TPTP Problem File: SLH0712^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Finite_Fields/0005_Ring_Characteristic/prob_00674_022566__18207692_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1487 ( 517 unt; 212 typ; 0 def)
% Number of atoms : 3582 (1038 equ; 0 cnn)
% Maximal formula atoms : 11 ( 2 avg)
% Number of connectives : 12826 ( 190 ~; 14 |; 229 &;10741 @)
% ( 0 <=>;1652 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 7 avg)
% Number of types : 20 ( 19 usr)
% Number of type conns : 1352 (1352 >; 0 *; 0 +; 0 <<)
% Number of symbols : 196 ( 193 usr; 10 con; 0-4 aty)
% Number of variables : 4114 ( 347 ^;3610 !; 157 ?;4114 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 13:21:38.355
%------------------------------------------------------------------------------
% Could-be-implicit typings (19)
thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_Itf__a_Mt__Group__Omonoid__Omonoid____ext_Itf__a_Mt__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J_J_J,type,
partia2175431115845679010xt_a_b: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J,type,
set_set_set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J_J_J,type,
set_set_set_nat_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_Mtf__a_J_J,type,
set_set_set_nat_a2: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
set_set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J_J,type,
set_set_nat_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Set__Oset_It__Nat__Onat_J_Mtf__a_J_J,type,
set_set_nat_a2: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_It__Nat__Onat_Mtf__a_J_Mtf__a_J_J,type,
set_nat_a_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
set_set_set_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Set__Oset_Itf__a_J_Mtf__a_J_J,type,
set_set_a_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
set_nat_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
set_a_a: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Set__Oset_I_Eo_J,type,
set_o: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (193)
thf(sy_c_AbelCoset_OA__RCOSETS_001tf__a_001tf__b,type,
a_RCOSETS_a_b: partia2175431115845679010xt_a_b > set_a > set_set_a ).
thf(sy_c_AbelCoset_Oa__l__coset_001tf__a_001tf__b,type,
a_l_coset_a_b: partia2175431115845679010xt_a_b > a > set_a > set_a ).
thf(sy_c_AbelCoset_Oadditive__subgroup_001tf__a_001tf__b,type,
additi2834746164131130830up_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_Eo,type,
complete_Sup_Sup_o: set_o > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Nat__Onat,type,
complete_Sup_Sup_nat: set_nat > nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
comple3545767860446109490_nat_a: set_set_nat_a > set_nat_a ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Nat__Onat_J,type,
comple7399068483239264473et_nat: set_set_nat > set_nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J_J,type,
comple2312494275411564946_nat_a: set_set_set_nat_a > set_set_nat_a ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
comple548664676211718543et_nat: set_set_set_nat > set_set_nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
comple6569609367425551173et_nat: set_set_set_set_nat > set_set_set_nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
comple3958522678809307947_set_a: set_set_set_a > set_set_a ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_Itf__a_J,type,
comple2307003609928055243_set_a: set_set_a > set_a ).
thf(sy_c_Congruence_Opartial__object_Ocarrier_001tf__a_001t__Group__Omonoid__Omonoid____ext_Itf__a_Mt__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J_J,type,
partia707051561876973205xt_a_b: partia2175431115845679010xt_a_b > set_a ).
thf(sy_c_Coset_Oorder_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
order_a_ring_ext_a_b: partia2175431115845679010xt_a_b > nat ).
thf(sy_c_Divisibility_Omonoid__cancel_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
monoid5798828371819920185xt_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Embedded__Algebras_Oring_Ofinite__dimension_001tf__a_001tf__b,type,
embedd8708762675212832759on_a_b: partia2175431115845679010xt_a_b > set_a > set_a > $o ).
thf(sy_c_Embedded__Algebras_Oring_Oline__extension_001tf__a_001tf__b,type,
embedd971793762689825387on_a_b: partia2175431115845679010xt_a_b > set_a > a > set_a > set_a ).
thf(sy_c_Embedded__Algebras_Osubalgebra_001tf__a_001tf__b,type,
embedd9027525575939734154ra_a_b: set_a > set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_FiniteProduct_Ofinprod_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J_001t__Nat__Onat,type,
finpro1280035270526425175_b_nat: partia2175431115845679010xt_a_b > ( nat > a ) > set_nat > a ).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_Itf__a_J,type,
finite_card_set_a: set_set_a > nat ).
thf(sy_c_Finite__Set_Ocard_001tf__a,type,
finite_card_a: set_a > nat ).
thf(sy_c_Finite__Set_Ofinite_001_062_It__Nat__Onat_Mtf__a_J,type,
finite_finite_nat_a: set_nat_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
finite1152437895449049373et_nat: set_set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
finite6739761609112101331et_nat: set_set_set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_Itf__a_J,type,
finite_finite_set_a: set_set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
finite_finite_a: set_a > $o ).
thf(sy_c_FuncSet_OPi_001_062_It__Nat__Onat_Mtf__a_J_001tf__a,type,
pi_nat_a_a: set_nat_a > ( ( nat > a ) > set_a ) > set_nat_a_a ).
thf(sy_c_FuncSet_OPi_001t__Nat__Onat_001tf__a,type,
pi_nat_a: set_nat > ( nat > set_a ) > set_nat_a ).
thf(sy_c_FuncSet_OPi_001t__Set__Oset_It__Nat__Onat_J_001tf__a,type,
pi_set_nat_a: set_set_nat > ( set_nat > set_a ) > set_set_nat_a2 ).
thf(sy_c_FuncSet_OPi_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001tf__a,type,
pi_set_set_nat_a: set_set_set_nat > ( set_set_nat > set_a ) > set_set_set_nat_a2 ).
thf(sy_c_FuncSet_OPi_001t__Set__Oset_Itf__a_J_001tf__a,type,
pi_set_a_a: set_set_a > ( set_a > set_a ) > set_set_a_a ).
thf(sy_c_FuncSet_OPi_001tf__a_001tf__a,type,
pi_a_a: set_a > ( a > set_a ) > set_a_a ).
thf(sy_c_Group_Omonoid_Omult_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
mult_a_ring_ext_a_b: partia2175431115845679010xt_a_b > a > a > a ).
thf(sy_c_Group_Omonoid_Oone_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
one_a_ring_ext_a_b: partia2175431115845679010xt_a_b > a ).
thf(sy_c_Group_Opow_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J_001t__Nat__Onat,type,
pow_a_1026414303147256608_b_nat: partia2175431115845679010xt_a_b > a > nat > a ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_I_062_It__Nat__Onat_Mtf__a_J_M_Eo_J,type,
minus_minus_nat_a_o: ( ( nat > a ) > $o ) > ( ( nat > a ) > $o ) > ( nat > a ) > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Nat__Onat_M_Eo_J,type,
minus_minus_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
minus_6910147592129066416_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > set_nat > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
minus_463385787819020154_nat_o: ( set_set_nat > $o ) > ( set_set_nat > $o ) > set_set_nat > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Set__Oset_Itf__a_J_M_Eo_J,type,
minus_minus_set_a_o: ( set_a > $o ) > ( set_a > $o ) > set_a > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_Itf__a_M_Eo_J,type,
minus_minus_a_o: ( a > $o ) > ( a > $o ) > a > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
minus_490503922182417452_nat_a: set_nat_a > set_nat_a > set_nat_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J_J,type,
minus_2199387171230727820_nat_a: set_set_nat_a > set_set_nat_a > set_set_nat_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
minus_2163939370556025621et_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
minus_2447799839930672331et_nat: set_set_set_nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
minus_5736297505244876581_set_a: set_set_a > set_set_a > set_set_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
minus_minus_set_a: set_a > set_a > set_a ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Ideal_Ocgenideal_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
cgenid547466209912283029xt_a_b: partia2175431115845679010xt_a_b > a > set_a ).
thf(sy_c_Ideal_Ogenideal_001tf__a_001tf__b,type,
genideal_a_b: partia2175431115845679010xt_a_b > set_a > set_a ).
thf(sy_c_Ideal_Oprincipalideal_001tf__a_001tf__b,type,
principalideal_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_If_001tf__a,type,
if_a: $o > a > a > a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__a_J,type,
inf_inf_set_a: set_a > set_a > set_a ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Oord__class_Oless_001_Eo,type,
ord_less_o: $o > $o > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_less_set_set_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
ord_less_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Nat__Onat_Mtf__a_J_M_Eo_J,type,
ord_less_eq_nat_a_o: ( ( nat > a ) > $o ) > ( ( nat > a ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
ord_le3964352015994296041_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
ord_le3616423863276227763_nat_o: ( set_set_nat > $o ) > ( set_set_nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Set__Oset_Itf__a_J_M_Eo_J,type,
ord_less_eq_set_a_o: ( set_a > $o ) > ( set_a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_Itf__a_M_Eo_J,type,
ord_less_eq_a_o: ( a > $o ) > ( a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_Eo,type,
ord_less_eq_o: $o > $o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
ord_le871467723717165285_nat_a: set_nat_a > set_nat_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_Eo_J,type,
ord_less_eq_set_o: set_o > set_o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J_J,type,
ord_le2390145808437456709_nat_a: set_set_nat_a > set_set_nat_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J_J_J,type,
ord_le2266563856530774437_nat_a: set_set_set_nat_a > set_set_set_nat_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
ord_le9131159989063066194et_nat: set_set_set_nat > set_set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J,type,
ord_le572741076514265352et_nat: set_set_set_set_nat > set_set_set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
ord_le5722252365846178494_set_a: set_set_set_a > set_set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
ord_le3724670747650509150_set_a: set_set_a > set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Ring_Oa__minus_001tf__a_001tf__b,type,
a_minus_a_b: partia2175431115845679010xt_a_b > a > a > a ).
thf(sy_c_Ring_Ocring_001tf__a_001tf__b,type,
cring_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring_Ofinsum_001tf__a_001tf__b_001_062_It__Nat__Onat_Mtf__a_J,type,
finsum_a_b_nat_a: partia2175431115845679010xt_a_b > ( ( nat > a ) > a ) > set_nat_a > a ).
thf(sy_c_Ring_Ofinsum_001tf__a_001tf__b_001t__Nat__Onat,type,
finsum_a_b_nat: partia2175431115845679010xt_a_b > ( nat > a ) > set_nat > a ).
thf(sy_c_Ring_Ofinsum_001tf__a_001tf__b_001t__Set__Oset_It__Nat__Onat_J,type,
finsum_a_b_set_nat: partia2175431115845679010xt_a_b > ( set_nat > a ) > set_set_nat > a ).
thf(sy_c_Ring_Ofinsum_001tf__a_001tf__b_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
finsum2649122254697571802et_nat: partia2175431115845679010xt_a_b > ( set_set_nat > a ) > set_set_set_nat > a ).
thf(sy_c_Ring_Ofinsum_001tf__a_001tf__b_001t__Set__Oset_Itf__a_J,type,
finsum_a_b_set_a: partia2175431115845679010xt_a_b > ( set_a > a ) > set_set_a > a ).
thf(sy_c_Ring_Ofinsum_001tf__a_001tf__b_001tf__a,type,
finsum_a_b_a: partia2175431115845679010xt_a_b > ( a > a ) > set_a > a ).
thf(sy_c_Ring_Oring_Oadd_001tf__a_001tf__b,type,
add_a_b: partia2175431115845679010xt_a_b > a > a > a ).
thf(sy_c_Ring_Oring_Ozero_001tf__a_001tf__b,type,
zero_a_b: partia2175431115845679010xt_a_b > a ).
thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_Mtf__a_J,type,
collect_nat_a: ( ( nat > a ) > $o ) > set_nat_a ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
collect_set_set_nat: ( set_set_nat > $o ) > set_set_set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
collect_set_set_a: ( set_set_a > $o ) > set_set_set_a ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__a_J,type,
collect_set_a: ( set_a > $o ) > set_set_a ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_OPow_001_062_It__Nat__Onat_Mtf__a_J,type,
pow_nat_a: set_nat_a > set_set_nat_a ).
thf(sy_c_Set_OPow_001t__Nat__Onat,type,
pow_nat: set_nat > set_set_nat ).
thf(sy_c_Set_OPow_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
pow_set_nat_a: set_set_nat_a > set_set_set_nat_a ).
thf(sy_c_Set_OPow_001t__Set__Oset_It__Nat__Onat_J,type,
pow_set_nat: set_set_nat > set_set_set_nat ).
thf(sy_c_Set_OPow_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
pow_set_set_nat: set_set_set_nat > set_set_set_set_nat ).
thf(sy_c_Set_OPow_001t__Set__Oset_Itf__a_J,type,
pow_set_a: set_set_a > set_set_set_a ).
thf(sy_c_Set_OPow_001tf__a,type,
pow_a: set_a > set_set_a ).
thf(sy_c_Set_Oimage_001_Eo_001_Eo,type,
image_o_o: ( $o > $o ) > set_o > set_o ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001_Eo,type,
image_nat_o: ( nat > $o ) > set_nat > set_o ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
image_nat_set_nat: ( nat > set_nat ) > set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
image_2194112158459175443et_nat: ( nat > set_set_nat ) > set_nat > set_set_set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
image_5738044413236618185et_nat: ( nat > set_set_set_nat ) > set_nat > set_set_set_set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
image_nat_set_set_a: ( nat > set_set_a ) > set_nat > set_set_set_a ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_Itf__a_J,type,
image_nat_set_a: ( nat > set_a ) > set_nat > set_set_a ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001tf__a,type,
image_nat_a: ( nat > a ) > set_nat > set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J_001_Eo,type,
image_set_nat_a_o: ( set_nat_a > $o ) > set_set_nat_a > set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
image_6965494298868581957_nat_a: ( set_nat_a > set_nat_a ) > set_set_nat_a > set_set_nat_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001_Eo,type,
image_set_nat_o: ( set_nat > $o ) > set_set_nat > set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
image_set_nat_nat: ( set_nat > nat ) > set_set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_7916887816326733075et_nat: ( set_nat > set_nat ) > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
image_6725021117256019401et_nat: ( set_nat > set_set_nat ) > set_set_nat > set_set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
image_4583741654806091647et_nat: ( set_nat > set_set_set_nat ) > set_set_nat > set_set_set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
image_8586572261461758321_set_a: ( set_nat > set_set_a ) > set_set_nat > set_set_set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_Itf__a_J,type,
image_set_nat_set_a: ( set_nat > set_a ) > set_set_nat > set_set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001tf__a,type,
image_set_nat_a: ( set_nat > a ) > set_set_nat > set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001_Eo,type,
image_set_set_nat_o: ( set_set_nat > $o ) > set_set_set_nat > set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_5842784325960735177et_nat: ( set_set_nat > set_nat ) > set_set_set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
image_7884819252390400639et_nat: ( set_set_nat > set_set_nat ) > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001t__Set__Oset_Itf__a_J,type,
image_6642697982911593691_set_a: ( set_set_nat > set_a ) > set_set_set_nat > set_set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_001_Eo,type,
image_3488003393078953823_nat_o: ( set_set_set_nat > $o ) > set_set_set_set_nat > set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_Itf__a_J_J_001_Eo,type,
image_set_set_a_o: ( set_set_a > $o ) > set_set_set_a > set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_Itf__a_J_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_6109939652141935103et_nat: ( set_set_a > set_nat ) > set_set_set_a > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_Itf__a_J_J_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
image_1042221919965026181_set_a: ( set_set_a > set_set_a ) > set_set_set_a > set_set_set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_Itf__a_J_J_001t__Set__Oset_Itf__a_J,type,
image_6061375613820669477_set_a: ( set_set_a > set_a ) > set_set_set_a > set_set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001_Eo,type,
image_set_a_o: ( set_a > $o ) > set_set_a > set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Nat__Onat,type,
image_set_a_nat: ( set_a > nat ) > set_set_a > set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
image_1117120162361407980_nat_a: ( set_a > set_nat_a ) > set_set_a > set_set_nat_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_set_a_set_nat: ( set_a > set_nat ) > set_set_a > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Set__Oset_It__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J_J,type,
image_7780357287214271564_nat_a: ( set_a > set_set_nat_a ) > set_set_a > set_set_set_nat_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
image_8216882647274671445et_nat: ( set_a > set_set_nat ) > set_set_a > set_set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
image_4955109552351689957_set_a: ( set_a > set_set_a ) > set_set_a > set_set_set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
image_set_a_set_a: ( set_a > set_a ) > set_set_a > set_set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001tf__a,type,
image_set_a_a: ( set_a > a ) > set_set_a > set_a ).
thf(sy_c_Set_Oimage_001tf__a_001_062_It__Nat__Onat_Mtf__a_J,type,
image_a_nat_a: ( a > nat > a ) > set_a > set_nat_a ).
thf(sy_c_Set_Oimage_001tf__a_001_Eo,type,
image_a_o: ( a > $o ) > set_a > set_o ).
thf(sy_c_Set_Oimage_001tf__a_001t__Nat__Onat,type,
image_a_nat: ( a > nat ) > set_a > set_nat ).
thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
image_a_set_nat_a: ( a > set_nat_a ) > set_a > set_set_nat_a ).
thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_It__Nat__Onat_J,type,
image_a_set_nat: ( a > set_nat ) > set_a > set_set_nat ).
thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_It__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J_J,type,
image_2266649148107646380_nat_a: ( a > set_set_nat_a ) > set_a > set_set_set_nat_a ).
thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
image_a_set_set_nat: ( a > set_set_nat ) > set_a > set_set_set_nat ).
thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
image_a_set_set_a: ( a > set_set_a ) > set_a > set_set_set_a ).
thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_Itf__a_J,type,
image_a_set_a: ( a > set_a ) > set_a > set_set_a ).
thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
image_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Nat__Onat,type,
set_or1269000886237332187st_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001_Eo,type,
set_ord_atMost_o: $o > set_o ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
set_ord_atMost_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
set_or2677650046130559372_nat_a: set_nat_a > set_set_nat_a ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Nat__Onat_J,type,
set_or4236626031148496127et_nat: set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_or7210490968680142261et_nat: set_set_nat > set_set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_or4016371710855203973_set_a: set_set_a > set_set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_Itf__a_J,type,
set_ord_atMost_set_a: set_a > set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
set_ord_lessThan_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Set__Oset_It__Nat__Onat_J,type,
set_or890127255671739683et_nat: set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_or6631954706645296601et_nat: set_set_nat > set_set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Set__Oset_Itf__a_J,type,
set_or5421148953861284865_set_a: set_a > set_set_a ).
thf(sy_c_Subrings_Osubfield_001tf__a_001tf__b,type,
subfield_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_UnivPoly_Obound_001tf__a,type,
bound_a: a > nat > ( nat > a ) > $o ).
thf(sy_c_UnivPoly_Oup_001tf__a_001tf__b,type,
up_a_b: partia2175431115845679010xt_a_b > set_nat_a ).
thf(sy_c_member_001_062_I_062_It__Nat__Onat_Mtf__a_J_Mtf__a_J,type,
member_nat_a_a: ( ( nat > a ) > a ) > set_nat_a_a > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mtf__a_J,type,
member_nat_a: ( nat > a ) > set_nat_a > $o ).
thf(sy_c_member_001_062_It__Set__Oset_It__Nat__Onat_J_Mtf__a_J,type,
member_set_nat_a: ( set_nat > a ) > set_set_nat_a2 > $o ).
thf(sy_c_member_001_062_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_Mtf__a_J,type,
member_set_set_nat_a: ( set_set_nat > a ) > set_set_set_nat_a2 > $o ).
thf(sy_c_member_001_062_It__Set__Oset_Itf__a_J_Mtf__a_J,type,
member_set_a_a: ( set_a > a ) > set_set_a_a > $o ).
thf(sy_c_member_001_062_Itf__a_Mtf__a_J,type,
member_a_a: ( a > a ) > set_a_a > $o ).
thf(sy_c_member_001_Eo,type,
member_o: $o > set_o > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
member_set_nat_a2: set_nat_a > set_set_nat_a > $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__Set__Oset_It__Nat__Onat_J_J,type,
member_set_set_nat: set_set_nat > set_set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
member2946998982187404937et_nat: set_set_set_nat > set_set_set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
member_set_set_a: set_set_a > set_set_set_a > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_A____,type,
a2: nat > set_set_nat ).
thf(sy_v_R,type,
r: partia2175431115845679010xt_a_b ).
thf(sy_v_n,type,
n: nat ).
thf(sy_v_x,type,
x: a ).
thf(sy_v_y,type,
y: a ).
% Relevant facts (1271)
thf(fact_0_card__A,axiom,
! [I: nat,B: set_nat] :
( ( member_nat @ I @ ( set_ord_atMost_nat @ n ) )
=> ( ( member_set_nat @ B @ ( a2 @ I ) )
=> ( ( finite_card_nat @ B )
= I ) ) ) ).
% card_A
thf(fact_1_assms_I2_J,axiom,
member_a @ y @ ( partia707051561876973205xt_a_b @ r ) ).
% assms(2)
thf(fact_2_assms_I1_J,axiom,
member_a @ x @ ( partia707051561876973205xt_a_b @ r ) ).
% assms(1)
thf(fact_3_is__cring,axiom,
cring_a_b @ r ).
% is_cring
thf(fact_4_image__set__eqI,axiom,
! [A: set_nat,F: nat > nat,B: set_nat,G: nat > nat] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member_nat @ ( F @ X ) @ B ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ B )
=> ( ( member_nat @ ( G @ X ) @ A )
& ( ( F @ ( G @ X ) )
= X ) ) )
=> ( ( image_nat_nat @ F @ A )
= B ) ) ) ).
% image_set_eqI
thf(fact_5_image__set__eqI,axiom,
! [A: set_nat,F: nat > a,B: set_a,G: a > nat] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member_a @ ( F @ X ) @ B ) )
=> ( ! [X: a] :
( ( member_a @ X @ B )
=> ( ( member_nat @ ( G @ X ) @ A )
& ( ( F @ ( G @ X ) )
= X ) ) )
=> ( ( image_nat_a @ F @ A )
= B ) ) ) ).
% image_set_eqI
thf(fact_6_image__set__eqI,axiom,
! [A: set_a,F: a > nat,B: set_nat,G: nat > a] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( member_nat @ ( F @ X ) @ B ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ B )
=> ( ( member_a @ ( G @ X ) @ A )
& ( ( F @ ( G @ X ) )
= X ) ) )
=> ( ( image_a_nat @ F @ A )
= B ) ) ) ).
% image_set_eqI
thf(fact_7_image__set__eqI,axiom,
! [A: set_a,F: a > a,B: set_a,G: a > a] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( member_a @ ( F @ X ) @ B ) )
=> ( ! [X: a] :
( ( member_a @ X @ B )
=> ( ( member_a @ ( G @ X ) @ A )
& ( ( F @ ( G @ X ) )
= X ) ) )
=> ( ( image_a_a @ F @ A )
= B ) ) ) ).
% image_set_eqI
thf(fact_8_image__set__eqI,axiom,
! [A: set_set_nat,F: set_nat > nat,B: set_nat,G: nat > set_nat] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ A )
=> ( member_nat @ ( F @ X ) @ B ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ B )
=> ( ( member_set_nat @ ( G @ X ) @ A )
& ( ( F @ ( G @ X ) )
= X ) ) )
=> ( ( image_set_nat_nat @ F @ A )
= B ) ) ) ).
% image_set_eqI
thf(fact_9_image__set__eqI,axiom,
! [A: set_set_nat,F: set_nat > a,B: set_a,G: a > set_nat] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ A )
=> ( member_a @ ( F @ X ) @ B ) )
=> ( ! [X: a] :
( ( member_a @ X @ B )
=> ( ( member_set_nat @ ( G @ X ) @ A )
& ( ( F @ ( G @ X ) )
= X ) ) )
=> ( ( image_set_nat_a @ F @ A )
= B ) ) ) ).
% image_set_eqI
thf(fact_10_image__set__eqI,axiom,
! [A: set_set_a,F: set_a > nat,B: set_nat,G: nat > set_a] :
( ! [X: set_a] :
( ( member_set_a @ X @ A )
=> ( member_nat @ ( F @ X ) @ B ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ B )
=> ( ( member_set_a @ ( G @ X ) @ A )
& ( ( F @ ( G @ X ) )
= X ) ) )
=> ( ( image_set_a_nat @ F @ A )
= B ) ) ) ).
% image_set_eqI
thf(fact_11_image__set__eqI,axiom,
! [A: set_set_a,F: set_a > a,B: set_a,G: a > set_a] :
( ! [X: set_a] :
( ( member_set_a @ X @ A )
=> ( member_a @ ( F @ X ) @ B ) )
=> ( ! [X: a] :
( ( member_a @ X @ B )
=> ( ( member_set_a @ ( G @ X ) @ A )
& ( ( F @ ( G @ X ) )
= X ) ) )
=> ( ( image_set_a_a @ F @ A )
= B ) ) ) ).
% image_set_eqI
thf(fact_12_image__set__eqI,axiom,
! [A: set_nat,F: nat > set_nat,B: set_set_nat,G: set_nat > nat] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member_set_nat @ ( F @ X ) @ B ) )
=> ( ! [X: set_nat] :
( ( member_set_nat @ X @ B )
=> ( ( member_nat @ ( G @ X ) @ A )
& ( ( F @ ( G @ X ) )
= X ) ) )
=> ( ( image_nat_set_nat @ F @ A )
= B ) ) ) ).
% image_set_eqI
thf(fact_13_image__set__eqI,axiom,
! [A: set_nat,F: nat > set_a,B: set_set_a,G: set_a > nat] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member_set_a @ ( F @ X ) @ B ) )
=> ( ! [X: set_a] :
( ( member_set_a @ X @ B )
=> ( ( member_nat @ ( G @ X ) @ A )
& ( ( F @ ( G @ X ) )
= X ) ) )
=> ( ( image_nat_set_a @ F @ A )
= B ) ) ) ).
% image_set_eqI
thf(fact_14_calculation,axiom,
( ( pow_a_1026414303147256608_b_nat @ r @ ( add_a_b @ r @ x @ y ) @ n )
= ( finsum_a_b_set_nat @ r
@ ^ [A2: set_nat] : ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ x @ ( finite_card_nat @ A2 ) ) @ ( pow_a_1026414303147256608_b_nat @ r @ y @ ( minus_minus_nat @ n @ ( finite_card_nat @ A2 ) ) ) )
@ ( pow_nat @ ( set_ord_lessThan_nat @ n ) ) ) ) ).
% calculation
thf(fact_15_UN__I,axiom,
! [A3: nat,A: set_nat,B2: nat,B: nat > set_nat] :
( ( member_nat @ A3 @ A )
=> ( ( member_nat @ B2 @ ( B @ A3 ) )
=> ( member_nat @ B2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_16_UN__I,axiom,
! [A3: nat,A: set_nat,B2: a,B: nat > set_a] :
( ( member_nat @ A3 @ A )
=> ( ( member_a @ B2 @ ( B @ A3 ) )
=> ( member_a @ B2 @ ( comple2307003609928055243_set_a @ ( image_nat_set_a @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_17_UN__I,axiom,
! [A3: a,A: set_a,B2: nat,B: a > set_nat] :
( ( member_a @ A3 @ A )
=> ( ( member_nat @ B2 @ ( B @ A3 ) )
=> ( member_nat @ B2 @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_18_UN__I,axiom,
! [A3: a,A: set_a,B2: a,B: a > set_a] :
( ( member_a @ A3 @ A )
=> ( ( member_a @ B2 @ ( B @ A3 ) )
=> ( member_a @ B2 @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_19_UN__I,axiom,
! [A3: set_nat,A: set_set_nat,B2: nat,B: set_nat > set_nat] :
( ( member_set_nat @ A3 @ A )
=> ( ( member_nat @ B2 @ ( B @ A3 ) )
=> ( member_nat @ B2 @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_20_UN__I,axiom,
! [A3: set_nat,A: set_set_nat,B2: a,B: set_nat > set_a] :
( ( member_set_nat @ A3 @ A )
=> ( ( member_a @ B2 @ ( B @ A3 ) )
=> ( member_a @ B2 @ ( comple2307003609928055243_set_a @ ( image_set_nat_set_a @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_21_UN__I,axiom,
! [A3: set_a,A: set_set_a,B2: nat,B: set_a > set_nat] :
( ( member_set_a @ A3 @ A )
=> ( ( member_nat @ B2 @ ( B @ A3 ) )
=> ( member_nat @ B2 @ ( comple7399068483239264473et_nat @ ( image_set_a_set_nat @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_22_UN__I,axiom,
! [A3: set_a,A: set_set_a,B2: a,B: set_a > set_a] :
( ( member_set_a @ A3 @ A )
=> ( ( member_a @ B2 @ ( B @ A3 ) )
=> ( member_a @ B2 @ ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_23_UN__I,axiom,
! [A3: nat,A: set_nat,B2: set_a,B: nat > set_set_a] :
( ( member_nat @ A3 @ A )
=> ( ( member_set_a @ B2 @ ( B @ A3 ) )
=> ( member_set_a @ B2 @ ( comple3958522678809307947_set_a @ ( image_nat_set_set_a @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_24_UN__I,axiom,
! [A3: a,A: set_a,B2: set_a,B: a > set_set_a] :
( ( member_a @ A3 @ A )
=> ( ( member_set_a @ B2 @ ( B @ A3 ) )
=> ( member_set_a @ B2 @ ( comple3958522678809307947_set_a @ ( image_a_set_set_a @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_25_UN__iff,axiom,
! [B2: a,B: a > set_a,A: set_a] :
( ( member_a @ B2 @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A ) ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A )
& ( member_a @ B2 @ ( B @ X2 ) ) ) ) ) ).
% UN_iff
thf(fact_26_UN__iff,axiom,
! [B2: nat > a,B: a > set_nat_a,A: set_a] :
( ( member_nat_a @ B2 @ ( comple3545767860446109490_nat_a @ ( image_a_set_nat_a @ B @ A ) ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A )
& ( member_nat_a @ B2 @ ( B @ X2 ) ) ) ) ) ).
% UN_iff
thf(fact_27_UN__iff,axiom,
! [B2: nat,B: a > set_nat,A: set_a] :
( ( member_nat @ B2 @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ B @ A ) ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A )
& ( member_nat @ B2 @ ( B @ X2 ) ) ) ) ) ).
% UN_iff
thf(fact_28_UN__iff,axiom,
! [B2: set_nat,B: nat > set_set_nat,A: set_nat] :
( ( member_set_nat @ B2 @ ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ B @ A ) ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( member_set_nat @ B2 @ ( B @ X2 ) ) ) ) ) ).
% UN_iff
thf(fact_29_A__def,axiom,
( a2
= ( ^ [K: nat] :
( collect_set_nat
@ ^ [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( set_ord_lessThan_nat @ n ) )
& ( ( finite_card_nat @ A2 )
= K ) ) ) ) ) ).
% A_def
thf(fact_30_SUP__identity__eq,axiom,
! [A: set_set_a] :
( ( comple2307003609928055243_set_a
@ ( image_set_a_set_a
@ ^ [X2: set_a] : X2
@ A ) )
= ( comple2307003609928055243_set_a @ A ) ) ).
% SUP_identity_eq
thf(fact_31_SUP__identity__eq,axiom,
! [A: set_set_nat_a] :
( ( comple3545767860446109490_nat_a
@ ( image_6965494298868581957_nat_a
@ ^ [X2: set_nat_a] : X2
@ A ) )
= ( comple3545767860446109490_nat_a @ A ) ) ).
% SUP_identity_eq
thf(fact_32_SUP__identity__eq,axiom,
! [A: set_set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_7916887816326733075et_nat
@ ^ [X2: set_nat] : X2
@ A ) )
= ( comple7399068483239264473et_nat @ A ) ) ).
% SUP_identity_eq
thf(fact_33_SUP__identity__eq,axiom,
! [A: set_set_set_a] :
( ( comple3958522678809307947_set_a
@ ( image_1042221919965026181_set_a
@ ^ [X2: set_set_a] : X2
@ A ) )
= ( comple3958522678809307947_set_a @ A ) ) ).
% SUP_identity_eq
thf(fact_34_SUP__identity__eq,axiom,
! [A: set_set_set_nat] :
( ( comple548664676211718543et_nat
@ ( image_7884819252390400639et_nat
@ ^ [X2: set_set_nat] : X2
@ A ) )
= ( comple548664676211718543et_nat @ A ) ) ).
% SUP_identity_eq
thf(fact_35_SUP__identity__eq,axiom,
! [A: set_o] :
( ( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [X2: $o] : X2
@ A ) )
= ( complete_Sup_Sup_o @ A ) ) ).
% SUP_identity_eq
thf(fact_36_SUP__identity__eq,axiom,
! [A: set_nat] :
( ( complete_Sup_Sup_nat
@ ( image_nat_nat
@ ^ [X2: nat] : X2
@ A ) )
= ( complete_Sup_Sup_nat @ A ) ) ).
% SUP_identity_eq
thf(fact_37_card__lessThan,axiom,
! [U: nat] :
( ( finite_card_nat @ ( set_ord_lessThan_nat @ U ) )
= U ) ).
% card_lessThan
thf(fact_38_Union__Pow__eq,axiom,
! [A: set_set_nat] :
( ( comple548664676211718543et_nat @ ( pow_set_nat @ A ) )
= A ) ).
% Union_Pow_eq
thf(fact_39_Union__Pow__eq,axiom,
! [A: set_a] :
( ( comple2307003609928055243_set_a @ ( pow_a @ A ) )
= A ) ).
% Union_Pow_eq
thf(fact_40_Union__Pow__eq,axiom,
! [A: set_nat_a] :
( ( comple3545767860446109490_nat_a @ ( pow_nat_a @ A ) )
= A ) ).
% Union_Pow_eq
thf(fact_41_Union__Pow__eq,axiom,
! [A: set_nat] :
( ( comple7399068483239264473et_nat @ ( pow_nat @ A ) )
= A ) ).
% Union_Pow_eq
thf(fact_42_Union__Pow__eq,axiom,
! [A: set_set_a] :
( ( comple3958522678809307947_set_a @ ( pow_set_a @ A ) )
= A ) ).
% Union_Pow_eq
thf(fact_43_UN__ball__bex__simps_I4_J,axiom,
! [B: nat > set_set_nat,A: set_nat,P: set_nat > $o] :
( ( ? [X2: set_nat] :
( ( member_set_nat @ X2 @ ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ B @ A ) ) )
& ( P @ X2 ) ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ? [Y: set_nat] :
( ( member_set_nat @ Y @ ( B @ X2 ) )
& ( P @ Y ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_44_UN__ball__bex__simps_I4_J,axiom,
! [B: a > set_a,A: set_a,P: a > $o] :
( ( ? [X2: a] :
( ( member_a @ X2 @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A ) ) )
& ( P @ X2 ) ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A )
& ? [Y: a] :
( ( member_a @ Y @ ( B @ X2 ) )
& ( P @ Y ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_45_UN__ball__bex__simps_I4_J,axiom,
! [B: a > set_nat_a,A: set_a,P: ( nat > a ) > $o] :
( ( ? [X2: nat > a] :
( ( member_nat_a @ X2 @ ( comple3545767860446109490_nat_a @ ( image_a_set_nat_a @ B @ A ) ) )
& ( P @ X2 ) ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A )
& ? [Y: nat > a] :
( ( member_nat_a @ Y @ ( B @ X2 ) )
& ( P @ Y ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_46_UN__ball__bex__simps_I4_J,axiom,
! [B: a > set_nat,A: set_a,P: nat > $o] :
( ( ? [X2: nat] :
( ( member_nat @ X2 @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ B @ A ) ) )
& ( P @ X2 ) ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A )
& ? [Y: nat] :
( ( member_nat @ Y @ ( B @ X2 ) )
& ( P @ Y ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_47_UN__ball__bex__simps_I2_J,axiom,
! [B: nat > set_set_nat,A: set_nat,P: set_nat > $o] :
( ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ B @ A ) ) )
=> ( P @ X2 ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ! [Y: set_nat] :
( ( member_set_nat @ Y @ ( B @ X2 ) )
=> ( P @ Y ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_48_UN__ball__bex__simps_I2_J,axiom,
! [B: a > set_a,A: set_a,P: a > $o] :
( ( ! [X2: a] :
( ( member_a @ X2 @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A ) ) )
=> ( P @ X2 ) ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ! [Y: a] :
( ( member_a @ Y @ ( B @ X2 ) )
=> ( P @ Y ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_49_UN__ball__bex__simps_I2_J,axiom,
! [B: a > set_nat_a,A: set_a,P: ( nat > a ) > $o] :
( ( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ ( comple3545767860446109490_nat_a @ ( image_a_set_nat_a @ B @ A ) ) )
=> ( P @ X2 ) ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ! [Y: nat > a] :
( ( member_nat_a @ Y @ ( B @ X2 ) )
=> ( P @ Y ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_50_UN__ball__bex__simps_I2_J,axiom,
! [B: a > set_nat,A: set_a,P: nat > $o] :
( ( ! [X2: nat] :
( ( member_nat @ X2 @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ B @ A ) ) )
=> ( P @ X2 ) ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ! [Y: nat] :
( ( member_nat @ Y @ ( B @ X2 ) )
=> ( P @ Y ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_51_bex__UN,axiom,
! [B: nat > set_set_nat,A: set_nat,P: set_nat > $o] :
( ( ? [X2: set_nat] :
( ( member_set_nat @ X2 @ ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ B @ A ) ) )
& ( P @ X2 ) ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ? [Y: set_nat] :
( ( member_set_nat @ Y @ ( B @ X2 ) )
& ( P @ Y ) ) ) ) ) ).
% bex_UN
thf(fact_52_bex__UN,axiom,
! [B: a > set_a,A: set_a,P: a > $o] :
( ( ? [X2: a] :
( ( member_a @ X2 @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A ) ) )
& ( P @ X2 ) ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A )
& ? [Y: a] :
( ( member_a @ Y @ ( B @ X2 ) )
& ( P @ Y ) ) ) ) ) ).
% bex_UN
thf(fact_53_bex__UN,axiom,
! [B: a > set_nat_a,A: set_a,P: ( nat > a ) > $o] :
( ( ? [X2: nat > a] :
( ( member_nat_a @ X2 @ ( comple3545767860446109490_nat_a @ ( image_a_set_nat_a @ B @ A ) ) )
& ( P @ X2 ) ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A )
& ? [Y: nat > a] :
( ( member_nat_a @ Y @ ( B @ X2 ) )
& ( P @ Y ) ) ) ) ) ).
% bex_UN
thf(fact_54_bex__UN,axiom,
! [B: a > set_nat,A: set_a,P: nat > $o] :
( ( ? [X2: nat] :
( ( member_nat @ X2 @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ B @ A ) ) )
& ( P @ X2 ) ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A )
& ? [Y: nat] :
( ( member_nat @ Y @ ( B @ X2 ) )
& ( P @ Y ) ) ) ) ) ).
% bex_UN
thf(fact_55_ball__UN,axiom,
! [B: nat > set_set_nat,A: set_nat,P: set_nat > $o] :
( ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ B @ A ) ) )
=> ( P @ X2 ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ! [Y: set_nat] :
( ( member_set_nat @ Y @ ( B @ X2 ) )
=> ( P @ Y ) ) ) ) ) ).
% ball_UN
thf(fact_56_ball__UN,axiom,
! [B: a > set_a,A: set_a,P: a > $o] :
( ( ! [X2: a] :
( ( member_a @ X2 @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A ) ) )
=> ( P @ X2 ) ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ! [Y: a] :
( ( member_a @ Y @ ( B @ X2 ) )
=> ( P @ Y ) ) ) ) ) ).
% ball_UN
thf(fact_57_ball__UN,axiom,
! [B: a > set_nat_a,A: set_a,P: ( nat > a ) > $o] :
( ( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ ( comple3545767860446109490_nat_a @ ( image_a_set_nat_a @ B @ A ) ) )
=> ( P @ X2 ) ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ! [Y: nat > a] :
( ( member_nat_a @ Y @ ( B @ X2 ) )
=> ( P @ Y ) ) ) ) ) ).
% ball_UN
thf(fact_58_ball__UN,axiom,
! [B: a > set_nat,A: set_a,P: nat > $o] :
( ( ! [X2: nat] :
( ( member_nat @ X2 @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ B @ A ) ) )
=> ( P @ X2 ) ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ! [Y: nat] :
( ( member_nat @ Y @ ( B @ X2 ) )
=> ( P @ Y ) ) ) ) ) ).
% ball_UN
thf(fact_59_a__lcomm,axiom,
! [X3: a,Y2: a,Z: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X3 @ ( add_a_b @ r @ Y2 @ Z ) )
= ( add_a_b @ r @ Y2 @ ( add_a_b @ r @ X3 @ Z ) ) ) ) ) ) ).
% a_lcomm
thf(fact_60_a__comm,axiom,
! [X3: a,Y2: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X3 @ Y2 )
= ( add_a_b @ r @ Y2 @ X3 ) ) ) ) ).
% a_comm
thf(fact_61_a__assoc,axiom,
! [X3: a,Y2: a,Z: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( add_a_b @ r @ X3 @ Y2 ) @ Z )
= ( add_a_b @ r @ X3 @ ( add_a_b @ r @ Y2 @ Z ) ) ) ) ) ) ).
% a_assoc
thf(fact_62_add_Or__cancel,axiom,
! [A3: a,C: a,B2: a] :
( ( ( add_a_b @ r @ A3 @ C )
= ( add_a_b @ r @ B2 @ C ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A3 = B2 ) ) ) ) ) ).
% add.r_cancel
thf(fact_63_add_Ol__cancel,axiom,
! [C: a,A3: a,B2: a] :
( ( ( add_a_b @ r @ C @ A3 )
= ( add_a_b @ r @ C @ B2 ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A3 = B2 ) ) ) ) ) ).
% add.l_cancel
thf(fact_64_m__lcomm,axiom,
! [X3: a,Y2: a,Z: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X3 @ ( mult_a_ring_ext_a_b @ r @ Y2 @ Z ) )
= ( mult_a_ring_ext_a_b @ r @ Y2 @ ( mult_a_ring_ext_a_b @ r @ X3 @ Z ) ) ) ) ) ) ).
% m_lcomm
thf(fact_65_m__comm,axiom,
! [X3: a,Y2: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X3 @ Y2 )
= ( mult_a_ring_ext_a_b @ r @ Y2 @ X3 ) ) ) ) ).
% m_comm
thf(fact_66_m__assoc,axiom,
! [X3: a,Y2: a,Z: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X3 @ Y2 ) @ Z )
= ( mult_a_ring_ext_a_b @ r @ X3 @ ( mult_a_ring_ext_a_b @ r @ Y2 @ Z ) ) ) ) ) ) ).
% m_assoc
thf(fact_67_add_Osurj__const__mult,axiom,
! [A3: a] :
( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( image_a_a @ ( add_a_b @ r @ A3 ) @ ( partia707051561876973205xt_a_b @ r ) )
= ( partia707051561876973205xt_a_b @ r ) ) ) ).
% add.surj_const_mult
thf(fact_68_r__distr,axiom,
! [X3: a,Y2: a,Z: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ Z @ ( add_a_b @ r @ X3 @ Y2 ) )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ Z @ X3 ) @ ( mult_a_ring_ext_a_b @ r @ Z @ Y2 ) ) ) ) ) ) ).
% r_distr
thf(fact_69_l__distr,axiom,
! [X3: a,Y2: a,Z: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( add_a_b @ r @ X3 @ Y2 ) @ Z )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X3 @ Z ) @ ( mult_a_ring_ext_a_b @ r @ Y2 @ Z ) ) ) ) ) ) ).
% l_distr
thf(fact_70_pow__mult__distrib,axiom,
! [X3: a,Y2: a,N: nat] :
( ( ( mult_a_ring_ext_a_b @ r @ X3 @ Y2 )
= ( mult_a_ring_ext_a_b @ r @ Y2 @ X3 ) )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ ( mult_a_ring_ext_a_b @ r @ X3 @ Y2 ) @ N )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X3 @ N ) @ ( pow_a_1026414303147256608_b_nat @ r @ Y2 @ N ) ) ) ) ) ) ).
