_int @ Q2 ) ) ) ).
% Collect_cong
thf(fact_90_fst__eqD,axiom,
! [X2: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
( ( ( produc4486455499028916547at_nat @ ( produc9060074326276436823at_nat @ X2 @ Y3 ) )
= A )
=> ( X2 = A ) ) ).
% fst_eqD
thf(fact_91_fst__eqD,axiom,
! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
( ( ( produc6011863565407846595at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y3 ) )
= A )
=> ( X2 = A ) ) ).
% fst_eqD
thf(fact_92_fst__eqD,axiom,
! [X2: nat,Y3: num,A: nat] :
( ( ( product_fst_nat_num @ ( product_Pair_nat_num @ X2 @ Y3 ) )
= A )
=> ( X2 = A ) ) ).
% fst_eqD
thf(fact_93_fst__eqD,axiom,
! [X2: nat,Y3: nat,A: nat] :
( ( ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) )
= A )
=> ( X2 = A ) ) ).
% fst_eqD
thf(fact_94_fst__eqD,axiom,
! [X2: int,Y3: int,A: int] :
( ( ( product_fst_int_int @ ( product_Pair_int_int @ X2 @ Y3 ) )
= A )
=> ( X2 = A ) ) ).
% fst_eqD
thf(fact_95_fst__eqD,axiom,
! [X2: ref_a,Y3: heap_e7401611519738050253t_unit,A: ref_a] :
( ( ( produc473321116828791851t_unit @ ( produc6544324848051098391t_unit @ X2 @ Y3 ) )
= A )
=> ( X2 = A ) ) ).
% fst_eqD
thf(fact_96_fst__conv,axiom,
! [X1: set_Pr4329608150637261639at_nat,X22: set_Pr4329608150637261639at_nat] :
( ( produc4486455499028916547at_nat @ ( produc9060074326276436823at_nat @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_97_fst__conv,axiom,
! [X1: set_Pr1261947904930325089at_nat,X22: set_Pr1261947904930325089at_nat] :
( ( produc6011863565407846595at_nat @ ( produc2922128104949294807at_nat @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_98_fst__conv,axiom,
! [X1: nat,X22: num] :
( ( product_fst_nat_num @ ( product_Pair_nat_num @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_99_fst__conv,axiom,
! [X1: nat,X22: nat] :
( ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_100_fst__conv,axiom,
! [X1: int,X22: int] :
( ( product_fst_int_int @ ( product_Pair_int_int @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_101_fst__conv,axiom,
! [X1: ref_a,X22: heap_e7401611519738050253t_unit] :
( ( produc473321116828791851t_unit @ ( produc6544324848051098391t_unit @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_102_snd__eqD,axiom,
! [X2: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
( ( ( produc826222791318859909at_nat @ ( produc9060074326276436823at_nat @ X2 @ Y3 ) )
= A )
=> ( Y3 = A ) ) ).
% snd_eqD
thf(fact_103_snd__eqD,axiom,
! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
( ( ( produc2937545259423399429at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y3 ) )
= A )
=> ( Y3 = A ) ) ).
% snd_eqD
thf(fact_104_snd__eqD,axiom,
! [X2: nat,Y3: num,A: num] :
( ( ( product_snd_nat_num @ ( product_Pair_nat_num @ X2 @ Y3 ) )
= A )
=> ( Y3 = A ) ) ).
% snd_eqD
thf(fact_105_snd__eqD,axiom,
! [X2: nat,Y3: nat,A: nat] :
( ( ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) )
= A )
=> ( Y3 = A ) ) ).
% snd_eqD
thf(fact_106_snd__eqD,axiom,
! [X2: int,Y3: int,A: int] :
( ( ( product_snd_int_int @ ( product_Pair_int_int @ X2 @ Y3 ) )
= A )
=> ( Y3 = A ) ) ).
% snd_eqD
thf(fact_107_snd__eqD,axiom,
! [X2: ref_a,Y3: heap_e7401611519738050253t_unit,A: heap_e7401611519738050253t_unit] :
( ( ( produc1126150082675451753t_unit @ ( produc6544324848051098391t_unit @ X2 @ Y3 ) )
= A )
=> ( Y3 = A ) ) ).
% snd_eqD
thf(fact_108_snd__conv,axiom,
! [X1: set_Pr4329608150637261639at_nat,X22: set_Pr4329608150637261639at_nat] :
( ( produc826222791318859909at_nat @ ( produc9060074326276436823at_nat @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_109_snd__conv,axiom,
! [X1: set_Pr1261947904930325089at_nat,X22: set_Pr1261947904930325089at_nat] :
( ( produc2937545259423399429at_nat @ ( produc2922128104949294807at_nat @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_110_snd__conv,axiom,
! [X1: nat,X22: num] :
( ( product_snd_nat_num @ ( product_Pair_nat_num @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_111_snd__conv,axiom,
! [X1: nat,X22: nat] :
( ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_112_snd__conv,axiom,
! [X1: int,X22: int] :
( ( product_snd_int_int @ ( product_Pair_int_int @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_113_snd__conv,axiom,
! [X1: ref_a,X22: heap_e7401611519738050253t_unit] :
( ( produc1126150082675451753t_unit @ ( produc6544324848051098391t_unit @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_114_prod_Oswap__def,axiom,
( produc5396952528824381687at_nat
= ( ^ [P3: produc1319942482725812455at_nat] : ( produc9060074326276436823at_nat @ ( produc826222791318859909at_nat @ P3 ) @ ( produc4486455499028916547at_nat @ P3 ) ) ) ) ).
% prod.swap_def
thf(fact_115_prod_Oswap__def,axiom,
( produc3946833401584108663at_nat
= ( ^ [P3: produc3843707927480180839at_nat] : ( produc2922128104949294807at_nat @ ( produc2937545259423399429at_nat @ P3 ) @ ( produc6011863565407846595at_nat @ P3 ) ) ) ) ).
% prod.swap_def
thf(fact_116_prod_Oswap__def,axiom,
( product_swap_num_nat
= ( ^ [P3: product_prod_num_nat] : ( product_Pair_nat_num @ ( product_snd_num_nat @ P3 ) @ ( product_fst_num_nat @ P3 ) ) ) ) ).
% prod.swap_def
thf(fact_117_prod_Oswap__def,axiom,
( product_swap_nat_nat
= ( ^ [P3: product_prod_nat_nat] : ( product_Pair_nat_nat @ ( product_snd_nat_nat @ P3 ) @ ( product_fst_nat_nat @ P3 ) ) ) ) ).
% prod.swap_def
thf(fact_118_prod_Oswap__def,axiom,
( product_swap_int_int
= ( ^ [P3: product_prod_int_int] : ( product_Pair_int_int @ ( product_snd_int_int @ P3 ) @ ( product_fst_int_int @ P3 ) ) ) ) ).
% prod.swap_def
thf(fact_119_prod_Oswap__def,axiom,
( produc970745769005243767t_unit
= ( ^ [P3: produc7812382424243267045t_unit] : ( produc2564672488694626579_ref_a @ ( produc1126150082675451753t_unit @ P3 ) @ ( produc473321116828791851t_unit @ P3 ) ) ) ) ).
% prod.swap_def
thf(fact_120_prod_Oexhaust__sel,axiom,
! [Prod: produc1319942482725812455at_nat] :
( Prod
= ( produc9060074326276436823at_nat @ ( produc4486455499028916547at_nat @ Prod ) @ ( produc826222791318859909at_nat @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_121_prod_Oexhaust__sel,axiom,
! [Prod: produc3843707927480180839at_nat] :
( Prod
= ( produc2922128104949294807at_nat @ ( produc6011863565407846595at_nat @ Prod ) @ ( produc2937545259423399429at_nat @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_122_prod_Oexhaust__sel,axiom,
! [Prod: product_prod_nat_num] :
( Prod
= ( product_Pair_nat_num @ ( product_fst_nat_num @ Prod ) @ ( product_snd_nat_num @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_123_prod_Oexhaust__sel,axiom,
! [Prod: product_prod_nat_nat] :
( Prod
= ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_124_prod_Oexhaust__sel,axiom,
! [Prod: product_prod_int_int] :
( Prod
= ( product_Pair_int_int @ ( product_fst_int_int @ Prod ) @ ( product_snd_int_int @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_125_prod_Oexhaust__sel,axiom,
! [Prod: produc7812382424243267045t_unit] :
( Prod
= ( produc6544324848051098391t_unit @ ( produc473321116828791851t_unit @ Prod ) @ ( produc1126150082675451753t_unit @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_126_exI__realizer,axiom,
! [P2: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o,Y3: set_Pr4329608150637261639at_nat,X2: set_Pr4329608150637261639at_nat] :
( ( P2 @ Y3 @ X2 )
=> ( P2 @ ( produc826222791318859909at_nat @ ( produc9060074326276436823at_nat @ X2 @ Y3 ) ) @ ( produc4486455499028916547at_nat @ ( produc9060074326276436823at_nat @ X2 @ Y3 ) ) ) ) ).
% exI_realizer
thf(fact_127_exI__realizer,axiom,
! [P2: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o,Y3: set_Pr1261947904930325089at_nat,X2: set_Pr1261947904930325089at_nat] :
( ( P2 @ Y3 @ X2 )
=> ( P2 @ ( produc2937545259423399429at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y3 ) ) @ ( produc6011863565407846595at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y3 ) ) ) ) ).
% exI_realizer
thf(fact_128_exI__realizer,axiom,
! [P2: num > nat > $o,Y3: num,X2: nat] :
( ( P2 @ Y3 @ X2 )
=> ( P2 @ ( product_snd_nat_num @ ( product_Pair_nat_num @ X2 @ Y3 ) ) @ ( product_fst_nat_num @ ( product_Pair_nat_num @ X2 @ Y3 ) ) ) ) ).
% exI_realizer
thf(fact_129_exI__realizer,axiom,
! [P2: nat > nat > $o,Y3: nat,X2: nat] :
( ( P2 @ Y3 @ X2 )
=> ( P2 @ ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) ) @ ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) ) ) ) ).
% exI_realizer
thf(fact_130_exI__realizer,axiom,
! [P2: int > int > $o,Y3: int,X2: int] :
( ( P2 @ Y3 @ X2 )
=> ( P2 @ ( product_snd_int_int @ ( product_Pair_int_int @ X2 @ Y3 ) ) @ ( product_fst_int_int @ ( product_Pair_int_int @ X2 @ Y3 ) ) ) ) ).
% exI_realizer
thf(fact_131_exI__realizer,axiom,
! [P2: heap_e7401611519738050253t_unit > ref_a > $o,Y3: heap_e7401611519738050253t_unit,X2: ref_a] :
( ( P2 @ Y3 @ X2 )
=> ( P2 @ ( produc1126150082675451753t_unit @ ( produc6544324848051098391t_unit @ X2 @ Y3 ) ) @ ( produc473321116828791851t_unit @ ( produc6544324848051098391t_unit @ X2 @ Y3 ) ) ) ) ).
% exI_realizer
thf(fact_132_conjI__realizer,axiom,
! [P2: set_Pr4329608150637261639at_nat > $o,P: set_Pr4329608150637261639at_nat,Q2: set_Pr4329608150637261639at_nat > $o,Q: set_Pr4329608150637261639at_nat] :
( ( P2 @ P )
=> ( ( Q2 @ Q )
=> ( ( P2 @ ( produc4486455499028916547at_nat @ ( produc9060074326276436823at_nat @ P @ Q ) ) )
& ( Q2 @ ( produc826222791318859909at_nat @ ( produc9060074326276436823at_nat @ P @ Q ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_133_conjI__realizer,axiom,
! [P2: set_Pr1261947904930325089at_nat > $o,P: set_Pr1261947904930325089at_nat,Q2: set_Pr1261947904930325089at_nat > $o,Q: set_Pr1261947904930325089at_nat] :
( ( P2 @ P )
=> ( ( Q2 @ Q )
=> ( ( P2 @ ( produc6011863565407846595at_nat @ ( produc2922128104949294807at_nat @ P @ Q ) ) )
& ( Q2 @ ( produc2937545259423399429at_nat @ ( produc2922128104949294807at_nat @ P @ Q ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_134_conjI__realizer,axiom,
! [P2: nat > $o,P: nat,Q2: num > $o,Q: num] :
( ( P2 @ P )
=> ( ( Q2 @ Q )
=> ( ( P2 @ ( product_fst_nat_num @ ( product_Pair_nat_num @ P @ Q ) ) )
& ( Q2 @ ( product_snd_nat_num @ ( product_Pair_nat_num @ P @ Q ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_135_conjI__realizer,axiom,
! [P2: nat > $o,P: nat,Q2: nat > $o,Q: nat] :
( ( P2 @ P )
=> ( ( Q2 @ Q )
=> ( ( P2 @ ( product_fst_nat_nat @ ( product_Pair_nat_nat @ P @ Q ) ) )
& ( Q2 @ ( product_snd_nat_nat @ ( product_Pair_nat_nat @ P @ Q ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_136_conjI__realizer,axiom,
! [P2: int > $o,P: int,Q2: int > $o,Q: int] :
( ( P2 @ P )
=> ( ( Q2 @ Q )
=> ( ( P2 @ ( product_fst_int_int @ ( product_Pair_int_int @ P @ Q ) ) )
& ( Q2 @ ( product_snd_int_int @ ( product_Pair_int_int @ P @ Q ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_137_conjI__realizer,axiom,
! [P2: ref_a > $o,P: ref_a,Q2: heap_e7401611519738050253t_unit > $o,Q: heap_e7401611519738050253t_unit] :
( ( P2 @ P )
=> ( ( Q2 @ Q )
=> ( ( P2 @ ( produc473321116828791851t_unit @ ( produc6544324848051098391t_unit @ P @ Q ) ) )
& ( Q2 @ ( produc1126150082675451753t_unit @ ( produc6544324848051098391t_unit @ P @ Q ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_138_surjective__pairing,axiom,
! [T2: produc1319942482725812455at_nat] :
( T2
= ( produc9060074326276436823at_nat @ ( produc4486455499028916547at_nat @ T2 ) @ ( produc826222791318859909at_nat @ T2 ) ) ) ).
% surjective_pairing
thf(fact_139_surjective__pairing,axiom,
! [T2: produc3843707927480180839at_nat] :
( T2
= ( produc2922128104949294807at_nat @ ( produc6011863565407846595at_nat @ T2 ) @ ( produc2937545259423399429at_nat @ T2 ) ) ) ).
% surjective_pairing
thf(fact_140_surjective__pairing,axiom,
! [T2: product_prod_nat_num] :
( T2
= ( product_Pair_nat_num @ ( product_fst_nat_num @ T2 ) @ ( product_snd_nat_num @ T2 ) ) ) ).
% surjective_pairing
thf(fact_141_surjective__pairing,axiom,
! [T2: product_prod_nat_nat] :
( T2
= ( product_Pair_nat_nat @ ( product_fst_nat_nat @ T2 ) @ ( product_snd_nat_nat @ T2 ) ) ) ).
% surjective_pairing
thf(fact_142_surjective__pairing,axiom,
! [T2: product_prod_int_int] :
( T2
= ( product_Pair_int_int @ ( product_fst_int_int @ T2 ) @ ( product_snd_int_int @ T2 ) ) ) ).
% surjective_pairing
thf(fact_143_surjective__pairing,axiom,
! [T2: produc7812382424243267045t_unit] :
( T2
= ( produc6544324848051098391t_unit @ ( produc473321116828791851t_unit @ T2 ) @ ( produc1126150082675451753t_unit @ T2 ) ) ) ).
% surjective_pairing
thf(fact_144_Ref_Oset__set__swap,axiom,
! [R: ref_a,R2: ref_a,X2: a,X3: a,H: heap_e7401611519738050253t_unit] :
( ( noteq_a_a @ R @ R2 )
=> ( ( set_a @ R @ X2 @ ( set_a @ R2 @ X3 @ H ) )
= ( set_a @ R2 @ X3 @ ( set_a @ R @ X2 @ H ) ) ) ) ).
% Ref.set_set_swap
thf(fact_145_next__fresh,axiom,
! [R: ref_a,H2: heap_e7401611519738050253t_unit,X2: a,H: heap_e7401611519738050253t_unit] :
( ( ( produc6544324848051098391t_unit @ R @ H2 )
= ( alloc_a @ X2 @ H ) )
=> ~ ( present_a @ H @ R ) ) ).
% next_fresh
thf(fact_146_next__present,axiom,
! [R: ref_a,H2: heap_e7401611519738050253t_unit,X2: a,H: heap_e7401611519738050253t_unit] :
( ( ( produc6544324848051098391t_unit @ R @ H2 )
= ( alloc_a @ X2 @ H ) )
=> ( present_a @ H2 @ R ) ) ).
% next_present
thf(fact_147_noteq__I,axiom,
! [H: heap_e7401611519738050253t_unit,R: ref_a,R2: ref_a] :
( ( present_a @ H @ R )
=> ( ~ ( present_a @ H @ R2 )
=> ( noteq_a_a @ R @ R2 ) ) ) ).
% noteq_I
thf(fact_148_effect__heapE,axiom,
! [F: heap_e7401611519738050253t_unit > produc7812382424243267045t_unit,H: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,R: ref_a] :
( ( heap_effect_ref_a @ ( heap_heap_ref_a @ F ) @ H @ H2 @ R )
=> ~ ( ( H2
= ( produc1126150082675451753t_unit @ ( F @ H ) ) )
=> ( R
!= ( produc473321116828791851t_unit @ ( F @ H ) ) ) ) ) ).
% effect_heapE
thf(fact_149_effect__heapI,axiom,
! [H2: heap_e7401611519738050253t_unit,F: heap_e7401611519738050253t_unit > produc7812382424243267045t_unit,H: heap_e7401611519738050253t_unit,R: ref_a] :
( ( H2
= ( produc1126150082675451753t_unit @ ( F @ H ) ) )
=> ( ( R
= ( produc473321116828791851t_unit @ ( F @ H ) ) )
=> ( heap_effect_ref_a @ ( heap_heap_ref_a @ F ) @ H @ H2 @ R ) ) ) ).
% effect_heapI
thf(fact_150_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P2: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o,X2: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat,A: produc1319942482725812455at_nat] :
( ( P2 @ X2 @ Y3 )
=> ( ( A
= ( produc9060074326276436823at_nat @ X2 @ Y3 ) )
=> ( P2 @ ( produc4486455499028916547at_nat @ A ) @ ( produc826222791318859909at_nat @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_151_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P2: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o,X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat,A: produc3843707927480180839at_nat] :
( ( P2 @ X2 @ Y3 )
=> ( ( A
= ( produc2922128104949294807at_nat @ X2 @ Y3 ) )
=> ( P2 @ ( produc6011863565407846595at_nat @ A ) @ ( produc2937545259423399429at_nat @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_152_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P2: nat > num > $o,X2: nat,Y3: num,A: product_prod_nat_num] :
( ( P2 @ X2 @ Y3 )
=> ( ( A
= ( product_Pair_nat_num @ X2 @ Y3 ) )
=> ( P2 @ ( product_fst_nat_num @ A ) @ ( product_snd_nat_num @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_153_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P2: nat > nat > $o,X2: nat,Y3: nat,A: product_prod_nat_nat] :
( ( P2 @ X2 @ Y3 )
=> ( ( A
= ( product_Pair_nat_nat @ X2 @ Y3 ) )
=> ( P2 @ ( product_fst_nat_nat @ A ) @ ( product_snd_nat_nat @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_154_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P2: int > int > $o,X2: int,Y3: int,A: product_prod_int_int] :
( ( P2 @ X2 @ Y3 )
=> ( ( A
= ( product_Pair_int_int @ X2 @ Y3 ) )
=> ( P2 @ ( product_fst_int_int @ A ) @ ( product_snd_int_int @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_155_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P2: ref_a > heap_e7401611519738050253t_unit > $o,X2: ref_a,Y3: heap_e7401611519738050253t_unit,A: produc7812382424243267045t_unit] :
( ( P2 @ X2 @ Y3 )
=> ( ( A
= ( produc6544324848051098391t_unit @ X2 @ Y3 ) )
=> ( P2 @ ( produc473321116828791851t_unit @ A ) @ ( produc1126150082675451753t_unit @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_156_sndI,axiom,
! [X2: produc1319942482725812455at_nat,Y3: set_Pr4329608150637261639at_nat,Z2: set_Pr4329608150637261639at_nat] :
( ( X2
= ( produc9060074326276436823at_nat @ Y3 @ Z2 ) )
=> ( ( produc826222791318859909at_nat @ X2 )
= Z2 ) ) ).
% sndI
thf(fact_157_sndI,axiom,
! [X2: produc3843707927480180839at_nat,Y3: set_Pr1261947904930325089at_nat,Z2: set_Pr1261947904930325089at_nat] :
( ( X2
= ( produc2922128104949294807at_nat @ Y3 @ Z2 ) )
=> ( ( produc2937545259423399429at_nat @ X2 )
= Z2 ) ) ).
% sndI
thf(fact_158_sndI,axiom,
! [X2: product_prod_nat_num,Y3: nat,Z2: num] :
( ( X2
= ( product_Pair_nat_num @ Y3 @ Z2 ) )
=> ( ( product_snd_nat_num @ X2 )
= Z2 ) ) ).
% sndI
thf(fact_159_sndI,axiom,
! [X2: product_prod_nat_nat,Y3: nat,Z2: nat] :
( ( X2
= ( product_Pair_nat_nat @ Y3 @ Z2 ) )
=> ( ( product_snd_nat_nat @ X2 )
= Z2 ) ) ).
% sndI
thf(fact_160_sndI,axiom,
! [X2: product_prod_int_int,Y3: int,Z2: int] :
( ( X2
= ( product_Pair_int_int @ Y3 @ Z2 ) )
=> ( ( product_snd_int_int @ X2 )
= Z2 ) ) ).
% sndI
thf(fact_161_sndI,axiom,
! [X2: produc7812382424243267045t_unit,Y3: ref_a,Z2: heap_e7401611519738050253t_unit] :
( ( X2
= ( produc6544324848051098391t_unit @ Y3 @ Z2 ) )
=> ( ( produc1126150082675451753t_unit @ X2 )
= Z2 ) ) ).
% sndI
thf(fact_162_eq__snd__iff,axiom,
! [B: set_Pr4329608150637261639at_nat,P: produc1319942482725812455at_nat] :
( ( B
= ( produc826222791318859909at_nat @ P ) )
= ( ? [A5: set_Pr4329608150637261639at_nat] :
( P
= ( produc9060074326276436823at_nat @ A5 @ B ) ) ) ) ).
% eq_snd_iff
thf(fact_163_eq__snd__iff,axiom,
! [B: set_Pr1261947904930325089at_nat,P: produc3843707927480180839at_nat] :
( ( B
= ( produc2937545259423399429at_nat @ P ) )
= ( ? [A5: set_Pr1261947904930325089at_nat] :
( P
= ( produc2922128104949294807at_nat @ A5 @ B ) ) ) ) ).
% eq_snd_iff
thf(fact_164_eq__snd__iff,axiom,
! [B: num,P: product_prod_nat_num] :
( ( B
= ( product_snd_nat_num @ P ) )
= ( ? [A5: nat] :
( P
= ( product_Pair_nat_num @ A5 @ B ) ) ) ) ).
% eq_snd_iff
thf(fact_165_eq__snd__iff,axiom,
! [B: nat,P: product_prod_nat_nat] :
( ( B
= ( product_snd_nat_nat @ P ) )
= ( ? [A5: nat] :
( P
= ( product_Pair_nat_nat @ A5 @ B ) ) ) ) ).
% eq_snd_iff
thf(fact_166_eq__snd__iff,axiom,
! [B: int,P: product_prod_int_int] :
( ( B
= ( product_snd_int_int @ P ) )
= ( ? [A5: int] :
( P
= ( product_Pair_int_int @ A5 @ B ) ) ) ) ).
% eq_snd_iff
thf(fact_167_eq__snd__iff,axiom,
! [B: heap_e7401611519738050253t_unit,P: produc7812382424243267045t_unit] :
( ( B
= ( produc1126150082675451753t_unit @ P ) )
= ( ? [A5: ref_a] :
( P
= ( produc6544324848051098391t_unit @ A5 @ B ) ) ) ) ).
% eq_snd_iff
thf(fact_168_fstI,axiom,
! [X2: produc1319942482725812455at_nat,Y3: set_Pr4329608150637261639at_nat,Z2: set_Pr4329608150637261639at_nat] :
( ( X2
= ( produc9060074326276436823at_nat @ Y3 @ Z2 ) )
=> ( ( produc4486455499028916547at_nat @ X2 )
= Y3 ) ) ).
% fstI
thf(fact_169_fstI,axiom,
! [X2: produc3843707927480180839at_nat,Y3: set_Pr1261947904930325089at_nat,Z2: set_Pr1261947904930325089at_nat] :
( ( X2
= ( produc2922128104949294807at_nat @ Y3 @ Z2 ) )
=> ( ( produc6011863565407846595at_nat @ X2 )
= Y3 ) ) ).
% fstI
thf(fact_170_fstI,axiom,
! [X2: product_prod_nat_num,Y3: nat,Z2: num] :
( ( X2
= ( product_Pair_nat_num @ Y3 @ Z2 ) )
=> ( ( product_fst_nat_num @ X2 )
= Y3 ) ) ).
% fstI
thf(fact_171_fstI,axiom,
! [X2: product_prod_nat_nat,Y3: nat,Z2: nat] :
( ( X2
= ( product_Pair_nat_nat @ Y3 @ Z2 ) )
=> ( ( product_fst_nat_nat @ X2 )
= Y3 ) ) ).
% fstI
thf(fact_172_fstI,axiom,
! [X2: product_prod_int_int,Y3: int,Z2: int] :
( ( X2
= ( product_Pair_int_int @ Y3 @ Z2 ) )
=> ( ( product_fst_int_int @ X2 )
= Y3 ) ) ).
% fstI
thf(fact_173_fstI,axiom,
! [X2: produc7812382424243267045t_unit,Y3: ref_a,Z2: heap_e7401611519738050253t_unit] :
( ( X2
= ( produc6544324848051098391t_unit @ Y3 @ Z2 ) )
=> ( ( produc473321116828791851t_unit @ X2 )
= Y3 ) ) ).
% fstI
thf(fact_174_eq__fst__iff,axiom,
! [A: set_Pr4329608150637261639at_nat,P: produc1319942482725812455at_nat] :
( ( A
= ( produc4486455499028916547at_nat @ P ) )
= ( ? [B4: set_Pr4329608150637261639at_nat] :
( P
= ( produc9060074326276436823at_nat @ A @ B4 ) ) ) ) ).
% eq_fst_iff
thf(fact_175_eq__fst__iff,axiom,
! [A: set_Pr1261947904930325089at_nat,P: produc3843707927480180839at_nat] :
( ( A
= ( produc6011863565407846595at_nat @ P ) )
= ( ? [B4: set_Pr1261947904930325089at_nat] :
( P
= ( produc2922128104949294807at_nat @ A @ B4 ) ) ) ) ).
% eq_fst_iff
thf(fact_176_eq__fst__iff,axiom,
! [A: nat,P: product_prod_nat_num] :
( ( A
= ( product_fst_nat_num @ P ) )
= ( ? [B4: num] :
( P
= ( product_Pair_nat_num @ A @ B4 ) ) ) ) ).
% eq_fst_iff
thf(fact_177_eq__fst__iff,axiom,
! [A: nat,P: product_prod_nat_nat] :
( ( A
= ( product_fst_nat_nat @ P ) )
= ( ? [B4: nat] :
( P
= ( product_Pair_nat_nat @ A @ B4 ) ) ) ) ).
% eq_fst_iff
thf(fact_178_eq__fst__iff,axiom,
! [A: int,P: product_prod_int_int] :
( ( A
= ( product_fst_int_int @ P ) )
= ( ? [B4: int] :
( P
= ( product_Pair_int_int @ A @ B4 ) ) ) ) ).
% eq_fst_iff
thf(fact_179_eq__fst__iff,axiom,
! [A: ref_a,P: produc7812382424243267045t_unit] :
( ( A
= ( produc473321116828791851t_unit @ P ) )
= ( ? [B4: heap_e7401611519738050253t_unit] :
( P
= ( produc6544324848051098391t_unit @ A @ B4 ) ) ) ) ).
% eq_fst_iff
thf(fact_180_effect__guardE,axiom,
! [P2: heap_e7401611519738050253t_unit > $o,F: heap_e7401611519738050253t_unit > produc7812382424243267045t_unit,H: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,R: ref_a] :
( ( heap_effect_ref_a @ ( heap_guard_ref_a @ P2 @ F ) @ H @ H2 @ R )
=> ~ ( ( H2
= ( produc1126150082675451753t_unit @ ( F @ H ) ) )
=> ( ( R
= ( produc473321116828791851t_unit @ ( F @ H ) ) )
=> ~ ( P2 @ H ) ) ) ) ).
% effect_guardE
thf(fact_181_effect__guardI,axiom,
! [P2: heap_e7401611519738050253t_unit > $o,H: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,F: heap_e7401611519738050253t_unit > produc7812382424243267045t_unit,R: ref_a] :
( ( P2 @ H )
=> ( ( H2
= ( produc1126150082675451753t_unit @ ( F @ H ) ) )
=> ( ( R
= ( produc473321116828791851t_unit @ ( F @ H ) ) )
=> ( heap_effect_ref_a @ ( heap_guard_ref_a @ P2 @ F ) @ H @ H2 @ R ) ) ) ) ).
% effect_guardI
thf(fact_182_effect__ifI,axiom,
! [C: $o,T2: heap_Heap_ref_a,H: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,R: ref_a,E: heap_Heap_ref_a] :
( ( C
=> ( heap_effect_ref_a @ T2 @ H @ H2 @ R ) )
=> ( ( ~ C
=> ( heap_effect_ref_a @ E @ H @ H2 @ R ) )
=> ( heap_effect_ref_a @ ( if_Heap_Heap_ref_a @ C @ T2 @ E ) @ H @ H2 @ R ) ) ) ).
% effect_ifI
thf(fact_183_effect__deterministic_I2_J,axiom,
! [F: heap_Heap_ref_a,H: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,A: ref_a,H3: heap_e7401611519738050253t_unit,B: ref_a] :
( ( heap_effect_ref_a @ F @ H @ H2 @ A )
=> ( ( heap_effect_ref_a @ F @ H @ H3 @ B )
=> ( H2 = H3 ) ) ) ).
% effect_deterministic(2)
thf(fact_184_effect__deterministic_I1_J,axiom,
! [F: heap_Heap_ref_a,H: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,A: ref_a,H3: heap_e7401611519738050253t_unit,B: ref_a] :
( ( heap_effect_ref_a @ F @ H @ H2 @ A )
=> ( ( heap_effect_ref_a @ F @ H @ H3 @ B )
=> ( A = B ) ) ) ).
% effect_deterministic(1)
thf(fact_185_effect__ifE,axiom,
! [C: $o,T2: heap_Heap_ref_a,E: heap_Heap_ref_a,H: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,R: ref_a] :
( ( heap_effect_ref_a @ ( if_Heap_Heap_ref_a @ C @ T2 @ E ) @ H @ H2 @ R )
=> ( ( C
=> ~ ( heap_effect_ref_a @ T2 @ H @ H2 @ R ) )
=> ~ ( ~ C
=> ~ ( heap_effect_ref_a @ E @ H @ H2 @ R ) ) ) ) ).
% effect_ifE
thf(fact_186_image2__eqI,axiom,
! [B: nat,F: $o > nat,X2: $o,C: num,G: $o > num,A4: set_o] :
( ( B
= ( F @ X2 ) )
=> ( ( C
= ( G @ X2 ) )
=> ( ( member_o @ X2 @ A4 )
=> ( member9148766508732265716at_num @ ( product_Pair_nat_num @ B @ C ) @ ( bNF_Gr7727518618234538026at_num @ A4 @ F @ G ) ) ) ) ) ).
% image2_eqI
thf(fact_187_image2__eqI,axiom,
! [B: nat,F: nat > nat,X2: nat,C: num,G: nat > num,A4: set_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( C
= ( G @ X2 ) )
=> ( ( member_nat @ X2 @ A4 )
=> ( member9148766508732265716at_num @ ( product_Pair_nat_num @ B @ C ) @ ( bNF_Gr3875488559241464598at_num @ A4 @ F @ G ) ) ) ) ) ).
% image2_eqI
thf(fact_188_image2__eqI,axiom,
! [B: nat,F: int > nat,X2: int,C: num,G: int > num,A4: set_int] :
( ( B
= ( F @ X2 ) )
=> ( ( C
= ( G @ X2 ) )
=> ( ( member_int @ X2 @ A4 )
=> ( member9148766508732265716at_num @ ( product_Pair_nat_num @ B @ C ) @ ( bNF_Gr4247142119478309234at_num @ A4 @ F @ G ) ) ) ) ) ).
% image2_eqI
thf(fact_189_image2__eqI,axiom,
! [B: nat,F: $o > nat,X2: $o,C: nat,G: $o > nat,A4: set_o] :
( ( B
= ( F @ X2 ) )
=> ( ( C
= ( G @ X2 ) )
=> ( ( member_o @ X2 @ A4 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C ) @ ( bNF_Gr1946814795223983584at_nat @ A4 @ F @ G ) ) ) ) ) ).
% image2_eqI
thf(fact_190_image2__eqI,axiom,
! [B: nat,F: nat > nat,X2: nat,C: nat,G: nat > nat,A4: set_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( C
= ( G @ X2 ) )
=> ( ( member_nat @ X2 @ A4 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C ) @ ( bNF_Gr7318156773085685964at_nat @ A4 @ F @ G ) ) ) ) ) ).
% image2_eqI
thf(fact_191_image2__eqI,axiom,
! [B: nat,F: int > nat,X2: int,C: nat,G: int > nat,A4: set_int] :
( ( B
= ( F @ X2 ) )
=> ( ( C
= ( G @ X2 ) )
=> ( ( member_int @ X2 @ A4 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C ) @ ( bNF_Gr7689810333322530600at_nat @ A4 @ F @ G ) ) ) ) ) ).
% image2_eqI
thf(fact_192_image2__eqI,axiom,
! [B: int,F: $o > int,X2: $o,C: int,G: $o > int,A4: set_o] :
( ( B
= ( F @ X2 ) )
=> ( ( C
= ( G @ X2 ) )
=> ( ( member_o @ X2 @ A4 )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ C ) @ ( bNF_Gr2943678036615987864nt_int @ A4 @ F @ G ) ) ) ) ) ).
% image2_eqI
thf(fact_193_image2__eqI,axiom,
! [B: int,F: nat > int,X2: nat,C: int,G: nat > int,A4: set_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( C
= ( G @ X2 ) )
=> ( ( member_nat @ X2 @ A4 )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ C ) @ ( bNF_Gr8315020014477690244nt_int @ A4 @ F @ G ) ) ) ) ) ).
% image2_eqI
thf(fact_194_image2__eqI,axiom,
! [B: int,F: int > int,X2: int,C: int,G: int > int,A4: set_int] :
( ( B
= ( F @ X2 ) )
=> ( ( C
= ( G @ X2 ) )
=> ( ( member_int @ X2 @ A4 )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ C ) @ ( bNF_Gr8686673574714534880nt_int @ A4 @ F @ G ) ) ) ) ) ).
% image2_eqI
thf(fact_195_image2__eqI,axiom,
! [B: nat,F: product_prod_nat_nat > nat,X2: product_prod_nat_nat,C: num,G: product_prod_nat_nat > num,A4: set_Pr1261947904930325089at_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( C
= ( G @ X2 ) )
=> ( ( member8440522571783428010at_nat @ X2 @ A4 )
=> ( member9148766508732265716at_num @ ( product_Pair_nat_num @ B @ C ) @ ( bNF_Gr5511461681843259283at_num @ A4 @ F @ G ) ) ) ) ) ).
% image2_eqI
thf(fact_196_lim__set,axiom,
! [R: ref_a,V: a,H: heap_e7401611519738050253t_unit] :
( ( lim_Product_unit @ ( set_a @ R @ V @ H ) )
= ( lim_Product_unit @ H ) ) ).
% lim_set
thf(fact_197_prod_Osplit__sel__asm,axiom,
! [P2: ( product_prod_nat_nat > product_prod_nat_nat ) > $o,F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,Prod: product_prod_nat_nat] :
( ( P2 @ ( produc27273713700761075at_nat @ F @ Prod ) )
= ( ~ ( ( Prod
= ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) )
& ~ ( P2 @ ( F @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) ) ) ) ) ).
% prod.split_sel_asm
thf(fact_198_prod_Osplit__sel__asm,axiom,
! [P2: ( product_prod_nat_nat > $o ) > $o,F: nat > nat > product_prod_nat_nat > $o,Prod: product_prod_nat_nat] :
( ( P2 @ ( produc8739625826339149834_nat_o @ F @ Prod ) )
= ( ~ ( ( Prod
= ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) )
& ~ ( P2 @ ( F @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) ) ) ) ) ).
% prod.split_sel_asm
thf(fact_199_prod_Osplit__sel__asm,axiom,
! [P2: product_prod_int_int > $o,F: int > int > product_prod_int_int,Prod: product_prod_int_int] :
( ( P2 @ ( produc4245557441103728435nt_int @ F @ Prod ) )
= ( ~ ( ( Prod
= ( product_Pair_int_int @ ( product_fst_int_int @ Prod ) @ ( product_snd_int_int @ Prod ) ) )
& ~ ( P2 @ ( F @ ( product_fst_int_int @ Prod ) @ ( product_snd_int_int @ Prod ) ) ) ) ) ) ).
% prod.split_sel_asm
thf(fact_200_prod_Osplit__sel__asm,axiom,
! [P2: $o > $o,F: int > int > $o,Prod: product_prod_int_int] :
( ( P2 @ ( produc4947309494688390418_int_o @ F @ Prod ) )
= ( ~ ( ( Prod
= ( product_Pair_int_int @ ( product_fst_int_int @ Prod ) @ ( product_snd_int_int @ Prod ) ) )
& ~ ( P2 @ ( F @ ( product_fst_int_int @ Prod ) @ ( product_snd_int_int @ Prod ) ) ) ) ) ) ).
% prod.split_sel_asm
thf(fact_201_prod_Osplit__sel__asm,axiom,
! [P2: int > $o,F: int > int > int,Prod: product_prod_int_int] :
( ( P2 @ ( produc8211389475949308722nt_int @ F @ Prod ) )
= ( ~ ( ( Prod
= ( product_Pair_int_int @ ( product_fst_int_int @ Prod ) @ ( product_snd_int_int @ Prod ) ) )
& ~ ( P2 @ ( F @ ( product_fst_int_int @ Prod ) @ ( product_snd_int_int @ Prod ) ) ) ) ) ) ).
% prod.split_sel_asm
thf(fact_202_prod_Osplit__sel,axiom,
! [P2: ( product_prod_nat_nat > product_prod_nat_nat ) > $o,F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,Prod: product_prod_nat_nat] :
( ( P2 @ ( produc27273713700761075at_nat @ F @ Prod ) )
= ( ( Prod
= ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) )
=> ( P2 @ ( F @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) ) ) ) ).
% prod.split_sel
thf(fact_203_prod_Osplit__sel,axiom,
! [P2: ( product_prod_nat_nat > $o ) > $o,F: nat > nat > product_prod_nat_nat > $o,Prod: product_prod_nat_nat] :
( ( P2 @ ( produc8739625826339149834_nat_o @ F @ Prod ) )
= ( ( Prod
= ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) )
=> ( P2 @ ( F @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) ) ) ) ).
% prod.split_sel
thf(fact_204_prod_Osplit__sel,axiom,
! [P2: product_prod_int_int > $o,F: int > int > product_prod_int_int,Prod: product_prod_int_int] :
( ( P2 @ ( produc4245557441103728435nt_int @ F @ Prod ) )
= ( ( Prod
= ( product_Pair_int_int @ ( product_fst_int_int @ Prod ) @ ( product_snd_int_int @ Prod ) ) )
=> ( P2 @ ( F @ ( product_fst_int_int @ Prod ) @ ( product_snd_int_int @ Prod ) ) ) ) ) ).
% prod.split_sel
thf(fact_205_prod_Osplit__sel,axiom,
! [P2: $o > $o,F: int > int > $o,Prod: product_prod_int_int] :
( ( P2 @ ( produc4947309494688390418_int_o @ F @ Prod ) )
= ( ( Prod
= ( product_Pair_int_int @ ( product_fst_int_int @ Prod ) @ ( product_snd_int_int @ Prod ) ) )
=> ( P2 @ ( F @ ( product_fst_int_int @ Prod ) @ ( product_snd_int_int @ Prod ) ) ) ) ) ).
% prod.split_sel
thf(fact_206_prod_Osplit__sel,axiom,
! [P2: int > $o,F: int > int > int,Prod: product_prod_int_int] :
( ( P2 @ ( produc8211389475949308722nt_int @ F @ Prod ) )
= ( ( Prod
= ( product_Pair_int_int @ ( product_fst_int_int @ Prod ) @ ( product_snd_int_int @ Prod ) ) )
=> ( P2 @ ( F @ ( product_fst_int_int @ Prod ) @ ( product_snd_int_int @ Prod ) ) ) ) ) ).
% prod.split_sel
thf(fact_207_ssubst__Pair__rhs,axiom,
! [R: set_Pr4329608150637261639at_nat,S2: set_Pr4329608150637261639at_nat,R3: set_Pr7459493094073627847at_nat,S3: set_Pr4329608150637261639at_nat] :
( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R @ S2 ) @ R3 )
=> ( ( S3 = S2 )
=> ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R @ S3 ) @ R3 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_208_ssubst__Pair__rhs,axiom,
! [R: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat,R3: set_Pr4329608150637261639at_nat,S3: set_Pr1261947904930325089at_nat] :
( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ S2 ) @ R3 )
=> ( ( S3 = S2 )
=> ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ S3 ) @ R3 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_209_ssubst__Pair__rhs,axiom,
! [R: nat,S2: num,R3: set_Pr6200539531224447659at_num,S3: num] :
( ( member9148766508732265716at_num @ ( product_Pair_nat_num @ R @ S2 ) @ R3 )
=> ( ( S3 = S2 )
=> ( member9148766508732265716at_num @ ( product_Pair_nat_num @ R @ S3 ) @ R3 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_210_ssubst__Pair__rhs,axiom,
! [R: nat,S2: nat,R3: set_Pr1261947904930325089at_nat,S3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R @ S2 ) @ R3 )
=> ( ( S3 = S2 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R @ S3 ) @ R3 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_211_ssubst__Pair__rhs,axiom,
! [R: int,S2: int,R3: set_Pr958786334691620121nt_int,S3: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ R @ S2 ) @ R3 )
=> ( ( S3 = S2 )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ R @ S3 ) @ R3 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_212_effect__raiseE,axiom,
! [X2: literal,H: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,R: ref_a] :
~ ( heap_effect_ref_a @ ( heap_raise_ref_a @ X2 ) @ H @ H2 @ R ) ).
% effect_raiseE
thf(fact_213_effect__updateE,axiom,
! [R2: ref_a,V: a,H: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,R: product_unit] :
( ( heap_e2830298082929533060t_unit @ ( update_a @ R2 @ V ) @ H @ H2 @ R )
=> ( H2
= ( set_a @ R2 @ V @ H ) ) ) ).
% effect_updateE
thf(fact_214_curryI,axiom,
! [F: produc1319942482725812455at_nat > $o,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
( ( F @ ( produc9060074326276436823at_nat @ A @ B ) )
=> ( produc447603880154497357_nat_o @ F @ A @ B ) ) ).
% curryI
thf(fact_215_curryI,axiom,
! [F: produc3843707927480180839at_nat > $o,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
( ( F @ ( produc2922128104949294807at_nat @ A @ B ) )
=> ( produc7820828192191229133_nat_o @ F @ A @ B ) ) ).
% curryI
thf(fact_216_curryI,axiom,
! [F: product_prod_nat_num > $o,A: nat,B: num] :
( ( F @ ( product_Pair_nat_num @ A @ B ) )
=> ( produc156083480235303841_num_o @ F @ A @ B ) ) ).
% curryI
thf(fact_217_curryI,axiom,
! [F: product_prod_nat_nat > $o,A: nat,B: nat] :
( ( F @ ( product_Pair_nat_nat @ A @ B ) )
=> ( produc1310100445399344235_nat_o @ F @ A @ B ) ) ).
% curryI
thf(fact_218_curryI,axiom,
! [F: product_prod_int_int > $o,A: int,B: int] :
( ( F @ ( product_Pair_int_int @ A @ B ) )
=> ( produc175634133007206835_int_o @ F @ A @ B ) ) ).
% curryI
thf(fact_219_case__prod__curry,axiom,
! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat] :
( ( produc27273713700761075at_nat @ ( produc3610074571335452306at_nat @ F ) )
= F ) ).
% case_prod_curry
thf(fact_220_case__prod__curry,axiom,
! [F: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ( produc8739625826339149834_nat_o @ ( produc3704529784387675049_nat_o @ F ) )
= F ) ).
% case_prod_curry
thf(fact_221_case__prod__curry,axiom,
! [F: product_prod_int_int > product_prod_int_int] :
( ( produc4245557441103728435nt_int @ ( produc8249235968001453780nt_int @ F ) )
= F ) ).
% case_prod_curry
thf(fact_222_case__prod__curry,axiom,
! [F: product_prod_int_int > $o] :
( ( produc4947309494688390418_int_o @ ( produc175634133007206835_int_o @ F ) )
= F ) ).
% case_prod_curry
thf(fact_223_case__prod__curry,axiom,
! [F: product_prod_int_int > int] :
( ( produc8211389475949308722nt_int @ ( produc1016772743285680337nt_int @ F ) )
= F ) ).
% case_prod_curry
thf(fact_224_curry__case__prod,axiom,
! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat] :
( ( produc3610074571335452306at_nat @ ( produc27273713700761075at_nat @ F ) )
= F ) ).
% curry_case_prod
thf(fact_225_curry__case__prod,axiom,
! [F: nat > nat > product_prod_nat_nat > $o] :
( ( produc3704529784387675049_nat_o @ ( produc8739625826339149834_nat_o @ F ) )
= F ) ).
% curry_case_prod
thf(fact_226_curry__case__prod,axiom,
! [F: int > int > product_prod_int_int] :
( ( produc8249235968001453780nt_int @ ( produc4245557441103728435nt_int @ F ) )
= F ) ).
% curry_case_prod
thf(fact_227_curry__case__prod,axiom,
! [F: int > int > $o] :
( ( produc175634133007206835_int_o @ ( produc4947309494688390418_int_o @ F ) )
= F ) ).
% curry_case_prod
thf(fact_228_curry__case__prod,axiom,
! [F: int > int > int] :
( ( produc1016772743285680337nt_int @ ( produc8211389475949308722nt_int @ F ) )
= F ) ).
% curry_case_prod
thf(fact_229_mem__case__prodE,axiom,
! [Z2: $o,C: nat > num > set_o,P: product_prod_nat_num] :
( ( member_o @ Z2 @ ( produc836176033315069408_set_o @ C @ P ) )
=> ~ ! [X: nat,Y: num] :
( ( P
= ( product_Pair_nat_num @ X @ Y ) )
=> ~ ( member_o @ Z2 @ ( C @ X @ Y ) ) ) ) ).
% mem_case_prodE
thf(fact_230_mem__case__prodE,axiom,
! [Z2: nat,C: nat > num > set_nat,P: product_prod_nat_num] :
( ( member_nat @ Z2 @ ( produc4130284055270567454et_nat @ C @ P ) )
=> ~ ! [X: nat,Y: num] :
( ( P
= ( product_Pair_nat_num @ X @ Y ) )
=> ~ ( member_nat @ Z2 @ ( C @ X @ Y ) ) ) ) ).
% mem_case_prodE
thf(fact_231_mem__case__prodE,axiom,
! [Z2: int,C: nat > num > set_int,P: product_prod_nat_num] :
( ( member_int @ Z2 @ ( produc9175805072616146554et_int @ C @ P ) )
=> ~ ! [X: nat,Y: num] :
( ( P
= ( product_Pair_nat_num @ X @ Y ) )
=> ~ ( member_int @ Z2 @ ( C @ X @ Y ) ) ) ) ).
% mem_case_prodE
thf(fact_232_mem__case__prodE,axiom,
! [Z2: $o,C: nat > nat > set_o,P: product_prod_nat_nat] :
( ( member_o @ Z2 @ ( produc59986286002894506_set_o @ C @ P ) )
=> ~ ! [X: nat,Y: nat] :
( ( P
= ( product_Pair_nat_nat @ X @ Y ) )
=> ~ ( member_o @ Z2 @ ( C @ X @ Y ) ) ) ) ).
% mem_case_prodE
thf(fact_233_mem__case__prodE,axiom,
! [Z2: nat,C: nat > nat > set_nat,P: product_prod_nat_nat] :
( ( member_nat @ Z2 @ ( produc6189476227299908564et_nat @ C @ P ) )
=> ~ ! [X: nat,Y: nat] :
( ( P
= ( product_Pair_nat_nat @ X @ Y ) )
=> ~ ( member_nat @ Z2 @ ( C @ X @ Y ) ) ) ) ).
% mem_case_prodE
thf(fact_234_mem__case__prodE,axiom,
! [Z2: int,C: nat > nat > set_int,P: product_prod_nat_nat] :
( ( member_int @ Z2 @ ( produc2011625207790711856et_int @ C @ P ) )
=> ~ ! [X: nat,Y: nat] :
( ( P
= ( product_Pair_nat_nat @ X @ Y ) )
=> ~ ( member_int @ Z2 @ ( C @ X @ Y ) ) ) ) ).
% mem_case_prodE
thf(fact_235_mem__case__prodE,axiom,
! [Z2: $o,C: int > int > set_o,P: product_prod_int_int] :
( ( member_o @ Z2 @ ( produc4257766111578684402_set_o @ C @ P ) )
=> ~ ! [X: int,Y: int] :
( ( P
= ( product_Pair_int_int @ X @ Y ) )
=> ~ ( member_o @ Z2 @ ( C @ X @ Y ) ) ) ) ).
% mem_case_prodE
thf(fact_236_mem__case__prodE,axiom,
! [Z2: nat,C: int > int > set_nat,P: product_prod_int_int] :
( ( member_nat @ Z2 @ ( produc4251311855443802252et_nat @ C @ P ) )
=> ~ ! [X: int,Y: int] :
( ( P
= ( product_Pair_int_int @ X @ Y ) )
=> ~ ( member_nat @ Z2 @ ( C @ X @ Y ) ) ) ) ).
% mem_case_prodE
thf(fact_237_mem__case__prodE,axiom,
! [Z2: int,C: int > int > set_int,P: product_prod_int_int] :
( ( member_int @ Z2 @ ( produc73460835934605544et_int @ C @ P ) )
=> ~ ! [X: int,Y: int] :
( ( P
= ( product_Pair_int_int @ X @ Y ) )
=> ~ ( member_int @ Z2 @ ( C @ X @ Y ) ) ) ) ).
% mem_case_prodE
thf(fact_238_mem__case__prodE,axiom,
! [Z2: product_prod_nat_nat,C: nat > num > set_Pr1261947904930325089at_nat,P: product_prod_nat_num] :
( ( member8440522571783428010at_nat @ Z2 @ ( produc859037500352796105at_nat @ C @ P ) )
=> ~ ! [X: nat,Y: num] :
( ( P
= ( product_Pair_nat_num @ X @ Y ) )
=> ~ ( member8440522571783428010at_nat @ Z2 @ ( C @ X @ Y ) ) ) ) ).
% mem_case_prodE
thf(fact_239_old_Oprod_Ocase,axiom,
! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,X1: nat,X22: nat] :
( ( produc27273713700761075at_nat @ F @ ( product_Pair_nat_nat @ X1 @ X22 ) )
= ( F @ X1 @ X22 ) ) ).
% old.prod.case
thf(fact_240_old_Oprod_Ocase,axiom,
! [F: nat > nat > product_prod_nat_nat > $o,X1: nat,X22: nat] :
( ( produc8739625826339149834_nat_o @ F @ ( product_Pair_nat_nat @ X1 @ X22 ) )
= ( F @ X1 @ X22 ) ) ).
% old.prod.case
thf(fact_241_old_Oprod_Ocase,axiom,
! [F: int > int > product_prod_int_int,X1: int,X22: int] :
( ( produc4245557441103728435nt_int @ F @ ( product_Pair_int_int @ X1 @ X22 ) )
= ( F @ X1 @ X22 ) ) ).
% old.prod.case
thf(fact_242_old_Oprod_Ocase,axiom,
! [F: int > int > $o,X1: int,X22: int] :
( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ X1 @ X22 ) )
= ( F @ X1 @ X22 ) ) ).
% old.prod.case
thf(fact_243_old_Oprod_Ocase,axiom,
! [F: int > int > int,X1: int,X22: int] :
( ( produc8211389475949308722nt_int @ F @ ( product_Pair_int_int @ X1 @ X22 ) )
= ( F @ X1 @ X22 ) ) ).
% old.prod.case
thf(fact_244_curryD,axiom,
! [F: produc1319942482725812455at_nat > $o,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
( ( produc447603880154497357_nat_o @ F @ A @ B )
=> ( F @ ( produc9060074326276436823at_nat @ A @ B ) ) ) ).
% curryD
thf(fact_245_curryD,axiom,
! [F: produc3843707927480180839at_nat > $o,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
( ( produc7820828192191229133_nat_o @ F @ A @ B )
=> ( F @ ( produc2922128104949294807at_nat @ A @ B ) ) ) ).
% curryD
thf(fact_246_curryD,axiom,
! [F: product_prod_nat_num > $o,A: nat,B: num] :
( ( produc156083480235303841_num_o @ F @ A @ B )
=> ( F @ ( product_Pair_nat_num @ A @ B ) ) ) ).
% curryD
thf(fact_247_curryD,axiom,
! [F: product_prod_nat_nat > $o,A: nat,B: nat] :
( ( produc1310100445399344235_nat_o @ F @ A @ B )
=> ( F @ ( product_Pair_nat_nat @ A @ B ) ) ) ).
% curryD
thf(fact_248_curryD,axiom,
! [F: product_prod_int_int > $o,A: int,B: int] :
( ( produc175634133007206835_int_o @ F @ A @ B )
=> ( F @ ( product_Pair_int_int @ A @ B ) ) ) ).
% curryD
thf(fact_249_curryE,axiom,
! [F: produc1319942482725812455at_nat > $o,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
( ( produc447603880154497357_nat_o @ F @ A @ B )
=> ( F @ ( produc9060074326276436823at_nat @ A @ B ) ) ) ).
% curryE
thf(fact_250_curryE,axiom,
! [F: produc3843707927480180839at_nat > $o,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
( ( produc7820828192191229133_nat_o @ F @ A @ B )
=> ( F @ ( produc2922128104949294807at_nat @ A @ B ) ) ) ).
% curryE
thf(fact_251_curryE,axiom,
! [F: product_prod_nat_num > $o,A: nat,B: num] :
( ( produc156083480235303841_num_o @ F @ A @ B )
=> ( F @ ( product_Pair_nat_num @ A @ B ) ) ) ).
% curryE
thf(fact_252_curryE,axiom,
! [F: product_prod_nat_nat > $o,A: nat,B: nat] :
( ( produc1310100445399344235_nat_o @ F @ A @ B )
=> ( F @ ( product_Pair_nat_nat @ A @ B ) ) ) ).
% curryE
thf(fact_253_curryE,axiom,
! [F: product_prod_int_int > $o,A: int,B: int] :
( ( produc175634133007206835_int_o @ F @ A @ B )
=> ( F @ ( product_Pair_int_int @ A @ B ) ) ) ).
% curryE
thf(fact_254_Product__Type_OCollect__case__prodD,axiom,
! [X2: produc3843707927480180839at_nat,A4: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o] :
( ( member8757157785044589968at_nat @ X2 @ ( collec6321179662152712658at_nat @ ( produc410239310623530412_nat_o @ A4 ) ) )
=> ( A4 @ ( produc6011863565407846595at_nat @ X2 ) @ ( produc2937545259423399429at_nat @ X2 ) ) ) ).
% Product_Type.Collect_case_prodD
thf(fact_255_Product__Type_OCollect__case__prodD,axiom,
! [X2: product_prod_nat_nat,A4: nat > nat > $o] :
( ( member8440522571783428010at_nat @ X2 @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ A4 ) ) )
=> ( A4 @ ( product_fst_nat_nat @ X2 ) @ ( product_snd_nat_nat @ X2 ) ) ) ).
% Product_Type.Collect_case_prodD
thf(fact_256_Product__Type_OCollect__case__prodD,axiom,
! [X2: produc7812382424243267045t_unit,A4: ref_a > heap_e7401611519738050253t_unit > $o] :
( ( member7849944260330398460t_unit @ X2 @ ( collec3252306575900995386t_unit @ ( produc4924168586796184916unit_o @ A4 ) ) )
=> ( A4 @ ( produc473321116828791851t_unit @ X2 ) @ ( produc1126150082675451753t_unit @ X2 ) ) ) ).
% Product_Type.Collect_case_prodD
thf(fact_257_Product__Type_OCollect__case__prodD,axiom,
! [X2: product_prod_int_int,A4: int > int > $o] :
( ( member5262025264175285858nt_int @ X2 @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ A4 ) ) )
=> ( A4 @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ X2 ) ) ) ).
% Product_Type.Collect_case_prodD
thf(fact_258_effect__returnI,axiom,
! [H: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,X2: ref_a] :
( ( H = H2 )
=> ( heap_effect_ref_a @ ( heap_return_ref_a @ X2 ) @ H @ H2 @ X2 ) ) ).
% effect_returnI
thf(fact_259_effect__returnE,axiom,
! [X2: ref_a,H: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,R: ref_a] :
( ( heap_effect_ref_a @ ( heap_return_ref_a @ X2 ) @ H @ H2 @ R )
=> ~ ( ( R = X2 )
=> ( H2 != H ) ) ) ).
% effect_returnE
thf(fact_260_split__beta,axiom,
( produc27273713700761075at_nat
= ( ^ [F2: nat > nat > product_prod_nat_nat > product_prod_nat_nat,Prod3: product_prod_nat_nat] : ( F2 @ ( product_fst_nat_nat @ Prod3 ) @ ( product_snd_nat_nat @ Prod3 ) ) ) ) ).
% split_beta
thf(fact_261_split__beta,axiom,
( produc8739625826339149834_nat_o
= ( ^ [F2: nat > nat > product_prod_nat_nat > $o,Prod3: product_prod_nat_nat] : ( F2 @ ( product_fst_nat_nat @ Prod3 ) @ ( product_snd_nat_nat @ Prod3 ) ) ) ) ).
% split_beta
thf(fact_262_split__beta,axiom,
( produc4245557441103728435nt_int
= ( ^ [F2: int > int > product_prod_int_int,Prod3: product_prod_int_int] : ( F2 @ ( product_fst_int_int @ Prod3 ) @ ( product_snd_int_int @ Prod3 ) ) ) ) ).
% split_beta
thf(fact_263_split__beta,axiom,
( produc4947309494688390418_int_o
= ( ^ [F2: int > int > $o,Prod3: product_prod_int_int] : ( F2 @ ( product_fst_int_int @ Prod3 ) @ ( product_snd_int_int @ Prod3 ) ) ) ) ).
% split_beta
thf(fact_264_split__beta,axiom,
( produc8211389475949308722nt_int
= ( ^ [F2: int > int > int,Prod3: product_prod_int_int] : ( F2 @ ( product_fst_int_int @ Prod3 ) @ ( product_snd_int_int @ Prod3 ) ) ) ) ).
% split_beta
thf(fact_265_case__prod__beta,axiom,
( produc27273713700761075at_nat
= ( ^ [F2: nat > nat > product_prod_nat_nat > product_prod_nat_nat,P3: product_prod_nat_nat] : ( F2 @ ( product_fst_nat_nat @ P3 ) @ ( product_snd_nat_nat @ P3 ) ) ) ) ).
% case_prod_beta
thf(fact_266_case__prod__beta,axiom,
( produc8739625826339149834_nat_o
= ( ^ [F2: nat > nat > product_prod_nat_nat > $o,P3: product_prod_nat_nat] : ( F2 @ ( product_fst_nat_nat @ P3 ) @ ( product_snd_nat_nat @ P3 ) ) ) ) ).
% case_prod_beta
thf(fact_267_case__prod__beta,axiom,
( produc4245557441103728435nt_int
= ( ^ [F2: int > int > product_prod_int_int,P3: product_prod_int_int] : ( F2 @ ( product_fst_int_int @ P3 ) @ ( product_snd_int_int @ P3 ) ) ) ) ).
% case_prod_beta
thf(fact_268_case__prod__beta,axiom,
( produc4947309494688390418_int_o
= ( ^ [F2: int > int > $o,P3: product_prod_int_int] : ( F2 @ ( product_fst_int_int @ P3 ) @ ( product_snd_int_int @ P3 ) ) ) ) ).
% case_prod_beta
thf(fact_269_case__prod__beta,axiom,
( produc8211389475949308722nt_int
= ( ^ [F2: int > int > int,P3: product_prod_int_int] : ( F2 @ ( product_fst_int_int @ P3 ) @ ( product_snd_int_int @ P3 ) ) ) ) ).
% case_prod_beta
thf(fact_270_internal__case__prod__def,axiom,
produc1854806715440696265at_nat = produc27273713700761075at_nat ).
% internal_case_prod_def
thf(fact_271_internal__case__prod__def,axiom,
produc4780622933104268256_nat_o = produc8739625826339149834_nat_o ).
% internal_case_prod_def
thf(fact_272_internal__case__prod__def,axiom,
produc297006045350968285nt_int = produc4245557441103728435nt_int ).
% internal_case_prod_def
thf(fact_273_internal__case__prod__def,axiom,
produc8005341501107743676_int_o = produc4947309494688390418_int_o ).
% internal_case_prod_def
thf(fact_274_internal__case__prod__def,axiom,
produc7926200574084438792nt_int = produc8211389475949308722nt_int ).
% internal_case_prod_def
thf(fact_275_effect__updateI,axiom,
! [H2: heap_e7401611519738050253t_unit,R: ref_a,V: a,H: heap_e7401611519738050253t_unit,X2: product_unit] :
( ( H2
= ( set_a @ R @ V @ H ) )
=> ( heap_e2830298082929533060t_unit @ ( update_a @ R @ V ) @ H @ H2 @ X2 ) ) ).
% effect_updateI
thf(fact_276_split__cong,axiom,
! [Q: product_prod_nat_nat,F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,G: nat > nat > product_prod_nat_nat > product_prod_nat_nat,P: product_prod_nat_nat] :
( ! [X: nat,Y: nat] :
( ( ( product_Pair_nat_nat @ X @ Y )
= Q )
=> ( ( F @ X @ Y )
= ( G @ X @ Y ) ) )
=> ( ( P = Q )
=> ( ( produc27273713700761075at_nat @ F @ P )
= ( produc27273713700761075at_nat @ G @ Q ) ) ) ) ).
% split_cong
thf(fact_277_split__cong,axiom,
! [Q: product_prod_nat_nat,F: nat > nat > product_prod_nat_nat > $o,G: nat > nat > product_prod_nat_nat > $o,P: product_prod_nat_nat] :
( ! [X: nat,Y: nat] :
( ( ( product_Pair_nat_nat @ X @ Y )
= Q )
=> ( ( F @ X @ Y )
= ( G @ X @ Y ) ) )
=> ( ( P = Q )
=> ( ( produc8739625826339149834_nat_o @ F @ P )
= ( produc8739625826339149834_nat_o @ G @ Q ) ) ) ) ).
% split_cong
thf(fact_278_split__cong,axiom,
! [Q: product_prod_int_int,F: int > int > product_prod_int_int,G: int > int > product_prod_int_int,P: product_prod_int_int] :
( ! [X: int,Y: int] :
( ( ( product_Pair_int_int @ X @ Y )
= Q )
=> ( ( F @ X @ Y )
= ( G @ X @ Y ) ) )
=> ( ( P = Q )
=> ( ( produc4245557441103728435nt_int @ F @ P )
= ( produc4245557441103728435nt_int @ G @ Q ) ) ) ) ).
% split_cong
thf(fact_279_split__cong,axiom,
! [Q: product_prod_int_int,F: int > int > $o,G: int > int > $o,P: product_prod_int_int] :
( ! [X: int,Y: int] :
( ( ( product_Pair_int_int @ X @ Y )
= Q )
=> ( ( F @ X @ Y )
= ( G @ X @ Y ) ) )
=> ( ( P = Q )
=> ( ( produc4947309494688390418_int_o @ F @ P )
= ( produc4947309494688390418_int_o @ G @ Q ) ) ) ) ).
% split_cong
thf(fact_280_split__cong,axiom,
! [Q: product_prod_int_int,F: int > int > int,G: int > int > int,P: product_prod_int_int] :
( ! [X: int,Y: int] :
( ( ( product_Pair_int_int @ X @ Y )
= Q )
=> ( ( F @ X @ Y )
= ( G @ X @ Y ) ) )
=> ( ( P = Q )
=> ( ( produc8211389475949308722nt_int @ F @ P )
= ( produc8211389475949308722nt_int @ G @ Q ) ) ) ) ).
% split_cong
thf(fact_281_csquare__fstOp__sndOp,axiom,
! [F: ( product_prod_nat_nat > int > $o ) > produc4196048018000912732at_int > $o,P2: product_prod_nat_nat > product_prod_nat_nat > $o,Q2: product_prod_nat_nat > int > $o] : ( bNF_cs7194553038297458973at_int @ ( collec2851339944984316337at_int @ ( F @ ( relcom8911704206367525917at_int @ P2 @ Q2 ) ) ) @ produc6408287024330202629at_nat @ produc4619704898428564714at_int @ ( bNF_fs6342258632037972510at_int @ P2 @ Q2 ) @ ( bNF_sn649997494946348640at_int @ P2 @ Q2 ) ) ).
% csquare_fstOp_sndOp
thf(fact_282_csquare__fstOp__sndOp,axiom,
! [F: ( int > code_integer > $o ) > produc2963269191615333738nteger > $o,P2: int > int > $o,Q2: int > code_integer > $o] : ( bNF_cs4943907971211754724nteger @ ( collec7215431352809750357nteger @ ( F @ ( relcom967497705283160040nteger @ P2 @ Q2 ) ) ) @ product_snd_int_int @ produc45500017171379470nteger @ ( bNF_fs4543836202474805097nteger @ P2 @ Q2 ) @ ( bNF_sn8000174127721201323nteger @ P2 @ Q2 ) ) ).
% csquare_fstOp_sndOp
thf(fact_283_effect__tapI,axiom,
! [H2: heap_e7401611519738050253t_unit,H: heap_e7401611519738050253t_unit,R: ref_a,F: heap_e7401611519738050253t_unit > ref_a] :
( ( H2 = H )
=> ( ( R
= ( F @ H ) )
=> ( heap_effect_ref_a @ ( heap_tap_ref_a @ F ) @ H @ H2 @ R ) ) ) ).
% effect_tapI
thf(fact_284_effect__tapE,axiom,
! [F: heap_e7401611519738050253t_unit > ref_a,H: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,R: ref_a] :
( ( heap_effect_ref_a @ ( heap_tap_ref_a @ F ) @ H @ H2 @ R )
=> ~ ( ( H2 = H )
=> ( R
!= ( F @ H ) ) ) ) ).
% effect_tapE
thf(fact_285_pair__in__swap__image,axiom,
! [Y3: num,X2: nat,A4: set_Pr6200539531224447659at_num] :
( ( member6570852975090897396um_nat @ ( product_Pair_num_nat @ Y3 @ X2 ) @ ( image_2347970371937200581um_nat @ product_swap_nat_num @ A4 ) )
= ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X2 @ Y3 ) @ A4 ) ) ).
% pair_in_swap_image
thf(fact_286_pair__in__swap__image,axiom,
! [Y3: set_Pr4329608150637261639at_nat,X2: set_Pr4329608150637261639at_nat,A4: set_Pr7459493094073627847at_nat] :
( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y3 @ X2 ) @ ( image_3313167189156628421at_nat @ produc5396952528824381687at_nat @ A4 ) )
= ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X2 @ Y3 ) @ A4 ) ) ).
% pair_in_swap_image
thf(fact_287_pair__in__swap__image,axiom,
! [Y3: set_Pr1261947904930325089at_nat,X2: set_Pr1261947904930325089at_nat,A4: set_Pr4329608150637261639at_nat] :
( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y3 @ X2 ) @ ( image_960689587819713477at_nat @ produc3946833401584108663at_nat @ A4 ) )
= ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y3 ) @ A4 ) ) ).
% pair_in_swap_image
thf(fact_288_pair__in__swap__image,axiom,
! [Y3: nat,X2: num,A4: set_Pr3280342998896498603um_nat] :
( ( member9148766508732265716at_num @ ( product_Pair_nat_num @ Y3 @ X2 ) @ ( image_7762062833379733445at_num @ product_swap_num_nat @ A4 ) )
= ( member6570852975090897396um_nat @ ( product_Pair_num_nat @ X2 @ Y3 ) @ A4 ) ) ).
% pair_in_swap_image
thf(fact_289_pair__in__swap__image,axiom,
! [Y3: nat,X2: nat,A4: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X2 ) @ ( image_5168914502847457605at_nat @ product_swap_nat_nat @ A4 ) )
= ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ A4 ) ) ).
% pair_in_swap_image
thf(fact_290_pair__in__swap__image,axiom,
! [Y3: int,X2: int,A4: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y3 @ X2 ) @ ( image_2653370878348428101nt_int @ product_swap_int_int @ A4 ) )
= ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y3 ) @ A4 ) ) ).
% pair_in_swap_image
thf(fact_291_snd__sndOp,axiom,
! [P2: ref_a > ref_a > $o,Q2: ref_a > heap_e7401611519738050253t_unit > $o] :
( produc1126150082675451753t_unit
= ( comp_P8019343326919024234t_unit @ produc1126150082675451753t_unit @ ( bNF_sn8323453914685745821t_unit @ P2 @ Q2 ) ) ) ).
% snd_sndOp
thf(fact_292_comp__cong,axiom,
! [F: $o > $o,G: nat > $o,X2: nat,F3: $o > $o,G2: nat > $o,X3: nat] :
( ( ( F @ ( G @ X2 ) )
= ( F3 @ ( G2 @ X3 ) ) )
=> ( ( comp_o_o_nat @ F @ G @ X2 )
= ( comp_o_o_nat @ F3 @ G2 @ X3 ) ) ) ).
% comp_cong
thf(fact_293_comp__cong,axiom,
! [F: nat > num,G: int > nat,X2: int,F3: nat > num,G2: int > nat,X3: int] :
( ( ( F @ ( G @ X2 ) )
= ( F3 @ ( G2 @ X3 ) ) )
=> ( ( comp_nat_num_int @ F @ G @ X2 )
= ( comp_nat_num_int @ F3 @ G2 @ X3 ) ) ) ).
% comp_cong
thf(fact_294_comp__cong,axiom,
! [F: nat > nat,G: nat > nat,X2: nat,F3: nat > nat,G2: nat > nat,X3: nat] :
( ( ( F @ ( G @ X2 ) )
= ( F3 @ ( G2 @ X3 ) ) )
=> ( ( comp_nat_nat_nat @ F @ G @ X2 )
= ( comp_nat_nat_nat @ F3 @ G2 @ X3 ) ) ) ).
% comp_cong
thf(fact_295_comp__cong,axiom,
! [F: nat > nat,G: nat > nat,X2: nat,F3: int > nat,G2: int > int,X3: int] :
( ( ( F @ ( G @ X2 ) )
= ( F3 @ ( G2 @ X3 ) ) )
=> ( ( comp_nat_nat_nat @ F @ G @ X2 )
= ( comp_int_nat_int @ F3 @ G2 @ X3 ) ) ) ).
% comp_cong
thf(fact_296_comp__cong,axiom,
! [F: int > nat,G: int > int,X2: int,F3: nat > nat,G2: nat > nat,X3: nat] :
( ( ( F @ ( G @ X2 ) )
= ( F3 @ ( G2 @ X3 ) ) )
=> ( ( comp_int_nat_int @ F @ G @ X2 )
= ( comp_nat_nat_nat @ F3 @ G2 @ X3 ) ) ) ).
% comp_cong
thf(fact_297_comp__cong,axiom,
! [F: int > nat,G: int > int,X2: int,F3: int > nat,G2: int > int,X3: int] :
( ( ( F @ ( G @ X2 ) )
= ( F3 @ ( G2 @ X3 ) ) )
=> ( ( comp_int_nat_int @ F @ G @ X2 )
= ( comp_int_nat_int @ F3 @ G2 @ X3 ) ) ) ).
% comp_cong
thf(fact_298_comp__cong,axiom,
! [F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer > code_integer,X2: code_integer,F3: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G2: code_integer > code_integer > code_integer,X3: code_integer] :
( ( ( F @ ( G @ X2 ) )
= ( F3 @ ( G2 @ X3 ) ) )
=> ( ( comp_C8797469213163452608nteger @ F @ G @ X2 )
= ( comp_C8797469213163452608nteger @ F3 @ G2 @ X3 ) ) ) ).
% comp_cong
thf(fact_299_comp__apply__eq,axiom,
! [F: $o > $o,G: nat > $o,X2: nat,H: $o > $o,K: nat > $o] :
( ( ( F @ ( G @ X2 ) )
= ( H @ ( K @ X2 ) ) )
=> ( ( comp_o_o_nat @ F @ G @ X2 )
= ( comp_o_o_nat @ H @ K @ X2 ) ) ) ).
% comp_apply_eq
thf(fact_300_comp__apply__eq,axiom,
! [F: nat > num,G: int > nat,X2: int,H: nat > num,K: int > nat] :
( ( ( F @ ( G @ X2 ) )
= ( H @ ( K @ X2 ) ) )
=> ( ( comp_nat_num_int @ F @ G @ X2 )
= ( comp_nat_num_int @ H @ K @ X2 ) ) ) ).
% comp_apply_eq
thf(fact_301_comp__apply__eq,axiom,
! [F: nat > nat,G: nat > nat,X2: nat,H: nat > nat,K: nat > nat] :
( ( ( F @ ( G @ X2 ) )
= ( H @ ( K @ X2 ) ) )
=> ( ( comp_nat_nat_nat @ F @ G @ X2 )
= ( comp_nat_nat_nat @ H @ K @ X2 ) ) ) ).
% comp_apply_eq
thf(fact_302_comp__apply__eq,axiom,
! [F: int > nat,G: int > int,X2: int,H: int > nat,K: int > int] :
( ( ( F @ ( G @ X2 ) )
= ( H @ ( K @ X2 ) ) )
=> ( ( comp_int_nat_int @ F @ G @ X2 )
= ( comp_int_nat_int @ H @ K @ X2 ) ) ) ).
% comp_apply_eq
thf(fact_303_comp__apply__eq,axiom,
! [F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer > code_integer,X2: code_integer,H: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,K: code_integer > code_integer > code_integer] :
( ( ( F @ ( G @ X2 ) )
= ( H @ ( K @ X2 ) ) )
=> ( ( comp_C8797469213163452608nteger @ F @ G @ X2 )
= ( comp_C8797469213163452608nteger @ H @ K @ X2 ) ) ) ).
% comp_apply_eq
thf(fact_304_pick__middlep,axiom,
! [P2: product_prod_nat_nat > product_prod_nat_nat > $o,Q2: product_prod_nat_nat > int > $o,A: product_prod_nat_nat,C: int] :
( ( relcom8911704206367525917at_int @ P2 @ Q2 @ A @ C )
=> ( ( P2 @ A @ ( bNF_pi770922605693484880at_int @ P2 @ Q2 @ A @ C ) )
& ( Q2 @ ( bNF_pi770922605693484880at_int @ P2 @ Q2 @ A @ C ) @ C ) ) ) ).
% pick_middlep
thf(fact_305_pick__middlep,axiom,
! [P2: int > int > $o,Q2: int > code_integer > $o,A: int,C: code_integer] :
( ( relcom967497705283160040nteger @ P2 @ Q2 @ A @ C )
=> ( ( P2 @ A @ ( bNF_pi4515785505327359899nteger @ P2 @ Q2 @ A @ C ) )
& ( Q2 @ ( bNF_pi4515785505327359899nteger @ P2 @ Q2 @ A @ C ) @ C ) ) ) ).
% pick_middlep
thf(fact_306_sndOp__in,axiom,
! [Ac: produc3843707927480180839at_nat,P2: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o,Q2: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o] :
( ( member8757157785044589968at_nat @ Ac @ ( collec6321179662152712658at_nat @ ( produc410239310623530412_nat_o @ ( relcom8363394528208558896at_nat @ P2 @ Q2 ) ) ) )
=> ( member8757157785044589968at_nat @ ( bNF_sn1857951156667551021at_nat @ P2 @ Q2 @ Ac ) @ ( collec6321179662152712658at_nat @ ( produc410239310623530412_nat_o @ Q2 ) ) ) ) ).
% sndOp_in
thf(fact_307_sndOp__in,axiom,
! [Ac: product_prod_nat_nat,P2: nat > nat > $o,Q2: nat > nat > $o] :
( ( member8440522571783428010at_nat @ Ac @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ( relcompp_nat_nat_nat @ P2 @ Q2 ) ) ) )
=> ( member8440522571783428010at_nat @ ( bNF_sn8916957246138178918at_nat @ P2 @ Q2 @ Ac ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ Q2 ) ) ) ) ).
% sndOp_in
thf(fact_308_sndOp__in,axiom,
! [Ac: produc4196048018000912732at_int,P2: product_prod_nat_nat > product_prod_nat_nat > $o,Q2: product_prod_nat_nat > int > $o] :
( ( member8394280151737582451at_int @ Ac @ ( collec2851339944984316337at_int @ ( produc6937227050501860053_int_o @ ( relcom8911704206367525917at_int @ P2 @ Q2 ) ) ) )
=> ( member8394280151737582451at_int @ ( bNF_sn649997494946348640at_int @ P2 @ Q2 @ Ac ) @ ( collec2851339944984316337at_int @ ( produc6937227050501860053_int_o @ Q2 ) ) ) ) ).
% sndOp_in
thf(fact_309_sndOp__in,axiom,
! [Ac: produc2963269191615333738nteger,P2: int > int > $o,Q2: int > code_integer > $o] :
( ( member4345257455949424659nteger @ Ac @ ( collec7215431352809750357nteger @ ( produc540600059291021473eger_o @ ( relcom967497705283160040nteger @ P2 @ Q2 ) ) ) )
=> ( member4345257455949424659nteger @ ( bNF_sn8000174127721201323nteger @ P2 @ Q2 @ Ac ) @ ( collec7215431352809750357nteger @ ( produc540600059291021473eger_o @ Q2 ) ) ) ) ).
% sndOp_in
thf(fact_310_sndOp__in,axiom,
! [Ac: product_prod_int_int,P2: int > int > $o,Q2: int > int > $o] :
( ( member5262025264175285858nt_int @ Ac @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ ( relcompp_int_int_int @ P2 @ Q2 ) ) ) )
=> ( member5262025264175285858nt_int @ ( bNF_sn1062102010912252026nt_int @ P2 @ Q2 @ Ac ) @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ Q2 ) ) ) ) ).
% sndOp_in
thf(fact_311_fstOp__in,axiom,
! [Ac: produc3843707927480180839at_nat,P2: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o,Q2: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o] :
( ( member8757157785044589968at_nat @ Ac @ ( collec6321179662152712658at_nat @ ( produc410239310623530412_nat_o @ ( relcom8363394528208558896at_nat @ P2 @ Q2 ) ) ) )
=> ( member8757157785044589968at_nat @ ( bNF_fs1946732770515719151at_nat @ P2 @ Q2 @ Ac ) @ ( collec6321179662152712658at_nat @ ( produc410239310623530412_nat_o @ P2 ) ) ) ) ).
% fstOp_in
thf(fact_312_fstOp__in,axiom,
! [Ac: product_prod_nat_nat,P2: nat > nat > $o,Q2: nat > nat > $o] :
( ( member8440522571783428010at_nat @ Ac @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ( relcompp_nat_nat_nat @ P2 @ Q2 ) ) ) )
=> ( member8440522571783428010at_nat @ ( bNF_fs6799373675401686564at_nat @ P2 @ Q2 @ Ac ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ P2 ) ) ) ) ).
% fstOp_in
thf(fact_313_fstOp__in,axiom,
! [Ac: produc4196048018000912732at_int,P2: product_prod_nat_nat > product_prod_nat_nat > $o,Q2: product_prod_nat_nat > int > $o] :
( ( member8394280151737582451at_int @ Ac @ ( collec2851339944984316337at_int @ ( produc6937227050501860053_int_o @ ( relcom8911704206367525917at_int @ P2 @ Q2 ) ) ) )
=> ( member8206827879206165904at_nat @ ( bNF_fs6342258632037972510at_int @ P2 @ Q2 @ Ac ) @ ( collec7088162979684241874at_nat @ ( produc6590410687421337004_nat_o @ P2 ) ) ) ) ).
% fstOp_in
thf(fact_314_fstOp__in,axiom,
! [Ac: produc2963269191615333738nteger,P2: int > int > $o,Q2: int > code_integer > $o] :
( ( member4345257455949424659nteger @ Ac @ ( collec7215431352809750357nteger @ ( produc540600059291021473eger_o @ ( relcom967497705283160040nteger @ P2 @ Q2 ) ) ) )
=> ( member5262025264175285858nt_int @ ( bNF_fs4543836202474805097nteger @ P2 @ Q2 @ Ac ) @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ P2 ) ) ) ) ).
% fstOp_in
thf(fact_315_fstOp__in,axiom,
! [Ac: product_prod_int_int,P2: int > int > $o,Q2: int > int > $o] :
( ( member5262025264175285858nt_int @ Ac @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ ( relcompp_int_int_int @ P2 @ Q2 ) ) ) )
=> ( member5262025264175285858nt_int @ ( bNF_fs8167890477030535480nt_int @ P2 @ Q2 @ Ac ) @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ P2 ) ) ) ) ).
% fstOp_in
thf(fact_316_effect__bindE,axiom,
! [F: heap_Heap_ref_a,G: ref_a > heap_Heap_ref_a,H: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,R2: ref_a] :
( ( heap_effect_ref_a @ ( heap_b146640354529102224_ref_a @ F @ G ) @ H @ H3 @ R2 )
=> ~ ! [H4: heap_e7401611519738050253t_unit,R4: ref_a] :
( ( heap_effect_ref_a @ F @ H @ H4 @ R4 )
=> ~ ( heap_effect_ref_a @ ( G @ R4 ) @ H4 @ H3 @ R2 ) ) ) ).
% effect_bindE
thf(fact_317_effect__bindI,axiom,
! [F: heap_Heap_ref_a,H: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,R: ref_a,G: ref_a > heap_Heap_ref_a,H3: heap_e7401611519738050253t_unit,R2: ref_a] :
( ( heap_effect_ref_a @ F @ H @ H2 @ R )
=> ( ( heap_effect_ref_a @ ( G @ R ) @ H2 @ H3 @ R2 )
=> ( heap_effect_ref_a @ ( heap_b146640354529102224_ref_a @ F @ G ) @ H @ H3 @ R2 ) ) ) ).
% effect_bindI
thf(fact_318_fst__fstOp,axiom,
! [P2: ref_a > heap_e7401611519738050253t_unit > $o,Q2: heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit > $o] :
( produc473321116828791851t_unit
= ( comp_P4044372497395451596t_unit @ produc473321116828791851t_unit @ ( bNF_fs409234558581511823t_unit @ P2 @ Q2 ) ) ) ).
% fst_fstOp
thf(fact_319_comp__apply,axiom,
( comp_o_o_nat
= ( ^ [F2: $o > $o,G3: nat > $o,X4: nat] : ( F2 @ ( G3 @ X4 ) ) ) ) ).
% comp_apply
thf(fact_320_comp__apply,axiom,
( comp_nat_num_int
= ( ^ [F2: nat > num,G3: int > nat,X4: int] : ( F2 @ ( G3 @ X4 ) ) ) ) ).
% comp_apply
thf(fact_321_comp__apply,axiom,
( comp_nat_nat_nat
= ( ^ [F2: nat > nat,G3: nat > nat,X4: nat] : ( F2 @ ( G3 @ X4 ) ) ) ) ).
% comp_apply
thf(fact_322_comp__apply,axiom,
( comp_int_nat_int
= ( ^ [F2: int > nat,G3: int > int,X4: int] : ( F2 @ ( G3 @ X4 ) ) ) ) ).
% comp_apply
thf(fact_323_comp__apply,axiom,
( comp_C8797469213163452608nteger
= ( ^ [F2: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G3: code_integer > code_integer > code_integer,X4: code_integer] : ( F2 @ ( G3 @ X4 ) ) ) ) ).
% comp_apply
thf(fact_324_image__eqI,axiom,
! [B: $o,F: $o > $o,X2: $o,A4: set_o] :
( ( B
= ( F @ X2 ) )
=> ( ( member_o @ X2 @ A4 )
=> ( member_o @ B @ ( image_o_o @ F @ A4 ) ) ) ) ).
% image_eqI
thf(fact_325_image__eqI,axiom,
! [B: nat,F: $o > nat,X2: $o,A4: set_o] :
( ( B
= ( F @ X2 ) )
=> ( ( member_o @ X2 @ A4 )
=> ( member_nat @ B @ ( image_o_nat @ F @ A4 ) ) ) ) ).
% image_eqI
thf(fact_326_image__eqI,axiom,
! [B: int,F: $o > int,X2: $o,A4: set_o] :
( ( B
= ( F @ X2 ) )
=> ( ( member_o @ X2 @ A4 )
=> ( member_int @ B @ ( image_o_int @ F @ A4 ) ) ) ) ).
% image_eqI
thf(fact_327_image__eqI,axiom,
! [B: rat,F: nat > rat,X2: nat,A4: set_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A4 )
=> ( member_rat @ B @ ( image_nat_rat @ F @ A4 ) ) ) ) ).
% image_eqI
thf(fact_328_image__eqI,axiom,
! [B: $o,F: nat > $o,X2: nat,A4: set_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A4 )
=> ( member_o @ B @ ( image_nat_o @ F @ A4 ) ) ) ) ).
% image_eqI
thf(fact_329_image__eqI,axiom,
! [B: nat,F: nat > nat,X2: nat,A4: set_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A4 )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A4 ) ) ) ) ).
% image_eqI
thf(fact_330_image__eqI,axiom,
! [B: int,F: nat > int,X2: nat,A4: set_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A4 )
=> ( member_int @ B @ ( image_nat_int @ F @ A4 ) ) ) ) ).
% image_eqI
thf(fact_331_image__eqI,axiom,
! [B: $o,F: int > $o,X2: int,A4: set_int] :
( ( B
= ( F @ X2 ) )
=> ( ( member_int @ X2 @ A4 )
=> ( member_o @ B @ ( image_int_o @ F @ A4 ) ) ) ) ).
% image_eqI
thf(fact_332_image__eqI,axiom,
! [B: nat,F: int > nat,X2: int,A4: set_int] :
( ( B
= ( F @ X2 ) )
=> ( ( member_int @ X2 @ A4 )
=> ( member_nat @ B @ ( image_int_nat @ F @ A4 ) ) ) ) ).
% image_eqI
thf(fact_333_image__eqI,axiom,
! [B: int,F: int > int,X2: int,A4: set_int] :
( ( B
= ( F @ X2 ) )
=> ( ( member_int @ X2 @ A4 )
=> ( member_int @ B @ ( image_int_int @ F @ A4 ) ) ) ) ).
% image_eqI
thf(fact_334_Inf_OINF__image,axiom,
! [Inf: set_o > $o,G: $o > $o,F: nat > $o,A4: set_nat] :
( ( Inf @ ( image_o_o @ G @ ( image_nat_o @ F @ A4 ) ) )
= ( Inf @ ( image_nat_o @ ( comp_o_o_nat @ G @ F ) @ A4 ) ) ) ).
% Inf.INF_image
thf(fact_335_Inf_OINF__image,axiom,
! [Inf: set_rat > rat,G: rat > rat,F: nat > rat,A4: set_nat] :
( ( Inf @ ( image_rat_rat @ G @ ( image_nat_rat @ F @ A4 ) ) )
= ( Inf @ ( image_nat_rat @ ( comp_rat_rat_nat @ G @ F ) @ A4 ) ) ) ).
% Inf.INF_image
thf(fact_336_Inf_OINF__image,axiom,
! [Inf: set_nat > nat,G: rat > nat,F: nat > rat,A4: set_nat] :
( ( Inf @ ( image_rat_nat @ G @ ( image_nat_rat @ F @ A4 ) ) )
= ( Inf @ ( image_nat_nat @ ( comp_rat_nat_nat @ G @ F ) @ A4 ) ) ) ).
% Inf.INF_image
thf(fact_337_Inf_OINF__image,axiom,
! [Inf: set_int > int,G: rat > int,F: nat > rat,A4: set_nat] :
( ( Inf @ ( image_rat_int @ G @ ( image_nat_rat @ F @ A4 ) ) )
= ( Inf @ ( image_nat_int @ ( comp_rat_int_nat @ G @ F ) @ A4 ) ) ) ).
% Inf.INF_image
thf(fact_338_Inf_OINF__image,axiom,
! [Inf: set_rat > rat,G: int > rat,F: nat > int,A4: set_nat] :
( ( Inf @ ( image_int_rat @ G @ ( image_nat_int @ F @ A4 ) ) )
= ( Inf @ ( image_nat_rat @ ( comp_int_rat_nat @ G @ F ) @ A4 ) ) ) ).
% Inf.INF_image
thf(fact_339_Inf_OINF__image,axiom,
! [Inf: set_num > num,G: nat > num,F: int > nat,A4: set_int] :
( ( Inf @ ( image_nat_num @ G @ ( image_int_nat @ F @ A4 ) ) )
= ( Inf @ ( image_int_num @ ( comp_nat_num_int @ G @ F ) @ A4 ) ) ) ).
% Inf.INF_image
thf(fact_340_Inf_OINF__image,axiom,
! [Inf: set_rat > rat,G: nat > rat,F: nat > nat,A4: set_nat] :
( ( Inf @ ( image_nat_rat @ G @ ( image_nat_nat @ F @ A4 ) ) )
= ( Inf @ ( image_nat_rat @ ( comp_nat_rat_nat @ G @ F ) @ A4 ) ) ) ).
% Inf.INF_image
thf(fact_341_Inf_OINF__image,axiom,
! [Inf: set_rat > rat,G: nat > rat,F: int > nat,A4: set_int] :
( ( Inf @ ( image_nat_rat @ G @ ( image_int_nat @ F @ A4 ) ) )
= ( Inf @ ( image_int_rat @ ( comp_nat_rat_int @ G @ F ) @ A4 ) ) ) ).
% Inf.INF_image
thf(fact_342_Inf_OINF__image,axiom,
! [Inf: set_nat > nat,G: nat > nat,F: nat > nat,A4: set_nat] :
( ( Inf @ ( image_nat_nat @ G @ ( image_nat_nat @ F @ A4 ) ) )
= ( Inf @ ( image_nat_nat @ ( comp_nat_nat_nat @ G @ F ) @ A4 ) ) ) ).
% Inf.INF_image
thf(fact_343_Inf_OINF__image,axiom,
! [Inf: set_nat > nat,G: nat > nat,F: int > nat,A4: set_int] :
( ( Inf @ ( image_nat_nat @ G @ ( image_int_nat @ F @ A4 ) ) )
= ( Inf @ ( image_int_nat @ ( comp_nat_nat_int @ G @ F ) @ A4 ) ) ) ).
% Inf.INF_image
thf(fact_344_Sup_OSUP__image,axiom,
! [Sup: set_o > $o,G: $o > $o,F: nat > $o,A4: set_nat] :
( ( Sup @ ( image_o_o @ G @ ( image_nat_o @ F @ A4 ) ) )
= ( Sup @ ( image_nat_o @ ( comp_o_o_nat @ G @ F ) @ A4 ) ) ) ).
% Sup.SUP_image
thf(fact_345_Sup_OSUP__image,axiom,
! [Sup: set_rat > rat,G: rat > rat,F: nat > rat,A4: set_nat] :
( ( Sup @ ( image_rat_rat @ G @ ( image_nat_rat @ F @ A4 ) ) )
= ( Sup @ ( image_nat_rat @ ( comp_rat_rat_nat @ G @ F ) @ A4 ) ) ) ).
% Sup.SUP_image
thf(fact_346_Sup_OSUP__image,axiom,
! [Sup: set_nat > nat,G: rat > nat,F: nat > rat,A4: set_nat] :
( ( Sup @ ( image_rat_nat @ G @ ( image_nat_rat @ F @ A4 ) ) )
= ( Sup @ ( image_nat_nat @ ( comp_rat_nat_nat @ G @ F ) @ A4 ) ) ) ).
% Sup.SUP_image
thf(fact_347_Sup_OSUP__image,axiom,
! [Sup: set_int > int,G: rat > int,F: nat > rat,A4: set_nat] :
( ( Sup @ ( image_rat_int @ G @ ( image_nat_rat @ F @ A4 ) ) )
= ( Sup @ ( image_nat_int @ ( comp_rat_int_nat @ G @ F ) @ A4 ) ) ) ).
% Sup.SUP_image
thf(fact_348_Sup_OSUP__image,axiom,
! [Sup: set_rat > rat,G: int > rat,F: nat > int,A4: set_nat] :
( ( Sup @ ( image_int_rat @ G @ ( image_nat_int @ F @ A4 ) ) )
= ( Sup @ ( image_nat_rat @ ( comp_int_rat_nat @ G @ F ) @ A4 ) ) ) ).
% Sup.SUP_image
thf(fact_349_Sup_OSUP__image,axiom,
! [Sup: set_num > num,G: nat > num,F: int > nat,A4: set_int] :
( ( Sup @ ( image_nat_num @ G @ ( image_int_nat @ F @ A4 ) ) )
= ( Sup @ ( image_int_num @ ( comp_nat_num_int @ G @ F ) @ A4 ) ) ) ).
% Sup.SUP_image
thf(fact_350_Sup_OSUP__image,axiom,
! [Sup: set_rat > rat,G: nat > rat,F: nat > nat,A4: set_nat] :
( ( Sup @ ( image_nat_rat @ G @ ( image_nat_nat @ F @ A4 ) ) )
= ( Sup @ ( image_nat_rat @ ( comp_nat_rat_nat @ G @ F ) @ A4 ) ) ) ).
% Sup.SUP_image
thf(fact_351_Sup_OSUP__image,axiom,
! [Sup: set_rat > rat,G: nat > rat,F: int > nat,A4: set_int] :
( ( Sup @ ( image_nat_rat @ G @ ( image_int_nat @ F @ A4 ) ) )
= ( Sup @ ( image_int_rat @ ( comp_nat_rat_int @ G @ F ) @ A4 ) ) ) ).
% Sup.SUP_image
thf(fact_352_Sup_OSUP__image,axiom,
! [Sup: set_nat > nat,G: nat > nat,F: nat > nat,A4: set_nat] :
( ( Sup @ ( image_nat_nat @ G @ ( image_nat_nat @ F @ A4 ) ) )
= ( Sup @ ( image_nat_nat @ ( comp_nat_nat_nat @ G @ F ) @ A4 ) ) ) ).
% Sup.SUP_image
thf(fact_353_Sup_OSUP__image,axiom,
! [Sup: set_nat > nat,G: nat > nat,F: int > nat,A4: set_int] :
( ( Sup @ ( image_nat_nat @ G @ ( image_int_nat @ F @ A4 ) ) )
= ( Sup @ ( image_int_nat @ ( comp_nat_nat_int @ G @ F ) @ A4 ) ) ) ).
% Sup.SUP_image
thf(fact_354_image__comp,axiom,
! [F: $o > $o,G: nat > $o,R: set_nat] :
( ( image_o_o @ F @ ( image_nat_o @ G @ R ) )
= ( image_nat_o @ ( comp_o_o_nat @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_355_image__comp,axiom,
! [F: rat > rat,G: nat > rat,R: set_nat] :
( ( image_rat_rat @ F @ ( image_nat_rat @ G @ R ) )
= ( image_nat_rat @ ( comp_rat_rat_nat @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_356_image__comp,axiom,
! [F: rat > nat,G: nat > rat,R: set_nat] :
( ( image_rat_nat @ F @ ( image_nat_rat @ G @ R ) )
= ( image_nat_nat @ ( comp_rat_nat_nat @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_357_image__comp,axiom,
! [F: rat > int,G: nat > rat,R: set_nat] :
( ( image_rat_int @ F @ ( image_nat_rat @ G @ R ) )
= ( image_nat_int @ ( comp_rat_int_nat @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_358_image__comp,axiom,
! [F: int > rat,G: nat > int,R: set_nat] :
( ( image_int_rat @ F @ ( image_nat_int @ G @ R ) )
= ( image_nat_rat @ ( comp_int_rat_nat @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_359_image__comp,axiom,
! [F: nat > num,G: int > nat,R: set_int] :
( ( image_nat_num @ F @ ( image_int_nat @ G @ R ) )
= ( image_int_num @ ( comp_nat_num_int @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_360_image__comp,axiom,
! [F: nat > rat,G: nat > nat,R: set_nat] :
( ( image_nat_rat @ F @ ( image_nat_nat @ G @ R ) )
= ( image_nat_rat @ ( comp_nat_rat_nat @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_361_image__comp,axiom,
! [F: nat > rat,G: int > nat,R: set_int] :
( ( image_nat_rat @ F @ ( image_int_nat @ G @ R ) )
= ( image_int_rat @ ( comp_nat_rat_int @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_362_image__comp,axiom,
! [F: nat > nat,G: nat > nat,R: set_nat] :
( ( image_nat_nat @ F @ ( image_nat_nat @ G @ R ) )
= ( image_nat_nat @ ( comp_nat_nat_nat @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_363_image__comp,axiom,
! [F: nat > nat,G: int > nat,R: set_int] :
( ( image_nat_nat @ F @ ( image_int_nat @ G @ R ) )
= ( image_int_nat @ ( comp_nat_nat_int @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_364_image__eq__imp__comp,axiom,
! [F: nat > $o,A4: set_nat,G: nat > $o,B5: set_nat,H: $o > $o] :
( ( ( image_nat_o @ F @ A4 )
= ( image_nat_o @ G @ B5 ) )
=> ( ( image_nat_o @ ( comp_o_o_nat @ H @ F ) @ A4 )
= ( image_nat_o @ ( comp_o_o_nat @ H @ G ) @ B5 ) ) ) ).
% image_eq_imp_comp
thf(fact_365_image__eq__imp__comp,axiom,
! [F: int > rat,A4: set_int,G: nat > rat,B5: set_nat,H: rat > nat] :
( ( ( image_int_rat @ F @ A4 )
= ( image_nat_rat @ G @ B5 ) )
=> ( ( image_int_nat @ ( comp_rat_nat_int @ H @ F ) @ A4 )
= ( image_nat_nat @ ( comp_rat_nat_nat @ H @ G ) @ B5 ) ) ) ).
% image_eq_imp_comp
thf(fact_366_image__eq__imp__comp,axiom,
! [F: int > rat,A4: set_int,G: nat > rat,B5: set_nat,H: rat > int] :
( ( ( image_int_rat @ F @ A4 )
= ( image_nat_rat @ G @ B5 ) )
=> ( ( image_int_int @ ( comp_rat_int_int @ H @ F ) @ A4 )
= ( image_nat_int @ ( comp_rat_int_nat @ H @ G ) @ B5 ) ) ) ).
% image_eq_imp_comp
thf(fact_367_image__eq__imp__comp,axiom,
! [F: nat > rat,A4: set_nat,G: int > rat,B5: set_int,H: rat > nat] :
( ( ( image_nat_rat @ F @ A4 )
= ( image_int_rat @ G @ B5 ) )
=> ( ( image_nat_nat @ ( comp_rat_nat_nat @ H @ F ) @ A4 )
= ( image_int_nat @ ( comp_rat_nat_int @ H @ G ) @ B5 ) ) ) ).
% image_eq_imp_comp
thf(fact_368_image__eq__imp__comp,axiom,
! [F: nat > rat,A4: set_nat,G: int > rat,B5: set_int,H: rat > int] :
( ( ( image_nat_rat @ F @ A4 )
= ( image_int_rat @ G @ B5 ) )
=> ( ( image_nat_int @ ( comp_rat_int_nat @ H @ F ) @ A4 )
= ( image_int_int @ ( comp_rat_int_int @ H @ G ) @ B5 ) ) ) ).
% image_eq_imp_comp
thf(fact_369_image__eq__imp__comp,axiom,
! [F: nat > rat,A4: set_nat,G: nat > rat,B5: set_nat,H: rat > rat] :
( ( ( image_nat_rat @ F @ A4 )
= ( image_nat_rat @ G @ B5 ) )
=> ( ( image_nat_rat @ ( comp_rat_rat_nat @ H @ F ) @ A4 )
= ( image_nat_rat @ ( comp_rat_rat_nat @ H @ G ) @ B5 ) ) ) ).
% image_eq_imp_comp
thf(fact_370_image__eq__imp__comp,axiom,
! [F: nat > rat,A4: set_nat,G: nat > rat,B5: set_nat,H: rat > nat] :
( ( ( image_nat_rat @ F @ A4 )
= ( image_nat_rat @ G @ B5 ) )
=> ( ( image_nat_nat @ ( comp_rat_nat_nat @ H @ F ) @ A4 )
= ( image_nat_nat @ ( comp_rat_nat_nat @ H @ G ) @ B5 ) ) ) ).
% image_eq_imp_comp
thf(fact_371_image__eq__imp__comp,axiom,
! [F: nat > rat,A4: set_nat,G: nat > rat,B5: set_nat,H: rat > int] :
( ( ( image_nat_rat @ F @ A4 )
= ( image_nat_rat @ G @ B5 ) )
=> ( ( image_nat_int @ ( comp_rat_int_nat @ H @ F ) @ A4 )
= ( image_nat_int @ ( comp_rat_int_nat @ H @ G ) @ B5 ) ) ) ).
% image_eq_imp_comp
thf(fact_372_image__eq__imp__comp,axiom,
! [F: nat > nat,A4: set_nat,G: nat > nat,B5: set_nat,H: nat > rat] :
( ( ( image_nat_nat @ F @ A4 )
= ( image_nat_nat @ G @ B5 ) )
=> ( ( image_nat_rat @ ( comp_nat_rat_nat @ H @ F ) @ A4 )
= ( image_nat_rat @ ( comp_nat_rat_nat @ H @ G ) @ B5 ) ) ) ).
% image_eq_imp_comp
thf(fact_373_image__eq__imp__comp,axiom,
! [F: nat > nat,A4: set_nat,G: nat > nat,B5: set_nat,H: nat > nat] :
( ( ( image_nat_nat @ F @ A4 )
= ( image_nat_nat @ G @ B5 ) )
=> ( ( image_nat_nat @ ( comp_nat_nat_nat @ H @ F ) @ A4 )
= ( image_nat_nat @ ( comp_nat_nat_nat @ H @ G ) @ B5 ) ) ) ).
% image_eq_imp_comp
thf(fact_374_snd__comp__apsnd,axiom,
! [F: code_integer > code_integer] :
( ( comp_P3428181540658402820nteger @ produc6174133586879617921nteger @ ( produc6499014454317279255nteger @ F ) )
= ( comp_C1043215000170864188nteger @ F @ produc6174133586879617921nteger ) ) ).
% snd_comp_apsnd
thf(fact_375_snd__comp__apsnd,axiom,
! [F: heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit] :
( ( comp_P8019343326919024234t_unit @ produc1126150082675451753t_unit @ ( produc3038839249364385243_ref_a @ F ) )
= ( comp_h4837689053573563144t_unit @ F @ produc1126150082675451753t_unit ) ) ).
% snd_comp_apsnd
thf(fact_376_fst__comp__apfst,axiom,
! [F: ref_a > ref_a] :
( ( comp_P4044372497395451596t_unit @ produc473321116828791851t_unit @ ( produc4508741384315284437t_unit @ F ) )
= ( comp_r2143332150499412868t_unit @ F @ produc473321116828791851t_unit ) ) ).
% fst_comp_apfst
thf(fact_377_Collect__case__prod__Grp__eqD,axiom,
! [Z2: produc3843707927480180839at_nat,A4: set_se7855581050983116737at_nat,F: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat] :
( ( member8757157785044589968at_nat @ Z2 @ ( collec6321179662152712658at_nat @ ( produc410239310623530412_nat_o @ ( bNF_Gr6677838525845560078at_nat @ A4 @ F ) ) ) )
=> ( ( comp_s7355055891894327172at_nat @ F @ produc6011863565407846595at_nat @ Z2 )
= ( produc2937545259423399429at_nat @ Z2 ) ) ) ).
% Collect_case_prod_Grp_eqD
thf(fact_378_Collect__case__prod__Grp__eqD,axiom,
! [Z2: product_prod_nat_nat,A4: set_nat,F: nat > nat] :
( ( member8440522571783428010at_nat @ Z2 @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ( bNF_Grp_nat_nat @ A4 @ F ) ) ) )
=> ( ( comp_n2711240574518091068at_nat @ F @ product_fst_nat_nat @ Z2 )
= ( product_snd_nat_nat @ Z2 ) ) ) ).
% Collect_case_prod_Grp_eqD
thf(fact_379_Collect__case__prod__Grp__eqD,axiom,
! [Z2: produc7812382424243267045t_unit,A4: set_ref_a,F: ref_a > heap_e7401611519738050253t_unit] :
( ( member7849944260330398460t_unit @ Z2 @ ( collec3252306575900995386t_unit @ ( produc4924168586796184916unit_o @ ( bNF_Gr6510375311308651680t_unit @ A4 @ F ) ) ) )
=> ( ( comp_r3604920825623061426t_unit @ F @ produc473321116828791851t_unit @ Z2 )
= ( produc1126150082675451753t_unit @ Z2 ) ) ) ).
% Collect_case_prod_Grp_eqD
thf(fact_380_Collect__case__prod__Grp__eqD,axiom,
! [Z2: product_prod_int_int,A4: set_int,F: int > int] :
( ( member5262025264175285858nt_int @ Z2 @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ ( bNF_Grp_int_int @ A4 @ F ) ) ) )
=> ( ( comp_i4330622015127193916nt_int @ F @ product_fst_int_int @ Z2 )
= ( product_snd_int_int @ Z2 ) ) ) ).
% Collect_case_prod_Grp_eqD
thf(fact_381_Collect__split__mono__strong,axiom,
! [X5: set_nat,A4: set_Pr1261947904930325089at_nat,Y4: set_nat,P2: nat > nat > $o,Q2: nat > nat > $o] :
( ( X5
= ( image_2486076414777270412at_nat @ product_fst_nat_nat @ A4 ) )
=> ( ( Y4
= ( image_2486076414777270412at_nat @ product_snd_nat_nat @ A4 ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ X5 )
=> ! [Xa: nat] :
( ( member_nat @ Xa @ Y4 )
=> ( ( P2 @ X @ Xa )
=> ( Q2 @ X @ Xa ) ) ) )
=> ( ( ord_le3146513528884898305at_nat @ A4 @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ P2 ) ) )
=> ( ord_le3146513528884898305at_nat @ A4 @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ Q2 ) ) ) ) ) ) ) ).
% Collect_split_mono_strong
thf(fact_382_Collect__split__mono__strong,axiom,
! [X5: set_ref_a,A4: set_Pr1133966238863718811t_unit,Y4: set_he8644174446132254637t_unit,P2: ref_a > heap_e7401611519738050253t_unit > $o,Q2: ref_a > heap_e7401611519738050253t_unit > $o] :
( ( X5
= ( image_7136766188234240905_ref_a @ produc473321116828791851t_unit @ A4 ) )
=> ( ( Y4
= ( image_17660834435998945t_unit @ produc1126150082675451753t_unit @ A4 ) )
=> ( ! [X: ref_a] :
( ( member_ref_a @ X @ X5 )
=> ! [Xa: heap_e7401611519738050253t_unit] :
( ( member242389114213480694t_unit @ Xa @ Y4 )
=> ( ( P2 @ X @ Xa )
=> ( Q2 @ X @ Xa ) ) ) )
=> ( ( ord_le5525191956872776187t_unit @ A4 @ ( collec3252306575900995386t_unit @ ( produc4924168586796184916unit_o @ P2 ) ) )
=> ( ord_le5525191956872776187t_unit @ A4 @ ( collec3252306575900995386t_unit @ ( produc4924168586796184916unit_o @ Q2 ) ) ) ) ) ) ) ).
% Collect_split_mono_strong
thf(fact_383_Collect__split__mono__strong,axiom,
! [X5: set_int,A4: set_Pr958786334691620121nt_int,Y4: set_int,P2: int > int > $o,Q2: int > int > $o] :
( ( X5
= ( image_5042161079198086560nt_int @ product_fst_int_int @ A4 ) )
=> ( ( Y4
= ( image_5042161079198086560nt_int @ product_snd_int_int @ A4 ) )
=> ( ! [X: int] :
( ( member_int @ X @ X5 )
=> ! [Xa: int] :
( ( member_int @ Xa @ Y4 )
=> ( ( P2 @ X @ Xa )
=> ( Q2 @ X @ Xa ) ) ) )
=> ( ( ord_le2843351958646193337nt_int @ A4 @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ P2 ) ) )
=> ( ord_le2843351958646193337nt_int @ A4 @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ Q2 ) ) ) ) ) ) ) ).
% Collect_split_mono_strong
thf(fact_384_snd__comp__map__prod,axiom,
! [F: ref_a > ref_a,G: heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit] :
( ( comp_P8019343326919024234t_unit @ produc1126150082675451753t_unit @ ( produc3456481717235892722t_unit @ F @ G ) )
= ( comp_h4837689053573563144t_unit @ G @ produc1126150082675451753t_unit ) ) ).
% snd_comp_map_prod
thf(fact_385_fst__comp__map__prod,axiom,
! [F: ref_a > ref_a,G: heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit] :
( ( comp_P4044372497395451596t_unit @ produc473321116828791851t_unit @ ( produc3456481717235892722t_unit @ F @ G ) )
= ( comp_r2143332150499412868t_unit @ F @ produc473321116828791851t_unit ) ) ).
% fst_comp_map_prod
thf(fact_386_map__prod__simp,axiom,
! [F: nat > nat,G: num > num,A: nat,B: num] :
( ( produc1665548884793878990um_num @ F @ G @ ( product_Pair_nat_num @ A @ B ) )
= ( product_Pair_nat_num @ ( F @ A ) @ ( G @ B ) ) ) ).
% map_prod_simp
thf(fact_387_map__prod__simp,axiom,
! [F: nat > nat,G: num > nat,A: nat,B: num] :
( ( produc5108217098638100356um_nat @ F @ G @ ( product_Pair_nat_num @ A @ B ) )
= ( product_Pair_nat_nat @ ( F @ A ) @ ( G @ B ) ) ) ).
% map_prod_simp
thf(fact_388_map__prod__simp,axiom,
! [F: nat > int,G: num > int,A: nat,B: num] :
( ( produc5477380188365894716um_int @ F @ G @ ( product_Pair_nat_num @ A @ B ) )
= ( product_Pair_int_int @ ( F @ A ) @ ( G @ B ) ) ) ).
% map_prod_simp
thf(fact_389_map__prod__simp,axiom,
! [F: nat > nat,G: nat > num,A: nat,B: nat] :
( ( produc3535218481486409604at_num @ F @ G @ ( product_Pair_nat_nat @ A @ B ) )
= ( product_Pair_nat_num @ ( F @ A ) @ ( G @ B ) ) ) ).
% map_prod_simp
thf(fact_390_map__prod__simp,axiom,
! [F: nat > nat,G: nat > nat,A: nat,B: nat] :
( ( produc6977886695330630970at_nat @ F @ G @ ( product_Pair_nat_nat @ A @ B ) )
= ( product_Pair_nat_nat @ ( F @ A ) @ ( G @ B ) ) ) ).
% map_prod_simp
thf(fact_391_map__prod__simp,axiom,
! [F: nat > int,G: nat > int,A: nat,B: nat] :
( ( produc7347049785058425330at_int @ F @ G @ ( product_Pair_nat_nat @ A @ B ) )
= ( product_Pair_int_int @ ( F @ A ) @ ( G @ B ) ) ) ).
% map_prod_simp
thf(fact_392_map__prod__simp,axiom,
! [F: int > nat,G: int > num,A: int,B: int] :
( ( produc2224754261294513212nt_num @ F @ G @ ( product_Pair_int_int @ A @ B ) )
= ( product_Pair_nat_num @ ( F @ A ) @ ( G @ B ) ) ) ).
% map_prod_simp
thf(fact_393_map__prod__simp,axiom,
! [F: int > nat,G: int > nat,A: int,B: int] :
( ( produc5667422475138734578nt_nat @ F @ G @ ( product_Pair_int_int @ A @ B ) )
= ( product_Pair_nat_nat @ ( F @ A ) @ ( G @ B ) ) ) ).
% map_prod_simp
thf(fact_394_map__prod__simp,axiom,
! [F: int > int,G: int > int,A: int,B: int] :
( ( produc6036585564866528938nt_int @ F @ G @ ( product_Pair_int_int @ A @ B ) )
= ( product_Pair_int_int @ ( F @ A ) @ ( G @ B ) ) ) ).
% map_prod_simp
thf(fact_395_map__prod__simp,axiom,
! [F: set_Pr1261947904930325089at_nat > nat,G: set_Pr1261947904930325089at_nat > num,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
( ( produc3932168101365401472at_num @ F @ G @ ( produc2922128104949294807at_nat @ A @ B ) )
= ( product_Pair_nat_num @ ( F @ A ) @ ( G @ B ) ) ) ).
% map_prod_simp
thf(fact_396_fst__map__prod,axiom,
! [F: ref_a > ref_a,G: heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit,X2: produc7812382424243267045t_unit] :
( ( produc473321116828791851t_unit @ ( produc3456481717235892722t_unit @ F @ G @ X2 ) )
= ( F @ ( produc473321116828791851t_unit @ X2 ) ) ) ).
% fst_map_prod
thf(fact_397_snd__map__prod,axiom,
! [F: ref_a > ref_a,G: heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit,X2: produc7812382424243267045t_unit] :
( ( produc1126150082675451753t_unit @ ( produc3456481717235892722t_unit @ F @ G @ X2 ) )
= ( G @ ( produc1126150082675451753t_unit @ X2 ) ) ) ).
% snd_map_prod
thf(fact_398_apfst__conv,axiom,
! [F: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat,X2: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] :
( ( produc6529295065807390923at_nat @ F @ ( produc9060074326276436823at_nat @ X2 @ Y3 ) )
= ( produc9060074326276436823at_nat @ ( F @ X2 ) @ Y3 ) ) ).
% apfst_conv
thf(fact_399_apfst__conv,axiom,
! [F: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat,X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
( ( produc5380969780118646373at_nat @ F @ ( produc2922128104949294807at_nat @ X2 @ Y3 ) )
= ( produc2922128104949294807at_nat @ ( F @ X2 ) @ Y3 ) ) ).
% apfst_conv
thf(fact_400_apfst__conv,axiom,
! [F: nat > nat,X2: nat,Y3: num] :
( ( produc6767424583952363640at_num @ F @ ( product_Pair_nat_num @ X2 @ Y3 ) )
= ( product_Pair_nat_num @ ( F @ X2 ) @ Y3 ) ) ).
% apfst_conv
thf(fact_401_apfst__conv,axiom,
! [F: nat > nat,X2: nat,Y3: nat] :
( ( produc986720760941809198at_nat @ F @ ( product_Pair_nat_nat @ X2 @ Y3 ) )
= ( product_Pair_nat_nat @ ( F @ X2 ) @ Y3 ) ) ).
% apfst_conv
thf(fact_402_apfst__conv,axiom,
! [F: int > int,X2: int,Y3: int] :
( ( produc2355237562570658114nt_int @ F @ ( product_Pair_int_int @ X2 @ Y3 ) )
= ( product_Pair_int_int @ ( F @ X2 ) @ Y3 ) ) ).
% apfst_conv
thf(fact_403_apsnd__conv,axiom,
! [F: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat,X2: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] :
( ( produc8057050257526108681at_nat @ F @ ( produc9060074326276436823at_nat @ X2 @ Y3 ) )
= ( produc9060074326276436823at_nat @ X2 @ ( F @ Y3 ) ) ) ).
% apsnd_conv
thf(fact_404_apsnd__conv,axiom,
! [F: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat,X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
( ( produc6468969725218530979at_nat @ F @ ( produc2922128104949294807at_nat @ X2 @ Y3 ) )
= ( produc2922128104949294807at_nat @ X2 @ ( F @ Y3 ) ) ) ).
% apsnd_conv
thf(fact_405_apsnd__conv,axiom,
! [F: num > num,X2: nat,Y3: num] :
( ( produc8389275579088931588um_nat @ F @ ( product_Pair_nat_num @ X2 @ Y3 ) )
= ( product_Pair_nat_num @ X2 @ ( F @ Y3 ) ) ) ).
% apsnd_conv
thf(fact_406_apsnd__conv,axiom,
! [F: num > nat,X2: nat,Y3: num] :
( ( produc1035573138926686394at_nat @ F @ ( product_Pair_nat_num @ X2 @ Y3 ) )
= ( product_Pair_nat_nat @ X2 @ ( F @ Y3 ) ) ) ).
% apsnd_conv
thf(fact_407_apsnd__conv,axiom,
! [F: nat > num,X2: nat,Y3: nat] :
( ( produc1225095714263496890um_nat @ F @ ( product_Pair_nat_nat @ X2 @ Y3 ) )
= ( product_Pair_nat_num @ X2 @ ( F @ Y3 ) ) ) ).
% apsnd_conv
thf(fact_408_apsnd__conv,axiom,
! [F: nat > nat,X2: nat,Y3: nat] :
( ( produc3094765310956027504at_nat @ F @ ( product_Pair_nat_nat @ X2 @ Y3 ) )
= ( product_Pair_nat_nat @ X2 @ ( F @ Y3 ) ) ) ).
% apsnd_conv
thf(fact_409_apsnd__conv,axiom,
! [F: int > int,X2: int,Y3: int] :
( ( produc4463282112584876420nt_int @ F @ ( product_Pair_int_int @ X2 @ Y3 ) )
= ( product_Pair_int_int @ X2 @ ( F @ Y3 ) ) ) ).
% apsnd_conv
thf(fact_410_apsnd__conv,axiom,
! [F: code_integer > code_integer,X2: code_integer,Y3: code_integer] :
( ( produc6499014454317279255nteger @ F @ ( produc1086072967326762835nteger @ X2 @ Y3 ) )
= ( produc1086072967326762835nteger @ X2 @ ( F @ Y3 ) ) ) ).
% apsnd_conv
thf(fact_411_fst__apfst,axiom,
! [F: ref_a > ref_a,X2: produc7812382424243267045t_unit] :
( ( produc473321116828791851t_unit @ ( produc4508741384315284437t_unit @ F @ X2 ) )
= ( F @ ( produc473321116828791851t_unit @ X2 ) ) ) ).
% fst_apfst
thf(fact_412_snd__apfst,axiom,
! [F: ref_a > ref_a,X2: produc7812382424243267045t_unit] :
( ( produc1126150082675451753t_unit @ ( produc4508741384315284437t_unit @ F @ X2 ) )
= ( produc1126150082675451753t_unit @ X2 ) ) ).
% snd_apfst
thf(fact_413_fst__apsnd,axiom,
! [F: code_integer > code_integer,X2: produc8923325533196201883nteger] :
( ( produc8508995932063986495nteger @ ( produc6499014454317279255nteger @ F @ X2 ) )
= ( produc8508995932063986495nteger @ X2 ) ) ).
% fst_apsnd
thf(fact_414_fst__apsnd,axiom,
! [F: heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit,X2: produc7812382424243267045t_unit] :
( ( produc473321116828791851t_unit @ ( produc3038839249364385243_ref_a @ F @ X2 ) )
= ( produc473321116828791851t_unit @ X2 ) ) ).
% fst_apsnd
thf(fact_415_snd__apsnd,axiom,
! [F: code_integer > code_integer,X2: produc8923325533196201883nteger] :
( ( produc6174133586879617921nteger @ ( produc6499014454317279255nteger @ F @ X2 ) )
= ( F @ ( produc6174133586879617921nteger @ X2 ) ) ) ).
% snd_apsnd
thf(fact_416_snd__apsnd,axiom,
! [F: heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit,X2: produc7812382424243267045t_unit] :
( ( produc1126150082675451753t_unit @ ( produc3038839249364385243_ref_a @ F @ X2 ) )
= ( F @ ( produc1126150082675451753t_unit @ X2 ) ) ) ).
% snd_apsnd
thf(fact_417_apsnd__eq__conv,axiom,
! [F: code_integer > code_integer,X2: produc8923325533196201883nteger,G: code_integer > code_integer] :
( ( ( produc6499014454317279255nteger @ F @ X2 )
= ( produc6499014454317279255nteger @ G @ X2 ) )
= ( ( F @ ( produc6174133586879617921nteger @ X2 ) )
= ( G @ ( produc6174133586879617921nteger @ X2 ) ) ) ) ).
% apsnd_eq_conv
thf(fact_418_map__prod__imageI,axiom,
! [A: nat,B: num,R3: set_Pr6200539531224447659at_num,F: nat > nat,G: num > num] :
( ( member9148766508732265716at_num @ ( product_Pair_nat_num @ A @ B ) @ R3 )
=> ( member9148766508732265716at_num @ ( product_Pair_nat_num @ ( F @ A ) @ ( G @ B ) ) @ ( image_4925883905578568901at_num @ ( produc1665548884793878990um_num @ F @ G ) @ R3 ) ) ) ).
% map_prod_imageI
thf(fact_419_map__prod__imageI,axiom,
! [A: nat,B: num,R3: set_Pr6200539531224447659at_num,F: nat > nat,G: num > nat] :
( ( member9148766508732265716at_num @ ( product_Pair_nat_num @ A @ B ) @ R3 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ A ) @ ( G @ B ) ) @ ( image_4217639968629731195at_nat @ ( produc5108217098638100356um_nat @ F @ G ) @ R3 ) ) ) ).
% map_prod_imageI
thf(fact_420_map__prod__imageI,axiom,
! [A: nat,B: num,R3: set_Pr6200539531224447659at_num,F: nat > int,G: num > int] :
( ( member9148766508732265716at_num @ ( product_Pair_nat_num @ A @ B ) @ R3 )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( F @ A ) @ ( G @ B ) ) @ ( image_1039142661021589043nt_int @ ( produc5477380188365894716um_int @ F @ G ) @ R3 ) ) ) ).
% map_prod_imageI
thf(fact_421_map__prod__imageI,axiom,
! [A: nat,B: nat,R3: set_Pr1261947904930325089at_nat,F: nat > nat,G: nat > num] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R3 )
=> ( member9148766508732265716at_num @ ( product_Pair_nat_num @ ( F @ A ) @ ( G @ B ) ) @ ( image_5877158439796295311at_num @ ( produc3535218481486409604at_num @ F @ G ) @ R3 ) ) ) ).
% map_prod_imageI
thf(fact_422_map__prod__imageI,axiom,
! [A: nat,B: nat,R3: set_Pr1261947904930325089at_nat,F: nat > nat,G: nat > nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R3 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ A ) @ ( G @ B ) ) @ ( image_5168914502847457605at_nat @ ( produc6977886695330630970at_nat @ F @ G ) @ R3 ) ) ) ).
% map_prod_imageI
thf(fact_423_map__prod__imageI,axiom,
! [A: nat,B: nat,R3: set_Pr1261947904930325089at_nat,F: nat > int,G: nat > int] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R3 )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( F @ A ) @ ( G @ B ) ) @ ( image_1990417195239315453nt_int @ ( produc7347049785058425330at_int @ F @ G ) @ R3 ) ) ) ).
% map_prod_imageI
thf(fact_424_map__prod__imageI,axiom,
! [A: int,B: int,R3: set_Pr958786334691620121nt_int,F: int > nat,G: int > num] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ R3 )
=> ( member9148766508732265716at_num @ ( product_Pair_nat_num @ ( F @ A ) @ ( G @ B ) ) @ ( image_6540112122905407959at_num @ ( produc2224754261294513212nt_num @ F @ G ) @ R3 ) ) ) ).
% map_prod_imageI
thf(fact_425_map__prod__imageI,axiom,
! [A: int,B: int,R3: set_Pr958786334691620121nt_int,F: int > nat,G: int > nat] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ R3 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ A ) @ ( G @ B ) ) @ ( image_5831868185956570253at_nat @ ( produc5667422475138734578nt_nat @ F @ G ) @ R3 ) ) ) ).
% map_prod_imageI
thf(fact_426_map__prod__imageI,axiom,
! [A: int,B: int,R3: set_Pr958786334691620121nt_int,F: int > int,G: int > int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ R3 )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( F @ A ) @ ( G @ B ) ) @ ( image_2653370878348428101nt_int @ ( produc6036585564866528938nt_int @ F @ G ) @ R3 ) ) ) ).
% map_prod_imageI
thf(fact_427_map__prod__imageI,axiom,
! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,R3: set_Pr4329608150637261639at_nat,F: set_Pr1261947904930325089at_nat > nat,G: set_Pr1261947904930325089at_nat > num] :
( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R3 )
=> ( member9148766508732265716at_num @ ( product_Pair_nat_num @ ( F @ A ) @ ( G @ B ) ) @ ( image_8033574621161215657at_num @ ( produc3932168101365401472at_num @ F @ G ) @ R3 ) ) ) ).
% map_prod_imageI
thf(fact_428_snd__comp__apfst,axiom,
! [F: ref_a > ref_a] :
( ( comp_P8019343326919024234t_unit @ produc1126150082675451753t_unit @ ( produc4508741384315284437t_unit @ F ) )
= produc1126150082675451753t_unit ) ).
% snd_comp_apfst
thf(fact_429_fst__comp__apsnd,axiom,
! [F: code_integer > code_integer] :
( ( comp_P3428181540658402820nteger @ produc8508995932063986495nteger @ ( produc6499014454317279255nteger @ F ) )
= produc8508995932063986495nteger ) ).
% fst_comp_apsnd
thf(fact_430_fst__comp__apsnd,axiom,
! [F: heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit] :
( ( comp_P4044372497395451596t_unit @ produc473321116828791851t_unit @ ( produc3038839249364385243_ref_a @ F ) )
= produc473321116828791851t_unit ) ).
% fst_comp_apsnd
thf(fact_431_apfst__apsnd,axiom,
! [F: ref_a > set_Pr4329608150637261639at_nat,G: heap_e7401611519738050253t_unit > set_Pr4329608150637261639at_nat,X2: produc7812382424243267045t_unit] :
( ( produc2316621926130398489at_nat @ F @ ( produc5080965012319039829_ref_a @ G @ X2 ) )
= ( produc9060074326276436823at_nat @ ( F @ ( produc473321116828791851t_unit @ X2 ) ) @ ( G @ ( produc1126150082675451753t_unit @ X2 ) ) ) ) ).
% apfst_apsnd
thf(fact_432_apfst__apsnd,axiom,
! [F: ref_a > set_Pr1261947904930325089at_nat,G: heap_e7401611519738050253t_unit > set_Pr1261947904930325089at_nat,X2: produc7812382424243267045t_unit] :
( ( produc4016808307611786905at_nat @ F @ ( produc2928627021203296111_ref_a @ G @ X2 ) )
= ( produc2922128104949294807at_nat @ ( F @ ( produc473321116828791851t_unit @ X2 ) ) @ ( G @ ( produc1126150082675451753t_unit @ X2 ) ) ) ) ).
% apfst_apsnd
thf(fact_433_apfst__apsnd,axiom,
! [F: ref_a > nat,G: heap_e7401611519738050253t_unit > num,X2: produc7812382424243267045t_unit] :
( ( produc1704413584773205829at_num @ F @ ( produc7633249859560231688_ref_a @ G @ X2 ) )
= ( product_Pair_nat_num @ ( F @ ( produc473321116828791851t_unit @ X2 ) ) @ ( G @ ( produc1126150082675451753t_unit @ X2 ) ) ) ) ).
% apfst_apsnd
thf(fact_434_apfst__apsnd,axiom,
! [F: ref_a > nat,G: heap_e7401611519738050253t_unit > nat,X2: produc7812382424243267045t_unit] :
( ( produc5147081798617427195at_nat @ F @ ( produc8366372614399963582_ref_a @ G @ X2 ) )
= ( product_Pair_nat_nat @ ( F @ ( produc473321116828791851t_unit @ X2 ) ) @ ( G @ ( produc1126150082675451753t_unit @ X2 ) ) ) ) ).
% apfst_apsnd
thf(fact_435_apfst__apsnd,axiom,
! [F: ref_a > int,G: heap_e7401611519738050253t_unit > int,X2: produc7812382424243267045t_unit] :
( ( produc6143945040009431475nt_int @ F @ ( produc3558794870892339738_ref_a @ G @ X2 ) )
= ( product_Pair_int_int @ ( F @ ( produc473321116828791851t_unit @ X2 ) ) @ ( G @ ( produc1126150082675451753t_unit @ X2 ) ) ) ) ).
% apfst_apsnd
thf(fact_436_apsnd__apfst,axiom,
! [F: heap_e7401611519738050253t_unit > set_Pr4329608150637261639at_nat,G: ref_a > set_Pr4329608150637261639at_nat,X2: produc7812382424243267045t_unit] :
( ( produc5990651871820846223at_nat @ F @ ( produc5988837738927868575t_unit @ G @ X2 ) )
= ( produc9060074326276436823at_nat @ ( G @ ( produc473321116828791851t_unit @ X2 ) ) @ ( F @ ( produc1126150082675451753t_unit @ X2 ) ) ) ) ).
% apsnd_apfst
thf(fact_437_apsnd__apfst,axiom,
! [F: heap_e7401611519738050253t_unit > set_Pr1261947904930325089at_nat,G: ref_a > set_Pr1261947904930325089at_nat,X2: produc7812382424243267045t_unit] :
( ( produc8705118745468743695at_nat @ F @ ( produc5323809043756447493t_unit @ G @ X2 ) )
= ( produc2922128104949294807at_nat @ ( G @ ( produc473321116828791851t_unit @ X2 ) ) @ ( F @ ( produc1126150082675451753t_unit @ X2 ) ) ) ) ).
% apsnd_apfst
thf(fact_438_apsnd__apfst,axiom,
! [F: heap_e7401611519738050253t_unit > num,G: ref_a > nat,X2: produc7812382424243267045t_unit] :
( ( produc3902377510303101371um_nat @ F @ ( produc6942186945380262242t_unit @ G @ X2 ) )
= ( product_Pair_nat_num @ ( G @ ( produc473321116828791851t_unit @ X2 ) ) @ ( F @ ( produc1126150082675451753t_unit @ X2 ) ) ) ) ).
% apsnd_apfst
thf(fact_439_apsnd__apfst,axiom,
! [F: heap_e7401611519738050253t_unit > nat,G: ref_a > nat,X2: produc7812382424243267045t_unit] :
( ( produc5772047106995631985at_nat @ F @ ( produc6942186945380262242t_unit @ G @ X2 ) )
= ( product_Pair_nat_nat @ ( G @ ( produc473321116828791851t_unit @ X2 ) ) @ ( F @ ( produc1126150082675451753t_unit @ X2 ) ) ) ) ).
% apsnd_apfst
thf(fact_440_apsnd__apfst,axiom,
! [F: heap_e7401611519738050253t_unit > int,G: ref_a > int,X2: produc7812382424243267045t_unit] :
( ( produc6768910348387636265nt_int @ F @ ( produc5845103964736263686t_unit @ G @ X2 ) )
= ( product_Pair_int_int @ ( G @ ( produc473321116828791851t_unit @ X2 ) ) @ ( F @ ( produc1126150082675451753t_unit @ X2 ) ) ) ) ).
% apsnd_apfst
thf(fact_441_apsnd__apfst__commute,axiom,
! [F: code_integer > code_integer,G: code_integer > code_integer,P: produc8923325533196201883nteger] :
( ( produc6499014454317279255nteger @ F @ ( produc3195571438000033749nteger @ G @ P ) )
= ( produc3195571438000033749nteger @ G @ ( produc6499014454317279255nteger @ F @ P ) ) ) ).
% apsnd_apfst_commute
thf(fact_442_subset__image__iff,axiom,
! [B5: set_rat,F: nat > rat,A4: set_nat] :
( ( ord_less_eq_set_rat @ B5 @ ( image_nat_rat @ F @ A4 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A4 )
& ( B5
= ( image_nat_rat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_443_subset__image__iff,axiom,
! [B5: set_nat,F: nat > nat,A4: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat @ F @ A4 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A4 )
& ( B5
= ( image_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_444_subset__image__iff,axiom,
! [B5: set_int,F: nat > int,A4: set_nat] :
( ( ord_less_eq_set_int @ B5 @ ( image_nat_int @ F @ A4 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A4 )
& ( B5
= ( image_nat_int @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_445_subset__image__iff,axiom,
! [B5: set_nat,F: int > nat,A4: set_int] :
( ( ord_less_eq_set_nat @ B5 @ ( image_int_nat @ F @ A4 ) )
= ( ? [AA: set_int] :
( ( ord_less_eq_set_int @ AA @ A4 )
& ( B5
= ( image_int_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_446_subset__image__iff,axiom,
! [B5: set_int,F: int > int,A4: set_int] :
( ( ord_less_eq_set_int @ B5 @ ( image_int_int @ F @ A4 ) )
= ( ? [AA: set_int] :
( ( ord_less_eq_set_int @ AA @ A4 )
& ( B5
= ( image_int_int @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_447_image__subset__iff,axiom,
! [F: nat > rat,A4: set_nat,B5: set_rat] :
( ( ord_less_eq_set_rat @ ( image_nat_rat @ F @ A4 ) @ B5 )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( member_rat @ ( F @ X4 ) @ B5 ) ) ) ) ).
% image_subset_iff
thf(fact_448_image__subset__iff,axiom,
! [F: nat > nat,A4: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A4 ) @ B5 )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( member_nat @ ( F @ X4 ) @ B5 ) ) ) ) ).
% image_subset_iff
thf(fact_449_image__subset__iff,axiom,
! [F: int > nat,A4: set_int,B5: set_nat] :
( ( ord_less_eq_set_nat @ ( image_int_nat @ F @ A4 ) @ B5 )
= ( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( member_nat @ ( F @ X4 ) @ B5 ) ) ) ) ).
% image_subset_iff
thf(fact_450_image__subset__iff,axiom,
! [F: nat > int,A4: set_nat,B5: set_int] :
( ( ord_less_eq_set_int @ ( image_nat_int @ F @ A4 ) @ B5 )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( member_int @ ( F @ X4 ) @ B5 ) ) ) ) ).
% image_subset_iff
thf(fact_451_image__subset__iff,axiom,
! [F: int > int,A4: set_int,B5: set_int] :
( ( ord_less_eq_set_int @ ( image_int_int @ F @ A4 ) @ B5 )
= ( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( member_int @ ( F @ X4 ) @ B5 ) ) ) ) ).
% image_subset_iff
thf(fact_452_subset__imageE,axiom,
! [B5: set_rat,F: nat > rat,A4: set_nat] :
( ( ord_less_eq_set_rat @ B5 @ ( image_nat_rat @ F @ A4 ) )
=> ~ ! [C2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ A4 )
=> ( B5
!= ( image_nat_rat @ F @ C2 ) ) ) ) ).
% subset_imageE
thf(fact_453_subset__imageE,axiom,
! [B5: set_nat,F: nat > nat,A4: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat @ F @ A4 ) )
=> ~ ! [C2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ A4 )
=> ( B5
!= ( image_nat_nat @ F @ C2 ) ) ) ) ).
% subset_imageE
thf(fact_454_subset__imageE,axiom,
! [B5: set_int,F: nat > int,A4: set_nat] :
( ( ord_less_eq_set_int @ B5 @ ( image_nat_int @ F @ A4 ) )
=> ~ ! [C2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ A4 )
=> ( B5
!= ( image_nat_int @ F @ C2 ) ) ) ) ).
% subset_imageE
thf(fact_455_subset__imageE,axiom,
! [B5: set_nat,F: int > nat,A4: set_int] :
( ( ord_less_eq_set_nat @ B5 @ ( image_int_nat @ F @ A4 ) )
=> ~ ! [C2: set_int] :
( ( ord_less_eq_set_int @ C2 @ A4 )
=> ( B5
!= ( image_int_nat @ F @ C2 ) ) ) ) ).
% subset_imageE
thf(fact_456_subset__imageE,axiom,
! [B5: set_int,F: int > int,A4: set_int] :
( ( ord_less_eq_set_int @ B5 @ ( image_int_int @ F @ A4 ) )
=> ~ ! [C2: set_int] :
( ( ord_less_eq_set_int @ C2 @ A4 )
=> ( B5
!= ( image_int_int @ F @ C2 ) ) ) ) ).
% subset_imageE
thf(fact_457_image__subsetI,axiom,
! [A4: set_o,F: $o > $o,B5: set_o] :
( ! [X: $o] :
( ( member_o @ X @ A4 )
=> ( member_o @ ( F @ X ) @ B5 ) )
=> ( ord_less_eq_set_o @ ( image_o_o @ F @ A4 ) @ B5 ) ) ).
% image_subsetI
thf(fact_458_image__subsetI,axiom,
! [A4: set_o,F: $o > nat,B5: set_nat] :
( ! [X: $o] :
( ( member_o @ X @ A4 )
=> ( member_nat @ ( F @ X ) @ B5 ) )
=> ( ord_less_eq_set_nat @ ( image_o_nat @ F @ A4 ) @ B5 ) ) ).
% image_subsetI
thf(fact_459_image__subsetI,axiom,
! [A4: set_o,F: $o > int,B5: set_int] :
( ! [X: $o] :
( ( member_o @ X @ A4 )
=> ( member_int @ ( F @ X ) @ B5 ) )
=> ( ord_less_eq_set_int @ ( image_o_int @ F @ A4 ) @ B5 ) ) ).
% image_subsetI
thf(fact_460_image__subsetI,axiom,
! [A4: set_nat,F: nat > rat,B5: set_rat] :
( ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ( member_rat @ ( F @ X ) @ B5 ) )
=> ( ord_less_eq_set_rat @ ( image_nat_rat @ F @ A4 ) @ B5 ) ) ).
% image_subsetI
thf(fact_461_image__subsetI,axiom,
! [A4: set_nat,F: nat > $o,B5: set_o] :
( ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ( member_o @ ( F @ X ) @ B5 ) )
=> ( ord_less_eq_set_o @ ( image_nat_o @ F @ A4 ) @ B5 ) ) ).
% image_subsetI
thf(fact_462_image__subsetI,axiom,
! [A4: set_nat,F: nat > nat,B5: set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ( member_nat @ ( F @ X ) @ B5 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A4 ) @ B5 ) ) ).
% image_subsetI
thf(fact_463_image__subsetI,axiom,
! [A4: set_nat,F: nat > int,B5: set_int] :
( ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ( member_int @ ( F @ X ) @ B5 ) )
=> ( ord_less_eq_set_int @ ( image_nat_int @ F @ A4 ) @ B5 ) ) ).
% image_subsetI
thf(fact_464_image__subsetI,axiom,
! [A4: set_int,F: int > $o,B5: set_o] :
( ! [X: int] :
( ( member_int @ X @ A4 )
=> ( member_o @ ( F @ X ) @ B5 ) )
=> ( ord_less_eq_set_o @ ( image_int_o @ F @ A4 ) @ B5 ) ) ).
% image_subsetI
thf(fact_465_image__subsetI,axiom,
! [A4: set_int,F: int > nat,B5: set_nat] :
( ! [X: int] :
( ( member_int @ X @ A4 )
=> ( member_nat @ ( F @ X ) @ B5 ) )
=> ( ord_less_eq_set_nat @ ( image_int_nat @ F @ A4 ) @ B5 ) ) ).
% image_subsetI
thf(fact_466_image__subsetI,axiom,
! [A4: set_int,F: int > int,B5: set_int] :
( ! [X: int] :
( ( member_int @ X @ A4 )
=> ( member_int @ ( F @ X ) @ B5 ) )
=> ( ord_less_eq_set_int @ ( image_int_int @ F @ A4 ) @ B5 ) ) ).
% image_subsetI
thf(fact_467_image__mono,axiom,
! [A4: set_nat,B5: set_nat,F: nat > rat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ord_less_eq_set_rat @ ( image_nat_rat @ F @ A4 ) @ ( image_nat_rat @ F @ B5 ) ) ) ).
% image_mono
thf(fact_468_image__mono,axiom,
! [A4: set_nat,B5: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A4 ) @ ( image_nat_nat @ F @ B5 ) ) ) ).
% image_mono
thf(fact_469_image__mono,axiom,
! [A4: set_nat,B5: set_nat,F: nat > int] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ord_less_eq_set_int @ ( image_nat_int @ F @ A4 ) @ ( image_nat_int @ F @ B5 ) ) ) ).
% image_mono
thf(fact_470_image__mono,axiom,
! [A4: set_int,B5: set_int,F: int > nat] :
( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( ord_less_eq_set_nat @ ( image_int_nat @ F @ A4 ) @ ( image_int_nat @ F @ B5 ) ) ) ).
% image_mono
thf(fact_471_image__mono,axiom,
! [A4: set_int,B5: set_int,F: int > int] :
( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( ord_less_eq_set_int @ ( image_int_int @ F @ A4 ) @ ( image_int_int @ F @ B5 ) ) ) ).
% image_mono
thf(fact_472_map__prod_Ocomp,axiom,
! [F: $o > $o,G: $o > $o,H: nat > $o,I: nat > $o] :
( ( comp_P4785046977597218846at_nat @ ( produc1928436952130685430_o_o_o @ F @ G ) @ ( produc6957950696844810486_nat_o @ H @ I ) )
= ( produc6957950696844810486_nat_o @ ( comp_o_o_nat @ F @ H ) @ ( comp_o_o_nat @ G @ I ) ) ) ).
% map_prod.comp
thf(fact_473_map__prod_Ocomp,axiom,
! [F: $o > $o,G: nat > num,H: nat > $o,I: int > nat] :
( ( comp_P9057111047297513408at_int @ ( produc3628557804203378146at_num @ F @ G ) @ ( produc8250590009387450446nt_nat @ H @ I ) )
= ( produc4807921795543229080nt_num @ ( comp_o_o_nat @ F @ H ) @ ( comp_nat_num_int @ G @ I ) ) ) ).
% map_prod.comp
thf(fact_474_map__prod_Ocomp,axiom,
! [F: $o > $o,G: nat > nat,H: nat > $o,I: nat > nat] :
( ( comp_P4962864564169660718at_nat @ ( produc7071226018047599512at_nat @ F @ G ) @ ( produc7251236297486395890at_nat @ H @ I ) )
= ( produc7251236297486395890at_nat @ ( comp_o_o_nat @ F @ H ) @ ( comp_nat_nat_nat @ G @ I ) ) ) ).
% map_prod.comp
thf(fact_475_map__prod_Ocomp,axiom,
! [F: $o > $o,G: int > nat,H: nat > $o,I: int > int] :
( ( comp_P465755969289462830at_int @ ( produc8070579729948654068nt_nat @ F @ G ) @ ( produc8248099538878400170nt_int @ H @ I ) )
= ( produc8250590009387450446nt_nat @ ( comp_o_o_nat @ F @ H ) @ ( comp_int_nat_int @ G @ I ) ) ) ).
% map_prod.comp
thf(fact_476_map__prod_Ocomp,axiom,
! [F: nat > num,G: $o > $o,H: int > nat,I: nat > $o] :
( ( comp_P5807473265676440968nt_nat @ ( produc3264868786340801506um_o_o @ F @ G ) @ ( produc8298419091753553490_nat_o @ H @ I ) )
= ( produc9074608839065728392_nat_o @ ( comp_nat_num_int @ F @ H ) @ ( comp_o_o_nat @ G @ I ) ) ) ).
% map_prod.comp
thf(fact_477_map__prod_Ocomp,axiom,
! [F: nat > num,G: nat > num,H: int > nat,I: int > nat] :
( ( comp_P9218946435557410242nt_int @ ( produc1476026309457068494at_num @ F @ G ) @ ( produc5667422475138734578nt_nat @ H @ I ) )
= ( produc165562089265172102nt_num @ ( comp_nat_num_int @ F @ H ) @ ( comp_nat_num_int @ G @ I ) ) ) ).
% map_prod.comp
thf(fact_478_map__prod_Ocomp,axiom,
! [F: nat > num,G: nat > nat,H: int > nat,I: nat > nat] :
( ( comp_P5124699952429557552nt_nat @ ( produc4918694523301289860at_nat @ F @ G ) @ ( produc4668068763237680022at_nat @ H @ I ) )
= ( produc2608876591208338912at_nat @ ( comp_nat_num_int @ F @ H ) @ ( comp_nat_nat_nat @ G @ I ) ) ) ).
% map_prod.comp
thf(fact_479_map__prod_Ocomp,axiom,
! [F: nat > num,G: int > nat,H: int > nat,I: int > int] :
( ( comp_P3772045228931366376nt_int @ ( produc5918048235202344416nt_nat @ F @ G ) @ ( produc5664932004629684302nt_int @ H @ I ) )
= ( produc3608230303109393468nt_nat @ ( comp_nat_num_int @ F @ H ) @ ( comp_int_nat_int @ G @ I ) ) ) ).
% map_prod.comp
thf(fact_480_map__prod_Ocomp,axiom,
! [F: nat > nat,G: $o > $o,H: nat > nat,I: nat > $o] :
( ( comp_P8901546994122883554at_nat @ ( produc2001155298098703512at_o_o @ F @ G ) @ ( produc4787215299218314158_nat_o @ H @ I ) )
= ( produc4787215299218314158_nat_o @ ( comp_nat_nat_nat @ F @ H ) @ ( comp_o_o_nat @ G @ I ) ) ) ).
% map_prod.comp
thf(fact_481_map__prod_Ocomp,axiom,
! [F: nat > nat,G: nat > num,H: nat > nat,I: int > nat] :
( ( comp_P6334688330072917552at_int @ ( produc3535218481486409604at_num @ F @ G ) @ ( produc7977240407231685526nt_nat @ H @ I ) )
= ( produc4534572193387464160nt_num @ ( comp_nat_nat_nat @ F @ H ) @ ( comp_nat_num_int @ G @ I ) ) ) ).
% map_prod.comp
thf(fact_482_map__prod_Ocompositionality,axiom,
! [F: $o > $o,G: $o > $o,H: nat > $o,I: nat > $o,Prod: product_prod_nat_nat] :
( ( produc1928436952130685430_o_o_o @ F @ G @ ( produc6957950696844810486_nat_o @ H @ I @ Prod ) )
= ( produc6957950696844810486_nat_o @ ( comp_o_o_nat @ F @ H ) @ ( comp_o_o_nat @ G @ I ) @ Prod ) ) ).
% map_prod.compositionality
thf(fact_483_map__prod_Ocompositionality,axiom,
! [F: $o > $o,G: nat > num,H: nat > $o,I: int > nat,Prod: product_prod_nat_int] :
( ( produc3628557804203378146at_num @ F @ G @ ( produc8250590009387450446nt_nat @ H @ I @ Prod ) )
= ( produc4807921795543229080nt_num @ ( comp_o_o_nat @ F @ H ) @ ( comp_nat_num_int @ G @ I ) @ Prod ) ) ).
% map_prod.compositionality
thf(fact_484_map__prod_Ocompositionality,axiom,
! [F: $o > $o,G: nat > nat,H: nat > $o,I: nat > nat,Prod: product_prod_nat_nat] :
( ( produc7071226018047599512at_nat @ F @ G @ ( produc7251236297486395890at_nat @ H @ I @ Prod ) )
= ( produc7251236297486395890at_nat @ ( comp_o_o_nat @ F @ H ) @ ( comp_nat_nat_nat @ G @ I ) @ Prod ) ) ).
% map_prod.compositionality
thf(fact_485_map__prod_Ocompositionality,axiom,
! [F: $o > $o,G: int > nat,H: nat > $o,I: int > int,Prod: product_prod_nat_int] :
( ( produc8070579729948654068nt_nat @ F @ G @ ( produc8248099538878400170nt_int @ H @ I @ Prod ) )
= ( produc8250590009387450446nt_nat @ ( comp_o_o_nat @ F @ H ) @ ( comp_int_nat_int @ G @ I ) @ Prod ) ) ).
% map_prod.compositionality
thf(fact_486_map__prod_Ocompositionality,axiom,
! [F: nat > num,G: $o > $o,H: int > nat,I: nat > $o,Prod: product_prod_int_nat] :
( ( produc3264868786340801506um_o_o @ F @ G @ ( produc8298419091753553490_nat_o @ H @ I @ Prod ) )
= ( produc9074608839065728392_nat_o @ ( comp_nat_num_int @ F @ H ) @ ( comp_o_o_nat @ G @ I ) @ Prod ) ) ).
% map_prod.compositionality
thf(fact_487_map__prod_Ocompositionality,axiom,
! [F: nat > num,G: nat > num,H: int > nat,I: int > nat,Prod: product_prod_int_int] :
( ( produc1476026309457068494at_num @ F @ G @ ( produc5667422475138734578nt_nat @ H @ I @ Prod ) )
= ( produc165562089265172102nt_num @ ( comp_nat_num_int @ F @ H ) @ ( comp_nat_num_int @ G @ I ) @ Prod ) ) ).
% map_prod.compositionality
thf(fact_488_map__prod_Ocompositionality,axiom,
! [F: nat > num,G: nat > nat,H: int > nat,I: nat > nat,Prod: product_prod_int_nat] :
( ( produc4918694523301289860at_nat @ F @ G @ ( produc4668068763237680022at_nat @ H @ I @ Prod ) )
= ( produc2608876591208338912at_nat @ ( comp_nat_num_int @ F @ H ) @ ( comp_nat_nat_nat @ G @ I ) @ Prod ) ) ).
% map_prod.compositionality
thf(fact_489_map__prod_Ocompositionality,axiom,
! [F: nat > num,G: int > nat,H: int > nat,I: int > int,Prod: product_prod_int_int] :
( ( produc5918048235202344416nt_nat @ F @ G @ ( produc5664932004629684302nt_int @ H @ I @ Prod ) )
= ( produc3608230303109393468nt_nat @ ( comp_nat_num_int @ F @ H ) @ ( comp_int_nat_int @ G @ I ) @ Prod ) ) ).
% map_prod.compositionality
thf(fact_490_map__prod_Ocompositionality,axiom,
! [F: nat > nat,G: $o > $o,H: nat > nat,I: nat > $o,Prod: product_prod_nat_nat] :
( ( produc2001155298098703512at_o_o @ F @ G @ ( produc4787215299218314158_nat_o @ H @ I @ Prod ) )
= ( produc4787215299218314158_nat_o @ ( comp_nat_nat_nat @ F @ H ) @ ( comp_o_o_nat @ G @ I ) @ Prod ) ) ).
% map_prod.compositionality
thf(fact_491_map__prod_Ocompositionality,axiom,
! [F: nat > nat,G: nat > num,H: nat > nat,I: int > nat,Prod: product_prod_nat_int] :
( ( produc3535218481486409604at_num @ F @ G @ ( produc7977240407231685526nt_nat @ H @ I @ Prod ) )
= ( produc4534572193387464160nt_num @ ( comp_nat_nat_nat @ F @ H ) @ ( comp_nat_num_int @ G @ I ) @ Prod ) ) ).
% map_prod.compositionality
thf(fact_492_map__prod__compose,axiom,
! [F1: $o > $o,F22: nat > $o,G1: $o > $o,G22: nat > $o] :
( ( produc6957950696844810486_nat_o @ ( comp_o_o_nat @ F1 @ F22 ) @ ( comp_o_o_nat @ G1 @ G22 ) )
= ( comp_P4785046977597218846at_nat @ ( produc1928436952130685430_o_o_o @ F1 @ G1 ) @ ( produc6957950696844810486_nat_o @ F22 @ G22 ) ) ) ).
% map_prod_compose
thf(fact_493_map__prod__compose,axiom,
! [F1: $o > $o,F22: nat > $o,G1: nat > num,G22: int > nat] :
( ( produc4807921795543229080nt_num @ ( comp_o_o_nat @ F1 @ F22 ) @ ( comp_nat_num_int @ G1 @ G22 ) )
= ( comp_P9057111047297513408at_int @ ( produc3628557804203378146at_num @ F1 @ G1 ) @ ( produc8250590009387450446nt_nat @ F22 @ G22 ) ) ) ).
% map_prod_compose
thf(fact_494_map__prod__compose,axiom,
! [F1: $o > $o,F22: nat > $o,G1: nat > nat,G22: nat > nat] :
( ( produc7251236297486395890at_nat @ ( comp_o_o_nat @ F1 @ F22 ) @ ( comp_nat_nat_nat @ G1 @ G22 ) )
= ( comp_P4962864564169660718at_nat @ ( produc7071226018047599512at_nat @ F1 @ G1 ) @ ( produc7251236297486395890at_nat @ F22 @ G22 ) ) ) ).
% map_prod_compose
thf(fact_495_map__prod__compose,axiom,
! [F1: $o > $o,F22: nat > $o,G1: int > nat,G22: int > int] :
( ( produc8250590009387450446nt_nat @ ( comp_o_o_nat @ F1 @ F22 ) @ ( comp_int_nat_int @ G1 @ G22 ) )
= ( comp_P465755969289462830at_int @ ( produc8070579729948654068nt_nat @ F1 @ G1 ) @ ( produc8248099538878400170nt_int @ F22 @ G22 ) ) ) ).
% map_prod_compose
thf(fact_496_map__prod__compose,axiom,
! [F1: nat > num,F22: int > nat,G1: $o > $o,G22: nat > $o] :
( ( produc9074608839065728392_nat_o @ ( comp_nat_num_int @ F1 @ F22 ) @ ( comp_o_o_nat @ G1 @ G22 ) )
= ( comp_P5807473265676440968nt_nat @ ( produc3264868786340801506um_o_o @ F1 @ G1 ) @ ( produc8298419091753553490_nat_o @ F22 @ G22 ) ) ) ).
% map_prod_compose
thf(fact_497_map__prod__compose,axiom,
! [F1: nat > num,F22: int > nat,G1: nat > num,G22: int > nat] :
( ( produc165562089265172102nt_num @ ( comp_nat_num_int @ F1 @ F22 ) @ ( comp_nat_num_int @ G1 @ G22 ) )
= ( comp_P9218946435557410242nt_int @ ( produc1476026309457068494at_num @ F1 @ G1 ) @ ( produc5667422475138734578nt_nat @ F22 @ G22 ) ) ) ).
% map_prod_compose
thf(fact_498_map__prod__compose,axiom,
! [F1: nat > num,F22: int > nat,G1: nat > nat,G22: nat > nat] :
( ( produc2608876591208338912at_nat @ ( comp_nat_num_int @ F1 @ F22 ) @ ( comp_nat_nat_nat @ G1 @ G22 ) )
= ( comp_P5124699952429557552nt_nat @ ( produc4918694523301289860at_nat @ F1 @ G1 ) @ ( produc4668068763237680022at_nat @ F22 @ G22 ) ) ) ).
% map_prod_compose
thf(fact_499_map__prod__compose,axiom,
! [F1: nat > num,F22: int > nat,G1: int > nat,G22: int > int] :
( ( produc3608230303109393468nt_nat @ ( comp_nat_num_int @ F1 @ F22 ) @ ( comp_int_nat_int @ G1 @ G22 ) )
= ( comp_P3772045228931366376nt_int @ ( produc5918048235202344416nt_nat @ F1 @ G1 ) @ ( produc5664932004629684302nt_int @ F22 @ G22 ) ) ) ).
% map_prod_compose
thf(fact_500_map__prod__compose,axiom,
! [F1: nat > nat,F22: nat > nat,G1: $o > $o,G22: nat > $o] :
( ( produc4787215299218314158_nat_o @ ( comp_nat_nat_nat @ F1 @ F22 ) @ ( comp_o_o_nat @ G1 @ G22 ) )
= ( comp_P8901546994122883554at_nat @ ( produc2001155298098703512at_o_o @ F1 @ G1 ) @ ( produc4787215299218314158_nat_o @ F22 @ G22 ) ) ) ).
% map_prod_compose
thf(fact_501_map__prod__compose,axiom,
! [F1: nat > nat,F22: nat > nat,G1: nat > num,G22: int > nat] :
( ( produc4534572193387464160nt_num @ ( comp_nat_nat_nat @ F1 @ F22 ) @ ( comp_nat_num_int @ G1 @ G22 ) )
= ( comp_P6334688330072917552at_int @ ( produc3535218481486409604at_num @ F1 @ G1 ) @ ( produc7977240407231685526nt_nat @ F22 @ G22 ) ) ) ).
% map_prod_compose
thf(fact_502_apsnd__compose,axiom,
! [F: code_integer > code_integer,G: code_integer > code_integer,X2: produc8923325533196201883nteger] :
( ( produc6499014454317279255nteger @ F @ ( produc6499014454317279255nteger @ G @ X2 ) )
= ( produc6499014454317279255nteger @ ( comp_C7449957260575251196nteger @ F @ G ) @ X2 ) ) ).
% apsnd_compose
thf(fact_503_Collect__case__prod__Grp__in,axiom,
! [Z2: produc3843707927480180839at_nat,A4: set_se7855581050983116737at_nat,F: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat] :
( ( member8757157785044589968at_nat @ Z2 @ ( collec6321179662152712658at_nat @ ( produc410239310623530412_nat_o @ ( bNF_Gr6677838525845560078at_nat @ A4 @ F ) ) ) )
=> ( member2643936169264416010at_nat @ ( produc6011863565407846595at_nat @ Z2 ) @ A4 ) ) ).
% Collect_case_prod_Grp_in
thf(fact_504_Collect__case__prod__Grp__in,axiom,
! [Z2: product_prod_nat_nat,A4: set_nat,F: nat > nat] :
( ( member8440522571783428010at_nat @ Z2 @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ( bNF_Grp_nat_nat @ A4 @ F ) ) ) )
=> ( member_nat @ ( product_fst_nat_nat @ Z2 ) @ A4 ) ) ).
% Collect_case_prod_Grp_in
thf(fact_505_Collect__case__prod__Grp__in,axiom,
! [Z2: produc7812382424243267045t_unit,A4: set_ref_a,F: ref_a > heap_e7401611519738050253t_unit] :
( ( member7849944260330398460t_unit @ Z2 @ ( collec3252306575900995386t_unit @ ( produc4924168586796184916unit_o @ ( bNF_Gr6510375311308651680t_unit @ A4 @ F ) ) ) )
=> ( member_ref_a @ ( produc473321116828791851t_unit @ Z2 ) @ A4 ) ) ).
% Collect_case_prod_Grp_in
thf(fact_506_Collect__case__prod__Grp__in,axiom,
! [Z2: product_prod_int_int,A4: set_int,F: int > int] :
( ( member5262025264175285858nt_int @ Z2 @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ ( bNF_Grp_int_int @ A4 @ F ) ) ) )
=> ( member_int @ ( product_fst_int_int @ Z2 ) @ A4 ) ) ).
% Collect_case_prod_Grp_in
thf(fact_507_prod__fun__imageE,axiom,
! [C: product_prod_nat_num,F: nat > nat,G: num > num,R3: set_Pr6200539531224447659at_num] :
( ( member9148766508732265716at_num @ C @ ( image_4925883905578568901at_num @ ( produc1665548884793878990um_num @ F @ G ) @ R3 ) )
=> ~ ! [X: nat,Y: num] :
( ( C
= ( product_Pair_nat_num @ ( F @ X ) @ ( G @ Y ) ) )
=> ~ ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X @ Y ) @ R3 ) ) ) ).
% prod_fun_imageE
thf(fact_508_prod__fun__imageE,axiom,
! [C: product_prod_nat_num,F: nat > nat,G: nat > num,R3: set_Pr1261947904930325089at_nat] :
( ( member9148766508732265716at_num @ C @ ( image_5877158439796295311at_num @ ( produc3535218481486409604at_num @ F @ G ) @ R3 ) )
=> ~ ! [X: nat,Y: nat] :
( ( C
= ( product_Pair_nat_num @ ( F @ X ) @ ( G @ Y ) ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R3 ) ) ) ).
% prod_fun_imageE
thf(fact_509_prod__fun__imageE,axiom,
! [C: product_prod_nat_num,F: int > nat,G: int > num,R3: set_Pr958786334691620121nt_int] :
( ( member9148766508732265716at_num @ C @ ( image_6540112122905407959at_num @ ( produc2224754261294513212nt_num @ F @ G ) @ R3 ) )
=> ~ ! [X: int,Y: int] :
( ( C
= ( product_Pair_nat_num @ ( F @ X ) @ ( G @ Y ) ) )
=> ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R3 ) ) ) ).
% prod_fun_imageE
thf(fact_510_prod__fun__imageE,axiom,
! [C: product_prod_nat_nat,F: nat > nat,G: num > nat,R3: set_Pr6200539531224447659at_num] :
( ( member8440522571783428010at_nat @ C @ ( image_4217639968629731195at_nat @ ( produc5108217098638100356um_nat @ F @ G ) @ R3 ) )
=> ~ ! [X: nat,Y: num] :
( ( C
= ( product_Pair_nat_nat @ ( F @ X ) @ ( G @ Y ) ) )
=> ~ ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X @ Y ) @ R3 ) ) ) ).
% prod_fun_imageE
thf(fact_511_prod__fun__imageE,axiom,
! [C: product_prod_nat_nat,F: nat > nat,G: nat > nat,R3: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ C @ ( image_5168914502847457605at_nat @ ( produc6977886695330630970at_nat @ F @ G ) @ R3 ) )
=> ~ ! [X: nat,Y: nat] :
( ( C
= ( product_Pair_nat_nat @ ( F @ X ) @ ( G @ Y ) ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R3 ) ) ) ).
% prod_fun_imageE
thf(fact_512_prod__fun__imageE,axiom,
! [C: product_prod_nat_nat,F: int > nat,G: int > nat,R3: set_Pr958786334691620121nt_int] :
( ( member8440522571783428010at_nat @ C @ ( image_5831868185956570253at_nat @ ( produc5667422475138734578nt_nat @ F @ G ) @ R3 ) )
=> ~ ! [X: int,Y: int] :
( ( C
= ( product_Pair_nat_nat @ ( F @ X ) @ ( G @ Y ) ) )
=> ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R3 ) ) ) ).
% prod_fun_imageE
thf(fact_513_prod__fun__imageE,axiom,
! [C: product_prod_int_int,F: nat > int,G: num > int,R3: set_Pr6200539531224447659at_num] :
( ( member5262025264175285858nt_int @ C @ ( image_1039142661021589043nt_int @ ( produc5477380188365894716um_int @ F @ G ) @ R3 ) )
=> ~ ! [X: nat,Y: num] :
( ( C
= ( product_Pair_int_int @ ( F @ X ) @ ( G @ Y ) ) )
=> ~ ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X @ Y ) @ R3 ) ) ) ).
% prod_fun_imageE
thf(fact_514_prod__fun__imageE,axiom,
! [C: product_prod_int_int,F: nat > int,G: nat > int,R3: set_Pr1261947904930325089at_nat] :
( ( member5262025264175285858nt_int @ C @ ( image_1990417195239315453nt_int @ ( produc7347049785058425330at_int @ F @ G ) @ R3 ) )
=> ~ ! [X: nat,Y: nat] :
( ( C
= ( product_Pair_int_int @ ( F @ X ) @ ( G @ Y ) ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R3 ) ) ) ).
% prod_fun_imageE
thf(fact_515_prod__fun__imageE,axiom,
! [C: product_prod_int_int,F: int > int,G: int > int,R3: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ C @ ( image_2653370878348428101nt_int @ ( produc6036585564866528938nt_int @ F @ G ) @ R3 ) )
=> ~ ! [X: int,Y: int] :
( ( C
= ( product_Pair_int_int @ ( F @ X ) @ ( G @ Y ) ) )
=> ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R3 ) ) ) ).
% prod_fun_imageE
thf(fact_516_prod__fun__imageE,axiom,
! [C: produc3843707927480180839at_nat,F: nat > set_Pr1261947904930325089at_nat,G: num > set_Pr1261947904930325089at_nat,R3: set_Pr6200539531224447659at_num] :
( ( member8757157785044589968at_nat @ C @ ( image_2708129733610033121at_nat @ ( produc170006858110003436at_nat @ F @ G ) @ R3 ) )
=> ~ ! [X: nat,Y: num] :
( ( C
= ( produc2922128104949294807at_nat @ ( F @ X ) @ ( G @ Y ) ) )
=> ~ ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X @ Y ) @ R3 ) ) ) ).
% prod_fun_imageE
thf(fact_517_rev__image__eqI,axiom,
! [X2: $o,A4: set_o,B: $o,F: $o > $o] :
( ( member_o @ X2 @ A4 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_o @ B @ ( image_o_o @ F @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_518_rev__image__eqI,axiom,
! [X2: $o,A4: set_o,B: nat,F: $o > nat] :
( ( member_o @ X2 @ A4 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_nat @ B @ ( image_o_nat @ F @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_519_rev__image__eqI,axiom,
! [X2: $o,A4: set_o,B: int,F: $o > int] :
( ( member_o @ X2 @ A4 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_int @ B @ ( image_o_int @ F @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_520_rev__image__eqI,axiom,
! [X2: nat,A4: set_nat,B: rat,F: nat > rat] :
( ( member_nat @ X2 @ A4 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_rat @ B @ ( image_nat_rat @ F @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_521_rev__image__eqI,axiom,
! [X2: nat,A4: set_nat,B: $o,F: nat > $o] :
( ( member_nat @ X2 @ A4 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_o @ B @ ( image_nat_o @ F @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_522_rev__image__eqI,axiom,
! [X2: nat,A4: set_nat,B: nat,F: nat > nat] :
( ( member_nat @ X2 @ A4 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_523_rev__image__eqI,axiom,
! [X2: nat,A4: set_nat,B: int,F: nat > int] :
( ( member_nat @ X2 @ A4 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_int @ B @ ( image_nat_int @ F @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_524_rev__image__eqI,axiom,
! [X2: int,A4: set_int,B: $o,F: int > $o] :
( ( member_int @ X2 @ A4 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_o @ B @ ( image_int_o @ F @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_525_rev__image__eqI,axiom,
! [X2: int,A4: set_int,B: nat,F: int > nat] :
( ( member_int @ X2 @ A4 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_nat @ B @ ( image_int_nat @ F @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_526_rev__image__eqI,axiom,
! [X2: int,A4: set_int,B: int,F: int > int] :
( ( member_int @ X2 @ A4 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_int @ B @ ( image_int_int @ F @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_527_ball__imageD,axiom,
! [F: nat > rat,A4: set_nat,P2: rat > $o] :
( ! [X: rat] :
( ( member_rat @ X @ ( image_nat_rat @ F @ A4 ) )
=> ( P2 @ X ) )
=> ! [X6: nat] :
( ( member_nat @ X6 @ A4 )
=> ( P2 @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_528_ball__imageD,axiom,
! [F: nat > nat,A4: set_nat,P2: nat > $o] :
( ! [X: nat] :
( ( member_nat @ X @ ( image_nat_nat @ F @ A4 ) )
=> ( P2 @ X ) )
=> ! [X6: nat] :
( ( member_nat @ X6 @ A4 )
=> ( P2 @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_529_ball__imageD,axiom,
! [F: nat > int,A4: set_nat,P2: int > $o] :
( ! [X: int] :
( ( member_int @ X @ ( image_nat_int @ F @ A4 ) )
=> ( P2 @ X ) )
=> ! [X6: nat] :
( ( member_nat @ X6 @ A4 )
=> ( P2 @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_530_ball__imageD,axiom,
! [F: int > nat,A4: set_int,P2: nat > $o] :
( ! [X: nat] :
( ( member_nat @ X @ ( image_int_nat @ F @ A4 ) )
=> ( P2 @ X ) )
=> ! [X6: int] :
( ( member_int @ X6 @ A4 )
=> ( P2 @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_531_ball__imageD,axiom,
! [F: int > int,A4: set_int,P2: int > $o] :
( ! [X: int] :
( ( member_int @ X @ ( image_int_int @ F @ A4 ) )
=> ( P2 @ X ) )
=> ! [X6: int] :
( ( member_int @ X6 @ A4 )
=> ( P2 @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_532_image__cong,axiom,
! [M: set_nat,N: set_nat,F: nat > rat,G: nat > rat] :
( ( M = N )
=> ( ! [X: nat] :
( ( member_nat @ X @ N )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image_nat_rat @ F @ M )
= ( image_nat_rat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_533_image__cong,axiom,
! [M: set_nat,N: set_nat,F: nat > nat,G: nat > nat] :
( ( M = N )
=> ( ! [X: nat] :
( ( member_nat @ X @ N )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image_nat_nat @ F @ M )
= ( image_nat_nat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_534_image__cong,axiom,
! [M: set_nat,N: set_nat,F: nat > int,G: nat > int] :
( ( M = N )
=> ( ! [X: nat] :
( ( member_nat @ X @ N )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image_nat_int @ F @ M )
= ( image_nat_int @ G @ N ) ) ) ) ).
% image_cong
thf(fact_535_image__cong,axiom,
! [M: set_int,N: set_int,F: int > nat,G: int > nat] :
( ( M = N )
=> ( ! [X: int] :
( ( member_int @ X @ N )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image_int_nat @ F @ M )
= ( image_int_nat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_536_image__cong,axiom,
! [M: set_int,N: set_int,F: int > int,G: int > int] :
( ( M = N )
=> ( ! [X: int] :
( ( member_int @ X @ N )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image_int_int @ F @ M )
= ( image_int_int @ G @ N ) ) ) ) ).
% image_cong
thf(fact_537_bex__imageD,axiom,
! [F: nat > rat,A4: set_nat,P2: rat > $o] :
( ? [X6: rat] :
( ( member_rat @ X6 @ ( image_nat_rat @ F @ A4 ) )
& ( P2 @ X6 ) )
=> ? [X: nat] :
( ( member_nat @ X @ A4 )
& ( P2 @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_538_bex__imageD,axiom,
! [F: nat > nat,A4: set_nat,P2: nat > $o] :
( ? [X6: nat] :
( ( member_nat @ X6 @ ( image_nat_nat @ F @ A4 ) )
& ( P2 @ X6 ) )
=> ? [X: nat] :
( ( member_nat @ X @ A4 )
& ( P2 @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_539_bex__imageD,axiom,
! [F: nat > int,A4: set_nat,P2: int > $o] :
( ? [X6: int] :
( ( member_int @ X6 @ ( image_nat_int @ F @ A4 ) )
& ( P2 @ X6 ) )
=> ? [X: nat] :
( ( member_nat @ X @ A4 )
& ( P2 @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_540_bex__imageD,axiom,
! [F: int > nat,A4: set_int,P2: nat > $o] :
( ? [X6: nat] :
( ( member_nat @ X6 @ ( image_int_nat @ F @ A4 ) )
& ( P2 @ X6 ) )
=> ? [X: int] :
( ( member_int @ X @ A4 )
& ( P2 @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_541_bex__imageD,axiom,
! [F: int > int,A4: set_int,P2: int > $o] :
( ? [X6: int] :
( ( member_int @ X6 @ ( image_int_int @ F @ A4 ) )
& ( P2 @ X6 ) )
=> ? [X: int] :
( ( member_int @ X @ A4 )
& ( P2 @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_542_image__iff,axiom,
! [Z2: rat,F: nat > rat,A4: set_nat] :
( ( member_rat @ Z2 @ ( image_nat_rat @ F @ A4 ) )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ A4 )
& ( Z2
= ( F @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_543_image__iff,axiom,
! [Z2: nat,F: nat > nat,A4: set_nat] :
( ( member_nat @ Z2 @ ( image_nat_nat @ F @ A4 ) )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ A4 )
& ( Z2
= ( F @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_544_image__iff,axiom,
! [Z2: nat,F: int > nat,A4: set_int] :
( ( member_nat @ Z2 @ ( image_int_nat @ F @ A4 ) )
= ( ? [X4: int] :
( ( member_int @ X4 @ A4 )
& ( Z2
= ( F @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_545_image__iff,axiom,
! [Z2: int,F: nat > int,A4: set_nat] :
( ( member_int @ Z2 @ ( image_nat_int @ F @ A4 ) )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ A4 )
& ( Z2
= ( F @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_546_image__iff,axiom,
! [Z2: int,F: int > int,A4: set_int] :
( ( member_int @ Z2 @ ( image_int_int @ F @ A4 ) )
= ( ? [X4: int] :
( ( member_int @ X4 @ A4 )
& ( Z2
= ( F @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_547_imageI,axiom,
! [X2: $o,A4: set_o,F: $o > $o] :
( ( member_o @ X2 @ A4 )
=> ( member_o @ ( F @ X2 ) @ ( image_o_o @ F @ A4 ) ) ) ).
% imageI
thf(fact_548_imageI,axiom,
! [X2: $o,A4: set_o,F: $o > nat] :
( ( member_o @ X2 @ A4 )
=> ( member_nat @ ( F @ X2 ) @ ( image_o_nat @ F @ A4 ) ) ) ).
% imageI
thf(fact_549_imageI,axiom,
! [X2: $o,A4: set_o,F: $o > int] :
( ( member_o @ X2 @ A4 )
=> ( member_int @ ( F @ X2 ) @ ( image_o_int @ F @ A4 ) ) ) ).
% imageI
thf(fact_550_imageI,axiom,
! [X2: nat,A4: set_nat,F: nat > rat] :
( ( member_nat @ X2 @ A4 )
=> ( member_rat @ ( F @ X2 ) @ ( image_nat_rat @ F @ A4 ) ) ) ).
% imageI
thf(fact_551_imageI,axiom,
! [X2: nat,A4: set_nat,F: nat > $o] :
( ( member_nat @ X2 @ A4 )
=> ( member_o @ ( F @ X2 ) @ ( image_nat_o @ F @ A4 ) ) ) ).
% imageI
thf(fact_552_imageI,axiom,
! [X2: nat,A4: set_nat,F: nat > nat] :
( ( member_nat @ X2 @ A4 )
=> ( member_nat @ ( F @ X2 ) @ ( image_nat_nat @ F @ A4 ) ) ) ).
% imageI
thf(fact_553_imageI,axiom,
! [X2: nat,A4: set_nat,F: nat > int] :
( ( member_nat @ X2 @ A4 )
=> ( member_int @ ( F @ X2 ) @ ( image_nat_int @ F @ A4 ) ) ) ).
% imageI
thf(fact_554_imageI,axiom,
! [X2: int,A4: set_int,F: int > $o] :
( ( member_int @ X2 @ A4 )
=> ( member_o @ ( F @ X2 ) @ ( image_int_o @ F @ A4 ) ) ) ).
% imageI
thf(fact_555_imageI,axiom,
! [X2: int,A4: set_int,F: int > nat] :
( ( member_int @ X2 @ A4 )
=> ( member_nat @ ( F @ X2 ) @ ( image_int_nat @ F @ A4 ) ) ) ).
% imageI
thf(fact_556_imageI,axiom,
! [X2: int,A4: set_int,F: int > int] :
( ( member_int @ X2 @ A4 )
=> ( member_int @ ( F @ X2 ) @ ( image_int_int @ F @ A4 ) ) ) ).
% imageI
thf(fact_557_Sup_OSUP__cong,axiom,
! [A4: set_nat,B5: set_nat,C3: nat > rat,D: nat > rat,Sup: set_rat > rat] :
( ( A4 = B5 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B5 )
=> ( ( C3 @ X )
= ( D @ X ) ) )
=> ( ( Sup @ ( image_nat_rat @ C3 @ A4 ) )
= ( Sup @ ( image_nat_rat @ D @ B5 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_558_Sup_OSUP__cong,axiom,
! [A4: set_nat,B5: set_nat,C3: nat > nat,D: nat > nat,Sup: set_nat > nat] :
( ( A4 = B5 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B5 )
=> ( ( C3 @ X )
= ( D @ X ) ) )
=> ( ( Sup @ ( image_nat_nat @ C3 @ A4 ) )
= ( Sup @ ( image_nat_nat @ D @ B5 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_559_Sup_OSUP__cong,axiom,
! [A4: set_nat,B5: set_nat,C3: nat > int,D: nat > int,Sup: set_int > int] :
( ( A4 = B5 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B5 )
=> ( ( C3 @ X )
= ( D @ X ) ) )
=> ( ( Sup @ ( image_nat_int @ C3 @ A4 ) )
= ( Sup @ ( image_nat_int @ D @ B5 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_560_Sup_OSUP__cong,axiom,
! [A4: set_int,B5: set_int,C3: int > nat,D: int > nat,Sup: set_nat > nat] :
( ( A4 = B5 )
=> ( ! [X: int] :
( ( member_int @ X @ B5 )
=> ( ( C3 @ X )
= ( D @ X ) ) )
=> ( ( Sup @ ( image_int_nat @ C3 @ A4 ) )
= ( Sup @ ( image_int_nat @ D @ B5 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_561_Sup_OSUP__cong,axiom,
! [A4: set_int,B5: set_int,C3: int > int,D: int > int,Sup: set_int > int] :
( ( A4 = B5 )
=> ( ! [X: int] :
( ( member_int @ X @ B5 )
=> ( ( C3 @ X )
= ( D @ X ) ) )
=> ( ( Sup @ ( image_int_int @ C3 @ A4 ) )
= ( Sup @ ( image_int_int @ D @ B5 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_562_Inf_OINF__cong,axiom,
! [A4: set_nat,B5: set_nat,C3: nat > rat,D: nat > rat,Inf: set_rat > rat] :
( ( A4 = B5 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B5 )
=> ( ( C3 @ X )
= ( D @ X ) ) )
=> ( ( Inf @ ( image_nat_rat @ C3 @ A4 ) )
= ( Inf @ ( image_nat_rat @ D @ B5 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_563_Inf_OINF__cong,axiom,
! [A4: set_nat,B5: set_nat,C3: nat > nat,D: nat > nat,Inf: set_nat > nat] :
( ( A4 = B5 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B5 )
=> ( ( C3 @ X )
= ( D @ X ) ) )
=> ( ( Inf @ ( image_nat_nat @ C3 @ A4 ) )
= ( Inf @ ( image_nat_nat @ D @ B5 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_564_Inf_OINF__cong,axiom,
! [A4: set_nat,B5: set_nat,C3: nat > int,D: nat > int,Inf: set_int > int] :
( ( A4 = B5 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B5 )
=> ( ( C3 @ X )
= ( D @ X ) ) )
=> ( ( Inf @ ( image_nat_int @ C3 @ A4 ) )
= ( Inf @ ( image_nat_int @ D @ B5 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_565_Inf_OINF__cong,axiom,
! [A4: set_int,B5: set_int,C3: int > nat,D: int > nat,Inf: set_nat > nat] :
( ( A4 = B5 )
=> ( ! [X: int] :
( ( member_int @ X @ B5 )
=> ( ( C3 @ X )
= ( D @ X ) ) )
=> ( ( Inf @ ( image_int_nat @ C3 @ A4 ) )
= ( Inf @ ( image_int_nat @ D @ B5 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_566_Inf_OINF__cong,axiom,
! [A4: set_int,B5: set_int,C3: int > int,D: int > int,Inf: set_int > int] :
( ( A4 = B5 )
=> ( ! [X: int] :
( ( member_int @ X @ B5 )
=> ( ( C3 @ X )
= ( D @ X ) ) )
=> ( ( Inf @ ( image_int_int @ C3 @ A4 ) )
= ( Inf @ ( image_int_int @ D @ B5 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_567_comp__eq__dest__lhs,axiom,
! [A: $o > $o,B: nat > $o,C: nat > $o,V: nat] :
( ( ( comp_o_o_nat @ A @ B )
= C )
=> ( ( A @ ( B @ V ) )
= ( C @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_568_comp__eq__dest__lhs,axiom,
! [A: nat > num,B: int > nat,C: int > num,V: int] :
( ( ( comp_nat_num_int @ A @ B )
= C )
=> ( ( A @ ( B @ V ) )
= ( C @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_569_comp__eq__dest__lhs,axiom,
! [A: nat > nat,B: nat > nat,C: nat > nat,V: nat] :
( ( ( comp_nat_nat_nat @ A @ B )
= C )
=> ( ( A @ ( B @ V ) )
= ( C @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_570_comp__eq__dest__lhs,axiom,
! [A: int > nat,B: int > int,C: int > nat,V: int] :
( ( ( comp_int_nat_int @ A @ B )
= C )
=> ( ( A @ ( B @ V ) )
= ( C @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_571_comp__eq__dest__lhs,axiom,
! [A: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,B: code_integer > code_integer > code_integer,C: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,V: code_integer] :
( ( ( comp_C8797469213163452608nteger @ A @ B )
= C )
=> ( ( A @ ( B @ V ) )
= ( C @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_572_comp__eq__elim,axiom,
! [A: $o > $o,B: nat > $o,C: $o > $o,D2: nat > $o] :
( ( ( comp_o_o_nat @ A @ B )
= ( comp_o_o_nat @ C @ D2 ) )
=> ! [V4: nat] :
( ( A @ ( B @ V4 ) )
= ( C @ ( D2 @ V4 ) ) ) ) ).
% comp_eq_elim
thf(fact_573_comp__eq__elim,axiom,
! [A: nat > num,B: int > nat,C: nat > num,D2: int > nat] :
( ( ( comp_nat_num_int @ A @ B )
= ( comp_nat_num_int @ C @ D2 ) )
=> ! [V4: int] :
( ( A @ ( B @ V4 ) )
= ( C @ ( D2 @ V4 ) ) ) ) ).
% comp_eq_elim
thf(fact_574_comp__eq__elim,axiom,
! [A: nat > nat,B: nat > nat,C: nat > nat,D2: nat > nat] :
( ( ( comp_nat_nat_nat @ A @ B )
= ( comp_nat_nat_nat @ C @ D2 ) )
=> ! [V4: nat] :
( ( A @ ( B @ V4 ) )
= ( C @ ( D2 @ V4 ) ) ) ) ).
% comp_eq_elim
thf(fact_575_comp__eq__elim,axiom,
! [A: int > nat,B: int > int,C: int > nat,D2: int > int] :
( ( ( comp_int_nat_int @ A @ B )
= ( comp_int_nat_int @ C @ D2 ) )
=> ! [V4: int] :
( ( A @ ( B @ V4 ) )
= ( C @ ( D2 @ V4 ) ) ) ) ).
% comp_eq_elim
thf(fact_576_comp__eq__elim,axiom,
! [A: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,B: code_integer > code_integer > code_integer,C: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,D2: code_integer > code_integer > code_integer] :
( ( ( comp_C8797469213163452608nteger @ A @ B )
= ( comp_C8797469213163452608nteger @ C @ D2 ) )
=> ! [V4: code_integer] :
( ( A @ ( B @ V4 ) )
= ( C @ ( D2 @ V4 ) ) ) ) ).
% comp_eq_elim
thf(fact_577_comp__eq__dest,axiom,
! [A: $o > $o,B: nat > $o,C: $o > $o,D2: nat > $o,V: nat] :
( ( ( comp_o_o_nat @ A @ B )
= ( comp_o_o_nat @ C @ D2 ) )
=> ( ( A @ ( B @ V ) )
= ( C @ ( D2 @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_578_comp__eq__dest,axiom,
! [A: nat > num,B: int > nat,C: nat > num,D2: int > nat,V: int] :
( ( ( comp_nat_num_int @ A @ B )
= ( comp_nat_num_int @ C @ D2 ) )
=> ( ( A @ ( B @ V ) )
= ( C @ ( D2 @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_579_comp__eq__dest,axiom,
! [A: nat > nat,B: nat > nat,C: nat > nat,D2: nat > nat,V: nat] :
( ( ( comp_nat_nat_nat @ A @ B )
= ( comp_nat_nat_nat @ C @ D2 ) )
=> ( ( A @ ( B @ V ) )
= ( C @ ( D2 @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_580_comp__eq__dest,axiom,
! [A: int > nat,B: int > int,C: int > nat,D2: int > int,V: int] :
( ( ( comp_int_nat_int @ A @ B )
= ( comp_int_nat_int @ C @ D2 ) )
=> ( ( A @ ( B @ V ) )
= ( C @ ( D2 @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_581_comp__eq__dest,axiom,
! [A: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,B: code_integer > code_integer > code_integer,C: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,D2: code_integer > code_integer > code_integer,V: code_integer] :
( ( ( comp_C8797469213163452608nteger @ A @ B )
= ( comp_C8797469213163452608nteger @ C @ D2 ) )
=> ( ( A @ ( B @ V ) )
= ( C @ ( D2 @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_582_comp__assoc,axiom,
! [F: $o > $o,G: nat > $o,H: nat > nat] :
( ( comp_nat_o_nat @ ( comp_o_o_nat @ F @ G ) @ H )
= ( comp_o_o_nat @ F @ ( comp_nat_o_nat @ G @ H ) ) ) ).
% comp_assoc
thf(fact_583_comp__assoc,axiom,
! [F: nat > num,G: int > nat,H: int > int] :
( ( comp_int_num_int @ ( comp_nat_num_int @ F @ G ) @ H )
= ( comp_nat_num_int @ F @ ( comp_int_nat_int @ G @ H ) ) ) ).
% comp_assoc
thf(fact_584_comp__assoc,axiom,
! [F: $o > $o,G: $o > $o,H: nat > $o] :
( ( comp_o_o_nat @ ( comp_o_o_o @ F @ G ) @ H )
= ( comp_o_o_nat @ F @ ( comp_o_o_nat @ G @ H ) ) ) ).
% comp_assoc
thf(fact_585_comp__assoc,axiom,
! [F: num > num,G: nat > num,H: int > nat] :
( ( comp_nat_num_int @ ( comp_num_num_nat @ F @ G ) @ H )
= ( comp_num_num_int @ F @ ( comp_nat_num_int @ G @ H ) ) ) ).
% comp_assoc
thf(fact_586_comp__assoc,axiom,
! [F: nat > num,G: nat > nat,H: int > nat] :
( ( comp_nat_num_int @ ( comp_nat_num_nat @ F @ G ) @ H )
= ( comp_nat_num_int @ F @ ( comp_nat_nat_int @ G @ H ) ) ) ).
% comp_assoc
thf(fact_587_comp__assoc,axiom,
! [F: nat > nat,G: nat > nat,H: nat > nat] :
( ( comp_nat_nat_nat @ ( comp_nat_nat_nat @ F @ G ) @ H )
= ( comp_nat_nat_nat @ F @ ( comp_nat_nat_nat @ G @ H ) ) ) ).
% comp_assoc
thf(fact_588_comp__assoc,axiom,
! [F: nat > nat,G: int > nat,H: int > int] :
( ( comp_int_nat_int @ ( comp_nat_nat_int @ F @ G ) @ H )
= ( comp_nat_nat_int @ F @ ( comp_int_nat_int @ G @ H ) ) ) ).
% comp_assoc
thf(fact_589_comp__assoc,axiom,
! [F: int > nat,G: int > int,H: int > int] :
( ( comp_int_nat_int @ ( comp_int_nat_int @ F @ G ) @ H )
= ( comp_int_nat_int @ F @ ( comp_int_int_int @ G @ H ) ) ) ).
% comp_assoc
thf(fact_590_comp__assoc,axiom,
! [F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer > code_integer,H: code_integer > code_integer] :
( ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ F @ G ) @ H )
= ( comp_C8797469213163452608nteger @ F @ ( comp_C2931745410781319416nteger @ G @ H ) ) ) ).
% comp_assoc
thf(fact_591_comp__assoc,axiom,
! [F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G: ( code_integer > code_integer ) > code_integer > code_integer,H: code_integer > code_integer > code_integer] :
( ( comp_C8797469213163452608nteger @ ( comp_C3983183376822107068nteger @ F @ G ) @ H )
= ( comp_C8797469213163452608nteger @ F @ ( comp_C2822359487432382964nteger @ G @ H ) ) ) ).
% comp_assoc
thf(fact_592_comp__def,axiom,
( comp_o_o_nat
= ( ^ [F2: $o > $o,G3: nat > $o,X4: nat] : ( F2 @ ( G3 @ X4 ) ) ) ) ).
% comp_def
thf(fact_593_comp__def,axiom,
( comp_nat_num_int
= ( ^ [F2: nat > num,G3: int > nat,X4: int] : ( F2 @ ( G3 @ X4 ) ) ) ) ).
% comp_def
thf(fact_594_comp__def,axiom,
( comp_nat_nat_nat
= ( ^ [F2: nat > nat,G3: nat > nat,X4: nat] : ( F2 @ ( G3 @ X4 ) ) ) ) ).
% comp_def
thf(fact_595_comp__def,axiom,
( comp_int_nat_int
= ( ^ [F2: int > nat,G3: int > int,X4: int] : ( F2 @ ( G3 @ X4 ) ) ) ) ).
% comp_def
thf(fact_596_comp__def,axiom,
( comp_C8797469213163452608nteger
= ( ^ [F2: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G3: code_integer > code_integer > code_integer,X4: code_integer] : ( F2 @ ( G3 @ X4 ) ) ) ) ).
% comp_def
thf(fact_597_apfst__convE,axiom,
! [Q: produc1319942482725812455at_nat,F: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat,P: produc1319942482725812455at_nat] :
( ( Q
= ( produc6529295065807390923at_nat @ F @ P ) )
=> ~ ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
( ( P
= ( produc9060074326276436823at_nat @ X @ Y ) )
=> ( Q
!= ( produc9060074326276436823at_nat @ ( F @ X ) @ Y ) ) ) ) ).
% apfst_convE
thf(fact_598_apfst__convE,axiom,
! [Q: produc3843707927480180839at_nat,F: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat,P: produc3843707927480180839at_nat] :
( ( Q
= ( produc5380969780118646373at_nat @ F @ P ) )
=> ~ ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
( ( P
= ( produc2922128104949294807at_nat @ X @ Y ) )
=> ( Q
!= ( produc2922128104949294807at_nat @ ( F @ X ) @ Y ) ) ) ) ).
% apfst_convE
thf(fact_599_apfst__convE,axiom,
! [Q: product_prod_nat_num,F: nat > nat,P: product_prod_nat_num] :
( ( Q
= ( produc6767424583952363640at_num @ F @ P ) )
=> ~ ! [X: nat,Y: num] :
( ( P
= ( product_Pair_nat_num @ X @ Y ) )
=> ( Q
!= ( product_Pair_nat_num @ ( F @ X ) @ Y ) ) ) ) ).
% apfst_convE
thf(fact_600_apfst__convE,axiom,
! [Q: product_prod_nat_nat,F: nat > nat,P: product_prod_nat_nat] :
( ( Q
= ( produc986720760941809198at_nat @ F @ P ) )
=> ~ ! [X: nat,Y: nat] :
( ( P
= ( product_Pair_nat_nat @ X @ Y ) )
=> ( Q
!= ( product_Pair_nat_nat @ ( F @ X ) @ Y ) ) ) ) ).
% apfst_convE
thf(fact_601_apfst__convE,axiom,
! [Q: product_prod_int_int,F: int > int,P: product_prod_int_int] :
( ( Q
= ( produc2355237562570658114nt_int @ F @ P ) )
=> ~ ! [X: int,Y: int] :
( ( P
= ( product_Pair_int_int @ X @ Y ) )
=> ( Q
!= ( product_Pair_int_int @ ( F @ X ) @ Y ) ) ) ) ).
% apfst_convE
thf(fact_602_prod_Omap__comp,axiom,
! [G1: $o > $o,G22: $o > $o,F1: nat > $o,F22: nat > $o,V: product_prod_nat_nat] :
( ( produc1928436952130685430_o_o_o @ G1 @ G22 @ ( produc6957950696844810486_nat_o @ F1 @ F22 @ V ) )
= ( produc6957950696844810486_nat_o @ ( comp_o_o_nat @ G1 @ F1 ) @ ( comp_o_o_nat @ G22 @ F22 ) @ V ) ) ).
% prod.map_comp
thf(fact_603_prod_Omap__comp,axiom,
! [G1: $o > $o,G22: nat > num,F1: nat > $o,F22: int > nat,V: product_prod_nat_int] :
( ( produc3628557804203378146at_num @ G1 @ G22 @ ( produc8250590009387450446nt_nat @ F1 @ F22 @ V ) )
= ( produc4807921795543229080nt_num @ ( comp_o_o_nat @ G1 @ F1 ) @ ( comp_nat_num_int @ G22 @ F22 ) @ V ) ) ).
% prod.map_comp
thf(fact_604_prod_Omap__comp,axiom,
! [G1: $o > $o,G22: nat > nat,F1: nat > $o,F22: nat > nat,V: product_prod_nat_nat] :
( ( produc7071226018047599512at_nat @ G1 @ G22 @ ( produc7251236297486395890at_nat @ F1 @ F22 @ V ) )
= ( produc7251236297486395890at_nat @ ( comp_o_o_nat @ G1 @ F1 ) @ ( comp_nat_nat_nat @ G22 @ F22 ) @ V ) ) ).
% prod.map_comp
thf(fact_605_prod_Omap__comp,axiom,
! [G1: $o > $o,G22: int > nat,F1: nat > $o,F22: int > int,V: product_prod_nat_int] :
( ( produc8070579729948654068nt_nat @ G1 @ G22 @ ( produc8248099538878400170nt_int @ F1 @ F22 @ V ) )
= ( produc8250590009387450446nt_nat @ ( comp_o_o_nat @ G1 @ F1 ) @ ( comp_int_nat_int @ G22 @ F22 ) @ V ) ) ).
% prod.map_comp
thf(fact_606_prod_Omap__comp,axiom,
! [G1: nat > num,G22: $o > $o,F1: int > nat,F22: nat > $o,V: product_prod_int_nat] :
( ( produc3264868786340801506um_o_o @ G1 @ G22 @ ( produc8298419091753553490_nat_o @ F1 @ F22 @ V ) )
= ( produc9074608839065728392_nat_o @ ( comp_nat_num_int @ G1 @ F1 ) @ ( comp_o_o_nat @ G22 @ F22 ) @ V ) ) ).
% prod.map_comp
thf(fact_607_prod_Omap__comp,axiom,
! [G1: nat > num,G22: nat > num,F1: int > nat,F22: int > nat,V: product_prod_int_int] :
( ( produc1476026309457068494at_num @ G1 @ G22 @ ( produc5667422475138734578nt_nat @ F1 @ F22 @ V ) )
= ( produc165562089265172102nt_num @ ( comp_nat_num_int @ G1 @ F1 ) @ ( comp_nat_num_int @ G22 @ F22 ) @ V ) ) ).
% prod.map_comp
thf(fact_608_prod_Omap__comp,axiom,
! [G1: nat > num,G22: nat > nat,F1: int > nat,F22: nat > nat,V: product_prod_int_nat] :
( ( produc4918694523301289860at_nat @ G1 @ G22 @ ( produc4668068763237680022at_nat @ F1 @ F22 @ V ) )
= ( produc2608876591208338912at_nat @ ( comp_nat_num_int @ G1 @ F1 ) @ ( comp_nat_nat_nat @ G22 @ F22 ) @ V ) ) ).
% prod.map_comp
thf(fact_609_prod_Omap__comp,axiom,
! [G1: nat > num,G22: int > nat,F1: int > nat,F22: int > int,V: product_prod_int_int] :
( ( produc5918048235202344416nt_nat @ G1 @ G22 @ ( produc5664932004629684302nt_int @ F1 @ F22 @ V ) )
= ( produc3608230303109393468nt_nat @ ( comp_nat_num_int @ G1 @ F1 ) @ ( comp_int_nat_int @ G22 @ F22 ) @ V ) ) ).
% prod.map_comp
thf(fact_610_prod_Omap__comp,axiom,
! [G1: nat > nat,G22: $o > $o,F1: nat > nat,F22: nat > $o,V: product_prod_nat_nat] :
( ( produc2001155298098703512at_o_o @ G1 @ G22 @ ( produc4787215299218314158_nat_o @ F1 @ F22 @ V ) )
= ( produc4787215299218314158_nat_o @ ( comp_nat_nat_nat @ G1 @ F1 ) @ ( comp_o_o_nat @ G22 @ F22 ) @ V ) ) ).
% prod.map_comp
thf(fact_611_prod_Omap__comp,axiom,
! [G1: nat > nat,G22: nat > num,F1: nat > nat,F22: int > nat,V: product_prod_nat_int] :
( ( produc3535218481486409604at_num @ G1 @ G22 @ ( produc7977240407231685526nt_nat @ F1 @ F22 @ V ) )
= ( produc4534572193387464160nt_num @ ( comp_nat_nat_nat @ G1 @ F1 ) @ ( comp_nat_num_int @ G22 @ F22 ) @ V ) ) ).
% prod.map_comp
thf(fact_612_subrelI,axiom,
! [R: set_Pr7459493094073627847at_nat,S2: set_Pr7459493094073627847at_nat] :
( ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ R )
=> ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ S2 ) )
=> ( ord_le5997549366648089703at_nat @ R @ S2 ) ) ).
% subrelI
thf(fact_613_subrelI,axiom,
! [R: set_Pr4329608150637261639at_nat,S2: set_Pr4329608150637261639at_nat] :
( ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R )
=> ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ S2 ) )
=> ( ord_le1268244103169919719at_nat @ R @ S2 ) ) ).
% subrelI
thf(fact_614_subrelI,axiom,
! [R: set_Pr6200539531224447659at_num,S2: set_Pr6200539531224447659at_num] :
( ! [X: nat,Y: num] :
( ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X @ Y ) @ R )
=> ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X @ Y ) @ S2 ) )
=> ( ord_le8085105155179020875at_num @ R @ S2 ) ) ).
% subrelI
thf(fact_615_subrelI,axiom,
! [R: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
( ! [X: nat,Y: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ S2 ) )
=> ( ord_le3146513528884898305at_nat @ R @ S2 ) ) ).
% subrelI
thf(fact_616_subrelI,axiom,
! [R: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
( ! [X: int,Y: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ S2 ) )
=> ( ord_le2843351958646193337nt_int @ R @ S2 ) ) ).
% subrelI
thf(fact_617_all__subset__image,axiom,
! [F: nat > rat,A4: set_nat,P2: set_rat > $o] :
( ( ! [B6: set_rat] :
( ( ord_less_eq_set_rat @ B6 @ ( image_nat_rat @ F @ A4 ) )
=> ( P2 @ B6 ) ) )
= ( ! [B6: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ A4 )
=> ( P2 @ ( image_nat_rat @ F @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_618_all__subset__image,axiom,
! [F: nat > nat,A4: set_nat,P2: set_nat > $o] :
( ( ! [B6: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ ( image_nat_nat @ F @ A4 ) )
=> ( P2 @ B6 ) ) )
= ( ! [B6: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ A4 )
=> ( P2 @ ( image_nat_nat @ F @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_619_all__subset__image,axiom,
! [F: nat > int,A4: set_nat,P2: set_int > $o] :
( ( ! [B6: set_int] :
( ( ord_less_eq_set_int @ B6 @ ( image_nat_int @ F @ A4 ) )
=> ( P2 @ B6 ) ) )
= ( ! [B6: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ A4 )
=> ( P2 @ ( image_nat_int @ F @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_620_all__subset__image,axiom,
! [F: int > nat,A4: set_int,P2: set_nat > $o] :
( ( ! [B6: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ ( image_int_nat @ F @ A4 ) )
=> ( P2 @ B6 ) ) )
= ( ! [B6: set_int] :
( ( ord_less_eq_set_int @ B6 @ A4 )
=> ( P2 @ ( image_int_nat @ F @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_621_all__subset__image,axiom,
! [F: int > int,A4: set_int,P2: set_int > $o] :
( ( ! [B6: set_int] :
( ( ord_less_eq_set_int @ B6 @ ( image_int_int @ F @ A4 ) )
=> ( P2 @ B6 ) ) )
= ( ! [B6: set_int] :
( ( ord_less_eq_set_int @ B6 @ A4 )
=> ( P2 @ ( image_int_int @ F @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_622_rel__fun__Collect__case__prodD,axiom,
! [A4: product_prod_nat_nat > int > $o,B5: nat > nat > $o,F: product_prod_nat_nat > nat,G: int > nat,X5: set_Pr49084382684228242at_int,X2: produc4196048018000912732at_int] :
( ( bNF_re4555766996558763186at_nat @ A4 @ B5 @ F @ G )
=> ( ( ord_le4604737605482584818at_int @ X5 @ ( collec2851339944984316337at_int @ ( produc6937227050501860053_int_o @ A4 ) ) )
=> ( ( member8394280151737582451at_int @ X2 @ X5 )
=> ( B5 @ ( comp_P8635049891247748876at_int @ F @ produc4619704898428564714at_int @ X2 ) @ ( comp_i5627883460353289149at_int @ G @ produc3064508320661873192at_int @ X2 ) ) ) ) ) ).
% rel_fun_Collect_case_prodD
thf(fact_623_rel__fun__Collect__case__prodD,axiom,
! [A4: product_prod_nat_nat > int > $o,B5: ( product_prod_nat_nat > product_prod_nat_nat ) > ( int > int ) > $o,F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,G: int > int > int,X5: set_Pr49084382684228242at_int,X2: produc4196048018000912732at_int] :
( ( bNF_re7408651293131936558nt_int @ A4 @ B5 @ F @ G )
=> ( ( ord_le4604737605482584818at_int @ X5 @ ( collec2851339944984316337at_int @ ( produc6937227050501860053_int_o @ A4 ) ) )
=> ( ( member8394280151737582451at_int @ X2 @ X5 )
=> ( B5 @ ( comp_P4524073163769703009at_int @ F @ produc4619704898428564714at_int @ X2 ) @ ( comp_i4519014998770822500at_int @ G @ produc3064508320661873192at_int @ X2 ) ) ) ) ) ).
% rel_fun_Collect_case_prodD
thf(fact_624_rel__fun__Collect__case__prodD,axiom,
! [A4: product_prod_nat_nat > int > $o,B5: ( product_prod_nat_nat > $o ) > ( int > $o ) > $o,F: product_prod_nat_nat > product_prod_nat_nat > $o,G: int > int > $o,X5: set_Pr49084382684228242at_int,X2: produc4196048018000912732at_int] :
( ( bNF_re717283939379294677_int_o @ A4 @ B5 @ F @ G )
=> ( ( ord_le4604737605482584818at_int @ X5 @ ( collec2851339944984316337at_int @ ( produc6937227050501860053_int_o @ A4 ) ) )
=> ( ( member8394280151737582451at_int @ X2 @ X5 )
=> ( B5 @ ( comp_P7900071416159368952at_int @ F @ produc4619704898428564714at_int @ X2 ) @ ( comp_i653165354545908656at_int @ G @ produc3064508320661873192at_int @ X2 ) ) ) ) ) ).
% rel_fun_Collect_case_prodD
thf(fact_625_rel__fun__Collect__case__prodD,axiom,
! [A4: int > code_integer > $o,B5: num > num > $o,F: int > num,G: code_integer > num,X5: set_Pr8415741480598773322nteger,X2: produc2963269191615333738nteger] :
( ( bNF_re6718328864250387230um_num @ A4 @ B5 @ F @ G )
=> ( ( ord_le2638047946463454698nteger @ X5 @ ( collec7215431352809750357nteger @ ( produc540600059291021473eger_o @ A4 ) ) )
=> ( ( member4345257455949424659nteger @ X2 @ X5 )
=> ( B5 @ ( comp_i3846818780818535807nteger @ F @ produc45500017171379470nteger @ X2 ) @ ( comp_C8580465762518603278nteger @ G @ produc2153544567185597776nteger @ X2 ) ) ) ) ) ).
% rel_fun_Collect_case_prodD
thf(fact_626_rel__fun__Collect__case__prodD,axiom,
! [A4: int > int > $o,B5: num > num > $o,F: int > num,G: int > num,X5: set_Pr958786334691620121nt_int,X2: product_prod_int_int] :
( ( bNF_re7626690874201225453um_num @ A4 @ B5 @ F @ G )
=> ( ( ord_le2843351958646193337nt_int @ X5 @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ A4 ) ) )
=> ( ( member5262025264175285858nt_int @ X2 @ X5 )
=> ( B5 @ ( comp_i7231038107349025742nt_int @ F @ product_fst_int_int @ X2 ) @ ( comp_i7231038107349025742nt_int @ G @ product_snd_int_int @ X2 ) ) ) ) ) ).
% rel_fun_Collect_case_prodD
thf(fact_627_convol__image__vimage2p,axiom,
! [F: nat > nat,G: nat > nat,R3: nat > nat > $o] : ( ord_le3146513528884898305at_nat @ ( image_5168914502847457605at_nat @ ( bNF_co8276296344278409722at_nat @ ( comp_n2711240574518091068at_nat @ F @ product_fst_nat_nat ) @ ( comp_n2711240574518091068at_nat @ G @ product_snd_nat_nat ) ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ( bNF_vi6667456707345531737_nat_o @ F @ G @ R3 ) ) ) ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R3 ) ) ) ).
% convol_image_vimage2p
thf(fact_628_convol__image__vimage2p,axiom,
! [F: ref_a > nat,G: heap_e7401611519738050253t_unit > nat,R3: nat > nat > $o] : ( ord_le3146513528884898305at_nat @ ( image_4710180989675653397at_nat @ ( bNF_co5610825919353547504at_nat @ ( comp_r8155216869944865719t_unit @ F @ produc473321116828791851t_unit ) @ ( comp_h8354993580257045729t_unit @ G @ produc1126150082675451753t_unit ) ) @ ( collec3252306575900995386t_unit @ ( produc4924168586796184916unit_o @ ( bNF_vi4623496019404053219_nat_o @ F @ G @ R3 ) ) ) ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R3 ) ) ) ).
% convol_image_vimage2p
thf(fact_629_convol__image__vimage2p,axiom,
! [F: nat > int,G: nat > int,R3: int > int > $o] : ( ord_le2843351958646193337nt_int @ ( image_1990417195239315453nt_int @ ( bNF_co49787548815638194nt_int @ ( comp_n5749082658982224864at_nat @ F @ product_fst_nat_nat ) @ ( comp_n5749082658982224864at_nat @ G @ product_snd_nat_nat ) ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ( bNF_vi8357618154448083105_int_o @ F @ G @ R3 ) ) ) ) @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ R3 ) ) ) ).
% convol_image_vimage2p
thf(fact_630_convol__image__vimage2p,axiom,
! [F: ref_a > int,G: heap_e7401611519738050253t_unit > int,R3: int > int > $o] : ( ord_le2843351958646193337nt_int @ ( image_1531683682067511245nt_int @ ( bNF_co6607689160745551784nt_int @ ( comp_r7249815618783853587t_unit @ F @ produc473321116828791851t_unit ) @ ( comp_h7449592329096033597t_unit @ G @ produc1126150082675451753t_unit ) ) @ ( collec3252306575900995386t_unit @ ( produc4924168586796184916unit_o @ ( bNF_vi7828673959370436579_int_o @ F @ G @ R3 ) ) ) ) @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ R3 ) ) ) ).
% convol_image_vimage2p
thf(fact_631_convol__image__vimage2p,axiom,
! [F: int > nat,G: int > nat,R3: nat > nat > $o] : ( ord_le3146513528884898305at_nat @ ( image_5831868185956570253at_nat @ ( bNF_co2751916026326257842at_nat @ ( comp_i1292779930663060120nt_int @ F @ product_fst_int_int ) @ ( comp_i1292779930663060120nt_int @ G @ product_snd_int_int ) ) @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ ( bNF_vi2193923016096921761_nat_o @ F @ G @ R3 ) ) ) ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R3 ) ) ) ).
% convol_image_vimage2p
thf(fact_632_convol__image__vimage2p,axiom,
! [F: int > int,G: int > int,R3: int > int > $o] : ( ord_le2843351958646193337nt_int @ ( image_2653370878348428101nt_int @ ( bNF_co3748779267718262122nt_int @ ( comp_i4330622015127193916nt_int @ F @ product_fst_int_int ) @ ( comp_i4330622015127193916nt_int @ G @ product_snd_int_int ) ) @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ ( bNF_vi3884084463199473129_int_o @ F @ G @ R3 ) ) ) ) @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ R3 ) ) ) ).
% convol_image_vimage2p
thf(fact_633_prod_Osize__gen__o__map,axiom,
! [F: nat > nat,Fa: nat > nat,G: nat > nat,Ga: nat > nat] :
( ( comp_P1567445206330693457at_nat @ ( basic_876126793109182934at_nat @ F @ Fa ) @ ( produc6977886695330630970at_nat @ G @ Ga ) )
= ( basic_876126793109182934at_nat @ ( comp_nat_nat_nat @ F @ G ) @ ( comp_nat_nat_nat @ Fa @ Ga ) ) ) ).
% prod.size_gen_o_map
thf(fact_634_prod_Osize__gen__o__map,axiom,
! [F: nat > nat,Fa: int > nat,G: nat > nat,Ga: int > int] :
( ( comp_P397146161216181329at_int @ ( basic_873636322600132658at_int @ F @ Fa ) @ ( produc7974749936722635250nt_int @ G @ Ga ) )
= ( basic_873636322600132658at_int @ ( comp_nat_nat_nat @ F @ G ) @ ( comp_int_nat_int @ Fa @ Ga ) ) ) ).
% prod.size_gen_o_map
thf(fact_635_prod_Osize__gen__o__map,axiom,
! [F: int > nat,Fa: nat > nat,G: int > int,Ga: nat > nat] :
( ( comp_P3488689526429045841nt_nat @ ( basic_1875480505010237490nt_nat @ F @ Fa ) @ ( produc5039722323474524658at_nat @ G @ Ga ) )
= ( basic_1875480505010237490nt_nat @ ( comp_int_nat_int @ F @ G ) @ ( comp_nat_nat_nat @ Fa @ Ga ) ) ) ).
% prod.size_gen_o_map
thf(fact_636_prod_Osize__gen__o__map,axiom,
! [F: int > nat,Fa: int > nat,G: int > int,Ga: int > int] :
( ( comp_P2318390481314533713nt_int @ ( basic_1872990034501187214nt_int @ F @ Fa ) @ ( produc6036585564866528938nt_int @ G @ Ga ) )
= ( basic_1872990034501187214nt_int @ ( comp_int_nat_int @ F @ G ) @ ( comp_int_nat_int @ Fa @ Ga ) ) ) ).
% prod.size_gen_o_map
thf(fact_637_relChain__def,axiom,
( bNF_Ca968750328013420230at_nat
= ( ^ [R5: set_Pr1261947904930325089at_nat,As: nat > nat] :
! [I2: nat,J: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I2 @ J ) @ R5 )
=> ( ord_less_eq_nat @ ( As @ I2 ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_638_relChain__def,axiom,
( bNF_Ca1968104039914474786nt_nat
= ( ^ [R5: set_Pr958786334691620121nt_int,As: int > nat] :
! [I2: int,J: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ I2 @ J ) @ R5 )
=> ( ord_less_eq_nat @ ( As @ I2 ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_639_relChain__def,axiom,
( bNF_Ca966259857504369954at_int
= ( ^ [R5: set_Pr1261947904930325089at_nat,As: nat > int] :
! [I2: nat,J: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I2 @ J ) @ R5 )
=> ( ord_less_eq_int @ ( As @ I2 ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_640_relChain__def,axiom,
( bNF_Ca1965613569405424510nt_int
= ( ^ [R5: set_Pr958786334691620121nt_int,As: int > int] :
! [I2: int,J: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ I2 @ J ) @ R5 )
=> ( ord_less_eq_int @ ( As @ I2 ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_641_relChain__def,axiom,
( bNF_Ca333620267926924494at_rat
= ( ^ [R5: set_Pr1261947904930325089at_nat,As: nat > rat] :
! [I2: nat,J: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I2 @ J ) @ R5 )
=> ( ord_less_eq_rat @ ( As @ I2 ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_642_relChain__def,axiom,
( bNF_Ca1332973979827979050nt_rat
= ( ^ [R5: set_Pr958786334691620121nt_int,As: int > rat] :
! [I2: int,J: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ I2 @ J ) @ R5 )
=> ( ord_less_eq_rat @ ( As @ I2 ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_643_relChain__def,axiom,
( bNF_Ca30959104405961555nteger
= ( ^ [R5: set_Pr1261947904930325089at_nat,As: nat > code_integer] :
! [I2: nat,J: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I2 @ J ) @ R5 )
=> ( ord_le3102999989581377725nteger @ ( As @ I2 ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_644_relChain__def,axiom,
( bNF_Ca402612664642806191nteger
= ( ^ [R5: set_Pr958786334691620121nt_int,As: int > code_integer] :
! [I2: int,J: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ I2 @ J ) @ R5 )
=> ( ord_le3102999989581377725nteger @ ( As @ I2 ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_645_relChain__def,axiom,
( bNF_Ca4949091444211332621at_nat
= ( ^ [R5: set_Pr4329608150637261639at_nat,As: set_Pr1261947904930325089at_nat > nat] :
! [I2: set_Pr1261947904930325089at_nat,J: set_Pr1261947904930325089at_nat] :
( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ I2 @ J ) @ R5 )
=> ( ord_less_eq_nat @ ( As @ I2 ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_646_relChain__def,axiom,
( bNF_Ca4946600973702282345at_int
= ( ^ [R5: set_Pr4329608150637261639at_nat,As: set_Pr1261947904930325089at_nat > int] :
! [I2: set_Pr1261947904930325089at_nat,J: set_Pr1261947904930325089at_nat] :
( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ I2 @ J ) @ R5 )
=> ( ord_less_eq_int @ ( As @ I2 ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_647_Collect__case__prod__in__rel__leI,axiom,
! [X5: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ X5 @ Y4 )
=> ( ord_le3146513528884898305at_nat @ X5 @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ( fun_in_rel_nat_nat @ Y4 ) ) ) ) ) ).
% Collect_case_prod_in_rel_leI
thf(fact_648_Collect__case__prod__in__rel__leI,axiom,
! [X5: set_Pr958786334691620121nt_int,Y4: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ X5 @ Y4 )
=> ( ord_le2843351958646193337nt_int @ X5 @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ ( fun_in_rel_int_int @ Y4 ) ) ) ) ) ).
% Collect_case_prod_in_rel_leI
thf(fact_649_Collect__case__prod__in__rel__leE,axiom,
! [X5: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ X5 @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ( fun_in_rel_nat_nat @ Y4 ) ) ) )
=> ( ord_le3146513528884898305at_nat @ X5 @ Y4 ) ) ).
% Collect_case_prod_in_rel_leE
thf(fact_650_Collect__case__prod__in__rel__leE,axiom,
! [X5: set_Pr958786334691620121nt_int,Y4: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ X5 @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ ( fun_in_rel_int_int @ Y4 ) ) ) )
=> ( ord_le2843351958646193337nt_int @ X5 @ Y4 ) ) ).
% Collect_case_prod_in_rel_leE
thf(fact_651_relcompp__assoc,axiom,
! [R: product_prod_nat_nat > product_prod_nat_nat > $o,S2: product_prod_nat_nat > int > $o,T2: int > int > $o] :
( ( relcom2949942408940504938nt_int @ ( relcom8911704206367525917at_int @ R @ S2 ) @ T2 )
= ( relcom8911704206367525917at_int @ R @ ( relcom2949942408940504938nt_int @ S2 @ T2 ) ) ) ).
% relcompp_assoc
thf(fact_652_relcompp__assoc,axiom,
! [R: int > int > $o,S2: int > code_integer > $o,T2: code_integer > code_integer > $o] :
( ( relcom8320293257703906841nteger @ ( relcom967497705283160040nteger @ R @ S2 ) @ T2 )
= ( relcom967497705283160040nteger @ R @ ( relcom8320293257703906841nteger @ S2 @ T2 ) ) ) ).
% relcompp_assoc
thf(fact_653_relcompp__assoc,axiom,
! [R: product_prod_nat_nat > product_prod_nat_nat > $o,S2: product_prod_nat_nat > product_prod_nat_nat > $o,T2: product_prod_nat_nat > int > $o] :
( ( relcom8911704206367525917at_int @ ( relcom282571730187242960at_nat @ R @ S2 ) @ T2 )
= ( relcom8911704206367525917at_int @ R @ ( relcom8911704206367525917at_int @ S2 @ T2 ) ) ) ).
% relcompp_assoc
thf(fact_654_relcompp__assoc,axiom,
! [R: int > int > $o,S2: int > int > $o,T2: int > code_integer > $o] :
( ( relcom967497705283160040nteger @ ( relcompp_int_int_int @ R @ S2 ) @ T2 )
= ( relcom967497705283160040nteger @ R @ ( relcom967497705283160040nteger @ S2 @ T2 ) ) ) ).
% relcompp_assoc
thf(fact_655_relcompp__apply,axiom,
( relcom8911704206367525917at_int
= ( ^ [R6: product_prod_nat_nat > product_prod_nat_nat > $o,S4: product_prod_nat_nat > int > $o,A5: product_prod_nat_nat,C4: int] :
? [B4: product_prod_nat_nat] :
( ( R6 @ A5 @ B4 )
& ( S4 @ B4 @ C4 ) ) ) ) ).
% relcompp_apply
thf(fact_656_relcompp__apply,axiom,
( relcom967497705283160040nteger
= ( ^ [R6: int > int > $o,S4: int > code_integer > $o,A5: int,C4: code_integer] :
? [B4: int] :
( ( R6 @ A5 @ B4 )
& ( S4 @ B4 @ C4 ) ) ) ) ).
% relcompp_apply
thf(fact_657_rel__funI,axiom,
! [A4: product_prod_nat_nat > int > $o,B5: nat > nat > $o,F: product_prod_nat_nat > nat,G: int > nat] :
( ! [X: product_prod_nat_nat,Y: int] :
( ( A4 @ X @ Y )
=> ( B5 @ ( F @ X ) @ ( G @ Y ) ) )
=> ( bNF_re4555766996558763186at_nat @ A4 @ B5 @ F @ G ) ) ).
% rel_funI
thf(fact_658_rel__funI,axiom,
! [A4: product_prod_nat_nat > int > $o,B5: ( product_prod_nat_nat > product_prod_nat_nat ) > ( int > int ) > $o,F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,G: int > int > int] :
( ! [X: product_prod_nat_nat,Y: int] :
( ( A4 @ X @ Y )
=> ( B5 @ ( F @ X ) @ ( G @ Y ) ) )
=> ( bNF_re7408651293131936558nt_int @ A4 @ B5 @ F @ G ) ) ).
% rel_funI
thf(fact_659_rel__funI,axiom,
! [A4: product_prod_nat_nat > int > $o,B5: ( product_prod_nat_nat > $o ) > ( int > $o ) > $o,F: product_prod_nat_nat > product_prod_nat_nat > $o,G: int > int > $o] :
( ! [X: product_prod_nat_nat,Y: int] :
( ( A4 @ X @ Y )
=> ( B5 @ ( F @ X ) @ ( G @ Y ) ) )
=> ( bNF_re717283939379294677_int_o @ A4 @ B5 @ F @ G ) ) ).
% rel_funI
thf(fact_660_rel__funI,axiom,
! [A4: int > code_integer > $o,B5: num > num > $o,F: int > num,G: code_integer > num] :
( ! [X: int,Y: code_integer] :
( ( A4 @ X @ Y )
=> ( B5 @ ( F @ X ) @ ( G @ Y ) ) )
=> ( bNF_re6718328864250387230um_num @ A4 @ B5 @ F @ G ) ) ).
% rel_funI
thf(fact_661_rel__funI,axiom,
! [A4: int > int > $o,B5: num > num > $o,F: int > num,G: int > num] :
( ! [X: int,Y: int] :
( ( A4 @ X @ Y )
=> ( B5 @ ( F @ X ) @ ( G @ Y ) ) )
=> ( bNF_re7626690874201225453um_num @ A4 @ B5 @ F @ G ) ) ).
% rel_funI
thf(fact_662_fun_Omap__transfer,axiom,
! [Rb: $o > $o > $o,Sd: $o > $o > $o] :
( bNF_re6580757339198446453_nat_o @ ( bNF_rel_fun_o_o_o_o @ Rb @ Sd )
@ ( bNF_re2697311982885662685_nat_o
@ ( bNF_re4705727531993890431at_o_o
@ ^ [Y2: nat,Z: nat] : Y2 = Z
@ Rb )
@ ( bNF_re4705727531993890431at_o_o
@ ^ [Y2: nat,Z: nat] : Y2 = Z
@ Sd ) )
@ comp_o_o_nat
@ comp_o_o_nat ) ).
% fun.map_transfer
thf(fact_663_fun_Omap__transfer,axiom,
! [Rb: $o > nat > $o,Sd: $o > nat > $o] :
( bNF_re1515921786380824775at_nat @ ( bNF_re4209105892566519389_o_nat @ Rb @ Sd )
@ ( bNF_re2144779315220482269at_nat
@ ( bNF_re7523573913982466217_o_nat
@ ^ [Y2: nat,Z: nat] : Y2 = Z
@ Rb )
@ ( bNF_re7523573913982466217_o_nat
@ ^ [Y2: nat,Z: nat] : Y2 = Z
@ Sd ) )
@ comp_o_o_nat
@ comp_nat_nat_nat ) ).
% fun.map_transfer
thf(fact_664_fun_Omap__transfer,axiom,
! [Rb: nat > int > $o,Sd: num > nat > $o] :
( bNF_re5843449549409830101nt_nat @ ( bNF_re4155804983283621959um_nat @ Rb @ Sd )
@ ( bNF_re3642433447925381491nt_nat
@ ( bNF_re3713166177374151349at_int
@ ^ [Y2: int,Z: int] : Y2 = Z
@ Rb )
@ ( bNF_re1845987051190671011um_nat
@ ^ [Y2: int,Z: int] : Y2 = Z
@ Sd ) )
@ comp_nat_num_int
@ comp_int_nat_int ) ).
% fun.map_transfer
thf(fact_665_fun_Omap__transfer,axiom,
! [Rb: nat > $o > $o,Sd: nat > $o > $o] :
( bNF_re3930744952143938759_nat_o @ ( bNF_re439150893885221597_nat_o @ Rb @ Sd )
@ ( bNF_re7744223787735169245_nat_o
@ ( bNF_re1402873238235875719_nat_o
@ ^ [Y2: nat,Z: nat] : Y2 = Z
@ Rb )
@ ( bNF_re1402873238235875719_nat_o
@ ^ [Y2: nat,Z: nat] : Y2 = Z
@ Sd ) )
@ comp_nat_nat_nat
@ comp_o_o_nat ) ).
% fun.map_transfer
thf(fact_666_fun_Omap__transfer,axiom,
! [Rb: nat > nat > $o,Sd: nat > nat > $o] :
( bNF_re3262823321055862553at_nat @ ( bNF_re5653821019739307937at_nat @ Rb @ Sd )
@ ( bNF_re239970166668089693at_nat
@ ( bNF_re5653821019739307937at_nat
@ ^ [Y2: nat,Z: nat] : Y2 = Z
@ Rb )
@ ( bNF_re5653821019739307937at_nat
@ ^ [Y2: nat,Z: nat] : Y2 = Z
@ Sd ) )
@ comp_nat_nat_nat
@ comp_nat_nat_nat ) ).
% fun.map_transfer
thf(fact_667_fun_Omap__transfer,axiom,
! [Rb: int > nat > $o,Sd: nat > num > $o] :
( bNF_re4310571189260822741nt_num @ ( bNF_re9124706910656911431at_num @ Rb @ Sd )
@ ( bNF_re7492190849051111027nt_num
@ ( bNF_re4715010359784256181nt_nat
@ ^ [Y2: int,Z: int] : Y2 = Z
@ Rb )
@ ( bNF_re272988434038980259at_num
@ ^ [Y2: int,Z: int] : Y2 = Z
@ Sd ) )
@ comp_int_nat_int
@ comp_nat_num_int ) ).
% fun.map_transfer
thf(fact_668_fun_Omap__transfer,axiom,
! [Rb: int > int > $o,Sd: nat > nat > $o] :
( bNF_re6307232852808291833nt_nat @ ( bNF_re3715656647883201625at_nat @ Rb @ Sd )
@ ( bNF_re2034674868506727813nt_nat
@ ( bNF_re4712519889275205905nt_int
@ ^ [Y2: int,Z: int] : Y2 = Z
@ Rb )
@ ( bNF_re3715656647883201625at_nat
@ ^ [Y2: int,Z: int] : Y2 = Z
@ Sd ) )
@ comp_int_nat_int
@ comp_int_nat_int ) ).
% fun.map_transfer
thf(fact_669_fun_Omap__transfer,axiom,
! [Rb: nat > nat > $o,Sd: num > num > $o] :
( bNF_re3846787885862361009nt_num @ ( bNF_re341483209202555957um_num @ Rb @ Sd )
@ ( bNF_re9099949428469764705nt_num
@ ( bNF_re3715656647883201625at_nat
@ ^ [Y2: int,Z: int] : Y2 = Z
@ Rb )
@ ( bNF_re7626690874201225453um_num
@ ^ [Y2: int,Z: int] : Y2 = Z
@ Sd ) )
@ comp_nat_num_int
@ comp_nat_num_int ) ).
% fun.map_transfer
thf(fact_670_fun_Omap__transfer,axiom,
! [Rb: num > num > $o,Sd: num > num > $o] :
( bNF_re5446066262214944985nt_num @ ( bNF_re4736500879113541833um_num @ Rb @ Sd )
@ ( bNF_re4768042542643474933nt_num
@ ( bNF_re7626690874201225453um_num
@ ^ [Y2: int,Z: int] : Y2 = Z
@ Rb )
@ ( bNF_re7626690874201225453um_num
@ ^ [Y2: int,Z: int] : Y2 = Z
@ Sd ) )
@ comp_num_num_int
@ comp_num_num_int ) ).
% fun.map_transfer
thf(fact_671_fun_Omap__transfer,axiom,
! [Rb: int > code_integer > $o,Sd: num > num > $o] :
( bNF_re7839584876883444355nt_num @ ( bNF_re6718328864250387230um_num @ Rb @ Sd )
@ ( bNF_re2961086658867094474nt_num
@ ( bNF_re982302072995117890nteger
@ ^ [Y2: int,Z: int] : Y2 = Z
@ Rb )
@ ( bNF_re7626690874201225453um_num
@ ^ [Y2: int,Z: int] : Y2 = Z
@ Sd ) )
@ comp_int_num_int
@ comp_C2177396528461469320um_int ) ).
% fun.map_transfer
thf(fact_672_rel__fun__iff__leq__vimage2p,axiom,
( bNF_re4555766996558763186at_nat
= ( ^ [R6: product_prod_nat_nat > int > $o,S4: nat > nat > $o,F2: product_prod_nat_nat > nat,G3: int > nat] : ( ord_le129227817499904167_int_o @ R6 @ ( bNF_vi5493721401438417730_nat_o @ F2 @ G3 @ S4 ) ) ) ) ).
% rel_fun_iff_leq_vimage2p
thf(fact_673_rel__fun__iff__leq__vimage2p,axiom,
( bNF_re7408651293131936558nt_int
= ( ^ [R6: product_prod_nat_nat > int > $o,S4: ( product_prod_nat_nat > product_prod_nat_nat ) > ( int > int ) > $o,F2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,G3: int > int > int] : ( ord_le129227817499904167_int_o @ R6 @ ( bNF_vi8966365312794588230_int_o @ F2 @ G3 @ S4 ) ) ) ) ).
% rel_fun_iff_leq_vimage2p
thf(fact_674_rel__fun__iff__leq__vimage2p,axiom,
( bNF_re717283939379294677_int_o
= ( ^ [R6: product_prod_nat_nat > int > $o,S4: ( product_prod_nat_nat > $o ) > ( int > $o ) > $o,F2: product_prod_nat_nat > product_prod_nat_nat > $o,G3: int > int > $o] : ( ord_le129227817499904167_int_o @ R6 @ ( bNF_vi5082502771524357669nt_o_o @ F2 @ G3 @ S4 ) ) ) ) ).
% rel_fun_iff_leq_vimage2p
thf(fact_675_rel__fun__iff__leq__vimage2p,axiom,
( bNF_re6718328864250387230um_num
= ( ^ [R6: int > code_integer > $o,S4: num > num > $o,F2: int > num,G3: code_integer > num] : ( ord_le996066559646421213eger_o @ R6 @ ( bNF_vi2326032003723069084_num_o @ F2 @ G3 @ S4 ) ) ) ) ).
% rel_fun_iff_leq_vimage2p
thf(fact_676_rel__fun__iff__leq__vimage2p,axiom,
( bNF_re7626690874201225453um_num
= ( ^ [R6: int > int > $o,S4: num > num > $o,F2: int > num,G3: int > num] : ( ord_le6741204236512500942_int_o @ R6 @ ( bNF_vi2107963681178519309_num_o @ F2 @ G3 @ S4 ) ) ) ) ).
% rel_fun_iff_leq_vimage2p
thf(fact_677_rel__fun__mono_H,axiom,
! [Y4: product_prod_nat_nat > int > $o,X5: product_prod_nat_nat > int > $o,A4: nat > nat > $o,B5: nat > nat > $o,F: product_prod_nat_nat > nat,G: int > nat] :
( ! [X: product_prod_nat_nat,Y: int] :
( ( Y4 @ X @ Y )
=> ( X5 @ X @ Y ) )
=> ( ! [X: nat,Y: nat] :
( ( A4 @ X @ Y )
=> ( B5 @ X @ Y ) )
=> ( ( bNF_re4555766996558763186at_nat @ X5 @ A4 @ F @ G )
=> ( bNF_re4555766996558763186at_nat @ Y4 @ B5 @ F @ G ) ) ) ) ).
% rel_fun_mono'
thf(fact_678_rel__fun__mono_H,axiom,
! [Y4: product_prod_nat_nat > int > $o,X5: product_prod_nat_nat > int > $o,A4: ( product_prod_nat_nat > product_prod_nat_nat ) > ( int > int ) > $o,B5: ( product_prod_nat_nat > product_prod_nat_nat ) > ( int > int ) > $o,F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,G: int > int > int] :
( ! [X: product_prod_nat_nat,Y: int] :
( ( Y4 @ X @ Y )
=> ( X5 @ X @ Y ) )
=> ( ! [X: product_prod_nat_nat > product_prod_nat_nat,Y: int > int] :
( ( A4 @ X @ Y )
=> ( B5 @ X @ Y ) )
=> ( ( bNF_re7408651293131936558nt_int @ X5 @ A4 @ F @ G )
=> ( bNF_re7408651293131936558nt_int @ Y4 @ B5 @ F @ G ) ) ) ) ).
% rel_fun_mono'
thf(fact_679_rel__fun__mono_H,axiom,
! [Y4: product_prod_nat_nat > int > $o,X5: product_prod_nat_nat > int > $o,A4: ( product_prod_nat_nat > $o ) > ( int > $o ) > $o,B5: ( product_prod_nat_nat > $o ) > ( int > $o ) > $o,F: product_prod_nat_nat > product_prod_nat_nat > $o,G: int > int > $o] :
( ! [X: product_prod_nat_nat,Y: int] :
( ( Y4 @ X @ Y )
=> ( X5 @ X @ Y ) )
=> ( ! [X: product_prod_nat_nat > $o,Y: int > $o] :
( ( A4 @ X @ Y )
=> ( B5 @ X @ Y ) )
=> ( ( bNF_re717283939379294677_int_o @ X5 @ A4 @ F @ G )
=> ( bNF_re717283939379294677_int_o @ Y4 @ B5 @ F @ G ) ) ) ) ).
% rel_fun_mono'
thf(fact_680_rel__fun__mono_H,axiom,
! [Y4: int > code_integer > $o,X5: int > code_integer > $o,A4: num > num > $o,B5: num > num > $o,F: int > num,G: code_integer > num] :
( ! [X: int,Y: code_integer] :
( ( Y4 @ X @ Y )
=> ( X5 @ X @ Y ) )
=> ( ! [X: num,Y: num] :
( ( A4 @ X @ Y )
=> ( B5 @ X @ Y ) )
=> ( ( bNF_re6718328864250387230um_num @ X5 @ A4 @ F @ G )
=> ( bNF_re6718328864250387230um_num @ Y4 @ B5 @ F @ G ) ) ) ) ).
% rel_fun_mono'
thf(fact_681_rel__fun__mono_H,axiom,
! [Y4: int > int > $o,X5: int > int > $o,A4: num > num > $o,B5: num > num > $o,F: int > num,G: int > num] :
( ! [X: int,Y: int] :
( ( Y4 @ X @ Y )
=> ( X5 @ X @ Y ) )
=> ( ! [X: num,Y: num] :
( ( A4 @ X @ Y )
=> ( B5 @ X @ Y ) )
=> ( ( bNF_re7626690874201225453um_num @ X5 @ A4 @ F @ G )
=> ( bNF_re7626690874201225453um_num @ Y4 @ B5 @ F @ G ) ) ) ) ).
% rel_fun_mono'
thf(fact_682_rel__fun__mono,axiom,
! [X5: product_prod_nat_nat > int > $o,A4: nat > nat > $o,F: product_prod_nat_nat > nat,G: int > nat,Y4: product_prod_nat_nat > int > $o,B5: nat > nat > $o] :
( ( bNF_re4555766996558763186at_nat @ X5 @ A4 @ F @ G )
=> ( ! [X: product_prod_nat_nat,Y: int] :
( ( Y4 @ X @ Y )
=> ( X5 @ X @ Y ) )
=> ( ! [X: nat,Y: nat] :
( ( A4 @ X @ Y )
=> ( B5 @ X @ Y ) )
=> ( bNF_re4555766996558763186at_nat @ Y4 @ B5 @ F @ G ) ) ) ) ).
% rel_fun_mono
thf(fact_683_rel__fun__mono,axiom,
! [X5: product_prod_nat_nat > int > $o,A4: ( product_prod_nat_nat > product_prod_nat_nat ) > ( int > int ) > $o,F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,G: int > int > int,Y4: product_prod_nat_nat > int > $o,B5: ( product_prod_nat_nat > product_prod_nat_nat ) > ( int > int ) > $o] :
( ( bNF_re7408651293131936558nt_int @ X5 @ A4 @ F @ G )
=> ( ! [X: product_prod_nat_nat,Y: int] :
( ( Y4 @ X @ Y )
=> ( X5 @ X @ Y ) )
=> ( ! [X: product_prod_nat_nat > product_prod_nat_nat,Y: int > int] :
( ( A4 @ X @ Y )
=> ( B5 @ X @ Y ) )
=> ( bNF_re7408651293131936558nt_int @ Y4 @ B5 @ F @ G ) ) ) ) ).
% rel_fun_mono
thf(fact_684_rel__fun__mono,axiom,
! [X5: product_prod_nat_nat > int > $o,A4: ( product_prod_nat_nat > $o ) > ( int > $o ) > $o,F: product_prod_nat_nat > product_prod_nat_nat > $o,G: int > int > $o,Y4: product_prod_nat_nat > int > $o,B5: ( product_prod_nat_nat > $o ) > ( int > $o ) > $o] :
( ( bNF_re717283939379294677_int_o @ X5 @ A4 @ F @ G )
=> ( ! [X: product_prod_nat_nat,Y: int] :
( ( Y4 @ X @ Y )
=> ( X5 @ X @ Y ) )
=> ( ! [X: product_prod_nat_nat > $o,Y: int > $o] :
( ( A4 @ X @ Y )
=> ( B5 @ X @ Y ) )
=> ( bNF_re717283939379294677_int_o @ Y4 @ B5 @ F @ G ) ) ) ) ).
% rel_fun_mono
thf(fact_685_rel__fun__mono,axiom,
! [X5: int > code_integer > $o,A4: num > num > $o,F: int > num,G: code_integer > num,Y4: int > code_integer > $o,B5: num > num > $o] :
( ( bNF_re6718328864250387230um_num @ X5 @ A4 @ F @ G )
=> ( ! [X: int,Y: code_integer] :
( ( Y4 @ X @ Y )
=> ( X5 @ X @ Y ) )
=> ( ! [X: num,Y: num] :
( ( A4 @ X @ Y )
=> ( B5 @ X @ Y ) )
=> ( bNF_re6718328864250387230um_num @ Y4 @ B5 @ F @ G ) ) ) ) ).
% rel_fun_mono
thf(fact_686_rel__fun__mono,axiom,
! [X5: int > int > $o,A4: num > num > $o,F: int > num,G: int > num,Y4: int > int > $o,B5: num > num > $o] :
( ( bNF_re7626690874201225453um_num @ X5 @ A4 @ F @ G )
=> ( ! [X: int,Y: int] :
( ( Y4 @ X @ Y )
=> ( X5 @ X @ Y ) )
=> ( ! [X: num,Y: num] :
( ( A4 @ X @ Y )
=> ( B5 @ X @ Y ) )
=> ( bNF_re7626690874201225453um_num @ Y4 @ B5 @ F @ G ) ) ) ) ).
% rel_fun_mono
thf(fact_687_rel__funD,axiom,
! [A4: product_prod_nat_nat > int > $o,B5: nat > nat > $o,F: product_prod_nat_nat > nat,G: int > nat,X2: product_prod_nat_nat,Y3: int] :
( ( bNF_re4555766996558763186at_nat @ A4 @ B5 @ F @ G )
=> ( ( A4 @ X2 @ Y3 )
=> ( B5 @ ( F @ X2 ) @ ( G @ Y3 ) ) ) ) ).
% rel_funD
thf(fact_688_rel__funD,axiom,
! [A4: product_prod_nat_nat > int > $o,B5: ( product_prod_nat_nat > product_prod_nat_nat ) > ( int > int ) > $o,F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,G: int > int > int,X2: product_prod_nat_nat,Y3: int] :
( ( bNF_re7408651293131936558nt_int @ A4 @ B5 @ F @ G )
=> ( ( A4 @ X2 @ Y3 )
=> ( B5 @ ( F @ X2 ) @ ( G @ Y3 ) ) ) ) ).
% rel_funD
thf(fact_689_rel__funD,axiom,
! [A4: product_prod_nat_nat > int > $o,B5: ( product_prod_nat_nat > $o ) > ( int > $o ) > $o,F: product_prod_nat_nat > product_prod_nat_nat > $o,G: int > int > $o,X2: product_prod_nat_nat,Y3: int] :
( ( bNF_re717283939379294677_int_o @ A4 @ B5 @ F @ G )
=> ( ( A4 @ X2 @ Y3 )
=> ( B5 @ ( F @ X2 ) @ ( G @ Y3 ) ) ) ) ).
% rel_funD
thf(fact_690_rel__funD,axiom,
! [A4: int > code_integer > $o,B5: num > num > $o,F: int > num,G: code_integer > num,X2: int,Y3: code_integer] :
( ( bNF_re6718328864250387230um_num @ A4 @ B5 @ F @ G )
=> ( ( A4 @ X2 @ Y3 )
=> ( B5 @ ( F @ X2 ) @ ( G @ Y3 ) ) ) ) ).
% rel_funD
thf(fact_691_rel__funD,axiom,
! [A4: int > int > $o,B5: num > num > $o,F: int > num,G: int > num,X2: int,Y3: int] :
( ( bNF_re7626690874201225453um_num @ A4 @ B5 @ F @ G )
=> ( ( A4 @ X2 @ Y3 )
=> ( B5 @ ( F @ X2 ) @ ( G @ Y3 ) ) ) ) ).
% rel_funD
thf(fact_692_fun_Orel__compp,axiom,
! [R3: num > num > $o,S5: num > num > $o] :
( ( bNF_re7626690874201225453um_num
@ ^ [Y2: int,Z: int] : Y2 = Z
@ ( relcompp_num_num_num @ R3 @ S5 ) )
= ( relcom9025577121137075810nt_num
@ ( bNF_re7626690874201225453um_num
@ ^ [Y2: int,Z: int] : Y2 = Z
@ R3 )
@ ( bNF_re7626690874201225453um_num
@ ^ [Y2: int,Z: int] : Y2 = Z
@ S5 ) ) ) ).
% fun.rel_compp
thf(fact_693_relcompp__mono,axiom,
! [R2: product_prod_nat_nat > product_prod_nat_nat > $o,R: product_prod_nat_nat > product_prod_nat_nat > $o,S3: product_prod_nat_nat > int > $o,S2: product_prod_nat_nat > int > $o] :
( ( ord_le5604493270027003598_nat_o @ R2 @ R )
=> ( ( ord_le129227817499904167_int_o @ S3 @ S2 )
=> ( ord_le129227817499904167_int_o @ ( relcom8911704206367525917at_int @ R2 @ S3 ) @ ( relcom8911704206367525917at_int @ R @ S2 ) ) ) ) ).
% relcompp_mono
thf(fact_694_relcompp__mono,axiom,
! [R2: int > int > $o,R: int > int > $o,S3: int > code_integer > $o,S2: int > code_integer > $o] :
( ( ord_le6741204236512500942_int_o @ R2 @ R )
=> ( ( ord_le996066559646421213eger_o @ S3 @ S2 )
=> ( ord_le996066559646421213eger_o @ ( relcom967497705283160040nteger @ R2 @ S3 ) @ ( relcom967497705283160040nteger @ R @ S2 ) ) ) ) ).
% relcompp_mono
thf(fact_695_in__rel__def,axiom,
( fun_in6812141508599261570at_nat
= ( ^ [R6: set_Pr7459493094073627847at_nat,X4: set_Pr4329608150637261639at_nat,Y5: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X4 @ Y5 ) @ R6 ) ) ) ).
% in_rel_def
thf(fact_696_in__rel__def,axiom,
( fun_in8056734985859589634at_nat
= ( ^ [R6: set_Pr4329608150637261639at_nat,X4: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y5 ) @ R6 ) ) ) ).
% in_rel_def
thf(fact_697_in__rel__def,axiom,
( fun_in_rel_nat_num
= ( ^ [R6: set_Pr6200539531224447659at_num,X4: nat,Y5: num] : ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X4 @ Y5 ) @ R6 ) ) ) ).
% in_rel_def
thf(fact_698_in__rel__def,axiom,
( fun_in_rel_nat_nat
= ( ^ [R6: set_Pr1261947904930325089at_nat,X4: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ R6 ) ) ) ).
% in_rel_def
thf(fact_699_in__rel__def,axiom,
( fun_in_rel_int_int
= ( ^ [R6: set_Pr958786334691620121nt_int,X4: int,Y5: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y5 ) @ R6 ) ) ) ).
% in_rel_def
thf(fact_700_in__rel__Collect__case__prod__eq,axiom,
! [X5: nat > nat > $o] :
( ( fun_in_rel_nat_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ X5 ) ) )
= X5 ) ).
% in_rel_Collect_case_prod_eq
thf(fact_701_in__rel__Collect__case__prod__eq,axiom,
! [X5: int > int > $o] :
( ( fun_in_rel_int_int @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ X5 ) ) )
= X5 ) ).
% in_rel_Collect_case_prod_eq
thf(fact_702_Collect__case__prod__mono,axiom,
! [A4: nat > nat > $o,B5: nat > nat > $o] :
( ( ord_le2646555220125990790_nat_o @ A4 @ B5 )
=> ( ord_le3146513528884898305at_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ A4 ) ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ B5 ) ) ) ) ).
% Collect_case_prod_mono
thf(fact_703_Collect__case__prod__mono,axiom,
! [A4: int > int > $o,B5: int > int > $o] :
( ( ord_le6741204236512500942_int_o @ A4 @ B5 )
=> ( ord_le2843351958646193337nt_int @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ A4 ) ) @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ B5 ) ) ) ) ).
% Collect_case_prod_mono
thf(fact_704_fun_Omap__comp,axiom,
! [G: num > num,F: nat > num,V: int > nat] :
( ( comp_num_num_int @ G @ ( comp_nat_num_int @ F @ V ) )
= ( comp_nat_num_int @ ( comp_num_num_nat @ G @ F ) @ V ) ) ).
% fun.map_comp
thf(fact_705_fun_Omap__comp,axiom,
! [G: nat > nat,F: int > nat,V: int > int] :
( ( comp_nat_nat_int @ G @ ( comp_int_nat_int @ F @ V ) )
= ( comp_int_nat_int @ ( comp_nat_nat_int @ G @ F ) @ V ) ) ).
% fun.map_comp
thf(fact_706_fun_Omap__comp,axiom,
! [G: $o > $o,F: nat > $o,V: nat > nat] :
( ( comp_o_o_nat @ G @ ( comp_nat_o_nat @ F @ V ) )
= ( comp_nat_o_nat @ ( comp_o_o_nat @ G @ F ) @ V ) ) ).
% fun.map_comp
thf(fact_707_fun_Omap__comp,axiom,
! [G: $o > $o,F: $o > $o,V: nat > $o] :
( ( comp_o_o_nat @ G @ ( comp_o_o_nat @ F @ V ) )
= ( comp_o_o_nat @ ( comp_o_o_o @ G @ F ) @ V ) ) ).
% fun.map_comp
thf(fact_708_fun_Omap__comp,axiom,
! [G: nat > num,F: nat > nat,V: int > nat] :
( ( comp_nat_num_int @ G @ ( comp_nat_nat_int @ F @ V ) )
= ( comp_nat_num_int @ ( comp_nat_num_nat @ G @ F ) @ V ) ) ).
% fun.map_comp
thf(fact_709_fun_Omap__comp,axiom,
! [G: nat > num,F: int > nat,V: int > int] :
( ( comp_nat_num_int @ G @ ( comp_int_nat_int @ F @ V ) )
= ( comp_int_num_int @ ( comp_nat_num_int @ G @ F ) @ V ) ) ).
% fun.map_comp
thf(fact_710_fun_Omap__comp,axiom,
! [G: nat > nat,F: nat > nat,V: nat > nat] :
( ( comp_nat_nat_nat @ G @ ( comp_nat_nat_nat @ F @ V ) )
= ( comp_nat_nat_nat @ ( comp_nat_nat_nat @ G @ F ) @ V ) ) ).
% fun.map_comp
thf(fact_711_fun_Omap__comp,axiom,
! [G: int > nat,F: int > int,V: int > int] :
( ( comp_int_nat_int @ G @ ( comp_int_int_int @ F @ V ) )
= ( comp_int_nat_int @ ( comp_int_nat_int @ G @ F ) @ V ) ) ).
% fun.map_comp
thf(fact_712_fun_Omap__comp,axiom,
! [G: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,F: code_integer > code_integer > code_integer,V: code_integer > code_integer] :
( ( comp_C8797469213163452608nteger @ G @ ( comp_C2931745410781319416nteger @ F @ V ) )
= ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ G @ F ) @ V ) ) ).
% fun.map_comp
thf(fact_713_fun_Omap__comp,axiom,
! [G: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,F: ( code_integer > code_integer ) > code_integer > code_integer,V: code_integer > code_integer > code_integer] :
( ( comp_C8797469213163452608nteger @ G @ ( comp_C2822359487432382964nteger @ F @ V ) )
= ( comp_C8797469213163452608nteger @ ( comp_C3983183376822107068nteger @ G @ F ) @ V ) ) ).
% fun.map_comp
thf(fact_714_relcompp_Ocases,axiom,
! [R: product_prod_nat_nat > product_prod_nat_nat > $o,S2: product_prod_nat_nat > int > $o,A1: product_prod_nat_nat,A22: int] :
( ( relcom8911704206367525917at_int @ R @ S2 @ A1 @ A22 )
=> ~ ! [B3: product_prod_nat_nat] :
( ( R @ A1 @ B3 )
=> ~ ( S2 @ B3 @ A22 ) ) ) ).
% relcompp.cases
thf(fact_715_relcompp_Ocases,axiom,
! [R: int > int > $o,S2: int > code_integer > $o,A1: int,A22: code_integer] :
( ( relcom967497705283160040nteger @ R @ S2 @ A1 @ A22 )
=> ~ ! [B3: int] :
( ( R @ A1 @ B3 )
=> ~ ( S2 @ B3 @ A22 ) ) ) ).
