TPTP Problem File: SLH0005^1.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 : LP_Duality/0000_Minimum_Maximum/prob_00039_001169__28670152_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1429 ( 354 unt; 152 typ; 0 def)
% Number of atoms : 4413 (1076 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 12364 ( 410 ~; 56 |; 214 &;9262 @)
% ( 0 <=>;2422 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 8 avg)
% Number of types : 9 ( 8 usr)
% Number of type conns : 1232 (1232 >; 0 *; 0 +; 0 <<)
% Number of symbols : 147 ( 144 usr; 13 con; 0-4 aty)
% Number of variables : 3763 ( 228 ^;3420 !; 115 ?;3763 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 13:49:21.538
%------------------------------------------------------------------------------
% Could-be-implicit typings (8)
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_I_Eo_Mt__Nat__Onat_J_J,type,
set_o_nat: $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_I_Eo_Mtf__a_J_J,type,
set_o_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__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (144)
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Nat__Onat,type,
complete_Inf_Inf_nat: set_nat > nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Nat__Onat,type,
complete_Sup_Sup_nat: set_nat > nat ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001t__Nat__Onat,type,
condit2214826472909112428ve_nat: set_nat > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001tf__a,type,
condit5209368051240477026bove_a: set_a > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__below_001t__Nat__Onat,type,
condit1738341127787009408ow_nat: set_nat > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__below_001tf__a,type,
condit5901475214736682318elow_a: set_a > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd_001t__Nat__Onat,type,
condit7935552474144124665dd_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd_001tf__a,type,
condit4103000493307248661_bdd_a: ( a > a > $o ) > ( a > a > $o ) > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd_Obdd_001t__Nat__Onat,type,
condit4013746787832047771dd_nat: ( nat > nat > $o ) > set_nat > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd_Obdd_001tf__a,type,
condit6541519642617408243_bdd_a: ( a > a > $o ) > set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
finite_finite_a: set_a > $o ).
thf(sy_c_Finite__Set_Ofold_001t__Nat__Onat_001t__Nat__Onat,type,
finite_fold_nat_nat: ( nat > nat > nat ) > nat > set_nat > nat ).
thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Nat__Onat,type,
bij_betw_nat_nat: ( nat > nat ) > set_nat > set_nat > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Nat__Onat_001t__Nat__Onat,type,
monotone_on_nat_nat: set_nat > ( nat > nat > $o ) > ( nat > nat > $o ) > ( nat > nat ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Nat__Onat_001tf__a,type,
monotone_on_nat_a: set_nat > ( nat > nat > $o ) > ( a > a > $o ) > ( nat > a ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
monoto1748750089227133045et_nat: set_set_nat > ( set_nat > set_nat > $o ) > ( set_nat > set_nat > $o ) > ( set_nat > set_nat ) > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
monoto7172710143293369831_set_a: set_set_a > ( set_a > set_a > $o ) > ( set_a > set_a > $o ) > ( set_a > set_a ) > $o ).
thf(sy_c_Fun_Omonotone__on_001tf__a_001t__Nat__Onat,type,
monotone_on_a_nat: set_a > ( a > a > $o ) > ( nat > nat > $o ) > ( a > nat ) > $o ).
thf(sy_c_Fun_Omonotone__on_001tf__a_001tf__a,type,
monotone_on_a_a: set_a > ( a > a > $o ) > ( a > a > $o ) > ( a > a ) > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_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_Itf__a_J,type,
minus_minus_set_a: set_a > set_a > set_a ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Nat__Onat_J,type,
uminus5710092332889474511et_nat: set_nat > set_nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001tf__a,type,
if_a: $o > a > a > a ).
thf(sy_c_Inductive_Ocomplete__lattice__class_Ogfp_001t__Set__Oset_It__Nat__Onat_J,type,
comple1596078789208929544et_nat: ( set_nat > set_nat ) > set_nat ).
thf(sy_c_Inductive_Ocomplete__lattice__class_Ogfp_001t__Set__Oset_Itf__a_J,type,
comple3341859861669737308_set_a: ( set_a > set_a ) > set_a ).
thf(sy_c_Inductive_Ocomplete__lattice__class_Olfp_001t__Set__Oset_It__Nat__Onat_J,type,
comple7975543026063415949et_nat: ( set_nat > set_nat ) > set_nat ).
thf(sy_c_Inductive_Ocomplete__lattice__class_Olfp_001t__Set__Oset_Itf__a_J,type,
comple1558298011288954135_set_a: ( set_a > set_a ) > set_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
inf_inf_nat: nat > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J,type,
inf_inf_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Osemilattice__neutr__order_001t__Set__Oset_It__Nat__Onat_J,type,
semila1667268886620078168et_nat: ( set_nat > set_nat > set_nat ) > set_nat > ( set_nat > set_nat > $o ) > ( set_nat > set_nat > $o ) > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J,type,
sup_sup_nat_nat_o: ( nat > nat > $o ) > ( nat > nat > $o ) > nat > nat > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_Itf__a_M_062_Itf__a_M_Eo_J_J,type,
sup_sup_a_a_o: ( a > a > $o ) > ( a > a > $o ) > a > a > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
sup_sup_nat: nat > nat > nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
sup_sup_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
sup_sup_set_a: set_a > set_a > set_a ).
thf(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Nat__Onat,type,
lattic8265883725875713057ax_nat: set_nat > nat ).
thf(sy_c_Lattices__Big_Olinorder__class_OMax_001tf__a,type,
lattic6529028001545966829_Max_a: set_a > a ).
thf(sy_c_Lattices__Big_Olinorder__class_OMin_001t__Nat__Onat,type,
lattic8721135487736765967in_nat: set_nat > nat ).
thf(sy_c_Lattices__Big_Olinorder__class_OMin_001tf__a,type,
lattic8372110310405614207_Min_a: set_a > a ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Nat__Onat,type,
lattic7446932960582359483at_nat: ( nat > nat ) > set_nat > nat ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001tf__a,type,
lattic1148846883994911187_nat_a: ( nat > a ) > set_nat > nat ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001tf__a_001t__Nat__Onat,type,
lattic6340287419671400565_a_nat: ( a > nat ) > set_a > a ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001tf__a_001tf__a,type,
lattic3288624042836100505on_a_a: ( a > a ) > set_a > a ).
thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Nat__Onat,type,
lattic5238388535129920115in_nat: set_nat > nat ).
thf(sy_c_Lattices__Big_Osemilattice__order__set_001t__Nat__Onat,type,
lattic6009151579333465974et_nat: ( nat > nat > nat ) > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
thf(sy_c_Lattices__Big_Osemilattice__order__set_001tf__a,type,
lattic5078705180708912344_set_a: ( a > a > a ) > ( a > a > $o ) > ( a > a > $o ) > $o ).
thf(sy_c_Lattices__Big_Osemilattice__set_001t__Nat__Onat,type,
lattic1029310888574255042et_nat: ( nat > nat > nat ) > $o ).
thf(sy_c_Lattices__Big_Osemilattice__set_001tf__a,type,
lattic5961991414251573132_set_a: ( a > a > a ) > $o ).
thf(sy_c_Lattices__Big_Osemilattice__set_OF_001t__Nat__Onat,type,
lattic7742739596368939638_F_nat: ( nat > nat > nat ) > set_nat > nat ).
thf(sy_c_Lattices__Big_Osemilattice__set_OF_001tf__a,type,
lattic5116578512385870296ce_F_a: ( a > a > a ) > set_a > a ).
thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Nat__Onat,type,
lattic1093996805478795353in_nat: set_nat > nat ).
thf(sy_c_Minimum__Maximum_OMaximum_001t__Nat__Onat,type,
minimum_Maximum_nat: set_nat > nat ).
thf(sy_c_Minimum__Maximum_OMaximum_001tf__a,type,
minimum_Maximum_a: set_a > a ).
thf(sy_c_Minimum__Maximum_OMinimum_001t__Nat__Onat,type,
minimum_Minimum_nat: set_nat > nat ).
thf(sy_c_Minimum__Maximum_OMinimum_001tf__a,type,
minimum_Minimum_a: set_a > a ).
thf(sy_c_Minimum__Maximum_Ohas__Maximum_001_062_I_Eo_Mt__Nat__Onat_J,type,
minimu2432920480775036308_o_nat: set_o_nat > $o ).
thf(sy_c_Minimum__Maximum_Ohas__Maximum_001_062_I_Eo_Mtf__a_J,type,
minimu315547183909508560um_o_a: set_o_a > $o ).
thf(sy_c_Minimum__Maximum_Ohas__Maximum_001t__Nat__Onat,type,
minimu5597823600188243285um_nat: set_nat > $o ).
thf(sy_c_Minimum__Maximum_Ohas__Maximum_001t__Set__Oset_It__Nat__Onat_J,type,
minimu2833918411073049867et_nat: set_set_nat > $o ).
thf(sy_c_Minimum__Maximum_Ohas__Maximum_001t__Set__Oset_Itf__a_J,type,
minimu8775777210878807577_set_a: set_set_a > $o ).
thf(sy_c_Minimum__Maximum_Ohas__Maximum_001tf__a,type,
minimu6197867597544231097imum_a: set_a > $o ).
thf(sy_c_Minimum__Maximum_Ohas__Minimum_001_062_I_Eo_Mt__Nat__Onat_J,type,
minimu7521381655477896998_o_nat: set_o_nat > $o ).
thf(sy_c_Minimum__Maximum_Ohas__Minimum_001_062_I_Eo_Mtf__a_J,type,
minimu4657282916794952894um_o_a: set_o_a > $o ).
thf(sy_c_Minimum__Maximum_Ohas__Minimum_001t__Nat__Onat,type,
minimu6712762941901723971um_nat: set_nat > $o ).
thf(sy_c_Minimum__Maximum_Ohas__Minimum_001t__Set__Oset_It__Nat__Onat_J,type,
minimu8045924777645095161et_nat: set_set_nat > $o ).
thf(sy_c_Minimum__Maximum_Ohas__Minimum_001t__Set__Oset_Itf__a_J,type,
minimu6896447672505010603_set_a: set_set_a > $o ).
thf(sy_c_Minimum__Maximum_Ohas__Minimum_001tf__a,type,
minimu7473987258551571531imum_a: set_a > $o ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Ocompow_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
compow_nat_nat: nat > ( nat > nat ) > nat > nat ).
thf(sy_c_Nat_Ocompow_001_062_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
compow8708494347934031032et_nat: nat > ( set_nat > set_nat ) > set_nat > set_nat ).
thf(sy_c_Nat_Ocompow_001_062_Itf__a_Mtf__a_J,type,
compow_a_a: nat > ( a > a ) > a > a ).
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_OLeast_001_062_I_Eo_Mt__Nat__Onat_J,type,
ord_Least_o_nat: ( ( $o > nat ) > $o ) > $o > nat ).
thf(sy_c_Orderings_Oord__class_OLeast_001_062_I_Eo_Mtf__a_J,type,
ord_Least_o_a: ( ( $o > a ) > $o ) > $o > a ).
thf(sy_c_Orderings_Oord__class_OLeast_001t__Nat__Onat,type,
ord_Least_nat: ( nat > $o ) > nat ).
thf(sy_c_Orderings_Oord__class_OLeast_001t__Set__Oset_It__Nat__Onat_J,type,
ord_Least_set_nat: ( set_nat > $o ) > set_nat ).
thf(sy_c_Orderings_Oord__class_OLeast_001t__Set__Oset_Itf__a_J,type,
ord_Least_set_a: ( set_a > $o ) > set_a ).
thf(sy_c_Orderings_Oord__class_OLeast_001tf__a,type,
ord_Least_a: ( a > $o ) > a ).
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_001tf__a,type,
ord_less_a: a > a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_M_062_I_Eo_Mt__Nat__Onat_J_J,type,
ord_less_eq_o_o_nat: ( $o > $o > nat ) > ( $o > $o > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_M_062_I_Eo_Mtf__a_J_J,type,
ord_less_eq_o_o_a: ( $o > $o > a ) > ( $o > $o > a ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Nat__Onat_J,type,
ord_less_eq_o_nat: ( $o > nat ) > ( $o > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Set__Oset_It__Nat__Onat_J_J,type,
ord_le7022414076629706543et_nat: ( $o > set_nat ) > ( $o > set_nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Set__Oset_Itf__a_J_J,type,
ord_less_eq_o_set_a: ( $o > set_a ) > ( $o > set_a ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mtf__a_J,type,
ord_less_eq_o_a: ( $o > a ) > ( $o > a ) > $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_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_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001tf__a,type,
ord_less_eq_a: a > a > $o ).
thf(sy_c_Orderings_Oord__class_Omax_001t__Nat__Onat,type,
ord_max_nat: nat > nat > nat ).
thf(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Nat__Onat_J,type,
ord_max_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Orderings_Oord__class_Omax_001tf__a,type,
ord_max_a: a > a > a ).
thf(sy_c_Orderings_Oord__class_Omin_001t__Nat__Onat,type,
ord_min_nat: nat > nat > nat ).
thf(sy_c_Orderings_Oord__class_Omin_001t__Set__Oset_It__Nat__Onat_J,type,
ord_min_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Orderings_Oord__class_Omin_001tf__a,type,
ord_min_a: a > a > a ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001_062_I_Eo_Mt__Nat__Onat_J,type,
order_Greatest_o_nat: ( ( $o > nat ) > $o ) > $o > nat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001_062_I_Eo_Mtf__a_J,type,
order_Greatest_o_a: ( ( $o > a ) > $o ) > $o > a ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
order_Greatest_nat: ( nat > $o ) > nat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_It__Nat__Onat_J,type,
order_5724808138429204845et_nat: ( set_nat > $o ) > set_nat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_Itf__a_J,type,
order_Greatest_set_a: ( set_a > $o ) > set_a ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001tf__a,type,
order_Greatest_a: ( a > $o ) > a ).
thf(sy_c_Orderings_Oordering_001t__Nat__Onat,type,
ordering_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
thf(sy_c_Orderings_Oordering_001tf__a,type,
ordering_a: ( a > a > $o ) > ( a > a > $o ) > $o ).
thf(sy_c_Orderings_Oordering__top_001t__Set__Oset_It__Nat__Onat_J,type,
ordering_top_set_nat: ( set_nat > set_nat > $o ) > ( set_nat > set_nat > $o ) > set_nat > $o ).
thf(sy_c_Orderings_Opartial__preordering_001_062_I_Eo_Mt__Nat__Onat_J,type,
partia6907330574123654159_o_nat: ( ( $o > nat ) > ( $o > nat ) > $o ) > $o ).
thf(sy_c_Orderings_Opartial__preordering_001_062_I_Eo_Mtf__a_J,type,
partia5423788306336055317ng_o_a: ( ( $o > a ) > ( $o > a ) > $o ) > $o ).
thf(sy_c_Orderings_Opartial__preordering_001t__Nat__Onat,type,
partia6822818058636336922ng_nat: ( nat > nat > $o ) > $o ).
thf(sy_c_Orderings_Opartial__preordering_001t__Set__Oset_It__Nat__Onat_J,type,
partia5623167761149600464et_nat: ( set_nat > set_nat > $o ) > $o ).
thf(sy_c_Orderings_Opartial__preordering_001t__Set__Oset_Itf__a_J,type,
partia6602192050731689876_set_a: ( set_a > set_a > $o ) > $o ).
thf(sy_c_Orderings_Opartial__preordering_001tf__a,type,
partia125584492769400372ring_a: ( a > a > $o ) > $o ).
thf(sy_c_Orderings_Opreordering_001t__Nat__Onat,type,
preordering_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
thf(sy_c_Orderings_Opreordering_001tf__a,type,
preordering_a: ( a > a > $o ) > ( a > a > $o ) > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
top_top_set_nat: set_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
top_top_set_set_nat: set_set_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
top_top_set_set_a: set_set_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_Itf__a_J,type,
top_top_set_a: set_a ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001tf__a,type,
image_nat_a: ( nat > a ) > set_nat > set_a ).
thf(sy_c_Set_Oimage_001tf__a_001t__Nat__Onat,type,
image_a_nat: ( a > nat ) > set_a > set_nat ).
thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
image_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Zorn_Opred__on_Ochain_001t__Nat__Onat,type,
pred_chain_nat: set_nat > ( nat > nat > $o ) > set_nat > $o ).
thf(sy_c_Zorn_Opred__on_Ochain_001tf__a,type,
pred_chain_a: set_a > ( a > a > $o ) > set_a > $o ).
thf(sy_c_member_001_062_I_Eo_Mt__Nat__Onat_J,type,
member_o_nat: ( $o > nat ) > set_o_nat > $o ).
thf(sy_c_member_001_062_I_Eo_Mtf__a_J,type,
member_o_a: ( $o > a ) > set_o_a > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $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_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_S,type,
s: set_a ).
thf(sy_v_thesis____,type,
thesis: $o ).
% Relevant facts (1270)
thf(fact_0_assms,axiom,
minimu6197867597544231097imum_a @ s ).
% assms
thf(fact_1__092_060open_062_092_060exists_062x_O_Ax_A_092_060in_062_AS_A_092_060and_062_A_I_092_060forall_062y_092_060in_062S_O_Ay_A_092_060le_062_Ax_J_092_060close_062,axiom,
? [X: a] :
( ( member_a @ X @ s )
& ! [Xa: a] :
( ( member_a @ Xa @ s )
=> ( ord_less_eq_a @ Xa @ X ) ) ) ).
% \<open>\<exists>x. x \<in> S \<and> (\<forall>y\<in>S. y \<le> x)\<close>
thf(fact_2_order__refl,axiom,
! [X2: $o > nat] : ( ord_less_eq_o_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_3_order__refl,axiom,
! [X2: $o > a] : ( ord_less_eq_o_a @ X2 @ X2 ) ).
% order_refl
thf(fact_4_order__refl,axiom,
! [X2: set_nat] : ( ord_less_eq_set_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_5_order__refl,axiom,
! [X2: set_a] : ( ord_less_eq_set_a @ X2 @ X2 ) ).
% order_refl
thf(fact_6_order__refl,axiom,
! [X2: a] : ( ord_less_eq_a @ X2 @ X2 ) ).
% order_refl
thf(fact_7_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_8_dual__order_Orefl,axiom,
! [A: $o > nat] : ( ord_less_eq_o_nat @ A @ A ) ).
% dual_order.refl
thf(fact_9_dual__order_Orefl,axiom,
! [A: $o > a] : ( ord_less_eq_o_a @ A @ A ) ).
% dual_order.refl
thf(fact_10_dual__order_Orefl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% dual_order.refl
thf(fact_11_dual__order_Orefl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_12_dual__order_Orefl,axiom,
! [A: a] : ( ord_less_eq_a @ A @ A ) ).
% dual_order.refl
thf(fact_13_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_14_has__Maximum__def,axiom,
( minimu2432920480775036308_o_nat
= ( ^ [S: set_o_nat] :
? [X3: $o > nat] :
( ( member_o_nat @ X3 @ S )
& ! [Y: $o > nat] :
( ( member_o_nat @ Y @ S )
=> ( ord_less_eq_o_nat @ Y @ X3 ) ) ) ) ) ).
% has_Maximum_def
thf(fact_15_has__Maximum__def,axiom,
( minimu315547183909508560um_o_a
= ( ^ [S: set_o_a] :
? [X3: $o > a] :
( ( member_o_a @ X3 @ S )
& ! [Y: $o > a] :
( ( member_o_a @ Y @ S )
=> ( ord_less_eq_o_a @ Y @ X3 ) ) ) ) ) ).
% has_Maximum_def
thf(fact_16_has__Maximum__def,axiom,
( minimu2833918411073049867et_nat
= ( ^ [S: set_set_nat] :
? [X3: set_nat] :
( ( member_set_nat @ X3 @ S )
& ! [Y: set_nat] :
( ( member_set_nat @ Y @ S )
=> ( ord_less_eq_set_nat @ Y @ X3 ) ) ) ) ) ).
% has_Maximum_def
thf(fact_17_has__Maximum__def,axiom,
( minimu8775777210878807577_set_a
= ( ^ [S: set_set_a] :
? [X3: set_a] :
( ( member_set_a @ X3 @ S )
& ! [Y: set_a] :
( ( member_set_a @ Y @ S )
=> ( ord_less_eq_set_a @ Y @ X3 ) ) ) ) ) ).
% has_Maximum_def
thf(fact_18_has__Maximum__def,axiom,
( minimu6197867597544231097imum_a
= ( ^ [S: set_a] :
? [X3: a] :
( ( member_a @ X3 @ S )
& ! [Y: a] :
( ( member_a @ Y @ S )
=> ( ord_less_eq_a @ Y @ X3 ) ) ) ) ) ).
% has_Maximum_def
thf(fact_19_has__Maximum__def,axiom,
( minimu5597823600188243285um_nat
= ( ^ [S: set_nat] :
? [X3: nat] :
( ( member_nat @ X3 @ S )
& ! [Y: nat] :
( ( member_nat @ Y @ S )
=> ( ord_less_eq_nat @ Y @ X3 ) ) ) ) ) ).
% has_Maximum_def
thf(fact_20_has__Minimum__def,axiom,
( minimu7521381655477896998_o_nat
= ( ^ [S: set_o_nat] :
? [X3: $o > nat] :
( ( member_o_nat @ X3 @ S )
& ! [Y: $o > nat] :
( ( member_o_nat @ Y @ S )
=> ( ord_less_eq_o_nat @ X3 @ Y ) ) ) ) ) ).
% has_Minimum_def
thf(fact_21_has__Minimum__def,axiom,
( minimu4657282916794952894um_o_a
= ( ^ [S: set_o_a] :
? [X3: $o > a] :
( ( member_o_a @ X3 @ S )
& ! [Y: $o > a] :
( ( member_o_a @ Y @ S )
=> ( ord_less_eq_o_a @ X3 @ Y ) ) ) ) ) ).
% has_Minimum_def
thf(fact_22_has__Minimum__def,axiom,
( minimu8045924777645095161et_nat
= ( ^ [S: set_set_nat] :
? [X3: set_nat] :
( ( member_set_nat @ X3 @ S )
& ! [Y: set_nat] :
( ( member_set_nat @ Y @ S )
=> ( ord_less_eq_set_nat @ X3 @ Y ) ) ) ) ) ).
% has_Minimum_def
thf(fact_23_has__Minimum__def,axiom,
( minimu6896447672505010603_set_a
= ( ^ [S: set_set_a] :
? [X3: set_a] :
( ( member_set_a @ X3 @ S )
& ! [Y: set_a] :
( ( member_set_a @ Y @ S )
=> ( ord_less_eq_set_a @ X3 @ Y ) ) ) ) ) ).
% has_Minimum_def
thf(fact_24_has__Minimum__def,axiom,
( minimu7473987258551571531imum_a
= ( ^ [S: set_a] :
? [X3: a] :
( ( member_a @ X3 @ S )
& ! [Y: a] :
( ( member_a @ Y @ S )
=> ( ord_less_eq_a @ X3 @ Y ) ) ) ) ) ).
% has_Minimum_def
thf(fact_25_has__Minimum__def,axiom,
( minimu6712762941901723971um_nat
= ( ^ [S: set_nat] :
? [X3: nat] :
( ( member_nat @ X3 @ S )
& ! [Y: nat] :
( ( member_nat @ Y @ S )
=> ( ord_less_eq_nat @ X3 @ Y ) ) ) ) ) ).
% has_Minimum_def
thf(fact_26_eqMaximumI,axiom,
! [X2: a,S2: set_a] :
( ( member_a @ X2 @ S2 )
=> ( ! [Y2: a] :
( ( member_a @ Y2 @ S2 )
=> ( ord_less_eq_a @ Y2 @ X2 ) )
=> ( ( minimum_Maximum_a @ S2 )
= X2 ) ) ) ).
% eqMaximumI
thf(fact_27_eqMaximumI,axiom,
! [X2: nat,S2: set_nat] :
( ( member_nat @ X2 @ S2 )
=> ( ! [Y2: nat] :
( ( member_nat @ Y2 @ S2 )
=> ( ord_less_eq_nat @ Y2 @ X2 ) )
=> ( ( minimum_Maximum_nat @ S2 )
= X2 ) ) ) ).
% eqMaximumI
thf(fact_28_eqMinimumI,axiom,
! [X2: a,S2: set_a] :
( ( member_a @ X2 @ S2 )
=> ( ! [Y2: a] :
( ( member_a @ Y2 @ S2 )
=> ( ord_less_eq_a @ X2 @ Y2 ) )
=> ( ( minimum_Minimum_a @ S2 )
= X2 ) ) ) ).
% eqMinimumI
thf(fact_29_eqMinimumI,axiom,
! [X2: nat,S2: set_nat] :
( ( member_nat @ X2 @ S2 )
=> ( ! [Y2: nat] :
( ( member_nat @ Y2 @ S2 )
=> ( ord_less_eq_nat @ X2 @ Y2 ) )
=> ( ( minimum_Minimum_nat @ S2 )
= X2 ) ) ) ).
% eqMinimumI
thf(fact_30_verit__comp__simplify1_I2_J,axiom,
! [A: $o > nat] : ( ord_less_eq_o_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_31_verit__comp__simplify1_I2_J,axiom,
! [A: $o > a] : ( ord_less_eq_o_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_32_verit__comp__simplify1_I2_J,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_33_verit__comp__simplify1_I2_J,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_34_verit__comp__simplify1_I2_J,axiom,
! [A: a] : ( ord_less_eq_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_35_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_36_nle__le,axiom,
! [A: a,B: a] :
( ( ~ ( ord_less_eq_a @ A @ B ) )
= ( ( ord_less_eq_a @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_37_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_38_le__cases3,axiom,
! [X2: a,Y3: a,Z: a] :
( ( ( ord_less_eq_a @ X2 @ Y3 )
=> ~ ( ord_less_eq_a @ Y3 @ Z ) )
=> ( ( ( ord_less_eq_a @ Y3 @ X2 )
=> ~ ( ord_less_eq_a @ X2 @ Z ) )
=> ( ( ( ord_less_eq_a @ X2 @ Z )
=> ~ ( ord_less_eq_a @ Z @ Y3 ) )
=> ( ( ( ord_less_eq_a @ Z @ Y3 )
=> ~ ( ord_less_eq_a @ Y3 @ X2 ) )
=> ( ( ( ord_less_eq_a @ Y3 @ Z )
=> ~ ( ord_less_eq_a @ Z @ X2 ) )
=> ~ ( ( ord_less_eq_a @ Z @ X2 )
=> ~ ( ord_less_eq_a @ X2 @ Y3 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_39_le__cases3,axiom,
! [X2: nat,Y3: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y3 )
=> ~ ( ord_less_eq_nat @ Y3 @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y3 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y3 ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y3 )
=> ~ ( ord_less_eq_nat @ Y3 @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y3 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y3 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_40_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: $o > nat,Z2: $o > nat] : ( Y4 = Z2 ) )
= ( ^ [X3: $o > nat,Y: $o > nat] :
( ( ord_less_eq_o_nat @ X3 @ Y )
& ( ord_less_eq_o_nat @ Y @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_41_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: $o > a,Z2: $o > a] : ( Y4 = Z2 ) )
= ( ^ [X3: $o > a,Y: $o > a] :
( ( ord_less_eq_o_a @ X3 @ Y )
& ( ord_less_eq_o_a @ Y @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_42_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_nat,Z2: set_nat] : ( Y4 = Z2 ) )
= ( ^ [X3: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y )
& ( ord_less_eq_set_nat @ Y @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_43_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 ) )
= ( ^ [X3: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y )
& ( ord_less_eq_set_a @ Y @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_44_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: a,Z2: a] : ( Y4 = Z2 ) )
= ( ^ [X3: a,Y: a] :
( ( ord_less_eq_a @ X3 @ Y )
& ( ord_less_eq_a @ Y @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_45_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
& ( ord_less_eq_nat @ Y @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_46_ord__eq__le__trans,axiom,
! [A: $o > nat,B: $o > nat,C: $o > nat] :
( ( A = B )
=> ( ( ord_less_eq_o_nat @ B @ C )
=> ( ord_less_eq_o_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_47_ord__eq__le__trans,axiom,
! [A: $o > a,B: $o > a,C: $o > a] :
( ( A = B )
=> ( ( ord_less_eq_o_a @ B @ C )
=> ( ord_less_eq_o_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_48_ord__eq__le__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( A = B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_49_ord__eq__le__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( A = B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_50_ord__eq__le__trans,axiom,
! [A: a,B: a,C: a] :
( ( A = B )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ord_less_eq_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_51_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_52_ord__le__eq__trans,axiom,
! [A: $o > nat,B: $o > nat,C: $o > nat] :
( ( ord_less_eq_o_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_o_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_53_ord__le__eq__trans,axiom,
! [A: $o > a,B: $o > a,C: $o > a] :
( ( ord_less_eq_o_a @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_o_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_54_ord__le__eq__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_55_ord__le__eq__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_56_ord__le__eq__trans,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_57_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_58_order__antisym__conv,axiom,
! [Y3: $o > nat,X2: $o > nat] :
( ( ord_less_eq_o_nat @ Y3 @ X2 )
=> ( ( ord_less_eq_o_nat @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_59_order__antisym__conv,axiom,
! [Y3: $o > a,X2: $o > a] :
( ( ord_less_eq_o_a @ Y3 @ X2 )
=> ( ( ord_less_eq_o_a @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_60_order__antisym__conv,axiom,
! [Y3: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ Y3 @ X2 )
=> ( ( ord_less_eq_set_nat @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_61_order__antisym__conv,axiom,
! [Y3: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ Y3 @ X2 )
=> ( ( ord_less_eq_set_a @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_62_order__antisym__conv,axiom,
! [Y3: a,X2: a] :
( ( ord_less_eq_a @ Y3 @ X2 )
=> ( ( ord_less_eq_a @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_63_order__antisym__conv,axiom,
! [Y3: nat,X2: nat] :
( ( ord_less_eq_nat @ Y3 @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_64_linorder__le__cases,axiom,
! [X2: a,Y3: a] :
( ~ ( ord_less_eq_a @ X2 @ Y3 )
=> ( ord_less_eq_a @ Y3 @ X2 ) ) ).
% linorder_le_cases
thf(fact_65_linorder__le__cases,axiom,
! [X2: nat,Y3: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X2 ) ) ).
% linorder_le_cases
thf(fact_66_ord__le__eq__subst,axiom,
! [A: a,B: a,F: a > a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_67_ord__le__eq__subst,axiom,
! [A: a,B: a,F: a > nat,C: nat] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_68_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > a,C: a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_69_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_70_ord__le__eq__subst,axiom,
! [A: a,B: a,F: a > set_nat,C: set_nat] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_71_ord__le__eq__subst,axiom,
! [A: a,B: a,F: a > set_a,C: set_a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_72_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_73_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_74_ord__le__eq__subst,axiom,
! [A: set_nat,B: set_nat,F: set_nat > a,C: a] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_75_ord__le__eq__subst,axiom,
! [A: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_76_ord__eq__le__subst,axiom,
! [A: a,F: a > a,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_77_ord__eq__le__subst,axiom,
! [A: nat,F: a > nat,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_78_ord__eq__le__subst,axiom,
! [A: a,F: nat > a,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_79_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_80_ord__eq__le__subst,axiom,
! [A: set_nat,F: a > set_nat,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_81_ord__eq__le__subst,axiom,
! [A: set_a,F: a > set_a,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_82_ord__eq__le__subst,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_83_ord__eq__le__subst,axiom,
! [A: set_a,F: nat > set_a,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_84_ord__eq__le__subst,axiom,
! [A: a,F: set_nat > a,B: set_nat,C: set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_85_ord__eq__le__subst,axiom,
! [A: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_86_linorder__linear,axiom,
! [X2: a,Y3: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
| ( ord_less_eq_a @ Y3 @ X2 ) ) ).
% linorder_linear
thf(fact_87_linorder__linear,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
| ( ord_less_eq_nat @ Y3 @ X2 ) ) ).
% linorder_linear
thf(fact_88_verit__la__disequality,axiom,
! [A: a,B: a] :
( ( A = B )
| ~ ( ord_less_eq_a @ A @ B )
| ~ ( ord_less_eq_a @ B @ A ) ) ).
% verit_la_disequality
thf(fact_89_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_90_order__eq__refl,axiom,
! [X2: $o > nat,Y3: $o > nat] :
( ( X2 = Y3 )
=> ( ord_less_eq_o_nat @ X2 @ Y3 ) ) ).
% order_eq_refl
thf(fact_91_order__eq__refl,axiom,
! [X2: $o > a,Y3: $o > a] :
( ( X2 = Y3 )
=> ( ord_less_eq_o_a @ X2 @ Y3 ) ) ).
% order_eq_refl
thf(fact_92_order__eq__refl,axiom,
! [X2: set_nat,Y3: set_nat] :
( ( X2 = Y3 )
=> ( ord_less_eq_set_nat @ X2 @ Y3 ) ) ).
% order_eq_refl
thf(fact_93_order__eq__refl,axiom,
! [X2: set_a,Y3: set_a] :
( ( X2 = Y3 )
=> ( ord_less_eq_set_a @ X2 @ Y3 ) ) ).
% order_eq_refl
thf(fact_94_order__eq__refl,axiom,
! [X2: a,Y3: a] :
( ( X2 = Y3 )
=> ( ord_less_eq_a @ X2 @ Y3 ) ) ).
% order_eq_refl
thf(fact_95_order__eq__refl,axiom,
! [X2: nat,Y3: nat] :
( ( X2 = Y3 )
=> ( ord_less_eq_nat @ X2 @ Y3 ) ) ).
% order_eq_refl
thf(fact_96_order__subst2,axiom,
! [A: a,B: a,F: a > a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ ( F @ B ) @ C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_97_order__subst2,axiom,
! [A: a,B: a,F: a > nat,C: nat] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_98_order__subst2,axiom,
! [A: nat,B: nat,F: nat > a,C: a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_a @ ( F @ B ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_99_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_100_order__subst2,axiom,
! [A: a,B: a,F: a > set_nat,C: set_nat] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_101_order__subst2,axiom,
! [A: a,B: a,F: a > set_a,C: set_a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_102_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_103_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_104_order__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > a,C: a] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_a @ ( F @ B ) @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_105_order__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_106_order__subst1,axiom,
! [A: a,F: a > a,B: a,C: a] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_107_order__subst1,axiom,
! [A: a,F: nat > a,B: nat,C: nat] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_108_order__subst1,axiom,
! [A: nat,F: a > nat,B: a,C: a] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_109_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_110_order__subst1,axiom,
! [A: a,F: set_nat > a,B: set_nat,C: set_nat] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_111_order__subst1,axiom,
! [A: a,F: set_a > a,B: set_a,C: set_a] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_112_order__subst1,axiom,
! [A: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_113_order__subst1,axiom,
! [A: nat,F: set_a > nat,B: set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_114_order__subst1,axiom,
! [A: set_nat,F: a > set_nat,B: a,C: a] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_115_order__subst1,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_116_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: $o > nat,Z2: $o > nat] : ( Y4 = Z2 ) )
= ( ^ [A2: $o > nat,B2: $o > nat] :
( ( ord_less_eq_o_nat @ A2 @ B2 )
& ( ord_less_eq_o_nat @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_117_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: $o > a,Z2: $o > a] : ( Y4 = Z2 ) )
= ( ^ [A2: $o > a,B2: $o > a] :
( ( ord_less_eq_o_a @ A2 @ B2 )
& ( ord_less_eq_o_a @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_118_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_nat,Z2: set_nat] : ( Y4 = Z2 ) )
= ( ^ [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
& ( ord_less_eq_set_nat @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_119_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 ) )
= ( ^ [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
& ( ord_less_eq_set_a @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_120_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: a,Z2: a] : ( Y4 = Z2 ) )
= ( ^ [A2: a,B2: a] :
( ( ord_less_eq_a @ A2 @ B2 )
& ( ord_less_eq_a @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_121_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_122_le__fun__def,axiom,
( ord_less_eq_o_nat
= ( ^ [F2: $o > nat,G: $o > nat] :
! [X3: $o] : ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ).
% le_fun_def
thf(fact_123_le__fun__def,axiom,
( ord_less_eq_o_a
= ( ^ [F2: $o > a,G: $o > a] :
! [X3: $o] : ( ord_less_eq_a @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ).
% le_fun_def
thf(fact_124_le__funI,axiom,
! [F: $o > a,G2: $o > a] :
( ! [X: $o] : ( ord_less_eq_a @ ( F @ X ) @ ( G2 @ X ) )
=> ( ord_less_eq_o_a @ F @ G2 ) ) ).
% le_funI
thf(fact_125_le__funI,axiom,
! [F: $o > nat,G2: $o > nat] :
( ! [X: $o] : ( ord_less_eq_nat @ ( F @ X ) @ ( G2 @ X ) )
=> ( ord_less_eq_o_nat @ F @ G2 ) ) ).
% le_funI
thf(fact_126_le__funE,axiom,
! [F: $o > nat,G2: $o > nat,X2: $o] :
( ( ord_less_eq_o_nat @ F @ G2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( G2 @ X2 ) ) ) ).
% le_funE
thf(fact_127_le__funE,axiom,
! [F: $o > a,G2: $o > a,X2: $o] :
( ( ord_less_eq_o_a @ F @ G2 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( G2 @ X2 ) ) ) ).
% le_funE
thf(fact_128_le__funD,axiom,
! [F: $o > nat,G2: $o > nat,X2: $o] :
( ( ord_less_eq_o_nat @ F @ G2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( G2 @ X2 ) ) ) ).
% le_funD
thf(fact_129_le__funD,axiom,
! [F: $o > a,G2: $o > a,X2: $o] :
( ( ord_less_eq_o_a @ F @ G2 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( G2 @ X2 ) ) ) ).
% le_funD
thf(fact_130_antisym,axiom,
! [A: $o > nat,B: $o > nat] :
( ( ord_less_eq_o_nat @ A @ B )
=> ( ( ord_less_eq_o_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_131_antisym,axiom,
! [A: $o > a,B: $o > a] :
( ( ord_less_eq_o_a @ A @ B )
=> ( ( ord_less_eq_o_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_132_antisym,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_133_antisym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_134_antisym,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_135_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_136_dual__order_Otrans,axiom,
! [B: $o > nat,A: $o > nat,C: $o > nat] :
( ( ord_less_eq_o_nat @ B @ A )
=> ( ( ord_less_eq_o_nat @ C @ B )
=> ( ord_less_eq_o_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_137_dual__order_Otrans,axiom,
! [B: $o > a,A: $o > a,C: $o > a] :
( ( ord_less_eq_o_a @ B @ A )
=> ( ( ord_less_eq_o_a @ C @ B )
=> ( ord_less_eq_o_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_138_dual__order_Otrans,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_eq_set_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_139_dual__order_Otrans,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_eq_set_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_140_dual__order_Otrans,axiom,
! [B: a,A: a,C: a] :
( ( ord_less_eq_a @ B @ A )
=> ( ( ord_less_eq_a @ C @ B )
=> ( ord_less_eq_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_141_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_142_dual__order_Oantisym,axiom,
! [B: $o > nat,A: $o > nat] :
( ( ord_less_eq_o_nat @ B @ A )
=> ( ( ord_less_eq_o_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_143_dual__order_Oantisym,axiom,
! [B: $o > a,A: $o > a] :
( ( ord_less_eq_o_a @ B @ A )
=> ( ( ord_less_eq_o_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_144_dual__order_Oantisym,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_145_dual__order_Oantisym,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_146_dual__order_Oantisym,axiom,
! [B: a,A: a] :
( ( ord_less_eq_a @ B @ A )
=> ( ( ord_less_eq_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_147_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_148_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: $o > nat,Z2: $o > nat] : ( Y4 = Z2 ) )
= ( ^ [A2: $o > nat,B2: $o > nat] :
( ( ord_less_eq_o_nat @ B2 @ A2 )
& ( ord_less_eq_o_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_149_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: $o > a,Z2: $o > a] : ( Y4 = Z2 ) )
= ( ^ [A2: $o > a,B2: $o > a] :
( ( ord_less_eq_o_a @ B2 @ A2 )
& ( ord_less_eq_o_a @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_150_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_nat,Z2: set_nat] : ( Y4 = Z2 ) )
= ( ^ [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
& ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_151_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 ) )
= ( ^ [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
& ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_152_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: a,Z2: a] : ( Y4 = Z2 ) )
= ( ^ [A2: a,B2: a] :
( ( ord_less_eq_a @ B2 @ A2 )
& ( ord_less_eq_a @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_153_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_154_linorder__wlog,axiom,
! [P: a > a > $o,A: a,B: a] :
( ! [A3: a,B3: a] :
( ( ord_less_eq_a @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: a,B3: a] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_155_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: nat,B3: nat] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_156_order__trans,axiom,
! [X2: $o > nat,Y3: $o > nat,Z: $o > nat] :
( ( ord_less_eq_o_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_o_nat @ Y3 @ Z )
=> ( ord_less_eq_o_nat @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_157_order__trans,axiom,
! [X2: $o > a,Y3: $o > a,Z: $o > a] :
( ( ord_less_eq_o_a @ X2 @ Y3 )
=> ( ( ord_less_eq_o_a @ Y3 @ Z )
=> ( ord_less_eq_o_a @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_158_order__trans,axiom,
! [X2: set_nat,Y3: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_set_nat @ Y3 @ Z )
=> ( ord_less_eq_set_nat @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_159_order__trans,axiom,
! [X2: set_a,Y3: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ( ord_less_eq_set_a @ Y3 @ Z )
=> ( ord_less_eq_set_a @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_160_order__trans,axiom,
! [X2: a,Y3: a,Z: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
=> ( ( ord_less_eq_a @ Y3 @ Z )
=> ( ord_less_eq_a @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_161_order__trans,axiom,
! [X2: nat,Y3: nat,Z: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ Z )
=> ( ord_less_eq_nat @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_162_order_Otrans,axiom,
! [A: $o > nat,B: $o > nat,C: $o > nat] :
( ( ord_less_eq_o_nat @ A @ B )
=> ( ( ord_less_eq_o_nat @ B @ C )
=> ( ord_less_eq_o_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_163_order_Otrans,axiom,
! [A: $o > a,B: $o > a,C: $o > a] :
( ( ord_less_eq_o_a @ A @ B )
=> ( ( ord_less_eq_o_a @ B @ C )
=> ( ord_less_eq_o_a @ A @ C ) ) ) ).
% order.trans
thf(fact_164_order_Otrans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_165_order_Otrans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% order.trans
thf(fact_166_order_Otrans,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ord_less_eq_a @ A @ C ) ) ) ).