% pow_mult_distrib
thf(fact_71_nat__pow__distrib,axiom,
! [X3: a,Y2: a,N: nat] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ ( mult_a_ring_ext_a_b @ r @ X3 @ Y2 ) @ N )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X3 @ N ) @ ( pow_a_1026414303147256608_b_nat @ r @ Y2 @ N ) ) ) ) ) ).
% nat_pow_distrib
thf(fact_72_nat__pow__comm,axiom,
! [X3: a,N: nat,M: nat] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X3 @ N ) @ ( pow_a_1026414303147256608_b_nat @ r @ X3 @ M ) )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X3 @ M ) @ ( pow_a_1026414303147256608_b_nat @ r @ X3 @ N ) ) ) ) ).
% nat_pow_comm
thf(fact_73_group__commutes__pow,axiom,
! [X3: a,Y2: a,N: nat] :
( ( ( mult_a_ring_ext_a_b @ r @ X3 @ Y2 )
= ( mult_a_ring_ext_a_b @ r @ Y2 @ X3 ) )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X3 @ N ) @ Y2 )
= ( mult_a_ring_ext_a_b @ r @ Y2 @ ( pow_a_1026414303147256608_b_nat @ r @ X3 @ N ) ) ) ) ) ) ).
% group_commutes_pow
thf(fact_74_Union__iff,axiom,
! [A: set_set_nat,C2: set_set_set_set_nat] :
( ( member_set_set_nat @ A @ ( comple6569609367425551173et_nat @ C2 ) )
= ( ? [X2: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X2 @ C2 )
& ( member_set_set_nat @ A @ X2 ) ) ) ) ).
% Union_iff
thf(fact_75_Union__iff,axiom,
! [A: set_nat,C2: set_set_set_nat] :
( ( member_set_nat @ A @ ( comple548664676211718543et_nat @ C2 ) )
= ( ? [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ C2 )
& ( member_set_nat @ A @ X2 ) ) ) ) ).
% Union_iff
thf(fact_76_Union__iff,axiom,
! [A: a,C2: set_set_a] :
( ( member_a @ A @ ( comple2307003609928055243_set_a @ C2 ) )
= ( ? [X2: set_a] :
( ( member_set_a @ X2 @ C2 )
& ( member_a @ A @ X2 ) ) ) ) ).
% Union_iff
thf(fact_77_Union__iff,axiom,
! [A: nat > a,C2: set_set_nat_a] :
( ( member_nat_a @ A @ ( comple3545767860446109490_nat_a @ C2 ) )
= ( ? [X2: set_nat_a] :
( ( member_set_nat_a2 @ X2 @ C2 )
& ( member_nat_a @ A @ X2 ) ) ) ) ).
% Union_iff
thf(fact_78_Union__iff,axiom,
! [A: nat,C2: set_set_nat] :
( ( member_nat @ A @ ( comple7399068483239264473et_nat @ C2 ) )
= ( ? [X2: set_nat] :
( ( member_set_nat @ X2 @ C2 )
& ( member_nat @ A @ X2 ) ) ) ) ).
% Union_iff
thf(fact_79_Union__iff,axiom,
! [A: set_a,C2: set_set_set_a] :
( ( member_set_a @ A @ ( comple3958522678809307947_set_a @ C2 ) )
= ( ? [X2: set_set_a] :
( ( member_set_set_a @ X2 @ C2 )
& ( member_set_a @ A @ X2 ) ) ) ) ).
% Union_iff
thf(fact_80_UnionI,axiom,
! [X4: set_set_set_nat,C2: set_set_set_set_nat,A: set_set_nat] :
( ( member2946998982187404937et_nat @ X4 @ C2 )
=> ( ( member_set_set_nat @ A @ X4 )
=> ( member_set_set_nat @ A @ ( comple6569609367425551173et_nat @ C2 ) ) ) ) ).
% UnionI
thf(fact_81_UnionI,axiom,
! [X4: set_set_nat,C2: set_set_set_nat,A: set_nat] :
( ( member_set_set_nat @ X4 @ C2 )
=> ( ( member_set_nat @ A @ X4 )
=> ( member_set_nat @ A @ ( comple548664676211718543et_nat @ C2 ) ) ) ) ).
% UnionI
thf(fact_82_UnionI,axiom,
! [X4: set_a,C2: set_set_a,A: a] :
( ( member_set_a @ X4 @ C2 )
=> ( ( member_a @ A @ X4 )
=> ( member_a @ A @ ( comple2307003609928055243_set_a @ C2 ) ) ) ) ).
% UnionI
thf(fact_83_UnionI,axiom,
! [X4: set_nat_a,C2: set_set_nat_a,A: nat > a] :
( ( member_set_nat_a2 @ X4 @ C2 )
=> ( ( member_nat_a @ A @ X4 )
=> ( member_nat_a @ A @ ( comple3545767860446109490_nat_a @ C2 ) ) ) ) ).
% UnionI
thf(fact_84_UnionI,axiom,
! [X4: set_nat,C2: set_set_nat,A: nat] :
( ( member_set_nat @ X4 @ C2 )
=> ( ( member_nat @ A @ X4 )
=> ( member_nat @ A @ ( comple7399068483239264473et_nat @ C2 ) ) ) ) ).
% UnionI
thf(fact_85_UnionI,axiom,
! [X4: set_set_a,C2: set_set_set_a,A: set_a] :
( ( member_set_set_a @ X4 @ C2 )
=> ( ( member_set_a @ A @ X4 )
=> ( member_set_a @ A @ ( comple3958522678809307947_set_a @ C2 ) ) ) ) ).
% UnionI
thf(fact_86_UN__ball__bex__simps_I1_J,axiom,
! [A: set_set_set_nat,P: set_nat > $o] :
( ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ ( comple548664676211718543et_nat @ A ) )
=> ( P @ X2 ) ) )
= ( ! [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ A )
=> ! [Y: set_nat] :
( ( member_set_nat @ Y @ X2 )
=> ( P @ Y ) ) ) ) ) ).
% UN_ball_bex_simps(1)
thf(fact_87_UN__ball__bex__simps_I1_J,axiom,
! [A: set_set_a,P: a > $o] :
( ( ! [X2: a] :
( ( member_a @ X2 @ ( comple2307003609928055243_set_a @ A ) )
=> ( P @ X2 ) ) )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ! [Y: a] :
( ( member_a @ Y @ X2 )
=> ( P @ Y ) ) ) ) ) ).
% UN_ball_bex_simps(1)
thf(fact_88_UN__ball__bex__simps_I1_J,axiom,
! [A: set_set_nat_a,P: ( nat > a ) > $o] :
( ( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ ( comple3545767860446109490_nat_a @ A ) )
=> ( P @ X2 ) ) )
= ( ! [X2: set_nat_a] :
( ( member_set_nat_a2 @ X2 @ A )
=> ! [Y: nat > a] :
( ( member_nat_a @ Y @ X2 )
=> ( P @ Y ) ) ) ) ) ).
% UN_ball_bex_simps(1)
thf(fact_89_UN__ball__bex__simps_I1_J,axiom,
! [A: set_set_nat,P: nat > $o] :
( ( ! [X2: nat] :
( ( member_nat @ X2 @ ( comple7399068483239264473et_nat @ A ) )
=> ( P @ X2 ) ) )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ! [Y: nat] :
( ( member_nat @ Y @ X2 )
=> ( P @ Y ) ) ) ) ) ).
% UN_ball_bex_simps(1)
thf(fact_90_UN__ball__bex__simps_I1_J,axiom,
! [A: set_set_set_a,P: set_a > $o] :
( ( ! [X2: set_a] :
( ( member_set_a @ X2 @ ( comple3958522678809307947_set_a @ A ) )
=> ( P @ X2 ) ) )
= ( ! [X2: set_set_a] :
( ( member_set_set_a @ X2 @ A )
=> ! [Y: set_a] :
( ( member_set_a @ Y @ X2 )
=> ( P @ Y ) ) ) ) ) ).
% UN_ball_bex_simps(1)
thf(fact_91_UN__ball__bex__simps_I3_J,axiom,
! [A: set_set_set_nat,P: set_nat > $o] :
( ( ? [X2: set_nat] :
( ( member_set_nat @ X2 @ ( comple548664676211718543et_nat @ A ) )
& ( P @ X2 ) ) )
= ( ? [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ A )
& ? [Y: set_nat] :
( ( member_set_nat @ Y @ X2 )
& ( P @ Y ) ) ) ) ) ).
% UN_ball_bex_simps(3)
thf(fact_92_UN__ball__bex__simps_I3_J,axiom,
! [A: set_set_a,P: a > $o] :
( ( ? [X2: a] :
( ( member_a @ X2 @ ( comple2307003609928055243_set_a @ A ) )
& ( P @ X2 ) ) )
= ( ? [X2: set_a] :
( ( member_set_a @ X2 @ A )
& ? [Y: a] :
( ( member_a @ Y @ X2 )
& ( P @ Y ) ) ) ) ) ).
% UN_ball_bex_simps(3)
thf(fact_93_UN__ball__bex__simps_I3_J,axiom,
! [A: set_set_nat_a,P: ( nat > a ) > $o] :
( ( ? [X2: nat > a] :
( ( member_nat_a @ X2 @ ( comple3545767860446109490_nat_a @ A ) )
& ( P @ X2 ) ) )
= ( ? [X2: set_nat_a] :
( ( member_set_nat_a2 @ X2 @ A )
& ? [Y: nat > a] :
( ( member_nat_a @ Y @ X2 )
& ( P @ Y ) ) ) ) ) ).
% UN_ball_bex_simps(3)
thf(fact_94_UN__ball__bex__simps_I3_J,axiom,
! [A: set_set_nat,P: nat > $o] :
( ( ? [X2: nat] :
( ( member_nat @ X2 @ ( comple7399068483239264473et_nat @ A ) )
& ( P @ X2 ) ) )
= ( ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
& ? [Y: nat] :
( ( member_nat @ Y @ X2 )
& ( P @ Y ) ) ) ) ) ).
% UN_ball_bex_simps(3)
thf(fact_95_UN__ball__bex__simps_I3_J,axiom,
! [A: set_set_set_a,P: set_a > $o] :
( ( ? [X2: set_a] :
( ( member_set_a @ X2 @ ( comple3958522678809307947_set_a @ A ) )
& ( P @ X2 ) ) )
= ( ? [X2: set_set_a] :
( ( member_set_set_a @ X2 @ A )
& ? [Y: set_a] :
( ( member_set_a @ Y @ X2 )
& ( P @ Y ) ) ) ) ) ).
% UN_ball_bex_simps(3)
thf(fact_96_atMost__eq__iff,axiom,
! [X3: nat,Y2: nat] :
( ( ( set_ord_atMost_nat @ X3 )
= ( set_ord_atMost_nat @ Y2 ) )
= ( X3 = Y2 ) ) ).
% atMost_eq_iff
thf(fact_97_lessThan__eq__iff,axiom,
! [X3: nat,Y2: nat] :
( ( ( set_ord_lessThan_nat @ X3 )
= ( set_ord_lessThan_nat @ Y2 ) )
= ( X3 = Y2 ) ) ).
% lessThan_eq_iff
thf(fact_98_atMost__iff,axiom,
! [I: set_set_nat,K2: set_set_nat] :
( ( member_set_set_nat @ I @ ( set_or7210490968680142261et_nat @ K2 ) )
= ( ord_le6893508408891458716et_nat @ I @ K2 ) ) ).
% atMost_iff
thf(fact_99_atMost__iff,axiom,
! [I: set_nat,K2: set_nat] :
( ( member_set_nat @ I @ ( set_or4236626031148496127et_nat @ K2 ) )
= ( ord_less_eq_set_nat @ I @ K2 ) ) ).
% atMost_iff
thf(fact_100_atMost__iff,axiom,
! [I: set_a,K2: set_a] :
( ( member_set_a @ I @ ( set_ord_atMost_set_a @ K2 ) )
= ( ord_less_eq_set_a @ I @ K2 ) ) ).
% atMost_iff
thf(fact_101_atMost__iff,axiom,
! [I: set_set_a,K2: set_set_a] :
( ( member_set_set_a @ I @ ( set_or4016371710855203973_set_a @ K2 ) )
= ( ord_le3724670747650509150_set_a @ I @ K2 ) ) ).
% atMost_iff
thf(fact_102_atMost__iff,axiom,
! [I: nat,K2: nat] :
( ( member_nat @ I @ ( set_ord_atMost_nat @ K2 ) )
= ( ord_less_eq_nat @ I @ K2 ) ) ).
% atMost_iff
thf(fact_103_Sup__atMost,axiom,
! [Y2: set_set_nat] :
( ( comple548664676211718543et_nat @ ( set_or7210490968680142261et_nat @ Y2 ) )
= Y2 ) ).
% Sup_atMost
thf(fact_104_Sup__atMost,axiom,
! [Y2: $o] :
( ( complete_Sup_Sup_o @ ( set_ord_atMost_o @ Y2 ) )
= Y2 ) ).
% Sup_atMost
thf(fact_105_Sup__atMost,axiom,
! [Y2: set_a] :
( ( comple2307003609928055243_set_a @ ( set_ord_atMost_set_a @ Y2 ) )
= Y2 ) ).
% Sup_atMost
thf(fact_106_Sup__atMost,axiom,
! [Y2: set_nat_a] :
( ( comple3545767860446109490_nat_a @ ( set_or2677650046130559372_nat_a @ Y2 ) )
= Y2 ) ).
% Sup_atMost
thf(fact_107_Sup__atMost,axiom,
! [Y2: set_nat] :
( ( comple7399068483239264473et_nat @ ( set_or4236626031148496127et_nat @ Y2 ) )
= Y2 ) ).
% Sup_atMost
thf(fact_108_Sup__atMost,axiom,
! [Y2: set_set_a] :
( ( comple3958522678809307947_set_a @ ( set_or4016371710855203973_set_a @ Y2 ) )
= Y2 ) ).
% Sup_atMost
thf(fact_109_atMost__subset__iff,axiom,
! [X3: set_nat,Y2: set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_or4236626031148496127et_nat @ X3 ) @ ( set_or4236626031148496127et_nat @ Y2 ) )
= ( ord_less_eq_set_nat @ X3 @ Y2 ) ) ).
% atMost_subset_iff
thf(fact_110_atMost__subset__iff,axiom,
! [X3: set_set_a,Y2: set_set_a] :
( ( ord_le5722252365846178494_set_a @ ( set_or4016371710855203973_set_a @ X3 ) @ ( set_or4016371710855203973_set_a @ Y2 ) )
= ( ord_le3724670747650509150_set_a @ X3 @ Y2 ) ) ).
% atMost_subset_iff
thf(fact_111_atMost__subset__iff,axiom,
! [X3: set_a,Y2: set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_ord_atMost_set_a @ X3 ) @ ( set_ord_atMost_set_a @ Y2 ) )
= ( ord_less_eq_set_a @ X3 @ Y2 ) ) ).
% atMost_subset_iff
thf(fact_112_atMost__subset__iff,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ X3 ) @ ( set_ord_atMost_nat @ Y2 ) )
= ( ord_less_eq_nat @ X3 @ Y2 ) ) ).
% atMost_subset_iff
thf(fact_113_lessThan__subset__iff,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X3 ) @ ( set_ord_lessThan_nat @ Y2 ) )
= ( ord_less_eq_nat @ X3 @ Y2 ) ) ).
% lessThan_subset_iff
thf(fact_114_a__closed,axiom,
! [X3: a,Y2: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( add_a_b @ r @ X3 @ Y2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% a_closed
thf(fact_115_mem__Collect__eq,axiom,
! [A3: set_a,P: set_a > $o] :
( ( member_set_a @ A3 @ ( collect_set_a @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_116_mem__Collect__eq,axiom,
! [A3: nat > a,P: ( nat > a ) > $o] :
( ( member_nat_a @ A3 @ ( collect_nat_a @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_117_mem__Collect__eq,axiom,
! [A3: a,P: a > $o] :
( ( member_a @ A3 @ ( collect_a @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_118_mem__Collect__eq,axiom,
! [A3: set_set_nat,P: set_set_nat > $o] :
( ( member_set_set_nat @ A3 @ ( collect_set_set_nat @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_119_mem__Collect__eq,axiom,
! [A3: set_nat,P: set_nat > $o] :
( ( member_set_nat @ A3 @ ( collect_set_nat @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_120_mem__Collect__eq,axiom,
! [A3: nat,P: nat > $o] :
( ( member_nat @ A3 @ ( collect_nat @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_121_Collect__mem__eq,axiom,
! [A: set_set_a] :
( ( collect_set_a
@ ^ [X2: set_a] : ( member_set_a @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_122_Collect__mem__eq,axiom,
! [A: set_nat_a] :
( ( collect_nat_a
@ ^ [X2: nat > a] : ( member_nat_a @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_123_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_124_Collect__mem__eq,axiom,
! [A: set_set_set_nat] :
( ( collect_set_set_nat
@ ^ [X2: set_set_nat] : ( member_set_set_nat @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_125_Collect__mem__eq,axiom,
! [A: set_set_nat] :
( ( collect_set_nat
@ ^ [X2: set_nat] : ( member_set_nat @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_126_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_127_Collect__cong,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ! [X: set_nat] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_set_nat @ P )
= ( collect_set_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_128_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X: nat] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_129_local_Oadd_Oright__cancel,axiom,
! [X3: a,Y2: a,Z: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ Y2 @ X3 )
= ( add_a_b @ r @ Z @ X3 ) )
= ( Y2 = Z ) ) ) ) ) ).
% local.add.right_cancel
thf(fact_130_m__closed,axiom,
! [X3: a,Y2: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ X3 @ Y2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% m_closed
thf(fact_131_nat__pow__closed,axiom,
! [X3: a,N: nat] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( pow_a_1026414303147256608_b_nat @ r @ X3 @ N ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% nat_pow_closed
thf(fact_132_Sup__subset__mono,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ord_le6893508408891458716et_nat @ ( comple548664676211718543et_nat @ A ) @ ( comple548664676211718543et_nat @ B ) ) ) ).
% Sup_subset_mono
thf(fact_133_Sup__subset__mono,axiom,
! [A: set_o,B: set_o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A ) @ ( complete_Sup_Sup_o @ B ) ) ) ).
% Sup_subset_mono
thf(fact_134_Sup__subset__mono,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A ) @ ( comple2307003609928055243_set_a @ B ) ) ) ).
% Sup_subset_mono
thf(fact_135_Sup__subset__mono,axiom,
! [A: set_set_nat_a,B: set_set_nat_a] :
( ( ord_le2390145808437456709_nat_a @ A @ B )
=> ( ord_le871467723717165285_nat_a @ ( comple3545767860446109490_nat_a @ A ) @ ( comple3545767860446109490_nat_a @ B ) ) ) ).
% Sup_subset_mono
thf(fact_136_Sup__subset__mono,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B ) ) ) ).
% Sup_subset_mono
thf(fact_137_Sup__subset__mono,axiom,
! [A: set_set_set_a,B: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A @ B )
=> ( ord_le3724670747650509150_set_a @ ( comple3958522678809307947_set_a @ A ) @ ( comple3958522678809307947_set_a @ B ) ) ) ).
% Sup_subset_mono
thf(fact_138_Union__mono,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ord_le6893508408891458716et_nat @ ( comple548664676211718543et_nat @ A ) @ ( comple548664676211718543et_nat @ B ) ) ) ).
% Union_mono
thf(fact_139_Union__mono,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A ) @ ( comple2307003609928055243_set_a @ B ) ) ) ).
% Union_mono
thf(fact_140_Union__mono,axiom,
! [A: set_set_nat_a,B: set_set_nat_a] :
( ( ord_le2390145808437456709_nat_a @ A @ B )
=> ( ord_le871467723717165285_nat_a @ ( comple3545767860446109490_nat_a @ A ) @ ( comple3545767860446109490_nat_a @ B ) ) ) ).
% Union_mono
thf(fact_141_Union__mono,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B ) ) ) ).
% Union_mono
thf(fact_142_Union__mono,axiom,
! [A: set_set_set_a,B: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A @ B )
=> ( ord_le3724670747650509150_set_a @ ( comple3958522678809307947_set_a @ A ) @ ( comple3958522678809307947_set_a @ B ) ) ) ).
% Union_mono
thf(fact_143_Sup__set__def,axiom,
( comple6569609367425551173et_nat
= ( ^ [A2: set_set_set_set_nat] :
( collect_set_set_nat
@ ^ [X2: set_set_nat] : ( complete_Sup_Sup_o @ ( image_3488003393078953823_nat_o @ ( member_set_set_nat @ X2 ) @ A2 ) ) ) ) ) ).
% Sup_set_def
thf(fact_144_Sup__set__def,axiom,
( comple548664676211718543et_nat
= ( ^ [A2: set_set_set_nat] :
( collect_set_nat
@ ^ [X2: set_nat] : ( complete_Sup_Sup_o @ ( image_set_set_nat_o @ ( member_set_nat @ X2 ) @ A2 ) ) ) ) ) ).
% Sup_set_def
thf(fact_145_Sup__set__def,axiom,
( comple2307003609928055243_set_a
= ( ^ [A2: set_set_a] :
( collect_a
@ ^ [X2: a] : ( complete_Sup_Sup_o @ ( image_set_a_o @ ( member_a @ X2 ) @ A2 ) ) ) ) ) ).
% Sup_set_def
thf(fact_146_Sup__set__def,axiom,
( comple3545767860446109490_nat_a
= ( ^ [A2: set_set_nat_a] :
( collect_nat_a
@ ^ [X2: nat > a] : ( complete_Sup_Sup_o @ ( image_set_nat_a_o @ ( member_nat_a @ X2 ) @ A2 ) ) ) ) ) ).
% Sup_set_def
thf(fact_147_Sup__set__def,axiom,
( comple7399068483239264473et_nat
= ( ^ [A2: set_set_nat] :
( collect_nat
@ ^ [X2: nat] : ( complete_Sup_Sup_o @ ( image_set_nat_o @ ( member_nat @ X2 ) @ A2 ) ) ) ) ) ).
% Sup_set_def
thf(fact_148_Sup__set__def,axiom,
( comple3958522678809307947_set_a
= ( ^ [A2: set_set_set_a] :
( collect_set_a
@ ^ [X2: set_a] : ( complete_Sup_Sup_o @ ( image_set_set_a_o @ ( member_set_a @ X2 ) @ A2 ) ) ) ) ) ).
% Sup_set_def
thf(fact_149_Sup__upper2,axiom,
! [U: set_set_nat,A: set_set_set_nat,V: set_set_nat] :
( ( member_set_set_nat @ U @ A )
=> ( ( ord_le6893508408891458716et_nat @ V @ U )
=> ( ord_le6893508408891458716et_nat @ V @ ( comple548664676211718543et_nat @ A ) ) ) ) ).
% Sup_upper2
thf(fact_150_Sup__upper2,axiom,
! [U: $o,A: set_o,V: $o] :
( ( member_o @ U @ A )
=> ( ( ord_less_eq_o @ V @ U )
=> ( ord_less_eq_o @ V @ ( complete_Sup_Sup_o @ A ) ) ) ) ).
% Sup_upper2
thf(fact_151_Sup__upper2,axiom,
! [U: set_a,A: set_set_a,V: set_a] :
( ( member_set_a @ U @ A )
=> ( ( ord_less_eq_set_a @ V @ U )
=> ( ord_less_eq_set_a @ V @ ( comple2307003609928055243_set_a @ A ) ) ) ) ).
% Sup_upper2
thf(fact_152_Sup__upper2,axiom,
! [U: set_nat_a,A: set_set_nat_a,V: set_nat_a] :
( ( member_set_nat_a2 @ U @ A )
=> ( ( ord_le871467723717165285_nat_a @ V @ U )
=> ( ord_le871467723717165285_nat_a @ V @ ( comple3545767860446109490_nat_a @ A ) ) ) ) ).
% Sup_upper2
thf(fact_153_Sup__upper2,axiom,
! [U: set_nat,A: set_set_nat,V: set_nat] :
( ( member_set_nat @ U @ A )
=> ( ( ord_less_eq_set_nat @ V @ U )
=> ( ord_less_eq_set_nat @ V @ ( comple7399068483239264473et_nat @ A ) ) ) ) ).
% Sup_upper2
thf(fact_154_Sup__upper2,axiom,
! [U: set_set_a,A: set_set_set_a,V: set_set_a] :
( ( member_set_set_a @ U @ A )
=> ( ( ord_le3724670747650509150_set_a @ V @ U )
=> ( ord_le3724670747650509150_set_a @ V @ ( comple3958522678809307947_set_a @ A ) ) ) ) ).
% Sup_upper2
thf(fact_155_Sup__le__iff,axiom,
! [A: set_set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( comple548664676211718543et_nat @ A ) @ B2 )
= ( ! [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ A )
=> ( ord_le6893508408891458716et_nat @ X2 @ B2 ) ) ) ) ).
% Sup_le_iff
thf(fact_156_Sup__le__iff,axiom,
! [A: set_o,B2: $o] :
( ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A ) @ B2 )
= ( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( ord_less_eq_o @ X2 @ B2 ) ) ) ) ).
% Sup_le_iff
thf(fact_157_Sup__le__iff,axiom,
! [A: set_set_a,B2: set_a] :
( ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A ) @ B2 )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ( ord_less_eq_set_a @ X2 @ B2 ) ) ) ) ).
% Sup_le_iff
thf(fact_158_Sup__le__iff,axiom,
! [A: set_set_nat_a,B2: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ ( comple3545767860446109490_nat_a @ A ) @ B2 )
= ( ! [X2: set_nat_a] :
( ( member_set_nat_a2 @ X2 @ A )
=> ( ord_le871467723717165285_nat_a @ X2 @ B2 ) ) ) ) ).
% Sup_le_iff
thf(fact_159_Sup__le__iff,axiom,
! [A: set_set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ B2 )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( ord_less_eq_set_nat @ X2 @ B2 ) ) ) ) ).
% Sup_le_iff
thf(fact_160_Sup__le__iff,axiom,
! [A: set_set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( comple3958522678809307947_set_a @ A ) @ B2 )
= ( ! [X2: set_set_a] :
( ( member_set_set_a @ X2 @ A )
=> ( ord_le3724670747650509150_set_a @ X2 @ B2 ) ) ) ) ).
% Sup_le_iff
thf(fact_161_Sup__upper,axiom,
! [X3: set_set_nat,A: set_set_set_nat] :
( ( member_set_set_nat @ X3 @ A )
=> ( ord_le6893508408891458716et_nat @ X3 @ ( comple548664676211718543et_nat @ A ) ) ) ).
% Sup_upper
thf(fact_162_Sup__upper,axiom,
! [X3: $o,A: set_o] :
( ( member_o @ X3 @ A )
=> ( ord_less_eq_o @ X3 @ ( complete_Sup_Sup_o @ A ) ) ) ).
% Sup_upper
thf(fact_163_Sup__upper,axiom,
! [X3: set_a,A: set_set_a] :
( ( member_set_a @ X3 @ A )
=> ( ord_less_eq_set_a @ X3 @ ( comple2307003609928055243_set_a @ A ) ) ) ).
% Sup_upper
thf(fact_164_Sup__upper,axiom,
! [X3: set_nat_a,A: set_set_nat_a] :
( ( member_set_nat_a2 @ X3 @ A )
=> ( ord_le871467723717165285_nat_a @ X3 @ ( comple3545767860446109490_nat_a @ A ) ) ) ).
% Sup_upper
thf(fact_165_Sup__upper,axiom,
! [X3: set_nat,A: set_set_nat] :
( ( member_set_nat @ X3 @ A )
=> ( ord_less_eq_set_nat @ X3 @ ( comple7399068483239264473et_nat @ A ) ) ) ).
% Sup_upper
thf(fact_166_Sup__upper,axiom,
! [X3: set_set_a,A: set_set_set_a] :
( ( member_set_set_a @ X3 @ A )
=> ( ord_le3724670747650509150_set_a @ X3 @ ( comple3958522678809307947_set_a @ A ) ) ) ).
% Sup_upper
thf(fact_167_Sup__least,axiom,
! [A: set_set_set_nat,Z: set_set_nat] :
( ! [X: set_set_nat] :
( ( member_set_set_nat @ X @ A )
=> ( ord_le6893508408891458716et_nat @ X @ Z ) )
=> ( ord_le6893508408891458716et_nat @ ( comple548664676211718543et_nat @ A ) @ Z ) ) ).
% Sup_least
thf(fact_168_Sup__least,axiom,
! [A: set_o,Z: $o] :
( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ord_less_eq_o @ X @ Z ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A ) @ Z ) ) ).
% Sup_least
thf(fact_169_Sup__least,axiom,
! [A: set_set_a,Z: set_a] :
( ! [X: set_a] :
( ( member_set_a @ X @ A )
=> ( ord_less_eq_set_a @ X @ Z ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A ) @ Z ) ) ).
% Sup_least
thf(fact_170_Sup__least,axiom,
! [A: set_set_nat_a,Z: set_nat_a] :
( ! [X: set_nat_a] :
( ( member_set_nat_a2 @ X @ A )
=> ( ord_le871467723717165285_nat_a @ X @ Z ) )
=> ( ord_le871467723717165285_nat_a @ ( comple3545767860446109490_nat_a @ A ) @ Z ) ) ).
% Sup_least
thf(fact_171_Sup__least,axiom,
! [A: set_set_nat,Z: set_nat] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ A )
=> ( ord_less_eq_set_nat @ X @ Z ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ Z ) ) ).
% Sup_least
thf(fact_172_Sup__least,axiom,
! [A: set_set_set_a,Z: set_set_a] :
( ! [X: set_set_a] :
( ( member_set_set_a @ X @ A )
=> ( ord_le3724670747650509150_set_a @ X @ Z ) )
=> ( ord_le3724670747650509150_set_a @ ( comple3958522678809307947_set_a @ A ) @ Z ) ) ).
% Sup_least
thf(fact_173_Sup__mono,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ! [A4: set_set_nat] :
( ( member_set_set_nat @ A4 @ A )
=> ? [X5: set_set_nat] :
( ( member_set_set_nat @ X5 @ B )
& ( ord_le6893508408891458716et_nat @ A4 @ X5 ) ) )
=> ( ord_le6893508408891458716et_nat @ ( comple548664676211718543et_nat @ A ) @ ( comple548664676211718543et_nat @ B ) ) ) ).
% Sup_mono
thf(fact_174_Sup__mono,axiom,
! [A: set_o,B: set_o] :
( ! [A4: $o] :
( ( member_o @ A4 @ A )
=> ? [X5: $o] :
( ( member_o @ X5 @ B )
& ( ord_less_eq_o @ A4 @ X5 ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A ) @ ( complete_Sup_Sup_o @ B ) ) ) ).
% Sup_mono
thf(fact_175_Sup__mono,axiom,
! [A: set_set_a,B: set_set_a] :
( ! [A4: set_a] :
( ( member_set_a @ A4 @ A )
=> ? [X5: set_a] :
( ( member_set_a @ X5 @ B )
& ( ord_less_eq_set_a @ A4 @ X5 ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A ) @ ( comple2307003609928055243_set_a @ B ) ) ) ).
% Sup_mono
thf(fact_176_Sup__mono,axiom,
! [A: set_set_nat_a,B: set_set_nat_a] :
( ! [A4: set_nat_a] :
( ( member_set_nat_a2 @ A4 @ A )
=> ? [X5: set_nat_a] :
( ( member_set_nat_a2 @ X5 @ B )
& ( ord_le871467723717165285_nat_a @ A4 @ X5 ) ) )
=> ( ord_le871467723717165285_nat_a @ ( comple3545767860446109490_nat_a @ A ) @ ( comple3545767860446109490_nat_a @ B ) ) ) ).
% Sup_mono
thf(fact_177_Sup__mono,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ! [A4: set_nat] :
( ( member_set_nat @ A4 @ A )
=> ? [X5: set_nat] :
( ( member_set_nat @ X5 @ B )
& ( ord_less_eq_set_nat @ A4 @ X5 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B ) ) ) ).
% Sup_mono
thf(fact_178_Sup__mono,axiom,
! [A: set_set_set_a,B: set_set_set_a] :
( ! [A4: set_set_a] :
( ( member_set_set_a @ A4 @ A )
=> ? [X5: set_set_a] :
( ( member_set_set_a @ X5 @ B )
& ( ord_le3724670747650509150_set_a @ A4 @ X5 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( comple3958522678809307947_set_a @ A ) @ ( comple3958522678809307947_set_a @ B ) ) ) ).
% Sup_mono
thf(fact_179_Sup__eqI,axiom,
! [A: set_set_set_nat,X3: set_set_nat] :
( ! [Y3: set_set_nat] :
( ( member_set_set_nat @ Y3 @ A )
=> ( ord_le6893508408891458716et_nat @ Y3 @ X3 ) )
=> ( ! [Y3: set_set_nat] :
( ! [Z2: set_set_nat] :
( ( member_set_set_nat @ Z2 @ A )
=> ( ord_le6893508408891458716et_nat @ Z2 @ Y3 ) )
=> ( ord_le6893508408891458716et_nat @ X3 @ Y3 ) )
=> ( ( comple548664676211718543et_nat @ A )
= X3 ) ) ) ).
% Sup_eqI
thf(fact_180_Sup__eqI,axiom,
! [A: set_o,X3: $o] :
( ! [Y3: $o] :
( ( member_o @ Y3 @ A )
=> ( ord_less_eq_o @ Y3 @ X3 ) )
=> ( ! [Y3: $o] :
( ! [Z2: $o] :
( ( member_o @ Z2 @ A )
=> ( ord_less_eq_o @ Z2 @ Y3 ) )
=> ( ord_less_eq_o @ X3 @ Y3 ) )
=> ( ( complete_Sup_Sup_o @ A )
= X3 ) ) ) ).
% Sup_eqI
thf(fact_181_Sup__eqI,axiom,
! [A: set_set_a,X3: set_a] :
( ! [Y3: set_a] :
( ( member_set_a @ Y3 @ A )
=> ( ord_less_eq_set_a @ Y3 @ X3 ) )
=> ( ! [Y3: set_a] :
( ! [Z2: set_a] :
( ( member_set_a @ Z2 @ A )
=> ( ord_less_eq_set_a @ Z2 @ Y3 ) )
=> ( ord_less_eq_set_a @ X3 @ Y3 ) )
=> ( ( comple2307003609928055243_set_a @ A )
= X3 ) ) ) ).
% Sup_eqI
thf(fact_182_Sup__eqI,axiom,
! [A: set_set_nat_a,X3: set_nat_a] :
( ! [Y3: set_nat_a] :
( ( member_set_nat_a2 @ Y3 @ A )
=> ( ord_le871467723717165285_nat_a @ Y3 @ X3 ) )
=> ( ! [Y3: set_nat_a] :
( ! [Z2: set_nat_a] :
( ( member_set_nat_a2 @ Z2 @ A )
=> ( ord_le871467723717165285_nat_a @ Z2 @ Y3 ) )
=> ( ord_le871467723717165285_nat_a @ X3 @ Y3 ) )
=> ( ( comple3545767860446109490_nat_a @ A )
= X3 ) ) ) ).
% Sup_eqI
thf(fact_183_Sup__eqI,axiom,
! [A: set_set_nat,X3: set_nat] :
( ! [Y3: set_nat] :
( ( member_set_nat @ Y3 @ A )
=> ( ord_less_eq_set_nat @ Y3 @ X3 ) )
=> ( ! [Y3: set_nat] :
( ! [Z2: set_nat] :
( ( member_set_nat @ Z2 @ A )
=> ( ord_less_eq_set_nat @ Z2 @ Y3 ) )
=> ( ord_less_eq_set_nat @ X3 @ Y3 ) )
=> ( ( comple7399068483239264473et_nat @ A )
= X3 ) ) ) ).
% Sup_eqI
thf(fact_184_Sup__eqI,axiom,
! [A: set_set_set_a,X3: set_set_a] :
( ! [Y3: set_set_a] :
( ( member_set_set_a @ Y3 @ A )
=> ( ord_le3724670747650509150_set_a @ Y3 @ X3 ) )
=> ( ! [Y3: set_set_a] :
( ! [Z2: set_set_a] :
( ( member_set_set_a @ Z2 @ A )
=> ( ord_le3724670747650509150_set_a @ Z2 @ Y3 ) )
=> ( ord_le3724670747650509150_set_a @ X3 @ Y3 ) )
=> ( ( comple3958522678809307947_set_a @ A )
= X3 ) ) ) ).
% Sup_eqI
thf(fact_185_Union__subsetI,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ! [X: set_set_nat] :
( ( member_set_set_nat @ X @ A )
=> ? [Y4: set_set_nat] :
( ( member_set_set_nat @ Y4 @ B )
& ( ord_le6893508408891458716et_nat @ X @ Y4 ) ) )
=> ( ord_le6893508408891458716et_nat @ ( comple548664676211718543et_nat @ A ) @ ( comple548664676211718543et_nat @ B ) ) ) ).
% Union_subsetI
thf(fact_186_Union__subsetI,axiom,
! [A: set_set_a,B: set_set_a] :
( ! [X: set_a] :
( ( member_set_a @ X @ A )
=> ? [Y4: set_a] :
( ( member_set_a @ Y4 @ B )
& ( ord_less_eq_set_a @ X @ Y4 ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A ) @ ( comple2307003609928055243_set_a @ B ) ) ) ).
% Union_subsetI
thf(fact_187_Union__subsetI,axiom,
! [A: set_set_nat_a,B: set_set_nat_a] :
( ! [X: set_nat_a] :
( ( member_set_nat_a2 @ X @ A )
=> ? [Y4: set_nat_a] :
( ( member_set_nat_a2 @ Y4 @ B )
& ( ord_le871467723717165285_nat_a @ X @ Y4 ) ) )
=> ( ord_le871467723717165285_nat_a @ ( comple3545767860446109490_nat_a @ A ) @ ( comple3545767860446109490_nat_a @ B ) ) ) ).
% Union_subsetI
thf(fact_188_Union__subsetI,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ A )
=> ? [Y4: set_nat] :
( ( member_set_nat @ Y4 @ B )
& ( ord_less_eq_set_nat @ X @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B ) ) ) ).
% Union_subsetI
thf(fact_189_Union__subsetI,axiom,
! [A: set_set_set_a,B: set_set_set_a] :
( ! [X: set_set_a] :
( ( member_set_set_a @ X @ A )
=> ? [Y4: set_set_a] :
( ( member_set_set_a @ Y4 @ B )
& ( ord_le3724670747650509150_set_a @ X @ Y4 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( comple3958522678809307947_set_a @ A ) @ ( comple3958522678809307947_set_a @ B ) ) ) ).
% Union_subsetI
thf(fact_190_Union__upper,axiom,
! [B: set_set_nat,A: set_set_set_nat] :
( ( member_set_set_nat @ B @ A )
=> ( ord_le6893508408891458716et_nat @ B @ ( comple548664676211718543et_nat @ A ) ) ) ).
% Union_upper
thf(fact_191_Union__upper,axiom,
! [B: set_a,A: set_set_a] :
( ( member_set_a @ B @ A )
=> ( ord_less_eq_set_a @ B @ ( comple2307003609928055243_set_a @ A ) ) ) ).
% Union_upper
thf(fact_192_Union__upper,axiom,
! [B: set_nat_a,A: set_set_nat_a] :
( ( member_set_nat_a2 @ B @ A )
=> ( ord_le871467723717165285_nat_a @ B @ ( comple3545767860446109490_nat_a @ A ) ) ) ).
% Union_upper
thf(fact_193_Union__upper,axiom,
! [B: set_nat,A: set_set_nat] :
( ( member_set_nat @ B @ A )
=> ( ord_less_eq_set_nat @ B @ ( comple7399068483239264473et_nat @ A ) ) ) ).
% Union_upper
thf(fact_194_Union__upper,axiom,
! [B: set_set_a,A: set_set_set_a] :
( ( member_set_set_a @ B @ A )
=> ( ord_le3724670747650509150_set_a @ B @ ( comple3958522678809307947_set_a @ A ) ) ) ).
% Union_upper
thf(fact_195_Union__least,axiom,
! [A: set_set_set_nat,C2: set_set_nat] :
( ! [X6: set_set_nat] :
( ( member_set_set_nat @ X6 @ A )
=> ( ord_le6893508408891458716et_nat @ X6 @ C2 ) )
=> ( ord_le6893508408891458716et_nat @ ( comple548664676211718543et_nat @ A ) @ C2 ) ) ).
% Union_least
thf(fact_196_Union__least,axiom,
! [A: set_set_a,C2: set_a] :
( ! [X6: set_a] :
( ( member_set_a @ X6 @ A )
=> ( ord_less_eq_set_a @ X6 @ C2 ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A ) @ C2 ) ) ).
% Union_least
thf(fact_197_Union__least,axiom,
! [A: set_set_nat_a,C2: set_nat_a] :
( ! [X6: set_nat_a] :
( ( member_set_nat_a2 @ X6 @ A )
=> ( ord_le871467723717165285_nat_a @ X6 @ C2 ) )
=> ( ord_le871467723717165285_nat_a @ ( comple3545767860446109490_nat_a @ A ) @ C2 ) ) ).
% Union_least
thf(fact_198_Union__least,axiom,
! [A: set_set_nat,C2: set_nat] :
( ! [X6: set_nat] :
( ( member_set_nat @ X6 @ A )
=> ( ord_less_eq_set_nat @ X6 @ C2 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ C2 ) ) ).
% Union_least
thf(fact_199_Union__least,axiom,
! [A: set_set_set_a,C2: set_set_a] :
( ! [X6: set_set_a] :
( ( member_set_set_a @ X6 @ A )
=> ( ord_le3724670747650509150_set_a @ X6 @ C2 ) )
=> ( ord_le3724670747650509150_set_a @ ( comple3958522678809307947_set_a @ A ) @ C2 ) ) ).