% relcompp.cases
thf(fact_716_relcompp_Osimps,axiom,
( relcom8911704206367525917at_int
= ( ^ [R5: product_prod_nat_nat > product_prod_nat_nat > $o,S: product_prod_nat_nat > int > $o,A12: product_prod_nat_nat,A23: int] :
? [A5: product_prod_nat_nat,B4: product_prod_nat_nat,C4: int] :
( ( A12 = A5 )
& ( A23 = C4 )
& ( R5 @ A5 @ B4 )
& ( S @ B4 @ C4 ) ) ) ) ).
% relcompp.simps
thf(fact_717_relcompp_Osimps,axiom,
( relcom967497705283160040nteger
= ( ^ [R5: int > int > $o,S: int > code_integer > $o,A12: int,A23: code_integer] :
? [A5: int,B4: int,C4: code_integer] :
( ( A12 = A5 )
& ( A23 = C4 )
& ( R5 @ A5 @ B4 )
& ( S @ B4 @ C4 ) ) ) ) ).
% relcompp.simps
thf(fact_718_eq__OO,axiom,
! [R3: product_prod_nat_nat > int > $o] :
( ( relcom8911704206367525917at_int
@ ^ [Y2: product_prod_nat_nat,Z: product_prod_nat_nat] : Y2 = Z
@ R3 )
= R3 ) ).
% eq_OO
thf(fact_719_eq__OO,axiom,
! [R3: int > code_integer > $o] :
( ( relcom967497705283160040nteger
@ ^ [Y2: int,Z: int] : Y2 = Z
@ R3 )
= R3 ) ).
% eq_OO
thf(fact_720_relcomppE,axiom,
! [R: product_prod_nat_nat > product_prod_nat_nat > $o,S2: product_prod_nat_nat > int > $o,A: product_prod_nat_nat,C: int] :
( ( relcom8911704206367525917at_int @ R @ S2 @ A @ C )
=> ~ ! [B3: product_prod_nat_nat] :
( ( R @ A @ B3 )
=> ~ ( S2 @ B3 @ C ) ) ) ).
% relcomppE
thf(fact_721_relcomppE,axiom,
! [R: int > int > $o,S2: int > code_integer > $o,A: int,C: code_integer] :
( ( relcom967497705283160040nteger @ R @ S2 @ A @ C )
=> ~ ! [B3: int] :
( ( R @ A @ B3 )
=> ~ ( S2 @ B3 @ C ) ) ) ).
% relcomppE
thf(fact_722_relcomppI,axiom,
! [R: product_prod_nat_nat > product_prod_nat_nat > $o,A: product_prod_nat_nat,B: product_prod_nat_nat,S2: product_prod_nat_nat > int > $o,C: int] :
( ( R @ A @ B )
=> ( ( S2 @ B @ C )
=> ( relcom8911704206367525917at_int @ R @ S2 @ A @ C ) ) ) ).
% relcomppI
thf(fact_723_relcomppI,axiom,
! [R: int > int > $o,A: int,B: int,S2: int > code_integer > $o,C: code_integer] :
( ( R @ A @ B )
=> ( ( S2 @ B @ C )
=> ( relcom967497705283160040nteger @ R @ S2 @ A @ C ) ) ) ).
% relcomppI
thf(fact_724_map__prod__o__convol,axiom,
! [H1: $o > $o,H22: $o > $o,F: nat > $o,G: nat > $o] :
( ( comp_P7034811116587396339_o_nat @ ( produc1928436952130685430_o_o_o @ H1 @ H22 ) @ ( bNF_convol_nat_o_o @ F @ G ) )
= ( bNF_convol_nat_o_o @ ( comp_o_o_nat @ H1 @ F ) @ ( comp_o_o_nat @ H22 @ G ) ) ) ).
% map_prod_o_convol
thf(fact_725_map__prod__o__convol,axiom,
! [H1: $o > $o,H22: nat > nat,F: nat > $o,G: nat > nat] :
( ( comp_P1391661485614827491at_nat @ ( produc7071226018047599512at_nat @ H1 @ H22 ) @ ( bNF_convol_nat_o_nat @ F @ G ) )
= ( bNF_convol_nat_o_nat @ ( comp_o_o_nat @ H1 @ F ) @ ( comp_nat_nat_nat @ H22 @ G ) ) ) ).
% map_prod_o_convol
thf(fact_726_map__prod__o__convol,axiom,
! [H1: nat > num,H22: nat > num,F: int > nat,G: int > nat] :
( ( comp_P7387482301570536291um_int @ ( produc1476026309457068494at_num @ H1 @ H22 ) @ ( bNF_co1177303703936631735at_nat @ F @ G ) )
= ( bNF_co5088337930254655563um_num @ ( comp_nat_num_int @ H1 @ F ) @ ( comp_nat_num_int @ H22 @ G ) ) ) ).
% map_prod_o_convol
thf(fact_727_map__prod__o__convol,axiom,
! [H1: nat > num,H22: int > nat,F: int > nat,G: int > int] :
( ( comp_P8376639181999047357at_int @ ( produc5918048235202344416nt_nat @ H1 @ H22 ) @ ( bNF_co1174813233427581459at_int @ F @ G ) )
= ( bNF_co8531006144098876929um_nat @ ( comp_nat_num_int @ H1 @ F ) @ ( comp_int_nat_int @ H22 @ G ) ) ) ).
% map_prod_o_convol
thf(fact_728_map__prod__o__convol,axiom,
! [H1: nat > nat,H22: $o > $o,F: nat > nat,G: nat > $o] :
( ( comp_P5799945282029801519_o_nat @ ( produc2001155298098703512at_o_o @ H1 @ H22 ) @ ( bNF_convol_nat_nat_o @ F @ G ) )
= ( bNF_convol_nat_nat_o @ ( comp_nat_nat_nat @ H1 @ F ) @ ( comp_o_o_nat @ H22 @ G ) ) ) ).
% map_prod_o_convol
thf(fact_729_map__prod__o__convol,axiom,
! [H1: nat > nat,H22: nat > nat,F: nat > nat,G: nat > nat] :
( ( comp_P7430769850142754163at_nat @ ( produc6977886695330630970at_nat @ H1 @ H22 ) @ ( bNF_co805650143699787099at_nat @ F @ G ) )
= ( bNF_co805650143699787099at_nat @ ( comp_nat_nat_nat @ H1 @ F ) @ ( comp_nat_nat_nat @ H22 @ G ) ) ) ).
% map_prod_o_convol
thf(fact_730_map__prod__o__convol,axiom,
! [H1: int > nat,H22: nat > num,F: int > int,G: int > nat] :
( ( comp_P1145193044942399421um_int @ ( produc1225400549393458656at_num @ H1 @ H22 ) @ ( bNF_co2176657415837686291nt_nat @ F @ G ) )
= ( bNF_co6958007526947186177at_num @ ( comp_int_nat_int @ H1 @ F ) @ ( comp_nat_num_int @ H22 @ G ) ) ) ).
% map_prod_o_convol
thf(fact_731_map__prod__o__convol,axiom,
! [H1: int > nat,H22: int > nat,F: int > int,G: int > int] :
( ( comp_P2134349925370910487at_int @ ( produc5667422475138734578nt_nat @ H1 @ H22 ) @ ( bNF_co2174166945328636015nt_int @ F @ G ) )
= ( bNF_co1177303703936631735at_nat @ ( comp_int_nat_int @ H1 @ F ) @ ( comp_int_nat_int @ H22 @ G ) ) ) ).
% map_prod_o_convol
thf(fact_732_map__prod__o__convol,axiom,
! [H1: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,H22: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,F: code_integer > code_integer > code_integer,G: code_integer > code_integer > code_integer] :
( ( comp_P8564252314792299288nteger @ ( produc4724772261320961782nteger @ H1 @ H22 ) @ ( bNF_co6169604788974073338nteger @ F @ G ) )
= ( bNF_co3156833790671340178nteger @ ( comp_C8797469213163452608nteger @ H1 @ F ) @ ( comp_C8797469213163452608nteger @ H22 @ G ) ) ) ).
% map_prod_o_convol
thf(fact_733_convol__o,axiom,
! [F: $o > $o,G: $o > $o,H: nat > $o] :
( ( comp_o7658611401268416066_o_nat @ ( bNF_convol_o_o_o @ F @ G ) @ H )
= ( bNF_convol_nat_o_o @ ( comp_o_o_nat @ F @ H ) @ ( comp_o_o_nat @ G @ H ) ) ) ).
% convol_o
thf(fact_734_convol__o,axiom,
! [F: nat > num,G: nat > num,H: int > nat] :
( ( comp_n8531277669757933902um_int @ ( bNF_co4716684370017810927um_num @ F @ G ) @ H )
= ( bNF_co5088337930254655563um_num @ ( comp_nat_num_int @ F @ H ) @ ( comp_nat_num_int @ G @ H ) ) ) ).
% convol_o
thf(fact_735_convol__o,axiom,
! [F: nat > nat,G: nat > nat,H: nat > nat] :
( ( comp_n8574565218330151774at_nat @ ( bNF_co805650143699787099at_nat @ F @ G ) @ H )
= ( bNF_co805650143699787099at_nat @ ( comp_nat_nat_nat @ F @ H ) @ ( comp_nat_nat_nat @ G @ H ) ) ) ).
% convol_o
thf(fact_736_convol__o,axiom,
! [F: int > nat,G: int > nat,H: int > int] :
( ( comp_i1108739374719436894at_int @ ( bNF_co1177303703936631735at_nat @ F @ G ) @ H )
= ( bNF_co1177303703936631735at_nat @ ( comp_int_nat_int @ F @ H ) @ ( comp_int_nat_int @ G @ H ) ) ) ).
% convol_o
thf(fact_737_convol__o,axiom,
! [F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,H: code_integer > code_integer > code_integer] :
( ( comp_C6484513282182537548nteger @ ( bNF_co1764293801198077070nteger @ F @ G ) @ H )
= ( bNF_co3156833790671340178nteger @ ( comp_C8797469213163452608nteger @ F @ H ) @ ( comp_C8797469213163452608nteger @ G @ H ) ) ) ).
% convol_o
thf(fact_738_vimage2p__relcompp__mono,axiom,
! [R3: product_prod_nat_nat > product_prod_nat_nat > $o,S5: product_prod_nat_nat > int > $o,T3: product_prod_nat_nat > int > $o,F: product_prod_nat_nat > product_prod_nat_nat,G: product_prod_nat_nat > product_prod_nat_nat,H: int > int] :
( ( ord_le129227817499904167_int_o @ ( relcom8911704206367525917at_int @ R3 @ S5 ) @ T3 )
=> ( ord_le129227817499904167_int_o @ ( relcom8911704206367525917at_int @ ( bNF_vi763171854521451549_nat_o @ F @ G @ R3 ) @ ( bNF_vi5426221348228882691_int_o @ G @ H @ S5 ) ) @ ( bNF_vi5426221348228882691_int_o @ F @ H @ T3 ) ) ) ).
% vimage2p_relcompp_mono
thf(fact_739_vimage2p__relcompp__mono,axiom,
! [R3: product_prod_nat_nat > product_prod_nat_nat > $o,S5: product_prod_nat_nat > int > $o,T3: product_prod_nat_nat > int > $o,F: int > product_prod_nat_nat,G: int > product_prod_nat_nat,H: code_integer > int] :
( ( ord_le129227817499904167_int_o @ ( relcom8911704206367525917at_int @ R3 @ S5 ) @ T3 )
=> ( ord_le996066559646421213eger_o @ ( relcom967497705283160040nteger @ ( bNF_vi1204582724742824157_nat_o @ F @ G @ R3 ) @ ( bNF_vi2881549544236934259_int_o @ G @ H @ S5 ) ) @ ( bNF_vi2881549544236934259_int_o @ F @ H @ T3 ) ) ) ).
% vimage2p_relcompp_mono
thf(fact_740_vimage2p__relcompp__mono,axiom,
! [R3: int > int > $o,S5: int > code_integer > $o,T3: int > code_integer > $o,F: product_prod_nat_nat > int,G: product_prod_nat_nat > int,H: int > code_integer] :
( ( ord_le996066559646421213eger_o @ ( relcom967497705283160040nteger @ R3 @ S5 ) @ T3 )
=> ( ord_le129227817499904167_int_o @ ( relcom8911704206367525917at_int @ ( bNF_vi2682369418337388893_int_o @ F @ G @ R3 ) @ ( bNF_vi33923956287962137eger_o @ G @ H @ S5 ) ) @ ( bNF_vi33923956287962137eger_o @ F @ H @ T3 ) ) ) ).
% vimage2p_relcompp_mono
thf(fact_741_vimage2p__relcompp__mono,axiom,
! [R3: int > int > $o,S5: int > code_integer > $o,T3: int > code_integer > $o,F: int > int,G: int > int,H: code_integer > code_integer] :
( ( ord_le996066559646421213eger_o @ ( relcom967497705283160040nteger @ R3 @ S5 ) @ T3 )
=> ( ord_le996066559646421213eger_o @ ( relcom967497705283160040nteger @ ( bNF_vi3884084463199473129_int_o @ F @ G @ R3 ) @ ( bNF_vi699694178825965831eger_o @ G @ H @ S5 ) ) @ ( bNF_vi699694178825965831eger_o @ F @ H @ T3 ) ) ) ).
% vimage2p_relcompp_mono
thf(fact_742_pos__fun__distr,axiom,
! [R3: int > code_integer > $o,S5: num > num > $o,R7: code_integer > code_integer > $o,S6: num > num > $o] : ( ord_le6931201860925936491_num_o @ ( relcom1848932251921554756er_num @ ( bNF_re6718328864250387230um_num @ R3 @ S5 ) @ ( bNF_re40212662316476751um_num @ R7 @ S6 ) ) @ ( bNF_re6718328864250387230um_num @ ( relcom8320293257703906841nteger @ R3 @ R7 ) @ ( relcompp_num_num_num @ S5 @ S6 ) ) ) ).
% pos_fun_distr
thf(fact_743_pos__fun__distr,axiom,
! [R3: int > code_integer > $o,S5: num > num > $o,R7: code_integer > int > $o,S6: num > num > $o] : ( ord_le4541714130423454556_num_o @ ( relcom7779292856395800339nt_num @ ( bNF_re6718328864250387230um_num @ R3 @ S5 ) @ ( bNF_re8313794964846168862um_num @ R7 @ S6 ) ) @ ( bNF_re7626690874201225453um_num @ ( relcom328798225758222312er_int @ R3 @ R7 ) @ ( relcompp_num_num_num @ S5 @ S6 ) ) ) ).
% pos_fun_distr
thf(fact_744_pos__fun__distr,axiom,
! [R3: int > int > $o,S5: num > num > $o,R7: int > int > $o,S6: num > num > $o] : ( ord_le4541714130423454556_num_o @ ( relcom9025577121137075810nt_num @ ( bNF_re7626690874201225453um_num @ R3 @ S5 ) @ ( bNF_re7626690874201225453um_num @ R7 @ S6 ) ) @ ( bNF_re7626690874201225453um_num @ ( relcompp_int_int_int @ R3 @ R7 ) @ ( relcompp_num_num_num @ S5 @ S6 ) ) ) ).
% pos_fun_distr
thf(fact_745_pos__fun__distr,axiom,
! [R3: int > int > $o,S5: num > num > $o,R7: int > code_integer > $o,S6: num > num > $o] : ( ord_le6931201860925936491_num_o @ ( relcom1533066287727470739er_num @ ( bNF_re7626690874201225453um_num @ R3 @ S5 ) @ ( bNF_re6718328864250387230um_num @ R7 @ S6 ) ) @ ( bNF_re6718328864250387230um_num @ ( relcom967497705283160040nteger @ R3 @ R7 ) @ ( relcompp_num_num_num @ S5 @ S6 ) ) ) ).
% pos_fun_distr
thf(fact_746_pos__fun__distr,axiom,
! [R3: int > int > $o,S5: int > int > $o,R7: int > code_integer > $o,S6: int > code_integer > $o] : ( ord_le7522433683449569238eger_o @ ( relcom7212586982944386682nteger @ ( bNF_re4712519889275205905nt_int @ R3 @ S5 ) @ ( bNF_re3379532845092657523nteger @ R7 @ S6 ) ) @ ( bNF_re3379532845092657523nteger @ ( relcom967497705283160040nteger @ R3 @ R7 ) @ ( relcom967497705283160040nteger @ S5 @ S6 ) ) ) ).
% pos_fun_distr
thf(fact_747_pos__fun__distr,axiom,
! [R3: product_prod_nat_nat > int > $o,S5: nat > nat > $o,R7: int > int > $o,S6: nat > nat > $o] : ( ord_le1714508135640884727_nat_o @ ( relcom4272412204205018633nt_nat @ ( bNF_re4555766996558763186at_nat @ R3 @ S5 ) @ ( bNF_re3715656647883201625at_nat @ R7 @ S6 ) ) @ ( bNF_re4555766996558763186at_nat @ ( relcom2949942408940504938nt_int @ R3 @ R7 ) @ ( relcompp_nat_nat_nat @ S5 @ S6 ) ) ) ).
% pos_fun_distr
thf(fact_748_pos__fun__distr,axiom,
! [R3: product_prod_nat_nat > product_prod_nat_nat > $o,S5: nat > nat > $o,R7: product_prod_nat_nat > int > $o,S6: nat > nat > $o] : ( ord_le1714508135640884727_nat_o @ ( relcom531901107642208332nt_nat @ ( bNF_re8246922863344978751at_nat @ R3 @ S5 ) @ ( bNF_re4555766996558763186at_nat @ R7 @ S6 ) ) @ ( bNF_re4555766996558763186at_nat @ ( relcom8911704206367525917at_int @ R3 @ R7 ) @ ( relcompp_nat_nat_nat @ S5 @ S6 ) ) ) ).
% pos_fun_distr
thf(fact_749_pos__fun__distr,axiom,
! [R3: product_prod_nat_nat > product_prod_nat_nat > $o,S5: int > int > $o,R7: product_prod_nat_nat > int > $o,S6: int > code_integer > $o] : ( ord_le6141271677531915726eger_o @ ( relcom4574826104943343961nteger @ ( bNF_re20414067882207223nt_int @ R3 @ S5 ) @ ( bNF_re6202762543029211035nteger @ R7 @ S6 ) ) @ ( bNF_re6202762543029211035nteger @ ( relcom8911704206367525917at_int @ R3 @ R7 ) @ ( relcom967497705283160040nteger @ S5 @ S6 ) ) ) ).
% pos_fun_distr
thf(fact_750_pos__fun__distr,axiom,
! [R3: int > int > $o,S5: product_prod_nat_nat > product_prod_nat_nat > $o,R7: int > code_integer > $o,S6: product_prod_nat_nat > int > $o] : ( ord_le8615040745438789472_int_o @ ( relcom2127790875031262553er_int @ ( bNF_re8766404587435608311at_nat @ R3 @ S5 ) @ ( bNF_re2828929484557863493at_int @ R7 @ S6 ) ) @ ( bNF_re2828929484557863493at_int @ ( relcom967497705283160040nteger @ R3 @ R7 ) @ ( relcom8911704206367525917at_int @ S5 @ S6 ) ) ) ).
% pos_fun_distr
thf(fact_751_pos__fun__distr,axiom,
! [R3: product_prod_nat_nat > product_prod_nat_nat > $o,S5: product_prod_nat_nat > product_prod_nat_nat > $o,R7: product_prod_nat_nat > int > $o,S6: product_prod_nat_nat > int > $o] : ( ord_le2484784447047653514_int_o @ ( relcom4982215633375716084nt_int @ ( bNF_re2241393799969408733at_nat @ R3 @ S5 ) @ ( bNF_re7400052026677387805at_int @ R7 @ S6 ) ) @ ( bNF_re7400052026677387805at_int @ ( relcom8911704206367525917at_int @ R3 @ R7 ) @ ( relcom8911704206367525917at_int @ S5 @ S6 ) ) ) ).
% pos_fun_distr
thf(fact_752_o__rsp_I2_J,axiom,
! [R1: nat > nat > $o] :
( bNF_re6580757339198446453_nat_o
@ ^ [Y2: $o > $o,Z: $o > $o] : Y2 = Z
@ ( bNF_re2697311982885662685_nat_o
@ ( bNF_re4705727531993890431at_o_o @ R1
@ ^ [Y2: $o,Z: $o] : Y2 = Z )
@ ( bNF_re4705727531993890431at_o_o @ R1
@ ^ [Y2: $o,Z: $o] : Y2 = Z ) )
@ comp_o_o_nat
@ comp_o_o_nat ) ).
% o_rsp(2)
thf(fact_753_o__rsp_I2_J,axiom,
! [R1: nat > nat > $o] :
( bNF_re3262823321055862553at_nat
@ ^ [Y2: nat > nat,Z: nat > nat] : Y2 = Z
@ ( bNF_re239970166668089693at_nat
@ ( bNF_re5653821019739307937at_nat @ R1
@ ^ [Y2: nat,Z: nat] : Y2 = Z )
@ ( bNF_re5653821019739307937at_nat @ R1
@ ^ [Y2: nat,Z: nat] : Y2 = Z ) )
@ comp_nat_nat_nat
@ comp_nat_nat_nat ) ).
% o_rsp(2)
thf(fact_754_o__rsp_I2_J,axiom,
! [R1: int > int > $o] :
( bNF_re6307232852808291833nt_nat
@ ^ [Y2: int > nat,Z: int > nat] : Y2 = Z
@ ( bNF_re2034674868506727813nt_nat
@ ( bNF_re4712519889275205905nt_int @ R1
@ ^ [Y2: int,Z: int] : Y2 = Z )
@ ( bNF_re3715656647883201625at_nat @ R1
@ ^ [Y2: nat,Z: nat] : Y2 = Z ) )
@ comp_int_nat_int
@ comp_int_nat_int ) ).
% o_rsp(2)
thf(fact_755_o__rsp_I2_J,axiom,
! [R1: int > code_integer > $o] :
( bNF_re2655444144887082515er_num
@ ^ [Y2: nat > num,Z: nat > num] : Y2 = Z
@ ( bNF_re5502197115626545091er_num
@ ( bNF_re2807294637932363402at_nat @ R1
@ ^ [Y2: nat,Z: nat] : Y2 = Z )
@ ( bNF_re6718328864250387230um_num @ R1
@ ^ [Y2: num,Z: num] : Y2 = Z ) )
@ comp_nat_num_int
@ comp_n2103536230552569900nteger ) ).
% o_rsp(2)
thf(fact_756_o__rsp_I2_J,axiom,
! [R1: int > int > $o] :
( bNF_re3846787885862361009nt_num
@ ^ [Y2: nat > num,Z: nat > num] : Y2 = Z
@ ( bNF_re9099949428469764705nt_num
@ ( bNF_re3715656647883201625at_nat @ R1
@ ^ [Y2: nat,Z: nat] : Y2 = Z )
@ ( bNF_re7626690874201225453um_num @ R1
@ ^ [Y2: num,Z: num] : Y2 = Z ) )
@ comp_nat_num_int
@ comp_nat_num_int ) ).
% o_rsp(2)
thf(fact_757_o__rsp_I2_J,axiom,
! [R1: int > code_integer > $o] :
( bNF_re4105200416192800571er_num
@ ^ [Y2: num > num,Z: num > num] : Y2 = Z
@ ( bNF_re6292805115969292119er_num
@ ( bNF_re6718328864250387230um_num @ R1
@ ^ [Y2: num,Z: num] : Y2 = Z )
@ ( bNF_re6718328864250387230um_num @ R1
@ ^ [Y2: num,Z: num] : Y2 = Z ) )
@ comp_num_num_int
@ comp_n8557746072492896886nteger ) ).
% o_rsp(2)
thf(fact_758_o__rsp_I2_J,axiom,
! [R1: int > int > $o] :
( bNF_re5446066262214944985nt_num
@ ^ [Y2: num > num,Z: num > num] : Y2 = Z
@ ( bNF_re4768042542643474933nt_num
@ ( bNF_re7626690874201225453um_num @ R1
@ ^ [Y2: num,Z: num] : Y2 = Z )
@ ( bNF_re7626690874201225453um_num @ R1
@ ^ [Y2: num,Z: num] : Y2 = Z ) )
@ comp_num_num_int
@ comp_num_num_int ) ).
% o_rsp(2)
thf(fact_759_o__rsp_I2_J,axiom,
! [R1: product_prod_nat_nat > int > $o] :
( bNF_re4099335476084980305nt_nat
@ ^ [Y2: int > nat,Z: int > nat] : Y2 = Z
@ ( bNF_re3284331099615978085nt_nat
@ ( bNF_re5552630237950767466nt_int @ R1
@ ^ [Y2: int,Z: int] : Y2 = Z )
@ ( bNF_re4555766996558763186at_nat @ R1
@ ^ [Y2: nat,Z: nat] : Y2 = Z ) )
@ comp_i4471277238271202272at_nat
@ comp_int_nat_int ) ).
% o_rsp(2)
thf(fact_760_o__rsp_I2_J,axiom,
! [R1: product_prod_nat_nat > int > $o] :
( bNF_re2742285375668209089nt_num
@ ^ [Y2: nat > num,Z: nat > num] : Y2 = Z
@ ( bNF_re2682547871993127601nt_num
@ ( bNF_re4555766996558763186at_nat @ R1
@ ^ [Y2: nat,Z: nat] : Y2 = Z )
@ ( bNF_re8466801222876787014um_num @ R1
@ ^ [Y2: num,Z: num] : Y2 = Z ) )
@ comp_n8649498751204056690at_nat
@ comp_nat_num_int ) ).
% o_rsp(2)
thf(fact_761_o__rsp_I2_J,axiom,
! [R1: product_prod_nat_nat > int > $o] :
( bNF_re1160762158786734233nt_nat
@ ^ [Y2: nat > nat,Z: nat > nat] : Y2 = Z
@ ( bNF_re8419181437172746013nt_nat
@ ( bNF_re4555766996558763186at_nat @ R1
@ ^ [Y2: nat,Z: nat] : Y2 = Z )
@ ( bNF_re4555766996558763186at_nat @ R1
@ ^ [Y2: nat,Z: nat] : Y2 = Z ) )
@ comp_n2711240574518091068at_nat
@ comp_nat_nat_int ) ).
% o_rsp(2)
thf(fact_762_o__rsp_I1_J,axiom,
! [R22: $o > $o > $o,R32: $o > $o > $o,R1: nat > nat > $o] : ( bNF_re6580757339198446453_nat_o @ ( bNF_rel_fun_o_o_o_o @ R22 @ R32 ) @ ( bNF_re2697311982885662685_nat_o @ ( bNF_re4705727531993890431at_o_o @ R1 @ R22 ) @ ( bNF_re4705727531993890431at_o_o @ R1 @ R32 ) ) @ comp_o_o_nat @ comp_o_o_nat ) ).
% o_rsp(1)
thf(fact_763_o__rsp_I1_J,axiom,
! [R22: $o > nat > $o,R32: $o > num > $o,R1: nat > int > $o] : ( bNF_re7387973359137000979nt_num @ ( bNF_re766437678722298023_o_num @ R22 @ R32 ) @ ( bNF_re4102725726183926567nt_num @ ( bNF_re8210149947023016781_o_nat @ R1 @ R22 ) @ ( bNF_re4767481733178795415_o_num @ R1 @ R32 ) ) @ comp_o_o_nat @ comp_nat_num_int ) ).
% o_rsp(1)
thf(fact_764_o__rsp_I1_J,axiom,
! [R22: $o > nat > $o,R32: $o > nat > $o,R1: nat > nat > $o] : ( bNF_re1515921786380824775at_nat @ ( bNF_re4209105892566519389_o_nat @ R22 @ R32 ) @ ( bNF_re2144779315220482269at_nat @ ( bNF_re7523573913982466217_o_nat @ R1 @ R22 ) @ ( bNF_re7523573913982466217_o_nat @ R1 @ R32 ) ) @ comp_o_o_nat @ comp_nat_nat_nat ) ).
% o_rsp(1)
thf(fact_765_o__rsp_I1_J,axiom,
! [R22: $o > int > $o,R32: $o > nat > $o,R1: nat > int > $o] : ( bNF_re7413876800493218871nt_nat @ ( bNF_re4895681925607069953_o_nat @ R22 @ R32 ) @ ( bNF_re1982657550219213441nt_nat @ ( bNF_re8207659476513966505_o_int @ R1 @ R22 ) @ ( bNF_re8210149947023016781_o_nat @ R1 @ R32 ) ) @ comp_o_o_nat @ comp_int_nat_int ) ).
% o_rsp(1)
thf(fact_766_o__rsp_I1_J,axiom,
! [R22: nat > $o > $o,R32: num > $o > $o,R1: int > nat > $o] : ( bNF_re3931674169038344211_nat_o @ ( bNF_re8508505965575957011_num_o @ R22 @ R32 ) @ ( bNF_re7435575058205011731_nat_o @ ( bNF_re4914077030771115051_nat_o @ R1 @ R22 ) @ ( bNF_re3760060065607074657_num_o @ R1 @ R32 ) ) @ comp_nat_num_int @ comp_o_o_nat ) ).
% o_rsp(1)
thf(fact_767_o__rsp_I1_J,axiom,
! [R22: nat > nat > $o,R32: num > nat > $o,R1: int > nat > $o] : ( bNF_re6524732562925676389at_nat @ ( bNF_re3784151423046777323um_nat @ R22 @ R32 ) @ ( bNF_re717418264989850719at_nat @ ( bNF_re3344003087646356989at_nat @ R1 @ R22 ) @ ( bNF_re1474333490953826375um_nat @ R1 @ R32 ) ) @ comp_nat_num_int @ comp_nat_nat_nat ) ).
% o_rsp(1)
thf(fact_768_o__rsp_I1_J,axiom,
! [R22: nat > int > $o,R32: num > nat > $o,R1: int > int > $o] : ( bNF_re5843449549409830101nt_nat @ ( bNF_re4155804983283621959um_nat @ R22 @ R32 ) @ ( bNF_re3642433447925381491nt_nat @ ( bNF_re3713166177374151349at_int @ R1 @ R22 ) @ ( bNF_re1845987051190671011um_nat @ R1 @ R32 ) ) @ comp_nat_num_int @ comp_int_nat_int ) ).
% o_rsp(1)
thf(fact_769_o__rsp_I1_J,axiom,
! [R22: nat > $o > $o,R32: nat > $o > $o,R1: nat > nat > $o] : ( bNF_re3930744952143938759_nat_o @ ( bNF_re439150893885221597_nat_o @ R22 @ R32 ) @ ( bNF_re7744223787735169245_nat_o @ ( bNF_re1402873238235875719_nat_o @ R1 @ R22 ) @ ( bNF_re1402873238235875719_nat_o @ R1 @ R32 ) ) @ comp_nat_nat_nat @ comp_o_o_nat ) ).
% o_rsp(1)
thf(fact_770_o__rsp_I1_J,axiom,
! [R22: nat > nat > $o,R32: nat > num > $o,R1: nat > int > $o] : ( bNF_re584878643992547173nt_num @ ( bNF_re2211152805895086571at_num @ R22 @ R32 ) @ ( bNF_re8622501330148003679nt_num @ ( bNF_re6025474579976152573at_nat @ R1 @ R22 ) @ ( bNF_re2582806366131931207at_num @ R1 @ R32 ) ) @ comp_nat_nat_nat @ comp_nat_num_int ) ).
% o_rsp(1)
thf(fact_771_o__rsp_I1_J,axiom,
! [R22: nat > nat > $o,R32: nat > nat > $o,R1: nat > nat > $o] : ( bNF_re3262823321055862553at_nat @ ( bNF_re5653821019739307937at_nat @ R22 @ R32 ) @ ( bNF_re239970166668089693at_nat @ ( bNF_re5653821019739307937at_nat @ R1 @ R22 ) @ ( bNF_re5653821019739307937at_nat @ R1 @ R32 ) ) @ comp_nat_nat_nat @ comp_nat_nat_nat ) ).
% o_rsp(1)
thf(fact_772_rewriteR__comp__comp2,axiom,
! [G: int > int,H: nat > int,R12: nat > int,R23: nat > nat,F: int > nat,L: nat > nat] :
( ( ( comp_int_int_nat @ G @ H )
= ( comp_nat_int_nat @ R12 @ R23 ) )
=> ( ( ( comp_int_nat_nat @ F @ R12 )
= L )
=> ( ( comp_int_nat_nat @ ( comp_int_nat_int @ F @ G ) @ H )
= ( comp_nat_nat_nat @ L @ R23 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_773_rewriteR__comp__comp2,axiom,
! [G: nat > int,H: nat > nat,R12: int > int,R23: nat > int,F: int > nat,L: int > nat] :
( ( ( comp_nat_int_nat @ G @ H )
= ( comp_int_int_nat @ R12 @ R23 ) )
=> ( ( ( comp_int_nat_int @ F @ R12 )
= L )
=> ( ( comp_nat_nat_nat @ ( comp_int_nat_nat @ F @ G ) @ H )
= ( comp_int_nat_nat @ L @ R23 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_774_rewriteR__comp__comp2,axiom,
! [G: int > int,H: int > int,R12: int > int,R23: int > int,F: int > nat,L: int > nat] :
( ( ( comp_int_int_int @ G @ H )
= ( comp_int_int_int @ R12 @ R23 ) )
=> ( ( ( comp_int_nat_int @ F @ R12 )
= L )
=> ( ( comp_int_nat_int @ ( comp_int_nat_int @ F @ G ) @ H )
= ( comp_int_nat_int @ L @ R23 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_775_rewriteR__comp__comp2,axiom,
! [G: nat > $o,H: nat > nat,R12: $o > $o,R23: nat > $o,F: $o > $o,L: $o > $o] :
( ( ( comp_nat_o_nat @ G @ H )
= ( comp_o_o_nat @ R12 @ R23 ) )
=> ( ( ( comp_o_o_o @ F @ R12 )
= L )
=> ( ( comp_nat_o_nat @ ( comp_o_o_nat @ F @ G ) @ H )
= ( comp_o_o_nat @ L @ R23 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_776_rewriteR__comp__comp2,axiom,
! [G: nat > $o,H: nat > nat,R12: $o > $o,R23: nat > $o,F: $o > nat,L: $o > nat] :
( ( ( comp_nat_o_nat @ G @ H )
= ( comp_o_o_nat @ R12 @ R23 ) )
=> ( ( ( comp_o_nat_o @ F @ R12 )
= L )
=> ( ( comp_nat_nat_nat @ ( comp_o_nat_nat @ F @ G ) @ H )
= ( comp_o_nat_nat @ L @ R23 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_777_rewriteR__comp__comp2,axiom,
! [G: int > num,H: int > int,R12: nat > num,R23: int > nat,F: num > nat,L: nat > nat] :
( ( ( comp_int_num_int @ G @ H )
= ( comp_nat_num_int @ R12 @ R23 ) )
=> ( ( ( comp_num_nat_nat @ F @ R12 )
= L )
=> ( ( comp_int_nat_int @ ( comp_num_nat_int @ F @ G ) @ H )
= ( comp_nat_nat_int @ L @ R23 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_778_rewriteR__comp__comp2,axiom,
! [G: int > nat,H: nat > int,R12: nat > nat,R23: nat > nat,F: nat > num,L: nat > num] :
( ( ( comp_int_nat_nat @ G @ H )
= ( comp_nat_nat_nat @ R12 @ R23 ) )
=> ( ( ( comp_nat_num_nat @ F @ R12 )
= L )
=> ( ( comp_int_num_nat @ ( comp_nat_num_int @ F @ G ) @ H )
= ( comp_nat_num_nat @ L @ R23 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_779_rewriteR__comp__comp2,axiom,
! [G: $o > nat,H: nat > $o,R12: nat > nat,R23: nat > nat,F: nat > $o,L: nat > $o] :
( ( ( comp_o_nat_nat @ G @ H )
= ( comp_nat_nat_nat @ R12 @ R23 ) )
=> ( ( ( comp_nat_o_nat @ F @ R12 )
= L )
=> ( ( comp_o_o_nat @ ( comp_nat_o_o @ F @ G ) @ H )
= ( comp_nat_o_nat @ L @ R23 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_780_rewriteR__comp__comp2,axiom,
! [G: nat > nat,H: int > nat,R12: int > nat,R23: int > int,F: nat > nat,L: int > nat] :
( ( ( comp_nat_nat_int @ G @ H )
= ( comp_int_nat_int @ R12 @ R23 ) )
=> ( ( ( comp_nat_nat_int @ F @ R12 )
= L )
=> ( ( comp_nat_nat_int @ ( comp_nat_nat_nat @ F @ G ) @ H )
= ( comp_int_nat_int @ L @ R23 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_781_rewriteR__comp__comp2,axiom,
! [G: nat > nat,H: int > nat,R12: int > nat,R23: int > int,F: nat > num,L: int > num] :
( ( ( comp_nat_nat_int @ G @ H )
= ( comp_int_nat_int @ R12 @ R23 ) )
=> ( ( ( comp_nat_num_int @ F @ R12 )
= L )
=> ( ( comp_nat_num_int @ ( comp_nat_num_nat @ F @ G ) @ H )
= ( comp_int_num_int @ L @ R23 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_782_rewriteL__comp__comp2,axiom,
! [F: $o > nat,G: $o > $o,L1: nat > nat,L2: $o > nat,H: nat > $o,R: nat > nat] :
( ( ( comp_o_nat_o @ F @ G )
= ( comp_nat_nat_o @ L1 @ L2 ) )
=> ( ( ( comp_o_nat_nat @ L2 @ H )
= R )
=> ( ( comp_o_nat_nat @ F @ ( comp_o_o_nat @ G @ H ) )
= ( comp_nat_nat_nat @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_783_rewriteL__comp__comp2,axiom,
! [F: num > num,G: nat > num,L1: nat > num,L2: nat > nat,H: int > nat,R: int > nat] :
( ( ( comp_num_num_nat @ F @ G )
= ( comp_nat_num_nat @ L1 @ L2 ) )
=> ( ( ( comp_nat_nat_int @ L2 @ H )
= R )
=> ( ( comp_num_num_int @ F @ ( comp_nat_num_int @ G @ H ) )
= ( comp_nat_num_int @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_784_rewriteL__comp__comp2,axiom,
! [F: num > nat,G: nat > num,L1: int > nat,L2: nat > int,H: int > nat,R: int > int] :
( ( ( comp_num_nat_nat @ F @ G )
= ( comp_int_nat_nat @ L1 @ L2 ) )
=> ( ( ( comp_nat_int_int @ L2 @ H )
= R )
=> ( ( comp_num_nat_int @ F @ ( comp_nat_num_int @ G @ H ) )
= ( comp_int_nat_int @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_785_rewriteL__comp__comp2,axiom,
! [F: $o > $o,G: $o > $o,L1: $o > $o,L2: $o > $o,H: nat > $o,R: nat > $o] :
( ( ( comp_o_o_o @ F @ G )
= ( comp_o_o_o @ L1 @ L2 ) )
=> ( ( ( comp_o_o_nat @ L2 @ H )
= R )
=> ( ( comp_o_o_nat @ F @ ( comp_o_o_nat @ G @ H ) )
= ( comp_o_o_nat @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_786_rewriteL__comp__comp2,axiom,
! [F: nat > nat,G: $o > nat,L1: $o > nat,L2: $o > $o,H: nat > $o,R: nat > $o] :
( ( ( comp_nat_nat_o @ F @ G )
= ( comp_o_nat_o @ L1 @ L2 ) )
=> ( ( ( comp_o_o_nat @ L2 @ H )
= R )
=> ( ( comp_nat_nat_nat @ F @ ( comp_o_nat_nat @ G @ H ) )
= ( comp_o_nat_nat @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_787_rewriteL__comp__comp2,axiom,
! [F: nat > num,G: nat > nat,L1: num > num,L2: nat > num,H: int > nat,R: int > num] :
( ( ( comp_nat_num_nat @ F @ G )
= ( comp_num_num_nat @ L1 @ L2 ) )
=> ( ( ( comp_nat_num_int @ L2 @ H )
= R )
=> ( ( comp_nat_num_int @ F @ ( comp_nat_nat_int @ G @ H ) )
= ( comp_num_num_int @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_788_rewriteL__comp__comp2,axiom,
! [F: int > nat,G: nat > int,L1: num > nat,L2: nat > num,H: int > nat,R: int > num] :
( ( ( comp_int_nat_nat @ F @ G )
= ( comp_num_nat_nat @ L1 @ L2 ) )
=> ( ( ( comp_nat_num_int @ L2 @ H )
= R )
=> ( ( comp_int_nat_int @ F @ ( comp_nat_int_int @ G @ H ) )
= ( comp_num_nat_int @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_789_rewriteL__comp__comp2,axiom,
! [F: num > $o,G: nat > num,L1: $o > $o,L2: nat > $o,H: int > nat,R: int > $o] :
( ( ( comp_num_o_nat @ F @ G )
= ( comp_o_o_nat @ L1 @ L2 ) )
=> ( ( ( comp_nat_o_int @ L2 @ H )
= R )
=> ( ( comp_num_o_int @ F @ ( comp_nat_num_int @ G @ H ) )
= ( comp_o_o_int @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_790_rewriteL__comp__comp2,axiom,
! [F: nat > $o,G: nat > nat,L1: $o > $o,L2: nat > $o,H: nat > nat,R: nat > $o] :
( ( ( comp_nat_o_nat @ F @ G )
= ( comp_o_o_nat @ L1 @ L2 ) )
=> ( ( ( comp_nat_o_nat @ L2 @ H )
= R )
=> ( ( comp_nat_o_nat @ F @ ( comp_nat_nat_nat @ G @ H ) )
= ( comp_o_o_nat @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_791_rewriteL__comp__comp2,axiom,
! [F: num > nat,G: nat > num,L1: nat > nat,L2: nat > nat,H: int > nat,R: int > nat] :
( ( ( comp_num_nat_nat @ F @ G )
= ( comp_nat_nat_nat @ L1 @ L2 ) )
=> ( ( ( comp_nat_nat_int @ L2 @ H )
= R )
=> ( ( comp_num_nat_int @ F @ ( comp_nat_num_int @ G @ H ) )
= ( comp_nat_nat_int @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_792_rewriteR__comp__comp,axiom,
! [G: nat > $o,H: nat > nat,R: nat > $o,F: $o > $o] :
( ( ( comp_nat_o_nat @ G @ H )
= R )
=> ( ( comp_nat_o_nat @ ( comp_o_o_nat @ F @ G ) @ H )
= ( comp_o_o_nat @ F @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_793_rewriteR__comp__comp,axiom,
! [G: nat > nat,H: int > nat,R: int > nat,F: nat > num] :
( ( ( comp_nat_nat_int @ G @ H )
= R )
=> ( ( comp_nat_num_int @ ( comp_nat_num_nat @ F @ G ) @ H )
= ( comp_nat_num_int @ F @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_794_rewriteR__comp__comp,axiom,
! [G: int > int,H: int > int,R: int > int,F: int > nat] :
( ( ( comp_int_int_int @ G @ H )
= R )
=> ( ( comp_int_nat_int @ ( comp_int_nat_int @ F @ G ) @ H )
= ( comp_int_nat_int @ F @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_795_rewriteR__comp__comp,axiom,
! [G: $o > $o,H: nat > $o,R: nat > $o,F: $o > $o] :
( ( ( comp_o_o_nat @ G @ H )
= R )
=> ( ( comp_o_o_nat @ ( comp_o_o_o @ F @ G ) @ H )
= ( comp_o_o_nat @ F @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_796_rewriteR__comp__comp,axiom,
! [G: nat > num,H: int > nat,R: int > num,F: num > num] :
( ( ( comp_nat_num_int @ G @ H )
= R )
=> ( ( comp_nat_num_int @ ( comp_num_num_nat @ F @ G ) @ H )
= ( comp_num_num_int @ F @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_797_rewriteR__comp__comp,axiom,
! [G: nat > nat,H: nat > nat,R: nat > nat,F: nat > nat] :
( ( ( comp_nat_nat_nat @ G @ H )
= R )
=> ( ( comp_nat_nat_nat @ ( comp_nat_nat_nat @ F @ G ) @ H )
= ( comp_nat_nat_nat @ F @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_798_rewriteR__comp__comp,axiom,
! [G: int > nat,H: int > int,R: int > nat,F: nat > num] :
( ( ( comp_int_nat_int @ G @ H )
= R )
=> ( ( comp_int_num_int @ ( comp_nat_num_int @ F @ G ) @ H )
= ( comp_nat_num_int @ F @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_799_rewriteR__comp__comp,axiom,
! [G: int > nat,H: int > int,R: int > nat,F: nat > nat] :
( ( ( comp_int_nat_int @ G @ H )
= R )
=> ( ( comp_int_nat_int @ ( comp_nat_nat_int @ F @ G ) @ H )
= ( comp_nat_nat_int @ F @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_800_rewriteR__comp__comp,axiom,
! [G: code_integer > code_integer > code_integer,H: code_integer > code_integer,R: code_integer > code_integer > code_integer,F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger] :
( ( ( comp_C2931745410781319416nteger @ G @ H )
= R )
=> ( ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ F @ G ) @ H )
= ( comp_C8797469213163452608nteger @ F @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_801_rewriteR__comp__comp,axiom,
! [G: ( code_integer > code_integer ) > code_integer > code_integer,H: code_integer > code_integer > code_integer,R: code_integer > code_integer > code_integer,F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger] :
( ( ( comp_C2822359487432382964nteger @ G @ H )
= R )
=> ( ( comp_C8797469213163452608nteger @ ( comp_C3983183376822107068nteger @ F @ G ) @ H )
= ( comp_C8797469213163452608nteger @ F @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_802_rewriteL__comp__comp,axiom,
! [F: num > num,G: nat > num,L: nat > num,H: int > nat] :
( ( ( comp_num_num_nat @ F @ G )
= L )
=> ( ( comp_num_num_int @ F @ ( comp_nat_num_int @ G @ H ) )
= ( comp_nat_num_int @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_803_rewriteL__comp__comp,axiom,
! [F: nat > nat,G: int > nat,L: int > nat,H: int > int] :
( ( ( comp_nat_nat_int @ F @ G )
= L )
=> ( ( comp_nat_nat_int @ F @ ( comp_int_nat_int @ G @ H ) )
= ( comp_int_nat_int @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_804_rewriteL__comp__comp,axiom,
! [F: $o > $o,G: $o > $o,L: $o > $o,H: nat > $o] :
( ( ( comp_o_o_o @ F @ G )
= L )
=> ( ( comp_o_o_nat @ F @ ( comp_o_o_nat @ G @ H ) )
= ( comp_o_o_nat @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_805_rewriteL__comp__comp,axiom,
! [F: nat > num,G: nat > nat,L: nat > num,H: int > nat] :
( ( ( comp_nat_num_nat @ F @ G )
= L )
=> ( ( comp_nat_num_int @ F @ ( comp_nat_nat_int @ G @ H ) )
= ( comp_nat_num_int @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_806_rewriteL__comp__comp,axiom,
! [F: $o > $o,G: nat > $o,L: nat > $o,H: nat > nat] :
( ( ( comp_o_o_nat @ F @ G )
= L )
=> ( ( comp_o_o_nat @ F @ ( comp_nat_o_nat @ G @ H ) )
= ( comp_nat_o_nat @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_807_rewriteL__comp__comp,axiom,
! [F: nat > num,G: int > nat,L: int > num,H: int > int] :
( ( ( comp_nat_num_int @ F @ G )
= L )
=> ( ( comp_nat_num_int @ F @ ( comp_int_nat_int @ G @ H ) )
= ( comp_int_num_int @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_808_rewriteL__comp__comp,axiom,
! [F: nat > nat,G: nat > nat,L: nat > nat,H: nat > nat] :
( ( ( comp_nat_nat_nat @ F @ G )
= L )
=> ( ( comp_nat_nat_nat @ F @ ( comp_nat_nat_nat @ G @ H ) )
= ( comp_nat_nat_nat @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_809_rewriteL__comp__comp,axiom,
! [F: int > nat,G: int > int,L: int > nat,H: int > int] :
( ( ( comp_int_nat_int @ F @ G )
= L )
=> ( ( comp_int_nat_int @ F @ ( comp_int_int_int @ G @ H ) )
= ( comp_int_nat_int @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_810_rewriteL__comp__comp,axiom,
! [F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer > code_integer,L: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,H: code_integer > code_integer] :
( ( ( comp_C8797469213163452608nteger @ F @ G )
= L )
=> ( ( comp_C8797469213163452608nteger @ F @ ( comp_C2931745410781319416nteger @ G @ H ) )
= ( comp_C1593894019821074884nteger @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_811_rewriteL__comp__comp,axiom,
! [F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G: ( code_integer > code_integer ) > code_integer > code_integer,L: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,H: code_integer > code_integer > code_integer] :
( ( ( comp_C3983183376822107068nteger @ F @ G )
= L )
=> ( ( comp_C8797469213163452608nteger @ F @ ( comp_C2822359487432382964nteger @ G @ H ) )
= ( comp_C8797469213163452608nteger @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_812_type__copy__map__cong0,axiom,
! [M: nat > $o,G: nat > nat,X2: nat,N: $o > $o,H: nat > $o,F: $o > $o] :
( ( ( M @ ( G @ X2 ) )
= ( N @ ( H @ X2 ) ) )
=> ( ( comp_nat_o_nat @ ( comp_o_o_nat @ F @ M ) @ G @ X2 )
= ( comp_o_o_nat @ ( comp_o_o_o @ F @ N ) @ H @ X2 ) ) ) ).
% type_copy_map_cong0
thf(fact_813_type__copy__map__cong0,axiom,
! [M: int > nat,G: int > int,X2: int,N: nat > nat,H: int > nat,F: nat > num] :
( ( ( M @ ( G @ X2 ) )
= ( N @ ( H @ X2 ) ) )
=> ( ( comp_int_num_int @ ( comp_nat_num_int @ F @ M ) @ G @ X2 )
= ( comp_nat_num_int @ ( comp_nat_num_nat @ F @ N ) @ H @ X2 ) ) ) ).
% type_copy_map_cong0
thf(fact_814_type__copy__map__cong0,axiom,
! [M: nat > nat,G: int > nat,X2: int,N: int > nat,H: int > int,F: nat > nat] :
( ( ( M @ ( G @ X2 ) )
= ( N @ ( H @ X2 ) ) )
=> ( ( comp_nat_nat_int @ ( comp_nat_nat_nat @ F @ M ) @ G @ X2 )
= ( comp_int_nat_int @ ( comp_nat_nat_int @ F @ N ) @ H @ X2 ) ) ) ).
% type_copy_map_cong0
thf(fact_815_type__copy__map__cong0,axiom,
! [M: int > int,G: nat > int,X2: nat,N: nat > int,H: nat > nat,F: int > nat] :
( ( ( M @ ( G @ X2 ) )
= ( N @ ( H @ X2 ) ) )
=> ( ( comp_int_nat_nat @ ( comp_int_nat_int @ F @ M ) @ G @ X2 )
= ( comp_nat_nat_nat @ ( comp_int_nat_nat @ F @ N ) @ H @ X2 ) ) ) ).
% type_copy_map_cong0
thf(fact_816_type__copy__map__cong0,axiom,
! [M: $o > $o,G: nat > $o,X2: nat,N: nat > $o,H: nat > nat,F: $o > $o] :
( ( ( M @ ( G @ X2 ) )
= ( N @ ( H @ X2 ) ) )
=> ( ( comp_o_o_nat @ ( comp_o_o_o @ F @ M ) @ G @ X2 )
= ( comp_nat_o_nat @ ( comp_o_o_nat @ F @ N ) @ H @ X2 ) ) ) ).
% type_copy_map_cong0
thf(fact_817_type__copy__map__cong0,axiom,
! [M: nat > nat,G: int > nat,X2: int,N: int > nat,H: int > int,F: nat > num] :
( ( ( M @ ( G @ X2 ) )
= ( N @ ( H @ X2 ) ) )
=> ( ( comp_nat_num_int @ ( comp_nat_num_nat @ F @ M ) @ G @ X2 )
= ( comp_int_num_int @ ( comp_nat_num_int @ F @ N ) @ H @ X2 ) ) ) ).
% type_copy_map_cong0
thf(fact_818_type__copy__map__cong0,axiom,
! [M: nat > int,G: nat > nat,X2: nat,N: int > int,H: nat > int,F: int > nat] :
( ( ( M @ ( G @ X2 ) )
= ( N @ ( H @ X2 ) ) )
=> ( ( comp_nat_nat_nat @ ( comp_int_nat_nat @ F @ M ) @ G @ X2 )
= ( comp_int_nat_nat @ ( comp_int_nat_int @ F @ N ) @ H @ X2 ) ) ) ).
% type_copy_map_cong0
thf(fact_819_type__copy__map__cong0,axiom,
! [M: nat > nat,G: nat > nat,X2: nat,N: nat > nat,H: nat > nat,F: nat > nat] :
( ( ( M @ ( G @ X2 ) )
= ( N @ ( H @ X2 ) ) )
=> ( ( comp_nat_nat_nat @ ( comp_nat_nat_nat @ F @ M ) @ G @ X2 )
= ( comp_nat_nat_nat @ ( comp_nat_nat_nat @ F @ N ) @ H @ X2 ) ) ) ).
% type_copy_map_cong0
thf(fact_820_type__copy__map__cong0,axiom,
! [M: int > nat,G: int > int,X2: int,N: nat > nat,H: int > nat,F: nat > nat] :
( ( ( M @ ( G @ X2 ) )
= ( N @ ( H @ X2 ) ) )
=> ( ( comp_int_nat_int @ ( comp_nat_nat_int @ F @ M ) @ G @ X2 )
= ( comp_nat_nat_int @ ( comp_nat_nat_nat @ F @ N ) @ H @ X2 ) ) ) ).
% type_copy_map_cong0
thf(fact_821_type__copy__map__cong0,axiom,
! [M: int > int,G: int > int,X2: int,N: int > int,H: int > int,F: int > nat] :
( ( ( M @ ( G @ X2 ) )
= ( N @ ( H @ X2 ) ) )
=> ( ( comp_int_nat_int @ ( comp_int_nat_int @ F @ M ) @ G @ X2 )
= ( comp_int_nat_int @ ( comp_int_nat_int @ F @ N ) @ H @ X2 ) ) ) ).
% type_copy_map_cong0
thf(fact_822_comp__transfer,axiom,
! [B5: $o > $o > $o,C3: $o > $o > $o,A4: nat > nat > $o] : ( bNF_re6580757339198446453_nat_o @ ( bNF_rel_fun_o_o_o_o @ B5 @ C3 ) @ ( bNF_re2697311982885662685_nat_o @ ( bNF_re4705727531993890431at_o_o @ A4 @ B5 ) @ ( bNF_re4705727531993890431at_o_o @ A4 @ C3 ) ) @ comp_o_o_nat @ comp_o_o_nat ) ).
% comp_transfer
thf(fact_823_comp__transfer,axiom,
! [B5: $o > nat > $o,C3: $o > num > $o,A4: nat > int > $o] : ( bNF_re7387973359137000979nt_num @ ( bNF_re766437678722298023_o_num @ B5 @ C3 ) @ ( bNF_re4102725726183926567nt_num @ ( bNF_re8210149947023016781_o_nat @ A4 @ B5 ) @ ( bNF_re4767481733178795415_o_num @ A4 @ C3 ) ) @ comp_o_o_nat @ comp_nat_num_int ) ).
% comp_transfer
thf(fact_824_comp__transfer,axiom,
! [B5: $o > nat > $o,C3: $o > nat > $o,A4: nat > nat > $o] : ( bNF_re1515921786380824775at_nat @ ( bNF_re4209105892566519389_o_nat @ B5 @ C3 ) @ ( bNF_re2144779315220482269at_nat @ ( bNF_re7523573913982466217_o_nat @ A4 @ B5 ) @ ( bNF_re7523573913982466217_o_nat @ A4 @ C3 ) ) @ comp_o_o_nat @ comp_nat_nat_nat ) ).
% comp_transfer
thf(fact_825_comp__transfer,axiom,
! [B5: $o > int > $o,C3: $o > nat > $o,A4: nat > int > $o] : ( bNF_re7413876800493218871nt_nat @ ( bNF_re4895681925607069953_o_nat @ B5 @ C3 ) @ ( bNF_re1982657550219213441nt_nat @ ( bNF_re8207659476513966505_o_int @ A4 @ B5 ) @ ( bNF_re8210149947023016781_o_nat @ A4 @ C3 ) ) @ comp_o_o_nat @ comp_int_nat_int ) ).
% comp_transfer
thf(fact_826_comp__transfer,axiom,
! [B5: nat > $o > $o,C3: num > $o > $o,A4: int > nat > $o] : ( bNF_re3931674169038344211_nat_o @ ( bNF_re8508505965575957011_num_o @ B5 @ C3 ) @ ( bNF_re7435575058205011731_nat_o @ ( bNF_re4914077030771115051_nat_o @ A4 @ B5 ) @ ( bNF_re3760060065607074657_num_o @ A4 @ C3 ) ) @ comp_nat_num_int @ comp_o_o_nat ) ).
% comp_transfer
thf(fact_827_comp__transfer,axiom,
! [B5: nat > nat > $o,C3: num > nat > $o,A4: int > nat > $o] : ( bNF_re6524732562925676389at_nat @ ( bNF_re3784151423046777323um_nat @ B5 @ C3 ) @ ( bNF_re717418264989850719at_nat @ ( bNF_re3344003087646356989at_nat @ A4 @ B5 ) @ ( bNF_re1474333490953826375um_nat @ A4 @ C3 ) ) @ comp_nat_num_int @ comp_nat_nat_nat ) ).
% comp_transfer
thf(fact_828_comp__transfer,axiom,
! [B5: nat > int > $o,C3: num > nat > $o,A4: int > int > $o] : ( bNF_re5843449549409830101nt_nat @ ( bNF_re4155804983283621959um_nat @ B5 @ C3 ) @ ( bNF_re3642433447925381491nt_nat @ ( bNF_re3713166177374151349at_int @ A4 @ B5 ) @ ( bNF_re1845987051190671011um_nat @ A4 @ C3 ) ) @ comp_nat_num_int @ comp_int_nat_int ) ).
% comp_transfer
thf(fact_829_comp__transfer,axiom,
! [B5: nat > $o > $o,C3: nat > $o > $o,A4: nat > nat > $o] : ( bNF_re3930744952143938759_nat_o @ ( bNF_re439150893885221597_nat_o @ B5 @ C3 ) @ ( bNF_re7744223787735169245_nat_o @ ( bNF_re1402873238235875719_nat_o @ A4 @ B5 ) @ ( bNF_re1402873238235875719_nat_o @ A4 @ C3 ) ) @ comp_nat_nat_nat @ comp_o_o_nat ) ).
% comp_transfer
thf(fact_830_comp__transfer,axiom,
! [B5: nat > nat > $o,C3: nat > num > $o,A4: nat > int > $o] : ( bNF_re584878643992547173nt_num @ ( bNF_re2211152805895086571at_num @ B5 @ C3 ) @ ( bNF_re8622501330148003679nt_num @ ( bNF_re6025474579976152573at_nat @ A4 @ B5 ) @ ( bNF_re2582806366131931207at_num @ A4 @ C3 ) ) @ comp_nat_nat_nat @ comp_nat_num_int ) ).
% comp_transfer
thf(fact_831_comp__transfer,axiom,
! [B5: nat > nat > $o,C3: nat > nat > $o,A4: nat > nat > $o] : ( bNF_re3262823321055862553at_nat @ ( bNF_re5653821019739307937at_nat @ B5 @ C3 ) @ ( bNF_re239970166668089693at_nat @ ( bNF_re5653821019739307937at_nat @ A4 @ B5 ) @ ( bNF_re5653821019739307937at_nat @ A4 @ C3 ) ) @ comp_nat_nat_nat @ comp_nat_nat_nat ) ).
% comp_transfer
thf(fact_832_rel__fun__iff__geq__image2p,axiom,
( bNF_re4555766996558763186at_nat
= ( ^ [R6: product_prod_nat_nat > int > $o,S4: nat > nat > $o,F2: product_prod_nat_nat > nat,G3: int > nat] : ( ord_le2646555220125990790_nat_o @ ( bNF_Gr6955563364113978072nt_nat @ F2 @ G3 @ R6 ) @ S4 ) ) ) ).
% rel_fun_iff_geq_image2p
thf(fact_833_rel__fun__iff__geq__image2p,axiom,
( bNF_re7408651293131936558nt_int
= ( ^ [R6: product_prod_nat_nat > int > $o,S4: ( product_prod_nat_nat > product_prod_nat_nat ) > ( int > int ) > $o,F2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,G3: int > int > int] : ( ord_le2484784447047653514_int_o @ ( bNF_Gr1354378531936274772nt_int @ F2 @ G3 @ R6 ) @ S4 ) ) ) ).
% rel_fun_iff_geq_image2p
thf(fact_834_rel__fun__iff__geq__image2p,axiom,
( bNF_re717283939379294677_int_o
= ( ^ [R6: product_prod_nat_nat > int > $o,S4: ( product_prod_nat_nat > $o ) > ( int > $o ) > $o,F2: product_prod_nat_nat > product_prod_nat_nat > $o,G3: int > int > $o] : ( ord_le2932013042137311317nt_o_o @ ( bNF_Gr364077372107685167_int_o @ F2 @ G3 @ R6 ) @ S4 ) ) ) ).
% rel_fun_iff_geq_image2p
thf(fact_835_rel__fun__iff__geq__image2p,axiom,
( bNF_re6718328864250387230um_num
= ( ^ [R6: int > code_integer > $o,S4: num > num > $o,F2: int > num,G3: code_integer > num] : ( ord_le6124364862034508274_num_o @ ( bNF_Gr166583433173494904er_num @ F2 @ G3 @ R6 ) @ S4 ) ) ) ).
% rel_fun_iff_geq_image2p
thf(fact_836_rel__fun__iff__geq__image2p,axiom,
( bNF_re7626690874201225453um_num
= ( ^ [R6: int > int > $o,S4: num > num > $o,F2: int > num,G3: int > num] : ( ord_le6124364862034508274_num_o @ ( bNF_Gr7020624548710275655nt_num @ F2 @ G3 @ R6 ) @ S4 ) ) ) ).
% rel_fun_iff_geq_image2p
thf(fact_837_map__prod__o__convol__id,axiom,
! [F: nat > num,G: nat > nat,X2: nat] :
( ( comp_P5371577678113413053at_nat @ ( produc4918694523301289860at_nat @ F @ id_nat ) @ ( bNF_co805650143699787099at_nat @ id_nat @ G ) @ X2 )
= ( bNF_co8159352583862032293um_nat @ ( comp_num_num_nat @ id_num @ F ) @ G @ X2 ) ) ).
% map_prod_o_convol_id
thf(fact_838_map__prod__o__convol__id,axiom,
! [F: $o > nat,G: $o > nat,X2: $o] :
( ( comp_P196761593691424337_nat_o @ ( produc6587185857518356756at_nat @ F @ id_nat ) @ ( bNF_convol_o_o_nat @ id_o @ G ) @ X2 )
= ( bNF_convol_o_nat_nat @ ( comp_nat_nat_o @ id_nat @ F ) @ G @ X2 ) ) ).
% map_prod_o_convol_id
thf(fact_839_map__prod__o__convol__id,axiom,
! [F: $o > $o,G: $o > nat,X2: $o] :
( ( comp_P1422407636550552773_nat_o @ ( produc7071226018047599512at_nat @ F @ id_nat ) @ ( bNF_convol_o_o_nat @ id_o @ G ) @ X2 )
= ( bNF_convol_o_o_nat @ ( comp_o_o_o @ id_o @ F ) @ G @ X2 ) ) ).
% map_prod_o_convol_id
thf(fact_840_map__prod__o__convol__id,axiom,
! [F: $o > num,G: $o > nat,X2: $o] :
( ( comp_P972951341003599239_nat_o @ ( produc4527993685489015646at_nat @ F @ id_nat ) @ ( bNF_convol_o_o_nat @ id_o @ G ) @ X2 )
= ( bNF_convol_o_num_nat @ ( comp_num_num_o @ id_num @ F ) @ G @ X2 ) ) ).
% map_prod_o_convol_id
thf(fact_841_map__prod__o__convol__id,axiom,
! [F: num > nat,G: num > nat,X2: num] :
( ( comp_P2539519431309506803at_num @ ( produc4208724500416182148at_nat @ F @ id_nat ) @ ( bNF_co6100160411832691183um_nat @ id_num @ G ) @ X2 )
= ( bNF_co7969830008525221797at_nat @ ( comp_nat_nat_num @ id_nat @ F ) @ G @ X2 ) ) ).
% map_prod_o_convol_id
thf(fact_842_map__prod__o__convol__id,axiom,
! [F: num > $o,G: num > nat,X2: num] :
( ( comp_P3192427084145116251at_num @ ( produc8319293927732033832at_nat @ F @ id_nat ) @ ( bNF_co6100160411832691183um_nat @ id_num @ G ) @ X2 )
= ( bNF_convol_num_o_nat @ ( comp_o_o_num @ id_o @ F ) @ G @ X2 ) ) ).
% map_prod_o_convol_id
thf(fact_843_map__prod__o__convol__id,axiom,
! [F: num > num,G: num > nat,X2: num] :
( ( comp_P480327259280165693at_num @ ( produc2149532328386841038at_nat @ F @ id_nat ) @ ( bNF_co6100160411832691183um_nat @ id_num @ G ) @ X2 )
= ( bNF_co6100160411832691183um_nat @ ( comp_num_num_num @ id_num @ F ) @ G @ X2 ) ) ).
% map_prod_o_convol_id
thf(fact_844_map__prod__o__convol__id,axiom,
! [F: nat > num,G: nat > $o,X2: nat] :
( ( comp_P6576135029341976421_o_nat @ ( produc3264868786340801506um_o_o @ F @ id_o ) @ ( bNF_convol_nat_nat_o @ id_nat @ G ) @ X2 )
= ( bNF_convol_nat_num_o @ ( comp_num_num_nat @ id_num @ F ) @ G @ X2 ) ) ).
% map_prod_o_convol_id
thf(fact_845_map__prod__o__convol__id,axiom,
! [F: $o > nat,G: $o > $o,X2: $o] :
( ( comp_P1471321874221585045at_o_o @ ( produc9141321880750005618at_o_o @ F @ id_o ) @ ( bNF_convol_o_o_o @ id_o @ G ) @ X2 )
= ( bNF_convol_o_nat_o @ ( comp_nat_nat_o @ id_nat @ F ) @ G @ X2 ) ) ).
% map_prod_o_convol_id
thf(fact_846_map__prod__o__convol__id,axiom,
! [F: $o > $o,G: $o > $o,X2: $o] :
( ( comp_P551574746845053877_o_o_o @ ( produc1928436952130685430_o_o_o @ F @ id_o ) @ ( bNF_convol_o_o_o @ id_o @ G ) @ X2 )
= ( bNF_convol_o_o_o @ ( comp_o_o_o @ id_o @ F ) @ G @ X2 ) ) ).
% map_prod_o_convol_id
thf(fact_847_image__Fpow__mono,axiom,
! [F: nat > rat,A4: set_nat,B5: set_rat] :
( ( ord_less_eq_set_rat @ ( image_nat_rat @ F @ A4 ) @ B5 )
=> ( ord_le513522071413781156et_rat @ ( image_4408659257933336347et_rat @ ( image_nat_rat @ F ) @ ( finite_Fpow_nat @ A4 ) ) @ ( finite_Fpow_rat @ B5 ) ) ) ).
% image_Fpow_mono
thf(fact_848_image__Fpow__mono,axiom,
! [F: nat > nat,A4: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A4 ) @ B5 )
=> ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat @ F ) @ ( finite_Fpow_nat @ A4 ) ) @ ( finite_Fpow_nat @ B5 ) ) ) ).
% image_Fpow_mono
thf(fact_849_image__Fpow__mono,axiom,
! [F: nat > int,A4: set_nat,B5: set_int] :
( ( ord_less_eq_set_int @ ( image_nat_int @ F @ A4 ) @ B5 )
=> ( ord_le4403425263959731960et_int @ ( image_3739036796817536367et_int @ ( image_nat_int @ F ) @ ( finite_Fpow_nat @ A4 ) ) @ ( finite_Fpow_int @ B5 ) ) ) ).
% image_Fpow_mono
thf(fact_850_image__Fpow__mono,axiom,
! [F: int > nat,A4: set_int,B5: set_nat] :
( ( ord_less_eq_set_nat @ ( image_int_nat @ F @ A4 ) @ B5 )
=> ( ord_le6893508408891458716et_nat @ ( image_4702325430467532143et_nat @ ( image_int_nat @ F ) @ ( finite_Fpow_int @ A4 ) ) @ ( finite_Fpow_nat @ B5 ) ) ) ).
% image_Fpow_mono
thf(fact_851_image__Fpow__mono,axiom,
! [F: int > int,A4: set_int,B5: set_int] :
( ( ord_less_eq_set_int @ ( image_int_int @ F @ A4 ) @ B5 )
=> ( ord_le4403425263959731960et_int @ ( image_524474410958335435et_int @ ( image_int_int @ F ) @ ( finite_Fpow_int @ A4 ) ) @ ( finite_Fpow_int @ B5 ) ) ) ).
% image_Fpow_mono
thf(fact_852_convol__mem__GrpI,axiom,
! [X2: set_Pr1261947904930325089at_nat,A4: set_se7855581050983116737at_nat,G: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ X2 @ A4 )
=> ( member8757157785044589968at_nat @ ( bNF_co147202357524637304at_nat @ id_set2334057505560298834at_nat @ G @ X2 ) @ ( collec6321179662152712658at_nat @ ( produc410239310623530412_nat_o @ ( bNF_Gr6677838525845560078at_nat @ A4 @ G ) ) ) ) ) ).
% convol_mem_GrpI
thf(fact_853_convol__mem__GrpI,axiom,
! [X2: nat,A4: set_nat,G: nat > nat] :
( ( member_nat @ X2 @ A4 )
=> ( member8440522571783428010at_nat @ ( bNF_co805650143699787099at_nat @ id_nat @ G @ X2 ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ( bNF_Grp_nat_nat @ A4 @ G ) ) ) ) ) ).
% convol_mem_GrpI
thf(fact_854_convol__mem__GrpI,axiom,
! [X2: int,A4: set_int,G: int > int] :
( ( member_int @ X2 @ A4 )
=> ( member5262025264175285858nt_int @ ( bNF_co2174166945328636015nt_int @ id_int @ G @ X2 ) @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ ( bNF_Grp_int_int @ A4 @ G ) ) ) ) ) ).
% convol_mem_GrpI
thf(fact_855_image__id,axiom,
( ( image_o_o @ id_o )
= id_set_o ) ).
% image_id
thf(fact_856_image__id,axiom,
( ( image_num_num @ id_num )
= id_set_num ) ).
% image_id
thf(fact_857_image__id,axiom,
( ( image_nat_nat @ id_nat )
= id_set_nat ) ).
% image_id
thf(fact_858_image__id,axiom,
( ( image_int_int @ id_int )
= id_set_int ) ).
% image_id
thf(fact_859_fun_Omap__id,axiom,
! [T2: nat > $o] :
( ( comp_o_o_nat @ id_o @ T2 )
= T2 ) ).
% fun.map_id
thf(fact_860_fun_Omap__id,axiom,
! [T2: nat > nat] :
( ( comp_nat_nat_nat @ id_nat @ T2 )
= T2 ) ).
% fun.map_id
thf(fact_861_comp__id,axiom,
! [F: nat > nat] :
( ( comp_nat_nat_nat @ F @ id_nat )
= F ) ).
% comp_id
thf(fact_862_comp__id,axiom,
! [F: int > nat] :
( ( comp_int_nat_int @ F @ id_int )
= F ) ).
% comp_id
thf(fact_863_id__comp,axiom,
! [G: nat > $o] :
( ( comp_o_o_nat @ id_o @ G )
= G ) ).
% id_comp
thf(fact_864_id__comp,axiom,
! [G: nat > nat] :
( ( comp_nat_nat_nat @ id_nat @ G )
= G ) ).
% id_comp
thf(fact_865_case__prod__Pair,axiom,
( ( produc5456455677105495163at_nat @ produc9060074326276436823at_nat )
= id_Pro6043886709289754456at_nat ) ).
% case_prod_Pair
thf(fact_866_case__prod__Pair,axiom,
( ( produc7548845685434208379at_nat @ produc2922128104949294807at_nat )
= id_Pro8722402698821300696at_nat ) ).
% case_prod_Pair
thf(fact_867_case__prod__Pair,axiom,
( ( produc49306077274653107at_num @ product_Pair_nat_num )
= id_Pro8267972854940751164at_num ) ).
% case_prod_Pair
thf(fact_868_case__prod__Pair,axiom,
( ( produc2626176000494625587at_nat @ product_Pair_nat_nat )
= id_Pro7559728917991913458at_nat ) ).
% case_prod_Pair
thf(fact_869_case__prod__Pair,axiom,
( ( produc4245557441103728435nt_int @ product_Pair_int_int )
= id_Pro4381231610383771306nt_int ) ).
% case_prod_Pair
thf(fact_870_apsnd__id,axiom,
( ( produc6499014454317279255nteger @ id_Code_integer )
= id_Pro2309048762231606924nteger ) ).
% apsnd_id
thf(fact_871_range__fst,axiom,
( ( image_7136766188234240905_ref_a @ produc473321116828791851t_unit @ top_to2503719809363509323t_unit )
= top_top_set_ref_a ) ).
% range_fst
thf(fact_872_range__snd,axiom,
( ( image_17660834435998945t_unit @ produc1126150082675451753t_unit @ top_to2503719809363509323t_unit )
= top_to3956321331435058429t_unit ) ).
% range_snd
thf(fact_873_fun_Omap__id0,axiom,
( ( comp_o_o_nat @ id_o )
= id_nat_o ) ).
% fun.map_id0
thf(fact_874_fun_Omap__id0,axiom,
( ( comp_nat_nat_nat @ id_nat )
= id_nat_nat ) ).
% fun.map_id0
thf(fact_875_surj__id,axiom,
( ( image_o_o @ id_o @ top_top_set_o )
= top_top_set_o ) ).
% surj_id
thf(fact_876_surj__id,axiom,
( ( image_num_num @ id_num @ top_top_set_num )
= top_top_set_num ) ).
% surj_id
thf(fact_877_surj__id,axiom,
( ( image_char_char @ id_char @ top_top_set_char )
= top_top_set_char ) ).
% surj_id
thf(fact_878_surj__id,axiom,
( ( image_7976474329151083847st_nat @ id_list_nat @ top_top_set_list_nat )
= top_top_set_list_nat ) ).
% surj_id
thf(fact_879_surj__id,axiom,
( ( image_rat_rat @ id_rat @ top_top_set_rat )
= top_top_set_rat ) ).
% surj_id
thf(fact_880_surj__id,axiom,
( ( image_nat_nat @ id_nat @ top_top_set_nat )
= top_top_set_nat ) ).
% surj_id
thf(fact_881_surj__id,axiom,
( ( image_int_int @ id_int @ top_top_set_int )
= top_top_set_int ) ).
% surj_id
thf(fact_882_Grp__UNIV__idI,axiom,
! [X2: $o,Y3: $o] :
( ( X2 = Y3 )
=> ( bNF_Grp_o_o @ top_top_set_o @ id_o @ X2 @ Y3 ) ) ).
% Grp_UNIV_idI
thf(fact_883_Grp__UNIV__idI,axiom,
! [X2: num,Y3: num] :
( ( X2 = Y3 )
=> ( bNF_Grp_num_num @ top_top_set_num @ id_num @ X2 @ Y3 ) ) ).
% Grp_UNIV_idI
thf(fact_884_Grp__UNIV__idI,axiom,
! [X2: char,Y3: char] :
( ( X2 = Y3 )
=> ( bNF_Grp_char_char @ top_top_set_char @ id_char @ X2 @ Y3 ) ) ).
% Grp_UNIV_idI
thf(fact_885_Grp__UNIV__idI,axiom,
! [X2: list_nat,Y3: list_nat] :
( ( X2 = Y3 )
=> ( bNF_Gr600725020315365264st_nat @ top_top_set_list_nat @ id_list_nat @ X2 @ Y3 ) ) ).
% Grp_UNIV_idI
thf(fact_886_Grp__UNIV__idI,axiom,
! [X2: rat,Y3: rat] :
( ( X2 = Y3 )
=> ( bNF_Grp_rat_rat @ top_top_set_rat @ id_rat @ X2 @ Y3 ) ) ).
% Grp_UNIV_idI
thf(fact_887_Grp__UNIV__idI,axiom,
! [X2: nat,Y3: nat] :
( ( X2 = Y3 )
=> ( bNF_Grp_nat_nat @ top_top_set_nat @ id_nat @ X2 @ Y3 ) ) ).
% Grp_UNIV_idI
thf(fact_888_Grp__UNIV__idI,axiom,
! [X2: int,Y3: int] :
( ( X2 = Y3 )
=> ( bNF_Grp_int_int @ top_top_set_int @ id_int @ X2 @ Y3 ) ) ).
% Grp_UNIV_idI
thf(fact_889_eq__alt,axiom,
( ( ^ [Y2: $o,Z: $o] : Y2 = Z )
= ( bNF_Grp_o_o @ top_top_set_o @ id_o ) ) ).
% eq_alt
thf(fact_890_eq__alt,axiom,
( ( ^ [Y2: num,Z: num] : Y2 = Z )
= ( bNF_Grp_num_num @ top_top_set_num @ id_num ) ) ).
% eq_alt
thf(fact_891_eq__alt,axiom,
( ( ^ [Y2: char,Z: char] : Y2 = Z )
= ( bNF_Grp_char_char @ top_top_set_char @ id_char ) ) ).
% eq_alt
thf(fact_892_eq__alt,axiom,
( ( ^ [Y2: list_nat,Z: list_nat] : Y2 = Z )
= ( bNF_Gr600725020315365264st_nat @ top_top_set_list_nat @ id_list_nat ) ) ).
% eq_alt
thf(fact_893_eq__alt,axiom,
( ( ^ [Y2: rat,Z: rat] : Y2 = Z )
= ( bNF_Grp_rat_rat @ top_top_set_rat @ id_rat ) ) ).
% eq_alt
thf(fact_894_eq__alt,axiom,
( ( ^ [Y2: nat,Z: nat] : Y2 = Z )
= ( bNF_Grp_nat_nat @ top_top_set_nat @ id_nat ) ) ).
% eq_alt
thf(fact_895_eq__alt,axiom,
( ( ^ [Y2: int,Z: int] : Y2 = Z )
= ( bNF_Grp_int_int @ top_top_set_int @ id_int ) ) ).
% eq_alt
thf(fact_896_Inf_OINF__id__eq,axiom,
! [Inf: set_o > $o,A4: set_o] :
( ( Inf @ ( image_o_o @ id_o @ A4 ) )
= ( Inf @ A4 ) ) ).
% Inf.INF_id_eq
thf(fact_897_Inf_OINF__id__eq,axiom,
! [Inf: set_num > num,A4: set_num] :
( ( Inf @ ( image_num_num @ id_num @ A4 ) )
= ( Inf @ A4 ) ) ).
% Inf.INF_id_eq
thf(fact_898_Inf_OINF__id__eq,axiom,
! [Inf: set_nat > nat,A4: set_nat] :
( ( Inf @ ( image_nat_nat @ id_nat @ A4 ) )
= ( Inf @ A4 ) ) ).
% Inf.INF_id_eq
thf(fact_899_Inf_OINF__id__eq,axiom,
! [Inf: set_int > int,A4: set_int] :
( ( Inf @ ( image_int_int @ id_int @ A4 ) )
= ( Inf @ A4 ) ) ).
% Inf.INF_id_eq
thf(fact_900_Sup_OSUP__id__eq,axiom,
! [Sup: set_o > $o,A4: set_o] :
( ( Sup @ ( image_o_o @ id_o @ A4 ) )
= ( Sup @ A4 ) ) ).
% Sup.SUP_id_eq
thf(fact_901_Sup_OSUP__id__eq,axiom,
! [Sup: set_num > num,A4: set_num] :
( ( Sup @ ( image_num_num @ id_num @ A4 ) )
= ( Sup @ A4 ) ) ).
% Sup.SUP_id_eq
thf(fact_902_Sup_OSUP__id__eq,axiom,
! [Sup: set_nat > nat,A4: set_nat] :
( ( Sup @ ( image_nat_nat @ id_nat @ A4 ) )
= ( Sup @ A4 ) ) ).
% Sup.SUP_id_eq
thf(fact_903_Sup_OSUP__id__eq,axiom,
! [Sup: set_int > int,A4: set_int] :
( ( Sup @ ( image_int_int @ id_int @ A4 ) )
= ( Sup @ A4 ) ) ).
% Sup.SUP_id_eq
thf(fact_904_comp__eq__id__dest,axiom,
! [A: $o > $o,B: nat > $o,C: nat > $o,V: nat] :
( ( ( comp_o_o_nat @ A @ B )
= ( comp_o_o_nat @ id_o @ C ) )
=> ( ( A @ ( B @ V ) )
= ( C @ V ) ) ) ).
% comp_eq_id_dest
thf(fact_905_comp__eq__id__dest,axiom,
! [A: nat > num,B: int > nat,C: int > num,V: int] :
( ( ( comp_nat_num_int @ A @ B )
= ( comp_num_num_int @ id_num @ C ) )
=> ( ( A @ ( B @ V ) )
= ( C @ V ) ) ) ).
% comp_eq_id_dest
thf(fact_906_comp__eq__id__dest,axiom,
! [A: nat > nat,B: nat > nat,C: nat > nat,V: nat] :
( ( ( comp_nat_nat_nat @ A @ B )
= ( comp_nat_nat_nat @ id_nat @ C ) )
=> ( ( A @ ( B @ V ) )
= ( C @ V ) ) ) ).
% comp_eq_id_dest
thf(fact_907_comp__eq__id__dest,axiom,
! [A: int > nat,B: int > int,C: int > nat,V: int] :
( ( ( comp_int_nat_int @ A @ B )
= ( comp_nat_nat_int @ id_nat @ C ) )
=> ( ( A @ ( B @ V ) )
= ( C @ V ) ) ) ).
% comp_eq_id_dest
thf(fact_908_comp__eq__id__dest,axiom,
! [A: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,B: code_integer > code_integer > code_integer,C: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,V: code_integer] :
( ( ( comp_C8797469213163452608nteger @ A @ B )
= ( comp_P3928908779438215820nteger @ id_Pro496765588584934260nteger @ C ) )
=> ( ( A @ ( B @ V ) )
= ( C @ V ) ) ) ).
% comp_eq_id_dest
thf(fact_909_pointfree__idE,axiom,
! [F: nat > nat,G: nat > nat,X2: nat] :
( ( ( comp_nat_nat_nat @ F @ G )
= id_nat )
=> ( ( F @ ( G @ X2 ) )
= X2 ) ) ).
% pointfree_idE
thf(fact_910_surjD,axiom,
! [F: char > char,Y3: char] :
( ( ( image_char_char @ F @ top_top_set_char )
= top_top_set_char )
=> ? [X: char] :
( Y3
= ( F @ X ) ) ) ).
% surjD
thf(fact_911_surjD,axiom,
! [F: char > rat,Y3: rat] :
( ( ( image_char_rat @ F @ top_top_set_char )
= top_top_set_rat )
=> ? [X: char] :
( Y3
= ( F @ X ) ) ) ).
% surjD
thf(fact_912_surjD,axiom,
! [F: char > nat,Y3: nat] :
( ( ( image_char_nat @ F @ top_top_set_char )
= top_top_set_nat )
=> ? [X: char] :
( Y3
= ( F @ X ) ) ) ).
% surjD
thf(fact_913_surjD,axiom,
! [F: char > int,Y3: int] :
( ( ( image_char_int @ F @ top_top_set_char )
= top_top_set_int )
=> ? [X: char] :
( Y3
= ( F @ X ) ) ) ).
% surjD
thf(fact_914_surjD,axiom,
! [F: rat > char,Y3: char] :
( ( ( image_rat_char @ F @ top_top_set_rat )
= top_top_set_char )
=> ? [X: rat] :
( Y3
= ( F @ X ) ) ) ).
% surjD
thf(fact_915_surjD,axiom,
! [F: rat > rat,Y3: rat] :
( ( ( image_rat_rat @ F @ top_top_set_rat )
= top_top_set_rat )
=> ? [X: rat] :
( Y3
= ( F @ X ) ) ) ).
% surjD
thf(fact_916_surjD,axiom,
! [F: rat > nat,Y3: nat] :
( ( ( image_rat_nat @ F @ top_top_set_rat )
= top_top_set_nat )
=> ? [X: rat] :
( Y3
= ( F @ X ) ) ) ).
% surjD
thf(fact_917_surjD,axiom,
! [F: rat > int,Y3: int] :
( ( ( image_rat_int @ F @ top_top_set_rat )
= top_top_set_int )
=> ? [X: rat] :
( Y3
= ( F @ X ) ) ) ).
% surjD
thf(fact_918_surjD,axiom,
! [F: nat > char,Y3: char] :
( ( ( image_nat_char @ F @ top_top_set_nat )
= top_top_set_char )
=> ? [X: nat] :
( Y3
= ( F @ X ) ) ) ).
% surjD
thf(fact_919_surjD,axiom,
! [F: nat > rat,Y3: rat] :
( ( ( image_nat_rat @ F @ top_top_set_nat )
= top_top_set_rat )
=> ? [X: nat] :
( Y3
= ( F @ X ) ) ) ).
% surjD
thf(fact_920_surjE,axiom,
! [F: char > char,Y3: char] :
( ( ( image_char_char @ F @ top_top_set_char )
= top_top_set_char )
=> ~ ! [X: char] :
( Y3
!= ( F @ X ) ) ) ).
% surjE
thf(fact_921_surjE,axiom,
! [F: char > rat,Y3: rat] :
( ( ( image_char_rat @ F @ top_top_set_char )
= top_top_set_rat )
=> ~ ! [X: char] :
( Y3
!= ( F @ X ) ) ) ).
% surjE
thf(fact_922_surjE,axiom,
! [F: char > nat,Y3: nat] :
( ( ( image_char_nat @ F @ top_top_set_char )
= top_top_set_nat )
=> ~ ! [X: char] :
( Y3
!= ( F @ X ) ) ) ).
% surjE
thf(fact_923_surjE,axiom,
! [F: char > int,Y3: int] :
( ( ( image_char_int @ F @ top_top_set_char )
= top_top_set_int )
=> ~ ! [X: char] :
( Y3
!= ( F @ X ) ) ) ).
% surjE
thf(fact_924_surjE,axiom,
! [F: rat > char,Y3: char] :
( ( ( image_rat_char @ F @ top_top_set_rat )
= top_top_set_char )
=> ~ ! [X: rat] :
( Y3
!= ( F @ X ) ) ) ).
% surjE
thf(fact_925_surjE,axiom,
! [F: rat > rat,Y3: rat] :
( ( ( image_rat_rat @ F @ top_top_set_rat )
= top_top_set_rat )
=> ~ ! [X: rat] :
( Y3
!= ( F @ X ) ) ) ).
% surjE
thf(fact_926_surjE,axiom,
! [F: rat > nat,Y3: nat] :
( ( ( image_rat_nat @ F @ top_top_set_rat )
= top_top_set_nat )
=> ~ ! [X: rat] :
( Y3
!= ( F @ X ) ) ) ).
% surjE
thf(fact_927_surjE,axiom,
! [F: rat > int,Y3: int] :
( ( ( image_rat_int @ F @ top_top_set_rat )
= top_top_set_int )
=> ~ ! [X: rat] :
( Y3
!= ( F @ X ) ) ) ).
% surjE
thf(fact_928_surjE,axiom,
! [F: nat > char,Y3: char] :
( ( ( image_nat_char @ F @ top_top_set_nat )
= top_top_set_char )
=> ~ ! [X: nat] :
( Y3
!= ( F @ X ) ) ) ).
% surjE
thf(fact_929_surjE,axiom,
! [F: nat > rat,Y3: rat] :
( ( ( image_nat_rat @ F @ top_top_set_nat )
= top_top_set_rat )
=> ~ ! [X: nat] :
( Y3
!= ( F @ X ) ) ) ).
% surjE
thf(fact_930_surjI,axiom,
! [G: char > char,F: char > char] :
( ! [X: char] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_char_char @ G @ top_top_set_char )
= top_top_set_char ) ) ).
% surjI
thf(fact_931_surjI,axiom,
! [G: char > rat,F: rat > char] :
( ! [X: rat] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_char_rat @ G @ top_top_set_char )
= top_top_set_rat ) ) ).
% surjI
thf(fact_932_surjI,axiom,
! [G: char > nat,F: nat > char] :
( ! [X: nat] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_char_nat @ G @ top_top_set_char )
= top_top_set_nat ) ) ).
% surjI
thf(fact_933_surjI,axiom,
! [G: char > int,F: int > char] :
( ! [X: int] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_char_int @ G @ top_top_set_char )
= top_top_set_int ) ) ).
% surjI
thf(fact_934_surjI,axiom,
! [G: rat > char,F: char > rat] :
( ! [X: char] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_rat_char @ G @ top_top_set_rat )
= top_top_set_char ) ) ).
% surjI
thf(fact_935_surjI,axiom,
! [G: rat > rat,F: rat > rat] :
( ! [X: rat] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_rat_rat @ G @ top_top_set_rat )
= top_top_set_rat ) ) ).
% surjI
thf(fact_936_surjI,axiom,
! [G: rat > nat,F: nat > rat] :
( ! [X: nat] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_rat_nat @ G @ top_top_set_rat )
= top_top_set_nat ) ) ).
% surjI
thf(fact_937_surjI,axiom,
! [G: rat > int,F: int > rat] :
( ! [X: int] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_rat_int @ G @ top_top_set_rat )
= top_top_set_int ) ) ).
% surjI
thf(fact_938_surjI,axiom,
! [G: nat > char,F: char > nat] :
( ! [X: char] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_nat_char @ G @ top_top_set_nat )
= top_top_set_char ) ) ).
% surjI
thf(fact_939_surjI,axiom,
! [G: nat > rat,F: rat > nat] :
( ! [X: rat] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_nat_rat @ G @ top_top_set_nat )
= top_top_set_rat ) ) ).
% surjI
thf(fact_940_rangeI,axiom,
! [F: char > $o,X2: char] : ( member_o @ ( F @ X2 ) @ ( image_char_o @ F @ top_top_set_char ) ) ).
% rangeI
thf(fact_941_rangeI,axiom,
! [F: char > nat,X2: char] : ( member_nat @ ( F @ X2 ) @ ( image_char_nat @ F @ top_top_set_char ) ) ).
% rangeI
thf(fact_942_rangeI,axiom,
! [F: char > int,X2: char] : ( member_int @ ( F @ X2 ) @ ( image_char_int @ F @ top_top_set_char ) ) ).
% rangeI
thf(fact_943_rangeI,axiom,
! [F: rat > $o,X2: rat] : ( member_o @ ( F @ X2 ) @ ( image_rat_o @ F @ top_top_set_rat ) ) ).
% rangeI
thf(fact_944_rangeI,axiom,
! [F: rat > nat,X2: rat] : ( member_nat @ ( F @ X2 ) @ ( image_rat_nat @ F @ top_top_set_rat ) ) ).
% rangeI
thf(fact_945_rangeI,axiom,
! [F: rat > int,X2: rat] : ( member_int @ ( F @ X2 ) @ ( image_rat_int @ F @ top_top_set_rat ) ) ).
% rangeI
thf(fact_946_rangeI,axiom,
! [F: nat > rat,X2: nat] : ( member_rat @ ( F @ X2 ) @ ( image_nat_rat @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_947_rangeI,axiom,
! [F: nat > $o,X2: nat] : ( member_o @ ( F @ X2 ) @ ( image_nat_o @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_948_rangeI,axiom,
! [F: nat > nat,X2: nat] : ( member_nat @ ( F @ X2 ) @ ( image_nat_nat @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_949_rangeI,axiom,
! [F: nat > int,X2: nat] : ( member_int @ ( F @ X2 ) @ ( image_nat_int @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_950_surj__def,axiom,
! [F: char > char] :
( ( ( image_char_char @ F @ top_top_set_char )
= top_top_set_char )
= ( ! [Y5: char] :
? [X4: char] :
( Y5
= ( F @ X4 ) ) ) ) ).
% surj_def
thf(fact_951_surj__def,axiom,
! [F: char > rat] :
( ( ( image_char_rat @ F @ top_top_set_char )
= top_top_set_rat )
= ( ! [Y5: rat] :
? [X4: char] :
( Y5
= ( F @ X4 ) ) ) ) ).
% surj_def
thf(fact_952_surj__def,axiom,
! [F: char > nat] :
( ( ( image_char_nat @ F @ top_top_set_char )
= top_top_set_nat )
= ( ! [Y5: nat] :
? [X4: char] :
( Y5
= ( F @ X4 ) ) ) ) ).
% surj_def
thf(fact_953_surj__def,axiom,
! [F: char > int] :
( ( ( image_char_int @ F @ top_top_set_char )
= top_top_set_int )
= ( ! [Y5: int] :
? [X4: char] :
( Y5
= ( F @ X4 ) ) ) ) ).
% surj_def
thf(fact_954_surj__def,axiom,
! [F: rat > char] :
( ( ( image_rat_char @ F @ top_top_set_rat )
= top_top_set_char )
= ( ! [Y5: char] :
? [X4: rat] :
( Y5
= ( F @ X4 ) ) ) ) ).
% surj_def
thf(fact_955_surj__def,axiom,
! [F: rat > rat] :
( ( ( image_rat_rat @ F @ top_top_set_rat )
= top_top_set_rat )
= ( ! [Y5: rat] :
? [X4: rat] :
( Y5
= ( F @ X4 ) ) ) ) ).
% surj_def
thf(fact_956_surj__def,axiom,
! [F: rat > nat] :
( ( ( image_rat_nat @ F @ top_top_set_rat )
= top_top_set_nat )
= ( ! [Y5: nat] :
? [X4: rat] :
( Y5
= ( F @ X4 ) ) ) ) ).
% surj_def
thf(fact_957_surj__def,axiom,
! [F: rat > int] :
( ( ( image_rat_int @ F @ top_top_set_rat )
= top_top_set_int )
= ( ! [Y5: int] :
? [X4: rat] :
( Y5
= ( F @ X4 ) ) ) ) ).
% surj_def
thf(fact_958_surj__def,axiom,
! [F: nat > char] :
( ( ( image_nat_char @ F @ top_top_set_nat )
= top_top_set_char )
= ( ! [Y5: char] :
? [X4: nat] :
( Y5
= ( F @ X4 ) ) ) ) ).
% surj_def
thf(fact_959_surj__def,axiom,
! [F: nat > rat] :
( ( ( image_nat_rat @ F @ top_top_set_nat )
= top_top_set_rat )
= ( ! [Y5: rat] :
? [X4: nat] :
( Y5
= ( F @ X4 ) ) ) ) ).
% surj_def
thf(fact_960_range__eqI,axiom,
! [B: $o,F: char > $o,X2: char] :
( ( B
= ( F @ X2 ) )
=> ( member_o @ B @ ( image_char_o @ F @ top_top_set_char ) ) ) ).
% range_eqI
thf(fact_961_range__eqI,axiom,
! [B: nat,F: char > nat,X2: char] :
( ( B
= ( F @ X2 ) )
=> ( member_nat @ B @ ( image_char_nat @ F @ top_top_set_char ) ) ) ).
% range_eqI
thf(fact_962_range__eqI,axiom,
! [B: int,F: char > int,X2: char] :
( ( B
= ( F @ X2 ) )
=> ( member_int @ B @ ( image_char_int @ F @ top_top_set_char ) ) ) ).
% range_eqI
thf(fact_963_range__eqI,axiom,
! [B: $o,F: rat > $o,X2: rat] :
( ( B
= ( F @ X2 ) )
=> ( member_o @ B @ ( image_rat_o @ F @ top_top_set_rat ) ) ) ).
% range_eqI
thf(fact_964_range__eqI,axiom,
! [B: nat,F: rat > nat,X2: rat] :
( ( B
= ( F @ X2 ) )
=> ( member_nat @ B @ ( image_rat_nat @ F @ top_top_set_rat ) ) ) ).
% range_eqI
thf(fact_965_range__eqI,axiom,
! [B: int,F: rat > int,X2: rat] :
( ( B
= ( F @ X2 ) )
=> ( member_int @ B @ ( image_rat_int @ F @ top_top_set_rat ) ) ) ).
% range_eqI
thf(fact_966_range__eqI,axiom,
! [B: rat,F: nat > rat,X2: nat] :
( ( B
= ( F @ X2 ) )
=> ( member_rat @ B @ ( image_nat_rat @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_967_range__eqI,axiom,
! [B: $o,F: nat > $o,X2: nat] :
( ( B
= ( F @ X2 ) )
=> ( member_o @ B @ ( image_nat_o @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_968_range__eqI,axiom,
! [B: nat,F: nat > nat,X2: nat] :
( ( B
= ( F @ X2 ) )
=> ( member_nat @ B @ ( image_nat_nat @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_969_range__eqI,axiom,
! [B: int,F: nat > int,X2: nat] :
( ( B
= ( F @ X2 ) )
=> ( member_int @ B @ ( image_nat_int @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_970_effect__success,axiom,
! [C: heap_Heap_ref_a,H: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,R: ref_a] :
( ( heap_effect_ref_a @ C @ H @ H2 @ R )
=> ( heap_success_ref_a @ C @ H ) ) ).
% effect_success
thf(fact_971_success__effectE,axiom,
! [C: heap_Heap_ref_a,H: heap_e7401611519738050253t_unit] :
( ( heap_success_ref_a @ C @ H )
=> ~ ! [R4: ref_a,H4: heap_e7401611519738050253t_unit] :
~ ( heap_effect_ref_a @ C @ H @ H4 @ R4 ) ) ).
% success_effectE
thf(fact_972_map__prod__surj,axiom,
! [F: char > char,G: char > char] :
( ( ( image_char_char @ F @ top_top_set_char )
= top_top_set_char )
=> ( ( ( image_char_char @ G @ top_top_set_char )
= top_top_set_char )
=> ( ( image_7676512070627423941r_char @ ( produc7830753070125841726r_char @ F @ G ) @ top_to1235241553633177651r_char )
= top_to1235241553633177651r_char ) ) ) ).
% map_prod_surj
thf(fact_973_map__prod__surj,axiom,
! [F: char > char,G: char > rat] :
( ( ( image_char_char @ F @ top_top_set_char )
= top_top_set_char )
=> ( ( ( image_char_rat @ G @ top_top_set_char )
= top_top_set_rat )
=> ( ( image_7275072250160737868ar_rat @ ( produc5209943433214700613ar_rat @ F @ G ) @ top_to1235241553633177651r_char )
= top_to8437673547576850234ar_rat ) ) ) ).
% map_prod_surj
thf(fact_974_map__prod__surj,axiom,
! [F: char > char,G: char > nat] :
( ( ( image_char_char @ F @ top_top_set_char )
= top_top_set_char )
=> ( ( ( image_char_nat @ G @ top_top_set_char )
= top_top_set_nat )
=> ( ( image_1559928771699358788ar_nat @ ( produc5845073493301196349ar_nat @ F @ G ) @ top_to1235241553633177651r_char )
= top_to5594287848199751986ar_nat ) ) ) ).
% map_prod_surj
thf(fact_975_map__prod__surj,axiom,
! [F: char > char,G: char > int] :
( ( ( image_char_char @ F @ top_top_set_char )
= top_top_set_char )
=> ( ( ( image_char_int @ G @ top_top_set_char )
= top_top_set_int )
=> ( ( image_6605449789044937888ar_int @ ( produc5842583022792146073ar_int @ F @ G ) @ top_to1235241553633177651r_char )
= top_to3104204703268025230ar_int ) ) ) ).
% map_prod_surj
thf(fact_976_map__prod__surj,axiom,
! [F: char > char,G: rat > char] :
( ( ( image_char_char @ F @ top_top_set_char )
= top_top_set_char )
=> ( ( ( image_rat_char @ G @ top_top_set_rat )
= top_top_set_char )
=> ( ( image_8036680598759533694r_char @ ( produc5432746246116964165t_char @ F @ G ) @ top_to8437673547576850234ar_rat )
= top_to1235241553633177651r_char ) ) ) ).
% map_prod_surj
thf(fact_977_map__prod__surj,axiom,
! [F: char > char,G: rat > rat] :
( ( ( image_char_char @ F @ top_top_set_char )
= top_top_set_char )
=> ( ( ( image_rat_rat @ G @ top_top_set_rat )
= top_top_set_rat )
=> ( ( image_5644637116751559941ar_rat @ ( produc1230515432254573900at_rat @ F @ G ) @ top_to8437673547576850234ar_rat )
= top_to8437673547576850234ar_rat ) ) ) ).
% map_prod_surj
thf(fact_978_map__prod__surj,axiom,
! [F: char > char,G: rat > nat] :
( ( ( image_char_char @ F @ top_top_set_char )
= top_top_set_char )
=> ( ( ( image_rat_nat @ G @ top_top_set_rat )
= top_top_set_nat )
=> ( ( image_9152865675144956669ar_nat @ ( produc1865645492341069636at_nat @ F @ G ) @ top_to8437673547576850234ar_rat )
= top_to5594287848199751986ar_nat ) ) ) ).
% map_prod_surj
thf(fact_979_map__prod__surj,axiom,
! [F: char > char,G: rat > int] :
( ( ( image_char_char @ F @ top_top_set_char )
= top_top_set_char )
=> ( ( ( image_rat_int @ G @ top_top_set_rat )
= top_top_set_int )
=> ( ( image_4975014655635759961ar_int @ ( produc1863155021832019360at_int @ F @ G ) @ top_to8437673547576850234ar_rat )
= top_to3104204703268025230ar_int ) ) ) ).
% map_prod_surj
thf(fact_980_map__prod__surj,axiom,
! [F: char > char,G: nat > char] :
( ( ( image_char_char @ F @ top_top_set_char )
= top_top_set_char )
=> ( ( ( image_nat_char @ G @ top_top_set_nat )
= top_top_set_char )
=> ( ( image_5973708714899544198r_char @ ( produc8759894717825771837t_char @ F @ G ) @ top_to5594287848199751986ar_nat )
= top_to1235241553633177651r_char ) ) ) ).
% map_prod_surj
thf(fact_981_map__prod__surj,axiom,
! [F: char > char,G: nat > rat] :
( ( ( image_char_char @ F @ top_top_set_char )
= top_top_set_char )
=> ( ( ( image_nat_rat @ G @ top_top_set_nat )
= top_top_set_rat )
=> ( ( image_2699825501378146573ar_rat @ ( produc1547070076189439812at_rat @ F @ G ) @ top_to5594287848199751986ar_nat )
= top_to8437673547576850234ar_rat ) ) ) ).
% map_prod_surj
thf(fact_982_GrD1,axiom,
! [X2: set_Pr4329608150637261639at_nat,Fx: set_Pr4329608150637261639at_nat,A4: set_se5479201105039950375at_nat,F: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat] :
( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X2 @ Fx ) @ ( bNF_Gr7888059597227825506at_nat @ A4 @ F ) )
=> ( member6273874620674182256at_nat @ X2 @ A4 ) ) ).
% GrD1
thf(fact_983_GrD1,axiom,
! [X2: set_Pr1261947904930325089at_nat,Fx: set_Pr1261947904930325089at_nat,A4: set_se7855581050983116737at_nat,F: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat] :
( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Fx ) @ ( bNF_Gr1527221351080490466at_nat @ A4 @ F ) )
=> ( member2643936169264416010at_nat @ X2 @ A4 ) ) ).
% GrD1
thf(fact_984_GrD1,axiom,
! [X2: nat,Fx: num,A4: set_nat,F: nat > num] :
( ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X2 @ Fx ) @ ( bNF_Gr_nat_num @ A4 @ F ) )
=> ( member_nat @ X2 @ A4 ) ) ).
% GrD1
thf(fact_985_GrD1,axiom,
! [X2: nat,Fx: nat,A4: set_nat,F: nat > nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Fx ) @ ( bNF_Gr_nat_nat @ A4 @ F ) )
=> ( member_nat @ X2 @ A4 ) ) ).
% GrD1
thf(fact_986_GrD1,axiom,
! [X2: int,Fx: int,A4: set_int,F: int > int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Fx ) @ ( bNF_Gr_int_int @ A4 @ F ) )
=> ( member_int @ X2 @ A4 ) ) ).
% GrD1
thf(fact_987_GrD2,axiom,
! [X2: set_Pr4329608150637261639at_nat,Fx: set_Pr4329608150637261639at_nat,A4: set_se5479201105039950375at_nat,F: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat] :
( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X2 @ Fx ) @ ( bNF_Gr7888059597227825506at_nat @ A4 @ F ) )
=> ( ( F @ X2 )
= Fx ) ) ).
% GrD2
thf(fact_988_GrD2,axiom,
! [X2: set_Pr1261947904930325089at_nat,Fx: set_Pr1261947904930325089at_nat,A4: set_se7855581050983116737at_nat,F: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat] :
( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Fx ) @ ( bNF_Gr1527221351080490466at_nat @ A4 @ F ) )
=> ( ( F @ X2 )
= Fx ) ) ).
% GrD2
thf(fact_989_GrD2,axiom,
! [X2: nat,Fx: num,A4: set_nat,F: nat > num] :
( ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X2 @ Fx ) @ ( bNF_Gr_nat_num @ A4 @ F ) )
=> ( ( F @ X2 )
= Fx ) ) ).
% GrD2
thf(fact_990_GrD2,axiom,
! [X2: nat,Fx: nat,A4: set_nat,F: nat > nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Fx ) @ ( bNF_Gr_nat_nat @ A4 @ F ) )
=> ( ( F @ X2 )
= Fx ) ) ).
% GrD2
thf(fact_991_GrD2,axiom,
! [X2: int,Fx: int,A4: set_int,F: int > int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Fx ) @ ( bNF_Gr_int_int @ A4 @ F ) )
=> ( ( F @ X2 )
= Fx ) ) ).
% GrD2
thf(fact_992_success__refI,axiom,
! [V: a,H: heap_e7401611519738050253t_unit] : ( heap_success_ref_a @ ( ref_a2 @ V ) @ H ) ).
% success_refI
thf(fact_993_fun_Orel__cong,axiom,
! [X2: char > $o,Ya: char > $o,Y3: char > $o,Xa2: char > $o,R3: $o > $o > $o,Ra: $o > $o > $o] :
( ( X2 = Ya )
=> ( ( Y3 = Xa2 )
=> ( ! [Z3: $o,Yb: $o] :
( ( member_o @ Z3 @ ( image_char_o @ Ya @ top_top_set_char ) )
=> ( ( member_o @ Yb @ ( image_char_o @ Xa2 @ top_top_set_char ) )
=> ( ( R3 @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( bNF_re1692451286490762625ar_o_o
@ ^ [Y2: char,Z: char] : Y2 = Z
@ R3
@ X2
@ Y3 )
= ( bNF_re1692451286490762625ar_o_o
@ ^ [Y2: char,Z: char] : Y2 = Z
@ Ra
@ Ya
@ Xa2 ) ) ) ) ) ).
% fun.rel_cong
thf(fact_994_fun_Orel__cong,axiom,
! [X2: char > $o,Ya: char > $o,Y3: char > nat,Xa2: char > nat,R3: $o > nat > $o,Ra: $o > nat > $o] :
( ( X2 = Ya )
=> ( ( Y3 = Xa2 )
=> ( ! [Z3: $o,Yb: nat] :
( ( member_o @ Z3 @ ( image_char_o @ Ya @ top_top_set_char ) )
=> ( ( member_nat @ Yb @ ( image_char_nat @ Xa2 @ top_top_set_char ) )
=> ( ( R3 @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( bNF_re4782998131235581479_o_nat
@ ^ [Y2: char,Z: char] : Y2 = Z
@ R3
@ X2
@ Y3 )
= ( bNF_re4782998131235581479_o_nat
@ ^ [Y2: char,Z: char] : Y2 = Z
@ Ra
@ Ya
@ Xa2 ) ) ) ) ) ).
% fun.rel_cong
thf(fact_995_fun_Orel__cong,axiom,
! [X2: char > $o,Ya: char > $o,Y3: char > int,Xa2: char > int,R3: $o > int > $o,Ra: $o > int > $o] :
( ( X2 = Ya )
=> ( ( Y3 = Xa2 )
=> ( ! [Z3: $o,Yb: int] :
( ( member_o @ Z3 @ ( image_char_o @ Ya @ top_top_set_char ) )
=> ( ( member_int @ Yb @ ( image_char_int @ Xa2 @ top_top_set_char ) )
=> ( ( R3 @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( bNF_re4780507660726531203_o_int
@ ^ [Y2: char,Z: char] : Y2 = Z
@ R3
@ X2
@ Y3 )
= ( bNF_re4780507660726531203_o_int
@ ^ [Y2: char,Z: char] : Y2 = Z
@ Ra
@ Ya
@ Xa2 ) ) ) ) ) ).
% fun.rel_cong
thf(fact_996_fun_Orel__cong,axiom,
! [X2: char > nat,Ya: char > nat,Y3: char > $o,Xa2: char > $o,R3: nat > $o > $o,Ra: nat > $o > $o] :
( ( X2 = Ya )
=> ( ( Y3 = Xa2 )
=> ( ! [Z3: nat,Yb: $o] :
( ( member_nat @ Z3 @ ( image_char_nat @ Ya @ top_top_set_char ) )
=> ( ( member_o @ Yb @ ( image_char_o @ Xa2 @ top_top_set_char ) )
=> ( ( R3 @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( bNF_re7885669492343766789_nat_o
@ ^ [Y2: char,Z: char] : Y2 = Z
@ R3
@ X2
@ Y3 )
= ( bNF_re7885669492343766789_nat_o
@ ^ [Y2: char,Z: char] : Y2 = Z
@ Ra
@ Ya
@ Xa2 ) ) ) ) ) ).
% fun.rel_cong
thf(fact_997_fun_Orel__cong,axiom,
! [X2: char > nat,Ya: char > nat,Y3: char > nat,Xa2: char > nat,R3: nat > nat > $o,Ra: nat > nat > $o] :
( ( X2 = Ya )
=> ( ( Y3 = Xa2 )
=> ( ! [Z3: nat,Yb: nat] :
( ( member_nat @ Z3 @ ( image_char_nat @ Ya @ top_top_set_char ) )
=> ( ( member_nat @ Yb @ ( image_char_nat @ Xa2 @ top_top_set_char ) )
=> ( ( R3 @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( bNF_re3734335887435638179at_nat
@ ^ [Y2: char,Z: char] : Y2 = Z
@ R3
@ X2
@ Y3 )
= ( bNF_re3734335887435638179at_nat
@ ^ [Y2: char,Z: char] : Y2 = Z
@ Ra
@ Ya
@ Xa2 ) ) ) ) ) ).
% fun.rel_cong
thf(fact_998_fun_Orel__cong,axiom,
! [X2: char > nat,Ya: char > nat,Y3: char > int,Xa2: char > int,R3: nat > int > $o,Ra: nat > int > $o] :
( ( X2 = Ya )
=> ( ( Y3 = Xa2 )
=> ( ! [Z3: nat,Yb: int] :
( ( member_nat @ Z3 @ ( image_char_nat @ Ya @ top_top_set_char ) )
=> ( ( member_int @ Yb @ ( image_char_int @ Xa2 @ top_top_set_char ) )
=> ( ( R3 @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( bNF_re3731845416926587903at_int
@ ^ [Y2: char,Z: char] : Y2 = Z
@ R3
@ X2
@ Y3 )
= ( bNF_re3731845416926587903at_int
@ ^ [Y2: char,Z: char] : Y2 = Z
@ Ra
@ Ya
@ Xa2 ) ) ) ) ) ).
% fun.rel_cong
thf(fact_999_fun_Orel__cong,axiom,
! [X2: char > int,Ya: char > int,Y3: char > $o,Xa2: char > $o,R3: int > $o > $o,Ra: int > $o > $o] :
( ( X2 = Ya )
=> ( ( Y3 = Xa2 )
=> ( ! [Z3: int,Yb: $o] :
( ( member_int @ Z3 @ ( image_char_int @ Ya @ top_top_set_char ) )
=> ( ( member_o @ Yb @ ( image_char_o @ Xa2 @ top_top_set_char ) )
=> ( ( R3 @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( bNF_re6064627146911078825_int_o
@ ^ [Y2: char,Z: char] : Y2 = Z
@ R3
@ X2
@ Y3 )
= ( bNF_re6064627146911078825_int_o
@ ^ [Y2: char,Z: char] : Y2 = Z
@ Ra
@ Ya
@ Xa2 ) ) ) ) ) ).
% fun.rel_cong
thf(fact_1000_fun_Orel__cong,axiom,
! [X2: char > int,Ya: char > int,Y3: char > nat,Xa2: char > nat,R3: int > nat > $o,Ra: int > nat > $o] :
( ( X2 = Ya )
=> ( ( Y3 = Xa2 )
=> ( ! [Z3: int,Yb: nat] :
( ( member_int @ Z3 @ ( image_char_int @ Ya @ top_top_set_char ) )
=> ( ( member_nat @ Yb @ ( image_char_nat @ Xa2 @ top_top_set_char ) )
=> ( ( R3 @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( bNF_re4733689599336692735nt_nat
@ ^ [Y2: char,Z: char] : Y2 = Z
@ R3
@ X2
@ Y3 )
= ( bNF_re4733689599336692735nt_nat
@ ^ [Y2: char,Z: char] : Y2 = Z
@ Ra
@ Ya
@ Xa2 ) ) ) ) ) ).
% fun.rel_cong
thf(fact_1001_fun_Orel__cong,axiom,
! [X2: char > int,Ya: char > int,Y3: char > int,Xa2: char > int,R3: int > int > $o,Ra: int > int > $o] :
( ( X2 = Ya )
=> ( ( Y3 = Xa2 )
=> ( ! [Z3: int,Yb: int] :
( ( member_int @ Z3 @ ( image_char_int @ Ya @ top_top_set_char ) )
=> ( ( member_int @ Yb @ ( image_char_int @ Xa2 @ top_top_set_char ) )
=> ( ( R3 @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( bNF_re4731199128827642459nt_int
@ ^ [Y2: char,Z: char] : Y2 = Z
@ R3
@ X2
@ Y3 )
= ( bNF_re4731199128827642459nt_int
@ ^ [Y2: char,Z: char] : Y2 = Z
@ Ra
@ Ya
@ Xa2 ) ) ) ) ) ).
% fun.rel_cong
thf(fact_1002_fun_Orel__cong,axiom,
! [X2: rat > $o,Ya: rat > $o,Y3: rat > $o,Xa2: rat > $o,R3: $o > $o > $o,Ra: $o > $o > $o] :
( ( X2 = Ya )
=> ( ( Y3 = Xa2 )
=> ( ! [Z3: $o,Yb: $o] :
( ( member_o @ Z3 @ ( image_rat_o @ Ya @ top_top_set_rat ) )
=> ( ( member_o @ Yb @ ( image_rat_o @ Xa2 @ top_top_set_rat ) )
=> ( ( R3 @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( bNF_re1865807574169326223at_o_o
@ ^ [Y2: rat,Z: rat] : Y2 = Z
@ R3
@ X2
@ Y3 )
= ( bNF_re1865807574169326223at_o_o
@ ^ [Y2: rat,Z: rat] : Y2 = Z
@ Ra
@ Ya
@ Xa2 ) ) ) ) ) ).
% fun.rel_cong
thf(fact_1003_fun_Orel__mono__strong,axiom,
! [R3: $o > $o > $o,X2: char > $o,Y3: char > $o,Ra: $o > $o > $o] :
( ( bNF_re1692451286490762625ar_o_o
@ ^ [Y2: char,Z: char] : Y2 = Z
@ R3
@ X2
@ Y3 )
=> ( ! [Z3: $o,Yb: $o] :
( ( member_o @ Z3 @ ( image_char_o @ X2 @ top_top_set_char ) )
=> ( ( member_o @ Yb @ ( image_char_o @ Y3 @ top_top_set_char ) )
=> ( ( R3 @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( bNF_re1692451286490762625ar_o_o
@ ^ [Y2: char,Z: char] : Y2 = Z
@ Ra
@ X2
@ Y3 ) ) ) ).
% fun.rel_mono_strong
thf(fact_1004_fun_Orel__mono__strong,axiom,
! [R3: $o > nat > $o,X2: char > $o,Y3: char > nat,Ra: $o > nat > $o] :
( ( bNF_re4782998131235581479_o_nat
@ ^ [Y2: char,Z: char] : Y2 = Z
@ R3
@ X2
@ Y3 )
=> ( ! [Z3: $o,Yb: nat] :
( ( member_o @ Z3 @ ( image_char_o @ X2 @ top_top_set_char ) )
=> ( ( member_nat @ Yb @ ( image_char_nat @ Y3 @ top_top_set_char ) )
=> ( ( R3 @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( bNF_re4782998131235581479_o_nat
@ ^ [Y2: char,Z: char] : Y2 = Z
@ Ra
@ X2
@ Y3 ) ) ) ).
% fun.rel_mono_strong
thf(fact_1005_fun_Orel__mono__strong,axiom,
! [R3: $o > int > $o,X2: char > $o,Y3: char > int,Ra: $o > int > $o] :
( ( bNF_re4780507660726531203_o_int
@ ^ [Y2: char,Z: char] : Y2 = Z
@ R3
@ X2
@ Y3 )
=> ( ! [Z3: $o,Yb: int] :
( ( member_o @ Z3 @ ( image_char_o @ X2 @ top_top_set_char ) )
=> ( ( member_int @ Yb @ ( image_char_int @ Y3 @ top_top_set_char ) )
=> ( ( R3 @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( bNF_re4780507660726531203_o_int
@ ^ [Y2: char,Z: char] : Y2 = Z
@ Ra
@ X2
@ Y3 ) ) ) ).
% fun.rel_mono_strong
thf(fact_1006_fun_Orel__mono__strong,axiom,
! [R3: nat > $o > $o,X2: char > nat,Y3: char > $o,Ra: nat > $o > $o] :
( ( bNF_re7885669492343766789_nat_o
@ ^ [Y2: char,Z: char] : Y2 = Z
@ R3
@ X2
@ Y3 )
=> ( ! [Z3: nat,Yb: $o] :
( ( member_nat @ Z3 @ ( image_char_nat @ X2 @ top_top_set_char ) )
=> ( ( member_o @ Yb @ ( image_char_o @ Y3 @ top_top_set_char ) )
=> ( ( R3 @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( bNF_re7885669492343766789_nat_o
@ ^ [Y2: char,Z: char] : Y2 = Z
@ Ra
@ X2
@ Y3 ) ) ) ).
% fun.rel_mono_strong
thf(fact_1007_fun_Orel__mono__strong,axiom,
! [R3: nat > nat > $o,X2: char > nat,Y3: char > nat,Ra: nat > nat > $o] :
( ( bNF_re3734335887435638179at_nat
@ ^ [Y2: char,Z: char] : Y2 = Z
@ R3
@ X2
@ Y3 )
=> ( ! [Z3: nat,Yb: nat] :
( ( member_nat @ Z3 @ ( image_char_nat @ X2 @ top_top_set_char ) )
=> ( ( member_nat @ Yb @ ( image_char_nat @ Y3 @ top_top_set_char ) )
=> ( ( R3 @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( bNF_re3734335887435638179at_nat
@ ^ [Y2: char,Z: char] : Y2 = Z
@ Ra
@ X2
@ Y3 ) ) ) ).
% fun.rel_mono_strong
thf(fact_1008_fun_Orel__mono__strong,axiom,
! [R3: nat > int > $o,X2: char > nat,Y3: char > int,Ra: nat > int > $o] :
( ( bNF_re3731845416926587903at_int
@ ^ [Y2: char,Z: char] : Y2 = Z
@ R3
@ X2
@ Y3 )
=> ( ! [Z3: nat,Yb: int] :
( ( member_nat @ Z3 @ ( image_char_nat @ X2 @ top_top_set_char ) )
=> ( ( member_int @ Yb @ ( image_char_int @ Y3 @ top_top_set_char ) )
=> ( ( R3 @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( bNF_re3731845416926587903at_int
@ ^ [Y2: char,Z: char] : Y2 = Z
@ Ra
@ X2
@ Y3 ) ) ) ).
% fun.rel_mono_strong
thf(fact_1009_fun_Orel__mono__strong,axiom,
! [R3: int > $o > $o,X2: char > int,Y3: char > $o,Ra: int > $o > $o] :
( ( bNF_re6064627146911078825_int_o
@ ^ [Y2: char,Z: char] : Y2 = Z
@ R3
@ X2
@ Y3 )
=> ( ! [Z3: int,Yb: $o] :
( ( member_int @ Z3 @ ( image_char_int @ X2 @ top_top_set_char ) )
=> ( ( member_o @ Yb @ ( image_char_o @ Y3 @ top_top_set_char ) )
=> ( ( R3 @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( bNF_re6064627146911078825_int_o
@ ^ [Y2: char,Z: char] : Y2 = Z
@ Ra
@ X2
@ Y3 ) ) ) ).
% fun.rel_mono_strong
thf(fact_1010_fun_Orel__mono__strong,axiom,
! [R3: int > nat > $o,X2: char > int,Y3: char > nat,Ra: int > nat > $o] :
( ( bNF_re4733689599336692735nt_nat
@ ^ [Y2: char,Z: char] : Y2 = Z
@ R3
@ X2
@ Y3 )
=> ( ! [Z3: int,Yb: nat] :
( ( member_int @ Z3 @ ( image_char_int @ X2 @ top_top_set_char ) )
=> ( ( member_nat @ Yb @ ( image_char_nat @ Y3 @ top_top_set_char ) )
=> ( ( R3 @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( bNF_re4733689599336692735nt_nat
@ ^ [Y2: char,Z: char] : Y2 = Z
@ Ra
@ X2
@ Y3 ) ) ) ).
% fun.rel_mono_strong
thf(fact_1011_fun_Orel__mono__strong,axiom,
! [R3: int > int > $o,X2: char > int,Y3: char > int,Ra: int > int > $o] :
( ( bNF_re4731199128827642459nt_int
@ ^ [Y2: char,Z: char] : Y2 = Z
@ R3
@ X2
@ Y3 )
=> ( ! [Z3: int,Yb: int] :
( ( member_int @ Z3 @ ( image_char_int @ X2 @ top_top_set_char ) )
=> ( ( member_int @ Yb @ ( image_char_int @ Y3 @ top_top_set_char ) )
=> ( ( R3 @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( bNF_re4731199128827642459nt_int
@ ^ [Y2: char,Z: char] : Y2 = Z
@ Ra
@ X2
@ Y3 ) ) ) ).
% fun.rel_mono_strong
thf(fact_1012_fun_Orel__mono__strong,axiom,
! [R3: $o > $o > $o,X2: rat > $o,Y3: rat > $o,Ra: $o > $o > $o] :
( ( bNF_re1865807574169326223at_o_o
@ ^ [Y2: rat,Z: rat] : Y2 = Z
@ R3
@ X2
@ Y3 )
=> ( ! [Z3: $o,Yb: $o] :
( ( member_o @ Z3 @ ( image_rat_o @ X2 @ top_top_set_rat ) )
=> ( ( member_o @ Yb @ ( image_rat_o @ Y3 @ top_top_set_rat ) )
=> ( ( R3 @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( bNF_re1865807574169326223at_o_o
@ ^ [Y2: rat,Z: rat] : Y2 = Z
@ Ra
@ X2
@ Y3 ) ) ) ).
% fun.rel_mono_strong
thf(fact_1013_fun_Orel__refl__strong,axiom,
! [X2: char > $o,Ra: $o > $o > $o] :
( ! [Z3: $o] :
( ( member_o @ Z3 @ ( image_char_o @ X2 @ top_top_set_char ) )
=> ( Ra @ Z3 @ Z3 ) )
=> ( bNF_re1692451286490762625ar_o_o
@ ^ [Y2: char,Z: char] : Y2 = Z
@ Ra
@ X2
@ X2 ) ) ).
% fun.rel_refl_strong
thf(fact_1014_fun_Orel__refl__strong,axiom,
! [X2: char > nat,Ra: nat > nat > $o] :
( ! [Z3: nat] :
( ( member_nat @ Z3 @ ( image_char_nat @ X2 @ top_top_set_char ) )
=> ( Ra @ Z3 @ Z3 ) )
=> ( bNF_re3734335887435638179at_nat
@ ^ [Y2: char,Z: char] : Y2 = Z
@ Ra
@ X2
@ X2 ) ) ).
% fun.rel_refl_strong
thf(fact_1015_fun_Orel__refl__strong,axiom,
! [X2: char > int,Ra: int > int > $o] :
( ! [Z3: int] :
( ( member_int @ Z3 @ ( image_char_int @ X2 @ top_top_set_char ) )
=> ( Ra @ Z3 @ Z3 ) )
=> ( bNF_re4731199128827642459nt_int
@ ^ [Y2: char,Z: char] : Y2 = Z
@ Ra
@ X2
@ X2 ) ) ).
% fun.rel_refl_strong
thf(fact_1016_fun_Orel__refl__strong,axiom,
! [X2: rat > $o,Ra: $o > $o > $o] :
( ! [Z3: $o] :
( ( member_o @ Z3 @ ( image_rat_o @ X2 @ top_top_set_rat ) )
=> ( Ra @ Z3 @ Z3 ) )
=> ( bNF_re1865807574169326223at_o_o
@ ^ [Y2: rat,Z: rat] : Y2 = Z
@ Ra
@ X2
@ X2 ) ) ).
% fun.rel_refl_strong
thf(fact_1017_fun_Orel__refl__strong,axiom,
! [X2: rat > nat,Ra: nat > nat > $o] :
( ! [Z3: nat] :
( ( member_nat @ Z3 @ ( image_rat_nat @ X2 @ top_top_set_rat ) )
=> ( Ra @ Z3 @ Z3 ) )
=> ( bNF_re6557955094579809201at_nat
@ ^ [Y2: rat,Z: rat] : Y2 = Z
@ Ra
@ X2
@ X2 ) ) ).
% fun.rel_refl_strong
thf(fact_1018_fun_Orel__refl__strong,axiom,
! [X2: rat > int,Ra: int > int > $o] :
( ! [Z3: int] :
( ( member_int @ Z3 @ ( image_rat_int @ X2 @ top_top_set_rat ) )
=> ( Ra @ Z3 @ Z3 ) )
=> ( bNF_re7554818335971813481nt_int
@ ^ [Y2: rat,Z: rat] : Y2 = Z
@ Ra
@ X2
@ X2 ) ) ).
% fun.rel_refl_strong
thf(fact_1019_fun_Orel__refl__strong,axiom,
! [X2: nat > rat,Ra: rat > rat > $o] :
( ! [Z3: rat] :
( ( member_rat @ Z3 @ ( image_nat_rat @ X2 @ top_top_set_nat ) )
=> ( Ra @ Z3 @ Z3 ) )
=> ( bNF_re4702136315717946289at_rat
@ ^ [Y2: nat,Z: nat] : Y2 = Z
@ Ra
@ X2
@ X2 ) ) ).
% fun.rel_refl_strong
thf(fact_1020_fun_Orel__refl__strong,axiom,
! [X2: nat > $o,Ra: $o > $o > $o] :
( ! [Z3: $o] :
( ( member_o @ Z3 @ ( image_nat_o @ X2 @ top_top_set_nat ) )
=> ( Ra @ Z3 @ Z3 ) )
=> ( bNF_re4705727531993890431at_o_o
@ ^ [Y2: nat,Z: nat] : Y2 = Z
@ Ra
@ X2
@ X2 ) ) ).
% fun.rel_refl_strong
thf(fact_1021_fun_Orel__refl__strong,axiom,
! [X2: nat > nat,Ra: nat > nat > $o] :
( ! [Z3: nat] :
( ( member_nat @ Z3 @ ( image_nat_nat @ X2 @ top_top_set_nat ) )
=> ( Ra @ Z3 @ Z3 ) )
=> ( bNF_re5653821019739307937at_nat
@ ^ [Y2: nat,Z: nat] : Y2 = Z
@ Ra
@ X2
@ X2 ) ) ).
% fun.rel_refl_strong
thf(fact_1022_fun_Orel__refl__strong,axiom,
! [X2: nat > int,Ra: int > int > $o] :
( ! [Z3: int] :
( ( member_int @ Z3 @ ( image_nat_int @ X2 @ top_top_set_nat ) )
=> ( Ra @ Z3 @ Z3 ) )
=> ( bNF_re6650684261131312217nt_int
@ ^ [Y2: nat,Z: nat] : Y2 = Z
@ Ra
@ X2
@ X2 ) ) ).
% fun.rel_refl_strong
thf(fact_1023_range__subsetD,axiom,
! [F: char > $o,B5: set_o,I: char] :
( ( ord_less_eq_set_o @ ( image_char_o @ F @ top_top_set_char ) @ B5 )
=> ( member_o @ ( F @ I ) @ B5 ) ) ).
% range_subsetD
thf(fact_1024_range__subsetD,axiom,
! [F: char > nat,B5: set_nat,I: char] :
( ( ord_less_eq_set_nat @ ( image_char_nat @ F @ top_top_set_char ) @ B5 )
=> ( member_nat @ ( F @ I ) @ B5 ) ) ).
% range_subsetD
thf(fact_1025_range__subsetD,axiom,
! [F: char > int,B5: set_int,I: char] :
( ( ord_less_eq_set_int @ ( image_char_int @ F @ top_top_set_char ) @ B5 )
=> ( member_int @ ( F @ I ) @ B5 ) ) ).
% range_subsetD
thf(fact_1026_range__subsetD,axiom,
! [F: rat > $o,B5: set_o,I: rat] :
( ( ord_less_eq_set_o @ ( image_rat_o @ F @ top_top_set_rat ) @ B5 )
=> ( member_o @ ( F @ I ) @ B5 ) ) ).
% range_subsetD
thf(fact_1027_range__subsetD,axiom,
! [F: rat > nat,B5: set_nat,I: rat] :
( ( ord_less_eq_set_nat @ ( image_rat_nat @ F @ top_top_set_rat ) @ B5 )
=> ( member_nat @ ( F @ I ) @ B5 ) ) ).
% range_subsetD
thf(fact_1028_range__subsetD,axiom,
! [F: rat > int,B5: set_int,I: rat] :
( ( ord_less_eq_set_int @ ( image_rat_int @ F @ top_top_set_rat ) @ B5 )
=> ( member_int @ ( F @ I ) @ B5 ) ) ).
% range_subsetD
thf(fact_1029_range__subsetD,axiom,
! [F: nat > rat,B5: set_rat,I: nat] :
( ( ord_less_eq_set_rat @ ( image_nat_rat @ F @ top_top_set_nat ) @ B5 )
=> ( member_rat @ ( F @ I ) @ B5 ) ) ).
% range_subsetD
thf(fact_1030_range__subsetD,axiom,
! [F: nat > $o,B5: set_o,I: nat] :
( ( ord_less_eq_set_o @ ( image_nat_o @ F @ top_top_set_nat ) @ B5 )
=> ( member_o @ ( F @ I ) @ B5 ) ) ).
% range_subsetD
thf(fact_1031_range__subsetD,axiom,
! [F: nat > nat,B5: set_nat,I: nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ top_top_set_nat ) @ B5 )
=> ( member_nat @ ( F @ I ) @ B5 ) ) ).
% range_subsetD
thf(fact_1032_range__subsetD,axiom,
! [F: nat > int,B5: set_int,I: nat] :
( ( ord_less_eq_set_int @ ( image_nat_int @ F @ top_top_set_nat ) @ B5 )
=> ( member_int @ ( F @ I ) @ B5 ) ) ).
% range_subsetD
thf(fact_1033_fun_Oset__map,axiom,
! [F: nat > rat,V: char > nat] :
( ( image_char_rat @ ( comp_nat_rat_char @ F @ V ) @ top_top_set_char )
= ( image_nat_rat @ F @ ( image_char_nat @ V @ top_top_set_char ) ) ) ).
% fun.set_map
thf(fact_1034_fun_Oset__map,axiom,
! [F: nat > nat,V: char > nat] :
( ( image_char_nat @ ( comp_nat_nat_char @ F @ V ) @ top_top_set_char )
= ( image_nat_nat @ F @ ( image_char_nat @ V @ top_top_set_char ) ) ) ).
% fun.set_map
thf(fact_1035_fun_Oset__map,axiom,
! [F: nat > int,V: char > nat] :
( ( image_char_int @ ( comp_nat_int_char @ F @ V ) @ top_top_set_char )
= ( image_nat_int @ F @ ( image_char_nat @ V @ top_top_set_char ) ) ) ).
% fun.set_map
thf(fact_1036_fun_Oset__map,axiom,
! [F: int > nat,V: char > int] :
( ( image_char_nat @ ( comp_int_nat_char @ F @ V ) @ top_top_set_char )
= ( image_int_nat @ F @ ( image_char_int @ V @ top_top_set_char ) ) ) ).
% fun.set_map
thf(fact_1037_fun_Oset__map,axiom,
! [F: int > int,V: char > int] :
( ( image_char_int @ ( comp_int_int_char @ F @ V ) @ top_top_set_char )
= ( image_int_int @ F @ ( image_char_int @ V @ top_top_set_char ) ) ) ).
% fun.set_map
thf(fact_1038_fun_Oset__map,axiom,
! [F: nat > rat,V: rat > nat] :
( ( image_rat_rat @ ( comp_nat_rat_rat @ F @ V ) @ top_top_set_rat )
= ( image_nat_rat @ F @ ( image_rat_nat @ V @ top_top_set_rat ) ) ) ).
% fun.set_map
thf(fact_1039_fun_Oset__map,axiom,
! [F: nat > nat,V: rat > nat] :
( ( image_rat_nat @ ( comp_nat_nat_rat @ F @ V ) @ top_top_set_rat )
= ( image_nat_nat @ F @ ( image_rat_nat @ V @ top_top_set_rat ) ) ) ).
% fun.set_map
thf(fact_1040_fun_Oset__map,axiom,
! [F: nat > int,V: rat > nat] :
( ( image_rat_int @ ( comp_nat_int_rat @ F @ V ) @ top_top_set_rat )
= ( image_nat_int @ F @ ( image_rat_nat @ V @ top_top_set_rat ) ) ) ).
% fun.set_map
thf(fact_1041_fun_Oset__map,axiom,
! [F: int > nat,V: rat > int] :
( ( image_rat_nat @ ( comp_int_nat_rat @ F @ V ) @ top_top_set_rat )
= ( image_int_nat @ F @ ( image_rat_int @ V @ top_top_set_rat ) ) ) ).
% fun.set_map
thf(fact_1042_fun_Oset__map,axiom,
! [F: int > int,V: rat > int] :
( ( image_rat_int @ ( comp_int_int_rat @ F @ V ) @ top_top_set_rat )
= ( image_int_int @ F @ ( image_rat_int @ V @ top_top_set_rat ) ) ) ).
% fun.set_map
thf(fact_1043_fun_Omap__cong,axiom,
! [X2: code_integer > code_integer > code_integer,Ya: code_integer > code_integer > code_integer,F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger] :
( ( X2 = Ya )
=> ( ! [Z3: code_integer > code_integer] :
( ( member2790140394383616496nteger @ Z3 @ ( image_2177946750938214461nteger @ Ya @ top_to4645266643341252675nteger ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( comp_C8797469213163452608nteger @ F @ X2 )
= ( comp_C8797469213163452608nteger @ G @ Ya ) ) ) ) ).
% fun.map_cong
thf(fact_1044_fun_Omap__cong,axiom,
! [X2: nat > $o,Ya: nat > $o,F: $o > $o,G: $o > $o] :
( ( X2 = Ya )
=> ( ! [Z3: $o] :
( ( member_o @ Z3 @ ( image_nat_o @ Ya @ top_top_set_nat ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( comp_o_o_nat @ F @ X2 )
= ( comp_o_o_nat @ G @ Ya ) ) ) ) ).
% fun.map_cong
thf(fact_1045_fun_Omap__cong,axiom,
! [X2: nat > nat,Ya: nat > nat,F: nat > nat,G: nat > nat] :
( ( X2 = Ya )
=> ( ! [Z3: nat] :
( ( member_nat @ Z3 @ ( image_nat_nat @ Ya @ top_top_set_nat ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( comp_nat_nat_nat @ F @ X2 )
= ( comp_nat_nat_nat @ G @ Ya ) ) ) ) ).
% fun.map_cong
thf(fact_1046_fun_Omap__cong,axiom,
! [X2: int > nat,Ya: int > nat,F: nat > num,G: nat > num] :
( ( X2 = Ya )
=> ( ! [Z3: nat] :
( ( member_nat @ Z3 @ ( image_int_nat @ Ya @ top_top_set_int ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( comp_nat_num_int @ F @ X2 )
= ( comp_nat_num_int @ G @ Ya ) ) ) ) ).
% fun.map_cong
thf(fact_1047_fun_Omap__cong,axiom,
! [X2: int > int,Ya: int > int,F: int > nat,G: int > nat] :
( ( X2 = Ya )
=> ( ! [Z3: int] :
( ( member_int @ Z3 @ ( image_int_int @ Ya @ top_top_set_int ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( comp_int_nat_int @ F @ X2 )
= ( comp_int_nat_int @ G @ Ya ) ) ) ) ).
% fun.map_cong
thf(fact_1048_fun_Omap__cong0,axiom,
! [X2: code_integer > code_integer > code_integer,F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger] :
( ! [Z3: code_integer > code_integer] :
( ( member2790140394383616496nteger @ Z3 @ ( image_2177946750938214461nteger @ X2 @ top_to4645266643341252675nteger ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( comp_C8797469213163452608nteger @ F @ X2 )
= ( comp_C8797469213163452608nteger @ G @ X2 ) ) ) ).
% fun.map_cong0
thf(fact_1049_fun_Omap__cong0,axiom,
! [X2: nat > $o,F: $o > $o,G: $o > $o] :
( ! [Z3: $o] :
( ( member_o @ Z3 @ ( image_nat_o @ X2 @ top_top_set_nat ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( comp_o_o_nat @ F @ X2 )
= ( comp_o_o_nat @ G @ X2 ) ) ) ).
% fun.map_cong0
thf(fact_1050_fun_Omap__cong0,axiom,
! [X2: nat > nat,F: nat > nat,G: nat > nat] :
( ! [Z3: nat] :
( ( member_nat @ Z3 @ ( image_nat_nat @ X2 @ top_top_set_nat ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( comp_nat_nat_nat @ F @ X2 )
= ( comp_nat_nat_nat @ G @ X2 ) ) ) ).
% fun.map_cong0
thf(fact_1051_fun_Omap__cong0,axiom,
! [X2: int > nat,F: nat > num,G: nat > num] :
( ! [Z3: nat] :
( ( member_nat @ Z3 @ ( image_int_nat @ X2 @ top_top_set_int ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( comp_nat_num_int @ F @ X2 )
= ( comp_nat_num_int @ G @ X2 ) ) ) ).
% fun.map_cong0
thf(fact_1052_fun_Omap__cong0,axiom,
! [X2: int > int,F: int > nat,G: int > nat] :
( ! [Z3: int] :
( ( member_int @ Z3 @ ( image_int_int @ X2 @ top_top_set_int ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( comp_int_nat_int @ F @ X2 )
= ( comp_int_nat_int @ G @ X2 ) ) ) ).
% fun.map_cong0
thf(fact_1053_fun_Oinj__map__strong,axiom,
! [X2: code_integer > code_integer > code_integer,Xa2: code_integer > code_integer > code_integer,F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,Fa: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger] :
( ! [Z3: code_integer > code_integer,Za: code_integer > code_integer] :
( ( member2790140394383616496nteger @ Z3 @ ( image_2177946750938214461nteger @ X2 @ top_to4645266643341252675nteger ) )
=> ( ( member2790140394383616496nteger @ Za @ ( image_2177946750938214461nteger @ Xa2 @ top_to4645266643341252675nteger ) )
=> ( ( ( F @ Z3 )
= ( Fa @ Za ) )
=> ( Z3 = Za ) ) ) )
=> ( ( ( comp_C8797469213163452608nteger @ F @ X2 )
= ( comp_C8797469213163452608nteger @ Fa @ Xa2 ) )
=> ( X2 = Xa2 ) ) ) ).
% fun.inj_map_strong
thf(fact_1054_fun_Oinj__map__strong,axiom,
! [X2: nat > $o,Xa2: nat > $o,F: $o > $o,Fa: $o > $o] :
( ! [Z3: $o,Za: $o] :
( ( member_o @ Z3 @ ( image_nat_o @ X2 @ top_top_set_nat ) )
=> ( ( member_o @ Za @ ( image_nat_o @ Xa2 @ top_top_set_nat ) )
=> ( ( ( F @ Z3 )
= ( Fa @ Za ) )
=> ( Z3 = Za ) ) ) )
=> ( ( ( comp_o_o_nat @ F @ X2 )
= ( comp_o_o_nat @ Fa @ Xa2 ) )
=> ( X2 = Xa2 ) ) ) ).
% fun.inj_map_strong
thf(fact_1055_fun_Oinj__map__strong,axiom,
! [X2: nat > nat,Xa2: nat > nat,F: nat > nat,Fa: nat > nat] :
( ! [Z3: nat,Za: nat] :
( ( member_nat @ Z3 @ ( image_nat_nat @ X2 @ top_top_set_nat ) )
=> ( ( member_nat @ Za @ ( image_nat_nat @ Xa2 @ top_top_set_nat ) )
=> ( ( ( F @ Z3 )
= ( Fa @ Za ) )
=> ( Z3 = Za ) ) ) )
=> ( ( ( comp_nat_nat_nat @ F @ X2 )
= ( comp_nat_nat_nat @ Fa @ Xa2 ) )
=> ( X2 = Xa2 ) ) ) ).
% fun.inj_map_strong
thf(fact_1056_fun_Oinj__map__strong,axiom,
! [X2: int > nat,Xa2: int > nat,F: nat > num,Fa: nat > num] :
( ! [Z3: nat,Za: nat] :
( ( member_nat @ Z3 @ ( image_int_nat @ X2 @ top_top_set_int ) )
=> ( ( member_nat @ Za @ ( image_int_nat @ Xa2 @ top_top_set_int ) )
=> ( ( ( F @ Z3 )
= ( Fa @ Za ) )
=> ( Z3 = Za ) ) ) )
=> ( ( ( comp_nat_num_int @ F @ X2 )
= ( comp_nat_num_int @ Fa @ Xa2 ) )
=> ( X2 = Xa2 ) ) ) ).
% fun.inj_map_strong
thf(fact_1057_fun_Oinj__map__strong,axiom,
! [X2: int > int,Xa2: int > int,F: int > nat,Fa: int > nat] :
( ! [Z3: int,Za: int] :
( ( member_int @ Z3 @ ( image_int_int @ X2 @ top_top_set_int ) )
=> ( ( member_int @ Za @ ( image_int_int @ Xa2 @ top_top_set_int ) )
=> ( ( ( F @ Z3 )
= ( Fa @ Za ) )
=> ( Z3 = Za ) ) ) )
=> ( ( ( comp_int_nat_int @ F @ X2 )
= ( comp_int_nat_int @ Fa @ Xa2 ) )
=> ( X2 = Xa2 ) ) ) ).
% fun.inj_map_strong
thf(fact_1058_surj__fun__eq,axiom,
! [F: nat > $o,X5: set_nat,G1: $o > $o,G22: $o > $o] :
( ( ( image_nat_o @ F @ X5 )
= top_top_set_o )
=> ( ! [X: nat] :
( ( member_nat @ X @ X5 )
=> ( ( comp_o_o_nat @ G1 @ F @ X )
= ( comp_o_o_nat @ G22 @ F @ X ) ) )
=> ( G1 = G22 ) ) ) ).
% surj_fun_eq
thf(fact_1059_surj__fun__eq,axiom,
! [F: code_integer > code_integer > code_integer,X5: set_Code_integer,G1: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G22: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger] :
( ( ( image_2177946750938214461nteger @ F @ X5 )
= top_to3804862627464599871nteger )
=> ( ! [X: code_integer] :
( ( member_Code_integer @ X @ X5 )
=> ( ( comp_C8797469213163452608nteger @ G1 @ F @ X )
= ( comp_C8797469213163452608nteger @ G22 @ F @ X ) ) )
=> ( G1 = G22 ) ) ) ).
% surj_fun_eq
thf(fact_1060_surj__fun__eq,axiom,
! [F: nat > nat,X5: set_nat,G1: nat > nat,G22: nat > nat] :
( ( ( image_nat_nat @ F @ X5 )
= top_top_set_nat )
=> ( ! [X: nat] :
( ( member_nat @ X @ X5 )
=> ( ( comp_nat_nat_nat @ G1 @ F @ X )
= ( comp_nat_nat_nat @ G22 @ F @ X ) ) )
=> ( G1 = G22 ) ) ) ).
% surj_fun_eq
thf(fact_1061_surj__fun__eq,axiom,
! [F: int > nat,X5: set_int,G1: nat > num,G22: nat > num] :
( ( ( image_int_nat @ F @ X5 )
= top_top_set_nat )
=> ( ! [X: int] :
( ( member_int @ X @ X5 )
=> ( ( comp_nat_num_int @ G1 @ F @ X )
= ( comp_nat_num_int @ G22 @ F @ X ) ) )
=> ( G1 = G22 ) ) ) ).
% surj_fun_eq
thf(fact_1062_surj__fun__eq,axiom,
! [F: int > int,X5: set_int,G1: int > nat,G22: int > nat] :
( ( ( image_int_int @ F @ X5 )
= top_top_set_int )
=> ( ! [X: int] :
( ( member_int @ X @ X5 )
=> ( ( comp_int_nat_int @ G1 @ F @ X )
= ( comp_int_nat_int @ G22 @ F @ X ) ) )
=> ( G1 = G22 ) ) ) ).
% surj_fun_eq
thf(fact_1063_comp__surj,axiom,
! [F: char > char,G: char > char] :
( ( ( image_char_char @ F @ top_top_set_char )
= top_top_set_char )
=> ( ( ( image_char_char @ G @ top_top_set_char )
= top_top_set_char )
=> ( ( image_char_char @ ( comp_char_char_char @ G @ F ) @ top_top_set_char )
= top_top_set_char ) ) ) ).
% comp_surj
thf(fact_1064_comp__surj,axiom,
! [F: char > char,G: char > rat] :
( ( ( image_char_char @ F @ top_top_set_char )
= top_top_set_char )
=> ( ( ( image_char_rat @ G @ top_top_set_char )
= top_top_set_rat )
=> ( ( image_char_rat @ ( comp_char_rat_char @ G @ F ) @ top_top_set_char )
= top_top_set_rat ) ) ) ).
% comp_surj
thf(fact_1065_comp__surj,axiom,
! [F: char > char,G: char > nat] :
( ( ( image_char_char @ F @ top_top_set_char )
= top_top_set_char )
=> ( ( ( image_char_nat @ G @ top_top_set_char )
= top_top_set_nat )
=> ( ( image_char_nat @ ( comp_char_nat_char @ G @ F ) @ top_top_set_char )
= top_top_set_nat ) ) ) ).
% comp_surj
thf(fact_1066_comp__surj,axiom,
! [F: char > char,G: char > int] :
( ( ( image_char_char @ F @ top_top_set_char )
= top_top_set_char )
=> ( ( ( image_char_int @ G @ top_top_set_char )
= top_top_set_int )
=> ( ( image_char_int @ ( comp_char_int_char @ G @ F ) @ top_top_set_char )
= top_top_set_int ) ) ) ).
% comp_surj
thf(fact_1067_comp__surj,axiom,
! [F: char > rat,G: rat > char] :
( ( ( image_char_rat @ F @ top_top_set_char )
= top_top_set_rat )
=> ( ( ( image_rat_char @ G @ top_top_set_rat )
= top_top_set_char )
=> ( ( image_char_char @ ( comp_rat_char_char @ G @ F ) @ top_top_set_char )
= top_top_set_char ) ) ) ).
% comp_surj
thf(fact_1068_comp__surj,axiom,
! [F: char > rat,G: rat > rat] :
( ( ( image_char_rat @ F @ top_top_set_char )
= top_top_set_rat )
=> ( ( ( image_rat_rat @ G @ top_top_set_rat )
= top_top_set_rat )
=> ( ( image_char_rat @ ( comp_rat_rat_char @ G @ F ) @ top_top_set_char )
= top_top_set_rat ) ) ) ).
% comp_surj
thf(fact_1069_comp__surj,axiom,
! [F: char > rat,G: rat > nat] :
( ( ( image_char_rat @ F @ top_top_set_char )
= top_top_set_rat )
=> ( ( ( image_rat_nat @ G @ top_top_set_rat )
= top_top_set_nat )
=> ( ( image_char_nat @ ( comp_rat_nat_char @ G @ F ) @ top_top_set_char )
= top_top_set_nat ) ) ) ).
% comp_surj
thf(fact_1070_comp__surj,axiom,
! [F: char > rat,G: rat > int] :
( ( ( image_char_rat @ F @ top_top_set_char )
= top_top_set_rat )
=> ( ( ( image_rat_int @ G @ top_top_set_rat )
= top_top_set_int )
=> ( ( image_char_int @ ( comp_rat_int_char @ G @ F ) @ top_top_set_char )
= top_top_set_int ) ) ) ).
% comp_surj
thf(fact_1071_comp__surj,axiom,
! [F: char > nat,G: nat > char] :
( ( ( image_char_nat @ F @ top_top_set_char )
= top_top_set_nat )
=> ( ( ( image_nat_char @ G @ top_top_set_nat )
= top_top_set_char )
=> ( ( image_char_char @ ( comp_nat_char_char @ G @ F ) @ top_top_set_char )
= top_top_set_char ) ) ) ).
% comp_surj
thf(fact_1072_comp__surj,axiom,
! [F: char > nat,G: nat > rat] :
( ( ( image_char_nat @ F @ top_top_set_char )
= top_top_set_nat )
=> ( ( ( image_nat_rat @ G @ top_top_set_nat )
= top_top_set_rat )
=> ( ( image_char_rat @ ( comp_nat_rat_char @ G @ F ) @ top_top_set_char )
= top_top_set_rat ) ) ) ).
% comp_surj
thf(fact_1073_apsnd__def,axiom,
( produc6499014454317279255nteger
= ( produc1802105962441033582nteger @ id_Code_integer ) ) ).
% apsnd_def
thf(fact_1074_rel__fun__image2p,axiom,
! [R3: product_prod_nat_nat > int > $o,F: product_prod_nat_nat > nat,G: int > nat] : ( bNF_re4555766996558763186at_nat @ R3 @ ( bNF_Gr6955563364113978072nt_nat @ F @ G @ R3 ) @ F @ G ) ).
% rel_fun_image2p
thf(fact_1075_rel__fun__image2p,axiom,
! [R3: product_prod_nat_nat > int > $o,F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,G: int > int > int] : ( bNF_re7408651293131936558nt_int @ R3 @ ( bNF_Gr1354378531936274772nt_int @ F @ G @ R3 ) @ F @ G ) ).
% rel_fun_image2p
thf(fact_1076_rel__fun__image2p,axiom,
! [R3: product_prod_nat_nat > int > $o,F: product_prod_nat_nat > product_prod_nat_nat > $o,G: int > int > $o] : ( bNF_re717283939379294677_int_o @ R3 @ ( bNF_Gr364077372107685167_int_o @ F @ G @ R3 ) @ F @ G ) ).
% rel_fun_image2p
thf(fact_1077_rel__fun__image2p,axiom,
! [R3: int > code_integer > $o,F: int > num,G: code_integer > num] : ( bNF_re6718328864250387230um_num @ R3 @ ( bNF_Gr166583433173494904er_num @ F @ G @ R3 ) @ F @ G ) ).
% rel_fun_image2p
thf(fact_1078_rel__fun__image2p,axiom,
! [R3: int > int > $o,F: int > num,G: int > num] : ( bNF_re7626690874201225453um_num @ R3 @ ( bNF_Gr7020624548710275655nt_num @ F @ G @ R3 ) @ F @ G ) ).
% rel_fun_image2p
thf(fact_1079_comp__fun__idem__on_Ocomp__comp__fun__idem__on,axiom,
! [S5: set_Co726186806682698383nteger,F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer > code_integer,R3: set_Code_integer] :
( ( finite954763878425697269nteger @ S5 @ F )
=> ( ( ord_le348599935723844335nteger @ ( image_2177946750938214461nteger @ G @ top_to4645266643341252675nteger ) @ S5 )
=> ( finite917065715488314481nteger @ R3 @ ( comp_C8797469213163452608nteger @ F @ G ) ) ) ) ).
% comp_fun_idem_on.comp_comp_fun_idem_on
thf(fact_1080_comp__fun__commute__on_Ocomp__comp__fun__commute__on,axiom,
! [S5: set_Co726186806682698383nteger,F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer > code_integer,R3: set_Code_integer] :
( ( finite6756937295472016048nteger @ S5 @ F )
=> ( ( ord_le348599935723844335nteger @ ( image_2177946750938214461nteger @ G @ top_to4645266643341252675nteger ) @ S5 )
=> ( finite366418920864469932nteger @ R3 @ ( comp_C8797469213163452608nteger @ F @ G ) ) ) ) ).
% comp_fun_commute_on.comp_comp_fun_commute_on
thf(fact_1081_image__Pow__mono,axiom,
! [F: nat > rat,A4: set_nat,B5: set_rat] :
( ( ord_less_eq_set_rat @ ( image_nat_rat @ F @ A4 ) @ B5 )
=> ( ord_le513522071413781156et_rat @ ( image_4408659257933336347et_rat @ ( image_nat_rat @ F ) @ ( pow_nat @ A4 ) ) @ ( pow_rat @ B5 ) ) ) ).
% image_Pow_mono
thf(fact_1082_image__Pow__mono,axiom,
! [F: nat > nat,A4: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A4 ) @ B5 )
=> ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat @ F ) @ ( pow_nat @ A4 ) ) @ ( pow_nat @ B5 ) ) ) ).
% image_Pow_mono
thf(fact_1083_image__Pow__mono,axiom,
! [F: nat > int,A4: set_nat,B5: set_int] :
( ( ord_less_eq_set_int @ ( image_nat_int @ F @ A4 ) @ B5 )
=> ( ord_le4403425263959731960et_int @ ( image_3739036796817536367et_int @ ( image_nat_int @ F ) @ ( pow_nat @ A4 ) ) @ ( pow_int @ B5 ) ) ) ).
% image_Pow_mono
thf(fact_1084_image__Pow__mono,axiom,
! [F: int > nat,A4: set_int,B5: set_nat] :
( ( ord_less_eq_set_nat @ ( image_int_nat @ F @ A4 ) @ B5 )
=> ( ord_le6893508408891458716et_nat @ ( image_4702325430467532143et_nat @ ( image_int_nat @ F ) @ ( pow_int @ A4 ) ) @ ( pow_nat @ B5 ) ) ) ).
% image_Pow_mono
thf(fact_1085_image__Pow__mono,axiom,
! [F: int > int,A4: set_int,B5: set_int] :
( ( ord_less_eq_set_int @ ( image_int_int @ F @ A4 ) @ B5 )
=> ( ord_le4403425263959731960et_int @ ( image_524474410958335435et_int @ ( image_int_int @ F ) @ ( pow_int @ A4 ) ) @ ( pow_int @ B5 ) ) ) ).
% image_Pow_mono
thf(fact_1086_vimage2p__Grp,axiom,
( bNF_vi7262943168399475603_int_o
= ( ^ [F2: char > int,G3: code_integer > int,P4: int > int > $o] : ( relcom5819909836510906509nteger @ ( bNF_Grp_char_int @ top_top_set_char @ F2 ) @ ( relcom967497705283160040nteger @ P4 @ ( conver1462408618118490240er_int @ ( bNF_Gr6045950513751025625er_int @ top_to4645266643341252675nteger @ G3 ) ) ) ) ) ) ).
% vimage2p_Grp
thf(fact_1087_vimage2p__Grp,axiom,
( bNF_vi508398028314858188_int_o
= ( ^ [F2: rat > int,G3: code_integer > int,P4: int > int > $o] : ( relcom3409376902655092628nteger @ ( bNF_Grp_rat_int @ top_top_set_rat @ F2 ) @ ( relcom967497705283160040nteger @ P4 @ ( conver1462408618118490240er_int @ ( bNF_Gr6045950513751025625er_int @ top_to4645266643341252675nteger @ G3 ) ) ) ) ) ) ).
% vimage2p_Grp
thf(fact_1088_vimage2p__Grp,axiom,
( bNF_vi7331807916455492308_int_o
= ( ^ [F2: nat > int,G3: code_integer > int,P4: int > int > $o] : ( relcom3277315637376110988nteger @ ( bNF_Grp_nat_int @ top_top_set_nat @ F2 ) @ ( relcom967497705283160040nteger @ P4 @ ( conver1462408618118490240er_int @ ( bNF_Gr6045950513751025625er_int @ top_to4645266643341252675nteger @ G3 ) ) ) ) ) ) ).
% vimage2p_Grp
thf(fact_1089_vimage2p__Grp,axiom,
( bNF_vi2125572119366604152_int_o
= ( ^ [F2: int > int,G3: code_integer > int,P4: int > int > $o] : ( relcom967497705283160040nteger @ ( bNF_Grp_int_int @ top_top_set_int @ F2 ) @ ( relcom967497705283160040nteger @ P4 @ ( conver1462408618118490240er_int @ ( bNF_Gr6045950513751025625er_int @ top_to4645266643341252675nteger @ G3 ) ) ) ) ) ) ).
% vimage2p_Grp
thf(fact_1090_vimage2p__Grp,axiom,
( bNF_vi467646977814820164_int_o
= ( ^ [F2: list_nat > int,G3: code_integer > int,P4: int > int > $o] : ( relcom7887202796854296604nteger @ ( bNF_Grp_list_nat_int @ top_top_set_list_nat @ F2 ) @ ( relcom967497705283160040nteger @ P4 @ ( conver1462408618118490240er_int @ ( bNF_Gr6045950513751025625er_int @ top_to4645266643341252675nteger @ G3 ) ) ) ) ) ) ).
% vimage2p_Grp
thf(fact_1091_vimage2p__Grp,axiom,
( bNF_vi728936465572800120_nat_o
= ( ^ [F2: char > product_prod_nat_nat,G3: int > product_prod_nat_nat,P4: product_prod_nat_nat > product_prod_nat_nat > $o] : ( relcom7879471502873553579at_int @ ( bNF_Gr8982933507515605408at_nat @ top_top_set_char @ F2 ) @ ( relcom8911704206367525917at_int @ P4 @ ( conver1866313717161370910at_nat @ ( bNF_Gr6591314147588017797at_nat @ top_top_set_int @ G3 ) ) ) ) ) ) ).
% vimage2p_Grp
thf(fact_1092_vimage2p__Grp,axiom,
( bNF_vi2997284877290766385_nat_o
= ( ^ [F2: rat > product_prod_nat_nat,G3: int > product_prod_nat_nat,P4: product_prod_nat_nat > product_prod_nat_nat > $o] : ( relcom1211213461163567844at_int @ ( bNF_Gr6883301533820405721at_nat @ top_top_set_rat @ F2 ) @ ( relcom8911704206367525917at_int @ P4 @ ( conver1866313717161370910at_nat @ ( bNF_Gr6591314147588017797at_nat @ top_top_set_int @ G3 ) ) ) ) ) ) ).
% vimage2p_Grp
thf(fact_1093_vimage2p__Grp,axiom,
( bNF_vi190728152058600505_nat_o
= ( ^ [F2: nat > product_prod_nat_nat,G3: int > product_prod_nat_nat,P4: product_prod_nat_nat > product_prod_nat_nat > $o] : ( relcom8569543703982574316at_int @ ( bNF_Gr3553472063123884001at_nat @ top_top_set_nat @ F2 ) @ ( relcom8911704206367525917at_int @ P4 @ ( conver1866313717161370910at_nat @ ( bNF_Gr6591314147588017797at_nat @ top_top_set_int @ G3 ) ) ) ) ) ) ).
% vimage2p_Grp
thf(fact_1094_vimage2p__Grp,axiom,
( bNF_vi1204582724742824157_nat_o
= ( ^ [F2: int > product_prod_nat_nat,G3: int > product_prod_nat_nat,P4: product_prod_nat_nat > product_prod_nat_nat > $o] : ( relcom1106208330880909712at_int @ ( bNF_Gr6591314147588017797at_nat @ top_top_set_int @ F2 ) @ ( relcom8911704206367525917at_int @ P4 @ ( conver1866313717161370910at_nat @ ( bNF_Gr6591314147588017797at_nat @ top_top_set_int @ G3 ) ) ) ) ) ) ).
% vimage2p_Grp
thf(fact_1095_vimage2p__Grp,axiom,
( bNF_vi5202502398764838569_nat_o
= ( ^ [F2: list_nat > product_prod_nat_nat,G3: int > product_prod_nat_nat,P4: product_prod_nat_nat > product_prod_nat_nat > $o] : ( relcom170568170053003356at_int @ ( bNF_Gr4119388860389678161at_nat @ top_top_set_list_nat @ F2 ) @ ( relcom8911704206367525917at_int @ P4 @ ( conver1866313717161370910at_nat @ ( bNF_Gr6591314147588017797at_nat @ top_top_set_int @ G3 ) ) ) ) ) ) ).
% vimage2p_Grp
thf(fact_1096_fun_Oin__rel,axiom,
! [R3: nat > nat > $o,A: char > nat,B: char > nat] :
( ( bNF_re3734335887435638179at_nat
@ ^ [Y2: char,Z: char] : Y2 = Z
@ R3
@ A
@ B )
= ( ? [Z4: char > product_prod_nat_nat] :
( ( member3295615932019376976at_nat @ Z4
@ ( collec1683818095804863890at_nat
@ ^ [X4: char > product_prod_nat_nat] : ( ord_le3146513528884898305at_nat @ ( image_8382823969816519465at_nat @ X4 @ top_top_set_char ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R3 ) ) ) ) )
& ( ( comp_P1491501995297412481t_char @ product_fst_nat_nat @ Z4 )
= A )
& ( ( comp_P1491501995297412481t_char @ product_snd_nat_nat @ Z4 )
= B ) ) ) ) ).
% fun.in_rel
thf(fact_1097_fun_Oin__rel,axiom,
! [R3: nat > nat > $o,A: rat > nat,B: rat > nat] :
( ( bNF_re6557955094579809201at_nat
@ ^ [Y2: rat,Z: rat] : Y2 = Z
@ R3
@ A
@ B )
= ( ? [Z4: rat > product_prod_nat_nat] :
( ( member849810989832120969at_nat @ Z4
@ ( collec2824469931397938891at_nat
@ ^ [X4: rat > product_prod_nat_nat] : ( ord_le3146513528884898305at_nat @ ( image_9175953278516507234at_nat @ X4 @ top_top_set_rat ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R3 ) ) ) ) )
& ( ( comp_P1320480151300532104at_rat @ product_fst_nat_nat @ Z4 )
= A )
& ( ( comp_P1320480151300532104at_rat @ product_snd_nat_nat @ Z4 )
= B ) ) ) ) ).
% fun.in_rel
thf(fact_1098_fun_Oin__rel,axiom,
! [R3: nat > nat > $o,A: nat > nat,B: nat > nat] :
( ( bNF_re5653821019739307937at_nat
@ ^ [Y2: nat,Z: nat] : Y2 = Z
@ R3
@ A
@ B )
= ( ? [Z4: nat > product_prod_nat_nat] :
( ( member6743353555990375057at_nat @ Z4
@ ( collec8718012497556192979at_nat
@ ^ [X4: nat > product_prod_nat_nat] : ( ord_le3146513528884898305at_nat @ ( image_5846123807819985514at_nat @ X4 @ top_top_set_nat ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R3 ) ) ) ) )
& ( ( comp_P1955610211387027840at_nat @ product_fst_nat_nat @ Z4 )
= A )
& ( ( comp_P1955610211387027840at_nat @ product_snd_nat_nat @ Z4 )
= B ) ) ) ) ).
% fun.in_rel
thf(fact_1099_fun_Oin__rel,axiom,
! [R3: nat > nat > $o,A: int > nat,B: int > nat] :
( ( bNF_re3715656647883201625at_nat
@ ^ [Y2: int,Z: int] : Y2 = Z
@ R3
@ A
@ B )
= ( ? [Z4: int > product_prod_nat_nat] :
( ( member557823603599733045at_nat @ Z4
@ ( collec2532482545165550967at_nat
@ ^ [X4: int > product_prod_nat_nat] : ( ord_le3146513528884898305at_nat @ ( image_8883965892284119310at_nat @ X4 @ top_top_set_int ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R3 ) ) ) ) )
& ( ( comp_P1953119740877977564at_int @ product_fst_nat_nat @ Z4 )
= A )
& ( ( comp_P1953119740877977564at_int @ product_snd_nat_nat @ Z4 )
= B ) ) ) ) ).
% fun.in_rel
thf(fact_1100_fun_Oin__rel,axiom,
! [R3: num > num > $o,A: int > num,B: int > num] :
( ( bNF_re7626690874201225453um_num
@ ^ [Y2: int,Z: int] : Y2 = Z
@ R3
@ A
@ B )
= ( ? [Z4: int > product_prod_num_num] :
( ( member7514810323860029129um_num @ Z4
@ ( collec266097228571071243um_num
@ ^ [X4: int > product_prod_num_num] : ( ord_le880128212290418581um_num @ ( image_7722540232540426402um_num @ X4 @ top_top_set_int ) @ ( collec2230928802738392704um_num @ ( produc5703948589228662326_num_o @ R3 ) ) ) ) )
& ( ( comp_P4305060815297604026um_int @ product_fst_num_num @ Z4 )
= A )
& ( ( comp_P4305060815297604026um_int @ product_snd_num_num @ Z4 )
= B ) ) ) ) ).
% fun.in_rel
thf(fact_1101_fun_Oin__rel,axiom,
! [R3: int > int > $o,A: char > int,B: char > int] :
( ( bNF_re4731199128827642459nt_int
@ ^ [Y2: char,Z: char] : Y2 = Z
@ R3
@ A
@ B )
= ( ? [Z4: char > product_prod_int_int] :
( ( member2992454361780672008nt_int @ Z4
@ ( collec1380656525566158922nt_int
@ ^ [X4: char > product_prod_int_int] : ( ord_le2843351958646193337nt_int @ ( image_5204326662208377313nt_int @ X4 @ top_top_set_char ) @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ R3 ) ) ) ) )
& ( ( comp_P4161745170962941589t_char @ product_fst_int_int @ Z4 )
= A )
& ( ( comp_P4161745170962941589t_char @ product_snd_int_int @ Z4 )
= B ) ) ) ) ).
% fun.in_rel
thf(fact_1102_fun_Oin__rel,axiom,
! [R3: int > int > $o,A: rat > int,B: rat > int] :
( ( bNF_re7554818335971813481nt_int
@ ^ [Y2: rat,Z: rat] : Y2 = Z
@ R3
@ A
@ B )
= ( ? [Z4: rat > product_prod_int_int] :
( ( member546649419593416001nt_int @ Z4
@ ( collec2521308361159233923nt_int
@ ^ [X4: rat > product_prod_int_int] : ( ord_le2843351958646193337nt_int @ ( image_5997455970908365082nt_int @ X4 @ top_top_set_rat ) @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ R3 ) ) ) ) )
& ( ( comp_P6018825582104210588nt_rat @ product_fst_int_int @ Z4 )
= A )
& ( ( comp_P6018825582104210588nt_rat @ product_snd_int_int @ Z4 )
= B ) ) ) ) ).
% fun.in_rel
thf(fact_1103_fun_Oin__rel,axiom,
! [R3: int > int > $o,A: nat > int,B: nat > int] :
( ( bNF_re6650684261131312217nt_int
@ ^ [Y2: nat,Z: nat] : Y2 = Z
@ R3
@ A
@ B )
= ( ? [Z4: nat > product_prod_int_int] :
( ( member6440191985751670089nt_int @ Z4
@ ( collec8414850927317488011nt_int
@ ^ [X4: nat > product_prod_int_int] : ( ord_le2843351958646193337nt_int @ ( image_2667626500211843362nt_int @ X4 @ top_top_set_nat ) @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ R3 ) ) ) ) )
& ( ( comp_P6653955642190706324nt_nat @ product_fst_int_int @ Z4 )
= A )
& ( ( comp_P6653955642190706324nt_nat @ product_snd_int_int @ Z4 )
= B ) ) ) ) ).
% fun.in_rel
thf(fact_1104_fun_Oin__rel,axiom,
! [R3: int > int > $o,A: int > int,B: int > int] :
( ( bNF_re4712519889275205905nt_int
@ ^ [Y2: int,Z: int] : Y2 = Z
@ R3
@ A
@ B )
= ( ? [Z4: int > product_prod_int_int] :
( ( member254662033361028077nt_int @ Z4
@ ( collec2229320974926845999nt_int
@ ^ [X4: int > product_prod_int_int] : ( ord_le2843351958646193337nt_int @ ( image_5705468584675977158nt_int @ X4 @ top_top_set_int ) @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ R3 ) ) ) ) )
& ( ( comp_P6651465171681656048nt_int @ product_fst_int_int @ Z4 )
= A )
& ( ( comp_P6651465171681656048nt_int @ product_snd_int_int @ Z4 )
= B ) ) ) ) ).
% fun.in_rel
thf(fact_1105_fun_Oin__rel,axiom,
! [R3: nat > nat > $o,A: list_nat > nat,B: list_nat > nat] :
( ( bNF_re4629543205785817921at_nat
@ ^ [Y2: list_nat,Z: list_nat] : Y2 = Z
@ R3
@ A
@ B )
= ( ? [Z4: list_nat > product_prod_nat_nat] :
( ( member3963856813518191233at_nat @ Z4
@ ( collec3092956710985254339at_nat
@ ^ [X4: list_nat > product_prod_nat_nat] : ( ord_le3146513528884898305at_nat @ ( image_2795870212791037530at_nat @ X4 @ top_top_set_list_nat ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R3 ) ) ) ) )
& ( ( comp_P5292260908303971472st_nat @ product_fst_nat_nat @ Z4 )
= A )
& ( ( comp_P5292260908303971472st_nat @ product_snd_nat_nat @ Z4 )
= B ) ) ) ) ).
% fun.in_rel
thf(fact_1106_Grp__UNIV__id,axiom,
! [F: $o > $o] :
( ( F = id_o )
=> ( ( relcompp_o_o_o @ ( conversep_o_o @ ( bNF_Grp_o_o @ top_top_set_o @ F ) ) @ ( bNF_Grp_o_o @ top_top_set_o @ F ) )
= ( bNF_Grp_o_o @ top_top_set_o @ F ) ) ) ).
% Grp_UNIV_id
thf(fact_1107_Grp__UNIV__id,axiom,
! [F: num > num] :
( ( F = id_num )
=> ( ( relcompp_num_num_num @ ( conversep_num_num @ ( bNF_Grp_num_num @ top_top_set_num @ F ) ) @ ( bNF_Grp_num_num @ top_top_set_num @ F ) )
= ( bNF_Grp_num_num @ top_top_set_num @ F ) ) ) ).
% Grp_UNIV_id
thf(fact_1108_Grp__UNIV__id,axiom,
! [F: char > char] :
( ( F = id_char )
=> ( ( relcom5872928454965100966r_char @ ( conversep_char_char @ ( bNF_Grp_char_char @ top_top_set_char @ F ) ) @ ( bNF_Grp_char_char @ top_top_set_char @ F ) )
= ( bNF_Grp_char_char @ top_top_set_char @ F ) ) ) ).
% Grp_UNIV_id
thf(fact_1109_Grp__UNIV__id,axiom,
! [F: list_nat > list_nat] :
( ( F = id_list_nat )
=> ( ( relcom8030397737370142931st_nat @ ( conver909888920752868663st_nat @ ( bNF_Gr600725020315365264st_nat @ top_top_set_list_nat @ F ) ) @ ( bNF_Gr600725020315365264st_nat @ top_top_set_list_nat @ F ) )
= ( bNF_Gr600725020315365264st_nat @ top_top_set_list_nat @ F ) ) ) ).
% Grp_UNIV_id
thf(fact_1110_Grp__UNIV__id,axiom,
! [F: rat > rat] :
( ( F = id_rat )
=> ( ( relcompp_rat_rat_rat @ ( conversep_rat_rat @ ( bNF_Grp_rat_rat @ top_top_set_rat @ F ) ) @ ( bNF_Grp_rat_rat @ top_top_set_rat @ F ) )
= ( bNF_Grp_rat_rat @ top_top_set_rat @ F ) ) ) ).
% Grp_UNIV_id
thf(fact_1111_Grp__UNIV__id,axiom,
! [F: nat > nat] :
( ( F = id_nat )
=> ( ( relcompp_nat_nat_nat @ ( conversep_nat_nat @ ( bNF_Grp_nat_nat @ top_top_set_nat @ F ) ) @ ( bNF_Grp_nat_nat @ top_top_set_nat @ F ) )
= ( bNF_Grp_nat_nat @ top_top_set_nat @ F ) ) ) ).
% Grp_UNIV_id
thf(fact_1112_Grp__UNIV__id,axiom,
! [F: int > int] :
( ( F = id_int )
=> ( ( relcompp_int_int_int @ ( conversep_int_int @ ( bNF_Grp_int_int @ top_top_set_int @ F ) ) @ ( bNF_Grp_int_int @ top_top_set_int @ F ) )
= ( bNF_Grp_int_int @ top_top_set_int @ F ) ) ) ).
% Grp_UNIV_id
thf(fact_1113_strict__mono__inv,axiom,
! [F: rat > rat,G: rat > rat] :
( ( order_4774338944571509483at_rat @ F )
=> ( ( ( image_rat_rat @ F @ top_top_set_rat )
= top_top_set_rat )
=> ( ! [X: rat] :
( ( G @ ( F @ X ) )
= X )
=> ( order_4774338944571509483at_rat @ G ) ) ) ) ).
% strict_mono_inv
thf(fact_1114_strict__mono__inv,axiom,
! [F: rat > nat,G: nat > rat] :
( ( order_5409469004658005219at_nat @ F )
=> ( ( ( image_rat_nat @ F @ top_top_set_rat )
= top_top_set_nat )
=> ( ! [X: rat] :
( ( G @ ( F @ X ) )
= X )
=> ( order_5090893588506375395at_rat @ G ) ) ) ) ).
% strict_mono_inv
thf(fact_1115_strict__mono__inv,axiom,
! [F: rat > int,G: int > rat] :
( ( order_5406978534148954943at_int @ F )
=> ( ( ( image_rat_int @ F @ top_top_set_rat )
= top_top_set_int )
=> ( ! [X: rat] :
( ( G @ ( F @ X ) )
= X )
=> ( order_6090247300407429951nt_rat @ G ) ) ) ) ).
% strict_mono_inv
thf(fact_1116_strict__mono__inv,axiom,
! [F: nat > rat,G: rat > nat] :
( ( order_5090893588506375395at_rat @ F )
=> ( ( ( image_nat_rat @ F @ top_top_set_nat )
= top_top_set_rat )
=> ( ! [X: nat] :
( ( G @ ( F @ X ) )
= X )
=> ( order_5409469004658005219at_nat @ G ) ) ) ) ).
% strict_mono_inv
thf(fact_1117_strict__mono__inv,axiom,
! [F: nat > nat,G: nat > nat] :
( ( order_5726023648592871131at_nat @ F )
=> ( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( ! [X: nat] :
( ( G @ ( F @ X ) )
= X )
=> ( order_5726023648592871131at_nat @ G ) ) ) ) ).
% strict_mono_inv
thf(fact_1118_strict__mono__inv,axiom,
! [F: nat > int,G: int > nat] :
( ( order_5723533178083820855at_int @ F )
=> ( ( ( image_nat_int @ F @ top_top_set_nat )
= top_top_set_int )
=> ( ! [X: nat] :
( ( G @ ( F @ X ) )
= X )
=> ( order_6725377360493925687nt_nat @ G ) ) ) ) ).
% strict_mono_inv
thf(fact_1119_strict__mono__inv,axiom,
! [F: int > rat,G: rat > int] :
( ( order_6090247300407429951nt_rat @ F )
=> ( ( ( image_int_rat @ F @ top_top_set_int )
= top_top_set_rat )
=> ( ! [X: int] :
( ( G @ ( F @ X ) )
= X )
=> ( order_5406978534148954943at_int @ G ) ) ) ) ).
% strict_mono_inv
thf(fact_1120_strict__mono__inv,axiom,
! [F: int > nat,G: nat > int] :
( ( order_6725377360493925687nt_nat @ F )
=> ( ( ( image_int_nat @ F @ top_top_set_int )
= top_top_set_nat )
=> ( ! [X: int] :
( ( G @ ( F @ X ) )
= X )
=> ( order_5723533178083820855at_int @ G ) ) ) ) ).
% strict_mono_inv
thf(fact_1121_strict__mono__inv,axiom,
! [F: int > int,G: int > int] :
( ( order_6722886889984875411nt_int @ F )
=> ( ( ( image_int_int @ F @ top_top_set_int )
= top_top_set_int )
=> ( ! [X: int] :
( ( G @ ( F @ X ) )
= X )
=> ( order_6722886889984875411nt_int @ G ) ) ) ) ).
% strict_mono_inv
thf(fact_1122_Id__on__Gr,axiom,
( id_on_nat
= ( ^ [A6: set_nat] : ( bNF_Gr_nat_nat @ A6 @ id_nat ) ) ) ).
% Id_on_Gr
thf(fact_1123_Id__on__Gr,axiom,
( id_on_o
= ( ^ [A6: set_o] : ( bNF_Gr_o_o @ A6 @ id_o ) ) ) ).
% Id_on_Gr
thf(fact_1124_Id__on__Gr,axiom,
( id_on_num
= ( ^ [A6: set_num] : ( bNF_Gr_num_num @ A6 @ id_num ) ) ) ).
% Id_on_Gr
thf(fact_1125_type__copy__map__id0,axiom,
! [Rep: nat > $o,Abs: $o > nat,M: $o > $o] :
( ( type_d1000680970699258650_nat_o @ Rep @ Abs @ top_top_set_o )
=> ( ( M = id_o )
=> ( ( comp_o_nat_nat @ ( comp_o_nat_o @ Abs @ M ) @ Rep )
= id_nat ) ) ) ).
% type_copy_map_id0
thf(fact_1126_type__copy__map__id0,axiom,
! [Rep: $o > $o,Abs: $o > $o,M: $o > $o] :
( ( type_definition_o_o @ Rep @ Abs @ top_top_set_o )
=> ( ( M = id_o )
=> ( ( comp_o_o_o @ ( comp_o_o_o @ Abs @ M ) @ Rep )
= id_o ) ) ) ).
% type_copy_map_id0
thf(fact_1127_type__copy__map__id0,axiom,
! [Rep: num > $o,Abs: $o > num,M: $o > $o] :
( ( type_d9070036042389994064_num_o @ Rep @ Abs @ top_top_set_o )
=> ( ( M = id_o )
=> ( ( comp_o_num_num @ ( comp_o_num_o @ Abs @ M ) @ Rep )
= id_num ) ) ) ).
% type_copy_map_id0
thf(fact_1128_type__copy__map__id0,axiom,
! [Rep: nat > num,Abs: num > nat,M: num > num] :
( ( type_d2807825734933527320at_num @ Rep @ Abs @ top_top_set_num )
=> ( ( M = id_num )
=> ( ( comp_num_nat_nat @ ( comp_num_nat_num @ Abs @ M ) @ Rep )
= id_nat ) ) ) ).
% type_copy_map_id0
thf(fact_1129_type__copy__map__id0,axiom,
! [Rep: $o > num,Abs: num > $o,M: num > num] :
( ( type_d3678713432601627782_o_num @ Rep @ Abs @ top_top_set_num )
=> ( ( M = id_num )
=> ( ( comp_num_o_o @ ( comp_num_o_num @ Abs @ M ) @ Rep )
= id_o ) ) ) ).
% type_copy_map_id0
thf(fact_1130_type__copy__map__id0,axiom,
! [Rep: num > num,Abs: num > num,M: num > num] :
( ( type_d938156138240996706um_num @ Rep @ Abs @ top_top_set_num )
=> ( ( M = id_num )
=> ( ( comp_num_num_num @ ( comp_num_num_num @ Abs @ M ) @ Rep )
= id_num ) ) ) ).
% type_copy_map_id0
thf(fact_1131_type__copy__map__id0,axiom,
! [Rep: nat > char,Abs: char > nat,M: char > char] :
( ( type_d5945782884131516111t_char @ Rep @ Abs @ top_top_set_char )
=> ( ( M = id_char )
=> ( ( comp_char_nat_nat @ ( comp_char_nat_char @ Abs @ M ) @ Rep )
= id_nat ) ) ) ).
% type_copy_map_id0
thf(fact_1132_type__copy__map__id0,axiom,
! [Rep: $o > char,Abs: char > $o,M: char > char] :
( ( type_d9076672964931703357o_char @ Rep @ Abs @ top_top_set_char )
=> ( ( M = id_char )
=> ( ( comp_char_o_o @ ( comp_char_o_char @ Abs @ M ) @ Rep )
= id_o ) ) ) ).
% type_copy_map_id0
thf(fact_1133_type__copy__map__id0,axiom,
! [Rep: num > char,Abs: char > num,M: char > char] :
( ( type_d4396077731765706521m_char @ Rep @ Abs @ top_top_set_char )
=> ( ( M = id_char )
=> ( ( comp_char_num_num @ ( comp_char_num_char @ Abs @ M ) @ Rep )
= id_num ) ) ) ).
% type_copy_map_id0
thf(fact_1134_type__copy__map__id0,axiom,
! [Rep: nat > rat,Abs: rat > nat,M: rat > rat] :
( ( type_d5615363888691252950at_rat @ Rep @ Abs @ top_top_set_rat )
=> ( ( M = id_rat )
=> ( ( comp_rat_nat_nat @ ( comp_rat_nat_rat @ Abs @ M ) @ Rep )
= id_nat ) ) ) ).
% type_copy_map_id0
thf(fact_1135_type__copy__Abs__o__Rep,axiom,
! [Rep: nat > char,Abs: char > nat] :
( ( type_d5945782884131516111t_char @ Rep @ Abs @ top_top_set_char )
=> ( ( comp_char_nat_nat @ Abs @ Rep )
= id_nat ) ) ).
% type_copy_Abs_o_Rep
thf(fact_1136_type__copy__Abs__o__Rep,axiom,
! [Rep: $o > char,Abs: char > $o] :
( ( type_d9076672964931703357o_char @ Rep @ Abs @ top_top_set_char )
=> ( ( comp_char_o_o @ Abs @ Rep )
= id_o ) ) ).
% type_copy_Abs_o_Rep
thf(fact_1137_type__copy__Abs__o__Rep,axiom,
! [Rep: num > char,Abs: char > num] :
( ( type_d4396077731765706521m_char @ Rep @ Abs @ top_top_set_char )
=> ( ( comp_char_num_num @ Abs @ Rep )
= id_num ) ) ).
% type_copy_Abs_o_Rep
thf(fact_1138_type__copy__Abs__o__Rep,axiom,
! [Rep: nat > rat,Abs: rat > nat] :
( ( type_d5615363888691252950at_rat @ Rep @ Abs @ top_top_set_rat )
=> ( ( comp_rat_nat_nat @ Abs @ Rep )
= id_nat ) ) ).
% type_copy_Abs_o_Rep
thf(fact_1139_type__copy__Abs__o__Rep,axiom,
! [Rep: $o > rat,Abs: rat > $o] :
( ( type_d6486251586359353412_o_rat @ Rep @ Abs @ top_top_set_rat )
=> ( ( comp_rat_o_o @ Abs @ Rep )
= id_o ) ) ).
% type_copy_Abs_o_Rep
thf(fact_1140_type__copy__Abs__o__Rep,axiom,
! [Rep: num > rat,Abs: rat > num] :
( ( type_d3745694291998722336um_rat @ Rep @ Abs @ top_top_set_rat )
=> ( ( comp_rat_num_num @ Abs @ Rep )
= id_num ) ) ).
% type_copy_Abs_o_Rep
thf(fact_1141_type__copy__Abs__o__Rep,axiom,
! [Rep: $o > nat,Abs: nat > $o] :
( ( type_d7121381646445849148_o_nat @ Rep @ Abs @ top_top_set_nat )
=> ( ( comp_nat_o_o @ Abs @ Rep )
= id_o ) ) ).
% type_copy_Abs_o_Rep
thf(fact_1142_type__copy__Abs__o__Rep,axiom,
! [Rep: num > nat,Abs: nat > num] :
( ( type_d4380824352085218072um_nat @ Rep @ Abs @ top_top_set_nat )
=> ( ( comp_nat_num_num @ Abs @ Rep )
= id_num ) ) ).
% type_copy_Abs_o_Rep
thf(fact_1143_type__copy__Abs__o__Rep,axiom,
! [Rep: nat > nat,Abs: nat > nat] :
( ( type_d6250493948777748686at_nat @ Rep @ Abs @ top_top_set_nat )
=> ( ( comp_nat_nat_nat @ Abs @ Rep )
= id_nat ) ) ).
% type_copy_Abs_o_Rep
thf(fact_1144_type__copy__Abs__o__Rep,axiom,
! [Rep: nat > int,Abs: int > nat] :
( ( type_d6248003478268698410at_int @ Rep @ Abs @ top_top_set_int )
=> ( ( comp_int_nat_nat @ Abs @ Rep )
= id_nat ) ) ).
% type_copy_Abs_o_Rep
thf(fact_1145_type__copy__Rep__o__Abs,axiom,
! [Rep: nat > nat,Abs: nat > nat] :
( ( type_d6250493948777748686at_nat @ Rep @ Abs @ top_top_set_nat )
=> ( ( comp_nat_nat_nat @ Rep @ Abs )
= id_nat ) ) ).
% type_copy_Rep_o_Abs
thf(fact_1146_type__copy__Rep__o__Abs,axiom,
! [Rep: product_unit > $o,Abs: $o > product_unit] :
( ( type_d6188575255521822967unit_o @ Rep @ Abs @ top_top_set_o )
=> ( ( comp_P5156358013004227690it_o_o @ Rep @ Abs )
= id_o ) ) ).
% type_copy_Rep_o_Abs
thf(fact_1147_case__prodI2_H,axiom,
! [P: product_prod_nat_nat,C: nat > nat > product_prod_nat_nat > $o,X2: product_prod_nat_nat] :
( ! [A3: nat,B3: nat] :
( ( ( product_Pair_nat_nat @ A3 @ B3 )
= P )
=> ( C @ A3 @ B3 @ X2 ) )
=> ( produc8739625826339149834_nat_o @ C @ P @ X2 ) ) ).
% case_prodI2'
thf(fact_1148_image__ident,axiom,
! [Y4: set_nat] :
( ( image_nat_nat
@ ^ [X4: nat] : X4
@ Y4 )
= Y4 ) ).
% image_ident
thf(fact_1149_image__ident,axiom,
! [Y4: set_int] :
( ( image_int_int
@ ^ [X4: int] : X4
@ Y4 )
= Y4 ) ).
% image_ident
thf(fact_1150_split__part,axiom,
! [P2: $o,Q2: int > int > $o] :
( ( produc4947309494688390418_int_o
@ ^ [A5: int,B4: int] :
( P2
& ( Q2 @ A5 @ B4 ) ) )
= ( ^ [Ab: product_prod_int_int] :
( P2
& ( produc4947309494688390418_int_o @ Q2 @ Ab ) ) ) ) ).
% split_part
thf(fact_1151_Id__onI,axiom,
! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
( ( member8757157785044589968at_nat @ A @ A4 )
=> ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ A ) @ ( id_on_7869290023980668164at_nat @ A4 ) ) ) ).
% Id_onI
thf(fact_1152_Id__onI,axiom,
! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ A @ A4 )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ A ) @ ( id_on_2554058798563519774at_nat @ A4 ) ) ) ).
% Id_onI
thf(fact_1153_Id__onI,axiom,
! [A: $o,A4: set_o] :
( ( member_o @ A @ A4 )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ A ) @ ( id_on_o @ A4 ) ) ) ).
% Id_onI
thf(fact_1154_Id__onI,axiom,
! [A: set_Pr4329608150637261639at_nat,A4: set_se5479201105039950375at_nat] :
( ( member6273874620674182256at_nat @ A @ A4 )
=> ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ A ) @ ( id_on_3275838214495744996at_nat @ A4 ) ) ) ).
% Id_onI
thf(fact_1155_Id__onI,axiom,
! [A: set_Pr1261947904930325089at_nat,A4: set_se7855581050983116737at_nat] :
( ( member2643936169264416010at_nat @ A @ A4 )
=> ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ A ) @ ( id_on_6713504163309144702at_nat @ A4 ) ) ) ).
% Id_onI
thf(fact_1156_Id__onI,axiom,
! [A: nat,A4: set_nat] :
( ( member_nat @ A @ A4 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A ) @ ( id_on_nat @ A4 ) ) ) ).
% Id_onI
thf(fact_1157_Id__onI,axiom,
! [A: int,A4: set_int] :
( ( member_int @ A @ A4 )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ A ) @ ( id_on_int @ A4 ) ) ) ).
% Id_onI
thf(fact_1158_case__prod__conv,axiom,
! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,A: nat,B: nat] :
( ( produc27273713700761075at_nat @ F @ ( product_Pair_nat_nat @ A @ B ) )
= ( F @ A @ B ) ) ).
% case_prod_conv
thf(fact_1159_case__prod__conv,axiom,
! [F: nat > nat > product_prod_nat_nat > $o,A: nat,B: nat] :
( ( produc8739625826339149834_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) )
= ( F @ A @ B ) ) ).
% case_prod_conv
thf(fact_1160_case__prod__conv,axiom,
! [F: int > int > product_prod_int_int,A: int,B: int] :
( ( produc4245557441103728435nt_int @ F @ ( product_Pair_int_int @ A @ B ) )
= ( F @ A @ B ) ) ).
% case_prod_conv
thf(fact_1161_case__prod__conv,axiom,
! [F: int > int > $o,A: int,B: int] :
( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B ) )
= ( F @ A @ B ) ) ).
% case_prod_conv
thf(fact_1162_case__prod__conv,axiom,
! [F: int > int > int,A: int,B: int] :
( ( produc8211389475949308722nt_int @ F @ ( product_Pair_int_int @ A @ B ) )
= ( F @ A @ B ) ) ).
% case_prod_conv
thf(fact_1163_case__prodI,axiom,
! [F: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
( ( F @ A @ B )
=> ( produc9020218426428501292_nat_o @ F @ ( produc9060074326276436823at_nat @ A @ B ) ) ) ).
% case_prodI
thf(fact_1164_case__prodI,axiom,
! [F: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
( ( F @ A @ B )
=> ( produc410239310623530412_nat_o @ F @ ( produc2922128104949294807at_nat @ A @ B ) ) ) ).
% case_prodI
thf(fact_1165_case__prodI,axiom,
! [F: nat > num > $o,A: nat,B: num] :
( ( F @ A @ B )
=> ( produc4927758841916487424_num_o @ F @ ( product_Pair_nat_num @ A @ B ) ) ) ).
% case_prodI
thf(fact_1166_case__prodI,axiom,
! [F: nat > nat > $o,A: nat,B: nat] :
( ( F @ A @ B )
=> ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) ) ) ).
% case_prodI
thf(fact_1167_case__prodI,axiom,
! [F: int > int > $o,A: int,B: int] :
( ( F @ A @ B )
=> ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B ) ) ) ).
% case_prodI
thf(fact_1168_case__prodI2,axiom,
! [P: produc1319942482725812455at_nat,C: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o] :
( ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
( ( P
= ( produc9060074326276436823at_nat @ A3 @ B3 ) )
=> ( C @ A3 @ B3 ) )
=> ( produc9020218426428501292_nat_o @ C @ P ) ) ).
% case_prodI2
thf(fact_1169_case__prodI2,axiom,
! [P: produc3843707927480180839at_nat,C: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o] :
( ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
( ( P
= ( produc2922128104949294807at_nat @ A3 @ B3 ) )
=> ( C @ A3 @ B3 ) )
=> ( produc410239310623530412_nat_o @ C @ P ) ) ).
% case_prodI2
thf(fact_1170_case__prodI2,axiom,
! [P: product_prod_nat_num,C: nat > num > $o] :
( ! [A3: nat,B3: num] :
( ( P
= ( product_Pair_nat_num @ A3 @ B3 ) )
=> ( C @ A3 @ B3 ) )
=> ( produc4927758841916487424_num_o @ C @ P ) ) ).
% case_prodI2
thf(fact_1171_case__prodI2,axiom,
! [P: product_prod_nat_nat,C: nat > nat > $o] :
( ! [A3: nat,B3: nat] :
( ( P
= ( product_Pair_nat_nat @ A3 @ B3 ) )
=> ( C @ A3 @ B3 ) )
=> ( produc6081775807080527818_nat_o @ C @ P ) ) ).
% case_prodI2
thf(fact_1172_case__prodI2,axiom,
! [P: product_prod_int_int,C: int > int > $o] :
( ! [A3: int,B3: int] :
( ( P
= ( product_Pair_int_int @ A3 @ B3 ) )
=> ( C @ A3 @ B3 ) )
=> ( produc4947309494688390418_int_o @ C @ P ) ) ).
% case_prodI2
thf(fact_1173_mem__case__prodI,axiom,
! [Z2: $o,C: nat > num > set_o,A: nat,B: num] :
( ( member_o @ Z2 @ ( C @ A @ B ) )
=> ( member_o @ Z2 @ ( produc836176033315069408_set_o @ C @ ( product_Pair_nat_num @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_1174_mem__case__prodI,axiom,
! [Z2: nat,C: nat > num > set_nat,A: nat,B: num] :
( ( member_nat @ Z2 @ ( C @ A @ B ) )
=> ( member_nat @ Z2 @ ( produc4130284055270567454et_nat @ C @ ( product_Pair_nat_num @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_1175_mem__case__prodI,axiom,
! [Z2: int,C: nat > num > set_int,A: nat,B: num] :
( ( member_int @ Z2 @ ( C @ A @ B ) )
=> ( member_int @ Z2 @ ( produc9175805072616146554et_int @ C @ ( product_Pair_nat_num @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_1176_mem__case__prodI,axiom,
! [Z2: $o,C: nat > nat > set_o,A: nat,B: nat] :
( ( member_o @ Z2 @ ( C @ A @ B ) )
=> ( member_o @ Z2 @ ( produc59986286002894506_set_o @ C @ ( product_Pair_nat_nat @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_1177_mem__case__prodI,axiom,
! [Z2: nat,C: nat > nat > set_nat,A: nat,B: nat] :
( ( member_nat @ Z2 @ ( C @ A @ B ) )
=> ( member_nat @ Z2 @ ( produc6189476227299908564et_nat @ C @ ( product_Pair_nat_nat @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_1178_mem__case__prodI,axiom,
! [Z2: int,C: nat > nat > set_int,A: nat,B: nat] :
( ( member_int @ Z2 @ ( C @ A @ B ) )
=> ( member_int @ Z2 @ ( produc2011625207790711856et_int @ C @ ( product_Pair_nat_nat @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_1179_mem__case__prodI,axiom,
! [Z2: $o,C: int > int > set_o,A: int,B: int] :
( ( member_o @ Z2 @ ( C @ A @ B ) )
=> ( member_o @ Z2 @ ( produc4257766111578684402_set_o @ C @ ( product_Pair_int_int @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_1180_mem__case__prodI,axiom,
! [Z2: nat,C: int > int > set_nat,A: int,B: int] :
( ( member_nat @ Z2 @ ( C @ A @ B ) )
=> ( member_nat @ Z2 @ ( produc4251311855443802252et_nat @ C @ ( product_Pair_int_int @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_1181_mem__case__prodI,axiom,
! [Z2: int,C: int > int > set_int,A: int,B: int] :
( ( member_int @ Z2 @ ( C @ A @ B ) )
=> ( member_int @ Z2 @ ( produc73460835934605544et_int @ C @ ( product_Pair_int_int @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_1182_mem__case__prodI,axiom,
! [Z2: product_prod_nat_nat,C: nat > num > set_Pr1261947904930325089at_nat,A: nat,B: num] :
( ( member8440522571783428010at_nat @ Z2 @ ( C @ A @ B ) )
=> ( member8440522571783428010at_nat @ Z2 @ ( produc859037500352796105at_nat @ C @ ( product_Pair_nat_num @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_1183_mem__case__prodI2,axiom,
! [P: product_prod_nat_num,Z2: $o,C: nat > num > set_o] :
( ! [A3: nat,B3: num] :
( ( P
= ( product_Pair_nat_num @ A3 @ B3 ) )
=> ( member_o @ Z2 @ ( C @ A3 @ B3 ) ) )
=> ( member_o @ Z2 @ ( produc836176033315069408_set_o @ C @ P ) ) ) ).
% mem_case_prodI2
thf(fact_1184_mem__case__prodI2,axiom,
! [P: product_prod_nat_num,Z2: nat,C: nat > num > set_nat] :
( ! [A3: nat,B3: num] :
( ( P
= ( product_Pair_nat_num @ A3 @ B3 ) )
=> ( member_nat @ Z2 @ ( C @ A3 @ B3 ) ) )
=> ( member_nat @ Z2 @ ( produc4130284055270567454et_nat @ C @ P ) ) ) ).
% mem_case_prodI2
thf(fact_1185_mem__case__prodI2,axiom,
! [P: product_prod_nat_num,Z2: int,C: nat > num > set_int] :
( ! [A3: nat,B3: num] :
( ( P
= ( product_Pair_nat_num @ A3 @ B3 ) )
=> ( member_int @ Z2 @ ( C @ A3 @ B3 ) ) )
=> ( member_int @ Z2 @ ( produc9175805072616146554et_int @ C @ P ) ) ) ).
% mem_case_prodI2
thf(fact_1186_mem__case__prodI2,axiom,
! [P: product_prod_nat_nat,Z2: $o,C: nat > nat > set_o] :
( ! [A3: nat,B3: nat] :
( ( P
= ( product_Pair_nat_nat @ A3 @ B3 ) )
=> ( member_o @ Z2 @ ( C @ A3 @ B3 ) ) )
=> ( member_o @ Z2 @ ( produc59986286002894506_set_o @ C @ P ) ) ) ).
% mem_case_prodI2
thf(fact_1187_mem__case__prodI2,axiom,
! [P: product_prod_nat_nat,Z2: nat,C: nat > nat > set_nat] :
( ! [A3: nat,B3: nat] :
( ( P
= ( product_Pair_nat_nat @ A3 @ B3 ) )
=> ( member_nat @ Z2 @ ( C @ A3 @ B3 ) ) )
=> ( member_nat @ Z2 @ ( produc6189476227299908564et_nat @ C @ P ) ) ) ).
% mem_case_prodI2
thf(fact_1188_mem__case__prodI2,axiom,
! [P: product_prod_nat_nat,Z2: int,C: nat > nat > set_int] :
( ! [A3: nat,B3: nat] :
( ( P
= ( product_Pair_nat_nat @ A3 @ B3 ) )
=> ( member_int @ Z2 @ ( C @ A3 @ B3 ) ) )
=> ( member_int @ Z2 @ ( produc2011625207790711856et_int @ C @ P ) ) ) ).
% mem_case_prodI2
thf(fact_1189_mem__case__prodI2,axiom,
! [P: product_prod_int_int,Z2: $o,C: int > int > set_o] :
( ! [A3: int,B3: int] :
( ( P
= ( product_Pair_int_int @ A3 @ B3 ) )
=> ( member_o @ Z2 @ ( C @ A3 @ B3 ) ) )
=> ( member_o @ Z2 @ ( produc4257766111578684402_set_o @ C @ P ) ) ) ).
% mem_case_prodI2
thf(fact_1190_mem__case__prodI2,axiom,
! [P: product_prod_int_int,Z2: nat,C: int > int > set_nat] :
( ! [A3: int,B3: int] :
( ( P
= ( product_Pair_int_int @ A3 @ B3 ) )
=> ( member_nat @ Z2 @ ( C @ A3 @ B3 ) ) )
=> ( member_nat @ Z2 @ ( produc4251311855443802252et_nat @ C @ P ) ) ) ).
% mem_case_prodI2
thf(fact_1191_mem__case__prodI2,axiom,
! [P: product_prod_int_int,Z2: int,C: int > int > set_int] :
( ! [A3: int,B3: int] :
( ( P
= ( product_Pair_int_int @ A3 @ B3 ) )
=> ( member_int @ Z2 @ ( C @ A3 @ B3 ) ) )
=> ( member_int @ Z2 @ ( produc73460835934605544et_int @ C @ P ) ) ) ).
% mem_case_prodI2
thf(fact_1192_mem__case__prodI2,axiom,
! [P: product_prod_nat_num,Z2: product_prod_nat_nat,C: nat > num > set_Pr1261947904930325089at_nat] :
( ! [A3: nat,B3: num] :
( ( P
= ( product_Pair_nat_num @ A3 @ B3 ) )
=> ( member8440522571783428010at_nat @ Z2 @ ( C @ A3 @ B3 ) ) )
=> ( member8440522571783428010at_nat @ Z2 @ ( produc859037500352796105at_nat @ C @ P ) ) ) ).
% mem_case_prodI2
thf(fact_1193_case__swap,axiom,
! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,P: product_prod_nat_nat] :
( ( produc27273713700761075at_nat
@ ^ [Y5: nat,X4: nat] : ( F @ X4 @ Y5 )
@ ( product_swap_nat_nat @ P ) )
= ( produc27273713700761075at_nat @ F @ P ) ) ).
% case_swap
thf(fact_1194_case__swap,axiom,
! [F: nat > nat > product_prod_nat_nat > $o,P: product_prod_nat_nat] :
( ( produc8739625826339149834_nat_o
@ ^ [Y5: nat,X4: nat] : ( F @ X4 @ Y5 )
@ ( product_swap_nat_nat @ P ) )
= ( produc8739625826339149834_nat_o @ F @ P ) ) ).
% case_swap
thf(fact_1195_case__swap,axiom,
! [F: int > int > product_prod_int_int,P: product_prod_int_int] :
( ( produc4245557441103728435nt_int
@ ^ [Y5: int,X4: int] : ( F @ X4 @ Y5 )
@ ( product_swap_int_int @ P ) )
= ( produc4245557441103728435nt_int @ F @ P ) ) ).
% case_swap
thf(fact_1196_case__swap,axiom,
! [F: int > int > $o,P: product_prod_int_int] :
( ( produc4947309494688390418_int_o
@ ^ [Y5: int,X4: int] : ( F @ X4 @ Y5 )
@ ( product_swap_int_int @ P ) )
= ( produc4947309494688390418_int_o @ F @ P ) ) ).
% case_swap
thf(fact_1197_case__swap,axiom,
! [F: int > int > int,P: product_prod_int_int] :
( ( produc8211389475949308722nt_int
@ ^ [Y5: int,X4: int] : ( F @ X4 @ Y5 )
@ ( product_swap_int_int @ P ) )
= ( produc8211389475949308722nt_int @ F @ P ) ) ).
% case_swap
thf(fact_1198_pair__imageI,axiom,
! [A: nat,B: num,A4: set_Pr6200539531224447659at_num,F: nat > num > $o] :
( ( member9148766508732265716at_num @ ( product_Pair_nat_num @ A @ B ) @ A4 )
=> ( member_o @ ( F @ A @ B ) @ ( image_4451812852358507538_num_o @ ( produc4927758841916487424_num_o @ F ) @ A4 ) ) ) ).
% pair_imageI
thf(fact_1199_pair__imageI,axiom,
! [A: nat,B: num,A4: set_Pr6200539531224447659at_num,F: nat > num > nat] :
( ( member9148766508732265716at_num @ ( product_Pair_nat_num @ A @ B ) @ A4 )
=> ( member_nat @ ( F @ A @ B ) @ ( image_4504471508743443926um_nat @ ( produc4973203077627929192um_nat @ F ) @ A4 ) ) ) ).
% pair_imageI
thf(fact_1200_pair__imageI,axiom,
! [A: nat,B: num,A4: set_Pr6200539531224447659at_num,F: nat > num > int] :
( ( member9148766508732265716at_num @ ( product_Pair_nat_num @ A @ B ) @ A4 )
=> ( member_int @ ( F @ A @ B ) @ ( image_4501981038234393650um_int @ ( produc4970712607118878916um_int @ F ) @ A4 ) ) ) ).
% pair_imageI
thf(fact_1201_pair__imageI,axiom,
! [A: nat,B: nat,A4: set_Pr1261947904930325089at_nat,F: nat > nat > $o] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ A4 )
=> ( member_o @ ( F @ A @ B ) @ ( image_3693632289388996572_nat_o @ ( produc6081775807080527818_nat_o @ F ) @ A4 ) ) ) ).
% pair_imageI
thf(fact_1202_pair__imageI,axiom,
! [A: nat,B: nat,A4: set_Pr1261947904930325089at_nat,F: nat > nat > nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ A4 )
=> ( member_nat @ ( F @ A @ B ) @ ( image_2486076414777270412at_nat @ ( produc6842872674320459806at_nat @ F ) @ A4 ) ) ) ).
% pair_imageI
thf(fact_1203_pair__imageI,axiom,
! [A: nat,B: nat,A4: set_Pr1261947904930325089at_nat,F: nat > nat > int] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ A4 )
=> ( member_int @ ( F @ A @ B ) @ ( image_2483585944268220136at_int @ ( produc6840382203811409530at_int @ F ) @ A4 ) ) ) ).
% pair_imageI
thf(fact_1204_pair__imageI,axiom,
! [A: int,B: int,A4: set_Pr958786334691620121nt_int,F: int > int > nat] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ A4 )
=> ( member_nat @ ( F @ A @ B ) @ ( image_5044651549707136836nt_nat @ ( produc8213879946458358998nt_nat @ F ) @ A4 ) ) ) ).
% pair_imageI
thf(fact_1205_pair__imageI,axiom,
! [A: int,B: int,A4: set_Pr958786334691620121nt_int,F: int > int > $o] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ A4 )
=> ( member_o @ ( F @ A @ B ) @ ( image_2135063354759101220_int_o @ ( produc4947309494688390418_int_o @ F ) @ A4 ) ) ) ).
% pair_imageI
thf(fact_1206_pair__imageI,axiom,
! [A: int,B: int,A4: set_Pr958786334691620121nt_int,F: int > int > int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ A4 )
=> ( member_int @ ( F @ A @ B ) @ ( image_5042161079198086560nt_int @ ( produc8211389475949308722nt_int @ F ) @ A4 ) ) ) ).
% pair_imageI
thf(fact_1207_pair__imageI,axiom,
! [A: nat,B: num,A4: set_Pr6200539531224447659at_num,F: nat > num > product_prod_nat_nat] :
( ( member9148766508732265716at_num @ ( product_Pair_nat_num @ A @ B ) @ A4 )
=> ( member8440522571783428010at_nat @ ( F @ A @ B ) @ ( image_4217639968629731195at_nat @ ( produc8564434177180591209at_nat @ F ) @ A4 ) ) ) ).
% pair_imageI
thf(fact_1208_Inf_OINF__identity__eq,axiom,
! [Inf: set_nat > nat,A4: set_nat] :
( ( Inf
@ ( image_nat_nat
@ ^ [X4: nat] : X4
@ A4 ) )
= ( Inf @ A4 ) ) ).
% Inf.INF_identity_eq
thf(fact_1209_Inf_OINF__identity__eq,axiom,
! [Inf: set_int > int,A4: set_int] :
( ( Inf
@ ( image_int_int
@ ^ [X4: int] : X4
@ A4 ) )
= ( Inf @ A4 ) ) ).
% Inf.INF_identity_eq
thf(fact_1210_Sup_OSUP__identity__eq,axiom,
! [Sup: set_nat > nat,A4: set_nat] :
( ( Sup
@ ( image_nat_nat
@ ^ [X4: nat] : X4
@ A4 ) )
= ( Sup @ A4 ) ) ).
% Sup.SUP_identity_eq
thf(fact_1211_Sup_OSUP__identity__eq,axiom,
! [Sup: set_int > int,A4: set_int] :
( ( Sup
@ ( image_int_int
@ ^ [X4: int] : X4
@ A4 ) )
= ( Sup @ A4 ) ) ).
% Sup.SUP_identity_eq
thf(fact_1212_imageE,axiom,
! [B: rat,F: nat > rat,A4: set_nat] :
( ( member_rat @ B @ ( image_nat_rat @ F @ A4 ) )
=> ~ ! [X: nat] :
( ( B
= ( F @ X ) )
=> ~ ( member_nat @ X @ A4 ) ) ) ).
% imageE
thf(fact_1213_imageE,axiom,
! [B: $o,F: $o > $o,A4: set_o] :
( ( member_o @ B @ ( image_o_o @ F @ A4 ) )
=> ~ ! [X: $o] :
( ( B
= ( F @ X ) )
=> ~ ( member_o @ X @ A4 ) ) ) ).
% imageE
thf(fact_1214_imageE,axiom,
! [B: $o,F: nat > $o,A4: set_nat] :
( ( member_o @ B @ ( image_nat_o @ F @ A4 ) )
=> ~ ! [X: nat] :
( ( B
= ( F @ X ) )
=> ~ ( member_nat @ X @ A4 ) ) ) ).
% imageE
thf(fact_1215_imageE,axiom,
! [B: $o,F: int > $o,A4: set_int] :
( ( member_o @ B @ ( image_int_o @ F @ A4 ) )
=> ~ ! [X: int] :
( ( B
= ( F @ X ) )
=> ~ ( member_int @ X @ A4 ) ) ) ).
% imageE
thf(fact_1216_imageE,axiom,
! [B: nat,F: $o > nat,A4: set_o] :
( ( member_nat @ B @ ( image_o_nat @ F @ A4 ) )
=> ~ ! [X: $o] :
( ( B
= ( F @ X ) )
=> ~ ( member_o @ X @ A4 ) ) ) ).
% imageE
thf(fact_1217_imageE,axiom,
! [B: nat,F: nat > nat,A4: set_nat] :
( ( member_nat @ B @ ( image_nat_nat @ F @ A4 ) )
=> ~ ! [X: nat] :
( ( B
= ( F @ X ) )
=> ~ ( member_nat @ X @ A4 ) ) ) ).
% imageE
thf(fact_1218_imageE,axiom,
! [B: nat,F: int > nat,A4: set_int] :
( ( member_nat @ B @ ( image_int_nat @ F @ A4 ) )
=> ~ ! [X: int] :
( ( B
= ( F @ X ) )
=> ~ ( member_int @ X @ A4 ) ) ) ).
% imageE
thf(fact_1219_imageE,axiom,
! [B: int,F: $o > int,A4: set_o] :
( ( member_int @ B @ ( image_o_int @ F @ A4 ) )
=> ~ ! [X: $o] :
( ( B
= ( F @ X ) )
=> ~ ( member_o @ X @ A4 ) ) ) ).
% imageE
thf(fact_1220_imageE,axiom,
! [B: int,F: nat > int,A4: set_nat] :
( ( member_int @ B @ ( image_nat_int @ F @ A4 ) )
=> ~ ! [X: nat] :
( ( B
= ( F @ X ) )
=> ~ ( member_nat @ X @ A4 ) ) ) ).
% imageE
thf(fact_1221_imageE,axiom,
! [B: int,F: int > int,A4: set_int] :
( ( member_int @ B @ ( image_int_int @ F @ A4 ) )
=> ~ ! [X: int] :
( ( B
= ( F @ X ) )
=> ~ ( member_int @ X @ A4 ) ) ) ).
% imageE
thf(fact_1222_image__image,axiom,
! [F: rat > rat,G: nat > rat,A4: set_nat] :
( ( image_rat_rat @ F @ ( image_nat_rat @ G @ A4 ) )
= ( image_nat_rat
@ ^ [X4: nat] : ( F @ ( G @ X4 ) )
@ A4 ) ) ).
% image_image
thf(fact_1223_image__image,axiom,
! [F: rat > nat,G: nat > rat,A4: set_nat] :
( ( image_rat_nat @ F @ ( image_nat_rat @ G @ A4 ) )
= ( image_nat_nat
@ ^ [X4: nat] : ( F @ ( G @ X4 ) )
@ A4 ) ) ).
% image_image
thf(fact_1224_image__image,axiom,
! [F: rat > int,G: nat > rat,A4: set_nat] :
( ( image_rat_int @ F @ ( image_nat_rat @ G @ A4 ) )
= ( image_nat_int
@ ^ [X4: nat] : ( F @ ( G @ X4 ) )
@ A4 ) ) ).
% image_image
thf(fact_1225_image__image,axiom,
! [F: int > rat,G: nat > int,A4: set_nat] :
( ( image_int_rat @ F @ ( image_nat_int @ G @ A4 ) )
= ( image_nat_rat
@ ^ [X4: nat] : ( F @ ( G @ X4 ) )
@ A4 ) ) ).
% image_image
thf(fact_1226_image__image,axiom,
! [F: nat > rat,G: nat > nat,A4: set_nat] :
( ( image_nat_rat @ F @ ( image_nat_nat @ G @ A4 ) )
= ( image_nat_rat
@ ^ [X4: nat] : ( F @ ( G @ X4 ) )
@ A4 ) ) ).
% image_image
thf(fact_1227_image__image,axiom,
! [F: nat > rat,G: int > nat,A4: set_int] :
( ( image_nat_rat @ F @ ( image_int_nat @ G @ A4 ) )
= ( image_int_rat
@ ^ [X4: int] : ( F @ ( G @ X4 ) )
@ A4 ) ) ).
% image_image
thf(fact_1228_image__image,axiom,
! [F: nat > nat,G: nat > nat,A4: set_nat] :
( ( image_nat_nat @ F @ ( image_nat_nat @ G @ A4 ) )
= ( image_nat_nat
@ ^ [X4: nat] : ( F @ ( G @ X4 ) )
@ A4 ) ) ).
% image_image
thf(fact_1229_image__image,axiom,
! [F: nat > nat,G: int > nat,A4: set_int] :
( ( image_nat_nat @ F @ ( image_int_nat @ G @ A4 ) )
= ( image_int_nat
@ ^ [X4: int] : ( F @ ( G @ X4 ) )
@ A4 ) ) ).
% image_image
thf(fact_1230_image__image,axiom,
! [F: nat > int,G: nat > nat,A4: set_nat] :
( ( image_nat_int @ F @ ( image_nat_nat @ G @ A4 ) )
= ( image_nat_int
@ ^ [X4: nat] : ( F @ ( G @ X4 ) )
@ A4 ) ) ).
% image_image
thf(fact_1231_image__image,axiom,
! [F: nat > int,G: int > nat,A4: set_int] :
( ( image_nat_int @ F @ ( image_int_nat @ G @ A4 ) )
= ( image_int_int
@ ^ [X4: int] : ( F @ ( G @ X4 ) )
@ A4 ) ) ).
% image_image
thf(fact_1232_Compr__image__eq,axiom,
! [F: $o > $o,A4: set_o,P2: $o > $o] :
( ( collect_o
@ ^ [X4: $o] :
( ( member_o @ X4 @ ( image_o_o @ F @ A4 ) )
& ( P2 @ X4 ) ) )
= ( image_o_o @ F
@ ( collect_o
@ ^ [X4: $o] :
( ( member_o @ X4 @ A4 )
& ( P2 @ ( F @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1233_Compr__image__eq,axiom,
! [F: int > $o,A4: set_int,P2: $o > $o] :
( ( collect_o
@ ^ [X4: $o] :
( ( member_o @ X4 @ ( image_int_o @ F @ A4 ) )
& ( P2 @ X4 ) ) )
= ( image_int_o @ F
@ ( collect_int
@ ^ [X4: int] :
( ( member_int @ X4 @ A4 )
& ( P2 @ ( F @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1234_Compr__image__eq,axiom,
! [F: $o > int,A4: set_o,P2: int > $o] :
( ( collect_int
@ ^ [X4: int] :
( ( member_int @ X4 @ ( image_o_int @ F @ A4 ) )
& ( P2 @ X4 ) ) )
= ( image_o_int @ F
@ ( collect_o
@ ^ [X4: $o] :
( ( member_o @ X4 @ A4 )
& ( P2 @ ( F @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1235_Compr__image__eq,axiom,
! [F: int > int,A4: set_int,P2: int > $o] :
( ( collect_int
@ ^ [X4: int] :
( ( member_int @ X4 @ ( image_int_int @ F @ A4 ) )
& ( P2 @ X4 ) ) )
= ( image_int_int @ F
@ ( collect_int
@ ^ [X4: int] :
( ( member_int @ X4 @ A4 )
& ( P2 @ ( F @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1236_Compr__image__eq,axiom,
! [F: nat > rat,A4: set_nat,P2: rat > $o] :
( ( collect_rat
@ ^ [X4: rat] :
( ( member_rat @ X4 @ ( image_nat_rat @ F @ A4 ) )
& ( P2 @ X4 ) ) )
= ( image_nat_rat @ F
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A4 )
& ( P2 @ ( F @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1237_Compr__image__eq,axiom,
! [F: nat > $o,A4: set_nat,P2: $o > $o] :
( ( collect_o
@ ^ [X4: $o] :
( ( member_o @ X4 @ ( image_nat_o @ F @ A4 ) )
& ( P2 @ X4 ) ) )
= ( image_nat_o @ F
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A4 )
& ( P2 @ ( F @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1238_Compr__image__eq,axiom,
! [F: nat > int,A4: set_nat,P2: int > $o] :
( ( collect_int
@ ^ [X4: int] :
( ( member_int @ X4 @ ( image_nat_int @ F @ A4 ) )
& ( P2 @ X4 ) ) )
= ( image_nat_int @ F
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A4 )
& ( P2 @ ( F @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1239_Compr__image__eq,axiom,
! [F: $o > nat,A4: set_o,P2: nat > $o] :
( ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ ( image_o_nat @ F @ A4 ) )
& ( P2 @ X4 ) ) )
= ( image_o_nat @ F
@ ( collect_o
@ ^ [X4: $o] :
( ( member_o @ X4 @ A4 )
& ( P2 @ ( F @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1240_Compr__image__eq,axiom,
! [F: int > nat,A4: set_int,P2: nat > $o] :
( ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ ( image_int_nat @ F @ A4 ) )
& ( P2 @ X4 ) ) )
= ( image_int_nat @ F
@ ( collect_int
@ ^ [X4: int] :
( ( member_int @ X4 @ A4 )
& ( P2 @ ( F @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1241_Compr__image__eq,axiom,
! [F: nat > nat,A4: set_nat,P2: nat > $o] :
( ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ ( image_nat_nat @ F @ A4 ) )
& ( P2 @ X4 ) ) )
= ( image_nat_nat @ F
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A4 )
& ( P2 @ ( F @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1242_Id__on__def_H,axiom,
! [A4: product_prod_nat_nat > $o] :
( ( id_on_2554058798563519774at_nat @ ( collec3392354462482085612at_nat @ A4 ) )
= ( collec7088162979684241874at_nat
@ ( produc6590410687421337004_nat_o
@ ^ [X4: product_prod_nat_nat,Y5: product_prod_nat_nat] :
( ( X4 = Y5 )
& ( A4 @ X4 ) ) ) ) ) ).
% Id_on_def'
thf(fact_1243_Id__on__def_H,axiom,
! [A4: nat > $o] :
( ( id_on_nat @ ( collect_nat @ A4 ) )
= ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [X4: nat,Y5: nat] :
( ( X4 = Y5 )
& ( A4 @ X4 ) ) ) ) ) ).
% Id_on_def'
thf(fact_1244_Id__on__def_H,axiom,
! [A4: product_prod_int_int > $o] :
( ( id_on_8598933527810153430nt_int @ ( collec213857154873943460nt_int @ A4 ) )
= ( collec7447955092554649554nt_int
@ ( produc1676490119946744748_int_o
@ ^ [X4: product_prod_int_int,Y5: product_prod_int_int] :
( ( X4 = Y5 )
& ( A4 @ X4 ) ) ) ) ) ).
% Id_on_def'
thf(fact_1245_Id__on__def_H,axiom,
! [A4: int > $o] :
( ( id_on_int @ ( collect_int @ A4 ) )
= ( collec213857154873943460nt_int
@ ( produc4947309494688390418_int_o
@ ^ [X4: int,Y5: int] :
( ( X4 = Y5 )
& ( A4 @ X4 ) ) ) ) ) ).
% Id_on_def'
thf(fact_1246_pred__equals__eq2,axiom,
! [R3: set_Pr7459493094073627847at_nat,S5: set_Pr7459493094073627847at_nat] :
( ( ( ^ [X4: set_Pr4329608150637261639at_nat,Y5: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X4 @ Y5 ) @ R3 ) )
= ( ^ [X4: set_Pr4329608150637261639at_nat,Y5: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X4 @ Y5 ) @ S5 ) ) )
= ( R3 = S5 ) ) ).
% pred_equals_eq2
thf(fact_1247_pred__equals__eq2,axiom,
! [R3: set_Pr4329608150637261639at_nat,S5: set_Pr4329608150637261639at_nat] :
( ( ( ^ [X4: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y5 ) @ R3 ) )
= ( ^ [X4: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y5 ) @ S5 ) ) )
= ( R3 = S5 ) ) ).
% pred_equals_eq2
thf(fact_1248_pred__equals__eq2,axiom,
! [R3: set_Pr6200539531224447659at_num,S5: set_Pr6200539531224447659at_num] :
( ( ( ^ [X4: nat,Y5: num] : ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X4 @ Y5 ) @ R3 ) )
= ( ^ [X4: nat,Y5: num] : ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X4 @ Y5 ) @ S5 ) ) )
= ( R3 = S5 ) ) ).
% pred_equals_eq2
thf(fact_1249_pred__equals__eq2,axiom,
! [R3: set_Pr1261947904930325089at_nat,S5: set_Pr1261947904930325089at_nat] :
( ( ( ^ [X4: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ R3 ) )
= ( ^ [X4: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ S5 ) ) )
= ( R3 = S5 ) ) ).
% pred_equals_eq2
thf(fact_1250_pred__equals__eq2,axiom,
! [R3: set_Pr958786334691620121nt_int,S5: set_Pr958786334691620121nt_int] :
( ( ( ^ [X4: int,Y5: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y5 ) @ R3 ) )
= ( ^ [X4: int,Y5: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y5 ) @ S5 ) ) )
= ( R3 = S5 ) ) ).
% pred_equals_eq2
thf(fact_1251_case__prodE_H,axiom,
! [C: nat > nat > product_prod_nat_nat > $o,P: product_prod_nat_nat,Z2: product_prod_nat_nat] :
( ( produc8739625826339149834_nat_o @ C @ P @ Z2 )
=> ~ ! [X: nat,Y: nat] :
( ( P
= ( product_Pair_nat_nat @ X @ Y ) )
=> ~ ( C @ X @ Y @ Z2 ) ) ) ).
% case_prodE'
thf(fact_1252_case__prodD_H,axiom,
! [R3: nat > nat > product_prod_nat_nat > $o,A: nat,B: nat,C: product_prod_nat_nat] :
( ( produc8739625826339149834_nat_o @ R3 @ ( product_Pair_nat_nat @ A @ B ) @ C )
=> ( R3 @ A @ B @ C ) ) ).
% case_prodD'
thf(fact_1253_case__prod__Pair__iden,axiom,
! [P: produc1319942482725812455at_nat] :
( ( produc5456455677105495163at_nat @ produc9060074326276436823at_nat @ P )
= P ) ).
% case_prod_Pair_iden
thf(fact_1254_case__prod__Pair__iden,axiom,
! [P: produc3843707927480180839at_nat] :
( ( produc7548845685434208379at_nat @ produc2922128104949294807at_nat @ P )
= P ) ).
% case_prod_Pair_iden
thf(fact_1255_case__prod__Pair__iden,axiom,
! [P: product_prod_nat_num] :
( ( produc49306077274653107at_num @ product_Pair_nat_num @ P )
= P ) ).
% case_prod_Pair_iden
thf(fact_1256_case__prod__Pair__iden,axiom,
! [P: product_prod_nat_nat] :
( ( produc2626176000494625587at_nat @ product_Pair_nat_nat @ P )
= P ) ).
% case_prod_Pair_iden
thf(fact_1257_case__prod__Pair__iden,axiom,
! [P: product_prod_int_int] :
( ( produc4245557441103728435nt_int @ product_Pair_int_int @ P )
= P ) ).
% case_prod_Pair_iden
thf(fact_1258_subset__CollectI,axiom,
! [B5: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat,Q2: produc3843707927480180839at_nat > $o,P2: produc3843707927480180839at_nat > $o] :
( ( ord_le1268244103169919719at_nat @ B5 @ A4 )
=> ( ! [X: produc3843707927480180839at_nat] :
( ( member8757157785044589968at_nat @ X @ B5 )
=> ( ( Q2 @ X )
=> ( P2 @ X ) ) )
=> ( ord_le1268244103169919719at_nat
@ ( collec6321179662152712658at_nat
@ ^ [X4: produc3843707927480180839at_nat] :
( ( member8757157785044589968at_nat @ X4 @ B5 )
& ( Q2 @ X4 ) ) )
@ ( collec6321179662152712658at_nat
@ ^ [X4: produc3843707927480180839at_nat] :
( ( member8757157785044589968at_nat @ X4 @ A4 )
& ( P2 @ X4 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_1259_subset__CollectI,axiom,
! [B5: set_o,A4: set_o,Q2: $o > $o,P2: $o > $o] :
( ( ord_less_eq_set_o @ B5 @ A4 )
=> ( ! [X: $o] :
( ( member_o @ X @ B5 )
=> ( ( Q2 @ X )
=> ( P2 @ X ) ) )
=> ( ord_less_eq_set_o
@ ( collect_o
@ ^ [X4: $o] :
( ( member_o @ X4 @ B5 )
& ( Q2 @ X4 ) ) )
@ ( collect_o
@ ^ [X4: $o] :
( ( member_o @ X4 @ A4 )
& ( P2 @ X4 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_1260_subset__CollectI,axiom,
! [B5: set_int,A4: set_int,Q2: int > $o,P2: int > $o] :
( ( ord_less_eq_set_int @ B5 @ A4 )
=> ( ! [X: int] :
( ( member_int @ X @ B5 )
=> ( ( Q2 @ X )
=> ( P2 @ X ) ) )
=> ( ord_less_eq_set_int
@ ( collect_int
@ ^ [X4: int] :
( ( member_int @ X4 @ B5 )
& ( Q2 @ X4 ) ) )
@ ( collect_int
@ ^ [X4: int] :
( ( member_int @ X4 @ A4 )
& ( P2 @ X4 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_1261_subset__CollectI,axiom,
! [B5: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,Q2: product_prod_nat_nat > $o,P2: product_prod_nat_nat > $o] :
( ( ord_le3146513528884898305at_nat @ B5 @ A4 )
=> ( ! [X: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X @ B5 )
=> ( ( Q2 @ X )
=> ( P2 @ X ) ) )
=> ( ord_le3146513528884898305at_nat
@ ( collec3392354462482085612at_nat
@ ^ [X4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X4 @ B5 )
& ( Q2 @ X4 ) ) )
@ ( collec3392354462482085612at_nat
@ ^ [X4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X4 @ A4 )
& ( P2 @ X4 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_1262_subset__CollectI,axiom,
! [B5: set_nat,A4: set_nat,Q2: nat > $o,P2: nat > $o] :
( ( ord_less_eq_set_nat @ B5 @ A4 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B5 )
=> ( ( Q2 @ X )
=> ( P2 @ X ) ) )
=> ( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ B5 )
& ( Q2 @ X4 ) ) )
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A4 )
& ( P2 @ X4 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_1263_subset__CollectI,axiom,
! [B5: set_Pr958786334691620121nt_int,A4: set_Pr958786334691620121nt_int,Q2: product_prod_int_int > $o,P2: product_prod_int_int > $o] :
( ( ord_le2843351958646193337nt_int @ B5 @ A4 )
=> ( ! [X: product_prod_int_int] :
( ( member5262025264175285858nt_int @ X @ B5 )
=> ( ( Q2 @ X )
=> ( P2 @ X ) ) )
=> ( ord_le2843351958646193337nt_int
@ ( collec213857154873943460nt_int
@ ^ [X4: product_prod_int_int] :
( ( member5262025264175285858nt_int @ X4 @ B5 )
& ( Q2 @ X4 ) ) )
@ ( collec213857154873943460nt_int
@ ^ [X4: product_prod_int_int] :
( ( member5262025264175285858nt_int @ X4 @ A4 )
& ( P2 @ X4 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_1264_subset__Collect__iff,axiom,
! [B5: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat,P2: produc3843707927480180839at_nat > $o] :
( ( ord_le1268244103169919719at_nat @ B5 @ A4 )
=> ( ( ord_le1268244103169919719at_nat @ B5
@ ( collec6321179662152712658at_nat
@ ^ [X4: produc3843707927480180839at_nat] :
( ( member8757157785044589968at_nat @ X4 @ A4 )
& ( P2 @ X4 ) ) ) )
= ( ! [X4: produc3843707927480180839at_nat] :
( ( member8757157785044589968at_nat @ X4 @ B5 )
=> ( P2 @ X4 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_1265_subset__Collect__iff,axiom,
! [B5: set_o,A4: set_o,P2: $o > $o] :
( ( ord_less_eq_set_o @ B5 @ A4 )
=> ( ( ord_less_eq_set_o @ B5
@ ( collect_o
@ ^ [X4: $o] :
( ( member_o @ X4 @ A4 )
& ( P2 @ X4 ) ) ) )
= ( ! [X4: $o] :
( ( member_o @ X4 @ B5 )
=> ( P2 @ X4 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_1266_subset__Collect__iff,axiom,
! [B5: set_int,A4: set_int,P2: int > $o] :
( ( ord_less_eq_set_int @ B5 @ A4 )
=> ( ( ord_less_eq_set_int @ B5
@ ( collect_int
@ ^ [X4: int] :
( ( member_int @ X4 @ A4 )
& ( P2 @ X4 ) ) ) )
= ( ! [X4: int] :
( ( member_int @ X4 @ B5 )
=> ( P2 @ X4 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_1267_subset__Collect__iff,axiom,
! [B5: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,P2: product_prod_nat_nat > $o] :
( ( ord_le3146513528884898305at_nat @ B5 @ A4 )
=> ( ( ord_le3146513528884898305at_nat @ B5
@ ( collec3392354462482085612at_nat
@ ^ [X4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X4 @ A4 )
& ( P2 @ X4 ) ) ) )
= ( ! [X4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X4 @ B5 )
=> ( P2 @ X4 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_1268_subset__Collect__iff,axiom,
! [B5: set_nat,A4: set_nat,P2: nat > $o] :
( ( ord_less_eq_set_nat @ B5 @ A4 )
=> ( ( ord_less_eq_set_nat @ B5
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A4 )
& ( P2 @ X4 ) ) ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ B5 )
=> ( P2 @ X4 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_1269_subset__Collect__iff,axiom,
! [B5: set_Pr958786334691620121nt_int,A4: set_Pr958786334691620121nt_int,P2: product_prod_int_int > $o] :
( ( ord_le2843351958646193337nt_int @ B5 @ A4 )
=> ( ( ord_le2843351958646193337nt_int @ B5
@ ( collec213857154873943460nt_int
@ ^ [X4: product_prod_int_int] :
( ( member5262025264175285858nt_int @ X4 @ A4 )
& ( P2 @ X4 ) ) ) )
= ( ! [X4: product_prod_int_int] :
( ( member5262025264175285858nt_int @ X4 @ B5 )
=> ( P2 @ X4 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_1270_prop__restrict,axiom,
! [X2: produc3843707927480180839at_nat,Z5: set_Pr4329608150637261639at_nat,X5: set_Pr4329608150637261639at_nat,P2: produc3843707927480180839at_nat > $o] :
( ( member8757157785044589968at_nat @ X2 @ Z5 )
=> ( ( ord_le1268244103169919719at_nat @ Z5
@ ( collec6321179662152712658at_nat
@ ^ [X4: produc3843707927480180839at_nat] :
( ( member8757157785044589968at_nat @ X4 @ X5 )
& ( P2 @ X4 ) ) ) )
=> ( P2 @ X2 ) ) ) ).
% prop_restrict
thf(fact_1271_prop__restrict,axiom,
! [X2: $o,Z5: set_o,X5: set_o,P2: $o > $o] :
( ( member_o @ X2 @ Z5 )
=> ( ( ord_less_eq_set_o @ Z5
@ ( collect_o
@ ^ [X4: $o] :
( ( member_o @ X4 @ X5 )
& ( P2 @ X4 ) ) ) )
=> ( P2 @ X2 ) ) ) ).
% prop_restrict
thf(fact_1272_prop__restrict,axiom,
! [X2: int,Z5: set_int,X5: set_int,P2: int > $o] :
( ( member_int @ X2 @ Z5 )
=> ( ( ord_less_eq_set_int @ Z5
@ ( collect_int
@ ^ [X4: int] :
( ( member_int @ X4 @ X5 )
& ( P2 @ X4 ) ) ) )
=> ( P2 @ X2 ) ) ) ).
% prop_restrict
thf(fact_1273_prop__restrict,axiom,
! [X2: product_prod_nat_nat,Z5: set_Pr1261947904930325089at_nat,X5: set_Pr1261947904930325089at_nat,P2: product_prod_nat_nat > $o] :
( ( member8440522571783428010at_nat @ X2 @ Z5 )
=> ( ( ord_le3146513528884898305at_nat @ Z5
@ ( collec3392354462482085612at_nat
@ ^ [X4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X4 @ X5 )
& ( P2 @ X4 ) ) ) )
=> ( P2 @ X2 ) ) ) ).
% prop_restrict
thf(fact_1274_prop__restrict,axiom,
! [X2: nat,Z5: set_nat,X5: set_nat,P2: nat > $o] :
( ( member_nat @ X2 @ Z5 )
=> ( ( ord_less_eq_set_nat @ Z5
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ X5 )
& ( P2 @ X4 ) ) ) )
=> ( P2 @ X2 ) ) ) ).
% prop_restrict
thf(fact_1275_prop__restrict,axiom,
! [X2: product_prod_int_int,Z5: set_Pr958786334691620121nt_int,X5: set_Pr958786334691620121nt_int,P2: product_prod_int_int > $o] :
( ( member5262025264175285858nt_int @ X2 @ Z5 )
=> ( ( ord_le2843351958646193337nt_int @ Z5
@ ( collec213857154873943460nt_int
@ ^ [X4: product_prod_int_int] :
( ( member5262025264175285858nt_int @ X4 @ X5 )
& ( P2 @ X4 ) ) ) )
=> ( P2 @ X2 ) ) ) ).
% prop_restrict
thf(fact_1276_Collect__restrict,axiom,
! [X5: set_Pr4329608150637261639at_nat,P2: produc3843707927480180839at_nat > $o] :
( ord_le1268244103169919719at_nat
@ ( collec6321179662152712658at_nat
@ ^ [X4: produc3843707927480180839at_nat] :
( ( member8757157785044589968at_nat @ X4 @ X5 )
& ( P2 @ X4 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_1277_Collect__restrict,axiom,
! [X5: set_o,P2: $o > $o] :
( ord_less_eq_set_o
@ ( collect_o
@ ^ [X4: $o] :
( ( member_o @ X4 @ X5 )
& ( P2 @ X4 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_1278_Collect__restrict,axiom,
! [X5: set_int,P2: int > $o] :
( ord_less_eq_set_int
@ ( collect_int
@ ^ [X4: int] :
( ( member_int @ X4 @ X5 )
& ( P2 @ X4 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_1279_Collect__restrict,axiom,
! [X5: set_Pr1261947904930325089at_nat,P2: product_prod_nat_nat > $o] :
( ord_le3146513528884898305at_nat
@ ( collec3392354462482085612at_nat
@ ^ [X4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X4 @ X5 )
& ( P2 @ X4 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_1280_Collect__restrict,axiom,
! [X5: set_nat,P2: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ X5 )
& ( P2 @ X4 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_1281_Collect__restrict,axiom,
! [X5: set_Pr958786334691620121nt_int,P2: product_prod_int_int > $o] :
( ord_le2843351958646193337nt_int
@ ( collec213857154873943460nt_int
@ ^ [X4: product_prod_int_int] :
( ( member5262025264175285858nt_int @ X4 @ X5 )
& ( P2 @ X4 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_1282_fun_Omap__ident,axiom,
! [T2: nat > $o] :
( ( comp_o_o_nat
@ ^ [X4: $o] : X4
@ T2 )
= T2 ) ).
% fun.map_ident
thf(fact_1283_fun_Omap__ident,axiom,
! [T2: nat > nat] :
( ( comp_nat_nat_nat
@ ^ [X4: nat] : X4
@ T2 )
= T2 ) ).
% fun.map_ident
thf(fact_1284_case__prod__app,axiom,
( produc27273713700761075at_nat
= ( ^ [F2: nat > nat > product_prod_nat_nat > product_prod_nat_nat,X4: product_prod_nat_nat,Y5: product_prod_nat_nat] :
( produc2626176000494625587at_nat
@ ^ [L3: nat,R5: nat] : ( F2 @ L3 @ R5 @ Y5 )
@ X4 ) ) ) ).
% case_prod_app
thf(fact_1285_case__prod__app,axiom,
( produc8739625826339149834_nat_o
= ( ^ [F2: nat > nat > product_prod_nat_nat > $o,X4: product_prod_nat_nat,Y5: product_prod_nat_nat] :
( produc6081775807080527818_nat_o
@ ^ [L3: nat,R5: nat] : ( F2 @ L3 @ R5 @ Y5 )
@ X4 ) ) ) ).
% case_prod_app
thf(fact_1286_prod_Ocase__distrib,axiom,
! [H: $o > $o,F: int > int > $o,Prod: product_prod_int_int] :
( ( H @ ( produc4947309494688390418_int_o @ F @ Prod ) )
= ( produc4947309494688390418_int_o
@ ^ [X12: int,X23: int] : ( H @ ( F @ X12 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_1287_prod_Ocase__distrib,axiom,
! [H: $o > int,F: int > int > $o,Prod: product_prod_int_int] :
( ( H @ ( produc4947309494688390418_int_o @ F @ Prod ) )
= ( produc8211389475949308722nt_int
@ ^ [X12: int,X23: int] : ( H @ ( F @ X12 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_1288_prod_Ocase__distrib,axiom,
! [H: int > $o,F: int > int > int,Prod: product_prod_int_int] :
( ( H @ ( produc8211389475949308722nt_int @ F @ Prod ) )
= ( produc4947309494688390418_int_o
@ ^ [X12: int,X23: int] : ( H @ ( F @ X12 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_1289_prod_Ocase__distrib,axiom,
! [H: int > int,F: int > int > int,Prod: product_prod_int_int] :
( ( H @ ( produc8211389475949308722nt_int @ F @ Prod ) )
= ( produc8211389475949308722nt_int
@ ^ [X12: int,X23: int] : ( H @ ( F @ X12 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_1290_prod_Ocase__distrib,axiom,
! [H: product_prod_int_int > $o,F: int > int > product_prod_int_int,Prod: product_prod_int_int] :
( ( H @ ( produc4245557441103728435nt_int @ F @ Prod ) )
= ( produc4947309494688390418_int_o
@ ^ [X12: int,X23: int] : ( H @ ( F @ X12 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_1291_prod_Ocase__distrib,axiom,
! [H: product_prod_int_int > int,F: int > int > product_prod_int_int,Prod: product_prod_int_int] :
( ( H @ ( produc4245557441103728435nt_int @ F @ Prod ) )
= ( produc8211389475949308722nt_int
@ ^ [X12: int,X23: int] : ( H @ ( F @ X12 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_1292_prod_Ocase__distrib,axiom,
! [H: $o > product_prod_int_int,F: int > int > $o,Prod: product_prod_int_int] :
( ( H @ ( produc4947309494688390418_int_o @ F @ Prod ) )
= ( produc4245557441103728435nt_int
@ ^ [X12: int,X23: int] : ( H @ ( F @ X12 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_1293_prod_Ocase__distrib,axiom,
! [H: int > product_prod_int_int,F: int > int > int,Prod: product_prod_int_int] :
( ( H @ ( produc8211389475949308722nt_int @ F @ Prod ) )
= ( produc4245557441103728435nt_int
@ ^ [X12: int,X23: int] : ( H @ ( F @ X12 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_1294_prod_Ocase__distrib,axiom,
! [H: product_prod_int_int > product_prod_int_int,F: int > int > product_prod_int_int,Prod: product_prod_int_int] :
( ( H @ ( produc4245557441103728435nt_int @ F @ Prod ) )
= ( produc4245557441103728435nt_int
@ ^ [X12: int,X23: int] : ( H @ ( F @ X12 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_1295_prod_Ocase__distrib,axiom,
! [H: ( product_prod_nat_nat > $o ) > product_prod_nat_nat > $o,F: nat > nat > product_prod_nat_nat > $o,Prod: product_prod_nat_nat] :
( ( H @ ( produc8739625826339149834_nat_o @ F @ Prod ) )
= ( produc8739625826339149834_nat_o
@ ^ [X12: nat,X23: nat] : ( H @ ( F @ X12 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_1296_prod_Odisc__eq__case,axiom,
! [Prod: product_prod_int_int] :
( produc4947309494688390418_int_o
@ ^ [Uu: int,Uv: int] : $true
@ Prod ) ).
% prod.disc_eq_case
thf(fact_1297_image__Collect__subsetI,axiom,
! [P2: int > $o,F: int > nat,B5: set_nat] :
( ! [X: int] :
( ( P2 @ X )
=> ( member_nat @ ( F @ X ) @ B5 ) )
=> ( ord_less_eq_set_nat @ ( image_int_nat @ F @ ( collect_int @ P2 ) ) @ B5 ) ) ).
% image_Collect_subsetI
thf(fact_1298_image__Collect__subsetI,axiom,
! [P2: int > $o,F: int > int,B5: set_int] :
( ! [X: int] :
( ( P2 @ X )
=> ( member_int @ ( F @ X ) @ B5 ) )
=> ( ord_less_eq_set_int @ ( image_int_int @ F @ ( collect_int @ P2 ) ) @ B5 ) ) ).
% image_Collect_subsetI
thf(fact_1299_image__Collect__subsetI,axiom,
! [P2: nat > $o,F: nat > rat,B5: set_rat] :
( ! [X: nat] :
( ( P2 @ X )
=> ( member_rat @ ( F @ X ) @ B5 ) )
=> ( ord_less_eq_set_rat @ ( image_nat_rat @ F @ ( collect_nat @ P2 ) ) @ B5 ) ) ).
% image_Collect_subsetI
thf(fact_1300_image__Collect__subsetI,axiom,
! [P2: nat > $o,F: nat > $o,B5: set_o] :
( ! [X: nat] :
( ( P2 @ X )
=> ( member_o @ ( F @ X ) @ B5 ) )
=> ( ord_less_eq_set_o @ ( image_nat_o @ F @ ( collect_nat @ P2 ) ) @ B5 ) ) ).
% image_Collect_subsetI
thf(fact_1301_image__Collect__subsetI,axiom,
! [P2: nat > $o,F: nat > nat,B5: set_nat] :
( ! [X: nat] :
( ( P2 @ X )
=> ( member_nat @ ( F @ X ) @ B5 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ ( collect_nat @ P2 ) ) @ B5 ) ) ).
% image_Collect_subsetI
thf(fact_1302_image__Collect__subsetI,axiom,
! [P2: nat > $o,F: nat > int,B5: set_int] :
( ! [X: nat] :
( ( P2 @ X )
=> ( member_int @ ( F @ X ) @ B5 ) )
=> ( ord_less_eq_set_int @ ( image_nat_int @ F @ ( collect_nat @ P2 ) ) @ B5 ) ) ).
% image_Collect_subsetI
thf(fact_1303_image__Collect__subsetI,axiom,
! [P2: product_prod_nat_nat > $o,F: product_prod_nat_nat > $o,B5: set_o] :
( ! [X: product_prod_nat_nat] :
( ( P2 @ X )
=> ( member_o @ ( F @ X ) @ B5 ) )
=> ( ord_less_eq_set_o @ ( image_3693632289388996572_nat_o @ F @ ( collec3392354462482085612at_nat @ P2 ) ) @ B5 ) ) ).
% image_Collect_subsetI
thf(fact_1304_image__Collect__subsetI,axiom,
! [P2: product_prod_nat_nat > $o,F: product_prod_nat_nat > nat,B5: set_nat] :
( ! [X: product_prod_nat_nat] :
( ( P2 @ X )
=> ( member_nat @ ( F @ X ) @ B5 ) )
=> ( ord_less_eq_set_nat @ ( image_2486076414777270412at_nat @ F @ ( collec3392354462482085612at_nat @ P2 ) ) @ B5 ) ) ).
% image_Collect_subsetI
thf(fact_1305_image__Collect__subsetI,axiom,
! [P2: product_prod_nat_nat > $o,F: product_prod_nat_nat > int,B5: set_int] :
( ! [X: product_prod_nat_nat] :
( ( P2 @ X )
=> ( member_int @ ( F @ X ) @ B5 ) )
=> ( ord_less_eq_set_int @ ( image_2483585944268220136at_int @ F @ ( collec3392354462482085612at_nat @ P2 ) ) @ B5 ) ) ).
% image_Collect_subsetI
thf(fact_1306_image__Collect__subsetI,axiom,
! [P2: nat > $o,F: nat > product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
( ! [X: nat] :
( ( P2 @ X )
=> ( member8440522571783428010at_nat @ ( F @ X ) @ B5 ) )
=> ( ord_le3146513528884898305at_nat @ ( image_5846123807819985514at_nat @ F @ ( collect_nat @ P2 ) ) @ B5 ) ) ).
% image_Collect_subsetI
thf(fact_1307_range__composition,axiom,
! [F: nat > rat,G: char > nat] :
( ( image_char_rat
@ ^ [X4: char] : ( F @ ( G @ X4 ) )
@ top_top_set_char )
= ( image_nat_rat @ F @ ( image_char_nat @ G @ top_top_set_char ) ) ) ).
% range_composition
thf(fact_1308_range__composition,axiom,
! [F: nat > nat,G: char > nat] :
( ( image_char_nat
@ ^ [X4: char] : ( F @ ( G @ X4 ) )
@ top_top_set_char )
= ( image_nat_nat @ F @ ( image_char_nat @ G @ top_top_set_char ) ) ) ).
% range_composition
thf(fact_1309_range__composition,axiom,
! [F: nat > int,G: char > nat] :
( ( image_char_int
@ ^ [X4: char] : ( F @ ( G @ X4 ) )
@ top_top_set_char )
= ( image_nat_int @ F @ ( image_char_nat @ G @ top_top_set_char ) ) ) ).
% range_composition
thf(fact_1310_range__composition,axiom,
! [F: int > nat,G: char > int] :
( ( image_char_nat
@ ^ [X4: char] : ( F @ ( G @ X4 ) )
@ top_top_set_char )
= ( image_int_nat @ F @ ( image_char_int @ G @ top_top_set_char ) ) ) ).
% range_composition
thf(fact_1311_range__composition,axiom,
! [F: int > int,G: char > int] :
( ( image_char_int
@ ^ [X4: char] : ( F @ ( G @ X4 ) )
@ top_top_set_char )
= ( image_int_int @ F @ ( image_char_int @ G @ top_top_set_char ) ) ) ).