% order.trans
thf(fact_167_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_168_order__antisym,axiom,
! [X2: $o > nat,Y3: $o > nat] :
( ( ord_less_eq_o_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_o_nat @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ).
% order_antisym
thf(fact_169_order__antisym,axiom,
! [X2: $o > a,Y3: $o > a] :
( ( ord_less_eq_o_a @ X2 @ Y3 )
=> ( ( ord_less_eq_o_a @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ).
% order_antisym
thf(fact_170_order__antisym,axiom,
! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_set_nat @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ).
% order_antisym
thf(fact_171_order__antisym,axiom,
! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ( ord_less_eq_set_a @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ).
% order_antisym
thf(fact_172_order__antisym,axiom,
! [X2: a,Y3: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
=> ( ( ord_less_eq_a @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ).
% order_antisym
thf(fact_173_order__antisym,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ).
% order_antisym
thf(fact_174_Greatest__equality,axiom,
! [P: ( $o > nat ) > $o,X2: $o > nat] :
( ( P @ X2 )
=> ( ! [Y2: $o > nat] :
( ( P @ Y2 )
=> ( ord_less_eq_o_nat @ Y2 @ X2 ) )
=> ( ( order_Greatest_o_nat @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_175_Greatest__equality,axiom,
! [P: ( $o > a ) > $o,X2: $o > a] :
( ( P @ X2 )
=> ( ! [Y2: $o > a] :
( ( P @ Y2 )
=> ( ord_less_eq_o_a @ Y2 @ X2 ) )
=> ( ( order_Greatest_o_a @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_176_Greatest__equality,axiom,
! [P: set_nat > $o,X2: set_nat] :
( ( P @ X2 )
=> ( ! [Y2: set_nat] :
( ( P @ Y2 )
=> ( ord_less_eq_set_nat @ Y2 @ X2 ) )
=> ( ( order_5724808138429204845et_nat @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_177_Greatest__equality,axiom,
! [P: set_a > $o,X2: set_a] :
( ( P @ X2 )
=> ( ! [Y2: set_a] :
( ( P @ Y2 )
=> ( ord_less_eq_set_a @ Y2 @ X2 ) )
=> ( ( order_Greatest_set_a @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_178_Greatest__equality,axiom,
! [P: a > $o,X2: a] :
( ( P @ X2 )
=> ( ! [Y2: a] :
( ( P @ Y2 )
=> ( ord_less_eq_a @ Y2 @ X2 ) )
=> ( ( order_Greatest_a @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_179_Greatest__equality,axiom,
! [P: nat > $o,X2: nat] :
( ( P @ X2 )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X2 ) )
=> ( ( order_Greatest_nat @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_180_GreatestI2__order,axiom,
! [P: ( $o > nat ) > $o,X2: $o > nat,Q: ( $o > nat ) > $o] :
( ( P @ X2 )
=> ( ! [Y2: $o > nat] :
( ( P @ Y2 )
=> ( ord_less_eq_o_nat @ Y2 @ X2 ) )
=> ( ! [X: $o > nat] :
( ( P @ X )
=> ( ! [Y5: $o > nat] :
( ( P @ Y5 )
=> ( ord_less_eq_o_nat @ Y5 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_Greatest_o_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_181_GreatestI2__order,axiom,
! [P: ( $o > a ) > $o,X2: $o > a,Q: ( $o > a ) > $o] :
( ( P @ X2 )
=> ( ! [Y2: $o > a] :
( ( P @ Y2 )
=> ( ord_less_eq_o_a @ Y2 @ X2 ) )
=> ( ! [X: $o > a] :
( ( P @ X )
=> ( ! [Y5: $o > a] :
( ( P @ Y5 )
=> ( ord_less_eq_o_a @ Y5 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_Greatest_o_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_182_GreatestI2__order,axiom,
! [P: set_nat > $o,X2: set_nat,Q: set_nat > $o] :
( ( P @ X2 )
=> ( ! [Y2: set_nat] :
( ( P @ Y2 )
=> ( ord_less_eq_set_nat @ Y2 @ X2 ) )
=> ( ! [X: set_nat] :
( ( P @ X )
=> ( ! [Y5: set_nat] :
( ( P @ Y5 )
=> ( ord_less_eq_set_nat @ Y5 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_5724808138429204845et_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_183_GreatestI2__order,axiom,
! [P: set_a > $o,X2: set_a,Q: set_a > $o] :
( ( P @ X2 )
=> ( ! [Y2: set_a] :
( ( P @ Y2 )
=> ( ord_less_eq_set_a @ Y2 @ X2 ) )
=> ( ! [X: set_a] :
( ( P @ X )
=> ( ! [Y5: set_a] :
( ( P @ Y5 )
=> ( ord_less_eq_set_a @ Y5 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_Greatest_set_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_184_GreatestI2__order,axiom,
! [P: a > $o,X2: a,Q: a > $o] :
( ( P @ X2 )
=> ( ! [Y2: a] :
( ( P @ Y2 )
=> ( ord_less_eq_a @ Y2 @ X2 ) )
=> ( ! [X: a] :
( ( P @ X )
=> ( ! [Y5: a] :
( ( P @ Y5 )
=> ( ord_less_eq_a @ Y5 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_Greatest_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_185_GreatestI2__order,axiom,
! [P: nat > $o,X2: nat,Q: nat > $o] :
( ( P @ X2 )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X2 ) )
=> ( ! [X: nat] :
( ( P @ X )
=> ( ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_Greatest_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_186_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_o_nat
= ( ^ [X4: $o > $o > nat,Y6: $o > $o > nat] :
( ( ord_less_eq_o_nat @ ( X4 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_o_nat @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_187_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_o_a
= ( ^ [X4: $o > $o > a,Y6: $o > $o > a] :
( ( ord_less_eq_o_a @ ( X4 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_o_a @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_188_le__rel__bool__arg__iff,axiom,
( ord_le7022414076629706543et_nat
= ( ^ [X4: $o > set_nat,Y6: $o > set_nat] :
( ( ord_less_eq_set_nat @ ( X4 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_set_nat @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_189_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_set_a
= ( ^ [X4: $o > set_a,Y6: $o > set_a] :
( ( ord_less_eq_set_a @ ( X4 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_set_a @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_190_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_a
= ( ^ [X4: $o > a,Y6: $o > a] :
( ( ord_less_eq_a @ ( X4 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_a @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_191_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_nat
= ( ^ [X4: $o > nat,Y6: $o > nat] :
( ( ord_less_eq_nat @ ( X4 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_nat @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_192_order_Opartial__preordering__axioms,axiom,
partia6907330574123654159_o_nat @ ord_less_eq_o_nat ).
% order.partial_preordering_axioms
thf(fact_193_order_Opartial__preordering__axioms,axiom,
partia5423788306336055317ng_o_a @ ord_less_eq_o_a ).
% order.partial_preordering_axioms
thf(fact_194_order_Opartial__preordering__axioms,axiom,
partia5623167761149600464et_nat @ ord_less_eq_set_nat ).
% order.partial_preordering_axioms
thf(fact_195_order_Opartial__preordering__axioms,axiom,
partia6602192050731689876_set_a @ ord_less_eq_set_a ).
% order.partial_preordering_axioms
thf(fact_196_order_Opartial__preordering__axioms,axiom,
partia125584492769400372ring_a @ ord_less_eq_a ).
% order.partial_preordering_axioms
thf(fact_197_order_Opartial__preordering__axioms,axiom,
partia6822818058636336922ng_nat @ ord_less_eq_nat ).
% order.partial_preordering_axioms
thf(fact_198_LeastI2__wellorder__ex,axiom,
! [P: nat > $o,Q: nat > $o] :
( ? [X_1: nat] : ( P @ X_1 )
=> ( ! [A3: nat] :
( ( P @ A3 )
=> ( ! [B4: nat] :
( ( P @ B4 )
=> ( ord_less_eq_nat @ A3 @ B4 ) )
=> ( Q @ A3 ) ) )
=> ( Q @ ( ord_Least_nat @ P ) ) ) ) ).
% LeastI2_wellorder_ex
thf(fact_199_LeastI2__wellorder,axiom,
! [P: nat > $o,A: nat,Q: nat > $o] :
( ( P @ A )
=> ( ! [A3: nat] :
( ( P @ A3 )
=> ( ! [B4: nat] :
( ( P @ B4 )
=> ( ord_less_eq_nat @ A3 @ B4 ) )
=> ( Q @ A3 ) ) )
=> ( Q @ ( ord_Least_nat @ P ) ) ) ) ).
% LeastI2_wellorder
thf(fact_200_Least__equality,axiom,
! [P: ( $o > nat ) > $o,X2: $o > nat] :
( ( P @ X2 )
=> ( ! [Y2: $o > nat] :
( ( P @ Y2 )
=> ( ord_less_eq_o_nat @ X2 @ Y2 ) )
=> ( ( ord_Least_o_nat @ P )
= X2 ) ) ) ).
% Least_equality
thf(fact_201_Least__equality,axiom,
! [P: ( $o > a ) > $o,X2: $o > a] :
( ( P @ X2 )
=> ( ! [Y2: $o > a] :
( ( P @ Y2 )
=> ( ord_less_eq_o_a @ X2 @ Y2 ) )
=> ( ( ord_Least_o_a @ P )
= X2 ) ) ) ).
% Least_equality
thf(fact_202_Least__equality,axiom,
! [P: set_nat > $o,X2: set_nat] :
( ( P @ X2 )
=> ( ! [Y2: set_nat] :
( ( P @ Y2 )
=> ( ord_less_eq_set_nat @ X2 @ Y2 ) )
=> ( ( ord_Least_set_nat @ P )
= X2 ) ) ) ).
% Least_equality
thf(fact_203_Least__equality,axiom,
! [P: set_a > $o,X2: set_a] :
( ( P @ X2 )
=> ( ! [Y2: set_a] :
( ( P @ Y2 )
=> ( ord_less_eq_set_a @ X2 @ Y2 ) )
=> ( ( ord_Least_set_a @ P )
= X2 ) ) ) ).
% Least_equality
thf(fact_204_Least__equality,axiom,
! [P: a > $o,X2: a] :
( ( P @ X2 )
=> ( ! [Y2: a] :
( ( P @ Y2 )
=> ( ord_less_eq_a @ X2 @ Y2 ) )
=> ( ( ord_Least_a @ P )
= X2 ) ) ) ).
% Least_equality
thf(fact_205_Least__equality,axiom,
! [P: nat > $o,X2: nat] :
( ( P @ X2 )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ X2 @ Y2 ) )
=> ( ( ord_Least_nat @ P )
= X2 ) ) ) ).
% Least_equality
thf(fact_206_LeastI2__order,axiom,
! [P: ( $o > nat ) > $o,X2: $o > nat,Q: ( $o > nat ) > $o] :
( ( P @ X2 )
=> ( ! [Y2: $o > nat] :
( ( P @ Y2 )
=> ( ord_less_eq_o_nat @ X2 @ Y2 ) )
=> ( ! [X: $o > nat] :
( ( P @ X )
=> ( ! [Y5: $o > nat] :
( ( P @ Y5 )
=> ( ord_less_eq_o_nat @ X @ Y5 ) )
=> ( Q @ X ) ) )
=> ( Q @ ( ord_Least_o_nat @ P ) ) ) ) ) ).
% LeastI2_order
thf(fact_207_LeastI2__order,axiom,
! [P: ( $o > a ) > $o,X2: $o > a,Q: ( $o > a ) > $o] :
( ( P @ X2 )
=> ( ! [Y2: $o > a] :
( ( P @ Y2 )
=> ( ord_less_eq_o_a @ X2 @ Y2 ) )
=> ( ! [X: $o > a] :
( ( P @ X )
=> ( ! [Y5: $o > a] :
( ( P @ Y5 )
=> ( ord_less_eq_o_a @ X @ Y5 ) )
=> ( Q @ X ) ) )
=> ( Q @ ( ord_Least_o_a @ P ) ) ) ) ) ).
% LeastI2_order
thf(fact_208_LeastI2__order,axiom,
! [P: set_nat > $o,X2: set_nat,Q: set_nat > $o] :
( ( P @ X2 )
=> ( ! [Y2: set_nat] :
( ( P @ Y2 )
=> ( ord_less_eq_set_nat @ X2 @ Y2 ) )
=> ( ! [X: set_nat] :
( ( P @ X )
=> ( ! [Y5: set_nat] :
( ( P @ Y5 )
=> ( ord_less_eq_set_nat @ X @ Y5 ) )
=> ( Q @ X ) ) )
=> ( Q @ ( ord_Least_set_nat @ P ) ) ) ) ) ).
% LeastI2_order
thf(fact_209_LeastI2__order,axiom,
! [P: set_a > $o,X2: set_a,Q: set_a > $o] :
( ( P @ X2 )
=> ( ! [Y2: set_a] :
( ( P @ Y2 )
=> ( ord_less_eq_set_a @ X2 @ Y2 ) )
=> ( ! [X: set_a] :
( ( P @ X )
=> ( ! [Y5: set_a] :
( ( P @ Y5 )
=> ( ord_less_eq_set_a @ X @ Y5 ) )
=> ( Q @ X ) ) )
=> ( Q @ ( ord_Least_set_a @ P ) ) ) ) ) ).
% LeastI2_order
thf(fact_210_LeastI2__order,axiom,
! [P: a > $o,X2: a,Q: a > $o] :
( ( P @ X2 )
=> ( ! [Y2: a] :
( ( P @ Y2 )
=> ( ord_less_eq_a @ X2 @ Y2 ) )
=> ( ! [X: a] :
( ( P @ X )
=> ( ! [Y5: a] :
( ( P @ Y5 )
=> ( ord_less_eq_a @ X @ Y5 ) )
=> ( Q @ X ) ) )
=> ( Q @ ( ord_Least_a @ P ) ) ) ) ) ).
% LeastI2_order
thf(fact_211_LeastI2__order,axiom,
! [P: nat > $o,X2: nat,Q: nat > $o] :
( ( P @ X2 )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ X2 @ Y2 ) )
=> ( ! [X: nat] :
( ( P @ X )
=> ( ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ X @ Y5 ) )
=> ( Q @ X ) ) )
=> ( Q @ ( ord_Least_nat @ P ) ) ) ) ) ).
% LeastI2_order
thf(fact_212_Least1__le,axiom,
! [P: ( $o > nat ) > $o,Z: $o > nat] :
( ? [X5: $o > nat] :
( ( P @ X5 )
& ! [Y2: $o > nat] :
( ( P @ Y2 )
=> ( ord_less_eq_o_nat @ X5 @ Y2 ) )
& ! [Y2: $o > nat] :
( ( ( P @ Y2 )
& ! [Ya: $o > nat] :
( ( P @ Ya )
=> ( ord_less_eq_o_nat @ Y2 @ Ya ) ) )
=> ( Y2 = X5 ) ) )
=> ( ( P @ Z )
=> ( ord_less_eq_o_nat @ ( ord_Least_o_nat @ P ) @ Z ) ) ) ).
% Least1_le
thf(fact_213_Least1__le,axiom,
! [P: ( $o > a ) > $o,Z: $o > a] :
( ? [X5: $o > a] :
( ( P @ X5 )
& ! [Y2: $o > a] :
( ( P @ Y2 )
=> ( ord_less_eq_o_a @ X5 @ Y2 ) )
& ! [Y2: $o > a] :
( ( ( P @ Y2 )
& ! [Ya: $o > a] :
( ( P @ Ya )
=> ( ord_less_eq_o_a @ Y2 @ Ya ) ) )
=> ( Y2 = X5 ) ) )
=> ( ( P @ Z )
=> ( ord_less_eq_o_a @ ( ord_Least_o_a @ P ) @ Z ) ) ) ).
% Least1_le
thf(fact_214_Least1__le,axiom,
! [P: set_nat > $o,Z: set_nat] :
( ? [X5: set_nat] :
( ( P @ X5 )
& ! [Y2: set_nat] :
( ( P @ Y2 )
=> ( ord_less_eq_set_nat @ X5 @ Y2 ) )
& ! [Y2: set_nat] :
( ( ( P @ Y2 )
& ! [Ya: set_nat] :
( ( P @ Ya )
=> ( ord_less_eq_set_nat @ Y2 @ Ya ) ) )
=> ( Y2 = X5 ) ) )
=> ( ( P @ Z )
=> ( ord_less_eq_set_nat @ ( ord_Least_set_nat @ P ) @ Z ) ) ) ).
% Least1_le
thf(fact_215_Least1__le,axiom,
! [P: set_a > $o,Z: set_a] :
( ? [X5: set_a] :
( ( P @ X5 )
& ! [Y2: set_a] :
( ( P @ Y2 )
=> ( ord_less_eq_set_a @ X5 @ Y2 ) )
& ! [Y2: set_a] :
( ( ( P @ Y2 )
& ! [Ya: set_a] :
( ( P @ Ya )
=> ( ord_less_eq_set_a @ Y2 @ Ya ) ) )
=> ( Y2 = X5 ) ) )
=> ( ( P @ Z )
=> ( ord_less_eq_set_a @ ( ord_Least_set_a @ P ) @ Z ) ) ) ).
% Least1_le
thf(fact_216_Least1__le,axiom,
! [P: a > $o,Z: a] :
( ? [X5: a] :
( ( P @ X5 )
& ! [Y2: a] :
( ( P @ Y2 )
=> ( ord_less_eq_a @ X5 @ Y2 ) )
& ! [Y2: a] :
( ( ( P @ Y2 )
& ! [Ya: a] :
( ( P @ Ya )
=> ( ord_less_eq_a @ Y2 @ Ya ) ) )
=> ( Y2 = X5 ) ) )
=> ( ( P @ Z )
=> ( ord_less_eq_a @ ( ord_Least_a @ P ) @ Z ) ) ) ).
% Least1_le
thf(fact_217_Least1__le,axiom,
! [P: nat > $o,Z: nat] :
( ? [X5: nat] :
( ( P @ X5 )
& ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ X5 @ Y2 ) )
& ! [Y2: nat] :
( ( ( P @ Y2 )
& ! [Ya: nat] :
( ( P @ Ya )
=> ( ord_less_eq_nat @ Y2 @ Ya ) ) )
=> ( Y2 = X5 ) ) )
=> ( ( P @ Z )
=> ( ord_less_eq_nat @ ( ord_Least_nat @ P ) @ Z ) ) ) ).
% Least1_le
thf(fact_218_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_219_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_220_Collect__mem__eq,axiom,
! [A4: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_221_Collect__mem__eq,axiom,
! [A4: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_222_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_223_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X: a] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_224_Least1I,axiom,
! [P: ( $o > nat ) > $o] :
( ? [X5: $o > nat] :
( ( P @ X5 )
& ! [Y2: $o > nat] :
( ( P @ Y2 )
=> ( ord_less_eq_o_nat @ X5 @ Y2 ) )
& ! [Y2: $o > nat] :
( ( ( P @ Y2 )
& ! [Ya: $o > nat] :
( ( P @ Ya )
=> ( ord_less_eq_o_nat @ Y2 @ Ya ) ) )
=> ( Y2 = X5 ) ) )
=> ( P @ ( ord_Least_o_nat @ P ) ) ) ).
% Least1I
thf(fact_225_Least1I,axiom,
! [P: ( $o > a ) > $o] :
( ? [X5: $o > a] :
( ( P @ X5 )
& ! [Y2: $o > a] :
( ( P @ Y2 )
=> ( ord_less_eq_o_a @ X5 @ Y2 ) )
& ! [Y2: $o > a] :
( ( ( P @ Y2 )
& ! [Ya: $o > a] :
( ( P @ Ya )
=> ( ord_less_eq_o_a @ Y2 @ Ya ) ) )
=> ( Y2 = X5 ) ) )
=> ( P @ ( ord_Least_o_a @ P ) ) ) ).
% Least1I
thf(fact_226_Least1I,axiom,
! [P: set_nat > $o] :
( ? [X5: set_nat] :
( ( P @ X5 )
& ! [Y2: set_nat] :
( ( P @ Y2 )
=> ( ord_less_eq_set_nat @ X5 @ Y2 ) )
& ! [Y2: set_nat] :
( ( ( P @ Y2 )
& ! [Ya: set_nat] :
( ( P @ Ya )
=> ( ord_less_eq_set_nat @ Y2 @ Ya ) ) )
=> ( Y2 = X5 ) ) )
=> ( P @ ( ord_Least_set_nat @ P ) ) ) ).
% Least1I
thf(fact_227_Least1I,axiom,
! [P: set_a > $o] :
( ? [X5: set_a] :
( ( P @ X5 )
& ! [Y2: set_a] :
( ( P @ Y2 )
=> ( ord_less_eq_set_a @ X5 @ Y2 ) )
& ! [Y2: set_a] :
( ( ( P @ Y2 )
& ! [Ya: set_a] :
( ( P @ Ya )
=> ( ord_less_eq_set_a @ Y2 @ Ya ) ) )
=> ( Y2 = X5 ) ) )
=> ( P @ ( ord_Least_set_a @ P ) ) ) ).
% Least1I
thf(fact_228_Least1I,axiom,
! [P: a > $o] :
( ? [X5: a] :
( ( P @ X5 )
& ! [Y2: a] :
( ( P @ Y2 )
=> ( ord_less_eq_a @ X5 @ Y2 ) )
& ! [Y2: a] :
( ( ( P @ Y2 )
& ! [Ya: a] :
( ( P @ Ya )
=> ( ord_less_eq_a @ Y2 @ Ya ) ) )
=> ( Y2 = X5 ) ) )
=> ( P @ ( ord_Least_a @ P ) ) ) ).
% Least1I
thf(fact_229_Least1I,axiom,
! [P: nat > $o] :
( ? [X5: nat] :
( ( P @ X5 )
& ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ X5 @ Y2 ) )
& ! [Y2: nat] :
( ( ( P @ Y2 )
& ! [Ya: nat] :
( ( P @ Ya )
=> ( ord_less_eq_nat @ Y2 @ Ya ) ) )
=> ( Y2 = X5 ) ) )
=> ( P @ ( ord_Least_nat @ P ) ) ) ).
% Least1I
thf(fact_230_LeastI2,axiom,
! [P: nat > $o,A: nat,Q: nat > $o] :
( ( P @ A )
=> ( ! [X: nat] :
( ( P @ X )
=> ( Q @ X ) )
=> ( Q @ ( ord_Least_nat @ P ) ) ) ) ).
% LeastI2
thf(fact_231_LeastI__ex,axiom,
! [P: nat > $o] :
( ? [X_1: nat] : ( P @ X_1 )
=> ( P @ ( ord_Least_nat @ P ) ) ) ).
% LeastI_ex
thf(fact_232_LeastI2__ex,axiom,
! [P: nat > $o,Q: nat > $o] :
( ? [X_1: nat] : ( P @ X_1 )
=> ( ! [X: nat] :
( ( P @ X )
=> ( Q @ X ) )
=> ( Q @ ( ord_Least_nat @ P ) ) ) ) ).
% LeastI2_ex
thf(fact_233_partial__preordering__def,axiom,
( partia6822818058636336922ng_nat
= ( ^ [Less_eq: nat > nat > $o] :
( ! [A2: nat] : ( Less_eq @ A2 @ A2 )
& ! [A2: nat,B2: nat,C2: nat] :
( ( Less_eq @ A2 @ B2 )
=> ( ( Less_eq @ B2 @ C2 )
=> ( Less_eq @ A2 @ C2 ) ) ) ) ) ) ).
% partial_preordering_def
thf(fact_234_partial__preordering__def,axiom,
( partia125584492769400372ring_a
= ( ^ [Less_eq: a > a > $o] :
( ! [A2: a] : ( Less_eq @ A2 @ A2 )
& ! [A2: a,B2: a,C2: a] :
( ( Less_eq @ A2 @ B2 )
=> ( ( Less_eq @ B2 @ C2 )
=> ( Less_eq @ A2 @ C2 ) ) ) ) ) ) ).
% partial_preordering_def
thf(fact_235_partial__preordering_Otrans,axiom,
! [Less_eq2: nat > nat > $o,A: nat,B: nat,C: nat] :
( ( partia6822818058636336922ng_nat @ Less_eq2 )
=> ( ( Less_eq2 @ A @ B )
=> ( ( Less_eq2 @ B @ C )
=> ( Less_eq2 @ A @ C ) ) ) ) ).
% partial_preordering.trans
thf(fact_236_partial__preordering_Otrans,axiom,
! [Less_eq2: a > a > $o,A: a,B: a,C: a] :
( ( partia125584492769400372ring_a @ Less_eq2 )
=> ( ( Less_eq2 @ A @ B )
=> ( ( Less_eq2 @ B @ C )
=> ( Less_eq2 @ A @ C ) ) ) ) ).
% partial_preordering.trans
thf(fact_237_partial__preordering_Ointro,axiom,
! [Less_eq2: nat > nat > $o] :
( ! [A3: nat] : ( Less_eq2 @ A3 @ A3 )
=> ( ! [A3: nat,B3: nat,C3: nat] :
( ( Less_eq2 @ A3 @ B3 )
=> ( ( Less_eq2 @ B3 @ C3 )
=> ( Less_eq2 @ A3 @ C3 ) ) )
=> ( partia6822818058636336922ng_nat @ Less_eq2 ) ) ) ).
% partial_preordering.intro
thf(fact_238_partial__preordering_Ointro,axiom,
! [Less_eq2: a > a > $o] :
( ! [A3: a] : ( Less_eq2 @ A3 @ A3 )
=> ( ! [A3: a,B3: a,C3: a] :
( ( Less_eq2 @ A3 @ B3 )
=> ( ( Less_eq2 @ B3 @ C3 )
=> ( Less_eq2 @ A3 @ C3 ) ) )
=> ( partia125584492769400372ring_a @ Less_eq2 ) ) ) ).
% partial_preordering.intro
thf(fact_239_partial__preordering_Orefl,axiom,
! [Less_eq2: nat > nat > $o,A: nat] :
( ( partia6822818058636336922ng_nat @ Less_eq2 )
=> ( Less_eq2 @ A @ A ) ) ).
% partial_preordering.refl
thf(fact_240_partial__preordering_Orefl,axiom,
! [Less_eq2: a > a > $o,A: a] :
( ( partia125584492769400372ring_a @ Less_eq2 )
=> ( Less_eq2 @ A @ A ) ) ).
% partial_preordering.refl
thf(fact_241_ordering_Oaxioms_I1_J,axiom,
! [Less_eq2: nat > nat > $o,Less: nat > nat > $o] :
( ( ordering_nat @ Less_eq2 @ Less )
=> ( partia6822818058636336922ng_nat @ Less_eq2 ) ) ).
% ordering.axioms(1)
thf(fact_242_ordering_Oaxioms_I1_J,axiom,
! [Less_eq2: a > a > $o,Less: a > a > $o] :
( ( ordering_a @ Less_eq2 @ Less )
=> ( partia125584492769400372ring_a @ Less_eq2 ) ) ).
% ordering.axioms(1)
thf(fact_243_preordering_Oaxioms_I1_J,axiom,
! [Less_eq2: nat > nat > $o,Less: nat > nat > $o] :
( ( preordering_nat @ Less_eq2 @ Less )
=> ( partia6822818058636336922ng_nat @ Less_eq2 ) ) ).
% preordering.axioms(1)
thf(fact_244_preordering_Oaxioms_I1_J,axiom,
! [Less_eq2: a > a > $o,Less: a > a > $o] :
( ( preordering_a @ Less_eq2 @ Less )
=> ( partia125584492769400372ring_a @ Less_eq2 ) ) ).
% preordering.axioms(1)
thf(fact_245_lfp__greatest,axiom,
! [F: set_a > set_a,A4: set_a] :
( ! [U: set_a] :
( ( ord_less_eq_set_a @ ( F @ U ) @ U )
=> ( ord_less_eq_set_a @ A4 @ U ) )
=> ( ord_less_eq_set_a @ A4 @ ( comple1558298011288954135_set_a @ F ) ) ) ).
% lfp_greatest
thf(fact_246_lfp__greatest,axiom,
! [F: set_nat > set_nat,A4: set_nat] :
( ! [U: set_nat] :
( ( ord_less_eq_set_nat @ ( F @ U ) @ U )
=> ( ord_less_eq_set_nat @ A4 @ U ) )
=> ( ord_less_eq_set_nat @ A4 @ ( comple7975543026063415949et_nat @ F ) ) ) ).
% lfp_greatest
thf(fact_247_lfp__lowerbound,axiom,
! [F: set_a > set_a,A4: set_a] :
( ( ord_less_eq_set_a @ ( F @ A4 ) @ A4 )
=> ( ord_less_eq_set_a @ ( comple1558298011288954135_set_a @ F ) @ A4 ) ) ).
% lfp_lowerbound
thf(fact_248_lfp__lowerbound,axiom,
! [F: set_nat > set_nat,A4: set_nat] :
( ( ord_less_eq_set_nat @ ( F @ A4 ) @ A4 )
=> ( ord_less_eq_set_nat @ ( comple7975543026063415949et_nat @ F ) @ A4 ) ) ).
% lfp_lowerbound
thf(fact_249_lfp__mono,axiom,
! [F: set_a > set_a,G2: set_a > set_a] :
( ! [Z3: set_a] : ( ord_less_eq_set_a @ ( F @ Z3 ) @ ( G2 @ Z3 ) )
=> ( ord_less_eq_set_a @ ( comple1558298011288954135_set_a @ F ) @ ( comple1558298011288954135_set_a @ G2 ) ) ) ).
% lfp_mono
thf(fact_250_lfp__mono,axiom,
! [F: set_nat > set_nat,G2: set_nat > set_nat] :
( ! [Z3: set_nat] : ( ord_less_eq_set_nat @ ( F @ Z3 ) @ ( G2 @ Z3 ) )
=> ( ord_less_eq_set_nat @ ( comple7975543026063415949et_nat @ F ) @ ( comple7975543026063415949et_nat @ G2 ) ) ) ).
% lfp_mono
thf(fact_251_gfp__least,axiom,
! [F: set_nat > set_nat,X6: set_nat] :
( ! [U: set_nat] :
( ( ord_less_eq_set_nat @ U @ ( F @ U ) )
=> ( ord_less_eq_set_nat @ U @ X6 ) )
=> ( ord_less_eq_set_nat @ ( comple1596078789208929544et_nat @ F ) @ X6 ) ) ).
% gfp_least
thf(fact_252_gfp__least,axiom,
! [F: set_a > set_a,X6: set_a] :
( ! [U: set_a] :
( ( ord_less_eq_set_a @ U @ ( F @ U ) )
=> ( ord_less_eq_set_a @ U @ X6 ) )
=> ( ord_less_eq_set_a @ ( comple3341859861669737308_set_a @ F ) @ X6 ) ) ).
% gfp_least
thf(fact_253_order__less__imp__not__less,axiom,
! [X2: a,Y3: a] :
( ( ord_less_a @ X2 @ Y3 )
=> ~ ( ord_less_a @ Y3 @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_254_order__less__imp__not__less,axiom,
! [X2: set_nat,Y3: set_nat] :
( ( ord_less_set_nat @ X2 @ Y3 )
=> ~ ( ord_less_set_nat @ Y3 @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_255_order__less__imp__not__less,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ~ ( ord_less_nat @ Y3 @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_256_order__less__imp__not__eq2,axiom,
! [X2: a,Y3: a] :
( ( ord_less_a @ X2 @ Y3 )
=> ( Y3 != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_257_order__less__imp__not__eq2,axiom,
! [X2: set_nat,Y3: set_nat] :
( ( ord_less_set_nat @ X2 @ Y3 )
=> ( Y3 != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_258_order__less__imp__not__eq2,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( Y3 != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_259_order__less__imp__not__eq,axiom,
! [X2: a,Y3: a] :
( ( ord_less_a @ X2 @ Y3 )
=> ( X2 != Y3 ) ) ).
% order_less_imp_not_eq
thf(fact_260_order__less__imp__not__eq,axiom,
! [X2: set_nat,Y3: set_nat] :
( ( ord_less_set_nat @ X2 @ Y3 )
=> ( X2 != Y3 ) ) ).
% order_less_imp_not_eq
thf(fact_261_order__less__imp__not__eq,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( X2 != Y3 ) ) ).
% order_less_imp_not_eq
thf(fact_262_linorder__less__linear,axiom,
! [X2: a,Y3: a] :
( ( ord_less_a @ X2 @ Y3 )
| ( X2 = Y3 )
| ( ord_less_a @ Y3 @ X2 ) ) ).
% linorder_less_linear
thf(fact_263_linorder__less__linear,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
| ( X2 = Y3 )
| ( ord_less_nat @ Y3 @ X2 ) ) ).
% linorder_less_linear
thf(fact_264_preordering__strictI,axiom,
! [Less_eq2: nat > nat > $o,Less: nat > nat > $o] :
( ! [A3: nat,B3: nat] :
( ( Less_eq2 @ A3 @ B3 )
= ( ( Less @ A3 @ B3 )
| ( A3 = B3 ) ) )
=> ( ! [A3: nat,B3: nat] :
( ( Less @ A3 @ B3 )
=> ~ ( Less @ B3 @ A3 ) )
=> ( ! [A3: nat] :
~ ( Less @ A3 @ A3 )
=> ( ! [A3: nat,B3: nat,C3: nat] :
( ( Less @ A3 @ B3 )
=> ( ( Less @ B3 @ C3 )
=> ( Less @ A3 @ C3 ) ) )
=> ( preordering_nat @ Less_eq2 @ Less ) ) ) ) ) ).
% preordering_strictI
thf(fact_265_preordering__strictI,axiom,
! [Less_eq2: a > a > $o,Less: a > a > $o] :
( ! [A3: a,B3: a] :
( ( Less_eq2 @ A3 @ B3 )
= ( ( Less @ A3 @ B3 )
| ( A3 = B3 ) ) )
=> ( ! [A3: a,B3: a] :
( ( Less @ A3 @ B3 )
=> ~ ( Less @ B3 @ A3 ) )
=> ( ! [A3: a] :
~ ( Less @ A3 @ A3 )
=> ( ! [A3: a,B3: a,C3: a] :
( ( Less @ A3 @ B3 )
=> ( ( Less @ B3 @ C3 )
=> ( Less @ A3 @ C3 ) ) )
=> ( preordering_a @ Less_eq2 @ Less ) ) ) ) ) ).
% preordering_strictI
thf(fact_266_order__less__imp__triv,axiom,
! [X2: a,Y3: a,P: $o] :
( ( ord_less_a @ X2 @ Y3 )
=> ( ( ord_less_a @ Y3 @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_267_order__less__imp__triv,axiom,
! [X2: set_nat,Y3: set_nat,P: $o] :
( ( ord_less_set_nat @ X2 @ Y3 )
=> ( ( ord_less_set_nat @ Y3 @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_268_order__less__imp__triv,axiom,
! [X2: nat,Y3: nat,P: $o] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ( ord_less_nat @ Y3 @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_269_order__less__not__sym,axiom,
! [X2: a,Y3: a] :
( ( ord_less_a @ X2 @ Y3 )
=> ~ ( ord_less_a @ Y3 @ X2 ) ) ).
% order_less_not_sym
thf(fact_270_order__less__not__sym,axiom,
! [X2: set_nat,Y3: set_nat] :
( ( ord_less_set_nat @ X2 @ Y3 )
=> ~ ( ord_less_set_nat @ Y3 @ X2 ) ) ).
% order_less_not_sym
thf(fact_271_order__less__not__sym,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ~ ( ord_less_nat @ Y3 @ X2 ) ) ).
% order_less_not_sym
thf(fact_272_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > a,C: a] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_a @ ( F @ B ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_273_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_274_order__less__subst2,axiom,
! [A: a,B: a,F: a > nat,C: nat] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_a @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_275_order__less__subst2,axiom,
! [A: a,B: a,F: a > a,C: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_a @ ( F @ B ) @ C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_a @ X @ Y2 )
=> ( ord_less_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_276_order__less__subst2,axiom,
! [A: a,B: a,F: a > set_nat,C: set_nat] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_a @ X @ Y2 )
=> ( ord_less_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_277_order__less__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_278_order__less__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > a,C: a] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ord_less_a @ ( F @ B ) @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X @ Y2 )
=> ( ord_less_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_279_order__less__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X @ Y2 )
=> ( ord_less_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_280_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_281_order__less__subst1,axiom,
! [A: nat,F: a > nat,B: a,C: a] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_a @ B @ C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_a @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_282_order__less__subst1,axiom,
! [A: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_283_order__less__subst1,axiom,
! [A: a,F: nat > a,B: nat,C: nat] :
( ( ord_less_a @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_284_order__less__subst1,axiom,
! [A: a,F: a > a,B: a,C: a] :
( ( ord_less_a @ A @ ( F @ B ) )
=> ( ( ord_less_a @ B @ C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_a @ X @ Y2 )
=> ( ord_less_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_285_order__less__subst1,axiom,
! [A: a,F: set_nat > a,B: set_nat,C: set_nat] :
( ( ord_less_a @ A @ ( F @ B ) )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X @ Y2 )
=> ( ord_less_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_286_order__less__subst1,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_287_order__less__subst1,axiom,
! [A: set_nat,F: a > set_nat,B: a,C: a] :
( ( ord_less_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_a @ B @ C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_a @ X @ Y2 )
=> ( ord_less_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_288_order__less__subst1,axiom,
! [A: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X @ Y2 )
=> ( ord_less_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_289_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_290_order__less__irrefl,axiom,
! [X2: a] :
~ ( ord_less_a @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_291_order__less__irrefl,axiom,
! [X2: set_nat] :
~ ( ord_less_set_nat @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_292_order__less__irrefl,axiom,
! [X2: nat] :
~ ( ord_less_nat @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_293_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > a,C: a] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_294_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_295_ord__less__eq__subst,axiom,
! [A: a,B: a,F: a > nat,C: nat] :
( ( ord_less_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_a @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_296_ord__less__eq__subst,axiom,
! [A: a,B: a,F: a > a,C: a] :
( ( ord_less_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_a @ X @ Y2 )
=> ( ord_less_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_297_ord__less__eq__subst,axiom,
! [A: a,B: a,F: a > set_nat,C: set_nat] :
( ( ord_less_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_a @ X @ Y2 )
=> ( ord_less_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_298_ord__less__eq__subst,axiom,
! [A: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_299_ord__less__eq__subst,axiom,
! [A: set_nat,B: set_nat,F: set_nat > a,C: a] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X @ Y2 )
=> ( ord_less_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_300_ord__less__eq__subst,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X @ Y2 )
=> ( ord_less_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_301_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_302_ord__eq__less__subst,axiom,
! [A: a,F: nat > a,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_303_ord__eq__less__subst,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_304_ord__eq__less__subst,axiom,
! [A: nat,F: a > nat,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_a @ B @ C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_a @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_305_ord__eq__less__subst,axiom,
! [A: a,F: a > a,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_a @ B @ C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_a @ X @ Y2 )
=> ( ord_less_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_306_ord__eq__less__subst,axiom,
! [A: set_nat,F: a > set_nat,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_a @ B @ C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_a @ X @ Y2 )
=> ( ord_less_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_307_ord__eq__less__subst,axiom,
! [A: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_308_ord__eq__less__subst,axiom,
! [A: a,F: set_nat > a,B: set_nat,C: set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X @ Y2 )
=> ( ord_less_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_309_ord__eq__less__subst,axiom,
! [A: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X @ Y2 )
=> ( ord_less_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_310_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_311_order__less__trans,axiom,
! [X2: nat,Y3: nat,Z: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ( ord_less_nat @ Y3 @ Z )
=> ( ord_less_nat @ X2 @ Z ) ) ) ).
% order_less_trans
thf(fact_312_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_313_linorder__neq__iff,axiom,
! [X2: nat,Y3: nat] :
( ( X2 != Y3 )
= ( ( ord_less_nat @ X2 @ Y3 )
| ( ord_less_nat @ Y3 @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_314_order__less__asym,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ~ ( ord_less_nat @ Y3 @ X2 ) ) ).
% order_less_asym
thf(fact_315_linorder__neqE,axiom,
! [X2: nat,Y3: nat] :
( ( X2 != Y3 )
=> ( ~ ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ Y3 @ X2 ) ) ) ).
% linorder_neqE
thf(fact_316_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_317_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_318_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_319_not__less__iff__gr__or__eq,axiom,
! [X2: nat,Y3: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
= ( ( ord_less_nat @ Y3 @ X2 )
| ( X2 = Y3 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_320_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_321_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: nat] : ( P @ A3 @ A3 )
=> ( ! [A3: nat,B3: nat] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_322_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X7: nat] : ( P2 @ X7 ) )
= ( ^ [P3: nat > $o] :
? [N: nat] :
( ( P3 @ N )
& ! [M: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ( P3 @ M ) ) ) ) ) ).
% exists_least_iff
thf(fact_323_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_324_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_325_linorder__cases,axiom,
! [X2: nat,Y3: nat] :
( ~ ( ord_less_nat @ X2 @ Y3 )
=> ( ( X2 != Y3 )
=> ( ord_less_nat @ Y3 @ X2 ) ) ) ).
% linorder_cases
thf(fact_326_antisym__conv3,axiom,
! [Y3: nat,X2: nat] :
( ~ ( ord_less_nat @ Y3 @ X2 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% antisym_conv3
thf(fact_327_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X )
=> ( P @ Y5 ) )
=> ( P @ X ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_328_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_329_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_330_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_331_less__imp__neq,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( X2 != Y3 ) ) ).
% less_imp_neq
thf(fact_332_gt__ex,axiom,
! [X2: nat] :
? [X_12: nat] : ( ord_less_nat @ X2 @ X_12 ) ).
% gt_ex
thf(fact_333_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_334_order_Opreordering__axioms,axiom,
preordering_a @ ord_less_eq_a @ ord_less_a ).
% order.preordering_axioms
thf(fact_335_order_Opreordering__axioms,axiom,
preordering_nat @ ord_less_eq_nat @ ord_less_nat ).
% order.preordering_axioms
thf(fact_336_order_Oordering__axioms,axiom,
ordering_a @ ord_less_eq_a @ ord_less_a ).
% order.ordering_axioms
thf(fact_337_order_Oordering__axioms,axiom,
ordering_nat @ ord_less_eq_nat @ ord_less_nat ).