% Union_least
thf(fact_200_SUP__subset__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > $o,G: nat > $o] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_less_eq_o @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A ) ) @ ( complete_Sup_Sup_o @ ( image_nat_o @ G @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_201_SUP__subset__mono,axiom,
! [A: set_a,B: set_a,F: a > $o,G: a > $o] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_less_eq_o @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_a_o @ F @ A ) ) @ ( complete_Sup_Sup_o @ ( image_a_o @ G @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_202_SUP__subset__mono,axiom,
! [A: set_set_nat,B: set_set_nat,F: set_nat > $o,G: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ! [X: set_nat] :
( ( member_set_nat @ X @ A )
=> ( ord_less_eq_o @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_set_nat_o @ F @ A ) ) @ ( complete_Sup_Sup_o @ ( image_set_nat_o @ G @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_203_SUP__subset__mono,axiom,
! [A: set_set_a,B: set_set_a,F: set_a > $o,G: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ! [X: set_a] :
( ( member_set_a @ X @ A )
=> ( ord_less_eq_o @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_set_a_o @ F @ A ) ) @ ( complete_Sup_Sup_o @ ( image_set_a_o @ G @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_204_SUP__subset__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > set_a,G: nat > set_a] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_nat_set_a @ F @ A ) ) @ ( comple2307003609928055243_set_a @ ( image_nat_set_a @ G @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_205_SUP__subset__mono,axiom,
! [A: set_a,B: set_a,F: a > set_a,G: a > set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A ) ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ G @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_206_SUP__subset__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > set_nat,G: nat > set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_207_SUP__subset__mono,axiom,
! [A: set_a,B: set_a,F: a > set_nat,G: a > set_nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ G @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_208_SUP__subset__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > set_set_nat,G: nat > set_set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_le6893508408891458716et_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_le6893508408891458716et_nat @ ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ F @ A ) ) @ ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ G @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_209_SUP__subset__mono,axiom,
! [A: set_a,B: set_a,F: a > set_set_nat,G: a > set_set_nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_le6893508408891458716et_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_le6893508408891458716et_nat @ ( comple548664676211718543et_nat @ ( image_a_set_set_nat @ F @ A ) ) @ ( comple548664676211718543et_nat @ ( image_a_set_set_nat @ G @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_210_atMost__def,axiom,
( set_or4236626031148496127et_nat
= ( ^ [U2: set_nat] :
( collect_set_nat
@ ^ [X2: set_nat] : ( ord_less_eq_set_nat @ X2 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_211_atMost__def,axiom,
( set_ord_atMost_set_a
= ( ^ [U2: set_a] :
( collect_set_a
@ ^ [X2: set_a] : ( ord_less_eq_set_a @ X2 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_212_atMost__def,axiom,
( set_or4016371710855203973_set_a
= ( ^ [U2: set_set_a] :
( collect_set_set_a
@ ^ [X2: set_set_a] : ( ord_le3724670747650509150_set_a @ X2 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_213_atMost__def,axiom,
( set_ord_atMost_nat
= ( ^ [U2: nat] :
( collect_nat
@ ^ [X2: nat] : ( ord_less_eq_nat @ X2 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_214_SUP__eq,axiom,
! [A: set_nat,B: set_nat,F: nat > $o,G: nat > $o] :
( ! [I2: nat] :
( ( member_nat @ I2 @ A )
=> ? [X5: nat] :
( ( member_nat @ X5 @ B )
& ( ord_less_eq_o @ ( F @ I2 ) @ ( G @ X5 ) ) ) )
=> ( ! [J: nat] :
( ( member_nat @ J @ B )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A )
& ( ord_less_eq_o @ ( G @ J ) @ ( F @ X5 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_nat_o @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_215_SUP__eq,axiom,
! [A: set_nat,B: set_a,F: nat > $o,G: a > $o] :
( ! [I2: nat] :
( ( member_nat @ I2 @ A )
=> ? [X5: a] :
( ( member_a @ X5 @ B )
& ( ord_less_eq_o @ ( F @ I2 ) @ ( G @ X5 ) ) ) )
=> ( ! [J: a] :
( ( member_a @ J @ B )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A )
& ( ord_less_eq_o @ ( G @ J ) @ ( F @ X5 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_a_o @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_216_SUP__eq,axiom,
! [A: set_a,B: set_nat,F: a > $o,G: nat > $o] :
( ! [I2: a] :
( ( member_a @ I2 @ A )
=> ? [X5: nat] :
( ( member_nat @ X5 @ B )
& ( ord_less_eq_o @ ( F @ I2 ) @ ( G @ X5 ) ) ) )
=> ( ! [J: nat] :
( ( member_nat @ J @ B )
=> ? [X5: a] :
( ( member_a @ X5 @ A )
& ( ord_less_eq_o @ ( G @ J ) @ ( F @ X5 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_a_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_nat_o @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_217_SUP__eq,axiom,
! [A: set_a,B: set_a,F: a > $o,G: a > $o] :
( ! [I2: a] :
( ( member_a @ I2 @ A )
=> ? [X5: a] :
( ( member_a @ X5 @ B )
& ( ord_less_eq_o @ ( F @ I2 ) @ ( G @ X5 ) ) ) )
=> ( ! [J: a] :
( ( member_a @ J @ B )
=> ? [X5: a] :
( ( member_a @ X5 @ A )
& ( ord_less_eq_o @ ( G @ J ) @ ( F @ X5 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_a_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_a_o @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_218_SUP__eq,axiom,
! [A: set_set_nat,B: set_nat,F: set_nat > $o,G: nat > $o] :
( ! [I2: set_nat] :
( ( member_set_nat @ I2 @ A )
=> ? [X5: nat] :
( ( member_nat @ X5 @ B )
& ( ord_less_eq_o @ ( F @ I2 ) @ ( G @ X5 ) ) ) )
=> ( ! [J: nat] :
( ( member_nat @ J @ B )
=> ? [X5: set_nat] :
( ( member_set_nat @ X5 @ A )
& ( ord_less_eq_o @ ( G @ J ) @ ( F @ X5 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_set_nat_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_nat_o @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_219_SUP__eq,axiom,
! [A: set_set_nat,B: set_a,F: set_nat > $o,G: a > $o] :
( ! [I2: set_nat] :
( ( member_set_nat @ I2 @ A )
=> ? [X5: a] :
( ( member_a @ X5 @ B )
& ( ord_less_eq_o @ ( F @ I2 ) @ ( G @ X5 ) ) ) )
=> ( ! [J: a] :
( ( member_a @ J @ B )
=> ? [X5: set_nat] :
( ( member_set_nat @ X5 @ A )
& ( ord_less_eq_o @ ( G @ J ) @ ( F @ X5 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_set_nat_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_a_o @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_220_SUP__eq,axiom,
! [A: set_set_a,B: set_nat,F: set_a > $o,G: nat > $o] :
( ! [I2: set_a] :
( ( member_set_a @ I2 @ A )
=> ? [X5: nat] :
( ( member_nat @ X5 @ B )
& ( ord_less_eq_o @ ( F @ I2 ) @ ( G @ X5 ) ) ) )
=> ( ! [J: nat] :
( ( member_nat @ J @ B )
=> ? [X5: set_a] :
( ( member_set_a @ X5 @ A )
& ( ord_less_eq_o @ ( G @ J ) @ ( F @ X5 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_set_a_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_nat_o @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_221_SUP__eq,axiom,
! [A: set_set_a,B: set_a,F: set_a > $o,G: a > $o] :
( ! [I2: set_a] :
( ( member_set_a @ I2 @ A )
=> ? [X5: a] :
( ( member_a @ X5 @ B )
& ( ord_less_eq_o @ ( F @ I2 ) @ ( G @ X5 ) ) ) )
=> ( ! [J: a] :
( ( member_a @ J @ B )
=> ? [X5: set_a] :
( ( member_set_a @ X5 @ A )
& ( ord_less_eq_o @ ( G @ J ) @ ( F @ X5 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_set_a_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_a_o @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_222_SUP__eq,axiom,
! [A: set_nat,B: set_set_nat,F: nat > $o,G: set_nat > $o] :
( ! [I2: nat] :
( ( member_nat @ I2 @ A )
=> ? [X5: set_nat] :
( ( member_set_nat @ X5 @ B )
& ( ord_less_eq_o @ ( F @ I2 ) @ ( G @ X5 ) ) ) )
=> ( ! [J: set_nat] :
( ( member_set_nat @ J @ B )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A )
& ( ord_less_eq_o @ ( G @ J ) @ ( F @ X5 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_set_nat_o @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_223_SUP__eq,axiom,
! [A: set_nat,B: set_set_a,F: nat > $o,G: set_a > $o] :
( ! [I2: nat] :
( ( member_nat @ I2 @ A )
=> ? [X5: set_a] :
( ( member_set_a @ X5 @ B )
& ( ord_less_eq_o @ ( F @ I2 ) @ ( G @ X5 ) ) ) )
=> ( ! [J: set_a] :
( ( member_set_a @ J @ B )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A )
& ( ord_less_eq_o @ ( G @ J ) @ ( F @ X5 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_set_a_o @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_224_SUP__upper2,axiom,
! [I: nat,A: set_nat,U: $o,F: nat > $o] :
( ( member_nat @ I @ A )
=> ( ( ord_less_eq_o @ U @ ( F @ I ) )
=> ( ord_less_eq_o @ U @ ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_225_SUP__upper2,axiom,
! [I: a,A: set_a,U: $o,F: a > $o] :
( ( member_a @ I @ A )
=> ( ( ord_less_eq_o @ U @ ( F @ I ) )
=> ( ord_less_eq_o @ U @ ( complete_Sup_Sup_o @ ( image_a_o @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_226_SUP__upper2,axiom,
! [I: set_nat,A: set_set_nat,U: $o,F: set_nat > $o] :
( ( member_set_nat @ I @ A )
=> ( ( ord_less_eq_o @ U @ ( F @ I ) )
=> ( ord_less_eq_o @ U @ ( complete_Sup_Sup_o @ ( image_set_nat_o @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_227_SUP__upper2,axiom,
! [I: set_a,A: set_set_a,U: $o,F: set_a > $o] :
( ( member_set_a @ I @ A )
=> ( ( ord_less_eq_o @ U @ ( F @ I ) )
=> ( ord_less_eq_o @ U @ ( complete_Sup_Sup_o @ ( image_set_a_o @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_228_SUP__upper2,axiom,
! [I: nat,A: set_nat,U: set_a,F: nat > set_a] :
( ( member_nat @ I @ A )
=> ( ( ord_less_eq_set_a @ U @ ( F @ I ) )
=> ( ord_less_eq_set_a @ U @ ( comple2307003609928055243_set_a @ ( image_nat_set_a @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_229_SUP__upper2,axiom,
! [I: a,A: set_a,U: set_a,F: a > set_a] :
( ( member_a @ I @ A )
=> ( ( ord_less_eq_set_a @ U @ ( F @ I ) )
=> ( ord_less_eq_set_a @ U @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_230_SUP__upper2,axiom,
! [I: nat,A: set_nat,U: set_nat,F: nat > set_nat] :
( ( member_nat @ I @ A )
=> ( ( ord_less_eq_set_nat @ U @ ( F @ I ) )
=> ( ord_less_eq_set_nat @ U @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_231_SUP__upper2,axiom,
! [I: a,A: set_a,U: set_nat,F: a > set_nat] :
( ( member_a @ I @ A )
=> ( ( ord_less_eq_set_nat @ U @ ( F @ I ) )
=> ( ord_less_eq_set_nat @ U @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_232_SUP__upper2,axiom,
! [I: nat,A: set_nat,U: set_set_nat,F: nat > set_set_nat] :
( ( member_nat @ I @ A )
=> ( ( ord_le6893508408891458716et_nat @ U @ ( F @ I ) )
=> ( ord_le6893508408891458716et_nat @ U @ ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_233_SUP__upper2,axiom,
! [I: a,A: set_a,U: set_set_nat,F: a > set_set_nat] :
( ( member_a @ I @ A )
=> ( ( ord_le6893508408891458716et_nat @ U @ ( F @ I ) )
=> ( ord_le6893508408891458716et_nat @ U @ ( comple548664676211718543et_nat @ ( image_a_set_set_nat @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_234_SUP__le__iff,axiom,
! [F: nat > set_set_nat,A: set_nat,U: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ F @ A ) ) @ U )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_le6893508408891458716et_nat @ ( F @ X2 ) @ U ) ) ) ) ).
% SUP_le_iff
thf(fact_235_SUP__le__iff,axiom,
! [F: a > set_a,A: set_a,U: set_a] :
( ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A ) ) @ U )
= ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ U ) ) ) ) ).
% SUP_le_iff
thf(fact_236_SUP__le__iff,axiom,
! [F: a > set_nat_a,A: set_a,U: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ ( comple3545767860446109490_nat_a @ ( image_a_set_nat_a @ F @ A ) ) @ U )
= ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ord_le871467723717165285_nat_a @ ( F @ X2 ) @ U ) ) ) ) ).
% SUP_le_iff
thf(fact_237_SUP__le__iff,axiom,
! [F: a > set_nat,A: set_a,U: set_nat] :
( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ F @ A ) ) @ U )
= ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ U ) ) ) ) ).
% SUP_le_iff
thf(fact_238_SUP__upper,axiom,
! [I: nat,A: set_nat,F: nat > $o] :
( ( member_nat @ I @ A )
=> ( ord_less_eq_o @ ( F @ I ) @ ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_239_SUP__upper,axiom,
! [I: a,A: set_a,F: a > $o] :
( ( member_a @ I @ A )
=> ( ord_less_eq_o @ ( F @ I ) @ ( complete_Sup_Sup_o @ ( image_a_o @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_240_SUP__upper,axiom,
! [I: set_nat,A: set_set_nat,F: set_nat > $o] :
( ( member_set_nat @ I @ A )
=> ( ord_less_eq_o @ ( F @ I ) @ ( complete_Sup_Sup_o @ ( image_set_nat_o @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_241_SUP__upper,axiom,
! [I: set_a,A: set_set_a,F: set_a > $o] :
( ( member_set_a @ I @ A )
=> ( ord_less_eq_o @ ( F @ I ) @ ( complete_Sup_Sup_o @ ( image_set_a_o @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_242_SUP__upper,axiom,
! [I: nat,A: set_nat,F: nat > set_a] :
( ( member_nat @ I @ A )
=> ( ord_less_eq_set_a @ ( F @ I ) @ ( comple2307003609928055243_set_a @ ( image_nat_set_a @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_243_SUP__upper,axiom,
! [I: a,A: set_a,F: a > set_a] :
( ( member_a @ I @ A )
=> ( ord_less_eq_set_a @ ( F @ I ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_244_SUP__upper,axiom,
! [I: nat,A: set_nat,F: nat > set_nat] :
( ( member_nat @ I @ A )
=> ( ord_less_eq_set_nat @ ( F @ I ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_245_SUP__upper,axiom,
! [I: a,A: set_a,F: a > set_nat] :
( ( member_a @ I @ A )
=> ( ord_less_eq_set_nat @ ( F @ I ) @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_246_SUP__upper,axiom,
! [I: nat,A: set_nat,F: nat > set_set_nat] :
( ( member_nat @ I @ A )
=> ( ord_le6893508408891458716et_nat @ ( F @ I ) @ ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_247_SUP__upper,axiom,
! [I: a,A: set_a,F: a > set_set_nat] :
( ( member_a @ I @ A )
=> ( ord_le6893508408891458716et_nat @ ( F @ I ) @ ( comple548664676211718543et_nat @ ( image_a_set_set_nat @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_248_SUP__mono_H,axiom,
! [F: nat > set_set_nat,G: nat > set_set_nat,A: set_nat] :
( ! [X: nat] : ( ord_le6893508408891458716et_nat @ ( F @ X ) @ ( G @ X ) )
=> ( ord_le6893508408891458716et_nat @ ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ F @ A ) ) @ ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ G @ A ) ) ) ) ).
% SUP_mono'
thf(fact_249_SUP__mono_H,axiom,
! [F: a > set_a,G: a > set_a,A: set_a] :
( ! [X: a] : ( ord_less_eq_set_a @ ( F @ X ) @ ( G @ X ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A ) ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ G @ A ) ) ) ) ).
% SUP_mono'
thf(fact_250_SUP__mono_H,axiom,
! [F: a > set_nat_a,G: a > set_nat_a,A: set_a] :
( ! [X: a] : ( ord_le871467723717165285_nat_a @ ( F @ X ) @ ( G @ X ) )
=> ( ord_le871467723717165285_nat_a @ ( comple3545767860446109490_nat_a @ ( image_a_set_nat_a @ F @ A ) ) @ ( comple3545767860446109490_nat_a @ ( image_a_set_nat_a @ G @ A ) ) ) ) ).
% SUP_mono'
thf(fact_251_SUP__mono_H,axiom,
! [F: a > set_nat,G: a > set_nat,A: set_a] :
( ! [X: a] : ( ord_less_eq_set_nat @ ( F @ X ) @ ( G @ X ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ G @ A ) ) ) ) ).
% SUP_mono'
thf(fact_252_SUP__least,axiom,
! [A: set_nat,F: nat > $o,U: $o] :
( ! [I2: nat] :
( ( member_nat @ I2 @ A )
=> ( ord_less_eq_o @ ( F @ I2 ) @ U ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A ) ) @ U ) ) ).
% SUP_least
thf(fact_253_SUP__least,axiom,
! [A: set_a,F: a > $o,U: $o] :
( ! [I2: a] :
( ( member_a @ I2 @ A )
=> ( ord_less_eq_o @ ( F @ I2 ) @ U ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_a_o @ F @ A ) ) @ U ) ) ).
% SUP_least
thf(fact_254_SUP__least,axiom,
! [A: set_set_nat,F: set_nat > $o,U: $o] :
( ! [I2: set_nat] :
( ( member_set_nat @ I2 @ A )
=> ( ord_less_eq_o @ ( F @ I2 ) @ U ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_set_nat_o @ F @ A ) ) @ U ) ) ).
% SUP_least
thf(fact_255_SUP__least,axiom,
! [A: set_set_a,F: set_a > $o,U: $o] :
( ! [I2: set_a] :
( ( member_set_a @ I2 @ A )
=> ( ord_less_eq_o @ ( F @ I2 ) @ U ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_set_a_o @ F @ A ) ) @ U ) ) ).
% SUP_least
thf(fact_256_SUP__least,axiom,
! [A: set_nat,F: nat > set_a,U: set_a] :
( ! [I2: nat] :
( ( member_nat @ I2 @ A )
=> ( ord_less_eq_set_a @ ( F @ I2 ) @ U ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_nat_set_a @ F @ A ) ) @ U ) ) ).
% SUP_least
thf(fact_257_SUP__least,axiom,
! [A: set_a,F: a > set_a,U: set_a] :
( ! [I2: a] :
( ( member_a @ I2 @ A )
=> ( ord_less_eq_set_a @ ( F @ I2 ) @ U ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A ) ) @ U ) ) ).
% SUP_least
thf(fact_258_SUP__least,axiom,
! [A: set_nat,F: nat > set_nat,U: set_nat] :
( ! [I2: nat] :
( ( member_nat @ I2 @ A )
=> ( ord_less_eq_set_nat @ ( F @ I2 ) @ U ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A ) ) @ U ) ) ).
% SUP_least
thf(fact_259_SUP__least,axiom,
! [A: set_a,F: a > set_nat,U: set_nat] :
( ! [I2: a] :
( ( member_a @ I2 @ A )
=> ( ord_less_eq_set_nat @ ( F @ I2 ) @ U ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ F @ A ) ) @ U ) ) ).
% SUP_least
thf(fact_260_SUP__least,axiom,
! [A: set_nat,F: nat > set_set_nat,U: set_set_nat] :
( ! [I2: nat] :
( ( member_nat @ I2 @ A )
=> ( ord_le6893508408891458716et_nat @ ( F @ I2 ) @ U ) )
=> ( ord_le6893508408891458716et_nat @ ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ F @ A ) ) @ U ) ) ).
% SUP_least
thf(fact_261_SUP__least,axiom,
! [A: set_a,F: a > set_set_nat,U: set_set_nat] :
( ! [I2: a] :
( ( member_a @ I2 @ A )
=> ( ord_le6893508408891458716et_nat @ ( F @ I2 ) @ U ) )
=> ( ord_le6893508408891458716et_nat @ ( comple548664676211718543et_nat @ ( image_a_set_set_nat @ F @ A ) ) @ U ) ) ).
% SUP_least
thf(fact_262_SUP__mono,axiom,
! [A: set_nat,B: set_a,F: nat > set_a,G: a > set_a] :
( ! [N2: nat] :
( ( member_nat @ N2 @ A )
=> ? [X5: a] :
( ( member_a @ X5 @ B )
& ( ord_less_eq_set_a @ ( F @ N2 ) @ ( G @ X5 ) ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_nat_set_a @ F @ A ) ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ G @ B ) ) ) ) ).
% SUP_mono
thf(fact_263_SUP__mono,axiom,
! [A: set_a,B: set_a,F: a > set_a,G: a > set_a] :
( ! [N2: a] :
( ( member_a @ N2 @ A )
=> ? [X5: a] :
( ( member_a @ X5 @ B )
& ( ord_less_eq_set_a @ ( F @ N2 ) @ ( G @ X5 ) ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A ) ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ G @ B ) ) ) ) ).
% SUP_mono
thf(fact_264_SUP__mono,axiom,
! [A: set_nat,B: set_a,F: nat > set_nat,G: a > set_nat] :
( ! [N2: nat] :
( ( member_nat @ N2 @ A )
=> ? [X5: a] :
( ( member_a @ X5 @ B )
& ( ord_less_eq_set_nat @ ( F @ N2 ) @ ( G @ X5 ) ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ G @ B ) ) ) ) ).
% SUP_mono
thf(fact_265_SUP__mono,axiom,
! [A: set_a,B: set_a,F: a > set_nat,G: a > set_nat] :
( ! [N2: a] :
( ( member_a @ N2 @ A )
=> ? [X5: a] :
( ( member_a @ X5 @ B )
& ( ord_less_eq_set_nat @ ( F @ N2 ) @ ( G @ X5 ) ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ G @ B ) ) ) ) ).
% SUP_mono
thf(fact_266_SUP__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > set_set_nat,G: nat > set_set_nat] :
( ! [N2: nat] :
( ( member_nat @ N2 @ A )
=> ? [X5: nat] :
( ( member_nat @ X5 @ B )
& ( ord_le6893508408891458716et_nat @ ( F @ N2 ) @ ( G @ X5 ) ) ) )
=> ( ord_le6893508408891458716et_nat @ ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ F @ A ) ) @ ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ G @ B ) ) ) ) ).
% SUP_mono
thf(fact_267_SUP__mono,axiom,
! [A: set_a,B: set_nat,F: a > set_set_nat,G: nat > set_set_nat] :
( ! [N2: a] :
( ( member_a @ N2 @ A )
=> ? [X5: nat] :
( ( member_nat @ X5 @ B )
& ( ord_le6893508408891458716et_nat @ ( F @ N2 ) @ ( G @ X5 ) ) ) )
=> ( ord_le6893508408891458716et_nat @ ( comple548664676211718543et_nat @ ( image_a_set_set_nat @ F @ A ) ) @ ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ G @ B ) ) ) ) ).
% SUP_mono
thf(fact_268_SUP__mono,axiom,
! [A: set_set_nat,B: set_a,F: set_nat > set_a,G: a > set_a] :
( ! [N2: set_nat] :
( ( member_set_nat @ N2 @ A )
=> ? [X5: a] :
( ( member_a @ X5 @ B )
& ( ord_less_eq_set_a @ ( F @ N2 ) @ ( G @ X5 ) ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_set_nat_set_a @ F @ A ) ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ G @ B ) ) ) ) ).
% SUP_mono
thf(fact_269_SUP__mono,axiom,
! [A: set_set_a,B: set_a,F: set_a > set_a,G: a > set_a] :
( ! [N2: set_a] :
( ( member_set_a @ N2 @ A )
=> ? [X5: a] :
( ( member_a @ X5 @ B )
& ( ord_less_eq_set_a @ ( F @ N2 ) @ ( G @ X5 ) ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ F @ A ) ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ G @ B ) ) ) ) ).
% SUP_mono
thf(fact_270_SUP__mono,axiom,
! [A: set_set_nat,B: set_a,F: set_nat > set_nat,G: a > set_nat] :
( ! [N2: set_nat] :
( ( member_set_nat @ N2 @ A )
=> ? [X5: a] :
( ( member_a @ X5 @ B )
& ( ord_less_eq_set_nat @ ( F @ N2 ) @ ( G @ X5 ) ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ G @ B ) ) ) ) ).
% SUP_mono
thf(fact_271_SUP__mono,axiom,
! [A: set_set_a,B: set_a,F: set_a > set_nat,G: a > set_nat] :
( ! [N2: set_a] :
( ( member_set_a @ N2 @ A )
=> ? [X5: a] :
( ( member_a @ X5 @ B )
& ( ord_less_eq_set_nat @ ( F @ N2 ) @ ( G @ X5 ) ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_set_a_set_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ G @ B ) ) ) ) ).
% SUP_mono
thf(fact_272_SUP__eqI,axiom,
! [A: set_nat,F: nat > $o,X3: $o] :
( ! [I2: nat] :
( ( member_nat @ I2 @ A )
=> ( ord_less_eq_o @ ( F @ I2 ) @ X3 ) )
=> ( ! [Y3: $o] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ( ord_less_eq_o @ ( F @ I3 ) @ Y3 ) )
=> ( ord_less_eq_o @ X3 @ Y3 ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_273_SUP__eqI,axiom,
! [A: set_a,F: a > $o,X3: $o] :
( ! [I2: a] :
( ( member_a @ I2 @ A )
=> ( ord_less_eq_o @ ( F @ I2 ) @ X3 ) )
=> ( ! [Y3: $o] :
( ! [I3: a] :
( ( member_a @ I3 @ A )
=> ( ord_less_eq_o @ ( F @ I3 ) @ Y3 ) )
=> ( ord_less_eq_o @ X3 @ Y3 ) )
=> ( ( complete_Sup_Sup_o @ ( image_a_o @ F @ A ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_274_SUP__eqI,axiom,
! [A: set_set_nat,F: set_nat > $o,X3: $o] :
( ! [I2: set_nat] :
( ( member_set_nat @ I2 @ A )
=> ( ord_less_eq_o @ ( F @ I2 ) @ X3 ) )
=> ( ! [Y3: $o] :
( ! [I3: set_nat] :
( ( member_set_nat @ I3 @ A )
=> ( ord_less_eq_o @ ( F @ I3 ) @ Y3 ) )
=> ( ord_less_eq_o @ X3 @ Y3 ) )
=> ( ( complete_Sup_Sup_o @ ( image_set_nat_o @ F @ A ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_275_SUP__eqI,axiom,
! [A: set_set_a,F: set_a > $o,X3: $o] :
( ! [I2: set_a] :
( ( member_set_a @ I2 @ A )
=> ( ord_less_eq_o @ ( F @ I2 ) @ X3 ) )
=> ( ! [Y3: $o] :
( ! [I3: set_a] :
( ( member_set_a @ I3 @ A )
=> ( ord_less_eq_o @ ( F @ I3 ) @ Y3 ) )
=> ( ord_less_eq_o @ X3 @ Y3 ) )
=> ( ( complete_Sup_Sup_o @ ( image_set_a_o @ F @ A ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_276_SUP__eqI,axiom,
! [A: set_nat,F: nat > set_a,X3: set_a] :
( ! [I2: nat] :
( ( member_nat @ I2 @ A )
=> ( ord_less_eq_set_a @ ( F @ I2 ) @ X3 ) )
=> ( ! [Y3: set_a] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ( ord_less_eq_set_a @ ( F @ I3 ) @ Y3 ) )
=> ( ord_less_eq_set_a @ X3 @ Y3 ) )
=> ( ( comple2307003609928055243_set_a @ ( image_nat_set_a @ F @ A ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_277_SUP__eqI,axiom,
! [A: set_a,F: a > set_a,X3: set_a] :
( ! [I2: a] :
( ( member_a @ I2 @ A )
=> ( ord_less_eq_set_a @ ( F @ I2 ) @ X3 ) )
=> ( ! [Y3: set_a] :
( ! [I3: a] :
( ( member_a @ I3 @ A )
=> ( ord_less_eq_set_a @ ( F @ I3 ) @ Y3 ) )
=> ( ord_less_eq_set_a @ X3 @ Y3 ) )
=> ( ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_278_SUP__eqI,axiom,
! [A: set_nat,F: nat > set_nat,X3: set_nat] :
( ! [I2: nat] :
( ( member_nat @ I2 @ A )
=> ( ord_less_eq_set_nat @ ( F @ I2 ) @ X3 ) )
=> ( ! [Y3: set_nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ( ord_less_eq_set_nat @ ( F @ I3 ) @ Y3 ) )
=> ( ord_less_eq_set_nat @ X3 @ Y3 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_279_SUP__eqI,axiom,
! [A: set_a,F: a > set_nat,X3: set_nat] :
( ! [I2: a] :
( ( member_a @ I2 @ A )
=> ( ord_less_eq_set_nat @ ( F @ I2 ) @ X3 ) )
=> ( ! [Y3: set_nat] :
( ! [I3: a] :
( ( member_a @ I3 @ A )
=> ( ord_less_eq_set_nat @ ( F @ I3 ) @ Y3 ) )
=> ( ord_less_eq_set_nat @ X3 @ Y3 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_a_set_nat @ F @ A ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_280_SUP__eqI,axiom,
! [A: set_nat,F: nat > set_set_nat,X3: set_set_nat] :
( ! [I2: nat] :
( ( member_nat @ I2 @ A )
=> ( ord_le6893508408891458716et_nat @ ( F @ I2 ) @ X3 ) )
=> ( ! [Y3: set_set_nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ( ord_le6893508408891458716et_nat @ ( F @ I3 ) @ Y3 ) )
=> ( ord_le6893508408891458716et_nat @ X3 @ Y3 ) )
=> ( ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ F @ A ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_281_SUP__eqI,axiom,
! [A: set_a,F: a > set_set_nat,X3: set_set_nat] :
( ! [I2: a] :
( ( member_a @ I2 @ A )
=> ( ord_le6893508408891458716et_nat @ ( F @ I2 ) @ X3 ) )
=> ( ! [Y3: set_set_nat] :
( ! [I3: a] :
( ( member_a @ I3 @ A )
=> ( ord_le6893508408891458716et_nat @ ( F @ I3 ) @ Y3 ) )
=> ( ord_le6893508408891458716et_nat @ X3 @ Y3 ) )
=> ( ( comple548664676211718543et_nat @ ( image_a_set_set_nat @ F @ A ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_282_UN__subset__iff,axiom,
! [A: nat > set_set_nat,I4: set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ A @ I4 ) ) @ B )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ I4 )
=> ( ord_le6893508408891458716et_nat @ ( A @ X2 ) @ B ) ) ) ) ).
% UN_subset_iff
thf(fact_283_UN__subset__iff,axiom,
! [A: a > set_a,I4: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ A @ I4 ) ) @ B )
= ( ! [X2: a] :
( ( member_a @ X2 @ I4 )
=> ( ord_less_eq_set_a @ ( A @ X2 ) @ B ) ) ) ) ).
% UN_subset_iff
thf(fact_284_UN__subset__iff,axiom,
! [A: a > set_nat_a,I4: set_a,B: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ ( comple3545767860446109490_nat_a @ ( image_a_set_nat_a @ A @ I4 ) ) @ B )
= ( ! [X2: a] :
( ( member_a @ X2 @ I4 )
=> ( ord_le871467723717165285_nat_a @ ( A @ X2 ) @ B ) ) ) ) ).
% UN_subset_iff
thf(fact_285_UN__subset__iff,axiom,
! [A: a > set_nat,I4: set_a,B: set_nat] :
( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ A @ I4 ) ) @ B )
= ( ! [X2: a] :
( ( member_a @ X2 @ I4 )
=> ( ord_less_eq_set_nat @ ( A @ X2 ) @ B ) ) ) ) ).
% UN_subset_iff
thf(fact_286_UN__upper,axiom,
! [A3: nat,A: set_nat,B: nat > set_a] :
( ( member_nat @ A3 @ A )
=> ( ord_less_eq_set_a @ ( B @ A3 ) @ ( comple2307003609928055243_set_a @ ( image_nat_set_a @ B @ A ) ) ) ) ).
% UN_upper
thf(fact_287_UN__upper,axiom,
! [A3: a,A: set_a,B: a > set_a] :
( ( member_a @ A3 @ A )
=> ( ord_less_eq_set_a @ ( B @ A3 ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A ) ) ) ) ).
% UN_upper
thf(fact_288_UN__upper,axiom,
! [A3: nat,A: set_nat,B: nat > set_nat] :
( ( member_nat @ A3 @ A )
=> ( ord_less_eq_set_nat @ ( B @ A3 ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A ) ) ) ) ).
% UN_upper
thf(fact_289_UN__upper,axiom,
! [A3: a,A: set_a,B: a > set_nat] :
( ( member_a @ A3 @ A )
=> ( ord_less_eq_set_nat @ ( B @ A3 ) @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ B @ A ) ) ) ) ).
% UN_upper
thf(fact_290_UN__upper,axiom,
! [A3: nat,A: set_nat,B: nat > set_set_nat] :
( ( member_nat @ A3 @ A )
=> ( ord_le6893508408891458716et_nat @ ( B @ A3 ) @ ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ B @ A ) ) ) ) ).
% UN_upper
thf(fact_291_UN__upper,axiom,
! [A3: a,A: set_a,B: a > set_set_nat] :
( ( member_a @ A3 @ A )
=> ( ord_le6893508408891458716et_nat @ ( B @ A3 ) @ ( comple548664676211718543et_nat @ ( image_a_set_set_nat @ B @ A ) ) ) ) ).
% UN_upper
thf(fact_292_UN__upper,axiom,
! [A3: set_nat,A: set_set_nat,B: set_nat > set_a] :
( ( member_set_nat @ A3 @ A )
=> ( ord_less_eq_set_a @ ( B @ A3 ) @ ( comple2307003609928055243_set_a @ ( image_set_nat_set_a @ B @ A ) ) ) ) ).
% UN_upper
thf(fact_293_UN__upper,axiom,
! [A3: set_a,A: set_set_a,B: set_a > set_a] :
( ( member_set_a @ A3 @ A )
=> ( ord_less_eq_set_a @ ( B @ A3 ) @ ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ B @ A ) ) ) ) ).
% UN_upper
thf(fact_294_UN__upper,axiom,
! [A3: set_nat,A: set_set_nat,B: set_nat > set_nat] :
( ( member_set_nat @ A3 @ A )
=> ( ord_less_eq_set_nat @ ( B @ A3 ) @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ B @ A ) ) ) ) ).
% UN_upper
thf(fact_295_UN__upper,axiom,
! [A3: set_a,A: set_set_a,B: set_a > set_nat] :
( ( member_set_a @ A3 @ A )
=> ( ord_less_eq_set_nat @ ( B @ A3 ) @ ( comple7399068483239264473et_nat @ ( image_set_a_set_nat @ B @ A ) ) ) ) ).
% UN_upper
thf(fact_296_UN__least,axiom,
! [A: set_nat,B: nat > set_a,C2: set_a] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_less_eq_set_a @ ( B @ X ) @ C2 ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_nat_set_a @ B @ A ) ) @ C2 ) ) ).
% UN_least
thf(fact_297_UN__least,axiom,
! [A: set_a,B: a > set_a,C2: set_a] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_less_eq_set_a @ ( B @ X ) @ C2 ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A ) ) @ C2 ) ) ).
% UN_least
thf(fact_298_UN__least,axiom,
! [A: set_nat,B: nat > set_nat,C2: set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_less_eq_set_nat @ ( B @ X ) @ C2 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A ) ) @ C2 ) ) ).
% UN_least
thf(fact_299_UN__least,axiom,
! [A: set_a,B: a > set_nat,C2: set_nat] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_less_eq_set_nat @ ( B @ X ) @ C2 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ B @ A ) ) @ C2 ) ) ).
% UN_least
thf(fact_300_UN__least,axiom,
! [A: set_nat,B: nat > set_set_nat,C2: set_set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_le6893508408891458716et_nat @ ( B @ X ) @ C2 ) )
=> ( ord_le6893508408891458716et_nat @ ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ B @ A ) ) @ C2 ) ) ).
% UN_least
thf(fact_301_UN__least,axiom,
! [A: set_a,B: a > set_set_nat,C2: set_set_nat] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_le6893508408891458716et_nat @ ( B @ X ) @ C2 ) )
=> ( ord_le6893508408891458716et_nat @ ( comple548664676211718543et_nat @ ( image_a_set_set_nat @ B @ A ) ) @ C2 ) ) ).
% UN_least
thf(fact_302_UN__least,axiom,
! [A: set_set_nat,B: set_nat > set_a,C2: set_a] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ A )
=> ( ord_less_eq_set_a @ ( B @ X ) @ C2 ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_set_nat_set_a @ B @ A ) ) @ C2 ) ) ).
% UN_least
thf(fact_303_UN__least,axiom,
! [A: set_set_a,B: set_a > set_a,C2: set_a] :
( ! [X: set_a] :
( ( member_set_a @ X @ A )
=> ( ord_less_eq_set_a @ ( B @ X ) @ C2 ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ B @ A ) ) @ C2 ) ) ).
% UN_least
thf(fact_304_UN__least,axiom,
! [A: set_set_nat,B: set_nat > set_nat,C2: set_nat] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ A )
=> ( ord_less_eq_set_nat @ ( B @ X ) @ C2 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ B @ A ) ) @ C2 ) ) ).
% UN_least
thf(fact_305_UN__least,axiom,
! [A: set_set_a,B: set_a > set_nat,C2: set_nat] :
( ! [X: set_a] :
( ( member_set_a @ X @ A )
=> ( ord_less_eq_set_nat @ ( B @ X ) @ C2 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_set_a_set_nat @ B @ A ) ) @ C2 ) ) ).
% UN_least
thf(fact_306_UN__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > set_a,G: nat > set_a] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_nat_set_a @ F @ A ) ) @ ( comple2307003609928055243_set_a @ ( image_nat_set_a @ G @ B ) ) ) ) ) ).
% UN_mono
thf(fact_307_UN__mono,axiom,
! [A: set_a,B: set_a,F: a > set_a,G: a > set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A ) ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ G @ B ) ) ) ) ) ).
% UN_mono
thf(fact_308_UN__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > set_nat,G: nat > set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B ) ) ) ) ) ).
% UN_mono
thf(fact_309_UN__mono,axiom,
! [A: set_a,B: set_a,F: a > set_nat,G: a > set_nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ G @ B ) ) ) ) ) ).
% UN_mono
thf(fact_310_UN__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > set_set_nat,G: nat > set_set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_le6893508408891458716et_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_le6893508408891458716et_nat @ ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ F @ A ) ) @ ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ G @ B ) ) ) ) ) ).
% UN_mono
thf(fact_311_UN__mono,axiom,
! [A: set_a,B: set_a,F: a > set_set_nat,G: a > set_set_nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_le6893508408891458716et_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_le6893508408891458716et_nat @ ( comple548664676211718543et_nat @ ( image_a_set_set_nat @ F @ A ) ) @ ( comple548664676211718543et_nat @ ( image_a_set_set_nat @ G @ B ) ) ) ) ) ).
% UN_mono
thf(fact_312_UN__mono,axiom,
! [A: set_set_nat,B: set_set_nat,F: set_nat > set_a,G: set_nat > set_a] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ! [X: set_nat] :
( ( member_set_nat @ X @ A )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_set_nat_set_a @ F @ A ) ) @ ( comple2307003609928055243_set_a @ ( image_set_nat_set_a @ G @ B ) ) ) ) ) ).
% UN_mono
thf(fact_313_UN__mono,axiom,
! [A: set_set_a,B: set_set_a,F: set_a > set_a,G: set_a > set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ! [X: set_a] :
( ( member_set_a @ X @ A )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ F @ A ) ) @ ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ G @ B ) ) ) ) ) ).
% UN_mono
thf(fact_314_UN__mono,axiom,
! [A: set_set_nat,B: set_set_nat,F: set_nat > set_nat,G: set_nat > set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ! [X: set_nat] :
( ( member_set_nat @ X @ A )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ G @ B ) ) ) ) ) ).
% UN_mono
thf(fact_315_UN__mono,axiom,
! [A: set_set_a,B: set_set_a,F: set_a > set_nat,G: set_a > set_nat] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ! [X: set_a] :
( ( member_set_a @ X @ A )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_set_a_set_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_set_a_set_nat @ G @ B ) ) ) ) ) ).
% UN_mono
thf(fact_316_Sup_OSUP__cong,axiom,
! [A: set_nat,B: set_nat,C2: nat > set_set_nat,D: nat > set_set_nat,Sup: set_set_set_nat > set_set_nat] :
( ( A = B )
=> ( ! [X: nat] :
( ( member_nat @ X @ B )
=> ( ( C2 @ X )
= ( D @ X ) ) )
=> ( ( Sup @ ( image_2194112158459175443et_nat @ C2 @ A ) )
= ( Sup @ ( image_2194112158459175443et_nat @ D @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_317_Sup_OSUP__cong,axiom,
! [A: set_nat,B: set_nat,C2: nat > nat,D: nat > nat,Sup: set_nat > nat] :
( ( A = B )
=> ( ! [X: nat] :
( ( member_nat @ X @ B )
=> ( ( C2 @ X )
= ( D @ X ) ) )
=> ( ( Sup @ ( image_nat_nat @ C2 @ A ) )
= ( Sup @ ( image_nat_nat @ D @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_318_Sup_OSUP__cong,axiom,
! [A: set_a,B: set_a,C2: a > a,D: a > a,Sup: set_a > a] :
( ( A = B )
=> ( ! [X: a] :
( ( member_a @ X @ B )
=> ( ( C2 @ X )
= ( D @ X ) ) )
=> ( ( Sup @ ( image_a_a @ C2 @ A ) )
= ( Sup @ ( image_a_a @ D @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_319_Sup_OSUP__cong,axiom,
! [A: set_a,B: set_a,C2: a > set_nat,D: a > set_nat,Sup: set_set_nat > set_nat] :
( ( A = B )
=> ( ! [X: a] :
( ( member_a @ X @ B )
=> ( ( C2 @ X )
= ( D @ X ) ) )
=> ( ( Sup @ ( image_a_set_nat @ C2 @ A ) )
= ( Sup @ ( image_a_set_nat @ D @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_320_Sup_OSUP__cong,axiom,
! [A: set_a,B: set_a,C2: a > set_nat_a,D: a > set_nat_a,Sup: set_set_nat_a > set_nat_a] :
( ( A = B )
=> ( ! [X: a] :
( ( member_a @ X @ B )
=> ( ( C2 @ X )
= ( D @ X ) ) )
=> ( ( Sup @ ( image_a_set_nat_a @ C2 @ A ) )
= ( Sup @ ( image_a_set_nat_a @ D @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_321_Sup_OSUP__cong,axiom,
! [A: set_a,B: set_a,C2: a > set_a,D: a > set_a,Sup: set_set_a > set_a] :
( ( A = B )
=> ( ! [X: a] :
( ( member_a @ X @ B )
=> ( ( C2 @ X )
= ( D @ X ) ) )
=> ( ( Sup @ ( image_a_set_a @ C2 @ A ) )
= ( Sup @ ( image_a_set_a @ D @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_322_Sup_OSUP__cong,axiom,
! [A: set_a,B: set_a,C2: a > nat,D: a > nat,Sup: set_nat > nat] :
( ( A = B )
=> ( ! [X: a] :
( ( member_a @ X @ B )
=> ( ( C2 @ X )
= ( D @ X ) ) )
=> ( ( Sup @ ( image_a_nat @ C2 @ A ) )
= ( Sup @ ( image_a_nat @ D @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_323_Sup_OSUP__cong,axiom,
! [A: set_a,B: set_a,C2: a > nat > a,D: a > nat > a,Sup: set_nat_a > nat > a] :
( ( A = B )
=> ( ! [X: a] :
( ( member_a @ X @ B )
=> ( ( C2 @ X )
= ( D @ X ) ) )
=> ( ( Sup @ ( image_a_nat_a @ C2 @ A ) )
= ( Sup @ ( image_a_nat_a @ D @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_324_Inf_OINF__cong,axiom,
! [A: set_nat,B: set_nat,C2: nat > set_set_nat,D: nat > set_set_nat,Inf: set_set_set_nat > set_set_nat] :
( ( A = B )
=> ( ! [X: nat] :
( ( member_nat @ X @ B )
=> ( ( C2 @ X )
= ( D @ X ) ) )
=> ( ( Inf @ ( image_2194112158459175443et_nat @ C2 @ A ) )
= ( Inf @ ( image_2194112158459175443et_nat @ D @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_325_Inf_OINF__cong,axiom,
! [A: set_nat,B: set_nat,C2: nat > nat,D: nat > nat,Inf: set_nat > nat] :
( ( A = B )
=> ( ! [X: nat] :
( ( member_nat @ X @ B )
=> ( ( C2 @ X )
= ( D @ X ) ) )
=> ( ( Inf @ ( image_nat_nat @ C2 @ A ) )
= ( Inf @ ( image_nat_nat @ D @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_326_Inf_OINF__cong,axiom,
! [A: set_a,B: set_a,C2: a > a,D: a > a,Inf: set_a > a] :
( ( A = B )
=> ( ! [X: a] :
( ( member_a @ X @ B )
=> ( ( C2 @ X )
= ( D @ X ) ) )
=> ( ( Inf @ ( image_a_a @ C2 @ A ) )
= ( Inf @ ( image_a_a @ D @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_327_Inf_OINF__cong,axiom,
! [A: set_a,B: set_a,C2: a > set_nat,D: a > set_nat,Inf: set_set_nat > set_nat] :
( ( A = B )
=> ( ! [X: a] :
( ( member_a @ X @ B )
=> ( ( C2 @ X )
= ( D @ X ) ) )
=> ( ( Inf @ ( image_a_set_nat @ C2 @ A ) )
= ( Inf @ ( image_a_set_nat @ D @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_328_Inf_OINF__cong,axiom,
! [A: set_a,B: set_a,C2: a > set_nat_a,D: a > set_nat_a,Inf: set_set_nat_a > set_nat_a] :
( ( A = B )
=> ( ! [X: a] :
( ( member_a @ X @ B )
=> ( ( C2 @ X )
= ( D @ X ) ) )
=> ( ( Inf @ ( image_a_set_nat_a @ C2 @ A ) )
= ( Inf @ ( image_a_set_nat_a @ D @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_329_Inf_OINF__cong,axiom,
! [A: set_a,B: set_a,C2: a > set_a,D: a > set_a,Inf: set_set_a > set_a] :
( ( A = B )
=> ( ! [X: a] :
( ( member_a @ X @ B )
=> ( ( C2 @ X )
= ( D @ X ) ) )
=> ( ( Inf @ ( image_a_set_a @ C2 @ A ) )
= ( Inf @ ( image_a_set_a @ D @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_330_Inf_OINF__cong,axiom,
! [A: set_a,B: set_a,C2: a > nat,D: a > nat,Inf: set_nat > nat] :
( ( A = B )
=> ( ! [X: a] :
( ( member_a @ X @ B )
=> ( ( C2 @ X )
= ( D @ X ) ) )
=> ( ( Inf @ ( image_a_nat @ C2 @ A ) )
= ( Inf @ ( image_a_nat @ D @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_331_Inf_OINF__cong,axiom,
! [A: set_a,B: set_a,C2: a > nat > a,D: a > nat > a,Inf: set_nat_a > nat > a] :
( ( A = B )
=> ( ! [X: a] :
( ( member_a @ X @ B )
=> ( ( C2 @ X )
= ( D @ X ) ) )
=> ( ( Inf @ ( image_a_nat_a @ C2 @ A ) )
= ( Inf @ ( image_a_nat_a @ D @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_332_UnionE,axiom,
! [A: set_set_nat,C2: set_set_set_set_nat] :
( ( member_set_set_nat @ A @ ( comple6569609367425551173et_nat @ C2 ) )
=> ~ ! [X6: set_set_set_nat] :
( ( member_set_set_nat @ A @ X6 )
=> ~ ( member2946998982187404937et_nat @ X6 @ C2 ) ) ) ).