% range_composition
thf(fact_1312_range__composition,axiom,
! [F: nat > rat,G: rat > nat] :
( ( image_rat_rat
@ ^ [X4: rat] : ( F @ ( G @ X4 ) )
@ top_top_set_rat )
= ( image_nat_rat @ F @ ( image_rat_nat @ G @ top_top_set_rat ) ) ) ).
% range_composition
thf(fact_1313_range__composition,axiom,
! [F: nat > nat,G: rat > nat] :
( ( image_rat_nat
@ ^ [X4: rat] : ( F @ ( G @ X4 ) )
@ top_top_set_rat )
= ( image_nat_nat @ F @ ( image_rat_nat @ G @ top_top_set_rat ) ) ) ).
% range_composition
thf(fact_1314_range__composition,axiom,
! [F: nat > int,G: rat > nat] :
( ( image_rat_int
@ ^ [X4: rat] : ( F @ ( G @ X4 ) )
@ top_top_set_rat )
= ( image_nat_int @ F @ ( image_rat_nat @ G @ top_top_set_rat ) ) ) ).
% range_composition
thf(fact_1315_range__composition,axiom,
! [F: int > nat,G: rat > int] :
( ( image_rat_nat
@ ^ [X4: rat] : ( F @ ( G @ X4 ) )
@ top_top_set_rat )
= ( image_int_nat @ F @ ( image_rat_int @ G @ top_top_set_rat ) ) ) ).
% range_composition
thf(fact_1316_range__composition,axiom,
! [F: int > int,G: rat > int] :
( ( image_rat_int
@ ^ [X4: rat] : ( F @ ( G @ X4 ) )
@ top_top_set_rat )
= ( image_int_int @ F @ ( image_rat_int @ G @ top_top_set_rat ) ) ) ).
% range_composition
thf(fact_1317_rangeE,axiom,
! [B: $o,F: char > $o] :
( ( member_o @ B @ ( image_char_o @ F @ top_top_set_char ) )
=> ~ ! [X: char] :
( B
= ( ~ ( F @ X ) ) ) ) ).
% rangeE
thf(fact_1318_rangeE,axiom,
! [B: nat,F: char > nat] :
( ( member_nat @ B @ ( image_char_nat @ F @ top_top_set_char ) )
=> ~ ! [X: char] :
( B
!= ( F @ X ) ) ) ).
% rangeE
thf(fact_1319_rangeE,axiom,
! [B: int,F: char > int] :
( ( member_int @ B @ ( image_char_int @ F @ top_top_set_char ) )
=> ~ ! [X: char] :
( B
!= ( F @ X ) ) ) ).
% rangeE
thf(fact_1320_rangeE,axiom,
! [B: $o,F: rat > $o] :
( ( member_o @ B @ ( image_rat_o @ F @ top_top_set_rat ) )
=> ~ ! [X: rat] :
( B
= ( ~ ( F @ X ) ) ) ) ).
% rangeE
thf(fact_1321_rangeE,axiom,
! [B: nat,F: rat > nat] :
( ( member_nat @ B @ ( image_rat_nat @ F @ top_top_set_rat ) )
=> ~ ! [X: rat] :
( B
!= ( F @ X ) ) ) ).
% rangeE
thf(fact_1322_rangeE,axiom,
! [B: int,F: rat > int] :
( ( member_int @ B @ ( image_rat_int @ F @ top_top_set_rat ) )
=> ~ ! [X: rat] :
( B
!= ( F @ X ) ) ) ).
% rangeE
thf(fact_1323_rangeE,axiom,
! [B: rat,F: nat > rat] :
( ( member_rat @ B @ ( image_nat_rat @ F @ top_top_set_nat ) )
=> ~ ! [X: nat] :
( B
!= ( F @ X ) ) ) ).
% rangeE
thf(fact_1324_rangeE,axiom,
! [B: $o,F: nat > $o] :
( ( member_o @ B @ ( image_nat_o @ F @ top_top_set_nat ) )
=> ~ ! [X: nat] :
( B
= ( ~ ( F @ X ) ) ) ) ).
% rangeE
thf(fact_1325_rangeE,axiom,
! [B: nat,F: nat > nat] :
( ( member_nat @ B @ ( image_nat_nat @ F @ top_top_set_nat ) )
=> ~ ! [X: nat] :
( B
!= ( F @ X ) ) ) ).
% rangeE
thf(fact_1326_rangeE,axiom,
! [B: int,F: nat > int] :
( ( member_int @ B @ ( image_nat_int @ F @ top_top_set_nat ) )
=> ~ ! [X: nat] :
( B
!= ( F @ X ) ) ) ).
% rangeE
thf(fact_1327_fun_Orel__map_I1_J,axiom,
! [Sb: num > num > $o,I: nat > num,X2: int > nat,Y3: int > num] :
( ( bNF_re7626690874201225453um_num
@ ^ [Y2: int,Z: int] : Y2 = Z
@ Sb
@ ( comp_nat_num_int @ I @ X2 )
@ Y3 )
= ( bNF_re272988434038980259at_num
@ ^ [Y2: int,Z: int] : Y2 = Z
@ ^ [X4: nat] : ( Sb @ ( I @ X4 ) )
@ X2
@ Y3 ) ) ).
% fun.rel_map(1)
thf(fact_1328_fun_Orel__map_I1_J,axiom,
! [Sb: num > num > $o,I: num > num,X2: int > num,Y3: int > num] :
( ( bNF_re7626690874201225453um_num
@ ^ [Y2: int,Z: int] : Y2 = Z
@ Sb
@ ( comp_num_num_int @ I @ X2 )
@ Y3 )
= ( bNF_re7626690874201225453um_num
@ ^ [Y2: int,Z: int] : Y2 = Z
@ ^ [X4: num] : ( Sb @ ( I @ X4 ) )
@ X2
@ Y3 ) ) ).
% fun.rel_map(1)
thf(fact_1329_fun_Orel__map_I2_J,axiom,
! [Sa: num > num > $o,X2: int > num,G: nat > num,Y3: int > nat] :
( ( bNF_re7626690874201225453um_num
@ ^ [Y2: int,Z: int] : Y2 = Z
@ Sa
@ X2
@ ( comp_nat_num_int @ G @ Y3 ) )
= ( bNF_re1845987051190671011um_nat
@ ^ [Y2: int,Z: int] : Y2 = Z
@ ^ [X4: num,Y5: nat] : ( Sa @ X4 @ ( G @ Y5 ) )
@ X2
@ Y3 ) ) ).
% fun.rel_map(2)
thf(fact_1330_fun_Orel__map_I2_J,axiom,
! [Sa: num > num > $o,X2: int > num,G: num > num,Y3: int > num] :
( ( bNF_re7626690874201225453um_num
@ ^ [Y2: int,Z: int] : Y2 = Z
@ Sa
@ X2
@ ( comp_num_num_int @ G @ Y3 ) )
= ( bNF_re7626690874201225453um_num
@ ^ [Y2: int,Z: int] : Y2 = Z
@ ^ [X4: num,Y5: num] : ( Sa @ X4 @ ( G @ Y5 ) )
@ X2
@ Y3 ) ) ).
% fun.rel_map(2)
thf(fact_1331_case__prodE2,axiom,
! [Q2: ( product_prod_nat_nat > product_prod_nat_nat ) > $o,P2: nat > nat > product_prod_nat_nat > product_prod_nat_nat,Z2: product_prod_nat_nat] :
( ( Q2 @ ( produc27273713700761075at_nat @ P2 @ Z2 ) )
=> ~ ! [X: nat,Y: nat] :
( ( Z2
= ( product_Pair_nat_nat @ X @ Y ) )
=> ~ ( Q2 @ ( P2 @ X @ Y ) ) ) ) ).
% case_prodE2
thf(fact_1332_case__prodE2,axiom,
! [Q2: ( product_prod_nat_nat > $o ) > $o,P2: nat > nat > product_prod_nat_nat > $o,Z2: product_prod_nat_nat] :
( ( Q2 @ ( produc8739625826339149834_nat_o @ P2 @ Z2 ) )
=> ~ ! [X: nat,Y: nat] :
( ( Z2
= ( product_Pair_nat_nat @ X @ Y ) )
=> ~ ( Q2 @ ( P2 @ X @ Y ) ) ) ) ).
% case_prodE2
thf(fact_1333_case__prodE2,axiom,
! [Q2: product_prod_int_int > $o,P2: int > int > product_prod_int_int,Z2: product_prod_int_int] :
( ( Q2 @ ( produc4245557441103728435nt_int @ P2 @ Z2 ) )
=> ~ ! [X: int,Y: int] :
( ( Z2
= ( product_Pair_int_int @ X @ Y ) )
=> ~ ( Q2 @ ( P2 @ X @ Y ) ) ) ) ).
% case_prodE2
thf(fact_1334_case__prodE2,axiom,
! [Q2: $o > $o,P2: int > int > $o,Z2: product_prod_int_int] :
( ( Q2 @ ( produc4947309494688390418_int_o @ P2 @ Z2 ) )
=> ~ ! [X: int,Y: int] :
( ( Z2
= ( product_Pair_int_int @ X @ Y ) )
=> ~ ( Q2 @ ( P2 @ X @ Y ) ) ) ) ).
% case_prodE2
thf(fact_1335_case__prodE2,axiom,
! [Q2: int > $o,P2: int > int > int,Z2: product_prod_int_int] :
( ( Q2 @ ( produc8211389475949308722nt_int @ P2 @ Z2 ) )
=> ~ ! [X: int,Y: int] :
( ( Z2
= ( product_Pair_int_int @ X @ Y ) )
=> ~ ( Q2 @ ( P2 @ X @ Y ) ) ) ) ).
% case_prodE2
thf(fact_1336_case__prod__eta,axiom,
! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat] :
( ( produc27273713700761075at_nat
@ ^ [X4: nat,Y5: nat] : ( F @ ( product_Pair_nat_nat @ X4 @ Y5 ) ) )
= F ) ).
% case_prod_eta
thf(fact_1337_case__prod__eta,axiom,
! [F: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ( produc8739625826339149834_nat_o
@ ^ [X4: nat,Y5: nat] : ( F @ ( product_Pair_nat_nat @ X4 @ Y5 ) ) )
= F ) ).
% case_prod_eta
thf(fact_1338_case__prod__eta,axiom,
! [F: product_prod_int_int > product_prod_int_int] :
( ( produc4245557441103728435nt_int
@ ^ [X4: int,Y5: int] : ( F @ ( product_Pair_int_int @ X4 @ Y5 ) ) )
= F ) ).
% case_prod_eta
thf(fact_1339_case__prod__eta,axiom,
! [F: product_prod_int_int > $o] :
( ( produc4947309494688390418_int_o
@ ^ [X4: int,Y5: int] : ( F @ ( product_Pair_int_int @ X4 @ Y5 ) ) )
= F ) ).
% case_prod_eta
thf(fact_1340_case__prod__eta,axiom,
! [F: product_prod_int_int > int] :
( ( produc8211389475949308722nt_int
@ ^ [X4: int,Y5: int] : ( F @ ( product_Pair_int_int @ X4 @ Y5 ) ) )
= F ) ).
% case_prod_eta
thf(fact_1341_cond__case__prod__eta,axiom,
! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,G: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat] :
( ! [X: nat,Y: nat] :
( ( F @ X @ Y )
= ( G @ ( product_Pair_nat_nat @ X @ Y ) ) )
=> ( ( produc27273713700761075at_nat @ F )
= G ) ) ).
% cond_case_prod_eta
thf(fact_1342_cond__case__prod__eta,axiom,
! [F: nat > nat > product_prod_nat_nat > $o,G: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ! [X: nat,Y: nat] :
( ( F @ X @ Y )
= ( G @ ( product_Pair_nat_nat @ X @ Y ) ) )
=> ( ( produc8739625826339149834_nat_o @ F )
= G ) ) ).
% cond_case_prod_eta
thf(fact_1343_cond__case__prod__eta,axiom,
! [F: int > int > product_prod_int_int,G: product_prod_int_int > product_prod_int_int] :
( ! [X: int,Y: int] :
( ( F @ X @ Y )
= ( G @ ( product_Pair_int_int @ X @ Y ) ) )
=> ( ( produc4245557441103728435nt_int @ F )
= G ) ) ).
% cond_case_prod_eta
thf(fact_1344_cond__case__prod__eta,axiom,
! [F: int > int > $o,G: product_prod_int_int > $o] :
( ! [X: int,Y: int] :
( ( F @ X @ Y )
= ( G @ ( product_Pair_int_int @ X @ Y ) ) )
=> ( ( produc4947309494688390418_int_o @ F )
= G ) ) ).
% cond_case_prod_eta
thf(fact_1345_cond__case__prod__eta,axiom,
! [F: int > int > int,G: product_prod_int_int > int] :
( ! [X: int,Y: int] :
( ( F @ X @ Y )
= ( G @ ( product_Pair_int_int @ X @ Y ) ) )
=> ( ( produc8211389475949308722nt_int @ F )
= G ) ) ).
% cond_case_prod_eta
thf(fact_1346_case__prodD,axiom,
! [F: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
( ( produc9020218426428501292_nat_o @ F @ ( produc9060074326276436823at_nat @ A @ B ) )
=> ( F @ A @ B ) ) ).
% case_prodD
thf(fact_1347_case__prodD,axiom,
! [F: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
( ( produc410239310623530412_nat_o @ F @ ( produc2922128104949294807at_nat @ A @ B ) )
=> ( F @ A @ B ) ) ).
% case_prodD
thf(fact_1348_case__prodD,axiom,
! [F: nat > num > $o,A: nat,B: num] :
( ( produc4927758841916487424_num_o @ F @ ( product_Pair_nat_num @ A @ B ) )
=> ( F @ A @ B ) ) ).
% case_prodD
thf(fact_1349_case__prodD,axiom,
! [F: nat > nat > $o,A: nat,B: nat] :
( ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) )
=> ( F @ A @ B ) ) ).
% case_prodD
thf(fact_1350_case__prodD,axiom,
! [F: int > int > $o,A: int,B: int] :
( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B ) )
=> ( F @ A @ B ) ) ).
% case_prodD
thf(fact_1351_case__prodE,axiom,
! [C: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o,P: produc1319942482725812455at_nat] :
( ( produc9020218426428501292_nat_o @ C @ P )
=> ~ ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
( ( P
= ( produc9060074326276436823at_nat @ X @ Y ) )
=> ~ ( C @ X @ Y ) ) ) ).
% case_prodE
thf(fact_1352_case__prodE,axiom,
! [C: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o,P: produc3843707927480180839at_nat] :
( ( produc410239310623530412_nat_o @ C @ P )
=> ~ ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
( ( P
= ( produc2922128104949294807at_nat @ X @ Y ) )
=> ~ ( C @ X @ Y ) ) ) ).
% case_prodE
thf(fact_1353_case__prodE,axiom,
! [C: nat > num > $o,P: product_prod_nat_num] :
( ( produc4927758841916487424_num_o @ C @ P )
=> ~ ! [X: nat,Y: num] :
( ( P
= ( product_Pair_nat_num @ X @ Y ) )
=> ~ ( C @ X @ Y ) ) ) ).
% case_prodE
thf(fact_1354_case__prodE,axiom,
! [C: nat > nat > $o,P: product_prod_nat_nat] :
( ( produc6081775807080527818_nat_o @ C @ P )
=> ~ ! [X: nat,Y: nat] :
( ( P
= ( product_Pair_nat_nat @ X @ Y ) )
=> ~ ( C @ X @ Y ) ) ) ).
% case_prodE
thf(fact_1355_case__prodE,axiom,
! [C: int > int > $o,P: product_prod_int_int] :
( ( produc4947309494688390418_int_o @ C @ P )
=> ~ ! [X: int,Y: int] :
( ( P
= ( product_Pair_int_int @ X @ Y ) )
=> ~ ( C @ X @ Y ) ) ) ).
% case_prodE
thf(fact_1356_fst__def,axiom,
( produc473321116828791851t_unit
= ( produc4846201513286513633_ref_a
@ ^ [X12: ref_a,X23: heap_e7401611519738050253t_unit] : X12 ) ) ).
% fst_def
thf(fact_1357_fst__def,axiom,
( product_fst_int_int
= ( produc8211389475949308722nt_int
@ ^ [X12: int,X23: int] : X12 ) ) ).
% fst_def
thf(fact_1358_snd__def,axiom,
( produc1126150082675451753t_unit
= ( produc2552614499379209865t_unit
@ ^ [X12: ref_a,X23: heap_e7401611519738050253t_unit] : X23 ) ) ).
% snd_def
thf(fact_1359_snd__def,axiom,
( product_snd_int_int
= ( produc8211389475949308722nt_int
@ ^ [X12: int,X23: int] : X23 ) ) ).
% snd_def
thf(fact_1360_map__prod__def,axiom,
( produc6036585564866528938nt_int
= ( ^ [F2: int > int,G3: int > int] :
( produc4245557441103728435nt_int
@ ^ [X4: int,Y5: int] : ( product_Pair_int_int @ ( F2 @ X4 ) @ ( G3 @ Y5 ) ) ) ) ) ).
% map_prod_def
thf(fact_1361_case__prod__map__prod,axiom,
! [H: nat > nat > product_prod_nat_nat > product_prod_nat_nat,F: nat > nat,G: nat > nat,X2: product_prod_nat_nat] :
( ( produc27273713700761075at_nat @ H @ ( produc6977886695330630970at_nat @ F @ G @ X2 ) )
= ( produc27273713700761075at_nat
@ ^ [L3: nat,R5: nat] : ( H @ ( F @ L3 ) @ ( G @ R5 ) )
@ X2 ) ) ).
% case_prod_map_prod
thf(fact_1362_case__prod__map__prod,axiom,
! [H: nat > nat > product_prod_nat_nat > $o,F: nat > nat,G: nat > nat,X2: product_prod_nat_nat] :
( ( produc8739625826339149834_nat_o @ H @ ( produc6977886695330630970at_nat @ F @ G @ X2 ) )
= ( produc8739625826339149834_nat_o
@ ^ [L3: nat,R5: nat] : ( H @ ( F @ L3 ) @ ( G @ R5 ) )
@ X2 ) ) ).
% case_prod_map_prod
thf(fact_1363_case__prod__map__prod,axiom,
! [H: int > int > product_prod_int_int,F: int > int,G: int > int,X2: product_prod_int_int] :
( ( produc4245557441103728435nt_int @ H @ ( produc6036585564866528938nt_int @ F @ G @ X2 ) )
= ( produc4245557441103728435nt_int
@ ^ [L3: int,R5: int] : ( H @ ( F @ L3 ) @ ( G @ R5 ) )
@ X2 ) ) ).
% case_prod_map_prod
thf(fact_1364_case__prod__map__prod,axiom,
! [H: int > int > $o,F: int > int,G: int > int,X2: product_prod_int_int] :
( ( produc4947309494688390418_int_o @ H @ ( produc6036585564866528938nt_int @ F @ G @ X2 ) )
= ( produc4947309494688390418_int_o
@ ^ [L3: int,R5: int] : ( H @ ( F @ L3 ) @ ( G @ R5 ) )
@ X2 ) ) ).
% case_prod_map_prod
thf(fact_1365_case__prod__map__prod,axiom,
! [H: int > int > int,F: int > int,G: int > int,X2: product_prod_int_int] :
( ( produc8211389475949308722nt_int @ H @ ( produc6036585564866528938nt_int @ F @ G @ X2 ) )
= ( produc8211389475949308722nt_int
@ ^ [L3: int,R5: int] : ( H @ ( F @ L3 ) @ ( G @ R5 ) )
@ X2 ) ) ).
% case_prod_map_prod
thf(fact_1366_converse__relcompp,axiom,
! [R: int > product_prod_nat_nat > $o,S2: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ( conver1866313717161370910at_nat @ ( relcom716700242784427165at_nat @ R @ S2 ) )
= ( relcom8911704206367525917at_int @ ( conver6965824823119931701at_nat @ S2 ) @ ( conver1866313717161370910at_nat @ R ) ) ) ).
% converse_relcompp
thf(fact_1367_converse__relcompp,axiom,
! [R: code_integer > int > $o,S2: int > int > $o] :
( ( conver1462408618118490240er_int @ ( relcom3789353511612409832nt_int @ R @ S2 ) )
= ( relcom967497705283160040nteger @ ( conversep_int_int @ S2 ) @ ( conver1462408618118490240er_int @ R ) ) ) ).
% converse_relcompp
thf(fact_1368_converse__relcompp,axiom,
! [R: product_prod_nat_nat > product_prod_nat_nat > $o,S2: product_prod_nat_nat > int > $o] :
( ( conver4689305806000247544at_int @ ( relcom8911704206367525917at_int @ R @ S2 ) )
= ( relcom716700242784427165at_nat @ ( conver4689305806000247544at_int @ S2 ) @ ( conver6965824823119931701at_nat @ R ) ) ) ).
% converse_relcompp
thf(fact_1369_converse__relcompp,axiom,
! [R: int > int > $o,S2: int > code_integer > $o] :
( ( conver2101108097643427968nteger @ ( relcom967497705283160040nteger @ R @ S2 ) )
= ( relcom3789353511612409832nt_int @ ( conver2101108097643427968nteger @ S2 ) @ ( conversep_int_int @ R ) ) ) ).
% converse_relcompp
thf(fact_1370_flip__pred,axiom,
! [A4: set_Pr7459493094073627847at_nat,R3: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o] :
( ( ord_le5997549366648089703at_nat @ A4 @ ( collec7872792499061297490at_nat @ ( produc9020218426428501292_nat_o @ ( conver9095369822871743413at_nat @ R3 ) ) ) )
=> ( ord_le5997549366648089703at_nat
@ ( image_3313167189156628421at_nat
@ ( produc5456455677105495163at_nat
@ ^ [X4: set_Pr4329608150637261639at_nat,Y5: set_Pr4329608150637261639at_nat] : ( produc9060074326276436823at_nat @ Y5 @ X4 ) )
@ A4 )
@ ( collec7872792499061297490at_nat @ ( produc9020218426428501292_nat_o @ R3 ) ) ) ) ).
% flip_pred
thf(fact_1371_flip__pred,axiom,
! [A4: set_Pr4329608150637261639at_nat,R3: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o] :
( ( ord_le1268244103169919719at_nat @ A4 @ ( collec6321179662152712658at_nat @ ( produc410239310623530412_nat_o @ ( conver5243090411617913141at_nat @ R3 ) ) ) )
=> ( ord_le1268244103169919719at_nat
@ ( image_960689587819713477at_nat
@ ( produc7548845685434208379at_nat
@ ^ [X4: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat] : ( produc2922128104949294807at_nat @ Y5 @ X4 ) )
@ A4 )
@ ( collec6321179662152712658at_nat @ ( produc410239310623530412_nat_o @ R3 ) ) ) ) ).
% flip_pred
thf(fact_1372_flip__pred,axiom,
! [A4: set_Pr3280342998896498603um_nat,R3: nat > num > $o] :
( ( ord_le5164908622851071819um_nat @ A4 @ ( collec1522684865789554998um_nat @ ( produc6857965554392702720_nat_o @ ( conversep_nat_num @ R3 ) ) ) )
=> ( ord_le8085105155179020875at_num
@ ( image_7762062833379733445at_num
@ ( produc5219324331026901427at_num
@ ^ [X4: num,Y5: nat] : ( product_Pair_nat_num @ Y5 @ X4 ) )
@ A4 )
@ ( collec4100598399430923318at_num @ ( produc4927758841916487424_num_o @ R3 ) ) ) ) ).
% flip_pred
thf(fact_1373_flip__pred,axiom,
! [A4: set_Pr1261947904930325089at_nat,R3: nat > nat > $o] :
( ( ord_le3146513528884898305at_nat @ A4 @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ( conversep_nat_nat @ R3 ) ) ) )
=> ( ord_le3146513528884898305at_nat
@ ( image_5168914502847457605at_nat
@ ( produc2626176000494625587at_nat
@ ^ [X4: nat,Y5: nat] : ( product_Pair_nat_nat @ Y5 @ X4 ) )
@ A4 )
@ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R3 ) ) ) ) ).
% flip_pred
thf(fact_1374_flip__pred,axiom,
! [A4: set_Pr958786334691620121nt_int,R3: int > int > $o] :
( ( ord_le2843351958646193337nt_int @ A4 @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ ( conversep_int_int @ R3 ) ) ) )
=> ( ord_le2843351958646193337nt_int
@ ( image_2653370878348428101nt_int
@ ( produc4245557441103728435nt_int
@ ^ [X4: int,Y5: int] : ( product_Pair_int_int @ Y5 @ X4 ) )
@ A4 )
@ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ R3 ) ) ) ) ).
% flip_pred
thf(fact_1375_type__copy__wit,axiom,
! [Rep: product_unit > $o,Abs: $o > product_unit,X2: produc3843707927480180839at_nat,S5: $o > set_Pr4329608150637261639at_nat,Y3: $o] :
( ( type_d6188575255521822967unit_o @ Rep @ Abs @ top_top_set_o )
=> ( ( member8757157785044589968at_nat @ X2 @ ( comp_o4913564575355137019t_unit @ S5 @ Rep @ ( Abs @ Y3 ) ) )
=> ( member8757157785044589968at_nat @ X2 @ ( S5 @ Y3 ) ) ) ) ).
% type_copy_wit
thf(fact_1376_type__copy__wit,axiom,
! [Rep: product_unit > $o,Abs: $o > product_unit,X2: product_prod_nat_nat,S5: $o > set_Pr1261947904930325089at_nat,Y3: $o] :
( ( type_d6188575255521822967unit_o @ Rep @ Abs @ top_top_set_o )
=> ( ( member8440522571783428010at_nat @ X2 @ ( comp_o2254869825208929t_unit @ S5 @ Rep @ ( Abs @ Y3 ) ) )
=> ( member8440522571783428010at_nat @ X2 @ ( S5 @ Y3 ) ) ) ) ).
% type_copy_wit
thf(fact_1377_type__copy__wit,axiom,
! [Rep: product_unit > $o,Abs: $o > product_unit,X2: $o,S5: $o > set_o,Y3: $o] :
( ( type_d6188575255521822967unit_o @ Rep @ Abs @ top_top_set_o )
=> ( ( member_o @ X2 @ ( comp_o6895741221744486154t_unit @ S5 @ Rep @ ( Abs @ Y3 ) ) )
=> ( member_o @ X2 @ ( S5 @ Y3 ) ) ) ) ).
% type_copy_wit
thf(fact_1378_type__copy__wit,axiom,
! [Rep: product_unit > $o,Abs: $o > product_unit,X2: nat,S5: $o > set_nat,Y3: $o] :
( ( type_d6188575255521822967unit_o @ Rep @ Abs @ top_top_set_o )
=> ( ( member_nat @ X2 @ ( comp_o9190160103024335696t_unit @ S5 @ Rep @ ( Abs @ Y3 ) ) )
=> ( member_nat @ X2 @ ( S5 @ Y3 ) ) ) ) ).
% type_copy_wit
thf(fact_1379_type__copy__wit,axiom,
! [Rep: product_unit > $o,Abs: $o > product_unit,X2: int,S5: $o > set_int,Y3: $o] :
( ( type_d6188575255521822967unit_o @ Rep @ Abs @ top_top_set_o )
=> ( ( member_int @ X2 @ ( comp_o1521574261646630388t_unit @ S5 @ Rep @ ( Abs @ Y3 ) ) )
=> ( member_int @ X2 @ ( S5 @ Y3 ) ) ) ) ).
% type_copy_wit
thf(fact_1380_type__copy__vimage2p__Grp__Rep,axiom,
! [Rep: $o > $o,Abs: $o > $o,F: nat > nat,P2: nat > $o,H: nat > $o] :
( ( type_definition_o_o @ Rep @ Abs @ top_top_set_o )
=> ( ( bNF_vi6818137723729380987_o_o_o @ F @ Rep @ ( bNF_Grp_nat_o @ ( collect_nat @ P2 ) @ H ) )
= ( bNF_Grp_nat_o
@ ( collect_nat
@ ^ [X4: nat] : ( P2 @ ( F @ X4 ) ) )
@ ( comp_nat_o_nat @ ( comp_o_o_nat @ Abs @ H ) @ F ) ) ) ) ).
% type_copy_vimage2p_Grp_Rep
thf(fact_1381_type__copy__vimage2p__Grp__Rep,axiom,
! [Rep: nat > char,Abs: char > nat,F: int > int,P2: int > $o,H: int > char] :
( ( type_d5945782884131516111t_char @ Rep @ Abs @ top_top_set_char )
=> ( ( bNF_vi9149698133064535136char_o @ F @ Rep @ ( bNF_Grp_int_char @ ( collect_int @ P2 ) @ H ) )
= ( bNF_Grp_int_nat
@ ( collect_int
@ ^ [X4: int] : ( P2 @ ( F @ X4 ) ) )
@ ( comp_int_nat_int @ ( comp_char_nat_int @ Abs @ H ) @ F ) ) ) ) ).
% type_copy_vimage2p_Grp_Rep
thf(fact_1382_type__copy__vimage2p__Grp__Rep,axiom,
! [Rep: $o > char,Abs: char > $o,F: nat > $o,P2: $o > $o,H: $o > char] :
( ( type_d9076672964931703357o_char @ Rep @ Abs @ top_top_set_char )
=> ( ( bNF_vi8816436626763797690char_o @ F @ Rep @ ( bNF_Grp_o_char @ ( collect_o @ P2 ) @ H ) )
= ( bNF_Grp_nat_o
@ ( collect_nat
@ ^ [X4: nat] : ( P2 @ ( F @ X4 ) ) )
@ ( comp_o_o_nat @ ( comp_char_o_o @ Abs @ H ) @ F ) ) ) ) ).
% type_copy_vimage2p_Grp_Rep
thf(fact_1383_type__copy__vimage2p__Grp__Rep,axiom,
! [Rep: num > char,Abs: char > num,F: int > nat,P2: nat > $o,H: nat > char] :
( ( type_d4396077731765706521m_char @ Rep @ Abs @ top_top_set_char )
=> ( ( bNF_vi1300075849944819442char_o @ F @ Rep @ ( bNF_Grp_nat_char @ ( collect_nat @ P2 ) @ H ) )
= ( bNF_Grp_int_num
@ ( collect_int
@ ^ [X4: int] : ( P2 @ ( F @ X4 ) ) )
@ ( comp_nat_num_int @ ( comp_char_num_nat @ Abs @ H ) @ F ) ) ) ) ).
% type_copy_vimage2p_Grp_Rep
thf(fact_1384_type__copy__vimage2p__Grp__Rep,axiom,
! [Rep: nat > char,Abs: char > nat,F: nat > nat,P2: nat > $o,H: nat > char] :
( ( type_d5945782884131516111t_char @ Rep @ Abs @ top_top_set_char )
=> ( ( bNF_vi8943400373106913048char_o @ F @ Rep @ ( bNF_Grp_nat_char @ ( collect_nat @ P2 ) @ H ) )
= ( bNF_Grp_nat_nat
@ ( collect_nat
@ ^ [X4: nat] : ( P2 @ ( F @ X4 ) ) )
@ ( comp_nat_nat_nat @ ( comp_char_nat_nat @ Abs @ H ) @ F ) ) ) ) ).
% type_copy_vimage2p_Grp_Rep
thf(fact_1385_type__copy__vimage2p__Grp__Rep,axiom,
! [Rep: nat > rat,Abs: rat > nat,F: int > int,P2: int > $o,H: int > rat] :
( ( type_d5615363888691252950at_rat @ Rep @ Abs @ top_top_set_rat )
=> ( ( bNF_vi1281217848275042457_rat_o @ F @ Rep @ ( bNF_Grp_int_rat @ ( collect_int @ P2 ) @ H ) )
= ( bNF_Grp_int_nat
@ ( collect_int
@ ^ [X4: int] : ( P2 @ ( F @ X4 ) ) )
@ ( comp_int_nat_int @ ( comp_rat_nat_int @ Abs @ H ) @ F ) ) ) ) ).
% type_copy_vimage2p_Grp_Rep
thf(fact_1386_type__copy__vimage2p__Grp__Rep,axiom,
! [Rep: $o > rat,Abs: rat > $o,F: nat > $o,P2: $o > $o,H: $o > rat] :
( ( type_d6486251586359353412_o_rat @ Rep @ Abs @ top_top_set_rat )
=> ( ( bNF_vi3868162310295888627_rat_o @ F @ Rep @ ( bNF_Grp_o_rat @ ( collect_o @ P2 ) @ H ) )
= ( bNF_Grp_nat_o
@ ( collect_nat
@ ^ [X4: nat] : ( P2 @ ( F @ X4 ) ) )
@ ( comp_o_o_nat @ ( comp_rat_o_o @ Abs @ H ) @ F ) ) ) ) ).
% type_copy_vimage2p_Grp_Rep
thf(fact_1387_type__copy__vimage2p__Grp__Rep,axiom,
! [Rep: num > rat,Abs: rat > num,F: int > nat,P2: nat > $o,H: nat > rat] :
( ( type_d3745694291998722336um_rat @ Rep @ Abs @ top_top_set_rat )
=> ( ( bNF_vi7769575839906753835_rat_o @ F @ Rep @ ( bNF_Grp_nat_rat @ ( collect_nat @ P2 ) @ H ) )
= ( bNF_Grp_int_num
@ ( collect_int
@ ^ [X4: int] : ( P2 @ ( F @ X4 ) ) )
@ ( comp_nat_num_int @ ( comp_rat_num_nat @ Abs @ H ) @ F ) ) ) ) ).
% type_copy_vimage2p_Grp_Rep
thf(fact_1388_type__copy__vimage2p__Grp__Rep,axiom,
! [Rep: nat > rat,Abs: rat > nat,F: nat > nat,P2: nat > $o,H: nat > rat] :
( ( type_d5615363888691252950at_rat @ Rep @ Abs @ top_top_set_rat )
=> ( ( bNF_vi2930123780028963665_rat_o @ F @ Rep @ ( bNF_Grp_nat_rat @ ( collect_nat @ P2 ) @ H ) )
= ( bNF_Grp_nat_nat
@ ( collect_nat
@ ^ [X4: nat] : ( P2 @ ( F @ X4 ) ) )
@ ( comp_nat_nat_nat @ ( comp_rat_nat_nat @ Abs @ H ) @ F ) ) ) ) ).
% type_copy_vimage2p_Grp_Rep
thf(fact_1389_type__copy__vimage2p__Grp__Rep,axiom,
! [Rep: nat > nat,Abs: nat > nat,F: int > int,P2: int > $o,H: int > nat] :
( ( type_d6250493948777748686at_nat @ Rep @ Abs @ top_top_set_nat )
=> ( ( bNF_vi5018550775591610529_nat_o @ F @ Rep @ ( bNF_Grp_int_nat @ ( collect_int @ P2 ) @ H ) )
= ( bNF_Grp_int_nat
@ ( collect_int
@ ^ [X4: int] : ( P2 @ ( F @ X4 ) ) )
@ ( comp_int_nat_int @ ( comp_nat_nat_int @ Abs @ H ) @ F ) ) ) ) ).
% type_copy_vimage2p_Grp_Rep
thf(fact_1390_type__copy__vimage2p__Grp__Abs,axiom,
! [Rep: nat > num,Abs: num > nat,G: nat > int,P2: int > $o,H: int > nat] :
( ( type_d2807825734933527320at_num @ Rep @ Abs @ top_top_set_num )
=> ( ( bNF_vi1731478210338170163_nat_o @ G @ Abs @ ( bNF_Grp_int_nat @ ( collect_int @ P2 ) @ H ) )
= ( bNF_Grp_nat_num
@ ( collect_nat
@ ^ [X4: nat] : ( P2 @ ( G @ X4 ) ) )
@ ( comp_int_num_nat @ ( comp_nat_num_int @ Rep @ H ) @ G ) ) ) ) ).
% type_copy_vimage2p_Grp_Abs
thf(fact_1391_type__copy__vimage2p__Grp__Abs,axiom,
! [Rep: $o > $o,Abs: $o > $o,G: nat > nat,P2: nat > $o,H: nat > $o] :
( ( type_definition_o_o @ Rep @ Abs @ top_top_set_o )
=> ( ( bNF_vi6818137723729380987_o_o_o @ G @ Abs @ ( bNF_Grp_nat_o @ ( collect_nat @ P2 ) @ H ) )
= ( bNF_Grp_nat_o
@ ( collect_nat
@ ^ [X4: nat] : ( P2 @ ( G @ X4 ) ) )
@ ( comp_nat_o_nat @ ( comp_o_o_nat @ Rep @ H ) @ G ) ) ) ) ).
% type_copy_vimage2p_Grp_Abs
thf(fact_1392_type__copy__vimage2p__Grp__Abs,axiom,
! [Rep: int > nat,Abs: nat > int,G: int > int,P2: int > $o,H: int > int] :
( ( type_d7249847660678803242nt_nat @ Rep @ Abs @ top_top_set_nat )
=> ( ( bNF_vi3197508430158922565_int_o @ G @ Abs @ ( bNF_Grp_int_int @ ( collect_int @ P2 ) @ H ) )
= ( bNF_Grp_int_nat
@ ( collect_int
@ ^ [X4: int] : ( P2 @ ( G @ X4 ) ) )
@ ( comp_int_nat_int @ ( comp_int_nat_int @ Rep @ H ) @ G ) ) ) ) ).
% type_copy_vimage2p_Grp_Abs
thf(fact_1393_type__copy__vimage2p__Grp__Abs,axiom,
! [Rep: int > nat,Abs: nat > int,G: nat > int,P2: int > $o,H: int > int] :
( ( type_d7249847660678803242nt_nat @ Rep @ Abs @ top_top_set_nat )
=> ( ( bNF_vi8357618154448083105_int_o @ G @ Abs @ ( bNF_Grp_int_int @ ( collect_int @ P2 ) @ H ) )
= ( bNF_Grp_nat_nat
@ ( collect_nat
@ ^ [X4: nat] : ( P2 @ ( G @ X4 ) ) )
@ ( comp_int_nat_nat @ ( comp_int_nat_int @ Rep @ H ) @ G ) ) ) ) ).
% type_copy_vimage2p_Grp_Abs
thf(fact_1394_type__copy__vimage2p__Grp__Abs,axiom,
! [Rep: nat > nat,Abs: nat > nat,G: nat > nat,P2: nat > $o,H: nat > nat] :
( ( type_d6250493948777748686at_nat @ Rep @ Abs @ top_top_set_nat )
=> ( ( bNF_vi6667456707345531737_nat_o @ G @ Abs @ ( bNF_Grp_nat_nat @ ( collect_nat @ P2 ) @ H ) )
= ( bNF_Grp_nat_nat
@ ( collect_nat
@ ^ [X4: nat] : ( P2 @ ( G @ X4 ) ) )
@ ( comp_nat_nat_nat @ ( comp_nat_nat_nat @ Rep @ H ) @ G ) ) ) ) ).
% type_copy_vimage2p_Grp_Abs
thf(fact_1395_type__copy__vimage2p__Grp__Abs,axiom,
! [Rep: product_unit > $o,Abs: $o > product_unit,G: nat > $o,P2: $o > $o,H: $o > product_unit] :
( ( type_d6188575255521822967unit_o @ Rep @ Abs @ top_top_set_o )
=> ( ( bNF_vi5856767400800367702unit_o @ G @ Abs @ ( bNF_Gr395642070819736771t_unit @ ( collect_o @ P2 ) @ H ) )
= ( bNF_Grp_nat_o
@ ( collect_nat
@ ^ [X4: nat] : ( P2 @ ( G @ X4 ) ) )
@ ( comp_o_o_nat @ ( comp_P5156358013004227690it_o_o @ Rep @ H ) @ G ) ) ) ) ).
% type_copy_vimage2p_Grp_Abs
thf(fact_1396_type__copy__vimage2p__Grp__Abs,axiom,
! [Rep: product_unit > $o,Abs: $o > product_unit,G: nat > nat,P2: nat > $o,H: nat > product_unit] :
( ( type_d6188575255521822967unit_o @ Rep @ Abs @ top_top_set_o )
=> ( ( bNF_vi1443435967014148806unit_o @ G @ Abs @ ( bNF_Gr1993012052113314029t_unit @ ( collect_nat @ P2 ) @ H ) )
= ( bNF_Grp_nat_o
@ ( collect_nat
@ ^ [X4: nat] : ( P2 @ ( G @ X4 ) ) )
@ ( comp_nat_o_nat @ ( comp_P4985893458130756478_o_nat @ Rep @ H ) @ G ) ) ) ) ).
% type_copy_vimage2p_Grp_Abs
thf(fact_1397_type__copy__vimage2p__Grp__Abs,axiom,
! [Rep: nat > num,Abs: num > nat,G: product_prod_nat_nat > int,P2: int > $o,H: int > nat] :
( ( type_d2807825734933527320at_num @ Rep @ Abs @ top_top_set_num )
=> ( ( bNF_vi9094538908245281400_nat_o @ G @ Abs @ ( bNF_Grp_int_nat @ ( collect_int @ P2 ) @ H ) )
= ( bNF_Gr5974128493091723341at_num
@ ( collec3392354462482085612at_nat
@ ^ [X4: product_prod_nat_nat] : ( P2 @ ( G @ X4 ) ) )
@ ( comp_i1186163378102392086at_nat @ ( comp_nat_num_int @ Rep @ H ) @ G ) ) ) ) ).
% type_copy_vimage2p_Grp_Abs
thf(fact_1398_type__copy__vimage2p__Grp__Abs,axiom,
! [Rep: nat > num,Abs: num > nat,G: product_prod_int_int > int,P2: int > $o,H: int > nat] :
( ( type_d2807825734933527320at_num @ Rep @ Abs @ top_top_set_num )
=> ( ( bNF_vi1633389952966819264_nat_o @ G @ Abs @ ( bNF_Grp_int_nat @ ( collect_int @ P2 ) @ H ) )
= ( bNF_Gr8532703628021589765nt_num
@ ( collec213857154873943460nt_int
@ ^ [X4: product_prod_int_int] : ( P2 @ ( G @ X4 ) ) )
@ ( comp_i7231038107349025742nt_int @ ( comp_nat_num_int @ Rep @ H ) @ G ) ) ) ) ).
% type_copy_vimage2p_Grp_Abs
thf(fact_1399_type__copy__vimage2p__Grp__Abs,axiom,
! [Rep: $o > $o,Abs: $o > $o,G: product_prod_nat_nat > nat,P2: nat > $o,H: nat > $o] :
( ( type_definition_o_o @ Rep @ Abs @ top_top_set_o )
=> ( ( bNF_vi9025507403824386624_o_o_o @ G @ Abs @ ( bNF_Grp_nat_o @ ( collect_nat @ P2 ) @ H ) )
= ( bNF_Gr8859634586176591525_nat_o
@ ( collec3392354462482085612at_nat
@ ^ [X4: product_prod_nat_nat] : ( P2 @ ( G @ X4 ) ) )
@ ( comp_n1662990198670838940at_nat @ ( comp_o_o_nat @ Rep @ H ) @ G ) ) ) ) ).
% type_copy_vimage2p_Grp_Abs
thf(fact_1400_exE__realizer,axiom,
! [P2: nat > nat > $o,P: product_prod_nat_nat,Q2: ( product_prod_nat_nat > product_prod_nat_nat ) > $o,F: nat > nat > product_prod_nat_nat > product_prod_nat_nat] :
( ( P2 @ ( product_snd_nat_nat @ P ) @ ( product_fst_nat_nat @ P ) )
=> ( ! [X: nat,Y: nat] :
( ( P2 @ Y @ X )
=> ( Q2 @ ( F @ X @ Y ) ) )
=> ( Q2 @ ( produc27273713700761075at_nat @ F @ P ) ) ) ) ).
% exE_realizer
thf(fact_1401_exE__realizer,axiom,
! [P2: nat > nat > $o,P: product_prod_nat_nat,Q2: ( product_prod_nat_nat > $o ) > $o,F: nat > nat > product_prod_nat_nat > $o] :
( ( P2 @ ( product_snd_nat_nat @ P ) @ ( product_fst_nat_nat @ P ) )
=> ( ! [X: nat,Y: nat] :
( ( P2 @ Y @ X )
=> ( Q2 @ ( F @ X @ Y ) ) )
=> ( Q2 @ ( produc8739625826339149834_nat_o @ F @ P ) ) ) ) ).
% exE_realizer
thf(fact_1402_exE__realizer,axiom,
! [P2: int > int > $o,P: product_prod_int_int,Q2: product_prod_int_int > $o,F: int > int > product_prod_int_int] :
( ( P2 @ ( product_snd_int_int @ P ) @ ( product_fst_int_int @ P ) )
=> ( ! [X: int,Y: int] :
( ( P2 @ Y @ X )
=> ( Q2 @ ( F @ X @ Y ) ) )
=> ( Q2 @ ( produc4245557441103728435nt_int @ F @ P ) ) ) ) ).
% exE_realizer
thf(fact_1403_exE__realizer,axiom,
! [P2: int > int > $o,P: product_prod_int_int,Q2: $o > $o,F: int > int > $o] :
( ( P2 @ ( product_snd_int_int @ P ) @ ( product_fst_int_int @ P ) )
=> ( ! [X: int,Y: int] :
( ( P2 @ Y @ X )
=> ( Q2 @ ( F @ X @ Y ) ) )
=> ( Q2 @ ( produc4947309494688390418_int_o @ F @ P ) ) ) ) ).
% exE_realizer
thf(fact_1404_exE__realizer,axiom,
! [P2: int > int > $o,P: product_prod_int_int,Q2: int > $o,F: int > int > int] :
( ( P2 @ ( product_snd_int_int @ P ) @ ( product_fst_int_int @ P ) )
=> ( ! [X: int,Y: int] :
( ( P2 @ Y @ X )
=> ( Q2 @ ( F @ X @ Y ) ) )
=> ( Q2 @ ( produc8211389475949308722nt_int @ F @ P ) ) ) ) ).
% exE_realizer
thf(fact_1405_case__prod__beta_H,axiom,
( produc27273713700761075at_nat
= ( ^ [F2: nat > nat > product_prod_nat_nat > product_prod_nat_nat,X4: product_prod_nat_nat] : ( F2 @ ( product_fst_nat_nat @ X4 ) @ ( product_snd_nat_nat @ X4 ) ) ) ) ).
% case_prod_beta'
thf(fact_1406_case__prod__beta_H,axiom,
( produc8739625826339149834_nat_o
= ( ^ [F2: nat > nat > product_prod_nat_nat > $o,X4: product_prod_nat_nat] : ( F2 @ ( product_fst_nat_nat @ X4 ) @ ( product_snd_nat_nat @ X4 ) ) ) ) ).
% case_prod_beta'
thf(fact_1407_case__prod__beta_H,axiom,
( produc4245557441103728435nt_int
= ( ^ [F2: int > int > product_prod_int_int,X4: product_prod_int_int] : ( F2 @ ( product_fst_int_int @ X4 ) @ ( product_snd_int_int @ X4 ) ) ) ) ).
% case_prod_beta'
thf(fact_1408_case__prod__beta_H,axiom,
( produc4947309494688390418_int_o
= ( ^ [F2: int > int > $o,X4: product_prod_int_int] : ( F2 @ ( product_fst_int_int @ X4 ) @ ( product_snd_int_int @ X4 ) ) ) ) ).
% case_prod_beta'
thf(fact_1409_case__prod__beta_H,axiom,
( produc8211389475949308722nt_int
= ( ^ [F2: int > int > int,X4: product_prod_int_int] : ( F2 @ ( product_fst_int_int @ X4 ) @ ( product_snd_int_int @ X4 ) ) ) ) ).
% case_prod_beta'
thf(fact_1410_case__prod__unfold,axiom,
( produc27273713700761075at_nat
= ( ^ [C4: nat > nat > product_prod_nat_nat > product_prod_nat_nat,P3: product_prod_nat_nat] : ( C4 @ ( product_fst_nat_nat @ P3 ) @ ( product_snd_nat_nat @ P3 ) ) ) ) ).
% case_prod_unfold
thf(fact_1411_case__prod__unfold,axiom,
( produc8739625826339149834_nat_o
= ( ^ [C4: nat > nat > product_prod_nat_nat > $o,P3: product_prod_nat_nat] : ( C4 @ ( product_fst_nat_nat @ P3 ) @ ( product_snd_nat_nat @ P3 ) ) ) ) ).
% case_prod_unfold
thf(fact_1412_case__prod__unfold,axiom,
( produc4245557441103728435nt_int
= ( ^ [C4: int > int > product_prod_int_int,P3: product_prod_int_int] : ( C4 @ ( product_fst_int_int @ P3 ) @ ( product_snd_int_int @ P3 ) ) ) ) ).
% case_prod_unfold
thf(fact_1413_case__prod__unfold,axiom,
( produc4947309494688390418_int_o
= ( ^ [C4: int > int > $o,P3: product_prod_int_int] : ( C4 @ ( product_fst_int_int @ P3 ) @ ( product_snd_int_int @ P3 ) ) ) ) ).
% case_prod_unfold
thf(fact_1414_case__prod__unfold,axiom,
( produc8211389475949308722nt_int
= ( ^ [C4: int > int > int,P3: product_prod_int_int] : ( C4 @ ( product_fst_int_int @ P3 ) @ ( product_snd_int_int @ P3 ) ) ) ) ).
% case_prod_unfold
thf(fact_1415_pred__subset__eq2,axiom,
! [R3: set_Pr7459493094073627847at_nat,S5: set_Pr7459493094073627847at_nat] :
( ( ord_le3072208448688395470_nat_o
@ ^ [X4: set_Pr4329608150637261639at_nat,Y5: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X4 @ Y5 ) @ R3 )
@ ^ [X4: set_Pr4329608150637261639at_nat,Y5: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X4 @ Y5 ) @ S5 ) )
= ( ord_le5997549366648089703at_nat @ R3 @ S5 ) ) ).
% pred_subset_eq2
thf(fact_1416_pred__subset__eq2,axiom,
! [R3: set_Pr4329608150637261639at_nat,S5: set_Pr4329608150637261639at_nat] :
( ( ord_le3935385432712749774_nat_o
@ ^ [X4: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y5 ) @ R3 )
@ ^ [X4: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y5 ) @ S5 ) )
= ( ord_le1268244103169919719at_nat @ R3 @ S5 ) ) ).
% pred_subset_eq2
thf(fact_1417_pred__subset__eq2,axiom,
! [R3: set_Pr6200539531224447659at_num,S5: set_Pr6200539531224447659at_num] :
( ( ord_le3404735783095501756_num_o
@ ^ [X4: nat,Y5: num] : ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X4 @ Y5 ) @ R3 )
@ ^ [X4: nat,Y5: num] : ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X4 @ Y5 ) @ S5 ) )
= ( ord_le8085105155179020875at_num @ R3 @ S5 ) ) ).
% pred_subset_eq2
thf(fact_1418_pred__subset__eq2,axiom,
! [R3: set_Pr1261947904930325089at_nat,S5: set_Pr1261947904930325089at_nat] :
( ( ord_le2646555220125990790_nat_o
@ ^ [X4: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ R3 )
@ ^ [X4: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ S5 ) )
= ( ord_le3146513528884898305at_nat @ R3 @ S5 ) ) ).
% pred_subset_eq2
thf(fact_1419_pred__subset__eq2,axiom,
! [R3: set_Pr958786334691620121nt_int,S5: set_Pr958786334691620121nt_int] :
( ( ord_le6741204236512500942_int_o
@ ^ [X4: int,Y5: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y5 ) @ R3 )
@ ^ [X4: int,Y5: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y5 ) @ S5 ) )
= ( ord_le2843351958646193337nt_int @ R3 @ S5 ) ) ).
% pred_subset_eq2
thf(fact_1420_comp__fun__commute__on__def,axiom,
( finite8619596206551965927at_nat
= ( ^ [S4: set_Pr4329608150637261639at_nat,F2: produc3843707927480180839at_nat > nat > nat] :
! [X4: produc3843707927480180839at_nat,Y5: produc3843707927480180839at_nat] :
( ( member8757157785044589968at_nat @ X4 @ S4 )
=> ( ( member8757157785044589968at_nat @ Y5 @ S4 )
=> ( ( comp_nat_nat_nat @ ( F2 @ Y5 ) @ ( F2 @ X4 ) )
= ( comp_nat_nat_nat @ ( F2 @ X4 ) @ ( F2 @ Y5 ) ) ) ) ) ) ) ).
% comp_fun_commute_on_def
thf(fact_1421_comp__fun__commute__on__def,axiom,
( finite2110315540964234241at_nat
= ( ^ [S4: set_Pr1261947904930325089at_nat,F2: product_prod_nat_nat > nat > nat] :
! [X4: product_prod_nat_nat,Y5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X4 @ S4 )
=> ( ( member8440522571783428010at_nat @ Y5 @ S4 )
=> ( ( comp_nat_nat_nat @ ( F2 @ Y5 ) @ ( F2 @ X4 ) )
= ( comp_nat_nat_nat @ ( F2 @ X4 ) @ ( F2 @ Y5 ) ) ) ) ) ) ) ).
% comp_fun_commute_on_def
thf(fact_1422_comp__fun__commute__on__def,axiom,
( finite4703585013658510744_o_nat
= ( ^ [S4: set_o,F2: $o > nat > nat] :
! [X4: $o,Y5: $o] :
( ( member_o @ X4 @ S4 )
=> ( ( member_o @ Y5 @ S4 )
=> ( ( comp_nat_nat_nat @ ( F2 @ Y5 ) @ ( F2 @ X4 ) )
= ( comp_nat_nat_nat @ ( F2 @ X4 ) @ ( F2 @ Y5 ) ) ) ) ) ) ) ).
% comp_fun_commute_on_def
thf(fact_1423_comp__fun__commute__on__def,axiom,
( finite3582905537739598962at_nat
= ( ^ [S4: set_nat,F2: nat > nat > nat] :
! [X4: nat,Y5: nat] :
( ( member_nat @ X4 @ S4 )
=> ( ( member_nat @ Y5 @ S4 )
=> ( ( comp_nat_nat_nat @ ( F2 @ Y5 ) @ ( F2 @ X4 ) )
= ( comp_nat_nat_nat @ ( F2 @ X4 ) @ ( F2 @ Y5 ) ) ) ) ) ) ) ).
% comp_fun_commute_on_def
thf(fact_1424_comp__fun__commute__on__def,axiom,
( finite4582259249640653518nt_nat
= ( ^ [S4: set_int,F2: int > nat > nat] :
! [X4: int,Y5: int] :
( ( member_int @ X4 @ S4 )
=> ( ( member_int @ Y5 @ S4 )
=> ( ( comp_nat_nat_nat @ ( F2 @ Y5 ) @ ( F2 @ X4 ) )
= ( comp_nat_nat_nat @ ( F2 @ X4 ) @ ( F2 @ Y5 ) ) ) ) ) ) ) ).
% comp_fun_commute_on_def
thf(fact_1425_comp__fun__commute__on_Ocomp__fun__commute__on,axiom,
! [S5: set_Pr4329608150637261639at_nat,F: produc3843707927480180839at_nat > nat > nat,X2: produc3843707927480180839at_nat,Y3: produc3843707927480180839at_nat] :
( ( finite8619596206551965927at_nat @ S5 @ F )
=> ( ( member8757157785044589968at_nat @ X2 @ S5 )
=> ( ( member8757157785044589968at_nat @ Y3 @ S5 )
=> ( ( comp_nat_nat_nat @ ( F @ Y3 ) @ ( F @ X2 ) )
= ( comp_nat_nat_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ) ) ) ).
% comp_fun_commute_on.comp_fun_commute_on
thf(fact_1426_comp__fun__commute__on_Ocomp__fun__commute__on,axiom,
! [S5: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat > nat,X2: product_prod_nat_nat,Y3: product_prod_nat_nat] :
( ( finite2110315540964234241at_nat @ S5 @ F )
=> ( ( member8440522571783428010at_nat @ X2 @ S5 )
=> ( ( member8440522571783428010at_nat @ Y3 @ S5 )
=> ( ( comp_nat_nat_nat @ ( F @ Y3 ) @ ( F @ X2 ) )
= ( comp_nat_nat_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ) ) ) ).
% comp_fun_commute_on.comp_fun_commute_on
thf(fact_1427_comp__fun__commute__on_Ocomp__fun__commute__on,axiom,
! [S5: set_o,F: $o > nat > nat,X2: $o,Y3: $o] :
( ( finite4703585013658510744_o_nat @ S5 @ F )
=> ( ( member_o @ X2 @ S5 )
=> ( ( member_o @ Y3 @ S5 )
=> ( ( comp_nat_nat_nat @ ( F @ Y3 ) @ ( F @ X2 ) )
= ( comp_nat_nat_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ) ) ) ).
% comp_fun_commute_on.comp_fun_commute_on
thf(fact_1428_comp__fun__commute__on_Ocomp__fun__commute__on,axiom,
! [S5: set_nat,F: nat > nat > nat,X2: nat,Y3: nat] :
( ( finite3582905537739598962at_nat @ S5 @ F )
=> ( ( member_nat @ X2 @ S5 )
=> ( ( member_nat @ Y3 @ S5 )
=> ( ( comp_nat_nat_nat @ ( F @ Y3 ) @ ( F @ X2 ) )
= ( comp_nat_nat_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ) ) ) ).
% comp_fun_commute_on.comp_fun_commute_on
thf(fact_1429_comp__fun__commute__on_Ocomp__fun__commute__on,axiom,
! [S5: set_int,F: int > nat > nat,X2: int,Y3: int] :
( ( finite4582259249640653518nt_nat @ S5 @ F )
=> ( ( member_int @ X2 @ S5 )
=> ( ( member_int @ Y3 @ S5 )
=> ( ( comp_nat_nat_nat @ ( F @ Y3 ) @ ( F @ X2 ) )
= ( comp_nat_nat_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ) ) ) ).
% comp_fun_commute_on.comp_fun_commute_on
thf(fact_1430_comp__fun__commute__on_Ocommute__left__comp,axiom,
! [S5: set_Pr4329608150637261639at_nat,F: produc3843707927480180839at_nat > $o > $o,X2: produc3843707927480180839at_nat,Y3: produc3843707927480180839at_nat,G: nat > $o] :
( ( finite2818816984019561153_nat_o @ S5 @ F )
=> ( ( member8757157785044589968at_nat @ X2 @ S5 )
=> ( ( member8757157785044589968at_nat @ Y3 @ S5 )
=> ( ( comp_o_o_nat @ ( F @ Y3 ) @ ( comp_o_o_nat @ ( F @ X2 ) @ G ) )
= ( comp_o_o_nat @ ( F @ X2 ) @ ( comp_o_o_nat @ ( F @ Y3 ) @ G ) ) ) ) ) ) ).
% comp_fun_commute_on.commute_left_comp
thf(fact_1431_comp__fun__commute__on_Ocommute__left__comp,axiom,
! [S5: set_Pr4329608150637261639at_nat,F: produc3843707927480180839at_nat > nat > nat,X2: produc3843707927480180839at_nat,Y3: produc3843707927480180839at_nat,G: nat > nat] :
( ( finite8619596206551965927at_nat @ S5 @ F )
=> ( ( member8757157785044589968at_nat @ X2 @ S5 )
=> ( ( member8757157785044589968at_nat @ Y3 @ S5 )
=> ( ( comp_nat_nat_nat @ ( F @ Y3 ) @ ( comp_nat_nat_nat @ ( F @ X2 ) @ G ) )
= ( comp_nat_nat_nat @ ( F @ X2 ) @ ( comp_nat_nat_nat @ ( F @ Y3 ) @ G ) ) ) ) ) ) ).
% comp_fun_commute_on.commute_left_comp
thf(fact_1432_comp__fun__commute__on_Ocommute__left__comp,axiom,
! [S5: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > $o > $o,X2: product_prod_nat_nat,Y3: product_prod_nat_nat,G: nat > $o] :
( ( finite1496024664758132007_nat_o @ S5 @ F )
=> ( ( member8440522571783428010at_nat @ X2 @ S5 )
=> ( ( member8440522571783428010at_nat @ Y3 @ S5 )
=> ( ( comp_o_o_nat @ ( F @ Y3 ) @ ( comp_o_o_nat @ ( F @ X2 ) @ G ) )
= ( comp_o_o_nat @ ( F @ X2 ) @ ( comp_o_o_nat @ ( F @ Y3 ) @ G ) ) ) ) ) ) ).
% comp_fun_commute_on.commute_left_comp
thf(fact_1433_comp__fun__commute__on_Ocommute__left__comp,axiom,
! [S5: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat > nat,X2: product_prod_nat_nat,Y3: product_prod_nat_nat,G: nat > nat] :
( ( finite2110315540964234241at_nat @ S5 @ F )
=> ( ( member8440522571783428010at_nat @ X2 @ S5 )
=> ( ( member8440522571783428010at_nat @ Y3 @ S5 )
=> ( ( comp_nat_nat_nat @ ( F @ Y3 ) @ ( comp_nat_nat_nat @ ( F @ X2 ) @ G ) )
= ( comp_nat_nat_nat @ ( F @ X2 ) @ ( comp_nat_nat_nat @ ( F @ Y3 ) @ G ) ) ) ) ) ) ).
% comp_fun_commute_on.commute_left_comp
thf(fact_1434_comp__fun__commute__on_Ocommute__left__comp,axiom,
! [S5: set_o,F: $o > $o > $o,X2: $o,Y3: $o,G: nat > $o] :
( ( finite7905178416053733328on_o_o @ S5 @ F )
=> ( ( member_o @ X2 @ S5 )
=> ( ( member_o @ Y3 @ S5 )
=> ( ( comp_o_o_nat @ ( F @ Y3 ) @ ( comp_o_o_nat @ ( F @ X2 ) @ G ) )
= ( comp_o_o_nat @ ( F @ X2 ) @ ( comp_o_o_nat @ ( F @ Y3 ) @ G ) ) ) ) ) ) ).
% comp_fun_commute_on.commute_left_comp
thf(fact_1435_comp__fun__commute__on_Ocommute__left__comp,axiom,
! [S5: set_o,F: $o > nat > nat,X2: $o,Y3: $o,G: nat > nat] :
( ( finite4703585013658510744_o_nat @ S5 @ F )
=> ( ( member_o @ X2 @ S5 )
=> ( ( member_o @ Y3 @ S5 )
=> ( ( comp_nat_nat_nat @ ( F @ Y3 ) @ ( comp_nat_nat_nat @ ( F @ X2 ) @ G ) )
= ( comp_nat_nat_nat @ ( F @ X2 ) @ ( comp_nat_nat_nat @ ( F @ Y3 ) @ G ) ) ) ) ) ) ).
% comp_fun_commute_on.commute_left_comp
thf(fact_1436_comp__fun__commute__on_Ocommute__left__comp,axiom,
! [S5: set_nat,F: nat > $o > $o,X2: nat,Y3: nat,G: nat > $o] :
( ( finite7806256374766696054_nat_o @ S5 @ F )
=> ( ( member_nat @ X2 @ S5 )
=> ( ( member_nat @ Y3 @ S5 )
=> ( ( comp_o_o_nat @ ( F @ Y3 ) @ ( comp_o_o_nat @ ( F @ X2 ) @ G ) )
= ( comp_o_o_nat @ ( F @ X2 ) @ ( comp_o_o_nat @ ( F @ Y3 ) @ G ) ) ) ) ) ) ).
% comp_fun_commute_on.commute_left_comp
thf(fact_1437_comp__fun__commute__on_Ocommute__left__comp,axiom,
! [S5: set_nat,F: nat > nat > nat,X2: nat,Y3: nat,G: nat > nat] :
( ( finite3582905537739598962at_nat @ S5 @ F )
=> ( ( member_nat @ X2 @ S5 )
=> ( ( member_nat @ Y3 @ S5 )
=> ( ( comp_nat_nat_nat @ ( F @ Y3 ) @ ( comp_nat_nat_nat @ ( F @ X2 ) @ G ) )
= ( comp_nat_nat_nat @ ( F @ X2 ) @ ( comp_nat_nat_nat @ ( F @ Y3 ) @ G ) ) ) ) ) ) ).
% comp_fun_commute_on.commute_left_comp
thf(fact_1438_comp__fun__commute__on_Ocommute__left__comp,axiom,
! [S5: set_int,F: int > $o > $o,X2: int,Y3: int,G: nat > $o] :
( ( finite5985214029334008090_int_o @ S5 @ F )
=> ( ( member_int @ X2 @ S5 )
=> ( ( member_int @ Y3 @ S5 )
=> ( ( comp_o_o_nat @ ( F @ Y3 ) @ ( comp_o_o_nat @ ( F @ X2 ) @ G ) )
= ( comp_o_o_nat @ ( F @ X2 ) @ ( comp_o_o_nat @ ( F @ Y3 ) @ G ) ) ) ) ) ) ).
% comp_fun_commute_on.commute_left_comp
thf(fact_1439_comp__fun__commute__on_Ocommute__left__comp,axiom,
! [S5: set_int,F: int > nat > nat,X2: int,Y3: int,G: nat > nat] :
( ( finite4582259249640653518nt_nat @ S5 @ F )
=> ( ( member_int @ X2 @ S5 )
=> ( ( member_int @ Y3 @ S5 )
=> ( ( comp_nat_nat_nat @ ( F @ Y3 ) @ ( comp_nat_nat_nat @ ( F @ X2 ) @ G ) )
= ( comp_nat_nat_nat @ ( F @ X2 ) @ ( comp_nat_nat_nat @ ( F @ Y3 ) @ G ) ) ) ) ) ) ).
% comp_fun_commute_on.commute_left_comp
thf(fact_1440_comp__fun__commute__on_Ointro,axiom,
! [S5: set_Pr4329608150637261639at_nat,F: produc3843707927480180839at_nat > nat > nat] :
( ! [X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
( ( member8757157785044589968at_nat @ X @ S5 )
=> ( ( member8757157785044589968at_nat @ Y @ S5 )
=> ( ( comp_nat_nat_nat @ ( F @ Y ) @ ( F @ X ) )
= ( comp_nat_nat_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) )
=> ( finite8619596206551965927at_nat @ S5 @ F ) ) ).
% comp_fun_commute_on.intro
thf(fact_1441_comp__fun__commute__on_Ointro,axiom,
! [S5: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat > nat] :
( ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X @ S5 )
=> ( ( member8440522571783428010at_nat @ Y @ S5 )
=> ( ( comp_nat_nat_nat @ ( F @ Y ) @ ( F @ X ) )
= ( comp_nat_nat_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) )
=> ( finite2110315540964234241at_nat @ S5 @ F ) ) ).
% comp_fun_commute_on.intro
thf(fact_1442_comp__fun__commute__on_Ointro,axiom,
! [S5: set_o,F: $o > nat > nat] :
( ! [X: $o,Y: $o] :
( ( member_o @ X @ S5 )
=> ( ( member_o @ Y @ S5 )
=> ( ( comp_nat_nat_nat @ ( F @ Y ) @ ( F @ X ) )
= ( comp_nat_nat_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) )
=> ( finite4703585013658510744_o_nat @ S5 @ F ) ) ).
% comp_fun_commute_on.intro
thf(fact_1443_comp__fun__commute__on_Ointro,axiom,
! [S5: set_nat,F: nat > nat > nat] :
( ! [X: nat,Y: nat] :
( ( member_nat @ X @ S5 )
=> ( ( member_nat @ Y @ S5 )
=> ( ( comp_nat_nat_nat @ ( F @ Y ) @ ( F @ X ) )
= ( comp_nat_nat_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) )
=> ( finite3582905537739598962at_nat @ S5 @ F ) ) ).
% comp_fun_commute_on.intro
thf(fact_1444_comp__fun__commute__on_Ointro,axiom,
! [S5: set_int,F: int > nat > nat] :
( ! [X: int,Y: int] :
( ( member_int @ X @ S5 )
=> ( ( member_int @ Y @ S5 )
=> ( ( comp_nat_nat_nat @ ( F @ Y ) @ ( F @ X ) )
= ( comp_nat_nat_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) )
=> ( finite4582259249640653518nt_nat @ S5 @ F ) ) ).
% comp_fun_commute_on.intro
thf(fact_1445_case__prod__o__map__prod,axiom,
! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,G1: nat > nat,G22: nat > nat] :
( ( comp_P3913276677246710332at_nat @ ( produc27273713700761075at_nat @ F ) @ ( produc6977886695330630970at_nat @ G1 @ G22 ) )
= ( produc27273713700761075at_nat
@ ^ [L3: nat,R5: nat] : ( F @ ( G1 @ L3 ) @ ( G22 @ R5 ) ) ) ) ).
% case_prod_o_map_prod
thf(fact_1446_case__prod__o__map__prod,axiom,
! [F: nat > nat > product_prod_nat_nat > $o,G1: nat > nat,G22: nat > nat] :
( ( comp_P2002127591014248037at_nat @ ( produc8739625826339149834_nat_o @ F ) @ ( produc6977886695330630970at_nat @ G1 @ G22 ) )
= ( produc8739625826339149834_nat_o
@ ^ [L3: nat,R5: nat] : ( F @ ( G1 @ L3 ) @ ( G22 @ R5 ) ) ) ) ).
% case_prod_o_map_prod
thf(fact_1447_case__prod__o__map__prod,axiom,
! [F: int > int > product_prod_int_int,G1: int > int,G22: int > int] :
( ( comp_P2788356708182965334nt_int @ ( produc4245557441103728435nt_int @ F ) @ ( produc6036585564866528938nt_int @ G1 @ G22 ) )
= ( produc4245557441103728435nt_int
@ ^ [L3: int,R5: int] : ( F @ ( G1 @ L3 ) @ ( G22 @ R5 ) ) ) ) ).
% case_prod_o_map_prod
thf(fact_1448_case__prod__o__map__prod,axiom,
! [F: int > int > $o,G1: int > int,G22: int > int] :
( ( comp_P9034173210191959223nt_int @ ( produc4947309494688390418_int_o @ F ) @ ( produc6036585564866528938nt_int @ G1 @ G22 ) )
= ( produc4947309494688390418_int_o
@ ^ [L3: int,R5: int] : ( F @ ( G1 @ L3 ) @ ( G22 @ R5 ) ) ) ) ).
% case_prod_o_map_prod
thf(fact_1449_case__prod__o__map__prod,axiom,
! [F: int > int > int,G1: int > int,G22: int > int] :
( ( comp_P5356232565778667509nt_int @ ( produc8211389475949308722nt_int @ F ) @ ( produc6036585564866528938nt_int @ G1 @ G22 ) )
= ( produc8211389475949308722nt_int
@ ^ [L3: int,R5: int] : ( F @ ( G1 @ L3 ) @ ( G22 @ R5 ) ) ) ) ).
% case_prod_o_map_prod
thf(fact_1450_vimage2p__relcompp__converse,axiom,
! [Rep: product_prod_nat_nat > product_prod_nat_nat,Abs: product_prod_nat_nat > product_prod_nat_nat,F: product_prod_nat_nat > product_prod_nat_nat,G: int > int,R3: product_prod_nat_nat > product_prod_nat_nat > $o,S5: product_prod_nat_nat > int > $o] :
( ( type_d281131942595027436at_nat @ Rep @ Abs @ top_to4669805908274784177at_nat )
=> ( ( bNF_vi5426221348228882691_int_o @ F @ G @ ( relcom8911704206367525917at_int @ ( conver6965824823119931701at_nat @ R3 ) @ S5 ) )
= ( relcom8911704206367525917at_int @ ( conver6965824823119931701at_nat @ ( bNF_vi763171854521451549_nat_o @ Rep @ F @ R3 ) ) @ ( bNF_vi5426221348228882691_int_o @ Rep @ G @ S5 ) ) ) ) ).
% vimage2p_relcompp_converse
thf(fact_1451_vimage2p__relcompp__converse,axiom,
! [Rep: int > product_prod_nat_nat,Abs: product_prod_nat_nat > int,F: int > product_prod_nat_nat,G: code_integer > int,R3: product_prod_nat_nat > product_prod_nat_nat > $o,S5: product_prod_nat_nat > int > $o] :
( ( type_d7732240953928457511at_nat @ Rep @ Abs @ top_to4669805908274784177at_nat )
=> ( ( bNF_vi2881549544236934259_int_o @ F @ G @ ( relcom8911704206367525917at_int @ ( conver6965824823119931701at_nat @ R3 ) @ S5 ) )
= ( relcom967497705283160040nteger @ ( conversep_int_int @ ( bNF_vi1204582724742824157_nat_o @ Rep @ F @ R3 ) ) @ ( bNF_vi2881549544236934259_int_o @ Rep @ G @ S5 ) ) ) ) ).
% vimage2p_relcompp_converse
thf(fact_1452_vimage2p__relcompp__converse,axiom,
! [Rep: product_prod_nat_nat > int,Abs: int > product_prod_nat_nat,F: product_prod_nat_nat > int,G: int > code_integer,R3: int > int > $o,S5: int > code_integer > $o] :
( ( type_d1331861005912558337at_int @ Rep @ Abs @ top_top_set_int )
=> ( ( bNF_vi33923956287962137eger_o @ F @ G @ ( relcom967497705283160040nteger @ ( conversep_int_int @ R3 ) @ S5 ) )
= ( relcom8911704206367525917at_int @ ( conver6965824823119931701at_nat @ ( bNF_vi2682369418337388893_int_o @ Rep @ F @ R3 ) ) @ ( bNF_vi33923956287962137eger_o @ Rep @ G @ S5 ) ) ) ) ).
% vimage2p_relcompp_converse
thf(fact_1453_vimage2p__relcompp__converse,axiom,
! [Rep: int > int,Abs: int > int,F: int > int,G: code_integer > code_integer,R3: int > int > $o,S5: int > code_integer > $o] :
( ( type_d7247357190169752966nt_int @ Rep @ Abs @ top_top_set_int )
=> ( ( bNF_vi699694178825965831eger_o @ F @ G @ ( relcom967497705283160040nteger @ ( conversep_int_int @ R3 ) @ S5 ) )
= ( relcom967497705283160040nteger @ ( conversep_int_int @ ( bNF_vi3884084463199473129_int_o @ Rep @ F @ R3 ) ) @ ( bNF_vi699694178825965831eger_o @ Rep @ G @ S5 ) ) ) ) ).
% vimage2p_relcompp_converse
thf(fact_1454_Id__on__iff,axiom,
! [X2: produc3843707927480180839at_nat,Y3: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X2 @ Y3 ) @ ( id_on_7869290023980668164at_nat @ A4 ) )
= ( ( X2 = Y3 )
& ( member8757157785044589968at_nat @ X2 @ A4 ) ) ) ).
% Id_on_iff
thf(fact_1455_Id__on__iff,axiom,
! [X2: product_prod_nat_nat,Y3: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y3 ) @ ( id_on_2554058798563519774at_nat @ A4 ) )
= ( ( X2 = Y3 )
& ( member8440522571783428010at_nat @ X2 @ A4 ) ) ) ).
% Id_on_iff
thf(fact_1456_Id__on__iff,axiom,
! [X2: $o,Y3: $o,A4: set_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ Y3 ) @ ( id_on_o @ A4 ) )
= ( ( X2 = Y3 )
& ( member_o @ X2 @ A4 ) ) ) ).
% Id_on_iff
thf(fact_1457_Id__on__iff,axiom,
! [X2: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat,A4: set_se5479201105039950375at_nat] :
( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X2 @ Y3 ) @ ( id_on_3275838214495744996at_nat @ A4 ) )
= ( ( X2 = Y3 )
& ( member6273874620674182256at_nat @ X2 @ A4 ) ) ) ).
% Id_on_iff
thf(fact_1458_Id__on__iff,axiom,
! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat,A4: set_se7855581050983116737at_nat] :
( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y3 ) @ ( id_on_6713504163309144702at_nat @ A4 ) )
= ( ( X2 = Y3 )
& ( member2643936169264416010at_nat @ X2 @ A4 ) ) ) ).
% Id_on_iff
thf(fact_1459_Id__on__iff,axiom,
! [X2: nat,Y3: nat,A4: set_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ ( id_on_nat @ A4 ) )
= ( ( X2 = Y3 )
& ( member_nat @ X2 @ A4 ) ) ) ).
% Id_on_iff
thf(fact_1460_Id__on__iff,axiom,
! [X2: int,Y3: int,A4: set_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y3 ) @ ( id_on_int @ A4 ) )
= ( ( X2 = Y3 )
& ( member_int @ X2 @ A4 ) ) ) ).
% Id_on_iff
thf(fact_1461_Id__on__eqI,axiom,
! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
( ( A = B )
=> ( ( member8757157785044589968at_nat @ A @ A4 )
=> ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ B ) @ ( id_on_7869290023980668164at_nat @ A4 ) ) ) ) ).
% Id_on_eqI
thf(fact_1462_Id__on__eqI,axiom,
! [A: product_prod_nat_nat,B: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
( ( A = B )
=> ( ( member8440522571783428010at_nat @ A @ A4 )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( id_on_2554058798563519774at_nat @ A4 ) ) ) ) ).
% Id_on_eqI
thf(fact_1463_Id__on__eqI,axiom,
! [A: $o,B: $o,A4: set_o] :
( ( A = B )
=> ( ( member_o @ A @ A4 )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ ( id_on_o @ A4 ) ) ) ) ).
% Id_on_eqI
thf(fact_1464_Id__on__eqI,axiom,
! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A4: set_se5479201105039950375at_nat] :
( ( A = B )
=> ( ( member6273874620674182256at_nat @ A @ A4 )
=> ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ B ) @ ( id_on_3275838214495744996at_nat @ A4 ) ) ) ) ).
% Id_on_eqI
thf(fact_1465_Id__on__eqI,axiom,
! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A4: set_se7855581050983116737at_nat] :
( ( A = B )
=> ( ( member2643936169264416010at_nat @ A @ A4 )
=> ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ ( id_on_6713504163309144702at_nat @ A4 ) ) ) ) ).
% Id_on_eqI
thf(fact_1466_Id__on__eqI,axiom,
! [A: nat,B: nat,A4: set_nat] :
( ( A = B )
=> ( ( member_nat @ A @ A4 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( id_on_nat @ A4 ) ) ) ) ).
% Id_on_eqI
thf(fact_1467_Id__on__eqI,axiom,
! [A: int,B: int,A4: set_int] :
( ( A = B )
=> ( ( member_int @ A @ A4 )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ ( id_on_int @ A4 ) ) ) ) ).
% Id_on_eqI
thf(fact_1468_Id__onE,axiom,
! [C: produc1906917888121415911at_nat,A4: set_Pr4329608150637261639at_nat] :
( ( member6009668390874684176at_nat @ C @ ( id_on_7869290023980668164at_nat @ A4 ) )
=> ~ ! [X: produc3843707927480180839at_nat] :
( ( member8757157785044589968at_nat @ X @ A4 )
=> ( C
!= ( produc2335810615602371415at_nat @ X @ X ) ) ) ) ).
% Id_onE
thf(fact_1469_Id__onE,axiom,
! [C: produc859450856879609959at_nat,A4: set_Pr1261947904930325089at_nat] :
( ( member8206827879206165904at_nat @ C @ ( id_on_2554058798563519774at_nat @ A4 ) )
=> ~ ! [X: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X @ A4 )
=> ( C
!= ( produc6161850002892822231at_nat @ X @ X ) ) ) ) ).
% Id_onE
thf(fact_1470_Id__onE,axiom,
! [C: product_prod_o_o,A4: set_o] :
( ( member7466972457876170832od_o_o @ C @ ( id_on_o @ A4 ) )
=> ~ ! [X: $o] :
( ( member_o @ X @ A4 )
=> ( C
!= ( product_Pair_o_o @ X @ X ) ) ) ) ).
% Id_onE
thf(fact_1471_Id__onE,axiom,
! [C: produc1319942482725812455at_nat,A4: set_se5479201105039950375at_nat] :
( ( member1466754251312161552at_nat @ C @ ( id_on_3275838214495744996at_nat @ A4 ) )
=> ~ ! [X: set_Pr4329608150637261639at_nat] :
( ( member6273874620674182256at_nat @ X @ A4 )
=> ( C
!= ( produc9060074326276436823at_nat @ X @ X ) ) ) ) ).
% Id_onE
thf(fact_1472_Id__onE,axiom,
! [C: produc3843707927480180839at_nat,A4: set_se7855581050983116737at_nat] :
( ( member8757157785044589968at_nat @ C @ ( id_on_6713504163309144702at_nat @ A4 ) )
=> ~ ! [X: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ X @ A4 )
=> ( C
!= ( produc2922128104949294807at_nat @ X @ X ) ) ) ) ).
% Id_onE
thf(fact_1473_Id__onE,axiom,
! [C: product_prod_nat_nat,A4: set_nat] :
( ( member8440522571783428010at_nat @ C @ ( id_on_nat @ A4 ) )
=> ~ ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ( C
!= ( product_Pair_nat_nat @ X @ X ) ) ) ) ).
% Id_onE
thf(fact_1474_Id__onE,axiom,
! [C: product_prod_int_int,A4: set_int] :
( ( member5262025264175285858nt_int @ C @ ( id_on_int @ A4 ) )
=> ~ ! [X: int] :
( ( member_int @ X @ A4 )
=> ( C
!= ( product_Pair_int_int @ X @ X ) ) ) ) ).
% Id_onE
thf(fact_1475_predicate2__transferD,axiom,
! [R1: nat > nat > $o,R22: nat > nat > $o,P2: nat > nat > $o,Q2: nat > nat > $o,A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
( ( bNF_re578469030762574527_nat_o @ R1
@ ( bNF_re4705727531993890431at_o_o @ R22
@ ^ [Y2: $o,Z: $o] : Y2 = Z )
@ P2
@ Q2 )
=> ( ( member8440522571783428010at_nat @ A @ A4 )
=> ( ( member8440522571783428010at_nat @ B @ B5 )
=> ( ( ord_le3146513528884898305at_nat @ A4 @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R1 ) ) )
=> ( ( ord_le3146513528884898305at_nat @ B5 @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R22 ) ) )
=> ( ( P2 @ ( product_fst_nat_nat @ A ) @ ( product_fst_nat_nat @ B ) )
= ( Q2 @ ( product_snd_nat_nat @ A ) @ ( product_snd_nat_nat @ B ) ) ) ) ) ) ) ) ).
% predicate2_transferD
thf(fact_1476_predicate2__transferD,axiom,
! [R1: nat > nat > $o,R22: int > int > $o,P2: nat > int > $o,Q2: nat > int > $o,A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B: product_prod_int_int,B5: set_Pr958786334691620121nt_int] :
( ( bNF_re605504016728627135_int_o @ R1
@ ( bNF_re5089333283451836215nt_o_o @ R22
@ ^ [Y2: $o,Z: $o] : Y2 = Z )
@ P2
@ Q2 )
=> ( ( member8440522571783428010at_nat @ A @ A4 )
=> ( ( member5262025264175285858nt_int @ B @ B5 )
=> ( ( ord_le3146513528884898305at_nat @ A4 @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R1 ) ) )
=> ( ( ord_le2843351958646193337nt_int @ B5 @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ R22 ) ) )
=> ( ( P2 @ ( product_fst_nat_nat @ A ) @ ( product_fst_int_int @ B ) )
= ( Q2 @ ( product_snd_nat_nat @ A ) @ ( product_snd_int_int @ B ) ) ) ) ) ) ) ) ).
% predicate2_transferD
thf(fact_1477_predicate2__transferD,axiom,
! [R1: int > int > $o,R22: nat > nat > $o,P2: int > nat > $o,Q2: int > nat > $o,A: product_prod_int_int,A4: set_Pr958786334691620121nt_int,B: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
( ( bNF_re3376528473927230327_nat_o @ R1
@ ( bNF_re4705727531993890431at_o_o @ R22
@ ^ [Y2: $o,Z: $o] : Y2 = Z )
@ P2
@ Q2 )
=> ( ( member5262025264175285858nt_int @ A @ A4 )
=> ( ( member8440522571783428010at_nat @ B @ B5 )
=> ( ( ord_le2843351958646193337nt_int @ A4 @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ R1 ) ) )
=> ( ( ord_le3146513528884898305at_nat @ B5 @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R22 ) ) )
=> ( ( P2 @ ( product_fst_int_int @ A ) @ ( product_fst_nat_nat @ B ) )
= ( Q2 @ ( product_snd_int_int @ A ) @ ( product_snd_nat_nat @ B ) ) ) ) ) ) ) ) ).
% predicate2_transferD
thf(fact_1478_predicate2__transferD,axiom,
! [R1: int > int > $o,R22: int > int > $o,P2: int > int > $o,Q2: int > int > $o,A: product_prod_int_int,A4: set_Pr958786334691620121nt_int,B: product_prod_int_int,B5: set_Pr958786334691620121nt_int] :
( ( bNF_re3403563459893282935_int_o @ R1
@ ( bNF_re5089333283451836215nt_o_o @ R22
@ ^ [Y2: $o,Z: $o] : Y2 = Z )
@ P2
@ Q2 )
=> ( ( member5262025264175285858nt_int @ A @ A4 )
=> ( ( member5262025264175285858nt_int @ B @ B5 )
=> ( ( ord_le2843351958646193337nt_int @ A4 @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ R1 ) ) )
=> ( ( ord_le2843351958646193337nt_int @ B5 @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ R22 ) ) )
=> ( ( P2 @ ( product_fst_int_int @ A ) @ ( product_fst_int_int @ B ) )
= ( Q2 @ ( product_snd_int_int @ A ) @ ( product_snd_int_int @ B ) ) ) ) ) ) ) ) ).
% predicate2_transferD
thf(fact_1479_predicate2__transferD,axiom,
! [R1: nat > nat > $o,R22: ref_a > heap_e7401611519738050253t_unit > $o,P2: nat > ref_a > $o,Q2: nat > heap_e7401611519738050253t_unit > $o,A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B: produc7812382424243267045t_unit,B5: set_Pr1133966238863718811t_unit] :
( ( bNF_re5655699643857186083unit_o @ R1
@ ( bNF_re1850668159662942673it_o_o @ R22
@ ^ [Y2: $o,Z: $o] : Y2 = Z )
@ P2
@ Q2 )
=> ( ( member8440522571783428010at_nat @ A @ A4 )
=> ( ( member7849944260330398460t_unit @ B @ B5 )
=> ( ( ord_le3146513528884898305at_nat @ A4 @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R1 ) ) )
=> ( ( ord_le5525191956872776187t_unit @ B5 @ ( collec3252306575900995386t_unit @ ( produc4924168586796184916unit_o @ R22 ) ) )
=> ( ( P2 @ ( product_fst_nat_nat @ A ) @ ( produc473321116828791851t_unit @ B ) )
= ( Q2 @ ( product_snd_nat_nat @ A ) @ ( produc1126150082675451753t_unit @ B ) ) ) ) ) ) ) ) ).
% predicate2_transferD
thf(fact_1480_predicate2__transferD,axiom,
! [R1: ref_a > heap_e7401611519738050253t_unit > $o,R22: nat > nat > $o,P2: ref_a > nat > $o,Q2: heap_e7401611519738050253t_unit > nat > $o,A: produc7812382424243267045t_unit,A4: set_Pr1133966238863718811t_unit,B: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
( ( bNF_re9202112420002600977_nat_o @ R1
@ ( bNF_re4705727531993890431at_o_o @ R22
@ ^ [Y2: $o,Z: $o] : Y2 = Z )
@ P2
@ Q2 )
=> ( ( member7849944260330398460t_unit @ A @ A4 )
=> ( ( member8440522571783428010at_nat @ B @ B5 )
=> ( ( ord_le5525191956872776187t_unit @ A4 @ ( collec3252306575900995386t_unit @ ( produc4924168586796184916unit_o @ R1 ) ) )
=> ( ( ord_le3146513528884898305at_nat @ B5 @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R22 ) ) )
=> ( ( P2 @ ( produc473321116828791851t_unit @ A ) @ ( product_fst_nat_nat @ B ) )
= ( Q2 @ ( produc1126150082675451753t_unit @ A ) @ ( product_snd_nat_nat @ B ) ) ) ) ) ) ) ) ).
% predicate2_transferD
thf(fact_1481_predicate2__transferD,axiom,
! [R1: ref_a > heap_e7401611519738050253t_unit > $o,R22: int > int > $o,P2: ref_a > int > $o,Q2: heap_e7401611519738050253t_unit > int > $o,A: produc7812382424243267045t_unit,A4: set_Pr1133966238863718811t_unit,B: product_prod_int_int,B5: set_Pr958786334691620121nt_int] :
( ( bNF_re5775369113877777_int_o @ R1
@ ( bNF_re5089333283451836215nt_o_o @ R22
@ ^ [Y2: $o,Z: $o] : Y2 = Z )
@ P2
@ Q2 )
=> ( ( member7849944260330398460t_unit @ A @ A4 )
=> ( ( member5262025264175285858nt_int @ B @ B5 )
=> ( ( ord_le5525191956872776187t_unit @ A4 @ ( collec3252306575900995386t_unit @ ( produc4924168586796184916unit_o @ R1 ) ) )
=> ( ( ord_le2843351958646193337nt_int @ B5 @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ R22 ) ) )
=> ( ( P2 @ ( produc473321116828791851t_unit @ A ) @ ( product_fst_int_int @ B ) )
= ( Q2 @ ( produc1126150082675451753t_unit @ A ) @ ( product_snd_int_int @ B ) ) ) ) ) ) ) ) ).
% predicate2_transferD
thf(fact_1482_predicate2__transferD,axiom,
! [R1: int > int > $o,R22: ref_a > heap_e7401611519738050253t_unit > $o,P2: int > ref_a > $o,Q2: int > heap_e7401611519738050253t_unit > $o,A: product_prod_int_int,A4: set_Pr958786334691620121nt_int,B: produc7812382424243267045t_unit,B5: set_Pr1133966238863718811t_unit] :
( ( bNF_re5404797796594736747unit_o @ R1
@ ( bNF_re1850668159662942673it_o_o @ R22
@ ^ [Y2: $o,Z: $o] : Y2 = Z )
@ P2
@ Q2 )
=> ( ( member5262025264175285858nt_int @ A @ A4 )
=> ( ( member7849944260330398460t_unit @ B @ B5 )
=> ( ( ord_le2843351958646193337nt_int @ A4 @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ R1 ) ) )
=> ( ( ord_le5525191956872776187t_unit @ B5 @ ( collec3252306575900995386t_unit @ ( produc4924168586796184916unit_o @ R22 ) ) )
=> ( ( P2 @ ( product_fst_int_int @ A ) @ ( produc473321116828791851t_unit @ B ) )
= ( Q2 @ ( product_snd_int_int @ A ) @ ( produc1126150082675451753t_unit @ B ) ) ) ) ) ) ) ) ).
% predicate2_transferD
thf(fact_1483_predicate2__transferD,axiom,
! [R1: product_prod_nat_nat > int > $o,R22: product_prod_nat_nat > int > $o,P2: product_prod_nat_nat > product_prod_nat_nat > $o,Q2: int > int > $o,A: produc4196048018000912732at_int,A4: set_Pr49084382684228242at_int,B: produc4196048018000912732at_int,B5: set_Pr49084382684228242at_int] :
( ( bNF_re717283939379294677_int_o @ R1
@ ( bNF_re6644619430987730960nt_o_o @ R22
@ ^ [Y2: $o,Z: $o] : Y2 = Z )
@ P2
@ Q2 )
=> ( ( member8394280151737582451at_int @ A @ A4 )
=> ( ( member8394280151737582451at_int @ B @ B5 )
=> ( ( ord_le4604737605482584818at_int @ A4 @ ( collec2851339944984316337at_int @ ( produc6937227050501860053_int_o @ R1 ) ) )
=> ( ( ord_le4604737605482584818at_int @ B5 @ ( collec2851339944984316337at_int @ ( produc6937227050501860053_int_o @ R22 ) ) )
=> ( ( P2 @ ( produc4619704898428564714at_int @ A ) @ ( produc4619704898428564714at_int @ B ) )
= ( Q2 @ ( produc3064508320661873192at_int @ A ) @ ( produc3064508320661873192at_int @ B ) ) ) ) ) ) ) ) ).
% predicate2_transferD
thf(fact_1484_predicate2__transferD,axiom,
! [R1: ref_a > heap_e7401611519738050253t_unit > $o,R22: ref_a > heap_e7401611519738050253t_unit > $o,P2: ref_a > ref_a > $o,Q2: heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit > $o,A: produc7812382424243267045t_unit,A4: set_Pr1133966238863718811t_unit,B: produc7812382424243267045t_unit,B5: set_Pr1133966238863718811t_unit] :
( ( bNF_re1366391616432607057unit_o @ R1
@ ( bNF_re1850668159662942673it_o_o @ R22
@ ^ [Y2: $o,Z: $o] : Y2 = Z )
@ P2
@ Q2 )
=> ( ( member7849944260330398460t_unit @ A @ A4 )
=> ( ( member7849944260330398460t_unit @ B @ B5 )
=> ( ( ord_le5525191956872776187t_unit @ A4 @ ( collec3252306575900995386t_unit @ ( produc4924168586796184916unit_o @ R1 ) ) )
=> ( ( ord_le5525191956872776187t_unit @ B5 @ ( collec3252306575900995386t_unit @ ( produc4924168586796184916unit_o @ R22 ) ) )
=> ( ( P2 @ ( produc473321116828791851t_unit @ A ) @ ( produc473321116828791851t_unit @ B ) )
= ( Q2 @ ( produc1126150082675451753t_unit @ A ) @ ( produc1126150082675451753t_unit @ B ) ) ) ) ) ) ) ) ).
% predicate2_transferD
thf(fact_1485_fst__diag__id,axiom,
! [Z2: $o] :
( ( comp_P2137996956111167876_o_o_o @ product_fst_o_o
@ ^ [X4: $o] : ( product_Pair_o_o @ X4 @ X4 )
@ Z2 )
= ( id_o @ Z2 ) ) ).
% fst_diag_id
thf(fact_1486_fst__diag__id,axiom,
! [Z2: num] :
( ( comp_P864883071962432936um_num @ product_fst_num_num
@ ^ [X4: num] : ( product_Pair_num_num @ X4 @ X4 )
@ Z2 )
= ( id_num @ Z2 ) ) ).
% fst_diag_id
thf(fact_1487_fst__diag__id,axiom,
! [Z2: set_Pr4329608150637261639at_nat] :
( ( comp_P8235817900121468292at_nat @ produc4486455499028916547at_nat
@ ^ [X4: set_Pr4329608150637261639at_nat] : ( produc9060074326276436823at_nat @ X4 @ X4 )
@ Z2 )
= ( id_set7849613212452405432at_nat @ Z2 ) ) ).
% fst_diag_id
thf(fact_1488_fst__diag__id,axiom,
! [Z2: set_Pr1261947904930325089at_nat] :
( ( comp_P4528037747288561284at_nat @ produc6011863565407846595at_nat
@ ^ [X4: set_Pr1261947904930325089at_nat] : ( produc2922128104949294807at_nat @ X4 @ X4 )
@ Z2 )
= ( id_set2334057505560298834at_nat @ Z2 ) ) ).
% fst_diag_id
thf(fact_1489_fst__diag__id,axiom,
! [Z2: nat] :
( ( comp_P1955610211387027840at_nat @ product_fst_nat_nat
@ ^ [X4: nat] : ( product_Pair_nat_nat @ X4 @ X4 )
@ Z2 )
= ( id_nat @ Z2 ) ) ).
% fst_diag_id
thf(fact_1490_fst__diag__id,axiom,
! [Z2: int] :
( ( comp_P6651465171681656048nt_int @ product_fst_int_int
@ ^ [X4: int] : ( product_Pair_int_int @ X4 @ X4 )
@ Z2 )
= ( id_int @ Z2 ) ) ).
% fst_diag_id
thf(fact_1491_snd__diag__id,axiom,
! [Z2: $o] :
( ( comp_P2137996956111167876_o_o_o @ product_snd_o_o
@ ^ [X4: $o] : ( product_Pair_o_o @ X4 @ X4 )
@ Z2 )
= ( id_o @ Z2 ) ) ).
% snd_diag_id
thf(fact_1492_snd__diag__id,axiom,
! [Z2: num] :
( ( comp_P864883071962432936um_num @ product_snd_num_num
@ ^ [X4: num] : ( product_Pair_num_num @ X4 @ X4 )
@ Z2 )
= ( id_num @ Z2 ) ) ).
% snd_diag_id
thf(fact_1493_snd__diag__id,axiom,
! [Z2: set_Pr4329608150637261639at_nat] :
( ( comp_P8235817900121468292at_nat @ produc826222791318859909at_nat
@ ^ [X4: set_Pr4329608150637261639at_nat] : ( produc9060074326276436823at_nat @ X4 @ X4 )
@ Z2 )
= ( id_set7849613212452405432at_nat @ Z2 ) ) ).
% snd_diag_id
thf(fact_1494_snd__diag__id,axiom,
! [Z2: set_Pr1261947904930325089at_nat] :
( ( comp_P4528037747288561284at_nat @ produc2937545259423399429at_nat
@ ^ [X4: set_Pr1261947904930325089at_nat] : ( produc2922128104949294807at_nat @ X4 @ X4 )
@ Z2 )
= ( id_set2334057505560298834at_nat @ Z2 ) ) ).
% snd_diag_id
thf(fact_1495_snd__diag__id,axiom,
! [Z2: nat] :
( ( comp_P1955610211387027840at_nat @ product_snd_nat_nat
@ ^ [X4: nat] : ( product_Pair_nat_nat @ X4 @ X4 )
@ Z2 )
= ( id_nat @ Z2 ) ) ).
% snd_diag_id
thf(fact_1496_snd__diag__id,axiom,
! [Z2: int] :
( ( comp_P6651465171681656048nt_int @ product_snd_int_int
@ ^ [X4: int] : ( product_Pair_int_int @ X4 @ X4 )
@ Z2 )
= ( id_int @ Z2 ) ) ).
% snd_diag_id
thf(fact_1497_case__prod__comp,axiom,
! [F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer > code_integer,X2: produc3346328345660746739nteger] :
( ( produc4281629032016041417nteger @ ( comp_C8797469213163452608nteger @ F @ G ) @ X2 )
= ( F @ ( G @ ( produc1861665112190112313nteger @ X2 ) ) @ ( produc7955729907432593271nteger @ X2 ) ) ) ).
% case_prod_comp
thf(fact_1498_lookup__def,axiom,
( lookup_a
= ( ^ [R5: ref_a] :
( heap_tap_a
@ ^ [H5: heap_e7401611519738050253t_unit] : ( get_a @ H5 @ R5 ) ) ) ) ).
% lookup_def
thf(fact_1499_fst__snd__flip,axiom,
( produc4486455499028916547at_nat
= ( comp_P3728553842126764068at_nat @ produc826222791318859909at_nat
@ ( produc5456455677105495163at_nat
@ ^ [X4: set_Pr4329608150637261639at_nat,Y5: set_Pr4329608150637261639at_nat] : ( produc9060074326276436823at_nat @ Y5 @ X4 ) ) ) ) ).
% fst_snd_flip
thf(fact_1500_fst__snd__flip,axiom,
( produc6011863565407846595at_nat
= ( comp_P4974406650942564362at_nat @ produc2937545259423399429at_nat
@ ( produc7548845685434208379at_nat
@ ^ [X4: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat] : ( produc2922128104949294807at_nat @ Y5 @ X4 ) ) ) ) ).
% fst_snd_flip
thf(fact_1501_fst__snd__flip,axiom,
( product_fst_num_nat
= ( comp_P5051845089628979463um_nat @ product_snd_nat_num
@ ( produc5219324331026901427at_num
@ ^ [X4: num,Y5: nat] : ( product_Pair_nat_num @ Y5 @ X4 ) ) ) ) ).
% fst_snd_flip
thf(fact_1502_fst__snd__flip,axiom,
( product_fst_nat_nat
= ( comp_P1567445206330693457at_nat @ product_snd_nat_nat
@ ( produc2626176000494625587at_nat
@ ^ [X4: nat,Y5: nat] : ( product_Pair_nat_nat @ Y5 @ X4 ) ) ) ) ).
% fst_snd_flip
thf(fact_1503_fst__snd__flip,axiom,
( produc473321116828791851t_unit
= ( comp_P1287623958910499840t_unit @ produc6369869760173755749_ref_a
@ ( produc2058451029528183389_ref_a
@ ^ [X4: ref_a,Y5: heap_e7401611519738050253t_unit] : ( produc2564672488694626579_ref_a @ Y5 @ X4 ) ) ) ) ).
% fst_snd_flip
thf(fact_1504_fst__snd__flip,axiom,
( produc5717040794327095847_ref_a
= ( comp_P2006160480335490974_ref_a @ produc1126150082675451753t_unit
@ ( produc4426278547552487717t_unit
@ ^ [X4: heap_e7401611519738050253t_unit,Y5: ref_a] : ( produc6544324848051098391t_unit @ Y5 @ X4 ) ) ) ) ).
% fst_snd_flip
thf(fact_1505_fst__snd__flip,axiom,
( product_fst_int_int
= ( comp_P5356232565778667509nt_int @ product_snd_int_int
@ ( produc4245557441103728435nt_int
@ ^ [X4: int,Y5: int] : ( product_Pair_int_int @ Y5 @ X4 ) ) ) ) ).
% fst_snd_flip
thf(fact_1506_snd__fst__flip,axiom,
( produc826222791318859909at_nat
= ( comp_P3728553842126764068at_nat @ produc4486455499028916547at_nat
@ ( produc5456455677105495163at_nat
@ ^ [X4: set_Pr4329608150637261639at_nat,Y5: set_Pr4329608150637261639at_nat] : ( produc9060074326276436823at_nat @ Y5 @ X4 ) ) ) ) ).
% snd_fst_flip
thf(fact_1507_snd__fst__flip,axiom,
( produc2937545259423399429at_nat
= ( comp_P4974406650942564362at_nat @ produc6011863565407846595at_nat
@ ( produc7548845685434208379at_nat
@ ^ [X4: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat] : ( produc2922128104949294807at_nat @ Y5 @ X4 ) ) ) ) ).
% snd_fst_flip
thf(fact_1508_snd__fst__flip,axiom,
( product_snd_num_nat
= ( comp_P8336958949797789649um_nat @ product_fst_nat_num
@ ( produc5219324331026901427at_num
@ ^ [X4: num,Y5: nat] : ( product_Pair_nat_num @ Y5 @ X4 ) ) ) ) ).
% snd_fst_flip
thf(fact_1509_snd__fst__flip,axiom,
( product_snd_nat_nat
= ( comp_P1567445206330693457at_nat @ product_fst_nat_nat
@ ( produc2626176000494625587at_nat
@ ^ [X4: nat,Y5: nat] : ( product_Pair_nat_nat @ Y5 @ X4 ) ) ) ) ).
% snd_fst_flip
thf(fact_1510_snd__fst__flip,axiom,
( produc6369869760173755749_ref_a
= ( comp_P7254561687666694144_ref_a @ produc473321116828791851t_unit
@ ( produc4426278547552487717t_unit
@ ^ [X4: heap_e7401611519738050253t_unit,Y5: ref_a] : ( produc6544324848051098391t_unit @ Y5 @ X4 ) ) ) ) ).
% snd_fst_flip
thf(fact_1511_snd__fst__flip,axiom,
( produc1126150082675451753t_unit
= ( comp_P3239167524078694454t_unit @ produc5717040794327095847_ref_a
@ ( produc2058451029528183389_ref_a
@ ^ [X4: ref_a,Y5: heap_e7401611519738050253t_unit] : ( produc2564672488694626579_ref_a @ Y5 @ X4 ) ) ) ) ).
% snd_fst_flip
thf(fact_1512_snd__fst__flip,axiom,
( product_snd_int_int
= ( comp_P5356232565778667509nt_int @ product_fst_int_int
@ ( produc4245557441103728435nt_int
@ ^ [X4: int,Y5: int] : ( product_Pair_int_int @ Y5 @ X4 ) ) ) ) ).
% snd_fst_flip
thf(fact_1513_comp__fun__idem__on_Ocomp__fun__idem__on,axiom,
! [S5: set_Pr4329608150637261639at_nat,F: produc3843707927480180839at_nat > nat > nat,X2: produc3843707927480180839at_nat] :
( ( finite352855791390415404at_nat @ S5 @ F )
=> ( ( member8757157785044589968at_nat @ X2 @ S5 )
=> ( ( comp_nat_nat_nat @ ( F @ X2 ) @ ( F @ X2 ) )
= ( F @ X2 ) ) ) ) ).
% comp_fun_idem_on.comp_fun_idem_on
thf(fact_1514_comp__fun__idem__on_Ocomp__fun__idem__on,axiom,
! [S5: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat > nat,X2: product_prod_nat_nat] :
( ( finite5150387867909132486at_nat @ S5 @ F )
=> ( ( member8440522571783428010at_nat @ X2 @ S5 )
=> ( ( comp_nat_nat_nat @ ( F @ X2 ) @ ( F @ X2 ) )
= ( F @ X2 ) ) ) ) ).
% comp_fun_idem_on.comp_fun_idem_on
thf(fact_1515_comp__fun__idem__on_Ocomp__fun__idem__on,axiom,
! [S5: set_o,F: $o > nat > nat,X2: $o] :
( ( finite2426395113773319005_o_nat @ S5 @ F )
=> ( ( member_o @ X2 @ S5 )
=> ( ( comp_nat_nat_nat @ ( F @ X2 ) @ ( F @ X2 ) )
= ( F @ X2 ) ) ) ) ).
% comp_fun_idem_on.comp_fun_idem_on
thf(fact_1516_comp__fun__idem__on_Ocomp__fun__idem__on,axiom,
! [S5: set_nat,F: nat > nat > nat,X2: nat] :
( ( finite7982400111564556781at_nat @ S5 @ F )
=> ( ( member_nat @ X2 @ S5 )
=> ( ( comp_nat_nat_nat @ ( F @ X2 ) @ ( F @ X2 ) )
= ( F @ X2 ) ) ) ) ).
% comp_fun_idem_on.comp_fun_idem_on
thf(fact_1517_comp__fun__idem__on_Ocomp__fun__idem__on,axiom,
! [S5: set_int,F: int > nat > nat,X2: int] :
( ( finite8981753823465611337nt_nat @ S5 @ F )
=> ( ( member_int @ X2 @ S5 )
=> ( ( comp_nat_nat_nat @ ( F @ X2 ) @ ( F @ X2 ) )
= ( F @ X2 ) ) ) ) ).
% comp_fun_idem_on.comp_fun_idem_on
thf(fact_1518_OO__Grp__cong,axiom,
! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > product_prod_nat_nat,G: product_prod_nat_nat > int] :
( ( A4 = B5 )
=> ( ( relcom8911704206367525917at_int @ ( conver6965824823119931701at_nat @ ( bNF_Gr7275288873855648526at_nat @ A4 @ F ) ) @ ( bNF_Gr190934199572118623at_int @ A4 @ G ) )
= ( relcom8911704206367525917at_int @ ( conver6965824823119931701at_nat @ ( bNF_Gr7275288873855648526at_nat @ B5 @ F ) ) @ ( bNF_Gr190934199572118623at_int @ B5 @ G ) ) ) ) ).
% OO_Grp_cong
thf(fact_1519_OO__Grp__cong,axiom,
! [A4: set_int,B5: set_int,F: int > int,G: int > code_integer] :
( ( A4 = B5 )
=> ( ( relcom967497705283160040nteger @ ( conversep_int_int @ ( bNF_Grp_int_int @ A4 @ F ) ) @ ( bNF_Gr6684649993275963353nteger @ A4 @ G ) )
= ( relcom967497705283160040nteger @ ( conversep_int_int @ ( bNF_Grp_int_int @ B5 @ F ) ) @ ( bNF_Gr6684649993275963353nteger @ B5 @ G ) ) ) ) ).
% OO_Grp_cong
thf(fact_1520_fst__diag__fst,axiom,
( ( comp_P9134237804721252988t_unit @ produc6942380573175061695_ref_a
@ ( comp_r6154141770783548544t_unit
@ ^ [X4: ref_a] : ( produc1340070678782110931_ref_a @ X4 @ X4 )
@ produc473321116828791851t_unit ) )
= produc473321116828791851t_unit ) ).
% fst_diag_fst
thf(fact_1521_snd__diag__snd,axiom,
( ( comp_P8237986404257982958t_unit @ produc1692618790031806469t_unit
@ ( comp_h5940001671827863714t_unit
@ ^ [X4: heap_e7401611519738050253t_unit] : ( produc8256076736985659095t_unit @ X4 @ X4 )
@ produc1126150082675451753t_unit ) )
= produc1126150082675451753t_unit ) ).
% snd_diag_snd
thf(fact_1522_type__copy__map__comp0,axiom,
! [Rep: $o > $o,Abs: $o > $o,M: nat > nat,M1: $o > nat,M2: nat > $o,F: nat > nat,G: nat > nat] :
( ( type_definition_o_o @ Rep @ Abs @ top_top_set_o )
=> ( ( M
= ( comp_o_nat_nat @ M1 @ M2 ) )
=> ( ( comp_nat_nat_nat @ ( comp_nat_nat_nat @ F @ M ) @ G )
= ( comp_o_nat_nat @ ( comp_o_nat_o @ ( comp_nat_nat_o @ F @ M1 ) @ Rep ) @ ( comp_nat_o_nat @ ( comp_o_o_nat @ Abs @ M2 ) @ G ) ) ) ) ) ).
% type_copy_map_comp0
thf(fact_1523_type__copy__map__comp0,axiom,
! [Rep: int > $o,Abs: $o > int,M: nat > $o,M1: $o > $o,M2: nat > $o,F: $o > nat,G: int > nat] :
( ( type_d8403010662121346494_int_o @ Rep @ Abs @ top_top_set_o )
=> ( ( M
= ( comp_o_o_nat @ M1 @ M2 ) )
=> ( ( comp_nat_nat_int @ ( comp_o_nat_nat @ F @ M ) @ G )
= ( comp_int_nat_int @ ( comp_o_nat_int @ ( comp_o_nat_o @ F @ M1 ) @ Rep ) @ ( comp_nat_int_int @ ( comp_o_int_nat @ Abs @ M2 ) @ G ) ) ) ) ) ).
% type_copy_map_comp0
thf(fact_1524_type__copy__map__comp0,axiom,
! [Rep: num > $o,Abs: $o > num,M: nat > $o,M1: $o > $o,M2: nat > $o,F: $o > $o,G: int > nat] :
( ( type_d9070036042389994064_num_o @ Rep @ Abs @ top_top_set_o )
=> ( ( M
= ( comp_o_o_nat @ M1 @ M2 ) )
=> ( ( comp_nat_o_int @ ( comp_o_o_nat @ F @ M ) @ G )
= ( comp_num_o_int @ ( comp_o_o_num @ ( comp_o_o_o @ F @ M1 ) @ Rep ) @ ( comp_nat_num_int @ ( comp_o_num_nat @ Abs @ M2 ) @ G ) ) ) ) ) ).
% type_copy_map_comp0
thf(fact_1525_type__copy__map__comp0,axiom,
! [Rep: nat > $o,Abs: $o > nat,M: nat > $o,M1: $o > $o,M2: nat > $o,F: $o > $o,G: nat > nat] :
( ( type_d1000680970699258650_nat_o @ Rep @ Abs @ top_top_set_o )
=> ( ( M
= ( comp_o_o_nat @ M1 @ M2 ) )
=> ( ( comp_nat_o_nat @ ( comp_o_o_nat @ F @ M ) @ G )
= ( comp_nat_o_nat @ ( comp_o_o_nat @ ( comp_o_o_o @ F @ M1 ) @ Rep ) @ ( comp_nat_nat_nat @ ( comp_o_nat_nat @ Abs @ M2 ) @ G ) ) ) ) ) ).
% type_copy_map_comp0
thf(fact_1526_type__copy__map__comp0,axiom,
! [Rep: $o > $o,Abs: $o > $o,M: nat > $o,M1: $o > $o,M2: nat > $o,F: $o > $o,G: nat > nat] :
( ( type_definition_o_o @ Rep @ Abs @ top_top_set_o )
=> ( ( M
= ( comp_o_o_nat @ M1 @ M2 ) )
=> ( ( comp_nat_o_nat @ ( comp_o_o_nat @ F @ M ) @ G )
= ( comp_o_o_nat @ ( comp_o_o_o @ ( comp_o_o_o @ F @ M1 ) @ Rep ) @ ( comp_nat_o_nat @ ( comp_o_o_nat @ Abs @ M2 ) @ G ) ) ) ) ) ).
% type_copy_map_comp0
thf(fact_1527_type__copy__map__comp0,axiom,
! [Rep: $o > $o,Abs: $o > $o,M: nat > $o,M1: $o > $o,M2: nat > $o,F: $o > num,G: int > nat] :
( ( type_definition_o_o @ Rep @ Abs @ top_top_set_o )
=> ( ( M
= ( comp_o_o_nat @ M1 @ M2 ) )
=> ( ( comp_nat_num_int @ ( comp_o_num_nat @ F @ M ) @ G )
= ( comp_o_num_int @ ( comp_o_num_o @ ( comp_o_num_o @ F @ M1 ) @ Rep ) @ ( comp_nat_o_int @ ( comp_o_o_nat @ Abs @ M2 ) @ G ) ) ) ) ) ).
% type_copy_map_comp0
thf(fact_1528_type__copy__map__comp0,axiom,
! [Rep: num > $o,Abs: $o > num,M: nat > $o,M1: $o > $o,M2: nat > $o,F: $o > num,G: int > nat] :
( ( type_d9070036042389994064_num_o @ Rep @ Abs @ top_top_set_o )
=> ( ( M
= ( comp_o_o_nat @ M1 @ M2 ) )
=> ( ( comp_nat_num_int @ ( comp_o_num_nat @ F @ M ) @ G )
= ( comp_num_num_int @ ( comp_o_num_num @ ( comp_o_num_o @ F @ M1 ) @ Rep ) @ ( comp_nat_num_int @ ( comp_o_num_nat @ Abs @ M2 ) @ G ) ) ) ) ) ).
% type_copy_map_comp0
thf(fact_1529_type__copy__map__comp0,axiom,
! [Rep: nat > $o,Abs: $o > nat,M: nat > $o,M1: $o > $o,M2: nat > $o,F: $o > num,G: int > nat] :
( ( type_d1000680970699258650_nat_o @ Rep @ Abs @ top_top_set_o )
=> ( ( M
= ( comp_o_o_nat @ M1 @ M2 ) )
=> ( ( comp_nat_num_int @ ( comp_o_num_nat @ F @ M ) @ G )
= ( comp_nat_num_int @ ( comp_o_num_nat @ ( comp_o_num_o @ F @ M1 ) @ Rep ) @ ( comp_nat_nat_int @ ( comp_o_nat_nat @ Abs @ M2 ) @ G ) ) ) ) ) ).
% type_copy_map_comp0
thf(fact_1530_type__copy__map__comp0,axiom,
! [Rep: $o > $o,Abs: $o > $o,M: nat > $o,M1: $o > $o,M2: nat > $o,F: $o > nat,G: nat > nat] :
( ( type_definition_o_o @ Rep @ Abs @ top_top_set_o )
=> ( ( M
= ( comp_o_o_nat @ M1 @ M2 ) )
=> ( ( comp_nat_nat_nat @ ( comp_o_nat_nat @ F @ M ) @ G )
= ( comp_o_nat_nat @ ( comp_o_nat_o @ ( comp_o_nat_o @ F @ M1 ) @ Rep ) @ ( comp_nat_o_nat @ ( comp_o_o_nat @ Abs @ M2 ) @ G ) ) ) ) ) ).
% type_copy_map_comp0
thf(fact_1531_type__copy__map__comp0,axiom,
! [Rep: nat > $o,Abs: $o > nat,M: nat > $o,M1: $o > $o,M2: nat > $o,F: $o > nat,G: nat > nat] :
( ( type_d1000680970699258650_nat_o @ Rep @ Abs @ top_top_set_o )
=> ( ( M
= ( comp_o_o_nat @ M1 @ M2 ) )
=> ( ( comp_nat_nat_nat @ ( comp_o_nat_nat @ F @ M ) @ G )
= ( comp_nat_nat_nat @ ( comp_o_nat_nat @ ( comp_o_nat_o @ F @ M1 ) @ Rep ) @ ( comp_nat_nat_nat @ ( comp_o_nat_nat @ Abs @ M2 ) @ G ) ) ) ) ) ).
% type_copy_map_comp0
thf(fact_1532_type__copy__map__comp0__undo,axiom,
! [Rep: int > $o,Abs: $o > int,Rep2: nat > $o,Abs2: $o > nat,Rep3: int > nat,Abs3: nat > int,M: nat > $o,M1: $o > $o,M2: nat > $o] :
( ( type_d8403010662121346494_int_o @ Rep @ Abs @ top_top_set_o )
=> ( ( type_d1000680970699258650_nat_o @ Rep2 @ Abs2 @ top_top_set_o )
=> ( ( type_d7249847660678803242nt_nat @ Rep3 @ Abs3 @ top_top_set_nat )
=> ( ( ( comp_nat_nat_int @ ( comp_o_nat_nat @ Abs2 @ M ) @ Rep3 )
= ( comp_int_nat_int @ ( comp_o_nat_int @ ( comp_o_nat_o @ Abs2 @ M1 ) @ Rep ) @ ( comp_nat_int_int @ ( comp_o_int_nat @ Abs @ M2 ) @ Rep3 ) ) )
=> ( ( comp_o_o_nat @ M1 @ M2 )
= M ) ) ) ) ) ).
% type_copy_map_comp0_undo
thf(fact_1533_type__copy__map__comp0__undo,axiom,
! [Rep: num > $o,Abs: $o > num,Rep2: $o > $o,Abs2: $o > $o,Rep3: int > nat,Abs3: nat > int,M: nat > $o,M1: $o > $o,M2: nat > $o] :
( ( type_d9070036042389994064_num_o @ Rep @ Abs @ top_top_set_o )
=> ( ( type_definition_o_o @ Rep2 @ Abs2 @ top_top_set_o )
=> ( ( type_d7249847660678803242nt_nat @ Rep3 @ Abs3 @ top_top_set_nat )
=> ( ( ( comp_nat_o_int @ ( comp_o_o_nat @ Abs2 @ M ) @ Rep3 )
= ( comp_num_o_int @ ( comp_o_o_num @ ( comp_o_o_o @ Abs2 @ M1 ) @ Rep ) @ ( comp_nat_num_int @ ( comp_o_num_nat @ Abs @ M2 ) @ Rep3 ) ) )
=> ( ( comp_o_o_nat @ M1 @ M2 )
= M ) ) ) ) ) ).
% type_copy_map_comp0_undo
thf(fact_1534_type__copy__map__comp0__undo,axiom,
! [Rep: nat > $o,Abs: $o > nat,Rep2: $o > $o,Abs2: $o > $o,Rep3: nat > nat,Abs3: nat > nat,M: nat > $o,M1: $o > $o,M2: nat > $o] :
( ( type_d1000680970699258650_nat_o @ Rep @ Abs @ top_top_set_o )
=> ( ( type_definition_o_o @ Rep2 @ Abs2 @ top_top_set_o )
=> ( ( type_d6250493948777748686at_nat @ Rep3 @ Abs3 @ top_top_set_nat )
=> ( ( ( comp_nat_o_nat @ ( comp_o_o_nat @ Abs2 @ M ) @ Rep3 )
= ( comp_nat_o_nat @ ( comp_o_o_nat @ ( comp_o_o_o @ Abs2 @ M1 ) @ Rep ) @ ( comp_nat_nat_nat @ ( comp_o_nat_nat @ Abs @ M2 ) @ Rep3 ) ) )
=> ( ( comp_o_o_nat @ M1 @ M2 )
= M ) ) ) ) ) ).
% type_copy_map_comp0_undo
thf(fact_1535_type__copy__map__comp0__undo,axiom,
! [Rep: $o > $o,Abs: $o > $o,Rep2: $o > $o,Abs2: $o > $o,Rep3: nat > nat,Abs3: nat > nat,M: nat > $o,M1: $o > $o,M2: nat > $o] :
( ( type_definition_o_o @ Rep @ Abs @ top_top_set_o )
=> ( ( type_definition_o_o @ Rep2 @ Abs2 @ top_top_set_o )
=> ( ( type_d6250493948777748686at_nat @ Rep3 @ Abs3 @ top_top_set_nat )
=> ( ( ( comp_nat_o_nat @ ( comp_o_o_nat @ Abs2 @ M ) @ Rep3 )
= ( comp_o_o_nat @ ( comp_o_o_o @ ( comp_o_o_o @ Abs2 @ M1 ) @ Rep ) @ ( comp_nat_o_nat @ ( comp_o_o_nat @ Abs @ M2 ) @ Rep3 ) ) )
=> ( ( comp_o_o_nat @ M1 @ M2 )
= M ) ) ) ) ) ).
% type_copy_map_comp0_undo
thf(fact_1536_type__copy__map__comp0__undo,axiom,
! [Rep: $o > $o,Abs: $o > $o,Rep2: num > $o,Abs2: $o > num,Rep3: int > nat,Abs3: nat > int,M: nat > $o,M1: $o > $o,M2: nat > $o] :
( ( type_definition_o_o @ Rep @ Abs @ top_top_set_o )
=> ( ( type_d9070036042389994064_num_o @ Rep2 @ Abs2 @ top_top_set_o )
=> ( ( type_d7249847660678803242nt_nat @ Rep3 @ Abs3 @ top_top_set_nat )
=> ( ( ( comp_nat_num_int @ ( comp_o_num_nat @ Abs2 @ M ) @ Rep3 )
= ( comp_o_num_int @ ( comp_o_num_o @ ( comp_o_num_o @ Abs2 @ M1 ) @ Rep ) @ ( comp_nat_o_int @ ( comp_o_o_nat @ Abs @ M2 ) @ Rep3 ) ) )
=> ( ( comp_o_o_nat @ M1 @ M2 )
= M ) ) ) ) ) ).
% type_copy_map_comp0_undo
thf(fact_1537_type__copy__map__comp0__undo,axiom,
! [Rep: num > $o,Abs: $o > num,Rep2: num > $o,Abs2: $o > num,Rep3: int > nat,Abs3: nat > int,M: nat > $o,M1: $o > $o,M2: nat > $o] :
( ( type_d9070036042389994064_num_o @ Rep @ Abs @ top_top_set_o )
=> ( ( type_d9070036042389994064_num_o @ Rep2 @ Abs2 @ top_top_set_o )
=> ( ( type_d7249847660678803242nt_nat @ Rep3 @ Abs3 @ top_top_set_nat )
=> ( ( ( comp_nat_num_int @ ( comp_o_num_nat @ Abs2 @ M ) @ Rep3 )
= ( comp_num_num_int @ ( comp_o_num_num @ ( comp_o_num_o @ Abs2 @ M1 ) @ Rep ) @ ( comp_nat_num_int @ ( comp_o_num_nat @ Abs @ M2 ) @ Rep3 ) ) )
=> ( ( comp_o_o_nat @ M1 @ M2 )
= M ) ) ) ) ) ).
% type_copy_map_comp0_undo
thf(fact_1538_type__copy__map__comp0__undo,axiom,
! [Rep: nat > $o,Abs: $o > nat,Rep2: num > $o,Abs2: $o > num,Rep3: int > nat,Abs3: nat > int,M: nat > $o,M1: $o > $o,M2: nat > $o] :
( ( type_d1000680970699258650_nat_o @ Rep @ Abs @ top_top_set_o )
=> ( ( type_d9070036042389994064_num_o @ Rep2 @ Abs2 @ top_top_set_o )
=> ( ( type_d7249847660678803242nt_nat @ Rep3 @ Abs3 @ top_top_set_nat )
=> ( ( ( comp_nat_num_int @ ( comp_o_num_nat @ Abs2 @ M ) @ Rep3 )
= ( comp_nat_num_int @ ( comp_o_num_nat @ ( comp_o_num_o @ Abs2 @ M1 ) @ Rep ) @ ( comp_nat_nat_int @ ( comp_o_nat_nat @ Abs @ M2 ) @ Rep3 ) ) )
=> ( ( comp_o_o_nat @ M1 @ M2 )
= M ) ) ) ) ) ).
% type_copy_map_comp0_undo
thf(fact_1539_type__copy__map__comp0__undo,axiom,
! [Rep: $o > $o,Abs: $o > $o,Rep2: nat > $o,Abs2: $o > nat,Rep3: nat > nat,Abs3: nat > nat,M: nat > $o,M1: $o > $o,M2: nat > $o] :
( ( type_definition_o_o @ Rep @ Abs @ top_top_set_o )
=> ( ( type_d1000680970699258650_nat_o @ Rep2 @ Abs2 @ top_top_set_o )
=> ( ( type_d6250493948777748686at_nat @ Rep3 @ Abs3 @ top_top_set_nat )
=> ( ( ( comp_nat_nat_nat @ ( comp_o_nat_nat @ Abs2 @ M ) @ Rep3 )
= ( comp_o_nat_nat @ ( comp_o_nat_o @ ( comp_o_nat_o @ Abs2 @ M1 ) @ Rep ) @ ( comp_nat_o_nat @ ( comp_o_o_nat @ Abs @ M2 ) @ Rep3 ) ) )
=> ( ( comp_o_o_nat @ M1 @ M2 )
= M ) ) ) ) ) ).
% type_copy_map_comp0_undo
thf(fact_1540_type__copy__map__comp0__undo,axiom,
! [Rep: nat > $o,Abs: $o > nat,Rep2: nat > $o,Abs2: $o > nat,Rep3: nat > nat,Abs3: nat > nat,M: nat > $o,M1: $o > $o,M2: nat > $o] :
( ( type_d1000680970699258650_nat_o @ Rep @ Abs @ top_top_set_o )
=> ( ( type_d1000680970699258650_nat_o @ Rep2 @ Abs2 @ top_top_set_o )
=> ( ( type_d6250493948777748686at_nat @ Rep3 @ Abs3 @ top_top_set_nat )
=> ( ( ( comp_nat_nat_nat @ ( comp_o_nat_nat @ Abs2 @ M ) @ Rep3 )
= ( comp_nat_nat_nat @ ( comp_o_nat_nat @ ( comp_o_nat_o @ Abs2 @ M1 ) @ Rep ) @ ( comp_nat_nat_nat @ ( comp_o_nat_nat @ Abs @ M2 ) @ Rep3 ) ) )
=> ( ( comp_o_o_nat @ M1 @ M2 )
= M ) ) ) ) ) ).
% type_copy_map_comp0_undo
thf(fact_1541_type__copy__map__comp0__undo,axiom,
! [Rep: nat > $o,Abs: $o > nat,Rep2: $o > char,Abs2: char > $o,Rep3: nat > $o,Abs3: $o > nat,M: $o > char,M1: $o > char,M2: $o > $o] :
( ( type_d1000680970699258650_nat_o @ Rep @ Abs @ top_top_set_o )
=> ( ( type_d9076672964931703357o_char @ Rep2 @ Abs2 @ top_top_set_char )
=> ( ( type_d1000680970699258650_nat_o @ Rep3 @ Abs3 @ top_top_set_o )
=> ( ( ( comp_o_o_nat @ ( comp_char_o_o @ Abs2 @ M ) @ Rep3 )
= ( comp_nat_o_nat @ ( comp_o_o_nat @ ( comp_char_o_o @ Abs2 @ M1 ) @ Rep ) @ ( comp_o_nat_nat @ ( comp_o_nat_o @ Abs @ M2 ) @ Rep3 ) ) )
=> ( ( comp_o_char_o @ M1 @ M2 )
= M ) ) ) ) ) ).
% type_copy_map_comp0_undo
thf(fact_1542_fun_Orel__compp__Grp,axiom,
! [R3: nat > nat > $o] :
( ( bNF_re3734335887435638179at_nat
@ ^ [Y2: char,Z: char] : Y2 = Z
@ R3 )
= ( relcom8141925468094106422ar_nat
@ ( conver5210974729819302706ar_nat
@ ( bNF_Gr1779741795384719769ar_nat
@ ( collec1683818095804863890at_nat
@ ^ [X4: char > product_prod_nat_nat] : ( ord_le3146513528884898305at_nat @ ( image_8382823969816519465at_nat @ X4 @ top_top_set_char ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R3 ) ) ) )
@ ( comp_P1491501995297412481t_char @ product_fst_nat_nat ) ) )
@ ( bNF_Gr1779741795384719769ar_nat
@ ( collec1683818095804863890at_nat
@ ^ [X4: char > product_prod_nat_nat] : ( ord_le3146513528884898305at_nat @ ( image_8382823969816519465at_nat @ X4 @ top_top_set_char ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R3 ) ) ) )
@ ( comp_P1491501995297412481t_char @ product_snd_nat_nat ) ) ) ) ).
% fun.rel_compp_Grp
thf(fact_1543_fun_Orel__compp__Grp,axiom,
! [R3: nat > nat > $o] :
( ( bNF_re6557955094579809201at_nat
@ ^ [Y2: rat,Z: rat] : Y2 = Z
@ R3 )
= ( relcom7707146719550107439at_nat
@ ( conver7478146827351332082at_nat
@ ( bNF_Gr7368557922277781849at_nat
@ ( collec2824469931397938891at_nat
@ ^ [X4: rat > product_prod_nat_nat] : ( ord_le3146513528884898305at_nat @ ( image_9175953278516507234at_nat @ X4 @ top_top_set_rat ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R3 ) ) ) )
@ ( comp_P1320480151300532104at_rat @ product_fst_nat_nat ) ) )
@ ( bNF_Gr7368557922277781849at_nat
@ ( collec2824469931397938891at_nat
@ ^ [X4: rat > product_prod_nat_nat] : ( ord_le3146513528884898305at_nat @ ( image_9175953278516507234at_nat @ X4 @ top_top_set_rat ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R3 ) ) ) )
@ ( comp_P1320480151300532104at_rat @ product_snd_nat_nat ) ) ) ) ).
% fun.rel_compp_Grp
thf(fact_1544_fun_Orel__compp__Grp,axiom,
! [R3: nat > nat > $o] :
( ( bNF_re5653821019739307937at_nat
@ ^ [Y2: nat,Z: nat] : Y2 = Z
@ R3 )
= ( relcom8186637903969514807at_nat
@ ( conver3498139987538047730at_nat
@ ( bNF_Gr3388551082464497497at_nat
@ ( collec8718012497556192979at_nat
@ ^ [X4: nat > product_prod_nat_nat] : ( ord_le3146513528884898305at_nat @ ( image_5846123807819985514at_nat @ X4 @ top_top_set_nat ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R3 ) ) ) )
@ ( comp_P1955610211387027840at_nat @ product_fst_nat_nat ) ) )
@ ( bNF_Gr3388551082464497497at_nat
@ ( collec8718012497556192979at_nat
@ ^ [X4: nat > product_prod_nat_nat] : ( ord_le3146513528884898305at_nat @ ( image_5846123807819985514at_nat @ X4 @ top_top_set_nat ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R3 ) ) ) )
@ ( comp_P1955610211387027840at_nat @ product_snd_nat_nat ) ) ) ) ).
% fun.rel_compp_Grp
thf(fact_1545_fun_Orel__compp__Grp,axiom,
! [R3: nat > nat > $o] :
( ( bNF_re3715656647883201625at_nat
@ ^ [Y2: int,Z: int] : Y2 = Z
@ R3 )
= ( relcom2406850561174050523nt_nat
@ ( conver1572571282669976050nt_nat
@ ( bNF_Gr1462982377596425817nt_nat
@ ( collec2532482545165550967at_nat
@ ^ [X4: int > product_prod_nat_nat] : ( ord_le3146513528884898305at_nat @ ( image_8883965892284119310at_nat @ X4 @ top_top_set_int ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R3 ) ) ) )
@ ( comp_P1953119740877977564at_int @ product_fst_nat_nat ) ) )
@ ( bNF_Gr1462982377596425817nt_nat
@ ( collec2532482545165550967at_nat
@ ^ [X4: int > product_prod_nat_nat] : ( ord_le3146513528884898305at_nat @ ( image_8883965892284119310at_nat @ X4 @ top_top_set_int ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R3 ) ) ) )
@ ( comp_P1953119740877977564at_int @ product_snd_nat_nat ) ) ) ) ).
% fun.rel_compp_Grp
thf(fact_1546_fun_Orel__compp__Grp,axiom,
! [R3: num > num > $o] :
( ( bNF_re7626690874201225453um_num
@ ^ [Y2: int,Z: int] : Y2 = Z
@ R3 )
= ( relcom1697847500783689711nt_num
@ ( conver5699786650113435344nt_num
@ ( bNF_Gr5590197745039885111nt_num
@ ( collec266097228571071243um_num
@ ^ [X4: int > product_prod_num_num] : ( ord_le880128212290418581um_num @ ( image_7722540232540426402um_num @ X4 @ top_top_set_int ) @ ( collec2230928802738392704um_num @ ( produc5703948589228662326_num_o @ R3 ) ) ) )
@ ( comp_P4305060815297604026um_int @ product_fst_num_num ) ) )
@ ( bNF_Gr5590197745039885111nt_num
@ ( collec266097228571071243um_num
@ ^ [X4: int > product_prod_num_num] : ( ord_le880128212290418581um_num @ ( image_7722540232540426402um_num @ X4 @ top_top_set_int ) @ ( collec2230928802738392704um_num @ ( produc5703948589228662326_num_o @ R3 ) ) ) )
@ ( comp_P4305060815297604026um_int @ product_snd_num_num ) ) ) ) ).
% fun.rel_compp_Grp
thf(fact_1547_fun_Orel__compp__Grp,axiom,
! [R3: int > int > $o] :
( ( bNF_re4731199128827642459nt_int
@ ^ [Y2: char,Z: char] : Y2 = Z
@ R3 )
= ( relcom4417706183330134766ar_int
@ ( conver1155128050257070662ar_int
@ ( bNF_Gr6947267152677263533ar_int
@ ( collec1380656525566158922nt_int
@ ^ [X4: char > product_prod_int_int] : ( ord_le2843351958646193337nt_int @ ( image_5204326662208377313nt_int @ X4 @ top_top_set_char ) @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ R3 ) ) ) )
@ ( comp_P4161745170962941589t_char @ product_fst_int_int ) ) )
@ ( bNF_Gr6947267152677263533ar_int
@ ( collec1380656525566158922nt_int
@ ^ [X4: char > product_prod_int_int] : ( ord_le2843351958646193337nt_int @ ( image_5204326662208377313nt_int @ X4 @ top_top_set_char ) @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ R3 ) ) ) )
@ ( comp_P4161745170962941589t_char @ product_snd_int_int ) ) ) ) ).
% fun.rel_compp_Grp
thf(fact_1548_fun_Orel__compp__Grp,axiom,
! [R3: int > int > $o] :
( ( bNF_re7554818335971813481nt_int
@ ^ [Y2: rat,Z: rat] : Y2 = Z
@ R3 )
= ( relcom2810802720173317863at_int
@ ( conver6967689361887734790at_int
@ ( bNF_Gr6858100456814184557at_int
@ ( collec2521308361159233923nt_int
@ ^ [X4: rat > product_prod_int_int] : ( ord_le2843351958646193337nt_int @ ( image_5997455970908365082nt_int @ X4 @ top_top_set_rat ) @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ R3 ) ) ) )
@ ( comp_P6018825582104210588nt_rat @ product_fst_int_int ) ) )
@ ( bNF_Gr6858100456814184557at_int
@ ( collec2521308361159233923nt_int
@ ^ [X4: rat > product_prod_int_int] : ( ord_le2843351958646193337nt_int @ ( image_5997455970908365082nt_int @ X4 @ top_top_set_rat ) @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ R3 ) ) ) )
@ ( comp_P6018825582104210588nt_rat @ product_snd_int_int ) ) ) ) ).
% fun.rel_compp_Grp
thf(fact_1549_fun_Orel__compp__Grp,axiom,
! [R3: int > int > $o] :
( ( bNF_re6650684261131312217nt_int
@ ^ [Y2: nat,Z: nat] : Y2 = Z
@ R3 )
= ( relcom3290293904592725231at_int
@ ( conver2987682522074450438at_int
@ ( bNF_Gr2878093617000900205at_int
@ ( collec8414850927317488011nt_int
@ ^ [X4: nat > product_prod_int_int] : ( ord_le2843351958646193337nt_int @ ( image_2667626500211843362nt_int @ X4 @ top_top_set_nat ) @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ R3 ) ) ) )
@ ( comp_P6653955642190706324nt_nat @ product_fst_int_int ) ) )
@ ( bNF_Gr2878093617000900205at_int
@ ( collec8414850927317488011nt_int
@ ^ [X4: nat > product_prod_int_int] : ( ord_le2843351958646193337nt_int @ ( image_2667626500211843362nt_int @ X4 @ top_top_set_nat ) @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ R3 ) ) ) )
@ ( comp_P6653955642190706324nt_nat @ product_snd_int_int ) ) ) ) ).
% fun.rel_compp_Grp
thf(fact_1550_fun_Orel__compp__Grp,axiom,
! [R3: int > int > $o] :
( ( bNF_re4712519889275205905nt_int
@ ^ [Y2: int,Z: int] : Y2 = Z
@ R3 )
= ( relcom6733878598652036755nt_int
@ ( conver1062113817206378758nt_int
@ ( bNF_Gr952524912132828525nt_int
@ ( collec2229320974926845999nt_int
@ ^ [X4: int > product_prod_int_int] : ( ord_le2843351958646193337nt_int @ ( image_5705468584675977158nt_int @ X4 @ top_top_set_int ) @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ R3 ) ) ) )
@ ( comp_P6651465171681656048nt_int @ product_fst_int_int ) ) )
@ ( bNF_Gr952524912132828525nt_int
@ ( collec2229320974926845999nt_int
@ ^ [X4: int > product_prod_int_int] : ( ord_le2843351958646193337nt_int @ ( image_5705468584675977158nt_int @ X4 @ top_top_set_int ) @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ R3 ) ) ) )
@ ( comp_P6651465171681656048nt_int @ product_snd_int_int ) ) ) ) ).
% fun.rel_compp_Grp
thf(fact_1551_fun_Orel__compp__Grp,axiom,
! [R3: nat > nat > $o] :
( ( bNF_re4629543205785817921at_nat
@ ^ [Y2: list_nat,Z: list_nat] : Y2 = Z
@ R3 )
= ( relcom5920149085449455271at_nat
@ ( conver6066198818478722162at_nat
@ ( bNF_Gr1532745923208276441at_nat
@ ( collec3092956710985254339at_nat
@ ^ [X4: list_nat > product_prod_nat_nat] : ( ord_le3146513528884898305at_nat @ ( image_2795870212791037530at_nat @ X4 @ top_top_set_list_nat ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R3 ) ) ) )
@ ( comp_P5292260908303971472st_nat @ product_fst_nat_nat ) ) )
@ ( bNF_Gr1532745923208276441at_nat
@ ( collec3092956710985254339at_nat
@ ^ [X4: list_nat > product_prod_nat_nat] : ( ord_le3146513528884898305at_nat @ ( image_2795870212791037530at_nat @ X4 @ top_top_set_list_nat ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R3 ) ) ) )
@ ( comp_P5292260908303971472st_nat @ product_snd_nat_nat ) ) ) ) ).
% fun.rel_compp_Grp
thf(fact_1552_image__Pow__surj,axiom,
! [F: nat > rat,A4: set_nat,B5: set_rat] :
( ( ( image_nat_rat @ F @ A4 )
= B5 )
=> ( ( image_4408659257933336347et_rat @ ( image_nat_rat @ F ) @ ( pow_nat @ A4 ) )
= ( pow_rat @ B5 ) ) ) ).
% image_Pow_surj
thf(fact_1553_image__Pow__surj,axiom,
! [F: nat > nat,A4: set_nat,B5: set_nat] :
( ( ( image_nat_nat @ F @ A4 )
= B5 )
=> ( ( image_7916887816326733075et_nat @ ( image_nat_nat @ F ) @ ( pow_nat @ A4 ) )
= ( pow_nat @ B5 ) ) ) ).
% image_Pow_surj
thf(fact_1554_image__Pow__surj,axiom,
! [F: nat > int,A4: set_nat,B5: set_int] :
( ( ( image_nat_int @ F @ A4 )
= B5 )
=> ( ( image_3739036796817536367et_int @ ( image_nat_int @ F ) @ ( pow_nat @ A4 ) )
= ( pow_int @ B5 ) ) ) ).
% image_Pow_surj
thf(fact_1555_image__Pow__surj,axiom,
! [F: int > nat,A4: set_int,B5: set_nat] :
( ( ( image_int_nat @ F @ A4 )
= B5 )
=> ( ( image_4702325430467532143et_nat @ ( image_int_nat @ F ) @ ( pow_int @ A4 ) )
= ( pow_nat @ B5 ) ) ) ).
% image_Pow_surj
thf(fact_1556_image__Pow__surj,axiom,
! [F: int > int,A4: set_int,B5: set_int] :
( ( ( image_int_int @ F @ A4 )
= B5 )
=> ( ( image_524474410958335435et_int @ ( image_int_int @ F ) @ ( pow_int @ A4 ) )
= ( pow_int @ B5 ) ) ) ).
% image_Pow_surj
thf(fact_1557_fst__diag__snd,axiom,
( ( comp_P8237986404257982958t_unit @ produc2762368996035445955t_unit
@ ( comp_h5940001671827863714t_unit
@ ^ [X4: heap_e7401611519738050253t_unit] : ( produc8256076736985659095t_unit @ X4 @ X4 )
@ produc1126150082675451753t_unit ) )
= produc1126150082675451753t_unit ) ).
% fst_diag_snd
thf(fact_1558_snd__diag__fst,axiom,
( ( comp_P9134237804721252988t_unit @ produc8185483580310833921_ref_a
@ ( comp_r6154141770783548544t_unit
@ ^ [X4: ref_a] : ( produc1340070678782110931_ref_a @ X4 @ X4 )
@ produc473321116828791851t_unit ) )
= produc473321116828791851t_unit ) ).
% snd_diag_fst
thf(fact_1559_fun_Orel__Grp,axiom,
! [A4: set_Co726186806682698383nteger,F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger] :
( ( bNF_re1181178334029315097nteger
@ ^ [Y2: code_integer,Z: code_integer] : Y2 = Z
@ ( bNF_Gr4092539093814444494nteger @ A4 @ F ) )
= ( bNF_Gr1453458339979195462nteger
@ ( collec5651646645946169514nteger
@ ^ [X4: code_integer > code_integer > code_integer] : ( ord_le348599935723844335nteger @ ( image_2177946750938214461nteger @ X4 @ top_to4645266643341252675nteger ) @ A4 ) )
@ ( comp_C8797469213163452608nteger @ F ) ) ) ).
% fun.rel_Grp
thf(fact_1560_fun_Orel__Grp,axiom,
! [A4: set_o,F: $o > $o] :
( ( bNF_re4705727531993890431at_o_o
@ ^ [Y2: nat,Z: nat] : Y2 = Z
@ ( bNF_Grp_o_o @ A4 @ F ) )
= ( bNF_Grp_nat_o_nat_o
@ ( collect_nat_o
@ ^ [X4: nat > $o] : ( ord_less_eq_set_o @ ( image_nat_o @ X4 @ top_top_set_nat ) @ A4 ) )
@ ( comp_o_o_nat @ F ) ) ) ).
% fun.rel_Grp
thf(fact_1561_fun_Orel__Grp,axiom,
! [A4: set_nat,F: nat > nat] :
( ( bNF_re5653821019739307937at_nat
@ ^ [Y2: nat,Z: nat] : Y2 = Z
@ ( bNF_Grp_nat_nat @ A4 @ F ) )
= ( bNF_Gr3847987472475283150at_nat
@ ( collect_nat_nat
@ ^ [X4: nat > nat] : ( ord_less_eq_set_nat @ ( image_nat_nat @ X4 @ top_top_set_nat ) @ A4 ) )
@ ( comp_nat_nat_nat @ F ) ) ) ).
% fun.rel_Grp
thf(fact_1562_fun_Orel__Grp,axiom,
! [A4: set_nat,F: nat > num] :
( ( bNF_re272988434038980259at_num
@ ^ [Y2: int,Z: int] : Y2 = Z
@ ( bNF_Grp_nat_num @ A4 @ F ) )
= ( bNF_Gr6763702254253555408nt_num
@ ( collect_int_nat
@ ^ [X4: int > nat] : ( ord_less_eq_set_nat @ ( image_int_nat @ X4 @ top_top_set_int ) @ A4 ) )
@ ( comp_nat_num_int @ F ) ) ) ).
% fun.rel_Grp
thf(fact_1563_fun_Orel__Grp,axiom,
! [A4: set_int,F: int > nat] :
( ( bNF_re4715010359784256181nt_nat
@ ^ [Y2: int,Z: int] : Y2 = Z
@ ( bNF_Grp_int_nat @ A4 @ F ) )
= ( bNF_Gr2243168462153404386nt_nat
@ ( collect_int_int
@ ^ [X4: int > int] : ( ord_less_eq_set_int @ ( image_int_int @ X4 @ top_top_set_int ) @ A4 ) )
@ ( comp_int_nat_int @ F ) ) ) ).
% fun.rel_Grp
thf(fact_1564_fun_Orel__Grp,axiom,
! [A4: set_num,F: num > num] :
( ( bNF_re7626690874201225453um_num
@ ^ [Y2: int,Z: int] : Y2 = Z
@ ( bNF_Grp_num_num @ A4 @ F ) )
= ( bNF_Gr318824752125099290nt_num
@ ( collect_int_num
@ ^ [X4: int > num] : ( ord_less_eq_set_num @ ( image_int_num @ X4 @ top_top_set_int ) @ A4 ) )
@ ( comp_num_num_int @ F ) ) ) ).
% fun.rel_Grp
thf(fact_1565_Grp__fst__snd,axiom,
! [R3: nat > nat > $o] :
( ( relcom8572034174491624592at_nat @ ( conver4691796276509297820at_nat @ ( bNF_Gr193424670081168899at_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R3 ) ) @ product_fst_nat_nat ) ) @ ( bNF_Gr193424670081168899at_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R3 ) ) @ product_snd_nat_nat ) )
= R3 ) ).
% Grp_fst_snd
thf(fact_1566_Grp__fst__snd,axiom,
! [R3: ref_a > heap_e7401611519738050253t_unit > $o] :
( ( relcom3136146828236442456t_unit @ ( conver4368408235330554745_ref_a @ ( bNF_Gr5964920315653434322_ref_a @ ( collec3252306575900995386t_unit @ ( produc4924168586796184916unit_o @ R3 ) ) @ produc473321116828791851t_unit ) ) @ ( bNF_Gr5237703264515865432t_unit @ ( collec3252306575900995386t_unit @ ( produc4924168586796184916unit_o @ R3 ) ) @ produc1126150082675451753t_unit ) )
= R3 ) ).
% Grp_fst_snd
thf(fact_1567_Grp__fst__snd,axiom,
! [R3: int > int > $o] :
( ( relcom3664783465810776136nt_int @ ( conver7247880940930113968nt_int @ ( bNF_Gr2749509334501985047nt_int @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ R3 ) ) @ product_fst_int_int ) ) @ ( bNF_Gr2749509334501985047nt_int @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ R3 ) ) @ product_snd_int_int ) )
= R3 ) ).
% Grp_fst_snd
thf(fact_1568_type__copy__set__map0,axiom,
! [Rep: $o > char,Abs: char > $o,S5: char > set_rat,M: $o > char,F: nat > rat,S6: $o > set_nat,G: nat > $o] :
( ( type_d9076672964931703357o_char @ Rep @ Abs @ top_top_set_char )
=> ( ( ( comp_char_set_rat_o @ S5 @ M )
= ( comp_s8507027185284273247_rat_o @ ( image_nat_rat @ F ) @ S6 ) )
=> ( ( comp_o_set_rat_nat @ ( comp_char_set_rat_o @ S5 @ Rep ) @ ( comp_o_o_nat @ ( comp_char_o_o @ Abs @ M ) @ G ) )
= ( comp_s8255929034757298889at_nat @ ( image_nat_rat @ F ) @ ( comp_o_set_nat_nat @ S6 @ G ) ) ) ) ) ).
% type_copy_set_map0
thf(fact_1569_type__copy__set__map0,axiom,
! [Rep: num > char,Abs: char > num,S5: char > set_rat,M: nat > char,F: nat > rat,S6: nat > set_nat,G: int > nat] :
( ( type_d4396077731765706521m_char @ Rep @ Abs @ top_top_set_char )
=> ( ( ( comp_c5738158305436223508at_nat @ S5 @ M )
= ( comp_s8255929034757298889at_nat @ ( image_nat_rat @ F ) @ S6 ) )
=> ( ( comp_num_set_rat_int @ ( comp_c2295490091592002142at_num @ S5 @ Rep ) @ ( comp_nat_num_int @ ( comp_char_num_nat @ Abs @ M ) @ G ) )
= ( comp_s8253438564248248613at_int @ ( image_nat_rat @ F ) @ ( comp_nat_set_nat_int @ S6 @ G ) ) ) ) ) ).
% type_copy_set_map0
thf(fact_1570_type__copy__set__map0,axiom,
! [Rep: nat > char,Abs: char > nat,S5: char > set_rat,M: nat > char,F: nat > rat,S6: nat > set_nat,G: nat > nat] :
( ( type_d5945782884131516111t_char @ Rep @ Abs @ top_top_set_char )
=> ( ( ( comp_c5738158305436223508at_nat @ S5 @ M )
= ( comp_s8255929034757298889at_nat @ ( image_nat_rat @ F ) @ S6 ) )
=> ( ( comp_nat_set_rat_nat @ ( comp_c5738158305436223508at_nat @ S5 @ Rep ) @ ( comp_nat_nat_nat @ ( comp_char_nat_nat @ Abs @ M ) @ G ) )
= ( comp_s8255929034757298889at_nat @ ( image_nat_rat @ F ) @ ( comp_nat_set_nat_nat @ S6 @ G ) ) ) ) ) ).
% type_copy_set_map0
thf(fact_1571_type__copy__set__map0,axiom,
! [Rep: nat > char,Abs: char > nat,S5: char > set_rat,M: int > char,F: nat > rat,S6: int > set_nat,G: int > int] :
( ( type_d5945782884131516111t_char @ Rep @ Abs @ top_top_set_char )
=> ( ( ( comp_c5735667834927173232at_int @ S5 @ M )
= ( comp_s8253438564248248613at_int @ ( image_nat_rat @ F ) @ S6 ) )
=> ( ( comp_nat_set_rat_int @ ( comp_c5738158305436223508at_nat @ S5 @ Rep ) @ ( comp_int_nat_int @ ( comp_char_nat_int @ Abs @ M ) @ G ) )
= ( comp_s8253438564248248613at_int @ ( image_nat_rat @ F ) @ ( comp_int_set_nat_int @ S6 @ G ) ) ) ) ) ).
% type_copy_set_map0
thf(fact_1572_type__copy__set__map0,axiom,
! [Rep: $o > char,Abs: char > $o,S5: char > set_nat,M: $o > char,F: nat > nat,S6: $o > set_nat,G: nat > $o] :
( ( type_d9076672964931703357o_char @ Rep @ Abs @ top_top_set_char )
=> ( ( ( comp_char_set_nat_o @ S5 @ M )
= ( comp_s33089643564333159_nat_o @ ( image_nat_nat @ F ) @ S6 ) )
=> ( ( comp_o_set_nat_nat @ ( comp_char_set_nat_o @ S5 @ Rep ) @ ( comp_o_o_nat @ ( comp_char_o_o @ Abs @ M ) @ G ) )
= ( comp_s3433241188411525313at_nat @ ( image_nat_nat @ F ) @ ( comp_o_set_nat_nat @ S6 @ G ) ) ) ) ) ).
% type_copy_set_map0
thf(fact_1573_type__copy__set__map0,axiom,
! [Rep: num > char,Abs: char > num,S5: char > set_nat,M: nat > char,F: nat > nat,S6: nat > set_nat,G: int > nat] :
( ( type_d4396077731765706521m_char @ Rep @ Abs @ top_top_set_char )
=> ( ( ( comp_c915470459090449932at_nat @ S5 @ M )
= ( comp_s3433241188411525313at_nat @ ( image_nat_nat @ F ) @ S6 ) )
=> ( ( comp_num_set_nat_int @ ( comp_c6696174282101004374at_num @ S5 @ Rep ) @ ( comp_nat_num_int @ ( comp_char_num_nat @ Abs @ M ) @ G ) )
= ( comp_s3430750717902475037at_int @ ( image_nat_nat @ F ) @ ( comp_nat_set_nat_int @ S6 @ G ) ) ) ) ) ).
% type_copy_set_map0
thf(fact_1574_type__copy__set__map0,axiom,
! [Rep: nat > char,Abs: char > nat,S5: char > set_nat,M: nat > char,F: nat > nat,S6: nat > set_nat,G: nat > nat] :
( ( type_d5945782884131516111t_char @ Rep @ Abs @ top_top_set_char )
=> ( ( ( comp_c915470459090449932at_nat @ S5 @ M )
= ( comp_s3433241188411525313at_nat @ ( image_nat_nat @ F ) @ S6 ) )
=> ( ( comp_nat_set_nat_nat @ ( comp_c915470459090449932at_nat @ S5 @ Rep ) @ ( comp_nat_nat_nat @ ( comp_char_nat_nat @ Abs @ M ) @ G ) )
= ( comp_s3433241188411525313at_nat @ ( image_nat_nat @ F ) @ ( comp_nat_set_nat_nat @ S6 @ G ) ) ) ) ) ).
% type_copy_set_map0
thf(fact_1575_type__copy__set__map0,axiom,
! [Rep: nat > char,Abs: char > nat,S5: char > set_nat,M: int > char,F: nat > nat,S6: int > set_nat,G: int > int] :
( ( type_d5945782884131516111t_char @ Rep @ Abs @ top_top_set_char )
=> ( ( ( comp_c912979988581399656at_int @ S5 @ M )
= ( comp_s3430750717902475037at_int @ ( image_nat_nat @ F ) @ S6 ) )
=> ( ( comp_nat_set_nat_int @ ( comp_c915470459090449932at_nat @ S5 @ Rep ) @ ( comp_int_nat_int @ ( comp_char_nat_int @ Abs @ M ) @ G ) )
= ( comp_s3430750717902475037at_int @ ( image_nat_nat @ F ) @ ( comp_int_set_nat_int @ S6 @ G ) ) ) ) ) ).
% type_copy_set_map0
thf(fact_1576_type__copy__set__map0,axiom,
! [Rep: $o > char,Abs: char > $o,S5: char > set_int,M: $o > char,F: nat > int,S6: $o > set_nat,G: nat > $o] :
( ( type_d9076672964931703357o_char @ Rep @ Abs @ top_top_set_char )
=> ( ( ( comp_char_set_int_o @ S5 @ M )
= ( comp_s7011316712748663051_int_o @ ( image_nat_int @ F ) @ S6 ) )
=> ( ( comp_o_set_int_nat @ ( comp_char_set_int_o @ S5 @ Rep ) @ ( comp_o_o_nat @ ( comp_char_o_o @ Abs @ M ) @ G ) )
= ( comp_s5620162763104547101nt_nat @ ( image_nat_int @ F ) @ ( comp_o_set_nat_nat @ S6 @ G ) ) ) ) ) ).
% type_copy_set_map0
thf(fact_1577_type__copy__set__map0,axiom,
! [Rep: num > char,Abs: char > num,S5: char > set_int,M: nat > char,F: nat > int,S6: nat > set_nat,G: int > nat] :
( ( type_d4396077731765706521m_char @ Rep @ Abs @ top_top_set_char )
=> ( ( ( comp_c3102392033783471720nt_nat @ S5 @ M )
= ( comp_s5620162763104547101nt_nat @ ( image_nat_int @ F ) @ S6 ) )
=> ( ( comp_num_set_int_int @ ( comp_c8883095856794026162nt_num @ S5 @ Rep ) @ ( comp_nat_num_int @ ( comp_char_num_nat @ Abs @ M ) @ G ) )
= ( comp_s5617672292595496825nt_int @ ( image_nat_int @ F ) @ ( comp_nat_set_nat_int @ S6 @ G ) ) ) ) ) ).
% type_copy_set_map0
thf(fact_1578_type__definition_ORep__range,axiom,
! [Rep: nat > rat,Abs: rat > nat,A4: set_rat] :
( ( type_d5615363888691252950at_rat @ Rep @ Abs @ A4 )
=> ( ( image_nat_rat @ Rep @ top_top_set_nat )
= A4 ) ) ).
% type_definition.Rep_range
thf(fact_1579_type__definition_ORep__range,axiom,
! [Rep: nat > nat,Abs: nat > nat,A4: set_nat] :
( ( type_d6250493948777748686at_nat @ Rep @ Abs @ A4 )
=> ( ( image_nat_nat @ Rep @ top_top_set_nat )
= A4 ) ) ).
% type_definition.Rep_range
thf(fact_1580_type__definition_ORep__range,axiom,
! [Rep: nat > int,Abs: int > nat,A4: set_int] :
( ( type_d6248003478268698410at_int @ Rep @ Abs @ A4 )
=> ( ( image_nat_int @ Rep @ top_top_set_nat )
= A4 ) ) ).
% type_definition.Rep_range
thf(fact_1581_type__definition_ORep__range,axiom,
! [Rep: int > nat,Abs: nat > int,A4: set_nat] :
( ( type_d7249847660678803242nt_nat @ Rep @ Abs @ A4 )
=> ( ( image_int_nat @ Rep @ top_top_set_int )
= A4 ) ) ).
% type_definition.Rep_range
thf(fact_1582_type__definition_ORep__range,axiom,
! [Rep: int > int,Abs: int > int,A4: set_int] :
( ( type_d7247357190169752966nt_int @ Rep @ Abs @ A4 )
=> ( ( image_int_int @ Rep @ top_top_set_int )
= A4 ) ) ).
% type_definition.Rep_range
thf(fact_1583_type__definition_ORep__range,axiom,
! [Rep: product_unit > $o,Abs: $o > product_unit,A4: set_o] :
( ( type_d6188575255521822967unit_o @ Rep @ Abs @ A4 )
=> ( ( image_Product_unit_o @ Rep @ top_to1996260823553986621t_unit )
= A4 ) ) ).
% type_definition.Rep_range
thf(fact_1584_type__definition_OAbs__image,axiom,
! [Rep: rat > nat,Abs: nat > rat,A4: set_nat] :
( ( type_d5933939304842882774at_nat @ Rep @ Abs @ A4 )
=> ( ( image_nat_rat @ Abs @ A4 )
= top_top_set_rat ) ) ).
% type_definition.Abs_image
thf(fact_1585_type__definition_OAbs__image,axiom,
! [Rep: nat > nat,Abs: nat > nat,A4: set_nat] :
( ( type_d6250493948777748686at_nat @ Rep @ Abs @ A4 )
=> ( ( image_nat_nat @ Abs @ A4 )
= top_top_set_nat ) ) ).
% type_definition.Abs_image
thf(fact_1586_type__definition_OAbs__image,axiom,
! [Rep: nat > int,Abs: int > nat,A4: set_int] :
( ( type_d6248003478268698410at_int @ Rep @ Abs @ A4 )
=> ( ( image_int_nat @ Abs @ A4 )
= top_top_set_nat ) ) ).
% type_definition.Abs_image
thf(fact_1587_type__definition_OAbs__image,axiom,
! [Rep: int > nat,Abs: nat > int,A4: set_nat] :
( ( type_d7249847660678803242nt_nat @ Rep @ Abs @ A4 )
=> ( ( image_nat_int @ Abs @ A4 )
= top_top_set_int ) ) ).
% type_definition.Abs_image
thf(fact_1588_type__definition_OAbs__image,axiom,
! [Rep: int > int,Abs: int > int,A4: set_int] :
( ( type_d7247357190169752966nt_int @ Rep @ Abs @ A4 )
=> ( ( image_int_int @ Abs @ A4 )
= top_top_set_int ) ) ).
% type_definition.Abs_image
thf(fact_1589_type__definition_OAbs__image,axiom,
! [Rep: product_unit > $o,Abs: $o > product_unit,A4: set_o] :
( ( type_d6188575255521822967unit_o @ Rep @ Abs @ A4 )
=> ( ( image_o_Product_unit @ Abs @ A4 )
= top_to1996260823553986621t_unit ) ) ).
% type_definition.Abs_image
thf(fact_1590_Quotient__alt__def5,axiom,
( quotient_char_char
= ( ^ [R6: char > char > $o,Abs4: char > char,Rep4: char > char,T4: char > char > $o] :
( ( ord_le2540245887939449860char_o @ T4 @ ( bNF_Grp_char_char @ top_top_set_char @ Abs4 ) )
& ( ord_le2540245887939449860char_o @ ( bNF_Grp_char_char @ top_top_set_char @ Rep4 ) @ ( conversep_char_char @ T4 ) )
& ( R6
= ( relcom5872928454965100966r_char @ T4 @ ( conversep_char_char @ T4 ) ) ) ) ) ) ).
% Quotient_alt_def5
thf(fact_1591_Quotient__alt__def5,axiom,
( quotient_char_rat
= ( ^ [R6: char > char > $o,Abs4: char > rat,Rep4: rat > char,T4: char > rat > $o] :
( ( ord_le4891902658761385405_rat_o @ T4 @ ( bNF_Grp_char_rat @ top_top_set_char @ Abs4 ) )
& ( ord_le6546876390139131069char_o @ ( bNF_Grp_rat_char @ top_top_set_rat @ Rep4 ) @ ( conversep_char_rat @ T4 ) )
& ( R6
= ( relcom3428143170894750637t_char @ T4 @ ( conversep_char_rat @ T4 ) ) ) ) ) ) ).
% Quotient_alt_def5
thf(fact_1592_Quotient__alt__def5,axiom,
( quotient_char_nat
= ( ^ [R6: char > char > $o,Abs4: char > nat,Rep4: nat > char,T4: char > nat > $o] :
( ( ord_le5641337153896221125_nat_o @ T4 @ ( bNF_Grp_char_nat @ top_top_set_char @ Abs4 ) )
& ( ord_le4878496333587278021char_o @ ( bNF_Grp_nat_char @ top_top_set_nat @ Rep4 ) @ ( conversep_char_nat @ T4 ) )
& ( R6
= ( relcom6755291642603558309t_char @ T4 @ ( conversep_char_nat @ T4 ) ) ) ) ) ) ).
% Quotient_alt_def5
thf(fact_1593_Quotient__alt__def5,axiom,
( quotient_char_int
= ( ^ [R6: char > char > $o,Abs4: char > int,Rep4: int > char,T4: char > int > $o] :
( ( ord_le3396192186225775209_int_o @ T4 @ ( bNF_Grp_char_int @ top_top_set_char @ Abs4 ) )
& ( ord_le7608100903186094441char_o @ ( bNF_Grp_int_char @ top_top_set_int @ Rep4 ) @ ( conversep_char_int @ T4 ) )
& ( R6
= ( relcom750623199913152513t_char @ T4 @ ( conversep_char_int @ T4 ) ) ) ) ) ) ).
% Quotient_alt_def5
thf(fact_1594_Quotient__alt__def5,axiom,
( quotient_rat_char
= ( ^ [R6: rat > rat > $o,Abs4: rat > char,Rep4: char > rat,T4: rat > char > $o] :
( ( ord_le6546876390139131069char_o @ T4 @ ( bNF_Grp_rat_char @ top_top_set_rat @ Abs4 ) )
& ( ord_le4891902658761385405_rat_o @ ( bNF_Grp_char_rat @ top_top_set_char @ Rep4 ) @ ( conversep_rat_char @ T4 ) )
& ( R6
= ( relcom1505247221262953652ar_rat @ T4 @ ( conversep_rat_char @ T4 ) ) ) ) ) ) ).
% Quotient_alt_def5
thf(fact_1595_Quotient__alt__def5,axiom,
( quotient_rat_rat
= ( ^ [R6: rat > rat > $o,Abs4: rat > rat,Rep4: rat > rat,T4: rat > rat > $o] :
( ( ord_le4717968354871517046_rat_o @ T4 @ ( bNF_Grp_rat_rat @ top_top_set_rat @ Abs4 ) )
& ( ord_le4717968354871517046_rat_o @ ( bNF_Grp_rat_rat @ top_top_set_rat @ Rep4 ) @ ( conversep_rat_rat @ T4 ) )
& ( R6
= ( relcompp_rat_rat_rat @ T4 @ ( conversep_rat_rat @ T4 ) ) ) ) ) ) ).
% Quotient_alt_def5
thf(fact_1596_Quotient__alt__def5,axiom,
( quotient_rat_nat
= ( ^ [R6: rat > rat > $o,Abs4: rat > nat,Rep4: nat > rat,T4: rat > nat > $o] :
( ( ord_le5467402850006352766_nat_o @ T4 @ ( bNF_Grp_rat_nat @ top_top_set_rat @ Abs4 ) )
& ( ord_le1897120724991155070_rat_o @ ( bNF_Grp_nat_rat @ top_top_set_nat @ Rep4 ) @ ( conversep_rat_nat @ T4 ) )
& ( R6
= ( relcompp_rat_nat_rat @ T4 @ ( conversep_rat_nat @ T4 ) ) ) ) ) ) ).
% Quotient_alt_def5
thf(fact_1597_Quotient__alt__def5,axiom,
( quotient_rat_int
= ( ^ [R6: rat > rat > $o,Abs4: rat > int,Rep4: int > rat,T4: rat > int > $o] :
( ( ord_le3222257882335906850_int_o @ T4 @ ( bNF_Grp_rat_int @ top_top_set_rat @ Abs4 ) )
& ( ord_le8236914709048111138_rat_o @ ( bNF_Grp_int_rat @ top_top_set_int @ Rep4 ) @ ( conversep_rat_int @ T4 ) )
& ( R6
= ( relcompp_rat_int_rat @ T4 @ ( conversep_rat_int @ T4 ) ) ) ) ) ) ).
% Quotient_alt_def5
thf(fact_1598_Quotient__alt__def5,axiom,
( quotient_nat_char
= ( ^ [R6: nat > nat > $o,Abs4: nat > char,Rep4: char > nat,T4: nat > char > $o] :
( ( ord_le4878496333587278021char_o @ T4 @ ( bNF_Grp_nat_char @ top_top_set_nat @ Abs4 ) )
& ( ord_le5641337153896221125_nat_o @ ( bNF_Grp_char_nat @ top_top_set_char @ Rep4 ) @ ( conversep_nat_char @ T4 ) )
& ( R6
= ( relcom8112501251453913252ar_nat @ T4 @ ( conversep_nat_char @ T4 ) ) ) ) ) ) ).
% Quotient_alt_def5
thf(fact_1599_Quotient__alt__def5,axiom,
( quotient_nat_rat
= ( ^ [R6: nat > nat > $o,Abs4: nat > rat,Rep4: rat > nat,T4: nat > rat > $o] :
( ( ord_le1897120724991155070_rat_o @ T4 @ ( bNF_Grp_nat_rat @ top_top_set_nat @ Abs4 ) )
& ( ord_le5467402850006352766_nat_o @ ( bNF_Grp_rat_nat @ top_top_set_rat @ Rep4 ) @ ( conversep_nat_rat @ T4 ) )
& ( R6
= ( relcompp_nat_rat_nat @ T4 @ ( conversep_nat_rat @ T4 ) ) ) ) ) ) ).
% Quotient_alt_def5
thf(fact_1600_K__record__comp,axiom,
! [C: $o,F: nat > $o] :
( ( comp_o_o_nat
@ ^ [X4: $o] : C
@ F )
= ( ^ [X4: nat] : C ) ) ).
% K_record_comp
thf(fact_1601_K__record__comp,axiom,
! [C: num,F: int > nat] :
( ( comp_nat_num_int
@ ^ [X4: nat] : C
@ F )
= ( ^ [X4: int] : C ) ) ).
% K_record_comp
thf(fact_1602_K__record__comp,axiom,
! [C: nat,F: nat > nat] :
( ( comp_nat_nat_nat
@ ^ [X4: nat] : C
@ F )
= ( ^ [X4: nat] : C ) ) ).
% K_record_comp
thf(fact_1603_K__record__comp,axiom,
! [C: nat,F: int > int] :
( ( comp_int_nat_int
@ ^ [X4: int] : C
@ F )
= ( ^ [X4: int] : C ) ) ).
% K_record_comp
thf(fact_1604_K__record__comp,axiom,
! [C: produc8923325533196201883nteger > produc8923325533196201883nteger,F: code_integer > code_integer > code_integer] :
( ( comp_C8797469213163452608nteger
@ ^ [X4: code_integer > code_integer] : C
@ F )
= ( ^ [X4: code_integer] : C ) ) ).
% K_record_comp
thf(fact_1605_image__paired__Times,axiom,
! [F: nat > rat,G: int > nat,A4: set_nat,B5: set_int] :
( ( image_3755276878268593137at_nat
@ ( produc5347463441023893471at_nat
@ ^ [X4: nat,Y5: int] : ( product_Pair_rat_nat @ ( F @ X4 ) @ ( G @ Y5 ) ) )
@ ( produc454536836294682310at_int @ A4
@ ^ [Uu: nat] : B5 ) )
= ( produc140472662868866674at_nat @ ( image_nat_rat @ F @ A4 )
@ ^ [Uu: rat] : ( image_int_nat @ G @ B5 ) ) ) ).
% image_paired_Times
thf(fact_1606_image__paired__Times,axiom,
! [F: nat > rat,G: int > int,A4: set_nat,B5: set_int] :
( ( image_8800797895614172237at_int
@ ( produc1169612421514696763at_int
@ ^ [X4: nat,Y5: int] : ( product_Pair_rat_int @ ( F @ X4 ) @ ( G @ Y5 ) ) )
@ ( produc454536836294682310at_int @ A4
@ ^ [Uu: nat] : B5 ) )
= ( produc137982192359816398at_int @ ( image_nat_rat @ F @ A4 )
@ ^ [Uu: rat] : ( image_int_int @ G @ B5 ) ) ) ).
% image_paired_Times
thf(fact_1607_image__paired__Times,axiom,
! [F: nat > nat,G: int > int,A4: set_nat,B5: set_int] :
( ( image_9117352539549038149at_int
@ ( produc1486167065449562675at_int
@ ^ [X4: nat,Y5: int] : ( product_Pair_nat_int @ ( F @ X4 ) @ ( G @ Y5 ) ) )
@ ( produc454536836294682310at_int @ A4
@ ^ [Uu: nat] : B5 ) )
= ( produc454536836294682310at_int @ ( image_nat_nat @ F @ A4 )
@ ^ [Uu: nat] : ( image_int_int @ G @ B5 ) ) ) ).
% image_paired_Times
thf(fact_1608_image__paired__Times,axiom,
! [F: nat > int,G: int > nat,A4: set_nat,B5: set_int] :
( ( image_5071185234104513605nt_nat
@ ( produc6663371796859813939nt_nat
@ ^ [X4: nat,Y5: int] : ( product_Pair_int_nat @ ( F @ X4 ) @ ( G @ Y5 ) ) )
@ ( produc454536836294682310at_int @ A4
@ ^ [Uu: nat] : B5 ) )
= ( produc1456381018704787142nt_nat @ ( image_nat_int @ F @ A4 )
@ ^ [Uu: int] : ( image_int_nat @ G @ B5 ) ) ) ).
% image_paired_Times
thf(fact_1609_image__paired__Times,axiom,
! [F: int > nat,G: nat > rat,A4: set_int,B5: set_nat] :
( ( image_3420722608207172081at_rat
@ ( produc877984105854340063at_rat
@ ^ [X4: int,Y5: nat] : ( product_Pair_nat_rat @ ( F @ X4 ) @ ( G @ Y5 ) ) )
@ ( produc1456381018704787142nt_nat @ A4
@ ^ [Uu: int] : B5 ) )
= ( produc9045269283572012658at_rat @ ( image_int_nat @ F @ A4 )
@ ^ [Uu: nat] : ( image_nat_rat @ G @ B5 ) ) ) ).
% image_paired_Times
thf(fact_1610_image__paired__Times,axiom,
! [F: int > nat,G: nat > int,A4: set_int,B5: set_nat] :
( ( image_2751100147091372101at_int
@ ( produc208361644738540083at_int
@ ^ [X4: int,Y5: nat] : ( product_Pair_nat_int @ ( F @ X4 ) @ ( G @ Y5 ) ) )
@ ( produc1456381018704787142nt_nat @ A4
@ ^ [Uu: int] : B5 ) )
= ( produc454536836294682310at_int @ ( image_int_nat @ F @ A4 )
@ ^ [Uu: nat] : ( image_nat_int @ G @ B5 ) ) ) ).
% image_paired_Times
thf(fact_1611_image__paired__Times,axiom,
! [F: int > nat,G: int > int,A4: set_int,B5: set_int] :
( ( image_1654017166447373545at_int
@ ( produc3246203729202673879at_int
@ ^ [X4: int,Y5: int] : ( product_Pair_nat_int @ ( F @ X4 ) @ ( G @ Y5 ) ) )
@ ( produc1453890548195736866nt_int @ A4
@ ^ [Uu: int] : B5 ) )
= ( produc454536836294682310at_int @ ( image_int_nat @ F @ A4 )
@ ^ [Uu: nat] : ( image_int_int @ G @ B5 ) ) ) ).
% image_paired_Times
thf(fact_1612_image__paired__Times,axiom,
! [F: int > int,G: nat > rat,A4: set_int,B5: set_nat] :
( ( image_4420076320108226637nt_rat
@ ( produc1877337817755394619nt_rat
@ ^ [X4: int,Y5: nat] : ( product_Pair_int_rat @ ( F @ X4 ) @ ( G @ Y5 ) ) )
@ ( produc1456381018704787142nt_nat @ A4
@ ^ [Uu: int] : B5 ) )
= ( produc821250958618291406nt_rat @ ( image_int_int @ F @ A4 )
@ ^ [Uu: int] : ( image_nat_rat @ G @ B5 ) ) ) ).
% image_paired_Times
thf(fact_1613_image__paired__Times,axiom,
! [F: int > int,G: nat > nat,A4: set_int,B5: set_nat] :
( ( image_7928304878501623365nt_nat
@ ( produc5385566376148791347nt_nat
@ ^ [X4: int,Y5: nat] : ( product_Pair_int_nat @ ( F @ X4 ) @ ( G @ Y5 ) ) )
@ ( produc1456381018704787142nt_nat @ A4
@ ^ [Uu: int] : B5 ) )
= ( produc1456381018704787142nt_nat @ ( image_int_int @ F @ A4 )
@ ^ [Uu: int] : ( image_nat_nat @ G @ B5 ) ) ) ).
% image_paired_Times
thf(fact_1614_image__paired__Times,axiom,
! [F: int > int,G: int > nat,A4: set_int,B5: set_int] :
( ( image_6831221897857624809nt_nat
@ ( produc8423408460612925143nt_nat
@ ^ [X4: int,Y5: int] : ( product_Pair_int_nat @ ( F @ X4 ) @ ( G @ Y5 ) ) )
@ ( produc1453890548195736866nt_int @ A4
@ ^ [Uu: int] : B5 ) )
= ( produc1456381018704787142nt_nat @ ( image_int_int @ F @ A4 )
@ ^ [Uu: int] : ( image_int_nat @ G @ B5 ) ) ) ).
% image_paired_Times
thf(fact_1615_Gr__incl,axiom,
! [A4: set_nat,F: nat > rat,B5: set_rat] :
( ( ord_le5989899228261996553at_rat @ ( bNF_Gr_nat_rat @ A4 @ F )
@ ( produc9045269283572012658at_rat @ A4
@ ^ [Uu: nat] : B5 ) )
= ( ord_less_eq_set_rat @ ( image_nat_rat @ F @ A4 ) @ B5 ) ) ).
% Gr_incl
thf(fact_1616_Gr__incl,axiom,
! [A4: set_nat,F: nat > int,B5: set_int] :
( ( ord_le656430383953171549at_int @ ( bNF_Gr_nat_int @ A4 @ F )
@ ( produc454536836294682310at_int @ A4
@ ^ [Uu: nat] : B5 ) )
= ( ord_less_eq_set_int @ ( image_nat_int @ F @ A4 ) @ B5 ) ) ).
% Gr_incl
thf(fact_1617_Gr__incl,axiom,
! [A4: set_int,F: int > nat,B5: set_nat] :
( ( ord_le5333435103577920093nt_nat @ ( bNF_Gr_int_nat @ A4 @ F )
@ ( produc1456381018704787142nt_nat @ A4
@ ^ [Uu: int] : B5 ) )
= ( ord_less_eq_set_nat @ ( image_int_nat @ F @ A4 ) @ B5 ) ) ).
% Gr_incl
thf(fact_1618_Gr__incl,axiom,
! [A4: set_int,F: int > int,B5: set_int] :
( ( ord_le2843351958646193337nt_int @ ( bNF_Gr_int_int @ A4 @ F )
@ ( produc1453890548195736866nt_int @ A4
@ ^ [Uu: int] : B5 ) )
= ( ord_less_eq_set_int @ ( image_int_int @ F @ A4 ) @ B5 ) ) ).
% Gr_incl
thf(fact_1619_Gr__incl,axiom,
! [A4: set_nat,F: nat > nat,B5: set_nat] :
( ( ord_le3146513528884898305at_nat @ ( bNF_Gr_nat_nat @ A4 @ F )
@ ( produc457027306803732586at_nat @ A4
@ ^ [Uu: nat] : B5 ) )
= ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A4 ) @ B5 ) ) ).
% Gr_incl
thf(fact_1620_subset__fst__snd,axiom,
! [A4: set_Pr1261947904930325089at_nat] :
( ord_le3146513528884898305at_nat @ A4
@ ( produc457027306803732586at_nat @ ( image_2486076414777270412at_nat @ product_fst_nat_nat @ A4 )
@ ^ [Uu: nat] : ( image_2486076414777270412at_nat @ product_snd_nat_nat @ A4 ) ) ) ).
% subset_fst_snd
thf(fact_1621_subset__fst__snd,axiom,
! [A4: set_Pr1133966238863718811t_unit] :
( ord_le5525191956872776187t_unit @ A4
@ ( produc3946373962484743846t_unit @ ( image_7136766188234240905_ref_a @ produc473321116828791851t_unit @ A4 )
@ ^ [Uu: ref_a] : ( image_17660834435998945t_unit @ produc1126150082675451753t_unit @ A4 ) ) ) ).
% subset_fst_snd
thf(fact_1622_inv__image__def,axiom,
( inv_im2903009620893676101at_nat
= ( ^ [R5: set_Pr7459493094073627847at_nat,F2: nat > set_Pr4329608150637261639at_nat] :
( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [X4: nat,Y5: nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ ( F2 @ X4 ) @ ( F2 @ Y5 ) ) @ R5 ) ) ) ) ) ).
% inv_image_def
thf(fact_1623_inv__image__def,axiom,
( inv_im4437241714082187487at_nat
= ( ^ [R5: set_Pr4329608150637261639at_nat,F2: nat > set_Pr1261947904930325089at_nat] :
( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [X4: nat,Y5: nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( F2 @ X4 ) @ ( F2 @ Y5 ) ) @ R5 ) ) ) ) ) ).
% inv_image_def
thf(fact_1624_inv__image__def,axiom,
( inv_image_nat_nat
= ( ^ [R5: set_Pr1261947904930325089at_nat,F2: nat > nat] :
( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [X4: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ X4 ) @ ( F2 @ Y5 ) ) @ R5 ) ) ) ) ) ).
% inv_image_def
thf(fact_1625_inv__image__def,axiom,
( inv_image_int_nat
= ( ^ [R5: set_Pr958786334691620121nt_int,F2: nat > int] :
( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [X4: nat,Y5: nat] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( F2 @ X4 ) @ ( F2 @ Y5 ) ) @ R5 ) ) ) ) ) ).
% inv_image_def
thf(fact_1626_inv__image__def,axiom,
( inv_im2900519150384625825at_int
= ( ^ [R5: set_Pr7459493094073627847at_nat,F2: int > set_Pr4329608150637261639at_nat] :
( collec213857154873943460nt_int
@ ( produc4947309494688390418_int_o
@ ^ [X4: int,Y5: int] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ ( F2 @ X4 ) @ ( F2 @ Y5 ) ) @ R5 ) ) ) ) ) ).
% inv_image_def
thf(fact_1627_inv__image__def,axiom,
( inv_im4434751243573137211at_int
= ( ^ [R5: set_Pr4329608150637261639at_nat,F2: int > set_Pr1261947904930325089at_nat] :
( collec213857154873943460nt_int
@ ( produc4947309494688390418_int_o
@ ^ [X4: int,Y5: int] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( F2 @ X4 ) @ ( F2 @ Y5 ) ) @ R5 ) ) ) ) ) ).
% inv_image_def
thf(fact_1628_inv__image__def,axiom,
( inv_image_nat_int
= ( ^ [R5: set_Pr1261947904930325089at_nat,F2: int > nat] :
( collec213857154873943460nt_int
@ ( produc4947309494688390418_int_o
@ ^ [X4: int,Y5: int] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ X4 ) @ ( F2 @ Y5 ) ) @ R5 ) ) ) ) ) ).
% inv_image_def
thf(fact_1629_inv__image__def,axiom,
( inv_image_int_int
= ( ^ [R5: set_Pr958786334691620121nt_int,F2: int > int] :
( collec213857154873943460nt_int
@ ( produc4947309494688390418_int_o
@ ^ [X4: int,Y5: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( F2 @ X4 ) @ ( F2 @ Y5 ) ) @ R5 ) ) ) ) ) ).
% inv_image_def
thf(fact_1630_The__case__prod,axiom,
! [P2: ref_a > heap_e7401611519738050253t_unit > $o] :
( ( the_Pr5012129486825234846t_unit @ ( produc4924168586796184916unit_o @ P2 ) )
= ( the_Pr5012129486825234846t_unit
@ ^ [Xy: produc7812382424243267045t_unit] : ( P2 @ ( produc473321116828791851t_unit @ Xy ) @ ( produc1126150082675451753t_unit @ Xy ) ) ) ) ).
% The_case_prod
thf(fact_1631_The__case__prod,axiom,
! [P2: int > int > $o] :
( ( the_Pr4378521158711661632nt_int @ ( produc4947309494688390418_int_o @ P2 ) )
= ( the_Pr4378521158711661632nt_int
@ ^ [Xy: product_prod_int_int] : ( P2 @ ( product_fst_int_int @ Xy ) @ ( product_snd_int_int @ Xy ) ) ) ) ).
% The_case_prod
thf(fact_1632_Quotient__composition__ge__eq,axiom,
! [T3: int > product_prod_nat_nat > $o,R3: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ( left_t8261381459436684205at_nat @ T3 )
=> ( ( ord_le5604493270027003598_nat_o
@ ^ [Y2: product_prod_nat_nat,Z: product_prod_nat_nat] : Y2 = Z
@ R3 )
=> ( ord_le6741204236512500942_int_o
@ ^ [Y2: int,Z: int] : Y2 = Z
@ ( relcom1106208330880909712at_int @ T3 @ ( relcom8911704206367525917at_int @ R3 @ ( conver1866313717161370910at_nat @ T3 ) ) ) ) ) ) ).
% Quotient_composition_ge_eq
thf(fact_1633_Quotient__composition__ge__eq,axiom,
! [T3: code_integer > int > $o,R3: int > int > $o] :
( ( left_t1735413524555719857er_int @ T3 )
=> ( ( ord_le6741204236512500942_int_o
@ ^ [Y2: int,Z: int] : Y2 = Z
@ R3 )
=> ( ord_le3602516367967493612eger_o
@ ^ [Y2: code_integer,Z: code_integer] : Y2 = Z
@ ( relcom692387321444912665nteger @ T3 @ ( relcom967497705283160040nteger @ R3 @ ( conver1462408618118490240er_int @ T3 ) ) ) ) ) ) ).
% Quotient_composition_ge_eq
thf(fact_1634_Quotient__composition__le__eq,axiom,
! [T3: int > product_prod_nat_nat > $o,R3: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ( left_u1182257597767569412at_nat @ T3 )
=> ( ( ord_le5604493270027003598_nat_o @ R3
@ ^ [Y2: product_prod_nat_nat,Z: product_prod_nat_nat] : Y2 = Z )
=> ( ord_le6741204236512500942_int_o @ ( relcom1106208330880909712at_int @ T3 @ ( relcom8911704206367525917at_int @ R3 @ ( conver1866313717161370910at_nat @ T3 ) ) )
@ ^ [Y2: int,Z: int] : Y2 = Z ) ) ) ).
% Quotient_composition_le_eq
thf(fact_1635_Quotient__composition__le__eq,axiom,
! [T3: code_integer > int > $o,R3: int > int > $o] :
( ( left_u6156896403773009178er_int @ T3 )
=> ( ( ord_le6741204236512500942_int_o @ R3
@ ^ [Y2: int,Z: int] : Y2 = Z )
=> ( ord_le3602516367967493612eger_o @ ( relcom692387321444912665nteger @ T3 @ ( relcom967497705283160040nteger @ R3 @ ( conver1462408618118490240er_int @ T3 ) ) )
@ ^ [Y2: code_integer,Z: code_integer] : Y2 = Z ) ) ) ).
% Quotient_composition_le_eq
thf(fact_1636_mem__Sigma__iff,axiom,
! [A: $o,B: $o,A4: set_o,B5: $o > set_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ ( product_Sigma_o_o @ A4 @ B5 ) )
= ( ( member_o @ A @ A4 )
& ( member_o @ B @ ( B5 @ A ) ) ) ) ).
% mem_Sigma_iff
thf(fact_1637_mem__Sigma__iff,axiom,
! [A: $o,B: nat,A4: set_o,B5: $o > set_nat] :
( ( member2802428098988154798_o_nat @ ( product_Pair_o_nat @ A @ B ) @ ( product_Sigma_o_nat @ A4 @ B5 ) )
= ( ( member_o @ A @ A4 )
& ( member_nat @ B @ ( B5 @ A ) ) ) ) ).
% mem_Sigma_iff
thf(fact_1638_mem__Sigma__iff,axiom,
! [A: $o,B: int,A4: set_o,B5: $o > set_int] :
( ( member7847949116333733898_o_int @ ( product_Pair_o_int @ A @ B ) @ ( product_Sigma_o_int @ A4 @ B5 ) )
= ( ( member_o @ A @ A4 )
& ( member_int @ B @ ( B5 @ A ) ) ) ) ).
% mem_Sigma_iff
thf(fact_1639_mem__Sigma__iff,axiom,
! [A: nat,B: $o,A4: set_nat,B5: nat > set_o] :
( ( member6310962623043647828_nat_o @ ( product_Pair_nat_o @ A @ B ) @ ( product_Sigma_nat_o @ A4 @ B5 ) )
= ( ( member_nat @ A @ A4 )
& ( member_o @ B @ ( B5 @ A ) ) ) ) ).
% mem_Sigma_iff
thf(fact_1640_mem__Sigma__iff,axiom,
! [A: nat,B: int,A4: set_nat,B5: nat > set_int] :
( ( member4262671552274231302at_int @ ( product_Pair_nat_int @ A @ B ) @ ( produc454536836294682310at_int @ A4 @ B5 ) )
= ( ( member_nat @ A @ A4 )
& ( member_int @ B @ ( B5 @ A ) ) ) ) ).
% mem_Sigma_iff
thf(fact_1641_mem__Sigma__iff,axiom,
! [A: int,B: $o,A4: set_int,B5: int > set_o] :
( ( member4489920277610959864_int_o @ ( product_Pair_int_o @ A @ B ) @ ( product_Sigma_int_o @ A4 @ B5 ) )
= ( ( member_int @ A @ A4 )
& ( member_o @ B @ ( B5 @ A ) ) ) ) ).
% mem_Sigma_iff
thf(fact_1642_mem__Sigma__iff,axiom,
! [A: int,B: nat,A4: set_int,B5: int > set_nat] :
( ( member216504246829706758nt_nat @ ( product_Pair_int_nat @ A @ B ) @ ( produc1456381018704787142nt_nat @ A4 @ B5 ) )
= ( ( member_int @ A @ A4 )
& ( member_nat @ B @ ( B5 @ A ) ) ) ) ).
% mem_Sigma_iff
thf(fact_1643_mem__Sigma__iff,axiom,
! [A: nat,B: num,A4: set_nat,B5: nat > set_num] :
( ( member9148766508732265716at_num @ ( product_Pair_nat_num @ A @ B ) @ ( produc6237731129814287028at_num @ A4 @ B5 ) )
= ( ( member_nat @ A @ A4 )
& ( member_num @ B @ ( B5 @ A ) ) ) ) ).
% mem_Sigma_iff
thf(fact_1644_mem__Sigma__iff,axiom,
! [A: int,B: int,A4: set_int,B5: int > set_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ ( produc1453890548195736866nt_int @ A4 @ B5 ) )
= ( ( member_int @ A @ A4 )
& ( member_int @ B @ ( B5 @ A ) ) ) ) ).
% mem_Sigma_iff
thf(fact_1645_mem__Sigma__iff,axiom,
! [A: nat,B: nat,A4: set_nat,B5: nat > set_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( produc457027306803732586at_nat @ A4 @ B5 ) )
= ( ( member_nat @ A @ A4 )
& ( member_nat @ B @ ( B5 @ A ) ) ) ) ).
% mem_Sigma_iff
thf(fact_1646_SigmaI,axiom,
! [A: $o,A4: set_o,B: $o,B5: $o > set_o] :
( ( member_o @ A @ A4 )
=> ( ( member_o @ B @ ( B5 @ A ) )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ ( product_Sigma_o_o @ A4 @ B5 ) ) ) ) ).
% SigmaI
thf(fact_1647_SigmaI,axiom,
! [A: $o,A4: set_o,B: nat,B5: $o > set_nat] :
( ( member_o @ A @ A4 )
=> ( ( member_nat @ B @ ( B5 @ A ) )
=> ( member2802428098988154798_o_nat @ ( product_Pair_o_nat @ A @ B ) @ ( product_Sigma_o_nat @ A4 @ B5 ) ) ) ) ).
% SigmaI
thf(fact_1648_SigmaI,axiom,
! [A: $o,A4: set_o,B: int,B5: $o > set_int] :
( ( member_o @ A @ A4 )
=> ( ( member_int @ B @ ( B5 @ A ) )
=> ( member7847949116333733898_o_int @ ( product_Pair_o_int @ A @ B ) @ ( product_Sigma_o_int @ A4 @ B5 ) ) ) ) ).
% SigmaI
thf(fact_1649_SigmaI,axiom,
! [A: nat,A4: set_nat,B: $o,B5: nat > set_o] :
( ( member_nat @ A @ A4 )
=> ( ( member_o @ B @ ( B5 @ A ) )
=> ( member6310962623043647828_nat_o @ ( product_Pair_nat_o @ A @ B ) @ ( product_Sigma_nat_o @ A4 @ B5 ) ) ) ) ).
% SigmaI
thf(fact_1650_SigmaI,axiom,
! [A: nat,A4: set_nat,B: int,B5: nat > set_int] :
( ( member_nat @ A @ A4 )
=> ( ( member_int @ B @ ( B5 @ A ) )
=> ( member4262671552274231302at_int @ ( product_Pair_nat_int @ A @ B ) @ ( produc454536836294682310at_int @ A4 @ B5 ) ) ) ) ).
% SigmaI
thf(fact_1651_SigmaI,axiom,
! [A: int,A4: set_int,B: $o,B5: int > set_o] :
( ( member_int @ A @ A4 )
=> ( ( member_o @ B @ ( B5 @ A ) )
=> ( member4489920277610959864_int_o @ ( product_Pair_int_o @ A @ B ) @ ( product_Sigma_int_o @ A4 @ B5 ) ) ) ) ).
% SigmaI
thf(fact_1652_SigmaI,axiom,
! [A: int,A4: set_int,B: nat,B5: int > set_nat] :
( ( member_int @ A @ A4 )
=> ( ( member_nat @ B @ ( B5 @ A ) )
=> ( member216504246829706758nt_nat @ ( product_Pair_int_nat @ A @ B ) @ ( produc1456381018704787142nt_nat @ A4 @ B5 ) ) ) ) ).
% SigmaI
thf(fact_1653_SigmaI,axiom,
! [A: nat,A4: set_nat,B: num,B5: nat > set_num] :
( ( member_nat @ A @ A4 )
=> ( ( member_num @ B @ ( B5 @ A ) )
=> ( member9148766508732265716at_num @ ( product_Pair_nat_num @ A @ B ) @ ( produc6237731129814287028at_num @ A4 @ B5 ) ) ) ) ).
% SigmaI
thf(fact_1654_SigmaI,axiom,
! [A: int,A4: set_int,B: int,B5: int > set_int] :
( ( member_int @ A @ A4 )
=> ( ( member_int @ B @ ( B5 @ A ) )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ ( produc1453890548195736866nt_int @ A4 @ B5 ) ) ) ) ).
% SigmaI
thf(fact_1655_SigmaI,axiom,
! [A: nat,A4: set_nat,B: nat,B5: nat > set_nat] :
( ( member_nat @ A @ A4 )
=> ( ( member_nat @ B @ ( B5 @ A ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( produc457027306803732586at_nat @ A4 @ B5 ) ) ) ) ).
% SigmaI
thf(fact_1656_in__inv__image,axiom,
! [X2: nat,Y3: nat,R: set_Pr1261947904930325089at_nat,F: nat > nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ ( inv_image_nat_nat @ R @ F ) )
= ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) @ R ) ) ).
% in_inv_image
thf(fact_1657_in__inv__image,axiom,
! [X2: nat,Y3: nat,R: set_Pr958786334691620121nt_int,F: nat > int] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ ( inv_image_int_nat @ R @ F ) )
= ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( F @ X2 ) @ ( F @ Y3 ) ) @ R ) ) ).
% in_inv_image
thf(fact_1658_in__inv__image,axiom,
! [X2: int,Y3: int,R: set_Pr1261947904930325089at_nat,F: int > nat] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y3 ) @ ( inv_image_nat_int @ R @ F ) )
= ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) @ R ) ) ).
% in_inv_image
thf(fact_1659_in__inv__image,axiom,
! [X2: int,Y3: int,R: set_Pr958786334691620121nt_int,F: int > int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y3 ) @ ( inv_image_int_int @ R @ F ) )
= ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( F @ X2 ) @ ( F @ Y3 ) ) @ R ) ) ).
% in_inv_image
thf(fact_1660_in__inv__image,axiom,
! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat,R: set_Pr1261947904930325089at_nat,F: set_Pr1261947904930325089at_nat > nat] :
( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y3 ) @ ( inv_im6750873321330482877at_nat @ R @ F ) )
= ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) @ R ) ) ).
% in_inv_image
thf(fact_1661_in__inv__image,axiom,
! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat,R: set_Pr958786334691620121nt_int,F: set_Pr1261947904930325089at_nat > int] :
( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y3 ) @ ( inv_im8510909985083594081at_nat @ R @ F ) )
= ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( F @ X2 ) @ ( F @ Y3 ) ) @ R ) ) ).
% in_inv_image
thf(fact_1662_in__inv__image,axiom,
! [X2: nat,Y3: nat,R: set_Pr4329608150637261639at_nat,F: nat > set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ ( inv_im4437241714082187487at_nat @ R @ F ) )
= ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) @ R ) ) ).
% in_inv_image
thf(fact_1663_in__inv__image,axiom,
! [X2: int,Y3: int,R: set_Pr4329608150637261639at_nat,F: int > set_Pr1261947904930325089at_nat] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y3 ) @ ( inv_im4434751243573137211at_int @ R @ F ) )
= ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) @ R ) ) ).
% in_inv_image
thf(fact_1664_in__inv__image,axiom,
! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat,F: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat] :
( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y3 ) @ ( inv_im7023619800666061458at_nat @ R @ F ) )
= ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) @ R ) ) ).
% in_inv_image
thf(fact_1665_in__inv__image,axiom,
! [X2: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat,R: set_Pr1261947904930325089at_nat,F: set_Pr4329608150637261639at_nat > nat] :
( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X2 @ Y3 ) @ ( inv_im2354225701593503139at_nat @ R @ F ) )
= ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) @ R ) ) ).
% in_inv_image
thf(fact_1666_Collect__case__prod,axiom,
! [P2: product_prod_nat_nat > $o,Q2: product_prod_nat_nat > $o] :
( ( collec7088162979684241874at_nat
@ ( produc6590410687421337004_nat_o
@ ^ [A5: product_prod_nat_nat,B4: product_prod_nat_nat] :
( ( P2 @ A5 )
& ( Q2 @ B4 ) ) ) )
= ( produc2761391749766926216at_nat @ ( collec3392354462482085612at_nat @ P2 )
@ ^ [Uu: product_prod_nat_nat] : ( collec3392354462482085612at_nat @ Q2 ) ) ) ).
% Collect_case_prod
thf(fact_1667_Collect__case__prod,axiom,
! [P2: product_prod_nat_nat > $o,Q2: nat > $o] :
( ( collec7029190964493513045at_nat
@ ( produc8758269395934548017_nat_o
@ ^ [A5: product_prod_nat_nat,B4: nat] :
( ( P2 @ A5 )
& ( Q2 @ B4 ) ) ) )
= ( produc7672662199629908489at_nat @ ( collec3392354462482085612at_nat @ P2 )
@ ^ [Uu: product_prod_nat_nat] : ( collect_nat @ Q2 ) ) ) ).
% Collect_case_prod
thf(fact_1668_Collect__case__prod,axiom,
! [P2: product_prod_nat_nat > $o,Q2: product_prod_int_int > $o] :
( ( collec6785001409445536906nt_int
@ ( produc5031841752791441652_int_o
@ ^ [A5: product_prod_nat_nat,B4: product_prod_int_int] :
( ( P2 @ A5 )
& ( Q2 @ B4 ) ) ) )
= ( produc8806266479013559872nt_int @ ( collec3392354462482085612at_nat @ P2 )
@ ^ [Uu: product_prod_nat_nat] : ( collec213857154873943460nt_int @ Q2 ) ) ) ).
% Collect_case_prod
thf(fact_1669_Collect__case__prod,axiom,
! [P2: nat > $o,Q2: product_prod_nat_nat > $o] :
( ( collec5903703980526211963at_nat
@ ( produc5864757623865647827_nat_o
@ ^ [A5: nat,B4: product_prod_nat_nat] :
( ( P2 @ A5 )
& ( Q2 @ B4 ) ) ) )
= ( produc1809337555817847783at_nat @ ( collect_nat @ P2 )
@ ^ [Uu: nat] : ( collec3392354462482085612at_nat @ Q2 ) ) ) ).
% Collect_case_prod
thf(fact_1670_Collect__case__prod,axiom,
! [P2: nat > $o,Q2: product_prod_int_int > $o] :
( ( collec5600542410287506995nt_int
@ ( produc4306188689235752475_int_o
@ ^ [A5: nat,B4: product_prod_int_int] :
( ( P2 @ A5 )
& ( Q2 @ B4 ) ) ) )
= ( produc7854212285064481439nt_int @ ( collect_nat @ P2 )
@ ^ [Uu: nat] : ( collec213857154873943460nt_int @ Q2 ) ) ) ).
% Collect_case_prod
thf(fact_1671_Collect__case__prod,axiom,
! [P2: product_prod_int_int > $o,Q2: product_prod_nat_nat > $o] :
( ( collec7751116662793354522at_nat
@ ( produc3235059054576640100_nat_o
@ ^ [A5: product_prod_int_int,B4: product_prod_nat_nat] :
( ( P2 @ A5 )
& ( Q2 @ B4 ) ) ) )
= ( produc3424345432876038864at_nat @ ( collec213857154873943460nt_int @ P2 )
@ ^ [Uu: product_prod_int_int] : ( collec3392354462482085612at_nat @ Q2 ) ) ) ).
% Collect_case_prod
thf(fact_1672_Collect__case__prod,axiom,
! [P2: product_prod_int_int > $o,Q2: nat > $o] :
( ( collec364394062568603661nt_nat
@ ( produc5441891513209881465_nat_o
@ ^ [A5: product_prod_int_int,B4: nat] :
( ( P2 @ A5 )
& ( Q2 @ B4 ) ) ) )
= ( produc1007865297704999105nt_nat @ ( collec213857154873943460nt_int @ P2 )
@ ^ [Uu: product_prod_int_int] : ( collect_nat @ Q2 ) ) ) ).
% Collect_case_prod
thf(fact_1673_Collect__case__prod,axiom,
! [P2: product_prod_int_int > $o,Q2: product_prod_int_int > $o] :
( ( collec7447955092554649554nt_int
@ ( produc1676490119946744748_int_o
@ ^ [A5: product_prod_int_int,B4: product_prod_int_int] :
( ( P2 @ A5 )
& ( Q2 @ B4 ) ) ) )
= ( produc245848125267896712nt_int @ ( collec213857154873943460nt_int @ P2 )
@ ^ [Uu: product_prod_int_int] : ( collec213857154873943460nt_int @ Q2 ) ) ) ).
% Collect_case_prod
thf(fact_1674_Collect__case__prod,axiom,
! [P2: nat > $o,Q2: nat > $o] :
( ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [A5: nat,B4: nat] :
( ( P2 @ A5 )
& ( Q2 @ B4 ) ) ) )
= ( produc457027306803732586at_nat @ ( collect_nat @ P2 )
@ ^ [Uu: nat] : ( collect_nat @ Q2 ) ) ) ).
% Collect_case_prod
thf(fact_1675_Collect__case__prod,axiom,
! [P2: int > $o,Q2: int > $o] :
( ( collec213857154873943460nt_int
@ ( produc4947309494688390418_int_o
@ ^ [A5: int,B4: int] :
( ( P2 @ A5 )
& ( Q2 @ B4 ) ) ) )
= ( produc1453890548195736866nt_int @ ( collect_int @ P2 )
@ ^ [Uu: int] : ( collect_int @ Q2 ) ) ) ).
% Collect_case_prod
thf(fact_1676_UNIV__Times__UNIV,axiom,
( ( produc2936396906406871148r_char @ top_top_set_char
@ ^ [Uu: char] : top_top_set_char )
= top_to1235241553633177651r_char ) ).
% UNIV_Times_UNIV
thf(fact_1677_UNIV__Times__UNIV,axiom,
( ( produc1204236661673008499ar_rat @ top_top_set_char
@ ^ [Uu: char] : top_top_set_rat )
= top_to8437673547576850234ar_rat ) ).
% UNIV_Times_UNIV
thf(fact_1678_UNIV__Times__UNIV,axiom,
( ( produc1839366721759504235ar_nat @ top_top_set_char
@ ^ [Uu: char] : top_top_set_nat )
= top_to5594287848199751986ar_nat ) ).
% UNIV_Times_UNIV
thf(fact_1679_UNIV__Times__UNIV,axiom,
( ( produc1836876251250453959ar_int @ top_top_set_char
@ ^ [Uu: char] : top_top_set_int )
= top_to3104204703268025230ar_int ) ).
% UNIV_Times_UNIV
thf(fact_1680_UNIV__Times__UNIV,axiom,
( ( produc1427039474575272051t_char @ top_top_set_rat
@ ^ [Uu: rat] : top_top_set_char )
= top_to3221206985699244474t_char ) ).
% UNIV_Times_UNIV
thf(fact_1681_UNIV__Times__UNIV,axiom,
( ( produc8728714639637146746at_rat @ top_top_set_rat
@ ^ [Uu: rat] : top_top_set_rat )
= top_to3112507417142880193at_rat ) ).
% UNIV_Times_UNIV
thf(fact_1682_UNIV__Times__UNIV,axiom,
( ( produc140472662868866674at_nat @ top_top_set_rat
@ ^ [Uu: rat] : top_top_set_nat )
= top_to269121717765781945at_nat ) ).
% UNIV_Times_UNIV
thf(fact_1683_UNIV__Times__UNIV,axiom,
( ( produc137982192359816398at_int @ top_top_set_rat
@ ^ [Uu: rat] : top_top_set_int )
= top_to7002410609688830997at_int ) ).
% UNIV_Times_UNIV
thf(fact_1684_UNIV__Times__UNIV,axiom,
( ( produc4754187946284079723t_char @ top_top_set_nat
@ ^ [Uu: nat] : top_top_set_char )
= top_to7107842681480589234t_char ) ).
% UNIV_Times_UNIV
thf(fact_1685_UNIV__Times__UNIV,axiom,
( ( produc9045269283572012658at_rat @ top_top_set_nat
@ ^ [Uu: nat] : top_top_set_rat )
= top_to7513191607651882425at_rat ) ).
% UNIV_Times_UNIV
thf(fact_1686_The__split__eq,axiom,
! [X2: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] :
( ( the_Pr4877703158196064750at_nat
@ ( produc9020218426428501292_nat_o
@ ^ [X7: set_Pr4329608150637261639at_nat,Y6: set_Pr4329608150637261639at_nat] :
( ( X2 = X7 )
& ( Y3 = Y6 ) ) ) )
= ( produc9060074326276436823at_nat @ X2 @ Y3 ) ) ).
% The_split_eq
thf(fact_1687_The__split__eq,axiom,
! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
( ( the_Pr4604535484834969198at_nat
@ ( produc410239310623530412_nat_o
@ ^ [X7: set_Pr1261947904930325089at_nat,Y6: set_Pr1261947904930325089at_nat] :
( ( X2 = X7 )
& ( Y3 = Y6 ) ) ) )
= ( produc2922128104949294807at_nat @ X2 @ Y3 ) ) ).
% The_split_eq
thf(fact_1688_The__split__eq,axiom,
! [X2: nat,Y3: num] :
( ( the_Pr8265262403268641490at_num
@ ( produc4927758841916487424_num_o
@ ^ [X7: nat,Y6: num] :
( ( X2 = X7 )
& ( Y3 = Y6 ) ) ) )
= ( product_Pair_nat_num @ X2 @ Y3 ) ) ).
% The_split_eq
thf(fact_1689_The__split__eq,axiom,
! [X2: nat,Y3: nat] :
( ( the_Pr7557018466319803784at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [X7: nat,Y6: nat] :
( ( X2 = X7 )
& ( Y3 = Y6 ) ) ) )
= ( product_Pair_nat_nat @ X2 @ Y3 ) ) ).
% The_split_eq
thf(fact_1690_The__split__eq,axiom,
! [X2: int,Y3: int] :
( ( the_Pr4378521158711661632nt_int
@ ( produc4947309494688390418_int_o
@ ^ [X7: int,Y6: int] :
( ( X2 = X7 )
& ( Y3 = Y6 ) ) ) )
= ( product_Pair_int_int @ X2 @ Y3 ) ) ).
% The_split_eq
thf(fact_1691_Times__eq__cancel2,axiom,
! [X2: nat,C3: set_nat,A4: set_nat,B5: set_nat] :
( ( member_nat @ X2 @ C3 )
=> ( ( ( produc457027306803732586at_nat @ A4
@ ^ [Uu: nat] : C3 )
= ( produc457027306803732586at_nat @ B5
@ ^ [Uu: nat] : C3 ) )
= ( A4 = B5 ) ) ) ).
% Times_eq_cancel2
thf(fact_1692_Sigma__cong,axiom,
! [A4: set_nat,B5: set_nat,C3: nat > set_nat,D: nat > set_nat] :
( ( A4 = B5 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B5 )
=> ( ( C3 @ X )
= ( D @ X ) ) )
=> ( ( produc457027306803732586at_nat @ A4 @ C3 )
= ( produc457027306803732586at_nat @ B5 @ D ) ) ) ) ).
% Sigma_cong
thf(fact_1693_SigmaE2,axiom,
! [A: $o,B: $o,A4: set_o,B5: $o > set_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ ( product_Sigma_o_o @ A4 @ B5 ) )
=> ~ ( ( member_o @ A @ A4 )
=> ~ ( member_o @ B @ ( B5 @ A ) ) ) ) ).
% SigmaE2
thf(fact_1694_SigmaE2,axiom,
! [A: $o,B: nat,A4: set_o,B5: $o > set_nat] :
( ( member2802428098988154798_o_nat @ ( product_Pair_o_nat @ A @ B ) @ ( product_Sigma_o_nat @ A4 @ B5 ) )
=> ~ ( ( member_o @ A @ A4 )
=> ~ ( member_nat @ B @ ( B5 @ A ) ) ) ) ).
% SigmaE2
thf(fact_1695_SigmaE2,axiom,
! [A: $o,B: int,A4: set_o,B5: $o > set_int] :
( ( member7847949116333733898_o_int @ ( product_Pair_o_int @ A @ B ) @ ( product_Sigma_o_int @ A4 @ B5 ) )
=> ~ ( ( member_o @ A @ A4 )
=> ~ ( member_int @ B @ ( B5 @ A ) ) ) ) ).
% SigmaE2
thf(fact_1696_SigmaE2,axiom,
! [A: nat,B: $o,A4: set_nat,B5: nat > set_o] :
( ( member6310962623043647828_nat_o @ ( product_Pair_nat_o @ A @ B ) @ ( product_Sigma_nat_o @ A4 @ B5 ) )
=> ~ ( ( member_nat @ A @ A4 )
=> ~ ( member_o @ B @ ( B5 @ A ) ) ) ) ).
% SigmaE2
thf(fact_1697_SigmaE2,axiom,
! [A: nat,B: int,A4: set_nat,B5: nat > set_int] :
( ( member4262671552274231302at_int @ ( product_Pair_nat_int @ A @ B ) @ ( produc454536836294682310at_int @ A4 @ B5 ) )
=> ~ ( ( member_nat @ A @ A4 )
=> ~ ( member_int @ B @ ( B5 @ A ) ) ) ) ).
% SigmaE2
thf(fact_1698_SigmaE2,axiom,
! [A: int,B: $o,A4: set_int,B5: int > set_o] :
( ( member4489920277610959864_int_o @ ( product_Pair_int_o @ A @ B ) @ ( product_Sigma_int_o @ A4 @ B5 ) )
=> ~ ( ( member_int @ A @ A4 )
=> ~ ( member_o @ B @ ( B5 @ A ) ) ) ) ).
% SigmaE2
thf(fact_1699_SigmaE2,axiom,
! [A: int,B: nat,A4: set_int,B5: int > set_nat] :
( ( member216504246829706758nt_nat @ ( product_Pair_int_nat @ A @ B ) @ ( produc1456381018704787142nt_nat @ A4 @ B5 ) )
=> ~ ( ( member_int @ A @ A4 )
=> ~ ( member_nat @ B @ ( B5 @ A ) ) ) ) ).
% SigmaE2
thf(fact_1700_SigmaE2,axiom,
! [A: nat,B: num,A4: set_nat,B5: nat > set_num] :
( ( member9148766508732265716at_num @ ( product_Pair_nat_num @ A @ B ) @ ( produc6237731129814287028at_num @ A4 @ B5 ) )
=> ~ ( ( member_nat @ A @ A4 )
=> ~ ( member_num @ B @ ( B5 @ A ) ) ) ) ).
% SigmaE2
thf(fact_1701_SigmaE2,axiom,
! [A: int,B: int,A4: set_int,B5: int > set_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ ( produc1453890548195736866nt_int @ A4 @ B5 ) )
=> ~ ( ( member_int @ A @ A4 )
=> ~ ( member_int @ B @ ( B5 @ A ) ) ) ) ).
% SigmaE2
thf(fact_1702_SigmaE2,axiom,
! [A: nat,B: nat,A4: set_nat,B5: nat > set_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( produc457027306803732586at_nat @ A4 @ B5 ) )
=> ~ ( ( member_nat @ A @ A4 )
=> ~ ( member_nat @ B @ ( B5 @ A ) ) ) ) ).
% SigmaE2
thf(fact_1703_SigmaD2,axiom,
! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A4: set_se5479201105039950375at_nat,B5: set_Pr4329608150637261639at_nat > set_se5479201105039950375at_nat] :
( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ B ) @ ( produc6258361011560675592at_nat @ A4 @ B5 ) )
=> ( member6273874620674182256at_nat @ B @ ( B5 @ A ) ) ) ).
% SigmaD2
thf(fact_1704_SigmaD2,axiom,
! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A4: set_se7855581050983116737at_nat,B5: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat] :
( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ ( produc7726050920695093128at_nat @ A4 @ B5 ) )
=> ( member2643936169264416010at_nat @ B @ ( B5 @ A ) ) ) ).
% SigmaD2
thf(fact_1705_SigmaD2,axiom,
! [A: nat,B: num,A4: set_nat,B5: nat > set_num] :
( ( member9148766508732265716at_num @ ( product_Pair_nat_num @ A @ B ) @ ( produc6237731129814287028at_num @ A4 @ B5 ) )
=> ( member_num @ B @ ( B5 @ A ) ) ) ).
% SigmaD2
thf(fact_1706_SigmaD2,axiom,
! [A: int,B: int,A4: set_int,B5: int > set_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ ( produc1453890548195736866nt_int @ A4 @ B5 ) )
=> ( member_int @ B @ ( B5 @ A ) ) ) ).
% SigmaD2
thf(fact_1707_SigmaD2,axiom,
! [A: nat,B: nat,A4: set_nat,B5: nat > set_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( produc457027306803732586at_nat @ A4 @ B5 ) )
=> ( member_nat @ B @ ( B5 @ A ) ) ) ).
% SigmaD2
thf(fact_1708_SigmaD1,axiom,
! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A4: set_se5479201105039950375at_nat,B5: set_Pr4329608150637261639at_nat > set_se5479201105039950375at_nat] :
( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ B ) @ ( produc6258361011560675592at_nat @ A4 @ B5 ) )
=> ( member6273874620674182256at_nat @ A @ A4 ) ) ).
% SigmaD1
thf(fact_1709_SigmaD1,axiom,
! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A4: set_se7855581050983116737at_nat,B5: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat] :
( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ ( produc7726050920695093128at_nat @ A4 @ B5 ) )
=> ( member2643936169264416010at_nat @ A @ A4 ) ) ).
% SigmaD1
thf(fact_1710_SigmaD1,axiom,
! [A: nat,B: num,A4: set_nat,B5: nat > set_num] :
( ( member9148766508732265716at_num @ ( product_Pair_nat_num @ A @ B ) @ ( produc6237731129814287028at_num @ A4 @ B5 ) )
=> ( member_nat @ A @ A4 ) ) ).
% SigmaD1
thf(fact_1711_SigmaD1,axiom,
! [A: int,B: int,A4: set_int,B5: int > set_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ ( produc1453890548195736866nt_int @ A4 @ B5 ) )
=> ( member_int @ A @ A4 ) ) ).
% SigmaD1
thf(fact_1712_SigmaD1,axiom,
! [A: nat,B: nat,A4: set_nat,B5: nat > set_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( produc457027306803732586at_nat @ A4 @ B5 ) )
=> ( member_nat @ A @ A4 ) ) ).
% SigmaD1
thf(fact_1713_SigmaE,axiom,
! [C: product_prod_o_o,A4: set_o,B5: $o > set_o] :
( ( member7466972457876170832od_o_o @ C @ ( product_Sigma_o_o @ A4 @ B5 ) )
=> ~ ! [X: $o] :
( ( member_o @ X @ A4 )
=> ! [Y: $o] :
( ( member_o @ Y @ ( B5 @ X ) )
=> ( C
!= ( product_Pair_o_o @ X @ Y ) ) ) ) ) ).
% SigmaE
thf(fact_1714_SigmaE,axiom,
! [C: product_prod_o_nat,A4: set_o,B5: $o > set_nat] :
( ( member2802428098988154798_o_nat @ C @ ( product_Sigma_o_nat @ A4 @ B5 ) )
=> ~ ! [X: $o] :
( ( member_o @ X @ A4 )
=> ! [Y: nat] :
( ( member_nat @ Y @ ( B5 @ X ) )
=> ( C
!= ( product_Pair_o_nat @ X @ Y ) ) ) ) ) ).
% SigmaE
thf(fact_1715_SigmaE,axiom,
! [C: product_prod_o_int,A4: set_o,B5: $o > set_int] :
( ( member7847949116333733898_o_int @ C @ ( product_Sigma_o_int @ A4 @ B5 ) )
=> ~ ! [X: $o] :
( ( member_o @ X @ A4 )
=> ! [Y: int] :
( ( member_int @ Y @ ( B5 @ X ) )
=> ( C
!= ( product_Pair_o_int @ X @ Y ) ) ) ) ) ).
% SigmaE
thf(fact_1716_SigmaE,axiom,
! [C: product_prod_nat_o,A4: set_nat,B5: nat > set_o] :
( ( member6310962623043647828_nat_o @ C @ ( product_Sigma_nat_o @ A4 @ B5 ) )
=> ~ ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ! [Y: $o] :
( ( member_o @ Y @ ( B5 @ X ) )
=> ( C
!= ( product_Pair_nat_o @ X @ Y ) ) ) ) ) ).
% SigmaE
thf(fact_1717_SigmaE,axiom,
! [C: product_prod_nat_int,A4: set_nat,B5: nat > set_int] :
( ( member4262671552274231302at_int @ C @ ( produc454536836294682310at_int @ A4 @ B5 ) )
=> ~ ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ! [Y: int] :
( ( member_int @ Y @ ( B5 @ X ) )
=> ( C
!= ( product_Pair_nat_int @ X @ Y ) ) ) ) ) ).
% SigmaE
thf(fact_1718_SigmaE,axiom,
! [C: product_prod_int_o,A4: set_int,B5: int > set_o] :
( ( member4489920277610959864_int_o @ C @ ( product_Sigma_int_o @ A4 @ B5 ) )
=> ~ ! [X: int] :
( ( member_int @ X @ A4 )
=> ! [Y: $o] :
( ( member_o @ Y @ ( B5 @ X ) )
=> ( C
!= ( product_Pair_int_o @ X @ Y ) ) ) ) ) ).
% SigmaE
thf(fact_1719_SigmaE,axiom,
! [C: product_prod_int_nat,A4: set_int,B5: int > set_nat] :
( ( member216504246829706758nt_nat @ C @ ( produc1456381018704787142nt_nat @ A4 @ B5 ) )
=> ~ ! [X: int] :
( ( member_int @ X @ A4 )
=> ! [Y: nat] :
( ( member_nat @ Y @ ( B5 @ X ) )
=> ( C
!= ( product_Pair_int_nat @ X @ Y ) ) ) ) ) ).
% SigmaE
thf(fact_1720_SigmaE,axiom,
! [C: product_prod_nat_num,A4: set_nat,B5: nat > set_num] :
( ( member9148766508732265716at_num @ C @ ( produc6237731129814287028at_num @ A4 @ B5 ) )
=> ~ ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ! [Y: num] :
( ( member_num @ Y @ ( B5 @ X ) )
=> ( C
!= ( product_Pair_nat_num @ X @ Y ) ) ) ) ) ).
% SigmaE
thf(fact_1721_SigmaE,axiom,
! [C: product_prod_int_int,A4: set_int,B5: int > set_int] :
( ( member5262025264175285858nt_int @ C @ ( produc1453890548195736866nt_int @ A4 @ B5 ) )
=> ~ ! [X: int] :
( ( member_int @ X @ A4 )
=> ! [Y: int] :
( ( member_int @ Y @ ( B5 @ X ) )
=> ( C
!= ( product_Pair_int_int @ X @ Y ) ) ) ) ) ).
% SigmaE
thf(fact_1722_SigmaE,axiom,
! [C: product_prod_nat_nat,A4: set_nat,B5: nat > set_nat] :
( ( member8440522571783428010at_nat @ C @ ( produc457027306803732586at_nat @ A4 @ B5 ) )
=> ~ ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ! [Y: nat] :
( ( member_nat @ Y @ ( B5 @ X ) )
=> ( C
!= ( product_Pair_nat_nat @ X @ Y ) ) ) ) ) ).
% SigmaE
thf(fact_1723_Collect__case__prod__Sigma,axiom,
! [P2: product_prod_nat_nat > $o,Q2: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ( collec7088162979684241874at_nat
@ ( produc6590410687421337004_nat_o
@ ^ [X4: product_prod_nat_nat,Y5: product_prod_nat_nat] :
( ( P2 @ X4 )
& ( Q2 @ X4 @ Y5 ) ) ) )
= ( produc2761391749766926216at_nat @ ( collec3392354462482085612at_nat @ P2 )
@ ^ [X4: product_prod_nat_nat] : ( collec3392354462482085612at_nat @ ( Q2 @ X4 ) ) ) ) ).
% Collect_case_prod_Sigma
thf(fact_1724_Collect__case__prod__Sigma,axiom,
! [P2: product_prod_nat_nat > $o,Q2: product_prod_nat_nat > nat > $o] :
( ( collec7029190964493513045at_nat
@ ( produc8758269395934548017_nat_o
@ ^ [X4: product_prod_nat_nat,Y5: nat] :
( ( P2 @ X4 )
& ( Q2 @ X4 @ Y5 ) ) ) )
= ( produc7672662199629908489at_nat @ ( collec3392354462482085612at_nat @ P2 )
@ ^ [X4: product_prod_nat_nat] : ( collect_nat @ ( Q2 @ X4 ) ) ) ) ).
% Collect_case_prod_Sigma
thf(fact_1725_Collect__case__prod__Sigma,axiom,
! [P2: product_prod_nat_nat > $o,Q2: product_prod_nat_nat > product_prod_int_int > $o] :
( ( collec6785001409445536906nt_int
@ ( produc5031841752791441652_int_o
@ ^ [X4: product_prod_nat_nat,Y5: product_prod_int_int] :
( ( P2 @ X4 )
& ( Q2 @ X4 @ Y5 ) ) ) )
= ( produc8806266479013559872nt_int @ ( collec3392354462482085612at_nat @ P2 )
@ ^ [X4: product_prod_nat_nat] : ( collec213857154873943460nt_int @ ( Q2 @ X4 ) ) ) ) ).
% Collect_case_prod_Sigma
thf(fact_1726_Collect__case__prod__Sigma,axiom,
! [P2: nat > $o,Q2: nat > product_prod_nat_nat > $o] :
( ( collec5903703980526211963at_nat
@ ( produc5864757623865647827_nat_o
@ ^ [X4: nat,Y5: product_prod_nat_nat] :
( ( P2 @ X4 )
& ( Q2 @ X4 @ Y5 ) ) ) )
= ( produc1809337555817847783at_nat @ ( collect_nat @ P2 )
@ ^ [X4: nat] : ( collec3392354462482085612at_nat @ ( Q2 @ X4 ) ) ) ) ).
% Collect_case_prod_Sigma
thf(fact_1727_Collect__case__prod__Sigma,axiom,
! [P2: nat > $o,Q2: nat > product_prod_int_int > $o] :
( ( collec5600542410287506995nt_int
@ ( produc4306188689235752475_int_o
@ ^ [X4: nat,Y5: product_prod_int_int] :
( ( P2 @ X4 )
& ( Q2 @ X4 @ Y5 ) ) ) )
= ( produc7854212285064481439nt_int @ ( collect_nat @ P2 )
@ ^ [X4: nat] : ( collec213857154873943460nt_int @ ( Q2 @ X4 ) ) ) ) ).
% Collect_case_prod_Sigma
thf(fact_1728_Collect__case__prod__Sigma,axiom,
! [P2: product_prod_int_int > $o,Q2: product_prod_int_int > product_prod_nat_nat > $o] :
( ( collec7751116662793354522at_nat
@ ( produc3235059054576640100_nat_o
@ ^ [X4: product_prod_int_int,Y5: product_prod_nat_nat] :
( ( P2 @ X4 )
& ( Q2 @ X4 @ Y5 ) ) ) )
= ( produc3424345432876038864at_nat @ ( collec213857154873943460nt_int @ P2 )
@ ^ [X4: product_prod_int_int] : ( collec3392354462482085612at_nat @ ( Q2 @ X4 ) ) ) ) ).
% Collect_case_prod_Sigma
thf(fact_1729_Collect__case__prod__Sigma,axiom,
! [P2: product_prod_int_int > $o,Q2: product_prod_int_int > nat > $o] :
( ( collec364394062568603661nt_nat
@ ( produc5441891513209881465_nat_o
@ ^ [X4: product_prod_int_int,Y5: nat] :
( ( P2 @ X4 )
& ( Q2 @ X4 @ Y5 ) ) ) )
= ( produc1007865297704999105nt_nat @ ( collec213857154873943460nt_int @ P2 )
@ ^ [X4: product_prod_int_int] : ( collect_nat @ ( Q2 @ X4 ) ) ) ) ).
% Collect_case_prod_Sigma
thf(fact_1730_Collect__case__prod__Sigma,axiom,
! [P2: product_prod_int_int > $o,Q2: product_prod_int_int > product_prod_int_int > $o] :
( ( collec7447955092554649554nt_int
@ ( produc1676490119946744748_int_o
@ ^ [X4: product_prod_int_int,Y5: product_prod_int_int] :
( ( P2 @ X4 )
& ( Q2 @ X4 @ Y5 ) ) ) )
= ( produc245848125267896712nt_int @ ( collec213857154873943460nt_int @ P2 )
@ ^ [X4: product_prod_int_int] : ( collec213857154873943460nt_int @ ( Q2 @ X4 ) ) ) ) ).
% Collect_case_prod_Sigma
thf(fact_1731_Collect__case__prod__Sigma,axiom,
! [P2: nat > $o,Q2: nat > nat > $o] :
( ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [X4: nat,Y5: nat] :
( ( P2 @ X4 )
& ( Q2 @ X4 @ Y5 ) ) ) )
= ( produc457027306803732586at_nat @ ( collect_nat @ P2 )
@ ^ [X4: nat] : ( collect_nat @ ( Q2 @ X4 ) ) ) ) ).
% Collect_case_prod_Sigma
thf(fact_1732_Collect__case__prod__Sigma,axiom,
! [P2: int > $o,Q2: int > int > $o] :
( ( collec213857154873943460nt_int
@ ( produc4947309494688390418_int_o
@ ^ [X4: int,Y5: int] :
( ( P2 @ X4 )
& ( Q2 @ X4 @ Y5 ) ) ) )
= ( produc1453890548195736866nt_int @ ( collect_int @ P2 )
@ ^ [X4: int] : ( collect_int @ ( Q2 @ X4 ) ) ) ) ).
% Collect_case_prod_Sigma
thf(fact_1733_Sigma__mono,axiom,
! [A4: set_nat,C3: set_nat,B5: nat > set_nat,D: nat > set_nat] :
( ( ord_less_eq_set_nat @ A4 @ C3 )
=> ( ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ( ord_less_eq_set_nat @ ( B5 @ X ) @ ( D @ X ) ) )
=> ( ord_le3146513528884898305at_nat @ ( produc457027306803732586at_nat @ A4 @ B5 ) @ ( produc457027306803732586at_nat @ C3 @ D ) ) ) ) ).
% Sigma_mono
thf(fact_1734_Times__subset__cancel2,axiom,
! [X2: nat,C3: set_nat,A4: set_nat,B5: set_nat] :
( ( member_nat @ X2 @ C3 )
=> ( ( ord_le3146513528884898305at_nat
@ ( produc457027306803732586at_nat @ A4
@ ^ [Uu: nat] : C3 )
@ ( produc457027306803732586at_nat @ B5
@ ^ [Uu: nat] : C3 ) )
= ( ord_less_eq_set_nat @ A4 @ B5 ) ) ) ).
% Times_subset_cancel2
thf(fact_1735_mem__Times__iff,axiom,
! [X2: product_prod_o_o,A4: set_o,B5: set_o] :
( ( member7466972457876170832od_o_o @ X2
@ ( product_Sigma_o_o @ A4
@ ^ [Uu: $o] : B5 ) )
= ( ( member_o @ ( product_fst_o_o @ X2 ) @ A4 )
& ( member_o @ ( product_snd_o_o @ X2 ) @ B5 ) ) ) ).
% mem_Times_iff
thf(fact_1736_mem__Times__iff,axiom,
! [X2: product_prod_o_nat,A4: set_o,B5: set_nat] :
( ( member2802428098988154798_o_nat @ X2
@ ( product_Sigma_o_nat @ A4
@ ^ [Uu: $o] : B5 ) )
= ( ( member_o @ ( product_fst_o_nat @ X2 ) @ A4 )
& ( member_nat @ ( product_snd_o_nat @ X2 ) @ B5 ) ) ) ).
% mem_Times_iff
thf(fact_1737_mem__Times__iff,axiom,
! [X2: product_prod_o_int,A4: set_o,B5: set_int] :
( ( member7847949116333733898_o_int @ X2
@ ( product_Sigma_o_int @ A4
@ ^ [Uu: $o] : B5 ) )
= ( ( member_o @ ( product_fst_o_int @ X2 ) @ A4 )
& ( member_int @ ( product_snd_o_int @ X2 ) @ B5 ) ) ) ).
% mem_Times_iff
thf(fact_1738_mem__Times__iff,axiom,
! [X2: product_prod_nat_o,A4: set_nat,B5: set_o] :
( ( member6310962623043647828_nat_o @ X2
@ ( product_Sigma_nat_o @ A4
@ ^ [Uu: nat] : B5 ) )
= ( ( member_nat @ ( product_fst_nat_o @ X2 ) @ A4 )
& ( member_o @ ( product_snd_nat_o @ X2 ) @ B5 ) ) ) ).
% mem_Times_iff
thf(fact_1739_mem__Times__iff,axiom,
! [X2: product_prod_nat_int,A4: set_nat,B5: set_int] :
( ( member4262671552274231302at_int @ X2
@ ( produc454536836294682310at_int @ A4
@ ^ [Uu: nat] : B5 ) )
= ( ( member_nat @ ( product_fst_nat_int @ X2 ) @ A4 )
& ( member_int @ ( product_snd_nat_int @ X2 ) @ B5 ) ) ) ).
% mem_Times_iff
thf(fact_1740_mem__Times__iff,axiom,
! [X2: product_prod_int_o,A4: set_int,B5: set_o] :
( ( member4489920277610959864_int_o @ X2
@ ( product_Sigma_int_o @ A4
@ ^ [Uu: int] : B5 ) )
= ( ( member_int @ ( product_fst_int_o @ X2 ) @ A4 )
& ( member_o @ ( product_snd_int_o @ X2 ) @ B5 ) ) ) ).
% mem_Times_iff
thf(fact_1741_mem__Times__iff,axiom,
! [X2: product_prod_int_nat,A4: set_int,B5: set_nat] :
( ( member216504246829706758nt_nat @ X2
@ ( produc1456381018704787142nt_nat @ A4
@ ^ [Uu: int] : B5 ) )
= ( ( member_int @ ( product_fst_int_nat @ X2 ) @ A4 )
& ( member_nat @ ( product_snd_int_nat @ X2 ) @ B5 ) ) ) ).
% mem_Times_iff
thf(fact_1742_mem__Times__iff,axiom,
! [X2: product_prod_int_int,A4: set_int,B5: set_int] :
( ( member5262025264175285858nt_int @ X2
@ ( produc1453890548195736866nt_int @ A4
@ ^ [Uu: int] : B5 ) )
= ( ( member_int @ ( product_fst_int_int @ X2 ) @ A4 )
& ( member_int @ ( product_snd_int_int @ X2 ) @ B5 ) ) ) ).
% mem_Times_iff
thf(fact_1743_mem__Times__iff,axiom,
! [X2: product_prod_nat_nat,A4: set_nat,B5: set_nat] :
( ( member8440522571783428010at_nat @ X2
@ ( produc457027306803732586at_nat @ A4
@ ^ [Uu: nat] : B5 ) )
= ( ( member_nat @ ( product_fst_nat_nat @ X2 ) @ A4 )
& ( member_nat @ ( product_snd_nat_nat @ X2 ) @ B5 ) ) ) ).
% mem_Times_iff
thf(fact_1744_mem__Times__iff,axiom,
! [X2: produc652263959640188542_nat_o,A4: set_Pr1261947904930325089at_nat,B5: set_o] :
( ( member2200362271989506215_nat_o @ X2
@ ( produc203973861852599583_nat_o @ A4
@ ^ [Uu: product_prod_nat_nat] : B5 ) )
= ( ( member8440522571783428010at_nat @ ( produc6435950281062462810_nat_o @ X2 ) @ A4 )
& ( member_o @ ( produc668916169454083228_nat_o @ X2 ) @ B5 ) ) ) ).
% mem_Times_iff
thf(fact_1745_product__swap,axiom,
! [A4: set_nat,B5: set_nat] :
( ( image_5168914502847457605at_nat @ product_swap_nat_nat
@ ( produc457027306803732586at_nat @ A4
@ ^ [Uu: nat] : B5 ) )
= ( produc457027306803732586at_nat @ B5
@ ^ [Uu: nat] : A4 ) ) ).
% product_swap
thf(fact_1746_Quotient__compose,axiom,
! [R1: $o > $o > $o,Abs1: $o > $o,Rep1: $o > $o,T1: $o > $o > $o,R22: $o > $o > $o,Abs22: $o > nat,Rep22: nat > $o,T22: $o > nat > $o] :
( ( quotient_o_o @ R1 @ Abs1 @ Rep1 @ T1 )
=> ( ( quotient_o_nat @ R22 @ Abs22 @ Rep22 @ T22 )
=> ( quotient_o_nat @ ( relcompp_o_o_o @ T1 @ ( relcompp_o_o_o @ R22 @ ( conversep_o_o @ T1 ) ) ) @ ( comp_o_nat_o @ Abs22 @ Abs1 ) @ ( comp_o_o_nat @ Rep1 @ Rep22 ) @ ( relcompp_o_o_nat @ T1 @ T22 ) ) ) ) ).
% Quotient_compose
thf(fact_1747_Quotient__compose,axiom,
! [R1: num > num > $o,Abs1: num > nat,Rep1: nat > num,T1: num > nat > $o,R22: nat > nat > $o,Abs22: nat > int,Rep22: int > nat,T22: nat > int > $o] :
( ( quotient_num_nat @ R1 @ Abs1 @ Rep1 @ T1 )
=> ( ( quotient_nat_int @ R22 @ Abs22 @ Rep22 @ T22 )
=> ( quotient_num_int @ ( relcompp_num_nat_num @ T1 @ ( relcompp_nat_nat_num @ R22 @ ( conversep_num_nat @ T1 ) ) ) @ ( comp_nat_int_num @ Abs22 @ Abs1 ) @ ( comp_nat_num_int @ Rep1 @ Rep22 ) @ ( relcompp_num_nat_int @ T1 @ T22 ) ) ) ) ).
% Quotient_compose
thf(fact_1748_Quotient__compose,axiom,
! [R1: nat > nat > $o,Abs1: nat > int,Rep1: int > nat,T1: nat > int > $o,R22: int > int > $o,Abs22: int > int,Rep22: int > int,T22: int > int > $o] :
( ( quotient_nat_int @ R1 @ Abs1 @ Rep1 @ T1 )
=> ( ( quotient_int_int @ R22 @ Abs22 @ Rep22 @ T22 )
=> ( quotient_nat_int @ ( relcompp_nat_int_nat @ T1 @ ( relcompp_int_int_nat @ R22 @ ( conversep_nat_int @ T1 ) ) ) @ ( comp_int_int_nat @ Abs22 @ Abs1 ) @ ( comp_int_nat_int @ Rep1 @ Rep22 ) @ ( relcompp_nat_int_int @ T1 @ T22 ) ) ) ) ).
% Quotient_compose
thf(fact_1749_Quotient__compose,axiom,
! [R1: nat > nat > $o,Abs1: nat > $o,Rep1: $o > nat,T1: nat > $o > $o,R22: $o > $o > $o,Abs22: $o > $o,Rep22: $o > $o,T22: $o > $o > $o] :
( ( quotient_nat_o @ R1 @ Abs1 @ Rep1 @ T1 )
=> ( ( quotient_o_o @ R22 @ Abs22 @ Rep22 @ T22 )
=> ( quotient_nat_o @ ( relcompp_nat_o_nat @ T1 @ ( relcompp_o_o_nat @ R22 @ ( conversep_nat_o @ T1 ) ) ) @ ( comp_o_o_nat @ Abs22 @ Abs1 ) @ ( comp_o_nat_o @ Rep1 @ Rep22 ) @ ( relcompp_nat_o_o @ T1 @ T22 ) ) ) ) ).
% Quotient_compose
thf(fact_1750_Quotient__compose,axiom,
! [R1: int > int > $o,Abs1: int > nat,Rep1: nat > int,T1: int > nat > $o,R22: nat > nat > $o,Abs22: nat > num,Rep22: num > nat,T22: nat > num > $o] :
( ( quotient_int_nat @ R1 @ Abs1 @ Rep1 @ T1 )
=> ( ( quotient_nat_num @ R22 @ Abs22 @ Rep22 @ T22 )
=> ( quotient_int_num @ ( relcompp_int_nat_int @ T1 @ ( relcompp_nat_nat_int @ R22 @ ( conversep_int_nat @ T1 ) ) ) @ ( comp_nat_num_int @ Abs22 @ Abs1 ) @ ( comp_nat_int_num @ Rep1 @ Rep22 ) @ ( relcompp_int_nat_num @ T1 @ T22 ) ) ) ) ).
% Quotient_compose
thf(fact_1751_Quotient__compose,axiom,
! [R1: nat > nat > $o,Abs1: nat > nat,Rep1: nat > nat,T1: nat > nat > $o,R22: nat > nat > $o,Abs22: nat > nat,Rep22: nat > nat,T22: nat > nat > $o] :
( ( quotient_nat_nat @ R1 @ Abs1 @ Rep1 @ T1 )
=> ( ( quotient_nat_nat @ R22 @ Abs22 @ Rep22 @ T22 )
=> ( quotient_nat_nat @ ( relcompp_nat_nat_nat @ T1 @ ( relcompp_nat_nat_nat @ R22 @ ( conversep_nat_nat @ T1 ) ) ) @ ( comp_nat_nat_nat @ Abs22 @ Abs1 ) @ ( comp_nat_nat_nat @ Rep1 @ Rep22 ) @ ( relcompp_nat_nat_nat @ T1 @ T22 ) ) ) ) ).
% Quotient_compose
thf(fact_1752_Quotient__compose,axiom,
! [R1: int > int > $o,Abs1: int > int,Rep1: int > int,T1: int > int > $o,R22: int > int > $o,Abs22: int > nat,Rep22: nat > int,T22: int > nat > $o] :
( ( quotient_int_int @ R1 @ Abs1 @ Rep1 @ T1 )
=> ( ( quotient_int_nat @ R22 @ Abs22 @ Rep22 @ T22 )
=> ( quotient_int_nat @ ( relcompp_int_int_int @ T1 @ ( relcompp_int_int_int @ R22 @ ( conversep_int_int @ T1 ) ) ) @ ( comp_int_nat_int @ Abs22 @ Abs1 ) @ ( comp_int_int_nat @ Rep1 @ Rep22 ) @ ( relcompp_int_int_nat @ T1 @ T22 ) ) ) ) ).
% Quotient_compose
thf(fact_1753_Quotient__compose,axiom,
! [R1: int > int > $o,Abs1: int > int,Rep1: int > int,T1: int > int > $o,R22: int > int > $o,Abs22: int > code_integer,Rep22: code_integer > int,T22: int > code_integer > $o] :
( ( quotient_int_int @ R1 @ Abs1 @ Rep1 @ T1 )
=> ( ( quotie2291698863146621601nteger @ R22 @ Abs22 @ Rep22 @ T22 )
=> ( quotie2291698863146621601nteger @ ( relcompp_int_int_int @ T1 @ ( relcompp_int_int_int @ R22 @ ( conversep_int_int @ T1 ) ) ) @ ( comp_i1585864551200866970er_int @ Abs22 @ Abs1 ) @ ( comp_i2224564030725804698nteger @ Rep1 @ Rep22 ) @ ( relcom967497705283160040nteger @ T1 @ T22 ) ) ) ) ).
% Quotient_compose
thf(fact_1754_Quotient__compose,axiom,
! [R1: product_prod_nat_nat > product_prod_nat_nat > $o,Abs1: product_prod_nat_nat > product_prod_nat_nat,Rep1: product_prod_nat_nat > product_prod_nat_nat,T1: product_prod_nat_nat > product_prod_nat_nat > $o,R22: product_prod_nat_nat > product_prod_nat_nat > $o,Abs22: product_prod_nat_nat > int,Rep22: int > product_prod_nat_nat,T22: product_prod_nat_nat > int > $o] :
( ( quotie4166942682936818134at_nat @ R1 @ Abs1 @ Rep1 @ T1 )
=> ( ( quotie1194848508323700631at_int @ R22 @ Abs22 @ Rep22 @ T22 )
=> ( quotie1194848508323700631at_int @ ( relcom282571730187242960at_nat @ T1 @ ( relcom282571730187242960at_nat @ R22 @ ( conver6965824823119931701at_nat @ T1 ) ) ) @ ( comp_P4605287290794827253at_nat @ Abs22 @ Abs1 ) @ ( comp_P7428279379633703887at_int @ Rep1 @ Rep22 ) @ ( relcom8911704206367525917at_int @ T1 @ T22 ) ) ) ) ).
% Quotient_compose
thf(fact_1755_Quotient__compose,axiom,
! [R1: ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > $o,Abs1: ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > code_integer > code_integer,Rep1: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,T1: ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer ) > $o,R22: ( code_integer > code_integer ) > ( code_integer > code_integer ) > $o,Abs22: ( code_integer > code_integer ) > code_integer,Rep22: code_integer > code_integer > code_integer,T22: ( code_integer > code_integer ) > code_integer > $o] :
( ( quotie8207154290130328982nteger @ R1 @ Abs1 @ Rep1 @ T1 )
=> ( ( quotie9062115205763428814nteger @ R22 @ Abs22 @ Rep22 @ T22 )
=> ( quotie8803226531959296410nteger @ ( relcom458776940982682326nteger @ T1 @ ( relcom1820845089908643850nteger @ R22 @ ( conver3229387634306789365nteger @ T1 ) ) ) @ ( comp_C7826480081638475456nteger @ Abs22 @ Abs1 ) @ ( comp_C8797469213163452608nteger @ Rep1 @ Rep22 ) @ ( relcom4829277440228594702nteger @ T1 @ T22 ) ) ) ) ).
% Quotient_compose
thf(fact_1756_swap__product,axiom,
! [A4: set_se5479201105039950375at_nat,B5: set_se5479201105039950375at_nat] :
( ( image_3313167189156628421at_nat
@ ( produc5456455677105495163at_nat
@ ^ [I2: set_Pr4329608150637261639at_nat,J: set_Pr4329608150637261639at_nat] : ( produc9060074326276436823at_nat @ J @ I2 ) )
@ ( produc6258361011560675592at_nat @ A4
@ ^ [Uu: set_Pr4329608150637261639at_nat] : B5 ) )
= ( produc6258361011560675592at_nat @ B5
@ ^ [Uu: set_Pr4329608150637261639at_nat] : A4 ) ) ).
% swap_product
thf(fact_1757_swap__product,axiom,
! [A4: set_se7855581050983116737at_nat,B5: set_se7855581050983116737at_nat] :
( ( image_960689587819713477at_nat
@ ( produc7548845685434208379at_nat
@ ^ [I2: set_Pr1261947904930325089at_nat,J: set_Pr1261947904930325089at_nat] : ( produc2922128104949294807at_nat @ J @ I2 ) )
@ ( produc7726050920695093128at_nat @ A4
@ ^ [Uu: set_Pr1261947904930325089at_nat] : B5 ) )
= ( produc7726050920695093128at_nat @ B5
@ ^ [Uu: set_Pr1261947904930325089at_nat] : A4 ) ) ).
% swap_product
thf(fact_1758_swap__product,axiom,
! [A4: set_num,B5: set_nat] :
( ( image_7762062833379733445at_num
@ ( produc5219324331026901427at_num
@ ^ [I2: num,J: nat] : ( product_Pair_nat_num @ J @ I2 ) )
@ ( produc7810729746965977780um_nat @ A4
@ ^ [Uu: num] : B5 ) )
= ( produc6237731129814287028at_num @ B5
@ ^ [Uu: nat] : A4 ) ) ).
% swap_product
thf(fact_1759_swap__product,axiom,
! [A4: set_nat,B5: set_nat] :
( ( image_5168914502847457605at_nat
@ ( produc2626176000494625587at_nat
@ ^ [I2: nat,J: nat] : ( product_Pair_nat_nat @ J @ I2 ) )
@ ( produc457027306803732586at_nat @ A4
@ ^ [Uu: nat] : B5 ) )
= ( produc457027306803732586at_nat @ B5
@ ^ [Uu: nat] : A4 ) ) ).
% swap_product
thf(fact_1760_swap__product,axiom,
! [A4: set_int,B5: set_int] :
( ( image_2653370878348428101nt_int
@ ( produc4245557441103728435nt_int
@ ^ [I2: int,J: int] : ( product_Pair_int_int @ J @ I2 ) )
@ ( produc1453890548195736866nt_int @ A4
@ ^ [Uu: int] : B5 ) )
= ( produc1453890548195736866nt_int @ B5
@ ^ [Uu: int] : A4 ) ) ).
% swap_product
thf(fact_1761_map__prod__surj__on,axiom,
! [F: nat > rat,A4: set_nat,A7: set_rat,G: int > nat,B5: set_int,B7: set_nat] :
( ( ( image_nat_rat @ F @ A4 )
= A7 )
=> ( ( ( image_int_nat @ G @ B5 )
= B7 )
=> ( ( image_3755276878268593137at_nat @ ( produc8749313216793205150nt_nat @ F @ G )
@ ( produc454536836294682310at_int @ A4
@ ^ [Uu: nat] : B5 ) )
= ( produc140472662868866674at_nat @ A7
@ ^ [Uu: rat] : B7 ) ) ) ) ).
% map_prod_surj_on
thf(fact_1762_map__prod__surj__on,axiom,
! [F: nat > rat,A4: set_nat,A7: set_rat,G: int > int,B5: set_int,B7: set_int] :
( ( ( image_nat_rat @ F @ A4 )
= A7 )
=> ( ( ( image_int_int @ G @ B5 )
= B7 )
=> ( ( image_8800797895614172237at_int @ ( produc8746822746284154874nt_int @ F @ G )
@ ( produc454536836294682310at_int @ A4
@ ^ [Uu: nat] : B5 ) )
= ( produc137982192359816398at_int @ A7
@ ^ [Uu: rat] : B7 ) ) ) ) ).
% map_prod_surj_on
thf(fact_1763_map__prod__surj__on,axiom,
! [F: nat > nat,A4: set_nat,A7: set_nat,G: int > int,B5: set_int,B7: set_int] :
( ( ( image_nat_nat @ F @ A4 )
= A7 )
=> ( ( ( image_int_int @ G @ B5 )
= B7 )
=> ( ( image_9117352539549038149at_int @ ( produc7974749936722635250nt_int @ F @ G )
@ ( produc454536836294682310at_int @ A4
@ ^ [Uu: nat] : B5 ) )
= ( produc454536836294682310at_int @ A7
@ ^ [Uu: nat] : B7 ) ) ) ) ).
% map_prod_surj_on
thf(fact_1764_map__prod__surj__on,axiom,
! [F: nat > int,A4: set_nat,A7: set_int,G: int > nat,B5: set_int,B7: set_nat] :
( ( ( image_nat_int @ F @ A4 )
= A7 )
=> ( ( ( image_int_nat @ G @ B5 )
= B7 )
=> ( ( image_5071185234104513605nt_nat @ ( produc8348893967468530162nt_nat @ F @ G )
@ ( produc454536836294682310at_int @ A4
@ ^ [Uu: nat] : B5 ) )
= ( produc1456381018704787142nt_nat @ A7
@ ^ [Uu: int] : B7 ) ) ) ) ).
% map_prod_surj_on
thf(fact_1765_map__prod__surj__on,axiom,
! [F: nat > int,A4: set_nat,A7: set_int,G: int > int,B5: set_int,B7: set_int] :
( ( ( image_nat_int @ F @ A4 )
= A7 )
=> ( ( ( image_int_int @ G @ B5 )
= B7 )
=> ( ( image_893334214595316897nt_int @ ( produc8346403496959479886nt_int @ F @ G )
@ ( produc454536836294682310at_int @ A4
@ ^ [Uu: nat] : B5 ) )
= ( produc1453890548195736866nt_int @ A7
@ ^ [Uu: int] : B7 ) ) ) ) ).