% order.ordering_axioms
thf(fact_338_order__le__imp__less__or__eq,axiom,
! [X2: a,Y3: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
=> ( ( ord_less_a @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_339_order__le__imp__less__or__eq,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( ord_less_nat @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_340_linorder__le__less__linear,axiom,
! [X2: a,Y3: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
| ( ord_less_a @ Y3 @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_341_linorder__le__less__linear,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
| ( ord_less_nat @ Y3 @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_342_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > a,C: a] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_a @ ( F @ B ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_343_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_344_order__less__le__subst1,axiom,
! [A: a,F: a > a,B: a,C: a] :
( ( ord_less_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_345_order__less__le__subst1,axiom,
! [A: nat,F: a > nat,B: a,C: a] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_346_order__less__le__subst1,axiom,
! [A: a,F: nat > a,B: nat,C: nat] :
( ( ord_less_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_347_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_348_order__le__less__subst2,axiom,
! [A: a,B: a,F: a > a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_a @ ( F @ B ) @ C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_349_order__le__less__subst2,axiom,
! [A: a,B: a,F: a > nat,C: nat] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_350_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > a,C: a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_a @ ( F @ B ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_351_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_352_order__le__less__subst1,axiom,
! [A: a,F: nat > a,B: nat,C: nat] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_353_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_354_order__less__le__trans,axiom,
! [X2: a,Y3: a,Z: a] :
( ( ord_less_a @ X2 @ Y3 )
=> ( ( ord_less_eq_a @ Y3 @ Z )
=> ( ord_less_a @ X2 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_355_order__less__le__trans,axiom,
! [X2: nat,Y3: nat,Z: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ Z )
=> ( ord_less_nat @ X2 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_356_order__le__less__trans,axiom,
! [X2: a,Y3: a,Z: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
=> ( ( ord_less_a @ Y3 @ Z )
=> ( ord_less_a @ X2 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_357_order__le__less__trans,axiom,
! [X2: nat,Y3: nat,Z: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( ord_less_nat @ Y3 @ Z )
=> ( ord_less_nat @ X2 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_358_order__neq__le__trans,axiom,
! [A: a,B: a] :
( ( A != B )
=> ( ( ord_less_eq_a @ A @ B )
=> ( ord_less_a @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_359_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_360_order__le__neq__trans,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_a @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_361_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_362_order__less__imp__le,axiom,
! [X2: a,Y3: a] :
( ( ord_less_a @ X2 @ Y3 )
=> ( ord_less_eq_a @ X2 @ Y3 ) ) ).
% order_less_imp_le
thf(fact_363_order__less__imp__le,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ X2 @ Y3 ) ) ).
% order_less_imp_le
thf(fact_364_linorder__not__less,axiom,
! [X2: a,Y3: a] :
( ( ~ ( ord_less_a @ X2 @ Y3 ) )
= ( ord_less_eq_a @ Y3 @ X2 ) ) ).
% linorder_not_less
thf(fact_365_linorder__not__less,axiom,
! [X2: nat,Y3: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
= ( ord_less_eq_nat @ Y3 @ X2 ) ) ).
% linorder_not_less
thf(fact_366_linorder__not__le,axiom,
! [X2: a,Y3: a] :
( ( ~ ( ord_less_eq_a @ X2 @ Y3 ) )
= ( ord_less_a @ Y3 @ X2 ) ) ).
% linorder_not_le
thf(fact_367_linorder__not__le,axiom,
! [X2: nat,Y3: nat] :
( ( ~ ( ord_less_eq_nat @ X2 @ Y3 ) )
= ( ord_less_nat @ Y3 @ X2 ) ) ).
% linorder_not_le
thf(fact_368_order__less__le,axiom,
( ord_less_a
= ( ^ [X3: a,Y: a] :
( ( ord_less_eq_a @ X3 @ Y )
& ( X3 != Y ) ) ) ) ).
% order_less_le
thf(fact_369_order__less__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
& ( X3 != Y ) ) ) ) ).
% order_less_le
thf(fact_370_order__le__less,axiom,
( ord_less_eq_a
= ( ^ [X3: a,Y: a] :
( ( ord_less_a @ X3 @ Y )
| ( X3 = Y ) ) ) ) ).
% order_le_less
thf(fact_371_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
| ( X3 = Y ) ) ) ) ).
% order_le_less
thf(fact_372_dual__order_Ostrict__implies__order,axiom,
! [B: a,A: a] :
( ( ord_less_a @ B @ A )
=> ( ord_less_eq_a @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_373_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_374_order_Ostrict__implies__order,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ B )
=> ( ord_less_eq_a @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_375_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_376_dual__order_Ostrict__iff__not,axiom,
( ord_less_a
= ( ^ [B2: a,A2: a] :
( ( ord_less_eq_a @ B2 @ A2 )
& ~ ( ord_less_eq_a @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_377_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ~ ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_378_dual__order_Ostrict__trans2,axiom,
! [B: a,A: a,C: a] :
( ( ord_less_a @ B @ A )
=> ( ( ord_less_eq_a @ C @ B )
=> ( ord_less_a @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_379_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_380_dual__order_Ostrict__trans1,axiom,
! [B: a,A: a,C: a] :
( ( ord_less_eq_a @ B @ A )
=> ( ( ord_less_a @ C @ B )
=> ( ord_less_a @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_381_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_382_dual__order_Ostrict__iff__order,axiom,
( ord_less_a
= ( ^ [B2: a,A2: a] :
( ( ord_less_eq_a @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_383_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_384_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_a
= ( ^ [B2: a,A2: a] :
( ( ord_less_a @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_385_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_386_order_Ostrict__iff__not,axiom,
( ord_less_a
= ( ^ [A2: a,B2: a] :
( ( ord_less_eq_a @ A2 @ B2 )
& ~ ( ord_less_eq_a @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_387_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ~ ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_388_order_Ostrict__trans2,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ord_less_a @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_389_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_390_order_Ostrict__trans1,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_a @ B @ C )
=> ( ord_less_a @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_391_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_392_order_Ostrict__iff__order,axiom,
( ord_less_a
= ( ^ [A2: a,B2: a] :
( ( ord_less_eq_a @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_393_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_394_order_Oorder__iff__strict,axiom,
( ord_less_eq_a
= ( ^ [A2: a,B2: a] :
( ( ord_less_a @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_395_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_396_not__le__imp__less,axiom,
! [Y3: a,X2: a] :
( ~ ( ord_less_eq_a @ Y3 @ X2 )
=> ( ord_less_a @ X2 @ Y3 ) ) ).
% not_le_imp_less
thf(fact_397_not__le__imp__less,axiom,
! [Y3: nat,X2: nat] :
( ~ ( ord_less_eq_nat @ Y3 @ X2 )
=> ( ord_less_nat @ X2 @ Y3 ) ) ).
% not_le_imp_less
thf(fact_398_less__le__not__le,axiom,
( ord_less_a
= ( ^ [X3: a,Y: a] :
( ( ord_less_eq_a @ X3 @ Y )
& ~ ( ord_less_eq_a @ Y @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_399_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
& ~ ( ord_less_eq_nat @ Y @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_400_antisym__conv2,axiom,
! [X2: a,Y3: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
=> ( ( ~ ( ord_less_a @ X2 @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% antisym_conv2
thf(fact_401_antisym__conv2,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% antisym_conv2
thf(fact_402_antisym__conv1,axiom,
! [X2: a,Y3: a] :
( ~ ( ord_less_a @ X2 @ Y3 )
=> ( ( ord_less_eq_a @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% antisym_conv1
thf(fact_403_antisym__conv1,axiom,
! [X2: nat,Y3: nat] :
( ~ ( ord_less_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_nat @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% antisym_conv1
thf(fact_404_nless__le,axiom,
! [A: a,B: a] :
( ( ~ ( ord_less_a @ A @ B ) )
= ( ~ ( ord_less_eq_a @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_405_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_406_leI,axiom,
! [X2: a,Y3: a] :
( ~ ( ord_less_a @ X2 @ Y3 )
=> ( ord_less_eq_a @ Y3 @ X2 ) ) ).
% leI
thf(fact_407_leI,axiom,
! [X2: nat,Y3: nat] :
( ~ ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X2 ) ) ).
% leI
thf(fact_408_leD,axiom,
! [Y3: a,X2: a] :
( ( ord_less_eq_a @ Y3 @ X2 )
=> ~ ( ord_less_a @ X2 @ Y3 ) ) ).
% leD
thf(fact_409_leD,axiom,
! [Y3: nat,X2: nat] :
( ( ord_less_eq_nat @ Y3 @ X2 )
=> ~ ( ord_less_nat @ X2 @ Y3 ) ) ).
% leD
thf(fact_410_verit__comp__simplify1_I3_J,axiom,
! [B5: a,A5: a] :
( ( ~ ( ord_less_eq_a @ B5 @ A5 ) )
= ( ord_less_a @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_411_verit__comp__simplify1_I3_J,axiom,
! [B5: nat,A5: nat] :
( ( ~ ( ord_less_eq_nat @ B5 @ A5 ) )
= ( ord_less_nat @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_412_complete__interval,axiom,
! [A: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C3: nat] :
( ( ord_less_eq_nat @ A @ C3 )
& ( ord_less_eq_nat @ C3 @ B )
& ! [X5: nat] :
( ( ( ord_less_eq_nat @ A @ X5 )
& ( ord_less_nat @ X5 @ C3 ) )
=> ( P @ X5 ) )
& ! [D: nat] :
( ! [X: nat] :
( ( ( ord_less_eq_nat @ A @ X )
& ( ord_less_nat @ X @ D ) )
=> ( P @ X ) )
=> ( ord_less_eq_nat @ D @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_413_pinf_I6_J,axiom,
! [T: a] :
? [Z4: a] :
! [X5: a] :
( ( ord_less_a @ Z4 @ X5 )
=> ~ ( ord_less_eq_a @ X5 @ T ) ) ).
% pinf(6)
thf(fact_414_pinf_I6_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ~ ( ord_less_eq_nat @ X5 @ T ) ) ).
% pinf(6)
thf(fact_415_pinf_I8_J,axiom,
! [T: a] :
? [Z4: a] :
! [X5: a] :
( ( ord_less_a @ Z4 @ X5 )
=> ( ord_less_eq_a @ T @ X5 ) ) ).
% pinf(8)
thf(fact_416_pinf_I8_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( ord_less_eq_nat @ T @ X5 ) ) ).
% pinf(8)
thf(fact_417_minf_I6_J,axiom,
! [T: a] :
? [Z4: a] :
! [X5: a] :
( ( ord_less_a @ X5 @ Z4 )
=> ( ord_less_eq_a @ X5 @ T ) ) ).
% minf(6)
thf(fact_418_minf_I6_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( ord_less_eq_nat @ X5 @ T ) ) ).
% minf(6)
thf(fact_419_minf_I8_J,axiom,
! [T: a] :
? [Z4: a] :
! [X5: a] :
( ( ord_less_a @ X5 @ Z4 )
=> ~ ( ord_less_eq_a @ T @ X5 ) ) ).
% minf(8)
thf(fact_420_minf_I8_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ~ ( ord_less_eq_nat @ T @ X5 ) ) ).
% minf(8)
thf(fact_421_pinf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X: nat] :
( ( ord_less_nat @ Z5 @ X )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z5: nat] :
! [X: nat] :
( ( ord_less_nat @ Z5 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_422_pinf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X: nat] :
( ( ord_less_nat @ Z5 @ X )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z5: nat] :
! [X: nat] :
( ( ord_less_nat @ Z5 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_423_pinf_I3_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_424_pinf_I4_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_425_pinf_I5_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ~ ( ord_less_nat @ X5 @ T ) ) ).
% pinf(5)
thf(fact_426_pinf_I7_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( ord_less_nat @ T @ X5 ) ) ).
% pinf(7)
thf(fact_427_minf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z5 )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z5: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z5 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_428_minf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z5 )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z5: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z5 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_429_minf_I3_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_430_minf_I4_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_431_minf_I5_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( ord_less_nat @ X5 @ T ) ) ).
% minf(5)
thf(fact_432_minf_I7_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ~ ( ord_less_nat @ T @ X5 ) ) ).
% minf(7)
thf(fact_433_bdd__above_Opreordering__bdd__axioms,axiom,
condit4103000493307248661_bdd_a @ ord_less_eq_a @ ord_less_a ).
% bdd_above.preordering_bdd_axioms
thf(fact_434_bdd__above_Opreordering__bdd__axioms,axiom,
condit7935552474144124665dd_nat @ ord_less_eq_nat @ ord_less_nat ).
% bdd_above.preordering_bdd_axioms
thf(fact_435_top_Oordering__top__axioms,axiom,
ordering_top_set_nat @ ord_less_eq_set_nat @ ord_less_set_nat @ top_top_set_nat ).
% top.ordering_top_axioms
thf(fact_436_preordering__bdd_OI,axiom,
! [Less_eq2: a > a > $o,Less: a > a > $o,A4: set_a,M2: a] :
( ( condit4103000493307248661_bdd_a @ Less_eq2 @ Less )
=> ( ! [X: a] :
( ( member_a @ X @ A4 )
=> ( Less_eq2 @ X @ M2 ) )
=> ( condit6541519642617408243_bdd_a @ Less_eq2 @ A4 ) ) ) ).
% preordering_bdd.I
thf(fact_437_preordering__bdd_OI,axiom,
! [Less_eq2: nat > nat > $o,Less: nat > nat > $o,A4: set_nat,M2: nat] :
( ( condit7935552474144124665dd_nat @ Less_eq2 @ Less )
=> ( ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ( Less_eq2 @ X @ M2 ) )
=> ( condit4013746787832047771dd_nat @ Less_eq2 @ A4 ) ) ) ).
% preordering_bdd.I
thf(fact_438_preordering__bdd_OE,axiom,
! [Less_eq2: a > a > $o,Less: a > a > $o,A4: set_a] :
( ( condit4103000493307248661_bdd_a @ Less_eq2 @ Less )
=> ( ( condit6541519642617408243_bdd_a @ Less_eq2 @ A4 )
=> ~ ! [M3: a] :
~ ! [X5: a] :
( ( member_a @ X5 @ A4 )
=> ( Less_eq2 @ X5 @ M3 ) ) ) ) ).
% preordering_bdd.E
thf(fact_439_preordering__bdd_OE,axiom,
! [Less_eq2: nat > nat > $o,Less: nat > nat > $o,A4: set_nat] :
( ( condit7935552474144124665dd_nat @ Less_eq2 @ Less )
=> ( ( condit4013746787832047771dd_nat @ Less_eq2 @ A4 )
=> ~ ! [M3: nat] :
~ ! [X5: nat] :
( ( member_nat @ X5 @ A4 )
=> ( Less_eq2 @ X5 @ M3 ) ) ) ) ).
% preordering_bdd.E
thf(fact_440_top__greatest,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ top_top_set_nat ) ).
% top_greatest
thf(fact_441_top_Oextremum__unique,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A )
= ( A = top_top_set_nat ) ) ).
% top.extremum_unique
thf(fact_442_top_Oextremum__uniqueI,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A )
=> ( A = top_top_set_nat ) ) ).
% top.extremum_uniqueI
thf(fact_443_top_Oextremum__strict,axiom,
! [A: set_nat] :
~ ( ord_less_set_nat @ top_top_set_nat @ A ) ).
% top.extremum_strict
thf(fact_444_top_Onot__eq__extremum,axiom,
! [A: set_nat] :
( ( A != top_top_set_nat )
= ( ord_less_set_nat @ A @ top_top_set_nat ) ) ).
% top.not_eq_extremum
thf(fact_445_preordering__bdd_Oempty,axiom,
! [Less_eq2: nat > nat > $o,Less: nat > nat > $o] :
( ( condit7935552474144124665dd_nat @ Less_eq2 @ Less )
=> ( condit4013746787832047771dd_nat @ Less_eq2 @ bot_bot_set_nat ) ) ).
% preordering_bdd.empty
thf(fact_446_preordering__bdd_OI2,axiom,
! [Less_eq2: nat > nat > $o,Less: nat > nat > $o,A4: set_nat,F: nat > nat,M2: nat] :
( ( condit7935552474144124665dd_nat @ Less_eq2 @ Less )
=> ( ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ( Less_eq2 @ ( F @ X ) @ M2 ) )
=> ( condit4013746787832047771dd_nat @ Less_eq2 @ ( image_nat_nat @ F @ A4 ) ) ) ) ).
% preordering_bdd.I2
thf(fact_447_bot_Oextremum__uniqueI,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
=> ( A = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_448_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_449_bot_Oextremum__unique,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_450_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_451_bot_Oextremum,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% bot.extremum
thf(fact_452_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_453_bot_Onot__eq__extremum,axiom,
! [A: set_nat] :
( ( A != bot_bot_set_nat )
= ( ord_less_set_nat @ bot_bot_set_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_454_bot_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_455_bot_Oextremum__strict,axiom,
! [A: set_nat] :
~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).
% bot.extremum_strict
thf(fact_456_bot_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_457_weak__coinduct__image,axiom,
! [A: a,X6: set_a,G2: a > a,F: set_a > set_a] :
( ( member_a @ A @ X6 )
=> ( ( ord_less_eq_set_a @ ( image_a_a @ G2 @ X6 ) @ ( F @ ( image_a_a @ G2 @ X6 ) ) )
=> ( member_a @ ( G2 @ A ) @ ( comple3341859861669737308_set_a @ F ) ) ) ) ).
% weak_coinduct_image
thf(fact_458_weak__coinduct__image,axiom,
! [A: a,X6: set_a,G2: a > nat,F: set_nat > set_nat] :
( ( member_a @ A @ X6 )
=> ( ( ord_less_eq_set_nat @ ( image_a_nat @ G2 @ X6 ) @ ( F @ ( image_a_nat @ G2 @ X6 ) ) )
=> ( member_nat @ ( G2 @ A ) @ ( comple1596078789208929544et_nat @ F ) ) ) ) ).
% weak_coinduct_image
thf(fact_459_weak__coinduct__image,axiom,
! [A: nat,X6: set_nat,G2: nat > a,F: set_a > set_a] :
( ( member_nat @ A @ X6 )
=> ( ( ord_less_eq_set_a @ ( image_nat_a @ G2 @ X6 ) @ ( F @ ( image_nat_a @ G2 @ X6 ) ) )
=> ( member_a @ ( G2 @ A ) @ ( comple3341859861669737308_set_a @ F ) ) ) ) ).
% weak_coinduct_image
thf(fact_460_weak__coinduct__image,axiom,
! [A: nat,X6: set_nat,G2: nat > nat,F: set_nat > set_nat] :
( ( member_nat @ A @ X6 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ G2 @ X6 ) @ ( F @ ( image_nat_nat @ G2 @ X6 ) ) )
=> ( member_nat @ ( G2 @ A ) @ ( comple1596078789208929544et_nat @ F ) ) ) ) ).
% weak_coinduct_image
thf(fact_461_inf_Obounded__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) )
= ( ( ord_less_eq_nat @ A @ B )
& ( ord_less_eq_nat @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_462_le__inf__iff,axiom,
! [X2: nat,Y3: nat,Z: nat] :
( ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ Y3 @ Z ) )
= ( ( ord_less_eq_nat @ X2 @ Y3 )
& ( ord_less_eq_nat @ X2 @ Z ) ) ) ).
% le_inf_iff
thf(fact_463_mono__inf,axiom,
! [F: nat > nat,A4: nat,B6: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ord_less_eq_nat @ ( F @ ( inf_inf_nat @ A4 @ B6 ) ) @ ( inf_inf_nat @ ( F @ A4 ) @ ( F @ B6 ) ) ) ) ).
% mono_inf
thf(fact_464_strict__mono__inv,axiom,
! [F: nat > nat,G2: nat > nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F )
=> ( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( ! [X: nat] :
( ( G2 @ ( F @ X ) )
= X )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ G2 ) ) ) ) ).
% strict_mono_inv
thf(fact_465_strict__mono__less__eq,axiom,
! [F: a > a,X2: a,Y3: a] :
( ( monotone_on_a_a @ top_top_set_a @ ord_less_a @ ord_less_a @ F )
=> ( ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y3 ) )
= ( ord_less_eq_a @ X2 @ Y3 ) ) ) ).
% strict_mono_less_eq
thf(fact_466_strict__mono__less__eq,axiom,
! [F: nat > a,X2: nat,Y3: nat] :
( ( monotone_on_nat_a @ top_top_set_nat @ ord_less_nat @ ord_less_a @ F )
=> ( ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y3 ) )
= ( ord_less_eq_nat @ X2 @ Y3 ) ) ) ).
% strict_mono_less_eq
thf(fact_467_strict__mono__less__eq,axiom,
! [F: a > nat,X2: a,Y3: a] :
( ( monotone_on_a_nat @ top_top_set_a @ ord_less_a @ ord_less_nat @ F )
=> ( ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) )
= ( ord_less_eq_a @ X2 @ Y3 ) ) ) ).
% strict_mono_less_eq
thf(fact_468_strict__mono__less__eq,axiom,
! [F: nat > nat,X2: nat,Y3: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F )
=> ( ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) )
= ( ord_less_eq_nat @ X2 @ Y3 ) ) ) ).
% strict_mono_less_eq
thf(fact_469_mono__strict__invE,axiom,
! [F: a > a,X2: a,Y3: a] :
( ( monotone_on_a_a @ top_top_set_a @ ord_less_eq_a @ ord_less_eq_a @ F )
=> ( ( ord_less_a @ ( F @ X2 ) @ ( F @ Y3 ) )
=> ( ord_less_a @ X2 @ Y3 ) ) ) ).
% mono_strict_invE
thf(fact_470_mono__strict__invE,axiom,
! [F: a > nat,X2: a,Y3: a] :
( ( monotone_on_a_nat @ top_top_set_a @ ord_less_eq_a @ ord_less_eq_nat @ F )
=> ( ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) )
=> ( ord_less_a @ X2 @ Y3 ) ) ) ).
% mono_strict_invE
thf(fact_471_mono__strict__invE,axiom,
! [F: nat > a,X2: nat,Y3: nat] :
( ( monotone_on_nat_a @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_a @ F )
=> ( ( ord_less_a @ ( F @ X2 ) @ ( F @ Y3 ) )
=> ( ord_less_nat @ X2 @ Y3 ) ) ) ).
% mono_strict_invE
thf(fact_472_mono__strict__invE,axiom,
! [F: nat > nat,X2: nat,Y3: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) )
=> ( ord_less_nat @ X2 @ Y3 ) ) ) ).
% mono_strict_invE
thf(fact_473_strict__mono__mono,axiom,
! [F: a > a] :
( ( monotone_on_a_a @ top_top_set_a @ ord_less_a @ ord_less_a @ F )
=> ( monotone_on_a_a @ top_top_set_a @ ord_less_eq_a @ ord_less_eq_a @ F ) ) ).
% strict_mono_mono
thf(fact_474_strict__mono__mono,axiom,
! [F: a > nat] :
( ( monotone_on_a_nat @ top_top_set_a @ ord_less_a @ ord_less_nat @ F )
=> ( monotone_on_a_nat @ top_top_set_a @ ord_less_eq_a @ ord_less_eq_nat @ F ) ) ).
% strict_mono_mono
thf(fact_475_strict__mono__mono,axiom,
! [F: nat > a] :
( ( monotone_on_nat_a @ top_top_set_nat @ ord_less_nat @ ord_less_a @ F )
=> ( monotone_on_nat_a @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_a @ F ) ) ).
% strict_mono_mono
thf(fact_476_strict__mono__mono,axiom,
! [F: nat > nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ).
% strict_mono_mono
thf(fact_477_inf__sup__ord_I2_J,axiom,
! [X2: nat,Y3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y3 ) @ Y3 ) ).
% inf_sup_ord(2)
thf(fact_478_inf__sup__ord_I1_J,axiom,
! [X2: nat,Y3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y3 ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_479_inf__le1,axiom,
! [X2: nat,Y3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y3 ) @ X2 ) ).
% inf_le1
thf(fact_480_inf__le2,axiom,
! [X2: nat,Y3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y3 ) @ Y3 ) ).
% inf_le2
thf(fact_481_le__infE,axiom,
! [X2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ A @ B ) )
=> ~ ( ( ord_less_eq_nat @ X2 @ A )
=> ~ ( ord_less_eq_nat @ X2 @ B ) ) ) ).
% le_infE
thf(fact_482_le__infI,axiom,
! [X2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ X2 @ A )
=> ( ( ord_less_eq_nat @ X2 @ B )
=> ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ A @ B ) ) ) ) ).
% le_infI
thf(fact_483_inf__mono,axiom,
! [A: nat,C: nat,B: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ D2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ ( inf_inf_nat @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_484_le__infI1,axiom,
! [A: nat,X2: nat,B: nat] :
( ( ord_less_eq_nat @ A @ X2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ X2 ) ) ).
% le_infI1
thf(fact_485_le__infI2,axiom,
! [B: nat,X2: nat,A: nat] :
( ( ord_less_eq_nat @ B @ X2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ X2 ) ) ).
% le_infI2
thf(fact_486_inf_OorderE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( A
= ( inf_inf_nat @ A @ B ) ) ) ).
% inf.orderE
thf(fact_487_inf_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( inf_inf_nat @ A @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% inf.orderI
thf(fact_488_inf__unique,axiom,
! [F: nat > nat > nat,X2: nat,Y3: nat] :
( ! [X: nat,Y2: nat] : ( ord_less_eq_nat @ ( F @ X @ Y2 ) @ X )
=> ( ! [X: nat,Y2: nat] : ( ord_less_eq_nat @ ( F @ X @ Y2 ) @ Y2 )
=> ( ! [X: nat,Y2: nat,Z4: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ord_less_eq_nat @ X @ Z4 )
=> ( ord_less_eq_nat @ X @ ( F @ Y2 @ Z4 ) ) ) )
=> ( ( inf_inf_nat @ X2 @ Y3 )
= ( F @ X2 @ Y3 ) ) ) ) ) ).
% inf_unique
thf(fact_489_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y: nat] :
( ( inf_inf_nat @ X3 @ Y )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_490_inf_Oabsorb1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( inf_inf_nat @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_491_inf_Oabsorb2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( inf_inf_nat @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_492_inf__absorb1,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( inf_inf_nat @ X2 @ Y3 )
= X2 ) ) ).
% inf_absorb1
thf(fact_493_inf__absorb2,axiom,
! [Y3: nat,X2: nat] :
( ( ord_less_eq_nat @ Y3 @ X2 )
=> ( ( inf_inf_nat @ X2 @ Y3 )
= Y3 ) ) ).
% inf_absorb2
thf(fact_494_inf_OboundedE,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) )
=> ~ ( ( ord_less_eq_nat @ A @ B )
=> ~ ( ord_less_eq_nat @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_495_inf_OboundedI,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ C )
=> ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_496_inf__greatest,axiom,
! [X2: nat,Y3: nat,Z: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_nat @ X2 @ Z )
=> ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ Y3 @ Z ) ) ) ) ).
% inf_greatest
thf(fact_497_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
( A2
= ( inf_inf_nat @ A2 @ B2 ) ) ) ) ).
% inf.order_iff
thf(fact_498_inf_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_499_inf_Ocobounded2,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_500_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
( ( inf_inf_nat @ A2 @ B2 )
= A2 ) ) ) ).
% inf.absorb_iff1
thf(fact_501_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A2: nat] :
( ( inf_inf_nat @ A2 @ B2 )
= B2 ) ) ) ).
% inf.absorb_iff2
thf(fact_502_inf_OcoboundedI1,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_503_inf_OcoboundedI2,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_504_less__infI1,axiom,
! [A: nat,X2: nat,B: nat] :
( ( ord_less_nat @ A @ X2 )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ X2 ) ) ).
% less_infI1
thf(fact_505_less__infI2,axiom,
! [B: nat,X2: nat,A: nat] :
( ( ord_less_nat @ B @ X2 )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ X2 ) ) ).
% less_infI2
thf(fact_506_inf_Oabsorb3,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( inf_inf_nat @ A @ B )
= A ) ) ).
% inf.absorb3
thf(fact_507_inf_Oabsorb4,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( inf_inf_nat @ A @ B )
= B ) ) ).
% inf.absorb4
thf(fact_508_inf_Ostrict__boundedE,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( inf_inf_nat @ B @ C ) )
=> ~ ( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ A @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_509_inf_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] :
( ( A2
= ( inf_inf_nat @ A2 @ B2 ) )
& ( A2 != B2 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_510_inf_Ostrict__coboundedI1,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ A @ C )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_511_inf_Ostrict__coboundedI2,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_512_ord_Omono__onD,axiom,
! [A4: set_a,Less_eq2: a > a > $o,F: a > a,R: a,S3: a] :
( ( monotone_on_a_a @ A4 @ Less_eq2 @ ord_less_eq_a @ F )
=> ( ( member_a @ R @ A4 )
=> ( ( member_a @ S3 @ A4 )
=> ( ( Less_eq2 @ R @ S3 )
=> ( ord_less_eq_a @ ( F @ R ) @ ( F @ S3 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_513_ord_Omono__onD,axiom,
! [A4: set_nat,Less_eq2: nat > nat > $o,F: nat > a,R: nat,S3: nat] :
( ( monotone_on_nat_a @ A4 @ Less_eq2 @ ord_less_eq_a @ F )
=> ( ( member_nat @ R @ A4 )
=> ( ( member_nat @ S3 @ A4 )
=> ( ( Less_eq2 @ R @ S3 )
=> ( ord_less_eq_a @ ( F @ R ) @ ( F @ S3 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_514_ord_Omono__onD,axiom,
! [A4: set_a,Less_eq2: a > a > $o,F: a > nat,R: a,S3: a] :
( ( monotone_on_a_nat @ A4 @ Less_eq2 @ ord_less_eq_nat @ F )
=> ( ( member_a @ R @ A4 )
=> ( ( member_a @ S3 @ A4 )
=> ( ( Less_eq2 @ R @ S3 )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S3 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_515_ord_Omono__onD,axiom,
! [A4: set_nat,Less_eq2: nat > nat > $o,F: nat > nat,R: nat,S3: nat] :
( ( monotone_on_nat_nat @ A4 @ Less_eq2 @ ord_less_eq_nat @ F )
=> ( ( member_nat @ R @ A4 )
=> ( ( member_nat @ S3 @ A4 )
=> ( ( Less_eq2 @ R @ S3 )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S3 ) ) ) ) ) ) ).
% ord.mono_onD
thf(fact_516_ord_Omono__onI,axiom,
! [A4: set_a,Less_eq2: a > a > $o,F: a > a] :
( ! [R2: a,S4: a] :
( ( member_a @ R2 @ A4 )
=> ( ( member_a @ S4 @ A4 )
=> ( ( Less_eq2 @ R2 @ S4 )
=> ( ord_less_eq_a @ ( F @ R2 ) @ ( F @ S4 ) ) ) ) )
=> ( monotone_on_a_a @ A4 @ Less_eq2 @ ord_less_eq_a @ F ) ) ).
% ord.mono_onI
thf(fact_517_ord_Omono__onI,axiom,
! [A4: set_nat,Less_eq2: nat > nat > $o,F: nat > a] :
( ! [R2: nat,S4: nat] :
( ( member_nat @ R2 @ A4 )
=> ( ( member_nat @ S4 @ A4 )
=> ( ( Less_eq2 @ R2 @ S4 )
=> ( ord_less_eq_a @ ( F @ R2 ) @ ( F @ S4 ) ) ) ) )
=> ( monotone_on_nat_a @ A4 @ Less_eq2 @ ord_less_eq_a @ F ) ) ).
% ord.mono_onI
thf(fact_518_ord_Omono__onI,axiom,
! [A4: set_a,Less_eq2: a > a > $o,F: a > nat] :
( ! [R2: a,S4: a] :
( ( member_a @ R2 @ A4 )
=> ( ( member_a @ S4 @ A4 )
=> ( ( Less_eq2 @ R2 @ S4 )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S4 ) ) ) ) )
=> ( monotone_on_a_nat @ A4 @ Less_eq2 @ ord_less_eq_nat @ F ) ) ).
% ord.mono_onI
thf(fact_519_ord_Omono__onI,axiom,
! [A4: set_nat,Less_eq2: nat > nat > $o,F: nat > nat] :
( ! [R2: nat,S4: nat] :
( ( member_nat @ R2 @ A4 )
=> ( ( member_nat @ S4 @ A4 )
=> ( ( Less_eq2 @ R2 @ S4 )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S4 ) ) ) ) )
=> ( monotone_on_nat_nat @ A4 @ Less_eq2 @ ord_less_eq_nat @ F ) ) ).
% ord.mono_onI
thf(fact_520_ord_Omono__on__def,axiom,
! [A4: set_a,Less_eq2: a > a > $o,F: a > a] :
( ( monotone_on_a_a @ A4 @ Less_eq2 @ ord_less_eq_a @ F )
= ( ! [R3: a,S5: a] :
( ( ( member_a @ R3 @ A4 )
& ( member_a @ S5 @ A4 )
& ( Less_eq2 @ R3 @ S5 ) )
=> ( ord_less_eq_a @ ( F @ R3 ) @ ( F @ S5 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_521_ord_Omono__on__def,axiom,
! [A4: set_nat,Less_eq2: nat > nat > $o,F: nat > a] :
( ( monotone_on_nat_a @ A4 @ Less_eq2 @ ord_less_eq_a @ F )
= ( ! [R3: nat,S5: nat] :
( ( ( member_nat @ R3 @ A4 )
& ( member_nat @ S5 @ A4 )
& ( Less_eq2 @ R3 @ S5 ) )
=> ( ord_less_eq_a @ ( F @ R3 ) @ ( F @ S5 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_522_ord_Omono__on__def,axiom,
! [A4: set_a,Less_eq2: a > a > $o,F: a > nat] :
( ( monotone_on_a_nat @ A4 @ Less_eq2 @ ord_less_eq_nat @ F )
= ( ! [R3: a,S5: a] :
( ( ( member_a @ R3 @ A4 )
& ( member_a @ S5 @ A4 )
& ( Less_eq2 @ R3 @ S5 ) )
=> ( ord_less_eq_nat @ ( F @ R3 ) @ ( F @ S5 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_523_ord_Omono__on__def,axiom,
! [A4: set_nat,Less_eq2: nat > nat > $o,F: nat > nat] :
( ( monotone_on_nat_nat @ A4 @ Less_eq2 @ ord_less_eq_nat @ F )
= ( ! [R3: nat,S5: nat] :
( ( ( member_nat @ R3 @ A4 )
& ( member_nat @ S5 @ A4 )
& ( Less_eq2 @ R3 @ S5 ) )
=> ( ord_less_eq_nat @ ( F @ R3 ) @ ( F @ S5 ) ) ) ) ) ).
% ord.mono_on_def
thf(fact_524_mono__onD,axiom,
! [A4: set_a,F: a > a,R: a,S3: a] :
( ( monotone_on_a_a @ A4 @ ord_less_eq_a @ ord_less_eq_a @ F )
=> ( ( member_a @ R @ A4 )
=> ( ( member_a @ S3 @ A4 )
=> ( ( ord_less_eq_a @ R @ S3 )
=> ( ord_less_eq_a @ ( F @ R ) @ ( F @ S3 ) ) ) ) ) ) ).
% mono_onD
thf(fact_525_mono__onD,axiom,
! [A4: set_a,F: a > nat,R: a,S3: a] :
( ( monotone_on_a_nat @ A4 @ ord_less_eq_a @ ord_less_eq_nat @ F )
=> ( ( member_a @ R @ A4 )
=> ( ( member_a @ S3 @ A4 )
=> ( ( ord_less_eq_a @ R @ S3 )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S3 ) ) ) ) ) ) ).
% mono_onD
thf(fact_526_mono__onD,axiom,
! [A4: set_nat,F: nat > a,R: nat,S3: nat] :
( ( monotone_on_nat_a @ A4 @ ord_less_eq_nat @ ord_less_eq_a @ F )
=> ( ( member_nat @ R @ A4 )
=> ( ( member_nat @ S3 @ A4 )
=> ( ( ord_less_eq_nat @ R @ S3 )
=> ( ord_less_eq_a @ ( F @ R ) @ ( F @ S3 ) ) ) ) ) ) ).
% mono_onD
thf(fact_527_mono__onD,axiom,
! [A4: set_nat,F: nat > nat,R: nat,S3: nat] :
( ( monotone_on_nat_nat @ A4 @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( member_nat @ R @ A4 )
=> ( ( member_nat @ S3 @ A4 )
=> ( ( ord_less_eq_nat @ R @ S3 )
=> ( ord_less_eq_nat @ ( F @ R ) @ ( F @ S3 ) ) ) ) ) ) ).
% mono_onD
thf(fact_528_mono__onI,axiom,
! [A4: set_a,F: a > a] :
( ! [R2: a,S4: a] :
( ( member_a @ R2 @ A4 )
=> ( ( member_a @ S4 @ A4 )
=> ( ( ord_less_eq_a @ R2 @ S4 )
=> ( ord_less_eq_a @ ( F @ R2 ) @ ( F @ S4 ) ) ) ) )
=> ( monotone_on_a_a @ A4 @ ord_less_eq_a @ ord_less_eq_a @ F ) ) ).
% mono_onI
thf(fact_529_mono__onI,axiom,
! [A4: set_a,F: a > nat] :
( ! [R2: a,S4: a] :
( ( member_a @ R2 @ A4 )
=> ( ( member_a @ S4 @ A4 )
=> ( ( ord_less_eq_a @ R2 @ S4 )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S4 ) ) ) ) )
=> ( monotone_on_a_nat @ A4 @ ord_less_eq_a @ ord_less_eq_nat @ F ) ) ).
% mono_onI
thf(fact_530_mono__onI,axiom,
! [A4: set_nat,F: nat > a] :
( ! [R2: nat,S4: nat] :
( ( member_nat @ R2 @ A4 )
=> ( ( member_nat @ S4 @ A4 )
=> ( ( ord_less_eq_nat @ R2 @ S4 )
=> ( ord_less_eq_a @ ( F @ R2 ) @ ( F @ S4 ) ) ) ) )
=> ( monotone_on_nat_a @ A4 @ ord_less_eq_nat @ ord_less_eq_a @ F ) ) ).
% mono_onI
thf(fact_531_mono__onI,axiom,
! [A4: set_nat,F: nat > nat] :
( ! [R2: nat,S4: nat] :
( ( member_nat @ R2 @ A4 )
=> ( ( member_nat @ S4 @ A4 )
=> ( ( ord_less_eq_nat @ R2 @ S4 )
=> ( ord_less_eq_nat @ ( F @ R2 ) @ ( F @ S4 ) ) ) ) )
=> ( monotone_on_nat_nat @ A4 @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ).
% mono_onI
thf(fact_532_ord_Ostrict__mono__onD,axiom,
! [A4: set_a,Less: a > a > $o,F: a > nat,R: a,S3: a] :
( ( monotone_on_a_nat @ A4 @ Less @ ord_less_nat @ F )
=> ( ( member_a @ R @ A4 )
=> ( ( member_a @ S3 @ A4 )
=> ( ( Less @ R @ S3 )
=> ( ord_less_nat @ ( F @ R ) @ ( F @ S3 ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_533_ord_Ostrict__mono__onD,axiom,
! [A4: set_nat,Less: nat > nat > $o,F: nat > nat,R: nat,S3: nat] :
( ( monotone_on_nat_nat @ A4 @ Less @ ord_less_nat @ F )
=> ( ( member_nat @ R @ A4 )
=> ( ( member_nat @ S3 @ A4 )
=> ( ( Less @ R @ S3 )
=> ( ord_less_nat @ ( F @ R ) @ ( F @ S3 ) ) ) ) ) ) ).
% ord.strict_mono_onD
thf(fact_534_ord_Ostrict__mono__onI,axiom,
! [A4: set_a,Less: a > a > $o,F: a > nat] :
( ! [R2: a,S4: a] :
( ( member_a @ R2 @ A4 )
=> ( ( member_a @ S4 @ A4 )
=> ( ( Less @ R2 @ S4 )
=> ( ord_less_nat @ ( F @ R2 ) @ ( F @ S4 ) ) ) ) )
=> ( monotone_on_a_nat @ A4 @ Less @ ord_less_nat @ F ) ) ).
% ord.strict_mono_onI
thf(fact_535_ord_Ostrict__mono__onI,axiom,
! [A4: set_nat,Less: nat > nat > $o,F: nat > nat] :
( ! [R2: nat,S4: nat] :
( ( member_nat @ R2 @ A4 )
=> ( ( member_nat @ S4 @ A4 )
=> ( ( Less @ R2 @ S4 )
=> ( ord_less_nat @ ( F @ R2 ) @ ( F @ S4 ) ) ) ) )
=> ( monotone_on_nat_nat @ A4 @ Less @ ord_less_nat @ F ) ) ).
% ord.strict_mono_onI
thf(fact_536_ord_Ostrict__mono__on__def,axiom,
! [A4: set_a,Less: a > a > $o,F: a > nat] :
( ( monotone_on_a_nat @ A4 @ Less @ ord_less_nat @ F )
= ( ! [R3: a,S5: a] :
( ( ( member_a @ R3 @ A4 )
& ( member_a @ S5 @ A4 )
& ( Less @ R3 @ S5 ) )
=> ( ord_less_nat @ ( F @ R3 ) @ ( F @ S5 ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_537_ord_Ostrict__mono__on__def,axiom,
! [A4: set_nat,Less: nat > nat > $o,F: nat > nat] :
( ( monotone_on_nat_nat @ A4 @ Less @ ord_less_nat @ F )
= ( ! [R3: nat,S5: nat] :
( ( ( member_nat @ R3 @ A4 )
& ( member_nat @ S5 @ A4 )
& ( Less @ R3 @ S5 ) )
=> ( ord_less_nat @ ( F @ R3 ) @ ( F @ S5 ) ) ) ) ) ).
% ord.strict_mono_on_def
thf(fact_538_strict__mono__onD,axiom,
! [A4: set_a,F: a > nat,R: a,S3: a] :
( ( monotone_on_a_nat @ A4 @ ord_less_a @ ord_less_nat @ F )
=> ( ( member_a @ R @ A4 )
=> ( ( member_a @ S3 @ A4 )
=> ( ( ord_less_a @ R @ S3 )
=> ( ord_less_nat @ ( F @ R ) @ ( F @ S3 ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_539_strict__mono__onD,axiom,
! [A4: set_nat,F: nat > nat,R: nat,S3: nat] :
( ( monotone_on_nat_nat @ A4 @ ord_less_nat @ ord_less_nat @ F )
=> ( ( member_nat @ R @ A4 )
=> ( ( member_nat @ S3 @ A4 )
=> ( ( ord_less_nat @ R @ S3 )
=> ( ord_less_nat @ ( F @ R ) @ ( F @ S3 ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_540_strict__mono__onI,axiom,
! [A4: set_a,F: a > nat] :
( ! [R2: a,S4: a] :
( ( member_a @ R2 @ A4 )
=> ( ( member_a @ S4 @ A4 )
=> ( ( ord_less_a @ R2 @ S4 )
=> ( ord_less_nat @ ( F @ R2 ) @ ( F @ S4 ) ) ) ) )
=> ( monotone_on_a_nat @ A4 @ ord_less_a @ ord_less_nat @ F ) ) ).