% UnionE
thf(fact_333_UnionE,axiom,
! [A: set_nat,C2: set_set_set_nat] :
( ( member_set_nat @ A @ ( comple548664676211718543et_nat @ C2 ) )
=> ~ ! [X6: set_set_nat] :
( ( member_set_nat @ A @ X6 )
=> ~ ( member_set_set_nat @ X6 @ C2 ) ) ) ).
% UnionE
thf(fact_334_UnionE,axiom,
! [A: a,C2: set_set_a] :
( ( member_a @ A @ ( comple2307003609928055243_set_a @ C2 ) )
=> ~ ! [X6: set_a] :
( ( member_a @ A @ X6 )
=> ~ ( member_set_a @ X6 @ C2 ) ) ) ).
% UnionE
thf(fact_335_UnionE,axiom,
! [A: nat > a,C2: set_set_nat_a] :
( ( member_nat_a @ A @ ( comple3545767860446109490_nat_a @ C2 ) )
=> ~ ! [X6: set_nat_a] :
( ( member_nat_a @ A @ X6 )
=> ~ ( member_set_nat_a2 @ X6 @ C2 ) ) ) ).
% UnionE
thf(fact_336_UnionE,axiom,
! [A: nat,C2: set_set_nat] :
( ( member_nat @ A @ ( comple7399068483239264473et_nat @ C2 ) )
=> ~ ! [X6: set_nat] :
( ( member_nat @ A @ X6 )
=> ~ ( member_set_nat @ X6 @ C2 ) ) ) ).
% UnionE
thf(fact_337_UnionE,axiom,
! [A: set_a,C2: set_set_set_a] :
( ( member_set_a @ A @ ( comple3958522678809307947_set_a @ C2 ) )
=> ~ ! [X6: set_set_a] :
( ( member_set_a @ A @ X6 )
=> ~ ( member_set_set_a @ X6 @ C2 ) ) ) ).
% UnionE
thf(fact_338_Sup_OSUP__identity__eq,axiom,
! [Sup: set_a > a,A: set_a] :
( ( Sup
@ ( image_a_a
@ ^ [X2: a] : X2
@ A ) )
= ( Sup @ A ) ) ).
% Sup.SUP_identity_eq
thf(fact_339_Sup_OSUP__identity__eq,axiom,
! [Sup: set_nat > nat,A: set_nat] :
( ( Sup
@ ( image_nat_nat
@ ^ [X2: nat] : X2
@ A ) )
= ( Sup @ A ) ) ).
% Sup.SUP_identity_eq
thf(fact_340_Inf_OINF__identity__eq,axiom,
! [Inf: set_a > a,A: set_a] :
( ( Inf
@ ( image_a_a
@ ^ [X2: a] : X2
@ A ) )
= ( Inf @ A ) ) ).
% Inf.INF_identity_eq
thf(fact_341_Inf_OINF__identity__eq,axiom,
! [Inf: set_nat > nat,A: set_nat] :
( ( Inf
@ ( image_nat_nat
@ ^ [X2: nat] : X2
@ A ) )
= ( Inf @ A ) ) ).
% Inf.INF_identity_eq
thf(fact_342_SUP__cong,axiom,
! [A: set_nat,B: set_nat,C2: nat > $o,D: nat > $o] :
( ( A = B )
=> ( ! [X: nat] :
( ( member_nat @ X @ B )
=> ( ( C2 @ X )
= ( D @ X ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ C2 @ A ) )
= ( complete_Sup_Sup_o @ ( image_nat_o @ D @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_343_SUP__cong,axiom,
! [A: set_a,B: set_a,C2: a > $o,D: a > $o] :
( ( A = B )
=> ( ! [X: a] :
( ( member_a @ X @ B )
=> ( ( C2 @ X )
= ( D @ X ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_a_o @ C2 @ A ) )
= ( complete_Sup_Sup_o @ ( image_a_o @ D @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_344_SUP__cong,axiom,
! [A: set_nat,B: set_nat,C2: nat > nat,D: nat > nat] :
( ( A = B )
=> ( ! [X: nat] :
( ( member_nat @ X @ B )
=> ( ( C2 @ X )
= ( D @ X ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_nat_nat @ C2 @ A ) )
= ( complete_Sup_Sup_nat @ ( image_nat_nat @ D @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_345_SUP__cong,axiom,
! [A: set_a,B: set_a,C2: a > nat,D: a > nat] :
( ( A = B )
=> ( ! [X: a] :
( ( member_a @ X @ B )
=> ( ( C2 @ X )
= ( D @ X ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_a_nat @ C2 @ A ) )
= ( complete_Sup_Sup_nat @ ( image_a_nat @ D @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_346_SUP__cong,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_nat > $o,D: set_nat > $o] :
( ( A = B )
=> ( ! [X: set_nat] :
( ( member_set_nat @ X @ B )
=> ( ( C2 @ X )
= ( D @ X ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_set_nat_o @ C2 @ A ) )
= ( complete_Sup_Sup_o @ ( image_set_nat_o @ D @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_347_SUP__cong,axiom,
! [A: set_set_a,B: set_set_a,C2: set_a > $o,D: set_a > $o] :
( ( A = B )
=> ( ! [X: set_a] :
( ( member_set_a @ X @ B )
=> ( ( C2 @ X )
= ( D @ X ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_set_a_o @ C2 @ A ) )
= ( complete_Sup_Sup_o @ ( image_set_a_o @ D @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_348_SUP__cong,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_nat > nat,D: set_nat > nat] :
( ( A = B )
=> ( ! [X: set_nat] :
( ( member_set_nat @ X @ B )
=> ( ( C2 @ X )
= ( D @ X ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_set_nat_nat @ C2 @ A ) )
= ( complete_Sup_Sup_nat @ ( image_set_nat_nat @ D @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_349_SUP__cong,axiom,
! [A: set_set_a,B: set_set_a,C2: set_a > nat,D: set_a > nat] :
( ( A = B )
=> ( ! [X: set_a] :
( ( member_set_a @ X @ B )
=> ( ( C2 @ X )
= ( D @ X ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_set_a_nat @ C2 @ A ) )
= ( complete_Sup_Sup_nat @ ( image_set_a_nat @ D @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_350_SUP__cong,axiom,
! [A: set_nat,B: set_nat,C2: nat > set_a,D: nat > set_a] :
( ( A = B )
=> ( ! [X: nat] :
( ( member_nat @ X @ B )
=> ( ( C2 @ X )
= ( D @ X ) ) )
=> ( ( comple2307003609928055243_set_a @ ( image_nat_set_a @ C2 @ A ) )
= ( comple2307003609928055243_set_a @ ( image_nat_set_a @ D @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_351_SUP__cong,axiom,
! [A: set_a,B: set_a,C2: a > set_a,D: a > set_a] :
( ( A = B )
=> ( ! [X: a] :
( ( member_a @ X @ B )
=> ( ( C2 @ X )
= ( D @ X ) ) )
=> ( ( comple2307003609928055243_set_a @ ( image_a_set_a @ C2 @ A ) )
= ( comple2307003609928055243_set_a @ ( image_a_set_a @ D @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_352_SUP__commute,axiom,
! [F: nat > nat > set_set_nat,B: set_nat,A: set_nat] :
( ( comple548664676211718543et_nat
@ ( image_2194112158459175443et_nat
@ ^ [I5: nat] : ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ ( F @ I5 ) @ B ) )
@ A ) )
= ( comple548664676211718543et_nat
@ ( image_2194112158459175443et_nat
@ ^ [J2: nat] :
( comple548664676211718543et_nat
@ ( image_2194112158459175443et_nat
@ ^ [I5: nat] : ( F @ I5 @ J2 )
@ A ) )
@ B ) ) ) ).
% SUP_commute
thf(fact_353_SUP__commute,axiom,
! [F: a > a > set_a,B: set_a,A: set_a] :
( ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [I5: a] : ( comple2307003609928055243_set_a @ ( image_a_set_a @ ( F @ I5 ) @ B ) )
@ A ) )
= ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [J2: a] :
( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [I5: a] : ( F @ I5 @ J2 )
@ A ) )
@ B ) ) ) ).
% SUP_commute
thf(fact_354_SUP__commute,axiom,
! [F: a > a > set_nat_a,B: set_a,A: set_a] :
( ( comple3545767860446109490_nat_a
@ ( image_a_set_nat_a
@ ^ [I5: a] : ( comple3545767860446109490_nat_a @ ( image_a_set_nat_a @ ( F @ I5 ) @ B ) )
@ A ) )
= ( comple3545767860446109490_nat_a
@ ( image_a_set_nat_a
@ ^ [J2: a] :
( comple3545767860446109490_nat_a
@ ( image_a_set_nat_a
@ ^ [I5: a] : ( F @ I5 @ J2 )
@ A ) )
@ B ) ) ) ).
% SUP_commute
thf(fact_355_SUP__commute,axiom,
! [F: a > a > set_nat,B: set_a,A: set_a] :
( ( comple7399068483239264473et_nat
@ ( image_a_set_nat
@ ^ [I5: a] : ( comple7399068483239264473et_nat @ ( image_a_set_nat @ ( F @ I5 ) @ B ) )
@ A ) )
= ( comple7399068483239264473et_nat
@ ( image_a_set_nat
@ ^ [J2: a] :
( comple7399068483239264473et_nat
@ ( image_a_set_nat
@ ^ [I5: a] : ( F @ I5 @ J2 )
@ A ) )
@ B ) ) ) ).
% SUP_commute
thf(fact_356_image__Union,axiom,
! [F: a > a,S: set_set_a] :
( ( image_a_a @ F @ ( comple2307003609928055243_set_a @ S ) )
= ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ ( image_a_a @ F ) @ S ) ) ) ).
% image_Union
thf(fact_357_image__Union,axiom,
! [F: a > nat,S: set_set_a] :
( ( image_a_nat @ F @ ( comple2307003609928055243_set_a @ S ) )
= ( comple7399068483239264473et_nat @ ( image_set_a_set_nat @ ( image_a_nat @ F ) @ S ) ) ) ).
% image_Union
thf(fact_358_image__Union,axiom,
! [F: nat > a,S: set_set_nat] :
( ( image_nat_a @ F @ ( comple7399068483239264473et_nat @ S ) )
= ( comple2307003609928055243_set_a @ ( image_set_nat_set_a @ ( image_nat_a @ F ) @ S ) ) ) ).
% image_Union
thf(fact_359_image__Union,axiom,
! [F: nat > nat,S: set_set_nat] :
( ( image_nat_nat @ F @ ( comple7399068483239264473et_nat @ S ) )
= ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat @ F ) @ S ) ) ) ).
% image_Union
thf(fact_360_image__Union,axiom,
! [F: set_nat > a,S: set_set_set_nat] :
( ( image_set_nat_a @ F @ ( comple548664676211718543et_nat @ S ) )
= ( comple2307003609928055243_set_a @ ( image_6642697982911593691_set_a @ ( image_set_nat_a @ F ) @ S ) ) ) ).
% image_Union
thf(fact_361_image__Union,axiom,
! [F: set_nat > nat,S: set_set_set_nat] :
( ( image_set_nat_nat @ F @ ( comple548664676211718543et_nat @ S ) )
= ( comple7399068483239264473et_nat @ ( image_5842784325960735177et_nat @ ( image_set_nat_nat @ F ) @ S ) ) ) ).
% image_Union
thf(fact_362_image__Union,axiom,
! [F: a > set_nat,S: set_set_a] :
( ( image_a_set_nat @ F @ ( comple2307003609928055243_set_a @ S ) )
= ( comple548664676211718543et_nat @ ( image_8216882647274671445et_nat @ ( image_a_set_nat @ F ) @ S ) ) ) ).
% image_Union
thf(fact_363_image__Union,axiom,
! [F: a > set_a,S: set_set_a] :
( ( image_a_set_a @ F @ ( comple2307003609928055243_set_a @ S ) )
= ( comple3958522678809307947_set_a @ ( image_4955109552351689957_set_a @ ( image_a_set_a @ F ) @ S ) ) ) ).
% image_Union
thf(fact_364_image__Union,axiom,
! [F: nat > set_nat,S: set_set_nat] :
( ( image_nat_set_nat @ F @ ( comple7399068483239264473et_nat @ S ) )
= ( comple548664676211718543et_nat @ ( image_6725021117256019401et_nat @ ( image_nat_set_nat @ F ) @ S ) ) ) ).
% image_Union
thf(fact_365_image__Union,axiom,
! [F: nat > set_a,S: set_set_nat] :
( ( image_nat_set_a @ F @ ( comple7399068483239264473et_nat @ S ) )
= ( comple3958522678809307947_set_a @ ( image_8586572261461758321_set_a @ ( image_nat_set_a @ F ) @ S ) ) ) ).
% image_Union
thf(fact_366_UN__UN__flatten,axiom,
! [C2: a > set_a,B: a > set_a,A: set_a] :
( ( comple2307003609928055243_set_a @ ( image_a_set_a @ C2 @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A ) ) ) )
= ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [Y: a] : ( comple2307003609928055243_set_a @ ( image_a_set_a @ C2 @ ( B @ Y ) ) )
@ A ) ) ) ).
% UN_UN_flatten
thf(fact_367_UN__UN__flatten,axiom,
! [C2: nat > set_a,B: a > set_nat,A: set_a] :
( ( comple2307003609928055243_set_a @ ( image_nat_set_a @ C2 @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ B @ A ) ) ) )
= ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [Y: a] : ( comple2307003609928055243_set_a @ ( image_nat_set_a @ C2 @ ( B @ Y ) ) )
@ A ) ) ) ).
% UN_UN_flatten
thf(fact_368_UN__UN__flatten,axiom,
! [C2: a > set_nat,B: a > set_a,A: set_a] :
( ( comple7399068483239264473et_nat @ ( image_a_set_nat @ C2 @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A ) ) ) )
= ( comple7399068483239264473et_nat
@ ( image_a_set_nat
@ ^ [Y: a] : ( comple7399068483239264473et_nat @ ( image_a_set_nat @ C2 @ ( B @ Y ) ) )
@ A ) ) ) ).
% UN_UN_flatten
thf(fact_369_UN__UN__flatten,axiom,
! [C2: nat > set_nat,B: a > set_nat,A: set_a] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C2 @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ B @ A ) ) ) )
= ( comple7399068483239264473et_nat
@ ( image_a_set_nat
@ ^ [Y: a] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C2 @ ( B @ Y ) ) )
@ A ) ) ) ).
% UN_UN_flatten
thf(fact_370_UN__UN__flatten,axiom,
! [C2: a > set_set_nat,B: nat > set_a,A: set_nat] :
( ( comple548664676211718543et_nat @ ( image_a_set_set_nat @ C2 @ ( comple2307003609928055243_set_a @ ( image_nat_set_a @ B @ A ) ) ) )
= ( comple548664676211718543et_nat
@ ( image_2194112158459175443et_nat
@ ^ [Y: nat] : ( comple548664676211718543et_nat @ ( image_a_set_set_nat @ C2 @ ( B @ Y ) ) )
@ A ) ) ) ).
% UN_UN_flatten
thf(fact_371_UN__UN__flatten,axiom,
! [C2: a > set_set_nat,B: a > set_a,A: set_a] :
( ( comple548664676211718543et_nat @ ( image_a_set_set_nat @ C2 @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A ) ) ) )
= ( comple548664676211718543et_nat
@ ( image_a_set_set_nat
@ ^ [Y: a] : ( comple548664676211718543et_nat @ ( image_a_set_set_nat @ C2 @ ( B @ Y ) ) )
@ A ) ) ) ).
% UN_UN_flatten
thf(fact_372_UN__UN__flatten,axiom,
! [C2: nat > set_set_nat,B: nat > set_nat,A: set_nat] :
( ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ C2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A ) ) ) )
= ( comple548664676211718543et_nat
@ ( image_2194112158459175443et_nat
@ ^ [Y: nat] : ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ C2 @ ( B @ Y ) ) )
@ A ) ) ) ).
% UN_UN_flatten
thf(fact_373_UN__UN__flatten,axiom,
! [C2: nat > set_set_nat,B: a > set_nat,A: set_a] :
( ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ C2 @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ B @ A ) ) ) )
= ( comple548664676211718543et_nat
@ ( image_a_set_set_nat
@ ^ [Y: a] : ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ C2 @ ( B @ Y ) ) )
@ A ) ) ) ).
% UN_UN_flatten
thf(fact_374_UN__UN__flatten,axiom,
! [C2: set_nat > set_a,B: a > set_set_nat,A: set_a] :
( ( comple2307003609928055243_set_a @ ( image_set_nat_set_a @ C2 @ ( comple548664676211718543et_nat @ ( image_a_set_set_nat @ B @ A ) ) ) )
= ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [Y: a] : ( comple2307003609928055243_set_a @ ( image_set_nat_set_a @ C2 @ ( B @ Y ) ) )
@ A ) ) ) ).
% UN_UN_flatten
thf(fact_375_UN__UN__flatten,axiom,
! [C2: set_nat > set_a,B: nat > set_set_nat,A: set_nat] :
( ( comple2307003609928055243_set_a @ ( image_set_nat_set_a @ C2 @ ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ B @ A ) ) ) )
= ( comple2307003609928055243_set_a
@ ( image_nat_set_a
@ ^ [Y: nat] : ( comple2307003609928055243_set_a @ ( image_set_nat_set_a @ C2 @ ( B @ Y ) ) )
@ A ) ) ) ).
% UN_UN_flatten
thf(fact_376_UN__E,axiom,
! [B2: a,B: nat > set_a,A: set_nat] :
( ( member_a @ B2 @ ( comple2307003609928055243_set_a @ ( image_nat_set_a @ B @ A ) ) )
=> ~ ! [X: nat] :
( ( member_nat @ X @ A )
=> ~ ( member_a @ B2 @ ( B @ X ) ) ) ) ).
% UN_E
thf(fact_377_UN__E,axiom,
! [B2: a,B: a > set_a,A: set_a] :
( ( member_a @ B2 @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A ) ) )
=> ~ ! [X: a] :
( ( member_a @ X @ A )
=> ~ ( member_a @ B2 @ ( B @ X ) ) ) ) ).
% UN_E
thf(fact_378_UN__E,axiom,
! [B2: nat,B: nat > set_nat,A: set_nat] :
( ( member_nat @ B2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A ) ) )
=> ~ ! [X: nat] :
( ( member_nat @ X @ A )
=> ~ ( member_nat @ B2 @ ( B @ X ) ) ) ) ).
% UN_E
thf(fact_379_UN__E,axiom,
! [B2: nat,B: a > set_nat,A: set_a] :
( ( member_nat @ B2 @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ B @ A ) ) )
=> ~ ! [X: a] :
( ( member_a @ X @ A )
=> ~ ( member_nat @ B2 @ ( B @ X ) ) ) ) ).
% UN_E
thf(fact_380_UN__E,axiom,
! [B2: set_nat,B: nat > set_set_nat,A: set_nat] :
( ( member_set_nat @ B2 @ ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ B @ A ) ) )
=> ~ ! [X: nat] :
( ( member_nat @ X @ A )
=> ~ ( member_set_nat @ B2 @ ( B @ X ) ) ) ) ).
% UN_E
thf(fact_381_UN__E,axiom,
! [B2: set_nat,B: a > set_set_nat,A: set_a] :
( ( member_set_nat @ B2 @ ( comple548664676211718543et_nat @ ( image_a_set_set_nat @ B @ A ) ) )
=> ~ ! [X: a] :
( ( member_a @ X @ A )
=> ~ ( member_set_nat @ B2 @ ( B @ X ) ) ) ) ).
% UN_E
thf(fact_382_UN__E,axiom,
! [B2: a,B: set_nat > set_a,A: set_set_nat] :
( ( member_a @ B2 @ ( comple2307003609928055243_set_a @ ( image_set_nat_set_a @ B @ A ) ) )
=> ~ ! [X: set_nat] :
( ( member_set_nat @ X @ A )
=> ~ ( member_a @ B2 @ ( B @ X ) ) ) ) ).
% UN_E
thf(fact_383_UN__E,axiom,
! [B2: a,B: set_a > set_a,A: set_set_a] :
( ( member_a @ B2 @ ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ B @ A ) ) )
=> ~ ! [X: set_a] :
( ( member_set_a @ X @ A )
=> ~ ( member_a @ B2 @ ( B @ X ) ) ) ) ).
% UN_E
thf(fact_384_UN__E,axiom,
! [B2: nat,B: set_nat > set_nat,A: set_set_nat] :
( ( member_nat @ B2 @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ B @ A ) ) )
=> ~ ! [X: set_nat] :
( ( member_set_nat @ X @ A )
=> ~ ( member_nat @ B2 @ ( B @ X ) ) ) ) ).
% UN_E
thf(fact_385_UN__E,axiom,
! [B2: nat,B: set_a > set_nat,A: set_set_a] :
( ( member_nat @ B2 @ ( comple7399068483239264473et_nat @ ( image_set_a_set_nat @ B @ A ) ) )
=> ~ ! [X: set_a] :
( ( member_set_a @ X @ A )
=> ~ ( member_nat @ B2 @ ( B @ X ) ) ) ) ).
% UN_E
thf(fact_386_UN__extend__simps_I8_J,axiom,
! [B: a > set_a,A: set_set_a] :
( ( comple2307003609928055243_set_a
@ ( image_set_a_set_a
@ ^ [Y: set_a] : ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ Y ) )
@ A ) )
= ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ ( comple2307003609928055243_set_a @ A ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_387_UN__extend__simps_I8_J,axiom,
! [B: nat > set_a,A: set_set_nat] :
( ( comple2307003609928055243_set_a
@ ( image_set_nat_set_a
@ ^ [Y: set_nat] : ( comple2307003609928055243_set_a @ ( image_nat_set_a @ B @ Y ) )
@ A ) )
= ( comple2307003609928055243_set_a @ ( image_nat_set_a @ B @ ( comple7399068483239264473et_nat @ A ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_388_UN__extend__simps_I8_J,axiom,
! [B: a > set_nat,A: set_set_a] :
( ( comple7399068483239264473et_nat
@ ( image_set_a_set_nat
@ ^ [Y: set_a] : ( comple7399068483239264473et_nat @ ( image_a_set_nat @ B @ Y ) )
@ A ) )
= ( comple7399068483239264473et_nat @ ( image_a_set_nat @ B @ ( comple2307003609928055243_set_a @ A ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_389_UN__extend__simps_I8_J,axiom,
! [B: nat > set_nat,A: set_set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_7916887816326733075et_nat
@ ^ [Y: set_nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ Y ) )
@ A ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ ( comple7399068483239264473et_nat @ A ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_390_UN__extend__simps_I8_J,axiom,
! [B: a > set_set_nat,A: set_set_a] :
( ( comple548664676211718543et_nat
@ ( image_8216882647274671445et_nat
@ ^ [Y: set_a] : ( comple548664676211718543et_nat @ ( image_a_set_set_nat @ B @ Y ) )
@ A ) )
= ( comple548664676211718543et_nat @ ( image_a_set_set_nat @ B @ ( comple2307003609928055243_set_a @ A ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_391_UN__extend__simps_I8_J,axiom,
! [B: nat > set_set_nat,A: set_set_nat] :
( ( comple548664676211718543et_nat
@ ( image_6725021117256019401et_nat
@ ^ [Y: set_nat] : ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ B @ Y ) )
@ A ) )
= ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ B @ ( comple7399068483239264473et_nat @ A ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_392_UN__extend__simps_I8_J,axiom,
! [B: set_nat > set_a,A: set_set_set_nat] :
( ( comple2307003609928055243_set_a
@ ( image_6642697982911593691_set_a
@ ^ [Y: set_set_nat] : ( comple2307003609928055243_set_a @ ( image_set_nat_set_a @ B @ Y ) )
@ A ) )
= ( comple2307003609928055243_set_a @ ( image_set_nat_set_a @ B @ ( comple548664676211718543et_nat @ A ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_393_UN__extend__simps_I8_J,axiom,
! [B: set_a > set_a,A: set_set_set_a] :
( ( comple2307003609928055243_set_a
@ ( image_6061375613820669477_set_a
@ ^ [Y: set_set_a] : ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ B @ Y ) )
@ A ) )
= ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ B @ ( comple3958522678809307947_set_a @ A ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_394_UN__extend__simps_I8_J,axiom,
! [B: set_nat > set_nat,A: set_set_set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_5842784325960735177et_nat
@ ^ [Y: set_set_nat] : ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ B @ Y ) )
@ A ) )
= ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ B @ ( comple548664676211718543et_nat @ A ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_395_UN__extend__simps_I8_J,axiom,
! [B: set_a > set_nat,A: set_set_set_a] :
( ( comple7399068483239264473et_nat
@ ( image_6109939652141935103et_nat
@ ^ [Y: set_set_a] : ( comple7399068483239264473et_nat @ ( image_set_a_set_nat @ B @ Y ) )
@ A ) )
= ( comple7399068483239264473et_nat @ ( image_set_a_set_nat @ B @ ( comple3958522678809307947_set_a @ A ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_396_UN__extend__simps_I9_J,axiom,
! [C2: a > set_a,B: a > set_a,A: set_a] :
( ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [X2: a] : ( comple2307003609928055243_set_a @ ( image_a_set_a @ C2 @ ( B @ X2 ) ) )
@ A ) )
= ( comple2307003609928055243_set_a @ ( image_a_set_a @ C2 @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_397_UN__extend__simps_I9_J,axiom,
! [C2: nat > set_a,B: a > set_nat,A: set_a] :
( ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [X2: a] : ( comple2307003609928055243_set_a @ ( image_nat_set_a @ C2 @ ( B @ X2 ) ) )
@ A ) )
= ( comple2307003609928055243_set_a @ ( image_nat_set_a @ C2 @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ B @ A ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_398_UN__extend__simps_I9_J,axiom,
! [C2: a > set_nat,B: a > set_a,A: set_a] :
( ( comple7399068483239264473et_nat
@ ( image_a_set_nat
@ ^ [X2: a] : ( comple7399068483239264473et_nat @ ( image_a_set_nat @ C2 @ ( B @ X2 ) ) )
@ A ) )
= ( comple7399068483239264473et_nat @ ( image_a_set_nat @ C2 @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_399_UN__extend__simps_I9_J,axiom,
! [C2: nat > set_nat,B: a > set_nat,A: set_a] :
( ( comple7399068483239264473et_nat
@ ( image_a_set_nat
@ ^ [X2: a] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C2 @ ( B @ X2 ) ) )
@ A ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C2 @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ B @ A ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_400_UN__extend__simps_I9_J,axiom,
! [C2: a > set_set_nat,B: a > set_a,A: set_a] :
( ( comple548664676211718543et_nat
@ ( image_a_set_set_nat
@ ^ [X2: a] : ( comple548664676211718543et_nat @ ( image_a_set_set_nat @ C2 @ ( B @ X2 ) ) )
@ A ) )
= ( comple548664676211718543et_nat @ ( image_a_set_set_nat @ C2 @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_401_UN__extend__simps_I9_J,axiom,
! [C2: a > set_set_nat,B: nat > set_a,A: set_nat] :
( ( comple548664676211718543et_nat
@ ( image_2194112158459175443et_nat
@ ^ [X2: nat] : ( comple548664676211718543et_nat @ ( image_a_set_set_nat @ C2 @ ( B @ X2 ) ) )
@ A ) )
= ( comple548664676211718543et_nat @ ( image_a_set_set_nat @ C2 @ ( comple2307003609928055243_set_a @ ( image_nat_set_a @ B @ A ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_402_UN__extend__simps_I9_J,axiom,
! [C2: nat > set_set_nat,B: a > set_nat,A: set_a] :
( ( comple548664676211718543et_nat
@ ( image_a_set_set_nat
@ ^ [X2: a] : ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ C2 @ ( B @ X2 ) ) )
@ A ) )
= ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ C2 @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ B @ A ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_403_UN__extend__simps_I9_J,axiom,
! [C2: nat > set_set_nat,B: nat > set_nat,A: set_nat] :
( ( comple548664676211718543et_nat
@ ( image_2194112158459175443et_nat
@ ^ [X2: nat] : ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ C2 @ ( B @ X2 ) ) )
@ A ) )
= ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ C2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_404_UN__extend__simps_I9_J,axiom,
! [C2: set_nat > set_a,B: nat > set_set_nat,A: set_nat] :
( ( comple2307003609928055243_set_a
@ ( image_nat_set_a
@ ^ [X2: nat] : ( comple2307003609928055243_set_a @ ( image_set_nat_set_a @ C2 @ ( B @ X2 ) ) )
@ A ) )
= ( comple2307003609928055243_set_a @ ( image_set_nat_set_a @ C2 @ ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ B @ A ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_405_UN__extend__simps_I9_J,axiom,
! [C2: set_nat > set_a,B: a > set_set_nat,A: set_a] :
( ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [X2: a] : ( comple2307003609928055243_set_a @ ( image_set_nat_set_a @ C2 @ ( B @ X2 ) ) )
@ A ) )
= ( comple2307003609928055243_set_a @ ( image_set_nat_set_a @ C2 @ ( comple548664676211718543et_nat @ ( image_a_set_set_nat @ B @ A ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_406_image__UN,axiom,
! [F: a > a,B: a > set_a,A: set_a] :
( ( image_a_a @ F @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A ) ) )
= ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [X2: a] : ( image_a_a @ F @ ( B @ X2 ) )
@ A ) ) ) ).
% image_UN
thf(fact_407_image__UN,axiom,
! [F: a > nat,B: a > set_a,A: set_a] :
( ( image_a_nat @ F @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A ) ) )
= ( comple7399068483239264473et_nat
@ ( image_a_set_nat
@ ^ [X2: a] : ( image_a_nat @ F @ ( B @ X2 ) )
@ A ) ) ) ).
% image_UN
thf(fact_408_image__UN,axiom,
! [F: nat > a,B: a > set_nat,A: set_a] :
( ( image_nat_a @ F @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ B @ A ) ) )
= ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [X2: a] : ( image_nat_a @ F @ ( B @ X2 ) )
@ A ) ) ) ).
% image_UN
thf(fact_409_image__UN,axiom,
! [F: nat > nat,B: a > set_nat,A: set_a] :
( ( image_nat_nat @ F @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ B @ A ) ) )
= ( comple7399068483239264473et_nat
@ ( image_a_set_nat
@ ^ [X2: a] : ( image_nat_nat @ F @ ( B @ X2 ) )
@ A ) ) ) ).
% image_UN
thf(fact_410_image__UN,axiom,
! [F: set_nat > a,B: a > set_set_nat,A: set_a] :
( ( image_set_nat_a @ F @ ( comple548664676211718543et_nat @ ( image_a_set_set_nat @ B @ A ) ) )
= ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [X2: a] : ( image_set_nat_a @ F @ ( B @ X2 ) )
@ A ) ) ) ).
% image_UN
thf(fact_411_image__UN,axiom,
! [F: set_nat > a,B: nat > set_set_nat,A: set_nat] :
( ( image_set_nat_a @ F @ ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ B @ A ) ) )
= ( comple2307003609928055243_set_a
@ ( image_nat_set_a
@ ^ [X2: nat] : ( image_set_nat_a @ F @ ( B @ X2 ) )
@ A ) ) ) ).
% image_UN
thf(fact_412_image__UN,axiom,
! [F: set_nat > nat,B: a > set_set_nat,A: set_a] :
( ( image_set_nat_nat @ F @ ( comple548664676211718543et_nat @ ( image_a_set_set_nat @ B @ A ) ) )
= ( comple7399068483239264473et_nat
@ ( image_a_set_nat
@ ^ [X2: a] : ( image_set_nat_nat @ F @ ( B @ X2 ) )
@ A ) ) ) ).
% image_UN
thf(fact_413_image__UN,axiom,
! [F: set_nat > nat,B: nat > set_set_nat,A: set_nat] :
( ( image_set_nat_nat @ F @ ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ B @ A ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X2: nat] : ( image_set_nat_nat @ F @ ( B @ X2 ) )
@ A ) ) ) ).
% image_UN
thf(fact_414_image__UN,axiom,
! [F: a > set_nat,B: nat > set_a,A: set_nat] :
( ( image_a_set_nat @ F @ ( comple2307003609928055243_set_a @ ( image_nat_set_a @ B @ A ) ) )
= ( comple548664676211718543et_nat
@ ( image_2194112158459175443et_nat
@ ^ [X2: nat] : ( image_a_set_nat @ F @ ( B @ X2 ) )
@ A ) ) ) ).
% image_UN
thf(fact_415_image__UN,axiom,
! [F: a > set_nat,B: a > set_a,A: set_a] :
( ( image_a_set_nat @ F @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A ) ) )
= ( comple548664676211718543et_nat
@ ( image_a_set_set_nat
@ ^ [X2: a] : ( image_a_set_nat @ F @ ( B @ X2 ) )
@ A ) ) ) ).
% image_UN
thf(fact_416_UN__extend__simps_I10_J,axiom,
! [B: nat > set_a,F: nat > nat,A: set_nat] :
( ( comple2307003609928055243_set_a
@ ( image_nat_set_a
@ ^ [A5: nat] : ( B @ ( F @ A5 ) )
@ A ) )
= ( comple2307003609928055243_set_a @ ( image_nat_set_a @ B @ ( image_nat_nat @ F @ A ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_417_UN__extend__simps_I10_J,axiom,
! [B: nat > set_a,F: a > nat,A: set_a] :
( ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [A5: a] : ( B @ ( F @ A5 ) )
@ A ) )
= ( comple2307003609928055243_set_a @ ( image_nat_set_a @ B @ ( image_a_nat @ F @ A ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_418_UN__extend__simps_I10_J,axiom,
! [B: a > set_a,F: a > a,A: set_a] :
( ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [A5: a] : ( B @ ( F @ A5 ) )
@ A ) )
= ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ ( image_a_a @ F @ A ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_419_UN__extend__simps_I10_J,axiom,
! [B: nat > set_nat,F: nat > nat,A: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [A5: nat] : ( B @ ( F @ A5 ) )
@ A ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ ( image_nat_nat @ F @ A ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_420_UN__extend__simps_I10_J,axiom,
! [B: nat > set_nat,F: a > nat,A: set_a] :
( ( comple7399068483239264473et_nat
@ ( image_a_set_nat
@ ^ [A5: a] : ( B @ ( F @ A5 ) )
@ A ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ ( image_a_nat @ F @ A ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_421_UN__extend__simps_I10_J,axiom,
! [B: a > set_nat,F: a > a,A: set_a] :
( ( comple7399068483239264473et_nat
@ ( image_a_set_nat
@ ^ [A5: a] : ( B @ ( F @ A5 ) )
@ A ) )
= ( comple7399068483239264473et_nat @ ( image_a_set_nat @ B @ ( image_a_a @ F @ A ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_422_UN__extend__simps_I10_J,axiom,
! [B: a > set_set_nat,F: a > a,A: set_a] :
( ( comple548664676211718543et_nat
@ ( image_a_set_set_nat
@ ^ [A5: a] : ( B @ ( F @ A5 ) )
@ A ) )
= ( comple548664676211718543et_nat @ ( image_a_set_set_nat @ B @ ( image_a_a @ F @ A ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_423_UN__extend__simps_I10_J,axiom,
! [B: nat > set_set_nat,F: a > nat,A: set_a] :
( ( comple548664676211718543et_nat
@ ( image_a_set_set_nat
@ ^ [A5: a] : ( B @ ( F @ A5 ) )
@ A ) )
= ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ B @ ( image_a_nat @ F @ A ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_424_UN__extend__simps_I10_J,axiom,
! [B: nat > set_set_nat,F: nat > nat,A: set_nat] :
( ( comple548664676211718543et_nat
@ ( image_2194112158459175443et_nat
@ ^ [A5: nat] : ( B @ ( F @ A5 ) )
@ A ) )
= ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ B @ ( image_nat_nat @ F @ A ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_425_UN__extend__simps_I10_J,axiom,
! [B: set_nat > set_a,F: a > set_nat,A: set_a] :
( ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [A5: a] : ( B @ ( F @ A5 ) )
@ A ) )
= ( comple2307003609928055243_set_a @ ( image_set_nat_set_a @ B @ ( image_a_set_nat @ F @ A ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_426_SUP__UNION,axiom,
! [F: a > $o,G: a > set_a,A: set_a] :
( ( complete_Sup_Sup_o @ ( image_a_o @ F @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ G @ A ) ) ) )
= ( complete_Sup_Sup_o
@ ( image_a_o
@ ^ [Y: a] : ( complete_Sup_Sup_o @ ( image_a_o @ F @ ( G @ Y ) ) )
@ A ) ) ) ).
% SUP_UNION
thf(fact_427_SUP__UNION,axiom,
! [F: nat > $o,G: a > set_nat,A: set_a] :
( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ G @ A ) ) ) )
= ( complete_Sup_Sup_o
@ ( image_a_o
@ ^ [Y: a] : ( complete_Sup_Sup_o @ ( image_nat_o @ F @ ( G @ Y ) ) )
@ A ) ) ) ).
% SUP_UNION
thf(fact_428_SUP__UNION,axiom,
! [F: set_nat > $o,G: nat > set_set_nat,A: set_nat] :
( ( complete_Sup_Sup_o @ ( image_set_nat_o @ F @ ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ G @ A ) ) ) )
= ( complete_Sup_Sup_o
@ ( image_nat_o
@ ^ [Y: nat] : ( complete_Sup_Sup_o @ ( image_set_nat_o @ F @ ( G @ Y ) ) )
@ A ) ) ) ).
% SUP_UNION
thf(fact_429_SUP__UNION,axiom,
! [F: a > set_a,G: a > set_a,A: set_a] :
( ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ G @ A ) ) ) )
= ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [Y: a] : ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ ( G @ Y ) ) )
@ A ) ) ) ).
% SUP_UNION
thf(fact_430_SUP__UNION,axiom,
! [F: nat > set_a,G: a > set_nat,A: set_a] :
( ( comple2307003609928055243_set_a @ ( image_nat_set_a @ F @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ G @ A ) ) ) )
= ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [Y: a] : ( comple2307003609928055243_set_a @ ( image_nat_set_a @ F @ ( G @ Y ) ) )
@ A ) ) ) ).
% SUP_UNION
thf(fact_431_SUP__UNION,axiom,
! [F: a > set_nat,G: a > set_a,A: set_a] :
( ( comple7399068483239264473et_nat @ ( image_a_set_nat @ F @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ G @ A ) ) ) )
= ( comple7399068483239264473et_nat
@ ( image_a_set_nat
@ ^ [Y: a] : ( comple7399068483239264473et_nat @ ( image_a_set_nat @ F @ ( G @ Y ) ) )
@ A ) ) ) ).
% SUP_UNION
thf(fact_432_SUP__UNION,axiom,
! [F: nat > set_nat,G: a > set_nat,A: set_a] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ G @ A ) ) ) )
= ( comple7399068483239264473et_nat
@ ( image_a_set_nat
@ ^ [Y: a] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ ( G @ Y ) ) )
@ A ) ) ) ).
% SUP_UNION
thf(fact_433_SUP__UNION,axiom,
! [F: a > set_set_nat,G: nat > set_a,A: set_nat] :
( ( comple548664676211718543et_nat @ ( image_a_set_set_nat @ F @ ( comple2307003609928055243_set_a @ ( image_nat_set_a @ G @ A ) ) ) )
= ( comple548664676211718543et_nat
@ ( image_2194112158459175443et_nat
@ ^ [Y: nat] : ( comple548664676211718543et_nat @ ( image_a_set_set_nat @ F @ ( G @ Y ) ) )
@ A ) ) ) ).
% SUP_UNION
thf(fact_434_SUP__UNION,axiom,
! [F: a > set_set_nat,G: a > set_a,A: set_a] :
( ( comple548664676211718543et_nat @ ( image_a_set_set_nat @ F @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ G @ A ) ) ) )
= ( comple548664676211718543et_nat
@ ( image_a_set_set_nat
@ ^ [Y: a] : ( comple548664676211718543et_nat @ ( image_a_set_set_nat @ F @ ( G @ Y ) ) )
@ A ) ) ) ).
% SUP_UNION
thf(fact_435_SUP__UNION,axiom,
! [F: nat > set_set_nat,G: nat > set_nat,A: set_nat] :
( ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ F @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ A ) ) ) )
= ( comple548664676211718543et_nat
@ ( image_2194112158459175443et_nat
@ ^ [Y: nat] : ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ F @ ( G @ Y ) ) )
@ A ) ) ) ).
% SUP_UNION
thf(fact_436_onepideal,axiom,
principalideal_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% onepideal
thf(fact_437_monoid__cancelI,axiom,
( ! [A4: a,B3: a,C3: a] :
( ( ( mult_a_ring_ext_a_b @ r @ C3 @ A4 )
= ( mult_a_ring_ext_a_b @ r @ C3 @ B3 ) )
=> ( ( member_a @ A4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A4 = B3 ) ) ) ) )
=> ( ! [A4: a,B3: a,C3: a] :
( ( ( mult_a_ring_ext_a_b @ r @ A4 @ C3 )
= ( mult_a_ring_ext_a_b @ r @ B3 @ C3 ) )
=> ( ( member_a @ A4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A4 = B3 ) ) ) ) )
=> ( monoid5798828371819920185xt_a_b @ r ) ) ) ).