% map_prod_surj_on
thf(fact_1766_map__prod__surj__on,axiom,
! [F: int > nat,A4: set_int,A7: set_nat,G: nat > rat,B5: set_nat,B7: set_rat] :
( ( ( image_int_nat @ F @ A4 )
= A7 )
=> ( ( ( image_nat_rat @ G @ B5 )
= B7 )
=> ( ( image_3420722608207172081at_rat @ ( produc4032938703151184286at_rat @ F @ G )
@ ( produc1456381018704787142nt_nat @ A4
@ ^ [Uu: int] : B5 ) )
= ( produc9045269283572012658at_rat @ A7
@ ^ [Uu: nat] : B7 ) ) ) ) ).
% map_prod_surj_on
thf(fact_1767_map__prod__surj__on,axiom,
! [F: int > nat,A4: set_int,A7: set_nat,G: nat > int,B5: set_nat,B7: set_int] :
( ( ( image_int_nat @ F @ A4 )
= A7 )
=> ( ( ( image_nat_int @ G @ B5 )
= B7 )
=> ( ( image_2751100147091372101at_int @ ( produc4665578292728629746at_int @ F @ G )
@ ( produc1456381018704787142nt_nat @ A4
@ ^ [Uu: int] : B5 ) )
= ( produc454536836294682310at_int @ A7
@ ^ [Uu: nat] : B7 ) ) ) ) ).
% map_prod_surj_on
thf(fact_1768_map__prod__surj__on,axiom,
! [F: int > nat,A4: set_int,A7: set_nat,G: int > int,B5: set_int,B7: set_int] :
( ( ( image_int_nat @ F @ A4 )
= A7 )
=> ( ( ( image_int_int @ G @ B5 )
= B7 )
=> ( ( image_1654017166447373545at_int @ ( produc5664932004629684302nt_int @ F @ G )
@ ( produc1453890548195736866nt_int @ A4
@ ^ [Uu: int] : B5 ) )
= ( produc454536836294682310at_int @ A7
@ ^ [Uu: nat] : B7 ) ) ) ) ).
% map_prod_surj_on
thf(fact_1769_map__prod__surj__on,axiom,
! [F: int > int,A4: set_int,A7: set_int,G: nat > rat,B5: set_nat,B7: set_rat] :
( ( ( image_int_int @ F @ A4 )
= A7 )
=> ( ( ( image_nat_rat @ G @ B5 )
= B7 )
=> ( ( image_4420076320108226637nt_rat @ ( produc4404592263388028922at_rat @ F @ G )
@ ( produc1456381018704787142nt_nat @ A4
@ ^ [Uu: int] : B5 ) )
= ( produc821250958618291406nt_rat @ A7
@ ^ [Uu: int] : B7 ) ) ) ) ).
% map_prod_surj_on
thf(fact_1770_map__prod__surj__on,axiom,
! [F: int > int,A4: set_int,A7: set_int,G: nat > nat,B5: set_nat,B7: set_nat] :
( ( ( image_int_int @ F @ A4 )
= A7 )
=> ( ( ( image_nat_nat @ G @ B5 )
= B7 )
=> ( ( image_7928304878501623365nt_nat @ ( produc5039722323474524658at_nat @ F @ G )
@ ( produc1456381018704787142nt_nat @ A4
@ ^ [Uu: int] : B5 ) )
= ( produc1456381018704787142nt_nat @ A7
@ ^ [Uu: int] : B7 ) ) ) ) ).
% map_prod_surj_on
thf(fact_1771_left__total__OO,axiom,
! [R3: product_prod_nat_nat > product_prod_nat_nat > $o,S5: product_prod_nat_nat > int > $o] :
( ( left_t5646938096895998950at_nat @ R3 )
=> ( ( left_t1861001511420785031at_int @ S5 )
=> ( left_t1861001511420785031at_int @ ( relcom8911704206367525917at_int @ R3 @ S5 ) ) ) ) ).
% left_total_OO
thf(fact_1772_left__total__OO,axiom,
! [R3: int > int > $o,S5: int > code_integer > $o] :
( ( left_total_int_int @ R3 )
=> ( ( left_t2374113004080657585nteger @ S5 )
=> ( left_t2374113004080657585nteger @ ( relcom967497705283160040nteger @ R3 @ S5 ) ) ) ) ).
% left_total_OO
thf(fact_1773_left__unique__OO,axiom,
! [R3: product_prod_nat_nat > product_prod_nat_nat > $o,S5: product_prod_nat_nat > int > $o] :
( ( left_u7296345956935681743at_nat @ R3 )
=> ( ( left_u4005249686606446046at_int @ S5 )
=> ( left_u4005249686606446046at_int @ ( relcom8911704206367525917at_int @ R3 @ S5 ) ) ) ) ).
% left_unique_OO
thf(fact_1774_left__unique__OO,axiom,
! [R3: int > int > $o,S5: int > code_integer > $o] :
( ( left_unique_int_int @ R3 )
=> ( ( left_u6795595883297946906nteger @ S5 )
=> ( left_u6795595883297946906nteger @ ( relcom967497705283160040nteger @ R3 @ S5 ) ) ) ) ).
% left_unique_OO
thf(fact_1775_Product__Type_Oproduct__def,axiom,
( produc929334515565554804at_nat
= ( ^ [A6: set_nat,B6: set_nat] :
( produc457027306803732586at_nat @ A6
@ ^ [Uu: nat] : B6 ) ) ) ).
% Product_Type.product_def
thf(fact_1776_member__product,axiom,
! [X2: produc3843707927480180839at_nat,A4: set_se7855581050983116737at_nat,B5: set_se7855581050983116737at_nat] :
( ( member8757157785044589968at_nat @ X2 @ ( produc3241716407206042770at_nat @ A4 @ B5 ) )
= ( member8757157785044589968at_nat @ X2
@ ( produc7726050920695093128at_nat @ A4
@ ^ [Uu: set_Pr1261947904930325089at_nat] : B5 ) ) ) ).
% member_product
thf(fact_1777_member__product,axiom,
! [X2: product_prod_nat_nat,A4: set_nat,B5: set_nat] :
( ( member8440522571783428010at_nat @ X2 @ ( produc929334515565554804at_nat @ A4 @ B5 ) )
= ( member8440522571783428010at_nat @ X2
@ ( produc457027306803732586at_nat @ A4
@ ^ [Uu: nat] : B5 ) ) ) ).
% member_product
thf(fact_1778_snd__image__times,axiom,
! [A4: set_Product_unit,B5: set_Product_unit] :
( ( ( A4 = bot_bo3957492148770167129t_unit )
=> ( ( image_262127665112854123t_unit @ produc6786084904866402309t_unit
@ ( produc6121607364029222536t_unit @ A4
@ ^ [Uu: product_unit] : B5 ) )
= bot_bo3957492148770167129t_unit ) )
& ( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ( image_262127665112854123t_unit @ produc6786084904866402309t_unit
@ ( produc6121607364029222536t_unit @ A4
@ ^ [Uu: product_unit] : B5 ) )
= B5 ) ) ) ).
% snd_image_times
thf(fact_1779_snd__image__times,axiom,
! [A4: set_Product_unit,B5: set_o] :
( ( ( A4 = bot_bo3957492148770167129t_unit )
=> ( ( image_1801747192096393899it_o_o @ produc7461459235754980752unit_o
@ ( produc4996980317674386579unit_o @ A4
@ ^ [Uu: product_unit] : B5 ) )
= bot_bot_set_o ) )
& ( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ( image_1801747192096393899it_o_o @ produc7461459235754980752unit_o
@ ( produc4996980317674386579unit_o @ A4
@ ^ [Uu: product_unit] : B5 ) )
= B5 ) ) ) ).
% snd_image_times
thf(fact_1780_snd__image__times,axiom,
! [A4: set_Product_unit,B5: set_nat] :
( ( ( A4 = bot_bo3957492148770167129t_unit )
=> ( ( image_6725568594985674851at_nat @ produc996414825743149016it_nat
@ ( produc6875377002296012693it_nat @ A4
@ ^ [Uu: product_unit] : B5 ) )
= bot_bot_set_nat ) )
& ( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ( image_6725568594985674851at_nat @ produc996414825743149016it_nat
@ ( produc6875377002296012693it_nat @ A4
@ ^ [Uu: product_unit] : B5 ) )
= B5 ) ) ) ).
% snd_image_times
thf(fact_1781_snd__image__times,axiom,
! [A4: set_o,B5: set_Product_unit] :
( ( ( A4 = bot_bot_set_o )
=> ( ( image_4419268941131634678t_unit @ produc2875488738488844090t_unit
@ ( produc411009820408249917t_unit @ A4
@ ^ [Uu: $o] : B5 ) )
= bot_bo3957492148770167129t_unit ) )
& ( ( A4 != bot_bot_set_o )
=> ( ( image_4419268941131634678t_unit @ produc2875488738488844090t_unit
@ ( produc411009820408249917t_unit @ A4
@ ^ [Uu: $o] : B5 ) )
= B5 ) ) ) ).
% snd_image_times
thf(fact_1782_snd__image__times,axiom,
! [A4: set_o,B5: set_o] :
( ( ( A4 = bot_bot_set_o )
=> ( ( image_7896445794123959606_o_o_o @ product_snd_o_o
@ ( product_Sigma_o_o @ A4
@ ^ [Uu: $o] : B5 ) )
= bot_bot_set_o ) )
& ( ( A4 != bot_bot_set_o )
=> ( ( image_7896445794123959606_o_o_o @ product_snd_o_o
@ ( product_Sigma_o_o @ A4
@ ^ [Uu: $o] : B5 ) )
= B5 ) ) ) ).
% snd_image_times
thf(fact_1783_snd__image__times,axiom,
! [A4: set_o,B5: set_nat] :
( ( ( A4 = bot_bot_set_o )
=> ( ( image_4888532181312399726at_nat @ product_snd_o_nat
@ ( product_Sigma_o_nat @ A4
@ ^ [Uu: $o] : B5 ) )
= bot_bot_set_nat ) )
& ( ( A4 != bot_bot_set_o )
=> ( ( image_4888532181312399726at_nat @ product_snd_o_nat
@ ( product_Sigma_o_nat @ A4
@ ^ [Uu: $o] : B5 ) )
= B5 ) ) ) ).
% snd_image_times
thf(fact_1784_snd__image__times,axiom,
! [A4: set_nat,B5: set_Product_unit] :
( ( ( A4 = bot_bot_set_nat )
=> ( ( image_5511522805742971924t_unit @ produc8850949007409916470t_unit
@ ( produc5506539147108004339t_unit @ A4
@ ^ [Uu: nat] : B5 ) )
= bot_bo3957492148770167129t_unit ) )
& ( ( A4 != bot_bot_set_nat )
=> ( ( image_5511522805742971924t_unit @ produc8850949007409916470t_unit
@ ( produc5506539147108004339t_unit @ A4
@ ^ [Uu: nat] : B5 ) )
= B5 ) ) ) ).
% snd_image_times
thf(fact_1785_snd__image__times,axiom,
! [A4: set_nat,B5: set_o] :
( ( ( A4 = bot_bot_set_nat )
=> ( ( image_7478392465989118804at_o_o @ product_snd_nat_o
@ ( product_Sigma_nat_o @ A4
@ ^ [Uu: nat] : B5 ) )
= bot_bot_set_o ) )
& ( ( A4 != bot_bot_set_nat )
=> ( ( image_7478392465989118804at_o_o @ product_snd_nat_o
@ ( product_Sigma_nat_o @ A4
@ ^ [Uu: nat] : B5 ) )
= B5 ) ) ) ).
% snd_image_times
thf(fact_1786_snd__image__times,axiom,
! [A4: set_nat,B5: set_nat] :
( ( ( A4 = bot_bot_set_nat )
=> ( ( image_2486076414777270412at_nat @ product_snd_nat_nat
@ ( produc457027306803732586at_nat @ A4
@ ^ [Uu: nat] : B5 ) )
= bot_bot_set_nat ) )
& ( ( A4 != bot_bot_set_nat )
=> ( ( image_2486076414777270412at_nat @ product_snd_nat_nat
@ ( produc457027306803732586at_nat @ A4
@ ^ [Uu: nat] : B5 ) )
= B5 ) ) ) ).
% snd_image_times
thf(fact_1787_snd__image__times,axiom,
! [A4: set_ref_a,B5: set_he8644174446132254637t_unit] :
( ( ( A4 = bot_bot_set_ref_a )
=> ( ( image_17660834435998945t_unit @ produc1126150082675451753t_unit
@ ( produc3946373962484743846t_unit @ A4
@ ^ [Uu: ref_a] : B5 ) )
= bot_bo3686245532977126937t_unit ) )
& ( ( A4 != bot_bot_set_ref_a )
=> ( ( image_17660834435998945t_unit @ produc1126150082675451753t_unit
@ ( produc3946373962484743846t_unit @ A4
@ ^ [Uu: ref_a] : B5 ) )
= B5 ) ) ) ).
% snd_image_times
thf(fact_1788_fst__image__times,axiom,
! [B5: set_Product_unit,A4: set_Product_unit] :
( ( ( B5 = bot_bo3957492148770167129t_unit )
=> ( ( image_262127665112854123t_unit @ produc6963936691090040003t_unit
@ ( produc6121607364029222536t_unit @ A4
@ ^ [Uu: product_unit] : B5 ) )
= bot_bo3957492148770167129t_unit ) )
& ( ( B5 != bot_bo3957492148770167129t_unit )
=> ( ( image_262127665112854123t_unit @ produc6963936691090040003t_unit
@ ( produc6121607364029222536t_unit @ A4
@ ^ [Uu: product_unit] : B5 ) )
= A4 ) ) ) ).
% fst_image_times
thf(fact_1789_fst__image__times,axiom,
! [B5: set_Product_unit,A4: set_o] :
( ( ( B5 = bot_bo3957492148770167129t_unit )
=> ( ( image_8742399529945250561unit_o @ produc5219655352205527288t_unit
@ ( produc411009820408249917t_unit @ A4
@ ^ [Uu: $o] : B5 ) )
= bot_bot_set_o ) )
& ( ( B5 != bot_bo3957492148770167129t_unit )
=> ( ( image_8742399529945250561unit_o @ produc5219655352205527288t_unit
@ ( produc411009820408249917t_unit @ A4
@ ^ [Uu: $o] : B5 ) )
= A4 ) ) ) ).
% fst_image_times
thf(fact_1790_fst__image__times,axiom,
! [B5: set_Product_unit,A4: set_nat] :
( ( ( B5 = bot_bo3957492148770167129t_unit )
=> ( ( image_3884712837920099081it_nat @ produc3800628931346975736t_unit
@ ( produc5506539147108004339t_unit @ A4
@ ^ [Uu: nat] : B5 ) )
= bot_bot_set_nat ) )
& ( ( B5 != bot_bo3957492148770167129t_unit )
=> ( ( image_3884712837920099081it_nat @ produc3800628931346975736t_unit
@ ( produc5506539147108004339t_unit @ A4
@ ^ [Uu: nat] : B5 ) )
= A4 ) ) ) ).
% fst_image_times
thf(fact_1791_fst__image__times,axiom,
! [B5: set_o,A4: set_Product_unit] :
( ( ( B5 = bot_bot_set_o )
=> ( ( image_4246157487713968288t_unit @ produc582253812616888142unit_o
@ ( produc4996980317674386579unit_o @ A4
@ ^ [Uu: product_unit] : B5 ) )
= bot_bo3957492148770167129t_unit ) )
& ( ( B5 != bot_bot_set_o )
=> ( ( image_4246157487713968288t_unit @ produc582253812616888142unit_o
@ ( produc4996980317674386579unit_o @ A4
@ ^ [Uu: product_unit] : B5 ) )
= A4 ) ) ) ).
% fst_image_times
thf(fact_1792_fst__image__times,axiom,
! [B5: set_o,A4: set_o] :
( ( ( B5 = bot_bot_set_o )
=> ( ( image_7896445794123959606_o_o_o @ product_fst_o_o
@ ( product_Sigma_o_o @ A4
@ ^ [Uu: $o] : B5 ) )
= bot_bot_set_o ) )
& ( ( B5 != bot_bot_set_o )
=> ( ( image_7896445794123959606_o_o_o @ product_fst_o_o
@ ( product_Sigma_o_o @ A4
@ ^ [Uu: $o] : B5 ) )
= A4 ) ) ) ).
% fst_image_times
thf(fact_1793_fst__image__times,axiom,
! [B5: set_o,A4: set_nat] :
( ( ( B5 = bot_bot_set_o )
=> ( ( image_8629803608462650132_o_nat @ product_fst_nat_o
@ ( product_Sigma_nat_o @ A4
@ ^ [Uu: nat] : B5 ) )
= bot_bot_set_nat ) )
& ( ( B5 != bot_bot_set_o )
=> ( ( image_8629803608462650132_o_nat @ product_fst_nat_o
@ ( product_Sigma_nat_o @ A4
@ ^ [Uu: nat] : B5 ) )
= A4 ) ) ) ).
% fst_image_times
thf(fact_1794_fst__image__times,axiom,
! [B5: set_nat,A4: set_Product_unit] :
( ( ( B5 = bot_bot_set_nat )
=> ( ( image_7474191388529947450t_unit @ produc5169466786534984090it_nat
@ ( produc6875377002296012693it_nat @ A4
@ ^ [Uu: product_unit] : B5 ) )
= bot_bo3957492148770167129t_unit ) )
& ( ( B5 != bot_bot_set_nat )
=> ( ( image_7474191388529947450t_unit @ produc5169466786534984090it_nat
@ ( produc6875377002296012693it_nat @ A4
@ ^ [Uu: product_unit] : B5 ) )
= A4 ) ) ) ).
% fst_image_times
thf(fact_1795_fst__image__times,axiom,
! [B5: set_nat,A4: set_o] :
( ( ( B5 = bot_bot_set_nat )
=> ( ( image_8105710389281276538_nat_o @ product_fst_o_nat
@ ( product_Sigma_o_nat @ A4
@ ^ [Uu: $o] : B5 ) )
= bot_bot_set_o ) )
& ( ( B5 != bot_bot_set_nat )
=> ( ( image_8105710389281276538_nat_o @ product_fst_o_nat
@ ( product_Sigma_o_nat @ A4
@ ^ [Uu: $o] : B5 ) )
= A4 ) ) ) ).
% fst_image_times
thf(fact_1796_fst__image__times,axiom,
! [B5: set_nat,A4: set_nat] :
( ( ( B5 = bot_bot_set_nat )
=> ( ( image_2486076414777270412at_nat @ product_fst_nat_nat
@ ( produc457027306803732586at_nat @ A4
@ ^ [Uu: nat] : B5 ) )
= bot_bot_set_nat ) )
& ( ( B5 != bot_bot_set_nat )
=> ( ( image_2486076414777270412at_nat @ product_fst_nat_nat
@ ( produc457027306803732586at_nat @ A4
@ ^ [Uu: nat] : B5 ) )
= A4 ) ) ) ).
% fst_image_times
thf(fact_1797_fst__image__times,axiom,
! [B5: set_he8644174446132254637t_unit,A4: set_ref_a] :
( ( ( B5 = bot_bo3686245532977126937t_unit )
=> ( ( image_7136766188234240905_ref_a @ produc473321116828791851t_unit
@ ( produc3946373962484743846t_unit @ A4
@ ^ [Uu: ref_a] : B5 ) )
= bot_bot_set_ref_a ) )
& ( ( B5 != bot_bo3686245532977126937t_unit )
=> ( ( image_7136766188234240905_ref_a @ produc473321116828791851t_unit
@ ( produc3946373962484743846t_unit @ A4
@ ^ [Uu: ref_a] : B5 ) )
= A4 ) ) ) ).
% fst_image_times
thf(fact_1798_image__is__empty,axiom,
! [F: int > int,A4: set_int] :
( ( ( image_int_int @ F @ A4 )
= bot_bot_set_int )
= ( A4 = bot_bot_set_int ) ) ).
% image_is_empty
thf(fact_1799_image__is__empty,axiom,
! [F: nat > rat,A4: set_nat] :
( ( ( image_nat_rat @ F @ A4 )
= bot_bot_set_rat )
= ( A4 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_1800_image__is__empty,axiom,
! [F: nat > int,A4: set_nat] :
( ( ( image_nat_int @ F @ A4 )
= bot_bot_set_int )
= ( A4 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_1801_image__is__empty,axiom,
! [F: product_unit > product_unit,A4: set_Product_unit] :
( ( ( image_405062704495631173t_unit @ F @ A4 )
= bot_bo3957492148770167129t_unit )
= ( A4 = bot_bo3957492148770167129t_unit ) ) ).
% image_is_empty
thf(fact_1802_image__is__empty,axiom,
! [F: $o > product_unit,A4: set_o] :
( ( ( image_o_Product_unit @ F @ A4 )
= bot_bo3957492148770167129t_unit )
= ( A4 = bot_bot_set_o ) ) ).
% image_is_empty
thf(fact_1803_image__is__empty,axiom,
! [F: nat > product_unit,A4: set_nat] :
( ( ( image_8730104196221521654t_unit @ F @ A4 )
= bot_bo3957492148770167129t_unit )
= ( A4 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_1804_image__is__empty,axiom,
! [F: product_unit > $o,A4: set_Product_unit] :
( ( ( image_Product_unit_o @ F @ A4 )
= bot_bot_set_o )
= ( A4 = bot_bo3957492148770167129t_unit ) ) ).
% image_is_empty
thf(fact_1805_image__is__empty,axiom,
! [F: $o > $o,A4: set_o] :
( ( ( image_o_o @ F @ A4 )
= bot_bot_set_o )
= ( A4 = bot_bot_set_o ) ) ).
% image_is_empty
thf(fact_1806_image__is__empty,axiom,
! [F: nat > $o,A4: set_nat] :
( ( ( image_nat_o @ F @ A4 )
= bot_bot_set_o )
= ( A4 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_1807_image__is__empty,axiom,
! [F: int > nat,A4: set_int] :
( ( ( image_int_nat @ F @ A4 )
= bot_bot_set_nat )
= ( A4 = bot_bot_set_int ) ) ).
% image_is_empty
thf(fact_1808_empty__is__image,axiom,
! [F: int > int,A4: set_int] :
( ( bot_bot_set_int
= ( image_int_int @ F @ A4 ) )
= ( A4 = bot_bot_set_int ) ) ).
% empty_is_image
thf(fact_1809_empty__is__image,axiom,
! [F: nat > rat,A4: set_nat] :
( ( bot_bot_set_rat
= ( image_nat_rat @ F @ A4 ) )
= ( A4 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_1810_empty__is__image,axiom,
! [F: nat > int,A4: set_nat] :
( ( bot_bot_set_int
= ( image_nat_int @ F @ A4 ) )
= ( A4 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_1811_empty__is__image,axiom,
! [F: product_unit > product_unit,A4: set_Product_unit] :
( ( bot_bo3957492148770167129t_unit
= ( image_405062704495631173t_unit @ F @ A4 ) )
= ( A4 = bot_bo3957492148770167129t_unit ) ) ).
% empty_is_image
thf(fact_1812_empty__is__image,axiom,
! [F: $o > product_unit,A4: set_o] :
( ( bot_bo3957492148770167129t_unit
= ( image_o_Product_unit @ F @ A4 ) )
= ( A4 = bot_bot_set_o ) ) ).
% empty_is_image
thf(fact_1813_empty__is__image,axiom,
! [F: nat > product_unit,A4: set_nat] :
( ( bot_bo3957492148770167129t_unit
= ( image_8730104196221521654t_unit @ F @ A4 ) )
= ( A4 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_1814_empty__is__image,axiom,
! [F: product_unit > $o,A4: set_Product_unit] :
( ( bot_bot_set_o
= ( image_Product_unit_o @ F @ A4 ) )
= ( A4 = bot_bo3957492148770167129t_unit ) ) ).
% empty_is_image
thf(fact_1815_empty__is__image,axiom,
! [F: $o > $o,A4: set_o] :
( ( bot_bot_set_o
= ( image_o_o @ F @ A4 ) )
= ( A4 = bot_bot_set_o ) ) ).
% empty_is_image
thf(fact_1816_empty__is__image,axiom,
! [F: nat > $o,A4: set_nat] :
( ( bot_bot_set_o
= ( image_nat_o @ F @ A4 ) )
= ( A4 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_1817_empty__is__image,axiom,
! [F: int > nat,A4: set_int] :
( ( bot_bot_set_nat
= ( image_int_nat @ F @ A4 ) )
= ( A4 = bot_bot_set_int ) ) ).
% empty_is_image
thf(fact_1818_image__empty,axiom,
! [F: int > int] :
( ( image_int_int @ F @ bot_bot_set_int )
= bot_bot_set_int ) ).
% image_empty
thf(fact_1819_image__empty,axiom,
! [F: int > nat] :
( ( image_int_nat @ F @ bot_bot_set_int )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_1820_image__empty,axiom,
! [F: product_unit > product_unit] :
( ( image_405062704495631173t_unit @ F @ bot_bo3957492148770167129t_unit )
= bot_bo3957492148770167129t_unit ) ).
% image_empty
thf(fact_1821_image__empty,axiom,
! [F: product_unit > $o] :
( ( image_Product_unit_o @ F @ bot_bo3957492148770167129t_unit )
= bot_bot_set_o ) ).
% image_empty
thf(fact_1822_image__empty,axiom,
! [F: product_unit > nat] :
( ( image_875570014554754200it_nat @ F @ bot_bo3957492148770167129t_unit )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_1823_image__empty,axiom,
! [F: $o > product_unit] :
( ( image_o_Product_unit @ F @ bot_bot_set_o )
= bot_bo3957492148770167129t_unit ) ).
% image_empty
thf(fact_1824_image__empty,axiom,
! [F: $o > $o] :
( ( image_o_o @ F @ bot_bot_set_o )
= bot_bot_set_o ) ).
% image_empty
thf(fact_1825_image__empty,axiom,
! [F: $o > nat] :
( ( image_o_nat @ F @ bot_bot_set_o )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_1826_image__empty,axiom,
! [F: nat > rat] :
( ( image_nat_rat @ F @ bot_bot_set_nat )
= bot_bot_set_rat ) ).
% image_empty
thf(fact_1827_image__empty,axiom,
! [F: nat > int] :
( ( image_nat_int @ F @ bot_bot_set_nat )
= bot_bot_set_int ) ).
% image_empty
thf(fact_1828_Sigma__empty1,axiom,
! [B5: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat] :
( ( produc7726050920695093128at_nat @ bot_bo3083307316010499117at_nat @ B5 )
= bot_bo228742789529271731at_nat ) ).
% Sigma_empty1
thf(fact_1829_Sigma__empty1,axiom,
! [B5: nat > set_nat] :
( ( produc457027306803732586at_nat @ bot_bot_set_nat @ B5 )
= bot_bo2099793752762293965at_nat ) ).
% Sigma_empty1
thf(fact_1830_Sigma__empty2,axiom,
! [A4: set_se7855581050983116737at_nat] :
( ( produc7726050920695093128at_nat @ A4
@ ^ [Uu: set_Pr1261947904930325089at_nat] : bot_bo3083307316010499117at_nat )
= bot_bo228742789529271731at_nat ) ).
% Sigma_empty2
thf(fact_1831_Sigma__empty2,axiom,
! [A4: set_nat] :
( ( produc457027306803732586at_nat @ A4
@ ^ [Uu: nat] : bot_bot_set_nat )
= bot_bo2099793752762293965at_nat ) ).
% Sigma_empty2
thf(fact_1832_Times__empty,axiom,
! [A4: set_Product_unit,B5: set_Product_unit] :
( ( ( produc6121607364029222536t_unit @ A4
@ ^ [Uu: product_unit] : B5 )
= bot_bo1654022762305692595t_unit )
= ( ( A4 = bot_bo3957492148770167129t_unit )
| ( B5 = bot_bo3957492148770167129t_unit ) ) ) ).
% Times_empty
thf(fact_1833_Times__empty,axiom,
! [A4: set_Product_unit,B5: set_o] :
( ( ( produc4996980317674386579unit_o @ A4
@ ^ [Uu: product_unit] : B5 )
= bot_bo3195362268743713342unit_o )
= ( ( A4 = bot_bo3957492148770167129t_unit )
| ( B5 = bot_bot_set_o ) ) ) ).
% Times_empty
thf(fact_1834_Times__empty,axiom,
! [A4: set_Product_unit,B5: set_nat] :
( ( ( produc6875377002296012693it_nat @ A4
@ ^ [Uu: product_unit] : B5 )
= bot_bo2472676621365177558it_nat )
= ( ( A4 = bot_bo3957492148770167129t_unit )
| ( B5 = bot_bot_set_nat ) ) ) ).
% Times_empty
thf(fact_1835_Times__empty,axiom,
! [A4: set_o,B5: set_Product_unit] :
( ( ( produc411009820408249917t_unit @ A4
@ ^ [Uu: $o] : B5 )
= bot_bo4402188677660116584t_unit )
= ( ( A4 = bot_bot_set_o )
| ( B5 = bot_bo3957492148770167129t_unit ) ) ) ).
% Times_empty
thf(fact_1836_Times__empty,axiom,
! [A4: set_o,B5: set_o] :
( ( ( product_Sigma_o_o @ A4
@ ^ [Uu: $o] : B5 )
= bot_bo7073875226086086771od_o_o )
= ( ( A4 = bot_bot_set_o )
| ( B5 = bot_bot_set_o ) ) ) ).
% Times_empty
thf(fact_1837_Times__empty,axiom,
! [A4: set_o,B5: set_nat] :
( ( ( product_Sigma_o_nat @ A4
@ ^ [Uu: $o] : B5 )
= bot_bo66526543525301729_o_nat )
= ( ( A4 = bot_bot_set_o )
| ( B5 = bot_bot_set_nat ) ) ) ).
% Times_empty
thf(fact_1838_Times__empty,axiom,
! [A4: set_nat,B5: set_Product_unit] :
( ( ( produc5506539147108004339t_unit @ A4
@ ^ [Uu: nat] : B5 )
= bot_bo5043309098482578556t_unit )
= ( ( A4 = bot_bot_set_nat )
| ( B5 = bot_bo3957492148770167129t_unit ) ) ) ).
% Times_empty
thf(fact_1839_Times__empty,axiom,
! [A4: set_nat,B5: set_o] :
( ( ( product_Sigma_nat_o @ A4
@ ^ [Uu: nat] : B5 )
= bot_bo1114129665703605383_nat_o )
= ( ( A4 = bot_bot_set_nat )
| ( B5 = bot_bot_set_o ) ) ) ).
% Times_empty
thf(fact_1840_Times__empty,axiom,
! [A4: set_nat,B5: set_nat] :
( ( ( produc457027306803732586at_nat @ A4
@ ^ [Uu: nat] : B5 )
= bot_bo2099793752762293965at_nat )
= ( ( A4 = bot_bot_set_nat )
| ( B5 = bot_bot_set_nat ) ) ) ).
% Times_empty
thf(fact_1841_Times__empty,axiom,
! [A4: set_Product_unit,B5: set_Pr1261947904930325089at_nat] :
( ( ( produc3904733337175673596at_nat @ A4
@ ^ [Uu: product_unit] : B5 )
= bot_bo6659885642593197735at_nat )
= ( ( A4 = bot_bo3957492148770167129t_unit )
| ( B5 = bot_bo2099793752762293965at_nat ) ) ) ).
% Times_empty
thf(fact_1842_Collect__const__case__prod,axiom,
! [P2: $o] :
( ( P2
=> ( ( collec6321179662152712658at_nat
@ ( produc410239310623530412_nat_o
@ ^ [A5: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] : P2 ) )
= top_to6833984726390702231at_nat ) )
& ( ~ P2
=> ( ( collec6321179662152712658at_nat
@ ( produc410239310623530412_nat_o
@ ^ [A5: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] : P2 ) )
= bot_bo228742789529271731at_nat ) ) ) ).
% Collect_const_case_prod
thf(fact_1843_Collect__const__case__prod,axiom,
! [P2: $o] :
( ( P2
=> ( ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [A5: nat,B4: nat] : P2 ) )
= top_to4669805908274784177at_nat ) )
& ( ~ P2
=> ( ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [A5: nat,B4: nat] : P2 ) )
= bot_bo2099793752762293965at_nat ) ) ) ).
% Collect_const_case_prod
thf(fact_1844_Collect__const__case__prod,axiom,
! [P2: $o] :
( ( P2
=> ( ( collec213857154873943460nt_int
@ ( produc4947309494688390418_int_o
@ ^ [A5: int,B4: int] : P2 ) )
= top_to4366644338036079209nt_int ) )
& ( ~ P2
=> ( ( collec213857154873943460nt_int
@ ( produc4947309494688390418_int_o
@ ^ [A5: int,B4: int] : P2 ) )
= bot_bo1796632182523588997nt_int ) ) ) ).
% Collect_const_case_prod
thf(fact_1845_Sigma__empty__iff,axiom,
! [I3: set_se7855581050983116737at_nat,X5: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat] :
( ( ( produc7726050920695093128at_nat @ I3 @ X5 )
= bot_bo228742789529271731at_nat )
= ( ! [X4: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ X4 @ I3 )
=> ( ( X5 @ X4 )
= bot_bo3083307316010499117at_nat ) ) ) ) ).
% Sigma_empty_iff
thf(fact_1846_Sigma__empty__iff,axiom,
! [I3: set_nat,X5: nat > set_nat] :
( ( ( produc457027306803732586at_nat @ I3 @ X5 )
= bot_bo2099793752762293965at_nat )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ I3 )
=> ( ( X5 @ X4 )
= bot_bot_set_nat ) ) ) ) ).
% Sigma_empty_iff
thf(fact_1847_subset__emptyI,axiom,
! [A4: set_int] :
( ! [X: int] :
~ ( member_int @ X @ A4 )
=> ( ord_less_eq_set_int @ A4 @ bot_bot_set_int ) ) ).
% subset_emptyI
thf(fact_1848_subset__emptyI,axiom,
! [A4: set_Product_unit] :
( ! [X: product_unit] :
~ ( member_Product_unit @ X @ A4 )
=> ( ord_le3507040750410214029t_unit @ A4 @ bot_bo3957492148770167129t_unit ) ) ).
% subset_emptyI
thf(fact_1849_subset__emptyI,axiom,
! [A4: set_Pr4329608150637261639at_nat] :
( ! [X: produc3843707927480180839at_nat] :
~ ( member8757157785044589968at_nat @ X @ A4 )
=> ( ord_le1268244103169919719at_nat @ A4 @ bot_bo228742789529271731at_nat ) ) ).
% subset_emptyI
thf(fact_1850_subset__emptyI,axiom,
! [A4: set_Pr1261947904930325089at_nat] :
( ! [X: product_prod_nat_nat] :
~ ( member8440522571783428010at_nat @ X @ A4 )
=> ( ord_le3146513528884898305at_nat @ A4 @ bot_bo2099793752762293965at_nat ) ) ).
% subset_emptyI
thf(fact_1851_subset__emptyI,axiom,
! [A4: set_o] :
( ! [X: $o] :
~ ( member_o @ X @ A4 )
=> ( ord_less_eq_set_o @ A4 @ bot_bot_set_o ) ) ).
% subset_emptyI
thf(fact_1852_subset__emptyI,axiom,
! [A4: set_nat] :
( ! [X: nat] :
~ ( member_nat @ X @ A4 )
=> ( ord_less_eq_set_nat @ A4 @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_1853_less__by__empty,axiom,
! [A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
( ( A4 = bot_bo228742789529271731at_nat )
=> ( ord_le1268244103169919719at_nat @ A4 @ B5 ) ) ).
% less_by_empty
thf(fact_1854_less__by__empty,axiom,
! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
( ( A4 = bot_bo2099793752762293965at_nat )
=> ( ord_le3146513528884898305at_nat @ A4 @ B5 ) ) ).
% less_by_empty
thf(fact_1855_times__eq__iff,axiom,
! [A4: set_Product_unit,B5: set_Product_unit,C3: set_Product_unit,D: set_Product_unit] :
( ( ( produc6121607364029222536t_unit @ A4
@ ^ [Uu: product_unit] : B5 )
= ( produc6121607364029222536t_unit @ C3
@ ^ [Uu: product_unit] : D ) )
= ( ( ( A4 = C3 )
& ( B5 = D ) )
| ( ( ( A4 = bot_bo3957492148770167129t_unit )
| ( B5 = bot_bo3957492148770167129t_unit ) )
& ( ( C3 = bot_bo3957492148770167129t_unit )
| ( D = bot_bo3957492148770167129t_unit ) ) ) ) ) ).
% times_eq_iff
thf(fact_1856_times__eq__iff,axiom,
! [A4: set_Product_unit,B5: set_o,C3: set_Product_unit,D: set_o] :
( ( ( produc4996980317674386579unit_o @ A4
@ ^ [Uu: product_unit] : B5 )
= ( produc4996980317674386579unit_o @ C3
@ ^ [Uu: product_unit] : D ) )
= ( ( ( A4 = C3 )
& ( B5 = D ) )
| ( ( ( A4 = bot_bo3957492148770167129t_unit )
| ( B5 = bot_bot_set_o ) )
& ( ( C3 = bot_bo3957492148770167129t_unit )
| ( D = bot_bot_set_o ) ) ) ) ) ).
% times_eq_iff
thf(fact_1857_times__eq__iff,axiom,
! [A4: set_Product_unit,B5: set_nat,C3: set_Product_unit,D: set_nat] :
( ( ( produc6875377002296012693it_nat @ A4
@ ^ [Uu: product_unit] : B5 )
= ( produc6875377002296012693it_nat @ C3
@ ^ [Uu: product_unit] : D ) )
= ( ( ( A4 = C3 )
& ( B5 = D ) )
| ( ( ( A4 = bot_bo3957492148770167129t_unit )
| ( B5 = bot_bot_set_nat ) )
& ( ( C3 = bot_bo3957492148770167129t_unit )
| ( D = bot_bot_set_nat ) ) ) ) ) ).
% times_eq_iff
thf(fact_1858_times__eq__iff,axiom,
! [A4: set_o,B5: set_Product_unit,C3: set_o,D: set_Product_unit] :
( ( ( produc411009820408249917t_unit @ A4
@ ^ [Uu: $o] : B5 )
= ( produc411009820408249917t_unit @ C3
@ ^ [Uu: $o] : D ) )
= ( ( ( A4 = C3 )
& ( B5 = D ) )
| ( ( ( A4 = bot_bot_set_o )
| ( B5 = bot_bo3957492148770167129t_unit ) )
& ( ( C3 = bot_bot_set_o )
| ( D = bot_bo3957492148770167129t_unit ) ) ) ) ) ).
% times_eq_iff
thf(fact_1859_times__eq__iff,axiom,
! [A4: set_o,B5: set_o,C3: set_o,D: set_o] :
( ( ( product_Sigma_o_o @ A4
@ ^ [Uu: $o] : B5 )
= ( product_Sigma_o_o @ C3
@ ^ [Uu: $o] : D ) )
= ( ( ( A4 = C3 )
& ( B5 = D ) )
| ( ( ( A4 = bot_bot_set_o )
| ( B5 = bot_bot_set_o ) )
& ( ( C3 = bot_bot_set_o )
| ( D = bot_bot_set_o ) ) ) ) ) ).
% times_eq_iff
thf(fact_1860_times__eq__iff,axiom,
! [A4: set_o,B5: set_nat,C3: set_o,D: set_nat] :
( ( ( product_Sigma_o_nat @ A4
@ ^ [Uu: $o] : B5 )
= ( product_Sigma_o_nat @ C3
@ ^ [Uu: $o] : D ) )
= ( ( ( A4 = C3 )
& ( B5 = D ) )
| ( ( ( A4 = bot_bot_set_o )
| ( B5 = bot_bot_set_nat ) )
& ( ( C3 = bot_bot_set_o )
| ( D = bot_bot_set_nat ) ) ) ) ) ).
% times_eq_iff
thf(fact_1861_times__eq__iff,axiom,
! [A4: set_nat,B5: set_Product_unit,C3: set_nat,D: set_Product_unit] :
( ( ( produc5506539147108004339t_unit @ A4
@ ^ [Uu: nat] : B5 )
= ( produc5506539147108004339t_unit @ C3
@ ^ [Uu: nat] : D ) )
= ( ( ( A4 = C3 )
& ( B5 = D ) )
| ( ( ( A4 = bot_bot_set_nat )
| ( B5 = bot_bo3957492148770167129t_unit ) )
& ( ( C3 = bot_bot_set_nat )
| ( D = bot_bo3957492148770167129t_unit ) ) ) ) ) ).
% times_eq_iff
thf(fact_1862_times__eq__iff,axiom,
! [A4: set_nat,B5: set_o,C3: set_nat,D: set_o] :
( ( ( product_Sigma_nat_o @ A4
@ ^ [Uu: nat] : B5 )
= ( product_Sigma_nat_o @ C3
@ ^ [Uu: nat] : D ) )
= ( ( ( A4 = C3 )
& ( B5 = D ) )
| ( ( ( A4 = bot_bot_set_nat )
| ( B5 = bot_bot_set_o ) )
& ( ( C3 = bot_bot_set_nat )
| ( D = bot_bot_set_o ) ) ) ) ) ).
% times_eq_iff
thf(fact_1863_times__eq__iff,axiom,
! [A4: set_nat,B5: set_nat,C3: set_nat,D: set_nat] :
( ( ( produc457027306803732586at_nat @ A4
@ ^ [Uu: nat] : B5 )
= ( produc457027306803732586at_nat @ C3
@ ^ [Uu: nat] : D ) )
= ( ( ( A4 = C3 )
& ( B5 = D ) )
| ( ( ( A4 = bot_bot_set_nat )
| ( B5 = bot_bot_set_nat ) )
& ( ( C3 = bot_bot_set_nat )
| ( D = bot_bot_set_nat ) ) ) ) ) ).
% times_eq_iff
thf(fact_1864_times__eq__iff,axiom,
! [A4: set_Product_unit,B5: set_Pr1261947904930325089at_nat,C3: set_Product_unit,D: set_Pr1261947904930325089at_nat] :
( ( ( produc3904733337175673596at_nat @ A4
@ ^ [Uu: product_unit] : B5 )
= ( produc3904733337175673596at_nat @ C3
@ ^ [Uu: product_unit] : D ) )
= ( ( ( A4 = C3 )
& ( B5 = D ) )
| ( ( ( A4 = bot_bo3957492148770167129t_unit )
| ( B5 = bot_bo2099793752762293965at_nat ) )
& ( ( C3 = bot_bo3957492148770167129t_unit )
| ( D = bot_bo2099793752762293965at_nat ) ) ) ) ) ).
% times_eq_iff
thf(fact_1865_times__subset__iff,axiom,
! [A4: set_Product_unit,C3: set_Product_unit,B5: set_Product_unit,D: set_Product_unit] :
( ( ord_le930853871177014503t_unit
@ ( produc6121607364029222536t_unit @ A4
@ ^ [Uu: product_unit] : C3 )
@ ( produc6121607364029222536t_unit @ B5
@ ^ [Uu: product_unit] : D ) )
= ( ( A4 = bot_bo3957492148770167129t_unit )
| ( C3 = bot_bo3957492148770167129t_unit )
| ( ( ord_le3507040750410214029t_unit @ A4 @ B5 )
& ( ord_le3507040750410214029t_unit @ C3 @ D ) ) ) ) ).
% times_subset_iff
thf(fact_1866_times__subset__iff,axiom,
! [A4: set_Product_unit,C3: set_o,B5: set_Product_unit,D: set_o] :
( ( ord_le8534741403851954546unit_o
@ ( produc4996980317674386579unit_o @ A4
@ ^ [Uu: product_unit] : C3 )
@ ( produc4996980317674386579unit_o @ B5
@ ^ [Uu: product_unit] : D ) )
= ( ( A4 = bot_bo3957492148770167129t_unit )
| ( C3 = bot_bot_set_o )
| ( ( ord_le3507040750410214029t_unit @ A4 @ B5 )
& ( ord_less_eq_set_o @ C3 @ D ) ) ) ) ).
% times_subset_iff
thf(fact_1867_times__subset__iff,axiom,
! [A4: set_Product_unit,C3: set_nat,B5: set_Product_unit,D: set_nat] :
( ( ord_le8121013740822022818it_nat
@ ( produc6875377002296012693it_nat @ A4
@ ^ [Uu: product_unit] : C3 )
@ ( produc6875377002296012693it_nat @ B5
@ ^ [Uu: product_unit] : D ) )
= ( ( A4 = bot_bo3957492148770167129t_unit )
| ( C3 = bot_bot_set_nat )
| ( ( ord_le3507040750410214029t_unit @ A4 @ B5 )
& ( ord_less_eq_set_nat @ C3 @ D ) ) ) ) ).
% times_subset_iff
thf(fact_1868_times__subset__iff,axiom,
! [A4: set_o,C3: set_Product_unit,B5: set_o,D: set_Product_unit] :
( ( ord_le518195775913581980t_unit
@ ( produc411009820408249917t_unit @ A4
@ ^ [Uu: $o] : C3 )
@ ( produc411009820408249917t_unit @ B5
@ ^ [Uu: $o] : D ) )
= ( ( A4 = bot_bot_set_o )
| ( C3 = bot_bo3957492148770167129t_unit )
| ( ( ord_less_eq_set_o @ A4 @ B5 )
& ( ord_le3507040750410214029t_unit @ C3 @ D ) ) ) ) ).
% times_subset_iff
thf(fact_1869_times__subset__iff,axiom,
! [A4: set_o,C3: set_o,B5: set_o,D: set_o] :
( ( ord_le192195778054999463od_o_o
@ ( product_Sigma_o_o @ A4
@ ^ [Uu: $o] : C3 )
@ ( product_Sigma_o_o @ B5
@ ^ [Uu: $o] : D ) )
= ( ( A4 = bot_bot_set_o )
| ( C3 = bot_bot_set_o )
| ( ( ord_less_eq_set_o @ A4 @ B5 )
& ( ord_less_eq_set_o @ C3 @ D ) ) ) ) ).
% times_subset_iff
thf(fact_1870_times__subset__iff,axiom,
! [A4: set_o,C3: set_nat,B5: set_o,D: set_nat] :
( ( ord_le6472378536984071597_o_nat
@ ( product_Sigma_o_nat @ A4
@ ^ [Uu: $o] : C3 )
@ ( product_Sigma_o_nat @ B5
@ ^ [Uu: $o] : D ) )
= ( ( A4 = bot_bot_set_o )
| ( C3 = bot_bot_set_nat )
| ( ( ord_less_eq_set_o @ A4 @ B5 )
& ( ord_less_eq_set_nat @ C3 @ D ) ) ) ) ).
% times_subset_iff
thf(fact_1871_times__subset__iff,axiom,
! [A4: set_nat,C3: set_Product_unit,B5: set_nat,D: set_Product_unit] :
( ( ord_le1468274181084648008t_unit
@ ( produc5506539147108004339t_unit @ A4
@ ^ [Uu: nat] : C3 )
@ ( produc5506539147108004339t_unit @ B5
@ ^ [Uu: nat] : D ) )
= ( ( A4 = bot_bot_set_nat )
| ( C3 = bot_bo3957492148770167129t_unit )
| ( ( ord_less_eq_set_nat @ A4 @ B5 )
& ( ord_le3507040750410214029t_unit @ C3 @ D ) ) ) ) ).
% times_subset_iff
thf(fact_1872_times__subset__iff,axiom,
! [A4: set_nat,C3: set_o,B5: set_nat,D: set_o] :
( ( ord_le7519981659162375251_nat_o
@ ( product_Sigma_nat_o @ A4
@ ^ [Uu: nat] : C3 )
@ ( product_Sigma_nat_o @ B5
@ ^ [Uu: nat] : D ) )
= ( ( A4 = bot_bot_set_nat )
| ( C3 = bot_bot_set_o )
| ( ( ord_less_eq_set_nat @ A4 @ B5 )
& ( ord_less_eq_set_o @ C3 @ D ) ) ) ) ).
% times_subset_iff
thf(fact_1873_times__subset__iff,axiom,
! [A4: set_nat,C3: set_nat,B5: set_nat,D: set_nat] :
( ( ord_le3146513528884898305at_nat
@ ( produc457027306803732586at_nat @ A4
@ ^ [Uu: nat] : C3 )
@ ( produc457027306803732586at_nat @ B5
@ ^ [Uu: nat] : D ) )
= ( ( A4 = bot_bot_set_nat )
| ( C3 = bot_bot_set_nat )
| ( ( ord_less_eq_set_nat @ A4 @ B5 )
& ( ord_less_eq_set_nat @ C3 @ D ) ) ) ) ).
% times_subset_iff
thf(fact_1874_times__subset__iff,axiom,
! [A4: set_Product_unit,C3: set_Pr1261947904930325089at_nat,B5: set_Product_unit,D: set_Pr1261947904930325089at_nat] :
( ( ord_le8681960759543468507at_nat
@ ( produc3904733337175673596at_nat @ A4
@ ^ [Uu: product_unit] : C3 )
@ ( produc3904733337175673596at_nat @ B5
@ ^ [Uu: product_unit] : D ) )
= ( ( A4 = bot_bo3957492148770167129t_unit )
| ( C3 = bot_bo2099793752762293965at_nat )
| ( ( ord_le3507040750410214029t_unit @ A4 @ B5 )
& ( ord_le3146513528884898305at_nat @ C3 @ D ) ) ) ) ).
% times_subset_iff
thf(fact_1875_fst__image__Sigma,axiom,
! [A4: set_o,B5: $o > set_Product_unit] :
( ( image_8742399529945250561unit_o @ produc5219655352205527288t_unit @ ( produc411009820408249917t_unit @ A4 @ B5 ) )
= ( collect_o
@ ^ [X4: $o] :
( ( member_o @ X4 @ A4 )
& ( ( B5 @ X4 )
!= bot_bo3957492148770167129t_unit ) ) ) ) ).
% fst_image_Sigma
thf(fact_1876_fst__image__Sigma,axiom,
! [A4: set_int,B5: int > set_Product_unit] :
( ( image_7716461468295075337it_int @ produc917050878549155996t_unit @ ( produc2622961094310184599t_unit @ A4 @ B5 ) )
= ( collect_int
@ ^ [X4: int] :
( ( member_int @ X4 @ A4 )
& ( ( B5 @ X4 )
!= bot_bo3957492148770167129t_unit ) ) ) ) ).
% fst_image_Sigma
thf(fact_1877_fst__image__Sigma,axiom,
! [A4: set_nat,B5: nat > set_Product_unit] :
( ( image_3884712837920099081it_nat @ produc3800628931346975736t_unit @ ( produc5506539147108004339t_unit @ A4 @ B5 ) )
= ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A4 )
& ( ( B5 @ X4 )
!= bot_bo3957492148770167129t_unit ) ) ) ) ).
% fst_image_Sigma
thf(fact_1878_fst__image__Sigma,axiom,
! [A4: set_o,B5: $o > set_o] :
( ( image_7896445794123959606_o_o_o @ product_fst_o_o @ ( product_Sigma_o_o @ A4 @ B5 ) )
= ( collect_o
@ ^ [X4: $o] :
( ( member_o @ X4 @ A4 )
& ( ( B5 @ X4 )
!= bot_bot_set_o ) ) ) ) ).
% fst_image_Sigma
thf(fact_1879_fst__image__Sigma,axiom,
! [A4: set_int,B5: int > set_o] :
( ( image_90517134139374612_o_int @ product_fst_int_o @ ( product_Sigma_int_o @ A4 @ B5 ) )
= ( collect_int
@ ^ [X4: int] :
( ( member_int @ X4 @ A4 )
& ( ( B5 @ X4 )
!= bot_bot_set_o ) ) ) ) ).
% fst_image_Sigma
thf(fact_1880_fst__image__Sigma,axiom,
! [A4: set_nat,B5: nat > set_o] :
( ( image_8629803608462650132_o_nat @ product_fst_nat_o @ ( product_Sigma_nat_o @ A4 @ B5 ) )
= ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A4 )
& ( ( B5 @ X4 )
!= bot_bot_set_o ) ) ) ) ).
% fst_image_Sigma
thf(fact_1881_fst__image__Sigma,axiom,
! [A4: set_o,B5: $o > set_nat] :
( ( image_8105710389281276538_nat_o @ product_fst_o_nat @ ( product_Sigma_o_nat @ A4 @ B5 ) )
= ( collect_o
@ ^ [X4: $o] :
( ( member_o @ X4 @ A4 )
& ( ( B5 @ X4 )
!= bot_bot_set_nat ) ) ) ) ).
% fst_image_Sigma
thf(fact_1882_fst__image__Sigma,axiom,
! [A4: set_int,B5: int > set_nat] :
( ( image_2855239504505064772at_int @ product_fst_int_nat @ ( produc1456381018704787142nt_nat @ A4 @ B5 ) )
= ( collect_int
@ ^ [X4: int] :
( ( member_int @ X4 @ A4 )
& ( ( B5 @ X4 )
!= bot_bot_set_nat ) ) ) ) ).
% fst_image_Sigma
thf(fact_1883_fst__image__Sigma,axiom,
! [A4: set_nat,B5: nat > set_nat] :
( ( image_2486076414777270412at_nat @ product_fst_nat_nat @ ( produc457027306803732586at_nat @ A4 @ B5 ) )
= ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A4 )
& ( ( B5 @ X4 )
!= bot_bot_set_nat ) ) ) ) ).
% fst_image_Sigma
thf(fact_1884_fst__image__Sigma,axiom,
! [A4: set_ref_a,B5: ref_a > set_he8644174446132254637t_unit] :
( ( image_7136766188234240905_ref_a @ produc473321116828791851t_unit @ ( produc3946373962484743846t_unit @ A4 @ B5 ) )
= ( collect_ref_a
@ ^ [X4: ref_a] :
( ( member_ref_a @ X4 @ A4 )
& ( ( B5 @ X4 )
!= bot_bo3686245532977126937t_unit ) ) ) ) ).
% fst_image_Sigma
thf(fact_1885_rp__inv__image__rp,axiom,
! [P2: produc1319942482725812455at_nat,F: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat] :
( ( fun_re2478310338295953701at_nat @ P2 )
=> ( fun_re2478310338295953701at_nat @ ( fun_rp8445993449185644698at_nat @ P2 @ F ) ) ) ).
% rp_inv_image_rp
thf(fact_1886_the__elem__image__unique,axiom,
! [A4: set_int,F: int > nat,X2: int] :
( ( A4 != bot_bot_set_int )
=> ( ! [Y: int] :
( ( member_int @ Y @ A4 )
=> ( ( F @ Y )
= ( F @ X2 ) ) )
=> ( ( the_elem_nat @ ( image_int_nat @ F @ A4 ) )
= ( F @ X2 ) ) ) ) ).
% the_elem_image_unique
thf(fact_1887_the__elem__image__unique,axiom,
! [A4: set_int,F: int > int,X2: int] :
( ( A4 != bot_bot_set_int )
=> ( ! [Y: int] :
( ( member_int @ Y @ A4 )
=> ( ( F @ Y )
= ( F @ X2 ) ) )
=> ( ( the_elem_int @ ( image_int_int @ F @ A4 ) )
= ( F @ X2 ) ) ) ) ).
% the_elem_image_unique
thf(fact_1888_the__elem__image__unique,axiom,
! [A4: set_nat,F: nat > rat,X2: nat] :
( ( A4 != bot_bot_set_nat )
=> ( ! [Y: nat] :
( ( member_nat @ Y @ A4 )
=> ( ( F @ Y )
= ( F @ X2 ) ) )
=> ( ( the_elem_rat @ ( image_nat_rat @ F @ A4 ) )
= ( F @ X2 ) ) ) ) ).
% the_elem_image_unique
thf(fact_1889_the__elem__image__unique,axiom,
! [A4: set_nat,F: nat > nat,X2: nat] :
( ( A4 != bot_bot_set_nat )
=> ( ! [Y: nat] :
( ( member_nat @ Y @ A4 )
=> ( ( F @ Y )
= ( F @ X2 ) ) )
=> ( ( the_elem_nat @ ( image_nat_nat @ F @ A4 ) )
= ( F @ X2 ) ) ) ) ).
% the_elem_image_unique
thf(fact_1890_the__elem__image__unique,axiom,
! [A4: set_nat,F: nat > int,X2: nat] :
( ( A4 != bot_bot_set_nat )
=> ( ! [Y: nat] :
( ( member_nat @ Y @ A4 )
=> ( ( F @ Y )
= ( F @ X2 ) ) )
=> ( ( the_elem_int @ ( image_nat_int @ F @ A4 ) )
= ( F @ X2 ) ) ) ) ).
% the_elem_image_unique
thf(fact_1891_snd__image__Sigma,axiom,
! [A4: set_ref_a,B5: ref_a > set_he8644174446132254637t_unit] :
( ( image_17660834435998945t_unit @ produc1126150082675451753t_unit @ ( produc3946373962484743846t_unit @ A4 @ B5 ) )
= ( comple6989579708249578522t_unit @ ( image_5570573867351727369t_unit @ B5 @ A4 ) ) ) ).
% snd_image_Sigma
thf(fact_1892_snd__image__Sigma,axiom,
! [A4: set_nat,B5: nat > set_nat] :
( ( image_2486076414777270412at_nat @ product_snd_nat_nat @ ( produc457027306803732586at_nat @ A4 @ B5 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ A4 ) ) ) ).
% snd_image_Sigma
thf(fact_1893_range__constant,axiom,
! [X2: product_unit] :
( ( image_4337914935673063477t_unit
@ ^ [Uu: char] : X2
@ top_top_set_char )
= ( insert_Product_unit @ X2 @ bot_bo3957492148770167129t_unit ) ) ).
% range_constant
thf(fact_1894_range__constant,axiom,
! [X2: $o] :
( ( image_char_o
@ ^ [Uu: char] : X2
@ top_top_set_char )
= ( insert_o @ X2 @ bot_bot_set_o ) ) ).
% range_constant
thf(fact_1895_range__constant,axiom,
! [X2: nat] :
( ( image_char_nat
@ ^ [Uu: char] : X2
@ top_top_set_char )
= ( insert_nat @ X2 @ bot_bot_set_nat ) ) ).
% range_constant
thf(fact_1896_range__constant,axiom,
! [X2: product_unit] :
( ( image_2327579789247107822t_unit
@ ^ [Uu: rat] : X2
@ top_top_set_rat )
= ( insert_Product_unit @ X2 @ bot_bo3957492148770167129t_unit ) ) ).
% range_constant
thf(fact_1897_range__constant,axiom,
! [X2: $o] :
( ( image_rat_o
@ ^ [Uu: rat] : X2
@ top_top_set_rat )
= ( insert_o @ X2 @ bot_bot_set_o ) ) ).
% range_constant
thf(fact_1898_range__constant,axiom,
! [X2: nat] :
( ( image_rat_nat
@ ^ [Uu: rat] : X2
@ top_top_set_rat )
= ( insert_nat @ X2 @ bot_bot_set_nat ) ) ).
% range_constant
thf(fact_1899_range__constant,axiom,
! [X2: rat] :
( ( image_nat_rat
@ ^ [Uu: nat] : X2
@ top_top_set_nat )
= ( insert_rat @ X2 @ bot_bot_set_rat ) ) ).
% range_constant
thf(fact_1900_range__constant,axiom,
! [X2: int] :
( ( image_nat_int
@ ^ [Uu: nat] : X2
@ top_top_set_nat )
= ( insert_int @ X2 @ bot_bot_set_int ) ) ).
% range_constant
thf(fact_1901_range__constant,axiom,
! [X2: product_unit] :
( ( image_8730104196221521654t_unit
@ ^ [Uu: nat] : X2
@ top_top_set_nat )
= ( insert_Product_unit @ X2 @ bot_bo3957492148770167129t_unit ) ) ).
% range_constant
thf(fact_1902_range__constant,axiom,
! [X2: $o] :
( ( image_nat_o
@ ^ [Uu: nat] : X2
@ top_top_set_nat )
= ( insert_o @ X2 @ bot_bot_set_o ) ) ).
% range_constant
thf(fact_1903_INT__constant,axiom,
! [A4: set_Product_unit,C: set_char] :
( ( ( A4 = bot_bo3957492148770167129t_unit )
=> ( ( comple7965528767983648192t_char
@ ( image_5325685772115642575t_char
@ ^ [Y5: product_unit] : C
@ A4 ) )
= top_top_set_char ) )
& ( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ( comple7965528767983648192t_char
@ ( image_5325685772115642575t_char
@ ^ [Y5: product_unit] : C
@ A4 ) )
= C ) ) ) ).
% INT_constant
thf(fact_1904_INT__constant,axiom,
! [A4: set_o,C: set_char] :
( ( ( A4 = bot_bot_set_o )
=> ( ( comple7965528767983648192t_char
@ ( image_o_set_char
@ ^ [Y5: $o] : C
@ A4 ) )
= top_top_set_char ) )
& ( ( A4 != bot_bot_set_o )
=> ( ( comple7965528767983648192t_char
@ ( image_o_set_char
@ ^ [Y5: $o] : C
@ A4 ) )
= C ) ) ) ).
% INT_constant
thf(fact_1905_INT__constant,axiom,
! [A4: set_nat,C: set_char] :
( ( ( A4 = bot_bot_set_nat )
=> ( ( comple7965528767983648192t_char
@ ( image_nat_set_char
@ ^ [Y5: nat] : C
@ A4 ) )
= top_top_set_char ) )
& ( ( A4 != bot_bot_set_nat )
=> ( ( comple7965528767983648192t_char
@ ( image_nat_set_char
@ ^ [Y5: nat] : C
@ A4 ) )
= C ) ) ) ).
% INT_constant
thf(fact_1906_INT__constant,axiom,
! [A4: set_Product_unit,C: set_rat] :
( ( ( A4 = bot_bo3957492148770167129t_unit )
=> ( ( comple4298007329820168263et_rat
@ ( image_4665621433434589782et_rat
@ ^ [Y5: product_unit] : C
@ A4 ) )
= top_top_set_rat ) )
& ( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ( comple4298007329820168263et_rat
@ ( image_4665621433434589782et_rat
@ ^ [Y5: product_unit] : C
@ A4 ) )
= C ) ) ) ).
% INT_constant
thf(fact_1907_INT__constant,axiom,
! [A4: set_o,C: set_rat] :
( ( ( A4 = bot_bot_set_o )
=> ( ( comple4298007329820168263et_rat
@ ( image_o_set_rat
@ ^ [Y5: $o] : C
@ A4 ) )
= top_top_set_rat ) )
& ( ( A4 != bot_bot_set_o )
=> ( ( comple4298007329820168263et_rat
@ ( image_o_set_rat
@ ^ [Y5: $o] : C
@ A4 ) )
= C ) ) ) ).
% INT_constant
thf(fact_1908_INT__constant,axiom,
! [A4: set_nat,C: set_rat] :
( ( ( A4 = bot_bot_set_nat )
=> ( ( comple4298007329820168263et_rat
@ ( image_nat_set_rat
@ ^ [Y5: nat] : C
@ A4 ) )
= top_top_set_rat ) )
& ( ( A4 != bot_bot_set_nat )
=> ( ( comple4298007329820168263et_rat
@ ( image_nat_set_rat
@ ^ [Y5: nat] : C
@ A4 ) )
= C ) ) ) ).
% INT_constant
thf(fact_1909_INT__constant,axiom,
! [A4: set_Product_unit,C: set_int] :
( ( ( A4 = bot_bo3957492148770167129t_unit )
=> ( ( comple3628384868704368283et_int
@ ( image_3995998972318789802et_int
@ ^ [Y5: product_unit] : C
@ A4 ) )
= top_top_set_int ) )
& ( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ( comple3628384868704368283et_int
@ ( image_3995998972318789802et_int
@ ^ [Y5: product_unit] : C
@ A4 ) )
= C ) ) ) ).
% INT_constant
thf(fact_1910_INT__constant,axiom,
! [A4: set_o,C: set_int] :
( ( ( A4 = bot_bot_set_o )
=> ( ( comple3628384868704368283et_int
@ ( image_o_set_int
@ ^ [Y5: $o] : C
@ A4 ) )
= top_top_set_int ) )
& ( ( A4 != bot_bot_set_o )
=> ( ( comple3628384868704368283et_int
@ ( image_o_set_int
@ ^ [Y5: $o] : C
@ A4 ) )
= C ) ) ) ).
% INT_constant
thf(fact_1911_INT__constant,axiom,
! [A4: set_nat,C: set_int] :
( ( ( A4 = bot_bot_set_nat )
=> ( ( comple3628384868704368283et_int
@ ( image_nat_set_int
@ ^ [Y5: nat] : C
@ A4 ) )
= top_top_set_int ) )
& ( ( A4 != bot_bot_set_nat )
=> ( ( comple3628384868704368283et_int
@ ( image_nat_set_int
@ ^ [Y5: nat] : C
@ A4 ) )
= C ) ) ) ).
% INT_constant
thf(fact_1912_INT__constant,axiom,
! [A4: set_Product_unit,C: set_nat] :
( ( ( A4 = bot_bo3957492148770167129t_unit )
=> ( ( comple7806235888213564991et_nat
@ ( image_8173849991827986510et_nat
@ ^ [Y5: product_unit] : C
@ A4 ) )
= top_top_set_nat ) )
& ( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ( comple7806235888213564991et_nat
@ ( image_8173849991827986510et_nat
@ ^ [Y5: product_unit] : C
@ A4 ) )
= C ) ) ) ).
% INT_constant
thf(fact_1913_neg__fun__distr1,axiom,
! [R3: int > code_integer > $o,R7: code_integer > code_integer > $o,S5: num > num > $o,S6: num > num > $o] :
( ( left_u6795595883297946906nteger @ R3 )
=> ( ( right_7281159819619925802nteger @ R3 )
=> ( ( right_5866438497372881682nteger @ R7 )
=> ( ( left_t2884206969071330nteger @ R7 )
=> ( ord_le6931201860925936491_num_o @ ( bNF_re6718328864250387230um_num @ ( relcom8320293257703906841nteger @ R3 @ R7 ) @ ( relcompp_num_num_num @ S5 @ S6 ) ) @ ( relcom1848932251921554756er_num @ ( bNF_re6718328864250387230um_num @ R3 @ S5 ) @ ( bNF_re40212662316476751um_num @ R7 @ S6 ) ) ) ) ) ) ) ).
% neg_fun_distr1
thf(fact_1914_neg__fun__distr1,axiom,
! [R3: int > code_integer > $o,R7: code_integer > int > $o,S5: num > num > $o,S6: num > num > $o] :
( ( left_u6795595883297946906nteger @ R3 )
=> ( ( right_7281159819619925802nteger @ R3 )
=> ( ( right_7836862732553076961er_int @ R7 )
=> ( ( left_t1735413524555719857er_int @ R7 )
=> ( ord_le4541714130423454556_num_o @ ( bNF_re7626690874201225453um_num @ ( relcom328798225758222312er_int @ R3 @ R7 ) @ ( relcompp_num_num_num @ S5 @ S6 ) ) @ ( relcom7779292856395800339nt_num @ ( bNF_re6718328864250387230um_num @ R3 @ S5 ) @ ( bNF_re8313794964846168862um_num @ R7 @ S6 ) ) ) ) ) ) ) ).
% neg_fun_distr1
thf(fact_1915_neg__fun__distr1,axiom,
! [R3: int > int > $o,R7: int > int > $o,S5: num > num > $o,S6: num > num > $o] :
( ( left_unique_int_int @ R3 )
=> ( ( right_total_int_int @ R3 )
=> ( ( right_unique_int_int @ R7 )
=> ( ( left_total_int_int @ R7 )
=> ( ord_le4541714130423454556_num_o @ ( bNF_re7626690874201225453um_num @ ( relcompp_int_int_int @ R3 @ R7 ) @ ( relcompp_num_num_num @ S5 @ S6 ) ) @ ( relcom9025577121137075810nt_num @ ( bNF_re7626690874201225453um_num @ R3 @ S5 ) @ ( bNF_re7626690874201225453um_num @ R7 @ S6 ) ) ) ) ) ) ) ).
% neg_fun_distr1
thf(fact_1916_neg__fun__distr1,axiom,
! [R3: int > int > $o,R7: int > code_integer > $o,S5: num > num > $o,S6: num > num > $o] :
( ( left_unique_int_int @ R3 )
=> ( ( right_total_int_int @ R3 )
=> ( ( right_8475562212078014689nteger @ R7 )
=> ( ( left_t2374113004080657585nteger @ R7 )
=> ( ord_le6931201860925936491_num_o @ ( bNF_re6718328864250387230um_num @ ( relcom967497705283160040nteger @ R3 @ R7 ) @ ( relcompp_num_num_num @ S5 @ S6 ) ) @ ( relcom1533066287727470739er_num @ ( bNF_re7626690874201225453um_num @ R3 @ S5 ) @ ( bNF_re6718328864250387230um_num @ R7 @ S6 ) ) ) ) ) ) ) ).
% neg_fun_distr1
thf(fact_1917_neg__fun__distr1,axiom,
! [R3: int > int > $o,R7: int > code_integer > $o,S5: int > int > $o,S6: int > code_integer > $o] :
( ( left_unique_int_int @ R3 )
=> ( ( right_total_int_int @ R3 )
=> ( ( right_8475562212078014689nteger @ R7 )
=> ( ( left_t2374113004080657585nteger @ R7 )
=> ( ord_le7522433683449569238eger_o @ ( bNF_re3379532845092657523nteger @ ( relcom967497705283160040nteger @ R3 @ R7 ) @ ( relcom967497705283160040nteger @ S5 @ S6 ) ) @ ( relcom7212586982944386682nteger @ ( bNF_re4712519889275205905nt_int @ R3 @ S5 ) @ ( bNF_re3379532845092657523nteger @ R7 @ S6 ) ) ) ) ) ) ) ).
% neg_fun_distr1
thf(fact_1918_neg__fun__distr1,axiom,
! [R3: product_prod_nat_nat > int > $o,R7: int > int > $o,S5: nat > nat > $o,S6: nat > nat > $o] :
( ( left_u4005249686606446046at_int @ R3 )
=> ( ( right_1157779049177982926at_int @ R3 )
=> ( ( right_unique_int_int @ R7 )
=> ( ( left_total_int_int @ R7 )
=> ( ord_le1714508135640884727_nat_o @ ( bNF_re4555766996558763186at_nat @ ( relcom2949942408940504938nt_int @ R3 @ R7 ) @ ( relcompp_nat_nat_nat @ S5 @ S6 ) ) @ ( relcom4272412204205018633nt_nat @ ( bNF_re4555766996558763186at_nat @ R3 @ S5 ) @ ( bNF_re3715656647883201625at_nat @ R7 @ S6 ) ) ) ) ) ) ) ).
% neg_fun_distr1
thf(fact_1919_neg__fun__distr1,axiom,
! [R3: product_prod_nat_nat > product_prod_nat_nat > $o,R7: product_prod_nat_nat > int > $o,S5: nat > nat > $o,S6: nat > nat > $o] :
( ( left_u7296345956935681743at_nat @ R3 )
=> ( ( right_4094035864313884895at_nat @ R3 )
=> ( ( right_8621877368619751767at_int @ R7 )
=> ( ( left_t1861001511420785031at_int @ R7 )
=> ( ord_le1714508135640884727_nat_o @ ( bNF_re4555766996558763186at_nat @ ( relcom8911704206367525917at_int @ R3 @ R7 ) @ ( relcompp_nat_nat_nat @ S5 @ S6 ) ) @ ( relcom531901107642208332nt_nat @ ( bNF_re8246922863344978751at_nat @ R3 @ S5 ) @ ( bNF_re4555766996558763186at_nat @ R7 @ S6 ) ) ) ) ) ) ) ).
% neg_fun_distr1
thf(fact_1920_neg__fun__distr1,axiom,
! [R3: product_prod_nat_nat > product_prod_nat_nat > $o,R7: product_prod_nat_nat > int > $o,S5: int > int > $o,S6: int > code_integer > $o] :
( ( left_u7296345956935681743at_nat @ R3 )
=> ( ( right_4094035864313884895at_nat @ R3 )
=> ( ( right_8621877368619751767at_int @ R7 )
=> ( ( left_t1861001511420785031at_int @ R7 )
=> ( ord_le6141271677531915726eger_o @ ( bNF_re6202762543029211035nteger @ ( relcom8911704206367525917at_int @ R3 @ R7 ) @ ( relcom967497705283160040nteger @ S5 @ S6 ) ) @ ( relcom4574826104943343961nteger @ ( bNF_re20414067882207223nt_int @ R3 @ S5 ) @ ( bNF_re6202762543029211035nteger @ R7 @ S6 ) ) ) ) ) ) ) ).
% neg_fun_distr1
thf(fact_1921_neg__fun__distr1,axiom,
! [R3: int > int > $o,R7: int > code_integer > $o,S5: product_prod_nat_nat > product_prod_nat_nat > $o,S6: product_prod_nat_nat > int > $o] :
( ( left_unique_int_int @ R3 )
=> ( ( right_total_int_int @ R3 )
=> ( ( right_8475562212078014689nteger @ R7 )
=> ( ( left_t2374113004080657585nteger @ R7 )
=> ( ord_le8615040745438789472_int_o @ ( bNF_re2828929484557863493at_int @ ( relcom967497705283160040nteger @ R3 @ R7 ) @ ( relcom8911704206367525917at_int @ S5 @ S6 ) ) @ ( relcom2127790875031262553er_int @ ( bNF_re8766404587435608311at_nat @ R3 @ S5 ) @ ( bNF_re2828929484557863493at_int @ R7 @ S6 ) ) ) ) ) ) ) ).
% neg_fun_distr1
thf(fact_1922_neg__fun__distr1,axiom,
! [R3: product_prod_nat_nat > product_prod_nat_nat > $o,R7: product_prod_nat_nat > int > $o,S5: product_prod_nat_nat > product_prod_nat_nat > $o,S6: product_prod_nat_nat > int > $o] :
( ( left_u7296345956935681743at_nat @ R3 )
=> ( ( right_4094035864313884895at_nat @ R3 )
=> ( ( right_8621877368619751767at_int @ R7 )
=> ( ( left_t1861001511420785031at_int @ R7 )
=> ( ord_le2484784447047653514_int_o @ ( bNF_re7400052026677387805at_int @ ( relcom8911704206367525917at_int @ R3 @ R7 ) @ ( relcom8911704206367525917at_int @ S5 @ S6 ) ) @ ( relcom4982215633375716084nt_int @ ( bNF_re2241393799969408733at_nat @ R3 @ S5 ) @ ( bNF_re7400052026677387805at_int @ R7 @ S6 ) ) ) ) ) ) ) ).
% neg_fun_distr1
thf(fact_1923_neg__fun__distr2,axiom,
! [R7: code_integer > code_integer > $o,S6: num > num > $o,R3: int > code_integer > $o,S5: num > num > $o] :
( ( right_5866438497372881682nteger @ R7 )
=> ( ( left_t2884206969071330nteger @ R7 )
=> ( ( left_unique_num_num @ S6 )
=> ( ( right_total_num_num @ S6 )
=> ( ord_le6931201860925936491_num_o @ ( bNF_re6718328864250387230um_num @ ( relcom8320293257703906841nteger @ R3 @ R7 ) @ ( relcompp_num_num_num @ S5 @ S6 ) ) @ ( relcom1848932251921554756er_num @ ( bNF_re6718328864250387230um_num @ R3 @ S5 ) @ ( bNF_re40212662316476751um_num @ R7 @ S6 ) ) ) ) ) ) ) ).
% neg_fun_distr2
thf(fact_1924_neg__fun__distr2,axiom,
! [R7: code_integer > int > $o,S6: num > num > $o,R3: int > code_integer > $o,S5: num > num > $o] :
( ( right_7836862732553076961er_int @ R7 )
=> ( ( left_t1735413524555719857er_int @ R7 )
=> ( ( left_unique_num_num @ S6 )
=> ( ( right_total_num_num @ S6 )
=> ( ord_le4541714130423454556_num_o @ ( bNF_re7626690874201225453um_num @ ( relcom328798225758222312er_int @ R3 @ R7 ) @ ( relcompp_num_num_num @ S5 @ S6 ) ) @ ( relcom7779292856395800339nt_num @ ( bNF_re6718328864250387230um_num @ R3 @ S5 ) @ ( bNF_re8313794964846168862um_num @ R7 @ S6 ) ) ) ) ) ) ) ).
% neg_fun_distr2
thf(fact_1925_neg__fun__distr2,axiom,
! [R7: int > int > $o,S6: num > num > $o,R3: int > int > $o,S5: num > num > $o] :
( ( right_unique_int_int @ R7 )
=> ( ( left_total_int_int @ R7 )
=> ( ( left_unique_num_num @ S6 )
=> ( ( right_total_num_num @ S6 )
=> ( ord_le4541714130423454556_num_o @ ( bNF_re7626690874201225453um_num @ ( relcompp_int_int_int @ R3 @ R7 ) @ ( relcompp_num_num_num @ S5 @ S6 ) ) @ ( relcom9025577121137075810nt_num @ ( bNF_re7626690874201225453um_num @ R3 @ S5 ) @ ( bNF_re7626690874201225453um_num @ R7 @ S6 ) ) ) ) ) ) ) ).
% neg_fun_distr2
thf(fact_1926_neg__fun__distr2,axiom,
! [R7: int > code_integer > $o,S6: num > num > $o,R3: int > int > $o,S5: num > num > $o] :
( ( right_8475562212078014689nteger @ R7 )
=> ( ( left_t2374113004080657585nteger @ R7 )
=> ( ( left_unique_num_num @ S6 )
=> ( ( right_total_num_num @ S6 )
=> ( ord_le6931201860925936491_num_o @ ( bNF_re6718328864250387230um_num @ ( relcom967497705283160040nteger @ R3 @ R7 ) @ ( relcompp_num_num_num @ S5 @ S6 ) ) @ ( relcom1533066287727470739er_num @ ( bNF_re7626690874201225453um_num @ R3 @ S5 ) @ ( bNF_re6718328864250387230um_num @ R7 @ S6 ) ) ) ) ) ) ) ).
% neg_fun_distr2
thf(fact_1927_neg__fun__distr2,axiom,
! [R7: int > code_integer > $o,S6: int > code_integer > $o,R3: int > int > $o,S5: int > int > $o] :
( ( right_8475562212078014689nteger @ R7 )
=> ( ( left_t2374113004080657585nteger @ R7 )
=> ( ( left_u6795595883297946906nteger @ S6 )
=> ( ( right_7281159819619925802nteger @ S6 )
=> ( ord_le7522433683449569238eger_o @ ( bNF_re3379532845092657523nteger @ ( relcom967497705283160040nteger @ R3 @ R7 ) @ ( relcom967497705283160040nteger @ S5 @ S6 ) ) @ ( relcom7212586982944386682nteger @ ( bNF_re4712519889275205905nt_int @ R3 @ S5 ) @ ( bNF_re3379532845092657523nteger @ R7 @ S6 ) ) ) ) ) ) ) ).
% neg_fun_distr2
thf(fact_1928_neg__fun__distr2,axiom,
! [R7: int > int > $o,S6: nat > nat > $o,R3: product_prod_nat_nat > int > $o,S5: nat > nat > $o] :
( ( right_unique_int_int @ R7 )
=> ( ( left_total_int_int @ R7 )
=> ( ( left_unique_nat_nat @ S6 )
=> ( ( right_total_nat_nat @ S6 )
=> ( ord_le1714508135640884727_nat_o @ ( bNF_re4555766996558763186at_nat @ ( relcom2949942408940504938nt_int @ R3 @ R7 ) @ ( relcompp_nat_nat_nat @ S5 @ S6 ) ) @ ( relcom4272412204205018633nt_nat @ ( bNF_re4555766996558763186at_nat @ R3 @ S5 ) @ ( bNF_re3715656647883201625at_nat @ R7 @ S6 ) ) ) ) ) ) ) ).
% neg_fun_distr2
thf(fact_1929_neg__fun__distr2,axiom,
! [R7: product_prod_nat_nat > int > $o,S6: nat > nat > $o,R3: product_prod_nat_nat > product_prod_nat_nat > $o,S5: nat > nat > $o] :
( ( right_8621877368619751767at_int @ R7 )
=> ( ( left_t1861001511420785031at_int @ R7 )
=> ( ( left_unique_nat_nat @ S6 )
=> ( ( right_total_nat_nat @ S6 )
=> ( ord_le1714508135640884727_nat_o @ ( bNF_re4555766996558763186at_nat @ ( relcom8911704206367525917at_int @ R3 @ R7 ) @ ( relcompp_nat_nat_nat @ S5 @ S6 ) ) @ ( relcom531901107642208332nt_nat @ ( bNF_re8246922863344978751at_nat @ R3 @ S5 ) @ ( bNF_re4555766996558763186at_nat @ R7 @ S6 ) ) ) ) ) ) ) ).
% neg_fun_distr2
thf(fact_1930_neg__fun__distr2,axiom,
! [R7: product_prod_nat_nat > int > $o,S6: int > code_integer > $o,R3: product_prod_nat_nat > product_prod_nat_nat > $o,S5: int > int > $o] :
( ( right_8621877368619751767at_int @ R7 )
=> ( ( left_t1861001511420785031at_int @ R7 )
=> ( ( left_u6795595883297946906nteger @ S6 )
=> ( ( right_7281159819619925802nteger @ S6 )
=> ( ord_le6141271677531915726eger_o @ ( bNF_re6202762543029211035nteger @ ( relcom8911704206367525917at_int @ R3 @ R7 ) @ ( relcom967497705283160040nteger @ S5 @ S6 ) ) @ ( relcom4574826104943343961nteger @ ( bNF_re20414067882207223nt_int @ R3 @ S5 ) @ ( bNF_re6202762543029211035nteger @ R7 @ S6 ) ) ) ) ) ) ) ).
% neg_fun_distr2
thf(fact_1931_neg__fun__distr2,axiom,
! [R7: int > code_integer > $o,S6: product_prod_nat_nat > int > $o,R3: int > int > $o,S5: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ( right_8475562212078014689nteger @ R7 )
=> ( ( left_t2374113004080657585nteger @ R7 )
=> ( ( left_u4005249686606446046at_int @ S6 )
=> ( ( right_1157779049177982926at_int @ S6 )
=> ( ord_le8615040745438789472_int_o @ ( bNF_re2828929484557863493at_int @ ( relcom967497705283160040nteger @ R3 @ R7 ) @ ( relcom8911704206367525917at_int @ S5 @ S6 ) ) @ ( relcom2127790875031262553er_int @ ( bNF_re8766404587435608311at_nat @ R3 @ S5 ) @ ( bNF_re2828929484557863493at_int @ R7 @ S6 ) ) ) ) ) ) ) ).
% neg_fun_distr2
thf(fact_1932_neg__fun__distr2,axiom,
! [R7: product_prod_nat_nat > int > $o,S6: product_prod_nat_nat > int > $o,R3: product_prod_nat_nat > product_prod_nat_nat > $o,S5: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ( right_8621877368619751767at_int @ R7 )
=> ( ( left_t1861001511420785031at_int @ R7 )
=> ( ( left_u4005249686606446046at_int @ S6 )
=> ( ( right_1157779049177982926at_int @ S6 )
=> ( ord_le2484784447047653514_int_o @ ( bNF_re7400052026677387805at_int @ ( relcom8911704206367525917at_int @ R3 @ R7 ) @ ( relcom8911704206367525917at_int @ S5 @ S6 ) ) @ ( relcom4982215633375716084nt_int @ ( bNF_re2241393799969408733at_nat @ R3 @ S5 ) @ ( bNF_re7400052026677387805at_int @ R7 @ S6 ) ) ) ) ) ) ) ).
% neg_fun_distr2
thf(fact_1933_refl__onI,axiom,
! [R: set_Pr5564308138774400199at_nat,A4: set_Pr4329608150637261639at_nat] :
( ( ord_le3091953645718357095at_nat @ R
@ ( produc4441300235185154312at_nat @ A4
@ ^ [Uu: produc3843707927480180839at_nat] : A4 ) )
=> ( ! [X: produc3843707927480180839at_nat] :
( ( member8757157785044589968at_nat @ X @ A4 )
=> ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X @ X ) @ R ) )
=> ( refl_o7446464703481863270at_nat @ A4 @ R ) ) ) ).
% refl_onI
thf(fact_1934_refl__onI,axiom,
! [R: set_Pr8693737435421807431at_nat,A4: set_Pr1261947904930325089at_nat] :
( ( ord_le3000389064537975527at_nat @ R
@ ( produc2761391749766926216at_nat @ A4
@ ^ [Uu: product_prod_nat_nat] : A4 ) )
=> ( ! [X: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X @ A4 )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ X ) @ R ) )
=> ( refl_o7122697186039111552at_nat @ A4 @ R ) ) ) ).
% refl_onI
thf(fact_1935_refl__onI,axiom,
! [R: set_Product_prod_o_o,A4: set_o] :
( ( ord_le192195778054999463od_o_o @ R
@ ( product_Sigma_o_o @ A4
@ ^ [Uu: $o] : A4 ) )
=> ( ! [X: $o] :
( ( member_o @ X @ A4 )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ X ) @ R ) )
=> ( refl_on_o @ A4 @ R ) ) ) ).
% refl_onI
thf(fact_1936_refl__onI,axiom,
! [R: set_Pr7459493094073627847at_nat,A4: set_se5479201105039950375at_nat] :
( ( ord_le5997549366648089703at_nat @ R
@ ( produc6258361011560675592at_nat @ A4
@ ^ [Uu: set_Pr4329608150637261639at_nat] : A4 ) )
=> ( ! [X: set_Pr4329608150637261639at_nat] :
( ( member6273874620674182256at_nat @ X @ A4 )
=> ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ X ) @ R ) )
=> ( refl_o6272981002401706822at_nat @ A4 @ R ) ) ) ).
% refl_onI
thf(fact_1937_refl__onI,axiom,
! [R: set_Pr4329608150637261639at_nat,A4: set_se7855581050983116737at_nat] :
( ( ord_le1268244103169919719at_nat @ R
@ ( produc7726050920695093128at_nat @ A4
@ ^ [Uu: set_Pr1261947904930325089at_nat] : A4 ) )
=> ( ! [X: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ X @ A4 )
=> ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ X ) @ R ) )
=> ( refl_o6955724340487615712at_nat @ A4 @ R ) ) ) ).
% refl_onI
thf(fact_1938_refl__onI,axiom,
! [R: set_Pr958786334691620121nt_int,A4: set_int] :
( ( ord_le2843351958646193337nt_int @ R
@ ( produc1453890548195736866nt_int @ A4
@ ^ [Uu: int] : A4 ) )
=> ( ! [X: int] :
( ( member_int @ X @ A4 )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ X ) @ R ) )
=> ( refl_on_int @ A4 @ R ) ) ) ).
% refl_onI
thf(fact_1939_refl__onI,axiom,
! [R: set_Pr1261947904930325089at_nat,A4: set_nat] :
( ( ord_le3146513528884898305at_nat @ R
@ ( produc457027306803732586at_nat @ A4
@ ^ [Uu: nat] : A4 ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ X ) @ R ) )
=> ( refl_on_nat @ A4 @ R ) ) ) ).
% refl_onI
thf(fact_1940_refl__on__def,axiom,
( refl_o6272981002401706822at_nat
= ( ^ [A6: set_se5479201105039950375at_nat,R5: set_Pr7459493094073627847at_nat] :
( ( ord_le5997549366648089703at_nat @ R5
@ ( produc6258361011560675592at_nat @ A6
@ ^ [Uu: set_Pr4329608150637261639at_nat] : A6 ) )
& ! [X4: set_Pr4329608150637261639at_nat] :
( ( member6273874620674182256at_nat @ X4 @ A6 )
=> ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X4 @ X4 ) @ R5 ) ) ) ) ) ).
% refl_on_def
thf(fact_1941_refl__on__def,axiom,
( refl_o6955724340487615712at_nat
= ( ^ [A6: set_se7855581050983116737at_nat,R5: set_Pr4329608150637261639at_nat] :
( ( ord_le1268244103169919719at_nat @ R5
@ ( produc7726050920695093128at_nat @ A6
@ ^ [Uu: set_Pr1261947904930325089at_nat] : A6 ) )
& ! [X4: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ X4 @ A6 )
=> ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ X4 ) @ R5 ) ) ) ) ) ).
% refl_on_def
thf(fact_1942_refl__on__def,axiom,
( refl_on_int
= ( ^ [A6: set_int,R5: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ R5
@ ( produc1453890548195736866nt_int @ A6
@ ^ [Uu: int] : A6 ) )
& ! [X4: int] :
( ( member_int @ X4 @ A6 )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ X4 ) @ R5 ) ) ) ) ) ).
% refl_on_def
thf(fact_1943_refl__on__def,axiom,
( refl_on_nat
= ( ^ [A6: set_nat,R5: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ R5
@ ( produc457027306803732586at_nat @ A6
@ ^ [Uu: nat] : A6 ) )
& ! [X4: nat] :
( ( member_nat @ X4 @ A6 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ X4 ) @ R5 ) ) ) ) ) ).
% refl_on_def
thf(fact_1944_image__insert,axiom,
! [F: int > int,A: int,B5: set_int] :
( ( image_int_int @ F @ ( insert_int @ A @ B5 ) )
= ( insert_int @ ( F @ A ) @ ( image_int_int @ F @ B5 ) ) ) ).
% image_insert
thf(fact_1945_image__insert,axiom,
! [F: int > nat,A: int,B5: set_int] :
( ( image_int_nat @ F @ ( insert_int @ A @ B5 ) )
= ( insert_nat @ ( F @ A ) @ ( image_int_nat @ F @ B5 ) ) ) ).
% image_insert
thf(fact_1946_image__insert,axiom,
! [F: nat > rat,A: nat,B5: set_nat] :
( ( image_nat_rat @ F @ ( insert_nat @ A @ B5 ) )
= ( insert_rat @ ( F @ A ) @ ( image_nat_rat @ F @ B5 ) ) ) ).
% image_insert
thf(fact_1947_image__insert,axiom,
! [F: nat > int,A: nat,B5: set_nat] :
( ( image_nat_int @ F @ ( insert_nat @ A @ B5 ) )
= ( insert_int @ ( F @ A ) @ ( image_nat_int @ F @ B5 ) ) ) ).
% image_insert
thf(fact_1948_image__insert,axiom,
! [F: nat > nat,A: nat,B5: set_nat] :
( ( image_nat_nat @ F @ ( insert_nat @ A @ B5 ) )
= ( insert_nat @ ( F @ A ) @ ( image_nat_nat @ F @ B5 ) ) ) ).
% image_insert
thf(fact_1949_image__insert,axiom,
! [F: nat > product_unit,A: nat,B5: set_nat] :
( ( image_8730104196221521654t_unit @ F @ ( insert_nat @ A @ B5 ) )
= ( insert_Product_unit @ ( F @ A ) @ ( image_8730104196221521654t_unit @ F @ B5 ) ) ) ).
% image_insert
thf(fact_1950_image__insert,axiom,
! [F: nat > $o,A: nat,B5: set_nat] :
( ( image_nat_o @ F @ ( insert_nat @ A @ B5 ) )
= ( insert_o @ ( F @ A ) @ ( image_nat_o @ F @ B5 ) ) ) ).
% image_insert
thf(fact_1951_image__insert,axiom,
! [F: product_unit > nat,A: product_unit,B5: set_Product_unit] :
( ( image_875570014554754200it_nat @ F @ ( insert_Product_unit @ A @ B5 ) )
= ( insert_nat @ ( F @ A ) @ ( image_875570014554754200it_nat @ F @ B5 ) ) ) ).
% image_insert
thf(fact_1952_image__insert,axiom,
! [F: product_unit > product_unit,A: product_unit,B5: set_Product_unit] :
( ( image_405062704495631173t_unit @ F @ ( insert_Product_unit @ A @ B5 ) )
= ( insert_Product_unit @ ( F @ A ) @ ( image_405062704495631173t_unit @ F @ B5 ) ) ) ).
% image_insert
thf(fact_1953_image__insert,axiom,
! [F: product_unit > $o,A: product_unit,B5: set_Product_unit] :
( ( image_Product_unit_o @ F @ ( insert_Product_unit @ A @ B5 ) )
= ( insert_o @ ( F @ A ) @ ( image_Product_unit_o @ F @ B5 ) ) ) ).
% image_insert
thf(fact_1954_insert__image,axiom,
! [X2: $o,A4: set_o,F: $o > nat] :
( ( member_o @ X2 @ A4 )
=> ( ( insert_nat @ ( F @ X2 ) @ ( image_o_nat @ F @ A4 ) )
= ( image_o_nat @ F @ A4 ) ) ) ).
% insert_image
thf(fact_1955_insert__image,axiom,
! [X2: $o,A4: set_o,F: $o > product_unit] :
( ( member_o @ X2 @ A4 )
=> ( ( insert_Product_unit @ ( F @ X2 ) @ ( image_o_Product_unit @ F @ A4 ) )
= ( image_o_Product_unit @ F @ A4 ) ) ) ).
% insert_image
thf(fact_1956_insert__image,axiom,
! [X2: $o,A4: set_o,F: $o > $o] :
( ( member_o @ X2 @ A4 )
=> ( ( insert_o @ ( F @ X2 ) @ ( image_o_o @ F @ A4 ) )
= ( image_o_o @ F @ A4 ) ) ) ).
% insert_image
thf(fact_1957_insert__image,axiom,
! [X2: nat,A4: set_nat,F: nat > rat] :
( ( member_nat @ X2 @ A4 )
=> ( ( insert_rat @ ( F @ X2 ) @ ( image_nat_rat @ F @ A4 ) )
= ( image_nat_rat @ F @ A4 ) ) ) ).
% insert_image
thf(fact_1958_insert__image,axiom,
! [X2: nat,A4: set_nat,F: nat > int] :
( ( member_nat @ X2 @ A4 )
=> ( ( insert_int @ ( F @ X2 ) @ ( image_nat_int @ F @ A4 ) )
= ( image_nat_int @ F @ A4 ) ) ) ).
% insert_image
thf(fact_1959_insert__image,axiom,
! [X2: nat,A4: set_nat,F: nat > nat] :
( ( member_nat @ X2 @ A4 )
=> ( ( insert_nat @ ( F @ X2 ) @ ( image_nat_nat @ F @ A4 ) )
= ( image_nat_nat @ F @ A4 ) ) ) ).
% insert_image
thf(fact_1960_insert__image,axiom,
! [X2: nat,A4: set_nat,F: nat > product_unit] :
( ( member_nat @ X2 @ A4 )
=> ( ( insert_Product_unit @ ( F @ X2 ) @ ( image_8730104196221521654t_unit @ F @ A4 ) )
= ( image_8730104196221521654t_unit @ F @ A4 ) ) ) ).
% insert_image
thf(fact_1961_insert__image,axiom,
! [X2: nat,A4: set_nat,F: nat > $o] :
( ( member_nat @ X2 @ A4 )
=> ( ( insert_o @ ( F @ X2 ) @ ( image_nat_o @ F @ A4 ) )
= ( image_nat_o @ F @ A4 ) ) ) ).
% insert_image
thf(fact_1962_insert__image,axiom,
! [X2: int,A4: set_int,F: int > int] :
( ( member_int @ X2 @ A4 )
=> ( ( insert_int @ ( F @ X2 ) @ ( image_int_int @ F @ A4 ) )
= ( image_int_int @ F @ A4 ) ) ) ).
% insert_image
thf(fact_1963_insert__image,axiom,
! [X2: int,A4: set_int,F: int > nat] :
( ( member_int @ X2 @ A4 )
=> ( ( insert_nat @ ( F @ X2 ) @ ( image_int_nat @ F @ A4 ) )
= ( image_int_nat @ F @ A4 ) ) ) ).
% insert_image
thf(fact_1964_SUP__identity__eq,axiom,
! [A4: set_nat] :
( ( complete_Sup_Sup_nat
@ ( image_nat_nat
@ ^ [X4: nat] : X4
@ A4 ) )
= ( complete_Sup_Sup_nat @ A4 ) ) ).
% SUP_identity_eq
thf(fact_1965_SUP__identity__eq,axiom,
! [A4: set_set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_7916887816326733075et_nat
@ ^ [X4: set_nat] : X4
@ A4 ) )
= ( comple7399068483239264473et_nat @ A4 ) ) ).
% SUP_identity_eq
thf(fact_1966_SUP__identity__eq,axiom,
! [A4: set_int] :
( ( complete_Sup_Sup_int
@ ( image_int_int
@ ^ [X4: int] : X4
@ A4 ) )
= ( complete_Sup_Sup_int @ A4 ) ) ).
% SUP_identity_eq
thf(fact_1967_SUP__identity__eq,axiom,
! [A4: set_Product_unit] :
( ( comple4687483117567863418t_unit
@ ( image_405062704495631173t_unit
@ ^ [X4: product_unit] : X4
@ A4 ) )
= ( comple4687483117567863418t_unit @ A4 ) ) ).
% SUP_identity_eq
thf(fact_1968_INF__identity__eq,axiom,
! [A4: set_nat] :
( ( complete_Inf_Inf_nat
@ ( image_nat_nat
@ ^ [X4: nat] : X4
@ A4 ) )
= ( complete_Inf_Inf_nat @ A4 ) ) ).
% INF_identity_eq
thf(fact_1969_INF__identity__eq,axiom,
! [A4: set_set_nat] :
( ( comple7806235888213564991et_nat
@ ( image_7916887816326733075et_nat
@ ^ [X4: set_nat] : X4
@ A4 ) )
= ( comple7806235888213564991et_nat @ A4 ) ) ).
% INF_identity_eq
thf(fact_1970_INF__identity__eq,axiom,
! [A4: set_int] :
( ( complete_Inf_Inf_int
@ ( image_int_int
@ ^ [X4: int] : X4
@ A4 ) )
= ( complete_Inf_Inf_int @ A4 ) ) ).
% INF_identity_eq
thf(fact_1971_INF__identity__eq,axiom,
! [A4: set_Product_unit] :
( ( comple2584293577114468500t_unit
@ ( image_405062704495631173t_unit
@ ^ [X4: product_unit] : X4
@ A4 ) )
= ( comple2584293577114468500t_unit @ A4 ) ) ).
% INF_identity_eq
thf(fact_1972_UN__I,axiom,
! [A: $o,A4: set_o,B: $o,B5: $o > set_o] :
( ( member_o @ A @ A4 )
=> ( ( member_o @ B @ ( B5 @ A ) )
=> ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B5 @ A4 ) ) ) ) ) ).
% UN_I
thf(fact_1973_UN__I,axiom,
! [A: $o,A4: set_o,B: int,B5: $o > set_int] :
( ( member_o @ A @ A4 )
=> ( ( member_int @ B @ ( B5 @ A ) )
=> ( member_int @ B @ ( comple3221217463730067765et_int @ ( image_o_set_int @ B5 @ A4 ) ) ) ) ) ).
% UN_I
thf(fact_1974_UN__I,axiom,
! [A: nat,A4: set_nat,B: $o,B5: nat > set_o] :
( ( member_nat @ A @ A4 )
=> ( ( member_o @ B @ ( B5 @ A ) )
=> ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ B5 @ A4 ) ) ) ) ) ).
% UN_I
thf(fact_1975_UN__I,axiom,
! [A: nat,A4: set_nat,B: int,B5: nat > set_int] :
( ( member_nat @ A @ A4 )
=> ( ( member_int @ B @ ( B5 @ A ) )
=> ( member_int @ B @ ( comple3221217463730067765et_int @ ( image_nat_set_int @ B5 @ A4 ) ) ) ) ) ).
% UN_I
thf(fact_1976_UN__I,axiom,
! [A: int,A4: set_int,B: $o,B5: int > set_o] :
( ( member_int @ A @ A4 )
=> ( ( member_o @ B @ ( B5 @ A ) )
=> ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_int_set_o @ B5 @ A4 ) ) ) ) ) ).
% UN_I
thf(fact_1977_UN__I,axiom,
! [A: int,A4: set_int,B: int,B5: int > set_int] :
( ( member_int @ A @ A4 )
=> ( ( member_int @ B @ ( B5 @ A ) )
=> ( member_int @ B @ ( comple3221217463730067765et_int @ ( image_int_set_int @ B5 @ A4 ) ) ) ) ) ).
% UN_I
thf(fact_1978_UN__I,axiom,
! [A: $o,A4: set_o,B: nat,B5: $o > set_nat] :
( ( member_o @ A @ A4 )
=> ( ( member_nat @ B @ ( B5 @ A ) )
=> ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B5 @ A4 ) ) ) ) ) ).
% UN_I
thf(fact_1979_UN__I,axiom,
! [A: nat,A4: set_nat,B: nat,B5: nat > set_nat] :
( ( member_nat @ A @ A4 )
=> ( ( member_nat @ B @ ( B5 @ A ) )
=> ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ A4 ) ) ) ) ) ).
% UN_I
thf(fact_1980_UN__I,axiom,
! [A: int,A4: set_int,B: nat,B5: int > set_nat] :
( ( member_int @ A @ A4 )
=> ( ( member_nat @ B @ ( B5 @ A ) )
=> ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_int_set_nat @ B5 @ A4 ) ) ) ) ) ).
% UN_I
thf(fact_1981_UN__I,axiom,
! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B: $o,B5: product_prod_nat_nat > set_o] :
( ( member8440522571783428010at_nat @ A @ A4 )
=> ( ( member_o @ B @ ( B5 @ A ) )
=> ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_501298686397505212_set_o @ B5 @ A4 ) ) ) ) ) ).
% UN_I
thf(fact_1982_INT__I,axiom,
! [A4: set_o,B: $o,B5: $o > set_o] :
( ! [X: $o] :
( ( member_o @ X @ A4 )
=> ( member_o @ B @ ( B5 @ X ) ) )
=> ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ B5 @ A4 ) ) ) ) ).
% INT_I
thf(fact_1983_INT__I,axiom,
! [A4: set_o,B: int,B5: $o > set_int] :
( ! [X: $o] :
( ( member_o @ X @ A4 )
=> ( member_int @ B @ ( B5 @ X ) ) )
=> ( member_int @ B @ ( comple3628384868704368283et_int @ ( image_o_set_int @ B5 @ A4 ) ) ) ) ).
% INT_I
thf(fact_1984_INT__I,axiom,
! [A4: set_nat,B: $o,B5: nat > set_o] :
( ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ( member_o @ B @ ( B5 @ X ) ) )
=> ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_nat_set_o @ B5 @ A4 ) ) ) ) ).
% INT_I
thf(fact_1985_INT__I,axiom,
! [A4: set_nat,B: int,B5: nat > set_int] :
( ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ( member_int @ B @ ( B5 @ X ) ) )
=> ( member_int @ B @ ( comple3628384868704368283et_int @ ( image_nat_set_int @ B5 @ A4 ) ) ) ) ).
% INT_I
thf(fact_1986_INT__I,axiom,
! [A4: set_int,B: $o,B5: int > set_o] :
( ! [X: int] :
( ( member_int @ X @ A4 )
=> ( member_o @ B @ ( B5 @ X ) ) )
=> ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_int_set_o @ B5 @ A4 ) ) ) ) ).
% INT_I
thf(fact_1987_INT__I,axiom,
! [A4: set_int,B: int,B5: int > set_int] :
( ! [X: int] :
( ( member_int @ X @ A4 )
=> ( member_int @ B @ ( B5 @ X ) ) )
=> ( member_int @ B @ ( comple3628384868704368283et_int @ ( image_int_set_int @ B5 @ A4 ) ) ) ) ).
% INT_I
thf(fact_1988_INT__I,axiom,
! [A4: set_o,B: nat,B5: $o > set_nat] :
( ! [X: $o] :
( ( member_o @ X @ A4 )
=> ( member_nat @ B @ ( B5 @ X ) ) )
=> ( member_nat @ B @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ B5 @ A4 ) ) ) ) ).
% INT_I
thf(fact_1989_INT__I,axiom,
! [A4: set_nat,B: nat,B5: nat > set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ( member_nat @ B @ ( B5 @ X ) ) )
=> ( member_nat @ B @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B5 @ A4 ) ) ) ) ).
% INT_I
thf(fact_1990_INT__I,axiom,
! [A4: set_int,B: nat,B5: int > set_nat] :
( ! [X: int] :
( ( member_int @ X @ A4 )
=> ( member_nat @ B @ ( B5 @ X ) ) )
=> ( member_nat @ B @ ( comple7806235888213564991et_nat @ ( image_int_set_nat @ B5 @ A4 ) ) ) ) ).
% INT_I
thf(fact_1991_INT__I,axiom,
! [A4: set_Pr1261947904930325089at_nat,B: $o,B5: product_prod_nat_nat > set_o] :
( ! [X: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X @ A4 )
=> ( member_o @ B @ ( B5 @ X ) ) )
=> ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_501298686397505212_set_o @ B5 @ A4 ) ) ) ) ).
% INT_I
thf(fact_1992_SUP__id__eq,axiom,
! [A4: set_o] :
( ( complete_Sup_Sup_o @ ( image_o_o @ id_o @ A4 ) )
= ( complete_Sup_Sup_o @ A4 ) ) ).
% SUP_id_eq
thf(fact_1993_SUP__id__eq,axiom,
! [A4: set_nat] :
( ( complete_Sup_Sup_nat @ ( image_nat_nat @ id_nat @ A4 ) )
= ( complete_Sup_Sup_nat @ A4 ) ) ).
% SUP_id_eq
thf(fact_1994_SUP__id__eq,axiom,
! [A4: set_set_nat] :
( ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ id_set_nat @ A4 ) )
= ( comple7399068483239264473et_nat @ A4 ) ) ).
% SUP_id_eq
thf(fact_1995_SUP__id__eq,axiom,
! [A4: set_int] :
( ( complete_Sup_Sup_int @ ( image_int_int @ id_int @ A4 ) )
= ( complete_Sup_Sup_int @ A4 ) ) ).
% SUP_id_eq
thf(fact_1996_SUP__id__eq,axiom,
! [A4: set_Product_unit] :
( ( comple4687483117567863418t_unit @ ( image_405062704495631173t_unit @ id_Product_unit @ A4 ) )
= ( comple4687483117567863418t_unit @ A4 ) ) ).
% SUP_id_eq
thf(fact_1997_INF__id__eq,axiom,
! [A4: set_o] :
( ( complete_Inf_Inf_o @ ( image_o_o @ id_o @ A4 ) )
= ( complete_Inf_Inf_o @ A4 ) ) ).
% INF_id_eq
thf(fact_1998_INF__id__eq,axiom,
! [A4: set_nat] :
( ( complete_Inf_Inf_nat @ ( image_nat_nat @ id_nat @ A4 ) )
= ( complete_Inf_Inf_nat @ A4 ) ) ).
% INF_id_eq
thf(fact_1999_INF__id__eq,axiom,
! [A4: set_set_nat] :
( ( comple7806235888213564991et_nat @ ( image_7916887816326733075et_nat @ id_set_nat @ A4 ) )
= ( comple7806235888213564991et_nat @ A4 ) ) ).
% INF_id_eq
thf(fact_2000_INF__id__eq,axiom,
! [A4: set_int] :
( ( complete_Inf_Inf_int @ ( image_int_int @ id_int @ A4 ) )
= ( complete_Inf_Inf_int @ A4 ) ) ).
% INF_id_eq
thf(fact_2001_INF__id__eq,axiom,
! [A4: set_Product_unit] :
( ( comple2584293577114468500t_unit @ ( image_405062704495631173t_unit @ id_Product_unit @ A4 ) )
= ( comple2584293577114468500t_unit @ A4 ) ) ).
% INF_id_eq
thf(fact_2002_SUP__const,axiom,
! [A4: set_Product_unit,F: set_nat] :
( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ( comple7399068483239264473et_nat
@ ( image_8173849991827986510et_nat
@ ^ [I2: product_unit] : F
@ A4 ) )
= F ) ) ).
% SUP_const
thf(fact_2003_SUP__const,axiom,
! [A4: set_Pr4329608150637261639at_nat,F: set_nat] :
( ( A4 != bot_bo228742789529271731at_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_3160451284896210088et_nat
@ ^ [I2: produc3843707927480180839at_nat] : F
@ A4 ) )
= F ) ) ).
% SUP_const
thf(fact_2004_SUP__const,axiom,
! [A4: set_Pr1261947904930325089at_nat,F: set_nat] :
( ( A4 != bot_bo2099793752762293965at_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_15824709712370754et_nat
@ ^ [I2: product_prod_nat_nat] : F
@ A4 ) )
= F ) ) ).
% SUP_const
thf(fact_2005_SUP__const,axiom,
! [A4: set_o,F: set_nat] :
( ( A4 != bot_bot_set_o )
=> ( ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [I2: $o] : F
@ A4 ) )
= F ) ) ).
% SUP_const
thf(fact_2006_SUP__const,axiom,
! [A4: set_nat,F: set_nat] :
( ( A4 != bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [I2: nat] : F
@ A4 ) )
= F ) ) ).
% SUP_const
thf(fact_2007_SUP__const,axiom,
! [A4: set_Product_unit,F: product_unit] :
( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ( comple4687483117567863418t_unit
@ ( image_405062704495631173t_unit
@ ^ [I2: product_unit] : F
@ A4 ) )
= F ) ) ).
% SUP_const
thf(fact_2008_SUP__const,axiom,
! [A4: set_Pr4329608150637261639at_nat,F: product_unit] :
( ( A4 != bot_bo228742789529271731at_nat )
=> ( ( comple4687483117567863418t_unit
@ ( image_7082456443403096939t_unit
@ ^ [I2: produc3843707927480180839at_nat] : F
@ A4 ) )
= F ) ) ).
% SUP_const
thf(fact_2009_SUP__const,axiom,
! [A4: set_Pr1261947904930325089at_nat,F: product_unit] :
( ( A4 != bot_bo2099793752762293965at_nat )
=> ( ( comple4687483117567863418t_unit
@ ( image_6169909464789311569t_unit
@ ^ [I2: product_prod_nat_nat] : F
@ A4 ) )
= F ) ) ).
% SUP_const
thf(fact_2010_SUP__const,axiom,
! [A4: set_o,F: product_unit] :
( ( A4 != bot_bot_set_o )
=> ( ( comple4687483117567863418t_unit
@ ( image_o_Product_unit
@ ^ [I2: $o] : F
@ A4 ) )
= F ) ) ).
% SUP_const
thf(fact_2011_SUP__const,axiom,
! [A4: set_nat,F: product_unit] :
( ( A4 != bot_bot_set_nat )
=> ( ( comple4687483117567863418t_unit
@ ( image_8730104196221521654t_unit
@ ^ [I2: nat] : F
@ A4 ) )
= F ) ) ).
% SUP_const
thf(fact_2012_INF__const,axiom,
! [A4: set_Product_unit,F: set_nat] :
( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ( comple7806235888213564991et_nat
@ ( image_8173849991827986510et_nat
@ ^ [I2: product_unit] : F
@ A4 ) )
= F ) ) ).
% INF_const
thf(fact_2013_INF__const,axiom,
! [A4: set_Pr4329608150637261639at_nat,F: set_nat] :
( ( A4 != bot_bo228742789529271731at_nat )
=> ( ( comple7806235888213564991et_nat
@ ( image_3160451284896210088et_nat
@ ^ [I2: produc3843707927480180839at_nat] : F
@ A4 ) )
= F ) ) ).
% INF_const
thf(fact_2014_INF__const,axiom,
! [A4: set_Pr1261947904930325089at_nat,F: set_nat] :
( ( A4 != bot_bo2099793752762293965at_nat )
=> ( ( comple7806235888213564991et_nat
@ ( image_15824709712370754et_nat
@ ^ [I2: product_prod_nat_nat] : F
@ A4 ) )
= F ) ) ).
% INF_const
thf(fact_2015_INF__const,axiom,
! [A4: set_o,F: set_nat] :
( ( A4 != bot_bot_set_o )
=> ( ( comple7806235888213564991et_nat
@ ( image_o_set_nat
@ ^ [I2: $o] : F
@ A4 ) )
= F ) ) ).
% INF_const
thf(fact_2016_INF__const,axiom,
! [A4: set_nat,F: set_nat] :
( ( A4 != bot_bot_set_nat )
=> ( ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [I2: nat] : F
@ A4 ) )
= F ) ) ).
% INF_const
thf(fact_2017_INF__const,axiom,
! [A4: set_Product_unit,F: product_unit] :
( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ( comple2584293577114468500t_unit
@ ( image_405062704495631173t_unit
@ ^ [I2: product_unit] : F
@ A4 ) )
= F ) ) ).
% INF_const
thf(fact_2018_INF__const,axiom,
! [A4: set_Pr4329608150637261639at_nat,F: product_unit] :
( ( A4 != bot_bo228742789529271731at_nat )
=> ( ( comple2584293577114468500t_unit
@ ( image_7082456443403096939t_unit
@ ^ [I2: produc3843707927480180839at_nat] : F
@ A4 ) )
= F ) ) ).
% INF_const
thf(fact_2019_INF__const,axiom,
! [A4: set_Pr1261947904930325089at_nat,F: product_unit] :
( ( A4 != bot_bo2099793752762293965at_nat )
=> ( ( comple2584293577114468500t_unit
@ ( image_6169909464789311569t_unit
@ ^ [I2: product_prod_nat_nat] : F
@ A4 ) )
= F ) ) ).
% INF_const
thf(fact_2020_INF__const,axiom,
! [A4: set_o,F: product_unit] :
( ( A4 != bot_bot_set_o )
=> ( ( comple2584293577114468500t_unit
@ ( image_o_Product_unit
@ ^ [I2: $o] : F
@ A4 ) )
= F ) ) ).
% INF_const
thf(fact_2021_INF__const,axiom,
! [A4: set_nat,F: product_unit] :
( ( A4 != bot_bot_set_nat )
=> ( ( comple2584293577114468500t_unit
@ ( image_8730104196221521654t_unit
@ ^ [I2: nat] : F
@ A4 ) )
= F ) ) ).
% INF_const
thf(fact_2022_UN__constant,axiom,
! [A4: set_Product_unit,C: set_Product_unit] :
( ( ( A4 = bot_bo3957492148770167129t_unit )
=> ( ( comple3349607820504768346t_unit
@ ( image_7778121751606633509t_unit
@ ^ [Y5: product_unit] : C
@ A4 ) )
= bot_bo3957492148770167129t_unit ) )
& ( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ( comple3349607820504768346t_unit
@ ( image_7778121751606633509t_unit
@ ^ [Y5: product_unit] : C
@ A4 ) )
= C ) ) ) ).
% UN_constant
thf(fact_2023_UN__constant,axiom,
! [A4: set_Product_unit,C: set_o] :
( ( ( A4 = bot_bo3957492148770167129t_unit )
=> ( ( comple90263536869209701_set_o
@ ( image_6930536745178679216_set_o
@ ^ [Y5: product_unit] : C
@ A4 ) )
= bot_bot_set_o ) )
& ( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ( comple90263536869209701_set_o
@ ( image_6930536745178679216_set_o
@ ^ [Y5: product_unit] : C
@ A4 ) )
= C ) ) ) ).
% UN_constant
thf(fact_2024_UN__constant,axiom,
! [A4: set_o,C: set_Product_unit] :
( ( ( A4 = bot_bot_set_o )
=> ( ( comple3349607820504768346t_unit
@ ( image_663195431845683034t_unit
@ ^ [Y5: $o] : C
@ A4 ) )
= bot_bo3957492148770167129t_unit ) )
& ( ( A4 != bot_bot_set_o )
=> ( ( comple3349607820504768346t_unit
@ ( image_663195431845683034t_unit
@ ^ [Y5: $o] : C
@ A4 ) )
= C ) ) ) ).
% UN_constant
thf(fact_2025_UN__constant,axiom,
! [A4: set_o,C: set_o] :
( ( ( A4 = bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [Y5: $o] : C
@ A4 ) )
= bot_bot_set_o ) )
& ( ( A4 != bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [Y5: $o] : C
@ A4 ) )
= C ) ) ) ).
% UN_constant
thf(fact_2026_UN__constant,axiom,
! [A4: set_nat,C: set_Product_unit] :
( ( ( A4 = bot_bot_set_nat )
=> ( ( comple3349607820504768346t_unit
@ ( image_4824033694279209942t_unit
@ ^ [Y5: nat] : C
@ A4 ) )
= bot_bo3957492148770167129t_unit ) )
& ( ( A4 != bot_bot_set_nat )
=> ( ( comple3349607820504768346t_unit
@ ( image_4824033694279209942t_unit
@ ^ [Y5: nat] : C
@ A4 ) )
= C ) ) ) ).
% UN_constant
thf(fact_2027_UN__constant,axiom,
! [A4: set_nat,C: set_o] :
( ( ( A4 = bot_bot_set_nat )
=> ( ( comple90263536869209701_set_o
@ ( image_nat_set_o
@ ^ [Y5: nat] : C
@ A4 ) )
= bot_bot_set_o ) )
& ( ( A4 != bot_bot_set_nat )
=> ( ( comple90263536869209701_set_o
@ ( image_nat_set_o
@ ^ [Y5: nat] : C
@ A4 ) )
= C ) ) ) ).
% UN_constant
thf(fact_2028_UN__constant,axiom,
! [A4: set_Product_unit,C: set_nat] :
( ( ( A4 = bot_bo3957492148770167129t_unit )
=> ( ( comple7399068483239264473et_nat
@ ( image_8173849991827986510et_nat
@ ^ [Y5: product_unit] : C
@ A4 ) )
= bot_bot_set_nat ) )
& ( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ( comple7399068483239264473et_nat
@ ( image_8173849991827986510et_nat
@ ^ [Y5: product_unit] : C
@ A4 ) )
= C ) ) ) ).
% UN_constant
thf(fact_2029_UN__constant,axiom,
! [A4: set_o,C: set_nat] :
( ( ( A4 = bot_bot_set_o )
=> ( ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [Y5: $o] : C
@ A4 ) )
= bot_bot_set_nat ) )
& ( ( A4 != bot_bot_set_o )
=> ( ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [Y5: $o] : C
@ A4 ) )
= C ) ) ) ).
% UN_constant
thf(fact_2030_UN__constant,axiom,
! [A4: set_nat,C: set_nat] :
( ( ( A4 = bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y5: nat] : C
@ A4 ) )
= bot_bot_set_nat ) )
& ( ( A4 != bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y5: nat] : C
@ A4 ) )
= C ) ) ) ).
% UN_constant
thf(fact_2031_UN__constant,axiom,
! [A4: set_Product_unit,C: set_Pr1261947904930325089at_nat] :
( ( ( A4 = bot_bo3957492148770167129t_unit )
=> ( ( comple5685304695842803022at_nat
@ ( image_3617315916088684441at_nat
@ ^ [Y5: product_unit] : C
@ A4 ) )
= bot_bo2099793752762293965at_nat ) )
& ( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ( comple5685304695842803022at_nat
@ ( image_3617315916088684441at_nat
@ ^ [Y5: product_unit] : C
@ A4 ) )
= C ) ) ) ).
% UN_constant
thf(fact_2032_UN__simps_I1_J,axiom,
! [C3: set_Product_unit,A: product_unit,B5: product_unit > set_Product_unit] :
( ( ( C3 = bot_bo3957492148770167129t_unit )
=> ( ( comple3349607820504768346t_unit
@ ( image_7778121751606633509t_unit
@ ^ [X4: product_unit] : ( insert_Product_unit @ A @ ( B5 @ X4 ) )
@ C3 ) )
= bot_bo3957492148770167129t_unit ) )
& ( ( C3 != bot_bo3957492148770167129t_unit )
=> ( ( comple3349607820504768346t_unit
@ ( image_7778121751606633509t_unit
@ ^ [X4: product_unit] : ( insert_Product_unit @ A @ ( B5 @ X4 ) )
@ C3 ) )
= ( insert_Product_unit @ A @ ( comple3349607820504768346t_unit @ ( image_7778121751606633509t_unit @ B5 @ C3 ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_2033_UN__simps_I1_J,axiom,
! [C3: set_Product_unit,A: $o,B5: product_unit > set_o] :
( ( ( C3 = bot_bo3957492148770167129t_unit )
=> ( ( comple90263536869209701_set_o
@ ( image_6930536745178679216_set_o
@ ^ [X4: product_unit] : ( insert_o @ A @ ( B5 @ X4 ) )
@ C3 ) )
= bot_bot_set_o ) )
& ( ( C3 != bot_bo3957492148770167129t_unit )
=> ( ( comple90263536869209701_set_o
@ ( image_6930536745178679216_set_o
@ ^ [X4: product_unit] : ( insert_o @ A @ ( B5 @ X4 ) )
@ C3 ) )
= ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_6930536745178679216_set_o @ B5 @ C3 ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_2034_UN__simps_I1_J,axiom,
! [C3: set_o,A: product_unit,B5: $o > set_Product_unit] :
( ( ( C3 = bot_bot_set_o )
=> ( ( comple3349607820504768346t_unit
@ ( image_663195431845683034t_unit
@ ^ [X4: $o] : ( insert_Product_unit @ A @ ( B5 @ X4 ) )
@ C3 ) )
= bot_bo3957492148770167129t_unit ) )
& ( ( C3 != bot_bot_set_o )
=> ( ( comple3349607820504768346t_unit
@ ( image_663195431845683034t_unit
@ ^ [X4: $o] : ( insert_Product_unit @ A @ ( B5 @ X4 ) )
@ C3 ) )
= ( insert_Product_unit @ A @ ( comple3349607820504768346t_unit @ ( image_663195431845683034t_unit @ B5 @ C3 ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_2035_UN__simps_I1_J,axiom,
! [C3: set_o,A: $o,B5: $o > set_o] :
( ( ( C3 = bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X4: $o] : ( insert_o @ A @ ( B5 @ X4 ) )
@ C3 ) )
= bot_bot_set_o ) )
& ( ( C3 != bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X4: $o] : ( insert_o @ A @ ( B5 @ X4 ) )
@ C3 ) )
= ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B5 @ C3 ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_2036_UN__simps_I1_J,axiom,
! [C3: set_nat,A: product_unit,B5: nat > set_Product_unit] :
( ( ( C3 = bot_bot_set_nat )
=> ( ( comple3349607820504768346t_unit
@ ( image_4824033694279209942t_unit
@ ^ [X4: nat] : ( insert_Product_unit @ A @ ( B5 @ X4 ) )
@ C3 ) )
= bot_bo3957492148770167129t_unit ) )
& ( ( C3 != bot_bot_set_nat )
=> ( ( comple3349607820504768346t_unit
@ ( image_4824033694279209942t_unit
@ ^ [X4: nat] : ( insert_Product_unit @ A @ ( B5 @ X4 ) )
@ C3 ) )
= ( insert_Product_unit @ A @ ( comple3349607820504768346t_unit @ ( image_4824033694279209942t_unit @ B5 @ C3 ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_2037_UN__simps_I1_J,axiom,
! [C3: set_nat,A: $o,B5: nat > set_o] :
( ( ( C3 = bot_bot_set_nat )
=> ( ( comple90263536869209701_set_o
@ ( image_nat_set_o
@ ^ [X4: nat] : ( insert_o @ A @ ( B5 @ X4 ) )
@ C3 ) )
= bot_bot_set_o ) )
& ( ( C3 != bot_bot_set_nat )
=> ( ( comple90263536869209701_set_o
@ ( image_nat_set_o
@ ^ [X4: nat] : ( insert_o @ A @ ( B5 @ X4 ) )
@ C3 ) )
= ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ B5 @ C3 ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_2038_UN__simps_I1_J,axiom,
! [C3: set_Product_unit,A: nat,B5: product_unit > set_nat] :
( ( ( C3 = bot_bo3957492148770167129t_unit )
=> ( ( comple7399068483239264473et_nat
@ ( image_8173849991827986510et_nat
@ ^ [X4: product_unit] : ( insert_nat @ A @ ( B5 @ X4 ) )
@ C3 ) )
= bot_bot_set_nat ) )
& ( ( C3 != bot_bo3957492148770167129t_unit )
=> ( ( comple7399068483239264473et_nat
@ ( image_8173849991827986510et_nat
@ ^ [X4: product_unit] : ( insert_nat @ A @ ( B5 @ X4 ) )
@ C3 ) )
= ( insert_nat @ A @ ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ B5 @ C3 ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_2039_UN__simps_I1_J,axiom,
! [C3: set_o,A: nat,B5: $o > set_nat] :
( ( ( C3 = bot_bot_set_o )
=> ( ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [X4: $o] : ( insert_nat @ A @ ( B5 @ X4 ) )
@ C3 ) )
= bot_bot_set_nat ) )
& ( ( C3 != bot_bot_set_o )
=> ( ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [X4: $o] : ( insert_nat @ A @ ( B5 @ X4 ) )
@ C3 ) )
= ( insert_nat @ A @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B5 @ C3 ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_2040_UN__simps_I1_J,axiom,
! [C3: set_nat,A: nat,B5: nat > set_nat] :
( ( ( C3 = bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X4: nat] : ( insert_nat @ A @ ( B5 @ X4 ) )
@ C3 ) )
= bot_bot_set_nat ) )
& ( ( C3 != bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X4: nat] : ( insert_nat @ A @ ( B5 @ X4 ) )
@ C3 ) )
= ( insert_nat @ A @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ C3 ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_2041_UN__simps_I1_J,axiom,
! [C3: set_Product_unit,A: product_prod_nat_nat,B5: product_unit > set_Pr1261947904930325089at_nat] :
( ( ( C3 = bot_bo3957492148770167129t_unit )
=> ( ( comple5685304695842803022at_nat
@ ( image_3617315916088684441at_nat
@ ^ [X4: product_unit] : ( insert8211810215607154385at_nat @ A @ ( B5 @ X4 ) )
@ C3 ) )
= bot_bo2099793752762293965at_nat ) )
& ( ( C3 != bot_bo3957492148770167129t_unit )
=> ( ( comple5685304695842803022at_nat
@ ( image_3617315916088684441at_nat
@ ^ [X4: product_unit] : ( insert8211810215607154385at_nat @ A @ ( B5 @ X4 ) )
@ C3 ) )
= ( insert8211810215607154385at_nat @ A @ ( comple5685304695842803022at_nat @ ( image_3617315916088684441at_nat @ B5 @ C3 ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_2042_UN__singleton,axiom,
! [A4: set_Product_unit] :
( ( comple3349607820504768346t_unit
@ ( image_7778121751606633509t_unit
@ ^ [X4: product_unit] : ( insert_Product_unit @ X4 @ bot_bo3957492148770167129t_unit )
@ A4 ) )
= A4 ) ).
% UN_singleton
thf(fact_2043_UN__singleton,axiom,
! [A4: set_Pr4329608150637261639at_nat] :
( ( comple4364388969068068532at_nat
@ ( image_7364710425340685989at_nat
@ ^ [X4: produc3843707927480180839at_nat] : ( insert9069300056098147895at_nat @ X4 @ bot_bo228742789529271731at_nat )
@ A4 ) )
= A4 ) ).
% UN_singleton
thf(fact_2044_UN__singleton,axiom,
! [A4: set_Pr1261947904930325089at_nat] :
( ( comple5685304695842803022at_nat
@ ( image_5423882813909314213at_nat
@ ^ [X4: product_prod_nat_nat] : ( insert8211810215607154385at_nat @ X4 @ bot_bo2099793752762293965at_nat )
@ A4 ) )
= A4 ) ).
% UN_singleton
thf(fact_2045_UN__singleton,axiom,
! [A4: set_o] :
( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X4: $o] : ( insert_o @ X4 @ bot_bot_set_o )
@ A4 ) )
= A4 ) ).
% UN_singleton
thf(fact_2046_UN__singleton,axiom,
! [A4: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X4: nat] : ( insert_nat @ X4 @ bot_bot_set_nat )
@ A4 ) )
= A4 ) ).
% UN_singleton
thf(fact_2047_refl__on__singleton,axiom,
! [X2: set_Pr4329608150637261639at_nat] : ( refl_o6272981002401706822at_nat @ ( insert2290795007606087959at_nat @ X2 @ bot_bo1534825418459847827at_nat ) @ ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ X2 @ X2 ) @ bot_bo4948859079157340979at_nat ) ) ).
% refl_on_singleton
thf(fact_2048_refl__on__singleton,axiom,
! [X2: int] : ( refl_on_int @ ( insert_int @ X2 @ bot_bot_set_int ) @ ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ X2 @ X2 ) @ bot_bo1796632182523588997nt_int ) ) ).
% refl_on_singleton
thf(fact_2049_refl__on__singleton,axiom,
! [X2: set_Pr1261947904930325089at_nat] : ( refl_o6955724340487615712at_nat @ ( insert9200635055090092081at_nat @ X2 @ bot_bo3083307316010499117at_nat ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ X2 @ X2 ) @ bot_bo228742789529271731at_nat ) ) ).
% refl_on_singleton
thf(fact_2050_refl__on__singleton,axiom,
! [X2: product_unit] : ( refl_on_Product_unit @ ( insert_Product_unit @ X2 @ bot_bo3957492148770167129t_unit ) @ ( insert4616891024742550839t_unit @ ( produc5692694829384537815t_unit @ X2 @ X2 ) @ bot_bo1654022762305692595t_unit ) ) ).
% refl_on_singleton
thf(fact_2051_refl__on__singleton,axiom,
! [X2: produc3843707927480180839at_nat] : ( refl_o7446464703481863270at_nat @ ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat ) @ ( insert398606708087465655at_nat @ ( produc2335810615602371415at_nat @ X2 @ X2 ) @ bot_bo7193250159860175667at_nat ) ) ).
% refl_on_singleton
thf(fact_2052_refl__on__singleton,axiom,
! [X2: product_prod_nat_nat] : ( refl_o7122697186039111552at_nat @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) @ ( insert5050368324300391991at_nat @ ( produc6161850002892822231at_nat @ X2 @ X2 ) @ bot_bo5327735625951526323at_nat ) ) ).
% refl_on_singleton
thf(fact_2053_refl__on__singleton,axiom,
! [X2: $o] : ( refl_on_o @ ( insert_o @ X2 @ bot_bot_set_o ) @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ X2 @ X2 ) @ bot_bo7073875226086086771od_o_o ) ) ).
% refl_on_singleton
thf(fact_2054_refl__on__singleton,axiom,
! [X2: nat] : ( refl_on_nat @ ( insert_nat @ X2 @ bot_bot_set_nat ) @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X2 @ X2 ) @ bot_bo2099793752762293965at_nat ) ) ).
% refl_on_singleton
thf(fact_2055_INT__insert__distrib,axiom,
! [U: $o,A4: set_o,A: product_unit,B5: $o > set_Product_unit] :
( ( member_o @ U @ A4 )
=> ( ( comple7216939928495492468t_unit
@ ( image_663195431845683034t_unit
@ ^ [X4: $o] : ( insert_Product_unit @ A @ ( B5 @ X4 ) )
@ A4 ) )
= ( insert_Product_unit @ A @ ( comple7216939928495492468t_unit @ ( image_663195431845683034t_unit @ B5 @ A4 ) ) ) ) ) ).
% INT_insert_distrib
thf(fact_2056_INT__insert__distrib,axiom,
! [U: $o,A4: set_o,A: $o,B5: $o > set_o] :
( ( member_o @ U @ A4 )
=> ( ( comple3063163877087187839_set_o
@ ( image_o_set_o
@ ^ [X4: $o] : ( insert_o @ A @ ( B5 @ X4 ) )
@ A4 ) )
= ( insert_o @ A @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ B5 @ A4 ) ) ) ) ) ).
% INT_insert_distrib
thf(fact_2057_INT__insert__distrib,axiom,
! [U: nat,A4: set_nat,A: product_unit,B5: nat > set_Product_unit] :
( ( member_nat @ U @ A4 )
=> ( ( comple7216939928495492468t_unit
@ ( image_4824033694279209942t_unit
@ ^ [X4: nat] : ( insert_Product_unit @ A @ ( B5 @ X4 ) )
@ A4 ) )
= ( insert_Product_unit @ A @ ( comple7216939928495492468t_unit @ ( image_4824033694279209942t_unit @ B5 @ A4 ) ) ) ) ) ).
% INT_insert_distrib
thf(fact_2058_INT__insert__distrib,axiom,
! [U: nat,A4: set_nat,A: $o,B5: nat > set_o] :
( ( member_nat @ U @ A4 )
=> ( ( comple3063163877087187839_set_o
@ ( image_nat_set_o
@ ^ [X4: nat] : ( insert_o @ A @ ( B5 @ X4 ) )
@ A4 ) )
= ( insert_o @ A @ ( comple3063163877087187839_set_o @ ( image_nat_set_o @ B5 @ A4 ) ) ) ) ) ).
% INT_insert_distrib
thf(fact_2059_INT__insert__distrib,axiom,
! [U: int,A4: set_int,A: product_unit,B5: int > set_Product_unit] :
( ( member_int @ U @ A4 )
=> ( ( comple7216939928495492468t_unit
@ ( image_8658272795163236474t_unit
@ ^ [X4: int] : ( insert_Product_unit @ A @ ( B5 @ X4 ) )
@ A4 ) )
= ( insert_Product_unit @ A @ ( comple7216939928495492468t_unit @ ( image_8658272795163236474t_unit @ B5 @ A4 ) ) ) ) ) ).
% INT_insert_distrib
thf(fact_2060_INT__insert__distrib,axiom,
! [U: int,A4: set_int,A: $o,B5: int > set_o] :
( ( member_int @ U @ A4 )
=> ( ( comple3063163877087187839_set_o
@ ( image_int_set_o
@ ^ [X4: int] : ( insert_o @ A @ ( B5 @ X4 ) )
@ A4 ) )
= ( insert_o @ A @ ( comple3063163877087187839_set_o @ ( image_int_set_o @ B5 @ A4 ) ) ) ) ) ).
% INT_insert_distrib
thf(fact_2061_INT__insert__distrib,axiom,
! [U: $o,A4: set_o,A: nat,B5: $o > set_nat] :
( ( member_o @ U @ A4 )
=> ( ( comple7806235888213564991et_nat
@ ( image_o_set_nat
@ ^ [X4: $o] : ( insert_nat @ A @ ( B5 @ X4 ) )
@ A4 ) )
= ( insert_nat @ A @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ B5 @ A4 ) ) ) ) ) ).
% INT_insert_distrib
thf(fact_2062_INT__insert__distrib,axiom,
! [U: nat,A4: set_nat,A: nat,B5: nat > set_nat] :
( ( member_nat @ U @ A4 )
=> ( ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [X4: nat] : ( insert_nat @ A @ ( B5 @ X4 ) )
@ A4 ) )
= ( insert_nat @ A @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B5 @ A4 ) ) ) ) ) ).
% INT_insert_distrib
thf(fact_2063_INT__insert__distrib,axiom,
! [U: int,A4: set_int,A: nat,B5: int > set_nat] :
( ( member_int @ U @ A4 )
=> ( ( comple7806235888213564991et_nat
@ ( image_int_set_nat
@ ^ [X4: int] : ( insert_nat @ A @ ( B5 @ X4 ) )
@ A4 ) )
= ( insert_nat @ A @ ( comple7806235888213564991et_nat @ ( image_int_set_nat @ B5 @ A4 ) ) ) ) ) ).
% INT_insert_distrib
thf(fact_2064_INT__insert__distrib,axiom,
! [U: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,A: product_unit,B5: product_prod_nat_nat > set_Product_unit] :
( ( member8440522571783428010at_nat @ U @ A4 )
=> ( ( comple7216939928495492468t_unit
@ ( image_121980125215734065t_unit
@ ^ [X4: product_prod_nat_nat] : ( insert_Product_unit @ A @ ( B5 @ X4 ) )
@ A4 ) )
= ( insert_Product_unit @ A @ ( comple7216939928495492468t_unit @ ( image_121980125215734065t_unit @ B5 @ A4 ) ) ) ) ) ).
% INT_insert_distrib
thf(fact_2065_UN__insert__distrib,axiom,
! [U: $o,A4: set_o,A: product_unit,B5: $o > set_Product_unit] :
( ( member_o @ U @ A4 )
=> ( ( comple3349607820504768346t_unit
@ ( image_663195431845683034t_unit
@ ^ [X4: $o] : ( insert_Product_unit @ A @ ( B5 @ X4 ) )
@ A4 ) )
= ( insert_Product_unit @ A @ ( comple3349607820504768346t_unit @ ( image_663195431845683034t_unit @ B5 @ A4 ) ) ) ) ) ).
% UN_insert_distrib
thf(fact_2066_UN__insert__distrib,axiom,
! [U: $o,A4: set_o,A: $o,B5: $o > set_o] :
( ( member_o @ U @ A4 )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X4: $o] : ( insert_o @ A @ ( B5 @ X4 ) )
@ A4 ) )
= ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B5 @ A4 ) ) ) ) ) ).
% UN_insert_distrib
thf(fact_2067_UN__insert__distrib,axiom,
! [U: nat,A4: set_nat,A: product_unit,B5: nat > set_Product_unit] :
( ( member_nat @ U @ A4 )
=> ( ( comple3349607820504768346t_unit
@ ( image_4824033694279209942t_unit
@ ^ [X4: nat] : ( insert_Product_unit @ A @ ( B5 @ X4 ) )
@ A4 ) )
= ( insert_Product_unit @ A @ ( comple3349607820504768346t_unit @ ( image_4824033694279209942t_unit @ B5 @ A4 ) ) ) ) ) ).
% UN_insert_distrib
thf(fact_2068_UN__insert__distrib,axiom,
! [U: nat,A4: set_nat,A: $o,B5: nat > set_o] :
( ( member_nat @ U @ A4 )
=> ( ( comple90263536869209701_set_o
@ ( image_nat_set_o
@ ^ [X4: nat] : ( insert_o @ A @ ( B5 @ X4 ) )
@ A4 ) )
= ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ B5 @ A4 ) ) ) ) ) ).
% UN_insert_distrib
thf(fact_2069_UN__insert__distrib,axiom,
! [U: int,A4: set_int,A: product_unit,B5: int > set_Product_unit] :
( ( member_int @ U @ A4 )
=> ( ( comple3349607820504768346t_unit
@ ( image_8658272795163236474t_unit
@ ^ [X4: int] : ( insert_Product_unit @ A @ ( B5 @ X4 ) )
@ A4 ) )
= ( insert_Product_unit @ A @ ( comple3349607820504768346t_unit @ ( image_8658272795163236474t_unit @ B5 @ A4 ) ) ) ) ) ).
% UN_insert_distrib
thf(fact_2070_UN__insert__distrib,axiom,
! [U: int,A4: set_int,A: $o,B5: int > set_o] :
( ( member_int @ U @ A4 )
=> ( ( comple90263536869209701_set_o
@ ( image_int_set_o
@ ^ [X4: int] : ( insert_o @ A @ ( B5 @ X4 ) )
@ A4 ) )
= ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_int_set_o @ B5 @ A4 ) ) ) ) ) ).
% UN_insert_distrib
thf(fact_2071_UN__insert__distrib,axiom,
! [U: $o,A4: set_o,A: nat,B5: $o > set_nat] :
( ( member_o @ U @ A4 )
=> ( ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [X4: $o] : ( insert_nat @ A @ ( B5 @ X4 ) )
@ A4 ) )
= ( insert_nat @ A @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B5 @ A4 ) ) ) ) ) ).
% UN_insert_distrib
thf(fact_2072_UN__insert__distrib,axiom,
! [U: nat,A4: set_nat,A: nat,B5: nat > set_nat] :
( ( member_nat @ U @ A4 )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X4: nat] : ( insert_nat @ A @ ( B5 @ X4 ) )
@ A4 ) )
= ( insert_nat @ A @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ A4 ) ) ) ) ) ).
% UN_insert_distrib
thf(fact_2073_UN__insert__distrib,axiom,
! [U: int,A4: set_int,A: nat,B5: int > set_nat] :
( ( member_int @ U @ A4 )
=> ( ( comple7399068483239264473et_nat
@ ( image_int_set_nat
@ ^ [X4: int] : ( insert_nat @ A @ ( B5 @ X4 ) )
@ A4 ) )
= ( insert_nat @ A @ ( comple7399068483239264473et_nat @ ( image_int_set_nat @ B5 @ A4 ) ) ) ) ) ).
% UN_insert_distrib
thf(fact_2074_UN__insert__distrib,axiom,
! [U: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,A: product_unit,B5: product_prod_nat_nat > set_Product_unit] :
( ( member8440522571783428010at_nat @ U @ A4 )
=> ( ( comple3349607820504768346t_unit
@ ( image_121980125215734065t_unit
@ ^ [X4: product_prod_nat_nat] : ( insert_Product_unit @ A @ ( B5 @ X4 ) )
@ A4 ) )
= ( insert_Product_unit @ A @ ( comple3349607820504768346t_unit @ ( image_121980125215734065t_unit @ B5 @ A4 ) ) ) ) ) ).
% UN_insert_distrib
thf(fact_2075_Sigma__Union,axiom,
! [X5: set_set_nat,B5: nat > set_nat] :
( ( produc457027306803732586at_nat @ ( comple7399068483239264473et_nat @ X5 ) @ B5 )
= ( comple5685304695842803022at_nat
@ ( image_6521575952113991828at_nat
@ ^ [A6: set_nat] : ( produc457027306803732586at_nat @ A6 @ B5 )
@ X5 ) ) ) ).
% Sigma_Union
thf(fact_2076_INF__cong,axiom,
! [A4: set_nat,B5: set_nat,C3: nat > nat,D: nat > nat] :
( ( A4 = B5 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B5 )
=> ( ( C3 @ X )
= ( D @ X ) ) )
=> ( ( complete_Inf_Inf_nat @ ( image_nat_nat @ C3 @ A4 ) )
= ( complete_Inf_Inf_nat @ ( image_nat_nat @ D @ B5 ) ) ) ) ) ).
% INF_cong
thf(fact_2077_INF__cong,axiom,
! [A4: set_int,B5: set_int,C3: int > nat,D: int > nat] :
( ( A4 = B5 )
=> ( ! [X: int] :
( ( member_int @ X @ B5 )
=> ( ( C3 @ X )
= ( D @ X ) ) )
=> ( ( complete_Inf_Inf_nat @ ( image_int_nat @ C3 @ A4 ) )
= ( complete_Inf_Inf_nat @ ( image_int_nat @ D @ B5 ) ) ) ) ) ).
% INF_cong
thf(fact_2078_INF__cong,axiom,
! [A4: set_o,B5: set_o,C3: $o > int,D: $o > int] :
( ( A4 = B5 )
=> ( ! [X: $o] :
( ( member_o @ X @ B5 )
=> ( ( C3 @ X )
= ( D @ X ) ) )
=> ( ( complete_Inf_Inf_int @ ( image_o_int @ C3 @ A4 ) )
= ( complete_Inf_Inf_int @ ( image_o_int @ D @ B5 ) ) ) ) ) ).
% INF_cong
thf(fact_2079_INF__cong,axiom,
! [A4: set_nat,B5: set_nat,C3: nat > int,D: nat > int] :
( ( A4 = B5 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B5 )
=> ( ( C3 @ X )
= ( D @ X ) ) )
=> ( ( complete_Inf_Inf_int @ ( image_nat_int @ C3 @ A4 ) )
= ( complete_Inf_Inf_int @ ( image_nat_int @ D @ B5 ) ) ) ) ) ).
% INF_cong
thf(fact_2080_INF__cong,axiom,
! [A4: set_int,B5: set_int,C3: int > int,D: int > int] :
( ( A4 = B5 )
=> ( ! [X: int] :
( ( member_int @ X @ B5 )
=> ( ( C3 @ X )
= ( D @ X ) ) )
=> ( ( complete_Inf_Inf_int @ ( image_int_int @ C3 @ A4 ) )
= ( complete_Inf_Inf_int @ ( image_int_int @ D @ B5 ) ) ) ) ) ).
% INF_cong
thf(fact_2081_INF__cong,axiom,
! [A4: set_o,B5: set_o,C3: $o > product_unit,D: $o > product_unit] :
( ( A4 = B5 )
=> ( ! [X: $o] :
( ( member_o @ X @ B5 )
=> ( ( C3 @ X )
= ( D @ X ) ) )
=> ( ( comple2584293577114468500t_unit @ ( image_o_Product_unit @ C3 @ A4 ) )
= ( comple2584293577114468500t_unit @ ( image_o_Product_unit @ D @ B5 ) ) ) ) ) ).
% INF_cong
thf(fact_2082_INF__cong,axiom,
! [A4: set_nat,B5: set_nat,C3: nat > product_unit,D: nat > product_unit] :
( ( A4 = B5 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B5 )
=> ( ( C3 @ X )
= ( D @ X ) ) )
=> ( ( comple2584293577114468500t_unit @ ( image_8730104196221521654t_unit @ C3 @ A4 ) )
= ( comple2584293577114468500t_unit @ ( image_8730104196221521654t_unit @ D @ B5 ) ) ) ) ) ).
% INF_cong
thf(fact_2083_INF__cong,axiom,
! [A4: set_int,B5: set_int,C3: int > product_unit,D: int > product_unit] :
( ( A4 = B5 )
=> ( ! [X: int] :
( ( member_int @ X @ B5 )
=> ( ( C3 @ X )
= ( D @ X ) ) )
=> ( ( comple2584293577114468500t_unit @ ( image_5846526143423701914t_unit @ C3 @ A4 ) )
= ( comple2584293577114468500t_unit @ ( image_5846526143423701914t_unit @ D @ B5 ) ) ) ) ) ).
% INF_cong
thf(fact_2084_INF__cong,axiom,
! [A4: set_o,B5: set_o,C3: $o > set_nat,D: $o > set_nat] :
( ( A4 = B5 )
=> ( ! [X: $o] :
( ( member_o @ X @ B5 )
=> ( ( C3 @ X )
= ( D @ X ) ) )
=> ( ( comple7806235888213564991et_nat @ ( image_o_set_nat @ C3 @ A4 ) )
= ( comple7806235888213564991et_nat @ ( image_o_set_nat @ D @ B5 ) ) ) ) ) ).
% INF_cong
thf(fact_2085_INF__cong,axiom,
! [A4: set_nat,B5: set_nat,C3: nat > set_nat,D: nat > set_nat] :
( ( A4 = B5 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B5 )
=> ( ( C3 @ X )
= ( D @ X ) ) )
=> ( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ C3 @ A4 ) )
= ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ D @ B5 ) ) ) ) ) ).
% INF_cong
thf(fact_2086_SUP__cong,axiom,
! [A4: set_nat,B5: set_nat,C3: nat > nat,D: nat > nat] :
( ( A4 = B5 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B5 )
=> ( ( C3 @ X )
= ( D @ X ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_nat_nat @ C3 @ A4 ) )
= ( complete_Sup_Sup_nat @ ( image_nat_nat @ D @ B5 ) ) ) ) ) ).
% SUP_cong
thf(fact_2087_SUP__cong,axiom,
! [A4: set_int,B5: set_int,C3: int > nat,D: int > nat] :
( ( A4 = B5 )
=> ( ! [X: int] :
( ( member_int @ X @ B5 )
=> ( ( C3 @ X )
= ( D @ X ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_int_nat @ C3 @ A4 ) )
= ( complete_Sup_Sup_nat @ ( image_int_nat @ D @ B5 ) ) ) ) ) ).
% SUP_cong
thf(fact_2088_SUP__cong,axiom,
! [A4: set_o,B5: set_o,C3: $o > int,D: $o > int] :
( ( A4 = B5 )
=> ( ! [X: $o] :
( ( member_o @ X @ B5 )
=> ( ( C3 @ X )
= ( D @ X ) ) )
=> ( ( complete_Sup_Sup_int @ ( image_o_int @ C3 @ A4 ) )
= ( complete_Sup_Sup_int @ ( image_o_int @ D @ B5 ) ) ) ) ) ).
% SUP_cong
thf(fact_2089_SUP__cong,axiom,
! [A4: set_nat,B5: set_nat,C3: nat > int,D: nat > int] :
( ( A4 = B5 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B5 )
=> ( ( C3 @ X )
= ( D @ X ) ) )
=> ( ( complete_Sup_Sup_int @ ( image_nat_int @ C3 @ A4 ) )
= ( complete_Sup_Sup_int @ ( image_nat_int @ D @ B5 ) ) ) ) ) ).
% SUP_cong
thf(fact_2090_SUP__cong,axiom,
! [A4: set_int,B5: set_int,C3: int > int,D: int > int] :
( ( A4 = B5 )
=> ( ! [X: int] :
( ( member_int @ X @ B5 )
=> ( ( C3 @ X )
= ( D @ X ) ) )
=> ( ( complete_Sup_Sup_int @ ( image_int_int @ C3 @ A4 ) )
= ( complete_Sup_Sup_int @ ( image_int_int @ D @ B5 ) ) ) ) ) ).
% SUP_cong
thf(fact_2091_SUP__cong,axiom,
! [A4: set_o,B5: set_o,C3: $o > product_unit,D: $o > product_unit] :
( ( A4 = B5 )
=> ( ! [X: $o] :
( ( member_o @ X @ B5 )
=> ( ( C3 @ X )
= ( D @ X ) ) )
=> ( ( comple4687483117567863418t_unit @ ( image_o_Product_unit @ C3 @ A4 ) )
= ( comple4687483117567863418t_unit @ ( image_o_Product_unit @ D @ B5 ) ) ) ) ) ).
% SUP_cong
thf(fact_2092_SUP__cong,axiom,
! [A4: set_nat,B5: set_nat,C3: nat > product_unit,D: nat > product_unit] :
( ( A4 = B5 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B5 )
=> ( ( C3 @ X )
= ( D @ X ) ) )
=> ( ( comple4687483117567863418t_unit @ ( image_8730104196221521654t_unit @ C3 @ A4 ) )
= ( comple4687483117567863418t_unit @ ( image_8730104196221521654t_unit @ D @ B5 ) ) ) ) ) ).
% SUP_cong
thf(fact_2093_SUP__cong,axiom,
! [A4: set_int,B5: set_int,C3: int > product_unit,D: int > product_unit] :
( ( A4 = B5 )
=> ( ! [X: int] :
( ( member_int @ X @ B5 )
=> ( ( C3 @ X )
= ( D @ X ) ) )
=> ( ( comple4687483117567863418t_unit @ ( image_5846526143423701914t_unit @ C3 @ A4 ) )
= ( comple4687483117567863418t_unit @ ( image_5846526143423701914t_unit @ D @ B5 ) ) ) ) ) ).
% SUP_cong
thf(fact_2094_SUP__cong,axiom,
! [A4: set_o,B5: set_o,C3: $o > set_nat,D: $o > set_nat] :
( ( A4 = B5 )
=> ( ! [X: $o] :
( ( member_o @ X @ B5 )
=> ( ( C3 @ X )
= ( D @ X ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ C3 @ A4 ) )
= ( comple7399068483239264473et_nat @ ( image_o_set_nat @ D @ B5 ) ) ) ) ) ).
% SUP_cong
thf(fact_2095_SUP__cong,axiom,
! [A4: set_nat,B5: set_nat,C3: nat > set_nat,D: nat > set_nat] :
( ( A4 = B5 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B5 )
=> ( ( C3 @ X )
= ( D @ X ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C3 @ A4 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ D @ B5 ) ) ) ) ) ).
% SUP_cong
thf(fact_2096_insert__subsetI,axiom,
! [X2: product_unit,A4: set_Product_unit,X5: set_Product_unit] :
( ( member_Product_unit @ X2 @ A4 )
=> ( ( ord_le3507040750410214029t_unit @ X5 @ A4 )
=> ( ord_le3507040750410214029t_unit @ ( insert_Product_unit @ X2 @ X5 ) @ A4 ) ) ) ).
% insert_subsetI
thf(fact_2097_insert__subsetI,axiom,
! [X2: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,X5: set_Pr4329608150637261639at_nat] :
( ( member8757157785044589968at_nat @ X2 @ A4 )
=> ( ( ord_le1268244103169919719at_nat @ X5 @ A4 )
=> ( ord_le1268244103169919719at_nat @ ( insert9069300056098147895at_nat @ X2 @ X5 ) @ A4 ) ) ) ).
% insert_subsetI
thf(fact_2098_insert__subsetI,axiom,
! [X2: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,X5: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ X2 @ A4 )
=> ( ( ord_le3146513528884898305at_nat @ X5 @ A4 )
=> ( ord_le3146513528884898305at_nat @ ( insert8211810215607154385at_nat @ X2 @ X5 ) @ A4 ) ) ) ).
% insert_subsetI
thf(fact_2099_insert__subsetI,axiom,
! [X2: $o,A4: set_o,X5: set_o] :
( ( member_o @ X2 @ A4 )
=> ( ( ord_less_eq_set_o @ X5 @ A4 )
=> ( ord_less_eq_set_o @ ( insert_o @ X2 @ X5 ) @ A4 ) ) ) ).
% insert_subsetI
thf(fact_2100_insert__subsetI,axiom,
! [X2: nat,A4: set_nat,X5: set_nat] :
( ( member_nat @ X2 @ A4 )
=> ( ( ord_less_eq_set_nat @ X5 @ A4 )
=> ( ord_less_eq_set_nat @ ( insert_nat @ X2 @ X5 ) @ A4 ) ) ) ).
% insert_subsetI
thf(fact_2101_insert__subsetI,axiom,
! [X2: int,A4: set_int,X5: set_int] :
( ( member_int @ X2 @ A4 )
=> ( ( ord_less_eq_set_int @ X5 @ A4 )
=> ( ord_less_eq_set_int @ ( insert_int @ X2 @ X5 ) @ A4 ) ) ) ).
% insert_subsetI
thf(fact_2102_UN__extend__simps_I1_J,axiom,
! [C3: set_Product_unit,A: product_unit,B5: product_unit > set_Product_unit] :
( ( ( C3 = bot_bo3957492148770167129t_unit )
=> ( ( insert_Product_unit @ A @ ( comple3349607820504768346t_unit @ ( image_7778121751606633509t_unit @ B5 @ C3 ) ) )
= ( insert_Product_unit @ A @ bot_bo3957492148770167129t_unit ) ) )
& ( ( C3 != bot_bo3957492148770167129t_unit )
=> ( ( insert_Product_unit @ A @ ( comple3349607820504768346t_unit @ ( image_7778121751606633509t_unit @ B5 @ C3 ) ) )
= ( comple3349607820504768346t_unit
@ ( image_7778121751606633509t_unit
@ ^ [X4: product_unit] : ( insert_Product_unit @ A @ ( B5 @ X4 ) )
@ C3 ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_2103_UN__extend__simps_I1_J,axiom,
! [C3: set_Product_unit,A: $o,B5: product_unit > set_o] :
( ( ( C3 = bot_bo3957492148770167129t_unit )
=> ( ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_6930536745178679216_set_o @ B5 @ C3 ) ) )
= ( insert_o @ A @ bot_bot_set_o ) ) )
& ( ( C3 != bot_bo3957492148770167129t_unit )
=> ( ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_6930536745178679216_set_o @ B5 @ C3 ) ) )
= ( comple90263536869209701_set_o
@ ( image_6930536745178679216_set_o
@ ^ [X4: product_unit] : ( insert_o @ A @ ( B5 @ X4 ) )
@ C3 ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_2104_UN__extend__simps_I1_J,axiom,
! [C3: set_o,A: product_unit,B5: $o > set_Product_unit] :
( ( ( C3 = bot_bot_set_o )
=> ( ( insert_Product_unit @ A @ ( comple3349607820504768346t_unit @ ( image_663195431845683034t_unit @ B5 @ C3 ) ) )
= ( insert_Product_unit @ A @ bot_bo3957492148770167129t_unit ) ) )
& ( ( C3 != bot_bot_set_o )
=> ( ( insert_Product_unit @ A @ ( comple3349607820504768346t_unit @ ( image_663195431845683034t_unit @ B5 @ C3 ) ) )
= ( comple3349607820504768346t_unit
@ ( image_663195431845683034t_unit
@ ^ [X4: $o] : ( insert_Product_unit @ A @ ( B5 @ X4 ) )
@ C3 ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_2105_UN__extend__simps_I1_J,axiom,
! [C3: set_o,A: $o,B5: $o > set_o] :
( ( ( C3 = bot_bot_set_o )
=> ( ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B5 @ C3 ) ) )
= ( insert_o @ A @ bot_bot_set_o ) ) )
& ( ( C3 != bot_bot_set_o )
=> ( ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B5 @ C3 ) ) )
= ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X4: $o] : ( insert_o @ A @ ( B5 @ X4 ) )
@ C3 ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_2106_UN__extend__simps_I1_J,axiom,
! [C3: set_nat,A: product_unit,B5: nat > set_Product_unit] :
( ( ( C3 = bot_bot_set_nat )
=> ( ( insert_Product_unit @ A @ ( comple3349607820504768346t_unit @ ( image_4824033694279209942t_unit @ B5 @ C3 ) ) )
= ( insert_Product_unit @ A @ bot_bo3957492148770167129t_unit ) ) )
& ( ( C3 != bot_bot_set_nat )
=> ( ( insert_Product_unit @ A @ ( comple3349607820504768346t_unit @ ( image_4824033694279209942t_unit @ B5 @ C3 ) ) )
= ( comple3349607820504768346t_unit
@ ( image_4824033694279209942t_unit
@ ^ [X4: nat] : ( insert_Product_unit @ A @ ( B5 @ X4 ) )
@ C3 ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_2107_UN__extend__simps_I1_J,axiom,
! [C3: set_nat,A: $o,B5: nat > set_o] :
( ( ( C3 = bot_bot_set_nat )
=> ( ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ B5 @ C3 ) ) )
= ( insert_o @ A @ bot_bot_set_o ) ) )
& ( ( C3 != bot_bot_set_nat )
=> ( ( insert_o @ A @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ B5 @ C3 ) ) )
= ( comple90263536869209701_set_o
@ ( image_nat_set_o
@ ^ [X4: nat] : ( insert_o @ A @ ( B5 @ X4 ) )
@ C3 ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_2108_UN__extend__simps_I1_J,axiom,
! [C3: set_Product_unit,A: nat,B5: product_unit > set_nat] :
( ( ( C3 = bot_bo3957492148770167129t_unit )
=> ( ( insert_nat @ A @ ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ B5 @ C3 ) ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) )
& ( ( C3 != bot_bo3957492148770167129t_unit )
=> ( ( insert_nat @ A @ ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ B5 @ C3 ) ) )
= ( comple7399068483239264473et_nat
@ ( image_8173849991827986510et_nat
@ ^ [X4: product_unit] : ( insert_nat @ A @ ( B5 @ X4 ) )
@ C3 ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_2109_UN__extend__simps_I1_J,axiom,
! [C3: set_o,A: nat,B5: $o > set_nat] :
( ( ( C3 = bot_bot_set_o )
=> ( ( insert_nat @ A @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B5 @ C3 ) ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) )
& ( ( C3 != bot_bot_set_o )
=> ( ( insert_nat @ A @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B5 @ C3 ) ) )
= ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [X4: $o] : ( insert_nat @ A @ ( B5 @ X4 ) )
@ C3 ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_2110_UN__extend__simps_I1_J,axiom,
! [C3: set_nat,A: nat,B5: nat > set_nat] :
( ( ( C3 = bot_bot_set_nat )
=> ( ( insert_nat @ A @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ C3 ) ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) )
& ( ( C3 != bot_bot_set_nat )
=> ( ( insert_nat @ A @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ C3 ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X4: nat] : ( insert_nat @ A @ ( B5 @ X4 ) )
@ C3 ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_2111_UN__extend__simps_I1_J,axiom,
! [C3: set_Product_unit,A: product_prod_nat_nat,B5: product_unit > set_Pr1261947904930325089at_nat] :
( ( ( C3 = bot_bo3957492148770167129t_unit )
=> ( ( insert8211810215607154385at_nat @ A @ ( comple5685304695842803022at_nat @ ( image_3617315916088684441at_nat @ B5 @ C3 ) ) )
= ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
& ( ( C3 != bot_bo3957492148770167129t_unit )
=> ( ( insert8211810215607154385at_nat @ A @ ( comple5685304695842803022at_nat @ ( image_3617315916088684441at_nat @ B5 @ C3 ) ) )
= ( comple5685304695842803022at_nat
@ ( image_3617315916088684441at_nat
@ ^ [X4: product_unit] : ( insert8211810215607154385at_nat @ A @ ( B5 @ X4 ) )
@ C3 ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_2112_Sigma__def,axiom,
( produc6258361011560675592at_nat
= ( ^ [A6: set_se5479201105039950375at_nat,B6: set_Pr4329608150637261639at_nat > set_se5479201105039950375at_nat] :
( comple7342073961973170484at_nat
@ ( image_1124969593608450885at_nat
@ ^ [X4: set_Pr4329608150637261639at_nat] :
( comple7342073961973170484at_nat
@ ( image_1124969593608450885at_nat
@ ^ [Y5: set_Pr4329608150637261639at_nat] : ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ X4 @ Y5 ) @ bot_bo4948859079157340979at_nat )
@ ( B6 @ X4 ) ) )
@ A6 ) ) ) ) ).
% Sigma_def
thf(fact_2113_Sigma__def,axiom,
( produc6237731129814287028at_num
= ( ^ [A6: set_nat,B6: nat > set_num] :
( comple1400524285282149784at_num
@ ( image_2893549341467670548at_num
@ ^ [X4: nat] :
( comple1400524285282149784at_num
@ ( image_4778453735051108682at_num
@ ^ [Y5: num] : ( insert8920054152555992091at_num @ ( product_Pair_nat_num @ X4 @ Y5 ) @ bot_bo7038385379056416535at_num )
@ ( B6 @ X4 ) ) )
@ A6 ) ) ) ) ).
% Sigma_def
thf(fact_2114_Sigma__def,axiom,
( produc1453890548195736866nt_int
= ( ^ [A6: set_int,B6: int > set_int] :
( comple5382143125604098054nt_int
@ ( image_8635204845542730022nt_int
@ ^ [X4: int] :
( comple5382143125604098054nt_int
@ ( image_8635204845542730022nt_int
@ ^ [Y5: int] : ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ X4 @ Y5 ) @ bot_bo1796632182523588997nt_int )
@ ( B6 @ X4 ) ) )
@ A6 ) ) ) ) ).
% Sigma_def
thf(fact_2115_Sigma__def,axiom,
( produc7726050920695093128at_nat
= ( ^ [A6: set_se7855581050983116737at_nat,B6: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat] :
( comple4364388969068068532at_nat
@ ( image_6898982319499645995at_nat
@ ^ [X4: set_Pr1261947904930325089at_nat] :
( comple4364388969068068532at_nat
@ ( image_6898982319499645995at_nat
@ ^ [Y5: set_Pr1261947904930325089at_nat] : ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y5 ) @ bot_bo228742789529271731at_nat )
@ ( B6 @ X4 ) ) )
@ A6 ) ) ) ) ).
% Sigma_def
thf(fact_2116_Sigma__def,axiom,
( produc457027306803732586at_nat
= ( ^ [A6: set_nat,B6: nat > set_nat] :
( comple5685304695842803022at_nat
@ ( image_7178329752028323786at_nat
@ ^ [X4: nat] :
( comple5685304695842803022at_nat
@ ( image_7178329752028323786at_nat
@ ^ [Y5: nat] : ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ bot_bo2099793752762293965at_nat )
@ ( B6 @ X4 ) ) )
@ A6 ) ) ) ) ).
% Sigma_def
thf(fact_2117_UN__Times__distrib,axiom,
! [E2: nat > set_nat,F4: nat > set_nat,A4: set_nat,B5: set_nat] :
( ( comple5685304695842803022at_nat
@ ( image_5423882813909314213at_nat
@ ( produc8197505143624133779at_nat
@ ^ [A5: nat,B4: nat] :
( produc457027306803732586at_nat @ ( E2 @ A5 )
@ ^ [Uu: nat] : ( F4 @ B4 ) ) )
@ ( produc457027306803732586at_nat @ A4
@ ^ [Uu: nat] : B5 ) ) )
= ( produc457027306803732586at_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ E2 @ A4 ) )
@ ^ [Uu: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F4 @ B5 ) ) ) ) ).
% UN_Times_distrib
thf(fact_2118_right__total__OO,axiom,
! [A4: product_prod_nat_nat > product_prod_nat_nat > $o,B5: product_prod_nat_nat > int > $o] :
( ( right_4094035864313884895at_nat @ A4 )
=> ( ( right_1157779049177982926at_int @ B5 )
=> ( right_1157779049177982926at_int @ ( relcom8911704206367525917at_int @ A4 @ B5 ) ) ) ) ).
% right_total_OO
thf(fact_2119_right__total__OO,axiom,
! [A4: int > int > $o,B5: int > code_integer > $o] :
( ( right_total_int_int @ A4 )
=> ( ( right_7281159819619925802nteger @ B5 )
=> ( right_7281159819619925802nteger @ ( relcom967497705283160040nteger @ A4 @ B5 ) ) ) ) ).
% right_total_OO
thf(fact_2120_refl__onD,axiom,
! [A4: set_Pr4329608150637261639at_nat,R: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat] :
( ( refl_o7446464703481863270at_nat @ A4 @ R )
=> ( ( member8757157785044589968at_nat @ A @ A4 )
=> ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ A ) @ R ) ) ) ).
% refl_onD
thf(fact_2121_refl__onD,axiom,
! [A4: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat,A: product_prod_nat_nat] :
( ( refl_o7122697186039111552at_nat @ A4 @ R )
=> ( ( member8440522571783428010at_nat @ A @ A4 )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ A ) @ R ) ) ) ).
% refl_onD
thf(fact_2122_refl__onD,axiom,
! [A4: set_o,R: set_Product_prod_o_o,A: $o] :
( ( refl_on_o @ A4 @ R )
=> ( ( member_o @ A @ A4 )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ A ) @ R ) ) ) ).
% refl_onD
thf(fact_2123_refl__onD,axiom,
! [A4: set_se5479201105039950375at_nat,R: set_Pr7459493094073627847at_nat,A: set_Pr4329608150637261639at_nat] :
( ( refl_o6272981002401706822at_nat @ A4 @ R )
=> ( ( member6273874620674182256at_nat @ A @ A4 )
=> ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ A ) @ R ) ) ) ).
% refl_onD
thf(fact_2124_refl__onD,axiom,
! [A4: set_se7855581050983116737at_nat,R: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat] :
( ( refl_o6955724340487615712at_nat @ A4 @ R )
=> ( ( member2643936169264416010at_nat @ A @ A4 )
=> ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ A ) @ R ) ) ) ).
% refl_onD
thf(fact_2125_refl__onD,axiom,
! [A4: set_nat,R: set_Pr1261947904930325089at_nat,A: nat] :
( ( refl_on_nat @ A4 @ R )
=> ( ( member_nat @ A @ A4 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A ) @ R ) ) ) ).
% refl_onD
thf(fact_2126_refl__onD,axiom,
! [A4: set_int,R: set_Pr958786334691620121nt_int,A: int] :
( ( refl_on_int @ A4 @ R )
=> ( ( member_int @ A @ A4 )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ A ) @ R ) ) ) ).
% refl_onD
thf(fact_2127_refl__onD1,axiom,
! [A4: set_Pr4329608150637261639at_nat,R: set_Pr5564308138774400199at_nat,X2: produc3843707927480180839at_nat,Y3: produc3843707927480180839at_nat] :
( ( refl_o7446464703481863270at_nat @ A4 @ R )
=> ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X2 @ Y3 ) @ R )
=> ( member8757157785044589968at_nat @ X2 @ A4 ) ) ) ).
% refl_onD1
thf(fact_2128_refl__onD1,axiom,
! [A4: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat,X2: product_prod_nat_nat,Y3: product_prod_nat_nat] :
( ( refl_o7122697186039111552at_nat @ A4 @ R )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y3 ) @ R )
=> ( member8440522571783428010at_nat @ X2 @ A4 ) ) ) ).
% refl_onD1
thf(fact_2129_refl__onD1,axiom,
! [A4: set_o,R: set_Product_prod_o_o,X2: $o,Y3: $o] :
( ( refl_on_o @ A4 @ R )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ Y3 ) @ R )
=> ( member_o @ X2 @ A4 ) ) ) ).
% refl_onD1
thf(fact_2130_refl__onD1,axiom,
! [A4: set_se5479201105039950375at_nat,R: set_Pr7459493094073627847at_nat,X2: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] :
( ( refl_o6272981002401706822at_nat @ A4 @ R )
=> ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X2 @ Y3 ) @ R )
=> ( member6273874620674182256at_nat @ X2 @ A4 ) ) ) ).
% refl_onD1
thf(fact_2131_refl__onD1,axiom,
! [A4: set_se7855581050983116737at_nat,R: set_Pr4329608150637261639at_nat,X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
( ( refl_o6955724340487615712at_nat @ A4 @ R )
=> ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y3 ) @ R )
=> ( member2643936169264416010at_nat @ X2 @ A4 ) ) ) ).
% refl_onD1
thf(fact_2132_refl__onD1,axiom,
! [A4: set_nat,R: set_Pr1261947904930325089at_nat,X2: nat,Y3: nat] :
( ( refl_on_nat @ A4 @ R )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ R )
=> ( member_nat @ X2 @ A4 ) ) ) ).
% refl_onD1
thf(fact_2133_refl__onD1,axiom,
! [A4: set_int,R: set_Pr958786334691620121nt_int,X2: int,Y3: int] :
( ( refl_on_int @ A4 @ R )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y3 ) @ R )
=> ( member_int @ X2 @ A4 ) ) ) ).
% refl_onD1
thf(fact_2134_refl__onD2,axiom,
! [A4: set_Pr4329608150637261639at_nat,R: set_Pr5564308138774400199at_nat,X2: produc3843707927480180839at_nat,Y3: produc3843707927480180839at_nat] :
( ( refl_o7446464703481863270at_nat @ A4 @ R )
=> ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X2 @ Y3 ) @ R )
=> ( member8757157785044589968at_nat @ Y3 @ A4 ) ) ) ).
% refl_onD2
thf(fact_2135_refl__onD2,axiom,
! [A4: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat,X2: product_prod_nat_nat,Y3: product_prod_nat_nat] :
( ( refl_o7122697186039111552at_nat @ A4 @ R )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y3 ) @ R )
=> ( member8440522571783428010at_nat @ Y3 @ A4 ) ) ) ).
% refl_onD2
thf(fact_2136_refl__onD2,axiom,
! [A4: set_o,R: set_Product_prod_o_o,X2: $o,Y3: $o] :
( ( refl_on_o @ A4 @ R )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ Y3 ) @ R )
=> ( member_o @ Y3 @ A4 ) ) ) ).
% refl_onD2
thf(fact_2137_refl__onD2,axiom,
! [A4: set_se5479201105039950375at_nat,R: set_Pr7459493094073627847at_nat,X2: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] :
( ( refl_o6272981002401706822at_nat @ A4 @ R )
=> ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X2 @ Y3 ) @ R )
=> ( member6273874620674182256at_nat @ Y3 @ A4 ) ) ) ).
% refl_onD2
thf(fact_2138_refl__onD2,axiom,
! [A4: set_se7855581050983116737at_nat,R: set_Pr4329608150637261639at_nat,X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
( ( refl_o6955724340487615712at_nat @ A4 @ R )
=> ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y3 ) @ R )
=> ( member2643936169264416010at_nat @ Y3 @ A4 ) ) ) ).
% refl_onD2
thf(fact_2139_refl__onD2,axiom,
! [A4: set_nat,R: set_Pr1261947904930325089at_nat,X2: nat,Y3: nat] :
( ( refl_on_nat @ A4 @ R )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ R )
=> ( member_nat @ Y3 @ A4 ) ) ) ).
% refl_onD2
thf(fact_2140_refl__onD2,axiom,
! [A4: set_int,R: set_Pr958786334691620121nt_int,X2: int,Y3: int] :
( ( refl_on_int @ A4 @ R )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y3 ) @ R )
=> ( member_int @ Y3 @ A4 ) ) ) ).
% refl_onD2
thf(fact_2141_INT__extend__simps_I8_J,axiom,
! [B5: nat > set_nat,A4: set_set_nat] :
( ( comple7806235888213564991et_nat
@ ( image_7916887816326733075et_nat
@ ^ [Y5: set_nat] : ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B5 @ Y5 ) )
@ A4 ) )
= ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B5 @ ( comple7399068483239264473et_nat @ A4 ) ) ) ) ).
% INT_extend_simps(8)
thf(fact_2142_Id__on__def,axiom,
( id_on_3275838214495744996at_nat
= ( ^ [A6: set_se5479201105039950375at_nat] :
( comple7342073961973170484at_nat
@ ( image_1124969593608450885at_nat
@ ^ [X4: set_Pr4329608150637261639at_nat] : ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ X4 @ X4 ) @ bot_bo4948859079157340979at_nat )
@ A6 ) ) ) ) ).
% Id_on_def
thf(fact_2143_Id__on__def,axiom,
( id_on_int
= ( ^ [A6: set_int] :
( comple5382143125604098054nt_int
@ ( image_8635204845542730022nt_int
@ ^ [X4: int] : ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ X4 @ X4 ) @ bot_bo1796632182523588997nt_int )
@ A6 ) ) ) ) ).
% Id_on_def
thf(fact_2144_Id__on__def,axiom,
( id_on_6713504163309144702at_nat
= ( ^ [A6: set_se7855581050983116737at_nat] :
( comple4364388969068068532at_nat
@ ( image_6898982319499645995at_nat
@ ^ [X4: set_Pr1261947904930325089at_nat] : ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ X4 @ X4 ) @ bot_bo228742789529271731at_nat )
@ A6 ) ) ) ) ).
% Id_on_def
thf(fact_2145_Id__on__def,axiom,
( id_on_nat
= ( ^ [A6: set_nat] :
( comple5685304695842803022at_nat
@ ( image_7178329752028323786at_nat
@ ^ [X4: nat] : ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X4 @ X4 ) @ bot_bo2099793752762293965at_nat )
@ A6 ) ) ) ) ).
% Id_on_def
thf(fact_2146_INF__le__SUP,axiom,
! [A4: set_Product_unit,F: product_unit > set_nat] :
( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_8173849991827986510et_nat @ F @ A4 ) ) @ ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ F @ A4 ) ) ) ) ).
% INF_le_SUP
thf(fact_2147_INF__le__SUP,axiom,
! [A4: set_Pr4329608150637261639at_nat,F: produc3843707927480180839at_nat > set_nat] :
( ( A4 != bot_bo228742789529271731at_nat )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_3160451284896210088et_nat @ F @ A4 ) ) @ ( comple7399068483239264473et_nat @ ( image_3160451284896210088et_nat @ F @ A4 ) ) ) ) ).
% INF_le_SUP
thf(fact_2148_INF__le__SUP,axiom,
! [A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > set_nat] :
( ( A4 != bot_bo2099793752762293965at_nat )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_15824709712370754et_nat @ F @ A4 ) ) @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ F @ A4 ) ) ) ) ).
% INF_le_SUP
thf(fact_2149_INF__le__SUP,axiom,
! [A4: set_o,F: $o > set_nat] :
( ( A4 != bot_bot_set_o )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ F @ A4 ) ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A4 ) ) ) ) ).
% INF_le_SUP
thf(fact_2150_INF__le__SUP,axiom,
! [A4: set_nat,F: nat > set_nat] :
( ( A4 != bot_bot_set_nat )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ A4 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A4 ) ) ) ) ).
% INF_le_SUP
thf(fact_2151_INF__le__SUP,axiom,
! [A4: set_Product_unit,F: product_unit > product_unit] :
( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ord_le3221252021190050221t_unit @ ( comple2584293577114468500t_unit @ ( image_405062704495631173t_unit @ F @ A4 ) ) @ ( comple4687483117567863418t_unit @ ( image_405062704495631173t_unit @ F @ A4 ) ) ) ) ).
% INF_le_SUP
thf(fact_2152_INF__le__SUP,axiom,
! [A4: set_Pr4329608150637261639at_nat,F: produc3843707927480180839at_nat > product_unit] :
( ( A4 != bot_bo228742789529271731at_nat )
=> ( ord_le3221252021190050221t_unit @ ( comple2584293577114468500t_unit @ ( image_7082456443403096939t_unit @ F @ A4 ) ) @ ( comple4687483117567863418t_unit @ ( image_7082456443403096939t_unit @ F @ A4 ) ) ) ) ).
% INF_le_SUP
thf(fact_2153_INF__le__SUP,axiom,
! [A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > product_unit] :
( ( A4 != bot_bo2099793752762293965at_nat )
=> ( ord_le3221252021190050221t_unit @ ( comple2584293577114468500t_unit @ ( image_6169909464789311569t_unit @ F @ A4 ) ) @ ( comple4687483117567863418t_unit @ ( image_6169909464789311569t_unit @ F @ A4 ) ) ) ) ).
% INF_le_SUP
thf(fact_2154_INF__le__SUP,axiom,
! [A4: set_o,F: $o > product_unit] :
( ( A4 != bot_bot_set_o )
=> ( ord_le3221252021190050221t_unit @ ( comple2584293577114468500t_unit @ ( image_o_Product_unit @ F @ A4 ) ) @ ( comple4687483117567863418t_unit @ ( image_o_Product_unit @ F @ A4 ) ) ) ) ).
% INF_le_SUP
thf(fact_2155_INF__le__SUP,axiom,
! [A4: set_nat,F: nat > product_unit] :
( ( A4 != bot_bot_set_nat )
=> ( ord_le3221252021190050221t_unit @ ( comple2584293577114468500t_unit @ ( image_8730104196221521654t_unit @ F @ A4 ) ) @ ( comple4687483117567863418t_unit @ ( image_8730104196221521654t_unit @ F @ A4 ) ) ) ) ).
% INF_le_SUP
thf(fact_2156_right__unique__OO,axiom,
! [A4: product_prod_nat_nat > product_prod_nat_nat > $o,B5: product_prod_nat_nat > int > $o] :
( ( right_1383565826979811350at_nat @ A4 )
=> ( ( right_8621877368619751767at_int @ B5 )
=> ( right_8621877368619751767at_int @ ( relcom8911704206367525917at_int @ A4 @ B5 ) ) ) ) ).
% right_unique_OO
thf(fact_2157_right__unique__OO,axiom,
! [A4: int > int > $o,B5: int > code_integer > $o] :
( ( right_unique_int_int @ A4 )
=> ( ( right_8475562212078014689nteger @ B5 )
=> ( right_8475562212078014689nteger @ ( relcom967497705283160040nteger @ A4 @ B5 ) ) ) ) ).
% right_unique_OO
thf(fact_2158_INT__E,axiom,
! [B: $o,B5: $o > set_o,A4: set_o,A: $o] :
( ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ B5 @ A4 ) ) )
=> ( ~ ( member_o @ B @ ( B5 @ A ) )
=> ~ ( member_o @ A @ A4 ) ) ) ).
% INT_E
thf(fact_2159_INT__E,axiom,
! [B: $o,B5: nat > set_o,A4: set_nat,A: nat] :
( ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_nat_set_o @ B5 @ A4 ) ) )
=> ( ~ ( member_o @ B @ ( B5 @ A ) )
=> ~ ( member_nat @ A @ A4 ) ) ) ).
% INT_E
thf(fact_2160_INT__E,axiom,
! [B: $o,B5: int > set_o,A4: set_int,A: int] :
( ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_int_set_o @ B5 @ A4 ) ) )
=> ( ~ ( member_o @ B @ ( B5 @ A ) )
=> ~ ( member_int @ A @ A4 ) ) ) ).
% INT_E
thf(fact_2161_INT__E,axiom,
! [B: int,B5: $o > set_int,A4: set_o,A: $o] :
( ( member_int @ B @ ( comple3628384868704368283et_int @ ( image_o_set_int @ B5 @ A4 ) ) )
=> ( ~ ( member_int @ B @ ( B5 @ A ) )
=> ~ ( member_o @ A @ A4 ) ) ) ).
% INT_E
thf(fact_2162_INT__E,axiom,
! [B: int,B5: nat > set_int,A4: set_nat,A: nat] :
( ( member_int @ B @ ( comple3628384868704368283et_int @ ( image_nat_set_int @ B5 @ A4 ) ) )
=> ( ~ ( member_int @ B @ ( B5 @ A ) )
=> ~ ( member_nat @ A @ A4 ) ) ) ).
% INT_E
thf(fact_2163_INT__E,axiom,
! [B: int,B5: int > set_int,A4: set_int,A: int] :
( ( member_int @ B @ ( comple3628384868704368283et_int @ ( image_int_set_int @ B5 @ A4 ) ) )
=> ( ~ ( member_int @ B @ ( B5 @ A ) )
=> ~ ( member_int @ A @ A4 ) ) ) ).
% INT_E
thf(fact_2164_INT__E,axiom,
! [B: nat,B5: $o > set_nat,A4: set_o,A: $o] :
( ( member_nat @ B @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ B5 @ A4 ) ) )
=> ( ~ ( member_nat @ B @ ( B5 @ A ) )
=> ~ ( member_o @ A @ A4 ) ) ) ).
% INT_E
thf(fact_2165_INT__E,axiom,
! [B: nat,B5: nat > set_nat,A4: set_nat,A: nat] :
( ( member_nat @ B @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B5 @ A4 ) ) )
=> ( ~ ( member_nat @ B @ ( B5 @ A ) )
=> ~ ( member_nat @ A @ A4 ) ) ) ).
% INT_E
thf(fact_2166_INT__E,axiom,
! [B: nat,B5: int > set_nat,A4: set_int,A: int] :
( ( member_nat @ B @ ( comple7806235888213564991et_nat @ ( image_int_set_nat @ B5 @ A4 ) ) )
=> ( ~ ( member_nat @ B @ ( B5 @ A ) )
=> ~ ( member_int @ A @ A4 ) ) ) ).
% INT_E
thf(fact_2167_INT__E,axiom,
! [B: product_prod_nat_nat,B5: $o > set_Pr1261947904930325089at_nat,A4: set_o,A: $o] :
( ( member8440522571783428010at_nat @ B @ ( comple1930127330929219688at_nat @ ( image_7077428525161401678at_nat @ B5 @ A4 ) ) )
=> ( ~ ( member8440522571783428010at_nat @ B @ ( B5 @ A ) )
=> ~ ( member_o @ A @ A4 ) ) ) ).
% INT_E
thf(fact_2168_INT__D,axiom,
! [B: $o,B5: $o > set_o,A4: set_o,A: $o] :
( ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ B5 @ A4 ) ) )
=> ( ( member_o @ A @ A4 )
=> ( member_o @ B @ ( B5 @ A ) ) ) ) ).
% INT_D
thf(fact_2169_INT__D,axiom,
! [B: $o,B5: nat > set_o,A4: set_nat,A: nat] :
( ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_nat_set_o @ B5 @ A4 ) ) )
=> ( ( member_nat @ A @ A4 )
=> ( member_o @ B @ ( B5 @ A ) ) ) ) ).
% INT_D
thf(fact_2170_INT__D,axiom,
! [B: $o,B5: int > set_o,A4: set_int,A: int] :
( ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_int_set_o @ B5 @ A4 ) ) )
=> ( ( member_int @ A @ A4 )
=> ( member_o @ B @ ( B5 @ A ) ) ) ) ).
% INT_D
thf(fact_2171_INT__D,axiom,
! [B: int,B5: $o > set_int,A4: set_o,A: $o] :
( ( member_int @ B @ ( comple3628384868704368283et_int @ ( image_o_set_int @ B5 @ A4 ) ) )
=> ( ( member_o @ A @ A4 )
=> ( member_int @ B @ ( B5 @ A ) ) ) ) ).
% INT_D
thf(fact_2172_INT__D,axiom,
! [B: int,B5: nat > set_int,A4: set_nat,A: nat] :
( ( member_int @ B @ ( comple3628384868704368283et_int @ ( image_nat_set_int @ B5 @ A4 ) ) )
=> ( ( member_nat @ A @ A4 )
=> ( member_int @ B @ ( B5 @ A ) ) ) ) ).
% INT_D
thf(fact_2173_INT__D,axiom,
! [B: int,B5: int > set_int,A4: set_int,A: int] :
( ( member_int @ B @ ( comple3628384868704368283et_int @ ( image_int_set_int @ B5 @ A4 ) ) )
=> ( ( member_int @ A @ A4 )
=> ( member_int @ B @ ( B5 @ A ) ) ) ) ).
% INT_D
thf(fact_2174_INT__D,axiom,
! [B: nat,B5: $o > set_nat,A4: set_o,A: $o] :
( ( member_nat @ B @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ B5 @ A4 ) ) )
=> ( ( member_o @ A @ A4 )
=> ( member_nat @ B @ ( B5 @ A ) ) ) ) ).
% INT_D
thf(fact_2175_INT__D,axiom,
! [B: nat,B5: nat > set_nat,A4: set_nat,A: nat] :
( ( member_nat @ B @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B5 @ A4 ) ) )
=> ( ( member_nat @ A @ A4 )
=> ( member_nat @ B @ ( B5 @ A ) ) ) ) ).
% INT_D
thf(fact_2176_INT__D,axiom,
! [B: nat,B5: int > set_nat,A4: set_int,A: int] :
( ( member_nat @ B @ ( comple7806235888213564991et_nat @ ( image_int_set_nat @ B5 @ A4 ) ) )
=> ( ( member_int @ A @ A4 )
=> ( member_nat @ B @ ( B5 @ A ) ) ) ) ).
% INT_D
thf(fact_2177_INT__D,axiom,
! [B: product_prod_nat_nat,B5: $o > set_Pr1261947904930325089at_nat,A4: set_o,A: $o] :
( ( member8440522571783428010at_nat @ B @ ( comple1930127330929219688at_nat @ ( image_7077428525161401678at_nat @ B5 @ A4 ) ) )
=> ( ( member_o @ A @ A4 )
=> ( member8440522571783428010at_nat @ B @ ( B5 @ A ) ) ) ) ).
% INT_D
thf(fact_2178_UN__E,axiom,
! [B: $o,B5: $o > set_o,A4: set_o] :
( ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B5 @ A4 ) ) )
=> ~ ! [X: $o] :
( ( member_o @ X @ A4 )
=> ~ ( member_o @ B @ ( B5 @ X ) ) ) ) ).
% UN_E
thf(fact_2179_UN__E,axiom,
! [B: $o,B5: nat > set_o,A4: set_nat] :
( ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ B5 @ A4 ) ) )
=> ~ ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ~ ( member_o @ B @ ( B5 @ X ) ) ) ) ).
% UN_E
thf(fact_2180_UN__E,axiom,
! [B: $o,B5: int > set_o,A4: set_int] :
( ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_int_set_o @ B5 @ A4 ) ) )
=> ~ ! [X: int] :
( ( member_int @ X @ A4 )
=> ~ ( member_o @ B @ ( B5 @ X ) ) ) ) ).
% UN_E
thf(fact_2181_UN__E,axiom,
! [B: int,B5: $o > set_int,A4: set_o] :
( ( member_int @ B @ ( comple3221217463730067765et_int @ ( image_o_set_int @ B5 @ A4 ) ) )
=> ~ ! [X: $o] :
( ( member_o @ X @ A4 )
=> ~ ( member_int @ B @ ( B5 @ X ) ) ) ) ).
% UN_E
thf(fact_2182_UN__E,axiom,
! [B: int,B5: nat > set_int,A4: set_nat] :
( ( member_int @ B @ ( comple3221217463730067765et_int @ ( image_nat_set_int @ B5 @ A4 ) ) )
=> ~ ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ~ ( member_int @ B @ ( B5 @ X ) ) ) ) ).
% UN_E
thf(fact_2183_UN__E,axiom,
! [B: int,B5: int > set_int,A4: set_int] :
( ( member_int @ B @ ( comple3221217463730067765et_int @ ( image_int_set_int @ B5 @ A4 ) ) )
=> ~ ! [X: int] :
( ( member_int @ X @ A4 )
=> ~ ( member_int @ B @ ( B5 @ X ) ) ) ) ).
% UN_E
thf(fact_2184_UN__E,axiom,
! [B: nat,B5: $o > set_nat,A4: set_o] :
( ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B5 @ A4 ) ) )
=> ~ ! [X: $o] :
( ( member_o @ X @ A4 )
=> ~ ( member_nat @ B @ ( B5 @ X ) ) ) ) ).
% UN_E
thf(fact_2185_UN__E,axiom,
! [B: nat,B5: nat > set_nat,A4: set_nat] :
( ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ A4 ) ) )
=> ~ ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ~ ( member_nat @ B @ ( B5 @ X ) ) ) ) ).
% UN_E
thf(fact_2186_UN__E,axiom,
! [B: nat,B5: int > set_nat,A4: set_int] :
( ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_int_set_nat @ B5 @ A4 ) ) )
=> ~ ! [X: int] :
( ( member_int @ X @ A4 )
=> ~ ( member_nat @ B @ ( B5 @ X ) ) ) ) ).
% UN_E
thf(fact_2187_UN__E,axiom,
! [B: product_prod_nat_nat,B5: $o > set_Pr1261947904930325089at_nat,A4: set_o] :
( ( member8440522571783428010at_nat @ B @ ( comple5685304695842803022at_nat @ ( image_7077428525161401678at_nat @ B5 @ A4 ) ) )
=> ~ ! [X: $o] :
( ( member_o @ X @ A4 )
=> ~ ( member8440522571783428010at_nat @ B @ ( B5 @ X ) ) ) ) ).
% UN_E
thf(fact_2188_UNION__singleton__eq__range,axiom,
! [F: nat > rat,A4: set_nat] :
( ( comple3890839924845867745et_rat
@ ( image_nat_set_rat
@ ^ [X4: nat] : ( insert_rat @ ( F @ X4 ) @ bot_bot_set_rat )
@ A4 ) )
= ( image_nat_rat @ F @ A4 ) ) ).
% UNION_singleton_eq_range
thf(fact_2189_UNION__singleton__eq__range,axiom,
! [F: nat > int,A4: set_nat] :
( ( comple3221217463730067765et_int
@ ( image_nat_set_int
@ ^ [X4: nat] : ( insert_int @ ( F @ X4 ) @ bot_bot_set_int )
@ A4 ) )
= ( image_nat_int @ F @ A4 ) ) ).
% UNION_singleton_eq_range
thf(fact_2190_UNION__singleton__eq__range,axiom,
! [F: int > int,A4: set_int] :
( ( comple3221217463730067765et_int
@ ( image_int_set_int
@ ^ [X4: int] : ( insert_int @ ( F @ X4 ) @ bot_bot_set_int )
@ A4 ) )
= ( image_int_int @ F @ A4 ) ) ).
% UNION_singleton_eq_range
thf(fact_2191_UNION__singleton__eq__range,axiom,
! [F: nat > nat,A4: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X4: nat] : ( insert_nat @ ( F @ X4 ) @ bot_bot_set_nat )
@ A4 ) )
= ( image_nat_nat @ F @ A4 ) ) ).
% UNION_singleton_eq_range
thf(fact_2192_UNION__singleton__eq__range,axiom,
! [F: int > nat,A4: set_int] :
( ( comple7399068483239264473et_nat
@ ( image_int_set_nat
@ ^ [X4: int] : ( insert_nat @ ( F @ X4 ) @ bot_bot_set_nat )
@ A4 ) )
= ( image_int_nat @ F @ A4 ) ) ).
% UNION_singleton_eq_range
thf(fact_2193_INF__eq,axiom,
! [A4: set_o,B5: set_o,G: $o > product_unit,F: $o > product_unit] :
( ! [I4: $o] :
( ( member_o @ I4 @ A4 )
=> ? [X6: $o] :
( ( member_o @ X6 @ B5 )
& ( ord_le3221252021190050221t_unit @ ( G @ X6 ) @ ( F @ I4 ) ) ) )
=> ( ! [J2: $o] :
( ( member_o @ J2 @ B5 )
=> ? [X6: $o] :
( ( member_o @ X6 @ A4 )
& ( ord_le3221252021190050221t_unit @ ( F @ X6 ) @ ( G @ J2 ) ) ) )
=> ( ( comple2584293577114468500t_unit @ ( image_o_Product_unit @ F @ A4 ) )
= ( comple2584293577114468500t_unit @ ( image_o_Product_unit @ G @ B5 ) ) ) ) ) ).
% INF_eq
thf(fact_2194_INF__eq,axiom,
! [A4: set_o,B5: set_nat,G: nat > product_unit,F: $o > product_unit] :
( ! [I4: $o] :
( ( member_o @ I4 @ A4 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ B5 )
& ( ord_le3221252021190050221t_unit @ ( G @ X6 ) @ ( F @ I4 ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat @ J2 @ B5 )
=> ? [X6: $o] :
( ( member_o @ X6 @ A4 )
& ( ord_le3221252021190050221t_unit @ ( F @ X6 ) @ ( G @ J2 ) ) ) )
=> ( ( comple2584293577114468500t_unit @ ( image_o_Product_unit @ F @ A4 ) )
= ( comple2584293577114468500t_unit @ ( image_8730104196221521654t_unit @ G @ B5 ) ) ) ) ) ).
% INF_eq
thf(fact_2195_INF__eq,axiom,
! [A4: set_o,B5: set_int,G: int > product_unit,F: $o > product_unit] :
( ! [I4: $o] :
( ( member_o @ I4 @ A4 )
=> ? [X6: int] :
( ( member_int @ X6 @ B5 )
& ( ord_le3221252021190050221t_unit @ ( G @ X6 ) @ ( F @ I4 ) ) ) )
=> ( ! [J2: int] :
( ( member_int @ J2 @ B5 )
=> ? [X6: $o] :
( ( member_o @ X6 @ A4 )
& ( ord_le3221252021190050221t_unit @ ( F @ X6 ) @ ( G @ J2 ) ) ) )
=> ( ( comple2584293577114468500t_unit @ ( image_o_Product_unit @ F @ A4 ) )
= ( comple2584293577114468500t_unit @ ( image_5846526143423701914t_unit @ G @ B5 ) ) ) ) ) ).
% INF_eq
thf(fact_2196_INF__eq,axiom,
! [A4: set_nat,B5: set_o,G: $o > product_unit,F: nat > product_unit] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A4 )
=> ? [X6: $o] :
( ( member_o @ X6 @ B5 )
& ( ord_le3221252021190050221t_unit @ ( G @ X6 ) @ ( F @ I4 ) ) ) )
=> ( ! [J2: $o] :
( ( member_o @ J2 @ B5 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A4 )
& ( ord_le3221252021190050221t_unit @ ( F @ X6 ) @ ( G @ J2 ) ) ) )
=> ( ( comple2584293577114468500t_unit @ ( image_8730104196221521654t_unit @ F @ A4 ) )
= ( comple2584293577114468500t_unit @ ( image_o_Product_unit @ G @ B5 ) ) ) ) ) ).
% INF_eq
thf(fact_2197_INF__eq,axiom,
! [A4: set_nat,B5: set_nat,G: nat > product_unit,F: nat > product_unit] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A4 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ B5 )
& ( ord_le3221252021190050221t_unit @ ( G @ X6 ) @ ( F @ I4 ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat @ J2 @ B5 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A4 )
& ( ord_le3221252021190050221t_unit @ ( F @ X6 ) @ ( G @ J2 ) ) ) )
=> ( ( comple2584293577114468500t_unit @ ( image_8730104196221521654t_unit @ F @ A4 ) )
= ( comple2584293577114468500t_unit @ ( image_8730104196221521654t_unit @ G @ B5 ) ) ) ) ) ).
% INF_eq
thf(fact_2198_INF__eq,axiom,
! [A4: set_nat,B5: set_int,G: int > product_unit,F: nat > product_unit] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A4 )
=> ? [X6: int] :
( ( member_int @ X6 @ B5 )
& ( ord_le3221252021190050221t_unit @ ( G @ X6 ) @ ( F @ I4 ) ) ) )
=> ( ! [J2: int] :
( ( member_int @ J2 @ B5 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A4 )
& ( ord_le3221252021190050221t_unit @ ( F @ X6 ) @ ( G @ J2 ) ) ) )
=> ( ( comple2584293577114468500t_unit @ ( image_8730104196221521654t_unit @ F @ A4 ) )
= ( comple2584293577114468500t_unit @ ( image_5846526143423701914t_unit @ G @ B5 ) ) ) ) ) ).
% INF_eq
thf(fact_2199_INF__eq,axiom,
! [A4: set_int,B5: set_o,G: $o > product_unit,F: int > product_unit] :
( ! [I4: int] :
( ( member_int @ I4 @ A4 )
=> ? [X6: $o] :
( ( member_o @ X6 @ B5 )
& ( ord_le3221252021190050221t_unit @ ( G @ X6 ) @ ( F @ I4 ) ) ) )
=> ( ! [J2: $o] :
( ( member_o @ J2 @ B5 )
=> ? [X6: int] :
( ( member_int @ X6 @ A4 )
& ( ord_le3221252021190050221t_unit @ ( F @ X6 ) @ ( G @ J2 ) ) ) )
=> ( ( comple2584293577114468500t_unit @ ( image_5846526143423701914t_unit @ F @ A4 ) )
= ( comple2584293577114468500t_unit @ ( image_o_Product_unit @ G @ B5 ) ) ) ) ) ).
% INF_eq
thf(fact_2200_INF__eq,axiom,
! [A4: set_int,B5: set_nat,G: nat > product_unit,F: int > product_unit] :
( ! [I4: int] :
( ( member_int @ I4 @ A4 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ B5 )
& ( ord_le3221252021190050221t_unit @ ( G @ X6 ) @ ( F @ I4 ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat @ J2 @ B5 )
=> ? [X6: int] :
( ( member_int @ X6 @ A4 )
& ( ord_le3221252021190050221t_unit @ ( F @ X6 ) @ ( G @ J2 ) ) ) )
=> ( ( comple2584293577114468500t_unit @ ( image_5846526143423701914t_unit @ F @ A4 ) )
= ( comple2584293577114468500t_unit @ ( image_8730104196221521654t_unit @ G @ B5 ) ) ) ) ) ).
% INF_eq
thf(fact_2201_INF__eq,axiom,
! [A4: set_int,B5: set_int,G: int > product_unit,F: int > product_unit] :
( ! [I4: int] :
( ( member_int @ I4 @ A4 )
=> ? [X6: int] :
( ( member_int @ X6 @ B5 )
& ( ord_le3221252021190050221t_unit @ ( G @ X6 ) @ ( F @ I4 ) ) ) )
=> ( ! [J2: int] :
( ( member_int @ J2 @ B5 )
=> ? [X6: int] :
( ( member_int @ X6 @ A4 )
& ( ord_le3221252021190050221t_unit @ ( F @ X6 ) @ ( G @ J2 ) ) ) )
=> ( ( comple2584293577114468500t_unit @ ( image_5846526143423701914t_unit @ F @ A4 ) )
= ( comple2584293577114468500t_unit @ ( image_5846526143423701914t_unit @ G @ B5 ) ) ) ) ) ).
% INF_eq
thf(fact_2202_INF__eq,axiom,
! [A4: set_o,B5: set_o,G: $o > set_nat,F: $o > set_nat] :
( ! [I4: $o] :
( ( member_o @ I4 @ A4 )
=> ? [X6: $o] :
( ( member_o @ X6 @ B5 )
& ( ord_less_eq_set_nat @ ( G @ X6 ) @ ( F @ I4 ) ) ) )
=> ( ! [J2: $o] :
( ( member_o @ J2 @ B5 )
=> ? [X6: $o] :
( ( member_o @ X6 @ A4 )
& ( ord_less_eq_set_nat @ ( F @ X6 ) @ ( G @ J2 ) ) ) )
=> ( ( comple7806235888213564991et_nat @ ( image_o_set_nat @ F @ A4 ) )
= ( comple7806235888213564991et_nat @ ( image_o_set_nat @ G @ B5 ) ) ) ) ) ).
% INF_eq
thf(fact_2203_SUP__eq,axiom,
! [A4: set_o,B5: set_o,F: $o > product_unit,G: $o > product_unit] :
( ! [I4: $o] :
( ( member_o @ I4 @ A4 )
=> ? [X6: $o] :
( ( member_o @ X6 @ B5 )
& ( ord_le3221252021190050221t_unit @ ( F @ I4 ) @ ( G @ X6 ) ) ) )
=> ( ! [J2: $o] :
( ( member_o @ J2 @ B5 )
=> ? [X6: $o] :
( ( member_o @ X6 @ A4 )
& ( ord_le3221252021190050221t_unit @ ( G @ J2 ) @ ( F @ X6 ) ) ) )
=> ( ( comple4687483117567863418t_unit @ ( image_o_Product_unit @ F @ A4 ) )
= ( comple4687483117567863418t_unit @ ( image_o_Product_unit @ G @ B5 ) ) ) ) ) ).
% SUP_eq
thf(fact_2204_SUP__eq,axiom,
! [A4: set_o,B5: set_nat,F: $o > product_unit,G: nat > product_unit] :
( ! [I4: $o] :
( ( member_o @ I4 @ A4 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ B5 )
& ( ord_le3221252021190050221t_unit @ ( F @ I4 ) @ ( G @ X6 ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat @ J2 @ B5 )
=> ? [X6: $o] :
( ( member_o @ X6 @ A4 )
& ( ord_le3221252021190050221t_unit @ ( G @ J2 ) @ ( F @ X6 ) ) ) )
=> ( ( comple4687483117567863418t_unit @ ( image_o_Product_unit @ F @ A4 ) )
= ( comple4687483117567863418t_unit @ ( image_8730104196221521654t_unit @ G @ B5 ) ) ) ) ) ).
% SUP_eq
thf(fact_2205_SUP__eq,axiom,
! [A4: set_o,B5: set_int,F: $o > product_unit,G: int > product_unit] :
( ! [I4: $o] :
( ( member_o @ I4 @ A4 )
=> ? [X6: int] :
( ( member_int @ X6 @ B5 )
& ( ord_le3221252021190050221t_unit @ ( F @ I4 ) @ ( G @ X6 ) ) ) )
=> ( ! [J2: int] :
( ( member_int @ J2 @ B5 )
=> ? [X6: $o] :
( ( member_o @ X6 @ A4 )
& ( ord_le3221252021190050221t_unit @ ( G @ J2 ) @ ( F @ X6 ) ) ) )
=> ( ( comple4687483117567863418t_unit @ ( image_o_Product_unit @ F @ A4 ) )
= ( comple4687483117567863418t_unit @ ( image_5846526143423701914t_unit @ G @ B5 ) ) ) ) ) ).
% SUP_eq
thf(fact_2206_SUP__eq,axiom,
! [A4: set_nat,B5: set_o,F: nat > product_unit,G: $o > product_unit] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A4 )
=> ? [X6: $o] :
( ( member_o @ X6 @ B5 )
& ( ord_le3221252021190050221t_unit @ ( F @ I4 ) @ ( G @ X6 ) ) ) )
=> ( ! [J2: $o] :
( ( member_o @ J2 @ B5 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A4 )
& ( ord_le3221252021190050221t_unit @ ( G @ J2 ) @ ( F @ X6 ) ) ) )
=> ( ( comple4687483117567863418t_unit @ ( image_8730104196221521654t_unit @ F @ A4 ) )
= ( comple4687483117567863418t_unit @ ( image_o_Product_unit @ G @ B5 ) ) ) ) ) ).
% SUP_eq
thf(fact_2207_SUP__eq,axiom,
! [A4: set_nat,B5: set_nat,F: nat > product_unit,G: nat > product_unit] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A4 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ B5 )
& ( ord_le3221252021190050221t_unit @ ( F @ I4 ) @ ( G @ X6 ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat @ J2 @ B5 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A4 )
& ( ord_le3221252021190050221t_unit @ ( G @ J2 ) @ ( F @ X6 ) ) ) )
=> ( ( comple4687483117567863418t_unit @ ( image_8730104196221521654t_unit @ F @ A4 ) )
= ( comple4687483117567863418t_unit @ ( image_8730104196221521654t_unit @ G @ B5 ) ) ) ) ) ).
% SUP_eq
thf(fact_2208_SUP__eq,axiom,
! [A4: set_nat,B5: set_int,F: nat > product_unit,G: int > product_unit] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A4 )
=> ? [X6: int] :
( ( member_int @ X6 @ B5 )
& ( ord_le3221252021190050221t_unit @ ( F @ I4 ) @ ( G @ X6 ) ) ) )
=> ( ! [J2: int] :
( ( member_int @ J2 @ B5 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A4 )
& ( ord_le3221252021190050221t_unit @ ( G @ J2 ) @ ( F @ X6 ) ) ) )
=> ( ( comple4687483117567863418t_unit @ ( image_8730104196221521654t_unit @ F @ A4 ) )
= ( comple4687483117567863418t_unit @ ( image_5846526143423701914t_unit @ G @ B5 ) ) ) ) ) ).
% SUP_eq
thf(fact_2209_SUP__eq,axiom,
! [A4: set_int,B5: set_o,F: int > product_unit,G: $o > product_unit] :
( ! [I4: int] :
( ( member_int @ I4 @ A4 )
=> ? [X6: $o] :
( ( member_o @ X6 @ B5 )
& ( ord_le3221252021190050221t_unit @ ( F @ I4 ) @ ( G @ X6 ) ) ) )
=> ( ! [J2: $o] :
( ( member_o @ J2 @ B5 )
=> ? [X6: int] :
( ( member_int @ X6 @ A4 )
& ( ord_le3221252021190050221t_unit @ ( G @ J2 ) @ ( F @ X6 ) ) ) )
=> ( ( comple4687483117567863418t_unit @ ( image_5846526143423701914t_unit @ F @ A4 ) )
= ( comple4687483117567863418t_unit @ ( image_o_Product_unit @ G @ B5 ) ) ) ) ) ).
% SUP_eq
thf(fact_2210_SUP__eq,axiom,
! [A4: set_int,B5: set_nat,F: int > product_unit,G: nat > product_unit] :
( ! [I4: int] :
( ( member_int @ I4 @ A4 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ B5 )
& ( ord_le3221252021190050221t_unit @ ( F @ I4 ) @ ( G @ X6 ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat @ J2 @ B5 )
=> ? [X6: int] :
( ( member_int @ X6 @ A4 )
& ( ord_le3221252021190050221t_unit @ ( G @ J2 ) @ ( F @ X6 ) ) ) )
=> ( ( comple4687483117567863418t_unit @ ( image_5846526143423701914t_unit @ F @ A4 ) )
= ( comple4687483117567863418t_unit @ ( image_8730104196221521654t_unit @ G @ B5 ) ) ) ) ) ).
% SUP_eq
thf(fact_2211_SUP__eq,axiom,
! [A4: set_int,B5: set_int,F: int > product_unit,G: int > product_unit] :
( ! [I4: int] :
( ( member_int @ I4 @ A4 )
=> ? [X6: int] :
( ( member_int @ X6 @ B5 )
& ( ord_le3221252021190050221t_unit @ ( F @ I4 ) @ ( G @ X6 ) ) ) )
=> ( ! [J2: int] :
( ( member_int @ J2 @ B5 )
=> ? [X6: int] :
( ( member_int @ X6 @ A4 )
& ( ord_le3221252021190050221t_unit @ ( G @ J2 ) @ ( F @ X6 ) ) ) )
=> ( ( comple4687483117567863418t_unit @ ( image_5846526143423701914t_unit @ F @ A4 ) )
= ( comple4687483117567863418t_unit @ ( image_5846526143423701914t_unit @ G @ B5 ) ) ) ) ) ).
% SUP_eq
thf(fact_2212_SUP__eq,axiom,
! [A4: set_o,B5: set_o,F: $o > set_nat,G: $o > set_nat] :
( ! [I4: $o] :
( ( member_o @ I4 @ A4 )
=> ? [X6: $o] :
( ( member_o @ X6 @ B5 )
& ( ord_less_eq_set_nat @ ( F @ I4 ) @ ( G @ X6 ) ) ) )
=> ( ! [J2: $o] :
( ( member_o @ J2 @ B5 )
=> ? [X6: $o] :
( ( member_o @ X6 @ A4 )
& ( ord_less_eq_set_nat @ ( G @ J2 ) @ ( F @ X6 ) ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A4 ) )
= ( comple7399068483239264473et_nat @ ( image_o_set_nat @ G @ B5 ) ) ) ) ) ).
% SUP_eq
thf(fact_2213_INF__eq__const,axiom,
! [I3: set_int,F: int > product_unit,X2: product_unit] :
( ( I3 != bot_bot_set_int )
=> ( ! [I4: int] :
( ( member_int @ I4 @ I3 )
=> ( ( F @ I4 )
= X2 ) )
=> ( ( comple2584293577114468500t_unit @ ( image_5846526143423701914t_unit @ F @ I3 ) )
= X2 ) ) ) ).
% INF_eq_const
thf(fact_2214_INF__eq__const,axiom,
! [I3: set_Product_unit,F: product_unit > product_unit,X2: product_unit] :
( ( I3 != bot_bo3957492148770167129t_unit )
=> ( ! [I4: product_unit] :
( ( member_Product_unit @ I4 @ I3 )
=> ( ( F @ I4 )
= X2 ) )
=> ( ( comple2584293577114468500t_unit @ ( image_405062704495631173t_unit @ F @ I3 ) )
= X2 ) ) ) ).
% INF_eq_const
thf(fact_2215_INF__eq__const,axiom,
! [I3: set_o,F: $o > product_unit,X2: product_unit] :
( ( I3 != bot_bot_set_o )
=> ( ! [I4: $o] :
( ( member_o @ I4 @ I3 )
=> ( ( F @ I4 )
= X2 ) )
=> ( ( comple2584293577114468500t_unit @ ( image_o_Product_unit @ F @ I3 ) )
= X2 ) ) ) ).
% INF_eq_const
thf(fact_2216_INF__eq__const,axiom,
! [I3: set_nat,F: nat > product_unit,X2: product_unit] :
( ( I3 != bot_bot_set_nat )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I3 )
=> ( ( F @ I4 )
= X2 ) )
=> ( ( comple2584293577114468500t_unit @ ( image_8730104196221521654t_unit @ F @ I3 ) )
= X2 ) ) ) ).
% INF_eq_const
thf(fact_2217_INF__eq__const,axiom,
! [I3: set_int,F: int > set_nat,X2: set_nat] :
( ( I3 != bot_bot_set_int )
=> ( ! [I4: int] :
( ( member_int @ I4 @ I3 )
=> ( ( F @ I4 )
= X2 ) )
=> ( ( comple7806235888213564991et_nat @ ( image_int_set_nat @ F @ I3 ) )
= X2 ) ) ) ).
% INF_eq_const
thf(fact_2218_INF__eq__const,axiom,
! [I3: set_Product_unit,F: product_unit > set_nat,X2: set_nat] :
( ( I3 != bot_bo3957492148770167129t_unit )
=> ( ! [I4: product_unit] :
( ( member_Product_unit @ I4 @ I3 )
=> ( ( F @ I4 )
= X2 ) )
=> ( ( comple7806235888213564991et_nat @ ( image_8173849991827986510et_nat @ F @ I3 ) )
= X2 ) ) ) ).
% INF_eq_const
thf(fact_2219_INF__eq__const,axiom,
! [I3: set_o,F: $o > set_nat,X2: set_nat] :
( ( I3 != bot_bot_set_o )
=> ( ! [I4: $o] :
( ( member_o @ I4 @ I3 )
=> ( ( F @ I4 )
= X2 ) )
=> ( ( comple7806235888213564991et_nat @ ( image_o_set_nat @ F @ I3 ) )
= X2 ) ) ) ).
% INF_eq_const
thf(fact_2220_INF__eq__const,axiom,
! [I3: set_nat,F: nat > set_nat,X2: set_nat] :
( ( I3 != bot_bot_set_nat )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I3 )
=> ( ( F @ I4 )
= X2 ) )
=> ( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ I3 ) )
= X2 ) ) ) ).
% INF_eq_const
thf(fact_2221_INF__eq__const,axiom,
! [I3: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > product_unit,X2: product_unit] :
( ( I3 != bot_bo2099793752762293965at_nat )
=> ( ! [I4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ I4 @ I3 )
=> ( ( F @ I4 )
= X2 ) )
=> ( ( comple2584293577114468500t_unit @ ( image_6169909464789311569t_unit @ F @ I3 ) )
= X2 ) ) ) ).
% INF_eq_const
thf(fact_2222_INF__eq__const,axiom,
! [I3: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > set_nat,X2: set_nat] :
( ( I3 != bot_bo2099793752762293965at_nat )
=> ( ! [I4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ I4 @ I3 )
=> ( ( F @ I4 )
= X2 ) )
=> ( ( comple7806235888213564991et_nat @ ( image_15824709712370754et_nat @ F @ I3 ) )
= X2 ) ) ) ).
% INF_eq_const
thf(fact_2223_SUP__eq__const,axiom,
! [I3: set_int,F: int > product_unit,X2: product_unit] :
( ( I3 != bot_bot_set_int )
=> ( ! [I4: int] :
( ( member_int @ I4 @ I3 )
=> ( ( F @ I4 )
= X2 ) )
=> ( ( comple4687483117567863418t_unit @ ( image_5846526143423701914t_unit @ F @ I3 ) )
= X2 ) ) ) ).
% SUP_eq_const
thf(fact_2224_SUP__eq__const,axiom,
! [I3: set_Product_unit,F: product_unit > product_unit,X2: product_unit] :
( ( I3 != bot_bo3957492148770167129t_unit )
=> ( ! [I4: product_unit] :
( ( member_Product_unit @ I4 @ I3 )
=> ( ( F @ I4 )
= X2 ) )
=> ( ( comple4687483117567863418t_unit @ ( image_405062704495631173t_unit @ F @ I3 ) )
= X2 ) ) ) ).
% SUP_eq_const
thf(fact_2225_SUP__eq__const,axiom,
! [I3: set_o,F: $o > product_unit,X2: product_unit] :
( ( I3 != bot_bot_set_o )
=> ( ! [I4: $o] :
( ( member_o @ I4 @ I3 )
=> ( ( F @ I4 )
= X2 ) )
=> ( ( comple4687483117567863418t_unit @ ( image_o_Product_unit @ F @ I3 ) )
= X2 ) ) ) ).
% SUP_eq_const
thf(fact_2226_SUP__eq__const,axiom,
! [I3: set_nat,F: nat > product_unit,X2: product_unit] :
( ( I3 != bot_bot_set_nat )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I3 )
=> ( ( F @ I4 )
= X2 ) )
=> ( ( comple4687483117567863418t_unit @ ( image_8730104196221521654t_unit @ F @ I3 ) )
= X2 ) ) ) ).
% SUP_eq_const
thf(fact_2227_SUP__eq__const,axiom,
! [I3: set_int,F: int > set_nat,X2: set_nat] :
( ( I3 != bot_bot_set_int )
=> ( ! [I4: int] :
( ( member_int @ I4 @ I3 )
=> ( ( F @ I4 )
= X2 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_int_set_nat @ F @ I3 ) )
= X2 ) ) ) ).
% SUP_eq_const
thf(fact_2228_SUP__eq__const,axiom,
! [I3: set_Product_unit,F: product_unit > set_nat,X2: set_nat] :
( ( I3 != bot_bo3957492148770167129t_unit )
=> ( ! [I4: product_unit] :
( ( member_Product_unit @ I4 @ I3 )
=> ( ( F @ I4 )
= X2 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ F @ I3 ) )
= X2 ) ) ) ).
% SUP_eq_const
thf(fact_2229_SUP__eq__const,axiom,
! [I3: set_o,F: $o > set_nat,X2: set_nat] :
( ( I3 != bot_bot_set_o )
=> ( ! [I4: $o] :
( ( member_o @ I4 @ I3 )
=> ( ( F @ I4 )
= X2 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ I3 ) )
= X2 ) ) ) ).
% SUP_eq_const
thf(fact_2230_SUP__eq__const,axiom,
! [I3: set_nat,F: nat > set_nat,X2: set_nat] :
( ( I3 != bot_bot_set_nat )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I3 )
=> ( ( F @ I4 )
= X2 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ I3 ) )
= X2 ) ) ) ).
% SUP_eq_const
thf(fact_2231_SUP__eq__const,axiom,
! [I3: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > product_unit,X2: product_unit] :
( ( I3 != bot_bo2099793752762293965at_nat )
=> ( ! [I4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ I4 @ I3 )
=> ( ( F @ I4 )
= X2 ) )
=> ( ( comple4687483117567863418t_unit @ ( image_6169909464789311569t_unit @ F @ I3 ) )
= X2 ) ) ) ).
% SUP_eq_const
thf(fact_2232_SUP__eq__const,axiom,
! [I3: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > set_nat,X2: set_nat] :
( ( I3 != bot_bo2099793752762293965at_nat )
=> ( ! [I4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ I4 @ I3 )
=> ( ( F @ I4 )
= X2 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ F @ I3 ) )
= X2 ) ) ) ).
% SUP_eq_const
thf(fact_2233_INF__image,axiom,
! [G: $o > $o,F: nat > $o,A4: set_nat] :
( ( complete_Inf_Inf_o @ ( image_o_o @ G @ ( image_nat_o @ F @ A4 ) ) )
= ( complete_Inf_Inf_o @ ( image_nat_o @ ( comp_o_o_nat @ G @ F ) @ A4 ) ) ) ).
% INF_image
thf(fact_2234_INF__image,axiom,
! [G: rat > nat,F: nat > rat,A4: set_nat] :
( ( complete_Inf_Inf_nat @ ( image_rat_nat @ G @ ( image_nat_rat @ F @ A4 ) ) )
= ( complete_Inf_Inf_nat @ ( image_nat_nat @ ( comp_rat_nat_nat @ G @ F ) @ A4 ) ) ) ).
% INF_image
thf(fact_2235_INF__image,axiom,
! [G: nat > nat,F: nat > nat,A4: set_nat] :
( ( complete_Inf_Inf_nat @ ( image_nat_nat @ G @ ( image_nat_nat @ F @ A4 ) ) )
= ( complete_Inf_Inf_nat @ ( image_nat_nat @ ( comp_nat_nat_nat @ G @ F ) @ A4 ) ) ) ).
% INF_image
thf(fact_2236_INF__image,axiom,
! [G: nat > nat,F: int > nat,A4: set_int] :
( ( complete_Inf_Inf_nat @ ( image_nat_nat @ G @ ( image_int_nat @ F @ A4 ) ) )
= ( complete_Inf_Inf_nat @ ( image_int_nat @ ( comp_nat_nat_int @ G @ F ) @ A4 ) ) ) ).
% INF_image
thf(fact_2237_INF__image,axiom,
! [G: int > nat,F: nat > int,A4: set_nat] :
( ( complete_Inf_Inf_nat @ ( image_int_nat @ G @ ( image_nat_int @ F @ A4 ) ) )
= ( complete_Inf_Inf_nat @ ( image_nat_nat @ ( comp_int_nat_nat @ G @ F ) @ A4 ) ) ) ).
% INF_image
thf(fact_2238_INF__image,axiom,
! [G: int > nat,F: int > int,A4: set_int] :
( ( complete_Inf_Inf_nat @ ( image_int_nat @ G @ ( image_int_int @ F @ A4 ) ) )
= ( complete_Inf_Inf_nat @ ( image_int_nat @ ( comp_int_nat_int @ G @ F ) @ A4 ) ) ) ).
% INF_image
thf(fact_2239_INF__image,axiom,
! [G: rat > int,F: nat > rat,A4: set_nat] :
( ( complete_Inf_Inf_int @ ( image_rat_int @ G @ ( image_nat_rat @ F @ A4 ) ) )
= ( complete_Inf_Inf_int @ ( image_nat_int @ ( comp_rat_int_nat @ G @ F ) @ A4 ) ) ) ).
% INF_image
thf(fact_2240_INF__image,axiom,
! [G: nat > int,F: nat > nat,A4: set_nat] :
( ( complete_Inf_Inf_int @ ( image_nat_int @ G @ ( image_nat_nat @ F @ A4 ) ) )
= ( complete_Inf_Inf_int @ ( image_nat_int @ ( comp_nat_int_nat @ G @ F ) @ A4 ) ) ) ).
% INF_image
thf(fact_2241_INF__image,axiom,
! [G: nat > int,F: int > nat,A4: set_int] :
( ( complete_Inf_Inf_int @ ( image_nat_int @ G @ ( image_int_nat @ F @ A4 ) ) )
= ( complete_Inf_Inf_int @ ( image_int_int @ ( comp_nat_int_int @ G @ F ) @ A4 ) ) ) ).
% INF_image
thf(fact_2242_INF__image,axiom,
! [G: int > int,F: nat > int,A4: set_nat] :
( ( complete_Inf_Inf_int @ ( image_int_int @ G @ ( image_nat_int @ F @ A4 ) ) )
= ( complete_Inf_Inf_int @ ( image_nat_int @ ( comp_int_int_nat @ G @ F ) @ A4 ) ) ) ).
% INF_image
thf(fact_2243_SUP__image,axiom,
! [G: $o > $o,F: nat > $o,A4: set_nat] :
( ( complete_Sup_Sup_o @ ( image_o_o @ G @ ( image_nat_o @ F @ A4 ) ) )
= ( complete_Sup_Sup_o @ ( image_nat_o @ ( comp_o_o_nat @ G @ F ) @ A4 ) ) ) ).
% SUP_image
thf(fact_2244_SUP__image,axiom,
! [G: rat > nat,F: nat > rat,A4: set_nat] :
( ( complete_Sup_Sup_nat @ ( image_rat_nat @ G @ ( image_nat_rat @ F @ A4 ) ) )
= ( complete_Sup_Sup_nat @ ( image_nat_nat @ ( comp_rat_nat_nat @ G @ F ) @ A4 ) ) ) ).
% SUP_image
thf(fact_2245_SUP__image,axiom,
! [G: nat > nat,F: nat > nat,A4: set_nat] :
( ( complete_Sup_Sup_nat @ ( image_nat_nat @ G @ ( image_nat_nat @ F @ A4 ) ) )
= ( complete_Sup_Sup_nat @ ( image_nat_nat @ ( comp_nat_nat_nat @ G @ F ) @ A4 ) ) ) ).
% SUP_image
thf(fact_2246_SUP__image,axiom,
! [G: nat > nat,F: int > nat,A4: set_int] :
( ( complete_Sup_Sup_nat @ ( image_nat_nat @ G @ ( image_int_nat @ F @ A4 ) ) )
= ( complete_Sup_Sup_nat @ ( image_int_nat @ ( comp_nat_nat_int @ G @ F ) @ A4 ) ) ) ).
% SUP_image
thf(fact_2247_SUP__image,axiom,
! [G: int > nat,F: nat > int,A4: set_nat] :
( ( complete_Sup_Sup_nat @ ( image_int_nat @ G @ ( image_nat_int @ F @ A4 ) ) )
= ( complete_Sup_Sup_nat @ ( image_nat_nat @ ( comp_int_nat_nat @ G @ F ) @ A4 ) ) ) ).
% SUP_image
thf(fact_2248_SUP__image,axiom,
! [G: int > nat,F: int > int,A4: set_int] :
( ( complete_Sup_Sup_nat @ ( image_int_nat @ G @ ( image_int_int @ F @ A4 ) ) )
= ( complete_Sup_Sup_nat @ ( image_int_nat @ ( comp_int_nat_int @ G @ F ) @ A4 ) ) ) ).
% SUP_image
thf(fact_2249_SUP__image,axiom,
! [G: rat > int,F: nat > rat,A4: set_nat] :
( ( complete_Sup_Sup_int @ ( image_rat_int @ G @ ( image_nat_rat @ F @ A4 ) ) )
= ( complete_Sup_Sup_int @ ( image_nat_int @ ( comp_rat_int_nat @ G @ F ) @ A4 ) ) ) ).
% SUP_image
thf(fact_2250_SUP__image,axiom,
! [G: nat > int,F: nat > nat,A4: set_nat] :
( ( complete_Sup_Sup_int @ ( image_nat_int @ G @ ( image_nat_nat @ F @ A4 ) ) )
= ( complete_Sup_Sup_int @ ( image_nat_int @ ( comp_nat_int_nat @ G @ F ) @ A4 ) ) ) ).
% SUP_image
thf(fact_2251_SUP__image,axiom,
! [G: nat > int,F: int > nat,A4: set_int] :
( ( complete_Sup_Sup_int @ ( image_nat_int @ G @ ( image_int_nat @ F @ A4 ) ) )
= ( complete_Sup_Sup_int @ ( image_int_int @ ( comp_nat_int_int @ G @ F ) @ A4 ) ) ) ).
% SUP_image
thf(fact_2252_SUP__image,axiom,
! [G: int > int,F: nat > int,A4: set_nat] :
( ( complete_Sup_Sup_int @ ( image_int_int @ G @ ( image_nat_int @ F @ A4 ) ) )
= ( complete_Sup_Sup_int @ ( image_nat_int @ ( comp_int_int_nat @ G @ F ) @ A4 ) ) ) ).
% SUP_image
thf(fact_2253_INF__greatest,axiom,
! [A4: set_Pr4329608150637261639at_nat,U: set_nat,F: produc3843707927480180839at_nat > set_nat] :
( ! [I4: produc3843707927480180839at_nat] :
( ( member8757157785044589968at_nat @ I4 @ A4 )
=> ( ord_less_eq_set_nat @ U @ ( F @ I4 ) ) )
=> ( ord_less_eq_set_nat @ U @ ( comple7806235888213564991et_nat @ ( image_3160451284896210088et_nat @ F @ A4 ) ) ) ) ).
% INF_greatest
thf(fact_2254_INF__greatest,axiom,
! [A4: set_Pr1261947904930325089at_nat,U: set_nat,F: product_prod_nat_nat > set_nat] :
( ! [I4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ I4 @ A4 )
=> ( ord_less_eq_set_nat @ U @ ( F @ I4 ) ) )
=> ( ord_less_eq_set_nat @ U @ ( comple7806235888213564991et_nat @ ( image_15824709712370754et_nat @ F @ A4 ) ) ) ) ).
% INF_greatest
thf(fact_2255_INF__greatest,axiom,
! [A4: set_o,U: set_nat,F: $o > set_nat] :
( ! [I4: $o] :
( ( member_o @ I4 @ A4 )
=> ( ord_less_eq_set_nat @ U @ ( F @ I4 ) ) )
=> ( ord_less_eq_set_nat @ U @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ F @ A4 ) ) ) ) ).
% INF_greatest
thf(fact_2256_INF__greatest,axiom,
! [A4: set_nat,U: set_nat,F: nat > set_nat] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A4 )
=> ( ord_less_eq_set_nat @ U @ ( F @ I4 ) ) )
=> ( ord_less_eq_set_nat @ U @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ A4 ) ) ) ) ).
% INF_greatest
thf(fact_2257_INF__greatest,axiom,
! [A4: set_int,U: set_nat,F: int > set_nat] :
( ! [I4: int] :
( ( member_int @ I4 @ A4 )
=> ( ord_less_eq_set_nat @ U @ ( F @ I4 ) ) )
=> ( ord_less_eq_set_nat @ U @ ( comple7806235888213564991et_nat @ ( image_int_set_nat @ F @ A4 ) ) ) ) ).
% INF_greatest
thf(fact_2258_INF__greatest,axiom,
! [A4: set_Pr4329608150637261639at_nat,U: product_unit,F: produc3843707927480180839at_nat > product_unit] :
( ! [I4: produc3843707927480180839at_nat] :
( ( member8757157785044589968at_nat @ I4 @ A4 )
=> ( ord_le3221252021190050221t_unit @ U @ ( F @ I4 ) ) )
=> ( ord_le3221252021190050221t_unit @ U @ ( comple2584293577114468500t_unit @ ( image_7082456443403096939t_unit @ F @ A4 ) ) ) ) ).
% INF_greatest
thf(fact_2259_INF__greatest,axiom,
! [A4: set_Pr1261947904930325089at_nat,U: product_unit,F: product_prod_nat_nat > product_unit] :
( ! [I4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ I4 @ A4 )
=> ( ord_le3221252021190050221t_unit @ U @ ( F @ I4 ) ) )
=> ( ord_le3221252021190050221t_unit @ U @ ( comple2584293577114468500t_unit @ ( image_6169909464789311569t_unit @ F @ A4 ) ) ) ) ).
% INF_greatest
thf(fact_2260_INF__greatest,axiom,
! [A4: set_o,U: product_unit,F: $o > product_unit] :
( ! [I4: $o] :
( ( member_o @ I4 @ A4 )
=> ( ord_le3221252021190050221t_unit @ U @ ( F @ I4 ) ) )
=> ( ord_le3221252021190050221t_unit @ U @ ( comple2584293577114468500t_unit @ ( image_o_Product_unit @ F @ A4 ) ) ) ) ).
% INF_greatest
thf(fact_2261_INF__greatest,axiom,
! [A4: set_nat,U: product_unit,F: nat > product_unit] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A4 )
=> ( ord_le3221252021190050221t_unit @ U @ ( F @ I4 ) ) )
=> ( ord_le3221252021190050221t_unit @ U @ ( comple2584293577114468500t_unit @ ( image_8730104196221521654t_unit @ F @ A4 ) ) ) ) ).
% INF_greatest
thf(fact_2262_INF__greatest,axiom,
! [A4: set_int,U: product_unit,F: int > product_unit] :
( ! [I4: int] :
( ( member_int @ I4 @ A4 )
=> ( ord_le3221252021190050221t_unit @ U @ ( F @ I4 ) ) )
=> ( ord_le3221252021190050221t_unit @ U @ ( comple2584293577114468500t_unit @ ( image_5846526143423701914t_unit @ F @ A4 ) ) ) ) ).
% INF_greatest
thf(fact_2263_INF__lower2,axiom,
! [I: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,F: produc3843707927480180839at_nat > set_nat,U: set_nat] :
( ( member8757157785044589968at_nat @ I @ A4 )
=> ( ( ord_less_eq_set_nat @ ( F @ I ) @ U )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_3160451284896210088et_nat @ F @ A4 ) ) @ U ) ) ) ).
% INF_lower2
thf(fact_2264_INF__lower2,axiom,
! [I: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > set_nat,U: set_nat] :
( ( member8440522571783428010at_nat @ I @ A4 )
=> ( ( ord_less_eq_set_nat @ ( F @ I ) @ U )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_15824709712370754et_nat @ F @ A4 ) ) @ U ) ) ) ).
% INF_lower2
thf(fact_2265_INF__lower2,axiom,
! [I: $o,A4: set_o,F: $o > set_nat,U: set_nat] :
( ( member_o @ I @ A4 )
=> ( ( ord_less_eq_set_nat @ ( F @ I ) @ U )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ F @ A4 ) ) @ U ) ) ) ).
% INF_lower2
thf(fact_2266_INF__lower2,axiom,
! [I: nat,A4: set_nat,F: nat > set_nat,U: set_nat] :
( ( member_nat @ I @ A4 )
=> ( ( ord_less_eq_set_nat @ ( F @ I ) @ U )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ A4 ) ) @ U ) ) ) ).
% INF_lower2
thf(fact_2267_INF__lower2,axiom,
! [I: int,A4: set_int,F: int > set_nat,U: set_nat] :
( ( member_int @ I @ A4 )
=> ( ( ord_less_eq_set_nat @ ( F @ I ) @ U )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_int_set_nat @ F @ A4 ) ) @ U ) ) ) ).
% INF_lower2
thf(fact_2268_INF__lower2,axiom,
! [I: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,F: produc3843707927480180839at_nat > product_unit,U: product_unit] :
( ( member8757157785044589968at_nat @ I @ A4 )
=> ( ( ord_le3221252021190050221t_unit @ ( F @ I ) @ U )
=> ( ord_le3221252021190050221t_unit @ ( comple2584293577114468500t_unit @ ( image_7082456443403096939t_unit @ F @ A4 ) ) @ U ) ) ) ).
% INF_lower2
thf(fact_2269_INF__lower2,axiom,
! [I: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > product_unit,U: product_unit] :
( ( member8440522571783428010at_nat @ I @ A4 )
=> ( ( ord_le3221252021190050221t_unit @ ( F @ I ) @ U )
=> ( ord_le3221252021190050221t_unit @ ( comple2584293577114468500t_unit @ ( image_6169909464789311569t_unit @ F @ A4 ) ) @ U ) ) ) ).
% INF_lower2
thf(fact_2270_INF__lower2,axiom,
! [I: $o,A4: set_o,F: $o > product_unit,U: product_unit] :
( ( member_o @ I @ A4 )
=> ( ( ord_le3221252021190050221t_unit @ ( F @ I ) @ U )
=> ( ord_le3221252021190050221t_unit @ ( comple2584293577114468500t_unit @ ( image_o_Product_unit @ F @ A4 ) ) @ U ) ) ) ).
% INF_lower2
thf(fact_2271_INF__lower2,axiom,
! [I: nat,A4: set_nat,F: nat > product_unit,U: product_unit] :
( ( member_nat @ I @ A4 )
=> ( ( ord_le3221252021190050221t_unit @ ( F @ I ) @ U )
=> ( ord_le3221252021190050221t_unit @ ( comple2584293577114468500t_unit @ ( image_8730104196221521654t_unit @ F @ A4 ) ) @ U ) ) ) ).
% INF_lower2
thf(fact_2272_INF__lower2,axiom,
! [I: int,A4: set_int,F: int > product_unit,U: product_unit] :
( ( member_int @ I @ A4 )
=> ( ( ord_le3221252021190050221t_unit @ ( F @ I ) @ U )
=> ( ord_le3221252021190050221t_unit @ ( comple2584293577114468500t_unit @ ( image_5846526143423701914t_unit @ F @ A4 ) ) @ U ) ) ) ).
% INF_lower2
thf(fact_2273_INF__lower,axiom,
! [I: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,F: produc3843707927480180839at_nat > set_nat] :
( ( member8757157785044589968at_nat @ I @ A4 )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_3160451284896210088et_nat @ F @ A4 ) ) @ ( F @ I ) ) ) ).
% INF_lower
thf(fact_2274_INF__lower,axiom,
! [I: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > set_nat] :
( ( member8440522571783428010at_nat @ I @ A4 )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_15824709712370754et_nat @ F @ A4 ) ) @ ( F @ I ) ) ) ).
% INF_lower
thf(fact_2275_INF__lower,axiom,
! [I: $o,A4: set_o,F: $o > set_nat] :
( ( member_o @ I @ A4 )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ F @ A4 ) ) @ ( F @ I ) ) ) ).
% INF_lower
thf(fact_2276_INF__lower,axiom,
! [I: nat,A4: set_nat,F: nat > set_nat] :
( ( member_nat @ I @ A4 )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ A4 ) ) @ ( F @ I ) ) ) ).
% INF_lower
thf(fact_2277_INF__lower,axiom,
! [I: int,A4: set_int,F: int > set_nat] :
( ( member_int @ I @ A4 )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_int_set_nat @ F @ A4 ) ) @ ( F @ I ) ) ) ).
% INF_lower
thf(fact_2278_INF__lower,axiom,
! [I: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,F: produc3843707927480180839at_nat > product_unit] :
( ( member8757157785044589968at_nat @ I @ A4 )
=> ( ord_le3221252021190050221t_unit @ ( comple2584293577114468500t_unit @ ( image_7082456443403096939t_unit @ F @ A4 ) ) @ ( F @ I ) ) ) ).
% INF_lower
thf(fact_2279_INF__lower,axiom,
! [I: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > product_unit] :
( ( member8440522571783428010at_nat @ I @ A4 )
=> ( ord_le3221252021190050221t_unit @ ( comple2584293577114468500t_unit @ ( image_6169909464789311569t_unit @ F @ A4 ) ) @ ( F @ I ) ) ) ).
% INF_lower
thf(fact_2280_INF__lower,axiom,
! [I: $o,A4: set_o,F: $o > product_unit] :
( ( member_o @ I @ A4 )
=> ( ord_le3221252021190050221t_unit @ ( comple2584293577114468500t_unit @ ( image_o_Product_unit @ F @ A4 ) ) @ ( F @ I ) ) ) ).
% INF_lower
thf(fact_2281_INF__lower,axiom,
! [I: nat,A4: set_nat,F: nat > product_unit] :
( ( member_nat @ I @ A4 )
=> ( ord_le3221252021190050221t_unit @ ( comple2584293577114468500t_unit @ ( image_8730104196221521654t_unit @ F @ A4 ) ) @ ( F @ I ) ) ) ).
% INF_lower
thf(fact_2282_INF__lower,axiom,
! [I: int,A4: set_int,F: int > product_unit] :
( ( member_int @ I @ A4 )
=> ( ord_le3221252021190050221t_unit @ ( comple2584293577114468500t_unit @ ( image_5846526143423701914t_unit @ F @ A4 ) ) @ ( F @ I ) ) ) ).
% INF_lower
thf(fact_2283_INF__eqI,axiom,
! [A4: set_Pr4329608150637261639at_nat,X2: set_nat,F: produc3843707927480180839at_nat > set_nat] :
( ! [I4: produc3843707927480180839at_nat] :
( ( member8757157785044589968at_nat @ I4 @ A4 )
=> ( ord_less_eq_set_nat @ X2 @ ( F @ I4 ) ) )
=> ( ! [Y: set_nat] :
( ! [I5: produc3843707927480180839at_nat] :
( ( member8757157785044589968at_nat @ I5 @ A4 )
=> ( ord_less_eq_set_nat @ Y @ ( F @ I5 ) ) )
=> ( ord_less_eq_set_nat @ Y @ X2 ) )
=> ( ( comple7806235888213564991et_nat @ ( image_3160451284896210088et_nat @ F @ A4 ) )
= X2 ) ) ) ).
% INF_eqI
thf(fact_2284_INF__eqI,axiom,
! [A4: set_Pr1261947904930325089at_nat,X2: set_nat,F: product_prod_nat_nat > set_nat] :
( ! [I4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ I4 @ A4 )
=> ( ord_less_eq_set_nat @ X2 @ ( F @ I4 ) ) )
=> ( ! [Y: set_nat] :
( ! [I5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ I5 @ A4 )
=> ( ord_less_eq_set_nat @ Y @ ( F @ I5 ) ) )
=> ( ord_less_eq_set_nat @ Y @ X2 ) )
=> ( ( comple7806235888213564991et_nat @ ( image_15824709712370754et_nat @ F @ A4 ) )
= X2 ) ) ) ).
% INF_eqI
thf(fact_2285_INF__eqI,axiom,
! [A4: set_o,X2: set_nat,F: $o > set_nat] :
( ! [I4: $o] :
( ( member_o @ I4 @ A4 )
=> ( ord_less_eq_set_nat @ X2 @ ( F @ I4 ) ) )
=> ( ! [Y: set_nat] :
( ! [I5: $o] :
( ( member_o @ I5 @ A4 )
=> ( ord_less_eq_set_nat @ Y @ ( F @ I5 ) ) )
=> ( ord_less_eq_set_nat @ Y @ X2 ) )
=> ( ( comple7806235888213564991et_nat @ ( image_o_set_nat @ F @ A4 ) )
= X2 ) ) ) ).
% INF_eqI
thf(fact_2286_INF__eqI,axiom,
! [A4: set_nat,X2: set_nat,F: nat > set_nat] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A4 )
=> ( ord_less_eq_set_nat @ X2 @ ( F @ I4 ) ) )
=> ( ! [Y: set_nat] :
( ! [I5: nat] :
( ( member_nat @ I5 @ A4 )
=> ( ord_less_eq_set_nat @ Y @ ( F @ I5 ) ) )
=> ( ord_less_eq_set_nat @ Y @ X2 ) )
=> ( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ A4 ) )
= X2 ) ) ) ).
% INF_eqI
thf(fact_2287_INF__eqI,axiom,
! [A4: set_int,X2: set_nat,F: int > set_nat] :
( ! [I4: int] :
( ( member_int @ I4 @ A4 )
=> ( ord_less_eq_set_nat @ X2 @ ( F @ I4 ) ) )
=> ( ! [Y: set_nat] :
( ! [I5: int] :
( ( member_int @ I5 @ A4 )
=> ( ord_less_eq_set_nat @ Y @ ( F @ I5 ) ) )
=> ( ord_less_eq_set_nat @ Y @ X2 ) )
=> ( ( comple7806235888213564991et_nat @ ( image_int_set_nat @ F @ A4 ) )
= X2 ) ) ) ).
% INF_eqI
thf(fact_2288_INF__eqI,axiom,
! [A4: set_Pr4329608150637261639at_nat,X2: product_unit,F: produc3843707927480180839at_nat > product_unit] :
( ! [I4: produc3843707927480180839at_nat] :
( ( member8757157785044589968at_nat @ I4 @ A4 )
=> ( ord_le3221252021190050221t_unit @ X2 @ ( F @ I4 ) ) )
=> ( ! [Y: product_unit] :
( ! [I5: produc3843707927480180839at_nat] :
( ( member8757157785044589968at_nat @ I5 @ A4 )
=> ( ord_le3221252021190050221t_unit @ Y @ ( F @ I5 ) ) )
=> ( ord_le3221252021190050221t_unit @ Y @ X2 ) )
=> ( ( comple2584293577114468500t_unit @ ( image_7082456443403096939t_unit @ F @ A4 ) )
= X2 ) ) ) ).
% INF_eqI
thf(fact_2289_INF__eqI,axiom,
! [A4: set_Pr1261947904930325089at_nat,X2: product_unit,F: product_prod_nat_nat > product_unit] :
( ! [I4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ I4 @ A4 )
=> ( ord_le3221252021190050221t_unit @ X2 @ ( F @ I4 ) ) )
=> ( ! [Y: product_unit] :
( ! [I5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ I5 @ A4 )
=> ( ord_le3221252021190050221t_unit @ Y @ ( F @ I5 ) ) )
=> ( ord_le3221252021190050221t_unit @ Y @ X2 ) )
=> ( ( comple2584293577114468500t_unit @ ( image_6169909464789311569t_unit @ F @ A4 ) )
= X2 ) ) ) ).
% INF_eqI
thf(fact_2290_INF__eqI,axiom,
! [A4: set_o,X2: product_unit,F: $o > product_unit] :
( ! [I4: $o] :
( ( member_o @ I4 @ A4 )
=> ( ord_le3221252021190050221t_unit @ X2 @ ( F @ I4 ) ) )
=> ( ! [Y: product_unit] :
( ! [I5: $o] :
( ( member_o @ I5 @ A4 )
=> ( ord_le3221252021190050221t_unit @ Y @ ( F @ I5 ) ) )
=> ( ord_le3221252021190050221t_unit @ Y @ X2 ) )
=> ( ( comple2584293577114468500t_unit @ ( image_o_Product_unit @ F @ A4 ) )
= X2 ) ) ) ).
% INF_eqI
thf(fact_2291_INF__eqI,axiom,
! [A4: set_nat,X2: product_unit,F: nat > product_unit] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A4 )
=> ( ord_le3221252021190050221t_unit @ X2 @ ( F @ I4 ) ) )
=> ( ! [Y: product_unit] :
( ! [I5: nat] :
( ( member_nat @ I5 @ A4 )
=> ( ord_le3221252021190050221t_unit @ Y @ ( F @ I5 ) ) )
=> ( ord_le3221252021190050221t_unit @ Y @ X2 ) )
=> ( ( comple2584293577114468500t_unit @ ( image_8730104196221521654t_unit @ F @ A4 ) )
= X2 ) ) ) ).
% INF_eqI
thf(fact_2292_INF__eqI,axiom,
! [A4: set_int,X2: product_unit,F: int > product_unit] :
( ! [I4: int] :
( ( member_int @ I4 @ A4 )
=> ( ord_le3221252021190050221t_unit @ X2 @ ( F @ I4 ) ) )
=> ( ! [Y: product_unit] :
( ! [I5: int] :
( ( member_int @ I5 @ A4 )
=> ( ord_le3221252021190050221t_unit @ Y @ ( F @ I5 ) ) )
=> ( ord_le3221252021190050221t_unit @ Y @ X2 ) )
=> ( ( comple2584293577114468500t_unit @ ( image_5846526143423701914t_unit @ F @ A4 ) )
= X2 ) ) ) ).
% INF_eqI
thf(fact_2293_SUP__upper2,axiom,
! [I: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,U: set_nat,F: produc3843707927480180839at_nat > set_nat] :
( ( member8757157785044589968at_nat @ I @ A4 )
=> ( ( ord_less_eq_set_nat @ U @ ( F @ I ) )
=> ( ord_less_eq_set_nat @ U @ ( comple7399068483239264473et_nat @ ( image_3160451284896210088et_nat @ F @ A4 ) ) ) ) ) ).
% SUP_upper2
thf(fact_2294_SUP__upper2,axiom,
! [I: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,U: set_nat,F: product_prod_nat_nat > set_nat] :
( ( member8440522571783428010at_nat @ I @ A4 )
=> ( ( ord_less_eq_set_nat @ U @ ( F @ I ) )
=> ( ord_less_eq_set_nat @ U @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ F @ A4 ) ) ) ) ) ).
% SUP_upper2
thf(fact_2295_SUP__upper2,axiom,
! [I: $o,A4: set_o,U: set_nat,F: $o > set_nat] :
( ( member_o @ I @ A4 )
=> ( ( ord_less_eq_set_nat @ U @ ( F @ I ) )
=> ( ord_less_eq_set_nat @ U @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A4 ) ) ) ) ) ).
% SUP_upper2
thf(fact_2296_SUP__upper2,axiom,
! [I: nat,A4: set_nat,U: set_nat,F: nat > set_nat] :
( ( member_nat @ I @ A4 )
=> ( ( ord_less_eq_set_nat @ U @ ( F @ I ) )
=> ( ord_less_eq_set_nat @ U @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A4 ) ) ) ) ) ).
% SUP_upper2
thf(fact_2297_SUP__upper2,axiom,
! [I: int,A4: set_int,U: set_nat,F: int > set_nat] :
( ( member_int @ I @ A4 )
=> ( ( ord_less_eq_set_nat @ U @ ( F @ I ) )
=> ( ord_less_eq_set_nat @ U @ ( comple7399068483239264473et_nat @ ( image_int_set_nat @ F @ A4 ) ) ) ) ) ).
% SUP_upper2
thf(fact_2298_SUP__upper2,axiom,
! [I: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,U: product_unit,F: produc3843707927480180839at_nat > product_unit] :
( ( member8757157785044589968at_nat @ I @ A4 )
=> ( ( ord_le3221252021190050221t_unit @ U @ ( F @ I ) )
=> ( ord_le3221252021190050221t_unit @ U @ ( comple4687483117567863418t_unit @ ( image_7082456443403096939t_unit @ F @ A4 ) ) ) ) ) ).
% SUP_upper2
thf(fact_2299_SUP__upper2,axiom,
! [I: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,U: product_unit,F: product_prod_nat_nat > product_unit] :
( ( member8440522571783428010at_nat @ I @ A4 )
=> ( ( ord_le3221252021190050221t_unit @ U @ ( F @ I ) )
=> ( ord_le3221252021190050221t_unit @ U @ ( comple4687483117567863418t_unit @ ( image_6169909464789311569t_unit @ F @ A4 ) ) ) ) ) ).
% SUP_upper2
thf(fact_2300_SUP__upper2,axiom,
! [I: $o,A4: set_o,U: product_unit,F: $o > product_unit] :
( ( member_o @ I @ A4 )
=> ( ( ord_le3221252021190050221t_unit @ U @ ( F @ I ) )
=> ( ord_le3221252021190050221t_unit @ U @ ( comple4687483117567863418t_unit @ ( image_o_Product_unit @ F @ A4 ) ) ) ) ) ).
% SUP_upper2
thf(fact_2301_SUP__upper2,axiom,
! [I: nat,A4: set_nat,U: product_unit,F: nat > product_unit] :
( ( member_nat @ I @ A4 )
=> ( ( ord_le3221252021190050221t_unit @ U @ ( F @ I ) )
=> ( ord_le3221252021190050221t_unit @ U @ ( comple4687483117567863418t_unit @ ( image_8730104196221521654t_unit @ F @ A4 ) ) ) ) ) ).
% SUP_upper2
thf(fact_2302_SUP__upper2,axiom,
! [I: int,A4: set_int,U: product_unit,F: int > product_unit] :
( ( member_int @ I @ A4 )
=> ( ( ord_le3221252021190050221t_unit @ U @ ( F @ I ) )
=> ( ord_le3221252021190050221t_unit @ U @ ( comple4687483117567863418t_unit @ ( image_5846526143423701914t_unit @ F @ A4 ) ) ) ) ) ).