% strict_mono_onI
thf(fact_541_strict__mono__onI,axiom,
! [A4: set_nat,F: nat > nat] :
( ! [R2: nat,S4: nat] :
( ( member_nat @ R2 @ A4 )
=> ( ( member_nat @ S4 @ A4 )
=> ( ( ord_less_nat @ R2 @ S4 )
=> ( ord_less_nat @ ( F @ R2 ) @ ( F @ S4 ) ) ) ) )
=> ( monotone_on_nat_nat @ A4 @ ord_less_nat @ ord_less_nat @ F ) ) ).
% strict_mono_onI
thf(fact_542_strict__mono__on__eqD,axiom,
! [A4: set_a,F: a > nat,X2: a,Y3: a] :
( ( monotone_on_a_nat @ A4 @ ord_less_a @ ord_less_nat @ F )
=> ( ( ( F @ X2 )
= ( F @ Y3 ) )
=> ( ( member_a @ X2 @ A4 )
=> ( ( member_a @ Y3 @ A4 )
=> ( Y3 = X2 ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_543_strict__mono__on__eqD,axiom,
! [A4: set_nat,F: nat > nat,X2: nat,Y3: nat] :
( ( monotone_on_nat_nat @ A4 @ ord_less_nat @ ord_less_nat @ F )
=> ( ( ( F @ X2 )
= ( F @ Y3 ) )
=> ( ( member_nat @ X2 @ A4 )
=> ( ( member_nat @ Y3 @ A4 )
=> ( Y3 = X2 ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_544_strict__mono__on__imp__mono__on,axiom,
! [A4: set_a,F: a > a] :
( ( monotone_on_a_a @ A4 @ ord_less_a @ ord_less_a @ F )
=> ( monotone_on_a_a @ A4 @ ord_less_eq_a @ ord_less_eq_a @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_545_strict__mono__on__imp__mono__on,axiom,
! [A4: set_a,F: a > nat] :
( ( monotone_on_a_nat @ A4 @ ord_less_a @ ord_less_nat @ F )
=> ( monotone_on_a_nat @ A4 @ ord_less_eq_a @ ord_less_eq_nat @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_546_strict__mono__on__imp__mono__on,axiom,
! [A4: set_nat,F: nat > a] :
( ( monotone_on_nat_a @ A4 @ ord_less_nat @ ord_less_a @ F )
=> ( monotone_on_nat_a @ A4 @ ord_less_eq_nat @ ord_less_eq_a @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_547_strict__mono__on__imp__mono__on,axiom,
! [A4: set_nat,F: nat > nat] :
( ( monotone_on_nat_nat @ A4 @ ord_less_nat @ ord_less_nat @ F )
=> ( monotone_on_nat_nat @ A4 @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_548_strict__mono__on__leD,axiom,
! [A4: set_a,F: a > a,X2: a,Y3: a] :
( ( monotone_on_a_a @ A4 @ ord_less_a @ ord_less_a @ F )
=> ( ( member_a @ X2 @ A4 )
=> ( ( member_a @ Y3 @ A4 )
=> ( ( ord_less_eq_a @ X2 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_549_strict__mono__on__leD,axiom,
! [A4: set_a,F: a > nat,X2: a,Y3: a] :
( ( monotone_on_a_nat @ A4 @ ord_less_a @ ord_less_nat @ F )
=> ( ( member_a @ X2 @ A4 )
=> ( ( member_a @ Y3 @ A4 )
=> ( ( ord_less_eq_a @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_550_strict__mono__on__leD,axiom,
! [A4: set_nat,F: nat > a,X2: nat,Y3: nat] :
( ( monotone_on_nat_a @ A4 @ ord_less_nat @ ord_less_a @ F )
=> ( ( member_nat @ X2 @ A4 )
=> ( ( member_nat @ Y3 @ A4 )
=> ( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_551_strict__mono__on__leD,axiom,
! [A4: set_nat,F: nat > nat,X2: nat,Y3: nat] :
( ( monotone_on_nat_nat @ A4 @ ord_less_nat @ ord_less_nat @ F )
=> ( ( member_nat @ X2 @ A4 )
=> ( ( member_nat @ Y3 @ A4 )
=> ( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_552_mono__on__greaterD,axiom,
! [A4: set_a,G2: a > a,X2: a,Y3: a] :
( ( monotone_on_a_a @ A4 @ ord_less_eq_a @ ord_less_eq_a @ G2 )
=> ( ( member_a @ X2 @ A4 )
=> ( ( member_a @ Y3 @ A4 )
=> ( ( ord_less_a @ ( G2 @ Y3 ) @ ( G2 @ X2 ) )
=> ( ord_less_a @ Y3 @ X2 ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_553_mono__on__greaterD,axiom,
! [A4: set_a,G2: a > nat,X2: a,Y3: a] :
( ( monotone_on_a_nat @ A4 @ ord_less_eq_a @ ord_less_eq_nat @ G2 )
=> ( ( member_a @ X2 @ A4 )
=> ( ( member_a @ Y3 @ A4 )
=> ( ( ord_less_nat @ ( G2 @ Y3 ) @ ( G2 @ X2 ) )
=> ( ord_less_a @ Y3 @ X2 ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_554_mono__on__greaterD,axiom,
! [A4: set_nat,G2: nat > a,X2: nat,Y3: nat] :
( ( monotone_on_nat_a @ A4 @ ord_less_eq_nat @ ord_less_eq_a @ G2 )
=> ( ( member_nat @ X2 @ A4 )
=> ( ( member_nat @ Y3 @ A4 )
=> ( ( ord_less_a @ ( G2 @ Y3 ) @ ( G2 @ X2 ) )
=> ( ord_less_nat @ Y3 @ X2 ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_555_mono__on__greaterD,axiom,
! [A4: set_nat,G2: nat > nat,X2: nat,Y3: nat] :
( ( monotone_on_nat_nat @ A4 @ ord_less_eq_nat @ ord_less_eq_nat @ G2 )
=> ( ( member_nat @ X2 @ A4 )
=> ( ( member_nat @ Y3 @ A4 )
=> ( ( ord_less_nat @ ( G2 @ Y3 ) @ ( G2 @ X2 ) )
=> ( ord_less_nat @ Y3 @ X2 ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_556_monoD,axiom,
! [F: a > a,X2: a,Y3: a] :
( ( monotone_on_a_a @ top_top_set_a @ ord_less_eq_a @ ord_less_eq_a @ F )
=> ( ( ord_less_eq_a @ X2 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ) ).
% monoD
thf(fact_557_monoD,axiom,
! [F: a > nat,X2: a,Y3: a] :
( ( monotone_on_a_nat @ top_top_set_a @ ord_less_eq_a @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_a @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ) ).
% monoD
thf(fact_558_monoD,axiom,
! [F: nat > a,X2: nat,Y3: nat] :
( ( monotone_on_nat_a @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_a @ F )
=> ( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ) ).
% monoD
thf(fact_559_monoD,axiom,
! [F: nat > nat,X2: nat,Y3: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ) ).
% monoD
thf(fact_560_monoE,axiom,
! [F: a > a,X2: a,Y3: a] :
( ( monotone_on_a_a @ top_top_set_a @ ord_less_eq_a @ ord_less_eq_a @ F )
=> ( ( ord_less_eq_a @ X2 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ) ).
% monoE
thf(fact_561_monoE,axiom,
! [F: a > nat,X2: a,Y3: a] :
( ( monotone_on_a_nat @ top_top_set_a @ ord_less_eq_a @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_a @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ) ).
% monoE
thf(fact_562_monoE,axiom,
! [F: nat > a,X2: nat,Y3: nat] :
( ( monotone_on_nat_a @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_a @ F )
=> ( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_a @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ) ).
% monoE
thf(fact_563_monoE,axiom,
! [F: nat > nat,X2: nat,Y3: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ) ).
% monoE
thf(fact_564_monoI,axiom,
! [F: a > a] :
( ! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( monotone_on_a_a @ top_top_set_a @ ord_less_eq_a @ ord_less_eq_a @ F ) ) ).
% monoI
thf(fact_565_monoI,axiom,
! [F: a > nat] :
( ! [X: a,Y2: a] :
( ( ord_less_eq_a @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( monotone_on_a_nat @ top_top_set_a @ ord_less_eq_a @ ord_less_eq_nat @ F ) ) ).
% monoI
thf(fact_566_monoI,axiom,
! [F: nat > a] :
( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( monotone_on_nat_a @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_a @ F ) ) ).
% monoI
thf(fact_567_monoI,axiom,
! [F: nat > nat] :
( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ).
% monoI
thf(fact_568_mono__imp__mono__on,axiom,
! [F: a > a,A4: set_a] :
( ( monotone_on_a_a @ top_top_set_a @ ord_less_eq_a @ ord_less_eq_a @ F )
=> ( monotone_on_a_a @ A4 @ ord_less_eq_a @ ord_less_eq_a @ F ) ) ).
% mono_imp_mono_on
thf(fact_569_mono__imp__mono__on,axiom,
! [F: a > nat,A4: set_a] :
( ( monotone_on_a_nat @ top_top_set_a @ ord_less_eq_a @ ord_less_eq_nat @ F )
=> ( monotone_on_a_nat @ A4 @ ord_less_eq_a @ ord_less_eq_nat @ F ) ) ).
% mono_imp_mono_on
thf(fact_570_mono__imp__mono__on,axiom,
! [F: nat > a,A4: set_nat] :
( ( monotone_on_nat_a @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_a @ F )
=> ( monotone_on_nat_a @ A4 @ ord_less_eq_nat @ ord_less_eq_a @ F ) ) ).
% mono_imp_mono_on
thf(fact_571_mono__imp__mono__on,axiom,
! [F: nat > nat,A4: set_nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( monotone_on_nat_nat @ A4 @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ).
% mono_imp_mono_on
thf(fact_572_mono__on__subset,axiom,
! [A4: set_a,F: a > a,B6: set_a] :
( ( monotone_on_a_a @ A4 @ ord_less_eq_a @ ord_less_eq_a @ F )
=> ( ( ord_less_eq_set_a @ B6 @ A4 )
=> ( monotone_on_a_a @ B6 @ ord_less_eq_a @ ord_less_eq_a @ F ) ) ) ).
% mono_on_subset
thf(fact_573_mono__on__subset,axiom,
! [A4: set_a,F: a > nat,B6: set_a] :
( ( monotone_on_a_nat @ A4 @ ord_less_eq_a @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_set_a @ B6 @ A4 )
=> ( monotone_on_a_nat @ B6 @ ord_less_eq_a @ ord_less_eq_nat @ F ) ) ) ).
% mono_on_subset
thf(fact_574_mono__on__subset,axiom,
! [A4: set_nat,F: nat > a,B6: set_nat] :
( ( monotone_on_nat_a @ A4 @ ord_less_eq_nat @ ord_less_eq_a @ F )
=> ( ( ord_less_eq_set_nat @ B6 @ A4 )
=> ( monotone_on_nat_a @ B6 @ ord_less_eq_nat @ ord_less_eq_a @ F ) ) ) ).
% mono_on_subset
thf(fact_575_mono__on__subset,axiom,
! [A4: set_nat,F: nat > nat,B6: set_nat] :
( ( monotone_on_nat_nat @ A4 @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_set_nat @ B6 @ A4 )
=> ( monotone_on_nat_nat @ B6 @ ord_less_eq_nat @ ord_less_eq_nat @ F ) ) ) ).
% mono_on_subset
thf(fact_576_ord_Omono__on__subset,axiom,
! [A4: set_nat,Less_eq2: nat > nat > $o,F: nat > nat,B6: set_nat] :
( ( monotone_on_nat_nat @ A4 @ Less_eq2 @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_set_nat @ B6 @ A4 )
=> ( monotone_on_nat_nat @ B6 @ Less_eq2 @ ord_less_eq_nat @ F ) ) ) ).
% ord.mono_on_subset
thf(fact_577_strict__monoD,axiom,
! [F: nat > nat,X2: nat,Y3: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F )
=> ( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ) ).
% strict_monoD
thf(fact_578_strict__monoI,axiom,
! [F: nat > nat] :
( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F ) ) ).
% strict_monoI
thf(fact_579_strict__mono__eq,axiom,
! [F: nat > nat,X2: nat,Y3: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F )
=> ( ( ( F @ X2 )
= ( F @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% strict_mono_eq
thf(fact_580_strict__mono__less,axiom,
! [F: nat > nat,X2: nat,Y3: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F )
=> ( ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) )
= ( ord_less_nat @ X2 @ Y3 ) ) ) ).
% strict_mono_less
thf(fact_581_mono__invE,axiom,
! [F: a > a,X2: a,Y3: a] :
( ( monotone_on_a_a @ top_top_set_a @ ord_less_eq_a @ ord_less_eq_a @ F )
=> ( ( ord_less_a @ ( F @ X2 ) @ ( F @ Y3 ) )
=> ( ord_less_eq_a @ X2 @ Y3 ) ) ) ).
% mono_invE
thf(fact_582_mono__invE,axiom,
! [F: a > nat,X2: a,Y3: a] :
( ( monotone_on_a_nat @ top_top_set_a @ ord_less_eq_a @ ord_less_eq_nat @ F )
=> ( ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) )
=> ( ord_less_eq_a @ X2 @ Y3 ) ) ) ).
% mono_invE
thf(fact_583_mono__invE,axiom,
! [F: nat > a,X2: nat,Y3: nat] :
( ( monotone_on_nat_a @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_a @ F )
=> ( ( ord_less_a @ ( F @ X2 ) @ ( F @ Y3 ) )
=> ( ord_less_eq_nat @ X2 @ Y3 ) ) ) ).
% mono_invE
thf(fact_584_mono__invE,axiom,
! [F: nat > nat,X2: nat,Y3: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) )
=> ( ord_less_eq_nat @ X2 @ Y3 ) ) ) ).
% mono_invE
thf(fact_585_inf__top_Osemilattice__neutr__order__axioms,axiom,
semila1667268886620078168et_nat @ inf_inf_set_nat @ top_top_set_nat @ ord_less_eq_set_nat @ ord_less_set_nat ).
% inf_top.semilattice_neutr_order_axioms
thf(fact_586_Inf__fin_Osemilattice__order__set__axioms,axiom,
lattic6009151579333465974et_nat @ inf_inf_nat @ ord_less_eq_nat @ ord_less_nat ).
% Inf_fin.semilattice_order_set_axioms
thf(fact_587_sup_Obounded__iff,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A )
= ( ( ord_less_eq_nat @ B @ A )
& ( ord_less_eq_nat @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_588_le__sup__iff,axiom,
! [X2: nat,Y3: nat,Z: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X2 @ Y3 ) @ Z )
= ( ( ord_less_eq_nat @ X2 @ Z )
& ( ord_less_eq_nat @ Y3 @ Z ) ) ) ).
% le_sup_iff
thf(fact_589_cSup__eq__maximum,axiom,
! [Z: nat,X6: set_nat] :
( ( member_nat @ Z @ X6 )
=> ( ! [X: nat] :
( ( member_nat @ X @ X6 )
=> ( ord_less_eq_nat @ X @ Z ) )
=> ( ( complete_Sup_Sup_nat @ X6 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_590_sup_OcoboundedI2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_591_sup_OcoboundedI1,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_592_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
( ( sup_sup_nat @ A2 @ B2 )
= B2 ) ) ) ).
% sup.absorb_iff2
thf(fact_593_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A2: nat] :
( ( sup_sup_nat @ A2 @ B2 )
= A2 ) ) ) ).
% sup.absorb_iff1
thf(fact_594_sup_Ocobounded2,axiom,
! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( sup_sup_nat @ A @ B ) ) ).
% sup.cobounded2
thf(fact_595_sup_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( sup_sup_nat @ A @ B ) ) ).
% sup.cobounded1
thf(fact_596_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A2: nat] :
( A2
= ( sup_sup_nat @ A2 @ B2 ) ) ) ) ).
% sup.order_iff
thf(fact_597_sup_OboundedI,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_598_sup_OboundedE,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_nat @ B @ A )
=> ~ ( ord_less_eq_nat @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_599_sup__absorb2,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( sup_sup_nat @ X2 @ Y3 )
= Y3 ) ) ).
% sup_absorb2
thf(fact_600_sup__absorb1,axiom,
! [Y3: nat,X2: nat] :
( ( ord_less_eq_nat @ Y3 @ X2 )
=> ( ( sup_sup_nat @ X2 @ Y3 )
= X2 ) ) ).
% sup_absorb1
thf(fact_601_sup_Oabsorb2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( sup_sup_nat @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_602_sup_Oabsorb1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( sup_sup_nat @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_603_sup__unique,axiom,
! [F: nat > nat > nat,X2: nat,Y3: nat] :
( ! [X: nat,Y2: nat] : ( ord_less_eq_nat @ X @ ( F @ X @ Y2 ) )
=> ( ! [X: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ ( F @ X @ Y2 ) )
=> ( ! [X: nat,Y2: nat,Z4: nat] :
( ( ord_less_eq_nat @ Y2 @ X )
=> ( ( ord_less_eq_nat @ Z4 @ X )
=> ( ord_less_eq_nat @ ( F @ Y2 @ Z4 ) @ X ) ) )
=> ( ( sup_sup_nat @ X2 @ Y3 )
= ( F @ X2 @ Y3 ) ) ) ) ) ).
% sup_unique
thf(fact_604_sup_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( sup_sup_nat @ A @ B ) )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% sup.orderI
thf(fact_605_sup_OorderE,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( A
= ( sup_sup_nat @ A @ B ) ) ) ).
% sup.orderE
thf(fact_606_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y: nat] :
( ( sup_sup_nat @ X3 @ Y )
= Y ) ) ) ).
% le_iff_sup
thf(fact_607_sup__least,axiom,
! [Y3: nat,X2: nat,Z: nat] :
( ( ord_less_eq_nat @ Y3 @ X2 )
=> ( ( ord_less_eq_nat @ Z @ X2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y3 @ Z ) @ X2 ) ) ) ).
% sup_least
thf(fact_608_sup__mono,axiom,
! [A: nat,C: nat,B: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ D2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ ( sup_sup_nat @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_609_sup_Omono,axiom,
! [C: nat,A: nat,D2: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ( ord_less_eq_nat @ D2 @ B )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D2 ) @ ( sup_sup_nat @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_610_le__supI2,axiom,
! [X2: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ X2 @ B )
=> ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ A @ B ) ) ) ).
% le_supI2
thf(fact_611_le__supI1,axiom,
! [X2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ X2 @ A )
=> ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ A @ B ) ) ) ).
% le_supI1
thf(fact_612_sup__ge2,axiom,
! [Y3: nat,X2: nat] : ( ord_less_eq_nat @ Y3 @ ( sup_sup_nat @ X2 @ Y3 ) ) ).
% sup_ge2
thf(fact_613_sup__ge1,axiom,
! [X2: nat,Y3: nat] : ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ X2 @ Y3 ) ) ).
% sup_ge1
thf(fact_614_le__supI,axiom,
! [A: nat,X2: nat,B: nat] :
( ( ord_less_eq_nat @ A @ X2 )
=> ( ( ord_less_eq_nat @ B @ X2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X2 ) ) ) ).
% le_supI
thf(fact_615_le__supE,axiom,
! [A: nat,B: nat,X2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X2 )
=> ~ ( ( ord_less_eq_nat @ A @ X2 )
=> ~ ( ord_less_eq_nat @ B @ X2 ) ) ) ).
% le_supE
thf(fact_616_inf__sup__ord_I3_J,axiom,
! [X2: nat,Y3: nat] : ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ X2 @ Y3 ) ) ).
% inf_sup_ord(3)
thf(fact_617_inf__sup__ord_I4_J,axiom,
! [Y3: nat,X2: nat] : ( ord_less_eq_nat @ Y3 @ ( sup_sup_nat @ X2 @ Y3 ) ) ).
% inf_sup_ord(4)
thf(fact_618_sup_Ostrict__coboundedI2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_619_sup_Ostrict__coboundedI1,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ C @ A )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_620_sup_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [B2: nat,A2: nat] :
( ( A2
= ( sup_sup_nat @ A2 @ B2 ) )
& ( A2 != B2 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_621_sup_Ostrict__boundedE,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_nat @ ( sup_sup_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ C @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_622_sup_Oabsorb4,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( sup_sup_nat @ A @ B )
= B ) ) ).
% sup.absorb4
thf(fact_623_sup_Oabsorb3,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( sup_sup_nat @ A @ B )
= A ) ) ).
% sup.absorb3
thf(fact_624_less__supI2,axiom,
! [X2: nat,B: nat,A: nat] :
( ( ord_less_nat @ X2 @ B )
=> ( ord_less_nat @ X2 @ ( sup_sup_nat @ A @ B ) ) ) ).
% less_supI2
thf(fact_625_less__supI1,axiom,
! [X2: nat,A: nat,B: nat] :
( ( ord_less_nat @ X2 @ A )
=> ( ord_less_nat @ X2 @ ( sup_sup_nat @ A @ B ) ) ) ).
% less_supI1
thf(fact_626_cSup__least,axiom,
! [X6: set_nat,Z: nat] :
( ( X6 != bot_bot_set_nat )
=> ( ! [X: nat] :
( ( member_nat @ X @ X6 )
=> ( ord_less_eq_nat @ X @ Z ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ X6 ) @ Z ) ) ) ).
% cSup_least
thf(fact_627_cSup__eq__non__empty,axiom,
! [X6: set_nat,A: nat] :
( ( X6 != bot_bot_set_nat )
=> ( ! [X: nat] :
( ( member_nat @ X @ X6 )
=> ( ord_less_eq_nat @ X @ A ) )
=> ( ! [Y2: nat] :
( ! [X5: nat] :
( ( member_nat @ X5 @ X6 )
=> ( ord_less_eq_nat @ X5 @ Y2 ) )
=> ( ord_less_eq_nat @ A @ Y2 ) )
=> ( ( complete_Sup_Sup_nat @ X6 )
= A ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_628_less__cSupD,axiom,
! [X6: set_nat,Z: nat] :
( ( X6 != bot_bot_set_nat )
=> ( ( ord_less_nat @ Z @ ( complete_Sup_Sup_nat @ X6 ) )
=> ? [X: nat] :
( ( member_nat @ X @ X6 )
& ( ord_less_nat @ Z @ X ) ) ) ) ).
% less_cSupD
thf(fact_629_less__cSupE,axiom,
! [Y3: nat,X6: set_nat] :
( ( ord_less_nat @ Y3 @ ( complete_Sup_Sup_nat @ X6 ) )
=> ( ( X6 != bot_bot_set_nat )
=> ~ ! [X: nat] :
( ( member_nat @ X @ X6 )
=> ~ ( ord_less_nat @ Y3 @ X ) ) ) ) ).
% less_cSupE
thf(fact_630_distrib__sup__le,axiom,
! [X2: nat,Y3: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X2 @ ( inf_inf_nat @ Y3 @ Z ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X2 @ Y3 ) @ ( sup_sup_nat @ X2 @ Z ) ) ) ).
% distrib_sup_le
thf(fact_631_distrib__inf__le,axiom,
! [X2: nat,Y3: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X2 @ Y3 ) @ ( inf_inf_nat @ X2 @ Z ) ) @ ( inf_inf_nat @ X2 @ ( sup_sup_nat @ Y3 @ Z ) ) ) ).
% distrib_inf_le
thf(fact_632_coinduct__set,axiom,
! [F: set_a > set_a,A: a,X6: set_a] :
( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ( member_a @ A @ X6 )
=> ( ( ord_less_eq_set_a @ X6 @ ( F @ ( sup_sup_set_a @ X6 @ ( comple3341859861669737308_set_a @ F ) ) ) )
=> ( member_a @ A @ ( comple3341859861669737308_set_a @ F ) ) ) ) ) ).
% coinduct_set
thf(fact_633_coinduct__set,axiom,
! [F: set_nat > set_nat,A: nat,X6: set_nat] :
( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F )
=> ( ( member_nat @ A @ X6 )
=> ( ( ord_less_eq_set_nat @ X6 @ ( F @ ( sup_sup_set_nat @ X6 @ ( comple1596078789208929544et_nat @ F ) ) ) )
=> ( member_nat @ A @ ( comple1596078789208929544et_nat @ F ) ) ) ) ) ).
% coinduct_set
thf(fact_634_gfp__fun__UnI2,axiom,
! [F: set_a > set_a,A: a,X6: set_a] :
( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ( member_a @ A @ ( comple3341859861669737308_set_a @ F ) )
=> ( member_a @ A @ ( F @ ( sup_sup_set_a @ X6 @ ( comple3341859861669737308_set_a @ F ) ) ) ) ) ) ).
% gfp_fun_UnI2
thf(fact_635_gfp__fun__UnI2,axiom,
! [F: set_nat > set_nat,A: nat,X6: set_nat] :
( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F )
=> ( ( member_nat @ A @ ( comple1596078789208929544et_nat @ F ) )
=> ( member_nat @ A @ ( F @ ( sup_sup_set_nat @ X6 @ ( comple1596078789208929544et_nat @ F ) ) ) ) ) ) ).
% gfp_fun_UnI2
thf(fact_636_def__coinduct__set,axiom,
! [A4: set_a,F: set_a > set_a,A: a,X6: set_a] :
( ( A4
= ( comple3341859861669737308_set_a @ F ) )
=> ( ( monoto7172710143293369831_set_a @ top_top_set_set_a @ ord_less_eq_set_a @ ord_less_eq_set_a @ F )
=> ( ( member_a @ A @ X6 )
=> ( ( ord_less_eq_set_a @ X6 @ ( F @ ( sup_sup_set_a @ X6 @ A4 ) ) )
=> ( member_a @ A @ A4 ) ) ) ) ) ).
% def_coinduct_set
thf(fact_637_def__coinduct__set,axiom,
! [A4: set_nat,F: set_nat > set_nat,A: nat,X6: set_nat] :
( ( A4
= ( comple1596078789208929544et_nat @ F ) )
=> ( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F )
=> ( ( member_nat @ A @ X6 )
=> ( ( ord_less_eq_set_nat @ X6 @ ( F @ ( sup_sup_set_nat @ X6 @ A4 ) ) )
=> ( member_nat @ A @ A4 ) ) ) ) ) ).
% def_coinduct_set
thf(fact_638_cSUP__least,axiom,
! [A4: set_a,F: a > nat,M2: nat] :
( ( A4 != bot_bot_set_a )
=> ( ! [X: a] :
( ( member_a @ X @ A4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ M2 ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_a_nat @ F @ A4 ) ) @ M2 ) ) ) ).
% cSUP_least
thf(fact_639_cSUP__least,axiom,
! [A4: set_nat,F: nat > nat,M2: nat] :
( ( A4 != bot_bot_set_nat )
=> ( ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ M2 ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ A4 ) ) @ M2 ) ) ) ).
% cSUP_least
thf(fact_640_mono__sup,axiom,
! [F: nat > nat,A4: nat,B6: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ ( F @ A4 ) @ ( F @ B6 ) ) @ ( F @ ( sup_sup_nat @ A4 @ B6 ) ) ) ) ).
% mono_sup
thf(fact_641_cSUP__union,axiom,
! [A4: set_nat,F: nat > nat,B6: set_nat] :
( ( A4 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ F @ A4 ) )
=> ( ( B6 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ F @ B6 ) )
=> ( ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ ( sup_sup_set_nat @ A4 @ B6 ) ) )
= ( sup_sup_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ A4 ) ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ B6 ) ) ) ) ) ) ) ) ).
% cSUP_union
thf(fact_642_cSup__inter__less__eq,axiom,
! [A4: set_nat,B6: set_nat] :
( ( condit2214826472909112428ve_nat @ A4 )
=> ( ( condit2214826472909112428ve_nat @ B6 )
=> ( ( ( inf_inf_set_nat @ A4 @ B6 )
!= bot_bot_set_nat )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( inf_inf_set_nat @ A4 @ B6 ) ) @ ( sup_sup_nat @ ( complete_Sup_Sup_nat @ A4 ) @ ( complete_Sup_Sup_nat @ B6 ) ) ) ) ) ) ).
% cSup_inter_less_eq
thf(fact_643_cSUP__subset__mono,axiom,
! [A4: set_a,G2: a > nat,B6: set_a,F: a > nat] :
( ( A4 != bot_bot_set_a )
=> ( ( condit2214826472909112428ve_nat @ ( image_a_nat @ G2 @ B6 ) )
=> ( ( ord_less_eq_set_a @ A4 @ B6 )
=> ( ! [X: a] :
( ( member_a @ X @ A4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( G2 @ X ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_a_nat @ F @ A4 ) ) @ ( complete_Sup_Sup_nat @ ( image_a_nat @ G2 @ B6 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_644_cSUP__subset__mono,axiom,
! [A4: set_nat,G2: nat > nat,B6: set_nat,F: nat > nat] :
( ( A4 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ G2 @ B6 ) )
=> ( ( ord_less_eq_set_nat @ A4 @ B6 )
=> ( ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( G2 @ X ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ A4 ) ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ G2 @ B6 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_645_bdd__above_OI,axiom,
! [A4: set_a,M2: a] :
( ! [X: a] :
( ( member_a @ X @ A4 )
=> ( ord_less_eq_a @ X @ M2 ) )
=> ( condit5209368051240477026bove_a @ A4 ) ) ).
% bdd_above.I
thf(fact_646_bdd__above_OI,axiom,
! [A4: set_nat,M2: nat] :
( ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ( ord_less_eq_nat @ X @ M2 ) )
=> ( condit2214826472909112428ve_nat @ A4 ) ) ).
% bdd_above.I
thf(fact_647_bdd__above__empty,axiom,
condit2214826472909112428ve_nat @ bot_bot_set_nat ).
% bdd_above_empty
thf(fact_648_bdd__above_OE,axiom,
! [A4: set_a] :
( ( condit5209368051240477026bove_a @ A4 )
=> ~ ! [M3: a] :
~ ! [X5: a] :
( ( member_a @ X5 @ A4 )
=> ( ord_less_eq_a @ X5 @ M3 ) ) ) ).
% bdd_above.E
thf(fact_649_bdd__above_OE,axiom,
! [A4: set_nat] :
( ( condit2214826472909112428ve_nat @ A4 )
=> ~ ! [M3: nat] :
~ ! [X5: nat] :
( ( member_nat @ X5 @ A4 )
=> ( ord_less_eq_nat @ X5 @ M3 ) ) ) ).
% bdd_above.E
thf(fact_650_bdd__above_Ounfold,axiom,
( condit5209368051240477026bove_a
= ( ^ [A6: set_a] :
? [M4: a] :
! [X3: a] :
( ( member_a @ X3 @ A6 )
=> ( ord_less_eq_a @ X3 @ M4 ) ) ) ) ).
% bdd_above.unfold
thf(fact_651_bdd__above_Ounfold,axiom,
( condit2214826472909112428ve_nat
= ( ^ [A6: set_nat] :
? [M4: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A6 )
=> ( ord_less_eq_nat @ X3 @ M4 ) ) ) ) ).
% bdd_above.unfold
thf(fact_652_bdd__above_OI2,axiom,
! [A4: set_a,F: a > a,M2: a] :
( ! [X: a] :
( ( member_a @ X @ A4 )
=> ( ord_less_eq_a @ ( F @ X ) @ M2 ) )
=> ( condit5209368051240477026bove_a @ ( image_a_a @ F @ A4 ) ) ) ).
% bdd_above.I2
thf(fact_653_bdd__above_OI2,axiom,
! [A4: set_nat,F: nat > a,M2: a] :
( ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ( ord_less_eq_a @ ( F @ X ) @ M2 ) )
=> ( condit5209368051240477026bove_a @ ( image_nat_a @ F @ A4 ) ) ) ).
% bdd_above.I2
thf(fact_654_bdd__above_OI2,axiom,
! [A4: set_a,F: a > nat,M2: nat] :
( ! [X: a] :
( ( member_a @ X @ A4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ M2 ) )
=> ( condit2214826472909112428ve_nat @ ( image_a_nat @ F @ A4 ) ) ) ).
% bdd_above.I2
thf(fact_655_bdd__above_OI2,axiom,
! [A4: set_nat,F: nat > nat,M2: nat] :
( ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ M2 ) )
=> ( condit2214826472909112428ve_nat @ ( image_nat_nat @ F @ A4 ) ) ) ).
% bdd_above.I2
thf(fact_656_cSup__upper2,axiom,
! [X2: nat,X6: set_nat,Y3: nat] :
( ( member_nat @ X2 @ X6 )
=> ( ( ord_less_eq_nat @ Y3 @ X2 )
=> ( ( condit2214826472909112428ve_nat @ X6 )
=> ( ord_less_eq_nat @ Y3 @ ( complete_Sup_Sup_nat @ X6 ) ) ) ) ) ).
% cSup_upper2
thf(fact_657_cSup__upper,axiom,
! [X2: nat,X6: set_nat] :
( ( member_nat @ X2 @ X6 )
=> ( ( condit2214826472909112428ve_nat @ X6 )
=> ( ord_less_eq_nat @ X2 @ ( complete_Sup_Sup_nat @ X6 ) ) ) ) ).
% cSup_upper
thf(fact_658_bdd__above__primitive__def,axiom,
( condit5209368051240477026bove_a
= ( condit6541519642617408243_bdd_a @ ord_less_eq_a ) ) ).
% bdd_above_primitive_def
thf(fact_659_bdd__above__primitive__def,axiom,
( condit2214826472909112428ve_nat
= ( condit4013746787832047771dd_nat @ ord_less_eq_nat ) ) ).
% bdd_above_primitive_def
thf(fact_660_cSUP__upper,axiom,
! [X2: a,A4: set_a,F: a > nat] :
( ( member_a @ X2 @ A4 )
=> ( ( condit2214826472909112428ve_nat @ ( image_a_nat @ F @ A4 ) )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( complete_Sup_Sup_nat @ ( image_a_nat @ F @ A4 ) ) ) ) ) ).
% cSUP_upper
thf(fact_661_cSUP__upper,axiom,
! [X2: nat,A4: set_nat,F: nat > nat] :
( ( member_nat @ X2 @ A4 )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ F @ A4 ) )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ A4 ) ) ) ) ) ).
% cSUP_upper
thf(fact_662_cSUP__upper2,axiom,
! [F: a > nat,A4: set_a,X2: a,U2: nat] :
( ( condit2214826472909112428ve_nat @ ( image_a_nat @ F @ A4 ) )
=> ( ( member_a @ X2 @ A4 )
=> ( ( ord_less_eq_nat @ U2 @ ( F @ X2 ) )
=> ( ord_less_eq_nat @ U2 @ ( complete_Sup_Sup_nat @ ( image_a_nat @ F @ A4 ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_663_cSUP__upper2,axiom,
! [F: nat > nat,A4: set_nat,X2: nat,U2: nat] :
( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ F @ A4 ) )
=> ( ( member_nat @ X2 @ A4 )
=> ( ( ord_less_eq_nat @ U2 @ ( F @ X2 ) )
=> ( ord_less_eq_nat @ U2 @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ A4 ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_664_cSup__mono,axiom,
! [B6: set_nat,A4: set_nat] :
( ( B6 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ A4 )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ B6 )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A4 )
& ( ord_less_eq_nat @ B3 @ X5 ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ B6 ) @ ( complete_Sup_Sup_nat @ A4 ) ) ) ) ) ).
% cSup_mono
thf(fact_665_cSup__le__iff,axiom,
! [S2: set_nat,A: nat] :
( ( S2 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ S2 )
=> ( ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ S2 ) @ A )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ S2 )
=> ( ord_less_eq_nat @ X3 @ A ) ) ) ) ) ) ).
% cSup_le_iff
thf(fact_666_less__cSup__iff,axiom,
! [X6: set_nat,Y3: nat] :
( ( X6 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ X6 )
=> ( ( ord_less_nat @ Y3 @ ( complete_Sup_Sup_nat @ X6 ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ X6 )
& ( ord_less_nat @ Y3 @ X3 ) ) ) ) ) ) ).
% less_cSup_iff
thf(fact_667_cSUP__mono,axiom,
! [A4: set_a,G2: nat > nat,B6: set_nat,F: a > nat] :
( ( A4 != bot_bot_set_a )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ G2 @ B6 ) )
=> ( ! [N2: a] :
( ( member_a @ N2 @ A4 )
=> ? [X5: nat] :
( ( member_nat @ X5 @ B6 )
& ( ord_less_eq_nat @ ( F @ N2 ) @ ( G2 @ X5 ) ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_a_nat @ F @ A4 ) ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ G2 @ B6 ) ) ) ) ) ) ).
% cSUP_mono
thf(fact_668_cSUP__mono,axiom,
! [A4: set_nat,G2: nat > nat,B6: set_nat,F: nat > nat] :
( ( A4 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ G2 @ B6 ) )
=> ( ! [N2: nat] :
( ( member_nat @ N2 @ A4 )
=> ? [X5: nat] :
( ( member_nat @ X5 @ B6 )
& ( ord_less_eq_nat @ ( F @ N2 ) @ ( G2 @ X5 ) ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ A4 ) ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ G2 @ B6 ) ) ) ) ) ) ).
% cSUP_mono
thf(fact_669_cSUP__le__iff,axiom,
! [A4: set_nat,F: nat > nat,U2: nat] :
( ( A4 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ F @ A4 ) )
=> ( ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ A4 ) ) @ U2 )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ U2 ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_670_cSup__subset__mono,axiom,
! [A4: set_nat,B6: set_nat] :
( ( A4 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ B6 )
=> ( ( ord_less_eq_set_nat @ A4 @ B6 )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ A4 ) @ ( complete_Sup_Sup_nat @ B6 ) ) ) ) ) ).
% cSup_subset_mono
thf(fact_671_bdd__above__image__mono,axiom,
! [F: a > a,A4: set_a] :
( ( monotone_on_a_a @ top_top_set_a @ ord_less_eq_a @ ord_less_eq_a @ F )
=> ( ( condit5209368051240477026bove_a @ A4 )
=> ( condit5209368051240477026bove_a @ ( image_a_a @ F @ A4 ) ) ) ) ).
% bdd_above_image_mono
thf(fact_672_bdd__above__image__mono,axiom,
! [F: a > nat,A4: set_a] :
( ( monotone_on_a_nat @ top_top_set_a @ ord_less_eq_a @ ord_less_eq_nat @ F )
=> ( ( condit5209368051240477026bove_a @ A4 )
=> ( condit2214826472909112428ve_nat @ ( image_a_nat @ F @ A4 ) ) ) ) ).
% bdd_above_image_mono
thf(fact_673_bdd__above__image__mono,axiom,
! [F: nat > a,A4: set_nat] :
( ( monotone_on_nat_a @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_a @ F )
=> ( ( condit2214826472909112428ve_nat @ A4 )
=> ( condit5209368051240477026bove_a @ ( image_nat_a @ F @ A4 ) ) ) ) ).
% bdd_above_image_mono
thf(fact_674_bdd__above__image__mono,axiom,
! [F: nat > nat,A4: set_nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( condit2214826472909112428ve_nat @ A4 )
=> ( condit2214826472909112428ve_nat @ ( image_nat_nat @ F @ A4 ) ) ) ) ).
% bdd_above_image_mono
thf(fact_675_cSup__union__distrib,axiom,
! [A4: set_nat,B6: set_nat] :
( ( A4 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ A4 )
=> ( ( B6 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ B6 )
=> ( ( complete_Sup_Sup_nat @ ( sup_sup_set_nat @ A4 @ B6 ) )
= ( sup_sup_nat @ ( complete_Sup_Sup_nat @ A4 ) @ ( complete_Sup_Sup_nat @ B6 ) ) ) ) ) ) ) ).
% cSup_union_distrib
thf(fact_676_cSUP__insert,axiom,
! [A4: set_nat,F: nat > nat,A: nat] :
( ( A4 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ F @ A4 ) )
=> ( ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ ( insert_nat @ A @ A4 ) ) )
= ( sup_sup_nat @ ( F @ A ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ A4 ) ) ) ) ) ) ).
% cSUP_insert
thf(fact_677_cSup__insert__If,axiom,
! [X6: set_nat,A: nat] :
( ( condit2214826472909112428ve_nat @ X6 )
=> ( ( ( X6 = bot_bot_set_nat )
=> ( ( complete_Sup_Sup_nat @ ( insert_nat @ A @ X6 ) )
= A ) )
& ( ( X6 != bot_bot_set_nat )
=> ( ( complete_Sup_Sup_nat @ ( insert_nat @ A @ X6 ) )
= ( sup_sup_nat @ A @ ( complete_Sup_Sup_nat @ X6 ) ) ) ) ) ) ).
% cSup_insert_If
thf(fact_678_cSup__insert,axiom,
! [X6: set_nat,A: nat] :
( ( X6 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ X6 )
=> ( ( complete_Sup_Sup_nat @ ( insert_nat @ A @ X6 ) )
= ( sup_sup_nat @ A @ ( complete_Sup_Sup_nat @ X6 ) ) ) ) ) ).
% cSup_insert
thf(fact_679_cSup__singleton,axiom,
! [X2: nat] :
( ( complete_Sup_Sup_nat @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
= X2 ) ).
% cSup_singleton
thf(fact_680_cInf__singleton,axiom,
! [X2: nat] :
( ( complete_Inf_Inf_nat @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
= X2 ) ).
% cInf_singleton
thf(fact_681_wellorder__InfI,axiom,
! [K: nat,A4: set_nat] :
( ( member_nat @ K @ A4 )
=> ( member_nat @ ( complete_Inf_Inf_nat @ A4 ) @ A4 ) ) ).
% wellorder_InfI
thf(fact_682_wellorder__Inf__le1,axiom,
! [K: nat,A4: set_nat] :
( ( member_nat @ K @ A4 )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ A4 ) @ K ) ) ).
% wellorder_Inf_le1
thf(fact_683_cInf__eq,axiom,
! [X6: set_nat,A: nat] :
( ! [X: nat] :
( ( member_nat @ X @ X6 )
=> ( ord_less_eq_nat @ A @ X ) )
=> ( ! [Y2: nat] :
( ! [X5: nat] :
( ( member_nat @ X5 @ X6 )
=> ( ord_less_eq_nat @ Y2 @ X5 ) )
=> ( ord_less_eq_nat @ Y2 @ A ) )
=> ( ( complete_Inf_Inf_nat @ X6 )
= A ) ) ) ).
% cInf_eq
thf(fact_684_cInf__eq__minimum,axiom,
! [Z: nat,X6: set_nat] :
( ( member_nat @ Z @ X6 )
=> ( ! [X: nat] :
( ( member_nat @ X @ X6 )
=> ( ord_less_eq_nat @ Z @ X ) )
=> ( ( complete_Inf_Inf_nat @ X6 )
= Z ) ) ) ).