% monoid_cancelI
thf(fact_438_Pow__iff,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( member_set_set_nat @ A @ ( pow_set_nat @ B ) )
= ( ord_le6893508408891458716et_nat @ A @ B ) ) ).
% Pow_iff
thf(fact_439_Pow__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( member_set_nat @ A @ ( pow_nat @ B ) )
= ( ord_less_eq_set_nat @ A @ B ) ) ).
% Pow_iff
thf(fact_440_Pow__iff,axiom,
! [A: set_a,B: set_a] :
( ( member_set_a @ A @ ( pow_a @ B ) )
= ( ord_less_eq_set_a @ A @ B ) ) ).
% Pow_iff
thf(fact_441_Pow__iff,axiom,
! [A: set_set_a,B: set_set_a] :
( ( member_set_set_a @ A @ ( pow_set_a @ B ) )
= ( ord_le3724670747650509150_set_a @ A @ B ) ) ).
% Pow_iff
thf(fact_442_PowI,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( member_set_set_nat @ A @ ( pow_set_nat @ B ) ) ) ).
% PowI
thf(fact_443_PowI,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( member_set_nat @ A @ ( pow_nat @ B ) ) ) ).
% PowI
thf(fact_444_PowI,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( member_set_a @ A @ ( pow_a @ B ) ) ) ).
% PowI
thf(fact_445_PowI,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( member_set_set_a @ A @ ( pow_set_a @ B ) ) ) ).
% PowI
thf(fact_446_cSup__atMost,axiom,
! [X3: set_set_nat] :
( ( comple548664676211718543et_nat @ ( set_or7210490968680142261et_nat @ X3 ) )
= X3 ) ).
% cSup_atMost
thf(fact_447_cSup__atMost,axiom,
! [X3: $o] :
( ( complete_Sup_Sup_o @ ( set_ord_atMost_o @ X3 ) )
= X3 ) ).
% cSup_atMost
thf(fact_448_cSup__atMost,axiom,
! [X3: nat] :
( ( complete_Sup_Sup_nat @ ( set_ord_atMost_nat @ X3 ) )
= X3 ) ).
% cSup_atMost
thf(fact_449_cSup__atMost,axiom,
! [X3: set_a] :
( ( comple2307003609928055243_set_a @ ( set_ord_atMost_set_a @ X3 ) )
= X3 ) ).
% cSup_atMost
thf(fact_450_cSup__atMost,axiom,
! [X3: set_nat_a] :
( ( comple3545767860446109490_nat_a @ ( set_or2677650046130559372_nat_a @ X3 ) )
= X3 ) ).
% cSup_atMost
thf(fact_451_cSup__atMost,axiom,
! [X3: set_nat] :
( ( comple7399068483239264473et_nat @ ( set_or4236626031148496127et_nat @ X3 ) )
= X3 ) ).
% cSup_atMost
thf(fact_452_cSup__atMost,axiom,
! [X3: set_set_a] :
( ( comple3958522678809307947_set_a @ ( set_or4016371710855203973_set_a @ X3 ) )
= X3 ) ).
% cSup_atMost
thf(fact_453_up__smult__closed,axiom,
! [A3: a,P2: nat > a] :
( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_nat_a @ P2 @ ( up_a_b @ r ) )
=> ( member_nat_a
@ ^ [I5: nat] : ( mult_a_ring_ext_a_b @ r @ A3 @ ( P2 @ I5 ) )
@ ( up_a_b @ r ) ) ) ) ).
% up_smult_closed
thf(fact_454_cring_Ocring__simprules_I25_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y2: a,Z: a] :
( ( cring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ Z @ ( add_a_b @ R @ X3 @ Y2 ) )
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ Z @ X3 ) @ ( mult_a_ring_ext_a_b @ R @ Z @ Y2 ) ) ) ) ) ) ) ).
% cring.cring_simprules(25)
thf(fact_455_cring_Ocring__simprules_I13_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y2: a,Z: a] :
( ( cring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( add_a_b @ R @ X3 @ Y2 ) @ Z )
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X3 @ Z ) @ ( mult_a_ring_ext_a_b @ R @ Y2 @ Z ) ) ) ) ) ) ) ).
% cring.cring_simprules(13)
thf(fact_456_line__extension__mem__iff,axiom,
! [U: a,K3: set_a,A3: a,E: set_a] :
( ( member_a @ U @ ( embedd971793762689825387on_a_b @ r @ K3 @ A3 @ E ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ K3 )
& ? [Y: a] :
( ( member_a @ Y @ E )
& ( U
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X2 @ A3 ) @ Y ) ) ) ) ) ) ).
% line_extension_mem_iff
thf(fact_457_nat__pow__mult,axiom,
! [X3: a,N: nat,M: nat] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X3 @ N ) @ ( pow_a_1026414303147256608_b_nat @ r @ X3 @ M ) )
= ( pow_a_1026414303147256608_b_nat @ r @ X3 @ ( plus_plus_nat @ N @ M ) ) ) ) ).
% nat_pow_mult
thf(fact_458_nat__pow__Suc2,axiom,
! [X3: a,N: nat] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ X3 @ ( suc @ N ) )
= ( mult_a_ring_ext_a_b @ r @ X3 @ ( pow_a_1026414303147256608_b_nat @ r @ X3 @ N ) ) ) ) ).
% nat_pow_Suc2
thf(fact_459_card__bound,axiom,
! [A: set_nat,N: nat] :
( ( ord_less_eq_set_nat @ A @ ( set_ord_lessThan_nat @ N ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ N ) ) ).
% card_bound
thf(fact_460_line__extension__in__carrier,axiom,
! [K3: set_a,A3: a,E: set_a] :
( ( ord_less_eq_set_a @ K3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ E @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( embedd971793762689825387on_a_b @ r @ K3 @ A3 @ E ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% line_extension_in_carrier
thf(fact_461_up__add__closed,axiom,
! [P2: nat > a,Q2: nat > a] :
( ( member_nat_a @ P2 @ ( up_a_b @ r ) )
=> ( ( member_nat_a @ Q2 @ ( up_a_b @ r ) )
=> ( member_nat_a
@ ^ [I5: nat] : ( add_a_b @ r @ ( P2 @ I5 ) @ ( Q2 @ I5 ) )
@ ( up_a_b @ r ) ) ) ) ).
% up_add_closed
thf(fact_462_subset__antisym,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_463_subset__antisym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_464_subset__antisym,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_465_subsetI,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ A )
=> ( member_set_nat @ X @ B ) )
=> ( ord_le6893508408891458716et_nat @ A @ B ) ) ).
% subsetI
thf(fact_466_subsetI,axiom,
! [A: set_nat_a,B: set_nat_a] :
( ! [X: nat > a] :
( ( member_nat_a @ X @ A )
=> ( member_nat_a @ X @ B ) )
=> ( ord_le871467723717165285_nat_a @ A @ B ) ) ).
% subsetI
thf(fact_467_subsetI,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ! [X: set_set_nat] :
( ( member_set_set_nat @ X @ A )
=> ( member_set_set_nat @ X @ B ) )
=> ( ord_le9131159989063066194et_nat @ A @ B ) ) ).
% subsetI
thf(fact_468_subsetI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member_nat @ X @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% subsetI
thf(fact_469_subsetI,axiom,
! [A: set_a,B: set_a] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( member_a @ X @ B ) )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% subsetI
thf(fact_470_subsetI,axiom,
! [A: set_set_a,B: set_set_a] :
( ! [X: set_a] :
( ( member_set_a @ X @ A )
=> ( member_set_a @ X @ B ) )
=> ( ord_le3724670747650509150_set_a @ A @ B ) ) ).
% subsetI
thf(fact_471_image__eqI,axiom,
! [B2: nat,F: nat > nat,X3: nat,A: set_nat] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_nat @ X3 @ A )
=> ( member_nat @ B2 @ ( image_nat_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_472_image__eqI,axiom,
! [B2: a,F: nat > a,X3: nat,A: set_nat] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_nat @ X3 @ A )
=> ( member_a @ B2 @ ( image_nat_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_473_image__eqI,axiom,
! [B2: nat,F: a > nat,X3: a,A: set_a] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_a @ X3 @ A )
=> ( member_nat @ B2 @ ( image_a_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_474_image__eqI,axiom,
! [B2: a,F: a > a,X3: a,A: set_a] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_a @ X3 @ A )
=> ( member_a @ B2 @ ( image_a_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_475_image__eqI,axiom,
! [B2: nat,F: set_nat > nat,X3: set_nat,A: set_set_nat] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_set_nat @ X3 @ A )
=> ( member_nat @ B2 @ ( image_set_nat_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_476_image__eqI,axiom,
! [B2: a,F: set_nat > a,X3: set_nat,A: set_set_nat] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_set_nat @ X3 @ A )
=> ( member_a @ B2 @ ( image_set_nat_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_477_image__eqI,axiom,
! [B2: nat,F: set_a > nat,X3: set_a,A: set_set_a] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_set_a @ X3 @ A )
=> ( member_nat @ B2 @ ( image_set_a_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_478_image__eqI,axiom,
! [B2: a,F: set_a > a,X3: set_a,A: set_set_a] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_set_a @ X3 @ A )
=> ( member_a @ B2 @ ( image_set_a_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_479_image__eqI,axiom,
! [B2: set_nat,F: nat > set_nat,X3: nat,A: set_nat] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_nat @ X3 @ A )
=> ( member_set_nat @ B2 @ ( image_nat_set_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_480_image__eqI,axiom,
! [B2: set_a,F: nat > set_a,X3: nat,A: set_nat] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_nat @ X3 @ A )
=> ( member_set_a @ B2 @ ( image_nat_set_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_481_image__ident,axiom,
! [Y5: set_a] :
( ( image_a_a
@ ^ [X2: a] : X2
@ Y5 )
= Y5 ) ).
% image_ident
thf(fact_482_image__ident,axiom,
! [Y5: set_nat] :
( ( image_nat_nat
@ ^ [X2: nat] : X2
@ Y5 )
= Y5 ) ).
% image_ident
thf(fact_483_card__atMost,axiom,
! [U: nat] :
( ( finite_card_nat @ ( set_ord_atMost_nat @ U ) )
= ( suc @ U ) ) ).
% card_atMost
thf(fact_484_local_Onat__pow__Suc,axiom,
! [X3: a,N: nat] :
( ( pow_a_1026414303147256608_b_nat @ r @ X3 @ ( suc @ N ) )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X3 @ N ) @ X3 ) ) ).
% local.nat_pow_Suc
thf(fact_485_Sup__bool__def,axiom,
( complete_Sup_Sup_o
= ( member_o @ $true ) ) ).
% Sup_bool_def
thf(fact_486_less__eq__set__def,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A2: set_set_nat,B4: set_set_nat] :
( ord_le3964352015994296041_nat_o
@ ^ [X2: set_nat] : ( member_set_nat @ X2 @ A2 )
@ ^ [X2: set_nat] : ( member_set_nat @ X2 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_487_less__eq__set__def,axiom,
( ord_le871467723717165285_nat_a
= ( ^ [A2: set_nat_a,B4: set_nat_a] :
( ord_less_eq_nat_a_o
@ ^ [X2: nat > a] : ( member_nat_a @ X2 @ A2 )
@ ^ [X2: nat > a] : ( member_nat_a @ X2 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_488_less__eq__set__def,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A2: set_set_set_nat,B4: set_set_set_nat] :
( ord_le3616423863276227763_nat_o
@ ^ [X2: set_set_nat] : ( member_set_set_nat @ X2 @ A2 )
@ ^ [X2: set_set_nat] : ( member_set_set_nat @ X2 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_489_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A2: set_nat,B4: set_nat] :
( ord_less_eq_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A2 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_490_less__eq__set__def,axiom,
( ord_less_eq_set_a
= ( ^ [A2: set_a,B4: set_a] :
( ord_less_eq_a_o
@ ^ [X2: a] : ( member_a @ X2 @ A2 )
@ ^ [X2: a] : ( member_a @ X2 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_491_less__eq__set__def,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A2: set_set_a,B4: set_set_a] :
( ord_less_eq_set_a_o
@ ^ [X2: set_a] : ( member_set_a @ X2 @ A2 )
@ ^ [X2: set_a] : ( member_set_a @ X2 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_492_lessThan__Suc__atMost,axiom,
! [K2: nat] :
( ( set_ord_lessThan_nat @ ( suc @ K2 ) )
= ( set_ord_atMost_nat @ K2 ) ) ).
% lessThan_Suc_atMost
thf(fact_493_subset__Pow__Union,axiom,
! [A: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ A @ ( pow_set_nat @ ( comple548664676211718543et_nat @ A ) ) ) ).
% subset_Pow_Union
thf(fact_494_subset__Pow__Union,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ A @ ( pow_a @ ( comple2307003609928055243_set_a @ A ) ) ) ).
% subset_Pow_Union
thf(fact_495_subset__Pow__Union,axiom,
! [A: set_set_nat_a] : ( ord_le2390145808437456709_nat_a @ A @ ( pow_nat_a @ ( comple3545767860446109490_nat_a @ A ) ) ) ).
% subset_Pow_Union
thf(fact_496_subset__Pow__Union,axiom,
! [A: set_set_nat] : ( ord_le6893508408891458716et_nat @ A @ ( pow_nat @ ( comple7399068483239264473et_nat @ A ) ) ) ).
% subset_Pow_Union
thf(fact_497_subset__Pow__Union,axiom,
! [A: set_set_set_a] : ( ord_le5722252365846178494_set_a @ A @ ( pow_set_a @ ( comple3958522678809307947_set_a @ A ) ) ) ).
% subset_Pow_Union
thf(fact_498_Collect__mono__iff,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) )
= ( ! [X2: set_nat] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_499_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X2: nat] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_500_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X2: a] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_501_Collect__mono__iff,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) )
= ( ! [X2: set_a] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_502_set__eq__subset,axiom,
( ( ^ [Y6: set_nat,Z3: set_nat] : ( Y6 = Z3 ) )
= ( ^ [A2: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A2 ) ) ) ) ).
% set_eq_subset
thf(fact_503_set__eq__subset,axiom,
( ( ^ [Y6: set_a,Z3: set_a] : ( Y6 = Z3 ) )
= ( ^ [A2: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A2 @ B4 )
& ( ord_less_eq_set_a @ B4 @ A2 ) ) ) ) ).
% set_eq_subset
thf(fact_504_set__eq__subset,axiom,
( ( ^ [Y6: set_set_a,Z3: set_set_a] : ( Y6 = Z3 ) )
= ( ^ [A2: set_set_a,B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B4 )
& ( ord_le3724670747650509150_set_a @ B4 @ A2 ) ) ) ) ).
% set_eq_subset
thf(fact_505_subset__trans,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_506_subset__trans,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_507_subset__trans,axiom,
! [A: set_set_a,B: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ C2 )
=> ( ord_le3724670747650509150_set_a @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_508_Collect__mono,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ! [X: set_nat] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_509_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X: nat] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_510_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X: a] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_511_Collect__mono,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ! [X: set_a] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) ) ) ).
% Collect_mono
thf(fact_512_subset__refl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% subset_refl
thf(fact_513_subset__refl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% subset_refl
thf(fact_514_subset__refl,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ A @ A ) ).
% subset_refl
thf(fact_515_subset__iff,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A2: set_set_nat,B4: set_set_nat] :
! [T: set_nat] :
( ( member_set_nat @ T @ A2 )
=> ( member_set_nat @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_516_subset__iff,axiom,
( ord_le871467723717165285_nat_a
= ( ^ [A2: set_nat_a,B4: set_nat_a] :
! [T: nat > a] :
( ( member_nat_a @ T @ A2 )
=> ( member_nat_a @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_517_subset__iff,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A2: set_set_set_nat,B4: set_set_set_nat] :
! [T: set_set_nat] :
( ( member_set_set_nat @ T @ A2 )
=> ( member_set_set_nat @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_518_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A2: set_nat,B4: set_nat] :
! [T: nat] :
( ( member_nat @ T @ A2 )
=> ( member_nat @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_519_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A2: set_a,B4: set_a] :
! [T: a] :
( ( member_a @ T @ A2 )
=> ( member_a @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_520_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A2: set_set_a,B4: set_set_a] :
! [T: set_a] :
( ( member_set_a @ T @ A2 )
=> ( member_set_a @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_521_equalityD2,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ B @ A ) ) ).
% equalityD2
thf(fact_522_equalityD2,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ( ord_less_eq_set_a @ B @ A ) ) ).
% equalityD2
thf(fact_523_equalityD2,axiom,
! [A: set_set_a,B: set_set_a] :
( ( A = B )
=> ( ord_le3724670747650509150_set_a @ B @ A ) ) ).
% equalityD2
thf(fact_524_equalityD1,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% equalityD1
thf(fact_525_equalityD1,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% equalityD1
thf(fact_526_equalityD1,axiom,
! [A: set_set_a,B: set_set_a] :
( ( A = B )
=> ( ord_le3724670747650509150_set_a @ A @ B ) ) ).
% equalityD1
thf(fact_527_subset__eq,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A2: set_set_nat,B4: set_set_nat] :
! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( member_set_nat @ X2 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_528_subset__eq,axiom,
( ord_le871467723717165285_nat_a
= ( ^ [A2: set_nat_a,B4: set_nat_a] :
! [X2: nat > a] :
( ( member_nat_a @ X2 @ A2 )
=> ( member_nat_a @ X2 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_529_subset__eq,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A2: set_set_set_nat,B4: set_set_set_nat] :
! [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ A2 )
=> ( member_set_set_nat @ X2 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_530_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A2: set_nat,B4: set_nat] :
! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat @ X2 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_531_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A2: set_a,B4: set_a] :
! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_a @ X2 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_532_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A2: set_set_a,B4: set_set_a] :
! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ( member_set_a @ X2 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_533_equalityE,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_nat @ A @ B )
=> ~ ( ord_less_eq_set_nat @ B @ A ) ) ) ).
% equalityE
thf(fact_534_equalityE,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_a @ A @ B )
=> ~ ( ord_less_eq_set_a @ B @ A ) ) ) ).
% equalityE
thf(fact_535_equalityE,axiom,
! [A: set_set_a,B: set_set_a] :
( ( A = B )
=> ~ ( ( ord_le3724670747650509150_set_a @ A @ B )
=> ~ ( ord_le3724670747650509150_set_a @ B @ A ) ) ) ).
% equalityE
thf(fact_536_subsetD,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( member_set_nat @ C @ A )
=> ( member_set_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_537_subsetD,axiom,
! [A: set_nat_a,B: set_nat_a,C: nat > a] :
( ( ord_le871467723717165285_nat_a @ A @ B )
=> ( ( member_nat_a @ C @ A )
=> ( member_nat_a @ C @ B ) ) ) ).
% subsetD
thf(fact_538_subsetD,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C: set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ( member_set_set_nat @ C @ A )
=> ( member_set_set_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_539_subsetD,axiom,
! [A: set_nat,B: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_540_subsetD,axiom,
! [A: set_a,B: set_a,C: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ) ).
% subsetD
thf(fact_541_subsetD,axiom,
! [A: set_set_a,B: set_set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( member_set_a @ C @ A )
=> ( member_set_a @ C @ B ) ) ) ).
% subsetD
thf(fact_542_in__mono,axiom,
! [A: set_set_nat,B: set_set_nat,X3: set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( member_set_nat @ X3 @ A )
=> ( member_set_nat @ X3 @ B ) ) ) ).
% in_mono
thf(fact_543_in__mono,axiom,
! [A: set_nat_a,B: set_nat_a,X3: nat > a] :
( ( ord_le871467723717165285_nat_a @ A @ B )
=> ( ( member_nat_a @ X3 @ A )
=> ( member_nat_a @ X3 @ B ) ) ) ).
% in_mono
thf(fact_544_in__mono,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,X3: set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ( member_set_set_nat @ X3 @ A )
=> ( member_set_set_nat @ X3 @ B ) ) ) ).
% in_mono
thf(fact_545_in__mono,axiom,
! [A: set_nat,B: set_nat,X3: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ X3 @ A )
=> ( member_nat @ X3 @ B ) ) ) ).
% in_mono
thf(fact_546_in__mono,axiom,
! [A: set_a,B: set_a,X3: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ X3 @ A )
=> ( member_a @ X3 @ B ) ) ) ).
% in_mono
thf(fact_547_in__mono,axiom,
! [A: set_set_a,B: set_set_a,X3: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( member_set_a @ X3 @ A )
=> ( member_set_a @ X3 @ B ) ) ) ).
% in_mono
thf(fact_548_imageI,axiom,
! [X3: nat,A: set_nat,F: nat > nat] :
( ( member_nat @ X3 @ A )
=> ( member_nat @ ( F @ X3 ) @ ( image_nat_nat @ F @ A ) ) ) ).
% imageI
thf(fact_549_imageI,axiom,
! [X3: nat,A: set_nat,F: nat > a] :
( ( member_nat @ X3 @ A )
=> ( member_a @ ( F @ X3 ) @ ( image_nat_a @ F @ A ) ) ) ).
% imageI
thf(fact_550_imageI,axiom,
! [X3: a,A: set_a,F: a > nat] :
( ( member_a @ X3 @ A )
=> ( member_nat @ ( F @ X3 ) @ ( image_a_nat @ F @ A ) ) ) ).
% imageI
thf(fact_551_imageI,axiom,
! [X3: a,A: set_a,F: a > a] :
( ( member_a @ X3 @ A )
=> ( member_a @ ( F @ X3 ) @ ( image_a_a @ F @ A ) ) ) ).
% imageI
thf(fact_552_imageI,axiom,
! [X3: set_nat,A: set_set_nat,F: set_nat > nat] :
( ( member_set_nat @ X3 @ A )
=> ( member_nat @ ( F @ X3 ) @ ( image_set_nat_nat @ F @ A ) ) ) ).
% imageI
thf(fact_553_imageI,axiom,
! [X3: set_nat,A: set_set_nat,F: set_nat > a] :
( ( member_set_nat @ X3 @ A )
=> ( member_a @ ( F @ X3 ) @ ( image_set_nat_a @ F @ A ) ) ) ).
% imageI
thf(fact_554_imageI,axiom,
! [X3: set_a,A: set_set_a,F: set_a > nat] :
( ( member_set_a @ X3 @ A )
=> ( member_nat @ ( F @ X3 ) @ ( image_set_a_nat @ F @ A ) ) ) ).
% imageI
thf(fact_555_imageI,axiom,
! [X3: set_a,A: set_set_a,F: set_a > a] :
( ( member_set_a @ X3 @ A )
=> ( member_a @ ( F @ X3 ) @ ( image_set_a_a @ F @ A ) ) ) ).
% imageI
thf(fact_556_imageI,axiom,
! [X3: nat,A: set_nat,F: nat > set_nat] :
( ( member_nat @ X3 @ A )
=> ( member_set_nat @ ( F @ X3 ) @ ( image_nat_set_nat @ F @ A ) ) ) ).
% imageI
thf(fact_557_imageI,axiom,
! [X3: nat,A: set_nat,F: nat > set_a] :
( ( member_nat @ X3 @ A )
=> ( member_set_a @ ( F @ X3 ) @ ( image_nat_set_a @ F @ A ) ) ) ).
% imageI
thf(fact_558_image__iff,axiom,
! [Z: set_nat_a,F: a > set_nat_a,A: set_a] :
( ( member_set_nat_a2 @ Z @ ( image_a_set_nat_a @ F @ A ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_559_image__iff,axiom,
! [Z: set_nat,F: a > set_nat,A: set_a] :
( ( member_set_nat @ Z @ ( image_a_set_nat @ F @ A ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_560_image__iff,axiom,
! [Z: set_a,F: a > set_a,A: set_a] :
( ( member_set_a @ Z @ ( image_a_set_a @ F @ A ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_561_image__iff,axiom,
! [Z: nat,F: nat > nat,A: set_nat] :
( ( member_nat @ Z @ ( image_nat_nat @ F @ A ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_562_image__iff,axiom,
! [Z: nat,F: a > nat,A: set_a] :
( ( member_nat @ Z @ ( image_a_nat @ F @ A ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_563_image__iff,axiom,
! [Z: nat > a,F: a > nat > a,A: set_a] :
( ( member_nat_a @ Z @ ( image_a_nat_a @ F @ A ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_564_image__iff,axiom,
! [Z: a,F: a > a,A: set_a] :
( ( member_a @ Z @ ( image_a_a @ F @ A ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_565_image__iff,axiom,
! [Z: set_set_nat,F: nat > set_set_nat,A: set_nat] :
( ( member_set_set_nat @ Z @ ( image_2194112158459175443et_nat @ F @ A ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_566_bex__imageD,axiom,
! [F: nat > set_set_nat,A: set_nat,P: set_set_nat > $o] :
( ? [X5: set_set_nat] :
( ( member_set_set_nat @ X5 @ ( image_2194112158459175443et_nat @ F @ A ) )
& ( P @ X5 ) )
=> ? [X: nat] :
( ( member_nat @ X @ A )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_567_bex__imageD,axiom,
! [F: a > a,A: set_a,P: a > $o] :
( ? [X5: a] :
( ( member_a @ X5 @ ( image_a_a @ F @ A ) )
& ( P @ X5 ) )
=> ? [X: a] :
( ( member_a @ X @ A )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_568_bex__imageD,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ? [X5: nat] :
( ( member_nat @ X5 @ ( image_nat_nat @ F @ A ) )
& ( P @ X5 ) )
=> ? [X: nat] :
( ( member_nat @ X @ A )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_569_bex__imageD,axiom,
! [F: a > set_nat,A: set_a,P: set_nat > $o] :
( ? [X5: set_nat] :
( ( member_set_nat @ X5 @ ( image_a_set_nat @ F @ A ) )
& ( P @ X5 ) )
=> ? [X: a] :
( ( member_a @ X @ A )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_570_bex__imageD,axiom,
! [F: a > set_nat_a,A: set_a,P: set_nat_a > $o] :
( ? [X5: set_nat_a] :
( ( member_set_nat_a2 @ X5 @ ( image_a_set_nat_a @ F @ A ) )
& ( P @ X5 ) )
=> ? [X: a] :
( ( member_a @ X @ A )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_571_bex__imageD,axiom,
! [F: a > set_a,A: set_a,P: set_a > $o] :
( ? [X5: set_a] :
( ( member_set_a @ X5 @ ( image_a_set_a @ F @ A ) )
& ( P @ X5 ) )
=> ? [X: a] :
( ( member_a @ X @ A )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_572_bex__imageD,axiom,
! [F: a > nat,A: set_a,P: nat > $o] :
( ? [X5: nat] :
( ( member_nat @ X5 @ ( image_a_nat @ F @ A ) )
& ( P @ X5 ) )
=> ? [X: a] :
( ( member_a @ X @ A )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_573_bex__imageD,axiom,
! [F: a > nat > a,A: set_a,P: ( nat > a ) > $o] :
( ? [X5: nat > a] :
( ( member_nat_a @ X5 @ ( image_a_nat_a @ F @ A ) )
& ( P @ X5 ) )
=> ? [X: a] :
( ( member_a @ X @ A )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_574_image__cong,axiom,
! [M2: set_nat,N3: set_nat,F: nat > set_set_nat,G: nat > set_set_nat] :
( ( M2 = N3 )
=> ( ! [X: nat] :
( ( member_nat @ X @ N3 )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image_2194112158459175443et_nat @ F @ M2 )
= ( image_2194112158459175443et_nat @ G @ N3 ) ) ) ) ).
% image_cong
thf(fact_575_image__cong,axiom,
! [M2: set_nat,N3: set_nat,F: nat > nat,G: nat > nat] :
( ( M2 = N3 )
=> ( ! [X: nat] :
( ( member_nat @ X @ N3 )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image_nat_nat @ F @ M2 )
= ( image_nat_nat @ G @ N3 ) ) ) ) ).
% image_cong
thf(fact_576_image__cong,axiom,
! [M2: set_a,N3: set_a,F: a > a,G: a > a] :
( ( M2 = N3 )
=> ( ! [X: a] :
( ( member_a @ X @ N3 )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image_a_a @ F @ M2 )
= ( image_a_a @ G @ N3 ) ) ) ) ).
% image_cong
thf(fact_577_image__cong,axiom,
! [M2: set_a,N3: set_a,F: a > set_nat,G: a > set_nat] :
( ( M2 = N3 )
=> ( ! [X: a] :
( ( member_a @ X @ N3 )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image_a_set_nat @ F @ M2 )
= ( image_a_set_nat @ G @ N3 ) ) ) ) ).
% image_cong
thf(fact_578_image__cong,axiom,
! [M2: set_a,N3: set_a,F: a > set_nat_a,G: a > set_nat_a] :
( ( M2 = N3 )
=> ( ! [X: a] :
( ( member_a @ X @ N3 )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image_a_set_nat_a @ F @ M2 )
= ( image_a_set_nat_a @ G @ N3 ) ) ) ) ).
% image_cong
thf(fact_579_image__cong,axiom,
! [M2: set_a,N3: set_a,F: a > set_a,G: a > set_a] :
( ( M2 = N3 )
=> ( ! [X: a] :
( ( member_a @ X @ N3 )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image_a_set_a @ F @ M2 )
= ( image_a_set_a @ G @ N3 ) ) ) ) ).
% image_cong
thf(fact_580_image__cong,axiom,
! [M2: set_a,N3: set_a,F: a > nat,G: a > nat] :
( ( M2 = N3 )
=> ( ! [X: a] :
( ( member_a @ X @ N3 )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image_a_nat @ F @ M2 )
= ( image_a_nat @ G @ N3 ) ) ) ) ).
% image_cong
thf(fact_581_image__cong,axiom,
! [M2: set_a,N3: set_a,F: a > nat > a,G: a > nat > a] :
( ( M2 = N3 )
=> ( ! [X: a] :
( ( member_a @ X @ N3 )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image_a_nat_a @ F @ M2 )
= ( image_a_nat_a @ G @ N3 ) ) ) ) ).
% image_cong
thf(fact_582_ball__imageD,axiom,
! [F: nat > set_set_nat,A: set_nat,P: set_set_nat > $o] :
( ! [X: set_set_nat] :
( ( member_set_set_nat @ X @ ( image_2194112158459175443et_nat @ F @ A ) )
=> ( P @ X ) )
=> ! [X5: nat] :
( ( member_nat @ X5 @ A )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_583_ball__imageD,axiom,
! [F: a > a,A: set_a,P: a > $o] :
( ! [X: a] :
( ( member_a @ X @ ( image_a_a @ F @ A ) )
=> ( P @ X ) )
=> ! [X5: a] :
( ( member_a @ X5 @ A )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_584_ball__imageD,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ! [X: nat] :
( ( member_nat @ X @ ( image_nat_nat @ F @ A ) )
=> ( P @ X ) )
=> ! [X5: nat] :
( ( member_nat @ X5 @ A )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_585_ball__imageD,axiom,
! [F: a > set_nat,A: set_a,P: set_nat > $o] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ ( image_a_set_nat @ F @ A ) )
=> ( P @ X ) )
=> ! [X5: a] :
( ( member_a @ X5 @ A )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_586_ball__imageD,axiom,
! [F: a > set_nat_a,A: set_a,P: set_nat_a > $o] :
( ! [X: set_nat_a] :
( ( member_set_nat_a2 @ X @ ( image_a_set_nat_a @ F @ A ) )
=> ( P @ X ) )
=> ! [X5: a] :
( ( member_a @ X5 @ A )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_587_ball__imageD,axiom,
! [F: a > set_a,A: set_a,P: set_a > $o] :
( ! [X: set_a] :
( ( member_set_a @ X @ ( image_a_set_a @ F @ A ) )
=> ( P @ X ) )
=> ! [X5: a] :
( ( member_a @ X5 @ A )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_588_ball__imageD,axiom,
! [F: a > nat,A: set_a,P: nat > $o] :
( ! [X: nat] :
( ( member_nat @ X @ ( image_a_nat @ F @ A ) )
=> ( P @ X ) )
=> ! [X5: a] :
( ( member_a @ X5 @ A )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_589_ball__imageD,axiom,
! [F: a > nat > a,A: set_a,P: ( nat > a ) > $o] :
( ! [X: nat > a] :
( ( member_nat_a @ X @ ( image_a_nat_a @ F @ A ) )
=> ( P @ X ) )
=> ! [X5: a] :
( ( member_a @ X5 @ A )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_590_rev__image__eqI,axiom,
! [X3: nat,A: set_nat,B2: nat,F: nat > nat] :
( ( member_nat @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_nat @ B2 @ ( image_nat_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_591_rev__image__eqI,axiom,
! [X3: nat,A: set_nat,B2: a,F: nat > a] :
( ( member_nat @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_a @ B2 @ ( image_nat_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_592_rev__image__eqI,axiom,
! [X3: a,A: set_a,B2: nat,F: a > nat] :
( ( member_a @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_nat @ B2 @ ( image_a_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_593_rev__image__eqI,axiom,
! [X3: a,A: set_a,B2: a,F: a > a] :
( ( member_a @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_a @ B2 @ ( image_a_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_594_rev__image__eqI,axiom,
! [X3: set_nat,A: set_set_nat,B2: nat,F: set_nat > nat] :
( ( member_set_nat @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_nat @ B2 @ ( image_set_nat_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_595_rev__image__eqI,axiom,
! [X3: set_nat,A: set_set_nat,B2: a,F: set_nat > a] :
( ( member_set_nat @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_a @ B2 @ ( image_set_nat_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_596_rev__image__eqI,axiom,
! [X3: set_a,A: set_set_a,B2: nat,F: set_a > nat] :
( ( member_set_a @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_nat @ B2 @ ( image_set_a_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_597_rev__image__eqI,axiom,
! [X3: set_a,A: set_set_a,B2: a,F: set_a > a] :
( ( member_set_a @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_a @ B2 @ ( image_set_a_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_598_rev__image__eqI,axiom,
! [X3: nat,A: set_nat,B2: set_nat,F: nat > set_nat] :
( ( member_nat @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_set_nat @ B2 @ ( image_nat_set_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_599_rev__image__eqI,axiom,
! [X3: nat,A: set_nat,B2: set_a,F: nat > set_a] :
( ( member_nat @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_set_a @ B2 @ ( image_nat_set_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_600_cring_Ois__cring,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( cring_a_b @ R )
=> ( cring_a_b @ R ) ) ).
% cring.is_cring
thf(fact_601_Pow__top,axiom,
! [A: set_nat] : ( member_set_nat @ A @ ( pow_nat @ A ) ) ).
% Pow_top
thf(fact_602_Pow__top,axiom,
! [A: set_a] : ( member_set_a @ A @ ( pow_a @ A ) ) ).
% Pow_top
thf(fact_603_Pow__top,axiom,
! [A: set_set_nat] : ( member_set_set_nat @ A @ ( pow_set_nat @ A ) ) ).
% Pow_top
thf(fact_604_Collect__subset,axiom,
! [A: set_nat_a,P: ( nat > a ) > $o] :
( ord_le871467723717165285_nat_a
@ ( collect_nat_a
@ ^ [X2: nat > a] :
( ( member_nat_a @ X2 @ A )
& ( P @ X2 ) ) )
@ A ) ).
% Collect_subset
thf(fact_605_Collect__subset,axiom,
! [A: set_set_set_nat,P: set_set_nat > $o] :
( ord_le9131159989063066194et_nat
@ ( collect_set_set_nat
@ ^ [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ A )
& ( P @ X2 ) ) )
@ A ) ).
% Collect_subset
thf(fact_606_Collect__subset,axiom,
! [A: set_set_nat,P: set_nat > $o] :
( ord_le6893508408891458716et_nat
@ ( collect_set_nat
@ ^ [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
& ( P @ X2 ) ) )
@ A ) ).
% Collect_subset
thf(fact_607_Collect__subset,axiom,
! [A: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P @ X2 ) ) )
@ A ) ).
% Collect_subset
thf(fact_608_Collect__subset,axiom,
! [A: set_a,P: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A )
& ( P @ X2 ) ) )
@ A ) ).
% Collect_subset
thf(fact_609_Collect__subset,axiom,
! [A: set_set_a,P: set_a > $o] :
( ord_le3724670747650509150_set_a
@ ( collect_set_a
@ ^ [X2: set_a] :
( ( member_set_a @ X2 @ A )
& ( P @ X2 ) ) )
@ A ) ).
% Collect_subset
thf(fact_610_imageE,axiom,
! [B2: nat,F: nat > nat,A: set_nat] :
( ( member_nat @ B2 @ ( image_nat_nat @ F @ A ) )
=> ~ ! [X: nat] :
( ( B2
= ( F @ X ) )
=> ~ ( member_nat @ X @ A ) ) ) ).
% imageE
thf(fact_611_imageE,axiom,
! [B2: nat,F: a > nat,A: set_a] :
( ( member_nat @ B2 @ ( image_a_nat @ F @ A ) )
=> ~ ! [X: a] :
( ( B2
= ( F @ X ) )
=> ~ ( member_a @ X @ A ) ) ) ).
% imageE
thf(fact_612_imageE,axiom,
! [B2: a,F: nat > a,A: set_nat] :
( ( member_a @ B2 @ ( image_nat_a @ F @ A ) )
=> ~ ! [X: nat] :
( ( B2
= ( F @ X ) )
=> ~ ( member_nat @ X @ A ) ) ) ).
% imageE
thf(fact_613_imageE,axiom,
! [B2: a,F: a > a,A: set_a] :
( ( member_a @ B2 @ ( image_a_a @ F @ A ) )
=> ~ ! [X: a] :
( ( B2
= ( F @ X ) )
=> ~ ( member_a @ X @ A ) ) ) ).
% imageE
thf(fact_614_imageE,axiom,
! [B2: set_nat,F: nat > set_nat,A: set_nat] :
( ( member_set_nat @ B2 @ ( image_nat_set_nat @ F @ A ) )
=> ~ ! [X: nat] :
( ( B2
= ( F @ X ) )
=> ~ ( member_nat @ X @ A ) ) ) ).
% imageE
thf(fact_615_imageE,axiom,
! [B2: set_nat,F: a > set_nat,A: set_a] :
( ( member_set_nat @ B2 @ ( image_a_set_nat @ F @ A ) )
=> ~ ! [X: a] :
( ( B2
= ( F @ X ) )
=> ~ ( member_a @ X @ A ) ) ) ).
% imageE
thf(fact_616_imageE,axiom,
! [B2: set_a,F: nat > set_a,A: set_nat] :
( ( member_set_a @ B2 @ ( image_nat_set_a @ F @ A ) )
=> ~ ! [X: nat] :
( ( B2
= ( F @ X ) )
=> ~ ( member_nat @ X @ A ) ) ) ).
% imageE
thf(fact_617_imageE,axiom,
! [B2: set_a,F: a > set_a,A: set_a] :
( ( member_set_a @ B2 @ ( image_a_set_a @ F @ A ) )
=> ~ ! [X: a] :
( ( B2
= ( F @ X ) )
=> ~ ( member_a @ X @ A ) ) ) ).
% imageE
thf(fact_618_imageE,axiom,
! [B2: nat,F: set_nat > nat,A: set_set_nat] :
( ( member_nat @ B2 @ ( image_set_nat_nat @ F @ A ) )
=> ~ ! [X: set_nat] :
( ( B2
= ( F @ X ) )
=> ~ ( member_set_nat @ X @ A ) ) ) ).
% imageE
thf(fact_619_imageE,axiom,
! [B2: nat,F: set_a > nat,A: set_set_a] :
( ( member_nat @ B2 @ ( image_set_a_nat @ F @ A ) )
=> ~ ! [X: set_a] :
( ( B2
= ( F @ X ) )
=> ~ ( member_set_a @ X @ A ) ) ) ).
% imageE
thf(fact_620_image__image,axiom,
! [F: nat > a,G: a > nat,A: set_a] :
( ( image_nat_a @ F @ ( image_a_nat @ G @ A ) )
= ( image_a_a
@ ^ [X2: a] : ( F @ ( G @ X2 ) )
@ A ) ) ).
% image_image
thf(fact_621_image__image,axiom,
! [F: a > a,G: a > a,A: set_a] :
( ( image_a_a @ F @ ( image_a_a @ G @ A ) )
= ( image_a_a
@ ^ [X2: a] : ( F @ ( G @ X2 ) )
@ A ) ) ).
% image_image
thf(fact_622_image__image,axiom,
! [F: nat > nat,G: nat > nat,A: set_nat] :
( ( image_nat_nat @ F @ ( image_nat_nat @ G @ A ) )
= ( image_nat_nat
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ A ) ) ).
% image_image
thf(fact_623_image__image,axiom,
! [F: nat > nat,G: a > nat,A: set_a] :
( ( image_nat_nat @ F @ ( image_a_nat @ G @ A ) )
= ( image_a_nat
@ ^ [X2: a] : ( F @ ( G @ X2 ) )
@ A ) ) ).
% image_image
thf(fact_624_image__image,axiom,
! [F: a > nat,G: nat > a,A: set_nat] :
( ( image_a_nat @ F @ ( image_nat_a @ G @ A ) )
= ( image_nat_nat
@ ^ [X2: nat] : ( F @ ( G @ X2 ) )
@ A ) ) ).
% image_image
thf(fact_625_image__image,axiom,
! [F: a > nat,G: a > a,A: set_a] :
( ( image_a_nat @ F @ ( image_a_a @ G @ A ) )
= ( image_a_nat
@ ^ [X2: a] : ( F @ ( G @ X2 ) )
@ A ) ) ).
% image_image
thf(fact_626_image__image,axiom,
! [F: set_nat > a,G: a > set_nat,A: set_a] :
( ( image_set_nat_a @ F @ ( image_a_set_nat @ G @ A ) )
= ( image_a_a
@ ^ [X2: a] : ( F @ ( G @ X2 ) )
@ A ) ) ).
% image_image
thf(fact_627_image__image,axiom,
! [F: set_nat > nat,G: a > set_nat,A: set_a] :
( ( image_set_nat_nat @ F @ ( image_a_set_nat @ G @ A ) )
= ( image_a_nat
@ ^ [X2: a] : ( F @ ( G @ X2 ) )
@ A ) ) ).
% image_image
thf(fact_628_image__image,axiom,
! [F: set_a > a,G: a > set_a,A: set_a] :
( ( image_set_a_a @ F @ ( image_a_set_a @ G @ A ) )
= ( image_a_a
@ ^ [X2: a] : ( F @ ( G @ X2 ) )
@ A ) ) ).
% image_image
thf(fact_629_image__image,axiom,
! [F: set_a > nat,G: a > set_a,A: set_a] :
( ( image_set_a_nat @ F @ ( image_a_set_a @ G @ A ) )
= ( image_a_nat
@ ^ [X2: a] : ( F @ ( G @ X2 ) )
@ A ) ) ).