% SUP_upper2
thf(fact_2303_SUP__upper,axiom,
! [I: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,F: produc3843707927480180839at_nat > set_nat] :
( ( member8757157785044589968at_nat @ I @ A4 )
=> ( ord_less_eq_set_nat @ ( F @ I ) @ ( comple7399068483239264473et_nat @ ( image_3160451284896210088et_nat @ F @ A4 ) ) ) ) ).
% SUP_upper
thf(fact_2304_SUP__upper,axiom,
! [I: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > set_nat] :
( ( member8440522571783428010at_nat @ I @ A4 )
=> ( ord_less_eq_set_nat @ ( F @ I ) @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ F @ A4 ) ) ) ) ).
% SUP_upper
thf(fact_2305_SUP__upper,axiom,
! [I: $o,A4: set_o,F: $o > set_nat] :
( ( member_o @ I @ A4 )
=> ( ord_less_eq_set_nat @ ( F @ I ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A4 ) ) ) ) ).
% SUP_upper
thf(fact_2306_SUP__upper,axiom,
! [I: nat,A4: set_nat,F: nat > set_nat] :
( ( member_nat @ I @ A4 )
=> ( ord_less_eq_set_nat @ ( F @ I ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A4 ) ) ) ) ).
% SUP_upper
thf(fact_2307_SUP__upper,axiom,
! [I: int,A4: set_int,F: int > set_nat] :
( ( member_int @ I @ A4 )
=> ( ord_less_eq_set_nat @ ( F @ I ) @ ( comple7399068483239264473et_nat @ ( image_int_set_nat @ F @ A4 ) ) ) ) ).
% SUP_upper
thf(fact_2308_SUP__upper,axiom,
! [I: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,F: produc3843707927480180839at_nat > product_unit] :
( ( member8757157785044589968at_nat @ I @ A4 )
=> ( ord_le3221252021190050221t_unit @ ( F @ I ) @ ( comple4687483117567863418t_unit @ ( image_7082456443403096939t_unit @ F @ A4 ) ) ) ) ).
% SUP_upper
thf(fact_2309_SUP__upper,axiom,
! [I: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > product_unit] :
( ( member8440522571783428010at_nat @ I @ A4 )
=> ( ord_le3221252021190050221t_unit @ ( F @ I ) @ ( comple4687483117567863418t_unit @ ( image_6169909464789311569t_unit @ F @ A4 ) ) ) ) ).
% SUP_upper
thf(fact_2310_SUP__upper,axiom,
! [I: $o,A4: set_o,F: $o > product_unit] :
( ( member_o @ I @ A4 )
=> ( ord_le3221252021190050221t_unit @ ( F @ I ) @ ( comple4687483117567863418t_unit @ ( image_o_Product_unit @ F @ A4 ) ) ) ) ).
% SUP_upper
thf(fact_2311_SUP__upper,axiom,
! [I: nat,A4: set_nat,F: nat > product_unit] :
( ( member_nat @ I @ A4 )
=> ( ord_le3221252021190050221t_unit @ ( F @ I ) @ ( comple4687483117567863418t_unit @ ( image_8730104196221521654t_unit @ F @ A4 ) ) ) ) ).
% SUP_upper
thf(fact_2312_SUP__upper,axiom,
! [I: int,A4: set_int,F: int > product_unit] :
( ( member_int @ I @ A4 )
=> ( ord_le3221252021190050221t_unit @ ( F @ I ) @ ( comple4687483117567863418t_unit @ ( image_5846526143423701914t_unit @ F @ A4 ) ) ) ) ).
% SUP_upper
thf(fact_2313_SUP__least,axiom,
! [A4: set_Pr4329608150637261639at_nat,F: produc3843707927480180839at_nat > set_nat,U: set_nat] :
( ! [I4: produc3843707927480180839at_nat] :
( ( member8757157785044589968at_nat @ I4 @ A4 )
=> ( ord_less_eq_set_nat @ ( F @ I4 ) @ U ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_3160451284896210088et_nat @ F @ A4 ) ) @ U ) ) ).
% SUP_least
thf(fact_2314_SUP__least,axiom,
! [A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > set_nat,U: set_nat] :
( ! [I4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ I4 @ A4 )
=> ( ord_less_eq_set_nat @ ( F @ I4 ) @ U ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ F @ A4 ) ) @ U ) ) ).
% SUP_least
thf(fact_2315_SUP__least,axiom,
! [A4: set_o,F: $o > set_nat,U: set_nat] :
( ! [I4: $o] :
( ( member_o @ I4 @ A4 )
=> ( ord_less_eq_set_nat @ ( F @ I4 ) @ U ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A4 ) ) @ U ) ) ).
% SUP_least
thf(fact_2316_SUP__least,axiom,
! [A4: set_nat,F: nat > set_nat,U: set_nat] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A4 )
=> ( ord_less_eq_set_nat @ ( F @ I4 ) @ U ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A4 ) ) @ U ) ) ).
% SUP_least
thf(fact_2317_SUP__least,axiom,
! [A4: set_int,F: int > set_nat,U: set_nat] :
( ! [I4: int] :
( ( member_int @ I4 @ A4 )
=> ( ord_less_eq_set_nat @ ( F @ I4 ) @ U ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_int_set_nat @ F @ A4 ) ) @ U ) ) ).
% SUP_least
thf(fact_2318_SUP__least,axiom,
! [A4: set_Pr4329608150637261639at_nat,F: produc3843707927480180839at_nat > product_unit,U: product_unit] :
( ! [I4: produc3843707927480180839at_nat] :
( ( member8757157785044589968at_nat @ I4 @ A4 )
=> ( ord_le3221252021190050221t_unit @ ( F @ I4 ) @ U ) )
=> ( ord_le3221252021190050221t_unit @ ( comple4687483117567863418t_unit @ ( image_7082456443403096939t_unit @ F @ A4 ) ) @ U ) ) ).
% SUP_least
thf(fact_2319_SUP__least,axiom,
! [A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > product_unit,U: product_unit] :
( ! [I4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ I4 @ A4 )
=> ( ord_le3221252021190050221t_unit @ ( F @ I4 ) @ U ) )
=> ( ord_le3221252021190050221t_unit @ ( comple4687483117567863418t_unit @ ( image_6169909464789311569t_unit @ F @ A4 ) ) @ U ) ) ).
% SUP_least
thf(fact_2320_SUP__least,axiom,
! [A4: set_o,F: $o > product_unit,U: product_unit] :
( ! [I4: $o] :
( ( member_o @ I4 @ A4 )
=> ( ord_le3221252021190050221t_unit @ ( F @ I4 ) @ U ) )
=> ( ord_le3221252021190050221t_unit @ ( comple4687483117567863418t_unit @ ( image_o_Product_unit @ F @ A4 ) ) @ U ) ) ).
% SUP_least
thf(fact_2321_SUP__least,axiom,
! [A4: set_nat,F: nat > product_unit,U: product_unit] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A4 )
=> ( ord_le3221252021190050221t_unit @ ( F @ I4 ) @ U ) )
=> ( ord_le3221252021190050221t_unit @ ( comple4687483117567863418t_unit @ ( image_8730104196221521654t_unit @ F @ A4 ) ) @ U ) ) ).
% SUP_least
thf(fact_2322_SUP__least,axiom,
! [A4: set_int,F: int > product_unit,U: product_unit] :
( ! [I4: int] :
( ( member_int @ I4 @ A4 )
=> ( ord_le3221252021190050221t_unit @ ( F @ I4 ) @ U ) )
=> ( ord_le3221252021190050221t_unit @ ( comple4687483117567863418t_unit @ ( image_5846526143423701914t_unit @ F @ A4 ) ) @ U ) ) ).
% SUP_least
thf(fact_2323_SUP__eqI,axiom,
! [A4: set_Pr4329608150637261639at_nat,F: produc3843707927480180839at_nat > set_nat,X2: set_nat] :
( ! [I4: produc3843707927480180839at_nat] :
( ( member8757157785044589968at_nat @ I4 @ A4 )
=> ( ord_less_eq_set_nat @ ( F @ I4 ) @ X2 ) )
=> ( ! [Y: set_nat] :
( ! [I5: produc3843707927480180839at_nat] :
( ( member8757157785044589968at_nat @ I5 @ A4 )
=> ( ord_less_eq_set_nat @ ( F @ I5 ) @ Y ) )
=> ( ord_less_eq_set_nat @ X2 @ Y ) )
=> ( ( comple7399068483239264473et_nat @ ( image_3160451284896210088et_nat @ F @ A4 ) )
= X2 ) ) ) ).
% SUP_eqI
thf(fact_2324_SUP__eqI,axiom,
! [A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > set_nat,X2: set_nat] :
( ! [I4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ I4 @ A4 )
=> ( ord_less_eq_set_nat @ ( F @ I4 ) @ X2 ) )
=> ( ! [Y: set_nat] :
( ! [I5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ I5 @ A4 )
=> ( ord_less_eq_set_nat @ ( F @ I5 ) @ Y ) )
=> ( ord_less_eq_set_nat @ X2 @ Y ) )
=> ( ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ F @ A4 ) )
= X2 ) ) ) ).
% SUP_eqI
thf(fact_2325_SUP__eqI,axiom,
! [A4: set_o,F: $o > set_nat,X2: set_nat] :
( ! [I4: $o] :
( ( member_o @ I4 @ A4 )
=> ( ord_less_eq_set_nat @ ( F @ I4 ) @ X2 ) )
=> ( ! [Y: set_nat] :
( ! [I5: $o] :
( ( member_o @ I5 @ A4 )
=> ( ord_less_eq_set_nat @ ( F @ I5 ) @ Y ) )
=> ( ord_less_eq_set_nat @ X2 @ Y ) )
=> ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A4 ) )
= X2 ) ) ) ).
% SUP_eqI
thf(fact_2326_SUP__eqI,axiom,
! [A4: set_nat,F: nat > set_nat,X2: set_nat] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A4 )
=> ( ord_less_eq_set_nat @ ( F @ I4 ) @ X2 ) )
=> ( ! [Y: set_nat] :
( ! [I5: nat] :
( ( member_nat @ I5 @ A4 )
=> ( ord_less_eq_set_nat @ ( F @ I5 ) @ Y ) )
=> ( ord_less_eq_set_nat @ X2 @ Y ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A4 ) )
= X2 ) ) ) ).
% SUP_eqI
thf(fact_2327_SUP__eqI,axiom,
! [A4: set_int,F: int > set_nat,X2: set_nat] :
( ! [I4: int] :
( ( member_int @ I4 @ A4 )
=> ( ord_less_eq_set_nat @ ( F @ I4 ) @ X2 ) )
=> ( ! [Y: set_nat] :
( ! [I5: int] :
( ( member_int @ I5 @ A4 )
=> ( ord_less_eq_set_nat @ ( F @ I5 ) @ Y ) )
=> ( ord_less_eq_set_nat @ X2 @ Y ) )
=> ( ( comple7399068483239264473et_nat @ ( image_int_set_nat @ F @ A4 ) )
= X2 ) ) ) ).
% SUP_eqI
thf(fact_2328_SUP__eqI,axiom,
! [A4: set_Pr4329608150637261639at_nat,F: produc3843707927480180839at_nat > product_unit,X2: product_unit] :
( ! [I4: produc3843707927480180839at_nat] :
( ( member8757157785044589968at_nat @ I4 @ A4 )
=> ( ord_le3221252021190050221t_unit @ ( F @ I4 ) @ X2 ) )
=> ( ! [Y: product_unit] :
( ! [I5: produc3843707927480180839at_nat] :
( ( member8757157785044589968at_nat @ I5 @ A4 )
=> ( ord_le3221252021190050221t_unit @ ( F @ I5 ) @ Y ) )
=> ( ord_le3221252021190050221t_unit @ X2 @ Y ) )
=> ( ( comple4687483117567863418t_unit @ ( image_7082456443403096939t_unit @ F @ A4 ) )
= X2 ) ) ) ).
% SUP_eqI
thf(fact_2329_SUP__eqI,axiom,
! [A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > product_unit,X2: product_unit] :
( ! [I4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ I4 @ A4 )
=> ( ord_le3221252021190050221t_unit @ ( F @ I4 ) @ X2 ) )
=> ( ! [Y: product_unit] :
( ! [I5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ I5 @ A4 )
=> ( ord_le3221252021190050221t_unit @ ( F @ I5 ) @ Y ) )
=> ( ord_le3221252021190050221t_unit @ X2 @ Y ) )
=> ( ( comple4687483117567863418t_unit @ ( image_6169909464789311569t_unit @ F @ A4 ) )
= X2 ) ) ) ).
% SUP_eqI
thf(fact_2330_SUP__eqI,axiom,
! [A4: set_o,F: $o > product_unit,X2: product_unit] :
( ! [I4: $o] :
( ( member_o @ I4 @ A4 )
=> ( ord_le3221252021190050221t_unit @ ( F @ I4 ) @ X2 ) )
=> ( ! [Y: product_unit] :
( ! [I5: $o] :
( ( member_o @ I5 @ A4 )
=> ( ord_le3221252021190050221t_unit @ ( F @ I5 ) @ Y ) )
=> ( ord_le3221252021190050221t_unit @ X2 @ Y ) )
=> ( ( comple4687483117567863418t_unit @ ( image_o_Product_unit @ F @ A4 ) )
= X2 ) ) ) ).
% SUP_eqI
thf(fact_2331_SUP__eqI,axiom,
! [A4: set_nat,F: nat > product_unit,X2: product_unit] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A4 )
=> ( ord_le3221252021190050221t_unit @ ( F @ I4 ) @ X2 ) )
=> ( ! [Y: product_unit] :
( ! [I5: nat] :
( ( member_nat @ I5 @ A4 )
=> ( ord_le3221252021190050221t_unit @ ( F @ I5 ) @ Y ) )
=> ( ord_le3221252021190050221t_unit @ X2 @ Y ) )
=> ( ( comple4687483117567863418t_unit @ ( image_8730104196221521654t_unit @ F @ A4 ) )
= X2 ) ) ) ).
% SUP_eqI
thf(fact_2332_SUP__eqI,axiom,
! [A4: set_int,F: int > product_unit,X2: product_unit] :
( ! [I4: int] :
( ( member_int @ I4 @ A4 )
=> ( ord_le3221252021190050221t_unit @ ( F @ I4 ) @ X2 ) )
=> ( ! [Y: product_unit] :
( ! [I5: int] :
( ( member_int @ I5 @ A4 )
=> ( ord_le3221252021190050221t_unit @ ( F @ I5 ) @ Y ) )
=> ( ord_le3221252021190050221t_unit @ X2 @ Y ) )
=> ( ( comple4687483117567863418t_unit @ ( image_5846526143423701914t_unit @ F @ A4 ) )
= X2 ) ) ) ).
% SUP_eqI
thf(fact_2333_INT__extend__simps_I10_J,axiom,
! [B5: rat > set_nat,F: nat > rat,A4: set_nat] :
( ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [A5: nat] : ( B5 @ ( F @ A5 ) )
@ A4 ) )
= ( comple7806235888213564991et_nat @ ( image_rat_set_nat @ B5 @ ( image_nat_rat @ F @ A4 ) ) ) ) ).
% INT_extend_simps(10)
thf(fact_2334_INT__extend__simps_I10_J,axiom,
! [B5: nat > set_nat,F: nat > nat,A4: set_nat] :
( ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [A5: nat] : ( B5 @ ( F @ A5 ) )
@ A4 ) )
= ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B5 @ ( image_nat_nat @ F @ A4 ) ) ) ) ).
% INT_extend_simps(10)
thf(fact_2335_INT__extend__simps_I10_J,axiom,
! [B5: int > set_nat,F: nat > int,A4: set_nat] :
( ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [A5: nat] : ( B5 @ ( F @ A5 ) )
@ A4 ) )
= ( comple7806235888213564991et_nat @ ( image_int_set_nat @ B5 @ ( image_nat_int @ F @ A4 ) ) ) ) ).
% INT_extend_simps(10)
thf(fact_2336_INT__extend__simps_I10_J,axiom,
! [B5: nat > set_nat,F: int > nat,A4: set_int] :
( ( comple7806235888213564991et_nat
@ ( image_int_set_nat
@ ^ [A5: int] : ( B5 @ ( F @ A5 ) )
@ A4 ) )
= ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B5 @ ( image_int_nat @ F @ A4 ) ) ) ) ).
% INT_extend_simps(10)
thf(fact_2337_INT__extend__simps_I10_J,axiom,
! [B5: int > set_nat,F: int > int,A4: set_int] :
( ( comple7806235888213564991et_nat
@ ( image_int_set_nat
@ ^ [A5: int] : ( B5 @ ( F @ A5 ) )
@ A4 ) )
= ( comple7806235888213564991et_nat @ ( image_int_set_nat @ B5 @ ( image_int_int @ F @ A4 ) ) ) ) ).
% INT_extend_simps(10)
thf(fact_2338_INT__lower,axiom,
! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B5: produc3843707927480180839at_nat > set_nat] :
( ( member8757157785044589968at_nat @ A @ A4 )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_3160451284896210088et_nat @ B5 @ A4 ) ) @ ( B5 @ A ) ) ) ).
% INT_lower
thf(fact_2339_INT__lower,axiom,
! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B5: product_prod_nat_nat > set_nat] :
( ( member8440522571783428010at_nat @ A @ A4 )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_15824709712370754et_nat @ B5 @ A4 ) ) @ ( B5 @ A ) ) ) ).
% INT_lower
thf(fact_2340_INT__lower,axiom,
! [A: $o,A4: set_o,B5: $o > set_nat] :
( ( member_o @ A @ A4 )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ B5 @ A4 ) ) @ ( B5 @ A ) ) ) ).
% INT_lower
thf(fact_2341_INT__lower,axiom,
! [A: nat,A4: set_nat,B5: nat > set_nat] :
( ( member_nat @ A @ A4 )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B5 @ A4 ) ) @ ( B5 @ A ) ) ) ).
% INT_lower
thf(fact_2342_INT__lower,axiom,
! [A: int,A4: set_int,B5: int > set_nat] :
( ( member_int @ A @ A4 )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_int_set_nat @ B5 @ A4 ) ) @ ( B5 @ A ) ) ) ).
% INT_lower
thf(fact_2343_INT__greatest,axiom,
! [A4: set_Pr4329608150637261639at_nat,C3: set_nat,B5: produc3843707927480180839at_nat > set_nat] :
( ! [X: produc3843707927480180839at_nat] :
( ( member8757157785044589968at_nat @ X @ A4 )
=> ( ord_less_eq_set_nat @ C3 @ ( B5 @ X ) ) )
=> ( ord_less_eq_set_nat @ C3 @ ( comple7806235888213564991et_nat @ ( image_3160451284896210088et_nat @ B5 @ A4 ) ) ) ) ).
% INT_greatest
thf(fact_2344_INT__greatest,axiom,
! [A4: set_Pr1261947904930325089at_nat,C3: set_nat,B5: product_prod_nat_nat > set_nat] :
( ! [X: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X @ A4 )
=> ( ord_less_eq_set_nat @ C3 @ ( B5 @ X ) ) )
=> ( ord_less_eq_set_nat @ C3 @ ( comple7806235888213564991et_nat @ ( image_15824709712370754et_nat @ B5 @ A4 ) ) ) ) ).
% INT_greatest
thf(fact_2345_INT__greatest,axiom,
! [A4: set_o,C3: set_nat,B5: $o > set_nat] :
( ! [X: $o] :
( ( member_o @ X @ A4 )
=> ( ord_less_eq_set_nat @ C3 @ ( B5 @ X ) ) )
=> ( ord_less_eq_set_nat @ C3 @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ B5 @ A4 ) ) ) ) ).
% INT_greatest
thf(fact_2346_INT__greatest,axiom,
! [A4: set_nat,C3: set_nat,B5: nat > set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ( ord_less_eq_set_nat @ C3 @ ( B5 @ X ) ) )
=> ( ord_less_eq_set_nat @ C3 @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B5 @ A4 ) ) ) ) ).
% INT_greatest
thf(fact_2347_INT__greatest,axiom,
! [A4: set_int,C3: set_nat,B5: int > set_nat] :
( ! [X: int] :
( ( member_int @ X @ A4 )
=> ( ord_less_eq_set_nat @ C3 @ ( B5 @ X ) ) )
=> ( ord_less_eq_set_nat @ C3 @ ( comple7806235888213564991et_nat @ ( image_int_set_nat @ B5 @ A4 ) ) ) ) ).
% INT_greatest
thf(fact_2348_INT__anti__mono,axiom,
! [A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat,F: produc3843707927480180839at_nat > set_nat,G: produc3843707927480180839at_nat > set_nat] :
( ( ord_le1268244103169919719at_nat @ A4 @ B5 )
=> ( ! [X: produc3843707927480180839at_nat] :
( ( member8757157785044589968at_nat @ X @ A4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_3160451284896210088et_nat @ F @ B5 ) ) @ ( comple7806235888213564991et_nat @ ( image_3160451284896210088et_nat @ G @ A4 ) ) ) ) ) ).
% INT_anti_mono
thf(fact_2349_INT__anti__mono,axiom,
! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > set_nat,G: product_prod_nat_nat > set_nat] :
( ( ord_le3146513528884898305at_nat @ A4 @ B5 )
=> ( ! [X: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X @ A4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_15824709712370754et_nat @ F @ B5 ) ) @ ( comple7806235888213564991et_nat @ ( image_15824709712370754et_nat @ G @ A4 ) ) ) ) ) ).
% INT_anti_mono
thf(fact_2350_INT__anti__mono,axiom,
! [A4: set_o,B5: set_o,F: $o > set_nat,G: $o > set_nat] :
( ( ord_less_eq_set_o @ A4 @ B5 )
=> ( ! [X: $o] :
( ( member_o @ X @ A4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ F @ B5 ) ) @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ G @ A4 ) ) ) ) ) ).
% INT_anti_mono
thf(fact_2351_INT__anti__mono,axiom,
! [A4: set_nat,B5: set_nat,F: nat > set_nat,G: nat > set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ B5 ) ) @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ G @ A4 ) ) ) ) ) ).
% INT_anti_mono
thf(fact_2352_INT__anti__mono,axiom,
! [A4: set_int,B5: set_int,F: int > set_nat,G: int > set_nat] :
( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( ! [X: int] :
( ( member_int @ X @ A4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_int_set_nat @ F @ B5 ) ) @ ( comple7806235888213564991et_nat @ ( image_int_set_nat @ G @ A4 ) ) ) ) ) ).
% INT_anti_mono
thf(fact_2353_image__constant__conv,axiom,
! [A4: set_int,C: int] :
( ( ( A4 = bot_bot_set_int )
=> ( ( image_int_int
@ ^ [X4: int] : C
@ A4 )
= bot_bot_set_int ) )
& ( ( A4 != bot_bot_set_int )
=> ( ( image_int_int
@ ^ [X4: int] : C
@ A4 )
= ( insert_int @ C @ bot_bot_set_int ) ) ) ) ).
% image_constant_conv
thf(fact_2354_image__constant__conv,axiom,
! [A4: set_int,C: nat] :
( ( ( A4 = bot_bot_set_int )
=> ( ( image_int_nat
@ ^ [X4: int] : C
@ A4 )
= bot_bot_set_nat ) )
& ( ( A4 != bot_bot_set_int )
=> ( ( image_int_nat
@ ^ [X4: int] : C
@ A4 )
= ( insert_nat @ C @ bot_bot_set_nat ) ) ) ) ).
% image_constant_conv
thf(fact_2355_image__constant__conv,axiom,
! [A4: set_Product_unit,C: product_unit] :
( ( ( A4 = bot_bo3957492148770167129t_unit )
=> ( ( image_405062704495631173t_unit
@ ^ [X4: product_unit] : C
@ A4 )
= bot_bo3957492148770167129t_unit ) )
& ( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ( image_405062704495631173t_unit
@ ^ [X4: product_unit] : C
@ A4 )
= ( insert_Product_unit @ C @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% image_constant_conv
thf(fact_2356_image__constant__conv,axiom,
! [A4: set_Product_unit,C: $o] :
( ( ( A4 = bot_bo3957492148770167129t_unit )
=> ( ( image_Product_unit_o
@ ^ [X4: product_unit] : C
@ A4 )
= bot_bot_set_o ) )
& ( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ( image_Product_unit_o
@ ^ [X4: product_unit] : C
@ A4 )
= ( insert_o @ C @ bot_bot_set_o ) ) ) ) ).
% image_constant_conv
thf(fact_2357_image__constant__conv,axiom,
! [A4: set_Product_unit,C: nat] :
( ( ( A4 = bot_bo3957492148770167129t_unit )
=> ( ( image_875570014554754200it_nat
@ ^ [X4: product_unit] : C
@ A4 )
= bot_bot_set_nat ) )
& ( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ( image_875570014554754200it_nat
@ ^ [X4: product_unit] : C
@ A4 )
= ( insert_nat @ C @ bot_bot_set_nat ) ) ) ) ).
% image_constant_conv
thf(fact_2358_image__constant__conv,axiom,
! [A4: set_o,C: product_unit] :
( ( ( A4 = bot_bot_set_o )
=> ( ( image_o_Product_unit
@ ^ [X4: $o] : C
@ A4 )
= bot_bo3957492148770167129t_unit ) )
& ( ( A4 != bot_bot_set_o )
=> ( ( image_o_Product_unit
@ ^ [X4: $o] : C
@ A4 )
= ( insert_Product_unit @ C @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% image_constant_conv
thf(fact_2359_image__constant__conv,axiom,
! [A4: set_o,C: $o] :
( ( ( A4 = bot_bot_set_o )
=> ( ( image_o_o
@ ^ [X4: $o] : C
@ A4 )
= bot_bot_set_o ) )
& ( ( A4 != bot_bot_set_o )
=> ( ( image_o_o
@ ^ [X4: $o] : C
@ A4 )
= ( insert_o @ C @ bot_bot_set_o ) ) ) ) ).
% image_constant_conv
thf(fact_2360_image__constant__conv,axiom,
! [A4: set_o,C: nat] :
( ( ( A4 = bot_bot_set_o )
=> ( ( image_o_nat
@ ^ [X4: $o] : C
@ A4 )
= bot_bot_set_nat ) )
& ( ( A4 != bot_bot_set_o )
=> ( ( image_o_nat
@ ^ [X4: $o] : C
@ A4 )
= ( insert_nat @ C @ bot_bot_set_nat ) ) ) ) ).
% image_constant_conv
thf(fact_2361_image__constant__conv,axiom,
! [A4: set_nat,C: rat] :
( ( ( A4 = bot_bot_set_nat )
=> ( ( image_nat_rat
@ ^ [X4: nat] : C
@ A4 )
= bot_bot_set_rat ) )
& ( ( A4 != bot_bot_set_nat )
=> ( ( image_nat_rat
@ ^ [X4: nat] : C
@ A4 )
= ( insert_rat @ C @ bot_bot_set_rat ) ) ) ) ).
% image_constant_conv
thf(fact_2362_image__constant__conv,axiom,
! [A4: set_nat,C: int] :
( ( ( A4 = bot_bot_set_nat )
=> ( ( image_nat_int
@ ^ [X4: nat] : C
@ A4 )
= bot_bot_set_int ) )
& ( ( A4 != bot_bot_set_nat )
=> ( ( image_nat_int
@ ^ [X4: nat] : C
@ A4 )
= ( insert_int @ C @ bot_bot_set_int ) ) ) ) ).
% image_constant_conv
thf(fact_2363_image__constant,axiom,
! [X2: nat,A4: set_nat,C: rat] :
( ( member_nat @ X2 @ A4 )
=> ( ( image_nat_rat
@ ^ [X4: nat] : C
@ A4 )
= ( insert_rat @ C @ bot_bot_set_rat ) ) ) ).
% image_constant
thf(fact_2364_image__constant,axiom,
! [X2: nat,A4: set_nat,C: int] :
( ( member_nat @ X2 @ A4 )
=> ( ( image_nat_int
@ ^ [X4: nat] : C
@ A4 )
= ( insert_int @ C @ bot_bot_set_int ) ) ) ).
% image_constant
thf(fact_2365_image__constant,axiom,
! [X2: int,A4: set_int,C: int] :
( ( member_int @ X2 @ A4 )
=> ( ( image_int_int
@ ^ [X4: int] : C
@ A4 )
= ( insert_int @ C @ bot_bot_set_int ) ) ) ).
% image_constant
thf(fact_2366_image__constant,axiom,
! [X2: $o,A4: set_o,C: product_unit] :
( ( member_o @ X2 @ A4 )
=> ( ( image_o_Product_unit
@ ^ [X4: $o] : C
@ A4 )
= ( insert_Product_unit @ C @ bot_bo3957492148770167129t_unit ) ) ) ).
% image_constant
thf(fact_2367_image__constant,axiom,
! [X2: nat,A4: set_nat,C: product_unit] :
( ( member_nat @ X2 @ A4 )
=> ( ( image_8730104196221521654t_unit
@ ^ [X4: nat] : C
@ A4 )
= ( insert_Product_unit @ C @ bot_bo3957492148770167129t_unit ) ) ) ).
% image_constant
thf(fact_2368_image__constant,axiom,
! [X2: int,A4: set_int,C: product_unit] :
( ( member_int @ X2 @ A4 )
=> ( ( image_5846526143423701914t_unit
@ ^ [X4: int] : C
@ A4 )
= ( insert_Product_unit @ C @ bot_bo3957492148770167129t_unit ) ) ) ).
% image_constant
thf(fact_2369_image__constant,axiom,
! [X2: $o,A4: set_o,C: $o] :
( ( member_o @ X2 @ A4 )
=> ( ( image_o_o
@ ^ [X4: $o] : C
@ A4 )
= ( insert_o @ C @ bot_bot_set_o ) ) ) ).
% image_constant
thf(fact_2370_image__constant,axiom,
! [X2: nat,A4: set_nat,C: $o] :
( ( member_nat @ X2 @ A4 )
=> ( ( image_nat_o
@ ^ [X4: nat] : C
@ A4 )
= ( insert_o @ C @ bot_bot_set_o ) ) ) ).
% image_constant
thf(fact_2371_image__constant,axiom,
! [X2: int,A4: set_int,C: $o] :
( ( member_int @ X2 @ A4 )
=> ( ( image_int_o
@ ^ [X4: int] : C
@ A4 )
= ( insert_o @ C @ bot_bot_set_o ) ) ) ).
% image_constant
thf(fact_2372_image__constant,axiom,
! [X2: $o,A4: set_o,C: nat] :
( ( member_o @ X2 @ A4 )
=> ( ( image_o_nat
@ ^ [X4: $o] : C
@ A4 )
= ( insert_nat @ C @ bot_bot_set_nat ) ) ) ).
% image_constant
thf(fact_2373_UN__extend__simps_I10_J,axiom,
! [B5: rat > set_nat,F: nat > rat,A4: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [A5: nat] : ( B5 @ ( F @ A5 ) )
@ A4 ) )
= ( comple7399068483239264473et_nat @ ( image_rat_set_nat @ B5 @ ( image_nat_rat @ F @ A4 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_2374_UN__extend__simps_I10_J,axiom,
! [B5: nat > set_nat,F: nat > nat,A4: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [A5: nat] : ( B5 @ ( F @ A5 ) )
@ A4 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ ( image_nat_nat @ F @ A4 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_2375_UN__extend__simps_I10_J,axiom,
! [B5: int > set_nat,F: nat > int,A4: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [A5: nat] : ( B5 @ ( F @ A5 ) )
@ A4 ) )
= ( comple7399068483239264473et_nat @ ( image_int_set_nat @ B5 @ ( image_nat_int @ F @ A4 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_2376_UN__extend__simps_I10_J,axiom,
! [B5: nat > set_nat,F: int > nat,A4: set_int] :
( ( comple7399068483239264473et_nat
@ ( image_int_set_nat
@ ^ [A5: int] : ( B5 @ ( F @ A5 ) )
@ A4 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ ( image_int_nat @ F @ A4 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_2377_UN__extend__simps_I10_J,axiom,
! [B5: int > set_nat,F: int > int,A4: set_int] :
( ( comple7399068483239264473et_nat
@ ( image_int_set_nat
@ ^ [A5: int] : ( B5 @ ( F @ A5 ) )
@ A4 ) )
= ( comple7399068483239264473et_nat @ ( image_int_set_nat @ B5 @ ( image_int_int @ F @ A4 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_2378_UN__empty,axiom,
! [B5: product_unit > set_Product_unit] :
( ( comple3349607820504768346t_unit @ ( image_7778121751606633509t_unit @ B5 @ bot_bo3957492148770167129t_unit ) )
= bot_bo3957492148770167129t_unit ) ).
% UN_empty
thf(fact_2379_UN__empty,axiom,
! [B5: product_unit > set_o] :
( ( comple90263536869209701_set_o @ ( image_6930536745178679216_set_o @ B5 @ bot_bo3957492148770167129t_unit ) )
= bot_bot_set_o ) ).
% UN_empty
thf(fact_2380_UN__empty,axiom,
! [B5: $o > set_Product_unit] :
( ( comple3349607820504768346t_unit @ ( image_663195431845683034t_unit @ B5 @ bot_bot_set_o ) )
= bot_bo3957492148770167129t_unit ) ).
% UN_empty
thf(fact_2381_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_2382_UN__empty,axiom,
! [B5: nat > set_Product_unit] :
( ( comple3349607820504768346t_unit @ ( image_4824033694279209942t_unit @ B5 @ bot_bot_set_nat ) )
= bot_bo3957492148770167129t_unit ) ).
% UN_empty
thf(fact_2383_UN__empty,axiom,
! [B5: nat > set_o] :
( ( comple90263536869209701_set_o @ ( image_nat_set_o @ B5 @ bot_bot_set_nat ) )
= bot_bot_set_o ) ).
% UN_empty
thf(fact_2384_UN__empty,axiom,
! [B5: product_unit > set_nat] :
( ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ B5 @ bot_bo3957492148770167129t_unit ) )
= bot_bot_set_nat ) ).
% UN_empty
thf(fact_2385_UN__empty,axiom,
! [B5: $o > set_nat] :
( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B5 @ bot_bot_set_o ) )
= bot_bot_set_nat ) ).
% UN_empty
thf(fact_2386_UN__empty,axiom,
! [B5: nat > set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ bot_bot_set_nat ) )
= bot_bot_set_nat ) ).
% UN_empty
thf(fact_2387_UN__empty,axiom,
! [B5: product_unit > set_Pr1261947904930325089at_nat] :
( ( comple5685304695842803022at_nat @ ( image_3617315916088684441at_nat @ B5 @ bot_bo3957492148770167129t_unit ) )
= bot_bo2099793752762293965at_nat ) ).
% UN_empty
thf(fact_2388_UN__mono,axiom,
! [A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat,F: produc3843707927480180839at_nat > set_nat,G: produc3843707927480180839at_nat > set_nat] :
( ( ord_le1268244103169919719at_nat @ A4 @ B5 )
=> ( ! [X: produc3843707927480180839at_nat] :
( ( member8757157785044589968at_nat @ X @ A4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_3160451284896210088et_nat @ F @ A4 ) ) @ ( comple7399068483239264473et_nat @ ( image_3160451284896210088et_nat @ G @ B5 ) ) ) ) ) ).
% UN_mono
thf(fact_2389_UN__mono,axiom,
! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > set_nat,G: product_prod_nat_nat > set_nat] :
( ( ord_le3146513528884898305at_nat @ A4 @ B5 )
=> ( ! [X: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X @ A4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ F @ A4 ) ) @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ G @ B5 ) ) ) ) ) ).
% UN_mono
thf(fact_2390_UN__mono,axiom,
! [A4: set_o,B5: set_o,F: $o > set_nat,G: $o > set_nat] :
( ( ord_less_eq_set_o @ A4 @ B5 )
=> ( ! [X: $o] :
( ( member_o @ X @ A4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A4 ) ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ G @ B5 ) ) ) ) ) ).
% UN_mono
thf(fact_2391_UN__mono,axiom,
! [A4: set_nat,B5: set_nat,F: nat > set_nat,G: nat > set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A4 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B5 ) ) ) ) ) ).
% UN_mono
thf(fact_2392_UN__mono,axiom,
! [A4: set_int,B5: set_int,F: int > set_nat,G: int > set_nat] :
( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( ! [X: int] :
( ( member_int @ X @ A4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_int_set_nat @ F @ A4 ) ) @ ( comple7399068483239264473et_nat @ ( image_int_set_nat @ G @ B5 ) ) ) ) ) ).
% UN_mono
thf(fact_2393_UN__least,axiom,
! [A4: set_Pr4329608150637261639at_nat,B5: produc3843707927480180839at_nat > set_nat,C3: set_nat] :
( ! [X: produc3843707927480180839at_nat] :
( ( member8757157785044589968at_nat @ X @ A4 )
=> ( ord_less_eq_set_nat @ ( B5 @ X ) @ C3 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_3160451284896210088et_nat @ B5 @ A4 ) ) @ C3 ) ) ).
% UN_least
thf(fact_2394_UN__least,axiom,
! [A4: set_Pr1261947904930325089at_nat,B5: product_prod_nat_nat > set_nat,C3: set_nat] :
( ! [X: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X @ A4 )
=> ( ord_less_eq_set_nat @ ( B5 @ X ) @ C3 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ B5 @ A4 ) ) @ C3 ) ) ).
% UN_least
thf(fact_2395_UN__least,axiom,
! [A4: set_o,B5: $o > set_nat,C3: set_nat] :
( ! [X: $o] :
( ( member_o @ X @ A4 )
=> ( ord_less_eq_set_nat @ ( B5 @ X ) @ C3 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B5 @ A4 ) ) @ C3 ) ) ).
% UN_least
thf(fact_2396_UN__least,axiom,
! [A4: set_nat,B5: nat > set_nat,C3: set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ( ord_less_eq_set_nat @ ( B5 @ X ) @ C3 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ A4 ) ) @ C3 ) ) ).
% UN_least
thf(fact_2397_UN__least,axiom,
! [A4: set_int,B5: int > set_nat,C3: set_nat] :
( ! [X: int] :
( ( member_int @ X @ A4 )
=> ( ord_less_eq_set_nat @ ( B5 @ X ) @ C3 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_int_set_nat @ B5 @ A4 ) ) @ C3 ) ) ).
% UN_least
thf(fact_2398_UN__upper,axiom,
! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B5: produc3843707927480180839at_nat > set_nat] :
( ( member8757157785044589968at_nat @ A @ A4 )
=> ( ord_less_eq_set_nat @ ( B5 @ A ) @ ( comple7399068483239264473et_nat @ ( image_3160451284896210088et_nat @ B5 @ A4 ) ) ) ) ).
% UN_upper
thf(fact_2399_UN__upper,axiom,
! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B5: product_prod_nat_nat > set_nat] :
( ( member8440522571783428010at_nat @ A @ A4 )
=> ( ord_less_eq_set_nat @ ( B5 @ A ) @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ B5 @ A4 ) ) ) ) ).
% UN_upper
thf(fact_2400_UN__upper,axiom,
! [A: $o,A4: set_o,B5: $o > set_nat] :
( ( member_o @ A @ A4 )
=> ( ord_less_eq_set_nat @ ( B5 @ A ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B5 @ A4 ) ) ) ) ).
% UN_upper
thf(fact_2401_UN__upper,axiom,
! [A: nat,A4: set_nat,B5: nat > set_nat] :
( ( member_nat @ A @ A4 )
=> ( ord_less_eq_set_nat @ ( B5 @ A ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ A4 ) ) ) ) ).
% UN_upper
thf(fact_2402_UN__upper,axiom,
! [A: int,A4: set_int,B5: int > set_nat] :
( ( member_int @ A @ A4 )
=> ( ord_less_eq_set_nat @ ( B5 @ A ) @ ( comple7399068483239264473et_nat @ ( image_int_set_nat @ B5 @ A4 ) ) ) ) ).
% UN_upper
thf(fact_2403_Union__natural,axiom,
! [F: int > int] :
( ( comp_s8709038167382007543et_int @ comple3221217463730067765et_int @ ( image_524474410958335435et_int @ ( image_int_int @ F ) ) )
= ( comp_s2487324123861929025et_int @ ( image_int_int @ F ) @ comple3221217463730067765et_int ) ) ).
% Union_natural
thf(fact_2404_Union__natural,axiom,
! [F: nat > rat] :
( ( comp_s703482200030188019et_nat @ comple3890839924845867745et_rat @ ( image_4408659257933336347et_rat @ ( image_nat_rat @ F ) ) )
= ( comp_s6435139744279249717et_nat @ ( image_nat_rat @ F ) @ comple7399068483239264473et_nat ) ) ).
% Union_natural
thf(fact_2405_Union__natural,axiom,
! [F: nat > int] :
( ( comp_s1975749275458958491et_nat @ comple3221217463730067765et_int @ ( image_3739036796817536367et_int @ ( image_nat_int @ F ) ) )
= ( comp_s3768223441818958217et_nat @ ( image_nat_int @ F ) @ comple7399068483239264473et_nat ) ) ).
% Union_natural
thf(fact_2406_Union__natural,axiom,
! [F: int > nat] :
( ( comp_s6907669228194913343et_int @ comple7399068483239264473et_nat @ ( image_4702325430467532143et_nat @ ( image_int_nat @ F ) ) )
= ( comp_s5712175470073906149et_int @ ( image_int_nat @ F ) @ comple3221217463730067765et_int ) ) ).
% Union_natural
thf(fact_2407_Union__natural,axiom,
! [F: nat > nat] :
( ( comp_s174380336271864291et_nat @ comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat @ F ) ) )
= ( comp_s6993074788030935341et_nat @ ( image_nat_nat @ F ) @ comple7399068483239264473et_nat ) ) ).
% Union_natural
thf(fact_2408_image__Union,axiom,
! [F: int > int,S5: set_set_int] :
( ( image_int_int @ F @ ( comple3221217463730067765et_int @ S5 ) )
= ( comple3221217463730067765et_int @ ( image_524474410958335435et_int @ ( image_int_int @ F ) @ S5 ) ) ) ).
% image_Union
thf(fact_2409_image__Union,axiom,
! [F: int > nat,S5: set_set_int] :
( ( image_int_nat @ F @ ( comple3221217463730067765et_int @ S5 ) )
= ( comple7399068483239264473et_nat @ ( image_4702325430467532143et_nat @ ( image_int_nat @ F ) @ S5 ) ) ) ).
% image_Union
thf(fact_2410_image__Union,axiom,
! [F: nat > rat,S5: set_set_nat] :
( ( image_nat_rat @ F @ ( comple7399068483239264473et_nat @ S5 ) )
= ( comple3890839924845867745et_rat @ ( image_4408659257933336347et_rat @ ( image_nat_rat @ F ) @ S5 ) ) ) ).
% image_Union
thf(fact_2411_image__Union,axiom,
! [F: nat > int,S5: set_set_nat] :
( ( image_nat_int @ F @ ( comple7399068483239264473et_nat @ S5 ) )
= ( comple3221217463730067765et_int @ ( image_3739036796817536367et_int @ ( image_nat_int @ F ) @ S5 ) ) ) ).
% image_Union
thf(fact_2412_image__Union,axiom,
! [F: nat > nat,S5: set_set_nat] :
( ( image_nat_nat @ F @ ( comple7399068483239264473et_nat @ S5 ) )
= ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat @ F ) @ S5 ) ) ) ).
% image_Union
thf(fact_2413_UN__extend__simps_I8_J,axiom,
! [B5: nat > set_nat,A4: set_set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_7916887816326733075et_nat
@ ^ [Y5: set_nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ Y5 ) )
@ A4 ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ ( comple7399068483239264473et_nat @ A4 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_2414_INF__eq__iff,axiom,
! [I3: set_int,F: int > product_unit,C: product_unit] :
( ( I3 != bot_bot_set_int )
=> ( ! [I4: int] :
( ( member_int @ I4 @ I3 )
=> ( ord_le3221252021190050221t_unit @ ( F @ I4 ) @ C ) )
=> ( ( ( comple2584293577114468500t_unit @ ( image_5846526143423701914t_unit @ F @ I3 ) )
= C )
= ( ! [X4: int] :
( ( member_int @ X4 @ I3 )
=> ( ( F @ X4 )
= C ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_2415_INF__eq__iff,axiom,
! [I3: set_Product_unit,F: product_unit > product_unit,C: product_unit] :
( ( I3 != bot_bo3957492148770167129t_unit )
=> ( ! [I4: product_unit] :
( ( member_Product_unit @ I4 @ I3 )
=> ( ord_le3221252021190050221t_unit @ ( F @ I4 ) @ C ) )
=> ( ( ( comple2584293577114468500t_unit @ ( image_405062704495631173t_unit @ F @ I3 ) )
= C )
= ( ! [X4: product_unit] :
( ( member_Product_unit @ X4 @ I3 )
=> ( ( F @ X4 )
= C ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_2416_INF__eq__iff,axiom,
! [I3: set_o,F: $o > product_unit,C: product_unit] :
( ( I3 != bot_bot_set_o )
=> ( ! [I4: $o] :
( ( member_o @ I4 @ I3 )
=> ( ord_le3221252021190050221t_unit @ ( F @ I4 ) @ C ) )
=> ( ( ( comple2584293577114468500t_unit @ ( image_o_Product_unit @ F @ I3 ) )
= C )
= ( ! [X4: $o] :
( ( member_o @ X4 @ I3 )
=> ( ( F @ X4 )
= C ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_2417_INF__eq__iff,axiom,
! [I3: set_nat,F: nat > product_unit,C: product_unit] :
( ( I3 != bot_bot_set_nat )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I3 )
=> ( ord_le3221252021190050221t_unit @ ( F @ I4 ) @ C ) )
=> ( ( ( comple2584293577114468500t_unit @ ( image_8730104196221521654t_unit @ F @ I3 ) )
= C )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ I3 )
=> ( ( F @ X4 )
= C ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_2418_INF__eq__iff,axiom,
! [I3: set_int,F: int > set_nat,C: set_nat] :
( ( I3 != bot_bot_set_int )
=> ( ! [I4: int] :
( ( member_int @ I4 @ I3 )
=> ( ord_less_eq_set_nat @ ( F @ I4 ) @ C ) )
=> ( ( ( comple7806235888213564991et_nat @ ( image_int_set_nat @ F @ I3 ) )
= C )
= ( ! [X4: int] :
( ( member_int @ X4 @ I3 )
=> ( ( F @ X4 )
= C ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_2419_INF__eq__iff,axiom,
! [I3: set_Product_unit,F: product_unit > set_nat,C: set_nat] :
( ( I3 != bot_bo3957492148770167129t_unit )
=> ( ! [I4: product_unit] :
( ( member_Product_unit @ I4 @ I3 )
=> ( ord_less_eq_set_nat @ ( F @ I4 ) @ C ) )
=> ( ( ( comple7806235888213564991et_nat @ ( image_8173849991827986510et_nat @ F @ I3 ) )
= C )
= ( ! [X4: product_unit] :
( ( member_Product_unit @ X4 @ I3 )
=> ( ( F @ X4 )
= C ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_2420_INF__eq__iff,axiom,
! [I3: set_o,F: $o > set_nat,C: set_nat] :
( ( I3 != bot_bot_set_o )
=> ( ! [I4: $o] :
( ( member_o @ I4 @ I3 )
=> ( ord_less_eq_set_nat @ ( F @ I4 ) @ C ) )
=> ( ( ( comple7806235888213564991et_nat @ ( image_o_set_nat @ F @ I3 ) )
= C )
= ( ! [X4: $o] :
( ( member_o @ X4 @ I3 )
=> ( ( F @ X4 )
= C ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_2421_INF__eq__iff,axiom,
! [I3: set_nat,F: nat > set_nat,C: set_nat] :
( ( I3 != bot_bot_set_nat )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I3 )
=> ( ord_less_eq_set_nat @ ( F @ I4 ) @ C ) )
=> ( ( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ I3 ) )
= C )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ I3 )
=> ( ( F @ X4 )
= C ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_2422_INF__eq__iff,axiom,
! [I3: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > product_unit,C: product_unit] :
( ( I3 != bot_bo2099793752762293965at_nat )
=> ( ! [I4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ I4 @ I3 )
=> ( ord_le3221252021190050221t_unit @ ( F @ I4 ) @ C ) )
=> ( ( ( comple2584293577114468500t_unit @ ( image_6169909464789311569t_unit @ F @ I3 ) )
= C )
= ( ! [X4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X4 @ I3 )
=> ( ( F @ X4 )
= C ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_2423_INF__eq__iff,axiom,
! [I3: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > set_nat,C: set_nat] :
( ( I3 != bot_bo2099793752762293965at_nat )
=> ( ! [I4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ I4 @ I3 )
=> ( ord_less_eq_set_nat @ ( F @ I4 ) @ C ) )
=> ( ( ( comple7806235888213564991et_nat @ ( image_15824709712370754et_nat @ F @ I3 ) )
= C )
= ( ! [X4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X4 @ I3 )
=> ( ( F @ X4 )
= C ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_2424_SUP__eq__iff,axiom,
! [I3: set_int,C: product_unit,F: int > product_unit] :
( ( I3 != bot_bot_set_int )
=> ( ! [I4: int] :
( ( member_int @ I4 @ I3 )
=> ( ord_le3221252021190050221t_unit @ C @ ( F @ I4 ) ) )
=> ( ( ( comple4687483117567863418t_unit @ ( image_5846526143423701914t_unit @ F @ I3 ) )
= C )
= ( ! [X4: int] :
( ( member_int @ X4 @ I3 )
=> ( ( F @ X4 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_2425_SUP__eq__iff,axiom,
! [I3: set_Product_unit,C: product_unit,F: product_unit > product_unit] :
( ( I3 != bot_bo3957492148770167129t_unit )
=> ( ! [I4: product_unit] :
( ( member_Product_unit @ I4 @ I3 )
=> ( ord_le3221252021190050221t_unit @ C @ ( F @ I4 ) ) )
=> ( ( ( comple4687483117567863418t_unit @ ( image_405062704495631173t_unit @ F @ I3 ) )
= C )
= ( ! [X4: product_unit] :
( ( member_Product_unit @ X4 @ I3 )
=> ( ( F @ X4 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_2426_SUP__eq__iff,axiom,
! [I3: set_o,C: product_unit,F: $o > product_unit] :
( ( I3 != bot_bot_set_o )
=> ( ! [I4: $o] :
( ( member_o @ I4 @ I3 )
=> ( ord_le3221252021190050221t_unit @ C @ ( F @ I4 ) ) )
=> ( ( ( comple4687483117567863418t_unit @ ( image_o_Product_unit @ F @ I3 ) )
= C )
= ( ! [X4: $o] :
( ( member_o @ X4 @ I3 )
=> ( ( F @ X4 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_2427_SUP__eq__iff,axiom,
! [I3: set_nat,C: product_unit,F: nat > product_unit] :
( ( I3 != bot_bot_set_nat )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I3 )
=> ( ord_le3221252021190050221t_unit @ C @ ( F @ I4 ) ) )
=> ( ( ( comple4687483117567863418t_unit @ ( image_8730104196221521654t_unit @ F @ I3 ) )
= C )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ I3 )
=> ( ( F @ X4 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_2428_SUP__eq__iff,axiom,
! [I3: set_int,C: set_nat,F: int > set_nat] :
( ( I3 != bot_bot_set_int )
=> ( ! [I4: int] :
( ( member_int @ I4 @ I3 )
=> ( ord_less_eq_set_nat @ C @ ( F @ I4 ) ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_int_set_nat @ F @ I3 ) )
= C )
= ( ! [X4: int] :
( ( member_int @ X4 @ I3 )
=> ( ( F @ X4 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_2429_SUP__eq__iff,axiom,
! [I3: set_Product_unit,C: set_nat,F: product_unit > set_nat] :
( ( I3 != bot_bo3957492148770167129t_unit )
=> ( ! [I4: product_unit] :
( ( member_Product_unit @ I4 @ I3 )
=> ( ord_less_eq_set_nat @ C @ ( F @ I4 ) ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ F @ I3 ) )
= C )
= ( ! [X4: product_unit] :
( ( member_Product_unit @ X4 @ I3 )
=> ( ( F @ X4 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_2430_SUP__eq__iff,axiom,
! [I3: set_o,C: set_nat,F: $o > set_nat] :
( ( I3 != bot_bot_set_o )
=> ( ! [I4: $o] :
( ( member_o @ I4 @ I3 )
=> ( ord_less_eq_set_nat @ C @ ( F @ I4 ) ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ I3 ) )
= C )
= ( ! [X4: $o] :
( ( member_o @ X4 @ I3 )
=> ( ( F @ X4 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_2431_SUP__eq__iff,axiom,
! [I3: set_nat,C: set_nat,F: nat > set_nat] :
( ( I3 != bot_bot_set_nat )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I3 )
=> ( ord_less_eq_set_nat @ C @ ( F @ I4 ) ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ I3 ) )
= C )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ I3 )
=> ( ( F @ X4 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_2432_SUP__eq__iff,axiom,
! [I3: set_Pr1261947904930325089at_nat,C: product_unit,F: product_prod_nat_nat > product_unit] :
( ( I3 != bot_bo2099793752762293965at_nat )
=> ( ! [I4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ I4 @ I3 )
=> ( ord_le3221252021190050221t_unit @ C @ ( F @ I4 ) ) )
=> ( ( ( comple4687483117567863418t_unit @ ( image_6169909464789311569t_unit @ F @ I3 ) )
= C )
= ( ! [X4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X4 @ I3 )
=> ( ( F @ X4 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_2433_SUP__eq__iff,axiom,
! [I3: set_Pr1261947904930325089at_nat,C: set_nat,F: product_prod_nat_nat > set_nat] :
( ( I3 != bot_bo2099793752762293965at_nat )
=> ( ! [I4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ I4 @ I3 )
=> ( ord_less_eq_set_nat @ C @ ( F @ I4 ) ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ F @ I3 ) )
= C )
= ( ! [X4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X4 @ I3 )
=> ( ( F @ X4 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_2434_range__eq__singletonD,axiom,
! [F: char > product_unit,A: product_unit,X2: char] :
( ( ( image_4337914935673063477t_unit @ F @ top_top_set_char )
= ( insert_Product_unit @ A @ bot_bo3957492148770167129t_unit ) )
=> ( ( F @ X2 )
= A ) ) ).
% range_eq_singletonD
thf(fact_2435_range__eq__singletonD,axiom,
! [F: char > $o,A: $o,X2: char] :
( ( ( image_char_o @ F @ top_top_set_char )
= ( insert_o @ A @ bot_bot_set_o ) )
=> ( ( F @ X2 )
= A ) ) ).
% range_eq_singletonD
thf(fact_2436_range__eq__singletonD,axiom,
! [F: char > nat,A: nat,X2: char] :
( ( ( image_char_nat @ F @ top_top_set_char )
= ( insert_nat @ A @ bot_bot_set_nat ) )
=> ( ( F @ X2 )
= A ) ) ).
% range_eq_singletonD
thf(fact_2437_range__eq__singletonD,axiom,
! [F: rat > product_unit,A: product_unit,X2: rat] :
( ( ( image_2327579789247107822t_unit @ F @ top_top_set_rat )
= ( insert_Product_unit @ A @ bot_bo3957492148770167129t_unit ) )
=> ( ( F @ X2 )
= A ) ) ).
% range_eq_singletonD
thf(fact_2438_range__eq__singletonD,axiom,
! [F: rat > $o,A: $o,X2: rat] :
( ( ( image_rat_o @ F @ top_top_set_rat )
= ( insert_o @ A @ bot_bot_set_o ) )
=> ( ( F @ X2 )
= A ) ) ).
% range_eq_singletonD
thf(fact_2439_range__eq__singletonD,axiom,
! [F: rat > nat,A: nat,X2: rat] :
( ( ( image_rat_nat @ F @ top_top_set_rat )
= ( insert_nat @ A @ bot_bot_set_nat ) )
=> ( ( F @ X2 )
= A ) ) ).
% range_eq_singletonD
thf(fact_2440_range__eq__singletonD,axiom,
! [F: nat > rat,A: rat,X2: nat] :
( ( ( image_nat_rat @ F @ top_top_set_nat )
= ( insert_rat @ A @ bot_bot_set_rat ) )
=> ( ( F @ X2 )
= A ) ) ).
% range_eq_singletonD
thf(fact_2441_range__eq__singletonD,axiom,
! [F: nat > int,A: int,X2: nat] :
( ( ( image_nat_int @ F @ top_top_set_nat )
= ( insert_int @ A @ bot_bot_set_int ) )
=> ( ( F @ X2 )
= A ) ) ).
% range_eq_singletonD
thf(fact_2442_range__eq__singletonD,axiom,
! [F: nat > product_unit,A: product_unit,X2: nat] :
( ( ( image_8730104196221521654t_unit @ F @ top_top_set_nat )
= ( insert_Product_unit @ A @ bot_bo3957492148770167129t_unit ) )
=> ( ( F @ X2 )
= A ) ) ).
% range_eq_singletonD
thf(fact_2443_range__eq__singletonD,axiom,
! [F: nat > $o,A: $o,X2: nat] :
( ( ( image_nat_o @ F @ top_top_set_nat )
= ( insert_o @ A @ bot_bot_set_o ) )
=> ( ( F @ X2 )
= A ) ) ).
% range_eq_singletonD
thf(fact_2444_INF__superset__mono,axiom,
! [B5: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat,F: produc3843707927480180839at_nat > set_nat,G: produc3843707927480180839at_nat > set_nat] :
( ( ord_le1268244103169919719at_nat @ B5 @ A4 )
=> ( ! [X: produc3843707927480180839at_nat] :
( ( member8757157785044589968at_nat @ X @ B5 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_3160451284896210088et_nat @ F @ A4 ) ) @ ( comple7806235888213564991et_nat @ ( image_3160451284896210088et_nat @ G @ B5 ) ) ) ) ) ).
% INF_superset_mono
thf(fact_2445_INF__superset__mono,axiom,
! [B5: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > set_nat,G: product_prod_nat_nat > set_nat] :
( ( ord_le3146513528884898305at_nat @ B5 @ A4 )
=> ( ! [X: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X @ B5 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_15824709712370754et_nat @ F @ A4 ) ) @ ( comple7806235888213564991et_nat @ ( image_15824709712370754et_nat @ G @ B5 ) ) ) ) ) ).
% INF_superset_mono
thf(fact_2446_INF__superset__mono,axiom,
! [B5: set_o,A4: set_o,F: $o > set_nat,G: $o > set_nat] :
( ( ord_less_eq_set_o @ B5 @ A4 )
=> ( ! [X: $o] :
( ( member_o @ X @ B5 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ F @ A4 ) ) @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ G @ B5 ) ) ) ) ) ).
% INF_superset_mono
thf(fact_2447_INF__superset__mono,axiom,
! [B5: set_nat,A4: set_nat,F: nat > set_nat,G: nat > set_nat] :
( ( ord_less_eq_set_nat @ B5 @ A4 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B5 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ A4 ) ) @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ G @ B5 ) ) ) ) ) ).
% INF_superset_mono
thf(fact_2448_INF__superset__mono,axiom,
! [B5: set_int,A4: set_int,F: int > set_nat,G: int > set_nat] :
( ( ord_less_eq_set_int @ B5 @ A4 )
=> ( ! [X: int] :
( ( member_int @ X @ B5 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_int_set_nat @ F @ A4 ) ) @ ( comple7806235888213564991et_nat @ ( image_int_set_nat @ G @ B5 ) ) ) ) ) ).
% INF_superset_mono
thf(fact_2449_INF__superset__mono,axiom,
! [B5: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat,F: produc3843707927480180839at_nat > product_unit,G: produc3843707927480180839at_nat > product_unit] :
( ( ord_le1268244103169919719at_nat @ B5 @ A4 )
=> ( ! [X: produc3843707927480180839at_nat] :
( ( member8757157785044589968at_nat @ X @ B5 )
=> ( ord_le3221252021190050221t_unit @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_le3221252021190050221t_unit @ ( comple2584293577114468500t_unit @ ( image_7082456443403096939t_unit @ F @ A4 ) ) @ ( comple2584293577114468500t_unit @ ( image_7082456443403096939t_unit @ G @ B5 ) ) ) ) ) ).
% INF_superset_mono
thf(fact_2450_INF__superset__mono,axiom,
! [B5: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > product_unit,G: product_prod_nat_nat > product_unit] :
( ( ord_le3146513528884898305at_nat @ B5 @ A4 )
=> ( ! [X: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X @ B5 )
=> ( ord_le3221252021190050221t_unit @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_le3221252021190050221t_unit @ ( comple2584293577114468500t_unit @ ( image_6169909464789311569t_unit @ F @ A4 ) ) @ ( comple2584293577114468500t_unit @ ( image_6169909464789311569t_unit @ G @ B5 ) ) ) ) ) ).
% INF_superset_mono
thf(fact_2451_INF__superset__mono,axiom,
! [B5: set_o,A4: set_o,F: $o > product_unit,G: $o > product_unit] :
( ( ord_less_eq_set_o @ B5 @ A4 )
=> ( ! [X: $o] :
( ( member_o @ X @ B5 )
=> ( ord_le3221252021190050221t_unit @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_le3221252021190050221t_unit @ ( comple2584293577114468500t_unit @ ( image_o_Product_unit @ F @ A4 ) ) @ ( comple2584293577114468500t_unit @ ( image_o_Product_unit @ G @ B5 ) ) ) ) ) ).
% INF_superset_mono
thf(fact_2452_INF__superset__mono,axiom,
! [B5: set_nat,A4: set_nat,F: nat > product_unit,G: nat > product_unit] :
( ( ord_less_eq_set_nat @ B5 @ A4 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B5 )
=> ( ord_le3221252021190050221t_unit @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_le3221252021190050221t_unit @ ( comple2584293577114468500t_unit @ ( image_8730104196221521654t_unit @ F @ A4 ) ) @ ( comple2584293577114468500t_unit @ ( image_8730104196221521654t_unit @ G @ B5 ) ) ) ) ) ).
% INF_superset_mono
thf(fact_2453_INF__superset__mono,axiom,
! [B5: set_int,A4: set_int,F: int > product_unit,G: int > product_unit] :
( ( ord_less_eq_set_int @ B5 @ A4 )
=> ( ! [X: int] :
( ( member_int @ X @ B5 )
=> ( ord_le3221252021190050221t_unit @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_le3221252021190050221t_unit @ ( comple2584293577114468500t_unit @ ( image_5846526143423701914t_unit @ F @ A4 ) ) @ ( comple2584293577114468500t_unit @ ( image_5846526143423701914t_unit @ G @ B5 ) ) ) ) ) ).
% INF_superset_mono
thf(fact_2454_SUP__subset__mono,axiom,
! [A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat,F: produc3843707927480180839at_nat > set_nat,G: produc3843707927480180839at_nat > set_nat] :
( ( ord_le1268244103169919719at_nat @ A4 @ B5 )
=> ( ! [X: produc3843707927480180839at_nat] :
( ( member8757157785044589968at_nat @ X @ A4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_3160451284896210088et_nat @ F @ A4 ) ) @ ( comple7399068483239264473et_nat @ ( image_3160451284896210088et_nat @ G @ B5 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_2455_SUP__subset__mono,axiom,
! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > set_nat,G: product_prod_nat_nat > set_nat] :
( ( ord_le3146513528884898305at_nat @ A4 @ B5 )
=> ( ! [X: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X @ A4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ F @ A4 ) ) @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ G @ B5 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_2456_SUP__subset__mono,axiom,
! [A4: set_o,B5: set_o,F: $o > set_nat,G: $o > set_nat] :
( ( ord_less_eq_set_o @ A4 @ B5 )
=> ( ! [X: $o] :
( ( member_o @ X @ A4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A4 ) ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ G @ B5 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_2457_SUP__subset__mono,axiom,
! [A4: set_nat,B5: set_nat,F: nat > set_nat,G: nat > set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A4 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B5 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_2458_SUP__subset__mono,axiom,
! [A4: set_int,B5: set_int,F: int > set_nat,G: int > set_nat] :
( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( ! [X: int] :
( ( member_int @ X @ A4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_int_set_nat @ F @ A4 ) ) @ ( comple7399068483239264473et_nat @ ( image_int_set_nat @ G @ B5 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_2459_SUP__subset__mono,axiom,
! [A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat,F: produc3843707927480180839at_nat > product_unit,G: produc3843707927480180839at_nat > product_unit] :
( ( ord_le1268244103169919719at_nat @ A4 @ B5 )
=> ( ! [X: produc3843707927480180839at_nat] :
( ( member8757157785044589968at_nat @ X @ A4 )
=> ( ord_le3221252021190050221t_unit @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_le3221252021190050221t_unit @ ( comple4687483117567863418t_unit @ ( image_7082456443403096939t_unit @ F @ A4 ) ) @ ( comple4687483117567863418t_unit @ ( image_7082456443403096939t_unit @ G @ B5 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_2460_SUP__subset__mono,axiom,
! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > product_unit,G: product_prod_nat_nat > product_unit] :
( ( ord_le3146513528884898305at_nat @ A4 @ B5 )
=> ( ! [X: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X @ A4 )
=> ( ord_le3221252021190050221t_unit @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_le3221252021190050221t_unit @ ( comple4687483117567863418t_unit @ ( image_6169909464789311569t_unit @ F @ A4 ) ) @ ( comple4687483117567863418t_unit @ ( image_6169909464789311569t_unit @ G @ B5 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_2461_SUP__subset__mono,axiom,
! [A4: set_o,B5: set_o,F: $o > product_unit,G: $o > product_unit] :
( ( ord_less_eq_set_o @ A4 @ B5 )
=> ( ! [X: $o] :
( ( member_o @ X @ A4 )
=> ( ord_le3221252021190050221t_unit @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_le3221252021190050221t_unit @ ( comple4687483117567863418t_unit @ ( image_o_Product_unit @ F @ A4 ) ) @ ( comple4687483117567863418t_unit @ ( image_o_Product_unit @ G @ B5 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_2462_SUP__subset__mono,axiom,
! [A4: set_nat,B5: set_nat,F: nat > product_unit,G: nat > product_unit] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ( ord_le3221252021190050221t_unit @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_le3221252021190050221t_unit @ ( comple4687483117567863418t_unit @ ( image_8730104196221521654t_unit @ F @ A4 ) ) @ ( comple4687483117567863418t_unit @ ( image_8730104196221521654t_unit @ G @ B5 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_2463_SUP__subset__mono,axiom,
! [A4: set_int,B5: set_int,F: int > product_unit,G: int > product_unit] :
( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( ! [X: int] :
( ( member_int @ X @ A4 )
=> ( ord_le3221252021190050221t_unit @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_le3221252021190050221t_unit @ ( comple4687483117567863418t_unit @ ( image_5846526143423701914t_unit @ F @ A4 ) ) @ ( comple4687483117567863418t_unit @ ( image_5846526143423701914t_unit @ G @ B5 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_2464_INF__constant,axiom,
! [A4: set_Product_unit,C: product_unit] :
( ( ( A4 = bot_bo3957492148770167129t_unit )
=> ( ( comple2584293577114468500t_unit
@ ( image_405062704495631173t_unit
@ ^ [Y5: product_unit] : C
@ A4 ) )
= top_top_Product_unit ) )
& ( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ( comple2584293577114468500t_unit
@ ( image_405062704495631173t_unit
@ ^ [Y5: product_unit] : C
@ A4 ) )
= C ) ) ) ).
% INF_constant
thf(fact_2465_INF__constant,axiom,
! [A4: set_o,C: product_unit] :
( ( ( A4 = bot_bot_set_o )
=> ( ( comple2584293577114468500t_unit
@ ( image_o_Product_unit
@ ^ [Y5: $o] : C
@ A4 ) )
= top_top_Product_unit ) )
& ( ( A4 != bot_bot_set_o )
=> ( ( comple2584293577114468500t_unit
@ ( image_o_Product_unit
@ ^ [Y5: $o] : C
@ A4 ) )
= C ) ) ) ).
% INF_constant
thf(fact_2466_INF__constant,axiom,
! [A4: set_nat,C: product_unit] :
( ( ( A4 = bot_bot_set_nat )
=> ( ( comple2584293577114468500t_unit
@ ( image_8730104196221521654t_unit
@ ^ [Y5: nat] : C
@ A4 ) )
= top_top_Product_unit ) )
& ( ( A4 != bot_bot_set_nat )
=> ( ( comple2584293577114468500t_unit
@ ( image_8730104196221521654t_unit
@ ^ [Y5: nat] : C
@ A4 ) )
= C ) ) ) ).
% INF_constant
thf(fact_2467_INF__constant,axiom,
! [A4: set_Product_unit,C: set_char] :
( ( ( A4 = bot_bo3957492148770167129t_unit )
=> ( ( comple7965528767983648192t_char
@ ( image_5325685772115642575t_char
@ ^ [Y5: product_unit] : C
@ A4 ) )
= top_top_set_char ) )
& ( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ( comple7965528767983648192t_char
@ ( image_5325685772115642575t_char
@ ^ [Y5: product_unit] : C
@ A4 ) )
= C ) ) ) ).
% INF_constant
thf(fact_2468_INF__constant,axiom,
! [A4: set_o,C: set_char] :
( ( ( A4 = bot_bot_set_o )
=> ( ( comple7965528767983648192t_char
@ ( image_o_set_char
@ ^ [Y5: $o] : C
@ A4 ) )
= top_top_set_char ) )
& ( ( A4 != bot_bot_set_o )
=> ( ( comple7965528767983648192t_char
@ ( image_o_set_char
@ ^ [Y5: $o] : C
@ A4 ) )
= C ) ) ) ).
% INF_constant
thf(fact_2469_INF__constant,axiom,
! [A4: set_nat,C: set_char] :
( ( ( A4 = bot_bot_set_nat )
=> ( ( comple7965528767983648192t_char
@ ( image_nat_set_char
@ ^ [Y5: nat] : C
@ A4 ) )
= top_top_set_char ) )
& ( ( A4 != bot_bot_set_nat )
=> ( ( comple7965528767983648192t_char
@ ( image_nat_set_char
@ ^ [Y5: nat] : C
@ A4 ) )
= C ) ) ) ).
% INF_constant
thf(fact_2470_INF__constant,axiom,
! [A4: set_Product_unit,C: set_rat] :
( ( ( A4 = bot_bo3957492148770167129t_unit )
=> ( ( comple4298007329820168263et_rat
@ ( image_4665621433434589782et_rat
@ ^ [Y5: product_unit] : C
@ A4 ) )
= top_top_set_rat ) )
& ( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ( comple4298007329820168263et_rat
@ ( image_4665621433434589782et_rat
@ ^ [Y5: product_unit] : C
@ A4 ) )
= C ) ) ) ).
% INF_constant
thf(fact_2471_INF__constant,axiom,
! [A4: set_o,C: set_rat] :
( ( ( A4 = bot_bot_set_o )
=> ( ( comple4298007329820168263et_rat
@ ( image_o_set_rat
@ ^ [Y5: $o] : C
@ A4 ) )
= top_top_set_rat ) )
& ( ( A4 != bot_bot_set_o )
=> ( ( comple4298007329820168263et_rat
@ ( image_o_set_rat
@ ^ [Y5: $o] : C
@ A4 ) )
= C ) ) ) ).
% INF_constant
thf(fact_2472_INF__constant,axiom,
! [A4: set_nat,C: set_rat] :
( ( ( A4 = bot_bot_set_nat )
=> ( ( comple4298007329820168263et_rat
@ ( image_nat_set_rat
@ ^ [Y5: nat] : C
@ A4 ) )
= top_top_set_rat ) )
& ( ( A4 != bot_bot_set_nat )
=> ( ( comple4298007329820168263et_rat
@ ( image_nat_set_rat
@ ^ [Y5: nat] : C
@ A4 ) )
= C ) ) ) ).
% INF_constant
thf(fact_2473_INF__constant,axiom,
! [A4: set_Product_unit,C: set_int] :
( ( ( A4 = bot_bo3957492148770167129t_unit )
=> ( ( comple3628384868704368283et_int
@ ( image_3995998972318789802et_int
@ ^ [Y5: product_unit] : C
@ A4 ) )
= top_top_set_int ) )
& ( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ( comple3628384868704368283et_int
@ ( image_3995998972318789802et_int
@ ^ [Y5: product_unit] : C
@ A4 ) )
= C ) ) ) ).
% INF_constant
thf(fact_2474_INF__empty,axiom,
! [F: product_unit > product_unit] :
( ( comple2584293577114468500t_unit @ ( image_405062704495631173t_unit @ F @ bot_bo3957492148770167129t_unit ) )
= top_top_Product_unit ) ).
% INF_empty
thf(fact_2475_INF__empty,axiom,
! [F: $o > product_unit] :
( ( comple2584293577114468500t_unit @ ( image_o_Product_unit @ F @ bot_bot_set_o ) )
= top_top_Product_unit ) ).
% INF_empty
thf(fact_2476_INF__empty,axiom,
! [F: nat > product_unit] :
( ( comple2584293577114468500t_unit @ ( image_8730104196221521654t_unit @ F @ bot_bot_set_nat ) )
= top_top_Product_unit ) ).
% INF_empty
thf(fact_2477_INF__empty,axiom,
! [F: product_unit > set_char] :
( ( comple7965528767983648192t_char @ ( image_5325685772115642575t_char @ F @ bot_bo3957492148770167129t_unit ) )
= top_top_set_char ) ).
% INF_empty
thf(fact_2478_INF__empty,axiom,
! [F: $o > set_char] :
( ( comple7965528767983648192t_char @ ( image_o_set_char @ F @ bot_bot_set_o ) )
= top_top_set_char ) ).
% INF_empty
thf(fact_2479_INF__empty,axiom,
! [F: nat > set_char] :
( ( comple7965528767983648192t_char @ ( image_nat_set_char @ F @ bot_bot_set_nat ) )
= top_top_set_char ) ).
% INF_empty
thf(fact_2480_INF__empty,axiom,
! [F: product_unit > set_rat] :
( ( comple4298007329820168263et_rat @ ( image_4665621433434589782et_rat @ F @ bot_bo3957492148770167129t_unit ) )
= top_top_set_rat ) ).
% INF_empty
thf(fact_2481_INF__empty,axiom,
! [F: $o > set_rat] :
( ( comple4298007329820168263et_rat @ ( image_o_set_rat @ F @ bot_bot_set_o ) )
= top_top_set_rat ) ).
% INF_empty
thf(fact_2482_INF__empty,axiom,
! [F: nat > set_rat] :
( ( comple4298007329820168263et_rat @ ( image_nat_set_rat @ F @ bot_bot_set_nat ) )
= top_top_set_rat ) ).
% INF_empty
thf(fact_2483_INF__empty,axiom,
! [F: product_unit > set_int] :
( ( comple3628384868704368283et_int @ ( image_3995998972318789802et_int @ F @ bot_bo3957492148770167129t_unit ) )
= top_top_set_int ) ).
% INF_empty
thf(fact_2484_SUP__constant,axiom,
! [A4: set_Product_unit,C: product_unit] :
( ( ( A4 = bot_bo3957492148770167129t_unit )
=> ( ( comple4687483117567863418t_unit
@ ( image_405062704495631173t_unit
@ ^ [Y5: product_unit] : C
@ A4 ) )
= bot_bot_Product_unit ) )
& ( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ( comple4687483117567863418t_unit
@ ( image_405062704495631173t_unit
@ ^ [Y5: product_unit] : C
@ A4 ) )
= C ) ) ) ).
% SUP_constant
thf(fact_2485_SUP__constant,axiom,
! [A4: set_o,C: product_unit] :
( ( ( A4 = bot_bot_set_o )
=> ( ( comple4687483117567863418t_unit
@ ( image_o_Product_unit
@ ^ [Y5: $o] : C
@ A4 ) )
= bot_bot_Product_unit ) )
& ( ( A4 != bot_bot_set_o )
=> ( ( comple4687483117567863418t_unit
@ ( image_o_Product_unit
@ ^ [Y5: $o] : C
@ A4 ) )
= C ) ) ) ).
% SUP_constant
thf(fact_2486_SUP__constant,axiom,
! [A4: set_nat,C: product_unit] :
( ( ( A4 = bot_bot_set_nat )
=> ( ( comple4687483117567863418t_unit
@ ( image_8730104196221521654t_unit
@ ^ [Y5: nat] : C
@ A4 ) )
= bot_bot_Product_unit ) )
& ( ( A4 != bot_bot_set_nat )
=> ( ( comple4687483117567863418t_unit
@ ( image_8730104196221521654t_unit
@ ^ [Y5: nat] : C
@ A4 ) )
= C ) ) ) ).
% SUP_constant
thf(fact_2487_SUP__constant,axiom,
! [A4: set_Product_unit,C: set_Product_unit] :
( ( ( A4 = bot_bo3957492148770167129t_unit )
=> ( ( comple3349607820504768346t_unit
@ ( image_7778121751606633509t_unit
@ ^ [Y5: product_unit] : C
@ A4 ) )
= bot_bo3957492148770167129t_unit ) )
& ( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ( comple3349607820504768346t_unit
@ ( image_7778121751606633509t_unit
@ ^ [Y5: product_unit] : C
@ A4 ) )
= C ) ) ) ).
% SUP_constant
thf(fact_2488_SUP__constant,axiom,
! [A4: set_Product_unit,C: set_o] :
( ( ( A4 = bot_bo3957492148770167129t_unit )
=> ( ( comple90263536869209701_set_o
@ ( image_6930536745178679216_set_o
@ ^ [Y5: product_unit] : C
@ A4 ) )
= bot_bot_set_o ) )
& ( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ( comple90263536869209701_set_o
@ ( image_6930536745178679216_set_o
@ ^ [Y5: product_unit] : C
@ A4 ) )
= C ) ) ) ).
% SUP_constant
thf(fact_2489_SUP__constant,axiom,
! [A4: set_o,C: set_Product_unit] :
( ( ( A4 = bot_bot_set_o )
=> ( ( comple3349607820504768346t_unit
@ ( image_663195431845683034t_unit
@ ^ [Y5: $o] : C
@ A4 ) )
= bot_bo3957492148770167129t_unit ) )
& ( ( A4 != bot_bot_set_o )
=> ( ( comple3349607820504768346t_unit
@ ( image_663195431845683034t_unit
@ ^ [Y5: $o] : C
@ A4 ) )
= C ) ) ) ).
% SUP_constant
thf(fact_2490_SUP__constant,axiom,
! [A4: set_o,C: set_o] :
( ( ( A4 = bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [Y5: $o] : C
@ A4 ) )
= bot_bot_set_o ) )
& ( ( A4 != bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [Y5: $o] : C
@ A4 ) )
= C ) ) ) ).
% SUP_constant
thf(fact_2491_SUP__constant,axiom,
! [A4: set_nat,C: set_Product_unit] :
( ( ( A4 = bot_bot_set_nat )
=> ( ( comple3349607820504768346t_unit
@ ( image_4824033694279209942t_unit
@ ^ [Y5: nat] : C
@ A4 ) )
= bot_bo3957492148770167129t_unit ) )
& ( ( A4 != bot_bot_set_nat )
=> ( ( comple3349607820504768346t_unit
@ ( image_4824033694279209942t_unit
@ ^ [Y5: nat] : C
@ A4 ) )
= C ) ) ) ).
% SUP_constant
thf(fact_2492_SUP__constant,axiom,
! [A4: set_nat,C: set_o] :
( ( ( A4 = bot_bot_set_nat )
=> ( ( comple90263536869209701_set_o
@ ( image_nat_set_o
@ ^ [Y5: nat] : C
@ A4 ) )
= bot_bot_set_o ) )
& ( ( A4 != bot_bot_set_nat )
=> ( ( comple90263536869209701_set_o
@ ( image_nat_set_o
@ ^ [Y5: nat] : C
@ A4 ) )
= C ) ) ) ).
% SUP_constant
thf(fact_2493_SUP__constant,axiom,
! [A4: set_Product_unit,C: set_nat] :
( ( ( A4 = bot_bo3957492148770167129t_unit )
=> ( ( comple7399068483239264473et_nat
@ ( image_8173849991827986510et_nat
@ ^ [Y5: product_unit] : C
@ A4 ) )
= bot_bot_set_nat ) )
& ( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ( comple7399068483239264473et_nat
@ ( image_8173849991827986510et_nat
@ ^ [Y5: product_unit] : C
@ A4 ) )
= C ) ) ) ).
% SUP_constant
thf(fact_2494_SUP__empty,axiom,
! [F: product_unit > product_unit] :
( ( comple4687483117567863418t_unit @ ( image_405062704495631173t_unit @ F @ bot_bo3957492148770167129t_unit ) )
= bot_bot_Product_unit ) ).
% SUP_empty
thf(fact_2495_SUP__empty,axiom,
! [F: $o > product_unit] :
( ( comple4687483117567863418t_unit @ ( image_o_Product_unit @ F @ bot_bot_set_o ) )
= bot_bot_Product_unit ) ).
% SUP_empty
thf(fact_2496_SUP__empty,axiom,
! [F: nat > product_unit] :
( ( comple4687483117567863418t_unit @ ( image_8730104196221521654t_unit @ F @ bot_bot_set_nat ) )
= bot_bot_Product_unit ) ).
% SUP_empty
thf(fact_2497_SUP__empty,axiom,
! [F: product_unit > set_Product_unit] :
( ( comple3349607820504768346t_unit @ ( image_7778121751606633509t_unit @ F @ bot_bo3957492148770167129t_unit ) )
= bot_bo3957492148770167129t_unit ) ).
% SUP_empty
thf(fact_2498_SUP__empty,axiom,
! [F: product_unit > set_o] :
( ( comple90263536869209701_set_o @ ( image_6930536745178679216_set_o @ F @ bot_bo3957492148770167129t_unit ) )
= bot_bot_set_o ) ).
% SUP_empty
thf(fact_2499_SUP__empty,axiom,
! [F: $o > set_Product_unit] :
( ( comple3349607820504768346t_unit @ ( image_663195431845683034t_unit @ F @ bot_bot_set_o ) )
= bot_bo3957492148770167129t_unit ) ).
% SUP_empty
thf(fact_2500_SUP__empty,axiom,
! [F: $o > set_o] :
( ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ bot_bot_set_o ) )
= bot_bot_set_o ) ).
% SUP_empty
thf(fact_2501_SUP__empty,axiom,
! [F: nat > set_Product_unit] :
( ( comple3349607820504768346t_unit @ ( image_4824033694279209942t_unit @ F @ bot_bot_set_nat ) )
= bot_bo3957492148770167129t_unit ) ).
% SUP_empty
thf(fact_2502_SUP__empty,axiom,
! [F: nat > set_o] :
( ( comple90263536869209701_set_o @ ( image_nat_set_o @ F @ bot_bot_set_nat ) )
= bot_bot_set_o ) ).
% SUP_empty
thf(fact_2503_SUP__empty,axiom,
! [F: product_unit > set_nat] :
( ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ F @ bot_bo3957492148770167129t_unit ) )
= bot_bot_set_nat ) ).
% SUP_empty
thf(fact_2504_INT__empty,axiom,
! [B5: product_unit > set_char] :
( ( comple7965528767983648192t_char @ ( image_5325685772115642575t_char @ B5 @ bot_bo3957492148770167129t_unit ) )
= top_top_set_char ) ).
% INT_empty
thf(fact_2505_INT__empty,axiom,
! [B5: $o > set_char] :
( ( comple7965528767983648192t_char @ ( image_o_set_char @ B5 @ bot_bot_set_o ) )
= top_top_set_char ) ).
% INT_empty
thf(fact_2506_INT__empty,axiom,
! [B5: nat > set_char] :
( ( comple7965528767983648192t_char @ ( image_nat_set_char @ B5 @ bot_bot_set_nat ) )
= top_top_set_char ) ).
% INT_empty
thf(fact_2507_INT__empty,axiom,
! [B5: product_unit > set_rat] :
( ( comple4298007329820168263et_rat @ ( image_4665621433434589782et_rat @ B5 @ bot_bo3957492148770167129t_unit ) )
= top_top_set_rat ) ).
% INT_empty
thf(fact_2508_INT__empty,axiom,
! [B5: $o > set_rat] :
( ( comple4298007329820168263et_rat @ ( image_o_set_rat @ B5 @ bot_bot_set_o ) )
= top_top_set_rat ) ).
% INT_empty
thf(fact_2509_INT__empty,axiom,
! [B5: nat > set_rat] :
( ( comple4298007329820168263et_rat @ ( image_nat_set_rat @ B5 @ bot_bot_set_nat ) )
= top_top_set_rat ) ).
% INT_empty
thf(fact_2510_INT__empty,axiom,
! [B5: product_unit > set_int] :
( ( comple3628384868704368283et_int @ ( image_3995998972318789802et_int @ B5 @ bot_bo3957492148770167129t_unit ) )
= top_top_set_int ) ).
% INT_empty
thf(fact_2511_INT__empty,axiom,
! [B5: $o > set_int] :
( ( comple3628384868704368283et_int @ ( image_o_set_int @ B5 @ bot_bot_set_o ) )
= top_top_set_int ) ).
% INT_empty
thf(fact_2512_INT__empty,axiom,
! [B5: nat > set_int] :
( ( comple3628384868704368283et_int @ ( image_nat_set_int @ B5 @ bot_bot_set_nat ) )
= top_top_set_int ) ).
% INT_empty
thf(fact_2513_INT__empty,axiom,
! [B5: product_unit > set_nat] :
( ( comple7806235888213564991et_nat @ ( image_8173849991827986510et_nat @ B5 @ bot_bo3957492148770167129t_unit ) )
= top_top_set_nat ) ).
% INT_empty
thf(fact_2514_cINF__const,axiom,
! [A4: set_int,C: nat] :
( ( A4 != bot_bot_set_int )
=> ( ( complete_Inf_Inf_nat
@ ( image_int_nat
@ ^ [X4: int] : C
@ A4 ) )
= C ) ) ).
% cINF_const
thf(fact_2515_cINF__const,axiom,
! [A4: set_nat,C: nat] :
( ( A4 != bot_bot_set_nat )
=> ( ( complete_Inf_Inf_nat
@ ( image_nat_nat
@ ^ [X4: nat] : C
@ A4 ) )
= C ) ) ).
% cINF_const
thf(fact_2516_cINF__const,axiom,
! [A4: set_int,C: int] :
( ( A4 != bot_bot_set_int )
=> ( ( complete_Inf_Inf_int
@ ( image_int_int
@ ^ [X4: int] : C
@ A4 ) )
= C ) ) ).
% cINF_const
thf(fact_2517_cINF__const,axiom,
! [A4: set_Product_unit,C: int] :
( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ( complete_Inf_Inf_int
@ ( image_873079544045703924it_int
@ ^ [X4: product_unit] : C
@ A4 ) )
= C ) ) ).
% cINF_const
thf(fact_2518_cINF__const,axiom,
! [A4: set_o,C: int] :
( ( A4 != bot_bot_set_o )
=> ( ( complete_Inf_Inf_int
@ ( image_o_int
@ ^ [X4: $o] : C
@ A4 ) )
= C ) ) ).
% cINF_const
thf(fact_2519_cINF__const,axiom,
! [A4: set_nat,C: int] :
( ( A4 != bot_bot_set_nat )
=> ( ( complete_Inf_Inf_int
@ ( image_nat_int
@ ^ [X4: nat] : C
@ A4 ) )
= C ) ) ).
% cINF_const
thf(fact_2520_cINF__const,axiom,
! [A4: set_Product_unit,C: product_unit] :
( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ( comple2584293577114468500t_unit
@ ( image_405062704495631173t_unit
@ ^ [X4: product_unit] : C
@ A4 ) )
= C ) ) ).
% cINF_const
thf(fact_2521_cINF__const,axiom,
! [A4: set_o,C: product_unit] :
( ( A4 != bot_bot_set_o )
=> ( ( comple2584293577114468500t_unit
@ ( image_o_Product_unit
@ ^ [X4: $o] : C
@ A4 ) )
= C ) ) ).
% cINF_const
thf(fact_2522_cINF__const,axiom,
! [A4: set_nat,C: product_unit] :
( ( A4 != bot_bot_set_nat )
=> ( ( comple2584293577114468500t_unit
@ ( image_8730104196221521654t_unit
@ ^ [X4: nat] : C
@ A4 ) )
= C ) ) ).
% cINF_const
thf(fact_2523_cINF__const,axiom,
! [A4: set_Product_unit,C: set_nat] :
( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ( comple7806235888213564991et_nat
@ ( image_8173849991827986510et_nat
@ ^ [X4: product_unit] : C
@ A4 ) )
= C ) ) ).
% cINF_const
thf(fact_2524_cSUP__const,axiom,
! [A4: set_int,C: nat] :
( ( A4 != bot_bot_set_int )
=> ( ( complete_Sup_Sup_nat
@ ( image_int_nat
@ ^ [X4: int] : C
@ A4 ) )
= C ) ) ).
% cSUP_const
thf(fact_2525_cSUP__const,axiom,
! [A4: set_nat,C: nat] :
( ( A4 != bot_bot_set_nat )
=> ( ( complete_Sup_Sup_nat
@ ( image_nat_nat
@ ^ [X4: nat] : C
@ A4 ) )
= C ) ) ).
% cSUP_const
thf(fact_2526_cSUP__const,axiom,
! [A4: set_int,C: int] :
( ( A4 != bot_bot_set_int )
=> ( ( complete_Sup_Sup_int
@ ( image_int_int
@ ^ [X4: int] : C
@ A4 ) )
= C ) ) ).
% cSUP_const
thf(fact_2527_cSUP__const,axiom,
! [A4: set_Product_unit,C: int] :
( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ( complete_Sup_Sup_int
@ ( image_873079544045703924it_int
@ ^ [X4: product_unit] : C
@ A4 ) )
= C ) ) ).
% cSUP_const
thf(fact_2528_cSUP__const,axiom,
! [A4: set_o,C: int] :
( ( A4 != bot_bot_set_o )
=> ( ( complete_Sup_Sup_int
@ ( image_o_int
@ ^ [X4: $o] : C
@ A4 ) )
= C ) ) ).
% cSUP_const
thf(fact_2529_cSUP__const,axiom,
! [A4: set_nat,C: int] :
( ( A4 != bot_bot_set_nat )
=> ( ( complete_Sup_Sup_int
@ ( image_nat_int
@ ^ [X4: nat] : C
@ A4 ) )
= C ) ) ).
% cSUP_const
thf(fact_2530_cSUP__const,axiom,
! [A4: set_Product_unit,C: product_unit] :
( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ( comple4687483117567863418t_unit
@ ( image_405062704495631173t_unit
@ ^ [X4: product_unit] : C
@ A4 ) )
= C ) ) ).
% cSUP_const
thf(fact_2531_cSUP__const,axiom,
! [A4: set_o,C: product_unit] :
( ( A4 != bot_bot_set_o )
=> ( ( comple4687483117567863418t_unit
@ ( image_o_Product_unit
@ ^ [X4: $o] : C
@ A4 ) )
= C ) ) ).
% cSUP_const
thf(fact_2532_cSUP__const,axiom,
! [A4: set_nat,C: product_unit] :
( ( A4 != bot_bot_set_nat )
=> ( ( comple4687483117567863418t_unit
@ ( image_8730104196221521654t_unit
@ ^ [X4: nat] : C
@ A4 ) )
= C ) ) ).
% cSUP_const
thf(fact_2533_cSUP__const,axiom,
! [A4: set_Product_unit,C: set_nat] :
( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ( comple7399068483239264473et_nat
@ ( image_8173849991827986510et_nat
@ ^ [X4: product_unit] : C
@ A4 ) )
= C ) ) ).
% cSUP_const
thf(fact_2534_SUP__INF,axiom,
! [P2: char > char > product_unit] :
( ( comple4687483117567863418t_unit
@ ( image_4337914935673063477t_unit
@ ^ [Y5: char] :
( comple2584293577114468500t_unit
@ ( image_4337914935673063477t_unit
@ ^ [X4: char] : ( P2 @ X4 @ Y5 )
@ top_top_set_char ) )
@ top_top_set_char ) )
= ( comple2584293577114468500t_unit
@ ( image_8077363282794994245t_unit
@ ^ [X4: char > char] :
( comple4687483117567863418t_unit
@ ( image_4337914935673063477t_unit
@ ^ [Y5: char] : ( P2 @ ( X4 @ Y5 ) @ Y5 )
@ top_top_set_char ) )
@ top_to8079350907815874599r_char ) ) ) ).
% SUP_INF
thf(fact_2535_SUP__INF,axiom,
! [P2: char > rat > product_unit] :
( ( comple4687483117567863418t_unit
@ ( image_2327579789247107822t_unit
@ ^ [Y5: rat] :
( comple2584293577114468500t_unit
@ ( image_4337914935673063477t_unit
@ ^ [X4: char] : ( P2 @ X4 @ Y5 )
@ top_top_set_char ) )
@ top_top_set_rat ) )
= ( comple2584293577114468500t_unit
@ ( image_6940606515012899070t_unit
@ ^ [X4: rat > char] :
( comple4687483117567863418t_unit
@ ( image_2327579789247107822t_unit
@ ^ [Y5: rat] : ( P2 @ ( X4 @ Y5 ) @ Y5 )
@ top_top_set_rat ) )
@ top_top_set_rat_char ) ) ) ).
% SUP_INF
thf(fact_2536_SUP__INF,axiom,
! [P2: char > nat > product_unit] :
( ( comple4687483117567863418t_unit
@ ( image_8730104196221521654t_unit
@ ^ [Y5: nat] :
( comple2584293577114468500t_unit
@ ( image_4337914935673063477t_unit
@ ^ [X4: char] : ( P2 @ X4 @ Y5 )
@ top_top_set_char ) )
@ top_top_set_nat ) )
= ( comple2584293577114468500t_unit
@ ( image_4600008085730780422t_unit
@ ^ [X4: nat > char] :
( comple4687483117567863418t_unit
@ ( image_8730104196221521654t_unit
@ ^ [Y5: nat] : ( P2 @ ( X4 @ Y5 ) @ Y5 )
@ top_top_set_nat ) )
@ top_top_set_nat_char ) ) ) ).
% SUP_INF
thf(fact_2537_SUP__INF,axiom,
! [P2: char > int > product_unit] :
( ( comple4687483117567863418t_unit
@ ( image_5846526143423701914t_unit
@ ^ [Y5: int] :
( comple2584293577114468500t_unit
@ ( image_4337914935673063477t_unit
@ ^ [X4: char] : ( P2 @ X4 @ Y5 )
@ top_top_set_char ) )
@ top_top_set_int ) )
= ( comple2584293577114468500t_unit
@ ( image_4409302170498516394t_unit
@ ^ [X4: int > char] :
( comple4687483117567863418t_unit
@ ( image_5846526143423701914t_unit
@ ^ [Y5: int] : ( P2 @ ( X4 @ Y5 ) @ Y5 )
@ top_top_set_int ) )
@ top_top_set_int_char ) ) ) ).
% SUP_INF
thf(fact_2538_SUP__INF,axiom,
! [P2: rat > char > product_unit] :
( ( comple4687483117567863418t_unit
@ ( image_4337914935673063477t_unit
@ ^ [Y5: char] :
( comple2584293577114468500t_unit
@ ( image_2327579789247107822t_unit
@ ^ [X4: rat] : ( P2 @ X4 @ Y5 )
@ top_top_set_rat ) )
@ top_top_set_char ) )
= ( comple2584293577114468500t_unit
@ ( image_7615136836863079294t_unit
@ ^ [X4: char > rat] :
( comple4687483117567863418t_unit
@ ( image_4337914935673063477t_unit
@ ^ [Y5: char] : ( P2 @ ( X4 @ Y5 ) @ Y5 )
@ top_top_set_char ) )
@ top_top_set_char_rat ) ) ) ).
% SUP_INF
thf(fact_2539_SUP__INF,axiom,
! [P2: rat > rat > product_unit] :
( ( comple4687483117567863418t_unit
@ ( image_2327579789247107822t_unit
@ ^ [Y5: rat] :
( comple2584293577114468500t_unit
@ ( image_2327579789247107822t_unit
@ ^ [X4: rat] : ( P2 @ X4 @ Y5 )
@ top_top_set_rat ) )
@ top_top_set_rat ) )
= ( comple2584293577114468500t_unit
@ ( image_1193274288206594871t_unit
@ ^ [X4: rat > rat] :
( comple4687483117567863418t_unit
@ ( image_2327579789247107822t_unit
@ ^ [Y5: rat] : ( P2 @ ( X4 @ Y5 ) @ Y5 )
@ top_top_set_rat ) )
@ top_top_set_rat_rat ) ) ) ).
% SUP_INF
thf(fact_2540_SUP__INF,axiom,
! [P2: rat > nat > product_unit] :
( ( comple4687483117567863418t_unit
@ ( image_8730104196221521654t_unit
@ ^ [Y5: nat] :
( comple2584293577114468500t_unit
@ ( image_2327579789247107822t_unit
@ ^ [X4: rat] : ( P2 @ X4 @ Y5 )
@ top_top_set_rat ) )
@ top_top_set_nat ) )
= ( comple2584293577114468500t_unit
@ ( image_340470173791133503t_unit
@ ^ [X4: nat > rat] :
( comple4687483117567863418t_unit
@ ( image_8730104196221521654t_unit
@ ^ [Y5: nat] : ( P2 @ ( X4 @ Y5 ) @ Y5 )
@ top_top_set_nat ) )
@ top_top_set_nat_rat ) ) ) ).
% SUP_INF
thf(fact_2541_SUP__INF,axiom,
! [P2: rat > int > product_unit] :
( ( comple4687483117567863418t_unit
@ ( image_5846526143423701914t_unit
@ ^ [Y5: int] :
( comple2584293577114468500t_unit
@ ( image_2327579789247107822t_unit
@ ^ [X4: rat] : ( P2 @ X4 @ Y5 )
@ top_top_set_rat ) )
@ top_top_set_int ) )
= ( comple2584293577114468500t_unit
@ ( image_4174709274675160035t_unit
@ ^ [X4: int > rat] :
( comple4687483117567863418t_unit
@ ( image_5846526143423701914t_unit
@ ^ [Y5: int] : ( P2 @ ( X4 @ Y5 ) @ Y5 )
@ top_top_set_int ) )
@ top_top_set_int_rat ) ) ) ).
% SUP_INF
thf(fact_2542_SUP__INF,axiom,
! [P2: nat > char > product_unit] :
( ( comple4687483117567863418t_unit
@ ( image_4337914935673063477t_unit
@ ^ [Y5: char] :
( comple2584293577114468500t_unit
@ ( image_8730104196221521654t_unit
@ ^ [X4: nat] : ( P2 @ X4 @ Y5 )
@ top_top_set_nat ) )
@ top_top_set_char ) )
= ( comple2584293577114468500t_unit
@ ( image_108929247839792006t_unit
@ ^ [X4: char > nat] :
( comple4687483117567863418t_unit
@ ( image_4337914935673063477t_unit
@ ^ [Y5: char] : ( P2 @ ( X4 @ Y5 ) @ Y5 )
@ top_top_set_char ) )
@ top_top_set_char_nat ) ) ) ).
% SUP_INF
thf(fact_2543_SUP__INF,axiom,
! [P2: nat > rat > product_unit] :
( ( comple4687483117567863418t_unit
@ ( image_2327579789247107822t_unit
@ ^ [Y5: rat] :
( comple2584293577114468500t_unit
@ ( image_8730104196221521654t_unit
@ ^ [X4: nat] : ( P2 @ X4 @ Y5 )
@ top_top_set_nat ) )
@ top_top_set_rat ) )
= ( comple2584293577114468500t_unit
@ ( image_2910438736038083391t_unit
@ ^ [X4: rat > nat] :
( comple4687483117567863418t_unit
@ ( image_2327579789247107822t_unit
@ ^ [Y5: rat] : ( P2 @ ( X4 @ Y5 ) @ Y5 )
@ top_top_set_rat ) )
@ top_top_set_rat_nat ) ) ) ).
% SUP_INF
thf(fact_2544_INF__SUP,axiom,
! [P2: char > char > product_unit] :
( ( comple2584293577114468500t_unit
@ ( image_4337914935673063477t_unit
@ ^ [Y5: char] :
( comple4687483117567863418t_unit
@ ( image_4337914935673063477t_unit
@ ^ [X4: char] : ( P2 @ X4 @ Y5 )
@ top_top_set_char ) )
@ top_top_set_char ) )
= ( comple4687483117567863418t_unit
@ ( image_8077363282794994245t_unit
@ ^ [F2: char > char] :
( comple2584293577114468500t_unit
@ ( image_4337914935673063477t_unit
@ ^ [X4: char] : ( P2 @ ( F2 @ X4 ) @ X4 )
@ top_top_set_char ) )
@ top_to8079350907815874599r_char ) ) ) ).
% INF_SUP
thf(fact_2545_INF__SUP,axiom,
! [P2: char > rat > product_unit] :
( ( comple2584293577114468500t_unit
@ ( image_2327579789247107822t_unit
@ ^ [Y5: rat] :
( comple4687483117567863418t_unit
@ ( image_4337914935673063477t_unit
@ ^ [X4: char] : ( P2 @ X4 @ Y5 )
@ top_top_set_char ) )
@ top_top_set_rat ) )
= ( comple4687483117567863418t_unit
@ ( image_6940606515012899070t_unit
@ ^ [F2: rat > char] :
( comple2584293577114468500t_unit
@ ( image_2327579789247107822t_unit
@ ^ [X4: rat] : ( P2 @ ( F2 @ X4 ) @ X4 )
@ top_top_set_rat ) )
@ top_top_set_rat_char ) ) ) ).
% INF_SUP
thf(fact_2546_INF__SUP,axiom,
! [P2: char > nat > product_unit] :
( ( comple2584293577114468500t_unit
@ ( image_8730104196221521654t_unit
@ ^ [Y5: nat] :
( comple4687483117567863418t_unit
@ ( image_4337914935673063477t_unit
@ ^ [X4: char] : ( P2 @ X4 @ Y5 )
@ top_top_set_char ) )
@ top_top_set_nat ) )
= ( comple4687483117567863418t_unit
@ ( image_4600008085730780422t_unit
@ ^ [F2: nat > char] :
( comple2584293577114468500t_unit
@ ( image_8730104196221521654t_unit
@ ^ [X4: nat] : ( P2 @ ( F2 @ X4 ) @ X4 )
@ top_top_set_nat ) )
@ top_top_set_nat_char ) ) ) ).
% INF_SUP
thf(fact_2547_INF__SUP,axiom,
! [P2: char > int > product_unit] :
( ( comple2584293577114468500t_unit
@ ( image_5846526143423701914t_unit
@ ^ [Y5: int] :
( comple4687483117567863418t_unit
@ ( image_4337914935673063477t_unit
@ ^ [X4: char] : ( P2 @ X4 @ Y5 )
@ top_top_set_char ) )
@ top_top_set_int ) )
= ( comple4687483117567863418t_unit
@ ( image_4409302170498516394t_unit
@ ^ [F2: int > char] :
( comple2584293577114468500t_unit
@ ( image_5846526143423701914t_unit
@ ^ [X4: int] : ( P2 @ ( F2 @ X4 ) @ X4 )
@ top_top_set_int ) )
@ top_top_set_int_char ) ) ) ).
% INF_SUP
thf(fact_2548_INF__SUP,axiom,
! [P2: rat > char > product_unit] :
( ( comple2584293577114468500t_unit
@ ( image_4337914935673063477t_unit
@ ^ [Y5: char] :
( comple4687483117567863418t_unit
@ ( image_2327579789247107822t_unit
@ ^ [X4: rat] : ( P2 @ X4 @ Y5 )
@ top_top_set_rat ) )
@ top_top_set_char ) )
= ( comple4687483117567863418t_unit
@ ( image_7615136836863079294t_unit
@ ^ [F2: char > rat] :
( comple2584293577114468500t_unit
@ ( image_4337914935673063477t_unit
@ ^ [X4: char] : ( P2 @ ( F2 @ X4 ) @ X4 )
@ top_top_set_char ) )
@ top_top_set_char_rat ) ) ) ).
% INF_SUP
thf(fact_2549_INF__SUP,axiom,
! [P2: rat > rat > product_unit] :
( ( comple2584293577114468500t_unit
@ ( image_2327579789247107822t_unit
@ ^ [Y5: rat] :
( comple4687483117567863418t_unit
@ ( image_2327579789247107822t_unit
@ ^ [X4: rat] : ( P2 @ X4 @ Y5 )
@ top_top_set_rat ) )
@ top_top_set_rat ) )
= ( comple4687483117567863418t_unit
@ ( image_1193274288206594871t_unit
@ ^ [F2: rat > rat] :
( comple2584293577114468500t_unit
@ ( image_2327579789247107822t_unit
@ ^ [X4: rat] : ( P2 @ ( F2 @ X4 ) @ X4 )
@ top_top_set_rat ) )
@ top_top_set_rat_rat ) ) ) ).
% INF_SUP
thf(fact_2550_INF__SUP,axiom,
! [P2: rat > nat > product_unit] :
( ( comple2584293577114468500t_unit
@ ( image_8730104196221521654t_unit
@ ^ [Y5: nat] :
( comple4687483117567863418t_unit
@ ( image_2327579789247107822t_unit
@ ^ [X4: rat] : ( P2 @ X4 @ Y5 )
@ top_top_set_rat ) )
@ top_top_set_nat ) )
= ( comple4687483117567863418t_unit
@ ( image_340470173791133503t_unit
@ ^ [F2: nat > rat] :
( comple2584293577114468500t_unit
@ ( image_8730104196221521654t_unit
@ ^ [X4: nat] : ( P2 @ ( F2 @ X4 ) @ X4 )
@ top_top_set_nat ) )
@ top_top_set_nat_rat ) ) ) ).
% INF_SUP
thf(fact_2551_INF__SUP,axiom,
! [P2: rat > int > product_unit] :
( ( comple2584293577114468500t_unit
@ ( image_5846526143423701914t_unit
@ ^ [Y5: int] :
( comple4687483117567863418t_unit
@ ( image_2327579789247107822t_unit
@ ^ [X4: rat] : ( P2 @ X4 @ Y5 )
@ top_top_set_rat ) )
@ top_top_set_int ) )
= ( comple4687483117567863418t_unit
@ ( image_4174709274675160035t_unit
@ ^ [F2: int > rat] :
( comple2584293577114468500t_unit
@ ( image_5846526143423701914t_unit
@ ^ [X4: int] : ( P2 @ ( F2 @ X4 ) @ X4 )
@ top_top_set_int ) )
@ top_top_set_int_rat ) ) ) ).
% INF_SUP
thf(fact_2552_INF__SUP,axiom,
! [P2: nat > char > product_unit] :
( ( comple2584293577114468500t_unit
@ ( image_4337914935673063477t_unit
@ ^ [Y5: char] :
( comple4687483117567863418t_unit
@ ( image_8730104196221521654t_unit
@ ^ [X4: nat] : ( P2 @ X4 @ Y5 )
@ top_top_set_nat ) )
@ top_top_set_char ) )
= ( comple4687483117567863418t_unit
@ ( image_108929247839792006t_unit
@ ^ [F2: char > nat] :
( comple2584293577114468500t_unit
@ ( image_4337914935673063477t_unit
@ ^ [X4: char] : ( P2 @ ( F2 @ X4 ) @ X4 )
@ top_top_set_char ) )
@ top_top_set_char_nat ) ) ) ).
% INF_SUP
thf(fact_2553_INF__SUP,axiom,
! [P2: nat > rat > product_unit] :
( ( comple2584293577114468500t_unit
@ ( image_2327579789247107822t_unit
@ ^ [Y5: rat] :
( comple4687483117567863418t_unit
@ ( image_8730104196221521654t_unit
@ ^ [X4: nat] : ( P2 @ X4 @ Y5 )
@ top_top_set_nat ) )
@ top_top_set_rat ) )
= ( comple4687483117567863418t_unit
@ ( image_2910438736038083391t_unit
@ ^ [F2: rat > nat] :
( comple2584293577114468500t_unit
@ ( image_2327579789247107822t_unit
@ ^ [X4: rat] : ( P2 @ ( F2 @ X4 ) @ X4 )
@ top_top_set_rat ) )
@ top_top_set_rat_nat ) ) ) ).
% INF_SUP
thf(fact_2554_cINF__greatest,axiom,
! [A4: set_int,M3: nat,F: int > nat] :
( ( A4 != bot_bot_set_int )
=> ( ! [X: int] :
( ( member_int @ X @ A4 )
=> ( ord_less_eq_nat @ M3 @ ( F @ X ) ) )
=> ( ord_less_eq_nat @ M3 @ ( complete_Inf_Inf_nat @ ( image_int_nat @ F @ A4 ) ) ) ) ) ).
% cINF_greatest
thf(fact_2555_cINF__greatest,axiom,
! [A4: set_Product_unit,M3: nat,F: product_unit > nat] :
( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ! [X: product_unit] :
( ( member_Product_unit @ X @ A4 )
=> ( ord_less_eq_nat @ M3 @ ( F @ X ) ) )
=> ( ord_less_eq_nat @ M3 @ ( complete_Inf_Inf_nat @ ( image_875570014554754200it_nat @ F @ A4 ) ) ) ) ) ).
% cINF_greatest
thf(fact_2556_cINF__greatest,axiom,
! [A4: set_o,M3: nat,F: $o > nat] :
( ( A4 != bot_bot_set_o )
=> ( ! [X: $o] :
( ( member_o @ X @ A4 )
=> ( ord_less_eq_nat @ M3 @ ( F @ X ) ) )
=> ( ord_less_eq_nat @ M3 @ ( complete_Inf_Inf_nat @ ( image_o_nat @ F @ A4 ) ) ) ) ) ).
% cINF_greatest
thf(fact_2557_cINF__greatest,axiom,
! [A4: set_nat,M3: nat,F: nat > nat] :
( ( A4 != bot_bot_set_nat )
=> ( ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ( ord_less_eq_nat @ M3 @ ( F @ X ) ) )
=> ( ord_less_eq_nat @ M3 @ ( complete_Inf_Inf_nat @ ( image_nat_nat @ F @ A4 ) ) ) ) ) ).
% cINF_greatest
thf(fact_2558_cINF__greatest,axiom,
! [A4: set_int,M3: int,F: int > int] :
( ( A4 != bot_bot_set_int )
=> ( ! [X: int] :
( ( member_int @ X @ A4 )
=> ( ord_less_eq_int @ M3 @ ( F @ X ) ) )
=> ( ord_less_eq_int @ M3 @ ( complete_Inf_Inf_int @ ( image_int_int @ F @ A4 ) ) ) ) ) ).
% cINF_greatest
thf(fact_2559_cINF__greatest,axiom,
! [A4: set_Product_unit,M3: int,F: product_unit > int] :
( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ! [X: product_unit] :
( ( member_Product_unit @ X @ A4 )
=> ( ord_less_eq_int @ M3 @ ( F @ X ) ) )
=> ( ord_less_eq_int @ M3 @ ( complete_Inf_Inf_int @ ( image_873079544045703924it_int @ F @ A4 ) ) ) ) ) ).
% cINF_greatest
thf(fact_2560_cINF__greatest,axiom,
! [A4: set_o,M3: int,F: $o > int] :
( ( A4 != bot_bot_set_o )
=> ( ! [X: $o] :
( ( member_o @ X @ A4 )
=> ( ord_less_eq_int @ M3 @ ( F @ X ) ) )
=> ( ord_less_eq_int @ M3 @ ( complete_Inf_Inf_int @ ( image_o_int @ F @ A4 ) ) ) ) ) ).
% cINF_greatest
thf(fact_2561_cINF__greatest,axiom,
! [A4: set_nat,M3: int,F: nat > int] :
( ( A4 != bot_bot_set_nat )
=> ( ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ( ord_less_eq_int @ M3 @ ( F @ X ) ) )
=> ( ord_less_eq_int @ M3 @ ( complete_Inf_Inf_int @ ( image_nat_int @ F @ A4 ) ) ) ) ) ).
% cINF_greatest
thf(fact_2562_cINF__greatest,axiom,
! [A4: set_int,M3: product_unit,F: int > product_unit] :
( ( A4 != bot_bot_set_int )
=> ( ! [X: int] :
( ( member_int @ X @ A4 )
=> ( ord_le3221252021190050221t_unit @ M3 @ ( F @ X ) ) )
=> ( ord_le3221252021190050221t_unit @ M3 @ ( comple2584293577114468500t_unit @ ( image_5846526143423701914t_unit @ F @ A4 ) ) ) ) ) ).
% cINF_greatest
thf(fact_2563_cINF__greatest,axiom,
! [A4: set_Product_unit,M3: product_unit,F: product_unit > product_unit] :
( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ! [X: product_unit] :
( ( member_Product_unit @ X @ A4 )
=> ( ord_le3221252021190050221t_unit @ M3 @ ( F @ X ) ) )
=> ( ord_le3221252021190050221t_unit @ M3 @ ( comple2584293577114468500t_unit @ ( image_405062704495631173t_unit @ F @ A4 ) ) ) ) ) ).
% cINF_greatest
thf(fact_2564_cSUP__least,axiom,
! [A4: set_int,F: int > nat,M: nat] :
( ( A4 != bot_bot_set_int )
=> ( ! [X: int] :
( ( member_int @ X @ A4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ M ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_int_nat @ F @ A4 ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_2565_cSUP__least,axiom,
! [A4: set_Product_unit,F: product_unit > nat,M: nat] :
( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ! [X: product_unit] :
( ( member_Product_unit @ X @ A4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ M ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_875570014554754200it_nat @ F @ A4 ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_2566_cSUP__least,axiom,
! [A4: set_o,F: $o > nat,M: nat] :
( ( A4 != bot_bot_set_o )
=> ( ! [X: $o] :
( ( member_o @ X @ A4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ M ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_o_nat @ F @ A4 ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_2567_cSUP__least,axiom,
! [A4: set_nat,F: nat > nat,M: nat] :
( ( A4 != bot_bot_set_nat )
=> ( ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ M ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ A4 ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_2568_cSUP__least,axiom,
! [A4: set_int,F: int > int,M: int] :
( ( A4 != bot_bot_set_int )
=> ( ! [X: int] :
( ( member_int @ X @ A4 )
=> ( ord_less_eq_int @ ( F @ X ) @ M ) )
=> ( ord_less_eq_int @ ( complete_Sup_Sup_int @ ( image_int_int @ F @ A4 ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_2569_cSUP__least,axiom,
! [A4: set_Product_unit,F: product_unit > int,M: int] :
( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ! [X: product_unit] :
( ( member_Product_unit @ X @ A4 )
=> ( ord_less_eq_int @ ( F @ X ) @ M ) )
=> ( ord_less_eq_int @ ( complete_Sup_Sup_int @ ( image_873079544045703924it_int @ F @ A4 ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_2570_cSUP__least,axiom,
! [A4: set_o,F: $o > int,M: int] :
( ( A4 != bot_bot_set_o )
=> ( ! [X: $o] :
( ( member_o @ X @ A4 )
=> ( ord_less_eq_int @ ( F @ X ) @ M ) )
=> ( ord_less_eq_int @ ( complete_Sup_Sup_int @ ( image_o_int @ F @ A4 ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_2571_cSUP__least,axiom,
! [A4: set_nat,F: nat > int,M: int] :
( ( A4 != bot_bot_set_nat )
=> ( ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ( ord_less_eq_int @ ( F @ X ) @ M ) )
=> ( ord_less_eq_int @ ( complete_Sup_Sup_int @ ( image_nat_int @ F @ A4 ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_2572_cSUP__least,axiom,
! [A4: set_int,F: int > product_unit,M: product_unit] :
( ( A4 != bot_bot_set_int )
=> ( ! [X: int] :
( ( member_int @ X @ A4 )
=> ( ord_le3221252021190050221t_unit @ ( F @ X ) @ M ) )
=> ( ord_le3221252021190050221t_unit @ ( comple4687483117567863418t_unit @ ( image_5846526143423701914t_unit @ F @ A4 ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_2573_cSUP__least,axiom,
! [A4: set_Product_unit,F: product_unit > product_unit,M: product_unit] :
( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ! [X: product_unit] :
( ( member_Product_unit @ X @ A4 )
=> ( ord_le3221252021190050221t_unit @ ( F @ X ) @ M ) )
=> ( ord_le3221252021190050221t_unit @ ( comple4687483117567863418t_unit @ ( image_405062704495631173t_unit @ F @ A4 ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_2574_UN__constant__eq,axiom,
! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,F: produc3843707927480180839at_nat > set_nat,C: set_nat] :
( ( member8757157785044589968at_nat @ A @ A4 )
=> ( ! [X: produc3843707927480180839at_nat] :
( ( member8757157785044589968at_nat @ X @ A4 )
=> ( ( F @ X )
= C ) )
=> ( ( comple7399068483239264473et_nat @ ( image_3160451284896210088et_nat @ F @ A4 ) )
= C ) ) ) ).
% UN_constant_eq
thf(fact_2575_UN__constant__eq,axiom,
! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > set_nat,C: set_nat] :
( ( member8440522571783428010at_nat @ A @ A4 )
=> ( ! [X: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X @ A4 )
=> ( ( F @ X )
= C ) )
=> ( ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ F @ A4 ) )
= C ) ) ) ).
% UN_constant_eq
thf(fact_2576_UN__constant__eq,axiom,
! [A: $o,A4: set_o,F: $o > set_nat,C: set_nat] :
( ( member_o @ A @ A4 )
=> ( ! [X: $o] :
( ( member_o @ X @ A4 )
=> ( ( F @ X )
= C ) )
=> ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A4 ) )
= C ) ) ) ).
% UN_constant_eq
thf(fact_2577_UN__constant__eq,axiom,
! [A: nat,A4: set_nat,F: nat > set_nat,C: set_nat] :
( ( member_nat @ A @ A4 )
=> ( ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ( ( F @ X )
= C ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A4 ) )
= C ) ) ) ).
% UN_constant_eq
thf(fact_2578_UN__constant__eq,axiom,
! [A: int,A4: set_int,F: int > set_nat,C: set_nat] :
( ( member_int @ A @ A4 )
=> ( ! [X: int] :
( ( member_int @ X @ A4 )
=> ( ( F @ X )
= C ) )
=> ( ( comple7399068483239264473et_nat @ ( image_int_set_nat @ F @ A4 ) )
= C ) ) ) ).
% UN_constant_eq
thf(fact_2579_refl__on__domain,axiom,
! [A4: set_Pr4329608150637261639at_nat,R: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
( ( refl_o7446464703481863270at_nat @ A4 @ R )
=> ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ B ) @ R )
=> ( ( member8757157785044589968at_nat @ A @ A4 )
& ( member8757157785044589968at_nat @ B @ A4 ) ) ) ) ).
% refl_on_domain
thf(fact_2580_refl__on__domain,axiom,
! [A4: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] :
( ( refl_o7122697186039111552at_nat @ A4 @ R )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R )
=> ( ( member8440522571783428010at_nat @ A @ A4 )
& ( member8440522571783428010at_nat @ B @ A4 ) ) ) ) ).
% refl_on_domain
thf(fact_2581_refl__on__domain,axiom,
! [A4: set_o,R: set_Product_prod_o_o,A: $o,B: $o] :
( ( refl_on_o @ A4 @ R )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ R )
=> ( ( member_o @ A @ A4 )
& ( member_o @ B @ A4 ) ) ) ) ).
% refl_on_domain
thf(fact_2582_refl__on__domain,axiom,
! [A4: set_se5479201105039950375at_nat,R: set_Pr7459493094073627847at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
( ( refl_o6272981002401706822at_nat @ A4 @ R )
=> ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ B ) @ R )
=> ( ( member6273874620674182256at_nat @ A @ A4 )
& ( member6273874620674182256at_nat @ B @ A4 ) ) ) ) ).
% refl_on_domain
thf(fact_2583_refl__on__domain,axiom,
! [A4: set_se7855581050983116737at_nat,R: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
( ( refl_o6955724340487615712at_nat @ A4 @ R )
=> ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R )
=> ( ( member2643936169264416010at_nat @ A @ A4 )
& ( member2643936169264416010at_nat @ B @ A4 ) ) ) ) ).
% refl_on_domain
thf(fact_2584_refl__on__domain,axiom,
! [A4: set_nat,R: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
( ( refl_on_nat @ A4 @ R )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
=> ( ( member_nat @ A @ A4 )
& ( member_nat @ B @ A4 ) ) ) ) ).
% refl_on_domain
thf(fact_2585_refl__on__domain,axiom,
! [A4: set_int,R: set_Pr958786334691620121nt_int,A: int,B: int] :
( ( refl_on_int @ A4 @ R )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ R )
=> ( ( member_int @ A @ A4 )
& ( member_int @ B @ A4 ) ) ) ) ).
% refl_on_domain
thf(fact_2586_INT__simps_I4_J,axiom,
! [C3: set_Product_unit,A4: set_char,B5: product_unit > set_char] :
( ( ( C3 = bot_bo3957492148770167129t_unit )
=> ( ( comple7965528767983648192t_char
@ ( image_5325685772115642575t_char
@ ^ [X4: product_unit] : ( minus_minus_set_char @ A4 @ ( B5 @ X4 ) )
@ C3 ) )
= top_top_set_char ) )
& ( ( C3 != bot_bo3957492148770167129t_unit )
=> ( ( comple7965528767983648192t_char
@ ( image_5325685772115642575t_char
@ ^ [X4: product_unit] : ( minus_minus_set_char @ A4 @ ( B5 @ X4 ) )
@ C3 ) )
= ( minus_minus_set_char @ A4 @ ( comple4338141846124676698t_char @ ( image_5325685772115642575t_char @ B5 @ C3 ) ) ) ) ) ) ).
% INT_simps(4)
thf(fact_2587_INT__simps_I4_J,axiom,
! [C3: set_o,A4: set_char,B5: $o > set_char] :
( ( ( C3 = bot_bot_set_o )
=> ( ( comple7965528767983648192t_char
@ ( image_o_set_char
@ ^ [X4: $o] : ( minus_minus_set_char @ A4 @ ( B5 @ X4 ) )
@ C3 ) )
= top_top_set_char ) )
& ( ( C3 != bot_bot_set_o )
=> ( ( comple7965528767983648192t_char
@ ( image_o_set_char
@ ^ [X4: $o] : ( minus_minus_set_char @ A4 @ ( B5 @ X4 ) )
@ C3 ) )
= ( minus_minus_set_char @ A4 @ ( comple4338141846124676698t_char @ ( image_o_set_char @ B5 @ C3 ) ) ) ) ) ) ).
% INT_simps(4)
thf(fact_2588_INT__simps_I4_J,axiom,
! [C3: set_nat,A4: set_char,B5: nat > set_char] :
( ( ( C3 = bot_bot_set_nat )
=> ( ( comple7965528767983648192t_char
@ ( image_nat_set_char
@ ^ [X4: nat] : ( minus_minus_set_char @ A4 @ ( B5 @ X4 ) )
@ C3 ) )
= top_top_set_char ) )
& ( ( C3 != bot_bot_set_nat )
=> ( ( comple7965528767983648192t_char
@ ( image_nat_set_char
@ ^ [X4: nat] : ( minus_minus_set_char @ A4 @ ( B5 @ X4 ) )
@ C3 ) )
= ( minus_minus_set_char @ A4 @ ( comple4338141846124676698t_char @ ( image_nat_set_char @ B5 @ C3 ) ) ) ) ) ) ).
% INT_simps(4)
thf(fact_2589_INT__simps_I4_J,axiom,
! [C3: set_Product_unit,A4: set_rat,B5: product_unit > set_rat] :
( ( ( C3 = bot_bo3957492148770167129t_unit )
=> ( ( comple4298007329820168263et_rat
@ ( image_4665621433434589782et_rat
@ ^ [X4: product_unit] : ( minus_minus_set_rat @ A4 @ ( B5 @ X4 ) )
@ C3 ) )
= top_top_set_rat ) )
& ( ( C3 != bot_bo3957492148770167129t_unit )
=> ( ( comple4298007329820168263et_rat
@ ( image_4665621433434589782et_rat
@ ^ [X4: product_unit] : ( minus_minus_set_rat @ A4 @ ( B5 @ X4 ) )
@ C3 ) )
= ( minus_minus_set_rat @ A4 @ ( comple3890839924845867745et_rat @ ( image_4665621433434589782et_rat @ B5 @ C3 ) ) ) ) ) ) ).
% INT_simps(4)
thf(fact_2590_INT__simps_I4_J,axiom,
! [C3: set_o,A4: set_rat,B5: $o > set_rat] :
( ( ( C3 = bot_bot_set_o )
=> ( ( comple4298007329820168263et_rat
@ ( image_o_set_rat
@ ^ [X4: $o] : ( minus_minus_set_rat @ A4 @ ( B5 @ X4 ) )
@ C3 ) )
= top_top_set_rat ) )
& ( ( C3 != bot_bot_set_o )
=> ( ( comple4298007329820168263et_rat
@ ( image_o_set_rat
@ ^ [X4: $o] : ( minus_minus_set_rat @ A4 @ ( B5 @ X4 ) )
@ C3 ) )
= ( minus_minus_set_rat @ A4 @ ( comple3890839924845867745et_rat @ ( image_o_set_rat @ B5 @ C3 ) ) ) ) ) ) ).
% INT_simps(4)
thf(fact_2591_INT__simps_I4_J,axiom,
! [C3: set_nat,A4: set_rat,B5: nat > set_rat] :
( ( ( C3 = bot_bot_set_nat )
=> ( ( comple4298007329820168263et_rat
@ ( image_nat_set_rat
@ ^ [X4: nat] : ( minus_minus_set_rat @ A4 @ ( B5 @ X4 ) )
@ C3 ) )
= top_top_set_rat ) )
& ( ( C3 != bot_bot_set_nat )
=> ( ( comple4298007329820168263et_rat
@ ( image_nat_set_rat
@ ^ [X4: nat] : ( minus_minus_set_rat @ A4 @ ( B5 @ X4 ) )
@ C3 ) )
= ( minus_minus_set_rat @ A4 @ ( comple3890839924845867745et_rat @ ( image_nat_set_rat @ B5 @ C3 ) ) ) ) ) ) ).
% INT_simps(4)
thf(fact_2592_INT__simps_I4_J,axiom,
! [C3: set_Product_unit,A4: set_int,B5: product_unit > set_int] :
( ( ( C3 = bot_bo3957492148770167129t_unit )
=> ( ( comple3628384868704368283et_int
@ ( image_3995998972318789802et_int
@ ^ [X4: product_unit] : ( minus_minus_set_int @ A4 @ ( B5 @ X4 ) )
@ C3 ) )
= top_top_set_int ) )
& ( ( C3 != bot_bo3957492148770167129t_unit )
=> ( ( comple3628384868704368283et_int
@ ( image_3995998972318789802et_int
@ ^ [X4: product_unit] : ( minus_minus_set_int @ A4 @ ( B5 @ X4 ) )
@ C3 ) )
= ( minus_minus_set_int @ A4 @ ( comple3221217463730067765et_int @ ( image_3995998972318789802et_int @ B5 @ C3 ) ) ) ) ) ) ).
% INT_simps(4)
thf(fact_2593_INT__simps_I4_J,axiom,
! [C3: set_o,A4: set_int,B5: $o > set_int] :
( ( ( C3 = bot_bot_set_o )
=> ( ( comple3628384868704368283et_int
@ ( image_o_set_int
@ ^ [X4: $o] : ( minus_minus_set_int @ A4 @ ( B5 @ X4 ) )
@ C3 ) )
= top_top_set_int ) )
& ( ( C3 != bot_bot_set_o )
=> ( ( comple3628384868704368283et_int
@ ( image_o_set_int
@ ^ [X4: $o] : ( minus_minus_set_int @ A4 @ ( B5 @ X4 ) )
@ C3 ) )
= ( minus_minus_set_int @ A4 @ ( comple3221217463730067765et_int @ ( image_o_set_int @ B5 @ C3 ) ) ) ) ) ) ).
% INT_simps(4)
thf(fact_2594_INT__simps_I4_J,axiom,
! [C3: set_nat,A4: set_int,B5: nat > set_int] :
( ( ( C3 = bot_bot_set_nat )
=> ( ( comple3628384868704368283et_int
@ ( image_nat_set_int
@ ^ [X4: nat] : ( minus_minus_set_int @ A4 @ ( B5 @ X4 ) )
@ C3 ) )
= top_top_set_int ) )
& ( ( C3 != bot_bot_set_nat )
=> ( ( comple3628384868704368283et_int
@ ( image_nat_set_int
@ ^ [X4: nat] : ( minus_minus_set_int @ A4 @ ( B5 @ X4 ) )
@ C3 ) )
= ( minus_minus_set_int @ A4 @ ( comple3221217463730067765et_int @ ( image_nat_set_int @ B5 @ C3 ) ) ) ) ) ) ).
% INT_simps(4)
thf(fact_2595_INT__simps_I4_J,axiom,
! [C3: set_Product_unit,A4: set_nat,B5: product_unit > set_nat] :
( ( ( C3 = bot_bo3957492148770167129t_unit )
=> ( ( comple7806235888213564991et_nat
@ ( image_8173849991827986510et_nat
@ ^ [X4: product_unit] : ( minus_minus_set_nat @ A4 @ ( B5 @ X4 ) )
@ C3 ) )
= top_top_set_nat ) )
& ( ( C3 != bot_bo3957492148770167129t_unit )
=> ( ( comple7806235888213564991et_nat
@ ( image_8173849991827986510et_nat
@ ^ [X4: product_unit] : ( minus_minus_set_nat @ A4 @ ( B5 @ X4 ) )
@ C3 ) )
= ( minus_minus_set_nat @ A4 @ ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ B5 @ C3 ) ) ) ) ) ) ).
% INT_simps(4)
thf(fact_2596_relcompp__bot2,axiom,
! [R3: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ( relcom8911704206367525917at_int @ R3 @ bot_bo5438875203632553179_int_o )
= bot_bo5438875203632553179_int_o ) ).
% relcompp_bot2
thf(fact_2597_relcompp__bot2,axiom,
! [R3: int > int > $o] :
( ( relcom967497705283160040nteger @ R3 @ bot_bo341146761387662761eger_o )
= bot_bo341146761387662761eger_o ) ).
% relcompp_bot2
thf(fact_2598_relcompp__bot1,axiom,
! [R3: product_prod_nat_nat > int > $o] :
( ( relcom8911704206367525917at_int @ bot_bo4898103413517107610_nat_o @ R3 )
= bot_bo5438875203632553179_int_o ) ).
% relcompp_bot1
thf(fact_2599_relcompp__bot1,axiom,
! [R3: int > code_integer > $o] :
( ( relcom967497705283160040nteger @ bot_bot_int_int_o @ R3 )
= bot_bo341146761387662761eger_o ) ).
% relcompp_bot1
thf(fact_2600_surj__diff__right,axiom,
! [A: int] :
( ( image_int_int
@ ^ [X4: int] : ( minus_minus_int @ X4 @ A )
@ top_top_set_int )
= top_top_set_int ) ).
% surj_diff_right
thf(fact_2601_surj__diff__right,axiom,
! [A: rat] :
( ( image_rat_rat
@ ^ [X4: rat] : ( minus_minus_rat @ X4 @ A )
@ top_top_set_rat )
= top_top_set_rat ) ).
% surj_diff_right
thf(fact_2602_surj__diff__right,axiom,
! [A: code_integer] :
( ( image_4470545334726330049nteger
@ ^ [X4: code_integer] : ( minus_8373710615458151222nteger @ X4 @ A )
@ top_to4645266643341252675nteger )
= top_to4645266643341252675nteger ) ).
% surj_diff_right
thf(fact_2603_INT__simps_I3_J,axiom,
! [C3: set_Product_unit,A4: product_unit > set_char,B5: set_char] :
( ( ( C3 = bot_bo3957492148770167129t_unit )
=> ( ( comple7965528767983648192t_char
@ ( image_5325685772115642575t_char
@ ^ [X4: product_unit] : ( minus_minus_set_char @ ( A4 @ X4 ) @ B5 )
@ C3 ) )
= top_top_set_char ) )
& ( ( C3 != bot_bo3957492148770167129t_unit )
=> ( ( comple7965528767983648192t_char
@ ( image_5325685772115642575t_char
@ ^ [X4: product_unit] : ( minus_minus_set_char @ ( A4 @ X4 ) @ B5 )
@ C3 ) )
= ( minus_minus_set_char @ ( comple7965528767983648192t_char @ ( image_5325685772115642575t_char @ A4 @ C3 ) ) @ B5 ) ) ) ) ).
% INT_simps(3)
thf(fact_2604_INT__simps_I3_J,axiom,
! [C3: set_o,A4: $o > set_char,B5: set_char] :
( ( ( C3 = bot_bot_set_o )
=> ( ( comple7965528767983648192t_char
@ ( image_o_set_char
@ ^ [X4: $o] : ( minus_minus_set_char @ ( A4 @ X4 ) @ B5 )
@ C3 ) )
= top_top_set_char ) )
& ( ( C3 != bot_bot_set_o )
=> ( ( comple7965528767983648192t_char
@ ( image_o_set_char
@ ^ [X4: $o] : ( minus_minus_set_char @ ( A4 @ X4 ) @ B5 )
@ C3 ) )
= ( minus_minus_set_char @ ( comple7965528767983648192t_char @ ( image_o_set_char @ A4 @ C3 ) ) @ B5 ) ) ) ) ).
% INT_simps(3)
thf(fact_2605_INT__simps_I3_J,axiom,
! [C3: set_nat,A4: nat > set_char,B5: set_char] :
( ( ( C3 = bot_bot_set_nat )
=> ( ( comple7965528767983648192t_char
@ ( image_nat_set_char
@ ^ [X4: nat] : ( minus_minus_set_char @ ( A4 @ X4 ) @ B5 )
@ C3 ) )
= top_top_set_char ) )
& ( ( C3 != bot_bot_set_nat )
=> ( ( comple7965528767983648192t_char
@ ( image_nat_set_char
@ ^ [X4: nat] : ( minus_minus_set_char @ ( A4 @ X4 ) @ B5 )
@ C3 ) )
= ( minus_minus_set_char @ ( comple7965528767983648192t_char @ ( image_nat_set_char @ A4 @ C3 ) ) @ B5 ) ) ) ) ).
% INT_simps(3)
thf(fact_2606_INT__simps_I3_J,axiom,
! [C3: set_Product_unit,A4: product_unit > set_rat,B5: set_rat] :
( ( ( C3 = bot_bo3957492148770167129t_unit )
=> ( ( comple4298007329820168263et_rat
@ ( image_4665621433434589782et_rat
@ ^ [X4: product_unit] : ( minus_minus_set_rat @ ( A4 @ X4 ) @ B5 )
@ C3 ) )
= top_top_set_rat ) )
& ( ( C3 != bot_bo3957492148770167129t_unit )
=> ( ( comple4298007329820168263et_rat
@ ( image_4665621433434589782et_rat
@ ^ [X4: product_unit] : ( minus_minus_set_rat @ ( A4 @ X4 ) @ B5 )
@ C3 ) )
= ( minus_minus_set_rat @ ( comple4298007329820168263et_rat @ ( image_4665621433434589782et_rat @ A4 @ C3 ) ) @ B5 ) ) ) ) ).
% INT_simps(3)
thf(fact_2607_INT__simps_I3_J,axiom,
! [C3: set_o,A4: $o > set_rat,B5: set_rat] :
( ( ( C3 = bot_bot_set_o )
=> ( ( comple4298007329820168263et_rat
@ ( image_o_set_rat
@ ^ [X4: $o] : ( minus_minus_set_rat @ ( A4 @ X4 ) @ B5 )
@ C3 ) )
= top_top_set_rat ) )
& ( ( C3 != bot_bot_set_o )
=> ( ( comple4298007329820168263et_rat
@ ( image_o_set_rat
@ ^ [X4: $o] : ( minus_minus_set_rat @ ( A4 @ X4 ) @ B5 )
@ C3 ) )
= ( minus_minus_set_rat @ ( comple4298007329820168263et_rat @ ( image_o_set_rat @ A4 @ C3 ) ) @ B5 ) ) ) ) ).
% INT_simps(3)
thf(fact_2608_INT__simps_I3_J,axiom,
! [C3: set_nat,A4: nat > set_rat,B5: set_rat] :
( ( ( C3 = bot_bot_set_nat )
=> ( ( comple4298007329820168263et_rat
@ ( image_nat_set_rat
@ ^ [X4: nat] : ( minus_minus_set_rat @ ( A4 @ X4 ) @ B5 )
@ C3 ) )
= top_top_set_rat ) )
& ( ( C3 != bot_bot_set_nat )
=> ( ( comple4298007329820168263et_rat
@ ( image_nat_set_rat
@ ^ [X4: nat] : ( minus_minus_set_rat @ ( A4 @ X4 ) @ B5 )
@ C3 ) )
= ( minus_minus_set_rat @ ( comple4298007329820168263et_rat @ ( image_nat_set_rat @ A4 @ C3 ) ) @ B5 ) ) ) ) ).
% INT_simps(3)
thf(fact_2609_INT__simps_I3_J,axiom,
! [C3: set_Product_unit,A4: product_unit > set_int,B5: set_int] :
( ( ( C3 = bot_bo3957492148770167129t_unit )
=> ( ( comple3628384868704368283et_int
@ ( image_3995998972318789802et_int
@ ^ [X4: product_unit] : ( minus_minus_set_int @ ( A4 @ X4 ) @ B5 )
@ C3 ) )
= top_top_set_int ) )
& ( ( C3 != bot_bo3957492148770167129t_unit )
=> ( ( comple3628384868704368283et_int
@ ( image_3995998972318789802et_int
@ ^ [X4: product_unit] : ( minus_minus_set_int @ ( A4 @ X4 ) @ B5 )
@ C3 ) )
= ( minus_minus_set_int @ ( comple3628384868704368283et_int @ ( image_3995998972318789802et_int @ A4 @ C3 ) ) @ B5 ) ) ) ) ).
% INT_simps(3)
thf(fact_2610_INT__simps_I3_J,axiom,
! [C3: set_o,A4: $o > set_int,B5: set_int] :
( ( ( C3 = bot_bot_set_o )
=> ( ( comple3628384868704368283et_int
@ ( image_o_set_int
@ ^ [X4: $o] : ( minus_minus_set_int @ ( A4 @ X4 ) @ B5 )
@ C3 ) )
= top_top_set_int ) )
& ( ( C3 != bot_bot_set_o )
=> ( ( comple3628384868704368283et_int
@ ( image_o_set_int
@ ^ [X4: $o] : ( minus_minus_set_int @ ( A4 @ X4 ) @ B5 )
@ C3 ) )
= ( minus_minus_set_int @ ( comple3628384868704368283et_int @ ( image_o_set_int @ A4 @ C3 ) ) @ B5 ) ) ) ) ).
% INT_simps(3)
thf(fact_2611_INT__simps_I3_J,axiom,
! [C3: set_nat,A4: nat > set_int,B5: set_int] :
( ( ( C3 = bot_bot_set_nat )
=> ( ( comple3628384868704368283et_int
@ ( image_nat_set_int
@ ^ [X4: nat] : ( minus_minus_set_int @ ( A4 @ X4 ) @ B5 )
@ C3 ) )
= top_top_set_int ) )
& ( ( C3 != bot_bot_set_nat )
=> ( ( comple3628384868704368283et_int
@ ( image_nat_set_int
@ ^ [X4: nat] : ( minus_minus_set_int @ ( A4 @ X4 ) @ B5 )
@ C3 ) )
= ( minus_minus_set_int @ ( comple3628384868704368283et_int @ ( image_nat_set_int @ A4 @ C3 ) ) @ B5 ) ) ) ) ).
% INT_simps(3)
thf(fact_2612_INT__simps_I3_J,axiom,
! [C3: set_Product_unit,A4: product_unit > set_nat,B5: set_nat] :
( ( ( C3 = bot_bo3957492148770167129t_unit )
=> ( ( comple7806235888213564991et_nat
@ ( image_8173849991827986510et_nat
@ ^ [X4: product_unit] : ( minus_minus_set_nat @ ( A4 @ X4 ) @ B5 )
@ C3 ) )
= top_top_set_nat ) )
& ( ( C3 != bot_bo3957492148770167129t_unit )
=> ( ( comple7806235888213564991et_nat
@ ( image_8173849991827986510et_nat
@ ^ [X4: product_unit] : ( minus_minus_set_nat @ ( A4 @ X4 ) @ B5 )
@ C3 ) )
= ( minus_minus_set_nat @ ( comple7806235888213564991et_nat @ ( image_8173849991827986510et_nat @ A4 @ C3 ) ) @ B5 ) ) ) ) ).
% INT_simps(3)
thf(fact_2613_Sigma__Diff__distrib1,axiom,
! [I3: set_nat,J3: set_nat,C3: nat > set_nat] :
( ( produc457027306803732586at_nat @ ( minus_minus_set_nat @ I3 @ J3 ) @ C3 )
= ( minus_1356011639430497352at_nat @ ( produc457027306803732586at_nat @ I3 @ C3 ) @ ( produc457027306803732586at_nat @ J3 @ C3 ) ) ) ).
% Sigma_Diff_distrib1
thf(fact_2614_translation__subtract__diff,axiom,
! [A: int,S2: set_int,T2: set_int] :
( ( image_int_int
@ ^ [X4: int] : ( minus_minus_int @ X4 @ A )
@ ( minus_minus_set_int @ S2 @ T2 ) )
= ( minus_minus_set_int
@ ( image_int_int
@ ^ [X4: int] : ( minus_minus_int @ X4 @ A )
@ S2 )
@ ( image_int_int
@ ^ [X4: int] : ( minus_minus_int @ X4 @ A )
@ T2 ) ) ) ).
% translation_subtract_diff
thf(fact_2615_translation__subtract__diff,axiom,
! [A: rat,S2: set_rat,T2: set_rat] :
( ( image_rat_rat
@ ^ [X4: rat] : ( minus_minus_rat @ X4 @ A )
@ ( minus_minus_set_rat @ S2 @ T2 ) )
= ( minus_minus_set_rat
@ ( image_rat_rat
@ ^ [X4: rat] : ( minus_minus_rat @ X4 @ A )
@ S2 )
@ ( image_rat_rat
@ ^ [X4: rat] : ( minus_minus_rat @ X4 @ A )
@ T2 ) ) ) ).
% translation_subtract_diff
thf(fact_2616_translation__subtract__diff,axiom,
! [A: code_integer,S2: set_Code_integer,T2: set_Code_integer] :
( ( image_4470545334726330049nteger
@ ^ [X4: code_integer] : ( minus_8373710615458151222nteger @ X4 @ A )
@ ( minus_2355218937544613996nteger @ S2 @ T2 ) )
= ( minus_2355218937544613996nteger
@ ( image_4470545334726330049nteger
@ ^ [X4: code_integer] : ( minus_8373710615458151222nteger @ X4 @ A )
@ S2 )
@ ( image_4470545334726330049nteger
@ ^ [X4: code_integer] : ( minus_8373710615458151222nteger @ X4 @ A )
@ T2 ) ) ) ).
% translation_subtract_diff
thf(fact_2617_Times__Diff__distrib1,axiom,
! [A4: set_nat,B5: set_nat,C3: set_nat] :
( ( produc457027306803732586at_nat @ ( minus_minus_set_nat @ A4 @ B5 )
@ ^ [Uu: nat] : C3 )
= ( minus_1356011639430497352at_nat
@ ( produc457027306803732586at_nat @ A4
@ ^ [Uu: nat] : C3 )
@ ( produc457027306803732586at_nat @ B5
@ ^ [Uu: nat] : C3 ) ) ) ).
% Times_Diff_distrib1
thf(fact_2618_Sigma__Diff__distrib2,axiom,
! [I3: set_nat,A4: nat > set_nat,B5: nat > set_nat] :
( ( produc457027306803732586at_nat @ I3
@ ^ [I2: nat] : ( minus_minus_set_nat @ ( A4 @ I2 ) @ ( B5 @ I2 ) ) )
= ( minus_1356011639430497352at_nat @ ( produc457027306803732586at_nat @ I3 @ A4 ) @ ( produc457027306803732586at_nat @ I3 @ B5 ) ) ) ).
% Sigma_Diff_distrib2
thf(fact_2619_Sup__SUP__eq,axiom,
( comple1177764198577243945_nat_o
= ( ^ [S4: set_Pr7010166363911110284_nat_o,X4: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X4 @ ( comple4364388969068068532at_nat @ ( image_4902389024228248874at_nat @ collec6321179662152712658at_nat @ S4 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_2620_Sup__SUP__eq,axiom,
( complete_Sup_Sup_o_o
= ( ^ [S4: set_o_o,X4: $o] : ( member_o @ X4 @ ( comple90263536869209701_set_o @ ( image_o_o_set_o @ collect_o @ S4 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_2621_Sup__SUP__eq,axiom,
( comple6496622788309502864_int_o
= ( ^ [S4: set_int_o,X4: int] : ( member_int @ X4 @ ( comple3221217463730067765et_int @ ( image_int_o_set_int @ collect_int @ S4 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_2622_Sup__SUP__eq,axiom,
( comple1901448594430626575_nat_o
= ( ^ [S4: set_Pr5582243495563764594_nat_o,X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ ( comple5685304695842803022at_nat @ ( image_7124889717316225246at_nat @ collec3392354462482085612at_nat @ S4 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_2623_Sup__SUP__eq,axiom,
( comple342879659800731223_int_o
= ( ^ [S4: set_Pr8049984349381595834_int_o,X4: product_prod_int_int] : ( member5262025264175285858nt_int @ X4 @ ( comple5382143125604098054nt_int @ ( image_4447441447610994254nt_int @ collec213857154873943460nt_int @ S4 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_2624_Sup__SUP__eq,axiom,
( comple8317665133742190828_nat_o
= ( ^ [S4: set_nat_o,X4: nat] : ( member_nat @ X4 @ ( comple7399068483239264473et_nat @ ( image_nat_o_set_nat @ collect_nat @ S4 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_2625_Inf__INT__eq,axiom,
( comple8183010123678664591_nat_o
= ( ^ [S4: set_Pr7010166363911110284_nat_o,X4: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X4 @ ( comple6751905152567503566at_nat @ ( image_4902389024228248874at_nat @ collec6321179662152712658at_nat @ S4 ) ) ) ) ) ).
% Inf_INT_eq
thf(fact_2626_Inf__INT__eq,axiom,
( complete_Inf_Inf_o_o
= ( ^ [S4: set_o_o,X4: $o] : ( member_o @ X4 @ ( comple3063163877087187839_set_o @ ( image_o_o_set_o @ collect_o @ S4 ) ) ) ) ) ).
% Inf_INT_eq
thf(fact_2627_Inf__INT__eq,axiom,
( comple4393433247856107946_int_o
= ( ^ [S4: set_int_o,X4: int] : ( member_int @ X4 @ ( comple3628384868704368283et_int @ ( image_int_o_set_int @ collect_int @ S4 ) ) ) ) ) ).
% Inf_INT_eq
thf(fact_2628_Inf__INT__eq,axiom,
( comple6265855634338933877_nat_o
= ( ^ [S4: set_Pr5582243495563764594_nat_o,X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ ( comple1930127330929219688at_nat @ ( image_7124889717316225246at_nat @ collec3392354462482085612at_nat @ S4 ) ) ) ) ) ).
% Inf_INT_eq
thf(fact_2629_Inf__INT__eq,axiom,
( comple4707286699709038525_int_o
= ( ^ [S4: set_Pr8049984349381595834_int_o,X4: product_prod_int_int] : ( member5262025264175285858nt_int @ X4 @ ( comple1626965760690514720nt_int @ ( image_4447441447610994254nt_int @ collec213857154873943460nt_int @ S4 ) ) ) ) ) ).
% Inf_INT_eq
thf(fact_2630_Inf__INT__eq,axiom,
( comple6214475593288795910_nat_o
= ( ^ [S4: set_nat_o,X4: nat] : ( member_nat @ X4 @ ( comple7806235888213564991et_nat @ ( image_nat_o_set_nat @ collect_nat @ S4 ) ) ) ) ) ).
% Inf_INT_eq
thf(fact_2631_INF__Int__eq2,axiom,
! [S5: set_se7592288975875140775at_nat] :
( ( comple5307621999477653941_nat_o
@ ( image_5337237954834410540_nat_o
@ ^ [I2: set_Pr7459493094073627847at_nat,X4: set_Pr4329608150637261639at_nat,Y5: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X4 @ Y5 ) @ I2 )
@ S5 ) )
= ( ^ [X4: set_Pr4329608150637261639at_nat,Y5: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X4 @ Y5 ) @ ( comple4403251139954110286at_nat @ S5 ) ) ) ) ).
% INF_Int_eq2
thf(fact_2632_INF__Int__eq2,axiom,
! [S5: set_se5479201105039950375at_nat] :
( ( comple8322679762649017525_nat_o
@ ( image_6595146019508340652_nat_o
@ ^ [I2: set_Pr4329608150637261639at_nat,X4: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y5 ) @ I2 )
@ S5 ) )
= ( ^ [X4: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y5 ) @ ( comple6751905152567503566at_nat @ S5 ) ) ) ) ).
% INF_Int_eq2
thf(fact_2633_INF__Int__eq2,axiom,
! [S5: set_se4826145725398303499at_num] :
( ( comple2433913294995299363_num_o
@ ( image_2055017250723459638_num_o
@ ^ [I2: set_Pr6200539531224447659at_num,X4: nat,Y5: num] : ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X4 @ Y5 ) @ I2 )
@ S5 ) )
= ( ^ [X4: nat,Y5: num] : ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X4 @ Y5 ) @ ( comple6868718957223342258at_num @ S5 ) ) ) ) ).
% INF_Int_eq2
thf(fact_2634_INF__Int__eq2,axiom,
! [S5: set_se7855581050983116737at_nat] :
( ( comple1675732732025788397_nat_o
@ ( image_6923511907955291850_nat_o
@ ^ [I2: set_Pr1261947904930325089at_nat,X4: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ I2 )
@ S5 ) )
= ( ^ [X4: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ ( comple1930127330929219688at_nat @ S5 ) ) ) ) ).
% INF_Int_eq2
thf(fact_2635_INF__Int__eq2,axiom,
! [S5: set_se6260736226359567993nt_int] :
( ( comple5770381748412298549_int_o
@ ( image_4446226961036766042_int_o
@ ^ [I2: set_Pr958786334691620121nt_int,X4: int,Y5: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y5 ) @ I2 )
@ S5 ) )
= ( ^ [X4: int,Y5: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y5 ) @ ( comple1626965760690514720nt_int @ S5 ) ) ) ) ).
% INF_Int_eq2
thf(fact_2636_SUP__Sup__eq2,axiom,
! [S5: set_se7592288975875140775at_nat] :
( ( comple4502628867940267931_nat_o
@ ( image_5337237954834410540_nat_o
@ ^ [I2: set_Pr7459493094073627847at_nat,X4: set_Pr4329608150637261639at_nat,Y5: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X4 @ Y5 ) @ I2 )
@ S5 ) )
= ( ^ [X4: set_Pr4329608150637261639at_nat,Y5: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X4 @ Y5 ) @ ( comple7342073961973170484at_nat @ S5 ) ) ) ) ).
% SUP_Sup_eq2
thf(fact_2637_SUP__Sup__eq2,axiom,
! [S5: set_se5479201105039950375at_nat] :
( ( comple493620192902636699_nat_o
@ ( image_6595146019508340652_nat_o
@ ^ [I2: set_Pr4329608150637261639at_nat,X4: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y5 ) @ I2 )
@ S5 ) )
= ( ^ [X4: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y5 ) @ ( comple4364388969068068532at_nat @ S5 ) ) ) ) ).
% SUP_Sup_eq2
thf(fact_2638_SUP__Sup__eq2,axiom,
! [S5: set_se4826145725398303499at_num] :
( ( comple4350791933526045961_num_o
@ ( image_2055017250723459638_num_o
@ ^ [I2: set_Pr6200539531224447659at_num,X4: nat,Y5: num] : ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X4 @ Y5 ) @ I2 )
@ S5 ) )
= ( ^ [X4: nat,Y5: num] : ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X4 @ Y5 ) @ ( comple1400524285282149784at_num @ S5 ) ) ) ) ).
% SUP_Sup_eq2
thf(fact_2639_SUP__Sup__eq2,axiom,
! [S5: set_se7855581050983116737at_nat] :
( ( comple3592611370556534995_nat_o
@ ( image_6923511907955291850_nat_o
@ ^ [I2: set_Pr1261947904930325089at_nat,X4: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ I2 )
@ S5 ) )
= ( ^ [X4: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ ( comple5685304695842803022at_nat @ S5 ) ) ) ) ).
% SUP_Sup_eq2
thf(fact_2640_SUP__Sup__eq2,axiom,
! [S5: set_se6260736226359567993nt_int] :
( ( comple7687260386943045147_int_o
@ ( image_4446226961036766042_int_o
@ ^ [I2: set_Pr958786334691620121nt_int,X4: int,Y5: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y5 ) @ I2 )
@ S5 ) )
= ( ^ [X4: int,Y5: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y5 ) @ ( comple5382143125604098054nt_int @ S5 ) ) ) ) ).
% SUP_Sup_eq2
thf(fact_2641_SUP__Sup__eq,axiom,
! [S5: set_se5479201105039950375at_nat] :
( ( comple1177764198577243945_nat_o
@ ( image_4859512832475012440_nat_o
@ ^ [I2: set_Pr4329608150637261639at_nat,X4: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X4 @ I2 )
@ S5 ) )
= ( ^ [X4: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X4 @ ( comple4364388969068068532at_nat @ S5 ) ) ) ) ).
% SUP_Sup_eq
thf(fact_2642_SUP__Sup__eq,axiom,
! [S5: set_se7855581050983116737at_nat] :
( ( comple1901448594430626575_nat_o
@ ( image_5253903537071781336_nat_o
@ ^ [I2: set_Pr1261947904930325089at_nat,X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ I2 )
@ S5 ) )
= ( ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ ( comple5685304695842803022at_nat @ S5 ) ) ) ) ).
% SUP_Sup_eq
thf(fact_2643_SUP__Sup__eq,axiom,
! [S5: set_set_o] :
( ( complete_Sup_Sup_o_o
@ ( image_set_o_o_o
@ ^ [I2: set_o,X4: $o] : ( member_o @ X4 @ I2 )
@ S5 ) )
= ( ^ [X4: $o] : ( member_o @ X4 @ ( comple90263536869209701_set_o @ S5 ) ) ) ) ).
% SUP_Sup_eq
thf(fact_2644_SUP__Sup__eq,axiom,
! [S5: set_set_int] :
( ( comple6496622788309502864_int_o
@ ( image_set_int_int_o
@ ^ [I2: set_int,X4: int] : ( member_int @ X4 @ I2 )
@ S5 ) )
= ( ^ [X4: int] : ( member_int @ X4 @ ( comple3221217463730067765et_int @ S5 ) ) ) ) ).
% SUP_Sup_eq
thf(fact_2645_SUP__Sup__eq,axiom,
! [S5: set_set_nat] :
( ( comple8317665133742190828_nat_o
@ ( image_set_nat_nat_o
@ ^ [I2: set_nat,X4: nat] : ( member_nat @ X4 @ I2 )
@ S5 ) )
= ( ^ [X4: nat] : ( member_nat @ X4 @ ( comple7399068483239264473et_nat @ S5 ) ) ) ) ).
% SUP_Sup_eq
thf(fact_2646_INF__Int__eq,axiom,
! [S5: set_se5479201105039950375at_nat] :
( ( comple8183010123678664591_nat_o
@ ( image_4859512832475012440_nat_o
@ ^ [I2: set_Pr4329608150637261639at_nat,X4: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X4 @ I2 )
@ S5 ) )
= ( ^ [X4: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X4 @ ( comple6751905152567503566at_nat @ S5 ) ) ) ) ).
% INF_Int_eq
thf(fact_2647_INF__Int__eq,axiom,
! [S5: set_se7855581050983116737at_nat] :
( ( comple6265855634338933877_nat_o
@ ( image_5253903537071781336_nat_o
@ ^ [I2: set_Pr1261947904930325089at_nat,X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ I2 )
@ S5 ) )
= ( ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ ( comple1930127330929219688at_nat @ S5 ) ) ) ) ).
% INF_Int_eq
thf(fact_2648_INF__Int__eq,axiom,
! [S5: set_set_o] :
( ( complete_Inf_Inf_o_o
@ ( image_set_o_o_o
@ ^ [I2: set_o,X4: $o] : ( member_o @ X4 @ I2 )
@ S5 ) )
= ( ^ [X4: $o] : ( member_o @ X4 @ ( comple3063163877087187839_set_o @ S5 ) ) ) ) ).
% INF_Int_eq
thf(fact_2649_INF__Int__eq,axiom,
! [S5: set_set_int] :
( ( comple4393433247856107946_int_o
@ ( image_set_int_int_o
@ ^ [I2: set_int,X4: int] : ( member_int @ X4 @ I2 )
@ S5 ) )
= ( ^ [X4: int] : ( member_int @ X4 @ ( comple3628384868704368283et_int @ S5 ) ) ) ) ).
% INF_Int_eq
thf(fact_2650_INF__Int__eq,axiom,
! [S5: set_set_nat] :
( ( comple6214475593288795910_nat_o
@ ( image_set_nat_nat_o
@ ^ [I2: set_nat,X4: nat] : ( member_nat @ X4 @ I2 )
@ S5 ) )
= ( ^ [X4: nat] : ( member_nat @ X4 @ ( comple7806235888213564991et_nat @ S5 ) ) ) ) ).
% INF_Int_eq
thf(fact_2651_Sup__set__def,axiom,
( comple4364388969068068532at_nat
= ( ^ [A6: set_se5479201105039950375at_nat] :
( collec6321179662152712658at_nat
@ ^ [X4: produc3843707927480180839at_nat] : ( complete_Sup_Sup_o @ ( image_3568060456209098518_nat_o @ ( member8757157785044589968at_nat @ X4 ) @ A6 ) ) ) ) ) ).
% Sup_set_def
thf(fact_2652_Sup__set__def,axiom,
( comple90263536869209701_set_o
= ( ^ [A6: set_set_o] :
( collect_o
@ ^ [X4: $o] : ( complete_Sup_Sup_o @ ( image_set_o_o @ ( member_o @ X4 ) @ A6 ) ) ) ) ) ).
% Sup_set_def
thf(fact_2653_Sup__set__def,axiom,
( comple3221217463730067765et_int
= ( ^ [A6: set_set_int] :
( collect_int
@ ^ [X4: int] : ( complete_Sup_Sup_o @ ( image_set_int_o @ ( member_int @ X4 ) @ A6 ) ) ) ) ) ).
% Sup_set_def
thf(fact_2654_Sup__set__def,axiom,
( comple5685304695842803022at_nat
= ( ^ [A6: set_se7855581050983116737at_nat] :
( collec3392354462482085612at_nat
@ ^ [X4: product_prod_nat_nat] : ( complete_Sup_Sup_o @ ( image_7457375456213313148_nat_o @ ( member8440522571783428010at_nat @ X4 ) @ A6 ) ) ) ) ) ).
% Sup_set_def
thf(fact_2655_Sup__set__def,axiom,
( comple5382143125604098054nt_int
= ( ^ [A6: set_se6260736226359567993nt_int] :
( collec213857154873943460nt_int
@ ^ [X4: product_prod_int_int] : ( complete_Sup_Sup_o @ ( image_701744273176368580_int_o @ ( member5262025264175285858nt_int @ X4 ) @ A6 ) ) ) ) ) ).
% Sup_set_def
thf(fact_2656_Sup__set__def,axiom,
( comple7399068483239264473et_nat
= ( ^ [A6: set_set_nat] :
( collect_nat
@ ^ [X4: nat] : ( complete_Sup_Sup_o @ ( image_set_nat_o @ ( member_nat @ X4 ) @ A6 ) ) ) ) ) ).
% Sup_set_def
thf(fact_2657_Inf__set__def,axiom,
( comple6751905152567503566at_nat
= ( ^ [A6: set_se5479201105039950375at_nat] :
( collec6321179662152712658at_nat
@ ^ [X4: produc3843707927480180839at_nat] : ( complete_Inf_Inf_o @ ( image_3568060456209098518_nat_o @ ( member8757157785044589968at_nat @ X4 ) @ A6 ) ) ) ) ) ).
% Inf_set_def
thf(fact_2658_Inf__set__def,axiom,
( comple3063163877087187839_set_o
= ( ^ [A6: set_set_o] :
( collect_o
@ ^ [X4: $o] : ( complete_Inf_Inf_o @ ( image_set_o_o @ ( member_o @ X4 ) @ A6 ) ) ) ) ) ).
% Inf_set_def
thf(fact_2659_Inf__set__def,axiom,
( comple3628384868704368283et_int
= ( ^ [A6: set_set_int] :
( collect_int
@ ^ [X4: int] : ( complete_Inf_Inf_o @ ( image_set_int_o @ ( member_int @ X4 ) @ A6 ) ) ) ) ) ).
% Inf_set_def
thf(fact_2660_Inf__set__def,axiom,
( comple1930127330929219688at_nat
= ( ^ [A6: set_se7855581050983116737at_nat] :
( collec3392354462482085612at_nat
@ ^ [X4: product_prod_nat_nat] : ( complete_Inf_Inf_o @ ( image_7457375456213313148_nat_o @ ( member8440522571783428010at_nat @ X4 ) @ A6 ) ) ) ) ) ).
% Inf_set_def
thf(fact_2661_Inf__set__def,axiom,
( comple1626965760690514720nt_int
= ( ^ [A6: set_se6260736226359567993nt_int] :
( collec213857154873943460nt_int
@ ^ [X4: product_prod_int_int] : ( complete_Inf_Inf_o @ ( image_701744273176368580_int_o @ ( member5262025264175285858nt_int @ X4 ) @ A6 ) ) ) ) ) ).
% Inf_set_def
thf(fact_2662_Inf__set__def,axiom,
( comple7806235888213564991et_nat
= ( ^ [A6: set_set_nat] :
( collect_nat
@ ^ [X4: nat] : ( complete_Inf_Inf_o @ ( image_set_nat_o @ ( member_nat @ X4 ) @ A6 ) ) ) ) ) ).
% Inf_set_def
thf(fact_2663_image__diff__subset,axiom,
! [F: nat > rat,A4: set_nat,B5: set_nat] : ( ord_less_eq_set_rat @ ( minus_minus_set_rat @ ( image_nat_rat @ F @ A4 ) @ ( image_nat_rat @ F @ B5 ) ) @ ( image_nat_rat @ F @ ( minus_minus_set_nat @ A4 @ B5 ) ) ) ).
% image_diff_subset
thf(fact_2664_image__diff__subset,axiom,
! [F: nat > nat,A4: set_nat,B5: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( image_nat_nat @ F @ A4 ) @ ( image_nat_nat @ F @ B5 ) ) @ ( image_nat_nat @ F @ ( minus_minus_set_nat @ A4 @ B5 ) ) ) ).
% image_diff_subset
thf(fact_2665_image__diff__subset,axiom,
! [F: nat > int,A4: set_nat,B5: set_nat] : ( ord_less_eq_set_int @ ( minus_minus_set_int @ ( image_nat_int @ F @ A4 ) @ ( image_nat_int @ F @ B5 ) ) @ ( image_nat_int @ F @ ( minus_minus_set_nat @ A4 @ B5 ) ) ) ).
% image_diff_subset
thf(fact_2666_image__diff__subset,axiom,
! [F: int > nat,A4: set_int,B5: set_int] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( image_int_nat @ F @ A4 ) @ ( image_int_nat @ F @ B5 ) ) @ ( image_int_nat @ F @ ( minus_minus_set_int @ A4 @ B5 ) ) ) ).
% image_diff_subset
thf(fact_2667_image__diff__subset,axiom,
! [F: int > int,A4: set_int,B5: set_int] : ( ord_less_eq_set_int @ ( minus_minus_set_int @ ( image_int_int @ F @ A4 ) @ ( image_int_int @ F @ B5 ) ) @ ( image_int_int @ F @ ( minus_minus_set_int @ A4 @ B5 ) ) ) ).
% image_diff_subset
thf(fact_2668_Inf__INT__eq2,axiom,
( comple5307621999477653941_nat_o
= ( ^ [S4: set_se7255092946006855566_nat_o,X4: set_Pr4329608150637261639at_nat,Y5: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X4 @ Y5 ) @ ( comple4403251139954110286at_nat @ ( image_5694205205332509994at_nat @ collec7872792499061297490at_nat @ ( image_5235725667318448191_nat_o @ produc9020218426428501292_nat_o @ S4 ) ) ) ) ) ) ).
% Inf_INT_eq2
thf(fact_2669_Inf__INT__eq2,axiom,
( comple8322679762649017525_nat_o
= ( ^ [S4: set_se9180647916752606350_nat_o,X4: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y5 ) @ ( comple6751905152567503566at_nat @ ( image_4902389024228248874at_nat @ collec6321179662152712658at_nat @ ( image_6226802281960588479_nat_o @ produc410239310623530412_nat_o @ S4 ) ) ) ) ) ) ).
% Inf_INT_eq2
thf(fact_2670_Inf__INT__eq2,axiom,
( comple2433913294995299363_num_o
= ( ^ [S4: set_nat_num_o,X4: nat,Y5: num] : ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X4 @ Y5 ) @ ( comple6868718957223342258at_num @ ( image_6580737572886966642at_num @ collec4100598399430923318at_num @ ( image_8484515203959327177_num_o @ produc4927758841916487424_num_o @ S4 ) ) ) ) ) ) ).
% Inf_INT_eq2
thf(fact_2671_Inf__INT__eq2,axiom,
( comple1675732732025788397_nat_o
= ( ^ [S4: set_nat_nat_o,X4: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ ( comple1930127330929219688at_nat @ ( image_7124889717316225246at_nat @ collec3392354462482085612at_nat @ ( image_7429393840292777309_nat_o @ produc6081775807080527818_nat_o @ S4 ) ) ) ) ) ) ).
% Inf_INT_eq2
thf(fact_2672_Inf__INT__eq2,axiom,
( comple5770381748412298549_int_o
= ( ^ [S4: set_int_int_o,X4: int,Y5: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y5 ) @ ( comple1626965760690514720nt_int @ ( image_4447441447610994254nt_int @ collec213857154873943460nt_int @ ( image_1365644804893650413_int_o @ produc4947309494688390418_int_o @ S4 ) ) ) ) ) ) ).
% Inf_INT_eq2
thf(fact_2673_Sup__SUP__eq2,axiom,
( comple4502628867940267931_nat_o
= ( ^ [S4: set_se7255092946006855566_nat_o,X4: set_Pr4329608150637261639at_nat,Y5: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X4 @ Y5 ) @ ( comple7342073961973170484at_nat @ ( image_5694205205332509994at_nat @ collec7872792499061297490at_nat @ ( image_5235725667318448191_nat_o @ produc9020218426428501292_nat_o @ S4 ) ) ) ) ) ) ).
% Sup_SUP_eq2
thf(fact_2674_Sup__SUP__eq2,axiom,
( comple493620192902636699_nat_o
= ( ^ [S4: set_se9180647916752606350_nat_o,X4: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y5 ) @ ( comple4364388969068068532at_nat @ ( image_4902389024228248874at_nat @ collec6321179662152712658at_nat @ ( image_6226802281960588479_nat_o @ produc410239310623530412_nat_o @ S4 ) ) ) ) ) ) ).
% Sup_SUP_eq2
thf(fact_2675_Sup__SUP__eq2,axiom,
( comple4350791933526045961_num_o
= ( ^ [S4: set_nat_num_o,X4: nat,Y5: num] : ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X4 @ Y5 ) @ ( comple1400524285282149784at_num @ ( image_6580737572886966642at_num @ collec4100598399430923318at_num @ ( image_8484515203959327177_num_o @ produc4927758841916487424_num_o @ S4 ) ) ) ) ) ) ).
% Sup_SUP_eq2
thf(fact_2676_Sup__SUP__eq2,axiom,
( comple3592611370556534995_nat_o
= ( ^ [S4: set_nat_nat_o,X4: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ ( comple5685304695842803022at_nat @ ( image_7124889717316225246at_nat @ collec3392354462482085612at_nat @ ( image_7429393840292777309_nat_o @ produc6081775807080527818_nat_o @ S4 ) ) ) ) ) ) ).
% Sup_SUP_eq2
thf(fact_2677_Sup__SUP__eq2,axiom,
( comple7687260386943045147_int_o
= ( ^ [S4: set_int_int_o,X4: int,Y5: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y5 ) @ ( comple5382143125604098054nt_int @ ( image_4447441447610994254nt_int @ collec213857154873943460nt_int @ ( image_1365644804893650413_int_o @ produc4947309494688390418_int_o @ S4 ) ) ) ) ) ) ).
% Sup_SUP_eq2
thf(fact_2678_bot__empty__eq2,axiom,
( bot_bo3364206721330744218_nat_o
= ( ^ [X4: set_Pr4329608150637261639at_nat,Y5: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X4 @ Y5 ) @ bot_bo4948859079157340979at_nat ) ) ) ).
% bot_empty_eq2
thf(fact_2679_bot__empty__eq2,axiom,
( bot_bot_nat_num_o
= ( ^ [X4: nat,Y5: num] : ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X4 @ Y5 ) @ bot_bo7038385379056416535at_num ) ) ) ).
% bot_empty_eq2
thf(fact_2680_bot__empty__eq2,axiom,
( bot_bot_int_int_o
= ( ^ [X4: int,Y5: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y5 ) @ bot_bo1796632182523588997nt_int ) ) ) ).
% bot_empty_eq2
thf(fact_2681_bot__empty__eq2,axiom,
( bot_bo394778441745866138_nat_o
= ( ^ [X4: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y5 ) @ bot_bo228742789529271731at_nat ) ) ) ).
% bot_empty_eq2
thf(fact_2682_bot__empty__eq2,axiom,
( bot_bot_nat_nat_o
= ( ^ [X4: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ bot_bo2099793752762293965at_nat ) ) ) ).
% bot_empty_eq2
thf(fact_2683_top__empty__eq2,axiom,
( top_to1326704617074743934_nat_o
= ( ^ [X4: set_Pr4329608150637261639at_nat,Y5: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X4 @ Y5 ) @ top_to5281548843312042519at_nat ) ) ) ).
% top_empty_eq2
thf(fact_2684_top__empty__eq2,axiom,
( top_to8903224817164384382_nat_o
= ( ^ [X4: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y5 ) @ top_to6833984726390702231at_nat ) ) ) ).
% top_empty_eq2
thf(fact_2685_top__empty__eq2,axiom,
( top_top_nat_num_o
= ( ^ [X4: nat,Y5: num] : ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X4 @ Y5 ) @ top_to385025497714130939at_num ) ) ) ).
% top_empty_eq2
thf(fact_2686_top__empty__eq2,axiom,
( top_top_nat_nat_o
= ( ^ [X4: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ top_to4669805908274784177at_nat ) ) ) ).
% top_empty_eq2
thf(fact_2687_top__empty__eq2,axiom,
( top_top_int_int_o
= ( ^ [X4: int,Y5: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y5 ) @ top_to4366644338036079209nt_int ) ) ) ).
% top_empty_eq2
thf(fact_2688_in__image__insert__iff,axiom,
! [B5: set_set_int,X2: int,A4: set_int] :
( ! [C2: set_int] :
( ( member_set_int @ C2 @ B5 )
=> ~ ( member_int @ X2 @ C2 ) )
=> ( ( member_set_int @ A4 @ ( image_524474410958335435et_int @ ( insert_int @ X2 ) @ B5 ) )
= ( ( member_int @ X2 @ A4 )
& ( member_set_int @ ( minus_minus_set_int @ A4 @ ( insert_int @ X2 @ bot_bot_set_int ) ) @ B5 ) ) ) ) ).
% in_image_insert_iff
thf(fact_2689_in__image__insert__iff,axiom,
! [B5: set_set_Product_unit,X2: product_unit,A4: set_Product_unit] :
( ! [C2: set_Product_unit] :
( ( member5877623283571906838t_unit @ C2 @ B5 )
=> ~ ( member_Product_unit @ X2 @ C2 ) )
=> ( ( member5877623283571906838t_unit @ A4 @ ( image_4019945301284682565t_unit @ ( insert_Product_unit @ X2 ) @ B5 ) )
= ( ( member_Product_unit @ X2 @ A4 )
& ( member5877623283571906838t_unit @ ( minus_6452836326544984404t_unit @ A4 @ ( insert_Product_unit @ X2 @ bot_bo3957492148770167129t_unit ) ) @ B5 ) ) ) ) ).
% in_image_insert_iff
thf(fact_2690_in__image__insert__iff,axiom,
! [B5: set_se5479201105039950375at_nat,X2: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
( ! [C2: set_Pr4329608150637261639at_nat] :
( ( member6273874620674182256at_nat @ C2 @ B5 )
=> ~ ( member8757157785044589968at_nat @ X2 @ C2 ) )
=> ( ( member6273874620674182256at_nat @ A4 @ ( image_4303274092540115909at_nat @ ( insert9069300056098147895at_nat @ X2 ) @ B5 ) )
= ( ( member8757157785044589968at_nat @ X2 @ A4 )
& ( member6273874620674182256at_nat @ ( minus_3314409938677909166at_nat @ A4 @ ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat ) ) @ B5 ) ) ) ) ).
% in_image_insert_iff
thf(fact_2691_in__image__insert__iff,axiom,
! [B5: set_se7855581050983116737at_nat,X2: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
( ! [C2: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ C2 @ B5 )
=> ~ ( member8440522571783428010at_nat @ X2 @ C2 ) )
=> ( ( member2643936169264416010at_nat @ A4 @ ( image_3684629450409544005at_nat @ ( insert8211810215607154385at_nat @ X2 ) @ B5 ) )
= ( ( member8440522571783428010at_nat @ X2 @ A4 )
& ( member2643936169264416010at_nat @ ( minus_1356011639430497352at_nat @ A4 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) @ B5 ) ) ) ) ).
% in_image_insert_iff
thf(fact_2692_in__image__insert__iff,axiom,
! [B5: set_set_o,X2: $o,A4: set_o] :
( ! [C2: set_o] :
( ( member_set_o @ C2 @ B5 )
=> ~ ( member_o @ X2 @ C2 ) )
=> ( ( member_set_o @ A4 @ ( image_set_o_set_o @ ( insert_o @ X2 ) @ B5 ) )
= ( ( member_o @ X2 @ A4 )
& ( member_set_o @ ( minus_minus_set_o @ A4 @ ( insert_o @ X2 @ bot_bot_set_o ) ) @ B5 ) ) ) ) ).
% in_image_insert_iff
thf(fact_2693_in__image__insert__iff,axiom,
! [B5: set_set_nat,X2: nat,A4: set_nat] :
( ! [C2: set_nat] :
( ( member_set_nat @ C2 @ B5 )
=> ~ ( member_nat @ X2 @ C2 ) )
=> ( ( member_set_nat @ A4 @ ( image_7916887816326733075et_nat @ ( insert_nat @ X2 ) @ B5 ) )
= ( ( member_nat @ X2 @ A4 )
& ( member_set_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) @ B5 ) ) ) ) ).
% in_image_insert_iff
thf(fact_2694_INT__extend__simps_I3_J,axiom,
! [C3: set_Product_unit,A4: product_unit > set_char,B5: set_char] :
( ( ( C3 = bot_bo3957492148770167129t_unit )
=> ( ( minus_minus_set_char @ ( comple7965528767983648192t_char @ ( image_5325685772115642575t_char @ A4 @ C3 ) ) @ B5 )
= ( minus_minus_set_char @ top_top_set_char @ B5 ) ) )
& ( ( C3 != bot_bo3957492148770167129t_unit )
=> ( ( minus_minus_set_char @ ( comple7965528767983648192t_char @ ( image_5325685772115642575t_char @ A4 @ C3 ) ) @ B5 )
= ( comple7965528767983648192t_char
@ ( image_5325685772115642575t_char
@ ^ [X4: product_unit] : ( minus_minus_set_char @ ( A4 @ X4 ) @ B5 )
@ C3 ) ) ) ) ) ).
% INT_extend_simps(3)
thf(fact_2695_INT__extend__simps_I3_J,axiom,
! [C3: set_o,A4: $o > set_char,B5: set_char] :
( ( ( C3 = bot_bot_set_o )
=> ( ( minus_minus_set_char @ ( comple7965528767983648192t_char @ ( image_o_set_char @ A4 @ C3 ) ) @ B5 )
= ( minus_minus_set_char @ top_top_set_char @ B5 ) ) )
& ( ( C3 != bot_bot_set_o )
=> ( ( minus_minus_set_char @ ( comple7965528767983648192t_char @ ( image_o_set_char @ A4 @ C3 ) ) @ B5 )
= ( comple7965528767983648192t_char
@ ( image_o_set_char
@ ^ [X4: $o] : ( minus_minus_set_char @ ( A4 @ X4 ) @ B5 )
@ C3 ) ) ) ) ) ).
% INT_extend_simps(3)
thf(fact_2696_INT__extend__simps_I3_J,axiom,
! [C3: set_nat,A4: nat > set_char,B5: set_char] :
( ( ( C3 = bot_bot_set_nat )
=> ( ( minus_minus_set_char @ ( comple7965528767983648192t_char @ ( image_nat_set_char @ A4 @ C3 ) ) @ B5 )
= ( minus_minus_set_char @ top_top_set_char @ B5 ) ) )
& ( ( C3 != bot_bot_set_nat )
=> ( ( minus_minus_set_char @ ( comple7965528767983648192t_char @ ( image_nat_set_char @ A4 @ C3 ) ) @ B5 )
= ( comple7965528767983648192t_char
@ ( image_nat_set_char
@ ^ [X4: nat] : ( minus_minus_set_char @ ( A4 @ X4 ) @ B5 )
@ C3 ) ) ) ) ) ).
% INT_extend_simps(3)
thf(fact_2697_INT__extend__simps_I3_J,axiom,
! [C3: set_Product_unit,A4: product_unit > set_rat,B5: set_rat] :
( ( ( C3 = bot_bo3957492148770167129t_unit )
=> ( ( minus_minus_set_rat @ ( comple4298007329820168263et_rat @ ( image_4665621433434589782et_rat @ A4 @ C3 ) ) @ B5 )
= ( minus_minus_set_rat @ top_top_set_rat @ B5 ) ) )
& ( ( C3 != bot_bo3957492148770167129t_unit )
=> ( ( minus_minus_set_rat @ ( comple4298007329820168263et_rat @ ( image_4665621433434589782et_rat @ A4 @ C3 ) ) @ B5 )
= ( comple4298007329820168263et_rat
@ ( image_4665621433434589782et_rat
@ ^ [X4: product_unit] : ( minus_minus_set_rat @ ( A4 @ X4 ) @ B5 )
@ C3 ) ) ) ) ) ).
% INT_extend_simps(3)
thf(fact_2698_INT__extend__simps_I3_J,axiom,
! [C3: set_o,A4: $o > set_rat,B5: set_rat] :
( ( ( C3 = bot_bot_set_o )
=> ( ( minus_minus_set_rat @ ( comple4298007329820168263et_rat @ ( image_o_set_rat @ A4 @ C3 ) ) @ B5 )
= ( minus_minus_set_rat @ top_top_set_rat @ B5 ) ) )
& ( ( C3 != bot_bot_set_o )
=> ( ( minus_minus_set_rat @ ( comple4298007329820168263et_rat @ ( image_o_set_rat @ A4 @ C3 ) ) @ B5 )
= ( comple4298007329820168263et_rat
@ ( image_o_set_rat
@ ^ [X4: $o] : ( minus_minus_set_rat @ ( A4 @ X4 ) @ B5 )
@ C3 ) ) ) ) ) ).
% INT_extend_simps(3)
thf(fact_2699_INT__extend__simps_I3_J,axiom,
! [C3: set_nat,A4: nat > set_rat,B5: set_rat] :
( ( ( C3 = bot_bot_set_nat )
=> ( ( minus_minus_set_rat @ ( comple4298007329820168263et_rat @ ( image_nat_set_rat @ A4 @ C3 ) ) @ B5 )
= ( minus_minus_set_rat @ top_top_set_rat @ B5 ) ) )
& ( ( C3 != bot_bot_set_nat )
=> ( ( minus_minus_set_rat @ ( comple4298007329820168263et_rat @ ( image_nat_set_rat @ A4 @ C3 ) ) @ B5 )
= ( comple4298007329820168263et_rat
@ ( image_nat_set_rat
@ ^ [X4: nat] : ( minus_minus_set_rat @ ( A4 @ X4 ) @ B5 )
@ C3 ) ) ) ) ) ).
% INT_extend_simps(3)
thf(fact_2700_INT__extend__simps_I3_J,axiom,
! [C3: set_Product_unit,A4: product_unit > set_int,B5: set_int] :
( ( ( C3 = bot_bo3957492148770167129t_unit )
=> ( ( minus_minus_set_int @ ( comple3628384868704368283et_int @ ( image_3995998972318789802et_int @ A4 @ C3 ) ) @ B5 )
= ( minus_minus_set_int @ top_top_set_int @ B5 ) ) )
& ( ( C3 != bot_bo3957492148770167129t_unit )
=> ( ( minus_minus_set_int @ ( comple3628384868704368283et_int @ ( image_3995998972318789802et_int @ A4 @ C3 ) ) @ B5 )
= ( comple3628384868704368283et_int
@ ( image_3995998972318789802et_int
@ ^ [X4: product_unit] : ( minus_minus_set_int @ ( A4 @ X4 ) @ B5 )
@ C3 ) ) ) ) ) ).
% INT_extend_simps(3)
thf(fact_2701_INT__extend__simps_I3_J,axiom,
! [C3: set_o,A4: $o > set_int,B5: set_int] :
( ( ( C3 = bot_bot_set_o )
=> ( ( minus_minus_set_int @ ( comple3628384868704368283et_int @ ( image_o_set_int @ A4 @ C3 ) ) @ B5 )
= ( minus_minus_set_int @ top_top_set_int @ B5 ) ) )
& ( ( C3 != bot_bot_set_o )
=> ( ( minus_minus_set_int @ ( comple3628384868704368283et_int @ ( image_o_set_int @ A4 @ C3 ) ) @ B5 )
= ( comple3628384868704368283et_int
@ ( image_o_set_int
@ ^ [X4: $o] : ( minus_minus_set_int @ ( A4 @ X4 ) @ B5 )
@ C3 ) ) ) ) ) ).
% INT_extend_simps(3)
thf(fact_2702_INT__extend__simps_I3_J,axiom,
! [C3: set_nat,A4: nat > set_int,B5: set_int] :
( ( ( C3 = bot_bot_set_nat )
=> ( ( minus_minus_set_int @ ( comple3628384868704368283et_int @ ( image_nat_set_int @ A4 @ C3 ) ) @ B5 )
= ( minus_minus_set_int @ top_top_set_int @ B5 ) ) )
& ( ( C3 != bot_bot_set_nat )
=> ( ( minus_minus_set_int @ ( comple3628384868704368283et_int @ ( image_nat_set_int @ A4 @ C3 ) ) @ B5 )
= ( comple3628384868704368283et_int
@ ( image_nat_set_int
@ ^ [X4: nat] : ( minus_minus_set_int @ ( A4 @ X4 ) @ B5 )
@ C3 ) ) ) ) ) ).
% INT_extend_simps(3)
thf(fact_2703_INT__extend__simps_I3_J,axiom,
! [C3: set_Product_unit,A4: product_unit > set_nat,B5: set_nat] :
( ( ( C3 = bot_bo3957492148770167129t_unit )
=> ( ( minus_minus_set_nat @ ( comple7806235888213564991et_nat @ ( image_8173849991827986510et_nat @ A4 @ C3 ) ) @ B5 )
= ( minus_minus_set_nat @ top_top_set_nat @ B5 ) ) )
& ( ( C3 != bot_bo3957492148770167129t_unit )
=> ( ( minus_minus_set_nat @ ( comple7806235888213564991et_nat @ ( image_8173849991827986510et_nat @ A4 @ C3 ) ) @ B5 )
= ( comple7806235888213564991et_nat
@ ( image_8173849991827986510et_nat
@ ^ [X4: product_unit] : ( minus_minus_set_nat @ ( A4 @ X4 ) @ B5 )
@ C3 ) ) ) ) ) ).
% INT_extend_simps(3)
thf(fact_2704_INT__extend__simps_I4_J,axiom,
! [C3: set_Product_unit,A4: set_nat,B5: product_unit > set_nat] :
( ( ( C3 = bot_bo3957492148770167129t_unit )
=> ( ( minus_minus_set_nat @ A4 @ ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ B5 @ C3 ) ) )
= A4 ) )
& ( ( C3 != bot_bo3957492148770167129t_unit )
=> ( ( minus_minus_set_nat @ A4 @ ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ B5 @ C3 ) ) )
= ( comple7806235888213564991et_nat
@ ( image_8173849991827986510et_nat
@ ^ [X4: product_unit] : ( minus_minus_set_nat @ A4 @ ( B5 @ X4 ) )
@ C3 ) ) ) ) ) ).
% INT_extend_simps(4)
thf(fact_2705_INT__extend__simps_I4_J,axiom,
! [C3: set_Pr4329608150637261639at_nat,A4: set_nat,B5: produc3843707927480180839at_nat > set_nat] :
( ( ( C3 = bot_bo228742789529271731at_nat )
=> ( ( minus_minus_set_nat @ A4 @ ( comple7399068483239264473et_nat @ ( image_3160451284896210088et_nat @ B5 @ C3 ) ) )
= A4 ) )
& ( ( C3 != bot_bo228742789529271731at_nat )
=> ( ( minus_minus_set_nat @ A4 @ ( comple7399068483239264473et_nat @ ( image_3160451284896210088et_nat @ B5 @ C3 ) ) )
= ( comple7806235888213564991et_nat
@ ( image_3160451284896210088et_nat
@ ^ [X4: produc3843707927480180839at_nat] : ( minus_minus_set_nat @ A4 @ ( B5 @ X4 ) )
@ C3 ) ) ) ) ) ).
% INT_extend_simps(4)
thf(fact_2706_INT__extend__simps_I4_J,axiom,
! [C3: set_Pr1261947904930325089at_nat,A4: set_nat,B5: product_prod_nat_nat > set_nat] :
( ( ( C3 = bot_bo2099793752762293965at_nat )
=> ( ( minus_minus_set_nat @ A4 @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ B5 @ C3 ) ) )
= A4 ) )
& ( ( C3 != bot_bo2099793752762293965at_nat )
=> ( ( minus_minus_set_nat @ A4 @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ B5 @ C3 ) ) )
= ( comple7806235888213564991et_nat
@ ( image_15824709712370754et_nat
@ ^ [X4: product_prod_nat_nat] : ( minus_minus_set_nat @ A4 @ ( B5 @ X4 ) )
@ C3 ) ) ) ) ) ).
% INT_extend_simps(4)
thf(fact_2707_INT__extend__simps_I4_J,axiom,
! [C3: set_o,A4: set_nat,B5: $o > set_nat] :
( ( ( C3 = bot_bot_set_o )
=> ( ( minus_minus_set_nat @ A4 @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B5 @ C3 ) ) )
= A4 ) )
& ( ( C3 != bot_bot_set_o )
=> ( ( minus_minus_set_nat @ A4 @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B5 @ C3 ) ) )
= ( comple7806235888213564991et_nat
@ ( image_o_set_nat
@ ^ [X4: $o] : ( minus_minus_set_nat @ A4 @ ( B5 @ X4 ) )
@ C3 ) ) ) ) ) ).
% INT_extend_simps(4)
thf(fact_2708_INT__extend__simps_I4_J,axiom,
! [C3: set_nat,A4: set_nat,B5: nat > set_nat] :
( ( ( C3 = bot_bot_set_nat )
=> ( ( minus_minus_set_nat @ A4 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ C3 ) ) )
= A4 ) )
& ( ( C3 != bot_bot_set_nat )
=> ( ( minus_minus_set_nat @ A4 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ C3 ) ) )
= ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [X4: nat] : ( minus_minus_set_nat @ A4 @ ( B5 @ X4 ) )
@ C3 ) ) ) ) ) ).
% INT_extend_simps(4)
thf(fact_2709_linear__order__on__singleton,axiom,
! [X2: set_Pr4329608150637261639at_nat] : ( order_1878278401035043452at_nat @ ( insert2290795007606087959at_nat @ X2 @ bot_bo1534825418459847827at_nat ) @ ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ X2 @ X2 ) @ bot_bo4948859079157340979at_nat ) ) ).
% linear_order_on_singleton
thf(fact_2710_linear__order__on__singleton,axiom,
! [X2: int] : ( order_4471489696718655927on_int @ ( insert_int @ X2 @ bot_bot_set_int ) @ ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ X2 @ X2 ) @ bot_bo1796632182523588997nt_int ) ) ).
% linear_order_on_singleton
thf(fact_2711_linear__order__on__singleton,axiom,
! [X2: set_Pr1261947904930325089at_nat] : ( order_5942792088423767318at_nat @ ( insert9200635055090092081at_nat @ X2 @ bot_bo3083307316010499117at_nat ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ X2 @ X2 ) @ bot_bo228742789529271731at_nat ) ) ).
% linear_order_on_singleton
thf(fact_2712_linear__order__on__singleton,axiom,
! [X2: product_unit] : ( order_888282469287196226t_unit @ ( insert_Product_unit @ X2 @ bot_bo3957492148770167129t_unit ) @ ( insert4616891024742550839t_unit @ ( produc5692694829384537815t_unit @ X2 @ X2 ) @ bot_bo1654022762305692595t_unit ) ) ).
% linear_order_on_singleton
thf(fact_2713_linear__order__on__singleton,axiom,
! [X2: produc3843707927480180839at_nat] : ( order_2711500226769404828at_nat @ ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat ) @ ( insert398606708087465655at_nat @ ( produc2335810615602371415at_nat @ X2 @ X2 ) @ bot_bo7193250159860175667at_nat ) ) ).
% linear_order_on_singleton
thf(fact_2714_linear__order__on__singleton,axiom,
! [X2: product_prod_nat_nat] : ( order_7480980593162297270at_nat @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) @ ( insert5050368324300391991at_nat @ ( produc6161850002892822231at_nat @ X2 @ X2 ) @ bot_bo5327735625951526323at_nat ) ) ).
% linear_order_on_singleton
thf(fact_2715_linear__order__on__singleton,axiom,
! [X2: $o] : ( order_6238756238976269133r_on_o @ ( insert_o @ X2 @ bot_bot_set_o ) @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ X2 @ X2 ) @ bot_bo7073875226086086771od_o_o ) ) ).
% linear_order_on_singleton
thf(fact_2716_linear__order__on__singleton,axiom,
! [X2: nat] : ( order_4473980167227706203on_nat @ ( insert_nat @ X2 @ bot_bot_set_nat ) @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X2 @ X2 ) @ bot_bo2099793752762293965at_nat ) ) ).
% linear_order_on_singleton
thf(fact_2717_total__on__singleton,axiom,
! [X2: set_Pr4329608150637261639at_nat] : ( total_5117575981256876747at_nat @ ( insert2290795007606087959at_nat @ X2 @ bot_bo1534825418459847827at_nat ) @ ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ X2 @ X2 ) @ bot_bo4948859079157340979at_nat ) ) ).
% total_on_singleton
thf(fact_2718_total__on__singleton,axiom,
! [X2: int] : ( total_on_int @ ( insert_int @ X2 @ bot_bot_set_int ) @ ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ X2 @ X2 ) @ bot_bo1796632182523588997nt_int ) ) ).
% total_on_singleton
thf(fact_2719_total__on__singleton,axiom,
! [X2: set_Pr1261947904930325089at_nat] : ( total_7212417256332265445at_nat @ ( insert9200635055090092081at_nat @ X2 @ bot_bo3083307316010499117at_nat ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ X2 @ X2 ) @ bot_bo228742789529271731at_nat ) ) ).
% total_on_singleton
thf(fact_2720_total__on__singleton,axiom,
! [X2: product_unit] : ( total_5665700600833547665t_unit @ ( insert_Product_unit @ X2 @ bot_bo3957492148770167129t_unit ) @ ( insert4616891024742550839t_unit @ ( produc5692694829384537815t_unit @ X2 @ X2 ) @ bot_bo1654022762305692595t_unit ) ) ).
% total_on_singleton
thf(fact_2721_total__on__singleton,axiom,
! [X2: produc3843707927480180839at_nat] : ( total_1513685709984341995at_nat @ ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat ) @ ( insert398606708087465655at_nat @ ( produc2335810615602371415at_nat @ X2 @ X2 ) @ bot_bo7193250159860175667at_nat ) ) ).
% total_on_singleton
thf(fact_2722_total__on__singleton,axiom,
! [X2: product_prod_nat_nat] : ( total_3592101749530773125at_nat @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) @ ( insert5050368324300391991at_nat @ ( produc6161850002892822231at_nat @ X2 @ X2 ) @ bot_bo5327735625951526323at_nat ) ) ).
% total_on_singleton
thf(fact_2723_total__on__singleton,axiom,
! [X2: $o] : ( total_on_o @ ( insert_o @ X2 @ bot_bot_set_o ) @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ X2 @ X2 ) @ bot_bo7073875226086086771od_o_o ) ) ).
% total_on_singleton
thf(fact_2724_total__on__singleton,axiom,
! [X2: nat] : ( total_on_nat @ ( insert_nat @ X2 @ bot_bot_set_nat ) @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X2 @ X2 ) @ bot_bo2099793752762293965at_nat ) ) ).
% total_on_singleton
thf(fact_2725_relation__of__def,axiom,
( order_8308734813930759573at_nat
= ( ^ [P4: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o,A6: set_se5479201105039950375at_nat] :
( collec7872792499061297490at_nat
@ ( produc9020218426428501292_nat_o
@ ^ [A5: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A5 @ B4 )
@ ( produc6258361011560675592at_nat @ A6
@ ^ [Uu: set_Pr4329608150637261639at_nat] : A6 ) )
& ( P4 @ A5 @ B4 ) ) ) ) ) ) ).
% relation_of_def
thf(fact_2726_relation__of__def,axiom,
( order_8809246210994205103at_nat
= ( ^ [P4: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o,A6: set_se7855581050983116737at_nat] :
( collec6321179662152712658at_nat
@ ( produc410239310623530412_nat_o
@ ^ [A5: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A5 @ B4 )
@ ( produc7726050920695093128at_nat @ A6
@ ^ [Uu: set_Pr1261947904930325089at_nat] : A6 ) )
& ( P4 @ A5 @ B4 ) ) ) ) ) ) ).
% relation_of_def
thf(fact_2727_relation__of__def,axiom,
( order_1962693441708344834of_nat
= ( ^ [P4: nat > nat > $o,A6: set_nat] :
( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [A5: nat,B4: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A5 @ B4 )
@ ( produc457027306803732586at_nat @ A6
@ ^ [Uu: nat] : A6 ) )
& ( P4 @ A5 @ B4 ) ) ) ) ) ) ).
% relation_of_def
thf(fact_2728_relation__of__def,axiom,
( order_1960202971199294558of_int
= ( ^ [P4: int > int > $o,A6: set_int] :
( collec213857154873943460nt_int
@ ( produc4947309494688390418_int_o
@ ^ [A5: int,B4: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A5 @ B4 )
@ ( produc1453890548195736866nt_int @ A6
@ ^ [Uu: int] : A6 ) )
& ( P4 @ A5 @ B4 ) ) ) ) ) ) ).
% relation_of_def
thf(fact_2729_insert__Times__insert,axiom,
! [A: nat,A4: set_nat,B: product_unit,B5: set_Product_unit] :
( ( produc5506539147108004339t_unit @ ( insert_nat @ A @ A4 )
@ ^ [Uu: nat] : ( insert_Product_unit @ B @ B5 ) )
= ( insert6804906068341198690t_unit @ ( produc3434450474625521892t_unit @ A @ B )
@ ( sup_su4209953369189666580t_unit
@ ( produc5506539147108004339t_unit @ A4
@ ^ [Uu: nat] : ( insert_Product_unit @ B @ B5 ) )
@ ( produc5506539147108004339t_unit @ ( insert_nat @ A @ A4 )
@ ^ [Uu: nat] : B5 ) ) ) ) ).
% insert_Times_insert
thf(fact_2730_insert__Times__insert,axiom,
! [A: nat,A4: set_nat,B: $o,B5: set_o] :
( ( product_Sigma_nat_o @ ( insert_nat @ A @ A4 )
@ ^ [Uu: nat] : ( insert_o @ B @ B5 ) )
= ( insert2233377226759017069_nat_o @ ( product_Pair_nat_o @ A @ B )
@ ( sup_su503371503184139039_nat_o
@ ( product_Sigma_nat_o @ A4
@ ^ [Uu: nat] : ( insert_o @ B @ B5 ) )
@ ( product_Sigma_nat_o @ ( insert_nat @ A @ A4 )
@ ^ [Uu: nat] : B5 ) ) ) ) ).
% insert_Times_insert
thf(fact_2731_insert__Times__insert,axiom,
! [A: product_unit,A4: set_Product_unit,B: nat,B5: set_nat] :
( ( produc6875377002296012693it_nat @ ( insert_Product_unit @ A @ A4 )
@ ^ [Uu: product_unit] : ( insert_nat @ B @ B5 ) )
= ( insert6879346373837673660it_nat @ ( produc4803288329813530246it_nat @ A @ B )
@ ( sup_su1639320892072265582it_nat
@ ( produc6875377002296012693it_nat @ A4
@ ^ [Uu: product_unit] : ( insert_nat @ B @ B5 ) )
@ ( produc6875377002296012693it_nat @ ( insert_Product_unit @ A @ A4 )
@ ^ [Uu: product_unit] : B5 ) ) ) ) ).
% insert_Times_insert
thf(fact_2732_insert__Times__insert,axiom,
! [A: product_unit,A4: set_Product_unit,B: product_unit,B5: set_Product_unit] :
( ( produc6121607364029222536t_unit @ ( insert_Product_unit @ A @ A4 )
@ ^ [Uu: product_unit] : ( insert_Product_unit @ B @ B5 ) )
= ( insert4616891024742550839t_unit @ ( produc5692694829384537815t_unit @ A @ B )
@ ( sup_su6481247232241170203t_unit
@ ( produc6121607364029222536t_unit @ A4
@ ^ [Uu: product_unit] : ( insert_Product_unit @ B @ B5 ) )
@ ( produc6121607364029222536t_unit @ ( insert_Product_unit @ A @ A4 )
@ ^ [Uu: product_unit] : B5 ) ) ) ) ).
% insert_Times_insert
thf(fact_2733_insert__Times__insert,axiom,
! [A: product_unit,A4: set_Product_unit,B: $o,B5: set_o] :
( ( produc4996980317674386579unit_o @ ( insert_Product_unit @ A @ A4 )
@ ^ [Uu: product_unit] : ( insert_o @ B @ B5 ) )
= ( insert3586912282507785154unit_o @ ( produc8487917018914037602unit_o @ A @ B )
@ ( sup_su2851927382625014182unit_o
@ ( produc4996980317674386579unit_o @ A4
@ ^ [Uu: product_unit] : ( insert_o @ B @ B5 ) )
@ ( produc4996980317674386579unit_o @ ( insert_Product_unit @ A @ A4 )
@ ^ [Uu: product_unit] : B5 ) ) ) ) ).
% insert_Times_insert
thf(fact_2734_insert__Times__insert,axiom,
! [A: $o,A4: set_o,B: nat,B5: set_nat] :
( ( product_Sigma_o_nat @ ( insert_o @ A @ A4 )
@ ^ [Uu: $o] : ( insert_nat @ B @ B5 ) )
= ( insert7948214739558299847_o_nat @ ( product_Pair_o_nat @ A @ B )
@ ( sup_su8679140417860611193_o_nat
@ ( product_Sigma_o_nat @ A4
@ ^ [Uu: $o] : ( insert_nat @ B @ B5 ) )
@ ( product_Sigma_o_nat @ ( insert_o @ A @ A4 )
@ ^ [Uu: $o] : B5 ) ) ) ) ).
% insert_Times_insert
thf(fact_2735_insert__Times__insert,axiom,
! [A: $o,A4: set_o,B: product_unit,B5: set_Product_unit] :
( ( produc411009820408249917t_unit @ ( insert_o @ A @ A4 )
@ ^ [Uu: $o] : ( insert_Product_unit @ B @ B5 ) )
= ( insert9112848208840649964t_unit @ ( produc3901946521647900940t_unit @ A @ B )
@ ( sup_su4058753791541417424t_unit
@ ( produc411009820408249917t_unit @ A4
@ ^ [Uu: $o] : ( insert_Product_unit @ B @ B5 ) )
@ ( produc411009820408249917t_unit @ ( insert_o @ A @ A4 )
@ ^ [Uu: $o] : B5 ) ) ) ) ).
% insert_Times_insert
thf(fact_2736_insert__Times__insert,axiom,
! [A: $o,A4: set_o,B: $o,B5: set_o] :
( ( product_Sigma_o_o @ ( insert_o @ A @ A4 )
@ ^ [Uu: $o] : ( insert_o @ B @ B5 ) )
= ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ A @ B )
@ ( sup_su5769328420594410459od_o_o
@ ( product_Sigma_o_o @ A4
@ ^ [Uu: $o] : ( insert_o @ B @ B5 ) )
@ ( product_Sigma_o_o @ ( insert_o @ A @ A4 )
@ ^ [Uu: $o] : B5 ) ) ) ) ).
% insert_Times_insert
thf(fact_2737_insert__Times__insert,axiom,
! [A: nat,A4: set_nat,B: num,B5: set_num] :
( ( produc6237731129814287028at_num @ ( insert_nat @ A @ A4 )
@ ^ [Uu: nat] : ( insert_num @ B @ B5 ) )
= ( insert8920054152555992091at_num @ ( product_Pair_nat_num @ A @ B )
@ ( sup_su2042722026077122175at_num
@ ( produc6237731129814287028at_num @ A4
@ ^ [Uu: nat] : ( insert_num @ B @ B5 ) )
@ ( produc6237731129814287028at_num @ ( insert_nat @ A @ A4 )
@ ^ [Uu: nat] : B5 ) ) ) ) ).
% insert_Times_insert
thf(fact_2738_insert__Times__insert,axiom,
! [A: int,A4: set_int,B: int,B5: set_int] :
( ( produc1453890548195736866nt_int @ ( insert_int @ A @ A4 )
@ ^ [Uu: int] : ( insert_int @ B @ B5 ) )
= ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ A @ B )
@ ( sup_su6024340866399070445nt_int
@ ( produc1453890548195736866nt_int @ A4
@ ^ [Uu: int] : ( insert_int @ B @ B5 ) )
@ ( produc1453890548195736866nt_int @ ( insert_int @ A @ A4 )
@ ^ [Uu: int] : B5 ) ) ) ) ).
% insert_Times_insert
thf(fact_2739_INT__simps_I2_J,axiom,
! [C3: set_Product_unit,A4: set_char,B5: product_unit > set_char] :
( ( ( C3 = bot_bo3957492148770167129t_unit )
=> ( ( comple7965528767983648192t_char
@ ( image_5325685772115642575t_char
@ ^ [X4: product_unit] : ( inf_inf_set_char @ A4 @ ( B5 @ X4 ) )
@ C3 ) )
= top_top_set_char ) )
& ( ( C3 != bot_bo3957492148770167129t_unit )
=> ( ( comple7965528767983648192t_char
@ ( image_5325685772115642575t_char
@ ^ [X4: product_unit] : ( inf_inf_set_char @ A4 @ ( B5 @ X4 ) )
@ C3 ) )
= ( inf_inf_set_char @ A4 @ ( comple7965528767983648192t_char @ ( image_5325685772115642575t_char @ B5 @ C3 ) ) ) ) ) ) ).
% INT_simps(2)
thf(fact_2740_INT__simps_I2_J,axiom,
! [C3: set_o,A4: set_char,B5: $o > set_char] :
( ( ( C3 = bot_bot_set_o )
=> ( ( comple7965528767983648192t_char
@ ( image_o_set_char
@ ^ [X4: $o] : ( inf_inf_set_char @ A4 @ ( B5 @ X4 ) )
@ C3 ) )
= top_top_set_char ) )
& ( ( C3 != bot_bot_set_o )
=> ( ( comple7965528767983648192t_char
@ ( image_o_set_char
@ ^ [X4: $o] : ( inf_inf_set_char @ A4 @ ( B5 @ X4 ) )
@ C3 ) )
= ( inf_inf_set_char @ A4 @ ( comple7965528767983648192t_char @ ( image_o_set_char @ B5 @ C3 ) ) ) ) ) ) ).
% INT_simps(2)
thf(fact_2741_INT__simps_I2_J,axiom,
! [C3: set_nat,A4: set_char,B5: nat > set_char] :
( ( ( C3 = bot_bot_set_nat )
=> ( ( comple7965528767983648192t_char
@ ( image_nat_set_char
@ ^ [X4: nat] : ( inf_inf_set_char @ A4 @ ( B5 @ X4 ) )
@ C3 ) )
= top_top_set_char ) )
& ( ( C3 != bot_bot_set_nat )
=> ( ( comple7965528767983648192t_char
@ ( image_nat_set_char
@ ^ [X4: nat] : ( inf_inf_set_char @ A4 @ ( B5 @ X4 ) )
@ C3 ) )
= ( inf_inf_set_char @ A4 @ ( comple7965528767983648192t_char @ ( image_nat_set_char @ B5 @ C3 ) ) ) ) ) ) ).
% INT_simps(2)
thf(fact_2742_INT__simps_I2_J,axiom,
! [C3: set_Product_unit,A4: set_rat,B5: product_unit > set_rat] :
( ( ( C3 = bot_bo3957492148770167129t_unit )
=> ( ( comple4298007329820168263et_rat
@ ( image_4665621433434589782et_rat
@ ^ [X4: product_unit] : ( inf_inf_set_rat @ A4 @ ( B5 @ X4 ) )
@ C3 ) )
= top_top_set_rat ) )
& ( ( C3 != bot_bo3957492148770167129t_unit )
=> ( ( comple4298007329820168263et_rat
@ ( image_4665621433434589782et_rat
@ ^ [X4: product_unit] : ( inf_inf_set_rat @ A4 @ ( B5 @ X4 ) )
@ C3 ) )
= ( inf_inf_set_rat @ A4 @ ( comple4298007329820168263et_rat @ ( image_4665621433434589782et_rat @ B5 @ C3 ) ) ) ) ) ) ).
% INT_simps(2)
thf(fact_2743_INT__simps_I2_J,axiom,
! [C3: set_o,A4: set_rat,B5: $o > set_rat] :
( ( ( C3 = bot_bot_set_o )
=> ( ( comple4298007329820168263et_rat
@ ( image_o_set_rat
@ ^ [X4: $o] : ( inf_inf_set_rat @ A4 @ ( B5 @ X4 ) )
@ C3 ) )
= top_top_set_rat ) )
& ( ( C3 != bot_bot_set_o )
=> ( ( comple4298007329820168263et_rat
@ ( image_o_set_rat
@ ^ [X4: $o] : ( inf_inf_set_rat @ A4 @ ( B5 @ X4 ) )
@ C3 ) )
= ( inf_inf_set_rat @ A4 @ ( comple4298007329820168263et_rat @ ( image_o_set_rat @ B5 @ C3 ) ) ) ) ) ) ).
% INT_simps(2)
thf(fact_2744_INT__simps_I2_J,axiom,
! [C3: set_nat,A4: set_rat,B5: nat > set_rat] :
( ( ( C3 = bot_bot_set_nat )
=> ( ( comple4298007329820168263et_rat
@ ( image_nat_set_rat
@ ^ [X4: nat] : ( inf_inf_set_rat @ A4 @ ( B5 @ X4 ) )
@ C3 ) )
= top_top_set_rat ) )
& ( ( C3 != bot_bot_set_nat )
=> ( ( comple4298007329820168263et_rat
@ ( image_nat_set_rat
@ ^ [X4: nat] : ( inf_inf_set_rat @ A4 @ ( B5 @ X4 ) )
@ C3 ) )
= ( inf_inf_set_rat @ A4 @ ( comple4298007329820168263et_rat @ ( image_nat_set_rat @ B5 @ C3 ) ) ) ) ) ) ).
% INT_simps(2)
thf(fact_2745_INT__simps_I2_J,axiom,
! [C3: set_Product_unit,A4: set_int,B5: product_unit > set_int] :
( ( ( C3 = bot_bo3957492148770167129t_unit )
=> ( ( comple3628384868704368283et_int
@ ( image_3995998972318789802et_int
@ ^ [X4: product_unit] : ( inf_inf_set_int @ A4 @ ( B5 @ X4 ) )
@ C3 ) )
= top_top_set_int ) )
& ( ( C3 != bot_bo3957492148770167129t_unit )
=> ( ( comple3628384868704368283et_int
@ ( image_3995998972318789802et_int
@ ^ [X4: product_unit] : ( inf_inf_set_int @ A4 @ ( B5 @ X4 ) )
@ C3 ) )
= ( inf_inf_set_int @ A4 @ ( comple3628384868704368283et_int @ ( image_3995998972318789802et_int @ B5 @ C3 ) ) ) ) ) ) ).
% INT_simps(2)
thf(fact_2746_INT__simps_I2_J,axiom,
! [C3: set_o,A4: set_int,B5: $o > set_int] :
( ( ( C3 = bot_bot_set_o )
=> ( ( comple3628384868704368283et_int
@ ( image_o_set_int
@ ^ [X4: $o] : ( inf_inf_set_int @ A4 @ ( B5 @ X4 ) )
@ C3 ) )
= top_top_set_int ) )
& ( ( C3 != bot_bot_set_o )
=> ( ( comple3628384868704368283et_int
@ ( image_o_set_int
@ ^ [X4: $o] : ( inf_inf_set_int @ A4 @ ( B5 @ X4 ) )
@ C3 ) )
= ( inf_inf_set_int @ A4 @ ( comple3628384868704368283et_int @ ( image_o_set_int @ B5 @ C3 ) ) ) ) ) ) ).
% INT_simps(2)
thf(fact_2747_INT__simps_I2_J,axiom,
! [C3: set_nat,A4: set_int,B5: nat > set_int] :
( ( ( C3 = bot_bot_set_nat )
=> ( ( comple3628384868704368283et_int
@ ( image_nat_set_int
@ ^ [X4: nat] : ( inf_inf_set_int @ A4 @ ( B5 @ X4 ) )
@ C3 ) )
= top_top_set_int ) )
& ( ( C3 != bot_bot_set_nat )
=> ( ( comple3628384868704368283et_int
@ ( image_nat_set_int
@ ^ [X4: nat] : ( inf_inf_set_int @ A4 @ ( B5 @ X4 ) )
@ C3 ) )
= ( inf_inf_set_int @ A4 @ ( comple3628384868704368283et_int @ ( image_nat_set_int @ B5 @ C3 ) ) ) ) ) ) ).
% INT_simps(2)
thf(fact_2748_INT__simps_I2_J,axiom,
! [C3: set_Product_unit,A4: set_nat,B5: product_unit > set_nat] :
( ( ( C3 = bot_bo3957492148770167129t_unit )
=> ( ( comple7806235888213564991et_nat
@ ( image_8173849991827986510et_nat
@ ^ [X4: product_unit] : ( inf_inf_set_nat @ A4 @ ( B5 @ X4 ) )
@ C3 ) )
= top_top_set_nat ) )
& ( ( C3 != bot_bo3957492148770167129t_unit )
=> ( ( comple7806235888213564991et_nat
@ ( image_8173849991827986510et_nat
@ ^ [X4: product_unit] : ( inf_inf_set_nat @ A4 @ ( B5 @ X4 ) )
@ C3 ) )
= ( inf_inf_set_nat @ A4 @ ( comple7806235888213564991et_nat @ ( image_8173849991827986510et_nat @ B5 @ C3 ) ) ) ) ) ) ).
% INT_simps(2)
thf(fact_2749_INT__simps_I1_J,axiom,
! [C3: set_Product_unit,A4: product_unit > set_char,B5: set_char] :
( ( ( C3 = bot_bo3957492148770167129t_unit )
=> ( ( comple7965528767983648192t_char
@ ( image_5325685772115642575t_char
@ ^ [X4: product_unit] : ( inf_inf_set_char @ ( A4 @ X4 ) @ B5 )
@ C3 ) )
= top_top_set_char ) )
& ( ( C3 != bot_bo3957492148770167129t_unit )
=> ( ( comple7965528767983648192t_char
@ ( image_5325685772115642575t_char
@ ^ [X4: product_unit] : ( inf_inf_set_char @ ( A4 @ X4 ) @ B5 )
@ C3 ) )
= ( inf_inf_set_char @ ( comple7965528767983648192t_char @ ( image_5325685772115642575t_char @ A4 @ C3 ) ) @ B5 ) ) ) ) ).
% INT_simps(1)
thf(fact_2750_INT__simps_I1_J,axiom,
! [C3: set_o,A4: $o > set_char,B5: set_char] :
( ( ( C3 = bot_bot_set_o )
=> ( ( comple7965528767983648192t_char
@ ( image_o_set_char
@ ^ [X4: $o] : ( inf_inf_set_char @ ( A4 @ X4 ) @ B5 )
@ C3 ) )
= top_top_set_char ) )
& ( ( C3 != bot_bot_set_o )
=> ( ( comple7965528767983648192t_char
@ ( image_o_set_char
@ ^ [X4: $o] : ( inf_inf_set_char @ ( A4 @ X4 ) @ B5 )
@ C3 ) )
= ( inf_inf_set_char @ ( comple7965528767983648192t_char @ ( image_o_set_char @ A4 @ C3 ) ) @ B5 ) ) ) ) ).
% INT_simps(1)
thf(fact_2751_INT__simps_I1_J,axiom,
! [C3: set_nat,A4: nat > set_char,B5: set_char] :
( ( ( C3 = bot_bot_set_nat )
=> ( ( comple7965528767983648192t_char
@ ( image_nat_set_char
@ ^ [X4: nat] : ( inf_inf_set_char @ ( A4 @ X4 ) @ B5 )
@ C3 ) )
= top_top_set_char ) )
& ( ( C3 != bot_bot_set_nat )
=> ( ( comple7965528767983648192t_char
@ ( image_nat_set_char
@ ^ [X4: nat] : ( inf_inf_set_char @ ( A4 @ X4 ) @ B5 )
@ C3 ) )
= ( inf_inf_set_char @ ( comple7965528767983648192t_char @ ( image_nat_set_char @ A4 @ C3 ) ) @ B5 ) ) ) ) ).
% INT_simps(1)
thf(fact_2752_INT__simps_I1_J,axiom,
! [C3: set_Product_unit,A4: product_unit > set_rat,B5: set_rat] :
( ( ( C3 = bot_bo3957492148770167129t_unit )
=> ( ( comple4298007329820168263et_rat
@ ( image_4665621433434589782et_rat
@ ^ [X4: product_unit] : ( inf_inf_set_rat @ ( A4 @ X4 ) @ B5 )
@ C3 ) )
= top_top_set_rat ) )
& ( ( C3 != bot_bo3957492148770167129t_unit )
=> ( ( comple4298007329820168263et_rat
@ ( image_4665621433434589782et_rat
@ ^ [X4: product_unit] : ( inf_inf_set_rat @ ( A4 @ X4 ) @ B5 )
@ C3 ) )
= ( inf_inf_set_rat @ ( comple4298007329820168263et_rat @ ( image_4665621433434589782et_rat @ A4 @ C3 ) ) @ B5 ) ) ) ) ).
% INT_simps(1)
thf(fact_2753_INT__simps_I1_J,axiom,
! [C3: set_o,A4: $o > set_rat,B5: set_rat] :
( ( ( C3 = bot_bot_set_o )
=> ( ( comple4298007329820168263et_rat
@ ( image_o_set_rat
@ ^ [X4: $o] : ( inf_inf_set_rat @ ( A4 @ X4 ) @ B5 )
@ C3 ) )
= top_top_set_rat ) )
& ( ( C3 != bot_bot_set_o )
=> ( ( comple4298007329820168263et_rat
@ ( image_o_set_rat
@ ^ [X4: $o] : ( inf_inf_set_rat @ ( A4 @ X4 ) @ B5 )
@ C3 ) )
= ( inf_inf_set_rat @ ( comple4298007329820168263et_rat @ ( image_o_set_rat @ A4 @ C3 ) ) @ B5 ) ) ) ) ).
% INT_simps(1)
thf(fact_2754_INT__simps_I1_J,axiom,
! [C3: set_nat,A4: nat > set_rat,B5: set_rat] :
( ( ( C3 = bot_bot_set_nat )
=> ( ( comple4298007329820168263et_rat
@ ( image_nat_set_rat
@ ^ [X4: nat] : ( inf_inf_set_rat @ ( A4 @ X4 ) @ B5 )
@ C3 ) )
= top_top_set_rat ) )
& ( ( C3 != bot_bot_set_nat )
=> ( ( comple4298007329820168263et_rat
@ ( image_nat_set_rat
@ ^ [X4: nat] : ( inf_inf_set_rat @ ( A4 @ X4 ) @ B5 )
@ C3 ) )
= ( inf_inf_set_rat @ ( comple4298007329820168263et_rat @ ( image_nat_set_rat @ A4 @ C3 ) ) @ B5 ) ) ) ) ).
% INT_simps(1)
thf(fact_2755_INT__simps_I1_J,axiom,
! [C3: set_Product_unit,A4: product_unit > set_int,B5: set_int] :
( ( ( C3 = bot_bo3957492148770167129t_unit )
=> ( ( comple3628384868704368283et_int
@ ( image_3995998972318789802et_int
@ ^ [X4: product_unit] : ( inf_inf_set_int @ ( A4 @ X4 ) @ B5 )
@ C3 ) )
= top_top_set_int ) )
& ( ( C3 != bot_bo3957492148770167129t_unit )
=> ( ( comple3628384868704368283et_int
@ ( image_3995998972318789802et_int
@ ^ [X4: product_unit] : ( inf_inf_set_int @ ( A4 @ X4 ) @ B5 )
@ C3 ) )
= ( inf_inf_set_int @ ( comple3628384868704368283et_int @ ( image_3995998972318789802et_int @ A4 @ C3 ) ) @ B5 ) ) ) ) ).
% INT_simps(1)
thf(fact_2756_INT__simps_I1_J,axiom,
! [C3: set_o,A4: $o > set_int,B5: set_int] :
( ( ( C3 = bot_bot_set_o )
=> ( ( comple3628384868704368283et_int
@ ( image_o_set_int
@ ^ [X4: $o] : ( inf_inf_set_int @ ( A4 @ X4 ) @ B5 )
@ C3 ) )
= top_top_set_int ) )
& ( ( C3 != bot_bot_set_o )
=> ( ( comple3628384868704368283et_int
@ ( image_o_set_int
@ ^ [X4: $o] : ( inf_inf_set_int @ ( A4 @ X4 ) @ B5 )
@ C3 ) )
= ( inf_inf_set_int @ ( comple3628384868704368283et_int @ ( image_o_set_int @ A4 @ C3 ) ) @ B5 ) ) ) ) ).
% INT_simps(1)
thf(fact_2757_INT__simps_I1_J,axiom,
! [C3: set_nat,A4: nat > set_int,B5: set_int] :
( ( ( C3 = bot_bot_set_nat )
=> ( ( comple3628384868704368283et_int
@ ( image_nat_set_int
@ ^ [X4: nat] : ( inf_inf_set_int @ ( A4 @ X4 ) @ B5 )
@ C3 ) )
= top_top_set_int ) )
& ( ( C3 != bot_bot_set_nat )
=> ( ( comple3628384868704368283et_int
@ ( image_nat_set_int
@ ^ [X4: nat] : ( inf_inf_set_int @ ( A4 @ X4 ) @ B5 )
@ C3 ) )
= ( inf_inf_set_int @ ( comple3628384868704368283et_int @ ( image_nat_set_int @ A4 @ C3 ) ) @ B5 ) ) ) ) ).
% INT_simps(1)
thf(fact_2758_INT__simps_I1_J,axiom,
! [C3: set_Product_unit,A4: product_unit > set_nat,B5: set_nat] :
( ( ( C3 = bot_bo3957492148770167129t_unit )
=> ( ( comple7806235888213564991et_nat
@ ( image_8173849991827986510et_nat
@ ^ [X4: product_unit] : ( inf_inf_set_nat @ ( A4 @ X4 ) @ B5 )
@ C3 ) )
= top_top_set_nat ) )
& ( ( C3 != bot_bo3957492148770167129t_unit )
=> ( ( comple7806235888213564991et_nat
@ ( image_8173849991827986510et_nat
@ ^ [X4: product_unit] : ( inf_inf_set_nat @ ( A4 @ X4 ) @ B5 )
@ C3 ) )
= ( inf_inf_set_nat @ ( comple7806235888213564991et_nat @ ( image_8173849991827986510et_nat @ A4 @ C3 ) ) @ B5 ) ) ) ) ).
% INT_simps(1)
thf(fact_2759_cSUP__UNION,axiom,
! [A4: set_int,B5: int > set_int,F: int > int] :
( ( A4 != bot_bot_set_int )
=> ( ! [X: int] :
( ( member_int @ X @ A4 )
=> ( ( B5 @ X )
!= bot_bot_set_int ) )
=> ( ( condit2212336002400062152ve_int
@ ( comple3221217463730067765et_int
@ ( image_int_set_int
@ ^ [X4: int] : ( image_int_int @ F @ ( B5 @ X4 ) )
@ A4 ) ) )
=> ( ( complete_Sup_Sup_int @ ( image_int_int @ F @ ( comple3221217463730067765et_int @ ( image_int_set_int @ B5 @ A4 ) ) ) )
= ( complete_Sup_Sup_int
@ ( image_int_int
@ ^ [X4: int] : ( complete_Sup_Sup_int @ ( image_int_int @ F @ ( B5 @ X4 ) ) )
@ A4 ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_2760_cSUP__UNION,axiom,
! [A4: set_int,B5: int > set_Product_unit,F: product_unit > int] :
( ( A4 != bot_bot_set_int )
=> ( ! [X: int] :
( ( member_int @ X @ A4 )
=> ( ( B5 @ X )
!= bot_bo3957492148770167129t_unit ) )
=> ( ( condit2212336002400062152ve_int
@ ( comple3221217463730067765et_int
@ ( image_int_set_int
@ ^ [X4: int] : ( image_873079544045703924it_int @ F @ ( B5 @ X4 ) )
@ A4 ) ) )
=> ( ( complete_Sup_Sup_int @ ( image_873079544045703924it_int @ F @ ( comple3349607820504768346t_unit @ ( image_8658272795163236474t_unit @ B5 @ A4 ) ) ) )
= ( complete_Sup_Sup_int
@ ( image_int_int
@ ^ [X4: int] : ( complete_Sup_Sup_int @ ( image_873079544045703924it_int @ F @ ( B5 @ X4 ) ) )
@ A4 ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_2761_cSUP__UNION,axiom,
! [A4: set_int,B5: int > set_o,F: $o > int] :
( ( A4 != bot_bot_set_int )
=> ( ! [X: int] :
( ( member_int @ X @ A4 )
=> ( ( B5 @ X )
!= bot_bot_set_o ) )
=> ( ( condit2212336002400062152ve_int
@ ( comple3221217463730067765et_int
@ ( image_int_set_int
@ ^ [X4: int] : ( image_o_int @ F @ ( B5 @ X4 ) )
@ A4 ) ) )
=> ( ( complete_Sup_Sup_int @ ( image_o_int @ F @ ( comple90263536869209701_set_o @ ( image_int_set_o @ B5 @ A4 ) ) ) )
= ( complete_Sup_Sup_int
@ ( image_int_int
@ ^ [X4: int] : ( complete_Sup_Sup_int @ ( image_o_int @ F @ ( B5 @ X4 ) ) )
@ A4 ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_2762_cSUP__UNION,axiom,
! [A4: set_Product_unit,B5: product_unit > set_int,F: int > int] :
( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ! [X: product_unit] :
( ( member_Product_unit @ X @ A4 )
=> ( ( B5 @ X )
!= bot_bot_set_int ) )
=> ( ( condit2212336002400062152ve_int
@ ( comple3221217463730067765et_int
@ ( image_3995998972318789802et_int
@ ^ [X4: product_unit] : ( image_int_int @ F @ ( B5 @ X4 ) )
@ A4 ) ) )
=> ( ( complete_Sup_Sup_int @ ( image_int_int @ F @ ( comple3221217463730067765et_int @ ( image_3995998972318789802et_int @ B5 @ A4 ) ) ) )
= ( complete_Sup_Sup_int
@ ( image_873079544045703924it_int
@ ^ [X4: product_unit] : ( complete_Sup_Sup_int @ ( image_int_int @ F @ ( B5 @ X4 ) ) )
@ A4 ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_2763_cSUP__UNION,axiom,
! [A4: set_Product_unit,B5: product_unit > set_Product_unit,F: product_unit > int] :
( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ! [X: product_unit] :
( ( member_Product_unit @ X @ A4 )
=> ( ( B5 @ X )
!= bot_bo3957492148770167129t_unit ) )
=> ( ( condit2212336002400062152ve_int
@ ( comple3221217463730067765et_int
@ ( image_3995998972318789802et_int
@ ^ [X4: product_unit] : ( image_873079544045703924it_int @ F @ ( B5 @ X4 ) )
@ A4 ) ) )
=> ( ( complete_Sup_Sup_int @ ( image_873079544045703924it_int @ F @ ( comple3349607820504768346t_unit @ ( image_7778121751606633509t_unit @ B5 @ A4 ) ) ) )
= ( complete_Sup_Sup_int
@ ( image_873079544045703924it_int
@ ^ [X4: product_unit] : ( complete_Sup_Sup_int @ ( image_873079544045703924it_int @ F @ ( B5 @ X4 ) ) )
@ A4 ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_2764_cSUP__UNION,axiom,
! [A4: set_Product_unit,B5: product_unit > set_o,F: $o > int] :
( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ! [X: product_unit] :
( ( member_Product_unit @ X @ A4 )
=> ( ( B5 @ X )
!= bot_bot_set_o ) )
=> ( ( condit2212336002400062152ve_int
@ ( comple3221217463730067765et_int
@ ( image_3995998972318789802et_int
@ ^ [X4: product_unit] : ( image_o_int @ F @ ( B5 @ X4 ) )
@ A4 ) ) )
=> ( ( complete_Sup_Sup_int @ ( image_o_int @ F @ ( comple90263536869209701_set_o @ ( image_6930536745178679216_set_o @ B5 @ A4 ) ) ) )
= ( complete_Sup_Sup_int
@ ( image_873079544045703924it_int
@ ^ [X4: product_unit] : ( complete_Sup_Sup_int @ ( image_o_int @ F @ ( B5 @ X4 ) ) )
@ A4 ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_2765_cSUP__UNION,axiom,
! [A4: set_o,B5: $o > set_int,F: int > int] :
( ( A4 != bot_bot_set_o )
=> ( ! [X: $o] :
( ( member_o @ X @ A4 )
=> ( ( B5 @ X )
!= bot_bot_set_int ) )
=> ( ( condit2212336002400062152ve_int
@ ( comple3221217463730067765et_int
@ ( image_o_set_int
@ ^ [X4: $o] : ( image_int_int @ F @ ( B5 @ X4 ) )
@ A4 ) ) )
=> ( ( complete_Sup_Sup_int @ ( image_int_int @ F @ ( comple3221217463730067765et_int @ ( image_o_set_int @ B5 @ A4 ) ) ) )
= ( complete_Sup_Sup_int
@ ( image_o_int
@ ^ [X4: $o] : ( complete_Sup_Sup_int @ ( image_int_int @ F @ ( B5 @ X4 ) ) )
@ A4 ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_2766_cSUP__UNION,axiom,
! [A4: set_o,B5: $o > set_Product_unit,F: product_unit > int] :
( ( A4 != bot_bot_set_o )
=> ( ! [X: $o] :
( ( member_o @ X @ A4 )
=> ( ( B5 @ X )
!= bot_bo3957492148770167129t_unit ) )
=> ( ( condit2212336002400062152ve_int
@ ( comple3221217463730067765et_int
@ ( image_o_set_int
@ ^ [X4: $o] : ( image_873079544045703924it_int @ F @ ( B5 @ X4 ) )
@ A4 ) ) )
=> ( ( complete_Sup_Sup_int @ ( image_873079544045703924it_int @ F @ ( comple3349607820504768346t_unit @ ( image_663195431845683034t_unit @ B5 @ A4 ) ) ) )
= ( complete_Sup_Sup_int
@ ( image_o_int
@ ^ [X4: $o] : ( complete_Sup_Sup_int @ ( image_873079544045703924it_int @ F @ ( B5 @ X4 ) ) )
@ A4 ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_2767_cSUP__UNION,axiom,
! [A4: set_o,B5: $o > set_o,F: $o > int] :
( ( A4 != bot_bot_set_o )
=> ( ! [X: $o] :
( ( member_o @ X @ A4 )
=> ( ( B5 @ X )
!= bot_bot_set_o ) )
=> ( ( condit2212336002400062152ve_int
@ ( comple3221217463730067765et_int
@ ( image_o_set_int
@ ^ [X4: $o] : ( image_o_int @ F @ ( B5 @ X4 ) )
@ A4 ) ) )
=> ( ( complete_Sup_Sup_int @ ( image_o_int @ F @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B5 @ A4 ) ) ) )
= ( complete_Sup_Sup_int
@ ( image_o_int
@ ^ [X4: $o] : ( complete_Sup_Sup_int @ ( image_o_int @ F @ ( B5 @ X4 ) ) )
@ A4 ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_2768_cSUP__UNION,axiom,
! [A4: set_nat,B5: nat > set_int,F: int > int] :
( ( A4 != bot_bot_set_nat )
=> ( ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ( ( B5 @ X )
!= bot_bot_set_int ) )
=> ( ( condit2212336002400062152ve_int
@ ( comple3221217463730067765et_int
@ ( image_nat_set_int
@ ^ [X4: nat] : ( image_int_int @ F @ ( B5 @ X4 ) )
@ A4 ) ) )
=> ( ( complete_Sup_Sup_int @ ( image_int_int @ F @ ( comple3221217463730067765et_int @ ( image_nat_set_int @ B5 @ A4 ) ) ) )
= ( complete_Sup_Sup_int
@ ( image_nat_int
@ ^ [X4: nat] : ( complete_Sup_Sup_int @ ( image_int_int @ F @ ( B5 @ X4 ) ) )
@ A4 ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_2769_cINF__UNION,axiom,
! [A4: set_int,B5: int > set_int,F: int > int] :
( ( A4 != bot_bot_set_int )
=> ( ! [X: int] :
( ( member_int @ X @ A4 )
=> ( ( B5 @ X )
!= bot_bot_set_int ) )
=> ( ( condit1735850657277959132ow_int
@ ( comple3221217463730067765et_int
@ ( image_int_set_int
@ ^ [X4: int] : ( image_int_int @ F @ ( B5 @ X4 ) )
@ A4 ) ) )
=> ( ( complete_Inf_Inf_int @ ( image_int_int @ F @ ( comple3221217463730067765et_int @ ( image_int_set_int @ B5 @ A4 ) ) ) )
= ( complete_Inf_Inf_int
@ ( image_int_int
@ ^ [X4: int] : ( complete_Inf_Inf_int @ ( image_int_int @ F @ ( B5 @ X4 ) ) )
@ A4 ) ) ) ) ) ) ).
% cINF_UNION
thf(fact_2770_cINF__UNION,axiom,
! [A4: set_int,B5: int > set_Product_unit,F: product_unit > int] :
( ( A4 != bot_bot_set_int )
=> ( ! [X: int] :
( ( member_int @ X @ A4 )
=> ( ( B5 @ X )
!= bot_bo3957492148770167129t_unit ) )
=> ( ( condit1735850657277959132ow_int
@ ( comple3221217463730067765et_int
@ ( image_int_set_int
@ ^ [X4: int] : ( image_873079544045703924it_int @ F @ ( B5 @ X4 ) )
@ A4 ) ) )
=> ( ( complete_Inf_Inf_int @ ( image_873079544045703924it_int @ F @ ( comple3349607820504768346t_unit @ ( image_8658272795163236474t_unit @ B5 @ A4 ) ) ) )
= ( complete_Inf_Inf_int
@ ( image_int_int
@ ^ [X4: int] : ( complete_Inf_Inf_int @ ( image_873079544045703924it_int @ F @ ( B5 @ X4 ) ) )
@ A4 ) ) ) ) ) ) ).
% cINF_UNION
thf(fact_2771_cINF__UNION,axiom,
! [A4: set_int,B5: int > set_o,F: $o > int] :
( ( A4 != bot_bot_set_int )
=> ( ! [X: int] :
( ( member_int @ X @ A4 )
=> ( ( B5 @ X )
!= bot_bot_set_o ) )
=> ( ( condit1735850657277959132ow_int
@ ( comple3221217463730067765et_int
@ ( image_int_set_int
@ ^ [X4: int] : ( image_o_int @ F @ ( B5 @ X4 ) )
@ A4 ) ) )
=> ( ( complete_Inf_Inf_int @ ( image_o_int @ F @ ( comple90263536869209701_set_o @ ( image_int_set_o @ B5 @ A4 ) ) ) )
= ( complete_Inf_Inf_int
@ ( image_int_int
@ ^ [X4: int] : ( complete_Inf_Inf_int @ ( image_o_int @ F @ ( B5 @ X4 ) ) )
@ A4 ) ) ) ) ) ) ).
% cINF_UNION
thf(fact_2772_cINF__UNION,axiom,
! [A4: set_Product_unit,B5: product_unit > set_int,F: int > int] :
( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ! [X: product_unit] :
( ( member_Product_unit @ X @ A4 )
=> ( ( B5 @ X )
!= bot_bot_set_int ) )
=> ( ( condit1735850657277959132ow_int
@ ( comple3221217463730067765et_int
@ ( image_3995998972318789802et_int
@ ^ [X4: product_unit] : ( image_int_int @ F @ ( B5 @ X4 ) )
@ A4 ) ) )
=> ( ( complete_Inf_Inf_int @ ( image_int_int @ F @ ( comple3221217463730067765et_int @ ( image_3995998972318789802et_int @ B5 @ A4 ) ) ) )
= ( complete_Inf_Inf_int
@ ( image_873079544045703924it_int
@ ^ [X4: product_unit] : ( complete_Inf_Inf_int @ ( image_int_int @ F @ ( B5 @ X4 ) ) )
@ A4 ) ) ) ) ) ) ).
% cINF_UNION
thf(fact_2773_cINF__UNION,axiom,
! [A4: set_Product_unit,B5: product_unit > set_Product_unit,F: product_unit > int] :
( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ! [X: product_unit] :
( ( member_Product_unit @ X @ A4 )
=> ( ( B5 @ X )
!= bot_bo3957492148770167129t_unit ) )
=> ( ( condit1735850657277959132ow_int
@ ( comple3221217463730067765et_int
@ ( image_3995998972318789802et_int
@ ^ [X4: product_unit] : ( image_873079544045703924it_int @ F @ ( B5 @ X4 ) )
@ A4 ) ) )
=> ( ( complete_Inf_Inf_int @ ( image_873079544045703924it_int @ F @ ( comple3349607820504768346t_unit @ ( image_7778121751606633509t_unit @ B5 @ A4 ) ) ) )
= ( complete_Inf_Inf_int
@ ( image_873079544045703924it_int
@ ^ [X4: product_unit] : ( complete_Inf_Inf_int @ ( image_873079544045703924it_int @ F @ ( B5 @ X4 ) ) )
@ A4 ) ) ) ) ) ) ).
% cINF_UNION
thf(fact_2774_cINF__UNION,axiom,
! [A4: set_Product_unit,B5: product_unit > set_o,F: $o > int] :
( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ! [X: product_unit] :
( ( member_Product_unit @ X @ A4 )
=> ( ( B5 @ X )
!= bot_bot_set_o ) )
=> ( ( condit1735850657277959132ow_int
@ ( comple3221217463730067765et_int
@ ( image_3995998972318789802et_int
@ ^ [X4: product_unit] : ( image_o_int @ F @ ( B5 @ X4 ) )
@ A4 ) ) )
=> ( ( complete_Inf_Inf_int @ ( image_o_int @ F @ ( comple90263536869209701_set_o @ ( image_6930536745178679216_set_o @ B5 @ A4 ) ) ) )
= ( complete_Inf_Inf_int
@ ( image_873079544045703924it_int
@ ^ [X4: product_unit] : ( complete_Inf_Inf_int @ ( image_o_int @ F @ ( B5 @ X4 ) ) )
@ A4 ) ) ) ) ) ) ).
% cINF_UNION
thf(fact_2775_cINF__UNION,axiom,
! [A4: set_o,B5: $o > set_int,F: int > int] :
( ( A4 != bot_bot_set_o )
=> ( ! [X: $o] :
( ( member_o @ X @ A4 )
=> ( ( B5 @ X )
!= bot_bot_set_int ) )
=> ( ( condit1735850657277959132ow_int
@ ( comple3221217463730067765et_int
@ ( image_o_set_int
@ ^ [X4: $o] : ( image_int_int @ F @ ( B5 @ X4 ) )
@ A4 ) ) )
=> ( ( complete_Inf_Inf_int @ ( image_int_int @ F @ ( comple3221217463730067765et_int @ ( image_o_set_int @ B5 @ A4 ) ) ) )
= ( complete_Inf_Inf_int
@ ( image_o_int
@ ^ [X4: $o] : ( complete_Inf_Inf_int @ ( image_int_int @ F @ ( B5 @ X4 ) ) )
@ A4 ) ) ) ) ) ) ).
% cINF_UNION
thf(fact_2776_cINF__UNION,axiom,
! [A4: set_o,B5: $o > set_Product_unit,F: product_unit > int] :
( ( A4 != bot_bot_set_o )
=> ( ! [X: $o] :
( ( member_o @ X @ A4 )
=> ( ( B5 @ X )
!= bot_bo3957492148770167129t_unit ) )
=> ( ( condit1735850657277959132ow_int
@ ( comple3221217463730067765et_int
@ ( image_o_set_int
@ ^ [X4: $o] : ( image_873079544045703924it_int @ F @ ( B5 @ X4 ) )
@ A4 ) ) )
=> ( ( complete_Inf_Inf_int @ ( image_873079544045703924it_int @ F @ ( comple3349607820504768346t_unit @ ( image_663195431845683034t_unit @ B5 @ A4 ) ) ) )
= ( complete_Inf_Inf_int
@ ( image_o_int
@ ^ [X4: $o] : ( complete_Inf_Inf_int @ ( image_873079544045703924it_int @ F @ ( B5 @ X4 ) ) )
@ A4 ) ) ) ) ) ) ).
% cINF_UNION
thf(fact_2777_cINF__UNION,axiom,
! [A4: set_o,B5: $o > set_o,F: $o > int] :
( ( A4 != bot_bot_set_o )
=> ( ! [X: $o] :
( ( member_o @ X @ A4 )
=> ( ( B5 @ X )
!= bot_bot_set_o ) )
=> ( ( condit1735850657277959132ow_int
@ ( comple3221217463730067765et_int
@ ( image_o_set_int
@ ^ [X4: $o] : ( image_o_int @ F @ ( B5 @ X4 ) )
@ A4 ) ) )
=> ( ( complete_Inf_Inf_int @ ( image_o_int @ F @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B5 @ A4 ) ) ) )
= ( complete_Inf_Inf_int
@ ( image_o_int
@ ^ [X4: $o] : ( complete_Inf_Inf_int @ ( image_o_int @ F @ ( B5 @ X4 ) ) )
@ A4 ) ) ) ) ) ) ).
% cINF_UNION
thf(fact_2778_cINF__UNION,axiom,
! [A4: set_nat,B5: nat > set_int,F: int > int] :
( ( A4 != bot_bot_set_nat )
=> ( ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ( ( B5 @ X )
!= bot_bot_set_int ) )
=> ( ( condit1735850657277959132ow_int
@ ( comple3221217463730067765et_int
@ ( image_nat_set_int
@ ^ [X4: nat] : ( image_int_int @ F @ ( B5 @ X4 ) )
@ A4 ) ) )
=> ( ( complete_Inf_Inf_int @ ( image_int_int @ F @ ( comple3221217463730067765et_int @ ( image_nat_set_int @ B5 @ A4 ) ) ) )
= ( complete_Inf_Inf_int
@ ( image_nat_int
@ ^ [X4: nat] : ( complete_Inf_Inf_int @ ( image_int_int @ F @ ( B5 @ X4 ) ) )
@ A4 ) ) ) ) ) ) ).
% cINF_UNION
thf(fact_2779_Ex__inj__on__UNION__Sigma,axiom,
! [A4: nat > set_nat,I3: set_nat] :
? [F5: nat > product_prod_nat_nat] :
( ( inj_on5538052773655684606at_nat @ F5 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A4 @ I3 ) ) )
& ( ord_le3146513528884898305at_nat @ ( image_5846123807819985514at_nat @ F5 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A4 @ I3 ) ) ) @ ( produc457027306803732586at_nat @ I3 @ A4 ) ) ) ).
% Ex_inj_on_UNION_Sigma
thf(fact_2780_prod__set__simps_I1_J,axiom,
! [X2: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] :
( ( basic_1274512288150691911at_nat @ ( produc9060074326276436823at_nat @ X2 @ Y3 ) )
= ( insert2290795007606087959at_nat @ X2 @ bot_bo1534825418459847827at_nat ) ) ).
% prod_set_simps(1)
thf(fact_2781_prod__set__simps_I1_J,axiom,
! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
( ( basic_3661821009137050567at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y3 ) )
= ( insert9200635055090092081at_nat @ X2 @ bot_bo3083307316010499117at_nat ) ) ).
% prod_set_simps(1)
thf(fact_2782_prod__set__simps_I1_J,axiom,
! [X2: int,Y3: int] :
( ( basic_fsts_int_int @ ( product_Pair_int_int @ X2 @ Y3 ) )
= ( insert_int @ X2 @ bot_bot_set_int ) ) ).
% prod_set_simps(1)
thf(fact_2783_prod__set__simps_I1_J,axiom,
! [X2: nat,Y3: num] :
( ( basic_fsts_nat_num @ ( product_Pair_nat_num @ X2 @ Y3 ) )
= ( insert_nat @ X2 @ bot_bot_set_nat ) ) ).
% prod_set_simps(1)
thf(fact_2784_prod__set__simps_I1_J,axiom,
! [X2: nat,Y3: nat] :
( ( basic_fsts_nat_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) )
= ( insert_nat @ X2 @ bot_bot_set_nat ) ) ).
% prod_set_simps(1)
thf(fact_2785_if__image__distrib,axiom,
! [P2: int > $o,F: int > nat,G: int > nat,S5: set_int] :
( ( image_int_nat
@ ^ [X4: int] : ( if_nat @ ( P2 @ X4 ) @ ( F @ X4 ) @ ( G @ X4 ) )
@ S5 )
= ( sup_sup_set_nat @ ( image_int_nat @ F @ ( inf_inf_set_int @ S5 @ ( collect_int @ P2 ) ) )
@ ( image_int_nat @ G
@ ( inf_inf_set_int @ S5
@ ( collect_int
@ ^ [X4: int] :
~ ( P2 @ X4 ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_2786_if__image__distrib,axiom,
! [P2: int > $o,F: int > int,G: int > int,S5: set_int] :
( ( image_int_int
@ ^ [X4: int] : ( if_int @ ( P2 @ X4 ) @ ( F @ X4 ) @ ( G @ X4 ) )
@ S5 )
= ( sup_sup_set_int @ ( image_int_int @ F @ ( inf_inf_set_int @ S5 @ ( collect_int @ P2 ) ) )
@ ( image_int_int @ G
@ ( inf_inf_set_int @ S5
@ ( collect_int
@ ^ [X4: int] :
~ ( P2 @ X4 ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_2787_if__image__distrib,axiom,
! [P2: nat > $o,F: nat > rat,G: nat > rat,S5: set_nat] :
( ( image_nat_rat
@ ^ [X4: nat] : ( if_rat @ ( P2 @ X4 ) @ ( F @ X4 ) @ ( G @ X4 ) )
@ S5 )
= ( sup_sup_set_rat @ ( image_nat_rat @ F @ ( inf_inf_set_nat @ S5 @ ( collect_nat @ P2 ) ) )
@ ( image_nat_rat @ G
@ ( inf_inf_set_nat @ S5
@ ( collect_nat
@ ^ [X4: nat] :
~ ( P2 @ X4 ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_2788_if__image__distrib,axiom,
! [P2: nat > $o,F: nat > nat,G: nat > nat,S5: set_nat] :
( ( image_nat_nat
@ ^ [X4: nat] : ( if_nat @ ( P2 @ X4 ) @ ( F @ X4 ) @ ( G @ X4 ) )
@ S5 )
= ( sup_sup_set_nat @ ( image_nat_nat @ F @ ( inf_inf_set_nat @ S5 @ ( collect_nat @ P2 ) ) )
@ ( image_nat_nat @ G
@ ( inf_inf_set_nat @ S5
@ ( collect_nat
@ ^ [X4: nat] :
~ ( P2 @ X4 ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_2789_if__image__distrib,axiom,
! [P2: nat > $o,F: nat > int,G: nat > int,S5: set_nat] :
( ( image_nat_int
@ ^ [X4: nat] : ( if_int @ ( P2 @ X4 ) @ ( F @ X4 ) @ ( G @ X4 ) )
@ S5 )
= ( sup_sup_set_int @ ( image_nat_int @ F @ ( inf_inf_set_nat @ S5 @ ( collect_nat @ P2 ) ) )
@ ( image_nat_int @ G
@ ( inf_inf_set_nat @ S5
@ ( collect_nat
@ ^ [X4: nat] :
~ ( P2 @ X4 ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_2790_if__image__distrib,axiom,
! [P2: nat > $o,F: nat > produc859450856879609959at_nat,G: nat > produc859450856879609959at_nat,S5: set_nat] :
( ( image_3276603626957510736at_nat
@ ^ [X4: nat] : ( if_Pro4507677147265585453at_nat @ ( P2 @ X4 ) @ ( F @ X4 ) @ ( G @ X4 ) )
@ S5 )
= ( sup_su718114333110466843at_nat @ ( image_3276603626957510736at_nat @ F @ ( inf_inf_set_nat @ S5 @ ( collect_nat @ P2 ) ) )
@ ( image_3276603626957510736at_nat @ G
@ ( inf_inf_set_nat @ S5
@ ( collect_nat
@ ^ [X4: nat] :
~ ( P2 @ X4 ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_2791_if__image__distrib,axiom,
! [P2: product_prod_int_int > $o,F: product_prod_int_int > produc859450856879609959at_nat,G: product_prod_int_int > produc859450856879609959at_nat,S5: set_Pr958786334691620121nt_int] :
( ( image_8858061497167895027at_nat
@ ^ [X4: product_prod_int_int] : ( if_Pro4507677147265585453at_nat @ ( P2 @ X4 ) @ ( F @ X4 ) @ ( G @ X4 ) )
@ S5 )
= ( sup_su718114333110466843at_nat @ ( image_8858061497167895027at_nat @ F @ ( inf_in2269163501485487111nt_int @ S5 @ ( collec213857154873943460nt_int @ P2 ) ) )
@ ( image_8858061497167895027at_nat @ G
@ ( inf_in2269163501485487111nt_int @ S5
@ ( collec213857154873943460nt_int
@ ^ [X4: product_prod_int_int] :
~ ( P2 @ X4 ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_2792_if__image__distrib,axiom,
! [P2: product_prod_nat_nat > $o,F: product_prod_nat_nat > produc859450856879609959at_nat,G: product_prod_nat_nat > produc859450856879609959at_nat,S5: set_Pr1261947904930325089at_nat] :
( ( image_1119892238107900587at_nat
@ ^ [X4: product_prod_nat_nat] : ( if_Pro4507677147265585453at_nat @ ( P2 @ X4 ) @ ( F @ X4 ) @ ( G @ X4 ) )
@ S5 )
= ( sup_su718114333110466843at_nat @ ( image_1119892238107900587at_nat @ F @ ( inf_in2572325071724192079at_nat @ S5 @ ( collec3392354462482085612at_nat @ P2 ) ) )
@ ( image_1119892238107900587at_nat @ G
@ ( inf_in2572325071724192079at_nat @ S5
@ ( collec3392354462482085612at_nat
@ ^ [X4: product_prod_nat_nat] :
~ ( P2 @ X4 ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_2793_if__image__distrib,axiom,
! [P2: nat > $o,F: nat > produc3843707927480180839at_nat,G: nat > produc3843707927480180839at_nat,S5: set_nat] :
( ( image_2499013659892586576at_nat
@ ^ [X4: nat] : ( if_Pro3594990065127090477at_nat @ ( P2 @ X4 ) @ ( F @ X4 ) @ ( G @ X4 ) )
@ S5 )
= ( sup_su5525570899277871387at_nat @ ( image_2499013659892586576at_nat @ F @ ( inf_inf_set_nat @ S5 @ ( collect_nat @ P2 ) ) )
@ ( image_2499013659892586576at_nat @ G
@ ( inf_inf_set_nat @ S5
@ ( collect_nat
@ ^ [X4: nat] :
~ ( P2 @ X4 ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_2794_if__image__distrib,axiom,
! [P2: product_prod_int_int > $o,F: product_prod_int_int > produc3843707927480180839at_nat,G: product_prod_int_int > produc3843707927480180839at_nat,S5: set_Pr958786334691620121nt_int] :
( ( image_6029125644611338739at_nat
@ ^ [X4: product_prod_int_int] : ( if_Pro3594990065127090477at_nat @ ( P2 @ X4 ) @ ( F @ X4 ) @ ( G @ X4 ) )
@ S5 )
= ( sup_su5525570899277871387at_nat @ ( image_6029125644611338739at_nat @ F @ ( inf_in2269163501485487111nt_int @ S5 @ ( collec213857154873943460nt_int @ P2 ) ) )
@ ( image_6029125644611338739at_nat @ G
@ ( inf_in2269163501485487111nt_int @ S5
@ ( collec213857154873943460nt_int
@ ^ [X4: product_prod_int_int] :
~ ( P2 @ X4 ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_2795_UN__Un,axiom,
! [M: produc859450856879609959at_nat > set_Pr8693737435421807431at_nat,A4: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] :
( ( comple2748534634209756852at_nat @ ( image_6654316230217496229at_nat @ M @ ( sup_su718114333110466843at_nat @ A4 @ B5 ) ) )
= ( sup_su718114333110466843at_nat @ ( comple2748534634209756852at_nat @ ( image_6654316230217496229at_nat @ M @ A4 ) ) @ ( comple2748534634209756852at_nat @ ( image_6654316230217496229at_nat @ M @ B5 ) ) ) ) ).