% cInf_eq_minimum
thf(fact_685_cInf__eq__non__empty,axiom,
! [X6: set_nat,A: nat] :
( ( X6 != bot_bot_set_nat )
=> ( ! [X: nat] :
( ( member_nat @ X @ X6 )
=> ( ord_less_eq_nat @ A @ X ) )
=> ( ! [Y2: nat] :
( ! [X5: nat] :
( ( member_nat @ X5 @ X6 )
=> ( ord_less_eq_nat @ Y2 @ X5 ) )
=> ( ord_less_eq_nat @ Y2 @ A ) )
=> ( ( complete_Inf_Inf_nat @ X6 )
= A ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_686_cInf__greatest,axiom,
! [X6: set_nat,Z: nat] :
( ( X6 != bot_bot_set_nat )
=> ( ! [X: nat] :
( ( member_nat @ X @ X6 )
=> ( ord_less_eq_nat @ Z @ X ) )
=> ( ord_less_eq_nat @ Z @ ( complete_Inf_Inf_nat @ X6 ) ) ) ) ).
% cInf_greatest
thf(fact_687_cInf__lessD,axiom,
! [X6: set_nat,Z: nat] :
( ( X6 != bot_bot_set_nat )
=> ( ( ord_less_nat @ ( complete_Inf_Inf_nat @ X6 ) @ Z )
=> ? [X: nat] :
( ( member_nat @ X @ X6 )
& ( ord_less_nat @ X @ Z ) ) ) ) ).
% cInf_lessD
thf(fact_688_cINF__greatest,axiom,
! [A4: set_a,M5: nat,F: a > nat] :
( ( A4 != bot_bot_set_a )
=> ( ! [X: a] :
( ( member_a @ X @ A4 )
=> ( ord_less_eq_nat @ M5 @ ( F @ X ) ) )
=> ( ord_less_eq_nat @ M5 @ ( complete_Inf_Inf_nat @ ( image_a_nat @ F @ A4 ) ) ) ) ) ).
% cINF_greatest
thf(fact_689_cINF__greatest,axiom,
! [A4: set_nat,M5: nat,F: nat > nat] :
( ( A4 != bot_bot_set_nat )
=> ( ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ( ord_less_eq_nat @ M5 @ ( F @ X ) ) )
=> ( ord_less_eq_nat @ M5 @ ( complete_Inf_Inf_nat @ ( image_nat_nat @ F @ A4 ) ) ) ) ) ).
% cINF_greatest
thf(fact_690_cInf__le__cSup,axiom,
! [A4: set_nat] :
( ( A4 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ A4 )
=> ( ( condit1738341127787009408ow_nat @ A4 )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ A4 ) @ ( complete_Sup_Sup_nat @ A4 ) ) ) ) ) ).
% cInf_le_cSup
thf(fact_691_cINF__union,axiom,
! [A4: set_nat,F: nat > nat,B6: set_nat] :
( ( A4 != bot_bot_set_nat )
=> ( ( condit1738341127787009408ow_nat @ ( image_nat_nat @ F @ A4 ) )
=> ( ( B6 != bot_bot_set_nat )
=> ( ( condit1738341127787009408ow_nat @ ( image_nat_nat @ F @ B6 ) )
=> ( ( complete_Inf_Inf_nat @ ( image_nat_nat @ F @ ( sup_sup_set_nat @ A4 @ B6 ) ) )
= ( inf_inf_nat @ ( complete_Inf_Inf_nat @ ( image_nat_nat @ F @ A4 ) ) @ ( complete_Inf_Inf_nat @ ( image_nat_nat @ F @ B6 ) ) ) ) ) ) ) ) ).
% cINF_union
thf(fact_692_cINF__insert,axiom,
! [A4: set_nat,F: nat > nat,A: nat] :
( ( A4 != bot_bot_set_nat )
=> ( ( condit1738341127787009408ow_nat @ ( image_nat_nat @ F @ A4 ) )
=> ( ( complete_Inf_Inf_nat @ ( image_nat_nat @ F @ ( insert_nat @ A @ A4 ) ) )
= ( inf_inf_nat @ ( F @ A ) @ ( complete_Inf_Inf_nat @ ( image_nat_nat @ F @ A4 ) ) ) ) ) ) ).
% cINF_insert
thf(fact_693_less__eq__cInf__inter,axiom,
! [A4: set_nat,B6: set_nat] :
( ( condit1738341127787009408ow_nat @ A4 )
=> ( ( condit1738341127787009408ow_nat @ B6 )
=> ( ( ( inf_inf_set_nat @ A4 @ B6 )
!= bot_bot_set_nat )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ ( complete_Inf_Inf_nat @ A4 ) @ ( complete_Inf_Inf_nat @ B6 ) ) @ ( complete_Inf_Inf_nat @ ( inf_inf_set_nat @ A4 @ B6 ) ) ) ) ) ) ).
% less_eq_cInf_inter
thf(fact_694_cINF__superset__mono,axiom,
! [A4: set_a,G2: a > nat,B6: set_a,F: a > nat] :
( ( A4 != bot_bot_set_a )
=> ( ( condit1738341127787009408ow_nat @ ( image_a_nat @ G2 @ B6 ) )
=> ( ( ord_less_eq_set_a @ A4 @ B6 )
=> ( ! [X: a] :
( ( member_a @ X @ B6 )
=> ( ord_less_eq_nat @ ( G2 @ X ) @ ( F @ X ) ) )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_a_nat @ G2 @ B6 ) ) @ ( complete_Inf_Inf_nat @ ( image_a_nat @ F @ A4 ) ) ) ) ) ) ) ).
% cINF_superset_mono
thf(fact_695_cINF__superset__mono,axiom,
! [A4: set_nat,G2: nat > nat,B6: set_nat,F: nat > nat] :
( ( A4 != bot_bot_set_nat )
=> ( ( condit1738341127787009408ow_nat @ ( image_nat_nat @ G2 @ B6 ) )
=> ( ( ord_less_eq_set_nat @ A4 @ B6 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B6 )
=> ( ord_less_eq_nat @ ( G2 @ X ) @ ( F @ X ) ) )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_nat_nat @ G2 @ B6 ) ) @ ( complete_Inf_Inf_nat @ ( image_nat_nat @ F @ A4 ) ) ) ) ) ) ) ).
% cINF_superset_mono
thf(fact_696_bdd__below_OI,axiom,
! [A4: set_a,M2: a] :
( ! [X: a] :
( ( member_a @ X @ A4 )
=> ( ord_less_eq_a @ M2 @ X ) )
=> ( condit5901475214736682318elow_a @ A4 ) ) ).
% bdd_below.I
thf(fact_697_bdd__below_OI,axiom,
! [A4: set_nat,M2: nat] :
( ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ( ord_less_eq_nat @ M2 @ X ) )
=> ( condit1738341127787009408ow_nat @ A4 ) ) ).
% bdd_below.I
thf(fact_698_bdd__belowI,axiom,
! [A4: set_a,M5: a] :
( ! [X: a] :
( ( member_a @ X @ A4 )
=> ( ord_less_eq_a @ M5 @ X ) )
=> ( condit5901475214736682318elow_a @ A4 ) ) ).
% bdd_belowI
thf(fact_699_bdd__belowI,axiom,
! [A4: set_nat,M5: nat] :
( ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ( ord_less_eq_nat @ M5 @ X ) )
=> ( condit1738341127787009408ow_nat @ A4 ) ) ).
% bdd_belowI
thf(fact_700_bdd__below__empty,axiom,
condit1738341127787009408ow_nat @ bot_bot_set_nat ).
% bdd_below_empty
thf(fact_701_bdd__below_OE,axiom,
! [A4: set_a] :
( ( condit5901475214736682318elow_a @ A4 )
=> ~ ! [M3: a] :
~ ! [X5: a] :
( ( member_a @ X5 @ A4 )
=> ( ord_less_eq_a @ M3 @ X5 ) ) ) ).
% bdd_below.E
thf(fact_702_bdd__below_OE,axiom,
! [A4: set_nat] :
( ( condit1738341127787009408ow_nat @ A4 )
=> ~ ! [M3: nat] :
~ ! [X5: nat] :
( ( member_nat @ X5 @ A4 )
=> ( ord_less_eq_nat @ M3 @ X5 ) ) ) ).
% bdd_below.E
thf(fact_703_bdd__below_Ounfold,axiom,
( condit5901475214736682318elow_a
= ( ^ [A6: set_a] :
? [M4: a] :
! [X3: a] :
( ( member_a @ X3 @ A6 )
=> ( ord_less_eq_a @ M4 @ X3 ) ) ) ) ).
% bdd_below.unfold
thf(fact_704_bdd__below_Ounfold,axiom,
( condit1738341127787009408ow_nat
= ( ^ [A6: set_nat] :
? [M4: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A6 )
=> ( ord_less_eq_nat @ M4 @ X3 ) ) ) ) ).
% bdd_below.unfold
thf(fact_705_bdd__belowI2,axiom,
! [A4: set_a,M5: a,F: a > a] :
( ! [X: a] :
( ( member_a @ X @ A4 )
=> ( ord_less_eq_a @ M5 @ ( F @ X ) ) )
=> ( condit5901475214736682318elow_a @ ( image_a_a @ F @ A4 ) ) ) ).
% bdd_belowI2
thf(fact_706_bdd__belowI2,axiom,
! [A4: set_nat,M5: a,F: nat > a] :
( ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ( ord_less_eq_a @ M5 @ ( F @ X ) ) )
=> ( condit5901475214736682318elow_a @ ( image_nat_a @ F @ A4 ) ) ) ).
% bdd_belowI2
thf(fact_707_bdd__belowI2,axiom,
! [A4: set_a,M5: nat,F: a > nat] :
( ! [X: a] :
( ( member_a @ X @ A4 )
=> ( ord_less_eq_nat @ M5 @ ( F @ X ) ) )
=> ( condit1738341127787009408ow_nat @ ( image_a_nat @ F @ A4 ) ) ) ).
% bdd_belowI2
thf(fact_708_bdd__belowI2,axiom,
! [A4: set_nat,M5: nat,F: nat > nat] :
( ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ( ord_less_eq_nat @ M5 @ ( F @ X ) ) )
=> ( condit1738341127787009408ow_nat @ ( image_nat_nat @ F @ A4 ) ) ) ).
% bdd_belowI2
thf(fact_709_bdd__below_OI2,axiom,
! [A4: set_a,M2: a,F: a > a] :
( ! [X: a] :
( ( member_a @ X @ A4 )
=> ( ord_less_eq_a @ M2 @ ( F @ X ) ) )
=> ( condit5901475214736682318elow_a @ ( image_a_a @ F @ A4 ) ) ) ).
% bdd_below.I2
thf(fact_710_bdd__below_OI2,axiom,
! [A4: set_nat,M2: a,F: nat > a] :
( ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ( ord_less_eq_a @ M2 @ ( F @ X ) ) )
=> ( condit5901475214736682318elow_a @ ( image_nat_a @ F @ A4 ) ) ) ).
% bdd_below.I2
thf(fact_711_bdd__below_OI2,axiom,
! [A4: set_a,M2: nat,F: a > nat] :
( ! [X: a] :
( ( member_a @ X @ A4 )
=> ( ord_less_eq_nat @ M2 @ ( F @ X ) ) )
=> ( condit1738341127787009408ow_nat @ ( image_a_nat @ F @ A4 ) ) ) ).
% bdd_below.I2
thf(fact_712_bdd__below_OI2,axiom,
! [A4: set_nat,M2: nat,F: nat > nat] :
( ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ( ord_less_eq_nat @ M2 @ ( F @ X ) ) )
=> ( condit1738341127787009408ow_nat @ ( image_nat_nat @ F @ A4 ) ) ) ).
% bdd_below.I2
thf(fact_713_cInf__lower2,axiom,
! [X2: nat,X6: set_nat,Y3: nat] :
( ( member_nat @ X2 @ X6 )
=> ( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( condit1738341127787009408ow_nat @ X6 )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ X6 ) @ Y3 ) ) ) ) ).
% cInf_lower2
thf(fact_714_cInf__lower,axiom,
! [X2: nat,X6: set_nat] :
( ( member_nat @ X2 @ X6 )
=> ( ( condit1738341127787009408ow_nat @ X6 )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ X6 ) @ X2 ) ) ) ).
% cInf_lower
thf(fact_715_cINF__lower2,axiom,
! [F: a > nat,A4: set_a,X2: a,U2: nat] :
( ( condit1738341127787009408ow_nat @ ( image_a_nat @ F @ A4 ) )
=> ( ( member_a @ X2 @ A4 )
=> ( ( ord_less_eq_nat @ ( F @ X2 ) @ U2 )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_a_nat @ F @ A4 ) ) @ U2 ) ) ) ) ).
% cINF_lower2
thf(fact_716_cINF__lower2,axiom,
! [F: nat > nat,A4: set_nat,X2: nat,U2: nat] :
( ( condit1738341127787009408ow_nat @ ( image_nat_nat @ F @ A4 ) )
=> ( ( member_nat @ X2 @ A4 )
=> ( ( ord_less_eq_nat @ ( F @ X2 ) @ U2 )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_nat_nat @ F @ A4 ) ) @ U2 ) ) ) ) ).
% cINF_lower2
thf(fact_717_cINF__lower,axiom,
! [F: a > nat,A4: set_a,X2: a] :
( ( condit1738341127787009408ow_nat @ ( image_a_nat @ F @ A4 ) )
=> ( ( member_a @ X2 @ A4 )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_a_nat @ F @ A4 ) ) @ ( F @ X2 ) ) ) ) ).
% cINF_lower
thf(fact_718_cINF__lower,axiom,
! [F: nat > nat,A4: set_nat,X2: nat] :
( ( condit1738341127787009408ow_nat @ ( image_nat_nat @ F @ A4 ) )
=> ( ( member_nat @ X2 @ A4 )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_nat_nat @ F @ A4 ) ) @ ( F @ X2 ) ) ) ) ).
% cINF_lower
thf(fact_719_le__cInf__iff,axiom,
! [S2: set_nat,A: nat] :
( ( S2 != bot_bot_set_nat )
=> ( ( condit1738341127787009408ow_nat @ S2 )
=> ( ( ord_less_eq_nat @ A @ ( complete_Inf_Inf_nat @ S2 ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ S2 )
=> ( ord_less_eq_nat @ A @ X3 ) ) ) ) ) ) ).
% le_cInf_iff
thf(fact_720_cInf__mono,axiom,
! [B6: set_nat,A4: set_nat] :
( ( B6 != bot_bot_set_nat )
=> ( ( condit1738341127787009408ow_nat @ A4 )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ B6 )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A4 )
& ( ord_less_eq_nat @ X5 @ B3 ) ) )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ A4 ) @ ( complete_Inf_Inf_nat @ B6 ) ) ) ) ) ).
% cInf_mono
thf(fact_721_cInf__less__iff,axiom,
! [X6: set_nat,Y3: nat] :
( ( X6 != bot_bot_set_nat )
=> ( ( condit1738341127787009408ow_nat @ X6 )
=> ( ( ord_less_nat @ ( complete_Inf_Inf_nat @ X6 ) @ Y3 )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ X6 )
& ( ord_less_nat @ X3 @ Y3 ) ) ) ) ) ) ).
% cInf_less_iff
thf(fact_722_cINF__mono,axiom,
! [B6: set_a,F: nat > nat,A4: set_nat,G2: a > nat] :
( ( B6 != bot_bot_set_a )
=> ( ( condit1738341127787009408ow_nat @ ( image_nat_nat @ F @ A4 ) )
=> ( ! [M6: a] :
( ( member_a @ M6 @ B6 )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A4 )
& ( ord_less_eq_nat @ ( F @ X5 ) @ ( G2 @ M6 ) ) ) )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_nat_nat @ F @ A4 ) ) @ ( complete_Inf_Inf_nat @ ( image_a_nat @ G2 @ B6 ) ) ) ) ) ) ).
% cINF_mono
thf(fact_723_cINF__mono,axiom,
! [B6: set_nat,F: nat > nat,A4: set_nat,G2: nat > nat] :
( ( B6 != bot_bot_set_nat )
=> ( ( condit1738341127787009408ow_nat @ ( image_nat_nat @ F @ A4 ) )
=> ( ! [M6: nat] :
( ( member_nat @ M6 @ B6 )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A4 )
& ( ord_less_eq_nat @ ( F @ X5 ) @ ( G2 @ M6 ) ) ) )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_nat_nat @ F @ A4 ) ) @ ( complete_Inf_Inf_nat @ ( image_nat_nat @ G2 @ B6 ) ) ) ) ) ) ).
% cINF_mono
thf(fact_724_le__cINF__iff,axiom,
! [A4: set_nat,F: nat > nat,U2: nat] :
( ( A4 != bot_bot_set_nat )
=> ( ( condit1738341127787009408ow_nat @ ( image_nat_nat @ F @ A4 ) )
=> ( ( ord_less_eq_nat @ U2 @ ( complete_Inf_Inf_nat @ ( image_nat_nat @ F @ A4 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( ord_less_eq_nat @ U2 @ ( F @ X3 ) ) ) ) ) ) ) ).
% le_cINF_iff
thf(fact_725_cInf__superset__mono,axiom,
! [A4: set_nat,B6: set_nat] :
( ( A4 != bot_bot_set_nat )
=> ( ( condit1738341127787009408ow_nat @ B6 )
=> ( ( ord_less_eq_set_nat @ A4 @ B6 )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ B6 ) @ ( complete_Inf_Inf_nat @ A4 ) ) ) ) ) ).
% cInf_superset_mono
thf(fact_726_bdd__below__image__mono,axiom,
! [F: a > a,A4: set_a] :
( ( monotone_on_a_a @ top_top_set_a @ ord_less_eq_a @ ord_less_eq_a @ F )
=> ( ( condit5901475214736682318elow_a @ A4 )
=> ( condit5901475214736682318elow_a @ ( image_a_a @ F @ A4 ) ) ) ) ).
% bdd_below_image_mono
thf(fact_727_bdd__below__image__mono,axiom,
! [F: a > nat,A4: set_a] :
( ( monotone_on_a_nat @ top_top_set_a @ ord_less_eq_a @ ord_less_eq_nat @ F )
=> ( ( condit5901475214736682318elow_a @ A4 )
=> ( condit1738341127787009408ow_nat @ ( image_a_nat @ F @ A4 ) ) ) ) ).
% bdd_below_image_mono
thf(fact_728_bdd__below__image__mono,axiom,
! [F: nat > a,A4: set_nat] :
( ( monotone_on_nat_a @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_a @ F )
=> ( ( condit1738341127787009408ow_nat @ A4 )
=> ( condit5901475214736682318elow_a @ ( image_nat_a @ F @ A4 ) ) ) ) ).
% bdd_below_image_mono
thf(fact_729_bdd__below__image__mono,axiom,
! [F: nat > nat,A4: set_nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( condit1738341127787009408ow_nat @ A4 )
=> ( condit1738341127787009408ow_nat @ ( image_nat_nat @ F @ A4 ) ) ) ) ).
% bdd_below_image_mono
thf(fact_730_cInf__insert__If,axiom,
! [X6: set_nat,A: nat] :
( ( condit1738341127787009408ow_nat @ X6 )
=> ( ( ( X6 = bot_bot_set_nat )
=> ( ( complete_Inf_Inf_nat @ ( insert_nat @ A @ X6 ) )
= A ) )
& ( ( X6 != bot_bot_set_nat )
=> ( ( complete_Inf_Inf_nat @ ( insert_nat @ A @ X6 ) )
= ( inf_inf_nat @ A @ ( complete_Inf_Inf_nat @ X6 ) ) ) ) ) ) ).
% cInf_insert_If
thf(fact_731_cInf__insert,axiom,
! [X6: set_nat,A: nat] :
( ( X6 != bot_bot_set_nat )
=> ( ( condit1738341127787009408ow_nat @ X6 )
=> ( ( complete_Inf_Inf_nat @ ( insert_nat @ A @ X6 ) )
= ( inf_inf_nat @ A @ ( complete_Inf_Inf_nat @ X6 ) ) ) ) ) ).
% cInf_insert
thf(fact_732_cInf__union__distrib,axiom,
! [A4: set_nat,B6: set_nat] :
( ( A4 != bot_bot_set_nat )
=> ( ( condit1738341127787009408ow_nat @ A4 )
=> ( ( B6 != bot_bot_set_nat )
=> ( ( condit1738341127787009408ow_nat @ B6 )
=> ( ( complete_Inf_Inf_nat @ ( sup_sup_set_nat @ A4 @ B6 ) )
= ( inf_inf_nat @ ( complete_Inf_Inf_nat @ A4 ) @ ( complete_Inf_Inf_nat @ B6 ) ) ) ) ) ) ) ).
% cInf_union_distrib
thf(fact_733_Kleene__iter__gpfp,axiom,
! [F: set_nat > set_nat,P5: set_nat,K: nat] :
( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F )
=> ( ( ord_less_eq_set_nat @ P5 @ ( F @ P5 ) )
=> ( ord_less_eq_set_nat @ P5 @ ( compow8708494347934031032et_nat @ K @ F @ top_top_set_nat ) ) ) ) ).
% Kleene_iter_gpfp
thf(fact_734_surj__fn,axiom,
! [F: nat > nat,N3: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( ( image_nat_nat @ ( compow_nat_nat @ N3 @ F ) @ top_top_set_nat )
= top_top_set_nat ) ) ).
% surj_fn
thf(fact_735_pred__on_Ochain__empty,axiom,
! [A4: set_nat,P: nat > nat > $o] : ( pred_chain_nat @ A4 @ P @ bot_bot_set_nat ) ).
% pred_on.chain_empty
thf(fact_736_pred__on_OchainI,axiom,
! [C4: set_a,A4: set_a,P: a > a > $o] :
( ( ord_less_eq_set_a @ C4 @ A4 )
=> ( ! [X: a,Y2: a] :
( ( member_a @ X @ C4 )
=> ( ( member_a @ Y2 @ C4 )
=> ( ( sup_sup_a_a_o @ P
@ ^ [Y4: a,Z2: a] : ( Y4 = Z2 )
@ X
@ Y2 )
| ( sup_sup_a_a_o @ P
@ ^ [Y4: a,Z2: a] : ( Y4 = Z2 )
@ Y2
@ X ) ) ) )
=> ( pred_chain_a @ A4 @ P @ C4 ) ) ) ).
% pred_on.chainI
thf(fact_737_pred__on_OchainI,axiom,
! [C4: set_nat,A4: set_nat,P: nat > nat > $o] :
( ( ord_less_eq_set_nat @ C4 @ A4 )
=> ( ! [X: nat,Y2: nat] :
( ( member_nat @ X @ C4 )
=> ( ( member_nat @ Y2 @ C4 )
=> ( ( sup_sup_nat_nat_o @ P
@ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
@ X
@ Y2 )
| ( sup_sup_nat_nat_o @ P
@ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
@ Y2
@ X ) ) ) )
=> ( pred_chain_nat @ A4 @ P @ C4 ) ) ) ).
% pred_on.chainI
thf(fact_738_chain__mono,axiom,
! [A4: set_a,P: a > a > $o,Q: a > a > $o,C4: set_a] :
( ! [X: a,Y2: a] :
( ( member_a @ X @ A4 )
=> ( ( member_a @ Y2 @ A4 )
=> ( ( P @ X @ Y2 )
=> ( Q @ X @ Y2 ) ) ) )
=> ( ( pred_chain_a @ A4 @ P @ C4 )
=> ( pred_chain_a @ A4 @ Q @ C4 ) ) ) ).
% chain_mono
thf(fact_739_chain__mono,axiom,
! [A4: set_nat,P: nat > nat > $o,Q: nat > nat > $o,C4: set_nat] :
( ! [X: nat,Y2: nat] :
( ( member_nat @ X @ A4 )
=> ( ( member_nat @ Y2 @ A4 )
=> ( ( P @ X @ Y2 )
=> ( Q @ X @ Y2 ) ) ) )
=> ( ( pred_chain_nat @ A4 @ P @ C4 )
=> ( pred_chain_nat @ A4 @ Q @ C4 ) ) ) ).
% chain_mono
thf(fact_740_pred__on_Ochain__total,axiom,
! [A4: set_a,P: a > a > $o,C4: set_a,X2: a,Y3: a] :
( ( pred_chain_a @ A4 @ P @ C4 )
=> ( ( member_a @ X2 @ C4 )
=> ( ( member_a @ Y3 @ C4 )
=> ( ( sup_sup_a_a_o @ P
@ ^ [Y4: a,Z2: a] : ( Y4 = Z2 )
@ X2
@ Y3 )
| ( sup_sup_a_a_o @ P
@ ^ [Y4: a,Z2: a] : ( Y4 = Z2 )
@ Y3
@ X2 ) ) ) ) ) ).
% pred_on.chain_total
thf(fact_741_pred__on_Ochain__total,axiom,
! [A4: set_nat,P: nat > nat > $o,C4: set_nat,X2: nat,Y3: nat] :
( ( pred_chain_nat @ A4 @ P @ C4 )
=> ( ( member_nat @ X2 @ C4 )
=> ( ( member_nat @ Y3 @ C4 )
=> ( ( sup_sup_nat_nat_o @ P
@ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
@ X2
@ Y3 )
| ( sup_sup_nat_nat_o @ P
@ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
@ Y3
@ X2 ) ) ) ) ) ).
% pred_on.chain_total
thf(fact_742_funpow__mono,axiom,
! [F: a > a,A4: a,B6: a,N3: nat] :
( ( monotone_on_a_a @ top_top_set_a @ ord_less_eq_a @ ord_less_eq_a @ F )
=> ( ( ord_less_eq_a @ A4 @ B6 )
=> ( ord_less_eq_a @ ( compow_a_a @ N3 @ F @ A4 ) @ ( compow_a_a @ N3 @ F @ B6 ) ) ) ) ).
% funpow_mono
thf(fact_743_funpow__mono,axiom,
! [F: nat > nat,A4: nat,B6: nat,N3: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_nat @ A4 @ B6 )
=> ( ord_less_eq_nat @ ( compow_nat_nat @ N3 @ F @ A4 ) @ ( compow_nat_nat @ N3 @ F @ B6 ) ) ) ) ).
% funpow_mono
thf(fact_744_funpow__mono2,axiom,
! [F: a > a,I: nat,J: nat,X2: a,Y3: a] :
( ( monotone_on_a_a @ top_top_set_a @ ord_less_eq_a @ ord_less_eq_a @ F )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_a @ X2 @ Y3 )
=> ( ( ord_less_eq_a @ X2 @ ( F @ X2 ) )
=> ( ord_less_eq_a @ ( compow_a_a @ I @ F @ X2 ) @ ( compow_a_a @ J @ F @ Y3 ) ) ) ) ) ) ).
% funpow_mono2
thf(fact_745_funpow__mono2,axiom,
! [F: nat > nat,I: nat,J: nat,X2: nat,Y3: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_nat @ X2 @ ( F @ X2 ) )
=> ( ord_less_eq_nat @ ( compow_nat_nat @ I @ F @ X2 ) @ ( compow_nat_nat @ J @ F @ Y3 ) ) ) ) ) ) ).
% funpow_mono2
thf(fact_746_pred__on_Ochain__extend,axiom,
! [A4: set_a,P: a > a > $o,C4: set_a,Z: a] :
( ( pred_chain_a @ A4 @ P @ C4 )
=> ( ( member_a @ Z @ A4 )
=> ( ! [X: a] :
( ( member_a @ X @ C4 )
=> ( sup_sup_a_a_o @ P
@ ^ [Y4: a,Z2: a] : ( Y4 = Z2 )
@ X
@ Z ) )
=> ( pred_chain_a @ A4 @ P @ ( sup_sup_set_a @ ( insert_a @ Z @ bot_bot_set_a ) @ C4 ) ) ) ) ) ).
% pred_on.chain_extend
thf(fact_747_pred__on_Ochain__extend,axiom,
! [A4: set_nat,P: nat > nat > $o,C4: set_nat,Z: nat] :
( ( pred_chain_nat @ A4 @ P @ C4 )
=> ( ( member_nat @ Z @ A4 )
=> ( ! [X: nat] :
( ( member_nat @ X @ C4 )
=> ( sup_sup_nat_nat_o @ P
@ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
@ X
@ Z ) )
=> ( pred_chain_nat @ A4 @ P @ ( sup_sup_set_nat @ ( insert_nat @ Z @ bot_bot_set_nat ) @ C4 ) ) ) ) ) ).
% pred_on.chain_extend
thf(fact_748_Kleene__iter__lpfp,axiom,
! [F: set_nat > set_nat,P5: set_nat,K: nat] :
( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F )
=> ( ( ord_less_eq_set_nat @ ( F @ P5 ) @ P5 )
=> ( ord_less_eq_set_nat @ ( compow8708494347934031032et_nat @ K @ F @ bot_bot_set_nat ) @ P5 ) ) ) ).
% Kleene_iter_lpfp
thf(fact_749_Kleene__iter__lpfp,axiom,
! [F: nat > nat,P5: nat,K: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_less_eq_nat @ ( F @ P5 ) @ P5 )
=> ( ord_less_eq_nat @ ( compow_nat_nat @ K @ F @ bot_bot_nat ) @ P5 ) ) ) ).
% Kleene_iter_lpfp
thf(fact_750_funpow__decreasing,axiom,
! [M5: nat,N3: nat,F: set_nat > set_nat] :
( ( ord_less_eq_nat @ M5 @ N3 )
=> ( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F )
=> ( ord_less_eq_set_nat @ ( compow8708494347934031032et_nat @ M5 @ F @ bot_bot_set_nat ) @ ( compow8708494347934031032et_nat @ N3 @ F @ bot_bot_set_nat ) ) ) ) ).
% funpow_decreasing
thf(fact_751_funpow__decreasing,axiom,
! [M5: nat,N3: nat,F: nat > nat] :
( ( ord_less_eq_nat @ M5 @ N3 )
=> ( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ord_less_eq_nat @ ( compow_nat_nat @ M5 @ F @ bot_bot_nat ) @ ( compow_nat_nat @ N3 @ F @ bot_bot_nat ) ) ) ) ).
% funpow_decreasing
thf(fact_752_funpow__increasing,axiom,
! [M5: nat,N3: nat,F: set_nat > set_nat] :
( ( ord_less_eq_nat @ M5 @ N3 )
=> ( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F )
=> ( ord_less_eq_set_nat @ ( compow8708494347934031032et_nat @ N3 @ F @ top_top_set_nat ) @ ( compow8708494347934031032et_nat @ M5 @ F @ top_top_set_nat ) ) ) ) ).
% funpow_increasing
thf(fact_753_strict__mono__imp__increasing,axiom,
! [F: nat > nat,N3: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F )
=> ( ord_less_eq_nat @ N3 @ ( F @ N3 ) ) ) ).
% strict_mono_imp_increasing
thf(fact_754_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_755_le__neq__implies__less,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_eq_nat @ M5 @ N3 )
=> ( ( M5 != N3 )
=> ( ord_less_nat @ M5 @ N3 ) ) ) ).
% le_neq_implies_less
thf(fact_756_less__or__eq__imp__le,axiom,
! [M5: nat,N3: nat] :
( ( ( ord_less_nat @ M5 @ N3 )
| ( M5 = N3 ) )
=> ( ord_less_eq_nat @ M5 @ N3 ) ) ).
% less_or_eq_imp_le
thf(fact_757_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_758_less__imp__le__nat,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_nat @ M5 @ N3 )
=> ( ord_less_eq_nat @ M5 @ N3 ) ) ).
% less_imp_le_nat
thf(fact_759_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
& ( M != N ) ) ) ) ).
% nat_less_le
thf(fact_760_le__refl,axiom,
! [N3: nat] : ( ord_less_eq_nat @ N3 @ N3 ) ).
% le_refl
thf(fact_761_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_762_eq__imp__le,axiom,
! [M5: nat,N3: nat] :
( ( M5 = N3 )
=> ( ord_less_eq_nat @ M5 @ N3 ) ) ).
% eq_imp_le
thf(fact_763_le__antisym,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_eq_nat @ M5 @ N3 )
=> ( ( ord_less_eq_nat @ N3 @ M5 )
=> ( M5 = N3 ) ) ) ).
% le_antisym
thf(fact_764_GreatestI__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% GreatestI_nat
thf(fact_765_nat__le__linear,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_eq_nat @ M5 @ N3 )
| ( ord_less_eq_nat @ N3 @ M5 ) ) ).
% nat_le_linear
thf(fact_766_Greatest__le__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).
% Greatest_le_nat
thf(fact_767_GreatestI__ex__nat,axiom,
! [P: nat > $o,B: nat] :
( ? [X_1: nat] : ( P @ X_1 )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% GreatestI_ex_nat
thf(fact_768_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ? [X: nat] :
( ( P @ X )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_769_inf__shunt,axiom,
! [X2: set_nat,Y3: set_nat] :
( ( ( inf_inf_set_nat @ X2 @ Y3 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ X2 @ ( uminus5710092332889474511et_nat @ Y3 ) ) ) ).
% inf_shunt
thf(fact_770_sup__shunt,axiom,
! [X2: set_nat,Y3: set_nat] :
( ( ( sup_sup_set_nat @ X2 @ Y3 )
= top_top_set_nat )
= ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X2 ) @ Y3 ) ) ).
% sup_shunt
thf(fact_771_gfp__Kleene__iter,axiom,
! [F: set_nat > set_nat,K: nat] :
( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F )
=> ( ( ( compow8708494347934031032et_nat @ ( suc @ K ) @ F @ top_top_set_nat )
= ( compow8708494347934031032et_nat @ K @ F @ top_top_set_nat ) )
=> ( ( comple1596078789208929544et_nat @ F )
= ( compow8708494347934031032et_nat @ K @ F @ top_top_set_nat ) ) ) ) ).
% gfp_Kleene_iter
thf(fact_772_lfp__Kleene__iter,axiom,
! [F: set_nat > set_nat,K: nat] :
( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F )
=> ( ( ( compow8708494347934031032et_nat @ ( suc @ K ) @ F @ bot_bot_set_nat )
= ( compow8708494347934031032et_nat @ K @ F @ bot_bot_set_nat ) )
=> ( ( comple7975543026063415949et_nat @ F )
= ( compow8708494347934031032et_nat @ K @ F @ bot_bot_set_nat ) ) ) ) ).
% lfp_Kleene_iter
thf(fact_773_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_774_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_775_lessI,axiom,
! [N3: nat] : ( ord_less_nat @ N3 @ ( suc @ N3 ) ) ).
% lessI
thf(fact_776_Suc__mono,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_nat @ M5 @ N3 )
=> ( ord_less_nat @ ( suc @ M5 ) @ ( suc @ N3 ) ) ) ).
% Suc_mono
thf(fact_777_Suc__less__eq,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_nat @ ( suc @ M5 ) @ ( suc @ N3 ) )
= ( ord_less_nat @ M5 @ N3 ) ) ).
% Suc_less_eq
thf(fact_778_Suc__le__mono,axiom,
! [N3: nat,M5: nat] :
( ( ord_less_eq_nat @ ( suc @ N3 ) @ ( suc @ M5 ) )
= ( ord_less_eq_nat @ N3 @ M5 ) ) ).
% Suc_le_mono
thf(fact_779_le__imp__less__Suc,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_eq_nat @ M5 @ N3 )
=> ( ord_less_nat @ M5 @ ( suc @ N3 ) ) ) ).
% le_imp_less_Suc
thf(fact_780_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N: nat] : ( ord_less_eq_nat @ ( suc @ N ) ) ) ) ).
% less_eq_Suc_le
thf(fact_781_less__Suc__eq__le,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_nat @ M5 @ ( suc @ N3 ) )
= ( ord_less_eq_nat @ M5 @ N3 ) ) ).
% less_Suc_eq_le
thf(fact_782_le__less__Suc__eq,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_eq_nat @ M5 @ N3 )
=> ( ( ord_less_nat @ N3 @ ( suc @ M5 ) )
= ( N3 = M5 ) ) ) ).
% le_less_Suc_eq
thf(fact_783_Suc__le__lessD,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_eq_nat @ ( suc @ M5 ) @ N3 )
=> ( ord_less_nat @ M5 @ N3 ) ) ).
% Suc_le_lessD
thf(fact_784_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_785_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_786_Suc__le__eq,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_eq_nat @ ( suc @ M5 ) @ N3 )
= ( ord_less_nat @ M5 @ N3 ) ) ).
% Suc_le_eq
thf(fact_787_Suc__leI,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_nat @ M5 @ N3 )
=> ( ord_less_eq_nat @ ( suc @ M5 ) @ N3 ) ) ).
% Suc_leI
thf(fact_788_Suc__leD,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_eq_nat @ ( suc @ M5 ) @ N3 )
=> ( ord_less_eq_nat @ M5 @ N3 ) ) ).
% Suc_leD
thf(fact_789_le__SucE,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_eq_nat @ M5 @ ( suc @ N3 ) )
=> ( ~ ( ord_less_eq_nat @ M5 @ N3 )
=> ( M5
= ( suc @ N3 ) ) ) ) ).
% le_SucE
thf(fact_790_le__SucI,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_eq_nat @ M5 @ N3 )
=> ( ord_less_eq_nat @ M5 @ ( suc @ N3 ) ) ) ).
% le_SucI
thf(fact_791_Suc__le__D,axiom,
! [N3: nat,M7: nat] :
( ( ord_less_eq_nat @ ( suc @ N3 ) @ M7 )
=> ? [M6: nat] :
( M7
= ( suc @ M6 ) ) ) ).
% Suc_le_D
thf(fact_792_le__Suc__eq,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_eq_nat @ M5 @ ( suc @ N3 ) )
= ( ( ord_less_eq_nat @ M5 @ N3 )
| ( M5
= ( suc @ N3 ) ) ) ) ).
% le_Suc_eq
thf(fact_793_Suc__n__not__le__n,axiom,
! [N3: nat] :
~ ( ord_less_eq_nat @ ( suc @ N3 ) @ N3 ) ).
% Suc_n_not_le_n
thf(fact_794_not__less__eq__eq,axiom,
! [M5: nat,N3: nat] :
( ( ~ ( ord_less_eq_nat @ M5 @ N3 ) )
= ( ord_less_eq_nat @ ( suc @ N3 ) @ M5 ) ) ).
% not_less_eq_eq
thf(fact_795_full__nat__induct,axiom,
! [P: nat > $o,N3: nat] :
( ! [N2: nat] :
( ! [M8: nat] :
( ( ord_less_eq_nat @ ( suc @ M8 ) @ N2 )
=> ( P @ M8 ) )
=> ( P @ N2 ) )
=> ( P @ N3 ) ) ).
% full_nat_induct
thf(fact_796_nat__induct__at__least,axiom,
! [M5: nat,N3: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M5 @ N3 )
=> ( ( P @ M5 )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M5 @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N3 ) ) ) ) ).
% nat_induct_at_least
thf(fact_797_transitive__stepwise__le,axiom,
! [M5: nat,N3: nat,R4: nat > nat > $o] :
( ( ord_less_eq_nat @ M5 @ N3 )
=> ( ! [X: nat] : ( R4 @ X @ X )
=> ( ! [X: nat,Y2: nat,Z4: nat] :
( ( R4 @ X @ Y2 )
=> ( ( R4 @ Y2 @ Z4 )
=> ( R4 @ X @ Z4 ) ) )
=> ( ! [N2: nat] : ( R4 @ N2 @ ( suc @ N2 ) )
=> ( R4 @ M5 @ N3 ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_798_mono__Suc,axiom,
monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ suc ).
% mono_Suc
thf(fact_799_n__not__Suc__n,axiom,
! [N3: nat] :
( N3
!= ( suc @ N3 ) ) ).
% n_not_Suc_n
thf(fact_800_Suc__inject,axiom,
! [X2: nat,Y3: nat] :
( ( ( suc @ X2 )
= ( suc @ Y3 ) )
=> ( X2 = Y3 ) ) ).
% Suc_inject
thf(fact_801_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N3: nat,M5: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ ( F @ N3 ) @ ( F @ M5 ) )
= ( ord_less_nat @ N3 @ M5 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_802_lift__Suc__mono__less,axiom,
! [F: nat > nat,N3: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N3 @ N4 )
=> ( ord_less_nat @ ( F @ N3 ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_803_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_804_Suc__lessD,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_nat @ ( suc @ M5 ) @ N3 )
=> ( ord_less_nat @ M5 @ N3 ) ) ).
% Suc_lessD
thf(fact_805_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_806_Suc__lessI,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_nat @ M5 @ N3 )
=> ( ( ( suc @ M5 )
!= N3 )
=> ( ord_less_nat @ ( suc @ M5 ) @ N3 ) ) ) ).
% Suc_lessI
thf(fact_807_less__SucE,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_nat @ M5 @ ( suc @ N3 ) )
=> ( ~ ( ord_less_nat @ M5 @ N3 )
=> ( M5 = N3 ) ) ) ).
% less_SucE
thf(fact_808_less__SucI,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_nat @ M5 @ N3 )
=> ( ord_less_nat @ M5 @ ( suc @ N3 ) ) ) ).
% less_SucI
thf(fact_809_Ex__less__Suc,axiom,
! [N3: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N3 ) )
& ( P @ I3 ) ) )
= ( ( P @ N3 )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N3 )
& ( P @ I3 ) ) ) ) ).
% Ex_less_Suc
thf(fact_810_less__Suc__eq,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_nat @ M5 @ ( suc @ N3 ) )
= ( ( ord_less_nat @ M5 @ N3 )
| ( M5 = N3 ) ) ) ).
% less_Suc_eq
thf(fact_811_nat__neq__iff,axiom,
! [M5: nat,N3: nat] :
( ( M5 != N3 )
= ( ( ord_less_nat @ M5 @ N3 )
| ( ord_less_nat @ N3 @ M5 ) ) ) ).
% nat_neq_iff
thf(fact_812_not__less__eq,axiom,
! [M5: nat,N3: nat] :
( ( ~ ( ord_less_nat @ M5 @ N3 ) )
= ( ord_less_nat @ N3 @ ( suc @ M5 ) ) ) ).
% not_less_eq
thf(fact_813_All__less__Suc,axiom,
! [N3: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N3 ) )
=> ( P @ I3 ) ) )
= ( ( P @ N3 )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N3 )
=> ( P @ I3 ) ) ) ) ).
% All_less_Suc
thf(fact_814_Suc__less__eq2,axiom,
! [N3: nat,M5: nat] :
( ( ord_less_nat @ ( suc @ N3 ) @ M5 )
= ( ? [M9: nat] :
( ( M5
= ( suc @ M9 ) )
& ( ord_less_nat @ N3 @ M9 ) ) ) ) ).