% image_image
thf(fact_630_Compr__image__eq,axiom,
! [F: a > a,A: set_a,P: a > $o] :
( ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ ( image_a_a @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_a_a @ F
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_631_Compr__image__eq,axiom,
! [F: nat > a,A: set_nat,P: a > $o] :
( ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ ( image_nat_a @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_nat_a @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_632_Compr__image__eq,axiom,
! [F: a > nat,A: set_a,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_a_nat @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_a_nat @ F
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_633_Compr__image__eq,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_nat_nat @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_nat_nat @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_634_Compr__image__eq,axiom,
! [F: a > set_a,A: set_a,P: set_a > $o] :
( ( collect_set_a
@ ^ [X2: set_a] :
( ( member_set_a @ X2 @ ( image_a_set_a @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_a_set_a @ F
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_635_Compr__image__eq,axiom,
! [F: set_a > a,A: set_set_a,P: a > $o] :
( ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ ( image_set_a_a @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_set_a_a @ F
@ ( collect_set_a
@ ^ [X2: set_a] :
( ( member_set_a @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_636_Compr__image__eq,axiom,
! [F: set_nat > a,A: set_set_nat,P: a > $o] :
( ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ ( image_set_nat_a @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_set_nat_a @ F
@ ( collect_set_nat
@ ^ [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_637_Compr__image__eq,axiom,
! [F: nat > set_a,A: set_nat,P: set_a > $o] :
( ( collect_set_a
@ ^ [X2: set_a] :
( ( member_set_a @ X2 @ ( image_nat_set_a @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_nat_set_a @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_638_Compr__image__eq,axiom,
! [F: a > set_nat,A: set_a,P: set_nat > $o] :
( ( collect_set_nat
@ ^ [X2: set_nat] :
( ( member_set_nat @ X2 @ ( image_a_set_nat @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_a_set_nat @ F
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_639_Compr__image__eq,axiom,
! [F: nat > set_nat,A: set_nat,P: set_nat > $o] :
( ( collect_set_nat
@ ^ [X2: set_nat] :
( ( member_set_nat @ X2 @ ( image_nat_set_nat @ F @ A ) )
& ( P @ X2 ) ) )
= ( image_nat_set_nat @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_640_UN__Pow__subset,axiom,
! [B: nat > set_set_nat,A: set_nat] :
( ord_le9131159989063066194et_nat
@ ( comple6569609367425551173et_nat
@ ( image_5738044413236618185et_nat
@ ^ [X2: nat] : ( pow_set_nat @ ( B @ X2 ) )
@ A ) )
@ ( pow_set_nat @ ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ B @ A ) ) ) ) ).
% UN_Pow_subset
thf(fact_641_UN__Pow__subset,axiom,
! [B: a > set_nat_a,A: set_a] :
( ord_le2390145808437456709_nat_a
@ ( comple2312494275411564946_nat_a
@ ( image_2266649148107646380_nat_a
@ ^ [X2: a] : ( pow_nat_a @ ( B @ X2 ) )
@ A ) )
@ ( pow_nat_a @ ( comple3545767860446109490_nat_a @ ( image_a_set_nat_a @ B @ A ) ) ) ) ).
% UN_Pow_subset
thf(fact_642_UN__Pow__subset,axiom,
! [B: a > set_nat,A: set_a] :
( ord_le6893508408891458716et_nat
@ ( comple548664676211718543et_nat
@ ( image_a_set_set_nat
@ ^ [X2: a] : ( pow_nat @ ( B @ X2 ) )
@ A ) )
@ ( pow_nat @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ B @ A ) ) ) ) ).
% UN_Pow_subset
thf(fact_643_UN__Pow__subset,axiom,
! [B: nat > set_nat,A: set_nat] :
( ord_le6893508408891458716et_nat
@ ( comple548664676211718543et_nat
@ ( image_2194112158459175443et_nat
@ ^ [X2: nat] : ( pow_nat @ ( B @ X2 ) )
@ A ) )
@ ( pow_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A ) ) ) ) ).
% UN_Pow_subset
thf(fact_644_UN__Pow__subset,axiom,
! [B: a > set_a,A: set_a] :
( ord_le3724670747650509150_set_a
@ ( comple3958522678809307947_set_a
@ ( image_a_set_set_a
@ ^ [X2: a] : ( pow_a @ ( B @ X2 ) )
@ A ) )
@ ( pow_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A ) ) ) ) ).
% UN_Pow_subset
thf(fact_645_image__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_646_image__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > a] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_set_a @ ( image_nat_a @ F @ A ) @ ( image_nat_a @ F @ B ) ) ) ).
% image_mono
thf(fact_647_image__mono,axiom,
! [A: set_a,B: set_a,F: a > nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_nat @ ( image_a_nat @ F @ A ) @ ( image_a_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_648_image__mono,axiom,
! [A: set_a,B: set_a,F: a > a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ ( image_a_a @ F @ B ) ) ) ).
% image_mono
thf(fact_649_image__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > set_a] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_le3724670747650509150_set_a @ ( image_nat_set_a @ F @ A ) @ ( image_nat_set_a @ F @ B ) ) ) ).
% image_mono
thf(fact_650_image__mono,axiom,
! [A: set_a,B: set_a,F: a > set_nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_le6893508408891458716et_nat @ ( image_a_set_nat @ F @ A ) @ ( image_a_set_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_651_image__mono,axiom,
! [A: set_a,B: set_a,F: a > set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ A ) @ ( image_a_set_a @ F @ B ) ) ) ).
% image_mono
thf(fact_652_image__mono,axiom,
! [A: set_set_a,B: set_set_a,F: set_a > nat] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ord_less_eq_set_nat @ ( image_set_a_nat @ F @ A ) @ ( image_set_a_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_653_image__mono,axiom,
! [A: set_set_a,B: set_set_a,F: set_a > a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ord_less_eq_set_a @ ( image_set_a_a @ F @ A ) @ ( image_set_a_a @ F @ B ) ) ) ).
% image_mono
thf(fact_654_image__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > set_set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_le9131159989063066194et_nat @ ( image_2194112158459175443et_nat @ F @ A ) @ ( image_2194112158459175443et_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_655_image__subsetI,axiom,
! [A: set_nat,F: nat > nat,B: set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member_nat @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_656_image__subsetI,axiom,
! [A: set_a,F: a > nat,B: set_nat] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( member_nat @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_a_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_657_image__subsetI,axiom,
! [A: set_nat,F: nat > a,B: set_a] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member_a @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_nat_a @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_658_image__subsetI,axiom,
! [A: set_a,F: a > a,B: set_a] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( member_a @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_659_image__subsetI,axiom,
! [A: set_nat,F: nat > set_nat,B: set_set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member_set_nat @ ( F @ X ) @ B ) )
=> ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_660_image__subsetI,axiom,
! [A: set_a,F: a > set_nat,B: set_set_nat] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( member_set_nat @ ( F @ X ) @ B ) )
=> ( ord_le6893508408891458716et_nat @ ( image_a_set_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_661_image__subsetI,axiom,
! [A: set_set_nat,F: set_nat > nat,B: set_nat] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ A )
=> ( member_nat @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_set_nat_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_662_image__subsetI,axiom,
! [A: set_set_a,F: set_a > nat,B: set_nat] :
( ! [X: set_a] :
( ( member_set_a @ X @ A )
=> ( member_nat @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_set_a_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_663_image__subsetI,axiom,
! [A: set_set_nat,F: set_nat > a,B: set_a] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ A )
=> ( member_a @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_set_nat_a @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_664_image__subsetI,axiom,
! [A: set_set_a,F: set_a > a,B: set_a] :
( ! [X: set_a] :
( ( member_set_a @ X @ A )
=> ( member_a @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_set_a_a @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_665_subset__imageE,axiom,
! [B: set_nat,F: nat > nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
=> ~ ! [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A )
=> ( B
!= ( image_nat_nat @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_666_subset__imageE,axiom,
! [B: set_nat,F: a > nat,A: set_a] :
( ( ord_less_eq_set_nat @ B @ ( image_a_nat @ F @ A ) )
=> ~ ! [C4: set_a] :
( ( ord_less_eq_set_a @ C4 @ A )
=> ( B
!= ( image_a_nat @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_667_subset__imageE,axiom,
! [B: set_a,F: nat > a,A: set_nat] :
( ( ord_less_eq_set_a @ B @ ( image_nat_a @ F @ A ) )
=> ~ ! [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A )
=> ( B
!= ( image_nat_a @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_668_subset__imageE,axiom,
! [B: set_a,F: a > a,A: set_a] :
( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A ) )
=> ~ ! [C4: set_a] :
( ( ord_less_eq_set_a @ C4 @ A )
=> ( B
!= ( image_a_a @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_669_subset__imageE,axiom,
! [B: set_set_nat,F: a > set_nat,A: set_a] :
( ( ord_le6893508408891458716et_nat @ B @ ( image_a_set_nat @ F @ A ) )
=> ~ ! [C4: set_a] :
( ( ord_less_eq_set_a @ C4 @ A )
=> ( B
!= ( image_a_set_nat @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_670_subset__imageE,axiom,
! [B: set_nat,F: set_a > nat,A: set_set_a] :
( ( ord_less_eq_set_nat @ B @ ( image_set_a_nat @ F @ A ) )
=> ~ ! [C4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C4 @ A )
=> ( B
!= ( image_set_a_nat @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_671_subset__imageE,axiom,
! [B: set_a,F: set_a > a,A: set_set_a] :
( ( ord_less_eq_set_a @ B @ ( image_set_a_a @ F @ A ) )
=> ~ ! [C4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C4 @ A )
=> ( B
!= ( image_set_a_a @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_672_subset__imageE,axiom,
! [B: set_set_a,F: nat > set_a,A: set_nat] :
( ( ord_le3724670747650509150_set_a @ B @ ( image_nat_set_a @ F @ A ) )
=> ~ ! [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A )
=> ( B
!= ( image_nat_set_a @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_673_subset__imageE,axiom,
! [B: set_set_a,F: a > set_a,A: set_a] :
( ( ord_le3724670747650509150_set_a @ B @ ( image_a_set_a @ F @ A ) )
=> ~ ! [C4: set_a] :
( ( ord_less_eq_set_a @ C4 @ A )
=> ( B
!= ( image_a_set_a @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_674_subset__imageE,axiom,
! [B: set_set_set_nat,F: nat > set_set_nat,A: set_nat] :
( ( ord_le9131159989063066194et_nat @ B @ ( image_2194112158459175443et_nat @ F @ A ) )
=> ~ ! [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A )
=> ( B
!= ( image_2194112158459175443et_nat @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_675_image__subset__iff,axiom,
! [F: a > set_nat_a,A: set_a,B: set_set_nat_a] :
( ( ord_le2390145808437456709_nat_a @ ( image_a_set_nat_a @ F @ A ) @ B )
= ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_set_nat_a2 @ ( F @ X2 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_676_image__subset__iff,axiom,
! [F: a > set_nat,A: set_a,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( image_a_set_nat @ F @ A ) @ B )
= ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_set_nat @ ( F @ X2 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_677_image__subset__iff,axiom,
! [F: a > nat > a,A: set_a,B: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ ( image_a_nat_a @ F @ A ) @ B )
= ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_nat_a @ ( F @ X2 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_678_image__subset__iff,axiom,
! [F: nat > set_set_nat,A: set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( image_2194112158459175443et_nat @ F @ A ) @ B )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_set_set_nat @ ( F @ X2 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_679_image__subset__iff,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_nat @ ( F @ X2 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_680_image__subset__iff,axiom,
! [F: a > nat,A: set_a,B: set_nat] :
( ( ord_less_eq_set_nat @ ( image_a_nat @ F @ A ) @ B )
= ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_nat @ ( F @ X2 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_681_image__subset__iff,axiom,
! [F: a > a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ B )
= ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_682_image__subset__iff,axiom,
! [F: a > set_a,A: set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ A ) @ B )
= ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_set_a @ ( F @ X2 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_683_subset__image__iff,axiom,
! [B: set_nat,F: nat > nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B
= ( image_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_684_subset__image__iff,axiom,
! [B: set_nat,F: a > nat,A: set_a] :
( ( ord_less_eq_set_nat @ B @ ( image_a_nat @ F @ A ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A )
& ( B
= ( image_a_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_685_subset__image__iff,axiom,
! [B: set_a,F: nat > a,A: set_nat] :
( ( ord_less_eq_set_a @ B @ ( image_nat_a @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B
= ( image_nat_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_686_subset__image__iff,axiom,
! [B: set_a,F: a > a,A: set_a] :
( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A )
& ( B
= ( image_a_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_687_subset__image__iff,axiom,
! [B: set_set_nat,F: a > set_nat,A: set_a] :
( ( ord_le6893508408891458716et_nat @ B @ ( image_a_set_nat @ F @ A ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A )
& ( B
= ( image_a_set_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_688_subset__image__iff,axiom,
! [B: set_nat,F: set_a > nat,A: set_set_a] :
( ( ord_less_eq_set_nat @ B @ ( image_set_a_nat @ F @ A ) )
= ( ? [AA: set_set_a] :
( ( ord_le3724670747650509150_set_a @ AA @ A )
& ( B
= ( image_set_a_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_689_subset__image__iff,axiom,
! [B: set_a,F: set_a > a,A: set_set_a] :
( ( ord_less_eq_set_a @ B @ ( image_set_a_a @ F @ A ) )
= ( ? [AA: set_set_a] :
( ( ord_le3724670747650509150_set_a @ AA @ A )
& ( B
= ( image_set_a_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_690_subset__image__iff,axiom,
! [B: set_set_a,F: nat > set_a,A: set_nat] :
( ( ord_le3724670747650509150_set_a @ B @ ( image_nat_set_a @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B
= ( image_nat_set_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_691_subset__image__iff,axiom,
! [B: set_set_a,F: a > set_a,A: set_a] :
( ( ord_le3724670747650509150_set_a @ B @ ( image_a_set_a @ F @ A ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A )
& ( B
= ( image_a_set_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_692_subset__image__iff,axiom,
! [B: set_set_set_nat,F: nat > set_set_nat,A: set_nat] :
( ( ord_le9131159989063066194et_nat @ B @ ( image_2194112158459175443et_nat @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B
= ( image_2194112158459175443et_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_693_cSup__eq__maximum,axiom,
! [Z: set_set_nat,X4: set_set_set_nat] :
( ( member_set_set_nat @ Z @ X4 )
=> ( ! [X: set_set_nat] :
( ( member_set_set_nat @ X @ X4 )
=> ( ord_le6893508408891458716et_nat @ X @ Z ) )
=> ( ( comple548664676211718543et_nat @ X4 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_694_cSup__eq__maximum,axiom,
! [Z: $o,X4: set_o] :
( ( member_o @ Z @ X4 )
=> ( ! [X: $o] :
( ( member_o @ X @ X4 )
=> ( ord_less_eq_o @ X @ Z ) )
=> ( ( complete_Sup_Sup_o @ X4 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_695_cSup__eq__maximum,axiom,
! [Z: nat,X4: set_nat] :
( ( member_nat @ Z @ X4 )
=> ( ! [X: nat] :
( ( member_nat @ X @ X4 )
=> ( ord_less_eq_nat @ X @ Z ) )
=> ( ( complete_Sup_Sup_nat @ X4 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_696_cSup__eq__maximum,axiom,
! [Z: set_a,X4: set_set_a] :
( ( member_set_a @ Z @ X4 )
=> ( ! [X: set_a] :
( ( member_set_a @ X @ X4 )
=> ( ord_less_eq_set_a @ X @ Z ) )
=> ( ( comple2307003609928055243_set_a @ X4 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_697_cSup__eq__maximum,axiom,
! [Z: set_nat_a,X4: set_set_nat_a] :
( ( member_set_nat_a2 @ Z @ X4 )
=> ( ! [X: set_nat_a] :
( ( member_set_nat_a2 @ X @ X4 )
=> ( ord_le871467723717165285_nat_a @ X @ Z ) )
=> ( ( comple3545767860446109490_nat_a @ X4 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_698_cSup__eq__maximum,axiom,
! [Z: set_nat,X4: set_set_nat] :
( ( member_set_nat @ Z @ X4 )
=> ( ! [X: set_nat] :
( ( member_set_nat @ X @ X4 )
=> ( ord_less_eq_set_nat @ X @ Z ) )
=> ( ( comple7399068483239264473et_nat @ X4 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_699_cSup__eq__maximum,axiom,
! [Z: set_set_a,X4: set_set_set_a] :
( ( member_set_set_a @ Z @ X4 )
=> ( ! [X: set_set_a] :
( ( member_set_set_a @ X @ X4 )
=> ( ord_le3724670747650509150_set_a @ X @ Z ) )
=> ( ( comple3958522678809307947_set_a @ X4 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_700_PowD,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( member_set_set_nat @ A @ ( pow_set_nat @ B ) )
=> ( ord_le6893508408891458716et_nat @ A @ B ) ) ).
% PowD
thf(fact_701_PowD,axiom,
! [A: set_nat,B: set_nat] :
( ( member_set_nat @ A @ ( pow_nat @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% PowD
thf(fact_702_PowD,axiom,
! [A: set_a,B: set_a] :
( ( member_set_a @ A @ ( pow_a @ B ) )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% PowD
thf(fact_703_PowD,axiom,
! [A: set_set_a,B: set_set_a] :
( ( member_set_set_a @ A @ ( pow_set_a @ B ) )
=> ( ord_le3724670747650509150_set_a @ A @ B ) ) ).
% PowD
thf(fact_704_Pow__mono,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_le6893508408891458716et_nat @ ( pow_nat @ A ) @ ( pow_nat @ B ) ) ) ).
% Pow_mono
thf(fact_705_Pow__mono,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_le3724670747650509150_set_a @ ( pow_a @ A ) @ ( pow_a @ B ) ) ) ).
% Pow_mono
thf(fact_706_Pow__mono,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ord_le5722252365846178494_set_a @ ( pow_set_a @ A ) @ ( pow_set_a @ B ) ) ) ).
% Pow_mono
thf(fact_707_image__Pow__surj,axiom,
! [F: nat > a,A: set_nat,B: set_a] :
( ( ( image_nat_a @ F @ A )
= B )
=> ( ( image_set_nat_set_a @ ( image_nat_a @ F ) @ ( pow_nat @ A ) )
= ( pow_a @ B ) ) ) ).
% image_Pow_surj
thf(fact_708_image__Pow__surj,axiom,
! [F: nat > set_set_nat,A: set_nat,B: set_set_set_nat] :
( ( ( image_2194112158459175443et_nat @ F @ A )
= B )
=> ( ( image_4583741654806091647et_nat @ ( image_2194112158459175443et_nat @ F ) @ ( pow_nat @ A ) )
= ( pow_set_set_nat @ B ) ) ) ).
% image_Pow_surj
thf(fact_709_image__Pow__surj,axiom,
! [F: a > a,A: set_a,B: set_a] :
( ( ( image_a_a @ F @ A )
= B )
=> ( ( image_set_a_set_a @ ( image_a_a @ F ) @ ( pow_a @ A ) )
= ( pow_a @ B ) ) ) ).
% image_Pow_surj
thf(fact_710_image__Pow__surj,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] :
( ( ( image_nat_nat @ F @ A )
= B )
=> ( ( image_7916887816326733075et_nat @ ( image_nat_nat @ F ) @ ( pow_nat @ A ) )
= ( pow_nat @ B ) ) ) ).
% image_Pow_surj
thf(fact_711_image__Pow__surj,axiom,
! [F: a > set_nat,A: set_a,B: set_set_nat] :
( ( ( image_a_set_nat @ F @ A )
= B )
=> ( ( image_8216882647274671445et_nat @ ( image_a_set_nat @ F ) @ ( pow_a @ A ) )
= ( pow_set_nat @ B ) ) ) ).
% image_Pow_surj
thf(fact_712_image__Pow__surj,axiom,
! [F: a > set_nat_a,A: set_a,B: set_set_nat_a] :
( ( ( image_a_set_nat_a @ F @ A )
= B )
=> ( ( image_7780357287214271564_nat_a @ ( image_a_set_nat_a @ F ) @ ( pow_a @ A ) )
= ( pow_set_nat_a @ B ) ) ) ).
% image_Pow_surj
thf(fact_713_image__Pow__surj,axiom,
! [F: a > set_a,A: set_a,B: set_set_a] :
( ( ( image_a_set_a @ F @ A )
= B )
=> ( ( image_4955109552351689957_set_a @ ( image_a_set_a @ F ) @ ( pow_a @ A ) )
= ( pow_set_a @ B ) ) ) ).
% image_Pow_surj
thf(fact_714_image__Pow__surj,axiom,
! [F: a > nat,A: set_a,B: set_nat] :
( ( ( image_a_nat @ F @ A )
= B )
=> ( ( image_set_a_set_nat @ ( image_a_nat @ F ) @ ( pow_a @ A ) )
= ( pow_nat @ B ) ) ) ).
% image_Pow_surj
thf(fact_715_image__Pow__surj,axiom,
! [F: a > nat > a,A: set_a,B: set_nat_a] :
( ( ( image_a_nat_a @ F @ A )
= B )
=> ( ( image_1117120162361407980_nat_a @ ( image_a_nat_a @ F ) @ ( pow_a @ A ) )
= ( pow_nat_a @ B ) ) ) ).
% image_Pow_surj
thf(fact_716_Cantors__paradox,axiom,
! [A: set_nat] :
~ ? [F2: nat > set_nat] :
( ( image_nat_set_nat @ F2 @ A )
= ( pow_nat @ A ) ) ).
% Cantors_paradox
thf(fact_717_Cantors__paradox,axiom,
! [A: set_a] :
~ ? [F2: a > set_a] :
( ( image_a_set_a @ F2 @ A )
= ( pow_a @ A ) ) ).
% Cantors_paradox
thf(fact_718_Pow__def,axiom,
( pow_nat
= ( ^ [A2: set_nat] :
( collect_set_nat
@ ^ [B4: set_nat] : ( ord_less_eq_set_nat @ B4 @ A2 ) ) ) ) ).
% Pow_def
thf(fact_719_Pow__def,axiom,
( pow_a
= ( ^ [A2: set_a] :
( collect_set_a
@ ^ [B4: set_a] : ( ord_less_eq_set_a @ B4 @ A2 ) ) ) ) ).
% Pow_def
thf(fact_720_Pow__def,axiom,
( pow_set_a
= ( ^ [A2: set_set_a] :
( collect_set_set_a
@ ^ [B4: set_set_a] : ( ord_le3724670747650509150_set_a @ B4 @ A2 ) ) ) ) ).
% Pow_def
thf(fact_721_cring_Ocring__simprules_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y2: a] :
( ( cring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( add_a_b @ R @ X3 @ Y2 ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% cring.cring_simprules(1)
thf(fact_722_cring_Ocring__simprules_I7_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y2: a,Z: a] :
( ( cring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( add_a_b @ R @ X3 @ Y2 ) @ Z )
= ( add_a_b @ R @ X3 @ ( add_a_b @ R @ Y2 @ Z ) ) ) ) ) ) ) ).
% cring.cring_simprules(7)
thf(fact_723_cring_Ocring__simprules_I10_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y2: a] :
( ( cring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X3 @ Y2 )
= ( add_a_b @ R @ Y2 @ X3 ) ) ) ) ) ).
% cring.cring_simprules(10)
thf(fact_724_cring_Ocring__simprules_I23_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y2: a,Z: a] :
( ( cring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X3 @ ( add_a_b @ R @ Y2 @ Z ) )
= ( add_a_b @ R @ Y2 @ ( add_a_b @ R @ X3 @ Z ) ) ) ) ) ) ) ).
% cring.cring_simprules(23)
thf(fact_725_cring_Ocring__simprules_I5_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y2: a] :
( ( cring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ X3 @ Y2 ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% cring.cring_simprules(5)
thf(fact_726_cring_Ocring__simprules_I11_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y2: a,Z: a] :
( ( cring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X3 @ Y2 ) @ Z )
= ( mult_a_ring_ext_a_b @ R @ X3 @ ( mult_a_ring_ext_a_b @ R @ Y2 @ Z ) ) ) ) ) ) ) ).
% cring.cring_simprules(11)
thf(fact_727_cring_Ocring__simprules_I14_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y2: a] :
( ( cring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ X3 @ Y2 )
= ( mult_a_ring_ext_a_b @ R @ Y2 @ X3 ) ) ) ) ) ).
% cring.cring_simprules(14)
thf(fact_728_cring_Ocring__simprules_I24_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y2: a,Z: a] :
( ( cring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ X3 @ ( mult_a_ring_ext_a_b @ R @ Y2 @ Z ) )
= ( mult_a_ring_ext_a_b @ R @ Y2 @ ( mult_a_ring_ext_a_b @ R @ X3 @ Z ) ) ) ) ) ) ) ).
% cring.cring_simprules(24)
thf(fact_729_image__Pow__mono,axiom,
! [F: nat > set_set_nat,A: set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( image_2194112158459175443et_nat @ F @ A ) @ B )
=> ( ord_le572741076514265352et_nat @ ( image_4583741654806091647et_nat @ ( image_2194112158459175443et_nat @ F ) @ ( pow_nat @ A ) ) @ ( pow_set_set_nat @ B ) ) ) ).
% image_Pow_mono
thf(fact_730_image__Pow__mono,axiom,
! [F: a > set_nat,A: set_a,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( image_a_set_nat @ F @ A ) @ B )
=> ( ord_le9131159989063066194et_nat @ ( image_8216882647274671445et_nat @ ( image_a_set_nat @ F ) @ ( pow_a @ A ) ) @ ( pow_set_nat @ B ) ) ) ).
% image_Pow_mono
thf(fact_731_image__Pow__mono,axiom,
! [F: a > set_nat_a,A: set_a,B: set_set_nat_a] :
( ( ord_le2390145808437456709_nat_a @ ( image_a_set_nat_a @ F @ A ) @ B )
=> ( ord_le2266563856530774437_nat_a @ ( image_7780357287214271564_nat_a @ ( image_a_set_nat_a @ F ) @ ( pow_a @ A ) ) @ ( pow_set_nat_a @ B ) ) ) ).
% image_Pow_mono
thf(fact_732_image__Pow__mono,axiom,
! [F: a > nat > a,A: set_a,B: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ ( image_a_nat_a @ F @ A ) @ B )
=> ( ord_le2390145808437456709_nat_a @ ( image_1117120162361407980_nat_a @ ( image_a_nat_a @ F ) @ ( pow_a @ A ) ) @ ( pow_nat_a @ B ) ) ) ).
% image_Pow_mono
thf(fact_733_image__Pow__mono,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B )
=> ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat @ F ) @ ( pow_nat @ A ) ) @ ( pow_nat @ B ) ) ) ).
% image_Pow_mono
thf(fact_734_image__Pow__mono,axiom,
! [F: a > nat,A: set_a,B: set_nat] :
( ( ord_less_eq_set_nat @ ( image_a_nat @ F @ A ) @ B )
=> ( ord_le6893508408891458716et_nat @ ( image_set_a_set_nat @ ( image_a_nat @ F ) @ ( pow_a @ A ) ) @ ( pow_nat @ B ) ) ) ).
% image_Pow_mono
thf(fact_735_image__Pow__mono,axiom,
! [F: nat > a,A: set_nat,B: set_a] :
( ( ord_less_eq_set_a @ ( image_nat_a @ F @ A ) @ B )
=> ( ord_le3724670747650509150_set_a @ ( image_set_nat_set_a @ ( image_nat_a @ F ) @ ( pow_nat @ A ) ) @ ( pow_a @ B ) ) ) ).
% image_Pow_mono
thf(fact_736_image__Pow__mono,axiom,
! [F: a > a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ B )
=> ( ord_le3724670747650509150_set_a @ ( image_set_a_set_a @ ( image_a_a @ F ) @ ( pow_a @ A ) ) @ ( pow_a @ B ) ) ) ).
% image_Pow_mono
thf(fact_737_image__Pow__mono,axiom,
! [F: nat > set_a,A: set_nat,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( image_nat_set_a @ F @ A ) @ B )
=> ( ord_le5722252365846178494_set_a @ ( image_8586572261461758321_set_a @ ( image_nat_set_a @ F ) @ ( pow_nat @ A ) ) @ ( pow_set_a @ B ) ) ) ).
% image_Pow_mono
thf(fact_738_image__Pow__mono,axiom,
! [F: a > set_a,A: set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ A ) @ B )
=> ( ord_le5722252365846178494_set_a @ ( image_4955109552351689957_set_a @ ( image_a_set_a @ F ) @ ( pow_a @ A ) ) @ ( pow_set_a @ B ) ) ) ).
% image_Pow_mono
thf(fact_739_Group_Onat__pow__Suc,axiom,
! [G2: partia2175431115845679010xt_a_b,X3: a,N: nat] :
( ( pow_a_1026414303147256608_b_nat @ G2 @ X3 @ ( suc @ N ) )
= ( mult_a_ring_ext_a_b @ G2 @ ( pow_a_1026414303147256608_b_nat @ G2 @ X3 @ N ) @ X3 ) ) ).
% Group.nat_pow_Suc
thf(fact_740_diff__Suc__diff__eq1,axiom,
! [K2: nat,J3: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J3 )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J3 @ K2 ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K2 ) @ ( suc @ J3 ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_741_diff__Suc__diff__eq2,axiom,
! [K2: nat,J3: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J3 )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J3 @ K2 ) ) @ I )
= ( minus_minus_nat @ ( suc @ J3 ) @ ( plus_plus_nat @ K2 @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_742_le__add__diff__inverse2,axiom,
! [B2: nat,A3: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A3 @ B2 ) @ B2 )
= A3 ) ) ).
% le_add_diff_inverse2
thf(fact_743_le__add__diff__inverse,axiom,
! [B2: nat,A3: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
=> ( ( plus_plus_nat @ B2 @ ( minus_minus_nat @ A3 @ B2 ) )
= A3 ) ) ).
% le_add_diff_inverse
thf(fact_744_card__Collect__le__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I5: nat] : ( ord_less_eq_nat @ I5 @ N ) ) )
= ( suc @ N ) ) ).
% card_Collect_le_nat
thf(fact_745_up__minus__closed,axiom,
! [P2: nat > a,Q2: nat > a] :
( ( member_nat_a @ P2 @ ( up_a_b @ r ) )
=> ( ( member_nat_a @ Q2 @ ( up_a_b @ r ) )
=> ( member_nat_a
@ ^ [I5: nat] : ( a_minus_a_b @ r @ ( P2 @ I5 ) @ ( Q2 @ I5 ) )
@ ( up_a_b @ r ) ) ) ) ).
% up_minus_closed
thf(fact_746_Nat_Oadd__diff__assoc,axiom,
! [K2: nat,J3: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J3 )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J3 @ K2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J3 ) @ K2 ) ) ) ).
% Nat.add_diff_assoc
thf(fact_747_Nat_Oadd__diff__assoc2,axiom,
! [K2: nat,J3: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J3 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J3 @ K2 ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J3 @ I ) @ K2 ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_748_Nat_Odiff__diff__right,axiom,
! [K2: nat,J3: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J3 )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J3 @ K2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K2 ) @ J3 ) ) ) ).
% Nat.diff_diff_right
thf(fact_749_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_750_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_751_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_752_Suc__diff__diff,axiom,
! [M: nat,N: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K2 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K2 ) ) ).
% Suc_diff_diff
thf(fact_753_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_754_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc_right
thf(fact_755_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_756_nat__add__left__cancel__le,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K2 @ M ) @ ( plus_plus_nat @ K2 @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_757_diff__diff__left,axiom,
! [I: nat,J3: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J3 ) @ K2 )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J3 @ K2 ) ) ) ).
% diff_diff_left
thf(fact_758_minus__closed,axiom,
! [X3: a,Y2: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( a_minus_a_b @ r @ X3 @ Y2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% minus_closed
thf(fact_759_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_760_le__trans,axiom,
! [I: nat,J3: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J3 )
=> ( ( ord_less_eq_nat @ J3 @ K2 )
=> ( ord_less_eq_nat @ I @ K2 ) ) ) ).
% le_trans
thf(fact_761_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_762_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_763_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_764_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K2: nat,B2: nat] :
( ( P @ K2 )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B2 ) )
=> ? [X: nat] :
( ( P @ X )
& ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_765_bounded__Max__nat,axiom,
! [P: nat > $o,X3: nat,M2: nat] :
( ( P @ X3 )
=> ( ! [X: nat] :
( ( P @ X )
=> ( ord_less_eq_nat @ X @ M2 ) )
=> ~ ! [M3: nat] :
( ( P @ M3 )
=> ~ ! [X5: nat] :
( ( P @ X5 )
=> ( ord_less_eq_nat @ X5 @ M3 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_766_Diff__mono,axiom,
! [A: set_nat,C2: set_nat,D: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ D @ B )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B ) @ ( minus_minus_set_nat @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_767_Diff__mono,axiom,
! [A: set_a,C2: set_a,D: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ D @ B )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ B ) @ ( minus_minus_set_a @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_768_Diff__mono,axiom,
! [A: set_set_a,C2: set_set_a,D: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ C2 )
=> ( ( ord_le3724670747650509150_set_a @ D @ B )
=> ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A @ B ) @ ( minus_5736297505244876581_set_a @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_769_Diff__subset,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B ) @ A ) ).
% Diff_subset
thf(fact_770_Diff__subset,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ B ) @ A ) ).
% Diff_subset
thf(fact_771_Diff__subset,axiom,
! [A: set_set_a,B: set_set_a] : ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A @ B ) @ A ) ).
% Diff_subset
thf(fact_772_double__diff,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ( minus_minus_set_nat @ B @ ( minus_minus_set_nat @ C2 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_773_double__diff,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ( minus_minus_set_a @ B @ ( minus_minus_set_a @ C2 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_774_double__diff,axiom,
! [A: set_set_a,B: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ C2 )
=> ( ( minus_5736297505244876581_set_a @ B @ ( minus_5736297505244876581_set_a @ C2 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_775_image__diff__subset,axiom,
! [F: nat > set_set_nat,A: set_nat,B: set_nat] : ( ord_le9131159989063066194et_nat @ ( minus_2447799839930672331et_nat @ ( image_2194112158459175443et_nat @ F @ A ) @ ( image_2194112158459175443et_nat @ F @ B ) ) @ ( image_2194112158459175443et_nat @ F @ ( minus_minus_set_nat @ A @ B ) ) ) ).
% image_diff_subset
thf(fact_776_image__diff__subset,axiom,
! [F: a > set_nat,A: set_a,B: set_a] : ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ ( image_a_set_nat @ F @ A ) @ ( image_a_set_nat @ F @ B ) ) @ ( image_a_set_nat @ F @ ( minus_minus_set_a @ A @ B ) ) ) ).
% image_diff_subset
thf(fact_777_image__diff__subset,axiom,
! [F: a > set_nat_a,A: set_a,B: set_a] : ( ord_le2390145808437456709_nat_a @ ( minus_2199387171230727820_nat_a @ ( image_a_set_nat_a @ F @ A ) @ ( image_a_set_nat_a @ F @ B ) ) @ ( image_a_set_nat_a @ F @ ( minus_minus_set_a @ A @ B ) ) ) ).
% image_diff_subset
thf(fact_778_image__diff__subset,axiom,
! [F: a > nat > a,A: set_a,B: set_a] : ( ord_le871467723717165285_nat_a @ ( minus_490503922182417452_nat_a @ ( image_a_nat_a @ F @ A ) @ ( image_a_nat_a @ F @ B ) ) @ ( image_a_nat_a @ F @ ( minus_minus_set_a @ A @ B ) ) ) ).
% image_diff_subset
thf(fact_779_image__diff__subset,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B ) ) @ ( image_nat_nat @ F @ ( minus_minus_set_nat @ A @ B ) ) ) ).
% image_diff_subset
thf(fact_780_image__diff__subset,axiom,
! [F: a > nat,A: set_a,B: set_a] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( image_a_nat @ F @ A ) @ ( image_a_nat @ F @ B ) ) @ ( image_a_nat @ F @ ( minus_minus_set_a @ A @ B ) ) ) ).
% image_diff_subset
thf(fact_781_image__diff__subset,axiom,
! [F: a > a,A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ ( image_a_a @ F @ A ) @ ( image_a_a @ F @ B ) ) @ ( image_a_a @ F @ ( minus_minus_set_a @ A @ B ) ) ) ).
% image_diff_subset
thf(fact_782_image__diff__subset,axiom,
! [F: a > set_a,A: set_a,B: set_a] : ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ ( image_a_set_a @ F @ A ) @ ( image_a_set_a @ F @ B ) ) @ ( image_a_set_a @ F @ ( minus_minus_set_a @ A @ B ) ) ) ).
% image_diff_subset
thf(fact_783_UN__extend__simps_I6_J,axiom,
! [A: nat > set_set_nat,C2: set_nat,B: set_set_nat] :
( ( minus_2163939370556025621et_nat @ ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ A @ C2 ) ) @ B )
= ( comple548664676211718543et_nat
@ ( image_2194112158459175443et_nat
@ ^ [X2: nat] : ( minus_2163939370556025621et_nat @ ( A @ X2 ) @ B )
@ C2 ) ) ) ).
% UN_extend_simps(6)
thf(fact_784_UN__extend__simps_I6_J,axiom,
! [A: a > set_a,C2: set_a,B: set_a] :
( ( minus_minus_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ A @ C2 ) ) @ B )
= ( comple2307003609928055243_set_a
@ ( image_a_set_a
@ ^ [X2: a] : ( minus_minus_set_a @ ( A @ X2 ) @ B )
@ C2 ) ) ) ).
% UN_extend_simps(6)
thf(fact_785_UN__extend__simps_I6_J,axiom,
! [A: a > set_nat_a,C2: set_a,B: set_nat_a] :
( ( minus_490503922182417452_nat_a @ ( comple3545767860446109490_nat_a @ ( image_a_set_nat_a @ A @ C2 ) ) @ B )
= ( comple3545767860446109490_nat_a
@ ( image_a_set_nat_a
@ ^ [X2: a] : ( minus_490503922182417452_nat_a @ ( A @ X2 ) @ B )
@ C2 ) ) ) ).
% UN_extend_simps(6)
thf(fact_786_UN__extend__simps_I6_J,axiom,
! [A: a > set_nat,C2: set_a,B: set_nat] :
( ( minus_minus_set_nat @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ A @ C2 ) ) @ B )
= ( comple7399068483239264473et_nat
@ ( image_a_set_nat
@ ^ [X2: a] : ( minus_minus_set_nat @ ( A @ X2 ) @ B )
@ C2 ) ) ) ).
% UN_extend_simps(6)
thf(fact_787_cring_Ocring__simprules_I4_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y2: a] :
( ( cring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( a_minus_a_b @ R @ X3 @ Y2 ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% cring.cring_simprules(4)
thf(fact_788_Suc__inject,axiom,
! [X3: nat,Y2: nat] :
( ( ( suc @ X3 )
= ( suc @ Y2 ) )
=> ( X3 = Y2 ) ) ).
% Suc_inject
thf(fact_789_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_790_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X: nat] : ( R @ X @ X )
=> ( ! [X: nat,Y3: nat,Z4: nat] :
( ( R @ X @ Y3 )
=> ( ( R @ Y3 @ Z4 )
=> ( R @ X @ Z4 ) ) )
=> ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
=> ( R @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_791_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_792_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M4: nat] :
( ( ord_less_eq_nat @ ( suc @ M4 ) @ N2 )
=> ( P @ M4 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_793_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_794_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_795_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_796_Suc__le__D,axiom,
! [N: nat,M5: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M5 )
=> ? [M3: nat] :
( M5
= ( suc @ M3 ) ) ) ).
% Suc_le_D
thf(fact_797_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_798_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_799_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_800_diff__commute,axiom,
! [I: nat,J3: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J3 ) @ K2 )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K2 ) @ J3 ) ) ).
% diff_commute
thf(fact_801_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_802_le__diff__iff_H,axiom,
! [A3: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ C )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A3 ) @ ( minus_minus_nat @ C @ B2 ) )
= ( ord_less_eq_nat @ B2 @ A3 ) ) ) ) ).
% le_diff_iff'
thf(fact_803_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_804_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_805_Nat_Odiff__diff__eq,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_806_le__diff__iff,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_807_eq__diff__iff,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ( minus_minus_nat @ M @ K2 )
= ( minus_minus_nat @ N @ K2 ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_808_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M6: nat,N4: nat] :
? [K: nat] :
( N4
= ( plus_plus_nat @ M6 @ K ) ) ) ) ).
% nat_le_iff_add
thf(fact_809_trans__le__add2,axiom,
! [I: nat,J3: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J3 )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J3 ) ) ) ).
% trans_le_add2
thf(fact_810_trans__le__add1,axiom,
! [I: nat,J3: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J3 )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J3 @ M ) ) ) ).
% trans_le_add1
thf(fact_811_add__le__mono1,axiom,
! [I: nat,J3: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J3 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J3 @ K2 ) ) ) ).
% add_le_mono1
thf(fact_812_add__le__mono,axiom,
! [I: nat,J3: nat,K2: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J3 )
=> ( ( ord_less_eq_nat @ K2 @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J3 @ L ) ) ) ) ).
% add_le_mono
thf(fact_813_le__Suc__ex,axiom,
! [K2: nat,L: nat] :
( ( ord_less_eq_nat @ K2 @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K2 @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_814_add__leD2,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K2 ) @ N )
=> ( ord_less_eq_nat @ K2 @ N ) ) ).
% add_leD2
thf(fact_815_add__leD1,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K2 ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_816_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_817_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_818_add__leE,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K2 ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K2 @ N ) ) ) ).
% add_leE
thf(fact_819_lift__Suc__mono__le,axiom,
! [F: nat > set_nat,N: nat,N5: nat] :
( ! [N2: nat] : ( ord_less_eq_set_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N5 )
=> ( ord_less_eq_set_nat @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_820_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N5: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N5 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_821_lift__Suc__mono__le,axiom,
! [F: nat > set_a,N: nat,N5: nat] :
( ! [N2: nat] : ( ord_less_eq_set_a @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N5 )
=> ( ord_less_eq_set_a @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_822_lift__Suc__mono__le,axiom,
! [F: nat > set_set_a,N: nat,N5: nat] :
( ! [N2: nat] : ( ord_le3724670747650509150_set_a @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N5 )
=> ( ord_le3724670747650509150_set_a @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_823_lift__Suc__antimono__le,axiom,
! [F: nat > set_nat,N: nat,N5: nat] :
( ! [N2: nat] : ( ord_less_eq_set_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N5 )
=> ( ord_less_eq_set_nat @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_824_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N5: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N5 )
=> ( ord_less_eq_nat @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_825_lift__Suc__antimono__le,axiom,
! [F: nat > set_a,N: nat,N5: nat] :
( ! [N2: nat] : ( ord_less_eq_set_a @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N5 )
=> ( ord_less_eq_set_a @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_826_lift__Suc__antimono__le,axiom,
! [F: nat > set_set_a,N: nat,N5: nat] :
( ! [N2: nat] : ( ord_le3724670747650509150_set_a @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N5 )
=> ( ord_le3724670747650509150_set_a @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_827_all__subset__image,axiom,
! [F: nat > nat,A: set_nat,P: set_nat > $o] :
( ( ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F @ A ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A )
=> ( P @ ( image_nat_nat @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_828_all__subset__image,axiom,
! [F: a > nat,A: set_a,P: set_nat > $o] :
( ( ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ ( image_a_nat @ F @ A ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A )
=> ( P @ ( image_a_nat @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_829_all__subset__image,axiom,
! [F: nat > a,A: set_nat,P: set_a > $o] :
( ( ! [B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ ( image_nat_a @ F @ A ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A )
=> ( P @ ( image_nat_a @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_830_all__subset__image,axiom,
! [F: a > a,A: set_a,P: set_a > $o] :
( ( ! [B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ ( image_a_a @ F @ A ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A )
=> ( P @ ( image_a_a @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_831_all__subset__image,axiom,
! [F: a > set_nat,A: set_a,P: set_set_nat > $o] :
( ( ! [B4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B4 @ ( image_a_set_nat @ F @ A ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A )
=> ( P @ ( image_a_set_nat @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_832_all__subset__image,axiom,
! [F: set_a > nat,A: set_set_a,P: set_nat > $o] :
( ( ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ ( image_set_a_nat @ F @ A ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B4 @ A )
=> ( P @ ( image_set_a_nat @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_833_all__subset__image,axiom,
! [F: set_a > a,A: set_set_a,P: set_a > $o] :
( ( ! [B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ ( image_set_a_a @ F @ A ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B4 @ A )
=> ( P @ ( image_set_a_a @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_834_all__subset__image,axiom,
! [F: nat > set_a,A: set_nat,P: set_set_a > $o] :
( ( ! [B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B4 @ ( image_nat_set_a @ F @ A ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A )
=> ( P @ ( image_nat_set_a @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_835_all__subset__image,axiom,
! [F: a > set_a,A: set_a,P: set_set_a > $o] :
( ( ! [B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B4 @ ( image_a_set_a @ F @ A ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A )
=> ( P @ ( image_a_set_a @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_836_all__subset__image,axiom,
! [F: nat > set_set_nat,A: set_nat,P: set_set_set_nat > $o] :
( ( ! [B4: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B4 @ ( image_2194112158459175443et_nat @ F @ A ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A )
=> ( P @ ( image_2194112158459175443et_nat @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_837_zero__induct__lemma,axiom,
! [P: nat > $o,K2: nat,I: nat] :
( ( P @ K2 )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus_nat @ K2 @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_838_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_839_nat__arith_Osuc1,axiom,
! [A: nat,K2: nat,A3: nat] :
( ( A
= ( plus_plus_nat @ K2 @ A3 ) )
=> ( ( suc @ A )
= ( plus_plus_nat @ K2 @ ( suc @ A3 ) ) ) ) ).