% UN_Un
thf(fact_2796_UN__Un,axiom,
! [M: produc859450856879609959at_nat > set_Pr4329608150637261639at_nat,A4: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] :
( ( comple4364388969068068532at_nat @ ( image_3876558564303283877at_nat @ M @ ( sup_su718114333110466843at_nat @ A4 @ B5 ) ) )
= ( sup_su5525570899277871387at_nat @ ( comple4364388969068068532at_nat @ ( image_3876558564303283877at_nat @ M @ A4 ) ) @ ( comple4364388969068068532at_nat @ ( image_3876558564303283877at_nat @ M @ B5 ) ) ) ) ).
% UN_Un
thf(fact_2797_UN__Un,axiom,
! [M: produc3843707927480180839at_nat > set_Pr8693737435421807431at_nat,A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
( ( comple2748534634209756852at_nat @ ( image_7260680879217540773at_nat @ M @ ( sup_su5525570899277871387at_nat @ A4 @ B5 ) ) )
= ( sup_su718114333110466843at_nat @ ( comple2748534634209756852at_nat @ ( image_7260680879217540773at_nat @ M @ A4 ) ) @ ( comple2748534634209756852at_nat @ ( image_7260680879217540773at_nat @ M @ B5 ) ) ) ) ).
% UN_Un
thf(fact_2798_UN__Un,axiom,
! [M: produc3843707927480180839at_nat > set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
( ( comple4364388969068068532at_nat @ ( image_7364710425340685989at_nat @ M @ ( sup_su5525570899277871387at_nat @ A4 @ B5 ) ) )
= ( sup_su5525570899277871387at_nat @ ( comple4364388969068068532at_nat @ ( image_7364710425340685989at_nat @ M @ A4 ) ) @ ( comple4364388969068068532at_nat @ ( image_7364710425340685989at_nat @ M @ B5 ) ) ) ) ).
% UN_Un
thf(fact_2799_UN__Un,axiom,
! [M: produc859450856879609959at_nat > set_nat,A4: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] :
( ( comple7399068483239264473et_nat @ ( image_2819269535336293544et_nat @ M @ ( sup_su718114333110466843at_nat @ A4 @ B5 ) ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_2819269535336293544et_nat @ M @ A4 ) ) @ ( comple7399068483239264473et_nat @ ( image_2819269535336293544et_nat @ M @ B5 ) ) ) ) ).
% UN_Un
thf(fact_2800_UN__Un,axiom,
! [M: produc3843707927480180839at_nat > set_nat,A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
( ( comple7399068483239264473et_nat @ ( image_3160451284896210088et_nat @ M @ ( sup_su5525570899277871387at_nat @ A4 @ B5 ) ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_3160451284896210088et_nat @ M @ A4 ) ) @ ( comple7399068483239264473et_nat @ ( image_3160451284896210088et_nat @ M @ B5 ) ) ) ) ).
% UN_Un
thf(fact_2801_bdd__below__image__inf,axiom,
! [F: nat > rat,G: nat > rat,A4: set_nat] :
( ( condit1103211067700513672ow_rat
@ ( image_nat_rat
@ ^ [X4: nat] : ( inf_inf_rat @ ( F @ X4 ) @ ( G @ X4 ) )
@ A4 ) )
= ( ( condit1103211067700513672ow_rat @ ( image_nat_rat @ F @ A4 ) )
& ( condit1103211067700513672ow_rat @ ( image_nat_rat @ G @ A4 ) ) ) ) ).
% bdd_below_image_inf
thf(fact_2802_bdd__below__image__inf,axiom,
! [F: nat > nat,G: nat > nat,A4: set_nat] :
( ( condit1738341127787009408ow_nat
@ ( image_nat_nat
@ ^ [X4: nat] : ( inf_inf_nat @ ( F @ X4 ) @ ( G @ X4 ) )
@ A4 ) )
= ( ( condit1738341127787009408ow_nat @ ( image_nat_nat @ F @ A4 ) )
& ( condit1738341127787009408ow_nat @ ( image_nat_nat @ G @ A4 ) ) ) ) ).
% bdd_below_image_inf
thf(fact_2803_bdd__below__image__inf,axiom,
! [F: nat > int,G: nat > int,A4: set_nat] :
( ( condit1735850657277959132ow_int
@ ( image_nat_int
@ ^ [X4: nat] : ( inf_inf_int @ ( F @ X4 ) @ ( G @ X4 ) )
@ A4 ) )
= ( ( condit1735850657277959132ow_int @ ( image_nat_int @ F @ A4 ) )
& ( condit1735850657277959132ow_int @ ( image_nat_int @ G @ A4 ) ) ) ) ).
% bdd_below_image_inf
thf(fact_2804_bdd__below__image__inf,axiom,
! [F: int > nat,G: int > nat,A4: set_int] :
( ( condit1738341127787009408ow_nat
@ ( image_int_nat
@ ^ [X4: int] : ( inf_inf_nat @ ( F @ X4 ) @ ( G @ X4 ) )
@ A4 ) )
= ( ( condit1738341127787009408ow_nat @ ( image_int_nat @ F @ A4 ) )
& ( condit1738341127787009408ow_nat @ ( image_int_nat @ G @ A4 ) ) ) ) ).
% bdd_below_image_inf
thf(fact_2805_bdd__below__image__inf,axiom,
! [F: int > int,G: int > int,A4: set_int] :
( ( condit1735850657277959132ow_int
@ ( image_int_int
@ ^ [X4: int] : ( inf_inf_int @ ( F @ X4 ) @ ( G @ X4 ) )
@ A4 ) )
= ( ( condit1735850657277959132ow_int @ ( image_int_int @ F @ A4 ) )
& ( condit1735850657277959132ow_int @ ( image_int_int @ G @ A4 ) ) ) ) ).
% bdd_below_image_inf
thf(fact_2806_bdd__above__image__sup,axiom,
! [F: nat > rat,G: nat > rat,A4: set_nat] :
( ( condit1579696412822616692ve_rat
@ ( image_nat_rat
@ ^ [X4: nat] : ( sup_sup_rat @ ( F @ X4 ) @ ( G @ X4 ) )
@ A4 ) )
= ( ( condit1579696412822616692ve_rat @ ( image_nat_rat @ F @ A4 ) )
& ( condit1579696412822616692ve_rat @ ( image_nat_rat @ G @ A4 ) ) ) ) ).
% bdd_above_image_sup
thf(fact_2807_bdd__above__image__sup,axiom,
! [F: nat > nat,G: nat > nat,A4: set_nat] :
( ( condit2214826472909112428ve_nat
@ ( image_nat_nat
@ ^ [X4: nat] : ( sup_sup_nat @ ( F @ X4 ) @ ( G @ X4 ) )
@ A4 ) )
= ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ F @ A4 ) )
& ( condit2214826472909112428ve_nat @ ( image_nat_nat @ G @ A4 ) ) ) ) ).
% bdd_above_image_sup
thf(fact_2808_bdd__above__image__sup,axiom,
! [F: nat > int,G: nat > int,A4: set_nat] :
( ( condit2212336002400062152ve_int
@ ( image_nat_int
@ ^ [X4: nat] : ( sup_sup_int @ ( F @ X4 ) @ ( G @ X4 ) )
@ A4 ) )
= ( ( condit2212336002400062152ve_int @ ( image_nat_int @ F @ A4 ) )
& ( condit2212336002400062152ve_int @ ( image_nat_int @ G @ A4 ) ) ) ) ).
% bdd_above_image_sup
thf(fact_2809_bdd__above__image__sup,axiom,
! [F: int > nat,G: int > nat,A4: set_int] :
( ( condit2214826472909112428ve_nat
@ ( image_int_nat
@ ^ [X4: int] : ( sup_sup_nat @ ( F @ X4 ) @ ( G @ X4 ) )
@ A4 ) )
= ( ( condit2214826472909112428ve_nat @ ( image_int_nat @ F @ A4 ) )
& ( condit2214826472909112428ve_nat @ ( image_int_nat @ G @ A4 ) ) ) ) ).
% bdd_above_image_sup
thf(fact_2810_bdd__above__image__sup,axiom,
! [F: int > int,G: int > int,A4: set_int] :
( ( condit2212336002400062152ve_int
@ ( image_int_int
@ ^ [X4: int] : ( sup_sup_int @ ( F @ X4 ) @ ( G @ X4 ) )
@ A4 ) )
= ( ( condit2212336002400062152ve_int @ ( image_int_int @ F @ A4 ) )
& ( condit2212336002400062152ve_int @ ( image_int_int @ G @ A4 ) ) ) ) ).
% bdd_above_image_sup
thf(fact_2811_UN__simps_I3_J,axiom,
! [C3: set_Product_unit,A4: set_Product_unit,B5: product_unit > set_Product_unit] :
( ( ( C3 = bot_bo3957492148770167129t_unit )
=> ( ( comple3349607820504768346t_unit
@ ( image_7778121751606633509t_unit
@ ^ [X4: product_unit] : ( sup_su793286257634532545t_unit @ A4 @ ( B5 @ X4 ) )
@ C3 ) )
= bot_bo3957492148770167129t_unit ) )
& ( ( C3 != bot_bo3957492148770167129t_unit )
=> ( ( comple3349607820504768346t_unit
@ ( image_7778121751606633509t_unit
@ ^ [X4: product_unit] : ( sup_su793286257634532545t_unit @ A4 @ ( B5 @ X4 ) )
@ C3 ) )
= ( sup_su793286257634532545t_unit @ A4 @ ( comple3349607820504768346t_unit @ ( image_7778121751606633509t_unit @ B5 @ C3 ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_2812_UN__simps_I3_J,axiom,
! [C3: set_Product_unit,A4: set_o,B5: product_unit > set_o] :
( ( ( C3 = bot_bo3957492148770167129t_unit )
=> ( ( comple90263536869209701_set_o
@ ( image_6930536745178679216_set_o
@ ^ [X4: product_unit] : ( sup_sup_set_o @ A4 @ ( B5 @ X4 ) )
@ C3 ) )
= bot_bot_set_o ) )
& ( ( C3 != bot_bo3957492148770167129t_unit )
=> ( ( comple90263536869209701_set_o
@ ( image_6930536745178679216_set_o
@ ^ [X4: product_unit] : ( sup_sup_set_o @ A4 @ ( B5 @ X4 ) )
@ C3 ) )
= ( sup_sup_set_o @ A4 @ ( comple90263536869209701_set_o @ ( image_6930536745178679216_set_o @ B5 @ C3 ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_2813_UN__simps_I3_J,axiom,
! [C3: set_o,A4: set_Product_unit,B5: $o > set_Product_unit] :
( ( ( C3 = bot_bot_set_o )
=> ( ( comple3349607820504768346t_unit
@ ( image_663195431845683034t_unit
@ ^ [X4: $o] : ( sup_su793286257634532545t_unit @ A4 @ ( B5 @ X4 ) )
@ C3 ) )
= bot_bo3957492148770167129t_unit ) )
& ( ( C3 != bot_bot_set_o )
=> ( ( comple3349607820504768346t_unit
@ ( image_663195431845683034t_unit
@ ^ [X4: $o] : ( sup_su793286257634532545t_unit @ A4 @ ( B5 @ X4 ) )
@ C3 ) )
= ( sup_su793286257634532545t_unit @ A4 @ ( comple3349607820504768346t_unit @ ( image_663195431845683034t_unit @ B5 @ C3 ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_2814_UN__simps_I3_J,axiom,
! [C3: set_o,A4: set_o,B5: $o > set_o] :
( ( ( C3 = bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X4: $o] : ( sup_sup_set_o @ A4 @ ( B5 @ X4 ) )
@ C3 ) )
= bot_bot_set_o ) )
& ( ( C3 != bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X4: $o] : ( sup_sup_set_o @ A4 @ ( B5 @ X4 ) )
@ C3 ) )
= ( sup_sup_set_o @ A4 @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B5 @ C3 ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_2815_UN__simps_I3_J,axiom,
! [C3: set_nat,A4: set_Product_unit,B5: nat > set_Product_unit] :
( ( ( C3 = bot_bot_set_nat )
=> ( ( comple3349607820504768346t_unit
@ ( image_4824033694279209942t_unit
@ ^ [X4: nat] : ( sup_su793286257634532545t_unit @ A4 @ ( B5 @ X4 ) )
@ C3 ) )
= bot_bo3957492148770167129t_unit ) )
& ( ( C3 != bot_bot_set_nat )
=> ( ( comple3349607820504768346t_unit
@ ( image_4824033694279209942t_unit
@ ^ [X4: nat] : ( sup_su793286257634532545t_unit @ A4 @ ( B5 @ X4 ) )
@ C3 ) )
= ( sup_su793286257634532545t_unit @ A4 @ ( comple3349607820504768346t_unit @ ( image_4824033694279209942t_unit @ B5 @ C3 ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_2816_UN__simps_I3_J,axiom,
! [C3: set_nat,A4: set_o,B5: nat > set_o] :
( ( ( C3 = bot_bot_set_nat )
=> ( ( comple90263536869209701_set_o
@ ( image_nat_set_o
@ ^ [X4: nat] : ( sup_sup_set_o @ A4 @ ( B5 @ X4 ) )
@ C3 ) )
= bot_bot_set_o ) )
& ( ( C3 != bot_bot_set_nat )
=> ( ( comple90263536869209701_set_o
@ ( image_nat_set_o
@ ^ [X4: nat] : ( sup_sup_set_o @ A4 @ ( B5 @ X4 ) )
@ C3 ) )
= ( sup_sup_set_o @ A4 @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ B5 @ C3 ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_2817_UN__simps_I3_J,axiom,
! [C3: set_Product_unit,A4: set_nat,B5: product_unit > set_nat] :
( ( ( C3 = bot_bo3957492148770167129t_unit )
=> ( ( comple7399068483239264473et_nat
@ ( image_8173849991827986510et_nat
@ ^ [X4: product_unit] : ( sup_sup_set_nat @ A4 @ ( B5 @ X4 ) )
@ C3 ) )
= bot_bot_set_nat ) )
& ( ( C3 != bot_bo3957492148770167129t_unit )
=> ( ( comple7399068483239264473et_nat
@ ( image_8173849991827986510et_nat
@ ^ [X4: product_unit] : ( sup_sup_set_nat @ A4 @ ( B5 @ X4 ) )
@ C3 ) )
= ( sup_sup_set_nat @ A4 @ ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ B5 @ C3 ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_2818_UN__simps_I3_J,axiom,
! [C3: set_o,A4: set_nat,B5: $o > set_nat] :
( ( ( C3 = bot_bot_set_o )
=> ( ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [X4: $o] : ( sup_sup_set_nat @ A4 @ ( B5 @ X4 ) )
@ C3 ) )
= bot_bot_set_nat ) )
& ( ( C3 != bot_bot_set_o )
=> ( ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [X4: $o] : ( sup_sup_set_nat @ A4 @ ( B5 @ X4 ) )
@ C3 ) )
= ( sup_sup_set_nat @ A4 @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B5 @ C3 ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_2819_UN__simps_I3_J,axiom,
! [C3: set_nat,A4: set_nat,B5: nat > set_nat] :
( ( ( C3 = bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X4: nat] : ( sup_sup_set_nat @ A4 @ ( B5 @ X4 ) )
@ C3 ) )
= bot_bot_set_nat ) )
& ( ( C3 != bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X4: nat] : ( sup_sup_set_nat @ A4 @ ( B5 @ X4 ) )
@ C3 ) )
= ( sup_sup_set_nat @ A4 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ C3 ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_2820_UN__simps_I3_J,axiom,
! [C3: set_Product_unit,A4: set_Pr1261947904930325089at_nat,B5: product_unit > set_Pr1261947904930325089at_nat] :
( ( ( C3 = bot_bo3957492148770167129t_unit )
=> ( ( comple5685304695842803022at_nat
@ ( image_3617315916088684441at_nat
@ ^ [X4: product_unit] : ( sup_su6327502436637775413at_nat @ A4 @ ( B5 @ X4 ) )
@ C3 ) )
= bot_bo2099793752762293965at_nat ) )
& ( ( C3 != bot_bo3957492148770167129t_unit )
=> ( ( comple5685304695842803022at_nat
@ ( image_3617315916088684441at_nat
@ ^ [X4: product_unit] : ( sup_su6327502436637775413at_nat @ A4 @ ( B5 @ X4 ) )
@ C3 ) )
= ( sup_su6327502436637775413at_nat @ A4 @ ( comple5685304695842803022at_nat @ ( image_3617315916088684441at_nat @ B5 @ C3 ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_2821_UN__simps_I2_J,axiom,
! [C3: set_Product_unit,A4: product_unit > set_Product_unit,B5: set_Product_unit] :
( ( ( C3 = bot_bo3957492148770167129t_unit )
=> ( ( comple3349607820504768346t_unit
@ ( image_7778121751606633509t_unit
@ ^ [X4: product_unit] : ( sup_su793286257634532545t_unit @ ( A4 @ X4 ) @ B5 )
@ C3 ) )
= bot_bo3957492148770167129t_unit ) )
& ( ( C3 != bot_bo3957492148770167129t_unit )
=> ( ( comple3349607820504768346t_unit
@ ( image_7778121751606633509t_unit
@ ^ [X4: product_unit] : ( sup_su793286257634532545t_unit @ ( A4 @ X4 ) @ B5 )
@ C3 ) )
= ( sup_su793286257634532545t_unit @ ( comple3349607820504768346t_unit @ ( image_7778121751606633509t_unit @ A4 @ C3 ) ) @ B5 ) ) ) ) ).
% UN_simps(2)
thf(fact_2822_UN__simps_I2_J,axiom,
! [C3: set_Product_unit,A4: product_unit > set_o,B5: set_o] :
( ( ( C3 = bot_bo3957492148770167129t_unit )
=> ( ( comple90263536869209701_set_o
@ ( image_6930536745178679216_set_o
@ ^ [X4: product_unit] : ( sup_sup_set_o @ ( A4 @ X4 ) @ B5 )
@ C3 ) )
= bot_bot_set_o ) )
& ( ( C3 != bot_bo3957492148770167129t_unit )
=> ( ( comple90263536869209701_set_o
@ ( image_6930536745178679216_set_o
@ ^ [X4: product_unit] : ( sup_sup_set_o @ ( A4 @ X4 ) @ B5 )
@ C3 ) )
= ( sup_sup_set_o @ ( comple90263536869209701_set_o @ ( image_6930536745178679216_set_o @ A4 @ C3 ) ) @ B5 ) ) ) ) ).
% UN_simps(2)
thf(fact_2823_UN__simps_I2_J,axiom,
! [C3: set_o,A4: $o > set_Product_unit,B5: set_Product_unit] :
( ( ( C3 = bot_bot_set_o )
=> ( ( comple3349607820504768346t_unit
@ ( image_663195431845683034t_unit
@ ^ [X4: $o] : ( sup_su793286257634532545t_unit @ ( A4 @ X4 ) @ B5 )
@ C3 ) )
= bot_bo3957492148770167129t_unit ) )
& ( ( C3 != bot_bot_set_o )
=> ( ( comple3349607820504768346t_unit
@ ( image_663195431845683034t_unit
@ ^ [X4: $o] : ( sup_su793286257634532545t_unit @ ( A4 @ X4 ) @ B5 )
@ C3 ) )
= ( sup_su793286257634532545t_unit @ ( comple3349607820504768346t_unit @ ( image_663195431845683034t_unit @ A4 @ C3 ) ) @ B5 ) ) ) ) ).
% UN_simps(2)
thf(fact_2824_UN__simps_I2_J,axiom,
! [C3: set_o,A4: $o > set_o,B5: set_o] :
( ( ( C3 = bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X4: $o] : ( sup_sup_set_o @ ( A4 @ X4 ) @ B5 )
@ C3 ) )
= bot_bot_set_o ) )
& ( ( C3 != bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X4: $o] : ( sup_sup_set_o @ ( A4 @ X4 ) @ B5 )
@ C3 ) )
= ( sup_sup_set_o @ ( comple90263536869209701_set_o @ ( image_o_set_o @ A4 @ C3 ) ) @ B5 ) ) ) ) ).
% UN_simps(2)
thf(fact_2825_UN__simps_I2_J,axiom,
! [C3: set_nat,A4: nat > set_Product_unit,B5: set_Product_unit] :
( ( ( C3 = bot_bot_set_nat )
=> ( ( comple3349607820504768346t_unit
@ ( image_4824033694279209942t_unit
@ ^ [X4: nat] : ( sup_su793286257634532545t_unit @ ( A4 @ X4 ) @ B5 )
@ C3 ) )
= bot_bo3957492148770167129t_unit ) )
& ( ( C3 != bot_bot_set_nat )
=> ( ( comple3349607820504768346t_unit
@ ( image_4824033694279209942t_unit
@ ^ [X4: nat] : ( sup_su793286257634532545t_unit @ ( A4 @ X4 ) @ B5 )
@ C3 ) )
= ( sup_su793286257634532545t_unit @ ( comple3349607820504768346t_unit @ ( image_4824033694279209942t_unit @ A4 @ C3 ) ) @ B5 ) ) ) ) ).
% UN_simps(2)
thf(fact_2826_UN__simps_I2_J,axiom,
! [C3: set_nat,A4: nat > set_o,B5: set_o] :
( ( ( C3 = bot_bot_set_nat )
=> ( ( comple90263536869209701_set_o
@ ( image_nat_set_o
@ ^ [X4: nat] : ( sup_sup_set_o @ ( A4 @ X4 ) @ B5 )
@ C3 ) )
= bot_bot_set_o ) )
& ( ( C3 != bot_bot_set_nat )
=> ( ( comple90263536869209701_set_o
@ ( image_nat_set_o
@ ^ [X4: nat] : ( sup_sup_set_o @ ( A4 @ X4 ) @ B5 )
@ C3 ) )
= ( sup_sup_set_o @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ A4 @ C3 ) ) @ B5 ) ) ) ) ).
% UN_simps(2)
thf(fact_2827_UN__simps_I2_J,axiom,
! [C3: set_Product_unit,A4: product_unit > set_nat,B5: set_nat] :
( ( ( C3 = bot_bo3957492148770167129t_unit )
=> ( ( comple7399068483239264473et_nat
@ ( image_8173849991827986510et_nat
@ ^ [X4: product_unit] : ( sup_sup_set_nat @ ( A4 @ X4 ) @ B5 )
@ C3 ) )
= bot_bot_set_nat ) )
& ( ( C3 != bot_bo3957492148770167129t_unit )
=> ( ( comple7399068483239264473et_nat
@ ( image_8173849991827986510et_nat
@ ^ [X4: product_unit] : ( sup_sup_set_nat @ ( A4 @ X4 ) @ B5 )
@ C3 ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ A4 @ C3 ) ) @ B5 ) ) ) ) ).
% UN_simps(2)
thf(fact_2828_UN__simps_I2_J,axiom,
! [C3: set_o,A4: $o > set_nat,B5: set_nat] :
( ( ( C3 = bot_bot_set_o )
=> ( ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [X4: $o] : ( sup_sup_set_nat @ ( A4 @ X4 ) @ B5 )
@ C3 ) )
= bot_bot_set_nat ) )
& ( ( C3 != bot_bot_set_o )
=> ( ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [X4: $o] : ( sup_sup_set_nat @ ( A4 @ X4 ) @ B5 )
@ C3 ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ A4 @ C3 ) ) @ B5 ) ) ) ) ).
% UN_simps(2)
thf(fact_2829_UN__simps_I2_J,axiom,
! [C3: set_nat,A4: nat > set_nat,B5: set_nat] :
( ( ( C3 = bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X4: nat] : ( sup_sup_set_nat @ ( A4 @ X4 ) @ B5 )
@ C3 ) )
= bot_bot_set_nat ) )
& ( ( C3 != bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X4: nat] : ( sup_sup_set_nat @ ( A4 @ X4 ) @ B5 )
@ C3 ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A4 @ C3 ) ) @ B5 ) ) ) ) ).
% UN_simps(2)
thf(fact_2830_UN__simps_I2_J,axiom,
! [C3: set_Product_unit,A4: product_unit > set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
( ( ( C3 = bot_bo3957492148770167129t_unit )
=> ( ( comple5685304695842803022at_nat
@ ( image_3617315916088684441at_nat
@ ^ [X4: product_unit] : ( sup_su6327502436637775413at_nat @ ( A4 @ X4 ) @ B5 )
@ C3 ) )
= bot_bo2099793752762293965at_nat ) )
& ( ( C3 != bot_bo3957492148770167129t_unit )
=> ( ( comple5685304695842803022at_nat
@ ( image_3617315916088684441at_nat
@ ^ [X4: product_unit] : ( sup_su6327502436637775413at_nat @ ( A4 @ X4 ) @ B5 )
@ C3 ) )
= ( sup_su6327502436637775413at_nat @ ( comple5685304695842803022at_nat @ ( image_3617315916088684441at_nat @ A4 @ C3 ) ) @ B5 ) ) ) ) ).
% UN_simps(2)
thf(fact_2831_UN__insert,axiom,
! [B5: nat > set_nat,A: nat,A4: set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ ( insert_nat @ A @ A4 ) ) )
= ( sup_sup_set_nat @ ( B5 @ A ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ A4 ) ) ) ) ).
% UN_insert
thf(fact_2832_UN__insert,axiom,
! [B5: product_unit > set_nat,A: product_unit,A4: set_Product_unit] :
( ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ B5 @ ( insert_Product_unit @ A @ A4 ) ) )
= ( sup_sup_set_nat @ ( B5 @ A ) @ ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ B5 @ A4 ) ) ) ) ).
% UN_insert
thf(fact_2833_UN__insert,axiom,
! [B5: $o > set_nat,A: $o,A4: set_o] :
( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B5 @ ( insert_o @ A @ A4 ) ) )
= ( sup_sup_set_nat @ ( B5 @ A ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B5 @ A4 ) ) ) ) ).
% UN_insert
thf(fact_2834_UN__insert,axiom,
! [B5: product_prod_nat_nat > set_nat,A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
( ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ B5 @ ( insert8211810215607154385at_nat @ A @ A4 ) ) )
= ( sup_sup_set_nat @ ( B5 @ A ) @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ B5 @ A4 ) ) ) ) ).
% UN_insert
thf(fact_2835_UN__insert,axiom,
! [B5: nat > set_Pr8693737435421807431at_nat,A: nat,A4: set_nat] :
( ( comple2748534634209756852at_nat @ ( image_4592875511192039216at_nat @ B5 @ ( insert_nat @ A @ A4 ) ) )
= ( sup_su718114333110466843at_nat @ ( B5 @ A ) @ ( comple2748534634209756852at_nat @ ( image_4592875511192039216at_nat @ B5 @ A4 ) ) ) ) ).
% UN_insert
thf(fact_2836_UN__insert,axiom,
! [B5: product_unit > set_Pr8693737435421807431at_nat,A: product_unit,A4: set_Product_unit] :
( ( comple2748534634209756852at_nat @ ( image_7288820656315755647at_nat @ B5 @ ( insert_Product_unit @ A @ A4 ) ) )
= ( sup_su718114333110466843at_nat @ ( B5 @ A ) @ ( comple2748534634209756852at_nat @ ( image_7288820656315755647at_nat @ B5 @ A4 ) ) ) ) ).
% UN_insert
thf(fact_2837_UN__insert,axiom,
! [B5: $o > set_Pr8693737435421807431at_nat,A: $o,A4: set_o] :
( ( comple2748534634209756852at_nat @ ( image_8493167801588156084at_nat @ B5 @ ( insert_o @ A @ A4 ) ) )
= ( sup_su718114333110466843at_nat @ ( B5 @ A ) @ ( comple2748534634209756852at_nat @ ( image_8493167801588156084at_nat @ B5 @ A4 ) ) ) ) ).
% UN_insert
thf(fact_2838_UN__insert,axiom,
! [B5: product_prod_nat_nat > set_Pr8693737435421807431at_nat,A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
( ( comple2748534634209756852at_nat @ ( image_5803077947706746251at_nat @ B5 @ ( insert8211810215607154385at_nat @ A @ A4 ) ) )
= ( sup_su718114333110466843at_nat @ ( B5 @ A ) @ ( comple2748534634209756852at_nat @ ( image_5803077947706746251at_nat @ B5 @ A4 ) ) ) ) ).
% UN_insert
thf(fact_2839_UN__insert,axiom,
! [B5: nat > set_Pr4329608150637261639at_nat,A: nat,A4: set_nat] :
( ( comple4364388969068068532at_nat @ ( image_1087905878993176368at_nat @ B5 @ ( insert_nat @ A @ A4 ) ) )
= ( sup_su5525570899277871387at_nat @ ( B5 @ A ) @ ( comple4364388969068068532at_nat @ ( image_1087905878993176368at_nat @ B5 @ A4 ) ) ) ) ).
% UN_insert
thf(fact_2840_UN__insert,axiom,
! [B5: product_unit > set_Pr4329608150637261639at_nat,A: product_unit,A4: set_Product_unit] :
( ( comple4364388969068068532at_nat @ ( image_5172358348707908735at_nat @ B5 @ ( insert_Product_unit @ A @ A4 ) ) )
= ( sup_su5525570899277871387at_nat @ ( B5 @ A ) @ ( comple4364388969068068532at_nat @ ( image_5172358348707908735at_nat @ B5 @ A4 ) ) ) ) ).
% UN_insert
thf(fact_2841_inj__apsnd,axiom,
! [F: code_integer > code_integer] :
( ( inj_on20428373856169009nteger @ ( produc6499014454317279255nteger @ F ) @ top_to2363841527343591755nteger )
= ( inj_on1158064722259180589nteger @ F @ top_to4645266643341252675nteger ) ) ).
% inj_apsnd
thf(fact_2842_INT__insert,axiom,
! [B5: nat > set_Pr1261947904930325089at_nat,A: nat,A4: set_nat] :
( ( comple1930127330929219688at_nat @ ( image_7178329752028323786at_nat @ B5 @ ( insert_nat @ A @ A4 ) ) )
= ( inf_in2572325071724192079at_nat @ ( B5 @ A ) @ ( comple1930127330929219688at_nat @ ( image_7178329752028323786at_nat @ B5 @ A4 ) ) ) ) ).
% INT_insert
thf(fact_2843_INT__insert,axiom,
! [B5: product_prod_nat_nat > set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
( ( comple1930127330929219688at_nat @ ( image_5423882813909314213at_nat @ B5 @ ( insert8211810215607154385at_nat @ A @ A4 ) ) )
= ( inf_in2572325071724192079at_nat @ ( B5 @ A ) @ ( comple1930127330929219688at_nat @ ( image_5423882813909314213at_nat @ B5 @ A4 ) ) ) ) ).
% INT_insert
thf(fact_2844_INT__insert,axiom,
! [B5: produc3843707927480180839at_nat > set_Pr1261947904930325089at_nat,A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
( ( comple1930127330929219688at_nat @ ( image_7979982168039674559at_nat @ B5 @ ( insert9069300056098147895at_nat @ A @ A4 ) ) )
= ( inf_in2572325071724192079at_nat @ ( B5 @ A ) @ ( comple1930127330929219688at_nat @ ( image_7979982168039674559at_nat @ B5 @ A4 ) ) ) ) ).
% INT_insert
thf(fact_2845_INT__insert,axiom,
! [B5: product_unit > set_Pr1261947904930325089at_nat,A: product_unit,A4: set_Product_unit] :
( ( comple1930127330929219688at_nat @ ( image_3617315916088684441at_nat @ B5 @ ( insert_Product_unit @ A @ A4 ) ) )
= ( inf_in2572325071724192079at_nat @ ( B5 @ A ) @ ( comple1930127330929219688at_nat @ ( image_3617315916088684441at_nat @ B5 @ A4 ) ) ) ) ).
% INT_insert
thf(fact_2846_INT__insert,axiom,
! [B5: $o > set_Pr1261947904930325089at_nat,A: $o,A4: set_o] :
( ( comple1930127330929219688at_nat @ ( image_7077428525161401678at_nat @ B5 @ ( insert_o @ A @ A4 ) ) )
= ( inf_in2572325071724192079at_nat @ ( B5 @ A ) @ ( comple1930127330929219688at_nat @ ( image_7077428525161401678at_nat @ B5 @ A4 ) ) ) ) ).
% INT_insert
thf(fact_2847_INT__insert,axiom,
! [B5: nat > set_nat,A: nat,A4: set_nat] :
( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B5 @ ( insert_nat @ A @ A4 ) ) )
= ( inf_inf_set_nat @ ( B5 @ A ) @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B5 @ A4 ) ) ) ) ).
% INT_insert
thf(fact_2848_INT__insert,axiom,
! [B5: product_prod_nat_nat > set_nat,A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
( ( comple7806235888213564991et_nat @ ( image_15824709712370754et_nat @ B5 @ ( insert8211810215607154385at_nat @ A @ A4 ) ) )
= ( inf_inf_set_nat @ ( B5 @ A ) @ ( comple7806235888213564991et_nat @ ( image_15824709712370754et_nat @ B5 @ A4 ) ) ) ) ).
% INT_insert
thf(fact_2849_INT__insert,axiom,
! [B5: produc3843707927480180839at_nat > set_nat,A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
( ( comple7806235888213564991et_nat @ ( image_3160451284896210088et_nat @ B5 @ ( insert9069300056098147895at_nat @ A @ A4 ) ) )
= ( inf_inf_set_nat @ ( B5 @ A ) @ ( comple7806235888213564991et_nat @ ( image_3160451284896210088et_nat @ B5 @ A4 ) ) ) ) ).
% INT_insert
thf(fact_2850_INT__insert,axiom,
! [B5: product_unit > set_nat,A: product_unit,A4: set_Product_unit] :
( ( comple7806235888213564991et_nat @ ( image_8173849991827986510et_nat @ B5 @ ( insert_Product_unit @ A @ A4 ) ) )
= ( inf_inf_set_nat @ ( B5 @ A ) @ ( comple7806235888213564991et_nat @ ( image_8173849991827986510et_nat @ B5 @ A4 ) ) ) ) ).
% INT_insert
thf(fact_2851_INT__insert,axiom,
! [B5: $o > set_nat,A: $o,A4: set_o] :
( ( comple7806235888213564991et_nat @ ( image_o_set_nat @ B5 @ ( insert_o @ A @ A4 ) ) )
= ( inf_inf_set_nat @ ( B5 @ A ) @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ B5 @ A4 ) ) ) ) ).
% INT_insert
thf(fact_2852_inj__on__apsnd,axiom,
! [F: code_integer > code_integer,A4: set_Code_integer] :
( ( inj_on20428373856169009nteger @ ( produc6499014454317279255nteger @ F )
@ ( produc5882431272649516420nteger @ top_to4645266643341252675nteger
@ ^ [Uu: code_integer] : A4 ) )
= ( inj_on1158064722259180589nteger @ F @ A4 ) ) ).
% inj_on_apsnd
thf(fact_2853_inj__on__insert,axiom,
! [F: int > nat,A: int,A4: set_int] :
( ( inj_on_int_nat @ F @ ( insert_int @ A @ A4 ) )
= ( ( inj_on_int_nat @ F @ A4 )
& ~ ( member_nat @ ( F @ A ) @ ( image_int_nat @ F @ ( minus_minus_set_int @ A4 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_2854_inj__on__insert,axiom,
! [F: int > int,A: int,A4: set_int] :
( ( inj_on_int_int @ F @ ( insert_int @ A @ A4 ) )
= ( ( inj_on_int_int @ F @ A4 )
& ~ ( member_int @ ( F @ A ) @ ( image_int_int @ F @ ( minus_minus_set_int @ A4 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_2855_inj__on__insert,axiom,
! [F: product_unit > $o,A: product_unit,A4: set_Product_unit] :
( ( inj_on8854722734121536060unit_o @ F @ ( insert_Product_unit @ A @ A4 ) )
= ( ( inj_on8854722734121536060unit_o @ F @ A4 )
& ~ ( member_o @ ( F @ A ) @ ( image_Product_unit_o @ F @ ( minus_6452836326544984404t_unit @ A4 @ ( insert_Product_unit @ A @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_2856_inj__on__insert,axiom,
! [F: product_unit > nat,A: product_unit,A4: set_Product_unit] :
( ( inj_on8430439091780834860it_nat @ F @ ( insert_Product_unit @ A @ A4 ) )
= ( ( inj_on8430439091780834860it_nat @ F @ A4 )
& ~ ( member_nat @ ( F @ A ) @ ( image_875570014554754200it_nat @ F @ ( minus_6452836326544984404t_unit @ A4 @ ( insert_Product_unit @ A @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_2857_inj__on__insert,axiom,
! [F: product_unit > int,A: product_unit,A4: set_Product_unit] :
( ( inj_on8427948621271784584it_int @ F @ ( insert_Product_unit @ A @ A4 ) )
= ( ( inj_on8427948621271784584it_int @ F @ A4 )
& ~ ( member_int @ ( F @ A ) @ ( image_873079544045703924it_int @ F @ ( minus_6452836326544984404t_unit @ A4 @ ( insert_Product_unit @ A @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_2858_inj__on__insert,axiom,
! [F: $o > $o,A: $o,A4: set_o] :
( ( inj_on_o_o @ F @ ( insert_o @ A @ A4 ) )
= ( ( inj_on_o_o @ F @ A4 )
& ~ ( member_o @ ( F @ A ) @ ( image_o_o @ F @ ( minus_minus_set_o @ A4 @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_2859_inj__on__insert,axiom,
! [F: $o > nat,A: $o,A4: set_o] :
( ( inj_on_o_nat @ F @ ( insert_o @ A @ A4 ) )
= ( ( inj_on_o_nat @ F @ A4 )
& ~ ( member_nat @ ( F @ A ) @ ( image_o_nat @ F @ ( minus_minus_set_o @ A4 @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_2860_inj__on__insert,axiom,
! [F: $o > int,A: $o,A4: set_o] :
( ( inj_on_o_int @ F @ ( insert_o @ A @ A4 ) )
= ( ( inj_on_o_int @ F @ A4 )
& ~ ( member_int @ ( F @ A ) @ ( image_o_int @ F @ ( minus_minus_set_o @ A4 @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_2861_inj__on__insert,axiom,
! [F: nat > rat,A: nat,A4: set_nat] :
( ( inj_on_nat_rat @ F @ ( insert_nat @ A @ A4 ) )
= ( ( inj_on_nat_rat @ F @ A4 )
& ~ ( member_rat @ ( F @ A ) @ ( image_nat_rat @ F @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_2862_inj__on__insert,axiom,
! [F: nat > $o,A: nat,A4: set_nat] :
( ( inj_on_nat_o @ F @ ( insert_nat @ A @ A4 ) )
= ( ( inj_on_nat_o @ F @ A4 )
& ~ ( member_o @ ( F @ A ) @ ( image_nat_o @ F @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_2863_inj__on__Un__image__eq__iff,axiom,
! [F: nat > rat,A4: set_nat,B5: set_nat] :
( ( inj_on_nat_rat @ F @ ( sup_sup_set_nat @ A4 @ B5 ) )
=> ( ( ( image_nat_rat @ F @ A4 )
= ( image_nat_rat @ F @ B5 ) )
= ( A4 = B5 ) ) ) ).
% inj_on_Un_image_eq_iff
thf(fact_2864_inj__on__Un__image__eq__iff,axiom,
! [F: nat > nat,A4: set_nat,B5: set_nat] :
( ( inj_on_nat_nat @ F @ ( sup_sup_set_nat @ A4 @ B5 ) )
=> ( ( ( image_nat_nat @ F @ A4 )
= ( image_nat_nat @ F @ B5 ) )
= ( A4 = B5 ) ) ) ).
% inj_on_Un_image_eq_iff
thf(fact_2865_inj__on__Un__image__eq__iff,axiom,
! [F: nat > int,A4: set_nat,B5: set_nat] :
( ( inj_on_nat_int @ F @ ( sup_sup_set_nat @ A4 @ B5 ) )
=> ( ( ( image_nat_int @ F @ A4 )
= ( image_nat_int @ F @ B5 ) )
= ( A4 = B5 ) ) ) ).
% inj_on_Un_image_eq_iff
thf(fact_2866_inj__on__Un__image__eq__iff,axiom,
! [F: int > nat,A4: set_int,B5: set_int] :
( ( inj_on_int_nat @ F @ ( sup_sup_set_int @ A4 @ B5 ) )
=> ( ( ( image_int_nat @ F @ A4 )
= ( image_int_nat @ F @ B5 ) )
= ( A4 = B5 ) ) ) ).
% inj_on_Un_image_eq_iff
thf(fact_2867_inj__on__Un__image__eq__iff,axiom,
! [F: int > int,A4: set_int,B5: set_int] :
( ( inj_on_int_int @ F @ ( sup_sup_set_int @ A4 @ B5 ) )
=> ( ( ( image_int_int @ F @ A4 )
= ( image_int_int @ F @ B5 ) )
= ( A4 = B5 ) ) ) ).
% inj_on_Un_image_eq_iff
thf(fact_2868_Sigma__Un__distrib1,axiom,
! [I3: set_nat,J3: set_nat,C3: nat > set_nat] :
( ( produc457027306803732586at_nat @ ( sup_sup_set_nat @ I3 @ J3 ) @ C3 )
= ( sup_su6327502436637775413at_nat @ ( produc457027306803732586at_nat @ I3 @ C3 ) @ ( produc457027306803732586at_nat @ J3 @ C3 ) ) ) ).
% Sigma_Un_distrib1
thf(fact_2869_Sigma__Un__distrib1,axiom,
! [I3: set_Pr1261947904930325089at_nat,J3: set_Pr1261947904930325089at_nat,C3: product_prod_nat_nat > set_Pr1261947904930325089at_nat] :
( ( produc2761391749766926216at_nat @ ( sup_su6327502436637775413at_nat @ I3 @ J3 ) @ C3 )
= ( sup_su718114333110466843at_nat @ ( produc2761391749766926216at_nat @ I3 @ C3 ) @ ( produc2761391749766926216at_nat @ J3 @ C3 ) ) ) ).
% Sigma_Un_distrib1
thf(fact_2870_Sigma__Un__distrib1,axiom,
! [I3: set_se7855581050983116737at_nat,J3: set_se7855581050983116737at_nat,C3: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat] :
( ( produc7726050920695093128at_nat @ ( sup_su3642409539654194069at_nat @ I3 @ J3 ) @ C3 )
= ( sup_su5525570899277871387at_nat @ ( produc7726050920695093128at_nat @ I3 @ C3 ) @ ( produc7726050920695093128at_nat @ J3 @ C3 ) ) ) ).
% Sigma_Un_distrib1
thf(fact_2871_Sigma__Int__distrib1,axiom,
! [I3: set_nat,J3: set_nat,C3: nat > set_nat] :
( ( produc457027306803732586at_nat @ ( inf_inf_set_nat @ I3 @ J3 ) @ C3 )
= ( inf_in2572325071724192079at_nat @ ( produc457027306803732586at_nat @ I3 @ C3 ) @ ( produc457027306803732586at_nat @ J3 @ C3 ) ) ) ).
% Sigma_Int_distrib1
thf(fact_2872_inj__on__disjoint__Un,axiom,
! [F: nat > rat,A4: set_nat,G: nat > rat,B5: set_nat] :
( ( inj_on_nat_rat @ F @ A4 )
=> ( ( inj_on_nat_rat @ G @ B5 )
=> ( ( ( inf_inf_set_rat @ ( image_nat_rat @ F @ A4 ) @ ( image_nat_rat @ G @ B5 ) )
= bot_bot_set_rat )
=> ( inj_on_nat_rat
@ ^ [X4: nat] : ( if_rat @ ( member_nat @ X4 @ A4 ) @ ( F @ X4 ) @ ( G @ X4 ) )
@ ( sup_sup_set_nat @ A4 @ B5 ) ) ) ) ) ).
% inj_on_disjoint_Un
thf(fact_2873_inj__on__disjoint__Un,axiom,
! [F: nat > int,A4: set_nat,G: nat > int,B5: set_nat] :
( ( inj_on_nat_int @ F @ A4 )
=> ( ( inj_on_nat_int @ G @ B5 )
=> ( ( ( inf_inf_set_int @ ( image_nat_int @ F @ A4 ) @ ( image_nat_int @ G @ B5 ) )
= bot_bot_set_int )
=> ( inj_on_nat_int
@ ^ [X4: nat] : ( if_int @ ( member_nat @ X4 @ A4 ) @ ( F @ X4 ) @ ( G @ X4 ) )
@ ( sup_sup_set_nat @ A4 @ B5 ) ) ) ) ) ).
% inj_on_disjoint_Un
thf(fact_2874_inj__on__disjoint__Un,axiom,
! [F: int > int,A4: set_int,G: int > int,B5: set_int] :
( ( inj_on_int_int @ F @ A4 )
=> ( ( inj_on_int_int @ G @ B5 )
=> ( ( ( inf_inf_set_int @ ( image_int_int @ F @ A4 ) @ ( image_int_int @ G @ B5 ) )
= bot_bot_set_int )
=> ( inj_on_int_int
@ ^ [X4: int] : ( if_int @ ( member_int @ X4 @ A4 ) @ ( F @ X4 ) @ ( G @ X4 ) )
@ ( sup_sup_set_int @ A4 @ B5 ) ) ) ) ) ).
% inj_on_disjoint_Un
thf(fact_2875_inj__on__disjoint__Un,axiom,
! [F: $o > product_unit,A4: set_o,G: $o > product_unit,B5: set_o] :
( ( inj_on4268752236855399398t_unit @ F @ A4 )
=> ( ( inj_on4268752236855399398t_unit @ G @ B5 )
=> ( ( ( inf_in4660618365625256667t_unit @ ( image_o_Product_unit @ F @ A4 ) @ ( image_o_Product_unit @ G @ B5 ) )
= bot_bo3957492148770167129t_unit )
=> ( inj_on4268752236855399398t_unit
@ ^ [X4: $o] : ( if_Product_unit @ ( member_o @ X4 @ A4 ) @ ( F @ X4 ) @ ( G @ X4 ) )
@ ( sup_sup_set_o @ A4 @ B5 ) ) ) ) ) ).
% inj_on_disjoint_Un
thf(fact_2876_inj__on__disjoint__Un,axiom,
! [F: nat > product_unit,A4: set_nat,G: nat > product_unit,B5: set_nat] :
( ( inj_on7061601236592826506t_unit @ F @ A4 )
=> ( ( inj_on7061601236592826506t_unit @ G @ B5 )
=> ( ( ( inf_in4660618365625256667t_unit @ ( image_8730104196221521654t_unit @ F @ A4 ) @ ( image_8730104196221521654t_unit @ G @ B5 ) )
= bot_bo3957492148770167129t_unit )
=> ( inj_on7061601236592826506t_unit
@ ^ [X4: nat] : ( if_Product_unit @ ( member_nat @ X4 @ A4 ) @ ( F @ X4 ) @ ( G @ X4 ) )
@ ( sup_sup_set_nat @ A4 @ B5 ) ) ) ) ) ).
% inj_on_disjoint_Un
thf(fact_2877_inj__on__disjoint__Un,axiom,
! [F: int > product_unit,A4: set_int,G: int > product_unit,B5: set_int] :
( ( inj_on4178023183795006766t_unit @ F @ A4 )
=> ( ( inj_on4178023183795006766t_unit @ G @ B5 )
=> ( ( ( inf_in4660618365625256667t_unit @ ( image_5846526143423701914t_unit @ F @ A4 ) @ ( image_5846526143423701914t_unit @ G @ B5 ) )
= bot_bo3957492148770167129t_unit )
=> ( inj_on4178023183795006766t_unit
@ ^ [X4: int] : ( if_Product_unit @ ( member_int @ X4 @ A4 ) @ ( F @ X4 ) @ ( G @ X4 ) )
@ ( sup_sup_set_int @ A4 @ B5 ) ) ) ) ) ).
% inj_on_disjoint_Un
thf(fact_2878_inj__on__disjoint__Un,axiom,
! [F: $o > $o,A4: set_o,G: $o > $o,B5: set_o] :
( ( inj_on_o_o @ F @ A4 )
=> ( ( inj_on_o_o @ G @ B5 )
=> ( ( ( inf_inf_set_o @ ( image_o_o @ F @ A4 ) @ ( image_o_o @ G @ B5 ) )
= bot_bot_set_o )
=> ( inj_on_o_o
@ ^ [X4: $o] :
( ( ( member_o @ X4 @ A4 )
=> ( F @ X4 ) )
& ( ~ ( member_o @ X4 @ A4 )
=> ( G @ X4 ) ) )
@ ( sup_sup_set_o @ A4 @ B5 ) ) ) ) ) ).
% inj_on_disjoint_Un
thf(fact_2879_inj__on__disjoint__Un,axiom,
! [F: nat > $o,A4: set_nat,G: nat > $o,B5: set_nat] :
( ( inj_on_nat_o @ F @ A4 )
=> ( ( inj_on_nat_o @ G @ B5 )
=> ( ( ( inf_inf_set_o @ ( image_nat_o @ F @ A4 ) @ ( image_nat_o @ G @ B5 ) )
= bot_bot_set_o )
=> ( inj_on_nat_o
@ ^ [X4: nat] :
( ( ( member_nat @ X4 @ A4 )
=> ( F @ X4 ) )
& ( ~ ( member_nat @ X4 @ A4 )
=> ( G @ X4 ) ) )
@ ( sup_sup_set_nat @ A4 @ B5 ) ) ) ) ) ).
% inj_on_disjoint_Un
thf(fact_2880_inj__on__disjoint__Un,axiom,
! [F: int > $o,A4: set_int,G: int > $o,B5: set_int] :
( ( inj_on_int_o @ F @ A4 )
=> ( ( inj_on_int_o @ G @ B5 )
=> ( ( ( inf_inf_set_o @ ( image_int_o @ F @ A4 ) @ ( image_int_o @ G @ B5 ) )
= bot_bot_set_o )
=> ( inj_on_int_o
@ ^ [X4: int] :
( ( ( member_int @ X4 @ A4 )
=> ( F @ X4 ) )
& ( ~ ( member_int @ X4 @ A4 )
=> ( G @ X4 ) ) )
@ ( sup_sup_set_int @ A4 @ B5 ) ) ) ) ) ).
% inj_on_disjoint_Un
thf(fact_2881_inj__on__disjoint__Un,axiom,
! [F: $o > nat,A4: set_o,G: $o > nat,B5: set_o] :
( ( inj_on_o_nat @ F @ A4 )
=> ( ( inj_on_o_nat @ G @ B5 )
=> ( ( ( inf_inf_set_nat @ ( image_o_nat @ F @ A4 ) @ ( image_o_nat @ G @ B5 ) )
= bot_bot_set_nat )
=> ( inj_on_o_nat
@ ^ [X4: $o] : ( if_nat @ ( member_o @ X4 @ A4 ) @ ( F @ X4 ) @ ( G @ X4 ) )
@ ( sup_sup_set_o @ A4 @ B5 ) ) ) ) ) ).
% inj_on_disjoint_Un
thf(fact_2882_Sigma__Un__distrib2,axiom,
! [I3: set_nat,A4: nat > set_nat,B5: nat > set_nat] :
( ( produc457027306803732586at_nat @ I3
@ ^ [I2: nat] : ( sup_sup_set_nat @ ( A4 @ I2 ) @ ( B5 @ I2 ) ) )
= ( sup_su6327502436637775413at_nat @ ( produc457027306803732586at_nat @ I3 @ A4 ) @ ( produc457027306803732586at_nat @ I3 @ B5 ) ) ) ).
% Sigma_Un_distrib2
thf(fact_2883_Sigma__Un__distrib2,axiom,
! [I3: set_Pr1261947904930325089at_nat,A4: product_prod_nat_nat > set_Pr1261947904930325089at_nat,B5: product_prod_nat_nat > set_Pr1261947904930325089at_nat] :
( ( produc2761391749766926216at_nat @ I3
@ ^ [I2: product_prod_nat_nat] : ( sup_su6327502436637775413at_nat @ ( A4 @ I2 ) @ ( B5 @ I2 ) ) )
= ( sup_su718114333110466843at_nat @ ( produc2761391749766926216at_nat @ I3 @ A4 ) @ ( produc2761391749766926216at_nat @ I3 @ B5 ) ) ) ).
% Sigma_Un_distrib2
thf(fact_2884_Sigma__Un__distrib2,axiom,
! [I3: set_se7855581050983116737at_nat,A4: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat,B5: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat] :
( ( produc7726050920695093128at_nat @ I3
@ ^ [I2: set_Pr1261947904930325089at_nat] : ( sup_su3642409539654194069at_nat @ ( A4 @ I2 ) @ ( B5 @ I2 ) ) )
= ( sup_su5525570899277871387at_nat @ ( produc7726050920695093128at_nat @ I3 @ A4 ) @ ( produc7726050920695093128at_nat @ I3 @ B5 ) ) ) ).
% Sigma_Un_distrib2
thf(fact_2885_Times__Un__distrib1,axiom,
! [A4: set_nat,B5: set_nat,C3: set_nat] :
( ( produc457027306803732586at_nat @ ( sup_sup_set_nat @ A4 @ B5 )
@ ^ [Uu: nat] : C3 )
= ( sup_su6327502436637775413at_nat
@ ( produc457027306803732586at_nat @ A4
@ ^ [Uu: nat] : C3 )
@ ( produc457027306803732586at_nat @ B5
@ ^ [Uu: nat] : C3 ) ) ) ).
% Times_Un_distrib1
thf(fact_2886_Times__Un__distrib1,axiom,
! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat] :
( ( produc2761391749766926216at_nat @ ( sup_su6327502436637775413at_nat @ A4 @ B5 )
@ ^ [Uu: product_prod_nat_nat] : C3 )
= ( sup_su718114333110466843at_nat
@ ( produc2761391749766926216at_nat @ A4
@ ^ [Uu: product_prod_nat_nat] : C3 )
@ ( produc2761391749766926216at_nat @ B5
@ ^ [Uu: product_prod_nat_nat] : C3 ) ) ) ).
% Times_Un_distrib1
thf(fact_2887_Times__Un__distrib1,axiom,
! [A4: set_se7855581050983116737at_nat,B5: set_se7855581050983116737at_nat,C3: set_se7855581050983116737at_nat] :
( ( produc7726050920695093128at_nat @ ( sup_su3642409539654194069at_nat @ A4 @ B5 )
@ ^ [Uu: set_Pr1261947904930325089at_nat] : C3 )
= ( sup_su5525570899277871387at_nat
@ ( produc7726050920695093128at_nat @ A4
@ ^ [Uu: set_Pr1261947904930325089at_nat] : C3 )
@ ( produc7726050920695093128at_nat @ B5
@ ^ [Uu: set_Pr1261947904930325089at_nat] : C3 ) ) ) ).
% Times_Un_distrib1
thf(fact_2888_Times__Int__Times,axiom,
! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat,D: set_Pr1261947904930325089at_nat] :
( ( inf_in4302113700860409141at_nat
@ ( produc2761391749766926216at_nat @ A4
@ ^ [Uu: product_prod_nat_nat] : B5 )
@ ( produc2761391749766926216at_nat @ C3
@ ^ [Uu: product_prod_nat_nat] : D ) )
= ( produc2761391749766926216at_nat @ ( inf_in2572325071724192079at_nat @ A4 @ C3 )
@ ^ [Uu: product_prod_nat_nat] : ( inf_in2572325071724192079at_nat @ B5 @ D ) ) ) ).
% Times_Int_Times
thf(fact_2889_Times__Int__Times,axiom,
! [A4: set_nat,B5: set_nat,C3: set_nat,D: set_nat] :
( ( inf_in2572325071724192079at_nat
@ ( produc457027306803732586at_nat @ A4
@ ^ [Uu: nat] : B5 )
@ ( produc457027306803732586at_nat @ C3
@ ^ [Uu: nat] : D ) )
= ( produc457027306803732586at_nat @ ( inf_inf_set_nat @ A4 @ C3 )
@ ^ [Uu: nat] : ( inf_inf_set_nat @ B5 @ D ) ) ) ).
% Times_Int_Times
thf(fact_2890_Sigma__Int__distrib2,axiom,
! [I3: set_nat,A4: nat > set_nat,B5: nat > set_nat] :
( ( produc457027306803732586at_nat @ I3
@ ^ [I2: nat] : ( inf_inf_set_nat @ ( A4 @ I2 ) @ ( B5 @ I2 ) ) )
= ( inf_in2572325071724192079at_nat @ ( produc457027306803732586at_nat @ I3 @ A4 ) @ ( produc457027306803732586at_nat @ I3 @ B5 ) ) ) ).
% Sigma_Int_distrib2
thf(fact_2891_Times__Int__distrib1,axiom,
! [A4: set_nat,B5: set_nat,C3: set_nat] :
( ( produc457027306803732586at_nat @ ( inf_inf_set_nat @ A4 @ B5 )
@ ^ [Uu: nat] : C3 )
= ( inf_in2572325071724192079at_nat
@ ( produc457027306803732586at_nat @ A4
@ ^ [Uu: nat] : C3 )
@ ( produc457027306803732586at_nat @ B5
@ ^ [Uu: nat] : C3 ) ) ) ).
% Times_Int_distrib1
thf(fact_2892_inj__on__Un,axiom,
! [F: nat > rat,A4: set_nat,B5: set_nat] :
( ( inj_on_nat_rat @ F @ ( sup_sup_set_nat @ A4 @ B5 ) )
= ( ( inj_on_nat_rat @ F @ A4 )
& ( inj_on_nat_rat @ F @ B5 )
& ( ( inf_inf_set_rat @ ( image_nat_rat @ F @ ( minus_minus_set_nat @ A4 @ B5 ) ) @ ( image_nat_rat @ F @ ( minus_minus_set_nat @ B5 @ A4 ) ) )
= bot_bot_set_rat ) ) ) ).
% inj_on_Un
thf(fact_2893_inj__on__Un,axiom,
! [F: nat > int,A4: set_nat,B5: set_nat] :
( ( inj_on_nat_int @ F @ ( sup_sup_set_nat @ A4 @ B5 ) )
= ( ( inj_on_nat_int @ F @ A4 )
& ( inj_on_nat_int @ F @ B5 )
& ( ( inf_inf_set_int @ ( image_nat_int @ F @ ( minus_minus_set_nat @ A4 @ B5 ) ) @ ( image_nat_int @ F @ ( minus_minus_set_nat @ B5 @ A4 ) ) )
= bot_bot_set_int ) ) ) ).
% inj_on_Un
thf(fact_2894_inj__on__Un,axiom,
! [F: int > int,A4: set_int,B5: set_int] :
( ( inj_on_int_int @ F @ ( sup_sup_set_int @ A4 @ B5 ) )
= ( ( inj_on_int_int @ F @ A4 )
& ( inj_on_int_int @ F @ B5 )
& ( ( inf_inf_set_int @ ( image_int_int @ F @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( image_int_int @ F @ ( minus_minus_set_int @ B5 @ A4 ) ) )
= bot_bot_set_int ) ) ) ).
% inj_on_Un
thf(fact_2895_inj__on__Un,axiom,
! [F: nat > nat,A4: set_nat,B5: set_nat] :
( ( inj_on_nat_nat @ F @ ( sup_sup_set_nat @ A4 @ B5 ) )
= ( ( inj_on_nat_nat @ F @ A4 )
& ( inj_on_nat_nat @ F @ B5 )
& ( ( inf_inf_set_nat @ ( image_nat_nat @ F @ ( minus_minus_set_nat @ A4 @ B5 ) ) @ ( image_nat_nat @ F @ ( minus_minus_set_nat @ B5 @ A4 ) ) )
= bot_bot_set_nat ) ) ) ).
% inj_on_Un
thf(fact_2896_inj__on__Un,axiom,
! [F: int > nat,A4: set_int,B5: set_int] :
( ( inj_on_int_nat @ F @ ( sup_sup_set_int @ A4 @ B5 ) )
= ( ( inj_on_int_nat @ F @ A4 )
& ( inj_on_int_nat @ F @ B5 )
& ( ( inf_inf_set_nat @ ( image_int_nat @ F @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( image_int_nat @ F @ ( minus_minus_set_int @ B5 @ A4 ) ) )
= bot_bot_set_nat ) ) ) ).
% inj_on_Un
thf(fact_2897_inj__on__Un,axiom,
! [F: produc859450856879609959at_nat > product_unit,A4: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] :
( ( inj_on1041401102088758999t_unit @ F @ ( sup_su718114333110466843at_nat @ A4 @ B5 ) )
= ( ( inj_on1041401102088758999t_unit @ F @ A4 )
& ( inj_on1041401102088758999t_unit @ F @ B5 )
& ( ( inf_in4660618365625256667t_unit @ ( image_8487516615895632747t_unit @ F @ ( minus_8321449233255521966at_nat @ A4 @ B5 ) ) @ ( image_8487516615895632747t_unit @ F @ ( minus_8321449233255521966at_nat @ B5 @ A4 ) ) )
= bot_bo3957492148770167129t_unit ) ) ) ).
% inj_on_Un
thf(fact_2898_inj__on__Un,axiom,
! [F: produc859450856879609959at_nat > $o,A4: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] :
( ( inj_on991952103320416098_nat_o @ F @ ( sup_su718114333110466843at_nat @ A4 @ B5 ) )
= ( ( inj_on991952103320416098_nat_o @ F @ A4 )
& ( inj_on991952103320416098_nat_o @ F @ B5 )
& ( ( inf_inf_set_o @ ( image_3275415580317762038_nat_o @ F @ ( minus_8321449233255521966at_nat @ A4 @ B5 ) ) @ ( image_3275415580317762038_nat_o @ F @ ( minus_8321449233255521966at_nat @ B5 @ A4 ) ) )
= bot_bot_set_o ) ) ) ).
% inj_on_Un
thf(fact_2899_inj__on__Un,axiom,
! [F: produc859450856879609959at_nat > nat,A4: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] :
( ( inj_on490725545064285830at_nat @ F @ ( sup_su718114333110466843at_nat @ A4 @ B5 ) )
= ( ( inj_on490725545064285830at_nat @ F @ A4 )
& ( inj_on490725545064285830at_nat @ F @ B5 )
& ( ( inf_inf_set_nat @ ( image_6160980552072806642at_nat @ F @ ( minus_8321449233255521966at_nat @ A4 @ B5 ) ) @ ( image_6160980552072806642at_nat @ F @ ( minus_8321449233255521966at_nat @ B5 @ A4 ) ) )
= bot_bot_set_nat ) ) ) ).
% inj_on_Un
thf(fact_2900_inj__on__Un,axiom,
! [F: produc3843707927480180839at_nat > product_unit,A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
( ( inj_on3108384905993948887t_unit @ F @ ( sup_su5525570899277871387at_nat @ A4 @ B5 ) )
= ( ( inj_on3108384905993948887t_unit @ F @ A4 )
& ( inj_on3108384905993948887t_unit @ F @ B5 )
& ( ( inf_in4660618365625256667t_unit @ ( image_7082456443403096939t_unit @ F @ ( minus_3314409938677909166at_nat @ A4 @ B5 ) ) @ ( image_7082456443403096939t_unit @ F @ ( minus_3314409938677909166at_nat @ B5 @ A4 ) ) )
= bot_bo3957492148770167129t_unit ) ) ) ).
% inj_on_Un
thf(fact_2901_inj__on__Un,axiom,
! [F: produc859450856879609959at_nat > product_prod_nat_nat,A4: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] :
( ( inj_on229527429751349963at_nat @ F @ ( sup_su718114333110466843at_nat @ A4 @ B5 ) )
= ( ( inj_on229527429751349963at_nat @ F @ A4 )
& ( inj_on229527429751349963at_nat @ F @ B5 )
& ( ( inf_in2572325071724192079at_nat @ ( image_6706873702294942047at_nat @ F @ ( minus_8321449233255521966at_nat @ A4 @ B5 ) ) @ ( image_6706873702294942047at_nat @ F @ ( minus_8321449233255521966at_nat @ B5 @ A4 ) ) )
= bot_bo2099793752762293965at_nat ) ) ) ).
% inj_on_Un
thf(fact_2902_inj__on__image,axiom,
! [F: int > nat,A4: set_set_int] :
( ( inj_on_int_nat @ F @ ( comple3221217463730067765et_int @ A4 ) )
=> ( inj_on1389844818000382683et_nat @ ( image_int_nat @ F ) @ A4 ) ) ).
% inj_on_image
thf(fact_2903_inj__on__image,axiom,
! [F: int > int,A4: set_set_int] :
( ( inj_on_int_int @ F @ ( comple3221217463730067765et_int @ A4 ) )
=> ( inj_on6435365835345961783et_int @ ( image_int_int @ F ) @ A4 ) ) ).
% inj_on_image
thf(fact_2904_inj__on__image,axiom,
! [F: nat > rat,A4: set_set_nat] :
( ( inj_on_nat_rat @ F @ ( comple7399068483239264473et_nat @ A4 ) )
=> ( inj_on1096178645466186887et_rat @ ( image_nat_rat @ F ) @ A4 ) ) ).
% inj_on_image
thf(fact_2905_inj__on__image,axiom,
! [F: nat > nat,A4: set_set_nat] :
( ( inj_on_nat_nat @ F @ ( comple7399068483239264473et_nat @ A4 ) )
=> ( inj_on4604407203859583615et_nat @ ( image_nat_nat @ F ) @ A4 ) ) ).
% inj_on_image
thf(fact_2906_inj__on__image,axiom,
! [F: nat > int,A4: set_set_nat] :
( ( inj_on_nat_int @ F @ ( comple7399068483239264473et_nat @ A4 ) )
=> ( inj_on426556184350386907et_int @ ( image_nat_int @ F ) @ A4 ) ) ).
% inj_on_image
thf(fact_2907_cSUP__union,axiom,
! [A4: set_int,F: int > nat,B5: set_int] :
( ( A4 != bot_bot_set_int )
=> ( ( condit2214826472909112428ve_nat @ ( image_int_nat @ F @ A4 ) )
=> ( ( B5 != bot_bot_set_int )
=> ( ( condit2214826472909112428ve_nat @ ( image_int_nat @ F @ B5 ) )
=> ( ( complete_Sup_Sup_nat @ ( image_int_nat @ F @ ( sup_sup_set_int @ A4 @ B5 ) ) )
= ( sup_sup_nat @ ( complete_Sup_Sup_nat @ ( image_int_nat @ F @ A4 ) ) @ ( complete_Sup_Sup_nat @ ( image_int_nat @ F @ B5 ) ) ) ) ) ) ) ) ).
% cSUP_union
thf(fact_2908_cSUP__union,axiom,
! [A4: set_nat,F: nat > nat,B5: set_nat] :
( ( A4 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ F @ A4 ) )
=> ( ( B5 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ F @ B5 ) )
=> ( ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ ( sup_sup_set_nat @ A4 @ B5 ) ) )
= ( sup_sup_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ A4 ) ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ B5 ) ) ) ) ) ) ) ) ).
% cSUP_union
thf(fact_2909_cSUP__union,axiom,
! [A4: set_int,F: int > int,B5: set_int] :
( ( A4 != bot_bot_set_int )
=> ( ( condit2212336002400062152ve_int @ ( image_int_int @ F @ A4 ) )
=> ( ( B5 != bot_bot_set_int )
=> ( ( condit2212336002400062152ve_int @ ( image_int_int @ F @ B5 ) )
=> ( ( complete_Sup_Sup_int @ ( image_int_int @ F @ ( sup_sup_set_int @ A4 @ B5 ) ) )
= ( sup_sup_int @ ( complete_Sup_Sup_int @ ( image_int_int @ F @ A4 ) ) @ ( complete_Sup_Sup_int @ ( image_int_int @ F @ B5 ) ) ) ) ) ) ) ) ).
% cSUP_union
thf(fact_2910_cSUP__union,axiom,
! [A4: set_Product_unit,F: product_unit > int,B5: set_Product_unit] :
( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ( condit2212336002400062152ve_int @ ( image_873079544045703924it_int @ F @ A4 ) )
=> ( ( B5 != bot_bo3957492148770167129t_unit )
=> ( ( condit2212336002400062152ve_int @ ( image_873079544045703924it_int @ F @ B5 ) )
=> ( ( complete_Sup_Sup_int @ ( image_873079544045703924it_int @ F @ ( sup_su793286257634532545t_unit @ A4 @ B5 ) ) )
= ( sup_sup_int @ ( complete_Sup_Sup_int @ ( image_873079544045703924it_int @ F @ A4 ) ) @ ( complete_Sup_Sup_int @ ( image_873079544045703924it_int @ F @ B5 ) ) ) ) ) ) ) ) ).
% cSUP_union
thf(fact_2911_cSUP__union,axiom,
! [A4: set_o,F: $o > int,B5: set_o] :
( ( A4 != bot_bot_set_o )
=> ( ( condit2212336002400062152ve_int @ ( image_o_int @ F @ A4 ) )
=> ( ( B5 != bot_bot_set_o )
=> ( ( condit2212336002400062152ve_int @ ( image_o_int @ F @ B5 ) )
=> ( ( complete_Sup_Sup_int @ ( image_o_int @ F @ ( sup_sup_set_o @ A4 @ B5 ) ) )
= ( sup_sup_int @ ( complete_Sup_Sup_int @ ( image_o_int @ F @ A4 ) ) @ ( complete_Sup_Sup_int @ ( image_o_int @ F @ B5 ) ) ) ) ) ) ) ) ).
% cSUP_union
thf(fact_2912_cSUP__union,axiom,
! [A4: set_nat,F: nat > int,B5: set_nat] :
( ( A4 != bot_bot_set_nat )
=> ( ( condit2212336002400062152ve_int @ ( image_nat_int @ F @ A4 ) )
=> ( ( B5 != bot_bot_set_nat )
=> ( ( condit2212336002400062152ve_int @ ( image_nat_int @ F @ B5 ) )
=> ( ( complete_Sup_Sup_int @ ( image_nat_int @ F @ ( sup_sup_set_nat @ A4 @ B5 ) ) )
= ( sup_sup_int @ ( complete_Sup_Sup_int @ ( image_nat_int @ F @ A4 ) ) @ ( complete_Sup_Sup_int @ ( image_nat_int @ F @ B5 ) ) ) ) ) ) ) ) ).
% cSUP_union
thf(fact_2913_cSUP__union,axiom,
! [A4: set_Product_unit,F: product_unit > product_unit,B5: set_Product_unit] :
( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ( condit2080475194861589617t_unit @ ( image_405062704495631173t_unit @ F @ A4 ) )
=> ( ( B5 != bot_bo3957492148770167129t_unit )
=> ( ( condit2080475194861589617t_unit @ ( image_405062704495631173t_unit @ F @ B5 ) )
=> ( ( comple4687483117567863418t_unit @ ( image_405062704495631173t_unit @ F @ ( sup_su793286257634532545t_unit @ A4 @ B5 ) ) )
= ( sup_sup_Product_unit @ ( comple4687483117567863418t_unit @ ( image_405062704495631173t_unit @ F @ A4 ) ) @ ( comple4687483117567863418t_unit @ ( image_405062704495631173t_unit @ F @ B5 ) ) ) ) ) ) ) ) ).
% cSUP_union
thf(fact_2914_cSUP__union,axiom,
! [A4: set_o,F: $o > product_unit,B5: set_o] :
( ( A4 != bot_bot_set_o )
=> ( ( condit2080475194861589617t_unit @ ( image_o_Product_unit @ F @ A4 ) )
=> ( ( B5 != bot_bot_set_o )
=> ( ( condit2080475194861589617t_unit @ ( image_o_Product_unit @ F @ B5 ) )
=> ( ( comple4687483117567863418t_unit @ ( image_o_Product_unit @ F @ ( sup_sup_set_o @ A4 @ B5 ) ) )
= ( sup_sup_Product_unit @ ( comple4687483117567863418t_unit @ ( image_o_Product_unit @ F @ A4 ) ) @ ( comple4687483117567863418t_unit @ ( image_o_Product_unit @ F @ B5 ) ) ) ) ) ) ) ) ).
% cSUP_union
thf(fact_2915_cSUP__union,axiom,
! [A4: set_nat,F: nat > product_unit,B5: set_nat] :
( ( A4 != bot_bot_set_nat )
=> ( ( condit2080475194861589617t_unit @ ( image_8730104196221521654t_unit @ F @ A4 ) )
=> ( ( B5 != bot_bot_set_nat )
=> ( ( condit2080475194861589617t_unit @ ( image_8730104196221521654t_unit @ F @ B5 ) )
=> ( ( comple4687483117567863418t_unit @ ( image_8730104196221521654t_unit @ F @ ( sup_sup_set_nat @ A4 @ B5 ) ) )
= ( sup_sup_Product_unit @ ( comple4687483117567863418t_unit @ ( image_8730104196221521654t_unit @ F @ A4 ) ) @ ( comple4687483117567863418t_unit @ ( image_8730104196221521654t_unit @ F @ B5 ) ) ) ) ) ) ) ) ).
% cSUP_union
thf(fact_2916_cSUP__union,axiom,
! [A4: set_Product_unit,F: product_unit > set_nat,B5: set_Product_unit] :
( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ( condit5477540289124974626et_nat @ ( image_8173849991827986510et_nat @ F @ A4 ) )
=> ( ( B5 != bot_bo3957492148770167129t_unit )
=> ( ( condit5477540289124974626et_nat @ ( image_8173849991827986510et_nat @ F @ B5 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ F @ ( sup_su793286257634532545t_unit @ A4 @ B5 ) ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ F @ A4 ) ) @ ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ F @ B5 ) ) ) ) ) ) ) ) ).
% cSUP_union
thf(fact_2917_cINF__union,axiom,
! [A4: set_int,F: int > nat,B5: set_int] :
( ( A4 != bot_bot_set_int )
=> ( ( condit1738341127787009408ow_nat @ ( image_int_nat @ F @ A4 ) )
=> ( ( B5 != bot_bot_set_int )
=> ( ( condit1738341127787009408ow_nat @ ( image_int_nat @ F @ B5 ) )
=> ( ( complete_Inf_Inf_nat @ ( image_int_nat @ F @ ( sup_sup_set_int @ A4 @ B5 ) ) )
= ( inf_inf_nat @ ( complete_Inf_Inf_nat @ ( image_int_nat @ F @ A4 ) ) @ ( complete_Inf_Inf_nat @ ( image_int_nat @ F @ B5 ) ) ) ) ) ) ) ) ).
% cINF_union
thf(fact_2918_cINF__union,axiom,
! [A4: set_nat,F: nat > nat,B5: set_nat] :
( ( A4 != bot_bot_set_nat )
=> ( ( condit1738341127787009408ow_nat @ ( image_nat_nat @ F @ A4 ) )
=> ( ( B5 != bot_bot_set_nat )
=> ( ( condit1738341127787009408ow_nat @ ( image_nat_nat @ F @ B5 ) )
=> ( ( complete_Inf_Inf_nat @ ( image_nat_nat @ F @ ( sup_sup_set_nat @ A4 @ B5 ) ) )
= ( inf_inf_nat @ ( complete_Inf_Inf_nat @ ( image_nat_nat @ F @ A4 ) ) @ ( complete_Inf_Inf_nat @ ( image_nat_nat @ F @ B5 ) ) ) ) ) ) ) ) ).
% cINF_union
thf(fact_2919_cINF__union,axiom,
! [A4: set_int,F: int > int,B5: set_int] :
( ( A4 != bot_bot_set_int )
=> ( ( condit1735850657277959132ow_int @ ( image_int_int @ F @ A4 ) )
=> ( ( B5 != bot_bot_set_int )
=> ( ( condit1735850657277959132ow_int @ ( image_int_int @ F @ B5 ) )
=> ( ( complete_Inf_Inf_int @ ( image_int_int @ F @ ( sup_sup_set_int @ A4 @ B5 ) ) )
= ( inf_inf_int @ ( complete_Inf_Inf_int @ ( image_int_int @ F @ A4 ) ) @ ( complete_Inf_Inf_int @ ( image_int_int @ F @ B5 ) ) ) ) ) ) ) ) ).
% cINF_union
thf(fact_2920_cINF__union,axiom,
! [A4: set_Product_unit,F: product_unit > int,B5: set_Product_unit] :
( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ( condit1735850657277959132ow_int @ ( image_873079544045703924it_int @ F @ A4 ) )
=> ( ( B5 != bot_bo3957492148770167129t_unit )
=> ( ( condit1735850657277959132ow_int @ ( image_873079544045703924it_int @ F @ B5 ) )
=> ( ( complete_Inf_Inf_int @ ( image_873079544045703924it_int @ F @ ( sup_su793286257634532545t_unit @ A4 @ B5 ) ) )
= ( inf_inf_int @ ( complete_Inf_Inf_int @ ( image_873079544045703924it_int @ F @ A4 ) ) @ ( complete_Inf_Inf_int @ ( image_873079544045703924it_int @ F @ B5 ) ) ) ) ) ) ) ) ).
% cINF_union
thf(fact_2921_cINF__union,axiom,
! [A4: set_o,F: $o > int,B5: set_o] :
( ( A4 != bot_bot_set_o )
=> ( ( condit1735850657277959132ow_int @ ( image_o_int @ F @ A4 ) )
=> ( ( B5 != bot_bot_set_o )
=> ( ( condit1735850657277959132ow_int @ ( image_o_int @ F @ B5 ) )
=> ( ( complete_Inf_Inf_int @ ( image_o_int @ F @ ( sup_sup_set_o @ A4 @ B5 ) ) )
= ( inf_inf_int @ ( complete_Inf_Inf_int @ ( image_o_int @ F @ A4 ) ) @ ( complete_Inf_Inf_int @ ( image_o_int @ F @ B5 ) ) ) ) ) ) ) ) ).
% cINF_union
thf(fact_2922_cINF__union,axiom,
! [A4: set_nat,F: nat > int,B5: set_nat] :
( ( A4 != bot_bot_set_nat )
=> ( ( condit1735850657277959132ow_int @ ( image_nat_int @ F @ A4 ) )
=> ( ( B5 != bot_bot_set_nat )
=> ( ( condit1735850657277959132ow_int @ ( image_nat_int @ F @ B5 ) )
=> ( ( complete_Inf_Inf_int @ ( image_nat_int @ F @ ( sup_sup_set_nat @ A4 @ B5 ) ) )
= ( inf_inf_int @ ( complete_Inf_Inf_int @ ( image_nat_int @ F @ A4 ) ) @ ( complete_Inf_Inf_int @ ( image_nat_int @ F @ B5 ) ) ) ) ) ) ) ) ).
% cINF_union
thf(fact_2923_cINF__union,axiom,
! [A4: set_Product_unit,F: product_unit > product_unit,B5: set_Product_unit] :
( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ( condit9161856309968323677t_unit @ ( image_405062704495631173t_unit @ F @ A4 ) )
=> ( ( B5 != bot_bo3957492148770167129t_unit )
=> ( ( condit9161856309968323677t_unit @ ( image_405062704495631173t_unit @ F @ B5 ) )
=> ( ( comple2584293577114468500t_unit @ ( image_405062704495631173t_unit @ F @ ( sup_su793286257634532545t_unit @ A4 @ B5 ) ) )
= ( inf_inf_Product_unit @ ( comple2584293577114468500t_unit @ ( image_405062704495631173t_unit @ F @ A4 ) ) @ ( comple2584293577114468500t_unit @ ( image_405062704495631173t_unit @ F @ B5 ) ) ) ) ) ) ) ) ).
% cINF_union
thf(fact_2924_cINF__union,axiom,
! [A4: set_o,F: $o > product_unit,B5: set_o] :
( ( A4 != bot_bot_set_o )
=> ( ( condit9161856309968323677t_unit @ ( image_o_Product_unit @ F @ A4 ) )
=> ( ( B5 != bot_bot_set_o )
=> ( ( condit9161856309968323677t_unit @ ( image_o_Product_unit @ F @ B5 ) )
=> ( ( comple2584293577114468500t_unit @ ( image_o_Product_unit @ F @ ( sup_sup_set_o @ A4 @ B5 ) ) )
= ( inf_inf_Product_unit @ ( comple2584293577114468500t_unit @ ( image_o_Product_unit @ F @ A4 ) ) @ ( comple2584293577114468500t_unit @ ( image_o_Product_unit @ F @ B5 ) ) ) ) ) ) ) ) ).
% cINF_union
thf(fact_2925_cINF__union,axiom,
! [A4: set_nat,F: nat > product_unit,B5: set_nat] :
( ( A4 != bot_bot_set_nat )
=> ( ( condit9161856309968323677t_unit @ ( image_8730104196221521654t_unit @ F @ A4 ) )
=> ( ( B5 != bot_bot_set_nat )
=> ( ( condit9161856309968323677t_unit @ ( image_8730104196221521654t_unit @ F @ B5 ) )
=> ( ( comple2584293577114468500t_unit @ ( image_8730104196221521654t_unit @ F @ ( sup_sup_set_nat @ A4 @ B5 ) ) )
= ( inf_inf_Product_unit @ ( comple2584293577114468500t_unit @ ( image_8730104196221521654t_unit @ F @ A4 ) ) @ ( comple2584293577114468500t_unit @ ( image_8730104196221521654t_unit @ F @ B5 ) ) ) ) ) ) ) ) ).
% cINF_union
thf(fact_2926_cINF__union,axiom,
! [A4: set_Product_unit,F: product_unit > set_nat,B5: set_Product_unit] :
( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ( condit68592940725977398et_nat @ ( image_8173849991827986510et_nat @ F @ A4 ) )
=> ( ( B5 != bot_bo3957492148770167129t_unit )
=> ( ( condit68592940725977398et_nat @ ( image_8173849991827986510et_nat @ F @ B5 ) )
=> ( ( comple7806235888213564991et_nat @ ( image_8173849991827986510et_nat @ F @ ( sup_su793286257634532545t_unit @ A4 @ B5 ) ) )
= ( inf_inf_set_nat @ ( comple7806235888213564991et_nat @ ( image_8173849991827986510et_nat @ F @ A4 ) ) @ ( comple7806235888213564991et_nat @ ( image_8173849991827986510et_nat @ F @ B5 ) ) ) ) ) ) ) ) ).
% cINF_union
thf(fact_2927_inj__on__image__Int,axiom,
! [F: nat > rat,C3: set_nat,A4: set_nat,B5: set_nat] :
( ( inj_on_nat_rat @ F @ C3 )
=> ( ( ord_less_eq_set_nat @ A4 @ C3 )
=> ( ( ord_less_eq_set_nat @ B5 @ C3 )
=> ( ( image_nat_rat @ F @ ( inf_inf_set_nat @ A4 @ B5 ) )
= ( inf_inf_set_rat @ ( image_nat_rat @ F @ A4 ) @ ( image_nat_rat @ F @ B5 ) ) ) ) ) ) ).
% inj_on_image_Int
thf(fact_2928_inj__on__image__Int,axiom,
! [F: nat > nat,C3: set_nat,A4: set_nat,B5: set_nat] :
( ( inj_on_nat_nat @ F @ C3 )
=> ( ( ord_less_eq_set_nat @ A4 @ C3 )
=> ( ( ord_less_eq_set_nat @ B5 @ C3 )
=> ( ( image_nat_nat @ F @ ( inf_inf_set_nat @ A4 @ B5 ) )
= ( inf_inf_set_nat @ ( image_nat_nat @ F @ A4 ) @ ( image_nat_nat @ F @ B5 ) ) ) ) ) ) ).
% inj_on_image_Int
thf(fact_2929_inj__on__image__Int,axiom,
! [F: nat > int,C3: set_nat,A4: set_nat,B5: set_nat] :
( ( inj_on_nat_int @ F @ C3 )
=> ( ( ord_less_eq_set_nat @ A4 @ C3 )
=> ( ( ord_less_eq_set_nat @ B5 @ C3 )
=> ( ( image_nat_int @ F @ ( inf_inf_set_nat @ A4 @ B5 ) )
= ( inf_inf_set_int @ ( image_nat_int @ F @ A4 ) @ ( image_nat_int @ F @ B5 ) ) ) ) ) ) ).
% inj_on_image_Int
thf(fact_2930_inj__on__image__Int,axiom,
! [F: int > nat,C3: set_int,A4: set_int,B5: set_int] :
( ( inj_on_int_nat @ F @ C3 )
=> ( ( ord_less_eq_set_int @ A4 @ C3 )
=> ( ( ord_less_eq_set_int @ B5 @ C3 )
=> ( ( image_int_nat @ F @ ( inf_inf_set_int @ A4 @ B5 ) )
= ( inf_inf_set_nat @ ( image_int_nat @ F @ A4 ) @ ( image_int_nat @ F @ B5 ) ) ) ) ) ) ).
% inj_on_image_Int
thf(fact_2931_inj__on__image__Int,axiom,
! [F: int > int,C3: set_int,A4: set_int,B5: set_int] :
( ( inj_on_int_int @ F @ C3 )
=> ( ( ord_less_eq_set_int @ A4 @ C3 )
=> ( ( ord_less_eq_set_int @ B5 @ C3 )
=> ( ( image_int_int @ F @ ( inf_inf_set_int @ A4 @ B5 ) )
= ( inf_inf_set_int @ ( image_int_int @ F @ A4 ) @ ( image_int_int @ F @ B5 ) ) ) ) ) ) ).
% inj_on_image_Int
thf(fact_2932_inj__on__image__Int,axiom,
! [F: product_prod_nat_nat > product_prod_nat_nat,C3: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
( ( inj_on8969904277767023793at_nat @ F @ C3 )
=> ( ( ord_le3146513528884898305at_nat @ A4 @ C3 )
=> ( ( ord_le3146513528884898305at_nat @ B5 @ C3 )
=> ( ( image_5168914502847457605at_nat @ F @ ( inf_in2572325071724192079at_nat @ A4 @ B5 ) )
= ( inf_in2572325071724192079at_nat @ ( image_5168914502847457605at_nat @ F @ A4 ) @ ( image_5168914502847457605at_nat @ F @ B5 ) ) ) ) ) ) ).
% inj_on_image_Int
thf(fact_2933_image__Int,axiom,
! [F: nat > rat,A4: set_nat,B5: set_nat] :
( ( inj_on_nat_rat @ F @ top_top_set_nat )
=> ( ( image_nat_rat @ F @ ( inf_inf_set_nat @ A4 @ B5 ) )
= ( inf_inf_set_rat @ ( image_nat_rat @ F @ A4 ) @ ( image_nat_rat @ F @ B5 ) ) ) ) ).
% image_Int
thf(fact_2934_image__Int,axiom,
! [F: nat > nat,A4: set_nat,B5: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( image_nat_nat @ F @ ( inf_inf_set_nat @ A4 @ B5 ) )
= ( inf_inf_set_nat @ ( image_nat_nat @ F @ A4 ) @ ( image_nat_nat @ F @ B5 ) ) ) ) ).
% image_Int
thf(fact_2935_image__Int,axiom,
! [F: nat > int,A4: set_nat,B5: set_nat] :
( ( inj_on_nat_int @ F @ top_top_set_nat )
=> ( ( image_nat_int @ F @ ( inf_inf_set_nat @ A4 @ B5 ) )
= ( inf_inf_set_int @ ( image_nat_int @ F @ A4 ) @ ( image_nat_int @ F @ B5 ) ) ) ) ).
% image_Int
thf(fact_2936_image__Int,axiom,
! [F: int > nat,A4: set_int,B5: set_int] :
( ( inj_on_int_nat @ F @ top_top_set_int )
=> ( ( image_int_nat @ F @ ( inf_inf_set_int @ A4 @ B5 ) )
= ( inf_inf_set_nat @ ( image_int_nat @ F @ A4 ) @ ( image_int_nat @ F @ B5 ) ) ) ) ).
% image_Int
thf(fact_2937_image__Int,axiom,
! [F: int > int,A4: set_int,B5: set_int] :
( ( inj_on_int_int @ F @ top_top_set_int )
=> ( ( image_int_int @ F @ ( inf_inf_set_int @ A4 @ B5 ) )
= ( inf_inf_set_int @ ( image_int_int @ F @ A4 ) @ ( image_int_int @ F @ B5 ) ) ) ) ).
% image_Int
thf(fact_2938_image__Int,axiom,
! [F: char > product_prod_nat_nat,A4: set_char,B5: set_char] :
( ( inj_on1021528964648911549at_nat @ F @ top_top_set_char )
=> ( ( image_8382823969816519465at_nat @ F @ ( inf_inf_set_char @ A4 @ B5 ) )
= ( inf_in2572325071724192079at_nat @ ( image_8382823969816519465at_nat @ F @ A4 ) @ ( image_8382823969816519465at_nat @ F @ B5 ) ) ) ) ).
% image_Int
thf(fact_2939_image__Int,axiom,
! [F: rat > product_prod_nat_nat,A4: set_rat,B5: set_rat] :
( ( inj_on8867882244352206326at_nat @ F @ top_top_set_rat )
=> ( ( image_9175953278516507234at_nat @ F @ ( inf_inf_set_rat @ A4 @ B5 ) )
= ( inf_in2572325071724192079at_nat @ ( image_9175953278516507234at_nat @ F @ A4 ) @ ( image_9175953278516507234at_nat @ F @ B5 ) ) ) ) ).
% image_Int
thf(fact_2940_image__Int,axiom,
! [F: nat > product_prod_nat_nat,A4: set_nat,B5: set_nat] :
( ( inj_on5538052773655684606at_nat @ F @ top_top_set_nat )
=> ( ( image_5846123807819985514at_nat @ F @ ( inf_inf_set_nat @ A4 @ B5 ) )
= ( inf_in2572325071724192079at_nat @ ( image_5846123807819985514at_nat @ F @ A4 ) @ ( image_5846123807819985514at_nat @ F @ B5 ) ) ) ) ).
% image_Int
thf(fact_2941_image__Int,axiom,
! [F: int > product_prod_nat_nat,A4: set_int,B5: set_int] :
( ( inj_on8575894858119818402at_nat @ F @ top_top_set_int )
=> ( ( image_8883965892284119310at_nat @ F @ ( inf_inf_set_int @ A4 @ B5 ) )
= ( inf_in2572325071724192079at_nat @ ( image_8883965892284119310at_nat @ F @ A4 ) @ ( image_8883965892284119310at_nat @ F @ B5 ) ) ) ) ).
% image_Int
thf(fact_2942_image__Int,axiom,
! [F: list_nat > product_prod_nat_nat,A4: set_list_nat,B5: set_list_nat] :
( ( inj_on7574789179980466158at_nat @ F @ top_top_set_list_nat )
=> ( ( image_2795870212791037530at_nat @ F @ ( inf_inf_set_list_nat @ A4 @ B5 ) )
= ( inf_in2572325071724192079at_nat @ ( image_2795870212791037530at_nat @ F @ A4 ) @ ( image_2795870212791037530at_nat @ F @ B5 ) ) ) ) ).
% image_Int
thf(fact_2943_inj__on__image__Pow,axiom,
! [F: nat > rat,A4: set_nat] :
( ( inj_on_nat_rat @ F @ A4 )
=> ( inj_on1096178645466186887et_rat @ ( image_nat_rat @ F ) @ ( pow_nat @ A4 ) ) ) ).
% inj_on_image_Pow
thf(fact_2944_inj__on__image__Pow,axiom,
! [F: nat > nat,A4: set_nat] :
( ( inj_on_nat_nat @ F @ A4 )
=> ( inj_on4604407203859583615et_nat @ ( image_nat_nat @ F ) @ ( pow_nat @ A4 ) ) ) ).
% inj_on_image_Pow
thf(fact_2945_inj__on__image__Pow,axiom,
! [F: nat > int,A4: set_nat] :
( ( inj_on_nat_int @ F @ A4 )
=> ( inj_on426556184350386907et_int @ ( image_nat_int @ F ) @ ( pow_nat @ A4 ) ) ) ).
% inj_on_image_Pow
thf(fact_2946_inj__on__image__Pow,axiom,
! [F: int > nat,A4: set_int] :
( ( inj_on_int_nat @ F @ A4 )
=> ( inj_on1389844818000382683et_nat @ ( image_int_nat @ F ) @ ( pow_int @ A4 ) ) ) ).
% inj_on_image_Pow
thf(fact_2947_inj__on__image__Pow,axiom,
! [F: int > int,A4: set_int] :
( ( inj_on_int_int @ F @ A4 )
=> ( inj_on6435365835345961783et_int @ ( image_int_int @ F ) @ ( pow_int @ A4 ) ) ) ).
% inj_on_image_Pow
thf(fact_2948_inj__on__image__Fpow,axiom,
! [F: nat > rat,A4: set_nat] :
( ( inj_on_nat_rat @ F @ A4 )
=> ( inj_on1096178645466186887et_rat @ ( image_nat_rat @ F ) @ ( finite_Fpow_nat @ A4 ) ) ) ).
% inj_on_image_Fpow
thf(fact_2949_inj__on__image__Fpow,axiom,
! [F: nat > nat,A4: set_nat] :
( ( inj_on_nat_nat @ F @ A4 )
=> ( inj_on4604407203859583615et_nat @ ( image_nat_nat @ F ) @ ( finite_Fpow_nat @ A4 ) ) ) ).
% inj_on_image_Fpow
thf(fact_2950_inj__on__image__Fpow,axiom,
! [F: nat > int,A4: set_nat] :
( ( inj_on_nat_int @ F @ A4 )
=> ( inj_on426556184350386907et_int @ ( image_nat_int @ F ) @ ( finite_Fpow_nat @ A4 ) ) ) ).
% inj_on_image_Fpow
thf(fact_2951_inj__on__image__Fpow,axiom,
! [F: int > nat,A4: set_int] :
( ( inj_on_int_nat @ F @ A4 )
=> ( inj_on1389844818000382683et_nat @ ( image_int_nat @ F ) @ ( finite_Fpow_int @ A4 ) ) ) ).
% inj_on_image_Fpow
thf(fact_2952_inj__on__image__Fpow,axiom,
! [F: int > int,A4: set_int] :
( ( inj_on_int_int @ F @ A4 )
=> ( inj_on6435365835345961783et_int @ ( image_int_int @ F ) @ ( finite_Fpow_int @ A4 ) ) ) ).
% inj_on_image_Fpow
thf(fact_2953_inj__on__imageI2,axiom,
! [F3: $o > $o,F: nat > $o,A4: set_nat] :
( ( inj_on_nat_o @ ( comp_o_o_nat @ F3 @ F ) @ A4 )
=> ( inj_on_nat_o @ F @ A4 ) ) ).
% inj_on_imageI2
thf(fact_2954_inj__on__imageI2,axiom,
! [F3: nat > num,F: int > nat,A4: set_int] :
( ( inj_on_int_num @ ( comp_nat_num_int @ F3 @ F ) @ A4 )
=> ( inj_on_int_nat @ F @ A4 ) ) ).
% inj_on_imageI2
thf(fact_2955_inj__on__imageI2,axiom,
! [F3: nat > nat,F: nat > nat,A4: set_nat] :
( ( inj_on_nat_nat @ ( comp_nat_nat_nat @ F3 @ F ) @ A4 )
=> ( inj_on_nat_nat @ F @ A4 ) ) ).
% inj_on_imageI2
thf(fact_2956_inj__on__imageI2,axiom,
! [F3: int > nat,F: int > int,A4: set_int] :
( ( inj_on_int_nat @ ( comp_int_nat_int @ F3 @ F ) @ A4 )
=> ( inj_on_int_int @ F @ A4 ) ) ).
% inj_on_imageI2
thf(fact_2957_inj__on__imageI2,axiom,
! [F3: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,F: code_integer > code_integer > code_integer,A4: set_Code_integer] :
( ( inj_on7031512951821619701nteger @ ( comp_C8797469213163452608nteger @ F3 @ F ) @ A4 )
=> ( inj_on108595395805848489nteger @ F @ A4 ) ) ).
% inj_on_imageI2
thf(fact_2958_SUP__union,axiom,
! [M: produc859450856879609959at_nat > set_Pr8693737435421807431at_nat,A4: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] :
( ( comple2748534634209756852at_nat @ ( image_6654316230217496229at_nat @ M @ ( sup_su718114333110466843at_nat @ A4 @ B5 ) ) )
= ( sup_su718114333110466843at_nat @ ( comple2748534634209756852at_nat @ ( image_6654316230217496229at_nat @ M @ A4 ) ) @ ( comple2748534634209756852at_nat @ ( image_6654316230217496229at_nat @ M @ B5 ) ) ) ) ).
% SUP_union
thf(fact_2959_SUP__union,axiom,
! [M: produc859450856879609959at_nat > set_Pr4329608150637261639at_nat,A4: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] :
( ( comple4364388969068068532at_nat @ ( image_3876558564303283877at_nat @ M @ ( sup_su718114333110466843at_nat @ A4 @ B5 ) ) )
= ( sup_su5525570899277871387at_nat @ ( comple4364388969068068532at_nat @ ( image_3876558564303283877at_nat @ M @ A4 ) ) @ ( comple4364388969068068532at_nat @ ( image_3876558564303283877at_nat @ M @ B5 ) ) ) ) ).
% SUP_union
thf(fact_2960_SUP__union,axiom,
! [M: produc3843707927480180839at_nat > set_Pr8693737435421807431at_nat,A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
( ( comple2748534634209756852at_nat @ ( image_7260680879217540773at_nat @ M @ ( sup_su5525570899277871387at_nat @ A4 @ B5 ) ) )
= ( sup_su718114333110466843at_nat @ ( comple2748534634209756852at_nat @ ( image_7260680879217540773at_nat @ M @ A4 ) ) @ ( comple2748534634209756852at_nat @ ( image_7260680879217540773at_nat @ M @ B5 ) ) ) ) ).
% SUP_union
thf(fact_2961_SUP__union,axiom,
! [M: produc3843707927480180839at_nat > set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
( ( comple4364388969068068532at_nat @ ( image_7364710425340685989at_nat @ M @ ( sup_su5525570899277871387at_nat @ A4 @ B5 ) ) )
= ( sup_su5525570899277871387at_nat @ ( comple4364388969068068532at_nat @ ( image_7364710425340685989at_nat @ M @ A4 ) ) @ ( comple4364388969068068532at_nat @ ( image_7364710425340685989at_nat @ M @ B5 ) ) ) ) ).
% SUP_union
thf(fact_2962_SUP__union,axiom,
! [M: produc859450856879609959at_nat > set_nat,A4: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] :
( ( comple7399068483239264473et_nat @ ( image_2819269535336293544et_nat @ M @ ( sup_su718114333110466843at_nat @ A4 @ B5 ) ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_2819269535336293544et_nat @ M @ A4 ) ) @ ( comple7399068483239264473et_nat @ ( image_2819269535336293544et_nat @ M @ B5 ) ) ) ) ).
% SUP_union
thf(fact_2963_SUP__union,axiom,
! [M: produc3843707927480180839at_nat > set_nat,A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
( ( comple7399068483239264473et_nat @ ( image_3160451284896210088et_nat @ M @ ( sup_su5525570899277871387at_nat @ A4 @ B5 ) ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_3160451284896210088et_nat @ M @ A4 ) ) @ ( comple7399068483239264473et_nat @ ( image_3160451284896210088et_nat @ M @ B5 ) ) ) ) ).
% SUP_union
thf(fact_2964_SUP__union,axiom,
! [M: produc859450856879609959at_nat > product_unit,A4: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] :
( ( comple4687483117567863418t_unit @ ( image_8487516615895632747t_unit @ M @ ( sup_su718114333110466843at_nat @ A4 @ B5 ) ) )
= ( sup_sup_Product_unit @ ( comple4687483117567863418t_unit @ ( image_8487516615895632747t_unit @ M @ A4 ) ) @ ( comple4687483117567863418t_unit @ ( image_8487516615895632747t_unit @ M @ B5 ) ) ) ) ).
% SUP_union
thf(fact_2965_SUP__union,axiom,
! [M: produc3843707927480180839at_nat > product_unit,A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
( ( comple4687483117567863418t_unit @ ( image_7082456443403096939t_unit @ M @ ( sup_su5525570899277871387at_nat @ A4 @ B5 ) ) )
= ( sup_sup_Product_unit @ ( comple4687483117567863418t_unit @ ( image_7082456443403096939t_unit @ M @ A4 ) ) @ ( comple4687483117567863418t_unit @ ( image_7082456443403096939t_unit @ M @ B5 ) ) ) ) ).
% SUP_union
thf(fact_2966_INF__union,axiom,
! [M: produc859450856879609959at_nat > set_Pr1261947904930325089at_nat,A4: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] :
( ( comple1930127330929219688at_nat @ ( image_1384942819424619199at_nat @ M @ ( sup_su718114333110466843at_nat @ A4 @ B5 ) ) )
= ( inf_in2572325071724192079at_nat @ ( comple1930127330929219688at_nat @ ( image_1384942819424619199at_nat @ M @ A4 ) ) @ ( comple1930127330929219688at_nat @ ( image_1384942819424619199at_nat @ M @ B5 ) ) ) ) ).
% INF_union
thf(fact_2967_INF__union,axiom,
! [M: produc3843707927480180839at_nat > set_Pr1261947904930325089at_nat,A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
( ( comple1930127330929219688at_nat @ ( image_7979982168039674559at_nat @ M @ ( sup_su5525570899277871387at_nat @ A4 @ B5 ) ) )
= ( inf_in2572325071724192079at_nat @ ( comple1930127330929219688at_nat @ ( image_7979982168039674559at_nat @ M @ A4 ) ) @ ( comple1930127330929219688at_nat @ ( image_7979982168039674559at_nat @ M @ B5 ) ) ) ) ).
% INF_union
thf(fact_2968_INF__union,axiom,
! [M: produc859450856879609959at_nat > set_nat,A4: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] :
( ( comple7806235888213564991et_nat @ ( image_2819269535336293544et_nat @ M @ ( sup_su718114333110466843at_nat @ A4 @ B5 ) ) )
= ( inf_inf_set_nat @ ( comple7806235888213564991et_nat @ ( image_2819269535336293544et_nat @ M @ A4 ) ) @ ( comple7806235888213564991et_nat @ ( image_2819269535336293544et_nat @ M @ B5 ) ) ) ) ).
% INF_union
thf(fact_2969_INF__union,axiom,
! [M: produc3843707927480180839at_nat > set_nat,A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
( ( comple7806235888213564991et_nat @ ( image_3160451284896210088et_nat @ M @ ( sup_su5525570899277871387at_nat @ A4 @ B5 ) ) )
= ( inf_inf_set_nat @ ( comple7806235888213564991et_nat @ ( image_3160451284896210088et_nat @ M @ A4 ) ) @ ( comple7806235888213564991et_nat @ ( image_3160451284896210088et_nat @ M @ B5 ) ) ) ) ).
% INF_union
thf(fact_2970_INF__union,axiom,
! [M: produc859450856879609959at_nat > product_unit,A4: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] :
( ( comple2584293577114468500t_unit @ ( image_8487516615895632747t_unit @ M @ ( sup_su718114333110466843at_nat @ A4 @ B5 ) ) )
= ( inf_inf_Product_unit @ ( comple2584293577114468500t_unit @ ( image_8487516615895632747t_unit @ M @ A4 ) ) @ ( comple2584293577114468500t_unit @ ( image_8487516615895632747t_unit @ M @ B5 ) ) ) ) ).
% INF_union
thf(fact_2971_INF__union,axiom,
! [M: produc3843707927480180839at_nat > product_unit,A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
( ( comple2584293577114468500t_unit @ ( image_7082456443403096939t_unit @ M @ ( sup_su5525570899277871387at_nat @ A4 @ B5 ) ) )
= ( inf_inf_Product_unit @ ( comple2584293577114468500t_unit @ ( image_7082456443403096939t_unit @ M @ A4 ) ) @ ( comple2584293577114468500t_unit @ ( image_7082456443403096939t_unit @ M @ B5 ) ) ) ) ).
% INF_union
thf(fact_2972_image__Un,axiom,
! [F: nat > rat,A4: set_nat,B5: set_nat] :
( ( image_nat_rat @ F @ ( sup_sup_set_nat @ A4 @ B5 ) )
= ( sup_sup_set_rat @ ( image_nat_rat @ F @ A4 ) @ ( image_nat_rat @ F @ B5 ) ) ) ).
% image_Un
thf(fact_2973_image__Un,axiom,
! [F: nat > nat,A4: set_nat,B5: set_nat] :
( ( image_nat_nat @ F @ ( sup_sup_set_nat @ A4 @ B5 ) )
= ( sup_sup_set_nat @ ( image_nat_nat @ F @ A4 ) @ ( image_nat_nat @ F @ B5 ) ) ) ).
% image_Un
thf(fact_2974_image__Un,axiom,
! [F: nat > int,A4: set_nat,B5: set_nat] :
( ( image_nat_int @ F @ ( sup_sup_set_nat @ A4 @ B5 ) )
= ( sup_sup_set_int @ ( image_nat_int @ F @ A4 ) @ ( image_nat_int @ F @ B5 ) ) ) ).
% image_Un
thf(fact_2975_image__Un,axiom,
! [F: int > nat,A4: set_int,B5: set_int] :
( ( image_int_nat @ F @ ( sup_sup_set_int @ A4 @ B5 ) )
= ( sup_sup_set_nat @ ( image_int_nat @ F @ A4 ) @ ( image_int_nat @ F @ B5 ) ) ) ).
% image_Un
thf(fact_2976_image__Un,axiom,
! [F: int > int,A4: set_int,B5: set_int] :
( ( image_int_int @ F @ ( sup_sup_set_int @ A4 @ B5 ) )
= ( sup_sup_set_int @ ( image_int_int @ F @ A4 ) @ ( image_int_int @ F @ B5 ) ) ) ).
% image_Un
thf(fact_2977_image__Un,axiom,
! [F: produc859450856879609959at_nat > produc859450856879609959at_nat,A4: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] :
( ( image_6321802852832981957at_nat @ F @ ( sup_su718114333110466843at_nat @ A4 @ B5 ) )
= ( sup_su718114333110466843at_nat @ ( image_6321802852832981957at_nat @ F @ A4 ) @ ( image_6321802852832981957at_nat @ F @ B5 ) ) ) ).
% image_Un
thf(fact_2978_image__Un,axiom,
! [F: produc859450856879609959at_nat > produc3843707927480180839at_nat,A4: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] :
( ( image_7637461350386011077at_nat @ F @ ( sup_su718114333110466843at_nat @ A4 @ B5 ) )
= ( sup_su5525570899277871387at_nat @ ( image_7637461350386011077at_nat @ F @ A4 ) @ ( image_7637461350386011077at_nat @ F @ B5 ) ) ) ).
% image_Un
thf(fact_2979_image__Un,axiom,
! [F: produc3843707927480180839at_nat > produc859450856879609959at_nat,A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
( ( image_5653576389946358725at_nat @ F @ ( sup_su5525570899277871387at_nat @ A4 @ B5 ) )
= ( sup_su718114333110466843at_nat @ ( image_5653576389946358725at_nat @ F @ A4 ) @ ( image_5653576389946358725at_nat @ F @ B5 ) ) ) ).
% image_Un
thf(fact_2980_image__Un,axiom,
! [F: produc3843707927480180839at_nat > produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
( ( image_960689587819713477at_nat @ F @ ( sup_su5525570899277871387at_nat @ A4 @ B5 ) )
= ( sup_su5525570899277871387at_nat @ ( image_960689587819713477at_nat @ F @ A4 ) @ ( image_960689587819713477at_nat @ F @ B5 ) ) ) ).
% image_Un
thf(fact_2981_INT__Un,axiom,
! [M: produc859450856879609959at_nat > set_Pr1261947904930325089at_nat,A4: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] :
( ( comple1930127330929219688at_nat @ ( image_1384942819424619199at_nat @ M @ ( sup_su718114333110466843at_nat @ A4 @ B5 ) ) )
= ( inf_in2572325071724192079at_nat @ ( comple1930127330929219688at_nat @ ( image_1384942819424619199at_nat @ M @ A4 ) ) @ ( comple1930127330929219688at_nat @ ( image_1384942819424619199at_nat @ M @ B5 ) ) ) ) ).
% INT_Un
thf(fact_2982_INT__Un,axiom,
! [M: produc3843707927480180839at_nat > set_Pr1261947904930325089at_nat,A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
( ( comple1930127330929219688at_nat @ ( image_7979982168039674559at_nat @ M @ ( sup_su5525570899277871387at_nat @ A4 @ B5 ) ) )
= ( inf_in2572325071724192079at_nat @ ( comple1930127330929219688at_nat @ ( image_7979982168039674559at_nat @ M @ A4 ) ) @ ( comple1930127330929219688at_nat @ ( image_7979982168039674559at_nat @ M @ B5 ) ) ) ) ).
% INT_Un
thf(fact_2983_INT__Un,axiom,
! [M: produc859450856879609959at_nat > set_nat,A4: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] :
( ( comple7806235888213564991et_nat @ ( image_2819269535336293544et_nat @ M @ ( sup_su718114333110466843at_nat @ A4 @ B5 ) ) )
= ( inf_inf_set_nat @ ( comple7806235888213564991et_nat @ ( image_2819269535336293544et_nat @ M @ A4 ) ) @ ( comple7806235888213564991et_nat @ ( image_2819269535336293544et_nat @ M @ B5 ) ) ) ) ).
% INT_Un
thf(fact_2984_INT__Un,axiom,
! [M: produc3843707927480180839at_nat > set_nat,A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
( ( comple7806235888213564991et_nat @ ( image_3160451284896210088et_nat @ M @ ( sup_su5525570899277871387at_nat @ A4 @ B5 ) ) )
= ( inf_inf_set_nat @ ( comple7806235888213564991et_nat @ ( image_3160451284896210088et_nat @ M @ A4 ) ) @ ( comple7806235888213564991et_nat @ ( image_3160451284896210088et_nat @ M @ B5 ) ) ) ) ).
% INT_Un
thf(fact_2985_inj__on__convol__ident,axiom,
! [F: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat,X5: set_se5479201105039950375at_nat] :
( inj_on1468925692805905105at_nat
@ ^ [X4: set_Pr4329608150637261639at_nat] : ( produc9060074326276436823at_nat @ X4 @ ( F @ X4 ) )
@ X5 ) ).
% inj_on_convol_ident
thf(fact_2986_inj__on__convol__ident,axiom,
! [F: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat,X5: set_se7855581050983116737at_nat] :
( inj_on699426629415580343at_nat
@ ^ [X4: set_Pr1261947904930325089at_nat] : ( produc2922128104949294807at_nat @ X4 @ ( F @ X4 ) )
@ X5 ) ).
% inj_on_convol_ident
thf(fact_2987_inj__on__convol__ident,axiom,
! [F: nat > num,X5: set_nat] :
( inj_on6246296710604522312at_num
@ ^ [X4: nat] : ( product_Pair_nat_num @ X4 @ ( F @ X4 ) )
@ X5 ) ).
% inj_on_convol_ident
thf(fact_2988_inj__on__convol__ident,axiom,
! [F: nat > nat,X5: set_nat] :
( inj_on5538052773655684606at_nat
@ ^ [X4: nat] : ( product_Pair_nat_nat @ X4 @ ( F @ X4 ) )
@ X5 ) ).
% inj_on_convol_ident
thf(fact_2989_inj__on__convol__ident,axiom,
! [F: int > int,X5: set_int] :
( inj_on5397397550511676250nt_int
@ ^ [X4: int] : ( product_Pair_int_int @ X4 @ ( F @ X4 ) )
@ X5 ) ).
% inj_on_convol_ident
thf(fact_2990_total__onI,axiom,
! [A4: set_Pr4329608150637261639at_nat,R: set_Pr5564308138774400199at_nat] :
( ! [X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
( ( member8757157785044589968at_nat @ X @ A4 )
=> ( ( member8757157785044589968at_nat @ Y @ A4 )
=> ( ( X != Y )
=> ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X @ Y ) @ R )
| ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ Y @ X ) @ R ) ) ) ) )
=> ( total_1513685709984341995at_nat @ A4 @ R ) ) ).
% total_onI
thf(fact_2991_total__onI,axiom,
! [A4: set_o,R: set_Product_prod_o_o] :
( ! [X: $o,Y: $o] :
( ( member_o @ X @ A4 )
=> ( ( member_o @ Y @ A4 )
=> ( ( X != Y )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ R )
| ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y @ X ) @ R ) ) ) ) )
=> ( total_on_o @ A4 @ R ) ) ).
% total_onI
thf(fact_2992_total__onI,axiom,
! [A4: set_se5479201105039950375at_nat,R: set_Pr7459493094073627847at_nat] :
( ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
( ( member6273874620674182256at_nat @ X @ A4 )
=> ( ( member6273874620674182256at_nat @ Y @ A4 )
=> ( ( X != Y )
=> ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ R )
| ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y @ X ) @ R ) ) ) ) )
=> ( total_5117575981256876747at_nat @ A4 @ R ) ) ).
% total_onI
thf(fact_2993_total__onI,axiom,
! [A4: set_se7855581050983116737at_nat,R: set_Pr4329608150637261639at_nat] :
( ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ X @ A4 )
=> ( ( member2643936169264416010at_nat @ Y @ A4 )
=> ( ( X != Y )
=> ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R )
| ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y @ X ) @ R ) ) ) ) )
=> ( total_7212417256332265445at_nat @ A4 @ R ) ) ).
% total_onI
thf(fact_2994_total__onI,axiom,
! [A4: set_nat,R: set_Pr1261947904930325089at_nat] :
( ! [X: nat,Y: nat] :
( ( member_nat @ X @ A4 )
=> ( ( member_nat @ Y @ A4 )
=> ( ( X != Y )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
| ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ R ) ) ) ) )
=> ( total_on_nat @ A4 @ R ) ) ).
% total_onI
thf(fact_2995_total__onI,axiom,
! [A4: set_int,R: set_Pr958786334691620121nt_int] :
( ! [X: int,Y: int] :
( ( member_int @ X @ A4 )
=> ( ( member_int @ Y @ A4 )
=> ( ( X != Y )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R )
| ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y @ X ) @ R ) ) ) ) )
=> ( total_on_int @ A4 @ R ) ) ).
% total_onI
thf(fact_2996_total__onI,axiom,
! [A4: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat] :
( ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X @ A4 )
=> ( ( member8440522571783428010at_nat @ Y @ A4 )
=> ( ( X != Y )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R )
| ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y @ X ) @ R ) ) ) ) )
=> ( total_3592101749530773125at_nat @ A4 @ R ) ) ).
% total_onI
thf(fact_2997_total__on__def,axiom,
( total_5117575981256876747at_nat
= ( ^ [A6: set_se5479201105039950375at_nat,R5: set_Pr7459493094073627847at_nat] :
! [X4: set_Pr4329608150637261639at_nat] :
( ( member6273874620674182256at_nat @ X4 @ A6 )
=> ! [Y5: set_Pr4329608150637261639at_nat] :
( ( member6273874620674182256at_nat @ Y5 @ A6 )
=> ( ( X4 != Y5 )
=> ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X4 @ Y5 ) @ R5 )
| ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y5 @ X4 ) @ R5 ) ) ) ) ) ) ) ).
% total_on_def
thf(fact_2998_total__on__def,axiom,
( total_7212417256332265445at_nat
= ( ^ [A6: set_se7855581050983116737at_nat,R5: set_Pr4329608150637261639at_nat] :
! [X4: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ X4 @ A6 )
=> ! [Y5: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ Y5 @ A6 )
=> ( ( X4 != Y5 )
=> ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y5 ) @ R5 )
| ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y5 @ X4 ) @ R5 ) ) ) ) ) ) ) ).
% total_on_def
thf(fact_2999_total__on__def,axiom,
( total_on_nat
= ( ^ [A6: set_nat,R5: set_Pr1261947904930325089at_nat] :
! [X4: nat] :
( ( member_nat @ X4 @ A6 )
=> ! [Y5: nat] :
( ( member_nat @ Y5 @ A6 )
=> ( ( X4 != Y5 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ R5 )
| ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ X4 ) @ R5 ) ) ) ) ) ) ) ).
% total_on_def
thf(fact_3000_total__on__def,axiom,
( total_on_int
= ( ^ [A6: set_int,R5: set_Pr958786334691620121nt_int] :
! [X4: int] :
( ( member_int @ X4 @ A6 )
=> ! [Y5: int] :
( ( member_int @ Y5 @ A6 )
=> ( ( X4 != Y5 )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y5 ) @ R5 )
| ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y5 @ X4 ) @ R5 ) ) ) ) ) ) ) ).
% total_on_def
thf(fact_3001_total__on__def,axiom,
( total_3592101749530773125at_nat
= ( ^ [A6: set_Pr1261947904930325089at_nat,R5: set_Pr8693737435421807431at_nat] :
! [X4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X4 @ A6 )
=> ! [Y5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ Y5 @ A6 )
=> ( ( X4 != Y5 )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y5 ) @ R5 )
| ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y5 @ X4 ) @ R5 ) ) ) ) ) ) ) ).
% total_on_def
thf(fact_3002_SUP__UNIV__bool__expand,axiom,
! [A4: $o > set_Pr8693737435421807431at_nat] :
( ( comple2748534634209756852at_nat @ ( image_8493167801588156084at_nat @ A4 @ top_top_set_o ) )
= ( sup_su718114333110466843at_nat @ ( A4 @ $true ) @ ( A4 @ $false ) ) ) ).
% SUP_UNIV_bool_expand
thf(fact_3003_SUP__UNIV__bool__expand,axiom,
! [A4: $o > set_Pr4329608150637261639at_nat] :
( ( comple4364388969068068532at_nat @ ( image_2496312346330513076at_nat @ A4 @ top_top_set_o ) )
= ( sup_su5525570899277871387at_nat @ ( A4 @ $true ) @ ( A4 @ $false ) ) ) ).
% SUP_UNIV_bool_expand
thf(fact_3004_SUP__UNIV__bool__expand,axiom,
! [A4: $o > set_nat] :
( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ A4 @ top_top_set_o ) )
= ( sup_sup_set_nat @ ( A4 @ $true ) @ ( A4 @ $false ) ) ) ).
% SUP_UNIV_bool_expand
thf(fact_3005_SUP__UNIV__bool__expand,axiom,
! [A4: $o > product_unit] :
( ( comple4687483117567863418t_unit @ ( image_o_Product_unit @ A4 @ top_top_set_o ) )
= ( sup_sup_Product_unit @ ( A4 @ $true ) @ ( A4 @ $false ) ) ) ).
% SUP_UNIV_bool_expand
thf(fact_3006_INF__UNIV__bool__expand,axiom,
! [A4: $o > set_Pr1261947904930325089at_nat] :
( ( comple1930127330929219688at_nat @ ( image_7077428525161401678at_nat @ A4 @ top_top_set_o ) )
= ( inf_in2572325071724192079at_nat @ ( A4 @ $true ) @ ( A4 @ $false ) ) ) ).
% INF_UNIV_bool_expand
thf(fact_3007_INF__UNIV__bool__expand,axiom,
! [A4: $o > set_nat] :
( ( comple7806235888213564991et_nat @ ( image_o_set_nat @ A4 @ top_top_set_o ) )
= ( inf_inf_set_nat @ ( A4 @ $true ) @ ( A4 @ $false ) ) ) ).
% INF_UNIV_bool_expand
thf(fact_3008_INF__UNIV__bool__expand,axiom,
! [A4: $o > product_unit] :
( ( comple2584293577114468500t_unit @ ( image_o_Product_unit @ A4 @ top_top_set_o ) )
= ( inf_inf_Product_unit @ ( A4 @ $true ) @ ( A4 @ $false ) ) ) ).
% INF_UNIV_bool_expand
thf(fact_3009_fun_Oinj__map,axiom,
! [F: $o > $o] :
( ( inj_on_o_o @ F @ top_top_set_o )
=> ( inj_on_nat_o_nat_o @ ( comp_o_o_nat @ F ) @ top_top_set_nat_o ) ) ).
% fun.inj_map
thf(fact_3010_fun_Oinj__map,axiom,
! [F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger] :
( ( inj_on3998011822705705585nteger @ F @ top_to3804862627464599871nteger )
=> ( inj_on5500977043710295017nteger @ ( comp_C8797469213163452608nteger @ F ) @ top_to3639204395728341691nteger ) ) ).
% fun.inj_map
thf(fact_3011_fun_Oinj__map,axiom,
! [F: nat > num] :
( ( inj_on_nat_num @ F @ top_top_set_nat )
=> ( inj_on5377432224680912883nt_num @ ( comp_nat_num_int @ F ) @ top_top_set_int_nat ) ) ).
% fun.inj_map
thf(fact_3012_fun_Oinj__map,axiom,
! [F: nat > nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( inj_on2461717442902640625at_nat @ ( comp_nat_nat_nat @ F ) @ top_top_set_nat_nat ) ) ).
% fun.inj_map
thf(fact_3013_fun_Oinj__map,axiom,
! [F: int > nat] :
( ( inj_on_int_nat @ F @ top_top_set_int )
=> ( inj_on856898432580761861nt_nat @ ( comp_int_nat_int @ F ) @ top_top_set_int_int ) ) ).
% fun.inj_map
thf(fact_3014_Un__eq__UN,axiom,
( sup_su718114333110466843at_nat
= ( ^ [A6: set_Pr8693737435421807431at_nat,B6: set_Pr8693737435421807431at_nat] :
( comple2748534634209756852at_nat
@ ( image_8493167801588156084at_nat
@ ^ [B4: $o] : ( if_set1508868558775481357at_nat @ B4 @ A6 @ B6 )
@ top_top_set_o ) ) ) ) ).
% Un_eq_UN
thf(fact_3015_Un__eq__UN,axiom,
( sup_su5525570899277871387at_nat
= ( ^ [A6: set_Pr4329608150637261639at_nat,B6: set_Pr4329608150637261639at_nat] :
( comple4364388969068068532at_nat
@ ( image_2496312346330513076at_nat
@ ^ [B4: $o] : ( if_set3208018901896979469at_nat @ B4 @ A6 @ B6 )
@ top_top_set_o ) ) ) ) ).
% Un_eq_UN
thf(fact_3016_Un__eq__UN,axiom,
( sup_sup_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [B4: $o] : ( if_set_nat @ B4 @ A6 @ B6 )
@ top_top_set_o ) ) ) ) ).
% Un_eq_UN
thf(fact_3017_UN__bool__eq,axiom,
! [A4: $o > set_Pr8693737435421807431at_nat] :
( ( comple2748534634209756852at_nat @ ( image_8493167801588156084at_nat @ A4 @ top_top_set_o ) )
= ( sup_su718114333110466843at_nat @ ( A4 @ $true ) @ ( A4 @ $false ) ) ) ).
% UN_bool_eq
thf(fact_3018_UN__bool__eq,axiom,
! [A4: $o > set_Pr4329608150637261639at_nat] :
( ( comple4364388969068068532at_nat @ ( image_2496312346330513076at_nat @ A4 @ top_top_set_o ) )
= ( sup_su5525570899277871387at_nat @ ( A4 @ $true ) @ ( A4 @ $false ) ) ) ).
% UN_bool_eq
thf(fact_3019_UN__bool__eq,axiom,
! [A4: $o > set_nat] :
( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ A4 @ top_top_set_o ) )
= ( sup_sup_set_nat @ ( A4 @ $true ) @ ( A4 @ $false ) ) ) ).
% UN_bool_eq
thf(fact_3020_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
! [A4: set_int,F: int > nat,G: int > nat] :
( ( A4 != bot_bot_set_int )
=> ( ( condit2214826472909112428ve_nat @ ( image_int_nat @ F @ A4 ) )
=> ( ( condit2214826472909112428ve_nat @ ( image_int_nat @ G @ A4 ) )
=> ( ( sup_sup_nat @ ( complete_Sup_Sup_nat @ ( image_int_nat @ F @ A4 ) ) @ ( complete_Sup_Sup_nat @ ( image_int_nat @ G @ A4 ) ) )
= ( complete_Sup_Sup_nat
@ ( image_int_nat
@ ^ [A5: int] : ( sup_sup_nat @ ( F @ A5 ) @ ( G @ A5 ) )
@ A4 ) ) ) ) ) ) ).
% conditionally_complete_lattice_class.SUP_sup_distrib
thf(fact_3021_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
! [A4: set_nat,F: nat > nat,G: nat > nat] :
( ( A4 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ F @ A4 ) )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ G @ A4 ) )
=> ( ( sup_sup_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ A4 ) ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ G @ A4 ) ) )
= ( complete_Sup_Sup_nat
@ ( image_nat_nat
@ ^ [A5: nat] : ( sup_sup_nat @ ( F @ A5 ) @ ( G @ A5 ) )
@ A4 ) ) ) ) ) ) ).
% conditionally_complete_lattice_class.SUP_sup_distrib
thf(fact_3022_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
! [A4: set_int,F: int > int,G: int > int] :
( ( A4 != bot_bot_set_int )
=> ( ( condit2212336002400062152ve_int @ ( image_int_int @ F @ A4 ) )
=> ( ( condit2212336002400062152ve_int @ ( image_int_int @ G @ A4 ) )
=> ( ( sup_sup_int @ ( complete_Sup_Sup_int @ ( image_int_int @ F @ A4 ) ) @ ( complete_Sup_Sup_int @ ( image_int_int @ G @ A4 ) ) )
= ( complete_Sup_Sup_int
@ ( image_int_int
@ ^ [A5: int] : ( sup_sup_int @ ( F @ A5 ) @ ( G @ A5 ) )
@ A4 ) ) ) ) ) ) ).
% conditionally_complete_lattice_class.SUP_sup_distrib
thf(fact_3023_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
! [A4: set_Product_unit,F: product_unit > int,G: product_unit > int] :
( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ( condit2212336002400062152ve_int @ ( image_873079544045703924it_int @ F @ A4 ) )
=> ( ( condit2212336002400062152ve_int @ ( image_873079544045703924it_int @ G @ A4 ) )
=> ( ( sup_sup_int @ ( complete_Sup_Sup_int @ ( image_873079544045703924it_int @ F @ A4 ) ) @ ( complete_Sup_Sup_int @ ( image_873079544045703924it_int @ G @ A4 ) ) )
= ( complete_Sup_Sup_int
@ ( image_873079544045703924it_int
@ ^ [A5: product_unit] : ( sup_sup_int @ ( F @ A5 ) @ ( G @ A5 ) )
@ A4 ) ) ) ) ) ) ).
% conditionally_complete_lattice_class.SUP_sup_distrib
thf(fact_3024_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
! [A4: set_o,F: $o > int,G: $o > int] :
( ( A4 != bot_bot_set_o )
=> ( ( condit2212336002400062152ve_int @ ( image_o_int @ F @ A4 ) )
=> ( ( condit2212336002400062152ve_int @ ( image_o_int @ G @ A4 ) )
=> ( ( sup_sup_int @ ( complete_Sup_Sup_int @ ( image_o_int @ F @ A4 ) ) @ ( complete_Sup_Sup_int @ ( image_o_int @ G @ A4 ) ) )
= ( complete_Sup_Sup_int
@ ( image_o_int
@ ^ [A5: $o] : ( sup_sup_int @ ( F @ A5 ) @ ( G @ A5 ) )
@ A4 ) ) ) ) ) ) ).
% conditionally_complete_lattice_class.SUP_sup_distrib
thf(fact_3025_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
! [A4: set_nat,F: nat > int,G: nat > int] :
( ( A4 != bot_bot_set_nat )
=> ( ( condit2212336002400062152ve_int @ ( image_nat_int @ F @ A4 ) )
=> ( ( condit2212336002400062152ve_int @ ( image_nat_int @ G @ A4 ) )
=> ( ( sup_sup_int @ ( complete_Sup_Sup_int @ ( image_nat_int @ F @ A4 ) ) @ ( complete_Sup_Sup_int @ ( image_nat_int @ G @ A4 ) ) )
= ( complete_Sup_Sup_int
@ ( image_nat_int
@ ^ [A5: nat] : ( sup_sup_int @ ( F @ A5 ) @ ( G @ A5 ) )
@ A4 ) ) ) ) ) ) ).
% conditionally_complete_lattice_class.SUP_sup_distrib
thf(fact_3026_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
! [A4: set_Product_unit,F: product_unit > product_unit,G: product_unit > product_unit] :
( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ( condit2080475194861589617t_unit @ ( image_405062704495631173t_unit @ F @ A4 ) )
=> ( ( condit2080475194861589617t_unit @ ( image_405062704495631173t_unit @ G @ A4 ) )
=> ( ( sup_sup_Product_unit @ ( comple4687483117567863418t_unit @ ( image_405062704495631173t_unit @ F @ A4 ) ) @ ( comple4687483117567863418t_unit @ ( image_405062704495631173t_unit @ G @ A4 ) ) )
= ( comple4687483117567863418t_unit
@ ( image_405062704495631173t_unit
@ ^ [A5: product_unit] : ( sup_sup_Product_unit @ ( F @ A5 ) @ ( G @ A5 ) )
@ A4 ) ) ) ) ) ) ).
% conditionally_complete_lattice_class.SUP_sup_distrib
thf(fact_3027_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
! [A4: set_o,F: $o > product_unit,G: $o > product_unit] :
( ( A4 != bot_bot_set_o )
=> ( ( condit2080475194861589617t_unit @ ( image_o_Product_unit @ F @ A4 ) )
=> ( ( condit2080475194861589617t_unit @ ( image_o_Product_unit @ G @ A4 ) )
=> ( ( sup_sup_Product_unit @ ( comple4687483117567863418t_unit @ ( image_o_Product_unit @ F @ A4 ) ) @ ( comple4687483117567863418t_unit @ ( image_o_Product_unit @ G @ A4 ) ) )
= ( comple4687483117567863418t_unit
@ ( image_o_Product_unit
@ ^ [A5: $o] : ( sup_sup_Product_unit @ ( F @ A5 ) @ ( G @ A5 ) )
@ A4 ) ) ) ) ) ) ).
% conditionally_complete_lattice_class.SUP_sup_distrib
thf(fact_3028_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
! [A4: set_nat,F: nat > product_unit,G: nat > product_unit] :
( ( A4 != bot_bot_set_nat )
=> ( ( condit2080475194861589617t_unit @ ( image_8730104196221521654t_unit @ F @ A4 ) )
=> ( ( condit2080475194861589617t_unit @ ( image_8730104196221521654t_unit @ G @ A4 ) )
=> ( ( sup_sup_Product_unit @ ( comple4687483117567863418t_unit @ ( image_8730104196221521654t_unit @ F @ A4 ) ) @ ( comple4687483117567863418t_unit @ ( image_8730104196221521654t_unit @ G @ A4 ) ) )
= ( comple4687483117567863418t_unit
@ ( image_8730104196221521654t_unit
@ ^ [A5: nat] : ( sup_sup_Product_unit @ ( F @ A5 ) @ ( G @ A5 ) )
@ A4 ) ) ) ) ) ) ).
% conditionally_complete_lattice_class.SUP_sup_distrib
thf(fact_3029_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
! [A4: set_Product_unit,F: product_unit > set_nat,G: product_unit > set_nat] :
( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ( condit5477540289124974626et_nat @ ( image_8173849991827986510et_nat @ F @ A4 ) )
=> ( ( condit5477540289124974626et_nat @ ( image_8173849991827986510et_nat @ G @ A4 ) )
=> ( ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ F @ A4 ) ) @ ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ G @ A4 ) ) )
= ( comple7399068483239264473et_nat
@ ( image_8173849991827986510et_nat
@ ^ [A5: product_unit] : ( sup_sup_set_nat @ ( F @ A5 ) @ ( G @ A5 ) )
@ A4 ) ) ) ) ) ) ).
% conditionally_complete_lattice_class.SUP_sup_distrib
thf(fact_3030_cINF__inf__distrib,axiom,
! [A4: set_int,F: int > nat,G: int > nat] :
( ( A4 != bot_bot_set_int )
=> ( ( condit1738341127787009408ow_nat @ ( image_int_nat @ F @ A4 ) )
=> ( ( condit1738341127787009408ow_nat @ ( image_int_nat @ G @ A4 ) )
=> ( ( inf_inf_nat @ ( complete_Inf_Inf_nat @ ( image_int_nat @ F @ A4 ) ) @ ( complete_Inf_Inf_nat @ ( image_int_nat @ G @ A4 ) ) )
= ( complete_Inf_Inf_nat
@ ( image_int_nat
@ ^ [A5: int] : ( inf_inf_nat @ ( F @ A5 ) @ ( G @ A5 ) )
@ A4 ) ) ) ) ) ) ).
% cINF_inf_distrib
thf(fact_3031_cINF__inf__distrib,axiom,
! [A4: set_nat,F: nat > nat,G: nat > nat] :
( ( A4 != bot_bot_set_nat )
=> ( ( condit1738341127787009408ow_nat @ ( image_nat_nat @ F @ A4 ) )
=> ( ( condit1738341127787009408ow_nat @ ( image_nat_nat @ G @ A4 ) )
=> ( ( inf_inf_nat @ ( complete_Inf_Inf_nat @ ( image_nat_nat @ F @ A4 ) ) @ ( complete_Inf_Inf_nat @ ( image_nat_nat @ G @ A4 ) ) )
= ( complete_Inf_Inf_nat
@ ( image_nat_nat
@ ^ [A5: nat] : ( inf_inf_nat @ ( F @ A5 ) @ ( G @ A5 ) )
@ A4 ) ) ) ) ) ) ).
% cINF_inf_distrib
thf(fact_3032_cINF__inf__distrib,axiom,
! [A4: set_int,F: int > int,G: int > int] :
( ( A4 != bot_bot_set_int )
=> ( ( condit1735850657277959132ow_int @ ( image_int_int @ F @ A4 ) )
=> ( ( condit1735850657277959132ow_int @ ( image_int_int @ G @ A4 ) )
=> ( ( inf_inf_int @ ( complete_Inf_Inf_int @ ( image_int_int @ F @ A4 ) ) @ ( complete_Inf_Inf_int @ ( image_int_int @ G @ A4 ) ) )
= ( complete_Inf_Inf_int
@ ( image_int_int
@ ^ [A5: int] : ( inf_inf_int @ ( F @ A5 ) @ ( G @ A5 ) )
@ A4 ) ) ) ) ) ) ).
% cINF_inf_distrib
thf(fact_3033_cINF__inf__distrib,axiom,
! [A4: set_Product_unit,F: product_unit > int,G: product_unit > int] :
( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ( condit1735850657277959132ow_int @ ( image_873079544045703924it_int @ F @ A4 ) )
=> ( ( condit1735850657277959132ow_int @ ( image_873079544045703924it_int @ G @ A4 ) )
=> ( ( inf_inf_int @ ( complete_Inf_Inf_int @ ( image_873079544045703924it_int @ F @ A4 ) ) @ ( complete_Inf_Inf_int @ ( image_873079544045703924it_int @ G @ A4 ) ) )
= ( complete_Inf_Inf_int
@ ( image_873079544045703924it_int
@ ^ [A5: product_unit] : ( inf_inf_int @ ( F @ A5 ) @ ( G @ A5 ) )
@ A4 ) ) ) ) ) ) ).
% cINF_inf_distrib
thf(fact_3034_cINF__inf__distrib,axiom,
! [A4: set_o,F: $o > int,G: $o > int] :
( ( A4 != bot_bot_set_o )
=> ( ( condit1735850657277959132ow_int @ ( image_o_int @ F @ A4 ) )
=> ( ( condit1735850657277959132ow_int @ ( image_o_int @ G @ A4 ) )
=> ( ( inf_inf_int @ ( complete_Inf_Inf_int @ ( image_o_int @ F @ A4 ) ) @ ( complete_Inf_Inf_int @ ( image_o_int @ G @ A4 ) ) )
= ( complete_Inf_Inf_int
@ ( image_o_int
@ ^ [A5: $o] : ( inf_inf_int @ ( F @ A5 ) @ ( G @ A5 ) )
@ A4 ) ) ) ) ) ) ).
% cINF_inf_distrib
thf(fact_3035_cINF__inf__distrib,axiom,
! [A4: set_nat,F: nat > int,G: nat > int] :
( ( A4 != bot_bot_set_nat )
=> ( ( condit1735850657277959132ow_int @ ( image_nat_int @ F @ A4 ) )
=> ( ( condit1735850657277959132ow_int @ ( image_nat_int @ G @ A4 ) )
=> ( ( inf_inf_int @ ( complete_Inf_Inf_int @ ( image_nat_int @ F @ A4 ) ) @ ( complete_Inf_Inf_int @ ( image_nat_int @ G @ A4 ) ) )
= ( complete_Inf_Inf_int
@ ( image_nat_int
@ ^ [A5: nat] : ( inf_inf_int @ ( F @ A5 ) @ ( G @ A5 ) )
@ A4 ) ) ) ) ) ) ).
% cINF_inf_distrib
thf(fact_3036_cINF__inf__distrib,axiom,
! [A4: set_Product_unit,F: product_unit > product_unit,G: product_unit > product_unit] :
( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ( condit9161856309968323677t_unit @ ( image_405062704495631173t_unit @ F @ A4 ) )
=> ( ( condit9161856309968323677t_unit @ ( image_405062704495631173t_unit @ G @ A4 ) )
=> ( ( inf_inf_Product_unit @ ( comple2584293577114468500t_unit @ ( image_405062704495631173t_unit @ F @ A4 ) ) @ ( comple2584293577114468500t_unit @ ( image_405062704495631173t_unit @ G @ A4 ) ) )
= ( comple2584293577114468500t_unit
@ ( image_405062704495631173t_unit
@ ^ [A5: product_unit] : ( inf_inf_Product_unit @ ( F @ A5 ) @ ( G @ A5 ) )
@ A4 ) ) ) ) ) ) ).
% cINF_inf_distrib
thf(fact_3037_cINF__inf__distrib,axiom,
! [A4: set_o,F: $o > product_unit,G: $o > product_unit] :
( ( A4 != bot_bot_set_o )
=> ( ( condit9161856309968323677t_unit @ ( image_o_Product_unit @ F @ A4 ) )
=> ( ( condit9161856309968323677t_unit @ ( image_o_Product_unit @ G @ A4 ) )
=> ( ( inf_inf_Product_unit @ ( comple2584293577114468500t_unit @ ( image_o_Product_unit @ F @ A4 ) ) @ ( comple2584293577114468500t_unit @ ( image_o_Product_unit @ G @ A4 ) ) )
= ( comple2584293577114468500t_unit
@ ( image_o_Product_unit
@ ^ [A5: $o] : ( inf_inf_Product_unit @ ( F @ A5 ) @ ( G @ A5 ) )
@ A4 ) ) ) ) ) ) ).
% cINF_inf_distrib
thf(fact_3038_cINF__inf__distrib,axiom,
! [A4: set_nat,F: nat > product_unit,G: nat > product_unit] :
( ( A4 != bot_bot_set_nat )
=> ( ( condit9161856309968323677t_unit @ ( image_8730104196221521654t_unit @ F @ A4 ) )
=> ( ( condit9161856309968323677t_unit @ ( image_8730104196221521654t_unit @ G @ A4 ) )
=> ( ( inf_inf_Product_unit @ ( comple2584293577114468500t_unit @ ( image_8730104196221521654t_unit @ F @ A4 ) ) @ ( comple2584293577114468500t_unit @ ( image_8730104196221521654t_unit @ G @ A4 ) ) )
= ( comple2584293577114468500t_unit
@ ( image_8730104196221521654t_unit
@ ^ [A5: nat] : ( inf_inf_Product_unit @ ( F @ A5 ) @ ( G @ A5 ) )
@ A4 ) ) ) ) ) ) ).
% cINF_inf_distrib
thf(fact_3039_cINF__inf__distrib,axiom,
! [A4: set_Product_unit,F: product_unit > set_nat,G: product_unit > set_nat] :
( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ( condit68592940725977398et_nat @ ( image_8173849991827986510et_nat @ F @ A4 ) )
=> ( ( condit68592940725977398et_nat @ ( image_8173849991827986510et_nat @ G @ A4 ) )
=> ( ( inf_inf_set_nat @ ( comple7806235888213564991et_nat @ ( image_8173849991827986510et_nat @ F @ A4 ) ) @ ( comple7806235888213564991et_nat @ ( image_8173849991827986510et_nat @ G @ A4 ) ) )
= ( comple7806235888213564991et_nat
@ ( image_8173849991827986510et_nat
@ ^ [A5: product_unit] : ( inf_inf_set_nat @ ( F @ A5 ) @ ( G @ A5 ) )
@ A4 ) ) ) ) ) ) ).
% cINF_inf_distrib
thf(fact_3040_fsts_Ointros,axiom,
! [P: produc7812382424243267045t_unit] : ( member_ref_a @ ( produc473321116828791851t_unit @ P ) @ ( basic_685971451221398055t_unit @ P ) ) ).
% fsts.intros
thf(fact_3041_fsts_Osimps,axiom,
! [A: ref_a,P: produc7812382424243267045t_unit] :
( ( member_ref_a @ A @ ( basic_685971451221398055t_unit @ P ) )
= ( A
= ( produc473321116828791851t_unit @ P ) ) ) ).
% fsts.simps
thf(fact_3042_fsts_Ocases,axiom,
! [A: ref_a,P: produc7812382424243267045t_unit] :
( ( member_ref_a @ A @ ( basic_685971451221398055t_unit @ P ) )
=> ( A
= ( produc473321116828791851t_unit @ P ) ) ) ).
% fsts.cases
thf(fact_3043_INT__bool__eq,axiom,
! [A4: $o > set_Pr1261947904930325089at_nat] :
( ( comple1930127330929219688at_nat @ ( image_7077428525161401678at_nat @ A4 @ top_top_set_o ) )
= ( inf_in2572325071724192079at_nat @ ( A4 @ $true ) @ ( A4 @ $false ) ) ) ).
% INT_bool_eq
thf(fact_3044_INT__bool__eq,axiom,
! [A4: $o > set_nat] :
( ( comple7806235888213564991et_nat @ ( image_o_set_nat @ A4 @ top_top_set_o ) )
= ( inf_inf_set_nat @ ( A4 @ $true ) @ ( A4 @ $false ) ) ) ).
% INT_bool_eq
thf(fact_3045_cSUP__eq__cINF__D,axiom,
! [F: nat > nat,A4: set_nat,A: nat] :
( ( ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ A4 ) )
= ( complete_Inf_Inf_nat @ ( image_nat_nat @ F @ A4 ) ) )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ F @ A4 ) )
=> ( ( condit1738341127787009408ow_nat @ ( image_nat_nat @ F @ A4 ) )
=> ( ( member_nat @ A @ A4 )
=> ( ( F @ A )
= ( complete_Inf_Inf_nat @ ( image_nat_nat @ F @ A4 ) ) ) ) ) ) ) ).
% cSUP_eq_cINF_D
thf(fact_3046_cSUP__eq__cINF__D,axiom,
! [F: int > nat,A4: set_int,A: int] :
( ( ( complete_Sup_Sup_nat @ ( image_int_nat @ F @ A4 ) )
= ( complete_Inf_Inf_nat @ ( image_int_nat @ F @ A4 ) ) )
=> ( ( condit2214826472909112428ve_nat @ ( image_int_nat @ F @ A4 ) )
=> ( ( condit1738341127787009408ow_nat @ ( image_int_nat @ F @ A4 ) )
=> ( ( member_int @ A @ A4 )
=> ( ( F @ A )
= ( complete_Inf_Inf_nat @ ( image_int_nat @ F @ A4 ) ) ) ) ) ) ) ).
% cSUP_eq_cINF_D
thf(fact_3047_cSUP__eq__cINF__D,axiom,
! [F: $o > int,A4: set_o,A: $o] :
( ( ( complete_Sup_Sup_int @ ( image_o_int @ F @ A4 ) )
= ( complete_Inf_Inf_int @ ( image_o_int @ F @ A4 ) ) )
=> ( ( condit2212336002400062152ve_int @ ( image_o_int @ F @ A4 ) )
=> ( ( condit1735850657277959132ow_int @ ( image_o_int @ F @ A4 ) )
=> ( ( member_o @ A @ A4 )
=> ( ( F @ A )
= ( complete_Inf_Inf_int @ ( image_o_int @ F @ A4 ) ) ) ) ) ) ) ).
% cSUP_eq_cINF_D
thf(fact_3048_cSUP__eq__cINF__D,axiom,
! [F: nat > int,A4: set_nat,A: nat] :
( ( ( complete_Sup_Sup_int @ ( image_nat_int @ F @ A4 ) )
= ( complete_Inf_Inf_int @ ( image_nat_int @ F @ A4 ) ) )
=> ( ( condit2212336002400062152ve_int @ ( image_nat_int @ F @ A4 ) )
=> ( ( condit1735850657277959132ow_int @ ( image_nat_int @ F @ A4 ) )
=> ( ( member_nat @ A @ A4 )
=> ( ( F @ A )
= ( complete_Inf_Inf_int @ ( image_nat_int @ F @ A4 ) ) ) ) ) ) ) ).
% cSUP_eq_cINF_D
thf(fact_3049_cSUP__eq__cINF__D,axiom,
! [F: int > int,A4: set_int,A: int] :
( ( ( complete_Sup_Sup_int @ ( image_int_int @ F @ A4 ) )
= ( complete_Inf_Inf_int @ ( image_int_int @ F @ A4 ) ) )
=> ( ( condit2212336002400062152ve_int @ ( image_int_int @ F @ A4 ) )
=> ( ( condit1735850657277959132ow_int @ ( image_int_int @ F @ A4 ) )
=> ( ( member_int @ A @ A4 )
=> ( ( F @ A )
= ( complete_Inf_Inf_int @ ( image_int_int @ F @ A4 ) ) ) ) ) ) ) ).
% cSUP_eq_cINF_D
thf(fact_3050_cSUP__eq__cINF__D,axiom,
! [F: $o > product_unit,A4: set_o,A: $o] :
( ( ( comple4687483117567863418t_unit @ ( image_o_Product_unit @ F @ A4 ) )
= ( comple2584293577114468500t_unit @ ( image_o_Product_unit @ F @ A4 ) ) )
=> ( ( condit2080475194861589617t_unit @ ( image_o_Product_unit @ F @ A4 ) )
=> ( ( condit9161856309968323677t_unit @ ( image_o_Product_unit @ F @ A4 ) )
=> ( ( member_o @ A @ A4 )
=> ( ( F @ A )
= ( comple2584293577114468500t_unit @ ( image_o_Product_unit @ F @ A4 ) ) ) ) ) ) ) ).
% cSUP_eq_cINF_D
thf(fact_3051_cSUP__eq__cINF__D,axiom,
! [F: nat > product_unit,A4: set_nat,A: nat] :
( ( ( comple4687483117567863418t_unit @ ( image_8730104196221521654t_unit @ F @ A4 ) )
= ( comple2584293577114468500t_unit @ ( image_8730104196221521654t_unit @ F @ A4 ) ) )
=> ( ( condit2080475194861589617t_unit @ ( image_8730104196221521654t_unit @ F @ A4 ) )
=> ( ( condit9161856309968323677t_unit @ ( image_8730104196221521654t_unit @ F @ A4 ) )
=> ( ( member_nat @ A @ A4 )
=> ( ( F @ A )
= ( comple2584293577114468500t_unit @ ( image_8730104196221521654t_unit @ F @ A4 ) ) ) ) ) ) ) ).
% cSUP_eq_cINF_D
thf(fact_3052_cSUP__eq__cINF__D,axiom,
! [F: int > product_unit,A4: set_int,A: int] :
( ( ( comple4687483117567863418t_unit @ ( image_5846526143423701914t_unit @ F @ A4 ) )
= ( comple2584293577114468500t_unit @ ( image_5846526143423701914t_unit @ F @ A4 ) ) )
=> ( ( condit2080475194861589617t_unit @ ( image_5846526143423701914t_unit @ F @ A4 ) )
=> ( ( condit9161856309968323677t_unit @ ( image_5846526143423701914t_unit @ F @ A4 ) )
=> ( ( member_int @ A @ A4 )
=> ( ( F @ A )
= ( comple2584293577114468500t_unit @ ( image_5846526143423701914t_unit @ F @ A4 ) ) ) ) ) ) ) ).
% cSUP_eq_cINF_D
thf(fact_3053_cSUP__eq__cINF__D,axiom,
! [F: $o > set_nat,A4: set_o,A: $o] :
( ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A4 ) )
= ( comple7806235888213564991et_nat @ ( image_o_set_nat @ F @ A4 ) ) )
=> ( ( condit5477540289124974626et_nat @ ( image_o_set_nat @ F @ A4 ) )
=> ( ( condit68592940725977398et_nat @ ( image_o_set_nat @ F @ A4 ) )
=> ( ( member_o @ A @ A4 )
=> ( ( F @ A )
= ( comple7806235888213564991et_nat @ ( image_o_set_nat @ F @ A4 ) ) ) ) ) ) ) ).
% cSUP_eq_cINF_D
thf(fact_3054_cSUP__eq__cINF__D,axiom,
! [F: nat > set_nat,A4: set_nat,A: nat] :
( ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A4 ) )
= ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ A4 ) ) )
=> ( ( condit5477540289124974626et_nat @ ( image_nat_set_nat @ F @ A4 ) )
=> ( ( condit68592940725977398et_nat @ ( image_nat_set_nat @ F @ A4 ) )
=> ( ( member_nat @ A @ A4 )
=> ( ( F @ A )
= ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ A4 ) ) ) ) ) ) ) ).
% cSUP_eq_cINF_D
thf(fact_3055_cSUP__insert,axiom,
! [A4: set_int,F: int > nat,A: int] :
( ( A4 != bot_bot_set_int )
=> ( ( condit2214826472909112428ve_nat @ ( image_int_nat @ F @ A4 ) )
=> ( ( complete_Sup_Sup_nat @ ( image_int_nat @ F @ ( insert_int @ A @ A4 ) ) )
= ( sup_sup_nat @ ( F @ A ) @ ( complete_Sup_Sup_nat @ ( image_int_nat @ F @ A4 ) ) ) ) ) ) ).
% cSUP_insert
thf(fact_3056_cSUP__insert,axiom,
! [A4: set_nat,F: nat > nat,A: nat] :
( ( A4 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ F @ A4 ) )
=> ( ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ ( insert_nat @ A @ A4 ) ) )
= ( sup_sup_nat @ ( F @ A ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ A4 ) ) ) ) ) ) ).
% cSUP_insert
thf(fact_3057_cSUP__insert,axiom,
! [A4: set_int,F: int > int,A: int] :
( ( A4 != bot_bot_set_int )
=> ( ( condit2212336002400062152ve_int @ ( image_int_int @ F @ A4 ) )
=> ( ( complete_Sup_Sup_int @ ( image_int_int @ F @ ( insert_int @ A @ A4 ) ) )
= ( sup_sup_int @ ( F @ A ) @ ( complete_Sup_Sup_int @ ( image_int_int @ F @ A4 ) ) ) ) ) ) ).
% cSUP_insert
thf(fact_3058_cSUP__insert,axiom,
! [A4: set_Product_unit,F: product_unit > int,A: product_unit] :
( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ( condit2212336002400062152ve_int @ ( image_873079544045703924it_int @ F @ A4 ) )
=> ( ( complete_Sup_Sup_int @ ( image_873079544045703924it_int @ F @ ( insert_Product_unit @ A @ A4 ) ) )
= ( sup_sup_int @ ( F @ A ) @ ( complete_Sup_Sup_int @ ( image_873079544045703924it_int @ F @ A4 ) ) ) ) ) ) ).
% cSUP_insert
thf(fact_3059_cSUP__insert,axiom,
! [A4: set_o,F: $o > int,A: $o] :
( ( A4 != bot_bot_set_o )
=> ( ( condit2212336002400062152ve_int @ ( image_o_int @ F @ A4 ) )
=> ( ( complete_Sup_Sup_int @ ( image_o_int @ F @ ( insert_o @ A @ A4 ) ) )
= ( sup_sup_int @ ( F @ A ) @ ( complete_Sup_Sup_int @ ( image_o_int @ F @ A4 ) ) ) ) ) ) ).
% cSUP_insert
thf(fact_3060_cSUP__insert,axiom,
! [A4: set_nat,F: nat > int,A: nat] :
( ( A4 != bot_bot_set_nat )
=> ( ( condit2212336002400062152ve_int @ ( image_nat_int @ F @ A4 ) )
=> ( ( complete_Sup_Sup_int @ ( image_nat_int @ F @ ( insert_nat @ A @ A4 ) ) )
= ( sup_sup_int @ ( F @ A ) @ ( complete_Sup_Sup_int @ ( image_nat_int @ F @ A4 ) ) ) ) ) ) ).
% cSUP_insert
thf(fact_3061_cSUP__insert,axiom,
! [A4: set_Product_unit,F: product_unit > product_unit,A: product_unit] :
( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ( condit2080475194861589617t_unit @ ( image_405062704495631173t_unit @ F @ A4 ) )
=> ( ( comple4687483117567863418t_unit @ ( image_405062704495631173t_unit @ F @ ( insert_Product_unit @ A @ A4 ) ) )
= ( sup_sup_Product_unit @ ( F @ A ) @ ( comple4687483117567863418t_unit @ ( image_405062704495631173t_unit @ F @ A4 ) ) ) ) ) ) ).
% cSUP_insert
thf(fact_3062_cSUP__insert,axiom,
! [A4: set_o,F: $o > product_unit,A: $o] :
( ( A4 != bot_bot_set_o )
=> ( ( condit2080475194861589617t_unit @ ( image_o_Product_unit @ F @ A4 ) )
=> ( ( comple4687483117567863418t_unit @ ( image_o_Product_unit @ F @ ( insert_o @ A @ A4 ) ) )
= ( sup_sup_Product_unit @ ( F @ A ) @ ( comple4687483117567863418t_unit @ ( image_o_Product_unit @ F @ A4 ) ) ) ) ) ) ).
% cSUP_insert
thf(fact_3063_cSUP__insert,axiom,
! [A4: set_nat,F: nat > product_unit,A: nat] :
( ( A4 != bot_bot_set_nat )
=> ( ( condit2080475194861589617t_unit @ ( image_8730104196221521654t_unit @ F @ A4 ) )
=> ( ( comple4687483117567863418t_unit @ ( image_8730104196221521654t_unit @ F @ ( insert_nat @ A @ A4 ) ) )
= ( sup_sup_Product_unit @ ( F @ A ) @ ( comple4687483117567863418t_unit @ ( image_8730104196221521654t_unit @ F @ A4 ) ) ) ) ) ) ).
% cSUP_insert
thf(fact_3064_cSUP__insert,axiom,
! [A4: set_Product_unit,F: product_unit > set_nat,A: product_unit] :
( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ( condit5477540289124974626et_nat @ ( image_8173849991827986510et_nat @ F @ A4 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ F @ ( insert_Product_unit @ A @ A4 ) ) )
= ( sup_sup_set_nat @ ( F @ A ) @ ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ F @ A4 ) ) ) ) ) ) ).
% cSUP_insert
thf(fact_3065_cINF__insert,axiom,
! [A4: set_int,F: int > nat,A: int] :
( ( A4 != bot_bot_set_int )
=> ( ( condit1738341127787009408ow_nat @ ( image_int_nat @ F @ A4 ) )
=> ( ( complete_Inf_Inf_nat @ ( image_int_nat @ F @ ( insert_int @ A @ A4 ) ) )
= ( inf_inf_nat @ ( F @ A ) @ ( complete_Inf_Inf_nat @ ( image_int_nat @ F @ A4 ) ) ) ) ) ) ).
% cINF_insert
thf(fact_3066_cINF__insert,axiom,
! [A4: set_nat,F: nat > nat,A: nat] :
( ( A4 != bot_bot_set_nat )
=> ( ( condit1738341127787009408ow_nat @ ( image_nat_nat @ F @ A4 ) )
=> ( ( complete_Inf_Inf_nat @ ( image_nat_nat @ F @ ( insert_nat @ A @ A4 ) ) )
= ( inf_inf_nat @ ( F @ A ) @ ( complete_Inf_Inf_nat @ ( image_nat_nat @ F @ A4 ) ) ) ) ) ) ).
% cINF_insert
thf(fact_3067_cINF__insert,axiom,
! [A4: set_int,F: int > int,A: int] :
( ( A4 != bot_bot_set_int )
=> ( ( condit1735850657277959132ow_int @ ( image_int_int @ F @ A4 ) )
=> ( ( complete_Inf_Inf_int @ ( image_int_int @ F @ ( insert_int @ A @ A4 ) ) )
= ( inf_inf_int @ ( F @ A ) @ ( complete_Inf_Inf_int @ ( image_int_int @ F @ A4 ) ) ) ) ) ) ).
% cINF_insert
thf(fact_3068_cINF__insert,axiom,
! [A4: set_Product_unit,F: product_unit > int,A: product_unit] :
( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ( condit1735850657277959132ow_int @ ( image_873079544045703924it_int @ F @ A4 ) )
=> ( ( complete_Inf_Inf_int @ ( image_873079544045703924it_int @ F @ ( insert_Product_unit @ A @ A4 ) ) )
= ( inf_inf_int @ ( F @ A ) @ ( complete_Inf_Inf_int @ ( image_873079544045703924it_int @ F @ A4 ) ) ) ) ) ) ).
% cINF_insert
thf(fact_3069_cINF__insert,axiom,
! [A4: set_o,F: $o > int,A: $o] :
( ( A4 != bot_bot_set_o )
=> ( ( condit1735850657277959132ow_int @ ( image_o_int @ F @ A4 ) )
=> ( ( complete_Inf_Inf_int @ ( image_o_int @ F @ ( insert_o @ A @ A4 ) ) )
= ( inf_inf_int @ ( F @ A ) @ ( complete_Inf_Inf_int @ ( image_o_int @ F @ A4 ) ) ) ) ) ) ).
% cINF_insert
thf(fact_3070_cINF__insert,axiom,
! [A4: set_nat,F: nat > int,A: nat] :
( ( A4 != bot_bot_set_nat )
=> ( ( condit1735850657277959132ow_int @ ( image_nat_int @ F @ A4 ) )
=> ( ( complete_Inf_Inf_int @ ( image_nat_int @ F @ ( insert_nat @ A @ A4 ) ) )
= ( inf_inf_int @ ( F @ A ) @ ( complete_Inf_Inf_int @ ( image_nat_int @ F @ A4 ) ) ) ) ) ) ).
% cINF_insert
thf(fact_3071_cINF__insert,axiom,
! [A4: set_Product_unit,F: product_unit > product_unit,A: product_unit] :
( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ( condit9161856309968323677t_unit @ ( image_405062704495631173t_unit @ F @ A4 ) )
=> ( ( comple2584293577114468500t_unit @ ( image_405062704495631173t_unit @ F @ ( insert_Product_unit @ A @ A4 ) ) )
= ( inf_inf_Product_unit @ ( F @ A ) @ ( comple2584293577114468500t_unit @ ( image_405062704495631173t_unit @ F @ A4 ) ) ) ) ) ) ).
% cINF_insert
thf(fact_3072_cINF__insert,axiom,
! [A4: set_o,F: $o > product_unit,A: $o] :
( ( A4 != bot_bot_set_o )
=> ( ( condit9161856309968323677t_unit @ ( image_o_Product_unit @ F @ A4 ) )
=> ( ( comple2584293577114468500t_unit @ ( image_o_Product_unit @ F @ ( insert_o @ A @ A4 ) ) )
= ( inf_inf_Product_unit @ ( F @ A ) @ ( comple2584293577114468500t_unit @ ( image_o_Product_unit @ F @ A4 ) ) ) ) ) ) ).
% cINF_insert
thf(fact_3073_cINF__insert,axiom,
! [A4: set_nat,F: nat > product_unit,A: nat] :
( ( A4 != bot_bot_set_nat )
=> ( ( condit9161856309968323677t_unit @ ( image_8730104196221521654t_unit @ F @ A4 ) )
=> ( ( comple2584293577114468500t_unit @ ( image_8730104196221521654t_unit @ F @ ( insert_nat @ A @ A4 ) ) )
= ( inf_inf_Product_unit @ ( F @ A ) @ ( comple2584293577114468500t_unit @ ( image_8730104196221521654t_unit @ F @ A4 ) ) ) ) ) ) ).
% cINF_insert
thf(fact_3074_cINF__insert,axiom,
! [A4: set_Product_unit,F: product_unit > set_nat,A: product_unit] :
( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ( condit68592940725977398et_nat @ ( image_8173849991827986510et_nat @ F @ A4 ) )
=> ( ( comple7806235888213564991et_nat @ ( image_8173849991827986510et_nat @ F @ ( insert_Product_unit @ A @ A4 ) ) )
= ( inf_inf_set_nat @ ( F @ A ) @ ( comple7806235888213564991et_nat @ ( image_8173849991827986510et_nat @ F @ A4 ) ) ) ) ) ) ).
% cINF_insert
thf(fact_3075_inj__on__image__mem__iff,axiom,
! [F: $o > $o,B5: set_o,A: $o,A4: set_o] :
( ( inj_on_o_o @ F @ B5 )
=> ( ( member_o @ A @ B5 )
=> ( ( ord_less_eq_set_o @ A4 @ B5 )
=> ( ( member_o @ ( F @ A ) @ ( image_o_o @ F @ A4 ) )
= ( member_o @ A @ A4 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_3076_inj__on__image__mem__iff,axiom,
! [F: $o > nat,B5: set_o,A: $o,A4: set_o] :
( ( inj_on_o_nat @ F @ B5 )
=> ( ( member_o @ A @ B5 )
=> ( ( ord_less_eq_set_o @ A4 @ B5 )
=> ( ( member_nat @ ( F @ A ) @ ( image_o_nat @ F @ A4 ) )
= ( member_o @ A @ A4 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_3077_inj__on__image__mem__iff,axiom,
! [F: $o > int,B5: set_o,A: $o,A4: set_o] :
( ( inj_on_o_int @ F @ B5 )
=> ( ( member_o @ A @ B5 )
=> ( ( ord_less_eq_set_o @ A4 @ B5 )
=> ( ( member_int @ ( F @ A ) @ ( image_o_int @ F @ A4 ) )
= ( member_o @ A @ A4 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_3078_inj__on__image__mem__iff,axiom,
! [F: nat > rat,B5: set_nat,A: nat,A4: set_nat] :
( ( inj_on_nat_rat @ F @ B5 )
=> ( ( member_nat @ A @ B5 )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( member_rat @ ( F @ A ) @ ( image_nat_rat @ F @ A4 ) )
= ( member_nat @ A @ A4 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_3079_inj__on__image__mem__iff,axiom,
! [F: nat > $o,B5: set_nat,A: nat,A4: set_nat] :
( ( inj_on_nat_o @ F @ B5 )
=> ( ( member_nat @ A @ B5 )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( member_o @ ( F @ A ) @ ( image_nat_o @ F @ A4 ) )
= ( member_nat @ A @ A4 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_3080_inj__on__image__mem__iff,axiom,
! [F: nat > nat,B5: set_nat,A: nat,A4: set_nat] :
( ( inj_on_nat_nat @ F @ B5 )
=> ( ( member_nat @ A @ B5 )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( member_nat @ ( F @ A ) @ ( image_nat_nat @ F @ A4 ) )
= ( member_nat @ A @ A4 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_3081_inj__on__image__mem__iff,axiom,
! [F: nat > int,B5: set_nat,A: nat,A4: set_nat] :
( ( inj_on_nat_int @ F @ B5 )
=> ( ( member_nat @ A @ B5 )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ( member_int @ ( F @ A ) @ ( image_nat_int @ F @ A4 ) )
= ( member_nat @ A @ A4 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_3082_inj__on__image__mem__iff,axiom,
! [F: int > $o,B5: set_int,A: int,A4: set_int] :
( ( inj_on_int_o @ F @ B5 )
=> ( ( member_int @ A @ B5 )
=> ( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( ( member_o @ ( F @ A ) @ ( image_int_o @ F @ A4 ) )
= ( member_int @ A @ A4 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_3083_inj__on__image__mem__iff,axiom,
! [F: int > nat,B5: set_int,A: int,A4: set_int] :
( ( inj_on_int_nat @ F @ B5 )
=> ( ( member_int @ A @ B5 )
=> ( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( ( member_nat @ ( F @ A ) @ ( image_int_nat @ F @ A4 ) )
= ( member_int @ A @ A4 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_3084_inj__on__image__mem__iff,axiom,
! [F: int > int,B5: set_int,A: int,A4: set_int] :
( ( inj_on_int_int @ F @ B5 )
=> ( ( member_int @ A @ B5 )
=> ( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( ( member_int @ ( F @ A ) @ ( image_int_int @ F @ A4 ) )
= ( member_int @ A @ A4 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_3085_inj__on__image__eq__iff,axiom,
! [F: nat > rat,C3: set_nat,A4: set_nat,B5: set_nat] :
( ( inj_on_nat_rat @ F @ C3 )
=> ( ( ord_less_eq_set_nat @ A4 @ C3 )
=> ( ( ord_less_eq_set_nat @ B5 @ C3 )
=> ( ( ( image_nat_rat @ F @ A4 )
= ( image_nat_rat @ F @ B5 ) )
= ( A4 = B5 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_3086_inj__on__image__eq__iff,axiom,
! [F: nat > nat,C3: set_nat,A4: set_nat,B5: set_nat] :
( ( inj_on_nat_nat @ F @ C3 )
=> ( ( ord_less_eq_set_nat @ A4 @ C3 )
=> ( ( ord_less_eq_set_nat @ B5 @ C3 )
=> ( ( ( image_nat_nat @ F @ A4 )
= ( image_nat_nat @ F @ B5 ) )
= ( A4 = B5 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_3087_inj__on__image__eq__iff,axiom,
! [F: nat > int,C3: set_nat,A4: set_nat,B5: set_nat] :
( ( inj_on_nat_int @ F @ C3 )
=> ( ( ord_less_eq_set_nat @ A4 @ C3 )
=> ( ( ord_less_eq_set_nat @ B5 @ C3 )
=> ( ( ( image_nat_int @ F @ A4 )
= ( image_nat_int @ F @ B5 ) )
= ( A4 = B5 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_3088_inj__on__image__eq__iff,axiom,
! [F: int > nat,C3: set_int,A4: set_int,B5: set_int] :
( ( inj_on_int_nat @ F @ C3 )
=> ( ( ord_less_eq_set_int @ A4 @ C3 )
=> ( ( ord_less_eq_set_int @ B5 @ C3 )
=> ( ( ( image_int_nat @ F @ A4 )
= ( image_int_nat @ F @ B5 ) )
= ( A4 = B5 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_3089_inj__on__image__eq__iff,axiom,
! [F: int > int,C3: set_int,A4: set_int,B5: set_int] :
( ( inj_on_int_int @ F @ C3 )
=> ( ( ord_less_eq_set_int @ A4 @ C3 )
=> ( ( ord_less_eq_set_int @ B5 @ C3 )
=> ( ( ( image_int_int @ F @ A4 )
= ( image_int_int @ F @ B5 ) )
= ( A4 = B5 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_3090_subset__image__inj,axiom,
! [S5: set_rat,F: nat > rat,T3: set_nat] :
( ( ord_less_eq_set_rat @ S5 @ ( image_nat_rat @ F @ T3 ) )
= ( ? [U2: set_nat] :
( ( ord_less_eq_set_nat @ U2 @ T3 )
& ( inj_on_nat_rat @ F @ U2 )
& ( S5
= ( image_nat_rat @ F @ U2 ) ) ) ) ) ).
% subset_image_inj
thf(fact_3091_subset__image__inj,axiom,
! [S5: set_nat,F: nat > nat,T3: set_nat] :
( ( ord_less_eq_set_nat @ S5 @ ( image_nat_nat @ F @ T3 ) )
= ( ? [U2: set_nat] :
( ( ord_less_eq_set_nat @ U2 @ T3 )
& ( inj_on_nat_nat @ F @ U2 )
& ( S5
= ( image_nat_nat @ F @ U2 ) ) ) ) ) ).
% subset_image_inj
thf(fact_3092_subset__image__inj,axiom,
! [S5: set_int,F: nat > int,T3: set_nat] :
( ( ord_less_eq_set_int @ S5 @ ( image_nat_int @ F @ T3 ) )
= ( ? [U2: set_nat] :
( ( ord_less_eq_set_nat @ U2 @ T3 )
& ( inj_on_nat_int @ F @ U2 )
& ( S5
= ( image_nat_int @ F @ U2 ) ) ) ) ) ).
% subset_image_inj
thf(fact_3093_subset__image__inj,axiom,
! [S5: set_nat,F: int > nat,T3: set_int] :
( ( ord_less_eq_set_nat @ S5 @ ( image_int_nat @ F @ T3 ) )
= ( ? [U2: set_int] :
( ( ord_less_eq_set_int @ U2 @ T3 )
& ( inj_on_int_nat @ F @ U2 )
& ( S5
= ( image_int_nat @ F @ U2 ) ) ) ) ) ).
% subset_image_inj
thf(fact_3094_subset__image__inj,axiom,
! [S5: set_int,F: int > int,T3: set_int] :
( ( ord_less_eq_set_int @ S5 @ ( image_int_int @ F @ T3 ) )
= ( ? [U2: set_int] :
( ( ord_less_eq_set_int @ U2 @ T3 )
& ( inj_on_int_int @ F @ U2 )
& ( S5
= ( image_int_int @ F @ U2 ) ) ) ) ) ).
% subset_image_inj
thf(fact_3095_inj__img__insertE,axiom,
! [F: product_unit > product_unit,A4: set_Product_unit,X2: product_unit,B5: set_Product_unit] :
( ( inj_on8151373323710067377t_unit @ F @ A4 )
=> ( ~ ( member_Product_unit @ X2 @ B5 )
=> ( ( ( insert_Product_unit @ X2 @ B5 )
= ( image_405062704495631173t_unit @ F @ A4 ) )
=> ~ ! [X8: product_unit,A8: set_Product_unit] :
( ~ ( member_Product_unit @ X8 @ A8 )
=> ( ( A4
= ( insert_Product_unit @ X8 @ A8 ) )
=> ( ( X2
= ( F @ X8 ) )
=> ( B5
!= ( image_405062704495631173t_unit @ F @ A8 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_3096_inj__img__insertE,axiom,
! [F: $o > product_unit,A4: set_o,X2: product_unit,B5: set_Product_unit] :
( ( inj_on4268752236855399398t_unit @ F @ A4 )
=> ( ~ ( member_Product_unit @ X2 @ B5 )
=> ( ( ( insert_Product_unit @ X2 @ B5 )
= ( image_o_Product_unit @ F @ A4 ) )
=> ~ ! [X8: $o,A8: set_o] :
( ~ ( member_o @ X8 @ A8 )
=> ( ( A4
= ( insert_o @ X8 @ A8 ) )
=> ( ( X2
= ( F @ X8 ) )
=> ( B5
!= ( image_o_Product_unit @ F @ A8 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_3097_inj__img__insertE,axiom,
! [F: nat > rat,A4: set_nat,X2: rat,B5: set_rat] :
( ( inj_on_nat_rat @ F @ A4 )
=> ( ~ ( member_rat @ X2 @ B5 )
=> ( ( ( insert_rat @ X2 @ B5 )
= ( image_nat_rat @ F @ A4 ) )
=> ~ ! [X8: nat,A8: set_nat] :
( ~ ( member_nat @ X8 @ A8 )
=> ( ( A4
= ( insert_nat @ X8 @ A8 ) )
=> ( ( X2
= ( F @ X8 ) )
=> ( B5
!= ( image_nat_rat @ F @ A8 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_3098_inj__img__insertE,axiom,
! [F: nat > product_unit,A4: set_nat,X2: product_unit,B5: set_Product_unit] :
( ( inj_on7061601236592826506t_unit @ F @ A4 )
=> ( ~ ( member_Product_unit @ X2 @ B5 )
=> ( ( ( insert_Product_unit @ X2 @ B5 )
= ( image_8730104196221521654t_unit @ F @ A4 ) )
=> ~ ! [X8: nat,A8: set_nat] :
( ~ ( member_nat @ X8 @ A8 )
=> ( ( A4
= ( insert_nat @ X8 @ A8 ) )
=> ( ( X2
= ( F @ X8 ) )
=> ( B5
!= ( image_8730104196221521654t_unit @ F @ A8 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_3099_inj__img__insertE,axiom,
! [F: int > product_unit,A4: set_int,X2: product_unit,B5: set_Product_unit] :
( ( inj_on4178023183795006766t_unit @ F @ A4 )
=> ( ~ ( member_Product_unit @ X2 @ B5 )
=> ( ( ( insert_Product_unit @ X2 @ B5 )
= ( image_5846526143423701914t_unit @ F @ A4 ) )
=> ~ ! [X8: int,A8: set_int] :
( ~ ( member_int @ X8 @ A8 )
=> ( ( A4
= ( insert_int @ X8 @ A8 ) )
=> ( ( X2
= ( F @ X8 ) )
=> ( B5
!= ( image_5846526143423701914t_unit @ F @ A8 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_3100_inj__img__insertE,axiom,
! [F: product_unit > $o,A4: set_Product_unit,X2: $o,B5: set_o] :
( ( inj_on8854722734121536060unit_o @ F @ A4 )
=> ( ~ ( member_o @ X2 @ B5 )
=> ( ( ( insert_o @ X2 @ B5 )
= ( image_Product_unit_o @ F @ A4 ) )
=> ~ ! [X8: product_unit,A8: set_Product_unit] :
( ~ ( member_Product_unit @ X8 @ A8 )
=> ( ( A4
= ( insert_Product_unit @ X8 @ A8 ) )
=> ( ( X2
= ( F @ X8 ) )
=> ( B5
!= ( image_Product_unit_o @ F @ A8 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_3101_inj__img__insertE,axiom,
! [F: $o > $o,A4: set_o,X2: $o,B5: set_o] :
( ( inj_on_o_o @ F @ A4 )
=> ( ~ ( member_o @ X2 @ B5 )
=> ( ( ( insert_o @ X2 @ B5 )
= ( image_o_o @ F @ A4 ) )
=> ~ ! [X8: $o,A8: set_o] :
( ~ ( member_o @ X8 @ A8 )
=> ( ( A4
= ( insert_o @ X8 @ A8 ) )
=> ( ( X2
= ( F @ X8 ) )
=> ( B5
!= ( image_o_o @ F @ A8 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_3102_inj__img__insertE,axiom,
! [F: nat > $o,A4: set_nat,X2: $o,B5: set_o] :
( ( inj_on_nat_o @ F @ A4 )
=> ( ~ ( member_o @ X2 @ B5 )
=> ( ( ( insert_o @ X2 @ B5 )
= ( image_nat_o @ F @ A4 ) )
=> ~ ! [X8: nat,A8: set_nat] :
( ~ ( member_nat @ X8 @ A8 )
=> ( ( A4
= ( insert_nat @ X8 @ A8 ) )
=> ( ( X2
= ( F @ X8 ) )
=> ( B5
!= ( image_nat_o @ F @ A8 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_3103_inj__img__insertE,axiom,
! [F: int > $o,A4: set_int,X2: $o,B5: set_o] :
( ( inj_on_int_o @ F @ A4 )
=> ( ~ ( member_o @ X2 @ B5 )
=> ( ( ( insert_o @ X2 @ B5 )
= ( image_int_o @ F @ A4 ) )
=> ~ ! [X8: int,A8: set_int] :
( ~ ( member_int @ X8 @ A8 )
=> ( ( A4
= ( insert_int @ X8 @ A8 ) )
=> ( ( X2
= ( F @ X8 ) )
=> ( B5
!= ( image_int_o @ F @ A8 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_3104_inj__img__insertE,axiom,
! [F: product_unit > nat,A4: set_Product_unit,X2: nat,B5: set_nat] :
( ( inj_on8430439091780834860it_nat @ F @ A4 )
=> ( ~ ( member_nat @ X2 @ B5 )
=> ( ( ( insert_nat @ X2 @ B5 )
= ( image_875570014554754200it_nat @ F @ A4 ) )
=> ~ ! [X8: product_unit,A8: set_Product_unit] :
( ~ ( member_Product_unit @ X8 @ A8 )
=> ( ( A4
= ( insert_Product_unit @ X8 @ A8 ) )
=> ( ( X2
= ( F @ X8 ) )
=> ( B5
!= ( image_875570014554754200it_nat @ F @ A8 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_3105_inj__image__mem__iff,axiom,
! [F: $o > $o,A: $o,A4: set_o] :
( ( inj_on_o_o @ F @ top_top_set_o )
=> ( ( member_o @ ( F @ A ) @ ( image_o_o @ F @ A4 ) )
= ( member_o @ A @ A4 ) ) ) ).
% inj_image_mem_iff
thf(fact_3106_inj__image__mem__iff,axiom,
! [F: $o > nat,A: $o,A4: set_o] :
( ( inj_on_o_nat @ F @ top_top_set_o )
=> ( ( member_nat @ ( F @ A ) @ ( image_o_nat @ F @ A4 ) )
= ( member_o @ A @ A4 ) ) ) ).
% inj_image_mem_iff
thf(fact_3107_inj__image__mem__iff,axiom,
! [F: $o > int,A: $o,A4: set_o] :
( ( inj_on_o_int @ F @ top_top_set_o )
=> ( ( member_int @ ( F @ A ) @ ( image_o_int @ F @ A4 ) )
= ( member_o @ A @ A4 ) ) ) ).
% inj_image_mem_iff
thf(fact_3108_inj__image__mem__iff,axiom,
! [F: char > $o,A: char,A4: set_char] :
( ( inj_on_char_o @ F @ top_top_set_char )
=> ( ( member_o @ ( F @ A ) @ ( image_char_o @ F @ A4 ) )
= ( member_char @ A @ A4 ) ) ) ).
% inj_image_mem_iff
thf(fact_3109_inj__image__mem__iff,axiom,
! [F: char > nat,A: char,A4: set_char] :
( ( inj_on_char_nat @ F @ top_top_set_char )
=> ( ( member_nat @ ( F @ A ) @ ( image_char_nat @ F @ A4 ) )
= ( member_char @ A @ A4 ) ) ) ).
% inj_image_mem_iff
thf(fact_3110_inj__image__mem__iff,axiom,
! [F: char > int,A: char,A4: set_char] :
( ( inj_on_char_int @ F @ top_top_set_char )
=> ( ( member_int @ ( F @ A ) @ ( image_char_int @ F @ A4 ) )
= ( member_char @ A @ A4 ) ) ) ).
% inj_image_mem_iff
thf(fact_3111_inj__image__mem__iff,axiom,
! [F: rat > $o,A: rat,A4: set_rat] :
( ( inj_on_rat_o @ F @ top_top_set_rat )
=> ( ( member_o @ ( F @ A ) @ ( image_rat_o @ F @ A4 ) )
= ( member_rat @ A @ A4 ) ) ) ).
% inj_image_mem_iff
thf(fact_3112_inj__image__mem__iff,axiom,
! [F: rat > nat,A: rat,A4: set_rat] :
( ( inj_on_rat_nat @ F @ top_top_set_rat )
=> ( ( member_nat @ ( F @ A ) @ ( image_rat_nat @ F @ A4 ) )
= ( member_rat @ A @ A4 ) ) ) ).
% inj_image_mem_iff
thf(fact_3113_inj__image__mem__iff,axiom,
! [F: rat > int,A: rat,A4: set_rat] :
( ( inj_on_rat_int @ F @ top_top_set_rat )
=> ( ( member_int @ ( F @ A ) @ ( image_rat_int @ F @ A4 ) )
= ( member_rat @ A @ A4 ) ) ) ).
% inj_image_mem_iff
thf(fact_3114_inj__image__mem__iff,axiom,
! [F: nat > rat,A: nat,A4: set_nat] :
( ( inj_on_nat_rat @ F @ top_top_set_nat )
=> ( ( member_rat @ ( F @ A ) @ ( image_nat_rat @ F @ A4 ) )
= ( member_nat @ A @ A4 ) ) ) ).
% inj_image_mem_iff
thf(fact_3115_inj__image__eq__iff,axiom,
! [F: nat > rat,A4: set_nat,B5: set_nat] :
( ( inj_on_nat_rat @ F @ top_top_set_nat )
=> ( ( ( image_nat_rat @ F @ A4 )
= ( image_nat_rat @ F @ B5 ) )
= ( A4 = B5 ) ) ) ).
% inj_image_eq_iff
thf(fact_3116_inj__image__eq__iff,axiom,
! [F: nat > nat,A4: set_nat,B5: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( ( image_nat_nat @ F @ A4 )
= ( image_nat_nat @ F @ B5 ) )
= ( A4 = B5 ) ) ) ).
% inj_image_eq_iff
thf(fact_3117_inj__image__eq__iff,axiom,
! [F: nat > int,A4: set_nat,B5: set_nat] :
( ( inj_on_nat_int @ F @ top_top_set_nat )
=> ( ( ( image_nat_int @ F @ A4 )
= ( image_nat_int @ F @ B5 ) )
= ( A4 = B5 ) ) ) ).
% inj_image_eq_iff
thf(fact_3118_inj__image__eq__iff,axiom,
! [F: int > nat,A4: set_int,B5: set_int] :
( ( inj_on_int_nat @ F @ top_top_set_int )
=> ( ( ( image_int_nat @ F @ A4 )
= ( image_int_nat @ F @ B5 ) )
= ( A4 = B5 ) ) ) ).
% inj_image_eq_iff
thf(fact_3119_inj__image__eq__iff,axiom,
! [F: int > int,A4: set_int,B5: set_int] :
( ( inj_on_int_int @ F @ top_top_set_int )
=> ( ( ( image_int_int @ F @ A4 )
= ( image_int_int @ F @ B5 ) )
= ( A4 = B5 ) ) ) ).
% inj_image_eq_iff
thf(fact_3120_range__ex1__eq,axiom,
! [F: char > $o,B: $o] :
( ( inj_on_char_o @ F @ top_top_set_char )
=> ( ( member_o @ B @ ( image_char_o @ F @ top_top_set_char ) )
= ( ? [X4: char] :
( ( B
= ( F @ X4 ) )
& ! [Y5: char] :
( ( B
= ( F @ Y5 ) )
=> ( Y5 = X4 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_3121_range__ex1__eq,axiom,
! [F: char > nat,B: nat] :
( ( inj_on_char_nat @ F @ top_top_set_char )
=> ( ( member_nat @ B @ ( image_char_nat @ F @ top_top_set_char ) )
= ( ? [X4: char] :
( ( B
= ( F @ X4 ) )
& ! [Y5: char] :
( ( B
= ( F @ Y5 ) )
=> ( Y5 = X4 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_3122_range__ex1__eq,axiom,
! [F: char > int,B: int] :
( ( inj_on_char_int @ F @ top_top_set_char )
=> ( ( member_int @ B @ ( image_char_int @ F @ top_top_set_char ) )
= ( ? [X4: char] :
( ( B
= ( F @ X4 ) )
& ! [Y5: char] :
( ( B
= ( F @ Y5 ) )
=> ( Y5 = X4 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_3123_range__ex1__eq,axiom,
! [F: rat > $o,B: $o] :
( ( inj_on_rat_o @ F @ top_top_set_rat )
=> ( ( member_o @ B @ ( image_rat_o @ F @ top_top_set_rat ) )
= ( ? [X4: rat] :
( ( B
= ( F @ X4 ) )
& ! [Y5: rat] :
( ( B
= ( F @ Y5 ) )
=> ( Y5 = X4 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_3124_range__ex1__eq,axiom,
! [F: rat > nat,B: nat] :
( ( inj_on_rat_nat @ F @ top_top_set_rat )
=> ( ( member_nat @ B @ ( image_rat_nat @ F @ top_top_set_rat ) )
= ( ? [X4: rat] :
( ( B
= ( F @ X4 ) )
& ! [Y5: rat] :
( ( B
= ( F @ Y5 ) )
=> ( Y5 = X4 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_3125_range__ex1__eq,axiom,
! [F: rat > int,B: int] :
( ( inj_on_rat_int @ F @ top_top_set_rat )
=> ( ( member_int @ B @ ( image_rat_int @ F @ top_top_set_rat ) )
= ( ? [X4: rat] :
( ( B
= ( F @ X4 ) )
& ! [Y5: rat] :
( ( B
= ( F @ Y5 ) )
=> ( Y5 = X4 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_3126_range__ex1__eq,axiom,
! [F: nat > rat,B: rat] :
( ( inj_on_nat_rat @ F @ top_top_set_nat )
=> ( ( member_rat @ B @ ( image_nat_rat @ F @ top_top_set_nat ) )
= ( ? [X4: nat] :
( ( B
= ( F @ X4 ) )
& ! [Y5: nat] :
( ( B
= ( F @ Y5 ) )
=> ( Y5 = X4 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_3127_range__ex1__eq,axiom,
! [F: nat > $o,B: $o] :
( ( inj_on_nat_o @ F @ top_top_set_nat )
=> ( ( member_o @ B @ ( image_nat_o @ F @ top_top_set_nat ) )
= ( ? [X4: nat] :
( ( B
= ( F @ X4 ) )
& ! [Y5: nat] :
( ( B
= ( F @ Y5 ) )
=> ( Y5 = X4 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_3128_range__ex1__eq,axiom,
! [F: nat > nat,B: nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( member_nat @ B @ ( image_nat_nat @ F @ top_top_set_nat ) )
= ( ? [X4: nat] :
( ( B
= ( F @ X4 ) )
& ! [Y5: nat] :
( ( B
= ( F @ Y5 ) )
=> ( Y5 = X4 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_3129_range__ex1__eq,axiom,
! [F: nat > int,B: int] :
( ( inj_on_nat_int @ F @ top_top_set_nat )
=> ( ( member_int @ B @ ( image_nat_int @ F @ top_top_set_nat ) )
= ( ? [X4: nat] :
( ( B
= ( F @ X4 ) )
& ! [Y5: nat] :
( ( B
= ( F @ Y5 ) )
=> ( Y5 = X4 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_3130_comp__inj__on,axiom,
! [F: nat > $o,A4: set_nat,G: $o > $o] :
( ( inj_on_nat_o @ F @ A4 )
=> ( ( inj_on_o_o @ G @ ( image_nat_o @ F @ A4 ) )
=> ( inj_on_nat_o @ ( comp_o_o_nat @ G @ F ) @ A4 ) ) ) ).
% comp_inj_on
thf(fact_3131_comp__inj__on,axiom,
! [F: code_integer > code_integer > code_integer,A4: set_Code_integer,G: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger] :
( ( inj_on108595395805848489nteger @ F @ A4 )
=> ( ( inj_on3998011822705705585nteger @ G @ ( image_2177946750938214461nteger @ F @ A4 ) )
=> ( inj_on7031512951821619701nteger @ ( comp_C8797469213163452608nteger @ G @ F ) @ A4 ) ) ) ).
% comp_inj_on
thf(fact_3132_comp__inj__on,axiom,
! [F: nat > nat,A4: set_nat,G: nat > nat] :
( ( inj_on_nat_nat @ F @ A4 )
=> ( ( inj_on_nat_nat @ G @ ( image_nat_nat @ F @ A4 ) )
=> ( inj_on_nat_nat @ ( comp_nat_nat_nat @ G @ F ) @ A4 ) ) ) ).
% comp_inj_on
thf(fact_3133_comp__inj__on,axiom,
! [F: int > nat,A4: set_int,G: nat > num] :
( ( inj_on_int_nat @ F @ A4 )
=> ( ( inj_on_nat_num @ G @ ( image_int_nat @ F @ A4 ) )
=> ( inj_on_int_num @ ( comp_nat_num_int @ G @ F ) @ A4 ) ) ) ).
% comp_inj_on
thf(fact_3134_comp__inj__on,axiom,
! [F: int > int,A4: set_int,G: int > nat] :
( ( inj_on_int_int @ F @ A4 )
=> ( ( inj_on_int_nat @ G @ ( image_int_int @ F @ A4 ) )
=> ( inj_on_int_nat @ ( comp_int_nat_int @ G @ F ) @ A4 ) ) ) ).
% comp_inj_on
thf(fact_3135_inj__on__imageI,axiom,
! [G: $o > $o,F: nat > $o,A4: set_nat] :
( ( inj_on_nat_o @ ( comp_o_o_nat @ G @ F ) @ A4 )
=> ( inj_on_o_o @ G @ ( image_nat_o @ F @ A4 ) ) ) ).
% inj_on_imageI
thf(fact_3136_inj__on__imageI,axiom,
! [G: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,F: code_integer > code_integer > code_integer,A4: set_Code_integer] :
( ( inj_on7031512951821619701nteger @ ( comp_C8797469213163452608nteger @ G @ F ) @ A4 )
=> ( inj_on3998011822705705585nteger @ G @ ( image_2177946750938214461nteger @ F @ A4 ) ) ) ).
% inj_on_imageI
thf(fact_3137_inj__on__imageI,axiom,
! [G: nat > nat,F: nat > nat,A4: set_nat] :
( ( inj_on_nat_nat @ ( comp_nat_nat_nat @ G @ F ) @ A4 )
=> ( inj_on_nat_nat @ G @ ( image_nat_nat @ F @ A4 ) ) ) ).
% inj_on_imageI
thf(fact_3138_inj__on__imageI,axiom,
! [G: nat > num,F: int > nat,A4: set_int] :
( ( inj_on_int_num @ ( comp_nat_num_int @ G @ F ) @ A4 )
=> ( inj_on_nat_num @ G @ ( image_int_nat @ F @ A4 ) ) ) ).
% inj_on_imageI
thf(fact_3139_inj__on__imageI,axiom,
! [G: int > nat,F: int > int,A4: set_int] :
( ( inj_on_int_nat @ ( comp_int_nat_int @ G @ F ) @ A4 )
=> ( inj_on_int_nat @ G @ ( image_int_int @ F @ A4 ) ) ) ).
% inj_on_imageI
thf(fact_3140_comp__inj__on__iff,axiom,
! [F: nat > $o,A4: set_nat,F3: $o > $o] :
( ( inj_on_nat_o @ F @ A4 )
=> ( ( inj_on_o_o @ F3 @ ( image_nat_o @ F @ A4 ) )
= ( inj_on_nat_o @ ( comp_o_o_nat @ F3 @ F ) @ A4 ) ) ) ).
% comp_inj_on_iff
thf(fact_3141_comp__inj__on__iff,axiom,
! [F: code_integer > code_integer > code_integer,A4: set_Code_integer,F3: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger] :
( ( inj_on108595395805848489nteger @ F @ A4 )
=> ( ( inj_on3998011822705705585nteger @ F3 @ ( image_2177946750938214461nteger @ F @ A4 ) )
= ( inj_on7031512951821619701nteger @ ( comp_C8797469213163452608nteger @ F3 @ F ) @ A4 ) ) ) ).
% comp_inj_on_iff
thf(fact_3142_comp__inj__on__iff,axiom,
! [F: nat > nat,A4: set_nat,F3: nat > nat] :
( ( inj_on_nat_nat @ F @ A4 )
=> ( ( inj_on_nat_nat @ F3 @ ( image_nat_nat @ F @ A4 ) )
= ( inj_on_nat_nat @ ( comp_nat_nat_nat @ F3 @ F ) @ A4 ) ) ) ).
% comp_inj_on_iff
thf(fact_3143_comp__inj__on__iff,axiom,
! [F: int > nat,A4: set_int,F3: nat > num] :
( ( inj_on_int_nat @ F @ A4 )
=> ( ( inj_on_nat_num @ F3 @ ( image_int_nat @ F @ A4 ) )
= ( inj_on_int_num @ ( comp_nat_num_int @ F3 @ F ) @ A4 ) ) ) ).
% comp_inj_on_iff
thf(fact_3144_comp__inj__on__iff,axiom,
! [F: int > int,A4: set_int,F3: int > nat] :
( ( inj_on_int_int @ F @ A4 )
=> ( ( inj_on_int_nat @ F3 @ ( image_int_int @ F @ A4 ) )
= ( inj_on_int_nat @ ( comp_int_nat_int @ F3 @ F ) @ A4 ) ) ) ).
% comp_inj_on_iff
thf(fact_3145_inj__compose,axiom,
! [F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer > code_integer] :
( ( inj_on3998011822705705585nteger @ F @ top_to3804862627464599871nteger )
=> ( ( inj_on108595395805848489nteger @ G @ top_to4645266643341252675nteger )
=> ( inj_on7031512951821619701nteger @ ( comp_C8797469213163452608nteger @ F @ G ) @ top_to4645266643341252675nteger ) ) ) ).
% inj_compose
thf(fact_3146_inj__compose,axiom,
! [F: $o > $o,G: nat > $o] :
( ( inj_on_o_o @ F @ top_top_set_o )
=> ( ( inj_on_nat_o @ G @ top_top_set_nat )
=> ( inj_on_nat_o @ ( comp_o_o_nat @ F @ G ) @ top_top_set_nat ) ) ) ).
% inj_compose
thf(fact_3147_inj__compose,axiom,
! [F: nat > nat,G: nat > nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( inj_on_nat_nat @ G @ top_top_set_nat )
=> ( inj_on_nat_nat @ ( comp_nat_nat_nat @ F @ G ) @ top_top_set_nat ) ) ) ).
% inj_compose
thf(fact_3148_inj__compose,axiom,
! [F: nat > num,G: int > nat] :
( ( inj_on_nat_num @ F @ top_top_set_nat )
=> ( ( inj_on_int_nat @ G @ top_top_set_int )
=> ( inj_on_int_num @ ( comp_nat_num_int @ F @ G ) @ top_top_set_int ) ) ) ).
% inj_compose
thf(fact_3149_inj__compose,axiom,
! [F: int > nat,G: int > int] :
( ( inj_on_int_nat @ F @ top_top_set_int )
=> ( ( inj_on_int_int @ G @ top_top_set_int )
=> ( inj_on_int_nat @ ( comp_int_nat_int @ F @ G ) @ top_top_set_int ) ) ) ).
% inj_compose
thf(fact_3150_inf__Sup,axiom,
! [A: set_Pr1261947904930325089at_nat,B5: set_se7855581050983116737at_nat] :
( ( inf_in2572325071724192079at_nat @ A @ ( comple5685304695842803022at_nat @ B5 ) )
= ( comple5685304695842803022at_nat @ ( image_3684629450409544005at_nat @ ( inf_in2572325071724192079at_nat @ A ) @ B5 ) ) ) ).
% inf_Sup
thf(fact_3151_inf__Sup,axiom,
! [A: set_nat,B5: set_set_nat] :
( ( inf_inf_set_nat @ A @ ( comple7399068483239264473et_nat @ B5 ) )
= ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ ( inf_inf_set_nat @ A ) @ B5 ) ) ) ).
% inf_Sup
thf(fact_3152_inf__Sup,axiom,
! [A: product_unit,B5: set_Product_unit] :
( ( inf_inf_Product_unit @ A @ ( comple4687483117567863418t_unit @ B5 ) )
= ( comple4687483117567863418t_unit @ ( image_405062704495631173t_unit @ ( inf_inf_Product_unit @ A ) @ B5 ) ) ) ).
% inf_Sup
thf(fact_3153_sup__Inf,axiom,
! [A: set_Pr8693737435421807431at_nat,B5: set_se6425085446898223655at_nat] :
( ( sup_su718114333110466843at_nat @ A @ ( comple6332534001959699150at_nat @ B5 ) )
= ( comple6332534001959699150at_nat @ ( image_1013614488123988933at_nat @ ( sup_su718114333110466843at_nat @ A ) @ B5 ) ) ) ).
% sup_Inf
thf(fact_3154_sup__Inf,axiom,
! [A: set_Pr4329608150637261639at_nat,B5: set_se5479201105039950375at_nat] :
( ( sup_su5525570899277871387at_nat @ A @ ( comple6751905152567503566at_nat @ B5 ) )
= ( comple6751905152567503566at_nat @ ( image_4303274092540115909at_nat @ ( sup_su5525570899277871387at_nat @ A ) @ B5 ) ) ) ).
% sup_Inf
thf(fact_3155_sup__Inf,axiom,
! [A: set_nat,B5: set_set_nat] :
( ( sup_sup_set_nat @ A @ ( comple7806235888213564991et_nat @ B5 ) )
= ( comple7806235888213564991et_nat @ ( image_7916887816326733075et_nat @ ( sup_sup_set_nat @ A ) @ B5 ) ) ) ).
% sup_Inf
thf(fact_3156_sup__Inf,axiom,
! [A: product_unit,B5: set_Product_unit] :
( ( sup_sup_Product_unit @ A @ ( comple2584293577114468500t_unit @ B5 ) )
= ( comple2584293577114468500t_unit @ ( image_405062704495631173t_unit @ ( sup_sup_Product_unit @ A ) @ B5 ) ) ) ).
% sup_Inf
thf(fact_3157_bdd__below_OI2,axiom,
! [A4: set_o,M: nat,F: $o > nat] :
( ! [X: $o] :
( ( member_o @ X @ A4 )
=> ( ord_less_eq_nat @ M @ ( F @ X ) ) )
=> ( condit1738341127787009408ow_nat @ ( image_o_nat @ F @ A4 ) ) ) ).
% bdd_below.I2
thf(fact_3158_bdd__below_OI2,axiom,
! [A4: set_nat,M: nat,F: nat > nat] :
( ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ( ord_less_eq_nat @ M @ ( F @ X ) ) )
=> ( condit1738341127787009408ow_nat @ ( image_nat_nat @ F @ A4 ) ) ) ).
% bdd_below.I2
thf(fact_3159_bdd__below_OI2,axiom,
! [A4: set_int,M: nat,F: int > nat] :
( ! [X: int] :
( ( member_int @ X @ A4 )
=> ( ord_less_eq_nat @ M @ ( F @ X ) ) )
=> ( condit1738341127787009408ow_nat @ ( image_int_nat @ F @ A4 ) ) ) ).
% bdd_below.I2
thf(fact_3160_bdd__below_OI2,axiom,
! [A4: set_o,M: int,F: $o > int] :
( ! [X: $o] :
( ( member_o @ X @ A4 )
=> ( ord_less_eq_int @ M @ ( F @ X ) ) )
=> ( condit1735850657277959132ow_int @ ( image_o_int @ F @ A4 ) ) ) ).
% bdd_below.I2
thf(fact_3161_bdd__below_OI2,axiom,
! [A4: set_nat,M: int,F: nat > int] :
( ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ( ord_less_eq_int @ M @ ( F @ X ) ) )
=> ( condit1735850657277959132ow_int @ ( image_nat_int @ F @ A4 ) ) ) ).
% bdd_below.I2
thf(fact_3162_bdd__below_OI2,axiom,
! [A4: set_int,M: int,F: int > int] :
( ! [X: int] :
( ( member_int @ X @ A4 )
=> ( ord_less_eq_int @ M @ ( F @ X ) ) )
=> ( condit1735850657277959132ow_int @ ( image_int_int @ F @ A4 ) ) ) ).
% bdd_below.I2
thf(fact_3163_bdd__below_OI2,axiom,
! [A4: set_o,M: rat,F: $o > rat] :
( ! [X: $o] :
( ( member_o @ X @ A4 )
=> ( ord_less_eq_rat @ M @ ( F @ X ) ) )
=> ( condit1103211067700513672ow_rat @ ( image_o_rat @ F @ A4 ) ) ) ).
% bdd_below.I2
thf(fact_3164_bdd__below_OI2,axiom,
! [A4: set_nat,M: rat,F: nat > rat] :
( ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ( ord_less_eq_rat @ M @ ( F @ X ) ) )
=> ( condit1103211067700513672ow_rat @ ( image_nat_rat @ F @ A4 ) ) ) ).
% bdd_below.I2
thf(fact_3165_bdd__below_OI2,axiom,
! [A4: set_int,M: rat,F: int > rat] :
( ! [X: int] :
( ( member_int @ X @ A4 )
=> ( ord_less_eq_rat @ M @ ( F @ X ) ) )
=> ( condit1103211067700513672ow_rat @ ( image_int_rat @ F @ A4 ) ) ) ).
% bdd_below.I2
thf(fact_3166_bdd__below_OI2,axiom,
! [A4: set_o,M: code_integer,F: $o > code_integer] :
( ! [X: $o] :
( ( member_o @ X @ A4 )
=> ( ord_le3102999989581377725nteger @ M @ ( F @ X ) ) )
=> ( condit4933233095834890765nteger @ ( image_o_Code_integer @ F @ A4 ) ) ) ).
% bdd_below.I2
thf(fact_3167_bdd__belowI2,axiom,
! [A4: set_o,M3: nat,F: $o > nat] :
( ! [X: $o] :
( ( member_o @ X @ A4 )
=> ( ord_less_eq_nat @ M3 @ ( F @ X ) ) )
=> ( condit1738341127787009408ow_nat @ ( image_o_nat @ F @ A4 ) ) ) ).
% bdd_belowI2
thf(fact_3168_bdd__belowI2,axiom,
! [A4: set_nat,M3: nat,F: nat > nat] :
( ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ( ord_less_eq_nat @ M3 @ ( F @ X ) ) )
=> ( condit1738341127787009408ow_nat @ ( image_nat_nat @ F @ A4 ) ) ) ).
% bdd_belowI2
thf(fact_3169_bdd__belowI2,axiom,
! [A4: set_int,M3: nat,F: int > nat] :
( ! [X: int] :
( ( member_int @ X @ A4 )
=> ( ord_less_eq_nat @ M3 @ ( F @ X ) ) )
=> ( condit1738341127787009408ow_nat @ ( image_int_nat @ F @ A4 ) ) ) ).
% bdd_belowI2
thf(fact_3170_bdd__belowI2,axiom,
! [A4: set_o,M3: int,F: $o > int] :
( ! [X: $o] :
( ( member_o @ X @ A4 )
=> ( ord_less_eq_int @ M3 @ ( F @ X ) ) )
=> ( condit1735850657277959132ow_int @ ( image_o_int @ F @ A4 ) ) ) ).
% bdd_belowI2
thf(fact_3171_bdd__belowI2,axiom,
! [A4: set_nat,M3: int,F: nat > int] :
( ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ( ord_less_eq_int @ M3 @ ( F @ X ) ) )
=> ( condit1735850657277959132ow_int @ ( image_nat_int @ F @ A4 ) ) ) ).
% bdd_belowI2
thf(fact_3172_bdd__belowI2,axiom,
! [A4: set_int,M3: int,F: int > int] :
( ! [X: int] :
( ( member_int @ X @ A4 )
=> ( ord_less_eq_int @ M3 @ ( F @ X ) ) )
=> ( condit1735850657277959132ow_int @ ( image_int_int @ F @ A4 ) ) ) ).
% bdd_belowI2
thf(fact_3173_bdd__belowI2,axiom,
! [A4: set_o,M3: rat,F: $o > rat] :
( ! [X: $o] :
( ( member_o @ X @ A4 )
=> ( ord_less_eq_rat @ M3 @ ( F @ X ) ) )
=> ( condit1103211067700513672ow_rat @ ( image_o_rat @ F @ A4 ) ) ) ).
% bdd_belowI2
thf(fact_3174_bdd__belowI2,axiom,
! [A4: set_nat,M3: rat,F: nat > rat] :
( ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ( ord_less_eq_rat @ M3 @ ( F @ X ) ) )
=> ( condit1103211067700513672ow_rat @ ( image_nat_rat @ F @ A4 ) ) ) ).
% bdd_belowI2
thf(fact_3175_bdd__belowI2,axiom,
! [A4: set_int,M3: rat,F: int > rat] :
( ! [X: int] :
( ( member_int @ X @ A4 )
=> ( ord_less_eq_rat @ M3 @ ( F @ X ) ) )
=> ( condit1103211067700513672ow_rat @ ( image_int_rat @ F @ A4 ) ) ) ).
% bdd_belowI2
thf(fact_3176_bdd__belowI2,axiom,
! [A4: set_o,M3: code_integer,F: $o > code_integer] :
( ! [X: $o] :
( ( member_o @ X @ A4 )
=> ( ord_le3102999989581377725nteger @ M3 @ ( F @ X ) ) )
=> ( condit4933233095834890765nteger @ ( image_o_Code_integer @ F @ A4 ) ) ) ).
% bdd_belowI2
thf(fact_3177_bdd__above_OI2,axiom,
! [A4: set_o,F: $o > nat,M: nat] :
( ! [X: $o] :
( ( member_o @ X @ A4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ M ) )
=> ( condit2214826472909112428ve_nat @ ( image_o_nat @ F @ A4 ) ) ) ).
% bdd_above.I2
thf(fact_3178_bdd__above_OI2,axiom,
! [A4: set_nat,F: nat > nat,M: nat] :
( ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ M ) )
=> ( condit2214826472909112428ve_nat @ ( image_nat_nat @ F @ A4 ) ) ) ).
% bdd_above.I2
thf(fact_3179_bdd__above_OI2,axiom,
! [A4: set_int,F: int > nat,M: nat] :
( ! [X: int] :
( ( member_int @ X @ A4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ M ) )
=> ( condit2214826472909112428ve_nat @ ( image_int_nat @ F @ A4 ) ) ) ).
% bdd_above.I2
thf(fact_3180_bdd__above_OI2,axiom,
! [A4: set_o,F: $o > int,M: int] :
( ! [X: $o] :
( ( member_o @ X @ A4 )
=> ( ord_less_eq_int @ ( F @ X ) @ M ) )
=> ( condit2212336002400062152ve_int @ ( image_o_int @ F @ A4 ) ) ) ).
% bdd_above.I2
thf(fact_3181_bdd__above_OI2,axiom,
! [A4: set_nat,F: nat > int,M: int] :
( ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ( ord_less_eq_int @ ( F @ X ) @ M ) )
=> ( condit2212336002400062152ve_int @ ( image_nat_int @ F @ A4 ) ) ) ).
% bdd_above.I2
thf(fact_3182_bdd__above_OI2,axiom,
! [A4: set_int,F: int > int,M: int] :
( ! [X: int] :
( ( member_int @ X @ A4 )
=> ( ord_less_eq_int @ ( F @ X ) @ M ) )
=> ( condit2212336002400062152ve_int @ ( image_int_int @ F @ A4 ) ) ) ).
% bdd_above.I2
thf(fact_3183_bdd__above_OI2,axiom,
! [A4: set_o,F: $o > rat,M: rat] :
( ! [X: $o] :
( ( member_o @ X @ A4 )
=> ( ord_less_eq_rat @ ( F @ X ) @ M ) )
=> ( condit1579696412822616692ve_rat @ ( image_o_rat @ F @ A4 ) ) ) ).
% bdd_above.I2
thf(fact_3184_bdd__above_OI2,axiom,
! [A4: set_nat,F: nat > rat,M: rat] :
( ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ( ord_less_eq_rat @ ( F @ X ) @ M ) )
=> ( condit1579696412822616692ve_rat @ ( image_nat_rat @ F @ A4 ) ) ) ).
% bdd_above.I2
thf(fact_3185_bdd__above_OI2,axiom,
! [A4: set_int,F: int > rat,M: rat] :
( ! [X: int] :
( ( member_int @ X @ A4 )
=> ( ord_less_eq_rat @ ( F @ X ) @ M ) )
=> ( condit1579696412822616692ve_rat @ ( image_int_rat @ F @ A4 ) ) ) ).
% bdd_above.I2
thf(fact_3186_bdd__above_OI2,axiom,
! [A4: set_o,F: $o > code_integer,M: code_integer] :
( ! [X: $o] :
( ( member_o @ X @ A4 )
=> ( ord_le3102999989581377725nteger @ ( F @ X ) @ M ) )
=> ( condit1118808407379112185nteger @ ( image_o_Code_integer @ F @ A4 ) ) ) ).
% bdd_above.I2
thf(fact_3187_image__Int__subset,axiom,
! [F: nat > rat,A4: set_nat,B5: set_nat] : ( ord_less_eq_set_rat @ ( image_nat_rat @ F @ ( inf_inf_set_nat @ A4 @ B5 ) ) @ ( inf_inf_set_rat @ ( image_nat_rat @ F @ A4 ) @ ( image_nat_rat @ F @ B5 ) ) ) ).
% image_Int_subset
thf(fact_3188_image__Int__subset,axiom,
! [F: nat > nat,A4: set_nat,B5: set_nat] : ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ ( inf_inf_set_nat @ A4 @ B5 ) ) @ ( inf_inf_set_nat @ ( image_nat_nat @ F @ A4 ) @ ( image_nat_nat @ F @ B5 ) ) ) ).
% image_Int_subset
thf(fact_3189_image__Int__subset,axiom,
! [F: nat > int,A4: set_nat,B5: set_nat] : ( ord_less_eq_set_int @ ( image_nat_int @ F @ ( inf_inf_set_nat @ A4 @ B5 ) ) @ ( inf_inf_set_int @ ( image_nat_int @ F @ A4 ) @ ( image_nat_int @ F @ B5 ) ) ) ).
% image_Int_subset
thf(fact_3190_image__Int__subset,axiom,
! [F: int > nat,A4: set_int,B5: set_int] : ( ord_less_eq_set_nat @ ( image_int_nat @ F @ ( inf_inf_set_int @ A4 @ B5 ) ) @ ( inf_inf_set_nat @ ( image_int_nat @ F @ A4 ) @ ( image_int_nat @ F @ B5 ) ) ) ).
% image_Int_subset
thf(fact_3191_image__Int__subset,axiom,
! [F: int > int,A4: set_int,B5: set_int] : ( ord_less_eq_set_int @ ( image_int_int @ F @ ( inf_inf_set_int @ A4 @ B5 ) ) @ ( inf_inf_set_int @ ( image_int_int @ F @ A4 ) @ ( image_int_int @ F @ B5 ) ) ) ).
% image_Int_subset
thf(fact_3192_image__Int__subset,axiom,
! [F: product_prod_nat_nat > product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( image_5168914502847457605at_nat @ F @ ( inf_in2572325071724192079at_nat @ A4 @ B5 ) ) @ ( inf_in2572325071724192079at_nat @ ( image_5168914502847457605at_nat @ F @ A4 ) @ ( image_5168914502847457605at_nat @ F @ B5 ) ) ) ).
% image_Int_subset
thf(fact_3193_swap__inj__on,axiom,
! [A4: set_Pr7459493094073627847at_nat] :
( inj_on3942866938762682161at_nat
@ ( produc5456455677105495163at_nat
@ ^ [I2: set_Pr4329608150637261639at_nat,J: set_Pr4329608150637261639at_nat] : ( produc9060074326276436823at_nat @ J @ I2 ) )
@ A4 ) ).
% swap_inj_on
thf(fact_3194_swap__inj__on,axiom,
! [A4: set_Pr4329608150637261639at_nat] :
( inj_on8430303779476321073at_nat
@ ( produc7548845685434208379at_nat
@ ^ [I2: set_Pr1261947904930325089at_nat,J: set_Pr1261947904930325089at_nat] : ( produc2922128104949294807at_nat @ J @ I2 ) )
@ A4 ) ).
% swap_inj_on
thf(fact_3195_swap__inj__on,axiom,
! [A4: set_Pr3280342998896498603um_nat] :
( inj_on2339680571444523825at_num
@ ( produc5219324331026901427at_num
@ ^ [I2: num,J: nat] : ( product_Pair_nat_num @ J @ I2 ) )
@ A4 ) ).
% swap_inj_on
thf(fact_3196_swap__inj__on,axiom,
! [A4: set_Pr1261947904930325089at_nat] :
( inj_on8969904277767023793at_nat
@ ( produc2626176000494625587at_nat
@ ^ [I2: nat,J: nat] : ( product_Pair_nat_nat @ J @ I2 ) )
@ A4 ) ).
% swap_inj_on
thf(fact_3197_swap__inj__on,axiom,
! [A4: set_Pr958786334691620121nt_int] :
( inj_on6454360653267994289nt_int
@ ( produc4245557441103728435nt_int
@ ^ [I2: int,J: int] : ( product_Pair_int_int @ J @ I2 ) )
@ A4 ) ).
% swap_inj_on
thf(fact_3198_SUP__absorb,axiom,
! [K: $o,I3: set_o,A4: $o > product_unit] :
( ( member_o @ K @ I3 )
=> ( ( sup_sup_Product_unit @ ( A4 @ K ) @ ( comple4687483117567863418t_unit @ ( image_o_Product_unit @ A4 @ I3 ) ) )
= ( comple4687483117567863418t_unit @ ( image_o_Product_unit @ A4 @ I3 ) ) ) ) ).
% SUP_absorb
thf(fact_3199_SUP__absorb,axiom,
! [K: nat,I3: set_nat,A4: nat > product_unit] :
( ( member_nat @ K @ I3 )
=> ( ( sup_sup_Product_unit @ ( A4 @ K ) @ ( comple4687483117567863418t_unit @ ( image_8730104196221521654t_unit @ A4 @ I3 ) ) )
= ( comple4687483117567863418t_unit @ ( image_8730104196221521654t_unit @ A4 @ I3 ) ) ) ) ).
% SUP_absorb
thf(fact_3200_SUP__absorb,axiom,
! [K: int,I3: set_int,A4: int > product_unit] :
( ( member_int @ K @ I3 )
=> ( ( sup_sup_Product_unit @ ( A4 @ K ) @ ( comple4687483117567863418t_unit @ ( image_5846526143423701914t_unit @ A4 @ I3 ) ) )
= ( comple4687483117567863418t_unit @ ( image_5846526143423701914t_unit @ A4 @ I3 ) ) ) ) ).
% SUP_absorb
thf(fact_3201_SUP__absorb,axiom,
! [K: $o,I3: set_o,A4: $o > set_nat] :
( ( member_o @ K @ I3 )
=> ( ( sup_sup_set_nat @ ( A4 @ K ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ A4 @ I3 ) ) )
= ( comple7399068483239264473et_nat @ ( image_o_set_nat @ A4 @ I3 ) ) ) ) ).
% SUP_absorb
thf(fact_3202_SUP__absorb,axiom,
! [K: nat,I3: set_nat,A4: nat > set_nat] :
( ( member_nat @ K @ I3 )
=> ( ( sup_sup_set_nat @ ( A4 @ K ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A4 @ I3 ) ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A4 @ I3 ) ) ) ) ).
% SUP_absorb
thf(fact_3203_SUP__absorb,axiom,
! [K: int,I3: set_int,A4: int > set_nat] :
( ( member_int @ K @ I3 )
=> ( ( sup_sup_set_nat @ ( A4 @ K ) @ ( comple7399068483239264473et_nat @ ( image_int_set_nat @ A4 @ I3 ) ) )
= ( comple7399068483239264473et_nat @ ( image_int_set_nat @ A4 @ I3 ) ) ) ) ).
% SUP_absorb
thf(fact_3204_SUP__absorb,axiom,
! [K: product_prod_nat_nat,I3: set_Pr1261947904930325089at_nat,A4: product_prod_nat_nat > product_unit] :
( ( member8440522571783428010at_nat @ K @ I3 )
=> ( ( sup_sup_Product_unit @ ( A4 @ K ) @ ( comple4687483117567863418t_unit @ ( image_6169909464789311569t_unit @ A4 @ I3 ) ) )
= ( comple4687483117567863418t_unit @ ( image_6169909464789311569t_unit @ A4 @ I3 ) ) ) ) ).
% SUP_absorb
thf(fact_3205_SUP__absorb,axiom,
! [K: product_prod_nat_nat,I3: set_Pr1261947904930325089at_nat,A4: product_prod_nat_nat > set_nat] :
( ( member8440522571783428010at_nat @ K @ I3 )
=> ( ( sup_sup_set_nat @ ( A4 @ K ) @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ A4 @ I3 ) ) )
= ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ A4 @ I3 ) ) ) ) ).
% SUP_absorb
thf(fact_3206_SUP__absorb,axiom,
! [K: $o,I3: set_o,A4: $o > set_Pr8693737435421807431at_nat] :
( ( member_o @ K @ I3 )
=> ( ( sup_su718114333110466843at_nat @ ( A4 @ K ) @ ( comple2748534634209756852at_nat @ ( image_8493167801588156084at_nat @ A4 @ I3 ) ) )
= ( comple2748534634209756852at_nat @ ( image_8493167801588156084at_nat @ A4 @ I3 ) ) ) ) ).
% SUP_absorb
thf(fact_3207_SUP__absorb,axiom,
! [K: nat,I3: set_nat,A4: nat > set_Pr8693737435421807431at_nat] :
( ( member_nat @ K @ I3 )
=> ( ( sup_su718114333110466843at_nat @ ( A4 @ K ) @ ( comple2748534634209756852at_nat @ ( image_4592875511192039216at_nat @ A4 @ I3 ) ) )
= ( comple2748534634209756852at_nat @ ( image_4592875511192039216at_nat @ A4 @ I3 ) ) ) ) ).
% SUP_absorb
thf(fact_3208_Sup__inf,axiom,
! [B5: set_se7855581050983116737at_nat,A: set_Pr1261947904930325089at_nat] :
( ( inf_in2572325071724192079at_nat @ ( comple5685304695842803022at_nat @ B5 ) @ A )
= ( comple5685304695842803022at_nat
@ ( image_3684629450409544005at_nat
@ ^ [B4: set_Pr1261947904930325089at_nat] : ( inf_in2572325071724192079at_nat @ B4 @ A )
@ B5 ) ) ) ).
% Sup_inf
thf(fact_3209_Sup__inf,axiom,
! [B5: set_set_nat,A: set_nat] :
( ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ B5 ) @ A )
= ( comple7399068483239264473et_nat
@ ( image_7916887816326733075et_nat
@ ^ [B4: set_nat] : ( inf_inf_set_nat @ B4 @ A )
@ B5 ) ) ) ).
% Sup_inf
thf(fact_3210_Sup__inf,axiom,
! [B5: set_Product_unit,A: product_unit] :
( ( inf_inf_Product_unit @ ( comple4687483117567863418t_unit @ B5 ) @ A )
= ( comple4687483117567863418t_unit
@ ( image_405062704495631173t_unit
@ ^ [B4: product_unit] : ( inf_inf_Product_unit @ B4 @ A )
@ B5 ) ) ) ).
% Sup_inf
thf(fact_3211_Inf__sup,axiom,
! [B5: set_se6425085446898223655at_nat,A: set_Pr8693737435421807431at_nat] :
( ( sup_su718114333110466843at_nat @ ( comple6332534001959699150at_nat @ B5 ) @ A )
= ( comple6332534001959699150at_nat
@ ( image_1013614488123988933at_nat
@ ^ [B4: set_Pr8693737435421807431at_nat] : ( sup_su718114333110466843at_nat @ B4 @ A )
@ B5 ) ) ) ).
% Inf_sup
thf(fact_3212_Inf__sup,axiom,
! [B5: set_se5479201105039950375at_nat,A: set_Pr4329608150637261639at_nat] :
( ( sup_su5525570899277871387at_nat @ ( comple6751905152567503566at_nat @ B5 ) @ A )
= ( comple6751905152567503566at_nat
@ ( image_4303274092540115909at_nat
@ ^ [B4: set_Pr4329608150637261639at_nat] : ( sup_su5525570899277871387at_nat @ B4 @ A )
@ B5 ) ) ) ).
% Inf_sup
thf(fact_3213_Inf__sup,axiom,
! [B5: set_set_nat,A: set_nat] :
( ( sup_sup_set_nat @ ( comple7806235888213564991et_nat @ B5 ) @ A )
= ( comple7806235888213564991et_nat
@ ( image_7916887816326733075et_nat
@ ^ [B4: set_nat] : ( sup_sup_set_nat @ B4 @ A )
@ B5 ) ) ) ).
% Inf_sup
thf(fact_3214_Inf__sup,axiom,
! [B5: set_Product_unit,A: product_unit] :
( ( sup_sup_Product_unit @ ( comple2584293577114468500t_unit @ B5 ) @ A )
= ( comple2584293577114468500t_unit
@ ( image_405062704495631173t_unit
@ ^ [B4: product_unit] : ( sup_sup_Product_unit @ B4 @ A )
@ B5 ) ) ) ).
% Inf_sup
thf(fact_3215_INF__absorb,axiom,
! [K: $o,I3: set_o,A4: $o > product_unit] :
( ( member_o @ K @ I3 )
=> ( ( inf_inf_Product_unit @ ( A4 @ K ) @ ( comple2584293577114468500t_unit @ ( image_o_Product_unit @ A4 @ I3 ) ) )
= ( comple2584293577114468500t_unit @ ( image_o_Product_unit @ A4 @ I3 ) ) ) ) ).
% INF_absorb
thf(fact_3216_INF__absorb,axiom,
! [K: nat,I3: set_nat,A4: nat > product_unit] :
( ( member_nat @ K @ I3 )
=> ( ( inf_inf_Product_unit @ ( A4 @ K ) @ ( comple2584293577114468500t_unit @ ( image_8730104196221521654t_unit @ A4 @ I3 ) ) )
= ( comple2584293577114468500t_unit @ ( image_8730104196221521654t_unit @ A4 @ I3 ) ) ) ) ).
% INF_absorb
thf(fact_3217_INF__absorb,axiom,
! [K: int,I3: set_int,A4: int > product_unit] :
( ( member_int @ K @ I3 )
=> ( ( inf_inf_Product_unit @ ( A4 @ K ) @ ( comple2584293577114468500t_unit @ ( image_5846526143423701914t_unit @ A4 @ I3 ) ) )
= ( comple2584293577114468500t_unit @ ( image_5846526143423701914t_unit @ A4 @ I3 ) ) ) ) ).
% INF_absorb
thf(fact_3218_INF__absorb,axiom,
! [K: $o,I3: set_o,A4: $o > set_nat] :
( ( member_o @ K @ I3 )
=> ( ( inf_inf_set_nat @ ( A4 @ K ) @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ A4 @ I3 ) ) )
= ( comple7806235888213564991et_nat @ ( image_o_set_nat @ A4 @ I3 ) ) ) ) ).
% INF_absorb
thf(fact_3219_INF__absorb,axiom,
! [K: nat,I3: set_nat,A4: nat > set_nat] :
( ( member_nat @ K @ I3 )
=> ( ( inf_inf_set_nat @ ( A4 @ K ) @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ A4 @ I3 ) ) )
= ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ A4 @ I3 ) ) ) ) ).
% INF_absorb
thf(fact_3220_INF__absorb,axiom,
! [K: int,I3: set_int,A4: int > set_nat] :
( ( member_int @ K @ I3 )
=> ( ( inf_inf_set_nat @ ( A4 @ K ) @ ( comple7806235888213564991et_nat @ ( image_int_set_nat @ A4 @ I3 ) ) )
= ( comple7806235888213564991et_nat @ ( image_int_set_nat @ A4 @ I3 ) ) ) ) ).
% INF_absorb
thf(fact_3221_INF__absorb,axiom,
! [K: product_prod_nat_nat,I3: set_Pr1261947904930325089at_nat,A4: product_prod_nat_nat > product_unit] :
( ( member8440522571783428010at_nat @ K @ I3 )
=> ( ( inf_inf_Product_unit @ ( A4 @ K ) @ ( comple2584293577114468500t_unit @ ( image_6169909464789311569t_unit @ A4 @ I3 ) ) )
= ( comple2584293577114468500t_unit @ ( image_6169909464789311569t_unit @ A4 @ I3 ) ) ) ) ).
% INF_absorb
thf(fact_3222_INF__absorb,axiom,
! [K: $o,I3: set_o,A4: $o > set_Pr1261947904930325089at_nat] :
( ( member_o @ K @ I3 )
=> ( ( inf_in2572325071724192079at_nat @ ( A4 @ K ) @ ( comple1930127330929219688at_nat @ ( image_7077428525161401678at_nat @ A4 @ I3 ) ) )
= ( comple1930127330929219688at_nat @ ( image_7077428525161401678at_nat @ A4 @ I3 ) ) ) ) ).
% INF_absorb
thf(fact_3223_INF__absorb,axiom,
! [K: nat,I3: set_nat,A4: nat > set_Pr1261947904930325089at_nat] :
( ( member_nat @ K @ I3 )
=> ( ( inf_in2572325071724192079at_nat @ ( A4 @ K ) @ ( comple1930127330929219688at_nat @ ( image_7178329752028323786at_nat @ A4 @ I3 ) ) )
= ( comple1930127330929219688at_nat @ ( image_7178329752028323786at_nat @ A4 @ I3 ) ) ) ) ).
% INF_absorb
thf(fact_3224_INF__absorb,axiom,
! [K: int,I3: set_int,A4: int > set_Pr1261947904930325089at_nat] :
( ( member_int @ K @ I3 )
=> ( ( inf_in2572325071724192079at_nat @ ( A4 @ K ) @ ( comple1930127330929219688at_nat @ ( image_8938366415781434990at_nat @ A4 @ I3 ) ) )
= ( comple1930127330929219688at_nat @ ( image_8938366415781434990at_nat @ A4 @ I3 ) ) ) ) ).
% INF_absorb
thf(fact_3225_translation__subtract__Int,axiom,
! [A: int,S2: set_int,T2: set_int] :
( ( image_int_int
@ ^ [X4: int] : ( minus_minus_int @ X4 @ A )
@ ( inf_inf_set_int @ S2 @ T2 ) )
= ( inf_inf_set_int
@ ( image_int_int
@ ^ [X4: int] : ( minus_minus_int @ X4 @ A )
@ S2 )
@ ( image_int_int
@ ^ [X4: int] : ( minus_minus_int @ X4 @ A )
@ T2 ) ) ) ).
% translation_subtract_Int
thf(fact_3226_translation__subtract__Int,axiom,
! [A: rat,S2: set_rat,T2: set_rat] :
( ( image_rat_rat
@ ^ [X4: rat] : ( minus_minus_rat @ X4 @ A )
@ ( inf_inf_set_rat @ S2 @ T2 ) )
= ( inf_inf_set_rat
@ ( image_rat_rat
@ ^ [X4: rat] : ( minus_minus_rat @ X4 @ A )
@ S2 )
@ ( image_rat_rat
@ ^ [X4: rat] : ( minus_minus_rat @ X4 @ A )
@ T2 ) ) ) ).
% translation_subtract_Int
thf(fact_3227_translation__subtract__Int,axiom,
! [A: code_integer,S2: set_Code_integer,T2: set_Code_integer] :
( ( image_4470545334726330049nteger
@ ^ [X4: code_integer] : ( minus_8373710615458151222nteger @ X4 @ A )
@ ( inf_in1364745209274528805nteger @ S2 @ T2 ) )
= ( inf_in1364745209274528805nteger
@ ( image_4470545334726330049nteger
@ ^ [X4: code_integer] : ( minus_8373710615458151222nteger @ X4 @ A )
@ S2 )
@ ( image_4470545334726330049nteger
@ ^ [X4: code_integer] : ( minus_8373710615458151222nteger @ X4 @ A )
@ T2 ) ) ) ).
% translation_subtract_Int
thf(fact_3228_UN__absorb,axiom,
! [K: $o,I3: set_o,A4: $o > set_nat] :
( ( member_o @ K @ I3 )
=> ( ( sup_sup_set_nat @ ( A4 @ K ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ A4 @ I3 ) ) )
= ( comple7399068483239264473et_nat @ ( image_o_set_nat @ A4 @ I3 ) ) ) ) ).
% UN_absorb
thf(fact_3229_UN__absorb,axiom,
! [K: nat,I3: set_nat,A4: nat > set_nat] :
( ( member_nat @ K @ I3 )
=> ( ( sup_sup_set_nat @ ( A4 @ K ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A4 @ I3 ) ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A4 @ I3 ) ) ) ) ).
% UN_absorb
thf(fact_3230_UN__absorb,axiom,
! [K: int,I3: set_int,A4: int > set_nat] :
( ( member_int @ K @ I3 )
=> ( ( sup_sup_set_nat @ ( A4 @ K ) @ ( comple7399068483239264473et_nat @ ( image_int_set_nat @ A4 @ I3 ) ) )
= ( comple7399068483239264473et_nat @ ( image_int_set_nat @ A4 @ I3 ) ) ) ) ).
% UN_absorb
thf(fact_3231_UN__absorb,axiom,
! [K: product_prod_nat_nat,I3: set_Pr1261947904930325089at_nat,A4: product_prod_nat_nat > set_nat] :
( ( member8440522571783428010at_nat @ K @ I3 )
=> ( ( sup_sup_set_nat @ ( A4 @ K ) @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ A4 @ I3 ) ) )
= ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ A4 @ I3 ) ) ) ) ).
% UN_absorb
thf(fact_3232_UN__absorb,axiom,
! [K: $o,I3: set_o,A4: $o > set_Pr8693737435421807431at_nat] :
( ( member_o @ K @ I3 )
=> ( ( sup_su718114333110466843at_nat @ ( A4 @ K ) @ ( comple2748534634209756852at_nat @ ( image_8493167801588156084at_nat @ A4 @ I3 ) ) )
= ( comple2748534634209756852at_nat @ ( image_8493167801588156084at_nat @ A4 @ I3 ) ) ) ) ).
% UN_absorb
thf(fact_3233_UN__absorb,axiom,
! [K: nat,I3: set_nat,A4: nat > set_Pr8693737435421807431at_nat] :
( ( member_nat @ K @ I3 )
=> ( ( sup_su718114333110466843at_nat @ ( A4 @ K ) @ ( comple2748534634209756852at_nat @ ( image_4592875511192039216at_nat @ A4 @ I3 ) ) )
= ( comple2748534634209756852at_nat @ ( image_4592875511192039216at_nat @ A4 @ I3 ) ) ) ) ).
% UN_absorb
thf(fact_3234_UN__absorb,axiom,
! [K: int,I3: set_int,A4: int > set_Pr8693737435421807431at_nat] :
( ( member_int @ K @ I3 )
=> ( ( sup_su718114333110466843at_nat @ ( A4 @ K ) @ ( comple2748534634209756852at_nat @ ( image_525487068563185620at_nat @ A4 @ I3 ) ) )
= ( comple2748534634209756852at_nat @ ( image_525487068563185620at_nat @ A4 @ I3 ) ) ) ) ).
% UN_absorb
thf(fact_3235_UN__absorb,axiom,
! [K: product_prod_nat_nat,I3: set_Pr1261947904930325089at_nat,A4: product_prod_nat_nat > set_Pr8693737435421807431at_nat] :
( ( member8440522571783428010at_nat @ K @ I3 )
=> ( ( sup_su718114333110466843at_nat @ ( A4 @ K ) @ ( comple2748534634209756852at_nat @ ( image_5803077947706746251at_nat @ A4 @ I3 ) ) )
= ( comple2748534634209756852at_nat @ ( image_5803077947706746251at_nat @ A4 @ I3 ) ) ) ) ).
% UN_absorb
thf(fact_3236_UN__absorb,axiom,
! [K: $o,I3: set_o,A4: $o > set_Pr4329608150637261639at_nat] :
( ( member_o @ K @ I3 )
=> ( ( sup_su5525570899277871387at_nat @ ( A4 @ K ) @ ( comple4364388969068068532at_nat @ ( image_2496312346330513076at_nat @ A4 @ I3 ) ) )
= ( comple4364388969068068532at_nat @ ( image_2496312346330513076at_nat @ A4 @ I3 ) ) ) ) ).
% UN_absorb
thf(fact_3237_UN__absorb,axiom,
! [K: nat,I3: set_nat,A4: nat > set_Pr4329608150637261639at_nat] :
( ( member_nat @ K @ I3 )
=> ( ( sup_su5525570899277871387at_nat @ ( A4 @ K ) @ ( comple4364388969068068532at_nat @ ( image_1087905878993176368at_nat @ A4 @ I3 ) ) )
= ( comple4364388969068068532at_nat @ ( image_1087905878993176368at_nat @ A4 @ I3 ) ) ) ) ).
% UN_absorb
thf(fact_3238_INT__absorb,axiom,
! [K: produc3843707927480180839at_nat,I3: set_Pr4329608150637261639at_nat,A4: produc3843707927480180839at_nat > set_Pr1261947904930325089at_nat] :
( ( member8757157785044589968at_nat @ K @ I3 )
=> ( ( inf_in2572325071724192079at_nat @ ( A4 @ K ) @ ( comple1930127330929219688at_nat @ ( image_7979982168039674559at_nat @ A4 @ I3 ) ) )
= ( comple1930127330929219688at_nat @ ( image_7979982168039674559at_nat @ A4 @ I3 ) ) ) ) ).
% INT_absorb
thf(fact_3239_INT__absorb,axiom,
! [K: product_prod_nat_nat,I3: set_Pr1261947904930325089at_nat,A4: product_prod_nat_nat > set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ K @ I3 )
=> ( ( inf_in2572325071724192079at_nat @ ( A4 @ K ) @ ( comple1930127330929219688at_nat @ ( image_5423882813909314213at_nat @ A4 @ I3 ) ) )
= ( comple1930127330929219688at_nat @ ( image_5423882813909314213at_nat @ A4 @ I3 ) ) ) ) ).
% INT_absorb
thf(fact_3240_INT__absorb,axiom,
! [K: $o,I3: set_o,A4: $o > set_Pr1261947904930325089at_nat] :
( ( member_o @ K @ I3 )
=> ( ( inf_in2572325071724192079at_nat @ ( A4 @ K ) @ ( comple1930127330929219688at_nat @ ( image_7077428525161401678at_nat @ A4 @ I3 ) ) )
= ( comple1930127330929219688at_nat @ ( image_7077428525161401678at_nat @ A4 @ I3 ) ) ) ) ).
% INT_absorb
thf(fact_3241_INT__absorb,axiom,
! [K: nat,I3: set_nat,A4: nat > set_Pr1261947904930325089at_nat] :
( ( member_nat @ K @ I3 )
=> ( ( inf_in2572325071724192079at_nat @ ( A4 @ K ) @ ( comple1930127330929219688at_nat @ ( image_7178329752028323786at_nat @ A4 @ I3 ) ) )
= ( comple1930127330929219688at_nat @ ( image_7178329752028323786at_nat @ A4 @ I3 ) ) ) ) ).
% INT_absorb
thf(fact_3242_INT__absorb,axiom,
! [K: int,I3: set_int,A4: int > set_Pr1261947904930325089at_nat] :
( ( member_int @ K @ I3 )
=> ( ( inf_in2572325071724192079at_nat @ ( A4 @ K ) @ ( comple1930127330929219688at_nat @ ( image_8938366415781434990at_nat @ A4 @ I3 ) ) )
= ( comple1930127330929219688at_nat @ ( image_8938366415781434990at_nat @ A4 @ I3 ) ) ) ) ).
% INT_absorb
thf(fact_3243_INT__absorb,axiom,
! [K: produc3843707927480180839at_nat,I3: set_Pr4329608150637261639at_nat,A4: produc3843707927480180839at_nat > set_nat] :
( ( member8757157785044589968at_nat @ K @ I3 )
=> ( ( inf_inf_set_nat @ ( A4 @ K ) @ ( comple7806235888213564991et_nat @ ( image_3160451284896210088et_nat @ A4 @ I3 ) ) )
= ( comple7806235888213564991et_nat @ ( image_3160451284896210088et_nat @ A4 @ I3 ) ) ) ) ).
% INT_absorb
thf(fact_3244_INT__absorb,axiom,
! [K: product_prod_nat_nat,I3: set_Pr1261947904930325089at_nat,A4: product_prod_nat_nat > set_nat] :
( ( member8440522571783428010at_nat @ K @ I3 )
=> ( ( inf_inf_set_nat @ ( A4 @ K ) @ ( comple7806235888213564991et_nat @ ( image_15824709712370754et_nat @ A4 @ I3 ) ) )
= ( comple7806235888213564991et_nat @ ( image_15824709712370754et_nat @ A4 @ I3 ) ) ) ) ).
% INT_absorb
thf(fact_3245_INT__absorb,axiom,
! [K: $o,I3: set_o,A4: $o > set_nat] :
( ( member_o @ K @ I3 )
=> ( ( inf_inf_set_nat @ ( A4 @ K ) @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ A4 @ I3 ) ) )
= ( comple7806235888213564991et_nat @ ( image_o_set_nat @ A4 @ I3 ) ) ) ) ).
% INT_absorb
thf(fact_3246_INT__absorb,axiom,
! [K: nat,I3: set_nat,A4: nat > set_nat] :
( ( member_nat @ K @ I3 )
=> ( ( inf_inf_set_nat @ ( A4 @ K ) @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ A4 @ I3 ) ) )
= ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ A4 @ I3 ) ) ) ) ).
% INT_absorb
thf(fact_3247_INT__absorb,axiom,
! [K: int,I3: set_int,A4: int > set_nat] :
( ( member_int @ K @ I3 )
=> ( ( inf_inf_set_nat @ ( A4 @ K ) @ ( comple7806235888213564991et_nat @ ( image_int_set_nat @ A4 @ I3 ) ) )
= ( comple7806235888213564991et_nat @ ( image_int_set_nat @ A4 @ I3 ) ) ) ) ).
% INT_absorb
thf(fact_3248_Int__Union2,axiom,
! [B5: set_se7855581050983116737at_nat,A4: set_Pr1261947904930325089at_nat] :
( ( inf_in2572325071724192079at_nat @ ( comple5685304695842803022at_nat @ B5 ) @ A4 )
= ( comple5685304695842803022at_nat
@ ( image_3684629450409544005at_nat
@ ^ [C5: set_Pr1261947904930325089at_nat] : ( inf_in2572325071724192079at_nat @ C5 @ A4 )
@ B5 ) ) ) ).
% Int_Union2
thf(fact_3249_Int__Union2,axiom,
! [B5: set_set_nat,A4: set_nat] :
( ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ B5 ) @ A4 )
= ( comple7399068483239264473et_nat
@ ( image_7916887816326733075et_nat
@ ^ [C5: set_nat] : ( inf_inf_set_nat @ C5 @ A4 )
@ B5 ) ) ) ).
% Int_Union2
thf(fact_3250_Int__Union,axiom,
! [A4: set_Pr1261947904930325089at_nat,B5: set_se7855581050983116737at_nat] :
( ( inf_in2572325071724192079at_nat @ A4 @ ( comple5685304695842803022at_nat @ B5 ) )
= ( comple5685304695842803022at_nat @ ( image_3684629450409544005at_nat @ ( inf_in2572325071724192079at_nat @ A4 ) @ B5 ) ) ) ).
% Int_Union
thf(fact_3251_Int__Union,axiom,
! [A4: set_nat,B5: set_set_nat] :
( ( inf_inf_set_nat @ A4 @ ( comple7399068483239264473et_nat @ B5 ) )
= ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ ( inf_inf_set_nat @ A4 ) @ B5 ) ) ) ).
% Int_Union
thf(fact_3252_Un__Inter,axiom,
! [A4: set_Pr8693737435421807431at_nat,B5: set_se6425085446898223655at_nat] :
( ( sup_su718114333110466843at_nat @ A4 @ ( comple6332534001959699150at_nat @ B5 ) )
= ( comple6332534001959699150at_nat @ ( image_1013614488123988933at_nat @ ( sup_su718114333110466843at_nat @ A4 ) @ B5 ) ) ) ).
% Un_Inter
thf(fact_3253_Un__Inter,axiom,
! [A4: set_Pr4329608150637261639at_nat,B5: set_se5479201105039950375at_nat] :
( ( sup_su5525570899277871387at_nat @ A4 @ ( comple6751905152567503566at_nat @ B5 ) )
= ( comple6751905152567503566at_nat @ ( image_4303274092540115909at_nat @ ( sup_su5525570899277871387at_nat @ A4 ) @ B5 ) ) ) ).
% Un_Inter
thf(fact_3254_Un__Inter,axiom,
! [A4: set_nat,B5: set_set_nat] :
( ( sup_sup_set_nat @ A4 @ ( comple7806235888213564991et_nat @ B5 ) )
= ( comple7806235888213564991et_nat @ ( image_7916887816326733075et_nat @ ( sup_sup_set_nat @ A4 ) @ B5 ) ) ) ).
% Un_Inter
thf(fact_3255_inj__on__iff__surj,axiom,
! [A4: set_rat,A7: set_nat] :
( ( A4 != bot_bot_set_rat )
=> ( ( ? [F2: rat > nat] :
( ( inj_on_rat_nat @ F2 @ A4 )
& ( ord_less_eq_set_nat @ ( image_rat_nat @ F2 @ A4 ) @ A7 ) ) )
= ( ? [G3: nat > rat] :
( ( image_nat_rat @ G3 @ A7 )
= A4 ) ) ) ) ).
% inj_on_iff_surj
thf(fact_3256_inj__on__iff__surj,axiom,
! [A4: set_int,A7: set_nat] :
( ( A4 != bot_bot_set_int )
=> ( ( ? [F2: int > nat] :
( ( inj_on_int_nat @ F2 @ A4 )
& ( ord_less_eq_set_nat @ ( image_int_nat @ F2 @ A4 ) @ A7 ) ) )
= ( ? [G3: nat > int] :
( ( image_nat_int @ G3 @ A7 )
= A4 ) ) ) ) ).
% inj_on_iff_surj
thf(fact_3257_inj__on__iff__surj,axiom,
! [A4: set_int,A7: set_int] :
( ( A4 != bot_bot_set_int )
=> ( ( ? [F2: int > int] :
( ( inj_on_int_int @ F2 @ A4 )
& ( ord_less_eq_set_int @ ( image_int_int @ F2 @ A4 ) @ A7 ) ) )
= ( ? [G3: int > int] :
( ( image_int_int @ G3 @ A7 )
= A4 ) ) ) ) ).
% inj_on_iff_surj
thf(fact_3258_inj__on__iff__surj,axiom,
! [A4: set_nat,A7: set_rat] :
( ( A4 != bot_bot_set_nat )
=> ( ( ? [F2: nat > rat] :
( ( inj_on_nat_rat @ F2 @ A4 )
& ( ord_less_eq_set_rat @ ( image_nat_rat @ F2 @ A4 ) @ A7 ) ) )
= ( ? [G3: rat > nat] :
( ( image_rat_nat @ G3 @ A7 )
= A4 ) ) ) ) ).
% inj_on_iff_surj
thf(fact_3259_inj__on__iff__surj,axiom,
! [A4: set_nat,A7: set_nat] :
( ( A4 != bot_bot_set_nat )
=> ( ( ? [F2: nat > nat] :
( ( inj_on_nat_nat @ F2 @ A4 )
& ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A4 ) @ A7 ) ) )
= ( ? [G3: nat > nat] :
( ( image_nat_nat @ G3 @ A7 )
= A4 ) ) ) ) ).
% inj_on_iff_surj
thf(fact_3260_inj__on__iff__surj,axiom,
! [A4: set_nat,A7: set_int] :
( ( A4 != bot_bot_set_nat )
=> ( ( ? [F2: nat > int] :
( ( inj_on_nat_int @ F2 @ A4 )
& ( ord_less_eq_set_int @ ( image_nat_int @ F2 @ A4 ) @ A7 ) ) )
= ( ? [G3: int > nat] :
( ( image_int_nat @ G3 @ A7 )
= A4 ) ) ) ) ).
% inj_on_iff_surj
thf(fact_3261_Int__Inter__eq_I1_J,axiom,
! [B8: set_se7855581050983116737at_nat,A4: set_Pr1261947904930325089at_nat] :
( ( ( B8 = bot_bo3083307316010499117at_nat )
=> ( ( inf_in2572325071724192079at_nat @ A4 @ ( comple1930127330929219688at_nat @ B8 ) )
= A4 ) )
& ( ( B8 != bot_bo3083307316010499117at_nat )
=> ( ( inf_in2572325071724192079at_nat @ A4 @ ( comple1930127330929219688at_nat @ B8 ) )
= ( comple1930127330929219688at_nat @ ( image_3684629450409544005at_nat @ ( inf_in2572325071724192079at_nat @ A4 ) @ B8 ) ) ) ) ) ).
% Int_Inter_eq(1)
thf(fact_3262_Int__Inter__eq_I1_J,axiom,
! [B8: set_set_nat,A4: set_nat] :
( ( ( B8 = bot_bot_set_set_nat )
=> ( ( inf_inf_set_nat @ A4 @ ( comple7806235888213564991et_nat @ B8 ) )
= A4 ) )
& ( ( B8 != bot_bot_set_set_nat )
=> ( ( inf_inf_set_nat @ A4 @ ( comple7806235888213564991et_nat @ B8 ) )
= ( comple7806235888213564991et_nat @ ( image_7916887816326733075et_nat @ ( inf_inf_set_nat @ A4 ) @ B8 ) ) ) ) ) ).
% Int_Inter_eq(1)
thf(fact_3263_Int__Inter__eq_I2_J,axiom,
! [B8: set_se7855581050983116737at_nat,A4: set_Pr1261947904930325089at_nat] :
( ( ( B8 = bot_bo3083307316010499117at_nat )
=> ( ( inf_in2572325071724192079at_nat @ ( comple1930127330929219688at_nat @ B8 ) @ A4 )
= A4 ) )
& ( ( B8 != bot_bo3083307316010499117at_nat )
=> ( ( inf_in2572325071724192079at_nat @ ( comple1930127330929219688at_nat @ B8 ) @ A4 )
= ( comple1930127330929219688at_nat
@ ( image_3684629450409544005at_nat
@ ^ [B6: set_Pr1261947904930325089at_nat] : ( inf_in2572325071724192079at_nat @ B6 @ A4 )
@ B8 ) ) ) ) ) ).
% Int_Inter_eq(2)
thf(fact_3264_Int__Inter__eq_I2_J,axiom,
! [B8: set_set_nat,A4: set_nat] :
( ( ( B8 = bot_bot_set_set_nat )
=> ( ( inf_inf_set_nat @ ( comple7806235888213564991et_nat @ B8 ) @ A4 )
= A4 ) )
& ( ( B8 != bot_bot_set_set_nat )
=> ( ( inf_inf_set_nat @ ( comple7806235888213564991et_nat @ B8 ) @ A4 )
= ( comple7806235888213564991et_nat
@ ( image_7916887816326733075et_nat
@ ^ [B6: set_nat] : ( inf_inf_set_nat @ B6 @ A4 )
@ B8 ) ) ) ) ) ).