% Suc_less_eq2
thf(fact_815_less__antisym,axiom,
! [N3: nat,M5: nat] :
( ~ ( ord_less_nat @ N3 @ M5 )
=> ( ( ord_less_nat @ N3 @ ( suc @ M5 ) )
=> ( M5 = N3 ) ) ) ).
% less_antisym
thf(fact_816_Suc__less__SucD,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_nat @ ( suc @ M5 ) @ ( suc @ N3 ) )
=> ( ord_less_nat @ M5 @ N3 ) ) ).
% Suc_less_SucD
thf(fact_817_less__not__refl,axiom,
! [N3: nat] :
~ ( ord_less_nat @ N3 @ N3 ) ).
% less_not_refl
thf(fact_818_less__not__refl2,axiom,
! [N3: nat,M5: nat] :
( ( ord_less_nat @ N3 @ M5 )
=> ( M5 != N3 ) ) ).
% less_not_refl2
thf(fact_819_less__not__refl3,axiom,
! [S3: nat,T: nat] :
( ( ord_less_nat @ S3 @ T )
=> ( S3 != T ) ) ).
% less_not_refl3
thf(fact_820_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_821_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P @ I2 @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I2 @ K2 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_822_less__irrefl__nat,axiom,
! [N3: nat] :
~ ( ord_less_nat @ N3 @ N3 ) ).
% less_irrefl_nat
thf(fact_823_nat__less__induct,axiom,
! [P: nat > $o,N3: nat] :
( ! [N2: nat] :
( ! [M8: nat] :
( ( ord_less_nat @ M8 @ N2 )
=> ( P @ M8 ) )
=> ( P @ N2 ) )
=> ( P @ N3 ) ) ).
% nat_less_induct
thf(fact_824_infinite__descent,axiom,
! [P: nat > $o,N3: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M8: nat] :
( ( ord_less_nat @ M8 @ N2 )
& ~ ( P @ M8 ) ) )
=> ( P @ N3 ) ) ).
% infinite_descent
thf(fact_825_linorder__neqE__nat,axiom,
! [X2: nat,Y3: nat] :
( ( X2 != Y3 )
=> ( ~ ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ Y3 @ X2 ) ) ) ).
% linorder_neqE_nat
thf(fact_826_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] :
( ( J
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_827_not__less__less__Suc__eq,axiom,
! [N3: nat,M5: nat] :
( ~ ( ord_less_nat @ N3 @ M5 )
=> ( ( ord_less_nat @ N3 @ ( suc @ M5 ) )
= ( N3 = M5 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_828_infinite__growing,axiom,
! [X6: set_a] :
( ( X6 != bot_bot_set_a )
=> ( ! [X: a] :
( ( member_a @ X @ X6 )
=> ? [Xa: a] :
( ( member_a @ Xa @ X6 )
& ( ord_less_a @ X @ Xa ) ) )
=> ~ ( finite_finite_a @ X6 ) ) ) ).
% infinite_growing
thf(fact_829_infinite__growing,axiom,
! [X6: set_nat] :
( ( X6 != bot_bot_set_nat )
=> ( ! [X: nat] :
( ( member_nat @ X @ X6 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ X6 )
& ( ord_less_nat @ X @ Xa ) ) )
=> ~ ( finite_finite_nat @ X6 ) ) ) ).
% infinite_growing
thf(fact_830_ex__min__if__finite,axiom,
! [S2: set_nat] :
( ( finite_finite_nat @ S2 )
=> ( ( S2 != bot_bot_set_nat )
=> ? [X: nat] :
( ( member_nat @ X @ S2 )
& ~ ? [Xa: nat] :
( ( member_nat @ Xa @ S2 )
& ( ord_less_nat @ Xa @ X ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_831_le__cSup__finite,axiom,
! [X6: set_nat,X2: nat] :
( ( finite_finite_nat @ X6 )
=> ( ( member_nat @ X2 @ X6 )
=> ( ord_less_eq_nat @ X2 @ ( complete_Sup_Sup_nat @ X6 ) ) ) ) ).
% le_cSup_finite
thf(fact_832_lift__Suc__antimono__le,axiom,
! [F: nat > a,N3: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_a @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N3 @ N4 )
=> ( ord_less_eq_a @ ( F @ N4 ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_833_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N3: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N3 @ N4 )
=> ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_834_lift__Suc__mono__le,axiom,
! [F: nat > a,N3: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_a @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N3 @ N4 )
=> ( ord_less_eq_a @ ( F @ N3 ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_835_lift__Suc__mono__le,axiom,
! [F: nat > nat,N3: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N3 @ N4 )
=> ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_836_mono__iff__le__Suc,axiom,
! [F: nat > a] :
( ( monotone_on_nat_a @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_a @ F )
= ( ! [N: nat] : ( ord_less_eq_a @ ( F @ N ) @ ( F @ ( suc @ N ) ) ) ) ) ).
% mono_iff_le_Suc
thf(fact_837_mono__iff__le__Suc,axiom,
! [F: nat > nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
= ( ! [N: nat] : ( ord_less_eq_nat @ ( F @ N ) @ ( F @ ( suc @ N ) ) ) ) ) ).
% mono_iff_le_Suc
thf(fact_838_finite__imp__Sup__less,axiom,
! [X6: set_nat,X2: nat,A: nat] :
( ( finite_finite_nat @ X6 )
=> ( ( member_nat @ X2 @ X6 )
=> ( ! [X: nat] :
( ( member_nat @ X @ X6 )
=> ( ord_less_nat @ X @ A ) )
=> ( ord_less_nat @ ( complete_Sup_Sup_nat @ X6 ) @ A ) ) ) ) ).
% finite_imp_Sup_less
thf(fact_839_cInf__le__finite,axiom,
! [X6: set_nat,X2: nat] :
( ( finite_finite_nat @ X6 )
=> ( ( member_nat @ X2 @ X6 )
=> ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ X6 ) @ X2 ) ) ) ).
% cInf_le_finite
thf(fact_840_finite__imp__less__Inf,axiom,
! [X6: set_nat,X2: nat,A: nat] :
( ( finite_finite_nat @ X6 )
=> ( ( member_nat @ X2 @ X6 )
=> ( ! [X: nat] :
( ( member_nat @ X @ X6 )
=> ( ord_less_nat @ A @ X ) )
=> ( ord_less_nat @ A @ ( complete_Inf_Inf_nat @ X6 ) ) ) ) ) ).
% finite_imp_less_Inf
thf(fact_841_finite__ranking__induct,axiom,
! [S2: set_a,P: set_a > $o,F: a > a] :
( ( finite_finite_a @ S2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X: a,S6: set_a] :
( ( finite_finite_a @ S6 )
=> ( ! [Y5: a] :
( ( member_a @ Y5 @ S6 )
=> ( ord_less_eq_a @ ( F @ Y5 ) @ ( F @ X ) ) )
=> ( ( P @ S6 )
=> ( P @ ( insert_a @ X @ S6 ) ) ) ) )
=> ( P @ S2 ) ) ) ) ).
% finite_ranking_induct
thf(fact_842_finite__ranking__induct,axiom,
! [S2: set_nat,P: set_nat > $o,F: nat > a] :
( ( finite_finite_nat @ S2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X: nat,S6: set_nat] :
( ( finite_finite_nat @ S6 )
=> ( ! [Y5: nat] :
( ( member_nat @ Y5 @ S6 )
=> ( ord_less_eq_a @ ( F @ Y5 ) @ ( F @ X ) ) )
=> ( ( P @ S6 )
=> ( P @ ( insert_nat @ X @ S6 ) ) ) ) )
=> ( P @ S2 ) ) ) ) ).
% finite_ranking_induct
thf(fact_843_finite__ranking__induct,axiom,
! [S2: set_a,P: set_a > $o,F: a > nat] :
( ( finite_finite_a @ S2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X: a,S6: set_a] :
( ( finite_finite_a @ S6 )
=> ( ! [Y5: a] :
( ( member_a @ Y5 @ S6 )
=> ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X ) ) )
=> ( ( P @ S6 )
=> ( P @ ( insert_a @ X @ S6 ) ) ) ) )
=> ( P @ S2 ) ) ) ) ).
% finite_ranking_induct
thf(fact_844_finite__ranking__induct,axiom,
! [S2: set_nat,P: set_nat > $o,F: nat > nat] :
( ( finite_finite_nat @ S2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X: nat,S6: set_nat] :
( ( finite_finite_nat @ S6 )
=> ( ! [Y5: nat] :
( ( member_nat @ Y5 @ S6 )
=> ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X ) ) )
=> ( ( P @ S6 )
=> ( P @ ( insert_nat @ X @ S6 ) ) ) ) )
=> ( P @ S2 ) ) ) ) ).
% finite_ranking_induct
thf(fact_845_finite__linorder__max__induct,axiom,
! [A4: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A4 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [B3: nat,A7: set_nat] :
( ( finite_finite_nat @ A7 )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A7 )
=> ( ord_less_nat @ X5 @ B3 ) )
=> ( ( P @ A7 )
=> ( P @ ( insert_nat @ B3 @ A7 ) ) ) ) )
=> ( P @ A4 ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_846_finite__linorder__min__induct,axiom,
! [A4: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A4 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [B3: nat,A7: set_nat] :
( ( finite_finite_nat @ A7 )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A7 )
=> ( ord_less_nat @ B3 @ X5 ) )
=> ( ( P @ A7 )
=> ( P @ ( insert_nat @ B3 @ A7 ) ) ) ) )
=> ( P @ A4 ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_847_finite__Sup__less__iff,axiom,
! [X6: set_nat,A: nat] :
( ( finite_finite_nat @ X6 )
=> ( ( X6 != bot_bot_set_nat )
=> ( ( ord_less_nat @ ( complete_Sup_Sup_nat @ X6 ) @ A )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ X6 )
=> ( ord_less_nat @ X3 @ A ) ) ) ) ) ) ).
% finite_Sup_less_iff
thf(fact_848_finite__less__Inf__iff,axiom,
! [X6: set_nat,A: nat] :
( ( finite_finite_nat @ X6 )
=> ( ( X6 != bot_bot_set_nat )
=> ( ( ord_less_nat @ A @ ( complete_Inf_Inf_nat @ X6 ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ X6 )
=> ( ord_less_nat @ A @ X3 ) ) ) ) ) ) ).
% finite_less_Inf_iff
thf(fact_849_finite__mono__remains__stable__implies__strict__prefix,axiom,
! [F: nat > a] :
( ( finite_finite_a @ ( image_nat_a @ F @ top_top_set_nat ) )
=> ( ( monotone_on_nat_a @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_a @ F )
=> ( ! [N2: nat] :
( ( ( F @ N2 )
= ( F @ ( suc @ N2 ) ) )
=> ( ( F @ ( suc @ N2 ) )
= ( F @ ( suc @ ( suc @ N2 ) ) ) ) )
=> ? [N5: nat] :
( ! [N6: nat] :
( ( ord_less_eq_nat @ N6 @ N5 )
=> ! [M8: nat] :
( ( ord_less_eq_nat @ M8 @ N5 )
=> ( ( ord_less_nat @ M8 @ N6 )
=> ( ord_less_a @ ( F @ M8 ) @ ( F @ N6 ) ) ) ) )
& ! [N6: nat] :
( ( ord_less_eq_nat @ N5 @ N6 )
=> ( ( F @ N5 )
= ( F @ N6 ) ) ) ) ) ) ) ).
% finite_mono_remains_stable_implies_strict_prefix
thf(fact_850_finite__mono__remains__stable__implies__strict__prefix,axiom,
! [F: nat > nat] :
( ( finite_finite_nat @ ( image_nat_nat @ F @ top_top_set_nat ) )
=> ( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ! [N2: nat] :
( ( ( F @ N2 )
= ( F @ ( suc @ N2 ) ) )
=> ( ( F @ ( suc @ N2 ) )
= ( F @ ( suc @ ( suc @ N2 ) ) ) ) )
=> ? [N5: nat] :
( ! [N6: nat] :
( ( ord_less_eq_nat @ N6 @ N5 )
=> ! [M8: nat] :
( ( ord_less_eq_nat @ M8 @ N5 )
=> ( ( ord_less_nat @ M8 @ N6 )
=> ( ord_less_nat @ ( F @ M8 ) @ ( F @ N6 ) ) ) ) )
& ! [N6: nat] :
( ( ord_less_eq_nat @ N5 @ N6 )
=> ( ( F @ N5 )
= ( F @ N6 ) ) ) ) ) ) ) ).
% finite_mono_remains_stable_implies_strict_prefix
thf(fact_851_finite__has__minimal,axiom,
! [A4: set_a] :
( ( finite_finite_a @ A4 )
=> ( ( A4 != bot_bot_set_a )
=> ? [X: a] :
( ( member_a @ X @ A4 )
& ! [Xa: a] :
( ( member_a @ Xa @ A4 )
=> ( ( ord_less_eq_a @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_852_finite__has__minimal,axiom,
! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ? [X: nat] :
( ( member_nat @ X @ A4 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A4 )
=> ( ( ord_less_eq_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_853_finite__has__maximal2,axiom,
! [A4: set_a,A: a] :
( ( finite_finite_a @ A4 )
=> ( ( member_a @ A @ A4 )
=> ? [X: a] :
( ( member_a @ X @ A4 )
& ( ord_less_eq_a @ A @ X )
& ! [Xa: a] :
( ( member_a @ Xa @ A4 )
=> ( ( ord_less_eq_a @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_854_finite__has__maximal2,axiom,
! [A4: set_nat,A: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ A @ A4 )
=> ? [X: nat] :
( ( member_nat @ X @ A4 )
& ( ord_less_eq_nat @ A @ X )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A4 )
=> ( ( ord_less_eq_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_855_finite__has__minimal2,axiom,
! [A4: set_a,A: a] :
( ( finite_finite_a @ A4 )
=> ( ( member_a @ A @ A4 )
=> ? [X: a] :
( ( member_a @ X @ A4 )
& ( ord_less_eq_a @ X @ A )
& ! [Xa: a] :
( ( member_a @ Xa @ A4 )
=> ( ( ord_less_eq_a @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_856_finite__has__minimal2,axiom,
! [A4: set_nat,A: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ A @ A4 )
=> ? [X: nat] :
( ( member_nat @ X @ A4 )
& ( ord_less_eq_nat @ X @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A4 )
=> ( ( ord_less_eq_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_857_finite__has__maximal,axiom,
! [A4: set_a] :
( ( finite_finite_a @ A4 )
=> ( ( A4 != bot_bot_set_a )
=> ? [X: a] :
( ( member_a @ X @ A4 )
& ! [Xa: a] :
( ( member_a @ Xa @ A4 )
=> ( ( ord_less_eq_a @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_858_finite__has__maximal,axiom,
! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ? [X: nat] :
( ( member_nat @ X @ A4 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A4 )
=> ( ( ord_less_eq_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_859_arg__min__if__finite_I2_J,axiom,
! [S2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ S2 )
=> ( ( S2 != bot_bot_set_nat )
=> ~ ? [X5: nat] :
( ( member_nat @ X5 @ S2 )
& ( ord_less_nat @ ( F @ X5 ) @ ( F @ ( lattic7446932960582359483at_nat @ F @ S2 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_860_arg__min__least,axiom,
! [S2: set_a,Y3: a,F: a > a] :
( ( finite_finite_a @ S2 )
=> ( ( S2 != bot_bot_set_a )
=> ( ( member_a @ Y3 @ S2 )
=> ( ord_less_eq_a @ ( F @ ( lattic3288624042836100505on_a_a @ F @ S2 ) ) @ ( F @ Y3 ) ) ) ) ) ).
% arg_min_least
thf(fact_861_arg__min__least,axiom,
! [S2: set_nat,Y3: nat,F: nat > a] :
( ( finite_finite_nat @ S2 )
=> ( ( S2 != bot_bot_set_nat )
=> ( ( member_nat @ Y3 @ S2 )
=> ( ord_less_eq_a @ ( F @ ( lattic1148846883994911187_nat_a @ F @ S2 ) ) @ ( F @ Y3 ) ) ) ) ) ).
% arg_min_least
thf(fact_862_arg__min__least,axiom,
! [S2: set_a,Y3: a,F: a > nat] :
( ( finite_finite_a @ S2 )
=> ( ( S2 != bot_bot_set_a )
=> ( ( member_a @ Y3 @ S2 )
=> ( ord_less_eq_nat @ ( F @ ( lattic6340287419671400565_a_nat @ F @ S2 ) ) @ ( F @ Y3 ) ) ) ) ) ).
% arg_min_least
thf(fact_863_arg__min__least,axiom,
! [S2: set_nat,Y3: nat,F: nat > nat] :
( ( finite_finite_nat @ S2 )
=> ( ( S2 != bot_bot_set_nat )
=> ( ( member_nat @ Y3 @ S2 )
=> ( ord_less_eq_nat @ ( F @ ( lattic7446932960582359483at_nat @ F @ S2 ) ) @ ( F @ Y3 ) ) ) ) ) ).
% arg_min_least
thf(fact_864_mono__Max__commute,axiom,
! [F: a > a,A4: set_a] :
( ( monotone_on_a_a @ top_top_set_a @ ord_less_eq_a @ ord_less_eq_a @ F )
=> ( ( finite_finite_a @ A4 )
=> ( ( A4 != bot_bot_set_a )
=> ( ( F @ ( lattic6529028001545966829_Max_a @ A4 ) )
= ( lattic6529028001545966829_Max_a @ ( image_a_a @ F @ A4 ) ) ) ) ) ) ).
% mono_Max_commute
thf(fact_865_mono__Max__commute,axiom,
! [F: a > nat,A4: set_a] :
( ( monotone_on_a_nat @ top_top_set_a @ ord_less_eq_a @ ord_less_eq_nat @ F )
=> ( ( finite_finite_a @ A4 )
=> ( ( A4 != bot_bot_set_a )
=> ( ( F @ ( lattic6529028001545966829_Max_a @ A4 ) )
= ( lattic8265883725875713057ax_nat @ ( image_a_nat @ F @ A4 ) ) ) ) ) ) ).
% mono_Max_commute
thf(fact_866_mono__Max__commute,axiom,
! [F: nat > a,A4: set_nat] :
( ( monotone_on_nat_a @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_a @ F )
=> ( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( F @ ( lattic8265883725875713057ax_nat @ A4 ) )
= ( lattic6529028001545966829_Max_a @ ( image_nat_a @ F @ A4 ) ) ) ) ) ) ).
% mono_Max_commute
thf(fact_867_mono__Max__commute,axiom,
! [F: nat > nat,A4: set_nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( F @ ( lattic8265883725875713057ax_nat @ A4 ) )
= ( lattic8265883725875713057ax_nat @ ( image_nat_nat @ F @ A4 ) ) ) ) ) ) ).
% mono_Max_commute
thf(fact_868_Max__singleton,axiom,
! [X2: nat] :
( ( lattic8265883725875713057ax_nat @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
= X2 ) ).
% Max_singleton
thf(fact_869_Max_Obounded__iff,axiom,
! [A4: set_a,X2: a] :
( ( finite_finite_a @ A4 )
=> ( ( A4 != bot_bot_set_a )
=> ( ( ord_less_eq_a @ ( lattic6529028001545966829_Max_a @ A4 ) @ X2 )
= ( ! [X3: a] :
( ( member_a @ X3 @ A4 )
=> ( ord_less_eq_a @ X3 @ X2 ) ) ) ) ) ) ).
% Max.bounded_iff
thf(fact_870_Max_Obounded__iff,axiom,
! [A4: set_nat,X2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A4 ) @ X2 )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( ord_less_eq_nat @ X3 @ X2 ) ) ) ) ) ) ).
% Max.bounded_iff
thf(fact_871_Max__less__iff,axiom,
! [A4: set_nat,X2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( ord_less_nat @ ( lattic8265883725875713057ax_nat @ A4 ) @ X2 )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( ord_less_nat @ X3 @ X2 ) ) ) ) ) ) ).
% Max_less_iff
thf(fact_872_Max_OcoboundedI,axiom,
! [A4: set_a,A: a] :
( ( finite_finite_a @ A4 )
=> ( ( member_a @ A @ A4 )
=> ( ord_less_eq_a @ A @ ( lattic6529028001545966829_Max_a @ A4 ) ) ) ) ).
% Max.coboundedI
thf(fact_873_Max_OcoboundedI,axiom,
! [A4: set_nat,A: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ A @ A4 )
=> ( ord_less_eq_nat @ A @ ( lattic8265883725875713057ax_nat @ A4 ) ) ) ) ).
% Max.coboundedI
thf(fact_874_Max__eq__if,axiom,
! [A4: set_a,B6: set_a] :
( ( finite_finite_a @ A4 )
=> ( ( finite_finite_a @ B6 )
=> ( ! [X: a] :
( ( member_a @ X @ A4 )
=> ? [Xa: a] :
( ( member_a @ Xa @ B6 )
& ( ord_less_eq_a @ X @ Xa ) ) )
=> ( ! [X: a] :
( ( member_a @ X @ B6 )
=> ? [Xa: a] :
( ( member_a @ Xa @ A4 )
& ( ord_less_eq_a @ X @ Xa ) ) )
=> ( ( lattic6529028001545966829_Max_a @ A4 )
= ( lattic6529028001545966829_Max_a @ B6 ) ) ) ) ) ) ).
% Max_eq_if
thf(fact_875_Max__eq__if,axiom,
! [A4: set_nat,B6: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( finite_finite_nat @ B6 )
=> ( ! [X: nat] :
( ( member_nat @ X @ A4 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B6 )
& ( ord_less_eq_nat @ X @ Xa ) ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ B6 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ A4 )
& ( ord_less_eq_nat @ X @ Xa ) ) )
=> ( ( lattic8265883725875713057ax_nat @ A4 )
= ( lattic8265883725875713057ax_nat @ B6 ) ) ) ) ) ) ).
% Max_eq_if
thf(fact_876_Max__eqI,axiom,
! [A4: set_a,X2: a] :
( ( finite_finite_a @ A4 )
=> ( ! [Y2: a] :
( ( member_a @ Y2 @ A4 )
=> ( ord_less_eq_a @ Y2 @ X2 ) )
=> ( ( member_a @ X2 @ A4 )
=> ( ( lattic6529028001545966829_Max_a @ A4 )
= X2 ) ) ) ) ).
% Max_eqI
thf(fact_877_Max__eqI,axiom,
! [A4: set_nat,X2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ! [Y2: nat] :
( ( member_nat @ Y2 @ A4 )
=> ( ord_less_eq_nat @ Y2 @ X2 ) )
=> ( ( member_nat @ X2 @ A4 )
=> ( ( lattic8265883725875713057ax_nat @ A4 )
= X2 ) ) ) ) ).
% Max_eqI
thf(fact_878_Max__ge,axiom,
! [A4: set_a,X2: a] :
( ( finite_finite_a @ A4 )
=> ( ( member_a @ X2 @ A4 )
=> ( ord_less_eq_a @ X2 @ ( lattic6529028001545966829_Max_a @ A4 ) ) ) ) ).
% Max_ge
thf(fact_879_Max__ge,axiom,
! [A4: set_nat,X2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ X2 @ A4 )
=> ( ord_less_eq_nat @ X2 @ ( lattic8265883725875713057ax_nat @ A4 ) ) ) ) ).
% Max_ge
thf(fact_880_Max__in,axiom,
! [A4: set_a] :
( ( finite_finite_a @ A4 )
=> ( ( A4 != bot_bot_set_a )
=> ( member_a @ ( lattic6529028001545966829_Max_a @ A4 ) @ A4 ) ) ) ).
% Max_in
thf(fact_881_Max__in,axiom,
! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( member_nat @ ( lattic8265883725875713057ax_nat @ A4 ) @ A4 ) ) ) ).
% Max_in
thf(fact_882_Max__eq__iff,axiom,
! [A4: set_a,M5: a] :
( ( finite_finite_a @ A4 )
=> ( ( A4 != bot_bot_set_a )
=> ( ( ( lattic6529028001545966829_Max_a @ A4 )
= M5 )
= ( ( member_a @ M5 @ A4 )
& ! [X3: a] :
( ( member_a @ X3 @ A4 )
=> ( ord_less_eq_a @ X3 @ M5 ) ) ) ) ) ) ).
% Max_eq_iff
thf(fact_883_Max__eq__iff,axiom,
! [A4: set_nat,M5: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( ( lattic8265883725875713057ax_nat @ A4 )
= M5 )
= ( ( member_nat @ M5 @ A4 )
& ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( ord_less_eq_nat @ X3 @ M5 ) ) ) ) ) ) ).
% Max_eq_iff
thf(fact_884_Max__ge__iff,axiom,
! [A4: set_a,X2: a] :
( ( finite_finite_a @ A4 )
=> ( ( A4 != bot_bot_set_a )
=> ( ( ord_less_eq_a @ X2 @ ( lattic6529028001545966829_Max_a @ A4 ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ A4 )
& ( ord_less_eq_a @ X2 @ X3 ) ) ) ) ) ) ).
% Max_ge_iff
thf(fact_885_Max__ge__iff,axiom,
! [A4: set_nat,X2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X2 @ ( lattic8265883725875713057ax_nat @ A4 ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A4 )
& ( ord_less_eq_nat @ X2 @ X3 ) ) ) ) ) ) ).
% Max_ge_iff
thf(fact_886_eq__Max__iff,axiom,
! [A4: set_a,M5: a] :
( ( finite_finite_a @ A4 )
=> ( ( A4 != bot_bot_set_a )
=> ( ( M5
= ( lattic6529028001545966829_Max_a @ A4 ) )
= ( ( member_a @ M5 @ A4 )
& ! [X3: a] :
( ( member_a @ X3 @ A4 )
=> ( ord_less_eq_a @ X3 @ M5 ) ) ) ) ) ) ).
% eq_Max_iff
thf(fact_887_eq__Max__iff,axiom,
! [A4: set_nat,M5: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( M5
= ( lattic8265883725875713057ax_nat @ A4 ) )
= ( ( member_nat @ M5 @ A4 )
& ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( ord_less_eq_nat @ X3 @ M5 ) ) ) ) ) ) ).
% eq_Max_iff
thf(fact_888_Max_OboundedE,axiom,
! [A4: set_a,X2: a] :
( ( finite_finite_a @ A4 )
=> ( ( A4 != bot_bot_set_a )
=> ( ( ord_less_eq_a @ ( lattic6529028001545966829_Max_a @ A4 ) @ X2 )
=> ! [A8: a] :
( ( member_a @ A8 @ A4 )
=> ( ord_less_eq_a @ A8 @ X2 ) ) ) ) ) ).
% Max.boundedE
thf(fact_889_Max_OboundedE,axiom,
! [A4: set_nat,X2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A4 ) @ X2 )
=> ! [A8: nat] :
( ( member_nat @ A8 @ A4 )
=> ( ord_less_eq_nat @ A8 @ X2 ) ) ) ) ) ).
% Max.boundedE
thf(fact_890_Max_OboundedI,axiom,
! [A4: set_a,X2: a] :
( ( finite_finite_a @ A4 )
=> ( ( A4 != bot_bot_set_a )
=> ( ! [A3: a] :
( ( member_a @ A3 @ A4 )
=> ( ord_less_eq_a @ A3 @ X2 ) )
=> ( ord_less_eq_a @ ( lattic6529028001545966829_Max_a @ A4 ) @ X2 ) ) ) ) ).
% Max.boundedI
thf(fact_891_Max_OboundedI,axiom,
! [A4: set_nat,X2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A4 )
=> ( ord_less_eq_nat @ A3 @ X2 ) )
=> ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A4 ) @ X2 ) ) ) ) ).
% Max.boundedI
thf(fact_892_Max__gr__iff,axiom,
! [A4: set_nat,X2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( ord_less_nat @ X2 @ ( lattic8265883725875713057ax_nat @ A4 ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A4 )
& ( ord_less_nat @ X2 @ X3 ) ) ) ) ) ) ).
% Max_gr_iff
thf(fact_893_Max__insert2,axiom,
! [A4: set_a,A: a] :
( ( finite_finite_a @ A4 )
=> ( ! [B3: a] :
( ( member_a @ B3 @ A4 )
=> ( ord_less_eq_a @ B3 @ A ) )
=> ( ( lattic6529028001545966829_Max_a @ ( insert_a @ A @ A4 ) )
= A ) ) ) ).
% Max_insert2
thf(fact_894_Max__insert2,axiom,
! [A4: set_nat,A: nat] :
( ( finite_finite_nat @ A4 )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ A4 )
=> ( ord_less_eq_nat @ B3 @ A ) )
=> ( ( lattic8265883725875713057ax_nat @ ( insert_nat @ A @ A4 ) )
= A ) ) ) ).
% Max_insert2
thf(fact_895_cSup__eq__Max,axiom,
! [X6: set_nat] :
( ( finite_finite_nat @ X6 )
=> ( ( X6 != bot_bot_set_nat )
=> ( ( complete_Sup_Sup_nat @ X6 )
= ( lattic8265883725875713057ax_nat @ X6 ) ) ) ) ).
% cSup_eq_Max
thf(fact_896_Max__mono,axiom,
! [M2: set_a,N7: set_a] :
( ( ord_less_eq_set_a @ M2 @ N7 )
=> ( ( M2 != bot_bot_set_a )
=> ( ( finite_finite_a @ N7 )
=> ( ord_less_eq_a @ ( lattic6529028001545966829_Max_a @ M2 ) @ ( lattic6529028001545966829_Max_a @ N7 ) ) ) ) ) ).
% Max_mono
thf(fact_897_Max__mono,axiom,
! [M2: set_nat,N7: set_nat] :
( ( ord_less_eq_set_nat @ M2 @ N7 )
=> ( ( M2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ N7 )
=> ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ M2 ) @ ( lattic8265883725875713057ax_nat @ N7 ) ) ) ) ) ).
% Max_mono
thf(fact_898_Max_Osubset__imp,axiom,
! [A4: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A4 @ B6 )
=> ( ( A4 != bot_bot_set_a )
=> ( ( finite_finite_a @ B6 )
=> ( ord_less_eq_a @ ( lattic6529028001545966829_Max_a @ A4 ) @ ( lattic6529028001545966829_Max_a @ B6 ) ) ) ) ) ).
% Max.subset_imp
thf(fact_899_Max_Osubset__imp,axiom,
! [A4: set_nat,B6: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B6 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B6 )
=> ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A4 ) @ ( lattic8265883725875713057ax_nat @ B6 ) ) ) ) ) ).
% Max.subset_imp
thf(fact_900_mono__Min__commute,axiom,
! [F: a > a,A4: set_a] :
( ( monotone_on_a_a @ top_top_set_a @ ord_less_eq_a @ ord_less_eq_a @ F )
=> ( ( finite_finite_a @ A4 )
=> ( ( A4 != bot_bot_set_a )
=> ( ( F @ ( lattic8372110310405614207_Min_a @ A4 ) )
= ( lattic8372110310405614207_Min_a @ ( image_a_a @ F @ A4 ) ) ) ) ) ) ).
% mono_Min_commute
thf(fact_901_mono__Min__commute,axiom,
! [F: a > nat,A4: set_a] :
( ( monotone_on_a_nat @ top_top_set_a @ ord_less_eq_a @ ord_less_eq_nat @ F )
=> ( ( finite_finite_a @ A4 )
=> ( ( A4 != bot_bot_set_a )
=> ( ( F @ ( lattic8372110310405614207_Min_a @ A4 ) )
= ( lattic8721135487736765967in_nat @ ( image_a_nat @ F @ A4 ) ) ) ) ) ) ).
% mono_Min_commute
thf(fact_902_mono__Min__commute,axiom,
! [F: nat > a,A4: set_nat] :
( ( monotone_on_nat_a @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_a @ F )
=> ( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( F @ ( lattic8721135487736765967in_nat @ A4 ) )
= ( lattic8372110310405614207_Min_a @ ( image_nat_a @ F @ A4 ) ) ) ) ) ) ).
% mono_Min_commute
thf(fact_903_mono__Min__commute,axiom,
! [F: nat > nat,A4: set_nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( F @ ( lattic8721135487736765967in_nat @ A4 ) )
= ( lattic8721135487736765967in_nat @ ( image_nat_nat @ F @ A4 ) ) ) ) ) ) ).
% mono_Min_commute
thf(fact_904_Min__singleton,axiom,
! [X2: nat] :
( ( lattic8721135487736765967in_nat @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
= X2 ) ).
% Min_singleton
thf(fact_905_Min_Obounded__iff,axiom,
! [A4: set_a,X2: a] :
( ( finite_finite_a @ A4 )
=> ( ( A4 != bot_bot_set_a )
=> ( ( ord_less_eq_a @ X2 @ ( lattic8372110310405614207_Min_a @ A4 ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ A4 )
=> ( ord_less_eq_a @ X2 @ X3 ) ) ) ) ) ) ).
% Min.bounded_iff
thf(fact_906_Min_Obounded__iff,axiom,
! [A4: set_nat,X2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X2 @ ( lattic8721135487736765967in_nat @ A4 ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( ord_less_eq_nat @ X2 @ X3 ) ) ) ) ) ) ).
% Min.bounded_iff
thf(fact_907_Min__gr__iff,axiom,
! [A4: set_nat,X2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( ord_less_nat @ X2 @ ( lattic8721135487736765967in_nat @ A4 ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( ord_less_nat @ X2 @ X3 ) ) ) ) ) ) ).
% Min_gr_iff
thf(fact_908_Min__le,axiom,
! [A4: set_a,X2: a] :
( ( finite_finite_a @ A4 )
=> ( ( member_a @ X2 @ A4 )
=> ( ord_less_eq_a @ ( lattic8372110310405614207_Min_a @ A4 ) @ X2 ) ) ) ).
% Min_le
thf(fact_909_Min__le,axiom,
! [A4: set_nat,X2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ X2 @ A4 )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ A4 ) @ X2 ) ) ) ).
% Min_le
thf(fact_910_Min__eqI,axiom,
! [A4: set_a,X2: a] :
( ( finite_finite_a @ A4 )
=> ( ! [Y2: a] :
( ( member_a @ Y2 @ A4 )
=> ( ord_less_eq_a @ X2 @ Y2 ) )
=> ( ( member_a @ X2 @ A4 )
=> ( ( lattic8372110310405614207_Min_a @ A4 )
= X2 ) ) ) ) ).
% Min_eqI
thf(fact_911_Min__eqI,axiom,
! [A4: set_nat,X2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ! [Y2: nat] :
( ( member_nat @ Y2 @ A4 )
=> ( ord_less_eq_nat @ X2 @ Y2 ) )
=> ( ( member_nat @ X2 @ A4 )
=> ( ( lattic8721135487736765967in_nat @ A4 )
= X2 ) ) ) ) ).
% Min_eqI
thf(fact_912_Min_OcoboundedI,axiom,
! [A4: set_a,A: a] :
( ( finite_finite_a @ A4 )
=> ( ( member_a @ A @ A4 )
=> ( ord_less_eq_a @ ( lattic8372110310405614207_Min_a @ A4 ) @ A ) ) ) ).
% Min.coboundedI
thf(fact_913_Min_OcoboundedI,axiom,
! [A4: set_nat,A: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ A @ A4 )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ A4 ) @ A ) ) ) ).
% Min.coboundedI
thf(fact_914_Min__in,axiom,
! [A4: set_a] :
( ( finite_finite_a @ A4 )
=> ( ( A4 != bot_bot_set_a )
=> ( member_a @ ( lattic8372110310405614207_Min_a @ A4 ) @ A4 ) ) ) ).
% Min_in
thf(fact_915_Min__in,axiom,
! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( member_nat @ ( lattic8721135487736765967in_nat @ A4 ) @ A4 ) ) ) ).
% Min_in
thf(fact_916_Min_OboundedI,axiom,
! [A4: set_a,X2: a] :
( ( finite_finite_a @ A4 )
=> ( ( A4 != bot_bot_set_a )
=> ( ! [A3: a] :
( ( member_a @ A3 @ A4 )
=> ( ord_less_eq_a @ X2 @ A3 ) )
=> ( ord_less_eq_a @ X2 @ ( lattic8372110310405614207_Min_a @ A4 ) ) ) ) ) ).
% Min.boundedI
thf(fact_917_Min_OboundedI,axiom,
! [A4: set_nat,X2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A4 )
=> ( ord_less_eq_nat @ X2 @ A3 ) )
=> ( ord_less_eq_nat @ X2 @ ( lattic8721135487736765967in_nat @ A4 ) ) ) ) ) ).
% Min.boundedI
thf(fact_918_Min_OboundedE,axiom,
! [A4: set_a,X2: a] :
( ( finite_finite_a @ A4 )
=> ( ( A4 != bot_bot_set_a )
=> ( ( ord_less_eq_a @ X2 @ ( lattic8372110310405614207_Min_a @ A4 ) )
=> ! [A8: a] :
( ( member_a @ A8 @ A4 )
=> ( ord_less_eq_a @ X2 @ A8 ) ) ) ) ) ).
% Min.boundedE
thf(fact_919_Min_OboundedE,axiom,
! [A4: set_nat,X2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X2 @ ( lattic8721135487736765967in_nat @ A4 ) )
=> ! [A8: nat] :
( ( member_nat @ A8 @ A4 )
=> ( ord_less_eq_nat @ X2 @ A8 ) ) ) ) ) ).
% Min.boundedE
thf(fact_920_eq__Min__iff,axiom,
! [A4: set_a,M5: a] :
( ( finite_finite_a @ A4 )
=> ( ( A4 != bot_bot_set_a )
=> ( ( M5
= ( lattic8372110310405614207_Min_a @ A4 ) )
= ( ( member_a @ M5 @ A4 )
& ! [X3: a] :
( ( member_a @ X3 @ A4 )
=> ( ord_less_eq_a @ M5 @ X3 ) ) ) ) ) ) ).
% eq_Min_iff
thf(fact_921_eq__Min__iff,axiom,
! [A4: set_nat,M5: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( M5
= ( lattic8721135487736765967in_nat @ A4 ) )
= ( ( member_nat @ M5 @ A4 )
& ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( ord_less_eq_nat @ M5 @ X3 ) ) ) ) ) ) ).
% eq_Min_iff
thf(fact_922_Min__le__iff,axiom,
! [A4: set_a,X2: a] :
( ( finite_finite_a @ A4 )
=> ( ( A4 != bot_bot_set_a )
=> ( ( ord_less_eq_a @ ( lattic8372110310405614207_Min_a @ A4 ) @ X2 )
= ( ? [X3: a] :
( ( member_a @ X3 @ A4 )
& ( ord_less_eq_a @ X3 @ X2 ) ) ) ) ) ) ).
% Min_le_iff
thf(fact_923_Min__le__iff,axiom,
! [A4: set_nat,X2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ A4 ) @ X2 )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A4 )
& ( ord_less_eq_nat @ X3 @ X2 ) ) ) ) ) ) ).
% Min_le_iff
thf(fact_924_Min__eq__iff,axiom,
! [A4: set_a,M5: a] :
( ( finite_finite_a @ A4 )
=> ( ( A4 != bot_bot_set_a )
=> ( ( ( lattic8372110310405614207_Min_a @ A4 )
= M5 )
= ( ( member_a @ M5 @ A4 )
& ! [X3: a] :
( ( member_a @ X3 @ A4 )
=> ( ord_less_eq_a @ M5 @ X3 ) ) ) ) ) ) ).
% Min_eq_iff
thf(fact_925_Min__eq__iff,axiom,
! [A4: set_nat,M5: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( ( lattic8721135487736765967in_nat @ A4 )
= M5 )
= ( ( member_nat @ M5 @ A4 )
& ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( ord_less_eq_nat @ M5 @ X3 ) ) ) ) ) ) ).
% Min_eq_iff
thf(fact_926_Min__less__iff,axiom,
! [A4: set_nat,X2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( ord_less_nat @ ( lattic8721135487736765967in_nat @ A4 ) @ X2 )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A4 )
& ( ord_less_nat @ X3 @ X2 ) ) ) ) ) ) ).
% Min_less_iff
thf(fact_927_Min__insert2,axiom,
! [A4: set_a,A: a] :
( ( finite_finite_a @ A4 )
=> ( ! [B3: a] :
( ( member_a @ B3 @ A4 )
=> ( ord_less_eq_a @ A @ B3 ) )
=> ( ( lattic8372110310405614207_Min_a @ ( insert_a @ A @ A4 ) )
= A ) ) ) ).
% Min_insert2
thf(fact_928_Min__insert2,axiom,
! [A4: set_nat,A: nat] :
( ( finite_finite_nat @ A4 )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ A4 )
=> ( ord_less_eq_nat @ A @ B3 ) )
=> ( ( lattic8721135487736765967in_nat @ ( insert_nat @ A @ A4 ) )
= A ) ) ) ).
% Min_insert2
thf(fact_929_cInf__eq__Min,axiom,
! [X6: set_nat] :
( ( finite_finite_nat @ X6 )
=> ( ( X6 != bot_bot_set_nat )
=> ( ( complete_Inf_Inf_nat @ X6 )
= ( lattic8721135487736765967in_nat @ X6 ) ) ) ) ).
% cInf_eq_Min
thf(fact_930_Min__antimono,axiom,
! [M2: set_a,N7: set_a] :
( ( ord_less_eq_set_a @ M2 @ N7 )
=> ( ( M2 != bot_bot_set_a )
=> ( ( finite_finite_a @ N7 )
=> ( ord_less_eq_a @ ( lattic8372110310405614207_Min_a @ N7 ) @ ( lattic8372110310405614207_Min_a @ M2 ) ) ) ) ) ).
% Min_antimono
thf(fact_931_Min__antimono,axiom,
! [M2: set_nat,N7: set_nat] :
( ( ord_less_eq_set_nat @ M2 @ N7 )
=> ( ( M2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ N7 )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ N7 ) @ ( lattic8721135487736765967in_nat @ M2 ) ) ) ) ) ).
% Min_antimono
thf(fact_932_Min_Osubset__imp,axiom,
! [A4: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A4 @ B6 )
=> ( ( A4 != bot_bot_set_a )
=> ( ( finite_finite_a @ B6 )
=> ( ord_less_eq_a @ ( lattic8372110310405614207_Min_a @ B6 ) @ ( lattic8372110310405614207_Min_a @ A4 ) ) ) ) ) ).
% Min.subset_imp
thf(fact_933_Min_Osubset__imp,axiom,
! [A4: set_nat,B6: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B6 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B6 )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ B6 ) @ ( lattic8721135487736765967in_nat @ A4 ) ) ) ) ) ).