% nat_arith.suc1
thf(fact_840_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_841_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_842_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_843_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_844_diff__cancel2,axiom,
! [M: nat,K2: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K2 ) @ ( plus_plus_nat @ N @ K2 ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_845_Nat_Odiff__cancel,axiom,
! [K2: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K2 @ M ) @ ( plus_plus_nat @ K2 @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_846_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J3: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J3 )
=> ( ( ( minus_minus_nat @ J3 @ I )
= K2 )
= ( J3
= ( plus_plus_nat @ K2 @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_847_Nat_Odiff__add__assoc2,axiom,
! [K2: nat,J3: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J3 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J3 @ I ) @ K2 )
= ( plus_plus_nat @ ( minus_minus_nat @ J3 @ K2 ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_848_Nat_Odiff__add__assoc,axiom,
! [K2: nat,J3: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J3 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J3 ) @ K2 )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J3 @ K2 ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_849_Nat_Ole__diff__conv2,axiom,
! [K2: nat,J3: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J3 )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J3 @ K2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ J3 ) ) ) ).
% Nat.le_diff_conv2
thf(fact_850_le__diff__conv,axiom,
! [J3: nat,K2: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J3 @ K2 ) @ I )
= ( ord_less_eq_nat @ J3 @ ( plus_plus_nat @ I @ K2 ) ) ) ).
% le_diff_conv
thf(fact_851_add__le__imp__le__diff,axiom,
! [I: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ N )
=> ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K2 ) ) ) ).
% add_le_imp_le_diff
thf(fact_852_add__le__add__imp__diff__le,axiom,
! [I: nat,K2: nat,N: nat,J3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J3 @ K2 ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J3 @ K2 ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K2 ) @ J3 ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_853_a__lcos__m__assoc,axiom,
! [M2: set_a,G: a,H: a] :
( ( ord_less_eq_set_a @ M2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ G @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_l_coset_a_b @ r @ G @ ( a_l_coset_a_b @ r @ H @ M2 ) )
= ( a_l_coset_a_b @ r @ ( add_a_b @ r @ G @ H ) @ M2 ) ) ) ) ) ).
% a_lcos_m_assoc
thf(fact_854_add__diff__cancel__right_H,axiom,
! [A3: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A3 @ B2 ) @ B2 )
= A3 ) ).
% add_diff_cancel_right'
thf(fact_855_add__diff__cancel__right,axiom,
! [A3: nat,C: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
= ( minus_minus_nat @ A3 @ B2 ) ) ).
% add_diff_cancel_right
thf(fact_856_add__diff__cancel__left_H,axiom,
! [A3: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A3 @ B2 ) @ A3 )
= B2 ) ).
% add_diff_cancel_left'
thf(fact_857_add__diff__cancel__left,axiom,
! [C: nat,A3: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B2 ) )
= ( minus_minus_nat @ A3 @ B2 ) ) ).
% add_diff_cancel_left
thf(fact_858_add__le__cancel__left,axiom,
! [C: nat,A3: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B2 ) )
= ( ord_less_eq_nat @ A3 @ B2 ) ) ).
% add_le_cancel_left
thf(fact_859_add__right__cancel,axiom,
! [B2: nat,A3: nat,C: nat] :
( ( ( plus_plus_nat @ B2 @ A3 )
= ( plus_plus_nat @ C @ A3 ) )
= ( B2 = C ) ) ).
% add_right_cancel
thf(fact_860_add__left__cancel,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ( plus_plus_nat @ A3 @ B2 )
= ( plus_plus_nat @ A3 @ C ) )
= ( B2 = C ) ) ).
% add_left_cancel
thf(fact_861_Diff__iff,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A @ B ) )
= ( ( member_set_nat @ C @ A )
& ~ ( member_set_nat @ C @ B ) ) ) ).
% Diff_iff
thf(fact_862_Diff__iff,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A @ B ) )
= ( ( member_set_a @ C @ A )
& ~ ( member_set_a @ C @ B ) ) ) ).
% Diff_iff
thf(fact_863_Diff__iff,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) )
= ( ( member_nat @ C @ A )
& ~ ( member_nat @ C @ B ) ) ) ).
% Diff_iff
thf(fact_864_Diff__iff,axiom,
! [C: nat > a,A: set_nat_a,B: set_nat_a] :
( ( member_nat_a @ C @ ( minus_490503922182417452_nat_a @ A @ B ) )
= ( ( member_nat_a @ C @ A )
& ~ ( member_nat_a @ C @ B ) ) ) ).
% Diff_iff
thf(fact_865_Diff__iff,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
= ( ( member_a @ C @ A )
& ~ ( member_a @ C @ B ) ) ) ).
% Diff_iff
thf(fact_866_Diff__iff,axiom,
! [C: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C @ ( minus_2447799839930672331et_nat @ A @ B ) )
= ( ( member_set_set_nat @ C @ A )
& ~ ( member_set_set_nat @ C @ B ) ) ) ).
% Diff_iff
thf(fact_867_DiffI,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ A )
=> ( ~ ( member_set_nat @ C @ B )
=> ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A @ B ) ) ) ) ).
% DiffI
thf(fact_868_DiffI,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ A )
=> ( ~ ( member_set_a @ C @ B )
=> ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A @ B ) ) ) ) ).
% DiffI
thf(fact_869_DiffI,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ A )
=> ( ~ ( member_nat @ C @ B )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) ) ) ) ).
% DiffI
thf(fact_870_DiffI,axiom,
! [C: nat > a,A: set_nat_a,B: set_nat_a] :
( ( member_nat_a @ C @ A )
=> ( ~ ( member_nat_a @ C @ B )
=> ( member_nat_a @ C @ ( minus_490503922182417452_nat_a @ A @ B ) ) ) ) ).
% DiffI
thf(fact_871_DiffI,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ A )
=> ( ~ ( member_a @ C @ B )
=> ( member_a @ C @ ( minus_minus_set_a @ A @ B ) ) ) ) ).
% DiffI
thf(fact_872_DiffI,axiom,
! [C: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C @ A )
=> ( ~ ( member_set_set_nat @ C @ B )
=> ( member_set_set_nat @ C @ ( minus_2447799839930672331et_nat @ A @ B ) ) ) ) ).
% DiffI
thf(fact_873_a__l__coset__subset__G,axiom,
! [H2: set_a,X3: a] :
( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( a_l_coset_a_b @ r @ X3 @ H2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% a_l_coset_subset_G
thf(fact_874_add__le__cancel__right,axiom,
! [A3: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
= ( ord_less_eq_nat @ A3 @ B2 ) ) ).
% add_le_cancel_right
thf(fact_875_minus__set__def,axiom,
( minus_5736297505244876581_set_a
= ( ^ [A2: set_set_a,B4: set_set_a] :
( collect_set_a
@ ( minus_minus_set_a_o
@ ^ [X2: set_a] : ( member_set_a @ X2 @ A2 )
@ ^ [X2: set_a] : ( member_set_a @ X2 @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_876_minus__set__def,axiom,
( minus_490503922182417452_nat_a
= ( ^ [A2: set_nat_a,B4: set_nat_a] :
( collect_nat_a
@ ( minus_minus_nat_a_o
@ ^ [X2: nat > a] : ( member_nat_a @ X2 @ A2 )
@ ^ [X2: nat > a] : ( member_nat_a @ X2 @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_877_minus__set__def,axiom,
( minus_minus_set_a
= ( ^ [A2: set_a,B4: set_a] :
( collect_a
@ ( minus_minus_a_o
@ ^ [X2: a] : ( member_a @ X2 @ A2 )
@ ^ [X2: a] : ( member_a @ X2 @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_878_minus__set__def,axiom,
( minus_2447799839930672331et_nat
= ( ^ [A2: set_set_set_nat,B4: set_set_set_nat] :
( collect_set_set_nat
@ ( minus_463385787819020154_nat_o
@ ^ [X2: set_set_nat] : ( member_set_set_nat @ X2 @ A2 )
@ ^ [X2: set_set_nat] : ( member_set_set_nat @ X2 @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_879_minus__set__def,axiom,
( minus_2163939370556025621et_nat
= ( ^ [A2: set_set_nat,B4: set_set_nat] :
( collect_set_nat
@ ( minus_6910147592129066416_nat_o
@ ^ [X2: set_nat] : ( member_set_nat @ X2 @ A2 )
@ ^ [X2: set_nat] : ( member_set_nat @ X2 @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_880_minus__set__def,axiom,
( minus_minus_set_nat
= ( ^ [A2: set_nat,B4: set_nat] :
( collect_nat
@ ( minus_minus_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A2 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_881_DiffD2,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A @ B ) )
=> ~ ( member_set_nat @ C @ B ) ) ).
% DiffD2
thf(fact_882_DiffD2,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A @ B ) )
=> ~ ( member_set_a @ C @ B ) ) ).
% DiffD2
thf(fact_883_DiffD2,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) )
=> ~ ( member_nat @ C @ B ) ) ).
% DiffD2
thf(fact_884_DiffD2,axiom,
! [C: nat > a,A: set_nat_a,B: set_nat_a] :
( ( member_nat_a @ C @ ( minus_490503922182417452_nat_a @ A @ B ) )
=> ~ ( member_nat_a @ C @ B ) ) ).
% DiffD2
thf(fact_885_DiffD2,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
=> ~ ( member_a @ C @ B ) ) ).
% DiffD2
thf(fact_886_DiffD2,axiom,
! [C: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C @ ( minus_2447799839930672331et_nat @ A @ B ) )
=> ~ ( member_set_set_nat @ C @ B ) ) ).
% DiffD2
thf(fact_887_DiffD1,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A @ B ) )
=> ( member_set_nat @ C @ A ) ) ).
% DiffD1
thf(fact_888_DiffD1,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A @ B ) )
=> ( member_set_a @ C @ A ) ) ).
% DiffD1
thf(fact_889_DiffD1,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) )
=> ( member_nat @ C @ A ) ) ).
% DiffD1
thf(fact_890_DiffD1,axiom,
! [C: nat > a,A: set_nat_a,B: set_nat_a] :
( ( member_nat_a @ C @ ( minus_490503922182417452_nat_a @ A @ B ) )
=> ( member_nat_a @ C @ A ) ) ).
% DiffD1
thf(fact_891_DiffD1,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
=> ( member_a @ C @ A ) ) ).
% DiffD1
thf(fact_892_DiffD1,axiom,
! [C: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C @ ( minus_2447799839930672331et_nat @ A @ B ) )
=> ( member_set_set_nat @ C @ A ) ) ).
% DiffD1
thf(fact_893_DiffE,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A @ B ) )
=> ~ ( ( member_set_nat @ C @ A )
=> ( member_set_nat @ C @ B ) ) ) ).
% DiffE
thf(fact_894_DiffE,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A @ B ) )
=> ~ ( ( member_set_a @ C @ A )
=> ( member_set_a @ C @ B ) ) ) ).
% DiffE
thf(fact_895_DiffE,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) )
=> ~ ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% DiffE
thf(fact_896_DiffE,axiom,
! [C: nat > a,A: set_nat_a,B: set_nat_a] :
( ( member_nat_a @ C @ ( minus_490503922182417452_nat_a @ A @ B ) )
=> ~ ( ( member_nat_a @ C @ A )
=> ( member_nat_a @ C @ B ) ) ) ).
% DiffE
thf(fact_897_DiffE,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
=> ~ ( ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ) ).
% DiffE
thf(fact_898_DiffE,axiom,
! [C: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C @ ( minus_2447799839930672331et_nat @ A @ B ) )
=> ~ ( ( member_set_set_nat @ C @ A )
=> ( member_set_set_nat @ C @ B ) ) ) ).
% DiffE
thf(fact_899_set__diff__eq,axiom,
( minus_5736297505244876581_set_a
= ( ^ [A2: set_set_a,B4: set_set_a] :
( collect_set_a
@ ^ [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ~ ( member_set_a @ X2 @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_900_set__diff__eq,axiom,
( minus_490503922182417452_nat_a
= ( ^ [A2: set_nat_a,B4: set_nat_a] :
( collect_nat_a
@ ^ [X2: nat > a] :
( ( member_nat_a @ X2 @ A2 )
& ~ ( member_nat_a @ X2 @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_901_set__diff__eq,axiom,
( minus_minus_set_a
= ( ^ [A2: set_a,B4: set_a] :
( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A2 )
& ~ ( member_a @ X2 @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_902_set__diff__eq,axiom,
( minus_2447799839930672331et_nat
= ( ^ [A2: set_set_set_nat,B4: set_set_set_nat] :
( collect_set_set_nat
@ ^ [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ A2 )
& ~ ( member_set_set_nat @ X2 @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_903_set__diff__eq,axiom,
( minus_2163939370556025621et_nat
= ( ^ [A2: set_set_nat,B4: set_set_nat] :
( collect_set_nat
@ ^ [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
& ~ ( member_set_nat @ X2 @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_904_set__diff__eq,axiom,
( minus_minus_set_nat
= ( ^ [A2: set_nat,B4: set_nat] :
( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ~ ( member_nat @ X2 @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_905_add__right__imp__eq,axiom,
! [B2: nat,A3: nat,C: nat] :
( ( ( plus_plus_nat @ B2 @ A3 )
= ( plus_plus_nat @ C @ A3 ) )
=> ( B2 = C ) ) ).
% add_right_imp_eq
thf(fact_906_add__left__imp__eq,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ( plus_plus_nat @ A3 @ B2 )
= ( plus_plus_nat @ A3 @ C ) )
=> ( B2 = C ) ) ).
% add_left_imp_eq
thf(fact_907_add_Oleft__commute,axiom,
! [B2: nat,A3: nat,C: nat] :
( ( plus_plus_nat @ B2 @ ( plus_plus_nat @ A3 @ C ) )
= ( plus_plus_nat @ A3 @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add.left_commute
thf(fact_908_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A5: nat,B5: nat] : ( plus_plus_nat @ B5 @ A5 ) ) ) ).
% add.commute
thf(fact_909_add_Oassoc,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A3 @ B2 ) @ C )
= ( plus_plus_nat @ A3 @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add.assoc
thf(fact_910_group__cancel_Oadd2,axiom,
! [B: nat,K2: nat,B2: nat,A3: nat] :
( ( B
= ( plus_plus_nat @ K2 @ B2 ) )
=> ( ( plus_plus_nat @ A3 @ B )
= ( plus_plus_nat @ K2 @ ( plus_plus_nat @ A3 @ B2 ) ) ) ) ).
% group_cancel.add2
thf(fact_911_group__cancel_Oadd1,axiom,
! [A: nat,K2: nat,A3: nat,B2: nat] :
( ( A
= ( plus_plus_nat @ K2 @ A3 ) )
=> ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ K2 @ ( plus_plus_nat @ A3 @ B2 ) ) ) ) ).
% group_cancel.add1
thf(fact_912_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J3: nat,K2: nat,L: nat] :
( ( ( I = J3 )
& ( K2 = L ) )
=> ( ( plus_plus_nat @ I @ K2 )
= ( plus_plus_nat @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_913_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A3 @ B2 ) @ C )
= ( plus_plus_nat @ A3 @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_914_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A3: nat,C: nat,B2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A3 @ C ) @ B2 )
= ( minus_minus_nat @ ( minus_minus_nat @ A3 @ B2 ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_915_add__le__imp__le__right,axiom,
! [A3: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
=> ( ord_less_eq_nat @ A3 @ B2 ) ) ).
% add_le_imp_le_right
thf(fact_916_add__le__imp__le__left,axiom,
! [C: nat,A3: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B2 ) )
=> ( ord_less_eq_nat @ A3 @ B2 ) ) ).
% add_le_imp_le_left
thf(fact_917_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] :
? [C5: nat] :
( B5
= ( plus_plus_nat @ A5 @ C5 ) ) ) ) ).
% le_iff_add
thf(fact_918_add__right__mono,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add_right_mono
thf(fact_919_less__eqE,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ~ ! [C3: nat] :
( B2
!= ( plus_plus_nat @ A3 @ C3 ) ) ) ).
% less_eqE
thf(fact_920_add__left__mono,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).
% add_left_mono
thf(fact_921_add__mono,axiom,
! [A3: nat,B2: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B2 @ D2 ) ) ) ) ).
% add_mono
thf(fact_922_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J3: nat,K2: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J3 )
& ( ord_less_eq_nat @ K2 @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_923_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J3: nat,K2: nat,L: nat] :
( ( ( I = J3 )
& ( ord_less_eq_nat @ K2 @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_924_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J3: nat,K2: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J3 )
& ( K2 = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_925_add__implies__diff,axiom,
! [C: nat,B2: nat,A3: nat] :
( ( ( plus_plus_nat @ C @ B2 )
= A3 )
=> ( C
= ( minus_minus_nat @ A3 @ B2 ) ) ) ).
% add_implies_diff
thf(fact_926_diff__diff__eq,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A3 @ B2 ) @ C )
= ( minus_minus_nat @ A3 @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% diff_diff_eq
thf(fact_927_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( ( minus_minus_nat @ B2 @ A3 )
= C )
= ( B2
= ( plus_plus_nat @ C @ A3 ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_928_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( plus_plus_nat @ A3 @ ( minus_minus_nat @ B2 @ A3 ) )
= B2 ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_929_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B2 @ A3 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A3 ) @ B2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_930_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A3 )
= ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A3 ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_931_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A3 ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A3 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_932_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B2 ) @ A3 )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B2 @ A3 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_933_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B2 @ A3 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B2 ) @ A3 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_934_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B2 @ A3 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A3 ) @ B2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_935_le__add__diff,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A3 ) ) ) ).
% le_add_diff
thf(fact_936_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A3 ) @ A3 )
= B2 ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_937_cgenideal__is__principalideal,axiom,
! [I: a] :
( ( member_a @ I @ ( partia707051561876973205xt_a_b @ r ) )
=> ( principalideal_a_b @ ( cgenid547466209912283029xt_a_b @ r @ I ) @ r ) ) ).
% cgenideal_is_principalideal
thf(fact_938_carrier__is__subalgebra,axiom,
! [K3: set_a] :
( ( ord_less_eq_set_a @ K3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( embedd9027525575939734154ra_a_b @ K3 @ ( partia707051561876973205xt_a_b @ r ) @ r ) ) ).
% carrier_is_subalgebra
thf(fact_939_subalgebra__in__carrier,axiom,
! [K3: set_a,V2: set_a] :
( ( embedd9027525575939734154ra_a_b @ K3 @ V2 @ r )
=> ( ord_less_eq_set_a @ V2 @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% subalgebra_in_carrier
thf(fact_940_line__extension__smult__closed,axiom,
! [K3: set_a,E: set_a,A3: a,K2: a,U: a] :
( ( subfield_a_b @ K3 @ r )
=> ( ! [K4: a,V3: a] :
( ( member_a @ K4 @ K3 )
=> ( ( member_a @ V3 @ E )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ K4 @ V3 ) @ E ) ) )
=> ( ( ord_less_eq_set_a @ E @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ K2 @ K3 )
=> ( ( member_a @ U @ ( embedd971793762689825387on_a_b @ r @ K3 @ A3 @ E ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ K2 @ U ) @ ( embedd971793762689825387on_a_b @ r @ K3 @ A3 @ E ) ) ) ) ) ) ) ) ).
% line_extension_smult_closed
thf(fact_941_genideal__self,axiom,
! [S: set_a] :
( ( ord_less_eq_set_a @ S @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ S @ ( genideal_a_b @ r @ S ) ) ) ).
% genideal_self
thf(fact_942_subset__Idl__subset,axiom,
! [I4: set_a,H2: set_a] :
( ( ord_less_eq_set_a @ I4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ H2 @ I4 )
=> ( ord_less_eq_set_a @ ( genideal_a_b @ r @ H2 ) @ ( genideal_a_b @ r @ I4 ) ) ) ) ).
% subset_Idl_subset
thf(fact_943_a__lcos__mult__one,axiom,
! [M2: set_a] :
( ( ord_less_eq_set_a @ M2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_l_coset_a_b @ r @ ( zero_a_b @ r ) @ M2 )
= M2 ) ) ).
% a_lcos_mult_one
thf(fact_944_subring__props_I2_J,axiom,
! [K3: set_a] :
( ( subfield_a_b @ K3 @ r )
=> ( member_a @ ( zero_a_b @ r ) @ K3 ) ) ).
% subring_props(2)
thf(fact_945_subring__props_I7_J,axiom,
! [K3: set_a,H1: a,H22: a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_a @ H1 @ K3 )
=> ( ( member_a @ H22 @ K3 )
=> ( member_a @ ( add_a_b @ r @ H1 @ H22 ) @ K3 ) ) ) ) ).
% subring_props(7)
thf(fact_946_add_Ofinprod__one__eqI,axiom,
! [A: set_set_a,F: set_a > a] :
( ! [X: set_a] :
( ( member_set_a @ X @ A )
=> ( ( F @ X )
= ( zero_a_b @ r ) ) )
=> ( ( finsum_a_b_set_a @ r @ F @ A )
= ( zero_a_b @ r ) ) ) ).
% add.finprod_one_eqI
thf(fact_947_add_Ofinprod__one__eqI,axiom,
! [A: set_nat_a,F: ( nat > a ) > a] :
( ! [X: nat > a] :
( ( member_nat_a @ X @ A )
=> ( ( F @ X )
= ( zero_a_b @ r ) ) )
=> ( ( finsum_a_b_nat_a @ r @ F @ A )
= ( zero_a_b @ r ) ) ) ).
% add.finprod_one_eqI
thf(fact_948_add_Ofinprod__one__eqI,axiom,
! [A: set_a,F: a > a] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( ( F @ X )
= ( zero_a_b @ r ) ) )
=> ( ( finsum_a_b_a @ r @ F @ A )
= ( zero_a_b @ r ) ) ) ).
% add.finprod_one_eqI
thf(fact_949_add_Ofinprod__one__eqI,axiom,
! [A: set_set_set_nat,F: set_set_nat > a] :
( ! [X: set_set_nat] :
( ( member_set_set_nat @ X @ A )
=> ( ( F @ X )
= ( zero_a_b @ r ) ) )
=> ( ( finsum2649122254697571802et_nat @ r @ F @ A )
= ( zero_a_b @ r ) ) ) ).
% add.finprod_one_eqI
thf(fact_950_add_Ofinprod__one__eqI,axiom,
! [A: set_set_nat,F: set_nat > a] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ A )
=> ( ( F @ X )
= ( zero_a_b @ r ) ) )
=> ( ( finsum_a_b_set_nat @ r @ F @ A )
= ( zero_a_b @ r ) ) ) ).
% add.finprod_one_eqI
thf(fact_951_add_Ofinprod__one__eqI,axiom,
! [A: set_nat,F: nat > a] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ( F @ X )
= ( zero_a_b @ r ) ) )
=> ( ( finsum_a_b_nat @ r @ F @ A )
= ( zero_a_b @ r ) ) ) ).
% add.finprod_one_eqI
thf(fact_952_subring__props_I6_J,axiom,
! [K3: set_a,H1: a,H22: a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( member_a @ H1 @ K3 )
=> ( ( member_a @ H22 @ K3 )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ H1 @ H22 ) @ K3 ) ) ) ) ).
% subring_props(6)
thf(fact_953_cgenideal__self,axiom,
! [I: a] :
( ( member_a @ I @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ I @ ( cgenid547466209912283029xt_a_b @ r @ I ) ) ) ).
% cgenideal_self
thf(fact_954_local_Ominus__unique,axiom,
! [Y2: a,X3: a,Y7: a] :
( ( ( add_a_b @ r @ Y2 @ X3 )
= ( zero_a_b @ r ) )
=> ( ( ( add_a_b @ r @ X3 @ Y7 )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y7 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( Y2 = Y7 ) ) ) ) ) ) ).
% local.minus_unique
thf(fact_955_add_Or__inv__ex,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ? [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( add_a_b @ r @ X3 @ X )
= ( zero_a_b @ r ) ) ) ) ).
% add.r_inv_ex
thf(fact_956_add_Oone__unique,axiom,
! [U: a] :
( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ U @ X )
= X ) )
=> ( U
= ( zero_a_b @ r ) ) ) ) ).
% add.one_unique
thf(fact_957_add_Ol__inv__ex,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ? [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( add_a_b @ r @ X @ X3 )
= ( zero_a_b @ r ) ) ) ) ).
% add.l_inv_ex
thf(fact_958_add_Oinv__comm,axiom,
! [X3: a,Y2: a] :
( ( ( add_a_b @ r @ X3 @ Y2 )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ Y2 @ X3 )
= ( zero_a_b @ r ) ) ) ) ) ).
% add.inv_comm
thf(fact_959_subring__props_I1_J,axiom,
! [K3: set_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ord_less_eq_set_a @ K3 @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% subring_props(1)
thf(fact_960_zero__closed,axiom,
member_a @ ( zero_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).
% zero_closed
thf(fact_961_finsum__zero,axiom,
! [A: set_set_nat] :
( ( finsum_a_b_set_nat @ r
@ ^ [I5: set_nat] : ( zero_a_b @ r )
@ A )
= ( zero_a_b @ r ) ) ).
% finsum_zero
thf(fact_962_finsum__zero,axiom,
! [A: set_nat] :
( ( finsum_a_b_nat @ r
@ ^ [I5: nat] : ( zero_a_b @ r )
@ A )
= ( zero_a_b @ r ) ) ).
% finsum_zero
thf(fact_963_r__zero,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X3 @ ( zero_a_b @ r ) )
= X3 ) ) ).
% r_zero
thf(fact_964_l__zero,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( zero_a_b @ r ) @ X3 )
= X3 ) ) ).
% l_zero
thf(fact_965_add_Or__cancel__one_H,axiom,
! [X3: a,A3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X3
= ( add_a_b @ r @ A3 @ X3 ) )
= ( A3
= ( zero_a_b @ r ) ) ) ) ) ).
% add.r_cancel_one'
thf(fact_966_add_Or__cancel__one,axiom,
! [X3: a,A3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ A3 @ X3 )
= X3 )
= ( A3
= ( zero_a_b @ r ) ) ) ) ) ).
% add.r_cancel_one
thf(fact_967_add_Ol__cancel__one_H,axiom,
! [X3: a,A3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X3
= ( add_a_b @ r @ X3 @ A3 ) )
= ( A3
= ( zero_a_b @ r ) ) ) ) ) ).
% add.l_cancel_one'
thf(fact_968_add_Ol__cancel__one,axiom,
! [X3: a,A3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ X3 @ A3 )
= X3 )
= ( A3
= ( zero_a_b @ r ) ) ) ) ) ).
% add.l_cancel_one
thf(fact_969_r__null,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X3 @ ( zero_a_b @ r ) )
= ( zero_a_b @ r ) ) ) ).
% r_null
thf(fact_970_l__null,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) @ X3 )
= ( zero_a_b @ r ) ) ) ).
% l_null
thf(fact_971_r__right__minus__eq,axiom,
! [A3: a,B2: a] :
( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( a_minus_a_b @ r @ A3 @ B2 )
= ( zero_a_b @ r ) )
= ( A3 = B2 ) ) ) ) ).
% r_right_minus_eq
thf(fact_972_cring_Ocring__simprules_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( cring_a_b @ R )
=> ( member_a @ ( zero_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% cring.cring_simprules(2)
thf(fact_973_cring_Ocring__simprules_I16_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a] :
( ( cring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X3 @ ( zero_a_b @ R ) )
= X3 ) ) ) ).
% cring.cring_simprules(16)
thf(fact_974_cring_Ocring__simprules_I8_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a] :
( ( cring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X3 )
= X3 ) ) ) ).
% cring.cring_simprules(8)
thf(fact_975_cring_Ocring__simprules_I27_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a] :
( ( cring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ X3 @ ( zero_a_b @ R ) )
= ( zero_a_b @ R ) ) ) ) ).
% cring.cring_simprules(27)
thf(fact_976_cring_Ocring__simprules_I26_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a] :
( ( cring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( zero_a_b @ R ) @ X3 )
= ( zero_a_b @ R ) ) ) ) ).
% cring.cring_simprules(26)
thf(fact_977_bound__upD,axiom,
! [F: nat > a] :
( ( member_nat_a @ F @ ( up_a_b @ r ) )
=> ? [N2: nat] : ( bound_a @ ( zero_a_b @ r ) @ N2 @ F ) ) ).
% bound_upD
thf(fact_978_subalbegra__incl__imp__finite__dimension,axiom,
! [K3: set_a,E: set_a,V2: set_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( embedd8708762675212832759on_a_b @ r @ K3 @ E )
=> ( ( embedd9027525575939734154ra_a_b @ K3 @ V2 @ r )
=> ( ( ord_less_eq_set_a @ V2 @ E )
=> ( embedd8708762675212832759on_a_b @ r @ K3 @ V2 ) ) ) ) ) ).
% subalbegra_incl_imp_finite_dimension
thf(fact_979_cring_Ocgenideal__is__principalideal,axiom,
! [R: partia2175431115845679010xt_a_b,I: a] :
( ( cring_a_b @ R )
=> ( ( member_a @ I @ ( partia707051561876973205xt_a_b @ R ) )
=> ( principalideal_a_b @ ( cgenid547466209912283029xt_a_b @ R @ I ) @ R ) ) ) ).
% cring.cgenideal_is_principalideal
thf(fact_980_finite__dimension__imp__subalgebra,axiom,
! [K3: set_a,E: set_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( embedd8708762675212832759on_a_b @ r @ K3 @ E )
=> ( embedd9027525575939734154ra_a_b @ K3 @ E @ r ) ) ) ).
% finite_dimension_imp_subalgebra
thf(fact_981_nat__pow__zero,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ ( zero_a_b @ r ) @ N )
= ( zero_a_b @ r ) ) ) ).
% nat_pow_zero
thf(fact_982_telescopic__base__dim_I1_J,axiom,
! [K3: set_a,F3: set_a,E: set_a] :
( ( subfield_a_b @ K3 @ r )
=> ( ( subfield_a_b @ F3 @ r )
=> ( ( embedd8708762675212832759on_a_b @ r @ K3 @ F3 )
=> ( ( embedd8708762675212832759on_a_b @ r @ F3 @ E )
=> ( embedd8708762675212832759on_a_b @ r @ K3 @ E ) ) ) ) ) ).
% telescopic_base_dim(1)
thf(fact_983_up__mult__closed,axiom,
! [P2: nat > a,Q2: nat > a] :
( ( member_nat_a @ P2 @ ( up_a_b @ r ) )
=> ( ( member_nat_a @ Q2 @ ( up_a_b @ r ) )
=> ( member_nat_a
@ ^ [N4: nat] :
( finsum_a_b_nat @ r
@ ^ [I5: nat] : ( mult_a_ring_ext_a_b @ r @ ( P2 @ I5 ) @ ( Q2 @ ( minus_minus_nat @ N4 @ I5 ) ) )
@ ( set_ord_atMost_nat @ N4 ) )
@ ( up_a_b @ r ) ) ) ) ).
% up_mult_closed
thf(fact_984_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_985_add__0,axiom,
! [A3: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A3 )
= A3 ) ).
% add_0
thf(fact_986_zero__eq__add__iff__both__eq__0,axiom,
! [X3: nat,Y2: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X3 @ Y2 ) )
= ( ( X3 = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_987_add__eq__0__iff__both__eq__0,axiom,
! [X3: nat,Y2: nat] :
( ( ( plus_plus_nat @ X3 @ Y2 )
= zero_zero_nat )
= ( ( X3 = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_988_add__cancel__right__right,axiom,
! [A3: nat,B2: nat] :
( ( A3
= ( plus_plus_nat @ A3 @ B2 ) )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_989_add__cancel__right__left,axiom,
! [A3: nat,B2: nat] :
( ( A3
= ( plus_plus_nat @ B2 @ A3 ) )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_990_add__cancel__left__right,axiom,
! [A3: nat,B2: nat] :
( ( ( plus_plus_nat @ A3 @ B2 )
= A3 )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_991_add__cancel__left__left,axiom,
! [B2: nat,A3: nat] :
( ( ( plus_plus_nat @ B2 @ A3 )
= A3 )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_992_add_Oright__neutral,axiom,
! [A3: nat] :
( ( plus_plus_nat @ A3 @ zero_zero_nat )
= A3 ) ).
% add.right_neutral
thf(fact_993_zero__diff,axiom,
! [A3: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A3 )
= zero_zero_nat ) ).
% zero_diff
thf(fact_994_diff__zero,axiom,
! [A3: nat] :
( ( minus_minus_nat @ A3 @ zero_zero_nat )
= A3 ) ).
% diff_zero
thf(fact_995_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A3: nat] :
( ( minus_minus_nat @ A3 @ A3 )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_996_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_997_bot__nat__0_Oextremum,axiom,
! [A3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A3 ) ).
% bot_nat_0.extremum
thf(fact_998_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_999_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1000_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_1001_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1002_add__le__same__cancel1,axiom,
! [B2: nat,A3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B2 @ A3 ) @ B2 )
= ( ord_less_eq_nat @ A3 @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_1003_add__le__same__cancel2,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ B2 ) @ B2 )
= ( ord_less_eq_nat @ A3 @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_1004_le__add__same__cancel1,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ ( plus_plus_nat @ A3 @ B2 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).
% le_add_same_cancel1
thf(fact_1005_le__add__same__cancel2,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ ( plus_plus_nat @ B2 @ A3 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).
% le_add_same_cancel2
thf(fact_1006_diff__add__zero,axiom,
! [A3: nat,B2: nat] :
( ( minus_minus_nat @ A3 @ ( plus_plus_nat @ A3 @ B2 ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_1007_image__add__0,axiom,
! [S: set_nat] :
( ( image_nat_nat @ ( plus_plus_nat @ zero_zero_nat ) @ S )
= S ) ).
% image_add_0
thf(fact_1008_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_1009_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1010_zero__reorient,axiom,
! [X3: nat] :
( ( zero_zero_nat = X3 )
= ( X3 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_1011_zero__le,axiom,
! [X3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X3 ) ).
% zero_le
thf(fact_1012_comm__monoid__add__class_Oadd__0,axiom,
! [A3: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A3 )
= A3 ) ).
% comm_monoid_add_class.add_0
thf(fact_1013_add_Ocomm__neutral,axiom,
! [A3: nat] :
( ( plus_plus_nat @ A3 @ zero_zero_nat )
= A3 ) ).
% add.comm_neutral
thf(fact_1014_zero__notin__Suc__image,axiom,
! [A: set_nat] :
~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A ) ) ).
% zero_notin_Suc_image
thf(fact_1015_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% not0_implies_Suc
thf(fact_1016_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_1017_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_1018_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_1019_zero__induct,axiom,
! [P: nat > $o,K2: nat] :
( ( P @ K2 )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_1020_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X: nat] : ( P @ X @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X: nat,Y3: nat] :
( ( P @ X @ Y3 )
=> ( P @ ( suc @ X ) @ ( suc @ Y3 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_1021_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_1022_old_Onat_Oexhaust,axiom,
! [Y2: nat] :
( ( Y2 != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y2
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_1023_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_1024_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_1025_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_1026_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_1027_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1028_bot__nat__0_Oextremum__uniqueI,axiom,
! [A3: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
=> ( A3 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_1029_bot__nat__0_Oextremum__unique,axiom,
! [A3: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
= ( A3 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_1030_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_1031_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_1032_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_1033_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1034_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1035_add__decreasing,axiom,
! [A3: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ B2 ) ) ) ).
% add_decreasing
thf(fact_1036_add__increasing,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A3 @ C ) ) ) ) ).
% add_increasing
thf(fact_1037_add__decreasing2,axiom,
! [C: nat,A3: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ B2 ) ) ) ).
% add_decreasing2
thf(fact_1038_add__increasing2,axiom,
! [C: nat,B2: nat,A3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B2 @ A3 )
=> ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A3 @ C ) ) ) ) ).
% add_increasing2
thf(fact_1039_add__nonneg__nonneg,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A3 @ B2 ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1040_add__nonpos__nonpos,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ B2 ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_1041_add__nonneg__eq__0__iff,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y2 )
=> ( ( ( plus_plus_nat @ X3 @ Y2 )
= zero_zero_nat )
= ( ( X3 = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1042_add__nonpos__eq__0__iff,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y2 @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X3 @ Y2 )
= zero_zero_nat )
= ( ( X3 = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1043_one__is__add,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_1044_add__is__1,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_1045_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1046_principalideal_Ois__principalideal,axiom,
! [I4: set_a,R: partia2175431115845679010xt_a_b] :
( ( principalideal_a_b @ I4 @ R )
=> ( principalideal_a_b @ I4 @ R ) ) ).
% principalideal.is_principalideal
thf(fact_1047_mem__upI,axiom,
! [F: nat > a,R: partia2175431115845679010xt_a_b] :
( ! [N2: nat] : ( member_a @ ( F @ N2 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ? [N6: nat] : ( bound_a @ ( zero_a_b @ R ) @ N6 @ F )
=> ( member_nat_a @ F @ ( up_a_b @ R ) ) ) ) ).
% mem_upI
thf(fact_1048_cauchy__product,axiom,
! [N: nat,F: nat > a,M: nat,G: nat > a] :
( ( bound_a @ ( zero_a_b @ r ) @ N @ F )
=> ( ( bound_a @ ( zero_a_b @ r ) @ M @ G )
=> ( ( member_nat_a @ F
@ ( pi_nat_a @ ( set_ord_atMost_nat @ N )
@ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_nat_a @ G
@ ( pi_nat_a @ ( set_ord_atMost_nat @ M )
@ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum_a_b_nat @ r
@ ^ [K: nat] :
( finsum_a_b_nat @ r
@ ^ [I5: nat] : ( mult_a_ring_ext_a_b @ r @ ( F @ I5 ) @ ( G @ ( minus_minus_nat @ K @ I5 ) ) )
@ ( set_ord_atMost_nat @ K ) )
@ ( set_ord_atMost_nat @ ( plus_plus_nat @ N @ M ) ) )
= ( mult_a_ring_ext_a_b @ r @ ( finsum_a_b_nat @ r @ F @ ( set_ord_atMost_nat @ N ) ) @ ( finsum_a_b_nat @ r @ G @ ( set_ord_atMost_nat @ M ) ) ) ) ) ) ) ) ).
% cauchy_product
thf(fact_1049_geom,axiom,
! [A3: a,Q2: nat] :
( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( a_minus_a_b @ r @ A3 @ ( one_a_ring_ext_a_b @ r ) ) @ ( finsum_a_b_nat @ r @ ( pow_a_1026414303147256608_b_nat @ r @ A3 ) @ ( set_ord_lessThan_nat @ Q2 ) ) )
= ( a_minus_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ A3 @ Q2 ) @ ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% geom
thf(fact_1050_boundD__carrier,axiom,
! [N: nat,F: nat > a,M: nat] :
( ( bound_a @ ( zero_a_b @ r ) @ N @ F )
=> ( ( ord_less_nat @ N @ M )
=> ( member_a @ ( F @ M ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% boundD_carrier
thf(fact_1051_diagonal__sum,axiom,
! [F: nat > a,N: nat,M: nat,G: nat > a] :
( ( member_nat_a @ F
@ ( pi_nat_a @ ( set_ord_atMost_nat @ ( plus_plus_nat @ N @ M ) )
@ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_nat_a @ G
@ ( pi_nat_a @ ( set_ord_atMost_nat @ ( plus_plus_nat @ N @ M ) )
@ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum_a_b_nat @ r
@ ^ [K: nat] :
( finsum_a_b_nat @ r
@ ^ [I5: nat] : ( mult_a_ring_ext_a_b @ r @ ( F @ I5 ) @ ( G @ ( minus_minus_nat @ K @ I5 ) ) )
@ ( set_ord_atMost_nat @ K ) )
@ ( set_ord_atMost_nat @ ( plus_plus_nat @ N @ M ) ) )
= ( finsum_a_b_nat @ r
@ ^ [K: nat] :
( finsum_a_b_nat @ r
@ ^ [I5: nat] : ( mult_a_ring_ext_a_b @ r @ ( F @ K ) @ ( G @ I5 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ K ) ) )
@ ( set_ord_atMost_nat @ ( plus_plus_nat @ N @ M ) ) ) ) ) ) ).
% diagonal_sum
thf(fact_1052_subring__props_I3_J,axiom,
! [K3: set_a] :
( ( subfield_a_b @ K3 @ r )
=> ( member_a @ ( one_a_ring_ext_a_b @ r ) @ K3 ) ) ).
% subring_props(3)
thf(fact_1053_one__unique,axiom,
! [U: a] :
( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ U @ X )
= X ) )
=> ( U
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% one_unique
thf(fact_1054_inv__unique,axiom,
! [Y2: a,X3: a,Y7: a] :
( ( ( mult_a_ring_ext_a_b @ r @ Y2 @ X3 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X3 @ Y7 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y7 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( Y2 = Y7 ) ) ) ) ) ) ).
% inv_unique
thf(fact_1055_finsum__cong_H,axiom,
! [A: set_set_a,B: set_set_a,G: set_a > a,F: set_a > a] :
( ( A = B )
=> ( ( member_set_a_a @ G
@ ( pi_set_a_a @ B
@ ^ [Uu: set_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ! [I2: set_a] :
( ( member_set_a @ I2 @ B )
=> ( ( F @ I2 )
= ( G @ I2 ) ) )
=> ( ( finsum_a_b_set_a @ r @ F @ A )
= ( finsum_a_b_set_a @ r @ G @ B ) ) ) ) ) ).
% finsum_cong'
thf(fact_1056_finsum__cong_H,axiom,
! [A: set_nat_a,B: set_nat_a,G: ( nat > a ) > a,F: ( nat > a ) > a] :
( ( A = B )
=> ( ( member_nat_a_a @ G
@ ( pi_nat_a_a @ B
@ ^ [Uu: nat > a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ! [I2: nat > a] :
( ( member_nat_a @ I2 @ B )
=> ( ( F @ I2 )
= ( G @ I2 ) ) )
=> ( ( finsum_a_b_nat_a @ r @ F @ A )
= ( finsum_a_b_nat_a @ r @ G @ B ) ) ) ) ) ).
% finsum_cong'
thf(fact_1057_finsum__cong_H,axiom,
! [A: set_a,B: set_a,G: a > a,F: a > a] :
( ( A = B )
=> ( ( member_a_a @ G
@ ( pi_a_a @ B
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ! [I2: a] :
( ( member_a @ I2 @ B )
=> ( ( F @ I2 )
= ( G @ I2 ) ) )
=> ( ( finsum_a_b_a @ r @ F @ A )
= ( finsum_a_b_a @ r @ G @ B ) ) ) ) ) ).