% Int_Inter_eq(2)
thf(fact_3265_inj__image__subset__iff,axiom,
! [F: nat > rat,A4: set_nat,B5: set_nat] :
( ( inj_on_nat_rat @ F @ top_top_set_nat )
=> ( ( ord_less_eq_set_rat @ ( image_nat_rat @ F @ A4 ) @ ( image_nat_rat @ F @ B5 ) )
= ( ord_less_eq_set_nat @ A4 @ B5 ) ) ) ).
% inj_image_subset_iff
thf(fact_3266_inj__image__subset__iff,axiom,
! [F: nat > nat,A4: set_nat,B5: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A4 ) @ ( image_nat_nat @ F @ B5 ) )
= ( ord_less_eq_set_nat @ A4 @ B5 ) ) ) ).
% inj_image_subset_iff
thf(fact_3267_inj__image__subset__iff,axiom,
! [F: nat > int,A4: set_nat,B5: set_nat] :
( ( inj_on_nat_int @ F @ top_top_set_nat )
=> ( ( ord_less_eq_set_int @ ( image_nat_int @ F @ A4 ) @ ( image_nat_int @ F @ B5 ) )
= ( ord_less_eq_set_nat @ A4 @ B5 ) ) ) ).
% inj_image_subset_iff
thf(fact_3268_inj__image__subset__iff,axiom,
! [F: int > nat,A4: set_int,B5: set_int] :
( ( inj_on_int_nat @ F @ top_top_set_int )
=> ( ( ord_less_eq_set_nat @ ( image_int_nat @ F @ A4 ) @ ( image_int_nat @ F @ B5 ) )
= ( ord_less_eq_set_int @ A4 @ B5 ) ) ) ).
% inj_image_subset_iff
thf(fact_3269_inj__image__subset__iff,axiom,
! [F: int > int,A4: set_int,B5: set_int] :
( ( inj_on_int_int @ F @ top_top_set_int )
=> ( ( ord_less_eq_set_int @ ( image_int_int @ F @ A4 ) @ ( image_int_int @ F @ B5 ) )
= ( ord_less_eq_set_int @ A4 @ B5 ) ) ) ).
% inj_image_subset_iff
thf(fact_3270_inj__on__image__set__diff,axiom,
! [F: nat > rat,C3: set_nat,A4: set_nat,B5: set_nat] :
( ( inj_on_nat_rat @ F @ C3 )
=> ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A4 @ B5 ) @ C3 )
=> ( ( ord_less_eq_set_nat @ B5 @ C3 )
=> ( ( image_nat_rat @ F @ ( minus_minus_set_nat @ A4 @ B5 ) )
= ( minus_minus_set_rat @ ( image_nat_rat @ F @ A4 ) @ ( image_nat_rat @ F @ B5 ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_3271_inj__on__image__set__diff,axiom,
! [F: nat > nat,C3: set_nat,A4: set_nat,B5: set_nat] :
( ( inj_on_nat_nat @ F @ C3 )
=> ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A4 @ B5 ) @ C3 )
=> ( ( ord_less_eq_set_nat @ B5 @ C3 )
=> ( ( image_nat_nat @ F @ ( minus_minus_set_nat @ A4 @ B5 ) )
= ( minus_minus_set_nat @ ( image_nat_nat @ F @ A4 ) @ ( image_nat_nat @ F @ B5 ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_3272_inj__on__image__set__diff,axiom,
! [F: nat > int,C3: set_nat,A4: set_nat,B5: set_nat] :
( ( inj_on_nat_int @ F @ C3 )
=> ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A4 @ B5 ) @ C3 )
=> ( ( ord_less_eq_set_nat @ B5 @ C3 )
=> ( ( image_nat_int @ F @ ( minus_minus_set_nat @ A4 @ B5 ) )
= ( minus_minus_set_int @ ( image_nat_int @ F @ A4 ) @ ( image_nat_int @ F @ B5 ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_3273_inj__on__image__set__diff,axiom,
! [F: int > nat,C3: set_int,A4: set_int,B5: set_int] :
( ( inj_on_int_nat @ F @ C3 )
=> ( ( ord_less_eq_set_int @ ( minus_minus_set_int @ A4 @ B5 ) @ C3 )
=> ( ( ord_less_eq_set_int @ B5 @ C3 )
=> ( ( image_int_nat @ F @ ( minus_minus_set_int @ A4 @ B5 ) )
= ( minus_minus_set_nat @ ( image_int_nat @ F @ A4 ) @ ( image_int_nat @ F @ B5 ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_3274_inj__on__image__set__diff,axiom,
! [F: int > int,C3: set_int,A4: set_int,B5: set_int] :
( ( inj_on_int_int @ F @ C3 )
=> ( ( ord_less_eq_set_int @ ( minus_minus_set_int @ A4 @ B5 ) @ C3 )
=> ( ( ord_less_eq_set_int @ B5 @ C3 )
=> ( ( image_int_int @ F @ ( minus_minus_set_int @ A4 @ B5 ) )
= ( minus_minus_set_int @ ( image_int_int @ F @ A4 ) @ ( image_int_int @ F @ B5 ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_3275_image__set__diff,axiom,
! [F: nat > rat,A4: set_nat,B5: set_nat] :
( ( inj_on_nat_rat @ F @ top_top_set_nat )
=> ( ( image_nat_rat @ F @ ( minus_minus_set_nat @ A4 @ B5 ) )
= ( minus_minus_set_rat @ ( image_nat_rat @ F @ A4 ) @ ( image_nat_rat @ F @ B5 ) ) ) ) ).
% image_set_diff
thf(fact_3276_image__set__diff,axiom,
! [F: nat > nat,A4: set_nat,B5: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( image_nat_nat @ F @ ( minus_minus_set_nat @ A4 @ B5 ) )
= ( minus_minus_set_nat @ ( image_nat_nat @ F @ A4 ) @ ( image_nat_nat @ F @ B5 ) ) ) ) ).
% image_set_diff
thf(fact_3277_image__set__diff,axiom,
! [F: nat > int,A4: set_nat,B5: set_nat] :
( ( inj_on_nat_int @ F @ top_top_set_nat )
=> ( ( image_nat_int @ F @ ( minus_minus_set_nat @ A4 @ B5 ) )
= ( minus_minus_set_int @ ( image_nat_int @ F @ A4 ) @ ( image_nat_int @ F @ B5 ) ) ) ) ).
% image_set_diff
thf(fact_3278_image__set__diff,axiom,
! [F: int > nat,A4: set_int,B5: set_int] :
( ( inj_on_int_nat @ F @ top_top_set_int )
=> ( ( image_int_nat @ F @ ( minus_minus_set_int @ A4 @ B5 ) )
= ( minus_minus_set_nat @ ( image_int_nat @ F @ A4 ) @ ( image_int_nat @ F @ B5 ) ) ) ) ).
% image_set_diff
thf(fact_3279_image__set__diff,axiom,
! [F: int > int,A4: set_int,B5: set_int] :
( ( inj_on_int_int @ F @ top_top_set_int )
=> ( ( image_int_int @ F @ ( minus_minus_set_int @ A4 @ B5 ) )
= ( minus_minus_set_int @ ( image_int_int @ F @ A4 ) @ ( image_int_int @ F @ B5 ) ) ) ) ).
% image_set_diff
thf(fact_3280_cINF__lower2,axiom,
! [F: $o > nat,A4: set_o,X2: $o,U: nat] :
( ( condit1738341127787009408ow_nat @ ( image_o_nat @ F @ A4 ) )
=> ( ( member_o @ X2 @ A4 )
=> ( ( ord_less_eq_nat @ ( F @ X2 ) @ U )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_o_nat @ F @ A4 ) ) @ U ) ) ) ) ).
% cINF_lower2
thf(fact_3281_cINF__lower2,axiom,
! [F: nat > nat,A4: set_nat,X2: nat,U: nat] :
( ( condit1738341127787009408ow_nat @ ( image_nat_nat @ F @ A4 ) )
=> ( ( member_nat @ X2 @ A4 )
=> ( ( ord_less_eq_nat @ ( F @ X2 ) @ U )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_nat_nat @ F @ A4 ) ) @ U ) ) ) ) ).
% cINF_lower2
thf(fact_3282_cINF__lower2,axiom,
! [F: int > nat,A4: set_int,X2: int,U: nat] :
( ( condit1738341127787009408ow_nat @ ( image_int_nat @ F @ A4 ) )
=> ( ( member_int @ X2 @ A4 )
=> ( ( ord_less_eq_nat @ ( F @ X2 ) @ U )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_int_nat @ F @ A4 ) ) @ U ) ) ) ) ).
% cINF_lower2
thf(fact_3283_cINF__lower2,axiom,
! [F: $o > int,A4: set_o,X2: $o,U: int] :
( ( condit1735850657277959132ow_int @ ( image_o_int @ F @ A4 ) )
=> ( ( member_o @ X2 @ A4 )
=> ( ( ord_less_eq_int @ ( F @ X2 ) @ U )
=> ( ord_less_eq_int @ ( complete_Inf_Inf_int @ ( image_o_int @ F @ A4 ) ) @ U ) ) ) ) ).
% cINF_lower2
thf(fact_3284_cINF__lower2,axiom,
! [F: nat > int,A4: set_nat,X2: nat,U: int] :
( ( condit1735850657277959132ow_int @ ( image_nat_int @ F @ A4 ) )
=> ( ( member_nat @ X2 @ A4 )
=> ( ( ord_less_eq_int @ ( F @ X2 ) @ U )
=> ( ord_less_eq_int @ ( complete_Inf_Inf_int @ ( image_nat_int @ F @ A4 ) ) @ U ) ) ) ) ).
% cINF_lower2
thf(fact_3285_cINF__lower2,axiom,
! [F: int > int,A4: set_int,X2: int,U: int] :
( ( condit1735850657277959132ow_int @ ( image_int_int @ F @ A4 ) )
=> ( ( member_int @ X2 @ A4 )
=> ( ( ord_less_eq_int @ ( F @ X2 ) @ U )
=> ( ord_less_eq_int @ ( complete_Inf_Inf_int @ ( image_int_int @ F @ A4 ) ) @ U ) ) ) ) ).
% cINF_lower2
thf(fact_3286_cINF__lower2,axiom,
! [F: $o > product_unit,A4: set_o,X2: $o,U: product_unit] :
( ( condit9161856309968323677t_unit @ ( image_o_Product_unit @ F @ A4 ) )
=> ( ( member_o @ X2 @ A4 )
=> ( ( ord_le3221252021190050221t_unit @ ( F @ X2 ) @ U )
=> ( ord_le3221252021190050221t_unit @ ( comple2584293577114468500t_unit @ ( image_o_Product_unit @ F @ A4 ) ) @ U ) ) ) ) ).
% cINF_lower2
thf(fact_3287_cINF__lower2,axiom,
! [F: nat > product_unit,A4: set_nat,X2: nat,U: product_unit] :
( ( condit9161856309968323677t_unit @ ( image_8730104196221521654t_unit @ F @ A4 ) )
=> ( ( member_nat @ X2 @ A4 )
=> ( ( ord_le3221252021190050221t_unit @ ( F @ X2 ) @ U )
=> ( ord_le3221252021190050221t_unit @ ( comple2584293577114468500t_unit @ ( image_8730104196221521654t_unit @ F @ A4 ) ) @ U ) ) ) ) ).
% cINF_lower2
thf(fact_3288_cINF__lower2,axiom,
! [F: int > product_unit,A4: set_int,X2: int,U: product_unit] :
( ( condit9161856309968323677t_unit @ ( image_5846526143423701914t_unit @ F @ A4 ) )
=> ( ( member_int @ X2 @ A4 )
=> ( ( ord_le3221252021190050221t_unit @ ( F @ X2 ) @ U )
=> ( ord_le3221252021190050221t_unit @ ( comple2584293577114468500t_unit @ ( image_5846526143423701914t_unit @ F @ A4 ) ) @ U ) ) ) ) ).
% cINF_lower2
thf(fact_3289_cINF__lower2,axiom,
! [F: $o > set_nat,A4: set_o,X2: $o,U: set_nat] :
( ( condit68592940725977398et_nat @ ( image_o_set_nat @ F @ A4 ) )
=> ( ( member_o @ X2 @ A4 )
=> ( ( ord_less_eq_set_nat @ ( F @ X2 ) @ U )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ F @ A4 ) ) @ U ) ) ) ) ).
% cINF_lower2
thf(fact_3290_cINF__lower,axiom,
! [F: $o > nat,A4: set_o,X2: $o] :
( ( condit1738341127787009408ow_nat @ ( image_o_nat @ F @ A4 ) )
=> ( ( member_o @ X2 @ A4 )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_o_nat @ F @ A4 ) ) @ ( F @ X2 ) ) ) ) ).
% cINF_lower
thf(fact_3291_cINF__lower,axiom,
! [F: nat > nat,A4: set_nat,X2: nat] :
( ( condit1738341127787009408ow_nat @ ( image_nat_nat @ F @ A4 ) )
=> ( ( member_nat @ X2 @ A4 )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_nat_nat @ F @ A4 ) ) @ ( F @ X2 ) ) ) ) ).
% cINF_lower
thf(fact_3292_cINF__lower,axiom,
! [F: int > nat,A4: set_int,X2: int] :
( ( condit1738341127787009408ow_nat @ ( image_int_nat @ F @ A4 ) )
=> ( ( member_int @ X2 @ A4 )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_int_nat @ F @ A4 ) ) @ ( F @ X2 ) ) ) ) ).
% cINF_lower
thf(fact_3293_cINF__lower,axiom,
! [F: $o > int,A4: set_o,X2: $o] :
( ( condit1735850657277959132ow_int @ ( image_o_int @ F @ A4 ) )
=> ( ( member_o @ X2 @ A4 )
=> ( ord_less_eq_int @ ( complete_Inf_Inf_int @ ( image_o_int @ F @ A4 ) ) @ ( F @ X2 ) ) ) ) ).
% cINF_lower
thf(fact_3294_cINF__lower,axiom,
! [F: nat > int,A4: set_nat,X2: nat] :
( ( condit1735850657277959132ow_int @ ( image_nat_int @ F @ A4 ) )
=> ( ( member_nat @ X2 @ A4 )
=> ( ord_less_eq_int @ ( complete_Inf_Inf_int @ ( image_nat_int @ F @ A4 ) ) @ ( F @ X2 ) ) ) ) ).
% cINF_lower
thf(fact_3295_cINF__lower,axiom,
! [F: int > int,A4: set_int,X2: int] :
( ( condit1735850657277959132ow_int @ ( image_int_int @ F @ A4 ) )
=> ( ( member_int @ X2 @ A4 )
=> ( ord_less_eq_int @ ( complete_Inf_Inf_int @ ( image_int_int @ F @ A4 ) ) @ ( F @ X2 ) ) ) ) ).
% cINF_lower
thf(fact_3296_cINF__lower,axiom,
! [F: $o > product_unit,A4: set_o,X2: $o] :
( ( condit9161856309968323677t_unit @ ( image_o_Product_unit @ F @ A4 ) )
=> ( ( member_o @ X2 @ A4 )
=> ( ord_le3221252021190050221t_unit @ ( comple2584293577114468500t_unit @ ( image_o_Product_unit @ F @ A4 ) ) @ ( F @ X2 ) ) ) ) ).
% cINF_lower
thf(fact_3297_cINF__lower,axiom,
! [F: nat > product_unit,A4: set_nat,X2: nat] :
( ( condit9161856309968323677t_unit @ ( image_8730104196221521654t_unit @ F @ A4 ) )
=> ( ( member_nat @ X2 @ A4 )
=> ( ord_le3221252021190050221t_unit @ ( comple2584293577114468500t_unit @ ( image_8730104196221521654t_unit @ F @ A4 ) ) @ ( F @ X2 ) ) ) ) ).
% cINF_lower
thf(fact_3298_cINF__lower,axiom,
! [F: int > product_unit,A4: set_int,X2: int] :
( ( condit9161856309968323677t_unit @ ( image_5846526143423701914t_unit @ F @ A4 ) )
=> ( ( member_int @ X2 @ A4 )
=> ( ord_le3221252021190050221t_unit @ ( comple2584293577114468500t_unit @ ( image_5846526143423701914t_unit @ F @ A4 ) ) @ ( F @ X2 ) ) ) ) ).
% cINF_lower
thf(fact_3299_cINF__lower,axiom,
! [F: $o > set_nat,A4: set_o,X2: $o] :
( ( condit68592940725977398et_nat @ ( image_o_set_nat @ F @ A4 ) )
=> ( ( member_o @ X2 @ A4 )
=> ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ F @ A4 ) ) @ ( F @ X2 ) ) ) ) ).
% cINF_lower
thf(fact_3300_cSUP__upper2,axiom,
! [F: $o > nat,A4: set_o,X2: $o,U: nat] :
( ( condit2214826472909112428ve_nat @ ( image_o_nat @ F @ A4 ) )
=> ( ( member_o @ X2 @ A4 )
=> ( ( ord_less_eq_nat @ U @ ( F @ X2 ) )
=> ( ord_less_eq_nat @ U @ ( complete_Sup_Sup_nat @ ( image_o_nat @ F @ A4 ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_3301_cSUP__upper2,axiom,
! [F: nat > nat,A4: set_nat,X2: nat,U: nat] :
( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ F @ A4 ) )
=> ( ( member_nat @ X2 @ A4 )
=> ( ( ord_less_eq_nat @ U @ ( F @ X2 ) )
=> ( ord_less_eq_nat @ U @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ A4 ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_3302_cSUP__upper2,axiom,
! [F: int > nat,A4: set_int,X2: int,U: nat] :
( ( condit2214826472909112428ve_nat @ ( image_int_nat @ F @ A4 ) )
=> ( ( member_int @ X2 @ A4 )
=> ( ( ord_less_eq_nat @ U @ ( F @ X2 ) )
=> ( ord_less_eq_nat @ U @ ( complete_Sup_Sup_nat @ ( image_int_nat @ F @ A4 ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_3303_cSUP__upper2,axiom,
! [F: $o > int,A4: set_o,X2: $o,U: int] :
( ( condit2212336002400062152ve_int @ ( image_o_int @ F @ A4 ) )
=> ( ( member_o @ X2 @ A4 )
=> ( ( ord_less_eq_int @ U @ ( F @ X2 ) )
=> ( ord_less_eq_int @ U @ ( complete_Sup_Sup_int @ ( image_o_int @ F @ A4 ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_3304_cSUP__upper2,axiom,
! [F: nat > int,A4: set_nat,X2: nat,U: int] :
( ( condit2212336002400062152ve_int @ ( image_nat_int @ F @ A4 ) )
=> ( ( member_nat @ X2 @ A4 )
=> ( ( ord_less_eq_int @ U @ ( F @ X2 ) )
=> ( ord_less_eq_int @ U @ ( complete_Sup_Sup_int @ ( image_nat_int @ F @ A4 ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_3305_cSUP__upper2,axiom,
! [F: int > int,A4: set_int,X2: int,U: int] :
( ( condit2212336002400062152ve_int @ ( image_int_int @ F @ A4 ) )
=> ( ( member_int @ X2 @ A4 )
=> ( ( ord_less_eq_int @ U @ ( F @ X2 ) )
=> ( ord_less_eq_int @ U @ ( complete_Sup_Sup_int @ ( image_int_int @ F @ A4 ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_3306_cSUP__upper2,axiom,
! [F: $o > product_unit,A4: set_o,X2: $o,U: product_unit] :
( ( condit2080475194861589617t_unit @ ( image_o_Product_unit @ F @ A4 ) )
=> ( ( member_o @ X2 @ A4 )
=> ( ( ord_le3221252021190050221t_unit @ U @ ( F @ X2 ) )
=> ( ord_le3221252021190050221t_unit @ U @ ( comple4687483117567863418t_unit @ ( image_o_Product_unit @ F @ A4 ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_3307_cSUP__upper2,axiom,
! [F: nat > product_unit,A4: set_nat,X2: nat,U: product_unit] :
( ( condit2080475194861589617t_unit @ ( image_8730104196221521654t_unit @ F @ A4 ) )
=> ( ( member_nat @ X2 @ A4 )
=> ( ( ord_le3221252021190050221t_unit @ U @ ( F @ X2 ) )
=> ( ord_le3221252021190050221t_unit @ U @ ( comple4687483117567863418t_unit @ ( image_8730104196221521654t_unit @ F @ A4 ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_3308_cSUP__upper2,axiom,
! [F: int > product_unit,A4: set_int,X2: int,U: product_unit] :
( ( condit2080475194861589617t_unit @ ( image_5846526143423701914t_unit @ F @ A4 ) )
=> ( ( member_int @ X2 @ A4 )
=> ( ( ord_le3221252021190050221t_unit @ U @ ( F @ X2 ) )
=> ( ord_le3221252021190050221t_unit @ U @ ( comple4687483117567863418t_unit @ ( image_5846526143423701914t_unit @ F @ A4 ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_3309_cSUP__upper2,axiom,
! [F: $o > set_nat,A4: set_o,X2: $o,U: set_nat] :
( ( condit5477540289124974626et_nat @ ( image_o_set_nat @ F @ A4 ) )
=> ( ( member_o @ X2 @ A4 )
=> ( ( ord_less_eq_set_nat @ U @ ( F @ X2 ) )
=> ( ord_less_eq_set_nat @ U @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A4 ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_3310_cSUP__upper,axiom,
! [X2: $o,A4: set_o,F: $o > nat] :
( ( member_o @ X2 @ A4 )
=> ( ( condit2214826472909112428ve_nat @ ( image_o_nat @ F @ A4 ) )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( complete_Sup_Sup_nat @ ( image_o_nat @ F @ A4 ) ) ) ) ) ).
% cSUP_upper
thf(fact_3311_cSUP__upper,axiom,
! [X2: nat,A4: set_nat,F: nat > nat] :
( ( member_nat @ X2 @ A4 )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ F @ A4 ) )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ A4 ) ) ) ) ) ).
% cSUP_upper
thf(fact_3312_cSUP__upper,axiom,
! [X2: int,A4: set_int,F: int > nat] :
( ( member_int @ X2 @ A4 )
=> ( ( condit2214826472909112428ve_nat @ ( image_int_nat @ F @ A4 ) )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( complete_Sup_Sup_nat @ ( image_int_nat @ F @ A4 ) ) ) ) ) ).
% cSUP_upper
thf(fact_3313_cSUP__upper,axiom,
! [X2: $o,A4: set_o,F: $o > int] :
( ( member_o @ X2 @ A4 )
=> ( ( condit2212336002400062152ve_int @ ( image_o_int @ F @ A4 ) )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( complete_Sup_Sup_int @ ( image_o_int @ F @ A4 ) ) ) ) ) ).
% cSUP_upper
thf(fact_3314_cSUP__upper,axiom,
! [X2: nat,A4: set_nat,F: nat > int] :
( ( member_nat @ X2 @ A4 )
=> ( ( condit2212336002400062152ve_int @ ( image_nat_int @ F @ A4 ) )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( complete_Sup_Sup_int @ ( image_nat_int @ F @ A4 ) ) ) ) ) ).
% cSUP_upper
thf(fact_3315_cSUP__upper,axiom,
! [X2: int,A4: set_int,F: int > int] :
( ( member_int @ X2 @ A4 )
=> ( ( condit2212336002400062152ve_int @ ( image_int_int @ F @ A4 ) )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( complete_Sup_Sup_int @ ( image_int_int @ F @ A4 ) ) ) ) ) ).
% cSUP_upper
thf(fact_3316_cSUP__upper,axiom,
! [X2: $o,A4: set_o,F: $o > product_unit] :
( ( member_o @ X2 @ A4 )
=> ( ( condit2080475194861589617t_unit @ ( image_o_Product_unit @ F @ A4 ) )
=> ( ord_le3221252021190050221t_unit @ ( F @ X2 ) @ ( comple4687483117567863418t_unit @ ( image_o_Product_unit @ F @ A4 ) ) ) ) ) ).
% cSUP_upper
thf(fact_3317_cSUP__upper,axiom,
! [X2: nat,A4: set_nat,F: nat > product_unit] :
( ( member_nat @ X2 @ A4 )
=> ( ( condit2080475194861589617t_unit @ ( image_8730104196221521654t_unit @ F @ A4 ) )
=> ( ord_le3221252021190050221t_unit @ ( F @ X2 ) @ ( comple4687483117567863418t_unit @ ( image_8730104196221521654t_unit @ F @ A4 ) ) ) ) ) ).
% cSUP_upper
thf(fact_3318_cSUP__upper,axiom,
! [X2: int,A4: set_int,F: int > product_unit] :
( ( member_int @ X2 @ A4 )
=> ( ( condit2080475194861589617t_unit @ ( image_5846526143423701914t_unit @ F @ A4 ) )
=> ( ord_le3221252021190050221t_unit @ ( F @ X2 ) @ ( comple4687483117567863418t_unit @ ( image_5846526143423701914t_unit @ F @ A4 ) ) ) ) ) ).
% cSUP_upper
thf(fact_3319_cSUP__upper,axiom,
! [X2: $o,A4: set_o,F: $o > set_nat] :
( ( member_o @ X2 @ A4 )
=> ( ( condit5477540289124974626et_nat @ ( image_o_set_nat @ F @ A4 ) )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A4 ) ) ) ) ) ).
% cSUP_upper
thf(fact_3320_Union__image__empty,axiom,
! [A4: set_nat,F: product_unit > set_nat] :
( ( sup_sup_set_nat @ A4 @ ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ F @ bot_bo3957492148770167129t_unit ) ) )
= A4 ) ).
% Union_image_empty
thf(fact_3321_Union__image__empty,axiom,
! [A4: set_nat,F: $o > set_nat] :
( ( sup_sup_set_nat @ A4 @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ bot_bot_set_o ) ) )
= A4 ) ).
% Union_image_empty
thf(fact_3322_Union__image__empty,axiom,
! [A4: set_nat,F: nat > set_nat] :
( ( sup_sup_set_nat @ A4 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ bot_bot_set_nat ) ) )
= A4 ) ).
% Union_image_empty
thf(fact_3323_Union__image__empty,axiom,
! [A4: set_nat,F: product_prod_nat_nat > set_nat] :
( ( sup_sup_set_nat @ A4 @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ F @ bot_bo2099793752762293965at_nat ) ) )
= A4 ) ).
% Union_image_empty
thf(fact_3324_Union__image__empty,axiom,
! [A4: set_Pr8693737435421807431at_nat,F: product_unit > set_Pr8693737435421807431at_nat] :
( ( sup_su718114333110466843at_nat @ A4 @ ( comple2748534634209756852at_nat @ ( image_7288820656315755647at_nat @ F @ bot_bo3957492148770167129t_unit ) ) )
= A4 ) ).
% Union_image_empty
thf(fact_3325_Union__image__empty,axiom,
! [A4: set_Pr8693737435421807431at_nat,F: $o > set_Pr8693737435421807431at_nat] :
( ( sup_su718114333110466843at_nat @ A4 @ ( comple2748534634209756852at_nat @ ( image_8493167801588156084at_nat @ F @ bot_bot_set_o ) ) )
= A4 ) ).
% Union_image_empty
thf(fact_3326_Union__image__empty,axiom,
! [A4: set_Pr8693737435421807431at_nat,F: nat > set_Pr8693737435421807431at_nat] :
( ( sup_su718114333110466843at_nat @ A4 @ ( comple2748534634209756852at_nat @ ( image_4592875511192039216at_nat @ F @ bot_bot_set_nat ) ) )
= A4 ) ).
% Union_image_empty
thf(fact_3327_Union__image__empty,axiom,
! [A4: set_Pr4329608150637261639at_nat,F: product_unit > set_Pr4329608150637261639at_nat] :
( ( sup_su5525570899277871387at_nat @ A4 @ ( comple4364388969068068532at_nat @ ( image_5172358348707908735at_nat @ F @ bot_bo3957492148770167129t_unit ) ) )
= A4 ) ).
% Union_image_empty
thf(fact_3328_Union__image__empty,axiom,
! [A4: set_Pr8693737435421807431at_nat,F: product_prod_nat_nat > set_Pr8693737435421807431at_nat] :
( ( sup_su718114333110466843at_nat @ A4 @ ( comple2748534634209756852at_nat @ ( image_5803077947706746251at_nat @ F @ bot_bo2099793752762293965at_nat ) ) )
= A4 ) ).
% Union_image_empty
thf(fact_3329_Union__image__empty,axiom,
! [A4: set_Pr4329608150637261639at_nat,F: $o > set_Pr4329608150637261639at_nat] :
( ( sup_su5525570899277871387at_nat @ A4 @ ( comple4364388969068068532at_nat @ ( image_2496312346330513076at_nat @ F @ bot_bot_set_o ) ) )
= A4 ) ).
% Union_image_empty
thf(fact_3330_Union__image__insert,axiom,
! [F: nat > set_nat,A: nat,B5: set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ ( insert_nat @ A @ B5 ) ) )
= ( sup_sup_set_nat @ ( F @ A ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ B5 ) ) ) ) ).
% Union_image_insert
thf(fact_3331_Union__image__insert,axiom,
! [F: product_unit > set_nat,A: product_unit,B5: set_Product_unit] :
( ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ F @ ( insert_Product_unit @ A @ B5 ) ) )
= ( sup_sup_set_nat @ ( F @ A ) @ ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ F @ B5 ) ) ) ) ).
% Union_image_insert
thf(fact_3332_Union__image__insert,axiom,
! [F: $o > set_nat,A: $o,B5: set_o] :
( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ ( insert_o @ A @ B5 ) ) )
= ( sup_sup_set_nat @ ( F @ A ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ B5 ) ) ) ) ).
% Union_image_insert
thf(fact_3333_Union__image__insert,axiom,
! [F: product_prod_nat_nat > set_nat,A: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
( ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ F @ ( insert8211810215607154385at_nat @ A @ B5 ) ) )
= ( sup_sup_set_nat @ ( F @ A ) @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ F @ B5 ) ) ) ) ).
% Union_image_insert
thf(fact_3334_Union__image__insert,axiom,
! [F: nat > set_Pr8693737435421807431at_nat,A: nat,B5: set_nat] :
( ( comple2748534634209756852at_nat @ ( image_4592875511192039216at_nat @ F @ ( insert_nat @ A @ B5 ) ) )
= ( sup_su718114333110466843at_nat @ ( F @ A ) @ ( comple2748534634209756852at_nat @ ( image_4592875511192039216at_nat @ F @ B5 ) ) ) ) ).
% Union_image_insert
thf(fact_3335_Union__image__insert,axiom,
! [F: product_unit > set_Pr8693737435421807431at_nat,A: product_unit,B5: set_Product_unit] :
( ( comple2748534634209756852at_nat @ ( image_7288820656315755647at_nat @ F @ ( insert_Product_unit @ A @ B5 ) ) )
= ( sup_su718114333110466843at_nat @ ( F @ A ) @ ( comple2748534634209756852at_nat @ ( image_7288820656315755647at_nat @ F @ B5 ) ) ) ) ).
% Union_image_insert
thf(fact_3336_Union__image__insert,axiom,
! [F: $o > set_Pr8693737435421807431at_nat,A: $o,B5: set_o] :
( ( comple2748534634209756852at_nat @ ( image_8493167801588156084at_nat @ F @ ( insert_o @ A @ B5 ) ) )
= ( sup_su718114333110466843at_nat @ ( F @ A ) @ ( comple2748534634209756852at_nat @ ( image_8493167801588156084at_nat @ F @ B5 ) ) ) ) ).
% Union_image_insert
thf(fact_3337_Union__image__insert,axiom,
! [F: product_prod_nat_nat > set_Pr8693737435421807431at_nat,A: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
( ( comple2748534634209756852at_nat @ ( image_5803077947706746251at_nat @ F @ ( insert8211810215607154385at_nat @ A @ B5 ) ) )
= ( sup_su718114333110466843at_nat @ ( F @ A ) @ ( comple2748534634209756852at_nat @ ( image_5803077947706746251at_nat @ F @ B5 ) ) ) ) ).
% Union_image_insert
thf(fact_3338_Union__image__insert,axiom,
! [F: nat > set_Pr4329608150637261639at_nat,A: nat,B5: set_nat] :
( ( comple4364388969068068532at_nat @ ( image_1087905878993176368at_nat @ F @ ( insert_nat @ A @ B5 ) ) )
= ( sup_su5525570899277871387at_nat @ ( F @ A ) @ ( comple4364388969068068532at_nat @ ( image_1087905878993176368at_nat @ F @ B5 ) ) ) ) ).
% Union_image_insert
thf(fact_3339_Union__image__insert,axiom,
! [F: product_unit > set_Pr4329608150637261639at_nat,A: product_unit,B5: set_Product_unit] :
( ( comple4364388969068068532at_nat @ ( image_5172358348707908735at_nat @ F @ ( insert_Product_unit @ A @ B5 ) ) )
= ( sup_su5525570899277871387at_nat @ ( F @ A ) @ ( comple4364388969068068532at_nat @ ( image_5172358348707908735at_nat @ F @ B5 ) ) ) ) ).
% Union_image_insert
thf(fact_3340_SUP__insert,axiom,
! [F: nat > product_unit,A: nat,A4: set_nat] :
( ( comple4687483117567863418t_unit @ ( image_8730104196221521654t_unit @ F @ ( insert_nat @ A @ A4 ) ) )
= ( sup_sup_Product_unit @ ( F @ A ) @ ( comple4687483117567863418t_unit @ ( image_8730104196221521654t_unit @ F @ A4 ) ) ) ) ).
% SUP_insert
thf(fact_3341_SUP__insert,axiom,
! [F: product_unit > product_unit,A: product_unit,A4: set_Product_unit] :
( ( comple4687483117567863418t_unit @ ( image_405062704495631173t_unit @ F @ ( insert_Product_unit @ A @ A4 ) ) )
= ( sup_sup_Product_unit @ ( F @ A ) @ ( comple4687483117567863418t_unit @ ( image_405062704495631173t_unit @ F @ A4 ) ) ) ) ).
% SUP_insert
thf(fact_3342_SUP__insert,axiom,
! [F: $o > product_unit,A: $o,A4: set_o] :
( ( comple4687483117567863418t_unit @ ( image_o_Product_unit @ F @ ( insert_o @ A @ A4 ) ) )
= ( sup_sup_Product_unit @ ( F @ A ) @ ( comple4687483117567863418t_unit @ ( image_o_Product_unit @ F @ A4 ) ) ) ) ).
% SUP_insert
thf(fact_3343_SUP__insert,axiom,
! [F: nat > set_nat,A: nat,A4: set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ ( insert_nat @ A @ A4 ) ) )
= ( sup_sup_set_nat @ ( F @ A ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A4 ) ) ) ) ).
% SUP_insert
thf(fact_3344_SUP__insert,axiom,
! [F: product_unit > set_nat,A: product_unit,A4: set_Product_unit] :
( ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ F @ ( insert_Product_unit @ A @ A4 ) ) )
= ( sup_sup_set_nat @ ( F @ A ) @ ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ F @ A4 ) ) ) ) ).
% SUP_insert
thf(fact_3345_SUP__insert,axiom,
! [F: $o > set_nat,A: $o,A4: set_o] :
( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ ( insert_o @ A @ A4 ) ) )
= ( sup_sup_set_nat @ ( F @ A ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A4 ) ) ) ) ).
% SUP_insert
thf(fact_3346_SUP__insert,axiom,
! [F: product_prod_nat_nat > product_unit,A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
( ( comple4687483117567863418t_unit @ ( image_6169909464789311569t_unit @ F @ ( insert8211810215607154385at_nat @ A @ A4 ) ) )
= ( sup_sup_Product_unit @ ( F @ A ) @ ( comple4687483117567863418t_unit @ ( image_6169909464789311569t_unit @ F @ A4 ) ) ) ) ).
% SUP_insert
thf(fact_3347_SUP__insert,axiom,
! [F: product_prod_nat_nat > set_nat,A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
( ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ F @ ( insert8211810215607154385at_nat @ A @ A4 ) ) )
= ( sup_sup_set_nat @ ( F @ A ) @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ F @ A4 ) ) ) ) ).
% SUP_insert
thf(fact_3348_SUP__insert,axiom,
! [F: nat > set_Pr8693737435421807431at_nat,A: nat,A4: set_nat] :
( ( comple2748534634209756852at_nat @ ( image_4592875511192039216at_nat @ F @ ( insert_nat @ A @ A4 ) ) )
= ( sup_su718114333110466843at_nat @ ( F @ A ) @ ( comple2748534634209756852at_nat @ ( image_4592875511192039216at_nat @ F @ A4 ) ) ) ) ).
% SUP_insert
thf(fact_3349_SUP__insert,axiom,
! [F: product_unit > set_Pr8693737435421807431at_nat,A: product_unit,A4: set_Product_unit] :
( ( comple2748534634209756852at_nat @ ( image_7288820656315755647at_nat @ F @ ( insert_Product_unit @ A @ A4 ) ) )
= ( sup_su718114333110466843at_nat @ ( F @ A ) @ ( comple2748534634209756852at_nat @ ( image_7288820656315755647at_nat @ F @ A4 ) ) ) ) ).
% SUP_insert
thf(fact_3350_INF__inf__const2,axiom,
! [I3: set_Product_unit,F: product_unit > product_unit,X2: product_unit] :
( ( I3 != bot_bo3957492148770167129t_unit )
=> ( ( comple2584293577114468500t_unit
@ ( image_405062704495631173t_unit
@ ^ [I2: product_unit] : ( inf_inf_Product_unit @ ( F @ I2 ) @ X2 )
@ I3 ) )
= ( inf_inf_Product_unit @ ( comple2584293577114468500t_unit @ ( image_405062704495631173t_unit @ F @ I3 ) ) @ X2 ) ) ) ).
% INF_inf_const2
thf(fact_3351_INF__inf__const2,axiom,
! [I3: set_o,F: $o > product_unit,X2: product_unit] :
( ( I3 != bot_bot_set_o )
=> ( ( comple2584293577114468500t_unit
@ ( image_o_Product_unit
@ ^ [I2: $o] : ( inf_inf_Product_unit @ ( F @ I2 ) @ X2 )
@ I3 ) )
= ( inf_inf_Product_unit @ ( comple2584293577114468500t_unit @ ( image_o_Product_unit @ F @ I3 ) ) @ X2 ) ) ) ).
% INF_inf_const2
thf(fact_3352_INF__inf__const2,axiom,
! [I3: set_nat,F: nat > product_unit,X2: product_unit] :
( ( I3 != bot_bot_set_nat )
=> ( ( comple2584293577114468500t_unit
@ ( image_8730104196221521654t_unit
@ ^ [I2: nat] : ( inf_inf_Product_unit @ ( F @ I2 ) @ X2 )
@ I3 ) )
= ( inf_inf_Product_unit @ ( comple2584293577114468500t_unit @ ( image_8730104196221521654t_unit @ F @ I3 ) ) @ X2 ) ) ) ).
% INF_inf_const2
thf(fact_3353_INF__inf__const2,axiom,
! [I3: set_Product_unit,F: product_unit > set_nat,X2: set_nat] :
( ( I3 != bot_bo3957492148770167129t_unit )
=> ( ( comple7806235888213564991et_nat
@ ( image_8173849991827986510et_nat
@ ^ [I2: product_unit] : ( inf_inf_set_nat @ ( F @ I2 ) @ X2 )
@ I3 ) )
= ( inf_inf_set_nat @ ( comple7806235888213564991et_nat @ ( image_8173849991827986510et_nat @ F @ I3 ) ) @ X2 ) ) ) ).
% INF_inf_const2
thf(fact_3354_INF__inf__const2,axiom,
! [I3: set_o,F: $o > set_nat,X2: set_nat] :
( ( I3 != bot_bot_set_o )
=> ( ( comple7806235888213564991et_nat
@ ( image_o_set_nat
@ ^ [I2: $o] : ( inf_inf_set_nat @ ( F @ I2 ) @ X2 )
@ I3 ) )
= ( inf_inf_set_nat @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ F @ I3 ) ) @ X2 ) ) ) ).
% INF_inf_const2
thf(fact_3355_INF__inf__const2,axiom,
! [I3: set_nat,F: nat > set_nat,X2: set_nat] :
( ( I3 != bot_bot_set_nat )
=> ( ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [I2: nat] : ( inf_inf_set_nat @ ( F @ I2 ) @ X2 )
@ I3 ) )
= ( inf_inf_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ I3 ) ) @ X2 ) ) ) ).
% INF_inf_const2
thf(fact_3356_INF__inf__const2,axiom,
! [I3: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > product_unit,X2: product_unit] :
( ( I3 != bot_bo2099793752762293965at_nat )
=> ( ( comple2584293577114468500t_unit
@ ( image_6169909464789311569t_unit
@ ^ [I2: product_prod_nat_nat] : ( inf_inf_Product_unit @ ( F @ I2 ) @ X2 )
@ I3 ) )
= ( inf_inf_Product_unit @ ( comple2584293577114468500t_unit @ ( image_6169909464789311569t_unit @ F @ I3 ) ) @ X2 ) ) ) ).
% INF_inf_const2
thf(fact_3357_INF__inf__const2,axiom,
! [I3: set_Product_unit,F: product_unit > set_Pr1261947904930325089at_nat,X2: set_Pr1261947904930325089at_nat] :
( ( I3 != bot_bo3957492148770167129t_unit )
=> ( ( comple1930127330929219688at_nat
@ ( image_3617315916088684441at_nat
@ ^ [I2: product_unit] : ( inf_in2572325071724192079at_nat @ ( F @ I2 ) @ X2 )
@ I3 ) )
= ( inf_in2572325071724192079at_nat @ ( comple1930127330929219688at_nat @ ( image_3617315916088684441at_nat @ F @ I3 ) ) @ X2 ) ) ) ).
% INF_inf_const2
thf(fact_3358_INF__inf__const2,axiom,
! [I3: set_o,F: $o > set_Pr1261947904930325089at_nat,X2: set_Pr1261947904930325089at_nat] :
( ( I3 != bot_bot_set_o )
=> ( ( comple1930127330929219688at_nat
@ ( image_7077428525161401678at_nat
@ ^ [I2: $o] : ( inf_in2572325071724192079at_nat @ ( F @ I2 ) @ X2 )
@ I3 ) )
= ( inf_in2572325071724192079at_nat @ ( comple1930127330929219688at_nat @ ( image_7077428525161401678at_nat @ F @ I3 ) ) @ X2 ) ) ) ).
% INF_inf_const2
thf(fact_3359_INF__inf__const2,axiom,
! [I3: set_nat,F: nat > set_Pr1261947904930325089at_nat,X2: set_Pr1261947904930325089at_nat] :
( ( I3 != bot_bot_set_nat )
=> ( ( comple1930127330929219688at_nat
@ ( image_7178329752028323786at_nat
@ ^ [I2: nat] : ( inf_in2572325071724192079at_nat @ ( F @ I2 ) @ X2 )
@ I3 ) )
= ( inf_in2572325071724192079at_nat @ ( comple1930127330929219688at_nat @ ( image_7178329752028323786at_nat @ F @ I3 ) ) @ X2 ) ) ) ).
% INF_inf_const2
thf(fact_3360_INF__inf__const1,axiom,
! [I3: set_Product_unit,X2: product_unit,F: product_unit > product_unit] :
( ( I3 != bot_bo3957492148770167129t_unit )
=> ( ( comple2584293577114468500t_unit
@ ( image_405062704495631173t_unit
@ ^ [I2: product_unit] : ( inf_inf_Product_unit @ X2 @ ( F @ I2 ) )
@ I3 ) )
= ( inf_inf_Product_unit @ X2 @ ( comple2584293577114468500t_unit @ ( image_405062704495631173t_unit @ F @ I3 ) ) ) ) ) ).
% INF_inf_const1
thf(fact_3361_INF__inf__const1,axiom,
! [I3: set_o,X2: product_unit,F: $o > product_unit] :
( ( I3 != bot_bot_set_o )
=> ( ( comple2584293577114468500t_unit
@ ( image_o_Product_unit
@ ^ [I2: $o] : ( inf_inf_Product_unit @ X2 @ ( F @ I2 ) )
@ I3 ) )
= ( inf_inf_Product_unit @ X2 @ ( comple2584293577114468500t_unit @ ( image_o_Product_unit @ F @ I3 ) ) ) ) ) ).
% INF_inf_const1
thf(fact_3362_INF__inf__const1,axiom,
! [I3: set_nat,X2: product_unit,F: nat > product_unit] :
( ( I3 != bot_bot_set_nat )
=> ( ( comple2584293577114468500t_unit
@ ( image_8730104196221521654t_unit
@ ^ [I2: nat] : ( inf_inf_Product_unit @ X2 @ ( F @ I2 ) )
@ I3 ) )
= ( inf_inf_Product_unit @ X2 @ ( comple2584293577114468500t_unit @ ( image_8730104196221521654t_unit @ F @ I3 ) ) ) ) ) ).
% INF_inf_const1
thf(fact_3363_INF__inf__const1,axiom,
! [I3: set_Product_unit,X2: set_nat,F: product_unit > set_nat] :
( ( I3 != bot_bo3957492148770167129t_unit )
=> ( ( comple7806235888213564991et_nat
@ ( image_8173849991827986510et_nat
@ ^ [I2: product_unit] : ( inf_inf_set_nat @ X2 @ ( F @ I2 ) )
@ I3 ) )
= ( inf_inf_set_nat @ X2 @ ( comple7806235888213564991et_nat @ ( image_8173849991827986510et_nat @ F @ I3 ) ) ) ) ) ).
% INF_inf_const1
thf(fact_3364_INF__inf__const1,axiom,
! [I3: set_o,X2: set_nat,F: $o > set_nat] :
( ( I3 != bot_bot_set_o )
=> ( ( comple7806235888213564991et_nat
@ ( image_o_set_nat
@ ^ [I2: $o] : ( inf_inf_set_nat @ X2 @ ( F @ I2 ) )
@ I3 ) )
= ( inf_inf_set_nat @ X2 @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ F @ I3 ) ) ) ) ) ).
% INF_inf_const1
thf(fact_3365_INF__inf__const1,axiom,
! [I3: set_nat,X2: set_nat,F: nat > set_nat] :
( ( I3 != bot_bot_set_nat )
=> ( ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [I2: nat] : ( inf_inf_set_nat @ X2 @ ( F @ I2 ) )
@ I3 ) )
= ( inf_inf_set_nat @ X2 @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ I3 ) ) ) ) ) ).
% INF_inf_const1
thf(fact_3366_INF__inf__const1,axiom,
! [I3: set_Pr1261947904930325089at_nat,X2: product_unit,F: product_prod_nat_nat > product_unit] :
( ( I3 != bot_bo2099793752762293965at_nat )
=> ( ( comple2584293577114468500t_unit
@ ( image_6169909464789311569t_unit
@ ^ [I2: product_prod_nat_nat] : ( inf_inf_Product_unit @ X2 @ ( F @ I2 ) )
@ I3 ) )
= ( inf_inf_Product_unit @ X2 @ ( comple2584293577114468500t_unit @ ( image_6169909464789311569t_unit @ F @ I3 ) ) ) ) ) ).
% INF_inf_const1
thf(fact_3367_INF__inf__const1,axiom,
! [I3: set_Product_unit,X2: set_Pr1261947904930325089at_nat,F: product_unit > set_Pr1261947904930325089at_nat] :
( ( I3 != bot_bo3957492148770167129t_unit )
=> ( ( comple1930127330929219688at_nat
@ ( image_3617315916088684441at_nat
@ ^ [I2: product_unit] : ( inf_in2572325071724192079at_nat @ X2 @ ( F @ I2 ) )
@ I3 ) )
= ( inf_in2572325071724192079at_nat @ X2 @ ( comple1930127330929219688at_nat @ ( image_3617315916088684441at_nat @ F @ I3 ) ) ) ) ) ).
% INF_inf_const1
thf(fact_3368_INF__inf__const1,axiom,
! [I3: set_o,X2: set_Pr1261947904930325089at_nat,F: $o > set_Pr1261947904930325089at_nat] :
( ( I3 != bot_bot_set_o )
=> ( ( comple1930127330929219688at_nat
@ ( image_7077428525161401678at_nat
@ ^ [I2: $o] : ( inf_in2572325071724192079at_nat @ X2 @ ( F @ I2 ) )
@ I3 ) )
= ( inf_in2572325071724192079at_nat @ X2 @ ( comple1930127330929219688at_nat @ ( image_7077428525161401678at_nat @ F @ I3 ) ) ) ) ) ).
% INF_inf_const1
thf(fact_3369_INF__inf__const1,axiom,
! [I3: set_nat,X2: set_Pr1261947904930325089at_nat,F: nat > set_Pr1261947904930325089at_nat] :
( ( I3 != bot_bot_set_nat )
=> ( ( comple1930127330929219688at_nat
@ ( image_7178329752028323786at_nat
@ ^ [I2: nat] : ( inf_in2572325071724192079at_nat @ X2 @ ( F @ I2 ) )
@ I3 ) )
= ( inf_in2572325071724192079at_nat @ X2 @ ( comple1930127330929219688at_nat @ ( image_7178329752028323786at_nat @ F @ I3 ) ) ) ) ) ).
% INF_inf_const1
thf(fact_3370_INF__insert,axiom,
! [F: nat > product_unit,A: nat,A4: set_nat] :
( ( comple2584293577114468500t_unit @ ( image_8730104196221521654t_unit @ F @ ( insert_nat @ A @ A4 ) ) )
= ( inf_inf_Product_unit @ ( F @ A ) @ ( comple2584293577114468500t_unit @ ( image_8730104196221521654t_unit @ F @ A4 ) ) ) ) ).
% INF_insert
thf(fact_3371_INF__insert,axiom,
! [F: product_unit > product_unit,A: product_unit,A4: set_Product_unit] :
( ( comple2584293577114468500t_unit @ ( image_405062704495631173t_unit @ F @ ( insert_Product_unit @ A @ A4 ) ) )
= ( inf_inf_Product_unit @ ( F @ A ) @ ( comple2584293577114468500t_unit @ ( image_405062704495631173t_unit @ F @ A4 ) ) ) ) ).
% INF_insert
thf(fact_3372_INF__insert,axiom,
! [F: $o > product_unit,A: $o,A4: set_o] :
( ( comple2584293577114468500t_unit @ ( image_o_Product_unit @ F @ ( insert_o @ A @ A4 ) ) )
= ( inf_inf_Product_unit @ ( F @ A ) @ ( comple2584293577114468500t_unit @ ( image_o_Product_unit @ F @ A4 ) ) ) ) ).
% INF_insert
thf(fact_3373_INF__insert,axiom,
! [F: nat > set_nat,A: nat,A4: set_nat] :
( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ ( insert_nat @ A @ A4 ) ) )
= ( inf_inf_set_nat @ ( F @ A ) @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ A4 ) ) ) ) ).
% INF_insert
thf(fact_3374_INF__insert,axiom,
! [F: product_unit > set_nat,A: product_unit,A4: set_Product_unit] :
( ( comple7806235888213564991et_nat @ ( image_8173849991827986510et_nat @ F @ ( insert_Product_unit @ A @ A4 ) ) )
= ( inf_inf_set_nat @ ( F @ A ) @ ( comple7806235888213564991et_nat @ ( image_8173849991827986510et_nat @ F @ A4 ) ) ) ) ).
% INF_insert
thf(fact_3375_INF__insert,axiom,
! [F: $o > set_nat,A: $o,A4: set_o] :
( ( comple7806235888213564991et_nat @ ( image_o_set_nat @ F @ ( insert_o @ A @ A4 ) ) )
= ( inf_inf_set_nat @ ( F @ A ) @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ F @ A4 ) ) ) ) ).
% INF_insert
thf(fact_3376_INF__insert,axiom,
! [F: product_prod_nat_nat > product_unit,A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
( ( comple2584293577114468500t_unit @ ( image_6169909464789311569t_unit @ F @ ( insert8211810215607154385at_nat @ A @ A4 ) ) )
= ( inf_inf_Product_unit @ ( F @ A ) @ ( comple2584293577114468500t_unit @ ( image_6169909464789311569t_unit @ F @ A4 ) ) ) ) ).
% INF_insert
thf(fact_3377_INF__insert,axiom,
! [F: nat > set_Pr1261947904930325089at_nat,A: nat,A4: set_nat] :
( ( comple1930127330929219688at_nat @ ( image_7178329752028323786at_nat @ F @ ( insert_nat @ A @ A4 ) ) )
= ( inf_in2572325071724192079at_nat @ ( F @ A ) @ ( comple1930127330929219688at_nat @ ( image_7178329752028323786at_nat @ F @ A4 ) ) ) ) ).
% INF_insert
thf(fact_3378_INF__insert,axiom,
! [F: product_unit > set_Pr1261947904930325089at_nat,A: product_unit,A4: set_Product_unit] :
( ( comple1930127330929219688at_nat @ ( image_3617315916088684441at_nat @ F @ ( insert_Product_unit @ A @ A4 ) ) )
= ( inf_in2572325071724192079at_nat @ ( F @ A ) @ ( comple1930127330929219688at_nat @ ( image_3617315916088684441at_nat @ F @ A4 ) ) ) ) ).
% INF_insert
thf(fact_3379_INF__insert,axiom,
! [F: $o > set_Pr1261947904930325089at_nat,A: $o,A4: set_o] :
( ( comple1930127330929219688at_nat @ ( image_7077428525161401678at_nat @ F @ ( insert_o @ A @ A4 ) ) )
= ( inf_in2572325071724192079at_nat @ ( F @ A ) @ ( comple1930127330929219688at_nat @ ( image_7077428525161401678at_nat @ F @ A4 ) ) ) ) ).
% INF_insert
thf(fact_3380_UN__extend__simps_I2_J,axiom,
! [C3: set_Product_unit,A4: product_unit > set_nat,B5: set_nat] :
( ( ( C3 = bot_bo3957492148770167129t_unit )
=> ( ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ A4 @ C3 ) ) @ B5 )
= B5 ) )
& ( ( C3 != bot_bo3957492148770167129t_unit )
=> ( ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ A4 @ C3 ) ) @ B5 )
= ( comple7399068483239264473et_nat
@ ( image_8173849991827986510et_nat
@ ^ [X4: product_unit] : ( sup_sup_set_nat @ ( A4 @ X4 ) @ B5 )
@ C3 ) ) ) ) ) ).
% UN_extend_simps(2)
thf(fact_3381_UN__extend__simps_I2_J,axiom,
! [C3: set_o,A4: $o > set_nat,B5: set_nat] :
( ( ( C3 = bot_bot_set_o )
=> ( ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ A4 @ C3 ) ) @ B5 )
= B5 ) )
& ( ( C3 != bot_bot_set_o )
=> ( ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ A4 @ C3 ) ) @ B5 )
= ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [X4: $o] : ( sup_sup_set_nat @ ( A4 @ X4 ) @ B5 )
@ C3 ) ) ) ) ) ).
% UN_extend_simps(2)
thf(fact_3382_UN__extend__simps_I2_J,axiom,
! [C3: set_nat,A4: nat > set_nat,B5: set_nat] :
( ( ( C3 = bot_bot_set_nat )
=> ( ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A4 @ C3 ) ) @ B5 )
= B5 ) )
& ( ( C3 != bot_bot_set_nat )
=> ( ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A4 @ C3 ) ) @ B5 )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X4: nat] : ( sup_sup_set_nat @ ( A4 @ X4 ) @ B5 )
@ C3 ) ) ) ) ) ).
% UN_extend_simps(2)
thf(fact_3383_UN__extend__simps_I2_J,axiom,
! [C3: set_Pr1261947904930325089at_nat,A4: product_prod_nat_nat > set_nat,B5: set_nat] :
( ( ( C3 = bot_bo2099793752762293965at_nat )
=> ( ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ A4 @ C3 ) ) @ B5 )
= B5 ) )
& ( ( C3 != bot_bo2099793752762293965at_nat )
=> ( ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ A4 @ C3 ) ) @ B5 )
= ( comple7399068483239264473et_nat
@ ( image_15824709712370754et_nat
@ ^ [X4: product_prod_nat_nat] : ( sup_sup_set_nat @ ( A4 @ X4 ) @ B5 )
@ C3 ) ) ) ) ) ).
% UN_extend_simps(2)
thf(fact_3384_UN__extend__simps_I2_J,axiom,
! [C3: set_Product_unit,A4: product_unit > set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] :
( ( ( C3 = bot_bo3957492148770167129t_unit )
=> ( ( sup_su718114333110466843at_nat @ ( comple2748534634209756852at_nat @ ( image_7288820656315755647at_nat @ A4 @ C3 ) ) @ B5 )
= B5 ) )
& ( ( C3 != bot_bo3957492148770167129t_unit )
=> ( ( sup_su718114333110466843at_nat @ ( comple2748534634209756852at_nat @ ( image_7288820656315755647at_nat @ A4 @ C3 ) ) @ B5 )
= ( comple2748534634209756852at_nat
@ ( image_7288820656315755647at_nat
@ ^ [X4: product_unit] : ( sup_su718114333110466843at_nat @ ( A4 @ X4 ) @ B5 )
@ C3 ) ) ) ) ) ).
% UN_extend_simps(2)
thf(fact_3385_UN__extend__simps_I2_J,axiom,
! [C3: set_o,A4: $o > set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] :
( ( ( C3 = bot_bot_set_o )
=> ( ( sup_su718114333110466843at_nat @ ( comple2748534634209756852at_nat @ ( image_8493167801588156084at_nat @ A4 @ C3 ) ) @ B5 )
= B5 ) )
& ( ( C3 != bot_bot_set_o )
=> ( ( sup_su718114333110466843at_nat @ ( comple2748534634209756852at_nat @ ( image_8493167801588156084at_nat @ A4 @ C3 ) ) @ B5 )
= ( comple2748534634209756852at_nat
@ ( image_8493167801588156084at_nat
@ ^ [X4: $o] : ( sup_su718114333110466843at_nat @ ( A4 @ X4 ) @ B5 )
@ C3 ) ) ) ) ) ).
% UN_extend_simps(2)
thf(fact_3386_UN__extend__simps_I2_J,axiom,
! [C3: set_nat,A4: nat > set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] :
( ( ( C3 = bot_bot_set_nat )
=> ( ( sup_su718114333110466843at_nat @ ( comple2748534634209756852at_nat @ ( image_4592875511192039216at_nat @ A4 @ C3 ) ) @ B5 )
= B5 ) )
& ( ( C3 != bot_bot_set_nat )
=> ( ( sup_su718114333110466843at_nat @ ( comple2748534634209756852at_nat @ ( image_4592875511192039216at_nat @ A4 @ C3 ) ) @ B5 )
= ( comple2748534634209756852at_nat
@ ( image_4592875511192039216at_nat
@ ^ [X4: nat] : ( sup_su718114333110466843at_nat @ ( A4 @ X4 ) @ B5 )
@ C3 ) ) ) ) ) ).
% UN_extend_simps(2)
thf(fact_3387_UN__extend__simps_I2_J,axiom,
! [C3: set_Product_unit,A4: product_unit > set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
( ( ( C3 = bot_bo3957492148770167129t_unit )
=> ( ( sup_su5525570899277871387at_nat @ ( comple4364388969068068532at_nat @ ( image_5172358348707908735at_nat @ A4 @ C3 ) ) @ B5 )
= B5 ) )
& ( ( C3 != bot_bo3957492148770167129t_unit )
=> ( ( sup_su5525570899277871387at_nat @ ( comple4364388969068068532at_nat @ ( image_5172358348707908735at_nat @ A4 @ C3 ) ) @ B5 )
= ( comple4364388969068068532at_nat
@ ( image_5172358348707908735at_nat
@ ^ [X4: product_unit] : ( sup_su5525570899277871387at_nat @ ( A4 @ X4 ) @ B5 )
@ C3 ) ) ) ) ) ).
% UN_extend_simps(2)
thf(fact_3388_UN__extend__simps_I2_J,axiom,
! [C3: set_Pr1261947904930325089at_nat,A4: product_prod_nat_nat > set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] :
( ( ( C3 = bot_bo2099793752762293965at_nat )
=> ( ( sup_su718114333110466843at_nat @ ( comple2748534634209756852at_nat @ ( image_5803077947706746251at_nat @ A4 @ C3 ) ) @ B5 )
= B5 ) )
& ( ( C3 != bot_bo2099793752762293965at_nat )
=> ( ( sup_su718114333110466843at_nat @ ( comple2748534634209756852at_nat @ ( image_5803077947706746251at_nat @ A4 @ C3 ) ) @ B5 )
= ( comple2748534634209756852at_nat
@ ( image_5803077947706746251at_nat
@ ^ [X4: product_prod_nat_nat] : ( sup_su718114333110466843at_nat @ ( A4 @ X4 ) @ B5 )
@ C3 ) ) ) ) ) ).
% UN_extend_simps(2)
thf(fact_3389_UN__extend__simps_I2_J,axiom,
! [C3: set_o,A4: $o > set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
( ( ( C3 = bot_bot_set_o )
=> ( ( sup_su5525570899277871387at_nat @ ( comple4364388969068068532at_nat @ ( image_2496312346330513076at_nat @ A4 @ C3 ) ) @ B5 )
= B5 ) )
& ( ( C3 != bot_bot_set_o )
=> ( ( sup_su5525570899277871387at_nat @ ( comple4364388969068068532at_nat @ ( image_2496312346330513076at_nat @ A4 @ C3 ) ) @ B5 )
= ( comple4364388969068068532at_nat
@ ( image_2496312346330513076at_nat
@ ^ [X4: $o] : ( sup_su5525570899277871387at_nat @ ( A4 @ X4 ) @ B5 )
@ C3 ) ) ) ) ) ).
% UN_extend_simps(2)
thf(fact_3390_UN__extend__simps_I3_J,axiom,
! [C3: set_Product_unit,A4: set_nat,B5: product_unit > set_nat] :
( ( ( C3 = bot_bo3957492148770167129t_unit )
=> ( ( sup_sup_set_nat @ A4 @ ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ B5 @ C3 ) ) )
= A4 ) )
& ( ( C3 != bot_bo3957492148770167129t_unit )
=> ( ( sup_sup_set_nat @ A4 @ ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ B5 @ C3 ) ) )
= ( comple7399068483239264473et_nat
@ ( image_8173849991827986510et_nat
@ ^ [X4: product_unit] : ( sup_sup_set_nat @ A4 @ ( B5 @ X4 ) )
@ C3 ) ) ) ) ) ).
% UN_extend_simps(3)
thf(fact_3391_UN__extend__simps_I3_J,axiom,
! [C3: set_o,A4: set_nat,B5: $o > set_nat] :
( ( ( C3 = bot_bot_set_o )
=> ( ( sup_sup_set_nat @ A4 @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B5 @ C3 ) ) )
= A4 ) )
& ( ( C3 != bot_bot_set_o )
=> ( ( sup_sup_set_nat @ A4 @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B5 @ C3 ) ) )
= ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [X4: $o] : ( sup_sup_set_nat @ A4 @ ( B5 @ X4 ) )
@ C3 ) ) ) ) ) ).
% UN_extend_simps(3)
thf(fact_3392_UN__extend__simps_I3_J,axiom,
! [C3: set_nat,A4: set_nat,B5: nat > set_nat] :
( ( ( C3 = bot_bot_set_nat )
=> ( ( sup_sup_set_nat @ A4 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ C3 ) ) )
= A4 ) )
& ( ( C3 != bot_bot_set_nat )
=> ( ( sup_sup_set_nat @ A4 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ C3 ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X4: nat] : ( sup_sup_set_nat @ A4 @ ( B5 @ X4 ) )
@ C3 ) ) ) ) ) ).
% UN_extend_simps(3)
thf(fact_3393_UN__extend__simps_I3_J,axiom,
! [C3: set_Pr1261947904930325089at_nat,A4: set_nat,B5: product_prod_nat_nat > set_nat] :
( ( ( C3 = bot_bo2099793752762293965at_nat )
=> ( ( sup_sup_set_nat @ A4 @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ B5 @ C3 ) ) )
= A4 ) )
& ( ( C3 != bot_bo2099793752762293965at_nat )
=> ( ( sup_sup_set_nat @ A4 @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ B5 @ C3 ) ) )
= ( comple7399068483239264473et_nat
@ ( image_15824709712370754et_nat
@ ^ [X4: product_prod_nat_nat] : ( sup_sup_set_nat @ A4 @ ( B5 @ X4 ) )
@ C3 ) ) ) ) ) ).
% UN_extend_simps(3)
thf(fact_3394_UN__extend__simps_I3_J,axiom,
! [C3: set_Product_unit,A4: set_Pr8693737435421807431at_nat,B5: product_unit > set_Pr8693737435421807431at_nat] :
( ( ( C3 = bot_bo3957492148770167129t_unit )
=> ( ( sup_su718114333110466843at_nat @ A4 @ ( comple2748534634209756852at_nat @ ( image_7288820656315755647at_nat @ B5 @ C3 ) ) )
= A4 ) )
& ( ( C3 != bot_bo3957492148770167129t_unit )
=> ( ( sup_su718114333110466843at_nat @ A4 @ ( comple2748534634209756852at_nat @ ( image_7288820656315755647at_nat @ B5 @ C3 ) ) )
= ( comple2748534634209756852at_nat
@ ( image_7288820656315755647at_nat
@ ^ [X4: product_unit] : ( sup_su718114333110466843at_nat @ A4 @ ( B5 @ X4 ) )
@ C3 ) ) ) ) ) ).
% UN_extend_simps(3)
thf(fact_3395_UN__extend__simps_I3_J,axiom,
! [C3: set_o,A4: set_Pr8693737435421807431at_nat,B5: $o > set_Pr8693737435421807431at_nat] :
( ( ( C3 = bot_bot_set_o )
=> ( ( sup_su718114333110466843at_nat @ A4 @ ( comple2748534634209756852at_nat @ ( image_8493167801588156084at_nat @ B5 @ C3 ) ) )
= A4 ) )
& ( ( C3 != bot_bot_set_o )
=> ( ( sup_su718114333110466843at_nat @ A4 @ ( comple2748534634209756852at_nat @ ( image_8493167801588156084at_nat @ B5 @ C3 ) ) )
= ( comple2748534634209756852at_nat
@ ( image_8493167801588156084at_nat
@ ^ [X4: $o] : ( sup_su718114333110466843at_nat @ A4 @ ( B5 @ X4 ) )
@ C3 ) ) ) ) ) ).
% UN_extend_simps(3)
thf(fact_3396_UN__extend__simps_I3_J,axiom,
! [C3: set_nat,A4: set_Pr8693737435421807431at_nat,B5: nat > set_Pr8693737435421807431at_nat] :
( ( ( C3 = bot_bot_set_nat )
=> ( ( sup_su718114333110466843at_nat @ A4 @ ( comple2748534634209756852at_nat @ ( image_4592875511192039216at_nat @ B5 @ C3 ) ) )
= A4 ) )
& ( ( C3 != bot_bot_set_nat )
=> ( ( sup_su718114333110466843at_nat @ A4 @ ( comple2748534634209756852at_nat @ ( image_4592875511192039216at_nat @ B5 @ C3 ) ) )
= ( comple2748534634209756852at_nat
@ ( image_4592875511192039216at_nat
@ ^ [X4: nat] : ( sup_su718114333110466843at_nat @ A4 @ ( B5 @ X4 ) )
@ C3 ) ) ) ) ) ).
% UN_extend_simps(3)
thf(fact_3397_UN__extend__simps_I3_J,axiom,
! [C3: set_Product_unit,A4: set_Pr4329608150637261639at_nat,B5: product_unit > set_Pr4329608150637261639at_nat] :
( ( ( C3 = bot_bo3957492148770167129t_unit )
=> ( ( sup_su5525570899277871387at_nat @ A4 @ ( comple4364388969068068532at_nat @ ( image_5172358348707908735at_nat @ B5 @ C3 ) ) )
= A4 ) )
& ( ( C3 != bot_bo3957492148770167129t_unit )
=> ( ( sup_su5525570899277871387at_nat @ A4 @ ( comple4364388969068068532at_nat @ ( image_5172358348707908735at_nat @ B5 @ C3 ) ) )
= ( comple4364388969068068532at_nat
@ ( image_5172358348707908735at_nat
@ ^ [X4: product_unit] : ( sup_su5525570899277871387at_nat @ A4 @ ( B5 @ X4 ) )
@ C3 ) ) ) ) ) ).
% UN_extend_simps(3)
thf(fact_3398_UN__extend__simps_I3_J,axiom,
! [C3: set_Pr1261947904930325089at_nat,A4: set_Pr8693737435421807431at_nat,B5: product_prod_nat_nat > set_Pr8693737435421807431at_nat] :
( ( ( C3 = bot_bo2099793752762293965at_nat )
=> ( ( sup_su718114333110466843at_nat @ A4 @ ( comple2748534634209756852at_nat @ ( image_5803077947706746251at_nat @ B5 @ C3 ) ) )
= A4 ) )
& ( ( C3 != bot_bo2099793752762293965at_nat )
=> ( ( sup_su718114333110466843at_nat @ A4 @ ( comple2748534634209756852at_nat @ ( image_5803077947706746251at_nat @ B5 @ C3 ) ) )
= ( comple2748534634209756852at_nat
@ ( image_5803077947706746251at_nat
@ ^ [X4: product_prod_nat_nat] : ( sup_su718114333110466843at_nat @ A4 @ ( B5 @ X4 ) )
@ C3 ) ) ) ) ) ).
% UN_extend_simps(3)
thf(fact_3399_UN__extend__simps_I3_J,axiom,
! [C3: set_o,A4: set_Pr4329608150637261639at_nat,B5: $o > set_Pr4329608150637261639at_nat] :
( ( ( C3 = bot_bot_set_o )
=> ( ( sup_su5525570899277871387at_nat @ A4 @ ( comple4364388969068068532at_nat @ ( image_2496312346330513076at_nat @ B5 @ C3 ) ) )
= A4 ) )
& ( ( C3 != bot_bot_set_o )
=> ( ( sup_su5525570899277871387at_nat @ A4 @ ( comple4364388969068068532at_nat @ ( image_2496312346330513076at_nat @ B5 @ C3 ) ) )
= ( comple4364388969068068532at_nat
@ ( image_2496312346330513076at_nat
@ ^ [X4: $o] : ( sup_su5525570899277871387at_nat @ A4 @ ( B5 @ X4 ) )
@ C3 ) ) ) ) ) ).
% UN_extend_simps(3)
thf(fact_3400_INT__extend__simps_I1_J,axiom,
! [C3: set_Product_unit,A4: product_unit > set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
( ( ( C3 = bot_bo3957492148770167129t_unit )
=> ( ( inf_in2572325071724192079at_nat @ ( comple1930127330929219688at_nat @ ( image_3617315916088684441at_nat @ A4 @ C3 ) ) @ B5 )
= B5 ) )
& ( ( C3 != bot_bo3957492148770167129t_unit )
=> ( ( inf_in2572325071724192079at_nat @ ( comple1930127330929219688at_nat @ ( image_3617315916088684441at_nat @ A4 @ C3 ) ) @ B5 )
= ( comple1930127330929219688at_nat
@ ( image_3617315916088684441at_nat
@ ^ [X4: product_unit] : ( inf_in2572325071724192079at_nat @ ( A4 @ X4 ) @ B5 )
@ C3 ) ) ) ) ) ).
% INT_extend_simps(1)
thf(fact_3401_INT__extend__simps_I1_J,axiom,
! [C3: set_Pr4329608150637261639at_nat,A4: produc3843707927480180839at_nat > set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
( ( ( C3 = bot_bo228742789529271731at_nat )
=> ( ( inf_in2572325071724192079at_nat @ ( comple1930127330929219688at_nat @ ( image_7979982168039674559at_nat @ A4 @ C3 ) ) @ B5 )
= B5 ) )
& ( ( C3 != bot_bo228742789529271731at_nat )
=> ( ( inf_in2572325071724192079at_nat @ ( comple1930127330929219688at_nat @ ( image_7979982168039674559at_nat @ A4 @ C3 ) ) @ B5 )
= ( comple1930127330929219688at_nat
@ ( image_7979982168039674559at_nat
@ ^ [X4: produc3843707927480180839at_nat] : ( inf_in2572325071724192079at_nat @ ( A4 @ X4 ) @ B5 )
@ C3 ) ) ) ) ) ).
% INT_extend_simps(1)
thf(fact_3402_INT__extend__simps_I1_J,axiom,
! [C3: set_Pr1261947904930325089at_nat,A4: product_prod_nat_nat > set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
( ( ( C3 = bot_bo2099793752762293965at_nat )
=> ( ( inf_in2572325071724192079at_nat @ ( comple1930127330929219688at_nat @ ( image_5423882813909314213at_nat @ A4 @ C3 ) ) @ B5 )
= B5 ) )
& ( ( C3 != bot_bo2099793752762293965at_nat )
=> ( ( inf_in2572325071724192079at_nat @ ( comple1930127330929219688at_nat @ ( image_5423882813909314213at_nat @ A4 @ C3 ) ) @ B5 )
= ( comple1930127330929219688at_nat
@ ( image_5423882813909314213at_nat
@ ^ [X4: product_prod_nat_nat] : ( inf_in2572325071724192079at_nat @ ( A4 @ X4 ) @ B5 )
@ C3 ) ) ) ) ) ).
% INT_extend_simps(1)
thf(fact_3403_INT__extend__simps_I1_J,axiom,
! [C3: set_o,A4: $o > set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
( ( ( C3 = bot_bot_set_o )
=> ( ( inf_in2572325071724192079at_nat @ ( comple1930127330929219688at_nat @ ( image_7077428525161401678at_nat @ A4 @ C3 ) ) @ B5 )
= B5 ) )
& ( ( C3 != bot_bot_set_o )
=> ( ( inf_in2572325071724192079at_nat @ ( comple1930127330929219688at_nat @ ( image_7077428525161401678at_nat @ A4 @ C3 ) ) @ B5 )
= ( comple1930127330929219688at_nat
@ ( image_7077428525161401678at_nat
@ ^ [X4: $o] : ( inf_in2572325071724192079at_nat @ ( A4 @ X4 ) @ B5 )
@ C3 ) ) ) ) ) ).
% INT_extend_simps(1)
thf(fact_3404_INT__extend__simps_I1_J,axiom,
! [C3: set_nat,A4: nat > set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
( ( ( C3 = bot_bot_set_nat )
=> ( ( inf_in2572325071724192079at_nat @ ( comple1930127330929219688at_nat @ ( image_7178329752028323786at_nat @ A4 @ C3 ) ) @ B5 )
= B5 ) )
& ( ( C3 != bot_bot_set_nat )
=> ( ( inf_in2572325071724192079at_nat @ ( comple1930127330929219688at_nat @ ( image_7178329752028323786at_nat @ A4 @ C3 ) ) @ B5 )
= ( comple1930127330929219688at_nat
@ ( image_7178329752028323786at_nat
@ ^ [X4: nat] : ( inf_in2572325071724192079at_nat @ ( A4 @ X4 ) @ B5 )
@ C3 ) ) ) ) ) ).
% INT_extend_simps(1)
thf(fact_3405_INT__extend__simps_I1_J,axiom,
! [C3: set_Product_unit,A4: product_unit > set_nat,B5: set_nat] :
( ( ( C3 = bot_bo3957492148770167129t_unit )
=> ( ( inf_inf_set_nat @ ( comple7806235888213564991et_nat @ ( image_8173849991827986510et_nat @ A4 @ C3 ) ) @ B5 )
= B5 ) )
& ( ( C3 != bot_bo3957492148770167129t_unit )
=> ( ( inf_inf_set_nat @ ( comple7806235888213564991et_nat @ ( image_8173849991827986510et_nat @ A4 @ C3 ) ) @ B5 )
= ( comple7806235888213564991et_nat
@ ( image_8173849991827986510et_nat
@ ^ [X4: product_unit] : ( inf_inf_set_nat @ ( A4 @ X4 ) @ B5 )
@ C3 ) ) ) ) ) ).
% INT_extend_simps(1)
thf(fact_3406_INT__extend__simps_I1_J,axiom,
! [C3: set_Pr4329608150637261639at_nat,A4: produc3843707927480180839at_nat > set_nat,B5: set_nat] :
( ( ( C3 = bot_bo228742789529271731at_nat )
=> ( ( inf_inf_set_nat @ ( comple7806235888213564991et_nat @ ( image_3160451284896210088et_nat @ A4 @ C3 ) ) @ B5 )
= B5 ) )
& ( ( C3 != bot_bo228742789529271731at_nat )
=> ( ( inf_inf_set_nat @ ( comple7806235888213564991et_nat @ ( image_3160451284896210088et_nat @ A4 @ C3 ) ) @ B5 )
= ( comple7806235888213564991et_nat
@ ( image_3160451284896210088et_nat
@ ^ [X4: produc3843707927480180839at_nat] : ( inf_inf_set_nat @ ( A4 @ X4 ) @ B5 )
@ C3 ) ) ) ) ) ).
% INT_extend_simps(1)
thf(fact_3407_INT__extend__simps_I1_J,axiom,
! [C3: set_Pr1261947904930325089at_nat,A4: product_prod_nat_nat > set_nat,B5: set_nat] :
( ( ( C3 = bot_bo2099793752762293965at_nat )
=> ( ( inf_inf_set_nat @ ( comple7806235888213564991et_nat @ ( image_15824709712370754et_nat @ A4 @ C3 ) ) @ B5 )
= B5 ) )
& ( ( C3 != bot_bo2099793752762293965at_nat )
=> ( ( inf_inf_set_nat @ ( comple7806235888213564991et_nat @ ( image_15824709712370754et_nat @ A4 @ C3 ) ) @ B5 )
= ( comple7806235888213564991et_nat
@ ( image_15824709712370754et_nat
@ ^ [X4: product_prod_nat_nat] : ( inf_inf_set_nat @ ( A4 @ X4 ) @ B5 )
@ C3 ) ) ) ) ) ).
% INT_extend_simps(1)
thf(fact_3408_INT__extend__simps_I1_J,axiom,
! [C3: set_o,A4: $o > set_nat,B5: set_nat] :
( ( ( C3 = bot_bot_set_o )
=> ( ( inf_inf_set_nat @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ A4 @ C3 ) ) @ B5 )
= B5 ) )
& ( ( C3 != bot_bot_set_o )
=> ( ( inf_inf_set_nat @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ A4 @ C3 ) ) @ B5 )
= ( comple7806235888213564991et_nat
@ ( image_o_set_nat
@ ^ [X4: $o] : ( inf_inf_set_nat @ ( A4 @ X4 ) @ B5 )
@ C3 ) ) ) ) ) ).
% INT_extend_simps(1)
thf(fact_3409_INT__extend__simps_I1_J,axiom,
! [C3: set_nat,A4: nat > set_nat,B5: set_nat] :
( ( ( C3 = bot_bot_set_nat )
=> ( ( inf_inf_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ A4 @ C3 ) ) @ B5 )
= B5 ) )
& ( ( C3 != bot_bot_set_nat )
=> ( ( inf_inf_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ A4 @ C3 ) ) @ B5 )
= ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [X4: nat] : ( inf_inf_set_nat @ ( A4 @ X4 ) @ B5 )
@ C3 ) ) ) ) ) ).
% INT_extend_simps(1)
thf(fact_3410_INT__extend__simps_I2_J,axiom,
! [C3: set_Product_unit,A4: set_Pr1261947904930325089at_nat,B5: product_unit > set_Pr1261947904930325089at_nat] :
( ( ( C3 = bot_bo3957492148770167129t_unit )
=> ( ( inf_in2572325071724192079at_nat @ A4 @ ( comple1930127330929219688at_nat @ ( image_3617315916088684441at_nat @ B5 @ C3 ) ) )
= A4 ) )
& ( ( C3 != bot_bo3957492148770167129t_unit )
=> ( ( inf_in2572325071724192079at_nat @ A4 @ ( comple1930127330929219688at_nat @ ( image_3617315916088684441at_nat @ B5 @ C3 ) ) )
= ( comple1930127330929219688at_nat
@ ( image_3617315916088684441at_nat
@ ^ [X4: product_unit] : ( inf_in2572325071724192079at_nat @ A4 @ ( B5 @ X4 ) )
@ C3 ) ) ) ) ) ).
% INT_extend_simps(2)
thf(fact_3411_INT__extend__simps_I2_J,axiom,
! [C3: set_Pr4329608150637261639at_nat,A4: set_Pr1261947904930325089at_nat,B5: produc3843707927480180839at_nat > set_Pr1261947904930325089at_nat] :
( ( ( C3 = bot_bo228742789529271731at_nat )
=> ( ( inf_in2572325071724192079at_nat @ A4 @ ( comple1930127330929219688at_nat @ ( image_7979982168039674559at_nat @ B5 @ C3 ) ) )
= A4 ) )
& ( ( C3 != bot_bo228742789529271731at_nat )
=> ( ( inf_in2572325071724192079at_nat @ A4 @ ( comple1930127330929219688at_nat @ ( image_7979982168039674559at_nat @ B5 @ C3 ) ) )
= ( comple1930127330929219688at_nat
@ ( image_7979982168039674559at_nat
@ ^ [X4: produc3843707927480180839at_nat] : ( inf_in2572325071724192079at_nat @ A4 @ ( B5 @ X4 ) )
@ C3 ) ) ) ) ) ).
% INT_extend_simps(2)
thf(fact_3412_INT__extend__simps_I2_J,axiom,
! [C3: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,B5: product_prod_nat_nat > set_Pr1261947904930325089at_nat] :
( ( ( C3 = bot_bo2099793752762293965at_nat )
=> ( ( inf_in2572325071724192079at_nat @ A4 @ ( comple1930127330929219688at_nat @ ( image_5423882813909314213at_nat @ B5 @ C3 ) ) )
= A4 ) )
& ( ( C3 != bot_bo2099793752762293965at_nat )
=> ( ( inf_in2572325071724192079at_nat @ A4 @ ( comple1930127330929219688at_nat @ ( image_5423882813909314213at_nat @ B5 @ C3 ) ) )
= ( comple1930127330929219688at_nat
@ ( image_5423882813909314213at_nat
@ ^ [X4: product_prod_nat_nat] : ( inf_in2572325071724192079at_nat @ A4 @ ( B5 @ X4 ) )
@ C3 ) ) ) ) ) ).
% INT_extend_simps(2)
thf(fact_3413_INT__extend__simps_I2_J,axiom,
! [C3: set_o,A4: set_Pr1261947904930325089at_nat,B5: $o > set_Pr1261947904930325089at_nat] :
( ( ( C3 = bot_bot_set_o )
=> ( ( inf_in2572325071724192079at_nat @ A4 @ ( comple1930127330929219688at_nat @ ( image_7077428525161401678at_nat @ B5 @ C3 ) ) )
= A4 ) )
& ( ( C3 != bot_bot_set_o )
=> ( ( inf_in2572325071724192079at_nat @ A4 @ ( comple1930127330929219688at_nat @ ( image_7077428525161401678at_nat @ B5 @ C3 ) ) )
= ( comple1930127330929219688at_nat
@ ( image_7077428525161401678at_nat
@ ^ [X4: $o] : ( inf_in2572325071724192079at_nat @ A4 @ ( B5 @ X4 ) )
@ C3 ) ) ) ) ) ).
% INT_extend_simps(2)
thf(fact_3414_INT__extend__simps_I2_J,axiom,
! [C3: set_nat,A4: set_Pr1261947904930325089at_nat,B5: nat > set_Pr1261947904930325089at_nat] :
( ( ( C3 = bot_bot_set_nat )
=> ( ( inf_in2572325071724192079at_nat @ A4 @ ( comple1930127330929219688at_nat @ ( image_7178329752028323786at_nat @ B5 @ C3 ) ) )
= A4 ) )
& ( ( C3 != bot_bot_set_nat )
=> ( ( inf_in2572325071724192079at_nat @ A4 @ ( comple1930127330929219688at_nat @ ( image_7178329752028323786at_nat @ B5 @ C3 ) ) )
= ( comple1930127330929219688at_nat
@ ( image_7178329752028323786at_nat
@ ^ [X4: nat] : ( inf_in2572325071724192079at_nat @ A4 @ ( B5 @ X4 ) )
@ C3 ) ) ) ) ) ).
% INT_extend_simps(2)
thf(fact_3415_INT__extend__simps_I2_J,axiom,
! [C3: set_Product_unit,A4: set_nat,B5: product_unit > set_nat] :
( ( ( C3 = bot_bo3957492148770167129t_unit )
=> ( ( inf_inf_set_nat @ A4 @ ( comple7806235888213564991et_nat @ ( image_8173849991827986510et_nat @ B5 @ C3 ) ) )
= A4 ) )
& ( ( C3 != bot_bo3957492148770167129t_unit )
=> ( ( inf_inf_set_nat @ A4 @ ( comple7806235888213564991et_nat @ ( image_8173849991827986510et_nat @ B5 @ C3 ) ) )
= ( comple7806235888213564991et_nat
@ ( image_8173849991827986510et_nat
@ ^ [X4: product_unit] : ( inf_inf_set_nat @ A4 @ ( B5 @ X4 ) )
@ C3 ) ) ) ) ) ).
% INT_extend_simps(2)
thf(fact_3416_INT__extend__simps_I2_J,axiom,
! [C3: set_Pr4329608150637261639at_nat,A4: set_nat,B5: produc3843707927480180839at_nat > set_nat] :
( ( ( C3 = bot_bo228742789529271731at_nat )
=> ( ( inf_inf_set_nat @ A4 @ ( comple7806235888213564991et_nat @ ( image_3160451284896210088et_nat @ B5 @ C3 ) ) )
= A4 ) )
& ( ( C3 != bot_bo228742789529271731at_nat )
=> ( ( inf_inf_set_nat @ A4 @ ( comple7806235888213564991et_nat @ ( image_3160451284896210088et_nat @ B5 @ C3 ) ) )
= ( comple7806235888213564991et_nat
@ ( image_3160451284896210088et_nat
@ ^ [X4: produc3843707927480180839at_nat] : ( inf_inf_set_nat @ A4 @ ( B5 @ X4 ) )
@ C3 ) ) ) ) ) ).
% INT_extend_simps(2)
thf(fact_3417_INT__extend__simps_I2_J,axiom,
! [C3: set_Pr1261947904930325089at_nat,A4: set_nat,B5: product_prod_nat_nat > set_nat] :
( ( ( C3 = bot_bo2099793752762293965at_nat )
=> ( ( inf_inf_set_nat @ A4 @ ( comple7806235888213564991et_nat @ ( image_15824709712370754et_nat @ B5 @ C3 ) ) )
= A4 ) )
& ( ( C3 != bot_bo2099793752762293965at_nat )
=> ( ( inf_inf_set_nat @ A4 @ ( comple7806235888213564991et_nat @ ( image_15824709712370754et_nat @ B5 @ C3 ) ) )
= ( comple7806235888213564991et_nat
@ ( image_15824709712370754et_nat
@ ^ [X4: product_prod_nat_nat] : ( inf_inf_set_nat @ A4 @ ( B5 @ X4 ) )
@ C3 ) ) ) ) ) ).
% INT_extend_simps(2)
thf(fact_3418_INT__extend__simps_I2_J,axiom,
! [C3: set_o,A4: set_nat,B5: $o > set_nat] :
( ( ( C3 = bot_bot_set_o )
=> ( ( inf_inf_set_nat @ A4 @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ B5 @ C3 ) ) )
= A4 ) )
& ( ( C3 != bot_bot_set_o )
=> ( ( inf_inf_set_nat @ A4 @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ B5 @ C3 ) ) )
= ( comple7806235888213564991et_nat
@ ( image_o_set_nat
@ ^ [X4: $o] : ( inf_inf_set_nat @ A4 @ ( B5 @ X4 ) )
@ C3 ) ) ) ) ) ).
% INT_extend_simps(2)
thf(fact_3419_INT__extend__simps_I2_J,axiom,
! [C3: set_nat,A4: set_nat,B5: nat > set_nat] :
( ( ( C3 = bot_bot_set_nat )
=> ( ( inf_inf_set_nat @ A4 @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B5 @ C3 ) ) )
= A4 ) )
& ( ( C3 != bot_bot_set_nat )
=> ( ( inf_inf_set_nat @ A4 @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B5 @ C3 ) ) )
= ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [X4: nat] : ( inf_inf_set_nat @ A4 @ ( B5 @ X4 ) )
@ C3 ) ) ) ) ) ).
% INT_extend_simps(2)
thf(fact_3420_le__cINF__iff,axiom,
! [A4: set_int,F: int > nat,U: nat] :
( ( A4 != bot_bot_set_int )
=> ( ( condit1738341127787009408ow_nat @ ( image_int_nat @ F @ A4 ) )
=> ( ( ord_less_eq_nat @ U @ ( complete_Inf_Inf_nat @ ( image_int_nat @ F @ A4 ) ) )
= ( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_eq_nat @ U @ ( F @ X4 ) ) ) ) ) ) ) ).
% le_cINF_iff
thf(fact_3421_le__cINF__iff,axiom,
! [A4: set_Product_unit,F: product_unit > nat,U: nat] :
( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ( condit1738341127787009408ow_nat @ ( image_875570014554754200it_nat @ F @ A4 ) )
=> ( ( ord_less_eq_nat @ U @ ( complete_Inf_Inf_nat @ ( image_875570014554754200it_nat @ F @ A4 ) ) )
= ( ! [X4: product_unit] :
( ( member_Product_unit @ X4 @ A4 )
=> ( ord_less_eq_nat @ U @ ( F @ X4 ) ) ) ) ) ) ) ).
% le_cINF_iff
thf(fact_3422_le__cINF__iff,axiom,
! [A4: set_o,F: $o > nat,U: nat] :
( ( A4 != bot_bot_set_o )
=> ( ( condit1738341127787009408ow_nat @ ( image_o_nat @ F @ A4 ) )
=> ( ( ord_less_eq_nat @ U @ ( complete_Inf_Inf_nat @ ( image_o_nat @ F @ A4 ) ) )
= ( ! [X4: $o] :
( ( member_o @ X4 @ A4 )
=> ( ord_less_eq_nat @ U @ ( F @ X4 ) ) ) ) ) ) ) ).
% le_cINF_iff
thf(fact_3423_le__cINF__iff,axiom,
! [A4: set_nat,F: nat > nat,U: nat] :
( ( A4 != bot_bot_set_nat )
=> ( ( condit1738341127787009408ow_nat @ ( image_nat_nat @ F @ A4 ) )
=> ( ( ord_less_eq_nat @ U @ ( complete_Inf_Inf_nat @ ( image_nat_nat @ F @ A4 ) ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ord_less_eq_nat @ U @ ( F @ X4 ) ) ) ) ) ) ) ).
% le_cINF_iff
thf(fact_3424_le__cINF__iff,axiom,
! [A4: set_int,F: int > int,U: int] :
( ( A4 != bot_bot_set_int )
=> ( ( condit1735850657277959132ow_int @ ( image_int_int @ F @ A4 ) )
=> ( ( ord_less_eq_int @ U @ ( complete_Inf_Inf_int @ ( image_int_int @ F @ A4 ) ) )
= ( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_eq_int @ U @ ( F @ X4 ) ) ) ) ) ) ) ).
% le_cINF_iff
thf(fact_3425_le__cINF__iff,axiom,
! [A4: set_Product_unit,F: product_unit > int,U: int] :
( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ( condit1735850657277959132ow_int @ ( image_873079544045703924it_int @ F @ A4 ) )
=> ( ( ord_less_eq_int @ U @ ( complete_Inf_Inf_int @ ( image_873079544045703924it_int @ F @ A4 ) ) )
= ( ! [X4: product_unit] :
( ( member_Product_unit @ X4 @ A4 )
=> ( ord_less_eq_int @ U @ ( F @ X4 ) ) ) ) ) ) ) ).
% le_cINF_iff
thf(fact_3426_le__cINF__iff,axiom,
! [A4: set_o,F: $o > int,U: int] :
( ( A4 != bot_bot_set_o )
=> ( ( condit1735850657277959132ow_int @ ( image_o_int @ F @ A4 ) )
=> ( ( ord_less_eq_int @ U @ ( complete_Inf_Inf_int @ ( image_o_int @ F @ A4 ) ) )
= ( ! [X4: $o] :
( ( member_o @ X4 @ A4 )
=> ( ord_less_eq_int @ U @ ( F @ X4 ) ) ) ) ) ) ) ).
% le_cINF_iff
thf(fact_3427_le__cINF__iff,axiom,
! [A4: set_nat,F: nat > int,U: int] :
( ( A4 != bot_bot_set_nat )
=> ( ( condit1735850657277959132ow_int @ ( image_nat_int @ F @ A4 ) )
=> ( ( ord_less_eq_int @ U @ ( complete_Inf_Inf_int @ ( image_nat_int @ F @ A4 ) ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ord_less_eq_int @ U @ ( F @ X4 ) ) ) ) ) ) ) ).
% le_cINF_iff
thf(fact_3428_le__cINF__iff,axiom,
! [A4: set_Product_unit,F: product_unit > product_unit,U: product_unit] :
( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ( condit9161856309968323677t_unit @ ( image_405062704495631173t_unit @ F @ A4 ) )
=> ( ( ord_le3221252021190050221t_unit @ U @ ( comple2584293577114468500t_unit @ ( image_405062704495631173t_unit @ F @ A4 ) ) )
= ( ! [X4: product_unit] :
( ( member_Product_unit @ X4 @ A4 )
=> ( ord_le3221252021190050221t_unit @ U @ ( F @ X4 ) ) ) ) ) ) ) ).
% le_cINF_iff
thf(fact_3429_le__cINF__iff,axiom,
! [A4: set_o,F: $o > product_unit,U: product_unit] :
( ( A4 != bot_bot_set_o )
=> ( ( condit9161856309968323677t_unit @ ( image_o_Product_unit @ F @ A4 ) )
=> ( ( ord_le3221252021190050221t_unit @ U @ ( comple2584293577114468500t_unit @ ( image_o_Product_unit @ F @ A4 ) ) )
= ( ! [X4: $o] :
( ( member_o @ X4 @ A4 )
=> ( ord_le3221252021190050221t_unit @ U @ ( F @ X4 ) ) ) ) ) ) ) ).
% le_cINF_iff
thf(fact_3430_cINF__mono,axiom,
! [B5: set_int,F: nat > nat,A4: set_nat,G: int > nat] :
( ( B5 != bot_bot_set_int )
=> ( ( condit1738341127787009408ow_nat @ ( image_nat_nat @ F @ A4 ) )
=> ( ! [M4: int] :
( ( member_int @ M4 @ B5 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A4 )
& ( ord_less_eq_nat @ ( F @ X6 ) @ ( G @ M4 ) ) ) )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_nat_nat @ F @ A4 ) ) @ ( complete_Inf_Inf_nat @ ( image_int_nat @ G @ B5 ) ) ) ) ) ) ).
% cINF_mono
thf(fact_3431_cINF__mono,axiom,
! [B5: set_int,F: int > nat,A4: set_int,G: int > nat] :
( ( B5 != bot_bot_set_int )
=> ( ( condit1738341127787009408ow_nat @ ( image_int_nat @ F @ A4 ) )
=> ( ! [M4: int] :
( ( member_int @ M4 @ B5 )
=> ? [X6: int] :
( ( member_int @ X6 @ A4 )
& ( ord_less_eq_nat @ ( F @ X6 ) @ ( G @ M4 ) ) ) )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_int_nat @ F @ A4 ) ) @ ( complete_Inf_Inf_nat @ ( image_int_nat @ G @ B5 ) ) ) ) ) ) ).
% cINF_mono
thf(fact_3432_cINF__mono,axiom,
! [B5: set_Product_unit,F: nat > nat,A4: set_nat,G: product_unit > nat] :
( ( B5 != bot_bo3957492148770167129t_unit )
=> ( ( condit1738341127787009408ow_nat @ ( image_nat_nat @ F @ A4 ) )
=> ( ! [M4: product_unit] :
( ( member_Product_unit @ M4 @ B5 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A4 )
& ( ord_less_eq_nat @ ( F @ X6 ) @ ( G @ M4 ) ) ) )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_nat_nat @ F @ A4 ) ) @ ( complete_Inf_Inf_nat @ ( image_875570014554754200it_nat @ G @ B5 ) ) ) ) ) ) ).
% cINF_mono
thf(fact_3433_cINF__mono,axiom,
! [B5: set_Product_unit,F: int > nat,A4: set_int,G: product_unit > nat] :
( ( B5 != bot_bo3957492148770167129t_unit )
=> ( ( condit1738341127787009408ow_nat @ ( image_int_nat @ F @ A4 ) )
=> ( ! [M4: product_unit] :
( ( member_Product_unit @ M4 @ B5 )
=> ? [X6: int] :
( ( member_int @ X6 @ A4 )
& ( ord_less_eq_nat @ ( F @ X6 ) @ ( G @ M4 ) ) ) )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_int_nat @ F @ A4 ) ) @ ( complete_Inf_Inf_nat @ ( image_875570014554754200it_nat @ G @ B5 ) ) ) ) ) ) ).
% cINF_mono
thf(fact_3434_cINF__mono,axiom,
! [B5: set_o,F: nat > nat,A4: set_nat,G: $o > nat] :
( ( B5 != bot_bot_set_o )
=> ( ( condit1738341127787009408ow_nat @ ( image_nat_nat @ F @ A4 ) )
=> ( ! [M4: $o] :
( ( member_o @ M4 @ B5 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A4 )
& ( ord_less_eq_nat @ ( F @ X6 ) @ ( G @ M4 ) ) ) )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_nat_nat @ F @ A4 ) ) @ ( complete_Inf_Inf_nat @ ( image_o_nat @ G @ B5 ) ) ) ) ) ) ).
% cINF_mono
thf(fact_3435_cINF__mono,axiom,
! [B5: set_o,F: int > nat,A4: set_int,G: $o > nat] :
( ( B5 != bot_bot_set_o )
=> ( ( condit1738341127787009408ow_nat @ ( image_int_nat @ F @ A4 ) )
=> ( ! [M4: $o] :
( ( member_o @ M4 @ B5 )
=> ? [X6: int] :
( ( member_int @ X6 @ A4 )
& ( ord_less_eq_nat @ ( F @ X6 ) @ ( G @ M4 ) ) ) )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_int_nat @ F @ A4 ) ) @ ( complete_Inf_Inf_nat @ ( image_o_nat @ G @ B5 ) ) ) ) ) ) ).
% cINF_mono
thf(fact_3436_cINF__mono,axiom,
! [B5: set_nat,F: nat > nat,A4: set_nat,G: nat > nat] :
( ( B5 != bot_bot_set_nat )
=> ( ( condit1738341127787009408ow_nat @ ( image_nat_nat @ F @ A4 ) )
=> ( ! [M4: nat] :
( ( member_nat @ M4 @ B5 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A4 )
& ( ord_less_eq_nat @ ( F @ X6 ) @ ( G @ M4 ) ) ) )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_nat_nat @ F @ A4 ) ) @ ( complete_Inf_Inf_nat @ ( image_nat_nat @ G @ B5 ) ) ) ) ) ) ).
% cINF_mono
thf(fact_3437_cINF__mono,axiom,
! [B5: set_nat,F: int > nat,A4: set_int,G: nat > nat] :
( ( B5 != bot_bot_set_nat )
=> ( ( condit1738341127787009408ow_nat @ ( image_int_nat @ F @ A4 ) )
=> ( ! [M4: nat] :
( ( member_nat @ M4 @ B5 )
=> ? [X6: int] :
( ( member_int @ X6 @ A4 )
& ( ord_less_eq_nat @ ( F @ X6 ) @ ( G @ M4 ) ) ) )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_int_nat @ F @ A4 ) ) @ ( complete_Inf_Inf_nat @ ( image_nat_nat @ G @ B5 ) ) ) ) ) ) ).
% cINF_mono
thf(fact_3438_cINF__mono,axiom,
! [B5: set_int,F: nat > int,A4: set_nat,G: int > int] :
( ( B5 != bot_bot_set_int )
=> ( ( condit1735850657277959132ow_int @ ( image_nat_int @ F @ A4 ) )
=> ( ! [M4: int] :
( ( member_int @ M4 @ B5 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A4 )
& ( ord_less_eq_int @ ( F @ X6 ) @ ( G @ M4 ) ) ) )
=> ( ord_less_eq_int @ ( complete_Inf_Inf_int @ ( image_nat_int @ F @ A4 ) ) @ ( complete_Inf_Inf_int @ ( image_int_int @ G @ B5 ) ) ) ) ) ) ).
% cINF_mono
thf(fact_3439_cINF__mono,axiom,
! [B5: set_int,F: int > int,A4: set_int,G: int > int] :
( ( B5 != bot_bot_set_int )
=> ( ( condit1735850657277959132ow_int @ ( image_int_int @ F @ A4 ) )
=> ( ! [M4: int] :
( ( member_int @ M4 @ B5 )
=> ? [X6: int] :
( ( member_int @ X6 @ A4 )
& ( ord_less_eq_int @ ( F @ X6 ) @ ( G @ M4 ) ) ) )
=> ( ord_less_eq_int @ ( complete_Inf_Inf_int @ ( image_int_int @ F @ A4 ) ) @ ( complete_Inf_Inf_int @ ( image_int_int @ G @ B5 ) ) ) ) ) ) ).
% cINF_mono
thf(fact_3440_cSUP__le__iff,axiom,
! [A4: set_int,F: int > nat,U: nat] :
( ( A4 != bot_bot_set_int )
=> ( ( condit2214826472909112428ve_nat @ ( image_int_nat @ F @ A4 ) )
=> ( ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_int_nat @ F @ A4 ) ) @ U )
= ( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ U ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_3441_cSUP__le__iff,axiom,
! [A4: set_Product_unit,F: product_unit > nat,U: nat] :
( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ( condit2214826472909112428ve_nat @ ( image_875570014554754200it_nat @ F @ A4 ) )
=> ( ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_875570014554754200it_nat @ F @ A4 ) ) @ U )
= ( ! [X4: product_unit] :
( ( member_Product_unit @ X4 @ A4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ U ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_3442_cSUP__le__iff,axiom,
! [A4: set_o,F: $o > nat,U: nat] :
( ( A4 != bot_bot_set_o )
=> ( ( condit2214826472909112428ve_nat @ ( image_o_nat @ F @ A4 ) )
=> ( ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_o_nat @ F @ A4 ) ) @ U )
= ( ! [X4: $o] :
( ( member_o @ X4 @ A4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ U ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_3443_cSUP__le__iff,axiom,
! [A4: set_nat,F: nat > nat,U: nat] :
( ( A4 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ F @ A4 ) )
=> ( ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ A4 ) ) @ U )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ U ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_3444_cSUP__le__iff,axiom,
! [A4: set_int,F: int > int,U: int] :
( ( A4 != bot_bot_set_int )
=> ( ( condit2212336002400062152ve_int @ ( image_int_int @ F @ A4 ) )
=> ( ( ord_less_eq_int @ ( complete_Sup_Sup_int @ ( image_int_int @ F @ A4 ) ) @ U )
= ( ! [X4: int] :
( ( member_int @ X4 @ A4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ U ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_3445_cSUP__le__iff,axiom,
! [A4: set_Product_unit,F: product_unit > int,U: int] :
( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ( condit2212336002400062152ve_int @ ( image_873079544045703924it_int @ F @ A4 ) )
=> ( ( ord_less_eq_int @ ( complete_Sup_Sup_int @ ( image_873079544045703924it_int @ F @ A4 ) ) @ U )
= ( ! [X4: product_unit] :
( ( member_Product_unit @ X4 @ A4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ U ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_3446_cSUP__le__iff,axiom,
! [A4: set_o,F: $o > int,U: int] :
( ( A4 != bot_bot_set_o )
=> ( ( condit2212336002400062152ve_int @ ( image_o_int @ F @ A4 ) )
=> ( ( ord_less_eq_int @ ( complete_Sup_Sup_int @ ( image_o_int @ F @ A4 ) ) @ U )
= ( ! [X4: $o] :
( ( member_o @ X4 @ A4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ U ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_3447_cSUP__le__iff,axiom,
! [A4: set_nat,F: nat > int,U: int] :
( ( A4 != bot_bot_set_nat )
=> ( ( condit2212336002400062152ve_int @ ( image_nat_int @ F @ A4 ) )
=> ( ( ord_less_eq_int @ ( complete_Sup_Sup_int @ ( image_nat_int @ F @ A4 ) ) @ U )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ U ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_3448_cSUP__le__iff,axiom,
! [A4: set_Product_unit,F: product_unit > product_unit,U: product_unit] :
( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ( condit2080475194861589617t_unit @ ( image_405062704495631173t_unit @ F @ A4 ) )
=> ( ( ord_le3221252021190050221t_unit @ ( comple4687483117567863418t_unit @ ( image_405062704495631173t_unit @ F @ A4 ) ) @ U )
= ( ! [X4: product_unit] :
( ( member_Product_unit @ X4 @ A4 )
=> ( ord_le3221252021190050221t_unit @ ( F @ X4 ) @ U ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_3449_cSUP__le__iff,axiom,
! [A4: set_o,F: $o > product_unit,U: product_unit] :
( ( A4 != bot_bot_set_o )
=> ( ( condit2080475194861589617t_unit @ ( image_o_Product_unit @ F @ A4 ) )
=> ( ( ord_le3221252021190050221t_unit @ ( comple4687483117567863418t_unit @ ( image_o_Product_unit @ F @ A4 ) ) @ U )
= ( ! [X4: $o] :
( ( member_o @ X4 @ A4 )
=> ( ord_le3221252021190050221t_unit @ ( F @ X4 ) @ U ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_3450_cSUP__mono,axiom,
! [A4: set_int,G: nat > nat,B5: set_nat,F: int > nat] :
( ( A4 != bot_bot_set_int )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ G @ B5 ) )
=> ( ! [N2: int] :
( ( member_int @ N2 @ A4 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ B5 )
& ( ord_less_eq_nat @ ( F @ N2 ) @ ( G @ X6 ) ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_int_nat @ F @ A4 ) ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ G @ B5 ) ) ) ) ) ) ).
% cSUP_mono
thf(fact_3451_cSUP__mono,axiom,
! [A4: set_int,G: int > nat,B5: set_int,F: int > nat] :
( ( A4 != bot_bot_set_int )
=> ( ( condit2214826472909112428ve_nat @ ( image_int_nat @ G @ B5 ) )
=> ( ! [N2: int] :
( ( member_int @ N2 @ A4 )
=> ? [X6: int] :
( ( member_int @ X6 @ B5 )
& ( ord_less_eq_nat @ ( F @ N2 ) @ ( G @ X6 ) ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_int_nat @ F @ A4 ) ) @ ( complete_Sup_Sup_nat @ ( image_int_nat @ G @ B5 ) ) ) ) ) ) ).
% cSUP_mono
thf(fact_3452_cSUP__mono,axiom,
! [A4: set_Product_unit,G: nat > nat,B5: set_nat,F: product_unit > nat] :
( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ G @ B5 ) )
=> ( ! [N2: product_unit] :
( ( member_Product_unit @ N2 @ A4 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ B5 )
& ( ord_less_eq_nat @ ( F @ N2 ) @ ( G @ X6 ) ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_875570014554754200it_nat @ F @ A4 ) ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ G @ B5 ) ) ) ) ) ) ).
% cSUP_mono
thf(fact_3453_cSUP__mono,axiom,
! [A4: set_Product_unit,G: int > nat,B5: set_int,F: product_unit > nat] :
( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ( condit2214826472909112428ve_nat @ ( image_int_nat @ G @ B5 ) )
=> ( ! [N2: product_unit] :
( ( member_Product_unit @ N2 @ A4 )
=> ? [X6: int] :
( ( member_int @ X6 @ B5 )
& ( ord_less_eq_nat @ ( F @ N2 ) @ ( G @ X6 ) ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_875570014554754200it_nat @ F @ A4 ) ) @ ( complete_Sup_Sup_nat @ ( image_int_nat @ G @ B5 ) ) ) ) ) ) ).
% cSUP_mono
thf(fact_3454_cSUP__mono,axiom,
! [A4: set_o,G: nat > nat,B5: set_nat,F: $o > nat] :
( ( A4 != bot_bot_set_o )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ G @ B5 ) )
=> ( ! [N2: $o] :
( ( member_o @ N2 @ A4 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ B5 )
& ( ord_less_eq_nat @ ( F @ N2 ) @ ( G @ X6 ) ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_o_nat @ F @ A4 ) ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ G @ B5 ) ) ) ) ) ) ).
% cSUP_mono
thf(fact_3455_cSUP__mono,axiom,
! [A4: set_o,G: int > nat,B5: set_int,F: $o > nat] :
( ( A4 != bot_bot_set_o )
=> ( ( condit2214826472909112428ve_nat @ ( image_int_nat @ G @ B5 ) )
=> ( ! [N2: $o] :
( ( member_o @ N2 @ A4 )
=> ? [X6: int] :
( ( member_int @ X6 @ B5 )
& ( ord_less_eq_nat @ ( F @ N2 ) @ ( G @ X6 ) ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_o_nat @ F @ A4 ) ) @ ( complete_Sup_Sup_nat @ ( image_int_nat @ G @ B5 ) ) ) ) ) ) ).
% cSUP_mono
thf(fact_3456_cSUP__mono,axiom,
! [A4: set_nat,G: nat > nat,B5: set_nat,F: nat > nat] :
( ( A4 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ G @ B5 ) )
=> ( ! [N2: nat] :
( ( member_nat @ N2 @ A4 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ B5 )
& ( ord_less_eq_nat @ ( F @ N2 ) @ ( G @ X6 ) ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ A4 ) ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ G @ B5 ) ) ) ) ) ) ).
% cSUP_mono
thf(fact_3457_cSUP__mono,axiom,
! [A4: set_nat,G: int > nat,B5: set_int,F: nat > nat] :
( ( A4 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ ( image_int_nat @ G @ B5 ) )
=> ( ! [N2: nat] :
( ( member_nat @ N2 @ A4 )
=> ? [X6: int] :
( ( member_int @ X6 @ B5 )
& ( ord_less_eq_nat @ ( F @ N2 ) @ ( G @ X6 ) ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ A4 ) ) @ ( complete_Sup_Sup_nat @ ( image_int_nat @ G @ B5 ) ) ) ) ) ) ).
% cSUP_mono
thf(fact_3458_cSUP__mono,axiom,
! [A4: set_int,G: nat > int,B5: set_nat,F: int > int] :
( ( A4 != bot_bot_set_int )
=> ( ( condit2212336002400062152ve_int @ ( image_nat_int @ G @ B5 ) )
=> ( ! [N2: int] :
( ( member_int @ N2 @ A4 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ B5 )
& ( ord_less_eq_int @ ( F @ N2 ) @ ( G @ X6 ) ) ) )
=> ( ord_less_eq_int @ ( complete_Sup_Sup_int @ ( image_int_int @ F @ A4 ) ) @ ( complete_Sup_Sup_int @ ( image_nat_int @ G @ B5 ) ) ) ) ) ) ).
% cSUP_mono
thf(fact_3459_cSUP__mono,axiom,
! [A4: set_int,G: int > int,B5: set_int,F: int > int] :
( ( A4 != bot_bot_set_int )
=> ( ( condit2212336002400062152ve_int @ ( image_int_int @ G @ B5 ) )
=> ( ! [N2: int] :
( ( member_int @ N2 @ A4 )
=> ? [X6: int] :
( ( member_int @ X6 @ B5 )
& ( ord_less_eq_int @ ( F @ N2 ) @ ( G @ X6 ) ) ) )
=> ( ord_less_eq_int @ ( complete_Sup_Sup_int @ ( image_int_int @ F @ A4 ) ) @ ( complete_Sup_Sup_int @ ( image_int_int @ G @ B5 ) ) ) ) ) ) ).
% cSUP_mono
thf(fact_3460_prod__set__defs_I1_J,axiom,
( basic_685971451221398055t_unit
= ( ^ [P3: produc7812382424243267045t_unit] : ( insert_ref_a @ ( produc473321116828791851t_unit @ P3 ) @ bot_bot_set_ref_a ) ) ) ).
% prod_set_defs(1)
thf(fact_3461_image__INT,axiom,
! [F: int > int,C3: set_int,A4: set_o,B5: $o > set_int,J4: $o] :
( ( inj_on_int_int @ F @ C3 )
=> ( ! [X: $o] :
( ( member_o @ X @ A4 )
=> ( ord_less_eq_set_int @ ( B5 @ X ) @ C3 ) )
=> ( ( member_o @ J4 @ A4 )
=> ( ( image_int_int @ F @ ( comple3628384868704368283et_int @ ( image_o_set_int @ B5 @ A4 ) ) )
= ( comple3628384868704368283et_int
@ ( image_o_set_int
@ ^ [X4: $o] : ( image_int_int @ F @ ( B5 @ X4 ) )
@ A4 ) ) ) ) ) ) ).
% image_INT
thf(fact_3462_image__INT,axiom,
! [F: int > int,C3: set_int,A4: set_nat,B5: nat > set_int,J4: nat] :
( ( inj_on_int_int @ F @ C3 )
=> ( ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ( ord_less_eq_set_int @ ( B5 @ X ) @ C3 ) )
=> ( ( member_nat @ J4 @ A4 )
=> ( ( image_int_int @ F @ ( comple3628384868704368283et_int @ ( image_nat_set_int @ B5 @ A4 ) ) )
= ( comple3628384868704368283et_int
@ ( image_nat_set_int
@ ^ [X4: nat] : ( image_int_int @ F @ ( B5 @ X4 ) )
@ A4 ) ) ) ) ) ) ).
% image_INT
thf(fact_3463_image__INT,axiom,
! [F: int > int,C3: set_int,A4: set_int,B5: int > set_int,J4: int] :
( ( inj_on_int_int @ F @ C3 )
=> ( ! [X: int] :
( ( member_int @ X @ A4 )
=> ( ord_less_eq_set_int @ ( B5 @ X ) @ C3 ) )
=> ( ( member_int @ J4 @ A4 )
=> ( ( image_int_int @ F @ ( comple3628384868704368283et_int @ ( image_int_set_int @ B5 @ A4 ) ) )
= ( comple3628384868704368283et_int
@ ( image_int_set_int
@ ^ [X4: int] : ( image_int_int @ F @ ( B5 @ X4 ) )
@ A4 ) ) ) ) ) ) ).
% image_INT
thf(fact_3464_image__INT,axiom,
! [F: int > nat,C3: set_int,A4: set_o,B5: $o > set_int,J4: $o] :
( ( inj_on_int_nat @ F @ C3 )
=> ( ! [X: $o] :
( ( member_o @ X @ A4 )
=> ( ord_less_eq_set_int @ ( B5 @ X ) @ C3 ) )
=> ( ( member_o @ J4 @ A4 )
=> ( ( image_int_nat @ F @ ( comple3628384868704368283et_int @ ( image_o_set_int @ B5 @ A4 ) ) )
= ( comple7806235888213564991et_nat
@ ( image_o_set_nat
@ ^ [X4: $o] : ( image_int_nat @ F @ ( B5 @ X4 ) )
@ A4 ) ) ) ) ) ) ).
% image_INT
thf(fact_3465_image__INT,axiom,
! [F: int > nat,C3: set_int,A4: set_nat,B5: nat > set_int,J4: nat] :
( ( inj_on_int_nat @ F @ C3 )
=> ( ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ( ord_less_eq_set_int @ ( B5 @ X ) @ C3 ) )
=> ( ( member_nat @ J4 @ A4 )
=> ( ( image_int_nat @ F @ ( comple3628384868704368283et_int @ ( image_nat_set_int @ B5 @ A4 ) ) )
= ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [X4: nat] : ( image_int_nat @ F @ ( B5 @ X4 ) )
@ A4 ) ) ) ) ) ) ).
% image_INT
thf(fact_3466_image__INT,axiom,
! [F: int > nat,C3: set_int,A4: set_int,B5: int > set_int,J4: int] :
( ( inj_on_int_nat @ F @ C3 )
=> ( ! [X: int] :
( ( member_int @ X @ A4 )
=> ( ord_less_eq_set_int @ ( B5 @ X ) @ C3 ) )
=> ( ( member_int @ J4 @ A4 )
=> ( ( image_int_nat @ F @ ( comple3628384868704368283et_int @ ( image_int_set_int @ B5 @ A4 ) ) )
= ( comple7806235888213564991et_nat
@ ( image_int_set_nat
@ ^ [X4: int] : ( image_int_nat @ F @ ( B5 @ X4 ) )
@ A4 ) ) ) ) ) ) ).
% image_INT
thf(fact_3467_image__INT,axiom,
! [F: nat > rat,C3: set_nat,A4: set_o,B5: $o > set_nat,J4: $o] :
( ( inj_on_nat_rat @ F @ C3 )
=> ( ! [X: $o] :
( ( member_o @ X @ A4 )
=> ( ord_less_eq_set_nat @ ( B5 @ X ) @ C3 ) )
=> ( ( member_o @ J4 @ A4 )
=> ( ( image_nat_rat @ F @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ B5 @ A4 ) ) )
= ( comple4298007329820168263et_rat
@ ( image_o_set_rat
@ ^ [X4: $o] : ( image_nat_rat @ F @ ( B5 @ X4 ) )
@ A4 ) ) ) ) ) ) ).
% image_INT
thf(fact_3468_image__INT,axiom,
! [F: nat > int,C3: set_nat,A4: set_o,B5: $o > set_nat,J4: $o] :
( ( inj_on_nat_int @ F @ C3 )
=> ( ! [X: $o] :
( ( member_o @ X @ A4 )
=> ( ord_less_eq_set_nat @ ( B5 @ X ) @ C3 ) )
=> ( ( member_o @ J4 @ A4 )
=> ( ( image_nat_int @ F @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ B5 @ A4 ) ) )
= ( comple3628384868704368283et_int
@ ( image_o_set_int
@ ^ [X4: $o] : ( image_nat_int @ F @ ( B5 @ X4 ) )
@ A4 ) ) ) ) ) ) ).
% image_INT
thf(fact_3469_image__INT,axiom,
! [F: nat > rat,C3: set_nat,A4: set_nat,B5: nat > set_nat,J4: nat] :
( ( inj_on_nat_rat @ F @ C3 )
=> ( ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ( ord_less_eq_set_nat @ ( B5 @ X ) @ C3 ) )
=> ( ( member_nat @ J4 @ A4 )
=> ( ( image_nat_rat @ F @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B5 @ A4 ) ) )
= ( comple4298007329820168263et_rat
@ ( image_nat_set_rat
@ ^ [X4: nat] : ( image_nat_rat @ F @ ( B5 @ X4 ) )
@ A4 ) ) ) ) ) ) ).
% image_INT
thf(fact_3470_image__INT,axiom,
! [F: nat > int,C3: set_nat,A4: set_nat,B5: nat > set_nat,J4: nat] :
( ( inj_on_nat_int @ F @ C3 )
=> ( ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ( ord_less_eq_set_nat @ ( B5 @ X ) @ C3 ) )
=> ( ( member_nat @ J4 @ A4 )
=> ( ( image_nat_int @ F @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B5 @ A4 ) ) )
= ( comple3628384868704368283et_int
@ ( image_nat_set_int
@ ^ [X4: nat] : ( image_nat_int @ F @ ( B5 @ X4 ) )
@ A4 ) ) ) ) ) ) ).
% image_INT
thf(fact_3471_cINF__superset__mono,axiom,
! [A4: set_int,G: int > nat,B5: set_int,F: int > nat] :
( ( A4 != bot_bot_set_int )
=> ( ( condit1738341127787009408ow_nat @ ( image_int_nat @ G @ B5 ) )
=> ( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( ! [X: int] :
( ( member_int @ X @ B5 )
=> ( ord_less_eq_nat @ ( G @ X ) @ ( F @ X ) ) )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_int_nat @ G @ B5 ) ) @ ( complete_Inf_Inf_nat @ ( image_int_nat @ F @ A4 ) ) ) ) ) ) ) ).
% cINF_superset_mono
thf(fact_3472_cINF__superset__mono,axiom,
! [A4: set_Product_unit,G: product_unit > nat,B5: set_Product_unit,F: product_unit > nat] :
( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ( condit1738341127787009408ow_nat @ ( image_875570014554754200it_nat @ G @ B5 ) )
=> ( ( ord_le3507040750410214029t_unit @ A4 @ B5 )
=> ( ! [X: product_unit] :
( ( member_Product_unit @ X @ B5 )
=> ( ord_less_eq_nat @ ( G @ X ) @ ( F @ X ) ) )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_875570014554754200it_nat @ G @ B5 ) ) @ ( complete_Inf_Inf_nat @ ( image_875570014554754200it_nat @ F @ A4 ) ) ) ) ) ) ) ).
% cINF_superset_mono
thf(fact_3473_cINF__superset__mono,axiom,
! [A4: set_o,G: $o > nat,B5: set_o,F: $o > nat] :
( ( A4 != bot_bot_set_o )
=> ( ( condit1738341127787009408ow_nat @ ( image_o_nat @ G @ B5 ) )
=> ( ( ord_less_eq_set_o @ A4 @ B5 )
=> ( ! [X: $o] :
( ( member_o @ X @ B5 )
=> ( ord_less_eq_nat @ ( G @ X ) @ ( F @ X ) ) )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_o_nat @ G @ B5 ) ) @ ( complete_Inf_Inf_nat @ ( image_o_nat @ F @ A4 ) ) ) ) ) ) ) ).
% cINF_superset_mono
thf(fact_3474_cINF__superset__mono,axiom,
! [A4: set_nat,G: nat > nat,B5: set_nat,F: nat > nat] :
( ( A4 != bot_bot_set_nat )
=> ( ( condit1738341127787009408ow_nat @ ( image_nat_nat @ G @ B5 ) )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B5 )
=> ( ord_less_eq_nat @ ( G @ X ) @ ( F @ X ) ) )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_nat_nat @ G @ B5 ) ) @ ( complete_Inf_Inf_nat @ ( image_nat_nat @ F @ A4 ) ) ) ) ) ) ) ).
% cINF_superset_mono
thf(fact_3475_cINF__superset__mono,axiom,
! [A4: set_int,G: int > int,B5: set_int,F: int > int] :
( ( A4 != bot_bot_set_int )
=> ( ( condit1735850657277959132ow_int @ ( image_int_int @ G @ B5 ) )
=> ( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( ! [X: int] :
( ( member_int @ X @ B5 )
=> ( ord_less_eq_int @ ( G @ X ) @ ( F @ X ) ) )
=> ( ord_less_eq_int @ ( complete_Inf_Inf_int @ ( image_int_int @ G @ B5 ) ) @ ( complete_Inf_Inf_int @ ( image_int_int @ F @ A4 ) ) ) ) ) ) ) ).
% cINF_superset_mono
thf(fact_3476_cINF__superset__mono,axiom,
! [A4: set_Product_unit,G: product_unit > int,B5: set_Product_unit,F: product_unit > int] :
( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ( condit1735850657277959132ow_int @ ( image_873079544045703924it_int @ G @ B5 ) )
=> ( ( ord_le3507040750410214029t_unit @ A4 @ B5 )
=> ( ! [X: product_unit] :
( ( member_Product_unit @ X @ B5 )
=> ( ord_less_eq_int @ ( G @ X ) @ ( F @ X ) ) )
=> ( ord_less_eq_int @ ( complete_Inf_Inf_int @ ( image_873079544045703924it_int @ G @ B5 ) ) @ ( complete_Inf_Inf_int @ ( image_873079544045703924it_int @ F @ A4 ) ) ) ) ) ) ) ).
% cINF_superset_mono
thf(fact_3477_cINF__superset__mono,axiom,
! [A4: set_o,G: $o > int,B5: set_o,F: $o > int] :
( ( A4 != bot_bot_set_o )
=> ( ( condit1735850657277959132ow_int @ ( image_o_int @ G @ B5 ) )
=> ( ( ord_less_eq_set_o @ A4 @ B5 )
=> ( ! [X: $o] :
( ( member_o @ X @ B5 )
=> ( ord_less_eq_int @ ( G @ X ) @ ( F @ X ) ) )
=> ( ord_less_eq_int @ ( complete_Inf_Inf_int @ ( image_o_int @ G @ B5 ) ) @ ( complete_Inf_Inf_int @ ( image_o_int @ F @ A4 ) ) ) ) ) ) ) ).
% cINF_superset_mono
thf(fact_3478_cINF__superset__mono,axiom,
! [A4: set_nat,G: nat > int,B5: set_nat,F: nat > int] :
( ( A4 != bot_bot_set_nat )
=> ( ( condit1735850657277959132ow_int @ ( image_nat_int @ G @ B5 ) )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B5 )
=> ( ord_less_eq_int @ ( G @ X ) @ ( F @ X ) ) )
=> ( ord_less_eq_int @ ( complete_Inf_Inf_int @ ( image_nat_int @ G @ B5 ) ) @ ( complete_Inf_Inf_int @ ( image_nat_int @ F @ A4 ) ) ) ) ) ) ) ).
% cINF_superset_mono
thf(fact_3479_cINF__superset__mono,axiom,
! [A4: set_int,G: int > product_unit,B5: set_int,F: int > product_unit] :
( ( A4 != bot_bot_set_int )
=> ( ( condit9161856309968323677t_unit @ ( image_5846526143423701914t_unit @ G @ B5 ) )
=> ( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( ! [X: int] :
( ( member_int @ X @ B5 )
=> ( ord_le3221252021190050221t_unit @ ( G @ X ) @ ( F @ X ) ) )
=> ( ord_le3221252021190050221t_unit @ ( comple2584293577114468500t_unit @ ( image_5846526143423701914t_unit @ G @ B5 ) ) @ ( comple2584293577114468500t_unit @ ( image_5846526143423701914t_unit @ F @ A4 ) ) ) ) ) ) ) ).
% cINF_superset_mono
thf(fact_3480_cINF__superset__mono,axiom,
! [A4: set_Product_unit,G: product_unit > product_unit,B5: set_Product_unit,F: product_unit > product_unit] :
( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ( condit9161856309968323677t_unit @ ( image_405062704495631173t_unit @ G @ B5 ) )
=> ( ( ord_le3507040750410214029t_unit @ A4 @ B5 )
=> ( ! [X: product_unit] :
( ( member_Product_unit @ X @ B5 )
=> ( ord_le3221252021190050221t_unit @ ( G @ X ) @ ( F @ X ) ) )
=> ( ord_le3221252021190050221t_unit @ ( comple2584293577114468500t_unit @ ( image_405062704495631173t_unit @ G @ B5 ) ) @ ( comple2584293577114468500t_unit @ ( image_405062704495631173t_unit @ F @ A4 ) ) ) ) ) ) ) ).
% cINF_superset_mono
thf(fact_3481_cSUP__subset__mono,axiom,
! [A4: set_int,G: int > nat,B5: set_int,F: int > nat] :
( ( A4 != bot_bot_set_int )
=> ( ( condit2214826472909112428ve_nat @ ( image_int_nat @ G @ B5 ) )
=> ( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( ! [X: int] :
( ( member_int @ X @ A4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_int_nat @ F @ A4 ) ) @ ( complete_Sup_Sup_nat @ ( image_int_nat @ G @ B5 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_3482_cSUP__subset__mono,axiom,
! [A4: set_Product_unit,G: product_unit > nat,B5: set_Product_unit,F: product_unit > nat] :
( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ( condit2214826472909112428ve_nat @ ( image_875570014554754200it_nat @ G @ B5 ) )
=> ( ( ord_le3507040750410214029t_unit @ A4 @ B5 )
=> ( ! [X: product_unit] :
( ( member_Product_unit @ X @ A4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_875570014554754200it_nat @ F @ A4 ) ) @ ( complete_Sup_Sup_nat @ ( image_875570014554754200it_nat @ G @ B5 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_3483_cSUP__subset__mono,axiom,
! [A4: set_o,G: $o > nat,B5: set_o,F: $o > nat] :
( ( A4 != bot_bot_set_o )
=> ( ( condit2214826472909112428ve_nat @ ( image_o_nat @ G @ B5 ) )
=> ( ( ord_less_eq_set_o @ A4 @ B5 )
=> ( ! [X: $o] :
( ( member_o @ X @ A4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_o_nat @ F @ A4 ) ) @ ( complete_Sup_Sup_nat @ ( image_o_nat @ G @ B5 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_3484_cSUP__subset__mono,axiom,
! [A4: set_nat,G: nat > nat,B5: set_nat,F: nat > nat] :
( ( A4 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ G @ B5 ) )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ A4 ) ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ G @ B5 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_3485_cSUP__subset__mono,axiom,
! [A4: set_int,G: int > int,B5: set_int,F: int > int] :
( ( A4 != bot_bot_set_int )
=> ( ( condit2212336002400062152ve_int @ ( image_int_int @ G @ B5 ) )
=> ( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( ! [X: int] :
( ( member_int @ X @ A4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_int @ ( complete_Sup_Sup_int @ ( image_int_int @ F @ A4 ) ) @ ( complete_Sup_Sup_int @ ( image_int_int @ G @ B5 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_3486_cSUP__subset__mono,axiom,
! [A4: set_Product_unit,G: product_unit > int,B5: set_Product_unit,F: product_unit > int] :
( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ( condit2212336002400062152ve_int @ ( image_873079544045703924it_int @ G @ B5 ) )
=> ( ( ord_le3507040750410214029t_unit @ A4 @ B5 )
=> ( ! [X: product_unit] :
( ( member_Product_unit @ X @ A4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_int @ ( complete_Sup_Sup_int @ ( image_873079544045703924it_int @ F @ A4 ) ) @ ( complete_Sup_Sup_int @ ( image_873079544045703924it_int @ G @ B5 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_3487_cSUP__subset__mono,axiom,
! [A4: set_o,G: $o > int,B5: set_o,F: $o > int] :
( ( A4 != bot_bot_set_o )
=> ( ( condit2212336002400062152ve_int @ ( image_o_int @ G @ B5 ) )
=> ( ( ord_less_eq_set_o @ A4 @ B5 )
=> ( ! [X: $o] :
( ( member_o @ X @ A4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_int @ ( complete_Sup_Sup_int @ ( image_o_int @ F @ A4 ) ) @ ( complete_Sup_Sup_int @ ( image_o_int @ G @ B5 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_3488_cSUP__subset__mono,axiom,
! [A4: set_nat,G: nat > int,B5: set_nat,F: nat > int] :
( ( A4 != bot_bot_set_nat )
=> ( ( condit2212336002400062152ve_int @ ( image_nat_int @ G @ B5 ) )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_int @ ( complete_Sup_Sup_int @ ( image_nat_int @ F @ A4 ) ) @ ( complete_Sup_Sup_int @ ( image_nat_int @ G @ B5 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_3489_cSUP__subset__mono,axiom,
! [A4: set_int,G: int > product_unit,B5: set_int,F: int > product_unit] :
( ( A4 != bot_bot_set_int )
=> ( ( condit2080475194861589617t_unit @ ( image_5846526143423701914t_unit @ G @ B5 ) )
=> ( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( ! [X: int] :
( ( member_int @ X @ A4 )
=> ( ord_le3221252021190050221t_unit @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_le3221252021190050221t_unit @ ( comple4687483117567863418t_unit @ ( image_5846526143423701914t_unit @ F @ A4 ) ) @ ( comple4687483117567863418t_unit @ ( image_5846526143423701914t_unit @ G @ B5 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_3490_cSUP__subset__mono,axiom,
! [A4: set_Product_unit,G: product_unit > product_unit,B5: set_Product_unit,F: product_unit > product_unit] :
( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ( condit2080475194861589617t_unit @ ( image_405062704495631173t_unit @ G @ B5 ) )
=> ( ( ord_le3507040750410214029t_unit @ A4 @ B5 )
=> ( ! [X: product_unit] :
( ( member_Product_unit @ X @ A4 )
=> ( ord_le3221252021190050221t_unit @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_le3221252021190050221t_unit @ ( comple4687483117567863418t_unit @ ( image_405062704495631173t_unit @ F @ A4 ) ) @ ( comple4687483117567863418t_unit @ ( image_405062704495631173t_unit @ G @ B5 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_3491_UNION__fun__upd,axiom,
! [A4: int > set_Product_unit,I: int,B5: set_Product_unit,J3: set_int] :
( ( comple3349607820504768346t_unit @ ( image_8658272795163236474t_unit @ ( fun_up5054257004045450754t_unit @ A4 @ I @ B5 ) @ J3 ) )
= ( sup_su793286257634532545t_unit @ ( comple3349607820504768346t_unit @ ( image_8658272795163236474t_unit @ A4 @ ( minus_minus_set_int @ J3 @ ( insert_int @ I @ bot_bot_set_int ) ) ) ) @ ( if_set_Product_unit @ ( member_int @ I @ J3 ) @ B5 @ bot_bo3957492148770167129t_unit ) ) ) ).
% UNION_fun_upd
thf(fact_3492_UNION__fun__upd,axiom,
! [A4: int > set_o,I: int,B5: set_o,J3: set_int] :
( ( comple90263536869209701_set_o @ ( image_int_set_o @ ( fun_upd_int_set_o @ A4 @ I @ B5 ) @ J3 ) )
= ( sup_sup_set_o @ ( comple90263536869209701_set_o @ ( image_int_set_o @ A4 @ ( minus_minus_set_int @ J3 @ ( insert_int @ I @ bot_bot_set_int ) ) ) ) @ ( if_set_o @ ( member_int @ I @ J3 ) @ B5 @ bot_bot_set_o ) ) ) ).
% UNION_fun_upd
thf(fact_3493_UNION__fun__upd,axiom,
! [A4: product_unit > set_Product_unit,I: product_unit,B5: set_Product_unit,J3: set_Product_unit] :
( ( comple3349607820504768346t_unit @ ( image_7778121751606633509t_unit @ ( fun_up488859312337516701t_unit @ A4 @ I @ B5 ) @ J3 ) )
= ( sup_su793286257634532545t_unit @ ( comple3349607820504768346t_unit @ ( image_7778121751606633509t_unit @ A4 @ ( minus_6452836326544984404t_unit @ J3 @ ( insert_Product_unit @ I @ bot_bo3957492148770167129t_unit ) ) ) ) @ ( if_set_Product_unit @ ( member_Product_unit @ I @ J3 ) @ B5 @ bot_bo3957492148770167129t_unit ) ) ) ).
% UNION_fun_upd
thf(fact_3494_UNION__fun__upd,axiom,
! [A4: product_unit > set_o,I: product_unit,B5: set_o,J3: set_Product_unit] :
( ( comple90263536869209701_set_o @ ( image_6930536745178679216_set_o @ ( fun_up1762901246852280360_set_o @ A4 @ I @ B5 ) @ J3 ) )
= ( sup_sup_set_o @ ( comple90263536869209701_set_o @ ( image_6930536745178679216_set_o @ A4 @ ( minus_6452836326544984404t_unit @ J3 @ ( insert_Product_unit @ I @ bot_bo3957492148770167129t_unit ) ) ) ) @ ( if_set_o @ ( member_Product_unit @ I @ J3 ) @ B5 @ bot_bot_set_o ) ) ) ).
% UNION_fun_upd
thf(fact_3495_UNION__fun__upd,axiom,
! [A4: $o > set_Product_unit,I: $o,B5: set_Product_unit,J3: set_o] :
( ( comple3349607820504768346t_unit @ ( image_663195431845683034t_unit @ ( fun_up4718931970374059986t_unit @ A4 @ I @ B5 ) @ J3 ) )
= ( sup_su793286257634532545t_unit @ ( comple3349607820504768346t_unit @ ( image_663195431845683034t_unit @ A4 @ ( minus_minus_set_o @ J3 @ ( insert_o @ I @ bot_bot_set_o ) ) ) ) @ ( if_set_Product_unit @ ( member_o @ I @ J3 ) @ B5 @ bot_bo3957492148770167129t_unit ) ) ) ).
% UNION_fun_upd
thf(fact_3496_UNION__fun__upd,axiom,
! [A4: $o > set_o,I: $o,B5: set_o,J3: set_o] :
( ( comple90263536869209701_set_o @ ( image_o_set_o @ ( fun_upd_o_set_o @ A4 @ I @ B5 ) @ J3 ) )
= ( sup_sup_set_o @ ( comple90263536869209701_set_o @ ( image_o_set_o @ A4 @ ( minus_minus_set_o @ J3 @ ( insert_o @ I @ bot_bot_set_o ) ) ) ) @ ( if_set_o @ ( member_o @ I @ J3 ) @ B5 @ bot_bot_set_o ) ) ) ).
% UNION_fun_upd
thf(fact_3497_UNION__fun__upd,axiom,
! [A4: nat > set_Product_unit,I: nat,B5: set_Product_unit,J3: set_nat] :
( ( comple3349607820504768346t_unit @ ( image_4824033694279209942t_unit @ ( fun_up1220017903161424222t_unit @ A4 @ I @ B5 ) @ J3 ) )
= ( sup_su793286257634532545t_unit @ ( comple3349607820504768346t_unit @ ( image_4824033694279209942t_unit @ A4 @ ( minus_minus_set_nat @ J3 @ ( insert_nat @ I @ bot_bot_set_nat ) ) ) ) @ ( if_set_Product_unit @ ( member_nat @ I @ J3 ) @ B5 @ bot_bo3957492148770167129t_unit ) ) ) ).
% UNION_fun_upd
thf(fact_3498_UNION__fun__upd,axiom,
! [A4: nat > set_o,I: nat,B5: set_o,J3: set_nat] :
( ( comple90263536869209701_set_o @ ( image_nat_set_o @ ( fun_upd_nat_set_o @ A4 @ I @ B5 ) @ J3 ) )
= ( sup_sup_set_o @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ A4 @ ( minus_minus_set_nat @ J3 @ ( insert_nat @ I @ bot_bot_set_nat ) ) ) ) @ ( if_set_o @ ( member_nat @ I @ J3 ) @ B5 @ bot_bot_set_o ) ) ) ).
% UNION_fun_upd
thf(fact_3499_UNION__fun__upd,axiom,
! [A4: int > set_nat,I: int,B5: set_nat,J3: set_int] :
( ( comple7399068483239264473et_nat @ ( image_int_set_nat @ ( fun_upd_int_set_nat @ A4 @ I @ B5 ) @ J3 ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_int_set_nat @ A4 @ ( minus_minus_set_int @ J3 @ ( insert_int @ I @ bot_bot_set_int ) ) ) ) @ ( if_set_nat @ ( member_int @ I @ J3 ) @ B5 @ bot_bot_set_nat ) ) ) ).
% UNION_fun_upd
thf(fact_3500_UNION__fun__upd,axiom,
! [A4: product_unit > set_nat,I: product_unit,B5: set_nat,J3: set_Product_unit] :
( ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ ( fun_up4569834200710200790et_nat @ A4 @ I @ B5 ) @ J3 ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ A4 @ ( minus_6452836326544984404t_unit @ J3 @ ( insert_Product_unit @ I @ bot_bo3957492148770167129t_unit ) ) ) ) @ ( if_set_nat @ ( member_Product_unit @ I @ J3 ) @ B5 @ bot_bot_set_nat ) ) ) ).
% UNION_fun_upd
thf(fact_3501_image__split__eq__Sigma,axiom,
! [F: nat > rat,G: nat > rat,A4: set_nat] :
( ( image_2021340605491722874at_rat
@ ^ [X4: nat] : ( product_Pair_rat_rat @ ( F @ X4 ) @ ( G @ X4 ) )
@ A4 )
= ( produc8728714639637146746at_rat @ ( image_nat_rat @ F @ A4 )
@ ^ [X4: rat] : ( image_nat_rat @ G @ ( inf_inf_set_nat @ ( vimage_nat_rat @ F @ ( insert_rat @ X4 @ bot_bot_set_rat ) ) @ A4 ) ) ) ) ).
% image_split_eq_Sigma
thf(fact_3502_image__split__eq__Sigma,axiom,
! [F: nat > rat,G: nat > nat,A4: set_nat] :
( ( image_5529569163885119602at_nat
@ ^ [X4: nat] : ( product_Pair_rat_nat @ ( F @ X4 ) @ ( G @ X4 ) )
@ A4 )
= ( produc140472662868866674at_nat @ ( image_nat_rat @ F @ A4 )
@ ^ [X4: rat] : ( image_nat_nat @ G @ ( inf_inf_set_nat @ ( vimage_nat_rat @ F @ ( insert_rat @ X4 @ bot_bot_set_rat ) ) @ A4 ) ) ) ) ).
% image_split_eq_Sigma
thf(fact_3503_image__split__eq__Sigma,axiom,
! [F: nat > rat,G: nat > int,A4: set_nat] :
( ( image_1351718144375922894at_int
@ ^ [X4: nat] : ( product_Pair_rat_int @ ( F @ X4 ) @ ( G @ X4 ) )
@ A4 )
= ( produc137982192359816398at_int @ ( image_nat_rat @ F @ A4 )
@ ^ [X4: rat] : ( image_nat_int @ G @ ( inf_inf_set_nat @ ( vimage_nat_rat @ F @ ( insert_rat @ X4 @ bot_bot_set_rat ) ) @ A4 ) ) ) ) ).
% image_split_eq_Sigma
thf(fact_3504_image__split__eq__Sigma,axiom,
! [F: nat > int,G: nat > rat,A4: set_nat] :
( ( image_3337248961327643342nt_rat
@ ^ [X4: nat] : ( product_Pair_int_rat @ ( F @ X4 ) @ ( G @ X4 ) )
@ A4 )
= ( produc821250958618291406nt_rat @ ( image_nat_int @ F @ A4 )
@ ^ [X4: int] : ( image_nat_rat @ G @ ( inf_inf_set_nat @ ( vimage_nat_int @ F @ ( insert_int @ X4 @ bot_bot_set_int ) ) @ A4 ) ) ) ) ).
% image_split_eq_Sigma
thf(fact_3505_image__split__eq__Sigma,axiom,
! [F: nat > int,G: nat > nat,A4: set_nat] :
( ( image_6845477519721040070nt_nat
@ ^ [X4: nat] : ( product_Pair_int_nat @ ( F @ X4 ) @ ( G @ X4 ) )
@ A4 )
= ( produc1456381018704787142nt_nat @ ( image_nat_int @ F @ A4 )
@ ^ [X4: int] : ( image_nat_nat @ G @ ( inf_inf_set_nat @ ( vimage_nat_int @ F @ ( insert_int @ X4 @ bot_bot_set_int ) ) @ A4 ) ) ) ) ).
% image_split_eq_Sigma
thf(fact_3506_image__split__eq__Sigma,axiom,
! [F: int > int,G: int > nat,A4: set_int] :
( ( image_659947567330398058nt_nat
@ ^ [X4: int] : ( product_Pair_int_nat @ ( F @ X4 ) @ ( G @ X4 ) )
@ A4 )
= ( produc1456381018704787142nt_nat @ ( image_int_int @ F @ A4 )
@ ^ [X4: int] : ( image_int_nat @ G @ ( inf_inf_set_int @ ( vimage_int_int @ F @ ( insert_int @ X4 @ bot_bot_set_int ) ) @ A4 ) ) ) ) ).
% image_split_eq_Sigma
thf(fact_3507_image__split__eq__Sigma,axiom,
! [F: nat > int,G: nat > int,A4: set_nat] :
( ( image_2667626500211843362nt_int
@ ^ [X4: nat] : ( product_Pair_int_int @ ( F @ X4 ) @ ( G @ X4 ) )
@ A4 )
= ( produc1453890548195736866nt_int @ ( image_nat_int @ F @ A4 )
@ ^ [X4: int] : ( image_nat_int @ G @ ( inf_inf_set_nat @ ( vimage_nat_int @ F @ ( insert_int @ X4 @ bot_bot_set_int ) ) @ A4 ) ) ) ) ).
% image_split_eq_Sigma
thf(fact_3508_image__split__eq__Sigma,axiom,
! [F: int > int,G: int > int,A4: set_int] :
( ( image_5705468584675977158nt_int
@ ^ [X4: int] : ( product_Pair_int_int @ ( F @ X4 ) @ ( G @ X4 ) )
@ A4 )
= ( produc1453890548195736866nt_int @ ( image_int_int @ F @ A4 )
@ ^ [X4: int] : ( image_int_int @ G @ ( inf_inf_set_int @ ( vimage_int_int @ F @ ( insert_int @ X4 @ bot_bot_set_int ) ) @ A4 ) ) ) ) ).
% image_split_eq_Sigma
thf(fact_3509_image__split__eq__Sigma,axiom,
! [F: nat > product_unit,G: nat > rat,A4: set_nat] :
( ( image_469905907365485675it_rat
@ ^ [X4: nat] : ( produc4168158269727034510it_rat @ ( F @ X4 ) @ ( G @ X4 ) )
@ A4 )
= ( produc6240246942209516957it_rat @ ( image_8730104196221521654t_unit @ F @ A4 )
@ ^ [X4: product_unit] : ( image_nat_rat @ G @ ( inf_inf_set_nat @ ( vimage4884490618288580032t_unit @ F @ ( insert_Product_unit @ X4 @ bot_bo3957492148770167129t_unit ) ) @ A4 ) ) ) ) ).
% image_split_eq_Sigma
thf(fact_3510_image__split__eq__Sigma,axiom,
! [F: nat > product_unit,G: nat > nat,A4: set_nat] :
( ( image_3978134465758882403it_nat
@ ^ [X4: nat] : ( produc4803288329813530246it_nat @ ( F @ X4 ) @ ( G @ X4 ) )
@ A4 )
= ( produc6875377002296012693it_nat @ ( image_8730104196221521654t_unit @ F @ A4 )
@ ^ [X4: product_unit] : ( image_nat_nat @ G @ ( inf_inf_set_nat @ ( vimage4884490618288580032t_unit @ F @ ( insert_Product_unit @ X4 @ bot_bo3957492148770167129t_unit ) ) @ A4 ) ) ) ) ).
% image_split_eq_Sigma
thf(fact_3511_If__the__inv__into__f__f,axiom,
! [I: num,C3: set_num,G: num > $o,X2: num] :
( ( member_num @ I @ C3 )
=> ( ( inj_on_num_o @ G @ C3 )
=> ( ( comp_o_num_num
@ ^ [I2: $o] : ( if_num @ ( member_o @ I2 @ ( image_num_o @ G @ C3 ) ) @ ( the_inv_into_num_o @ C3 @ G @ I2 ) @ X2 )
@ G
@ I )
= ( id_num @ I ) ) ) ) ).
% If_the_inv_into_f_f
thf(fact_3512_If__the__inv__into__f__f,axiom,
! [I: num,C3: set_num,G: num > nat,X2: num] :
( ( member_num @ I @ C3 )
=> ( ( inj_on_num_nat @ G @ C3 )
=> ( ( comp_nat_num_num
@ ^ [I2: nat] : ( if_num @ ( member_nat @ I2 @ ( image_num_nat @ G @ C3 ) ) @ ( the_inv_into_num_nat @ C3 @ G @ I2 ) @ X2 )
@ G
@ I )
= ( id_num @ I ) ) ) ) ).
% If_the_inv_into_f_f
thf(fact_3513_If__the__inv__into__f__f,axiom,
! [I: num,C3: set_num,G: num > int,X2: num] :
( ( member_num @ I @ C3 )
=> ( ( inj_on_num_int @ G @ C3 )
=> ( ( comp_int_num_num
@ ^ [I2: int] : ( if_num @ ( member_int @ I2 @ ( image_num_int @ G @ C3 ) ) @ ( the_inv_into_num_int @ C3 @ G @ I2 ) @ X2 )
@ G
@ I )
= ( id_num @ I ) ) ) ) ).
% If_the_inv_into_f_f
thf(fact_3514_If__the__inv__into__f__f,axiom,
! [I: $o,C3: set_o,G: $o > $o,X2: $o] :
( ( member_o @ I @ C3 )
=> ( ( inj_on_o_o @ G @ C3 )
=> ( ( comp_o_o_o
@ ^ [I2: $o] :
( ( ( member_o @ I2 @ ( image_o_o @ G @ C3 ) )
=> ( the_inv_into_o_o @ C3 @ G @ I2 ) )
& ( ~ ( member_o @ I2 @ ( image_o_o @ G @ C3 ) )
=> X2 ) )
@ G
@ I )
= ( id_o @ I ) ) ) ) ).
% If_the_inv_into_f_f
thf(fact_3515_If__the__inv__into__f__f,axiom,
! [I: $o,C3: set_o,G: $o > nat,X2: $o] :
( ( member_o @ I @ C3 )
=> ( ( inj_on_o_nat @ G @ C3 )
=> ( ( comp_nat_o_o
@ ^ [I2: nat] :
( ( ( member_nat @ I2 @ ( image_o_nat @ G @ C3 ) )
=> ( the_inv_into_o_nat @ C3 @ G @ I2 ) )
& ( ~ ( member_nat @ I2 @ ( image_o_nat @ G @ C3 ) )
=> X2 ) )
@ G
@ I )
= ( id_o @ I ) ) ) ) ).
% If_the_inv_into_f_f
thf(fact_3516_If__the__inv__into__f__f,axiom,
! [I: $o,C3: set_o,G: $o > int,X2: $o] :
( ( member_o @ I @ C3 )
=> ( ( inj_on_o_int @ G @ C3 )
=> ( ( comp_int_o_o
@ ^ [I2: int] :
( ( ( member_int @ I2 @ ( image_o_int @ G @ C3 ) )
=> ( the_inv_into_o_int @ C3 @ G @ I2 ) )
& ( ~ ( member_int @ I2 @ ( image_o_int @ G @ C3 ) )
=> X2 ) )
@ G
@ I )
= ( id_o @ I ) ) ) ) ).
% If_the_inv_into_f_f
thf(fact_3517_If__the__inv__into__f__f,axiom,
! [I: nat,C3: set_nat,G: nat > rat,X2: nat] :
( ( member_nat @ I @ C3 )
=> ( ( inj_on_nat_rat @ G @ C3 )
=> ( ( comp_rat_nat_nat
@ ^ [I2: rat] : ( if_nat @ ( member_rat @ I2 @ ( image_nat_rat @ G @ C3 ) ) @ ( the_inv_into_nat_rat @ C3 @ G @ I2 ) @ X2 )
@ G
@ I )
= ( id_nat @ I ) ) ) ) ).
% If_the_inv_into_f_f
thf(fact_3518_If__the__inv__into__f__f,axiom,
! [I: nat,C3: set_nat,G: nat > $o,X2: nat] :
( ( member_nat @ I @ C3 )
=> ( ( inj_on_nat_o @ G @ C3 )
=> ( ( comp_o_nat_nat
@ ^ [I2: $o] : ( if_nat @ ( member_o @ I2 @ ( image_nat_o @ G @ C3 ) ) @ ( the_inv_into_nat_o @ C3 @ G @ I2 ) @ X2 )
@ G
@ I )
= ( id_nat @ I ) ) ) ) ).
% If_the_inv_into_f_f
thf(fact_3519_If__the__inv__into__f__f,axiom,
! [I: nat,C3: set_nat,G: nat > nat,X2: nat] :
( ( member_nat @ I @ C3 )
=> ( ( inj_on_nat_nat @ G @ C3 )
=> ( ( comp_nat_nat_nat
@ ^ [I2: nat] : ( if_nat @ ( member_nat @ I2 @ ( image_nat_nat @ G @ C3 ) ) @ ( the_inv_into_nat_nat @ C3 @ G @ I2 ) @ X2 )
@ G
@ I )
= ( id_nat @ I ) ) ) ) ).
% If_the_inv_into_f_f
thf(fact_3520_If__the__inv__into__f__f,axiom,
! [I: nat,C3: set_nat,G: nat > int,X2: nat] :
( ( member_nat @ I @ C3 )
=> ( ( inj_on_nat_int @ G @ C3 )
=> ( ( comp_int_nat_nat
@ ^ [I2: int] : ( if_nat @ ( member_int @ I2 @ ( image_nat_int @ G @ C3 ) ) @ ( the_inv_into_nat_int @ C3 @ G @ I2 ) @ X2 )
@ G
@ I )
= ( id_nat @ I ) ) ) ) ).
% If_the_inv_into_f_f
thf(fact_3521_Field__insert,axiom,
! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,R: set_Pr7459493094073627847at_nat] :
( ( field_1428494348145443988at_nat @ ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ A @ B ) @ R ) )
= ( sup_su474284620922681339at_nat @ ( insert2290795007606087959at_nat @ A @ ( insert2290795007606087959at_nat @ B @ bot_bo1534825418459847827at_nat ) ) @ ( field_1428494348145443988at_nat @ R ) ) ) ).
% Field_insert
thf(fact_3522_Field__insert,axiom,
! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat] :
( ( field_3145993188331922222at_nat @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R ) )
= ( sup_su3642409539654194069at_nat @ ( insert9200635055090092081at_nat @ A @ ( insert9200635055090092081at_nat @ B @ bot_bo3083307316010499117at_nat ) ) @ ( field_3145993188331922222at_nat @ R ) ) ) ).
% Field_insert
thf(fact_3523_Field__insert,axiom,
! [A: int,B: int,R: set_Pr958786334691620121nt_int] :
( ( field_int @ ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ A @ B ) @ R ) )
= ( sup_sup_set_int @ ( insert_int @ A @ ( insert_int @ B @ bot_bot_set_int ) ) @ ( field_int @ R ) ) ) ).
% Field_insert
thf(fact_3524_Field__insert,axiom,
! [A: produc859450856879609959at_nat,B: produc859450856879609959at_nat,R: set_Pr553994874890374343at_nat] :
( ( field_3044959742847878580at_nat @ ( insert5475735164933346999at_nat @ ( produc4662710985925991255at_nat @ A @ B ) @ R ) )
= ( sup_su718114333110466843at_nat @ ( insert5050368324300391991at_nat @ A @ ( insert5050368324300391991at_nat @ B @ bot_bo5327735625951526323at_nat ) ) @ ( field_3044959742847878580at_nat @ R ) ) ) ).
% Field_insert
thf(fact_3525_Field__insert,axiom,
! [A: product_unit,B: product_unit,R: set_Pr5094982260447487303t_unit] :
( ( field_Product_unit @ ( insert4616891024742550839t_unit @ ( produc5692694829384537815t_unit @ A @ B ) @ R ) )
= ( sup_su793286257634532545t_unit @ ( insert_Product_unit @ A @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) @ ( field_Product_unit @ R ) ) ) ).
% Field_insert
thf(fact_3526_Field__insert,axiom,
! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat,R: set_Pr5564308138774400199at_nat] :
( ( field_2862844103579723188at_nat @ ( insert398606708087465655at_nat @ ( produc2335810615602371415at_nat @ A @ B ) @ R ) )
= ( sup_su5525570899277871387at_nat @ ( insert9069300056098147895at_nat @ A @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) ) @ ( field_2862844103579723188at_nat @ R ) ) ) ).
% Field_insert
thf(fact_3527_Field__insert,axiom,
! [A: product_prod_nat_nat,B: product_prod_nat_nat,R: set_Pr8693737435421807431at_nat] :
( ( field_4706345496283612622at_nat @ ( insert5050368324300391991at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R ) )
= ( sup_su6327502436637775413at_nat @ ( insert8211810215607154385at_nat @ A @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) ) @ ( field_4706345496283612622at_nat @ R ) ) ) ).
% Field_insert
thf(fact_3528_Field__insert,axiom,
! [A: $o,B: $o,R: set_Product_prod_o_o] :
( ( field_o @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ A @ B ) @ R ) )
= ( sup_sup_set_o @ ( insert_o @ A @ ( insert_o @ B @ bot_bot_set_o ) ) @ ( field_o @ R ) ) ) ).
% Field_insert
thf(fact_3529_Field__insert,axiom,
! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat] :
( ( field_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R ) )
= ( sup_sup_set_nat @ ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) ) @ ( field_nat @ R ) ) ) ).
% Field_insert
thf(fact_3530_bdd__below__image__antimono,axiom,
! [F: nat > rat,A4: set_nat] :
( ( order_2045666486212196060at_rat @ F )
=> ( ( condit2214826472909112428ve_nat @ A4 )
=> ( condit1103211067700513672ow_rat @ ( image_nat_rat @ F @ A4 ) ) ) ) ).
% bdd_below_image_antimono
thf(fact_3531_bdd__below__image__antimono,axiom,
! [F: nat > nat,A4: set_nat] :
( ( order_2680796546298691796at_nat @ F )
=> ( ( condit2214826472909112428ve_nat @ A4 )
=> ( condit1738341127787009408ow_nat @ ( image_nat_nat @ F @ A4 ) ) ) ) ).
% bdd_below_image_antimono
thf(fact_3532_bdd__below__image__antimono,axiom,
! [F: nat > int,A4: set_nat] :
( ( order_2678306075789641520at_int @ F )
=> ( ( condit2214826472909112428ve_nat @ A4 )
=> ( condit1735850657277959132ow_int @ ( image_nat_int @ F @ A4 ) ) ) ) ).
% bdd_below_image_antimono
thf(fact_3533_bdd__below__image__antimono,axiom,
! [F: int > nat,A4: set_int] :
( ( order_3680150258199746352nt_nat @ F )
=> ( ( condit2212336002400062152ve_int @ A4 )
=> ( condit1738341127787009408ow_nat @ ( image_int_nat @ F @ A4 ) ) ) ) ).
% bdd_below_image_antimono
thf(fact_3534_bdd__below__image__antimono,axiom,
! [F: int > int,A4: set_int] :
( ( order_3677659787690696076nt_int @ F )
=> ( ( condit2212336002400062152ve_int @ A4 )
=> ( condit1735850657277959132ow_int @ ( image_int_int @ F @ A4 ) ) ) ) ).
% bdd_below_image_antimono
thf(fact_3535_bdd__above__image__antimono,axiom,
! [F: nat > rat,A4: set_nat] :
( ( order_2045666486212196060at_rat @ F )
=> ( ( condit1738341127787009408ow_nat @ A4 )
=> ( condit1579696412822616692ve_rat @ ( image_nat_rat @ F @ A4 ) ) ) ) ).
% bdd_above_image_antimono
thf(fact_3536_bdd__above__image__antimono,axiom,
! [F: nat > nat,A4: set_nat] :
( ( order_2680796546298691796at_nat @ F )
=> ( ( condit1738341127787009408ow_nat @ A4 )
=> ( condit2214826472909112428ve_nat @ ( image_nat_nat @ F @ A4 ) ) ) ) ).
% bdd_above_image_antimono
thf(fact_3537_bdd__above__image__antimono,axiom,
! [F: nat > int,A4: set_nat] :
( ( order_2678306075789641520at_int @ F )
=> ( ( condit1738341127787009408ow_nat @ A4 )
=> ( condit2212336002400062152ve_int @ ( image_nat_int @ F @ A4 ) ) ) ) ).
% bdd_above_image_antimono
thf(fact_3538_bdd__above__image__antimono,axiom,
! [F: int > nat,A4: set_int] :
( ( order_3680150258199746352nt_nat @ F )
=> ( ( condit1735850657277959132ow_int @ A4 )
=> ( condit2214826472909112428ve_nat @ ( image_int_nat @ F @ A4 ) ) ) ) ).
% bdd_above_image_antimono
thf(fact_3539_bdd__above__image__antimono,axiom,
! [F: int > int,A4: set_int] :
( ( order_3677659787690696076nt_int @ F )
=> ( ( condit1735850657277959132ow_int @ A4 )
=> ( condit2212336002400062152ve_int @ ( image_int_int @ F @ A4 ) ) ) ) ).
% bdd_above_image_antimono
thf(fact_3540_the__inv__f__o__f__id,axiom,
! [F: nat > nat,Z2: nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( comp_nat_nat_nat @ ( the_inv_into_nat_nat @ top_top_set_nat @ F ) @ F @ Z2 )
= ( id_nat @ Z2 ) ) ) ).
% the_inv_f_o_f_id
thf(fact_3541_If__the__inv__into__in__Func,axiom,
! [G: product_unit > $o,C3: set_Product_unit,B5: set_Product_unit,X2: product_unit] :
( ( inj_on8854722734121536060unit_o @ G @ C3 )
=> ( ( ord_le3507040750410214029t_unit @ C3 @ ( sup_su793286257634532545t_unit @ B5 @ ( insert_Product_unit @ X2 @ bot_bo3957492148770167129t_unit ) ) )
=> ( member2370919827131729009t_unit
@ ^ [I2: $o] : ( if_Product_unit @ ( member_o @ I2 @ ( image_Product_unit_o @ G @ C3 ) ) @ ( the_in43325605794470270unit_o @ C3 @ G @ I2 ) @ X2 )
@ ( bNF_We8730861560085513481t_unit @ top_top_set_o @ ( sup_su793286257634532545t_unit @ B5 @ ( insert_Product_unit @ X2 @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ).
% If_the_inv_into_in_Func
thf(fact_3542_If__the__inv__into__in__Func,axiom,
! [G: $o > $o,C3: set_o,B5: set_o,X2: $o] :
( ( inj_on_o_o @ G @ C3 )
=> ( ( ord_less_eq_set_o @ C3 @ ( sup_sup_set_o @ B5 @ ( insert_o @ X2 @ bot_bot_set_o ) ) )
=> ( member_o_o
@ ^ [I2: $o] :
( ( ( member_o @ I2 @ ( image_o_o @ G @ C3 ) )
=> ( the_inv_into_o_o @ C3 @ G @ I2 ) )
& ( ~ ( member_o @ I2 @ ( image_o_o @ G @ C3 ) )
=> X2 ) )
@ ( bNF_We7690717809131995284nc_o_o @ top_top_set_o @ ( sup_sup_set_o @ B5 @ ( insert_o @ X2 @ bot_bot_set_o ) ) ) ) ) ) ).
% If_the_inv_into_in_Func
thf(fact_3543_If__the__inv__into__in__Func,axiom,
! [G: nat > $o,C3: set_nat,B5: set_nat,X2: nat] :
( ( inj_on_nat_o @ G @ C3 )
=> ( ( ord_less_eq_set_nat @ C3 @ ( sup_sup_set_nat @ B5 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) )
=> ( member_o_nat
@ ^ [I2: $o] : ( if_nat @ ( member_o @ I2 @ ( image_nat_o @ G @ C3 ) ) @ ( the_inv_into_nat_o @ C3 @ G @ I2 ) @ X2 )
@ ( bNF_We6830391624356636116_o_nat @ top_top_set_o @ ( sup_sup_set_nat @ B5 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ) ).
% If_the_inv_into_in_Func
thf(fact_3544_If__the__inv__into__in__Func,axiom,
! [G: product_unit > char,C3: set_Product_unit,B5: set_Product_unit,X2: product_unit] :
( ( inj_on1959294024837303853t_char @ G @ C3 )
=> ( ( ord_le3507040750410214029t_unit @ C3 @ ( sup_su793286257634532545t_unit @ B5 @ ( insert_Product_unit @ X2 @ bot_bo3957492148770167129t_unit ) ) )
=> ( member379592906918970460t_unit
@ ^ [I2: char] : ( if_Product_unit @ ( member_char @ I2 @ ( image_7248553556944771225t_char @ G @ C3 ) ) @ ( the_in3194898024646563179t_char @ C3 @ G @ I2 ) @ X2 )
@ ( bNF_We7396736742984212070t_unit @ top_top_set_char @ ( sup_su793286257634532545t_unit @ B5 @ ( insert_Product_unit @ X2 @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ).
% If_the_inv_into_in_Func
thf(fact_3545_If__the__inv__into__in__Func,axiom,
! [G: $o > char,C3: set_o,B5: set_o,X2: $o] :
( ( inj_on_o_char @ G @ C3 )
=> ( ( ord_less_eq_set_o @ C3 @ ( sup_sup_set_o @ B5 @ ( insert_o @ X2 @ bot_bot_set_o ) ) )
=> ( member_char_o
@ ^ [I2: char] :
( ( ( member_char @ I2 @ ( image_o_char @ G @ C3 ) )
=> ( the_inv_into_o_char @ C3 @ G @ I2 ) )
& ( ~ ( member_char @ I2 @ ( image_o_char @ G @ C3 ) )
=> X2 ) )
@ ( bNF_We6174688282833725809char_o @ top_top_set_char @ ( sup_sup_set_o @ B5 @ ( insert_o @ X2 @ bot_bot_set_o ) ) ) ) ) ) ).
% If_the_inv_into_in_Func
thf(fact_3546_If__the__inv__into__in__Func,axiom,
! [G: nat > char,C3: set_nat,B5: set_nat,X2: nat] :
( ( inj_on_nat_char @ G @ C3 )
=> ( ( ord_less_eq_set_nat @ C3 @ ( sup_sup_set_nat @ B5 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) )
=> ( member_char_nat
@ ^ [I2: char] : ( if_nat @ ( member_char @ I2 @ ( image_nat_char @ G @ C3 ) ) @ ( the_in9023905271258939222t_char @ C3 @ G @ I2 ) @ X2 )
@ ( bNF_We935846424581991991ar_nat @ top_top_set_char @ ( sup_sup_set_nat @ B5 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ) ).
% If_the_inv_into_in_Func
thf(fact_3547_If__the__inv__into__in__Func,axiom,
! [G: product_unit > rat,C3: set_Product_unit,B5: set_Product_unit,X2: product_unit] :
( ( inj_on7795309031694339124it_rat @ G @ C3 )
=> ( ( ord_le3507040750410214029t_unit @ C3 @ ( sup_su793286257634532545t_unit @ B5 @ ( insert_Product_unit @ X2 @ bot_bo3957492148770167129t_unit ) ) )
=> ( member7011034171605722901t_unit
@ ^ [I2: rat] : ( if_Product_unit @ ( member_rat @ I2 @ ( image_240439954468258464it_rat @ G @ C3 ) ) @ ( the_in3915408195775746418it_rat @ C3 @ G @ I2 ) @ X2 )
@ ( bNF_We8564238559827753503t_unit @ top_top_set_rat @ ( sup_su793286257634532545t_unit @ B5 @ ( insert_Product_unit @ X2 @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ).
% If_the_inv_into_in_Func
thf(fact_3548_If__the__inv__into__in__Func,axiom,
! [G: $o > rat,C3: set_o,B5: set_o,X2: $o] :
( ( inj_on_o_rat @ G @ C3 )
=> ( ( ord_less_eq_set_o @ C3 @ ( sup_sup_set_o @ B5 @ ( insert_o @ X2 @ bot_bot_set_o ) ) )
=> ( member_rat_o
@ ^ [I2: rat] :
( ( ( member_rat @ I2 @ ( image_o_rat @ G @ C3 ) )
=> ( the_inv_into_o_rat @ C3 @ G @ I2 ) )
& ( ~ ( member_rat @ I2 @ ( image_o_rat @ G @ C3 ) )
=> X2 ) )
@ ( bNF_We6195730058148253354_rat_o @ top_top_set_rat @ ( sup_sup_set_o @ B5 @ ( insert_o @ X2 @ bot_bot_set_o ) ) ) ) ) ) ).
% If_the_inv_into_in_Func
thf(fact_3549_If__the__inv__into__in__Func,axiom,
! [G: nat > rat,C3: set_nat,B5: set_nat,X2: nat] :
( ( inj_on_nat_rat @ G @ C3 )
=> ( ( ord_less_eq_set_nat @ C3 @ ( sup_sup_set_nat @ B5 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) )
=> ( member_rat_nat
@ ^ [I2: rat] : ( if_nat @ ( member_rat @ I2 @ ( image_nat_rat @ G @ C3 ) ) @ ( the_inv_into_nat_rat @ C3 @ G @ I2 ) @ X2 )
@ ( bNF_We7223962090624190782at_nat @ top_top_set_rat @ ( sup_sup_set_nat @ B5 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ) ).
% If_the_inv_into_in_Func
thf(fact_3550_If__the__inv__into__in__Func,axiom,
! [G: int > nat,C3: set_int,B5: set_int,X2: int] :
( ( inj_on_int_nat @ G @ C3 )
=> ( ( ord_less_eq_set_int @ C3 @ ( sup_sup_set_int @ B5 @ ( insert_int @ X2 @ bot_bot_set_int ) ) )
=> ( member_nat_int
@ ^ [I2: nat] : ( if_int @ ( member_nat @ I2 @ ( image_int_nat @ G @ C3 ) ) @ ( the_inv_into_int_nat @ C3 @ G @ I2 ) @ X2 )
@ ( bNF_We7538026264050006418at_int @ top_top_set_nat @ ( sup_sup_set_int @ B5 @ ( insert_int @ X2 @ bot_bot_set_int ) ) ) ) ) ) ).
% If_the_inv_into_in_Func
thf(fact_3551_mono__cInf,axiom,
! [F: $o > nat,A4: set_o] :
( ( order_mono_o_nat @ F )
=> ( ( condit5413489452508810728elow_o @ A4 )
=> ( ( A4 != bot_bot_set_o )
=> ( ord_less_eq_nat @ ( F @ ( complete_Inf_Inf_o @ A4 ) ) @ ( complete_Inf_Inf_nat @ ( image_o_nat @ F @ A4 ) ) ) ) ) ) ).
% mono_cInf
thf(fact_3552_mono__cInf,axiom,
! [F: nat > nat,A4: set_nat] :
( ( order_mono_nat_nat @ F )
=> ( ( condit1738341127787009408ow_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ord_less_eq_nat @ ( F @ ( complete_Inf_Inf_nat @ A4 ) ) @ ( complete_Inf_Inf_nat @ ( image_nat_nat @ F @ A4 ) ) ) ) ) ) ).
% mono_cInf
thf(fact_3553_mono__cInf,axiom,
! [F: $o > int,A4: set_o] :
( ( order_mono_o_int @ F )
=> ( ( condit5413489452508810728elow_o @ A4 )
=> ( ( A4 != bot_bot_set_o )
=> ( ord_less_eq_int @ ( F @ ( complete_Inf_Inf_o @ A4 ) ) @ ( complete_Inf_Inf_int @ ( image_o_int @ F @ A4 ) ) ) ) ) ) ).
% mono_cInf
thf(fact_3554_mono__cInf,axiom,
! [F: nat > int,A4: set_nat] :
( ( order_mono_nat_int @ F )
=> ( ( condit1738341127787009408ow_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ord_less_eq_int @ ( F @ ( complete_Inf_Inf_nat @ A4 ) ) @ ( complete_Inf_Inf_int @ ( image_nat_int @ F @ A4 ) ) ) ) ) ) ).
% mono_cInf
thf(fact_3555_mono__cInf,axiom,
! [F: $o > product_unit,A4: set_o] :
( ( order_7782127260464396005t_unit @ F )
=> ( ( condit5413489452508810728elow_o @ A4 )
=> ( ( A4 != bot_bot_set_o )
=> ( ord_le3221252021190050221t_unit @ ( F @ ( complete_Inf_Inf_o @ A4 ) ) @ ( comple2584293577114468500t_unit @ ( image_o_Product_unit @ F @ A4 ) ) ) ) ) ) ).
% mono_cInf
thf(fact_3556_mono__cInf,axiom,
! [F: nat > product_unit,A4: set_nat] :
( ( order_5727276828720722763t_unit @ F )
=> ( ( condit1738341127787009408ow_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ord_le3221252021190050221t_unit @ ( F @ ( complete_Inf_Inf_nat @ A4 ) ) @ ( comple2584293577114468500t_unit @ ( image_8730104196221521654t_unit @ F @ A4 ) ) ) ) ) ) ).
% mono_cInf
thf(fact_3557_mono__cInf,axiom,
! [F: int > nat,A4: set_int] :
( ( order_mono_int_nat @ F )
=> ( ( condit1735850657277959132ow_int @ A4 )
=> ( ( A4 != bot_bot_set_int )
=> ( ord_less_eq_nat @ ( F @ ( complete_Inf_Inf_int @ A4 ) ) @ ( complete_Inf_Inf_nat @ ( image_int_nat @ F @ A4 ) ) ) ) ) ) ).
% mono_cInf
thf(fact_3558_mono__cInf,axiom,
! [F: int > int,A4: set_int] :
( ( order_mono_int_int @ F )
=> ( ( condit1735850657277959132ow_int @ A4 )
=> ( ( A4 != bot_bot_set_int )
=> ( ord_less_eq_int @ ( F @ ( complete_Inf_Inf_int @ A4 ) ) @ ( complete_Inf_Inf_int @ ( image_int_int @ F @ A4 ) ) ) ) ) ) ).
% mono_cInf
thf(fact_3559_mono__cInf,axiom,
! [F: int > product_unit,A4: set_int] :
( ( order_2843698775922903023t_unit @ F )
=> ( ( condit1735850657277959132ow_int @ A4 )
=> ( ( A4 != bot_bot_set_int )
=> ( ord_le3221252021190050221t_unit @ ( F @ ( complete_Inf_Inf_int @ A4 ) ) @ ( comple2584293577114468500t_unit @ ( image_5846526143423701914t_unit @ F @ A4 ) ) ) ) ) ) ).
% mono_cInf
thf(fact_3560_mono__cInf,axiom,
! [F: product_unit > nat,A4: set_Product_unit] :
( ( order_7096114683908731117it_nat @ F )
=> ( ( condit9161856309968323677t_unit @ A4 )
=> ( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ord_less_eq_nat @ ( F @ ( comple2584293577114468500t_unit @ A4 ) ) @ ( complete_Inf_Inf_nat @ ( image_875570014554754200it_nat @ F @ A4 ) ) ) ) ) ) ).
% mono_cInf
thf(fact_3561_mono__cINF,axiom,
! [F: nat > nat,A4: int > nat,I3: set_int] :
( ( order_mono_nat_nat @ F )
=> ( ( condit1738341127787009408ow_nat @ ( image_int_nat @ A4 @ I3 ) )
=> ( ( I3 != bot_bot_set_int )
=> ( ord_less_eq_nat @ ( F @ ( complete_Inf_Inf_nat @ ( image_int_nat @ A4 @ I3 ) ) )
@ ( complete_Inf_Inf_nat
@ ( image_int_nat
@ ^ [X4: int] : ( F @ ( A4 @ X4 ) )
@ I3 ) ) ) ) ) ) ).
% mono_cINF
thf(fact_3562_mono__cINF,axiom,
! [F: nat > nat,A4: nat > nat,I3: set_nat] :
( ( order_mono_nat_nat @ F )
=> ( ( condit1738341127787009408ow_nat @ ( image_nat_nat @ A4 @ I3 ) )
=> ( ( I3 != bot_bot_set_nat )
=> ( ord_less_eq_nat @ ( F @ ( complete_Inf_Inf_nat @ ( image_nat_nat @ A4 @ I3 ) ) )
@ ( complete_Inf_Inf_nat
@ ( image_nat_nat
@ ^ [X4: nat] : ( F @ ( A4 @ X4 ) )
@ I3 ) ) ) ) ) ) ).
% mono_cINF
thf(fact_3563_mono__cINF,axiom,
! [F: nat > int,A4: int > nat,I3: set_int] :
( ( order_mono_nat_int @ F )
=> ( ( condit1738341127787009408ow_nat @ ( image_int_nat @ A4 @ I3 ) )
=> ( ( I3 != bot_bot_set_int )
=> ( ord_less_eq_int @ ( F @ ( complete_Inf_Inf_nat @ ( image_int_nat @ A4 @ I3 ) ) )
@ ( complete_Inf_Inf_int
@ ( image_int_int
@ ^ [X4: int] : ( F @ ( A4 @ X4 ) )
@ I3 ) ) ) ) ) ) ).
% mono_cINF
thf(fact_3564_mono__cINF,axiom,
! [F: nat > int,A4: nat > nat,I3: set_nat] :
( ( order_mono_nat_int @ F )
=> ( ( condit1738341127787009408ow_nat @ ( image_nat_nat @ A4 @ I3 ) )
=> ( ( I3 != bot_bot_set_nat )
=> ( ord_less_eq_int @ ( F @ ( complete_Inf_Inf_nat @ ( image_nat_nat @ A4 @ I3 ) ) )
@ ( complete_Inf_Inf_int
@ ( image_nat_int
@ ^ [X4: nat] : ( F @ ( A4 @ X4 ) )
@ I3 ) ) ) ) ) ) ).
% mono_cINF
thf(fact_3565_mono__cINF,axiom,
! [F: nat > product_unit,A4: int > nat,I3: set_int] :
( ( order_5727276828720722763t_unit @ F )
=> ( ( condit1738341127787009408ow_nat @ ( image_int_nat @ A4 @ I3 ) )
=> ( ( I3 != bot_bot_set_int )
=> ( ord_le3221252021190050221t_unit @ ( F @ ( complete_Inf_Inf_nat @ ( image_int_nat @ A4 @ I3 ) ) )
@ ( comple2584293577114468500t_unit
@ ( image_5846526143423701914t_unit
@ ^ [X4: int] : ( F @ ( A4 @ X4 ) )
@ I3 ) ) ) ) ) ) ).
% mono_cINF
thf(fact_3566_mono__cINF,axiom,
! [F: nat > product_unit,A4: nat > nat,I3: set_nat] :
( ( order_5727276828720722763t_unit @ F )
=> ( ( condit1738341127787009408ow_nat @ ( image_nat_nat @ A4 @ I3 ) )
=> ( ( I3 != bot_bot_set_nat )
=> ( ord_le3221252021190050221t_unit @ ( F @ ( complete_Inf_Inf_nat @ ( image_nat_nat @ A4 @ I3 ) ) )
@ ( comple2584293577114468500t_unit
@ ( image_8730104196221521654t_unit
@ ^ [X4: nat] : ( F @ ( A4 @ X4 ) )
@ I3 ) ) ) ) ) ) ).
% mono_cINF
thf(fact_3567_mono__cINF,axiom,
! [F: int > nat,A4: int > int,I3: set_int] :
( ( order_mono_int_nat @ F )
=> ( ( condit1735850657277959132ow_int @ ( image_int_int @ A4 @ I3 ) )
=> ( ( I3 != bot_bot_set_int )
=> ( ord_less_eq_nat @ ( F @ ( complete_Inf_Inf_int @ ( image_int_int @ A4 @ I3 ) ) )
@ ( complete_Inf_Inf_nat
@ ( image_int_nat
@ ^ [X4: int] : ( F @ ( A4 @ X4 ) )
@ I3 ) ) ) ) ) ) ).
% mono_cINF
thf(fact_3568_mono__cINF,axiom,
! [F: int > nat,A4: product_unit > int,I3: set_Product_unit] :
( ( order_mono_int_nat @ F )
=> ( ( condit1735850657277959132ow_int @ ( image_873079544045703924it_int @ A4 @ I3 ) )
=> ( ( I3 != bot_bo3957492148770167129t_unit )
=> ( ord_less_eq_nat @ ( F @ ( complete_Inf_Inf_int @ ( image_873079544045703924it_int @ A4 @ I3 ) ) )
@ ( complete_Inf_Inf_nat
@ ( image_875570014554754200it_nat
@ ^ [X4: product_unit] : ( F @ ( A4 @ X4 ) )
@ I3 ) ) ) ) ) ) ).
% mono_cINF
thf(fact_3569_mono__cINF,axiom,
! [F: int > nat,A4: $o > int,I3: set_o] :
( ( order_mono_int_nat @ F )
=> ( ( condit1735850657277959132ow_int @ ( image_o_int @ A4 @ I3 ) )
=> ( ( I3 != bot_bot_set_o )
=> ( ord_less_eq_nat @ ( F @ ( complete_Inf_Inf_int @ ( image_o_int @ A4 @ I3 ) ) )
@ ( complete_Inf_Inf_nat
@ ( image_o_nat
@ ^ [X4: $o] : ( F @ ( A4 @ X4 ) )
@ I3 ) ) ) ) ) ) ).
% mono_cINF
thf(fact_3570_mono__cINF,axiom,
! [F: int > nat,A4: nat > int,I3: set_nat] :
( ( order_mono_int_nat @ F )
=> ( ( condit1735850657277959132ow_int @ ( image_nat_int @ A4 @ I3 ) )
=> ( ( I3 != bot_bot_set_nat )
=> ( ord_less_eq_nat @ ( F @ ( complete_Inf_Inf_int @ ( image_nat_int @ A4 @ I3 ) ) )
@ ( complete_Inf_Inf_nat
@ ( image_nat_nat
@ ^ [X4: nat] : ( F @ ( A4 @ X4 ) )
@ I3 ) ) ) ) ) ) ).
% mono_cINF
thf(fact_3571_relcompp__distrib2,axiom,
! [S5: product_prod_nat_nat > product_prod_nat_nat > $o,T3: product_prod_nat_nat > product_prod_nat_nat > $o,R3: product_prod_nat_nat > int > $o] :
( ( relcom8911704206367525917at_int @ ( sup_su362511073950362882_nat_o @ S5 @ T3 ) @ R3 )
= ( sup_su5231854730092680051_int_o @ ( relcom8911704206367525917at_int @ S5 @ R3 ) @ ( relcom8911704206367525917at_int @ T3 @ R3 ) ) ) ).
% relcompp_distrib2
thf(fact_3572_relcompp__distrib2,axiom,
! [S5: int > int > $o,T3: int > int > $o,R3: int > code_integer > $o] :
( ( relcom967497705283160040nteger @ ( sup_sup_int_int_o @ S5 @ T3 ) @ R3 )
= ( sup_su1544571744061072657eger_o @ ( relcom967497705283160040nteger @ S5 @ R3 ) @ ( relcom967497705283160040nteger @ T3 @ R3 ) ) ) ).
% relcompp_distrib2
thf(fact_3573_relcompp__distrib,axiom,
! [R3: product_prod_nat_nat > product_prod_nat_nat > $o,S5: product_prod_nat_nat > int > $o,T3: product_prod_nat_nat > int > $o] :
( ( relcom8911704206367525917at_int @ R3 @ ( sup_su5231854730092680051_int_o @ S5 @ T3 ) )
= ( sup_su5231854730092680051_int_o @ ( relcom8911704206367525917at_int @ R3 @ S5 ) @ ( relcom8911704206367525917at_int @ R3 @ T3 ) ) ) ).
% relcompp_distrib
thf(fact_3574_relcompp__distrib,axiom,
! [R3: int > int > $o,S5: int > code_integer > $o,T3: int > code_integer > $o] :
( ( relcom967497705283160040nteger @ R3 @ ( sup_su1544571744061072657eger_o @ S5 @ T3 ) )
= ( sup_su1544571744061072657eger_o @ ( relcom967497705283160040nteger @ R3 @ S5 ) @ ( relcom967497705283160040nteger @ R3 @ T3 ) ) ) ).
% relcompp_distrib
thf(fact_3575_Field__Union,axiom,
! [R3: set_se7855581050983116737at_nat] :
( ( field_nat @ ( comple5685304695842803022at_nat @ R3 ) )
= ( comple7399068483239264473et_nat @ ( image_5853224409075402146et_nat @ field_nat @ R3 ) ) ) ).
% Field_Union
thf(fact_3576_the__inv__into__onto,axiom,
! [F: nat > rat,A4: set_nat] :
( ( inj_on_nat_rat @ F @ A4 )
=> ( ( image_rat_nat @ ( the_inv_into_nat_rat @ A4 @ F ) @ ( image_nat_rat @ F @ A4 ) )
= A4 ) ) ).
% the_inv_into_onto
thf(fact_3577_the__inv__into__onto,axiom,
! [F: rat > nat,A4: set_rat] :
( ( inj_on_rat_nat @ F @ A4 )
=> ( ( image_nat_rat @ ( the_inv_into_rat_nat @ A4 @ F ) @ ( image_rat_nat @ F @ A4 ) )
= A4 ) ) ).
% the_inv_into_onto
thf(fact_3578_the__inv__into__onto,axiom,
! [F: nat > nat,A4: set_nat] :
( ( inj_on_nat_nat @ F @ A4 )
=> ( ( image_nat_nat @ ( the_inv_into_nat_nat @ A4 @ F ) @ ( image_nat_nat @ F @ A4 ) )
= A4 ) ) ).
% the_inv_into_onto
thf(fact_3579_the__inv__into__onto,axiom,
! [F: int > nat,A4: set_int] :
( ( inj_on_int_nat @ F @ A4 )
=> ( ( image_nat_int @ ( the_inv_into_int_nat @ A4 @ F ) @ ( image_int_nat @ F @ A4 ) )
= A4 ) ) ).
% the_inv_into_onto
thf(fact_3580_the__inv__into__onto,axiom,
! [F: nat > int,A4: set_nat] :
( ( inj_on_nat_int @ F @ A4 )
=> ( ( image_int_nat @ ( the_inv_into_nat_int @ A4 @ F ) @ ( image_nat_int @ F @ A4 ) )
= A4 ) ) ).
% the_inv_into_onto
thf(fact_3581_the__inv__into__onto,axiom,
! [F: int > int,A4: set_int] :
( ( inj_on_int_int @ F @ A4 )
=> ( ( image_int_int @ ( the_inv_into_int_int @ A4 @ F ) @ ( image_int_int @ F @ A4 ) )
= A4 ) ) ).
% the_inv_into_onto
thf(fact_3582_image__vimage__eq,axiom,
! [F: char > product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
( ( image_8382823969816519465at_nat @ F @ ( vimage8942442433166402803at_nat @ F @ A4 ) )
= ( inf_in2572325071724192079at_nat @ A4 @ ( image_8382823969816519465at_nat @ F @ top_top_set_char ) ) ) ).
% image_vimage_eq
thf(fact_3583_image__vimage__eq,axiom,
! [F: list_nat > product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
( ( image_2795870212791037530at_nat @ F @ ( vimage8527374361057053988at_nat @ F @ A4 ) )
= ( inf_in2572325071724192079at_nat @ A4 @ ( image_2795870212791037530at_nat @ F @ top_top_set_list_nat ) ) ) ).
% image_vimage_eq
thf(fact_3584_image__vimage__eq,axiom,
! [F: rat > product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
( ( image_9175953278516507234at_nat @ F @ ( vimage2119786153496215084at_nat @ F @ A4 ) )
= ( inf_in2572325071724192079at_nat @ A4 @ ( image_9175953278516507234at_nat @ F @ top_top_set_rat ) ) ) ).
% image_vimage_eq
thf(fact_3585_image__vimage__eq,axiom,
! [F: nat > rat,A4: set_rat] :
( ( image_nat_rat @ F @ ( vimage_nat_rat @ F @ A4 ) )
= ( inf_inf_set_rat @ A4 @ ( image_nat_rat @ F @ top_top_set_nat ) ) ) ).
% image_vimage_eq
thf(fact_3586_image__vimage__eq,axiom,
! [F: nat > nat,A4: set_nat] :
( ( image_nat_nat @ F @ ( vimage_nat_nat @ F @ A4 ) )
= ( inf_inf_set_nat @ A4 @ ( image_nat_nat @ F @ top_top_set_nat ) ) ) ).
% image_vimage_eq
thf(fact_3587_image__vimage__eq,axiom,
! [F: nat > int,A4: set_int] :
( ( image_nat_int @ F @ ( vimage_nat_int @ F @ A4 ) )
= ( inf_inf_set_int @ A4 @ ( image_nat_int @ F @ top_top_set_nat ) ) ) ).
% image_vimage_eq
thf(fact_3588_image__vimage__eq,axiom,
! [F: nat > product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
( ( image_5846123807819985514at_nat @ F @ ( vimage8013328719654469172at_nat @ F @ A4 ) )
= ( inf_in2572325071724192079at_nat @ A4 @ ( image_5846123807819985514at_nat @ F @ top_top_set_nat ) ) ) ).
% image_vimage_eq
thf(fact_3589_image__vimage__eq,axiom,
! [F: int > nat,A4: set_nat] :
( ( image_int_nat @ F @ ( vimage_int_nat @ F @ A4 ) )
= ( inf_inf_set_nat @ A4 @ ( image_int_nat @ F @ top_top_set_int ) ) ) ).
% image_vimage_eq
thf(fact_3590_image__vimage__eq,axiom,
! [F: int > int,A4: set_int] :
( ( image_int_int @ F @ ( vimage_int_int @ F @ A4 ) )
= ( inf_inf_set_int @ A4 @ ( image_int_int @ F @ top_top_set_int ) ) ) ).
% image_vimage_eq
thf(fact_3591_image__vimage__eq,axiom,
! [F: int > product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
( ( image_8883965892284119310at_nat @ F @ ( vimage1827798767263827160at_nat @ F @ A4 ) )
= ( inf_in2572325071724192079at_nat @ A4 @ ( image_8883965892284119310at_nat @ F @ top_top_set_int ) ) ) ).
% image_vimage_eq
thf(fact_3592_vimage__comp,axiom,
! [F: nat > $o,G: $o > $o,X2: set_o] :
( ( vimage_nat_o @ F @ ( vimage_o_o @ G @ X2 ) )
= ( vimage_nat_o @ ( comp_o_o_nat @ G @ F ) @ X2 ) ) ).
% vimage_comp
thf(fact_3593_vimage__comp,axiom,
! [F: int > nat,G: nat > num,X2: set_num] :
( ( vimage_int_nat @ F @ ( vimage_nat_num @ G @ X2 ) )
= ( vimage_int_num @ ( comp_nat_num_int @ G @ F ) @ X2 ) ) ).
% vimage_comp
thf(fact_3594_vimage__comp,axiom,
! [F: nat > nat,G: nat > nat,X2: set_nat] :
( ( vimage_nat_nat @ F @ ( vimage_nat_nat @ G @ X2 ) )
= ( vimage_nat_nat @ ( comp_nat_nat_nat @ G @ F ) @ X2 ) ) ).
% vimage_comp
thf(fact_3595_vimage__comp,axiom,
! [F: int > int,G: int > nat,X2: set_nat] :
( ( vimage_int_int @ F @ ( vimage_int_nat @ G @ X2 ) )
= ( vimage_int_nat @ ( comp_int_nat_int @ G @ F ) @ X2 ) ) ).
% vimage_comp
thf(fact_3596_vimage__comp,axiom,
! [F: code_integer > code_integer > code_integer,G: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,X2: set_Pr4111642102156807771nteger] :
( ( vimage6029918387083761651nteger @ F @ ( vimage4289680085338140859nteger @ G @ X2 ) )
= ( vimage6839621282666897215nteger @ ( comp_C8797469213163452608nteger @ G @ F ) @ X2 ) ) ).
% vimage_comp
thf(fact_3597_set_Ocompositionality,axiom,
! [F: nat > $o,G: $o > $o,Set: set_o] :
( ( vimage_nat_o @ F @ ( vimage_o_o @ G @ Set ) )
= ( vimage_nat_o @ ( comp_o_o_nat @ G @ F ) @ Set ) ) ).
% set.compositionality
thf(fact_3598_set_Ocompositionality,axiom,
! [F: int > nat,G: nat > num,Set: set_num] :
( ( vimage_int_nat @ F @ ( vimage_nat_num @ G @ Set ) )
= ( vimage_int_num @ ( comp_nat_num_int @ G @ F ) @ Set ) ) ).
% set.compositionality
thf(fact_3599_set_Ocompositionality,axiom,
! [F: nat > nat,G: nat > nat,Set: set_nat] :
( ( vimage_nat_nat @ F @ ( vimage_nat_nat @ G @ Set ) )
= ( vimage_nat_nat @ ( comp_nat_nat_nat @ G @ F ) @ Set ) ) ).
% set.compositionality
thf(fact_3600_set_Ocompositionality,axiom,
! [F: int > int,G: int > nat,Set: set_nat] :
( ( vimage_int_int @ F @ ( vimage_int_nat @ G @ Set ) )
= ( vimage_int_nat @ ( comp_int_nat_int @ G @ F ) @ Set ) ) ).
% set.compositionality
thf(fact_3601_set_Ocompositionality,axiom,
! [F: code_integer > code_integer > code_integer,G: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,Set: set_Pr4111642102156807771nteger] :
( ( vimage6029918387083761651nteger @ F @ ( vimage4289680085338140859nteger @ G @ Set ) )
= ( vimage6839621282666897215nteger @ ( comp_C8797469213163452608nteger @ G @ F ) @ Set ) ) ).
% set.compositionality
thf(fact_3602_FieldI2,axiom,
! [I: produc3843707927480180839at_nat,J4: produc3843707927480180839at_nat,R3: set_Pr5564308138774400199at_nat] :
( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ I @ J4 ) @ R3 )
=> ( member8757157785044589968at_nat @ J4 @ ( field_2862844103579723188at_nat @ R3 ) ) ) ).
% FieldI2
thf(fact_3603_FieldI2,axiom,
! [I: product_prod_nat_nat,J4: product_prod_nat_nat,R3: set_Pr8693737435421807431at_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ I @ J4 ) @ R3 )
=> ( member8440522571783428010at_nat @ J4 @ ( field_4706345496283612622at_nat @ R3 ) ) ) ).
% FieldI2
thf(fact_3604_FieldI2,axiom,
! [I: $o,J4: $o,R3: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ I @ J4 ) @ R3 )
=> ( member_o @ J4 @ ( field_o @ R3 ) ) ) ).
% FieldI2
thf(fact_3605_FieldI2,axiom,
! [I: set_Pr4329608150637261639at_nat,J4: set_Pr4329608150637261639at_nat,R3: set_Pr7459493094073627847at_nat] :
( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ I @ J4 ) @ R3 )
=> ( member6273874620674182256at_nat @ J4 @ ( field_1428494348145443988at_nat @ R3 ) ) ) ).
% FieldI2
thf(fact_3606_FieldI2,axiom,
! [I: set_Pr1261947904930325089at_nat,J4: set_Pr1261947904930325089at_nat,R3: set_Pr4329608150637261639at_nat] :
( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ I @ J4 ) @ R3 )
=> ( member2643936169264416010at_nat @ J4 @ ( field_3145993188331922222at_nat @ R3 ) ) ) ).
% FieldI2
thf(fact_3607_FieldI2,axiom,
! [I: nat,J4: nat,R3: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ J4 ) @ R3 )
=> ( member_nat @ J4 @ ( field_nat @ R3 ) ) ) ).
% FieldI2
thf(fact_3608_FieldI2,axiom,
! [I: int,J4: int,R3: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ I @ J4 ) @ R3 )
=> ( member_int @ J4 @ ( field_int @ R3 ) ) ) ).
% FieldI2
thf(fact_3609_FieldI1,axiom,
! [I: produc3843707927480180839at_nat,J4: produc3843707927480180839at_nat,R3: set_Pr5564308138774400199at_nat] :
( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ I @ J4 ) @ R3 )
=> ( member8757157785044589968at_nat @ I @ ( field_2862844103579723188at_nat @ R3 ) ) ) ).
% FieldI1
thf(fact_3610_FieldI1,axiom,
! [I: product_prod_nat_nat,J4: product_prod_nat_nat,R3: set_Pr8693737435421807431at_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ I @ J4 ) @ R3 )
=> ( member8440522571783428010at_nat @ I @ ( field_4706345496283612622at_nat @ R3 ) ) ) ).
% FieldI1
thf(fact_3611_FieldI1,axiom,
! [I: $o,J4: $o,R3: set_Product_prod_o_o] :
( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ I @ J4 ) @ R3 )
=> ( member_o @ I @ ( field_o @ R3 ) ) ) ).
% FieldI1
thf(fact_3612_FieldI1,axiom,
! [I: set_Pr4329608150637261639at_nat,J4: set_Pr4329608150637261639at_nat,R3: set_Pr7459493094073627847at_nat] :
( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ I @ J4 ) @ R3 )
=> ( member6273874620674182256at_nat @ I @ ( field_1428494348145443988at_nat @ R3 ) ) ) ).
% FieldI1
thf(fact_3613_FieldI1,axiom,
! [I: set_Pr1261947904930325089at_nat,J4: set_Pr1261947904930325089at_nat,R3: set_Pr4329608150637261639at_nat] :
( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ I @ J4 ) @ R3 )
=> ( member2643936169264416010at_nat @ I @ ( field_3145993188331922222at_nat @ R3 ) ) ) ).
% FieldI1
thf(fact_3614_FieldI1,axiom,
! [I: nat,J4: nat,R3: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ J4 ) @ R3 )
=> ( member_nat @ I @ ( field_nat @ R3 ) ) ) ).
% FieldI1
thf(fact_3615_FieldI1,axiom,
! [I: int,J4: int,R3: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ I @ J4 ) @ R3 )
=> ( member_int @ I @ ( field_int @ R3 ) ) ) ).
% FieldI1
thf(fact_3616_inf__Int__eq2,axiom,
! [R3: set_Pr7459493094073627847at_nat,S5: set_Pr7459493094073627847at_nat] :
( ( inf_in6969519006101653788_nat_o
@ ^ [X4: set_Pr4329608150637261639at_nat,Y5: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X4 @ Y5 ) @ R3 )
@ ^ [X4: set_Pr4329608150637261639at_nat,Y5: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X4 @ Y5 ) @ S5 ) )
= ( ^ [X4: set_Pr4329608150637261639at_nat,Y5: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X4 @ Y5 ) @ ( inf_in2456762090751560885at_nat @ R3 @ S5 ) ) ) ) ).
% inf_Int_eq2
thf(fact_3617_inf__Int__eq2,axiom,
! [R3: set_Pr4329608150637261639at_nat,S5: set_Pr4329608150637261639at_nat] :
( ( inf_in6124848772414083356_nat_o
@ ^ [X4: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y5 ) @ R3 )
@ ^ [X4: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y5 ) @ S5 ) )
= ( ^ [X4: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y5 ) @ ( inf_in7913087082777306421at_nat @ R3 @ S5 ) ) ) ) ).
% inf_Int_eq2
thf(fact_3618_inf__Int__eq2,axiom,
! [R3: set_Pr6200539531224447659at_num,S5: set_Pr6200539531224447659at_num] :
( ( inf_inf_nat_num_o
@ ^ [X4: nat,Y5: num] : ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X4 @ Y5 ) @ R3 )
@ ^ [X4: nat,Y5: num] : ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X4 @ Y5 ) @ S5 ) )
= ( ^ [X4: nat,Y5: num] : ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X4 @ Y5 ) @ ( inf_in7510916698018314649at_num @ R3 @ S5 ) ) ) ) ).
% inf_Int_eq2
thf(fact_3619_inf__Int__eq2,axiom,
! [R3: set_Pr958786334691620121nt_int,S5: set_Pr958786334691620121nt_int] :
( ( inf_inf_int_int_o
@ ^ [X4: int,Y5: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y5 ) @ R3 )
@ ^ [X4: int,Y5: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y5 ) @ S5 ) )
= ( ^ [X4: int,Y5: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y5 ) @ ( inf_in2269163501485487111nt_int @ R3 @ S5 ) ) ) ) ).
% inf_Int_eq2
thf(fact_3620_inf__Int__eq2,axiom,
! [R3: set_Pr1261947904930325089at_nat,S5: set_Pr1261947904930325089at_nat] :
( ( inf_inf_nat_nat_o
@ ^ [X4: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ R3 )
@ ^ [X4: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ S5 ) )
= ( ^ [X4: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ ( inf_in2572325071724192079at_nat @ R3 @ S5 ) ) ) ) ).
% inf_Int_eq2
thf(fact_3621_fun__upd__comp,axiom,
! [F: $o > $o,G: nat > $o,X2: nat,Y3: $o] :
( ( comp_o_o_nat @ F @ ( fun_upd_nat_o @ G @ X2 @ Y3 ) )
= ( fun_upd_nat_o @ ( comp_o_o_nat @ F @ G ) @ X2 @ ( F @ Y3 ) ) ) ).
% fun_upd_comp
thf(fact_3622_fun__upd__comp,axiom,
! [F: nat > num,G: int > nat,X2: int,Y3: nat] :
( ( comp_nat_num_int @ F @ ( fun_upd_int_nat @ G @ X2 @ Y3 ) )
= ( fun_upd_int_num @ ( comp_nat_num_int @ F @ G ) @ X2 @ ( F @ Y3 ) ) ) ).
% fun_upd_comp
thf(fact_3623_fun__upd__comp,axiom,
! [F: nat > nat,G: nat > nat,X2: nat,Y3: nat] :
( ( comp_nat_nat_nat @ F @ ( fun_upd_nat_nat @ G @ X2 @ Y3 ) )
= ( fun_upd_nat_nat @ ( comp_nat_nat_nat @ F @ G ) @ X2 @ ( F @ Y3 ) ) ) ).
% fun_upd_comp
thf(fact_3624_fun__upd__comp,axiom,
! [F: int > nat,G: int > int,X2: int,Y3: int] :
( ( comp_int_nat_int @ F @ ( fun_upd_int_int @ G @ X2 @ Y3 ) )
= ( fun_upd_int_nat @ ( comp_int_nat_int @ F @ G ) @ X2 @ ( F @ Y3 ) ) ) ).
% fun_upd_comp
thf(fact_3625_fun__upd__comp,axiom,
! [F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer > code_integer,X2: code_integer,Y3: code_integer > code_integer] :
( ( comp_C8797469213163452608nteger @ F @ ( fun_up8056716234270145717nteger @ G @ X2 @ Y3 ) )
= ( fun_up8139772397987072257nteger @ ( comp_C8797469213163452608nteger @ F @ G ) @ X2 @ ( F @ Y3 ) ) ) ).
% fun_upd_comp
thf(fact_3626_image__vimage__subset,axiom,
! [F: nat > rat,A4: set_rat] : ( ord_less_eq_set_rat @ ( image_nat_rat @ F @ ( vimage_nat_rat @ F @ A4 ) ) @ A4 ) ).
% image_vimage_subset
thf(fact_3627_image__vimage__subset,axiom,
! [F: nat > nat,A4: set_nat] : ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ ( vimage_nat_nat @ F @ A4 ) ) @ A4 ) ).
% image_vimage_subset
thf(fact_3628_image__vimage__subset,axiom,
! [F: nat > int,A4: set_int] : ( ord_less_eq_set_int @ ( image_nat_int @ F @ ( vimage_nat_int @ F @ A4 ) ) @ A4 ) ).
% image_vimage_subset
thf(fact_3629_image__vimage__subset,axiom,
! [F: int > nat,A4: set_nat] : ( ord_less_eq_set_nat @ ( image_int_nat @ F @ ( vimage_int_nat @ F @ A4 ) ) @ A4 ) ).
% image_vimage_subset
thf(fact_3630_image__vimage__subset,axiom,
! [F: int > int,A4: set_int] : ( ord_less_eq_set_int @ ( image_int_int @ F @ ( vimage_int_int @ F @ A4 ) ) @ A4 ) ).
% image_vimage_subset
thf(fact_3631_image__subset__iff__subset__vimage,axiom,
! [F: nat > rat,A4: set_nat,B5: set_rat] :
( ( ord_less_eq_set_rat @ ( image_nat_rat @ F @ A4 ) @ B5 )
= ( ord_less_eq_set_nat @ A4 @ ( vimage_nat_rat @ F @ B5 ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_3632_image__subset__iff__subset__vimage,axiom,
! [F: nat > nat,A4: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A4 ) @ B5 )
= ( ord_less_eq_set_nat @ A4 @ ( vimage_nat_nat @ F @ B5 ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_3633_image__subset__iff__subset__vimage,axiom,
! [F: nat > int,A4: set_nat,B5: set_int] :
( ( ord_less_eq_set_int @ ( image_nat_int @ F @ A4 ) @ B5 )
= ( ord_less_eq_set_nat @ A4 @ ( vimage_nat_int @ F @ B5 ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_3634_image__subset__iff__subset__vimage,axiom,
! [F: int > nat,A4: set_int,B5: set_nat] :
( ( ord_less_eq_set_nat @ ( image_int_nat @ F @ A4 ) @ B5 )
= ( ord_less_eq_set_int @ A4 @ ( vimage_int_nat @ F @ B5 ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_3635_image__subset__iff__subset__vimage,axiom,
! [F: int > int,A4: set_int,B5: set_int] :
( ( ord_less_eq_set_int @ ( image_int_int @ F @ A4 ) @ B5 )
= ( ord_less_eq_set_int @ A4 @ ( vimage_int_int @ F @ B5 ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_3636_surj__image__vimage__eq,axiom,
! [F: char > char,A4: set_char] :
( ( ( image_char_char @ F @ top_top_set_char )
= top_top_set_char )
=> ( ( image_char_char @ F @ ( vimage_char_char @ F @ A4 ) )
= A4 ) ) ).
% surj_image_vimage_eq
thf(fact_3637_surj__image__vimage__eq,axiom,
! [F: char > rat,A4: set_rat] :
( ( ( image_char_rat @ F @ top_top_set_char )
= top_top_set_rat )
=> ( ( image_char_rat @ F @ ( vimage_char_rat @ F @ A4 ) )
= A4 ) ) ).
% surj_image_vimage_eq
thf(fact_3638_surj__image__vimage__eq,axiom,
! [F: char > nat,A4: set_nat] :
( ( ( image_char_nat @ F @ top_top_set_char )
= top_top_set_nat )
=> ( ( image_char_nat @ F @ ( vimage_char_nat @ F @ A4 ) )
= A4 ) ) ).
% surj_image_vimage_eq
thf(fact_3639_surj__image__vimage__eq,axiom,
! [F: char > int,A4: set_int] :
( ( ( image_char_int @ F @ top_top_set_char )
= top_top_set_int )
=> ( ( image_char_int @ F @ ( vimage_char_int @ F @ A4 ) )
= A4 ) ) ).
% surj_image_vimage_eq
thf(fact_3640_surj__image__vimage__eq,axiom,
! [F: rat > char,A4: set_char] :
( ( ( image_rat_char @ F @ top_top_set_rat )
= top_top_set_char )
=> ( ( image_rat_char @ F @ ( vimage_rat_char @ F @ A4 ) )
= A4 ) ) ).
% surj_image_vimage_eq
thf(fact_3641_surj__image__vimage__eq,axiom,
! [F: rat > rat,A4: set_rat] :
( ( ( image_rat_rat @ F @ top_top_set_rat )
= top_top_set_rat )
=> ( ( image_rat_rat @ F @ ( vimage_rat_rat @ F @ A4 ) )
= A4 ) ) ).
% surj_image_vimage_eq
thf(fact_3642_surj__image__vimage__eq,axiom,
! [F: rat > nat,A4: set_nat] :
( ( ( image_rat_nat @ F @ top_top_set_rat )
= top_top_set_nat )
=> ( ( image_rat_nat @ F @ ( vimage_rat_nat @ F @ A4 ) )
= A4 ) ) ).
% surj_image_vimage_eq
thf(fact_3643_surj__image__vimage__eq,axiom,
! [F: rat > int,A4: set_int] :
( ( ( image_rat_int @ F @ top_top_set_rat )
= top_top_set_int )
=> ( ( image_rat_int @ F @ ( vimage_rat_int @ F @ A4 ) )
= A4 ) ) ).
% surj_image_vimage_eq
thf(fact_3644_surj__image__vimage__eq,axiom,
! [F: nat > char,A4: set_char] :
( ( ( image_nat_char @ F @ top_top_set_nat )
= top_top_set_char )
=> ( ( image_nat_char @ F @ ( vimage_nat_char @ F @ A4 ) )
= A4 ) ) ).
% surj_image_vimage_eq
thf(fact_3645_surj__image__vimage__eq,axiom,
! [F: nat > rat,A4: set_rat] :
( ( ( image_nat_rat @ F @ top_top_set_nat )
= top_top_set_rat )
=> ( ( image_nat_rat @ F @ ( vimage_nat_rat @ F @ A4 ) )
= A4 ) ) ).
% surj_image_vimage_eq
thf(fact_3646_bdd__below__image__mono,axiom,
! [F: nat > rat,A4: set_nat] :
( ( order_mono_nat_rat @ F )
=> ( ( condit1738341127787009408ow_nat @ A4 )
=> ( condit1103211067700513672ow_rat @ ( image_nat_rat @ F @ A4 ) ) ) ) ).
% bdd_below_image_mono
thf(fact_3647_bdd__below__image__mono,axiom,
! [F: nat > nat,A4: set_nat] :
( ( order_mono_nat_nat @ F )
=> ( ( condit1738341127787009408ow_nat @ A4 )
=> ( condit1738341127787009408ow_nat @ ( image_nat_nat @ F @ A4 ) ) ) ) ).
% bdd_below_image_mono
thf(fact_3648_bdd__below__image__mono,axiom,
! [F: nat > int,A4: set_nat] :
( ( order_mono_nat_int @ F )
=> ( ( condit1738341127787009408ow_nat @ A4 )
=> ( condit1735850657277959132ow_int @ ( image_nat_int @ F @ A4 ) ) ) ) ).
% bdd_below_image_mono
thf(fact_3649_bdd__below__image__mono,axiom,
! [F: int > nat,A4: set_int] :
( ( order_mono_int_nat @ F )
=> ( ( condit1735850657277959132ow_int @ A4 )
=> ( condit1738341127787009408ow_nat @ ( image_int_nat @ F @ A4 ) ) ) ) ).
% bdd_below_image_mono
thf(fact_3650_bdd__below__image__mono,axiom,
! [F: int > int,A4: set_int] :
( ( order_mono_int_int @ F )
=> ( ( condit1735850657277959132ow_int @ A4 )
=> ( condit1735850657277959132ow_int @ ( image_int_int @ F @ A4 ) ) ) ) ).
% bdd_below_image_mono
thf(fact_3651_bdd__above__image__mono,axiom,
! [F: nat > rat,A4: set_nat] :
( ( order_mono_nat_rat @ F )
=> ( ( condit2214826472909112428ve_nat @ A4 )
=> ( condit1579696412822616692ve_rat @ ( image_nat_rat @ F @ A4 ) ) ) ) ).
% bdd_above_image_mono
thf(fact_3652_bdd__above__image__mono,axiom,
! [F: nat > nat,A4: set_nat] :
( ( order_mono_nat_nat @ F )
=> ( ( condit2214826472909112428ve_nat @ A4 )
=> ( condit2214826472909112428ve_nat @ ( image_nat_nat @ F @ A4 ) ) ) ) ).
% bdd_above_image_mono
thf(fact_3653_bdd__above__image__mono,axiom,
! [F: nat > int,A4: set_nat] :
( ( order_mono_nat_int @ F )
=> ( ( condit2214826472909112428ve_nat @ A4 )
=> ( condit2212336002400062152ve_int @ ( image_nat_int @ F @ A4 ) ) ) ) ).
% bdd_above_image_mono
thf(fact_3654_bdd__above__image__mono,axiom,
! [F: int > nat,A4: set_int] :
( ( order_mono_int_nat @ F )
=> ( ( condit2212336002400062152ve_int @ A4 )
=> ( condit2214826472909112428ve_nat @ ( image_int_nat @ F @ A4 ) ) ) ) ).
% bdd_above_image_mono
thf(fact_3655_bdd__above__image__mono,axiom,
! [F: int > int,A4: set_int] :
( ( order_mono_int_int @ F )
=> ( ( condit2212336002400062152ve_int @ A4 )
=> ( condit2212336002400062152ve_int @ ( image_int_int @ F @ A4 ) ) ) ) ).
% bdd_above_image_mono
thf(fact_3656_sup__Un__eq2,axiom,
! [R3: set_Pr7459493094073627847at_nat,S5: set_Pr7459493094073627847at_nat] :
( ( sup_su6164525874564267778_nat_o
@ ^ [X4: set_Pr4329608150637261639at_nat,Y5: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X4 @ Y5 ) @ R3 )
@ ^ [X4: set_Pr4329608150637261639at_nat,Y5: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X4 @ Y5 ) @ S5 ) )
= ( ^ [X4: set_Pr4329608150637261639at_nat,Y5: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X4 @ Y5 ) @ ( sup_su5395584912770621083at_nat @ R3 @ S5 ) ) ) ) ).
% sup_Un_eq2
thf(fact_3657_sup__Un__eq2,axiom,
! [R3: set_Pr6200539531224447659at_num,S5: set_Pr6200539531224447659at_num] :
( ( sup_sup_nat_num_o
@ ^ [X4: nat,Y5: num] : ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X4 @ Y5 ) @ R3 )
@ ^ [X4: nat,Y5: num] : ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X4 @ Y5 ) @ S5 ) )
= ( ^ [X4: nat,Y5: num] : ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X4 @ Y5 ) @ ( sup_su2042722026077122175at_num @ R3 @ S5 ) ) ) ) ).
% sup_Un_eq2
thf(fact_3658_sup__Un__eq2,axiom,
! [R3: set_Pr1261947904930325089at_nat,S5: set_Pr1261947904930325089at_nat] :
( ( sup_sup_nat_nat_o
@ ^ [X4: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ R3 )
@ ^ [X4: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ S5 ) )
= ( ^ [X4: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ ( sup_su6327502436637775413at_nat @ R3 @ S5 ) ) ) ) ).
% sup_Un_eq2
thf(fact_3659_sup__Un__eq2,axiom,
! [R3: set_Pr958786334691620121nt_int,S5: set_Pr958786334691620121nt_int] :
( ( sup_sup_int_int_o
@ ^ [X4: int,Y5: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y5 ) @ R3 )
@ ^ [X4: int,Y5: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y5 ) @ S5 ) )
= ( ^ [X4: int,Y5: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y5 ) @ ( sup_su6024340866399070445nt_int @ R3 @ S5 ) ) ) ) ).
% sup_Un_eq2
thf(fact_3660_sup__Un__eq2,axiom,
! [R3: set_Pr8693737435421807431at_nat,S5: set_Pr8693737435421807431at_nat] :
( ( sup_su362511073950362882_nat_o
@ ^ [X4: product_prod_nat_nat,Y5: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y5 ) @ R3 )
@ ^ [X4: product_prod_nat_nat,Y5: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y5 ) @ S5 ) )
= ( ^ [X4: product_prod_nat_nat,Y5: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y5 ) @ ( sup_su718114333110466843at_nat @ R3 @ S5 ) ) ) ) ).
% sup_Un_eq2
thf(fact_3661_sup__Un__eq2,axiom,
! [R3: set_Pr4329608150637261639at_nat,S5: set_Pr4329608150637261639at_nat] :
( ( sup_su7519161239522478338_nat_o
@ ^ [X4: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y5 ) @ R3 )
@ ^ [X4: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y5 ) @ S5 ) )
= ( ^ [X4: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y5 ) @ ( sup_su5525570899277871387at_nat @ R3 @ S5 ) ) ) ) ).
% sup_Un_eq2
thf(fact_3662_vimage__Union,axiom,
! [F: nat > nat,A4: set_set_nat] :
( ( vimage_nat_nat @ F @ ( comple7399068483239264473et_nat @ A4 ) )
= ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ ( vimage_nat_nat @ F ) @ A4 ) ) ) ).
% vimage_Union
thf(fact_3663_vimage__Times,axiom,
! [F: product_prod_nat_nat > product_prod_nat_nat,A4: set_nat,B5: set_nat] :
( ( vimage2449269961533847803at_nat @ F
@ ( produc457027306803732586at_nat @ A4
@ ^ [Uu: nat] : B5 ) )
= ( inf_in2572325071724192079at_nat @ ( vimage4653281326611754070at_nat @ ( comp_P1567445206330693457at_nat @ product_fst_nat_nat @ F ) @ A4 ) @ ( vimage4653281326611754070at_nat @ ( comp_P1567445206330693457at_nat @ product_snd_nat_nat @ F ) @ B5 ) ) ) ).
% vimage_Times
thf(fact_3664_vimage__Times,axiom,
! [F: product_prod_nat_nat > produc7812382424243267045t_unit,A4: set_ref_a,B5: set_he8644174446132254637t_unit] :
( ( vimage4308919176130087787t_unit @ F
@ ( produc3946373962484743846t_unit @ A4
@ ^ [Uu: ref_a] : B5 ) )
= ( inf_in2572325071724192079at_nat @ ( vimage7322659125625010723_ref_a @ ( comp_P7986581389989349338at_nat @ produc473321116828791851t_unit @ F ) @ A4 ) @ ( vimage1024995831485383111t_unit @ ( comp_P6390991081591698364at_nat @ produc1126150082675451753t_unit @ F ) @ B5 ) ) ) ).
% vimage_Times
thf(fact_3665_inj__on__fun__updI,axiom,
! [F: nat > rat,A4: set_nat,Y3: rat,X2: nat] :
( ( inj_on_nat_rat @ F @ A4 )
=> ( ~ ( member_rat @ Y3 @ ( image_nat_rat @ F @ A4 ) )
=> ( inj_on_nat_rat @ ( fun_upd_nat_rat @ F @ X2 @ Y3 ) @ A4 ) ) ) ).
% inj_on_fun_updI
thf(fact_3666_inj__on__fun__updI,axiom,
! [F: nat > nat,A4: set_nat,Y3: nat,X2: nat] :
( ( inj_on_nat_nat @ F @ A4 )
=> ( ~ ( member_nat @ Y3 @ ( image_nat_nat @ F @ A4 ) )
=> ( inj_on_nat_nat @ ( fun_upd_nat_nat @ F @ X2 @ Y3 ) @ A4 ) ) ) ).
% inj_on_fun_updI
thf(fact_3667_inj__on__fun__updI,axiom,
! [F: int > nat,A4: set_int,Y3: nat,X2: int] :
( ( inj_on_int_nat @ F @ A4 )
=> ( ~ ( member_nat @ Y3 @ ( image_int_nat @ F @ A4 ) )
=> ( inj_on_int_nat @ ( fun_upd_int_nat @ F @ X2 @ Y3 ) @ A4 ) ) ) ).
% inj_on_fun_updI
thf(fact_3668_inj__on__fun__updI,axiom,
! [F: nat > int,A4: set_nat,Y3: int,X2: nat] :
( ( inj_on_nat_int @ F @ A4 )
=> ( ~ ( member_int @ Y3 @ ( image_nat_int @ F @ A4 ) )
=> ( inj_on_nat_int @ ( fun_upd_nat_int @ F @ X2 @ Y3 ) @ A4 ) ) ) ).
% inj_on_fun_updI
thf(fact_3669_inj__on__fun__updI,axiom,
! [F: int > int,A4: set_int,Y3: int,X2: int] :
( ( inj_on_int_int @ F @ A4 )
=> ( ~ ( member_int @ Y3 @ ( image_int_int @ F @ A4 ) )
=> ( inj_on_int_int @ ( fun_upd_int_int @ F @ X2 @ Y3 ) @ A4 ) ) ) ).
% inj_on_fun_updI
thf(fact_3670_set_Ocomp,axiom,
! [F: nat > $o,G: $o > $o] :
( ( comp_s9189197539837705563_set_o @ ( vimage_nat_o @ F ) @ ( vimage_o_o @ G ) )
= ( vimage_nat_o @ ( comp_o_o_nat @ G @ F ) ) ) ).
% set.comp
thf(fact_3671_set_Ocomp,axiom,
! [F: int > nat,G: nat > num] :
( ( comp_s6458263553158497821et_num @ ( vimage_int_nat @ F ) @ ( vimage_nat_num @ G ) )
= ( vimage_int_num @ ( comp_nat_num_int @ G @ F ) ) ) ).
% set.comp
thf(fact_3672_set_Ocomp,axiom,
! [F: nat > nat,G: nat > nat] :
( ( comp_s8964582002068861047et_nat @ ( vimage_nat_nat @ F ) @ ( vimage_nat_nat @ G ) )
= ( vimage_nat_nat @ ( comp_nat_nat_nat @ G @ F ) ) ) ).
% set.comp
thf(fact_3673_set_Ocomp,axiom,
! [F: int > int,G: int > nat] :
( ( comp_s7717450105097109807et_nat @ ( vimage_int_int @ F ) @ ( vimage_int_nat @ G ) )
= ( vimage_int_nat @ ( comp_int_nat_int @ G @ F ) ) ) ).
% set.comp
thf(fact_3674_set_Ocomp,axiom,
! [F: code_integer > code_integer > code_integer,G: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger] :
( ( comp_s4786119426940707554nteger @ ( vimage6029918387083761651nteger @ F ) @ ( vimage4289680085338140859nteger @ G ) )
= ( vimage6839621282666897215nteger @ ( comp_C8797469213163452608nteger @ G @ F ) ) ) ).
% set.comp
thf(fact_3675_surj__vimage__empty,axiom,
! [F: product_unit > product_unit,A4: set_Product_unit] :
( ( ( image_405062704495631173t_unit @ F @ top_to1996260823553986621t_unit )
= top_to1996260823553986621t_unit )
=> ( ( ( vimage7995052115951654139t_unit @ F @ A4 )
= bot_bo3957492148770167129t_unit )
= ( A4 = bot_bo3957492148770167129t_unit ) ) ) ).
% surj_vimage_empty
thf(fact_3676_surj__vimage__empty,axiom,
! [F: product_unit > $o,A4: set_o] :
( ( ( image_Product_unit_o @ F @ top_to1996260823553986621t_unit )
= top_top_set_o )
=> ( ( ( vimage8808348054653608838unit_o @ F @ A4 )
= bot_bo3957492148770167129t_unit )
= ( A4 = bot_bot_set_o ) ) ) ).
% surj_vimage_empty
thf(fact_3677_surj__vimage__empty,axiom,
! [F: $o > product_unit,A4: set_Product_unit] :
( ( ( image_o_Product_unit @ F @ top_top_set_o )
= top_to1996260823553986621t_unit )
=> ( ( ( vimage4222377557387472176t_unit @ F @ A4 )
= bot_bot_set_o )
= ( A4 = bot_bo3957492148770167129t_unit ) ) ) ).
% surj_vimage_empty
thf(fact_3678_surj__vimage__empty,axiom,
! [F: $o > $o,A4: set_o] :
( ( ( image_o_o @ F @ top_top_set_o )
= top_top_set_o )
=> ( ( ( vimage_o_o @ F @ A4 )
= bot_bot_set_o )
= ( A4 = bot_bot_set_o ) ) ) ).
% surj_vimage_empty
thf(fact_3679_surj__vimage__empty,axiom,
! [F: product_unit > char,A4: set_char] :
( ( ( image_7248553556944771225t_char @ F @ top_to1996260823553986621t_unit )
= top_top_set_char )
=> ( ( ( vimage2648087219933717859t_char @ F @ A4 )
= bot_bo3957492148770167129t_unit )
= ( A4 = bot_bot_set_char ) ) ) ).
% surj_vimage_empty
thf(fact_3680_surj__vimage__empty,axiom,
! [F: $o > char,A4: set_char] :
( ( ( image_o_char @ F @ top_top_set_o )
= top_top_set_char )
=> ( ( ( vimage_o_char @ F @ A4 )
= bot_bot_set_o )
= ( A4 = bot_bot_set_char ) ) ) ).
% surj_vimage_empty
thf(fact_3681_surj__vimage__empty,axiom,
! [F: product_unit > rat,A4: set_rat] :
( ( ( image_240439954468258464it_rat @ F @ top_to1996260823553986621t_unit )
= top_top_set_rat )
=> ( ( ( vimage5618198413390092650it_rat @ F @ A4 )
= bot_bo3957492148770167129t_unit )
= ( A4 = bot_bot_set_rat ) ) ) ).
% surj_vimage_empty
thf(fact_3682_surj__vimage__empty,axiom,
! [F: $o > rat,A4: set_rat] :
( ( ( image_o_rat @ F @ top_top_set_o )
= top_top_set_rat )
=> ( ( ( vimage_o_rat @ F @ A4 )
= bot_bot_set_o )
= ( A4 = bot_bot_set_rat ) ) ) ).
% surj_vimage_empty
thf(fact_3683_surj__vimage__empty,axiom,
! [F: product_unit > nat,A4: set_nat] :
( ( ( image_875570014554754200it_nat @ F @ top_to1996260823553986621t_unit )
= top_top_set_nat )
=> ( ( ( vimage6253328473476588386it_nat @ F @ A4 )
= bot_bo3957492148770167129t_unit )
= ( A4 = bot_bot_set_nat ) ) ) ).
% surj_vimage_empty
thf(fact_3684_surj__vimage__empty,axiom,
! [F: $o > nat,A4: set_nat] :
( ( ( image_o_nat @ F @ top_top_set_o )
= top_top_set_nat )
=> ( ( ( vimage_o_nat @ F @ A4 )
= bot_bot_set_o )
= ( A4 = bot_bot_set_nat ) ) ) ).
% surj_vimage_empty
thf(fact_3685_vimage__subsetD,axiom,
! [F: char > char,B5: set_char,A4: set_char] :
( ( ( image_char_char @ F @ top_top_set_char )
= top_top_set_char )
=> ( ( ord_less_eq_set_char @ ( vimage_char_char @ F @ B5 ) @ A4 )
=> ( ord_less_eq_set_char @ B5 @ ( image_char_char @ F @ A4 ) ) ) ) ).
% vimage_subsetD
thf(fact_3686_vimage__subsetD,axiom,
! [F: char > rat,B5: set_rat,A4: set_char] :
( ( ( image_char_rat @ F @ top_top_set_char )
= top_top_set_rat )
=> ( ( ord_less_eq_set_char @ ( vimage_char_rat @ F @ B5 ) @ A4 )
=> ( ord_less_eq_set_rat @ B5 @ ( image_char_rat @ F @ A4 ) ) ) ) ).
% vimage_subsetD
thf(fact_3687_vimage__subsetD,axiom,
! [F: char > nat,B5: set_nat,A4: set_char] :
( ( ( image_char_nat @ F @ top_top_set_char )
= top_top_set_nat )
=> ( ( ord_less_eq_set_char @ ( vimage_char_nat @ F @ B5 ) @ A4 )
=> ( ord_less_eq_set_nat @ B5 @ ( image_char_nat @ F @ A4 ) ) ) ) ).
% vimage_subsetD
thf(fact_3688_vimage__subsetD,axiom,
! [F: char > int,B5: set_int,A4: set_char] :
( ( ( image_char_int @ F @ top_top_set_char )
= top_top_set_int )
=> ( ( ord_less_eq_set_char @ ( vimage_char_int @ F @ B5 ) @ A4 )
=> ( ord_less_eq_set_int @ B5 @ ( image_char_int @ F @ A4 ) ) ) ) ).
% vimage_subsetD
thf(fact_3689_vimage__subsetD,axiom,
! [F: rat > char,B5: set_char,A4: set_rat] :
( ( ( image_rat_char @ F @ top_top_set_rat )
= top_top_set_char )
=> ( ( ord_less_eq_set_rat @ ( vimage_rat_char @ F @ B5 ) @ A4 )
=> ( ord_less_eq_set_char @ B5 @ ( image_rat_char @ F @ A4 ) ) ) ) ).
% vimage_subsetD
thf(fact_3690_vimage__subsetD,axiom,
! [F: rat > rat,B5: set_rat,A4: set_rat] :
( ( ( image_rat_rat @ F @ top_top_set_rat )
= top_top_set_rat )
=> ( ( ord_less_eq_set_rat @ ( vimage_rat_rat @ F @ B5 ) @ A4 )
=> ( ord_less_eq_set_rat @ B5 @ ( image_rat_rat @ F @ A4 ) ) ) ) ).
% vimage_subsetD
thf(fact_3691_vimage__subsetD,axiom,
! [F: rat > nat,B5: set_nat,A4: set_rat] :
( ( ( image_rat_nat @ F @ top_top_set_rat )
= top_top_set_nat )
=> ( ( ord_less_eq_set_rat @ ( vimage_rat_nat @ F @ B5 ) @ A4 )
=> ( ord_less_eq_set_nat @ B5 @ ( image_rat_nat @ F @ A4 ) ) ) ) ).
% vimage_subsetD
thf(fact_3692_vimage__subsetD,axiom,
! [F: rat > int,B5: set_int,A4: set_rat] :
( ( ( image_rat_int @ F @ top_top_set_rat )
= top_top_set_int )
=> ( ( ord_less_eq_set_rat @ ( vimage_rat_int @ F @ B5 ) @ A4 )
=> ( ord_less_eq_set_int @ B5 @ ( image_rat_int @ F @ A4 ) ) ) ) ).
% vimage_subsetD
thf(fact_3693_vimage__subsetD,axiom,
! [F: nat > char,B5: set_char,A4: set_nat] :
( ( ( image_nat_char @ F @ top_top_set_nat )
= top_top_set_char )
=> ( ( ord_less_eq_set_nat @ ( vimage_nat_char @ F @ B5 ) @ A4 )
=> ( ord_less_eq_set_char @ B5 @ ( image_nat_char @ F @ A4 ) ) ) ) ).
% vimage_subsetD
thf(fact_3694_vimage__subsetD,axiom,
! [F: nat > rat,B5: set_rat,A4: set_nat] :
( ( ( image_nat_rat @ F @ top_top_set_nat )
= top_top_set_rat )
=> ( ( ord_less_eq_set_nat @ ( vimage_nat_rat @ F @ B5 ) @ A4 )
=> ( ord_less_eq_set_rat @ B5 @ ( image_nat_rat @ F @ A4 ) ) ) ) ).
% vimage_subsetD
thf(fact_3695_Pair__vimage__Sigma,axiom,
! [X2: nat,A4: set_nat,F: nat > set_num] :
( ( ( member_nat @ X2 @ A4 )
=> ( ( vimage5436458796434496692at_num @ ( product_Pair_nat_num @ X2 ) @ ( produc6237731129814287028at_num @ A4 @ F ) )
= ( F @ X2 ) ) )
& ( ~ ( member_nat @ X2 @ A4 )
=> ( ( vimage5436458796434496692at_num @ ( product_Pair_nat_num @ X2 ) @ ( produc6237731129814287028at_num @ A4 @ F ) )
= bot_bot_set_num ) ) ) ).
% Pair_vimage_Sigma
thf(fact_3696_Pair__vimage__Sigma,axiom,
! [X2: int,A4: set_int,F: int > set_int] :
( ( ( member_int @ X2 @ A4 )
=> ( ( vimage7872673496510460816nt_int @ ( product_Pair_int_int @ X2 ) @ ( produc1453890548195736866nt_int @ A4 @ F ) )
= ( F @ X2 ) ) )
& ( ~ ( member_int @ X2 @ A4 )
=> ( ( vimage7872673496510460816nt_int @ ( product_Pair_int_int @ X2 ) @ ( produc1453890548195736866nt_int @ A4 @ F ) )
= bot_bot_set_int ) ) ) ).
% Pair_vimage_Sigma
thf(fact_3697_Pair__vimage__Sigma,axiom,
! [X2: $o,A4: set_o,F: $o > set_Product_unit] :
( ( ( member_o @ X2 @ A4 )
=> ( ( vimage508909887551888138t_unit @ ( produc3901946521647900940t_unit @ X2 ) @ ( produc411009820408249917t_unit @ A4 @ F ) )
= ( F @ X2 ) ) )
& ( ~ ( member_o @ X2 @ A4 )
=> ( ( vimage508909887551888138t_unit @ ( produc3901946521647900940t_unit @ X2 ) @ ( produc411009820408249917t_unit @ A4 @ F ) )
= bot_bo3957492148770167129t_unit ) ) ) ).
% Pair_vimage_Sigma
thf(fact_3698_Pair__vimage__Sigma,axiom,
! [X2: nat,A4: set_nat,F: nat > set_Product_unit] :
( ( ( member_nat @ X2 @ A4 )
=> ( ( vimage7305857734363866308t_unit @ ( produc3434450474625521892t_unit @ X2 ) @ ( produc5506539147108004339t_unit @ A4 @ F ) )
= ( F @ X2 ) ) )
& ( ~ ( member_nat @ X2 @ A4 )
=> ( ( vimage7305857734363866308t_unit @ ( produc3434450474625521892t_unit @ X2 ) @ ( produc5506539147108004339t_unit @ A4 @ F ) )
= bot_bo3957492148770167129t_unit ) ) ) ).
% Pair_vimage_Sigma
thf(fact_3699_Pair__vimage__Sigma,axiom,
! [X2: int,A4: set_int,F: int > set_Product_unit] :
( ( ( member_int @ X2 @ A4 )
=> ( ( vimage4422279681566046568t_unit @ ( produc550872421827702152t_unit @ X2 ) @ ( produc2622961094310184599t_unit @ A4 @ F ) )
= ( F @ X2 ) ) )
& ( ~ ( member_int @ X2 @ A4 )
=> ( ( vimage4422279681566046568t_unit @ ( produc550872421827702152t_unit @ X2 ) @ ( produc2622961094310184599t_unit @ A4 @ F ) )
= bot_bo3957492148770167129t_unit ) ) ) ).
% Pair_vimage_Sigma
thf(fact_3700_Pair__vimage__Sigma,axiom,
! [X2: $o,A4: set_o,F: $o > set_o] :
( ( ( member_o @ X2 @ A4 )
=> ( ( vimage8945963521958007626od_o_o @ ( product_Pair_o_o @ X2 ) @ ( product_Sigma_o_o @ A4 @ F ) )
= ( F @ X2 ) ) )
& ( ~ ( member_o @ X2 @ A4 )
=> ( ( vimage8945963521958007626od_o_o @ ( product_Pair_o_o @ X2 ) @ ( product_Sigma_o_o @ A4 @ F ) )
= bot_bot_set_o ) ) ) ).
% Pair_vimage_Sigma
thf(fact_3701_Pair__vimage__Sigma,axiom,
! [X2: nat,A4: set_nat,F: nat > set_o] :
( ( ( member_nat @ X2 @ A4 )
=> ( ( vimage6125388955539849434_nat_o @ ( product_Pair_nat_o @ X2 ) @ ( product_Sigma_nat_o @ A4 @ F ) )
= ( F @ X2 ) ) )
& ( ~ ( member_nat @ X2 @ A4 )
=> ( ( vimage6125388955539849434_nat_o @ ( product_Pair_nat_o @ X2 ) @ ( product_Sigma_nat_o @ A4 @ F ) )
= bot_bot_set_o ) ) ) ).
% Pair_vimage_Sigma
thf(fact_3702_Pair__vimage__Sigma,axiom,
! [X2: int,A4: set_int,F: int > set_o] :
( ( ( member_int @ X2 @ A4 )
=> ( ( vimage4304346610107161470_int_o @ ( product_Pair_int_o @ X2 ) @ ( product_Sigma_int_o @ A4 @ F ) )
= ( F @ X2 ) ) )
& ( ~ ( member_int @ X2 @ A4 )
=> ( ( vimage4304346610107161470_int_o @ ( product_Pair_int_o @ X2 ) @ ( product_Sigma_int_o @ A4 @ F ) )
= bot_bot_set_o ) ) ) ).
% Pair_vimage_Sigma
thf(fact_3703_Pair__vimage__Sigma,axiom,
! [X2: $o,A4: set_o,F: $o > set_nat] :
( ( ( member_o @ X2 @ A4 )
=> ( ( vimage5656379182745576100_o_nat @ ( product_Pair_o_nat @ X2 ) @ ( product_Sigma_o_nat @ A4 @ F ) )
= ( F @ X2 ) ) )
& ( ~ ( member_o @ X2 @ A4 )
=> ( ( vimage5656379182745576100_o_nat @ ( product_Pair_o_nat @ X2 ) @ ( product_Sigma_o_nat @ A4 @ F ) )
= bot_bot_set_nat ) ) ) ).
% Pair_vimage_Sigma
thf(fact_3704_Pair__vimage__Sigma,axiom,
! [X2: int,A4: set_int,F: int > set_nat] :
( ( ( member_int @ X2 @ A4 )
=> ( ( vimage9012682431555523728nt_nat @ ( product_Pair_int_nat @ X2 ) @ ( produc1456381018704787142nt_nat @ A4 @ F ) )
= ( F @ X2 ) ) )
& ( ~ ( member_int @ X2 @ A4 )
=> ( ( vimage9012682431555523728nt_nat @ ( product_Pair_int_nat @ X2 ) @ ( produc1456381018704787142nt_nat @ A4 @ F ) )
= bot_bot_set_nat ) ) ) ).
% Pair_vimage_Sigma
thf(fact_3705_inj__vimage__image__eq,axiom,
! [F: nat > rat,A4: set_nat] :
( ( inj_on_nat_rat @ F @ top_top_set_nat )
=> ( ( vimage_nat_rat @ F @ ( image_nat_rat @ F @ A4 ) )
= A4 ) ) ).
% inj_vimage_image_eq
thf(fact_3706_inj__vimage__image__eq,axiom,
! [F: nat > nat,A4: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( vimage_nat_nat @ F @ ( image_nat_nat @ F @ A4 ) )
= A4 ) ) ).
% inj_vimage_image_eq
thf(fact_3707_inj__vimage__image__eq,axiom,
! [F: nat > int,A4: set_nat] :
( ( inj_on_nat_int @ F @ top_top_set_nat )
=> ( ( vimage_nat_int @ F @ ( image_nat_int @ F @ A4 ) )
= A4 ) ) ).
% inj_vimage_image_eq
thf(fact_3708_inj__vimage__image__eq,axiom,
! [F: int > nat,A4: set_int] :
( ( inj_on_int_nat @ F @ top_top_set_int )
=> ( ( vimage_int_nat @ F @ ( image_int_nat @ F @ A4 ) )
= A4 ) ) ).
% inj_vimage_image_eq
thf(fact_3709_inj__vimage__image__eq,axiom,
! [F: int > int,A4: set_int] :
( ( inj_on_int_int @ F @ top_top_set_int )
=> ( ( vimage_int_int @ F @ ( image_int_int @ F @ A4 ) )
= A4 ) ) ).
% inj_vimage_image_eq
thf(fact_3710_inj__graph,axiom,
( inj_on3412903767820855855at_nat
@ ^ [F2: nat > nat] :
( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [X4: nat,Y5: nat] :
( Y5
= ( F2 @ X4 ) ) ) )
@ top_top_set_nat_nat ) ).
% inj_graph
thf(fact_3711_inj__graph,axiom,
( inj_on7039184780174133295nt_int
@ ^ [F2: int > int] :
( collec213857154873943460nt_int
@ ( produc4947309494688390418_int_o
@ ^ [X4: int,Y5: int] :
( Y5
= ( F2 @ X4 ) ) ) )
@ top_top_set_int_int ) ).
% inj_graph
thf(fact_3712_f__the__inv__into__f,axiom,
! [F: nat > rat,A4: set_nat,Y3: rat] :
( ( inj_on_nat_rat @ F @ A4 )
=> ( ( member_rat @ Y3 @ ( image_nat_rat @ F @ A4 ) )
=> ( ( F @ ( the_inv_into_nat_rat @ A4 @ F @ Y3 ) )
= Y3 ) ) ) ).
% f_the_inv_into_f
thf(fact_3713_f__the__inv__into__f,axiom,
! [F: nat > nat,A4: set_nat,Y3: nat] :
( ( inj_on_nat_nat @ F @ A4 )
=> ( ( member_nat @ Y3 @ ( image_nat_nat @ F @ A4 ) )
=> ( ( F @ ( the_inv_into_nat_nat @ A4 @ F @ Y3 ) )
= Y3 ) ) ) ).
% f_the_inv_into_f
thf(fact_3714_f__the__inv__into__f,axiom,
! [F: int > nat,A4: set_int,Y3: nat] :
( ( inj_on_int_nat @ F @ A4 )
=> ( ( member_nat @ Y3 @ ( image_int_nat @ F @ A4 ) )
=> ( ( F @ ( the_inv_into_int_nat @ A4 @ F @ Y3 ) )
= Y3 ) ) ) ).
% f_the_inv_into_f
thf(fact_3715_f__the__inv__into__f,axiom,
! [F: nat > int,A4: set_nat,Y3: int] :
( ( inj_on_nat_int @ F @ A4 )
=> ( ( member_int @ Y3 @ ( image_nat_int @ F @ A4 ) )
=> ( ( F @ ( the_inv_into_nat_int @ A4 @ F @ Y3 ) )
= Y3 ) ) ) ).
% f_the_inv_into_f
thf(fact_3716_f__the__inv__into__f,axiom,
! [F: int > int,A4: set_int,Y3: int] :
( ( inj_on_int_int @ F @ A4 )
=> ( ( member_int @ Y3 @ ( image_int_int @ F @ A4 ) )
=> ( ( F @ ( the_inv_into_int_int @ A4 @ F @ Y3 ) )
= Y3 ) ) ) ).
% f_the_inv_into_f
thf(fact_3717_inj__on__the__inv__into,axiom,
! [F: nat > rat,A4: set_nat] :
( ( inj_on_nat_rat @ F @ A4 )
=> ( inj_on_rat_nat @ ( the_inv_into_nat_rat @ A4 @ F ) @ ( image_nat_rat @ F @ A4 ) ) ) ).
% inj_on_the_inv_into
thf(fact_3718_inj__on__the__inv__into,axiom,
! [F: nat > nat,A4: set_nat] :
( ( inj_on_nat_nat @ F @ A4 )
=> ( inj_on_nat_nat @ ( the_inv_into_nat_nat @ A4 @ F ) @ ( image_nat_nat @ F @ A4 ) ) ) ).
% inj_on_the_inv_into
thf(fact_3719_inj__on__the__inv__into,axiom,
! [F: nat > int,A4: set_nat] :
( ( inj_on_nat_int @ F @ A4 )
=> ( inj_on_int_nat @ ( the_inv_into_nat_int @ A4 @ F ) @ ( image_nat_int @ F @ A4 ) ) ) ).
% inj_on_the_inv_into
thf(fact_3720_inj__on__the__inv__into,axiom,
! [F: int > nat,A4: set_int] :
( ( inj_on_int_nat @ F @ A4 )
=> ( inj_on_nat_int @ ( the_inv_into_int_nat @ A4 @ F ) @ ( image_int_nat @ F @ A4 ) ) ) ).
% inj_on_the_inv_into
thf(fact_3721_inj__on__the__inv__into,axiom,
! [F: int > int,A4: set_int] :
( ( inj_on_int_int @ F @ A4 )
=> ( inj_on_int_int @ ( the_inv_into_int_int @ A4 @ F ) @ ( image_int_int @ F @ A4 ) ) ) ).
% inj_on_the_inv_into
thf(fact_3722_vimage__fst,axiom,
! [A4: set_ref_a] :
( ( vimage2910454108108208703_ref_a @ produc473321116828791851t_unit @ A4 )
= ( produc3946373962484743846t_unit @ A4
@ ^ [Uu: ref_a] : top_to3956321331435058429t_unit ) ) ).
% vimage_fst
thf(fact_3723_vimage__fst,axiom,
! [A4: set_nat] :
( ( vimage4653281326611754070at_nat @ product_fst_nat_nat @ A4 )
= ( produc457027306803732586at_nat @ A4
@ ^ [Uu: nat] : top_top_set_nat ) ) ).
% vimage_fst
thf(fact_3724_vimage__snd,axiom,
! [A4: set_he8644174446132254637t_unit] :
( ( vimage6116673589273818795t_unit @ produc1126150082675451753t_unit @ A4 )
= ( produc3946373962484743846t_unit @ top_top_set_ref_a
@ ^ [Uu: ref_a] : A4 ) ) ).
% vimage_snd
thf(fact_3725_vimage__snd,axiom,
! [A4: set_nat] :
( ( vimage4653281326611754070at_nat @ product_snd_nat_nat @ A4 )
= ( produc457027306803732586at_nat @ top_top_set_nat
@ ^ [Uu: nat] : A4 ) ) ).
% vimage_snd
thf(fact_3726_mono__Sup,axiom,
! [F: set_nat > set_nat,A4: set_set_nat] :
( ( order_1005506198798982270et_nat @ F )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ F @ A4 ) ) @ ( F @ ( comple7399068483239264473et_nat @ A4 ) ) ) ) ).
% mono_Sup
thf(fact_3727_mono__Sup,axiom,
! [F: product_unit > set_nat,A4: set_Product_unit] :
( ( order_509894322932672675et_nat @ F )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ F @ A4 ) ) @ ( F @ ( comple4687483117567863418t_unit @ A4 ) ) ) ) ).
% mono_Sup
thf(fact_3728_mono__Sup,axiom,
! [F: set_nat > product_unit,A4: set_set_nat] :
( ( order_7928700340835332885t_unit @ F )
=> ( ord_le3221252021190050221t_unit @ ( comple4687483117567863418t_unit @ ( image_6369283972875870912t_unit @ F @ A4 ) ) @ ( F @ ( comple7399068483239264473et_nat @ A4 ) ) ) ) ).
% mono_Sup
thf(fact_3729_mono__Sup,axiom,
! [F: product_unit > product_unit,A4: set_Product_unit] :
( ( order_4454465170704313136t_unit @ F )
=> ( ord_le3221252021190050221t_unit @ ( comple4687483117567863418t_unit @ ( image_405062704495631173t_unit @ F @ A4 ) ) @ ( F @ ( comple4687483117567863418t_unit @ A4 ) ) ) ) ).
% mono_Sup
thf(fact_3730_mono__Inf,axiom,
! [F: set_nat > set_nat,A4: set_set_nat] :
( ( order_1005506198798982270et_nat @ F )
=> ( ord_less_eq_set_nat @ ( F @ ( comple7806235888213564991et_nat @ A4 ) ) @ ( comple7806235888213564991et_nat @ ( image_7916887816326733075et_nat @ F @ A4 ) ) ) ) ).
% mono_Inf
thf(fact_3731_mono__Inf,axiom,
! [F: set_nat > product_unit,A4: set_set_nat] :
( ( order_7928700340835332885t_unit @ F )
=> ( ord_le3221252021190050221t_unit @ ( F @ ( comple7806235888213564991et_nat @ A4 ) ) @ ( comple2584293577114468500t_unit @ ( image_6369283972875870912t_unit @ F @ A4 ) ) ) ) ).
% mono_Inf
thf(fact_3732_mono__Inf,axiom,
! [F: product_unit > set_nat,A4: set_Product_unit] :
( ( order_509894322932672675et_nat @ F )
=> ( ord_less_eq_set_nat @ ( F @ ( comple2584293577114468500t_unit @ A4 ) ) @ ( comple7806235888213564991et_nat @ ( image_8173849991827986510et_nat @ F @ A4 ) ) ) ) ).
% mono_Inf
thf(fact_3733_mono__Inf,axiom,
! [F: product_unit > product_unit,A4: set_Product_unit] :
( ( order_4454465170704313136t_unit @ F )
=> ( ord_le3221252021190050221t_unit @ ( F @ ( comple2584293577114468500t_unit @ A4 ) ) @ ( comple2584293577114468500t_unit @ ( image_405062704495631173t_unit @ F @ A4 ) ) ) ) ).
% mono_Inf
thf(fact_3734_Pow__insert,axiom,
! [A: nat,A4: set_nat] :
( ( pow_nat @ ( insert_nat @ A @ A4 ) )
= ( sup_sup_set_set_nat @ ( pow_nat @ A4 ) @ ( image_7916887816326733075et_nat @ ( insert_nat @ A ) @ ( pow_nat @ A4 ) ) ) ) ).
% Pow_insert
thf(fact_3735_Pow__insert,axiom,
! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
( ( pow_Pr3466873192417945886at_nat @ ( insert8211810215607154385at_nat @ A @ A4 ) )
= ( sup_su3642409539654194069at_nat @ ( pow_Pr3466873192417945886at_nat @ A4 ) @ ( image_3684629450409544005at_nat @ ( insert8211810215607154385at_nat @ A ) @ ( pow_Pr3466873192417945886at_nat @ A4 ) ) ) ) ).
% Pow_insert
thf(fact_3736_Pow__insert,axiom,
! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
( ( pow_Pr1023649790151954692at_nat @ ( insert9069300056098147895at_nat @ A @ A4 ) )
= ( sup_su474284620922681339at_nat @ ( pow_Pr1023649790151954692at_nat @ A4 ) @ ( image_4303274092540115909at_nat @ ( insert9069300056098147895at_nat @ A ) @ ( pow_Pr1023649790151954692at_nat @ A4 ) ) ) ) ).
% Pow_insert
thf(fact_3737_Pow__insert,axiom,
! [A: product_unit,A4: set_Product_unit] :
( ( pow_Product_unit @ ( insert_Product_unit @ A @ A4 ) )
= ( sup_su5975557051335651745t_unit @ ( pow_Product_unit @ A4 ) @ ( image_4019945301284682565t_unit @ ( insert_Product_unit @ A ) @ ( pow_Product_unit @ A4 ) ) ) ) ).