% Min.subset_imp
thf(fact_934_Sup__fin_Oinsert,axiom,
! [A4: set_nat,X2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X2 @ A4 ) )
= ( sup_sup_nat @ X2 @ ( lattic1093996805478795353in_nat @ A4 ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_935_Inf__fin_Oinsert,axiom,
! [A4: set_nat,X2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X2 @ A4 ) )
= ( inf_inf_nat @ X2 @ ( lattic5238388535129920115in_nat @ A4 ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_936_semilattice__order__set_Osubset__imp,axiom,
! [F: nat > nat > nat,Less_eq2: nat > nat > $o,Less: nat > nat > $o,A4: set_nat,B6: set_nat] :
( ( lattic6009151579333465974et_nat @ F @ Less_eq2 @ Less )
=> ( ( ord_less_eq_set_nat @ A4 @ B6 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B6 )
=> ( Less_eq2 @ ( lattic7742739596368939638_F_nat @ F @ B6 ) @ ( lattic7742739596368939638_F_nat @ F @ A4 ) ) ) ) ) ) ).
% semilattice_order_set.subset_imp
thf(fact_937_Sup__fin_Osingleton,axiom,
! [X2: nat] :
( ( lattic1093996805478795353in_nat @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
= X2 ) ).
% Sup_fin.singleton
thf(fact_938_Inf__fin_Osingleton,axiom,
! [X2: nat] :
( ( lattic5238388535129920115in_nat @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
= X2 ) ).
% Inf_fin.singleton
thf(fact_939_inf__Sup__absorb,axiom,
! [A4: set_nat,A: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ A @ A4 )
=> ( ( inf_inf_nat @ A @ ( lattic1093996805478795353in_nat @ A4 ) )
= A ) ) ) ).
% inf_Sup_absorb
thf(fact_940_sup__Inf__absorb,axiom,
! [A4: set_nat,A: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ A @ A4 )
=> ( ( sup_sup_nat @ ( lattic5238388535129920115in_nat @ A4 ) @ A )
= A ) ) ) ).
% sup_Inf_absorb
thf(fact_941_Inf__fin__le__Sup__fin,axiom,
! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ A4 ) @ ( lattic1093996805478795353in_nat @ A4 ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_942_Sup__fin__def,axiom,
( lattic1093996805478795353in_nat
= ( lattic7742739596368939638_F_nat @ sup_sup_nat ) ) ).
% Sup_fin_def
thf(fact_943_Inf__fin__def,axiom,
( lattic5238388535129920115in_nat
= ( lattic7742739596368939638_F_nat @ inf_inf_nat ) ) ).
% Inf_fin_def
thf(fact_944_Sup__fin__Max,axiom,
lattic1093996805478795353in_nat = lattic8265883725875713057ax_nat ).
% Sup_fin_Max
thf(fact_945_Sup__fin_OcoboundedI,axiom,
! [A4: set_nat,A: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ A @ A4 )
=> ( ord_less_eq_nat @ A @ ( lattic1093996805478795353in_nat @ A4 ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_946_Inf__fin_OcoboundedI,axiom,
! [A4: set_nat,A: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ A @ A4 )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ A4 ) @ A ) ) ) ).
% Inf_fin.coboundedI
thf(fact_947_Inf__fin_Oin__idem,axiom,
! [A4: set_nat,X2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ X2 @ A4 )
=> ( ( inf_inf_nat @ X2 @ ( lattic5238388535129920115in_nat @ A4 ) )
= ( lattic5238388535129920115in_nat @ A4 ) ) ) ) ).
% Inf_fin.in_idem
thf(fact_948_Sup__fin_Oin__idem,axiom,
! [A4: set_nat,X2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ X2 @ A4 )
=> ( ( sup_sup_nat @ X2 @ ( lattic1093996805478795353in_nat @ A4 ) )
= ( lattic1093996805478795353in_nat @ A4 ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_949_semilattice__order__set_OcoboundedI,axiom,
! [F: a > a > a,Less_eq2: a > a > $o,Less: a > a > $o,A4: set_a,A: a] :
( ( lattic5078705180708912344_set_a @ F @ Less_eq2 @ Less )
=> ( ( finite_finite_a @ A4 )
=> ( ( member_a @ A @ A4 )
=> ( Less_eq2 @ ( lattic5116578512385870296ce_F_a @ F @ A4 ) @ A ) ) ) ) ).
% semilattice_order_set.coboundedI
thf(fact_950_semilattice__order__set_OcoboundedI,axiom,
! [F: nat > nat > nat,Less_eq2: nat > nat > $o,Less: nat > nat > $o,A4: set_nat,A: nat] :
( ( lattic6009151579333465974et_nat @ F @ Less_eq2 @ Less )
=> ( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ A @ A4 )
=> ( Less_eq2 @ ( lattic7742739596368939638_F_nat @ F @ A4 ) @ A ) ) ) ) ).
% semilattice_order_set.coboundedI
thf(fact_951_Inf__fin_OboundedE,axiom,
! [A4: set_nat,X2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X2 @ ( lattic5238388535129920115in_nat @ A4 ) )
=> ! [A8: nat] :
( ( member_nat @ A8 @ A4 )
=> ( ord_less_eq_nat @ X2 @ A8 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_952_Inf__fin_OboundedI,axiom,
! [A4: set_nat,X2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A4 )
=> ( ord_less_eq_nat @ X2 @ A3 ) )
=> ( ord_less_eq_nat @ X2 @ ( lattic5238388535129920115in_nat @ A4 ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_953_Sup__fin_OboundedE,axiom,
! [A4: set_nat,X2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A4 ) @ X2 )
=> ! [A8: nat] :
( ( member_nat @ A8 @ A4 )
=> ( ord_less_eq_nat @ A8 @ X2 ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_954_Sup__fin_OboundedI,axiom,
! [A4: set_nat,X2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A4 )
=> ( ord_less_eq_nat @ A3 @ X2 ) )
=> ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A4 ) @ X2 ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_955_Inf__fin_Obounded__iff,axiom,
! [A4: set_nat,X2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X2 @ ( lattic5238388535129920115in_nat @ A4 ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( ord_less_eq_nat @ X2 @ X3 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_956_Sup__fin_Obounded__iff,axiom,
! [A4: set_nat,X2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A4 ) @ X2 )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( ord_less_eq_nat @ X3 @ X2 ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_957_cSup__eq__Sup__fin,axiom,
! [X6: set_nat] :
( ( finite_finite_nat @ X6 )
=> ( ( X6 != bot_bot_set_nat )
=> ( ( complete_Sup_Sup_nat @ X6 )
= ( lattic1093996805478795353in_nat @ X6 ) ) ) ) ).
% cSup_eq_Sup_fin
thf(fact_958_cInf__eq__Inf__fin,axiom,
! [X6: set_nat] :
( ( finite_finite_nat @ X6 )
=> ( ( X6 != bot_bot_set_nat )
=> ( ( complete_Inf_Inf_nat @ X6 )
= ( lattic5238388535129920115in_nat @ X6 ) ) ) ) ).
% cInf_eq_Inf_fin
thf(fact_959_semilattice__order__set_Obounded__iff,axiom,
! [F: nat > nat > nat,Less_eq2: nat > nat > $o,Less: nat > nat > $o,A4: set_nat,X2: nat] :
( ( lattic6009151579333465974et_nat @ F @ Less_eq2 @ Less )
=> ( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( Less_eq2 @ X2 @ ( lattic7742739596368939638_F_nat @ F @ A4 ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( Less_eq2 @ X2 @ X3 ) ) ) ) ) ) ) ).
% semilattice_order_set.bounded_iff
thf(fact_960_semilattice__order__set_OboundedI,axiom,
! [F: a > a > a,Less_eq2: a > a > $o,Less: a > a > $o,A4: set_a,X2: a] :
( ( lattic5078705180708912344_set_a @ F @ Less_eq2 @ Less )
=> ( ( finite_finite_a @ A4 )
=> ( ( A4 != bot_bot_set_a )
=> ( ! [A3: a] :
( ( member_a @ A3 @ A4 )
=> ( Less_eq2 @ X2 @ A3 ) )
=> ( Less_eq2 @ X2 @ ( lattic5116578512385870296ce_F_a @ F @ A4 ) ) ) ) ) ) ).
% semilattice_order_set.boundedI
thf(fact_961_semilattice__order__set_OboundedI,axiom,
! [F: nat > nat > nat,Less_eq2: nat > nat > $o,Less: nat > nat > $o,A4: set_nat,X2: nat] :
( ( lattic6009151579333465974et_nat @ F @ Less_eq2 @ Less )
=> ( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A4 )
=> ( Less_eq2 @ X2 @ A3 ) )
=> ( Less_eq2 @ X2 @ ( lattic7742739596368939638_F_nat @ F @ A4 ) ) ) ) ) ) ).
% semilattice_order_set.boundedI
thf(fact_962_semilattice__order__set_OboundedE,axiom,
! [F: a > a > a,Less_eq2: a > a > $o,Less: a > a > $o,A4: set_a,X2: a] :
( ( lattic5078705180708912344_set_a @ F @ Less_eq2 @ Less )
=> ( ( finite_finite_a @ A4 )
=> ( ( A4 != bot_bot_set_a )
=> ( ( Less_eq2 @ X2 @ ( lattic5116578512385870296ce_F_a @ F @ A4 ) )
=> ! [A8: a] :
( ( member_a @ A8 @ A4 )
=> ( Less_eq2 @ X2 @ A8 ) ) ) ) ) ) ).
% semilattice_order_set.boundedE
thf(fact_963_semilattice__order__set_OboundedE,axiom,
! [F: nat > nat > nat,Less_eq2: nat > nat > $o,Less: nat > nat > $o,A4: set_nat,X2: nat] :
( ( lattic6009151579333465974et_nat @ F @ Less_eq2 @ Less )
=> ( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( Less_eq2 @ X2 @ ( lattic7742739596368939638_F_nat @ F @ A4 ) )
=> ! [A8: nat] :
( ( member_nat @ A8 @ A4 )
=> ( Less_eq2 @ X2 @ A8 ) ) ) ) ) ) ).
% semilattice_order_set.boundedE
thf(fact_964_Inf__fin_Osubset__imp,axiom,
! [A4: set_nat,B6: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B6 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B6 )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ B6 ) @ ( lattic5238388535129920115in_nat @ A4 ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_965_Sup__fin_Osubset__imp,axiom,
! [A4: set_nat,B6: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B6 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B6 )
=> ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A4 ) @ ( lattic1093996805478795353in_nat @ B6 ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_966_Inf__fin_Ohom__commute,axiom,
! [H: nat > nat,N7: set_nat] :
( ! [X: nat,Y2: nat] :
( ( H @ ( inf_inf_nat @ X @ Y2 ) )
= ( inf_inf_nat @ ( H @ X ) @ ( H @ Y2 ) ) )
=> ( ( finite_finite_nat @ N7 )
=> ( ( N7 != bot_bot_set_nat )
=> ( ( H @ ( lattic5238388535129920115in_nat @ N7 ) )
= ( lattic5238388535129920115in_nat @ ( image_nat_nat @ H @ N7 ) ) ) ) ) ) ).
% Inf_fin.hom_commute
thf(fact_967_Sup__fin_Ohom__commute,axiom,
! [H: nat > nat,N7: set_nat] :
( ! [X: nat,Y2: nat] :
( ( H @ ( sup_sup_nat @ X @ Y2 ) )
= ( sup_sup_nat @ ( H @ X ) @ ( H @ Y2 ) ) )
=> ( ( finite_finite_nat @ N7 )
=> ( ( N7 != bot_bot_set_nat )
=> ( ( H @ ( lattic1093996805478795353in_nat @ N7 ) )
= ( lattic1093996805478795353in_nat @ ( image_nat_nat @ H @ N7 ) ) ) ) ) ) ).
% Sup_fin.hom_commute
thf(fact_968_Inf__fin_Osubset,axiom,
! [A4: set_nat,B6: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( B6 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ B6 @ A4 )
=> ( ( inf_inf_nat @ ( lattic5238388535129920115in_nat @ B6 ) @ ( lattic5238388535129920115in_nat @ A4 ) )
= ( lattic5238388535129920115in_nat @ A4 ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_969_Sup__fin_Osubset,axiom,
! [A4: set_nat,B6: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( B6 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ B6 @ A4 )
=> ( ( sup_sup_nat @ ( lattic1093996805478795353in_nat @ B6 ) @ ( lattic1093996805478795353in_nat @ A4 ) )
= ( lattic1093996805478795353in_nat @ A4 ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_970_Inf__fin_Oinsert__not__elem,axiom,
! [A4: set_nat,X2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ~ ( member_nat @ X2 @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X2 @ A4 ) )
= ( inf_inf_nat @ X2 @ ( lattic5238388535129920115in_nat @ A4 ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_971_Inf__fin_Oclosed,axiom,
! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ! [X: nat,Y2: nat] : ( member_nat @ ( inf_inf_nat @ X @ Y2 ) @ ( insert_nat @ X @ ( insert_nat @ Y2 @ bot_bot_set_nat ) ) )
=> ( member_nat @ ( lattic5238388535129920115in_nat @ A4 ) @ A4 ) ) ) ) ).
% Inf_fin.closed
thf(fact_972_Sup__fin_Oinsert__not__elem,axiom,
! [A4: set_nat,X2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ~ ( member_nat @ X2 @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X2 @ A4 ) )
= ( sup_sup_nat @ X2 @ ( lattic1093996805478795353in_nat @ A4 ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_973_Sup__fin_Oclosed,axiom,
! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ! [X: nat,Y2: nat] : ( member_nat @ ( sup_sup_nat @ X @ Y2 ) @ ( insert_nat @ X @ ( insert_nat @ Y2 @ bot_bot_set_nat ) ) )
=> ( member_nat @ ( lattic1093996805478795353in_nat @ A4 ) @ A4 ) ) ) ) ).
% Sup_fin.closed
thf(fact_974_Inf__fin_Ounion,axiom,
! [A4: set_nat,B6: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B6 )
=> ( ( B6 != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( sup_sup_set_nat @ A4 @ B6 ) )
= ( inf_inf_nat @ ( lattic5238388535129920115in_nat @ A4 ) @ ( lattic5238388535129920115in_nat @ B6 ) ) ) ) ) ) ) ).
% Inf_fin.union
thf(fact_975_Sup__fin_Ounion,axiom,
! [A4: set_nat,B6: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B6 )
=> ( ( B6 != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( sup_sup_set_nat @ A4 @ B6 ) )
= ( sup_sup_nat @ ( lattic1093996805478795353in_nat @ A4 ) @ ( lattic1093996805478795353in_nat @ B6 ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_976_semilattice__set_Ounion,axiom,
! [F: nat > nat > nat,A4: set_nat,B6: set_nat] :
( ( lattic1029310888574255042et_nat @ F )
=> ( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B6 )
=> ( ( B6 != bot_bot_set_nat )
=> ( ( lattic7742739596368939638_F_nat @ F @ ( sup_sup_set_nat @ A4 @ B6 ) )
= ( F @ ( lattic7742739596368939638_F_nat @ F @ A4 ) @ ( lattic7742739596368939638_F_nat @ F @ B6 ) ) ) ) ) ) ) ) ).
% semilattice_set.union
thf(fact_977_semilattice__set_Oclosed,axiom,
! [F: a > a > a,A4: set_a] :
( ( lattic5961991414251573132_set_a @ F )
=> ( ( finite_finite_a @ A4 )
=> ( ( A4 != bot_bot_set_a )
=> ( ! [X: a,Y2: a] : ( member_a @ ( F @ X @ Y2 ) @ ( insert_a @ X @ ( insert_a @ Y2 @ bot_bot_set_a ) ) )
=> ( member_a @ ( lattic5116578512385870296ce_F_a @ F @ A4 ) @ A4 ) ) ) ) ) ).
% semilattice_set.closed
thf(fact_978_semilattice__set_Oclosed,axiom,
! [F: nat > nat > nat,A4: set_nat] :
( ( lattic1029310888574255042et_nat @ F )
=> ( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ! [X: nat,Y2: nat] : ( member_nat @ ( F @ X @ Y2 ) @ ( insert_nat @ X @ ( insert_nat @ Y2 @ bot_bot_set_nat ) ) )
=> ( member_nat @ ( lattic7742739596368939638_F_nat @ F @ A4 ) @ A4 ) ) ) ) ) ).
% semilattice_set.closed
thf(fact_979_semilattice__set_Oinsert,axiom,
! [F: nat > nat > nat,A4: set_nat,X2: nat] :
( ( lattic1029310888574255042et_nat @ F )
=> ( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( lattic7742739596368939638_F_nat @ F @ ( insert_nat @ X2 @ A4 ) )
= ( F @ X2 @ ( lattic7742739596368939638_F_nat @ F @ A4 ) ) ) ) ) ) ).
% semilattice_set.insert
thf(fact_980_Sup__fin_Osemilattice__set__axioms,axiom,
lattic1029310888574255042et_nat @ sup_sup_nat ).
% Sup_fin.semilattice_set_axioms
thf(fact_981_Inf__fin_Osemilattice__set__axioms,axiom,
lattic1029310888574255042et_nat @ inf_inf_nat ).
% Inf_fin.semilattice_set_axioms
thf(fact_982_semilattice__set_Oin__idem,axiom,
! [F: a > a > a,A4: set_a,X2: a] :
( ( lattic5961991414251573132_set_a @ F )
=> ( ( finite_finite_a @ A4 )
=> ( ( member_a @ X2 @ A4 )
=> ( ( F @ X2 @ ( lattic5116578512385870296ce_F_a @ F @ A4 ) )
= ( lattic5116578512385870296ce_F_a @ F @ A4 ) ) ) ) ) ).
% semilattice_set.in_idem
thf(fact_983_semilattice__set_Oin__idem,axiom,
! [F: nat > nat > nat,A4: set_nat,X2: nat] :
( ( lattic1029310888574255042et_nat @ F )
=> ( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ X2 @ A4 )
=> ( ( F @ X2 @ ( lattic7742739596368939638_F_nat @ F @ A4 ) )
= ( lattic7742739596368939638_F_nat @ F @ A4 ) ) ) ) ) ).
% semilattice_set.in_idem
thf(fact_984_semilattice__set_Osingleton,axiom,
! [F: nat > nat > nat,X2: nat] :
( ( lattic1029310888574255042et_nat @ F )
=> ( ( lattic7742739596368939638_F_nat @ F @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
= X2 ) ) ).
% semilattice_set.singleton
thf(fact_985_semilattice__set_Ohom__commute,axiom,
! [F: nat > nat > nat,H: nat > nat,N7: set_nat] :
( ( lattic1029310888574255042et_nat @ F )
=> ( ! [X: nat,Y2: nat] :
( ( H @ ( F @ X @ Y2 ) )
= ( F @ ( H @ X ) @ ( H @ Y2 ) ) )
=> ( ( finite_finite_nat @ N7 )
=> ( ( N7 != bot_bot_set_nat )
=> ( ( H @ ( lattic7742739596368939638_F_nat @ F @ N7 ) )
= ( lattic7742739596368939638_F_nat @ F @ ( image_nat_nat @ H @ N7 ) ) ) ) ) ) ) ).
% semilattice_set.hom_commute
thf(fact_986_semilattice__set_Osubset,axiom,
! [F: nat > nat > nat,A4: set_nat,B6: set_nat] :
( ( lattic1029310888574255042et_nat @ F )
=> ( ( finite_finite_nat @ A4 )
=> ( ( B6 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ B6 @ A4 )
=> ( ( F @ ( lattic7742739596368939638_F_nat @ F @ B6 ) @ ( lattic7742739596368939638_F_nat @ F @ A4 ) )
= ( lattic7742739596368939638_F_nat @ F @ A4 ) ) ) ) ) ) ).
% semilattice_set.subset
thf(fact_987_semilattice__set_Oinsert__not__elem,axiom,
! [F: a > a > a,A4: set_a,X2: a] :
( ( lattic5961991414251573132_set_a @ F )
=> ( ( finite_finite_a @ A4 )
=> ( ~ ( member_a @ X2 @ A4 )
=> ( ( A4 != bot_bot_set_a )
=> ( ( lattic5116578512385870296ce_F_a @ F @ ( insert_a @ X2 @ A4 ) )
= ( F @ X2 @ ( lattic5116578512385870296ce_F_a @ F @ A4 ) ) ) ) ) ) ) ).
% semilattice_set.insert_not_elem
thf(fact_988_semilattice__set_Oinsert__not__elem,axiom,
! [F: nat > nat > nat,A4: set_nat,X2: nat] :
( ( lattic1029310888574255042et_nat @ F )
=> ( ( finite_finite_nat @ A4 )
=> ( ~ ( member_nat @ X2 @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( lattic7742739596368939638_F_nat @ F @ ( insert_nat @ X2 @ A4 ) )
= ( F @ X2 @ ( lattic7742739596368939638_F_nat @ F @ A4 ) ) ) ) ) ) ) ).
% semilattice_set.insert_not_elem
thf(fact_989_semilattice__set_Oremove,axiom,
! [F: a > a > a,A4: set_a,X2: a] :
( ( lattic5961991414251573132_set_a @ F )
=> ( ( finite_finite_a @ A4 )
=> ( ( member_a @ X2 @ A4 )
=> ( ( ( ( minus_minus_set_a @ A4 @ ( insert_a @ X2 @ bot_bot_set_a ) )
= bot_bot_set_a )
=> ( ( lattic5116578512385870296ce_F_a @ F @ A4 )
= X2 ) )
& ( ( ( minus_minus_set_a @ A4 @ ( insert_a @ X2 @ bot_bot_set_a ) )
!= bot_bot_set_a )
=> ( ( lattic5116578512385870296ce_F_a @ F @ A4 )
= ( F @ X2 @ ( lattic5116578512385870296ce_F_a @ F @ ( minus_minus_set_a @ A4 @ ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ) ) ) ) ) ) ).
% semilattice_set.remove
thf(fact_990_semilattice__set_Oremove,axiom,
! [F: nat > nat > nat,A4: set_nat,X2: nat] :
( ( lattic1029310888574255042et_nat @ F )
=> ( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ X2 @ A4 )
=> ( ( ( ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic7742739596368939638_F_nat @ F @ A4 )
= X2 ) )
& ( ( ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic7742739596368939638_F_nat @ F @ A4 )
= ( F @ X2 @ ( lattic7742739596368939638_F_nat @ F @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ) ).
% semilattice_set.remove
thf(fact_991_semilattice__set_Oinsert__remove,axiom,
! [F: nat > nat > nat,A4: set_nat,X2: nat] :
( ( lattic1029310888574255042et_nat @ F )
=> ( ( finite_finite_nat @ A4 )
=> ( ( ( ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic7742739596368939638_F_nat @ F @ ( insert_nat @ X2 @ A4 ) )
= X2 ) )
& ( ( ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic7742739596368939638_F_nat @ F @ ( insert_nat @ X2 @ A4 ) )
= ( F @ X2 @ ( lattic7742739596368939638_F_nat @ F @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% semilattice_set.insert_remove
thf(fact_992_diff__shunt__var,axiom,
! [X2: set_nat,Y3: set_nat] :
( ( ( minus_minus_set_nat @ X2 @ Y3 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ X2 @ Y3 ) ) ).
% diff_shunt_var
thf(fact_993_Inf__fin_Oinsert__remove,axiom,
! [A4: set_nat,X2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( ( ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X2 @ A4 ) )
= X2 ) )
& ( ( ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X2 @ A4 ) )
= ( inf_inf_nat @ X2 @ ( lattic5238388535129920115in_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_994_Inf__fin_Oremove,axiom,
! [A4: set_nat,X2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ X2 @ A4 )
=> ( ( ( ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ A4 )
= X2 ) )
& ( ( ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ A4 )
= ( inf_inf_nat @ X2 @ ( lattic5238388535129920115in_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_995_Sup__fin_Oinsert__remove,axiom,
! [A4: set_nat,X2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( ( ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X2 @ A4 ) )
= X2 ) )
& ( ( ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X2 @ A4 ) )
= ( sup_sup_nat @ X2 @ ( lattic1093996805478795353in_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_996_Sup__fin_Oremove,axiom,
! [A4: set_nat,X2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ X2 @ A4 )
=> ( ( ( ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ A4 )
= X2 ) )
& ( ( ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ A4 )
= ( sup_sup_nat @ X2 @ ( lattic1093996805478795353in_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_997_Min_Oremove,axiom,
! [A4: set_a,X2: a] :
( ( finite_finite_a @ A4 )
=> ( ( member_a @ X2 @ A4 )
=> ( ( ( ( minus_minus_set_a @ A4 @ ( insert_a @ X2 @ bot_bot_set_a ) )
= bot_bot_set_a )
=> ( ( lattic8372110310405614207_Min_a @ A4 )
= X2 ) )
& ( ( ( minus_minus_set_a @ A4 @ ( insert_a @ X2 @ bot_bot_set_a ) )
!= bot_bot_set_a )
=> ( ( lattic8372110310405614207_Min_a @ A4 )
= ( ord_min_a @ X2 @ ( lattic8372110310405614207_Min_a @ ( minus_minus_set_a @ A4 @ ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ) ) ) ) ) ).
% Min.remove
thf(fact_998_Min_Oremove,axiom,
! [A4: set_nat,X2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ X2 @ A4 )
=> ( ( ( ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic8721135487736765967in_nat @ A4 )
= X2 ) )
& ( ( ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic8721135487736765967in_nat @ A4 )
= ( ord_min_nat @ X2 @ ( lattic8721135487736765967in_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% Min.remove
thf(fact_999_Min_Oinsert__remove,axiom,
! [A4: set_nat,X2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( ( ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic8721135487736765967in_nat @ ( insert_nat @ X2 @ A4 ) )
= X2 ) )
& ( ( ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic8721135487736765967in_nat @ ( insert_nat @ X2 @ A4 ) )
= ( ord_min_nat @ X2 @ ( lattic8721135487736765967in_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).
% Min.insert_remove
thf(fact_1000_diff__Suc__Suc,axiom,
! [M5: nat,N3: nat] :
( ( minus_minus_nat @ ( suc @ M5 ) @ ( suc @ N3 ) )
= ( minus_minus_nat @ M5 @ N3 ) ) ).
% diff_Suc_Suc
thf(fact_1001_Suc__diff__diff,axiom,
! [M5: nat,N3: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M5 ) @ N3 ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M5 @ N3 ) @ K ) ) ).
% Suc_diff_diff
thf(fact_1002_min__Suc__Suc,axiom,
! [M5: nat,N3: nat] :
( ( ord_min_nat @ ( suc @ M5 ) @ ( suc @ N3 ) )
= ( suc @ ( ord_min_nat @ M5 @ N3 ) ) ) ).
% min_Suc_Suc
thf(fact_1003_diff__diff__cancel,axiom,
! [I: nat,N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( minus_minus_nat @ N3 @ ( minus_minus_nat @ N3 @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1004_min_Oabsorb1,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_min_a @ A @ B )
= A ) ) ).
% min.absorb1
thf(fact_1005_min_Oabsorb1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_min_nat @ A @ B )
= A ) ) ).
% min.absorb1
thf(fact_1006_min_Oabsorb2,axiom,
! [B: a,A: a] :
( ( ord_less_eq_a @ B @ A )
=> ( ( ord_min_a @ A @ B )
= B ) ) ).
% min.absorb2
thf(fact_1007_min_Oabsorb2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_min_nat @ A @ B )
= B ) ) ).
% min.absorb2
thf(fact_1008_min_Obounded__iff,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ A @ ( ord_min_a @ B @ C ) )
= ( ( ord_less_eq_a @ A @ B )
& ( ord_less_eq_a @ A @ C ) ) ) ).
% min.bounded_iff
thf(fact_1009_min_Obounded__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( ord_min_nat @ B @ C ) )
= ( ( ord_less_eq_nat @ A @ B )
& ( ord_less_eq_nat @ A @ C ) ) ) ).
% min.bounded_iff
thf(fact_1010_min__less__iff__conj,axiom,
! [Z: nat,X2: nat,Y3: nat] :
( ( ord_less_nat @ Z @ ( ord_min_nat @ X2 @ Y3 ) )
= ( ( ord_less_nat @ Z @ X2 )
& ( ord_less_nat @ Z @ Y3 ) ) ) ).
% min_less_iff_conj
thf(fact_1011_min_Oabsorb4,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_min_nat @ A @ B )
= B ) ) ).
% min.absorb4
thf(fact_1012_min_Oabsorb3,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_min_nat @ A @ B )
= A ) ) ).
% min.absorb3
thf(fact_1013_min__bot2,axiom,
! [X2: set_nat] :
( ( ord_min_set_nat @ X2 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% min_bot2
thf(fact_1014_min__bot2,axiom,
! [X2: nat] :
( ( ord_min_nat @ X2 @ bot_bot_nat )
= bot_bot_nat ) ).
% min_bot2
thf(fact_1015_min__bot,axiom,
! [X2: set_nat] :
( ( ord_min_set_nat @ bot_bot_set_nat @ X2 )
= bot_bot_set_nat ) ).
% min_bot
thf(fact_1016_min__bot,axiom,
! [X2: nat] :
( ( ord_min_nat @ bot_bot_nat @ X2 )
= bot_bot_nat ) ).
% min_bot
thf(fact_1017_min__top,axiom,
! [X2: set_nat] :
( ( ord_min_set_nat @ top_top_set_nat @ X2 )
= X2 ) ).
% min_top
thf(fact_1018_min__top2,axiom,
! [X2: set_nat] :
( ( ord_min_set_nat @ X2 @ top_top_set_nat )
= X2 ) ).
% min_top2
thf(fact_1019_Min__insert,axiom,
! [A4: set_nat,X2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( lattic8721135487736765967in_nat @ ( insert_nat @ X2 @ A4 ) )
= ( ord_min_nat @ X2 @ ( lattic8721135487736765967in_nat @ A4 ) ) ) ) ) ).
% Min_insert
thf(fact_1020_diff__less__Suc,axiom,
! [M5: nat,N3: nat] : ( ord_less_nat @ ( minus_minus_nat @ M5 @ N3 ) @ ( suc @ M5 ) ) ).
% diff_less_Suc
thf(fact_1021_Suc__diff__Suc,axiom,
! [N3: nat,M5: nat] :
( ( ord_less_nat @ N3 @ M5 )
=> ( ( suc @ ( minus_minus_nat @ M5 @ ( suc @ N3 ) ) )
= ( minus_minus_nat @ M5 @ N3 ) ) ) ).
% Suc_diff_Suc
thf(fact_1022_Suc__diff__le,axiom,
! [N3: nat,M5: nat] :
( ( ord_less_eq_nat @ N3 @ M5 )
=> ( ( minus_minus_nat @ ( suc @ M5 ) @ N3 )
= ( suc @ ( minus_minus_nat @ M5 @ N3 ) ) ) ) ).
% Suc_diff_le
thf(fact_1023_less__diff__iff,axiom,
! [K: nat,M5: nat,N3: nat] :
( ( ord_less_eq_nat @ K @ M5 )
=> ( ( ord_less_eq_nat @ K @ N3 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M5 @ K ) @ ( minus_minus_nat @ N3 @ K ) )
= ( ord_less_nat @ M5 @ N3 ) ) ) ) ).
% less_diff_iff
thf(fact_1024_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1025_eq__diff__iff,axiom,
! [K: nat,M5: nat,N3: nat] :
( ( ord_less_eq_nat @ K @ M5 )
=> ( ( ord_less_eq_nat @ K @ N3 )
=> ( ( ( minus_minus_nat @ M5 @ K )
= ( minus_minus_nat @ N3 @ K ) )
= ( M5 = N3 ) ) ) ) ).
% eq_diff_iff
thf(fact_1026_le__diff__iff,axiom,
! [K: nat,M5: nat,N3: nat] :
( ( ord_less_eq_nat @ K @ M5 )
=> ( ( ord_less_eq_nat @ K @ N3 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M5 @ K ) @ ( minus_minus_nat @ N3 @ K ) )
= ( ord_less_eq_nat @ M5 @ N3 ) ) ) ) ).
% le_diff_iff
thf(fact_1027_Nat_Odiff__diff__eq,axiom,
! [K: nat,M5: nat,N3: nat] :
( ( ord_less_eq_nat @ K @ M5 )
=> ( ( ord_less_eq_nat @ K @ N3 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M5 @ K ) @ ( minus_minus_nat @ N3 @ K ) )
= ( minus_minus_nat @ M5 @ N3 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1028_diff__le__mono,axiom,
! [M5: nat,N3: nat,L: nat] :
( ( ord_less_eq_nat @ M5 @ N3 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M5 @ L ) @ ( minus_minus_nat @ N3 @ L ) ) ) ).
% diff_le_mono
thf(fact_1029_diff__le__self,axiom,
! [M5: nat,N3: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M5 @ N3 ) @ M5 ) ).
% diff_le_self
thf(fact_1030_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_1031_diff__le__mono2,axiom,
! [M5: nat,N3: nat,L: nat] :
( ( ord_less_eq_nat @ M5 @ N3 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N3 ) @ ( minus_minus_nat @ L @ M5 ) ) ) ).
% diff_le_mono2
thf(fact_1032_less__imp__diff__less,axiom,
! [J: nat,K: nat,N3: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N3 ) @ K ) ) ).
% less_imp_diff_less
thf(fact_1033_diff__less__mono2,axiom,
! [M5: nat,N3: nat,L: nat] :
( ( ord_less_nat @ M5 @ N3 )
=> ( ( ord_less_nat @ M5 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N3 ) @ ( minus_minus_nat @ L @ M5 ) ) ) ) ).
% diff_less_mono2
thf(fact_1034_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_1035_Min_Oin__idem,axiom,
! [A4: set_a,X2: a] :
( ( finite_finite_a @ A4 )
=> ( ( member_a @ X2 @ A4 )
=> ( ( ord_min_a @ X2 @ ( lattic8372110310405614207_Min_a @ A4 ) )
= ( lattic8372110310405614207_Min_a @ A4 ) ) ) ) ).
% Min.in_idem
thf(fact_1036_Min_Oin__idem,axiom,
! [A4: set_nat,X2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ X2 @ A4 )
=> ( ( ord_min_nat @ X2 @ ( lattic8721135487736765967in_nat @ A4 ) )
= ( lattic8721135487736765967in_nat @ A4 ) ) ) ) ).
% Min.in_idem
thf(fact_1037_min_Ostrict__coboundedI2,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ ( ord_min_nat @ A @ B ) @ C ) ) ).
% min.strict_coboundedI2
thf(fact_1038_min_Ostrict__coboundedI1,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ A @ C )
=> ( ord_less_nat @ ( ord_min_nat @ A @ B ) @ C ) ) ).
% min.strict_coboundedI1
thf(fact_1039_min_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] :
( ( A2
= ( ord_min_nat @ A2 @ B2 ) )
& ( A2 != B2 ) ) ) ) ).
% min.strict_order_iff
thf(fact_1040_min_Ostrict__boundedE,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( ord_min_nat @ B @ C ) )
=> ~ ( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ A @ C ) ) ) ).
% min.strict_boundedE
thf(fact_1041_min__less__iff__disj,axiom,
! [X2: nat,Y3: nat,Z: nat] :
( ( ord_less_nat @ ( ord_min_nat @ X2 @ Y3 ) @ Z )
= ( ( ord_less_nat @ X2 @ Z )
| ( ord_less_nat @ Y3 @ Z ) ) ) ).
% min_less_iff_disj
thf(fact_1042_min_Omono,axiom,
! [A: a,C: a,B: a,D2: a] :
( ( ord_less_eq_a @ A @ C )
=> ( ( ord_less_eq_a @ B @ D2 )
=> ( ord_less_eq_a @ ( ord_min_a @ A @ B ) @ ( ord_min_a @ C @ D2 ) ) ) ) ).
% min.mono
thf(fact_1043_min_Omono,axiom,
! [A: nat,C: nat,B: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ D2 )
=> ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ ( ord_min_nat @ C @ D2 ) ) ) ) ).
% min.mono
thf(fact_1044_min_OorderE,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ B )
=> ( A
= ( ord_min_a @ A @ B ) ) ) ).
% min.orderE
thf(fact_1045_min_OorderE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( A
= ( ord_min_nat @ A @ B ) ) ) ).
% min.orderE
thf(fact_1046_min_OorderI,axiom,
! [A: a,B: a] :
( ( A
= ( ord_min_a @ A @ B ) )
=> ( ord_less_eq_a @ A @ B ) ) ).
% min.orderI
thf(fact_1047_min_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( ord_min_nat @ A @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% min.orderI
thf(fact_1048_min_OboundedE,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ A @ ( ord_min_a @ B @ C ) )
=> ~ ( ( ord_less_eq_a @ A @ B )
=> ~ ( ord_less_eq_a @ A @ C ) ) ) ).
% min.boundedE
thf(fact_1049_min_OboundedE,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( ord_min_nat @ B @ C ) )
=> ~ ( ( ord_less_eq_nat @ A @ B )
=> ~ ( ord_less_eq_nat @ A @ C ) ) ) ).
% min.boundedE
thf(fact_1050_min_OboundedI,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ A @ C )
=> ( ord_less_eq_a @ A @ ( ord_min_a @ B @ C ) ) ) ) ).
% min.boundedI
thf(fact_1051_min_OboundedI,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ C )
=> ( ord_less_eq_nat @ A @ ( ord_min_nat @ B @ C ) ) ) ) ).
% min.boundedI
thf(fact_1052_min_Oorder__iff,axiom,
( ord_less_eq_a
= ( ^ [A2: a,B2: a] :
( A2
= ( ord_min_a @ A2 @ B2 ) ) ) ) ).
% min.order_iff
thf(fact_1053_min_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
( A2
= ( ord_min_nat @ A2 @ B2 ) ) ) ) ).
% min.order_iff
thf(fact_1054_min_Ocobounded1,axiom,
! [A: a,B: a] : ( ord_less_eq_a @ ( ord_min_a @ A @ B ) @ A ) ).
% min.cobounded1
thf(fact_1055_min_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ A ) ).
% min.cobounded1
thf(fact_1056_min_Ocobounded2,axiom,
! [A: a,B: a] : ( ord_less_eq_a @ ( ord_min_a @ A @ B ) @ B ) ).
% min.cobounded2
thf(fact_1057_min_Ocobounded2,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ B ) ).
% min.cobounded2
thf(fact_1058_min_Oabsorb__iff1,axiom,
( ord_less_eq_a
= ( ^ [A2: a,B2: a] :
( ( ord_min_a @ A2 @ B2 )
= A2 ) ) ) ).
% min.absorb_iff1
thf(fact_1059_min_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_min_nat @ A2 @ B2 )
= A2 ) ) ) ).
% min.absorb_iff1
thf(fact_1060_min_Oabsorb__iff2,axiom,
( ord_less_eq_a
= ( ^ [B2: a,A2: a] :
( ( ord_min_a @ A2 @ B2 )
= B2 ) ) ) ).
% min.absorb_iff2
thf(fact_1061_min_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_min_nat @ A2 @ B2 )
= B2 ) ) ) ).
% min.absorb_iff2
thf(fact_1062_min_OcoboundedI1,axiom,
! [A: a,C: a,B: a] :
( ( ord_less_eq_a @ A @ C )
=> ( ord_less_eq_a @ ( ord_min_a @ A @ B ) @ C ) ) ).
% min.coboundedI1
thf(fact_1063_min_OcoboundedI1,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ C ) ) ).
% min.coboundedI1
thf(fact_1064_min_OcoboundedI2,axiom,
! [B: a,C: a,A: a] :
( ( ord_less_eq_a @ B @ C )
=> ( ord_less_eq_a @ ( ord_min_a @ A @ B ) @ C ) ) ).
% min.coboundedI2
thf(fact_1065_min_OcoboundedI2,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ C ) ) ).
% min.coboundedI2
thf(fact_1066_min__le__iff__disj,axiom,
! [X2: a,Y3: a,Z: a] :
( ( ord_less_eq_a @ ( ord_min_a @ X2 @ Y3 ) @ Z )
= ( ( ord_less_eq_a @ X2 @ Z )
| ( ord_less_eq_a @ Y3 @ Z ) ) ) ).
% min_le_iff_disj
thf(fact_1067_min__le__iff__disj,axiom,
! [X2: nat,Y3: nat,Z: nat] :
( ( ord_less_eq_nat @ ( ord_min_nat @ X2 @ Y3 ) @ Z )
= ( ( ord_less_eq_nat @ X2 @ Z )
| ( ord_less_eq_nat @ Y3 @ Z ) ) ) ).
% min_le_iff_disj
thf(fact_1068_min__def,axiom,
( ord_min_a
= ( ^ [A2: a,B2: a] : ( if_a @ ( ord_less_eq_a @ A2 @ B2 ) @ A2 @ B2 ) ) ) ).
% min_def
thf(fact_1069_min__def,axiom,
( ord_min_nat
= ( ^ [A2: nat,B2: nat] : ( if_nat @ ( ord_less_eq_nat @ A2 @ B2 ) @ A2 @ B2 ) ) ) ).
% min_def
thf(fact_1070_min__absorb1,axiom,
! [X2: a,Y3: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
=> ( ( ord_min_a @ X2 @ Y3 )
= X2 ) ) ).
% min_absorb1
thf(fact_1071_min__absorb1,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( ord_min_nat @ X2 @ Y3 )
= X2 ) ) ).
% min_absorb1
thf(fact_1072_min__absorb2,axiom,
! [Y3: a,X2: a] :
( ( ord_less_eq_a @ Y3 @ X2 )
=> ( ( ord_min_a @ X2 @ Y3 )
= Y3 ) ) ).
% min_absorb2
thf(fact_1073_min__absorb2,axiom,
! [Y3: nat,X2: nat] :
( ( ord_less_eq_nat @ Y3 @ X2 )
=> ( ( ord_min_nat @ X2 @ Y3 )
= Y3 ) ) ).
% min_absorb2
thf(fact_1074_Min__def,axiom,
( lattic8721135487736765967in_nat
= ( lattic7742739596368939638_F_nat @ ord_min_nat ) ) ).
% Min_def
thf(fact_1075_Min_Osemilattice__set__axioms,axiom,
lattic1029310888574255042et_nat @ ord_min_nat ).
% Min.semilattice_set_axioms
thf(fact_1076_min__of__mono,axiom,
! [F: a > a,M5: a,N3: a] :
( ( monotone_on_a_a @ top_top_set_a @ ord_less_eq_a @ ord_less_eq_a @ F )
=> ( ( ord_min_a @ ( F @ M5 ) @ ( F @ N3 ) )
= ( F @ ( ord_min_a @ M5 @ N3 ) ) ) ) ).