% finsum_cong'
thf(fact_1058_finsum__cong_H,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,G: set_set_nat > a,F: set_set_nat > a] :
( ( A = B )
=> ( ( member_set_set_nat_a @ G
@ ( pi_set_set_nat_a @ B
@ ^ [Uu: set_set_nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ! [I2: set_set_nat] :
( ( member_set_set_nat @ I2 @ B )
=> ( ( F @ I2 )
= ( G @ I2 ) ) )
=> ( ( finsum2649122254697571802et_nat @ r @ F @ A )
= ( finsum2649122254697571802et_nat @ r @ G @ B ) ) ) ) ) ).
% finsum_cong'
thf(fact_1059_finsum__cong_H,axiom,
! [A: set_set_nat,B: set_set_nat,G: set_nat > a,F: set_nat > a] :
( ( A = B )
=> ( ( member_set_nat_a @ G
@ ( pi_set_nat_a @ B
@ ^ [Uu: set_nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ! [I2: set_nat] :
( ( member_set_nat @ I2 @ B )
=> ( ( F @ I2 )
= ( G @ I2 ) ) )
=> ( ( finsum_a_b_set_nat @ r @ F @ A )
= ( finsum_a_b_set_nat @ r @ G @ B ) ) ) ) ) ).
% finsum_cong'
thf(fact_1060_finsum__cong_H,axiom,
! [A: set_nat,B: set_nat,G: nat > a,F: nat > a] :
( ( A = B )
=> ( ( member_nat_a @ G
@ ( pi_nat_a @ B
@ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ B )
=> ( ( F @ I2 )
= ( G @ I2 ) ) )
=> ( ( finsum_a_b_nat @ r @ F @ A )
= ( finsum_a_b_nat @ r @ G @ B ) ) ) ) ) ).
% finsum_cong'
thf(fact_1061_finsum__closed,axiom,
! [F: set_nat > a,A: set_set_nat] :
( ( member_set_nat_a @ F
@ ( pi_set_nat_a @ A
@ ^ [Uu: set_nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( member_a @ ( finsum_a_b_set_nat @ r @ F @ A ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% finsum_closed
thf(fact_1062_finsum__closed,axiom,
! [F: nat > a,A: set_nat] :
( ( member_nat_a @ F
@ ( pi_nat_a @ A
@ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( member_a @ ( finsum_a_b_nat @ r @ F @ A ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% finsum_closed
thf(fact_1063_up__one__closed,axiom,
( member_nat_a
@ ^ [N4: nat] : ( if_a @ ( N4 = zero_zero_nat ) @ ( one_a_ring_ext_a_b @ r ) @ ( zero_a_b @ r ) )
@ ( up_a_b @ r ) ) ).
% up_one_closed
thf(fact_1064_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_1065_add__less__cancel__left,axiom,
! [C: nat,A3: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B2 ) )
= ( ord_less_nat @ A3 @ B2 ) ) ).
% add_less_cancel_left
thf(fact_1066_add__less__cancel__right,axiom,
! [A3: nat,C: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
= ( ord_less_nat @ A3 @ B2 ) ) ).
% add_less_cancel_right
thf(fact_1067_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_1068_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_1069_bot__nat__0_Onot__eq__extremum,axiom,
! [A3: nat] :
( ( A3 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A3 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_1070_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_1071_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_1072_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_1073_lessThan__iff,axiom,
! [I: set_nat,K2: set_nat] :
( ( member_set_nat @ I @ ( set_or890127255671739683et_nat @ K2 ) )
= ( ord_less_set_nat @ I @ K2 ) ) ).
% lessThan_iff
thf(fact_1074_lessThan__iff,axiom,
! [I: set_a,K2: set_a] :
( ( member_set_a @ I @ ( set_or5421148953861284865_set_a @ K2 ) )
= ( ord_less_set_a @ I @ K2 ) ) ).
% lessThan_iff
thf(fact_1075_lessThan__iff,axiom,
! [I: set_set_nat,K2: set_set_nat] :
( ( member_set_set_nat @ I @ ( set_or6631954706645296601et_nat @ K2 ) )
= ( ord_less_set_set_nat @ I @ K2 ) ) ).
% lessThan_iff
thf(fact_1076_lessThan__iff,axiom,
! [I: nat,K2: nat] :
( ( member_nat @ I @ ( set_ord_lessThan_nat @ K2 ) )
= ( ord_less_nat @ I @ K2 ) ) ).
% lessThan_iff
thf(fact_1077_nat__add__left__cancel__less,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K2 @ M ) @ ( plus_plus_nat @ K2 @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_1078_add_Ofinprod__Suc3,axiom,
! [F: nat > a,N: nat] :
( ( member_nat_a @ F
@ ( pi_nat_a @ ( set_ord_atMost_nat @ N )
@ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum_a_b_nat @ r @ F @ ( set_ord_atMost_nat @ N ) )
= ( add_a_b @ r @ ( F @ N ) @ ( finsum_a_b_nat @ r @ F @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% add.finprod_Suc3
thf(fact_1079_finsum__Suc2,axiom,
! [F: nat > a,N: nat] :
( ( member_nat_a @ F
@ ( pi_nat_a @ ( set_ord_atMost_nat @ ( suc @ N ) )
@ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum_a_b_nat @ r @ F @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( add_a_b @ r
@ ( finsum_a_b_nat @ r
@ ^ [I5: nat] : ( F @ ( suc @ I5 ) )
@ ( set_ord_atMost_nat @ N ) )
@ ( F @ zero_zero_nat ) ) ) ) ).
% finsum_Suc2
thf(fact_1080_card__Collect__less__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I5: nat] : ( ord_less_nat @ I5 @ N ) ) )
= N ) ).
% card_Collect_less_nat
thf(fact_1081_less__add__same__cancel2,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ ( plus_plus_nat @ B2 @ A3 ) )
= ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).
% less_add_same_cancel2
thf(fact_1082_less__add__same__cancel1,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ ( plus_plus_nat @ A3 @ B2 ) )
= ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).
% less_add_same_cancel1
thf(fact_1083_add__less__same__cancel2,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A3 @ B2 ) @ B2 )
= ( ord_less_nat @ A3 @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_1084_add__less__same__cancel1,axiom,
! [B2: nat,A3: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B2 @ A3 ) @ B2 )
= ( ord_less_nat @ A3 @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_1085_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1086_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_1087_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_1088_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_1089_Group_Onat__pow__0,axiom,
! [G2: partia2175431115845679010xt_a_b,X3: a] :
( ( pow_a_1026414303147256608_b_nat @ G2 @ X3 @ zero_zero_nat )
= ( one_a_ring_ext_a_b @ G2 ) ) ).
% Group.nat_pow_0
thf(fact_1090_one__closed,axiom,
member_a @ ( one_a_ring_ext_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).
% one_closed
thf(fact_1091_nat__pow__one,axiom,
! [N: nat] :
( ( pow_a_1026414303147256608_b_nat @ r @ ( one_a_ring_ext_a_b @ r ) @ N )
= ( one_a_ring_ext_a_b @ r ) ) ).
% nat_pow_one
thf(fact_1092_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_1093_r__one,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X3 @ ( one_a_ring_ext_a_b @ r ) )
= X3 ) ) ).
% r_one
thf(fact_1094_l__one,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( one_a_ring_ext_a_b @ r ) @ X3 )
= X3 ) ) ).
% l_one
thf(fact_1095_local_Onat__pow__0,axiom,
! [X3: a] :
( ( pow_a_1026414303147256608_b_nat @ r @ X3 @ zero_zero_nat )
= ( one_a_ring_ext_a_b @ r ) ) ).
% local.nat_pow_0
thf(fact_1096_finsum__addf,axiom,
! [F: set_nat > a,A: set_set_nat,G: set_nat > a] :
( ( member_set_nat_a @ F
@ ( pi_set_nat_a @ A
@ ^ [Uu: set_nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_set_nat_a @ G
@ ( pi_set_nat_a @ A
@ ^ [Uu: set_nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum_a_b_set_nat @ r
@ ^ [X2: set_nat] : ( add_a_b @ r @ ( F @ X2 ) @ ( G @ X2 ) )
@ A )
= ( add_a_b @ r @ ( finsum_a_b_set_nat @ r @ F @ A ) @ ( finsum_a_b_set_nat @ r @ G @ A ) ) ) ) ) ).
% finsum_addf
thf(fact_1097_finsum__addf,axiom,
! [F: nat > a,A: set_nat,G: nat > a] :
( ( member_nat_a @ F
@ ( pi_nat_a @ A
@ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_nat_a @ G
@ ( pi_nat_a @ A
@ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum_a_b_nat @ r
@ ^ [X2: nat] : ( add_a_b @ r @ ( F @ X2 ) @ ( G @ X2 ) )
@ A )
= ( add_a_b @ r @ ( finsum_a_b_nat @ r @ F @ A ) @ ( finsum_a_b_nat @ r @ G @ A ) ) ) ) ) ).
% finsum_addf
thf(fact_1098_finsum__Suc,axiom,
! [F: nat > a,N: nat] :
( ( member_nat_a @ F
@ ( pi_nat_a @ ( set_ord_atMost_nat @ ( suc @ N ) )
@ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum_a_b_nat @ r @ F @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( add_a_b @ r @ ( F @ ( suc @ N ) ) @ ( finsum_a_b_nat @ r @ F @ ( set_ord_atMost_nat @ N ) ) ) ) ) ).
% finsum_Suc
thf(fact_1099_lessThan__strict__subset__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M ) @ ( set_ord_lessThan_nat @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_1100_linorder__neqE__nat,axiom,
! [X3: nat,Y2: nat] :
( ( X3 != Y2 )
=> ( ~ ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ Y2 @ X3 ) ) ) ).
% linorder_neqE_nat
thf(fact_1101_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N2 )
& ~ ( P @ M4 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_1102_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M4: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( P @ M4 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_1103_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_1104_less__not__refl3,axiom,
! [S2: nat,T2: nat] :
( ( ord_less_nat @ S2 @ T2 )
=> ( S2 != T2 ) ) ).
% less_not_refl3
thf(fact_1105_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_1106_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_1107_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_1108_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N5: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N5 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_1109_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_1110_complete__interval,axiom,
! [A3: nat,B2: nat,P: nat > $o] :
( ( ord_less_nat @ A3 @ B2 )
=> ( ( P @ A3 )
=> ( ~ ( P @ B2 )
=> ? [C3: nat] :
( ( ord_less_eq_nat @ A3 @ C3 )
& ( ord_less_eq_nat @ C3 @ B2 )
& ! [X5: nat] :
( ( ( ord_less_eq_nat @ A3 @ X5 )
& ( ord_less_nat @ X5 @ C3 ) )
=> ( P @ X5 ) )
& ! [D3: nat] :
( ! [X: nat] :
( ( ( ord_less_eq_nat @ A3 @ X )
& ( ord_less_nat @ X @ D3 ) )
=> ( P @ X ) )
=> ( ord_less_eq_nat @ D3 @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_1111_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_1112_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_1113_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_1114_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_1115_add__less__imp__less__right,axiom,
! [A3: nat,C: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
=> ( ord_less_nat @ A3 @ B2 ) ) ).
% add_less_imp_less_right
thf(fact_1116_add__less__imp__less__left,axiom,
! [C: nat,A3: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B2 ) )
=> ( ord_less_nat @ A3 @ B2 ) ) ).
% add_less_imp_less_left
thf(fact_1117_add__strict__right__mono,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add_strict_right_mono
thf(fact_1118_add__strict__left__mono,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).
% add_strict_left_mono
thf(fact_1119_add__strict__mono,axiom,
! [A3: nat,B2: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B2 @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_1120_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J3: nat,K2: nat,L: nat] :
( ( ( ord_less_nat @ I @ J3 )
& ( K2 = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_1121_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J3: nat,K2: nat,L: nat] :
( ( ( I = J3 )
& ( ord_less_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_1122_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J3: nat,K2: nat,L: nat] :
( ( ( ord_less_nat @ I @ J3 )
& ( ord_less_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_1123_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N2 )
& ~ ( P @ M4 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_1124_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1125_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1126_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_1127_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1128_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_1129_bot__nat__0_Oextremum__strict,axiom,
! [A3: nat] :
~ ( ord_less_nat @ A3 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1130_Nat_OlessE,axiom,
! [I: nat,K2: nat] :
( ( ord_less_nat @ I @ K2 )
=> ( ( K2
!= ( suc @ I ) )
=> ~ ! [J: nat] :
( ( ord_less_nat @ I @ J )
=> ( K2
!= ( suc @ J ) ) ) ) ) ).
% Nat.lessE
thf(fact_1131_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_1132_Suc__lessE,axiom,
! [I: nat,K2: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K2 )
=> ~ ! [J: nat] :
( ( ord_less_nat @ I @ J )
=> ( K2
!= ( suc @ J ) ) ) ) ).
% Suc_lessE
thf(fact_1133_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_1134_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_1135_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_1136_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I5: nat] :
( ( ord_less_nat @ I5 @ ( suc @ N ) )
& ( P @ I5 ) ) )
= ( ( P @ N )
| ? [I5: nat] :
( ( ord_less_nat @ I5 @ N )
& ( P @ I5 ) ) ) ) ).
% Ex_less_Suc
thf(fact_1137_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_1138_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_1139_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I5: nat] :
( ( ord_less_nat @ I5 @ ( suc @ N ) )
=> ( P @ I5 ) ) )
= ( ( P @ N )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ N )
=> ( P @ I5 ) ) ) ) ).
% All_less_Suc
thf(fact_1140_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M7: nat] :
( ( M
= ( suc @ M7 ) )
& ( ord_less_nat @ N @ M7 ) ) ) ) ).
% Suc_less_eq2
thf(fact_1141_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_1142_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_1143_less__trans__Suc,axiom,
! [I: nat,J3: nat,K2: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( ( ord_less_nat @ J3 @ K2 )
=> ( ord_less_nat @ ( suc @ I ) @ K2 ) ) ) ).
% less_trans_Suc
thf(fact_1144_less__Suc__induct,axiom,
! [I: nat,J3: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J3 )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J: nat,K4: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ J @ K4 )
=> ( ( P @ I2 @ J )
=> ( ( P @ J @ K4 )
=> ( P @ I2 @ K4 ) ) ) ) )
=> ( P @ I @ J3 ) ) ) ) ).
% less_Suc_induct
thf(fact_1145_strict__inc__induct,axiom,
! [I: nat,J3: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J3 )
=> ( ! [I2: nat] :
( ( J3
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J3 )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_1146_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_1147_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M6: nat,N4: nat] :
( ( ord_less_eq_nat @ M6 @ N4 )
& ( M6 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_1148_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_1149_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M6: nat,N4: nat] :
( ( ord_less_nat @ M6 @ N4 )
| ( M6 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1150_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_1151_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_1152_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J3: nat] :
( ! [I2: nat,J: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J ) ) )
=> ( ( ord_less_eq_nat @ I @ J3 )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J3 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1153_less__imp__diff__less,axiom,
! [J3: nat,K2: nat,N: nat] :
( ( ord_less_nat @ J3 @ K2 )
=> ( ord_less_nat @ ( minus_minus_nat @ J3 @ N ) @ K2 ) ) ).
% less_imp_diff_less
thf(fact_1154_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_1155_add__lessD1,axiom,
! [I: nat,J3: nat,K2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J3 ) @ K2 )
=> ( ord_less_nat @ I @ K2 ) ) ).
% add_lessD1
thf(fact_1156_add__less__mono,axiom,
! [I: nat,J3: nat,K2: nat,L: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( ( ord_less_nat @ K2 @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J3 @ L ) ) ) ) ).
% add_less_mono
thf(fact_1157_not__add__less1,axiom,
! [I: nat,J3: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J3 ) @ I ) ).
% not_add_less1
thf(fact_1158_not__add__less2,axiom,
! [J3: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J3 @ I ) @ I ) ).
% not_add_less2
thf(fact_1159_add__less__mono1,axiom,
! [I: nat,J3: nat,K2: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J3 @ K2 ) ) ) ).
% add_less_mono1
thf(fact_1160_trans__less__add1,axiom,
! [I: nat,J3: nat,M: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J3 @ M ) ) ) ).
% trans_less_add1
thf(fact_1161_trans__less__add2,axiom,
! [I: nat,J3: nat,M: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J3 ) ) ) ).
% trans_less_add2
thf(fact_1162_less__add__eq__less,axiom,
! [K2: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K2 @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K2 @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_1163_lessThan__def,axiom,
( set_or890127255671739683et_nat
= ( ^ [U2: set_nat] :
( collect_set_nat
@ ^ [X2: set_nat] : ( ord_less_set_nat @ X2 @ U2 ) ) ) ) ).
% lessThan_def
thf(fact_1164_lessThan__def,axiom,
( set_ord_lessThan_nat
= ( ^ [U2: nat] :
( collect_nat
@ ^ [X2: nat] : ( ord_less_nat @ X2 @ U2 ) ) ) ) ).
% lessThan_def
thf(fact_1165_add__less__le__mono,axiom,
! [A3: nat,B2: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B2 @ D2 ) ) ) ) ).
% add_less_le_mono
thf(fact_1166_add__le__less__mono,axiom,
! [A3: nat,B2: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B2 @ D2 ) ) ) ) ).
% add_le_less_mono
thf(fact_1167_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J3: nat,K2: nat,L: nat] :
( ( ( ord_less_nat @ I @ J3 )
& ( ord_less_eq_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1168_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J3: nat,K2: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J3 )
& ( ord_less_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1169_pos__add__strict,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ B2 @ ( plus_plus_nat @ A3 @ C ) ) ) ) ).
% pos_add_strict
thf(fact_1170_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ~ ! [C3: nat] :
( ( B2
= ( plus_plus_nat @ A3 @ C3 ) )
=> ( C3 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_1171_add__pos__pos,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A3 @ B2 ) ) ) ) ).
% add_pos_pos
thf(fact_1172_add__neg__neg,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ B2 ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_1173_cring_Ocring__simprules_I6_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( cring_a_b @ R )
=> ( member_a @ ( one_a_ring_ext_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% cring.cring_simprules(6)
thf(fact_1174_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A3: nat,B2: nat] :
( ~ ( ord_less_nat @ A3 @ B2 )
=> ( ( plus_plus_nat @ B2 @ ( minus_minus_nat @ A3 @ B2 ) )
= A3 ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1175_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I5: nat] :
( ( ord_less_nat @ I5 @ ( suc @ N ) )
& ( P @ I5 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I5: nat] :
( ( ord_less_nat @ I5 @ N )
& ( P @ ( suc @ I5 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_1176_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M6: nat] :
( N
= ( suc @ M6 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1177_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I5: nat] :
( ( ord_less_nat @ I5 @ ( suc @ N ) )
=> ( P @ I5 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ N )
=> ( P @ ( suc @ I5 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1178_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% gr0_implies_Suc
thf(fact_1179_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J2: nat] :
( ( M
= ( suc @ J2 ) )
& ( ord_less_nat @ J2 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1180_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K4: nat] :
( ( ord_less_eq_nat @ K4 @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K4 )
=> ~ ( P @ I3 ) )
& ( P @ K4 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1181_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_1182_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_1183_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N4: nat] : ( ord_less_eq_nat @ ( suc @ N4 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1184_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_1185_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_1186_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_1187_inc__induct,axiom,
! [I: nat,J3: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J3 )
=> ( ( P @ J3 )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J3 )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_1188_dec__induct,axiom,
! [I: nat,J3: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J3 )
=> ( ( P @ I )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J3 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J3 ) ) ) ) ).
% dec_induct
thf(fact_1189_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_1190_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_1191_less__imp__add__positive,axiom,
! [I: nat,J3: nat] :
( ( ord_less_nat @ I @ J3 )
=> ? [K4: nat] :
( ( ord_less_nat @ zero_zero_nat @ K4 )
& ( ( plus_plus_nat @ I @ K4 )
= J3 ) ) ) ).
% less_imp_add_positive
thf(fact_1192_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
= ( minus_minus_nat @ M @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_1193_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_1194_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ! [Q3: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M @ Q3 ) ) ) ) ).
% less_natE
thf(fact_1195_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_1196_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_1197_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M6: nat,N4: nat] :
? [K: nat] :
( N4
= ( suc @ ( plus_plus_nat @ M6 @ K ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1198_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K4: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K4 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1199_diff__less__mono,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( ( ord_less_eq_nat @ C @ A3 )
=> ( ord_less_nat @ ( minus_minus_nat @ A3 @ C ) @ ( minus_minus_nat @ B2 @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1200_less__diff__iff,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1201_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K2: nat] :
( ! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( ord_less_nat @ ( F @ M3 ) @ ( F @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K2 ) @ ( F @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1202_less__diff__conv,axiom,
! [I: nat,J3: nat,K2: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J3 @ K2 ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ J3 ) ) ).
% less_diff_conv
thf(fact_1203_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_1204_SUP__lessD,axiom,
! [F: nat > $o,A: set_nat,Y2: $o,I: nat] :
( ( ord_less_o @ ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A ) ) @ Y2 )
=> ( ( member_nat @ I @ A )
=> ( ord_less_o @ ( F @ I ) @ Y2 ) ) ) ).
% SUP_lessD
thf(fact_1205_SUP__lessD,axiom,
! [F: a > $o,A: set_a,Y2: $o,I: a] :
( ( ord_less_o @ ( complete_Sup_Sup_o @ ( image_a_o @ F @ A ) ) @ Y2 )
=> ( ( member_a @ I @ A )
=> ( ord_less_o @ ( F @ I ) @ Y2 ) ) ) ).
% SUP_lessD
thf(fact_1206_SUP__lessD,axiom,
! [F: set_nat > $o,A: set_set_nat,Y2: $o,I: set_nat] :
( ( ord_less_o @ ( complete_Sup_Sup_o @ ( image_set_nat_o @ F @ A ) ) @ Y2 )
=> ( ( member_set_nat @ I @ A )
=> ( ord_less_o @ ( F @ I ) @ Y2 ) ) ) ).
% SUP_lessD
thf(fact_1207_SUP__lessD,axiom,
! [F: set_a > $o,A: set_set_a,Y2: $o,I: set_a] :
( ( ord_less_o @ ( complete_Sup_Sup_o @ ( image_set_a_o @ F @ A ) ) @ Y2 )
=> ( ( member_set_a @ I @ A )
=> ( ord_less_o @ ( F @ I ) @ Y2 ) ) ) ).
% SUP_lessD
thf(fact_1208_SUP__lessD,axiom,
! [F: nat > set_a,A: set_nat,Y2: set_a,I: nat] :
( ( ord_less_set_a @ ( comple2307003609928055243_set_a @ ( image_nat_set_a @ F @ A ) ) @ Y2 )
=> ( ( member_nat @ I @ A )
=> ( ord_less_set_a @ ( F @ I ) @ Y2 ) ) ) ).
% SUP_lessD
thf(fact_1209_SUP__lessD,axiom,
! [F: a > set_a,A: set_a,Y2: set_a,I: a] :
( ( ord_less_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A ) ) @ Y2 )
=> ( ( member_a @ I @ A )
=> ( ord_less_set_a @ ( F @ I ) @ Y2 ) ) ) ).
% SUP_lessD
thf(fact_1210_SUP__lessD,axiom,
! [F: nat > set_nat,A: set_nat,Y2: set_nat,I: nat] :
( ( ord_less_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A ) ) @ Y2 )
=> ( ( member_nat @ I @ A )
=> ( ord_less_set_nat @ ( F @ I ) @ Y2 ) ) ) ).
% SUP_lessD
thf(fact_1211_SUP__lessD,axiom,
! [F: a > set_nat,A: set_a,Y2: set_nat,I: a] :
( ( ord_less_set_nat @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ F @ A ) ) @ Y2 )
=> ( ( member_a @ I @ A )
=> ( ord_less_set_nat @ ( F @ I ) @ Y2 ) ) ) ).
% SUP_lessD
thf(fact_1212_SUP__lessD,axiom,
! [F: nat > set_set_nat,A: set_nat,Y2: set_set_nat,I: nat] :
( ( ord_less_set_set_nat @ ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ F @ A ) ) @ Y2 )
=> ( ( member_nat @ I @ A )
=> ( ord_less_set_set_nat @ ( F @ I ) @ Y2 ) ) ) ).
% SUP_lessD
thf(fact_1213_SUP__lessD,axiom,
! [F: a > set_set_nat,A: set_a,Y2: set_set_nat,I: a] :
( ( ord_less_set_set_nat @ ( comple548664676211718543et_nat @ ( image_a_set_set_nat @ F @ A ) ) @ Y2 )
=> ( ( member_a @ I @ A )
=> ( ord_less_set_set_nat @ ( F @ I ) @ Y2 ) ) ) ).
% SUP_lessD
thf(fact_1214_add__neg__nonpos,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ B2 ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_1215_add__nonneg__pos,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A3 @ B2 ) ) ) ) ).
% add_nonneg_pos
thf(fact_1216_add__nonpos__neg,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ B2 ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_1217_add__pos__nonneg,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A3 @ B2 ) ) ) ) ).
% add_pos_nonneg
thf(fact_1218_add__strict__increasing,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_nat @ B2 @ ( plus_plus_nat @ A3 @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_1219_add__strict__increasing2,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ B2 @ ( plus_plus_nat @ A3 @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_1220_cring_Ocring__simprules_I12_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a] :
( ( cring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( one_a_ring_ext_a_b @ R ) @ X3 )
= X3 ) ) ) ).
% cring.cring_simprules(12)
thf(fact_1221_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K4: nat] :
( ( ord_less_nat @ K4 @ N )
& ! [I3: nat] :
( ( ord_less_eq_nat @ I3 @ K4 )
=> ~ ( P @ I3 ) )
& ( P @ ( suc @ K4 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1222_diff__Suc__less,axiom,
! [N: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_1223_Iic__subset__Iio__iff,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ A3 ) @ ( set_ord_lessThan_nat @ B2 ) )
= ( ord_less_nat @ A3 @ B2 ) ) ).
% Iic_subset_Iio_iff
thf(fact_1224_nat__diff__split,axiom,
! [P: nat > $o,A3: nat,B2: nat] :
( ( P @ ( minus_minus_nat @ A3 @ B2 ) )
= ( ( ( ord_less_nat @ A3 @ B2 )
=> ( P @ zero_zero_nat ) )
& ! [D4: nat] :
( ( A3
= ( plus_plus_nat @ B2 @ D4 ) )
=> ( P @ D4 ) ) ) ) ).
% nat_diff_split
thf(fact_1225_nat__diff__split__asm,axiom,
! [P: nat > $o,A3: nat,B2: nat] :
( ( P @ ( minus_minus_nat @ A3 @ B2 ) )
= ( ~ ( ( ( ord_less_nat @ A3 @ B2 )
& ~ ( P @ zero_zero_nat ) )
| ? [D4: nat] :
( ( A3
= ( plus_plus_nat @ B2 @ D4 ) )
& ~ ( P @ D4 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1226_less__diff__conv2,axiom,
! [K2: nat,J3: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J3 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J3 @ K2 ) @ I )
= ( ord_less_nat @ J3 @ ( plus_plus_nat @ I @ K2 ) ) ) ) ).
% less_diff_conv2
thf(fact_1227_card__less,axiom,
! [M2: set_nat,I: nat] :
( ( member_nat @ zero_zero_nat @ M2 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [K: nat] :
( ( member_nat @ K @ M2 )
& ( ord_less_nat @ K @ ( suc @ I ) ) ) ) )
!= zero_zero_nat ) ) ).
% card_less
thf(fact_1228_card__less__Suc,axiom,
! [M2: set_nat,I: nat] :
( ( member_nat @ zero_zero_nat @ M2 )
=> ( ( suc
@ ( finite_card_nat
@ ( collect_nat
@ ^ [K: nat] :
( ( member_nat @ ( suc @ K ) @ M2 )
& ( ord_less_nat @ K @ I ) ) ) ) )
= ( finite_card_nat
@ ( collect_nat
@ ^ [K: nat] :
( ( member_nat @ K @ M2 )
& ( ord_less_nat @ K @ ( suc @ I ) ) ) ) ) ) ) ).
% card_less_Suc
thf(fact_1229_card__less__Suc2,axiom,
! [M2: set_nat,I: nat] :
( ~ ( member_nat @ zero_zero_nat @ M2 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [K: nat] :
( ( member_nat @ ( suc @ K ) @ M2 )
& ( ord_less_nat @ K @ I ) ) ) )
= ( finite_card_nat
@ ( collect_nat
@ ^ [K: nat] :
( ( member_nat @ K @ M2 )
& ( ord_less_nat @ K @ ( suc @ I ) ) ) ) ) ) ) ).
% card_less_Suc2
thf(fact_1230_mem__upD,axiom,
! [F: nat > a,R: partia2175431115845679010xt_a_b,N: nat] :
( ( member_nat_a @ F @ ( up_a_b @ R ) )
=> ( member_a @ ( F @ N ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% mem_upD
thf(fact_1231_bound__below,axiom,
! [Z: a,M: nat,F: nat > a,N: nat] :
( ( bound_a @ Z @ M @ F )
=> ( ( ( F @ N )
!= Z )
=> ( ord_less_eq_nat @ N @ M ) ) ) ).
% bound_below
thf(fact_1232_cring_Ogeom,axiom,
! [R: partia2175431115845679010xt_a_b,A3: a,Q2: nat] :
( ( cring_a_b @ R )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( a_minus_a_b @ R @ A3 @ ( one_a_ring_ext_a_b @ R ) ) @ ( finsum_a_b_nat @ R @ ( pow_a_1026414303147256608_b_nat @ R @ A3 ) @ ( set_ord_lessThan_nat @ Q2 ) ) )
= ( a_minus_a_b @ R @ ( pow_a_1026414303147256608_b_nat @ R @ A3 @ Q2 ) @ ( one_a_ring_ext_a_b @ R ) ) ) ) ) ).
% cring.geom
thf(fact_1233_add_Ofinprod__0_H,axiom,
! [F: nat > a,N: nat] :
( ( member_nat_a @ F
@ ( pi_nat_a @ ( set_ord_atMost_nat @ N )
@ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( add_a_b @ r @ ( F @ zero_zero_nat ) @ ( finsum_a_b_nat @ r @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
= ( finsum_a_b_nat @ r @ F @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% add.finprod_0'
thf(fact_1234_add_Ofinprod__mono__neutral__cong__right,axiom,
! [B: set_nat_a,A: set_nat_a,G: ( nat > a ) > a,H: ( nat > a ) > a] :
( ( finite_finite_nat_a @ B )
=> ( ( ord_le871467723717165285_nat_a @ A @ B )
=> ( ! [I2: nat > a] :
( ( member_nat_a @ I2 @ ( minus_490503922182417452_nat_a @ B @ A ) )
=> ( ( G @ I2 )
= ( zero_a_b @ r ) ) )
=> ( ! [X: nat > a] :
( ( member_nat_a @ X @ A )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( member_nat_a_a @ G
@ ( pi_nat_a_a @ B
@ ^ [Uu: nat > a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum_a_b_nat_a @ r @ G @ B )
= ( finsum_a_b_nat_a @ r @ H @ A ) ) ) ) ) ) ) ).
% add.finprod_mono_neutral_cong_right
thf(fact_1235_add_Ofinprod__mono__neutral__cong__right,axiom,
! [B: set_set_set_nat,A: set_set_set_nat,G: set_set_nat > a,H: set_set_nat > a] :
( ( finite6739761609112101331et_nat @ B )
=> ( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ! [I2: set_set_nat] :
( ( member_set_set_nat @ I2 @ ( minus_2447799839930672331et_nat @ B @ A ) )
=> ( ( G @ I2 )
= ( zero_a_b @ r ) ) )
=> ( ! [X: set_set_nat] :
( ( member_set_set_nat @ X @ A )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( member_set_set_nat_a @ G
@ ( pi_set_set_nat_a @ B
@ ^ [Uu: set_set_nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum2649122254697571802et_nat @ r @ G @ B )
= ( finsum2649122254697571802et_nat @ r @ H @ A ) ) ) ) ) ) ) ).
% add.finprod_mono_neutral_cong_right
thf(fact_1236_add_Ofinprod__mono__neutral__cong__right,axiom,
! [B: set_set_nat,A: set_set_nat,G: set_nat > a,H: set_nat > a] :
( ( finite1152437895449049373et_nat @ B )
=> ( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ! [I2: set_nat] :
( ( member_set_nat @ I2 @ ( minus_2163939370556025621et_nat @ B @ A ) )
=> ( ( G @ I2 )
= ( zero_a_b @ r ) ) )
=> ( ! [X: set_nat] :
( ( member_set_nat @ X @ A )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( member_set_nat_a @ G
@ ( pi_set_nat_a @ B
@ ^ [Uu: set_nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum_a_b_set_nat @ r @ G @ B )
= ( finsum_a_b_set_nat @ r @ H @ A ) ) ) ) ) ) ) ).
% add.finprod_mono_neutral_cong_right
thf(fact_1237_add_Ofinprod__mono__neutral__cong__right,axiom,
! [B: set_nat,A: set_nat,G: nat > a,H: nat > a] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ ( minus_minus_set_nat @ B @ A ) )
=> ( ( G @ I2 )
= ( zero_a_b @ r ) ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( member_nat_a @ G
@ ( pi_nat_a @ B
@ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum_a_b_nat @ r @ G @ B )
= ( finsum_a_b_nat @ r @ H @ A ) ) ) ) ) ) ) ).
% add.finprod_mono_neutral_cong_right
thf(fact_1238_add_Ofinprod__mono__neutral__cong__right,axiom,
! [B: set_a,A: set_a,G: a > a,H: a > a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( ! [I2: a] :
( ( member_a @ I2 @ ( minus_minus_set_a @ B @ A ) )
=> ( ( G @ I2 )
= ( zero_a_b @ r ) ) )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( member_a_a @ G
@ ( pi_a_a @ B
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum_a_b_a @ r @ G @ B )
= ( finsum_a_b_a @ r @ H @ A ) ) ) ) ) ) ) ).
% add.finprod_mono_neutral_cong_right
thf(fact_1239_add_Ofinprod__mono__neutral__cong__right,axiom,
! [B: set_set_a,A: set_set_a,G: set_a > a,H: set_a > a] :
( ( finite_finite_set_a @ B )
=> ( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ! [I2: set_a] :
( ( member_set_a @ I2 @ ( minus_5736297505244876581_set_a @ B @ A ) )
=> ( ( G @ I2 )
= ( zero_a_b @ r ) ) )
=> ( ! [X: set_a] :
( ( member_set_a @ X @ A )
=> ( ( G @ X )
= ( H @ X ) ) )
=> ( ( member_set_a_a @ G
@ ( pi_set_a_a @ B
@ ^ [Uu: set_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum_a_b_set_a @ r @ G @ B )
= ( finsum_a_b_set_a @ r @ H @ A ) ) ) ) ) ) ) ).
% add.finprod_mono_neutral_cong_right
thf(fact_1240_fin__A,axiom,
! [I: nat] : ( finite1152437895449049373et_nat @ ( a2 @ I ) ) ).
% fin_A
thf(fact_1241_finite__atMost,axiom,
! [K2: nat] : ( finite_finite_nat @ ( set_ord_atMost_nat @ K2 ) ) ).
% finite_atMost
thf(fact_1242_finite__atLeastAtMost,axiom,
! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or1269000886237332187st_nat @ L @ U ) ) ).
% finite_atLeastAtMost
thf(fact_1243_finite__lessThan,axiom,
! [K2: nat] : ( finite_finite_nat @ ( set_ord_lessThan_nat @ K2 ) ) ).
% finite_lessThan
thf(fact_1244_finite__Collect__less__nat,axiom,
! [K2: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N4: nat] : ( ord_less_nat @ N4 @ K2 ) ) ) ).
% finite_Collect_less_nat
thf(fact_1245_finite__Collect__le__nat,axiom,
! [K2: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N4: nat] : ( ord_less_eq_nat @ N4 @ K2 ) ) ) ).
% finite_Collect_le_nat
thf(fact_1246_image__Suc__atLeastAtMost,axiom,
! [I: nat,J3: nat] :
( ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ I @ J3 ) )
= ( set_or1269000886237332187st_nat @ ( suc @ I ) @ ( suc @ J3 ) ) ) ).
% image_Suc_atLeastAtMost
thf(fact_1247_card__atLeastAtMost,axiom,
! [L: nat,U: nat] :
( ( finite_card_nat @ ( set_or1269000886237332187st_nat @ L @ U ) )
= ( minus_minus_nat @ ( suc @ U ) @ L ) ) ).
% card_atLeastAtMost
thf(fact_1248_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N7: set_nat] :
? [M6: nat] :
! [X2: nat] :
( ( member_nat @ X2 @ N7 )
=> ( ord_less_nat @ X2 @ M6 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_1249_bounded__nat__set__is__finite,axiom,
! [N3: set_nat,N: nat] :
( ! [X: nat] :
( ( member_nat @ X @ N3 )
=> ( ord_less_nat @ X @ N ) )
=> ( finite_finite_nat @ N3 ) ) ).
% bounded_nat_set_is_finite
thf(fact_1250_subset__eq__atLeast0__atMost__finite,axiom,
! [N3: set_nat,N: nat] :
( ( ord_less_eq_set_nat @ N3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
=> ( finite_finite_nat @ N3 ) ) ).
% subset_eq_atLeast0_atMost_finite
thf(fact_1251_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N7: set_nat] :
? [M6: nat] :
! [X2: nat] :
( ( member_nat @ X2 @ N7 )
=> ( ord_less_eq_nat @ X2 @ M6 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_1252_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [K: nat] :
( ( P @ K )
& ( ord_less_nat @ K @ I ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_1253_finite__less__ub,axiom,
! [F: nat > nat,U: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ N2 @ ( F @ N2 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N4: nat] : ( ord_less_eq_nat @ ( F @ N4 ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_1254_atMost__atLeast0,axiom,
( set_ord_atMost_nat
= ( set_or1269000886237332187st_nat @ zero_zero_nat ) ) ).
% atMost_atLeast0
thf(fact_1255_order__gt__0__iff__finite,axiom,
( ( ord_less_nat @ zero_zero_nat @ ( order_a_ring_ext_a_b @ r ) )
= ( finite_finite_a @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% order_gt_0_iff_finite
thf(fact_1256_finprod__0_H,axiom,
! [F: nat > a,N: nat] :
( ( member_nat_a @ F
@ ( pi_nat_a @ ( set_ord_atMost_nat @ N )
@ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( F @ zero_zero_nat ) @ ( finpro1280035270526425175_b_nat @ r @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
= ( finpro1280035270526425175_b_nat @ r @ F @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% finprod_0'
thf(fact_1257_subalgebra__inter,axiom,
! [K3: set_a,V2: set_a,V4: set_a] :
( ( embedd9027525575939734154ra_a_b @ K3 @ V2 @ r )
=> ( ( embedd9027525575939734154ra_a_b @ K3 @ V4 @ r )
=> ( embedd9027525575939734154ra_a_b @ K3 @ ( inf_inf_set_a @ V2 @ V4 ) @ r ) ) ) ).
% subalgebra_inter
thf(fact_1258_finprod__Suc3,axiom,
! [F: nat > a,N: nat] :
( ( member_nat_a @ F
@ ( pi_nat_a @ ( set_ord_atMost_nat @ N )
@ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro1280035270526425175_b_nat @ r @ F @ ( set_ord_atMost_nat @ N ) )
= ( mult_a_ring_ext_a_b @ r @ ( F @ N ) @ ( finpro1280035270526425175_b_nat @ r @ F @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% finprod_Suc3
thf(fact_1259_finprod__Suc2,axiom,
! [F: nat > a,N: nat] :
( ( member_nat_a @ F
@ ( pi_nat_a @ ( set_ord_atMost_nat @ ( suc @ N ) )
@ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro1280035270526425175_b_nat @ r @ F @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( mult_a_ring_ext_a_b @ r
@ ( finpro1280035270526425175_b_nat @ r
@ ^ [I5: nat] : ( F @ ( suc @ I5 ) )
@ ( set_ord_atMost_nat @ N ) )
@ ( F @ zero_zero_nat ) ) ) ) ).
% finprod_Suc2
thf(fact_1260_finprod__Suc,axiom,
! [F: nat > a,N: nat] :
( ( member_nat_a @ F
@ ( pi_nat_a @ ( set_ord_atMost_nat @ ( suc @ N ) )
@ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro1280035270526425175_b_nat @ r @ F @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( mult_a_ring_ext_a_b @ r @ ( F @ ( suc @ N ) ) @ ( finpro1280035270526425175_b_nat @ r @ F @ ( set_ord_atMost_nat @ N ) ) ) ) ) ).
% finprod_Suc
thf(fact_1261_a__card__cosets__equal,axiom,
! [C: set_a,H2: set_a] :
( ( member_set_a @ C @ ( a_RCOSETS_a_b @ r @ H2 ) )
=> ( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( finite_finite_a @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( finite_card_a @ C )
= ( finite_card_a @ H2 ) ) ) ) ) ).
% a_card_cosets_equal
thf(fact_1262_carrier__not__empty,axiom,
( ( partia707051561876973205xt_a_b @ r )
!= bot_bot_set_a ) ).
% carrier_not_empty
thf(fact_1263_subring__props_I4_J,axiom,
! [K3: set_a] :
( ( subfield_a_b @ K3 @ r )
=> ( K3 != bot_bot_set_a ) ) ).
% subring_props(4)
thf(fact_1264_Sup__nat__empty,axiom,
( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% Sup_nat_empty
thf(fact_1265_lessThan__0,axiom,
( ( set_ord_lessThan_nat @ zero_zero_nat )
= bot_bot_set_nat ) ).
% lessThan_0
thf(fact_1266_lessThan__empty__iff,axiom,
! [N: nat] :
( ( ( set_ord_lessThan_nat @ N )
= bot_bot_set_nat )
= ( N = zero_zero_nat ) ) ).
% lessThan_empty_iff
thf(fact_1267_rcosets__subset__PowG,axiom,
! [H2: set_a] :
( ( additi2834746164131130830up_a_b @ H2 @ r )
=> ( ord_le3724670747650509150_set_a @ ( a_RCOSETS_a_b @ r @ H2 ) @ ( pow_a @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% rcosets_subset_PowG
thf(fact_1268_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1269_a__lagrange,axiom,
! [H2: set_a] :
( ( finite_finite_a @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( additi2834746164131130830up_a_b @ H2 @ r )
=> ( ( times_times_nat @ ( finite_card_set_a @ ( a_RCOSETS_a_b @ r @ H2 ) ) @ ( finite_card_a @ H2 ) )
= ( order_a_ring_ext_a_b @ r ) ) ) ) ).
% a_lagrange
thf(fact_1270_nat__pow__pow,axiom,
! [X3: a,N: nat,M: nat] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X3 @ N ) @ M )
= ( pow_a_1026414303147256608_b_nat @ r @ X3 @ ( times_times_nat @ N @ M ) ) ) ) ).
% nat_pow_pow
% Helper facts (3)
thf(help_If_3_1_If_001tf__a_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001tf__a_T,axiom,
! [X3: a,Y2: a] :
( ( if_a @ $false @ X3 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001tf__a_T,axiom,
! [X3: a,Y2: a] :
( ( if_a @ $true @ X3 @ Y2 )
= X3 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( finsum_a_b_set_nat @ r
@ ^ [A2: set_nat] : ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ x @ ( finite_card_nat @ A2 ) ) @ ( pow_a_1026414303147256608_b_nat @ r @ y @ ( minus_minus_nat @ n @ ( finite_card_nat @ A2 ) ) ) )
@ ( pow_nat @ ( set_ord_lessThan_nat @ n ) ) )
= ( finsum_a_b_set_nat @ r
@ ^ [A2: set_nat] : ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ x @ ( finite_card_nat @ A2 ) ) @ ( pow_a_1026414303147256608_b_nat @ r @ y @ ( minus_minus_nat @ n @ ( finite_card_nat @ A2 ) ) ) )
@ ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ a2 @ ( set_ord_atMost_nat @ n ) ) ) ) ) ).
%------------------------------------------------------------------------------