% Pow_insert
thf(fact_3738_Pow__insert,axiom,
! [A: $o,A4: set_o] :
( ( pow_o @ ( insert_o @ A @ A4 ) )
= ( sup_sup_set_set_o @ ( pow_o @ A4 ) @ ( image_set_o_set_o @ ( insert_o @ A ) @ ( pow_o @ A4 ) ) ) ) ).
% Pow_insert
thf(fact_3739_vimage__subsetI,axiom,
! [F: nat > rat,B5: set_rat,A4: set_nat] :
( ( inj_on_nat_rat @ F @ top_top_set_nat )
=> ( ( ord_less_eq_set_rat @ B5 @ ( image_nat_rat @ F @ A4 ) )
=> ( ord_less_eq_set_nat @ ( vimage_nat_rat @ F @ B5 ) @ A4 ) ) ) ).
% vimage_subsetI
thf(fact_3740_vimage__subsetI,axiom,
! [F: nat > nat,B5: set_nat,A4: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat @ F @ A4 ) )
=> ( ord_less_eq_set_nat @ ( vimage_nat_nat @ F @ B5 ) @ A4 ) ) ) ).
% vimage_subsetI
thf(fact_3741_vimage__subsetI,axiom,
! [F: nat > int,B5: set_int,A4: set_nat] :
( ( inj_on_nat_int @ F @ top_top_set_nat )
=> ( ( ord_less_eq_set_int @ B5 @ ( image_nat_int @ F @ A4 ) )
=> ( ord_less_eq_set_nat @ ( vimage_nat_int @ F @ B5 ) @ A4 ) ) ) ).
% vimage_subsetI
thf(fact_3742_vimage__subsetI,axiom,
! [F: int > nat,B5: set_nat,A4: set_int] :
( ( inj_on_int_nat @ F @ top_top_set_int )
=> ( ( ord_less_eq_set_nat @ B5 @ ( image_int_nat @ F @ A4 ) )
=> ( ord_less_eq_set_int @ ( vimage_int_nat @ F @ B5 ) @ A4 ) ) ) ).
% vimage_subsetI
thf(fact_3743_vimage__subsetI,axiom,
! [F: int > int,B5: set_int,A4: set_int] :
( ( inj_on_int_int @ F @ top_top_set_int )
=> ( ( ord_less_eq_set_int @ B5 @ ( image_int_int @ F @ A4 ) )
=> ( ord_less_eq_set_int @ ( vimage_int_int @ F @ B5 ) @ A4 ) ) ) ).
% vimage_subsetI
thf(fact_3744_vimage__eq__UN,axiom,
( vimage4884490618288580032t_unit
= ( ^ [F2: nat > product_unit,B6: set_Product_unit] :
( comple7399068483239264473et_nat
@ ( image_8173849991827986510et_nat
@ ^ [Y5: product_unit] : ( vimage4884490618288580032t_unit @ F2 @ ( insert_Product_unit @ Y5 @ bot_bo3957492148770167129t_unit ) )
@ B6 ) ) ) ) ).
% vimage_eq_UN
thf(fact_3745_vimage__eq__UN,axiom,
( vimage6435164912253009178at_nat
= ( ^ [F2: nat > produc3843707927480180839at_nat,B6: set_Pr4329608150637261639at_nat] :
( comple7399068483239264473et_nat
@ ( image_3160451284896210088et_nat
@ ^ [Y5: produc3843707927480180839at_nat] : ( vimage6435164912253009178at_nat @ F2 @ ( insert9069300056098147895at_nat @ Y5 @ bot_bo228742789529271731at_nat ) )
@ B6 ) ) ) ) ).
% vimage_eq_UN
thf(fact_3746_vimage__eq__UN,axiom,
( vimage8013328719654469172at_nat
= ( ^ [F2: nat > product_prod_nat_nat,B6: set_Pr1261947904930325089at_nat] :
( comple7399068483239264473et_nat
@ ( image_15824709712370754et_nat
@ ^ [Y5: product_prod_nat_nat] : ( vimage8013328719654469172at_nat @ F2 @ ( insert8211810215607154385at_nat @ Y5 @ bot_bo2099793752762293965at_nat ) )
@ B6 ) ) ) ) ).
% vimage_eq_UN
thf(fact_3747_vimage__eq__UN,axiom,
( vimage_nat_o
= ( ^ [F2: nat > $o,B6: set_o] :
( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [Y5: $o] : ( vimage_nat_o @ F2 @ ( insert_o @ Y5 @ bot_bot_set_o ) )
@ B6 ) ) ) ) ).
% vimage_eq_UN
thf(fact_3748_vimage__eq__UN,axiom,
( vimage_nat_nat
= ( ^ [F2: nat > nat,B6: set_nat] :
( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y5: nat] : ( vimage_nat_nat @ F2 @ ( insert_nat @ Y5 @ bot_bot_set_nat ) )
@ B6 ) ) ) ) ).
% vimage_eq_UN
thf(fact_3749_the__inv__into__into,axiom,
! [F: nat > rat,A4: set_nat,X2: rat,B5: set_nat] :
( ( inj_on_nat_rat @ F @ A4 )
=> ( ( member_rat @ X2 @ ( image_nat_rat @ F @ A4 ) )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( member_nat @ ( the_inv_into_nat_rat @ A4 @ F @ X2 ) @ B5 ) ) ) ) ).
% the_inv_into_into
thf(fact_3750_the__inv__into__into,axiom,
! [F: $o > $o,A4: set_o,X2: $o,B5: set_o] :
( ( inj_on_o_o @ F @ A4 )
=> ( ( member_o @ X2 @ ( image_o_o @ F @ A4 ) )
=> ( ( ord_less_eq_set_o @ A4 @ B5 )
=> ( member_o @ ( the_inv_into_o_o @ A4 @ F @ X2 ) @ B5 ) ) ) ) ).
% the_inv_into_into
thf(fact_3751_the__inv__into__into,axiom,
! [F: nat > $o,A4: set_nat,X2: $o,B5: set_nat] :
( ( inj_on_nat_o @ F @ A4 )
=> ( ( member_o @ X2 @ ( image_nat_o @ F @ A4 ) )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( member_nat @ ( the_inv_into_nat_o @ A4 @ F @ X2 ) @ B5 ) ) ) ) ).
% the_inv_into_into
thf(fact_3752_the__inv__into__into,axiom,
! [F: int > $o,A4: set_int,X2: $o,B5: set_int] :
( ( inj_on_int_o @ F @ A4 )
=> ( ( member_o @ X2 @ ( image_int_o @ F @ A4 ) )
=> ( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( member_int @ ( the_inv_into_int_o @ A4 @ F @ X2 ) @ B5 ) ) ) ) ).
% the_inv_into_into
thf(fact_3753_the__inv__into__into,axiom,
! [F: $o > nat,A4: set_o,X2: nat,B5: set_o] :
( ( inj_on_o_nat @ F @ A4 )
=> ( ( member_nat @ X2 @ ( image_o_nat @ F @ A4 ) )
=> ( ( ord_less_eq_set_o @ A4 @ B5 )
=> ( member_o @ ( the_inv_into_o_nat @ A4 @ F @ X2 ) @ B5 ) ) ) ) ).
% the_inv_into_into
thf(fact_3754_the__inv__into__into,axiom,
! [F: nat > nat,A4: set_nat,X2: nat,B5: set_nat] :
( ( inj_on_nat_nat @ F @ A4 )
=> ( ( member_nat @ X2 @ ( image_nat_nat @ F @ A4 ) )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( member_nat @ ( the_inv_into_nat_nat @ A4 @ F @ X2 ) @ B5 ) ) ) ) ).
% the_inv_into_into
thf(fact_3755_the__inv__into__into,axiom,
! [F: int > nat,A4: set_int,X2: nat,B5: set_int] :
( ( inj_on_int_nat @ F @ A4 )
=> ( ( member_nat @ X2 @ ( image_int_nat @ F @ A4 ) )
=> ( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( member_int @ ( the_inv_into_int_nat @ A4 @ F @ X2 ) @ B5 ) ) ) ) ).
% the_inv_into_into
thf(fact_3756_the__inv__into__into,axiom,
! [F: $o > int,A4: set_o,X2: int,B5: set_o] :
( ( inj_on_o_int @ F @ A4 )
=> ( ( member_int @ X2 @ ( image_o_int @ F @ A4 ) )
=> ( ( ord_less_eq_set_o @ A4 @ B5 )
=> ( member_o @ ( the_inv_into_o_int @ A4 @ F @ X2 ) @ B5 ) ) ) ) ).
% the_inv_into_into
thf(fact_3757_the__inv__into__into,axiom,
! [F: nat > int,A4: set_nat,X2: int,B5: set_nat] :
( ( inj_on_nat_int @ F @ A4 )
=> ( ( member_int @ X2 @ ( image_nat_int @ F @ A4 ) )
=> ( ( ord_less_eq_set_nat @ A4 @ B5 )
=> ( member_nat @ ( the_inv_into_nat_int @ A4 @ F @ X2 ) @ B5 ) ) ) ) ).
% the_inv_into_into
thf(fact_3758_the__inv__into__into,axiom,
! [F: int > int,A4: set_int,X2: int,B5: set_int] :
( ( inj_on_int_int @ F @ A4 )
=> ( ( member_int @ X2 @ ( image_int_int @ F @ A4 ) )
=> ( ( ord_less_eq_set_int @ A4 @ B5 )
=> ( member_int @ ( the_inv_into_int_int @ A4 @ F @ X2 ) @ B5 ) ) ) ) ).
% the_inv_into_into
thf(fact_3759_the__inv__into__comp,axiom,
! [F: nat > rat,G: nat > nat,A4: set_nat,X2: rat] :
( ( inj_on_nat_rat @ F @ ( image_nat_nat @ G @ A4 ) )
=> ( ( inj_on_nat_nat @ G @ A4 )
=> ( ( member_rat @ X2 @ ( image_nat_rat @ F @ ( image_nat_nat @ G @ A4 ) ) )
=> ( ( the_inv_into_nat_rat @ A4 @ ( comp_nat_rat_nat @ F @ G ) @ X2 )
= ( comp_nat_nat_rat @ ( the_inv_into_nat_nat @ A4 @ G ) @ ( the_inv_into_nat_rat @ ( image_nat_nat @ G @ A4 ) @ F ) @ X2 ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_3760_the__inv__into__comp,axiom,
! [F: nat > num,G: int > nat,A4: set_int,X2: num] :
( ( inj_on_nat_num @ F @ ( image_int_nat @ G @ A4 ) )
=> ( ( inj_on_int_nat @ G @ A4 )
=> ( ( member_num @ X2 @ ( image_nat_num @ F @ ( image_int_nat @ G @ A4 ) ) )
=> ( ( the_inv_into_int_num @ A4 @ ( comp_nat_num_int @ F @ G ) @ X2 )
= ( comp_nat_int_num @ ( the_inv_into_int_nat @ A4 @ G ) @ ( the_inv_into_nat_num @ ( image_int_nat @ G @ A4 ) @ F ) @ X2 ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_3761_the__inv__into__comp,axiom,
! [F: nat > rat,G: int > nat,A4: set_int,X2: rat] :
( ( inj_on_nat_rat @ F @ ( image_int_nat @ G @ A4 ) )
=> ( ( inj_on_int_nat @ G @ A4 )
=> ( ( member_rat @ X2 @ ( image_nat_rat @ F @ ( image_int_nat @ G @ A4 ) ) )
=> ( ( the_inv_into_int_rat @ A4 @ ( comp_nat_rat_int @ F @ G ) @ X2 )
= ( comp_nat_int_rat @ ( the_inv_into_int_nat @ A4 @ G ) @ ( the_inv_into_nat_rat @ ( image_int_nat @ G @ A4 ) @ F ) @ X2 ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_3762_the__inv__into__comp,axiom,
! [F: $o > $o,G: nat > $o,A4: set_nat,X2: $o] :
( ( inj_on_o_o @ F @ ( image_nat_o @ G @ A4 ) )
=> ( ( inj_on_nat_o @ G @ A4 )
=> ( ( member_o @ X2 @ ( image_o_o @ F @ ( image_nat_o @ G @ A4 ) ) )
=> ( ( the_inv_into_nat_o @ A4 @ ( comp_o_o_nat @ F @ G ) @ X2 )
= ( comp_o_nat_o @ ( the_inv_into_nat_o @ A4 @ G ) @ ( the_inv_into_o_o @ ( image_nat_o @ G @ A4 ) @ F ) @ X2 ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_3763_the__inv__into__comp,axiom,
! [F: rat > $o,G: nat > rat,A4: set_nat,X2: $o] :
( ( inj_on_rat_o @ F @ ( image_nat_rat @ G @ A4 ) )
=> ( ( inj_on_nat_rat @ G @ A4 )
=> ( ( member_o @ X2 @ ( image_rat_o @ F @ ( image_nat_rat @ G @ A4 ) ) )
=> ( ( the_inv_into_nat_o @ A4 @ ( comp_rat_o_nat @ F @ G ) @ X2 )
= ( comp_rat_nat_o @ ( the_inv_into_nat_rat @ A4 @ G ) @ ( the_inv_into_rat_o @ ( image_nat_rat @ G @ A4 ) @ F ) @ X2 ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_3764_the__inv__into__comp,axiom,
! [F: nat > $o,G: nat > nat,A4: set_nat,X2: $o] :
( ( inj_on_nat_o @ F @ ( image_nat_nat @ G @ A4 ) )
=> ( ( inj_on_nat_nat @ G @ A4 )
=> ( ( member_o @ X2 @ ( image_nat_o @ F @ ( image_nat_nat @ G @ A4 ) ) )
=> ( ( the_inv_into_nat_o @ A4 @ ( comp_nat_o_nat @ F @ G ) @ X2 )
= ( comp_nat_nat_o @ ( the_inv_into_nat_nat @ A4 @ G ) @ ( the_inv_into_nat_o @ ( image_nat_nat @ G @ A4 ) @ F ) @ X2 ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_3765_the__inv__into__comp,axiom,
! [F: int > $o,G: nat > int,A4: set_nat,X2: $o] :
( ( inj_on_int_o @ F @ ( image_nat_int @ G @ A4 ) )
=> ( ( inj_on_nat_int @ G @ A4 )
=> ( ( member_o @ X2 @ ( image_int_o @ F @ ( image_nat_int @ G @ A4 ) ) )
=> ( ( the_inv_into_nat_o @ A4 @ ( comp_int_o_nat @ F @ G ) @ X2 )
= ( comp_int_nat_o @ ( the_inv_into_nat_int @ A4 @ G ) @ ( the_inv_into_int_o @ ( image_nat_int @ G @ A4 ) @ F ) @ X2 ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_3766_the__inv__into__comp,axiom,
! [F: nat > $o,G: int > nat,A4: set_int,X2: $o] :
( ( inj_on_nat_o @ F @ ( image_int_nat @ G @ A4 ) )
=> ( ( inj_on_int_nat @ G @ A4 )
=> ( ( member_o @ X2 @ ( image_nat_o @ F @ ( image_int_nat @ G @ A4 ) ) )
=> ( ( the_inv_into_int_o @ A4 @ ( comp_nat_o_int @ F @ G ) @ X2 )
= ( comp_nat_int_o @ ( the_inv_into_int_nat @ A4 @ G ) @ ( the_inv_into_nat_o @ ( image_int_nat @ G @ A4 ) @ F ) @ X2 ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_3767_the__inv__into__comp,axiom,
! [F: int > $o,G: int > int,A4: set_int,X2: $o] :
( ( inj_on_int_o @ F @ ( image_int_int @ G @ A4 ) )
=> ( ( inj_on_int_int @ G @ A4 )
=> ( ( member_o @ X2 @ ( image_int_o @ F @ ( image_int_int @ G @ A4 ) ) )
=> ( ( the_inv_into_int_o @ A4 @ ( comp_int_o_int @ F @ G ) @ X2 )
= ( comp_int_int_o @ ( the_inv_into_int_int @ A4 @ G ) @ ( the_inv_into_int_o @ ( image_int_int @ G @ A4 ) @ F ) @ X2 ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_3768_the__inv__into__comp,axiom,
! [F: $o > nat,G: $o > $o,A4: set_o,X2: nat] :
( ( inj_on_o_nat @ F @ ( image_o_o @ G @ A4 ) )
=> ( ( inj_on_o_o @ G @ A4 )
=> ( ( member_nat @ X2 @ ( image_o_nat @ F @ ( image_o_o @ G @ A4 ) ) )
=> ( ( the_inv_into_o_nat @ A4 @ ( comp_o_nat_o @ F @ G ) @ X2 )
= ( comp_o_o_nat @ ( the_inv_into_o_o @ A4 @ G ) @ ( the_inv_into_o_nat @ ( image_o_o @ G @ A4 ) @ F ) @ X2 ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_3769_fun__upd__image,axiom,
! [X2: int,A4: set_int,F: int > int,Y3: int] :
( ( ( member_int @ X2 @ A4 )
=> ( ( image_int_int @ ( fun_upd_int_int @ F @ X2 @ Y3 ) @ A4 )
= ( insert_int @ Y3 @ ( image_int_int @ F @ ( minus_minus_set_int @ A4 @ ( insert_int @ X2 @ bot_bot_set_int ) ) ) ) ) )
& ( ~ ( member_int @ X2 @ A4 )
=> ( ( image_int_int @ ( fun_upd_int_int @ F @ X2 @ Y3 ) @ A4 )
= ( image_int_int @ F @ A4 ) ) ) ) ).
% fun_upd_image
thf(fact_3770_fun__upd__image,axiom,
! [X2: int,A4: set_int,F: int > nat,Y3: nat] :
( ( ( member_int @ X2 @ A4 )
=> ( ( image_int_nat @ ( fun_upd_int_nat @ F @ X2 @ Y3 ) @ A4 )
= ( insert_nat @ Y3 @ ( image_int_nat @ F @ ( minus_minus_set_int @ A4 @ ( insert_int @ X2 @ bot_bot_set_int ) ) ) ) ) )
& ( ~ ( member_int @ X2 @ A4 )
=> ( ( image_int_nat @ ( fun_upd_int_nat @ F @ X2 @ Y3 ) @ A4 )
= ( image_int_nat @ F @ A4 ) ) ) ) ).
% fun_upd_image
thf(fact_3771_fun__upd__image,axiom,
! [X2: int,A4: set_int,F: int > product_unit,Y3: product_unit] :
( ( ( member_int @ X2 @ A4 )
=> ( ( image_5846526143423701914t_unit @ ( fun_up2220634978854545698t_unit @ F @ X2 @ Y3 ) @ A4 )
= ( insert_Product_unit @ Y3 @ ( image_5846526143423701914t_unit @ F @ ( minus_minus_set_int @ A4 @ ( insert_int @ X2 @ bot_bot_set_int ) ) ) ) ) )
& ( ~ ( member_int @ X2 @ A4 )
=> ( ( image_5846526143423701914t_unit @ ( fun_up2220634978854545698t_unit @ F @ X2 @ Y3 ) @ A4 )
= ( image_5846526143423701914t_unit @ F @ A4 ) ) ) ) ).
% fun_upd_image
thf(fact_3772_fun__upd__image,axiom,
! [X2: int,A4: set_int,F: int > $o,Y3: $o] :
( ( ( member_int @ X2 @ A4 )
=> ( ( image_int_o @ ( fun_upd_int_o @ F @ X2 @ Y3 ) @ A4 )
= ( insert_o @ Y3 @ ( image_int_o @ F @ ( minus_minus_set_int @ A4 @ ( insert_int @ X2 @ bot_bot_set_int ) ) ) ) ) )
& ( ~ ( member_int @ X2 @ A4 )
=> ( ( image_int_o @ ( fun_upd_int_o @ F @ X2 @ Y3 ) @ A4 )
= ( image_int_o @ F @ A4 ) ) ) ) ).
% fun_upd_image
thf(fact_3773_fun__upd__image,axiom,
! [X2: product_unit,A4: set_Product_unit,F: product_unit > nat,Y3: nat] :
( ( ( member_Product_unit @ X2 @ A4 )
=> ( ( image_875570014554754200it_nat @ ( fun_up6473050886840373792it_nat @ F @ X2 @ Y3 ) @ A4 )
= ( insert_nat @ Y3 @ ( image_875570014554754200it_nat @ F @ ( minus_6452836326544984404t_unit @ A4 @ ( insert_Product_unit @ X2 @ bot_bo3957492148770167129t_unit ) ) ) ) ) )
& ( ~ ( member_Product_unit @ X2 @ A4 )
=> ( ( image_875570014554754200it_nat @ ( fun_up6473050886840373792it_nat @ F @ X2 @ Y3 ) @ A4 )
= ( image_875570014554754200it_nat @ F @ A4 ) ) ) ) ).
% fun_upd_image
thf(fact_3774_fun__upd__image,axiom,
! [X2: product_unit,A4: set_Product_unit,F: product_unit > product_unit,Y3: product_unit] :
( ( ( member_Product_unit @ X2 @ A4 )
=> ( ( image_405062704495631173t_unit @ ( fun_up6875000483049295805t_unit @ F @ X2 @ Y3 ) @ A4 )
= ( insert_Product_unit @ Y3 @ ( image_405062704495631173t_unit @ F @ ( minus_6452836326544984404t_unit @ A4 @ ( insert_Product_unit @ X2 @ bot_bo3957492148770167129t_unit ) ) ) ) ) )
& ( ~ ( member_Product_unit @ X2 @ A4 )
=> ( ( image_405062704495631173t_unit @ ( fun_up6875000483049295805t_unit @ F @ X2 @ Y3 ) @ A4 )
= ( image_405062704495631173t_unit @ F @ A4 ) ) ) ) ).
% fun_upd_image
thf(fact_3775_fun__upd__image,axiom,
! [X2: product_unit,A4: set_Product_unit,F: product_unit > $o,Y3: $o] :
( ( ( member_Product_unit @ X2 @ A4 )
=> ( ( image_Product_unit_o @ ( fun_up5432024876597553480unit_o @ F @ X2 @ Y3 ) @ A4 )
= ( insert_o @ Y3 @ ( image_Product_unit_o @ F @ ( minus_6452836326544984404t_unit @ A4 @ ( insert_Product_unit @ X2 @ bot_bo3957492148770167129t_unit ) ) ) ) ) )
& ( ~ ( member_Product_unit @ X2 @ A4 )
=> ( ( image_Product_unit_o @ ( fun_up5432024876597553480unit_o @ F @ X2 @ Y3 ) @ A4 )
= ( image_Product_unit_o @ F @ A4 ) ) ) ) ).
% fun_upd_image
thf(fact_3776_fun__upd__image,axiom,
! [X2: $o,A4: set_o,F: $o > nat,Y3: nat] :
( ( ( member_o @ X2 @ A4 )
=> ( ( image_o_nat @ ( fun_upd_o_nat @ F @ X2 @ Y3 ) @ A4 )
= ( insert_nat @ Y3 @ ( image_o_nat @ F @ ( minus_minus_set_o @ A4 @ ( insert_o @ X2 @ bot_bot_set_o ) ) ) ) ) )
& ( ~ ( member_o @ X2 @ A4 )
=> ( ( image_o_nat @ ( fun_upd_o_nat @ F @ X2 @ Y3 ) @ A4 )
= ( image_o_nat @ F @ A4 ) ) ) ) ).
% fun_upd_image
thf(fact_3777_fun__upd__image,axiom,
! [X2: $o,A4: set_o,F: $o > product_unit,Y3: product_unit] :
( ( ( member_o @ X2 @ A4 )
=> ( ( image_o_Product_unit @ ( fun_up846054379331416818t_unit @ F @ X2 @ Y3 ) @ A4 )
= ( insert_Product_unit @ Y3 @ ( image_o_Product_unit @ F @ ( minus_minus_set_o @ A4 @ ( insert_o @ X2 @ bot_bot_set_o ) ) ) ) ) )
& ( ~ ( member_o @ X2 @ A4 )
=> ( ( image_o_Product_unit @ ( fun_up846054379331416818t_unit @ F @ X2 @ Y3 ) @ A4 )
= ( image_o_Product_unit @ F @ A4 ) ) ) ) ).
% fun_upd_image
thf(fact_3778_fun__upd__image,axiom,
! [X2: $o,A4: set_o,F: $o > $o,Y3: $o] :
( ( ( member_o @ X2 @ A4 )
=> ( ( image_o_o @ ( fun_upd_o_o @ F @ X2 @ Y3 ) @ A4 )
= ( insert_o @ Y3 @ ( image_o_o @ F @ ( minus_minus_set_o @ A4 @ ( insert_o @ X2 @ bot_bot_set_o ) ) ) ) ) )
& ( ~ ( member_o @ X2 @ A4 )
=> ( ( image_o_o @ ( fun_upd_o_o @ F @ X2 @ Y3 ) @ A4 )
= ( image_o_o @ F @ A4 ) ) ) ) ).
% fun_upd_image
thf(fact_3779_mono__cSUP,axiom,
! [F: nat > nat,A4: int > nat,I3: set_int] :
( ( order_mono_nat_nat @ F )
=> ( ( condit2214826472909112428ve_nat @ ( image_int_nat @ A4 @ I3 ) )
=> ( ( I3 != bot_bot_set_int )
=> ( ord_less_eq_nat
@ ( complete_Sup_Sup_nat
@ ( image_int_nat
@ ^ [X4: int] : ( F @ ( A4 @ X4 ) )
@ I3 ) )
@ ( F @ ( complete_Sup_Sup_nat @ ( image_int_nat @ A4 @ I3 ) ) ) ) ) ) ) ).
% mono_cSUP
thf(fact_3780_mono__cSUP,axiom,
! [F: nat > nat,A4: nat > nat,I3: set_nat] :
( ( order_mono_nat_nat @ F )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ A4 @ I3 ) )
=> ( ( I3 != bot_bot_set_nat )
=> ( ord_less_eq_nat
@ ( complete_Sup_Sup_nat
@ ( image_nat_nat
@ ^ [X4: nat] : ( F @ ( A4 @ X4 ) )
@ I3 ) )
@ ( F @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ A4 @ I3 ) ) ) ) ) ) ) ).
% mono_cSUP
thf(fact_3781_mono__cSUP,axiom,
! [F: int > nat,A4: int > int,I3: set_int] :
( ( order_mono_int_nat @ F )
=> ( ( condit2212336002400062152ve_int @ ( image_int_int @ A4 @ I3 ) )
=> ( ( I3 != bot_bot_set_int )
=> ( ord_less_eq_nat
@ ( complete_Sup_Sup_nat
@ ( image_int_nat
@ ^ [X4: int] : ( F @ ( A4 @ X4 ) )
@ I3 ) )
@ ( F @ ( complete_Sup_Sup_int @ ( image_int_int @ A4 @ I3 ) ) ) ) ) ) ) ).
% mono_cSUP
thf(fact_3782_mono__cSUP,axiom,
! [F: int > nat,A4: product_unit > int,I3: set_Product_unit] :
( ( order_mono_int_nat @ F )
=> ( ( condit2212336002400062152ve_int @ ( image_873079544045703924it_int @ A4 @ I3 ) )
=> ( ( I3 != bot_bo3957492148770167129t_unit )
=> ( ord_less_eq_nat
@ ( complete_Sup_Sup_nat
@ ( image_875570014554754200it_nat
@ ^ [X4: product_unit] : ( F @ ( A4 @ X4 ) )
@ I3 ) )
@ ( F @ ( complete_Sup_Sup_int @ ( image_873079544045703924it_int @ A4 @ I3 ) ) ) ) ) ) ) ).
% mono_cSUP
thf(fact_3783_mono__cSUP,axiom,
! [F: int > nat,A4: $o > int,I3: set_o] :
( ( order_mono_int_nat @ F )
=> ( ( condit2212336002400062152ve_int @ ( image_o_int @ A4 @ I3 ) )
=> ( ( I3 != bot_bot_set_o )
=> ( ord_less_eq_nat
@ ( complete_Sup_Sup_nat
@ ( image_o_nat
@ ^ [X4: $o] : ( F @ ( A4 @ X4 ) )
@ I3 ) )
@ ( F @ ( complete_Sup_Sup_int @ ( image_o_int @ A4 @ I3 ) ) ) ) ) ) ) ).
% mono_cSUP
thf(fact_3784_mono__cSUP,axiom,
! [F: int > nat,A4: nat > int,I3: set_nat] :
( ( order_mono_int_nat @ F )
=> ( ( condit2212336002400062152ve_int @ ( image_nat_int @ A4 @ I3 ) )
=> ( ( I3 != bot_bot_set_nat )
=> ( ord_less_eq_nat
@ ( complete_Sup_Sup_nat
@ ( image_nat_nat
@ ^ [X4: nat] : ( F @ ( A4 @ X4 ) )
@ I3 ) )
@ ( F @ ( complete_Sup_Sup_int @ ( image_nat_int @ A4 @ I3 ) ) ) ) ) ) ) ).
% mono_cSUP
thf(fact_3785_mono__cSUP,axiom,
! [F: product_unit > nat,A4: int > product_unit,I3: set_int] :
( ( order_7096114683908731117it_nat @ F )
=> ( ( condit2080475194861589617t_unit @ ( image_5846526143423701914t_unit @ A4 @ I3 ) )
=> ( ( I3 != bot_bot_set_int )
=> ( ord_less_eq_nat
@ ( complete_Sup_Sup_nat
@ ( image_int_nat
@ ^ [X4: int] : ( F @ ( A4 @ X4 ) )
@ I3 ) )
@ ( F @ ( comple4687483117567863418t_unit @ ( image_5846526143423701914t_unit @ A4 @ I3 ) ) ) ) ) ) ) ).
% mono_cSUP
thf(fact_3786_mono__cSUP,axiom,
! [F: product_unit > nat,A4: product_unit > product_unit,I3: set_Product_unit] :
( ( order_7096114683908731117it_nat @ F )
=> ( ( condit2080475194861589617t_unit @ ( image_405062704495631173t_unit @ A4 @ I3 ) )
=> ( ( I3 != bot_bo3957492148770167129t_unit )
=> ( ord_less_eq_nat
@ ( complete_Sup_Sup_nat
@ ( image_875570014554754200it_nat
@ ^ [X4: product_unit] : ( F @ ( A4 @ X4 ) )
@ I3 ) )
@ ( F @ ( comple4687483117567863418t_unit @ ( image_405062704495631173t_unit @ A4 @ I3 ) ) ) ) ) ) ) ).
% mono_cSUP
thf(fact_3787_mono__cSUP,axiom,
! [F: product_unit > nat,A4: $o > product_unit,I3: set_o] :
( ( order_7096114683908731117it_nat @ F )
=> ( ( condit2080475194861589617t_unit @ ( image_o_Product_unit @ A4 @ I3 ) )
=> ( ( I3 != bot_bot_set_o )
=> ( ord_less_eq_nat
@ ( complete_Sup_Sup_nat
@ ( image_o_nat
@ ^ [X4: $o] : ( F @ ( A4 @ X4 ) )
@ I3 ) )
@ ( F @ ( comple4687483117567863418t_unit @ ( image_o_Product_unit @ A4 @ I3 ) ) ) ) ) ) ) ).
% mono_cSUP
thf(fact_3788_mono__cSUP,axiom,
! [F: product_unit > nat,A4: nat > product_unit,I3: set_nat] :
( ( order_7096114683908731117it_nat @ F )
=> ( ( condit2080475194861589617t_unit @ ( image_8730104196221521654t_unit @ A4 @ I3 ) )
=> ( ( I3 != bot_bot_set_nat )
=> ( ord_less_eq_nat
@ ( complete_Sup_Sup_nat
@ ( image_nat_nat
@ ^ [X4: nat] : ( F @ ( A4 @ X4 ) )
@ I3 ) )
@ ( F @ ( comple4687483117567863418t_unit @ ( image_8730104196221521654t_unit @ A4 @ I3 ) ) ) ) ) ) ) ).
% mono_cSUP
thf(fact_3789_mono__cSup,axiom,
! [F: $o > nat,A4: set_o] :
( ( order_mono_o_nat @ F )
=> ( ( condit5488710616941104124bove_o @ A4 )
=> ( ( A4 != bot_bot_set_o )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_o_nat @ F @ A4 ) ) @ ( F @ ( complete_Sup_Sup_o @ A4 ) ) ) ) ) ) ).
% mono_cSup
thf(fact_3790_mono__cSup,axiom,
! [F: nat > nat,A4: set_nat] :
( ( order_mono_nat_nat @ F )
=> ( ( condit2214826472909112428ve_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ A4 ) ) @ ( F @ ( complete_Sup_Sup_nat @ A4 ) ) ) ) ) ) ).
% mono_cSup
thf(fact_3791_mono__cSup,axiom,
! [F: int > nat,A4: set_int] :
( ( order_mono_int_nat @ F )
=> ( ( condit2212336002400062152ve_int @ A4 )
=> ( ( A4 != bot_bot_set_int )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_int_nat @ F @ A4 ) ) @ ( F @ ( complete_Sup_Sup_int @ A4 ) ) ) ) ) ) ).
% mono_cSup
thf(fact_3792_mono__cSup,axiom,
! [F: product_unit > nat,A4: set_Product_unit] :
( ( order_7096114683908731117it_nat @ F )
=> ( ( condit2080475194861589617t_unit @ A4 )
=> ( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_875570014554754200it_nat @ F @ A4 ) ) @ ( F @ ( comple4687483117567863418t_unit @ A4 ) ) ) ) ) ) ).
% mono_cSup
thf(fact_3793_mono__cSup,axiom,
! [F: $o > int,A4: set_o] :
( ( order_mono_o_int @ F )
=> ( ( condit5488710616941104124bove_o @ A4 )
=> ( ( A4 != bot_bot_set_o )
=> ( ord_less_eq_int @ ( complete_Sup_Sup_int @ ( image_o_int @ F @ A4 ) ) @ ( F @ ( complete_Sup_Sup_o @ A4 ) ) ) ) ) ) ).
% mono_cSup
thf(fact_3794_mono__cSup,axiom,
! [F: nat > int,A4: set_nat] :
( ( order_mono_nat_int @ F )
=> ( ( condit2214826472909112428ve_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ord_less_eq_int @ ( complete_Sup_Sup_int @ ( image_nat_int @ F @ A4 ) ) @ ( F @ ( complete_Sup_Sup_nat @ A4 ) ) ) ) ) ) ).
% mono_cSup
thf(fact_3795_mono__cSup,axiom,
! [F: int > int,A4: set_int] :
( ( order_mono_int_int @ F )
=> ( ( condit2212336002400062152ve_int @ A4 )
=> ( ( A4 != bot_bot_set_int )
=> ( ord_less_eq_int @ ( complete_Sup_Sup_int @ ( image_int_int @ F @ A4 ) ) @ ( F @ ( complete_Sup_Sup_int @ A4 ) ) ) ) ) ) ).
% mono_cSup
thf(fact_3796_mono__cSup,axiom,
! [F: product_unit > int,A4: set_Product_unit] :
( ( order_7093624213399680841it_int @ F )
=> ( ( condit2080475194861589617t_unit @ A4 )
=> ( ( A4 != bot_bo3957492148770167129t_unit )
=> ( ord_less_eq_int @ ( complete_Sup_Sup_int @ ( image_873079544045703924it_int @ F @ A4 ) ) @ ( F @ ( comple4687483117567863418t_unit @ A4 ) ) ) ) ) ) ).
% mono_cSup
thf(fact_3797_mono__cSup,axiom,
! [F: $o > product_unit,A4: set_o] :
( ( order_7782127260464396005t_unit @ F )
=> ( ( condit5488710616941104124bove_o @ A4 )
=> ( ( A4 != bot_bot_set_o )
=> ( ord_le3221252021190050221t_unit @ ( comple4687483117567863418t_unit @ ( image_o_Product_unit @ F @ A4 ) ) @ ( F @ ( complete_Sup_Sup_o @ A4 ) ) ) ) ) ) ).
% mono_cSup
thf(fact_3798_mono__cSup,axiom,
! [F: nat > product_unit,A4: set_nat] :
( ( order_5727276828720722763t_unit @ F )
=> ( ( condit2214826472909112428ve_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ord_le3221252021190050221t_unit @ ( comple4687483117567863418t_unit @ ( image_8730104196221521654t_unit @ F @ A4 ) ) @ ( F @ ( complete_Sup_Sup_nat @ A4 ) ) ) ) ) ) ).
% mono_cSup
thf(fact_3799_Func__map__surj,axiom,
! [F1: nat > rat,A13: set_nat,B1: set_rat,F22: int > int,B22: set_int,A24: set_int] :
( ( ( image_nat_rat @ F1 @ A13 )
= B1 )
=> ( ( inj_on_int_int @ F22 @ B22 )
=> ( ( ord_less_eq_set_int @ ( image_int_int @ F22 @ B22 ) @ A24 )
=> ( ( ( B22 = bot_bot_set_int )
=> ( A24 = bot_bot_set_int ) )
=> ( ( bNF_We7904740386373615514nt_rat @ B22 @ B1 )
= ( image_1904597124500147013nt_rat @ ( bNF_We576767280138975660at_int @ B22 @ F1 @ F22 ) @ ( bNF_We8539870446460111250nt_nat @ A24 @ A13 ) ) ) ) ) ) ) ).
% Func_map_surj
thf(fact_3800_Func__map__surj,axiom,
! [F1: nat > nat,A13: set_nat,B1: set_nat,F22: int > int,B22: set_int,A24: set_int] :
( ( ( image_nat_nat @ F1 @ A13 )
= B1 )
=> ( ( inj_on_int_int @ F22 @ B22 )
=> ( ( ord_less_eq_set_int @ ( image_int_int @ F22 @ B22 ) @ A24 )
=> ( ( ( B22 = bot_bot_set_int )
=> ( A24 = bot_bot_set_int ) )
=> ( ( bNF_We8539870446460111250nt_nat @ B22 @ B1 )
= ( image_5412825682893543741nt_nat @ ( bNF_We893321924073841572at_int @ B22 @ F1 @ F22 ) @ ( bNF_We8539870446460111250nt_nat @ A24 @ A13 ) ) ) ) ) ) ) ).
% Func_map_surj
thf(fact_3801_Func__map__surj,axiom,
! [F1: nat > int,A13: set_nat,B1: set_int,F22: int > int,B22: set_int,A24: set_int] :
( ( ( image_nat_int @ F1 @ A13 )
= B1 )
=> ( ( inj_on_int_int @ F22 @ B22 )
=> ( ( ord_less_eq_set_int @ ( image_int_int @ F22 @ B22 ) @ A24 )
=> ( ( ( B22 = bot_bot_set_int )
=> ( A24 = bot_bot_set_int ) )
=> ( ( bNF_We8537379975951060974nt_int @ B22 @ B1 )
= ( image_1234974663384347033nt_int @ ( bNF_We1892675635974896128nt_int @ B22 @ F1 @ F22 ) @ ( bNF_We8539870446460111250nt_nat @ A24 @ A13 ) ) ) ) ) ) ) ).
% Func_map_surj
thf(fact_3802_Func__map__surj,axiom,
! [F1: int > nat,A13: set_int,B1: set_nat,F22: int > int,B22: set_int,A24: set_int] :
( ( ( image_int_nat @ F1 @ A13 )
= B1 )
=> ( ( inj_on_int_int @ F22 @ B22 )
=> ( ( ord_less_eq_set_int @ ( image_int_int @ F22 @ B22 ) @ A24 )
=> ( ( ( B22 = bot_bot_set_int )
=> ( A24 = bot_bot_set_int ) )
=> ( ( bNF_We8539870446460111250nt_nat @ B22 @ B1 )
= ( image_1600535827742230425nt_nat @ ( bNF_We1264975484310686208at_int @ B22 @ F1 @ F22 ) @ ( bNF_We8537379975951060974nt_int @ A24 @ A13 ) ) ) ) ) ) ) ).
% Func_map_surj
thf(fact_3803_Func__map__surj,axiom,
! [F1: int > int,A13: set_int,B1: set_int,F22: int > int,B22: set_int,A24: set_int] :
( ( ( image_int_int @ F1 @ A13 )
= B1 )
=> ( ( inj_on_int_int @ F22 @ B22 )
=> ( ( ord_less_eq_set_int @ ( image_int_int @ F22 @ B22 ) @ A24 )
=> ( ( ( B22 = bot_bot_set_int )
=> ( A24 = bot_bot_set_int ) )
=> ( ( bNF_We8537379975951060974nt_int @ B22 @ B1 )
= ( image_6646056845087809525nt_int @ ( bNF_We2264329196211740764nt_int @ B22 @ F1 @ F22 ) @ ( bNF_We8537379975951060974nt_int @ A24 @ A13 ) ) ) ) ) ) ) ).
% Func_map_surj
thf(fact_3804_Func__map__surj,axiom,
! [F1: nat > rat,A13: set_nat,B1: set_rat,F22: int > nat,B22: set_int,A24: set_nat] :
( ( ( image_nat_rat @ F1 @ A13 )
= B1 )
=> ( ( inj_on_int_nat @ F22 @ B22 )
=> ( ( ord_less_eq_set_nat @ ( image_int_nat @ F22 @ B22 ) @ A24 )
=> ( ( ( B22 = bot_bot_set_int )
=> ( A24 = bot_bot_set_nat ) )
=> ( ( bNF_We7904740386373615514nt_rat @ B22 @ B1 )
= ( image_696479991571767017nt_rat @ ( bNF_We579257750648025936at_nat @ B22 @ F1 @ F22 ) @ ( bNF_We7540516734559056694at_nat @ A24 @ A13 ) ) ) ) ) ) ) ).
% Func_map_surj
thf(fact_3805_Func__map__surj,axiom,
! [F1: nat > nat,A13: set_nat,B1: set_nat,F22: int > nat,B22: set_int,A24: set_nat] :
( ( ( image_nat_nat @ F1 @ A13 )
= B1 )
=> ( ( inj_on_int_nat @ F22 @ B22 )
=> ( ( ord_less_eq_set_nat @ ( image_int_nat @ F22 @ B22 ) @ A24 )
=> ( ( ( B22 = bot_bot_set_int )
=> ( A24 = bot_bot_set_nat ) )
=> ( ( bNF_We8539870446460111250nt_nat @ B22 @ B1 )
= ( image_4204708549965163745nt_nat @ ( bNF_We895812394582891848at_nat @ B22 @ F1 @ F22 ) @ ( bNF_We7540516734559056694at_nat @ A24 @ A13 ) ) ) ) ) ) ) ).
% Func_map_surj
thf(fact_3806_Func__map__surj,axiom,
! [F1: nat > int,A13: set_nat,B1: set_int,F22: int > nat,B22: set_int,A24: set_nat] :
( ( ( image_nat_int @ F1 @ A13 )
= B1 )
=> ( ( inj_on_int_nat @ F22 @ B22 )
=> ( ( ord_less_eq_set_nat @ ( image_int_nat @ F22 @ B22 ) @ A24 )
=> ( ( ( B22 = bot_bot_set_int )
=> ( A24 = bot_bot_set_nat ) )
=> ( ( bNF_We8537379975951060974nt_int @ B22 @ B1 )
= ( image_26857530455967037nt_int @ ( bNF_We1895166106483946404nt_nat @ B22 @ F1 @ F22 ) @ ( bNF_We7540516734559056694at_nat @ A24 @ A13 ) ) ) ) ) ) ) ).
% Func_map_surj
thf(fact_3807_Func__map__surj,axiom,
! [F1: int > nat,A13: set_int,B1: set_nat,F22: int > nat,B22: set_int,A24: set_nat] :
( ( ( image_int_nat @ F1 @ A13 )
= B1 )
=> ( ( inj_on_int_nat @ F22 @ B22 )
=> ( ( ord_less_eq_set_nat @ ( image_int_nat @ F22 @ B22 ) @ A24 )
=> ( ( ( B22 = bot_bot_set_int )
=> ( A24 = bot_bot_set_nat ) )
=> ( ( bNF_We8539870446460111250nt_nat @ B22 @ B1 )
= ( image_392418694813850429nt_nat @ ( bNF_We1267465954819736484at_nat @ B22 @ F1 @ F22 ) @ ( bNF_We7538026264050006418at_int @ A24 @ A13 ) ) ) ) ) ) ) ).
% Func_map_surj
thf(fact_3808_Func__map__surj,axiom,
! [F1: int > int,A13: set_int,B1: set_int,F22: int > nat,B22: set_int,A24: set_nat] :
( ( ( image_int_int @ F1 @ A13 )
= B1 )
=> ( ( inj_on_int_nat @ F22 @ B22 )
=> ( ( ord_less_eq_set_nat @ ( image_int_nat @ F22 @ B22 ) @ A24 )
=> ( ( ( B22 = bot_bot_set_int )
=> ( A24 = bot_bot_set_nat ) )
=> ( ( bNF_We8537379975951060974nt_int @ B22 @ B1 )
= ( image_5437939712159429529nt_int @ ( bNF_We2266819666720791040nt_nat @ B22 @ F1 @ F22 ) @ ( bNF_We7538026264050006418at_int @ A24 @ A13 ) ) ) ) ) ) ) ).
% Func_map_surj
thf(fact_3809_cofinal__def,axiom,
( bNF_Ca6037544080215331608at_nat
= ( ^ [A6: set_se5479201105039950375at_nat,R5: set_Pr7459493094073627847at_nat] :
! [X4: set_Pr4329608150637261639at_nat] :
( ( member6273874620674182256at_nat @ X4 @ ( field_1428494348145443988at_nat @ R5 ) )
=> ? [Y5: set_Pr4329608150637261639at_nat] :
( ( member6273874620674182256at_nat @ Y5 @ A6 )
& ( X4 != Y5 )
& ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X4 @ Y5 ) @ R5 ) ) ) ) ) ).
% cofinal_def
thf(fact_3810_cofinal__def,axiom,
( bNF_Ca8448034394003515314at_nat
= ( ^ [A6: set_se7855581050983116737at_nat,R5: set_Pr4329608150637261639at_nat] :
! [X4: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ X4 @ ( field_3145993188331922222at_nat @ R5 ) )
=> ? [Y5: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ Y5 @ A6 )
& ( X4 != Y5 )
& ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y5 ) @ R5 ) ) ) ) ) ).
% cofinal_def
thf(fact_3811_cofinal__def,axiom,
( bNF_Ca5159621140400407487al_nat
= ( ^ [A6: set_nat,R5: set_Pr1261947904930325089at_nat] :
! [X4: nat] :
( ( member_nat @ X4 @ ( field_nat @ R5 ) )
=> ? [Y5: nat] :
( ( member_nat @ Y5 @ A6 )
& ( X4 != Y5 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ R5 ) ) ) ) ) ).
% cofinal_def
thf(fact_3812_cofinal__def,axiom,
( bNF_Ca5157130669891357211al_int
= ( ^ [A6: set_int,R5: set_Pr958786334691620121nt_int] :
! [X4: int] :
( ( member_int @ X4 @ ( field_int @ R5 ) )
=> ? [Y5: int] :
( ( member_int @ Y5 @ A6 )
& ( X4 != Y5 )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y5 ) @ R5 ) ) ) ) ) ).
% cofinal_def
thf(fact_3813_Total__subset__Id,axiom,
! [R: set_Pr7459493094073627847at_nat] :
( ( total_5117575981256876747at_nat @ ( field_1428494348145443988at_nat @ R ) @ R )
=> ( ( ord_le5997549366648089703at_nat @ R @ id_set4888139568991089827at_nat )
=> ( ( R = bot_bo4948859079157340979at_nat )
| ? [A3: set_Pr4329608150637261639at_nat] :
( R
= ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ A3 @ A3 ) @ bot_bo4948859079157340979at_nat ) ) ) ) ) ).
% Total_subset_Id
thf(fact_3814_Total__subset__Id,axiom,
! [R: set_Pr958786334691620121nt_int] :
( ( total_on_int @ ( field_int @ R ) @ R )
=> ( ( ord_le2843351958646193337nt_int @ R @ id_int2 )
=> ( ( R = bot_bo1796632182523588997nt_int )
| ? [A3: int] :
( R
= ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ A3 @ A3 ) @ bot_bo1796632182523588997nt_int ) ) ) ) ) ).
% Total_subset_Id
thf(fact_3815_Total__subset__Id,axiom,
! [R: set_Pr8693737435421807431at_nat] :
( ( total_3592101749530773125at_nat @ ( field_4706345496283612622at_nat @ R ) @ R )
=> ( ( ord_le3000389064537975527at_nat @ R @ id_Pro2258643101195443293at_nat )
=> ( ( R = bot_bo5327735625951526323at_nat )
| ? [A3: product_prod_nat_nat] :
( R
= ( insert5050368324300391991at_nat @ ( produc6161850002892822231at_nat @ A3 @ A3 ) @ bot_bo5327735625951526323at_nat ) ) ) ) ) ).
% Total_subset_Id
thf(fact_3816_Total__subset__Id,axiom,
! [R: set_Pr4329608150637261639at_nat] :
( ( total_7212417256332265445at_nat @ ( field_3145993188331922222at_nat @ R ) @ R )
=> ( ( ord_le1268244103169919719at_nat @ R @ id_set1796276052702428605at_nat )
=> ( ( R = bot_bo228742789529271731at_nat )
| ? [A3: set_Pr1261947904930325089at_nat] :
( R
= ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ A3 @ A3 ) @ bot_bo228742789529271731at_nat ) ) ) ) ) ).
% Total_subset_Id
thf(fact_3817_Total__subset__Id,axiom,
! [R: set_Pr1261947904930325089at_nat] :
( ( total_on_nat @ ( field_nat @ R ) @ R )
=> ( ( ord_le3146513528884898305at_nat @ R @ id_nat2 )
=> ( ( R = bot_bo2099793752762293965at_nat )
| ? [A3: nat] :
( R
= ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A3 @ A3 ) @ bot_bo2099793752762293965at_nat ) ) ) ) ) ).
% Total_subset_Id
thf(fact_3818_inf__img__fin__dom_H,axiom,
! [F: int > int,A4: set_int] :
( ( finite_finite_int @ ( image_int_int @ F @ A4 ) )
=> ( ~ ( finite_finite_int @ A4 )
=> ? [X: int] :
( ( member_int @ X @ ( image_int_int @ F @ A4 ) )
& ~ ( finite_finite_int @ ( inf_inf_set_int @ ( vimage_int_int @ F @ ( insert_int @ X @ bot_bot_set_int ) ) @ A4 ) ) ) ) ) ).
% inf_img_fin_dom'
thf(fact_3819_inf__img__fin__dom_H,axiom,
! [F: nat > rat,A4: set_nat] :
( ( finite_finite_rat @ ( image_nat_rat @ F @ A4 ) )
=> ( ~ ( finite_finite_nat @ A4 )
=> ? [X: rat] :
( ( member_rat @ X @ ( image_nat_rat @ F @ A4 ) )
& ~ ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_rat @ F @ ( insert_rat @ X @ bot_bot_set_rat ) ) @ A4 ) ) ) ) ) ).
% inf_img_fin_dom'
thf(fact_3820_inf__img__fin__dom_H,axiom,
! [F: nat > int,A4: set_nat] :
( ( finite_finite_int @ ( image_nat_int @ F @ A4 ) )
=> ( ~ ( finite_finite_nat @ A4 )
=> ? [X: int] :
( ( member_int @ X @ ( image_nat_int @ F @ A4 ) )
& ~ ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_int @ F @ ( insert_int @ X @ bot_bot_set_int ) ) @ A4 ) ) ) ) ) ).
% inf_img_fin_dom'
thf(fact_3821_inf__img__fin__dom_H,axiom,
! [F: nat > product_unit,A4: set_nat] :
( ( finite4290736615968046902t_unit @ ( image_8730104196221521654t_unit @ F @ A4 ) )
=> ( ~ ( finite_finite_nat @ A4 )
=> ? [X: product_unit] :
( ( member_Product_unit @ X @ ( image_8730104196221521654t_unit @ F @ A4 ) )
& ~ ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage4884490618288580032t_unit @ F @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) @ A4 ) ) ) ) ) ).
% inf_img_fin_dom'
thf(fact_3822_inf__img__fin__dom_H,axiom,
! [F: nat > $o,A4: set_nat] :
( ( finite_finite_o @ ( image_nat_o @ F @ A4 ) )
=> ( ~ ( finite_finite_nat @ A4 )
=> ? [X: $o] :
( ( member_o @ X @ ( image_nat_o @ F @ A4 ) )
& ~ ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_o @ F @ ( insert_o @ X @ bot_bot_set_o ) ) @ A4 ) ) ) ) ) ).
% inf_img_fin_dom'
thf(fact_3823_inf__img__fin__dom_H,axiom,
! [F: int > nat,A4: set_int] :
( ( finite_finite_nat @ ( image_int_nat @ F @ A4 ) )
=> ( ~ ( finite_finite_int @ A4 )
=> ? [X: nat] :
( ( member_nat @ X @ ( image_int_nat @ F @ A4 ) )
& ~ ( finite_finite_int @ ( inf_inf_set_int @ ( vimage_int_nat @ F @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ A4 ) ) ) ) ) ).
% inf_img_fin_dom'
thf(fact_3824_inf__img__fin__dom_H,axiom,
! [F: nat > nat,A4: set_nat] :
( ( finite_finite_nat @ ( image_nat_nat @ F @ A4 ) )
=> ( ~ ( finite_finite_nat @ A4 )
=> ? [X: nat] :
( ( member_nat @ X @ ( image_nat_nat @ F @ A4 ) )
& ~ ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_nat @ F @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ A4 ) ) ) ) ) ).
% inf_img_fin_dom'
thf(fact_3825_inf__img__fin__dom_H,axiom,
! [F: product_prod_nat_nat > product_unit,A4: set_Pr1261947904930325089at_nat] :
( ( finite4290736615968046902t_unit @ ( image_6169909464789311569t_unit @ F @ A4 ) )
=> ( ~ ( finite6177210948735845034at_nat @ A4 )
=> ? [X: product_unit] :
( ( member_Product_unit @ X @ ( image_6169909464789311569t_unit @ F @ A4 ) )
& ~ ( finite6177210948735845034at_nat @ ( inf_in2572325071724192079at_nat @ ( vimage3104651464403166471t_unit @ F @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) @ A4 ) ) ) ) ) ).
% inf_img_fin_dom'
thf(fact_3826_inf__img__fin__dom_H,axiom,
! [F: nat > product_prod_nat_nat,A4: set_nat] :
( ( finite6177210948735845034at_nat @ ( image_5846123807819985514at_nat @ F @ A4 ) )
=> ( ~ ( finite_finite_nat @ A4 )
=> ? [X: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X @ ( image_5846123807819985514at_nat @ F @ A4 ) )
& ~ ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage8013328719654469172at_nat @ F @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) @ A4 ) ) ) ) ) ).
% inf_img_fin_dom'
thf(fact_3827_inf__img__fin__dom_H,axiom,
! [F: product_prod_nat_nat > $o,A4: set_Pr1261947904930325089at_nat] :
( ( finite_finite_o @ ( image_3693632289388996572_nat_o @ F @ A4 ) )
=> ( ~ ( finite6177210948735845034at_nat @ A4 )
=> ? [X: $o] :
( ( member_o @ X @ ( image_3693632289388996572_nat_o @ F @ A4 ) )
& ~ ( finite6177210948735845034at_nat @ ( inf_in2572325071724192079at_nat @ ( vimage2444243245536687506_nat_o @ F @ ( insert_o @ X @ bot_bot_set_o ) ) @ A4 ) ) ) ) ) ).
% inf_img_fin_dom'
thf(fact_3828_inf__img__fin__domE_H,axiom,
! [F: int > int,A4: set_int] :
( ( finite_finite_int @ ( image_int_int @ F @ A4 ) )
=> ( ~ ( finite_finite_int @ A4 )
=> ~ ! [Y: int] :
( ( member_int @ Y @ ( image_int_int @ F @ A4 ) )
=> ( finite_finite_int @ ( inf_inf_set_int @ ( vimage_int_int @ F @ ( insert_int @ Y @ bot_bot_set_int ) ) @ A4 ) ) ) ) ) ).
% inf_img_fin_domE'
thf(fact_3829_inf__img__fin__domE_H,axiom,
! [F: nat > rat,A4: set_nat] :
( ( finite_finite_rat @ ( image_nat_rat @ F @ A4 ) )
=> ( ~ ( finite_finite_nat @ A4 )
=> ~ ! [Y: rat] :
( ( member_rat @ Y @ ( image_nat_rat @ F @ A4 ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_rat @ F @ ( insert_rat @ Y @ bot_bot_set_rat ) ) @ A4 ) ) ) ) ) ).
% inf_img_fin_domE'
thf(fact_3830_inf__img__fin__domE_H,axiom,
! [F: nat > int,A4: set_nat] :
( ( finite_finite_int @ ( image_nat_int @ F @ A4 ) )
=> ( ~ ( finite_finite_nat @ A4 )
=> ~ ! [Y: int] :
( ( member_int @ Y @ ( image_nat_int @ F @ A4 ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_int @ F @ ( insert_int @ Y @ bot_bot_set_int ) ) @ A4 ) ) ) ) ) ).
% inf_img_fin_domE'
thf(fact_3831_inf__img__fin__domE_H,axiom,
! [F: nat > product_unit,A4: set_nat] :
( ( finite4290736615968046902t_unit @ ( image_8730104196221521654t_unit @ F @ A4 ) )
=> ( ~ ( finite_finite_nat @ A4 )
=> ~ ! [Y: product_unit] :
( ( member_Product_unit @ Y @ ( image_8730104196221521654t_unit @ F @ A4 ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage4884490618288580032t_unit @ F @ ( insert_Product_unit @ Y @ bot_bo3957492148770167129t_unit ) ) @ A4 ) ) ) ) ) ).
% inf_img_fin_domE'
thf(fact_3832_inf__img__fin__domE_H,axiom,
! [F: nat > $o,A4: set_nat] :
( ( finite_finite_o @ ( image_nat_o @ F @ A4 ) )
=> ( ~ ( finite_finite_nat @ A4 )
=> ~ ! [Y: $o] :
( ( member_o @ Y @ ( image_nat_o @ F @ A4 ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_o @ F @ ( insert_o @ Y @ bot_bot_set_o ) ) @ A4 ) ) ) ) ) ).
% inf_img_fin_domE'
thf(fact_3833_inf__img__fin__domE_H,axiom,
! [F: int > nat,A4: set_int] :
( ( finite_finite_nat @ ( image_int_nat @ F @ A4 ) )
=> ( ~ ( finite_finite_int @ A4 )
=> ~ ! [Y: nat] :
( ( member_nat @ Y @ ( image_int_nat @ F @ A4 ) )
=> ( finite_finite_int @ ( inf_inf_set_int @ ( vimage_int_nat @ F @ ( insert_nat @ Y @ bot_bot_set_nat ) ) @ A4 ) ) ) ) ) ).
% inf_img_fin_domE'
thf(fact_3834_inf__img__fin__domE_H,axiom,
! [F: nat > nat,A4: set_nat] :
( ( finite_finite_nat @ ( image_nat_nat @ F @ A4 ) )
=> ( ~ ( finite_finite_nat @ A4 )
=> ~ ! [Y: nat] :
( ( member_nat @ Y @ ( image_nat_nat @ F @ A4 ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_nat @ F @ ( insert_nat @ Y @ bot_bot_set_nat ) ) @ A4 ) ) ) ) ) ).
% inf_img_fin_domE'
thf(fact_3835_inf__img__fin__domE_H,axiom,
! [F: product_prod_nat_nat > int,A4: set_Pr1261947904930325089at_nat] :
( ( finite_finite_int @ ( image_2483585944268220136at_int @ F @ A4 ) )
=> ( ~ ( finite6177210948735845034at_nat @ A4 )
=> ~ ! [Y: int] :
( ( member_int @ Y @ ( image_2483585944268220136at_int @ F @ A4 ) )
=> ( finite6177210948735845034at_nat @ ( inf_in2572325071724192079at_nat @ ( vimage4650790856102703794at_int @ F @ ( insert_int @ Y @ bot_bot_set_int ) ) @ A4 ) ) ) ) ) ).
% inf_img_fin_domE'
thf(fact_3836_inf__img__fin__domE_H,axiom,
! [F: product_prod_nat_nat > product_unit,A4: set_Pr1261947904930325089at_nat] :
( ( finite4290736615968046902t_unit @ ( image_6169909464789311569t_unit @ F @ A4 ) )
=> ( ~ ( finite6177210948735845034at_nat @ A4 )
=> ~ ! [Y: product_unit] :
( ( member_Product_unit @ Y @ ( image_6169909464789311569t_unit @ F @ A4 ) )
=> ( finite6177210948735845034at_nat @ ( inf_in2572325071724192079at_nat @ ( vimage3104651464403166471t_unit @ F @ ( insert_Product_unit @ Y @ bot_bo3957492148770167129t_unit ) ) @ A4 ) ) ) ) ) ).
% inf_img_fin_domE'
thf(fact_3837_inf__img__fin__domE_H,axiom,
! [F: nat > product_prod_nat_nat,A4: set_nat] :
( ( finite6177210948735845034at_nat @ ( image_5846123807819985514at_nat @ F @ A4 ) )
=> ( ~ ( finite_finite_nat @ A4 )
=> ~ ! [Y: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ Y @ ( image_5846123807819985514at_nat @ F @ A4 ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage8013328719654469172at_nat @ F @ ( insert8211810215607154385at_nat @ Y @ bot_bo2099793752762293965at_nat ) ) @ A4 ) ) ) ) ) ).
% inf_img_fin_domE'
thf(fact_3838_prod__set__simps_I2_J,axiom,
! [X2: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] :
( ( basic_5713354656523905413at_nat @ ( produc9060074326276436823at_nat @ X2 @ Y3 ) )
= ( insert2290795007606087959at_nat @ Y3 @ bot_bo1534825418459847827at_nat ) ) ).
% prod_set_simps(2)
thf(fact_3839_prod__set__simps_I2_J,axiom,
! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
( ( basic_484852565060450053at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y3 ) )
= ( insert9200635055090092081at_nat @ Y3 @ bot_bo3083307316010499117at_nat ) ) ).
% prod_set_simps(2)
thf(fact_3840_prod__set__simps_I2_J,axiom,
! [X2: nat,Y3: num] :
( ( basic_snds_nat_num @ ( product_Pair_nat_num @ X2 @ Y3 ) )
= ( insert_num @ Y3 @ bot_bot_set_num ) ) ).
% prod_set_simps(2)
thf(fact_3841_prod__set__simps_I2_J,axiom,
! [X2: int,Y3: int] :
( ( basic_snds_int_int @ ( product_Pair_int_int @ X2 @ Y3 ) )
= ( insert_int @ Y3 @ bot_bot_set_int ) ) ).
% prod_set_simps(2)
thf(fact_3842_prod__set__simps_I2_J,axiom,
! [X2: nat,Y3: nat] :
( ( basic_snds_nat_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) )
= ( insert_nat @ Y3 @ bot_bot_set_nat ) ) ).
% prod_set_simps(2)
thf(fact_3843_Linear__order__in__diff__Id,axiom,
! [R: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
( ( order_2711500226769404828at_nat @ ( field_2862844103579723188at_nat @ R ) @ R )
=> ( ( member8757157785044589968at_nat @ A @ ( field_2862844103579723188at_nat @ R ) )
=> ( ( member8757157785044589968at_nat @ B @ ( field_2862844103579723188at_nat @ R ) )
=> ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ B ) @ R )
= ( ~ ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ B @ A ) @ ( minus_8235382265540049710at_nat @ R @ id_Pro5428700290092274115at_nat ) ) ) ) ) ) ) ).
% Linear_order_in_diff_Id
thf(fact_3844_Linear__order__in__diff__Id,axiom,
! [R: set_Product_prod_o_o,A: $o,B: $o] :
( ( order_6238756238976269133r_on_o @ ( field_o @ R ) @ R )
=> ( ( member_o @ A @ ( field_o @ R ) )
=> ( ( member_o @ B @ ( field_o @ R ) )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ R )
= ( ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ A ) @ ( minus_7052513330764642158od_o_o @ R @ id_o2 ) ) ) ) ) ) ) ).
% Linear_order_in_diff_Id
thf(fact_3845_Linear__order__in__diff__Id,axiom,
! [R: set_Pr8693737435421807431at_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] :
( ( order_7480980593162297270at_nat @ ( field_4706345496283612622at_nat @ R ) @ R )
=> ( ( member8440522571783428010at_nat @ A @ ( field_4706345496283612622at_nat @ R ) )
=> ( ( member8440522571783428010at_nat @ B @ ( field_4706345496283612622at_nat @ R ) )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R )
= ( ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B @ A ) @ ( minus_8321449233255521966at_nat @ R @ id_Pro2258643101195443293at_nat ) ) ) ) ) ) ) ).
% Linear_order_in_diff_Id
thf(fact_3846_Linear__order__in__diff__Id,axiom,
! [R: set_Pr7459493094073627847at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
( ( order_1878278401035043452at_nat @ ( field_1428494348145443988at_nat @ R ) @ R )
=> ( ( member6273874620674182256at_nat @ A @ ( field_1428494348145443988at_nat @ R ) )
=> ( ( member6273874620674182256at_nat @ B @ ( field_1428494348145443988at_nat @ R ) )
=> ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ B ) @ R )
= ( ~ ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ B @ A ) @ ( minus_8674369445256047406at_nat @ R @ id_set4888139568991089827at_nat ) ) ) ) ) ) ) ).
% Linear_order_in_diff_Id
thf(fact_3847_Linear__order__in__diff__Id,axiom,
! [R: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
( ( order_5942792088423767318at_nat @ ( field_3145993188331922222at_nat @ R ) @ R )
=> ( ( member2643936169264416010at_nat @ A @ ( field_3145993188331922222at_nat @ R ) )
=> ( ( member2643936169264416010at_nat @ B @ ( field_3145993188331922222at_nat @ R ) )
=> ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R )
= ( ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B @ A ) @ ( minus_3314409938677909166at_nat @ R @ id_set1796276052702428605at_nat ) ) ) ) ) ) ) ).
% Linear_order_in_diff_Id
thf(fact_3848_Linear__order__in__diff__Id,axiom,
! [R: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
( ( order_4473980167227706203on_nat @ ( field_nat @ R ) @ R )
=> ( ( member_nat @ A @ ( field_nat @ R ) )
=> ( ( member_nat @ B @ ( field_nat @ R ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
= ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A ) @ ( minus_1356011639430497352at_nat @ R @ id_nat2 ) ) ) ) ) ) ) ).
% Linear_order_in_diff_Id
thf(fact_3849_Linear__order__in__diff__Id,axiom,
! [R: set_Pr958786334691620121nt_int,A: int,B: int] :
( ( order_4471489696718655927on_int @ ( field_int @ R ) @ R )
=> ( ( member_int @ A @ ( field_int @ R ) )
=> ( ( member_int @ B @ ( field_int @ R ) )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ R )
= ( ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ A ) @ ( minus_1052850069191792384nt_int @ R @ id_int2 ) ) ) ) ) ) ) ).
% Linear_order_in_diff_Id
thf(fact_3850_aboveS__def,axiom,
( order_4940263126897443436at_nat
= ( ^ [R5: set_Pr8693737435421807431at_nat,A5: product_prod_nat_nat] :
( collec3392354462482085612at_nat
@ ^ [B4: product_prod_nat_nat] :
( ( B4 != A5 )
& ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A5 @ B4 ) @ R5 ) ) ) ) ) ).
% aboveS_def
thf(fact_3851_aboveS__def,axiom,
( order_1761765819289301284nt_int
= ( ^ [R5: set_Pr2560585780119916871nt_int,A5: product_prod_int_int] :
( collec213857154873943460nt_int
@ ^ [B4: product_prod_int_int] :
( ( B4 != A5 )
& ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ A5 @ B4 ) @ R5 ) ) ) ) ) ).
% aboveS_def
thf(fact_3852_aboveS__def,axiom,
( order_2442354982230848562at_nat
= ( ^ [R5: set_Pr7459493094073627847at_nat,A5: set_Pr4329608150637261639at_nat] :
( collec4573900263915890866at_nat
@ ^ [B4: set_Pr4329608150637261639at_nat] :
( ( B4 != A5 )
& ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A5 @ B4 ) @ R5 ) ) ) ) ) ).
% aboveS_def
thf(fact_3853_aboveS__def,axiom,
( order_4440095836999870412at_nat
= ( ^ [R5: set_Pr4329608150637261639at_nat,A5: set_Pr1261947904930325089at_nat] :
( collec5514110066124741708at_nat
@ ^ [B4: set_Pr1261947904930325089at_nat] :
( ( B4 != A5 )
& ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A5 @ B4 ) @ R5 ) ) ) ) ) ).
% aboveS_def
thf(fact_3854_aboveS__def,axiom,
( order_aboveS_nat
= ( ^ [R5: set_Pr1261947904930325089at_nat,A5: nat] :
( collect_nat
@ ^ [B4: nat] :
( ( B4 != A5 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A5 @ B4 ) @ R5 ) ) ) ) ) ).
% aboveS_def
thf(fact_3855_aboveS__def,axiom,
( order_aboveS_int
= ( ^ [R5: set_Pr958786334691620121nt_int,A5: int] :
( collect_int
@ ^ [B4: int] :
( ( B4 != A5 )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A5 @ B4 ) @ R5 ) ) ) ) ) ).
% aboveS_def
thf(fact_3856_card__vimage__inj,axiom,
! [F: nat > rat,A4: set_rat] :
( ( inj_on_nat_rat @ F @ top_top_set_nat )
=> ( ( ord_less_eq_set_rat @ A4 @ ( image_nat_rat @ F @ top_top_set_nat ) )
=> ( ( finite_card_nat @ ( vimage_nat_rat @ F @ A4 ) )
= ( finite_card_rat @ A4 ) ) ) ) ).
% card_vimage_inj
thf(fact_3857_card__vimage__inj,axiom,
! [F: nat > nat,A4: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( ord_less_eq_set_nat @ A4 @ ( image_nat_nat @ F @ top_top_set_nat ) )
=> ( ( finite_card_nat @ ( vimage_nat_nat @ F @ A4 ) )
= ( finite_card_nat @ A4 ) ) ) ) ).
% card_vimage_inj
thf(fact_3858_card__vimage__inj,axiom,
! [F: nat > int,A4: set_int] :
( ( inj_on_nat_int @ F @ top_top_set_nat )
=> ( ( ord_less_eq_set_int @ A4 @ ( image_nat_int @ F @ top_top_set_nat ) )
=> ( ( finite_card_nat @ ( vimage_nat_int @ F @ A4 ) )
= ( finite_card_int @ A4 ) ) ) ) ).
% card_vimage_inj
thf(fact_3859_card__vimage__inj,axiom,
! [F: int > nat,A4: set_nat] :
( ( inj_on_int_nat @ F @ top_top_set_int )
=> ( ( ord_less_eq_set_nat @ A4 @ ( image_int_nat @ F @ top_top_set_int ) )
=> ( ( finite_card_int @ ( vimage_int_nat @ F @ A4 ) )
= ( finite_card_nat @ A4 ) ) ) ) ).
% card_vimage_inj
thf(fact_3860_card__vimage__inj,axiom,
! [F: int > int,A4: set_int] :
( ( inj_on_int_int @ F @ top_top_set_int )
=> ( ( ord_less_eq_set_int @ A4 @ ( image_int_int @ F @ top_top_set_int ) )
=> ( ( finite_card_int @ ( vimage_int_int @ F @ A4 ) )
= ( finite_card_int @ A4 ) ) ) ) ).
% card_vimage_inj
thf(fact_3861_Linear__order__Well__order__iff,axiom,
! [R: set_Pr7459493094073627847at_nat] :
( ( order_1878278401035043452at_nat @ ( field_1428494348145443988at_nat @ R ) @ R )
=> ( ( order_423636163458052581at_nat @ ( field_1428494348145443988at_nat @ R ) @ R )
= ( ! [A6: set_se5479201105039950375at_nat] :
( ( ord_le7966958417815640519at_nat @ A6 @ ( field_1428494348145443988at_nat @ R ) )
=> ( ( A6 != bot_bo1534825418459847827at_nat )
=> ? [X4: set_Pr4329608150637261639at_nat] :
( ( member6273874620674182256at_nat @ X4 @ A6 )
& ! [Y5: set_Pr4329608150637261639at_nat] :
( ( member6273874620674182256at_nat @ Y5 @ A6 )
=> ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X4 @ Y5 ) @ R ) ) ) ) ) ) ) ) ).
% Linear_order_Well_order_iff
thf(fact_3862_Linear__order__Well__order__iff,axiom,
! [R: set_Pr4329608150637261639at_nat] :
( ( order_5942792088423767318at_nat @ ( field_3145993188331922222at_nat @ R ) @ R )
=> ( ( order_1968183844437341695at_nat @ ( field_3145993188331922222at_nat @ R ) @ R )
= ( ! [A6: set_se7855581050983116737at_nat] :
( ( ord_le2077887516847798113at_nat @ A6 @ ( field_3145993188331922222at_nat @ R ) )
=> ( ( A6 != bot_bo3083307316010499117at_nat )
=> ? [X4: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ X4 @ A6 )
& ! [Y5: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ Y5 @ A6 )
=> ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y5 ) @ R ) ) ) ) ) ) ) ) ).
% Linear_order_Well_order_iff
thf(fact_3863_Linear__order__Well__order__iff,axiom,
! [R: set_Pr958786334691620121nt_int] :
( ( order_4471489696718655927on_int @ ( field_int @ R ) @ R )
=> ( ( order_2886507596567047182on_int @ ( field_int @ R ) @ R )
= ( ! [A6: set_int] :
( ( ord_less_eq_set_int @ A6 @ ( field_int @ R ) )
=> ( ( A6 != bot_bot_set_int )
=> ? [X4: int] :
( ( member_int @ X4 @ A6 )
& ! [Y5: int] :
( ( member_int @ Y5 @ A6 )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y5 ) @ R ) ) ) ) ) ) ) ) ).
% Linear_order_Well_order_iff
thf(fact_3864_Linear__order__Well__order__iff,axiom,
! [R: set_Pr5094982260447487303t_unit] :
( ( order_888282469287196226t_unit @ ( field_Product_unit @ R ) @ R )
=> ( ( order_3164716736300062379t_unit @ ( field_Product_unit @ R ) @ R )
= ( ! [A6: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A6 @ ( field_Product_unit @ R ) )
=> ( ( A6 != bot_bo3957492148770167129t_unit )
=> ? [X4: product_unit] :
( ( member_Product_unit @ X4 @ A6 )
& ! [Y5: product_unit] :
( ( member_Product_unit @ Y5 @ A6 )
=> ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X4 @ Y5 ) @ R ) ) ) ) ) ) ) ) ).
% Linear_order_Well_order_iff
thf(fact_3865_Linear__order__Well__order__iff,axiom,
! [R: set_Pr5564308138774400199at_nat] :
( ( order_2711500226769404828at_nat @ ( field_2862844103579723188at_nat @ R ) @ R )
=> ( ( order_2339240034117818117at_nat @ ( field_2862844103579723188at_nat @ R ) @ R )
= ( ! [A6: set_Pr4329608150637261639at_nat] :
( ( ord_le1268244103169919719at_nat @ A6 @ ( field_2862844103579723188at_nat @ R ) )
=> ( ( A6 != bot_bo228742789529271731at_nat )
=> ? [X4: produc3843707927480180839at_nat] :
( ( member8757157785044589968at_nat @ X4 @ A6 )
& ! [Y5: produc3843707927480180839at_nat] :
( ( member8757157785044589968at_nat @ Y5 @ A6 )
=> ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X4 @ Y5 ) @ R ) ) ) ) ) ) ) ) ).
% Linear_order_Well_order_iff
thf(fact_3866_Linear__order__Well__order__iff,axiom,
! [R: set_Pr8693737435421807431at_nat] :
( ( order_7480980593162297270at_nat @ ( field_4706345496283612622at_nat @ R ) @ R )
=> ( ( order_8331247224601271455at_nat @ ( field_4706345496283612622at_nat @ R ) @ R )
= ( ! [A6: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ A6 @ ( field_4706345496283612622at_nat @ R ) )
=> ( ( A6 != bot_bo2099793752762293965at_nat )
=> ? [X4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X4 @ A6 )
& ! [Y5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ Y5 @ A6 )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y5 ) @ R ) ) ) ) ) ) ) ) ).
% Linear_order_Well_order_iff
thf(fact_3867_Linear__order__Well__order__iff,axiom,
! [R: set_Product_prod_o_o] :
( ( order_6238756238976269133r_on_o @ ( field_o @ R ) @ R )
=> ( ( order_2615775362358406454r_on_o @ ( field_o @ R ) @ R )
= ( ! [A6: set_o] :
( ( ord_less_eq_set_o @ A6 @ ( field_o @ R ) )
=> ( ( A6 != bot_bot_set_o )
=> ? [X4: $o] :
( ( member_o @ X4 @ A6 )
& ! [Y5: $o] :
( ( member_o @ Y5 @ A6 )
=> ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X4 @ Y5 ) @ R ) ) ) ) ) ) ) ) ).
% Linear_order_Well_order_iff
thf(fact_3868_Linear__order__Well__order__iff,axiom,
! [R: set_Pr1261947904930325089at_nat] :
( ( order_4473980167227706203on_nat @ ( field_nat @ R ) @ R )
=> ( ( order_2888998067076097458on_nat @ ( field_nat @ R ) @ R )
= ( ! [A6: set_nat] :
( ( ord_less_eq_set_nat @ A6 @ ( field_nat @ R ) )
=> ( ( A6 != bot_bot_set_nat )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A6 )
& ! [Y5: nat] :
( ( member_nat @ Y5 @ A6 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ R ) ) ) ) ) ) ) ) ).
% Linear_order_Well_order_iff
thf(fact_3869_finite__imageI,axiom,
! [F4: set_int,H: int > int] :
( ( finite_finite_int @ F4 )
=> ( finite_finite_int @ ( image_int_int @ H @ F4 ) ) ) ).
% finite_imageI
thf(fact_3870_finite__imageI,axiom,
! [F4: set_int,H: int > nat] :
( ( finite_finite_int @ F4 )
=> ( finite_finite_nat @ ( image_int_nat @ H @ F4 ) ) ) ).
% finite_imageI
thf(fact_3871_finite__imageI,axiom,
! [F4: set_Pr1261947904930325089at_nat,H: product_prod_nat_nat > product_prod_nat_nat] :
( ( finite6177210948735845034at_nat @ F4 )
=> ( finite6177210948735845034at_nat @ ( image_5168914502847457605at_nat @ H @ F4 ) ) ) ).
% finite_imageI
thf(fact_3872_finite__imageI,axiom,
! [F4: set_Pr1261947904930325089at_nat,H: product_prod_nat_nat > nat] :
( ( finite6177210948735845034at_nat @ F4 )
=> ( finite_finite_nat @ ( image_2486076414777270412at_nat @ H @ F4 ) ) ) ).
% finite_imageI
thf(fact_3873_finite__imageI,axiom,
! [F4: set_nat,H: nat > rat] :
( ( finite_finite_nat @ F4 )
=> ( finite_finite_rat @ ( image_nat_rat @ H @ F4 ) ) ) ).
% finite_imageI
thf(fact_3874_finite__imageI,axiom,
! [F4: set_nat,H: nat > int] :
( ( finite_finite_nat @ F4 )
=> ( finite_finite_int @ ( image_nat_int @ H @ F4 ) ) ) ).
% finite_imageI
thf(fact_3875_finite__imageI,axiom,
! [F4: set_nat,H: nat > product_prod_nat_nat] :
( ( finite_finite_nat @ F4 )
=> ( finite6177210948735845034at_nat @ ( image_5846123807819985514at_nat @ H @ F4 ) ) ) ).
% finite_imageI
thf(fact_3876_finite__imageI,axiom,
! [F4: set_nat,H: nat > nat] :
( ( finite_finite_nat @ F4 )
=> ( finite_finite_nat @ ( image_nat_nat @ H @ F4 ) ) ) ).
% finite_imageI
thf(fact_3877_IdI,axiom,
! [A: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ A ) @ id_Pro2258643101195443293at_nat ) ).
% IdI
thf(fact_3878_IdI,axiom,
! [A: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ A ) @ id_set4888139568991089827at_nat ) ).
% IdI
thf(fact_3879_IdI,axiom,
! [A: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ A ) @ id_set1796276052702428605at_nat ) ).
% IdI
thf(fact_3880_IdI,axiom,
! [A: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A ) @ id_nat2 ) ).
% IdI
thf(fact_3881_IdI,axiom,
! [A: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ A ) @ id_int2 ) ).
% IdI
thf(fact_3882_pair__in__Id__conv,axiom,
! [A: product_prod_nat_nat,B: product_prod_nat_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ id_Pro2258643101195443293at_nat )
= ( A = B ) ) ).
% pair_in_Id_conv
thf(fact_3883_pair__in__Id__conv,axiom,
! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ B ) @ id_set4888139568991089827at_nat )
= ( A = B ) ) ).
% pair_in_Id_conv
thf(fact_3884_pair__in__Id__conv,axiom,
! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ id_set1796276052702428605at_nat )
= ( A = B ) ) ).
% pair_in_Id_conv
thf(fact_3885_pair__in__Id__conv,axiom,
! [A: nat,B: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ id_nat2 )
= ( A = B ) ) ).
% pair_in_Id_conv
thf(fact_3886_pair__in__Id__conv,axiom,
! [A: int,B: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ id_int2 )
= ( A = B ) ) ).
% pair_in_Id_conv
thf(fact_3887_finite__UN,axiom,
! [A4: set_Pr1261947904930325089at_nat,B5: product_prod_nat_nat > set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ A4 )
=> ( ( finite6177210948735845034at_nat @ ( comple5685304695842803022at_nat @ ( image_5423882813909314213at_nat @ B5 @ A4 ) ) )
= ( ! [X4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X4 @ A4 )
=> ( finite6177210948735845034at_nat @ ( B5 @ X4 ) ) ) ) ) ) ).
% finite_UN
thf(fact_3888_finite__UN,axiom,
! [A4: set_nat,B5: nat > set_Pr1261947904930325089at_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( finite6177210948735845034at_nat @ ( comple5685304695842803022at_nat @ ( image_7178329752028323786at_nat @ B5 @ A4 ) ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( finite6177210948735845034at_nat @ ( B5 @ X4 ) ) ) ) ) ) ).
% finite_UN
thf(fact_3889_finite__UN,axiom,
! [A4: set_Pr1261947904930325089at_nat,B5: product_prod_nat_nat > set_nat] :
( ( finite6177210948735845034at_nat @ A4 )
=> ( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ B5 @ A4 ) ) )
= ( ! [X4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X4 @ A4 )
=> ( finite_finite_nat @ ( B5 @ X4 ) ) ) ) ) ) ).
% finite_UN
thf(fact_3890_finite__UN,axiom,
! [A4: set_nat,B5: nat > set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ A4 ) ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( finite_finite_nat @ ( B5 @ X4 ) ) ) ) ) ) ).
% finite_UN
thf(fact_3891_finite__UN__I,axiom,
! [A4: set_Pr4329608150637261639at_nat,B5: produc3843707927480180839at_nat > set_Pr1261947904930325089at_nat] :
( ( finite4343798906461161616at_nat @ A4 )
=> ( ! [A3: produc3843707927480180839at_nat] :
( ( member8757157785044589968at_nat @ A3 @ A4 )
=> ( finite6177210948735845034at_nat @ ( B5 @ A3 ) ) )
=> ( finite6177210948735845034at_nat @ ( comple5685304695842803022at_nat @ ( image_7979982168039674559at_nat @ B5 @ A4 ) ) ) ) ) ).
% finite_UN_I
thf(fact_3892_finite__UN__I,axiom,
! [A4: set_o,B5: $o > set_Pr1261947904930325089at_nat] :
( ( finite_finite_o @ A4 )
=> ( ! [A3: $o] :
( ( member_o @ A3 @ A4 )
=> ( finite6177210948735845034at_nat @ ( B5 @ A3 ) ) )
=> ( finite6177210948735845034at_nat @ ( comple5685304695842803022at_nat @ ( image_7077428525161401678at_nat @ B5 @ A4 ) ) ) ) ) ).
% finite_UN_I
thf(fact_3893_finite__UN__I,axiom,
! [A4: set_int,B5: int > set_Pr1261947904930325089at_nat] :
( ( finite_finite_int @ A4 )
=> ( ! [A3: int] :
( ( member_int @ A3 @ A4 )
=> ( finite6177210948735845034at_nat @ ( B5 @ A3 ) ) )
=> ( finite6177210948735845034at_nat @ ( comple5685304695842803022at_nat @ ( image_8938366415781434990at_nat @ B5 @ A4 ) ) ) ) ) ).
% finite_UN_I
thf(fact_3894_finite__UN__I,axiom,
! [A4: set_Pr1261947904930325089at_nat,B5: product_prod_nat_nat > set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ A4 )
=> ( ! [A3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ A3 @ A4 )
=> ( finite6177210948735845034at_nat @ ( B5 @ A3 ) ) )
=> ( finite6177210948735845034at_nat @ ( comple5685304695842803022at_nat @ ( image_5423882813909314213at_nat @ B5 @ A4 ) ) ) ) ) ).
% finite_UN_I
thf(fact_3895_finite__UN__I,axiom,
! [A4: set_nat,B5: nat > set_Pr1261947904930325089at_nat] :
( ( finite_finite_nat @ A4 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A4 )
=> ( finite6177210948735845034at_nat @ ( B5 @ A3 ) ) )
=> ( finite6177210948735845034at_nat @ ( comple5685304695842803022at_nat @ ( image_7178329752028323786at_nat @ B5 @ A4 ) ) ) ) ) ).
% finite_UN_I
thf(fact_3896_finite__UN__I,axiom,
! [A4: set_Pr4329608150637261639at_nat,B5: produc3843707927480180839at_nat > set_nat] :
( ( finite4343798906461161616at_nat @ A4 )
=> ( ! [A3: produc3843707927480180839at_nat] :
( ( member8757157785044589968at_nat @ A3 @ A4 )
=> ( finite_finite_nat @ ( B5 @ A3 ) ) )
=> ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_3160451284896210088et_nat @ B5 @ A4 ) ) ) ) ) ).
% finite_UN_I
thf(fact_3897_finite__UN__I,axiom,
! [A4: set_o,B5: $o > set_nat] :
( ( finite_finite_o @ A4 )
=> ( ! [A3: $o] :
( ( member_o @ A3 @ A4 )
=> ( finite_finite_nat @ ( B5 @ A3 ) ) )
=> ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B5 @ A4 ) ) ) ) ) ).
% finite_UN_I
thf(fact_3898_finite__UN__I,axiom,
! [A4: set_int,B5: int > set_nat] :
( ( finite_finite_int @ A4 )
=> ( ! [A3: int] :
( ( member_int @ A3 @ A4 )
=> ( finite_finite_nat @ ( B5 @ A3 ) ) )
=> ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_int_set_nat @ B5 @ A4 ) ) ) ) ) ).
% finite_UN_I
thf(fact_3899_finite__UN__I,axiom,
! [A4: set_Pr1261947904930325089at_nat,B5: product_prod_nat_nat > set_nat] :
( ( finite6177210948735845034at_nat @ A4 )
=> ( ! [A3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ A3 @ A4 )
=> ( finite_finite_nat @ ( B5 @ A3 ) ) )
=> ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ B5 @ A4 ) ) ) ) ) ).
% finite_UN_I
thf(fact_3900_finite__UN__I,axiom,
! [A4: set_nat,B5: nat > set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A4 )
=> ( finite_finite_nat @ ( B5 @ A3 ) ) )
=> ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ A4 ) ) ) ) ) ).
% finite_UN_I
thf(fact_3901_bdd__below__UN,axiom,
! [I3: set_Pr1261947904930325089at_nat,A4: product_prod_nat_nat > set_nat] :
( ( finite6177210948735845034at_nat @ I3 )
=> ( ( condit1738341127787009408ow_nat @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ A4 @ I3 ) ) )
= ( ! [X4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X4 @ I3 )
=> ( condit1738341127787009408ow_nat @ ( A4 @ X4 ) ) ) ) ) ) ).
% bdd_below_UN
thf(fact_3902_bdd__below__UN,axiom,
! [I3: set_nat,A4: nat > set_nat] :
( ( finite_finite_nat @ I3 )
=> ( ( condit1738341127787009408ow_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A4 @ I3 ) ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ I3 )
=> ( condit1738341127787009408ow_nat @ ( A4 @ X4 ) ) ) ) ) ) ).
% bdd_below_UN
thf(fact_3903_bdd__above__UN,axiom,
! [I3: set_Pr1261947904930325089at_nat,A4: product_prod_nat_nat > set_nat] :
( ( finite6177210948735845034at_nat @ I3 )
=> ( ( condit2214826472909112428ve_nat @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ A4 @ I3 ) ) )
= ( ! [X4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X4 @ I3 )
=> ( condit2214826472909112428ve_nat @ ( A4 @ X4 ) ) ) ) ) ) ).
% bdd_above_UN
thf(fact_3904_bdd__above__UN,axiom,
! [I3: set_nat,A4: nat > set_nat] :
( ( finite_finite_nat @ I3 )
=> ( ( condit2214826472909112428ve_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A4 @ I3 ) ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ I3 )
=> ( condit2214826472909112428ve_nat @ ( A4 @ X4 ) ) ) ) ) ) ).
% bdd_above_UN
thf(fact_3905_card__image__le,axiom,
! [A4: set_int,F: int > nat] :
( ( finite_finite_int @ A4 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( image_int_nat @ F @ A4 ) ) @ ( finite_card_int @ A4 ) ) ) ).
% card_image_le
thf(fact_3906_card__image__le,axiom,
! [A4: set_int,F: int > int] :
( ( finite_finite_int @ A4 )
=> ( ord_less_eq_nat @ ( finite_card_int @ ( image_int_int @ F @ A4 ) ) @ ( finite_card_int @ A4 ) ) ) ).
% card_image_le
thf(fact_3907_card__image__le,axiom,
! [A4: set_nat,F: nat > rat] :
( ( finite_finite_nat @ A4 )
=> ( ord_less_eq_nat @ ( finite_card_rat @ ( image_nat_rat @ F @ A4 ) ) @ ( finite_card_nat @ A4 ) ) ) ).
% card_image_le
thf(fact_3908_card__image__le,axiom,
! [A4: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A4 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( image_nat_nat @ F @ A4 ) ) @ ( finite_card_nat @ A4 ) ) ) ).
% card_image_le
thf(fact_3909_card__image__le,axiom,
! [A4: set_nat,F: nat > int] :
( ( finite_finite_nat @ A4 )
=> ( ord_less_eq_nat @ ( finite_card_int @ ( image_nat_int @ F @ A4 ) ) @ ( finite_card_nat @ A4 ) ) ) ).
% card_image_le
thf(fact_3910_surj__card__le,axiom,
! [A4: set_int,B5: set_nat,F: int > nat] :
( ( finite_finite_int @ A4 )
=> ( ( ord_less_eq_set_nat @ B5 @ ( image_int_nat @ F @ A4 ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ B5 ) @ ( finite_card_int @ A4 ) ) ) ) ).
% surj_card_le
thf(fact_3911_surj__card__le,axiom,
! [A4: set_int,B5: set_int,F: int > int] :
( ( finite_finite_int @ A4 )
=> ( ( ord_less_eq_set_int @ B5 @ ( image_int_int @ F @ A4 ) )
=> ( ord_less_eq_nat @ ( finite_card_int @ B5 ) @ ( finite_card_int @ A4 ) ) ) ) ).
% surj_card_le
thf(fact_3912_surj__card__le,axiom,
! [A4: set_nat,B5: set_rat,F: nat > rat] :
( ( finite_finite_nat @ A4 )
=> ( ( ord_less_eq_set_rat @ B5 @ ( image_nat_rat @ F @ A4 ) )
=> ( ord_less_eq_nat @ ( finite_card_rat @ B5 ) @ ( finite_card_nat @ A4 ) ) ) ) ).
% surj_card_le
thf(fact_3913_surj__card__le,axiom,
! [A4: set_nat,B5: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A4 )
=> ( ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat @ F @ A4 ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ B5 ) @ ( finite_card_nat @ A4 ) ) ) ) ).
% surj_card_le
thf(fact_3914_surj__card__le,axiom,
! [A4: set_nat,B5: set_int,F: nat > int] :
( ( finite_finite_nat @ A4 )
=> ( ( ord_less_eq_set_int @ B5 @ ( image_nat_int @ F @ A4 ) )
=> ( ord_less_eq_nat @ ( finite_card_int @ B5 ) @ ( finite_card_nat @ A4 ) ) ) ) ).
% surj_card_le
thf(fact_3915_eq__card__imp__inj__on,axiom,
! [A4: set_int,F: int > nat] :
( ( finite_finite_int @ A4 )
=> ( ( ( finite_card_nat @ ( image_int_nat @ F @ A4 ) )
= ( finite_card_int @ A4 ) )
=> ( inj_on_int_nat @ F @ A4 ) ) ) ).
% eq_card_imp_inj_on
thf(fact_3916_eq__card__imp__inj__on,axiom,
! [A4: set_int,F: int > int] :
( ( finite_finite_int @ A4 )
=> ( ( ( finite_card_int @ ( image_int_int @ F @ A4 ) )
= ( finite_card_int @ A4 ) )
=> ( inj_on_int_int @ F @ A4 ) ) ) ).
% eq_card_imp_inj_on
thf(fact_3917_eq__card__imp__inj__on,axiom,
! [A4: set_nat,F: nat > rat] :
( ( finite_finite_nat @ A4 )
=> ( ( ( finite_card_rat @ ( image_nat_rat @ F @ A4 ) )
= ( finite_card_nat @ A4 ) )
=> ( inj_on_nat_rat @ F @ A4 ) ) ) ).
% eq_card_imp_inj_on
thf(fact_3918_eq__card__imp__inj__on,axiom,
! [A4: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A4 )
=> ( ( ( finite_card_nat @ ( image_nat_nat @ F @ A4 ) )
= ( finite_card_nat @ A4 ) )
=> ( inj_on_nat_nat @ F @ A4 ) ) ) ).
% eq_card_imp_inj_on
thf(fact_3919_eq__card__imp__inj__on,axiom,
! [A4: set_nat,F: nat > int] :
( ( finite_finite_nat @ A4 )
=> ( ( ( finite_card_int @ ( image_nat_int @ F @ A4 ) )
= ( finite_card_nat @ A4 ) )
=> ( inj_on_nat_int @ F @ A4 ) ) ) ).
% eq_card_imp_inj_on
thf(fact_3920_inj__on__iff__eq__card,axiom,
! [A4: set_int,F: int > nat] :
( ( finite_finite_int @ A4 )
=> ( ( inj_on_int_nat @ F @ A4 )
= ( ( finite_card_nat @ ( image_int_nat @ F @ A4 ) )
= ( finite_card_int @ A4 ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_3921_inj__on__iff__eq__card,axiom,
! [A4: set_int,F: int > int] :
( ( finite_finite_int @ A4 )
=> ( ( inj_on_int_int @ F @ A4 )
= ( ( finite_card_int @ ( image_int_int @ F @ A4 ) )
= ( finite_card_int @ A4 ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_3922_inj__on__iff__eq__card,axiom,
! [A4: set_nat,F: nat > rat] :
( ( finite_finite_nat @ A4 )
=> ( ( inj_on_nat_rat @ F @ A4 )
= ( ( finite_card_rat @ ( image_nat_rat @ F @ A4 ) )
= ( finite_card_nat @ A4 ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_3923_inj__on__iff__eq__card,axiom,
! [A4: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A4 )
=> ( ( inj_on_nat_nat @ F @ A4 )
= ( ( finite_card_nat @ ( image_nat_nat @ F @ A4 ) )
= ( finite_card_nat @ A4 ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_3924_inj__on__iff__eq__card,axiom,
! [A4: set_nat,F: nat > int] :
( ( finite_finite_nat @ A4 )
=> ( ( inj_on_nat_int @ F @ A4 )
= ( ( finite_card_int @ ( image_nat_int @ F @ A4 ) )
= ( finite_card_nat @ A4 ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_3925_well__order__on__domain,axiom,
! [A4: set_Pr4329608150637261639at_nat,R: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
( ( order_2339240034117818117at_nat @ A4 @ R )
=> ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ B ) @ R )
=> ( ( member8757157785044589968at_nat @ A @ A4 )
& ( member8757157785044589968at_nat @ B @ A4 ) ) ) ) ).
% well_order_on_domain
thf(fact_3926_well__order__on__domain,axiom,
! [A4: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] :
( ( order_8331247224601271455at_nat @ A4 @ R )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R )
=> ( ( member8440522571783428010at_nat @ A @ A4 )
& ( member8440522571783428010at_nat @ B @ A4 ) ) ) ) ).
% well_order_on_domain
thf(fact_3927_well__order__on__domain,axiom,
! [A4: set_o,R: set_Product_prod_o_o,A: $o,B: $o] :
( ( order_2615775362358406454r_on_o @ A4 @ R )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ R )
=> ( ( member_o @ A @ A4 )
& ( member_o @ B @ A4 ) ) ) ) ).
% well_order_on_domain
thf(fact_3928_well__order__on__domain,axiom,
! [A4: set_se5479201105039950375at_nat,R: set_Pr7459493094073627847at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
( ( order_423636163458052581at_nat @ A4 @ R )
=> ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ B ) @ R )
=> ( ( member6273874620674182256at_nat @ A @ A4 )
& ( member6273874620674182256at_nat @ B @ A4 ) ) ) ) ).
% well_order_on_domain
thf(fact_3929_well__order__on__domain,axiom,
! [A4: set_se7855581050983116737at_nat,R: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
( ( order_1968183844437341695at_nat @ A4 @ R )
=> ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R )
=> ( ( member2643936169264416010at_nat @ A @ A4 )
& ( member2643936169264416010at_nat @ B @ A4 ) ) ) ) ).
% well_order_on_domain
thf(fact_3930_well__order__on__domain,axiom,
! [A4: set_int,R: set_Pr958786334691620121nt_int,A: int,B: int] :
( ( order_2886507596567047182on_int @ A4 @ R )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ R )
=> ( ( member_int @ A @ A4 )
& ( member_int @ B @ A4 ) ) ) ) ).
% well_order_on_domain
thf(fact_3931_well__order__on__domain,axiom,
! [A4: set_nat,R: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
( ( order_2888998067076097458on_nat @ A4 @ R )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
=> ( ( member_nat @ A @ A4 )
& ( member_nat @ B @ A4 ) ) ) ) ).
% well_order_on_domain
thf(fact_3932_infinite__countable__subset,axiom,
! [S5: set_rat] :
( ~ ( finite_finite_rat @ S5 )
=> ? [F5: nat > rat] :
( ( inj_on_nat_rat @ F5 @ top_top_set_nat )
& ( ord_less_eq_set_rat @ ( image_nat_rat @ F5 @ top_top_set_nat ) @ S5 ) ) ) ).
% infinite_countable_subset
thf(fact_3933_infinite__countable__subset,axiom,
! [S5: set_int] :
( ~ ( finite_finite_int @ S5 )
=> ? [F5: nat > int] :
( ( inj_on_nat_int @ F5 @ top_top_set_nat )
& ( ord_less_eq_set_int @ ( image_nat_int @ F5 @ top_top_set_nat ) @ S5 ) ) ) ).
% infinite_countable_subset
thf(fact_3934_infinite__countable__subset,axiom,
! [S5: set_Pr1261947904930325089at_nat] :
( ~ ( finite6177210948735845034at_nat @ S5 )
=> ? [F5: nat > product_prod_nat_nat] :
( ( inj_on5538052773655684606at_nat @ F5 @ top_top_set_nat )
& ( ord_le3146513528884898305at_nat @ ( image_5846123807819985514at_nat @ F5 @ top_top_set_nat ) @ S5 ) ) ) ).
% infinite_countable_subset
thf(fact_3935_infinite__countable__subset,axiom,
! [S5: set_nat] :
( ~ ( finite_finite_nat @ S5 )
=> ? [F5: nat > nat] :
( ( inj_on_nat_nat @ F5 @ top_top_set_nat )
& ( ord_less_eq_set_nat @ ( image_nat_nat @ F5 @ top_top_set_nat ) @ S5 ) ) ) ).
% infinite_countable_subset
thf(fact_3936_infinite__iff__countable__subset,axiom,
! [S5: set_rat] :
( ( ~ ( finite_finite_rat @ S5 ) )
= ( ? [F2: nat > rat] :
( ( inj_on_nat_rat @ F2 @ top_top_set_nat )
& ( ord_less_eq_set_rat @ ( image_nat_rat @ F2 @ top_top_set_nat ) @ S5 ) ) ) ) ).
% infinite_iff_countable_subset
thf(fact_3937_infinite__iff__countable__subset,axiom,
! [S5: set_int] :
( ( ~ ( finite_finite_int @ S5 ) )
= ( ? [F2: nat > int] :
( ( inj_on_nat_int @ F2 @ top_top_set_nat )
& ( ord_less_eq_set_int @ ( image_nat_int @ F2 @ top_top_set_nat ) @ S5 ) ) ) ) ).
% infinite_iff_countable_subset
thf(fact_3938_infinite__iff__countable__subset,axiom,
! [S5: set_Pr1261947904930325089at_nat] :
( ( ~ ( finite6177210948735845034at_nat @ S5 ) )
= ( ? [F2: nat > product_prod_nat_nat] :
( ( inj_on5538052773655684606at_nat @ F2 @ top_top_set_nat )
& ( ord_le3146513528884898305at_nat @ ( image_5846123807819985514at_nat @ F2 @ top_top_set_nat ) @ S5 ) ) ) ) ).
% infinite_iff_countable_subset
thf(fact_3939_infinite__iff__countable__subset,axiom,
! [S5: set_nat] :
( ( ~ ( finite_finite_nat @ S5 ) )
= ( ? [F2: nat > nat] :
( ( inj_on_nat_nat @ F2 @ top_top_set_nat )
& ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ top_top_set_nat ) @ S5 ) ) ) ) ).
% infinite_iff_countable_subset
thf(fact_3940_IdE,axiom,
! [P: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ P @ id_Pro2258643101195443293at_nat )
=> ~ ! [X: product_prod_nat_nat] :
( P
!= ( produc6161850002892822231at_nat @ X @ X ) ) ) ).
% IdE
thf(fact_3941_IdE,axiom,
! [P: produc1319942482725812455at_nat] :
( ( member1466754251312161552at_nat @ P @ id_set4888139568991089827at_nat )
=> ~ ! [X: set_Pr4329608150637261639at_nat] :
( P
!= ( produc9060074326276436823at_nat @ X @ X ) ) ) ).
% IdE
thf(fact_3942_IdE,axiom,
! [P: produc3843707927480180839at_nat] :
( ( member8757157785044589968at_nat @ P @ id_set1796276052702428605at_nat )
=> ~ ! [X: set_Pr1261947904930325089at_nat] :
( P
!= ( produc2922128104949294807at_nat @ X @ X ) ) ) ).
% IdE
thf(fact_3943_IdE,axiom,
! [P: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ P @ id_nat2 )
=> ~ ! [X: nat] :
( P
!= ( product_Pair_nat_nat @ X @ X ) ) ) ).
% IdE
thf(fact_3944_IdE,axiom,
! [P: product_prod_int_int] :
( ( member5262025264175285858nt_int @ P @ id_int2 )
=> ~ ! [X: int] :
( P
!= ( product_Pair_int_int @ X @ X ) ) ) ).
% IdE
thf(fact_3945_IdD,axiom,
! [A: product_prod_nat_nat,B: product_prod_nat_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ id_Pro2258643101195443293at_nat )
=> ( A = B ) ) ).
% IdD
thf(fact_3946_IdD,axiom,
! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ B ) @ id_set4888139568991089827at_nat )
=> ( A = B ) ) ).
% IdD
thf(fact_3947_IdD,axiom,
! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ id_set1796276052702428605at_nat )
=> ( A = B ) ) ).
% IdD
thf(fact_3948_IdD,axiom,
! [A: nat,B: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ id_nat2 )
=> ( A = B ) ) ).
% IdD
thf(fact_3949_IdD,axiom,
! [A: int,B: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ id_int2 )
=> ( A = B ) ) ).
% IdD
thf(fact_3950_card__bij__eq,axiom,
! [F: int > int,A4: set_int,B5: set_int,G: int > int] :
( ( inj_on_int_int @ F @ A4 )
=> ( ( ord_less_eq_set_int @ ( image_int_int @ F @ A4 ) @ B5 )
=> ( ( inj_on_int_int @ G @ B5 )
=> ( ( ord_less_eq_set_int @ ( image_int_int @ G @ B5 ) @ A4 )
=> ( ( finite_finite_int @ A4 )
=> ( ( finite_finite_int @ B5 )
=> ( ( finite_card_int @ A4 )
= ( finite_card_int @ B5 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_3951_card__bij__eq,axiom,
! [F: rat > nat,A4: set_rat,B5: set_nat,G: nat > rat] :
( ( inj_on_rat_nat @ F @ A4 )
=> ( ( ord_less_eq_set_nat @ ( image_rat_nat @ F @ A4 ) @ B5 )
=> ( ( inj_on_nat_rat @ G @ B5 )
=> ( ( ord_less_eq_set_rat @ ( image_nat_rat @ G @ B5 ) @ A4 )
=> ( ( finite_finite_rat @ A4 )
=> ( ( finite_finite_nat @ B5 )
=> ( ( finite_card_rat @ A4 )
= ( finite_card_nat @ B5 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_3952_card__bij__eq,axiom,
! [F: int > nat,A4: set_int,B5: set_nat,G: nat > int] :
( ( inj_on_int_nat @ F @ A4 )
=> ( ( ord_less_eq_set_nat @ ( image_int_nat @ F @ A4 ) @ B5 )
=> ( ( inj_on_nat_int @ G @ B5 )
=> ( ( ord_less_eq_set_int @ ( image_nat_int @ G @ B5 ) @ A4 )
=> ( ( finite_finite_int @ A4 )
=> ( ( finite_finite_nat @ B5 )
=> ( ( finite_card_int @ A4 )
= ( finite_card_nat @ B5 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_3953_card__bij__eq,axiom,
! [F: product_prod_nat_nat > product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,G: product_prod_nat_nat > product_prod_nat_nat] :
( ( inj_on8969904277767023793at_nat @ F @ A4 )
=> ( ( ord_le3146513528884898305at_nat @ ( image_5168914502847457605at_nat @ F @ A4 ) @ B5 )
=> ( ( inj_on8969904277767023793at_nat @ G @ B5 )
=> ( ( ord_le3146513528884898305at_nat @ ( image_5168914502847457605at_nat @ G @ B5 ) @ A4 )
=> ( ( finite6177210948735845034at_nat @ A4 )
=> ( ( finite6177210948735845034at_nat @ B5 )
=> ( ( finite711546835091564841at_nat @ A4 )
= ( finite711546835091564841at_nat @ B5 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_3954_card__bij__eq,axiom,
! [F: product_prod_nat_nat > nat,A4: set_Pr1261947904930325089at_nat,B5: set_nat,G: nat > product_prod_nat_nat] :
( ( inj_on2178005380612969504at_nat @ F @ A4 )
=> ( ( ord_less_eq_set_nat @ ( image_2486076414777270412at_nat @ F @ A4 ) @ B5 )
=> ( ( inj_on5538052773655684606at_nat @ G @ B5 )
=> ( ( ord_le3146513528884898305at_nat @ ( image_5846123807819985514at_nat @ G @ B5 ) @ A4 )
=> ( ( finite6177210948735845034at_nat @ A4 )
=> ( ( finite_finite_nat @ B5 )
=> ( ( finite711546835091564841at_nat @ A4 )
= ( finite_card_nat @ B5 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_3955_card__bij__eq,axiom,
! [F: nat > rat,A4: set_nat,B5: set_rat,G: rat > nat] :
( ( inj_on_nat_rat @ F @ A4 )
=> ( ( ord_less_eq_set_rat @ ( image_nat_rat @ F @ A4 ) @ B5 )
=> ( ( inj_on_rat_nat @ G @ B5 )
=> ( ( ord_less_eq_set_nat @ ( image_rat_nat @ G @ B5 ) @ A4 )
=> ( ( finite_finite_nat @ A4 )
=> ( ( finite_finite_rat @ B5 )
=> ( ( finite_card_nat @ A4 )
= ( finite_card_rat @ B5 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_3956_card__bij__eq,axiom,
! [F: nat > int,A4: set_nat,B5: set_int,G: int > nat] :
( ( inj_on_nat_int @ F @ A4 )
=> ( ( ord_less_eq_set_int @ ( image_nat_int @ F @ A4 ) @ B5 )
=> ( ( inj_on_int_nat @ G @ B5 )
=> ( ( ord_less_eq_set_nat @ ( image_int_nat @ G @ B5 ) @ A4 )
=> ( ( finite_finite_nat @ A4 )
=> ( ( finite_finite_int @ B5 )
=> ( ( finite_card_nat @ A4 )
= ( finite_card_int @ B5 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_3957_card__bij__eq,axiom,
! [F: nat > product_prod_nat_nat,A4: set_nat,B5: set_Pr1261947904930325089at_nat,G: product_prod_nat_nat > nat] :
( ( inj_on5538052773655684606at_nat @ F @ A4 )
=> ( ( ord_le3146513528884898305at_nat @ ( image_5846123807819985514at_nat @ F @ A4 ) @ B5 )
=> ( ( inj_on2178005380612969504at_nat @ G @ B5 )
=> ( ( ord_less_eq_set_nat @ ( image_2486076414777270412at_nat @ G @ B5 ) @ A4 )
=> ( ( finite_finite_nat @ A4 )
=> ( ( finite6177210948735845034at_nat @ B5 )
=> ( ( finite_card_nat @ A4 )
= ( finite711546835091564841at_nat @ B5 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_3958_card__bij__eq,axiom,
! [F: nat > nat,A4: set_nat,B5: set_nat,G: nat > nat] :
( ( inj_on_nat_nat @ F @ A4 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A4 ) @ B5 )
=> ( ( inj_on_nat_nat @ G @ B5 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ G @ B5 ) @ A4 )
=> ( ( finite_finite_nat @ A4 )
=> ( ( finite_finite_nat @ B5 )
=> ( ( finite_card_nat @ A4 )
= ( finite_card_nat @ B5 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_3959_card__le__inj,axiom,
! [A4: set_int,B5: set_int] :
( ( finite_finite_int @ A4 )
=> ( ( finite_finite_int @ B5 )
=> ( ( ord_less_eq_nat @ ( finite_card_int @ A4 ) @ ( finite_card_int @ B5 ) )
=> ? [F5: int > int] :
( ( ord_less_eq_set_int @ ( image_int_int @ F5 @ A4 ) @ B5 )
& ( inj_on_int_int @ F5 @ A4 ) ) ) ) ) ).
% card_le_inj
thf(fact_3960_card__le__inj,axiom,
! [A4: set_int,B5: set_nat] :
( ( finite_finite_int @ A4 )
=> ( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_nat @ ( finite_card_int @ A4 ) @ ( finite_card_nat @ B5 ) )
=> ? [F5: int > nat] :
( ( ord_less_eq_set_nat @ ( image_int_nat @ F5 @ A4 ) @ B5 )
& ( inj_on_int_nat @ F5 @ A4 ) ) ) ) ) ).
% card_le_inj
thf(fact_3961_card__le__inj,axiom,
! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ A4 )
=> ( ( finite6177210948735845034at_nat @ B5 )
=> ( ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ A4 ) @ ( finite711546835091564841at_nat @ B5 ) )
=> ? [F5: product_prod_nat_nat > product_prod_nat_nat] :
( ( ord_le3146513528884898305at_nat @ ( image_5168914502847457605at_nat @ F5 @ A4 ) @ B5 )
& ( inj_on8969904277767023793at_nat @ F5 @ A4 ) ) ) ) ) ).
% card_le_inj
thf(fact_3962_card__le__inj,axiom,
! [A4: set_Pr1261947904930325089at_nat,B5: set_nat] :
( ( finite6177210948735845034at_nat @ A4 )
=> ( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ A4 ) @ ( finite_card_nat @ B5 ) )
=> ? [F5: product_prod_nat_nat > nat] :
( ( ord_less_eq_set_nat @ ( image_2486076414777270412at_nat @ F5 @ A4 ) @ B5 )
& ( inj_on2178005380612969504at_nat @ F5 @ A4 ) ) ) ) ) ).
% card_le_inj
thf(fact_3963_card__le__inj,axiom,
! [A4: set_nat,B5: set_rat] :
( ( finite_finite_nat @ A4 )
=> ( ( finite_finite_rat @ B5 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_rat @ B5 ) )
=> ? [F5: nat > rat] :
( ( ord_less_eq_set_rat @ ( image_nat_rat @ F5 @ A4 ) @ B5 )
& ( inj_on_nat_rat @ F5 @ A4 ) ) ) ) ) ).
% card_le_inj
thf(fact_3964_card__le__inj,axiom,
! [A4: set_nat,B5: set_int] :
( ( finite_finite_nat @ A4 )
=> ( ( finite_finite_int @ B5 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_int @ B5 ) )
=> ? [F5: nat > int] :
( ( ord_less_eq_set_int @ ( image_nat_int @ F5 @ A4 ) @ B5 )
& ( inj_on_nat_int @ F5 @ A4 ) ) ) ) ) ).
% card_le_inj
thf(fact_3965_card__le__inj,axiom,
! [A4: set_nat,B5: set_Pr1261947904930325089at_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( finite6177210948735845034at_nat @ B5 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite711546835091564841at_nat @ B5 ) )
=> ? [F5: nat > product_prod_nat_nat] :
( ( ord_le3146513528884898305at_nat @ ( image_5846123807819985514at_nat @ F5 @ A4 ) @ B5 )
& ( inj_on5538052773655684606at_nat @ F5 @ A4 ) ) ) ) ) ).
% card_le_inj
thf(fact_3966_card__le__inj,axiom,
! [A4: set_nat,B5: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B5 ) )
=> ? [F5: nat > nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F5 @ A4 ) @ B5 )
& ( inj_on_nat_nat @ F5 @ A4 ) ) ) ) ) ).
% card_le_inj
thf(fact_3967_card__inj__on__le,axiom,
! [F: nat > rat,A4: set_nat,B5: set_rat] :
( ( inj_on_nat_rat @ F @ A4 )
=> ( ( ord_less_eq_set_rat @ ( image_nat_rat @ F @ A4 ) @ B5 )
=> ( ( finite_finite_rat @ B5 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_rat @ B5 ) ) ) ) ) ).
% card_inj_on_le
thf(fact_3968_card__inj__on__le,axiom,
! [F: nat > int,A4: set_nat,B5: set_int] :
( ( inj_on_nat_int @ F @ A4 )
=> ( ( ord_less_eq_set_int @ ( image_nat_int @ F @ A4 ) @ B5 )
=> ( ( finite_finite_int @ B5 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_int @ B5 ) ) ) ) ) ).
% card_inj_on_le
thf(fact_3969_card__inj__on__le,axiom,
! [F: int > int,A4: set_int,B5: set_int] :
( ( inj_on_int_int @ F @ A4 )
=> ( ( ord_less_eq_set_int @ ( image_int_int @ F @ A4 ) @ B5 )
=> ( ( finite_finite_int @ B5 )
=> ( ord_less_eq_nat @ ( finite_card_int @ A4 ) @ ( finite_card_int @ B5 ) ) ) ) ) ).
% card_inj_on_le
thf(fact_3970_card__inj__on__le,axiom,
! [F: nat > nat,A4: set_nat,B5: set_nat] :
( ( inj_on_nat_nat @ F @ A4 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A4 ) @ B5 )
=> ( ( finite_finite_nat @ B5 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B5 ) ) ) ) ) ).
% card_inj_on_le
thf(fact_3971_card__inj__on__le,axiom,
! [F: int > nat,A4: set_int,B5: set_nat] :
( ( inj_on_int_nat @ F @ A4 )
=> ( ( ord_less_eq_set_nat @ ( image_int_nat @ F @ A4 ) @ B5 )
=> ( ( finite_finite_nat @ B5 )
=> ( ord_less_eq_nat @ ( finite_card_int @ A4 ) @ ( finite_card_nat @ B5 ) ) ) ) ) ).
% card_inj_on_le
thf(fact_3972_inj__on__iff__card__le,axiom,
! [A4: set_int,B5: set_int] :
( ( finite_finite_int @ A4 )
=> ( ( finite_finite_int @ B5 )
=> ( ( ? [F2: int > int] :
( ( inj_on_int_int @ F2 @ A4 )
& ( ord_less_eq_set_int @ ( image_int_int @ F2 @ A4 ) @ B5 ) ) )
= ( ord_less_eq_nat @ ( finite_card_int @ A4 ) @ ( finite_card_int @ B5 ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_3973_inj__on__iff__card__le,axiom,
! [A4: set_int,B5: set_nat] :
( ( finite_finite_int @ A4 )
=> ( ( finite_finite_nat @ B5 )
=> ( ( ? [F2: int > nat] :
( ( inj_on_int_nat @ F2 @ A4 )
& ( ord_less_eq_set_nat @ ( image_int_nat @ F2 @ A4 ) @ B5 ) ) )
= ( ord_less_eq_nat @ ( finite_card_int @ A4 ) @ ( finite_card_nat @ B5 ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_3974_inj__on__iff__card__le,axiom,
! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ A4 )
=> ( ( finite6177210948735845034at_nat @ B5 )
=> ( ( ? [F2: product_prod_nat_nat > product_prod_nat_nat] :
( ( inj_on8969904277767023793at_nat @ F2 @ A4 )
& ( ord_le3146513528884898305at_nat @ ( image_5168914502847457605at_nat @ F2 @ A4 ) @ B5 ) ) )
= ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ A4 ) @ ( finite711546835091564841at_nat @ B5 ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_3975_inj__on__iff__card__le,axiom,
! [A4: set_Pr1261947904930325089at_nat,B5: set_nat] :
( ( finite6177210948735845034at_nat @ A4 )
=> ( ( finite_finite_nat @ B5 )
=> ( ( ? [F2: product_prod_nat_nat > nat] :
( ( inj_on2178005380612969504at_nat @ F2 @ A4 )
& ( ord_less_eq_set_nat @ ( image_2486076414777270412at_nat @ F2 @ A4 ) @ B5 ) ) )
= ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ A4 ) @ ( finite_card_nat @ B5 ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_3976_inj__on__iff__card__le,axiom,
! [A4: set_nat,B5: set_rat] :
( ( finite_finite_nat @ A4 )
=> ( ( finite_finite_rat @ B5 )
=> ( ( ? [F2: nat > rat] :
( ( inj_on_nat_rat @ F2 @ A4 )
& ( ord_less_eq_set_rat @ ( image_nat_rat @ F2 @ A4 ) @ B5 ) ) )
= ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_rat @ B5 ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_3977_inj__on__iff__card__le,axiom,
! [A4: set_nat,B5: set_int] :
( ( finite_finite_nat @ A4 )
=> ( ( finite_finite_int @ B5 )
=> ( ( ? [F2: nat > int] :
( ( inj_on_nat_int @ F2 @ A4 )
& ( ord_less_eq_set_int @ ( image_nat_int @ F2 @ A4 ) @ B5 ) ) )
= ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_int @ B5 ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_3978_inj__on__iff__card__le,axiom,
! [A4: set_nat,B5: set_Pr1261947904930325089at_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( finite6177210948735845034at_nat @ B5 )
=> ( ( ? [F2: nat > product_prod_nat_nat] :
( ( inj_on5538052773655684606at_nat @ F2 @ A4 )
& ( ord_le3146513528884898305at_nat @ ( image_5846123807819985514at_nat @ F2 @ A4 ) @ B5 ) ) )
= ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite711546835091564841at_nat @ B5 ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_3979_inj__on__iff__card__le,axiom,
! [A4: set_nat,B5: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( finite_finite_nat @ B5 )
=> ( ( ? [F2: nat > nat] :
( ( inj_on_nat_nat @ F2 @ A4 )
& ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A4 ) @ B5 ) ) )
= ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B5 ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_3980_surjective__iff__injective__gen,axiom,
! [S5: set_int,T3: set_int,F: int > int] :
( ( finite_finite_int @ S5 )
=> ( ( finite_finite_int @ T3 )
=> ( ( ( finite_card_int @ S5 )
= ( finite_card_int @ T3 ) )
=> ( ( ord_less_eq_set_int @ ( image_int_int @ F @ S5 ) @ T3 )
=> ( ( ! [X4: int] :
( ( member_int @ X4 @ T3 )
=> ? [Y5: int] :
( ( member_int @ Y5 @ S5 )
& ( ( F @ Y5 )
= X4 ) ) ) )
= ( inj_on_int_int @ F @ S5 ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_3981_surjective__iff__injective__gen,axiom,
! [S5: set_int,T3: set_nat,F: int > nat] :
( ( finite_finite_int @ S5 )
=> ( ( finite_finite_nat @ T3 )
=> ( ( ( finite_card_int @ S5 )
= ( finite_card_nat @ T3 ) )
=> ( ( ord_less_eq_set_nat @ ( image_int_nat @ F @ S5 ) @ T3 )
=> ( ( ! [X4: nat] :
( ( member_nat @ X4 @ T3 )
=> ? [Y5: int] :
( ( member_int @ Y5 @ S5 )
& ( ( F @ Y5 )
= X4 ) ) ) )
= ( inj_on_int_nat @ F @ S5 ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_3982_surjective__iff__injective__gen,axiom,
! [S5: set_Pr1261947904930325089at_nat,T3: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > product_prod_nat_nat] :
( ( finite6177210948735845034at_nat @ S5 )
=> ( ( finite6177210948735845034at_nat @ T3 )
=> ( ( ( finite711546835091564841at_nat @ S5 )
= ( finite711546835091564841at_nat @ T3 ) )
=> ( ( ord_le3146513528884898305at_nat @ ( image_5168914502847457605at_nat @ F @ S5 ) @ T3 )
=> ( ( ! [X4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X4 @ T3 )
=> ? [Y5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ Y5 @ S5 )
& ( ( F @ Y5 )
= X4 ) ) ) )
= ( inj_on8969904277767023793at_nat @ F @ S5 ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_3983_surjective__iff__injective__gen,axiom,
! [S5: set_Pr1261947904930325089at_nat,T3: set_nat,F: product_prod_nat_nat > nat] :
( ( finite6177210948735845034at_nat @ S5 )
=> ( ( finite_finite_nat @ T3 )
=> ( ( ( finite711546835091564841at_nat @ S5 )
= ( finite_card_nat @ T3 ) )
=> ( ( ord_less_eq_set_nat @ ( image_2486076414777270412at_nat @ F @ S5 ) @ T3 )
=> ( ( ! [X4: nat] :
( ( member_nat @ X4 @ T3 )
=> ? [Y5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ Y5 @ S5 )
& ( ( F @ Y5 )
= X4 ) ) ) )
= ( inj_on2178005380612969504at_nat @ F @ S5 ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_3984_surjective__iff__injective__gen,axiom,
! [S5: set_nat,T3: set_rat,F: nat > rat] :
( ( finite_finite_nat @ S5 )
=> ( ( finite_finite_rat @ T3 )
=> ( ( ( finite_card_nat @ S5 )
= ( finite_card_rat @ T3 ) )
=> ( ( ord_less_eq_set_rat @ ( image_nat_rat @ F @ S5 ) @ T3 )
=> ( ( ! [X4: rat] :
( ( member_rat @ X4 @ T3 )
=> ? [Y5: nat] :
( ( member_nat @ Y5 @ S5 )
& ( ( F @ Y5 )
= X4 ) ) ) )
= ( inj_on_nat_rat @ F @ S5 ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_3985_surjective__iff__injective__gen,axiom,
! [S5: set_nat,T3: set_int,F: nat > int] :
( ( finite_finite_nat @ S5 )
=> ( ( finite_finite_int @ T3 )
=> ( ( ( finite_card_nat @ S5 )
= ( finite_card_int @ T3 ) )
=> ( ( ord_less_eq_set_int @ ( image_nat_int @ F @ S5 ) @ T3 )
=> ( ( ! [X4: int] :
( ( member_int @ X4 @ T3 )
=> ? [Y5: nat] :
( ( member_nat @ Y5 @ S5 )
& ( ( F @ Y5 )
= X4 ) ) ) )
= ( inj_on_nat_int @ F @ S5 ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_3986_surjective__iff__injective__gen,axiom,
! [S5: set_nat,T3: set_Pr1261947904930325089at_nat,F: nat > product_prod_nat_nat] :
( ( finite_finite_nat @ S5 )
=> ( ( finite6177210948735845034at_nat @ T3 )
=> ( ( ( finite_card_nat @ S5 )
= ( finite711546835091564841at_nat @ T3 ) )
=> ( ( ord_le3146513528884898305at_nat @ ( image_5846123807819985514at_nat @ F @ S5 ) @ T3 )
=> ( ( ! [X4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X4 @ T3 )
=> ? [Y5: nat] :
( ( member_nat @ Y5 @ S5 )
& ( ( F @ Y5 )
= X4 ) ) ) )
= ( inj_on5538052773655684606at_nat @ F @ S5 ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_3987_surjective__iff__injective__gen,axiom,
! [S5: set_nat,T3: set_nat,F: nat > nat] :
( ( finite_finite_nat @ S5 )
=> ( ( finite_finite_nat @ T3 )
=> ( ( ( finite_card_nat @ S5 )
= ( finite_card_nat @ T3 ) )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ S5 ) @ T3 )
=> ( ( ! [X4: nat] :
( ( member_nat @ X4 @ T3 )
=> ? [Y5: nat] :
( ( member_nat @ Y5 @ S5 )
& ( ( F @ Y5 )
= X4 ) ) ) )
= ( inj_on_nat_nat @ F @ S5 ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_3988_pigeonhole__infinite,axiom,
! [A4: set_int,F: int > int] :
( ~ ( finite_finite_int @ A4 )
=> ( ( finite_finite_int @ ( image_int_int @ F @ A4 ) )
=> ? [X: int] :
( ( member_int @ X @ A4 )
& ~ ( finite_finite_int
@ ( collect_int
@ ^ [A5: int] :
( ( member_int @ A5 @ A4 )
& ( ( F @ A5 )
= ( F @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_3989_pigeonhole__infinite,axiom,
! [A4: set_o,F: $o > nat] :
( ~ ( finite_finite_o @ A4 )
=> ( ( finite_finite_nat @ ( image_o_nat @ F @ A4 ) )
=> ? [X: $o] :
( ( member_o @ X @ A4 )
& ~ ( finite_finite_o
@ ( collect_o
@ ^ [A5: $o] :
( ( member_o @ A5 @ A4 )
& ( ( F @ A5 )
= ( F @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_3990_pigeonhole__infinite,axiom,
! [A4: set_int,F: int > nat] :
( ~ ( finite_finite_int @ A4 )
=> ( ( finite_finite_nat @ ( image_int_nat @ F @ A4 ) )
=> ? [X: int] :
( ( member_int @ X @ A4 )
& ~ ( finite_finite_int
@ ( collect_int
@ ^ [A5: int] :
( ( member_int @ A5 @ A4 )
& ( ( F @ A5 )
= ( F @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_3991_pigeonhole__infinite,axiom,
! [A4: set_nat,F: nat > rat] :
( ~ ( finite_finite_nat @ A4 )
=> ( ( finite_finite_rat @ ( image_nat_rat @ F @ A4 ) )
=> ? [X: nat] :
( ( member_nat @ X @ A4 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A5: nat] :
( ( member_nat @ A5 @ A4 )
& ( ( F @ A5 )
= ( F @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_3992_pigeonhole__infinite,axiom,
! [A4: set_nat,F: nat > int] :
( ~ ( finite_finite_nat @ A4 )
=> ( ( finite_finite_int @ ( image_nat_int @ F @ A4 ) )
=> ? [X: nat] :
( ( member_nat @ X @ A4 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A5: nat] :
( ( member_nat @ A5 @ A4 )
& ( ( F @ A5 )
= ( F @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_3993_pigeonhole__infinite,axiom,
! [A4: set_nat,F: nat > nat] :
( ~ ( finite_finite_nat @ A4 )
=> ( ( finite_finite_nat @ ( image_nat_nat @ F @ A4 ) )
=> ? [X: nat] :
( ( member_nat @ X @ A4 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A5: nat] :
( ( member_nat @ A5 @ A4 )
& ( ( F @ A5 )
= ( F @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_3994_pigeonhole__infinite,axiom,
! [A4: set_o,F: $o > product_prod_nat_nat] :
( ~ ( finite_finite_o @ A4 )
=> ( ( finite6177210948735845034at_nat @ ( image_3855930084881510382at_nat @ F @ A4 ) )
=> ? [X: $o] :
( ( member_o @ X @ A4 )
& ~ ( finite_finite_o
@ ( collect_o
@ ^ [A5: $o] :
( ( member_o @ A5 @ A4 )
& ( ( F @ A5 )
= ( F @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_3995_pigeonhole__infinite,axiom,
! [A4: set_int,F: int > product_prod_nat_nat] :
( ~ ( finite_finite_int @ A4 )
=> ( ( finite6177210948735845034at_nat @ ( image_8883965892284119310at_nat @ F @ A4 ) )
=> ? [X: int] :
( ( member_int @ X @ A4 )
& ~ ( finite_finite_int
@ ( collect_int
@ ^ [A5: int] :
( ( member_int @ A5 @ A4 )
& ( ( F @ A5 )
= ( F @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_3996_pigeonhole__infinite,axiom,
! [A4: set_Pr958786334691620121nt_int,F: product_prod_int_int > nat] :
( ~ ( finite2998713641127702882nt_int @ A4 )
=> ( ( finite_finite_nat @ ( image_5044651549707136836nt_nat @ F @ A4 ) )
=> ? [X: product_prod_int_int] :
( ( member5262025264175285858nt_int @ X @ A4 )
& ~ ( finite2998713641127702882nt_int
@ ( collec213857154873943460nt_int
@ ^ [A5: product_prod_int_int] :
( ( member5262025264175285858nt_int @ A5 @ A4 )
& ( ( F @ A5 )
= ( F @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_3997_pigeonhole__infinite,axiom,
! [A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
( ~ ( finite6177210948735845034at_nat @ A4 )
=> ( ( finite_finite_nat @ ( image_2486076414777270412at_nat @ F @ A4 ) )
=> ? [X: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X @ A4 )
& ~ ( finite6177210948735845034at_nat
@ ( collec3392354462482085612at_nat
@ ^ [A5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ A5 @ A4 )
& ( ( F @ A5 )
= ( F @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_3998_snds_Ointros,axiom,
! [P: produc7812382424243267045t_unit] : ( member242389114213480694t_unit @ ( produc1126150082675451753t_unit @ P ) @ ( basic_1496934909540232809t_unit @ P ) ) ).
% snds.intros
thf(fact_3999_snds_Osimps,axiom,
! [A: heap_e7401611519738050253t_unit,P: produc7812382424243267045t_unit] :
( ( member242389114213480694t_unit @ A @ ( basic_1496934909540232809t_unit @ P ) )
= ( A
= ( produc1126150082675451753t_unit @ P ) ) ) ).
% snds.simps
thf(fact_4000_snds_Ocases,axiom,
! [A: heap_e7401611519738050253t_unit,P: produc7812382424243267045t_unit] :
( ( member242389114213480694t_unit @ A @ ( basic_1496934909540232809t_unit @ P ) )
=> ( A
= ( produc1126150082675451753t_unit @ P ) ) ) ).
% snds.cases
thf(fact_4001_card__image,axiom,
! [F: nat > rat,A4: set_nat] :
( ( inj_on_nat_rat @ F @ A4 )
=> ( ( finite_card_rat @ ( image_nat_rat @ F @ A4 ) )
= ( finite_card_nat @ A4 ) ) ) ).
% card_image
thf(fact_4002_card__image,axiom,
! [F: nat > nat,A4: set_nat] :
( ( inj_on_nat_nat @ F @ A4 )
=> ( ( finite_card_nat @ ( image_nat_nat @ F @ A4 ) )
= ( finite_card_nat @ A4 ) ) ) ).
% card_image
thf(fact_4003_card__image,axiom,
! [F: nat > int,A4: set_nat] :
( ( inj_on_nat_int @ F @ A4 )
=> ( ( finite_card_int @ ( image_nat_int @ F @ A4 ) )
= ( finite_card_nat @ A4 ) ) ) ).
% card_image
thf(fact_4004_card__image,axiom,
! [F: int > nat,A4: set_int] :
( ( inj_on_int_nat @ F @ A4 )
=> ( ( finite_card_nat @ ( image_int_nat @ F @ A4 ) )
= ( finite_card_int @ A4 ) ) ) ).
% card_image
thf(fact_4005_card__image,axiom,
! [F: int > int,A4: set_int] :
( ( inj_on_int_int @ F @ A4 )
=> ( ( finite_card_int @ ( image_int_int @ F @ A4 ) )
= ( finite_card_int @ A4 ) ) ) ).
% card_image
thf(fact_4006_all__finite__subset__image,axiom,
! [F: int > int,A4: set_int,P2: set_int > $o] :
( ( ! [B6: set_int] :
( ( ( finite_finite_int @ B6 )
& ( ord_less_eq_set_int @ B6 @ ( image_int_int @ F @ A4 ) ) )
=> ( P2 @ B6 ) ) )
= ( ! [B6: set_int] :
( ( ( finite_finite_int @ B6 )
& ( ord_less_eq_set_int @ B6 @ A4 ) )
=> ( P2 @ ( image_int_int @ F @ B6 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_4007_all__finite__subset__image,axiom,
! [F: nat > rat,A4: set_nat,P2: set_rat > $o] :
( ( ! [B6: set_rat] :
( ( ( finite_finite_rat @ B6 )
& ( ord_less_eq_set_rat @ B6 @ ( image_nat_rat @ F @ A4 ) ) )
=> ( P2 @ B6 ) ) )
= ( ! [B6: set_nat] :
( ( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ A4 ) )
=> ( P2 @ ( image_nat_rat @ F @ B6 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_4008_all__finite__subset__image,axiom,
! [F: nat > int,A4: set_nat,P2: set_int > $o] :
( ( ! [B6: set_int] :
( ( ( finite_finite_int @ B6 )
& ( ord_less_eq_set_int @ B6 @ ( image_nat_int @ F @ A4 ) ) )
=> ( P2 @ B6 ) ) )
= ( ! [B6: set_nat] :
( ( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ A4 ) )
=> ( P2 @ ( image_nat_int @ F @ B6 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_4009_all__finite__subset__image,axiom,
! [F: product_prod_nat_nat > product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,P2: set_Pr1261947904930325089at_nat > $o] :
( ( ! [B6: set_Pr1261947904930325089at_nat] :
( ( ( finite6177210948735845034at_nat @ B6 )
& ( ord_le3146513528884898305at_nat @ B6 @ ( image_5168914502847457605at_nat @ F @ A4 ) ) )
=> ( P2 @ B6 ) ) )
= ( ! [B6: set_Pr1261947904930325089at_nat] :
( ( ( finite6177210948735845034at_nat @ B6 )
& ( ord_le3146513528884898305at_nat @ B6 @ A4 ) )
=> ( P2 @ ( image_5168914502847457605at_nat @ F @ B6 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_4010_all__finite__subset__image,axiom,
! [F: nat > product_prod_nat_nat,A4: set_nat,P2: set_Pr1261947904930325089at_nat > $o] :
( ( ! [B6: set_Pr1261947904930325089at_nat] :
( ( ( finite6177210948735845034at_nat @ B6 )
& ( ord_le3146513528884898305at_nat @ B6 @ ( image_5846123807819985514at_nat @ F @ A4 ) ) )
=> ( P2 @ B6 ) ) )
= ( ! [B6: set_nat] :
( ( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ A4 ) )
=> ( P2 @ ( image_5846123807819985514at_nat @ F @ B6 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_4011_all__finite__subset__image,axiom,
! [F: int > nat,A4: set_int,P2: set_nat > $o] :
( ( ! [B6: set_nat] :
( ( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ ( image_int_nat @ F @ A4 ) ) )
=> ( P2 @ B6 ) ) )
= ( ! [B6: set_int] :
( ( ( finite_finite_int @ B6 )
& ( ord_less_eq_set_int @ B6 @ A4 ) )
=> ( P2 @ ( image_int_nat @ F @ B6 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_4012_all__finite__subset__image,axiom,
! [F: product_prod_nat_nat > nat,A4: set_Pr1261947904930325089at_nat,P2: set_nat > $o] :
( ( ! [B6: set_nat] :
( ( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ ( image_2486076414777270412at_nat @ F @ A4 ) ) )
=> ( P2 @ B6 ) ) )
= ( ! [B6: set_Pr1261947904930325089at_nat] :
( ( ( finite6177210948735845034at_nat @ B6 )
& ( ord_le3146513528884898305at_nat @ B6 @ A4 ) )
=> ( P2 @ ( image_2486076414777270412at_nat @ F @ B6 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_4013_all__finite__subset__image,axiom,
! [F: nat > nat,A4: set_nat,P2: set_nat > $o] :
( ( ! [B6: set_nat] :
( ( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ ( image_nat_nat @ F @ A4 ) ) )
=> ( P2 @ B6 ) ) )
= ( ! [B6: set_nat] :
( ( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ A4 ) )
=> ( P2 @ ( image_nat_nat @ F @ B6 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_4014_ex__finite__subset__image,axiom,
! [F: int > int,A4: set_int,P2: set_int > $o] :
( ( ? [B6: set_int] :
( ( finite_finite_int @ B6 )
& ( ord_less_eq_set_int @ B6 @ ( image_int_int @ F @ A4 ) )
& ( P2 @ B6 ) ) )
= ( ? [B6: set_int] :
( ( finite_finite_int @ B6 )
& ( ord_less_eq_set_int @ B6 @ A4 )
& ( P2 @ ( image_int_int @ F @ B6 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_4015_ex__finite__subset__image,axiom,
! [F: nat > rat,A4: set_nat,P2: set_rat > $o] :
( ( ? [B6: set_rat] :
( ( finite_finite_rat @ B6 )
& ( ord_less_eq_set_rat @ B6 @ ( image_nat_rat @ F @ A4 ) )
& ( P2 @ B6 ) ) )
= ( ? [B6: set_nat] :
( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ A4 )
& ( P2 @ ( image_nat_rat @ F @ B6 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_4016_ex__finite__subset__image,axiom,
! [F: nat > int,A4: set_nat,P2: set_int > $o] :
( ( ? [B6: set_int] :
( ( finite_finite_int @ B6 )
& ( ord_less_eq_set_int @ B6 @ ( image_nat_int @ F @ A4 ) )
& ( P2 @ B6 ) ) )
= ( ? [B6: set_nat] :
( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ A4 )
& ( P2 @ ( image_nat_int @ F @ B6 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_4017_ex__finite__subset__image,axiom,
! [F: product_prod_nat_nat > product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,P2: set_Pr1261947904930325089at_nat > $o] :
( ( ? [B6: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ B6 )
& ( ord_le3146513528884898305at_nat @ B6 @ ( image_5168914502847457605at_nat @ F @ A4 ) )
& ( P2 @ B6 ) ) )
= ( ? [B6: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ B6 )
& ( ord_le3146513528884898305at_nat @ B6 @ A4 )
& ( P2 @ ( image_5168914502847457605at_nat @ F @ B6 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_4018_ex__finite__subset__image,axiom,
! [F: nat > product_prod_nat_nat,A4: set_nat,P2: set_Pr1261947904930325089at_nat > $o] :
( ( ? [B6: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ B6 )
& ( ord_le3146513528884898305at_nat @ B6 @ ( image_5846123807819985514at_nat @ F @ A4 ) )
& ( P2 @ B6 ) ) )
= ( ? [B6: set_nat] :
( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ A4 )
& ( P2 @ ( image_5846123807819985514at_nat @ F @ B6 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_4019_ex__finite__subset__image,axiom,
! [F: int > nat,A4: set_int,P2: set_nat > $o] :
( ( ? [B6: set_nat] :
( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ ( image_int_nat @ F @ A4 ) )
& ( P2 @ B6 ) ) )
= ( ? [B6: set_int] :
( ( finite_finite_int @ B6 )
& ( ord_less_eq_set_int @ B6 @ A4 )
& ( P2 @ ( image_int_nat @ F @ B6 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_4020_ex__finite__subset__image,axiom,
! [F: product_prod_nat_nat > nat,A4: set_Pr1261947904930325089at_nat,P2: set_nat > $o] :
( ( ? [B6: set_nat] :
( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ ( image_2486076414777270412at_nat @ F @ A4 ) )
& ( P2 @ B6 ) ) )
= ( ? [B6: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ B6 )
& ( ord_le3146513528884898305at_nat @ B6 @ A4 )
& ( P2 @ ( image_2486076414777270412at_nat @ F @ B6 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_4021_ex__finite__subset__image,axiom,
! [F: nat > nat,A4: set_nat,P2: set_nat > $o] :
( ( ? [B6: set_nat] :
( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ ( image_nat_nat @ F @ A4 ) )
& ( P2 @ B6 ) ) )
= ( ? [B6: set_nat] :
( ( finite_finite_nat @ B6 )
& ( ord_less_eq_set_nat @ B6 @ A4 )
& ( P2 @ ( image_nat_nat @ F @ B6 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_4022_finite__subset__image,axiom,
! [B5: set_int,F: int > int,A4: set_int] :
( ( finite_finite_int @ B5 )
=> ( ( ord_less_eq_set_int @ B5 @ ( image_int_int @ F @ A4 ) )
=> ? [C2: set_int] :
( ( ord_less_eq_set_int @ C2 @ A4 )
& ( finite_finite_int @ C2 )
& ( B5
= ( image_int_int @ F @ C2 ) ) ) ) ) ).
% finite_subset_image
thf(fact_4023_finite__subset__image,axiom,
! [B5: set_rat,F: nat > rat,A4: set_nat] :
( ( finite_finite_rat @ B5 )
=> ( ( ord_less_eq_set_rat @ B5 @ ( image_nat_rat @ F @ A4 ) )
=> ? [C2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ A4 )
& ( finite_finite_nat @ C2 )
& ( B5
= ( image_nat_rat @ F @ C2 ) ) ) ) ) ).
% finite_subset_image
thf(fact_4024_finite__subset__image,axiom,
! [B5: set_int,F: nat > int,A4: set_nat] :
( ( finite_finite_int @ B5 )
=> ( ( ord_less_eq_set_int @ B5 @ ( image_nat_int @ F @ A4 ) )
=> ? [C2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ A4 )
& ( finite_finite_nat @ C2 )
& ( B5
= ( image_nat_int @ F @ C2 ) ) ) ) ) ).
% finite_subset_image
thf(fact_4025_finite__subset__image,axiom,
! [B5: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ B5 )
=> ( ( ord_le3146513528884898305at_nat @ B5 @ ( image_5168914502847457605at_nat @ F @ A4 ) )
=> ? [C2: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ C2 @ A4 )
& ( finite6177210948735845034at_nat @ C2 )
& ( B5
= ( image_5168914502847457605at_nat @ F @ C2 ) ) ) ) ) ).
% finite_subset_image
thf(fact_4026_finite__subset__image,axiom,
! [B5: set_Pr1261947904930325089at_nat,F: nat > product_prod_nat_nat,A4: set_nat] :
( ( finite6177210948735845034at_nat @ B5 )
=> ( ( ord_le3146513528884898305at_nat @ B5 @ ( image_5846123807819985514at_nat @ F @ A4 ) )
=> ? [C2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ A4 )
& ( finite_finite_nat @ C2 )
& ( B5
= ( image_5846123807819985514at_nat @ F @ C2 ) ) ) ) ) ).
% finite_subset_image
thf(fact_4027_finite__subset__image,axiom,
! [B5: set_nat,F: int > nat,A4: set_int] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ B5 @ ( image_int_nat @ F @ A4 ) )
=> ? [C2: set_int] :
( ( ord_less_eq_set_int @ C2 @ A4 )
& ( finite_finite_int @ C2 )
& ( B5
= ( image_int_nat @ F @ C2 ) ) ) ) ) ).
% finite_subset_image
thf(fact_4028_finite__subset__image,axiom,
! [B5: set_nat,F: product_prod_nat_nat > nat,A4: set_Pr1261947904930325089at_nat] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ B5 @ ( image_2486076414777270412at_nat @ F @ A4 ) )
=> ? [C2: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ C2 @ A4 )
& ( finite6177210948735845034at_nat @ C2 )
& ( B5
= ( image_2486076414777270412at_nat @ F @ C2 ) ) ) ) ) ).
% finite_subset_image
thf(fact_4029_finite__subset__image,axiom,
! [B5: set_nat,F: nat > nat,A4: set_nat] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat @ F @ A4 ) )
=> ? [C2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ A4 )
& ( finite_finite_nat @ C2 )
& ( B5
= ( image_nat_nat @ F @ C2 ) ) ) ) ) ).
% finite_subset_image
thf(fact_4030_finite__surj,axiom,
! [A4: set_int,B5: set_int,F: int > int] :
( ( finite_finite_int @ A4 )
=> ( ( ord_less_eq_set_int @ B5 @ ( image_int_int @ F @ A4 ) )
=> ( finite_finite_int @ B5 ) ) ) ).
% finite_surj
thf(fact_4031_finite__surj,axiom,
! [A4: set_int,B5: set_nat,F: int > nat] :
( ( finite_finite_int @ A4 )
=> ( ( ord_less_eq_set_nat @ B5 @ ( image_int_nat @ F @ A4 ) )
=> ( finite_finite_nat @ B5 ) ) ) ).
% finite_surj
thf(fact_4032_finite__surj,axiom,
! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > product_prod_nat_nat] :
( ( finite6177210948735845034at_nat @ A4 )
=> ( ( ord_le3146513528884898305at_nat @ B5 @ ( image_5168914502847457605at_nat @ F @ A4 ) )
=> ( finite6177210948735845034at_nat @ B5 ) ) ) ).
% finite_surj
thf(fact_4033_finite__surj,axiom,
! [A4: set_Pr1261947904930325089at_nat,B5: set_nat,F: product_prod_nat_nat > nat] :
( ( finite6177210948735845034at_nat @ A4 )
=> ( ( ord_less_eq_set_nat @ B5 @ ( image_2486076414777270412at_nat @ F @ A4 ) )
=> ( finite_finite_nat @ B5 ) ) ) ).
% finite_surj
thf(fact_4034_finite__surj,axiom,
! [A4: set_nat,B5: set_rat,F: nat > rat] :
( ( finite_finite_nat @ A4 )
=> ( ( ord_less_eq_set_rat @ B5 @ ( image_nat_rat @ F @ A4 ) )
=> ( finite_finite_rat @ B5 ) ) ) ).
% finite_surj
thf(fact_4035_finite__surj,axiom,
! [A4: set_nat,B5: set_int,F: nat > int] :
( ( finite_finite_nat @ A4 )
=> ( ( ord_less_eq_set_int @ B5 @ ( image_nat_int @ F @ A4 ) )
=> ( finite_finite_int @ B5 ) ) ) ).
% finite_surj
thf(fact_4036_finite__surj,axiom,
! [A4: set_nat,B5: set_Pr1261947904930325089at_nat,F: nat > product_prod_nat_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( ord_le3146513528884898305at_nat @ B5 @ ( image_5846123807819985514at_nat @ F @ A4 ) )
=> ( finite6177210948735845034at_nat @ B5 ) ) ) ).
% finite_surj
thf(fact_4037_finite__surj,axiom,
! [A4: set_nat,B5: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A4 )
=> ( ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat @ F @ A4 ) )
=> ( finite_finite_nat @ B5 ) ) ) ).
% finite_surj
thf(fact_4038_finite__image__iff,axiom,
! [F: int > int,A4: set_int] :
( ( inj_on_int_int @ F @ A4 )
=> ( ( finite_finite_int @ ( image_int_int @ F @ A4 ) )
= ( finite_finite_int @ A4 ) ) ) ).
% finite_image_iff
thf(fact_4039_finite__image__iff,axiom,
! [F: nat > rat,A4: set_nat] :
( ( inj_on_nat_rat @ F @ A4 )
=> ( ( finite_finite_rat @ ( image_nat_rat @ F @ A4 ) )
= ( finite_finite_nat @ A4 ) ) ) ).
% finite_image_iff
thf(fact_4040_finite__image__iff,axiom,
! [F: nat > int,A4: set_nat] :
( ( inj_on_nat_int @ F @ A4 )
=> ( ( finite_finite_int @ ( image_nat_int @ F @ A4 ) )
= ( finite_finite_nat @ A4 ) ) ) ).
% finite_image_iff
thf(fact_4041_finite__image__iff,axiom,
! [F: product_prod_nat_nat > product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
( ( inj_on8969904277767023793at_nat @ F @ A4 )
=> ( ( finite6177210948735845034at_nat @ ( image_5168914502847457605at_nat @ F @ A4 ) )
= ( finite6177210948735845034at_nat @ A4 ) ) ) ).
% finite_image_iff
thf(fact_4042_finite__image__iff,axiom,
! [F: nat > product_prod_nat_nat,A4: set_nat] :
( ( inj_on5538052773655684606at_nat @ F @ A4 )
=> ( ( finite6177210948735845034at_nat @ ( image_5846123807819985514at_nat @ F @ A4 ) )
= ( finite_finite_nat @ A4 ) ) ) ).
% finite_image_iff
thf(fact_4043_finite__image__iff,axiom,
! [F: int > nat,A4: set_int] :
( ( inj_on_int_nat @ F @ A4 )
=> ( ( finite_finite_nat @ ( image_int_nat @ F @ A4 ) )
= ( finite_finite_int @ A4 ) ) ) ).
% finite_image_iff
thf(fact_4044_finite__image__iff,axiom,
! [F: product_prod_nat_nat > nat,A4: set_Pr1261947904930325089at_nat] :
( ( inj_on2178005380612969504at_nat @ F @ A4 )
=> ( ( finite_finite_nat @ ( image_2486076414777270412at_nat @ F @ A4 ) )
= ( finite6177210948735845034at_nat @ A4 ) ) ) ).
% finite_image_iff
thf(fact_4045_finite__image__iff,axiom,
! [F: nat > nat,A4: set_nat] :
( ( inj_on_nat_nat @ F @ A4 )
=> ( ( finite_finite_nat @ ( image_nat_nat @ F @ A4 ) )
= ( finite_finite_nat @ A4 ) ) ) ).
% finite_image_iff
thf(fact_4046_finite__imageD,axiom,
! [F: int > int,A4: set_int] :
( ( finite_finite_int @ ( image_int_int @ F @ A4 ) )
=> ( ( inj_on_int_int @ F @ A4 )
=> ( finite_finite_int @ A4 ) ) ) ).
% finite_imageD
thf(fact_4047_finite__imageD,axiom,
! [F: nat > rat,A4: set_nat] :
( ( finite_finite_rat @ ( image_nat_rat @ F @ A4 ) )
=> ( ( inj_on_nat_rat @ F @ A4 )
=> ( finite_finite_nat @ A4 ) ) ) ).
% finite_imageD
thf(fact_4048_finite__imageD,axiom,
! [F: nat > int,A4: set_nat] :
( ( finite_finite_int @ ( image_nat_int @ F @ A4 ) )
=> ( ( inj_on_nat_int @ F @ A4 )
=> ( finite_finite_nat @ A4 ) ) ) ).
% finite_imageD
thf(fact_4049_finite__imageD,axiom,
! [F: product_prod_nat_nat > product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ ( image_5168914502847457605at_nat @ F @ A4 ) )
=> ( ( inj_on8969904277767023793at_nat @ F @ A4 )
=> ( finite6177210948735845034at_nat @ A4 ) ) ) ).
% finite_imageD
thf(fact_4050_finite__imageD,axiom,
! [F: nat > product_prod_nat_nat,A4: set_nat] :
( ( finite6177210948735845034at_nat @ ( image_5846123807819985514at_nat @ F @ A4 ) )
=> ( ( inj_on5538052773655684606at_nat @ F @ A4 )
=> ( finite_finite_nat @ A4 ) ) ) ).
% finite_imageD
thf(fact_4051_finite__imageD,axiom,
! [F: int > nat,A4: set_int] :
( ( finite_finite_nat @ ( image_int_nat @ F @ A4 ) )
=> ( ( inj_on_int_nat @ F @ A4 )
=> ( finite_finite_int @ A4 ) ) ) ).
% finite_imageD
thf(fact_4052_finite__imageD,axiom,
! [F: product_prod_nat_nat > nat,A4: set_Pr1261947904930325089at_nat] :
( ( finite_finite_nat @ ( image_2486076414777270412at_nat @ F @ A4 ) )
=> ( ( inj_on2178005380612969504at_nat @ F @ A4 )
=> ( finite6177210948735845034at_nat @ A4 ) ) ) ).
% finite_imageD
thf(fact_4053_finite__imageD,axiom,
! [F: nat > nat,A4: set_nat] :
( ( finite_finite_nat @ ( image_nat_nat @ F @ A4 ) )
=> ( ( inj_on_nat_nat @ F @ A4 )
=> ( finite_finite_nat @ A4 ) ) ) ).
% finite_imageD
thf(fact_4054_finite__range__imageI,axiom,
! [G: char > nat,F: nat > nat] :
( ( finite_finite_nat @ ( image_char_nat @ G @ top_top_set_char ) )
=> ( finite_finite_nat
@ ( image_char_nat
@ ^ [X4: char] : ( F @ ( G @ X4 ) )
@ top_top_set_char ) ) ) ).
% finite_range_imageI
thf(fact_4055_finite__range__imageI,axiom,
! [G: rat > nat,F: nat > nat] :
( ( finite_finite_nat @ ( image_rat_nat @ G @ top_top_set_rat ) )
=> ( finite_finite_nat
@ ( image_rat_nat
@ ^ [X4: rat] : ( F @ ( G @ X4 ) )
@ top_top_set_rat ) ) ) ).
% finite_range_imageI
thf(fact_4056_finite__range__imageI,axiom,
! [G: nat > rat,F: rat > rat] :
( ( finite_finite_rat @ ( image_nat_rat @ G @ top_top_set_nat ) )
=> ( finite_finite_rat
@ ( image_nat_rat
@ ^ [X4: nat] : ( F @ ( G @ X4 ) )
@ top_top_set_nat ) ) ) ).
% finite_range_imageI
thf(fact_4057_finite__range__imageI,axiom,
! [G: nat > rat,F: rat > int] :
( ( finite_finite_rat @ ( image_nat_rat @ G @ top_top_set_nat ) )
=> ( finite_finite_int
@ ( image_nat_int
@ ^ [X4: nat] : ( F @ ( G @ X4 ) )
@ top_top_set_nat ) ) ) ).
% finite_range_imageI
thf(fact_4058_finite__range__imageI,axiom,
! [G: nat > int,F: int > rat] :
( ( finite_finite_int @ ( image_nat_int @ G @ top_top_set_nat ) )
=> ( finite_finite_rat
@ ( image_nat_rat
@ ^ [X4: nat] : ( F @ ( G @ X4 ) )
@ top_top_set_nat ) ) ) ).
% finite_range_imageI
thf(fact_4059_finite__range__imageI,axiom,
! [G: nat > int,F: int > int] :
( ( finite_finite_int @ ( image_nat_int @ G @ top_top_set_nat ) )
=> ( finite_finite_int
@ ( image_nat_int
@ ^ [X4: nat] : ( F @ ( G @ X4 ) )
@ top_top_set_nat ) ) ) ).
% finite_range_imageI
thf(fact_4060_finite__range__imageI,axiom,
! [G: nat > rat,F: rat > nat] :
( ( finite_finite_rat @ ( image_nat_rat @ G @ top_top_set_nat ) )
=> ( finite_finite_nat
@ ( image_nat_nat
@ ^ [X4: nat] : ( F @ ( G @ X4 ) )
@ top_top_set_nat ) ) ) ).
% finite_range_imageI
thf(fact_4061_finite__range__imageI,axiom,
! [G: nat > int,F: int > nat] :
( ( finite_finite_int @ ( image_nat_int @ G @ top_top_set_nat ) )
=> ( finite_finite_nat
@ ( image_nat_nat
@ ^ [X4: nat] : ( F @ ( G @ X4 ) )
@ top_top_set_nat ) ) ) ).
% finite_range_imageI
thf(fact_4062_finite__range__imageI,axiom,
! [G: nat > nat,F: nat > rat] :
( ( finite_finite_nat @ ( image_nat_nat @ G @ top_top_set_nat ) )
=> ( finite_finite_rat
@ ( image_nat_rat
@ ^ [X4: nat] : ( F @ ( G @ X4 ) )
@ top_top_set_nat ) ) ) ).
% finite_range_imageI
thf(fact_4063_finite__range__imageI,axiom,
! [G: nat > nat,F: nat > int] :
( ( finite_finite_nat @ ( image_nat_nat @ G @ top_top_set_nat ) )
=> ( finite_finite_int
@ ( image_nat_int
@ ^ [X4: nat] : ( F @ ( G @ X4 ) )
@ top_top_set_nat ) ) ) ).
% finite_range_imageI
thf(fact_4064_Id__fstsnd__eq,axiom,
( id_Pro2258643101195443293at_nat
= ( collec7088162979684241874at_nat
@ ^ [X4: produc859450856879609959at_nat] :
( ( produc3213797794245857475at_nat @ X4 )
= ( produc6408287024330202629at_nat @ X4 ) ) ) ) ).
% Id_fstsnd_eq
thf(fact_4065_Id__fstsnd__eq,axiom,
( id_nat2
= ( collec3392354462482085612at_nat
@ ^ [X4: product_prod_nat_nat] :
( ( product_fst_nat_nat @ X4 )
= ( product_snd_nat_nat @ X4 ) ) ) ) ).
% Id_fstsnd_eq
thf(fact_4066_Id__fstsnd__eq,axiom,
( id_int2
= ( collec213857154873943460nt_int
@ ^ [X4: product_prod_int_int] :
( ( product_fst_int_int @ X4 )
= ( product_snd_int_int @ X4 ) ) ) ) ).
% Id_fstsnd_eq
thf(fact_4067_finite__surj__inj,axiom,
! [A4: set_int,F: int > int] :
( ( finite_finite_int @ A4 )
=> ( ( ord_less_eq_set_int @ A4 @ ( image_int_int @ F @ A4 ) )
=> ( inj_on_int_int @ F @ A4 ) ) ) ).
% finite_surj_inj
thf(fact_4068_finite__surj__inj,axiom,
! [A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > product_prod_nat_nat] :
( ( finite6177210948735845034at_nat @ A4 )
=> ( ( ord_le3146513528884898305at_nat @ A4 @ ( image_5168914502847457605at_nat @ F @ A4 ) )
=> ( inj_on8969904277767023793at_nat @ F @ A4 ) ) ) ).
% finite_surj_inj
thf(fact_4069_finite__surj__inj,axiom,
! [A4: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A4 )
=> ( ( ord_less_eq_set_nat @ A4 @ ( image_nat_nat @ F @ A4 ) )
=> ( inj_on_nat_nat @ F @ A4 ) ) ) ).
% finite_surj_inj
thf(fact_4070_inj__on__finite,axiom,
! [F: int > int,A4: set_int,B5: set_int] :
( ( inj_on_int_int @ F @ A4 )
=> ( ( ord_less_eq_set_int @ ( image_int_int @ F @ A4 ) @ B5 )
=> ( ( finite_finite_int @ B5 )
=> ( finite_finite_int @ A4 ) ) ) ) ).
% inj_on_finite
thf(fact_4071_inj__on__finite,axiom,
! [F: nat > rat,A4: set_nat,B5: set_rat] :
( ( inj_on_nat_rat @ F @ A4 )
=> ( ( ord_less_eq_set_rat @ ( image_nat_rat @ F @ A4 ) @ B5 )
=> ( ( finite_finite_rat @ B5 )
=> ( finite_finite_nat @ A4 ) ) ) ) ).
% inj_on_finite
thf(fact_4072_inj__on__finite,axiom,
! [F: nat > int,A4: set_nat,B5: set_int] :
( ( inj_on_nat_int @ F @ A4 )
=> ( ( ord_less_eq_set_int @ ( image_nat_int @ F @ A4 ) @ B5 )
=> ( ( finite_finite_int @ B5 )
=> ( finite_finite_nat @ A4 ) ) ) ) ).
% inj_on_finite
thf(fact_4073_inj__on__finite,axiom,
! [F: product_prod_nat_nat > product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
( ( inj_on8969904277767023793at_nat @ F @ A4 )
=> ( ( ord_le3146513528884898305at_nat @ ( image_5168914502847457605at_nat @ F @ A4 ) @ B5 )
=> ( ( finite6177210948735845034at_nat @ B5 )
=> ( finite6177210948735845034at_nat @ A4 ) ) ) ) ).
% inj_on_finite
thf(fact_4074_inj__on__finite,axiom,
! [F: nat > product_prod_nat_nat,A4: set_nat,B5: set_Pr1261947904930325089at_nat] :
( ( inj_on5538052773655684606at_nat @ F @ A4 )
=> ( ( ord_le3146513528884898305at_nat @ ( image_5846123807819985514at_nat @ F @ A4 ) @ B5 )
=> ( ( finite6177210948735845034at_nat @ B5 )
=> ( finite_finite_nat @ A4 ) ) ) ) ).
% inj_on_finite
thf(fact_4075_inj__on__finite,axiom,
! [F: int > nat,A4: set_int,B5: set_nat] :
( ( inj_on_int_nat @ F @ A4 )
=> ( ( ord_less_eq_set_nat @ ( image_int_nat @ F @ A4 ) @ B5 )
=> ( ( finite_finite_nat @ B5 )
=> ( finite_finite_int @ A4 ) ) ) ) ).
% inj_on_finite
thf(fact_4076_inj__on__finite,axiom,
! [F: product_prod_nat_nat > nat,A4: set_Pr1261947904930325089at_nat,B5: set_nat] :
( ( inj_on2178005380612969504at_nat @ F @ A4 )
=> ( ( ord_less_eq_set_nat @ ( image_2486076414777270412at_nat @ F @ A4 ) @ B5 )
=> ( ( finite_finite_nat @ B5 )
=> ( finite6177210948735845034at_nat @ A4 ) ) ) ) ).
% inj_on_finite
thf(fact_4077_inj__on__finite,axiom,
! [F: nat > nat,A4: set_nat,B5: set_nat] :
( ( inj_on_nat_nat @ F @ A4 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A4 ) @ B5 )
=> ( ( finite_finite_nat @ B5 )
=> ( finite_finite_nat @ A4 ) ) ) ) ).
% inj_on_finite
thf(fact_4078_endo__inj__surj,axiom,
! [A4: set_int,F: int > int] :
( ( finite_finite_int @ A4 )
=> ( ( ord_less_eq_set_int @ ( image_int_int @ F @ A4 ) @ A4 )
=> ( ( inj_on_int_int @ F @ A4 )
=> ( ( image_int_int @ F @ A4 )
= A4 ) ) ) ) ).
% endo_inj_surj
thf(fact_4079_endo__inj__surj,axiom,
! [A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > product_prod_nat_nat] :
( ( finite6177210948735845034at_nat @ A4 )
=> ( ( ord_le3146513528884898305at_nat @ ( image_5168914502847457605at_nat @ F @ A4 ) @ A4 )
=> ( ( inj_on8969904277767023793at_nat @ F @ A4 )
=> ( ( image_5168914502847457605at_nat @ F @ A4 )
= A4 ) ) ) ) ).
% endo_inj_surj
thf(fact_4080_endo__inj__surj,axiom,
! [A4: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A4 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A4 ) @ A4 )
=> ( ( inj_on_nat_nat @ F @ A4 )
=> ( ( image_nat_nat @ F @ A4 )
= A4 ) ) ) ) ).
% endo_inj_surj
thf(fact_4081_finite__UNIV__surj__inj,axiom,
! [F: product_prod_nat_nat > product_prod_nat_nat] :
( ( finite6177210948735845034at_nat @ top_to4669805908274784177at_nat )
=> ( ( ( image_5168914502847457605at_nat @ F @ top_to4669805908274784177at_nat )
= top_to4669805908274784177at_nat )
=> ( inj_on8969904277767023793at_nat @ F @ top_to4669805908274784177at_nat ) ) ) ).
% finite_UNIV_surj_inj
thf(fact_4082_finite__UNIV__surj__inj,axiom,
! [F: char > char] :
( ( finite_finite_char @ top_top_set_char )
=> ( ( ( image_char_char @ F @ top_top_set_char )
= top_top_set_char )
=> ( inj_on_char_char @ F @ top_top_set_char ) ) ) ).
% finite_UNIV_surj_inj
thf(fact_4083_finite__UNIV__surj__inj,axiom,
! [F: list_nat > list_nat] :
( ( finite8100373058378681591st_nat @ top_top_set_list_nat )
=> ( ( ( image_7976474329151083847st_nat @ F @ top_top_set_list_nat )
= top_top_set_list_nat )
=> ( inj_on3049792774292151987st_nat @ F @ top_top_set_list_nat ) ) ) ).
% finite_UNIV_surj_inj
thf(fact_4084_finite__UNIV__surj__inj,axiom,
! [F: rat > rat] :
( ( finite_finite_rat @ top_top_set_rat )
=> ( ( ( image_rat_rat @ F @ top_top_set_rat )
= top_top_set_rat )
=> ( inj_on_rat_rat @ F @ top_top_set_rat ) ) ) ).
% finite_UNIV_surj_inj
thf(fact_4085_finite__UNIV__surj__inj,axiom,
! [F: nat > nat] :
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( inj_on_nat_nat @ F @ top_top_set_nat ) ) ) ).
% finite_UNIV_surj_inj
thf(fact_4086_finite__UNIV__surj__inj,axiom,
! [F: int > int] :
( ( finite_finite_int @ top_top_set_int )
=> ( ( ( image_int_int @ F @ top_top_set_int )
= top_top_set_int )
=> ( inj_on_int_int @ F @ top_top_set_int ) ) ) ).
% finite_UNIV_surj_inj
thf(fact_4087_finite__UNIV__inj__surj,axiom,
! [F: product_prod_nat_nat > product_prod_nat_nat] :
( ( finite6177210948735845034at_nat @ top_to4669805908274784177at_nat )
=> ( ( inj_on8969904277767023793at_nat @ F @ top_to4669805908274784177at_nat )
=> ( ( image_5168914502847457605at_nat @ F @ top_to4669805908274784177at_nat )
= top_to4669805908274784177at_nat ) ) ) ).
% finite_UNIV_inj_surj
thf(fact_4088_finite__UNIV__inj__surj,axiom,
! [F: char > char] :
( ( finite_finite_char @ top_top_set_char )
=> ( ( inj_on_char_char @ F @ top_top_set_char )
=> ( ( image_char_char @ F @ top_top_set_char )
= top_top_set_char ) ) ) ).
% finite_UNIV_inj_surj
thf(fact_4089_finite__UNIV__inj__surj,axiom,
! [F: list_nat > list_nat] :
( ( finite8100373058378681591st_nat @ top_top_set_list_nat )
=> ( ( inj_on3049792774292151987st_nat @ F @ top_top_set_list_nat )
=> ( ( image_7976474329151083847st_nat @ F @ top_top_set_list_nat )
= top_top_set_list_nat ) ) ) ).
% finite_UNIV_inj_surj
thf(fact_4090_finite__UNIV__inj__surj,axiom,
! [F: rat > rat] :
( ( finite_finite_rat @ top_top_set_rat )
=> ( ( inj_on_rat_rat @ F @ top_top_set_rat )
=> ( ( image_rat_rat @ F @ top_top_set_rat )
= top_top_set_rat ) ) ) ).
% finite_UNIV_inj_surj
thf(fact_4091_finite__UNIV__inj__surj,axiom,
! [F: nat > nat] :
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat ) ) ) ).
% finite_UNIV_inj_surj
thf(fact_4092_finite__UNIV__inj__surj,axiom,
! [F: int > int] :
( ( finite_finite_int @ top_top_set_int )
=> ( ( inj_on_int_int @ F @ top_top_set_int )
=> ( ( image_int_int @ F @ top_top_set_int )
= top_top_set_int ) ) ) ).
% finite_UNIV_inj_surj
thf(fact_4093_finite__vimageD,axiom,
! [H: char > char,F4: set_char] :
( ( finite_finite_char @ ( vimage_char_char @ H @ F4 ) )
=> ( ( ( image_char_char @ H @ top_top_set_char )
= top_top_set_char )
=> ( finite_finite_char @ F4 ) ) ) ).
% finite_vimageD
thf(fact_4094_finite__vimageD,axiom,
! [H: char > rat,F4: set_rat] :
( ( finite_finite_char @ ( vimage_char_rat @ H @ F4 ) )
=> ( ( ( image_char_rat @ H @ top_top_set_char )
= top_top_set_rat )
=> ( finite_finite_rat @ F4 ) ) ) ).
% finite_vimageD
thf(fact_4095_finite__vimageD,axiom,
! [H: char > nat,F4: set_nat] :
( ( finite_finite_char @ ( vimage_char_nat @ H @ F4 ) )
=> ( ( ( image_char_nat @ H @ top_top_set_char )
= top_top_set_nat )
=> ( finite_finite_nat @ F4 ) ) ) ).
% finite_vimageD
thf(fact_4096_finite__vimageD,axiom,
! [H: char > int,F4: set_int] :
( ( finite_finite_char @ ( vimage_char_int @ H @ F4 ) )
=> ( ( ( image_char_int @ H @ top_top_set_char )
= top_top_set_int )
=> ( finite_finite_int @ F4 ) ) ) ).
% finite_vimageD
thf(fact_4097_finite__vimageD,axiom,
! [H: rat > char,F4: set_char] :
( ( finite_finite_rat @ ( vimage_rat_char @ H @ F4 ) )
=> ( ( ( image_rat_char @ H @ top_top_set_rat )
= top_top_set_char )
=> ( finite_finite_char @ F4 ) ) ) ).
% finite_vimageD
thf(fact_4098_finite__vimageD,axiom,
! [H: rat > rat,F4: set_rat] :
( ( finite_finite_rat @ ( vimage_rat_rat @ H @ F4 ) )
=> ( ( ( image_rat_rat @ H @ top_top_set_rat )
= top_top_set_rat )
=> ( finite_finite_rat @ F4 ) ) ) ).
% finite_vimageD
thf(fact_4099_finite__vimageD,axiom,
! [H: rat > nat,F4: set_nat] :
( ( finite_finite_rat @ ( vimage_rat_nat @ H @ F4 ) )
=> ( ( ( image_rat_nat @ H @ top_top_set_rat )
= top_top_set_nat )
=> ( finite_finite_nat @ F4 ) ) ) ).
% finite_vimageD
thf(fact_4100_finite__vimageD,axiom,
! [H: rat > int,F4: set_int] :
( ( finite_finite_rat @ ( vimage_rat_int @ H @ F4 ) )
=> ( ( ( image_rat_int @ H @ top_top_set_rat )
= top_top_set_int )
=> ( finite_finite_int @ F4 ) ) ) ).
% finite_vimageD
thf(fact_4101_finite__vimageD,axiom,
! [H: nat > char,F4: set_char] :
( ( finite_finite_nat @ ( vimage_nat_char @ H @ F4 ) )
=> ( ( ( image_nat_char @ H @ top_top_set_nat )
= top_top_set_char )
=> ( finite_finite_char @ F4 ) ) ) ).
% finite_vimageD
thf(fact_4102_finite__vimageD,axiom,
! [H: nat > rat,F4: set_rat] :
( ( finite_finite_nat @ ( vimage_nat_rat @ H @ F4 ) )
=> ( ( ( image_nat_rat @ H @ top_top_set_nat )
= top_top_set_rat )
=> ( finite_finite_rat @ F4 ) ) ) ).
% finite_vimageD
thf(fact_4103_prod__set__defs_I2_J,axiom,
( basic_1496934909540232809t_unit
= ( ^ [P3: produc7812382424243267045t_unit] : ( insert8563697753282147485t_unit @ ( produc1126150082675451753t_unit @ P3 ) @ bot_bo3686245532977126937t_unit ) ) ) ).
% prod_set_defs(2)
thf(fact_4104_inf__img__fin__dom,axiom,
! [F: nat > product_unit,A4: set_nat] :
( ( finite4290736615968046902t_unit @ ( image_8730104196221521654t_unit @ F @ A4 ) )
=> ( ~ ( finite_finite_nat @ A4 )
=> ? [X: product_unit] :
( ( member_Product_unit @ X @ ( image_8730104196221521654t_unit @ F @ A4 ) )
& ~ ( finite_finite_nat @ ( vimage4884490618288580032t_unit @ F @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ).
% inf_img_fin_dom
thf(fact_4105_inf__img__fin__dom,axiom,
! [F: nat > $o,A4: set_nat] :
( ( finite_finite_o @ ( image_nat_o @ F @ A4 ) )
=> ( ~ ( finite_finite_nat @ A4 )
=> ? [X: $o] :
( ( member_o @ X @ ( image_nat_o @ F @ A4 ) )
& ~ ( finite_finite_nat @ ( vimage_nat_o @ F @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ).
% inf_img_fin_dom
thf(fact_4106_inf__img__fin__dom,axiom,
! [F: int > nat,A4: set_int] :
( ( finite_finite_nat @ ( image_int_nat @ F @ A4 ) )
=> ( ~ ( finite_finite_int @ A4 )
=> ? [X: nat] :
( ( member_nat @ X @ ( image_int_nat @ F @ A4 ) )
& ~ ( finite_finite_int @ ( vimage_int_nat @ F @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ).
% inf_img_fin_dom
thf(fact_4107_inf__img__fin__dom,axiom,
! [F: nat > nat,A4: set_nat] :
( ( finite_finite_nat @ ( image_nat_nat @ F @ A4 ) )
=> ( ~ ( finite_finite_nat @ A4 )
=> ? [X: nat] :
( ( member_nat @ X @ ( image_nat_nat @ F @ A4 ) )
& ~ ( finite_finite_nat @ ( vimage_nat_nat @ F @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ).
% inf_img_fin_dom
thf(fact_4108_inf__img__fin__dom,axiom,
! [F: product_prod_nat_nat > product_unit,A4: set_Pr1261947904930325089at_nat] :
( ( finite4290736615968046902t_unit @ ( image_6169909464789311569t_unit @ F @ A4 ) )
=> ( ~ ( finite6177210948735845034at_nat @ A4 )
=> ? [X: product_unit] :
( ( member_Product_unit @ X @ ( image_6169909464789311569t_unit @ F @ A4 ) )
& ~ ( finite6177210948735845034at_nat @ ( vimage3104651464403166471t_unit @ F @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ).
% inf_img_fin_dom
thf(fact_4109_inf__img__fin__dom,axiom,
! [F: nat > product_prod_nat_nat,A4: set_nat] :
( ( finite6177210948735845034at_nat @ ( image_5846123807819985514at_nat @ F @ A4 ) )
=> ( ~ ( finite_finite_nat @ A4 )
=> ? [X: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X @ ( image_5846123807819985514at_nat @ F @ A4 ) )
& ~ ( finite_finite_nat @ ( vimage8013328719654469172at_nat @ F @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ) ) ) ).
% inf_img_fin_dom
thf(fact_4110_inf__img__fin__dom,axiom,
! [F: product_prod_nat_nat > $o,A4: set_Pr1261947904930325089at_nat] :
( ( finite_finite_o @ ( image_3693632289388996572_nat_o @ F @ A4 ) )
=> ( ~ ( finite6177210948735845034at_nat @ A4 )
=> ? [X: $o] :
( ( member_o @ X @ ( image_3693632289388996572_nat_o @ F @ A4 ) )
& ~ ( finite6177210948735845034at_nat @ ( vimage2444243245536687506_nat_o @ F @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ).
% inf_img_fin_dom
thf(fact_4111_inf__img__fin__dom,axiom,
! [F: product_prod_nat_nat > nat,A4: set_Pr1261947904930325089at_nat] :
( ( finite_finite_nat @ ( image_2486076414777270412at_nat @ F @ A4 ) )
=> ( ~ ( finite6177210948735845034at_nat @ A4 )
=> ? [X: nat] :
( ( member_nat @ X @ ( image_2486076414777270412at_nat @ F @ A4 ) )
& ~ ( finite6177210948735845034at_nat @ ( vimage4653281326611754070at_nat @ F @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ).
% inf_img_fin_dom
thf(fact_4112_inf__img__fin__dom,axiom,
! [F: product_prod_nat_nat > product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ ( image_5168914502847457605at_nat @ F @ A4 ) )
=> ( ~ ( finite6177210948735845034at_nat @ A4 )
=> ? [X: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X @ ( image_5168914502847457605at_nat @ F @ A4 ) )
& ~ ( finite6177210948735845034at_nat @ ( vimage2449269961533847803at_nat @ F @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ) ) ) ).
% inf_img_fin_dom
thf(fact_4113_inf__img__fin__dom,axiom,
! [F: nat > produc3843707927480180839at_nat,A4: set_nat] :
( ( finite4343798906461161616at_nat @ ( image_2499013659892586576at_nat @ F @ A4 ) )
=> ( ~ ( finite_finite_nat @ A4 )
=> ? [X: produc3843707927480180839at_nat] :
( ( member8757157785044589968at_nat @ X @ ( image_2499013659892586576at_nat @ F @ A4 ) )
& ~ ( finite_finite_nat @ ( vimage6435164912253009178at_nat @ F @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ) ) ) ) ).
% inf_img_fin_dom
thf(fact_4114_nat__descend__induct,axiom,
! [N3: nat,P2: nat > $o,M3: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N3 @ K2 )
=> ( P2 @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N3 )
=> ( ! [I5: nat] :
( ( ord_less_nat @ K2 @ I5 )
=> ( P2 @ I5 ) )
=> ( P2 @ K2 ) ) )
=> ( P2 @ M3 ) ) ) ).
% nat_descend_induct
thf(fact_4115_Field__natLeq__on,axiom,
! [N3: nat] :
( ( field_nat
@ ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [X4: nat,Y5: nat] :
( ( ord_less_nat @ X4 @ N3 )
& ( ord_less_nat @ Y5 @ N3 )
& ( ord_less_eq_nat @ X4 @ Y5 ) ) ) ) )
= ( collect_nat
@ ^ [X4: nat] : ( ord_less_nat @ X4 @ N3 ) ) ) ).
% Field_natLeq_on
thf(fact_4116_natLeq__on__well__order__on,axiom,
! [N3: nat] :
( order_2888998067076097458on_nat
@ ( collect_nat
@ ^ [X4: nat] : ( ord_less_nat @ X4 @ N3 ) )
@ ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [X4: nat,Y5: nat] :
( ( ord_less_nat @ X4 @ N3 )
& ( ord_less_nat @ Y5 @ N3 )
& ( ord_less_eq_nat @ X4 @ Y5 ) ) ) ) ) ).
% natLeq_on_well_order_on
thf(fact_4117_natLeq__on__Well__order,axiom,
! [N3: nat] :
( order_2888998067076097458on_nat
@ ( field_nat
@ ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [X4: nat,Y5: nat] :
( ( ord_less_nat @ X4 @ N3 )
& ( ord_less_nat @ Y5 @ N3 )
& ( ord_less_eq_nat @ X4 @ Y5 ) ) ) ) )
@ ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [X4: nat,Y5: nat] :
( ( ord_less_nat @ X4 @ N3 )
& ( ord_less_nat @ Y5 @ N3 )
& ( ord_less_eq_nat @ X4 @ Y5 ) ) ) ) ) ).
% natLeq_on_Well_order
thf(fact_4118_natLeq__on__wo__rel,axiom,
! [N3: nat] :
( bNF_We3818239936649020644el_nat
@ ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [X4: nat,Y5: nat] :
( ( ord_less_nat @ X4 @ N3 )
& ( ord_less_nat @ Y5 @ N3 )
& ( ord_less_eq_nat @ X4 @ Y5 ) ) ) ) ) ).
% natLeq_on_wo_rel
thf(fact_4119_wf__less,axiom,
wf_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ).
% wf_less
thf(fact_4120_natLess__def,axiom,
( bNF_Ca8459412986667044542atLess
= ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ) ).
% natLess_def
thf(fact_4121_card_Ocomp__fun__commute__on,axiom,
( ( comp_nat_nat_nat @ suc @ suc )
= ( comp_nat_nat_nat @ suc @ suc ) ) ).
% card.comp_fun_commute_on
thf(fact_4122_pred__nat__def,axiom,
( pred_nat
= ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [M5: nat,N4: nat] :
( N4
= ( suc @ M5 ) ) ) ) ) ).
% pred_nat_def
thf(fact_4123_zero__notin__Suc__image,axiom,
! [A4: set_nat] :
~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A4 ) ) ).
% zero_notin_Suc_image
thf(fact_4124_Restr__natLeq2,axiom,
! [N3: nat] :
( ( inf_in2572325071724192079at_nat @ bNF_Ca8665028551170535155natLeq
@ ( produc457027306803732586at_nat @ ( order_underS_nat @ bNF_Ca8665028551170535155natLeq @ N3 )
@ ^ [Uu: nat] : ( order_underS_nat @ bNF_Ca8665028551170535155natLeq @ N3 ) ) )
= ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [X4: nat,Y5: nat] :
( ( ord_less_nat @ X4 @ N3 )
& ( ord_less_nat @ Y5 @ N3 )
& ( ord_less_eq_nat @ X4 @ Y5 ) ) ) ) ) ).
% Restr_natLeq2
thf(fact_4125_Restr__natLeq,axiom,
! [N3: nat] :
( ( inf_in2572325071724192079at_nat @ bNF_Ca8665028551170535155natLeq
@ ( produc457027306803732586at_nat
@ ( collect_nat
@ ^ [X4: nat] : ( ord_less_nat @ X4 @ N3 ) )
@ ^ [Uu: nat] :
( collect_nat
@ ^ [X4: nat] : ( ord_less_nat @ X4 @ N3 ) ) ) )
= ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [X4: nat,Y5: nat] :
( ( ord_less_nat @ X4 @ N3 )
& ( ord_less_nat @ Y5 @ N3 )
& ( ord_less_eq_nat @ X4 @ Y5 ) ) ) ) ) ).
% Restr_natLeq
thf(fact_4126_natLeq__def,axiom,
( bNF_Ca8665028551170535155natLeq
= ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_eq_nat ) ) ) ).
% natLeq_def
thf(fact_4127_rat__denum,axiom,
? [F5: nat > rat] :
( ( image_nat_rat @ F5 @ top_top_set_nat )
= top_top_set_rat ) ).
% rat_denum
thf(fact_4128_greaterThan__0,axiom,
( ( set_or1210151606488870762an_nat @ zero_zero_nat )
= ( image_nat_nat @ suc @ top_top_set_nat ) ) ).
% greaterThan_0
thf(fact_4129_image__Suc__atLeastLessThan,axiom,
! [I: nat,J4: nat] :
( ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ I @ J4 ) )
= ( set_or4665077453230672383an_nat @ ( suc @ I ) @ ( suc @ J4 ) ) ) ).
% image_Suc_atLeastLessThan
thf(fact_4130_INT__greaterThan__UNIV,axiom,
( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ set_or1210151606488870762an_nat @ top_top_set_nat ) )
= bot_bot_set_nat ) ).
% INT_greaterThan_UNIV
thf(fact_4131_atLeast0__lessThan__Suc__eq__insert__0,axiom,
! [N3: nat] :
( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N3 ) )
= ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N3 ) ) ) ) ).
% atLeast0_lessThan_Suc_eq_insert_0
thf(fact_4132_image__minus__const__atLeastLessThan__nat,axiom,
! [C: nat,Y3: nat,X2: nat] :
( ( ( ord_less_nat @ C @ Y3 )
=> ( ( image_nat_nat
@ ^ [I2: nat] : ( minus_minus_nat @ I2 @ C )
@ ( set_or4665077453230672383an_nat @ X2 @ Y3 ) )
= ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ X2 @ C ) @ ( minus_minus_nat @ Y3 @ C ) ) ) )
& ( ~ ( ord_less_nat @ C @ Y3 )
=> ( ( ( ord_less_nat @ X2 @ Y3 )
=> ( ( image_nat_nat
@ ^ [I2: nat] : ( minus_minus_nat @ I2 @ C )
@ ( set_or4665077453230672383an_nat @ X2 @ Y3 ) )
= ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) )
& ( ~ ( ord_less_nat @ X2 @ Y3 )
=> ( ( image_nat_nat
@ ^ [I2: nat] : ( minus_minus_nat @ I2 @ C )
@ ( set_or4665077453230672383an_nat @ X2 @ Y3 ) )
= bot_bot_set_nat ) ) ) ) ) ).
% image_minus_const_atLeastLessThan_nat
thf(fact_4133_surj__nat__to__rat__surj,axiom,
( ( image_nat_rat @ nat_to_rat_surj @ top_top_set_nat )
= top_top_set_rat ) ).
% surj_nat_to_rat_surj
thf(fact_4134_range__mod,axiom,
! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( image_nat_nat
@ ^ [M5: nat] : ( modulo_modulo_nat @ M5 @ N3 )
@ top_top_set_nat )
= ( set_or4665077453230672383an_nat @ zero_zero_nat @ N3 ) ) ) ).
% range_mod
thf(fact_4135_divmod__nat__if,axiom,
( divmod_nat
= ( ^ [M5: nat,N4: nat] :
( if_Pro6206227464963214023at_nat
@ ( ( N4 = zero_zero_nat )
| ( ord_less_nat @ M5 @ N4 ) )
@ ( product_Pair_nat_nat @ zero_zero_nat @ M5 )
@ ( produc2626176000494625587at_nat
@ ^ [Q3: nat] : ( product_Pair_nat_nat @ ( suc @ Q3 ) )
@ ( divmod_nat @ ( minus_minus_nat @ M5 @ N4 ) @ N4 ) ) ) ) ) ).
% divmod_nat_if
thf(fact_4136_Rats__eq__range__nat__to__rat__surj,axiom,
( field_6020823756834552118ts_rat
= ( image_nat_rat @ nat_to_rat_surj @ top_top_set_nat ) ) ).
% Rats_eq_range_nat_to_rat_surj
thf(fact_4137_image__Suc__atLeastAtMost,axiom,
! [I: nat,J4: nat] :
( ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ I @ J4 ) )
= ( set_or1269000886237332187st_nat @ ( suc @ I ) @ ( suc @ J4 ) ) ) ).
% image_Suc_atLeastAtMost
thf(fact_4138_UN__atMost__UNIV,axiom,
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_atMost_nat @ top_top_set_nat ) )
= top_top_set_nat ) ).
% UN_atMost_UNIV
thf(fact_4139_atMost__Suc__eq__insert__0,axiom,
! [N3: nat] :
( ( set_ord_atMost_nat @ ( suc @ N3 ) )
= ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N3 ) ) ) ) ).
% atMost_Suc_eq_insert_0
thf(fact_4140_atLeast0__atMost__Suc__eq__insert__0,axiom,
! [N3: nat] :
( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N3 ) )
= ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N3 ) ) ) ) ).
% atLeast0_atMost_Suc_eq_insert_0
thf(fact_4141_image__Suc__lessThan,axiom,
! [N3: nat] :
( ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N3 ) )
= ( set_or1269000886237332187st_nat @ one_one_nat @ N3 ) ) ).
% image_Suc_lessThan
thf(fact_4142_UN__lessThan__UNIV,axiom,
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_lessThan_nat @ top_top_set_nat ) )
= top_top_set_nat ) ).
% UN_lessThan_UNIV
thf(fact_4143_lessThan__Suc__eq__insert__0,axiom,
! [N3: nat] :
( ( set_ord_lessThan_nat @ ( suc @ N3 ) )
= ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N3 ) ) ) ) ).
% lessThan_Suc_eq_insert_0
thf(fact_4144_image__Suc__atMost,axiom,
! [N3: nat] :
( ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N3 ) )
= ( set_or1269000886237332187st_nat @ one_one_nat @ ( suc @ N3 ) ) ) ).
% image_Suc_atMost
thf(fact_4145_image__int__atLeastAtMost,axiom,
! [A: nat,B: nat] :
( ( image_nat_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% image_int_atLeastAtMost
thf(fact_4146_image__int__atLeastLessThan,axiom,
! [A: nat,B: nat] :
( ( image_nat_int @ semiri1314217659103216013at_int @ ( set_or4665077453230672383an_nat @ A @ B ) )
= ( set_or4662586982721622107an_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% image_int_atLeastLessThan
thf(fact_4147_image__add__int__atLeastLessThan,axiom,
! [L: int,U: int] :
( ( image_int_int
@ ^ [X4: int] : ( plus_plus_int @ X4 @ L )
@ ( set_or4662586982721622107an_int @ zero_zero_int @ ( minus_minus_int @ U @ L ) ) )
= ( set_or4662586982721622107an_int @ L @ U ) ) ).
% image_add_int_atLeastLessThan
thf(fact_4148_pair__lessI2,axiom,
! [A: nat,B: nat,S2: nat,T2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ S2 @ T2 )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S2 ) @ ( product_Pair_nat_nat @ B @ T2 ) ) @ fun_pair_less ) ) ) ).
% pair_lessI2
thf(fact_4149_pair__less__iff1,axiom,
! [X2: nat,Y3: nat,Z2: nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ ( product_Pair_nat_nat @ X2 @ Z2 ) ) @ fun_pair_less )
= ( ord_less_nat @ Y3 @ Z2 ) ) ).
% pair_less_iff1
thf(fact_4150_total__pair__less,axiom,
! [A4: set_Pr1261947904930325089at_nat] : ( total_3592101749530773125at_nat @ A4 @ fun_pair_less ) ).
% total_pair_less
thf(fact_4151_Inf__int__def,axiom,
( complete_Inf_Inf_int
= ( ^ [X9: set_int] : ( uminus_uminus_int @ ( complete_Sup_Sup_int @ ( image_int_int @ uminus_uminus_int @ X9 ) ) ) ) ) ).
% Inf_int_def
thf(fact_4152_wf__pair__less,axiom,
wf_Pro7803398752247294826at_nat @ fun_pair_less ).
% wf_pair_less
thf(fact_4153_pair__lessI1,axiom,
! [A: nat,B: nat,S2: nat,T2: nat] :
( ( ord_less_nat @ A @ B )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S2 ) @ ( product_Pair_nat_nat @ B @ T2 ) ) @ fun_pair_less ) ) ).
% pair_lessI1
thf(fact_4154_bij__betw__Suc,axiom,
! [M: set_nat,N: set_nat] :
( ( bij_betw_nat_nat @ suc @ M @ N )
= ( ( image_nat_nat @ suc @ M )
= N ) ) ).
% bij_betw_Suc
thf(fact_4155_pair__leqI2,axiom,
! [A: nat,B: nat,S2: nat,T2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ S2 @ T2 )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S2 ) @ ( product_Pair_nat_nat @ B @ T2 ) ) @ fun_pair_leq ) ) ) ).
% pair_leqI2
thf(fact_4156_pair__leqI1,axiom,
! [A: nat,B: nat,S2: nat,T2: nat] :
( ( ord_less_nat @ A @ B )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S2 ) @ ( product_Pair_nat_nat @ B @ T2 ) ) @ fun_pair_leq ) ) ).
% pair_leqI1
thf(fact_4157_pair__leq__def,axiom,
( fun_pair_leq
= ( sup_su718114333110466843at_nat @ fun_pair_less @ id_Pro2258643101195443293at_nat ) ) ).
% pair_leq_def
thf(fact_4158_wmax__insertI,axiom,
! [Y3: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ Y3 @ YS )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y3 ) @ fun_pair_leq )
=> ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_max_weak )
=> ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( insert8211810215607154385at_nat @ X2 @ XS ) @ YS ) @ fun_max_weak ) ) ) ) ).
% wmax_insertI
thf(fact_4159_wmin__insertI,axiom,
! [X2: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y3: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ X2 @ XS )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y3 ) @ fun_pair_leq )
=> ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_min_weak )
=> ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ ( insert8211810215607154385at_nat @ Y3 @ YS ) ) @ fun_min_weak ) ) ) ) ).
% wmin_insertI
thf(fact_4160_smax__insertI,axiom,
! [Y3: product_prod_nat_nat,Y4: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat,X5: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ Y3 @ Y4 )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y3 ) @ fun_pair_less )
=> ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X5 @ Y4 ) @ fun_max_strict )
=> ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( insert8211810215607154385at_nat @ X2 @ X5 ) @ Y4 ) @ fun_max_strict ) ) ) ) ).
% smax_insertI
thf(fact_4161_smin__insertI,axiom,
! [X2: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y3: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ X2 @ XS )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y3 ) @ fun_pair_less )
=> ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_min_strict )
=> ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ ( insert8211810215607154385at_nat @ Y3 @ YS ) ) @ fun_min_strict ) ) ) ) ).
% smin_insertI
thf(fact_4162_max__rpair__set,axiom,
fun_re2478310338295953701at_nat @ ( produc9060074326276436823at_nat @ fun_max_strict @ fun_max_weak ) ).
% max_rpair_set
thf(fact_4163_min__rpair__set,axiom,
fun_re2478310338295953701at_nat @ ( produc9060074326276436823at_nat @ fun_min_strict @ fun_min_weak ) ).
% min_rpair_set
thf(fact_4164_wmax__emptyI,axiom,
! [X5: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ X5 )
=> ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ X5 ) @ fun_max_weak ) ) ).
% wmax_emptyI
thf(fact_4165_smax__emptyI,axiom,
! [Y4: set_Pr1261947904930325089at_nat] :
( ( finite6177210948735845034at_nat @ Y4 )
=> ( ( Y4 != bot_bo2099793752762293965at_nat )
=> ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ Y4 ) @ fun_max_strict ) ) ) ).
% smax_emptyI
thf(fact_4166_smin__emptyI,axiom,
! [X5: set_Pr1261947904930325089at_nat] :
( ( X5 != bot_bo2099793752762293965at_nat )
=> ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X5 @ bot_bo2099793752762293965at_nat ) @ fun_min_strict ) ) ).
% smin_emptyI
thf(fact_4167_wmin__emptyI,axiom,
! [X5: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X5 @ bot_bo2099793752762293965at_nat ) @ fun_min_weak ) ).
% wmin_emptyI
thf(fact_4168_max__strict__def,axiom,
( fun_max_strict
= ( max_ex8135407076693332796at_nat @ fun_pair_less ) ) ).
% max_strict_def
thf(fact_4169_min__strict__def,axiom,
( fun_min_strict
= ( min_ex6901939911449802026at_nat @ fun_pair_less ) ) ).
% min_strict_def
thf(fact_4170_min__weak__def,axiom,
( fun_min_weak
= ( sup_su5525570899277871387at_nat @ ( min_ex6901939911449802026at_nat @ fun_pair_leq ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ bot_bo2099793752762293965at_nat ) @ bot_bo228742789529271731at_nat ) ) ) ).
% min_weak_def
thf(fact_4171_max__weak__def,axiom,
( fun_max_weak
= ( sup_su5525570899277871387at_nat @ ( max_ex8135407076693332796at_nat @ fun_pair_leq ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ bot_bo2099793752762293965at_nat ) @ bot_bo228742789529271731at_nat ) ) ) ).
% max_weak_def
thf(fact_4172_image__atLeastZeroLessThan__int,axiom,
! [U: int] :
( ( ord_less_eq_int @ zero_zero_int @ U )
=> ( ( set_or4662586982721622107an_int @ zero_zero_int @ U )
= ( image_nat_int @ semiri1314217659103216013at_int @ ( set_ord_lessThan_nat @ ( nat2 @ U ) ) ) ) ) ).
% image_atLeastZeroLessThan_int
thf(fact_4173_UN__atLeast__UNIV,axiom,
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_atLeast_nat @ top_top_set_nat ) )
= top_top_set_nat ) ).
% UN_atLeast_UNIV
thf(fact_4174_trans__pair__less,axiom,
trans_4347625901269045472at_nat @ fun_pair_less ).
% trans_pair_less
thf(fact_4175_less__eq__int_Orep__eq,axiom,
( ord_less_eq_int
= ( ^ [X4: int,Xa3: int] :
( produc8739625826339149834_nat_o
@ ^ [Y5: nat,Z4: nat] :
( produc6081775807080527818_nat_o
@ ^ [U3: nat,V3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ Y5 @ V3 ) @ ( plus_plus_nat @ U3 @ Z4 ) ) )
@ ( rep_Integ @ X4 )
@ ( rep_Integ @ Xa3 ) ) ) ) ).
% less_eq_int.rep_eq
thf(fact_4176_nat_Orep__eq,axiom,
( nat2
= ( ^ [X4: int] : ( produc6842872674320459806at_nat @ minus_minus_nat @ ( rep_Integ @ X4 ) ) ) ) ).
% nat.rep_eq
thf(fact_4177_less__int_Orep__eq,axiom,
( ord_less_int
= ( ^ [X4: int,Xa3: int] :
( produc8739625826339149834_nat_o
@ ^ [Y5: nat,Z4: nat] :
( produc6081775807080527818_nat_o
@ ^ [U3: nat,V3: nat] : ( ord_less_nat @ ( plus_plus_nat @ Y5 @ V3 ) @ ( plus_plus_nat @ U3 @ Z4 ) ) )
@ ( rep_Integ @ X4 )
@ ( rep_Integ @ Xa3 ) ) ) ) ).
% less_int.rep_eq
thf(fact_4178_less__eq__int_Otransfer,axiom,
( bNF_re717283939379294677_int_o @ pcr_int
@ ( bNF_re6644619430987730960nt_o_o @ pcr_int
@ ^ [Y2: $o,Z: $o] : Y2 = Z )
@ ( produc8739625826339149834_nat_o
@ ^ [X4: nat,Y5: nat] :
( produc6081775807080527818_nat_o
@ ^ [U3: nat,V3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X4 @ V3 ) @ ( plus_plus_nat @ U3 @ Y5 ) ) ) )
@ ord_less_eq_int ) ).
% less_eq_int.transfer
thf(fact_4179_less__eq__int_Oabs__eq,axiom,
! [Xa2: product_prod_nat_nat,X2: product_prod_nat_nat] :
( ( ord_less_eq_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X2 ) )
= ( produc8739625826339149834_nat_o
@ ^ [X4: nat,Y5: nat] :
( produc6081775807080527818_nat_o
@ ^ [U3: nat,V3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X4 @ V3 ) @ ( plus_plus_nat @ U3 @ Y5 ) ) )
@ Xa2
@ X2 ) ) ).
% less_eq_int.abs_eq
thf(fact_4180_int__ge__less__than2__def,axiom,
( int_ge_less_than2
= ( ^ [D3: int] :
( collec213857154873943460nt_int
@ ( produc4947309494688390418_int_o
@ ^ [Z6: int,Z4: int] :
( ( ord_less_eq_int @ D3 @ Z4 )
& ( ord_less_int @ Z6 @ Z4 ) ) ) ) ) ) ).
% int_ge_less_than2_def
thf(fact_4181_int__ge__less__than__def,axiom,
( int_ge_less_than
= ( ^ [D3: int] :
( collec213857154873943460nt_int
@ ( produc4947309494688390418_int_o
@ ^ [Z6: int,Z4: int] :
( ( ord_less_eq_int @ D3 @ Z6 )
& ( ord_less_int @ Z6 @ Z4 ) ) ) ) ) ) ).
% int_ge_less_than_def
thf(fact_4182_less__int_Otransfer,axiom,
( bNF_re717283939379294677_int_o @ pcr_int
@ ( bNF_re6644619430987730960nt_o_o @ pcr_int
@ ^ [Y2: $o,Z: $o] : Y2 = Z )
@ ( produc8739625826339149834_nat_o
@ ^ [X4: nat,Y5: nat] :
( produc6081775807080527818_nat_o
@ ^ [U3: nat,V3: nat] : ( ord_less_nat @ ( plus_plus_nat @ X4 @ V3 ) @ ( plus_plus_nat @ U3 @ Y5 ) ) ) )
@ ord_less_int ) ).
% less_int.transfer
thf(fact_4183_minus__int_Otransfer,axiom,
( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
@ ( produc27273713700761075at_nat
@ ^ [X4: nat,Y5: nat] :
( produc2626176000494625587at_nat
@ ^ [U3: nat,V3: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ V3 ) @ ( plus_plus_nat @ Y5 @ U3 ) ) ) )
@ minus_minus_int ) ).
% minus_int.transfer
thf(fact_4184_plus__int_Otransfer,axiom,
( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
@ ( produc27273713700761075at_nat
@ ^ [X4: nat,Y5: nat] :
( produc2626176000494625587at_nat
@ ^ [U3: nat,V3: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ U3 ) @ ( plus_plus_nat @ Y5 @ V3 ) ) ) )
@ plus_plus_int ) ).
% plus_int.transfer
thf(fact_4185_nat_Oabs__eq,axiom,
! [X2: product_prod_nat_nat] :
( ( nat2 @ ( abs_Integ @ X2 ) )
= ( produc6842872674320459806at_nat @ minus_minus_nat @ X2 ) ) ).
% nat.abs_eq
thf(fact_4186_nat_Otransfer,axiom,
( bNF_re4555766996558763186at_nat @ pcr_int
@ ^ [Y2: nat,Z: nat] : Y2 = Z
@ ( produc6842872674320459806at_nat @ minus_minus_nat )
@ nat2 ) ).
% nat.transfer
thf(fact_4187_uminus__int_Oabs__eq,axiom,
! [X2: product_prod_nat_nat] :
( ( uminus_uminus_int @ ( abs_Integ @ X2 ) )
= ( abs_Integ
@ ( produc2626176000494625587at_nat
@ ^ [X4: nat,Y5: nat] : ( product_Pair_nat_nat @ Y5 @ X4 )
@ X2 ) ) ) ).
% uminus_int.abs_eq
thf(fact_4188_uminus__int_Otransfer,axiom,
( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int
@ ( produc2626176000494625587at_nat
@ ^ [X4: nat,Y5: nat] : ( product_Pair_nat_nat @ Y5 @ X4 ) )
@ uminus_uminus_int ) ).
% uminus_int.transfer
thf(fact_4189_times__int_Oabs__eq,axiom,
! [Xa2: product_prod_nat_nat,X2: product_prod_nat_nat] :
( ( times_times_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X2 ) )
= ( abs_Integ
@ ( produc27273713700761075at_nat
@ ^ [X4: nat,Y5: nat] :
( produc2626176000494625587at_nat
@ ^ [U3: nat,V3: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X4 @ U3 ) @ ( times_times_nat @ Y5 @ V3 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X4 @ V3 ) @ ( times_times_nat @ Y5 @ U3 ) ) ) )
@ Xa2
@ X2 ) ) ) ).
% times_int.abs_eq
thf(fact_4190_plus__int_Oabs__eq,axiom,
! [Xa2: product_prod_nat_nat,X2: product_prod_nat_nat] :
( ( plus_plus_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X2 ) )
= ( abs_Integ
@ ( produc27273713700761075at_nat
@ ^ [X4: nat,Y5: nat] :
( produc2626176000494625587at_nat
@ ^ [U3: nat,V3: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ U3 ) @ ( plus_plus_nat @ Y5 @ V3 ) ) )
@ Xa2
@ X2 ) ) ) ).
% plus_int.abs_eq
thf(fact_4191_minus__int_Oabs__eq,axiom,
! [Xa2: product_prod_nat_nat,X2: product_prod_nat_nat] :
( ( minus_minus_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X2 ) )
= ( abs_Integ
@ ( produc27273713700761075at_nat
@ ^ [X4: nat,Y5: nat] :
( produc2626176000494625587at_nat
@ ^ [U3: nat,V3: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ V3 ) @ ( plus_plus_nat @ Y5 @ U3 ) ) )
@ Xa2
@ X2 ) ) ) ).
% minus_int.abs_eq
thf(fact_4192_times__int_Otransfer,axiom,
( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
@ ( produc27273713700761075at_nat
@ ^ [X4: nat,Y5: nat] :
( produc2626176000494625587at_nat
@ ^ [U3: nat,V3: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X4 @ U3 ) @ ( times_times_nat @ Y5 @ V3 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X4 @ V3 ) @ ( times_times_nat @ Y5 @ U3 ) ) ) ) )
@ times_times_int ) ).
% times_int.transfer
thf(fact_4193_less__int_Oabs__eq,axiom,
! [Xa2: product_prod_nat_nat,X2: product_prod_nat_nat] :
( ( ord_less_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X2 ) )
= ( produc8739625826339149834_nat_o
@ ^ [X4: nat,Y5: nat] :
( produc6081775807080527818_nat_o
@ ^ [U3: nat,V3: nat] : ( ord_less_nat @ ( plus_plus_nat @ X4 @ V3 ) @ ( plus_plus_nat @ U3 @ Y5 ) ) )
@ Xa2
@ X2 ) ) ).
% less_int.abs_eq
thf(fact_4194_uminus__int__def,axiom,
( uminus_uminus_int
= ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ
@ ( produc2626176000494625587at_nat
@ ^ [X4: nat,Y5: nat] : ( product_Pair_nat_nat @ Y5 @ X4 ) ) ) ) ).
% uminus_int_def
thf(fact_4195_times__int_Orsp,axiom,
( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
@ ( produc27273713700761075at_nat
@ ^ [X4: nat,Y5: nat] :
( produc2626176000494625587at_nat
@ ^ [U3: nat,V3: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X4 @ U3 ) @ ( times_times_nat @ Y5 @ V3 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X4 @ V3 ) @ ( times_times_nat @ Y5 @ U3 ) ) ) ) )
@ ( produc27273713700761075at_nat
@ ^ [X4: nat,Y5: nat] :
( produc2626176000494625587at_nat
@ ^ [U3: nat,V3: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X4 @ U3 ) @ ( times_times_nat @ Y5 @ V3 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X4 @ V3 ) @ ( times_times_nat @ Y5 @ U3 ) ) ) ) ) ) ).
% times_int.rsp
thf(fact_4196_nat__def,axiom,
( nat2
= ( map_fu2345160673673942751at_nat @ rep_Integ @ id_nat @ ( produc6842872674320459806at_nat @ minus_minus_nat ) ) ) ).
% nat_def
thf(fact_4197_minus__int_Orsp,axiom,
( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
@ ( produc27273713700761075at_nat
@ ^ [X4: nat,Y5: nat] :
( produc2626176000494625587at_nat
@ ^ [U3: nat,V3: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ V3 ) @ ( plus_plus_nat @ Y5 @ U3 ) ) ) )
@ ( produc27273713700761075at_nat
@ ^ [X4: nat,Y5: nat] :
( produc2626176000494625587at_nat
@ ^ [U3: nat,V3: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ V3 ) @ ( plus_plus_nat @ Y5 @ U3 ) ) ) ) ) ).
% minus_int.rsp
thf(fact_4198_plus__int_Orsp,axiom,
( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
@ ( produc27273713700761075at_nat
@ ^ [X4: nat,Y5: nat] :
( produc2626176000494625587at_nat
@ ^ [U3: nat,V3: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ U3 ) @ ( plus_plus_nat @ Y5 @ V3 ) ) ) )
@ ( produc27273713700761075at_nat
@ ^ [X4: nat,Y5: nat] :
( produc2626176000494625587at_nat
@ ^ [U3: nat,V3: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ U3 ) @ ( plus_plus_nat @ Y5 @ V3 ) ) ) ) ) ).
% plus_int.rsp
thf(fact_4199_less__eq__int_Orsp,axiom,
( bNF_re4202695980764964119_nat_o @ intrel
@ ( bNF_re3666534408544137501at_o_o @ intrel
@ ^ [Y2: $o,Z: $o] : Y2 = Z )
@ ( produc8739625826339149834_nat_o
@ ^ [X4: nat,Y5: nat] :
( produc6081775807080527818_nat_o
@ ^ [U3: nat,V3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X4 @ V3 ) @ ( plus_plus_nat @ U3 @ Y5 ) ) ) )
@ ( produc8739625826339149834_nat_o
@ ^ [X4: nat,Y5: nat] :
( produc6081775807080527818_nat_o
@ ^ [U3: nat,V3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X4 @ V3 ) @ ( plus_plus_nat @ U3 @ Y5 ) ) ) ) ) ).
% less_eq_int.rsp
thf(fact_4200_uminus__int_Orsp,axiom,
( bNF_re2241393799969408733at_nat @ intrel @ intrel
@ ( produc2626176000494625587at_nat
@ ^ [X4: nat,Y5: nat] : ( product_Pair_nat_nat @ Y5 @ X4 ) )
@ ( produc2626176000494625587at_nat
@ ^ [X4: nat,Y5: nat] : ( product_Pair_nat_nat @ Y5 @ X4 ) ) ) ).
% uminus_int.rsp
thf(fact_4201_nat_Orsp,axiom,
( bNF_re8246922863344978751at_nat @ intrel
@ ^ [Y2: nat,Z: nat] : Y2 = Z
@ ( produc6842872674320459806at_nat @ minus_minus_nat )
@ ( produc6842872674320459806at_nat @ minus_minus_nat ) ) ).
% nat.rsp
thf(fact_4202_intrel__def,axiom,
( intrel
= ( produc8739625826339149834_nat_o
@ ^ [X4: nat,Y5: nat] :
( produc6081775807080527818_nat_o
@ ^ [U3: nat,V3: nat] :
( ( plus_plus_nat @ X4 @ V3 )
= ( plus_plus_nat @ U3 @ Y5 ) ) ) ) ) ).
% intrel_def
thf(fact_4203_less__int_Orsp,axiom,
( bNF_re4202695980764964119_nat_o @ intrel
@ ( bNF_re3666534408544137501at_o_o @ intrel
@ ^ [Y2: $o,Z: $o] : Y2 = Z )
@ ( produc8739625826339149834_nat_o
@ ^ [X4: nat,Y5: nat] :
( produc6081775807080527818_nat_o
@ ^ [U3: nat,V3: nat] : ( ord_less_nat @ ( plus_plus_nat @ X4 @ V3 ) @ ( plus_plus_nat @ U3 @ Y5 ) ) ) )
@ ( produc8739625826339149834_nat_o
@ ^ [X4: nat,Y5: nat] :
( produc6081775807080527818_nat_o
@ ^ [U3: nat,V3: nat] : ( ord_less_nat @ ( plus_plus_nat @ X4 @ V3 ) @ ( plus_plus_nat @ U3 @ Y5 ) ) ) ) ) ).
% less_int.rsp
thf(fact_4204_times__int__def,axiom,
( times_times_int
= ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
@ ( produc27273713700761075at_nat
@ ^ [X4: nat,Y5: nat] :
( produc2626176000494625587at_nat
@ ^ [U3: nat,V3: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X4 @ U3 ) @ ( times_times_nat @ Y5 @ V3 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X4 @ V3 ) @ ( times_times_nat @ Y5 @ U3 ) ) ) ) ) ) ) ).
% times_int_def
thf(fact_4205_minus__int__def,axiom,
( minus_minus_int
= ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
@ ( produc27273713700761075at_nat
@ ^ [X4: nat,Y5: nat] :
( produc2626176000494625587at_nat
@ ^ [U3: nat,V3: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ V3 ) @ ( plus_plus_nat @ Y5 @ U3 ) ) ) ) ) ) ).
% minus_int_def
thf(fact_4206_plus__int__def,axiom,
( plus_plus_int
= ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
@ ( produc27273713700761075at_nat
@ ^ [X4: nat,Y5: nat] :
( produc2626176000494625587at_nat
@ ^ [U3: nat,V3: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ U3 ) @ ( plus_plus_nat @ Y5 @ V3 ) ) ) ) ) ) ).
% plus_int_def
thf(fact_4207_rat__less__code,axiom,
( ord_less_rat
= ( ^ [P3: rat,Q3: rat] :
( produc4947309494688390418_int_o
@ ^ [A5: int,C4: int] :
( produc4947309494688390418_int_o
@ ^ [B4: int,D3: int] : ( ord_less_int @ ( times_times_int @ A5 @ D3 ) @ ( times_times_int @ C4 @ B4 ) )
@ ( quotient_of @ Q3 ) )
@ ( quotient_of @ P3 ) ) ) ) ).
% rat_less_code
thf(fact_4208_rat__less__eq__code,axiom,
( ord_less_eq_rat
= ( ^ [P3: rat,Q3: rat] :
( produc4947309494688390418_int_o
@ ^ [A5: int,C4: int] :
( produc4947309494688390418_int_o
@ ^ [B4: int,D3: int] : ( ord_less_eq_int @ ( times_times_int @ A5 @ D3 ) @ ( times_times_int @ C4 @ B4 ) )
@ ( quotient_of @ Q3 ) )
@ ( quotient_of @ P3 ) ) ) ) ).
% rat_less_eq_code
thf(fact_4209_rat__uminus__code,axiom,
! [P: rat] :
( ( quotient_of @ ( uminus_uminus_rat @ P ) )
= ( produc4245557441103728435nt_int
@ ^ [A5: int] : ( product_Pair_int_int @ ( uminus_uminus_int @ A5 ) )
@ ( quotient_of @ P ) ) ) ).
% rat_uminus_code
thf(fact_4210_less__int__def,axiom,
( ord_less_int
= ( map_fu434086159418415080_int_o @ rep_Integ @ ( map_fu4826362097070443709at_o_o @ rep_Integ @ id_o )
@ ( produc8739625826339149834_nat_o
@ ^ [X4: nat,Y5: nat] :
( produc6081775807080527818_nat_o
@ ^ [U3: nat,V3: nat] : ( ord_less_nat @ ( plus_plus_nat @ X4 @ V3 ) @ ( plus_plus_nat @ U3 @ Y5 ) ) ) ) ) ) ).
% less_int_def
thf(fact_4211_less__eq__int__def,axiom,
( ord_less_eq_int
= ( map_fu434086159418415080_int_o @ rep_Integ @ ( map_fu4826362097070443709at_o_o @ rep_Integ @ id_o )
@ ( produc8739625826339149834_nat_o
@ ^ [X4: nat,Y5: nat] :
( produc6081775807080527818_nat_o
@ ^ [U3: nat,V3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X4 @ V3 ) @ ( plus_plus_nat @ U3 @ Y5 ) ) ) ) ) ) ).
% less_eq_int_def
thf(fact_4212_rat__minus__code,axiom,
! [P: rat,Q: rat] :
( ( quotient_of @ ( minus_minus_rat @ P @ Q ) )
= ( produc4245557441103728435nt_int
@ ^ [A5: int,C4: int] :
( produc4245557441103728435nt_int
@ ^ [B4: int,D3: int] : ( normalize @ ( product_Pair_int_int @ ( minus_minus_int @ ( times_times_int @ A5 @ D3 ) @ ( times_times_int @ B4 @ C4 ) ) @ ( times_times_int @ C4 @ D3 ) ) )
@ ( quotient_of @ Q ) )
@ ( quotient_of @ P ) ) ) ).
% rat_minus_code
thf(fact_4213_rat__plus__code,axiom,
! [P: rat,Q: rat] :
( ( quotient_of @ ( plus_plus_rat @ P @ Q ) )
= ( produc4245557441103728435nt_int
@ ^ [A5: int,C4: int] :
( produc4245557441103728435nt_int
@ ^ [B4: int,D3: int] : ( normalize @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ A5 @ D3 ) @ ( times_times_int @ B4 @ C4 ) ) @ ( times_times_int @ C4 @ D3 ) ) )
@ ( quotient_of @ Q ) )
@ ( quotient_of @ P ) ) ) ).
% rat_plus_code
thf(fact_4214_rat__divide__code,axiom,
! [P: rat,Q: rat] :
( ( quotient_of @ ( divide_divide_rat @ P @ Q ) )
= ( produc4245557441103728435nt_int
@ ^ [A5: int,C4: int] :
( produc4245557441103728435nt_int
@ ^ [B4: int,D3: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A5 @ D3 ) @ ( times_times_int @ C4 @ B4 ) ) )
@ ( quotient_of @ Q ) )
@ ( quotient_of @ P ) ) ) ).
% rat_divide_code
thf(fact_4215_rat__times__code,axiom,
! [P: rat,Q: rat] :
( ( quotient_of @ ( times_times_rat @ P @ Q ) )
= ( produc4245557441103728435nt_int
@ ^ [A5: int,C4: int] :
( produc4245557441103728435nt_int
@ ^ [B4: int,D3: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A5 @ B4 ) @ ( times_times_int @ C4 @ D3 ) ) )
@ ( quotient_of @ Q ) )
@ ( quotient_of @ P ) ) ) ).
% rat_times_code
thf(fact_4216_pcr__int__def,axiom,
( pcr_int
= ( relcom8911704206367525917at_int
@ ( basic_5328504652464829177at_nat
@ ^ [Y2: nat,Z: nat] : Y2 = Z
@ ^ [Y2: nat,Z: nat] : Y2 = Z )
@ cr_int ) ) ).
% pcr_int_def
thf(fact_4217_UNIV__unit,axiom,
( top_to1996260823553986621t_unit
= ( insert_Product_unit @ product_Unity @ bot_bo3957492148770167129t_unit ) ) ).
% UNIV_unit
thf(fact_4218_top__unit__def,axiom,
top_top_Product_unit = product_Unity ).
% top_unit_def
thf(fact_4219_uminus__unit__def,axiom,
( uminus2952777764628376836t_unit
= ( ^ [Uu2: product_unit] : product_Unity ) ) ).
% uminus_unit_def
thf(fact_4220_inf__unit__def,axiom,
( inf_inf_Product_unit
= ( ^ [Uu2: product_unit,Uv2: product_unit] : product_Unity ) ) ).
% inf_unit_def
thf(fact_4221_old_Ounit_Oexhaust,axiom,
! [Y3: product_unit] : Y3 = product_Unity ).
% old.unit.exhaust
thf(fact_4222_Inf__unit__def,axiom,
( comple2584293577114468500t_unit
= ( ^ [Uu2: set_Product_unit] : product_Unity ) ) ).
% Inf_unit_def
thf(fact_4223_Sup__unit__def,axiom,
( comple4687483117567863418t_unit
= ( ^ [Uu2: set_Product_unit] : product_Unity ) ) ).
% Sup_unit_def
thf(fact_4224_sup__unit__def,axiom,
( sup_sup_Product_unit
= ( ^ [Uu2: product_unit,Uv2: product_unit] : product_Unity ) ) ).
% sup_unit_def
thf(fact_4225_bot__unit__def,axiom,
bot_bot_Product_unit = product_Unity ).
% bot_unit_def
thf(fact_4226_default__unit__def,axiom,
defaul566961228789861419t_unit = product_Unity ).
% default_unit_def
thf(fact_4227_Gcd__eq__Max,axiom,
! [M: set_nat] :
( ( finite_finite_nat @ M )
=> ( ( M != bot_bot_set_nat )
=> ( ~ ( member_nat @ zero_zero_nat @ M )
=> ( ( gcd_Gcd_nat @ M )
= ( lattic8265883725875713057ax_nat
@ ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [M5: nat] :
( collect_nat
@ ^ [D3: nat] : ( dvd_dvd_nat @ D3 @ M5 ) )
@ M ) ) ) ) ) ) ) ).
% Gcd_eq_Max
thf(fact_4228_Gcd__int__eq,axiom,
! [N: set_nat] :
( ( gcd_Gcd_int @ ( image_nat_int @ semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ ( gcd_Gcd_nat @ N ) ) ) ).
% Gcd_int_eq
thf(fact_4229_Gcd__int__def,axiom,
( gcd_Gcd_int
= ( ^ [K3: set_int] : ( semiri1314217659103216013at_int @ ( gcd_Gcd_nat @ ( image_int_nat @ ( comp_int_nat_int @ nat2 @ abs_abs_int ) @ K3 ) ) ) ) ) ).
% Gcd_int_def
thf(fact_4230_Gcd__nat__abs__eq,axiom,
! [K4: set_int] :
( ( gcd_Gcd_nat
@ ( image_int_nat
@ ^ [K5: int] : ( nat2 @ ( abs_abs_int @ K5 ) )
@ K4 ) )
= ( nat2 @ ( gcd_Gcd_int @ K4 ) ) ) ).
% Gcd_nat_abs_eq
thf(fact_4231_range__abs__Nats,axiom,
( ( image_int_int @ abs_abs_int @ top_top_set_int )
= semiring_1_Nats_int ) ).
% range_abs_Nats
thf(fact_4232_Gcd__abs__eq,axiom,
! [K4: set_int] :
( ( gcd_Gcd_int @ ( image_int_int @ abs_abs_int @ K4 ) )
= ( gcd_Gcd_int @ K4 ) ) ).
% Gcd_abs_eq
thf(fact_4233_int__in__range__abs,axiom,
! [N3: nat] : ( member_int @ ( semiri1314217659103216013at_int @ N3 ) @ ( image_int_int @ abs_abs_int @ top_top_set_int ) ) ).
% int_in_range_abs
thf(fact_4234_measure__function__int,axiom,
fun_is_measure_int @ ( comp_int_nat_int @ nat2 @ abs_abs_int ) ).
% measure_function_int
thf(fact_4235_rat__abs__code,axiom,
! [P: rat] :
( ( quotient_of @ ( abs_abs_rat @ P ) )
= ( produc4245557441103728435nt_int
@ ^ [A5: int] : ( product_Pair_int_int @ ( abs_abs_int @ A5 ) )
@ ( quotient_of @ P ) ) ) ).
% rat_abs_code
thf(fact_4236_surj__int__encode,axiom,
( ( image_int_nat @ nat_int_encode @ top_top_set_int )
= top_top_set_nat ) ).
% surj_int_encode
thf(fact_4237_surj__int__decode,axiom,
( ( image_nat_int @ nat_int_decode @ top_top_set_nat )
= top_top_set_int ) ).
% surj_int_decode
thf(fact_4238_Lcm__abs__eq,axiom,
! [K4: set_int] :
( ( gcd_Lcm_int @ ( image_int_int @ abs_abs_int @ K4 ) )
= ( gcd_Lcm_int @ K4 ) ) ).
% Lcm_abs_eq
thf(fact_4239_Lcm__int__def,axiom,
( gcd_Lcm_int
= ( ^ [K3: set_int] : ( semiri1314217659103216013at_int @ ( gcd_Lcm_nat @ ( image_int_nat @ ( comp_int_nat_int @ nat2 @ abs_abs_int ) @ K3 ) ) ) ) ) ).
% Lcm_int_def
thf(fact_4240_Lcm__int__eq,axiom,
! [N: set_nat] :
( ( gcd_Lcm_int @ ( image_nat_int @ semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ ( gcd_Lcm_nat @ N ) ) ) ).
% Lcm_int_eq
thf(fact_4241_Lcm__nat__abs__eq,axiom,
! [K4: set_int] :
( ( gcd_Lcm_nat
@ ( image_int_nat
@ ^ [K5: int] : ( nat2 @ ( abs_abs_int @ K5 ) )
@ K4 ) )
= ( nat2 @ ( gcd_Lcm_int @ K4 ) ) ) ).
% Lcm_nat_abs_eq
thf(fact_4242_rat__inverse__code,axiom,
! [P: rat] :
( ( quotient_of @ ( inverse_inverse_rat @ P ) )
= ( produc4245557441103728435nt_int
@ ^ [A5: int,B4: int] : ( if_Pro3027730157355071871nt_int @ ( A5 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ ( times_times_int @ ( sgn_sgn_int @ A5 ) @ B4 ) @ ( abs_abs_int @ A5 ) ) )
@ ( quotient_of @ P ) ) ) ).
% rat_inverse_code
thf(fact_4243_euclidean__size__int__def,axiom,
( euclid4774559944035922753ze_int
= ( comp_int_nat_int @ nat2 @ abs_abs_int ) ) ).
% euclidean_size_int_def
thf(fact_4244_Unity__def,axiom,
( product_Unity
= ( product_Abs_unit @ $true ) ) ).
% Unity_def
thf(fact_4245_prod__encode__def,axiom,
( nat_prod_encode
= ( produc6842872674320459806at_nat
@ ^ [M5: nat,N4: nat] : ( plus_plus_nat @ ( nat_triangle @ ( plus_plus_nat @ M5 @ N4 ) ) @ M5 ) ) ) ).
% prod_encode_def
thf(fact_4246_Abs__unit__cases,axiom,
! [X2: product_unit] :
~ ! [Y: $o] :
( ( X2
= ( product_Abs_unit @ Y ) )
=> ~ ( member_o @ Y @ ( insert_o @ $true @ bot_bot_set_o ) ) ) ).
% Abs_unit_cases
thf(fact_4247_Abs__unit__induct,axiom,
! [P2: product_unit > $o,X2: product_unit] :
( ! [Y: $o] :
( ( member_o @ Y @ ( insert_o @ $true @ bot_bot_set_o ) )
=> ( P2 @ ( product_Abs_unit @ Y ) ) )
=> ( P2 @ X2 ) ) ).
% Abs_unit_induct
thf(fact_4248_Abs__unit__inject,axiom,
! [X2: $o,Y3: $o] :
( ( member_o @ X2 @ ( insert_o @ $true @ bot_bot_set_o ) )
=> ( ( member_o @ Y3 @ ( insert_o @ $true @ bot_bot_set_o ) )
=> ( ( ( product_Abs_unit @ X2 )
= ( product_Abs_unit @ Y3 ) )
= ( X2 = Y3 ) ) ) ) ).
% Abs_unit_inject
thf(fact_4249_surj__prod__encode,axiom,
( ( image_2486076414777270412at_nat @ nat_prod_encode @ top_to4669805908274784177at_nat )
= top_top_set_nat ) ).
% surj_prod_encode
thf(fact_4250_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_4251_Rep__unit__inject,axiom,
! [X2: product_unit,Y3: product_unit] :
( ( ( product_Rep_unit @ X2 )
= ( product_Rep_unit @ Y3 ) )
= ( X2 = Y3 ) ) ).
% Rep_unit_inject
thf(fact_4252_Rep__unit__induct,axiom,
! [Y3: $o,P2: $o > $o] :
( ( member_o @ Y3 @ ( insert_o @ $true @ bot_bot_set_o ) )
=> ( ! [X: product_unit] : ( P2 @ ( product_Rep_unit @ X ) )
=> ( P2 @ Y3 ) ) ) ).
% Rep_unit_induct
thf(fact_4253_Rep__unit__cases,axiom,
! [Y3: $o] :
( ( member_o @ Y3 @ ( insert_o @ $true @ bot_bot_set_o ) )
=> ~ ! [X: product_unit] :
( Y3
= ( ~ ( product_Rep_unit @ X ) ) ) ) ).
% Rep_unit_cases
thf(fact_4254_Rep__unit,axiom,
! [X2: product_unit] : ( member_o @ ( product_Rep_unit @ X2 ) @ ( insert_o @ $true @ bot_bot_set_o ) ) ).
% Rep_unit
thf(fact_4255_Rep__unit__inverse,axiom,
! [X2: product_unit] :
( ( product_Abs_unit @ ( product_Rep_unit @ X2 ) )
= X2 ) ).
% Rep_unit_inverse
thf(fact_4256_Abs__unit__inverse,axiom,
! [Y3: $o] :
( ( member_o @ Y3 @ ( insert_o @ $true @ bot_bot_set_o ) )
=> ( ( product_Rep_unit @ ( product_Abs_unit @ Y3 ) )
= Y3 ) ) ).
% Abs_unit_inverse
thf(fact_4257_surj__prod__decode,axiom,
( ( image_5846123807819985514at_nat @ nat_prod_decode @ top_top_set_nat )
= top_to4669805908274784177at_nat ) ).
% surj_prod_decode
thf(fact_4258_pair__less__def,axiom,
( fun_pair_less
= ( lex_prod_nat_nat @ less_than @ less_than ) ) ).
% pair_less_def
thf(fact_4259_list__decode_Oelims,axiom,
! [X2: nat,Y3: list_nat] :
( ( ( nat_list_decode @ X2 )
= Y3 )
=> ( ( ( X2 = zero_zero_nat )
=> ( Y3 != nil_nat ) )
=> ~ ! [N2: nat] :
( ( X2
= ( suc @ N2 ) )
=> ( Y3
!= ( produc2761476792215241774st_nat
@ ^ [X4: nat,Y5: nat] : ( cons_nat @ X4 @ ( nat_list_decode @ Y5 ) )
@ ( nat_prod_decode @ N2 ) ) ) ) ) ) ).
% list_decode.elims
thf(fact_4260_nat__to__rat__surj__def,axiom,
( nat_to_rat_surj
= ( ^ [N4: nat] :
( produc6207742614233964070at_rat
@ ^ [A5: nat,B4: nat] : ( fract @ ( nat_int_decode @ A5 ) @ ( nat_int_decode @ B4 ) )
@ ( nat_prod_decode @ N4 ) ) ) ) ).
% nat_to_rat_surj_def
thf(fact_4261_list__decode_Osimps_I2_J,axiom,
! [N3: nat] :
( ( nat_list_decode @ ( suc @ N3 ) )
= ( produc2761476792215241774st_nat
@ ^ [X4: nat,Y5: nat] : ( cons_nat @ X4 @ ( nat_list_decode @ Y5 ) )
@ ( nat_prod_decode @ N3 ) ) ) ).
% list_decode.simps(2)
thf(fact_4262_list__decode_Opelims,axiom,
! [X2: nat,Y3: list_nat] :
( ( ( nat_list_decode @ X2 )
= Y3 )
=> ( ( accp_nat @ nat_list_decode_rel @ X2 )
=> ( ( ( X2 = zero_zero_nat )
=> ( ( Y3 = nil_nat )
=> ~ ( accp_nat @ nat_list_decode_rel @ zero_zero_nat ) ) )
=> ~ ! [N2: nat] :
( ( X2
= ( suc @ N2 ) )
=> ( ( Y3
= ( produc2761476792215241774st_nat
@ ^ [X4: nat,Y5: nat] : ( cons_nat @ X4 @ ( nat_list_decode @ Y5 ) )
@ ( nat_prod_decode @ N2 ) ) )
=> ~ ( accp_nat @ nat_list_decode_rel @ ( suc @ N2 ) ) ) ) ) ) ) ).
% list_decode.pelims
thf(fact_4263_surj__list__decode,axiom,
( ( image_nat_list_nat @ nat_list_decode @ top_top_set_nat )
= top_top_set_list_nat ) ).
% surj_list_decode
thf(fact_4264_list__decode_Opsimps_I2_J,axiom,
! [N3: nat] :
( ( accp_nat @ nat_list_decode_rel @ ( suc @ N3 ) )
=> ( ( nat_list_decode @ ( suc @ N3 ) )
= ( produc2761476792215241774st_nat
@ ^ [X4: nat,Y5: nat] : ( cons_nat @ X4 @ ( nat_list_decode @ Y5 ) )
@ ( nat_prod_decode @ N3 ) ) ) ) ).
% list_decode.psimps(2)
thf(fact_4265_surj__sum__encode,axiom,
( ( image_1320371278474632150at_nat @ nat_sum_encode @ top_to6661820994512907621at_nat )
= top_top_set_nat ) ).
% surj_sum_encode
thf(fact_4266_surj__list__encode,axiom,
( ( image_list_nat_nat @ nat_list_encode @ top_top_set_list_nat )
= top_top_set_nat ) ).
% surj_list_encode
thf(fact_4267_surj__sum__decode,axiom,
( ( image_678696785212003926at_nat @ nat_sum_decode @ top_top_set_nat )
= top_to6661820994512907621at_nat ) ).
% surj_sum_decode
thf(fact_4268_divmod__step__int__def,axiom,
( unique5024387138958732305ep_int
= ( ^ [L3: num] :
( produc4245557441103728435nt_int
@ ^ [Q3: int,R5: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L3 ) @ R5 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q3 ) @ one_one_int ) @ ( minus_minus_int @ R5 @ ( numeral_numeral_int @ L3 ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q3 ) @ R5 ) ) ) ) ) ).
% divmod_step_int_def
thf(fact_4269_divmod__step__nat__def,axiom,
( unique5026877609467782581ep_nat
= ( ^ [L3: num] :
( produc2626176000494625587at_nat
@ ^ [Q3: nat,R5: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L3 ) @ R5 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q3 ) @ one_one_nat ) @ ( minus_minus_nat @ R5 @ ( numeral_numeral_nat @ L3 ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q3 ) @ R5 ) ) ) ) ) ).
% divmod_step_nat_def
thf(fact_4270_divmod__step__integer__def,axiom,
( unique4921790084139445826nteger
= ( ^ [L3: num] :
( produc6916734918728496179nteger
@ ^ [Q3: code_integer,R5: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L3 ) @ R5 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q3 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R5 @ ( numera6620942414471956472nteger @ L3 ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q3 ) @ R5 ) ) ) ) ) ).
% divmod_step_integer_def
thf(fact_4271_nat__of__integer__code,axiom,
( code_nat_of_integer
= ( ^ [K5: code_integer] :
( if_nat @ ( ord_le3102999989581377725nteger @ K5 @ zero_z3403309356797280102nteger ) @ zero_zero_nat
@ ( produc1555791787009142072er_nat
@ ^ [L3: code_integer,J: code_integer] : ( if_nat @ ( J = zero_z3403309356797280102nteger ) @ ( plus_plus_nat @ ( code_nat_of_integer @ L3 ) @ ( code_nat_of_integer @ L3 ) ) @ ( plus_plus_nat @ ( plus_plus_nat @ ( code_nat_of_integer @ L3 ) @ ( code_nat_of_integer @ L3 ) ) @ one_one_nat ) )
@ ( code_divmod_integer @ K5 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% nat_of_integer_code
thf(fact_4272_num__of__integer__code,axiom,
( code_num_of_integer
= ( ^ [K5: code_integer] :
( if_num @ ( ord_le3102999989581377725nteger @ K5 @ one_one_Code_integer ) @ one
@ ( produc7336495610019696514er_num
@ ^ [L3: code_integer,J: code_integer] : ( if_num @ ( J = zero_z3403309356797280102nteger ) @ ( plus_plus_num @ ( code_num_of_integer @ L3 ) @ ( code_num_of_integer @ L3 ) ) @ ( plus_plus_num @ ( plus_plus_num @ ( code_num_of_integer @ L3 ) @ ( code_num_of_integer @ L3 ) ) @ one ) )
@ ( code_divmod_integer @ K5 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% num_of_integer_code
thf(fact_4273_divmod__integer__code,axiom,
( code_divmod_integer
= ( ^ [K5: code_integer,L3: code_integer] :
( if_Pro6119634080678213985nteger @ ( K5 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
@ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ L3 )
@ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K5 ) @ ( code_divmod_abs @ K5 @ L3 )
@ ( produc6916734918728496179nteger
@ ^ [R5: code_integer,S: code_integer] : ( if_Pro6119634080678213985nteger @ ( S = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ L3 @ S ) ) )
@ ( code_divmod_abs @ K5 @ L3 ) ) )
@ ( if_Pro6119634080678213985nteger @ ( L3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K5 )
@ ( produc6499014454317279255nteger @ uminus1351360451143612070nteger
@ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ K5 @ zero_z3403309356797280102nteger ) @ ( code_divmod_abs @ K5 @ L3 )
@ ( produc6916734918728496179nteger
@ ^ [R5: code_integer,S: code_integer] : ( if_Pro6119634080678213985nteger @ ( S = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ L3 ) @ S ) ) )
@ ( code_divmod_abs @ K5 @ L3 ) ) ) ) ) ) ) ) ) ).
% divmod_integer_code
thf(fact_4274_int__of__integer__code,axiom,
( code_int_of_integer
= ( ^ [K5: code_integer] :
( if_int @ ( ord_le6747313008572928689nteger @ K5 @ zero_z3403309356797280102nteger ) @ ( uminus_uminus_int @ ( code_int_of_integer @ ( uminus1351360451143612070nteger @ K5 ) ) )
@ ( if_int @ ( K5 = zero_z3403309356797280102nteger ) @ zero_zero_int
@ ( produc1553301316500091796er_int
@ ^ [L3: code_integer,J: code_integer] : ( if_int @ ( J = zero_z3403309356797280102nteger ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L3 ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L3 ) ) @ one_one_int ) )
@ ( code_divmod_integer @ K5 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% int_of_integer_code
thf(fact_4275_Code__Numeral_Onegative__def,axiom,
( code_negative
= ( comp_C3531382070062128313er_num @ uminus1351360451143612070nteger @ numera6620942414471956472nteger ) ) ).
% Code_Numeral.negative_def
thf(fact_4276_bit__cut__integer__code,axiom,
( code_bit_cut_integer
= ( ^ [K5: code_integer] :
( if_Pro5737122678794959658eger_o @ ( K5 = zero_z3403309356797280102nteger ) @ ( produc6677183202524767010eger_o @ zero_z3403309356797280102nteger @ $false )
@ ( produc9125791028180074456eger_o
@ ^ [R5: code_integer,S: code_integer] : ( produc6677183202524767010eger_o @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K5 ) @ R5 @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ S ) ) @ ( S = one_one_Code_integer ) )
@ ( code_divmod_abs @ K5 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% bit_cut_integer_code
thf(fact_4277_divmod__integer__eq__cases,axiom,
( code_divmod_integer
= ( ^ [K5: code_integer,L3: code_integer] :
( if_Pro6119634080678213985nteger @ ( K5 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
@ ( if_Pro6119634080678213985nteger @ ( L3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K5 )
@ ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ produc6499014454317279255nteger @ times_3573771949741848930nteger ) @ sgn_sgn_Code_integer @ L3
@ ( if_Pro6119634080678213985nteger
@ ( ( sgn_sgn_Code_integer @ K5 )
= ( sgn_sgn_Code_integer @ L3 ) )
@ ( code_divmod_abs @ K5 @ L3 )
@ ( produc6916734918728496179nteger
@ ^ [R5: code_integer,S: code_integer] : ( if_Pro6119634080678213985nteger @ ( S = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ L3 ) @ S ) ) )
@ ( code_divmod_abs @ K5 @ L3 ) ) ) ) ) ) ) ) ).
% divmod_integer_eq_cases
thf(fact_4278_UNIV__char__of__nat,axiom,
( top_top_set_char
= ( image_nat_char @ unique3096191561947761185of_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).
% UNIV_char_of_nat
thf(fact_4279_num__of__integer__def,axiom,
( code_num_of_integer
= ( map_fu1227494855608507351um_num @ code_int_of_integer @ id_num @ ( comp_nat_num_int @ num_of_nat @ nat2 ) ) ) ).
% num_of_integer_def
thf(fact_4280_range__nat__of__char,axiom,
( ( image_char_nat @ comm_s629917340098488124ar_nat @ top_top_set_char )
= ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% range_nat_of_char
thf(fact_4281_num__of__integer_Orsp,axiom,
( bNF_re7626690874201225453um_num
@ ^ [Y2: int,Z: int] : Y2 = Z
@ ^ [Y2: num,Z: num] : Y2 = Z
@ ( comp_nat_num_int @ num_of_nat @ nat2 )
@ ( comp_nat_num_int @ num_of_nat @ nat2 ) ) ).
% num_of_integer.rsp
thf(fact_4282_num__of__integer_Otransfer,axiom,
( bNF_re6718328864250387230um_num @ code_pcr_integer
@ ^ [Y2: num,Z: num] : Y2 = Z
@ ( comp_nat_num_int @ num_of_nat @ nat2 )
@ code_num_of_integer ) ).
% num_of_integer.transfer
thf(fact_4283_pcr__integer__def,axiom,
( code_pcr_integer
= ( relcom967497705283160040nteger
@ ^ [Y2: int,Z: int] : Y2 = Z
@ code_cr_integer ) ) ).
% pcr_integer_def
thf(fact_4284_Random__Pred_Onot__randompred__def,axiom,
( random6974930770145893639ompred
= ( ^ [P4: produc7822875418678951345atural > produc4675096598859438275atural,S: produc7822875418678951345atural] :
( produc6665183775751917029atural
@ ^ [P5: pred_Product_unit,S7: produc7822875418678951345atural] : ( if_Pro3444522238938527101atural @ ( eval_Product_unit @ P5 @ product_Unity ) @ ( produc5069803637994805237atural @ bot_bo2717538794563056311t_unit @ S7 ) @ ( produc5069803637994805237atural @ ( single_Product_unit @ product_Unity ) @ S7 ) )
@ ( P4 @ S ) ) ) ) ).
% Random_Pred.not_randompred_def
thf(fact_4285_rat__floor__code,axiom,
( archim3151403230148437115or_rat
= ( ^ [P3: rat] : ( produc8211389475949308722nt_int @ divide_divide_int @ ( quotient_of @ P3 ) ) ) ) ).
% rat_floor_code
thf(fact_4286_Divides_Oadjust__div__def,axiom,
( adjust_div
= ( produc8211389475949308722nt_int
@ ^ [Q3: int,R5: int] : ( plus_plus_int @ Q3 @ ( zero_n2684676970156552555ol_int @ ( R5 != zero_zero_int ) ) ) ) ) ).
% Divides.adjust_div_def
thf(fact_4287_take__bit__num__code,axiom,
( bit_take_bit_num
= ( ^ [N4: nat,M5: num] :
( produc478579273971653890on_num
@ ^ [A5: nat,X4: num] :
( case_nat_option_num @ none_num
@ ^ [O: nat] :
( case_num_option_num @ ( some_num @ one )
@ ^ [P3: num] :
( case_o6005452278849405969um_num @ none_num
@ ^ [Q3: num] : ( some_num @ ( bit0 @ Q3 ) )
@ ( bit_take_bit_num @ O @ P3 ) )
@ ^ [P3: num] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ O @ P3 ) ) )
@ X4 )
@ A5 )
@ ( product_Pair_nat_num @ N4 @ M5 ) ) ) ) ).
% take_bit_num_code
thf(fact_4288_list__ex__iff__not__all__inverval__nat,axiom,
! [P2: nat > $o,I: nat,J4: nat] :
( ( list_ex_nat @ P2 @ ( upt @ I @ J4 ) )
= ( ~ ( all_interval_nat @ ( comp_o_o_nat @ (~) @ P2 ) @ I @ J4 ) ) ) ).
% list_ex_iff_not_all_inverval_nat
% Helper facts (39)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X2: int,Y3: int] :
( ( if_int @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X2: int,Y3: int] :
( ( if_int @ $true @ X2 @ Y3 )
= X2 ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y3: nat] :
( ( if_nat @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y3: nat] :
( ( if_nat @ $true @ X2 @ Y3 )
= X2 ) ).
thf(help_If_2_1_If_001t__Num__Onum_T,axiom,
! [X2: num,Y3: num] :
( ( if_num @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Num__Onum_T,axiom,
! [X2: num,Y3: num] :
( ( if_num @ $true @ X2 @ Y3 )
= X2 ) ).
thf(help_If_2_1_If_001t__Rat__Orat_T,axiom,
! [X2: rat,Y3: rat] :
( ( if_rat @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Rat__Orat_T,axiom,
! [X2: rat,Y3: rat] :
( ( if_rat @ $true @ X2 @ Y3 )
= X2 ) ).
thf(help_If_2_1_If_001t__Set__Oset_I_Eo_J_T,axiom,
! [X2: set_o,Y3: set_o] :
( ( if_set_o @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Set__Oset_I_Eo_J_T,axiom,
! [X2: set_o,Y3: set_o] :
( ( if_set_o @ $true @ X2 @ Y3 )
= X2 ) ).
thf(help_If_2_1_If_001t__Product____Type__Ounit_T,axiom,
! [X2: product_unit,Y3: product_unit] :
( ( if_Product_unit @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Product____Type__Ounit_T,axiom,
! [X2: product_unit,Y3: product_unit] :
( ( if_Product_unit @ $true @ X2 @ Y3 )
= X2 ) ).
thf(help_If_2_1_If_001t__Code____Numeral__Ointeger_T,axiom,
! [X2: code_integer,Y3: code_integer] :
( ( if_Code_integer @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Code____Numeral__Ointeger_T,axiom,
! [X2: code_integer,Y3: code_integer] :
( ( if_Code_integer @ $true @ X2 @ Y3 )
= X2 ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
! [X2: set_nat,Y3: set_nat] :
( ( if_set_nat @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
! [X2: set_nat,Y3: set_nat] :
( ( if_set_nat @ $true @ X2 @ Y3 )
= X2 ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Product____Type__Ounit_J_T,axiom,
! [X2: set_Product_unit,Y3: set_Product_unit] :
( ( if_set_Product_unit @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Product____Type__Ounit_J_T,axiom,
! [X2: set_Product_unit,Y3: set_Product_unit] :
( ( if_set_Product_unit @ $true @ X2 @ Y3 )
= X2 ) ).
thf(help_If_2_1_If_001t__Heap____Monad__OHeap_It__Heap__Oref_Itf__a_J_J_T,axiom,
! [X2: heap_Heap_ref_a,Y3: heap_Heap_ref_a] :
( ( if_Heap_Heap_ref_a @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Heap____Monad__OHeap_It__Heap__Oref_Itf__a_J_J_T,axiom,
! [X2: heap_Heap_ref_a,Y3: heap_Heap_ref_a] :
( ( if_Heap_Heap_ref_a @ $true @ X2 @ Y3 )
= X2 ) ).
thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
! [X2: product_prod_int_int,Y3: product_prod_int_int] :
( ( if_Pro3027730157355071871nt_int @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
! [X2: product_prod_int_int,Y3: product_prod_int_int] :
( ( if_Pro3027730157355071871nt_int @ $true @ X2 @ Y3 )
= X2 ) ).
thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [X2: product_prod_nat_nat,Y3: product_prod_nat_nat] :
( ( if_Pro6206227464963214023at_nat @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [X2: product_prod_nat_nat,Y3: product_prod_nat_nat] :
( ( if_Pro6206227464963214023at_nat @ $true @ X2 @ Y3 )
= X2 ) ).
thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
! [X2: produc6271795597528267376eger_o,Y3: produc6271795597528267376eger_o] :
( ( if_Pro5737122678794959658eger_o @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
! [X2: produc6271795597528267376eger_o,Y3: produc6271795597528267376eger_o] :
( ( if_Pro5737122678794959658eger_o @ $true @ X2 @ Y3 )
= X2 ) ).
thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
! [X2: produc8923325533196201883nteger,Y3: produc8923325533196201883nteger] :
( ( if_Pro6119634080678213985nteger @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
! [X2: produc8923325533196201883nteger,Y3: produc8923325533196201883nteger] :
( ( if_Pro6119634080678213985nteger @ $true @ X2 @ Y3 )
= X2 ) ).
thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
! [X2: produc859450856879609959at_nat,Y3: produc859450856879609959at_nat] :
( ( if_Pro4507677147265585453at_nat @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
! [X2: produc859450856879609959at_nat,Y3: produc859450856879609959at_nat] :
( ( if_Pro4507677147265585453at_nat @ $true @ X2 @ Y3 )
= X2 ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_T,axiom,
! [X2: set_Pr8693737435421807431at_nat,Y3: set_Pr8693737435421807431at_nat] :
( ( if_set1508868558775481357at_nat @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_T,axiom,
! [X2: set_Pr8693737435421807431at_nat,Y3: set_Pr8693737435421807431at_nat] :
( ( if_set1508868558775481357at_nat @ $true @ X2 @ Y3 )
= X2 ) ).
thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Predicate__Opred_It__Product____Type__Ounit_J_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J_T,axiom,
! [X2: produc4675096598859438275atural,Y3: produc4675096598859438275atural] :
( ( if_Pro3444522238938527101atural @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Predicate__Opred_It__Product____Type__Ounit_J_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J_T,axiom,
! [X2: produc4675096598859438275atural,Y3: produc4675096598859438275atural] :
( ( if_Pro3444522238938527101atural @ $true @ X2 @ Y3 )
= X2 ) ).
thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_T,axiom,
! [X2: produc3843707927480180839at_nat,Y3: produc3843707927480180839at_nat] :
( ( if_Pro3594990065127090477at_nat @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_T,axiom,
! [X2: produc3843707927480180839at_nat,Y3: produc3843707927480180839at_nat] :
( ( if_Pro3594990065127090477at_nat @ $true @ X2 @ Y3 )
= X2 ) ).
thf(help_If_3_1_If_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_T,axiom,
! [X2: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] :
( ( if_set3208018901896979469at_nat @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_T,axiom,
! [X2: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] :
( ( if_set3208018901896979469at_nat @ $true @ X2 @ Y3 )
= X2 ) ).
% Conjectures (2)
thf(conj_0,hypothesis,
$true ).
thf(conj_1,conjecture,
thesis ).
%------------------------------------------------------------------------------