% min_of_mono
thf(fact_1077_min__of__mono,axiom,
! [F: a > nat,M5: a,N3: a] :
( ( monotone_on_a_nat @ top_top_set_a @ ord_less_eq_a @ ord_less_eq_nat @ F )
=> ( ( ord_min_nat @ ( F @ M5 ) @ ( F @ N3 ) )
= ( F @ ( ord_min_a @ M5 @ N3 ) ) ) ) ).
% min_of_mono
thf(fact_1078_min__of__mono,axiom,
! [F: nat > a,M5: nat,N3: nat] :
( ( monotone_on_nat_a @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_a @ F )
=> ( ( ord_min_a @ ( F @ M5 ) @ ( F @ N3 ) )
= ( F @ ( ord_min_nat @ M5 @ N3 ) ) ) ) ).
% min_of_mono
thf(fact_1079_min__of__mono,axiom,
! [F: nat > nat,M5: nat,N3: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_min_nat @ ( F @ M5 ) @ ( F @ N3 ) )
= ( F @ ( ord_min_nat @ M5 @ N3 ) ) ) ) ).
% min_of_mono
thf(fact_1080_Min_Osemilattice__order__set__axioms,axiom,
lattic5078705180708912344_set_a @ ord_min_a @ ord_less_eq_a @ ord_less_a ).
% Min.semilattice_order_set_axioms
thf(fact_1081_Min_Osemilattice__order__set__axioms,axiom,
lattic6009151579333465974et_nat @ ord_min_nat @ ord_less_eq_nat @ ord_less_nat ).
% Min.semilattice_order_set_axioms
thf(fact_1082_Inf__insert__finite,axiom,
! [S2: set_nat,X2: nat] :
( ( finite_finite_nat @ S2 )
=> ( ( ( S2 = bot_bot_set_nat )
=> ( ( complete_Inf_Inf_nat @ ( insert_nat @ X2 @ S2 ) )
= X2 ) )
& ( ( S2 != bot_bot_set_nat )
=> ( ( complete_Inf_Inf_nat @ ( insert_nat @ X2 @ S2 ) )
= ( ord_min_nat @ X2 @ ( complete_Inf_Inf_nat @ S2 ) ) ) ) ) ) ).
% Inf_insert_finite
thf(fact_1083_hom__Min__commute,axiom,
! [H: nat > nat,N7: set_nat] :
( ! [X: nat,Y2: nat] :
( ( H @ ( ord_min_nat @ X @ Y2 ) )
= ( ord_min_nat @ ( H @ X ) @ ( H @ Y2 ) ) )
=> ( ( finite_finite_nat @ N7 )
=> ( ( N7 != bot_bot_set_nat )
=> ( ( H @ ( lattic8721135487736765967in_nat @ N7 ) )
= ( lattic8721135487736765967in_nat @ ( image_nat_nat @ H @ N7 ) ) ) ) ) ) ).
% hom_Min_commute
thf(fact_1084_Min_Osubset,axiom,
! [A4: set_nat,B6: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( B6 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ B6 @ A4 )
=> ( ( ord_min_nat @ ( lattic8721135487736765967in_nat @ B6 ) @ ( lattic8721135487736765967in_nat @ A4 ) )
= ( lattic8721135487736765967in_nat @ A4 ) ) ) ) ) ).
% Min.subset
thf(fact_1085_Min_Oclosed,axiom,
! [A4: set_a] :
( ( finite_finite_a @ A4 )
=> ( ( A4 != bot_bot_set_a )
=> ( ! [X: a,Y2: a] : ( member_a @ ( ord_min_a @ X @ Y2 ) @ ( insert_a @ X @ ( insert_a @ Y2 @ bot_bot_set_a ) ) )
=> ( member_a @ ( lattic8372110310405614207_Min_a @ A4 ) @ A4 ) ) ) ) ).
% Min.closed
thf(fact_1086_Min_Oclosed,axiom,
! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ! [X: nat,Y2: nat] : ( member_nat @ ( ord_min_nat @ X @ Y2 ) @ ( insert_nat @ X @ ( insert_nat @ Y2 @ bot_bot_set_nat ) ) )
=> ( member_nat @ ( lattic8721135487736765967in_nat @ A4 ) @ A4 ) ) ) ) ).
% Min.closed
thf(fact_1087_Min_Oinsert__not__elem,axiom,
! [A4: set_a,X2: a] :
( ( finite_finite_a @ A4 )
=> ( ~ ( member_a @ X2 @ A4 )
=> ( ( A4 != bot_bot_set_a )
=> ( ( lattic8372110310405614207_Min_a @ ( insert_a @ X2 @ A4 ) )
= ( ord_min_a @ X2 @ ( lattic8372110310405614207_Min_a @ A4 ) ) ) ) ) ) ).
% Min.insert_not_elem
thf(fact_1088_Min_Oinsert__not__elem,axiom,
! [A4: set_nat,X2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ~ ( member_nat @ X2 @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( lattic8721135487736765967in_nat @ ( insert_nat @ X2 @ A4 ) )
= ( ord_min_nat @ X2 @ ( lattic8721135487736765967in_nat @ A4 ) ) ) ) ) ) ).
% Min.insert_not_elem
thf(fact_1089_Min_Ounion,axiom,
! [A4: set_nat,B6: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B6 )
=> ( ( B6 != bot_bot_set_nat )
=> ( ( lattic8721135487736765967in_nat @ ( sup_sup_set_nat @ A4 @ B6 ) )
= ( ord_min_nat @ ( lattic8721135487736765967in_nat @ A4 ) @ ( lattic8721135487736765967in_nat @ B6 ) ) ) ) ) ) ) ).
% Min.union
thf(fact_1090_Max_Oinsert__remove,axiom,
! [A4: set_nat,X2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( ( ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic8265883725875713057ax_nat @ ( insert_nat @ X2 @ A4 ) )
= X2 ) )
& ( ( ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic8265883725875713057ax_nat @ ( insert_nat @ X2 @ A4 ) )
= ( ord_max_nat @ X2 @ ( lattic8265883725875713057ax_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).
% Max.insert_remove
thf(fact_1091_Max_Oremove,axiom,
! [A4: set_a,X2: a] :
( ( finite_finite_a @ A4 )
=> ( ( member_a @ X2 @ A4 )
=> ( ( ( ( minus_minus_set_a @ A4 @ ( insert_a @ X2 @ bot_bot_set_a ) )
= bot_bot_set_a )
=> ( ( lattic6529028001545966829_Max_a @ A4 )
= X2 ) )
& ( ( ( minus_minus_set_a @ A4 @ ( insert_a @ X2 @ bot_bot_set_a ) )
!= bot_bot_set_a )
=> ( ( lattic6529028001545966829_Max_a @ A4 )
= ( ord_max_a @ X2 @ ( lattic6529028001545966829_Max_a @ ( minus_minus_set_a @ A4 @ ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ) ) ) ) ) ).
% Max.remove
thf(fact_1092_Max_Oremove,axiom,
! [A4: set_nat,X2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ X2 @ A4 )
=> ( ( ( ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic8265883725875713057ax_nat @ A4 )
= X2 ) )
& ( ( ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic8265883725875713057ax_nat @ A4 )
= ( ord_max_nat @ X2 @ ( lattic8265883725875713057ax_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% Max.remove
thf(fact_1093_max__Suc__Suc,axiom,
! [M5: nat,N3: nat] :
( ( ord_max_nat @ ( suc @ M5 ) @ ( suc @ N3 ) )
= ( suc @ ( ord_max_nat @ M5 @ N3 ) ) ) ).
% max_Suc_Suc
thf(fact_1094_max_Obounded__iff,axiom,
! [B: a,C: a,A: a] :
( ( ord_less_eq_a @ ( ord_max_a @ B @ C ) @ A )
= ( ( ord_less_eq_a @ B @ A )
& ( ord_less_eq_a @ C @ A ) ) ) ).
% max.bounded_iff
thf(fact_1095_max_Obounded__iff,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A )
= ( ( ord_less_eq_nat @ B @ A )
& ( ord_less_eq_nat @ C @ A ) ) ) ).
% max.bounded_iff
thf(fact_1096_max_Oabsorb2,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_max_a @ A @ B )
= B ) ) ).
% max.absorb2
thf(fact_1097_max_Oabsorb2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_max_nat @ A @ B )
= B ) ) ).
% max.absorb2
thf(fact_1098_max_Oabsorb1,axiom,
! [B: a,A: a] :
( ( ord_less_eq_a @ B @ A )
=> ( ( ord_max_a @ A @ B )
= A ) ) ).
% max.absorb1
thf(fact_1099_max_Oabsorb1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_max_nat @ A @ B )
= A ) ) ).
% max.absorb1
thf(fact_1100_max_Oabsorb3,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_max_nat @ A @ B )
= A ) ) ).
% max.absorb3
thf(fact_1101_max_Oabsorb4,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_max_nat @ A @ B )
= B ) ) ).
% max.absorb4
thf(fact_1102_max__less__iff__conj,axiom,
! [X2: nat,Y3: nat,Z: nat] :
( ( ord_less_nat @ ( ord_max_nat @ X2 @ Y3 ) @ Z )
= ( ( ord_less_nat @ X2 @ Z )
& ( ord_less_nat @ Y3 @ Z ) ) ) ).
% max_less_iff_conj
thf(fact_1103_max__bot,axiom,
! [X2: set_nat] :
( ( ord_max_set_nat @ bot_bot_set_nat @ X2 )
= X2 ) ).
% max_bot
thf(fact_1104_max__bot,axiom,
! [X2: nat] :
( ( ord_max_nat @ bot_bot_nat @ X2 )
= X2 ) ).
% max_bot
thf(fact_1105_max__bot2,axiom,
! [X2: set_nat] :
( ( ord_max_set_nat @ X2 @ bot_bot_set_nat )
= X2 ) ).
% max_bot2
thf(fact_1106_max__bot2,axiom,
! [X2: nat] :
( ( ord_max_nat @ X2 @ bot_bot_nat )
= X2 ) ).
% max_bot2
thf(fact_1107_max__top2,axiom,
! [X2: set_nat] :
( ( ord_max_set_nat @ X2 @ top_top_set_nat )
= top_top_set_nat ) ).
% max_top2
thf(fact_1108_max__top,axiom,
! [X2: set_nat] :
( ( ord_max_set_nat @ top_top_set_nat @ X2 )
= top_top_set_nat ) ).
% max_top
thf(fact_1109_max__min__same_I4_J,axiom,
! [Y3: nat,X2: nat] :
( ( ord_max_nat @ Y3 @ ( ord_min_nat @ X2 @ Y3 ) )
= Y3 ) ).
% max_min_same(4)
thf(fact_1110_max__min__same_I3_J,axiom,
! [X2: nat,Y3: nat] :
( ( ord_max_nat @ ( ord_min_nat @ X2 @ Y3 ) @ Y3 )
= Y3 ) ).
% max_min_same(3)
thf(fact_1111_max__min__same_I2_J,axiom,
! [X2: nat,Y3: nat] :
( ( ord_max_nat @ ( ord_min_nat @ X2 @ Y3 ) @ X2 )
= X2 ) ).
% max_min_same(2)
thf(fact_1112_max__min__same_I1_J,axiom,
! [X2: nat,Y3: nat] :
( ( ord_max_nat @ X2 @ ( ord_min_nat @ X2 @ Y3 ) )
= X2 ) ).
% max_min_same(1)
thf(fact_1113_Max__insert,axiom,
! [A4: set_nat,X2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( lattic8265883725875713057ax_nat @ ( insert_nat @ X2 @ A4 ) )
= ( ord_max_nat @ X2 @ ( lattic8265883725875713057ax_nat @ A4 ) ) ) ) ) ).
% Max_insert
thf(fact_1114_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_1115_min__diff,axiom,
! [M5: nat,I: nat,N3: nat] :
( ( ord_min_nat @ ( minus_minus_nat @ M5 @ I ) @ ( minus_minus_nat @ N3 @ I ) )
= ( minus_minus_nat @ ( ord_min_nat @ M5 @ N3 ) @ I ) ) ).
% min_diff
thf(fact_1116_inf__nat__def,axiom,
inf_inf_nat = ord_min_nat ).
% inf_nat_def
thf(fact_1117_Max_Osemilattice__set__axioms,axiom,
lattic1029310888574255042et_nat @ ord_max_nat ).
% Max.semilattice_set_axioms
thf(fact_1118_Max__def,axiom,
( lattic8265883725875713057ax_nat
= ( lattic7742739596368939638_F_nat @ ord_max_nat ) ) ).
% Max_def
thf(fact_1119_max__absorb2,axiom,
! [X2: a,Y3: a] :
( ( ord_less_eq_a @ X2 @ Y3 )
=> ( ( ord_max_a @ X2 @ Y3 )
= Y3 ) ) ).
% max_absorb2
thf(fact_1120_max__absorb2,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( ord_max_nat @ X2 @ Y3 )
= Y3 ) ) ).
% max_absorb2
thf(fact_1121_max__absorb1,axiom,
! [Y3: a,X2: a] :
( ( ord_less_eq_a @ Y3 @ X2 )
=> ( ( ord_max_a @ X2 @ Y3 )
= X2 ) ) ).
% max_absorb1
thf(fact_1122_max__absorb1,axiom,
! [Y3: nat,X2: nat] :
( ( ord_less_eq_nat @ Y3 @ X2 )
=> ( ( ord_max_nat @ X2 @ Y3 )
= X2 ) ) ).
% max_absorb1
thf(fact_1123_max__def,axiom,
( ord_max_a
= ( ^ [A2: a,B2: a] : ( if_a @ ( ord_less_eq_a @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).
% max_def
thf(fact_1124_max__def,axiom,
( ord_max_nat
= ( ^ [A2: nat,B2: nat] : ( if_nat @ ( ord_less_eq_nat @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).
% max_def
thf(fact_1125_max_OcoboundedI2,axiom,
! [C: a,B: a,A: a] :
( ( ord_less_eq_a @ C @ B )
=> ( ord_less_eq_a @ C @ ( ord_max_a @ A @ B ) ) ) ).
% max.coboundedI2
thf(fact_1126_max_OcoboundedI2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% max.coboundedI2
thf(fact_1127_max_OcoboundedI1,axiom,
! [C: a,A: a,B: a] :
( ( ord_less_eq_a @ C @ A )
=> ( ord_less_eq_a @ C @ ( ord_max_a @ A @ B ) ) ) ).
% max.coboundedI1
thf(fact_1128_max_OcoboundedI1,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% max.coboundedI1
thf(fact_1129_max_Oabsorb__iff2,axiom,
( ord_less_eq_a
= ( ^ [A2: a,B2: a] :
( ( ord_max_a @ A2 @ B2 )
= B2 ) ) ) ).
% max.absorb_iff2
thf(fact_1130_max_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_max_nat @ A2 @ B2 )
= B2 ) ) ) ).
% max.absorb_iff2
thf(fact_1131_max_Oabsorb__iff1,axiom,
( ord_less_eq_a
= ( ^ [B2: a,A2: a] :
( ( ord_max_a @ A2 @ B2 )
= A2 ) ) ) ).
% max.absorb_iff1
thf(fact_1132_max_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_max_nat @ A2 @ B2 )
= A2 ) ) ) ).
% max.absorb_iff1
thf(fact_1133_le__max__iff__disj,axiom,
! [Z: a,X2: a,Y3: a] :
( ( ord_less_eq_a @ Z @ ( ord_max_a @ X2 @ Y3 ) )
= ( ( ord_less_eq_a @ Z @ X2 )
| ( ord_less_eq_a @ Z @ Y3 ) ) ) ).
% le_max_iff_disj
thf(fact_1134_le__max__iff__disj,axiom,
! [Z: nat,X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ Z @ ( ord_max_nat @ X2 @ Y3 ) )
= ( ( ord_less_eq_nat @ Z @ X2 )
| ( ord_less_eq_nat @ Z @ Y3 ) ) ) ).
% le_max_iff_disj
thf(fact_1135_max_Ocobounded2,axiom,
! [B: a,A: a] : ( ord_less_eq_a @ B @ ( ord_max_a @ A @ B ) ) ).
% max.cobounded2
thf(fact_1136_max_Ocobounded2,axiom,
! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( ord_max_nat @ A @ B ) ) ).
% max.cobounded2
thf(fact_1137_max_Ocobounded1,axiom,
! [A: a,B: a] : ( ord_less_eq_a @ A @ ( ord_max_a @ A @ B ) ) ).
% max.cobounded1
thf(fact_1138_max_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( ord_max_nat @ A @ B ) ) ).
% max.cobounded1
thf(fact_1139_max_Oorder__iff,axiom,
( ord_less_eq_a
= ( ^ [B2: a,A2: a] :
( A2
= ( ord_max_a @ A2 @ B2 ) ) ) ) ).
% max.order_iff
thf(fact_1140_max_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A2: nat] :
( A2
= ( ord_max_nat @ A2 @ B2 ) ) ) ) ).
% max.order_iff
thf(fact_1141_max_OboundedI,axiom,
! [B: a,A: a,C: a] :
( ( ord_less_eq_a @ B @ A )
=> ( ( ord_less_eq_a @ C @ A )
=> ( ord_less_eq_a @ ( ord_max_a @ B @ C ) @ A ) ) ) ).
% max.boundedI
thf(fact_1142_max_OboundedI,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A ) ) ) ).
% max.boundedI
thf(fact_1143_max_OboundedE,axiom,
! [B: a,C: a,A: a] :
( ( ord_less_eq_a @ ( ord_max_a @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_a @ B @ A )
=> ~ ( ord_less_eq_a @ C @ A ) ) ) ).
% max.boundedE
thf(fact_1144_max_OboundedE,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_nat @ B @ A )
=> ~ ( ord_less_eq_nat @ C @ A ) ) ) ).
% max.boundedE
thf(fact_1145_max_OorderI,axiom,
! [A: a,B: a] :
( ( A
= ( ord_max_a @ A @ B ) )
=> ( ord_less_eq_a @ B @ A ) ) ).
% max.orderI
thf(fact_1146_max_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( ord_max_nat @ A @ B ) )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% max.orderI
thf(fact_1147_max_OorderE,axiom,
! [B: a,A: a] :
( ( ord_less_eq_a @ B @ A )
=> ( A
= ( ord_max_a @ A @ B ) ) ) ).
% max.orderE
thf(fact_1148_max_OorderE,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( A
= ( ord_max_nat @ A @ B ) ) ) ).
% max.orderE
thf(fact_1149_max_Omono,axiom,
! [C: a,A: a,D2: a,B: a] :
( ( ord_less_eq_a @ C @ A )
=> ( ( ord_less_eq_a @ D2 @ B )
=> ( ord_less_eq_a @ ( ord_max_a @ C @ D2 ) @ ( ord_max_a @ A @ B ) ) ) ) ).
% max.mono
thf(fact_1150_max_Omono,axiom,
! [C: nat,A: nat,D2: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ( ord_less_eq_nat @ D2 @ B )
=> ( ord_less_eq_nat @ ( ord_max_nat @ C @ D2 ) @ ( ord_max_nat @ A @ B ) ) ) ) ).
% max.mono
thf(fact_1151_less__max__iff__disj,axiom,
! [Z: nat,X2: nat,Y3: nat] :
( ( ord_less_nat @ Z @ ( ord_max_nat @ X2 @ Y3 ) )
= ( ( ord_less_nat @ Z @ X2 )
| ( ord_less_nat @ Z @ Y3 ) ) ) ).
% less_max_iff_disj
thf(fact_1152_max_Ostrict__boundedE,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_nat @ ( ord_max_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ C @ A ) ) ) ).
% max.strict_boundedE
thf(fact_1153_max_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [B2: nat,A2: nat] :
( ( A2
= ( ord_max_nat @ A2 @ B2 ) )
& ( A2 != B2 ) ) ) ) ).
% max.strict_order_iff
thf(fact_1154_max_Ostrict__coboundedI1,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ C @ A )
=> ( ord_less_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% max.strict_coboundedI1
thf(fact_1155_max_Ostrict__coboundedI2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% max.strict_coboundedI2
thf(fact_1156_Max_Oin__idem,axiom,
! [A4: set_a,X2: a] :
( ( finite_finite_a @ A4 )
=> ( ( member_a @ X2 @ A4 )
=> ( ( ord_max_a @ X2 @ ( lattic6529028001545966829_Max_a @ A4 ) )
= ( lattic6529028001545966829_Max_a @ A4 ) ) ) ) ).
% Max.in_idem
thf(fact_1157_Max_Oin__idem,axiom,
! [A4: set_nat,X2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ X2 @ A4 )
=> ( ( ord_max_nat @ X2 @ ( lattic8265883725875713057ax_nat @ A4 ) )
= ( lattic8265883725875713057ax_nat @ A4 ) ) ) ) ).
% Max.in_idem
thf(fact_1158_max__of__mono,axiom,
! [F: a > a,M5: a,N3: a] :
( ( monotone_on_a_a @ top_top_set_a @ ord_less_eq_a @ ord_less_eq_a @ F )
=> ( ( ord_max_a @ ( F @ M5 ) @ ( F @ N3 ) )
= ( F @ ( ord_max_a @ M5 @ N3 ) ) ) ) ).
% max_of_mono
thf(fact_1159_max__of__mono,axiom,
! [F: a > nat,M5: a,N3: a] :
( ( monotone_on_a_nat @ top_top_set_a @ ord_less_eq_a @ ord_less_eq_nat @ F )
=> ( ( ord_max_nat @ ( F @ M5 ) @ ( F @ N3 ) )
= ( F @ ( ord_max_a @ M5 @ N3 ) ) ) ) ).
% max_of_mono
thf(fact_1160_max__of__mono,axiom,
! [F: nat > a,M5: nat,N3: nat] :
( ( monotone_on_nat_a @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_a @ F )
=> ( ( ord_max_a @ ( F @ M5 ) @ ( F @ N3 ) )
= ( F @ ( ord_max_nat @ M5 @ N3 ) ) ) ) ).
% max_of_mono
thf(fact_1161_max__of__mono,axiom,
! [F: nat > nat,M5: nat,N3: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F )
=> ( ( ord_max_nat @ ( F @ M5 ) @ ( F @ N3 ) )
= ( F @ ( ord_max_nat @ M5 @ N3 ) ) ) ) ).
% max_of_mono
thf(fact_1162_Sup__insert__finite,axiom,
! [S2: set_nat,X2: nat] :
( ( finite_finite_nat @ S2 )
=> ( ( ( S2 = bot_bot_set_nat )
=> ( ( complete_Sup_Sup_nat @ ( insert_nat @ X2 @ S2 ) )
= X2 ) )
& ( ( S2 != bot_bot_set_nat )
=> ( ( complete_Sup_Sup_nat @ ( insert_nat @ X2 @ S2 ) )
= ( ord_max_nat @ X2 @ ( complete_Sup_Sup_nat @ S2 ) ) ) ) ) ) ).
% Sup_insert_finite
thf(fact_1163_hom__Max__commute,axiom,
! [H: nat > nat,N7: set_nat] :
( ! [X: nat,Y2: nat] :
( ( H @ ( ord_max_nat @ X @ Y2 ) )
= ( ord_max_nat @ ( H @ X ) @ ( H @ Y2 ) ) )
=> ( ( finite_finite_nat @ N7 )
=> ( ( N7 != bot_bot_set_nat )
=> ( ( H @ ( lattic8265883725875713057ax_nat @ N7 ) )
= ( lattic8265883725875713057ax_nat @ ( image_nat_nat @ H @ N7 ) ) ) ) ) ) ).
% hom_Max_commute
thf(fact_1164_Max_Osubset,axiom,
! [A4: set_nat,B6: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( B6 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ B6 @ A4 )
=> ( ( ord_max_nat @ ( lattic8265883725875713057ax_nat @ B6 ) @ ( lattic8265883725875713057ax_nat @ A4 ) )
= ( lattic8265883725875713057ax_nat @ A4 ) ) ) ) ) ).
% Max.subset
thf(fact_1165_Max_Oinsert__not__elem,axiom,
! [A4: set_a,X2: a] :
( ( finite_finite_a @ A4 )
=> ( ~ ( member_a @ X2 @ A4 )
=> ( ( A4 != bot_bot_set_a )
=> ( ( lattic6529028001545966829_Max_a @ ( insert_a @ X2 @ A4 ) )
= ( ord_max_a @ X2 @ ( lattic6529028001545966829_Max_a @ A4 ) ) ) ) ) ) ).
% Max.insert_not_elem
thf(fact_1166_Max_Oinsert__not__elem,axiom,
! [A4: set_nat,X2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ~ ( member_nat @ X2 @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( lattic8265883725875713057ax_nat @ ( insert_nat @ X2 @ A4 ) )
= ( ord_max_nat @ X2 @ ( lattic8265883725875713057ax_nat @ A4 ) ) ) ) ) ) ).
% Max.insert_not_elem
thf(fact_1167_Max_Oclosed,axiom,
! [A4: set_a] :
( ( finite_finite_a @ A4 )
=> ( ( A4 != bot_bot_set_a )
=> ( ! [X: a,Y2: a] : ( member_a @ ( ord_max_a @ X @ Y2 ) @ ( insert_a @ X @ ( insert_a @ Y2 @ bot_bot_set_a ) ) )
=> ( member_a @ ( lattic6529028001545966829_Max_a @ A4 ) @ A4 ) ) ) ) ).
% Max.closed
thf(fact_1168_Max_Oclosed,axiom,
! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ! [X: nat,Y2: nat] : ( member_nat @ ( ord_max_nat @ X @ Y2 ) @ ( insert_nat @ X @ ( insert_nat @ Y2 @ bot_bot_set_nat ) ) )
=> ( member_nat @ ( lattic8265883725875713057ax_nat @ A4 ) @ A4 ) ) ) ) ).
% Max.closed
thf(fact_1169_Max_Ounion,axiom,
! [A4: set_nat,B6: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B6 )
=> ( ( B6 != bot_bot_set_nat )
=> ( ( lattic8265883725875713057ax_nat @ ( sup_sup_set_nat @ A4 @ B6 ) )
= ( ord_max_nat @ ( lattic8265883725875713057ax_nat @ A4 ) @ ( lattic8265883725875713057ax_nat @ B6 ) ) ) ) ) ) ) ).
% Max.union
thf(fact_1170_sup__nat__def,axiom,
sup_sup_nat = ord_max_nat ).
% sup_nat_def
thf(fact_1171_Inf__fin_Oeq__fold,axiom,
! [A4: set_nat,X2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X2 @ A4 ) )
= ( finite_fold_nat_nat @ inf_inf_nat @ X2 @ A4 ) ) ) ).
% Inf_fin.eq_fold
thf(fact_1172_Sup__fin_Oeq__fold,axiom,
! [A4: set_nat,X2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X2 @ A4 ) )
= ( finite_fold_nat_nat @ sup_sup_nat @ X2 @ A4 ) ) ) ).
% Sup_fin.eq_fold
thf(fact_1173_Max_Oeq__fold,axiom,
! [A4: set_nat,X2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( lattic8265883725875713057ax_nat @ ( insert_nat @ X2 @ A4 ) )
= ( finite_fold_nat_nat @ ord_max_nat @ X2 @ A4 ) ) ) ).
% Max.eq_fold
thf(fact_1174_Min_Oeq__fold,axiom,
! [A4: set_nat,X2: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( lattic8721135487736765967in_nat @ ( insert_nat @ X2 @ A4 ) )
= ( finite_fold_nat_nat @ ord_min_nat @ X2 @ A4 ) ) ) ).
% Min.eq_fold
thf(fact_1175_Suc__pred,axiom,
! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( suc @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) )
= N3 ) ) ).
% Suc_pred
thf(fact_1176_bij__betw__Suc,axiom,
! [M2: set_nat,N7: set_nat] :
( ( bij_betw_nat_nat @ suc @ M2 @ N7 )
= ( ( image_nat_nat @ suc @ M2 )
= N7 ) ) ).
% bij_betw_Suc
thf(fact_1177_Least__eq__0,axiom,
! [P: nat > $o] :
( ( P @ zero_zero_nat )
=> ( ( ord_Least_nat @ P )
= zero_zero_nat ) ) ).
% Least_eq_0
thf(fact_1178_Sup__nat__empty,axiom,
( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% Sup_nat_empty
thf(fact_1179_le__zero__eq,axiom,
! [N3: nat] :
( ( ord_less_eq_nat @ N3 @ zero_zero_nat )
= ( N3 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_1180_not__gr__zero,axiom,
! [N3: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N3 ) )
= ( N3 = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_1181_less__nat__zero__code,axiom,
! [N3: nat] :
~ ( ord_less_nat @ N3 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_1182_neq0__conv,axiom,
! [N3: nat] :
( ( N3 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N3 ) ) ).
% neq0_conv
thf(fact_1183_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_1184_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_1185_le0,axiom,
! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N3 ) ).
% le0
thf(fact_1186_diff__0__eq__0,axiom,
! [N3: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N3 )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1187_diff__self__eq__0,axiom,
! [M5: nat] :
( ( minus_minus_nat @ M5 @ M5 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1188_min__0R,axiom,
! [N3: nat] :
( ( ord_min_nat @ N3 @ zero_zero_nat )
= zero_zero_nat ) ).
% min_0R
thf(fact_1189_min__0L,axiom,
! [N3: nat] :
( ( ord_min_nat @ zero_zero_nat @ N3 )
= zero_zero_nat ) ).
% min_0L
thf(fact_1190_max__0R,axiom,
! [N3: nat] :
( ( ord_max_nat @ N3 @ zero_zero_nat )
= N3 ) ).
% max_0R
thf(fact_1191_max__0L,axiom,
! [N3: nat] :
( ( ord_max_nat @ zero_zero_nat @ N3 )
= N3 ) ).
% max_0L
thf(fact_1192_max__nat_Oright__neutral,axiom,
! [A: nat] :
( ( ord_max_nat @ A @ zero_zero_nat )
= A ) ).
% max_nat.right_neutral
thf(fact_1193_max__nat_Oneutr__eq__iff,axiom,
! [A: nat,B: nat] :
( ( zero_zero_nat
= ( ord_max_nat @ A @ B ) )
= ( ( A = zero_zero_nat )
& ( B = zero_zero_nat ) ) ) ).
% max_nat.neutr_eq_iff
thf(fact_1194_max__nat_Oleft__neutral,axiom,
! [A: nat] :
( ( ord_max_nat @ zero_zero_nat @ A )
= A ) ).
% max_nat.left_neutral
thf(fact_1195_max__nat_Oeq__neutr__iff,axiom,
! [A: nat,B: nat] :
( ( ( ord_max_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
& ( B = zero_zero_nat ) ) ) ).
% max_nat.eq_neutr_iff
thf(fact_1196_less__Suc0,axiom,
! [N3: nat] :
( ( ord_less_nat @ N3 @ ( suc @ zero_zero_nat ) )
= ( N3 = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1197_zero__less__Suc,axiom,
! [N3: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N3 ) ) ).
% zero_less_Suc
thf(fact_1198_zero__less__diff,axiom,
! [N3: nat,M5: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N3 @ M5 ) )
= ( ord_less_nat @ M5 @ N3 ) ) ).
% zero_less_diff
thf(fact_1199_diff__is__0__eq,axiom,
! [M5: nat,N3: nat] :
( ( ( minus_minus_nat @ M5 @ N3 )
= zero_zero_nat )
= ( ord_less_eq_nat @ M5 @ N3 ) ) ).
% diff_is_0_eq
thf(fact_1200_diff__is__0__eq_H,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_eq_nat @ M5 @ N3 )
=> ( ( minus_minus_nat @ M5 @ N3 )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1201_less__Suc__eq__0__disj,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_nat @ M5 @ ( suc @ N3 ) )
= ( ( M5 = zero_zero_nat )
| ? [J3: nat] :
( ( M5
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N3 ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1202_gr0__implies__Suc,axiom,
! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ? [M6: nat] :
( N3
= ( suc @ M6 ) ) ) ).
% gr0_implies_Suc
thf(fact_1203_All__less__Suc2,axiom,
! [N3: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N3 ) )
=> ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N3 )
=> ( P @ ( suc @ I3 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1204_gr0__conv__Suc,axiom,
! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
= ( ? [M: nat] :
( N3
= ( suc @ M ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1205_Ex__less__Suc2,axiom,
! [N3: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N3 ) )
& ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N3 )
& ( P @ ( suc @ I3 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_1206_ex__least__nat__le,axiom,
! [P: nat > $o,N3: nat] :
( ( P @ N3 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N3 )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1207_less__eq__nat_Osimps_I1_J,axiom,
! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N3 ) ).
% less_eq_nat.simps(1)
thf(fact_1208_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_1209_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_1210_le__0__eq,axiom,
! [N3: nat] :
( ( ord_less_eq_nat @ N3 @ zero_zero_nat )
= ( N3 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1211_zero__le,axiom,
! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% zero_le
thf(fact_1212_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1213_bij__betw__funpow,axiom,
! [F: nat > nat,S2: set_nat,N3: nat] :
( ( bij_betw_nat_nat @ F @ S2 @ S2 )
=> ( bij_betw_nat_nat @ ( compow_nat_nat @ N3 @ F ) @ S2 @ S2 ) ) ).
% bij_betw_funpow
thf(fact_1214_infinite__descent0,axiom,
! [P: nat > $o,N3: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M8: nat] :
( ( ord_less_nat @ M8 @ N2 )
& ~ ( P @ M8 ) ) ) )
=> ( P @ N3 ) ) ) ).
% infinite_descent0
thf(fact_1215_gr__implies__not0,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_nat @ M5 @ N3 )
=> ( N3 != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1216_less__zeroE,axiom,
! [N3: nat] :
~ ( ord_less_nat @ N3 @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1217_not__less0,axiom,
! [N3: nat] :
~ ( ord_less_nat @ N3 @ zero_zero_nat ) ).
% not_less0
thf(fact_1218_not__gr0,axiom,
! [N3: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N3 ) )
= ( N3 = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1219_gr0I,axiom,
! [N3: nat] :
( ( N3 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N3 ) ) ).
% gr0I
thf(fact_1220_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1221_Least__Suc2,axiom,
! [P: nat > $o,N3: nat,Q: nat > $o,M5: nat] :
( ( P @ N3 )
=> ( ( Q @ M5 )
=> ( ~ ( P @ zero_zero_nat )
=> ( ! [K2: nat] :
( ( P @ ( suc @ K2 ) )
= ( Q @ K2 ) )
=> ( ( ord_Least_nat @ P )
= ( suc @ ( ord_Least_nat @ Q ) ) ) ) ) ) ) ).
% Least_Suc2
thf(fact_1222_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_1223_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_1224_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_1225_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_1226_old_Onat_Oexhaust,axiom,
! [Y3: nat] :
( ( Y3 != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y3
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_1227_nat__induct,axiom,
! [P: nat > $o,N3: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) )
=> ( P @ N3 ) ) ) ).
% nat_induct
thf(fact_1228_diff__induct,axiom,
! [P: nat > nat > $o,M5: nat,N3: nat] :
( ! [X: nat] : ( P @ X @ zero_zero_nat )
=> ( ! [Y2: nat] : ( P @ zero_zero_nat @ ( suc @ Y2 ) )
=> ( ! [X: nat,Y2: nat] :
( ( P @ X @ Y2 )
=> ( P @ ( suc @ X ) @ ( suc @ Y2 ) ) )
=> ( P @ M5 @ N3 ) ) ) ) ).
% diff_induct
thf(fact_1229_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_1230_Suc__neq__Zero,axiom,
! [M5: nat] :
( ( suc @ M5 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_1231_Zero__neq__Suc,axiom,
! [M5: nat] :
( zero_zero_nat
!= ( suc @ M5 ) ) ).
% Zero_neq_Suc
thf(fact_1232_Zero__not__Suc,axiom,
! [M5: nat] :
( zero_zero_nat
!= ( suc @ M5 ) ) ).
% Zero_not_Suc
thf(fact_1233_not0__implies__Suc,axiom,
! [N3: nat] :
( ( N3 != zero_zero_nat )
=> ? [M6: nat] :
( N3
= ( suc @ M6 ) ) ) ).
% not0_implies_Suc
thf(fact_1234_gr__zeroI,axiom,
! [N3: nat] :
( ( N3 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N3 ) ) ).
% gr_zeroI
thf(fact_1235_not__less__zero,axiom,
! [N3: nat] :
~ ( ord_less_nat @ N3 @ zero_zero_nat ) ).
% not_less_zero
thf(fact_1236_gr__implies__not__zero,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_nat @ M5 @ N3 )
=> ( N3 != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_1237_zero__less__iff__neq__zero,axiom,
! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
= ( N3 != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_1238_bij__fn,axiom,
! [F: nat > nat,N3: nat] :
( ( bij_betw_nat_nat @ F @ top_top_set_nat @ top_top_set_nat )
=> ( bij_betw_nat_nat @ ( compow_nat_nat @ N3 @ F ) @ top_top_set_nat @ top_top_set_nat ) ) ).
% bij_fn
thf(fact_1239_Sup__nat__def,axiom,
( complete_Sup_Sup_nat
= ( ^ [X4: set_nat] : ( if_nat @ ( X4 = bot_bot_set_nat ) @ zero_zero_nat @ ( lattic8265883725875713057ax_nat @ X4 ) ) ) ) ).
% Sup_nat_def
thf(fact_1240_diff__less,axiom,
! [N3: nat,M5: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( ord_less_nat @ zero_zero_nat @ M5 )
=> ( ord_less_nat @ ( minus_minus_nat @ M5 @ N3 ) @ M5 ) ) ) ).
% diff_less
thf(fact_1241_minus__nat_Odiff__0,axiom,
! [M5: nat] :
( ( minus_minus_nat @ M5 @ zero_zero_nat )
= M5 ) ).
% minus_nat.diff_0
thf(fact_1242_diffs0__imp__equal,axiom,
! [M5: nat,N3: nat] :
( ( ( minus_minus_nat @ M5 @ N3 )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N3 @ M5 )
= zero_zero_nat )
=> ( M5 = N3 ) ) ) ).
% diffs0_imp_equal
thf(fact_1243_ex__least__nat__less,axiom,
! [P: nat > $o,N3: nat] :
( ( P @ N3 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N3 )
& ! [I4: nat] :
( ( ord_less_eq_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1244_diff__Suc__less,axiom,
! [N3: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ord_less_nat @ ( minus_minus_nat @ N3 @ ( suc @ I ) ) @ N3 ) ) ).
% diff_Suc_less
thf(fact_1245_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_1246_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_1247_Inf__nat__def1,axiom,
! [K3: set_nat] :
( ( K3 != bot_bot_set_nat )
=> ( member_nat @ ( complete_Inf_Inf_nat @ K3 ) @ K3 ) ) ).
% Inf_nat_def1
thf(fact_1248_mono__times__nat,axiom,
! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ ( times_times_nat @ N3 ) ) ) ).
% mono_times_nat
thf(fact_1249_Suc__diff__1,axiom,
! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( suc @ ( minus_minus_nat @ N3 @ one_one_nat ) )
= N3 ) ) ).
% Suc_diff_1
thf(fact_1250_mult__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_1251_mult__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_1252_mult__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_1253_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_1254_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_1255_mult__cancel2,axiom,
! [M5: nat,K: nat,N3: nat] :
( ( ( times_times_nat @ M5 @ K )
= ( times_times_nat @ N3 @ K ) )
= ( ( M5 = N3 )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1256_mult__cancel1,axiom,
! [K: nat,M5: nat,N3: nat] :
( ( ( times_times_nat @ K @ M5 )
= ( times_times_nat @ K @ N3 ) )
= ( ( M5 = N3 )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1257_mult__0__right,axiom,
! [M5: nat] :
( ( times_times_nat @ M5 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1258_mult__is__0,axiom,
! [M5: nat,N3: nat] :
( ( ( times_times_nat @ M5 @ N3 )
= zero_zero_nat )
= ( ( M5 = zero_zero_nat )
| ( N3 = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_1259_nat__mult__eq__1__iff,axiom,
! [M5: nat,N3: nat] :
( ( ( times_times_nat @ M5 @ N3 )
= one_one_nat )
= ( ( M5 = one_one_nat )
& ( N3 = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1260_nat__1__eq__mult__iff,axiom,
! [M5: nat,N3: nat] :
( ( one_one_nat
= ( times_times_nat @ M5 @ N3 ) )
= ( ( M5 = one_one_nat )
& ( N3 = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1261_mult__eq__1__iff,axiom,
! [M5: nat,N3: nat] :
( ( ( times_times_nat @ M5 @ N3 )
= ( suc @ zero_zero_nat ) )
= ( ( M5
= ( suc @ zero_zero_nat ) )
& ( N3
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_1262_one__eq__mult__iff,axiom,
! [M5: nat,N3: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M5 @ N3 ) )
= ( ( M5
= ( suc @ zero_zero_nat ) )
& ( N3
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_1263_nat__0__less__mult__iff,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M5 @ N3 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M5 )
& ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1264_mult__less__cancel2,axiom,
! [M5: nat,K: nat,N3: nat] :
( ( ord_less_nat @ ( times_times_nat @ M5 @ K ) @ ( times_times_nat @ N3 @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M5 @ N3 ) ) ) ).
% mult_less_cancel2
thf(fact_1265_less__one,axiom,
! [N3: nat] :
( ( ord_less_nat @ N3 @ one_one_nat )
= ( N3 = zero_zero_nat ) ) ).
% less_one
thf(fact_1266_diff__Suc__1,axiom,
! [N3: nat] :
( ( minus_minus_nat @ ( suc @ N3 ) @ one_one_nat )
= N3 ) ).
% diff_Suc_1
thf(fact_1267_one__le__mult__iff,axiom,
! [M5: nat,N3: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M5 @ N3 ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M5 )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N3 ) ) ) ).
% one_le_mult_iff
thf(fact_1268_mult__le__cancel2,axiom,
! [M5: nat,K: nat,N3: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M5 @ K ) @ ( times_times_nat @ N3 @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M5 @ N3 ) ) ) ).
% mult_le_cancel2
thf(fact_1269_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
% Helper facts (5)
thf(help_If_2_1_If_001tf__a_T,axiom,
! [X2: a,Y3: a] :
( ( if_a @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001tf__a_T,axiom,
! [X2: a,Y3: a] :
( ( if_a @ $true @ X2 @ Y3 )
= X2 ) ).
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y3: nat] :
( ( if_nat @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y3: nat] :
( ( if_nat @ $true @ X2 @ Y3 )
= X2 ) ).
% Conjectures (2)
thf(conj_0,hypothesis,
! [M8: a] :
( ( member_a @ M8 @ s )
=> ( ! [Y2: a] :
( ( member_a @ Y2 @ s )
=> ( ord_less_eq_a @ Y2 @ M8 ) )
=> thesis ) ) ).
thf(conj_1,conjecture,
thesis ).
%------------------------------------------------------------------------------